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+{"text":"\\section{Introduction}\n\\label{intro}\n\nAnti-de Sitter space is one of the Lorentz space forms which has rich geometric properties.\nIt is defined as a pseudo-sphere with negative curvature in semi-Euclidean space with index 2 which admits the biggest symmetry in  Riemannian or Lorentz space forms.\nAnti-de Sitter space can be naturally considered as a Lorentzian version (generalization) of Hyperbolic space. Recently we discovered interesting geometric properties of submanifolds in Hyperbolic space as an\napplication of the theory of Legendrian\/Lagrangian singularities\\cite{Izu2,Izu11,IPT04,IPRT05}.\nTherefore anti-de Sitter space might have rich geometric properties comparing with Hyperbolic space.\nThis is one of the motivations for the investigation of submanifolds in anti-de Sitter space from a mathematical viewpoint.\n\\par\nOn the other hand, anti-de Sitter space plays important roles in theoretical physics such as the theory of general relativity, the \nstring theory and the brane world scenario etc \\cite{BR02,M98,RS99,W98}.\nIt is well known that Lorentzian space forms are classified into three types depending on the value of the scalar curvature.\nOne of them is Lorentz-Minkowski space which has zero curvature.\nThe Lorentz space form with positive curvature is de Sitter space.\nAnti-de Sitter space is a Lorentzian space form with negative curvature.\nRecently, submanifolds in Lorentz-Minkowski space or de Sitter space have been well investigated (cf., \\cite{Izu3,Izu4,IzuSM,IzuAG,Kosso1,Kosso2}).\nHowever, there are not so many results on submanifolds in anti-de Sitter space, in particular  from the\nviewpoint of singularity theory.\nThe lightlike hypersurfaces (i.e. the light-sheets in physics) are important objects in theoretical physics because they provide good models for  different types of horizons \\cite{Ch, MTW}.\nA lightlike hypersurface is generally a ruled hypersurface along a spacelike submanifold with codimension two whose rulings are lightlike geodesics. In this paper we consider lightlike hypersurfaces along spacelike submanifolds with general codimension in anti-de Sitter space.\nMoreover, the lightlike hypersurface in anti-de Sitter $5$-space is a very important subject in the brane world scenario\\cite{KR01,BR02,RS99}.\n\\par\nIn the meantime, tools in the theory of singularities\nhave proven to be useful \ndescription of geometrical properties of submanifolds immersed in\ndifferent ambient spaces, from both the local and global viewpoint\n\\cite{Izu2,Izu3,IPRT05,IzuSM,IzuAG,Little}.\nThe natural connection between geometry and singularities relies on\nthe basic fact that the contacts of a submanifold with the models of\nthe ambient space can be described by means of the analysis of the\nsingularities of appropriate families of contact functions, or\nequivalently, of their associated  Legendrian maps\n(\\cite{Arnold1,Montaldi,Zak}).\nWhen working in a Lorentz space form, the properties associated to the\ncontacts of a given submanifold with  lightcones have a\nspecial relevance. \nIn \\cite{Izu3,IzuSM, Kasedou2}, it was constructed a framework for the study of\nspacelike submanifolds with codimension two in Lorentz-Minkowski space or de Sitter space and discovered a Lorentz invariant \nconcerning their contacts with models related to lightlike hyperplanes. \nThe geometry described in this framework is called the {\\it lightlike geometry} of spacelike submanifolds\nwith codimension two.\nBy using the invariants of lightlike geometry,\nthe singularities of lightlike hypersurfaces along spacelike submanifolds with codimension two in Lorentz-Minkowski space \nor de Sitter space\nwere investigated in \\cite{Izu4,Izu08,Kasedou2}.\nHowever, the situation is rather complicated for the general codimensional case.\nThe main difference from the Euclidean space (or, Hyperbolic space) case is the fiber of the canal hypersurface of a spacelike submanifold is neither connected nor\ncompact. \nIn order to avoid the above difficulty, we arbitrarily choose a timelike future directed unit normal vector field\nalong the spacelike submanifold which always exists for an orientable manifold (cf., \\S 3).\nThen we construct the unit spherical normal bundle relative to the above timelike unit normal vector field, which can be considered as a codimension two spacelike canal submanifold of\nthe ambient space.\nTherefore, we can apply the idea of the lightlike geometry of spacelike submanifolds of\nthe ambient space with codimension two. \nIn this paper we apply the idea of this framework and the theory of Legendrian singularities to\ninvestigate the singularities of lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension.\n\\par\nIn \\S 3 we construct the framework of the lightlike geometry of spacelike submanifolds with\ngeneral codimension analogous to \\cite{IPRT05}.\nThe notion of lightlike hypersurfaces along spacelike submanifolds is introduced and  the basic properties are investigated in \\S 4.\n The notion of the anti-de Sitter height functions families\nis useful for the study of lightlike hypersurfaces (cf., \\S 4).\nThe critical value set of the lightlike hypersurface along a spacelike submanifold is called\nthe {\\it lightlike focal set} of the submanifold.\nIn \\S 5 we show that the lightlike focal set of a spacelike submanifold is a point\nif and only if the lightlike hypersurface along the submanifold is a subset of a lightcone (Proposition 5.1). Therefore, an anti-de Sitter lightcone is \na model hypersurface of lightlike hypersurfaces.\nThe geometric meaning of the singularities of lightlike hypersurface is described by\nthe theory of contact of submanifolds with model hypersurfaces.\nMoreover, as an application of the theory of Legendrian singularities, we show that\ntwo lightlike hypersurfaces are locally diffeomorphic if and only if the types of the\ncontact of spacelike submanifolds with lightcones are the same in the sense of Montaldi\\cite{Montaldi} under some generic conditions (Theorem 5.5).\nIn \\S 6 we describe the case of codimension two as a special case.\nWe describe the detailed properties of lightlike focal sets of spacelike surfaces in \nanti-de Sitter $4$-space.\nWe also investigate spacelike curves in anti-de Sitter $4$-space as\nthe simplest case of a higher codimension in \\S 7.\n \\par\nWe shall assume throughout the whole paper that all the\nmaps and manifolds are $C^{\\infty}$ unless the contrary is explicitly stated.\n\n\\section{Basic facts and notations on semi-Euclidean space with index 2}\n\\label{sec:1}\n\\par\nIn this section we prepare the basic notions on semi-Euclidean\n(n+2)-space with index 2. For details of semi-Euclidean geometry,\nsee \\cite{Oneil}.\nLet ${\\mathbb R}^{n+2}=\\{(x_{-1}, x_0, x_1,\\cdots,x_{n})|x_i \\in {\\mathbb R}\\\n(i=-1, 0,\\cdots,n)\\ \\}$ be an (n+2)-dimensional vector space. For\nany vectors $\\mbox{\\boldmath $x$}=(x_{-1}, x_0, x_1,\\cdots,x_{n})$ and $\\mbox{\\boldmath $y$}\n=(y_{-1}, y_0, y_1,\\cdots,y_{n})$ in ${\\mathbb R}^{n+2},$ the {\\it\npseudo scalar product \\\/} of $\\mbox{\\boldmath $x$}$ and $\\mbox{\\boldmath $y$}$ is defined to be\n$\\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $y$}\\rangle =-x_{-1} y_{-1}-x_0 y_0 +\\sum_{i=1}^{n}x_i\ny_i$. We call $(\\Bbb R^{n+2}, \\langle ,\\rangle )$ {\\it\nsemi-Euclidean\\\/} (n+2)-{\\it space with index 2\\\/} and write $\\Bbb\nR^{n+2}_2$ instead of $(\\Bbb R^{n+2},\\langle ,\\rangle )$.\nWe say that a non-zero vector $\\mbox{\\boldmath $x$}$ in $\\Bbb R^{n+2}_2$ is {\\it\nspacelike\\\/}, {\\it null\\\/} or {\\it timelike\\\/} if\n$\\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $x$}\\rangle>0,\\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $x$}\\rangle =0$ or\n$\\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $x$}\\rangle <0$, respectively. The norm of the vector $\\mbox{\\boldmath $x$}\n\\in \\Bbb R^{n+2}_2$ is defined to be $\\|\\mbox{\\boldmath $x$}\\|=\\sqrt{|\\langle\\mbox{\\boldmath $x$},\n\\mbox{\\boldmath $x$}\\rangle|}$.\nWe define the signature of $\\mbox{\\boldmath $x$}$ by\n\\[{\\rm sign} (\\mbox{\\boldmath $x$})=\\left\\{\n  \\begin{array}{ccc}\n  1\\qquad\\quad $\\mbox{\\boldmath $x$}$\\ \\mbox{is\\ spacelike,}\\\\\n\n  \\mbox{}0\\qquad\\quad \\mbox{\\boldmath $x$}\\ \\mbox{is\\ null,} \\hspace*{\\fill}\\\\\n\n  -1\\qquad \\mbox{\\boldmath $x$}\\ \\mbox{is\\ timelike.}\n  \\end{array}\\right.\n  \\]\n For a non-zero vector $\\mbox{\\boldmath $n$}\\in \\Bbb R^{n+2}_2$ and a real number $c$, we define the\n {\\it hyperplane with pseudo-normal \\\/}$\\mbox{\\boldmath $n$}$ by\n$$\nHP(\\mbox{\\boldmath $n$},c)=\\{\\mbox{\\boldmath $x$} \\in \\Bbb R^{n+2}_2 |\\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $n$}\\rangle=c\\}.\n$$\nWe call $HP(\\mbox{\\boldmath $n$},c)$ a {\\it Lorentz hyperplane\\\/}, a {\\it\nsemi-Euclidean hyperplane with index 2\\\/} or a {\\it null\nhyperplane\\\/} if $\\mbox{\\boldmath $n$}$ is {\\it timelike, spacelike or null\n\\\/}respectively.\n\nWe now define an {\\it anti-de Sitter $(n+1)$-space \\\/} (briefly, {\\it AdS\n$(n+1)$-space\\\/}) by\n$$\nAdS^{n+1}=\\{\\mbox{\\boldmath $x$}\\in \\Bbb R_2^{n+2}\\ |\\ \\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $x$}\\rangle =-1\\}=H^{n+1}_1,\n$$\na {\\it unit pseudo $(n+1)$-sphere with index 2\\\/} by\n$$\nS^{n+1}_2=\\{\\mbox{\\boldmath $x$}\\in {\\mathbb R}_2^{n+2}\\ |\\ \\langle\\mbox{\\boldmath $x$},\\mbox{\\boldmath $x$}\\rangle =1\\}\n$$\nand a {\\it {\\rm (}closed{\\rm )} nullcone\\\/} with vertex $\\mbox{\\boldmath $a$}$ by\\\\\n$$\\Lambda_{a}^{n+1}=\\{\\mbox{\\boldmath $x$}\\in {\\mathbb R}_{2}^{n+2}|\\langle\\mbox{\\boldmath $x$}-\\mbox{\\boldmath $a$}, \\mbox{\\boldmath $x$}-\\mbox{\\boldmath $a$}\\rangle=0\\}.\n$$\nIn particular we denote that $\\Lambda ^*=\\Lambda ^{n+1}_0\\setminus \\{\\mbox{\\boldmath $0$}\\}$ and also call it\nthe {\\it {\\rm (}open{\\rm )} nullcone\\\/}. Our main subject in this paper is $AdS^{n+1}$.\nSince there are timelike closed curves in $AdS^{n+1}$, the causality of $AdS^{n+1}$ is violated.\nIn order to avoid such a situation, it is usually considered the universal covering\nspace $\\widetilde{AdS}^{n+1}$ of $AdS^{n+1}$ in physics which is called the {\\it universal Anti de Sitter space\\\/}.\nWe remark that the local structure of these spaces are the same.\n\\par\nFor any $\\mbox{\\boldmath $x$}_1,\\cdots, \\mbox{\\boldmath $x$}_{n} \\in {\\mathbb R}^{n+2}_2$. We define a\nvector $\\mbox{\\boldmath $x$}_1\\wedge \\cdots\\wedge \\mbox{\\boldmath $x$}_n$ by\n$$\n\\mbox{\\boldmath $x$}_1\\wedge\\cdots\\wedge \\mbox{\\boldmath $x$}_n= \\vmatrix\n-\\mbox{\\boldmath $e$}_{-1}&-\\mbox{\\boldmath $e$}_0&\\mbox{\\boldmath $e$}_1&\\cdots&\\mbox{\\boldmath $e$}_{n}\\vspace{2mm}\\\\\nx^1_{-1}&x^1_0&x^1_1&\\cdots&x^1_{n}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\nx^{n}_{-1}&x^{n}_{0}&x^{n}_1&\\cdots&x^{n}_{n}\n\\endvmatrix,\n$$\nwhere $\\{\\mbox{\\boldmath $e$}_{-1}, \\mbox{\\boldmath $e$}_0, \\mbox{\\boldmath $e$}_1,\\cdots,\\mbox{\\boldmath $e$}_{n}\\}$ is the canonical\nbasis of $\\Bbb R^{n+2}_2$ and $\\mbox{\\boldmath $x$}_i=(x^i_{-1}, x^i_0,\nx^i_1,\\cdots,x^i_{n})$. We can easily check that\n$$\n\\langle\\mbox{\\boldmath $x$},\\ \\mbox{\\boldmath $x$}_1\\wedge\\cdots\\wedge\n\\mbox{\\boldmath $x$}_{n}\\rangle=\\textrm{det}(\\mbox{\\boldmath $x$}, \\mbox{\\boldmath $x$}_1,\\cdots,\\mbox{\\boldmath $x$}_{n}),\n$$\nso that $\\mbox{\\boldmath $x$}_1\\wedge\\cdots\\wedge \\mbox{\\boldmath $x$}_{n}$ is pseudo-orthogonal to\nany $\\mbox{\\boldmath $x$}_i$\\ (for\\ $i=1,\\cdots,n$).\n\n\\section{Spacelike submanifolds in anti-de Sitter space}\n\\label{sec:2}\n We introduce in this section the basic geometrical\ntools for the study of spacelike submanifolds in\nthe anti-de Sitter $(n+1)$-space. \n\\par\n Consider the orientation of ${\\mathbb R}^{n+2}_2$\nprovided by the condition that $\\textrm{det}(\\mbox{\\boldmath $e$}_{-1}, \\mbox{\\boldmath $e$}_0,\\mbox{\\boldmath $e$} _1,\\cdots,\\mbox{\\boldmath $e$}_{n})>0.$\nThis orientation induces the orientation of $x_{-1}x_0$-plane,\nso that it gives a time\norientation on $AdS^{n+1}$. \nIf we consider the universal Anti de Sitter space $\\widetilde{AdS}^{n+1},$ we can determine the\nfuture direction.\n\\par\nWe consider a spacelike embedding $\\mbox{\\boldmath $X$}:U\\rightarrow AdS^{n+1}$ from an open subset $U\\subset {\\mathbb R}^s$\nwith $s+k=n+1.$ We write\n$M=\\mbox{\\boldmath $X$}(U)$ and identify $M$ and $U$ through the embedding $\\mbox{\\boldmath $X$}.$ We\nsay that $\\mbox{\\boldmath $X$}$ is {\\it spacelike} if the tangent space $T_p  M$\nconsists only spacelike vectors (i.e. spacelike subspace) for any point $p\\in M$. In this case, the pseudo-normal\nspace $N_p(M)$ in ${\\mathbb R}^{n+2}_2$ is a $k+1$-dimensional semi-Euclidean space with index $2$\n(cf. \\cite{Oneil}). We write $N(M)$ as the pseudo-normal bundle in ${\\mathbb R}^{n+2}_2$\nover $M.$ \nOn the pseudo-normal space $N_p(M),$ we have two kinds of pseudo spheres:\n\\begin{eqnarray*}\nN_p(M;-1)& = & \\{\\mbox{\\boldmath $v$}\\in N_p(M)\\ |\\ \\langle \\mbox{\\boldmath $v$},\\mbox{\\boldmath $v$}\\rangle =-1\\ \\} \\\\\nN_p(M;1)&= & \\{\\mbox{\\boldmath $v$}\\in N_p(M)\\ |\\ \\langle \\mbox{\\boldmath $v$},\\mbox{\\boldmath $v$}\\rangle =1\\ \\},\n\\end{eqnarray*}\nso that we have two unit spherical normal bundles over $M$:\n\\[\nN(M;-1)=\\bigcup _{p\\in M} N_p(M;-1)\\ \\mbox{and}\\  N(M;1)=\\bigcup _{p\\in M} N_p(M;1).\n\\]\nThen we have the Whitney sum decomposition\n\\[\nT{\\mathbb R}^{n+2}_2|M=TM\\oplus N(M).\n\\]\nBy definition $\\mbox{\\boldmath $X$}(u)$ is one of the timelike unit normal vectors of $M$ at $p=\\mbox{\\boldmath $X$}(u),$ so that\n$\\mbox{\\boldmath $X$} \\in N_p(M).$ Since $AdS^{n+1}$ is time oriented, we can arbitrarily choose an\nadopted unit timelike normal section $\\mbox{\\boldmath $n$} ^T(u)\\in N_p(M)$ pseudo-orthogonal to\n$\\mbox{\\boldmath $X$}(u)$ even globally. Here, we say that $\\mbox{\\boldmath $n$}^T$ is {\\it adopted}\nif \n\\[\n\\textrm{det}(\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $n$}^T(u),\\mbox{\\boldmath $e$} _1,\\dots ,\\mbox{\\boldmath $e$} _{n})>0.\n\\]\nTherefore we have the pseudo-orthonormal complement\n$(\\langle \\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $n$} ^T(u)\\rangle _{\\mathbb R})^\\perp$ in $N_p(M)$\nwhich is a $(k-1)$-dimensional subspace of $N_p(M).$\nWe define a $(k-2)$-dimensional spacelike unit sphere in $N_p(M)$ by\n\\[\nN_1(M)_p[\\mbox{\\boldmath $n$} ^T]=\\{\\mbox{\\boldmath $\\xi$} \\in N_p(M;1)\\ |\\ \\langle \\mbox{\\boldmath $\\xi$}, \\mbox{\\boldmath $n$} ^T(p)\\rangle =\\langle \\mbox{\\boldmath $\\xi$},\\mbox{\\boldmath $X$}(u)\\rangle=0,p=\\mbox{\\boldmath $X$}(u)\\ \\}.\n\\]\nThen we have a {\\it spacelike unit $k-2$-spherical bundle over $M$ with respect to $\\mbox{\\boldmath $n$} ^T$} defined by\n\\[\nN_1(M)[\\mbox{\\boldmath $n$} ^T]=\\bigcup _{p\\in M} N_1(M)_p[\\mbox{\\boldmath $n$} ^T].\n\\]\nSince we have\n$T_{(p,\\xi)}N_1(M)[\\mbox{\\boldmath $n$}^T]=T_pM\\times T_\\xi N_1(M)_p[\\mbox{\\boldmath $n$} ^T],$\nwe have the canonical Riemannian metric on $N_1(M)[\\mbox{\\boldmath $n$}^T].$\nWe denote the Riemannian metric on $N_1(M)[\\mbox{\\boldmath $n$}^T]$ by $(G_{ij}(p,\\mbox{\\boldmath $\\xi$}))_{1\\leqslant i,j\\leqslant n-1}.$\nWe now arbitrarily choose (at least locally) a unit\nspacelike normal vector field $\\mbox{\\boldmath $n$}^S$ with $\\mbox{\\boldmath $n$}^S(u)\\in N_1(M)_p[\\mbox{\\boldmath $n$}^T]$, where $p=\\mbox{\\boldmath $X$}(u).$\nWe call $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$ an {\\it adopted pair of normal vector fields\\\/} along $M.$\nClearly, the vectors\n$\\mbox{\\boldmath $n$}^T (u)\\pm \\mbox{\\boldmath $n$}^S(u)$ are null. \nWe define a mapping\n\\[\n\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S):U\\longrightarrow \\Lambda^*\n\\]\nby $\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u)=\\mbox{\\boldmath $n$}^T(u)+\\mbox{\\boldmath $n$}^S(u).$\nWe call it the {\\it nullcone Gauss image} of $M=\\mbox{\\boldmath $X$}(U)$ with respect to\n$(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S).$\nWith the identification of $M$ and $U$ through $\\mbox{\\boldmath $X$},$ we have the\nlinear mapping provided by the derivative of the nullcone Gauss image $\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$ at each point $p\\in M$ as follows:\n\\[\nd_p\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S):T_pM\\longrightarrow T_p{\\mathbb R}^{n+1}_1= T_pM\\oplus N_p(M).\n\\]\nConsider the orthogonal projections $\\pi ^t:T_pM\\oplus\nN_p(M)\\rightarrow T_p(M).$ We define\n\\[\nd_p\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)^t=\\pi ^t\\circ d_p(\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S).\n\\]\nWe call the\nlinear transformation $S_{p}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)=- d_p\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)^t$ the {\\it $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$-shape\noperator} of $M=\\mbox{\\boldmath $X$} (U)$ at $p=\\mbox{\\boldmath $X$} (u).$ \nLet $\\{\\kappa\n_{i}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(p)\\}_ {i=1}^s$ be the eigenvalues of $S_{p}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$, which are called the {\\it nullcone\nprincipal curvatures  with respect to $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S) $\\\/} at $p=\\mbox{\\boldmath $X$}(u)$.