diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzawtu" "b/data_all_eng_slimpj/shuffled/split2/finalzzawtu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzawtu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and results}\n\nLet us recall the objects we will deal with. Throughout the paper $\\DD$ denotes the unit open disc on the complex plane, $\\TT$ is the unit circle and $p$ --- the Poincar\\'e distance on $\\DD$.\n\nLet $D\\subset\\CC^{n}$ be a domain and let $z,w\\in D$, $v\\in\\CC^{n}$. The {\\it Lempert function}\\\/ is defined as\n\\begin{equation}\\label{lem}\n\\widetilde{k}_{D}(z,w):=\\inf\\{p(0,\\xi):\\xi\\in[0,1)\\textnormal{ and }\\exists f\\in \\mathcal{O}(\\mathbb{D},D):f(0)=z,\\ f(\\xi)=w\\}.\n\\end{equation} The {\\it Kobayashi-Royden \\emph{(}pseudo\\emph{)}metric}\\\/ we define as\n\\begin{equation}\\label{kob-roy}\n\\kappa_{D}(z;v):=\\inf\\{\\lambda^{-1}:\\lambda>0\\text{ and }\\exists f\\in\\mathcal{O}(\\mathbb{D},D):f(0)=z,\\ f'(0)=\\lambda v\\}.\n\\end{equation}\nNote that\n\\begin{equation}\\label{lem1}\n\\widetilde{k}_{D}(z,w)=\\inf\\{p(\\zeta,\\xi):\\zeta,\\xi\\in\\DD\\textnormal{ and }\\exists f\\in \\mathcal{O}(\\mathbb{D},D):f(\\zeta)=z,\\ f(\\xi)=w\\},\n\\end{equation}\n\\begin{multline}\\label{kob-roy1}\n\\kappa_{D}(z;v)=\\inf\\{|\\lambda|^{-1}\/(1-|\\zeta|^2):\\lambda\\in\\CC_*,\\,\\zeta\\in\\DD\\text{ and }\\\\ \\exists f\\in\\mathcal{O}(\\mathbb{D},D):f(\\zeta)=z,\\ f'(\\zeta)=\\lambda v\\}.\n\\end{multline}\n\nIf $z\\neq w$ (respectively $v\\neq 0$), a mapping $f$ for which the infimum in \\eqref{lem1} (resp. in \\eqref{kob-roy1}) is attained, we call a $\\wi{k}_D$-\\textit{extremal} (or a \\textit{Lempert extremal}) for $z,w$ (resp. a $\\kappa_D$-\\textit{extremal} for $z,v$). A mapping being a $\\wi k_D$-extremal or a $\\kappa_D$-extremal we will call just an \\textit{extremal} or an \\textit{extremal mapping}.\n\nWe shall say that $f:\\DD\\longrightarrow D$ is a unique $\\wi{k}_D$-extremal for $z,w$ (resp. a unique $\\kappa_D$-extremal for $z,v$) if any other $\\wi{k}_D$-extremal $g:\\DD\\longrightarrow D$ for $z,w$ (resp. $\\kappa_D$-extremal for $z,v$) satisfies $g=f\\circ a$ for some M\\\"obius function $a$.\n\nIn general, $\\wi{k}_{D}$ does not satisfy a triangle inequality --- take for example $D_{\\alpha}:=\\{(z,w)\\in\\CC^{2}:|z|,|w|<1,\\ |zw|<\\alpha\\}$, $\\alpha\\in(0,1)$. Therefore, it is natural to consider the so-called \\textit{Kobayashi \\emph{(}pseudo\\emph{)}distance} given by the formula \\begin{multline*}k_{D}(w,z):=\\sup\\{d_{D}(w,z):(d_{D})\\text{ is a family of holomorphically invariant} \\\\\\text{pseudodistances less than or equal to }\\widetilde{k}_{D}\\}.\\end{multline*}\nIt follows directly from the definition that $$k_{D}(z,w)=\\inf\\left\\{\\sum_{j=1}^{N}\\wi{k}_{D}(z_{j-1},z_{j}):N\\in\\NN,\\ z_{1},\\ldots,z_{N}\\in\nD,\\ z_{0}=z,\\ z_{N}=w\\right\\}.$$\n\nThe next objects we are dealing with, are the \\textit{Carath\\'eodory \\emph{(}pseudo\\emph{)}distance}\n$$c_{D}(z,w):=\\sup\\{p(F(z),F(w)):F\\in\\mathcal{O}(D,\\DD)\\}$$\nand the \\textit{Carath\\'eodory-Reiffen \\emph{(}pseudo\\emph{)}metric}\n$$\\gamma_D(z;v):=\\sup\\{|F'(z)v|:F\\in\\mathcal{O}(D,\\DD),\\ F(z)=0\\}.$$\n\nA holomorphic mapping $f:\\DD\\longrightarrow D$ is said to be a \\emph{complex geodesic} if $c_D(f(\\zeta),f(\\xi))=p(\\zeta,\\xi)$ for any $\\zeta,\\xi\\in\\DD$.\n\\bigskip\n\nHere is some notation. Let $z_1,\\ldots,z_n$ be the standard complex coordinates in $\\CC^n$ and $x_1,\\ldots,x_{2n}$ --- the standard real coordinates in $\\CC^n=\\RR^n+i\\RR^n\\simeq\\RR^{2n}$. We use $T_{D}^\\mathbb{R}(a)$, $T_{D}^\\mathbb{C}(a)$ to denote a real and a complex tangent space to a $\\cC^1$-smooth domain $D$ at a point $a\\in\\partial D$, i.e. the sets \\begin{align*}T_{D}^\\mathbb{R}(a):&=\\left\\{X\\in\\CC^{n}:\\re\\sum_{j=1}^n\\frac{\\partial r}{\\partial z_j}(a)X_{j}=0\\right\\},\\\\ T_{D}^\\mathbb{C}(a):&=\\left\\{X\\in\\CC^{n}:\\sum_{j=1}^n\\frac{\\partial r}{\\partial z_j}(a)X_{j}=0\\right\\},\\end{align*}\nwhere $r$ is a defining function of $D$. Let $\\nu_D(a)$ be the outward unit normal vector to $\\partial D$ at $a$.\n\nLet $\\mathcal{C}^{k}(\\CDD)$, where $k\\in(0,\\infty]$, denote a class of continuous functions on $\\CDD$, which are of class $\\cC^k$ on $\\DD$ and\n\\begin{itemize}\n\\item if $k\\in\\NN\\cup\\{\\infty\\}$ then derivatives up to the order $k$ extend continuously on~$\\CDD$;\n\\item if $k-[k]=:c>0$ then derivatives up to the order $[k]$ are $c$-H\\\"older continuous on $\\DD$.\n\\end{itemize}\nBy $\\mathcal{C}^\\omega$ class we shall denote real analytic functions. Further, saying that $f$ is of class $\\mathcal{C}^{k}(\\TT)$, $k\\in(0,\\infty]\\cup\\{\\omega\\}$, we mean that the function $t\\longmapsto f(e^{it})$, $t\\in\\RR$, is in $\\mathcal{C}^{k}(\\mathbb R)$. For a compact set $K\\su\\CC^n$ let $\\OO(K)$ denote the set of functions extending holomorphically on a neighborhood of $K$ (we assume that all neighborhoods are open). In that case we shall sometimes say that a given function is of class $\\OO(K)$. Note that $\\CLW(\\TT)=\\OO(\\TT)$. \n\nLet $|\\cdot|$ denote the Euclidean norm in $\\CC^{n}$ and let $\\dist(z,S):=\\inf\\{|z-s|:s\\in S\\}$ be a distance of the point $z\\in\\CC^n$ to the set $S\\su\\CC^n$. For such a set $S$ we define $S_*:=S\\setminus\\{0\\}$. Let $\\BB_n:=\\{z\\in\\CC^n:|z|=1\\}$ be the unit ball and $B_n(a,r):=\\{z\\in\\CC^n:|z-a|0$. Put $$z\\bullet w:=\\sum_{j=1}^nz_{j}{w}_{j}$$ for $z,w\\in\\CC^{n}$ and let $\\langle\\cdotp,-\\rangle$ be a hermitian inner product on $\\CC^n$. The real inner product on $\\CC^n$ is denoted by $\\langle\\cdotp,-\\rangle_{\\RR}=\\re\\langle\\cdotp,-\\rangle$.\n\nWe use $\\nabla$ to denote the gradient $(\\pa\/\\pa x_1,\\ldots,\\pa\/\\pa x_{2n})$. For real-valued functions the gradient is naturally identified with $2(\\pa\/\\pa\\ov z_1,\\ldots,\\pa\/\\pa\\ov z_n)$. Recall that $$\\nu_D(a)=\\frac{\\nabla r(a)}{|\\nabla r(a)|}.$$ Let $\\mathcal{H}$ be the Hessian matrix $$\\left[\\frac{\\pa^2}{\\pa x_j\\pa x_k}\\right]_{1\\leq j,k\\leq 2n}.$$ Sometimes, for a $\\cC^2$-smooth function $u$ and a vector $X\\in\\RR^{2n}$ the Hessian $$\\sum_{j,k=1}^{2n}\\frac{\\partial^2 u}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}=X^T\\HH u(a)X$$ will be denoted by $\\HH u(a;X)$. By $\\|\\cdot\\|$ we denote the operator norm.\n\\bigskip\n\\begin{df}\\label{29}\nLet $D\\subset\\CC^{n}$ be a domain.\n\nWe say that $D$ is \\emph{linearly convex} (resp. \\emph{weakly linearly convex}) if through any point $a\\in\\mathbb C^n\\setminus D$ (resp. $a\\in \\partial D$) there goes an $(n-1)$-dimensional complex hyperplane disjoint from $D$.\n\nA domain $D$ is said to be \\emph{strongly linearly convex} if\n\\begin{enumerate}\n\\item $D$ has $\\mathcal{C}^{2}$-smooth boundary;\n\\item there exists a defining function $r$ of $D$ such that\n\\begin{equation}\\label{48}\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial\\overline z_k}(a)X_{j}\\overline{X}_{k}>\\left|\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial z_k}(a)X_{j}X_{k}\\right|,\\ a\\in\\partial D,\\ X\\in T_{D}^\\mathbb{C}(a)_*.\\end{equation}\n\\end{enumerate}\n\nMore generally, any point $a\\in\\pa D$ for which there exists a defining function $r$ satisfying \\eqref{48}, is called a \\emph{point of the strong linear convexity} of $D$.\n\nFurthermore, we say that a domain $D$ has \\emph{real analytic boundary} if it possesses a real analytic defining function.\n\\end{df}\n\nNote that the condition \\eqref{48} does not depend on the choice of a defining function of $D$.\n\n\\begin{rem}\nLet $D\\subset\\CC^{n}$ be a strongly linearly convex domain. Then\n\\begin{enumerate}\n\\item any $(n-1)$-dimensional complex tangent hyperplane intersects $\\partial{D}$ at precisely one point; in other words $$\\overline D\\cap(a+T_{D}^\\mathbb{C}(a))=\\{a\\},\\ a\\in\\pa D;$$\n\\item for $a\\in\\pa D$ the equation $\\langle w-a, \\nu_D(a)\\rangle=0$ describes the $(n-1)$-dimensional complex tangent hyperplane $a+T_{D}^\\mathbb{C}(a)$, consequently $$\\langle z-a, \\nu_D(a)\\rangle\\neq 0,\\ z\\in D,\\ a\\in\\pa D.$$\n\\end{enumerate}\n\\end{rem}\n\\bigskip\nThe main aim of the paper is to present a detailed proof of the following\n\n\\begin{tw}[Lempert Theorem]\\label{lem-car}\nLet $D\\subset\\CC^{n}$, $n\\geq 2$, be a bounded strongly linearly convex domain. Then $$c_{D}=k_{D}=\\wi{k}_{D}\\text{\\,\\ and\\,\\, }\\gamma_D=\\kappa_D.$$\n\\end{tw}\n\n\\bigskip\n\nAn important role will be played by strongly convex domains and strongly convex functions.\n\\begin{df}\nA domain $D\\subset\\CC^{n}$ is called \\emph{strongly convex} if\n\\begin{enumerate}\n\\item $D$ has $\\mathcal{C}^{2}$-smooth boundary;\n\\item there exists a defining function $r$ of $D$ such that\n\\begin{equation}\\label{sc}\\sum_{j,k=1}^{2n}\\frac{\\partial^2 r}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}>0,\\ a\\in\\partial D,\\ X\\in T_{D}^\\mathbb{R}(a)_*.\\end{equation}\n\\end{enumerate}\nGenerally, any point $a\\in\\pa D$ for which there exists a defining function $r$ satisfying \\eqref{sc}, is called a \\emph{point of the strong convexity} of $D$.\n\\end{df}\n\\begin{rem}\nA strongly convex domain $D\\subset\\CC^{n}$ is convex and strongly linearly convex. Moreover, it is strictly convex, i.e. for any different points $a,b\\in\\overline D$ the interior of the segment $[a,b]=\\{ta+(1-t)b:t\\in [0,1]\\}$ is contained in $D$ (i.e. $ta+(1-t)b\\in D$ for any $t\\in(0,1)$).\n\nObserve also that any bounded convex domain with a real analytic boundary is strictly convex. Actually, if a domain $D$ with a real analytic boundary were not strictly convex, then we would be able to find two distinct points $a,b\\in\\pa D$ such that the segment $[a,b]$ lies entirely in $\\partial D$. On the other hand, the identity principle would imply that the set $\\{t\\in\\mathbb R:\\exists\\eps>0:sa+(1-s)b\\in\\pa D\\text{ for }|s-t|<\\eps\\}$ is open-closed in $\\mathbb R$. Therefore it has to be empty. This immediately gives a contradiction.\n\\end{rem}\n\n\\begin{rem}\nIt is well-known that for any convex domain $D\\su\\CC^{n}$ there is a sequence $\\{D_m\\}$ of bounded strongly convex domains with real analytic boundaries, such that $D_m\\su D_{m+1}$ and $\\bigcup_m D_m=D$. \n\nIn particular, Theorem~\\ref{lem-car} holds for convex domains.\n\\end{rem}\n\n\\begin{df}\nLet $U\\su\\CC^n$ be a domain. A function $u:U\\longrightarrow\\RR$ is called \\emph{strongly convex} if\n\\begin{enumerate}\n\\item $u$ is $\\mathcal{C}^{2}$-smooth;\n\\item $$\\sum_{j,k=1}^{2n}\\frac{\\partial^2 u}{\\partial x_j\\partial x_k}(a)X_{j}X_{k}>0,\\ a\\in U,\\ X\\in(\\RR^{2n})_*.$$\n\\end{enumerate}\n\\end{df}\n\n\\begin{df} A degree of a continuous function (treated as a curve) $:\\mathbb T\\longrightarrow\\mathbb T$ is called its winding number. The fundamental group is a homotopy invariant. Thus the definition of the \\emph{winding number of a continuous function} $\\phi:\\mathbb T\\longrightarrow\\mathbb C_*$ is the same. We denote it by $\\wind\\phi$. \n\nIn the case of a $\\cC^1$-smooth function $\\phi:\\TT\\longrightarrow\\CC_*$, its winding number is just the index of $\\phi$ at 0, i.e. $$\\wind\\phi=\\frac{1}{2\\pi i}\\int_{\\phi(\\TT)}\\frac{d\\zeta}{\\zeta}=\\frac{1}{2\\pi i}\\int_{0}^{2\\pi}\\frac{\\frac{d}{dt}\\phi(e^{it})}{\\phi(e^{it})}dt.$$\n\\end{df}\n\n\\begin{rem}\\label{49}\n\\begin{enumerate}\n\\item\\label{51} If $\\phi\\in\\cC(\\TT,\\CC_*)$ extends to a function $\\widetilde{\\phi}\\in\\OO(\\DD)\\cap \\mathcal C(\\CDD)$ then $\\wind\\phi$ is the number of zeroes of $\\widetilde{\\phi}$ in $\\DD$ counted with multiplicities;\n\\item\\label{52} $\\wind(\\phi\\psi)=\\wind\\phi+\\wind\\psi$, $\\phi,\\psi\\in\\cC(\\TT,\\CC_*)$;\n\\item\\label{53} $\\wind\\phi=0$ if $\\phi\\in\\cC(\\TT)$ and $\\re\\phi>0$.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{df}\nThe boundary of a domain $D$ of $\\mathbb C^n$ is \\emph{real analytic in a neighborhood} $U$ of the set $S\\su\\pa D$ if there exists a function $r\\in\\mathcal C^{\\omega}(U,\\RR)$ such that $D\\cap U=\\{z\\in U:r(z)<0\\}$ and $\\nabla r$ does not vanish in $U$.\n\\end{df}\n\n\n\n\\begin{df}\\label{21}\nLet $D\\subset\\CC^{n}$ be a domain. We call a holomorphic mapping $f:\\DD\\longrightarrow D$ a \\emph{stationary mapping} if\n\\begin{enumerate}\n\\item $f$ extends to a holomorphic mapping in a neighborhood od $\\CDD$ $($denoted by the same letter$)$;\n\\item $f(\\TT)\\subset\\partial D$;\n\\item there exists a real analytic function\n$\\rho:\\TT\\longrightarrow\\RR_{>0}$ such that the mapping $\\TT\\ni\\zeta\\longmapsto\\zeta\n\\rho(\\zeta)\\overline{\\nu_D(f(\\zeta))}\\in\\CC^{n}$ extends to a mapping holomorphic in a neighborhood of $\\CDD$ $($denoted by $\\widetilde{f}${$)$}.\n\\end{enumerate}\n\nFurthermore, we call a holomorphic mapping $f:\\DD\\longrightarrow D$ a \\emph{weak stationary mapping} if\n\\begin{enumerate}\n\\item[(1')] $f$ extends to a $\\cC^{1\/2}$-smooth mapping on $\\CDD$ $($denoted by the same letter$)$;\n\\item[(2')] $f(\\TT)\\subset\\partial D$;\n\\item[(3')] there exists a $\\cC^{1\/2}$-smooth function\n$\\rho:\\TT\\longrightarrow\\RR_{>0}$ such that the mapping $\\TT\\ni\\zeta\\longmapsto\\zeta\n\\rho(\\zeta)\\overline{\\nu_D(f(\\zeta))}\\in\\CC^{n}$ extends to a mapping $\\widetilde{f}\\in\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$.\n\\end{enumerate}\n\nThe definition of a $($weak$)$ stationary mapping $f:\\mathbb D\\longrightarrow D$ extends naturally to the case when $\\pa D$ is real analytic in a neighborhood of $f(\\TT)$.\n\\end{df}\n\n\nDirectly from the definition of a stationary mapping $f$, it follows that $f$ and $\\wi f$ extend holomorphically on some neighborhoods of $\\CDD$. By $\\DD_f$ we shall denote their intersection.\n\n\\begin{df}\\label{21e}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain with real analytic boundary. A holomorphic mapping $f:\\DD\\longrightarrow D$ is called a (\\emph{weak}) $E$-\\emph{mapping} if it is a (weak) stationary mapping and\n\\begin{enumerate}\n\\item[(4)] setting $\\varphi_z(\\zeta):=\\langle z-f(\\zeta),\\nu_D(f(\\zeta))\\rangle,\\ \\zeta\\in\\TT$, we have $\\wind\\phi_z=0$ for some $z\\in D$.\n\\end{enumerate}\n\\end{df}\n\n\\begin{rem}\nThe strong linear convexity of $D$ implies $\\varphi_z(\\zeta)\\neq 0$ for any $z\\in D$ and $\\zeta\\in\\TT$. Therefore, $\\wind\\phi_z$ vanishes for all $z\\in D$ if it vanishes for some $z\\in D$.\n\nAdditionally, any stationary mapping of a convex domain is an $E$-mapping (as $\\re \\varphi_z<0$).\n\\end{rem}\n\nWe shall prove that in a class of non-planar bounded strongly linearly convex domains with real analytic boundaries weak stationary mappings are just stationary mappings, so there is no difference between $E$-mappings and weak $E$-mappings. \n\nWe have the following result describing extremal mappings, which is very interesting in its own.\n\n\\begin{tw}\\label{main} Let $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain. \n\nThen a holomorphic mapping $f:\\DD\\longrightarrow D$ is an extremal if and only if $f$ is a weak $E$-mapping.\n\nFor a domain $D$ with real analytic boundary, a holomorphic mapping $f:\\mathbb D\\longrightarrow D$ is an extremal if and only if $f$ is an $E$-mapping.\n\nIf $\\pa D$ is of class $\\cC^k$, $k=3,4,\\ldots,\\infty$, then any weak $E$-mapping $f:\\DD\\longrightarrow D$ and its associated mappings $\\wi f,\\rho$ are $\\mathcal C^{k-1-\\eps}$-smooth for any $\\eps>0$.\n\n\\end{tw}\n\n\nThe idea of the proof of the Lempert Theorem is as follows. In real analytic case we shall show that $E$-mappings are complex geodesics (because they have left inverses). Then we shall prove that for any different points $z,w\\in D$ (resp. for a point $z\\in D$ and a vector $v\\in(\\CC^n)_*$) there is an $E$-mapping passing through $z,w$ (resp. such that $f(0)=z$ and $f'(0)=v$). This will give the equality between the Lempert function and the Carath\\'eodory distance. In the general case, we exhaust a $\\cC^2$-smooth domain by strongly linearly convex domains with real analytic boundaries.\n\nTo prove Theorem \\ref{main} we shall additionally observe that (weak) $E$-mappings are unique extremals.\n\\bigskip\n\n\\begin{center}{\\sc Real analytic case}\\end{center}\n\\bigskip\n\nIn what follows and if not mentioned otherwise, $D\\su\\CC^n$, $n\\geq 2$, is a \\textbf{bounded strongly linearly convex domain with real analytic boundary}.\n\\section{Weak stationary mappings of strongly linearly convex domains with real analytic boundaries are stationary mappings}\\label{55}\nLet $M\\subset\\CC^m$ be a totally real $\\CLW$ submanifold of the real dimension $m$. Fix a point $z\\in M$. There are neighborhoods $U,V\\su\\CC^m$ of $0$ and $z$ respectively and a biholomorphic mapping $\\Phi:U\\longrightarrow V$ such that $\\Phi(\\RR^m\\cap U)=M\\cap V$ (for the proof see Appendix).\n\n\n\\begin{prop}\\label{6}\nA weak stationary mapping of $D$ is a stationary mapping of $D$ with the same associated mappings.\n\\end{prop}\n\\begin{proof}\nLet $f:\\DD\\longrightarrow D$ be a weak stationary mapping. Our aim is to prove that $f,\\widetilde{f}\\in\\OO(\\CDD)$ and $\\rho\\in\\mathcal C^{\\omega}(\\TT)$. Choose a point $\\zeta_0\\in\\TT$. Since $\\widetilde{f}(\\zeta_0)\\neq 0$, we can assume that $\\widetilde{f}_1(\\zeta)\\neq 0$ in $\\CDD\\cap U_0$, where $U_0$ is a neighborhood of $\\zeta_0$. This implies\n$\\nu_{D,1}(f(\\zeta_0))\\neq 0$, so $\\nu_{D,1}$ does not vanish on some set $V_0\\su\\pa D$, relatively open in\n$\\pa D$, containing the point $f(\\zeta_0)$. Shrinking $U_0$, if necessary, we may assume that $f(\\TT\\cap U_0)\\subset V_0$.\n\nDefine $\\psi:V_0\\longrightarrow\\CC^{2n-1}$ by\n$$\\psi(z)=\\left(z_1,\\ldots,z_n,\n\\ov{\\left(\\frac{\\nu_{D,2}(z)}{\\nu_{D,1}(z)}\\right)},\\ldots,\\ov{\\left(\\frac{\\nu_{D,n}(z)}{\\nu_{D,1}(z)}\\right)}\\right).$$ The set $M:=\\psi(V_0)$ is the graph of a $\\CLW$ function defined on the local $\\CLW$ submanifold $V_0$, so it is a local $\\CLW$ submanifold in $\\CC^{2n-1}$ of the real dimension $2n-1$. Assume for a moment that $M$ is totally real.\n\nLet $$g(\\zeta):=\\left(f_1(\\zeta),\\ldots,f_n(\\zeta),\n\\frac{\\widetilde{f}_2(\\zeta)}{\\widetilde{f}_1(\\zeta)},\\ldots,\\frac{\\widetilde{f}_n(\\zeta)}{\\widetilde{f}_1(\\zeta)}\\right),\\ \\zeta\\in\\CDD\\cap U_0.$$ If $\\zeta\\in\\TT\\cap U_0$ then\n$\\widetilde{f}_k(\\zeta)\\widetilde{f}_1(\\zeta)^{-1} =\n\\overline{\\nu_{D,k}(f(\\zeta))}\\ \\overline{\\nu_{D,1}(f(\\zeta))}^{-1}$, so\n$g(\\zeta)=\\psi(f(\\zeta))$. Therefore, $g(\\TT\\cap U_0)\\subset M$. Thanks to the Reflection\nPrinciple (see Appendix), $g$ extends holomorphically past $\\TT\\cap U_0$, so $f$ extends holomorphically on a neighborhood of $\\zeta_0$.\n\nThe mapping $\\overline{\\nu_D\\circ f}$ is real analytic on $\\TT$, so it extends to a mapping $h$ holomorphic in a neighborhood $W$ of $\\TT$. For $\\zeta\\in\\TT\\cap U_0$ we have $$\\frac{\\zeta\nh_1(\\zeta)}{\\widetilde{f}_1(\\zeta)}=\\frac{1}{\\rho(\\zeta)}.$$ The function on the\nleft side is holomorphic in $\\DD\\cap U_0\\cap W$ and continuous in $\\CDD\\cap U_0\\cap W$. Since it\nhas real values on $\\TT\\cap U_0$, the Reflection Principle implies that it is holomorphic in a neighborhood of $\\TT\\cap U_0$. Hence $\\rho$ and $\\widetilde{f}$ are holomorphic in a neighborhood of $\\zeta_0$. Since $\\zeta_0$ is arbitrary, we get the assertion.\n\nIt remains to prove that $M$ is totally real. Let $r$ be a defining function of $D$. Recall that for any point $z\\in V_0$ $$\\frac{\\ov{\\nu_{D,k}(z)}}{\\ov{\\nu_{D,1}(z)}}=\\frac{\\partial r}{\\partial z_k}(z)\\left(\\frac{\\partial r}{\\partial z_1}(z)\\right)^{-1},\\,k=1,\\ldots,n.$$\nConsider the mapping $S=(S_1,\\ldots,S_n):V_0\\times\\CC^{n-1}\\longrightarrow\\RR\\times\\CC^{n-1}$\ngiven by $$S(z,w):=\\left(r(z),\\frac{\\partial r}{\\partial z_2}(z)-w_{1}\\frac{\\partial r}{\\partial z_1}(z),\\ldots,\\frac{\\partial r}{\\partial z_n}(z)-w_{n-1}\\frac{\\partial r}{\\partial z_1}(z)\\right).$$ Clearly, $M=S^{-1}(\\{0\\})$. Hence\n\\begin{equation}\\label{tan} T_{M}^{\\RR}(z,w)\\subset\\ker\\nabla S(z,w),\\ (z,w)\\in M,\\end{equation} where\n$\\nabla S:=(\\nabla S_1,\\ldots,\\nabla S_n)$.\n\nFix a point $(z,w)\\in M$. Our goal is to prove that $T_{M}^{\\CC}(z,w)=\\lbrace 0\\rbrace$. Take an arbitrary vector $(X,Y)=(X_1,\\ldots,X_n,Y_1,\\ldots,Y_{n-1})\\in T_{M}^{\\CC}(z,w)$. Then we infer from \\eqref{tan} that $$\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)X_k=0,$$ i.e. $X\\in T_{D}^{\\CC}(z)$. Denoting $v:=(z,w)$, $V:=(X,Y)$ and making use of \\eqref{tan} again we find that\n$$0=\\nabla S_k(v)(V)=\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa v_j}(v)V_j+\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j$$ for $k=2,\\ldots,n$.\nBut $V\\in T_{M}^{\\CC}(v)$, so $iV\\in T_{M}^{\\CC}(v)$. Thus $$0=\\nabla S_k(v)(iV)=i\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa v_j}(v)V_j-i\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j.$$ In particular, \\begin{multline*}0=\\sum_{j=1}^{2n-1}\\frac{\\pa S_k}{\\pa\\ov v_j}(v)\\ov V_j=\\sum_{j=1}^{n}\\frac{\\pa S_k}{\\pa\\ov z_j}(z,w)\\ov X_j+\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa\\ov w_j}(z,w)\\ov Y_j=\\\\=\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_j-w_{k-1}\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j.\n\\end{multline*}\nThe equality $M=S^{-1}(\\{0\\})$ gives $$w_{k-1}=\\frac{\\partial r}{\\partial z_k}(z)\\left(\\frac{\\partial r}{\\partial z_1}(z)\\right)^{-1},$$ so $$\\frac{\\partial r}{\\partial z_1}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_j=\\frac{\\partial r}{\\partial z_k}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j,\\ k=2,\\ldots,n.$$ Note that the last equality holds also for $k=1$. Therefore, \\begin{multline*}\n\\frac{\\partial r}{\\partial z_1}(z)\\sum_{j,k=1}^n\\frac{\\partial^2r}{\\partial z_k\\partial\\overline{z}_j}(z)\\overline X_jX_k=\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_jX_k =\\\\=\\left(\\sum_{k=1}^n\\frac{\\partial r}{\\partial z_k}(z)X_k\\right)\\left(\\sum_{j=1}^n\\frac{\\partial^2r}{\\partial z_1\\partial\\overline{z}_j}(z)\\overline X_j\\right)=0.\n\\end{multline*}\nBy the strong linear convexity of $D$ we have $X=0$. This implies $Y=0$, since $$0=\\nabla S_k(z,w)(0,Y)=\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa w_j}(v)Y_j+\\sum_{j=1}^{n-1}\\frac{\\pa S_k}{\\pa\\ov w_j}(v)\\ov Y_j=-\\frac{\\partial r}{\\partial z_1}(z)Y_{k-1}$$ for $k=2,\\ldots,n$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{(Weak) $E$-mappings vs. extremal mappings and complex geodesics}\n\nIn this section we will prove important properties of (weak) $E$-mappings. In particular, we will show that they are complex geodesics and unique extremals.\n\\subsection{Weak $E$-mappings are complex geodesics and unique extremals}\nThe results of this subsection are related to weak $E$-mappings of bounded strongly linearly convex domains $D\\su\\CC^n$, $n\\geq 2$.\n\nLet $$G(z,\\zeta):=(z-f(\\zeta))\\bullet\\widetilde{f}(\\zeta),\\ z\\in\\CC^n,\\ \\zeta\\in\\DD_f.$$\n\n\\begin{propp}\\label{1}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain and let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping. Then there exist an open set $W\\supset\\overline D\\setminus f(\\TT)$ and a holomorphic mapping $F:W\\longrightarrow\\DD$ such that for any $z\\in W$ the number $F(z)$ is a unique solution of the equation $G(z,\\zeta)=0,\\ \\zeta\\in\\DD$. In particular, $F\\circ f=\\id_{\\DD}$.\n\\end{propp}\n\nIn the sequel we will strengthen the above proposition for domains with real analytic boundaries (see Proposition~\\ref{34}).\n\n\\begin{proof}[Proof of Proposition~\\ref{1}]\nSet $A:=\\overline{D}\\setminus f(\\TT)$. Since $D$ is strongly linearly convex, $\\varphi_z$ does not vanish in $\\TT$ for any $z\\in A$, so by a continuity argument the condition (4) of Definition~\\ref{21e} holds for every $z$ in some open set $W\\supset A$. For a fixed $z\\in W$ we have $$G(z,\\zeta)=\\zeta\\rho(\\zeta)\\varphi_z(\\zeta),\\ \\zeta\\in\\TT,$$ so $\\wind G(z,\\cdotp)=1$. Since $G(z,\\cdotp)\\in\\OO(\\DD)$, it has in $\\DD$ exactly one simple root $F(z)$. Hence $G(z,F(z))=0$ and $\\frac{\\partial G}{\\partial\\zeta}(z,F(z))\\neq 0$. By the Implicit Function Theorem, $F$ is holomorphic in $W$. The equality $F(f(\\zeta))=\\zeta$ for $\\zeta\\in\\DD$ is clear.\n\\end{proof}\n\nFrom the proposition above we immediately get the following\n\\begin{corr}\\label{5}\nA weak $E$-mapping $f:\\DD\\longrightarrow D$ of a bounded strongly linearly convex domain $D\\su\\CC^n$, $n\\geq 2$, is a complex geodesic. In particular,\n$$c_{D}(f(\\zeta),f(\\xi))=\\wi k_D(f(\\zeta),f(\\xi))\\text{\\,\\ and\\,\\, }\\gamma_D(f(\\zeta);f'(\\zeta))=\\kappa_D(f(\\zeta);f'(\\zeta)),$$ for any $\\zeta,\\xi\\in\\DD$.\n\\end{corr}\n\nUsing left inverses of weak $E$-mappings we may prove the uniqueness of extremals.\n\\begin{propp}\\label{2}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain and let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping. Then for any $\\xi\\in(0,1)$ the mapping $f$ is a unique $\\wi{k}_D$-extremal for $z=f(0)$, $w=f(\\xi)$ \\emph{(}resp. a unique $\\kappa_D$-extremal for $z=f(0)$, $v=f'(0)$\\emph{)}.\n\\end{propp}\n\\begin{proof}\n\nSuppose that $g$ is a $\\wi{k}_D$-extremal for $z,w$ (resp. a $\\kappa_D$-extremal for $z,v$) such that $g(0)=z$, $g(\\xi)=w$ (resp. $g(0)=z$, $g'(0)=v$). Our aim is to show that $f=g$. Proposition~\\ref{1} provides us with the mapping $F$, which is a left inverse for $f$. By the Schwarz Lemma, $F$ is a left inverse for $g$, as well, that is $F\\circ g=\\text{id}_{\\DD}$. We claim that $\\lim_{\\DD\\ni\\zeta\\to\\zeta_0}g(\\zeta)=f(\\zeta_0)$ for any $\\zeta_0\\in\\TT$ (in particular, we shall show that the limit does exist).\n\nAssume the contrary. Then there are $\\zeta_0\\in\\TT$ and a sequence $\\{\\zeta_m\\}\\subset\\DD$ convergent to $\\zeta_0$ such that the limit $Z:=\\lim_{m\\to\\infty}g(\\zeta_m)\\in\\overline{D}$ exists and is not equal to $f(\\zeta_0)$. We have $G(z,F(z))=0$, so putting $z=g(\\zeta_m)$ we infer that $$0=(g(\\zeta_m)-f(F(g(\\zeta_m))))\\bullet \\widetilde{f}(F(g(\\zeta_m)))=(g(\\zeta_m)-f(\\zeta_m))\\bullet\\widetilde{f}(\\zeta_m).\n$$ Passing with $m$ to the infinity we get $$0=(Z-f(\\zeta_0))\\bullet \\widetilde{f}(\\zeta_0)=\\zeta_0\\rho(\\zeta_0)\\langle Z-f(\\zeta_0),\\nu_D(f(\\zeta_0))\\rangle.$$ This means that $Z-f(\\zeta_0)\\in T^{\\CC}_D(f(\\zeta_0))$. Since $D$ is strongly linearly convex, we deduce that $Z=f(\\zeta_0)$, which is a contradiction.\n\nHence $g$ extends continuously on $\\CDD$ and, by the maximum principle, $g=f$.\n\\end{proof}\n\n\n\\begin{propp}\\label{3}\nLet $D\\su\\CC^n$, $n\\geq 2$, be a bounded strongly linearly convex domain, let $f:\\DD\\longrightarrow D$ be a weak $E$-mapping and let $a$ be an automorphism of $\\DD$. Then $f\\circ a$ is a weak $E$-mapping of $D$.\n\\end{propp}\n\\begin{proof}\nSet $g:=f\\circ a$.\n\nClearly, the conditions (1') and (2') of Definition~\\ref{21} are satisfied by $g$.\n\nTo prove that $g$ satisfies the condition (4) of Definition~\\ref{21e} fix a point $z\\in D$. Let $\\varphi_{z,f}$, $\\varphi_{z,g}$ be the functions appearing in the condition (4) for $f$ and $g$ respectively. Then $\\varphi_{z,g}=\\varphi_{z,f}\\circ a$. Since $a$ maps $\\TT$ to $\\TT$ diffeomorphically, we have $\\wind\\varphi_{z,g}=\\pm\\wind\\varphi_{z,f}=0$.\n\nIt remains to show that the condition (3') of Definition~\\ref{21} is also satisfied by $g$. Note that the function $\\wi a(\\zeta):=\\zeta\/a(\\zeta)$ has a holomorphic branch of the logarithm in the neighborhood of $\\TT$. This follows from the fact that $\\wind \\wi a=0$, however the existence of the holomorphic branch may be shown quite elementary. Actually, it would suffices to prove that $\\wi a(\\TT)\\neq\\TT$. Expand $a$ as $$a(\\zeta)=e^{it}\\frac{\\zeta-b}{1-\\overline b\\zeta}$$ with some $t\\in\\RR$, $b\\in\\DD$ and observe that $\\widetilde a$ does not attain the value $-e^{-it}$. Indeed, if $\\zeta\/a(\\zeta)=-e^{-it}$ for some $\\zeta\\in\\TT$, then $$\\frac{1-\\overline b\\zeta}{1-b\\overline\\zeta}=-1,$$ so $2=2\\re(b\\overline\\zeta)\\leq 2|b|$, which is impossible.\n\nConcluding, there exists a function $v$ holomorphic in a neighborhood of $\\TT$ such that $$\\frac{\\zeta}{a(\\zeta)}=e^{i v(\\zeta)}.$$ Note that $v(\\TT)\\su\\RR$. Expanding $v$ in Laurent series $$v(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^k,\\ \\zeta\\text{ near }\\TT,$$ we infer that $a_{-k}=\\overline a_k$, $k\\in\\ZZ$. Therefore, $$v(\\zeta)=a_0+\\sum_{k=1}^\\infty 2\\re(a_k\\zeta^k)=\\re\\left(a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k\\right),\\ \\zeta\\in\\TT.$$ Hence, there is a function $h$ holomorphic in the neighborhood of $\\CDD$ such that $v=\\im h$. Put $u:=h-iv$. Then $u\\in\\OO(\\TT)$ and $u(\\TT)\\su\\RR$.\n\nTake $\\rho$ be as in the condition (3') of Definition~\\ref{21} for $f$ and define $$r(\\zeta):=\\rho(a(\\zeta))e^{u(\\zeta)},\\ \\zeta\\in\\TT.$$ Let us compute\n\\begin{eqnarray*}\\zeta r(\\zeta)\\overline{\\nu_D(g(\\zeta))}=\\zeta u^{u(\\zeta)}\\rho(a(\\zeta))\\overline{\\nu_D(f(a(\\zeta)))}&=&\\\\=a(\\zeta)h(\\zeta)\\rho(a(\\zeta))\\overline{\\nu_D(f(a(\\zeta)))}\n&=&h(\\zeta)\\widetilde{f}(a(\\zeta)),\\quad\\zeta\\in\\TT.\n\\end{eqnarray*} Thus $\\zeta\\longmapsto\\zeta r(\\zeta)\\overline{\\nu_D(g(\\zeta))}$ extends holomorphically to a function of class $\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$.\n\\end{proof}\n\n\n\\begin{corr}\\label{28}\nA weak $E$-mapping $f:\\DD\\longrightarrow D$ of a bounded strongly linearly convex domain $D\\su\\CC^n$, $n\\geq 2$, is a unique $\\wi{k}_D$-extremal for $f(\\zeta),f(\\xi)$ \\emph{(}resp. a unique $\\kappa_D$-extremal for $f(\\zeta),f'(\\zeta)$\\emph{)}, where $\\zeta,\\xi\\in\\DD$, $\\zeta\\neq\\xi$.\n\\end{corr}\n\n\\subsection{Generalization of Proposition~\\ref{1}}\nThe results obtained in this subsection will play an important role in the sequel.\n\nWe start with \n\\begin{propp}\\label{4}\nLet $f:\\DD\\longrightarrow D$ be an $E$-mapping. Then the function $f'\\bullet\\widetilde{f}$ is a positive constant.\n\\end{propp}\n\\begin{proof}\nConsider the curve $$\\RR\\ni t\\longmapsto f(e^{it})\\in\\partial D.$$ Its any tangent vector $ie^{it}f'(e^{it})$ belongs to $T_{D}^\\mathbb{R}(f(e^{it}))$, i.e. $$\\re\\langle ie^{it}f'(e^{it}),\\nu_D(f(e^{it}))\\rangle=0.$$ Thus for $\\zeta\\in\\TT$ $$0=\\rho(\\zeta)\\re\\langle i\\zeta f'(\\zeta),\\nu_D(f(\\zeta))\\rangle=-\\im f'(\\zeta)\\bullet\\widetilde{f}(\\zeta),$$ so the holomorphic function $f'\\bullet\\widetilde{f}$ is a real constant $C$.\n\nConsidering the curve $$[0,1+\\eps)\\ni t\\longmapsto f(t)\\in\\overline D$$ for small $\\eps>0$ and noting that $f([0,1))\\su D$, $f(1)\\in\\partial D$, we see that the derivative of $r\\circ f$ at a point $t=1$ is non-negative, where $r$ is a defining function of $D$. Hence $$0\\leq\\re\\langle f'(1),\\nu_D(f(1))\\rangle =\\frac{1}{\\rho(1)} \\re( f'(1)\\bullet\\widetilde{f}(1))=\n\\frac{C}{\\rho(1)},$$ i.e. $C\\geq 0$. For $\\zeta\\in\\TT$\n$$\\frac{f(\\zeta)-f(0)}{\\zeta}\\bullet\\widetilde{f}(\\zeta)=\\rho(\\zeta)\\langle f (\\zeta)-f(0),\\nu_D(f(\\zeta))\\rangle.$$ This function has the winding number equal to $0$. Therefore, the function $$g(\\zeta):=\\frac{f(\\zeta)-f(0)}{\\zeta}\\bullet\\widetilde{f}(\\zeta),$$ which is holomorphic in a neighborhood of $\\CDD$, does not vanish\nin $\\DD$. In particular, $C=g(0)\\neq 0$.\n\\end{proof}\nThe function $\\rho$ is defined up to a constant factor. \\textbf{We choose $\\rho$ so that $ f'\\bullet\\widetilde{f}\\equiv 1$}, i.e. \\begin{equation}\\label{rho}\\rho(\\zeta)^{-1}=\\langle\\zeta f'(\\zeta),\\nu_D(f(\\zeta))\\rangle,\\ \\zeta\\in\\TT.\\end{equation} In that way $\\widetilde{f}$ and $\\rho$ are uniquely determined by $f$.\n\n\\begin{propp}\nAn $E$-mapping $f:\\DD\\longrightarrow D$ is injective in $\\CDD$.\n\\end{propp}\n\n\\begin{proof}The function $f$ has the left-inverse in $\\DD$, so it suffices to check the injectivity on $\\TT$. Suppose that $f(\\zeta_1)=f(\\zeta_2)$ for some $\\zeta_1,\\zeta_2\\in\\TT$, $\\zeta_1\\neq\\zeta_2$, and consider the curves $$\\gamma_j:[0,1]\\ni t\\longmapsto f(t\\zeta_j)\\in\\overline D,\\ j=1,2.$$ Since $$\\re\\langle\\gamma_j'(1),\\nu_D(f(\\zeta_j))\\rangle=\\re\\langle\\zeta_jf'(\\zeta_j),\\nu_D(f(\\zeta_j))\\rangle\n=\\rho(\\zeta_j)^{-1}\\neq 0,$$ the curves $\\gamma_j$ hit $\\pa D$ transversally at their common point $f(\\zeta_1)$. We claim that there exists $C>0$ such that for $t\\in(0,1)$ close to $1$ there is $s_t\\in(0,1)$ satisfying $\\wi k_D(f(t\\zeta_1),f(s_t\\zeta_2))k|z|\\},\\quad k>0,$$ such that $\\gamma_1(t),\\gamma_2(t)\\in A\\cap B$ if $t\\in(0,1)$ is close to $1$. For $z\\in A$ let $k_z>k$ be a positive number satisfying the equality $$|z|=\\frac{-\\re z_1}{k_z}.$$ \n\nNote that for any $a\\in\\gamma_1((0,1))$ sufficiently close to $0$ one may find $b\\in\\gamma_2((0,1))\\cap A\\cap B$ such that $\\re b_1=\\re a_1$. To get a contradiction it suffices to show that $\\wi k_D(a,b)$ is bounded from above by a constant independent on $a$ and $b$. \n\nWe have the following estimate \\begin{multline*}\\wi k_D(a,b)\\leq\\wi k_{\\BB_n-e_1}(a,b)=\\wi k_{\\BB_n}(a+e_1,b+e_1)=\\\\=\\tanh^{-1}\\sqrt{1-\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1 \\rangle|^2}}.\\end{multline*} The last expression is bounded from above if and only if $$\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1\\rangle|^2}$$ is bounded from below by some positive constant. We estimate $$\\frac{(1-|a+e_1|^2)(1-|b+e_1|^2)}{|1-\\langle a+e_1,b+e_1\\rangle|^2}=\\frac{(2\\re a_1+|a|^2)(2\\re b_1+|b|^2)}{|\\langle a, b\\rangle+a_1+\\overline b_1|^2}=$$$$=\\frac{\\left(2\\re a_1+\\frac{(\\re a_1)^2}{k^2_a}\\right)\\left(2\\re a_1+\\frac{(\\re a_1)^2}{k^2_b}\\right)}{|\\langle a, b\\rangle+2\\re a_1+i\\im a_1-i\\im b_1|^2}\\geq\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2|\\langle a, b\\rangle+i\\im a_1-i\\im b_1|^2+2|2\\re a_1|^2}$$$$\\geq\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2(|a||b|+|a|+|b|)^2+8(\\re a_1)^2}=\\frac{(\\re a_1)^2\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2\\left(\\frac{(-\\re a_1)^2}{k^2_ak^2_b}-\\frac{\\re a_1}{k_a}-\\frac{\\re a_1}{k_b}\\right)^2+8(\\re a_1)^2}$$$$=\\frac{\\left(2+\\frac{\\re a_1}{k^2_a}\\right)\\left(2+\\frac{\\re a_1}{k^2_b}\\right)}{2\\left(\\frac{-\\re a_1}{k^2_ak^2_b}+\\frac{1}{k_a}+\\frac{1}{k_b}\\right)^2+8}>\\frac{1}{2(1+2\/k)^2+8}.$$ This finishes the proof.\n\\end{proof}\n\n\\medskip\n\nAssume that we are in the settings of Proposition~\\ref{1} and $D$ has real analytic boundary. Our aim is to replace $W$ with a neighborhood of $\\ov D$.\n\n\\begin{remm}\\label{przed34}\nFor $\\zeta_0\\in\\DD_f$ we have $G(f(\\zeta_0),\\zeta_0)=0$ and $\\frac{\\partial G}{\\partial\\zeta}(f(\\zeta_0),\\zeta_0)=-1$. By the Implicit Function Theorem there exist neighborhoods $U_{\\zeta_0},V_{\\zeta_0}$ of $f(\\zeta_0),\\zeta_0$ respectively and a holomorphic function $F_{\\zeta_0}:U_{\\zeta_0}\\longrightarrow V_{\\zeta_0}$ such that for any $z\\in U_{\\zeta_0}$ the point $F_{\\zeta_0}(\\zeta)$ is the unique solution of the equation $G(z,\\zeta)=0$, $\\zeta\\in V_{\\zeta_0}$.\n\nIn particular, if $\\zeta_0\\in\\DD$ then $F_{\\zeta_0}=F$ near $f(\\zeta_0)$.\n\\end{remm}\n\n\\begin{propp}\\label{34}\nLet $f:\\DD\\longrightarrow D$ be an $E$-mapping. Then there exist arbitrarily small neighborhoods $U$, $V$ of $\\overline D$, $\\CDD$ respectively such that for any $z\\in U$ the equation $G(z,\\zeta)=0$, $\\zeta\\in V$, has exactly one solution.\n\\end{propp}\n\\begin{proof} In view of Proposition~\\ref{1} and Remark~\\ref{przed34} it suffices to prove that there exist neighborhoods $U$, $V$ of $\\overline D$, $\\CDD$ respectively such that for any $z\\in U$ the equation $G(z,\\cdotp)=0$ has at most one solution $\\zeta\\in V$.\n\nAssume the contrary. Then for any neighborhoods $U$ of $\\overline D$ and $V$ of $\\CDD$ there are $z\\in U$, $\\zeta_1,\\zeta_2\\in V$, $\\zeta_1\\neq\\zeta_2$ such that $G(z,\\zeta_1)=G(z,\\zeta_2)=0$. For $m\\in\\NN$ put $$U_m:=\\{z\\in\\CC^n:\\dist(z,D)<1\/m\\},$$ $$V_m:=\\{\\zeta\\in\\CC:\\dist(\\zeta,\\DD)<1\/m\\}.$$ There exist $z_m\\in U_m$, $\\zeta_{m,1},\\zeta_{m,2}\\in V_m$, $\\zeta_{m,1}\\neq\\zeta_{m,2}$ such that $G(z_m,\\zeta_{m,1})=G(z_m,\\zeta_{m,2})=0$. Passing to a subsequence we may assume that $z_m\\to z_0\\in\\ov D$. Analogously we may assume $\\zeta_{m,1}\\to\\zeta_1\\in \\CDD$ and $\\zeta_{m,2}\\to\\zeta_2\\in\\CDD$. Clearly, $G(z_0,\\zeta_1)=G(z_0,\\zeta_2)=0$. Let us consider few cases.\n\n1) If $\\zeta_1,\\zeta_2\\in\\TT$, then $G(z_0,\\zeta_j)=0$ is equivalent to $$\\langle z_0-f(\\zeta_j), \\nu_D(f(\\zeta_j))\\rangle=0,\\ j=1,2,$$ consequently $z_0-f(\\zeta_j)\\in T^{\\CC}_D(f(\\zeta_j))$. By the strong linear convexity of $D$ we get $z_0=f(\\zeta_j)$. But $f$ is injective in $\\CDD$, so $\\zeta_1=\\zeta_2=:\\zeta_0$. It follows from Remark~\\ref{przed34} that in a sufficiently small neighborhood of $(z_0,\\zeta_0)$ all solutions of the equation $G(z,\\zeta)=0$ are of the form $(z,F_{\\zeta_0}(z))$. Points $(z_m,\\zeta_{m,1})$ and $(z_m,\\zeta_{m,2})$ belong to this neighborhood for large $m$, which gives a contradiction.\n\n2) If $\\zeta_1\\in\\TT$ and $\\zeta_2\\in\\DD$, then analogously as above we deduce that $z_0=f(\\zeta_1)$. Let us take an arbitrary sequence $\\{\\eta_m\\}\\su\\DD$ convergent to $\\zeta_1$. Then $f(\\eta_m) \\in D$ and $f(\\eta_m)\\to z_0$, so the sequence $G(f(\\eta_m),\\cdotp)$ converges to $G(z_0,\\cdotp)$ uniformly on $\\DD$. Since $G(z_0,\\cdotp)\\not\\equiv 0$, $G(z_0,\\zeta_2)=0$ and $\\zeta_2\\in\\DD$, we deduce from Hurwitz Theorem that for large $m$ the functions $G(f(\\eta_m),\\cdotp)$ have roots $\\theta_m\\in\\DD$ such that $\\theta_m\\to\\zeta_2$. Hence $G(f(\\eta_m),\\theta_m)=0$ and from the uniqueness of solutions in $D\\times\\DD$ (Proposition~\\ref{1}) we have $$\\theta_m=F(f(\\eta_m))=\\eta_m.$$ This is a contradiction, because the left side tends to $\\zeta_2$ and the right one to $\\zeta_1$, as $m\\to\\infty$.\n\n3) We are left with the case $\\zeta_1,\\zeta_2\\in\\DD$.\nIf $z_0\\in\\overline{D}\\setminus f(\\TT)$ then $z_0\\in W$. In $W\\times\\DD$ all solutions of the equation $G=0$ are of the form $(z,F(z))$, $z\\in W$. But for large $m$ the points $(z_m,\\zeta_{m,1})$, $(z_m,\\zeta_{m,2})$ belong to $W\\times\\DD$, which is a contradiction with the uniqueness.\n\nIf $z_0\\in f(\\TT)$, then $z_0=f(\\zeta_0)$ for some $\\zeta_0\\in\\TT$. Clearly, $G(f(\\zeta_0),\\zeta_0)=0$, whence $G(z_0,\\zeta_0)=G(z_0,\\zeta_1)=0$ and $\\zeta_0\\in\\TT$, $\\zeta_1\\in \\DD$. This is just the case 2), which has been already considered.\n\\end{proof}\n\n\n\\begin{corr} There are neighborhoods $U$, $V$ of $\\overline D$ and $\\CDD$ respectively with $V\\Subset\\DD_f$, such that the function $F$ extends holomorphically on $U$. Moreover, all solutions of the equation $G|_{U\\times V}=0$ are of the form $(z,F(z))$, $z\\in U$.\n\nIn particular, $F\\circ f=\\id_{V}$.\n\\end{corr}\n\n\n\n\n\n\n\n\n\n\n\n\\section{H\\\"older estimates}\\label{22}\n\n\\begin{df}\\label{30} For a given $c>0$ let the family $\\mathcal{D}(c)$ consist of all pairs $(D,z)$, where $D\\su\\CC^n$, $n\\geq 2$, is a bounded pseudoconvex domain with real $\\mathcal C^2$ boundary and $z\\in D$, satisfying\n\\begin{enumerate}\n\\item $\\dist(z,\\partial D)\\geq 1\/c$;\n\\item the diameter of $D$ is not greater than $c$ and $D$ satisfies the interior ball condition with a radius $1\/c$;\n\\item for any $x,y\\in D$ there exist $m\\leq 8 c^2$ and open balls $B_0,\\ldots,B_m\\subset D$ of radius $1\/(2c)$ such that $x\\in B_0$, $y\\in B_m$ and the distance between the centers of the balls $B_j$, $B_{j+1}$ is not greater than $1\/(4c)$ for $j=0,\\ldots,m-1$;\n\\item for any open ball $B\\subset\\mathbb{C}^n$ of radius not greater than $1\/c$, intersecting non-emptily with $\\pa D$, there exists a mapping $\\Phi\\in\\OO(\\overline{D},\\mathbb{C}^n)$ such that\n\\begin{enumerate}\n\\item for any $w\\in\\Phi(B\\cap\\partial D)$ there is a ball of radius $c$ containing $\\Phi(D)$ and tangent to $\\partial\\Phi(D)$ at $w$ (let us call it the ``exterior ball condition'' with a radius $c$);\n\\item $\\Phi$ is biholomorphic in a neighborhood of $\\ov B$ and $\\Phi^{-1}(\\Phi(B))=B$;\n\\item entries of all matrices $\\Phi'$ on $B\\cap\\ov D$ and $(\\Phi^{-1})'$ on $\\Phi(B\\cap\\overline{D})$ are bounded in modulus by $c$;\n\\item $\\dist(\\Phi(z),\\partial\\Phi(D))\\geq 1\/c$;\n\\end{enumerate}\n\\item the normal vector $\\nu_D$ is Lipschitz with a constant $2c$, that is $$|\\nu_D(a)-\\nu_D(b)|\\leq 2c|a-b|,\\ a,b\\in \\partial D;$$\n\\item the $\\eps$-hull of $D$, i.e. a domain $D_{\\eps}:=\\{w\\in\\mathbb C^n:\\dist (w,D)<\\eps\\}$, is strongly pseudoconvex for any $\\eps\\in (0,1\/c).$\n\\end{enumerate}\n\\end{df}\n\nRecall that the {\\it interior ball condition} with a radius $r>0$ means that for any point $a\\in\\pa D$ there is $a'\\in D$ and a ball $B_n(a',r)\\su D$ tangent to $\\pa D$ at $a$. Equivalently $$D=\\bigcup_{a'\\in D'}B_n(a',r)$$ for some set $D'\\su D$.\n\nIt may be shown that (2) and (5) may be expressed in terms of boundedness of the normal curvature, boundedness of a domain and the condition (3). This however lies beyond the scope of this paper and needs some very technical arguments so we omit the proof of this fact. The reasons why we decided to use (2) in such a form is its connection with the condition (3) (this allows us to simplify the proof in some places).\n\n\\begin{rem}\\label{con}\nNote that any convex domain satisfying conditions (1)-...-(4) of Definition~\\ref{30} satisfies conditions (5) and (6), as well.\n\nActually, it follows from (2) that for any $a\\in\\pa D$ there exists a ball $B_n(a',1\/c)\\su D$ tangent to $\\pa D$ at $a$. Then $$\\nu_D(a)=\\frac{a'-a}{|a'-a|}=c(a'-a).$$ Hence $$|\\nu_D(a)-\\nu_D(b)|=c|a'-a-b'+b|=c|a'-b'-(a-b)|\\leq c|a'-b'|+c|a-b|.$$ Since $D$ is convex, we have $|a'-b'|\\leq|a-b|$, which gives (5).\n\nThe condition (6) is also clear --- for any $\\eps>0$ an $\\eps$-hull of a strongly convex domain is strongly convex.\n\\end{rem}\n\n\\begin{rem}\nFor a convex domain $D$ the condition (3) of Definition \\ref{30} amounts to the condition (2).\n\nIndeed, for two points $x,y\\in D$ take two balls of radius $1\/(2c)$ containing them and contained in $D$. Then divide the interval between the centers of the balls into $[4c^2]+1$ equal parts and take balls of radius $1\/(2c)$ with centers at the points of the partition.\n\nNote also that if $D$ is strongly convex and satisfies the interior ball condition with a radius $1\/c$ and the exterior ball condition with a radius $c$, one can take $\\Phi:=\\id_{\\CC^n}$.\n\\end{rem}\n\n\n\\begin{rem}\\label{D(c),4}\nFor a strongly pseudoconvex domain $D$ and $c'>0$ and for any $z\\in D$ such that $\\dist(z,\\partial D)>1\/c'$ there exists $c=c(c')>0$ satisfying $(D,z)\\in\\mathcal{D}(c)$.\n\nIndeed, the conditions (1)-...-(3) and (5)-(6) are clear. Only (4) is non-trivial.\n\nThe construction of the mapping $\\Phi$ amounts to the construction of Forn\\ae ss peak functions. Actually, apply directly Proposition 1 from \\cite{For} to any boundary point of $\\partial D$ (obviously $D$ has a Stein neighborhood basis). This gives a covering of $\\partial D$ with a finite number of balls $B_j$, maps $\\Phi_j\\in\\OO(\\overline{D},\\mathbb{C}^n)$ and strongly convex $C^\\infty$-smooth domains $C_j$, $j=1,\\ldots, N$, such that\n\\begin{itemize}\\item $\\Phi_j(D)\\subset C_j$;\n\\item $\\Phi_j(\\ov D)\\subset\\ov C_j$;\n\\item $\\Phi_j(B_j\\setminus\\ov D)\\subset\\mathbb C^n\\setminus\\ov C_j$;\n\\item $\\Phi_j^{-1}(\\Phi_j(B_j))=B_j$;\n\\item $\\Phi_j|_{B_j}: B_j\\longrightarrow \\Phi_j(B_j)$ is biholomorphic.\n\\end{itemize} Therefore, one may choose $c>0$ such that every $C_j$ satisfies the exterior ball condition with $c$, i.e. for any $x\\in \\partial C_j$ there is a ball of radius $c$ containing $C_j$ and tangent to $\\partial C_j$ at $x$, every ball of radius $1\/c$ intersecting non-emptily with $\\pa D$ is contained in some $B_j$ (here one may use a standard argument invoking the Lebesgue number) and the conditions (c), (d) are also satisfied (with $\\Phi:=\\Phi_j$).\n\\end{rem}\n\n\nIn this section we use the words `uniform', `uniformly' if $(D,z)\\in \\mathcal D(c)$. This means that estimates will depend only on $c$ and will be independent on $D$ and $z$ if $(D,z)\\in\\mathcal{D}(c)$ and on $E$-mappings of $D$ mapping $0$ to $z$. Moreover, in what follows we assume that $D$ is a strongly linearlu convex domain with real-analytic boundary.\n\n\\begin{prop}\\label{7}\nLet $f:(\\mathbb{D},0)\\longrightarrow(D,z)$ be an $E$-mapping. Then $$\\dist(f(\\zeta),\\partial D)\\leq C(1-|\\zeta|),\\ \\zeta\\in\\CDD$$ with $C>0$ uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\\end{prop}\n\\begin{proof} There exists a uniform $C_1$ such that $$\\text{if }\\dist(w,\\partial D)\\geq 1\/c\\text{ then }k_D(w,z)\\varepsilon$ for some $\\varepsilon>0$ independent on $x$. Thus $$\\frac{\\delta(x)}{\\re x_1}=1+O(|x|)\\text{ as }x\\to 0\\text{ transversally. }$$ Consequently \\begin{equation}\\label{50}-\\re x_1\\leq 2\\dist(x,\\partial\\Phi(D))\\text{ as }x\\to 0\\text{ transversally. }\\end{equation}\n\nWe know that $t\\longmapsto f(t\\zeta_0)$ hits $\\partial D$ transversally. Therefore, $t\\longmapsto h(t\\zeta_0)$ hits $\\partial \\Phi(D)$ transversally, as well. Indeed, we have \\begin{multline}\\label{hf}\\left\\langle\\left.\\frac{d}{dt}h(t\\zeta_0)\\right|_{t=1},\\nu_{\\Phi(D)}(h(\\zeta_0))\\right\\rangle=\\left\\langle \\Phi'(0)f'(\\zeta_0)\\zeta_0,\\frac{(\\Phi^{-1})'(0)^*\\nabla r(0)}{|(\\Phi^{-1})'(0)^*\\nabla r(0)|}\\right\\rangle=\\\\=\\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nabla r(0)\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}=\\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nu_D(f(\\zeta_0))|\\nabla r(0)|\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}.\n\\end{multline}\nwhere $r$ is a defining function of $D$. In particular,\n\\begin{multline*} \\re \\left\\langle\\left.\\frac{d}{dt}h(t\\zeta_0)\\right|_{t=1},\\nu_{\\Phi(D)}(h(\\zeta_0))\\right\\rangle=\\re \\frac{\\langle\\zeta_0 f'(\\zeta_0),\\nu_D(f(\\zeta_0))|\\nabla r(0)|\\rangle}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}=\\\\=\\frac{\\rho(\\zeta_0)^{-1}|\\nabla r(0)|}{|(\\Phi'(0)^{-1})^*\\overline{\\nabla r(0)}|}\\neq 0.\\end{multline*} This proves that $t\\longmapsto h(t\\zeta_0)$ hits $\\partial\\Phi(D)$ transversally.\n\n\nConsequently, we may put $x=h(t\\zeta_0)$ into \\eqref{50} to get \\begin{equation}\\label{hf1}\\frac{-2\\re h_1(t\\zeta_0)}{1-|t\\zeta_0|^2}\\leq\\frac{4\\dist(h(t\\zeta_0),\\partial\\Phi(D))}\n{1-|t\\zeta_0|^2},\\ t\\to 1.\\end{equation}\nBut $\\Phi$ is a biholomorphism near $0$, so \\begin{equation}\\label{nfr}\\frac{4\\dist(h(t\\zeta_0),\\partial\\Phi(D))}{1-|t\\zeta_0|^2}\\leq C_3\\frac{\\dist(f(t\\zeta_0),\\partial D)}{1-|t\\zeta_0|},\\ t\\to 1,\\end{equation} where $C_3$ is a uniform constant depending only on $c$ (thanks to the condition (4)(c) of Definition~\\ref{30}). By Proposition \\ref{7}, the term on the right side of~\\eqref{nfr} does not exceed some uniform constant.\n\nIt follows from \\eqref{hf} that \\begin{multline*}\\rho(\\zeta_0)^{-1}=|\\langle f'(\\zeta_0)\\zeta_0,\\nu_D(f(\\zeta_0))\\rangle|\\leq C_4|\\langle h'(\\zeta_0), \\nu_{\\Phi(D)}(h(\\zeta_0))\\rangle|=\\\\=C_4|h_1'(\\zeta_0)|=\\lim_{t\\to 1}C_4|h_1'(t\\zeta_0)|\\end{multline*} with a uniform $C_4$ (here we use the condition (4)(c) of Definition~\\ref{30} again).\nCombining \\eqref{schh1}, \\eqref{hf1} and \\eqref{nfr} we get the upper estimate for $\\rho(\\zeta_0)^{-1}.$\n\nNow we are proving the lower estimate. Let $r$ be the signed boundary distance to $\\partial D$. For $\\varepsilon=1\/c$ the function $$\\varrho(w):=-\\log(\\varepsilon-r(w))+\\log\\varepsilon,\\ w\\in\nD_\\varepsilon,$$ where $D_\\varepsilon$ is an $\\varepsilon$-hull of $D$, is plurisubharmonic and defining for $D$. Indeed, we have $$-\\log(\\varepsilon-r(w))=-\\log\\dist(w,\\partial D_\\varepsilon),\\ w\\in D_\\varepsilon$$ and $D_\\varepsilon$ is pseudoconvex.\n\nTherefore, a function $$v:=\\varrho\\circ f:\\overline{\\mathbb{D}}\\longrightarrow(-\\infty,0]$$ is subharmonic on $\\DD$. Moreover, since $f$ maps $\\TT$ in $\\partial D$ we infer that $v=0$ on $\\TT$. Moreover, since $|f(\\lambda)-z|0$ such that $$|\\rho(\\zeta_1)-\\rho(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\TT,\\ |\\zeta_1-\\zeta_2|0$. There exists a function $\\psi\\in\\cC^1(\\TT,[0,1])$ such that $\\psi=1$ on $\\TT\\cap B_n(\\zeta_1,2C_1)$ and $\\psi=0$ on $\\TT\\setminus B_n(\\zeta_1,3C_1)$. Then the function $\\phi:\\TT\\longrightarrow\\CC$ defined by $$\\varphi:=(\\overline{\\nu_{D,1}\\circ f}-1)\\psi+1$$ satisfies\n\\begin{enumerate}\n\\item $\\varphi(\\zeta)=\\overline{\\nu_{D,1}(f(\\zeta))}$, $\\zeta\\in\\TT\\cap B_n(\\zeta_1,2C_1)$;\n\\item $|\\varphi(\\zeta)-1|<1\/2$, $\\zeta\\in\\TT$;\n\\item $\\phi$ is uniformly $1\/2$-H\\\"older continuous on $\\TT$, i.e. it is $1\/2$-H\\\"older continuous with a uniform constant (remember that $\\psi$ was chosen uniformly).\n\\end{enumerate}\n\nFirst observe that $\\log\\varphi$ is well-defined. Using using properties listed above we deduce that $\\log\\varphi$ and $\\im\\log\\varphi$ are uniformly $1\/2$-H\\\"older continuous on $\\TT$, as well. The function $\\im\\log\\varphi$ can be extended continuously to a function $v:\\CDD\\longrightarrow\\RR$, harmonic in $\\DD$. There is a function $h\\in\\mathcal O(\\DD)$ such that $v=\\im h$ in $\\DD$. Taking $h-\\re h(0)$ instead of $h$, one can assume that $\\re h(0)=0$. By Theorem \\ref{priv} applied to $ih$, we get that the function $h$ extends continuously on $\\CDD$ and $h$ is uniformly $1\/2$-H\\\"older continuous in $\\CDD$. Hence the function $u:=\\re h:\\CDD\\longrightarrow\\RR$ is uniformly $1\/2$-H\\\"older continuous in $\\CDD$ with a uniform constant $C_2$. Furthermore, $u$ is uniformly bounded in $\\CDD$, since $$|u(\\zeta)|=|u(\\zeta)-u(0)|\\leq C_2\\sqrt{|\\zeta|},\\ \\zeta\\in\\CDD.$$\n\nLet $g(\\zeta):=\\wi{f}_1(\\zeta)e^{-h(\\zeta)}$ and $G(\\zeta):=g(\\zeta)\/\\zeta$. Then $g\\in\\mathcal O(\\DD)\\cap\\mathcal C(\\overline{\\DD})$ and $G\\in\\mathcal O(\\DD_*)\\cap\\mathcal C((\\overline{\\DD})_*)$. Note that for $\\zeta\\in\\TT$ $$|g(\\zeta)|=|\\zeta\n\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-h(\\zeta)}|\\leq\\rho(\\zeta)e^{-u(\\zeta)},$$ which, combined with\nProposition \\ref{9}, the uniform boundedness of $u$ and the maximum principle, gives a uniform boundedness of $g$ in $\\CDD$. The function $G$ is uniformly bounded in $\\overline{\\DD}\\cap B_n(\\zeta_1,2C_1)$. Moreover, for $\\zeta\\in\\TT\\cap B_n(\\zeta_1,2C_1)$ \\begin{eqnarray*} G(\\zeta)&=&\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-u(\\zeta)-i\\im\\log \\phi(\\zeta)}=\\\\&=&\\rho(\\zeta)\\overline{\\nu_{D,1}(f(\\zeta))}e^{-u(\\zeta)+\\re\\log\\phi(\\zeta)}e^{-\\log\\phi(\\zeta)}\n=\\rho(\\zeta)e^{-u(\\zeta)+\\re\\log\\phi(\\zeta)}\\in\\mathbb{R}.\\end{eqnarray*} By the Reflection Principle one can extend $G$ holomorphically past $\\TT\\cap B_n(\\zeta_1,2C_1)$ to a function (denoted by the same letter) uniformly bounded in $B_n(\\zeta_1,2C_2)$, where a constant $C_2$ is uniform. Hence, from the Cauchy formula, $G$ is uniformly Lipschitz continuous in $B_n(\\zeta_1,C_2)$, consequently uniformly $1\/2$-H\\\"older continuous in $B_n(\\zeta_1,C_2)$.\n\nFinally, the functions $G$, $h$, $\\nu_{D,1}\\circ f$ are uniformly $1\/2$-H\\\"older continuous on $\\TT\\cap B_n(\\zeta_1,C_2)$, $|\\nu_{D,1}\\circ f|>1\/2$ on $\\TT\\cap B_n(\\zeta_1,C_2)$, so the function $\\rho=Ge^h\/\\overline{\\nu_{D,1}\\circ f}$ is uniformly $1\/2$-H\\\"older continuous on $\\TT\\cap B_n(\\zeta_1,C_2)$.\n\\end{proof}\n\n\n\\begin{prop}\\label{10b}\nLet $f:(\\DD,0)\\longrightarrow (D,z)$ be an $E$-mapping.\nThen $$|\\wi{f}(\\zeta_1)-\\wi{f}(\\zeta_2)|\\leq C\\sqrt{|\\zeta_1-\\zeta_2|},\\ \\zeta_1,\\zeta_2\\in\\overline{\\DD},$$ where\n$C$ is uniform if $(D,z)\\in\\mathcal{D}(c)$.\n\n\\end{prop}\n\\begin{proof}\nBy Propositions \\ref{8} and \\ref{10a} we have desired inequality for $\\zeta_1,\\zeta_2\\in\\TT$. Theorem \\ref{lit2} finishes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Openness of $E$-mappings' set}\\label{27}\nWe shall show that perturbing a little a domain $D$ equipped with an $E$-mapping, we obtain a domain which also has an $E$-mapping, being close to a given one.\n\n\\subsection{Preliminary results}\n\n\\begin{propp}\\label{11}\nLet $f:\\mathbb{D}\\longrightarrow D$ be an $E$-mapping. Then there exist domains $G,\\wi D,\\wi G\\subset\\CC^n$ and a biholomorphism $\\Phi:\\wi D\\longrightarrow\\wi G$ such that\n\\begin{enumerate}\n\\item $\\wi D,\\wi G$ are neighborhoods of $\\overline D,\\overline G$ respectively;\n\\item $\\Phi(D)=G$;\n\\item $g(\\zeta):=\\Phi(f(\\zeta))=(\\zeta,0,\\ldots,0),\\ \\zeta\\in\\CDD$;\n\\item $\\nu_G(g(\\zeta))=(\\zeta,0,\\ldots,0),\\ \\zeta\\in\\TT$;\n\\item for any $\\zeta\\in\\TT$, a point $g(\\zeta)$ is a point of the strong linear convexity of $G$.\n\\end{enumerate}\n\\end{propp}\n\\begin{proof}\nLet $U,V$ be the sets from Proposition \\ref{34}. We claim that after a linear change of coordinates one can assume that $\\widetilde{f}_1,\\widetilde{f}_2$ do not have common zeroes in $V$.\n\nSince $ f'\\bullet\\widetilde{f}=1$, at least one of the functions $\\wi f_1,\\ldots,\\wi f_n$, say $\\wi f_1$, is not identically equal to $0$. Let $\\lambda_1,\\ldots,\\lambda_m$ be all zeroes of $\\wi f_1$ in $V$. We may find $\\alpha\\in\\CC^n$ such that $$(\\alpha_1\\wi f_1+\\ldots+\\alpha_n\\wi f_n)(\\lambda_j)\\neq 0,\\ j=1,\\ldots,m.$$ Otherwise, for any $\\alpha\\in\\CC^n$ there would exist $j\\in\\{1,\\ldots,m\\}$ such that $\\alpha\\bullet\\wi f(\\lambda_j)=0$, hence $$\\CC^n=\\bigcup_{j=1}^m\\{\\alpha\\in\\CC^n:\\ \\alpha\\bullet\\wi f(\\lambda_j)=0\\}.$$ The sets $\\{\\alpha\\in\\CC^n:\\alpha \\bullet \\wi f(\\lambda_j)=0\\}$, $j=1,\\ldots,m$, are the $(n-1)$-dimensional complex hyperplanes, so their finite sum cannot be the space $\\CC^n$.\n\nOf course, at least one of the numbers $\\alpha_2,\\ldots,\\alpha_n$, say $\\alpha_2$, is non-zero. Let\n$$A:=\\left[\\begin{matrix}\n1 & 0 & 0 & \\cdots & 0\\\\\n\\alpha_1 & \\alpha_2 & \\alpha_3 &\\cdots & \\alpha_n\\\\\n0 & 0 & 1 & \\cdots & 0\\\\\n\\vdots & \\vdots & \\vdots &\\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 1\n\\end{matrix}\\right],\\quad B:=(A^T)^{-1}.$$ We claim that $B$ is a change of coordinates we are looking for. If $r$ is a defining function of $D$ then $r\\circ B^{-1}$ is a defining function of $B_n(D)$, so $B_n(D)$ is a bounded strongly linearly convex domain with real analytic boundary. Let us check that $Bf$ is an $E$-mapping of $B_n(D)$ with associated mappings \\begin{equation}\\label{56}A\\wi f\\in\\OO(\\CDD)\\text{\\ \\ and\\ \\ }\\rho\\frac{|A\\overline{\\nabla r\\circ f}|}{|\\nabla r\\circ f|}\\in\\CLW(\\TT).\\end{equation} The conditions (1) and (2) of Definition~\\ref{21} are clear. For $\\zeta\\in\\TT$ we have \\begin{equation}\\label{57}\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}=\\frac{\\overline{\\nabla(r\\circ B^{-1})(Bf(\\zeta))}}{|\\nabla(r\\circ B^{-1})(Bf(\\zeta))|}=\\frac{(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}=\\frac{A\\overline{\\nabla r(f(\\zeta))}}{|A\\overline{\\nabla r(f(\\zeta))}|},\\end{equation} so\n\\begin{equation}\\label{58}\\zeta\\rho(\\zeta)\\frac{|A\\overline{\\nabla r(f(\\zeta))}|}{|\\nabla r(f(\\zeta))|}\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}=\\zeta\\rho(\\zeta)A\\overline{\\nu_D(f(\\zeta))}=A\\wi f(\\zeta).\\end{equation} Moreover, for $\\zeta\\in\\TT$, $z\\in D$ \\begin{multline*}\\langle Bz-Bf(\\zeta), \\nu_{B_n(D)}(Bf(\\zeta))\\rangle=\\overline{\\nu_{B_n(D)}(Bf(\\zeta))}^T(Bz-Bf(\\zeta))=\\\\=\\frac{\\overline{\\nabla r(f(\\zeta))}^TB^{-1}B_n(z-f(\\zeta))}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}=\\frac{|\\nabla r(f(\\zeta))|}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}\\overline{\\nu_D(f(\\zeta))}^T(z-f(\\zeta))=\\\\=\\frac{|\\nabla r(f(\\zeta))|}{|(B^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}\\langle z-f(\\zeta), \\nu_D(f(\\zeta))\\rangle.\n\\end{multline*}\nTherefore, $B$ is a desired linear change of coordinates, as claimed.\n\nIf necessary, we shrink the sets $U,V$ associated with $f$ to sets associated with $Bf$. There exist holomorphic mappings $h_1,h_2:V\\longrightarrow\\mathbb{C}$ such that\n$$h_1\\widetilde{f}_1+h_2\\widetilde{f}_2\\equiv 1\\text{ in }V.$$ Generally, it is a well-known fact for functions on pseudoconvex domains, however in this case it may be shown quite elementarily. Indeed, if $\\widetilde{f}_1\\equiv 0$ or $\\widetilde{f}_2\\equiv 0$ then it is obvious. In the opposite case, let $\\widetilde{f}_j=F_jP_j$, $j=1,2$, where $F_j$ are holomorphic, non-zero in $V$ and $P_j$ are polynomials with all (finitely many) zeroes in $V$. Then $P_j$ are relatively prime, so there are polynomials $Q_j$, $j=1,2$, such that $$Q_1P_1+Q_2P_2\\equiv 1.$$ Hence $$\\frac{Q_1}{F_1}\\widetilde{f}_1+\\frac{Q_2}{F_2}\\widetilde{f}_2\\equiv 1\\ \\text{ in }V.$$\n\nConsider the mapping $\\Psi:V\\times\\mathbb{C}^{n-1}\\longrightarrow\\mathbb{C}^n$ given by\n\\begin{equation}\\label{et2}\n\\Psi_1(Z):=f_1(Z_1)-Z_2\\widetilde{f}_2(Z_1)-h_1(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1),\n\\end{equation}\n\\begin{equation}\\label{et3}\n\\Psi_2(Z):=f_2(Z_1)+Z_2\\widetilde{f}_1(Z_1)-h_2(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1),\n\\end{equation}\n\\begin{equation}\\label{et4}\n\\Psi_j(Z):=f_j(Z_1)+Z_j,\\ j=3,\\ldots,n.\n\\end{equation}\n\nWe claim that $\\Psi$ is biholomorphic in $\\Psi^{-1}(U)$. First of all observe that $\\Psi^{-1}(\\{z\\})\\neq\\emptyset$ for any $z\\in U$. Indeed, by Proposition \\ref{34} there exists (exactly one) $Z_1\\in V$ such that $$(z-f(Z_1))\\bullet\\widetilde{f}(Z_1)=0.$$ The numbers $Z_j\\in\\CC$, $j=3,\\ldots,n$ are determined uniquely by the equations $$Z_j=z_j-f_j(Z_1).$$ At least one of the numbers $\\wi f_1(Z_1),\\wi f_2(Z_1)$, say $\\wi f_1(Z_1)$, is non-zero. Let $$Z_2:=\\frac{z_2-f_2(Z_1)+h_2(Z_1)\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1)}{\\wi f_1(Z_1)}.$$ Then we easily check that the equality $$z_1=f_1(Z_1)-Z_2\\widetilde{f}_2(Z_1)-h_1(Z_1)\n\\sum_{j=3}^{n}Z_j\\widetilde{f}_j(Z_1)$$ is equivalent to $(z-f(Z_1))\\bullet\\widetilde{f}(Z_1)=0$, which is true.\n\nTo finish the proof of biholomorphicity of $\\Psi$ in $\\Psi^{-1}(U)$ it suffices to check that $\\Psi$ is injective in $\\Psi^{-1}(U)$. Let us take $Z,W$ such that $\\Psi(Z)=\\Psi(W)=z\\in U$. By a direct computation both $\\zeta=Z_1\\in V$ and $\\zeta=W_1\\in V$ solve the equation\n$$(z-f(\\zeta))\\bullet\\widetilde{f}(\\zeta)=0.$$ From Proposition \\ref{34} we infer that it has exactly one solution. Hence $Z_1=W_1$. By \\eqref{et4} we have $Z_j=W_j$ for $j=3,\\ldots,n$. Finally $Z_2=W_2$ follows from\none of the equations \\eqref{et2}, \\eqref{et3}. Let $G:=\\Psi^{-1}(D)$, $\\wi D:=U$, $\\wi G:=\\Psi^{-1}(U)$, $\\Phi:=\\Psi^{-1}$.\n\nNow we are proving that $\\Phi$ has desired properties. We have $$\\Psi_j(\\zeta,0,\\ldots,0)=f_j(\\zeta),\\ j=1,\\ldots,n,$$ so $\\Phi(f(\\zeta))=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\CDD$. Put $g(\\zeta):=\\Phi(f(\\zeta))$, $\\zeta\\in\\CDD$. Note that the entries of the matrix $\\Psi'(g(\\zeta))$ are $$\\frac{\\partial\\Psi_1}{\\partial Z_1}(g(\\zeta))=f_1'(\\zeta),\\ \\frac{\\partial\\Psi_1}{\\partial Z_2}(g(\\zeta))=-\\widetilde{f}_2(\\zeta),\\ \\frac{\\partial\\Psi_1}{\\partial Z_j}(g(\\zeta))=-h_1(\\zeta)\\widetilde{f}_j(\\zeta),\\ j\\geq 3,$$$$\\frac{\\partial\\Psi_2}{\\partial Z_1}(g(\\zeta))=f_2'(\\zeta),\\ \\frac{\\partial\\Psi_2}{\\partial Z_2}(g(\\zeta))=\\widetilde{f}_1(\\zeta),\\ \\frac{\\partial\\Psi_2}{\\partial Z_j}(g(\\zeta))=-h_2(\\zeta)\\widetilde{f}_j(\\zeta),\\ j\\geq 3,$$$$\\frac{\\partial\\Psi_k}{\\partial Z_1}(g(\\zeta))=f_k'(\\zeta),\\ \\frac{\\partial\\Psi_k}{\\partial Z_2}(g(\\zeta))=0,\\ \\frac{\\partial\\Psi_k}{\\partial Z_j}(g(\\zeta))=\\delta^{k}_{j},\\ j,k\\geq 3.$$ Thus $\\Psi '(g(\\zeta))^T\\wi f(\\zeta)=(1,0,\\ldots,0)$, $\\zeta\\in\\CDD$ (since $f'\\bullet\\wi f=1$). Let us take a defining function $r$ of $D$. Then $r\\circ\\Psi$ is a defining function of $G$. Therefore, \\begin{multline*}\\nu_G(g(\\zeta))=\\frac{\\nabla(r\\circ\\Psi)(g(\\zeta))}{|\\nabla(r\\circ\\Psi)(g(\\zeta))|}=\n\\frac{\\overline{\\Psi'(g(\\zeta))}^T\\nabla r(f(\\zeta))}{|\\overline{\\Psi'(g(\\zeta))}^T\\nabla r(f(\\zeta))|}=\\\\=\\frac{\\overline{\\Psi'(g(\\zeta))}^T\\ov{\\frac{\\wi f(\\zeta)}{\\zeta\\rho(\\zeta)}}|\\nabla r(f(\\zeta))|}{\\left|\\overline{\\Psi'(g(\\zeta))}^T\\ov{\\frac{\\wi f(\\zeta)}{\\zeta\\rho(\\zeta)}}|\\nabla r(f(\\zeta))|\\right|}=g(\\zeta),\\ \\zeta\\in\\TT.\\end{multline*}\n\nIt remains to prove the fifth condition. By Definition \\ref{29}(2) we have to show that \\begin{equation}\\label{sgf}\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial\\overline{z}_k}(g(\\zeta))X_{j}\\overline{X}_{k}>\\left|\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|\\end{equation} for $\\zeta\\in\\TT$ and $X\\in(\\CC^{n})_*$ with\n$$\\sum_{j=1}^n\\frac{\\partial(r\\circ\\Psi)}{\\partial z_j}(g(\\zeta))X_{j}=0,$$ i.e. $X_1=0$. We have $$\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial\\overline{z}_k}(g(\\zeta))X_{j}\\overline{X}_{k}=\\sum_{j,k,s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))\\overline{\\frac{\\partial\\Psi_t}{\\partial z_k}(g(\\zeta))}X_{j}\\overline{X}_{k}=$$$$=\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))Y_{s}\\overline{Y}_{t},$$ where $$Y:=\\Psi'(g(\\zeta))X.$$ Note that $Y\\neq 0$. Additionally $$\\sum_{s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))Y_{s}=\\sum_{j,s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))X_j=\\sum_{j=1}^n\\frac{\\partial(r\\circ\\Psi)}{\\partial z_j}(g(\\zeta))X_{j}=0.$$ Therefore, by the strong linear convexity of $D$ at $f(\\zeta)$ $$\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial\\overline{z}_t}(f(\\zeta))Y_{s}\\overline{Y}_{t}>\\left|\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))Y_{s}Y_{t}\\right|.$$ To finish the proof observe that $$\\left|\\sum_{j,k=1}^n\\frac{\\partial^2(r\\circ\\Psi)}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|=\\left|\\sum_{j,k,s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))\\frac{\\partial\\Psi_s}{\\partial z_j}(g(\\zeta))\\frac{\\partial\\Psi_t}{\\partial z_k}(g(\\zeta))X_{j}X_{k}+\\right.$$$$\\left.+\\sum_{j,k,s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|=$$$$=\\left|\\sum_{s,t=1}^n\\frac{\\partial^2 r}{\\partial z_s\\partial z_t}(f(\\zeta))Y_{s}Y_{t}+\\sum_{j,k=2}^n\\sum_{s=1}^n\\frac{\\partial r}{\\partial z_s}(f(\\zeta))\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))X_{j}X_{k}\\right|$$ and $$\\frac{\\partial^2\\Psi_s}{\\partial z_j\\partial z_k}(g(\\zeta))=0,\\ j,k\\geq 2,\\ s\\geq 1,$$ which gives \\eqref{sgf}.\n\\end{proof}\n\n\\begin{remm}\\label{rem:theta}\nLet $D$ be a bounded domain in $\\mathbb C^n$ and let $f:\\DD\\longrightarrow D$ be a (weak) stationary mapping such that $\\partial D$ is real analytic in a neighborhood of $f(\\TT)$. Assume moreover that there are a neighborhood $U$ of $f(\\CDD)$ and a mapping $\\Theta:U\\longrightarrow\\CC^n$ biholomorphic onto its image and the set $D\\cap U$ is connected. Then $\\Theta\\circ f$ is a (weak) stationary mapping of $G:=\\Theta(D\\cap U)$.\n\nIn particular, if $U_1$, $U_2$ are neighborhoods of the closures of domains $D_1$, $D_2$ with real analytic boundaries and $\\Theta:U_1\\longrightarrow U_2$ is a biholomorphism such that $\\Theta(D_1)=D_2$, then $\\Theta$ maps (weak) stationary mappings of $D_1$ onto (weak) stationary mappings of $D_2$.\n\\end{remm}\n\\begin{proof}\nActually, it is clear that two first conditions of the definition of (weak) stationary mappings are preserved by $\\Theta$. To show the third one we proceed similarly as in the equations \\eqref{56}, \\eqref{57}, \\eqref{58}. Let $f:\\DD\\longrightarrow D $ be a (weak) stationary mapping. The candidates for the mappings in condition (3) (resp. (3')) of Definition~\\ref{21} for $\\Theta\\circ f$ in the domain $G$ are $$((\\Theta'\\circ f)^{-1})^T\\wi f\\text{\\ \\ and\\ \\ }\\rho\\frac{|((\\Theta'\\circ f)^{-1})^T\\overline{\\nabla r\\circ f}|}{|\\nabla r\\circ f|}.$$ Indeed, for $\\zeta\\in\\TT$ \\begin{multline*}\\overline{\\nu_{G}(\\Theta(f(\\zeta)))}=\n\\frac{\\overline{\\nabla(r\\circ\\Theta^{-1})(\\Theta(f(\\zeta)))}}{|\\nabla(r\\circ\\Theta^{-1})(\\Theta(f(\\zeta)))|}=\\frac{[(\\Theta^{-1})'(\\Theta(f(\\zeta)))]^T\\overline{\\nabla r(f(\\zeta))}}{|[(\\Theta^{-1})'(\\Theta(f(\\zeta)))]^T\\overline{\\nabla r(f(\\zeta))}|}=\\\\\n=\\frac{(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}}{|(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}|},\n\\end{multline*}\nhence\n\\begin{multline*}\\zeta\\rho(\\zeta)\\frac{|(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nabla r(f(\\zeta))}|}{|\\nabla r(f(\\zeta))|}\\overline{\\nu_{G}(\\Theta(f(\\zeta)))}=\\\\\n=\\zeta\\rho(\\zeta)(\\Theta'(f(\\zeta))^{-1})^T\\overline{\\nu_{D}(f(\\zeta))}=\n(\\Theta'(f(\\zeta))^{-1})^T\\wi f(\\zeta).\n\\end{multline*}\n\\end{proof}\n\n\n\\subsection{Situation (\\dag)}\\label{dag}\nConsider the following situation, denoted by (\\dag) (with data $D_0$ and $U_0$):\n\\begin{itemize}\n\\item $D_0$ is a bounded domain in $\\CC^n$, $n\\geq 2$;\n\\item $f_0:\\CDD\\ni\\zeta\\longmapsto(\\zeta,0,\\ldots,0)\\in\\ov D_0$, $\\zeta\\in\\CDD$;\n\\item $f_0(\\DD)\\subset D_0$;\n\\item $f_0(\\TT)\\subset\\partial D_0$;\n\\item $\\nu_{D_0}(f_0(\\zeta))=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\TT$;\n\\item for any $\\zeta\\in\\TT$, a point $f_0(\\zeta)$ is a point of the strong linear convexity of $D_0$;\n\\item $\\partial D_0$ is real analytic in a neighborhood $U_0$ of $f_0(\\TT)$ with a function $r_0$;\n\\item $|\\nabla r_0|=1$ on $f_0(\\TT)$ (in particular, $r_{0z}(f_0(\\zeta))=(\\ov\\zeta\/2,0,\\ldots,0)$, $\\zeta\\in\\TT$).\n\\end{itemize}\n\nSince $r_0$ is real analytic on $U_0\\su\\RR^{2n}$, it extends in a natural way to a holomorphic function in a neighborhood $U_0^\\CC\\su\\mb{C}^{2n}$ of $U_0$. Without loss of generality we may assume that $r_0$ is bounded on $U_0^\\CC$. Set $$X_0=X_0(U_0,U_0^{\\mathbb C}):=\\{r\\in\\mc{O}(U_0^\\CC):\\text{$r(U_0)\\su\\mb{R}$ and $r$ is bounded}\\},$$ which equipped with the sup-norm is a (real) Banach space.\n\n\\begin{remm} Lempert considered the case when $U_0$ is a neighborhood of a boundary of a bounded domain $D_0$ with real analytic boundary. We shall need more general results to prove the `localization property'.\n\\end{remm}\n\n\\subsection{General lemmas}\\label{General lemmas}\nWe keep the notation from Subsection \\eqref{dag} and assume Situation (\\dag).\n\nLet us introduce some additional objects we shall be dealing with and let us prove more general lemmas (its generality will be useful in the next section).\n\nConsider the Sobolev space $W^{2,2}(\\TT)=W^{2,2}(\\TT,\\CC^m)$ of functions $f:\\TT\\longrightarrow\\CC^m$, whose first two derivatives (in the sense of distribution) are in $L^2(\\TT)$. The $W^{2,2}$-norm is denoted by $\\|\\cdot\\|_W$. For the basic properties of $W^{2,2}(\\TT)$ see Appendix.\n\nPut $$B:=\\{f\\in W^{2,2}(\\TT,\\CC^n):f\\text{ extends holomorphically on $\\mb{D}$ and $f(0)=0$}\\},$$$$B_0:=\\{f\\in B:f(\\TT)\\su U_0\\},\\quad B^*:=\\{\\overline{f}:f\\in B\\},$$$$Q:=\\{q\\in W^{2,2}(\\TT,\\CC):q(\\TT)\\su\\RR\\},\\quad Q_0:=\\{q\\in Q:q(1)=0\\}.$$\n\nIt is clear that $B$, $B^*$, $Q$ and $Q_0$ equipped with the norm $\\|\\cdot\\|_W$ are (real) Banach spaces. Note that $B_0$ is an open neighborhood of $f_0$. In what follows, we identify $f\\in B$ with its unique holomorphic extension on $\\mb{D}$.\n\nLet us define the projection $$\\pi:W^{2,2}(\\TT,\\CC^n)\\ni f=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}\\longmapsto\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}\\in{B^*}.$$ Note that $f\\in W^{2,2}(\\TT,\\CC^n)$ extends holomorphically on $\\mb{D}$ if and only if $\\pi(f)=0$ (and the extension is $\\mathcal C^{1\/2}$ on $\\TT$). Actually, it suffices to observe that\n$g(\\zeta):=\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}$, $\\zeta\\in\\TT$, extends holomorphically on $\\DD$ if and only if $a_k=0$ for $k<0$. This follows immediately from the fact that the mapping $\\TT\\ni\\zeta\\longmapsto g(\\ov\\zeta)\\in\\CC^n$ extends holomorphically on $\\DD$.\n\nConsider the mapping $\\Xi:X_0\\times\\mb{C}^n\\times B_0\\times\nQ_0\\times\\mb{R}\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$ defined by\n$$\\Xi(r,v,f,q,\\lambda):=(r\\circ f,\\pi(\\zeta(1+q)(r_z\\circ f)),f'(0)-\\lambda v),$$ where $\\zeta$ is treated as the identity function on $\\TT$.\n\n\nWe have the following\n\n\\begin{lemm}\\label{cruciallemma} There exist a neighborhood $V_0$ of $(r_0,f_0'(0))$ in $X_0\\times\\mb{C}^n$ and a real analytic mapping $\\Upsilon:V_0\\longrightarrow B_0\\times Q_0\\times\\mb{R}$ such that for any $(r,v)\\in V_0$ we have $\\Xi(r,v,\\Upsilon(r,v))=0$.\n\\end{lemm}\n\\bigskip\nLet $\\wi\\Xi:X_0\\times\\mb{C}^n\\times B_0\\times Q_0\\times(0,1)\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$ be defined as $$\\wi\\Xi(r,w,f,q,\\xi):=(r\\circ f,\\pi(\\zeta(1+q)(r_z\\circ f)),f(\\xi)-w).$$\n\n\nAnalogously we have\n\\begin{lemm}\\label{cruciallemma1} Let $\\xi_0\\in(0,1)$. Then there exist a neighborhood $W_0$ of $(r_0,f_0(\\xi_0))$ in $X_0\\times D_0$ and a real analytic mapping $\\wi\\Upsilon:W_0\\longrightarrow B_0\\times Q_0\\times(0,1)$ such that for any $(r,w)\\in W_0$ we have $\\wi\\Xi(r,w,\\wi\\Upsilon(r,w))=0$.\n\\end{lemm}\n\n\n\n\\begin{proof}[Proof of Lemmas \\ref{cruciallemma} and \\ref{cruciallemma1}]\n\n\nWe will prove the first lemma. Then we will see that a proof of the second one reduces to that proof.\n\nWe claim that $\\Xi$ is real analytic. The only problem is to show that the mapping $$T: X_0\\times B_0\\ni(r,f)\\longmapsto r\\circ f\\in Q$$ is real analytic (the real analyticity of the mapping $X_0\\times B_0\\ni(r,f)\\longmapsto r_z\\circ f\\in W^{2,2}(\\TT,\\CC^n)$ follows from this claim).\n\nFix $r\\in X_0$, $f\\in B_0$ and take $\\eps>0$ so that a $2n$-dimensional polydisc $P_{2n}(f(\\zeta),\\eps)$ is contained in $U_0^\\CC$ for any $\\zeta\\in\\TT$. Then any function $\\wi r\\in X_0$ is holomorphic in $U_0^\\CC$, so it may be expanded as a holomorphic series convergent in $P_{2n}(f(\\zeta),\\eps)$. Losing no generality we may assume that $n$-dimensional polydiscs $P_{n}(f(\\zeta),\\eps)$, $\\zeta\\in\\TT$, satisfy $P_{n}(f(\\zeta),\\eps)\\su U_0$. This gives an expansion of the function $\\wi r$ at any point $f(\\zeta)$, $\\zeta\\in\\TT$, into a series $$\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}(f(\\zeta))x^\\alpha$$ convergent to $\\wi r(f(\\zeta)+x)$, provided that $x=(x_1,\\ldots,x_{2n})\\in P_n(0,\\eps)$ (where $\\NN_0:=\\NN\\cup\\{0\\}$ and $|\\alpha|:=\\alpha_1+\\ldots+\\alpha_{2n}$). Hence \\begin{equation}\\label{69}T(r+\\varrho,f+h)=\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}r}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha+\\sum_{\\alpha\\in\\NN_0^{2n}}\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}\\varrho}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha\\end{equation} pointwise for $\\varrho\\in X_0$ and $h\\in W^{2,2}(\\TT,\\CC^n)$ with $\\|h\\|_{\\sup}<\\eps$.\n\nPut $P:=\\bigcup_{\\zeta\\in \\TT} P_{2n}(f(\\zeta),\\eps)$ and for $\\wi r\\in X_0$ put $||\\wi r||_P:=\\sup_P|\\wi r|$. Let $\\wi r$ be equal to $r$ or to $\\varrho$, where $\\varrho$ lies is in a neighborhood of $0$ in $X_0$. The Cauchy inequalities give\n\\begin{equation}\\label{series}\\left|\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}(f(\\zeta))\\right|\\leq\\frac{\\alpha!\\|\\wi r\\|_{P}}{\\eps^{|\\alpha|}},\\quad\\zeta\\in\\TT.\\end{equation}\nTherefore, $$\\left|\\left|\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}\\circ f\\right|\\right|_W\\leq C_1\\frac{\\alpha!\\|\\wi r\\|_{P}}{\\eps^{|\\alpha|}}$$ for some $C_1>0$.\n\nThere is $C_2>0$ such that $$\\|gh^\\alpha\\|_W\\leq C_2^{|\\alpha|+1}\\|g\\|_W\\|h_1\\|^{\\alpha_1}_W\\cdotp\\ldots\\cdotp\\|h_{2n}\\|^{\\alpha_{2n}}_W$$ for $g\\in W^{2,2}(\\TT,\\CC)$, $h\\in W^{2,2}(\\TT,\\CC^n)$, $\\alpha\\in\\NN_0^{2n}$ (see Appendix for a proof of this fact). Using the above inequalities we infer that $$\\sum_{\\alpha\\in\\NN_0^{2n}}\\left|\\left|\\frac{1}{\\alpha!}\\left(\\frac{\\pa^{|\\alpha|}\\wi r}{\\pa x^\\alpha}\\circ f\\right)h^\\alpha\\right|\\right|_W$$ is convergent if $h$ is small enough on the norm $\\|\\cdot\\|_W$. Therefore, the series~\\eqref{69} is absolutely convergent in the norm $\\|\\cdot\\|_W$, whence $T$ is real analytic.\n\n\nTo show the existence of $V_0$ and $\\Upsilon$ we will make use of the Implicit Function Theorem. More precisely, we shall show that the partial derivative $$\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1):B\\times Q_0\\times\\mb{R}\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$$ is an isomorphism.\nObserve that for any $(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})\\in B\\times Q_0\\times\\mb{R}$ the following equality holds\n\\begin{multline*}\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1)(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})=\\left.\\frac{d}{dt}\n\\Xi(r_0,f_0'(0),f_0+t\\widetilde{f},t\\widetilde{q},1+t\\widetilde{\\lambda})\\right|_{t=0}=\\\\\n=((r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}},\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta(r_{0zz} \\circ\nf_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}),\\widetilde{f}'(0)-\\widetilde{\\lambda}f_0'(0)),\n\\end{multline*}\nwhere we treat ${r_0}_z,{r_0}_{\\overline{z}}$ as row vectors, $\\widetilde{f},\\overline{\\widetilde{f}}$ as column vectors and $r_{0zz}=\\left[\\frac{\\partial^2r_0}{\\partial z_j\\partial z_k}\\right]_{j,k=1}^n$, $r_{0z\\overline{z}}=\\left[\\frac{\\partial^2r_0}{\\partial z_j\\partial\\overline z_k}\\right]_{j,k=1}^n$ as $n\\times n$ matrices.