\nThen the {\\it nullcone Gauss-Kronecker curvature with respect to\n$(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$\\\/} at $p=\\mbox{\\boldmath $X$} (u)$ is defined by\n\\[\nK_N(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(p)={\\rm det} S_{p}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S).\n\\]\nWe say that a point $p=\\mbox{\\boldmath $X$} (u)$ is an {\\it $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$-umbilical\npoint} if \n\\[\nS_{p}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)=\\kappa (\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(p) 1_{T_{p}M}.\n\\]\nWe say that $M=\\mbox{\\boldmath $X$} (U)$ is {\\it totally\n$(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$-umbilical} if all points on $M$ are\n$(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$-umbilical.\nMoreover, $M=\\mbox{\\boldmath $X$}(U)$ is said to be {\\it totally nullcone umbilical} if\nit is totally $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$-umbilical for any adopted pair $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S).$\n\\par\nWe deduce now the nullcone Weingarten formula. Since $\\mbox{\\boldmath $X$} _{u_i}$\n$(i=1,\\dots s)$ are spacelike vectors, we have a Riemannian metric\n(the {\\it first fundamental form \\\/}) on $M=\\mbox{\\boldmath $X$} (U)$\ndefined by $ds^2 =\\sum _{i=1}^{s} g_{ij}du_idu_j$,  where\n$g_{ij}(u) =\\langle \\mbox{\\boldmath $X$} _{u_i}(u ),\\mbox{\\boldmath $X$} _{u_j}(u)\\rangle$ for any\n$u\\in U.$ We also have the {\\it nullcone second fundamental invariant\nwith respect to the normal vector field $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$} ^S) $\\\/} defined\nby $h _{ij}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )(u)=\\langle -(\\mbox{\\boldmath $n$}^T +\\mbox{\\boldmath $n$}^S)\n_{u_i}(u),\\mbox{\\boldmath $X$}_{u_j}(u)\\rangle$ for any $u\\in U.$\nBy similar arguments to those in the proof of \\cite[Proposition 3.2]{IzuSM}, we have \nthe following proposition.\n\\begin{Pro}\nWe choose a pseudo-orthonormal frame $\\{\\mbox{\\boldmath $X$}, \\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S_1,\\dots ,\\mbox{\\boldmath $n$}^S_{k-1}\\}$ of $N(M)$ with $\\mbox{\\boldmath $n$}^S_{k-1}=\\mbox{\\boldmath $n$}^S.$ Then we have the following nullcone Weingarten formula {\\rm :}\n\\vskip1.5pt\n\\par\\noindent\n{\\rm (a)} $\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)_{u_i}=\\langle \\mbox{\\boldmath $n$} ^T_{u_i},\\mbox{\\boldmath $n$} ^S\\rangle(\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S)+\\sum _{\\ell =1}^{k-2}\\langle (\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S)_{u_i},\\mbox{\\boldmath $n$}^S_\\ell \\rangle\\mbox{\\boldmath $n$}^S_\\ell -\\sum_{j=1}^{s}\nh_i^j(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )\\mbox{\\boldmath $X$} _{u_j}$\n\\par\\noindent\n{\\rm (b)} $\n\\pi ^t\\circ \\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)_{u_i}=-\\sum_{j=1}^{s}\nh_i^j(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )\\mbox{\\boldmath $X$} _{u_j}.\n$\n\\smallskip\n\\par\\noindent\nHere $\\displaystyle{\\left(h_i^j(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )\\right)=\\left(h_{ik}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)\\right)\\left(g^{kj}\\right)}$\nand $\\displaystyle{\\left( g^{kj}\\right)=\\left(g_{kj}\\right)^{-1}}.$\n\\end{Pro}\n\\par\nAs a consequence of the above proposition, we have an explicit\nexpression of the nullcone curvature\nby\n$$\nK_N (\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )=\\frac{\\displaystyle{{\\rm det}\\left(h_{ij}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S )\\right)}}\n{\\displaystyle{{\\rm det}\\left(g_{\\alpha \\beta}\\right)}}.\n$$\nSince $\\langle -(\\mbox{\\boldmath $n$}^T +\\mbox{\\boldmath $n$}^S )(u),\\mbox{\\boldmath $X$} _{u_j}(u)\\rangle =0,$ we have\n$h_{ij}(\\mbox{\\boldmath $n$} ^T,\\mbox{\\boldmath $n$}^S)(u)=\\langle \\mbox{\\boldmath $n$}^T (u)+\\mbox{\\boldmath $n$}^S (u),\\mbox{\\boldmath $X$}\n_{u_iu_j}(u)\\rangle.$ Therefore the nullcone second fundamental\ninvariant at a point $p_0=\\mbox{\\boldmath $X$} (u_0)$ depends only on the values \n$\\mbox{\\boldmath $n$}^T (u_0)+\\mbox{\\boldmath $n$}^S (u_0)$ and $\\mbox{\\boldmath $X$} _{u_iu_j}(u_0)$, respectively.\nThus, the nullcone curvatures also depend only on\n$\\mbox{\\boldmath $n$}^T (u_0)+\\mbox{\\boldmath $n$}^S (u_0)$, $\\mbox{\\boldmath $X$}_{u_i}(u_0)$  and $\\mbox{\\boldmath $X$}\n_{u_iu_j}(u_0)$, which are independent of the derivations of the vector fields \n$\\mbox{\\boldmath $n$}^T$ and $\\mbox{\\boldmath $n$}^S .$ We write $\\kappa _i(\\mbox{\\boldmath $n$}^T_0,\\mbox{\\boldmath $n$}^S_0)(p_0)$ $(i=1,\\dots ,s)$\nand $K_N (\\mbox{\\boldmath $n$}\n^T_0,\\mbox{\\boldmath $n$}^S_0)(u_0)$ as the nullcone curvatures at $p_0=\\mbox{\\boldmath $X$} (u_0)$\nwith respect to $(\\mbox{\\boldmath $n$} ^T_0,\\mbox{\\boldmath $n$}^S_0)=(\\mbox{\\boldmath $n$}^T (u_0),\\mbox{\\boldmath $n$}^S(u_0)).$ We\nmight also say that a point $p_0=\\mbox{\\boldmath $X$} (u_0)$ is \n{\\it $(\\mbox{\\boldmath $n$}^T_0,\\mbox{\\boldmath $n$}^S_0)$-umbilical\\\/} because the nullcone $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S) $-shape\noperator at $p_0$ depends only on the normal vectors $(\\mbox{\\boldmath $n$}\n^T_0,\\mbox{\\boldmath $n$}^S_0).$\nSo we denote that $h_{ij}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $\\xi$})(u_0)=h_{ij}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)$ and \n $K_N(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $\\xi$})(p_0)=K_N(\\mbox{\\boldmath $n$}^T_0,\\mbox{\\boldmath $n$}^S_0)(p_0)$,\nwhere $\\mbox{\\boldmath $\\xi$} =\\mbox{\\boldmath $n$}^S(u_0)$ for some local extension $\\mbox{\\boldmath $n$}^T(u)$ of $\\mbox{\\boldmath $\\xi$}.$\nAnalogously, we say that a point $p_0=\\mbox{\\boldmath $X$} (u_0)$ is an {\\it $(\\mbox{\\boldmath $n$}\n^T_0,\\mbox{\\boldmath $n$}^S_0)$-parabolic point \\\/} of $\\mbox{\\boldmath $X$} :U\\longrightarrow {\\mathbb R}^{n+1}_1$ if\n$K_N (\\mbox{\\boldmath $n$} ^T_0,\\mbox{\\boldmath $n$}^S_0)(u_0)=0.$ We also say that a point $p_0=\\mbox{\\boldmath $X$}\n(u_0)$ is a {\\it $(\\mbox{\\boldmath $n$} ^T_0,\\mbox{\\boldmath $n$}^S_0)$-flat point \\\/} if it is an\n$(\\mbox{\\boldmath $n$} ^T_0,\\mbox{\\boldmath $n$}^S_0)$-umbilical point and $K_N(\\mbox{\\boldmath $n$}^T\n_0,\\mbox{\\boldmath $n$}^S_0)(u_0)=0.$\n\\par\nOn the other hand, we define a map\n$\n\\mathbb{NG}(\\mbox{\\boldmath $n$}^T):N_1(M)[\\mbox{\\boldmath $n$}^T]\\longrightarrow \\Lambda ^*\n$\nby\n$\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$})=\\mbox{\\boldmath $n$}^T(u)+\\mbox{\\boldmath $\\xi$},$\nwhich we call the {\\it nullcone Gauss image} of $N_1(M)[\\mbox{\\boldmath $n$}^T].$\nThis map leads us to the notions of curvatures.\nLet $T_{(p,\\xi)}N_1(M)[\\mbox{\\boldmath $n$}^T]$ be the tangent space of $N_1(M)[\\mbox{\\boldmath $n$}^T]$ at $(p,\\mbox{\\boldmath $\\xi$}).$\nUnder the canonical identification $(\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)^*T{\\mathbb R}^{n+2}_2)_{(p,\\mbox{\\scriptsize \\boldmath$\\xi$})}\n=T_{(\\mbox{\\scriptsize \\boldmath$n$}^T(p)+\\mbox{\\scriptsize \\boldmath$\\xi$})}{\\mathbb R}^{n+2}_2\\equiv T_p{\\mathbb R}^{n+2}_2,$\nwe have\n\\[\nT_{(p,\\mbox{\\scriptsize \\boldmath$\\xi$})}N_1(M)[\\mbox{\\boldmath $n$}^T]=T_pM\\oplus T_\\xi S^{k-2}\\subset T_pM\\oplus N_p(M)=T_p{\\mathbb R}^{n+2}_2,\n\\]  \nwhere $T_\\xi S^{k-2}\\subset T_\\xi N_p(M)\\equiv N_p(M)$ and $p=\\mbox{\\boldmath $X$}(u).$\nLet \n\\[\n\\Pi ^t :\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)^*T{\\mathbb R}^{n+2}_2=TN_1(M)[\\mbox{\\boldmath $n$}^T]\\oplus {\\mathbb R}^{k+1}\n\\longrightarrow TN_1(M)[\\mbox{\\boldmath $n$}^T]\n\\]\nbe the canonical projection.\nThen\nwe have a linear transformation\n\\[\nS_N (\\mbox{\\boldmath $n$}^T)_{(p,\\mbox{\\scriptsize \\boldmath$\\xi$})}=-\\Pi^t_{\\mathbb{LG}(n^T)(p,\\xi)}\\circ d_{(p,\\xi)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)\n: T_{(p,\\xi)}N_1(M)[\\mbox{\\boldmath $n$}^T]\\longrightarrow T_{(p,\\xi)}N_1(M)[\\mbox{\\boldmath $n$}^T],\n\\]\nwhich is called the {\\it nullcone shape operator} of $N_1(M)[\\mbox{\\boldmath $n$}^T]$ at $(p,\\mbox{\\boldmath $\\xi$}).$ \nLet $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$ be the eigenvalues of $S_N(\\mbox{\\boldmath $n$}^T) _{(p,\\mbox{\\scriptsize \\boldmath$\\xi$})}$, $(i=1,\\dots ,n-1)$. \nHere, we denote $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$, $(i=1,\\dots ,s)$ as the eigenvalues belonging to\nthe eigenvectors on $T_pM$\nand $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$, $(i=s+1,\\dots n-1)$ as the eigenvalues belonging to the eigenvectors on \nthe tangent space of the fiber  \nof $N_1(M)[\\mbox{\\boldmath $n$}^T].$  \nThen we have the following proposition.\n\\begin{Pro}\nWe choose a {\\rm (}local\\\/{\\rm )} pseudo-orthonormal frame $\\{\\mbox{\\boldmath $X$}, \\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S_1,\\dots ,\\mbox{\\boldmath $n$}^S_{k-1}\\}$ of $N(M)$ with $\\mbox{\\boldmath $n$}^S_{k-1}=\\mbox{\\boldmath $n$}^S.$\nFor $p_0=\\mbox{\\boldmath $X$}(u_0)$ and $\\mbox{\\boldmath $\\xi$}_0=\\mbox{\\boldmath $n$}^S(u_0),$ we have\n$\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0)=\\kappa _i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)$, $(i=1,\\dots ,s)$ and\n$\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0)=-1$, $(i=s+1,\\dots n-1).$\n\\end{Pro}\n\\par\\noindent{\\it Proof. \\\/}\\ \nSince $\\{\\mbox{\\boldmath $X$}, \\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S_1,\\dots ,\\mbox{\\boldmath $n$}^S_{k-1}\\}$ is a pseudo-orthonormal frame of$N(M),$ we have $\\langle \\mbox{\\boldmath $X$}(u_0),\\mbox{\\boldmath $\\xi$}_0\\rangle =\n\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $\\xi$}_0\\rangle =\\langle \\mbox{\\boldmath $n$}^S_i(u_0),\\mbox{\\boldmath $\\xi$}_0\\rangle =0.$\nTherefore, we have $T_{\\mbox{\\scriptsize \\boldmath$\\xi$}}S^{k-2}=\\langle \\mbox{\\boldmath $n$}^S_1(u_0),\\dots ,\\mbox{\\boldmath $n$}^S_{k-2}(u_0)\\rangle .$\nBy this orthonormal basis of $T_{\\mbox{\\scriptsize \\boldmath$\\xi$}_0}S^{k-2},$\nthe canonical Riemannian metric $G_{ij}(p_0,\\mbox{\\boldmath $\\xi$}_0)$ is represented by\n\\[\n(G_{ij}(p_0,\\mbox{\\boldmath $\\xi$}))=\\left(\n\\begin{array}{cc}\ng_{ij}(p_0)  & 0 \\\\\n0 & I_{k-2}\n\\end{array}\n\\right) ,\n\\]\nwhere $g_{ij}(p_0)=\\langle \\mbox{\\boldmath $X$}_{u_i}(u_0), \\mbox{\\boldmath $X$}_{u_j}(u_0)\\rangle $.\n\\par\nOn the other hand, by Proposition 3.1, we have\n\\[\n-\\sum_{j=1}^s h^j_i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)\\mbox{\\boldmath $X$}_{u_j}=\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)_{u_i}(u_0)=\nd_{p_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)\\left(\\frac{\\partial}{\\partial u_i}\\right),\n\\]\nso that we have\n\\[\nS_N(\\mbox{\\boldmath $n$}^T)_{(p_0,\\mbox{\\boldmath $\\xi$}_0)}\\left(\\frac{\\partial}{\\partial u_i}\\right)=\\sum_{j=1}^s h^j_i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)\\mbox{\\boldmath $X$}_{u_j}.\n\\]\nTherefore, the representation matrix of $S_N(\\mbox{\\boldmath $n$}^T)_{(p_0,\\mbox{\\boldmath $\\xi$}_0)}$ with respect to the basis\n$$\n\\{\\mbox{\\boldmath $X$}_{u_1}(u_0),\\dots ,\\mbox{\\boldmath $X$}_{u_s}(u_0),\\mbox{\\boldmath $n$}^S_1(u_0),\\dots ,\\mbox{\\boldmath $n$}^S_{k-2}(u_0)\\}\n$$ of $T_{(p_0,\\mbox{\\boldmath $\\xi$}_0)}(N_1(M)[\\mbox{\\boldmath $n$}^T])$\nis of the form\n\\[\n\\left(\n\\begin{array}{cc}\nh^j_i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)  & * \\\\\n0 & -I_{k-2}\n\\end{array}\n\\right).\n\\]\nThus, the eigenvalues of this matrix are $\\lambda _i=\\kappa _i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)$, $(i=1,\\dots ,s)$ and\n$\\lambda _i=-1$ , $(i=s+1,\\dots ,n-1)$.\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\nWe call $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$, $(i=1,\\dots ,s)$\nthe {\\it nullcone principal curvatures} of $M$ with respect to $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $\\xi$})$ at $p\\in M.$\nThe {\\it nullcone Lipschitz-Killing curvature} of $N_1(M)[\\mbox{\\boldmath $n$}^T]$ at $(p,\\mbox{\\boldmath $\\xi$})$\nis defined to be $K_N (\\mbox{\\boldmath $n$}^T)(p,\\mbox{\\boldmath $\\xi$})=\\det S_N (\\mbox{\\boldmath $n$}^T)_{(p,\\xi)}.$\n\n\n\n\n\n\\section{Lightlike hypersurfaces in anti-de Sitter space}\n We define a hypersurface\n$$\n\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T):N_1(M)[\\mbox{\\boldmath $n$}^T]\\times {\\mathbb R}\\longrightarrow AdS^{n+1}\n$$\nby\n$$\n\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)((p,\\mbox{\\boldmath $\\xi$}),\\mu)=\\mbox{\\boldmath $X$}(u)+\\mu (\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $\\xi$})(u)=\\mbox{\\boldmath $X$}(u)+\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$}),\n$$\nwhere $p=\\mbox{\\boldmath $X$} (u),$ which is called the {\\it lightlike hypersurface\\\/} along $M$ relative to $\\mbox{\\boldmath $n$}^T.$\nIn general, a hypersurface $H\\subset AdS^{n+1}$ is called a {\\it lightlike hypersurface\\\/} if it is tangent to\nthe lightcone at any regular point.\nWe remark that $\\mathbb{NH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$ is a lightlike hypersurface.\n \\par\n We introduce the notion of height\nfunctions on spacelike submanifold, which is useful for the study of\nsingularities of lightlike hypersurfaces.\nWe define a family of functions \n$$H: M\\times AdS^{n+1}\\longrightarrow {\\mathbb R}$$\n on a spacelike submanifold  $M=\\mbox{\\boldmath $X$} (U)$ \n by\n$\n H(p,\\mbox{\\boldmath $\\lambda$})=H(u,\\mbox{\\boldmath $\\lambda$} )=\\langle \\mbox{\\boldmath $X$} (u) ,\\mbox{\\boldmath $\\lambda$}\\rangle +1,\n$\nwhere $p=\\mbox{\\boldmath $X$}(u).$\nWe call\n$H$ the {\\it anti-de Sitter height function\\\/} (briefly, {\\it AdS-height function\\\/}) on the spacelike submanifold\n$M.$\nFor any fixed $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1},$ we write $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p)=H(p,\\mbox{\\boldmath $\\lambda$} _0)$\nand have\nthe following proposition.\n\\par\n\\begin{Pro}\nLet $M$ be a spacelike submanifold \nand\n$H: M\\times(AdS^{n+1}\\setminus M)\\to{\\mathbb R}$\nthe AdS-height function on $M.$\nSuppose that $p_0=\\mbox{\\boldmath $X$}(u_0)\\not=\\mbox{\\boldmath $\\lambda$} _0.$ Then we have the following\\\/$:$\n\\par\n{\\rm (1)}\n$h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p_0)=\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i(p_0)=0$, $(i=1,\\dots ,s)$\nif and only if\nthere exist $\\mbox{\\boldmath $\\xi$}_0 \\in N_1(M)_{p_0}[\\mbox{\\boldmath $n$}^T]$ and $\\mu_0\\in\n{\\mathbb R}\\setminus \\{0\\}$ such that \n$$\n\\mbox{\\boldmath $\\lambda$} _0 =\\mbox{\\boldmath $X$}(u_0)+\\mu_0\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(u_0,\\mbox{\\boldmath $\\xi$}_0)=\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)((p_0,\\mbox{\\boldmath $\\xi$}_0),\\mu_0).\n$$ \n\\par\n{\\rm (2)}\n$h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p_0)=\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i(p_0)=\n{\\rm det}{\\mathcal H}(h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0})(p_0)=0$ $(i=1,\\dots ,s)$\nif and only if\nthere exist $\\mbox{\\boldmath $\\xi$}_0 \\in N_1(M)_{p_0}[\\mbox{\\boldmath $n$}^T]$ and $\\mu_0\\in\n{\\mathbb R}\\setminus \\{0\\}$ such that \n$$\n\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)((p_0,\\mbox{\\boldmath $\\xi$}_0),\\mu_0)\n$$\nand \n$1\/\\mu$ is one of the non-zero nullcone\nprincipal curvatures \n$\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0), (i=1,\\dots ,s).$\n\\par\nHere, ${\\mathcal H}(h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0})(p_0)$\nis the Hessian matrix of $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}$ at $p_0.$\n\\par\n{\\rm (3)} With condition {\\rm (2)}, ${\\rm rank}\\, {\\mathcal H}(h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0})(p_0)=0$ if and only if\n$p_0=\\mbox{\\boldmath $X$}(u_0)$ is a non-flat $(\\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $\\xi$} _0)$-umbilical point.\n\\end{Pro}\n\\par\\noindent{\\it Proof. \\\/}\\ \n(1) For $p=\\mbox{\\boldmath $X$}(u),$ the condition $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p)=\\langle \\mbox{\\boldmath $X$}(u),{\\mbox{\\boldmath $\\lambda$}\n_0}\\rangle+1 =0$\nmeans that\n\\[\n\\langle \\mbox{\\boldmath $X$}(u)-\\lambda _0,\\mbox{\\boldmath $X$}(u)-\\mbox{\\boldmath $\\lambda$}_0\\rangle =\\langle\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $X$}(u)\\rangle-2\\langle\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\lambda$} _0\\rangle+\\langle\\mbox{\\boldmath $\\lambda$} _0,\\mbox{\\boldmath $\\lambda$}_0\\rangle=-2(1+\\langle\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\lambda$} _0\\rangle)=0,\n\\]\nso that\n$\\mbox{\\boldmath $X$}(u)-{\\mbox{\\boldmath $\\lambda$} _0}\\in \\Lambda^*.$\nSince $\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i(p)=\\langle \\mbox{\\boldmath $X$}_{u_i}(u), {\\mbox{\\boldmath $\\lambda$} _0}\\rangle $\nand $\\langle \\mbox{\\boldmath $X$}_{u_i},\\mbox{\\boldmath $X$}\\rangle=0,$\nwe have $\\langle \\mbox{\\boldmath $X$}_{u_i}(u),\\mbox{\\boldmath $\\lambda$} _0\\rangle=-\\langle \\mbox{\\boldmath $X$}_{u_i}(u)-\\mbox{\\boldmath $\\lambda$} _0\\rangle$.\nTherefore, $\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i(p)=0$\nif and only if\n$\\mbox{\\boldmath $X$} (u)-{\\mbox{\\boldmath $\\lambda$} _0}\\in N_pM.$\nOn the other hand, the condition $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0} (p)=\\langle \\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\lambda$} _0\\rangle+1=0$\nimplies that $\\langle\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $X$}(u)-\\mbox{\\boldmath $\\lambda$}_0\\rangle =0$.