\n\nBy the Bounded Inverse Theorem it suffices to show that $\\Xi_{(f,q,\\lambda)}(r_0,f_0'(0),f_0,0,1)$ is bijective, i.e. for $(\\eta,\\varphi,v)\\in Q\\times B^*\\times\\mb{C}^n$ there exists exactly one $(\\widetilde{f},\\widetilde{q},\\widetilde{\\lambda})\\in B\\times Q_0\\times\\mb{R}$ satisfying\n\\begin{equation}\n(r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}=\\eta,\n\\label{al1}\n\\end{equation}\n\\begin{equation}\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta (r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}})=\\varphi,\n\\label{al2}\n\\end{equation}\n\\begin{equation}\n\\widetilde{f}'(0)-\\widetilde{\\lambda} f_0'(0)=v.\n\\label{al3}\n\\end{equation}\nFirst we show that $\\wi\\lambda$ and $\\wi f_1$ are uniquely determined. Observe that, in view of assumptions, (\\ref{al1}) is just $$\\frac{1}{2}\\overline{\\zeta}\\widetilde{f}_1+\\frac{1}{2}\\zeta\\overline{\\widetilde{f}_1}=\\eta$$ or equivalently\n\\begin{equation}\n\\re(\\widetilde{f}_1\/\\zeta)=\\eta\\text{ (on }\\TT).\n\\label{al4}\n\\end{equation}\nNote that the equation (\\ref{al4}) uniquely determines $\\widetilde{f}_1\/\\zeta\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ up to an imaginary additive constant, which may be computed using (\\ref{al3}). Actually, $\\eta=\\re G$ on $\\TT$ for some function $G\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$. To see this, let us expand $\\eta(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}$, $\\zeta\\in\\TT$. From the equality $\\eta(\\zeta)=\\ov{\\eta(\\zeta)}$, $\\zeta\\in\\TT$, we get \\begin{equation}\\label{65}\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}=\\sum_{k=-\\infty}^{\\infty}\\ov a_k\\zeta^{-k}=\\sum_{k=-\\infty}^{\\infty}\\ov a_{-k}\\zeta^{k},\\ \\zeta\\in\\TT,\\end{equation} so $a_{-k}=\\ov a_k$, $k\\in\\ZZ$. Hence $$\\eta(\\zeta)=a_0+\\sum_{k=1}^\\infty 2\\re(a_k\\zeta^k)=\\re\\left(a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k\\right),\\ \\zeta\\in\\TT.$$ Set $$G(\\zeta):=a_0+2\\sum_{k=1}^\\infty a_k\\zeta^k,\\ \\zeta\\in\\DD.$$ This series is convergent for $\\zeta\\in\\DD$, so $G\\in\\OO(\\DD)$. Further, the function $G$ extends continuously on $\\CDD$ (to the function denoted by the same letter) and the extension lies in $W^{2,2}(\\TT,\\CC)$. Clearly, $\\eta=\\re G$ on $\\TT$.\n\nWe are searching $C\\in\\RR$ such that the functions $\\widetilde{f}_1:=\\zeta(G+iC)$ and $\\theta:=\\im(\\widetilde{f}_1\/\\zeta)$ satisfy $$\\eta(0)+i\\theta(0)=\\widetilde{f}_1'(0)$$ and\n$$\\eta(0)+i\\theta(0)-\\widetilde{\\lambda}\\re{f_{01}'(0)}-i\\widetilde{\\lambda}\\im{{f_{01}'(0)}}=\\re{v_1}+i\\im{v_1}.$$ But $$\\eta(0)-\\widetilde{\\lambda}\\re{f_{01}'(0)}=\\re{v_1},$$ which yields $\\widetilde{\\lambda}$ and then $\\theta(0)$, consequently the number $C$.\nHaving $\\widetilde{\\lambda}$ and once again using (\\ref{al3}), we find uniquely determined $\\widetilde{f}_2'(0),\\ldots,\\widetilde{f}_n'(0)$.\n\nTherefore, the equations $\\eqref{al1}$ and $\\eqref{al3}$ are satisfied by uniquely determined $\\wi f_1$, $\\wi\\lambda$ and $\\widetilde{f}_2'(0),\\ldots,\\widetilde{f}_n'(0)$.\n\nConsider (\\ref{al2}), which is the system of $n$ equations with unknown $\\widetilde{q},\\widetilde{f}_2,\\ldots,\\widetilde{f}_n$. Observe that $\\widetilde{q}$ appears only in the first of the equations and the remaining $n-1$ equations mean exactly that the mapping\n\\begin{equation}\n\\zeta(r_{0\\widehat{z}\\widehat{z}}\\circ f_0)\n\\widehat{\\widetilde{f}}+\\zeta(r_{0\\widehat{z}\\widehat{\\overline{z}}}\\circ f_0)\\widehat{\\overline{\\widetilde{f}}}-\\psi\n\\label{al5}\n\\end{equation}\nextends holomorphically on $\\mb{D}$, where $\\widehat{a}:=(a_{2},\\ldots,a_{n})$ and $\\psi\\in W^{2,2}(\\TT,\\mb{C}^{n-1})$ may be obtained from $\\varphi$ and $\\widetilde{f}_1$. Indeed, to see this, write (\\ref{al2}) in the form $$\\pi(F_{1}+\\zeta F_{2}+\\zeta F_{3})=(\\phi_1,\\ldots,\\phi_n),$$ where $$F_1:=(\\wi q,0,\\ldots,0),$$$$F_2:=(A_{j})_{j=1}^n,\\ A_{j}:=\\sum\\limits_{k=1}^n(r_{0z_jz_k}\\circ f_0)\\widetilde{f}_k,$$$$F_3=(B_{j})_{j=1}^n,\\ B_{j}:=\\sum\\limits_{k=1}^n(r_{0z_j\\ov z_k}\\circ f_0)\\overline{\\widetilde{f}_k}.$$ It follows that $$\\widetilde{q}+\\zeta A_1+\\zeta B_1-\\phi_1$$ and $$\\zeta A_j+\\zeta B_j-\\phi_j,\\ j=2,\\ldots,n,$$ extend holomorphically on $\\mb{D}$ and $$\\psi:=\\left(\\phi_j-\\zeta(r_{0z_jz_1}\\circ f_0)\\widetilde{f}_1-\\zeta(r_{0z_j\\ov z_1}\\circ f_0)\\overline{\\widetilde{f}_1}\\right)_{j=2}^n.$$\nPut $$g(\\zeta):=\\widehat{\\widetilde{f}}(\\zeta)\/\\zeta,\\quad\\alpha(\\zeta):=\\zeta^2r_{0\\widehat{z}\\widehat{z}}(f_0(\\zeta)),\n\\quad\\beta(\\zeta):=r_{0\\widehat{z}\\widehat{\\overline{z}}}(f_0(\\zeta)).$$\n\nObserve that $\\alpha(\\zeta)$, $\\beta(\\zeta)$ are the $(n-1)\\times(n-1)$ matrices depending real analytically on $\\zeta$ and $g(\\zeta)$ is a column vector in $\\mb{C}^{n-1}$. This allows us to reduce \\eqref{al5} to the following problem: we have to find a unique $g\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ such that \\begin{equation}\n\\alpha g+\\beta\\overline{g}-\\psi\\text{ extends holomorphically on $\\mb{D}$ and } g(0)={\\widehat{\\widetilde{f}'}}(0).\n\\label{al6}\n\\end{equation}\nThe fact that every $f_0(\\zeta)$ is a point of strong linear convexity of the domain $D_0$ may be written as\n\\begin{equation}\n|X^T\\alpha(\\zeta)X|0$ independent on $\\zeta$ and $X$. Thus $\\|\\gamma(\\zeta)\\|\\leq 1-\\wi\\eps$ by Proposition \\ref{59}.\n\nWe have to prove that there is a unique solution $h\\in W^{2,2}(\\TT,\\CC^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ of (\\ref{al9}) such that $h(0)=a$ with a given $a\\in\\CC^{n-1}$.\n\nDefine the operator $$P:W^{2,2}(\\TT,\\mb{C}^{n-1})\\ni\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^{k}\\longmapsto\\overline{\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}}\\in W^{2,2}(\\TT,\\mb{C}^{n-1}),$$ where $a_k\\in\\CC^{n-1}$, $k\\in\\ZZ$.\n\nWe will show that a mapping $h\\in\nW^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies (\\ref{al9}) and $h(0)=a$ if and only if it is a fixed point of the mapping $$K:W^{2,2}(\\TT,\\mb{C}^{n-1})\\ni h\\longmapsto P(H^{-1}\\psi-\\gamma h)+a\\in W^{2,2}(\\TT,\\mb{C}^{n-1}).$$\n\nIndeed, take $h\\in\nW^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ and suppose that $h(0)=a$ and $\\gamma h+\\overline{h}-H^{-1}\\psi$ extends holomorphically on $\\mb{D}$. Then $$h=a+\\sum_{k=1}^{\\infty}a_k\\zeta^{k},\\quad\\overline{h}=\\overline{a}+\\sum_{k=1}^{\\infty}\\overline a_k\\zeta^{-k}=\\sum_{k=-\\infty}^{-1}\\overline a_{-k}\\zeta^{k}+\\overline{a},$$ $$P(h)=0,\\quad P(\\overline{h})=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}=h-a$$ and $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=0,$$ which implies $$P(H^{-1}\\psi-\\gamma h)=h-a$$ and finally $K(h)=h$. Conversely, suppose that $K(h)=h$. Then $$P(H^{-1}\\psi-\\gamma h)=h-a=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}+a_1-a,\\quad P(h)=0$$ and\n$$P(\\overline{h})=\\sum_{k=1}^{\\infty}a_k\\zeta^{k}=h-a_1,$$ from which follows that $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=P(\\overline{h})-P(H^{-1}\\psi-\\gamma h)=a-a_1$$ and $$P(\\gamma h+\\overline{h}-H^{-1}\\psi)=0\\text{ iff }a=a_1.$$ Observe that $h(0)=K(h)(0)=P(H^{-1}\\psi-\\gamma h)(0)+a=a$.\n\nWe shall make use of the Banach Fixed Point Theorem. To do this, consider $W^{2,2}(\\TT,\\CC^{n-1})$ equipped with the following norm $$\\|h\\|_{\\varepsilon}:=\\|h\\|_L+\\varepsilon\\|h'\\|_L+\n\\varepsilon^2\\|h''\\|_L,$$ where $\\eps>0$ and $\\|\\cdot\\|_L$ is the $L^2$-norm (it is a Banach space). We will prove that $K$ is a contraction with respect to the norm $\\|\\cdot\\|_{\\varepsilon}$ for sufficiently small $\\eps$. Indeed, there is $\\wi\\eps>0$ such that for any $h_1,h_2\\in W^{2,2}(\\TT,\\CC^{n-1})$\n\\begin{equation}\n\\|K(h_1)-K(h_2)\\|_L=\\|P(\\gamma(h_2-h_1))\\|_L\\leq\\|\\gamma(h_2-h_1)\\|_L\\leq (1-\\wi\\eps)\\|h_2-h_1\\|_L.\n\\label{al10}\n\\end{equation}\nMoreover,\n\\begin{multline}\n\\|K(h_1)'-K(h_2)'\\|_L= \\|P(\\gamma h_2)'-P(\\gamma h_1)'\\|_L\\leq\\\\\n\\leq\\|(\\gamma h_2)'-(\\gamma h_1)'\\|_L= \\|\\gamma '(h_2-h_1)+\\gamma(h_2'-h_1')\\|_L.\n\\label{al11}\n\\end{multline} Furthermore,\n\\begin{equation}\n\\|K(h_1)''-K(h_2)''\\|_L\\leq\\|\\gamma ''(h_2-h_1)\\|_L+2\\|\\gamma '(h_2'-h_1')\\|_L+\\|\\gamma\n(h_1''-h_2'')\\|_L.\\label{al12}\n\\end{equation}\nUsing the finiteness of $\\|\\gamma '\\|$, $\\|\\gamma ''\\|$ and putting (\\ref{al10}), (\\ref{al11}), (\\ref{al12}) together we see that there exists $\\varepsilon>0$ such that $K$ is a contraction w.r.t. the norm $\\|\\cdot\\|_{\\varepsilon}$.\n\nWe have found $\\widetilde{f}$ and $\\widetilde{\\lambda}$ satisfying (\\ref{al1}), (\\ref{al3}) and the last $n-1$ equations from (\\ref{al2}) are satisfied. \n\nIt remains to show that there exists a unique $\\widetilde{q}\\in Q_0$ such that $\\widetilde{q}+\\zeta A_1+\\zeta B_1-\\varphi_1$ extends holomorphically on $\\mb{D}$.\n\nComparing the coefficients as in \\eqref{65}, we see that if $$\\pi(\\zeta A_1+\\zeta B_1-\\varphi_1)=\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}$$\nthen $\\widetilde{q}$ has to be taken as $$-\\sum_{k=-\\infty}^{-1}a_k\\zeta^{k}-\\sum_{k=0}^{\\infty}b_k\\zeta^{k}$$\nwith $b_k:=\\overline a_{-k}$ for $k\\geq 1$ and $b_0\\in\\RR$ uniquely determined by $\\widetilde{q}(1)=0$.\\\\\n\nLet us show that the proof of the second Lemma follows from the proof of the first one.\nSince $\\wi\\Xi$ is real analytic it suffices to prove that the derivative $$\\wi\\Xi_{(f,q,\\xi)}(r_0,f_0(\\xi_0),f_0,0,\\xi_0):B\\times Q_0\\times\\RR\\longrightarrow Q\\times{B^*}\\times\\mb{C}^n$$ is invertible.\nFor $(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})\\in B\\times Q_0\\times\\RR$ we get\n\\begin{multline*}\n\\wi\\Xi_{(f,q,\\xi)}(r_0,f_0(\\xi_0),f_0,0,\\xi_0)(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})=\\left.\\frac{d}{dt}\n\\wi\\Xi(r_0,f_0(\\xi_0),f_0+t\\widetilde{f},t\\widetilde{q},\\xi_0+t\\widetilde{\\xi})\\right|_{t=0}=\\\\\n=((r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}},\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta(r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}),\\widetilde{f}(\\xi_0)+\\wi\\xi f_0'(\\xi_0)).\n\\end{multline*}\nWe have to show that for $(\\eta,\\varphi,w)\\in Q\\times B^*\\times\\mb{C}^n$ there exists exactly one $(\\widetilde{f},\\widetilde{q},\\widetilde{\\xi})\\in B\\times Q_0\\times\\RR$ satisfying\n\\begin{equation}\n(r_{0z}\\circ f_0)\\widetilde{f}+(r_{0\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}}=\\eta,\n\\label{1al1}\n\\end{equation}\n\\begin{equation}\n\\pi(\\zeta\\widetilde{q}r_{0z}\\circ f_0+\\zeta (r_{0zz}\\circ f_0)\\widetilde{f}+\\zeta(r_{0z\\overline{z}}\\circ f_0)\\overline{\\widetilde{f}})=\\varphi,\n\\label{1al2}\n\\end{equation}\n\\begin{equation}\n\\wi f(\\xi_0)+\\wi\\xi f_0'(\\xi_0)=w.\n\\label{1al3}\n\\end{equation}\nThe equation (\\ref{1al1}) turns out to be\n\\begin{equation}\n\\re(\\widetilde{f}_1\/\\zeta)=\\eta\\text{ (on }\\TT).\n\\label{1al4}\n\\end{equation}\nThe equation above uniquely determines $\\widetilde{f}_1\/\\zeta\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ up to an imaginary additive constant, which may be computed using (\\ref{1al3}). Indeed, there exists $G\\in W^{2,2}(\\TT,\\CC)\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ such that $\\eta=\\re G$ on $\\TT$. We are searching $C\\in\\RR$ such that the functions $\\widetilde{f}_1:=\\zeta(G+iC)$ and $\\theta:=\\im(\\widetilde{f}_1\/\\zeta)$ satisfy $$\\xi_0\\eta(\\xi_0)+i\\xi_0\\theta(\\xi_0)=\\widetilde{f}_1(\\xi_0)$$ and $$\\xi_0(\\eta(\\xi_0)+i\\theta(\\xi_0))+\\widetilde{\\xi}\\re{f_{01}'(\\xi_0)}+i\\widetilde{\\xi}\\im{{f_{01}'(\\xi_0)}}=\n\\re{w_1}+i\\im{w_1}.$$ But $$\\xi_0\\eta(\\xi_0)+\\widetilde{\\xi}\\re{f_{01}'(\\xi_0)}=\\re{w_1},$$ which yields $\\widetilde{\\xi}$ and then $\\theta(\\xi_0)$, consequently the number $C$. Having $\\widetilde{\\xi}$ and once again using (\\ref{1al3}), we find uniquely determined $\\widetilde{f}_2(\\xi_0),\\ldots,\\widetilde{f}_n(\\xi_0)$.\n\nTherefore, the equations $\\eqref{1al1}$ and $\\eqref{1al3}$ are satisfied by uniquely determined $\\wi f_1$, $\\wi\\xi$ and $\\widetilde{f}_2(\\xi_0),\\ldots,\\widetilde{f}_n(\\xi_0)$.\n\nIn the remaining part of the proof we change the second condition of \\eqref{al6} to $$g(\\xi_0)={\\widehat{\\widetilde{f}}}(\\xi_0)\/\\xi_0$$ and we have to prove that there is a unique solution $h\\in W^{2,2}(\\TT,\\CC^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ of (\\ref{al9}) such that $h(\\xi_0)=a$ with a given $a\\in\\CC^{n-1}$. Let $\\tau$ be an automorphism of $\\DD$ (so it extends holomorphically near $\\CDD$), which maps $0$ to $\\xi_0$, i.e. $$\\tau(\\xi):=\\frac{\\xi_0-\\xi}{1-\\ov\\xi_0\\xi},\\ \\xi\\in\\DD.$$ Let the maps $P,K$ be as before. Then $h\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies\n(\\ref{al9}) and $h(\\xi_0)=a$ if and only if $h\\circ\\tau\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfies (\\ref{al9}) and $(h\\circ\\tau)(0)=a$. We already know that there is exactly one $\\wi h\\in W^{2,2}(\\TT,\\mb{C}^{n-1})\\cap\\OO(\\DD)\\cap\\cC(\\CDD)$ satisfying (\\ref{al9}) and $\\wi h(0)=a$. Setting $h:=\\wi h\\circ\\tau^{-1}$, we get the claim.\n\\end{proof}\n\n\\subsection{Topology in the class of domains with real analytic boundaries}\\label{topol}\n\nWe introduce a concept of a domain being close to some other domain. Let $D_0\\su\\mb{C}^n$ be a bounded domain with real analytic boundary. Then there exist a neighborhood $U_0$ of $\\partial D_0$ and a real analytic defining function $r_0:U_0\\longrightarrow\\mb{R}$ such that $\\nabla r_0$ does not vanish in $U_0$ and $$D_0\\cap U_0=\\{z\\in U_0:r_0(z)<0\\}.$$\n\n\\begin{dff}\nWe say that domains $D$ \\textit{tend to} $D_0$ $($or are \\textit{close to} $D_0${$)$} if one can choose their defining functions $r\\in X_0$ such that $r$ tend to $r_0$ in $X_0$.\n\\end{dff}\n\n\\begin{remm} If $r\\in X_0$ is near to $r_0$ with respect to the topology in $X_0$, then $\\{z\\in U_0:r(z)=0\\}$ is a compact real analytic hypersurface which bounds a bounded domain. We denote it by $D^{r}$.\n\nMoreover, if $D^{r_0}$ is strongly linearly convex then a domain $D^r$ is also strongly linearly convex provided that $r$ is near $r_0$.\n\\end{remm}\n\n\n\n\n\n\n\n\\subsection{Statement of the main result of this section}\n\n\\begin{remm}\\label{f} Assume that $D^r$ is a strongly linearly convex domain bounded by a real analytic hypersurface $\\{z\\in U_0:r(z)=0\\}$. Let $\\xi\\in(0,1)$ and $w\\in(\\CC^n)_*$.\n\nThen a function $f\\in B_0$ satisfies the conditions $$f\\text{ is a weak stationary mapping of }D^r,\\ f(0)=0,\\ f(\\xi)=w$$ if and only if there exists $q\\in Q_0$ such that $q>-1$ and $\\wi\\Xi(r,w,f,q,\\xi)=0$.\n\nActually, from $\\wi\\Xi(r,w,f,q,\\xi)=0$ we deduce immediately that $r\\circ f=0$ on $\\TT$, $f(\\xi)=w$ and $\\pi(\\zeta(1+q)(r_z\\circ f))=0$. From the first equality we get $f(\\TT)\\subset \\partial D^{r}$. From the last one we deduce that the condition (3') of Definition~\\ref{21} is satisfied (with $\\rho:=(1+q)|r_z\\circ f|$). Since $D^{r}$ is strongly linearly convex, $\\ov{D^r}$ is polynomially convex (use the fact that projections of $\\CC$-convex domains are $\\CC$-convex, as well, and the fact that $D^r$ is smooth). In particular, $$f(\\CDD)=f(\\widehat{\\TT})\\subset\\widehat{f(\\TT)}\\subset\\widehat{\\ov{D^r}}=\\ov{D^r},$$ where $\\wh S:=\\{z\\in\\CC^m:|P(z)|\\leq\\sup_S|P|\\text{ for any polynomial }P\\in\\CC[z_1,\\ldots,z_m]\\}$ is the polynomial hull of a set $S\\su\\CC^m$. \n\nNote that this implies $f(\\DD)\\su D^r$ --- this follows from the fact that $\\pa D^r$ does not contain non-constant analytic discs (as $D^r$ is strongly pseudoconvex).\n\nThe opposite implication is clear.\n\n\\bigskip\n\nIn a similar way we show that for any $v\\in(\\CC^n)_*$ and $\\lambda>0$, a function $f\\in B_0$ satisfies the conditions $$f\\text{ is a weak stationary mapping of }D^r,\\ f(0)=0,\\ f'(0)=\\lambda v$$ if and only if there exists $q\\in Q_0$ such that $q>-1$ and $\\Xi(r,v,f,q,\\lambda)=0$.\n\n\\end{remm}\n\n\n\\begin{propp}\\label{13} Let $D_0\\su\\CC^n$, $n\\geq 2$, be a strongly linearly convex domain with real analytic boundary and let $f_0:\\DD\\longrightarrow D_0$ be an $E$-mapping.\n\n$(1)$ Let $\\xi_0\\in(0,1)$. Then there exist a neighborhood $W_0$ of $(r_0,f_0(\\xi_0))$ in $X_0\\times D_0$ and real analytic mappings $$\\Lambda:W_0\\longrightarrow\\mc{C}^{1\/2}(\\overline{\\mb{D}}),\\ \\Omega:W_0\\longrightarrow(0,1)$$ such that $$\\Lambda(r_0,f_0(\\xi_0))=f_0,\\ \\Omega(r_0,f_0(\\xi_0))=\\xi_0$$ and for any $(r,w)\\in W_0$ the mapping\n$f:=\\Lambda(r,w)$ is an $E$-mapping of $D^{r}$ satisfying $$f(0)=f_0(0)\\text{ and }f(\\Omega(r,w))=w.$$\n\n$(2)$ There exist a neighborhood $V_0$ of $(r_0,f_0'(0))$ in $X_0\\times\\mb{C}^n$\nand a real analytic mapping $$\\Gamma:V_0\\longrightarrow\\mc{C}^{1\/2}(\\overline{\\mb{D}})$$ such that $$\\Gamma(r_0,f_0'(0))=f_0$$ and for any $(r,v)\\in V_0$ the mapping $f:=\\Gamma(r,v)$ is an $E$-mapping of $D^{r}$ satisfying $$f(0)=f_0(0)\\text{ and }f'(0)=\\lambda v\\text{ for some }\\lambda>0.$$\n\\end{propp}\n\n\\begin{proof}\n\nObserve that Proposition \\ref{11} provides us with a mapping $g_0=\\Phi\\circ f_0$ and a domain $G_0:=\\Phi(D_0)$ giving a data for situation (\\dag) (here $\\partial D_0$ is contained in $U_0$). Clearly, $\\rho_0:=r_0\\circ\\Phi^{-1}$ is a defining function of $G_0$.\n\nUsing Lemmas \\ref{cruciallemma}, \\ref{cruciallemma1} we get neighborhoods $V_0$, $W_0$ of $(\\rho_0, g_0'(0))$, $(\\rho_0,g_0(\\xi_0))$ respectively and real analytic mappings $\\Upsilon$, $\\wi\\Upsilon$ such that $ \\Xi(\\rho,v,\\Upsilon(\\rho,v))=0$ on $V_0$ and $ \\wi\\Xi(\\rho,w,\\wi\\Upsilon(\\rho,w))=0$ on $W_0$. Define $$\\wh\\Lambda:=\\pi_B\\circ\\wi\\Upsilon,\\quad\\Omega:=\\pi_\\RR\\circ\\wi\\Upsilon,\\quad\\wh\\Gamma:=\\pi_B\\circ\\Upsilon,$$ where $$\\pi_B:B\\times Q_0\\times\\mb{R}\\longrightarrow B,\\quad\\pi_\\RR:B\\times Q_0\\times\\mb{R}\\longrightarrow\\RR,\\ $$ are the projections.\n\nIf $\\rho$ is sufficiently close to $\\rho_0$, then the hypersurface $\\{\\rho=0\\}$ bounds a strongly linearly convex domain. Moreover, then $\\wh\\Lambda(\\rho,w)$ and $\\wh\\Gamma(\\rho,v)$ are extremal mappings in $G^{\\rho}$ (see Remark~\\ref{f}).\n\nComposing $\\wh\\Lambda(\\rho,w)$ and $\\wh\\Gamma(\\rho,v)$ with $\\Phi^{-1}$ and making use of Remark \\ref{rem:theta} we get weak stationary mappings in $D^r$, where $r:=\\rho\\circ\\Phi$. To show that they are $E$-mappings we proceed as follows. If $D^r$ is sufficiently close to $D_0$ (this depends on a distance between $\\rho$ and $\\rho_0$), the domain $D^r$ is strongly linearly convex, so by the results of Section \\ref{55} $$\\Lambda(r,w):=\\Phi^{-1}\\circ\\wh\\Lambda(\\rho,w)\\text{\\ and\\ }\\Gamma(r,v):=\\Phi^{-1}\\circ\\wh\\Gamma(\\rho,v)$$ are stationary mappings. Moreover, they are close to $f_0$ provided that $r$ is sufficiently close to $r_0$. Therefore, their winding numbers are equal. Thus $f$ satisfies condition (4) of Definition~\\ref{21e}, i.e. $f$ is an $E$-mapping.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Localization property}\n\n\\begin{prop}\\label{localization} Let $D\\su\\mathbb C^n$, $n\\geq 2$, be a domain. Assume that $a\\in\\partial D$ is such that $\\partial D$ is real analytic and strongly convex in a neighborhood of $a$. Then for any sufficiently small neighborhood $V_0$ of $a$ there is a weak stationary mapping of $D\\cap V_0$ such that $f(\\mathbb T)\\su\\partial D$.\n\nIn particular, $f$ is a weak stationary mapping of $D$.\n\\end{prop}\n\n\\begin{proof} Let $r$ be a real analytic defining function in a neighborhood of $a$. The problem we are dealing with has a local character, so replacing $r$ with $r\\circ\\Psi$, where $\\Psi$ is a local biholomorphism near $a$, we may assume that $a=(0,\\ldots,0,1)$ and a defining function of $D$ near $a$ is $r(z)=-1+|z|^2+h(z-a)$, where $h$ is real analytic in a neighborhood of $0$ and $h(z)=O(|z|^3)$ as $z\\to 0$ (cf. \\cite{Rud}, p. 321).\n\nFollowing \\cite{Lem2}, let us consider the mappings\n$$A_t(z):=\\left((1-t^2)^{1\/2}\\frac{z'}{1+tz_n},\\frac{z_n+t}{1+tz_n}\\right),\\quad z=(z',z_n)\\in\\CC^{n-1}\\times\\DD,\\,\\,t\\in(0,1),$$ which restricted to $\\BB_n$ are automorphisms. Let $$r_t(z):=\\begin{cases}\\frac{|1+tz_n|^2}{1-t^2}r(A_t(z)),&t\\in(0,1),\\\\-1+|z|^2,&t=1.\\end{cases}$$ It is clear that $f_{(1)}(\\zeta)=(\\zeta,0,\\ldots,0)$, $\\zeta\\in\\DD$ is a stationary mapping of $\\mathbb B_n$. We want to have the situation (\\dag) which will allow us to use Lemma \\ref{cruciallemma} (or Lemma \\ref{cruciallemma1}). Note that $r_t$ does not converge to $r_1$ as $t\\to 1$. However, $r_t\\to r_1$ in $X_0(U_0,U_0^{\\mathbb C})$, where $U_0$ is a neighborhood of $f_{(1)}(\\TT)$ contained in $\\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\}$ and $U_0^{\\mathbb C}$ is sufficiently small (remember that $h(z)=O(|z|^3)$).\n\nTherefore, making use of Lemma \\ref{cruciallemma} for $t$ sufficiently close to $1$ we obtain stationary mappings $f_{(t)}$ in $D_t:=\\{z\\in \\mathbb C^n: r_t(z)<0,\\ \\re z_n>-1\/2\\}$ such that $f_{(t)}\\to f_{(1)}$ in the $W^{2,2}$-norm (so also in the sup-norm). Actually, it follows from Lemma~\\ref{cruciallemma} that one may take $f_{(t)}:=\\pi_B\\circ\\Upsilon(r_t,f_{(1)}'(0))$ (keeping the notation from this lemma). The argument used in Remark~\\ref{f} gives that $f_{(t)}$ satisfies conditions (1'), (2') and (3') of Definition~\\ref{21}. Since the non-constant function $r\\circ A_t\\circ f_{(t)}$ is subharmonic on $\\DD$, continuous on $\\CDD$ and $r\\circ A_t\\circ f_{(t)}=0$ on $\\TT$, we see from the maximum principle that $f_{(t)}$ maps $\\DD$ in $D_t$. Therefore, $f_{(t)}$ are weak stationary mappings for $t$ close to $1$.\n\nIn particular, $$f_{(t)}(\\DD)\\subset 2\\mathbb B_n \\cap \\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\}$$ provided that $t$ is close to $1$. The mappings $A_t$ have the following important property $$A_t(2\\mathbb B_n\\cap\\{z\\in\\mathbb C^n:\\re z_n>-1\/2\\})\\to\\{a\\}$$ as $t\\to 1$ in the sense of the Hausdorff distance.\n\nTherefore, we find from Remark \\ref{rem:theta} that $g_{(t)}:=A_t\\circ f_{(t)}$ is a stationary mapping of $D$. Since $g_{(t)}$ maps $\\DD$ onto arbitrarily small neighborhood of $a$ provided that $t$ is sufficiently close to $1$, we immediately get the assertion.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs of Theorems \\ref{lem-car} and \\ref{main}}\n\nWe start this section with the following\n\\begin{lem}\\label{lemat} For any different $z,w\\in D$ $($resp. for any $z\\in D$, $v\\in(\\CC^n)_*${$)$} there exists an $E$-mapping $f:\\DD\\longrightarrow D$ such that $f(0)=z$, $f(\\xi)=w$ for some $\\xi\\in(0,1)$ $($resp. $f(0)=z$, $f'(0)=\\lambda v$ for some $\\lambda>0${$)$}.\n\\end{lem}\n\n\\begin{proof}\nFix different $z,w\\in D$ (resp. $z\\in D$, $v\\in(\\CC^{n})_*$).\n\nFirst, consider the case when $D$ is bounded strongly convex with real analytic boundary. Without loss of generality one may assume that $0\\in D\\Subset\\BB_n$. We need some properties of the Minkowski functionals.\n\nLet $\\mu_G$ be a Minkowski functional of a domain $G\\subset\\CC^n$ containing the origin, i.e. $$\\mu_G(x):=\\inf\\left\\{s>0:\\frac{x}{s}\\in G\\right\\},\\ x\\in\\CC^n.$$ Assume that $G$ is bounded strongly convex with real analytic boundary. We shall show that\n\\begin{itemize}\n\\item $\\mu_G-1$ is a real analytic outside $0$, defining function of $G$;\n\\item $\\mu^2_G-1$ is a real analytic outside $0$, strongly convex outside $0$, defining function of $G$.\n\\end{itemize}\nClearly, $G=\\{x\\in\\RR^{2n}:\\mu_G(x)<1\\}$. Setting $$q(x,s):=r\\left(\\frac{x}{s}\\right),\\ (x,s)\\in U_0\\times U_1,$$ where $r$ is a real analytic defining function of $G$ (defined near $\\pa G$) and $U_0\\su\\RR^{2n}$, $U_1\\su\\RR$ are neighborhoods of $\\pa G$ and $1$ respectively, we have $$\\frac{\\partial q}{\\partial s}(x,s)=-\\frac{1}{s^2}\\left\\langle\\nabla r\\left(\\frac{x}{s}\\right),x\\right\\rangle_{\\RR}\\neq 0$$ for $(x,s)$ such that $x\\in\\partial G$ and $s=\\mu_G(x)=1$ (since $0\\in G$, the vector $-x$ hooked at the point $x$ is inward $G$, so it is not orthogonal to the normal vector at $x$). By the Implicit Function Theorem for the equation $q=0$, the function $\\mu_G$ is real analytic in a neighborhood $V_0$ of $\\partial G$. To see that $\\mu_G$ is real analytic outside $0$, fix $x_0\\in(\\RR^{2n})_*$. Then the set $$W_0:=\\left\\{x\\in\\RR^{2n}:\\frac{x}{\\mu_G(x_0)}\\in V_0\\right\\}$$ is open and contains $x_0$. Since $$\\mu_G(x)=\\mu_G(x_0)\\mu_G\\left(\\frac{x}{\\mu_G(x_0)}\\right),\\ x\\in W_0,$$ the function $\\mu_G$ is real analytic in $W_0$. Therefore, we can take $d\/ds$ on both sides of $\\mu_G(sx)=s\\mu_G(x),\\ x\\neq 0,\\ s>0$ to obtain $$\\langle\\nabla\\mu_G(x),x\\rangle_{\\RR}=\\mu_G(x),\\ x\\neq 0,$$ so $\\nabla\\mu_G\\neq 0$ in $(\\RR^{2n})_*$.\n\nFurthermore, $\\nabla\\mu^2_G=2\\mu_G\\nabla\\mu_G$, so $\\mu^2_G-1$ is also a defining function of $G$.\nTo show that $u:=\\mu^2_G$ is strongly convex outside $0$ let us prove that $$X^T\\mathcal{H}_aX>0,\\quad a\\in\\pa G,\\ X\\in(\\RR^{2n})_*,$$ where $\\mathcal{H}_x:=\\mathcal{H}u(x)$ for $x\\in(\\RR^{2n})_*$. Taking $\\pa\/\\pa x_j$ on both sides of $$u(sx)=s^2u(x),\\ x,s\\neq 0,$$ we get \\begin{equation}\\label{62}\\frac{\\pa u}{\\pa x_j}(sx)=s\\frac{\\pa u}{\\pa x_j}(x)\\end{equation} and further taking $d\/ds$ $$\\sum_{k=1}^{2n}\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(sx)x_k=\\frac{\\pa u}{\\pa x_j}(x).$$ In particular, $$x^T\\mathcal{H}_xy=\\sum_{j,k=1}^{2n}\\frac{\\pa^2 u}{\\pa x_k\\pa x_j}(x)x_ky_j=\\langle\\nabla u(x),y\\rangle_{\\RR},\\ x\\in(\\RR^{2n})_*,\\ y\\in\\RR^{2n}.$$ Let $a\\in\\pa G$. Since $\\langle\\nabla\\mu_G(a),a\\rangle_{\\RR}=\\mu_G(a)=1$, we have $a\\notin T^\\RR_G(a)$. Any $X\\in(\\RR^{2n})_*$ can be represented as $\\alpha a+\\beta Y$, where $Y\\in T^\\RR_G(a)$, $\\alpha,\\beta\\in\\RR$, $(\\alpha,\\beta)\\neq(0,0)$. Then \\begin{eqnarray*}X^T\\mathcal{H}_aX&=&\\alpha^2a^T\\mathcal{H}_aa+2\\alpha\\beta a^T\\mathcal{H}_aY+\\beta^2Y^T\\mathcal{H}_aY=\\\\&=&\\alpha^2\\langle\\nabla u(a),a\\rangle_{\\RR} +2\\alpha\\beta\\langle\\nabla u(a),Y\\rangle_{\\RR} +\\beta^2Y^T\\mathcal{H}_aY= \\\\&=&\\alpha^22\\mu_G(a)\\langle\\nabla\\mu_G(a),a\\rangle_{\\RR} +\\beta^2Y^T\\mathcal{H}_aY=\n2\\alpha^2+\\beta^2Y^T\\mathcal{H}_aY.\\end{eqnarray*} Since $G$ is strongly convex, the Hessian of any defining function is strictly positive on the tangent space, i.e. $Y^T\\mathcal{H}_aY>0$ if $Y\\in(T^\\RR_G(a))_*$. Hence $X^T\\mathcal{H}_aX\\geq 0$. Note that it cannot be $X^T\\mathcal{H}_aX=0$, since then $\\alpha=0$, consequently $\\beta\\neq 0$ and $Y^T\\mathcal{H}_aY=0$. On the other side $Y=X\/\\beta\\neq 0$ --- a contradiction.\n\nTaking $\\pa\/\\pa x_k$ on both sides of \\eqref{62} we obtain $$\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(sx)=\\frac{\\pa^2 u}{\\pa x_j\\pa x_k}(x),\\ x,s\\neq 0$$ and for $a,X\\in(\\RR^{2n})_*$ $$X^T\\mathcal{H}_aX=X^T\\mathcal{H}_{a\/\\mu_G(a)}X>0.$$\n\nLet us consider the sets $$D_t:=\\{x\\in\\CC^n:t\\mu^2_D(x)+(1-t)\\mu^2_{\\BB_n}(x)<1\\},\\ t\\in[0,1].$$ The functions $t\\mu^2_D+(1-t)\\mu^2_{\\BB_n}$ are real analytic in $(\\CC^n)_*$ and strongly convex in $(\\CC^n)_*$, so $D_t$ are strongly convex domains with real analytic boundaries satisfying $$D=D_1\\Subset D_{t_2}\\Subset D_{t_1}\\Subset D_0=\\BB_n\\text{\\ if \\ }00$ such that $\\delta\\BB_n\\Subset D$. Further, $\\nabla\\mu_{D_t}^2\\neq 0$ in $(\\RR^{2n})_*$. Set $$M:=\\sup\\left\\{\\frac{\\mathcal{H}\\mu_{D_t}^2(x;X)}{|\\nabla\\mu_{D_t}^2(y)|}:\nt\\in[0,1],\\ x,y\\in 2\\ov{\\BB}_n\\setminus\\delta\\BB_n,\\ X\\in\\RR^{2n},\\ |X|=1\\right\\}.$$ It is a positive number since the functions $\\mu_{D_t}^2$ are strongly convex in $(\\RR^{2n})_*$ and the `sup' of the continuous, positive function is taken over a compact set. Let $$r:=\\min\\left\\{\\frac{1}{2M},\\frac{\\dist(\\pa D,\\delta\\BB_n)}{2}\\right\\}.$$ For fixed $t\\in[0,1]$ and $a\\in\\pa D_t$ put $a':=a-r\\nu_{D_t}(a)$. In particular, $\\ov{B_n(a',r)}\\su 2\\ov{\\BB}_n\\setminus\\delta\\BB_n$. Let us define $$h(x):=\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a'|}(|x-a'|^2-r^2),\\ x\\in 2\\ov{\\BB}_n\\setminus\\delta\\BB_n.$$ We have $h(a)=1$ and $$\\nabla h(x)=\\nabla\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{|a-a'|}(x-a').$$ For $x=a$, dividing the right side by $|\\nabla\\mu^2_{D_t}(a)|$, we get a difference of the same normal vectors $\\nu_{D_t}(a)$, so $\\nabla h(a)=0$. Moreover, for $|X|=1$ $$\\mathcal{H}h(x;X)=\\mathcal{H}\\mu^2_{D_t}(x;X)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{r}\\leq M|\\nabla\\mu^2_{D_t}(a)|-2M|\\nabla\\mu^2_{D_t}(a)|<0.$$ It follows that $h\\leq 1$ in any convex set $S$ such that $a\\in S\\su 2\\ov{\\BB}_n\\setminus\\delta\\BB_n$. Indeed, assume the contrary. Then there is $y\\in S$ such that $h(y)>1$. Let us join $a$ and $y$ with an interval $$g:[0,1]\\ni t\\longmapsto h(ta+(1-t)y)\\in S.$$ Since $a$ is a strong local maximum of $h$, the function $g$ has a local minimum at some point $t_0\\in(0,1)$. Hence $$0\\leq g''(t_0)=\\mathcal{H}h(t_0a+(1-t_0)y;a-y),$$ which is impossible.\n\nSetting $S:=\\ov{B_n(a',r)}$, we get $$\\mu^2_{D_t}(x)\\leq 1+\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a'|}(|x-a'|^2-r^2)<1$$ for $x\\in B_n(a',r)$, i.e. $x\\in D_t$.\n\nThe proof of the exterior ball condition is similar. Set $$m:=\\inf\\left\\{\\frac{\\mathcal{H}\\mu_{D_t}^2(x;X)}{|\\nabla\\mu_{D_t}^2(y)|}:\nt\\in[0,1],\\ x,y\\in(\\ov{\\BB}_n)_*,\\ X\\in\\RR^{2n},\\ |X|=1\\right\\}.$$ Note that the $m>0$. Actually, the homogeneity of $\\mu_{D_t}$ implies $\\mathcal{H}\\mu_{D_t}^2(sx;X)=\\mathcal{H}\\mu_{D_t}^2(x;X)$ and $\\nabla\\mu_{D_t}^2(sx)=s\\nabla\\mu_{D_t}^2(x)$ for $x\\neq 0$, $X\\in \\RR^{2n}$, $s>0$. Therefore, there are positive constants $C_1,C_2$ such that $C_1\\leq\\mathcal{H}\\mu_{D_t}^2(x;X)$ for $x\\neq 0$, $X\\in \\RR^{2n}$, $|X|=1$ and $|\\nabla\\mu_{D_t}^2(y)|\\leq C_2$ for $y\\in\\ov\\BB_n$. In particular, $m\\geq C_1\/C_2$.\n\nLet $R:=2\/m$. For fixed $t\\in[0,1]$ and $a\\in\\pa D_t$ put $a'':=a-R\\nu_{D_t}(a)$. Let us define $$\\wi h(x):=\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a''|}(|x-a''|^2-R^2),\\ x\\in\\ov{\\BB}_n.$$ We have $\\wi h(a)=1$ and $$\\nabla\\wi h(x)=\\nabla\\mu^2_{D_t}(x)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{|a-a''|}(x-a''),$$ so $\\nabla\\wi h(a)=0$. Moreover, for $x\\in(\\ov{\\BB}_n)_*$ and $|X|=1$ $$\\mathcal{H}\\wi h(x;X)=\\mathcal{H}\\mu^2_{D_t}(x;X)-\\frac{|\\nabla\\mu^2_{D_t}(a)|}{R}\\geq m|\\nabla\\mu^2_{D_t}(a)|-m\/2|\\nabla\\mu^2_{D_t}(a)|>0.$$ Therefore, $a$ is a strong local minimum of $\\wi h$.\n\nNow using the properties listed above we may deduce that $\\wi h\\geq 1$ in $\\ov\\BB_n$. We proceed similarly as before: seeking a contradiction suppose that there is $y\\in\\ov\\BB_n$ such that $\\wi h(y)<1$. Moving $y$ a little (if necessary) we may assume that $0$ does not lie on the interval joining $a$ and $y$. Then the mapping $\\wi g(t):=\\wi h(ta+ (1-t)y)$ attains its local maximum at some point $t_0\\in(0,1)$. The second derivative of $\\wi g$ at $t_0$ is non-positive, which gives a contradiction with a positivity of the Hessian of the function $\\wi h$. \n\nHence, we get $$\\frac{|\\nabla\\mu^2_{D_t}(a)|}{2|a-a''|}(|x-a''|^2-R^2)\\leq\\mu^2_{D_t}(x)-1<0,$$ for $x\\in D_t$, so $D_t \\subset B_n(a'',R)$.\n\nLet $T$ be the set of all $t\\in[0,1]$ such that there is an $E$-mapping $f_{t}:\\DD\\longrightarrow D_{t}$ with $f_{t}(0)=z$, $f_{t}(\\xi_{t})=w$ for some $\\xi_{t}\\in(0,1)$ (resp. $f_{t}(0)=z$, $f_{t}'(0)=\\lambda_{t}v$ for some $\\lambda_{t}>0$). We claim that $T=[0,1]$. To prove it we will use the open-close argument.\n\nClearly, $T\\neq\\emptyset$, as $0\\in T$. Moreover, $T$ is open in $[0,1]$. Indeed, let $t_{0}\\in T$. It follows from Proposition \\ref{13} that there is a neighborhood $T_{0}$ of $t_{0}$ such that there are $E$-mappings $f_{t}:\\DD\\longrightarrow D_{t}$ and $\\xi_{t}\\in(0,1)$ such that $f_{t}(0)=z$, $f_{t}(\\xi_{t})=w$ for all $t\\in T_{0}$ (resp. $\\lambda_{t}>0$ such that $f_{t}(0)=z$, $f_{t}'(0)=\\lambda_{t} v$ for all $t\\in T_{0}$).\n\nTo prove that $T$ is closed, choose a sequence $\\{t_{m}\\}\\su T$ convergent to some $t\\in[0,1]$. We want to show that $t\\in T$. Since $f_{t_m}$ are $E$-mappings, they are complex geodesics. Therefore, making use of the inclusions $D\\subset D_{t_m}\\subset\\mathbb B_n$ we find that there is a compact set $K\\su(0,1)$ (resp. a compact set $\\widetilde K\\subset(0,\\infty)$) such that $\\{\\xi_{t_m}\\}\\subset K$ (resp. $\\{\\lambda_{t_m}\\}\\subset\\widetilde K$). By Propositions \\ref{8} and \\ref{10b} the functions $f_{t_{m}}$ and $\\widetilde f_{t_{m}}$ are equicontinuous in $\\mathcal{C}^{1\/2}(\\overline{\\DD})$ and by Propositions \\ref{9} and \\ref{10a} the functions $\\rho_{t_{m}}$ are uniformly bounded from both sides by positive numbers and equicontinuous in $\\mathcal{C}^{1\/2}(\\TT)$. From the Arzela-Ascoli Theorem there are a subsequence $\\{s_{m}\\}\\subset\\{t_{m}\\}$ and mappings $f,\\wi f\\in\\OO(\\DD)\\cap\\mathcal C^{1\/2}(\\overline{\\mathbb D})$, $\\rho\\in\\cC^{1\/2}(\\TT)$ such that $f_{s_{m}}\\to f$, $\\widetilde{f}_{s_{m}}\\to\\wi f$ uniformly on $\\overline{\\DD}$, $\\rho_{s_{m}}\\to\\rho$ uniformly on $\\TT$ and $\\xi_{s_m}\\to\\xi\\in (0,1)$ (resp. $\\lambda_{s_m}\\to\\lambda>0$).\n\nClearly, $f(\\CDD)\\su\\overline{D}_{t}$, $f(\\TT)\\su\\partial D_{t}$ and $\\rho>0$. By the strong pseudoconvexity of $D_t$ we get $f(\\DD)\\su D_t$.\n\nThe conditions (3') and (4) of Definitions~\\ref{21} and \\ref{21e} follow from the uniform convergence of suitable functions. Therefore, $f$ is a weak $E$-mapping of $D_{t}$, consequently an $E$-mapping of $D_t$, satisfying $f(0)=z$, $f(\\xi)=w$ (resp. $f(0)=z$, $f'(0)=\\lambda v$).\n\nLet us go back to the general situation that is when a domain $D$ is bounded strongly linearly convex with real analytic boundary. Take a of point $\\eta\\in\\partial{D}$ such that $\\max_{\\zeta\\in\\partial{D}}|z-\\zeta|=|z-\\eta|$. Then $\\eta$ is a point of the strong convexity of $D$. Indeed, by the Implicit Function Theorem one can assume that in a neighborhood of $\\eta$ the defining functions of $D$ and $B:=B_n(z,|z-\\eta|)$ are of the form $r(x):=\\wi r(\\wi x)-x_{2n}$ and $q(x):=\\wi q(\\wi x)-x_{2n}$ respectively, where $x=(\\wi x,x_{2n})\\in\\RR^{2n}$ is sufficiently close to $\\eta$. From the inclusion $D\\su B$ it it follows that $r-q\\geq 0$ near $\\eta$ and $(r-q)(\\eta)=0$. Thus the Hessian $\\mathcal{H}(r-q)(\\eta)$ is weakly positive in $\\CC^n$. Since $\\mathcal{H}q(\\eta)$ is strictly positive on $T_B^\\RR(\\eta)_*=T_D^\\RR(\\eta)_*$, we find that $\\mathcal{H}r(\\eta)$ is strictly positive on $T_D^\\RR(\\eta)_*$, as well.\n\nBy a continuity argument, there is a convex neighborhood $V_0$ of $\\eta$ such that all points from $\\pa D\\cap V_0$ are points of the strong convexity of $D$. It follows from Proposition \\ref{localization} (after shrinking $V_0$ if necessary) that there is a weak stationary mapping $g:\\DD\\longrightarrow D\\cap V_0$ such that $g(\\TT)\\subset\\partial D$. In particular, $g$ is a weak stationary mapping of $D$. Since $D\\cap V_0$ is convex, the condition with the winding number is satisfied on $D\\cap V_0$ (and then on the whole $D$). Consequently $g$ is an $E$-mapping of $D$.\n\nIf $z=g(0)$, $w=g(\\xi)$ for some $\\xi\\in\\DD$ (resp. $z=g(0)$, $v=g'(0)$) then there is nothing to prove. In the other case let us take curves $\\alpha:[0,1]\\longrightarrow D$, $\\beta:[0,1]\\longrightarrow D$ joining $g(0)$ and $z$, $g(\\xi)$ and $w$ (resp. $g(0)$ and $z$, $g'(0)$ and $v$). We may assume that the images of $\\alpha$ and $\\beta$ are disjoint. Let $T$ be the set of all $t\\in[0,1]$ such that there is an $E$-mapping $g_{t}:\\DD\\longrightarrow D$ such that $g_{t}(0)=\\alpha(t)$, $g_{t}(\\xi_{t})=\\beta(t)$ for some $\\xi_{t}\\in(0,1)$ (resp. $g_{t}(0)=\\alpha(t)$, $g_{t}'(0)=\\lambda_{t}\\beta(t)$ for some $\\lambda_{t}>0$). Again $T\\neq\\emptyset$ since $0\\in T$. Using the results of Section \\ref{22} similarly as before (but for one domain), we see that $T$ is closed.\n\nSince $\\wi k_D$ is symmetric, it follows from Proposition \\ref{13}(1) that the set $T$ is open in $[0,1]$ (first we move along $\\alpha$, then by the symmetry we move along $\\beta$). Therefore, $g_1$ is the $E$-mapping for $z,w$.\n\nIn the case of $\\kappa_{D}$ we change a point and then we change a direction. To be more precise, consider the set $S$ of all $s\\in[0,1]$ such that there is an $E$-mapping $h_{s}:\\DD\\longrightarrow D$ such that $h_{s}(0)=\\alpha(s)$. Then $0\\in S$, by Proposition \\ref{13}(1) the set $S$ is open in $[0,1]$ and by results of Section~\\ref{22} again, it is closed. Hence $S=[0,1]$. Now we may join $h'_{1}(0)$ and $v$ with a curve $\\gamma:[0,1]\\longrightarrow \\mathbb C^n$. Let us define $R$ as the set of all $r\\in[0,1]$ such that there is an $E$-mapping $\\wi h_{r}:\\DD\\longrightarrow D$ such that $\\wi h_{r}(0)=h_1(0)$, $\\wi h'_{r}(0)=\\sigma_{r}\\gamma(1-r)$ for some $\\sigma_r>0$. Then $1\\in R$, by Proposition \\ref{13}(2) the set $R$ is open in $[0,1]$ and, by Section \\ref{22}, it is closed. Hence $R=[0,1]$, so $\\wi h_{0}$ is the $E$-mapping for $z,v$.\n\\end{proof}\n\nNow we are in position that allows us to prove the main results of the Lempert's paper.\n\n\\begin{proof}[Proof of Theorem \\ref{lem-car} $($real analytic case$)$] It follows from Lemma \\ref{lemat} that for any different points $z,w\\in D$ (resp. $z\\in D$, $v\\in(\\CC^n)_*$) one may find an $E$-mapping passing through them (resp. $f(0)=z$, $f'(0)=v$). On the other hand, it follows from Proposition \\ref{1} that $E$-mappings have left inverses, so they are complex geodesics.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main} $($real analytic case$)$] This is a direct consequence of Lemma \\ref{lemat} and Corollary \\ref{28}.\n\\end{proof}\n\n\\bigskip\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{center}{\\sc $\\cC^2$-smooth case}\\end{center}\n\\bigskip\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{un} Let $D\\su\\mathbb C^n$, $n\\geq 2$, be a bounded strongly pseudoconvex domain with $\\mathcal C^2$-smooth boundary. Take $z\\in D$ and let $r$ be a defining function of $D$ such that \n\\begin{itemize}\\item $r\\in \\mathcal C^2(\\mathbb C^n);$\n\\item $D=\\{x\\in \\mathbb C^n:r(x)<0\\}$;\n\\item $\\mathbb C^n\\setminus D=\\{x\\in \\mathbb C^n:r(x)>0\\}$;\n\\item $|\\nabla r|=1$ on $\\partial D;$\n\\item $\\sum_{j,k=1}^n\\frac{\\partial^2 r}{\\partial z_j\\partial\\overline z_k}(a)X_{j}\\overline{X}_{k}\\geq C|X|^2$ for any $a\\in \\partial D$ and $X\\in \\mathbb C^n$ with some constant $C>0$.\n\\end{itemize}\n\nSuppose that there is a sequence $\\{r_m\\}$ of $\\mathcal C^2$-smooth real-valued functions such that $D^{\\alpha}r_n$ converges to $D^{\\alpha}r$ locally uniformly for any $\\alpha\\in \\mathbb N_0^{2n}$ such that $|\\alpha|:=|\\alpha_1| +\\ldots+|\\alpha_n|\\leq 2$. Let $D_m$ be a connected component of the set $\\{x\\in\\mathbb C^n:r_m(x)<0\\}$, containing the point $z$.\n\nThen there is $c>0$ such that $(D_m,z)$ and $(D,z)$ belong to $\\mathcal D(c)$, $m>>1.$\n\\end{lem}\n\n\n\\begin{proof} Losing no generality assume that $D\\Subset\\mathbb B_n.$\nNote that the conditions (1), (5), (6) of Definition \\ref{30} are clearly satisfied. To find $c$ satisfying ($2$), we take $s>0$ such that $\\mathcal H r (x;X)< s |X|^2$ for $x\\in\\ov\\BB_n$ and $X\\in(\\mathbb R^{2n})_*$. Then $\\HH r_m (x;X)<2s|X|^2$ for $x\\in\\ov\\BB_n$, $X\\in(\\mathbb R^{2n})_*$ and $m>>1$. Let $U_0\\subset\\mathbb B_n$ be an open neighborhood of $\\pa D$ such that $|\\nabla r|$ is on $U_0$ between $3\/4$ and $5\/4$. Note that $\\partial D_m\\subset U_0$ and $|\\nabla r_m|\\in (1\/2, 3\/2)$ on $U_0$ for $m>>1$.\n\nFix $m$ and $a\\in \\partial D_m$ and put $b:=a-R\\nu_{D_m}(a)$, where a small number $R>0$ will be specified later. There is $t>0$ such that $\\nabla r_m(a)=2t(a-b)$. Note that $t$ may be arbitrarily large provided that $R$ was small enough. We take $t:=2s$ and $R:=|\\nabla r_m(a)|\/t$. Then we have $\\mathcal H r_m(x;X)<2t |X|^2$ for $x\\in\\ov\\BB_n$, $X\\in(\\mathbb R^{2n})_*$ and $m>>1$. Then a function $$h(x):=r_m(x)-t(|x-b|^2-R^2),\\ x\\in \\mathbb C^n,$$ attains at $a$ its global maximum on $\\ov\\BB_n$ ($a$ is a strong local maximum and the Hessian of $h$ is negative on the convex set $\\ov\\BB_n$, cf. the proof of Lemma \\ref{lemat}).\nThus $h\\leq 0$ on $\\mathbb B_n$. From this we immediately get (2).\n\nNote that it follows from (2) that $D_m=\\{x\\in\\mathbb C^n:r_m(x)<0\\}$ for $m$ big enough (i.e. $\\{x\\in \\mathbb C^n:\\ r_m(x)<0\\}$ is connected).\n\nMoreover, the condition (2) implies the condition (3) as follows. We infer from Remark~\\ref{D(c),4} that there is $c'>0$ such that $D$ satisfies (3) with $c'$. Let $m_0$ be such that the Hausdorff distance between $\\partial D$ and $\\partial D_m$ is smaller than $1\/c'$ for $m\\geq m_0$. There is $c''$ such that $D_{m_0}$ satisfies (3) with $c''$. Losing no generality we may assume that $c''c'$ such that every $D_m$ satisfies (4) with $c$ for $m$ big enough. To do it let us cover $\\partial D$ with a finite number of balls $B_j$, $j=1,\\ldots,N$, from condition (4) and let $B'_j$ be a ball contained relatively in $B_j$ such that $\\{B_j\\}$ covers $\\partial D$, as well. Let $\\Phi_j$ be mappings corresponding to $B_j$. Let $\\eps$ be such that any ball of radius $\\eps$ intersecting $\\partial D$ non-emptily is relatively contained in $B_j'$ for some $j$. Observe that any ball $B$ of radius $\\eps\/2$ intersecting non-emptily $\\partial D_m$ is contained in a ball of radius $\\eps$ intersecting non-emptily $\\partial D$; hence it is contained in $B_j'$ for some $j$. Then the pair $B$, $\\Phi_j$ satisfies the conditions (4) (b), (c) and (d). Therefore, it suffices to check that there is $c>2\/\\eps$ such that each pair $B_j'$, $\\Phi_j$ satisfies the condition (4) for $D_m$ with $c$ ($m>>1$). This is possible since $\\Phi_j(D_m)\\subset\\Phi_j(D)$, $D^\\alpha\\Phi_j(\\pa D_m\\cap B_j)$ converges to $D^\\alpha\\Phi_j(\\pa D\\cap B_j)$ for $|\\alpha|\\leq 2$ and for any $w\\in\\Phi(\\pa D\\cap B_j)$ there is a ball of radius $2\/\\eps$ containing $\\Phi_j(D)$ and tangent to $\\partial\\Phi_j(D)$ at $w$. To be precise, we proceed as follows. \n\nLet $a,b\\in\\CC^n$ and let $x\\in\\pa B_n(a,\\wi c)$, where $\\wi c>c'$. Then a ball $B_n(2a-x,2\\wi c)$ contains $B_n(a,\\wi c)$ and is tangent to $B_n(a,\\wi c)$ at $x$. There is a number $\\eta=\\eta(\\delta,\\wi c)>0$, independent of $a,b,x$, such that the diameter of the set $B_n(b,\\wi c)\\setminus B_n(2a-x,2\\wi c)$ is smaller than $\\delta>0$, whenever $|a-b|<\\eta$ (this is a simple consequence of the triangle inequality).\n\nLet $\\wi s>0$ be such that $\\mathcal H(r\\circ\\Phi_j^{-1})(x;X)\\geq 2\\wi s|X|^2$ for $x\\in U_j$, $j=1,\\ldots,N$, where $U_j$ is an open neighborhood of $\\Phi_j(\\partial D\\cap B_j)$. Then, for $m$ big enough, $\\mathcal H(r_m\\circ \\Phi_j^{-1})(x;X)\\geq\\wi s|X|^2$ for $x\\in U_j$ and $\\Phi_j(\\partial D_m\\cap B_j')\\subset U_j$, $j=1,\\ldots,N$. Repeating for the function $$x\\longmapsto(r_m\\circ\\Phi_j^{-1})(x)-\\wi t(|x-\\wi b|^2-\\wi R^2)$$ the argument used in the interior ball condition with suitable chosen $\\wi t$ and uniform $\\wi R>c$, we find that there is uniform $\\wi\\eps>0$ such that for any $j,m$ and $w\\in\\Phi_j(\\partial D_m\\cap B_j')$ there is a ball $B$ of radius $\\wi R$, tangent to $\\Phi_j(\\partial D_m\\cap B_j')$ at $w$, such that $\\Phi_j(\\partial D_m\\cap B_j')\\cap B_n(w,\\wi\\eps)\\subset B$. Let $a_{j,m}(w)$ denote its center.\n\nOn the other hand for any $w\\in \\Phi_j(\\partial D_m\\cap B_j')$ there is $t>0$ such that $w'=w+t\\nu (w)\\in \\Phi_j(\\partial D\\cap B_j)$, where $\\nu(w)$ is a normal vector to $\\Phi_j(\\partial D_m\\cap B_j')$ at $w$. Let $a_j(w')$ be a center of a ball of radius $\\wi R$ tangent to $\\Phi_j(\\partial D\\cap B_j)$ at $w'$. It follows that $|a_{j,m}(w)-a_j(w')|<\\eta(\\wi\\eps\/2,\\wi R)$ provided that $m$ is big enough. \n\nJoinining the facts presented above, we finish the proof of the exterior ball condition (with a radius dependent only on $\\wi\\eps$ and $\\wi R$).\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorems \\ref{lem-car} and \\ref{main} \\emph{(}$\\mathcal C^2$-smooth case$)$]\nLosing no generality assume that $0\\in D\\Subset\\BB_n$.\n\nIt follows from the Weierstrass Theorem that there is sequence $\\{P_k\\}$ of real polynomials on $\\CC^n\\simeq\\mathbb R^{2n}$ such that $$D^{\\alpha}P_{k}\\to D^{\\alpha}r \\text{ uniformly on }\\ov\\BB_n,$$ where $\\alpha=(\\alpha_1,\\ldots, \\alpha_{2n})\\in \\mathbb N_0^{2n}$ is such that $|\\alpha|=\\alpha_1+\\ldots +\\alpha_{2n}\\leq 2$. Consider the open set $$\\wi D_{k,\\eps}:=\\{x\\in \\mathbb C^n:P_{k}(x)+\\eps<0\\}.$$ Let $\\eps_{m}$ be a sequence of positive numbers converging to $0$ such that $3\\eps_{m+1}<\\eps_m.$\n\nFor any $m\\in \\mathbb N$ there is $k_{m}\\in\\NN$ such that $\\sup_{\\ov\\BB_n}|P_{k_{m}}-r|<\\eps_{m}$. Putting $r_{m}:=P_{k_{m}}+2\\eps_{m}$, we get $r+\\eps_{m}>1$. Therefore, for any $m>>1$ one may find an $E$-mapping $f_m$ of $D_m$ for $z,w$ (resp. for $z,v$). Since $(D_m,z)\\in \\mathcal D(c)$ for some uniform $c>0$ ($m>>1$) (Lemma~\\ref{un}), we find that $f_m$, $\\wi f_m$ and $\\rho_m$ satisfy the uniform estimates from Section~\\ref{22}. Thus, passing to a subsequence we may assume that $\\{f_m\\}$ converges uniformly on $\\CDD$ to a mapping $f\\in\\OO(\\DD)\\cap\\cC^{1\/2}(\\CDD)$ passing through $z,w$ (resp. such that $f(0)=z$, $f'(0)=\\lambda v$, $\\lambda>0$), $\\{\\wi f_m\\}$ converges uniformly on $\\CDD$ to a mapping $\\wi f\\in\\OO(\\DD)\\cap\\mathcal C^{1\/2}(\\overline{\\mathbb D})$ and $\\{\\rho_m\\}$ is convergent uniformly on $\\TT$ to a positive function $\\rho\\in\\cC^{1\/2}(\\TT)$ (in particular, $f'\\bullet\\wi f=1$ on $\\DD$, so $\\wi f$ has no zeroes in $\\CDD$). We already know that this implies that $f$ is a weak $E$-mapping of $D$.\n\nTo get $\\cC^{k-1-\\eps}$-smoothness of the extremal $f$ and its associated mappings for $k\\geq 3$, it suffices to repeat the proof of Proposition~5 of \\cite{Lem2}. This is just the Webster Lemma (we have proved it in the real analytic case --- see Proposition~\\ref{6}). Namely, let $$\\psi:\\partial D\\ni z\\longmapsto(z,T_{D}^\\mathbb{C}(z))\\in \\mathbb C^n\\times(\\mathbb P^{n-1})_*,$$ where $\\mathbb P^{n-1}$ is the $(n-1)$-dimensional complex projective space. Let $\\pi:(\\CC^n)_*\\longrightarrow\\mathbb P^{n-1}$ be the canonical projection. \n\nBy \\cite{Web}, $\\psi(\\partial D)$ is a totally real manifold of $\\mathcal C^{k-1}$ class. Observe that the mapping $(f,\\pi\\circ \\wi f):\\CDD\\longrightarrow\\CC^n\\times\\mathbb P^{n-1}$ is $1\/2$-H\\\"older continuous, is holomorphic on $\\mathbb D$ and maps $\\mathbb T$ into $\\psi(\\partial D)$. Therefore, it is $\\mathcal C^{k-1-\\eps}$-smooth for any $\\eps>0$, whence $f$ is $\\mathcal C^{k-1-\\eps}$-smooth. Since $\\nu_D\\circ f$ is of class $\\mathcal C^{k-1-\\eps}$, it suffices to proceed as in the proof of Proposition~\\ref{6}.\n\\end{proof}\n\n\n\n\\section{Appendix}\\label{Appendix}\n\\subsection{Totally real submanifolds}\nLet $M\\subset\\CC^m$ be a totally real local $\\CLW$ submanifold of the real dimension $m$. Fix a point $z\\in M$. There are neighborhoods $U_0\\su\\RR^m$, $V_0\\su\\CC^m$ of $0$ and $z$ and a $\\CLW$ diffeomorphism $\\widetilde{\\Phi}:U_0\\longrightarrow M\\cap V_0$ such that $\\widetilde{\\Phi}(0)=z$. The mapping $\\widetilde{\\Phi}$ can be extended in a natural way to a mapping $\\Phi$ holomorphic in a neighborhood of $0$ in $\\CC^m$. Note that this extension will be biholomorphic in a neighborhood of $0$. Actually, we have $$\\frac{\\partial\\Phi_j}{\\partial z_k}(0)=\\frac{\\partial\\Phi_j}{\\partial\nx_k}(0)=\\frac{\\partial\\widetilde{\\Phi}_j}{\\partial x_k}(0),\\ j,k=1,\\ldots,m,$$ where $x_k=\\re z_k$. Suppose that the complex derivative $\\Phi'(0)$ is not an isomorphism. Then there is $X\\in(\\CC^m)_*$ such that $\\Phi'(0)X=0$, so \\begin{multline*}0=\\sum_{k=1}^m\\frac{\\partial\\Phi}{\\partial z_k}(0)X_k=\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)(\\re X_k+i\\im X_k)=\\\\=\\underbrace{\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)\\re X_k}_{=:A}+i\\underbrace{\\sum_{k=1}^m\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0)\\im X_k}_{=:B}.\\end{multline*}\nThe vectors $$\\frac{\\partial\\wi\\Phi}{\\partial x_k}(0),\\ k=1,\\ldots,m$$ form a basis of $T^{\\RR}_M(z)$, so $A,B\\in T^{\\RR}_M(z)$, consequently $A,B\\in iT^{\\RR}_M(z)$. Since $M$ is totally real, i.e. $T^{\\RR}_M(z)\\cap iT^{\\RR}_M(z)=\\{0\\}$, we have $A=B=0$. By a property of the basis we get $\\re X_k=\\im X_k=0$, $k=1,\\ldots,m$ --- a contradiction.\n\nTherefore, $\\Phi$ in a neighborhood of $0$ is a biholomorphism of two open subsets of $\\CC^m$, which maps a neighborhood of $0$ in $\\RR^m$ to a neighborhood of $z$ in $M$.\n\n\n\\begin{lemm}[Reflection Principle]\\label{reflection}\nLet $M\\subset\\CC^m$ be a totally real local $\\CLW$ submanifold of the real\ndimension $m$. Let $V_0\\subset\\CC$ be a neighborhood of $\\zeta_0\\in\\TT$ and let $g:\\overline{\\DD}\\cap V_0\\longrightarrow\\CC^m$ be a continuous mapping. Suppose that $g\\in\\OO(\\DD\\cap V_0)$ and $g(\\TT\\cap V_0)\\subset M$. Then $g$ can be extended holomorphically past $\\TT\\cap V_0$.\n\\end{lemm}\n\\begin{proof}\nIn virtue of the identity principle it is sufficient to extend $g$ locally\npast an arbitrary point $\\zeta_0\\in\\TT\\cap V_0$. For a point $g(\\zeta_0)\\in M$ take $\\Phi$ as above. Let $V_1\\subset V_0$ be a neighborhood of $\\zeta_0$ such that $g(\\CDD\\cap V_1)$ is contained in the image\nof $\\Phi$. The mapping $\\Phi^{-1}\\circ g$ is holomorphic in $\\DD\\cap V_1$ and has\nreal values on $\\TT\\cap V_1$. By the ordinary Reflection Principle we can\nextend this mapping holomorphically past $\\TT\\cap V_1$. Denote this extension by\n$h$. Then $\\Phi\\circ h$ is an extension of $g$ in a neighborhood of $\\zeta_0$.\n\\end{proof}\n\n\n\n\n\\subsection{Schwarz Lemma for the unit ball}\n\\begin{lemm}[Schwarz Lemma]\\label{schw}\nLet $f\\in\\OO(\\DD,B_n(a,R))$ and $r:=|f(0)-a|$. Then $$|f'(0)|\\leq \\sqrt{R^2-r^2}.$$\n\\end{lemm}\n\n\n\\subsection{Some estimates of holomorphic functions of $\\cC^{\\alpha}$-class}\n\nLet us recall some theorems about functions holomorphic in $\\DD$ and continuous in $\\CDD$. Concrete values of constants $M,K$ are possible to calculate, seeing on the proofs. In fact, it is only important that they do not depend on functions.\n\\begin{tww}[Hardy, Littlewood, \\cite{Gol}, Theorem 3, p. 411]\\label{lit1}\nLet $f\\in\\OO(\\DD)\\cap\\cC(\\CDD)$. Then for $\\alpha\\in(0,1]$ the following conditions are equivalent\n\\begin{eqnarray}\\label{47}\\exists M>0:\\ |f(e^{i\\theta})-f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^{\\alpha},\\ \\theta,\\theta'\\in\\RR;\\\\\n\\label{45}\\exists K>0:\\ |f'(\\zeta)|\\leq K(1-|\\zeta|)^{\\alpha-1},\\ \\zeta\\in\\DD.\n\\end{eqnarray}\nMoreover, if there is given $M$ satisfying \\eqref{47} then $K$ can be chosen as $$2^{\\frac{1-3\\alpha}{2}}\\pi^\\alpha M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt$$ and if there is given $K$ satisfying \\eqref{45} then $M$ can be chosen as $(2\/\\alpha+1)K$.\n\\end{tww}\n\\begin{tww}[Hardy, Littlewood, \\cite{Gol}, Theorem 4, p. 413]\\label{lit2}\nLet $f\\in\\OO(\\DD)\\cap\\cC(\\CDD)$ be such that $$|f(e^{i\\theta})-f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^{\\alpha},\\ \\theta,\\theta'\\in\\RR,$$ for some $\\alpha\\in(0,1]$ and $M>0$. Then $$|f(\\zeta)-f(\\zeta')|\\leq K|\\zeta-\\zeta'|^{\\alpha},\\\n\\zeta,\\zeta'\\in\\CDD,$$ where $$K:=\\max\\left\\{2^{1-2\\alpha}\\pi^\\alpha M,2^{\\frac{3-5\\alpha}{2}}\\pi^\\alpha\\alpha^{-1} M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt\\right\\}.$$\n\\end{tww}\n\\begin{tww}[Privalov, \\cite{Gol}, Theorem 5, p. 414]\\label{priv}\nLet $f\\in\\OO(\\DD)$ be such that $\\re f$ extends continuously on $\\CDD$ and $$|\\re f(e^{i\\theta})-\\re f(e^{i\\theta'})|\\leq M|\\theta-\\theta'|^\\alpha,\\ \\theta,\\theta'\\in\\RR,$$ for some $\\alpha\\in(0,1)$ and $M>0$. Then $f$ extends continuously on $\\CDD$ and $$|f(\\zeta)-f(\\zeta')|\\leq K|\\zeta-\\zeta'|^\\alpha,\\ \\zeta,\\zeta'\\in\\CDD,$$ where $$K:=\\max\\left\\{2^{1-2\\alpha}\\pi^\\alpha,2^{\\frac{3-5\\alpha}{2}}\\pi^\\alpha\\alpha^{-1}\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt\\right\\}\\left(\\frac{2}{\\alpha}+1\\right)2^{\\frac{3-3\\alpha}{2}}\\pi^{\\alpha}M\\int_0^\\infty\\frac{t^\\alpha}{1+t^2}dt.$$\n\\end{tww}\n\\subsection{Sobolev space}\nThe Sobolev space $W^{2,2}(\\TT)=W^{2,2}(\\TT,\\CC^m)$ is a space of functions $f:\\TT\\longrightarrow\\CC^m$, whose first two derivatives (in the sense of distribution) are in $L^2(\\TT)$ (here we use a standard identification of functions on the unit circle and functions on the interval $[0,2\\pi]$). Then $f$ is $\\mathcal C^1$-smooth.\n\nIt is a complex Hilbert space with the following scalar product\n$$\\langle f,g\\rangle_W:=\\langle f,g\\rangle_{L}+\\langle f',g'\\rangle_{L}+\\langle f'',g''\\rangle_{L},$$\nwhere $$\\langle\\wi f,\\wi g\\rangle_{L}:=\\frac{1}{2\\pi}\\int_0^{2\\pi}\\langle\\wi f(e^{it}),\\wi g(e^{it})\\rangle dt.$$ Let $\\|\\cdot\\|_L$, $\\|\\cdot\\|_W$ denote the norms induced by $\\langle\\cdotp,-\\rangle_L$ and $\\langle\\cdotp,-\\rangle_W$. The following characterization simply follows from Parseval's identity $$W^{2,2}(\\TT)=\\left\\{f\\in L^2(\\TT):\\sum_{k=-\\infty}^{\\infty}(1+k^2+k^4)|a_k|^2<\\infty\\right\\},$$ where $a_k\\in\\CC^m$ are the $m$-dimensional Fourier coefficients of $f$, i.e. $$f(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a_k\\zeta^k,\\ \\zeta\\in\\TT.$$ More precisely, Parseval's identity gives $$\\|f\\|_W=\\sqrt{\\sum_{k=-\\infty}^{\\infty}(1+k^2+k^4)|a_k|^2},\\ f\\in W^{2,2}(\\TT).$$ Note that $W^{2,2}(\\TT)\\su\\mc{C}^{1\/2}(\\TT)\\su\\mc{C}(\\TT)$ and both inclusions are continuous (in particular, both inclusions are real analytic). Note also that\n \\begin{equation}\\label{67}\\|f\\|_{\\sup}\\leq\\sum_{k=-\\infty}^{\\infty}|a_k|\\leq\\sqrt{\\sum_{k=-\\infty}^{\\infty}\\frac{1}{1+k^2}\\sum_{k=-\\infty}^{\\infty}(1+k^2)|a_k|^2}\\leq\\frac{\\pi}{\\sqrt 3}\\|f\\|_W.\\end{equation}\\\\\n\nNow we want to show that there exists $C>0$ such that $$\\|h^\\alpha\\|_W\\leq C^{|\\alpha|}\\|h_1\\|^{\\alpha_1}_W\\cdotp\\ldots\\cdotp\\|h_{2n}\\|^{\\alpha_{2n}}_W,\\quad h\\in W^{2,2}(\\TT,\\CC^n),\\,\\alpha\\in\\NN_0^{2n}.$$ Thanks to the induction it suffices to prove that there is $\\wi C>0$ satisfying $$\\|h_1h_2\\|_W\\leq\\wi C\\|h_1\\|_W\\|h_2\\|_W,\\quad h_1,h_2\\in W^{2,2}(\\TT,\\CC).$$ Using \\eqref{67}, we estimate $$\\|h_1h_2\\|^2_W=\\|h_1h_2\\|^2_L+\\|h_1'h_2+h_1h_2'\\|^2_L+\\|h_1''h_2+2h_1'h_2'+h_1h_2''\\|^2_L\\leq$$$$\\leq C_1\\|h_1h_2\\|_{\\sup}^2+(\\|h_1'h_2\\|_L+\\|h_1h_2'\\|_L)^2+(\\|h_1''h_2\\|_L+\\|2h_1'h_2'\\|_L+\\|h_1h_2''\\|_L)^2\\leq$$\\begin{multline*}\\leq C_1\\|h_1\\|_{\\sup}^2\\|h_2\\|_{\\sup}^2+(C_2\\|h_1'\\|_L\\|h_2\\|_{\\sup}+C_2\\|h_1\\|_{\\sup}\\|h_2'\\|_L)^2+\\\\+(C_2\\|h_1''\\|_L\\|h_2\\|_{\\sup}+C_2\\|2h_1'h_2'\\|_{\\sup}+C_2\\|h_1\\|_{\\sup}\\|h_2''\\|_L)^2\\leq\\end{multline*}\\begin{multline*}\\leq C_3\\|h_1\\|_W^2\\|h_2\\|_W^2+(C_4\\|h_1\\|_W\\|h_2\\|_W+C_4\\|h_1\\|_W\\|h_2\\|_W)^2+\\\\+(C_4\\|h_1\\|_W\\|h_2\\|_W+2C_2\\|h_1'\\|_{\\sup}\\|h_2'\\|_{\\sup}+C_4\\|h_1\\|_W\\|h_2\\|_W)^2\\leq\\end{multline*}$$\\leq C_5\\|h_1\\|_W^2\\|h_2\\|_W^2+(2C_4\\|h_1\\|_W\\|h_2\\|_W+2C_2\\|h_1'\\|_{\\sup}\\|h_2'\\|_{\\sup})^2$$ with constants $C_1,\\ldots,C_5$. Expanding $h_j(\\zeta)=\\sum_{k=-\\infty}^{\\infty}a^{(j)}_k\\zeta^{k}$, $\\zeta\\in\\TT$, $j=1,2$, we obtain $$\\|h_j'\\|_{\\sup}\\leq\\sum_{k=-\\infty}^{\\infty}|k||a^{(j)}_k|\\leq\\sqrt{\\sum_{k\\in\\ZZ_*}\\frac{1}{k^2}\\sum_{k\\in\\ZZ_*}k^4|a^{(j)}_k|^2}\\leq\\frac{\\pi}{\\sqrt 3}\\|h_j\\|_W$$ and finally $\\|h_1h_2\\|^2_W\\leq C_6\\|h_1\\|_W^2\\|h_2\\|_W^2$ for some constant $C_6$.\n\\subsection{Matrices}\n\\begin{propp}[Lempert, \\cite{Lem2}, Th\\'eor\\`eme $B$]\\label{12}\nLet $A:\\TT\\longrightarrow\\CC^{n\\times n}$ be a matrix-valued real analytic mapping such\nthat $A(\\zeta)$ is self-adjoint and strictly positive for any $\\zeta\\in\\TT$. Then there exists $H\\in\\OO(\\CDD,\\CC^{(n-1)\\times(n-1)})$ such that $\\det H\\neq 0$ on $\\CDD$ and $HH^*=A$ on $\\TT$.\n\\end{propp}\nIn \\cite{Lem2}, the mapping $H$ was claimed to be real analytic in a neighborhood of $\\CDD$ and holomorphic in $\\DD$, but it is equivalent to $H\\in\\OO(\\CDD)$. Indeed, since $\\ov\\pa H$ is real analytic near $\\CDD$ and $\\ov\\pa H=0$ in $\\DD$, the identity principle for real analytic functions implies $\\ov\\pa H=0$ in a neighborhood of $\\CDD$.\n\\begin{propp}[\\cite{Tad}, Lemma $2.1$]\\label{59}\nLet $A$ be a complex symmetric $n\\times n$ matrix. Then $$\\|A\\|=\\sup\\{|z^TAz|:z\\in\\CC^n,\\,|z|=1\\}.$$\n\\end{propp}\n\\bigskip\n\\textsc{Acknowledgements.} We would like to thank Sylwester Zaj\\k ac for helpful discussions. We are also grateful to our friends for the participation in preparing some parts of the work.\n\\medskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction \\label{sec:introduction}}\nBrillouin scattering (BS) refers to the nonlinear interaction between optical and mechanical fields inside a material. BS has been widely exploited in optical fibers to implement a wide range of devices, including optical amplifiers, ultra-narrow linewidth lasers, radio-frequency (RF) signal generators, and distributed sensors \\cite{garmire2017perspectives}.\n\nBrillouin scattering was for long thought to be mediated by electrostrictive forces only. Thus, its spectrum was considered to be governed by material properties \\cite{wiederhecker_controlling_2009}. In 2006, microstructuration of optical fibers enabled shaping the BS spectrum \\cite{dainese2006stimulated}, opening a new path for geometric control of this effect \\cite{beugnot2007guided}. In 2012, a new theory \\cite{peter_t_rakich_giant_2012} predicted that Brillouin interactions could be greatly magnified by strong radiation\npressure on the boundaries of suspended silicon waveguides with nanometric-scale core sizes \\cite{qiu_stimulated_2013,wolff_stimulated_2015}. The simultaneous confinement of optical and mechanical modes is challenging in silicon-on-insulator (SOI) waveguides due to a strong phonon leakage towards the silica cladding \\cite{eggleton2019brillouin_LL,wiederhecker_brillouin_2019,safavi-naeini_controlling_2019}. However, this limitation can be circumvented by isolating the silicon waveguide core by complete or partial removal of the silica cladding \\cite{shin_tailorable_2013,laer_net_2015,peter_t_rakich_giant_2012}. Suspended or quasi-suspended structures such as silicon membrane rib waveguides \\cite{kittlaus_large_2016} and fully suspended silicon nanowires \\cite{laer_net_2015} have demonstrated large Brillouin gain. These results generated a great scientific interest for its potential for laser sources \\cite{otterstrom_silicon_2018}, microwave signal generation \\cite{li_microwave_2013} and processing \\cite{liu_chip-based_2018}, sensing applications \\cite{chow_distributed_2018, lai_earth_2020} and non-reciprocal optical devices \\cite{kittlaus_non-reciprocal_2018}.\nIn particular, pedestal waveguides \\cite{van_laer_interaction_2015} yield an experimental Brillouin gain of 3000 W$^{-1}$m$^{-1}$. \nHowever, the need for narrow-width pedestals to optimize the Brillouin gain complicates the fabrication process and may compromise the mechanical stability of the structures. On the other hand, a lower experimental Brillouin gain (1000 W$^{-1}$m$^{-1}$) was obtained for silicon membrane rib waveguides due to the very different confinement of optical and mechanical modes \\cite{kittlaus_large_2016}. Still, this comparatively modest Brillouin gain was compensated by achieving ultra-low optical propagation loss, allowing the demonstration of lasing effect \\cite{otterstrom_silicon_2018}. The use of photonic crystals with simultaneous photonic and phononic bandgaps \\cite{zhang2017design} (also referred to as phoxonic crystals) has been proposed to maximize the Brillouin gain in silicon membrane waveguides, achieving calculated values up to 8000 W$^{-1}$m$^{-1}$. Yet, the narrow bandwidth and high optical propagation loss, typically linked to bandgap confinement \\cite{baba_slow_2008}, may compromise the performance of these phoxonic crystals. \n\nSubwavelength grating silicon waveguides, with periods shorter than half of the wavelength of the guided light, exploit index-contrast confinement to yield low optical loss and wideband operation \\cite{halir_waveguide_2015,cheben2018subwavelength}. Interestingly, near-infrared photons and GHz phonons in nanoscale Si waveguides have comparable wavelengths (near 1 \\textmu m) \\cite{safavi-naeini_controlling_2019}. Thus, the same periodic structuration could operate in the subwavelength regime for both, photons and phonons. In addition, forward Brillouin scattering (FBS), used to demonstrate Brillouin gain in Si, relies on longitudinally propagating photons and transversally propagating phonons \\cite{eggleton2019brillouin_LL,wiederhecker_brillouin_2019, safavi-naeini_controlling_2019}. Hence, engineering the longitudinal and transversal subwavelength geometries would allow independent control of photonic and phononic modes. Brillouin optimization in silicon membranes has been proposed based on index-contrast confinement of photons (longitudinal subwavelength grating) and bandgap confinement of phonons (transversal phononic crystal) \\cite{schmidt2019suspended}, achieving a calculated gain of 1750 W$^{-1}$m$^{-1}$. More recently, the combination of subwavelength index-contrast and subwavelength softening has been proposed to optimize Brillouin gain in suspended Si waveguides, achieving a calculated value of 3000 W$^{-1}$m$^{-1}$, for a minimum feature size of 50 nm \\cite{zhang_subwavelength_2020}. Still, these two approaches require several etch steps of the silicon core, complicating the device's fabrication. In this work, we propose a novel subwavelength-structured Si membrane, illustrated in Fig. \\ref{fig:structure}, requiring only one etch step of silicon. We develop an optimization method to design the waveguide geometry, combining multi-physics optical and mechanical simulations with a genetic algorithm (GA) capable of handling a large number of parameters \\cite{hakansson_generating_2019}. The optimized geometry yields a calculated Brillouin gain of 3300 W$^{-1}$m$^{-1}$, with a minimum feature size of 50 nm, compatible with electron-beam lithography.\n\n\\section{Design and Results \\label{sec:Results}}\nThe proposed optomechanical waveguide geometry, depicted in Fig. \\ref{fig:structure}, comprises a suspended central strip of width $W_g=400$ nm that is anchored to the lateral silicon slabs by a lattice of arms with a longitudinal period ($z$-direction) of $\\Lambda=300$ nm. This period is shorter than half of the optical wavelength, ensuring optical operation in the subwavelength regime. The anchoring arms are symmetric with respect to the waveguide center. We split the arms into five different sections with widths and lengths of $W_i$ ($x$-direction) and $L_i$ ($z$-direction), respectively. The index $i=1$ refers to the section adjacent to the waveguide core, while the index $i=5$ refers to the outermost section (see Fig. \\ref{fig:structure}, inset). The fifth section has a fixed width of $W_5=500$ nm and length of $L_5=50$ nm to ensure proper guidance and localization of the optical mode. The widths and lengths of sections 1 to 4 are optimized using the genetic algorithm. The whole waveguide has a fixed silicon thickness of $t=220$ nm, allowing fabrication in a single-etch step.\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\columnwidth]{fig1_structure_single.png}\n \\caption{Proposed optomechanical waveguide. In the inset, the different sections of the anchoring arms are numbered from $1$ to $5$. The width of the waveguide core ($W_g=400$ nm), the period ($\\Lambda=300$ nm), and the dimensions of the outermost section ($L_5=50$ nm, $W_5=500$ nm) remain fixed throughout the optimization process. The thickness of the silicon slab is $t=220$ nm.}\n \\label{fig:structure}\n\\end{figure}\n\nWe focus on FBS, where only near-cut-off acoustic modes are involved. In the absence of optical absorption, which is the case of silicon at near-infrared wavelengths, the optical and mechanical mode equations describing FBS decouple and can be solved separately \\cite{safavi-naeini_controlling_2019}. We use here COMSOL Multiphysics software for the optomechanical simulations. For the calculation of optical and mechanical modes in the optimization process, we reduce the 3D structure to an equivalent 2D geometry. The effective index method \\cite{chen_foundations_2005} is considered for the computation of the transverse-electric (TE) polarized optical modes while the in-plane mechanical modes are calculated assuming the plane stress approximation \\cite{auld_acoustic_1973}. We compute the Brillouin gain, $G_\\mathrm{B}$, as \\cite{wiederhecker_brillouin_2019}\n\\begin{equation}\n G_\\mathrm{B}(\\Omega_\\mathrm{m}) = Q_\\mathrm{m} \\, \\frac{2 \\omega_\\mathrm{p}}{m_\\mathrm{eff} \\, \\Omega_\\mathrm{m}^2} \\, \\left| \\int f_\\mathrm{MB} \\,\\mathrm{d}\\ell + \\int f_\\mathrm{PE} \\,\\mathrm{d} A \\right|^2 ,\n\\label{eq:GB}\n\\end{equation} \nwhere $\\omega_\\textrm{p}$ is the frequency of the optical pump, $\\Omega_\\mathrm{m}$ is the mechanical frequency, $Q_\\mathrm{m}$ is the mechanical quality factor, $m_\\mathrm{eff} = \\int\\rho\\,|\\mathbf{u}_\\mathrm{m}|^2\/\\max|\\mathbf{u}_\\mathrm{m}|^2 \\,\\mathrm{d}A$ is the effective linear mass density of the mechanical mode with displacement profile $\\mathbf{u}_\\mathrm{m}$, and $f_\\mathrm{MB}$ and $f_\\mathrm{PE}$ are the linear and surface overlap of optical force density and deformation representing the moving boundaries effect (MB) and the photoelastic effect (PE), respectively,\n\\begin{align}\n & f_\\mathrm{MB} = \\frac{\\mathbf{u}_\\mathrm{m}^*\\cdot\\mathbf{n} \\, \\left(\\delta\\varepsilon_\\mathrm{MB} \\, \\mathbf{E}^*_\\mathrm{p,t}\\cdot \\mathbf{E}_\\mathrm{s,t} - \\delta\\varepsilon_\\mathrm{MB}^{-1} \\, \\mathbf{D}_\\mathrm{p,n}^*\\cdot\\mathbf{D}_\\mathrm{s,n}\\right)}{\\max|\\mathbf{u}_\\mathrm{m}| \\, P_\\mathrm{p} \\, P_\\mathrm{s}} \\nonumber \\\\\n & \\mathrm{and} \\quad f_\\mathrm{PE} = \\frac{\\mathbf{E}^*_\\mathrm{p}\\cdot \\delta\\varepsilon_\\mathrm{PE}^* \\cdot \\mathbf{E}_\\mathrm{s}}{\\max|\\mathbf{u}_m| \\, P_\\mathrm{p} \\, P_\\mathrm{s}} ,\n\\label{eq:MB_PE}\n\\end{align}\nwhere the permittivity differences due to the moving boundaries effects are given by $\\delta\\varepsilon_\\mathrm{MB} = \\varepsilon_1 - \\varepsilon_2$ and $\\delta\\varepsilon_\\mathrm{MB}^{-1} = 1\/\\varepsilon_1 - 1\/\\varepsilon_2$, with $\\varepsilon_i=\\varepsilon_0 n_i^2$ being the permittivities of the silicon ($i=1$) and air ($i=2$). The photoelastic tensor perturbation in the material permittivity is $\\delta\\varepsilon_\\mathrm{PE} = -\\varepsilon_0 \\, n^4 \\, \\mathbf{p}:\\mathbf{S}$, with $n$ being the material refractive index, $\\mathbf{p}$ the photoelastic tensor, and $\\mathbf{S}$ the mechanical stress tensor induced by the mechanical mode. The term $\\mathbf{u}_\\mathrm{m}\\cdot\\mathbf{n}$ is the normal component of the mechanical displacement and $\\mathbf{E}_{j,\\mathrm{t}}$ and $\\mathbf{D}_{j,\\mathrm{n}}$ are the tangential electric field and normal dielectric displacement for the pump ($j=\\mathrm{p}$) and the scattered field ($j=\\mathrm{s}$). The denominator represents the power normalization given by $P_j = [2 \\Re(\\int [\\mathbf{E}_j\\times\\mathbf{H}_j^*] \\cdot \\mathbf{z} \\, \\mathrm{d}A)]^{1\/2}$.\n\nThe symmetry directions $[100]$, $[010]$, and $[001]$ of the crystalline silicon are set to coincide with the $x$, $y$, and $z$ simulation axis, respectively. With this orientation, the photoelastic tensor \\cite{qiu_stimulated_2013,rakich_tailoring_2010} is $[p_{11},p_{12},p_{44}]=[-0.094,0.017,-0.051]$. The refractive index of silicon is $n=3.45$ and its density $\\rho=2329$ kg m$^{-3}$ while the corresponding values for the air are $n=1$ and $\\rho=1.293$ kg m$^{-3}$. \n\nThe quality factor of the mechanical mode, $Q_\\mathrm{m}$, is related to the full width at half maximum (FWHM) of the gain spectrum, $\\gamma_\\mathrm{m}$, through $Q_\\mathrm{m}=\\Omega_\\mathrm{m}\/\\gamma_\\mathrm{m}$ and it is limited by different loss mechanisms, \n\\begin{equation}\n \\frac{1}{Q_\\mathrm{m}} = \\frac{1}{Q_\\mathrm{TE}} + \\frac{1}{Q_\\mathrm{L}} + \\frac{1}{Q_\\mathrm{air}}.\n\\label{eq:Q}\n\\end{equation}\nHere, we consider the thermoelastic loss ($Q_\\mathrm{TE}$), the mechanical leakage towards the silica under-cladding ($Q_\\mathrm{L}$), and the viscous loss from surrounding air ($Q_\\mathrm{air}$). The thermoelastic loss yields mechanical quality factors of $Q_\\mathrm{TE}\\sim6\\cdot10^5$ \\cite{comsol_2018} for silicon nanostructures while the leakage loss is mainly governed by the geometries of the waveguide and the arms anchoring it to the lateral silicon slab. These two effects are directly considered in the mechanical-mode simulations performed in COMSOL Multiphysics. The viscous loss induced by the surrounding air is considered here by imposing a limiting value to the mechanical quality factor of $Q_\\mathrm{m}=4\\cdot10^3$, which is the highest expected value at atmospheric pressure and room temperature for phonon frequency in the order of GHz \\cite{ghaffari_quantum_2013}.\n\nBased on the resulting optomechanical coupling calculations, a genetic algorithm \\cite{xin-she_yang_chapter_2021} is used to maximize the FBS gain. Starting with randomly generated combinations of parameters $W_i$ and $L_i$ (individuals), optomechanical simulations are carried out and the individuals are ranked according to their Brillouin gain. Recombination is used to produce a successor set of individuals, the next generation. The best-performing individuals directly become part of the next generation (elitism). A large number of individuals of the new generation is obtained by combining the parameter of pairs of individuals from the current generation (crossover). Finally, the remaining individuals of the new generation are produced by randomly modifying the parameters of single individuals of the current generation (mutation). This process continues until the convergence criterion has been reached.\n\nIn our particular optimization problem, an individual is a possible geometry, represented by a set of 8 parameters (width and length of each of the arm sections). Each generation is composed of 50 individuals and the successive generations are obtained applying a rate of elitism and crossover of 6\\% and 80\\%, respectively, with the remaining elements obtained through mutation. The convergence criterion was defined in terms of the difference between the best and the average performance, $G_\\mathrm{B} - \\langle G_\\mathrm{B}\\rangle <$ 10 W$^{-1}$m$^{-1}$, over 10 generations. For this work, we have used a standard computer with the following specifications: a 64-bit operating system with an x64-based processor Intel\\textsuperscript{\\tiny\\textregistered} Core\\textsuperscript{\\tiny\\texttrademark} i7-4790 (4 total cores, 8 total threads, base-frequency of 3.60 GHz), and an installed RAM of 8.00 GB. Under these conditions, the optimization process was completed in 12h 35 min, comprising 1500 optomechanical simulations of 30 seconds each.\n\nThe method we propose here relies on a defined geometry whose parameters are allowed to vary within a specific range of values. Hence, the optimized structure will depend strongly on our initial guess.\n\nIn Fig. \\ref{fig:convergence}, we present the optimization process. Figures \\ref{fig:convergence}a and \\ref{fig:convergence}b show the Brillouin gain and mechanical frequency, respectively, as a function of the generation number. As a result of the evolution of the geometry, we observe an increase in the gain and a variation in the mechanical frequency. This result should be expected as the Brillouin shift in FBS is particularly sensitive to the waveguide dimensions. The optimum performance is achieved after 10 generations while 30 generations are required for convergence. The optimized geometry, whose dimensions are listed in Table \\ref{tab:geom}, is characterized by a Brillouin gain of $G_\\mathrm{B}=3350$ W$^{-1}$m$^{-1}$ for a mechanical mode with frequency of $\\Omega_\\mathrm{m}=14.357$ GHz and mechanical quality factor of\n$Q_\\mathrm{m}\\approx3.2\\cdot10^3$. The optical mode has a mode effective index of 2.36 and wavelength in vacuum of $\\lambda=1556.5$ nm ($\\omega_\\mathrm{p}=2\\pi\\cdot192.6$ THz in (\\ref{eq:GB})).\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\columnwidth]{fig2_convergence_double.png}\n \\caption{Optimization process. a) Best (in blue) and average (in orange) Brillouin gain as a function of the number of generations during genetic optimization. b) Evolution of the mechanical frequency as a function of the number of generations. During the optimization process, all possible mechanical losses are considered, including thermoelastic loss, mechanical leakage, and viscous loss due to air (operation in air ambient at room temperature).}\n \\label{fig:convergence}\n\\end{figure}\n\nIn terms of geometry, the first and fourth sections, with considerably larger widths, generate reflections that help localize the mechanical mode in the waveguide core. The frequency of the mechanical mode is governed by the interplay between the waveguide width and the length of the partial cavity formed by the fourth section on each side.\n\n\\begin{table}[htb]\n \\centering\n \\caption{Dimensions for the GA-optimized geometry when operating in air ambient at room temperature. In the table above, S$_i$ stands for section $i$ in Fig. \\ref{fig:structure}.} \n \\label{tab:geom}\n \\begin{tabular}{@{}ccccc}\n \\toprule\n & S1 & S2 & S3 & S4 \\\\\n \\midrule\n Width & 170 nm & 320 nm & 330 nm & 100 nm \\\\\n Length & 130 nm & 60 nm & 60 nm & 190 nm \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\nFull 3D simulations are realized to verify the performance of the optimized geometry. This structure provides a Brillouin gain of $G_\\mathrm{B}=3310$ W$^{-1}$m$^{-1}$ for a mechanical mode with a frequency of $\\Omega_\\mathrm{m}=14.579$ GHz. The optical mode has a mode effective index of 2.23 and wavelength in vacuum of $\\lambda=1557.2$ nm ($\\omega_\\mathrm{p}=2\\pi\\cdot 192.52$ THz in (\\ref{eq:GB})). Figure \\ref{fig:modes} shows the calculated field distribution for the mechanical and optical modes in the optimized geometry.\n\n\\begin{figure}[htbp] \n \\centering\n \\includegraphics[width=\\columnwidth]{fig3_modes_double.png}\n \\caption{Optical and mechanical modes of the optimized geometry operating in air ambient and room temperature (table \\ref{tab:geom}): a) Approximated 2D structure. The upper structure corresponds to the normalized mechanical displacement at 14.357 GHz and the lower figure to the $x$-component of the electric field at 1556.5~nm (mode effective index 2.36). b) Full 3D device. On the bottom left, $x$-component of the electric field at 1557.2 nm (mode effective index 2.23), and on the top right, normalized mechanical displacement at 14.579 GHz.}\n\\label{fig:modes}\n\\end{figure}\n\nThese results show a good agreement between the approximated 2D geometry used for the optimization and the full 3D structure. The small discrepancies in the optical mode index and mechanical frequency are due to the influence of the thickness. \n\nFinally, we study the fabrication tolerance of the proposed structure using again 3D simulations. We consider under- and over-etching errors that we model by a variation of all the waveguide lengths and widths by a factor $\\Delta$, measured in nm (Fig. \\ref{fig:fab_tolerance}a). Figure \\ref{fig:fab_tolerance}c shows the variation of the Brillouin gain (in blue) and mechanical frequency (in orange) as a function of $\\Delta$. The Brillouin gain remains above 2000 W$^{-1}$m$^{-1}$ for geometry variations of $\\pm 10$\\,nm. It should be noted that for the over-etch case ($\\Delta<0$ in Fig. \\ref{fig:fab_tolerance}c), the Brillouin gain is larger than the optimized case due to the larger optomechanical coupling resulting from a better overlap of the mechanical mode with the optical field. However, these smaller structures are incompatible with the target minimum feature size of 50 nm that was chosen to guarantee fabrication reliability. The mechanical frequency varies less than 2\\% (Fig.\\ref{fig:fab_tolerance}c, in orange) and the mechanical profile is not modified significantly.\n\nWe also study the effect of stitching errors, modeled by a deviation $\\zeta$ (in nm) of the arm axis at both sides of the waveguide core, hence breaking the symmetry of the structure (Fig. \\ref{fig:fab_tolerance}b). Figure \\ref{fig:fab_tolerance}d shows the variation of the Brillouin gain (in blue) and mechanical frequency (in orange) as a function of $\\zeta$. A non-perfectly symmetric structure is slightly detrimental to the Brillouin gain but does not affect the mechanical frequency or profile. Interestingly, both parameters (Brillouin gain and mechanical frequency) remain constant over a large range of stitching errors.\n\nLastly, we examine the effect of random fabrication errors affecting each section independently (Table \\ref{tab:geom_random}). We consider deviations of 5 to 20 nm, both in positive (enlargement) or negative (shrinking) directions. Our geometry exhibits a robust performance despite these errors with Brillouin gains above 2000 W$^{-1}$m$^{-1}$ (Fig. \\ref{fig:fab_tolerance}e, blue) and mechanical frequencies between 14 and 15 GHz (Fig. \\ref{fig:fab_tolerance}e, orange). It should be noted that the period remains constant, $\\Lambda = 300$ nm since it is controlled with high precision ($\\pm 2$ nm) in terms of fabrication.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{fig4_fab_double.png}\n \\caption{Fabrication tolerance of the optimized geometry. a) and b) Variation of the geometry due to fabrication errors. The solid black line corresponds to optimized geometry, dotted (solid) blue depicts a positive deviation from the nominal design, and dotted orange refers to a negative deviation from the expected design. c) and d) Evolution of the Brillouin gain (in blue, left axis) and the mechanical frequency (in orange, right axis) for different values of under- and over-etching (c), different values of stitching errors (d), and different structures with randomized geometrical parameters (e). In e), N stands for the nominal design obtained after the optimization problem and $i$ for the different geometries listed in Table \\ref{tab:geom_random}.}\n \\label{fig:fab_tolerance}\n\\end{figure}\n\n\\begin{table}\n \\centering\n \\caption{Dimensions for the different geometries used for studying the effect of randomization of the design parameters. In the table, S$_i$ stands for section $i$ in Fig. \\ref{fig:structure}, N stands for the nominal design as obtained from the optimization (Table 1), and $i$ stands for the different geometries in Fig. \\ref{fig:fab_tolerance}e. In all cases, the period, $\\Lambda = 300$ nm, remains constant.}\n \\label{tab:geom_random}\n \\begin{tabular}{@{}cccccccc}\n \\toprule\n Geometry & & S1 & S2 & S3 & S4 & S5 & $W_g$ \\\\\n \\midrule\n \\multirow{2}*{N} & Width & 170 nm & 320 nm & 330 nm & 100 nm & 500 nm & 400 nm \\\\\n & Length & 130 nm & 60 nm & 60 nm & 190 nm & 50 nm & \\\\\n \\midrule\n \\multirow{2}*{1} & Width & 165 nm & 305 nm & 345 nm & 90 nm & 510 nm & 405 nm \\\\\n & Length & 130 nm & 45 nm & 65 nm & 180 nm & 60 nm & \\\\\n \\midrule\n \\multirow{2}*{2} & Width & 165 nm & 320 nm & 340 nm & 115 nm & 495 nm & 400 nm \\\\\n & Length & 110 nm & 45 nm & 55 nm & 170 nm & 35 nm & \\\\\n \\midrule\n \\multirow{2}*{3} & Width & 155 nm & 340 nm & 340 nm & 100 nm & 485 nm & 405 nm \\\\\n & Length & 150 nm & 40 nm & 70 nm & 200 nm & 55 nm & \\\\\n \\midrule\n \\multirow{2}*{4} & Width & 185 nm & 300 nm & 325 nm & 95 nm & 480 nm & 385 nm \\\\\n & Length & 140 nm & 65 nm & 75 nm & 185 nm & 60 nm & \\\\\n \\midrule\n \\multirow{2}*{5} & Width & 160 nm & 320 nm & 330 nm & 95 nm & 510 nm & 390 nm \\\\\n & Length & 140 nm & 65 nm & 55 nm & 210 nm & 60 nm & \\\\\n \\midrule\n \\multirow{2}*{6} & Width & 185 nm & 340 nm & 315 nm & 120 nm & 520 nm & 420 nm \\\\\n & Length & 135 nm & 40 nm & 50 nm & 190 nm & 35 nm & \\\\\n \\midrule\n \\multirow{2}*{7} & Width & 185 nm & 340 nm & 340 nm & 110 nm & 480 nm & 410 nm \\\\\n & Length & 140 nm & 55 nm & 65 nm & 175 nm & 40 nm & \\\\\n \\midrule\n \\multirow{2}*{8} & Width & 150 nm & 300 nm & 345 nm & 110 nm & 510 nm & 395 nm \\\\\n & Length & 120 nm & 80 nm & 40 nm & 175 nm & 65 nm & \\\\\n \\midrule\n \\multirow{2}*{9} & Width & 170 nm & 340 nm & 325 nm & 105 nm & 520 nm & 410 nm \\\\\n & Length & 120 nm & 70 nm & 50 nm & 190 nm & 70 nm & \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Conclusions}\nIn summary, we have proposed a new approach to optimizing Brillouin gain in silicon membrane waveguides. We exploit genetic optimization to maximize Brillouin gain in subwavelength-structured Si waveguides, requiring only one etch step. Genetic algorithm is a well-known optimization technique capable of handling design spaces of moderate dimension \\cite{xin-she_yang_chapter_2021}. It has the main advantage over gradient-based algorithms in its capability to search the design space in many directions simultaneously. On the other hand, the genetic algorithms cannot guarantee a global optimum solution, being the final result strongly dependent on the initial population. Based on this strategy, a calculated Brillouin gain up to 3310 W$^{-1}$m$^{-1}$ is achieved for air environment. This result compares favorably to previously reported subwavelength-based Brillouin waveguides requiring several etching steps \\cite{schmidt2019suspended,zhang_subwavelength_2020}, with calculated Brillouin gain of 1750 W$^{-1}$m$^{-1}$ and 3000 W$^{-1}$m$^{-1}$. Our results show the potential of optimization for obtaining novel designs with improved performance in the context of Brillouin scattering. Moreover, they show the reliability of computationally efficient optimizations based on approximated 2D simulations.\n\n\n\\section*{Declaration of Competing Interest}\nThe authors declare that they have no known competing financial\ninterests or personal relationships that could have appeared to influence\nthe work reported in this paper.\n\n\\section*{Author Statement}\nPaula Nu\u00f1o Ruano, Jianhao Zhang, and Carlos Alonso Ramos proposed the concept. Paula Nu\u00f1o Ruano, Jianhao Zhang, and Daniele Melati developed the simulation framework. Paula Nu\u00f1o Ruano, Jianhao Zhang, Daniele Melati, David Gonz\u00e1lez Andrade, and Carlos Alonso Ramos optimized and analyzed the results. All authors contributed to the manuscript.\n\n\\section*{Data Availability Statement}\nThe data supporting this study's findings are available from the corresponding author upon reasonable request.\n\n\\section*{Acknowledgements}\nThe authors want to thank the Agence Nationale de la Recherche for supporting this work through BRIGHT ANR-18-CE24-0023-01 and MIRSPEC ANR-17-CE09-0041. P.N.R. acknowledges the support of Erasmus Mundus Grant: Erasmus+ Erasmus Mundus Europhotonics Master program (599098-EPP-1-2018-1-FR-EPPKA1-JMD-MOB) of the European Union. This project has received funding from the European Union's Horizon Europe research and innovation program under the Marie Sklodowska-Curie grant agreement N\u00ba 101062518.\n\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction} The idea of spatial\ncoupling emerged in the coding context from the study of Low-Density\nParity-Check Convolutional (LDPCC) codes. LDPCC codes were introduced\nby Felstr{\\\"{o}}m and Zigangirov \\cite{FeZ99}. We refer the reader\nto \\cite{EnZ99, ELZ99, LTZ01, TSSFC04} as well as to the introduction\nin \\cite{KRU10} which contains an extensive review. It has long\nbeen known that LDPCC codes outperform their block coding counterparts\n\\cite{SLCZ04, LSZC10, LSZC05}. Subsequent work isolated\nand identified the key system structure which is responsible for\nthis improvement.\n\nIn particular, it was conjectured in \\cite{KRU10} that spatially\ncoupled systems exhibit BP threshold behavior corresponding to the MAP threshold behavior of\n uncoupled component system. This phenomenon was termed ``threshold saturation\" \nand a\nrigorous proof of the threshold saturation phenomenon over the BEC\nand regular LDPC ensembles was given. The proof was generalized\nto all binary-input memoryless output-symmetric (BMS) channels in\n\\cite{KRU12}. From these results it follows that universal\ncapacity-achieving codes for BMS channels can be constructed by\nspatially coupling regular LDPC codes. Spatial coupling has also\nbeen successfully applied to the CDMA multiple-access channel\n\\cite{ScT11,TTK11}, to compressed sensing \\cite{KP10,KMSSZ11,DMM,DJM11},\nto the Slepian-Wolf coding problem \\cite{YPN11}, to models in\nstatistical physics \\cite{HMU11a,HMU11b}, and to many other problems in\ncommunications, statistical physics and computer science, see\n\\cite{KRU12} for a review.\n\nThe purpose of this paper is two-fold. First, we establish the existence of wave-like solutions to\nspatially coupled graphical models which, in the large size limit,\ncan be characterized by a one-dimensional real-valued state.\nThis is applied to give a rigorous\nproof of the threshold saturation phenomenon for all such\nmodels. This includes spatial\ncoupling of irregular LDPC codes over the BEC, but it also addresses\nother cases like hard-decision decoding for transmission over general\nchannels, and the CDMA multiple-access problem \\cite{ScT11,TTK11}\nand compressed sensing \\cite{DJM11}.\nAs mentioned above, transmission over the BEC using spatially-coupled\nregular LDPC codes was already solved in \\cite{KRU10}, but our\ncurrent set-up is more general. Whereas the proof in \\cite{KRU10}\ndepends on specific features of the BEC, here we derive a graphical\ncharacterization of the threshold saturation phenomena in terms of\nEXIT-like functions that define the spatial system. This broadens\nthe range of potential applications considerably.\n\nConsider the example of coding over the BEC. In\nthe traditional irregular LDPC EXIT chart setup the condition for successful decoding reduces\nto the two EXIT charts not crossing. We will show that the EXIT\ncondition for good performance of the spatially-coupled system is\nsignificantly relaxed and reduces to a condition on the area bounded\nbetween the component EXIT functions.\n\nThe criteria is best demonstrated by a simple example. Consider transmission\nover the BEC using the $(3, 6)$ ensemble.\nFigure~\\ref{fig:positivegapbec36} shows the corresponding EXIT\ncharts for $\\epsilon=0.45$ and $\\epsilon=0.53$. Note that both these\nchannel parameters are larger than the BP threshold which is\n$\\epsilon^{\\text{\\small BP}} \\simeq 0.4294$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/positivegapbec36}\n}\n\\caption{\\label{fig:positivegapbec36} Both pictures show the EXIT\ncurves for the $(3, 6)$ ensemble and transmission over the BEC.\nLeft: $\\epsilon=0.45$. In this case $A=0.03125>0$, i.e., the white\narea is larger than the dark gray area. Right: $\\epsilon=0.53$.\nIn this case $A=-0.0253749<0$, i.e., the white area is smaller than\nthe dark gray area. } \n\\end{figure}\nIf we consider the signed area bounded by the two\nEXIT charts and integrate from $0$ to $u$ then on the left hand side, with $\\epsilon=0.45,$ this\narea is positive for all $u \\in [0, 1]$. This property guarantees\nthat the decoder for the spatially coupled system succeeds for this\ncase. On the right-hand side with $\\epsilon=0.53$, however, the area becomes\nnegative at some point (the total area in white is smaller than the\ntotal area in dark gray) and by our condition this implies that the\ndecoder for the spatially coupled system does not succeed. The\nthreshold of the spatially coupled system is that channel\nparameter such that the area in white and the area in dark gray are\nexactly equal. \n\nThis simple graphical condition is the essence of our result and\napplies regardless whether we look at coding systems or other\ngraphical models. Given any system characterized by two EXIT functions,\nwe can plot these two functions and consider the signed area area bound between them,\nsay for the first coordinate ranging from $0$ to a point $u$. As long as this area is positive for all\n$u \\in (0, 1]$ the iterative process succeeds, i.e., it converges to $0.$ Indeed, we will even\nbe able to make predictions on the speed of the process based on\nthe ``excess\" area we have. \n\nA few conclusions can immediately be drawn from such a picture.\nFirst, if the threshold of the uncoupled system is determined by\nthe so-called stability condition, i.e., the behavior of the EXIT\ncharts for $u$ around $0$ then spatial coupling does not increase\nthe threshold. Indeed, if we increase the parameter beyond what is\nallowed according to the stability condition, the area will become\nnegative around $0$. Second, if the curves only have\none non-trivial crossing (besides the one at $0$ and at the right\nend point) then the threshold is given by a balance of the two\nenclosed areas.\n\nFor ``nice\" EXIT charts (e.g., continuous, and with a finite number of\ncrossings) the above picture contains all that is needed. But since\nwe want to develop the theory for the general case, some care is\nneeded when defining all relevant quantities. When reading the\ntechnical parts below, it is probably a good idea to keep the above\nsimple picture in mind. For readers familiar with the so-called\nMaxwell conjecture, it is worth pointing out that the above picture\nshows that this conjecture is generically correct for coupled\nsystems. To show that it is also correct for uncoupled systems one\nneeds to show in addition that under MAP decoding the coupled and\nthe uncoupled system behave identically. This can often be accomplished\nby using the so-called interpolation method. For e.g., regular\nensembles with no odd check degrees this second step was shown to\nbe correct in \\cite{GMU12}.\n\nLet us point out a few differences to the set-up in \\cite{KRU10}.\nFirst, rather than analyzing directly the spatially-discrete system,\nkey results are established in the limit of continuum spatial\ncomponents. We will see that for such systems the solution for the\ncoupled system is characterized in terms of traveling waves, especially fixed points. The\nspatially discrete version is then recovered as a sampling of the continuum\nsystem. The existence of traveling wave solutions and their\nrelationship to the EXIT charts of the underlying component systems\nis the essential technical content of the analysis and does not depend\non information theoretic aspects of the coding case.\n\nThe second purpose of this paper is to show that herein-developed\none-dimensional theory can model many higher-dimensional or even\ninfinite-dimensional systems to enable accurate prediction of their\nperformance. This is very much in the spirit of the use of EXIT\ncharts and Gaussian approximations for the the design of iterative\nsystems. \nUsing this interpretation,\nwe apply our method to channel coding over general channels. Even\nthough the method is no longer rigorous in these cases, we show\nthat our graphical characterization gives very good predictions on\nthe system performance and can therefore provide a convenient design\ntool.\n\nRecently several alternative approaches to the analysis of\nspatially-coupled systems have been developed independently by\nvarious authors \\cite{TTK11b, DJM11, YJNP12a, YJNP12b}. These\napproaches share some important aspects with our work but there are\nalso some important differences. Let us quickly discuss this. \n\nIn \\cite{DJM11} a proof was given that spatially coupled measurement\nmatrices, together with a suitable iterative decoding algorithm,\nthe so-called {\\em approximate message-passing} (AMP) algorithm,\nallows to achieve the information theoretic limits of compressive sensing\nin a wide array of settings. The key technical idea is to show that\nthe iterative system is characterized in the limit of large block\nsizes by a one-dimensional parameter (which in this case represents\nper-coordinate mean square error) and which can be tracked faithfully by\n{\\em state evolution equations}. To ease the analysis the authors\nconsider continuum state evolution equations. The discrete state\nevolution is then obtained by sampling the continuous state evolution\nequations. The most important ingredient of the proof is a construction\nof an appropriate free energy or potential function for the system such\nthat the fixed points of the state evolution are the stationary\npoints. It is then shown that if the under-sampling ratio is greater\nthan the information dimension, then the solution of the state evolution\nbecomes arbitrarily small. Suppose this did not happen. Then\nby perturbing slightly the non-trivial fixed point (the solution\nis ``moved'' inside) one can show that the potential strictly\ndecreases. However, since the fixed point is a stationary point of\nthe potential function, we get a contradiction. \n\nIn \\cite{YJNP12a, YJNP12b} the two main ingredients are also the\ncharacterization of the iterative system by a one-dimensional (or\nfinite-dimensional) parameter and the construction of a suitable\npotential function whose stationary points are the fixed points of\ndensity evolution. A significant innovation introduced in\n\\cite{YJNP12a, YJNP12b} is that it is shown how to {\\em systematically}\nconstruct such a potential function in a very general setting. This\nmakes it possible to apply the analysis to a wide array of setting\nand provides a systematic framework for the proof. In addition,\nthis framework allows not only to attack the scalar case but can\nbe carried over to vector-valued states.\n\nOur starting point is the set of EXIT functions, a familiar tool\nin the setting of iterative systems.\nWe also use a type of potential function for the underlying component system.\nUnlike the works mentioned above we retain the symmetry of the iterative system rather than\ncollapsing one of the equations.\nIn addition, the form of the potential function used is such that each step of the iteration \nminimizes the potential function for the variable being updated.\nThe potential function can be lifted to the spatially coupled system but that is not the approach we take in this paper.\nWe consider the spatially coupled system of infinite extent and analize solutions.\nSpatial fixed point solutions that interpolate between fixed points of the component system are of particular importance.\nWe show that the evaluation of the potential function of the component system at points determined by the spatial fixed point can be expressed in terms an integral accross space of the\nfixed point. Surprisingly, the underlying potential can be expressed using the spatial fixed point solution in a way that uses the portion of the solution local to the evaluation points.\nThis basic result yields structural information on the fixed point solution.\nThis result is used as a foundation to characterize and construct wave-like solutions for \nspatially coupled systems.\nPerhaps one of the strong points of the current paper is\nthat it gives a fairly detailed and complete picture of the system\nbehavior. I.e., we not only characterize the threshold(s) but\nwe also are able to characterize {\\em how} the system converges to\nthe various FPs (these are the wave solutions) and {\\em how fast}\nit does so.\n\nThe outline of the paper is as follows. In Section~\\ref{sec:main}\nwe consider an abstract system, characterized by two EXIT-like\nfunctions. In terms of these functions we state a graphical criterion\nfor the occurrence of threshold saturation. In\nSection~\\ref{sec:applications} we then apply the method to several\none-dimensional systems. We will see that in each case the analysis\nis accomplished in just a few paragraphs by applying the general\nframework to the specific setting. In Section~\\ref{sec:gaussapprox}\nwe develop a framework that can be used to analyze higher-dimensional\nsystems in a manner analogous to the way the Gaussian approximation\nis used together with EXIT charts in iterative system design. We\nalso show by means of several examples that this approach typically\ngives accurate predictions. In Section~\\ref{sec:proof} we give a\nproof of the main results. Many of the supporting lemmas and bounds\nare relegated to appendices.\n\n\\section{Threshold Saturation in One-Dimensional Systems}\\label{sec:main}\nIn\nthis section we develop and state the main ingredients which we\nwill later use to analyze various spatially coupled systems. Although\nin most cases we are ultimately interested in ``spatially discrete'' and\n``finite-length'' coupled systems, i.e., systems where we have a\nfinite number of ``components'' which are spatially coupled along\na line, it turns out that the theory is more elegant and simpler\nto derive if we start with spatially continuous and unterminated systems, i.e., stretching from\n$-\\infty$ to $\\infty$. Once a suitably defined continuous system\nis understood, one can make contact with the actual system at hand\nby spatially discretizing it and by imposing specific boundary\nconditions.\n\nThroughout this section we use the example of the spatially-coupled\n$(\\dl, \\dr)$-regular LDPC ensemble. \\bexample[$(\\dl, \\dr, w, L)$\nEnsemble]\\label{def:ensemble} The $(\\dl, \\dr, w, L)$ random ensemble\nis defined as follows, see \\cite{KRU10}. In the ensuing paragraphs\nwe use $[a, a+b]\\Delta$, for integers $a$ and $b$, $b\\geq0$, and\nthe real non-negative number $\\Delta$, to denote the set of points\n$a\\Delta, (a+1)\\Delta, \\dots, (a+b)\\Delta$.\n\nWe assume that the variable nodes are located at positions $[0,\nL] \\Delta$, where $L \\in \\naturals$ and $\\Delta >0$. At each position\nthere are $M$ variable nodes, $M \\in \\naturals$. Conceptually we\nthink of the check nodes as located at all positions $[- \\infty,\n\\infty] \\Delta$. Only some of these positions are used and contain check nodes\nthat are actually connected to\nvariable nodes. At each position there are $\\frac{\\dl}{\\dr}\nM$ check nodes. It remains to describe how the connections are\nchosen.\n\nWe assume that each of the $\\dl$ neighbors of a variable node at\nposition $i \\Delta$ is uniformly and independently chosen from the\nrange $[i-w, \\dots, i+w] \\Delta$, where $w$ is a ``smoothing''\nparameter.\\footnote{Full independence is not possible while satisfying the degree constraints.\nThis does not affect the analysis since we only need the independence to hold asymptotically in large block size over finite neighborhoods in the graph.}\n In the same way, we assume that each of the $\\dr$\nconnections of a check node at position $i$ is independently chosen\nfrom the range $[i-w, \\dots, i+w] \\Delta$. Note that this deviates\nfrom the definition in \\cite{KRU10} where the ranges were $[i,\n\\dots, i+w-1] \\Delta$ and $[i-w+1, \\dots, i] \\Delta$ respectively.\nIn our current setting the symmetry of the current definition\nsimplifies the presentation. The present definition is equivalent\nto the previous one with $w$ replaced by $2w+1.$\n\nThis ensemble is spatially discrete. As we mentioned earlier, it\nis somewhat simpler to start with a system which is spatially\ncontinuous. We will discuss later on in detail how to connect these\ntwo points of view. Just to get started -- how might one go from a\nspatially discrete system as the $(\\dl, \\dr, w, L)$ ensemble to a\nspatially continuous system? Assume that we let $\\Delta$ tend to\n$0$ while $L$ and $w$ tend to infinity so that $L \\Delta$ tends to $\\infty$ and\n$W=w\\Delta$ is held constant. In\nthis case we can imagine that in the limit there is a component\ncode at each location $x \\in (-\\infty, +\\infty)$ in space and that\na component at position $x$ ``interacts'' with all components in a\nparticular ``neighborhood'' of $x$ of width $2W.$ {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nConsider a system on $(-\\infty, +\\infty)$ (the spatial component)\nwhose ``state'' at each point (in space) is described by a scalar\n(more precisely an element of $[0, 1]$). This means, the state of\nthe system at time $t$, $t \\in \\naturals$, is described by a function\n$\\ff^t$, where $\\ff^t(x) \\in [0, 1]$, $x \\in (-\\infty, \\infty)$.\n\n\\bexample[Coding for the BEC]\nConsider transmission over a binary erasure channel (BEC) using the\n$(\\dl, \\dr, w, L)$ ensemble described in Definition~\\ref{def:ensemble}.\nThen the ``state'' of each component code at a particular point in\ntime is the fraction of erasure messages that are\nemitted by variable nodes at this iteration. Hence the state of\neach component is indeed an element of $[0, 1]$. {\\hfill\n$\\ensuremath{\\Box}$} \\eexample\n\n\\bdefinition\nWe denote the space of non-decreasing functions $[0,1] \\rightarrow [0,1]$ by $\\exitfns.$\nA function $h\\in\\exitfns$ has right limits $h(x+)$ for $x\\in (0,1]$ and left limits $h(x-)$ for $x\\in[0,1).$\nTo simplify some notation we define $h(0-)=0$ and $h(1+)=1.$\nThe function $h$ is continuous at $x$ if $h(x-) = h(x+).$ \n\nSimilarly, let $\\sptfns$ denote the space\nof non-decreasing functions on $(\\minfty, \\pinfty).$ \nWe denote $\\lim_{x\\rightarrow -\\infty} f(x)$ as $f(\\minfty)$\n$\\lim_{x\\rightarrow +\\infty} f(x)$ as \n$f(\\pinfty).$ \nWe call a function $f \\in \\sptfns$ {\\em$(a,b)$-interpolating} if\n$f(\\minfty) = a$\nand\n$f(\\pinfty) = b.$\nWe will generally use the term ``interpolating\" with the understanding that $b>a.$\nThe canonical case will be $(0,1)$-interpolating functions and we will also use the term\n``$(0,1)$-interpolating spatial fixed point'' to refer to a pair of $(0,1)$-interpolating functions.\nWe may also refer to a pair of functions $f,g$ interpolating over $[a,b]\\times[c,d]$ to mean\nthat $f$ is $(b,d)$-interpolating and $g$ is $(a,c)$-interpolating. \n\nIn general we work with discontinuous functions. Because of this we\noccasionally need to distinguish between functions in $\\exitfns$ or in $\\sptfns$\nthat differ only on a set of measure $0.$ \nWe say $h_1 \\equiv h_2$ if $h_1$ and $h_2$ differ on a set of measure $0.$\nThese functions are equivalent in the $L_1$ sense.\nWe still enforce monotonicity so equivalent functions can differ only\nat points of discontinuity.\n \\edefinition\n\nWe think of $\\hf$ and $\\hg$ as EXIT-like functions describing the\nevolution of the underlying component system under an iterative\noperation. Usually, we will have $(0,0)$ and $(1,1)$ as key fixed points.\n\nWe say that a sequence $h_i \\rightarrow h$ in $\\exitfns$ if $h_i(u)\n\\rightarrow h(u)$ for all points of continuity of $h.$ We use a\nsimilar definition of convergence in $\\sptfns.$ In general only the\nequivalence class of the limit is determined. I.e., if the limit $h$ is\ndiscontinuous then it is not uniquely determined.\n\nAny function $h \\in \\exitfns$ has a unique equivalence class of inverse functions in $\\exitfns$. \nFor $h \\in \\exitfns$ we will use $h^{-1}$ to denote any member of the equivalence class.\nFormally, we can set $h^{-1}(v)$ to any value $u$ such that\n$v \\in [ h(u-), h(u+) ].$ \nNote that $h^{-1}(v-)$ and $h^{-1}(v+)$ are uniquely determined\nfor each $v\\in [0,1].$ \nThus, we see that the function $h^{-1}$ is uniquely determined at all of its points of continuity and\nit is not uniquely determined at points of discontinuity.\nSimilarly, any function $f \\in \\sptfns$ has\na well defined monotonically non-decreasing inverse equivalence class\nand we use $f^{-1}:[0,1]\\rightarrow [-\\infty,\\infty]$ to denote any member.\nFor notational completeness we define $f^{-1}(0-)=-\\infty$ and $f^{-1}(1+)=+\\infty.$\n\nWe assume that the dynamics of the underlying component system is\ndescribed by iterative updates according to the two functions $\\hf,\\hg \\in \\exitfns.$\nIn deference to standard nomenclature in coding, we refer to these\niterative updates as the {\\em density evolution} (DE) equations.\nIf we assume that $\\xf$ and $\\xg$ are scalars describing the component\nsystem state then these update equations are given by\n\\begin{equation}\\label{eqn:DE}\n\\begin{split}\n{\\xg}^{t} & = \\hg (\\xf^t), \\\\\n{\\xf}^{t+1} & = \\hf (\\xg^{t})\\,.\n\\end{split}\n\\end{equation}\n\n\\bexample[DE for the BEC]\nConsider a $(\\dl, \\dr)$-regular ensemble.\nLet $\\lambda(u)=u^{\\dl-1}$ and $\\rho(v)=v^{\\dr-1}$. Let $\\xf^t$ be\nthe fraction of erasure messages emitted at variable nodes at time $t$\nand let $\\xg^t$ be the fraction of erasure messages emitted at\ncheck nodes at iteration $t$.\\footnote{Conventionally, in iterative coding these quantities are denoted by $x$ and $y$.\nBut since we soon will introduce a continuous spatial dimension, which naturally is denoted by $x$, we prefer\nto stick with this new notation to minimize confusion.}\nLet $\\epsilon$ be the\nchannel parameter. Then we have\n\\begin{equation}\\label{eqn:DEBEC}\n\\begin{split}\n\\xg^{t} & = 1-\\rho(1-\\xf^t), \\\\\n\\xf^{t+1} & = \\epsilon \\lambda(\\xg^{t})\\,.\n\\end{split}\n\\end{equation}\nIn words, we have the correspondences $\\hg(\\xf)=1-\\rho(1-\\xf)$, and\n$\\hf(\\xg)=\\epsilon \\lambda(\\xg)$. As written, the function $\\hf(\\xg)$\nis not continuous at $\\xg=1.$ More explicitly, $\\hf(1) = \\epsilon <1$, whereas\nwe defined the right limit at $1$ to be generically equal to $1$.\nWe will see shortly how to deal with this. {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nLet us now discuss DE for the spatial continuum version. Consider a spatially-coupled\nsystem with the following update equations:\n\\begin{equation}\\label{eqn:gfrecursion}\n\\begin{split}\n\\fg^{t}(x) & = \\hg ((\\ff^t \\otimes \\smthker) (x)), \\\\\n\\ff^{t+1}(x) & = \\hf ( (\\fg^{t} \\otimes \\smthker) (x) )\\,.\n\\end{split}\n\\end{equation}\nHere, $\\otimes$ denotes the standard convolution operator on $\\reals$\nand $\\smthker$ is an {\\em averaging kernel.}\n\\begin{definition}[Averaging Kernel]\nAn averaging kernel $\\smthker$ is a non-negative even function,\n$\\smthker(x)=\\smthker(-x)$, of bounded variation that integrates to $1,$\ni.e., $\\int \\smthker(x) \\text{d} x =1.$\nWe call $\\smthker$ {\\em regular} if there exists $W \\in (0,\\pinfty]$ such that\n$\\smthker(x) = 0$ for $x \\not\\in [-W,W]$ and \n$\\smthker(x) > 0$ for $x \\in (-W,W).$ Note that we do not require $W$ to be finite,\nwe may have $W=\\infty.$\n\\end{definition}\n\nWe will generally assume a regular averaging kernel. This assumption can largely be dropped when $\\hf$ and $\\hg$ are continuous.\n\n\\bexample[Continuous Version of DE for the BEC]\nIf we specialize the maps to the case of transmission over the BEC we get\nthe update equations:\n\\begin{equation}\\label{eqn:gfrecursionBECcont}\n\\begin{split}\n\\fg^{t}(x) & = 1-\\rho(1-(\\ff^t \\otimes \\smthker) (x)), \\\\\n\\ff^{t+1}(x) & = \\epsilon \\lambda( (\\fg^{t} \\otimes \\smthker) (x) )\\,.\n\\end{split}\n\\end{equation}\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\nFor compactness we will often use the notation\n$\\fS$ to denote $\\ff \\otimes \\smthker.$\n\nIn the usual manner of EXIT chart analysis, it is convenient to\nconsider simultaneously the plots of $\\hf$ and the reflected plot\nof $\\hg.$ More precisely, in the unit square $[0,1]^2,$ we consider\nthe monotonic curves\\footnote{If $\\hf$ or $\\hg$ is discontinuous then the curve interpolates\nthe jump with a line segment.} $(\\xg, \\hf(\\xg))$ and $(\\hg(\\xf), \\xf)$ for $\\xf, \\xg \\in [0,1].$ \nDensity evolution (DE)\nof the underlying (uncoupled) iterative system can then be viewed as a path drawn out by\nmoving alternately between these two curves (see Fig. \\ref{fig:exitbec36}).\nThis path has the characteristic ``staircase'' shape. \nWe will sometimes refer to the system being defined on $[0,1]\\times[0,1]$ with this picture in mind.\nThe fixed\npoints of DE of the uncoupled system correspond to the\npoints where these two curves meet or cross. Assuming continuity of $\\hf$ and $\\hg,$ they are the\npoints $(\\xg, \\xf)$ such that $(\\xg, \\hf(\\xg)) = (\\hg(\\xf), \\xf).$\n\nTo help with analysis in the discontinuous case we introduce the following notation.\nFor any $h \\in \\exitfns$ we write\n\\[\nu \\veq h(v)\n\\]\nto mean $u \\in [h(v-),h(v+)].$\n\\begin{definition}[Crossing Points]\nGiven $(\\hf,\\hg)\\in\\exitfns^2$ and we say that $(\\xg, \\xf)$ is a crossing point\nif \n\\[ \n\\xg \\veq \\hg(\\xf),\\text{ and } \\xf \\veq \\hf(\\xg)\\,.\n\\]\nThe following are three equivalent characterizations of crossing points.\n\\begin{itemize}\n\\item $u \\veq \\hfinv(v)$ and $v \\veq \\hginv(u),$\n\\item $u \\veq \\hg(v)$ and $u \\veq \\hfinv(v),$\n\\item $v \\veq \\hf(u)$ and $v \\veq \\hginv(u).$\n\\end{itemize}\n\nThe set of all crossing points will be denoted $\\cross(\\hf,\\hg).$ \nIt is easy to see that $\\cross(\\hf,\\hg)$ is closed as a subset of $[0,1]^2.$\nBy definition of $\\exitfns$,\nwe have $(0,0) \\in \\cross(\\hf,\\hf)$ and $(1,1) \\in \\cross(\\hf,\\hg).$ \nWe term $(0,0)$ and $(1,1)$\nthe {\\em trivial} crossing points and denote the non-trivial crossing points by\n\\[\n\\intcross (\\hf,\\hg) = \\cross(\\hf,\\hg)\\backslash \\{(0,0),(1,1)\\}.\n\\]\n\\end{definition}\n\nIf $(u,v)\\in\\cross(\\hf,\\hf)$ and $\\hf$ and $\\hg$ are continuous at\n$u$ and $v$ respectively then $(u,v)$ is a fixed point of density\nevolution. In general, if $(u,v)\\in\\cross(\\hf,\\hf)$ then $(u,v)$\nis a fixed point of density evolution for a pair of exit functions\nequivalent to the pair $(\\hf,\\hg).$ \n\n\\begin{lemma}\\label{lem:crossorder}\nFor any $\\hf,\\hg \\in \\exitfns$ the set $\\cross(\\hf,\\hg)$ is component-wise ordered,\ni.e., given $(u_1,v_1),(u_2,v_2) \\in \\cross(\\hf,\\hg)$ we have \n$(u_2-u_1)(v_2-v_1)\\ge 0.$\n\\end{lemma}\n\\begin{IEEEproof}\nLet $(u_1,v_1),(u_2,v_2) \\in \\cross(\\hf,\\hg).$\nIf $u_10,$ we have\n$\\cross(\\hf^i,\\hg^i) \\subset \\neigh{\\cross(\\hf,\\hg)}{\\delta}$ for\nall $i$ sufficiently large.\n\\end{lemma}\n\\begin{IEEEproof}\nAssume $(u^i,v^i)\\in\\cross(\\hf^i,\\hg^i)$ converges in $i$ to a limit point $(u,v).$ \nSince $\\hf^i(u) \\rightarrow \\hf(u)$ at points of continuity of $\\hf$\nit is easy to see that\n\\[\n\\liminf_{i\\rightarrow \\infty} \\hf^i(u^i-)\\ge \\hf(u-)\n\\]\nand\n\\[\n\\limsup_{i\\rightarrow \\infty} \\hf^i(u^i+)\\le \\hf(u+)\n\\]\nand it follows that $v \\veq \\hf(u).$\nSimilarly, $u \\veq\\hg(v).$\nHence, $(u,v) \\in \\cross(\\hf,\\hg).$\n \nSince $[0,1]^2 \\backslash \\neigh{\\cross(\\hf,\\hg)}{\\delta}$ is compact\nand the same argument applies to subsequences\nthe Lemma follows.\n\\end{IEEEproof}\n\n\\begin{lemma}\nConsider initialization of system \\eqref{eqn:DE} with an arbitrary choice of $u^0.$\nThen the sequence $(u_1,v_1),(u_2,v_2),\\ldots$ is monotonic (either non-increasing or non-decreasing)\nin both coordinates.\n\\end{lemma}\n\\begin{IEEEproof} \nIf $v^{t+1} = \\hf(u^t) \\ge v^t$ then $u^{t+1}=\\hg(v^{t+1}) \\ge \\hg(v^t) = u^t.$\nAnd if $u^{t+1} \\ge u^t$ then $v^{t+2}=\\hg(u^{t+1}) \\ge \\hg(u^t) = v^{t+1}.$\n\\end{IEEEproof}\nIt follows that the sequence $(u_i,v_i)$ converges and the limit point is clearly a crossing point of $(\\hf,\\hg).$\nThus, the limiting behavior of the scalar component system is governed by crossing points.\nIn the spatially coupled system the behavior is often involves a pair\nof crossing points. \n\nSuppose $(u_1,v_1) < (u_2,v_2)$ are fixed points of the component DE.\nIf $\\ff^0(x)\\in[v_1,v_2]$ for all $x$\nthen $\\ff^t(x)\\in[v_1,v_2]$ and\nthen $\\fg^t(x)\\in[u_1,u_2]$ \nfor all $x$ and $t.$ and \n$\\fg^t$ is in the range between $u_1$ and $u_2$ for all $t.$\nThus, in this situation the system is effectively confined to $[u_1,u_2]\\times [v_1,v_2].$\nThis circumstance occurs frequently but we can easily transform this into our canonical form.\nWe can introduce new coordinates $\\tilde{u},\\tilde{v}$ characterized by the inverse map\n\\begin{align*}\nu & = a \\tilde{u} + b \\\\\nv & = c \\tilde{v} + d\\,.\n\\end{align*}\nBy choosing $(b,d) = (u_1,v_1)$ and $(a,c) = (u_2-u_1,v_2-v_1)$\nwe map $(u,v) =(u_1,v_1)$ to $(\\tilde{u},\\tilde{v}) =(0,0)$ and $(u,v) =(u_1,u_2)$ to $(\\tilde{u},\\tilde{v}) =(1,1).$\nSimilarly, by choosing $(b,d) = (u_2,v_2)$ and $(a,c) = (u_1-u_2,v_1-v_2)$\nwe map $(u,v) = (u_2,v_2)$ to $(\\tilde{u},\\tilde{v}) = (0,0)$ and $(u,v)=(u_1,v_1)$ to $(\\tilde{u},\\tilde{v}) =(1,1).$\n(This shows the symmetry that allows us to occasionally exchange $(0,0)$ and $(1,1)$\nin the analysis.)\nBy rescaling the update functions appropriately we can thereby redefine the system on $[0,1]\\times[0,1].$\nIn particular we can define ${\\thf}(\\tilde{u}) = \\frac{1}{c}(\\hf( a \\tilde{u} + b) - d)$\nand ${\\thg}(\\tilde{v}) = \\frac{1}{a}(\\hf( c \\tilde{u} + d) - b).$\n\n\\bexample[EXIT Chart Analysis for the BEC] Figure~\\ref{fig:exitbec36}\nshows the EXIT chart analysis for the $(3, 6)$-regular ensemble\nwhen transmission takes place over the BEC. The left picture shows\nthe situation when the channel parameter is below the BP threshold.\nIn this case we only have the trivial FP at $(0, 0)$. According to our definition we\nhave $(1,1)$ as a crossing point, but it is not a fixed point because $\\hf(1)=\\epsilon<1.$ The right\npicture shows a situation when we transmit above\nthe BP threshold. We now see two further crossings of the EXIT curves\nand so $\\cross(\\hf,\\hg)$ is non-trivial.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitbec36}\n}\n\\caption{\\label{fig:exitbec36} Left: The figure shows the EXIT\nfunctions $\\hf(\\xg)=\\epsilon \\lambda(\\xg)$ and $\\hg(\\xf)=1-\\rho(1-\\xf)$ for the $(3, 6)$-regular\nensemble and $\\epsilon=0.35$. Note that the horizontal axis is $\\xg$\nand the vertical axis is $\\xf$ so that we effectively plot the inverse\nof the function $1-\\rho(1-\\xf)$. Since $0.35=\\epsilon < \\epsilon^{\\text{BP}}\n\\approx 0.4292$, the two curves do not cross. The dashed ``staircase''\nshaped curve indicates how DE proceeds. Right: In this figure the\nchannel parameter is $\\epsilon=0.5 > \\epsilon^{\\text{BP}}$. Hence,\nthe two EXIT curves cross. In fact, they cross exactly twice (besides\nthe trivial FP at $(0, 0)$), the first point corresponds to an\nunstable FP of DE, whereas the second one is a stable FP. }\n\\end{figure}\n\nIn this case we can renormalize the system according to our prescription as follows.\nConsider the DE\nequations stated in (\\ref{eqn:DEBEC}). If $(\\xg^*, \\xf^*)$ is the\nlargest (in both components) FP of the corresponding\nDE and if we can set $\\hf(\\xg)=\\epsilon\\lambda(\\xg \\xg^*)\/\\xf^*$\nand $\\hg(\\xf)=(1-\\rho(1 - \\xf \\xf^*))\/\\xg^*$ then system \\eqref{eqn:DE}\nis again equivalent to \\eqref{eqn:DEBEC} on the restricted domain but, in addition, the component\nfunctions are continuous at $0$ and $1$ and $(0,0)$ and $(1,1)$ are the relevant fixed points. This rescaling is indicated\nin the right picture of Figure~\\ref{fig:exitbec36} through the\ndashed gray box. Since the standard (unscaled) EXIT chart picture\nis very familiar in the coding context, we will continue to plot\nthe unscaled picture. But we will always indicate the scaled version\nby drawing a gray box as in the right picture of\nFigure~\\ref{fig:exitbec36}. This hopefully will not cause any\nconfusion. There is perhaps only one point of caution. The behavior\nof the coupled system depends on certain areas in this EXIT chart.\nThese areas are defined in the scaled version and are different by\na factor $\\xg^* \\xf^*$ in the unscaled version.\n{\\hfill $\\ensuremath{\\Box}$} \\eexample\n\n\n\nSo far we have considered the uncoupled system and seen that its\nbehavior can be characterized in terms of fixed points, or more generally crossing points. \nThe behavior of the\nspatially coupled system can also be characterized by its FPs. For the\nspatially coupled system a FP is not a pair of scalars, but a pair\nof functions $(\\tmplF(x), \\tmplG(x))$ such that if we set $\\ff^t(x)=\\tmplF(x)$\nand $\\fg^t(x)=\\tmplG(x)$, $t \\geq 0$, then these functions fulfill\n(\\ref{eqn:gfrecursion}). One set of FPs arise as the constant functions\ncorresponding to the fixed points of the underlying component system.\nThe crucial phenomena in spatial coupling is the emergence of interpolating spatial\nfixed points, i.e., non-constant monotonic fixed point solutions.\nFor the coupled system it is\nfruitful not only to look at interpolating FPs but slightly more general objects,\nnamely interpolating {\\em waves}. Here a wave is like a FP, except that it\n{\\em shifts}. I.e., for $(\\tmplF(x), \\tmplG(x))$ fixed and for some real\nvalue $\\ashift$, if we set $\\ff^t(x)=\\tmplF(x-\\ashift t)$ and $\\fg^t(x)=\\tmplG(x-\\ashift\nt)$, $t \\geq 0$, then these functions fulfill (\\ref{eqn:gfrecursion}).\nWe will see that the behavior of coupled systems is governed\nby the (non)existence of such waves and this\n(non)existence has a simple graphical characterization in terms of\nthe component-wise EXIT functions and their associated FPs. This\nis the main technical result of this paper. In fact, the {\\em\ndirection} of travel of the wave\ndepends in a simple way on the EXIT functions and the area bound\nby them. \nThe extremal values of spatial wave solutions are generally crossing points of the\nunderlying component system. One aspect of the analysis involves\ndetermining the crossing points that can appear as such an extremal value.\nThe solution is is formulated in terms of the following definition.\n\n\\begin{definition}[Component Potential Functions]\nFor any pair $(\\hf,\\hg) \\in \\exitfns^2$ and point $(u,v) \\in [0,1]^2,$ we define\n\\[\n\\altPhi(\\hf,\\hg;u,v) = \\int_0^u \\hginv(u')\\text{d}u' + \\int_0^v \\hfinv(v')\\text{d}v' \\, - uv\\,.\n\\]\n\\end{definition}\n\n{\\em Discussion:} The functional $\\altPhi$ serves as a potential function\nfor the scalar system. \nAssuming continuity of $\\hfinv$ at $v$ and continuity of $\\hginv$ at $u$ we have\n$\\nabla \\altPhi(\\hf,\\hg; u,v) = (\\hginv(u)-v,\\hfinv(v)-u).$\nThus, under some regularity conditions\na crossing point $(\\xg,\\xf)$ is a stationary point of $\\altPhi.$\ni.e., $ \\nabla \\altPhi(\\hf,\\hg; u,v) =0$\nfor $(u,v)\\in \\cross(\\hf,\\hg).$\n\nNote that in the definition of $\\altPhi$ we have used\n$(0,0)$ as an originating point. We can choose the origin arbitrarily.\nWe have\n\\begin{align}\\begin{split}\\label{eqn:potdiff}\n&\\altPhi(\\hf,\\hg;u,v) - \\altPhi(\\hf,\\hg;u_1,u_2)\\\\\n= &\n\\int_{u_1}^u \\hginv(u')\\text{d}u' + \\int_{v_1}^v \\hfinv(v')\\text{d}v' \\, - uv + u_1 v_1\n\\\\= &\n\\int_{u_1}^u (\\hginv(u')-v_1)\\text{d}u' + \\int_{v_1}^v (\\hfinv(v')-u_1)\\text{d}v' \\,\n\\\\&\\qquad - (u-u_1)(v-v_1)\n\\end{split}\n\\end{align}\nand we see that we can place the origin at $(u_1,u_2)$ while preserving differences.\n\nA straightforward calculation, noting that\n\\(\nu\\hf(u) = \\int_0^u \\hf(u')\\, du' + \\int_0^{\\hf(u)} \\hfinv(v)\\, dv\n\\)\nand\n\\(\nv\\hg(v) = \\int_0^v \\hg(y)\\, dy + \\int_0^{\\hg(v)} \\hginv(x)\\, dx\\,,\n\\)\nshows that for all $(\\hf,\\hg)$ we have\n\\begin{align}\n\\begin{split}\\label{eqn:altPhiPhi}\n&\\altPhi(\\hf,\\hg;\\hg(v),\\hf(u))\\\\\n=&\nuv - (u-\\hg(v))(v-\\hf(u)) \\\\\n& - \\int_0^u \\hf(u')\\text{d}u' - \\int_0^v \\hg(v')\\text{d}v'\\, .\n\\end{split}\n\\end{align}\nA similar potential function form, along the lines of \\eqref{eqn:altPhiPhi}, is\n\\begin{align*}\n\\Phi(\\hf,\\hg;u,v) = uv - \\int_0^u \\hf(u')\\text{d}u' - \\int_0^v \\hg(v')\\text{d}v'\\, .\n\\end{align*}\nThis functional is also stationary on the FPs of the component density evolution and is equal\nto $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ on the points $(u,\\hf(u))$ and $(v,\\hg(v)).$\nThis form underlies the work in \\cite{TTK11b, DJM11, YJNP12a, YJNP12b}.\nWe prefer $\\altPhi$ because of various properties developed below. The two forms are\nrelated through Legendre transforms, e.g., $\\int_0^u \\hf(u')\\text{d}u'$ is the Legendre transform\nof $\\int_0^v \\hfinv(v')\\text{d}v'.$ We see that on the graph of \n$\\hf$ or $\\hg,$ i.e., on the either the set with\n$v=\\hf(u)$ or $u=\\hg(v),$ the two functionals are equivalent up to reparameterization.\nIn terms of $\\altPhi$ we have\n\\begin{align}\n\\label{eqn:altPhiuhfu}\n\\altPhi(\\hf,\\hg;u,\\hf(u)) &= \\int_0^u (\\hginv(u')-\\hf(u'))\\,du'\\\\\n\\altPhi(\\hf,\\hg;\\hg(v),v) &= \\int_0^v (\\hfinv(v')-\\hg(v'))\\,dv' \\label{eqn:altPhivhgv}\n\\end{align}\n\n\n\\begin{lemma}\\label{lem:monotonic}\nThe function $\\altPhi(\\hf,\\hg;u,v)$ is convex in $u$ for fixed $v$ and convex in $v$ for fixed $u.$\nIn addition, for all $(u,v) \\in[0,1]^2$ we have\n\\begin{align*}\n\\altPhi(\\hf,\\hg;u,v) &\\ge \\altPhi(\\hf,\\hg;u,\\hf(u)) \\\\\n\\altPhi(\\hf,\\hg;u,v) &\\ge \\altPhi(\\hf,\\hg;\\hg(v),v) \n\\end{align*}\nwith equality holding in the first case if and only if $v\\veq\\hf(u)$\nand\n in the second case if and only if $u\\veq\\hg(v).$\n\\end{lemma}\n\\begin{IEEEproof}\nIt is easy to check that $\\altPhi(\\hf,\\hg;u,v)$ is Lipschitz (hence absolutely) continuous \nand we have almost everywhere\n\\begin{align}\n\\begin{split}\\label{eqn:altPhiderivatives}\n\\frac{\\partial}{\\partial u} \\altPhi(\\hf,\\hg;u,v) &= \\hginv(u) - v, \\\\\n\\frac{\\partial}{\\partial v} \\altPhi(\\hf,\\hg;u,v) &= \\hfinv(v) - u \\,.\n\\end{split}\n\\end{align}\nThe lemma now follows immediately from the monotonicity (non-decreasing) of $\\hginv$ and $\\hfinv.$\n\\end{IEEEproof}\nWe have immediately the following two results.\n\\begin{corollary}\\label{cor:monotonic}\nIf $(u^0,v^0) \\in [0,1]^2$ and we define $(u^t,v^t)$ for $t \\ge 1$ via\n\\eqref{eqn:DE} then $\\altPhi(\\hg,\\hg;u^t,v^t)$ is a non-increasing sequence in $t.$\n\\end{corollary}\n\\begin{lemma}\\label{lem:miniscross}\nIf $(u,v) \\in\\cross(\\hf,\\hg)$ if and only if\n$\\altPhi(\\hf,\\hg;u',v)$ is minimized at $u'=u$ and\n$\\altPhi(\\hf,\\hg;u,v')$ is minimized at $v'=v.$\n\\end{lemma}\n\nWe will be most interested in the value of $\\altPhi$\nat crossing points $(u,v) = \\cross(\\hf,\\hg).$\n\n\nOne of the key results on the existence of wave solutions, and especially spatial fixed points,\nis that the crossing points associated to the extremal values of the solution are extreme\n(minimizing) values of the the $\\altPhi$ over the range spanned by the solution. The following definition characterizes this.\n\n\\begin{definition}[Strictly Positive Gap Condition]\nWe say that the pair of functions $(\\hf,\\hg)$ satisfies the {\\em\nstrictly positive gap condition} if $\\cross(\\hf,\\hg)$ is {\\em\nnon-trivial} and if \n\\[\n(u,v) \\in \\intcross(\\hf,\\hg) \\Rightarrow \\altPhi(\\hf,\\hg;u,v) > \\max\\{0,A(\\hf,\\hg)\\}\n\\] \nwhere we define $A(\\hf,\\hg) =\\altPhi(\\hf,\\hg;1,1).$\nWe say that the pair of functions $(\\hf,\\hg)$ satisfies the {\\em\npositive gap condition} (no longer strict) if $\\cross(\\hf,\\hg)$ is {\\em\nnon-trivial} and \n\\[\n(u,v) \\in \\intcross(\\hf,\\hg) \\Rightarrow \\altPhi(\\hf,\\hg;u,v) \\ge \\max\\{0,A(\\hf,\\hg)\\}\\,.\n\\] \n {\\hfill $\\ensuremath{\\Box}$}\n\\end{definition}\n\n{\\em Discussion:} The (strictly) positive gap condition is related to the existence of interpolating spatial fixed point solutions.\nIn particular, we will see that systems possessing $(0,1)$-interpolating fixed point solutions must satisfy the positive gap condition\nand have $A(\\hf,\\hg)=0.$ Systems satisfying the strictly positive gap condition with $A(\\hf,\\hg)=0$ will be proven to\npossess $(0,1)$-interpolating spatial fixed point solutions. The cases where $A(\\hf,\\hg) \\neq 0$ correspond to $(0,1)$-interpolating traveling wave solutions.\n\n\\blemma[Trivial Behavior]\\label{lem:trivialbehavior}\nIf $\\cross(\\hf,\\hg)$ is trivial then the system behavior is simplified\nand under DE, i.e., under $\\eqref{eqn:gfrecursion}$, the only spatial fixed points are with\n$\\ff^t$ and $\\fg^t$ set to either the constant $0$ or the constant\n$1,$ one of which is stable and one of which is unstable.\nThe system converges for all initial values, other than the unstable spatial fixed point itself,\nto the stable spatial fixed point.\n\\elemma\n\nNow that we have covered the ``trivial'' cases, let us consider\nthe system behavior when $\\cross(\\hf,\\hg)$ is non-trivial. As we\nwill see, it is qualitatively different. \nThe value $A(\\hf,\\hg)=\\altPhi(\\hf,\\hg;1,1)$ plays an important role\nin the behavior of the system. This is why we \nintroduced a special notation. The strictly positive gap condition\nimplies that the value of $\\altPhi(\\hf,\\hg;\\xg,\\xf)$ for $(\\xg,\\xf)\\in \\intcross(\\hf,\\hg)$\nis strictly larger than\nthe values $0$ and $A(\\hf,\\hg)$ found at the two trivial fixed points. We will\nsee that this condition is related to the existence of wave-like\nsolutions that interpolate between the two trivial fixed\npoints.\n\n\\bexample[Positive Gap Condition for the BEC]\nFigure~\\ref{fig:positivegapbec36} illustrates the (strictly) positive\ngap condition for the $(3, 6)$-regular ensemble when transmission\ntakes place over the BEC. The left picture shows the situation when\nthe channel parameter is between the BP and the MAP threshold of\nthe underlying ensemble. The right picture shows the situation when\nthe channel parameter is above the MAP threshold of the underlying\nensemble. In both cases $\\cross(\\hf,\\hg)$ contains one non-trivial\nFP $(\\xg, \\xf)$ and for this FP $\\altPhi(\\hf,\\hg;\\xg,\\xf) > \\max\\{0,A\\}$, i.e.,\nboth cases fulfill the strictly positive gap condition.\n\nIn the first case $A>0$, whereas in the second case $A<0$. We\nwill see in Theorem~\\ref{thm:mainexist} below that this\nchange in the sign of $A$ leads to a reversal of direction of a wave-like\nsolution to the system and hence to fundamentally different asymptotic behavior.\nBoth pictures show the unscaled curve and the lightly shaded box\nshows what the picture would look like if we rescaled it so that\nthe largest FP appears at $(1, 1)$.\n\nIt is not hard to see that the strictly positive gap\ncondition is necessarily satisfied for any $\\hf,\\hg$ for which $\\cross(\\hf,\\hg)$\nhas a single non-trivial fixed point, and for which $(0,0)$ and\n$(1,1)$ are stable fixed points under the DE equations \\eqref{eqn:DE}.\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nWe are now ready to state the main result concerning the existence of interpolating wave solutions.\n\n\\begin{theorem}[Existence of Continuum Spatial Waves]\\label{thm:mainexist}\nAssume that $\\smthker$ is a regular averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of functions in $\\exitfns$ satisfying the strictly positive\ngap condition.\n\nThen there exist $(0,1)$-interpolating functions $\\tmplF,\\tmplG \\in\\sptfns$ and a\nreal-valued constant $\\ashift,$ satisfying $\\sgn(\\ashift) = \\sgn(A(\\hf,\\hg))$\nand $|\\ashift| \\ge |A(\\hf,\\hg)|\/\\|\\smthker\\|_\\infty$, such that setting\n$f^t(x) = \\tmplF(x - \\ashift t)$ and $g^t(x) = \\tmplG(x - \\ashift t)$ for\n$t=0,1,\\ldots$ solves \\eqref{eqn:gfrecursion}\\,.\n\\end{theorem}\n\nWe remark that we can relax the regularity condition on $\\smthker$ if\n$\\hf$ and $\\hg$ are continuous; cf. Lemma \\ref{lem:pathology}.\n\n\\bexample[Spatial wave for the BEC]\nFigure~\\ref{fig:spatialfpbec36} shows the spatial waves whose existence\nis guaranteed by Theorem~\\ref{thm:mainexist} for the $(3, 6)$\nensemble and transmission over the BEC. The top picture corresponds\nto the cases $\\epsilon=0.45$ and the bottom picture to the case\n$\\epsilon=0.53$. In both cases we used the smoothing kernel\n$\\omega(x)=\\frac12 \\mathbbm{1}_{\\{|x|\\leq 1\\}}$. As predicted, in\nthe first case the curve moves to the right by a value of\n$0.142 \\geq |A|\/\\|\\smthker\\|_\\infty = 0.03125 \\times 2 =0.06245$\nand in the second case the curve moves to the left by an amount\nof $0.101 \\geq |A|\/\\|\\smthker\\|_\\infty = 0.0253740 \\times 2 = 0.0507498$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/fpbec36}\n}\n\\caption{\\label{fig:spatialfpbec36}\nFPs whose existence is guaranteed by Theorem~\\ref{thm:mainexist}\nfor the $(3, 6)$ ensemble and transmission over the BEC. The top\npicture corresponds to the cases $\\epsilon=0.45$ and the bottom\npicture to the case $\\epsilon=0.53$. In both cases we used the\nsmoothing kernel $\\omega(x)=\\frac 12 \\mathbbm{1}_{\\{|x|\\leq 1\\}}$. The dashed curve\nis the result of applying one step of DE to the solid curve. As\npredicted, in the top picture the curve moves to the right (the\ncorresponding gap $A$ in Figure~\\ref{fig:positivegapbec36} is\npositive) whereas in bottom picture the curve moves to the left\n(the corresponding gap $A$ is negative). The shifts are $0.142$ and\n$-0.102$, respectively. } \\end{figure} {\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nOne consequence of Theorem \\ref{thm:mainexist} is that the existence of an\n$(0,1)$-interpolating fixed point implies $A(\\hf,\\hg) = 0.$ This is true even without\nregularity assumptions.\n\n\\begin{theorem}[Continuum Fixed Point Positivity]\\label{thm:FPAzero}\nLet $\\smthker$ be an averaging kernel (not necessarily regular) and\nassume that there exists a $(0,1)$-interpolating fixed point solution to \\eqref{eqn:gfrecursion}.\nThen $(\\hf,\\hg)$ satisfies the positive gap condition and $A(\\hf,\\hg) = 0.$\n\\end{theorem}\nA more general version of this result appears as Lemma \\ref{lem:FPAzero}.\n\nTheorem \\ref{thm:mainexist} is our most fundamental result concerning the spatially coupled system.\nOne limitation of the result arises in cases with infinitely many crossing points. In such a case it\ncan be difficult to extract asymptotic behavior since there may exist many wave-like solutions\nand the strictly positive gap condition may not hold globally.\nFor such cases we develop the following altered analysis.\n\nLet $\\hf$ and $\\hg$ be given and define\n\\[\nm(\\hf,\\hg)=\\min_{(u,v) \\in [0,1]^2} \\altPhi(\\hf,\\hg;u,v)\n\\]\nDefine\n\\[\n\\cross_m(\\hf,\\hg) = \\{(u,v)\\in\\cross(\\hf,\\hg) : \\altPhi(\\hf,\\hg;u,v) = m\\}\\,.\n\\]\nSince $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is continuous it follows that\n$\\cross_m(\\hf,\\hg)$ is closed. Since $\\cross(\\hf,\\hg)$ is component-wise linearly\nordered we can define\n\\begin{align*}\n(u',v') & = \\min \\cross_m(\\hf,\\hg)\\\\\n\\intertext{and}\n(u'',v'')& = \\max \\cross_m(\\hf,\\hg)\\\\\n\\end{align*}\nwhere $\\min$ and $\\max$ are component-wise.\n\n\\begin{theorem}[General Continuum Convergence]\\label{thm:globalconv}\nLet $(\\hf,\\hg)$ be given as above, let $\\smthker$ be regular,\nand assume $\\ff^0 \\in \\sptfns$ is given\nwith $\\ff^0(\\minfty)\\le v''$ and \n$\\ff^0(\\pinfty)\\ge v'.$\nThen in system \\ref{eqn:gfrecursion} we have for all $x\\in \\reals$\n\\begin{align*}\n\\liminf_{t\\rightarrow \\infty} f^t(x) &\\ge v' \\quad \\liminf_{t\\rightarrow \\infty} g^t(x) \\ge u' \\\\\n\\limsup_{t\\rightarrow \\infty} f^t(x) &\\le v'' \\quad \\limsup_{t\\rightarrow \\infty} g^t(x) \\le u'' \\,.\n\\end{align*}\n\\end{theorem}\nThe proof may be found in appendix \\ref{app:E}.\n\nNote, in particular, that if $\\altPhi$ is uniquely minimized at some crossing point $(u,v),$ then\nthis point is a fixed point of the component system and if the spatial system is initialized (either $f$ or $g$) with this point (the appropriate coordinate)\n in the closed range spanned by the initial condition, then the coupled system\nglobally converges to the constant function associated to this fixed point.\n\nIn many applications the problems are spatially discrete and finite length.\nThe analysis can be applied to these cases with suitable adjustments. As a first step we state a\nresult analogous to Theorem \\ref{thm:mainexist}\nfor a spatially discrete system. The DE equations\nfor the spatially discrete version can be written as in\n\\eqref{eqn:gfrecursion} with the following modifications:\nthe variable $x$ is discrete, the averaging kernel is a discrete sequence, and\nthe convolution\noperation is convolution of discrete sequences. The analysis views the\nspatially discrete problem as a sampled version of the continuum\nversion. In the limit of infinitely fine sampling the discrete\nversion converges to the continuum version.\n\n\\subsection{Discrete Spatial Sampling}\n\nLet $x_i = i \\Delta$ and let $\\discsmthker$ be a non-negative\nfunction over $\\integers$ that is even, $\\discsmthker_i =\n\\discsmthker_{-i},$ and sums to $1,$ $\\sum_i \\discsmthker_i = 1.$\nIt is convenient to interpret $\\discsmthker$ as a discretization of\n$\\smthker,$ i.e.,\n\\begin{equation}\\label{eqn:kerdiscretetosmth}\n\\discsmthker_i = \\int_{(i-\\frac{1}{2})\\Delta}^{(i+\\frac{1}{2})\\Delta} \\smthker(z)\\text{d}z.\n\\end{equation}\nThis relationship then makes it clear that the discrete ``width'' of\nspatial averaging is inversely proportional to $\\Delta.$\nA good example is the smoothing kernel\n$\\smthker(x) = \\frac{1}{2} \\mathbbm{1}_{\\{|x|\\le 1\\}}.$\nIf we set $\\Delta = \\frac{2}{2W+1}$ then $\\discsmthker_i =\n\\frac{1}{2W+1} \\mathbbm{1}_{\\{|i|\\le W\\}}.$\nGiven a real-valued function $\\fg$ defined on $\\Delta\\integers$ we will call \nthe function $\\tfg \\in \\sptfns,$ defined as \n$\\tfg(x)=\\fg(x_i)$ for $x\\in[x_i-\\Delta\/2,x_i+\\Delta\/2),$\nthe {\\em piecewise constant extension} of $\\fg.$ \nNote that by this definition, we have\n\\begin{align*}\n\\tfg^{\\smthker}(x_i)&=\\int_{-\\infty}^\\infty \\smthker(x_i-y)\\tfg(y)dy\n\\\\&=\n\\sum_{j=-\\infty}^\\infty\n\\int_{x_j-\\Delta\/2}^{x_j+\\Delta\/2} \\smthker(x_i-y)\\tfg(y)dy\n\\\\\n&=\n\\sum_{j=-\\infty}^\\infty \\discsmthker_{i-j}\\fg(x_j)dy\\,\n\\\\&=\n\\fg^{\\discsmthker}(x_i)\n\\end{align*}\n\nWith this framework in mind, we can write the spatially discrete DE equations as follows.\n\\begin{equation}\\label{eqn:discretegfrecursion}\n\\begin{split}\n\\fg^{t}(x_i) & = \\hg ((\\ff^t \\otimes \\discsmthker) (x_i)) \\\\\n\\ff^{t+1}(x_i) & = \\hf ( (\\fg^{t} \\otimes \\discsmthker) (x_i) )\\,.\n\\end{split}\n\\end{equation}\n\n\\bexample[Spatially Discrete DE for the BEC]\n\\begin{equation}\\label{eqn:gfrecursionBEC}\n\\begin{split}\n\\fg^{t}(x_i) & = 1-\\rho((\\ff^t \\otimes \\discsmthker) (x_i)), \\\\\n\\ff^{t+1}(x_i) & = \\epsilon \\lambda( (\\fg^{t} \\otimes \\discsmthker) (x_i) )\\,.\n\\end{split}\n\\end{equation}\n{\\hfill $\\ensuremath{\\Box}$}\n\\eexample\n\nAn elementary but critical result relating the spatially continuous case to the\ndiscrete case is the following.\n\\begin{lemma}\\label{lem:disccontbnd}\nLet $\\tmplF \\in \\sptfns$ and let $f$ be a real valued function defined on $\\Delta \\integers.$\nThen, if for all $i$ we have\n\\(\nf(x_i) \\le \\tmplF(x_i)\n\\)\nthen \n\\(\nf^{\\discsmthker}(x_i) \\le \\tmplF^{\\smthker}(x_i+\\half\\Delta)\n\\)\nand if\n\\(\nf(x_i) \\ge \\tmplF(x_i)\n\\)\nthen \n\\(\nf^{\\discsmthker}(x_i) \\ge \\tmplF^{\\smthker}(x_i-\\half\\Delta)\n\\)\n\\end{lemma}\n\\begin{IEEEproof}\nAssume\n$\\ff(x_i) \\le \\tmplF(x_i)$ (for all $i$).\nConsider the piecewise constant extension $\\tilde{\\ff}.$ \nIt follows that $\\tilde{\\ff}(x) \\le \\tmplF(x+\\half\\Delta)$ for all $x$\nand so $\\ff^{\\discsmthker}(x_i) = \\tilde{\\ff}^{\\smthker}(x_i) \\le \\tmplF^{\\smthker}(x_i+\\half\\Delta)$\nfor each $i.$\n\nThe opposite inequality is handled similarly.\n\\end{IEEEproof}\n\nApplying the above to system \\eqref{eqn:discretegfrecursion} we obtain the following.\n\\begin{theorem}[Continuum-Discrete Bounds] \\label{thm:mainquantize}\nAssume that $\\discsmthker$ is a discrete sequence\nrelated to a regular smoothing kernel $\\smthker$\nas indicated in \\eqref{eqn:kerdiscretetosmth}.\nLet $\\ff^t_c,\\fg^t_c \\in \\sptfns,\\, t=0,1,2,\\ldots$ denote spatially continuous functions determined\naccording to \\eqref{eqn:gfrecursion} and let\n$\\ff^t,\\fg^t$ denote spatially discrete functions determined\naccording to \\eqref{eqn:discretegfrecursion}.\nThen, if\n$\\ff^0(x_i) \\le \\ff^0_c (x_i)$ (for all $i$) then $\\ff^t(x_i) \\le \\ff^t_c(x_i - t\\Delta )$ and\nif $\\fg^t(x_i) \\ge \\fg_c^t(x_i - (t-\\half) \\Delta)$ for all $t.$\nSimilarly, if\n$\\ff^0(x_i) \\ge \\ff^0_c (x_i)$ (for all $i$) then $\\ff^t(x_i) \\ge \\ff^t_c(x_i + t\\Delta )$ and\nif $\\fg^t(x_i) \\ge \\fg_c^t(x_i + (t+\\half) \\Delta)$ for all $t.$\n\\end{theorem}\n\\begin{IEEEproof}\nAssume\n$\\ff^0(x_i) \\le \\ff_c^0(x_i)$ (for all $i$).\nBy Lemma \\ref{lem:disccontbnd} $\\ftdS{0}(x_i) = \\ff_c^{0,\\smthker}(x_i+\\half\\Delta)$ for each $i.$\nBy monotoniciy of $\\hg$ we have\n\\[\n\\fg^0(x_i) = \\hg(\\ff^0(x_i))\\le \\hg(\\ff_c^{0,\\smthker}(x_i+\\half\\Delta)) = \\fg_c^0(x_i+\\half\\Delta).\n\\]\nBy the same argument we obtain\n\\(\n\\gtdS{0}(x_i) \\le \\fg_c^{0,\\smthker}(x_i+\\Delta)\\,\n\\)\nand hence\n\\(\n\\ff^1(x_i)\\le \\ff_c^1(x_i+\\half\\Delta).\n\\)\nThe general result now follows by induction.\n\nThe reverse inequality can be handled similarly.\n\\end{IEEEproof}\n\nThis result is convenient when there exist wave-like solutions.\nFor example, if $\\ff_c^t(x) = \\tmplF(x-\\ashift t)$ with $\\ashift>0$ and\n$\\tmplF$ is a $(0,1)$-interpolating function, and\n$\\ff^0(x_i) \\le \\ff_c^0(x_i),$ then we have\n$\\ff^t(x_i) \\le \\tmplF(x-(\\ashift-\\Delta) t).$\nThus, if $\\ashift > \\Delta$ then we obtain asymptotic convergence for the\nspatially discrete case.\nThus we have the following result\n\n\\begin{theorem}[Discrete Spatial Convergence]\\label{thm:mainqqqqq}\nAssume that $\\smthker$ is a regular averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of functions in $\\exitfns$ satisfying the strictly positive\ngap condition.\nAssume $\\Delta < |A(\\hf,\\hg)|\/\\|\\smthker\\|_\\infty$\nand initialize system \\eqref{eqn:discretegfrecursion} with\nany $(0,1)$ interpolating $\\ff^0 \\in \\sptfns.$\nIf $A(\\hf,\\hg)>0$ then $\\ff^t(x_i) \\rightarrow 0$ and\nif $A(\\hf,\\hg)<0$ then $\\ff^t(x_i) \\rightarrow 1$ \nfor all $x_i$ \n\\end{theorem}\n\n\nThis result gives order ${\\Delta}$ convergence of the spatially discrete system to the continuum one (under positive gap assumptions).\nMuch faster convergence is observed in many situations. \nIn \\cite{HMU11b} a particular example is presented with a compelling heuristic argument for exponential convergence.\nIn general the rate of convergence appears to depend on the regularity of $\\hf$ and $\\hg$ and\n$\\smthker.$\nA $(0,1)$-interpolating spatial fixed point does not sample $\\hf$ and $\\hg$ at every value, so one cannot\nconclude that $A(\\hf,\\hg) = 0$ and, indeed, this generally does not hold. One can construct fixed point examples where\n$|A(\\hf,\\hg)|$ is of order $\\Delta.$\nAs a general result we have the following.\n\\begin{theorem}\\label{thm:discreteFPDelta}\nAssume $\\hf$ and $\\hg$ have a $(0,1)$-interpolating fixed point\nfor the spatially discrete system. Then,\n\\[\n|A(\\hf,\\hg) | \\le 2{\\Delta}\\|\\smthker\\|_\\infty\n\\]\n\\end{theorem}\nAs indicated, regularity assumptions on $\\hf,\\hg$ can lead to stronger results. \nIn this direction we have the following.\n\\begin{theorem}[$C^2$ Discrete Fixed Point Bound]\\label{thm:discreteFPsum}\nAssume $\\hf$ and $\\hg$ are $C^2$ and there exists an $(0,1)$-interpolating\nspatial fixed point for the spatially discrete system. Then\n\\[\n|A(\\hf,\\hg) | \\le \n\\frac{1}{2} (\\|\\hf''\\|_\\infty+\\|\\hg''\\|_\\infty)\\|\\smthker\\|_\\infty^2{\\Delta^2}\n\\]\n\\end{theorem}\nProofs for the above are presented in appendix \\ref{app:Aa}.\n\nFor discrete systems where gap conditions may be difficult to verify we may require more general\nresults.\nEspecially challenging are cases with an infinite number of crossing points\nclustering near the extremal ones. For such generic situations we have\nthe following spatially discrete version of Theorem \\ref{thm:globalconv}.\n\\begin{theorem}[General Discrete Convergence]\\label{thm:discreteglobalconv}\nLet $(\\hf,\\hg)$ be given as in Theorem \\ref{thm:globalconv}, let $\\smthker$ be regular,\nand assume $\\ff^0 \\in \\sptfns$ is given\nwith $\\ff^0(\\minfty)\\le v''$ and \n$\\ff^0(\\pinfty)\\ge v'.$\nThen, for any $\\epsilon>0,$ in system \\ref{eqn:gfrecursion} \nwith $\\Delta$ sufficiently small we have for all $x\\in \\reals$\n\\begin{align*}\n\\liminf_{t\\rightarrow \\infty} f^t(x) &\\ge v'-\\epsilon \\quad \\liminf_{t\\rightarrow \\infty} g^t(x) \\ge u'-\\epsilon \\\\\n\\limsup_{t\\rightarrow \\infty} f^t(x) &\\le v''+\\epsilon \\quad \\limsup_{t\\rightarrow \\infty} g^t(x) \\le u''+\\epsilon \\,.\n\\end{align*}\n\\end{theorem}\nThe proof may be found in appendix \\ref{app:E}.\n\n\nFinite length systems can be modeled by introducing spatial dependence\ninto the definition $\\hf$ and\/or $\\hg.$ For example, in the LDPC-BEC case termination corresponds\nto setting $\\hf = 0$ outside some finite region.\nThe analysis most closely follows the unterminated case\nwith one-sided termination, e.g., setting $\\hf=0$ for $x<0.$\nWhen $A(\\hf,\\hg)>0$ and the strictly positive gap condition holds we can apply Theorem \\ref{thm:mainexist}\nto conclude that the infinite length unterminated system has a wave-like solution\nthat converges point-wise to $0.$\nSuch a solution can often be used to bound from above the\nsolutions for terminated cases to show that\ntheir solutions also tend to $0.$\nAlternatively, we can apply Theorem \\ref{thm:globalconv} to conclude that even if we remove the termination\nafter initialization the system will converge to $0.$ \n\n\\subsection{One-sided Termination}\nSince setting $\\hf=0$ over some region reduces $f$ relative to the\nunterminated case, it is more difficult to obtain lower bounds for\nthe terminated case. It turns out for one-sided termination,\nhowever, that an analogy can be drawn between the spatial variation\nin $\\hf$ and a global perturbation in $\\hf$ that is spatially invariant\nand which then allows application of Theorem \\ref{thm:mainexist}.\n\nLet us formally define the one-sided termination version of\n\\eqref{eqn:gfrecursion} to be the system that follows\n\\eqref{eqn:gfrecursion}\nexcept that when $x<0$ we set $f^t(u)=0$ for all $u\\in [0,1].$\nThis is equivalent to redefining $\\hf = \\hf(x;u)$ to have spatial dependence so that\nwhen $x<0$ we have $\\hf(x;u)=0$ and for $x \\ge 0$ we have $\\hf(x;u) =\\hf(u)$ as before.\n\nSince this system is not translation invariant,\nit does not admit interpolating traveling wave-like solutions. It does, however,\nadmit interpolating spatial fixed points.\n\nLet us define\n\\[\n\\unitstep_a (x) = \\begin{cases}\n0 & x <0 \\\\\na & x = 0 \\\\\n1 & x > 0\n\\end{cases}\n\\]\nIn some cases the value of $a$ is immaterial and we may drop the subscript from the \nnotation. \n\n\\begin{theorem}[Continuum Terminated Fixed Point]\\label{thm:terminatedexist}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg)\\in \\exitfns^2$ and assume that $\\hg$ is continuous at $0$ and that\n$\\altPhi(\\hf,\\hg;u,v)$ is uniquely minimized at $(1,1)$\n(hence $A(\\hf,\\hg) < 0$ but we do not assume that the strictly positive gap condition holds).\nThen there exists $(0,1)$-interpolating $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a fixed point of the one-sided termination of\n\\eqref{eqn:gfrecursion}.\n\\end{theorem}\n\\begin{IEEEproof}\nDefine ${\\hf}(z;u) = \\hf(u)\\wedge \\unitstep(u-z)$ (where $a \\wedge b = \\min \\{a,b\\}$) and choose $z$ so that\n$A(\\hf(z;\\cdot),\\hg) =0.$\nIf $(u,v) \\in \\intcross(\\hf(z;\\cdot),\\hg)$ then, since $\\hg$ is continuous at $0,$ we have\n$u \\ge z.$ \nIf $v < \\hf(z+)$ then we have\n$\\altPhi(\\hf(z;\\cdot),\\hg;u,v) = \\int_0^{z}\\hginv(u') du' > 0$\nand if $v \\ge \\hf(z+)$ then we have\n$\\altPhi(\\hf(z;\\cdot),\\hg;u,v) = \\altPhi(\\hf,\\hg;u,v) - A(\\hf,\\hg) > 0.$\nIt follows that $(\\hf(z;\\cdot),\\hg)$ satisfies the strictly positive gap condition.\nBy Theorem \\ref{thm:mainexist} there exists $\\tmplF,\\tmplG \\in \\Psi_{[-\\infty,\\infty]}$\nthat form a $(0,1)$-interpolating spatial fixed point ($\\ashift=0$) for\n\\eqref{eqn:gfrecursion} with $\\hf(z;\\cdot)$ replacing $\\hf.$\nIt is easy to see that there is some finite maximal $y$ such that\n$\\tmplF(x)=0$ for $x < y.$ Translate $\\tmplF$ and $\\tmplG$ so that $y=0.$\nIt follows that the resulting $\\tmplF,\\tmplG$ pair is a fixed point of the one-sided termination version of\n\\eqref{eqn:gfrecursion}.\n\\end{IEEEproof}\nIt is interesting to note in the above construction that the fixed point solution\nhas $\\tmplG^{\\smthker}(0)=z$ and $\\tmplF(0+) = \\hf(z+).$ Hence the value of \nthe discontinuity at the boundary of the termination is determined by the condition\n$A=0.$\nIn the case where $\\hg$ is not continuous at $0,$ i.e., $\\hg(0+)>0$ we can construct a fixed point solution\nas above with $\\tmplG(\\minfty) =\\hg(0+).$\n\nFor the case $A(\\hf,\\hg) \\ge 0$ we have the following.\n\\begin{theorem}[Continuum Terminated Convergence]\\label{thm:terminatedzero}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg) \\in \\exitfns^2$ and assume that\n$\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\neq (0,0).$\nThen $\\ff^t \\rightarrow 0$ for the\none-sided termination of\n\\eqref{eqn:gfrecursion}\nfor any choice of $\\ff^0.$\nIf $\\hf(x) > 0$ and $\\hg(x)>0$ on $(0,1]$ then\n$\\ff^t \\rightarrow 0$ also when $\\altPhi(\\hf,\\hg;) \\ge 0.$\n\\end{theorem}\nThe proof is presented in Appendix \\ref{app:E}.\n\nWe can, of course, also terminate the spatially\ndiscrete versions of the system.\nThus, consider the one sided termination of\n\\eqref{eqn:discretegfrecursion}\nin which the equations are modified so that\nwe set $f^t(x_i)=0$ if $x_i < 0,$\nwhich is equivalent to redefining $\\hf$ to have spatial dependence so that\n$\\hf = 0$ if $x_i < 0.$\nWe assume that $\\discsmthker$ is related to $\\smthker$ (for a continuum version)\nas indicated in \\eqref{eqn:kerdiscretetosmth}.\nFor this case we have the following quantitative result.\n\\begin{theorem}[Discrete Fixed Point Positive Gap]\\label{thm:discreteterminatedexist}\nAssume $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) < -\\Delta\\|\\smthker\\|_\\infty.$\nThen there exists $(0,1)$ interpolating $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a spatial fixed point of the one-sided termination of\n\\eqref{eqn:discretegfrecursion}.\n\\end{theorem}\n\\begin{IEEEproof}\nDefine $\\hf(z;u) = \\hf(u) \\wedge \\unitstep_1(u-z)$\n with $z>0$ chosen sufficiently small\nso that $A(\\hf(z;\\cdot),\\hg) \\le -\\Delta\\|\\smthker\\|_\\infty.$\nBy Theorem \\ref{thm:mainexist}\nthere exists $\\tmplF,\\tmplG \\in \\Psi_{[-\\infty,\\infty]}$\nthat form a spatial wave solution for\n\\eqref{eqn:gfrecursion} with $\\hf(z;\\cdot)$ replacing $\\hf$\nand $\\ashift \\le -\\Delta.$\nBy Theorem \\ref{thm:mainquantize} we see that by setting\n$\\ff^0(x_i) = {\\tmplF}(x_i)$ in\n\\eqref{eqn:discretegfrecursion} (the non-terminated case)\nwe have $\\ff^1(x_i) \\ge {\\tmplF}(x_i).$\nBy translation, we can assume that ${\\tmplF}(x_i)=0$ for $x_i<0.$\nNow, the inequality $\\ff^1(x_i) \\ge {\\tmplF}(x_i)$ also holds in the one sided terminated case\nsince the values of $\\ff^1(x_i)$ are unchanged for $x_i\\ge 0.$\nThus, in the one-sided termination case the sequence $\\ff^t$ is monotonically non-decreasing\nfor each $x_i$ and must therefore have a limit $\\ff^{\\infty}.$\nIf $\\hf$ and $\\hg$ are continuous then the pair $\\ff^{\\infty},\\fg^{\\infty}$ then constitute a fixed point of the one-sided termination case.\nIf $\\hf$ and $\\hg$ are not continuous then it is possible that the pair $\\ff^{\\infty},\\fg^{\\infty}$ does not constitute a fixed point and that initializing with\n$\\ff^{\\infty}$ we obtain another non-decreasing sequence.