\nThis means that $\\mbox{\\boldmath $X$}(u)-\\mbox{\\boldmath $\\lambda$} _0\\in T_pAdS^{n+1}.$\nHence\n$h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p_0)=\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i((p_0)=0$ $(i=1,\\dots, s)$\nif and only if\n$\\mbox{\\boldmath $X$}(u_0)-{\\mbox{\\boldmath $\\lambda$} _0}\\in N_{p_0}M\\cap \\Lambda^*\\cap T_{p_0}AdS^{n+1}.$\nThen we denote that\n$\\mbox{\\boldmath $v$}=\\mbox{\\boldmath $X$} (u_0)-{\\mbox{\\boldmath $\\lambda$} _0}\\in N_{p_0}M\\cap \\Lambda^*\\cap T_{p_0}AdS^{n+1}.$\nIf $\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle =0,$ then $\\mbox{\\boldmath $n$}^T(u_0)$ belongs to\na lightlike hyperplane in the Lorentz space $T_{p_0}AdS^{n+1},$ so that $\\mbox{\\boldmath $n$}^T(u_0)$ is lightlike or spacelike.\nThis contradiction to the fact that $\\mbox{\\boldmath $n$}^T(u_0)$ is a timelike unit vector. Thus,\n$\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle \\not=0.$ \nWe set\n\\[\n\\mbox{\\boldmath $\\xi$}_0=\\frac{-1}{\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle}\\mbox{\\boldmath $v$} -\\mbox{\\boldmath $n$}^T(u_0).\n\\]\nThen we have\n\\begin{eqnarray*}\n\\langle \\mbox{\\boldmath $\\xi$}_0,\\mbox{\\boldmath $\\xi$}_0\\rangle &=& -2\\frac{-1}{\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle} \\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle-1=1 \\\\\n\\langle \\mbox{\\boldmath $\\xi$}_0,\\mbox{\\boldmath $n$}^T(u_0)\\rangle &=& \\frac{-1}{\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle} \\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle+1=0.\n\\end{eqnarray*}\nThis means that $\\mbox{\\boldmath $\\xi$}_0\\in N_1(M)_{p_0}(M)[\\mbox{\\boldmath $n$}^T].$\nSince $-\\mbox{\\boldmath $v$}=\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle(\\mbox{\\boldmath $n$}^T(u_0)+\\mbox{\\boldmath $\\xi$}_0),$\nwe have \n${\\mbox{\\boldmath $\\lambda$} _0}=\\mbox{\\boldmath $X$}(u_0)+\\mu_0\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(p_0,\\mbox{\\boldmath $\\xi$}_0)$, where\n$p_0=\\mbox{\\boldmath $X$}(u_0)$ and $\\mu_0=\\langle \\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $v$}\\rangle.$\nFor the converse assertion,  suppose that $\\mbox{\\boldmath $\\lambda$}_0=\\mbox{\\boldmath $X$}(u_0)+\\mu_0\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(p_0,\\mbox{\\boldmath $\\xi$}_0).$\nThen $\\mbox{\\boldmath $\\lambda$}_0-\\mbox{\\boldmath $X$}(u_0)\\in N_{p_0}(M)\\cap \\Lambda ^*$ and\n$\\langle\\mbox{\\boldmath $\\lambda$}_0-\\mbox{\\boldmath $X$}(u_0),\\mbox{\\boldmath $X$}(u_0)\\rangle=\\langle \\mu_0\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(p_0,\\mbox{\\boldmath $\\xi$}_0),\\mbox{\\boldmath $X$}(u_0)\\rangle=0.$\nThus we have $\\mbox{\\boldmath $\\lambda$}_0-\\mbox{\\boldmath $X$}(u_0)\\in N_{p_0}(M)\\cap \\Lambda ^*\\cap T_{p_0}AdS^{n+1}.$\nBy the previous arguments, these conditions are equivalent to the condition that\n$h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(p_0)=\\partial h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\/\\partial u_i((p_0)=0$ $(i=1,\\dots, s)$.\n\n\\par\n(2) By a straightforward calculation, we have\n\\[\n\\frac{\\partial ^2 h_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}}{\\partial u_i\\partial u_j}(u)\n=\\langle\\mbox{\\boldmath $X$} _{u_iu_j},\\mbox{\\boldmath $\\lambda$} _0\\rangle.\n\\]\nUnder the conditions ${\\mbox{\\boldmath $\\lambda$} _0}=\\mbox{\\boldmath $X$}(u_0)+\\mu_0(\\mbox{\\boldmath $n$}^T(u_0)+\\mbox{\\boldmath $\\xi$}_0)$,\nwe have\n\\[\n\\frac{\\partial ^2 h_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}}{\\partial u_i\\partial u_j}(u_0)\n=\\langle \\mbox{\\boldmath $X$} _{u_iu_j}(u_0),\\mbox{\\boldmath $X$}(u_0)\\rangle +\\mu_0\\langle\\mbox{\\boldmath $X$}_{u_iu_j}(u_0), (\\mbox{\\boldmath $n$}^T(u_0)+\\mbox{\\boldmath $\\xi$}_0)\\rangle .\n\\]\nSince $\\langle \\mbox{\\boldmath $X$}_{u_i},\\mbox{\\boldmath $X$}\\rangle =0,$ we have $\\langle\\mbox{\\boldmath $X$}_{u_iu_j},\\mbox{\\boldmath $X$}\\rangle=-\\langle \\mbox{\\boldmath $X$}_{u_i},\\mbox{\\boldmath $X$}_{u_j}\\rangle.$\nTherefore, we have\n\\[\n\\left(\\frac{\\partial ^2 h_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}}{\\partial u_i\\partial u_\\ell}(u_0)\\right)\\left(g^{j\\ell}(u_0)\\right)\n=\\left(\\mu_0 h^j_i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)-\\delta ^j_i\\right),\n\\]\nwhere $\\mbox{\\boldmath $n$}^S$ is the local section of $N_1(M)[\\mbox{\\boldmath $n$}^T]$ with $\\mbox{\\boldmath $n$}^S(u_0)=\\mbox{\\boldmath $\\xi$} _0.$\nIt follows that ${\\rm det}{\\mathcal H}(g)(p_0)=0$ if and only if \n$1\/\\mu_0$ is an eigenvalue of $(h^i_j(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)),$ which is equal to\none of the nullcone principal curvatures $\\kappa _i(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0)=\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0), (i=1,\\dots ,s)$.\n\\par\n(3) By the above calculation,  ${\\rm rank}\\, {\\mathcal H}(h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0})(p_0)=0$ if and only if\n$$\n(h^i_j(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)(u_0))=\\frac{1}{\\mu _0}(\\delta ^j_i),\n$$\nwhere $1\/\\mu_0=\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0),\\ (i=1,\\dots ,s)$. This means that $p_0=\\mbox{\\boldmath $X$}(u_0)$ is an $(\\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $\\xi$}_0)$-umbilical point.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nIn order to understand the geometric meaning of the assertions of Proposition 4.1, we briefly review the theory of Legendrian singularities.\nFor detailed expressions, see \\cite{Arnold1, Zak}.\nLet \n$\\pi :PT^*({\\mathbb R}^{n+1}) \\longrightarrow {\\mathbb R}^{n+1}$ be the projective cotangent bundle with its canonical contact structure.\nWe  next review the geometric properties of this bundle.\nConsider the tangent bundle\n$\n\\tau :TPT^*({\\mathbb R}^{n+})\\rightarrow PT^*({\\mathbb R}^{n+1})\n$\nand the differential map\n$\nd\\pi :TPT^*({\\mathbb R}^{n+1})\\rightarrow T{\\mathbb R}^{n+1}\n$\nof $\\pi .$\nFor any $X\\in TPT^*({\\mathbb R}^{n+1}),$ there exists an element\n$\\alpha\\in T^*({\\mathbb R}^{n+1}_1$ such that\n$\\tau (X)=[\\alpha ].$  For an element $V\\in T_x({\\mathbb R}^{n+1}),$\nthe property $\\alpha (V)=0$ does not depend on the choice of\nrepresentative of the class $[\\alpha ].$  Thus we can define the canonical\ncontact structure on $PT^*({\\mathbb R}^{n+1})$ by\n$$\nK=\\{X\\in TPT^*({\\mathbb R}^{n+1})\\ |\\ \\tau (X)(d\\pi (X))=0\\}.\n$$\nWe have a trivialization\n$\nPT^*({\\mathbb R}^{n+1})\\cong\n{\\mathbb R}^{n+1}\\times P^n({\\mathbb R})^*,\n$\nand call\n$$\n((v_0,v_1,\\dots ,v_n),[\\xi _0:\\xi _1:\\cdots :\\xi _n])\n$$\nthe {\\it  homogeneous coordinates} of $PT^*({\\mathbb R}^{n+1}),$ where\n$\n[\\xi _0:\\xi _1:\\cdots :\\xi _n]\n$\nare the homogeneous coordinates of the dual projective space\n$P^n({\\mathbb R})^*.$\nIt is easy to show that $X\\in K_{(x,[\\xi])}$ if and only if\n$\n\\sum_{i=0}^n \\mu _i\\xi _i=0,\n$\nwhere\n$\nd\\tilde\\pi (X)=\\sum_{i=0}^n \\mu _i\\partial\/\\partial v_i.\n$\nAn immersion  $i:L\\rightarrow PT^*({\\mathbb R}^{n+1})$ is said to be\n{\\it a Legendrian immersion} if $\\text{dim}\\, L=n$ and $di_q(T_qL)\\subset\nK_{i(q)}$\nfor any $q\\in L.$\nThe map $\\pi\\circ i$ is also called {\\it the Legendrian map} and the set\n$W(i)=\\text{image}\\, \\pi\\circ i$, the {\\it wave front set} of $i.$\nMoreover, $i$ (or, the image of $i$) is called the {\\it Legendrian lift }\nof $W(i).$\n\\par\nLet $F:({\\mathbb R}^k\\times{\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )\\longrightarrow ({\\mathbb\nR},\\mbox{\\boldmath $0$} )$ be a\nfunction germ.\nWe say that $F$ is {\\it a Morse family of hypersurfaces} if the map germ\n$$\n\\Delta^*F=\\left(F,\\frac{\\partial F}{\\partial q_1},\\dots ,\\frac{\\partial\nF}{\\partial q_k}\n\\right):({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )\\longrightarrow ({\\mathbb R}\\times\n{\\mathbb R}^k,\\mbox{\\boldmath $0$} )\n$$\nis submersive, where $(q,x)=(q_1,\\dots ,q_k,x_0,\\dots ,x_n)\\in  ({\\mathbb\nR}^k\\times\n{\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ).$\nIn this case we have a smooth $n$-dimensional submanifold\n$$\n\\Sigma _*(F)=\\Bigl\\{(q,x)\\in ({\\mathbb R}^k\\times{\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )\\  \\Bigm|\n\\ F(q,x)=\\frac{\\partial F}{\\partial q_1}(q,x)=\\cdots =\\frac{\\partial F}{\\partial q_k}(q,x)=0\n\\ \\Bigr\\}\n$$\nand the map germ $\\mathscr{L} _F:(\\Sigma _*(F), \\mbox{\\boldmath $0$})\\longrightarrow PT^*{\\mathbb R}^{n+1}$ defined by\n$$\n\\mathscr{L} _F(q,x)=\\left(x,\\left[\\frac{\\partial F}{\\partial x_0}(q,x):\\cdots :\\frac{\\partial F}{\\partial x_n}(q,x)\\right]\\right)\n$$\nis a Legendrian immersion. \nWe call $F$ {\\it a generating family} of $\\mathscr{L} _F(\\Sigma _*(F)),$\nand the wave front set is given by\n$\nW(\\mathscr{L} _F)\\! =\\pi _n(\\Sigma _*(F)),\n$\nwhere $\\pi _n:{\\mathbb R}^k\\times{\\mathbb R}^n\\longrightarrow {\\mathbb R}^n$ is the canonical projection.\nIn the theory of unfoldings of function germs, the wave front set $W(\\mathscr{L} _F)$ is called a\n{\\it discriminant set} of $F,$ which we also denote $\\mathcal{D}_F.$\nTherefore, Proposition 4.1 asserts that the discriminant set of the\nAdS-height function $H$ is given by\n\\[\n{\\mathcal D}_{H}=\\Bigl\\{\\mbox{\\boldmath $\\lambda$}\\in AdS^{n+1}  \\Bigm| \\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $X$} (u)+\\mu (\\mbox{\\boldmath $n$}^T\\pm \\mbox{\\boldmath $\\xi$})\n(u),\\ p=\\mbox{\\boldmath $X$}(u)\\in M, \\mbox{\\boldmath $\\xi$}\\in N_1(M)_p[\\mbox{\\boldmath $n$}^T], \\mu \\in {\\mathbb R}\\ \\Bigr\\},\n\\]\nwhich is the image of the lightlike hypersurface along $M$ relative to $\\mbox{\\boldmath $n$}^T.$\n\\par\nBy the assertion (2) of Proposition 4.1, a singular point of the lightlike hypersurface is\na point $\\mbox{\\boldmath $\\lambda$} _0=\\mbox{\\boldmath $X$}(u_0)+\\mu _0(\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $\\xi$}_0)(u_0)$\nfor\n$p_0=\\mbox{\\boldmath $X$}(u_0)$ and \n$\\mu _0 =1\/\\kappa  _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0),$ $i=1,\\dots .s).$\nThen we have the following corollary.\n\\begin{Co}\n The critical value of $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)$ is the point \n \\[\n\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $X$}(u)+\\frac{1}{\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})}\\mathbb{LG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$}),\n\\]\nwhere\n$p=\\mbox{\\boldmath $X$}(u)$ and \n$\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})\\not= 0.$\n\\end{Co}\n\\par\nFor a non-zero nullcone principal curvature $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0)\\not= 0,$ we\nhave an open subset $O_i\\subset N_1(M)[\\mbox{\\boldmath $n$}^T]$ such that $\\kappa _N(\\mbox{\\boldmath $n$}^T)(p,\\mbox{\\boldmath $\\xi$})\\not= 0$\nTherefore, we have a non-zero nullcone principal curvature function $\\kappa _N(\\mbox{\\boldmath $n$}^T):O_i\\longrightarrow {\\mathbb R}$.\nWe define a mapping\n$\n\\mathbb{LF}_{\\kappa_N(\\mbox{\\boldmath $n$}^T)_i} :O_i\\longrightarrow AdS^{n+1}\n$\nby\n\\[\n\\mathbb{LF}_{\\kappa_N(\\mbox{\\scriptsize \\boldmath$n$}^T)_i}(p,\\mbox{\\boldmath $\\xi$})=\\mbox{\\boldmath $X$}(u)+\\frac{1}{\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$}),\n\\]\nwhere $p=\\mbox{\\boldmath $X$}(u).$\nWe also  define\n\\[\n\\mathbb{LF}_M(\\mbox{\\boldmath $n$}^T)=\\bigcup\\left\\{\\mathbb{LF}_{\\kappa_N(\\mbox{\\scriptsize \\boldmath$n$}^T)_i}(p,\\mbox{\\boldmath $\\xi$})\\ |\\ (p,\\mbox{\\boldmath $\\xi$})\\in N_1(M)[\\mbox{\\boldmath $n$}^T]\\ \\mbox{s.t.}\\ \\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})\\not= 0,i=1,\\dots ,s \\right\\} .\n\\]\nWe call $\\mathbb{LF}_{M}(\\mbox{\\boldmath $n$}^T)$ the {\\it lightlike focal set} of $M=\\mbox{\\boldmath $X$}(U)$\nrelative to $\\mbox{\\boldmath $n$}^T.$\nBy definition, the lightlike focal set of $M=\\mbox{\\boldmath $X$}(U)$\nrelative to $\\mbox{\\boldmath $n$}^T$ is the critical values set of the lightlike hypersurface\n$\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$ along $M$ relative to $\\mbox{\\boldmath $n$}^T.$\n\n\\par\nWe can show that the image of the lightlike hypersurface along $M$ is independent of the choice of the future directed\ntimelike normal vector field $\\mbox{\\boldmath $n$}^T$ as a corollary of Proposition 4.1.\n\\begin{Co}\nLet $\\mbox{\\boldmath $n$}^T$ and $\\overline{\\mbox{\\boldmath $n$}}^T$ be future directed timelike unit normal fields along $M$.\nThen we have\n\\[\n\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})=\\mathbb{LH}_M(\\overline{\\mbox{\\boldmath $n$}}^T)(N_1(M)[\\overline{\\mbox{\\boldmath $n$}}^T]\\times{\\mathbb R})\\ \\mbox{and}\\ \\mathbb{LF}_M(\\mbox{\\boldmath $n$}^T)=\\mathbb{LF}_M(\\overline{\\mbox{\\boldmath $n$}}^T).\n\\]\n\\end{Co}\n\\par\\noindent{\\it Proof. \\\/}\\ \nBy Proposition 4.1, the images of the lightlike hypersurface along $M$ relative to $\\mbox{\\boldmath $n$}^T$ and $\\overline{\\mbox{\\boldmath $n$}}^T$ are\nthe discriminant sets of the  AdS-height function $H$ on $M$.\nMoreover, the focal set is the critical value set of the lightlike hypersurface\nalong $M$ relative to $\\mbox{\\boldmath $n$}^T.$\nSince $H$ is independent of the choice of $\\mbox{\\boldmath $n$}^T,$ we have the assertion.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nWe have the following proposition.\n\\begin{Pro}\nLet $H$ be the AdS-height function on $M.$\nFor any point $(u,\\mbox{\\boldmath $\\lambda$} )\\in \\Delta ^*H^{-1}(0),$ the germ of $H$ at\n$(u,\\mbox{\\boldmath $\\lambda$} )$\nis a Morse family of hypersurfaces. \n\\end{Pro}\n\\par\\noindent{\\it Proof. \\\/}\\ \nWe denote that\n$$\n\\mbox{\\boldmath $X$}(u)=(X_{-1}(u),X_0(u),X_1(u),\\dots ,X_n(u))\\ {\\rm and}\\\n\\mbox{\\boldmath $\\lambda$} =(\\lambda _{-1},\\lambda _0,\\lambda _1,\\dots ,\\lambda _n).\n$$\nWe define an open subset \n$U_{-1}^+=\\{\\mbox{\\boldmath $\\lambda$} \\in AdS^{n+1}\\ |\\ \\lambda _{-1}>0\\ \\}$.\nFor any $\\mbox{\\boldmath $\\lambda$} \\in U_{-1}^+,$ we have\n\\[\n\\lambda _{-1}=\\sqrt{1-\\lambda _0^2+\\lambda _1^2+\\cdots \\lambda _n^2}.\n\\]\nThus, we have a local coordinate of $AdS^{n+1}$ given by $(\\lambda _0,\\lambda _1,\\dots ,\\lambda _n)$ on\n$U_{-1}^+.$\nBy definition, we have\n$$\nH(u,\\mbox{\\boldmath $\\lambda$} )=-X_{-1}(u)\\sqrt{1-\\lambda _0^2+\\sum_{i=1}^n \\lambda _i^2}-X_0(u)\\lambda_0+\nX_1(u)\\lambda_1 +\\cdots \n+X_n(u)\\lambda _n.\n$$\nWe now prove that the mapping $$\n\\Delta^*H=\\left(H, \\frac{\\partial H}{\\partial u_1},\\dots ,\\frac{\\partial H}{\\partial\nu_s}\\right)\n$$\nis non-singular at $(u,\\mbox{\\boldmath $\\lambda$} )\\in \\Delta ^*H^{-1}(0).$\nIndeed, the Jacobian matrix of $\\Delta ^*H$ is given by\n\\newfont{\\bg}{cmr10 scaled\\magstep5}\n\\newcommand{\\smash{\\lower1.0ex\\hbox{\\bg A}}}{\\smash{\\lower1.0ex\\hbox{\\bg A}}}\n\\[\n\\left(\n\\begin{array}{ccccc}\n &\nX_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_0 & -X_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_1 &  \\cdots  &\n-X_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_n \\\\\n\\smash{\\lower1.0ex\\hbox{\\bg A}} & X_{-1u_1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_1} &\n-X_{-1u_1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_1}& \\cdots  &-X_{-1u_1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_1}\\\\\n&\\vdots & \\vdots & \\ddots & \\vdots \\\\\n& X_{-1u_s}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_s} &\n-X_{-1u_s}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_s}& \\cdots  &-X_{-1u_s}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_s}\n\\end{array}\n\\right) ,\n\\]\nwhere\n\\begin{eqnarray*}\n\\smash{\\lower1.0ex\\hbox{\\bg A}}=\n\\left(\\!\\!\n\\begin{array}{ccc}\n\\langle \\mbox{\\boldmath $X$}_{u_1} ,\\mbox{\\boldmath $\\lambda$}\\rangle & \\!\\! \\cdots\\!\\! & \\langle \\mbox{\\boldmath $X$}_{u_s},\\mbox{\\boldmath $\\lambda$} \\rangle \\\\\n\\langle \\mbox{\\boldmath $X$}_{u_1u_1},\\mbox{\\boldmath $\\lambda$}\\rangle & \\!\\! \\cdots\\!\\!  &\n\\langle \\mbox{\\boldmath $X$}_{u_1u_s},\\mbox{\\boldmath $\\lambda$} \\rangle \\\\\n\\vdots & \\!\\!\\ddots\\!\\! & \\vdots \\\\\n\\langle \\mbox{\\boldmath $X$}_{u_su_1},\\mbox{\\boldmath $\\lambda$} \\rangle &\\!\\! \\cdots\\!\\! &\n\\langle \\mbox{\\boldmath $X$}_{u_su_s},\\mbox{\\boldmath $\\lambda$}\\rangle \n\\end{array}\n\\!\\!\n\\right) .\n\\end{eqnarray*}\n\\newcommand{\\smash{\\lower1.0ex\\hbox{\\bg B}}}{\\smash{\\lower1.0ex\\hbox{\\bg B}}}\nWe now show that\nthe rank of \n\\[\n\\smash{\\lower1.0ex\\hbox{\\bg B}}=\n\\left(\n\\begin{array}{cccc}\nX_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_0 & -X_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_1 &  \\cdots  &\n-X_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_n \\\\\n X_{-1u_1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_1} &\n-X_{-1u_1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_1}& \\cdots  &-X_{-1u_1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_1}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n X_{-1u_s}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_s} &\n-X_{-1u_s}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_s}& \\cdots  &-X_{-1u_s}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_s}\n\\end{array}\n\\right) \n\\]\nis $s+1$ at $(u,\\mbox{\\boldmath $\\lambda$})\\in \\Sigma _*(H).