\nIn general we can use transfinite recursion together with monotonicity in $x$\nto conclude the existence of\n a fixed point at least as large point-wise as $ (\\ff^{\\infty},\\fg^{\\infty}).$ \n\\end{IEEEproof}\n\nThe previous result gives quantitative information on the discrete approximation but it requires the strictly positive gap assumption.\nThe following result, whose proof is in Appendix \\ref{app:E},\nremoves that requirement at the cost of the quantitative bound.\n\n\\begin{theorem}[Discrete Fixed Point General]\\label{thm:discreteterminatedexistB}\nAssume $\\smthker$ is regular.\nAssume that\n$\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is uniquely minimized at $(1,1)$\nwith $A(\\hf,\\hg) < 0.$ Then\nfor all $\\Delta$ sufficiently small\nthere exists $\\tmplF,\\tmplG \\in \\sptfns$\nthat form a spatial fixed point of the one-sided termination of\n\\eqref{eqn:discretegfrecursion} with $\\lim_{\\Delta\\rightarrow 0}\\tmplF(\\pinfty) =1.$\n\\end{theorem}\n\n\nFor the case $A(\\hf,\\hg) \\ge 0$ we have the following quantitative result.\n\\begin{theorem}[Discrete Terminated Convergence]\\label{thm:discreteterminatedzero}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) > \\Delta\\|\\smthker\\|_\\infty.$\nThen $\\ff^t \\rightarrow 0$ for the\none-sided termination of\n\\eqref{eqn:discretegfrecursion}\nfor any choice of $\\ff^0.$\n\\end{theorem}\n\\begin{IEEEproof}\nWe consider the initialization\n\\[\n\\ff^0(x_i)=\\unitstep_1(x_i)\\,\n\\] \nand show that $\\ff^t \\rightarrow 0.$\nDefine\n\\[\n\\hf(\\eta;u) =\\hf(u) \\vee \\unitstep_1(u-(1-\\eta))\n\\]\nwhere we assume $\\eta>0$ sufficiently small so that\n$A(\\hf(\\eta;\\cdot),\\hg) > \\Delta\\|\\smthker\\|_\\infty.$\n\nBy Theorem \\ref{thm:mainquantize} there exists $(0,1)$-interpolating $\\tmplF$ and\n$\\tmplG$ and $\\ashift>\\Delta$ such that, even for the unterminated case,\n$\\ff^0(x_i) \\le\\tmplF(x_i)$ implies $\\ff^t(x_i) \\le\\tmplF(x_i-(\\ashift-\\Delta)t)$\nfor all $t.$ The same clearly holds also in the terminated case.\nClearly, $\\tmplF(x)=1$ for all $x \\ge z$ for some finite $z,$ and by translation\nwe can take $z$ to be $0$ yielding the desired result.\n\\end{IEEEproof}\n\n\n\\subsection{Two-sided Termination}\nThe two-sided termination of system \\eqref{eqn:gfrecursion}\nis defined by setting $\\ff^t(x) = 0$ for\nall $x$ outside some finite region, say $[0,Z]$ for all $t.$\nThis can be understood as a spatial dependence of $\\hf =\\hf(x;u)$\nwhere $\\hf(x;u) =0$ for $x \\not\\in [0,Z]$ and $\\hf(x;u) =\\hf(u)$ as before otherwise.\nThis system can be bounded from above by the one-sided termination case.\nThus, Theorem \\ref{thm:terminatedzero}\nand Theorem \\ref{thm:discreteterminatedzero}\napply equally well to the two-sided terminated case.\nTheorem \\ref{thm:discreteterminatedexist} on the other hand does not\nimmediately generalize, but a similar statement holds.\n\n\\begin{theorem}[Two Sided Continuum Fixed Point]\\label{thm:twoterminatedexist}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and let\n$A(\\hf,\\hg) < 0.$\nThen, for any $\\epsilon > 0,$ and for all $Z$ sufficiently large,\nthere exists $\\ff,\\fg$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:gfrecursion} such that\n$\\ff$ and $\\fg$ are symmetric about $\\frac{Z}{2},$ monotonically non-decreasing on $(-\\infty,\\frac{Z}{2}]$ and have left and right limits at least\n$1-\\epsilon$ at $\\frac{Z}{2}.$\n\\end{theorem}\nThe proof is presented in appendix \\ref{app:C}.\n\nWe have also the following spatially discrete version of the above,\nwhose proof is also in appendix \\ref{app:C}.\nIn the discrete case the termination is taken to hold for $x_i < 0$ and\n$x_i > Z = L\\Delta$ where $L$ is an integer.\nSymmetry in the spatial dimension then takes the form\n$\\ff(x_i) = \\ff(x_{L-i}).$ \n\\begin{theorem}[Two Sided Discrete Fixed Point with Gap]\\label{thm:discretetwoterminatedexist}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition and assume that\n$A(\\hf,\\hg) < -\\Delta\\|\\smthker\\|_\\infty.$\nThen, for any $\\epsilon > 0,$ and for all $Z$ sufficiently large,\nthere exists $\\tmplF,\\tmplG$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:discretegfrecursion}\nsuch that\n$\\tmplF$ and $\\tmplG$ are spatially symmetric, monotonically non-decreasing on $(-\\infty,\\half Z]$ and satisfy $\\max_i \\{\\tmplF(x_i)\\} \\ge 1-\\epsilon$\nand $\\max \\{\\tmplG(x_i)\\} \\ge 1-\\epsilon.$ \n\\end{theorem}\n\n\nWe have also the following qualitative version that relaxes the strictly positive gap condition\nand\nwhose proof is in appendix \\ref{app:E}.\n\\begin{theorem}[Two Sided Discrete Fixed Point]\\label{thm:discretetwoterminatedexistGB}\nAssume that $\\smthker$ is regular.\nLet $(\\hf,\\hg)$ be given such that $\\altPhi(\\hf,\\hg;\\cdot,\\cdot)$ is\nuniquely minimized at $(1,1)$ and therefore \n$A(\\hf,\\hg) < 0.$\nThen, for any $\\epsilon > 0,$ and for all $Z=L\\Delta$ sufficiently large and $\\Delta$ sufficiently small,\nthere exists $\\tmplF,\\tmplG$\nthat form a fixed point of the two-sided termination of\n\\eqref{eqn:discretegfrecursion}\nsuch that\n$\\tmplF$ and $\\tmplG$ are spatially symmetric, monotonically non-decreasing on $(-\\infty,\\half Z)$ and satisfy $\\max_i \\{\\tmplF(x_i)\\} \\ge 1-\\epsilon$\nand $\\max_i \\{\\tmplG(x_i)\\} \\ge 1-\\epsilon.$ \n\\end{theorem}\n\n\n\n\n\n\\section{Examples of 1-D Systems}\\label{sec:applications}\n\n\\subsection{Binary Erasure Channel}\nLet us start by re-deriving a proof that for transmission over the\nBEC regular spatially-coupled ensembles achieve the MAP threshold\nof the underlying ensemble. By keeping the rate fixed and by\nincreasing the degrees it then follows that one can achieve capacity\nthis way. This was first shown in \\cite{KRU10}. Given the current\nframework, this can be accomplished in a few lines. Before we prove this\nlet us see a few more examples.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/capacitybec}\n}\n\\caption{\\label{fig:capacitybec}\nEXIT charts for the $(4, 8)$-regular (left) and the $(5, 10)$-regular (right)\ndegree distributions and transmission over the BEC. The respective coupled thresholds\nare\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(4, 8)=0.497741$, and\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(5, 10)=0.499486$.\n}\n\\end{figure}\nWe have already seen the corresponding EXIT charts for the $(3,\n6)$-regular case in Figure~\\ref{fig:exitbec36}. Figure~\\ref{fig:capacitybec}\nshows two more examples, namely the $(4, 8)$-regular as well as the\n$(5, 10)$-regular case. Numerically, the thresholds are\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(3, 6)=0.48814$,\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(4, 8)=0.497741$, and\n$\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}}(5, 10)=0.499486$. As\nwe see these thresholds quickly approach the Shannon limit of\none-half.\n\nConsider now a degree distribution pair $(\\lambda, \\rho)$. The BP\nthreshold of the uncoupled system is determined by the maximum\nchannel parameter $\\epsilon$ so that $\\epsilon \\lambda(x) \\leq\n1-\\rho^{-1}(1-x)$ for all $x \\in (0, 1]$. Therefore, dividing both\nsides by $\\lambda(x)$ we get for each $x \\in (0, 1]$ an upper bound\non the BP threshold. In other words, the BP threshold of the uncoupled\nensemble can be characterized as\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny uncoupled}} =\n\\inf_{x\\in(0,1]}\\frac{1-\\rho^{-1}(1-x)}{\\lambda(x)}\\,.\n\\end{align*}\nThe limiting spatially coupled threshold (when $L$ and $w$ tend to infinity)\ncan be characterized in a similar way. In this case the determining quantity is\nthe area enclosed by the curves. Therefore,\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}} =\n\\inf_{x\\in(0,1]}\\frac{\\int_0^x 1-\\rho^{-1}(1-u)\\,\\text{d}u}{\\int_0^x \\lambda(u)\\text{d}u}\\,.\n\\end{align*}\nIn the case where the BP threshold equals $\\frac{1}{\\lambda'(0)\\rho'(1)},$\ni.e., when the threshold equals the stability threshold, then the\nspatially coupled threshold equals the BP threshold.\n\nIn the regular case and in many other cases\n\\begin{align*}\n\\epsilon^{\\BPsmall}_{\\text{\\tiny coupled}} =\n\\frac{\\int_0^{x^*} 1-\\rho^{-1}(1-u)\\,\\text{d}u}{\\int_0^{x^*} \\lambda(u)\\text{d}u}\\,\n\\end{align*}\nwhere $x^*$ corresponds to the forward BP fixed point with channel\nparameter $\\epsilon^{\\small}_{\\text{\\tiny coupled}}.$ In this case\none can check that the threshold is exactly equal to the area\nthreshold. Further, we already know that the area threshold is an\nupper bound on the MAP threshold of the underlying ensemble and we\nknow that the MAP threshold of the underlying system is equal to\nthe MAP threshold of the coupled system when $L$ tends to infinity.\nWe therefore conclude that for all such underlying ensembles where\nthe area threshold satisfies the strictly positive gap condition, the area\nthreshold equals the MAP threshold.\n\nOur current framework can also be adapted to more complicated\ncases. The following example is from \\cite[Fig. 4.15]{Mea06}. Consider the\ndegree distribution $(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20},\n\\rho(x)=x^{15})$. The left picture in Figure~\\ref{fig:complicated}\nshows the BP EXIT curve of the whole code.\n\\begin{figure}[htp]\n\\centering\n\\input{ps\/complicated}\n\\caption{\\label{fig:complicated} BP EXIT curves for the ensemble\n$(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20}, \\rho(x)=x^{15})$ and\ntransmission over the BEC. Left: Determination of the BP threshold.\nRight: Determination of MAP behavior as conjectured by the Maxwell construction.}\n\\end{figure}\nAs one can see, the BP threshold of the uncoupled ensemble in this\ncase is $\\epsilon^{\\BPsmall}_{\\text{\\tiny uncoup.}} = 0.3531$ and\nthe BP EXIT curve has a single jump.\n\nThe right picture shows the MAP EXIT curve according to the Maxwell\nconstruction, see \\cite[Section 3.20]{RiU08}. According to this\nconstruction, the MAP EXIT curve has two jumps, namely at\n$\\epsilon=0.403174$, the conjectured MAP threshold, and at\n$\\epsilon=0.4855$. These two thresholds are determined by local\nbalances of areas. This is in particular easy to see for the threshold\nat $\\epsilon=0.4855$, where the two areas are quite large.\n\nLet us now show that for the coupled ensemble the Maxwell conjecture\nis indeed correct, i.e., we show that the asymptotic (in the coupling\nlength $L$) BP EXIT curve for the spatially-coupled ensemble indeed\nlooks as shown in the right-hand side of Figure~\\ref{fig:complicated}.\nTo show that the Maxwell conjecture is also correct for the uncoupled\nsystem requires a second step which we do not address here. This\nsecond step consists in showing that the MAP behavior of the uncoupled\nand coupled system is identical and is typical accomplished by using\nthe so called ``interpolation'' technique.\n\n\nThe left picture in Figure~\\ref{fig:complicatedgap} shows the\nindividual EXIT curves according to our framework for $\\epsilon=0.4855$.\nFor this channel parameter the two EXIT curves cross four times,\nnamely for $u=0$, $u=0.824784$, $u=0.967733$, and $u=0.999952$.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/complicatedgap}\n}\n\\caption{\\label{fig:complicatedgap}\nConfirmation of the Maxwell conjecture using the one-dimensional framework of\nspatial coupling for the ensemble $(\\lambda(x)=\\frac{3 x+3 x^2+14 x^{50}}{20}, \\rho(x)=x^{15})$\nand transmission over the BEC.\nThe two inlets show in a magnified way the behavior of the curves\ninside the two gray boxes.\n}\n\\end{figure}\nNote that for this channel parameter the curves do not fulfill the\npositive gap condition since initially the curve $\\epsilon \\lambda(\\xg)$\nis above the curve $1-\\rho(1-\\xf)$. Nevertheless we can use our\nformalism. Let us explain the idea informally. Let us first check\nthe behavior of the system for $\\epsilon=0.4855$. Let us shift both\ncurves and renormalize them in such a way that first (from the left)\nnon-trivial FP is mapped to zero and the last FP (on the right) is\nmapped to one. Then these curves {\\em do} fulfill the our conditions\nand our theory applies. This shows that once the channel parameter\nhas reached slightly below $0.4855$, the EXIT function drops as\nindicated in the righ-hand side of Figure~\\ref{fig:complicated}.\n\nNow where we know what the curve looks like above $\\epsilon=0.4855$\nwe can look at the remaining part. The right picture in\nFigure~\\ref{fig:complicatedgap} shows the individual EXIT curves\naccording to our framework for $\\epsilon=0.4032$. Again, we can\nredefine our curves above this parameter and reparametrize and then\nthey do fulfill the positive gap condition. So this marks the second\nthreshold. The inlet shows the curve magnified by 1.5 and 15 respectively.\nFrom this we see that the curves are quite well matched, so the areas\nare not so easy to see.\n\n\n\\subsection{Hard-Decision Decoding}\nLow-dimensional descriptions appear naturally when we investigate\nthe performance of quantized decoders. The perhaps simplest case\nis the Gallager decoder A, \\cite{Gal63} (see \\cite{RiU01} for an\nin-depth discussion). All messages in this case are from $\\{\\pm\n1\\}$. The initial message sent out by the variable nodes is the\nreceived message. At a check node, the outgoing message is the\nproduct of the incoming messages. At variable nodes, the outgoing\nmessage is the received message unless all incoming messages agree,\nin which case we forward this incoming message.\n\nLet $x^{(\\ell)}$, $\\ell \\in \\naturals$, be the state of the decoder,\nnamely the fraction of ``$-1$\"-messages sent out by the variable\nnodes in iteration $\\ell$. We have $x^{(0)} = \\epsilon$, and for\n$\\ell \\geq 1$, the DE equations read\n\\begin{align*}\ny^{(\\ell)} & = \\frac{1-\\rho(1-2 x^{(\\ell-1)})}{2}, \\\\\nx^{(\\ell)} & = \\epsilon (1- \\lambda(1-y^{(\\ell)}))+ (1-\\epsilon) \\lambda(y^{(\\ell)}).\n\\end{align*}\nSince the state of this system is a scalar, our theory can be applied\ndirectly. Unfortunately, as discussed in \\cite{BRU04}, for most\n(good) degree-distributions the threshold under the Gallager A\nalgorithm is determined by the behavior either at the very beginning\nof the decoding process or at the very end. In neither of those\ncases does spatial coupling improve the threshold.\n\nIn more detail, consider Figure~\\ref{fig:exitgalA}.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitgalA}\n}\n\\caption{\\label{fig:exitgalA}\nLeft: EXIT charts for the $(4, 8)$-regular degree distribution\nunder the Gallager algorithm A with $\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} = 0.0476$. The curves\ndo not cross. The threshold is determined by the stability condition. Right:\nEXIT charts for the $(3, 6)$-regular degree distribution\nunder the Gallager algorithm A with $\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} = 0.0395$. The\nthreshold is determined by the behavior at the start of the algorithm.\n}\n\\end{figure}\nThe left picture shows the two EXIT functions for the $(4, 8)$-regular\nensemble under the Gallager algorithm A and\n$\\epsilon^{\\GalAsmall}_{\\text{\\tiny uncoup}} = 0.0476$. As one can\nsee from this picture, this is the threshold for the uncoupled case.\nThis threshold is determined by the stability condition, i.e., the\nbehavior of the decoder towards the end of the decoding process.\nIn other words, the functions $\\hg(\\xf)$ and the inverse of $\\hf(\\xg)$\nhave the same derivative at $0$. If we increase the channel parameter\nthen the resulting EXIT curves no longer fulfill the positive gap\ncondition (since they cross already at $0$). This implies that the\nthreshold of the spatially coupled ensemble is the same as for the\nuncoupled one.\n\nThe right picture in Figure~\\ref{fig:exitgalA} shows the two EXIT\nfunctions for the $(3, 6)$-regular ensemble under the Gallager\nalgorithm A and $\\epsilon = 0.0395$, the threshold for the uncoupled\ncase. In this case the threshold is determined by the behavior at\nthe beginning of the decoding process. As one can see from the\npicture, there are two non-zero FPs. The ``smaller'' one is unstable\nand the ``larger'' one is stable. If the initial state of the system\nis below the small FP then the decoder converges to $0$, i.e., it\nsucceeds. But if it starts above the small FP, then the decoder\nconverges to the large and stable non-zero FP, i.e., it fails. As\none can see from the picture, already for the channel parameter\nwhich corresponds to the threshold of the uncoupled these two EXIT\ncurves do not fulfill the positive gap condition -- the total area\nenclosed by the two curves is negative. And if we increase the\nchannel parameter, the area would become even more negative. Hence,\nalso in this case spatial coupling does not help.\n\nLet us therefore consider the Gallager algorithm B, \\cite{Gal63,RiU01}.\nAs for the Gallager algorithm A, all messages are from the set\n$\\{\\pm 1\\}$. The initial message and the message-passing rule at\nthe check nodes are identical. But at variable nodes we have a\nparameter $b$, an integer. If at least $b$ of the incoming messages\nagree, then we send this value, otherwise we send the received\nvalue. This threshold $b$ can be a function of time. Initially the\ninternal messages are quite unreliable. Therefore, $b$ should be\nchosen large in this stage (if we choose $b$ to be the degree of\nthe node minus one we recover the Gallager algorithm A). But as\ntime goes on, the internal messages become more and more reliable\nand a simple majority of the internal nodes will be appropriate.\nThe DE equations for this case are\n\\begin{align*}\ny^{(\\ell)} & = \\frac{1-\\rho(1-2 x^{(\\ell-1)})}{2}, \\\\\nx^{(\\ell)} = & (1-\\epsilon) \\sum_{k=b}^{\\dl-1}\n\\binom{\\dl-1}{k} (y^{(\\ell)})^k (1-y^{(\\ell)})^{\\dl-1-k}\\\\\n& + \\epsilon \\sum_{\\dl-1-b}^{\\dl-1}\n\\binom{\\dl-1}{k} (y^{(\\ell)})^k (1-y^{(\\ell)})^{\\dl-1-k}.\n\\end{align*}\nAssume at first that we keep $b$ constant over time. Consider\nthe $(4, 10)$-regular ensemble and choose $b=3$.\nThe left picture in Figure~\\ref{fig:exitgalB} shows this example for\n$\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}} = 0.02454$. As we can see, this is the largest\nchannel parameter for which the two curves do not cross, i.e., this is\nthe threshold for the uncoupled case.\nThe right picture in Figure~\\ref{fig:exitgalB} shows the same example\nbut for $\\epsilon^{\\GalBsmall}_{\\tiny \\text{coup}} = 0.0333$. For\nthis channel parameter the strictly positive gap condition is fulfilled and\nthe two areas are exactly in balance, i.e., this is the threshold\nfor the coupled ensemble.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/exitgalB}\n}\n\\caption{\\label{fig:exitgalB}\nLeft: EXIT charts for the\nthe $(4, 10)$-regular ensemble and the Gallager algorithm B with $b=3$\nand $\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}} = 0.02454$. The curves do not cross.\nRight: The same example but with\n$\\epsilon^{\\GalBsmall}_{\\tiny \\text{coup}} = 0.0333$. For\nthis channel parameter the positive gap condition is fulfilled and\nthe two areas are in balance. In both cases, the inlets show a\nmagnified version of the gray box.}\n\\end{figure}\nWe see that the increase in the threshold is substantial for this\ncase.\n\n\nWe can do even better if we allow $b$ to vary as a function of the state of the system. The optimum\nchoice of $b$ as a function of the state $x$ was already determined\nby Gallager and we have\n\\[\nb(\\epsilon, x) = \\Big\\lceil \\Bigl(\\frac{\\log \\frac{1-\\epsilon}{\\epsilon}}{\\log \\frac{1-x}{x}} + (\\dr-1) \\Bigr)\/2 \\Big\\rceil.\n\\]\nAssume that at any point we pick the optimum $b$ value. For the\nEXIT charts this corresponds to looking at the minimum of the EXIT\nchart at the variable node over all admissible values of $b$. If\nwe apply this to the $(4, 10)$-regular ensemble then we get a\nthreshold of $\\epsilon^{\\GalBsmall, \\text{\\tiny opt}}_{\\tiny\n\\text{coup}}(4, 8) = 0.04085$, another marked improvement. As a\nsecond example, consider the $(6, 12)$-regular ensemble. For this\nensemble no fixed-$b$ decoding strategy improves the threshold under\nspatial coupling compared to the uncoupled case. But if we admit\nan optimization over $b$ then we get a substantially improved\nthreshold, namely the threshold is now $\\epsilon^{\\GalBsmall,\\text{\\tiny\nopt}}_{\\tiny \\text{\\tiny coup}}(6, 12) = 0.0555$. For comparison,\n$\\epsilon^{\\GalBsmall}_{\\text{\\tiny uncoup}}(6, 12) = 0.0341$.\n\n{\\em Discussion:} The optimum strategy assumes that at the\ndecoder we know at each iteration (at at each position if we consider\nspatially coupled ensembles) the current state of the system. Whether\nor not this is realistic depends somewhat on the circumstances. For\nvery large codes the evolution of the state is well predicted by\nDE and can hence be determined once and for all. For smaller systems\nthe evolution shows more variation. One option is to measure e.g.\nthe number unsatisfied check nodes given the current decisions and\nto estimate from this the state.\n\n\n\n\n\n\\subsection{CDMA Demodulation}\n\nSpatially coupling has been considered for CDMA demodulation in\n \\cite{ScT11} and \\cite{TTK11}.\nWe will follow \\cite{ScT11}.\n\nThe basic (uncoded) CDMA transmission model is\n\\[\ny=\\sum_{k=1}^Kd_k \\bold{a}_k + \\sigma \\bold{n}\n\\]\nwhere there are $K = \\alpha N$ users each transmitting a single bit $d_k = \\pm 1$ \nusing random spreading sequence $\\bold{a}_k$ of unit energy and length $N$,\nand $\\bold{n}$ is a vector of independent $N(0,1)$ random variables\n(for details see \\cite{ScT11}).\n\nIn \\cite{Tan2002} statistical mechanical methods were used to analyze randomly spread synchronous CDMA detectors over the additive white Gaussian noise channel.\nThe non-rigorous replica method predicted the optimal (asymptotic in system size) performance of various detectors.\nIn this setting the solution states that\nthe symbol-wise marginal-posterior-mode detector in the large $K$ and $N$ limit has posterior probabilities with signal to interference ratio $(1\/x)$ satisfying the equation\n\\begin{equation}\\label{eqn:cdmaFP}\nx= \n\\sigma^2 + \\alpha \\expectation \\Biggl(1-\\tanh\\Bigl(\\frac{1}{x}+\\sqrt{\\frac{1}{x}} \\xi\\Bigr) \\Biggr)^2 \n\\end{equation}\nwhere the expectation is over $\\xi \\sim N(0,1).$\nHere $x$ represents the variance of the posterior equivalent Gaussian channel $d_k + \\sqrt{x} n.$\n\nFor $\\alpha < \\alpha_{\\text{crit}} \\simeq 1.49$ (numerically determined) this\nequation has single solution\n(including the case $x=0$ for $\\sigma^2=0.$)\nFor $\\alpha \\ge \\alpha_{\\text{crit}}$ it is observed\nthat the equation has one, two, or three solutions depending \non $\\sigma^2.$\n\nIn \\cite{ScT11}, a message passing scheme was developed such that the associated density evolution gives rise to\n\\eqref{eqn:cdmaFP} as a fixed point equation.\nThe scheme requires a modification of the transmission setup.\nTo describe the scheme first consider repeating each bit $M$ times and scaling power accordingly.\n\\[\ny=\\sum_{k=1}^K \\frac{1}{M} \\sum_{m=1}^M d_{k,m} \\boldmath{a}_k + \\sigma \\boldmath{n}\n\\]\nNow, take $l=1,2,...,L$ instances of this system and permute the indices on a per-user basis to get\n\\[\ny_l=\\sum_{k=1}^K \\frac{1}{\\sqrt{M}} \\sum_{m=1}^M d_{k\\pi_k(m,l)} \\boldmath{a}_k + \\sigma \\boldmath{n}\n\\]\nwhere $\\pi_k$ is a (randomizing) permutation on $[M]\\times [L].$ Note the change in scaling with respect to $M$ due to\nnon-coherent addition of the bit values (take $L\\gg M$). Belief propagation is applied to this setup and the analysis\nleads to the density evolution equations. \n\nDefine \n$g:[0,\\infty] \\rightarrow [0,\\infty].$\n\\begin{align*}\ng(x) & = \\expectation{(1-\\tanh{(x+\\sqrt{x}\\xi))}^2} \n\\end{align*}\nwhere $\\xi \\sim N(0,1).$\nNow, further define\n\\begin{align*}\n\\hf(u)&=\\alpha g(u)+\\sigma^2\\\\\n\\hg(v)&=1\/v\n\\end{align*}\nwhere, here, $u,v \\in [0,+\\infty].$\nThe fixed point equation \\eqref{eqn:cdmaFP} can now be written\n\\[\nx = \\hf(\\hg(x))\\,.\n\\]\nThe function $\\hf$\ncorresponds to updating the LLRs of the bits taking into account the repetition of the bits and the function\n$\\hg$ corresponds to a soft cancellation step. In each case the resulting message LLR values are (symmetric) \nGaussian distributed and the density evolution update corresponds to the input-output map of the effective\nvariances of the equivalent AWGN channel. The iterations can be initialized with $x = \\infty.$\n\nThe DE corresponding to the message passing decoder will converge to the solution of\n\\eqref{eqn:cdmaFP} having the largest magnitude.\nHence for $\\alpha \\le \\alpha_{\\text{crit}}$ the BP decoder will not generally achieve optimal performance.\n\nIn \\cite{ScT11} the authors further introduce spatial coupling. The basic construction uses a chain of\ninstances of the above system and couples them by exchanging bits between neighboring instances\n\nThe spatially coupled version of \\eqref{eqn:cdmaFP} (corresponding to local uniform coupling of width $W$) \nappearing in \\cite{ScT11} reads\n\\begin{align*}\nx_i^t & = \\sigma^2 + \\frac{\\alpha}{2W+1}\\sum_{j=-W}^W g\\Bigl( \\frac{1}{2W+1} \\sum_{l=-W}^W \\frac{1}{x_{i-1}^{t+j+l}}\\Bigr)\\, \\\\\n&= \\frac{1}{2W+1}\\sum_{j=-W}^W \\Biggl(\\sigma^2 +\\alpha g\\Bigl( \\frac{1}{2W+1} \\sum_{l=-W}^W \\frac{1}{x_{i-1}^{t+j+l}}\\Bigr)\\Biggr)\\,.\n\\end{align*}\nTermination is accomplished by setting bits outside some finite region to be known.\n\nWe are now in the regime where our results may be applied.\nThe solutions to \\eqref{eqn:cdmaFP} correspond to crossing points of $\\hf,\\hg.$\nIf $(u,v) \\in \\cross(\\hf,\\hg)$ then the corresponding solution to \\eqref{eqn:cdmaFP}\nis $x = v = 1\/u.$\nLet $x_1$ be the smallest solution and when there are multple solutions let $x_2$ be the largest.\nThe system is initialized with $x=u=\\infty$ and terminated with $x=0.$\nFor definition of $\\altPhi$ consistent with our canonical form we can take $(u,v)=(x_1,1\/x_1)$ as the origin and invert the sign of $v.$.\nThus, we see that for $W$ large enough the spatially coupled\nsystem will converge to the solution $x_2$ (or better near the termination) if \n\\[\n\\int_{x_1}^{x_2} \\Bigl( \\hf(z) - \\frac{1}{z} \\Bigr) \\, dz < 0\\,.\n\\]\n\nThe case $\\sigma^2=0$ ($x_2 = \\infty$) is special.\nIn \\cite{ScT11} it is claimed that $x_i^t \\rightarrow 0$ in this case.\nThis is now an easy consequence and a special case of our results since $g(x)$ approaches $0$ exponentially for large $x.$\n\n\n\\subsection{Compressed Sensing}\n\nIn a typical variant of compressed sensing one observes a ``sparse\" vector $x$\nthrough a underdetermined linear system as\n\\[\ny = Ax +n\\,.\n\\]\nwhere $n$ is an additive noise vector.\nThe matrix $A$ is $m \\times n$ typically with \n$m \\ll n$ where $\\delta= m\/n$ is termed the undersampling ratio.\nThe vector $x$ is constrained to be sparse, or, alternatively, to have\nentries distributed according to a distribution $p_X$ with small R\\'{e}nyi information dimension\n\\cite{VWcs}. \nIn the setup we consider here the entries of $A$ are sampled independent zero mean Gaussians random variables.\nLetting $V$ denote the $m \\times n$ all-1 matrix, the\n variances of the entries of $A$ are component-wise given by $\\frac{1}{m} V$ so that columns of \n$A$ have (roughly) unit norm.\nThe problem is to estimate $x$ from knowledge of $y$ and $A.$\nHere we also assume knowledge of $p_X.$\nThe problem can be scaled up by letting $n$ and $m$ tend to infinity while keeping\n$\\delta$ fixed. \nAsymptotic performance is characterized in terms of the limit.\n\nOne can associate a bipartite graph to $A$ in which one set of nodes corresponds to \nthe columns (and the entries of $x$) and the other set of nodes corresponds to the rows.\nThe graphical representation suggests the use of message passing algorithms for\nthis problem and they have indeed been proposed and studied \\cite{DMM}[and references therein].\nIn \\cite{DMM} a reduced complexity variation, AMP (Approximate Message Passing,) is developed in which there are\nonly $n$ or $m$ distinct messages, depending on the direction.\nAn additional term, the so-called Onsager reaction term, is brought into the algorithm to compensate\nof the feedback inherent in AMP (due to the violation of the extrinsic information principle and the denseness of the graph).\nIn \\cite{DMM} an analysis of AMP is given that leads in the large system limit\nto an iterative function system called\nstate evolution, which is analogous to density evolution.\nThe large system limit analysis is quite different from the usual density evolution analysis in that,\nrather than relying on sparseness and tree-like limits, the state evolution analysis relies on the central limit theorem and the fact that contributions from single edges are asymptotically negligible.\nIn the large system limit, messages (or their errors) in the AMP algorithm are normally distributed \n(this is the important consequence of the including the Onsager reaction term)\nand state evolution captures the variance (SNR) associated to the messages.\nFor our current setup: a $m\\times n$ sensing matrix with independent $\\frac{1}{\\sqrt{m}}N(0,1)$ Gaussian entries \nand known $p_X,$\nthe state evolution equations take the form\n\\cite{DJM11}\n\\begin{align*}\n\\phi_{t+1} = \\sigma^2+\\frac{1}{\\delta}\\text{mmse}(\\phi_t^{-1})\n\\end{align*}\nwhere $\\phi$ is the estimation error variance. In this expression\n\\[\n\\text{mmse}(s) =\n\\expectation (X-\\expectation(X\\mid Y))^2\n\\]\nis the minimum mean square error of an estimator of $X$ given $Y$\nwhere $X$ is distributed as $p_X$ and $Y = \\sqrt{s}X+Z$ where $Z$ is $N(0,1)$\nand independent of $X.$\nThe main properties of $\\text{mmse}$ that are relevant here are\n\\begin{align*}\n\\limsup_{x\\rightarrow \\infty} s\\,\\text{mmse}(s) & = \\bar{D}_{p_X} \\\\\n\\intertext{and}\n\\limsup_{x\\rightarrow \\infty} \\int_0^s \\text{mmse}(s) ds &= \\log(s) \\bar{d}_{p_X}\n\\end{align*}\nwhere $\\bar{D}_{p_X}$ is termed the mmse dimension \\cite{VWcs} \nand $\\bar{d}_{p_X}$ is the upper information dimension \\cite{VWcs} of\n$p_X$ which can be defined by\n\\[\n\\bar{d}_{p_X}\n=\\limsup_{\\ell \\rightarrow \\infty} \\frac{H\\lfloor \\ell X \\rfloor}{\\log \\ell}\n\\]\nwhere, here, $H$ denotes the Shannon entropy.\n\nSpatial coupling can be introduced by imposing additional structure on $A.$\nLet us first consider a collection of parallel systems.\nThus, let $\\tilde{A}$ be a doubly infinite array of $m\\times n$ matrices in \nwhich one diagonal $\\tilde{A}_{i,i}$ is non-zero with each $\\tilde{A}_{i,i}$ \ni.i.d. Gaussian samples with entry-wise variance matrix $\\frac{1}{m}V.$\nThe variance matrix associated to matrix $\\tilde{A}$ is \n$\\tilde{V}$ with $\\tilde{V}_{i,i}= \\frac{1}{m}V$ and $\\tilde{V}_{i,j}=0$ for $i\\neq j.$\nSpatial coupling is achieved by setting $V_{i,j} = w_{i-j} \\frac{1}{m} V.$\nTermination can be effected by providing additional measurements for variables associated\nto the termination.\nSpatially coupled constructions of this type and resulting performance improvements were first presented\nin \\cite{KMSSZ11}. \nThe analytical results on information theoretic optimal results that we reproduce here were presented in\n\\cite{DJM11}.\n\nThe spatially coupled system can be understood within our framework as having the following exit functions.\n\\begin{align*}\n\\hf(u) &= \\sigma^2+\\frac{1}{\\delta} \\text{mmse}(u)\\\\\n\\hg (v) &= 1\/v \n\\end{align*}\nNote in this case that rather than $[0,1]^2$ the system is defined on $[0,+\\infty]^2.$\nThere is a crossing point $(u_1,v_1)$ where $u_1$ is minimal and $v_1$ is maximal.\nIt is easy to see that we have the bound $u_1 \\ge \\frac{\\delta}{ \\expectation(X^2)}$\nsince $\\hf$ is decreasing in $u$ and $\\hf(0) = \\frac{1}{\\delta} \\expectation(X^2).$\nTo apply our analysis we can use this point as our origin in defining $\\altPhi$\nand we can invert the sign of $v$ to recover our canonical ordering.\n\nWe can now easily recover the main results in \\cite{DJM11}.\nConsider first the noiseless case $\\sigma^2 = 0.$ The FP of interest in the\ncomponent system above occurs at $(\\infty,0).$\nIf $\\bar{d}_{p_X} < \\delta$ then we have\n\\[\n\\altPhi(\\hf,\\hg;\\infty,0)\n=\n\\int_{x_1}^{\\infty} (\\frac{1}{\\delta} \\text{mmse}(x)-\\frac{1}{x}) dx\n=\n-\\infty\\,.\n\\]\nThus we get convergence to the fixed point at $(\\infty,0).$ \n(Some simple adjustment of our arguments are needed to handle the unbounded case.)\n\nConsider now $\\sigma^2 > 0$ and let $(u^*(\\sigma^2),v^*(\\sigma^2))$ denote the crossing point\nwith maximal $u$ and minimal $v.$\nIn this case we have\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;u^*(\\sigma^2),v^*(\\sigma^2))\n\\\\= &\n\\int_{u_1}^{u^*} (\\sigma^2 + \\frac{1}{\\delta} \\text{mmse}(u)-\\frac{1}{u}) du\n\\\\ = &\n\\sigma^2 (u^*-u_1) +\\frac{1}{\\delta}\\int_{u_1}^{u^*} \\text{mmse}(u) dx -\\log(u^*\/u_1)\\,.\n\\end{align*}\nBy choosing $\\sigma^2$ small enough we can have $u^*(\\sigma^2)$ as large\nas desired.\nAssume $\\bar{d}_{p_X}<\\delta.$ Then,\nassuming $\\sigma^2$ small enough we have $\\altPhi(\\hf,\\hg;u',v')>\\altPhi(\\hf,\\hg;u^*,v^*)$ for\nany crossing point $(u',v')$ with $u' \\le z.$ It then follows that the crossing point that minimizes $\\altPhi(\\hf,\\hg)$\nhas $u$ value larger than $z.$\n\nAssume now that $\\bar{D}_{p_X}<\\delta$ then for all $\\sigma^2$ small enough we have\n$(u^*(\\sigma^2),v^*(\\sigma^2))$ is minimizing $\\altPhi(\\hf,\\hg).$\nFurthermore it follows that $u^*(\\sigma^2) > C \\sigma^{-2}$ for some constant $C.$\n\n\\section{Higher-Dimensional Systems and the Gaussian Approximation}\\label{sec:gaussapprox}\nWe have discussed in the previous section several scenarios where\nthe state of the system is one dimensional and the developed theory\ncan be applied directly and gives precise predictions on the threshold\nof coupled systems. But we can considerably expand the field of\napplications if we are content with {\\em approximations}. For\nuncoupled systems a good example is the use of EXIT functions. EXIT\nfunctions are equivalent to DE for the case of the BEC, where the\nstate is indeed one dimensional. For transmission over general BMS\nchannels they are no longer exact but they are very useful engineering\ntools which give accurate predictions and valuable insight into\nthe behavior of the system.\n\nThe idea of EXIT functions is to replace the unknown message densities\nappearing in DE by Gaussian densities. If one assumes that the\ndensities are symmetric (all densities appearing in DE are symmetric)\nthen each Gaussian density has only a single degree of freedom and\nwe are back to a one-dimensional system. Clearly, the same approach\ncan be applied to coupled systems. Let us now discuss several\nconcrete examples. We start with transmission over general BMS\nchannels.\n\n\\subsection{Coding and Transmission over General Channels}\nAs we have just discussed, for transmission over general BMS channels\nit is natural to use EXIT charts as a one-dimensional approximation\nof the DE process \\cite{teB99a,teB99b,teB00,teB01}. This strategy\nhas been used successfully in a wide array of settings to approximately\npredict the performance of the BP decoder. As we have seen, whereas\nfor the BP decoder the criterion of success is that the two EXIT\ncurves do not overlap, for the performance of spatially coupled\nsystems the criterion is the positive gap condition and the area condition.\n\nWe demonstrate the basic technique by considering the simple setting\nof point-to-point transmission using irregular LDPC ensembles. It\nis understood that the same ideas can be applied to any of the many\nother scenarios where EXIT charts have been used to predict the\nperformance of the BP decoder of uncoupled systems.\n\nIn the sequel, let $\\psi(m)$ denote the function which gives the\nentropy of a symmetric Gaussian of mean $m$ (and therefore standard\nvariation $\\sigma=\\sqrt{2\/m}$). Although there is no elementary\nexpression for this function, there are a variety of efficient\nnumerical methods to determine its value, see \\cite{RiU08}.\n\nDefine the two functions\n\\begin{align*}\n\\hg(\\xf) & = 1-\\sum_{i} \\rho_i \\psi\\bigl((i-1)\\psi^{-1}(1-\\xf)\\big), \\\\\n\\hf(\\xg) & = \\sum_{i} \\lambda_i \\psi\\bigl((i-1)\\psi^{-1}(\\xg)+\\psi^{-1}(c) \\big).\n\\end{align*}\nNote that $\\hg(\\xf)$ describes the entropy at the output of a check\nnode assuming that the input entropy is equal to $\\xf$ and $\\hf(\\xg)$\ndescribes the entropy at the output of a variable node assuming\nthat the input entropy is equal to $\\xg$ and that the entropy of the\nchannel is $c$. Both of these functions are computed under the\nassumption that all incoming densities are symmetric Gaussians (with\nthe corresponding entropy). In addition, for the computation of the\nfunction $\\hg(\\xf)$ we have used the so-called ``dual'' approximation,\nsee \\cite[p. 236]{RiU08}.\n\n\nFig.~\\ref{fig:positivegapbawgnc36} plots the EXIT charts for the\n$(3, 6)$-regular ensemble and transmission over the BAWGNC. The\nplot on the left shows the determination of the BP threshold for\nthe uncoupled system according to the EXIT chart paradigm. The\nthreshold is determined by the largest channel parameter so that\nthe two curves do not cross. This parameter is equal to $\\ent^{\\BPsmall,\n\\EXITsmall}=0.42915$. Note that according to DE the BP threshold\nis equal to $\\ent^{\\BPsmall} = 0.4293$, see \\cite[Table 4.115\n]{RiU08}, a good match.\n\nThe plot on the right show the determination of the BP threshold\nfor the coupled ensemble according to the positive gap condition.\nSince for this case we only have a single nontrivial FP, this\nthreshold is given by the maximum channel entropy so that the gap\nfor the largest FP is equal to $0$. This means, that for this\nchannel parameter the ``white'' and the ``dark gray'' area are\nequally large. This parameter is equal to $\\ent^{\\BPsmall,\n\\EXITsmall}_{\\text{\\tiny coupled}}=0.4758$. Note that according\nto DE, the BP threshold of the coupled system is equal to\n$\\ent^{\\BPsmall}_{\\text{\\tiny coupled}} = 0.4794$, see \\cite[Table\nII]{KRU12}, again a good match.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/positivegapbawgnc36}\n}\n\\caption{\\label{fig:positivegapbawgnc36} Left: Determination of the\nBP threshold according to the EXIT chart paradigm for the $(3,\n6)$-regular ensemble and transmission over the BAWGNC. The two\ncurves are shown for $\\ent^{\\BPsmall, \\EXITsmall}=0.42915$. As one\ncan see from this picture, the two curves touch but do not cross.\nRight: Determination of the BP threshold for the coupled ensemble\naccording to the EXIT chart paradigm and the positive gap condition.\nThe two curves are shown for $\\ent^{\\BPsmall, \\EXITsmall}_{\\text{\\tiny\ncoupled}}=0.4758$. For this parameter the ``white'' and the ``dark\ngray'' area are in balance. } \\end{figure}\n\n\\subsection{Min-Sum Decoder}\nAs a second application let us consider the min-sum decoder. The\nmessage-passing rule at the variable nodes is identical to the one\nused for the BP decoder. But at a check nodes the rule differs --\nfor the min-sum decoder the sign of the output is the product of\nthe signs of the incoming messages (just like for the BP decoder)\nbut the absolute value of the outgoing message is the minimum of\nthe absolute values of the incoming messages. \n\nFor, e.g., the $(3, 6)$-regular ensemble DE predicts a min-sum\ndecoding threshold on the BAWGNC of $\\ent^{\\MinSumsmall}_{\\text{\\tiny\nuncoup}}=0.381787$, \\cite{Chu00}. For the coupled case this threshold\njumps to $\\ent^{\\MinSumsmall}_{\\text{\\tiny\ncoupled}}=0.429$.\\footnote{Strictly speaking it is not known that\nmin-sum {\\em has} a threshold, i.e., that there exists a channel\nparameter so that for all better channels the decoder converges\nwith high probability in the large system limit and that for all\nworse channels it does not. Nevertheless, one can numerically compute\n``thresholds'' and check empirically that indeed they behave in the\nexpected way. }\n\nIn order to derive a one-dimensional representation of DE , we\nrestrict the class of densities to symmetric Gaussians. Of course,\nthis introduces some error. Contrary to BP decoding, the messages\nappearing in the min-sum decoding are not in general symmetric (and\nneither are they Gaussian).\n\nThe DE rule at variable nodes is identical to the one used when we\nmodeled the BP decoder. The DE rule for the check nodes is more\ndifficult to model but it is easy to compute numerically. \n\nRather than plotting EXIT charts using entropy, we use the error\nas our basic parameter. There are two reasons for this choice.\nFirst, our one-dimensional theory does not depend on the choice of\nparameters and so it is instructive see an example which uses a\nparameter other than entropy. Second, the min-sum decoder is\ninherently invariant to a scaling, whereas entropy is quite sensitive\nto such a scaling. Error probability on the other hand is also\ninvariant to scaling.\n\nFigure~\\ref{fig:minsum36} shows the predictions we get by applying our\none-dimensional model.\n\\begin{figure}[htp]\n{\n\\centering\n\\input{ps\/minsum36}\n}\n\\caption{\\label{fig:minsum36}\nLeft: Determination of the MinSum threshold according to the EXIT\nchart paradigm for the $(3, 6)$-regular ensemble and transmission\nover the BAWGNC. The two curves are shown for $\\ent^{\\BPsmall,\n\\EXITsmall}=0.401$. As one can see from this picture, the two curves\ntouch but do not cross. Right: Determination of the MinSum threshold\nfor the coupled ensemble according to the EXIT chart paradigm and\nthe positive gap condition. The two curves are shown for\n$\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny coupled}}=0.436$. For\nthis parameter the ``white'' and the ``dark gray'' area are in\nbalance. } \\end{figure} \nThe predicted thresholds are $\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny\nuncoup}}= 0.401$, $\\ent^{\\MinSumsmall, \\EXITsmall}_{\\text{\\tiny\ncoupled}}=0.436$. These predictions are less accurate than the\nequivalent predictions for the BP decoder. Most likely this is due\nto the lack of symmetry of the min-sum decoder. But the predictions\nstill show the right qualitative behavior.\n\n\n\n\n\n\n\n\n\n\n\\section{Analysis and Proofs}\\label{sec:proof}\n\nIn the analysis we allow discontinuous update (EXIT) functions.\nThis is not merely for generality\nbut also for modeling of termination and to allow discontinuous perturbations.\nWe will require some notation for dealing with this.\n\nGiven a monotonically non-decreasing function $\\ff$ we write\n\\[\nv \\veq \\ff(u)\n\\]\nto mean $v \\in [\\ff(u-),\\ff(u+)].$\nGiven $\\fg\\in\\sptfns,$ continuous $\\ff \\in \\sptfns,$ and $h\\in\\exitfns,$ we write\n\\[\n\\fg \\veq h\\circ\\ff\n\\]\nto mean $\\fg(x)\\in [h(\\ff(x)-),h(\\ff(x)+)],$ i.e.,\n$\\fg(x)\\veq h(\\ff(x)),$\nfor all $x\\in \\reals.$ We write\n\\[\n\\fg = h\\circ\\ff\n\\]\nto mean $\\fg(x) = h(\\ff(x))$ for all $x.$\nIn some contexts we may have equality holding\nup to a set of $x$ of measure $0.$\nTo distinguish this we write\n\\[\n\\fg \\equiv h\\circ\\ff\n\\]\nto mean $\\fg(x) = h(\\ff(x))$ for all $x$\nup to a set of measure $0.$\nNote that modifying $\\fg$ on a set of measure $0$ has no impact on $\\gS.$\nIn general we use $\\equiv$ to indicate equality up to sets of measure $0.$\n\nGiven a real number $\\ashift$ we use the notation\n$\\gSa$ to denote the reverse shift of $\\gS$ by $\\ashift,$ i.e.,\n\\[\n\\gSa(x) = \\gS(x+\\ashift) \\,.\n\\]\nUltimately we are interested in interpolating functions such that\n\\(\n\\fg = \\hg\\circ\\fS\\,,\n\\)\nand\n\\(\n\\ff = \\hf\\circ\\gSa\\,,\n\\)\nsince this represents a wave-like solution to system \\ref{eqn:gfrecursion}.\nThe mathematical arguments, however, sometimes only give rise to functions\n{\\em consistent} with the equations, i.e., such that\n\\(\n\\fg \\veq \\hg\\circ\\fS\\,,\n\\)\nand\n\\(\n\\ff \\veq \\hf\\circ\\gSa\\,.\n\\)\nMuch of the analysis works with this weaker condition and then strengthens it to obtain proper solutions.\n\n\n\\subsection{Spatial Fixed Points and Waves}\n\n\nEven when one exists, it is typically difficult to analytically determine an interpolating spatial fixed point solution $(\\ff,\\fg) \\in \\sptfns^2$\nfor a given pair $(\\hg,\\hf) \\in \\exitfns.$\nThe reverse direction, however, is relatively easy.\nIn particular, given a putative $(0,1)$-interpolating spatial fixed point $(\\ff,\\fg)$\nthe corresponding $(\\hf,\\hg)$ is essentially determined by the requirement that\n$\\fg(x) = \\hg(\\fS(x))$ and $f(x) = \\hf(\\gS(x)).$\nSome degeneracy is possible if, for example, $\\fS$ is constant over some interval on which $\\fg$ varies. \nEven in this degenerate case, however, the equivalence class of $\\hg$ is uniquely determined.\nThus, given $(0,1)$-interpolating $\\ff$ and $\\fg$ where $\\ff$ is continuous, we define\n$h_{[\\fg,\\ff]}$ to be any element of the uniquely determined equivalence class such that,\n\\[\n\\fg \\veq h_{[\\fg,\\ff]}\\circ\\ff\n\\]\ni.e., for each $x,$\n$\\fg(x) \\in [h_{[\\fg,\\ff]}(\\ff(x)-), h_{[\\fg,\\ff]}(\\ff(x)+)].$\n(A simple argument shows that the equivalence class is indeed uniquely determined.)\nIn general, if $f$ and $g$ are not $(0,1)$-interpolating, then we still consider \n$h_{[\\fg,\\ff]}$ to be defined on $[\\ff(\\minfty),\\ff(\\pinfty)]$ and the inverse to be define on\n$[\\fg(\\minfty),\\fg(\\pinfty)].$ By definition, $f$ and $g$ are\n$(\\ff(\\minfty),\\ff(\\pinfty))$-interpolating and \n$(\\fg(\\minfty),\\fg(\\pinfty))$-interpolating respectively.\n\nIf $(\\ff,\\fg)$ is a $(0,1)$-interpolating spatial fixed point solution to \\eqref{eqn:gfrecursion}\nthen we have $h_{[\\ff,\\gS]} \\equiv \\hf$ and $h_{[\\fg,\\fS]} \\equiv \\hg.$\nIn the reverse direction, $h_{[\\ff,\\gS]} \\equiv \\hf$ and $h_{[\\fg,\\fS]} \\equiv \\hg$ implies,\nand, (assuming $\\ff$ and $\\fg$ are $(0,1)$-interpolating) is in fact equivalent to,\n\\begin{align*}\n\\fg \\veq \\hg\\circ \\fS,\\quad\n\\ff \\veq \\hf\\circ \\gS \n\\end{align*}\nbut does not in general imply the stronger condition\n\\begin{align*}\n\\fg \\equiv \\hg\\circ \\fS,\\quad\n\\ff \\equiv \\hf\\circ \\gS \\,.\n\\end{align*}\nIf $\\hf$ and $\\hg$ are continuous then equivalence, and in fact equality, is implied.\nIn general, given the above equivalence we can achieve equality by replacing $\\fg$ with $\\hg \\circ \\fS$\nand $\\ff$ with $\\hf \\circ \\gS,$ since $\\fS$ and $\\gS$ are thereby unchanged.\n\n\\subsubsection{Sensitivity with Irregular Smoothing}\\label{sec:pathology}\n\nIn this section we illustrate by example some of the subtlety that\ncan arise with non-regular smoothing kernels. We also show how\nnon-uniqueness of fixed point solutions can occur when the positive\ngap condition is satisfied but the strictly positive gap condition is not satisfied.\n\nThe following example shows that changing $\\hf$ or $\\hg$ on a set of measure $0$\ncan, for some choices of $\\smthker,$ have a dramatic effect on the solution\nto \\eqref{eqn:gfrecursion}.\nAssume an averaging kernel $\\smthker$ that is positive everywhere on $\\reals$ except on\n$[-2,2],$ where it equals $0.$\nConsider\n\\[\n\\hf(u) = \\unitstep_a(u-\\frac{1}{2})\n\\]\nand\n\\[\n\\hg(u) = \\unitstep_b(u-\\frac{1}{2})\n\\]\nwhere $a$ and $b$ are specified below.\nLet $\\ff(x) = \\unitstep(x),$ then we have \nwe have \n$\\fS(x)<\\frac{1}{2}$ for $x \\in (\\infty,-2),$\n$\\fS(x)=\\frac{1}{2}$ for $x \\in [-2,2],$ and\n$\\fS(x)>\\frac{1}{2}$ for $x \\in (2,\\infty)\\,.$\nConsider initializing system \\eqref{eqn:gfrecursion} with $\\ff^0(x)=\\unitstep(x).$ \nIf $a=b=\\frac{1}{2}$ then the solution is the fixed point\n\\[\n\\ff^t(x)=\\fg^t(x) =\\frac{1}{2}(\\unitstep_1(x+2)+\\unitstep_0(x-2))\\,.\n\\]\nIf $a=b=1$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_1(x+4t)\n\\\\\n\\fg^t(x)&=\\unitstep_1(x+4t+2)\\,,\n\\end{align*}\nand $\\ff^t(x) \\rightarrow 1.$\nIf $a=b=0$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_0(x-4t)\n\\\\\n\\fg^t(x)&=\\unitstep_0(x-4t-2)\\,,\n\\end{align*}\nand $\\ff^t(x) \\rightarrow 0.$\nIf $a=0$ and $b=1$ then the solution is \n\\begin{align*}\n\\ff^t(x)&= \\unitstep_0(x)\n\\\\\n\\fg^t(x)&=\\unitstep_1(x-2)\\,,\n\\end{align*}\nanother fixed point.\n\nTo give a more general example, if we define $\\ff$ and $\\fg$ as any functions in $\\sptfns$ that\nequal $0$ on $(\\minfty,-1)$ and $1$ on $(1,\\pinfty)$\nthen we have \n$\\fg \\veq \\hg\\circ\\fS$ \nand\n$\\ff \\veq \\hf\\circ\\gS.$ \nIt follows in all such cases that \n$h_{[\\ff,\\gS]} \\equiv \\hf$ and\n$h_{[\\fg,\\fS]} \\equiv \\hg.$\nThis example shows that it is possible to have many solutions that are ``consistent\" with\nthe equation \\eqref{eqn:gfrecursion}, in that\n$\\fg \\veq \\hg\\circ\\fS$ \nand\n$\\ff \\veq \\hf\\circ\\gS.$ \n\\subsubsection{Non-Unique Solutions}\\label{sec:nonunique}\nNow assume $\\smthker = \\frac{1}{2}\\indicator_{|x|<1}.$\nLet $\\tilde{f}$ and $\\tilde{g}$ be any functions in $\\sptfns$ that\nequal $0$ on $(\\minfty,-1)$ and $1$ on $(1,\\pinfty)$ and take values in\n$(0,1)$ on $(-1,1).$\nNow consider\n\\begin{align*}\n\\ff_a(x) &= \\frac{1}{2} \\bigl(\\tilde{f}(x+a)+\\tilde{f}(x-a)\\bigr) \\\\\n\\fg_a(x) &= \\frac{1}{2} \\bigl(\\tilde{g}(x+a)+\\tilde{g}(x-a)\\bigr) \n\\end{align*}\nFor all $a>3$ we see that\n$(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]})$\ndoes not depend on $a$ and the given functions form a family\nof spatial fixed points for the system.\nThis gives an example where system \\eqref{eqn:DE} exhibits multiple\nspatial fixed point solutions.\nNote that $(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]})$ does not satisfy the strictly positive\ngap condition since $\\altPhi(h_{[\\ff_a,\\gS_a]},h_{[\\fg_a,\\fS_a]};\\frac{1}{2},\\frac{1}{2})=0.$\n\n\\subsection{Spatial fixed point integration.}\n\nConsider a $(0,1)$-interpolating spatial fixed point $(\\ff,\\fg).$\nThen, at $v=\\fS(z_1)$ the integral $\\int_0^v \\hg$\ncan be expressed as\n\\[\n\\int_0^v \\hg(z) \\text{d}z=\n\\int_{-\\infty}^{x_1} \\fg(x) \\bigl(\\frac{d}{dx} \\fS(x )\\bigr) \\text{d}x\n=\n\\int_{-\\infty}^{x_1} \\fg(x) \\text{d}\\fS(x)\\,.\n\\]\nSimilarly, at $u=\\gS(x_2)$ we have\n\\[\n\\int_0^u \\hf(z)\\text{d}z = \\int_{-\\infty}^{x_2} \\ff(x) \\text{d}\\gS(x)\\,.\n\\]\nThe product rule of calculus gives, under mild regularity assumptions, $\\fg(x) \\text{d}\\ff(x) + \\ff(x) \\text{d}\\fg(x) = \\text{d}(\\fg(x)\\ff(x))$ and, were it not for the\nspatial smoothing, this would solve directly the sum of the above two integrals in terms of the product\n$\\fg(x)\\ff(x).$ By properly handling the spatial smoothing we can accomplish something similar, and the result is\npresented in Lemma \\ref{lem:twofint}.\nWe obtain a succinct formula for the evaluation of\n$\\altPhi(\\hf,\\hg;\\fg(x_1),\\ff(x_2))$ that is {\\em local}\nin its dependence on $\\ff$ and $\\fg.$ \nThis formula captures a valuable information concerning\nthe $(0,1)$-interpolating spatial fixed point solution and its relation to $\\altPhi.$ In particular it relates local properties of fixed point solutions\nto corresponding values of $\\altPhi.$\n\nFor $(\\ff, \\fg) \\in \\sptfns^2$ and $\\smthker$ an even averaging kernel,\n\\begin{align}\\label{eqn:definexi}\n\\PhiSI (\\smthker;f,g;x_1,x_2) & :=\n(\\fS(x_1) - f(x_2+))\n(\\gS(x_2) - g(x_1+))\\nonumber\n\\\\ &\\quad\n+\n\\altPhiSI(\\smthker;f,g;x_1,x_2)\n\\end{align}\nwhere\n\\begin{align*}\n\\altPhiSI(\\smthker;& f,g;x_1,x_2) = \\\\\n& \\int_{0}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x))df(y)\n\\Bigr) dx\n\\\\ & +\n\\int_{0}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x))dg(y)\n\\Bigr) dx\n\\\\ & =\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x))df(y)\n\\Bigr) dx\n\\\\ & =\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x))dg(y)\n\\Bigr) dx\n\\end{align*}\nwhere the integrals are Lebesgue-Stieltjes integrals.\nIf $x'0$ the following inequalities hold\n\\begin{align*}\n\\altPhiSI(\\smthker,\\ff,\\fg;x,x)\n& \\le\n\\Delta_L f(x)\\Delta_L g(x)+ e_L \n\\end{align*}\n\\begin{align*}\n|\\fS(x)-f(x)|\n \\le\n\\Delta_L f(x) + e_L \n\\end{align*}\n\nFor $(0,1)$-interpolating $\\ff,\\fg \\in \\sptfns$ we have\n\\begin{align*}\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\fg(x),\\ff(x))\n& \\le\n\\Delta_L f(x)\\Delta_L g(x)+ e_L \n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\gS(x),\\ff(x)) & \\le\n2\\Delta_L g(x) + 2e_L\n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\fg(x),\\fS(x)) & \\le\n2\\Delta_L f(x) + 2e_L\n\\\\\n\\altPhi(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]};\\gS(x),\\fS(x)) & \\le\n2\\Delta_L f(x) + 2\\Delta_L g(x) + 3e_L\n\\end{align*}\n\\end{lemma}\nThe Lemma is proved in appendix \\ref{app:A}. \n\n\\subsubsection{Invariance of Fixed Point Potential}\n\n\\begin{lemma}\\label{lem:FPequal}\nAssume $(\\hf,\\hg) \\in \\exitfns^2$ and $\\smthker$ an averaging kernel.\nLet $(\\ff,\\fg) \\in \\sptfns^2$ be a consistent travelling wave solution (not necessarily interpolating) to\nthe system \\eqref{eqn:gfrecursion} with shift $\\ashift,$ i.e.,\n\\begin{align*}\n\\ff &\\veq \\hf \\circ \\fg^{\\smthker,\\ashift} \\\\\n\\fg &\\veq \\hg \\circ \\ff^{\\smthker} \\,.\n\\end{align*}\n Then\n\\begin{itemize}\n\\item[A.] $(\\ff(\\minfty),\\fg(\\minfty)) \\in \\cross(\\hf,\\hg).$\n\\item[B.] $(\\ff(\\pinfty),\\fg(\\pinfty)) \\in \\cross(\\hf,\\hg).$\n\\item[C.] If $\\ashift = 0$ then \n\\[\n\\altPhi(\\hf,\\hg;\\ff(\\minfty),\\fg(\\minfty)) =\n\\altPhi(\\hf,\\hg;\\ff(\\pinfty),\\fg(\\pinfty))\\,.\n\\]\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nBy definition we have $\\ff(x) \\veq \\hf(\\fg^{\\smthker,\\ashift}(x))$ for each $x \\in \\reals.$\nTaking limits we have $\\ff(\\minfty) \\veq \\hf(\\fg^{\\smthker,\\ashift}(\\minfty)).$\nSince $\\fg^{\\smthker,\\ashift}(\\minfty)=\\fg^{\\smthker}(\\minfty)=\\fg^{}(\\minfty)$ we have\n$\\ff(\\minfty) \\veq \\hf (\\fg(\\minfty)).$ Parts A and B now follow easily.\n\nIf $(\\ff(\\minfty),\\fg(\\minfty)) = \\ff(\\pinfty),\\fg(\\pinfty))$ then part C is immediate,\nso assume $(\\ff(\\minfty),\\fg(\\minfty)) < \\ff(\\pinfty),\\fg(\\pinfty)).$\nWe now apply \\eqref{eqn:potdiff} and Lemma \\ref{lem:twofint} to write\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;\\ff(x_2+),\\fg(x_1+)) -\n\\altPhi(\\hf,\\hg;\\ff(\\minfty),\\fg(\\minfty)) \\\\\n=& \\int_{\\fg(\\minfty)}^{\\fg(x_1+)} \\hg^{-1} (u)\\text{d}u\n+ \\int_{\\ff(\\minfty)}^{\\ff(x_2+)} \\hf^{-1} (v)\\text{d}v \\\\\n&-\\ff(x_2+)\\fg(x_1+) + \\ff(\\minfty)\\fg(\\minfty)\\\\\n=& \\altPhiSI(\\smthker;f,g;x_1,x_2) \n\\end{align*}\nLetting $x_1$ and $x_2$ tend to $+\\infty$ the result follows from\nLemma \\ref{lem:transitionPhiBounds}.\n\\end{IEEEproof}\n\n\\subsubsection{Transition length.}\n\nIn this section our aim is to show that fixed point solutions arising from systems satisfying\nthe strictly positive gap condition have bounded transition regions.\nWe show that the transition of solutions from one value to another\nis confined to a region whose width can be bound from above using properties of $\\altPhi$\n\n\\begin{lemma}\\label{lem:transitionBounds}\nLet $\\ff,\\fg$ be $(0,1)$-interpolating functions in $\\sptfns.$\nLet $\\hf \\equiv h_{[\\ff,\\gS]}$ and $\\hg \\equiv h_{[\\fg,\\fS]}.$\nLet $0 < a < b < 1$ and let $x_a,x_b$ satisfy $a=\\gS(x_a)$\nand $b=\\gS(x_b).$\nDefine\n\\begin{align*}\n\\delta &= \\inf \\{ \\altPhi(\\hf,\\hg;\\gS(x),\\ff(x)) : x\\in [x_a,x_b]\\}\n\\\\\n& = \\inf \\{ \\altPhi(\\hf,\\hg;u,\\hf(u)) : u\\in [a,b]\\}\n\\end{align*}\nthen\n\\[\n\\Bigl(\\frac{1}{2}\\delta-e_L\\Bigr)\\lfloor \\frac{x_b-x_a}{2L} \\rfloor \\le\n1\n\\]\nand\n\\begin{align*}\n\\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \n\\lfloor\n\\frac{x_b-x_a-2L}{2L}\n\\rfloor \n\\le b-a\\,.\n\\end{align*}\n\\end{lemma}\n\\begin{IEEEproof}\nFor any $x \\in [x_a,x_b]$ we have\n\\(\n\\Delta_L g(x) \\ge \\frac{1}{2}{\\delta- e_L}\\,\n\\)\nby Lemma \\ref{lem:transitionPhiBounds}.\nIn the interval $[x_a,x_b]$ we can find\n\\(\n\\lfloor\n\\frac{x_b-x_a}{2L}\n\\rfloor\n\\)\nnon-overlapping intervals of length $2L.$\nFrom this we obtain\n\\[\n\\lfloor\n\\frac{x_b-x_a}{2L}\n\\rfloor \\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \\le g(x_b-)-g(x_a+) \\le 1\\,.\n\\]\nA similar argument considering $x_a+L$ and $x_b-L$ gives\n\\begin{align*}\n\\lfloor\n&\\frac{x_b-x_a-2L}{2L}\n\\rfloor \\Bigl( \\frac{1}{2}{\\delta- e_L}\\Bigr) \n\\\\ \\le &\ng((x_b-L)-)-g((x_a+L)+)\n\\\\ \\le & b-a\\,.\n\\end{align*}\n\\end{IEEEproof}\n\n\\subsubsection{Discrete Spatial Integration}\n\nPerhaps somewhat surprisingly, a version of Lemma \\ref{lem:twofint} that\napplies to spatially discrete systems also holds.\nIf $\\ff,\\fg$ are spatially discrete functions and $\\tff,\\tfg$ are their\npiecewise constant extensions, then\nLemma \\ref{lem:twofint} can be applied to\nthese extensions. If we then restrict $x_1$ and $x_2$ to points in\n$\\Delta \\integers,$ then $\\altPhiSI$ can be written as discrete sums.\n\nLet $\\discsmthker$ be related to $\\smthker$ as in \\eqref{eqn:kerdiscretetosmth}\nand let $x_1,x_2 \\in \\Delta \\integers,$ denoted\n$x_{i_1},x_{i_2}.$\nThen \n\\begin{align*}\n&\\altPhiSI(\\smthker;\\tff,\\tfg;x_{i_1},x_{i_2}) \\\\&=\n\\frac{1}{2}\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_1-j,i_2]} (2\\dv{f}_{i_2}-\\dv{f}_{i}-\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\&=\\frac{1}{2}\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_2-j,i_1]} (2\\dv{g}_{i_1}-\\dv{g}_{i}-\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\n\nLemma \\ref{lem:twofint} continues to hold and a proof using entirely discrete\nsummation can be found in appendix \\ref{app:Aa}.\n\n{\\em Discussion:} The proof of Lemma \\ref{lem:twofint} as well as the spatially discrete version found in appendix \\ref{app:Aa} are entirely algebraic in character. Consequently, they apply to spatially coupled systems generally and not only those with a one dimensional state. In a follow-up paper we apply the result to the arbitrary binary memoryless symmetric channel case to obtain a new proof that spatially coupled regular ensembles achieve capacity universally on such channels.\n\n\n\n\n\\subsection{Bounds on Translation Rates}\n\n\\begin{lemma}\\label{lem:shiftlowerbound}\nLet $f,g \\in \\sptfns$ be $(0,1)$-interpolating and let $\\smthker$ be an averaging kernel.\nThen \n\\[\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]}) \\le |\\ashift|\\|\\smthker\\|_\\infty\\,.\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nWe have $A(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]})=0$ and hence\n\\begin{align*}\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})\n&=\nA(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})\n-\nA(h_{[\\ff,\\gS]},h_{[\\fg,\\fS]})\n \\\\\n&= \\int_0^1 h_{[\\ff,\\gSa]}(u) - h_{[\\ff,\\gS]}(u) \\,\\text{d}u \\\\\n&= \\int_{\\minfty}^{\\pinfty} (\\ff(x-\\ashift) - \\ff(x)) \\gS_x(x) dx \\,.\n\\end{align*}\nSince $|\\gS_x(x)| \\le \\| \\smthker \\|_{\\infty}$ we obtain\n$|A(h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})|\\le |\\ashift|\\|\\smthker\\|_\\infty.$\n\\end{IEEEproof}\n\nIn general this estimate can be weak. In Section \\ref{sec:pathology} we gave an example\nof a system with $A(\\hf,\\hg) =0$ and irregular $\\smthker$ that can exhibit both left\nand right moving waves by changing the value $\\hf$ and $\\hg$ at a point of discontinuity.\nFurther, given $(0,1)$-interpolating $\\ff,\\fg$ and positive $\\smthker$ the system\n $(h_{[\\ff,\\fg^{\\smthker,a+\\ashift}]},h_{[\\fg,\\ff^{\\smthker,-a}]})$ (with real parameter $a$)\nhas a \ntraveling solution with shift $\\ashift$ and yet\n$A(h_{[\\ff,\\fg^{\\smthker,a+\\ashift}]},h_{[\\fg,\\ff^{\\smthker,-a}]})$\ncan be made arbitrarily close to $0$ by choosing $a$ with large enough magnitude.\n\nNow we consider upper bounds on $|\\ashift|.$\nLet\n\\[\n\\intsmthker(x) =\\int_{-\\infty}^x \\smthker(y) \\text{d}y.\n\\]\nIf $\\smthker$ has compact support then\nthe width of the support is an upper bound. Consider a $\\smthker$ that is strictly positive\non $\\reals.$ Let $\\hf(x)=\\hg(x)=\\unitstep(x-(1-\\epsilon))$ for small positive $\\epsilon.$\nA traveling wave solution for this system is $\\ff^t(x)=\\unitstep(x-t\\ashift)$\nand $\\fg^t(x)=\\unitstep(x-t\\ashift-\\ashift\/2)$\nwhere $\\ashift$ is given by\n$\\intsmthker(-\\ashift\/2) = (1-\\epsilon).$\nThis example motivates the following bound.\n\n\\begin{lemma}\\label{lem:shiftupperbound}\nLet $\\ff,\\fg\\in\\sptfns$ be $(0,1)$-interpolating and let\n$\\hf\\equiv h_{[\\ff,\\gSa]},$ and $\\hg\\equiv h_{[\\fg,\\fS]}$\nGiven $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-)$ we have the bound\n\\[\n\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\\,.\n\\]\nand given $(u,v)$ with $v > \\hf(u+)$ and $u>\\hg(v+)$ we have the bound\n\\[\n-\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{1-v}{1-\\hf(u+)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{1-u}{1-\\hg(v+)}+\\Bigr)\\,.\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nWe will show the first bound, the second is similar.\nFor any $x_1,x_2\\in\\reals$ we have\n\\begin{align*}\n\\fS(x_1) &= \\int_{\\minfty}^{\\pinfty} \\ff(x)\\smthker(x_1-x) dx \\\\\n&\\ge \\int_{x_2}^{\\pinfty} \\ff(x)\\smthker(x_1-x) dx \\\\\n&\\ge \\ff(x_2+) \\int_{x_2}^{\\pinfty} \\smthker(x_1-x) dx \\\\\n&= \\ff(x_2+) \\int_{\\minfty}^{x_1-x_2} \\smthker(x) dx\\,.\n\\end{align*}\nThus, we obtain the inequality\n\\[\nx_1-x_2 \\le \\Omega^{-1} \\Bigl(\\frac{\\fS(x_1)}{\\ff(x_2+)}+\\Bigr)\n\\]\nChoose $x_1$ so that $\\fS(x_1)=v.$\nThen we have $\\fg(x_1+) \\ge \\fg(x_1) \\ge \\hg(v-).$\nChoose $x_2$ so that $\\gSa(x_2)=\\gS(x_2+\\ashift)=u.$\nThen we have $\\ff(x_2+) \\ge \\ff(x_2) \\ge \\hf(u-).$\n\nApplying the above inequality we obtain\n\\[\nx_1-x_2 \\le \\Omega^{-1} \\Bigl(\\frac{\\fS(x_1)}{\\ff(x_2+)}+\\Bigr)\n \\le \\Omega^{-1} \\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n\\]\nand\n\\[\nx_2+\\ashift-x_1 \\le \\Omega^{-1}\\Bigl(\\frac{\\gS(x_2+\\ashift)}{\\fg(x_1+)}+\\Bigr)\n\\le \\Omega^{-1}\\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\n\\]\nSumming, we obtain\n\\[\n\\ashift \\le \\Omega^{-1}\\Bigl(\\frac{v}{\\hf(u-)}+\\Bigr)\n+\n\\Omega^{-1} \\Bigl(\\frac{u}{\\hg(v-)}+\\Bigr)\\,.\n\\]\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:stposbound}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition.\nThen there exists $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-)$\nand $(u,v)$ with $v > \\hf(u+)$ and $u<\\hg(v+).$\n\\end{lemma}\n\\begin{IEEEproof}\nLet $G^+ = \\{(u,v):v > \\hf(u+) \\text{ and } u>\\hg(v+)\\}$\nand\n$G^- = \\{(u,v):v < \\hf(u-) \\text{ and } u<\\hg(v-)\\}.$\nThen\n\\[\nA(\\hf,\\hg) = \\mu(G^+) - \\mu(G^-)\n\\]\nwhere $\\mu(G)$ denotes the 2-D Lebesgue measure of $G.$\nSince the strictly positive gap condition is satisfied, there\nexists $(u^*,v^*) \\in \\intcross(\\hf,\\hg)$ and\n$\\altPhi(\\hf,\\hg;u^*,v^*) > \\max \\{0,A(\\hf,\\hg)\\}.$\nLet $R = [0,u^*]\\times[0,v^*].$ Now\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*)\n=\n \\mu(G^+ \\cap R ) - \\mu(G^- \\cap R )\n\\]\nhence $\\mu(G^+ \\cap R ) > \\max \\{0,A(\\hf,\\hg)\\}$\nand it follows that $\\mu(G^-) > 0.$\n\\end{IEEEproof}\n\\begin{corollary}\\label{cor:regshiftbound}\nLet $\\ff,\\fg \\in \\sptfns$ be $(0,1)$-interpolating.\nIf $( h_{[\\ff,\\gSa]},h_{[\\fg,\\fS]})$ satisfies the strictly positive gap condition\nand $\\smthker$ is regular\nthen $\\ashift < 2W.$\n\\end{corollary}\n\\begin{IEEEproof}\nThis combines Lemma \\ref{lem:shiftupperbound} with Lemma \\ref{lem:stposbound}.\n\\end{IEEEproof}\n\n\\subsection{Monotonicity of $\\altPhi$ and the Gap Conditions}\n\nIn this section we collect some basic results on $\\altPhi$ and the component DE that are useful for constructing spatial wave solutions.\n\n\\begin{lemma}\\label{lem:descend}\nLet $\\hf,\\hg \\in \\exitfns.$\nIf $(u,v) \\in [0,1]^2$ satisfies $v <\\hf(u-)$ and $u < \\hg(v-),$\nthen there exists a minimal element $(u^*,v^*) \\in \\cross(\\hf,\\hg)$ with\n$(u^*,v^*)>(u,v)$ component-wise and $\\altPhi(u^*,v^*) < \\altPhi(u,v).$\n\nSimilarly, if $(u,v)$ satisfies $v >\\hf(u+)$ and $u > \\hg(v+),$\nthen there exists a maximal element $(u^*,v^*) \\in \\cross(\\hf,\\hg),$\nwith $(u^*,v^*)<(u,v)$ (component-wise) and $\\altPhi(u^*,v^*) < \\altPhi(u,v).$\n\\end{lemma}\n\\begin{IEEEproof}\nWe show only the first case since the other case is analogous.\nAssuming $v <\\hf(u-)$ and $u < \\hg(v-)$ we have $\\hginv(u+) < v < \\hf(u-)$ and we\nsee that there is no crossing point $(u',v')$ with $u'=u.$ Similarly,\nthere is no crossing point with $v'=v.$ \nSince $\\cross(\\hf,\\hg)$ is closed, the set $(u,1]\\times(v,1] \\cap \\cross(\\hf,\\hg)$ is closed.\nBy Lemma \\ref{lem:crossorder} $\\cross(\\hf,\\hg)$ is ordered so there exists a minimal element $(u^*,v^*)$ in\n$(u,1]\\times(v,1] \\cap \\cross(\\hf,\\hg).$\nSet $(u^0,v^0)=(u,v)$ and consider the sequence of points\n$(u^0,v^0),(u^0,v^1),(u^1,v^1),(u^1,v^2),(u^2,v^2),\\ldots$ as determined by \\eqref{eqn:DE}.\nIt follows easily from \\eqref{eqn:DE} that this sequence is non-decreasing.\nIf $u^t < u^*$ then $v^{t+1} \\le v^*$\nand\nif $v^t < v^*$ then $u^{t} \\le u^*.$\nThus we have either $(u^t,v^t) < (u^*,v^*)$ or there is some minimal $t$ where at least one of the coordinates\nis equal.\nIf $u^t < u^*$ and $v^t < v^*$ for all $t$ then the sequence must converge to $(u^*,v^*)$\nsince the limit is in $\\cross(\\hf,\\hg)$ by continuity and $(u^*,v^*)$ is minimal.\nIt then follows by continuity of $\\altPhi(\\hf,\\hg;)$ and Lemma \\ref{lem:monotonic} that \n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) \\le \\altPhi(\\hf,\\hg;u^0,v^1) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nAssume now that $u^t = u^*$ for some $t.$ Then $t>0$ and Lemma \\ref{lem:monotonic} gives\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) = \\altPhi(\\hf,\\hg;u^t,v^{t+1}) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nFinally, assume that $v^t = v^*$ for some $t.$ Then $t>0$ and Lemma \\ref{lem:monotonic} gives\n\\[\n\\altPhi(\\hf,\\hg;u^*,v^*) = \\altPhi(\\hf,\\hg;u^t,v^{t}) < \\altPhi(\\hf,\\hg;u^0,v^0)\\,.\n\\]\nThis completes the proof.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:crosspointmono}\nLet $(\\hf,\\hg)\\in \\exitfns^2$ and \nlet $(u,v) \\in [0,1]^2.$\nWe then have the following trichotomy:\n\\begin{itemize}\n\\item\nIf $\\hg(\\hf(u))=u$ then $(u,\\hf(u))\\in\\cross(\\hf,\\hg).$\n\\item\nIf $\\hg(\\hf(u))>u$ then \n\\(\n\\altPhi(\\hf,\\hg;u^*,v^*)\n\\le\n\\altPhi(\\hf,\\hg;u,v)\\,\n\\)\nwhere $(u^*,v^*)\\in\\cross(\\hf,\\hg)$ is coordinate-wise minimal with\n $(u^*,v^*)\\ge(u,\\hf(u)).$ \n\\item\nIf $\\hg(\\hf(u))u,$ we now have $\\hg(\\hf(u)-)>u.$\nWe also have $\\hginv(u+) < \\hf(u).$\n\nLet $(u^*,v^*)\\in\\cross(\\hf,\\hg)$ be the minimal element such that $(u^*,v^*)\\ge(u,\\hf(u)).$\nIt follows that $u^* >u$ which implies $v^* \\ge \\hf(u+).$\nFor all $\\epsilon>0$ sufficiently small we have\n$u+\\epsilon < \\hg((\\hf(u)-\\epsilon)-)$ and we obviously have\n$\\hf(u)-\\epsilon <\\hf((u+\\epsilon)-).$\nAssuming $\\epsilon$ sufficiently small $(u^*,v^*)$ is the minimal element in $\\cross(\\hf,\\hg)$\nwith $(u^*,v^*)> (u+\\epsilon,\\hf(u)-\\epsilon)$ and\nby Lemma \\ref{lem:descend} we have\n$\\altPhi(u^*,v^*) < \\altPhi(u+\\epsilon,\\hf(u)-\\epsilon).$\nLetting $\\epsilon$ tend to $0$ we obtain\n$\\altPhi(u^*,v^*) \\le \\altPhi(u,\\hf(u)).$\n\nThe argument for the case $\\hg(\\hf(u))0.$\nBy \\eqref{eqn:altPhiderivatives} we see that $\\altPhi(\\hf,\\hg;0,v)=0$ for all\n$v \\in [0,\\hf(0+)).$\nIt follows that $\\hginv(0+)=0$ or we obtain a contradiction with the strictly positive gap condition.\nHence $\\hg(\\hf(0+))>0$ and\nthen for $u\\in(0,\\hg(\\hf(0+))$ we have $\\hg(\\hf(u))>u.$\nBy Lemma \\ref{lem:crosspointmono} the minimal crossing point $(u^*,v^*) \\ge (0,\\hf(u+))$\nsatisfies $\\altPhi(\\hf,\\hg;u^*,v*)<0.$ By the strictly positive gap condition\n$(u^*,v^*) < (1,1)$ and we obtain a contradiction.\nTherefore, we must have $\\hf(0+) = 0.$\n\nAll other conditions, $\\hg(0+) = 0, \\hf(1-) = 1,$ and $\\hg(1-) = 1$ can be shown similarly.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:Sstructure}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition.\nIf $A(\\hf,\\hg) \\ge 0$ then $\\altPhi(\\hf,\\hg;u,v) >0$ \nfor $(u,v)\\in [0,1]^2\\backslash\\{(0,0),(1,1)\\}.$\nIf $A(\\hf,\\hg) > 0$ then there exists a minimal point $(u^*,v^*) \\in \\intcross(\\hf,\\hg)$\nand the set\n\\[\nS(\\hf,\\hg) = \\{(u,v): \\altPhi(\\hf,\\hg;u,v) < A(\\hf,\\hg)\\}\n\\]\nis simply connected \nand $\\closure{S(\\hf,\\hg)} \\subset[0,u^*)\\times[0,v^*).$\nMoreover,\n\\[\n\\{(u,v): \\altPhi(\\hf,\\hg;u,v) \\le A(\\hf,\\hg)\\}\n= \\closure{S(\\hf,\\hg)}\\cup \\{(1,1)\\}.\n\\]\n\\end{lemma}\n\\begin{IEEEproof} \nAssume $(\\hf,\\hg) \\in \\exitfns^2$ satisfies the strictly positive gap condition and that\n$A(\\hf,\\hg) \\ge 0.$\nIt follows from Lemma \\ref{lem:miniscross} that $\\altPhi(\\hf,\\hg;u,v)$\nachieves its minimum on $\\cross(\\hf,\\hg),$\nhence, the strictly positive gap condition implies $\\altPhi(\\hf,\\hg;) \\ge 0.$\nLemma \\ref{lem:miniscross} now further implies that if there exists $(u,v)$ with\n$\\altPhi(\\hf,\\hg;u,v) = 0$ then $(u,v) \\in \\cross(\\hf,\\hg).$\nThus, we have $\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\not\\in \\{(0,0),(1,1)\\}.$\n\nAssume now that $A(\\hf,\\hg) > 0.$ \nLet $(u^*,v^*)$ be the infimum of $\\intcross(\\hf,\\hg).$\nThen $(u^*,v^*)\\in \\cross(\\hf,\\hg)$ and\n$(u^*,v^*) \\neq (0,0)$ \nand therefore $(u^*,v^*) \\in \\intcross(\\hf,\\hg).$\nBy Lemma \\ref{lem:zocontinuity} we also have $u^*>0$ and $v^*>0.$\n\nLet $(u,v) \\in S(\\hf,\\hg)$ and assume that $u \\neq 0$ and $v\\neq 0.$\nBy Lemma \\ref{lem:monotonic} and Lemma \\ref{lem:crosspointmono}\nwe see that we must have $\\hg(\\hf(u)) u^*,$ and, consequently, $v^* \\le \\hf(u).$ \nThen, by Lemma \\ref{lem:monotonic} we have\n$(u,\\hf(u)) \\in S(\\hf,\\hg).$ Lemma \\ref{lem:crosspointmono}\nnow implies the existence of $(u',v') \\in \\cross(\\hf,\\hg)$ with $(u',v') \\ge (u^*,v^*)$ and\n$\\altPhi(\\hf,\\hg;u',v') < A(\\hf,\\hg),$ which contradicts the strictly positive gap condition.\nHence $S(\\hf,\\hg) \\subset [0,u^*) \\times [0,v^*).$\nFurthermore, Lemma \\ref{lem:monotonic} implies that $\\altPhi(\\hf,\\hg;u^*,v) > A(\\hf,\\hg)$ for all $v\\in [0,1]$\nand $\\altPhi(\\hf,\\hg;u,v^*) > A(\\hf,\\hg)$ for all $u\\in [0,1]$\nso $\\closure{S(\\hf,\\hg)} \\subset [0,u^*) \\times [0,v^*).$\n\nAssume there exists $(u,v)\\not\\in \\closure{S(\\hf,\\hg)} \\cup \\{ (1,1)\\}$ with $\\altPhi(\\hf,\\hg;u,v) = A(\\hf,\\hg).$\nThen $(u,v)$ is a local minimum of $\\altPhi(\\hf,\\hg;u,v)$\nwhich, by Lemma \\ref{lem:miniscross}, implies $(u,v) \\in \\cross(\\hf,\\hg),$\ncontradicting the strictly positive gap condition.\n\\end{IEEEproof}\n\n\\subsection{Inverse Formulation.\\label{sect:inverse}}\n\nIt is instructive in to the analysis to view the system in terms of inverse functions.\nLet $\\fg(x) = h((\\ff\\otimes \\smthker) (x))$ with $f \\in \\sptfns.$\nThen, for almost all $u\\in [0,1]$ we have\n\\(\n\\hg^{-1}(u) = \\int_0^1 \\intsmthker (\\fg^{-1}(u)-\\ff^{-1}(v)) \\text{d}v\\,.\n\\)\nTo show this we first integrate by parts to write\n\\(\n(\\ff\\otimes \\smthker) (x) = \\int_{-\\infty}^\\infty \\intsmthker(x-y) d\\ff(y)\n\\)\nand then make the substitutions $v=f(y)$ and $u=g(x).$\nIt follows that, up to equivalence, the recursion \\eqref{eqn:gfrecursion} may be expressed as\n\\begin{equation}\\label{eqn:gfrecursionInv}\n\\begin{split}\n\\hginv(u) & = \\int_0^1 \\intsmthker ((\\fg^t)^{-1}(u)-(\\ff^t)^{-1}(v)) \\text{d}v, \\\\\n\\hfinv(v) & = \\int_0^1 \\intsmthker ((\\ff^{t+1})^{-1}(v)-(\\fg^t)^{-1}(u)) \\text{d}u\\,.\n\\end{split}\n\\end{equation}\nSince $\\smthker$ is even we have $\\Omega(x) = 1-\\Omega(-x),$ so we immediately observe\nthat if $(\\ff,\\fg)\\in\\sptfns^2$ is a $(0,1)$-interpolating fixed point of the above system then\n\\begin{align*}\n1 & =\n\\int_0^1 \\hginv(u) \\text{d}u + \\int_0^1 \\hfinv(v)\\text{d}v \\\\\n& =\n\\int_0^1 \\hg(u) \\text{d}u + \\int_0^1 \\hf(v)\\text{d}v \\,.\n\\end{align*}\nThis puts a very strong requirement on $(\\hf,\\hg)$ to admit a $(0,1)$-interpolating spatial fixed point.\n\nAssume that $\\ff$ and $\\fg,$ both in $\\sptfns,$ form a $(0,1)$-interpolating spatial fixed point.\nConsider perturbing the inverse functions by $\\delta \\ffinv$ and $\\delta \\fginv$ respectively.\nWe could then perturb $\\hfinv$ and $\\hginv,$ by $\\delta\\hfinv$ and $\\delta\\hginv$ respectively so that\nthe perturbed system would remain a fixed point. To first order we will have from \\eqref{eqn:gfrecursionInv},\n\\begin{equation}\\label{eqn:delgfrecursionInv}\n\\begin{split}\n\\delta\\hginv(u) & = \\int_0^1 \\smthker (\\fginv(u)-\\ffinv(v)) (\\delta\\fginv(u)-\\delta\\ffinv(v)) \\text{d}v, \\\\\n\\delta\\hfinv(v) & = \\int_0^1 \\smthker (\\ffinv(v)-\\fginv(u)) (\\delta\\ffinv(v)-\\delta\\fginv(u))\\text{d}u\\,.\n\\end{split}\n\\end{equation}\n\nThis formulation is at the heart of the analysis in the next section.\n\n\n\\subsection{Existence: The Piecewise Constant Case}\\label{sec:PCcase}\nIn this section we focus on the case where $\\hf$ and $\\hg$ are piecewise constant.\nIn this case the spatially coupled system is finite dimensional.\nWe further assume that $\\smthker$ is strictly positive on $\\reals.$\nThis ensures in a simple way that no degeneracy occurs when determining EXIT functions\nfrom spatial functions since $\\frac{d}{dx} \\gS(x) > 0$ for any non-constant $g \\in \\sptfns.$\n\nWe will write piecewise constant functions $\\hf,\\hg \\in \\exitfns$ as\n\\begin{align*}\n\\hf(u) &= \\sum_{j=1}^{\\Kf} \\delhf_j \\,\\unitstep(u - \\uf_j) \\\\\n\\hg(u) &= \\sum_{i=1}^{\\Kg} \\delhg_i \\,\\unitstep(u - \\ug_i)\n\\end{align*}\nwhere $\\unitstep$ is the unit step (Heaviside)\n function\\footnote{The regularity assumptions on $\\smthker$ ensure that the precise value of $\\hf$ and $\\hg$\nat points of discontinuity has no impact on the analysis.