$\nSince $(u,\\mbox{\\boldmath $\\lambda$})\\in \\Sigma _*(H),$ we have\n\\[\n\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $X$}(u)+\\mu\\left(\\mbox{\\boldmath $n$}^T(u)+\\sum _{i=1}^{k-1}\\xi _i\\mbox{\\boldmath $n$}_i(u)\\right)\n\\]\nwith $\\sum_{i=1}^{k-1}\\xi ^2_i=1,$ where \n$\\{\\mbox{\\boldmath $X$},\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S_1,\\dots ,\\mbox{\\boldmath $n$}^S_{k-1}\\}$ is a pseudo-orthonormal (local) frame of $N(M).$\nWithout the loss of generality, we assume that $\\mu\\not= 0$ and $\\xi_{k-1}\\not= 0.$\nWe denote that\n\\[\n\\mbox{\\boldmath $n$}^T(u)=^t\\!\\!(n^T_{-1}(u),n^T_0(u),\\dots n^T_n(u)),\\ \n\\mbox{\\boldmath $n$}_i(u)=^t\\!\\!(n^i_{-1}(u),n_0^i(u),\\dots n^i_n(u)).\n\\]\n\\newcommand{\\smash{\\lower1.0ex\\hbox{\\bg C}}}{\\smash{\\lower1.0ex\\hbox{\\bg C}}}\nIt is enough to show that the rank of \nthe matrix\n\\[\n\\smash{\\lower1.0ex\\hbox{\\bg C}} =\n\\left(\n\\begin{array}{cccc}\nX_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_0 & -X_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_1 &  \\cdots  &\n-X_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_n \\\\\n X_{-1u_1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_1} &\n-X_{-1u_1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_1}& \\cdots  &-X_{-1u_1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_1}\\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n X_{-1u_s}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_{0u_s} &\n-X_{-1u_s}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+X_{1u_s}& \\cdots  &-X_{-1u_s}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-X_{nu_s} \\\\\nn^T_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-n^T_0 & -n^T_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+n^T_1 &  \\cdots  &\n-n^T_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-n^T_n \\\\\nn^1_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-n^1_0 & -n^1_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+n^1_1 &  \\cdots  &\n-n^1_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-n^1_n \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\nn^{k-2}_{-1}\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-n^{k-2}_0 & -n^{k-2}_{-1}\\displaystyle{\\frac{\\lambda _1}{\\lambda _{-1}}}+n^{k-2}_1 &  \\cdots  &\n-n^{k-2}_{-1}\\displaystyle{\\frac{\\lambda _n}{\\lambda _{-1}}}-n^{k-2}_n \n\\end{array}\n\\right) \n\\]\nis $n+1$ at $(u,\\mbox{\\boldmath $\\lambda$})\\in \\Sigma _*(H).$\nWe denote that\n\\[\n\\mbox{\\boldmath $a$}_i=^t\\!\\!(x_i(u),x_{iu_1}(u),\\dots x_{iu_s}(u),n^T_i(u),n^1_i(u),\\dots ,n^{k-2}_i(u)).\n\\]\nThen we have\n\\[\n\\smash{\\lower1.0ex\\hbox{\\bg C}}=\\left(\\mbox{\\boldmath $a$}_{-1}\\frac{\\lambda_0}{\\lambda _{-1}}-\\mbox{\\boldmath $a$}_0,-\\mbox{\\boldmath $a$}_{-1}\\frac{\\lambda _1}{\\lambda _{-1}}+\\mbox{\\boldmath $a$}_1,\\dots ,\n-\\mbox{\\boldmath $a$} _{-1}\\frac{\\lambda _n}{\\lambda _{-1}}+\\mbox{\\boldmath $a$}_n\\right).\n\\]\nIt follows that\n\\begin{eqnarray*}\n\\det \\smash{\\lower1.0ex\\hbox{\\bg C}}\\!\\!\\!\\!\\!\\!\\!\n&{}&=\\frac{\\lambda _{-1}}{\\lambda _{-1}}\\det (\\mbox{\\boldmath $a$}_0,\\mbox{\\boldmath $a$}_1,\\dots,\\mbox{\\boldmath $a$}_n)+\\frac{\\lambda_0}{\\lambda_{-1}}\\det (\\mbox{\\boldmath $a$}_{-1}\\mbox{\\boldmath $a$}_{1},\\dots ,\\mbox{\\boldmath $a$}_n)\n\\\\\n&{}&-\\frac{\\lambda _1}{\\lambda _{-1}}(-1)\\det(\\mbox{\\boldmath $a$}_{-1},\\mbox{\\boldmath $a$}_0,\\mbox{\\boldmath $a$}_2,\\dots ,\\mbox{\\boldmath $a$}_n)-\\cdots \n-\\frac{\\lambda_n}{\\lambda _{-1}}(-1)^{n-1}\\det(\\mbox{\\boldmath $a$} _{-1}\\mbox{\\boldmath $a$}_0,\\mbox{\\boldmath $a$}_1,\\dots ,\\mbox{\\boldmath $a$}_{n-1}).\n\\end{eqnarray*}\nMoreover, we define\n$\\delta _i=\\det (\\mbox{\\boldmath $a$}_{-1},\\mbox{\\boldmath $a$}_0,\\mbox{\\boldmath $a$}_1,\\dots,\\mbox{\\boldmath $a$} _{i-1},\\mbox{\\boldmath $a$}_{i+1},\\dots ,\\mbox{\\boldmath $a$}_n)$ for $i=-1,0,1,\\dots ,n$\nand\n$\\mbox{\\boldmath $a$}=(-\\delta _{-1},-\\delta _0,-\\delta _1,(-1)^2\\delta _2,\\dots ,(-1)^{n-1}\\delta _n).$\nThen we have\n\\[\n\\mbox{\\boldmath $a$}=\\mbox{\\boldmath $X$}\\wedge\\mbox{\\boldmath $X$}_{u_1}\\wedge\\cdots \\wedge \\mbox{\\boldmath $X$}_{u_s}\\wedge\\mbox{\\boldmath $n$}^T\\wedge\\mbox{\\boldmath $n$}_1\\wedge\\cdots\\wedge\\mbox{\\boldmath $n$}_{k-2}.\n\\]\nWe remark that $\\mbox{\\boldmath $a$}\\not=0$ and $\\mbox{\\boldmath $a$}=\\pm\\|\\mbox{\\boldmath $a$}\\|\\mbox{\\boldmath $n$}_{k-1}.$\nBy the above calculation, we have\n\\begin{eqnarray*}\n\\det\\smash{\\lower1.0ex\\hbox{\\bg C}}\\!\\!\\!\\!\\!\\!\\!&{}&=\\left\\langle\\left(\\frac{\\lambda{-1}}{\\lambda_{-1}},\\frac{\\lambda_0}{\\lambda _{-1}},\\dots,\\frac{\\lambda_n}{\\lambda _{-1}}\\right), \\mbox{\\boldmath $a$}\\right\\rangle =\\frac{1}{\\lambda_{-1}}\\left\\langle \\mbox{\\boldmath $X$}(u)+\\mu\\left(\\mbox{\\boldmath $n$}^T(u)+\\sum_{i=1}^{k-1}\\xi_i\\mbox{\\boldmath $n$}_i(u)\\right),\\mbox{\\boldmath $a$}\\right\\rangle \\\\\n&{}&=\\frac{1}{\\lambda _{-1}}\\times\\pm\\mu\\xi_{k-1}\\|\\mbox{\\boldmath $a$}\\|=\\pm \\frac{\\mu\\xi_{k-1}\\|\\mbox{\\boldmath $a$}\\|}{\\lambda _{-1}}\\not= 0.\n\\end{eqnarray*}\nTherefore the Jacobi matrix of $\\Delta^*H$\nis non-singular at $(u,\\mbox{\\boldmath $\\lambda$} )\\in \\Delta ^*H^{-1}(0).$\n\\par\nFor other local coordinates of $AdS^{n+1}$, we can apply the same method for the proof as the above case.\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nHere we also consider the local coordinate \n$U^+_{-1}$.\nSince $H$ is a Morse family of hypersurfaces, we have a Legendrian immersion\n\\[\n\\mathscr{L} _H:\\Sigma _*(H)\\longrightarrow PT^*(AdS^{n+1})|U^+_{-1}\n\\]\nby the general theory of Legendrian singularities.\nBy definition, we have\n\\[\n\\frac{\\partial H}{\\partial \\lambda _0}(u,\\mbox{\\boldmath $\\lambda$}) \\\\\n=X_{-1}(u)\\displaystyle{\\frac{\\lambda _0}{\\lambda _{-1}}}-X_0(u),\\ \\frac{\\partial H}{\\partial \\lambda _i}(u,\\mbox{\\boldmath $\\lambda$})=-X_{-1}(u)\\displaystyle{\\frac{\\lambda _i}{\\lambda _{-1}}}+X_i(u),\\ (i=1,\\dots ,n).\n\\]\nIt follows that\n\\begin{eqnarray*}\n&{}&\\left[\\frac{\\partial H}{\\partial \\lambda _0}(u,\\mbox{\\boldmath $\\lambda$}):\\frac{\\partial H}{\\partial \\lambda _1}(u,\\mbox{\\boldmath $\\lambda$}):\\cdots :\\frac{\\partial H}{\\partial \\lambda _n}(u,\\mbox{\\boldmath $\\lambda$})\\right]\n\\\\\n&{}&\\qquad =[X_{-1}(u)\\lambda _0-X_0(u)\\lambda _{-1}:X_1(u)\\lambda _{-1}-X_{-1}(u)\\lambda _1:\\cdots :X_n(u)\\lambda _{-1}-X_{-1}(u)\\lambda _n].\n\\end{eqnarray*}\nTherefore, we have\n\\[\n\\mathscr{L}_H(u,\\mbox{\\boldmath $\\lambda$} )=(\\mbox{\\boldmath $\\lambda$}, [X_{-1}(u)\\lambda _0-X_0(u)\\lambda _{-1}:X_1(u)\\lambda _{-1}-X_{-1}(u)\\lambda _1:\\cdots :X_n(u)\\lambda _{-1}-X_{-1}(u)\\lambda _n]),\n\\]\nwhere\n\\[\n\\Sigma _*(H)=\\{(u,\\mbox{\\boldmath $\\lambda$})\\ |\\ \\mbox{\\boldmath $\\lambda$} =\\mathbb{LH} _M(\\mbox{\\boldmath $n$}^T)(p,\\mbox{\\boldmath $\\xi$},t)\\ ((p,\\mbox{\\boldmath $\\xi$}),t)\\in N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R}\\}.\n\\]\nWe observe that $H$ is a generating family of the Legendrian submanifold $\\mathscr{L}_H(\\Sigma _*(H))$ whose wave front is $\\mathbb{LH} _M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$.\nTherefore we say that the AdS-height \nfunction\n$H$ on $M$ gives an {\\it AdS-canonical generating family for the Legendrian lift\\\/} of $\\mathbb{LH} _M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$.\nFor other local coordinates of $AdS^{n+1},$ we have the similar results to the above case.\n\n\\section{Contact with lightcones}\n\\par\nIn this section we consider the geometric meaning of the singularities of lightlike hypersurfaces in Anti-de Sitter space from the view point of the theory of contact of submanifolds with model hypersurfaces in \\cite{Montaldi}.\nWe begin with the following basic observations.\n\\begin{Pro}\nLet $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1}$ and $M=\\mbox{\\boldmath $X$}(U)$ a spacelike submanifold without points\nsatisfying $K_N (\\mbox{\\boldmath $n$}^T)(p,\\mbox{\\boldmath $\\xi$})= 0.$\nThen $M\\subset\n\\Lambda ^{n+1}_{\\lambda _0}\\cap AdS^{n+1}$ \nif and only if $\\{\\mbox{\\boldmath $\\lambda$} _0\\}=\\mathbb{LF}_M(\\mbox{\\boldmath $n$}^T)$ is the lightcone focal set.\nIn this case we have $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T])\\subset \\Lambda ^{n+1}_{\\lambda _0}\\cap AdS^{n+1}$\nand $M=\\mbox{\\boldmath $X$}(U)$ is totally nullcone umbilical.\n\\end{Pro}\n\\par\\noindent{\\it Proof. \\\/}\\ \nBy Proposition 3.1, $K_N(\\mbox{\\boldmath $n$}^T)(p_0,\\mbox{\\boldmath $\\xi$}_0)\\not= 0$ if and only if\n\\[\n\\{(\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S), (\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S)_{u_1},\n\\dots , (\\mbox{\\boldmath $n$}^T+\\mbox{\\boldmath $n$}^S)_{u_{s}}\\}\n\\]\nis linearly independent for  $p_0=\\mbox{\\boldmath $X$}(u_0)\\in M$ and $\\mbox{\\boldmath $\\xi$}_0=\\mbox{\\boldmath $n$}^S(u_0),$\nwhere $\\mbox{\\boldmath $n$}^S:U\\longrightarrow N_1(M)[\\mbox{\\boldmath $n$}^T]$ is a local section.\nBy the proof of the assertion (1) of Proposition 4.1, \n$M\\subset \\Lambda ^{n+1}_{\\lambda _0}\\cap AdS^{n+1}$\nif and only if $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(u)= 0$ for any $u\\in U,$\nwhere $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(u)=H(u,\\mbox{\\boldmath $\\lambda$} _0)$ is the AdS-height function on $M.$\nIt also follows from  Proposition 4.1 that there exists a smooth function $\\eta \n:U\\times N_1(M)[\\mbox{\\boldmath $n$}^T]\\longrightarrow {\\mathbb R}$ and section $\\mbox{\\boldmath $n$}^S:U\\longrightarrow N_1(M)[\\mbox{\\boldmath $n$}^T]$ such that\n\\[\n\\mbox{\\boldmath $X$}(u)=\\mbox{\\boldmath $\\lambda$} _0+\\eta (u,\\mbox{\\boldmath $n$}^S(u))(\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u)).\n\\]\nIn fact, we have $\\eta (u,\\mbox{\\boldmath $n$}^S(u))=-1\/\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$ $i=1,\\dots, s$, where\n$p=\\mbox{\\boldmath $X$}(u)$ and $\\mbox{\\boldmath $\\xi$}=\\mbox{\\boldmath $n$}^S(u).$\nIt follows that $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})=\\kappa _N(\\mbox{\\boldmath $n$}^T)_j(p,\\mbox{\\boldmath $\\xi$}),$ so that\n$M=\\mbox{\\boldmath $X$}(U)$ is totally nullcone umbilical.\nTherefore we have\n\\[\n\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $n$}^S(u),\\mu)=\\mbox{\\boldmath $\\lambda$} _0+(\\mu +\\eta (u,\\mbox{\\boldmath $n$}^S(u))(\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u)).\n\\]\nHence we have\n$\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T) (N_1(M)[\\mbox{\\boldmath $n$}^T]\\times {\\mathbb R})\\subset \\Lambda ^{n+1}_{\\lambda _0}.$\nBy Corollary 4.2, the critical value set of $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T) (N_1(M)[\\mbox{\\boldmath $n$}^T]\\times {\\mathbb R})$ is the lightlike focal set\n$\\mathbb{LF}_M(\\mbox{\\boldmath $n$}^T).$\nHowever, it is equal to $\\lambda _0$ by the previous arguments. \n\\par\nFor the converse assertion, suppose that $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LF}_M(\\mbox{\\boldmath $n$}^T).$\nThen we have \n\\[\n\\mbox{\\boldmath $\\lambda$} _0=\\mbox{\\boldmath $X$}(u)+\\frac{1}{\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\xi$})}\\mathbb{LG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$}),\n\\]\nfor any $i=1,\\dots ,s$ and $(p,\\mbox{\\boldmath $\\xi$})\\in N_1(M)[\\mbox{\\boldmath $n$}^T],$ where $p=\\mbox{\\boldmath $X$}(u).$\nThus, we have \n\\[\n\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\xi$})=\\kappa _N(\\mbox{\\boldmath $n$}^T)_j(\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\xi$})\n\\]\nfor any $i,j=1,\\dots ,s,$ so that $M$ is totally nullcone umbilical.\nSince $\\mathbb{LG}(\\mbox{\\boldmath $n$}^T)(u,\\mbox{\\boldmath $\\xi$})$ is null, we have $\\mbox{\\boldmath $X$}(u)\\in \\Lambda ^{n+1}_{\\lambda_0}.$\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nAccording to the above proposition, $\\Lambda ^{n+1}_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\\cap AdS^{n+1}$ is regarded as a model lightlike hypersurface in $AdS^{n+1}.$\nWe define\n\\[\nT(AdS^{n+1})_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}=\\{\\mbox{\\boldmath $x$}\\in {\\mathbb R}^{n+2}_2\\ |\\ \\ \\mbox{\\boldmath $x$}-\\mbox{\\boldmath $\\lambda$} _0\\in T_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}AdS^{n+1}\\ \\},\n\\]\nwhere $T_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}AdS^{n+1}$ is the tangent space of $AdS^{n+1}$ at $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1}.$\nWe call $T(AdS^{n+1})_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}$ a {\\it tangent affine space} of $AdS^{n+1}$ at $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1}.$\nIt is easy to show that\n\\[\n\\Lambda ^{n+1}_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\\cap AdS^{n+1}=T(AdS^{n+1})_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\\cap AdS^{n+1}.\n\\]\nWe denote that $LC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(AdS^{n+1})=\\Lambda ^{n+1}_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\\cap AdS^{n+1}=T(AdS^{n+1})_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}\\cap AdS^{n+1}$,\nwhich is called an {\\it AdS-lightcone} with the vertex $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1}.$\nTherefore, the model lightlike hypersurface is an AdS-lightcone.\n\\par\nWe consider the contact of\nspacelike submanifolds with AdS-lightcones. \nLet \n\\[\n{\\mathcal H}:AdS^{n+1}\\times AdS^{n+1}\\longrightarrow {\\mathbb R}\n\\]\nbe a function defined\nby ${\\mathcal H}(\\mbox{\\boldmath $x$},\\mbox{\\boldmath $\\lambda$})=\\langle\\mbox{\\boldmath $x$} , \\mbox{\\boldmath $\\lambda$}\n\\rangle+1 .$\nGiven $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{n+1},$\nwe denote  ${\\mathfrak h}_{\\lambda _0}(\\mbox{\\boldmath $x$})={\\mathcal H}(\\mbox{\\boldmath $x$} ,\\mbox{\\boldmath $\\lambda$} _0)$,\nso that we have ${\\mathfrak h}_{\\lambda _0}^{-1}(0)=\nLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(AdS^{n+1}).$\nFor any $p_0=\\mbox{\\boldmath $X$}(u_0)\\in M$, $\\mu_0\\in {\\mathbb R}$ and $\\mbox{\\boldmath $\\xi$}_0\\in N_1(M)_p[\\mbox{\\boldmath $n$}^T],$ we consider the point\n$\\mbox{\\boldmath $\\lambda$} _0=\\mbox{\\boldmath $X$}(u_0)+\\mu_0(\\mbox{\\boldmath $n$}^T(u_0)+ \\mbox{\\boldmath $\\xi$}_0).$\nThen we have\n$$\n{\\mathfrak h}_{\\lambda _0}\\circ\\mbox{\\boldmath $X$} (u_0))={\\mathcal H}\\circ\n(\\mbox{\\boldmath $X$}\\times 1_{AdS^{n+1}})(u_0,\\mbox{\\boldmath $\\lambda$} _0)\n=H(p_0,\\mbox{\\boldmath $\\lambda$} _0)=0,\n$$\nwhere $\\mu_0=1\/\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0),$ $i=1,\\dots , s.$\nWe also have relations\n$$\n\\frac{\\partial {\\mathfrak h}_{\\lambda _0}\\circ\\mbox{\\boldmath $X$}}{\\partial u_i}(u_0)=\n\\frac{\\partial H}{\\partial u_i}(p_0,\\mbox{\\boldmath $\\lambda$} _0)=0,\\\ni=1,\\dots ,s.\n$$\nThese imply that the AdS-lightcone ${\\mathfrak h}_{\\lambda_0}^{-1}(0)=\nLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}(AdS^{n+1})$ is tangent to $M=\\mbox{\\boldmath $X$} (U)$  at  $p_0=\\mbox{\\boldmath $X$} (u_0).$\nIn this case, we call $LC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(AdS^{n+1})$ a {\\it tangent AdS-lightcone} of\n$M=\\mbox{\\boldmath $X$}(U)$ at $p_0=\\mbox{\\boldmath $X$}(u_0),$ which is denoted by $TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}.$\nMoreover, the tangent AdS-lightcone $TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}$ is called an\n{\\it osculating AdS-lightcone} if $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LF}_{\\kappa _N(\\mbox{\\scriptsize \\boldmath$n$}^T)_i(p_0,\\mbox{\\scriptsize \\boldmath$\\xi$}_0)}(u_0)\\in \\mathbb{LF}_M,$ for one of the nullcone principal curvature $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0).$\nIn this case, we call $\\mbox{\\boldmath $\\lambda$} _0$ the {\\it center of the nullcone principal curvature} $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0)(u_0).$\nTherefore, we can interpret that the lightlike focal set is the locus of the centers of \nnullcone principal curvatures. This fact is analogous to the notion of the focal sets of submanifolds in Euclidean space.\n\n\\par\nFirstly, we consider a special contact of $M=\\mbox{\\boldmath $X$}(U)$ with AdS-lightcones.\nWe say that \n$p_0=\\mbox{\\boldmath $X$}(u_0)$ is  an {\\it AdS-lightlike $k$-ridge  point} if $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}$ \nhas the $A_{k+2}$ singular point at $u_0$ for some $k\\ge 1,$\nwhere $\\mbox{\\boldmath $\\lambda$} _0 \\in \\mathcal{D}_{H}=\\mathbb{LH}_M(N_1(M)\\times {\\mathbb R}).$ \nWe simply say that  $p_0=\\mbox{\\boldmath $X$}(u_0)$ is  an {\\it AdS-lightlike ridge  point} if it is the AdS-lightlike $k$-ridge point \nfor some $k\\ge 1.$\nFor a function germ $f:({\\mathbb R}^{s},\\widetilde{u} _0) \\longrightarrow {\\mathbb R}$, \n$f$ has an {\\it $A_{k}$ singular point} at $\\widetilde{u} _0$ if $f$ is \n$\\mathcal{K}$-equivalent to the germ \n$ u^{k+1}_1 \\pm u^2_2 \\pm \\dots \\pm u^2_{s}$.\nWe say that two function germs $f_i:({\\mathbb R}^{s},\\widetilde{u} _i)\\longrightarrow {\\mathbb R}$ $(i=1,2)$ are \n${\\mathcal K}$-{\\it equivalent} if there exists a diffeomorphism germ \n$\\Phi :({\\mathbb R}^{s},\\widetilde{u} _1)\\longrightarrow  ({\\mathbb R}^{s},\\widetilde{u} _2)$ and  a non-zero function germ $\\rho:({\\mathbb R}^{s},\\widetilde{u}_1)\\longrightarrow {\\mathbb R}$ such that\n$f_2\\circ\\Phi (u)=\\rho (u)f_1(u).$ We consider the geometric meaning of AdS-lightlike ridge points.