}\nand where we assume $\\delhf_j,\\delhg_i > 0,$ and $\\sum_{j=1}^{\\Kf} \\delhf_j = 1$\nand $\\sum_{i=1}^{\\Kg} \\delhg_i = 1.$\n\nGenerally we will have\n$0 < \\uf_1 \\le \\uf_2 \\le \\cdots \\le \\uf_{\\Kf} < 1$\nand\n$0 < \\ug_1 \\le \\ug_2 \\le \\cdots \\le \\ug_{\\Kg} < 1$\nbut the ordering is actually not critical to the definition.\nWe generally view the vectors $\\delhf$ and $\\delhg$ as fixed and \nto indicate the dependence on $\\uf = (\\uf_1,\\ldots,\\uf_{\\Kf})$ and $\\ug$ we will\nwrite $\\hf(u;\\uf)$ and $\\hg(u;\\ug).$\n\nPiecewise constant $\\hf$ and $\\hg$ also have piecewise constant inverses.\nGiven $\\hf$ as above we have\n\\[\n\\hfinv (v) = \\sum_{j=1}^{\\Kf} (\\uf_j-\\uf_{j-1})\\unitstep(v-\\sum_{k=1}^{j} \\delhf_j)\n\\]\nwhere we set $\\uf_0=0.$\n\n\nIf $\\fg \\in \\sptfns$ is continuous, strictly increasing, and $(0,1)$-interpolating\nand $\\hf$ is piecewise constant as above then $\\ff \\in\\sptfns$ defined by\n$\\ff(x) = \\hf(\\fg(x))$ is also piecewise constant and\ncan be written as\n\\[\n\\ff(x) = \\sum_{i=1}^{\\Kf} \\delhf_i \\,\\unitstep(x - \\zf_i)\n\\]\nwith\n$-\\infty < \\zf_1 \\le \\zf_2 \\le \\cdots \\le \\zf_{\\Kf} < \\infty$\nsatisfying $\\uf_i = \\fg(\\zf_i).$ \nAs before, we have\n\\[\n\\ffinv(v) = \\sum_{j=1}^{\\Kf} (\\zf_j-\\zf_{j-1})\\unitstep(v-\\sum_{k=1}^{j} \\delhf_j) \\,.\n\\]\nwhere we set $\\zf_0 =0$\n\n\nThe purpose of this section is to prove a special case of Theorem\n\\ref{thm:mainexist} under piecewise constant assumptions on the EXIT functions and\nregularity conditions on\n$\\smthker.$ In this special case we obtain in addition uniqueness and continuous dependence of the solution. For convenience we state the main result here.\n\\begin{theorem}\\label{thm:PCexist}\nAssume $\\smthker$ is a strictly positive and $C^1$ averaging kernel.\nLet $(\\hf,\\hg)$ be a pair of piecewise constant functions in $\\exitfns$\nsatisfying the strictly positive gap condition.\nThen there exists unique (up to translations) $(0,1)$-interpolating functions\n$\\tmplF,\\tmplG \\in\\sptfns$ and $\\ashift \\in \\reals$ satisfying $\\sgn (\\ashift) = \\sgn (A(\\hf,\\hg)),$ such that\nsetting\n$\\ff^t(x) = \\tmplF(x-\\ashift t)$ and\n$\\fg^t(x) = \\tmplG(x-\\ashift t)$\nsolves \\eqref{eqn:gfrecursion}.\nFurther,\n$\\tmplF^{-1}(v)-\\tmplG^{-1}(u)$ depends continuously on the vectors $\\uf,\\ug.$\n\\end{theorem}\n\nThe remainder of this section is dedicated to the proof of this result.\nOur proof constructs the solutions $\\tmplF$ and $\\tmplG$ by a method of continuation.\nIn the case where $\\hf$ and $\\hg$ are unit step functions it is easy to find\nthe solution: $\\tmplF$ and $\\tmplG$ are also unit step functions.\nStarting there we deform the solution to\narrive at a solution for a given $\\hf$ and $\\hg.$\nWe do this in two stages where in the first stage $\\ashift = 0$ and in the second\nis $\\ashift$ varied while $\\hg$ is held fixed.\nThe deformation is obtained as a solution to a differential equation.\nTo set up the equation we require a detailed description of the dependence\nof $\\uf$ and $\\ug$ on $\\zf,\\zg$ and $\\ashift.$\n\n\nLet us first consider the case $\\ashift=0.$\nThus, let $\\ff(x;\\zf)$ and $\\fg(x;\\zg)$ be a piecewise constant functions\nparameterized by their jump point locations\n$\\zf$ and $\\zg$ as\n\\begin{align}\n\\begin{split}\\label{eqn:fgtdef}\ng(x;\\zg) & = \\sum_{i=1}^{\\Kg} \\delhg_i \\,\\unitstep(x - \\zg_{i}) \\\\\nf(x;\\zf) & = \\sum_{j=1}^{\\Kf} \\delhf_j \\,\\unitstep(x - \\zf_{j})\n\\end{split}\n\\end{align}\nThen, from \\eqref{eqn:gfrecursionInv} we have\n\\begin{align}\n\\begin{split}\\label{eqn:discreteInv}\n\\ug_i &= \\fS(\\zg_i) = \\sum_{j=1}^{\\Kf} \\delhf_j \\Omega (\\zg_i-\\zf_j) \\\\\n\\uf_j &= \\gS(\\zf_j) =\\sum_{i=1}^{\\Kg} \\delhg_i \\Omega (\\zf_j-\\zg_i)\n\\end{split}\n\\end{align}\nNow, suppose we introduce smooth dependence on a parameter $t,$\ni.e., we are given smooth functions $\\zf(t)$ and $\\zg(t)$\nand then determine $\\uf(t)$ and $\\ug(t)$ from the above.\nBy differentiating we obtain\n\\begin{align}\\label{eqn:zeroAdiffeq}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n&=\nH(\\zf(t),\\zg(t))\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n\\end{align}\nwhere $H(\\zf(t),\\zg(t))$ is a $(\\Kg+\\Kf)\\times(\\Kg+\\Kf)$ matrix\n\\begin{align}\n\\label{eqn:matrixdif}\nH(\\zf,\\zg)\n& =\n\\begin{bmatrix}\n \\Df & -\\Bf \\\\\n-\\Bg & \\Dg\n\\end{bmatrix},\n\\end{align}\nwhich we rewrite as $H=D(I-M),$\nand where\n\\[\nD =\n\\begin{bmatrix}\n \\Df & 0 \\\\\n 0 & \\Dg\n\\end{bmatrix}\n\\text{ and }\nM =\n\\begin{bmatrix}\n 0 & (\\Df)^{-1}\\Bf \\\\\n(\\Dg)^{-1}\\Bg & 0\n\\end{bmatrix}\n\\]\nand where \n\\begin{itemize}\n\\item $\\Dg$ is the $\\Kf \\times \\Kf$ diagonal matrix with\n\\[\n\\Dg_{i,i} = \\gS_x(\\zf_i) = \\sum_{j=1}^{\\Kg} \\smthker(\\zg_j-\\zf_i) \\delhg_j,\\]\n\\item $\\Df$ is the $\\Kg \\times \\Kg$ diagonal matrix with\n\\[\n\\Df_{i,i} = \\fS_x(\\zg_i) = \\sum_{j=1}^{\\Kf} \\smthker(\\zf_j-\\zg_i) \\delhf_j,\n\\]\n\\item $\\Bga$ is the $\\Kf \\times \\Kg$ matrix with \n\\[\n\\Bg_{i,j} = -\\frac{\\partial \\gS(\\zf_i;\\zg)}{\\partial \\zg_j} = \\smthker(\\zg_j-\\zf_i) \\delhg_j,\n\\]\n\\item $\\Bf$ is the $\\Kg \\times \\Kf$ matrix with \n\\[\n\\Bf_{i,j} = \n-\\frac{\\partial \\fS(\\zg_i;\\zf)}{\\partial \\zf_j} \n \\smthker(\\zf_j-\\zg_i) \\delhf_j.\n\\]\n\\end{itemize}\n\nSince $\\Dg_{i,i} = \\sum_{j=1}^{\\Kg} \\Bg_{i,j}$ and\n$\\Df_{i,i} = \\sum_{j=1}^{\\Kf} \\Bf_{i,j}$\nwe observe that $M$ is a stochastic matrix: $\\sum_{j=1}^{\\Kf+\\Kg} M_{i,j} = 1.$\n\nOur strategy for constructing fixed points for a given $\\hf,\\hg$ involves solving\n\\eqref{eqn:zeroAdiffeq} for $\\zg(t),\\zf(t)$ given $\\uf(t),\\ug(t).$\nThe main difficulty we face is that $H(\\zf,\\zg)$ is not invertible.\nIn particular, $(I-M)\\vec{1} = 0.$\nThis is a consequence of the fact that\ntranslating $\\zf$ and $\\zg$ together does not alter $\\ug$ and $\\uf$ as defined\nby \\eqref{eqn:discreteInv}.\nThe corresponding left null eigenvector of $H(\\zf,\\zg)$ \narises from the fixed point condition $\\int_0^1 \\hg(x) dx + \\int_0^1\\hf(x) dx = 1$ which here reduces to\n\\[\n1 = \\sum_{j=1}^{\\Kf} (1-\\uf_j) \\delhf_j\n+ \\sum_{i=1}^{\\Kg} (1-\\uf_i) \\delhg_i,\n\\]\nhence\n\\[\n\\sum_{j=1}^{\\Kf} \\delhf_j \\frac{d\\uf_j}{dt}\n+ \\sum_{i=1}^{\\Kg} \\delhg_i \\frac{d\\ug_i}{dt}= 0\n\\]\nas can be verified directly.\n\nLet us consider the matrix\n\\[\nH(\\zf,\\zg) + \\vec{1}\\vec{\\delta}^T\n\\]\nwhere $\\vec{1}$ is the column vector of all $1$s and $\\vec{\\delta}$ is the column\nvector obtained by stacking $\\delhg$ on $\\delhf.$ \nThis matrix is invertible, i.e., its determinant is non-zero. \nTo see this note that $M$ is a positive stochastic matrix \n$M\\vec{1} = \\vec{1}$ and by the Perron-Frobenious theorem all other eigenvalues of $M$\nhave magnitude strictly less than $1.$\nIt follows that $\\vec{1}$ is the unique right null vector of $H(\\zf,\\zg)$ (up to scaling)\nand that $\\vec{\\delta}$ is the corresponding left null vector.\nThe left subspace orthogonal to $\\vec{1}$ is invariant under $H(\\zf,\\zg).$\nIt now follows that $H(\\zf,\\zg) + \\vec{1}\\vec{\\delta}^T$ has no left null vector and\nis therefore invertible.\n\nNow, consider the differential equation\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n&=\n(H(\\zf(t),\\zg(t))+ \\vec{1}\\vec{\\delta}^T)^{-1}\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n\\label{eqn:diffEQR}\n\\end{align}\nIf \n\\(\n\\frac{d}{dt}\n\\vec{\\delta}^T \n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}=0\n\\)\nthen we obtain\n\\(\n\\frac{d}{dt}\n\\vec{\\delta}^T \n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}=0\n\\)\nand we see that \\eqref{eqn:zeroAdiffeq} is satisfied.\n\n\\begin{lemma} \\label{lem:PCexitcont}\nLet $\\smthker$ be a strictly positive smoothing kernel.\nLet $\\uf(t)$ and $\\ug(t)$ be $C^2$ ordered vector valued functions on $[0,1]$\nsuch that $(\\hf(;\\uf(t)),\\hg(;\\ug(t))$ are satisfy the strictly positive\ngap condition uniformly\nin the sense that $\\min_{t\\in [0,1]} \\altPhi(\\hf(;\\uf(t)),\\hg(;\\ug(t);u,v) > 0$ \nfor all $(u,v) \\in [0,1]^2\\backslash \\{(0,0),(1,1)\\}$\nand $A(\\hf(;\\uf(t)),\\hg(;\\ug(t))=0$ for all $t\\in [0,1].$\nAssume further that $\\zf(0)$ and $\\zg(0)$\nare given so that $\\ff(;\\zf(0))$ and $\\fg(;\\zg(0))$ as defined by\n\\eqref{eqn:fgtdef}\nsatisfies $\\fg(x;\\zg(0)) = \\hg(\\fS(x;\\zf(0));\\ug(0))$\nand $\\ff(x;\\zg(0)) = \\hf(\\gS(x;\\zg(0));\\uf(0))$ for all $x\\in\\reals,$\ni.e., the determined functions form a $(0,1)$-interpolating spatial fixed point for the $t=0$ system.\nThen there exist bounded $C^1$ ordered vector valued functions\n$\\zf(t)$ and $\\zg(t)$ on $[0,1],$\nwith $\\zf(0)$ and $\\zg(0)$ as specified,\nsuch that $\\fg(x;\\zg(t)) = \\hg(\\fS(x;\\zf(t));\\ug(t))$\nand $\\ff(x;\\zg(t)) = \\hf(\\gS(x;\\zg(t));\\uf(t))$\nfor all $x\\in \\reals$ and $t\\in [0,1].$\n\\end{lemma}\n\\begin{IEEEproof}\nConsider the differential equation \\eqref{eqn:diffEQR} with initial condition\ngiven by $\\zg(0)$ and $\\zf(0).$ By translating (adding a constant to both) we can assume\n$\\sum_{i=1}^{\\Kg} \\delhg_i \\zg_i(0) +\\sum_{j=1}^{\\Kf} \\delhf_j \\zf_j(0) = 0.$\nBy standard results on differential equations, the equation\nhas a unique $C^1$ solution $(\\zg(t),\\zf(t))$ on $[0,T)$ for some maximal $T>0.$\nSince $\\frac{d}{dt}(\\sum_{i=1}^{\\Kg} \\delhg_i \\ug_i(t) +\\sum_{j=1}^{\\Kf} \\delhf_j \\uf_j(t) ) = 0$\nwe have $\\frac{d}{dt}(\\sum_{i=1}^{\\Kg} \\delhg_i \\zg_j(t) +\\sum_{j=1}^{\\Kf} \\delhf_j \\zf_j(t)) = 0$\nand therefore $\\sum_{i=1}^{\\Kg} \\delhg_i \\zg(t) +\\sum_{j=1}^{\\Kf} \\delhf_j \\zf_j(t) = 0.$\nFurther, the solution satisfies \\eqref{eqn:zeroAdiffeq} and it follows that the determined\nfunctions $f(;\\zf(t))$ and $g(;\\zg(t))$ are corresponding $(0,1)$-interpolating spatial fixed points\nfor $\\hf(;\\uf(t)),\\hg(;\\ug(t)).$\nLemma \\ref{lem:transitionPhiBounds} implies that $\\zf(t)$ and $\\zg(t)$ are bounded\nand we can conclude that the solution exists for all $t\\in [0,1].$\n\\end{IEEEproof}\n\nNow we consider adding a shift to the model.\nWe generalize \\eqref{eqn:discreteInv} as follows\n\\begin{align}\n\\begin{split}\\label{eqn:discreteInvShift}\n\\ug_i &= \\fS(\\zg_i) = \\sum_{j=1}^{\\Kf} \\delhf_j \\Omega (\\zg_i-\\zf_j) \\\\\n\\uf_j &= \\gS(\\zf_j+\\ashift) =\\sum_{j=1}^{\\Kf} \\delhf_i \\Omega (\\zf_j+\\ashift-\\zg_i)\n\\end{split}\n\\end{align}\nNow, let $\\zg(t),\\zf(t),\\ashift(t)$ be smooth functions of time, then\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix}\n&=\nH(\\zf(t),\\zg(t))\n\\;\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+\n\\begin{bmatrix}\n0 \\\\\n\\Dga \\vec{1}\n\\end{bmatrix}\n\\frac{d}{dt}\\ashift(t)\n\\label{eqn:diffEQ}\n\\end{align}\nwhere $H(\\zf(t),\\zg(t))$ is a $(\\Kg+\\Kf)\\times(\\Kg+\\Kf)$ matrix\n\\begin{align}\n\\label{eqn:matrixdif}\nH(\\zf,\\zg)\n& =\n\\begin{bmatrix}\n \\Df & -\\Bf \\\\\n-\\Bga & \\Dga\n\\end{bmatrix}\n\\\\\n& =\nD\n\\begin{bmatrix}\nI-M\n\\end{bmatrix}\n\\end{align}\nwhere\n\\[\nD =\n\\begin{bmatrix}\n \\Df & 0 \\\\\n 0 & \\Dga\n\\end{bmatrix}\n\\text{ and }\nM =\n\\begin{bmatrix}\n 0 & (\\Df)^{-1}\\Bf \\\\\n(\\Dga)^{-1}\\Bga & 0\n\\end{bmatrix}\n\\]\n\n\n\nwhere $\\Df$ and $\\Bf$ are as before and\n\\begin{itemize}\n\\item $\\Dga$ is the $\\Kf \\times \\Kf$ diagonal matrix with\n\\[\n\\Dga_{i,i} = \\gS_x(\\zf_i+\\ashift) = \\sum_{j=1}^{\\Kg} \\smthker(\\zg_j-(\\zf_i+\\ashift)) \\delhg_j,\n\\]\n\\item $\\Bga$ is the $\\Kf \\times \\Kg$ matrix with \n\\[\n\\Bga_{i,j} = \n\\smthker(\\zg_j-(\\zf_i+\\ashift)) \\delhg_j,\n\\]\n\\end{itemize}\n\nSince $\\Dga_{i,i} = \\sum_{j=1}^{\\Kg} \\Bga_{i,j}$ and\n$\\Df_{i,i} = \\sum_{j=1}^{\\Kf} \\Bf_{i,j}$\nwe observe that $M$ is a stochastic matrix: $\\sum_{j=1}^{\\Kf+\\Kg} M_{i,j} = 1.$\n\n\nLet $P$ be the projection matrix which is the $(\\Kf+\\Kg) \\times (\\Kf+\\Kg)$\nidentity matrix except that $P_{\\Kf+\\Kg , \\Kf+\\Kg}=0.$\nIt follows that $I-PMP$ is invertible and $PMP$ has spectral radius less than one.\nIndeed, let\n$\\tilde{B_1}$ denote the matrix obtained from $(\\Df)^{-1}\\Bf$ be removing the rightmost column and\nlet $\\tilde{B_2}$ denote the matrix obtained from $(\\Dga)^{-1}\\Bga$ be removing the bottom row.\nLet $\\tilde{M}$ denote the upper left $\\Kf+\\Kg -1 \\times \\Kf+\\Kg-1$ submatrix of $M.$\nThen\n\\[\n\\tilde{M}^{2} =\n\\begin{bmatrix}\n(\\tilde{B_1}\\tilde{B_2})^{2k} & 0 \\\\\n0 & (\\tilde{B_2}\\tilde{B_1})^{2}\n\\end{bmatrix}\\,.\n\\]\nLet $\\xi < 1$ denote the maximum row sum from $\\tilde{B_1}.$\nBy the Perron-Frobenious theorem $\\tilde{B_2}\\tilde{B_1}$ has a maximal positive eigenvalue $\\lambda$\nwith positive left eigenvector $x.$ Then $x^T \\tilde{B_2}\\tilde{B_1} \\vec{1} = \\lambda x^T \\vec{1},$\nbut $\\tilde{B_2}\\tilde{B_1} \\vec{1} \\le \\xi \\vec{1}$ (component-wise) so $\\lambda \\le \\xi.$\nWe easily conclude that $\\| \\tilde{M}^2 \\|_2 \\le \\xi.$\nHence $(I-PMP)^{-1}$ exists and is strictly positive.\n\nGiven $\\zf(0),\\zg(0)$ we define $\\zg(t),\\zf(t)$ as the solution to\n\\begin{align}\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n&=\n-(I-PM(\\zf(t),\\zg(t))P)^{-1}\n\\;\nP\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\\,.\n\\label{eqn:diffEQRshift}\n\\end{align}\nNote that \n\\[\nP\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix} =\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\\,.\n\\]\nNow we substitute the solution into \\eqref{eqn:diffEQ}\nand we obtain\n\\begin{align*}\nP\\frac{d}{dt}\n\\begin{bmatrix}\n\\ug(t) \\\\\n\\uf(t)\n\\end{bmatrix} \n&=\nD\\Biggl[\nP(I-M)\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+P\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\n\\Biggr] \\\\\n&=\nD\\Biggl[\nP(I-PMP)\n\\frac{d}{dt}\n\\begin{bmatrix}\n\\zg(t) \\\\\n\\zf(t)\n\\end{bmatrix}\n+P\n\\begin{bmatrix}\n0 \\\\\n\\vec{1}\n\\end{bmatrix}\n\\Biggr] \\\\\n& = 0\\,.\n\\end{align*}\n\n\\begin{lemma}\\label{lem:PCshiftexist}\nLet $\\zf(0),\\zg(0)$ be given, thereby defining\n$f(;\\zf(0))$ and $g(;\\zg(0))$ in $\\sptfns$\nand set\n\\[\n\\hg = \\sum_{i=1}^{\\Kg} \\delhg_j \\,\\unitstep( x- \\fS(\\zg_{i}(0);\\zf(0)) )\\,.\n\\]\nLet $\\ashift(0) \\ge 0$ also be given and\nlet $\\hf(;r)$ be parametrized piecewise constant functions defined by\n$\\hf(x;r) = \\hf(x;\\uf(r))=\\sum_{i=1}^{\\Kf} \\delhf_i \\,\\unitstep( x - \\uf_i(r) )$\nwhere\n\\[\n\\uf_i(r) =\n\\begin{cases}\n\\gS(\\zf_i(0)+\\ashift(0);\\zg(0)) & i < \\Kf \\\\\n\\gS(\\zf_{\\Kf}(0)+\\ashift(0);\\zg(0)) +r & i = \\Kf\\,.\n\\end{cases}\n\\]\nNote that $f(;\\zf(0))$ and $g(;\\zg(0))$ form a traveling wave solution\nfor $\\hf(;0),\\hg$ with shift $\\ashift(0).$\n\nLet $r'>0$ be such that $\\uf_{\\Kf}( r') < 1$\nand assume that $(\\hf(;r'),\\hg)$ satisfies\nthe strictly positive gap condition\nwith $A(\\hf(;r'),\\hg) > 0.$\nDefine $\\ashift(t) = \\ashift(0) + t.$\n\nThen there exists $C^1$ functions $\\zf(t)$ and $\\zg(t)$\nfor $t\\in [0,T],$ where $T<\\infty,$\nwith $\\zf(0)$ and $\\zg(0)$ as given, such that \n$f(;\\zf(t)),g(;\\zg(t)) \\in \\sptfns,$\nform a spatial wave solution with shift $\\ashift(t)$ for\n$(\\hf(;r(t)),\\hg)$\nand where $r(t)$ is an increasing $C^1$ function\nwith $r(0)=0,$ and $r(T)=r'.$\n\\end{lemma}\n\\begin{IEEEproof}\nConsider the differential equation\n\\eqref{eqn:diffEQRshift}.\nBy standard results on differential equations\na unique solution exists on $[0,T')$ for some maximal $T'>0.$\nNote that $\\zf(t)$ and $\\zg(t)$ are component-wise decreasing in $t,$\nexcept for $\\zf_{\\Kf}(t)$ which is constant.\n\nIt follows that \n\\begin{align*}\n& \\frac{d}{dt} \\gS(\\zf_{\\Kf}+\\ashift(t);\\zg(t)) \\\\\n& =\n\\gS_x(\\zf_{\\Kf}+\\ashift(t);\\zg(t))\\cdot \\\\\n&\\qquad\\Bigl(1 - \\sum_{j=1}^{\\Kg}\n\\smthker(\\zf_{\\Kf}+\\ashift(t)-\\zg_j(t))\\, \\delhg_j\n\\frac{d}{dt} \\zg_j(t)\n\\Bigr)\n\\end{align*}\nwhich is strictly positive for all $t\\in [0,T')$ since $\\frac{d}{dt} \\zg_j(t) \\le 0.$\nIt follows that $\\ff(;\\zf(t)),\\fg(;\\zg(t))$ form a spatial wave solution with\nshift $\\ashift(t)$ for $(\\hf(;r(t)),\\hg)$ for all $t\\in [0,T')$\nand that $r(t)=\\gS(\\zf_{\\Kf} + \\ashift(t);\\zg(t)) - \\gS(\\zf_{\\Kf} + \\ashift(0);\\zg(0))$\nis monotonically increasing on $[0,T').$\n\nWe now show that $r(t) \\le r'$ implies that $Z(t)$ is bounded.\nAs a first step we show that $\\ashift(t)$ is bounded.\nIf $r \\le r'$ then there exists $(u,v)$ with $v < \\hf(u-)$ and $u<\\hg(v-).$\nIn particular if we take $u>r'$ and $v > \\fS(\\zg_{\\Zg}(0);\\zf(0)) = \\fS(\\zg_{\\Zg}(t);\\zf(t))$\nthen we obtain\na finite upper bound on $\\ashift(t)$ from Lemma \\ref{lem:shiftupperbound}.\n\nAssume there exists $t_i \\rightarrow T'' \\le T'$\nfrom below such that $f(;t_i) \\rightarrow f,$\n$g(;t_i) \\rightarrow g$ and $r(t_i) \\rightarrow r \\le r'.$\nFrom Theorem \\ref{thm:mainlimit}\nit follows that $(f(\\minfty),g(\\minfty)) \\in \\cross(\\hf(;r),\\hg)$\nand $\\altPhi(\\hf(;r),\\hg;g(\\minfty),f(\\minfty)) = 0.$\nBy the assumptions on\n$\\hg$ we see that the crossing point cannot be interior.\nThus, $(f(\\minfty),g(\\minfty)) = (0,0)$\nand we conclude that $\\zg(t)$ and $\\zf(t)$ are bounded.\nHence $T'' < T'.$\n\nFinally, an upper bound on $\\|\\zg(t)\\|$ yields a positive lower bound on\n$\\frac{d}{dt} r(t)$ so we conclude that\nthere exists $T < T'$ such that $r(T)=r'.$\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:PCexist}]\nLet us first consider the case $A(\\hf,\\hg)= 0.$\nLet the target EXIT functions be $\\hf=\\hf(;\\uf)$ and $\\hg=\\hg(;\\ug).$\nLet $B_{\\hf} = \\int_0^1 \\hfinv(x) dx = \\sum_j \\delhf_{j} \\uf_{j},$\nand $B_{\\hg} = \\int_0^1 \\hginv(x) dx = \\sum_i \\delhg_{i} \\ug_{i}= 1-B_{\\hf}.$\nFor $t \\in [0,1]$ define the vector valued functions\n\\begin{align}\n\\begin{split}\n\\uf (t) & = (1-t) B_{\\hf}\\vec{1} + t \\uf \\\\\n\\ug (t) & = (1-t) B_{\\hg}\\vec{1} + t \\ug\\,,\\label{eqn:pcscale}\n\\end{split}\n\\end{align}\nwhere $\\vec{1}$ denotes a vector of all $1$s (of appropriate length).\nNote that we have $\\uf(1)=\\uf$ and $\\ug(1)=\\ug.$\nNote that $\\hf(;\\uf(t))$ and $\\hg(;\\ug(t))$ are in $\\exitfns$ for all $t.$\nNote that $\\int_0^1 \\hfinv(v;\\uf(t)) dv = B_{\\hf}$\nand $\\int_0^1 \\hginv(u;\\ug(t)) du = B_{\\hg}$\nso $A (\\hf(;\\uf(t)),\\hg(;\\ug(t))) = 0$ for all $t \\in [0,1].$\n\nLet $h \\in \\exitfns$ and let $(u,v)$ be in the graph of $h.$ Then\n\\begin{align*}\n&\\altPhi(h,\\hg(;\\ug(t));u,v) - \\altPhi(h,\\hg(;\\ug(1));u,v)\\\\\n= &\n\\int_0^u (\\hginv(z;\\ug(t))-\\hginv(z;\\ug(1)))\\text{d}z \\\\\n= &\n(1-t)(u B_{\\hg} - \\int_0^u \\hginv(z;\\ug(1)))\\text{d}z \\\\\n\\ge & 0\n\\end{align*}\nwhere the last inequality holds since we have equality at $u=0$ and $u=1$\nand $(u B_{\\hg} - \\int_0^u \\hginv(z;\\ug(1)))\\text{d}z$ is concave.\nThis implies that if $(h,\\hg(;\\ug(1)))$ satisfies the strictly positive gap condition\nwith $A(h,\\hg(;\\ug(1))) = 0$ then\n$(h,\\hg(;\\ug(t)))$ also satisfies the strictly positive gap condition\nwith $A(h,\\hg(;\\ug(t))) = 0$ for all $t \\in [0,1].$\n\nThe above argument shows that $(\\hf(;\\uf(1)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $t\\in [0,1].$\nApplying the argument analogously to $\\hf$ we see that\n$(\\hf(;\\uf(s)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $s,t\\in [0,1],$\nand, in particular, \n$(\\hf(;\\uf(t)),\\hg(;\\ug(t)))$\nsatisfies the strictly positive gap condition for all $t\\in [0,1].$\n\nAll that remains to apply Lemma \\ref{lem:PCexitcont} over $t \\in [0,1]$ and\nconclude the proof for the case $A(\\hf,\\hg)=0$\nis to find $\\zf(0)$ and $\\zg(0).$\nSet $\\zf(0) = 0$ so that\n$f(x;\\zf(0)) = \\,\\unitstep(x).$\nLet $y$ be the unique point such that $\\fS(y;\\zf(0)) = B_{\\hg}$ and\nset each component of $\\zg(0)$ to $y$ so that $\\fg(x;\\zg(0)) = \\unitstep(x-y).$\nIt follows that $\\fg(x;\\zg(0)) = \\hg(\\fS(x;\\zf(0));\\ug(0))$ and that\n$\\ff(x;\\zf(0)) = \\hf(\\gS(x;\\zg(0));\\uf(0)).$\n\nApplying Lemma \\ref{lem:PCexitcont} for $t \\in [0,1]$ we obtain\n$\\ff(;\\zf(t))$ and $\\fg(;\\zg(t))$ such that\n$\\fg(x;\\zg(t)) = \\hg(\\fS(x;\\zf(t));\\ug(t))$ and\n$\\ff(x;\\zf(t)) = \\hf(\\gS(x;\\zg(t));\\uf(t))$\ncompleting the proof when $A(\\hf,\\hg)=0.$\n\nWe now consider the case $A(\\hf,\\hg) \\neq 0.$\nWithout loss of generality we assume $A(\\hf,\\hg) > 0.$\nThe case $A(\\hf,\\hg) < 0$ is equivalent to the case $A(\\hf,\\hg) > 0$\nunder symmetry.\n\nLet us introduce a modification of $\\hf(;\\uf(t))$ as follows.\nFor $r \\in (0,1)$ define $\\uf(t;r)$ by $\\uf_i (t;r) = \\min \\{ \\uf_{i}(t), r \\}.$\nThen $\\int_0^u \\hfinv(x;\\uf(t;r)) dx$ is non-decreasing in $r$ for all $u \\in [0,1].$\nSince $\\int_0^1 \\hfinv(x) dx > 1 -\\int_0^1 \\hginv(x) dx$ and\n$\\int_0^1 \\hfinv(x;\\uf(t;0)) dx = 0$\nthere exists a unique positive $r_0 < \\uf_{\\Kf}$ such that\n$\\int_0^1 \\hginv(x;\\uf(t;r_0)) dx = 1-\\int_0^1 \\hginv(x) dx.$\n\nWe claim that for all $r \\in [r_0,\\uf_{\\Kf}]$ the pair\n$(\\hf(;r),\\hg)$ satisfies the strictly positive gap condition.\nTo establish the claim we need to show that\n$\\altPhi(\\hf(;r),\\hg) > A(\\hf(;r),\\hg )$ on $\\intcross(\\hf(;r),\\hg ).$\nLet $ (u,v) \\in \\intcrossing (\\hf(;r),\\hg ).$\nNote that this implies $u \\le r$ since $\\hg$ is continuous at $1$ \nby Lemma \\ref{lem:zocontinuity}.\nIf $u < r$ then $(u,v) \\in \\intcrossing (\\hf,\\hg)$\nand we have\n\\(\n\\altPhi(\\hf(;r),\\hg;u,v) =\n\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg) \\ge A(\\hf(;r),\\hg)\n\\,.\n\\)\nIf $u = r$ then we have $v \\ge \\hf(u-;r)$ and\n$\\hfinv(v';r) = v$ for all $v' \\in (v,1].$\nApplying \\eqref{eqn:altPhiderivatives} we have\n$\\altPhi(\\hf(;r),\\hg;u,v) = \\altPhi(\\hf(;r),\\hg;u,1)$\nand applying \\eqref{eqn:altPhiderivatives} and using continuity of\n$\\hg$ at $1,$ we have\n$\\altPhi(\\hf(;r),\\hg;u,1) > \\altPhi(\\hf(;r),\\hg;1,1).$\nHence \n$\\altPhi(\\hf(;r),\\hg;u,v) > A(\\hf(;r),\\hg )$ and the claim is established.\n\nApplying our result for the $\\ashift = 0$ case\nwe can find $f,g \\in \\sptfns$ that\nform a $(0,1)$-interpolating spatial fixed point pair for $(\\hf, \\hg(x;r_0)).$\nWe now apply Lemma \\ref{lem:PCshiftexist} in a series of stages\nincreasing $\\ashift$ (linearly $\\ashift = t$) and adjusting\n$\\zf$ and $\\zg$ so that\n$h^{\\fS,g}$ is held fixed while\n$h^{\\gSa,f}$ tracks $\\hg(x;r(t)).$\nThe process can be decomposed into stages where in each stage\n$u_{2,i-1} \\le r(t) \\le u_{2,i}$ for some $i,$\nand where in the final stage $i = \\Kf.$\nIn each stage $\\ashift(t)$ and $r(t)$ is increasing until,\nat the end of the stage, $r(t) = u_{2,i}.$\nDuring the stage where $r(t)$ increases to $u_{2,i}$ we have\n$u_{2,j}(t) = r(t)$ for all $j \\ge i.$\nWe can, in principle, simplify the notation by collapsing these indices to a single value, i.e.,\nto assume that $i = \\Kf.$\nIn this way we reduce each stage to the case $i= \\Kf$\n(while admitting an initial condition $\\ashift_0 \\ge 0).$\nLemma \\ref{lem:PCshiftexist} therefore completes the proof.\n\\end{IEEEproof}\n\n\\subsubsection{Convergence}\n\nIn the piecewise constant case with strictly positive averaging kernel \nwe can also show convergence to the solution constructed above for all initial conditions.\nDefine $f_\\lambda$ by \n\\[\nf^{-1}_\\lambda = \\lambda f^{t+1,-1} + (1-\\lambda) f^{t,-1}\n\\]\nand set\n$g_\\lambda = \\hf \\circ f^\\smthker_\\lambda.$\nThen by applying \\eqref{eqn:delgfrecursionInv} we obtain\n\\begin{align*}\n&g^{t+1,-1}(u) - g^{t,-1}(u) \n\\\\\n=&\n\\int_0^1 \\int_0^1 M(\\lambda,u,v)(f^{t+1,-1}(v) - f^{t,-1}(v)) \\,dv\\,d\\lambda\n\\end{align*}\nwhere\n\\begin{align*}\nM(\\lambda,u,v)=\\frac{\\int_0^1 \\smthker(g^{-1}_\\lambda(u)-f^{-1}_\\lambda(v))}{\\int_0^1 \\smthker(g^{-1}_\\lambda(u)-f^{-1}_\\lambda(v)) \\,dv}\n\\end{align*}\nSince $\\smthker$ is strictly positive we have\n for each $\\lambda$ and $u$ that $M(\\lambda,u,v)>0$ and $\\int_0^1 M(\\lambda,u,v)\\,dv = 1.$\nSo we obtain\n\\begin{align*}\n\\sup_u (g^{t+1,-1}(u) - g^{t,-1}(u)) &\\le \\sup_v (f^{t+1,-1}(v) - f^{t,-1}(v))\\, ,\n\\\\\n\\inf_u (g^{t+1,-1}(u) - g^{t,-1}(u)) &\\ge \\inf_v (f^{t+1,-1}(v) - f^{t,-1}(v))\\,.\n\\end{align*}\nIn the piecewise constant case the inverse functions are bounded and with strictly positive $\\smthker$ we see that\nthe inequalities are strict unless $f^{t+1,-1}(v) - f^{t,-1}(v)$ is a constant.\nIt is easy to conclude in this case that $f^{t+1,-1}(v) - f^{t,-1}(v)$ converges in $t$ to a constant in $v.$\nFrom this it follows that the $f^t$ converges to the solution given above (with suitable translation).\n\n\n\\subsection{Limit Theorems \\label{sec:limitthms}}\n\nOne of the main tools we use to extend the existence results for the piecewise constant\ncase to the general case is taking limits. In this section we develop the basic results needed.\n\nLet us recall the notation $g^{{\\smthker},\\ashift}(x) = g^{\\smthker}(x+\\ashift).$\nThen, we have the bound\n\\begin{align*}\n&g^{{\\smthker},\\ashift}(x)\n-\ng^{\\smthker',\\ashift'}(x)\n=\n\\\\ &\n\\int_{-\\infty}^\\infty (g(y-\\ashift)-g(y-\\ashift')) \\smthker(x-y) \\,dx\n\\\\&+\n\\int_{-\\infty}^\\infty g(y-\\ashift') (\\smthker(x-y)-\\smthker'(x-y)) \\,dx\n\\end{align*}\n\\begin{align}\n\\begin{split}\\label{eqn:diffbound}\n|g^{{\\smthker},\\ashift}(x)\n-\ng^{\\smthker',\\ashift'}(x)|\n\\le\n|\\ashift-\\ashift'| \\|\\smthker\\|_\\infty\n+\n\\|\\smthker-\\smthker'\\|_1 \\,\n\\end{split}\n\\end{align}\n\n\n\n\\begin{theorem}\\label{thm:mainlimit}\nLet $f_i,g_i,\\ashift_i,\\smthker_i,\\; i=1,2,3,...$ be sequences where $(f_i,g_i) \\in \\sptfns^2$\nare $(0,1)$-interpolating, \n$\\ashift_i \\in \\reals,$ and $\\smthker_i$ are averaging kernels. \nAssume\n\\[\nf_i \\rightarrow f,\\;\ng_i \\rightarrow g,\\;\n\\ashift_i \\rightarrow \\ashift,\\;\\text{ and }\n\\smthker_i \\rightarrow \\smthker \\text{ (in $L_1$) },\n\\]\nwhere $|\\ashift| < \\infty$\nand $\\smthker$ is an averaging kernel.\n(Note that we do not assume $f$ and $g$ are interpolating or that $\\smthker$ is regular.)\nFurther assume\n\\[\nh_{[\\ff_i,\\gSiai_i]}\\rightarrow \\hf\n,\\quad\nh_{[\\fg_i,\\fSi_i]}\\rightarrow \\hg\n\\]\nfor some $\\hf,\\hg \\in \\exitfns$ respectively.\nThen we have the following\n\\begin{itemize}\n\\item[A.] \n\\begin{align*}\n\\ff \\veq \\hf\\circ\\gS\n\\text{ and }\n\\fg \\veq \\hg\\circ\\fS\n\\end{align*}\n\\item[B.]\n\\[\n(f(\\minfty),g(\\minfty)),(f(\\pinfty),g(\\pinfty)) \\in \\cross (\\hf,\\hg)\n\\]\n\\item[C.]\nIf $\\ashift=0$ then\n\\begin{align*}\n0 = &\\altPhi(\\hf,\\hg;g(\\minfty),f(\\minfty)) \\\\ \n = &\\altPhi(\\hf,\\hg;g(\\pinfty),f(\\pinfty)) \\,.\n\\end{align*}\nand, for all $x_1,x_2$\n\\[\n\\altPhi(\\hf,\\hg;\\fg(x_2+),\\ff(x_1+)) = \n\\altPhiSI (\\smthker;f,g;x_1,x_2).\n\\]\n\\item[D.]\nFor $(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}$\n\\[\n\\min\\{0,A(\\hf,\\hg)\\}\\le\\altPhi(\\hf,\\hg;u,v) \\le \\max\\{0,A(\\hf,\\hg)\\}.\n\\]\n\\end{itemize}\n\\end{theorem}\n\\begin{IEEEproof}\nSince $\\fg_i \\rightarrow \\fg,$ $\\smthker_i \\rightarrow \\smthker$\nand $\\ashift_i \\rightarrow \\ashift$ we have from \\eqref{eqn:diffbound}\nthat $\\fg^{{{\\smthker}_i},\\ashift_i}_i \\rightarrow \\fg^{{\\smthker},\\ashift}$ point-wise.\n\nIf $x$ is a point of continuity of $f$ then $f_i(x) \\rightarrow f(x)$ and we have\n$(g^{{\\smthker}_i}(x+\\ashift_i),f_i(x))\\rightarrow (g^{\\smthker}(x+\\ashift),f(x))$ which implies\n$f(x) \\veq \\hf( \\gSa(x)).$ \nSince $\\gS$ is continuous we can extend this to all $x$ by taking limits.\nThis shows part A.\n\nPart B follows from part A by Lemma \\ref{lem:FPequal}.\n\nNow we consider part C where we assume $\\ashift=0.$\nBy Lemma \\ref{lem:FPequal} we need only show that\n$\\altPhi(\\hf,\\hg;g(\\pinfty),f(\\pinfty))=0.$\nFor any $\\epsilon>0$ \nwe can find $L$ large enough so that $\\int_L^\\infty \\smthker_i(x) dx < \\epsilon$ for all $i$ since $\\smthker_i\\rightarrow\\smthker.$\nNow choose $z$ large enough so that \n$f^\\smthker(z-L),f(z-L) > f(\\minfty)-\\epsilon$\nand\n$g^{\\smthker}(z-L),g(z-L) > g(\\minfty)-\\epsilon.$ \nIt follows that \n$\\Delta_L g(z) < \\epsilon$ and $\\Delta_L f(z) < \\epsilon.$\nFor all $i$ large enough we have $\\Delta_L g_i^{\\smthker_i}(z) < 2\\epsilon$ and \n$\\Delta_L f_i^{\\smthker_i}(z) < 2\\epsilon.$\n\nBy Lemma \\ref{lem:transitionPhiBounds} this implies\n\\[\n \\altPhi(\\hf^i,\\hg^i; \\fg_i^{\\smthker_i}(z),\\ff_i^{\\smthker_i}(z)) < 11\\epsilon\n\\]\nIt follows from \\eqref{eqn:diffbound} that $\\ff_i^{\\smthker_i} \\rightarrow \\fS$ \nand $\\fg_i^{\\smthker_i} \\rightarrow \\gS$ \npoint-wise\nand since \n$h_{[\\fg_i,\\ff_i^{\\smthker_i}]} \\rightarrow \\hg$\nand\n$h_{[\\ff_i,\\fg_i^{\\smthker_i}]} \\rightarrow \\hf$\nwe have for all $x_1,x_2,$\n\\[\n\\begin{split}\n& \\altPhi(\\smthker_i;\\hf^i,\\hg^i; g^{\\smthker_i}_i(x_2),f^{\\smthker_i}_i(x_1))\n\\\\ &\n\\rightarrow\n\\altPhi(\\smthker;\\hf,\\hg;g^{\\smthker}(x_2),f^{\\smthker}(x_1))\\,.\n\\end{split}\n\\]\nWe now obtain\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(z),f^{\\smthker}(z)) \\le 11\\epsilon\\,.\n\\]\nBy Lipschitz continuity of $\\altPhi$ we have\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(\\pinfty),f^{\\smthker}(\\pinfty)) < 13\\epsilon\n\\]\nand since $\\epsilon$ is arbitrary we obtain\n\\[\n \\altPhi(\\hf,\\hg; g^{\\smthker}(\\pinfty),f^{\\smthker}(\\pinfty)) = 0\n\\]\nIt now follows from Lemma \\ref{lem:twofint} and part A that\n\\[\n\\altPhi(\\hf,\\hg;g(x_2+),f(x_1+)) = \\altPhiSI(\\smthker;\\ff,\\fg;x_1,x_2)\n\\]\nfor all $x_1,x_2.$\n\nFinally, we show part D. \nIf $\\ashift=0$ then part $C$ gives part $D.$\nBy choosing a subsequence if necessary, we can assume that\n\\(\nh_{[\\ff_i,\\gSi_i]}\n\\)\nconverges to some $\\thf \\in \\exitfns.$\n\nWe assume $\\ashift > 0,$ the case $\\ashift<0$ is analogous.\nSince $\\thfinv \\le \\hfinv$ almost everywhere we have\n\\(\n\\altPhi(\\thf,\\hg;u,v) - \\altPhi(\\hf,\\hg;u,v) =\n\\int_0^{v} (\\thfinv(x) -\\hfinv(x))dx \\le 0\\,.\n\\)\nFor $(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}$\nwe have $\\altPhi(\\thf,\\hg;u,v)=0$ \nby part C,\nand therefore $\\altPhi(\\hf,\\hg;u,v) \\ge 0.$\n\nNow \n\\(\nA(\\hf,\\hg)-A(\\thf,\\hg) =\n\\int_0^{1} (\\hfinv(x) -\\thfinv(x))dx\n\\ge \n \\int_0^{v} (\\hfinv(x) -\\thfinv(x))dx\n\\)\nand since $A(\\thf,\\hg)=0,$ we have\n$\\altPhi(\\hf,\\hg;u,v) \\le A(\\hf,\\hg)$ for\n$(u,v)\\in\\{(\\ff(\\minfty),\\fg(\\minfty)),(\\ff(\\pinfty),\\fg(\\pinfty))\\}.$\nThis completes the proof.\n\\end{IEEEproof}\n\nThe following result is largely a corollary of the above but it is\nmore convenient to apply.\n\n\\begin{lemma}\\label{lem:limitexist}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition.\nIf there exists a sequence of $(0,1)$-interpolating $f_i,g_i \\in \\sptfns$ and bounded $\\ashift_i$ such that \n$(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$\nand $\\smthker_i \\rightarrow \\smthker$ in $L_1,$\nwhere $\\hf^i \\equiv h_{[\\ff_i,\\fg^{\\smthker_i,\\ashift_i}_i]}$ and\n$\\hg^i \\equiv h_{[\\fg_i,\\ff^{\\smthker_i}_i]},$\nthen\nthere exists $(0,1)$-interpolating $\\ff,\\fg \\in \\sptfns$ and finite $\\ashift,$ all limits of some translated subsequence,\nsuch that $\\hf = h_{[\\ff,\\gSa]}$ and\n$\\hg = h_{[\\fg,\\fS]}.$\n\\end{lemma}\n\\begin{IEEEproof}\nSince $\\smthker_i \\rightarrow \\smthker$ in $L_1$ and\n $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ we conclude from \nLemma \\ref{lem:stposbound} and Lemma \\ref{lem:shiftupperbound}\nthat $|\\ashift_i|$ is bounded.\n\nBy translating $\\ff$ and $\\fg$ as necessary, we can assume that $\\ff^{\\smthker_i}(0) = 1\/2$ for each $i.$\nTaking subsequences as necessary, we can now assume that $f_i \\rarrowi f,$\n$g_i \\rarrowi g,$ and $\\ashift_i \\rarrowi \\ashift,$ for some finite $\\ashift.$\n\nWe claim that $\\ff$ and $\\fg$ are $(0,1)$-interpolating.\nFor all $(u,v) \\in \\intcross(\\hf,\\hg)$ we have\n$\\altPhi(\\hf,\\hg;u,v) >\\max \\{0,A(\\hf,\\hg)\\}$\nby assumption.\nBy Theorem \\ref{thm:mainlimit} parts B and D we now have\n$(\\ff(\\minfty),\\fg(\\minfty)) \\in \\cross(\\hf,\\hg) \\backslash \\intcross(\\hf,\\hg) = \\{ (0,0),(1,1) \\}.$\nSince $\\fS(0) = \\frac{1}{2}$ we must have $(\\ff(\\minfty),\\fg(\\minfty)) = (0,0)$\nand $(\\ff(\\pinfty),\\fg(\\pinfty)) = (1,1),$\nproving the claim.\n\\end{IEEEproof}\n\n\\subsection{Existence of Consistent Spatial Waves}\n\nIn Section \\ref{sec:PCcase} we proved Theorem\n\\ref{thm:PCexist},\na special case of Theorem \\ref{thm:mainexist}\nin which $\\hg$ and $\\hf$ are piecewise constant\nfunctions and $\\smthker$ is $C_1$ and strictly positive.\nIn this section we show how to remove the special conditions \nto arrive at the general results.\nWe make repeated use of the limit theorems of Section \\ref{sec:limitthms} and develop\nsome approximations for functions in $\\exitfns.$\nIt is quite simple to approximate $h \\in \\exitfns$ using piecewise constant functions.\nThe challenge is to approximate a pair $(\\hg,\\hf)$ so that the strictly positive gap\ncondition is preserved.\n\n\\subsubsection{Approximation by Tilting}\n\nIn a manner analogous to \\eqref{eqn:pcscale} we define a perturbation of\n$\\hf,\\hg$ as $\\hf(;t),\\hg(;t)$ for $t \\in [0,1]$ by\n\\begin{align}\n\\begin{split}\n\\hfinv (v;t) & = (1-t) B_{\\hf} + t \\hfinv(v) \\\\\n\\hginv (u;t) & = (1-t) B_{\\hg} + t \\hginv(u)\\,,\\label{eqn:genscale}\n\\end{split}\n\\end{align}\nwhere we recall $B_h = \\int_0^1 h^{-1}(x) \\,dx.$\nThis can also be expressed as\n\\begin{align}\n\\begin{split}\n\\hf (u;t) & = \\hf\\Bigl(\\frac{u-B_{\\hf}}{t} + B_{\\hf} \\Bigr) \\\\\n\\hg (v;t) & = \\hg\\Bigl(\\frac{u-B_{\\hg}}{t} + B_{\\hg}\\Bigr) \\,,\\label{eqn:genscalefor}\n\\end{split}\n\\end{align}\nwith appropriate extension of $\\hf$ and $\\hg$ outside of $[0,1],$\n$\\hf(x)=\\hg(x)=0$ for $x<0$ and\n$\\hf(x)=\\hg(x)=1$ for $x>1.$\n\nLetting $h$ denote either $\\hf$ or $\\hg,$ we clearly have\n\\begin{align}\n\\int_0^1 h^{-1} (x;t) dx = B_h\\label{eqn:slanteq}\n\\end{align}\nfor all $t.$\nNote also that \n\\(\nh^{-1}(v;t) -h^{-1} (v) =\n(1-t)(B_h-h^{-1} (v))\n\\)\nis non-increasing in $v.$\nIt follows that\n\\(\n\\int_0^v h^{-1}(x;t) dx \\ge \n\\int_0^v h^{-1} (x) dx \n\\)\nfor all $v\\in[0,1]$\nand we obtain\n\\begin{align}\n\\altPhi(\\hf(;t),\\hg(;t);)\\ge \\altPhi(\\hf,\\hg;)\\label{eqn:altslbound}\n\\end{align}\nfor all $t\\in [0,1].$\n\n\\begin{lemma}\\label{lem:smoothcompress}\nLet $(\\hg,\\hf) \\in \\exitfns^2$ \nsatisfy the strictly positive gap condition.\nThen, there exists $\\epsilon > 0$ such that\n$(\\slanta{\\hg},\\slanta{\\hf})$ \nsatisfies the strictly positive gap condition\nfor any $t \\in (1-\\epsilon,1].$\n\\end{lemma}\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:smoothcompress}]\nFor the case $A(\\hf,\\hg)=0$ equation \\eqref{eqn:slanteq}, inequality \\eqref{eqn:altslbound} and\nLemma \\ref{lem:monotonic} gives\nthe result immediately.\nBy symmetry we now need only consider the case $A(\\hf,\\hg) > 0.$\n\nBy Lemma \\ref{lem:Sstructure} and \\eqref{eqn:altslbound}\nit is sufficient to show that \n$\\intcross(\\hf(;t),\\hg(;t)) \\cap \\closure{S(\\hf,\\hg)}=\\emptyset$\nfor $t\\in[1-\\epsilon,1].$\n\nAlso by Lemma \\ref{lem:Sstructure}, there exists a\nminimal and positive element $(u^*,v^*) \\in \\intcross(\\hf,\\hg).$\nThere exists a neighborhood ${\\cal N}$ of $(0,0),$ which we take to be a subset of\n$\\subset [0,u^*)\\times [0,v^*),$\nin which\n$\\hginv(u;t) \\ge \\hginv(u)$ and \n$\\hfinv(v;t) \\ge \\hfinv(v).$ \nIt follows that \n$\\intcross(\\hf(;t),\\hg(;t)) \\cap {\\cal N} =\\emptyset$\nfor all $t.$\n\nLet $\\delta>0$ be small enough so that $\\neigh{(0,0)}{\\delta} \\subset {\\cal N}$\nand $\\neigh{(x^*,y^*)}{\\delta} \\cap \\closure{S(\\hf,\\hg)}=\\emptyset.$\nFor $\\epsilon$ small enough and $t\\in[1-\\epsilon,1]$ we have\n$\\cross(\\hf(;t),\\hg(;t))\\subset \\neigh{\\cross(\\hf,\\hg)}{\\delta}$\nby Lemma \\ref{lem:crosspointlimit}\nand it now follows that\n$\\intcross(\\hf(;t),\\hg(;t)) \\cap \\closure{S(\\hf,\\hg)}=\\emptyset.$\n\\end{IEEEproof}\n\n\\subsubsection{Piecewise Constant Approximation}\nGiven $h \\in \\exitfns$ let us define a sequence of piecewise constant approximations\n$Q_n(h),$ $n=1,2,...$ by\n\\begin{align*}\nQ_n(h) (x) &= \\sum_{j=1}^n \\frac{1}{n} \\,\\unitstep(x- u_{n,j}) \n\\end{align*}\nwhere we set \n\\begin{align*}\nu_{n,j} \n& = n\\int_{(j-1)\/n}^{j\/n}h^{-1}(v)dv\n\\end{align*}\nand we have\n\\begin{align*}\n\\int_0^1 Q_n(h) (x) dx &\n= \\sum_{j=1}^n \\frac{1-u_{n,j}}{n} \\\\ \n& = \\int_0^1 (1-h^{-1})(x) dx \\\\\n&= \\int_0^1 h (x) dx.\n\\end{align*}\nIn general, it holds that\n\\(\n\\int_0^1 Q_n(h) (x) dx = \\int_0^1 h (x) dx.\n\\)\nIt also follows that $\\int_0^z Q_n(h) (x) dx \\le \\int_0^z h (x) dx$ for all\n$z \\in [0,1].$\n\n\\begin{lemma}\\label{lem:PCapprox}\nLet $(\\hg,\\hf)$ be pair of functions in $\\exitfns$ satisfying the strictly positive gap condition \nsuch that for some $\\eta>0$ we have\n$\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nThen, for all $n$ sufficiently large\n$(Q_n(\\hg),Q_n(\\hf))$ satisfies the strictly positive gap condition.\n\\end{lemma}\n\\begin{IEEEproof}\nWe have\n\\(\nA(Q_n(\\hf),Q_n(\\hg)) =\nA(\\hf,\\hg)\n\\)\nand\n\\(\n\\altPhi(Q_n(\\hf),Q_n(\\hg);\\cdot,\\cdot) \\ge\n\\altPhi(\\hf,\\hg;\\cdot,\\cdot)\n\\)\nso it suffices to show that \n\\(\n\\intcrossing (Q_n(\\hf),Q_n(\\hg)) \\cap \\closure{S(\\hf,\\hg)}=\\emptyset.\n\\)\n\nSince $\\hg$ and $\\hf$ are $0$ on $[0,\\eta)$ and $1$ on $(1-\\eta,1]$ it follows that \n$\\intcrossing (\\hf,\\hg) \\subset [\\eta,1-\\eta]^2$ and\n$\\intcrossing (\\hf,\\hg)$ is closed and by Lemma \\ref{lem:Sstructure} it is disjoint from\n$\\closure{S(\\hf,\\hg)}.$\nThus, for $\\delta$ sufficiently small we have $\\neigh{\\intcrossing(\\hf,\\hg)}{\\delta}\\cap \\closure{S(\\hf,\\hg)}=\\emptyset.$\n\nBy Lemma \\ref{lem:crosspointlimit} we now have\n\\(\n\\intcrossing (Q_n(\\hf),Q_n(\\hg)) \\cap \n\\closure{S(\\hf,\\hg)}=\\emptyset\n\\)\nfor all $n$ sufficiently large.\n\\end{IEEEproof}\n\nWe are now ready to prove the main result of this section.\n\n\\begin{lemma}\\label{lem:weakexistence}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition.\nThen there exists $(0,1)$-interpolating $(\\ff,\\fg) \\in \\sptfns^2$ and $\\ashift$ such that\n$\\hf = h_{[\\ff,\\gSa]}$\nand\n$\\hg = h_{[\\fg,\\fS]}.$\n\\end{lemma}\n\\begin{IEEEproof}\nThe simplest case is already established in Theorem \\ref{thm:PCexist}\nand we first generalize to arbitrary $\\smthker.$\nAssume that $(\\hg,\\hf)$ are both piecewise constant.\nDefine $\\smthker_k = \\smthker \\otimes G_k$ where \n$G_i(x) = \\frac{k}{\\sqrt{2\\pi}} e^{- (kx)^2\/2}.$\nIt follows that $\\smthker_k \\rightarrow \\smthker$ in $L_1$\nand $\\| \\smthker_k \\|_\\infty \\le \\| \\smthker\\|_\\infty.$\nFor each $\\smthker_k$ we apply Theorem \\ref{thm:PCexist}\nto obtain piecewise constant $f_k,g_k \\in \\sptfns$ \n(with corresponding $z^{f_k},z^{g_k}$) and constants $\\ashift_k$\nsuch that $ h_{[\\fg_k,\\ff_k^{{\\smthker}_k}]} = \\hg$ and $h_{[\\ff_k,g_k^{{\\smthker}_k,\\ashift_k}]} = \\hf.$\nWe can now apply Lemma \\ref{lem:limitexist} to conclude\nthat the theorem holds for piecewise constant $\\hf,\\hg$ and general $\\smthker.$\n\nLet now assume first that for some $\\eta>0$ we have $\\hf(x)=\\hg(x)=0$ for $x\\in [0,\\eta)$\nand $\\hf(x)=\\hg(x)=0$ for $x\\in (1-\\eta,1].$\nConsider $Q_n(\\hf)$ and $Q_n(\\hg).$ \nWe apply Lemma \\ref{lem:PCapprox} and the preceding case already established\nto conclude that for all $n$ sufficiently large\nthere exists (piecewise constant) $(0,1)$-interpolating $f_n,g_n \\in \\sptfns$ and finite constants $\\ashift_n$ such that\n$h_{[\\fg_n,\\fS_n]} = Q_n(\\hg)$ and $h_{[\\ff_k,\\fg_k^{{\\smthker}_k,\\ashift_n}]}= Q_n(\\hf).$\nSince $Q_n(\\hg)$ and $Q_n(\\hf)$ converge to $\\hg$ and $\\hf$ respectively,\nwe can apply Lemma \\ref{lem:limitexist} to conclude that the theorem holds for this case.\n\nFor arbitrary $(\\hg,\\hf)$ we consider $(\\slanta{\\hg},\\slanta{\\hf}).$\n\nBy Lemma \\ref{lem:smoothcompress} we can find a sequence $t_i \\rightarrow 1$\nsuch $(\\hg(;t_i),\\hf(;t_i))$ satisfies the strictly positive gap condition for each $i.$\nBy the preceding case, there exists $f_{t_i},g_{t_i} \\in \\sptfns$ and finite constants $\\ashift_i$ such that\n$h_{[\\fg_{t_i},\\fS_{t_i}]} = \\hg(;t_i)$ and $h_{[\\ff_{t_i},\\fg_{t_i}^{{\\smthker},\\ashift_i}]} = \\hf(;t_i).$\nBy taking a subsequence if necessary we can assume that $\\ashift_i \\rightarrow \\ashift.$\nSince $(\\hf(;t_i),\\hg;(t_i))\\rightarrow (\\hf,\\hg)$ we can apply\nLemma \\ref{lem:limitexist} to obtain $(\\ff,\\fg) \\in \\sptfns^2$\nsuch that $\\hf = h_{[\\ff,\\gSa]}$ and $\\hg = h_{[\\fg,\\fS]}.$\n\\end{IEEEproof}\n\\subsection{Existence of Spatial Wave Solutions}\n\nIn the preceeding section we estabished the existence of consistent spatial waves\nunder general conditions. In this section we refine the results to obtain full\nspatial wave solutions. Thus, in this section we complete the proof of \n\\ref{thm:mainexist}.\n\n\\subsubsection{Analysis of Consistent Spatial Waves}\n\nFor $h \\in \\exitfns$ we use $\\jump{h}$ to denote the set of discontinuity points of\n$h,$ i.e.,\n\\[\n\\jump{h} = \\{ u\\in[0,1]:h(u-)0$ (where $\\mu$ is Lebesgue measure)\nwe have\n$\\mu (\\{x:\\ff(x) \\neq \\hf(\\fg(x))\\}\\cap [\\fginv(v-),\\fginv(v+)]) >0$\nfor some $v\\in \\jump{\\hf}.$\nThen $I = [\\fginv(v-),\\fginv(v+)] \\in \\flats{\\fg}.$\n\\end{IEEEproof}\n\n\n\n\n\\begin{lemma}\\label{lem:pathology}\nLet $\\ff,\\fg \\in \\sptfns$ be $(0,1)$-interpolating.\nIf $h_{[\\ff,\\gSa]} = \\hf$ and\n$h_{[\\fg,\\fS]} = \\hg$ \nthen $\\ff = \\hf\\circ\\gSa$\nand $\\fg = \\hg\\circ\\fS$\nin any of the following scenarios.\n\\begin{itemize}\n\\item[A.] $\\hf$ and $\\hg$ are continuous.\n\\item[B.] $\\smthker$ is positive on all $\\reals.$\n\\item[C.] $\\smthker$ is regular,\n$\\ashift = 0,$\nand $(\\hf,\\hg)$ satisfies the strictly positive gap condition.\n\\item[D.] $\\smthker$ is regular, \n$(\\hf,\\hg)$ satisfies the strictly positive gap condition\nand\n$\\jump{\\hf} \\cap \\jump{\\hginv}=\\emptyset$\nand\n$\\jump{\\hg} \\cap \\jump{\\hfinv}=\\emptyset.$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nIf $\\hf$ is continuous then $\\jump{\\hf}=\\emptyset$ and,\nby Lemma \\ref{lem:notequal},\n$\\ff \\veq\\hf \\circ \\gSa$ then implies\n$\\ff = \\hf \\circ \\gSa$\nThus, case A is clear.\n\nIf $\\smthker(x)>0$ for all $x$ then $\\gS(x)$ is increasing (strictly) on $\\reals$\nand takes values in $(0,1).$\nHence $\\gSinv(v-)=\\gSinv(v+)$ for all $v\\in (0,1)$ and we see by\nLemma \\ref{lem:notequal} that\n$\\ff \\veq\\hf \\circ \\gSa$ then implies\n$\\ff = \\hf \\circ \\gSa.$\nThis shows case B.\n\nAssume $\\smthker$ is regular\nand $\\ashift=0.$ \nIf $\\gSx(x_1) = 0$ then\nLemma \\ref{lem:transitionPhiBounds} yields\n$\\altPhiSI(\\smthker,\\ff,\\fg;x_1,x_1) = 0$\nwhich implies\n$\\altPhi(\\hf,\\hg;\\ff(x_1),\\fg(x_1))=0.$\nBy Lemma \\ref{lem:monotonic} this violates the strictly positive gap condition if $\\gS(x_1) \\in (0,1)$\nso the condition implies that $\\gS$ is strictly increasing on $\\{x: 0 < \\gS(x) < 1 \\}.$\nSince $\\hf$ is continuous at $0$ and $1$ by Lemma\n\\ref{lem:zocontinuity}, part C now follows from Lemma \\ref{lem:notequal}.\n\nTo show part D \nassume $\\smthker$ is regular and that\n$(\\hf,\\hg)$ satisfies the strictly positive gap condition.\nAssume $\\ff \\not\\equiv \\hf \\circ\\gSa.$\nWe have $\\ff \\veq \\hf \\circ\\gSa$ so\nwe apply Lemma \\ref{lem:notequal} to obtain $I-\\ashift \\in\\flats{\\gSa}$ (so $I \\in \\flats{\\gS}$)\nsuch that \n$\\gSa(I-\\ashift) \\in \\jump{\\hf}$\nand such that $\\ff(x) \\neq \\hf(U)$ on a set of positive measure in\n$I-\\ashift.$\nLet us denote $\\gSa(I-\\ashift) =\\gS(I)$ by $U.$\nWe claim that $\\fS$ is not constant on $\\neigh{I}{W}$ and hence we have\n$U \\in \\jump{\\hginv}.$\n\nTo prove the claim assume $\\fS$ is a constant $V$ on $\\neigh{I}{W}.$ Then $\\ff(x)=V$ on\n $\\neigh{I}{2W}$ which,\nsince $|\\ashift|<2W$ by Corollary \\ref{cor:regshiftbound},\n gives $\\ff(x)=V$ on $\\neigh{I-\\ashift}{\\delta}$\nfor some $\\delta>0.$ \nThis, however, implies $\\hf(U)=V$ and $U \\not\\in \\jump{\\hf},$ which is a contradiction.\n\nHence $\\ff \\neq \\hf \\circ\\gSa$ implies $\\jump{\\hf} \\cap \\jump{\\hginv}\\neq\\emptyset.$\nSimilarly,\n$\\fg \\neq \\hg \\circ\\fS$ implies $\\jump{\\hg} \\cap \\jump{\\hfinv}\\neq\\emptyset.$\n\\end{IEEEproof}\n\n\n\n\n\\subsubsection{Proof of Theorem \\ref{thm:mainexist}}\n\nSince we assume that $\\smthker$ is regular \nLemma \\ref{lem:pathology} shows that Lemma \\ref{lem:weakexistence}\nimplies Theorem \\ref{thm:mainexist} except in the case\n$\\jump{\\hf}\\cap\\jump{\\hginv} \\neq \\emptyset$ or\n$\\jump{\\hg}\\cap\\jump{\\hfinv} \\neq \\emptyset.$\nIt turns out that this case can be handled by constructing $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$\nwith certain properties. The argument is lengthy and is relegated to appendix \\ref{app:B}.\n\n\n\n\n\n\n\\appendices\n\n\\section{Continuum Spatial Fixed Point Integration}\\label{app:A}\n\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:twofint}]\n\nWe assume that the smoothing kernel $\\smthker$ has finite total variation hence \n$\\|\\smthker\\|_\\infty < \\infty.$ \nFor any $\\ff \\in \\sptfns$ a simple calculation shows that $\\fS(x)-\\fS(y) \\le \\|\\smthker\\|_\\infty |x-y|.$\nThis implies that and $\\fS$ is Lipschitz continuous with Lipschitz constant $\\|\\smthker\\|_\\infty.$ \n\nThus, $\\gSx$ and $\\fSx,$ the derivatives of $\\gS$ and $\\fS,$ exist for almost all $x$ and $\\gS$ and $\\fS$ are absolutely continuous.\nIf $\\smthker(x-y)$ and $g(y)$ do not have in common any points of discontinuity in $y$,\nthen $\\gSx(x) = \\int_{-\\infty}^\\infty \\smthker(x-y)\\,dg (y)$ where the right hand side is\na Lebesgue-Stieltjes integral. The integral is well defined as long as $\\smthker(x-y)$ and $g(y)$ do not have\nany discontinuity points (in $y$) in common. The set of $x$ at which this can occur is countable.\nMore generally in the sequal we will have integrals in the form \n$\\int_{(a,b]} g(x) df(x).$\nThe integral is well defined as long as $g(x)$ and $f(y)$ do not have\nany discontinuity points in common.\nWe define the integral so that \n$\\int_{(a,b]} df(x) = f(b+)-f(a+).$\nThis holds even if $a \\ge b.$\n\nWe now have\n\\begin{align*}\n& \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \\; \n\\\\ & = \n\\; \\int_{-\\infty}^{x_2} f(x) \\gSx(x) dx\n\\\\& =\n\\int_{-\\infty}^{x_2} f(x) \n\\Bigl(\n\\int_{-\\infty}^\\infty \\smthker(x-y)\\,\ndg (y)\n\\Bigr)\ndx\\,.\n\\end{align*}\nSince, $\\int_{-\\infty}^{x_2} f(x)\\smthker(x-y)dx \\le \\fS(y)$ we see that the Fubini theorem can\nbe applied \nand we obtain\n\\begin{align*}\n& \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \\; \n\\\\ & = \n\\int_{-\\infty}^\\infty \n\\Bigl(\n\\int_{-\\infty}^{x_2} f(x) \n\\smthker(x-y)\\,\ndx\\,\n\\Bigr)\ndg (y)\n\\\\ & = \n\\int_{-\\infty}^\\infty \n\\Bigl(\n\\int_{-\\infty}^{x_2-y} f(x+y) \n\\smthker(x)\\,\ndx\\,\n\\Bigr)\ndg (y)\n\\\\ & =\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2-x} \nf(y+x) \ndg(y)\\,\\Bigr)\ndx\\,\n\\end{align*}\nwhere the inner integral is defined for almost all $x.$\nSimilarly,\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \\; \n\\\\& = \n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\\,\\Bigr)\ndx\\,\n\\end{align*}\nWe can now exploit the evenness of $\\smthker$ to replace $x$ with $-x$ in the above\nand $\\smthker(-x)$ with $\\smthker(x)$ to write\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\&=\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2-x} \nf(y+x) \ndg(y)\n+\\int_{-\\infty}^{x_1+x} \ng(y-x) \ndf(y)\n\\,\\Bigr)\ndx\\,\n\\\\ \n&\n=\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{-\\infty}^{x_2+x} \nf(y-x) \ndg(y)\n+\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\n\\,\\Bigr)\ndx\\,\n\\end{align*}\nConsider the second form, in which we have the expression\n\\begin{align*}\n& \\quad \\int_{-\\infty}^{x_2+x} f(y-x) dg(y)\n+\n\\int_{-\\infty}^{x_1-x} g(y+x) df(y) \\,.\n\\end{align*}\nWith a slight abuse of notation, we may write \n\\(\n\\int_{-\\infty}^{x_1-x} g(y+x) df(y) \\,\n\\)\nas\n\\(\n\\int_{-\\infty}^{x_1} g(y) df(y-x) \\,\n\\)\nand we see that for almost all $x$ we have\n\\begin{align*}\n&\\int_{-\\infty}^{x_2+x} \nf(y-x) \ndg(y)\n+\\int_{-\\infty}^{x_1-x} \ng(y+x) \ndf(y)\n\\\\&=\ng(x_1+) f(x_1-x) - g(\\minfty)f(\\minfty) +\n\\int_{(x_1,x_2+x]} f(y-x) dg(y)\\,\n\\\\&=\ng(x_1+) f(x_1-x) + \nf(x_2+)(g(x_2+x)-g(x_1+))\n\\\\\n&\\quad - f(\\minfty)g(\\minfty) -\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,.\n\\end{align*}\nThus, we obtain\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\=&\ng(x_1+) \\fS(x_1) \n+f(x_2+)\\gS(x_2)\n\\\\&-f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nThe same analysis applied to the first form gives\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)}h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n\\\\= &\ng(x_1+) \\fS(x_1) \n+f(x_2+)\\gS(x_2)\n\\\\ & -f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{-\\infty}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x)) df(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nFinally, by combining the two forms and integrating from $0$ to $\\infty$ we obtain\n\\begin{align*}\n& \\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n+ \\int_{\\gS(\\minfty)}^{\\gS(x_2)}h_{[\\ff,\\gS]}(u) du \n\\\\= &\ng(x_1+) \\fS(x_1) \n+f(x_2+)\\gS(x_2)\n\\\\&-f(x_2+)g(x_1+)\n- f(\\minfty)g(\\minfty) \n\\\\& -\n\\int_{0}^\\infty\n\\smthker(x)\n\\Bigl(\n\\int_{(x_2,x_1+x]} (g(x_1+)-g(y-x)) df(y)\\,\n\\\\ & \\qquad +\n\\int_{(x_1,x_2+x]} (f(x_2+)-f(y-x)) dg(y)\\,\n\\,\\Bigr)\ndx\\,.\n\\end{align*}\nNow note that\n\\begin{align*}\n&\\int_{\\fS(\\minfty)}^{\\fS(x_1)} h_{[\\fg,\\fS]}(v) dv \n \\\\ = & \\fS(x_1)\\fg(x_1+) - \\fS(\\minfty)\\fg(\\minfty)\n\\\\ & -\\int_{\\fg(\\minfty)}^{\\fg(x_1+)} h^{-1}_{[\\fg,\\fS]}(u) du\n\\end{align*} \nand, similarly,\n\\begin{align*}\n&\\int_{\\gS(\\minfty)}^{\\gS(x_2)} h_{[\\ff,\\gS]}(u) du \n \\\\ = & \\gS(x_2)\\ff(x_2+) - \\gS(\\minfty)\\ff(\\minfty)\n\\\\ & -\\int_{\\ff(\\minfty)}^{\\ff(x_2+)} h^{-1}_{[\\ff,\\gS]}(u) du\n\\end{align*} \nand use the fact that $\\fS(\\minfty)=\\ff(\\minfty)$ and\n$\\gS(\\minfty)=\\fg(\\minfty)$\nand the desired result follows.\n\\end{IEEEproof}\n\n\\subsection{Fixed Point Bounds on Potentials}\n\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:transitionPhiBounds}]\n\nAs a first bound use the evenness of $\\smthker$ to write\n\\begin{align*}\n\\fS(x) &= \\int_{-\\infty}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&= \\int_{-\\infty}^{L} \\smthker(y)f(x+y)\\text{d}y\n+ \\int_{L}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\le (1-e_L) f((x+L)-) + e_L\n\\\\\n&\\le f((x+L)-) + e_L (1-f((x+L)-))\n\\\\\n&\\le f(x) + \\Delta_L f(x) + e_L (1- f(x))\n\\end{align*}\nand\n\\begin{align*}\n\\fS(x) &= \\int_{-\\infty}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\ge\n\\int_{-L}^{\\infty} \\smthker(y)f(x+y)\\text{d}y\n\\\\\n&\\ge (1-e_L) f((x-L)+)\n\\\\\n&\\ge f(x) - \\Delta_L f(x) - e_L f(x)\\,.\n\\end{align*}\n\nTo get a bound on\n$\\altPhiSI(\\smthker;f,g;x_1,x_1)$ we proceed similarly\n\\begin{align*}\n&\n\\int_{-\\infty}^\\infty \\smthker(x)\n\\Bigl(\n\\int_{(x_1,x_1+x]} (g(x_1)-g(y-x))df(y)\n\\Bigr) dx\n\\\\\n\\le&\n\\int_{-L}^L \\smthker(x)\n\\Bigl(\n\\Delta_L g(x_1) |f(x_1+x)-f(x_1)|\n\\Bigr)\n\\text{d}x\\\\\n&+\\int_L^\\infty\\smthker(x) ( f(x_1+x)-f(x_1-x)) dx\n\\\\\n\\le&\n\\int_{-L}^L \\smthker(x)\n\\Bigl(\n \\Delta_L g(x_1) \\Delta_L f(x_1)\n\\Bigr) dx +e_L\n\\\\\n\\le&\\\n \\Delta_L g(x_1) \\Delta_L f(x_1)\n+e_L\n\\end{align*}\n\nThus, we obtain the bound\n\\begin{align*}\n\\altPhi(\\fg(x),\\ff(x))\n&\\le\n\\Delta_L f(x)\\Delta_L g(x)\n+ e_L\n\\end{align*}\nand since, by \\eqref{eqn:altPhiderivatives},\n\\begin{align*}\n|\\altPhi(\\fg(x),\\fS(x))\n-\\altPhi(\\fg(x),f(x)) |\n&\\le\n|\\fS(x)-f(x)|\n\\\\\n&\\le\n\\Delta_L f(x) + e_L\n\\end{align*}\nthe other bounds follow easily.\n\\end{IEEEproof}\n\n\\section{Discrete Spatial Fixed Point Integration}\\label{app:Aa}\n\nIn this section we show how spatial integration can be done directly in the \nspatially discrete setting.\nFirst, an unusual point of notation.\nLet $\\dv{g}_i$ be defined for all integers $i.$\nWe will occasionally write a sum as\n\\(\n\\sum_{i\\in (a,b]} \\dv{g}_i\\,.\n\\)\nIn the case where $ab$ the sum is\n\\(\n-\\sum_{i\\in (b,a]} \\dv{g}_i= -\\sum_{i=b+1}^a \\dv{g}_i\\,.\n\\)\nNote that\n\\(\n\\sum_{i\\in (a,b]} (\\dv{g}_i-\\dv{g}_{i-1}) = \\dv{g}_b - \\dv{g}_a\n\\) \nfor all choices of $a$ and $b.$\nFurther, we can write\n\\begin{align*}\n&\\sum_{i=-\\infty}^a \\dv{g}_i + \\sum_{i=-\\infty}^b \\dv{f}_i\n\\\\ =&\n\\sum_{i=-\\infty}^a (\\dv{g}_i+\\dv{f}_i) + \\sum_{i\\in(a,b]} \\dv{f}_i\n\\end{align*}\nregardless of $a$ and $b.$\n\nThe spatially smoothed $\\dv{g}$ will be denoted $\\dv{\\gSdisc}$ and is defined by\n\\[\n\\dv{\\gSdisc}_i = \\sum_{j=-W}^W \\discsmthker_j \\dv{g}_{i-j}\n\\]\nwhere we require $\\discsmthker_j = \\discsmthker_{-j}$ and $\\discsmthker_j \\ge 0$ and $\\sum_j \\discsmthker_j =1.$\n\nFirst we note\n\\begin{align*}\n& \\sum_{i=-\\infty}^{k} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} )\n\\\\\n&\\qquad+\\sum_{i=-\\infty}^{k}( \\dv{g}_{i+j}+\\dv{g}_{i+j-1}) (\\dv{f}_{i} - \\dv{f}_{i-1} )\n\\\\ &\n= \\sum_{i=-\\infty}^{k} 2 \\dv{f}_i \\dv{g}_{i+j} - 2 \\dv{f}_{i-1} \\dv{g}_{i+j-1} \n- \\dv{f}_i \\dv{g}_{i+j-1} \n\\\\\n& \\qquad+ \\dv{f}_{i} \\dv{g}_{i+j-1} \n+ \\dv{f}_{i-1} \\dv{g}_{i+j} - \\dv{f}_{i-1} \\dv{g}_{i+j} \n\\\\ &\n= 2 (\\dv{f}_{k} \\dv{g}_{k+j}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\end{align*}\nand we obtain\n\\begin{align}\n\\begin{split}\\label{eqn:sumcutA}\n&\\sum_{i=-\\infty}^{i_2}( \\dv{f}_i + \\dv{f}_{i-1})(\\dv{g}_{i+j} - \\dv{g}_{i+j-1} )\n\\\\+& \\sum_{i=-\\infty}^{i_1}( \\dv{g}_i + \\dv{g}_{i-1}) (\\dv{f}_{i-j} - \\dv{f}_{i-j-1} )\n\\\\ \n=\\quad &\\qquad\n 2 (\\dv{f}_{i_1-j} \\dv{g}_{i_1}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+&\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{split}\n\\end{align}\n\n\nWe now can write\n\\begin{align*}\n& \\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\+ &\\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\n\\\\ =\n\\sum_{j=-W}^W \\discsmthker_j \n \\Bigl( &\n \\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1})\n\\\\+& \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1})\n\\Bigr)\n\\\\ =\n\\sum_{j=-W}^W \\discsmthker_j \n\\Bigl( &\n 2(\\dv{f}_{i_1-j} \\dv{g}_{i_1} - \\dv{f}_{-\\infty} \\dv{g}_{-\\infty} )\n\\\\+ &\n\\sum_{i\\in(i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\Bigr)\n\\\\\n=\\qquad\n&\n2( \\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1}\n- \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+\\sum_{j=-W}^W \\discsmthker_j &\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\\n=\\qquad\n&\n 2(\\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1}\n- \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+&2 \\dv{f}_{i_2}(\\dv{\\gSdisc_{i_2}} - \\dv{g}_{i_1})\n\\\\ +\\sum_{j=-W}^W \\discsmthker_j &\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}-2\\dv{f}_{i_2}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\\\\n=\\qquad\n&\n 2 (\\dv{\\fSdisc_{i_1}} \\dv{g}_{i_1} \n+ \\dv{\\gSdisc_{i_2}} \\dv{f}_{i_2} - \\dv{f}_{i_2}\\dv{g}_{i_1}\n- \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\+\\sum_{j=-W}^W \\discsmthker_j&\\sum_{i\\in (i_1-j,i_2]} (\\dv{f}_i+\\dv{f}_{i-1}-2\\dv{f}_{i_2}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{align*}\n\nAnd finally we have the result\n\\begin{align*}\n&2(\\dv{\\fSdisc_{i_1}} \\dv{\\gSdisc_{i_2}} - \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\-\n& \\Bigl(\\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\&+ \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\\Bigr)\n\\\\ =\\quad\n&\n2( \\dv{\\fSdisc_{i_1}} -\\dv{f}_{i_2}) \n( \\dv{\\gSdisc_{i_2}} - \\dv{g}_{i_1})\n\\\\&+\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_1-j,i_2]} (2\\dv{f}_{i_2}-\\dv{f}_{i}-\\dv{f}_{i-1}) (\\dv{g}_{i+j} - \\dv{g}_{i+j-1} ) \n\\end{align*}\n\nNow, if in place of \\eqref{eqn:sumcutA} we write,\n\\begin{align*}\n&\\sum_{i=-\\infty}^{i_2}( \\dv{f}_i + \\dv{f}_{i-1})(\\dv{g}_{i-j} - \\dv{g}_{i-j-1} )\n\\\\+& \\sum_{i=-\\infty}^{i_1}( \\dv{g}_i + \\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} )\n\\\\ \n=\\quad&\\qquad 2( \\dv{g}_{i_2-j} \\dv{f}_{i_2}-\\dv{f}_{-\\infty} \\dv{g}_{-\\infty}) \n\\\\+&\\sum_{i\\in (i_2-j,i_1]} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\nand subsequently proceed similarly, then we obtain\n\\begin{align*}\n&2(\\dv{\\fSdisc_{i_1}} \\dv{\\gSdisc_{i_2}} - \\dv{f}_{-\\infty} \\dv{g}_{-\\infty})\n\\\\-\n& \\Bigl(\\sum_{i=-\\infty}^{i_2} (\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{\\gSdisc}_i - \\dv{\\gSdisc}_{i-1})\n\\\\&+ \\sum_{i=-\\infty}^{i_1} (\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{\\fSdisc}_i - \\dv{\\fSdisc}_{i-1})\\Bigr)\n\\\\ =\\quad\n&\n2( \\dv{\\fSdisc_{i_1}} -\\dv{f}_{i_2}) \n( \\dv{\\gSdisc_{i_2}} - \\dv{g}_{i_1})\n\\\\&+\\sum_{j=-W}^W \\discsmthker_j\\sum_{i\\in (i_2-j,i_1]} (2\\dv{g}_{i_1}-\\dv{g}_{i}-\\dv{g}_{i-1}) (\\dv{f}_{i+j} - \\dv{f}_{i+j-1} ) \n\\end{align*}\n\n\\subsection{Discrete-Continuum Relation}\n\nIn this section we obtain tighter bounds on the dependence on $\\Delta.$\n\\begin{lemma}\\label{lem:discreteFPsum}\nFor a spatially discrete fixed point for the regular ensemble\nwe have\n\\[\n|A(\\dv{f}_\\infty,\\dv{g}_\\infty) - A(\\dv{f}_{-\\infty},\\dv{g}_{-\\infty})| \\le \n\\frac{1}{2} (\\|\\hf''\\|_\\infty+\\|\\hg''\\|_\\infty)\\|\\smthker\\|_\\infty^2{\\Delta^2}\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nAs before, we associate to the discrete spatial index $i$ the real valued point $x_i = i\\Delta.$\nWe assume that $\\smthker$ is the piecewise constant extension of\n$\\discsmthker.$ Thus, \\eqref{eqn:kerdiscretetosmth} holds trivially.\n\nAssume a spatially discrete fixed point $\\ff,\\fg.$\nLet $\\tff$ and $\\tfg$ be the piecewise extensions of $\\ff$ and $\\fg.$\nWe can now relate the discrete spatial EXIT sum to the corresponding continuum integral\nto arrive at approximate fixed point conditions for spatially discrete fixed points.\n\nThe discrete sum\n\\begin{align*}\n&\\frac{1}{2}\\sum_{i=-\\infty}^i\n(\\dv{f}_i+\\dv{f}_{i-1}) (\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\& =\n\\int_{-\\infty}^{x_i}\n\\dv{\\tff}(x) \\text{d} \\dv{\\tgS}(x)\n\\\\ & =\n\\int_{-\\infty}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\end{align*}\nand, similarly,\n\\begin{align*}\n&\\frac{1}{2}\\sum_{i=-\\infty}^i\n(\\dv{g}_i+\\dv{g}_{i-1}) (\\dv{f}_i^\\discsmthker-\\dv{f}_{i-1}^\\discsmthker)\n\\\\& =\n\\int_{-\\infty}^{x_i}\n\\dv{\\tfg}(x) \\text{d} \\dv{\\tfS}(x)\n\\\\\n& =\n\\int_{-\\infty}^{x_i}\nh_{[\\tfg,\\tfS]}\n(\\dv{\\tfS}(x)) \\text{d} \\dv{\\tfS}(x)\n\\end{align*}\nWe want to compare\n\\(\n\\int_{x_{i-1}}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\)\nto\n\\(\n\\int_{x_{i-1}}^{x_i}\n\\hf\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\\,.\n\\)\n\nWe have\n\\begin{align*}\n&\\int_{x_{i-1}}^{x_i}\nh_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\\\& =\n\\frac{1}{2}(\n\\hf(\\dv{g}_i^\\discsmthker)+\n\\hf(\\dv{g}^\\discsmthker_{i-1}))(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\ & = \n\\int_0^1 (\\alpha\n\\hf(\\dv{g}_i^\\discsmthker)+\n\\bar{\\alpha} \n\\hf(\\dv{g}^\\discsmthker_{i-1})) \\text{d}\\alpha \\,(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nand\n\\begin{align*}\n&\\int_{x_{i-1}}^{x_i}\n\\hf\n(\\dv{\\tgS}(x)) \\text{d} \\dv{\\tgS}(x)\n\\\\& =\n\\int_0^1 \n\\hf(\\alpha \\dv{g}_i^\\discsmthker +\n\\bar{\\alpha} \\dv{g}^\\discsmthker_{i-1}) \\text{d}\\alpha \\,(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nwhere $\\bar{\\alpha}$ denotes $1-\\alpha.$\n\n\nLet $\\dv{g}(\\alpha) = \\alpha \\dv{g}_i^\\discsmthker+ \\bar{\\alpha} \\dv{g}^\\discsmthker_{i-1}.$\nThen, assuming $\\hf$ is $C^2$ we have by a simple application of the remainder theorem\n\\begin{align*}\n& |\\alpha \\hf(\\dv{g}_i^\\discsmthker) + \\bar{\\alpha} \\hf(\\dv{g}_{i-1}^\\discsmthker)\n-\n\\hf (\\alpha \\dv{g}_i^\\discsmthker + \\bar{\\alpha} \\dv{g}_{i-1}^\\discsmthker)|\n\\\\\\le &\n\\frac{C_i}{2}(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)^2\n\\end{align*}\nwhere $C_i$ is the maximum of $|\\hf''(u)|$ for $u$ in $[\\dv{g}_{i-1}^\\discsmthker,\\dv{g}_{i}^\\discsmthker].$\n\nWe now have\n\\begin{align*}\n&\\Bigl|\\int_{x_{i-1}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n\\frac{C_i}{2}(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)^3\\,.\n\\end{align*}\nSince $\\sum_i (\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker) \\le 1$\nand\n\\[\n\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker\n\\le \\Delta \\|\\smthker\\|_\\infty\n\\]\nwe obtain\n\\begin{align*}\n&\\Bigl|\\int_{0}^{1}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n\\frac{\\|\\hf''\\|_\\infty}{2} ( \\|\\smthker\\|_\\infty)^2 \\Delta^2\n\\end{align*}\n\nA similar argument applies to $\\hg$ and $h_{[\\tfg,\\tfS]},$\nand the Lemma follows.\n\\end{IEEEproof}\n\nNote that a general inequality can also be derived based on\n\\begin{align*}\n& |\\alpha \\hf(\\dv{g}_i^\\discsmthker) + \\bar{\\alpha} \\hf(\\dv{g}_{i-1}^\\discsmthker)\n-\n\\hf (\\alpha \\dv{g}_i^\\discsmthker + \\bar{\\alpha} \\dv{g}_{i-1}^\\discsmthker)|\n\\\\\\le &\n\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker)\n\\end{align*}\nfrom which we obtain\n\\begin{align*}\n&\\Bigl|\\int_{x_{i-1}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\& \\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker))\n(\\dv{g}_i^\\discsmthker-\\dv{g}_{i-1}^\\discsmthker)\n\\\\&\\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-1}^\\discsmthker))\n\\Delta\\|\\omega\\|_\\infty\n\\end{align*}\nand for $k\\ge 0,$\n\\begin{align}\n\\begin{split}\\label{eqn:discIntegralbnd}\n&\\Bigl|\\int_{x_{i-k}}^{x_i}\n\\bigl(h_{[\\tff,\\tgS]}\n(\\dv{\\tgS}(x)) \n-\n\\hf(\\dv{\\tgS}(x))\\bigr)\n\\text{d} \\dv{\\tgS}(x)\\Bigr|\n\\\\&\\le\n(\\hf(\\dv{g}_i^\\discsmthker)- \\hf(\\dv{g}_{i-k}^\\discsmthker))\n\\Delta\\|\\omega\\|_\\infty\n\\end{split}\n\\end{align}\nThis inequality proves Theorem \\ref{thm:discreteFPDelta}.\n\n\\section{Existence of Travelling Wave Solution: Final Case}\\label{app:B}\n\nIn this section we prove Theorem \\ref{thm:mainexist} for the case where\n$\\jump{\\hf}\\cap \\jump{\\hginv} \\neq \\emptyset$ or\n$\\jump{\\hg}\\cap \\jump{\\hfinv} \\neq \\emptyset$\nand $\\ashift \\neq 0.$\nWithout loss of generality we assume $\\ashift>0.$\n\n\nGiven an interval $I$ let \n$\\upperx{I}$ denote its right end point and\nlet\n$\\lowerx{I}$ denote its left end point.\nFor two closed intervals $I,I'$ we say $I \\le I'$ \nif $\\upperx{I} \\le \\lowerx{I'}.$\nThe interval $I+x$ denotes the interval $I$ translated by $x.$ \nFor a non-empty interval $I$ and $\\epsilon>0$ by $\\neigh{I}{-\\epsilon}$ we mean\n$(\\lowerx{I}-\\epsilon,\\lowerx{I}+\\epsilon).$\n\n\\begin{lemma}\\label{lem:flatseparate}\nLet $I,I' \\in \\flats{\\fS}$ be distinct where $\\ff\\in \\sptfns$ \nand $\\smthker$ is regular.\nThen $I \\le I'$ implies $I+2W \\le I'.$\n\\end{lemma}\n\\begin{IEEEproof}\nSince $I$ and $I'$ are both maximal we have $\\fS(I) < \\fS(I').$\nSince $\\smthker$ is regular, we have $\\ff(x)=\\fS(I)$ for $x\\in\\neigh{I}{W}$ \nand $\\ff(x)=\\fS(I')$ for $x\\in\\neigh{I'}{W}.$\nIt follows that $\\neigh{I}{W}$ and $\\neigh{I'}{W}$ are disjoint.\\end{IEEEproof}\n\nGiven regular $\\smthker$ and shift $\\ashift >0$ \nwe say $I \\in \\flats{\\fS}$ is {\\em linked to}\n$I' \\in \\flats{\\gS}$ if $\\upperx{I}+W +\\ashift \\in I',$\nand\nwe say $I' \\in \\flats{\\gS}$ is {\\em linked to}\n$I'' \\in \\flats{\\fS}$ if $\\upperx{I'}+W \\in I''.$\nIf we have a sequence $I_1,I_2,\\ldots$\nsuch that $I_j$ is linked to $I_{j+1}$ then we call this a {\\em chain}.\nNote that all intervals in a chain in either $\\flats{\\fS}$\nor $\\flats{\\gS}$ must be distinct.\nThe chain {\\em terminates} if the last element in the chain is not linked to \nanother interval.\n\n\n\n\\begin{lemma}\\label{lem:linkterminate}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg)>0.$\nLet $(f,g)\\in\\sptfns^2$ be $(0,1)$-interpolating and let $\\smthker$ be regular.\nAssume $h_{[\\ff,\\gSa]} \\equiv\\hf$ and $h_{[\\fg,\\fS]} \\equiv\\hg$ \n(hence $\\ashift>0$),\nthen any chain in $\\flats{\\fS},\\flats{\\gS}$ terminates.\n\\end{lemma}\n\\begin{IEEEproof}\nLet $(u^*,v^*)$ be the minimal element in $\\intcross(\\hf,\\hg)$\nas guaranteed by Lemma \\ref{lem:Sstructure}.\nThere exists finite $y$ such that $g(y) \\ge u^*$ and \n$f(y) \\ge v^*.$\nBy Lemma \\ref{lem:phiflat} if $z \\in I \\in \\{\\flats{\\gS} \\cup \\flats{\\fS}\\}$\nthen $\\altPhi(\\hf,\\hg;g(z),f(z)) \\in [0,A(\\hf,\\hg)]$ and we therefore have\n$(g(z),f(z)) < (u^*,v^*)$ componentwise by Lemma \\ref{lem:Sstructure}.\nThus, we obtain $z 0$ (we cannot have $u_k=0$ or $u_k=1$ since $\\hf$ is continuous at $0$ and $1$ by Lemma \\ref{lem:zocontinuity}. \nFor each $i=1,2,\\ldots$ we define sequences $\\eta_{i,k},$\n$k=1,2,\\ldots$ such that \n\\[\n0 < \\eta_{i,k} < \\frac{1}{2}\\min \\{ 3^{-ik},d_k \\}\n\\] \nand such that \n\\[\n\\{ u_k \\pm \\eta_{i,k} \\} \\cap \\jump{ \\hginv } = \\emptyset\\,.\n\\]\nNote that $2\\sum_k \\eta_{i,k} \\le \\frac{1}{2^i}.$\n\nFor each $k$ we define $H_k = \\unitstep_{r_k}$ (which is a unit step function\nexcept that we set $H_k(0) =r_k$)\nwhere $r_k= \\frac{\\hf(u_k)-\\hf(u_k-)}{\\hf(u_k+)-\\hf(u_k-)}.$\nThis function represents the jump in $\\hf$ at $u_k.$\nWe will substitute for this a function continuous at $0$:\n\\[\nS_{i,k}(x) = \\begin{cases}\n0 & x < 1- \\eta_{i,k}\\\\\n0 \\vee (x-r_k) \\wedge 1 & |x| \\le \\eta_{i,k} \\\\\n1 & x > 1+ \\eta_{i,k}\n\\end{cases}\n\\]\nwhere $0 \\vee z \\wedge 1 = \\min\\{ \\max \\{ 0,z\\} ,1\\}.$\nDefine\n\\[\n\\hf^i(x) = \\hf(x) - \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) (H_k(x) - S_{i,k}(x))\\,.\n\\]\nNote that $\\sum_k ( \\hf(u_k+)-\\hf(u_k-)) \\le 1$ and $|H_k(x) - S_{i,k}(x)|\\le 1$\nso the sum is well defined.\nThe function $\\hf^i(x)$ can be expressed as the sum of two functions,\n\\[\nh_1(x)= \\hf(x) - \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) H_k(x) \\,\n\\]\nand\n\\[\nh_{2,i}(x)= \\sum_k ( \\hf(u_k+)-\\hf(u_k-)) S_{i,k}(x)\\,,\n\\]\nboth of which are in $\\exitfns,$ i.e., both of which are non-decreasing.\nThe function $h_1$ is continuous for all $u \\in \\jump{\\hginv} \\cap \\jump{\\hf}$\nsince $H_k(u+)-H_k(u-)=1$ if $u=u_k$ and $H_k(u+)-H_k(u-)=0$ if $u \\neq u_k.$\nIf $u \\in \\jump{\\hginv} \\backslash \\jump{\\hf}$ then $\\hf$ is continuous at $u$ and\ntherefore $h_1$ is continuous at $u.$\nIf follows that $\\hf^i \\in \\exitfns$ and $\\hf^i \\xrightarrow{i\\rightarrow\\infty} \\hf.$\nWe assume a similar definition of $\\hg^i.$\n\nWe will now show that properties A through E hold for this sequence.\nEach property has two essentially equivalent forms (through the symmetry of substitution of $f$ and $g$). In each case we will show the first form.\n\nConsider part A. Let $v \\in \\jump{\\hfiinv}.$ There is non-empty interval\n$I =(x_1,x_2)$ such that $\\hf^i$ is evaluates to $v$ on $I.$\nSince both $h_1$ and $h_{2,i}$ are non-decreasing it follows that both are constant\non $I.$ From the fact that $h_{2,i}$ is constant on $I$ we easily obtain that\n$\\sum_k ( \\hf(u_k+)-\\hf(u_k-)) H_{k}(x)\\,$ is also constant on $I$ and \nwe deduce that $\\hf$ is constant on $I.$\nHence $v \\in \\jump{\\hfinv}$\nand part A is proved.\n\nConsider part B. We have $S_{i,k}(u+) - S_{i,k}(u-) =0$ unless $u = u_j \\pm \\eta_{j,k}$ for some $j.$\nBy construction, $u_j \\pm \\eta_{j,k} \\not\\in \\jump{\\hginv}.$\nHence, $h_{2,i}(u)$ is continuous at all $u \\in \\jump{\\hginv}.$\nSince $h_1$ is continuous, $\\hf^i$ is continuous at all $u\\in \\jump{\\hginv}$ and part B is proved.\n\nConsider part C. Let $u \\in\\jump{\\hginv}\\cap \\jump{\\hf}\\,,$ i.e., $u=u_j$ for some $j.$\nWe prove part C by showing that\n$H_k(u_j) - S_{i,k}(u_j) =0$ for all $k$ for all $i$ large enough.\nFor $k=j$ we have $H_k(u_j) - S_{i,k}(u_j) =0$ by definition.\nFor $k \\eta_{i,k}.$\nThis proves part C.\n\nConsider part D.\nAssume $v \\in \\jump{\\hfinv},$ then $\\hfinv(v-)<\\hfinv(v+).$\nSince $(\\hfinv(v-),\\hfinv(v+)) \\cap \\jump{\\hf} = \\emptyset$ and $\\eta_{i,k} < 2^{-i}$ it follows\nthat for $u \\in (\\hfinv(v-)+2^{-i},\\hfinv(v+)-2^{-i})$ we have\n$\\hf^i(u) = \\hf(u) = v$ and for \n$u \\not\\in (\\hfinv(v-)-2^{-i},\\hfinv(v+)+2^{-i})$\nwe have\n$\\hf^i(u) \\neq v.$\nPart D now follows.\n\nConsider part E.\nLet $v \\in \\jump{\\hfinv}$ and set $u=\\hfinv(v+).$\nIf $u \\in \\jump{\\hginv}\\cap \\jump{\\hf}$ then $u=u_k$ for some $k$ and\nproperty C implies property E.\nOtherwise, we have $u = t_k$ for some $k.$\nFor $j\\ge k$ we have $ |u_j-t_k| > \\eta_{i,k}$ for all $i.$\nFir $j < k$ we have $\\frac{2}{3^i} < \\min_{j0.$\nThen\n\\begin{align*}\n&\\altPhi(\\hf,\\hg;u,v)-\n\\altPhi(\\hf^\\delta,\\hg^\\delta;u,v)\\\\\n=&\n\\int_0^u (\\hginv(x)-{(\\hg^\\delta)}^{-1}(x)) dx\n+\n\\int_0^v (\\hfinv(x)-{(\\hf^\\delta)}^{-1}(x)) dx\n\\end{align*}\nwhich is non-negative and non-decreasing in $u$ and $v.$\nHence for $(u,v) \\in \\intcross(\\hf^\\delta,\\hg^\\delta)\\subset \\intcross(\\hf,\\hg)$ we have\n\\begin{align*}\n&\\altPhi(\\hf^\\delta,\\hg^\\delta;u,v) - A(\\hf^\\delta,\\hg^\\delta)\\\\\n\\ge&\n\\altPhi(\\hf,\\hg;u,v) - A(\\hf,\\hg)\\\\\n>&0\n\\end{align*}\nwhich establishes the claim. \nNow for each $\\delta>0$ we can choose $\\eta$ sufficiently small so that\n\\begin{align*}\n\\unitstep_0(x-\\eta) &\\wedge \\hf^\\delta \\\\\n\\unitstep_0(x-\\eta) &\\wedge \\hg^\\delta\n\\end{align*}\nhas a non-trivial crossing point, and it follows easily that the pair\nsatisfies the strictly positive gap condition with $A >0.$\n(The values of $\\altPhi$ at crossing points and $A$ increase by identical amounts.)\n\nLet us define $\\delta_j \\rightarrow 0$ and $\\eta_j \\rightarrow 0$ with \n$1-\\delta_j, \\eta_j \\not\\in \\jump{\\hginv} \\cup \\jump{\\hfinv}$ so that, for each $j,$\n\\begin{align*}\n\\hf^j = \\unitstep_0(x-\\eta_j) \\wedge \\hf(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\\\\n\\hg^j = \\unitstep_0(x-\\eta_j) \\wedge \\hg(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \n\\end{align*}\nsatisfies the strictly positive gap condition with $A >0.$ Now, for each $i$ we define the sequence\n\\begin{align*}\n\\hf^{i,j}(x) &=\n\\unitstep_0(x-\\eta_j) \\wedge \\hf^i(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\\\\n\\hg^{i,j}(x) &=\n\\unitstep_0(x-\\eta_j) \\wedge \\hg^i(x) \\vee \\unitstep_1(x - (1-\\delta_j)) \\,.\n\\end{align*}\nThen we have\n\\begin{align*}\n\\hf^{i,j} &\\xrightarrow{\\i \\rightarrow \\infty} \\hf^j \\\\\n\\hg^{i,j} &\\xrightarrow{\\i \\rightarrow \\infty} \\hg^j \\,.\n\\end{align*}\nClearly $\\intcross(\\hf^j,\\hg^j) = \\intcross(\\hf,\\hg) \\cap [\\eta_j,1-\\delta_j]^2,$\nand it follows that, for each $j,$ $(\\hf^{i,j},\\hg^{i,j})$ satisfies the strictly positive gap condition\nwith $A(\\hf^{i,j},\\hg^{i,j}) >0$ for all $i$ large enough.\nProperties A and B still hold for all $i$ and $j.$\n\nFor each $j$ we can find $i(j)$ such that $(\\hf^{i,j},\\hg^{i,j})$\nsatisfies the strictly positive gap condition for all $i \\ge i(j).$\nWe can assume $i(j)$ is increasing in $j.$\nConsider the diagonal sequence \n$(\\hf^{i(j),j},\\hg^{i(j),j})\\, j=1,2,\\ldots.$\nLet us re-index this as \n$(\\hf^{i},\\hg^{i})\\, i=1,2,\\ldots$ with corresponding $\\delta_i,\\eta_i.$\nWe now show that properties C,D, and E continue to hold.\n\nProperty C holds since, by Lemma \\ref{lem:zocontinuity}, $u \\in \\jump{\\hf}$ implies $u \\in (0,1)$\nand $v \\in \\jump{\\hg}$ implies $v \\in (0,1).$\nNow we show property D.\nAssume $v \\in \\jump{\\hfinv}.$ If $v \\in (0,1)$ then\n$ [\\hfinv(v-),\\hfinv(v+)] \\subset (0,1)$ by Lemma \\ref{lem:zocontinuity} and property D clearly holds.\nIf $v=0$ then $\\hfinv(v-)=\\hfiinv(v-)=0$ and \n$\\hfinv(v+) < 1$ by Lemma \\ref{lem:zocontinuity}.\nSince $1-\\delta_i \\rightarrow 1$ we have $\\hfiinv(v+) \\rightarrow \\hfinv(v+).$ \nSimilarly, if $v=1$ then $\\hfinv(v+)=\\hfiinv(v+)=1$ and \n$\\hfinv(v-) >0$ and we have $\\hfiinv(v-) \\rightarrow \\hfinv(v-).$ \nThus, property D holds generally.\n\nFinally we consider property E.\nLet $v \\in \\jump{\\hfinv}$ and set $u=\\hfinv(v+).$\nThen $u>0$ and if $u<1$ then we clearly have\n$\\hf^i(u)=\\hf(u)$ for all $i$ large enough.\nIf $u=1$ then $\\hf(u)=1$\nand $\\hf^i(u)=1$ for all $i.$\nThus, property E holds.\n\\end{IEEEproof}\n\nFor $\\ff \\in \\sptfns$ we say that $\\ff$ is increasing to the right of $x$ if\n$z > x \\Rightarrow \\ff(z) > \\ff(x)$\nand\nwe say that $\\ff$ is increasing to the left of $x$ if\n$z < x \\Rightarrow \\ff(z) < \\ff(x).$\n\n\n\\begin{lemma}\\label{lem:rightincrease}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition with $A(\\hf,\\hg)>0$ and let $\\smthker$ be regular.\nLet $(\\ff,\\fg)$ be $(0,1)$-interpolating such that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:weakexistence}.\nIf $I \\in \\flats{\\fS},$ \nthen $\\gSa$ is increasing to the right of $\\lowerx{I}-W.$\nIf $I' \\in \\flats{\\gS},$ \nthen $\\fS$ is increasing to the right of $\\lowerx{I'}-W.$\n\\end{lemma}\n\\begin{IEEEproof}\nAssume $I \\in \\flats{\\fS}.$ \nAssume $\\gSa$ is not increasing to the right of $\\lowerx{I}-W.$\nThen $\\lowerx{I}-W \\in \\hat{I}-\\ashift$ for some $\\hat{I} \\in \\flats{\\gS}$ with \n\\[\n\\lowerx{\\hat{I}-\\ashift} \\le \\lowerx{I}-W < \\upperx{\\hat{I}-\\ashift}.\n\\]\nIt follows that there exists $x \\in \\neigh{{I}}{W} \\cap (\\hat{I}-\\ashift)$ and,\nsince $\\smthker$ is regular, we have $\\ff(x) = \\fS({I}).$ \nHence we obtain $\\fS({I}) \\veq \\hf(\\gSa(\\hat{I}-\\ashift))$\nwhich is equivalent to $\\fS({I}) \\veq \\hf(\\gS(\\hat{I})).$\n\nSince $\\ashift<2W$ by Corollary \\ref{cor:regshiftbound},\nwe have\n\\[\n\\lowerx{\\hat{I}}-W \\le\n\\lowerx{I}+\\ashift-2W <\n\\lowerx{I} <\n \\upperx{\\hat{I}}+W\\,.\n\\]\nIt follows that there exists $x \\in \\neigh{\\hat{I}}{W} \\cap {I}$ and,\nsince $\\fg(x) = \\gS(\\hat{I}),$ \nwe obtain $\\gS(\\hat{I}) \\veq \\hg(\\fS({I})).