\nLet $F: AdS^{n+1} \\longrightarrow {\\mathbb R}$ be a submersion and $\\mbox{\\boldmath $X$}:U \\longrightarrow AdS^{n+1}$ a spacelike embedding from an open set\n$U\\subset {\\mathbb R}^s.$\nWe say that $M=\\mbox{\\boldmath $X$}(U)$ and $F^{-1}(0)$ have a {\\it corank $r$ contact} at $p_0=\\mbox{\\boldmath $X$}(u_0)$ \nif the Hessian of the function $g(u) = F \\circ \\mbox{\\boldmath $X$}(u) $ has corank $r$ at $u_0$. \nWe also say that $M=\\mbox{\\boldmath $X$}(U)$ and $F^{-1}(0)$ have an {\\it $A_k$-type contact} at  $p_0=\\mbox{\\boldmath $X$}(u_0)$\nif the function $g(u) = F \\circ \\mbox{\\boldmath $X$}(u) $ has the $A_k$ singularity at $u_0.$\nBy definition, if $M=\\mbox{\\boldmath $X$}(U)$ and $F^{-1}(0)$ have an {\\it $A_k$-type contact} at  $p_0=\\mbox{\\boldmath $X$}(u_0),$\nthen these have a corank $1$ contact.\nFor a regular curve $\\mbox{\\boldmath $\\gamma$} :I\\longrightarrow AdS^{n+1},$ we say that $\\mbox{\\boldmath $\\gamma$}(I)$ and $F^{-1}(0)$ have\na {\\it contact of order $k$} if $F\\circ \\mbox{\\boldmath $\\gamma$}$ has the $A_k$ singularity at $s_0.$\nWe have the following simple proposition:\n\\begin{Pro}\nFor any $p_0=\\mbox{\\boldmath $X$}(u_0)$ and $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LF}_{\\kappa _N(\\mbox{\\scriptsize \\boldmath$n$}^T)_i(p_0,\\mbox{\\scriptsize \\boldmath$\\xi$}_0)}(u_0),$\nthere exists an integer $r $ with $1\\leq r\\leq s$ such that\n$M=\\mbox{\\boldmath $X$}(U)$ and $TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}$ have \ncorank $r$ contact at $p_0=\\mbox{\\boldmath $X$}(u_0).$\n\\end{Pro}\nBy Proposition 4.1, $M=\\mbox{\\boldmath $X$} (U)$ and the osculating hypersphere $TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}$\nhave corank $s$ contact at an $(\\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $\\xi$}_0)$-umbilical point.\nTherefore the AdS-lightlike ridge point is not an $(\\mbox{\\boldmath $n$}^T(u_0),\\mbox{\\boldmath $\\xi$}_0)$-umbilical point.\n\\par\nBy the general theory of unfoldings of function germs, the discriminant set ${\\mathcal D}_F$\nis non-singular at the origin if and only if the function $f=F|{\\mathbb R}^k\\times \\{0\\}$ has the $A_1$-type singularity (i.e., the\nMorse-type singularity).  \nTherefore we have the following proposition:\n\n\\begin{Pro} With the same notations as in the previous proposition, \nthe lightlike hypersurface $\\mathbb{LH}_M$ is non-singular at $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LH}_M((p_0,\\mbox{\\boldmath $\\xi$}_0),\\mu_0)$ if and only if\n$M=\\mbox{\\boldmath $X$}(U)$ and $TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}$ have the $A_1$-type contact at $p_0=\\mbox{\\boldmath $X$}(u_0).$\n\\end{Pro}\n\n\n\\par\nWe now consider the general contact of $M=\\mbox{\\boldmath $X$}(U)$ with AdS-lightcones as an application of  the theory of contact for submanifolds in Montaldi\\cite{Montaldi}.\nLet $X_i$ and $Y_i,$ $i=1,2,$ be submanifolds of ${\\mathbb R}^n$ with\n${\\rm dim}\\, X_1={\\rm dim}\\, X_2$ and ${\\rm dim}\\, Y_1={\\rm dim}\\, Y_2.$\nWe say that {\\it the contact of} $X_1$ and $Y_1$ at $y_1$ is same type as\n{\\it the contact of} $X_2$ and $Y_2$ at $y_2$ if there is a diffeomorphism germ\n$\\Phi :({\\mathbb R}^n,y_1)\\longrightarrow ({\\mathbb R}^n,y_2)$\nsuch that $\\Phi (X_1)=X_2$ and $\\Phi (Y_1)=Y_2.$\nIn this case we write\n$\nK(X_1,Y_1;y_1)=K(X_2,Y_2;y_2).\n$\nSince this definition of contact is local, we can replace ${\\mathbb R}^n$\nby arbitrary $n$-manifold.\nMontaldi gives in \\cite{Montaldi} the following characterization of\ncontact by using ${\\mathcal K}$-equivalence.\n\\begin{Th} Let $X_i$ and $Y_i,$ $i=1,2,$ be submanifolds of ${\\mathbb R}^n$ with\n${\\rm dim}\\, X_1={\\rm dim}\\, X_2$\nand ${\\rm dim}\\, Y_1={\\rm dim}\\, Y_2.$\nLet $g_i:(X_i,x_i)\\longrightarrow\n ({\\mathbb R}^n, y_i)$ be immersion germs and $f_i:({\\mathbb R}^n,y_i)\\longrightarrow ({\\mathbb R}^p,0)$\nbe submersion germs with $(Y_i,y_i)=(f_i^{-1}(0),y_i).$\nThen\n$$K(X_1,Y_1;y_1)=K(X_2,Y_2;y_2)$$ if and only if $ f_1\\circ g_1$ and\n$f_2\\circ g_2$ are ${\\mathcal K}$-equivalent.\n\\end{Th}\n\\par\nOn the other hand, we now return to the review on the theory of Legendrian singularities.\nWe introduce a natural equivalence relation among Legendrian submanifold germs.\nLet\n$F, G : ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ) \\longrightarrow  ({\\mathbb R},0)$  be\nMorse families of hypersurfaces. Then we say\nthat  $\\mathscr{L}_F(\\Sigma _*(F))$  and  $\\mathscr{L}_G(\\Sigma _*(G))$ are\n{\\it Legendrian equivalent} if there exists a contact diffeomorphism germ\n$H : (PT^*{\\mathbb R}^{n+1},z) \\longrightarrow  (PT^*{\\mathbb R}^{n+1},z')$  such that  $H$\npreserves fibers of  $\\pi$ and that $H(\\mathscr{L}_F(\\Sigma _*(F))) = \\mathscr{L}_G(\\Sigma _*(G))$, where $z=\\mathscr{L}_F(0), z'=\\mathscr{L}_G(0).$\nBy using the Legendrian equivalence, we can define the notion of Legendrian stability for\nLegendrian submanifold germs by the ordinary way (see, \\cite[Part III]{Arnold1}).\nWe can interpret the Legendrian equivalence  by using the notion of\ngenerating families.\nWe denote ${\\cal E}_k$ as the local ring of function germs $({\\mathbb\nR}^k,\\mbox{\\boldmath $0$} )\\longrightarrow\n{\\mathbb R}$\nwith the unique maximal ideal $\\mathfrak{M}_k=\\{h\\in {\\cal E}_k\\ |\\ h(0)=0\\ \\}.$\nLet  $F,G : ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )\n\\longrightarrow  ({\\mathbb R},\\mbox{\\boldmath $0$} )$  be function germs. We say that  $F$  and  $G$  are\n$ P$-${\\cal K}$-{\\it equivalent} if there exists a diffeomorphism germ\n$\\Psi  : ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ) \\longrightarrow\n({\\mathbb R}^k\\times\n{\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )$\nof the form  $\\Psi (x,u) = (\\psi _1(q,x), \\psi _{2}(x))$  for\n$(q,x) \\in  ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} )$  such that\n$\\Psi ^{*}(\\langle F\\rangle_{ {\\cal E}_{k+n+1}}) = \\langle G\\rangle_{{ \\cal\nE}_{k+n+1}}$.\nHere  $\\Psi ^{*} : {\\cal E}_{k+n+1} \\longrightarrow\n{\\cal E}_{k+n+1}$  is the pull back ${\\mathbb R}$-algebra isomorphism defined by\n$\\Psi ^{*}(h) =  h\\circ \\Psi$. We say that $F$ is an {\\it infinitesimally ${\\cal K}$-versal deformation of}\n$f = F\\vert\n{\\mathbb R}^k\\times\\{ \\mbox{\\boldmath $0$} \\}$ if\n$${\\cal E}_k =\nT_e({\\cal K})(f)\n+ \\left\\langle\n\\frac{\\partial F}{\\partial x_1}\\vert\n{\\mathbb R}^k\\times\\{ \\mbox{\\boldmath $0$} \\},\n\\dots ,\n\\frac{\\partial F}{\\partial x_{n+1}}\\vert\n{\\mathbb R}^k\\times\\{ \\mbox{\\boldmath $0$} \\} \\right\\rangle_{\\mathbb R},\n$$\nwhere\n$$\nT_e({\\cal K})(f) =\n\\left\\langle\n\\frac{\\partial f}{\\partial q_1}, \\dots,\n\\frac{\\partial f}{\\partial q_k}, f\\right\\rangle_{{\\cal E}_k}$$\n(see, \\cite{martine}).\nThe main result in the theory of Legendrian singularities (\\cite{Arnold1}, \\S 20.8 and \\cite{Zak}, THEOREM 2) is the\nfollowing:\n\\begin{Th}\n Let\n$F, G : ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ) \\longrightarrow  ({\\mathbb R},0)$  be\nMorse families of hypersurfaces.\nThen we have the following assertions:\n\\par\\noindent\n {\\rm (1)} $\\mathscr{L} _{F}(\\Sigma _*(F))$ and $\\mathscr{L} _{G}(\\Sigma _*(G))$  are Legendrian equivalent if and\nonly if\n $F$ and $G$  are  $P$-${\\cal K}$-equivalent, \n\\par\\noindent\n{\\rm (2)} $\\mathscr{L} _{F}(\\Sigma _*(F))$   is Legendrian stable if and only if  $F$  is an infinitesimally\n${\\cal K}$-versal deformation of  $f=F\\vert {\\mathbb R}^k\\times \\{\\mbox{\\boldmath $0$} \\}.$\n\n\\end{Th}\n\\par\nSince $F$ and $G$ are function germs on the common space germ $({\\mathbb\nR}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ),$\nwe do not need the notion of stably $P$-${\\cal K}$-equivalences in this\nsituation \\cite[page 27]{Zak}.\nFor any map germ $f:({\\mathbb R}^k,\\mbox{\\boldmath $0$})\\longrightarrow ({\\mathbb R}^p,\\mbox{\\boldmath $0$}),$\nwe define {\\it the local ring of} $f$ by\n$Q_r(f)={\\cal E}_k\/(f^*(\\mathfrak{M}_p){\\cal E}_n+\\mathfrak{M}_k^{r+1}).$\nWe have the following classification result of Legendrian stable germs (cf.\n\\cite[Proposition A.4]{Izu2}) which is the key for the purpose in this section.\n\\begin{Pro} Let\n$F, G : ({\\mathbb R}^k\\times {\\mathbb R}^{n+1},\\mbox{\\boldmath $0$} ) \\longrightarrow  ({\\mathbb R},0)$  be\nMorse families of hypersurfaces and $f=F|{\\mathbb R}^k\\times\\{\\mbox{\\boldmath $0$}\\},g=G|{\\mathbb R}^k\\times \\{\\mbox{\\boldmath $0$}\\}$.\nSuppose that $\\mathscr{L} _F(\\Sigma _*(F))$ and $\\mathscr{L} _G(\\Sigma _*(G))$ are Legendrian stable.\nThe the following conditions are equivalent\\\/{\\rm :}\n\\par\n{\\rm (1)} $(W(\\mathscr{L} _F),\\mbox{\\boldmath $0$})$ and $(W(\\mathscr{L} _G),\\mbox{\\boldmath $0$} )$ are diffeomorphic as set germs,\n\\par\n{\\rm (2)} $\\mathscr{L} _F(\\Sigma _*(F))$ and $\\mathscr{L} _G(\\Sigma _*(G))$ are Legendrian equivalent,\n\\par\n{\\rm (3)} $Q_{n+2}(f)$ and $Q_{n+2}(g)$ are isomorphic as ${\\mathbb R}$-algebras.\n\\end{Pro}\n\\par\nWe now describe the contacts of spacelike\nsubmanifolds in $AdS^{n+1}$ with AdS-lightcones.\nWe denote $Q   (\\mbox{\\boldmath $X$} ,u_0)$ as the  local ring of the function\ngerm $\\widetilde{h}_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}:(U,u_0)\n\\longrightarrow {\\mathbb R},$\nwhere $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LC}_M(u_0,\\mbox{\\boldmath $\\xi$}_0,\\mu_0).$\nWe remark that we can explicitly write the local ring as follows:\n$$\nQ_{n+2}(\\mbox{\\boldmath $X$} ,u_0)=\n\\frac{C^\\infty _{u_0}(U)}{\\displaystyle{\\langle \\langle \\mbox{\\boldmath $X$} (u),\\mbox{\\boldmath $\\lambda$} _0\\rangle +1 \\rangle _{C^\\infty\n_{u_0}(U)}}+\\mathfrak{M}_{u_0}(U)^{n+2}},\n$$\nwhere $C^\\infty _{u_0}(U)$ is the local ring of function germs on $U$ at\n$u_0.$\n\\par\nLet $\\mathbb{LH}_{M_i}(\\mbox{\\boldmath $n$}^T_i) :(N_1(M_i)[\\mbox{\\boldmath $n$}^T_i]\\times{\\mathbb R},(p_i,\\mbox{\\boldmath $\\xi$}_i,\\mu_i))\\longrightarrow (AdS^{n+1} ,\\mbox{\\boldmath $\\lambda$} _i),$\n$(i=1,2)$ be two\nlightlike hypersurface germs of spacelike submanifold germs \n$\\mbox{\\boldmath $X$} _i:(U,u^i)\\longrightarrow (AdS^{n+1},p_i).$ \nLet\n$H_i:(U\\times AdS^{n+1},(u^i,\\mbox{\\boldmath $\\lambda$}  _i))\\longrightarrow {\\mathbb\nR}$ be the AdS-height function germ of $\\mbox{\\boldmath $X$} _i.$\nThen we have the following theorem:\n\\begin{Th}\nLet $\\mbox{\\boldmath $X$} _i:(U,u^i)\\longrightarrow (AdS^{n+1},p_i),$ $i=1,2,$ be\nspacelike submanifold germs such that\nthe corresponding Legendrian submanifold germs $\\mathscr{L}_{H_i}(\\Sigma _*(H_i))$\nare Legendrian stable. We denote that $\\mbox{\\boldmath $X$}_i(U)=M_i.$\nThen\nthe following conditions are equivalent$:$\n\\par\n{\\rm (1)} $(\\mathbb{LH}_{M_1}(N_1(M_1)[\\mbox{\\boldmath $n$}^T_1]\\times{\\mathbb R}),\\mbox{\\boldmath $\\lambda$} _1) $ and\n$(\\mathbb{LH}_{M_2}(N_1(M_2)[\\mbox{\\boldmath $n$}^T_2]\\times{\\mathbb R}),\\mbox{\\boldmath $\\lambda$}_2) $\nare diffeomorphic,\n\\par\n{\\rm (2)} $(\\mathscr{L}_{H_1}(\\Sigma _*(H_1)), (u^1,\\lambda _1)$ and $(\\mathscr{L}_{H_2}(\\Sigma _*(H_2)), (u^2,\\lambda _2)$\nare Legendrian equivalent,\n\\par\n{\\rm (3)} $H_1$ and $H_2$ are $P$-${\\mathcal K}$-equivalent,\n\\par\n{\\rm (4)} $h_{1,\\lambda _1}$ and $h_{2,\\lambda _2}$ are ${\\cal K}$-equivalent,\n\\par\n{\\rm (5)} $K(M_1,TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _1}(M_1)_{p_1} ,p_1)=K(M_2,\nTLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$}  _2}(M_2)_{p_2},p_2).$\n\\par\n{\\rm (6)} $Q_{n+1}  (\\mbox{\\boldmath $X$} _1 ,u^1)$ and $Q_{n+1}  (\\mbox{\\boldmath $X$} _2\n,u^2)$ are isomorphic as ${\\mathbb R}$-algebras.\n\\end{Th}\n\n\\par\\noindent{\\it Proof. \\\/}\\ \nBy Proposition 5.6, the conditions (1), (2) and (6) are equivalent.\nThis condition is also equivalent to that\ntwo generating families\n$H_1$ and $H_2$ are $P$-${\\cal K}$-equivalent by Theorem 5.3. \nIf we denote $h_{i,\\mbox{\\scriptsize \\boldmath$\\lambda$}  _i}(u)=\nH_i(u,\\mbox{\\boldmath $\\lambda$}  _i),$ then\nwe have $h_{i,\\mbox{\\scriptsize \\boldmath$\\lambda$} _i}(u)=\\mathfrak{h}_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _i}\\circ\\mbox{\\boldmath $X$}\n_i(u).$\nBy Theorem 5.2, $K(\\mbox{\\boldmath $X$} _1(U),LC_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _1} ,p_1)=K(\\mbox{\\boldmath $x$} _2(U),LC{\\lambda _2},p_2)$\nif and only if $\\widetilde{h}_{1,\\mbox{\\scriptsize \\boldmath$\\lambda$} _1}$ and $\\widetilde{h}_{2,\\mbox{\\scriptsize \\boldmath$\\lambda$}\n_2}$ are ${\\mathcal K}$-equivalent.\nThis means that (4) and (5) are equivalent.\nBy definition, (3) implies (4).\nThe uniqueness of the infinitesimally $\\mathcal{K}$-versal deformation of $h_{i,\\mbox{\\scriptsize \\boldmath$\\lambda$} _i}$ (cf., \\cite{martine})\nleads that  (4) implies (3). This completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nFor a spacelike embedding germ $\\mbox{\\boldmath $X$}:(U,u_0)\\longrightarrow (AdS^{n+1},p_0)$, we consider\na set germ \n$\n(\\mbox{\\boldmath $X$} ^{-1}(TLC_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}(M)_{p_0}),u_0),\n$\nwhich is called the {\\it AdS-tangent lightcone indicatrix germ} of $\\mbox{\\boldmath $X$} ,$\nwhere $\\mbox{\\boldmath $\\lambda$}_0 =\\mathbb{LH}_M(p_0,\\mbox{\\boldmath $\\xi$} _0,\\mu_0)$ and\n$\\mu_0=-1\/\\kappa_N(\\mbox{\\boldmath $n$}^T)_i(p_0,\\mbox{\\boldmath $\\xi$}_0) (i=1,\\dots s).$\nWe have the following corollary of Theorem 5.7.\n\\begin{Co}\nWith the assumptions of Theorem {\\rm 5.7}, if the lightlike hypersurface germs\n$$\n(\\mathbb{LH}_{M_1}(N_1(M_1)[\\mbox{\\boldmath $n$}^T_1]\\times{\\mathbb R}),\\mbox{\\boldmath $\\lambda$} _1)\\ and\\\n(\\mathbb{LH}_{M_2}(N_1(M_2)[\\mbox{\\boldmath $n$}^T_2]\\times{\\mathbb R}),\\mbox{\\boldmath $\\lambda$}_2) \n$$ are diffeomorphic as set germs, then AdS-tangent lightcone indicatrix germs\n$$(\\mbox{\\boldmath $X$}_1 ^{-1}(TLC_{\\lambda_1}(M_1))_{p_1},u^1)\\ and \\\n(\\mbox{\\boldmath $X$} _2^{-1}(TLC_{\\lambda_2}(M_2)_{p_2}),u^2)\n$$ are diffeomorphic as set germs.\n\\end{Co}\n\\par\\noindent{\\it Proof. \\\/}\\ \nWe remark that the tangent lightcone indicatrix germ of $\\mbox{\\boldmath $X$}_i$ is the zero\nlevel set of\n$h_{i,\\lambda _i}.$\nSince ${\\mathcal K}$-equivalence among function germs preserves the\nzero-level sets of\nfunction germs, the assertion follows from Theorem 5.7.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nOn the other hand, we consider generic properties of lightlike hypersurfaces along spacelike submanifolds.\nLet ${\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1})$ be the space of spacelike embeddings with the \n$C^\\infty $-topology for an open set $U\\subset {\\mathbb R}^s.$\nWe consider the function\n$\n{\\mathcal H}:AdS^{n+1}\\times AdS^{n+1}\\longrightarrow {\\mathbb R}\n$\nagain.\nWe claim that $\\mathfrak{h}_{\\mbox{\\scriptsize \\boldmath$\\lambda$}}$ is a submersion at $\\mbox{\\boldmath $x$}\\not=\\mbox{\\boldmath $\\lambda$}$ for any $\\mbox{\\boldmath $\\lambda$} \\in AdS^{n+1}.$\nFor any $\\mbox{\\boldmath $X$}\\in {\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1})$, we have $H=\\mathcal{H}\\circ (\\mbox{\\boldmath $X$}\\times1_{AdS^{n+1}}).$ \nWe have the $r$-jet extension \n$j^r_1H:U\\times AdS^{n+1}\\longrightarrow J^r(U,{\\mathbb R})$ defined by $j^r_1H(u,\\mbox{\\boldmath $\\lambda$})=j^rh_{\\mbox{\\scriptsize \\boldmath$\\lambda$}}(u),$\nwhere $J^k(U,{\\mathbb R})$ is the $k$-jet space of functions on $U.$\nWe consider the trivialization\n$J^r(U,{\\mathbb R}) \\equiv U\\times{\\mathbb R}\\times J^r(s,1).$\nFor any submanifold $Q\\subset J^r(s,1),$ we denote that $\\widetilde{Q}=U\\times{\\mathbb R}\\times Q.$\nAs an application of \\cite[Lemma 6]{Wassermann}, the set\n\\[\nT_Q=\\{\\mbox{\\boldmath $X$}\\in {\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1})\\ |\\ j^r_1G\\ \\mbox{is transversal to}\\ \\widetilde{Q}\\ \\}\n\\]\nis a residual set of ${\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1}).$\nMoreover, if $Q$ is a closet subset , then $T_Q$ is open.\nIt is known \\cite{GWPL} that there exists a semi-algebraic set $W^r(s,1)\\subset J^k(s,1)$ and a stratification\n$\\mathcal{A}^r(s,1)$ of $J^k(s,1)\\setminus W^r(s,1)$ such that $\\lim _{k\\mapsto \\infty}{\\rm cod}\\, W^r(s,1)=+\\infty.$\nThe stratification $\\mathcal{A}^r(s,1)$ is called the {\\it canonical stratification}.\nWe define a stratification $\\mathcal{A}^r(U,{\\mathbb R})$ of $J^r(U,{\\mathbb R})\\setminus W^r(U,{\\mathbb R})$ by\n\\[\nU\\times({\\mathbb R}\\setminus \\{0\\})\\times (J^r(s,1)\\setminus W^r(s,1)),\\ U\\times \\{0\\}\\times \\mathcal{A}^r(s,1),\n\\]\nwhere $W^r(U,{\\mathbb R})=U\\times{\\mathbb R}\\times W^r(s,1).$\nIn \\cite{Wan}, it was shown  that if $j^r_1H(U\\times AdS^{n+1})\\cap W^r(U,{\\mathbb R})=\\emptyset$ and\n$j^r_1H$ is transversal to $\\mathcal{A}^r(U,{\\mathbb R})$, then the map $\\pi |H^{-1}(0):H^{-1}(0)\\longrightarrow AdS^{n+1}$ is MT-stable map-germ at each point,\nwhere $\\pi :U\\times{\\mathbb R}^{n+1}_1\\longrightarrow AdS^{n+1}$ is the canonical projection.\nHere, a map germ is said to be {\\it MT-stable} if the jet extension is transversal to the canonical stratification of the jet space of sufficiently higher order (cf., \\cite{GWPL,Mather2}).\nThe main result of the theory of Topological stability of Mather is that MT-stability implies topological stability.\nBy Proposition 4.1, the lightlike hypersurface $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$ is the discriminant set of\n$H$, which is equal to the critical value set of $\\pi|H^{-1}(0).