$\n\nWe now have $(\\gS(\\hat{I'}),\\fS(I')) \\in \\cross( \\hf,\\hg),$ but since\n$\\altPhi(\\hf,\\hg;\\gSa(I),\\fS(I')) \\in [0,A]$ by Lemma \\ref{lem:phiflat}, this contradicts the strictly positive gap condition.\n\nThe argument for the second case is similar. \n\nAssume $I' \\in \\flats{\\gS}$ which is equivalent to\n$I'-\\ashift \\in \\flats{\\gSa}.$\nAssume $\\fS$ is not increasing to the right of $\\lowerx{I'}-W.$\nThen $\\lowerx{I'}-W \\in \\hat{I'}$ for some $\\hat{I'} \\in \\flats{\\fS}$ with \n\\[\n\\lowerx{\\hat{I'}} \\le \\lowerx{I'}-W < \\upperx{\\hat{I'}}.\n\\]\nIt follows that there exists $x \\in \\neigh{{I'}}{W} \\cap \\hat{I'}$ and,\nsince $\\smthker$ is regular, we have $\\fg(x) = \\gS({I'}).$ \nHence we obtain $\\gS({I'}) \\veq \\hg(\\fS(\\hat{I'})).$\n\nSince $\\ashift<2W$ by Corollary \\ref{cor:regshiftbound},\nwe have\n\\[\n\\lowerx{\\hat{I'}}-W \\le\n\\lowerx{I'}-2W <\n\\lowerx{I'}-\\ashift <\n\\lowerx{I'}< \\upperx{\\hat{I'}}+W\\,.\n\\]\nIt follows that there exists $x \\in \\neigh{\\hat{I'}}{W} \\cap {(I'-\\ashift)}$ and,\nsince $\\ff(x) = \\fS(\\hat{I'}),$ \nwe obtain $\\fS(\\hat{I'}) \\veq \\hf(\\gSa({I'}-\\ashift))$\nwhich is equivalent to $\\fS(\\hat{I'}) \\veq \\hf(\\gS({I'})).$\n\nThe rest of the argument is as before.\n\\end{IEEEproof}\n\n\\begin{lemma}\\label{lem:convprop}\nLet $(\\hf,\\hg) \\in \\exitfns^2$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg) > 0$ and let $\\smthker$ be regular.\nAssume we have $\\ff,\\fg \\in \\sptfns$ such that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$\nand sequences\n$\\ff_i \\rightarrow \\ff$ and $\\fg_i\\rightarrow \\fg$ and $\\ashift_i \\rightarrow \\ashift$ \nwhere\n$\\ff_i = \\hf^i \\circ \\gSai_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i,$\nand\n$(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ is given as in Lemma \\ref{lem:regularapprox}.\n\nThen $\\ashift>0$ and \nthe following properties hold for any $I\\in\\flats{\\gS}.$\n\\begin{itemize}\n\\item[A.]\nIf $I$ is not linked to an $I'\\in\\flats{\\fS}$\nthen for any $\\epsilon >0$ we have \n$\\gS_i(x) =\\gS(I)$ for $x\\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough.\n\\item[B.]\nIf $I$ is linked to $I' \\in \\flats{\\fS}$ and\nfor any $\\delta>0$ we have\n$\\fS_i$ is a fixed constant, denoted $F,$ on $\\neigh{I'}{-\\delta}$ for all $i$ large enough,\nthen,\nfor any $\\epsilon >0$ we have \n$\\gS_i(x) =\\gS(I)$ for $x\\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough.\n\\item[C.]\nAssume that for any $\\epsilon>0$ we have\n$\\gSai_i(x) = \\gS(I)$ for $x \\in \\neigh{I-\\ashift}{-\\epsilon}$ for all $i$ large enough.\nThen we have $\\ff(x) = \\hf(\\gSa(I-\\ashift))$ for all $x \\in (\\lowerx{I-\\ashift},\\upperx{I-\\ashift}).$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nConsider part A.\nLet $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ be given as in Lemma \\ref{lem:regularapprox}.\nBy Lemma \\ref{lem:weakexistence} and Lemma \\ref{lem:pathology}\nthere exists $\\ff_i,\\fg_i$ such that\n$\\ff_i = \\hf^i \\circ \\gS_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i.$\nLet $\\ff$ and $\\fg$ be limits so that\n$\\ff \\veq \\hf \\circ \\gS$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:limitexist}.\n\nLet $I \\in \\flats{\\gS}.$\nWe have $\\fg(x)=\\gS(I)$ for all $x \\in \\neigh{I}{W}.$\nSince $\\fS$ is increasing to the right of $\\lowerx{I}-W$ by Lemma \\ref{lem:rightincrease}\nand increasing to the left of $\\upperx{I}+W$ by assumption (no linked interval),\nwe see by monotonicity of $\\hg$ that\n\\(\n\\hg(v) = \\gS(I)\n\\)\nfor all $v \\in (\\fS(\\lowerx{I}-W),\\fS(\\upperx{I}+W))$\nand $\\gS(I) \\in \\jump{\\hginv}.$\nGiven any $\\epsilon>0$ property D of Lemma \\ref{lem:regularapprox} now implies that\n\\(\n(\\fS(\\lowerx{I}-W+\\epsilon),\\fS(\\upperx{I}+W)-\\epsilon)\n\\subset\n[\\hgiinv(\\gS(I)-),\\hgiinv(\\gS(I)+)]\n\\)\nfor all $i$ large enough.\nWe conclude from this that $\\fg_i(x)=\\gS(I)$ for $x \\in \\neigh{I}{W-\\epsilon}$\nfor all $i$ large enough. \nThis implies that $\\gS_i(x) = \\gS(I)$ \nfor $x \\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough, proving part A.\n\nConsider part B.\nLet $I$ be linked to $I' \\in \\flats{\\fS}.$\nWe have $\\gS(I) \\in \\jump{\\hginv}$ since\n $\\fS$ is increasing to the right of $\\lowerx{I}-W.$\nAs in the proof of part A we have\n\\(\n\\hg(v) = \\gS(I)\n\\)\nfor all $v \\in (\\fS(\\lowerx{I}-W),F)$\n\nWe have $F = \\hginv(\\gS(I)+)$ since $I$ is maximal.\nWe now apply property E of Lemma \\ref{lem:regularapprox} to conclude that $\\hg^i(F) = \\gS(I)$\nfor all $i$ large enough.\nGiven $\\epsilon>0$ we combine this\n with property D of Lemma \\ref{lem:regularapprox} to obtain\n\\(\n(\\fS(x_1-W+\\epsilon),F]\n\\subset\n[\\hgiinv(\\gS(I)-),\\hgiinv(\\gS(I)+)]\n\\)\nfor all $i$ large enough.\nLet $\\delta = \\epsilon,$ then for all $i$ large enough\nwe also have $\\fS_i(x) = F$ for $x \\in\\neigh{I'}{-\\epsilon}.$\nWe conclude that $\\fg_i(x)=\\gS(I)$ for $x \\in \\neigh{I}{W-\\epsilon}$\nfor all $i$ large enough. \nThis implies that $\\gS_i(x) = \\gS(I)$ \nfor $x \\in \\neigh{I}{-\\epsilon}$\nfor all $i$ large enough, proving part B.\n\nConsider part C. \nIf $\\hf$ is continuous at $\\gS(I)$ then we must have\n$f(x)=\\hf(\\gS(I))$ on $I-\\ashift.$\nAssume now that $\\gS(I) \\in \\jump{\\hf}.$\nWe have $\\gS(I) \\in \\jump{\\hginv}$ by \nLemma \\ref{lem:rightincrease} since $\\hf,\\hg$ satisfies the strictly\npositive gap condition.\nProperty $C$ of Lemma \\ref{lem:regularapprox} now gives $\\hf^i(\\gS(I))=\\hf(\\gS(I))$ \nfor all $i$ large enough.\nThis implies that for any $\\epsilon>0$ we now have\n$\\ff_i(x)=\\hf(\\gS(I))$ for all $x \\in \\neigh{I-\\ashift}{-\\epsilon}$\nfor all $i$ large enough.\nSince $\\ff_i \\rightarrow \\ff$ this proves part C.\n\\end{IEEEproof}\n\nLemma \\ref{lem:convprop} essentially completes the proof of Theorem \\ref{thm:mainexist} and we state the main\nresult as the following.\n\\begin{corollary}\nLet $(\\hf,\\hg)$ satisfy the strictly positive gap condition \nwith $A(\\hf,\\hg) > 0$ and let $\\smthker$ be regular.\nThere exists $(0,1)$-interpolating $\\ff,\\fg$ such that \n$\\ff = \\hf \\circ \\gS$ and \n$\\fg = \\hg \\circ \\fSa.$\n\\end{corollary}\n\\begin{IEEEproof}\nLet $(\\hf^i,\\hg^i) \\rightarrow (\\hf,\\hg)$ be given as in Lemma \\ref{lem:regularapprox}.\nBy Lemma \\ref{lem:weakexistence} and Lemma \\ref{lem:pathology}\nthere exists $\\ff_i,\\fg_i$ such that\n$\\ff_i = \\hf^i \\circ \\gS_i$ and \n$\\fg_i = \\hg^i \\circ \\fS_i$ for each $i.$\nLet $\\ff$ and $\\fg$ be limits so that\n$\\ff \\veq \\hf \\circ \\gSa$ and \n$\\fg \\veq \\hg \\circ \\fS$ as guaranteed by Lemma \\ref{lem:limitexist}.\n\nLemma \\ref{lem:linkterminate} states that any element in $I\\in \\flats{\\gS}$ must be part of a terminating\nchain.\nParts A and B of Lemma \\ref{lem:convprop} show that for any $\\epsilon > 0$ we have\n$\\gS_i(x)=\\gS(I)$ for all $x\\in \\neigh{I}{-\\epsilon}$ for all $i$ large enough.\nPart C of Lemma \\ref{lem:convprop} then shows that $\\ff(x) = \\hf(\\gSa(x))$ for all\n$x$ in the interior of $I.$\nLemma \\ref{lem:notequal} states that if $\\ff \\neq \\hf \\circ \\gSa$ then there exists\n$I\\in\\flats{\\gS}$ such that $\\ff(x) \\neq \\hf(\\gSa(x))$ on a subset of positive measure in $I-\\ashift.$\nHence, $\\ff \\equiv \\hf \\circ \\gSa.$\nA similar argument shows that $\\fg \\equiv \\hg \\circ \\fS.$\nWe can obtain equality by modifying $\\ff$ and $\\fg$ on a set of measure $0.$\n\\end{IEEEproof}\n\\section{Two Sided Termination with Positive Gap}\\label{app:C}\n\n\n\nIn this section we prove Theorems\n\\ref{thm:twoterminatedexist} and\n\\ref{thm:discretetwoterminatedexist}.\nThe two results have much in common and we begin with some\nconstructions that apply to both.\n\nWe assume that $\\smthker$ is regular and that\n$(\\hf,\\hg)$ satisfies the strictly positive gap condition with $A(\\hf,\\hg) < 0.$\n\n\n\n\n\nConsider first the parametric modification of $(\\hf,\\hg)$ given by\n\\begin{align}\n\\begin{split}\n\\hf(\\eta;u) &=(\\hf(u)-\\eta)^+\\,\\\\ \\label{eqn:FirstMod}\n\\hg(\\eta;v) &= (\\hg(v)-\\eta)^+ \\,.\n\\end{split}\n\\end{align}\n\nThe minimum of $\\altPhi(\\hf(\\eta;u),\\hg(\\eta,\\cdot);u,v)$ for $(u,v)\\in [0,1]\\times[0,1]$\nis achieved in $\\cross(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot))\\backslash (1,1).$ \nLet $(u(\\eta),v(\\eta))$ denote the minimum point where this minimum is achieved.\nThen $(u(\\eta),v(\\eta)) \\in \\cross(\\hf(\\eta;u),\\hg(\\eta;\\cdot))$\nand we have\n$\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) \\rightarrow A(\\hf,\\hg)$ as $\\eta\\rightarrow 0.$\nIt follow that $(u(\\eta),v(\\eta)) \\rightarrow (1,1)$ as $\\eta \\rightarrow 0.$\nHence, given arbitrary $\\epsilon > 0$ we have for all $\\eta$ small enough that\n$u(\\eta),v(\\eta) > 1-\\epsilon$ and $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) < 0.$\nIf $\\hf(\\eta;u(\\eta)) > v(\\eta)$ then redefine $\\hf(\\eta;u(\\eta)) = v(\\eta)$\nand\nif $\\hg(\\eta; v(\\eta)) > u(\\eta)$ then redefine $\\hg(\\eta;v(\\eta)) = u(\\eta).$\n\nWe can now apply Theorem \\ref{thm:mainexist}\nto $(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot))$ on $[0,u(\\eta)]\\times[0,v(\\eta)]$ to obtain\n ${\\tmplF},{\\tmplG} \\in \\Psi_{[-\\infty,\\infty]}$ \ninterpolating over $[0,u(\\eta)]\\times[0,v(\\eta)]$\nand $\\ashift < 0$ so that\nsetting $\\ff^t(x) = {\\tmplF}(x-\\ashift t)$ and\n$\\fg^t(x) = {\\tmplG}(x-\\ashift t)$ solves\n\\eqref{eqn:gfrecursion} for $(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot)).$\nSince $\\hf$ and $\\hg$ are continuous at $0$\nwe have ${\\tmplF}(x) = 0$ on some maximal interval, we may take to be $[-\\infty,0),$\nand ${\\tmplG}(x) = 0$ on some maximal interval $[-\\infty,z_g).$\n\nNote that we have\n\\begin{equation}\\label{eqn:ADFGbnd}\n\\hg(\\tmplF^{\\smthker}(x))\n\\ge\n\\tmplG(x) + \\eta \\unitstep_0 (x-x_g)\n\\end{equation}\nand\n\\begin{equation}\\label{eqn:ADGFbnd}\n\\hf(\\tmplG^{\\smthker}(x+\\ashift))\n\\ge\n\\tmplF(x) + \\eta \\unitstep_0 (x)\\,.\n\\end{equation}\n\nApplying Lemmas \\ref{lem:shiftupperbound} with Lemma \\ref{lem:stposbound} we can\nassert the existence of a bound $S<2W$ such that $-\\ashift \\le S$ for all $\\eta$ sufficiently small. Furthermore, \n\nWe assume $Z = Z(\\eta)$ large enough so that\n\\begin{align}\n{\\tmplF}(\\tfrac{1}{4}Z+\\ashift-\\Delta) & > v(\\eta)-\\frac{\\eta}{4},\\label{eqn:topF}\\\\\n{\\tmplG}(\\tfrac{1}{4}Z+\\ashift-\\Delta) & > u(\\eta)-\\frac{\\eta}{4} \\label{eqn:topG}\\,.\n\\end{align}\nIn the discrete case we will assume $Z = L\\Delta$ for an integer $L.$\n\nLet us define \n\\begin{equation}\\label{eqn:f0initial}\n\\ff^{0}(x) = \\tmplF(x+\\ashift) + \\eta \\unitstep_0(x+\\ashift)\n\\end{equation}\nfor $x \\le \\half Z$ and for $x > \\half Z$\ninitialize symmetrically using $\\ff^{0}(x) = \\ff^{0}(Z-x).$\nClearly this is even about $\\half Z$ and we have\n$\\ff^0(x) \\le 1.$ \nFor $x \\in [\\tfrac{1}{4}Z,\\tfrac{1}{2}Z]$ we have\n${\\tmplF}(x+\\ashift) > v(\\eta)-\\frac{\\eta}{4}$ by \\eqref{eqn:topF}\nand for all $x$ we have ${\\tmplF}(x) \\le v(\\eta).$ This gives for all $x$ the bound\n\\begin{equation}\\label{eqn:f0initialbound}\n\\ff^{0}(x) \\ge \\tmplF(x+\\ashift) + \\tfrac{3}{4} \\eta \\unitstep_0(x+\\ashift)\n-\n\\unitstep_1(x-\\tfrac{3}{4}Z)\n\\end{equation}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:twoterminatedexist}]\n\nWe assume $Z$ large enough so that\n\\begin{align}\n\\tfrac{3}{4}\\eta \\Omega(0)\n- \\Omega(-\\tfrac{1}{4}Z) & \\ge 0 \\label{eqn:lm15Obnd} \\\\\n\\tfrac{3}{4}\\eta \\Omega(-x_g+\\ashift)\n- \\Omega(-\\tfrac{1}{4}Z) & \\ge 0 \\label{eqn:lm15Obbnd} \n\\end{align}\n\nLet us initialize the system \\eqref{eqn:gfrecursion} with \n\\(\n\\ff^{0}(x) \n\\)\nas given in \\eqref{eqn:f0initial}.\nBy \\eqref{eqn:f0initialbound} we have\n\\[\n\\ff^{0,\\smthker}(x) \\ge \\tmplF^{\\smthker} (x+\\ashift) + \\tfrac{3}{4} \\eta \\Omega(x+\\ashift)\n- \\Omega(x-\\tfrac{3}{4}Z)\n\\] \nand for $x \\in [x_g-\\ashift,\\half Z]$ we have by \\eqref{eqn:lm15Obnd}\n\\[\n\\ff^{0,\\smthker}(x) \\ge \\tmplF^{\\smthker} (x+\\ashift) \\,.\n\\] \n\nConsider now \n\\(\ng^{0}(x) = \\hg(\\ftS{0}(x)).\n\\)\nWe have for $x\\in [x_g-\\ashift,\\half Z]$\n\\begin{align*}\ng^{0}(x) &= \\hg(\\ftS{0}(x)) \\\\\n& \\ge \\hg(\\tmplF^{\\smthker} (x+\\ashift)) \n\\\\ & \\stackrel{\\eqref{eqn:ADFGbnd}}{\\ge}\n\\tmplG(x+\\ashift)+\\eta\\unitstep_0(x-x_g+\\ashift)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x < x_g-\\ashift$ the inequality holds for all $x \\le \\half Z.$\n\nBy the same argument that gave \\eqref{eqn:f0initialbound}\nwe have for all $x$ the bound\n\\begin{align*}\ng^{0}(x) & \\ge\n\\tmplG(x+\\ashift)+\\tfrac{3}{4}\\eta\\unitstep_0(x-x_g+\\ashift) - \\unitstep_1(x-\\tfrac{3}{4}Z)\n\\end{align*}\nand we obtain\n\\begin{align*}\ng^{0,\\smthker}(x) \\ge \\tmplG^{\\smthker}(x+\\ashift) + \\tfrac{3}{4}\\eta \\Omega(x-x_g+\\ashift) - \\Omega(x-\\tfrac{3}{4}Z)\n\\end{align*}\nWhich, by \\eqref{eqn:lm15Obbnd}, gives\n$g^{0,\\smthker}(x) \\ge \\tmplG^{\\smthker}(x+\\ashift)$\nfor $x\\in [0,\\half Z].$\n\nNow, define $\\ff^1$ by\n$\\ff^1(x) = \\hf(\\gtS{0}(x))$ for $x\\in[0,Z]$ and\n$\\ff^1(x) = 0$ otherwise. For $x\\in[0,\\half Z]$ we have\n\\begin{align*}\nf^{1}(x) &= \\hf(\\gtS{0}(x)) \\\\\n& \\ge \\hf(\\tmplG^{\\smthker} (x+\\ashift)) \n\\\\ & \\stackrel{\\eqref{eqn:ADGFbnd}}{\\ge}\n\\tmplF(x)+\\eta\\unitstep_0(x)\n\\\\ & \\ge\nf^{0}(x)\n\\end{align*}\nThis implies the existence of a fixed point lower bounded by\n$f^0,g^0,$ which completes the proof.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discretetwoterminatedexist}]\nThe proof is similar to the proof of Lemma \\ref{thm:twoterminatedexist}\nbut we require some stronger assumptions.\nFirst, we assume that $Z=L\\Delta$ for an integer $L.$\nIn addition we assume $\\eta$ small enough so that \n$\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta,\\cdot);u(\\eta),v(\\eta))< -\\Delta\\|\\smthker\\|_\\infty.$\nTheorem \\ref{thm:mainexist} now implies\n$\\ashift <-\\Delta$ (actually we have $\\ashift < -\\Delta\/(u(\\eta) v(\\eta)$).\nFinally, we assume $Z$ large enough so that\n\\begin{align}\n\\tfrac{3}{4}\\eta \\Omega(0)\n- \\Omega(-\\tfrac{1}{4}Z+\\half \\Delta) & \\ge 0 \\label{eqn:lm16Obnd}\n\\\\\n\\tfrac{3}{4}\\eta \\Omega(-x_g+\\ashift-\\Delta)\n- \\Omega(-\\tfrac{1}{4}Z+ \\Delta) & \\ge 0 \\label{eqn:lm16Obbnd}\n\\end{align}\n\nLet us initialize the system \\eqref{eqn:discretegfrecursion} with \n\\(\n\\ff^{0}(x) \n\\)\nas given in \\eqref{eqn:f0initial}.\nBy \\eqref{eqn:f0initialbound} and Lemma \\ref{lem:disccontbnd} we have\n\\begin{align*}\n\\ff^{0,\\smthker}(x_i) \\ge &\\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta) + \\tfrac{3}{4} \\eta \\Omega(x_i+\\ashift-\\half\\Delta)\\\\\n&- \\Omega(x_i-\\tfrac{3}{4}Z+\\half\\Delta)\n\\end{align*}\nand for $x_i \\in [x_g-\\ashift+\\half\\Delta,\\half Z]$ we have by \\eqref{eqn:lm16Obnd}\n\\[\n\\ff^{0,\\smthker}(x_i) \\ge \\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta) \\,.\n\\] \n\nConsider now \n\\(\ng^{0}(x_i) = \\hg(\\ftS{0}(x_i)).\n\\)\nWe have for $x_i\\in [x_g-\\ashift+\\half\\Delta,\\half Z]$\n\\begin{align*}\ng^{0}(x_i) &= \\hg(\\ftS{0}(x_i)) \\\\\n& \\ge \\hg(\\tmplF^{\\smthker} (x_i+\\ashift-\\half\\Delta)) \n\\\\ & \\stackrel{\\eqref{eqn:ADFGbnd}}{\\ge}\n\\tmplG(x_i+\\ashift-\\half\\Delta)+\\eta\\unitstep_0(x_i-x_g+\\ashift-\\half\\Delta)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x_i < x_g-\\ashift+\\half\\Delta$ the inequality holds for all $x_i \\le \\half Z.$\n\nBy the same argument that gave \\eqref{eqn:f0initialbound}\nwe have for all $x$ the bound\n\\begin{align*}\ng^{0}(x_i) \\ge &\n\\tmplG(x_i+\\ashift-\\half\\Delta)+\\tfrac{3}{4}\\eta\\unitstep_0(x_i-x_g+\\ashift-\\half\\Delta)\n\\\\& - \\unitstep_1(x_i-\\tfrac{3}{4}Z)\n\\end{align*}\nand, applying Lemma \\ref{lem:disccontbnd}, we obtain\n\\begin{align*}\ng^{0,\\discsmthker}(x_i) \\ge& \\tmplG^{\\smthker}(x_i+\\ashift-\\Delta) + \\tfrac{3}{4}\\eta \\Omega(x_i-x_g+\\ashift-\\Delta)\n\\\\& - \\Omega(x_i-\\tfrac{3}{4}Z+\\Delta)\n\\end{align*}\nWhich, by \\eqref{eqn:lm16Obbnd}, gives\n$g^{0,\\smthker}(x_i) \\ge \\tmplG^{\\smthker}(x_i+\\ashift-\\Delta)$\nfor $x_i\\in [0,\\half Z].$\n\nNow, define $\\ff^1$ by\n$\\ff^1(x_i) = \\hf(\\gtS{0}(x_i))$ for $x_i\\in[0,Z]$ and\n$\\ff^1(x_i) = 0$ otherwise. For $x_i\\in[0,\\half Z]$ we have\n\\begin{align*}\nf^{1}(x_i) &= \\hf(\\gtS{0}(x_i)) \\\\\n& \\ge \\hf(\\tmplG^{\\smthker} (x_i+\\ashift-\\Delta)) \n\\\\ & \\stackrel{\\eqref{eqn:ADGFbnd}}{\\ge}\n\\tmplF(x_i-\\Delta)+\\eta\\unitstep_0(x_i-\\Delta)\n\\\\ & \\ge\nf^{0}(x_i)\n\\end{align*}\nwhere the last inequality uses $\\ashift <-\\Delta.$\nThis implies the existence of a fixed point lower bounded by\n$f^0,g^0,$ which completes the proof.\n\n\n\\end{IEEEproof}\n\n\n\\section{General Convergence Results}\\label{app:E}\n\nThe existence of interpolating wave solutions often implies \nglobal convergence of the spatially coupled system. The structure of $\\cross(\\hf,\\hg)$ can potentially be complicated\nenough to prevent direct application of the existence results for wave-like solutions.\nTypically, the existence results for spatial fixed points are easier to apply.\nOur technique to prove the general convergence results largely consists of modifying the\nsystem monotonically to elicit the existence of a fixed point solution for the modified system\nthat then initiates a monotonic sequence for the original system.\nMonotonicity implies convergence and necessary conditions on interpolating fixed points\nprovide the leverage to get the desired results.\nWe start with a Lemma that uses this approach in a canonical way and that we can\nthen use for the more complicated statements.\n\n\\begin{lemma}\\label{lem:liminfgap}\nLet $(\\hf,\\hg)$ be given with $A(\\hf,\\hg)<0$ and \n$\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg)$ for $(u,v) \\neq (1,1).$\nConsider the spatially continuous system \\eqref{eqn:gfrecursion}.\nIf $\\ff^0 \\in \\sptfns$ \nsatisfies $\\ff^0(\\pinfty) = 1$ then for all $x\\in\\reals$ we have\n\\[\n\\lim_{t\\rightarrow \\infty} \\ff^t(x) =1\\,,\\quad\n\\lim_{t\\rightarrow \\infty} \\fg^t(x) =1\n\\]\n\\end{lemma}\n\\begin{IEEEproof}\nConsider first the parametric modification of $(\\hf,\\hg)$ given \nin \\eqref{eqn:FirstMod}, i.e.,\n\\begin{align*}\n\\hf(\\eta;u) &=(\\hf(u)-\\eta)^+\\,\\\\\n\\hg(\\eta;v) &= (\\hg(v)-\\eta)^+ \\,.\n\\end{align*}\n\nAs before, $(u(\\eta),v(\\eta)) \\in \\cross(\\hf(\\eta;u),\\hg(\\eta;\\cdot))$ is the minimum point\nwhere $\\altPhi(\\hf(\\eta;u),\\hg(\\eta,\\cdot);u,v)$ achieves its minimum\nin $[0,1]\\times[0,1]$\nand $(u(\\eta),v(\\eta)) \\rightarrow (1,1)$ as $\\eta \\rightarrow 0.$\nGiven arbitrary $\\epsilon > 0$ we choose $\\eta$ small enough so that\n$u(\\eta),v(\\eta) > 1-\\epsilon$ and $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);u(\\eta),v(\\eta)) < 0.$\nIf $\\hf(\\eta;u(\\eta)) > v(\\eta)$ then redefine $\\hf(\\eta;u(\\eta)) = v(\\eta)$\nand\nif $\\hg(\\eta; v(\\eta)) > u(\\eta)$ then redefine $\\hg(\\eta;v(\\eta)) = u(\\eta).$\n\nWe are interested in the modified pair $\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot)$ restricted to $[0,u(\\eta)] \\times [0,v(\\eta)].$\nNow consider a further parametric modification\n\\begin{align*}\n\\hf(\\eta',\\eta;u) &=v(\\eta)\\unitstep_1(u-(u(\\eta)-\\eta'))\\vee \\hf(\\eta;u)\\,\\\\\n\\hg(\\eta',\\eta;v) &=u(\\eta)\\unitstep_1(v-(v(\\eta)-\\eta'))\\vee \\hg(\\eta;v) \\,.\n\\end{align*}\nSince $\\hf(\\eta;u)$ is continuous at $u(\\eta)$ and\n$\\hg(\\eta;u)$ is continuous at $v(\\eta)$\n(by the minimization of $\\altPhi(\\hf(\\eta;\\cdot),\\hg(\\eta;\\cdot);\\cdot,\\cdot)$)\nwe can assume $\\eta' >0$ sufficiently small so that\n\\begin{align*}\n\\hf(u) &\\ge\\hf(\\eta',\\eta;u) \\,\\\\\n\\hg(v) &\\ge\\hg(\\eta',\\eta;v) \\,.\n\\end{align*}\n\nBy construction \n$\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u,v )$ is uniquely minimized on\n $[0,u(\\eta)] \\times [0,v(\\eta)]$\nat $(u,v) = (u(\\eta),v(\\eta))$ and there exists a positive $\\delta>0$ such\nthat\n\\(\n\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u,v ) >\n\\altPhi(\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot);u(\\eta),v(\\eta))+\\delta\n\\)\nfor all $(u,v)\\in \\cross (\\hf(\\eta',\\eta;\\cdot),\\hg(\\eta',\\eta;\\cdot))\\backslash (u(\\eta),v(\\eta)).$\n\nNow we make one further parametric modification, further reducing the functions,\n\\begin{align*}\n\\hf(z,\\eta',\\eta;u) &=\\unitstep_1(u-z)\\wedge \\hf(\\eta',\\eta;u)\\,\\\\\n\\hg(z,\\eta',\\eta;v) &=\\unitstep_1(v-z)\\wedge \\hg(\\eta',\\eta;v) \\,,\n\\end{align*}\nwhere we choose $z>0$ so that \n$(\\hf(z,\\eta',\\eta;u),\\hg(z,\\eta',\\eta;v))$ satisfies the strictly positive gap condition\non $[0,u(\\eta)] \\times [0,v(\\eta)].$\nTo do this we choose $z$ so that\n\\(\n\\altPhi(\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot);u(\\eta),v(\\eta)) = - \\delta\/2\n\\)\n\nBy Theorem \\ref{thm:mainexist} there exists $\\tmplF,\\tmplG \\in \\sptfns$ \ninterpolating over $[0,u(\\eta)] \\times [0,v(\\eta)]$ and $\\ashift \\le -(\\delta\/2)\/\\|\\omega\\|_\\infty$\nsuch that $f^t(x)=\\tmplF(x-\\ashift t)$ and\n$g^t(x)=\\tmplG(x-\\ashift t)$ solves \\eqref{eqn:gfrecursion} for\nthe pair $\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot).$\n\nSince $\\tmplF(\\pinfty)=v(\\eta)<1$ and $\\tmplF(x)=0$ for some finite $x,$ we see that \nfor any $(0,1)$-interpolating function $f^0 \\in \\sptfns$ we can find a $y$ so that\nsuch that $f^0(x) \\ge \\tmplF(x-y)$ for all $x.$\nIt now follows that under \\eqref{eqn:gfrecursion} for the original pair $(\\hf,\\hg)$ we have $\\liminf_{t\\rightarrow\\infty} f^t(x)\\ge v(\\eta)\\ge 1-\\epsilon$\nand\n$\\liminf_{t\\rightarrow\\infty} g^t(x)\\ge u(\\eta)\\ge 1-\\epsilon$\nfor all $x.$\n\nSince $\\epsilon$ is arbitrary the proof is complete.\n\\end{IEEEproof}\n\nThe above proof can be easily adapted to the spatially discrete case.\n\\begin{lemma}\\label{lem:discreteliminfgap}\nLet $(\\hf,\\hg)$ be given with $A(\\hf,\\hg)<0$ and \n$\\altPhi(\\hf,\\hg;u,v) > A(\\hf,\\hg)$ for $(u,v) \\neq (1,1).$\nConsider the spatially discrete system \\eqref{eqn:discretegfrecursion}.\nFor any $\\epsilon>0,$ if $\\Delta$ is sufficiently small then\nfor all $x\\in\\reals$ we have\n\\[\n\\lim_{t\\rightarrow \\infty} \\ff^t(x) =1-\\epsilon\\,,\\quad\n\\lim_{t\\rightarrow \\infty} \\fg^t(x) =1-\\epsilon\n\\]\nfor anny $\\ff^0 \\in \\sptfns$ satisfying $\\ff^0(\\pinfty) = 1.$\n\\end{lemma}\n\\begin{IEEEproof}\nWe use the construction from the proof of Lemma \\ref{lem:liminfgap}\nand recall\nthe existence of $\\tmplF,\\tmplG \\in \\sptfns$ \ninterpolating over $[0,u(\\eta)] \\times [0,v(\\eta)]$ and $\\ashift \\le -(\\delta\/2)\/\\|\\omega\\|_\\infty$\nsuch that $f^t(x)=\\tmplF(x-\\ashift t)$ and\n$g^t(x)=\\tmplG(x-\\ashift t)$ solves \\eqref{eqn:gfrecursion} for\nthe pair $\\hf(z,\\eta',\\eta;\\cdot),\\hg(z,\\eta',\\eta;\\cdot).$\nAssume $\\Delta \\le |\\ashift|.$ \n\nGiven $f^0$ satisfying $\\ff^0(\\pinfty) = 1$ we can find $y$ \nsuch that $f^0(x_i) \\ge \\tmplF(x_i-y)$ for all $x.$\nWe can apply Theorem \\ref{thm:mainquantize} and the inequalities\n$\\hf \\ge \\hf(z,\\eta',\\eta;\\cdot)$ and $\\hg \\ge \\hg(z,\\eta',\\eta;\\cdot)$\nto obtain\n$f^t(x_i) \\ge \\tmplF(x_i-y-(\\ashift+\\Delta)t)$ and\n$g^t(x_i) \\ge \\tmplG(x_i-y-(\\ashift+\\Delta)t).$\n\nThe Lemma now follows.\n\\end{IEEEproof}\n\n\n\nRecall that in the statement of Theorem \\ref{thm:globalconv} \nwe have $(0,0)\\le (u',v') \\le (u'',v'') \\le (1,1)$ and $\\altPhi$ is minimized on\n$(u',v')$ and $(u'',v'')$ where it takes the value $m(\\hf,\\hg).$\nFurthermore, $(u',v')$ and $(u'',v'')$ are the extreme points where the minimum is attained.\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:globalconv}]\nWe will prove the first statement in the Theorem, i.e.,\n\\(\n\\liminf_{t\\rightarrow \\infty} f^t(x) \\ge v'\\,,\n\\)\nthe other cases being similar.\n\nIf $u' =0$ or $v'=0$ then $m=0$ and $(u',v')=(0,0)$ and the result is immediate.\nLet us assume that $m<0$ and hence that $(u',v')>(0,0).$\nConsider the system restricted to $[0,u']\\times [0,v'].$\nIf $\\hf(u')>v'$ then let us redefine $\\hf(u')=v'$ and \nif $\\hg(v')>u'$ then let us redefine $\\hf(u')=u'.$\nThis makes $(u',v')$ a fixed point of the underlying system.\nThis reduction will not affect the remaining argument.\nLet us reduce $\\ff^0$ by saturating it at $v',$ i.e., replacing it with\n$\\ff^0 \\wedge v'.$\n\nWe can now apply Lemma \\ref{lem:liminfgap} to obtain $\\ff^t(x) \\rightarrow v'$\nand $\\fg^t(x) \\rightarrow u'.$\nSince $\\ff^t$ and $\\fg^t$ in the original system are only larger, the result follows.\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discreteglobalconv}]\nJust as Lemma \\ref{lem:liminfgap} proved Theorem \\ref{thm:globalconv},\nwe can use Lemma \\ref{lem:discreteliminfgap} proved Theorem \\ref{thm:discreteglobalconv}.\nThe argument is essentially the same so we omit it.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:terminatedzero}]\nThe case where $\\altPhi(\\hf,\\hg;u,v) > 0$ for $(u,v) \\neq (0,0)$\nfollows easily from Theorem \\ref{thm:globalconv}. We assume now that\n$\\altPhi(\\hf,\\hg;u,v) = 0$ and $\\hf$ and $\\hg$ are strictly positive on $(0,1].$\n\nDefine \n\\[\n\\ff^0(x)=\\unitstep_1(x)\n\\] \nWe will show that $\\ff^t\\rightarrow 0,$ which implies the same for arbitrary initial conditions.\nBy monotonicity in $t,$ $\\ff^t$ has a point-wise limit $\\ff^\\infty \\in \\sptfns$\nand $\\fg^t$ has a point-wise limit $\\fg^\\infty \\in \\sptfns$\nBy continuity we have $(\\ff^\\infty(\\pinfty),\\fg^\\infty(\\pinfty)) \\in \\cross(\\hf,\\hg).$\n\nIn general $h_{[\\ff^\\infty,\\fg^{\\smthker,\\infty}]}$\nis well defined on\n$[0,\\fg^{\\infty}(\\pinfty)]$ and\n$h_{[\\fg^\\infty,\\ff^{\\smthker,\\infty}]}$\nis well defined on\n$[0,\\ff^{\\infty}(\\pinfty)]$ \nand we have\n\\begin{align}\n\\begin{split}\\label{eqn:AineqB}\n0 \\le &\\altPhi(\\hf,\\hg;\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty))\n\\\\ \\le &\n\\altPhi(h_{[\\ff^\\infty,\\fg^{\\smthker,\\infty}]},h_{[\\fg^\\infty,\\ff^{\\smthker,\\infty}]};\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty))\\,.\n\\\\ = &0\\,.\n\\end{split}\n\\end{align}\n\nAssume that $\\ff^\\infty \\neq 0.$\nLet $z = \\sup \\{x:\\ff^\\infty(x)=0\\}.$\nWe have $\\ff^{\\smthker,\\infty}(x)>0$ on $\\neigh{z}{W}$ and therefore \n$\\fg^{\\infty}(x)>0$ on $\\neigh{z}{W}$ and\n$\\fg^{\\smthker,\\infty}(x)>0$ on $\\neigh{z}{2W}.$\nHence $\\ff^\\infty(x) > 0$ for $x \\in \\neigh{z}{2W} \\cap (0,\\infty)$\nbut $\\ff^\\infty(x)=0$ for $x 0$ and we easily conclude that \n$(\\fg^{\\infty}(\\pinfty),\\ff^{\\infty}(\\pinfty)) = (0,0).$\n\\end{IEEEproof}\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discreteterminatedexistB}]\nTheorem \\ref{thm:discreteterminatedexistB} can be proved along lines similar to\nLemma \\ref{lem:liminfgap} with some additional features introduced to handle the spatial discreteness.\n\nConsider a parametric modification of $(\\hf,\\hg)$ given by\n\\begin{align*}\n\\hf(z,\\eta;u) &=\\unitstep_1(u-z)\\wedge ( \\hf(u)-\\eta)^+ \\\\\n\\hg(z,\\eta;v) &=\\unitstep_1(v-z)\\wedge (\\hg(v)-\\eta)^+ \\,.\n\\end{align*}\nWe first fix $z=0.$\nDefine $m(\\eta) = \\min \\altPhi(\\hf(0,\\eta;\\cdot),\\hg(0,\\eta;\\cdot);\\cdot,\\cdot)$\nand let $(u(\\eta),v(\\eta))$ be the minimum point where $m(\\eta)$ is \nrealized.\nAs $\\eta\\rightarrow 0$ we have $(u(\\eta),v(\\eta))\\rightarrow (1,1)$\nand $m(\\eta)\\rightarrow A(\\hf,\\hg).$\nGiven $\\epsilon>0$ we may assume $\\eta$ small enough so that\n$(u(\\eta),v(\\eta)) \\ge (1-\\epsilon,1-\\epsilon)$ and we also assume\n$m(\\eta)<0.$\n\nNow we choose $z(\\eta) (>0)$ so that\n\\[\n \\altPhi\\bigl(\\hf(z(\\eta),\\eta;\\cdot),\\hg(z(\\eta),\\eta;\\cdot);u(\\eta),v(\\eta)\\bigr)=0\n\\]\nIf necessary we reduce $\\hf(z(\\eta),\\eta;u(\\eta))$ and $\\hg(z(\\eta),\\eta;v(\\eta))$\nso that they equal $v(\\eta)$ and $u(\\eta)$ respectively.\nThen \n$(\\hf(z(\\eta),\\eta;\\cdot),\\hg(z(\\eta),\\eta;\\cdot))$\nsatisfies the strictly positive gap condition on\n$[0,u(\\eta)]\\times[0,v(\\eta)].$\n\nBy Theorem \\ref{thm:mainexist} there exists\n $\\tmplF$ and $\\tmplG$ that form a fixed point for \n\\eqref{eqn:gfrecursion}\nwith $(\\tmplF(\\minfty),\\tmplG(\\minfty))=(0,0)$ and\n$(\\tmplF(\\pinfty),\\tmplG(\\pinfty))=(v(\\eta),u(\\eta)).$\n\nLet us translate the solution so that $0 =\\sup_x \\{\\tmplF(x)=0\\}$\nand let us then define $x_g = \\sup_x \\{\\tmplG(x)=0\\}.$\nIt follows that $|x_g| < W.$\nLet us choose $\\Delta$ sufficiently small so that\nthe following holds:\n\\begin{equation}\\label{eqn:9condB}\n\\eta\\intsmthker(-|x_g|-\\half\\Delta)\n\\ge\n\\half\\Delta \\|\\omega\\|_\\infty\\,.\n\\end{equation}\n\n\nConsider initializing \\eqref{eqn:discretegfrecursion} with\n\\begin{align*}\nf^{0}(x_i)&=\n\\tmplF(x_i) +\\eta \\unitstep_0(x_i)\\,.\n\\end{align*} \nApplying Lemma \\ref{lem:disccontbnd} this yields for $x_i\\ge x_g$\n\\begin{align*}\nf^{0,\\discsmthker}(x_i)&\\ge\n\\tmplF^\\smthker (x_i-\\half\\Delta) + \\eta\\intsmthker(x_i-\\half\\Delta) \n\\\\\n&\\stackrel{\\eqref{eqn:9condB}}{\\ge}\n\\tmplF^\\smthker (x_i-\\half\\Delta) +\\half\\Delta\\|\\omega\\|_\\infty\\,\n\\\\\n&{\\ge}\n\\tmplF^\\smthker (x_i) \\,.\n\\end{align*}\nWe now obtain for $x_i \\ge x_g$\n\\begin{align*}\ng^{0}(x_i)&=\\hg(f^{0,\\discsmthker}(x_i))\n\\\\\n&\\ge\n\\hg(\\tmplF^{\\smthker}(x_i))\n\\\\\n&\\ge\n\\hg(z,\\eta;\\tmplF^{\\smthker}(x_i))+\\eta\\unitstep_0(x_i-x_g)\n\\\\\n&\\ge\n\\tmplG(x_i)+\\eta\\unitstep_0(x_i-x_g)\n\\end{align*}\nand we observe that since $\\tmplG(x_i)=0$ for\n$x_i < x_g$ this bound holds for all $x_i.$\nWe now have\n\\begin{align*}\ng^{0,\\discsmthker}(x_i)&\\ge\n\\tmplG^\\smthker (x_i-\\half\\Delta)\n+\\eta\\intsmthker(x_i-x_g-\\half\\Delta)\\,.\n\\end{align*}\n For $x_i \\ge 0$ we obtain\n\\begin{align*}\ng^{0,\\discsmthker}(x_i)\n&\\stackrel{\\eqref{eqn:9condB}}{\\ge}\n\\tmplG^\\smthker (x_i-\\half\\Delta)\n+\\half\\Delta \\|\\omega\\|_\\infty\n\\\\ & \\ge \n\\tmplG^\\smthker (x_i)\\,.\n\\end{align*}\nThus we have \n\\begin{align*}\n\\ff^{1}(x_i)&=\\hf(\\fg^{0,\\discsmthker}(x_i))\n\\\\&\\ge \\hf(\\tmplG^{\\smthker}(x_i))\n\\\\&\\ge \\tmplF(x_i)+\\eta\\unitstep_0(x_i)\n\\\\&= \\ff^0(x_i)\n\\end{align*}\nand the consequently increasing sequence establishes the existence of the\ndesired fixed point.\n\\end{IEEEproof}\n\n\n\\begin{IEEEproof}[Proof of Theorem \\ref{thm:discretetwoterminatedexistGB}]\nWe assume $Z = L\\Delta$ for an integer $L.$\nthe termination $\\hf(x_i,\\cdot)=0$ holds for $x_i<0$ and $x_i > Z.$\nThis means that symmetry holds about $\\half Z.$\nThe proof follows that of Theorem \\ref{thm:discreteterminatedexistB}\nup to the point where requirements on $\\Delta$ are given.\nContinuing from there we\nchoose $\\Delta$ small enough and $Z$ large enough \nso that\nall of the following hold.\n\\begin{equation}\\label{eqn:14condB}\n\\tfrac{3}{4}\\eta\\intsmthker(-|x_g|-\\Delta)\n- \\intsmthker(-\\tfrac{1}{4}Z+\\half\\Delta )\n\\ge\n\\half\\Delta \\|\\omega\\|_\\infty\n\\end{equation}\n\\begin{equation}\\label{eqn:14condA}\n\\tmplF(\\tfrac{1}{4}Z) > v(\\eta)-\\eta\/4\n\\end{equation}\n\\begin{equation}\\label{eqn:14condE}\n\\tmplG(\\tfrac{1}{4}Z) > u(\\eta)-\\eta\/4\n\\end{equation}\n\nConsider initializing for $x_i \\le \\half Z$ with\n\\[\nf^{0}(x_i)=\n\\tmplF(x_i) +\\eta \\unitstep_0(x_i)\\,.\n\\]\nand for $x_i > \\half Z$ initializing symmetrially with $\\ff^{0}(x_i)=\\ff^{0}(x_{L-i}).$\nAs in the derivation of \\eqref{eqn:f0initialbound}, this, by \\eqref{eqn:14condA}, implies for all $x$\n\\[\nf^{0}(x_i)=\n\\tmplF(x_i) +\\tfrac{3}{4}\\eta \\unitstep_0(x_i)\n-\\unitstep_1(x_i - \\tfrac{3}{4}Z)\\,.\n\\]\nwhich gives by Lemma \\ref{lem:disccontbnd},\n\\[\nf^{0,\\discsmthker}(x_i)\\ge\n\\tmplF^{\\smthker}(x_i-\\half\\Delta) +\\tfrac{3}{4}\\eta \\intsmthker(x_i-\\half\\Delta)\n-\\intsmthker(x_i - \\tfrac{3}{4}Z+\\half\\Delta)\\,.\n\\]\n\nThis yields for $x_i\\in[x_g-\\half\\Delta,\\tfrac{1}{2} Z]$\n\\begin{align*}\nf^{0,\\discsmthker}(x_i)\n&\\ge\n\\tmplF^\\smthker (x_i) + \\tfrac{3}{4}\\eta\\intsmthker(x_g-\\Delta) - \\intsmthker(-\\tfrac{1}{4} Z+\\half\\Delta )\n\\\\\n&\\stackrel{\\eqref{eqn:14condB}}{\\ge}\n\\tmplF^\\smthker (x_i) +\\half\\Delta\\|\\omega\\|_\\infty\\,\n\\\\\n&{\\ge}\n\\tmplF^\\smthker (x_i+\\half\\Delta) \\,.\n\\end{align*}\n\nWe now obtain for $x_i \\in [x_g-\\half \\Delta, \\half Z]$\n\\begin{align*}\ng^{0}(x_i)&=\\hg(f^{0,\\discsmthker}(x_i))\n\\\\\n&\\ge\n\\hg(\\tmplF(x_i+\\half\\Delta))\n\\\\\n&\\ge\n\\tmplG(x_i+\\half\\Delta)+\\eta\\unitstep_0(x_i+\\half\\Delta-x_g)\n\\end{align*}\nand we observe that since the right hand side is $0$ for\n$x_i+\\half\\Delta < x_g$ this bound holds for all $x_i \\le \\half Z.$\nAs in the derivation of \\eqref{eqn:f0initialbound}, by \\eqref{eqn:14condE} we now have\n\\[\ng^{0}(x_i)\\ge\n\\tmplG(x_i+\\half\\Delta)+\\tfrac{3}{4}\\eta\\unitstep_0(x_i+\\half\\Delta-x_g)-\\unitstep_1(x_i-\\tfrac{3}{4}Z)\n\\]\nwhich gives by Lemma \\ref{lem:disccontbnd}\n\\[\ng^{0,\\discsmthker}(x_i)\\ge\n\\tmplG^\\smthker(x_i)+\\tfrac{3}{4}\\eta\\Omega(x_i-x_g)-\\Omega(x_i-\\tfrac{3}{4}Z+\\half\\Delta)\n\\]\nwhich by \\eqref{eqn:14condB} yields for $x_i \\in [0,\\half Z],$\n\\[\ng^{0,\\discsmthker}(x_i)\\ge\n\\tmplG^\\smthker(x_i)\\,.\n\\]\n\n\nThus we have for $x_i \\in [0,\\half Z],$\n\\begin{align*}\n\\ff^{1}(x_i)&=\\hf(\\fg^{0,\\discsmthker}(x_i))\n\\\\&\\ge \\hf(\\tmplG^{\\smthker}(x_i))\n\\\\&\\ge \\tmplF(x_i)+\\eta\\unitstep_0(x_i)\n\\\\&= \\ff^0(x_i)\n\\end{align*}\nand the consequently increasing sequence establishes the existence of the\ndesired fixed point.\n\\end{IEEEproof}\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof for the General Case\\label{sec:general}}\n\nIn Section \\ref{sec:PCcase} we proved Theorem\n\\ref{thm:PCexist},\na special case of Theorem \\ref{thm:mainexist}\nin which $\\hg$ and $\\hf$ are piecewise constant\nfunctions and $\\smthker$ is $C_1$ and strictly positive.\nIn this section we show how to remove the speical conditions \nto arrive at the general results.\nWe make repeated use of the limit theorems of Section \\ref{sec:limitthms} and develop\nsome approximations for functions in $\\exitfns.$\nIt is quite simple to approximate $h \\in \\exitfns$ using piecewise constant functions.\nThe only difficulty is to approximate a pair $(\\hg,\\hf)$ so that the strictly positive gap\ncondition is preserved.\n\n\\subsection{Approximation by Scaling}\nLet $h \\in \\exitfns$ and denote $\\int_0^1 h(x) \\, dx$ by $A.$\nFor $0< a < 1$ we define \n\\[\n\\slanta{h} (x) = h\\Bigl(\\frac{x - (1-A)}{1-a} + (1-A)\\Bigr)\n\\]\nwhere we assume $h(x)=0$ for $x<0$ and \n$h(x)=1$ for $x>1.$ A quick calculation verifies\n$\\int_0^1 \\slanta{h} (x) dx = A.$\nNote that $h(x) \\ge \\slanta{h}(x)$ for $u \\in [0,1-A]$ and \nthat $h(x) \\le \\slanta{h}(x)$ for $x \\in [1-A,1].$\nIt follows that \n$0 \\le \\int_0^z (h(x)-\\slanta{h}(x))\\, dx \\le \\frac{a}{1-a}\\int_0^{1-A} h(x)\\,dx$\nfor all $z \\in [0,1],$\nwhich yields\n$\\Phi(\\hg,\\hf) \\le \\Phi(\\slanta{\\hg},\\slanta{\\hf})$ and\n$E(\\hg,\\hf) = E(\\slanta{\\hg},\\slanta{\\hf}).$\n\n\\begin{lemma}\\label{lem:smoothcompress}\nLet $(\\hg,\\hf) \\in \\exitfns^2$ \nsatisfy the strictly positive gap condition.\nThen, there exists $\\epsilon > 0$ such that\n$(\\slanta{\\hg},\\slanta{\\hf})$ \nsatisfies the strictly positive gap condition\nfor any $a \\in (0,\\epsilon).$\n\\end{lemma}\n\\begin{IEEEproof}[Proof of Lemma \\ref{lem:smoothcompress}]\nLet $A_1 = \\int_0^1 \\hg(x) dx,$ $A_2 = \\int_0^1 \\hf(x) dx$ so $E(\\hg,\\hf) = 1-A_1-A_2.$\nBy symmetry we need only consider the case $E \\ge 0.$\n\nFirst, we note that \n$[0,a (1-A_1)]\\times [0,a (1-A_2)]\\subset \\gapptsplus(\\slanta{\\hg},\\slanta{\\hf})$ \nand\n$[1-a A_1,1]\\times [1-a A_2,1]\\subset \\gapptsminus(\\slanta{\\hg},\\slanta{\\hf}).$\nThus, it remains only to show that $\\Phi(\\slanta{\\hg},\\slanta{\\hf}) > E$ on \n$\\intcross(\\slanta{\\hg},\\slanta{\\hf}).$\nAssume $v \\in \\intcross(\\slanta{\\hg},\\slanta{\\hf}).$ \n\nLet $\\eta$ be chosen with $0 < \\eta < \\min\\{A_1,A_2,1-A_1,1-A_2\\}.$\nDefine\n\\[\n\\tilde{\\cross}(\\hg,\\hf) = \\cross(\\hg,\\hf) \\backslash (\\ball_{\\eta\/2}(0,0) \\cup \\ball_{\\eta\/2}(1,1)).\n\\]\nBy continuity, there exists $\\gamma >0,$ which we may take to be less than $2\\eta,$\nsuch that\n$\\Phi(\\hg,\\hf) > \\gamma + E$ on $\\tilde{\\cross}(\\hg,\\hf).$\nSince $\\Phi$ is Lipschitz-1 in its last two arguments it follows that\n$\\Phi(\\hg,\\hf) > \\gamma\/2 + E$ on $\\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}.$\nFor all $a$ sufficiently small we have\n$|\\Phi(\\hg,\\hf) - \\Phi(\\slanta{\\hg},\\slanta{\\hf})| < \\gamma\/2$ uniformly\nand it follows that \n$\\Phi(\\slanta{\\hg},\\slanta{\\hf}) > E$ on $\\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}.$\nBy Lemma \\ref{lem:crosspointlimit}, for all $a$ sufficiently small we have\n\\(\n{\\cross}(\\slanta{\\hg},\\slanta{\\hf})\n\\subset \\neigh{{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\)\nand since\n\\(\n\\neigh{{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\subset\n\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1) \\cup \\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}\n\\)\nwe have\n\\[\n{\\cross}(\\slanta{\\hg},\\slanta{\\hf})\n\\subset \n\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1) \\cup \\neigh{\\tilde{\\cross}(\\hg,\\hf)}{\\gamma\/4}\\,.\n\\]\nTherefore, it remains only to bound \n$\\Phi(\\slanta{\\hg},\\slanta{\\hf}; v)$ on $\\ball_{\\eta}(0,0) \\cup \\ball_{\\eta}(1,1).$\n\nSince $\\Phi(\\hg,\\hf) \\le \\Phi(\\slanta{\\hg},\\slanta{\\hf})$ it is sufficient to show\n$\\Phi(h_{1},h_{2};v_1,v_2) > E,$ or equivalently\n$\\hat{\\Phi}(h_{1},h_{2};v_1,v_2) > 0.$\nIf $v\\in \\cross(\\hg,\\hf)$ then the inequality is immediate, so we assume\n$v\\not\\in \\cross(\\hg,\\hf).$\nIf $(v_1,v_2) \\le (1-A_1,1-A_2)$ \nthen since $v_1 \\in \\closure{h}_{2,a}(v_2)$ we have\n$v_1 \\le \\hf(v_2^+)$ and, similarly, $v_2 \\le \\hg(v_1^+),$ hence\n$v \\in \\gapptsminus(\\hg,\\hf).$\nSimilarly, if $(v_1,v_2) \\ge (1-A_1,1-A_2)$ \nthen $v \\in \\gapptsplus(\\hg,\\hf).$\n\nSince $\\closeM[\\hg,\\hf](v) \\in \\cross(\\hg,\\hf)$ and \n$v\\not\\in \\cross(\\hg,\\hf),$ \nLemma \\ref{lem:verphigap} and Lemma \\ref{lem:Rvverify}\nyield\n${\\Phi}(\\hg,\\hf;\\closeM(v)) > {\\Phi}(\\hg,\\hf;v)$\nand, equivalently,\n$\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) < \\hat{\\Phi}(\\hg,\\hf;v).$\n\nAssume $v\\in \\ball_\\eta(0,0).$ \nThen $v \\in \\gapptsminus(\\hg,\\hf)$ and $\\closeM(v) \\le v.$\nWe cannot have $\\closeM(v)=(0,0)$ since ${\\Phi}(\\hg,\\hf;0,0)=0$ and\n${\\Phi}(\\hg,\\hf;v) > 0$ (by Lemma \\ref{lem:PhiEquiv}).\nHence $\\closeM[\\hg,\\hf](v) \\in \\intcross(\\hg,\\hf),$ \nwhich implies $\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) > 0,$ and we obtain\n$\\hat{\\Phi}(\\hg,\\hf;v) > 0.$\n\nNow consider the case $v\\in \\ball_\\eta(1,1).$ \nThen $v \\in \\gapptsplus(\\hg,\\hf)$ and $\\closeM(v) \\ge v.$\nIt follows directly that $\\hat{\\Phi}(\\hg,\\hf;\\closeM(v)) \\ge 0$ (although we could show\nthat $\\closeM(v) \\neq (1,1)$ and that the inequality is strict)\nand we obtain $\\hat{\\Phi}(\\hg,\\hf;v) > 0.$\n\\end{IEEEproof}\n\n\\subsection{Piecewise Constant Approximation}\nGiven any $h \\in \\exitfns$ let us define a sequence of piecewise constant approximations\n$Q_n(h),$ $n=1,2,...$ by\n\\(\nQ_n(h) (x) = \\sum_{j=1}^n \\frac{1}{n} \\,1_{x \\ge u_{n,j}}\n\\)\nwhere we set \n\\[\nu_{n,j} = \\int_0^1 \\max\\{ (n h(x) - (j-1))^+,1 \\} dx.\n\\]\nIf $h$ is invertible then we have $u_{n,j} = n\\int_{(j-1)\/n}^{j\/n}(1-h^{-1}(x))dx$\nand \n\\(\n\\int_0^1 Q_n(h) (x) dx = \\sum_{j=1}^n \\frac{1-u_{n,j}}{n} = \\int_0^1 (1-h^{-1})(x) dx = \\int_0^1 h (x) dx.\n\\)\nIn general, it holds that\n\\(\n\\int_0^1 Q_n(h) (x) dx = \\int_0^1 h (x) dx.\n\\)\nSince $h$ is non-decreasing, it also follows that $\\int_0^z Q_n(h) (x) dx \\le \\int_0^z h (x) dx$ for all\n$z \\in [0,1].$\n\n\\begin{lemma}\\label{lem:PCapprox}\nLet $(\\hg,\\hf)$ be pair of functions in $\\exitfns$ satisfying the strictly positive gap condition \nsuch that for some $\\eta>0$ we have\n$\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nThen, for all $n$ sufficiently large $n,$\n$(Q_n(\\hg),Q_n(\\hf))$ satisfies the strictly positive gap condition.\n\\end{lemma}\n\\begin{IEEEproof}\nBy symmetry we need only consider the case $E(\\hg,\\hf) \\ge 0.$\n\nSince $\\hg$ and $\\hf$ are $0$ on $[0,\\eta)$ and $(1-\\eta,1]$ it follows that \n$\\intcrossing (\\hg,\\hf) \\subset [\\eta,1-\\eta]^2.$\nFurther, $\\intcrossing (\\hg,\\hf)$ is closed and there exists $\\gamma > 0$ such that \n$\\Phi(\\hg,\\hf;v_1,v_2) \\ge \\gamma + E$ on $\\intcrossing (\\hg,\\hf).$\nSince $\\Phi(\\hg,\\hf)$ is Lipschitz-1 componentwise it follows that \n$\\Phi(\\hg,\\hf) > \\gamma\/2 + E$ on $\\neigh{\\intcrossing (\\hg,\\hf)}{\\gamma\/4}.$\nObserve that $\\Phi(Q_n(h_{1}),Q_n(h_{2}))$ converges in $n$ uniformly to $\\Phi(\\hg,\\hf).$\n(In fact we have $0 \\le |\\Phi(\\hg,\\hf)-\\Phi(Q_n(h_{1}),Q_n(h_{2}))| \\le \\frac{1}{n}.$)\nSo, for $n$ sufficiently large, we have $\\Phi(Q_n(h_{1}),Q_n(h_{2})) > E$ on \n$\\neigh{\\intcrossing (\\hg,\\hf)}{\\gamma\/4}.$\nBy construction we also have $\\intcrossing (Q_n(h_{1},Q_n(h_{2})) \\in [\\eta,1-\\eta]^2.$\nBy Lemma \\ref{lem:crosspointlimit} we now have \n\\(\n\\intcrossing (Q_n(h_{1}),Q_n(h_{2})) \\subset \n\\neigh{\\intcrossing (h_{1},h_{2})}{\\gamma\/4}\n\\)\nfor all $n$ sufficiently large so $\\Phi(Q_n(h_{1}),Q_n(h_{2})) > E$ on\n$\\intcrossing (Q_n(h_{1}),Q_n(h_{2})).$\n\nFinally, we note that $[0,\\eta]^2 \\subset \\gapptsplus(Q_n(h_{1}),Q_n(h_{2}))$\nand $[1-\\eta,1]^2 \\subset \\gapptsminus(Q_n(h_{1}),Q_n(h_{2})).$\nThis completes the proof.\n\\end{IEEEproof}\n\n\\subsection{Proof of Main Results}\n\nWe are now ready to prove our main results.\n\n\\begin{IEEEproof}[Proof Theorem \\ref{thm:mainexist}]\nThe proof proceeds by repeated application of Lemma \\ref{lem:limitexist}\nto establish existence of $f,g \\in \\Psi$ and constant $\\ashift$ such that\n$h^{\\fS,g} = \\hg$ and $h^{\\gSa,f} = \\hf$ for a series of \nincreasingly generalized cases of $(\\hg,\\hf)$ and $\\smthker.$\nThe simplest case is already establishted in Thereom \\ref{thm:PCexist}.\n\nWe first generalize to arbitrary $\\smthker.$\nAssume that $(\\hg,\\hf)$ are both piecewise constant.\nDefine $\\smthker_i = \\smthker \\otimes G_i$ where \n$G_i(x) = \\frac{i}{\\sqrt{2\\pi}} e^{- (ix)^2\/2}.$\nIt follows that $\\smthker_i \\rightarrow \\smthker$ in $L_1$\nand $\\| \\smthker_i \\|_\\infty \\le \\| \\smthker\\|_\\infty.$\nFor each $\\smthker_i$ we apply Theorem \\ref{thm:PCexist}\nto obtain piecewise constant $f_i,g_i \\in \\Psi$ \n(with corresponding $\\zfi,\\zgi$) and constants $\\ashift_i$\nsuch that $h^{f_i^{{\\smthker}_i},g_i} = \\hg$ and $h^{g_i^{{\\smthker}_i,\\ashift_i},f_i} = \\hf.$\nWe can now apply Lemma \\ref{lem:limitexist} as indicated above to conclude\nexistence for general $\\smthker.$\n\nWe now generalize $(\\hg,\\hf)$ and to\nrequire, beyond the conditions of the Theorem statement, only that there exists $\\eta > 0$ such that $\\hg (x) =\\hf(x)= 0$ for\n$x \\in [0,\\eta)$ and $\\hg (x) =\\hf(x)= 1$ for $x \\in (1-\\eta,1].$\nWe apply Lemma \\ref{lem:PCapprox} and the preceding case already established\nto conclude that for all $n$ sufficiently large\nthere exists (piecewise constant) $f_n,g_n \\in \\Psi$ and finite constants $\\ashift_n$ such that\n$h^{f_n^{{\\smthker}},g_n} = Q_n(\\hg)$ and $h^{g_n^{{\\smthker},\\ashift_n},f_n} = Q_n(\\hf).$\nSince $Q_n(\\hg)$ and $Q_n(\\hf)$ converge to $\\hg$ and $\\hf$ respectively,\nwe can apply Lemma \\ref{lem:limitexist} to conclude existence for this case.\n\nFor arbitrary $(\\hg,\\hf)$ we consider $(\\slanta{\\hg},\\slanta{\\hf}).$\nWe apply Lemma \\ref{lem:smoothcompress} and the preceding case to conclude that for all $a$ sufficiently small\nthere exists $f_a,g_a \\in \\Psi$ and finite constants $\\ashift_a$ such that\n$h^{f_a^{{\\smthker}},g_a} = \\slanta{\\hg}$ and $h^{g_a^{{\\smthker},\\ashift_a},f_a} = \\slanta{\\hf}.$\nWe make a final application of Lemma \\ref{lem:limitexist} to obtain a solution for the general case.\n\nIn all cases above the approximations used preserve $E.$\nIn particular, if $E(\\hg,\\hf) = 0$ then we have respectively,\n$\\ashift_i=0$ or $\\ashift_n=0$ of $\\ashift_a=0$ and we obtain $\\ashift=0$\nfrom the limit construction in Lemma \\ref{lem:limitexist}.\nIf $E(\\hg,\\hf) \\neq 0$ then we cannot have $\\ashift=0$ by Lemma \\ref{lem:PPhi}.\nMoreover, we must have $\\sgn (E) = \\sgn (\\ashift).$\nFinally, the bound $|\\ashift| \\ge |E|\/\\|\\smthker\\|_\\infty$ was proved in Lemma \\ref{lem:shiftbound}.\n\\end{IEEEproof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMachine learning at its core involves solving stochastic optimization (SO) problems of the form\n\\begin{equation}\n\\label{eqn:SO_prob}\n\t\\min_{\\mathbf{x} \\in X} \\psi(\\mathbf{x}) \\triangleq \\min_{\\mathbf{x} \\in X} E_\\xi[\\phi(\\mathbf{x},\\xi)]\n\\end{equation}\nto learn a ``model'' $\\mathbf{x} \\in X \\subset \\mathbb{R}^n$ that is then used for tasks such as dimensionality reduction, classification, clustering, regression, and\/or prediction. A primary challenge of machine learning is to find a solution to the SO problem \\eqref{eqn:SO_prob} without knowledge of the distribution $P(\\xi)$. This involves finding an approximate solution to \\eqref{eqn:SO_prob} using a sequence of $T$ training samples $\\{\\xi(t) \\in \\Upsilon\\}_{t=1}^T$ drawn independently from the distribution $P(\\xi)$, which is supported on a subset of $\\Upsilon$. There are, in particular, two main categorizations of training data that, in turn, determine the types of methods that can be used to find approximate solutions to the SO problem. These are ($i$) \\emph{batch} training data and ($ii$) \\emph{streaming} training data.\n\nIn the case of {\\em batch} training data, where all $T$ samples $\\{\\xi(t)\\}$ are pre-stored and simultaneously available, a common strategy is {\\em sample average approximation} (SAA) (also referred to as {\\em empirical risk minimization} (ERM)), in which one minimizes the empirical average of the ``risk'' function $\\phi(\\cdot,\\cdot)$ in lieu of the true expectation. In the case of {\\em streaming} data, by contrast, the samples $\\{\\xi(t)\\}$ arrive one-by-one, cannot be stored in memory for long, and should be processed as soon as possible. In this setting, {\\em stochastic approximation} (SA) methods---the most well known of which is stochastic gradient descent (SGD)---are more common. Both SAA and SA have a long history in the literature; see~\\cite{Kushner.Book2010} for a historical survey of SA methods, \\cite{Kim.etal.HSO2015} for a comparative review of SAA and SA techniques, and \\cite{Pereyra.etal.JSTSP16} for a recent survey of SO techniques.\n\nAmong other trends, the rapid proliferation of sensing and wearable devices, the emergence of the internet-of-things (IoT), and the storage of data across geographically-distributed data centers have spurred a renewed interest in development and analysis of new methods for learning from {\\em fast-streaming} and {\\em distributed} data. The goal of this paper is to find a fast and efficient solution to the SO problem \\eqref{eqn:SO_prob} in this setting of distributed, streaming data. In particular, we focus on geographically-distributed nodes that collaborate over {\\em rate-limited} communication links (e.g., wireless links within an IoT infrastructure) and obtain independent streams of training data arriving at a constant rate.\n\nThe relationship between the rate at which communication takes place between nodes and the rate at which streaming data arrive at individual nodes plays a critical role in this setting. If, for example, data samples arrive much faster than nodes can communicate among themselves, it is difficult for the nodes to exchange enough information to enable an SA iteration on existing data in the network before new data arrives, thereby overwhelming the network. In order to address the challenge of distributed SO in the presence of a mismatch between the communications and streaming rates, we propose and analyze two distributed SA techniques, each based on distributed averaging consensus and stochastic mirror descent. In particular, we present bounds on the convergence rates of these techniques and derive conditions---involving the number of nodes, network topology, the streaming rate, and the communications rate---under which our solutions achieve order-optimum convergence speed.\n\n\\subsection{Relationship to Prior Work}\nSA methods date back to the seminal work of Robbins and Monro~\\cite{Robbins.Monro.AMS51}, and recent work shows that, for stochastic convex optimization, SA methods can outperform SAA methods~\\cite{Nemirovski.etal.JOO09,Juditsky.etal.SS11}. Lan~\\cite{Lan.MP12} proposed {\\em accelerated stochastic mirror descent}, which achieves the best possible convergence rate for general stochastic convex problems. This method, which makes use of noisy subgradients of $\\psi(\\cdot)$ computed using incoming training samples, satisfies\n\\begin{equation}\\label{eqn:smd.rate}\n\tE[\\psi(\\mathbf{x}(T)) - \\psi(\\mathbf{x}^*)] \\leq O(1)\\left[\\frac{L}{T^2} + \\frac{\\mathcal{M}+\\sigma}{\\sqrt{T}} \\right],\n\\end{equation}\nwhere $\\mathbf{x}^*$ denotes the minimizer of \\eqref{eqn:SO_prob}, $\\sigma^2$ denotes variance of the subgradient noise, and $\\mathcal{M}$ and $L$ denote the Lipschitz constants associated with the non-smooth (convex) component of $\\psi$ and the gradient of the smooth (convex) component of $\\psi$, respectively. Further assumptions such as smoothness and strong convexity of $\\psi(\\cdot)$ and\/or presence of a structured regularizer term in $\\psi(\\cdot)$ can remove the dependence of the convergence rate on $\\mathcal{M}$ and\/or improve the convergence rate to $O(\\sigma\/T)$ \\cite{Hu.etal.NIPS09,Nemirovski.etal.JOO09,Xiao.JMLR10,Chen.etal.NIPS12}.\n\nThe problem of {\\em distributed} SO goes back to the seminal work of Tsitsiklis et al.~\\cite{Tsitsiklis.etal.ITAC1986}, which presents distributed first-order methods for SO and gives proofs of their asymptotic convergence. Myriad works since then have applied these ideas to other settings, each with different assumptions about the type of data, how the data are distributed across the network, and how distributed units process data and share information among themselves. In order to put our work in context, we review a representative sample of these works. A recent line of work was initiated by {\\em distributed gradient descent} (DGD) \\cite{Nedic.Ozdaglar.ITAC2009}, in which nodes descend using gradients of local data and collaborate via averaging consensus \\cite{Dimakis.etal:IEEE2010}. More recent works incorporate accelerated methods, time-varying or directed graphs, data structure, etc. \\cite{Srivastava.Nedic.IJSTSP2011,Tsianos.etal.Conf2012,Mokhtari.Ribeiro.JMLR2016,Bijra.etal.arxiv2016,LiChenEtAl.N16}. These works tend not to address the SA problem directly; instead, they suppose a linearly separable function consistent with SAA using local, independent and identically distributed (i.i.d.) data. The works \\cite{Ram.etal.JOTA2010,Duchi.etal.ITAC2012,RaginskyBouvrie.ConfCDC12,DuchiAgarwalEtAl.SJO12} do consider SA directly, but suppose that nodes engage in a single round of message passing per stochastic subgradient sample.\n\nWe conclude by discussing two lines of works \\cite{Dekel.etal.JMLR2012,Rabbat.ConfCAMSAP15,tsianos.rabbat.SIPN2016} that are most closely related to this work. In \\cite{Dekel.etal.JMLR2012}, nodes perform distributed SA by forming distributed mini-batch averages of stochastic gradients and using stochastic dual averaging.\nThe main assumption in this work is that nodes can compute {\\em exact} stochastic gradient averages (e.g., via {\\tt AllReduce} in parallel computing architectures). Under this assumption, it is shown in this work that there is an appropriate mini-batch size for which the nodes' iterates converge at the optimum (centralized) rate. However, the need for exact averages in this work is not suited to rate-limited (e.g., wireless) networks, in which mimicking the {\\tt AllReduce} functionality can be costly and challenging.\n\nThe need for exact stochastic gradient averages in~\\cite{Dekel.etal.JMLR2012} has been relaxed recently in~\\cite{tsianos.rabbat.SIPN2016}, in which nodes carry out distributed stochastic dual averaging by computing {\\em approximate} mini-batch averages of dual variables via distributed consensus. In addition, and similar to our work,~\\cite{tsianos.rabbat.SIPN2016} allows for a mismatch between the communications rate and the data streaming rate. Nonetheless, there are four main distinctions between \\cite{tsianos.rabbat.SIPN2016} and our work.\nFirst, we provide results for stochastic {\\em composite} optimization, whereas \\cite{Dekel.etal.JMLR2012,tsianos.rabbat.SIPN2016} suppose a differentiable objective. Second, we consider distributed {\\em mirror descent}, which allows for a limited generalization to non-Euclidean settings. Third, we explicitly examine the impact of slow communications rate on performance, in particular highlighting the need for large mini-batches and their impact on convergence speed when the communications rate is slow. In \\cite{tsianos.rabbat.SIPN2016}, the optimum mini-batch size is first derived from \\cite{Dekel.etal.JMLR2012}, after which the communications rate needed to facilitate distributed consensus at the optimum mini-batch size is specified. While it appears to be possible to derive some of our results from a re-framing of the results of \\cite{tsianos.rabbat.SIPN2016}, it is crucial to highlight the trade-offs necessary under slow communications, which is not done in prior works. Finally, this work also presents a distributed {\\em accelerated} mirror descent approach to distributed SA; a somewhat surprising outcome is that acceleration substantially improves the convergence rate in networks with slow communications.\n\n\\subsection{Our Contributions}\nIn this paper, we present two strategies for distributed SA over networks with fast streaming data and slow communications links: distributed stochastic approximation mirror descent (D-SAMD) and accelerated distributed stochastic approximation mirror descent (AD-SAMD). In both cases, nodes first locally compute mini-batch stochastic subgradient averages to accommodate a fast streaming rate (or, equivalently, a slow communications rate), and then they collaboratively compute approximate network subgradient averages via distributed consensus. Finally, nodes individually employ mirror descent and accelerated mirror descent, respectively, on the approximate averaged subgradients for the next set of iterates.\n\nOur main theoretical contribution is the derivation of \\textit{upper} bounds on the convergence rates of D-SAMD and AD-SAMD. These bounds involve a careful analysis of the impact of imperfect subgradient averaging on individual nodes' iterates. In addition, we derive sufficient conditions for order-optimum convergence of D-SAMD and AD-SAMD in terms of the streaming and communications rates, the size and topology of the network, and the data statistics.\n\nTwo key findings of this paper are that distributed methods can achieve order-optimum convergence with small communication rates, as long as the number of nodes in the network does not grow too quickly as a function of the number of data samples each node processes, and that accelerated methods seem to offer order-optimum convergence in a larger regime than D-SAMD, thus potentially accommodating slower communications links relative to the streaming rate. By contrast, the convergence speeds of {\\em centralized} stochastic mirror descent and accelerated stochastic mirror descent typically differ only in higher-order terms. We hasten to point out that we do {\\em not} claim superior performance of D-SAMD and AD-SAMD versus other distributed methods. Instead, the larger goal is to establish the existence of methods for order-optimum stochastic learning in the fast-streaming, rate-limited regimes. D-SAMD and AD-SAMD should be best regarded as a proof of concept towards this end.\n\n\\subsection{Notation and Organization}\nWe typically use boldfaced lowercase and boldfaced capital letters (e.g., $\\mathbf{x}$ and $\\mathbf{W}$) to denote (possibly random) vectors and matrices, respectively. Unless otherwise specified, all vectors are assumed to be column vectors. We use $(\\cdot)^T$ to denote the transpose operation and $\\mathbf{1}$ to denote the vector of all ones. Further, we denote the expectation operation by $E[\\cdot]$ and the field of real numbers by $\\mathbb{R}$. We use $\\nabla$ to denote the gradient operator, while $\\odot$ denotes the Hadamard product. Finally, given two functions $p(r)$ and $q(r)$, we write $p(r) = O(q(r))$ if there exists a constant $C$ such that $\\forall r, p(r) \\leq C q(r)$, and we write $p(r) = \\Omega(q(r))$ if $q(r) = O(p(r))$.\n\nThe rest of this paper is organized as follows. In Section~\\ref{sect:setting}, we formalize the problem of distributed stochastic composite optimization. In Sections~\\ref{sect:mirror.descent} and \\ref{sect:accelerated.mirror.descent}, we describe D-SAMD and AD-SAMD, respectively, and also derive performance guarantees for these two methods. We examine the empirical performance of the proposed methods via numerical experiments in Section~\\ref{sect:numerical}, and we conclude the paper in Section~\\ref{sect:conclusion}. Proofs are provided in the appendix.\n\n\\section{Problem Formulation}\\label{sect:setting}\nThe objective of this paper is order-optimal, distributed minimization of the composite function\n\\begin{equation}\n\t\\psi(\\mathbf{x}) = f(\\mathbf{x}) + h(\\mathbf{x}),\n\\end{equation}\nwhere $\\mathbf{x} \\in X \\subset \\mathbb{R}^n$ and $X$ is convex and compact. The space $\\mathbb{R}^n$ is endowed with an inner product $\\langle \\cdot , \\cdot \\rangle$ that need not be the usual one and a norm $\\norm{\\cdot}$ that need not be the one induced by the inner product. In the following, the minimizer and the minimum value of $\\psi$ are denoted as:\n\\begin{equation}\n\t\\mathbf{x}^* \\triangleq \\arg\\min_{\\mathbf{x} \\in X} \\psi(\\mathbf{x}), \\quad \\text{and} \\quad \\psi^* \\triangleq \\psi(\\mathbf{x}^*).\n\\end{equation}\n\nWe now make a few assumptions on the smooth ($f(\\cdot)$) and non-smooth ($h(\\cdot)$) components of $\\psi$. The function $f: X \\to \\mathbb{R}$ is convex with Lipschitz continuous gradients, i.e.,\n\\begin{equation}\n\t\\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y})}_* \\leq L\\norm{\\mathbf{x} - \\mathbf{x}}, \\ \\forall \\ \\mathbf{x},\\mathbf{y} \\in X,\n\\end{equation}\nwhere $\\norm{\\cdot}_*$ is the dual norm associated with $\\langle \\cdot, \\cdot \\rangle$ and $\\norm{\\cdot}$:\n\\begin{equation}\n\t\\norm{\\mathbf{g}}_* \\triangleq \\sup_{\\norm{\\mathbf{x}} \\leq 1} \\langle \\mathbf{g}, \\mathbf{x} \\rangle.\n\\end{equation}\nThe function $h: X \\to \\mathbb{R}$ is convex and Lipschitz continuous:\n\\begin{equation}\n\t\\norm{h(\\mathbf{x}) - h(\\mathbf{y})} \\leq \\mathcal{M}\\norm{\\mathbf{x} - \\mathbf{y}}, \\forall \\ \\mathbf{x},\\mathbf{y} \\in X.\n\\end{equation}\nNote that $h$ need not have gradients; however, since it is convex we can consider its {\\em subdifferential}, denoted by $\\partial h(\\mathbf{y})$:\n\\begin{equation}\n\t\\partial h(\\mathbf{y}) = \\{\\mathbf{g}: h(\\mathbf{z}) \\geq h(\\mathbf{y}) + \\mathbf{g}^T(\\mathbf{z} - \\mathbf{y}), \\forall \\ \\mathbf{z} \\in X\\}.\n\\end{equation}\n\nAn important fact that will be used in this paper is that the \\emph{subgradient} $\\mathbf{g} \\in \\partial h$ of a Lipschitz-continuous convex function $h$ is bounded~\\cite[Lemma~2.6]{Shalev-Shwartz.Book2012}:\n\\begin{equation}\n\t\\norm{\\mathbf{g}}_* \\leq \\mathcal{M}, \\ \\forall \\mathbf{g} \\in \\partial h(\\mathbf{y}), \\ \\mathbf{y} \\in X.\n\\end{equation}\nConsequently, the gap between the subgradients of $\\psi$ is bounded: $\\forall \\mathbf{x},\\mathbf{y} \\in X$ and $\\mathbf{g}_\\mathbf{x} \\in \\partial h(\\mathbf{x})$, $\\mathbf{g}_\\mathbf{y} \\in \\partial h(\\mathbf{y})$, we have\n\\begin{align}\n\t\\norm{\\partial \\psi(\\mathbf{x}) - \\partial \\psi(\\mathbf{y})}_* &= \\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y}) + \\mathbf{g}_\\mathbf{x} - \\mathbf{g}_\\mathbf{y}}_* \\notag \\\\\n &\\leq \\norm{\\nabla f(\\mathbf{x}) - \\nabla f(\\mathbf{y})}_* + \\norm{\\mathbf{g}_\\mathbf{x} - \\mathbf{g}_\\mathbf{y}}_* \\notag\\\\\n &\\leq L\\norm{\\mathbf{x} - \\mathbf{y}} + 2\\mathcal{M}. \\label{eqn:subgradient.bound}\n\\end{align}\n\n\\subsection{Distributed Stochastic Composite Optimization}\nOur focus in this paper is minimization of $\\psi(\\mathbf{x})$ over a network of $m$ nodes, represented by the undirected graph $G=(V,E)$. To this end, we suppose that nodes minimize $\\psi$ collaboratively by exchanging subgradient information with their neighbors at each communications round. Specifically, each node $i \\in V$ transmits a message at each communications round to each of its neighbors, defined as\n\\begin{equation}\n \\mathcal{N}_i = \\{j \\in V: (i,j) \\in E\\},\n\\end{equation}\nwhere we suppose that a node is in its own neighborhood, i.e., $i \\in \\mathcal{N}_i$. We assume that this message passing between nodes takes place without any error or distortion. Further, we constrain the messages between nodes to be members of the dual space of $X$ and to satisfy causality; i.e., messages transmitted by a node can depend only on its local data and previous messages received from its neighbors.\n\nNext, in terms of data generation, we suppose that each node $i \\in V$ queries a first-order stochastic ``oracle'' at a fixed rate---which may be different from the rate of message exchange---to obtain noisy estimates of the subgradient of $\\psi$ at different query points in $X$. Formally, we use `$t$' to index time according to {\\em data-acquisition} rounds and define $\\{\\xi_i(t) \\in \\Upsilon\\}_{t \\geq 1}$ to be a sequence of independent (with respect to $i$ and $t$) and identically distributed (i.i.d.) random variables with unknown probability distribution $P(\\xi)$. At each data-acquisition round $t$, node $i$ queries the oracle at search point $\\mathbf{x}_i(s)$ to obtain a point $G(\\mathbf{x}_i(s),\\xi_i(t))$ that is a noisy version of the subgradient of $\\psi$ at $\\mathbf{x}_i(s)$. Here, we use `$s$' to index time according to \\emph{search-point update} rounds, with possibly multiple data-acquisition rounds per search-point update. The reason for allowing the search-point update index $s$ to be different from the data-acquisition index $t$ is to accommodate the setting in which data (equivalently, subgradient estimates) arrive at a much faster rate than the rate at which nodes can communicate with each other; we will elaborate further on this in the next subsection.\n\nFormally, $G(\\mathbf{x},\\xi)$ is a Borel function that satisfies the following properties:\n\\begin{align}\n\tE[G(\\mathbf{x},\\xi)] &\\triangleq \\mathbf{g}(\\mathbf{x}) \\in \\partial \\psi(\\mathbf{x}), \\quad \\text{and}\\\\\n E[\\norm{G(\\mathbf{x},\\xi) - \\mathbf{g}(\\mathbf{x})}_*^2] &\\leq \\sigma^2,\n\\end{align}\nwhere the expectation is with respect to the distribution $P(\\xi)$. We emphasize that this formulation is equivalent to that in which the objective function is $\\psi(\\mathbf{x}) \\triangleq E[\\phi(\\mathbf{x},\\xi)]$, and where nodes in the network acquire data point $\\{\\xi_i(t)\\}_{i\\in V}$ at each data-acquisition round $t$ that are then used to compute the subgradients of $\\phi(\\mathbf{x},\\xi_i(t))$, which---in turn---are noisy subgradients of $\\psi(\\mathbf{x})$.\n\n\\subsection{Mini-batching for Rate-Limited Networks}\nA common technique to reduce the variance of the (sub)gradient noise and\/or reduce the computational burden in centralized SO is to average ``batches'' of oracle outputs into a single (sub)gradient estimate. This technique, which is referred to as \\emph{mini-batching}, is also used in this paper; however, its purpose in our distributed setting is to both reduce the subgradient noise variance \\emph{and} manage the potential mismatch between the communications rate and the data streaming rate. Before delving into the details of our mini-batch strategy, we present a simple model to parametrize the mismatch between the two rates. Specifically, let $\\rho >0$ be the {\\em communications ratio}, i.e. the fixed ratio between the rate of communications and the rate of data acquisition. That is, $\\rho \\geq 1$ implies nodes engage in $\\rho$ rounds of message exchanges for every data-acquisition round. Similarly, $\\rho < 1$ means there is one communications round for every $1\/\\rho$ data-acquisition rounds. We ignore rounding issues for simplicity.\n\nThe mini-batching in our distributed problem proceeds as follows. Each mini-batch round spans $b \\geq 1$ data-acquisition rounds and coincides with the search-point update round, i.e., each node $i$ updates its search point at the end of a mini-batch round. In each mini-batch round $s$, each node $i$ uses its current search point $\\mathbf{x}_i(s)$ to compute an average of oracle outputs\n\\begin{equation}\n\t\\theta_i(s) = \\frac{1}{b}\\sum_{t = (s-1)b +1}^{sb} G(\\mathbf{x}_i(s),\\xi_i(t)).\n\\end{equation}\nThis is followed by each node computing a new search point $\\mathbf{x}_i(s+1)$ using $\\theta_i(s)$ and messages received from its neighbors.\n\nIn order to analyze the mini-batching distributed SA techniques proposed in this work, we need to generalize the usual averaging property of variances to non-Euclidean norms.\n\\begin{lemma}\\label{lem:average.variance}\n\tLet $\\mathbf{z}_1,\\dots,\\mathbf{z}_k$ be i.i.d. random vectors in $\\mathbb{R}^n$ with $E[\\mathbf{z}_i] = 0$ and $E[\\norm{\\mathbf{z}_i}^2_*] \\leq \\sigma^2$. There exists a constant $C_* \\geq 0$, which depends only on $\\norm{\\cdot}$ and $\\langle \\cdot, \\cdot \\rangle$, such that\n \\begin{equation}\n \tE\\left[\\norm{\\frac{1}{k}\\sum_{i=1}^k \\mathbf{z}_i }_*^2\\right] \\leq \\frac{C_* \\sigma^2}{k}.\n \\end{equation}\n\\end{lemma}\n\\begin{IEEEproof}\n\tThis follows directly from the property of norm equivalence in finite-dimensional spaces.\n\\end{IEEEproof}\nIn order to illustrate Lemma~\\ref{lem:average.variance}, notice that when $\\norm{\\cdot} = \\norm{\\cdot}_1$, i.e., the $\\ell_1$ norm, and $\\langle \\cdot, \\cdot \\rangle$ is the standard inner product, the associated dual norm is the $\\ell_\\infty$ norm: $\\norm{\\cdot}_* = \\norm{\\cdot}_\\infty$. Since $\\norm{\\mathbf{x}}^2_\\infty \\leq \\norm{\\mathbf{x}}^2_2 \\leq n\\norm{\\mathbf{x}}_\\infty^2$, we have $C_* = n$ in this case. Thus, depending on the norm in use, the extent to which averaging reduces subgradient noise variance may depend on the dimension of the optimization space.\n\nIn the following, we will use the notation $\\mathbf{z}_i(s) \\triangleq \\theta_i(s) - \\mathbf{g}(\\mathbf{x}_i(s))$. Then, $E[\\norm{\\mathbf{z}_i(s)}_*^2] \\leq C_*\\sigma^2\/b$. We emphasize that the subgradient noise vectors $\\mathbf{z}_i(s)$ depend on the search points $\\mathbf{x}_i(s)$; we suppress this notation for brevity.\n\n\\subsection{Problem Statement}\nIt is straightforward to see that mini-batching induces a performance trade-off: Averaging reduces subgradient noise and processing time, but it also reduces the rate of search-point updates (and hence slows down convergence). This trade-off depends on the relationship between the streaming and communications rates. In order to carry out distributed SA in an order-optimal manner, we will require that the nodes collaborate by carrying out $r \\geq 1$ rounds of averaging consensus on their mini-batch averages $\\theta_i(s)$ in each mini-batch round $s$ (see Section~\\ref{sect:mirror.descent} for details). In order to complete the $r$ communication rounds in time for the next mini-batch round, we have the constraint\n\\begin{equation}\n\tr \\leq b \\rho.\n\\end{equation}\nIf communications is faster, or if the mini-batch rounds are longer, nodes can fit in more rounds of information exchange between each mini-batch round or, equivalently, between each search-point update. But when the mismatch factor $\\rho$ is small, the mini-batch size $b$ needed to enable sufficiently many consensus rounds may be so large that the reduction in subgradient noise is outstripped by the reduction in search-point updates and the resulting convergence speed is sub-optimum. In this context, our main goal is specification of sufficient conditions for $\\rho$ such that the resulting convergence speeds of the proposed distributed SA techniques are optimum.\n\n\\section{Distributed Stochastic Approximation\\\\Mirror Descent}\\label{sect:mirror.descent}\nIn this section we present our first distributed SA algorithm, called \\emph{distributed stochastic approximation mirror descent} (D-SAMD). This algorithm is based upon stochastic approximated mirror descent, which is a generalized version of stochastic subgradient descent. Before presenting D-SAMD, we review a few concepts that underlie mirror descent.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.95\\textwidth]{figures\/timing_diag.png}\n\t\\caption{The different time counters for the rate-limited framework of this paper, here with $\\rho = 1\/2$, $b=4$, and $r=2$. In this particular case, over $T=8$ total data-acquisition rounds, each node receives $8$ data samples and computes $8$ (sub)gradients; it averages those (sub)gradients into $S=2$ mini-batch (sub)gradients; it then engages in $r=2$ rounds of consensus averaging to produce $S=2$ {\\em locally averaged} (sub)gradients; each of those (sub)gradients is finally used to update the search points twice, one for each $1 \\leq s \\leq S$. Note that, while not explicitly shown in the figure, the search point $\\mathbf{x}_i(1)$ is used for all computations spanning the data-acquisition rounds $5 \\leq t \\leq 8$.}\\label{fig:timing_diag}\n\\end{figure*}\n\n\\subsection{Stochastic Mirror Descent Preliminaries}\nStochastic mirror descent, presented in \\cite{Lan.MP12}, is a generalization of stochastic subgradient descent. This generalization is characterized by a {\\em distance-generating function} $\\omega: X \\to \\mathbb{R}$ that generalizes the Euclidean norm. The distance-generating function must be continuously differentiable and strongly convex with modulus $\\alpha$, i.e.\n\\begin{equation}\n\t\\langle \\nabla \\omega(\\mathbf{x}) - \\nabla \\omega(\\mathbf{y}), \\mathbf{x} - \\mathbf{y} \\rangle \\geq \\alpha \\norm{\\mathbf{x} - \\mathbf{y}}^2, \\forall \\ \\mathbf{x},\\mathbf{y} \\in X.\n\\end{equation}\nIn the convergence analysis, we will require two measures of the ``radius'' of $X$ that will arise in the convergence analysis, defined as follows:\n\\begin{equation*}\n\tD_\\omega \\triangleq \\sqrt{\\max_{\\mathbf{x} \\in X}\\omega(x) - \\min_{\\mathbf{x} \\in X} \\omega(x)}, \\quad \\Omega_\\omega \\triangleq \\sqrt{\\frac{2}{\\alpha} D_\\omega}.\n\\end{equation*}\n\nThe distance-generating function induces the {\\em prox function}, or the Bregman divergence $V : X \\times X \\to \\mathbb{R}_+$, which generalizes the Euclidean distance:\n\\begin{equation}\n\tV(\\mathbf{x},\\mathbf{z}) = \\omega(\\mathbf{z}) - (\\omega(\\mathbf{x}) + \\langle \\nabla \\omega(\\mathbf{x}), \\mathbf{z}-\\mathbf{x} \\rangle).\n\\end{equation}\nThe prox function $V(\\mathbf{x},\\cdot)$ inherits strong convexity from $\\omega(\\cdot)$, but it need not be symmetric or satisfy the triangle inequality. We define the {\\em prox mapping} $P_{\\mathbf{x}}: \\mathbb{R}^n \\to X$ as\n\\begin{equation}\\label{eqn:prox.mapping}\n\tP_\\mathbf{x}(\\mathbf{y}) = \\argmin_{\\mathbf{z} \\in X} \\langle \\mathbf{y}, \\mathbf{z}-\\mathbf{x} \\rangle + V(\\mathbf{x},\\mathbf{z}).\n\\end{equation}\nThe prox mapping generalizes the usual subgradient descent step, in which one minimizes the local linearization of the objective function regularized by the Euclidean distance of the step taken. In (centralized) stochastic mirror descent, one computes iterates of the form\n\\begin{align*}\n\t\\mathbf{x}(s+1) &= P_{\\mathbf{x}(s)}(\\gamma_s \\mathbf{g}(s))\n\\end{align*}\nwhere $\\mathbf{g}(s)$ is a stochastic subgradient of $\\psi(\\mathbf{x}(s))$, and $\\gamma_s$ is a step size. These iterates have the same form as (stochastic) subgradient descent; indeed, choosing $\\omega(\\mathbf{x}) = \\frac{1}{2}\\norm{\\mathbf{x}}^2_2$ as well as $\\langle \\cdot, \\cdot \\rangle$ and $\\norm{\\cdot}$ to be the usual ones results in subgradient descent iterations.\n\nOne can speed up convergence by choosing $\\omega(\\cdot)$ to match the structure of $X$ and $\\psi$. For example, if the optimization space $X$ is the unit simplex over $\\mathbb{R}^n$, one can choose $\\omega(\\mathbf{x}) = \\sum_i x_i \\log(x_i)$ and $\\norm{\\cdot}$ to be the $\\ell_1$ norm. This leads to $V(\\mathbf{x},\\mathbf{z}) = D(\\mathbf{z} || \\mathbf{x}))$, where $D(\\cdot || \\cdot)$ denotes the Kullback-Leibler (K-L) divergence between $\\mathbf{z}$ and $\\mathbf{x}$. %\nThis choice speeds up convergence on the order of $O\\left(\\sqrt{n\/\\log(n)}\\right)$ over using the Euclidean norm throughout. Along a similar vein, when $\\psi$ includes an $\\ell_1$ regularizer to promote a sparse minimizer, one can speed up convergence by choosing $\\omega(\\cdot)$ to be a $p$-norm with $p = \\log(n)\/(\\log(n)-1)$.\n\nIn order to guarantee convergence for D-SAMD, we need to restrict further the distance-generating function $\\omega(\\cdot)$. In particular, we require that the resulting prox mapping be 1-Lipschitz continuous in $\\mathbf{x},\\mathbf{y}$ pairs, i.e., $\\forall \\ \\mathbf{x},\\mathbf{x}^\\prime,\\mathbf{y},\\mathbf{y}^\\prime \\in \\mathbb{R}^n$,\n\\begin{equation*}\n\t\\norm{P_{\\mathbf{x}}(\\mathbf{y}) - P_{\\mathbf{x}^\\prime}(\\mathbf{y}^\\prime)} \\leq \\norm{\\mathbf{x}-\\mathbf{x}^\\prime} + \\norm{\\mathbf{y} - \\mathbf{y}\\prime}.