$\nSince ${\\rm cod}\\, W^r(U,{\\mathbb R})> s+n+1$ for sufficiently large $k,$  the\nset\n\\[\n\\mathcal{O}_1=\\{\\mbox{\\boldmath $X$}\\in {\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1}_1)\\ |\\ j^r_1H(U\\times AdS^{n+1})\\cap W^r(U,{\\mathbb R})=\\emptyset\\ \\}\n\\]\nis a residual set.\nIt follows that the set\n\\[\n\\mathcal{O}=\\{\\mbox{\\boldmath $X$}\\in \\mathcal{O}_1\\ |\\ j^r_1H\\ \\mbox{is transversal to}\\ \\mathcal{A}^r(U,{\\mathbb R})\\ \\}\n\\]\nis a residual set. Therefore, we have the following theorem.\n\n\\begin{Th}\nThere exists a residual set $\\mathcal{O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1})$ such that\nfor any $\\mbox{\\boldmath $X$}\\in \\mathcal{O}$, the germ of the lightlike hypersurface  $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$ at any point\nis a germ of the critical value set of an MT-stable map germ.\n\\end{Th}\nIn the case when $n\\leq 5,$ by the classification results of the $\\mathcal{K}$-equivalence among function germs, the canonical stratification $\\mathcal{A}^k (s,1)$ is given by the finite collection of\nthe $\\mathcal{K}$-orbits. Moreover, if $j^r_1H$ is transversal to the $\\mathcal{K}$-orbit of $j^rh_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}(u_0)$\nfor sufficiently large $r,$ then $H$ is an infinitesimally $\\mathcal{K}$-versal deformation of $h_{\\mbox{\\scriptsize \\boldmath$\\lambda$}}$ at\n$(u_0,\\mbox{\\boldmath $\\lambda$} _0)$ \\cite{martine}. By Theorem 5.5, we have the following theorem. \n\\begin{Th} Suppose that $n\\leq 5.$ Then there exists a residual set ${\\mathcal O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^{n+1})$\nsuch that\nfor any\n$\\mbox{\\boldmath $X$} \\in {\\mathcal O},$ the germ of the lightlike\nhypersurface  $\\mathbb{LH}_M(\\mbox{\\boldmath $n$}^T)(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})$ at any point is the germ of the wave front set of a stable\nLegendrian submanifold germ $\\mathscr{L}_H(\\Sigma_*(H)).$\n\\end{Th}\n\n\\section{Spacelike submanifolds with codimension two}\nIn the case when $s=n-1,$ \n$N_1(M)[\\mbox{\\boldmath $n$}^T]$ is a double covering of $M=\\mbox{\\boldmath $X$}(U).$ \nWe can construct  a\nspacelike unit normal section $\\mbox{\\boldmath $n$}^S(u)\\in N_p(M)$ by\n\\[\n\\mbox{\\boldmath $n$} ^S(u)=\\frac{\\mbox{\\boldmath $X$}(u)\\wedge \\mbox{\\boldmath $n$}^T (u )\\wedge\\mbox{\\boldmath $X$} _{u_1}(u )\\wedge\\cdots \\wedge\n\\mbox{\\boldmath $X$} _{u_{n-1}}(u )}{\\|\\mbox{\\boldmath $X$}(u)\\wedge \\mbox{\\boldmath $n$}^T (u )\\wedge\\mbox{\\boldmath $X$} _{u_1}(u )\\wedge\\cdots\n\\wedge\\mbox{\\boldmath $X$} _{u_{n-1}}(u )\\|}.\n\\]\nThen $\\mbox{\\boldmath $\\sigma$} ^\\pm (u)=(\\mbox{\\boldmath $X$}(u),\\pm\\mbox{\\boldmath $n$}^S(u))$ are sections of \n$N_1(M)[\\mbox{\\boldmath $n$}^T]$.\nWe call $(\\mbox{\\boldmath $n$} ^T,\\mbox{\\boldmath $n$}^S)$ a {\\it adopted normal frame\\\/} along $M=\\mbox{\\boldmath $X$}(U).$ \nThe vectors\n$\\mbox{\\boldmath $n$}^T (u)\\pm \\mbox{\\boldmath $n$}^S(u)$ are null. Since\n$\\{\\mbox{\\boldmath $X$}_{u_1}(u),\\dots ,\\mbox{\\boldmath $X$}_{u_{n-1}}(u)\\}$ is a basis of $T_pM,$ the\nsystem $ \\{\\mbox{\\boldmath $X$}(u), \\mbox{\\boldmath $n$}^T (u),\\mbox{\\boldmath $n$} ^S(u),\\mbox{\\boldmath $X$}_{u_1}(u),\\dots\n,\\mbox{\\boldmath $X$}_{u_{n-1}}(u)\\} $ provides a basis for $T_p{\\mathbb R}^{n+2}_2$ such that\n$\\{\\mbox{\\boldmath $n$} ^T(u),\\mbox{\\boldmath $n$} ^S(u)\\}$ is a pseudo-orthonormal frame of the normal timelike plane $N_p(M)\\cap T_pAdS^{n+1}$\nin $AdS^{n+1}.$\n\\begin{Lem}\\label{parallel}\nGiven two adopted unit timelike normal sections $\\mbox{\\boldmath $n$}^T(u),\n\\bar{\\mbox{\\boldmath $n$}}^T(u)\\in N_p(M),$  the corresponding nullcone normal\nsections $\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u), \\bar{\\mbox{\\boldmath $n$}}^T(u)\\pm\\bar{\\mbox{\\boldmath $n$}}^S(u)$ are\nparallel.\n\\end{Lem}\n\\par\\noindent{\\it Proof. \\\/}\\  We consider the orientation and the timelike orientation on\nthe normal space $N_p(M)$ induced by the orientation and the\ntimelike orientation of ${\\mathbb R}^{n+1}_1$ and $\\{\\mbox{\\boldmath $X$}_{u_1}(u),\\dots\n,\\mbox{\\boldmath $X$}_{u_{n-1}}(u)\\}.$ By the construction, both the\npseudo-orthogonal basis $\\{\\mbox{\\boldmath $n$} ^T(u),\\mbox{\\boldmath $n$}^S(u)\\}$ and\n$\\{\\bar{\\mbox{\\boldmath $n$}}^T(u),\\bar{\\mbox{\\boldmath $n$}}^S(u)\\}$ of $N_p(M)\\cap T_pAdS^n$ correspond to the\nsame orientation and the same timelike orientation on $N_p(M)\\cap T_pAdS^n.$\nSince both of $\\mbox{\\boldmath $n$}^T(u)$ and $\\bar{\\mbox{\\boldmath $n$}}^T(u)$ are adopted\nand $\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u), \\bar{\\mbox{\\boldmath $n$}}^T(u)\\pm\\bar{\\mbox{\\boldmath $n$}}^S(u)$ are\nnull in the Lorentz plane $N_p(M)\\cap T_pAdS^{n+1}$, $\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u)$ and  $\\bar{\\mbox{\\boldmath $n$}}^T(u)\\pm\\bar{\\mbox{\\boldmath $n$}}^S(u)$ are\nparallel. This completes the proof. \\hfill $\\Box$\\vspace{3truemm} \\par\nTherefore, the null cone Gauss images of $M=\\mbox{\\boldmath $X$}(U)$ with respect to $(\\mbox{\\boldmath $n$}^T,\\mbox{\\boldmath $n$}^S)$ are given by\n$\n\\mathbb{NG}(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u)=\\mbox{\\boldmath $n$}^T(u)\\pm\\mbox{\\boldmath $n$}^S(u).\n$\nSince $N_p(M)[\\mbox{\\boldmath $n$}^T]$ is a spacelike line in $N_p(M),$ we have $\\mbox{\\boldmath $\\xi$}=\\mbox{\\boldmath $n$}^S(u)$ or $\\mbox{\\boldmath $\\xi$} =-\\mbox{\\boldmath $n$}^S(u)$ for any $(\\mbox{\\boldmath $X$}(u),\\mbox{\\boldmath $\\xi$})\\in N_1(M)[\\mbox{\\boldmath $n$}^T].$ \nWe denote that $\\kappa _i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S) (u) $, $i=1,\\dots , n-1$\ninstead of $\\kappa _N(\\mbox{\\boldmath $n$}^T)_i(p,\\mbox{\\boldmath $\\xi$})$, $i=1,\\dots , n-1$ for $\\mbox{\\boldmath $\\xi$} =\\pm\\mbox{\\boldmath $n$}^S(u)$ \nand $p=\\mbox{\\boldmath $X$}(u).$\nThen we have the following decomposition of the lightlike hypersurface along $M=\\mbox{\\boldmath $X$}(U)$:\n\\[\n\\mathbb{LH}_M(N_1(M)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})=\\mathbb{LH}_M^+(U\\times{\\mathbb R})\\cup \\mathbb{LH}_M^-(U\\times{\\mathbb R}),\n\\]\nwhere\n\\[\n\\mathbb{LH}_M^\\pm (u,\\mu)= \\mbox{\\boldmath $X$}(u)+\\mu (\\mbox{\\boldmath $n$}^T\\pm\\mbox{\\boldmath $n$}^S)(u).\n\\]\nIn this case the critical value of $\\mathbb{LH}_M^\\pm $ is the point where\n$\\kappa_i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(p)\\not= 0$ and\n\\[\n\\mbox{\\boldmath $\\lambda$}^\\pm =\\mbox{\\boldmath $X$}(u)+\\frac{1}{\\kappa_i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u)}(\\mbox{\\boldmath $n$}^T\\pm\\mbox{\\boldmath $n$}^S)(u).\n\\]\nFor each $i=1,\\dots ,n-1,$  we have a mapping\n$\n\\mathbb{LE}_{\\kappa_i(\\mbox{\\scriptsize \\boldmath$n$}^T,\\pm\\mbox{\\scriptsize \\boldmath$n$}^S)} :O_i\\longrightarrow AdS^{n+1}\n$\ndefined by\n\\[\n\\mathbb{LE}_{\\kappa_i(\\mbox{\\scriptsize \\boldmath$n$}^T,\\pm\\mbox{\\scriptsize \\boldmath$n$}^S)}(u)=\\mbox{\\boldmath $X$}(u)+\\frac{1}{\\kappa_i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u)}(\\mbox{\\boldmath $n$}^T\\pm\\mbox{\\boldmath $n$}^S)(u),\n\\]\nwhere $O_i=\\{u\\in U\\ |\\ \\kappa_i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u)\\not= 0\\ \\}.$\nThen we define\n\\[\n\\mathbb{LE}^\\pm_M=\\bigcup\\left\\{\\mathbb{LE}_{\\kappa_i(\\mbox{\\scriptsize \\boldmath$n$}^T,\\pm\\mbox{\\scriptsize \\boldmath$n$}^S)}(u)\\ |\\ u\\in U\\ \\mbox{such\\ that}\\ \\kappa_i(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u)\\not= 0,i=1,\\dots ,n-1. \\right\\}.\n\\]\nBy the above arguments, we know that $\\mathbb{LE}^\\pm_{M}$ is nothing but the AdS-lightlike focal set of $M=\\mbox{\\boldmath $X$}(U)$\nrelative to $\\mbox{\\boldmath $n$}^T.$\nHowever, we call it the {\\it AdS-lightlike evolute} of $M=\\mbox{\\boldmath $X$}(U)$ in the case when ${\\rm codim}\\, M=2.$\nFor any $p_0=\\mbox{\\boldmath $X$}(u_0),$ we have the tangent AdS-lightcones $TLC^\\pm_{\\mbox{\\scriptsize \\boldmath$\\lambda$}_0}(M)_{p_0},$\nwhere $\\mbox{\\boldmath $\\lambda$} ^\\pm_0=\\mathbb{LE}_{\\kappa_i(\\mbox{\\scriptsize \\boldmath$n$}^T,\\pm\\mbox{\\scriptsize \\boldmath$n$}^S)}(u_0).$\nIn the general codimension case, it depends on the choice of $\\mbox{\\boldmath $n$}^S$, so that there are infinitely many tangent AdS-lightcones of $M$ at\n$p_0=\\mbox{\\boldmath $X$}(u_0)$.\nHowever, the codimension two case, $\\mbox{\\boldmath $n$}^S$ is uniquely determined by $\\mbox{\\boldmath $n$}^T$.\nTherefore, we have two tangent AdS-lightcones of $M$ at $p_0=\\mbox{\\boldmath $X$}(u_0).$\nIn this case, each one of the tangent AdS-lightcones $TLC^\\pm _{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(M)_{p_0}$ is called\n an {\\it osculating AdS-lightcone} of $M=\\mbox{\\boldmath $x$} (U)$.\nThe AdS-lightlike evolutes are the loci of the vertices of osculating AdS-lightcones of $M.$\nAnalogous to the case of hypersurfaces in Euclidean space, we call $\\mathbb{LE}^\\pm_M$ the {\\it AdS-lightlike evolute} of $M=\\mbox{\\boldmath $X$}(U).$\n\n\\par\nWe now consider the low dimensions cases in the following subsections.\n\\subsection{Spacelike curves in $AdS^3$}\nWe consider spacelike curves in $AdS^3$ as a simplest case of the codimension two case.\nLet\n$\\mbox{\\boldmath $\\gamma$}:I\\longrightarrow AdS^3$ be a unit speed spacelike curve with $\\langle \\mbox{\\boldmath $\\gamma$} ''(s),\\gamma ''(s)\\rangle\\not= -1$, where $I$ is an open interval.\nWe denote that $C=\\gamma (I).$\nThen we define $\\mbox{\\boldmath $t$}(s)= \\mbox{\\boldmath $\\gamma$} '(s)$ and call $\\mbox{\\boldmath $t$}(s)$ a {\\it unit tangent vector\\\/ }\nof  $\\mbox{\\boldmath $\\gamma$}$ at $s.$\nThe {\\it curvature \\\/} of $\\mbox{\\boldmath $\\gamma$}$ at $s$ is defined to be\n$\\kappa _g(s)=\\sqrt{|\\langle \\mbox{\\boldmath $\\gamma$} (s)-\\mbox{\\boldmath $\\gamma$} ''(s),\\mbox{\\boldmath $\\gamma$} (s)-\\mbox{\\boldmath $\\gamma$}''(s)\\rangle | }.$\nSince $\\kappa _g(s)\\not= 0,$ the {\\it unit principal normal vector\\\/} $\\mbox{\\boldmath $n$}(s)$\nof the curve $\\mbox{\\boldmath $\\gamma$}$ at $s$ is defined by \n$\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$} (s)=\\kappa _g(s)\\mbox{\\boldmath $n$}(s).$\nWe denote that $\\delta{(\\mbox{\\boldmath $\\gamma$}(s))}={\\rm sign}(\\mbox{\\boldmath $n$}(s))$.\nThe unit vector $\\mbox{\\boldmath $b$}(s)=\\mbox{\\boldmath $\\gamma$} (s)\\wedge \\mbox{\\boldmath $t$}(s)\\wedge \\mbox{\\boldmath $n$}(s)$ is called a {\\it unit binormal vector}\nof the curve $\\mbox{\\boldmath $\\gamma$} $ at $s.$\nSince $\\mbox{\\boldmath $\\gamma$} (s)$ is timelike and $\\mbox{\\boldmath $t$}(s)$ is spacelike, we have $\\langle\\mbox{\\boldmath $b$}(s),\\mbox{\\boldmath $b$}(s)\\rangle =-\\delta{(\\mbox{\\boldmath $\\gamma$}(s))}$ and\n${\\rm sign}\\,({\\mbox{\\boldmath $\\gamma$}}'(s))=1$\nThen the following Frenet-Serret type formulae hold:\n$$ \\left\\{\n\\begin{array}{l}\n\\mbox{\\boldmath $\\gamma$} '(s)=\\mbox{\\boldmath $t$} (s) \\\\\n\\mbox{\\boldmath $t$}'{(s)}= \\kappa _g(s)\\mbox{\\boldmath $n$}(s)+\\mbox{\\boldmath $\\gamma$} (s), \\\\\n\\mbox{\\boldmath $n$}'{(s)}=\\delta(\\mbox{\\boldmath $\\gamma$}(s))(-\\kappa_g(s) \n\\mbox{\\boldmath $t$}(s)+ \\tau_g{(s)}\\mbox{\\boldmath $b$}(s)), \\\\\n\\mbox{\\boldmath $b$}'{(s)}= \\delta (\\mbox{\\boldmath $\\gamma$}(s))\\tau_g{(s)}\\mbox{\\boldmath $n$}(s),\n\\end{array}\n\\right.\n$$\nwhere $\\tau_g{(s)}=\\langle \\mbox{\\boldmath $b$}'(s),\\mbox{\\boldmath $n$}(s)\\rangle$ is the torsion of the curve $\\mbox{\\boldmath $\\gamma$}$ at $s$.\nIn \\cite{Chen-Han} the singularities of lightlike hypersurfaces along spacelike curves in $AdS^3$ are classified.\nWe define an invariant $\\sigma ^\\pm(s)=\\kappa _g'(s)\\mp\\kappa _g(s)\\tau _g(s).$\nThen we have the following theorem.\n\\begin{Th}[\\cite{Chen-Han}]\nLet\n$\\mbox{\\boldmath $\\gamma$}:I\\longrightarrow AdS^3$ be a unit speed spacelike curve with $\\langle \\mbox{\\boldmath $\\gamma$} ''(s),\\gamma ''(s)\\rangle\\not= -1$.\nThen we have the following\\\/{\\rm :}\n\\par\\noindent\n{\\rm (1)} The AdS-lightlike evolute of $\\gamma$ is\n$$\n\\mathbb{LE}_C=\\left\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$} (s)+\\frac{1}{\\delta (\\mbox{\\boldmath $\\gamma$} (s))\\kappa _g(s)}(\\mbox{\\boldmath $n$}(s)\\pm \\mbox{\\boldmath $b$}(s))\\right\\}.\n$$\n\\par\\noindent \n{\\rm (2)} The germ of $\\mathbb{LH}_M$ at $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LE}_{\\kappa _g}(s_0)$ is diffeomorphic to\nthe cuspidal edge $C(2,3)\\times{\\mathbb R}$ if and only if $\\sigma ^\\pm(s_0)\\not= 0.$\n\\par\\noindent\n{\\rm (3)} The germ of $\\mathbb{LH}_M$ at $\\mbox{\\boldmath $\\lambda$} _0=\\mathbb{LE}_{\\kappa _g}(s_0)$ is diffeomorphic to\nthe swallowtail $SW$ if and only if $\\sigma ^\\pm(s_0)= 0$ and $(\\sigma^\\pm) '(s_0)\\not= 0.$\n\\end{Th}\n\\par\nHere, $C(2,3)\\times {\\mathbb R}=\\{(x_1,x_2)\\ | x_1^2-x_2^3=0\\}\\times {\\mathbb R}$ is the {\\it cuspidal edge\\\/} and\n$SW=\\{(x_1,x_2,x_3)\\ | x_1 =3u^2+u^2v,x_2=4u^3+2uv,x_3=v, (u,v)\\in ({\\mathbb R}^2,0)\\}$\nis the {\\it swallowtail\\\/}.\n\\par\nIt has been also shown the following geometric characterizations of the singularities of\nAdS-lightlike hypersurfaces in \\cite{Chen-Han}.\n\\begin{Pro} Let\n$\\mbox{\\boldmath $\\gamma$}:I\\longrightarrow AdS^3$ be a unit speed spacelike curve with $\\langle \\mbox{\\boldmath $\\gamma$} ''(s),\\gamma ''(s)\\rangle\\not= -1$.\nThen we have the followings{\\rm :}\n\\par\\noindent\n{\\rm (1)} $\\kappa _g(s_0)\\not= 0$ and $\\sigma ^\\pm (s_0)\\not=0$ if and only if $C=\\mbox{\\boldmath $\\gamma$} (I)$ and\nthe osculating AdS-lightcone $TLC^\\pm_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(C)_{p_0}$ have contact of order $2.$\n\\par\\noindent\n{\\rm (2)} $\\kappa _g(s_0)\\not= 0$, $\\sigma ^\\pm (s_0)=0$ and $(\\sigma ^\\pm)'(s_0)\\not= 0$ if and only if $C=\\mbox{\\boldmath $\\gamma$} (I)$ and\nthe osculating AdS-lightcone $TLC^\\pm_{\\mbox{\\scriptsize \\boldmath$\\lambda$} _0}(C)_{p_0}$ have contact of order $3.$\n\\end{Pro}\n\\subsection{Spacelike surfaces in $AdS^4$}\nWe consider spacelike surfaces in $AdS^4$ here.\nLet $\\mbox{\\boldmath $X$}:U\\longrightarrow AdS^4$ be a spacelike embedding from an open subset $U\\subset {\\mathbb R}^2.$\nAs a corollary of Theorem 5.10, we have the following generic classification theorem.\nWe say that two map germs $f,g:({\\mathbb R}^n,0)\\longrightarrow ({\\mathbb R}^p,0)$ are {\\it $\\mathcal{A}$-equivalent} if\nthere exists diffeomorphism germs $\\phi :({\\mathbb R}^n,0)\\longrightarrow ({\\mathbb R}^n,0)$ and $\\psi :({\\mathbb R}^p,0)\\longrightarrow ({\\mathbb R}^p,0)$\nsuch that $f\\circ \\phi =\\psi\\circ g.$\n\n\\begin{Th}\nThere exists an open dense subset ${\\cal O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^4)$\nsuch that\nfor any\n$\\mbox{\\boldmath $X$} \\in {\\cal O},$ the germ of the corresponding lightlike hypersurfaces\n$\\mathbb{LH}^\\pm _M$ at any point $(u_0,\\mu_0)\\in U\\times {\\mathbb R}$\nis ${\\mathcal A}$-equivalent to one of the map germs $A_k$ $(1\\leq k\\leq 4)$ or\n$D_4^\\pm :$ where, $A_k,\\ D^\\pm_4$-map germ $f:({\\mathbb R}^3,0)\\longrightarrow ({\\mathbb R}^4,0)$ are given by\n\\par\n$A_1;\\  f(u_1,u_2,u_3)=(u_1,u_2,u_3,0),$\n\\par\n$A_2; \\  f(u_1,u_2,u_3)=(3u_1^2,2u_1^3,u_2,u_3),$\n\\par\n$A_3;\\   f(u_1,u_2,u_3)=(4u^3_1+2u_1u_2,3u^4_1+u_2u_1^2,u_2,u_3),$\n\\par\n$A_4;\\ \nf(u_1,u_2,u_3)=(5u_1^4+3u_2u_1^2+2u_1u_3,4u_1^5+2u_2u_1^3+u_3u_1^2,u_2,u_3),\n$\n\\par\n$D^+_4;\\ \nf(u_1,u_2,u_3)=(2(u_1^3+u_2^3)+u_1u_2u_3,3u_1^2+u_2u_3,3u_2^2+u_1u_3,u_3),$\n\\par\n$D^-_4;\\ \n\\displaystyle{f(u_1,u_2,u_3)=\\left(\\left(\\frac{u_1^3}{3}-u_1u_2^2\\right)+(u_1^2+u_2^2)u_3,u_2^2-u_1^2-2u_1u_3,2(u_1u\n_2-u_2u_3),u_3\\right).}$\n\\end{Th}\n\\par\\noindent{\\it Proof. \\\/}\\ \nBy Theorems 5.5 and 5.10,  the AdS-height function  $H$ on $M$ is a\n${\\mathcal K}$-versal  deformation of $h_{\\lambda _0}$ at each \n$(u_0,\\mbox{\\boldmath $\\lambda$} _0)\\in U\\times AdS^4.$\nTherefore we can apply the classification of ${\\mathcal K}$-versal\ndeformations $F(x,y,\\mbox{\\boldmath $\\lambda$})$ of function germs up to\n$4$-parameters \\cite{Arnold1}.\nFor any $F(x,y,\\mbox{\\boldmath $\\lambda$}),$ we have\n\\[\n\\Sigma _*(F)=\\Bigl\\{(x,y,\\mbox{\\boldmath $\\lambda$})\\in ({\\mathbb R}^2\\times{\\mathbb R}^4,0)\\ \\Bigm|\\ F(x,y,\\mbox{\\boldmath $\\lambda$})=\\frac{\\partial F}{\\partial x}(x,y,\\mbox{\\boldmath $\\lambda$})=\n\\frac{\\partial F}{\\partial y}(x,y,\\mbox{\\boldmath $\\lambda$})=0\\Bigr\\}.\n\\]\nThe normal forms are given by\n \\begin{eqnarray*}\n A_k;&{}& \\!\\!\\!\\!\\!\\!\\! F(x,y,\\mbox{\\boldmath $\\lambda$} )=x^{k+1}\\pm y^2+\\lambda _1+\\lambda _2x+\\cdots +\\lambda\n_{k-1}x^{k-1},\\ 1\\leq k\\leq 4, \\\\\nD^+_4;&{}& \\!\\!\\!\\!\\!\\!\\! F(x,y,\\mbox{\\boldmath $\\lambda$} )=x^3+y^3+\\lambda _1+\\lambda _2x+\\lambda _3y+\\lambda _4xy, \\\\\nD^-_4;&{}& \\!\\!\\!\\!\\!\\!\\! F(x,y,\\mbox{\\boldmath $\\lambda$} )= \\frac{x^3}{3}-xy^2+\\lambda _1+\\lambda _2x+\\lambda _3y+\\lambda\n_4(x^2+y^2).\n\\end{eqnarray*}\nFor example, if we consider the germ given by\n$$\nF(x,y,\\mbox{\\boldmath $\\lambda$} )=x^3+y^3+\\lambda _1+\\lambda _2x+\\lambda _3y+\\lambda _4xy.\n$$\nThen we get\n$$\n\\Sigma _*(F)=\\{(x,y,2(x^3+y^3)+\\lambda _4xy,-3x^2-\\lambda _4y,-3y^2-\\lambda_4x,\\lambda _4)\\ |\\ (x,y,\\lambda _4)\\in {\\mathbb R}^3\n\\}.\n$$\nTherefore the corresponding Legendrian map germ is\n$$\nf(u_1,u_2,u_3)=(2(u_1^2+u_2^2)+u_1u_2u_3,3u_1^2+u_2u_3,3u_2^2+u_1u_3,u_3)\\quad (D^+_4) .\n$$\nThe other cases follow from similar arguments, so that we omit the details.\n\\hfill $\\Box$\\vspace{3truemm} \\par\nAs a corollary of the above theorem, we have the following generic local classification of\nAdS-lightlike evolutes along spacelike surfaces.\nWe define that $C(2,3,4)=\\{(u_1^2,u_1^3,u_1^4)\\ |\\ u_1\\in {\\mathbb R}\\}$ which is called\na {\\it $(2,3,4)$-cusp}.\nWe also define that\n$C(BF)=\\{(10u_1^3+3u_2u_1,5u_1^4+u_2u_1^2,6u_1^5+u_2u_1^3,u_2)\\ |\\ (u_1,u_2)\\in {\\mathbb R}^2\\}$.\nWe call $C(BF)$ a {\\it C-butterfly} (i.e., the critical value set of the butterfly).\nFinally we define that where $C(2,3,4,5)=\\{(u_1^2,u_1^3,u_1^4,u_1^5)\\ |\\ u_1\\in {\\mathbb R}\\}$ which is called\na {\\it $(2,3,4,5)$-cusp}.\n\n\\begin{Co} There exists an open dense subset ${\\cal O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^4)$\nsuch that\nfor any\n$\\mbox{\\boldmath $X$} \\in {\\cal O},$ the germ of the corresponding AdS-lightlike evolute\n$\\mathbb{LE}^\\pm _M$ at any point $(u_0,\\mu_0)\\in U\\times {\\mathbb R}$ is diffeomorphic to one of \nthe following set germs at the origin in ${\\mathbb R}^4${\\rm:}\n\\par\\noindent\n$A_2; \\ \\{(0,0)\\}\\times{\\mathbb R}^2,$\n\\par\\noindent\n$A_3;\\  C(2,3,4)\\times {\\mathbb R},$\n\\par\\noindent\n$A_4;\\ C(BF),\n$\n\\par\\noindent\n$D^+_4;\\ \n\\{(2(u_1^3+u_2^3)+u_1u_2u_3,3u_1^2+u_2u_3,3u_2^2+u_1u_3,u_3)\\ |\\ u_3^2=36u_1u_2\\},$\n\\par\\noindent\n$D^-_4;\\ \n\\displaystyle{\\left\\{\\left(\\left(\\frac{u_1^3}{3}-u_1u_2^2\\right)+(u_1^2+u_2^2)u_3,u_2^2-u_1^2-2u_1u_3,2(u_1u\n_2-u_2u_3),u_3\\right)\\ \\Bigl|\\ u_3^2=u_1^2+u_2^2\\right\\}.}$\n\\end{Co} \n\\par\\noindent{\\it Proof. \\\/}\\ \nFor $A_3$, we can calculate the Jacobi matrix of the normal form $f$ in Theorem 6.4:\n\\[\nJ_f=\\left(\n\\begin{array}{ccc}\n12u^2_1+2u_1 & 2u_1 & 0 \\\\\n12u^3_1+2u_1u_2 & u^2_1 & 0 \\\\\n0 & 1 &0 \\\\\n0 & 0 & 1\n\\end{array}\n\\right),\n\\]\nso that ${\\rm rank}\\, J_f <3$ if and only if $6u^2_1+u_2=0.$\nThus, the critical value set of $f$ is\n$$C(f)=\\{(-8u^3_1,-3u^4_1,-6u^2_1,u_3)\\ |\\ (u_1,u_3)\\in {\\mathbb R}^2\\}.$$\nIt is $C(2,3,4)\\times {\\mathbb R}.$\nBy the similar calculation, we can show that the germ of $A_4$ is diffeomorphic to $C(BF).$\nFor $D^+_4,$ we can calculate the Jacobi matrix o the normal form $f$:\n\\[\nJ_f=\\left(\n\\begin{array}{ccc}\n6u^2_1+u_2u_3 & 6u_2^2+u_1u_3,u_1u_2 & 0 \\\\\n6u_1 & u_3 & u_2 \\\\\nu_3 & 6u_2 &u_1 \\\\\n0 & 0 & 1\n\\end{array}\n\\right).\n\\]\nTherefore, ${\\rm rank}\\,J_f<3$ if and only if\n\\[\n\\left|\\begin{array}{cc}\n6u^2_1+u_2u_3 & 6u_2^2+u_1u_3,u_1u_2 \\\\\n6u_1 & u_3  \\\\\n\\end{array}\n\\right|=\n\\left|\n\\begin{array}{cc}\n6u^2_1+u_2u_3 & 6u_2^2+u_1u_3,u_1u_2 \\\\\nu_3 & 6u_2\\\\\n\\end{array}\n\\right|=\n\\left|\n\\begin{array}{cc}\n6u_1 & u_3  \\\\\nu_3 & 6u_2 \\\\\n\\end{array}\n\\right|=0,\n\\]\nwhich is equivalent to the condition that \n$u_3^2=36u_1u_2.$\nFor $D^-_4,$ by the similar calculation to the above, we have the condition that\n$u_3^2=u_1^2+u_2^2.$\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nIn the list of the above corollary, \none of the projections of the image of $D^+_4$ into ${\\mathbb R}^3$ is $PU=\\{(3u_1^2+u_2u_3,3u_2^2+u_1u_3,u_3)\\ |\\ u_3^2=36u_1u_2\\}$ \nwhich is called a {\\it purse} and one of the projection of the image of $D^-_4$ is\ncalled a {\\it pyramid} given by \n\\[\nPY=\\left\\{\\left(u_2^2-u_1^2-2u_1u_3,2(u_1u\n_2-u_2u_3),u_3\\right)\\ \\Bigl|\\ u_3^2=u_1^2+u_2^2\\right\\}.\n\\]\nWe can draw these pictures as follows:\nWe denote that $\\widetilde{PU}$ as the set of $D^+_4$ and $\\widetilde{PY}$ as the set of $D^-_4$\nin the list of Corollary 6.5, respectively.\nWe also have a classification of the singularities of $\\mathbb{LE}^\\pm _M$ as a corollary of Theorem 6.4\nand Corollary 6.5.\nThe set of singularities\n of $\\mathbb{LE}^\\pm _M$ is denoted by $\\Sigma (\\mathbb{LE}^\\pm _M).$ \n\\begin{Co}\nThere exists an open dense subset ${\\cal O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^4)$\nsuch that\nfor any\n$\\mbox{\\boldmath $X$} \\in {\\cal O},$ the germ of the pair \n$(\\mathbb{LE}^\\pm _M,\\Sigma(\\mathbb{LE}^\\pm _M))$ at any point $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\Sigma(\\mathbb{LE}^\\pm_M)$ is diffeomorphic to one of \nthe following pairs of set germs at the origin in ${\\mathbb R}^4${\\rm:}\n\\par\\noindent\n$A_3;\\ (C(2,3,4)\\times {\\mathbb R}, \\{(0,0,0)\\}\\times {\\mathbb R}),$\n\\par\\noindent\n$A_4;\\ (C(BF),C(2,3,4,5)),\n$\n\\par\\noindent\n$D^+_4;\\ (\\widetilde{PU},\\Sigma (\\widetilde{PU})),\n$\nwhere\n$\\Sigma (\\widetilde{PU})=\\{(5u^3_3\/108,u^2_3\/4,u^2_3\/4,u_3)\\ |\\ u_3\\in {\\mathbb R}\\},$\n\\par\\noindent\n$D^-_4;\\ (\\widetilde{PY},\\Sigma (\\widetilde{PY})),\n$\nwhere\n\\par\\noindent\n$\n\\Sigma (\\widetilde{PY}))=\\{\\left(4u^3_3\/3,-3u^2_3,0,u_3\\right)\\ |\\ u_3\\in {\\mathbb R}\\}\n\\cup \\\\\n\\hfill \\{(4u^3_3\/3,3u^2_3\/2,-3\\sqrt{3}u^2_3\/2,u_3)\\ |\\ u_3\\in {\\mathbb R}\\}\\cup \n\\{(4u^3_3\/3,3u^2_3\/2,3\\sqrt{3}u^2_3\/2,u_3)\\ |\\ u_3\\in {\\mathbb R}\\}.\n$\n\\end{Co}\n\\par\\noindent{\\it Proof. \\\/}\\ \nFor $A_3$, in order to detect the singularities, we calculate the Jacobi matrix\nof $f(u_1,u_2)=(u_1^2,u_1^3,u_1^4,u_2).$ Then we have\n\\[\nJ_f=\\left(\n\\begin{array}{cc}\n2u_1 & 0 \\\\\n3u_1^2 & 0 \\\\\n4u_1^3 & 0 \\\\\n0 & 1\n\\end{array}\n\\right),\n\\]\nso that ${\\rm rank}\\, J_f<2$ if and only if $u_1=0.$\nThis means that $\\Sigma (C(2,3,4)\\times{\\mathbb R})=\\{(0,0,0)\\}\\times {\\mathbb R}.$\nSimilar to the case $A_3,$ we calculate the Jacobi matrix\nof $f(u_1,u_2)=(10u_1^3+3u_2u_1,5u_1^4+u_2u_1^2,6u_1^5+u_2u_1^3,u_2)$.\nThen we have\n\\[\nJ_f=\\left(\n\\begin{array}{cc}\n30u_1^2+3u_2 & 3u_1 \\\\\n20u_1^3+2u_2u_2 & u_1^2 \\\\\n30u_1^4+3u_2u_1^2 & u_1^3 \\\\\n0 & 1\n\\end{array}\n\\right),\n\\]\nso that ${\\rm rank}\\, J_f<2$ if and only if $u_2=-10u_1^2.$\nTherefore, we have\n\\[\nf(u_1,-10u_1^2)=(-20u_1^3,-5u_1^4,-4u_1^5,-10u_1^2).\n\\]\nThis means that $\\Sigma (C(BF))=C(2,3,4,5).$\nFor $D^+_4,$ we consider the following parameter transformation:\n\\[\nu_1+u_2=\\frac{u_3}{3}\\cosh\\phi,\\ u_1-u_2=\\frac{u_3}{3}\\sinh\\phi.\n\\]\nThen $\\widetilde{PU}$ is parametrized by\n\\[\nf(\\phi,u_3)=\\left(\\frac{4}{6^3}u_3^3\\left(\\cosh 3\\phi+\\frac{1}{4}\\right),\\frac{u_3^2}{6^2}\\left(3e^{2\\phi}+6e^{-\\phi}\\right),\n\\frac{u_3^2}{6^2}\\left(3e^{-2\\phi}+e^\\phi\\right),u_3\\right).\\]\nThus , the Jacobi matrix is \n\\[\nJ_f=\\left(\n\\begin{array}{cc}\n\\displaystyle{\\frac{1}{18}u_3^3\\sinh3\\phi} & \\displaystyle{\\frac{1}{18}u_3^2(\\cosh3\\phi +\\frac{1}{4})}\\\\\n\\displaystyle{\\frac{u_3^2}{6}(e^{2\\phi}-e^{-\\phi})} & \\displaystyle{\\frac{1}{6}(e^{2\\phi}+2e^{-\\phi})} \\\\\n\\displaystyle{\\frac{u_3^2}{6}(-e^{-2\\phi}+e^\\phi)} & \\displaystyle{\\frac{1}{6}(e^{-2\\phi}+2e^\\phi)}\\\\\n0 & 1\n\\end{array}\n\\right),\n\\]\nso that ${\\rm rank}\\, J_f<2$ if and only if $\\phi=0$ or $u_3=0.$\nThis means that $u_1=u_3\/6,\\ u_2=u_3\/6.$\nTherefore, we have\n\\[\\Sigma (\\widetilde{PU})=\\left\\{\\left(\\frac{5}{108}u^3_3,\\frac{1}{4}u^2_3,\\frac{1}{4}u^2_3,u_3\\right)\\ \\Bigl|\\ u_3\\in {\\mathbb R}\\right\\}.\n\\]\nFor $D^-_4,$ we also consider the following parameter transformation:\n\\[\nu_1=u_3\\cos\\theta,\\ u_2=u_3\\sin\\theta.\n\\]\nThen $\\widetilde{PY}$ is parametrized by\n\\[\nf(\\theta,u_3)=\\left(\\frac{1}{3}u_3^3\\left(\\cos3\\theta+1\\right),-u_3^2(\\cos2\\theta+2\\cos\\theta),\n2u_3^2(\\sin2\\theta -2\\sin\\theta),u_3\\right).\\]\nThus , the Jacobi matrix is \n\\[\nJ_f=\\left(\n\\begin{array}{cc}\n-u_3^3\\sin3\\theta & u_3^2(\\cos3\\theta +1) \\\\\n2u_3^2(\\sin2\\theta +\\sin\\theta) & -2u_3(\\cos2\\theta+2\\cos\\theta) \\\\\n2u_3^2(\\cos2\\theta -\\cos\\theta) & 2u_3(\\sin2\\theta-2\\sin\\theta) \\\\\n0 & 1\n\\end{array}\n\\right),\n\\]\nso that ${\\rm rank}\\, J_f<2$ if and only if $\\theta=0,\\ 2\\pi\/3, 5\\pi\/3$ or $u_3=0.$\nIf $\\theta=0,$ then we have $u_1=u_3,\\ u_2=0,$ so that\nwe have $\\{\\left(4u^3_3\/3,-3u^2_3,0,u_3\\right)| u_3\\in {\\mathbb R}\\}.$\nIf $\\theta =2\\pi\/3,$ then we have\n$u_1=-u_3\/2,\\ u_2=\\sqrt{3}u_3\/2,$ so that we have\n$\\{(4u^3_3\/3,3u^2_3\/2,-3\\sqrt{3}u^2_3\/2,u_3| u_3\\in {\\mathbb R}\\}.$\nFinally, if $\\theta =5\\pi\/3,$ then we have\n$u=-u_3\/2,\\ u_2=-\\sqrt{3}u_3\/2,$ so that we have\n$\\{(4u^3_3\/3,3u^2_3\/2,3\\sqrt{3}u^2_3\/2,u_3)| u_3\\in {\\mathbb R}\\}.$\nIt follows that $\\Sigma (\\widetilde{PY}))$ is the union of these three set germs.\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\\par\nWe can interpret the geometric meaning of the above normal forms of the singularities of\nlightlike hypersurfaces as the following theorem shows.\n\n\\begin{Th}\nLet $\\mathcal{O}\\subset {\\rm Emb}_{\\rm sp}\\, (U,AdS^4)$ be the open dense subset given in\nthe above theorem. For any $\\mbox{\\boldmath $X$} \\in {\\cal O},$ the germ of the corresponding lightlike hypersurfaces\n$(\\mathbb{LH}^\\pm _M,\\mbox{\\boldmath $\\lambda$} _0)$ at a point $(u_0,\\mu_0)\\in U\\times {\\mathbb R}$\nis characterized as follows{\\rm :}\n\\par\\noindent\n{\\rm (1)} $(A_1)$ if and only if $\\mbox{\\boldmath $\\lambda$} _0\\in \\mathbb{LH}^+_M(U\\times {\\mathbb R})\\cup \\mathbb{LH}^-_M(U\\times {\\mathbb R})$ is a non-singular point.\n\\par\\noindent\n{\\rm (2)} $(A_2)$ if and only if $p_0=\\mbox{\\boldmath $X$}(u_0)$ is not a AdS-lightlike ridge point and $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\mathbb{LE}^+_M\\cup\\mathbb{LE}^-_M.$\n\\par\\noindent\n{\\rm (3)} $(A_3)$ if and only if $p_0=\\mbox{\\boldmath $X$}(u_0)$ is an AdS-lightlike $1$-ridge point and $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\mathbb{LE}^+_M\\cup \\mathbb{LE}^-_M.$\n\\par\\noindent\n{\\rm (4)} $(A_4)$ if and only if $p_0=\\mbox{\\boldmath $X$}(u_0)$ is an AdS-lightlike $2$-ridge point, $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\mathbb{LE}^+_M\\cup \\mathbb{LE}^-_M$ and $u_0\\in U$ satisfies the following condition{\\rm :} For sufficiently small $\\varepsilon >0$, there exist two different AdS-lightlike $1$-ridge points $u^1,u^2\\in U$\nsuch that $|u_0-u^i|<\\varepsilon$, $(i=1,2)$.\n\\par\\noindent\n{\\rm (5)} $(D_4^+)$ if and only if $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\mathbb{LE}^+_M\\cup\\mathbb{LE}^-_M$, $p_0=\\mbox{\\boldmath $X$}(u_0)$ has corank two contact with the osculating lightcone and\n$u_0\\in U$ satisfies\nthe following condition{\\rm :} \n\\par\n{\\rm (a)} $\\kappa _1(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)= \\kappa _2(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)$.\n\\par\n{\\rm (b)}\n For sufficiently small $\\varepsilon >0$, there exist two different AdS-lightlike $1$-ridge points $u^1,u^2\\in U$\nsuch that $|u_0-u^i|<\\varepsilon$, $(i=1,2)$ and $\\kappa _1(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^i)\\not= \\kappa _2(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^i)$.\n\\par\\noindent\n{\\rm (6)} $(D_4^-)$ if and only if $\\mbox{\\boldmath $\\lambda$}_0\n\\in \\mathbb{LE}_M$, $p_0=\\mbox{\\boldmath $X$}(u_0)$ has corank two contact with the the osculating lightcone and\n$u_0\\in U$ satisfies\nthe following condition{\\rm :} \n\\par\n{\\rm (a)} $\\kappa _1(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)= \\kappa _2(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)$.\n\\par\n{\\rm (b)} For sufficiently small $\\varepsilon >0$, there exist three different AdS-lightlike $1$-ridge points $u^i\\in U$, $(i=1,2,3)$\nsuch that $|u_0-u^i|<\\varepsilon$ and\n$\\kappa _1(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^i)\\not=\\kappa _2(\\mbox{\\boldmath $n$}^T,\\pm \\mbox{\\boldmath $n$}^S)(u^i)$.\n\\end{Th}\n\n\\par\\noindent{\\it Proof. \\\/}\\ \nBy the normal form $(A_1)$ and $(A_2)$ in Theorem 6.4, the assertions (1) and (2) are trivial.\nFor the normal form $(A_3)$, the AdS-height function has the $A_3$ singularity at $p_0=\\mbox{\\boldmath $X$}(u_0)$, so that\nit is an AdS-lightlike $1$-ridge point.\n\\par\nHere, \nwe give a remark on the classification of $\\mathcal{K}$-simple singularities of function germs.\nIn the list of the classification, we say that a class of singularities $L$ is {\\it adjacent} to a class $K$\n(notation: $K\\leftarrow L$) if every function germ $f\\in L$ can be deformed into a function of $K$ by an\narbitrarily small perturbation.\nFor the class of $A_k,D^\\pm_k$ of the $\\mathcal{K}$-classification are adjacent to each other as follows \\cite[Page 243]{Arnold1}:\n\\[\n\\begin{array}{cccccccc}\nA_1 &\\leftarrow & A_2 &\\leftarrow & A_3 &\\leftarrow & A_4 &\\leftarrow \\\\\n{} & {} & {} & {} & \\uparrow & {} & \\uparrow & {} \\\\\n{} & {} & {} & {} & D^\\pm_4 &\\leftarrow & D^\\pm_5 &\\leftarrow  \\\\\n\\end{array}\n\\]\nBy the normal form $(A_4)$, the singularities of the AdS-lightcone evolute is a $(2,3,4,5)$-cusp.\nTherefore, two singular loci approach to the $(2,3,4,5)$-cusp point. \nSince $A_4$ is adjacent to $A_3$, such the singular loci \nconsist of AdS-lightlike $1$-ridge points except the origin.\nThus, the assertion (4) holds.\nFor $(D^+_4)$, the singularities of the AdS-lightcone evolute is $\\Sigma (\\widetilde{PU})$.\nBy the normal form of the generating family, the corank of the AdS-height function at $u^0\\in U$ is\ntwo, so that it is an $(\\mbox{\\boldmath $n$}^T(u_0),\\pm \\mbox{\\boldmath $n$}^S(u_0))$-umbilical point. Therefore, we have\n$\\kappa _1(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)= \\kappa _2(\\mbox{\\boldmath $n$}^T,\\pm\\mbox{\\boldmath $n$}^S)(u^0)$.\nBy the normal form of $(D^+_4)$ in Corollary 6.6, two singular loci approach to the origin\nfrom both the positive and the negative side of the parameter $u_3$.\nSince $D^+_4$ is adjacent to $A_3$, such the singular loci consist of AdS-lightlike $1$-ridge points\nexcept at the origin.\nOf course, an AdS-lightlike $1$-ridge point is not an $(\\mbox{\\boldmath $n$}^T,\\pm \\mbox{\\boldmath $n$}^S)$-umbilical point.\nSo that two nullcone principal curvatures are different at an AdS-lightlike $1$-ridge point.\nFor $(D^-_4)$, we have the assertion by the similar arguments to the case $(D^+_4).$\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\n\\section{Spacelike curves in $AdS^4$}\nIn this section we consider lightlike hypersurfaces along spacelike curves in $AdS^4$ as\nthe simplest case of higher codimension.\nLet $\\mbox{\\boldmath $\\gamma$}:I\\longrightarrow AdS^4$ be a spacelike curve with $\\| \\mbox{\\boldmath $\\gamma$}''(s)\\|\\neq-1.$ In this case we write $C=\\mbox{\\boldmath $\\gamma$} (I)$ instead of $M=\\mbox{\\boldmath $\\gamma$} (I).$ Since $\\|\\mbox{\\boldmath $\\gamma$}'(s)\\|>0,$ we can reparameterize it by the arc-length s. So we have the unit tangent vector $\\mbox{\\boldmath $t$}(s)=\\mbox{\\boldmath $\\gamma$}'(s)$ of $\\mbox{\\boldmath $\\gamma$}(s).$ Moreover we have two unit normal vectors $\\displaystyle{\\mbox{\\boldmath $n$}_1(s)=\\frac{\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s)}{\\|\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s)\\|}},$ $\\displaystyle{\\mbox{\\boldmath $n$}_2(s)=\\frac{\\mbox{\\boldmath $n$}'_1(s)+\\delta \\kappa _1(s)\\mbox{\\boldmath $t$}(s)}{\\| \\mbox{\\boldmath $n$}'_1(s)+\\delta \\kappa _1(s)\\mbox{\\boldmath $t$}(s)\\|}}$ under the conditions that $\\kappa_1(s)=\\|\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s)\\|\\neq0,$ $\\kappa _2(s)=\\| \\mbox{\\boldmath $n$}'_1(s)+\\delta k_1(s)\\mbox{\\boldmath $t$}(s)\\|\\neq0,$ where $\\delta_i={\\rm sign}(\\mbox{\\boldmath $n$}_i(s))$ and ${\\rm sign}(\\mbox{\\boldmath $n$}_i(s))$ is the signature of $\\mbox{\\boldmath $n$}_i(s)$ $(i=1,2,3).$ Then we have another unit normal vector field $\\mbox{\\boldmath $n$}_3(s)$ defined by $\\mbox{\\boldmath $n$}_3(s)=\\mbox{\\boldmath $\\gamma$}(s)\\wedge\\mbox{\\boldmath $t$}(s)\\wedge\\mbox{\\boldmath $n$}_1(s)\\wedge\\mbox{\\boldmath $n$}_2(s).$ Therefore we can construct a pseudo-orthogonal frame $\\{\\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $t$}(s), \\mbox{\\boldmath $n$}_1(s), \\mbox{\\boldmath $n$}_2(s), \\mbox{\\boldmath $n$}_3(s)\\},$ which satisfies the {\\it Frenet-Serret type formulae}:\n $$\\left\\{\n       \\begin{array}{ll}\n          \\mbox{\\boldmath $\\gamma$}'(s)=\\mbox{\\boldmath $t$}(s), \\\\\n          \\mbox{\\boldmath $t$}'(s)=\\mbox{\\boldmath $\\gamma$}(s)+\\kappa_1(s)\\mbox{\\boldmath $n$}_1(s),  \\\\\n          \\mbox{\\boldmath $n$}_1'(s)=-\\delta_1\\kappa_1(s)\\mbox{\\boldmath $t$}(s)+\\kappa_2(s)\\mbox{\\boldmath $n$}_2(s),\\\\\n          \\mbox{\\boldmath $n$}_2'(s)=\\delta_3\\kappa_2(s)\\mbox{\\boldmath $n$}_1(s)+\\kappa_3(s)\\mbox{\\boldmath $n$}_3(s),\\\\\n          \\mbox{\\boldmath $n$}_3'(s)=\\delta_1\\kappa_3(s)n_2(s),  \n       \\end{array}\n     \\right.$$   \nwhere $\\kappa_2(s)= \\delta_2\\langle \\mbox{\\boldmath $n$}_1'(s),\\mbox{\\boldmath $n$}_2(s)\\rangle$ and $\\kappa_3(s)=\\delta_3\\langle \\mbox{\\boldmath $n$}_2'(s),\\mbox{\\boldmath $n$}_3(s)\\rangle.$ \nSince $\\mbox{\\boldmath $\\gamma$}(s)$ is timelike and $\\mbox{\\boldmath $t$}(s)$ is spacelike, we distinguish the following three cases:\n\\par\n\\smallskip\nCase 1: $\\mbox{\\boldmath $n$}_1(s)$ is timelike, that is, $\\delta_1=-1$ and $\\delta_2=\\delta_3=1.$\n\\par\nCase 2: $\\mbox{\\boldmath $n$}_2(s)$ is timelike, that is, $\\delta_2=-1$ and $\\delta_1=\\delta_3=1.$\n\\par\nCase 3: $\\mbox{\\boldmath $n$}_3(s)$ is timelike, that is, $\\delta_3=-1$ and $\\delta_1=\\delta_2=1.$\n\\par\\noindent\nWe consider the lightlike hypersurface along $C,$ and calculate the anti-de Sitter height  function on $C$ which is useful for the study the singularities of lightlike hypersurfaces in the each case.