\n\\end{equation*}\nThis condition is in addition to the conditions one usually places on the Bregman divergence for stochastic optimization; we will use it to guarantee that imprecise gradient averages make a bounded perturbation in the iterates of stochastic mirror descent. The condition holds whenever the prox mapping is the projection of a 1-Lipschitz function of $\\mathbf{x},\\mathbf{y}$ onto $X$. For example, it is easy to verify that this condition holds in the Euclidean setting. One can also show that when the distance-generating function $\\omega(\\mathbf{x})$ is an $\\ell_p$ norm for $p > 1$, the resulting prox mapping is 1-Lipschitz continuous in $\\mathbf{x}$ and $\\mathbf{y}$ as required.\n\nHowever, not all Bregman divergences satisfy this condition. One can show that the K-L divergence results in a prox mapping that is not Lipschitz. Consequently, while we present our results in terms of general prox functions, we emphasize that the results do not apply in all cases.\\footnote{We note further that it is possible to relax the constraint that the best Lipschitz constant be no larger than unity. This worsens the scaling laws---in particular, the required communications ratio $\\rho$ grows in $T$ rather than decreases---and we omit this case for brevity's sake.} One can think of the results primarily in the setting of Euclidean (accelerated) stochastic subgradient descent---for which case they are guaranteed to hold---with the understanding that one can check on a case-by-case basis to see if they hold for a particular non-Euclidean setting.\n\n\\subsection{Description of D-SAMD}\\label{sect:dsamd.description}\nHere we present in detail D-SAMD, which generalizes stochastic mirror descent to the setting of distributed, streaming data. In D-SAMD, nodes carry out iterations similar to stochastic mirror descent as presented in \\cite{Lan.MP12}, but instead of using local stochastic subgradients associated with the local search points, they carry out approximate consensus to estimate the {\\em average} of stochastic subgradients across the network. This reduces the subgradient noise at each node and speeds up convergence.\n\nLet $\\mathbf{W}$ be a symmetric, doubly-stochastic matrix consistent with the network graph $G$, i.e., $[\\mathbf{W}]_{ij} \\triangleq w_{ij}= 0$ if $(i,j) \\notin E$. Further suppose that $\\mathbf{W} - \\mathbf{1}\\mathbf{1}^T\/n$ has spectral radius strictly less than one, i.e. the second-largest eigenvalue magnitude is strictly less than one. This condition is guaranteed by choosing the diagonal elements of $\\mathbf{W}$ to be strictly greater than zero.\n\nNext, we focus on the case of constant step size $\\gamma$ and set it as $0 < \\gamma \\leq \\alpha\/(2L)$.\\footnote{It is shown in \\cite{Lan.MP12} that a constant step size is sufficient for order-optimal performance, so we adopt such a rule here.} For simplicity, we suppose that there is a predetermined number of data-acquisition rounds $T$, which leads to $S=T\/b$ mini-batch rounds. We detail the steps of D-SAMD in Algorithm \\ref{alg:standard}. Further, in Figure \\ref{fig:timing_diag} we illustrate the data acquisition round, mini-batch round, communication round, and search point update counters and their role in the D-SAMD algorithm.\n\n\\begin{algorithm}[t]\n \\caption{Distributed stochastic approximation mirror descent (D-SAMD)\n \\label{alg:standard}}\n \\begin{algorithmic}[1]\n \\Require Doubly-stochastic matrix $\\mathbf{W}$, step size $\\gamma$, number of consensus rounds $r$, batch size $b$, and stream of mini-batched subgradients $\\theta_i(s)$.\n \\For{$i=1:m$}\n \t\\State $\\mathbf{x}_i(1) \\gets \\min_{\\mathbf{x} \\in X} \\omega(\\mathbf{x})$ \\Comment{Initialize search points}\n \\EndFor\n \\For {$s=1:S$}\n \\State $\\mathbf{h}_i^0(s) \\gets \\theta_i(s)$ \\Comment{Get mini-batched subgradients}\n \\For{$q=1:r$, $i=1:m$}\n \t\\State $\\mathbf{h}_i^q(s) \\gets \\sum_{j \\in \\mathcal{N}_i} w_{ij}\\mathbf{h}_j^{q-1}(s)$ \\Comment{Consensus rounds}\n \\EndFor\n \\For{$i=1:m$}\n \t\\State $\\mathbf{x}_i(s+1) \\gets P_{\\mathbf{x}_i(s)}(\\gamma \\mathbf{h}_i^r(s))$ \\Comment{Prox mapping}\n \\State $\\mathbf{x}_i^\\mathrm{av}(s+1) \\gets \\frac{1}{s}\\sum_{k=1}^s\\mathbf{x}_i(k)$ \\Comment{Average iterates}\n \\EndFor\n \\EndFor\n \\end{algorithmic}\n \\Return $\\mathbf{x}_i^{\\mathrm{av}}(S+1), i=1,\\dots,m.$\n\\end{algorithm}\n\nIn D-SAMD, each node $i$ initializes its iterate at the minimizer of $\\omega(\\cdot)$, which is guaranteed to be unique due to strong convexity. At each mini-batch round $s$, each node $i$ obtains its mini-batched subgradient and nodes engage in $r$ rounds of averaging consensus to produce the (approximate) average subgradients $\\mathbf{h}_i^r(s)$. Then, each node $i$ takes a mirror prox step, using $\\mathbf{h}^r_i(s)$ instead of its own mini-batched estimate. Finally, each node keeps a running average of its iterates, which is well-known to speed up convergence \\cite{Polyak.Juditsky.JCO1992}.\n\n\\subsection{Convergence Analysis}\nThe convergence rate of D-SAMD depends on the bias and variance of the approximate subgradient averages $\\mathbf{h}_i^r(s)$. In principle, averaging subgradients together reduces the noise variance and speeds up convergence. However, because averaging consensus using only $r$ communications rounds results in {\\em approximate} averages, each node takes a slightly different mirror prox step and therefore ends up with a different iterate. At each mini-batch round $s$, nodes then compute subgradients at different search points, leading to bias in the averages $\\mathbf{h}^r_i(s)$. This bias accumulates at a rate that depends on the subgradient noise variance, the topology of the network, and the number of consensus rounds per mini-batch round.\n\nTherefore, the first step in bounding the convergence speed of D-SAMD is to bound the bias and the variance of the subgradient estimates $\\mathbf{h}^r_i(s)$, which we do in the following lemma.\n\\begin{lemma}\\label{lem:consensus.error.norm}\n\tLet $0 \\leq \\lambda_2 < 1$ denote the magnitude of the second-largest (ordered by magnitude) eigenvalue of $\\mathbf{W}$. Define the matrices\n \\begin{align*}\n \t\\mathbf{H}(s) &\\triangleq [\\mathbf{h}_1^r(s), \\dots, \\mathbf{h}_m^r(s)], \\\\\n \\mathbf{G}(s) &\\triangleq [\\mathbf{g}(\\mathbf{x}_1(s)), \\dots, \\mathbf{g}(\\mathbf{x}_m(s))], \\text{ and} \\\\\n \\mathbf{Z}(s) &\\triangleq [\\mathbf{z}_1(s), \\dots, \\mathbf{z}_m(s)],\n \\end{align*}\n recalling that the subgradient noise $\\mathbf{z}_i(s)$ is defined with respect to the mini-batched subgradient $\\theta_i(s)$. Also define\n \\begin{align*}\n \t\\overline{\\mathbf{g}}(s) &\\triangleq \\frac{1}{m}\\sum_{i=1}^m \\mathbf{g}(\\mathbf{x}_i(s)), \\quad \\overline{\\mathbf{G}}(s) \\triangleq [\\overline{\\mathbf{g}}(s), \\dots, \\overline{\\mathbf{g}}(s)], \\text{ and}\\\\\n \\overline{\\mathbf{z}}(s) &\\triangleq \\frac{1}{m}\\sum_{i=1}^m\\mathbf{z}_i(s), \\quad \\overline{\\mathbf{Z}}(s) \\triangleq [\\overline{\\mathbf{z}}(s), \\cdots, \\overline{\\mathbf{z}}(s)],\n \\end{align*}\n where the matrices $\\overline{\\mathbf{G}}(s), \\overline{\\mathbf{Z}}(s) \\in \\mathbb{R}^{n \\times n}$ have identical columns. Finally, define the matrices\n \\begin{align*}\n \t\\mathbf{E}(s) &\\triangleq \\mathbf{G}(s)\\mathbf{W}^r - \\overline{\\mathbf{G}}(s) \\text{ and} \\\\\n \\tilde{\\mathbf{Z}}(s) &\\triangleq \\mathbf{Z}(s)\\mathbf{W}^r - \\overline{\\mathbf{Z}}(s)\n \\end{align*}\n of average consensus error on the subgradients and subgradient noise, respectively. Then, the following facts are true. First, one can write $\\mathbf{H}(s)$ as\n \\begin{equation}\\label{eqn:gradient.decomposition}\n \t\\mathbf{H}(s) = \\overline{\\mathbf{G}}(s) + \\mathbf{E}(s) + \\overline{\\mathbf{Z}}(s) + \\tilde{\\mathbf{Z}}(s).\n \\end{equation}\n Second, the columns of $\\overline{\\mathbf{Z}}(s)$ satisfy\n \\begin{equation}\n \tE[\\norm{\\overline{\\mathbf{z}}(s)}_*^2] \\leq \\frac{C_*\\sigma^2}{mb}.\n \\end{equation}\n Finally, the $i$th columns of $\\mathbf{E}(s)$ and $\\tilde{\\mathbf{Z}}(s)$, denoted by $\\mathbf{e}_i(s)$ and $\\tilde{\\mathbf{z}}_i(s)$, respectively, satisfy\n \\begin{equation}\\label{eqn:gradient.average.norm}\n \t\\norm{\\mathbf{e}_i(s)}_* \\leq \\max_{j,k} m^2\\sqrt{C_*}\\lambda_2^r \\norm{\\mathbf{g}_j(s) - \\mathbf{g}_k(s)}_*\n \\end{equation}\n and\n \\begin{equation}\\label{eqn:average.gradient.noise.variance}\n \tE[\\norm{\\tilde{\\mathbf{z}}_i(s)}_*^2] \\leq \\frac{\\lambda^{2r}m^2 C_* \\sigma^2}{b},\n \\end{equation}\n where we have used $\\mathbf{g}_j(s)$ as a shorthand for $\\mathbf{g}(\\mathbf{x}_j(s))$.\n\\end{lemma}\n\nThe next step in the convergence analysis is to bound the distance between iterates at different nodes. As long as iterates are not too far apart, the subgradients computed at different nodes have sufficiently similar means that averaging them together reduces the overall subgradient noise.\n\\begin{lemma}\\label{lem:iterate.gap}\n\tLet $a_s \\triangleq \\max_{i,j} \\norm{\\mathbf{x}_i(s) - \\mathbf{x}_j(s)}$. The moments of $a_s$ follow:\n \\begin{align}\n \tE[a_s] &\\leq \\frac{\\mathcal{M}+\\sigma\/\\sqrt{b}}{L}((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1), \\\\\n E[a_s^2] &\\leq \\frac{(\\mathcal{M}+\\sigma\/\\sqrt{b})^2}{L^2}((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1)^2.\n \\end{align}\n\\end{lemma}\n\nNow, we bound D-SAMD's expected gap to optimality.\n\\begin{theorem}\\label{thm:mirror.descent.convergence.rate}\n\tFor D-SAMD, the expected gap to optimality at each node $i$ satisfies\n \\begin{multline}\\label{eqn:DSAMD.convergence.rate}\n \tE[\\psi(\\mathbf{x}_i^{\\mathrm{av}}(S+1))] - \\psi^* \\leq \\\\ \\frac{2L\\Omega_\\omega^2}{\\alpha S} + \\sqrt{\\frac{2(4\\mathcal{M}^2 + 2\\Delta_S^2)}{\\alpha S}} + \\sqrt{\\frac{\\alpha}{2}}\\frac{\\Xi_S D_\\omega}{L},\n \\end{multline}\nwhere\n\\begin{align}\n\t\\Xi_s &\\triangleq \\left(\\mathcal{M}+\\frac{\\sigma}{\\sqrt{b}}\\right)(1+ m^2 \\sqrt{C_*} \\lambda_2^r)\\times\\nonumber\\\\\n&\\qquad\\qquad\\qquad ((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1) + 2\\mathcal{M}\n\\end{align}\nand\n\\begin{align}\n \\Delta_s^2 &\\triangleq 2 \\left(\\mathcal{M}+\\frac{\\sigma}{\\sqrt{b}}\\right)^2(1+m^4 C_* \\lambda_2^{2r})\\times\\nonumber\\\\\n &\\qquad\\qquad((1+\\alpha m^2 \\sqrt{C_*} \\lambda_2^r)^{s}-1)^2 + 4C_*\\sigma^2\/(mb) \\nonumber\\\\\n &\\qquad\\qquad\\qquad+ 4\\lambda_2^{2r}C_*\\sigma^2 m^2\/b +4\\mathcal{M}\n\\end{align}\nquantify the moments of the effective subgradient noise.\n\\end{theorem}\n\nThe convergence rate proven in Theorem \\ref{thm:mirror.descent.convergence.rate} is akin to that provided in \\cite{Lan.MP12}, with $\\Delta_s^2$ taking the role of the subgradient noise variance. A crucial difference is the presence of the final term involving $\\Xi_s$. In \\cite{Lan.MP12}, this term vanishes because the intrinsic subgradient noise has zero mean. However, the equivalent gradient error in D-SAMD does not have zero mean in general. As nodes' iterates diverge, their subgradients differ, and the nonlinear mapping between iterates and subgradients results in noise with nonzero mean.\n\nThe critical question is how fast communication needs to be for order-optimum convergence speed, i.e., the convergence speed that one would obtain if nodes had access to other nodes' subgradient estimates at each round. After $S$ mini-batch rounds, the network has processed $mT$ data samples. Centralized mirror descent, with access to all $mT$ data samples in sequence, achieves the convergence rate \\cite{Lan.MP12}\n\\begin{equation*}\n\tO(1)\\left[\\frac{L}{mT} + \\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right].\n\\end{equation*}\nThe final term dominates the error as a function of $m$ and $T$ if $\\sigma^2 > 0$. In the following corollary we derive conditions under which the convergence rate of D-SAMD matches this term.\n\\begin{corollary}\\label{cor:mirror.descent.consensus.rounds}\n\tThe optimality gap for D-SAMD satisfies\n \\begin{equation}\n E[\\psi(\\mathbf{x}_i^\\mathrm{av}(S+1))] - \\psi^* = O\\left(\\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right),\n \\end{equation}\n provided the mini-batch size $b$, the communications ratio $\\rho$, the number of users $m$, and the Lipschitz constant $\\mathcal{M}$ satisfy\n \\begin{align*}\n \tb &= \\Omega\\left(1 + \\frac{\\log(mT)}{\\rho\\log(1\/\\lambda_2)}\\right), \\quad b = O\\left(\\frac{\\sigma T^{1\/2}}{m^{1\/2}}\\right),\\\\\n \\rho &= \\Omega\\left(\\frac{m^{1\/2}\\log(mT)}{\\sigma T^{1\/2}\\log(1\/\\lambda_2)}\\right), \\quad T = \\Omega\\left(\\frac{m}{\\sigma^2}\\right), \\text{ and}\\\\\n \\mathcal{M} &= O\\left(\\min\\left\\{\\frac{1}{m},\\frac{1}{\\sqrt{ m \\sigma^2 T}}\\right\\}\\right).\n \\end{align*}\n\\end{corollary}\n\n\\subsection{Discussion}\nCorollary \\ref{cor:mirror.descent.consensus.rounds} gives new insights into influences of the communications and streaming rates, network topology, and mini-batch size on the convergence rate of distributed stochastic learning. In \\cite{Dekel.etal.JMLR2012}, a mini-batch size of $b=O(T^{1\/2})$ is prescribed---which is sufficient whenever gradient averages are perfect---and in \\cite{tsianos.rabbat.SIPN2016} the number of imperfect consensus rounds needed to facilitate the mini-batch size $b$ prescribed in \\cite{Dekel.etal.JMLR2012} is derived. By contrast, we derive a mini-batch condition sufficient to drive the effective noise variance to $O(\\sigma^2\/(mT))$ while taking into consideration the impact of imperfect subgradient averaging. This condition depends not only on $T$ but also on $m$, $\\rho$, $\\lambda_2$, and $\\sigma^2$---indeed, for all else constant, the optimum mini-batch size is merely $\\Omega(\\log(T))$. Then, the condition on $\\rho$ essentially ensures that $b = O(T^{1\/2})$ as specified in \\cite{Dekel.etal.JMLR2012}.\n\nWe note that Corollary \\ref{cor:mirror.descent.consensus.rounds} imposes a strict requirement on $\\mathcal{M}$, the Lipschitz constant of the non-smooth part of $\\psi$. Essentially the non-smooth part must vanish as $m$, $T$, or $\\sigma^2$ becomes large. This is because the contribution of $h(\\mathbf{x})$ to the convergence rate depends only on the number of iterations taken, not on the noise variance. Reducing the effective subgradient noise via mini-batching has no impact on this contribution, so we require the Lipschitz constant $\\mathcal{M}$ to be small to compensate.\n\nFinally, we note that Corollary \\ref{cor:mirror.descent.consensus.rounds} dictates the relationship between the size of the network and the number of data samples obtained at each node. Leaving the terms besides $m$ and $T$ constant, Corollary \\ref{cor:mirror.descent.consensus.rounds} requires $T = \\Omega(m)$, i.e. the number of nodes in the network should scale no faster than the number of data samples processed per node. This is a relatively mild condition for big data applications; many applications involve data streams that are large relative to the size of the network. Furthermore, ignoring the $\\log(mT)$ term and assuming $\\lambda_2$ and $\\sigma$ to be fixed, Corollary \\ref{cor:mirror.descent.consensus.rounds} indicates that a communication ratio of $\\rho = \\Omega\\big(\\sqrt{m\/T}\\big)$ is sufficient for order optimality; i.e., nodes need to communicate at least $\\Omega\\big(\\sqrt{m\/T}\\big)$ times per data sample. This means that if $T$ scales faster than $\\Omega(m)$ then the required communications ratio approaches zero in this case as $m,T \\to \\infty$. In particular, fast stochastic learning is possible in expander graphs, for which the spectral gap $1-\\lambda_2$ is bounded away from zero, even in communication rate-limited scenarios. For graph families that are poor expanders, however, the required communications ratio depends on the scaling of $\\lambda_2$ as a function of $m$.\n\n\\section{Accelerated Distributed Stochastic Approximation Mirror Descent}\\label{sect:accelerated.mirror.descent}\nIn this section, we present {\\em accelerated} distributed stochastic approximation mirror descent (AD-SAMD), which distributes the accelerated stochastic approximation mirror descent proposed in \\cite{Lan.MP12}. The centralized version of accelerated mirror descent achieves the optimum convergence rate of\n\\begin{equation*}\n\tO(1)\\left[\\frac{L}{T^2}+\\frac{\\mathcal{M} + \\sigma^2}{\\sqrt{T}} \\right].\n\\end{equation*}\nConsequently, we will see that the convergence rate of AD-SAMD has $1\/S^2$ as its first term. This faster convergence in $S$ allows for more aggressive mini-batching, and the resulting conditions for order-optimal convergence are less stringent.\n\n\\subsection{Description of AD-SAMD}\nThe setting for AD-SAMD is the same as in Section \\ref{sect:mirror.descent}. We again suppose a distance function $\\omega: X \\to \\mathbb{R}$, its associated prox function\/Bregman divergence $V: X \\times X \\to \\mathbb{R}$, and the resulting (Lipschitz) prox mapping $P_x: \\mathbb{R}^n \\to X$.\n\nAs in Section \\ref{sect:dsamd.description}, we suppose a mixing matrix $\\mathbf{W} \\in \\mathbb{R}^{m \\times m}$ that is symmetric, doubly stochastic, consistent with $G$, and has nonzero spectral gap. The main distinction between accelerated and standard mirror descent is the way one averages iterates. Rather than simply average the sequence of iterates, one maintains several distinct sequences of iterates, carefully averaging them along the way. This involves two sequences of step sizes $\\beta_s \\in [1,\\infty)$ and $\\gamma_s \\in \\mathbb{R}$, which are not held constant. Again we suppose that the number of mini-batch rounds $S=T\/b$ is predetermined. We detail the steps of AD-SAMD in Algorithm \\ref{alg:accelerated}.\n\n\\begin{algorithm}[t]\n \\caption{Accelerated distributed stochastic approximation mirror descent (AD-SAMD)\n \\label{alg:accelerated}}\n \\begin{algorithmic}[1]\n \\Require Doubly-stochastic matrix $\\mathbf{W}$, step size sequences $\\gamma_s$, $\\beta_s$, number of consensus rounds $r$, batch size $b$, and stream of mini-batched subgradients $\\theta_i(s)$.\n \\For{$i=1:m$}\n \t\\State $\\mathbf{x}_i(1),\\mathbf{x}^\\mathrm{md}_i(1),\\mathbf{x}^\\mathrm{ag}_i(1) \\gets \\min_{\\mathbf{x} \\in X} \\omega(\\mathbf{x})$ \\Comment{Initialize search points}\n \\EndFor\n \\For {$s=1:S$}\n \\For {$i=1:m$}\n \t\\State $\\mathbf{x}_i^\\mathrm{md}(s) \\gets \\beta_s^{-1}\\mathbf{x}_i(s) + (1-\\beta^{-1}_s)\\mathbf{x}_i^\\mathrm{ag}(s)$\n \\State $\\mathbf{h}_i^0(s) \\gets \\theta_i(s)$ \\Comment{Get mini-batched subgradients}\n \\EndFor\n\n \\For{$q=1:r$, $i=1:m$}\n \t\\State $\\mathbf{h}_i^q(s) \\gets \\sum_{j \\in \\mathcal{N}_i} w_{ij}\\mathbf{h}_j^{q-1}(s)$ \\Comment{Consensus rounds}\n \\EndFor\n \\For{$i=1:m$}\n \t\\State $\\mathbf{x}_i(s+1) \\gets P_{\\mathbf{x}_i(s)}(\\gamma_s \\mathbf{h}_i^r(s))$ \\Comment{Prox mapping}\n \\State $\\mathbf{x}^\\mathrm{ag}_i(s+1) \\gets \\beta_s^{-1}\\mathbf{x}_i(s+1) + (1-\\beta_s^{-1})\\mathbf{x}_i^\\mathrm{ag}(s)$\n \\EndFor\n \\EndFor\n \\end{algorithmic}\n \\Return $\\mathbf{x}_i^{\\mathrm{ag}}(S+1), i=1,\\dots,m.$\n\\end{algorithm}\n\nThe sequences of iterates $\\mathbf{x}_i(s)$, $\\mathbf{x}_i^{\\mathrm{md}}(s)$, and $\\mathbf{x}^\\mathrm{ag}_i(s)$ are interrelated in complicated ways; we refer the reader to \\cite{Lan.MP12} for an intuitive explanation of these iterations.\n\n\\subsection{Convergence Analysis}\nAs with D-SAMD, the convergence analysis relies on bounds on the bias and variance of the averaged subgradients. To this end, we note first that Lemma \\ref{lem:consensus.error.norm} also holds for AD-SAMD, where $\\mathbf{H}(s)$ has columns corresponding to noisy subgradients evaluated at $\\mathbf{x}_i^\\mathrm{md}(s)$. Next, we bound the distance between iterates at different nodes. This analysis is somewhat more complicated due to the relationships between the three iterate sequences.\n\\begin{lemma}\\label{lem:accelerated.iterate.gap}\n\tLet\n \\begin{align*}\n \ta_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}^\\mathrm{ag}_i(s) - \\mathbf{x}^\\mathrm{ag}_j(s)}, \\\\\n b_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}_i(s) - \\mathbf{x}_j(s)}, \\text{ and} \\\\\n c_s &\\triangleq \\max_{i,j}\\norm{\\mathbf{x}^\\mathrm{md}_i(s) - \\mathbf{x}^\\mathrm{md}_j(s)}.\n \\end{align*}\n Then, the moments of $a_s$, $b_s$, and $c_s$ satisfy:\n \\begin{align*}\n \tE[a_s],E[b_s],E[c_s] &\\leq \\frac{\\mathcal{M} \\!\\!+\\! \\sigma\/\\sqrt{b}}{L}((1 \\!\\!+\\! 2\\gamma_s m^2 \\sqrt{C_*}L\\lambda_2^r)^s \\!-\\! 1), \\\\\n E[a_s^2],E[b_s^2],E[c_s^2] &\\leq \\frac{(\\mathcal{M} \\!\\!+\\! \\sigma\/\\sqrt{b})^2}{L^2}(\\!(1 \\!\\!+\\! 2\\gamma_s m^2 \\!\\!\\sqrt{C_*} L\\lambda_2^r)^s \\!\\!-\\!\\! 1)^2.\n \\end{align*}\n\\end{lemma}\n\nNow, we bound the expected gap to optimality of the AD-SAMD iterates.\n\\begin{theorem}\\label{thm:accelerated.optimality.gap}\n\tFor AD-SAMD, there exist step size sequences $\\beta_s$ and $\\gamma_s$ such that the expected gap to optimality satisfies\n \\begin{multline}\n \tE[\\Psi(\\mathbf{x}_i^\\mathrm{ag}(S+1))] - \\Psi^* \\leq \\frac{8 L D_{\\omega,X}^2}{\\alpha S^2} + \\\\ 4 D_{\\omega,X}\\sqrt{\\frac{4M + \\Delta_S^2}{\\alpha S}} + \\sqrt{\\frac{32}{\\alpha}}D_{\\omega,X}\\Xi_S,\n \\end{multline}\n where\n \\begin{multline*}\n \t\\Delta_\\tau^2 = 2(\\mathcal{M}+\\sigma\/\\sqrt{b})^2((1+ 2\\gamma_\\tau m^2 \\sqrt{C_*} L\\lambda_2^r)^\\tau-1)^2 + \\\\ \\frac{4 C_* \\sigma^2}{b}(\\lambda_2^{2r}m^2 + 1\/m) + 4\\mathcal{M}.\n \\end{multline*}\n and\n \\begin{multline*}\n \\Xi_\\tau = (\\mathcal{M} + \\sigma\/\\sqrt{b})(1+\\sqrt{C_*}m^2\\lambda_2^r) \\times \\\\ ((1+2\\gamma_\\tau m^2\\sqrt{C_*}L\\lambda_2^r)^\\tau-1) + 2\\mathcal{M}.%\n \\end{multline*}\n\\end{theorem}\n\nAs with D-SAMD, we study the conditions under which AD-SAMD achieves order-optimum convergence speed. The centralized version of accelerated mirror descent, after processing the $mT$ data samples that the network sees after $S$ mini-batch rounds, achieves the convergence rate\n\\begin{equation*}\n\tO(1)\\left[ \\frac{L}{(mT)^2} + \\frac{\\mathcal{M}+\\sigma}{\\sqrt{mT}}\\right].\n\\end{equation*}\nThis is the optimum convergence rate under any circumstance. In the following corollary, we derive the conditions under which the convergence rate matches the second term, which usually dominates the error when $\\sigma^2 > 0$.\n\\begin{corollary}\\label{cor:accelerated.descent.consensus.rounds}\n\tThe optimality gap satisfies\n \\begin{equation*}\n \tE[\\psi(\\mathbf{x}^\\mathrm{ag}_i(S+1)] - \\psi^* = O\\left(\\frac{\\mathcal{M} + \\sigma}{\\sqrt{mT}} \\right),\n \\end{equation*}\n provided\n \\begin{align*}\n \tb &= \\Omega\\left(1 + \\frac{\\log(mT)}{\\rho\\log(1\/\\lambda_2)}\\right), \\quad b = O\\left(\\frac{\\sigma^{1\/2}T^{3\/4}}{m^{1\/4}}\\right), \\\\\n \\rho &= \\Omega\\left( \\frac{m^{1\/4}\\log(m T)}{\\sigma T^{3\/4}\\log(1\/\\lambda_2)} \\right), \\quad T = \\Omega\\left(\\frac{m^{1\/3}}{\\sigma^2}\\right), \\text{ and}\\\\\n \\mathcal{M} &= O\\left(\\min\\left\\{\\frac{1}{m},\\frac{1}{\\sqrt{ m \\sigma^2 T}}\\right\\}\\right).\n \\end{align*}\n\\end{corollary}\n\n\\subsection{Discussion}\nThe crucial difference between the two schemes is that AD-SAMD has a convergence rate of $1\/S^2$ in the absence of noise and non-smoothness. This faster term, which is often negligible in centralized mirror descent, means that AD-SAMD tolerates more aggressive mini-batching without impact on the order of the convergence rate. As a result, while the condition on the mini-batch size $b$ is the same in terms of $\\rho$, the condition on $\\rho$ is relaxed by $1\/4$ in the exponents of $m$ and $T$. This is because the condition $b = O(T^{1\/2})$, which holds for standard stochastic SO methods, is relaxed to $b = O(T^{3\/4})$ for accelerated stochastic mirror descent.\n\nSimilar to Corollary \\ref{cor:mirror.descent.consensus.rounds}, Corollary \\ref{cor:accelerated.descent.consensus.rounds} prescribes a relationship between $m$ and $T$, but the relationship for AD-SAMD is $T~=~\\Omega(m^{1\/3})$, holding all but $m,T$ constant. This again is due to the relaxed mini-batch condition $b = O(T^{3\/4})$ for accelerated stochastic mirror descent. Furthermore, ignoring the $\\log$ term, Corollary \\ref{cor:accelerated.descent.consensus.rounds} indicates that a communications ratio $\\rho = \\Omega\\left(\\frac{m^{1\/4}}{T^{3\/4}}\\right)$ is needed for well-connected graphs such as expander graphs. In this case, as long as $T$ grows faster than the cube root of $m$, order-optimum convergence rates can be obtained even for small communications ratio. Thus, the use of accelerated methods increases the domain in which order optimum rate-limited learning is guaranteed.\n\n\\section{Numerical Example: Logistic Regression}\\label{sect:numerical}\nTo demonstrate the scaling laws predicted by Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds} and to investigate the empirical performance of D-SAMD and AD-SAMD, we consider supervised learning via binary logistic regression. Specifically, we assume each node observes a stream of pairs $\\xi_i(t) = (y(t),l(t))$ of\ndata points $y_i(t) \\in \\mathbb{R}^d$ and their labels $l_i(t) \\in \\{0,1\\}$, from which it learns a classifier with the log-likelihood function\n\\begin{equation*}\n\tF(\\mathbf{x},x_0,\\mathbf{y},l) = l (\\mathbf{y}^T\\mathbf{x} + x_0) - \\log(1+\\exp(\\mathbf{y}^T\\mathbf{x} + x_0))\n\\end{equation*}\nwhere $\\mathbf{x} \\in \\mathbb{R}^d$ and $x_0 \\in \\mathbb{R}$ are regression coefficients.\n\nThe SO task is to learn the optimum regression coefficients $\\mathbf{x},x_0$. In terms of the framework of this paper, $\\Upsilon = (\\mathbb{R}^d \\times \\{0,1\\})$, and $X = \\mathbb{R}^{d+1}$ (i.e., $n = d+1$). We use the Euclidean norm, inner product, and distance-generating function to compute the prox mapping. The convex objective function is the negative of the log-likelihood function, averaged over the unknown distribution of the data, i.e.,\n\\begin{equation*}\n\t\\psi(\\mathbf{x}) = -E_{\\mathbf{y},l}[F(\\mathbf{x},x_0,\\mathbf{y},l)].\n\\end{equation*}\nMinimizing $\\psi$ is equivalent to performing maximum likelihood estimation of the regression coefficients \\cite{bishop:book06}.\n\nWe examine performance on synthetic data so that there exists a ``ground truth'' distribution with which to compute $\\psi(\\mathbf{x})$ and evaluate empirical performance. We suppose that the data follow a Gaussian distribution. For $l_i(t)~\\in~\\{0,1\\}$, we let $\\mathbf{y}_i(t)~\\sim~\\mathcal{N}(\\mu_{l_i(t)},\\sigma_r^2\\mathbf{I})$, where $\\mu_{l(t)}$ is one of two mean vectors, and $\\sigma_r^2 > 0$ is the noise variance.\\footnote{The variance $\\sigma_r^2$ is distinct from the resulting gradient noise variance $\\sigma^2$.} For this experiment, we draw the elements $\\mu_0$ and $\\mu_1$ randomly from the standard normal distribution, let $d=20$, and choose $\\sigma_r^2=2$. We consider several network topologies, as detailed in the next subsections.\n\nWe compare the performance of D-SAMD and AD-SAMD against several other schemes. As a best-case scenario, we consider {\\em centralized} mirror descent, meaning that at each data-acquisition round $t$ all $m$ data samples and their associated gradients are available at a single machine, which carries out stochastic mirror descent and {\\em accelerated} stochastic mirror descent. These algorithms naturally have the best average performance. As a baseline, we consider {\\em local} (accelerated) stochastic mirror descent, in which nodes simply perform mirror descent on their own data streams without collaboration. This scheme does benefit from an insensitivity to the communications ratio $\\rho$, and no mini-batching is required, but it represents a minimum standard for performance in the sense that it does not require collaboration among nodes.\n\nFinally, we consider a communications-constrained adaptation of {\\em distributed gradient descent} (DGD), introduced in \\cite{Nedic.Ozdaglar.ITAC2009}, where local subgradient updates are followed by a single round of consensus averaging on the search points $\\mathbf{x}_i(s)$. DGD implicitly supposes that $\\rho=1$. To handle the $\\rho < 1$ case, we consider two adaptations: {\\em naive} DGD, in which data samples that arrive between communications rounds are simply discarded, and {\\em mini-batched} DGD, in which nodes compute {\\em local} mini-batches of size $b=1\/\\rho$, take gradient updates with the local mini-batch, and carry out a consensus round. While it is not designed for the communications rate-limited scenario, DGD has good performance in general, so it represents a natural alternative against which to compare the performance of D-SAMD and AD-SAMD.\n\n\\subsection{Fully Connected Graphs}\nFirst, we consider the simple case of a fully connected graph, in which $E$ is the set of all possible edges, and the obvious mixing matrix is $\\mathbf{W}= \\mathbf{1}\\mathbf{1}^T\/n$, which has $\\lambda_2 = 0$. This represents a best-case scenario in which to validate the theoretical claims made above. We choose $\\rho = 1\/2$ to examine the regime of low communications ratio, and we let $m$ and $T$ grow according to two regimes: $T = m$, and $T = \\sqrt{m}$, which are the regimes in which D-SAMD and AD-SAMD are predicted to give order-optimum performance, respectively. The constraint on mini-batch size per Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds} is trivial, so we take $b=2$ to ensure that nodes can average each mini-batch gradient via (perfect) consensus. We select the following step-size parameter $\\gamma$: $0.5$ and $2$ for (local and centralized) stochastic mirror descent (MD) and accelerated stochastic mirror descent (A-MD), respectively; 5 for both variants of DGD; and $5$ and $20$ for D-SAMD and AD-SAMD, respectively.\\footnote{While the accelerated variant of stochastic mirror descent makes use of two\nsequences of step sizes, $\\beta_s$ and $\\gamma_s$, these two sequences can be expressed as a function of a single parameter $\\gamma$; see, e.g., the proof of Theorem~\\ref{thm:accelerated.optimality.gap}.} These values were selected via trial-and-error to give good performance for all algorithms; future work involves the use of adaptive step size rules such as AdaGrad and ADAM~\\cite{Duchi:JMLR2011,Kingma:ICLR2015}.\n\n\\begin{figure}[htb]\n \\centering\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/fullConnT_m.eps}\n \\caption{$T = m$}\n \\end{subfigure}\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/fullConnT_Sq_m.eps}\n \\caption{$T = \\sqrt{m}$}\n \\end{subfigure}\n \\caption{Performance of different schemes for a fully connected graph on $\\log$-$\\log$ scale. The dashed lines (without markers) in (a) and (b) correspond to the asymptotic performance upper bounds for D-SAMD and AD-SAMD predicted by the theoretical analysis.}\n\t\\label{fig:fully.connected}\n\\end{figure}\n\nIn Figure~\\ref{fig:fully.connected}(a) and Figure~\\ref{fig:fully.connected}(b), we plot the performance averaged over 1200 and 2400 independent instances of the problem, respectively. We also plot the order-wise theoretical performance $1\/\\sqrt{mT}$, which has a constant slope on the log-log axis. As expected, the distributed methods significantly outperform the local methods. The performance of the distributed methods is on par with the asymptotic theoretical predictions, as seen by the slope of the performance curves, with the possible exception of D-SAMD for $T = m$. However, we observe that D-SAMD performance is at least as good as predicted by theory for $T = \\sqrt{m}$, a regime in which optimality is not guaranteed for D-SAMD. This suggests the possibility that the requirement that $T = \\Omega(m)$ for D-SAMD is an artifact of the analysis, at least for this problem.\n\n\\subsection{Expander Graphs}\nFor a more realistic setting, we consider {\\em expander graphs}, which are families of graphs that have spectral gap $1-\\lambda_2$ bounded away from zero as $m \\to \\infty$. In particular, we use 6-regular graphs, i.e., regular graphs in which each node has six neighbors, drawn uniformly from the ensemble of such graphs. Because the spectral gap is bounded away from zero for expander graphs, one can more easily examine whether performance of D-SAMD and AD-SAMD agrees with the ideal scaling laws discussed in Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds}. At the same time, because D-SAMD and AD-SAMD make use of imperfect averaging, expander graphs also allow us to examine non-asymptotic behavior of the two schemes. Per Corollaries \\ref{cor:mirror.descent.consensus.rounds} and \\ref{cor:accelerated.descent.consensus.rounds}, we choose $b = \\frac{1}{10}\\frac{\\log(mT)}{\\rho \\log(1\/\\lambda_2)}$. While this choice is guaranteed to be sufficient for optimum asymptotic performance, we chose the multiplicative constant $1\/10$ via trial-and-error to give good non-asymptotic performance.\n\n\\begin{figure}[htb]\n \\centering\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/expanderT_m.eps}\n \\caption{$T = m$}\n \\end{subfigure}\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/expanderT_Sq_m.eps}\n \\caption{$T = \\sqrt{m}$}\n \\end{subfigure}\n \\caption{Performance of different schemes for $6$-regular expander graphs on $\\log$-$\\log$ scale, for $\\rho = 1\/2$. Dashed lines once again represent asymptotic theoretical upper bounds on performance.}\n\t\\label{fig:expander}\n\\end{figure}\n\nIn Figure \\ref{fig:expander} we plot the performance averaged over 600 problem instances. We again take $\\rho = 1\/2$, and consider the regimes $T = m$ and $T = \\sqrt{m}$. The step sizes are the same as in the previous subsection except that $\\gamma = 2.5$ for D-SAMD when $T = \\sqrt{m}$, $\\gamma = 28$ for AD-SAMD when $T = m$, and $\\gamma= 8$ for AD-SAMD when $T = \\sqrt{m}$. Again, we see that AD-SAMD and D-SAMD outperform local methods, while their performance is roughly in line with asymptotic theoretical predictions. The performance of DGD, on the other hand, depends on the regime: For $T = m$, it appears to have order-optimum performance, whereas for $T = \\sqrt{m}$ it has suboptimum performance on par with local methods. The reason for the dependency on regime is not immediately clear and suggests the need for further study into DGD-style methods in the case of rate-limited networks.\n\n\\subsection{Erd\\H{o}s-Renyi Graphs}\nFinally, we consider {\\em Erd\\H{o}s-Renyi} graphs, in which a random fraction (in this case $0.1$) of possible edges are chosen. These graphs are not expanders, and their spectral gaps are not bounded. Therefore, order-optimum performance is not easy to guarantee, since the conditions on the rate and the size of the network depend on $\\lambda_2$, which is not guaranteed to be well behaved. We again take $\\rho = 1\/2$, consider the regimes $T = m$ and $T = \\sqrt{m}$, and again we choose $b = \\frac{1}{10}\\frac{\\log(mT)}{\\rho \\log(1\/\\lambda_2)}$. The step sizes are chosen to be the same as for expander graphs in both regimes.\n\n\\begin{figure}[htb]\n \\centering\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/ER_T_m.eps}\n \\caption{$T = m$}\n \\end{subfigure}\n \\begin{subfigure}{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/ER_T_Sq_m.eps}\n \\caption{$T = \\sqrt{m}$}\n \\end{subfigure}\n \\caption{Performance of different schemes on Erd\\H{o}s-Renyi graphs, for $\\rho = 1\/2$, displayed using $\\log$-$\\log$ scale.}\n\t\\label{fig:erdos}\n\\end{figure}\n\nOnce again, we observe a clear distinction in performance between local and distributed methods; in particular, all distributed methods (including DGD) appear to show near-optimum performance in both regimes. However, as expected the performance is somewhat more volatile than in the case of expander graphs, especially for the case of $T = m$, and it is possible that the trends seen in these plots will change as $T$ and $m$ increase.\n\n\\section{Conclusion}\\label{sect:conclusion}\nWe have presented two distributed schemes, D-SAMD and AD-SAMD, for convex stochastic optimization over networks of nodes that collaborate via rate-limited links. Further, we have derived sufficient conditions for the order-optimum convergence of D-SAMD and AD-SAMD, showing that accelerated mirror descent provides a foundation for distributed SO that better tolerates slow communications links. These results characterize relationships between network communications speed and the convergence speed of stochastic optimization.\n\nA limitation of this work is that we are restricted to settings in which the prox mapping is Lipschitz continuous, which excludes important Bregman divergences such as the Kullbeck-Liebler divergence. Further, the conditions for optimum convergence restrict the Lipschitz constant of non-smooth component of the objective function to be small. Future work in this direction includes study of the limits on convergence speed for more general divergences and composite objective functions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecent developments, initiated in\n\\cite{Bagger:2006sk,Gustavsson:2007vu}, which led to important\nprogress in understanding the holographic duality between $D=3$\nsuperconformal theories and type IIA string\/M--theory on $AdS_4$\nhave revived an interest in studying strings and branes in\nsupergravity backgrounds whose bosonic subspace is $AdS_4\\times\nM^{6}$ and $AdS_4\\times M^{7}$, respectively, where $M^6$ is a\ncompactified manifold of $D=10$ type IIA supergravity and $M^7$ is\nits Hopf fibration counterpart in $D=11$ supergravity (or\nM--theory). Examples of interest include the supergravity solutions\nwith $M^6=CP^3$ and $M^7=S^7\/Z_k$ (with an integer $k$ being the\nChern--Simons theory level) and their squashings.\n\n\nIn particular, the ${\\cal N}=6$ Chern-Simons theory with the gauge\ngroup $U(N)_k\\times U(N)_{-k}$ \\cite{Aharony:2008ug} has been\nconjectured to describe, from the $CFT_3$ side, M--theory on $AdS_4\n\\times S^7\/Z_k$. In the limit of the parameter space of the ABJM\ntheory in which the 't Hooft coupling $\\lambda={N\/k}$ is\n$\\lambda^{5\/2}<>1$, the bulk description is given in\nterms of perturbative type IIA string theory on the $AdS_4\n\\times CP^3$ background. To analyze this new type of holographic\ncorrespondence from the bulk theory side, an explicit form of the\naction for the superstring in $AdS_4 \\times CP^3$ superspace is\nrequired.\n\nIn contrast to \\emph{e.g.} the case of type IIB string theory in\n$AdS_5 \\times S^5$ superspace which preserves the maximum\nnumber of 32 supersymmetries and is thus described by the supercoset\n$PSU(2,2|4)\/SO(1,4)\\times SO(5)$, the case of type IIA string theory\non $AdS_4 \\times CP^3$ is more complicated since $AdS_4 \\times CP^3$\npreserves only 24 of 32 supersymmetries. As a consequence, the\ncomplete type IIA superspace with 32 fermionic coordinates, that\nsolves the IIA supergravity constraints for the $AdS_4 \\times CP^3$\nvacuum solution, is not a coset superspace. This superspace has been\nconstructed in\n\\cite{Gomis:2008jt} by dimensional reduction\nof the $AdS_4\\times S^7\/Z_k$ solution of $D=11$ supergravity\ndescribed by the supercoset $OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$\nwith 32 fermionic coordinates. The construction of\n\\cite{Gomis:2008jt} has generalized to superspace the results of\n\\cite{Giani:1984wc,Nilsson:1984bj,Sorokin:1985ap} on the relation of\n$AdS_4 \\times M^6$ solutions of $D=10$ type IIA supergravity and\n$AdS_4\\times M^7$ solutions of $D=11$ supergravity by identifying the\ncompact manifolds $M^7$ as $S^1$ Hopf fibrations over corresponding\n$M^6$.\n\nIn \\cite{Gomis:2008jt} it has been shown that the supercoset space\n$OSp(6|4)\/U(3)\\times SO(1,3)$ with 24 fermionic directions, which\nhas been used in\n\\cite{Arutyunov:2008if}--\\cite{D'Auria:2008cw} to construct a superstring\nsigma model in $AdS_4 \\times CP^3$, is a subspace of the complete\nsuperspace and that the supercoset sigma--model action (being a\npartially gauge--fixed Green--Schwarz superstring action) describes\nonly a subsector of the complete type IIA superstring theory in\n$AdS_4 \\times CP^3$. The reason for this is that the\nkappa--symmetry gauge fixing condition which puts to zero eight\nfermionic modes corresponding to the 8 broken supersymmetries is not\nadmissible for all possible string configurations. So, in\nparticular, though the $OSp(6|4)\/U(3)\\times SO(1,3)$ sigma model\nsector of the theory is classically integrable\n\\cite{Arutyunov:2008if,Stefanski:2008ik} and there are\ngeneric arguments in favor of the integrability of the whole theory,\nthe direct proof of the integrability of the complete $AdS_4 \\times\nCP^3$ superstring still remains an open problem.\n\nThe knowledge of the explicit structure of the $AdS_4 \\times CP^3$\nsuperspace with 32 fermionic directions allows one to approach this\nand other problems. The form of the string action in the $AdS_4\n\\times CP^3$ superspace can be drastically simplified by choosing a\nsuitable description of the background supergeometry and an\nappropriate kappa--symmetry gauge, as was shown previously for the\ncases of the type IIB superstring, D3, M2 and M5--branes in the\ncorresponding $AdS\\times S$ backgrounds\n\\cite{Kallosh:1998qv}--\\cite{Pasti:1998tc}. A superconformal\nrealization and a kappa--symmetry gauge fixing of the $OSp(6|4)$\nsigma model sector of the $AdS_4 \\times CP^3$ superstring have been\nconsidered in \\cite{Uvarov:2008yi} and in a light--cone gauge in\n\\cite{Zarembo:2009au}.\n\n In this paper we perform an alternative\n$\\kappa$--symmetry gauge fixing of the complete $AdS_4\\times CP^3$\nsuperspace which is suitable for studying regions of the theory that\nare not reachable by the supercoset sigma model. In Subsection\n\\ref{FS} we apply this gauge fixing to simplify the superstring\naction in $AdS_4\\times CP^3$ and consider its T--dualization along a\n$3d$ translationally invariant subspace of $AdS_4$, similar to that\nperformed in\n\\cite{Kallosh:1998ji},\nwhich results in a simple action that contains fermions only up to\nthe fourth order. We also argue that, in contrast to the\n$AdS_5\\times S^5$ superstring\n\\cite{Ricci:2007eq,Berkovits:2008ic,Beisert:2008iq}, it is not possible to T--dualize\nthe fermionic sector of the superstring action in $AdS_4\\times\nCP^3$, which agrees with the conclusion of \\cite{Adam:2009kt}\nregarding the $OSp(6|4)$ supercoset subsector of the theory.\n\nIn addition to the superstring, also for certain configurations of\ntype IIA branes, \\emph{e.g.} D0-- and D2--branes considered in\nSection 4, the complete $AdS_4 \\times CP^3$ superspace should be\nused. An interesting example is a 1\/2 BPS probe D2--brane placed at\nthe $d=3$ Minkowski boundary of $AdS_4$. Upon gauge fixing\nworldvolume diffeomorphisms and kappa--symmetry, the effective\ntheory on the worldvolume of this D2--brane, which describes its\nfluctuations in $AdS_4\n\\times CP^3$, is an interacting $d=3$ gauge Born--Infeld--matter theory\npossessing the (spontaneously broken) superconformal symmetry\n$OSp(6|4)$. The model is superconformally invariant in spite of the\npresence on the $d=3$ worldvolume of the dynamical Abelian vector\nfield, since the latter is coupled to the $3d$ dilaton field\nassociated with the radial direction of $AdS_4$. The superconformal\ninvariance is spontaneously broken by a non--zero expectation value\nof the dilaton. This example is a type IIA counterpart of so called\nsingleton M2, tripleton M5 and doubleton D3--branes\n\\cite{deWit:1998tk,Claus:1998fh,Metsaev:1998hf,Pasti:1998tc} at the boundary of $AdS_{p+2}\\times S^{D-p-2}$ ($p=2,3$ and\n5), respectively, in $D=11$ supergravity and type IIB string theory\n(see\n\\cite{Duff:2008pa} for a corresponding brane scan and a review of\nrelated earlier work).\n\nAnother example of interest for the study of the $AdS_4\/CFT_3$\ncorrespondence is a D2--brane filling $AdS_2\\times S^1\\subset AdS_4$\n\\cite{Drukker:2008jm}. This BPS D2--brane configuration corresponds\nto a disorder loop operator in the ABJM theory. Other D--brane\nconfigurations, which are to be related to Wilson loop operators in\nthe ABJM theory, were considered \\emph{e.g.} in\n\\cite{Berenstein:2008dc}--\\cite{Rey:2008bh}.\nIn this paper we extend the bosonic action for a D2--brane wrapping\n$AdS_2\\times S^1$ to include the worldvolume fermionic modes.\n\nWe start our consideration with an overview of the geometry of the\n$AdS_4\\times CP^3$ superspace.\n\n\n\\section{$AdS_4 \\times CP^3$ superspace}\\label{superspace}\n\nThe superspace under consideration contains $AdS_4\\times CP^3$\nas its bosonic subspace and has 32 fermionic directions\n\\cite{Gomis:2008jt}. It is parametrized by the supercoordinates\n\\begin{equation}\\label{Z}\nZ^{\\mathcal M}=(x^{\\hat\nm},y^{m'},\\Theta^{\\underline\\alpha})=(x^{\\hat\nm},y^{m'},\\vartheta^{\\alpha a'},\\upsilon^{\\alpha i}),\n\\end{equation}\nwhere $x^{\\hat m}$ $(\\hat m=0,1,2,3)$ and $y^{m'}$ $(m'=1,\\cdots,6)$\nare, respectively, the coordinates of $AdS_4=SO(2,3)\/SO(1,3)$ and\n$CP^3=SU(4)\/SU(3)\\times U(1)$. $\\Theta^{\\underline\\alpha}$ are the\n32 fermionic coordinates which we split into the 24 coordinates\n$\\vartheta^{\\alpha a'}$, that correspond to the 24 unbroken\nsupersymmetries in the $AdS_4\\times CP^3$ background, and the 8\ncoordinates $\\upsilon^{\\alpha i}$ which correspond to the broken\nsupersymmetries. The indices $\\alpha=1,2,3,4$ are $AdS_4$ spinor\nindices, $a'=1,\\cdots,6$ correspond to a six--dimensional\nrepresentation of $SU(3)$ (note that the index $a'$ appearing on\nspinors is different from the same index appearing in bosonic\nquantities, see Appendix A.5) and $i=1,2$ are $SO(2)\\sim\nU(1)$ indices. For more details of our notation and conventions see\nAppendix A\n\\footnote{Our notation and conventions are close to those in\n\\cite{Gomis:2008jt}. The difference is that, in this paper we put a\n``hat\" on the $AdS_4$ vector indices and use a more conventional IIA\nsuperspace torsion constraint $T_{\\underline{\\alpha\\beta}}{}^A\n=-2i\\Gamma^A_{\\underline{\\alpha\\beta}}$ (instead of\n$T_{\\underline{\\alpha\\beta}}{}^A\n=2\\Gamma^A_{\\underline{\\alpha\\beta}}$) and corresponding constraints\non the gauge field strengths. We also restore the dependence of the\ngeometric quantities and fields on the $S^7$ radius $R$, the\neleven-dimensional Planck length\n$l_p=e^{\\frac{1}{3}<\\phi>}\\sqrt{\\alpha'}$ and the Chern--Simons\nlevel $k$, which were put equal to one in\n\\cite{Gomis:2008jt}.}. For\nthe reader's convenience, below we list some of the notation used in\nthe text:\n\\begin{enumerate}\n\\item $D=10$ $AdS_4\\times CP^3$ superspace with 24 fermions is the\nsupercoset $OSp(6|4)\/ U(3)\\times SO(1,3)$. The supervielbeins and\nconnections are denoted by\n\\begin{equation}\\label{notaA}\n\\Big(E^{\\hat a}, E^{a'}, E^{\\alpha a'}, \\Omega^{\\hat a\\hat b}, \\Omega^{a'b'}, A \\Big)\n\\end{equation}\nwhose expressions are given in Appendix B, eq. (B.1).\n\\item $D=11$ $AdS_4\\times S^7$ superspace with 24 fermions. This is obtained as a $U(1)$ bundle over\nthe $OSp(6|4)\/ U(3)\\times SO(1,3)$ supercoset with the fiber\ncoordinate denoted by $z$. It is the supercoset $OSp(6|4)\\times U(1)\n\/ U(3) \\times SO(1,3)$ whose supervielbeins and connections are\ndenoted by\n\\begin{equation}\\label{notaB}\n\\Big(\\hat E^{\\hat a}, \\hat E^{a'}, \\hat E^7, \\hat E^{\\alpha a'}, \\hat \\Omega^{\\hat a\\hat b}, \\hat\n\\Omega^{a'b'}\\Big)\\,.\n\\end{equation}\nThey are given in eqs. (\\ref{24thA}), see also\n\\cite{Gomis:2008jt}. ${\\hat E}^7$ stands for the 7th (fiber) direction of $S^7$\n(or, equivalently, the 11th direction in $D=11$).\n\\item $D=11$ $AdS_4\\times S^7$ superspace with 32 fermions.\nThis is the supercoset $OSp(8|4)\/SO(7)\\times SO(1,3)$. Its\nsupervielbeins and connections are denoted by\n\\begin{equation}\\label{notaC}\n\\Big(\\underline E^{\\hat a}, \\underline E^{a'}, \\underline E^7, \\underline E^{\\alpha a'},\\underline E^{\\alpha i},\n\\underline \\Omega^{\\hat a\\hat b},\n\\underline \\Omega^{a'b'}, \\underline \\Omega^{a' 7}\\Big)\\,.\n\\end{equation}\nTheir explicit expressions are given in (\\ref{upsilonfunctions}), (\\ref{ads4connection}) and (\\ref{so7connection}).\n\\item Finally, the $D=10$ $AdS_4\\times CP^3$ superspace\nwith 32 fermionic directions is obtained by performing a rotation of\n(\\ref{notaC}) in the $(\\hat a, 7)$--plane accompanied by the\ndimensional reduction to $D=10$ (see \\cite{Gomis:2008jt}). The\ngeometric quantities characterizing this superspace are denoted by\n\\begin{equation}\\label{notaD}\n\\Big({\\cal E}^{\\hat a}, {\\cal E}^{a'}, {\\cal E}^{\\alpha a'}, {\\cal E}^{\\alpha i},\n{\\mathcal O}^{\\hat a\\hat b}, {\\mathcal O}^{a'b'}, {\\cal A} \\Big).\n\\end{equation}\nThe supervielbeins have the following form\n\\end{enumerate}\n\\begin{equation}\\label{simplA}\n\\begin{aligned}\n{\\mathcal E}^{a'}(x,y,\\vartheta,\\upsilon)&=e^{\\frac{1}{3}\\phi(\\upsilon)}\\,\\left(E^{a'}(x,y,\\vartheta)+2i\\upsilon\\,{{\\sinh m}\\over\nm}\\gamma^{a'}\\gamma^5\\,E(x,y,\\vartheta)\\right) \\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\hat a}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over3}\\phi(\\upsilon)}\\,\\left(E^{\\hat\nb}(x,y,\\vartheta)+4i\\upsilon\\gamma^{\\hat b}\\,{{\\sinh^2{{\\mathcal M}\/\n2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right)\\Lambda_{\\hat b}{}^{\\hat\na}(\\upsilon)\n\\\\\n&{}\n\\hskip+1cm -e^{-{1\\over3}\\phi(\\upsilon)}\\,\\frac{R^2}{kl_p}\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right) E_7{}^{\\hat\na}(\\upsilon)\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over6}\\phi(\\upsilon)}\\,\\left({{\\sinh{\\mathcal\nM}}\\over{\\mathcal M}}\\,D\\upsilon\\right)^{\\beta j}\\,S_{\\beta\nj}{}^{\\alpha i}\\,(\\upsilon)\n-ie^{\\phi(\\upsilon)}{\\mathcal A}_1(x,y,\\vartheta,\\upsilon)\\,(\\gamma^5\\varepsilon\\lambda(\\upsilon))^{\\alpha\ni}\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\ne^{{1\\over6}\\phi(\\upsilon)}\\,E^{\\gamma b'}(x,y,\\vartheta)\\,\\left(\n\\delta_{\\gamma}{}^{\\beta}-\\frac{8}{R}\\,\\left(\\gamma^5\\,\\upsilon\\,{{\\sinh^2{{m}\/2}}\\over{m}^2}\\right)_{\\gamma\ni}\\upsilon^{\\beta i} \\right)S_{\\beta b'}{}^{\\alpha\na'}\\,(\\upsilon)\\,.\n\\end{aligned}\n\\end{equation}\nThe new objects appearing in these expressions, $m$, $\\mathcal M$,\n$\\Lambda_{\\hat a}{}^{\\hat b}$, $E_7{}^{\\hat a}$ and\n$S_{\\underline\\alpha}^{\\underline\\beta}$, are functions of $\\upsilon$\nand their explicit forms are given in Appendix B.1 while the dilaton\n$\\phi$, dilatino $\\lambda$ and RR one--form $\\mathcal A_1$ are given\nbelow. Contracted spinor indices have been suppressed, \\emph{e.g.}\n$(\\upsilon\\varepsilon\\gamma^5)_{\\alpha i}=\\upsilon^{\\beta\nj}\\varepsilon_{ji}\\gamma^5_{\\beta\\alpha}$, where\n$\\varepsilon_{ij}=-\\varepsilon_{ji}$, $\\varepsilon_{12}=1$ is the\n$SO(2)$ invariant tensor. The covariant derivative is defined as\n\\begin{eqnarray}\\label{D}\nD\\upsilon=\\left(d+\\frac{i}{R}E^{\\hat\na}(x,y,\\vartheta)\\,\\gamma^5\\gamma_{\\hat a}-\\frac{1}{4}\\Omega^{\\hat a\n\\hat b}(x,y,\\vartheta)\\,\\gamma_{\\hat a\\hat b}\\right)\\upsilon \\,.\n\\end{eqnarray}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{simplB}\n\\begin{aligned}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon) &=\nR\\,e^{-{4\\over3}\\phi(\\upsilon)}\\,\\left[\n\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal\nM}^2}\\,D\\upsilon\\right)\\frac{R}{kl_p}\\,\\Phi(\\upsilon)\n\\right.\\\\\n&\\left.\\hspace{40pt}+\\frac{1}{kl_p}\\left(E^{\\hat\na}(x,y,\\vartheta)+4i\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\right)E_{7\\hat a}(\\upsilon)\n\\right]\\,.\n\\end{aligned}\n\\end{equation}\n\nThe RR four-form and the NS--NS three-form superfield strengths are\ngiven by\n\\begin{equation}\\label{f4h3}\n\\begin{aligned}\nF_4&=d{\\mathcal A}_3-{\\mathcal A}_1\\,H_3=-\\frac{1}{4!}{\\mathcal\nE}^{\\hat d}{\\mathcal E}^{\\hat c}{\\mathcal E}^{\\hat b}{\\mathcal\nE}^{\\hat a}\\left(\\frac{6}{kl_p}\\,e^{-2\\phi}\\Phi\\varepsilon_{\\hat\na\\hat b\\hat c\\hat d}\\right) -\\frac{i}{2}{\\mathcal E}^{B}{\\mathcal\nE}^{A}{\\mathcal E}^{\\underline\\beta}\n{\\mathcal E}^{\\underline\\alpha}e^{-\\phi}(\\Gamma_{AB})_{\\underline{\\alpha\\beta}}\\,,\\\\\nH_3&=dB_2=-\\frac{1}{3!}{\\mathcal E}^{\\hat c}{\\mathcal E}^{\\hat\nb}{\\mathcal E}^{\\hat a}(\\frac{6}{kl_p}e^{-\\phi}\\varepsilon_{\\hat a\\hat b\\hat\nc\\hat d}E_7{}^{\\hat d}) -i{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_A\\Gamma_{11})_{\\underline{\\alpha\\beta}}\n+i{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB}\\Gamma^{11}\\lambda)_{\\underline\\alpha}\n\\end{aligned}\n\\end{equation}\nand the corresponding gauge potentials are\n\\begin{equation}\\label{B2}\nB_2=b_2+\\int_0^1\\,dt\\,i_\\Theta H_3(x,y,t\\Theta)\\,,\\qquad \\Theta=(\\vartheta,\\upsilon)\\,\\\\\n\\end{equation}\n\\begin{equation}\\label{A3}\n\\hskip+1.9cm{\\mathcal\nA}_3=a_3+\\int_0^1\\,dt\\,i_\\Theta\\left(F_4+\\mathcal{A}_1H_3\\right)(x,y,t\\Theta)\\,,\n\\end{equation}\nwhere $b_2$ and $a_3$ are the purely bosonic parts of the gauge\npotentials and $i_\\Theta$ means the inner product with\n$\\Theta^{\\underline\\alpha}$. Note that $b_2$ is pure gauge in the\n$AdS_4\\times CP^3$ solution while $a_3$ is the RR three-form\npotential of the bosonic background.\n\n The dilaton superfield $\\phi(\\upsilon)$, which depends only on\nthe eight fermionic coordinates corresponding to the broken\nsupersymmetries, has the following form in terms of $E_7{}^{\\hat\na}(\\upsilon)$ and $\\Phi(\\upsilon)$\n\\begin{equation}\\label{dilaton1}\ne^{{2\\over\n3}\\phi(\\upsilon)}={R\\over{kl_p}}\\,\\sqrt{\\Phi^2+E_7{}^{\\hat\na}\\,E_7{}^{\\hat b}\\,\\eta_{\\hat a\\hat b}}\\,.\n\\end{equation}\nThe value of the dilaton at $\\upsilon=0$ is\n\\begin{equation}\ne^{\\frac{2}{3}\\phi(\\upsilon)}|_{\\upsilon=0}=e^{\\frac{2}{3}\\phi_0}=\\frac{R}{kl_p}\\,.\n\\end{equation}\nThe fermionic field $\\lambda^{\\alpha i}(\\upsilon)$ describes the\nnon--zero components of the dilatino superfield and is given by the\nequation \\cite{Howe:2004ib}\n\\begin{equation}\\label{dilatino1}\n\\lambda_{\\alpha i}=-\\frac{i}{3}D_{\\alpha i}\\,\\phi(\\upsilon)\\,.\n\\end{equation}\n\nIn the above expressions $E^{\\hat a}(x,y,\\vartheta)$, $E^{\na'}(x,y,\\vartheta)$ and $\\Omega^{\\hat a\\hat b}(x,y,\\vartheta)$ are\nthe supervielbeins and the $AdS_4$ part of the spin connection of\nthe supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$ and $A(x,y,\\vartheta)$\nis the corresponding type IIA RR one--form gauge superfield, eq.\n(\\ref{notaA}), whose explicit form is given in Appendix B.\n\nAs mentioned above other quantities appearing in eqs.\n(\\ref{simplA})--(\\ref{dilatino1}), namely $\\mathcal M$, $m$,\n$\\Phi(\\upsilon)$, $E_7{}^{\\hat a}(\\upsilon)$, $\\Lambda_{\\hat a}{}^{\\hat b}(\\upsilon)$ and\n$S_{\\underline\\beta}{}^{\\underline\\alpha}(\\upsilon)$, whose geometrical and\ngroup--theoretical meaning has been explained in\n\\cite{Gomis:2008jt}, are also given in Appendix B.\n\n\\setcounter{equation}0\n\\section{Kappa--symmetry gauge fixing}\nWe shall now consider conditions for gauge fixing kappa--symmetry\nwhich are convenient for the description of configurations of\nsuperstrings and D-branes in the $AdS_4\\times CP^3$ superbackground\ndescribed above and for studying $AdS_4\/CFT_3$ correspondence\nproblems.\n\nSince the $AdS_4\/CFT_3$ holography is realized at the $3d$ Minkowski\nboundary of $AdS_4$ it is convenient to choose the $AdS_4\\times\nCP^3$ metric in the form\n\\begin{equation}\\label{ads4metric11}\nds^2=\\left(r\\over\n{R_{{CP^3}}}\\right)^4\\,dx^m\\,\\eta_{mn}\\,dx^n+\\left({R_{CP^3}}\\over\nr\\right)^2\\,dr^2+R_{CP^3}^2\\,ds^2_{_{CP^3}}\\,\n\\end{equation}\nwhere $m=0,1,2$ are indices corresponding to the coordinates of the\n$3d$ Minkowski boundary and $r$ is the 4th, radial, coordinate of\n$AdS_4$. So the $AdS_4$ coordinates are $x^{\\hat m}=(x^m,r)$. The\n$AdS_4$ radius is half of the $CP^3$ radius $R_{CP^3}$ which (in the\nstring frame) is related to the $S^7$ radius $R$ as follows\n\\begin{equation}\\label{R}\nR_{CP^3} = e^{\\frac{1}{3}\\phi_0}R =\\left(\\frac{R^3}{kl_p}\\right)^{1\/2}\\,.\n\\end{equation}\n\nIn the coordinate system associated with the metric\n(\\ref{ads4metric11}) (the bosonic part of) the RR field ${\\mathcal A}_3$, whose flux,\ntogether with $F_2=da_1={e^{-\\phi_0}\\over\n{R_{CP^3}}}\\,dy^{m'}dy^{n'}J_{m'n'}$ (where $dy^{m'}dy^{n'}J_{m'n'}$\nis the K\\\"ahler form on $CP^3$), ensures the compactification on\n$AdS_4\\times CP^{3}$\n\\cite{Watamura:1983hj,Nilsson:1984bj,Sorokin:1985ap}, has the\nfollowing form\n\\begin{equation}\\label{A31}\na_3=e^{-\\phi_0}\\left({r\\over\n{R_{CP^3}}}\\right)^6\\,dx^0\\,dx^1\\,dx^2\\,,\\qquad F_4={6\\over\nR_{CP^3}}e^{-\\phi_0}\\,\\left({r\\over\n{R_{CP^3}}}\\right)^5\\,dx^0\\,dx^1\\,dx^2\\,dr\\,.\n\\end{equation}\n(In our conventions the exterior derivative acts from the right.)\n\nInstead of the $AdS_4$ part of the metric (\\ref{ads4metric11}),\nwhich obscures a bit the fact that the metric of the conformal\nboundary is the flat Minkowski metric on $R^{1,2}$, one can use the\n$AdS_4$ metric in the conformally flat form\n\\begin{equation}\\label{ads4metric21}\nds^2_{_{AdS_4}}={1\\over\nu^2}(dx^m\\eta_{mn}dx^n+\\frac{R_{CP^3}^2}{4}\\,du^2)\\,,\n\\qquad u=\\left(R_{CP^3}\\over r\\right)^2\\,.\n\\end{equation}\nThis metric is associated with a simple coset representative $\ng=\\exp(x^m\\,\\Pi_m)$ $\\exp(R_{CP^3}\\,\\ln(u) D)$, where $\\Pi_m$ are\nthe generators of the Poincar\\'e translations along the Minkowski\nboundary $([\\Pi_m,\\,\\Pi_n]=0)$ and $D$ is the dilatation generator\n$[D,\\,\\Pi_m]=\\Pi_m$.\n\nNote that if the components of the vielbein associated with the\nmetric (\\ref{ads4metric11}) or (\\ref{ads4metric21}) are chosen to\nbe\\footnote{Note that the vielbeins $e^a$ and $e^3$ appearing in eq.\n(\\ref{ad4v}) correspond to the $AdS_4$ metric of the $D=11$\n$AdS_4\\times S^7$ solution characterized by the radius R which is\nrelated to the $CP^3$ radius in the string frame according to eq.\n(\\ref{R}). These bosonic vielbeins will appear in our explicit\nexpressions for the $AdS_4\\times CP^3$ supergeometry.}\n\\begin{equation}\\label{ad4v}\ne^{\\frac{\\phi_0}{3}}\\,e^a={r^2\\over\nR_{CP^3}^2}\\,dx^a=u^{-1}\\,dx^a\\,,\\qquad\ne^{\\frac{\\phi_0}{3}}\\,e^3=\\frac{R_{CP^3}}{r}\\,dr=-\\frac{R_{CP^3}}{2u}\\,du,\n\\end{equation}\nthe components of the $SO(1,3)$ spin connection are\n\\begin{equation}\\label{eaoa31}\n\\omega^{a3}=-\\frac{2}{R}\\,e^a\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{oab}\n\\omega^{ab}=0\\,.\n\\end{equation}\nWe shall use the relation (\\ref{eaoa31}) to simplify the form of the\ngauge fixed $AdS_4 \\times CP^3$ supergeometry. Note that the\ncondition (\\ref{eaoa31}) can always be imposed by performing an\nappropriate local $SO(1,3)$ transformations of the vielbein and\nconnection, though in general the $SO(1,2)$ components $\\omega^{ab}$\nof the connection will be non--zero.\n\nUsing the previous experience of gauge fixing kappa--symmetry of superstrings, D-branes and M-branes in AdS backgrounds\n\\cite{Kallosh:1998qv}--\\cite{Pasti:1998tc} we choose the\nkappa--symmetry gauge fixing condition in the form\n\\footnote{Such a gauge for fixing kappa--symmetry is analogous to\nthe so called Killing spinor gauge \\cite{Kallosh:1998qv}, or\nsupersolvable gauge\n\\cite{Dall'Agata:1998wz}, or the superconformal gauge \\cite{Pasti:1998tc}.}\n\\begin{equation}\\label{kappagauge1}\n\\Theta=\\frac{1}{2}(1\\pm\\gamma)\\Theta\\,\\quad \\Rightarrow\n\\quad \\vartheta^{a'}=\\frac{1}{2}(1\\pm\\gamma)\\vartheta^{a'}\\,,\\qquad\n\\upsilon^{i}=\\frac{1}{2}(1\\pm\\gamma)\\upsilon^{i},\n\\end{equation}\nwhere\n\\begin{equation}\\label{gamma}\n\\gamma=\\gamma^{012}\\qquad\\Rightarrow\\qquad\\gamma^2=1,\n\\quad\\{\\gamma,\\gamma^5\\}=[\\gamma,\\gamma^a]=0\\quad\\mbox{and}\\quad\\gamma\\gamma^3=-i\\gamma^5\\,,\n\\end{equation}\n$a=0,1,2$ are the indices of the 3d Minkowski boundary or of\n$AdS_2\\times S^1$ and $\\gamma^3$ is associated with the third\nspatial direction of $AdS_4$. Note that, in view of our definition\n(\\ref{Gamma10}) of the $D=10$ gamma--matrices the matrices defined\nin (\\ref{gamma}) can be regarded either as $4d$ gamma matrices or as\nthe $D=10$ matrices $\\Gamma^{\\hat a}=\\gamma^{\\hat a}\\otimes {\\bf 1}$\n$(\\hat a=0,1,2,3)$.\n\nThe condition (\\ref{kappagauge1}) is admissible for fixing\nkappa--symmetry if the projection matrix $\\frac{1}{2}(1\\mp\\gamma)$ either coincides or does not\ncommute with the kappa--symmetry projection matrix $\\frac{1}{2}(1+\\Gamma)$ of a given\nconfiguration of the superstring and D--branes.\nThis can be understood in the following way. To lowest order in fermions $\\Theta$ transforms under kappa--symmetry as\n\\begin{equation}\n\\delta_\\kappa\\Theta=\\frac{1}{2}(1+\\Gamma)\\kappa\\,,\n\\end{equation}\nwhere $\\frac{1}{2}(1+\\Gamma)$ is a projection matrix and\n$\\kappa(\\xi)$ is an arbitrary spinor parameter. It is then clear\nthat if the two projectors coincide, we can pick a $\\kappa$ such\nthat $\\frac{1}{2}(1+\\Gamma)\\Theta=0$, or equivalently\n$\\Theta=\\frac{1}{2}(1-\\Gamma)\\Theta$. In the case when the two\nprojection operators do not coincide a kappa--symmetry variation of\nthe gauge--fixing condition $\\frac{1}{2}(1\\mp\\gamma)\\Theta=0$ which\nleaves it intact gives\n\\begin{equation}\n0=\\frac{1}{4}(1\\mp\\gamma)(1+\\Gamma)\\kappa\n=\\frac{1}{8}(1+\\Gamma)(1\\mp\\gamma)(1+\\Gamma)\\kappa\\mp\\frac{1}{8}[\\gamma,\\Gamma](1+\\Gamma)\\kappa\n=\\mp\\frac{1}{8}[\\gamma,\\Gamma](1+\\Gamma)\\kappa\\,,\n\\end{equation}\nwhere in the last step we made use of the initial equation. This\nmeans that for the gauge--fixing to be complete, i.e. that the\nvariation of the gauge fixing condition vanishes if and only if all\nindependent kappa--symmetry parameters are put to zero, the\ncommutator $[\\gamma,\\Gamma]$ has to be an invertible matrix (when\nrestricted to the relevant subspace).\n\n As we shall see\nbelow, for any choice of the sign the condition (\\ref{kappagauge1})\n is an admissible gauge-fixing in the case of\n arbitrary motion of D0--branes in $AdS_4 \\times CP^3$, while in the\ncase of the superstring it is admissible (for both signs) for those\n configurations for which the projection of the string\nworldsheet on the $3d$ Minkowski boundary is a non--degenerate\ntwo--dimensional time--like surface. In the case of the D2--brane\nplaced at the Minkowski boundary of $AdS_4$, to gauge fix kappa--symmetry one must choose the condition (\\ref{kappagauge1}) with the\nlower sign\n\\cite{Pasti:1998tc}, while both signs are admissible when the\n D2--brane wraps an $AdS_2 \\times S^1$ subspace of $AdS_4$. However,\n the choice of (\\ref{kappagauge1}) with the upper sign yields\n the simplest gauge-fixed form of the string and\n brane actions in the $AdS_4\\times CP^3$ superbackground.\n\n\nWhen the fermionic coordinates are restricted by the condition\n(\\ref{kappagauge1}), the expressions for the supervielbeins and the\ngauge superfields of the $AdS_4 \\times CP^3$ superspace drastically\nsimplify due to the identities satisfied by the projected fermionic\ncoordinates given in Appendix C. In particular, the functions of\n$\\upsilon$ which enter the eqs. (\\ref{simplA})--(\\ref{dilaton1}),\nwhose explicit forms are given in Appendix B.1, reduce to\n\\begin{equation}\\label{Phi1}\n\\Phi(\\upsilon)=1+\\frac{8}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal\nM}^2}\\,\\varepsilon\\upsilon =1\\,,\n\\end{equation}\n\\begin{eqnarray}\nE_7{}^a(\\upsilon)&=&-\\frac{8i}{R}\\,\\upsilon\\gamma^a\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}=-\\frac{2i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,,\\\\\nE_7{}^3(\\upsilon)&=&-\\frac{8i}{R}\\,\\upsilon\\gamma^3\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}=0\\,.\n\\end{eqnarray}\nThe dilaton superfield (\\ref{dilaton1}) takes the form\n\\begin{equation}\\label{gfdialton}\ne^{{2\\over 3}\\phi(\\upsilon)}\n=\\frac{R}{kl_p}(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\quad \\Rightarrow \\quad\n\\phi(\\upsilon)=\\frac{3}{2}(\\log\\frac{R}{kl_p}-\\frac{6}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{equation}\nwhere $\\upsilon\\ups=\\delta_{ij}C_{\\alpha\\beta}\\upsilon^{\\alpha i}\\upsilon^{\\beta j}$, and the dilatino becomes\n\\begin{eqnarray}\n\\lambda^{\\alpha i}\n=\n\\frac{2i}{R}\\left(\\frac{R}{kl_p}\\right)^{-1\/4}((\\gamma^5\\upsilon)^{\\alpha i}+\\frac{3}{R}\\upsilon^{\\alpha i}\\,\\upsilon\\ups)\\,.\n\\end{eqnarray}\nWe also find that\n\\begin{equation}\n\\Lambda_a{}^\n= (1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\\delta_a{}^b\\,,\n\\qquad\n\\Lambda_3{}^a = \\Lambda_a{}^3=0\\,,\\qquad\n\\Lambda_3{}^3=1\\,,\n\\end{equation}\nand\n\\begin{eqnarray}\\label{S}\n&&S_{\\underline\\alpha}{}^{\\underline\\beta}=\n\\left(\\frac{R}{kl_p}\\right)^{1\/4}e^{-\\frac{1}{6}\\phi}\\,\\delta_{\\underline\\alpha}{}^{\\underline\\beta}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(\\Gamma_a\\Gamma_{11})_{\\underline\\alpha}{}^{\\underline\\beta}\\,.\n\\end{eqnarray}\n\n\n\n\\subsection{$AdS_4 \\times CP^3$ supergeometry with\n$\\Theta={1\\over 2}(1+\\gamma)\\Theta$}\\label{theta-}\n\nThe supervielbeins (\\ref{simplA}) and the gauge superfields\n(\\ref{simplB}), (\\ref{B2}) and (\\ref{A3}) take the simplest form\nwhen the kappa--symmetry gauge condition (\\ref{kappagauge1}) is\nchosen with the upper sign. In virtue of eqs.\n(\\ref{Phi1})--(\\ref{S}) and expressions given in Appendix C, the\nsupervielbeins reduce to\n\\begin{equation}\\label{simple+v}\n\\begin{aligned}\n{\\mathcal\nE}^{a'}(x,y,\\vartheta,\\upsilon)&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^{a'}(y)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^a(x,y,\\vartheta,\\upsilon) &=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\n(e^a(x)+i\\Theta\\gamma^aD\\Theta)(1-\\frac{1}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^3(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^3(x)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{simplesv}\n\\begin{aligned}\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_8\\upsilon)^{\\alpha\ni}-\\frac{1}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_8\\upsilon\\varepsilon\\gamma_a\\gamma_5)^{\\alpha\ni}\n-\\frac{4i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)(\\gamma_a\\upsilon)^{\\alpha\ni}\\,\\upsilon\\ups\n\\Big)\n\\,,\n\\nonumber\\\\\n\\nonumber\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_{_{24}}\\vartheta)^{\\alpha a'}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_{_{24}}\\vartheta\\gamma_a\\gamma_5\\gamma_7)^{\\alpha\na'}\\Big)\\,.\\nonumber\n\\end{aligned}\n\\end{equation}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{simple+A}\n\\begin{aligned}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon) &=kl_p\\Big(\nA(y)-\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)\\,\\upsilon\\varepsilon\\gamma_a\\upsilon\\Big)\\,,\n\\end{aligned}\n\\end{equation}\nwhere $A(y)$ is the potential for the K\\\"ahler form on $CP^3$,\n\\emph{i.e.} $dA(y)=\\frac{1}{R^2}\\,dy^{m'}dy^{n'}\\,J_{m'n'}$, and the covariant derivatives are\n\\begin{eqnarray}\\label{D81}\nD\\Theta&=&(D_8\\upsilon,\\,D_{24}\\vartheta)\\\\\nD_8\\upsilon&=&{\\mathcal P_2}\\,\\left(d-\\frac{1}{R}e^3-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}+2A(y)\\varepsilon\\right)\\upsilon\\nonumber\\\\\nD_{24}\\vartheta&=&{\\mathcal P}_6\\,(d-\\frac{1}{R}e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'} -\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,,\\nonumber\n\\end{eqnarray}\nwhere ${\\mathcal P}_2$ and ${\\mathcal P}_6$ are projectors that\nsingle out from 32 $\\Theta$, respectively, 8 $\\upsilon$ and 24\n$\\vartheta$ (see Appendix A.5). The appearance of the $U(1)$ gauge potential $A(y)$ in the covariant\nderivative of $\\upsilon$ (\\ref{D81}) reflects the fact that\n$\\upsilon$ has $U(1)$ charge equal to 2.\n\\\\\nNote that\n\\begin{equation}\nD_8={\\mathcal P}_2\\, {\\mathcal D}\\,{\\mathcal P}_2\\,,\\qquad\nD_{24}={\\mathcal P}_6\\, {\\mathcal D}\\,{\\mathcal P}_6\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n{\\mathcal D}=d-\\frac{1}{R}e^3+\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'}\\,.\n\\end{equation}\nThe NS--NS three-form, eq. (\\ref{f4h3}), becomes\n\\begin{eqnarray}\\label{H3ex}\nH_3&=&-\\frac{6i}{R^2} e^3\\, {\\mathcal E}^b\\,{\\mathcal\nE}^a\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n-\\frac{2R}{kl_p}\\,\\Big[\n\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)(e^a+i\\Theta\\gamma^aD\\Theta)\\,D_8\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{1}{R}(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\,\\gamma_{ab}\\,D\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n+\\frac{1}{2}\\,e^3\\,D\\Theta\\,\\gamma_7\\,D\\Theta\n+\\frac{2i}{R}\\,e^3\\,e^{a'}\\,D\\Theta\\gamma_{a'}\\gamma_7\\upsilon\n\\nonumber\\\\\n&&{} +\\frac{i}{2}\\,e^{a'}\\,D\\Theta\\,\\gamma_{a'}\\gamma_7\\,D\\Theta\n+\\frac{1}{R}\\,e^{b'}\\,e^{a'}\\,D\\Theta\\,\\gamma_{a'b'}\\gamma_7\\upsilon\\,\\Big]\\,,\n\\end{eqnarray}\nwhere $\\Theta=(\\vartheta,\\upsilon)$ and\n$D\\Theta=(D_{24}\\vartheta,\\,D_8\\upsilon)$.\n\\\\\nWe now want to determine the potential of $H_3=dB_2$ using eq.\n(\\ref{B2}). Taking into account that $i_\\Theta\\mathcal E^{A}=0$, and\nthe fact that with the plus sign in the projector\n(\\ref{kappagauge1})\n$\\Theta\\gamma_{ab}D\\Theta=\\varepsilon_{abc}\\,\\Theta\\gamma^cD\\Theta$ etc., we\nget\n\\begin{eqnarray}\ni_\\Theta H_3 &=& 2\\frac{R}{kl_p}\n\\Big(\n-\\frac{i}{R}e^be^a\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n+e^3\\,\\Theta\\gamma^7D\\Theta\n\\nonumber\\\\\n&&{} +ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n+\\frac{4}{R}e^b\\,\\Theta\\gamma^aD\\Theta\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n+\\frac{3i}{R}\\Theta\\gamma^bD\\Theta\\,\\Theta\\gamma^aD\\Theta\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{eqnarray}\nThis gives the NS--NS two-form potential (see eq. (\\ref{B2}))\n\\begin{eqnarray}\\label{simple+B}\nB_2&=&\n\\frac{R}{kl_p}\n\\Big[\n-\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)\\,(e^a+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{20pt}}\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n+e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n\\Big]\\,.\n\\end{eqnarray}\n\nNow we turn our attention to the RR four-form $F_4$ (\\ref{f4h3}) and\nits potential $\\mathcal A_3$ (\\ref{A3}). $F_4$ simplifies to\n\\begin{equation}\\label{f4}\nF_4= -\\frac{1}{kl_p}\\,e^{-2\\phi}\\,{\\mathcal E}^3{\\mathcal\nE}^c{\\mathcal E}^b{\\mathcal E}^a\\,\\varepsilon_{abc}\n-\\frac{i}{2}e^{-\\phi}{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta} {\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB})_{\\underline{\\alpha\\beta}}\\,,\n\\end{equation}\nwhich gives\n\\begin{eqnarray}\ni_\\Theta F_4 &=& -i(\\frac{R}{kl_p})^{1\/4}e^{-\\phi}{\\mathcal\nE}^{B}{\\mathcal E}^{A} ({\\mathcal E}\\Gamma_{AB}\\Theta\n+\\frac{i}{R}{\\mathcal\nE}\\Gamma_{AB}\\gamma_a\\Gamma_{11}\\Theta\\,\\upsilon\\gamma^a\\varepsilon\\upsilon)\n\\nonumber\\\\\n&=&\n-i(e^b+i\\Theta\\gamma^bD\\Theta)(e^a+i\\Theta\\gamma^aD\\Theta)\\,(1+\\frac{12}{R^2}(\\upsilon\\ups)^2)(D\\Theta\\gamma_{ab}\\Theta\n+\\frac{4i}{R^2}e^c\\varepsilon_{abc}\\,(\\upsilon\\ups)^2 )\n\\nonumber\\\\\n&&{}\n+\\frac{4i}{R}e^3(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\gamma^7\\Theta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n-\\frac{4}{R}e^{a'}(e^a+i\\Theta\\gamma^aD\\Theta)\\,D\\Theta\\gamma_{a'}\\gamma^7\\Theta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{} +2e^3e^{a'}(D\\Theta\\gamma_{a'}\\Theta\n+\\frac{4i}{R^2}(e^a+i\\Theta\\gamma^aD\\Theta)\\upsilon\\gamma_a\\gamma_{a'}\\vartheta\\,\\upsilon\\ups\n)\n\\nonumber\\\\\n&&{} -ie^{b'}e^{a'}(D\\Theta\\gamma_{a'b'}\\Theta\n+\\frac{4i}{R^2}(e^a+i\\Theta\\gamma^aD\\Theta)\\upsilon\\gamma_a\\gamma_{a'b'}\\vartheta\\,\\upsilon\\ups)\\,.\n\\end{eqnarray}\nSince $i_\\Theta\\mathcal A_1=0$ the RR three--form potential\n(\\ref{A3}) becomes\n\\begin{eqnarray}\\label{A3-}\n\\mathcal A_3&=&a_3+\\int_0^1\\,dt(i_\\Theta F_4+\\mathcal A_1i_\\Theta H_3)(x,y,t\\Theta)\n\\nonumber\\\\\n&=& a_3 -\\frac{i}{2}e^be^a\\,D\\Theta\\gamma_{ab}\\Theta\n+e^3e^{a'}\\,D\\Theta\\gamma_{a'}\\Theta\n-\\frac{i}{2}e^{b'}e^{a'}\\,D\\Theta\\gamma_{a'b'}\\Theta\n+\\frac{1}{2}e^b\\Theta\\gamma^aD\\Theta\\,D\\Theta\\gamma_{ab}\\Theta\n\\nonumber\\\\\n&&{}\n+\\frac{i}{6}\\Theta\\gamma^bD\\Theta\\,\\Theta\\gamma^aD\\Theta\\,D\\Theta\\gamma_{ab}\\Theta\n+k\\,l_p\\,A(y)\\,B_2\\,.\n\\end{eqnarray}\nLooking at the purely bosonic part of $F_4$, eq. (\\ref{f4}) it is\neasy to see (compare also with eqs. (\\ref{A31})) that we can take\n\\begin{equation}\na_3=\\frac{1}{3!}e^ce^be^a\\,\\varepsilon_{abc}\\,.\n\\end{equation}\nNote that in the above expressions for the supervielbeins\n(\\ref{simple+v}), the RR one--form (\\ref{simple+A}), the three--form\n(\\ref{A3-}) and the NS-NS two--form (\\ref{simple+B}) the maximum\norder of the fermions is six.\n\n\n\\subsection{$AdS_4 \\times CP^3$ supergeometry with\n$\\Theta={1\\over 2}(1-\\gamma)\\Theta$}\\label{theta++}\n\nWhen the condition (\\ref{kappagauge1}) is chosen with the lower\nsign, in view of eqs. (\\ref{Phi1})--(\\ref{S}) and expressions given\nin Appendix C, the supervielbeins (\\ref{simplA}) and the RR\none--form gauge superfield (\\ref{simplB}) reduce to a form which is\nmore complicated than their gauge--fixed counterparts of the\nprevious Subsection. But, as we have already mentioned, one cannot\nuse the gauge fixing condition of Subsection \\ref{theta-} to\ndescribe the D2--brane at the Minkowski boundary of $AdS_4$, and\nshould impose $\\Theta={1\\over 2}(1-\\gamma)\\Theta$ instead. In this\ncase the supervielbeins take the following form\n\\begin{equation}\\label{-bv}\n\\begin{aligned}\n{\\mathcal E}^{a'}(x,y,\\vartheta,\\upsilon)&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\n\\Big(e^{a'}(y)-\\frac{2}{R}e^a(x)\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta\\Big)\n(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^a(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}\\,\\Big( e^a(x) +i\\Theta\\gamma^a D\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta-\\upsilon\\ups)^2\n\\Big)(1-\\frac{1}{R^2}(\\upsilon\\ups)^2)\\,,\n\\\\\n\\\\\n{\\mathcal E}^3(x,y,\\vartheta,\\upsilon)\n&=\\Big(\\frac{R}{kl_p}\\Big)^{1\/2}e^3(x)(1-\\frac{3}{R^2}(\\upsilon\\ups)^2)\\,,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{-fv}\n\\begin{aligned}\n{\\mathcal E}^{\\alpha i}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_8\\upsilon)^{\\alpha i}\n-\\frac{1}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(D_8\\upsilon\\varepsilon\\gamma_a\\gamma_5)^{\\alpha\ni}\n\\\\\n&{}\n\\hskip+1cm\n-\\frac{4i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta-\\upsilon\\ups)^2\n)(\\gamma_a\\upsilon)^{\\alpha i}\\,\\upsilon\\ups\n\\Big)\n\\,,\n\\\\\n\\\\\n{\\mathcal E}^{\\alpha a'}(x,y,\\vartheta,\\upsilon) &=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}\\Big( (D_{_{24}}\\vartheta)^{\\alpha a'}\n+\\frac{i}{R}(D_{24}\\vartheta\\gamma_a\\gamma^5\\gamma^7)^{\\alpha\na'}\\,\\upsilon\\gamma^a\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{aligned}\n\\nonumber\n\\end{equation}\nThe type IIA RR one--form gauge superfield is\n\\begin{equation}\\label{A2-}\n{\\mathcal A}_1(x,y,\\vartheta,\\upsilon)=kl_p\\Big(A(y)\n-\\frac{2}{R^2}e^a(x)\\,\\Theta\\gamma^7\\gamma_a\\Theta\n -\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta\n+\\frac{1}{R^2}e^a(x)(\\vartheta\\vartheta)^2\n)\\upsilon\\varepsilon\\gamma_a\\upsilon\n\\Big)\\,.\n\\end{equation}\nIn the above expressions\n\\begin{eqnarray}\\label{D8}\nD\\Theta&=&(D_8\\upsilon\\,,D_{24}\\vartheta)\\,,\\nonumber\\\\\nD_8\\upsilon&=&(D-\\frac{2i}{R^2}\\upsilon\\ups\\,e^a\\,\\gamma_a+2A(x,y,\\vartheta)\\,\\varepsilon)\\upsilon\n\\\\\n&&\\hspace{-40pt}=\n\\Big(d+\\frac{2i}{R}e^a(\\gamma^5\\gamma_a+\\frac{1}{R}(\\vartheta\\vartheta-\\upsilon\\ups)\\gamma_a)\n+\\frac{1}{R}e^3\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n+(2A(y)-\\frac{4}{R^2}e^a\\vartheta\\gamma^7\\gamma_a\\vartheta)\\varepsilon\n\\Big)\\upsilon,\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\\label{D_{24}}\nD_{_{24}}\\vartheta\n&=&{\\mathcal P}_6\\,\\Big(d\n+\\frac{2i}{R}e^a(\\gamma^5\\gamma_a+\\frac{1}{R}(\\vartheta\\vartheta-\\upsilon\\ups)\\gamma_a)\n+\\frac{1}{R}e^3 +\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'}\\Big)\\vartheta\\,.\\nonumber\\\\\n\\end{eqnarray}\n(The shift of $D$ by $-\\frac{2i}{R^2}\\upsilon\\ups\\,e^a\\,\\gamma_a$ has\nbeen made for the expressions to have a nicer and more\ncovariant--looking form).\n\\\\\nThe NS--NS three-form, eq. (\\ref{f4h3}), becomes\n\\begin{equation}\nH_3= -\\frac{6i}{R^2}e^3{\\mathcal E}^b{\\mathcal\nE}^a\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon -i{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\beta}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_A\\Gamma_{11})_{\\underline{\\alpha\\beta}}\n+i{\\mathcal E}^{B}{\\mathcal E}^{A}{\\mathcal\nE}^{\\underline\\alpha}(\\Gamma_{AB}\\Gamma^{11}\\lambda)_{\\underline\\alpha}\n\\,.\n\\end{equation}\nWe now would like to determine its potential according to eq.\n(\\ref{B2}). Using the fact that\n\\begin{equation}\ni_\\Theta{\\mathcal E}^{\\underline\\alpha}=\n\\Big(\\frac{R}{kl_p}\\Big)^{1\/4}(\\Theta^{\\underline\\alpha}\n+\\frac{i}{R}\\upsilon\\gamma^a\\varepsilon\\upsilon\\,(\\Theta\\Gamma_a\\Gamma_{11})^{\\underline\\alpha})\n\\end{equation}\nand $i_\\Theta\\mathcal E^{A}=0$ we get\n\\begin{eqnarray}\\label{iH3-}\ni_\\Theta H_3&=&\n\\frac{R}{kl_p}\\Big(\n-\\frac{2}{R}(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\\upsilon\\gamma_{ab}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{4}{R}(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\nD\\Theta\\gamma_{ba}\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n+\\frac{8}{R^2}e^3e^a\\vartheta\\vartheta\\,\\upsilon\\gamma_a\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{4}{R}(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)e^b\\Theta\\gamma_{ba}\\gamma^7\\Theta\n+2e^3D\\Theta\\gamma^7\\Theta\n\\nonumber\\\\\n&&{}\n-2i(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'}\\gamma^7\\Theta\n+\\frac{4i}{R}e^3(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n-\\frac{2}{R}(e^{b'}-\\frac{2}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{8i}{R^2}(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)e^a\\Theta\\gamma_{a'}\\gamma_{ab}\\Theta\\,\\upsilon\\gamma^b\\varepsilon\\upsilon\n\\Big)\n\\end{eqnarray}\nand finally\n\\begin{eqnarray}\\label{B2-}\nB_2&=&\n\\frac{R}{kl_p}\\Big(\n\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta)^2)(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta)^2)\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{2}{R}(e^a+\\frac{i}{2}\\Theta\\gamma^aD\\Theta+\\frac{1}{3R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)e^b\\,\\varepsilon_{abc}\\,\\Theta\\gamma^c\\gamma^7\\Theta\n+\\frac{2i}{R}e^3(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{}\n-\\frac{1}{R}(e^{b'}-\\frac{1}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n-\\frac{1}{3R^3}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta\\,e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{} -e^3\\,\\Theta\\gamma^7D\\Theta\n+\\frac{i}{R^2}e^3e^a(\\vartheta\\gamma^7\\gamma_a\\vartheta-\\upsilon\\gamma_a\\gamma^7\\upsilon)\\,\\Theta\\Theta\n+i(e^{a'}-\\frac{1}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n\\nonumber\\\\\n&&{}\n+\\frac{2i}{R^2}(e^{a'}-\\frac{4}{3R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)\\,e^a\\,\\varepsilon_{abc}\\,\\Theta\\gamma_{a'}\\gamma^b\\Theta\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n+\\frac{2}{R^2}(e^{a'}-\\frac{2}{3R}e^b\\,\\Theta\\gamma^{a'}\\gamma_b\\Theta)e^a\\,\\vartheta\\gamma^7\\gamma_a\\vartheta\\,\\vartheta\\gamma_{a'}\\upsilon\n\\nonumber\\\\\n&&{}\n+\\frac{1}{R^2}(e^{a'}-\\frac{2}{3R}e^b\\,\\Theta\\gamma^{a'}\\gamma_b\\Theta)e^a\\,\\Theta\\gamma_{a'}\\gamma^7\\gamma_a\\Theta\\,(\\vartheta\\vartheta-\\upsilon\\ups)\n-\\frac{2}{3R^3}e^be^a\\,\\varepsilon_{abc}\\,\\Theta\\gamma^c\\gamma^7\\Theta\\,((\\vartheta\\vartheta)^2-(\\upsilon\\ups)^2)\n\\nonumber\\\\\n&&{}\n+\\frac{4i}{3R^3}e^be^d\\,\\vartheta\\gamma_d\\gamma^7\\vartheta\\,\\vartheta\\gamma^a\\gamma^7\\vartheta\\,\\varepsilon_{abc}\\,\\upsilon\\gamma^c\\varepsilon\\upsilon\n\\Big)\\,.\n\\end{eqnarray}\nNote that the maximum order of the fermions in the above expressions\nis ten.\n\nUsing the form of $F_4$ in (\\ref{f4h3}) as well as the expressions\n(\\ref{A2-}) for ${\\mathcal A}_1$ and (\\ref{iH3-}) for $i_\\Theta H_3$\nthe quantity relevant for computing the RR three-form potential\n$\\mathcal A_3$ becomes\n\\begin{eqnarray}\n\\lefteqn{i_\\Theta F_4+\\mathcal A_1i_\\Theta H_3}\n\\nonumber\\\\\n&=&\n-(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n(e^a+i\\Theta\\gamma^aD\\Theta+\\frac{1}{R^2}e^a(\\vartheta\\vartheta-\\upsilon\\ups)^2)\\varepsilon_{abc}\ni\\Theta\\gamma^cD\\Theta\n\\nonumber\\\\\n&&{}\n-2e^3(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'}\\Theta\n-\\frac{4}{R}e^3e^a\\Theta\\gamma_aD\\Theta\\,\\Theta\\Theta(1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\n\\\\\n&&{}\n\\hspace{-10pt}-\\frac{4}{R}(e^{a'}-\\frac{2}{R}e^c\\,\\Theta\\gamma^{a'}\\gamma_c\\Theta)e^a\n(e^b+i\\Theta\\gamma^bD\\Theta+\\frac{1}{R^2}e^b(\\vartheta\\vartheta-\\upsilon\\ups)^2)\n\\Theta\\gamma_{a'}\\gamma_{ab}\\Theta(1+\\frac{2}{R^2}(\\upsilon\\ups)^2)\n\\nonumber\\\\\n&&{}\n\\hspace{-10pt}-i(e^{b'}-\\frac{2}{R}e^b\\,\\Theta\\gamma^{b'}\\gamma_b\\Theta)(e^{a'}-\\frac{2}{R}e^a\\,\\Theta\\gamma^{a'}\\gamma_a\\Theta)D\\Theta\\gamma_{a'b'}\\Theta\n+kl_p(A(y)-\\frac{2}{R^2}e^a\\,\\Theta\\gamma^7\\gamma_a\\Theta)i_\\Theta\nH_3\\,.\\nonumber\n\\end{eqnarray}\nOne can now substitute this together with the expression for\n$i_\\Theta H_3$ (\\ref{iH3-}) into eq. (\\ref{A3}) and compute the\nexplicit form of the RR three--form potential ${\\mathcal A}_3$ in\nthis gauge. Since we have not got a reasonably simple expression for\n${\\mathcal A}_3$ we shall not present it here.\n\n\n\\setcounter{equation}0\n\\section{Applications}\nWe can now use the kappa--gauge fixed form of the $AdS_4 \\times\nCP^3$ superbackground of Subsections \\ref{theta-} and \\ref{theta++}\nto simplify the actions for the type IIA superstring and D--branes.\nLet us note that the gauge fixing conditions (\\ref{kappagauge1}) can\nalso be used to simplify the actions for the $D=11$ superparticle,\nM2-- and M5--branes in the $AdS_4\\times S^7\/Z_k$ superbackground\n(\\ref{notaC}). We shall consider the example of the $D=11$ superparticle\nbelow.\n\n\n\\subsection{$D=11$ superparticle}\n\\def\\mathcal M{\\mathcal M}\n\nLet us consider a massless superparticle in the $AdS_4\\times\nS^7\/Z_k$ supergravity background. Recall that when $k=1,2$, the\nsupergravity background preserves the maximum number of 32\nsupersymmetries, while for $k>2$ it preserves only 24.\nThe superparticle action in the complete superspace\nwith 32 $\\Theta$ is constructed using the supervielbeins of the\n$OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ supercoset derived in\n\\cite{Gomis:2008jt}\n\\begin{equation}\\label{upsilonfunctions}\n\\begin{aligned}\n\\underline E^{\\hat a}&=E^{\\hat a}(x,y,\\vartheta) + 4 i\n\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal M}^2}\\,D\\upsilon\n+\\frac{R}{k}\\,dz\\,E_7{}^{\\hat a}(\\upsilon)\\\\\n\\underline E^{a'}&=E^{a'}(x,y,\\vartheta)+2i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'}\\gamma^5\\,E(x,y,\\vartheta)\\\\\n\\underline E^7&=\\frac{R}{k}\\,dz\\,\\Phi(\\upsilon)+\nR\\,\\left(A(x,y,\\vartheta)-\\frac{4}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,D\\upsilon\\,\\right) \\\\\n\\underline E^{\\alpha i}&=\\left({{\\sinh{\\mathcal M}}\\over{\\mathcal\nM}}\\,(D\\upsilon-\\frac{2}{k}\\,dz\\,\\varepsilon\\upsilon)\\right)^{\\alpha i}\n\\\\\n\\underline E^{\\alpha a'}&=E^{\\alpha a'}(x,y,\\vartheta)-\\frac{8}{R}E^{\\beta\na'}\\left(\\gamma^5\\,\\upsilon\\,{{\\sinh^2{{m}\/2}}\\over{m}^2}\\right)_{\\beta\ni}\\upsilon^{\\alpha i}\\,,\n\\end{aligned}\n\\end{equation}\nwhere $z$ is the 7th, $U(1)$ fiber, coordinate of\n$S^7$, $D\\upsilon$ has been given in (\\ref{D}) and the eight fermionic\ncoordinates $\\upsilon^{\\alpha i}$ correspond to the eight\nsupersymmetries broken by orbifolding with $k>2$.\n\nThe explicit form of the fermionic supervielbeins in\n(\\ref{upsilonfunctions}) and of the connections on\\\\\n$OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ are not required for the\nconstruction of the Brink--Schwarz superparticle action but one\nneeds them for the construction of the pure--spinor superparticle\naction in curved superbackgrounds, so we present also the form of\nthe spin--connection below.\n\nThe $SO(1,3)$ connection is\n\\begin{equation}\\label{ads4connection}\n\\underline \\Omega^{\\hat a\\hat b}= \\Omega^{\\hat a\\hat b}(x,y,\\vartheta)\n+\\frac{8}{R} \\upsilon\\gamma^{\\hat a\\hat\nb}\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal\nM}^2}\\,\\left(D\\upsilon-\\frac{2}{k}dz\\,\\varepsilon\\upsilon\\right)\n\\end{equation}\nand the $SO(7)$ connection is\n\\begin{eqnarray}\n\\underline \\Omega^{a'b'}\n&=&\n\\Omega^{a'b'}(x,y,\\vartheta)-\\frac{1}{R}\\,\\underline E^7\\,J^{a'b'}\n-\\frac{2}{R}\\,\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'b'}\\gamma^5E\\,,\n\\nonumber\\\\\n\\label{so7connection}\n\\\\\n{\\underline\\Omega}^{a'7}\n&=&\n\\frac{1}{R}\\left(\\underline E^{b'}-4i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{b'}\\gamma^5E\\right)\\,J_{b'}{}^{a'}\\,.\\nonumber\n\\end{eqnarray}\nThe functions and forms appearing in\n(\\ref{upsilonfunctions})--(\\ref{so7connection}) are defined in\nAppendix B.\n\nThe first order form of the action for the massless superparticle in\nthe $OSp(8|4)\/SO(7)\\times SO(1,3)\\times Z_k$ superbackground is\n\\begin{eqnarray}\\label{11dsuper}\nS = \\int d\\tau \\left( P_{\\underline A} \\,{\\underline\nE}^{\\underline A}_\\tau +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\\,,\n\\end{eqnarray}\nwhere $ P_{\\underline A}$ ($\\underline A=0,1,\\dots,10$) is the\nparticle momentum, $e(\\tau)$ is the Lagrange multiplier which\nensures the mass shell condition $P^2=0$ and\n$$\n{\\underline E}^{\\underline A}_\\tau=\\partial_\\tau\nZ^{\\underline{\\mathcal M}}\\,{\\underline E}_{\\underline{\\mathcal\nM}}{}^{\\underline A}\\,, \\qquad Z^{\\underline{\\mathcal\nM}}=(x,y,z,\\vartheta,\\upsilon)\n$$\nis the pullback to the worldline of the supervielbeins\n(\\ref{upsilonfunctions}). The action is\ninvariant under local worldline diffeomorphisms and under the\nfermionic kappa--symmetry transformations\n\\begin{equation}\\label{kappaA}\n\\delta Z^{\\underline{\\mathcal M}} \\,{\\underline E}_{\\underline{\\mathcal M}}{}^{\\underline \\alpha}\n= P^{\\underline A} \\,(\\Gamma_{\\underline A}\\,\\kappa)^{\\underline\\alpha}\\,,\\qquad\n\\delta Z^{\\underline{\\mathcal M}} \\,{\\underline E}_{\\underline\n{\\mathcal M}}{}^{\\underline A} = 0\\,, \\qquad\n\\end{equation}\n\\begin{equation}\\label{kappaA1}\n\\delta e=-4i\\,{\\underline E}^{\\underline\n\\alpha}_\\tau\\,\\kappa_{\\underline\\alpha}\\,,\n\\qquad \\delta P_{\\underline A}=\n\\delta Z^{\\underline{\\mathcal M}}\\,\\underline\\Omega_{{\\underline{\\mathcal M}}\\underline A}{}^{\\underline B}\\,P_{\\underline B}.\n\\end{equation}\nInserting in the action the expressions for the vielbeins\n(\\ref{upsilonfunctions}), we get\n\\begin{eqnarray}\\label{11dsuper2}\nS = \\int d\\tau \\!\\!\\!\\!\\!\\!&& \\left[\nP_{\\hat a}\\left(\nE^{\\hat a}_\\tau + 4 i\n\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/ 2}}\\over{\\mathcal M}^2}\\,D_\\tau\\upsilon\n- 8 i \\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2} \\varepsilon {\\upsilon} \\,\n\\frac{\\partial_\\tau z}{k}\n\\right) \\right. \\nonumber \\\\\n&&\\left.+ P_{a'}\n\\left(E^{a'}_\\tau+2i\\upsilon\\,{{\\sinh m}\\over m}\\gamma^{a'}\\gamma^5\\,E_\\tau \\right)\\right. \\\\\n&&\\left. + P_7 \\left( R \\left( \\frac{\\partial_\\tau z}{k} + A\\right)\n-4\n\\upsilon \\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal\nM}^2}\\, (D_\\tau \\upsilon\\, -2 \\varepsilon\\upsilon \\frac{\\partial_\\tau z}{k})\n\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right]\n\\,.\n\\nonumber\n\\end{eqnarray}\n\nThe action (\\ref{11dsuper2}) can be simplified by eliminating some\nor all pure--gauge fermionic modes using the kappa--symmetry\ntransformations (\\ref{kappaA}). For instance, when the momentum of\nthe particle is non--zero along a $CP^3$ direction inside $S^7$, the\nprojectors ${\\mathcal P}_6$ and ${\\mathcal P}_2$, defined in eqs.\n(\\ref{p6}) and (\\ref{p2}), do not commute with the kappa--symmetry\nprojector (\\ref{kappaA}) and one can use \\emph{e.g.} the 16\nkappa--symmetry transformations to eliminate 16 of the 24\n$\\vartheta$. After such a gauge fixing the action will contain 8\nremaining $\\vartheta$ and 8 $\\upsilon$.\n\nAlternatively, by partially gauge fixing the kappa--symmetry one can\neliminate all eight $\\upsilon$ keeping 24 $\\vartheta$. In the latter\ncase the action reduces to the form in which it describes the\ndynamics of a superparticle in a superspace with 11 bosonic\ncoordinates and 24 fermionic ones. This superspace has been\nintroduced in\n\\cite{Gomis:2008jt} as a Hopf fibration of the supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$.\nIt is the supercoset\n\\begin{equation}\\label{24th}\n\\frac{OSp(6|4) \\times U(1)}{U(3) \\times SO(1,3)\\times\nZ_k}.\n\\end{equation}\nThe geometry of (\\ref{24th}) is described by the supervielbeins\n\\begin{eqnarray}\\label{24thA}\n&&\\hat E^{\\hat a} = E^{\\hat a}(x,y,\\vartheta)\\,, \\nonumber \\\\\n&&\\hat E^{a'} = E^{a'}(x,y,\\vartheta) \\,,\\\\\n&&\\hat E^{7} = R\\Big(\\frac{dz}{k} + A(x,y,\\vartheta) \\Big)\\,,\n\\nonumber \\\\\n&&\n\\hat E^{\\alpha a'} = E^{\\alpha a'}(x,y,\\vartheta)\\,,\\nonumber\n\\end{eqnarray}\nwhere (as already mentioned) the explicit form of the\nright--hand sides of (\\ref{24thA}) are given in (\\ref{cartan24}). Notice that now $z$ appears only in the vielbein $\\hat\nE^7$ along the $U(1)$-fiber direction of $S^7$.\n\nThe first order form of the superparticle action in the superspace\n(\\ref{24thA}) is\n\\begin{eqnarray}\\label{11dsuper24_A}\nS& = & \\int d\\tau \\left( P_{\\hat a} \\,{\\hat E}^{\\hat a}_\\tau +\n P_{a'} \\,{\\hat E}^{a'}_\\tau +\n P_{7} \\,{\\hat E}^{7}_\\tau +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\n\\,\\nonumber\\\\\n& =& \\int d\\tau \\left( P_{\\hat a} \\,{E}^{\\hat a}_\\tau +\n P_{a'} \\,{E}^{a'}_\\tau +\n P_{7} \\, R \\left( \\frac{\\partial_\\tau z}{k} + A_\\tau\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right)\\,,\n\\end{eqnarray}\nwhere now\n$$\n{\\hat E}^{\\underline A}_\\tau=\\partial_\\tau Z^{{\\mathcal M}}\\,{\\hat\nE}_{{\\mathcal M}}{}^{\\underline A}+\\partial_\\tau\\,z\\,{\\hat\nE}_z{}^{\\underline A}\\,,\n\\qquad Z^{{\\mathcal M}}=(x,y,\\vartheta)\n$$\nis the pullback to the worldline of the supervielbeins\n(\\ref{24thA}).\n\nIt is easy to reduce the action (\\ref{11dsuper24_A}) to $D=10$. Once it\nis done, one obtains the action for a D0--brane moving in the\nsupercoset $OSp(6|4)\/U(3)\\times SO(1,3)$.\n\nAs we have mentioned above, the action (\\ref{11dsuper24_A})\ndescribes a superparticle which has a non--zero momentum along the\n$CP^3$ base of the $S^7$ bundle. This is required by the consistency\nof the kappa--symmetry gauge fixing condition $\\upsilon=0$. To\ndescribe other possible classical motions of the superparticle,\n\\emph{ e.g.} when $P^{a'}=0$, one should chose a different\nkappa--symmetry gauge.\n\nFor instance, if the superparticle has a non--zero spacial momentum\nalong the 7th, fiber direction, of $S^7$, one can use the gauge\nfixing condition corresponding to that of Subsection\n\\ref{theta-}. In this case, in virtue of the gauge-fixed expressions of Appendix C,\n the action (\\ref{11dsuper2}) simplifies to\n\\begin{eqnarray}\\label{11dsuper-gauged}\nS = \\int d\\tau \\hspace{-.5cm}&&\\left[\nP_{a} \\,\\Big( e^a_\\tau(x)+i\\vartheta\\gamma^aD_{\\tau}\\vartheta +\ni \\upsilon \\gamma^aD_{\\tau} \\upsilon - 2 i\n\\upsilon \\gamma^a \\varepsilon \\upsilon \\frac{\\partial_\\tau z}{k} \\Big) \\right. \\nonumber \\\\\n&& \\left. +P_{a'} \\, {e}^{a'}_\\tau(y) + P_{3} \\,{e}^{3}_\\tau(x) +\nP_{7}\n\\, R\n\\left(\\frac{\\partial_\\tau z}{k} + A_\\tau(y)\\right) +\n\\frac{{e}}{2} \\, P_{\\underline A} P_{\\underline B}\\, \\eta^{\\underline{AB}} \\right]\\,.\n\\end{eqnarray}\nThe dimensional reduction of the $D=11$ superparticle action\n(\\ref{11dsuper-gauged}) along $z$ results in the kappa--symmetry\ngauge--fixed action which describes an arbitrary motion of the type\nIIA D0--brane in $AdS_4\\times CP^{3}$ superspace.\n\nBefore considering the D0--brane, let us note that the action\n(\\ref{11dsuper2}) is the most appropriate starting point for the\nconstruction of the pure--spinor formulation of the $D=11$\nsuperparticle in the $AdS_4\\times CP^3$ supergravity background. The\npure--spinor condition $\\lambda\\Gamma^{\\underline A}\\lambda=0$ in\n$D=11$ implies that the 32--component bosonic pure spinor\n$\\lambda^{\\underline \\alpha}$ has 23 independent components\n\\cite{Berkovits:2002uc,Fre':2006es}. This counting ensures the correct\nnumber of bosonic and fermionic degrees of freedom.\n\nIn the cases of the actions (\\ref{11dsuper24_A}) and\n(\\ref{11dsuper-gauged}) that describe a particle motion in the\nreduced superspaces, one can also develop pure--spinor formulations\nin which the pure spinor $\\lambda$, in addition, is subject to the\nsame constraint as the one imposed on $\\Theta$ by kappa--symmetry\ngauge fixing, e.g. ${\\mathcal P_2}\\lambda=0$ in the case\n$\\upsilon={\\mathcal P_2}\\Theta=0$. This guarantees the correct\ncounting of the degrees of freedom in the pure--spinor formulation\n(similar to the cases considered in\n\\cite{Fre:2008qc,Bonelli:2008us}). That is, the difference between\nthe bosonic and fermionic degrees of freedom remains the same.\nIndeed, in the case of the pure--spinor formulation of the massless\n$D=11$ superparticle\n\\cite{Berkovits:2002uc} there are 11 bosonic $X^{\\underline A}$ plus 23 pure spinor\ndegrees of freedom and 32 fermionic $\\Theta$, while in the above\nexample of the reduced pure spinor formulation the pure spinor\neffectively contains $23-8=15$ degrees of freedom against 24\nfermionic ones, while the number of $X$ remains the same.\n\n When the pure spinor formulations of the superparticle\nin reduced superspaces correspond to the kappa--gauge fixed versions\nof the Brink--Schwarz superparticle whose consistency is limited to\nparticular subsectors of the classical configuration space of the\nfull theory, one may expect that the former will also describe only\nsubsectors of the pure--spinor superparticle model formulated in the\ncomplete superspace with 32 fermionic coordinates. As in the case of\nthe pure--spinor type IIA superstring in $AdS_4\\times CP^3$\n\\cite{Fre:2008qc,Bonelli:2008us}, these issues require additional\nanalysis.\n\n\\subsection{$D0$-brane}\nTo obtain the action for the $D0$--brane by dimensional\nreduction of the $D=11$ superparticle action, one should first perform the\nappropriate Lorentz transformation of the $D=11$\nsupervielbeins (as was explained in \\cite{Gomis:2008jt}) and make a\ncorresponding redefinition of the particle momentum. We shall not\nperform this dimensional reduction procedure since the result is\nwell known. The $D0$--brane action has the following first order\nform in the type IIA superbackground in the string frame (see eqs.\n(\\ref{simplA}) and (\\ref{simplB}))\n\\begin{equation}\\label{d0-action}\nS = \\int d\\tau e^{- \\phi} \\,\\left(P_A {\\cal E}^A_\\tau +\n\\frac{e}{2} (P_A P_B \\eta^{AB}+ m^2) \\right) + m\\, \\int {\\cal\nA}_1\\,,\n\\end{equation}\nwhere $m$ is the mass of the particle and the second term describes\nits coupling to the RR one--form potential ${\\cal A}_1$.\n\nIntegrating out the momenta $P_A$ and the auxiliary field $e(\\tau)$\nwe arrive at the action\n\\begin{equation}\\label{d0-action2}\nS = -m\\,\\int d\\tau e^{- \\phi} \\,\\sqrt{-{\\cal E}^A_\\tau {\\cal\nE}^B_\\tau\\,\\eta_{AB}} + m\\int {\\cal A}_1\\,.\n\\end{equation}\nThe action (\\ref{d0-action2}) is invariant under worldline\ndiffeomorphisms and the kappa-symmetry transformations (to\nverify the kappa-symmetry one needs the superspace constraints\non the torsion $T^A$ and on $F_2$ given in Appendix A.4)\n\\begin{eqnarray}\n&&\\delta_\\kappa Z^{\\mathcal M} {\\cal E}_{\\cal\nM}^{~~\\underline\\alpha} =\n\\frac{1}{2}(1 + \\Gamma)^{\\underline\\alpha}{}_{\\underline\\beta} \\,\\kappa^{\\underline\\beta}(\\tau)\\,, ~~~\n{\\underline\\alpha} = 1,\\dots,32\\,,\\qquad Z^{\\mathcal M}=(x,y,\\vartheta,\\upsilon)\\,\\nonumber\\\\\n\\\\\n&&\\delta_\\kappa Z^{\\mathcal M} {\\cal E}_{\\cal M}^{~~A} = 0\\,, ~~~\nA=0,1,\\dots,9\\,\n\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{GammaD0}\n\\Gamma = \\frac{1}{\\sqrt{-\\mathcal E^2_\\tau}}\\,{\\mathcal E}_\\tau^{~A} \\Gamma_A \\Gamma_{11}\\,,\n\\,\\qquad \\Gamma^2=1\\,.\n\\end{equation}\nComparing the form of the kappa--symmetry projector matrix\n(\\ref{GammaD0}) with the kappa--symmetry gauge fixing condition of\nSubsection \\ref{theta-}, we see that\n$\\gamma=\\Gamma^0\\Gamma^1\\Gamma^2$ (introduced in eq. (\\ref{gamma}))\ndoes not commute with $\\Gamma$ in (\\ref{GammaD0}) provided that the\nenergy $P^0\\sim{{\\mathcal E}^0_\\tau}$ of the massive particle is\nnonzero, which is always the case. Thus to simplify, \\emph{e.g.} the first\norder action (\\ref{d0-action}) we can use the gauge fixed form of\nthe supervielbeins and the RR one--form of Subsection\n\\ref{theta-}. The action takes the following explicit form, with an\nappropriately rescaled Lagrange multiplier $e(\\tau)$,\n\\begin{eqnarray}\\label{simply}\nS &=& \\left(\\frac{R}{kl_p}\\right)^{-1} \\,\\int d\\tau \\,\n\\left[\\Big( P_{a'}e_\\tau^{a'}(y)\n+ P_{3} \\,e_\\tau^3(x)\\Big)\\,(1+\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\right. \\nonumber \\\\\n&+& \\left. P_{a} \\,\n(e_\\tau^a(x)+i\\Theta\\gamma^aD_{\\tau}\\Theta)\\,(1+\\frac{8}{R^2}(\\upsilon\\ups)^2)\n+\\frac{e(\\tau)}{2} \\,(P_A P_B \\eta^{AB}+ m^2) \\nonumber\\right] \\\\\n&+& m kl_p\\,\\int \\, \\Big(\nA(y)-\\frac{2i}{R^2}(e^a(x)+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\varepsilon\\gamma_a\\upsilon\\Big)\\,.\n\\end{eqnarray}\nThis action contains fermionic terms up to the 6th order in\n$\\Theta=(\\vartheta,\\upsilon)$.\n\n\\subsection{The fundamental string}\\label{FS}\n\nIn this section we use the geometry discussed above to construct\nthe Green-Schwarz model for the fundamental string. We will first\nreview the form of the superstring sigma model without gauge fixing\nand then impose the gauge fixing of the kappa--symmetry. This will\nprovide a calculable sigma model.\n\nThe action for the Green--Schwarz superstring has the following form\n\\begin{equation}\\label{cordaA}\nS = -\\frac{1}{4\\pi\\alpha'}\\,\\int d^2\\xi\\, \\sqrt {-h}\\, h^{IJ}\\,\n{\\cal E}_{I}^{A} {\\cal E}_{J}^{B} \\eta_{AB}\n-\\frac{1}{2\\pi\\alpha'}\\,\\int B_2\\,,\n\\end{equation}\nwhere $\\xi^I$ $(I,J=0,1)$ are the worldsheet coordinates,\n$h_{IJ}(\\xi)$ is a worldsheet metric and $B_2$ is the pull--back to\nthe worldsheet of the NS--NS 2--form.\n\nThe kappa--symmetry transformations which leave the superstring\naction (\\ref{cordaA}) invariant are\n\n\\begin{equation}\\label{kappastring}\n\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^{\\underline \\alpha}=\n{1\\over 2}(1+\\Gamma)^{\\underline \\alpha}_{~\\underline\\beta}\\,\n\\kappa^{\\underline\\beta}(\\xi),\\qquad {\\underline \\alpha}=1,\\cdots, 32\n\\end{equation}\n\\begin{equation}\\label{kA}\n\\hskip-2.5cm\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^A=0,\n\\qquad A=0,1,\\cdots,9\n\\end{equation}\nwhere $\\kappa^{\\underline\\alpha}(\\xi)$ is a 32--component spinor\nparameter, ${1\\over 2}(1+\\Gamma)^{\\underline\n\\alpha}_{~\\underline\\beta}$ is a spinor projection matrix with\n\\begin{equation}\\label{gbs}\n\\Gamma={1\\over {2\\,\\sqrt{-\\det{g_{IJ}}}}}\\,\\epsilon^{IJ}\\,{\\mathcal\nE}_{I}{}^A\\,{\\mathcal E}_{J}{}^B\\,\\Gamma_{AB}\\,\\Gamma_{11}, \\qquad\n\\Gamma^2=1\\,,\n\\end{equation}\nand the auxiliary worldsheet metric $h^{IJ}$ transforms as follows\n\\begin{eqnarray}\\label{deltah}\n\\lefteqn{\\delta_\\kappa\\,(\\sqrt{-h}\\,h^{IJ})}\n\\nonumber\\\\\n&=&2i\\,\\sqrt{-h}\\,(h^{IJ}\\,g^{KL}-2h^{K(I}\\,g^{J)L})\n\\left(\\delta_\\kappa\nZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}\\,\\Gamma_A\\,{\\mathcal\nE}_K\\,{\\mathcal E}^A_L +\\frac{1}{2}g_{KL}\\delta_\\kappa Z^{\\mathcal\nM}\\,{\\mathcal E}_{\\mathcal M}{}^{\\alpha i}\\lambda_{\\alpha i}\n\\right)\n\\\\\n&&\\hspace{-20pt}\n-2i\\,\\sqrt{-h}\\,\\,\\frac{h^{IK'}g_{K'L'}h^{L'J}-\\frac{1}{2}h^{IJ}\\,h^{K'L'}g_{K'L'}}\n{\\frac{1}{2}\\,h^{K'L'}\\,g_{K'L'}+\\sqrt{\\frac{g}{h}}}\\,g^{KL}\\,\\left(\\delta_\\kappa\nZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}\\,\\Gamma_A\\,{\\mathcal\nE}_K\\,{\\mathcal E}^A_L +\\frac{1}{2}g_{KL}\\delta_\\kappa Z^{\\mathcal\nM}\\,{\\mathcal E}_{\\mathcal M}{}^{\\alpha i}\\lambda_{\\alpha i}\n\\right)\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{im}\ng_{IJ}(\\xi)={\\mathcal E}_{I}{}^{A}\\, {\\mathcal\nE}_{J}{}^{B}\\,\\eta_{AB}\\,,\\qquad g^{IJ}\\equiv (g_{IJ})^{-1}\n\\end{equation}\nis the induced metric on the worldsheet of the string that on the\nmass shell coincides with the auxiliary metric $h_{IJ}(\\xi)$ modulo\na conformal factor. Finally, $g=\\det g_{IJ}$ and $h=\\det h_{IJ}$.\n\\\\\nUsing the identity\n\\begin{equation}\nh^{IJ}\\,g_{JK}\\,h^{KL}\\,g_{LI}-\\frac{1}{2}\\,(h^{IJ}\\,g_{IJ})^{2}\\equiv\n\\frac{1}{2}\\,(h^{IJ}\\,g_{IJ})^{2}-2\\,\\frac{g}{h}\\,\n\\end{equation}\none can check that eq. (\\ref{deltah}) multiplied by $g_{IJ}$ results\nin\n\\begin{equation}\\label{kappag}\n\\delta_\\kappa\\,(\\sqrt{-h}\\,h^{IJ})\\,g_{IJ}\n=4i(\\sqrt{-g}\\,g^{KL}-\\sqrt{-h}\\,h^{KL})\\,\n\\delta_\\kappa Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal M}{}^{\\underline\\alpha}(\\Gamma_A\\,{\\mathcal E}_K\\,{\\mathcal E}^A_L\n+\\frac{1}{2}g_{KL}\\lambda )_{\\underline\\alpha}\\,,\n\\end{equation}\nwhich together with the variation (\\ref{kappastring}) and (\\ref{kA})\nof the superspace coordinates insures the invariance of the action\n(\\ref{cordaA}).\n\n Comparing the form of the kappa--symmetry projector\n(\\ref{kappastring}) with the kappa--symmetry gauge fixing condition\nof Subsection\n\\ref{theta-} we see that this gauge choice is admissible when the\nstring moves in such a way that the projection of its worldsheet on\nthe $3d$ subspace along the directions $e^a$ $(a=0,1,2)$ of the\ntarget space is a non--degenerate two--dimensional time--like\nsurface. Thus, it can be used to analyze the string dynamics in the\nsector which is not reachable by the supercoset model of\n\\cite{Arutyunov:2008if,Stefanski:2008ik,Fre:2008qc}. The latter is\nobtained from the action (\\ref{cordaA}) by gauge fixing to zero the\neight fermions $\\upsilon$, which is only possible when the string\nworldsheet extends in the $CP^3$ directions.\n\nIn the gauge of Subsection \\ref{theta-} we insert into the action\n(\\ref{cordaA}) the expressions (\\ref{simple+v}) and (\\ref{simple+B})\nfor the supervielbeins and $B_2$. This results in an action that\ncontains fermionic terms only up to the 8th order in\n$\\Theta=(\\vartheta,\\upsilon)$\n\\footnote{The factor in front of the\naction is unconventional due to our normalization of the vielbeins,\nwhich comes from the dimensional reduction of the eleven-dimensional\ngeometry. More conventional, unit radius string frame vielbeins\n$(\\hat e^{\\hat a},\\hat e^{a'})$ can be introduced by the following\nrescaling\n$$\n(e^{\\hat a},e^{a'})=\\left(\\frac{R}{k\nl_p}\\right)^{-1\/2}\\left(\\frac{R}{2}\\,\\hat e^{\\hat a},\\,R\\,\\hat\ne^{a'}\\right)\\,.\n$$\nThen the factor in front of the action becomes\n$\\frac{R^2}{4\\pi\\alpha'}=\\frac{(R\/l_p)^3}{4\\pi k}$.}\n\\begin{eqnarray}\\label{cordaB}\nS &=&-\\frac{1}{4\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\sqrt{-h}\\,h^{IJ}\\left[\n\\left(e^{a'}_I e^{b'}_{J} \\delta_{a'b'} +\ne^3_I e^3_{J}\\right)\\,(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\right.\n\\nonumber \\\\\n&&{}\n\\hspace{4cm}\n+\\left.(e^a_I+i\\Theta\\gamma^aD_I\\Theta)\\,\n(e^b_{J}+i\\Theta\\gamma^bD_{J}\\Theta)\\,\\eta_{ab}\\,\n(1-\\frac{2}{R^2}(\\upsilon\\ups)^2)\\right]\n\\nonumber\\\\\n\\\\\n &-&\\frac{1}{2\\pi\\alpha'}\\frac{R}{kl_p}\\int\n\\Big[e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{60pt}}-\\frac{i}{R}(e^b+i\\Theta\\gamma^bD\\Theta)\\,(e^a+i\\Theta\\gamma^aD\\Theta)\\,\n\\upsilon\\gamma_{ab}\\,\\varepsilon\\upsilon\\,\n\\Big]\\,.\\nonumber\n\\end{eqnarray}\nTo avoid possible confusion, let us remind the reader that in eqs.\n(\\ref{cordaB})--(\\ref{cordaBT}) the covariant derivative\n$D\\Theta\\equiv (D_8\\upsilon,\\,D_{24}\\vartheta)$ is defined in eqs.\n(\\ref{D81}). Actually, in (\\ref{cordaB}) the vielbein\n$e^3$ does not contribute to the covariant derivative and the\nconnection $\\omega^{ab}$ is zero along the $3d$ Minkowski boundary\nof $AdS_4$, for the vielbeins chosen as in eqs. (\\ref{ad4v}). It is\nnot hard to check that the action (\\ref{cordaB}) is invariant\nunder twelve `linearly realized' supersymmetry transformations\n$$\n\\delta\\vartheta=\\epsilon, \\qquad \\delta\ne^a=-i\\epsilon\\,\\gamma^a\\,D_{24}\\vartheta\n$$\nwith parameters $\\epsilon=\\frac{1}{2}\\,(1+\\gamma)\\,\\epsilon$ being\n$CP^3$ Killing spinors\n$$\nD_{24}\\epsilon={\\mathcal P}_6\\,(d-\\frac{1}{R}e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'} -\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\epsilon=0\\,.\n$$\nThe other twelve supersymmetries of the $OSp(6|4)$ isometries of\n(\\ref{cordaB}) are non--linearly realized on the worldsheet fields\nand include compensating kappa--symmetry transformations required to\nmaintain the gauge $\\vartheta=\\frac{1}{2}\\,(1+\\gamma)\\,\\vartheta$.\n\nThe action (\\ref{cordaB}) is slightly more complicated than the\naction for the $AdS_5\\times S^5$ superstring in the analogous\nkappa--symmetry gauge\n\\cite{Pesando:1998fv,Kallosh:1998nx,Kallosh:1998ji},\nthat contains fermions only up to the fourth order, since\n$AdS_4\\times CP^3$ is less supersymmetric than $AdS_5\\times\nS^5$. The action (\\ref{cordaB}) takes a form similar to that of\n\\cite{Pesando:1998fv,Kallosh:1998nx,Kallosh:1998ji} when we formally\nput the broken supersymmetry fermions $\\upsilon^{\\alpha i}$ to zero.\n\nAs in the case of the string in $AdS_5\\times S^5$ it is possible to simplify the action further\nby performing a T--duality transformation on the worldsheet \\cite{Kallosh:1998ji}. Following\n\\cite{Kallosh:1998ji} we first rewrite the part of the action\n(\\ref{cordaB}) containing the vielbeins $e^a$ in the first order\nform\n\\begin{equation}\\label{first}\nS_1=\\frac{1}{2\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\left[P_a^I\\,(e^a_I+i\\Theta\\gamma^aD_I\\Theta)\n+\\frac{1-\\frac{6}{R^2}\\,(\\upsilon\\upsilon)^2}{2\\sqrt{-h}}\\,P_a^I\\,P_b^J\\,h_{IJ}\\,\\eta^{ab}\n-\\frac{i}{R}\\,\\frac{\\varepsilon_{IJ}}{-h}\\,P^I_a\\,P^J_b\\,\\upsilon\\gamma^{ab}\\varepsilon\\upsilon\\right]\\,.\n\\end{equation}\nThe equations of motion for the momenta $P_a^I$ imply that\n\\begin{equation}\\label{P}\nP^I_a=-\\sqrt{-h}\\,(1-\\frac{2}{R^2}(\\upsilon\\upsilon)^2)\\,\n\\Big(h^{IJ}\\eta_{ab}\n+\\frac{2i}{R\\sqrt{-h}}\\,\\varepsilon^{IJ}\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\\Big)\n\\,(e^b_J+i\\Theta\\gamma^bD_J\\Theta)\\,.\n\\end{equation}\nUsing the explicit form of the $AdS_4$ vielbeins given in eq.\n(\\ref{ad4v}) and varying the first order action (\\ref{first}) with\nrespect to $x^a$ we find that $P^I_a$ is proportional to the\nconserved current associated with translations along $x^a$\n\\begin{equation}\\label{trcurrent}\n\\partial_I\\,\\Big({r^2\\over R^2}\\,P^I_a\\Big)=0\\, \\qquad\\Rightarrow\n\\qquad P^I_a=\\frac{R^2}{r^2}\\,\\varepsilon^{IJ}\\partial_J\\,{\\tilde\nx}_a\\equiv \\varepsilon^{IJ}\\,{\\tilde e}_{Ja}\\,.\n\\end{equation}\nIf we now substitute eq. (\\ref{trcurrent}) into (\\ref{first}) the\nT--dualized version of the action (\\ref{cordaB}) for the string in\n$AdS_4\\times CP^3$ takes the form\n\\begin{eqnarray}\\label{cordaBT}\nS &=&-\\frac{1}{4\\pi\\alpha'}\\,\\frac{R}{k l_p}\n\\int\\,d^2\\xi\\,\\sqrt{-h}\\,h^{IJ}\n\\left(\n{\\tilde e}_I^a\\,{\\tilde e}_J^b\\,\\,\\eta_{ab}+e^3_I e^3_{J}\n+e^{a'}_I e^{b'}_{J} \\delta_{a'b'}\n\\right)\\,(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)\n\\nonumber\\\\\n\\\\\n&-&\\frac{1}{2\\pi\\alpha'}\\frac{R}{kl_p}\\int\n\\Big[e^3\\,\\Theta\\gamma^7D\\Theta\n+ie^{a'}\\,\\Theta\\gamma_{a'}\\gamma^7D\\Theta\n-\\frac{2i}{R}e^3e^{a'}\\,\\vartheta\\gamma_{a'}\\gamma^7\\upsilon\n-\\frac{1}{R}e^{b'}e^{a'}\\,\\Theta\\gamma_{a'b'}\\gamma^7\\upsilon\n\\nonumber\\\\\n&&{\\hspace{60pt}} +i{\\tilde e}^a\\,\\Theta\\gamma_aD\\Theta\n-\\frac{i}{R}\\,\\,{\\tilde e}^a\\,{\\tilde e}^b\\,\\upsilon\\gamma_{ab}\\varepsilon\\upsilon\n\\Big]\\,.\\nonumber\n\\end{eqnarray}\n\nNote that in the T--dualized action the fermionic kinetic terms\nappear only in the Wess--Zumino term and that there are now terms of\nat most fourth order in fermions. Note also that the first (induced\nmetric) term of (\\ref{cordaBT}) acquires a common factor\n$(1-\\frac{6}{R^2}(\\upsilon\\ups)^2)$ in contrast to the corresponding\nterms in the original action (\\ref{cordaB}).\n\nTo preserve the conformal invariance of the dual action at the\nquantum level one should add to it a dilaton term\n$\\int\\,R^{(2)}\\,\\tilde\n\\phi$ (where $R^{(2)}$ is the worldsheet curvature), which is\ninduced by the functional integration of $P_a^I$ when passing to the\ndual action (see \\cite{Buscher:1987qj,Schwarz:1992te,Kallosh:1998ji}\nfor details). Here we should point out that in our case the original\n$AdS_4\\times CP^3$ superbackground already has a non--trivial\ndilaton which depends on $\\upsilon$ (see eqs. (\\ref{dilaton1}) and\n(\\ref{gfdialton})).\n\nThe following comment is now in order. As in the $AdS_5\\times S^5$\ncase \\cite{Kallosh:1998ji,Ricci:2007eq,Beisert:2008iq}, upon the\nT--duality along the three translational directions $x^a$ of $AdS_4$\nthe purely bosonic (classically integrable) $AdS_4\\times CP^3$\nsector of the type IIA superstring sigma model maps into an\nequivalent sigma model on a dual $AdS_4$ space, both models sharing\nthe same integrable structure \\cite{Ricci:2007eq}. The situation\nwith the fermionic sector of the $AdS_4\\times CP^3$ superstring is,\nhowever, different due to the fact that there is less supersymmetry\nthan in the $AdS_5\\times S^5$ case.\n\n In the case of the\n$AdS_5\\times S^5$ superstring sigma model, one can accompany the\nabove bosonic T--duality transformation by a fermionic one along\nfermionic directions in (complexified) superspace which have\ntranslational isometries \\cite{Berkovits:2008ic,Beisert:2008iq}.\nThis compensates the dilaton term generated by the bosonic\nT--duality and maps the $AdS_5\\times S^5$ superstring action to an\nequivalent (dual) one, which is also integrable\\footnote{In\n\\cite{Ricci:2007eq,Berkovits:2008ic,Beisert:2008iq} it has been shown that this\nduality property of the $AdS_5\\times S^5$ superstring is related to\nearlier observed dual conformal symmetry of maximally helicity\nviolating amplitudes of the ${\\mathcal N}=4$ super--Yang--Mills\ntheory and to the relation between gluon scattering amplitudes and\nWilson loops at strong and weak coupling.}. However, in the\n$AdS_4\\times CP^3$ case under consideration the fermionic directions\nin superspace parametrized by $\\upsilon$ do not have translational\nisometries, since the action (\\ref{cordaB}) or (\\ref{cordaBT}) has\n$\\upsilon$--dependent fermionic terms which do not contain\nworldsheet derivatives. This just reflects the fact that the\nfermionic modes $\\upsilon$ correspond to the broken supersymmetries\nof the superbackground.\n\nAs far as the T--dualization of the supersymmetric fermionic modes\n$\\vartheta$ is concerned, it might be, in principle, possible (at\nleast in the absence of $\\upsilon$) if, as in the case of the\n$PSU(2,2|4)$ superstring sigma model\n\\cite{Berkovits:2008ic,Beisert:2008iq}, there existed a\nrealization of the $OSp(6|4)$ superalgebra in which 12 of the 24\n(complex conjugate) supersymmetry generators squared to zero and\nformed a representation of the bosonic subalgebra of $OSp(6|4)$. In\nother words the possibility of T--dualizing part of fermionic modes\n$\\vartheta$ (in the absence of $\\upsilon$) is related to the\nquestion of the existence of a chiral superspace representation of\nthe superalgebra $OSp(6|4)$. Such a realization of $OSp(6|4)$ seems\nnot to exist. In fact, it has been argued in \\cite{Adam:2009kt} that\nthe $OSp(6|4)$ supercoset subsector of the Green--Schwarz\nsuperstring in $AdS_4\\times CP^3$ does not have any fermionic\nT--duality symmetry since in $OSp(6|4)$ the dimension of the\nrepresentation of the supercharges under the R--symmetry is odd. The\nabsence of the fermionic T--duality of the superstring in\n$AdS_4\\times CP^3$ may have interesting manifestations in particular\nfeatures of the $AdS_4\/CFT_3$ holography.\n\nThe gauge--fixed actions (\\ref{cordaB}) or (\\ref{cordaBT}) can be\nused for studying different aspects of the $AdS_4\/CFT_3$\ncorrespondence and integrability on both of its sides, in\nparticular, for making two-- and higher--loop string computations\nfor testing the Bethe ansatz and the S--matrix\n\\cite{Minahan:2008hf}--\\cite{Minahan:2009te} in the dual planar ${\\mathcal N}=6$\nsuperconformal Chern--Simons--matter theory, which would extend the\nanalysis of\n\\cite{Nishioka:2008gz}--\\cite{Suzuki:2009sc},\\cite{Zarembo:2009au} and\nothers.\n\n\\subsection{D2--branes}\nLet us now consider the effective worldvolume theory of probe\nD2--branes moving in the $AdS_4 \\times CP^3$ superbackground. This\ncan be derived from the action for $D$--branes in a generic type IIA\nsuperbackground\n\\cite{Cederwall:1996ri,Aganagic:1996pe,Bergshoeff:1996tu} by\nsubstituting the explicit form of the $AdS_4 \\times CP^3$\nsupergeometry (\\ref{simplA})--(\\ref{A3}).\n\nThe action for a D2--brane in a generic type IIA supergravity\nbackground in the string frame has the following form\n\\begin{equation}\\label{DBIstring}\nS= -T\\int\\,d^{3}\\xi\\,e^{{ - } {\\phi}}\\sqrt{-\\det(g_{IJ}+{\\cal\nF}_{IJ})}+T \\int\\, ({\\mathcal A}_3 + {\\mathcal A}_1\\,{\\cal F}_2)\\,,\n\\end{equation}\nwhere $T$ is the tension of the D2--brane, $\\phi(Z)$ is the dilaton\nsuperfield,\n\\begin{equation}\\label{imb}\ng_{IJ}(\\xi)={\\mathcal E}_{I}{}^{A}\\, {\\mathcal\nE}_{J}{}^{B}\\,\\eta_{AB}\\qquad I,J=0,1,2;\\qquad A,B=0,1,\\cdots,9\n\\end{equation}\nis the induced metric on the D2--brane worldvolume with ${\\mathcal\nE}_{I}{}^{A}=\\partial_I\\,Z^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal\nM}{}^{A}$ being the pullbacks of the vector supervielbeins of the\ntype IIA $D=10$ superspace and\n\\begin{equation}\\label{deltaQstring}\n{\\cal F}_2 = d{\\mathcal V} - {B}_2\n\\end{equation}\nis the field strength of the worldvolume Born--Infeld gauge field\n${\\mathcal V}_I(\\xi)$ extended by the pullback of the NS--NS\ntwo--form. ${\\mathcal A}_1$ and ${\\mathcal A}_3$ are the pullbacks\nof the type IIA supergravity RR superforms (\\ref{simplB}) and\n(\\ref{A3}).\n\nProvided that the superbackground satisfies the IIA supergravity\nconstraints, the action (\\ref{DBIstring}) is invariant under\nkappa--symmetry transformations of the superstring coordinates\n$Z^{\\mathcal M}(\\xi)$ of the form\n(\\ref{kappastring}), (\\ref{kA}), together with\n\\begin{equation}\n\\delta_\\kappa\\mathcal F_{IJ}=-\\mathcal E_J^{\\mathcal B}\\,\\mathcal E_I^{\\mathcal A}\\,\\delta_\\kappa Z^{\\mathcal M}\\mathcal E_{\\mathcal M}{}^{\\underline\\alpha}\\,H_{\\underline\\alpha\\mathcal{AB}}\n=-4i\\mathcal E_{[I}^A\\,\\mathcal E_{J]}\\Gamma_A\\Gamma_{11}\\delta_\\kappa\\mathcal E\n-2i\\mathcal E_J^B\\,\\mathcal E_I^A\\,\\delta_\\kappa\\mathcal E\\Gamma_{AB}\\Gamma_{11}\\lambda\\,,\n\\end{equation}\nwhere $\\delta_\\kappa\\mathcal E=\\delta_\\kappa Z^{\\mathcal M}\\mathcal E_{\\mathcal M}{}^{\\underline\\alpha}$.\n\nIn the case of the $D2$--brane the matrix $\\Gamma$ has the form\n\\begin{eqnarray}\\label{bargamma}\n\\Gamma&=-&{{1}\\over {\\sqrt{-\\det(g+{\\cal\nF})}}}\\,\\,\\varepsilon^{IJK}\\,({1\\over 3!}\\,{\\mathcal\nE}_I^{A}\\,{\\mathcal E}_J^{B}\\,{\\mathcal E}_K^{C}\\Gamma_{ABC}+{1\\over\n2}\\,{\\mathcal F_{IJ}}\\,{\\mathcal\nE}_K^{A}\\,\\Gamma_A\\,\\Gamma_{11})\\nonumber\\\\\n&&\\\\\n&=&-{{1}\\over {3!\\sqrt{-\\det(g+{\\cal\nF})}}}\\,\\,\\varepsilon^{IJK}\\,{\\mathcal E}_I^{A}\\,{\\mathcal\nE}_J^{B}\\,{\\mathcal E}_K^{C}\\,\\Gamma_{ABC}\\,\\,(1+{1\\over\n2}\\,{\\mathcal F^{IJ}}\\,{\\mathcal E}_I^{A}\\,{\\mathcal\nE}_J^{B}\\,\\Gamma_{AB}\\,\\Gamma_{11})\\,.\\nonumber\n\\end{eqnarray}\n\n\\subsubsection{D2 filling $AdS_2\\times S^1$ inside of\n$AdS_4$}\\label{d2ads2s1}\n\nLet us consider the D2--brane configuration which corresponds to a\ndisorder loop operator in the ABJM theory\n\\cite{Drukker:2008jm}. The 1\/2 BPS static solution of the equations of motion of the D2--brane on\n$AdS_2\\times S^1$ in the metric\n\\begin{equation}\\label{ads4}\nds^2=\\frac{R_{CP^3}^2}{4u^2}(-dx^0\\,dx^0+dr^2+r^2d\\varphi^2+du^2)+R_{CP^3}^2ds^2_{CP^3}\\,,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nds^2_{CP^3}={1\\over 4}\\Big[d\\alpha^2+\\cos^2{\\alpha\\over\n2}(d\\vartheta_1^2+\\sin^2\\vartheta_1d\\varphi_1^2)+\\sin^2{\\alpha\\over\n2}(d\\vartheta_2^2+\\sin^2\\vartheta_2d\\varphi_2^2)\\cr\n+\\sin^2{\\alpha\\over 2}\\cos^2{\\alpha\\over 2}(d\\chi+\n\\cos\\vartheta_1d\\varphi_1-\\cos\\vartheta_2d\\varphi_2)^2\\Big]\\,,\\nonumber\n\\end{eqnarray}\nis characterized by the following embedding of the brane worldvolume\n\\begin{equation}\\label{ads2s11}\n\\xi^0=x^0,\\qquad \\xi_1=u,\\qquad \\xi_2=\\varphi\\,,\n\\qquad r=a\\,\\xi_1\n\\end{equation}\nwhich is supported by the non--zero (electric) Born--Infeld field\nstrength\n\\begin{equation}\nF=E {dx^0\\wedge du\\over u^2}\\,, \\qquad\nE=\\frac{R_{CP^3}^2}{4}\\sqrt{1+a^2}\\,,\n\\end{equation}\nwhere $a$ is an arbitrary constant. Note that the presence of the\nnon--zero DBI flux on the $AdS_2$ subspace of the D2--brane\nworldvolume is required to ensure the no--force condition, i.e.\nvanishing of the classical action (\\ref{DBIstring}) of this static\nD2--brane configuration, provided that also an additional BI flux\nboundary counterterm is added to the action (see\n\\cite{Drukker:2008jm} for more details). A natural explanation of\nthis boundary term is that it appears in the process of the\ndualization of the compactified 11th coordinate scalar field of the\nM2--brane into the BI vector field of the D2--brane.\n\nNote that in \\cite{Drukker:2008jm} this brane configuration was\nconsidered in a different coordinate system, in which $AdS_4$ is\nfoliated with $AdS_2\\times S^1$ slices instead of the flat $R^{1,2}$\nslices. This makes manifest the symmetries of the D2--brane\nconfiguration. An explicit form of the $AdS_4$ metric in this\nslicing is\n\\begin{equation}\\label{ads2s1}\nds^2_{_{AdS_4}}= {R_{CP^3}^2\\over 4}\\,(\\cosh^2 \\psi\n\\,ds^2_{_{AdS_2}}+d\\psi^2+\n\\sinh^2 \\psi\n\\, d\\varphi^2)\n\\end{equation}\nwhich is essentially a double analytic continuation of the usual\nglobal $AdS_4$ metric. The static D2--brane configuration is then\ncharacterized by the identification of the worldvolume coordinates\n$\\xi^a$ with those of $AdS_2$ and the $S^1$ angle $\\varphi$.\nHowever, for our choice of the kappa--symmetry gauge fixing\ncondition the use of the metric in the form (\\ref{ads4}) is more\nconvenient, since the associated $AdS_4$ vielbeins\n\\begin{equation}\\label{0123}\ne^0=\\frac{R_{CP^3}}{2u}\\,dx^0\\,,\\qquad\ne^1=\\frac{R_{CP^3}}{2u}\\,dr\\,,\\qquad\ne^2=\\frac{R_{CP^3}\\,r}{2u}\\,d\\varphi\\,,\\qquad\ne^3=-\\frac{R_{CP^3}}{2u}\\,du\\,\n\\end{equation}\nand the spin connection directly satisfy the relations\n(\\ref{eaoa31}) and (\\ref{oab}).\n\nOne can be interested in D2--brane bosonic and fermionic\nfluctuations around this 1\/2 BPS static D2--brane solution described\nby the action (\\ref{DBIstring}). To simplify the form of the\nfermionic terms, the kappa--symmetry gauge fixing for the D2--brane\nwrapping $AdS_2\n\\times S^1$ can be made in the simplest possible way considered in\nSubsection \\ref{theta-}. To get the gauge fixed D2--brane action in\nthis case one should substitute into (\\ref{DBIstring}) the\nexpressions for the vector supervielbeins (\\ref{simple+v}), the RR\none--form (\\ref{simple+A}) and the three--form (\\ref{A3-}), and the\nNS--NS two--form (\\ref{simple+B}).\n\n\\subsubsection{D2 at the Minkowski boundary of $AdS_4$}\nLet us now consider the supersymmetric effective worldvolume action\ndescribing a D2--brane placed at the Minkowski boundary of the\n$AdS_4$ space. In this case it is convenient to choose the\n$AdS_4\\times CP^3$ metric in the form (\\ref{ads4metric11}) or\n(\\ref{ads4metric21}).\n\n\nWhen the D2--brane is at the Minkowski boundary, we take the static\ngauge $\\xi^m=x^m$. The 1\/2 BPS ground state of the D2--brane is when\nits transverse scalar modes are constant and the Born--Infeld field\nand the fermionic modes are zero. As a consistency check, let us\nnote that with the choice of the background value of the RR 3--form\n(\\ref{A3}) and (\\ref{A31}) and of the corresponding (positive) $D2$--brane charge\n(characterized by the plus sign in front of the Wess--Zumino term\n(\\ref{DBIstring})), the action of the ground state of the D2--brane\nat the Minkowski boundary vanishes. This means that such a brane\nconfiguration is stable and does not experience any external force,\n\\emph{i.e.} it is a BPS state.\n\nIf, on the other hand, with the same choice of ${\\mathcal A}_3$\n(\\ref{A3}) and (\\ref{A31}), we considered an anti--$D2$--brane carrying a negative\n${\\mathcal A}_3$ charge (which would be characterized by a minus\nsign in front of the Wess--Zumino term in (\\ref{DBIstring})), the\nground state of this anti--$D2$--brane at the Minkowski boundary\nwould have a non--zero action\n$$\nS_{\\overline{D2}}=-2Te^{-\\phi_0}\\,\\int\\,d^{\\,3}x\\,\\left(r\\over {R_{CP^3}}\\right)^6\\,\n$$\nimplying that such a solution is unstable (as is well known to be\nthe case for a probe anti--D--brane in a background of D--branes).\nIt is, therefore, important for the consistency of the solution to\ntake care that the relative signs of the RR potential ${\\mathcal\nA}_3$ and the $D2$--brane charge (and, as a consequence, the sign of\nthe kappa--symmetry projector) ensure the no--force condition, i.e.\nvanishing of the static $D2$--brane action. In the case of M2, M5\nand D3--branes at the Minkowski boundary of $AdS$ this issue was\ndiscussed in detail in\n\\cite{Pasti:1998tc}.\n\nFor the static D2--brane configuration the kappa--symmetry projector\n(\\ref{bargamma}) reduces to\n\\begin{equation}\\label{kappa}\nP=\\frac{1}{2}(1+\\gamma), \\qquad\n\\gamma=\\gamma^0\\gamma^1\\gamma^2=-\\gamma_0\\gamma_1\\gamma_2\n\\end{equation}\nSo the natural choice of the kappa--symmetry gauge fixing condition\nis\n\\begin{equation}\\label{kappagauge}\n\\Theta=\\frac{1}{2}(1-\\gamma)\\Theta\\,,\n\\end{equation}\n\\emph{i.e.} the gauge choice considered in detail in Subsection\n\\ref{theta++}.\nNote that in the case of the D2--brane at the Minkowski boundary we\ncannot use the simpler condition\n$\\Theta=\\frac{1}{2}(1+\\gamma)\\Theta$ of Subsection \\ref{theta-},\nbecause the kappa--symmetry projector (\\ref{kappa}) has the same\nsign.\n\nPlugging the kappa--symmetry gauge--fixed quantities of Subsection\n3.2 into the action (\\ref{DBIstring}), one can study the properties\nof the $OSp(6|4)$ invariant effective $3d$ gauge--matter field\ntheory on the worldvolume of the $D2$--brane placed at the Minkowski\nboundary of $AdS_4$, which from the point of view of M--theory\ncorresponds to an $M2$--brane pulled out to a finite distance from a\nstack of $M2$--branes probing $R^8\/Z_k$.\n\n\nThe effective theory on the worldvolume of this D2--brane, which\ndescribes its fluctuations in $AdS_4\n\\times CP^3$, is an interacting $d=3$ gauge Born--Infeld--matter theory\npossessing the (spontaneously broken) superconformal symmetry\n$OSp(6|4)$. The model is superconformally invariant in spite of the\npresence on the $d=3$ worldvolume of the dynamical Abelian vector\nfield, since the latter is coupled to the $3d$ dilaton field\nassociated with the radial direction of $AdS_4$. The superconformal\ninvariance is spontaneously broken by a non--zero expectation value\nof the dilaton. An ${\\mathcal N=3}$ superfield model with similar\nsymmetry properties was considered in the Appendix of\n\\cite{Buchbinder:2008vi}. To establish the explicit relation between\n the two models one should extract from the\nsuperfield action of \\cite{Buchbinder:2008vi} the component terms\ndescribing its physical sector and compare the result with\ncorresponding terms in the D2--brane action.\n\n\\section{Conclusion}\nIn this paper we have considered the gauge--fixing of\nkappa--symmetry of the superparticle, superstring and D2--brane\nactions in the complete $AdS_4\\times CP^3$ superspace which is\nsuitable, in particular, for studying regions of these theories that\nare not reachable by partially kappa--symmetry gauge fixed models\nbased on the supercoset $OSp(6|4)\/U(3)\\times SO(1,3)$. The\nsimplified form of these actions can be used to approach various\nproblems of the $AdS_4\/CFT_3$ correspondence. The gauge fixed form\nof the $AdS_4\\times CP^3$ supergeometry can also be used to consider\nthe actions for higher dimensional D4--, D6-- and D8--branes.\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank Pietro Fr\\'e and Jaume Gomis for\ncollaboration at early stages of this project and for many fruitful\ndiscussions and comments. D.S. is also thankful to Soo--Jong Rey for\nuseful discussions. P.A.G. and D.S. are grateful to the Organizers\nof the Workshop Program ``Fundamental Aspects of Superstring Theory\"\nfor their hospitality at KITP, Santa Barbara, where their research\nwas supported in part by the National Science Foundation under Grant\nNo. PHY05-51164. Work of P.A.G., D.S. and L.W. was partially\nsupported by the INFN Special Initiative TV12. D.S. was also\npartially supported by the INTAS Project Grant 05-1000008-7928, an\nExcellence Grant of Fondazione Cariparo and the grant FIS2008-1980\nof the Spanish Ministry of Science and Innovation.\n\n\\def{}\n\\defC.\\arabic{equation}}\\label{C{A.\\arabic{equation}}\\label{A}\n\\section{Appendix A. Main notation and conventions}\n\\setcounter{equation}0\n\nThe convention for the ten and eleven dimensional metrics is the\n`almost plus' signature $(-,+,\\cdots,+)$. Generically, the tangent\nspace vector indices are labeled by letters from the beginning of\nthe Latin alphabet, while letters from the middle of the Latin\nalphabet stand for curved (world) indices. The spinor indices are\nlabeled by Greek letters.\n\n\\defC.2{A.1}\n\\subsection{$AdS_4$ space}\n\n$AdS_4$ is parametrized by the coordinates $x^{\\hat m}$ and its\nvielbeins are $e^{\\hat a}=dx^{\\hat m}\\,e_{\\hat m}{}^{\\hat a}(x)$,\n${\\hat m}=0,1,2,3;$ ${\\hat a}=0,1,2,3$. The $D=4$ gamma--matrices\nsatisfy:\n\\begin{equation}\\label{gammaa}\n\\{\\gamma^{\\hat a},\\gamma^{\\hat b}\\}=2\\,\\eta^{\\hat a\\hat b}\\,,\n\\qquad \\eta^{\\hat a\\hat b}={\\rm diag}\\,(-,+,+,+)\\,,\n\\end{equation}\n\\begin{equation}\\label{gamma5}\n\\gamma^5=i\\gamma^0\\,\\gamma^1\\,\\gamma^2\\,\\gamma^3, \\qquad\n\\gamma^5\\,\\gamma^5=1\\,.\n\\end{equation}\nThe charge conjugation matrix $C$ is antisymmetric, the matrices\n$(\\gamma^{\\hat a})_{\\alpha\\beta}\\equiv (C\\,\\gamma^{\\hat\na})_{\\alpha\\beta}$ and $(\\gamma^{\\hat a\\hat\nb})_{\\alpha\\beta}\\equiv(C\\,\\gamma^{\\hat a\\hat b})_{\\alpha\\beta}$ are\nsymmetric and $\\gamma^5_{\\alpha\\beta}\\equiv\n(C\\gamma^5)_{\\alpha\\beta}$ is antisymmetric, with\n$\\alpha,\\beta=1,2,3,4$ being the indices of a 4--dimensional spinor\nrepresentation of $SO(1,3)$ or $SO(2,3)$.\n\n\\defC.2{A.2}\n\\subsection{$CP^3$ space}\n\n$CP^3$ is parametrized by the coordinates $y^{m'}$ and its vielbeins\nare $e^{a'}=dy^{m'}e_{m'}{}^{a'}(y)$, ${m'}=1,\\cdots,6;$\n${a'}=1,\\cdots,6$. The $D=6$ gamma--matrices satisfy:\n\\begin{equation}\\label{gammaa'}\n\\{\\gamma^{a'},\\gamma^{b'}\\}=2\\,\\delta^{{a'}{b'}}\\,,\\qquad \\delta^{a'b'}={\\rm\ndiag}\\,(+,+,+,+,+,+)\\,,\n\\end{equation}\n\\begin{equation}\\label{gamma7}\n\\gamma^7={i\\over{6!}}\\,\\epsilon_{\\,a_1'a_2'a_3'a_4'a_5'a_6'}\\,\\gamma^{a_1'}\\cdots \\gamma^{a_6'} \\qquad\n\\gamma^7\\,\\gamma^7=1\\,.\n\\end{equation}\nThe charge conjugation matrix $C'$ is symmetric and the matrices\n$(\\gamma^{a'})_{\\alpha'\\beta'}\\equiv\n(C\\,\\gamma^{a'})_{\\alpha'\\beta'}$ and\n$(\\gamma^{a'b'})_{\\alpha'\\beta'}\\equiv(C'\\,\\gamma^{a'b'})_{\\alpha'\\beta'}$\nare antisymmetric, with $\\alpha',\\beta'=1,\\cdots,8$ being the\nindices of an 8--dimensional spinor representation of $SO(6)$.\n\n\\defC.2{A.3}\n\\subsection{ Type IIA $AdS_4\\times CP^3$ superspace}\n\nThe type IIA superspace whose bosonic body is $AdS_4\\times CP^3$ is\nparametrized by 10 bosonic coordinates $X^M=(x^{\\hat m},\\,y^{m'})$\nand 32-fermionic coordinates\n$\\Theta^{\\underline\\mu}=(\\Theta^{\\mu\\mu'})$\n($\\mu=1,2,3,4;\\,\\mu'=1,\\cdots,8$). These combine into the\nsuperspace supercoordinates $Z^{\\cal M}=(x^{\\hat\nm},\\,y^{m'},\\,\\Theta^{\\mu\\mu'})$. The type IIA supervielbeins are\n\\begin{equation}\\label{IIAsv}\n{\\mathcal E}^{\\mathcal A}=dZ^{\\mathcal M}\\,{\\mathcal E}_{\\mathcal\nM}{}^{\\mathcal A}(Z)=({\\mathcal E}^{A},\\,{\\mathcal\nE}^{\\underline\\alpha})\\,,\\qquad {\\mathcal E}^{A}(Z)=({\\mathcal\nE}^{\\hat a},\\,{\\mathcal E}^{a'})\\,,\\qquad {\\mathcal\nE}^{\\underline\\alpha}(Z)={\\mathcal E}^{\\alpha\\alpha'}\\,.\n\\end{equation}\n\\defC.2{A.4}\n\\subsection{Superspace constraints}\nIn our conventions the superspace constraint on the bosonic part of the torsion is\n\\begin{equation}\nT^A=-i\\mathcal E\\Gamma^A\\mathcal E+i\\mathcal E^A\\,\\mathcal\nE\\lambda+\\frac{1}{3}{\\mathcal E}^A\\,\\mathcal E^B\\nabla_B\\phi\\,,\n\\end{equation}\nwhile the constraints on the RR and NS--NS field strengths are\n\\begin{eqnarray}\nF_2&=&-i\\,e^{-\\phi}\\,\\mathcal E\\Gamma_{11}\\mathcal E\n+2i\\,e^{-\\phi}\\,\\mathcal E^A\\,\\mathcal E\\Gamma_A\\Gamma_{11}\\lambda+\\frac{1}{2}\\mathcal E^B\\mathcal E^A\\,F_{AB}\\,,\\\\\nF_4&=&-\\frac{i}{2}\\,e^{-\\phi}\\,{\\mathcal E}^B{\\mathcal E}^A\\,\\mathcal E\\Gamma_{AB}\\mathcal E\n+\\frac{1}{4!}{\\mathcal E}^D{\\mathcal E}^C{\\mathcal E}^B{\\mathcal E}^A\\,F_{ABCD}\n\\,,\\\\\nH_3&=&\n-i{\\mathcal E}^A\\,\\mathcal E\\Gamma_A\\Gamma_{11}\\mathcal E\n+i{\\mathcal E}^B{\\mathcal E}^A\\,\\mathcal E\\Gamma_{AB}\\Gamma^{11}\\lambda\n+\\frac{1}{3!}{\\mathcal E}^C{\\mathcal E}^B{\\mathcal E}^A\\,H_{ABC}\\,.\n\\end{eqnarray}\nThese differ from the conventional string frame constraints by the $\\lambda$--term in $T^A$ and related terms in $F_2$, $F_4$ and $H_3$. This is\na consequence of the dimensional reduction from eleven dimensions. They can be brought to a more conventional form by shifting the fermionic supervielbein $\\mathcal E^{\\underline\\alpha}$ by $-\\frac{1}{2}\\mathcal E^A(\\Gamma_A\\lambda)^{\\underline\\alpha}$ accompanied by a related shift in the connection.\n\\\\\n\\\\\n{\\bf The $D=10$ gamma--matrices $\\Gamma^A$} are given by\n\\begin{eqnarray}\\label{Gamma10}\n&\\{\\Gamma^A,\\,\\Gamma^B\\}=2\\eta^{AB},\\qquad\n\\Gamma^{A}=(\\Gamma^{\\hat a},\\,\\Gamma^{a'})\\,,\\nonumber\\\\\n&\\\\\n&\\Gamma^{\\hat a}=\\gamma^{\\hat a}\\,\\otimes\\,{\\bf 1},\\qquad\n\\Gamma^{a'}=\\gamma^5\\,\\otimes\\,\\gamma^{a'},\\qquad\n\\Gamma^{11}=\\gamma^5\\,\\otimes\\,\\gamma^7,\\qquad a=0,1,2,3;\\quad\na'=1,\\cdots,6\\,. \\nonumber\n\\end{eqnarray}\nThe charge conjugation matrix is ${\\mathcal C}=C\\otimes C'$.\n\nThe fermionic variables $\\Theta^{\\underline\\alpha}$ of IIA\nsupergravity carrying 32--component spinor indices of $Spin(1,9)$,\nin the $AdS_4\\times CP^3$ background and for the above choice of the\n$D=10$ gamma--matrices, naturally split into 4--dimensional\n$Spin(1,3)$ indices and 8--dimensional spinor indices of $Spin(6)$,\ni.e. $\\Theta^{\\underline\\alpha}=\\Theta^{\\alpha\\alpha'}$\n($\\alpha=1,2,3,4$; $\\alpha'=1,\\cdots,8$).\n\n\\defC.2{A.5}\n\\subsection{$24+8$ splitting of $32$ $\\Theta$}\n\n24 of $\\Theta^{\\underline\\alpha}=\\Theta^{\\alpha\\alpha'}$ correspond\nto the unbroken supersymmetries of the $AdS_4\\times CP^3$\nbackground. They are singled out by a projector introduced in\n\\cite{Nilsson:1984bj} which is constructed using the $CP^3$ K\\\"ahler\nform $J_{a'b'}$ and seven $8\\times 8$ antisymmetric gamma--matrices\n(\\ref{gammaa'}). The $8\\times 8$ projector matrix has the following\nform\n\\begin{equation}\\label{p6}\n{\\mathcal P}_{6}={1\\over 8}(6-J)\\,,\n\\end{equation}\nwhere the $8\\times 8$ matrix\n\\begin{equation}\\label{J}\nJ=-iJ_{a'b'}\\,\\gamma^{a'b'}\\,\\gamma^7 \\qquad {\\rm such~ that} \\qquad\nJ^2= 4J+12\n\\end{equation}\nhas six eigenvalues $-2$ and two eigenvalues $6$, \\emph{i.e.} its\ndiagonalization results in\n\\begin{equation}\\label{Jdia}\nJ=\\hbox{diag}(-2,-2,-2,-2,-2,-2,6,6)\\,.\n\\end{equation}\nTherefore, the projector (\\ref{p6}) when acting on an 8--dimensional\nspinor annihilates 2 and leaves 6 of its components, while the\ncomplementary projector\n\\begin{equation}\\label{p2}\n{\\mathcal P}_{2}={1\\over 8}(2+J)\\,,\\qquad\n\\mathcal{P}_2+\\mathcal{P}_6=\\mathbf 1\n\\end{equation}\nannihilates 6 and leaves 2 spinor components.\n\nThus the spinor\n\\begin{equation}\\label{24}\n\\vartheta^{\\alpha\\alpha'}=({\\mathcal P}_6\\,\\Theta)^{\\alpha\\alpha'} \\qquad \\Longleftrightarrow \\qquad\n\\vartheta^{\\alpha a'}\\, \\qquad a'=1,\\cdots, 6\n\\end{equation}\nhas 24 non--zero components and the spinor\n\\begin{equation}\\label{8}\n\\upsilon^{\\alpha\\alpha'}=({\\mathcal P}_2\\,\\Theta)^{\\alpha\\alpha'}\\qquad \\Longleftrightarrow \\qquad\n\\upsilon^{\\alpha i}\\, \\qquad i=1,2\n\\end{equation}\nhas 8 non--zero components. The latter corresponds to the eight\nsupersymmetries broken by the $AdS_4\\times CP^3$ background.\n\nTo avoid confusion, let us note that the index $a'$ on spinors is\ndifferent from the same index on bosonic quantities. They are\nrelated by the usual relation between vector and spinor\nrepresentations, \\emph{i.e.} given two $Spin(6)$ spinors\n$\\psi_1^{\\alpha'}$ and $\\psi_2^{\\alpha'}$, projected as in\n(\\ref{24}), their bilinear combination $v^{a'}=\\psi_1\\mathcal\nP_6\\gamma^{a'}\\mathcal P_6\\psi_2=\\psi_1^{b'}(\\mathcal\nP_6\\gamma^{a'}\\mathcal P_6)_{b'c'}\\psi_2^{c'}$ transforms as a\n6--dimensional 'vector'.\n\n\\defC.\\arabic{equation}}\\label{C{B.\\arabic{equation}}\\label{B}\n\\section{Appendix B. $OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset realization\nand other ingredients of the $(10|32)$--dimensional $AdS_4\\times\nCP^3$ superspace}\n\\setcounter{equation}0\n\nThe supervielbeins and the superconnections of the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset which appear in the\ndefinition of the geometric and gauge quantities of the $AdS_4\\times\nCP^3$ superspace in Section \\ref{superspace} are\n\\begin{equation}\\label{cartan24}\n\\begin{aligned}\nE^{\\hat a}&=e^{\\hat a}(x)+4i\\vartheta\\gamma^{\\hat\na}\\,{{\\sinh^2{{\\mathcal M}_{24}\/ 2}}\\over{\\mathcal M}^2_{24}}\\,\nD_{24}\\vartheta,\\\\\nE^{a'}&=e^{a'}(y)+4i\\vartheta\\gamma^{a'}\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}_{24}\/2}}\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\n\\\\\nE^{\\alpha a'}&=\\left({{\\sinh{\\mathcal M}_{24}}\\over{\\mathcal\nM}_{24}}D_{24}\\vartheta\\right)^{\\alpha a'},\\\\\n\\Omega^{\\hat a\\hat b}&=\\omega^{\\hat a\\hat b}(x)+\\frac{8}{R}\\vartheta\\gamma^{\\hat a\\hat b}\\gamma^5\\,\n{{\\sinh^2{{\\mathcal M}_{24}\/2}}\\over{\\mathcal M}_{24}^2}D_{24}\\vartheta\\,,\\\\\n\\Omega^{a'b'}&=\\omega^{a'b'}(y)-\\frac{4}{R}\n\\vartheta(\\gamma^{a'b'}-iJ^{a'b'}\\gamma^7)\\gamma^5\\,{{\\sinh^2{{\\mathcal M}_{24}\/2}}\n\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\\\\\nA&=\\frac{1}{8}J_{a'b'}\\Omega^{a'b'}=A(y)+\\frac{4i}{R}\\,\\vartheta\\gamma^7\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}_{24}\/2}}\\over{\\mathcal M}_{24}^2}\\,D_{24}\\vartheta\\,,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\\label{M24}\nR\\,({\\mathcal M}_{24}^2)^{\\alpha a'}{}_{\\beta b'}=\n4\\vartheta^{\\alpha}_ {b'}\\,(\\vartheta^{a'}\\gamma^5)_\\beta\n-4\\delta^{a'}_{b'}\\vartheta^{\\alpha c'}(\\vartheta\\gamma^5)_{\\beta\nc'} -2(\\gamma^5\\gamma^{\\hat a}\\vartheta)^{\\alpha\na'}(\\vartheta\\gamma_{\\hat a})_{\\beta b'} -(\\gamma^{\\hat a\\hat b}\\vartheta)^{\\alpha\na'}(\\vartheta\\gamma_{\\hat a\\hat b}\\gamma^5)_{\\beta b'}\\,.\n\\end{equation}\nThe derivative appearing in the above equations is defined as\n\\begin{equation}\\label{D24}\nD_{24}\\vartheta={\\mathcal P_6}\\,(d +\\frac{i}{R}\\,e^{\\hat\na}\\gamma^5\\gamma_{\\hat a}\n+\\frac{i}{R}e^{a'}\\gamma_{a'}-\\frac{1}{4}\\omega^{\\hat a\\hat\nb}\\gamma_{\\hat a\\hat b}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,,\n\\end{equation}\nwhere $e^{\\hat a}(x)$, $e^{a'}(y)$, $\\omega^{\\hat a\\hat b}(x)$, $\\omega^{a'b'}(y)$\nand $A(y)$ are the vielbeins and connections of the bosonic\nsolution. The $U(3)$--connection $\\Omega^{a'b'}$ satisfies the\ncondition\n\\begin{equation}\n{(P^{-})_{a'b'}}^{c'd'}\\Omega_{c'd'}=\\frac{1}{2}\\,({\\delta_{[a'}}^{c'}\\,{\\delta_{b']}}^{d'}\\,-\\,\n{J_{[a'}}^{c'}\\,{J_{b']}}^{d'})\\Omega_{c'd'}=0\\,,\n\\end{equation}\nwhere $J_{a'b'}$ is the K\\\"ahler form on $CP^3$.\n\n\\defC.2{B.1}\n\\subsection{Other quantities appearing in the definition of the\n$AdS_4\\times CP^3$ superspace of Section \\ref{superspace}}\n\n\\begin{equation}\\label{M}\nR\\,({\\mathcal M}^2)^{\\alpha i}{}_{\\beta j}= 4(\\varepsilon\\upsilon)^{\\alpha\ni}(\\upsilon\\varepsilon\\gamma^5)_{\\beta j} -2(\\gamma^5\\gamma^{\\hat\na}\\upsilon)^{\\alpha i}(\\upsilon\\gamma_{\\hat a})_{\\beta j} -(\\gamma^{\\hat\na\\hat b}\\upsilon)^{\\alpha i}(\\upsilon\\gamma_{\\hat a\\hat\nb}\\gamma^5)_{\\beta j}\\,,\n\\end{equation}\n\\begin{equation}\n(m^2)^{ij}=-\\frac{4}{R}\\upsilon^i\\,\\gamma^5\\,\\upsilon^j\\,,\n\\end{equation}\n\n\\begin{equation}\n\\begin{aligned}\n\\Lambda_{\\hat a}{}^{\\hat b}&=\n\\delta_{\\hat a}{}^{\\hat b}-\\frac{R^2}{k^2l_p^2}\\,\\cdot\\,\n\\frac{e^{-\\frac{2}{3}\\phi}}{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}\\,{E_{7\\hat a}}\\,E_7{}^{\\hat b}\\,,\n\\\\\n\\\\\nS_{\\underline\\beta}{}^{\\underline\\alpha}&=\n\\frac{e^{-\\frac{1}{3}\\phi}}{\\sqrt2}\\left(\\sqrt{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}-{R\\over{kl_p}}\\,\n\\frac{E_7{}^{\\hat a}\\,\\Gamma_{\\hat a}\\Gamma_{11}}{\\sqrt{e^{\\frac{2}{3}\\phi}\n+{R\\over{kl_p}}\\,\\Phi}}\n\\,\\right)_{\\underline\\beta}{}^{\\underline\\alpha}\n\\end{aligned}\n\\end{equation}\n\n\\begin{equation}\\label{phiE7}\\begin{aligned}\nE_7{}^{\\hat a}(\\upsilon)&=-\\frac{8i}{R}\\,\\upsilon\\gamma^{\\hat a}\\,{{\\sinh^2{{\\mathcal\nM}\/ 2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\,{\\upsilon}\\,,\n\\\\\n\\Phi(\\upsilon)&= 1+\\frac{8}{R}\\,\\upsilon\\,\\varepsilon\\gamma^5\\,{{\\sinh^2{{\\mathcal\nM}\/2}}\\over{\\mathcal M}^2}\\,\\varepsilon\\upsilon\\,.\n\\end{aligned}\n\\end{equation}\nLet us emphasise that the $SO(2)$ indices $i,j=1,2$ are raised and\nlowered with the unit matrices $\\delta^{ij}$ and $\\delta_{ij}$ so\nthat there is actually no difference between the upper and the\nlower $SO(2)$ indices, $\\varepsilon_{ij}=-\\varepsilon_{ji}$,\n$\\varepsilon^{ij}=-\\varepsilon^{ji}$ and\n$\\varepsilon^{12}=\\varepsilon_{12}=1$.\n\n\\defC.\\arabic{equation}}\\label{C{C.\\arabic{equation}}\\label{C}\n\\section{Appendix C. Identities for the kappa-projected fermions}\n\\setcounter{equation}0\n\n\nWhen the fermionic variables\n$\\Theta^{\\underline\\alpha}=(\\vartheta^{\\alpha a'},\\,\\upsilon^{\\alpha\ni})$ are subject to the constraint (\\ref{kappagauge1}), the\nfollowing identities hold.\n\n\\defC.2{C.1}\n\\subsection{Identities involving $\\upsilon^{\\alpha i}$}\n\\begin{equation}\\label{i1}\n\\upsilon^i\\gamma^5\\upsilon^j=\\upsilon^i\\gamma^3\\upsilon^j=0\\,,\\qquad \\upsilon^{\\alpha i}\\upsilon^{\\beta j}\\delta_{ij}=-\\frac{1}{4}((1\\pm\\gamma)C^{-1})^{\\alpha\\beta}\\upsilon\\ups\\,,\n\\end{equation}\nwhere $\\gamma=\\gamma^{012}$ and $\\upsilon\\ups=\\delta_{ij}\\upsilon^{\\alpha\ni}C_{\\alpha\\beta}\\upsilon^{\\beta j}$.\n\nAnother useful relation is ($\\varepsilon^{012}=-\\varepsilon_{012}=1$)\n\\begin{equation}\\label{gg}\n\\upsilon\\gamma_{ab}d\\upsilon=\\pm\\varepsilon_{abc}\\upsilon\\gamma^cd\\upsilon\\,,\n\\end{equation}\nwhich also holds for the kappa--projected $\\vartheta$ and\n$d\\vartheta$.\n\nUsing eqs. (\\ref{i1}) and (\\ref{gg}) we find that\n\\begin{equation}\n\\upsilon\\varepsilon\\gamma^a\\upsilon\\,\\upsilon\\varepsilon\\gamma_b\\upsilon=\\delta_b^a(\\upsilon\\ups)^2\\,,\n\\qquad \\upsilon\\varepsilon\\gamma^{ac}\\upsilon\\,\\upsilon\\varepsilon\\gamma_{cb}\\upsilon=2\\delta_b^a(\\upsilon\\ups)^2\\,,\n\\end{equation}\n\\begin{equation}\n(m^2)^{ij}=-\\frac{4}{R}\\upsilon^i\\,\\gamma^5\\,\\upsilon^j=0\n\\end{equation}\nand\n\\begin{eqnarray}\n({\\mathcal M}^2\\varepsilon\\upsilon)^{\\alpha i}=0\\,.\n\\end{eqnarray}\n A similar computation shows that\n\\begin{equation}\n\\upsilon\\varepsilon\\gamma^5{\\mathcal M}^2=0.\n\\end{equation}\nIt is also true in general (i.e. without fixing $\\kappa$--symmetry)\nthat\n\\begin{equation}\n{\\mathcal M}^2\\upsilon=0\\,,\\qquad \\upsilon\\gamma^5{\\mathcal M}^2=0.\n\\end{equation}\nUsing the above identities we find that for $\\upsilon$ satisfying\n(\\ref{kappagauge1})\n\\begin{equation}\\label{M2d}\n\\mathcal M^2D\\upsilon\n=\\frac{6i}{R^2}(E^a\\pm\\frac{R}{2}\\Omega^{a3})(\\gamma_a\\upsilon)\\,\\upsilon\\ups\n\\end{equation}\nwhich results in\n\\begin{equation}\\label{vmdv}\n4\\upsilon\\gamma^a \\frac{\\sinh^2(\\mathcal M\/2)}{\\mathcal\nM^2}D\\upsilon=\n\\upsilon\\gamma^a(1+\\frac{1}{12}\\mathcal M^2)D\\upsilon\n=\\upsilon\\gamma^a\\,(d-\\frac{1}{4}\\Omega^{bc}\\gamma_{bc})\\upsilon\n+\\frac{i}{2R^2}(E^a\\pm\\frac{R}{2}\\Omega^{a3})(\\upsilon\\ups)^2\\,,\n\\end{equation}\nwhere $E^a$, $\\Omega^{bc}$ and $\\Omega^{a3}$ are $AdS_4$ components\nof the supervielbein and connection of the supercoset\n$OSp(6|4)\/U(3)\\times SO(1,3)$ defined in eqs. (\\ref{cartan24}) and\nthe matrix ${\\mathcal M}^2$ is defined in eq. (\\ref{M}).\n\nWe also find that\n\\begin{equation}\n4\\upsilon\\varepsilon\\gamma^5{{\\sinh^2{{\\mathcal M}\/2}}\\over{\\mathcal M}^2}D\\upsilon\n=\\upsilon\\varepsilon\\gamma^5D\\upsilon=\\frac{i}{R}(E^a\\pm\\frac{R}{2}\\Omega^{a3})\\upsilon\\varepsilon\\gamma_a\\upsilon\\,.\n\\end{equation}\n\n\\defC.2{C.2}\n\\subsection{Identities involving $\\vartheta^{\\alpha a'}$ and the simplified\nform of the \\\\ $OSp(6|4)\/U(3)\\times SO(1,3)$ supergeometry}\n\nUsing the definition of $\\mathcal M_{_{24}}$, eq. (\\ref{M24}), and\nthe fact that\n\\begin{equation}\n[\\gamma^{012},\\gamma^{a'}]=0\n\\end{equation}\nwe find that\n\\begin{equation}\n(\\vartheta\\gamma'\\gamma^5{\\mathcal M}_{_{24}}^2)_{\\beta b'} =0\n\\qquad\n({\\mathcal M}_{_{24}}^2\\gamma'\\vartheta)^{\\alpha a'}=0\\,,\n\\end{equation}\nwhere $\\gamma'$ is any product of the gamma-matrices that commutes\nwith $\\gamma=\\gamma^{012}$,\n\\emph{e.g.} any product of $\\gamma^{a'}$ and $\\gamma^{a}$. A\nslightly longer computation, using the fact that\n\\begin{equation}\n\\gamma^3\\vartheta=\\pm\\gamma^3\\gamma^{012}\\vartheta=\\pm i\\gamma^5\\vartheta\\,,\\qquad\\vartheta\\gamma^3=\\mp i\\vartheta\\gamma^5\\qquad\\mbox{for}\\quad\\vartheta=\\frac{1}{2}(1\\pm\\gamma)\\vartheta\\,,\n\\end{equation}\nshows that with this projection of the $\\vartheta$s\n\\begin{equation}\n\\mathcal M_{_{24}}^4=0\\,.\n\\end{equation}\nUsing the identity\n\\begin{equation}\n\\vartheta^{\\alpha a'}\\vartheta^{\\beta b'}\\delta_{a'b'}=-\\frac{1}{4}((1\\pm\\gamma)C^{-1})^{\\alpha\\beta}\\vartheta\\vartheta\\,,\n\\end{equation}\nwhere $\\vartheta\\vartheta\\equiv\\vartheta^{\\alpha\na'}C_{\\alpha\\beta}\\vartheta^{\\beta b'}\\,\\delta_{a'b'}\\,,$\n one can further show that\n\\begin{equation}\n({\\mathcal M}^2_{_{24}}D_{_{24}}\\vartheta)^{\\alpha a'}\n=\\frac{6i}{R^2}(e^b\\pm\\frac{R}{2}\\omega^{b3})(\\gamma_b\\vartheta)^{\\alpha\na'}\\,\\vartheta\\vartheta,\n\\end{equation}\nwhere the covariant derivative $D_{24}$, defined in\n(\\ref{D24}), becomes\n\\begin{equation}\nD_{24}\\vartheta=\\mathcal P_6(d\n+\\frac{i}{R}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\gamma^5\\gamma_a\n\\mp\\frac{1}{R}\\,e^3\n+\\frac{i}{R}e^{a'}\\gamma_{a'}\n-\\frac{1}{4}\\omega^{ab}\\gamma_{ab}\n-\\frac{1}{4}\\omega^{a'b'}\\gamma_{a'b'})\\vartheta\\,.\n\\end{equation}\nThis gives\n\\begin{eqnarray}\n\\vartheta\\gamma^a(1+\\frac{1}{12}{\\mathcal M}^2_{_{24}})D_{_{24}}\\vartheta\n&=&\n\\vartheta\\gamma^aD_{24}\\vartheta+\\frac{i}{2R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2\\,.\n\\end{eqnarray}\nUsing the above expressions one finds that the form of the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ geometrical objects (\\ref{cartan24})\nsimplify to\n\\begin{equation}\n\\begin{aligned}\nE^a&=e^a(x)+i\\vartheta\\gamma^aD_{24}\\vartheta-\\frac{1}{2R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2,\n\\\\\nE^3&=e^3(x),\\\\\nE^{a'}&=e^{a'}(y)-\\frac{1}{R}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta\\gamma^{a'}\\gamma_a\\vartheta\n\\\\\nE^{\\alpha a'}&=\n\\left(D_{_{24}}\\vartheta\\right)^{\\alpha a'}\n+\\frac{i}{R^2}(e^b\\pm\\frac{R}{2}\\omega^{b3})(\\gamma_b\\vartheta)^{\\alpha\na'}\\,\\vartheta\\vartheta,\n\\\\\n\\Omega^{ab}&=\\omega^{ab}(x)+\\frac{2i}{R^2}(e^c\\pm\\frac{R}{2}\\omega^{c3})\\vartheta\\gamma^{ab}{}_c\\vartheta,\n\\\\\n\\Omega^{a3}&=\\omega^{a3}(x)\n\\mp\\frac{2i}{R}\\vartheta\\gamma^aD_{24}\\vartheta\n\\pm\\frac{1}{R^3}(e^a\\pm\\frac{R}{2}\\omega^{a3})(\\vartheta\\vartheta)^2,\n\\\\\n\\Omega^{a'b'}&=\\omega^{a'b'}(y)-\\frac{i}{R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta(\\gamma^{a'b'}-iJ^{a'b'}\\gamma^7)\\gamma_a\\vartheta,\n\\\\\nA&=A(y)-\\frac{1}{R^2}(e^a\\pm\\frac{R}{2}\\omega^{a3})\\vartheta\\gamma^7\\gamma_a\\vartheta\n\\,,\n\\end{aligned}\n\\end{equation}\nand in particular\n\\begin{equation}\nE^a\\pm\\frac{R}{2}\\Omega^{a3}=e^a(x)\\pm\\frac{R}{2}\\omega^{a3}(x)\\,.\n\\end{equation}\nThus, in the chosen $\\kappa$--symmetry gauge the\n$OSp(6|4)\/U(3)\\times SO(1,3)$ supercoset geometry depends on the\nfermionic coordinates only up to the 4th power.\n\nNote that in all the above expressions the components $e^a(x)$\n$(a=0,1,2)$ and ${R\/2}\\,\\omega^{a3}(x)$ of the $AdS_4$ vielbein and\nconnection appear only in the combination $e^a(x)\\pm\n{R\/2}\\,\\omega^{a3}(x)$. This combination has a very clear\ngeometrical meaning. In the case, when the indices $a=0,1,2$ label\nthe directions of the 3d Minkowski slice of the $AdS_4$, $e^a(x)\\pm\n{R\/2}\\,\\omega^{a3}(x)$ corresponds to the generator\n$\\Pi_a=P_a\\mp{1\/2}\\,M_{a3}$ of the Poincar\\'e translations\n($[\\Pi_a,\\Pi_b]=0$) along the 3d Minkowski boundary which is the\nlinear combination of boosts and Lorentz rotations in $AdS_4$ (see\n\\cite{Pasti:1998tc} for more details). More precisely, $e^a(x)-\n{R\/2}\\,\\omega^{a3}(x)$ corresponds to the Poincar\\'e translation,\nwhile $e^a(x)+{R\/2}\\,\\omega^{a3}(x)$ corresponds to the conformal\nboosts in $M_3$, or vice versa, depending on the orientation.\n\nWhen the $AdS_4$ metric is chosen in the form (\\ref{ads4metric11})\nthe vielbein $e^a(x)$ and the connection $\\omega^{a3}(x)$ are\nproportional to each other, namely,\n\\begin{equation}\\label{eaoa3}\ne^a=-{R\\over 2}\\,\\omega^{a3}.\n\\end{equation}\nActually, this relation can be imposed for any\nform of the metric by performing an appropriate $SO(1,3)$\ntransformation of the $AdS_4$ vielbein and connection.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}