\n\\subsection{Case 1}\nSuppose that $\\mbox{\\boldmath $n$}_1(s)$ is timelike. In this case we adopt  $\\mbox{\\boldmath $n$}^T(s)=\\mbox{\\boldmath $n$}_1(s)$ and denote that $\\mbox{\\boldmath $b$}_1(s)=\\mbox{\\boldmath $n$}_2(s), \\mbox{\\boldmath $b$}_2(s)=\\mbox{\\boldmath $n$}_3(s).$ Then we have the pseudo-orthogonal frame \n\\[\n\\{ \\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $t$}(s), \\mbox{\\boldmath $n$}^T(s), \\mbox{\\boldmath $b$}_1(s), \\mbox{\\boldmath $b$}_2(s)\\},\n\\]\n$\\delta_1=-1$ and $\\delta_2=\\delta_3=1$, which satisfies the following Frenet-Serret type formulae:\n$$\\left\\{\n       \\begin{array}{ll}\n          \\mbox{\\boldmath $\\gamma$}'(s)=\\mbox{\\boldmath $t$}(s), \\\\\n          \\mbox{\\boldmath $t$}'(s)=\\mbox{\\boldmath $\\gamma$}(s)+\\kappa_1(s)\\mbox{\\boldmath $n$}^T(s),  \\\\\n          {\\mbox{\\boldmath $n$}^T}'(s)=\\kappa_1(s)\\mbox{\\boldmath $t$}(s)+\\kappa_2(s)\\mbox{\\boldmath $b$}_1(s),\\\\\n          \\mbox{\\boldmath $b$}_1'(s)=\\kappa_2(s)\\mbox{\\boldmath $n$}^T(s)+\\kappa_3(s)\\mbox{\\boldmath $b$}_2(s),\\\\\n          \\mbox{\\boldmath $b$}_2'(s)=-\\kappa_3(s)\\mbox{\\boldmath $b$}_1(s). \n       \\end{array}\n     \\right.$$\nSince $N_1(C)[\\mbox{\\boldmath $n$}^T]$ is parametrized by \n\\[\nN_1(C)[\\mbox{\\boldmath $n$}^T]=\\{(\\mbox{\\boldmath $\\gamma$} (s),\\mbox{\\boldmath $\\xi$})\\in \\mbox{\\boldmath $\\gamma$} ^*T{\\mathbb R}^5_1\\ |\\ \\mbox{\\boldmath $\\xi$} =\\cos\\theta\\mbox{\\boldmath $b$}_1(s)+\\sin\\theta\\mbox{\\boldmath $b$}_2(s)\\in N_{\\mbox{\\scriptsize \\boldmath$\\gamma$}(s)}(C),\\ s\\in I \\},\n\\]  \nthe nullcone Gauss image of $N_1(C)_{p}[\\mbox{\\boldmath $n$}^T]$ is given by\n\\[\n\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s,\\theta)=\\mbox{\\boldmath $n$}^T(s)+\\cos\\theta\\mbox{\\boldmath $b$}_1(s)+\\sin\\theta\\mbox{\\boldmath $b$}_2(s).\n\\]\nThen we have the lightlike hypersurface along $C$ \n$$\n\\mathbb{LH}_C((s,\\theta),\\mu)=\\mbox{\\boldmath $\\gamma$}(s)+\\mu(\\mbox{\\boldmath $n$}^T(s)+\\cos\\theta\\mbox{\\boldmath $b$}_1(s)+\\sin\\theta\\mbox{\\boldmath $b$}_2(s))=\n\\mbox{\\boldmath $\\gamma$} (s)+\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s,\\theta).\n$$\n We remark that the image of this lightlike hypersurface along $C$ is independent of the choice of the future directed timelike normal vector field $\\mbox{\\boldmath $n$}^T$ by Corollary 4.3. \n\\par\\noindent\nNow we investigate the anti- de Sitter height functions $H: I\\times AdS^{4}\\longrightarrow {\\mathbb R}$ on a spacelike curve  $C=\\mbox{\\boldmath $\\gamma$}(I)$ defined by\n$$\n H(p,\\mbox{\\boldmath $\\lambda$})=H(s,\\mbox{\\boldmath $\\lambda$})=\\langle \\mbox{\\boldmath $\\gamma$}(s),\\mbox{\\boldmath $\\lambda$}\\rangle+1,\n$$\nwhere $p=\\mbox{\\boldmath $\\gamma$}(s).$ For any fixed $\\mbox{\\boldmath $\\lambda$} _0\\in AdS^{4},$ we write $h(p)=H_{\\lambda _0}(p)=H(p,\\mbox{\\boldmath $\\lambda$} _0).$\n\\par\\noindent\nBy Proposition 4.1, the discriminant set of the\nanti-de Sitter height function $H$ is given by\n\\[\n{\\mathcal D}_{H}=\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)+\\mu\\mathbb{NG}(s,\\theta)\\Bigm|\\ \\theta\\in [0,2\\pi), s\\in I,\\ \\mu\\in {\\mathbb R}\\ \\Bigr\\},\n\\]\nwhich is the image of the lightlike hypersurface along $C.$ \nWe also calculate that $h''(p)=\\langle \\mbox{\\boldmath $\\gamma$}''(s), \\lambda_0\\rangle=-\\mu\\kappa_1-1.$ Then $h''(p)=0$ if and only if $\\mu=-1\/\\kappa_1(s).$ \n It means that a singular point of the lightlike hypersurface is a point $\\mbox{\\boldmath $\\lambda$} _0=\\mbox{\\boldmath $\\gamma$}(s_0)+\\mu_0\\mathbb{NG}(\\theta _0,s_0)$\nfor $\\mu_0 =-1\/\\kappa_1(s_0).$ \nTherefore, the lightlike focal surface is\n\\[\n\\mathbb{LF}_C=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)-\\frac{1}{\\kappa _1(s)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)\n(s,\\theta)\\Bigm| \\ s\\in I,\\ \\theta\\in [0,2\\pi)\\ \\Bigr\\}.\n\\]\n\n\\par\nMoreover, if we calculate the third, 4th and 5th derivatives of $h(s),$ we have the following proposition.\n\\par\n\\begin{Pro}\nLet $C$ be a spacelike curve and\n$H: C\\times(AdS^{4}\\setminus C)\\to{\\mathbb R}$\nthe anti-de Sitter height function on $C.$\nSuppose that $p_0=\\mbox{\\boldmath $\\gamma$}(s_0)\\not=\\mbox{\\boldmath $\\lambda$} _0.$ Then we have the followings{\\rm :}\n\\par\\noindent\n{\\rm (1)}\n$h(p_0)=h'(p_0)=0$ if and only if\nthere exist $\\theta _0\\in [0,2\\pi)$ and $\\mu\\in\n{\\mathbb R}\\setminus \\{0\\}$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0 =\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0).\n$$ \n\\par\\noindent\n{\\rm (2)}\n$h(p_0)=h'(p_0)=h''(p_0)=0$ if and only if\nthere exists $\\theta _0\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0).\n$$\n\\par\\noindent\n{\\rm(3)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=0$ if and only if there exists $\\theta _0\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0)\n$$\nand $\\kappa_1'(s_0)-\\cos\\theta _0 \\kappa_1(s_0)\\kappa_2(s_0)=0,$\nso that we can write $\\theta _0=\\theta (s_0).$\n\\par\\noindent\n{\\rm(4)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=0$ if and only if there exists $\\theta _0=\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)-\\cos\\theta (s_0) \\kappa_1(s_0)\\kappa_2(s_0)=0$ and \n$(2\\kappa_1'(s_0)\\kappa_2(s_0)+\\kappa_1(s_0)\\kappa_2'(s_0))\\cos\\theta (s_0)-\\kappa_1''(s_0)-\\kappa_1(s_0)\\kappa_2^2(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta (s_0)=0.$\n\\par\\noindent\n{\\rm(5)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=h^{(5)}(p_0)=0$ if and only if there exists $\\theta _0=\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)-\\cos\\theta (s_0) \\kappa_1(s_0)\\kappa_2(s_0)=0$,\n$(2\\kappa_1'(s_0)\\kappa_2(s_0)+\\kappa_1(s_0)\\kappa_2'(s_0))\\cos\\theta (s_0)-\\kappa_1''(s_0)-\\kappa_1(s_0)\\kappa_2^2(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta (s_0)=0$\nand\n$((2\\kappa_1'(s_0)\\kappa_2(s_0)+\\kappa_1(s_0)\\kappa_2'(s_0))\\cos\\theta(s_0)-\\kappa_1''(s_0)-\\kappa_1(s_0)\\kappa_2^2(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta(s_0))'=0.$\n\\end{Pro}\n\\par\nTaking account of the above proposition, we denote that\n$\\rho _1(s,\\theta)=\\kappa_1'(s)-\\cos\\theta  \\kappa_1(s)\\kappa_2(s)$\nand\n$\\eta _1(s,\\theta)=(2\\kappa_1'(s)\\kappa_2(s)+\\kappa_1(s)\\kappa_2'(s))\\cos\\theta -\\kappa_1''(s)-\\kappa_1(s)\\kappa_2^2(s)+\\kappa_1(s)\\kappa_2(s)\\kappa_3(s)\\sin\\theta ,$\nwhich might be important invariants of $C=\\mbox{\\boldmath $\\gamma$} (I).$\nThen we can show that\n$\\rho_1(s,\\theta)=\\eta_1(s,\\theta)=0$ if and only if $\\rho _1(s,\\theta)=\\sigma _1(s)=0$,\nwhere\n\\[\n\\sigma _1(s)=\\left[\\kappa _1\\kappa _2(\\kappa _1''+\\kappa _1\\kappa _2^2)\n-\\kappa _1'(2\\kappa _1'\\kappa _2+\\kappa _1\\kappa '_2)\\mp\n\\kappa _1\\kappa _2\\kappa _3\\sqrt{(\\kappa _1\\kappa _2)^2-(\\kappa '_1)^2}\\right](s).\n\\]\n\n\\par\n\\subsection{Case 2}\nSuppose that $\\mbox{\\boldmath $n$}_2(s)$ is timelike. Then we adopt $\\mbox{\\boldmath $n$}^T(s)=\\mbox{\\boldmath $n$}_2(s)$ and denote that $\\mbox{\\boldmath $b$}_1(s)=\\mbox{\\boldmath $n$}_1(s), \\mbox{\\boldmath $b$}_2(s)=\\mbox{\\boldmath $n$}_3(s).$ We have  a pseudo-orthogonal frame $\\{ \\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $t$}(s), \\mbox{\\boldmath $n$}^T(s), \\mbox{\\boldmath $b$}_1(s), \\mbox{\\boldmath $b$}_2(s)\\}$, $\\delta_2=-1$ and $\\delta_1=\\delta_3=1,$ which satisfies the following Frenet-Serret type formulae:\n$$\\left\\{\n       \\begin{array}{ll}\n          \\mbox{\\boldmath $\\gamma$}'(s)=\\mbox{\\boldmath $t$}(s), \\\\\n          \\mbox{\\boldmath $t$}'(s)=\\mbox{\\boldmath $\\gamma$}(s)+\\kappa_1(s)\\mbox{\\boldmath $b$}_1(s),  \\\\\n          \\mbox{\\boldmath $b$}_1'(s)=-\\kappa_1(s)\\mbox{\\boldmath $t$}(s)+\\kappa_2(s)\\mbox{\\boldmath $n$}^T(s),\\\\\n          {\\mbox{\\boldmath $n$}^T}'(s)=\\kappa_2(s)\\mbox{\\boldmath $b$}_1(s)+\\kappa_3(s)\\mbox{\\boldmath $b$}_2(s),\\\\\n          \\mbox{\\boldmath $b$}_2'(s)=\\kappa_3(s)\\mbox{\\boldmath $n$}^T(s),  \n       \\end{array}\n     \\right.$$\nHere, $N_1(C)[\\mbox{\\boldmath $n$}^T]$ is parametrized by \n\\[\nN_1(C)[\\mbox{\\boldmath $n$}^T]=\\{(\\mbox{\\boldmath $\\gamma$} (s),\\mbox{\\boldmath $\\xi$})\\in \\mbox{\\boldmath $\\gamma$} ^*T{\\mathbb R}^5_1\\ |\\ \\mbox{\\boldmath $\\xi$} =\\cos\\theta\\mbox{\\boldmath $b$}_1(s)+\\sin\\theta\\mbox{\\boldmath $b$}_2(s)\\in N_{\\mbox{\\scriptsize \\boldmath$\\gamma$}(s)}(C),\\ s\\in I \\},\n\\]  so that we have the lightlike hypersurface along $C=\\mbox{\\boldmath $\\gamma$} (I)$:\n$$\n\\mathbb{LH}_C((s,\\theta),t)=\\mbox{\\boldmath $\\gamma$}(s)+\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s,\\theta).\n$$\n\\par\nWe consider the anti-de Sitter height function $H: I\\times AdS^{4}\\longrightarrow {\\mathbb R}$ on a spacelike curve  $C=\\mbox{\\boldmath $\\gamma$}(I)$. Under the similar notations to the case 1), we have the following proposition:\n\\begin{Pro}\nLet $C$ be a spacelike curve and\n$H: C\\times(AdS^{4}\\setminus C)\\to{\\mathbb R}$\nthe anti-de Sitter height function on $C.$\nSuppose that $p_0\\not=\\mbox{\\boldmath $\\lambda$} _0.$ Then we have the following$:$\n\\par\\noindent\n{\\rm (1)}\n$h(p_0)=h'(p_0)=0$ if and only if\nthere exist $\\theta _0 \\in [0,2\\pi)$ and $\\mu\\in\n{\\mathbb R}\\setminus \\{0\\}$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0 =\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0).\n$$ \n\\par\\noindent\n{\\rm (2)}\n$h(p_0)=h'(p_0)=h''(p_0)=0$ if and only if\nthere exists $\\theta _0 \\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$}(s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0).\n$$\n\\par\\noindent\n{\\rm(3)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=0$ if and only if there exists $\\theta _0 \\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$}(s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0)\n$$\nand $\\kappa_1'(s_0)\\cos\\theta_0-\\kappa_1(s_0)\\kappa_2(s_0)=0,$\nso that we can write $\\theta _0=\\theta (s_0).$\n\\par\\noindent\n{\\rm(4)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=0$ if and only if there exists $\\theta _0 =\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$}(s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta (s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)\\cos\\theta (s_0)-\\kappa_1(s_0)\\kappa_2(s_0)=0$ and\n$(\\kappa_1''(s_0)+\\kappa_1(s_0)\\kappa_2^2(s_0))\\cos\\theta (s_0)-2\\kappa_1'(s_0)\\kappa_2(s_0)-\\kappa_1(s_0)\\kappa_2'(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta (s_0)=0$.\n\\par\\noindent\n{\\rm(5)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=h^{(5)}(p_0)=0$ if and only if there exists $\\theta _0 =\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$}(s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta (s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)\\cos\\theta (s_0)-\\kappa_1(s_0)\\kappa_2(s_0)=0$,\n$(\\kappa_1''(s_0)+\\kappa_1(s_0)\\kappa_2^2(s_0))\\cos\\theta (s_0)-2\\kappa_1'(s_0)\\kappa_2(s_0)-\\kappa_1(s_0)\\kappa_2'(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta (s_0)=0$ \nand $((\\kappa_1''(s_0)+\\kappa_1(s_0)\\kappa_2^2(s_0))\\cos\\theta (s_0)-2\\kappa_1'(s_0)\\kappa_2(s_0)-\\kappa_1(s_0)\\kappa_2'(s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)\\sin\\theta (s_0))'=0.$\n\\end{Pro}\n\\par\\noindent\nThe above proposition asserts that the discriminant set of the Lorentzian distance-squared function $G$ is given by\n\\[\n{\\mathcal D}_{H}=\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)-\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)\n(s,\\theta)\\Bigm| \\ s\\in I,\\ \\theta\\in [0,2\\pi), \\mu\\in {\\mathbb R}\\ \\Bigr\\}.\n\\]\nMoreover, the lightlike focal surface is\n\\[\n\\mathbb{LF}_C=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)-\\frac{1}{\\kappa _1(s)\\cos\\theta}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)\n(s,\\theta)\\Bigm| \\ s\\in I,\\ \\theta\\in [0,2\\pi)\\ \\Bigr\\}.\n\\]\n\\par\nHere, we also denote that\n$\n\\rho_2(s,\\theta)=\\kappa '_1(s)\\cos\\theta -\\kappa _1(s)\\kappa _2(s)$ and\n\\[\n\\eta _2(s,\\theta)=(\\kappa ''_1(s)+\\kappa _1(s)\\kappa ^2_2(s))\\cos\\theta -2\\kappa '_1(s)\\kappa _2(s)-\\kappa _1(s)\\kappa '_2(s)+\\kappa _1(s)\\kappa _2(s)\\kappa _3(s)\\sin\\theta.\n\\]\nWe can also show that\n$\\rho_2(s,\\theta)=\\eta_2(s,\\theta)=0$ if and only if $\\rho _2(s,\\theta)=\\sigma _2(s)=0$,\nwhere\n\\[\n\\sigma _2(s)=\\left[\\kappa _1\\kappa _2(\\kappa _1''+\\kappa _1\\kappa _2^2)\n-\\kappa _1'(2\\kappa _1'\\kappa _2+\\kappa _1\\kappa '_2)\\pm\n\\kappa _1\\kappa _2\\kappa _3\\sqrt{-(\\kappa _1\\kappa _2)^2+(\\kappa '_1)^2}\\right](s).\n\\]\n\n\\par\n\\subsection{Case 3}\nSuppose that $\\mbox{\\boldmath $n$}_3(s)$ is timelike. Then we adopt $\\mbox{\\boldmath $n$}^T(s)=\\mbox{\\boldmath $n$}_3(s)$ and denote that \n$\\mbox{\\boldmath $b$}_1(s)=\\mbox{\\boldmath $n$}_1(s), \\mbox{\\boldmath $b$}_2(s)=\\mbox{\\boldmath $n$}_2(s).$ We have  a pseudo-orthogonal frame $\\{ \\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $t$}(s), \\mbox{\\boldmath $n$}^T(s), \\mbox{\\boldmath $b$}_1(s), \\mbox{\\boldmath $b$}_2(s)\\}$ and $\\delta_3=-1$ and $\\delta_1=\\delta_2=1,$which satisfies the following Frenet-Serret type formulae:\n$$\\left\\{\n       \\begin{array}{ll}\n          \\mbox{\\boldmath $\\gamma$}'(s)=\\mbox{\\boldmath $t$}(s), \\\\\n          \\mbox{\\boldmath $t$}'(s)=\\mbox{\\boldmath $\\gamma$}(s)+\\kappa_1(s)\\mbox{\\boldmath $b$}_1(s),  \\\\\n          \\mbox{\\boldmath $b$}_1'(s)=-\\kappa_1(s)\\mbox{\\boldmath $t$}(s)+\\kappa_2(s)\\mbox{\\boldmath $b$}_2(s),\\\\\n          \\mbox{\\boldmath $b$}_2'(s)=-\\kappa_2(s)\\mbox{\\boldmath $b$}_1(s)+\\kappa_3(s)\\mbox{\\boldmath $n$}^T(s),\\\\\n          {\\mbox{\\boldmath $n$}^T}'(s)=\\kappa_3(s)\\mbox{\\boldmath $b$}_2(s),  \n       \\end{array}\n     \\right.$$\nHere, $N_1(C)[\\mbox{\\boldmath $n$}^T]$ is parametrized by \n\\[\nN_1(C)[\\mbox{\\boldmath $n$}^T]=\\{(\\mbox{\\boldmath $\\gamma$} (s),\\mbox{\\boldmath $\\xi$})\\in \\mbox{\\boldmath $\\gamma$} ^*T{\\mathbb R}^5_1\\ |\\ \\mbox{\\boldmath $\\xi$} =\\cos\\theta\\mbox{\\boldmath $b$}_1(s)+\\sin\\theta\\mbox{\\boldmath $b$}_2(s)\\in N_{\\mbox{\\scriptsize \\boldmath$\\gamma$}(s)}(C),\\ s\\in I \\},\n\\]  \nso that we have the lightlike hypersurface along $C$:\n$$\n\\mathbb{LH}_C((s,\\theta),t)=\\mbox{\\boldmath $\\gamma$}(s)+t\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s,\\theta).\n$$\nWe investigate the anti-de Sitter function on a spacelike curve  $C=\\mbox{\\boldmath $\\gamma$}(I)$ \nBy the calculations similar to the cases 1 and 2, we have the following proposition:\n\\par\n\\begin{Pro}\nLet $C$ be a spacelike curve and\n$H: C\\times(AdS^4\\setminus C)\\to{\\mathbb R}$\nthe anti-de Sitter function on $C=\\mbox{\\boldmath $\\gamma$} (I).$\nSuppose that $p_0\\not=\\mbox{\\boldmath $\\lambda$} _0.$ Then we have the following$:$\n\\par\\noindent\n{\\rm (1)}\n$h(p_0)=h'(p_0)=0$ if and only if\nthere exist $\\theta _0\\in [0,2\\pi)$ and $\\mu\\in\n{\\mathbb R}\\setminus \\{0\\}$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0 =\\mu\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0).\n$$ \n\\par\\noindent\n{\\rm (2)}\n$h(p_0)=h'(p_0)=h''(p_0)=0$ if and only if\nthere exists $\\theta _0\\in [0,s\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0).\n$$\n\\par\\noindent\n{\\rm(3)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=0$ if and only if there exists $\\theta _0\\in [0,s\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0)\n$$\nand $\\kappa_1'(s_0)\\cos\\theta _0+\\kappa_1(s_0)\\kappa_2(s_0)\\sin\\theta _0=0,$\nso that we can write $\\theta _0=\\theta (s_0).$\n\\par\\noindent\n{\\rm(4)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=0$ if and only if there exists $\\theta _0=\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta (s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)\\cos\\theta (s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\sin\\theta (s_0)=0.$\n and  $(2\\kappa_1'(s_0)\\kappa_2(s_0)+\\kappa_1(s_0)\\kappa_2'(s_0))\\sin\\theta (s_0)+(\\kappa_1''(s_0)-\\kappa_1(s_0)\\kappa_2^2(s_0))\\cos\\theta (s_0)-\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0)=0.$\n\\par\\noindent\n{\\rm(5)}\n$h(p_0)=h'(p_0)=h''(p_0)=h'''(p_0)=h^{(4)}(p_0)=h^{(5)}(p_0)=0$ if and only if there exists $\\theta _0=\\theta (s_0)\\in [0,2\\pi)$ such that \n$$\n\\mbox{\\boldmath $\\gamma$} (s_0)-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta (s_0)),\n$$\n$\\kappa_1'(s_0)\\cos\\theta (s_0)+\\kappa_1(s_0)\\kappa_2(s_0)\\sin\\theta (s_0)=0,$\nand $((2\\kappa_1'(s_0)\\kappa_2(s_0)+\\kappa_1(s_0)\\kappa_2'(s_0))\\sin\\theta (s_0))+(\\kappa_1''(s_0)-\\kappa_1(s_0)\\kappa_2^2(s_0))\\cos\\theta (s_0))-\\kappa_1(s_0)\\kappa_2(s_0)\\kappa_3(s_0))'=0.$\n\\end{Pro}\nThe above proposition asserts that the discriminant set of the anti-de Sitter function $H$ is given by\n\\[\n{\\mathcal D}_{H}=\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R})=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)+t\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s,\\theta) \\Big|\\  s\\in I,\\ \\theta\\in [0,2\\pi), t\\in {\\mathbb R}\\ \\Bigr\\}.\n\\]\nMoreover, the lightlike focal surface is\n\\[\n\\mathbb{LF}_C=\\Bigl\\{\\mbox{\\boldmath $\\lambda$} =\\mbox{\\boldmath $\\gamma$}(s)-\\frac{1}{\\kappa _1(s)\\cos\\theta}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)\n(s,\\theta)\\Bigm| \\ s\\in I,\\ \\theta\\in [0,2\\pi)\\ \\Bigr\\}.\n\\]\n\\par\nHere, we also denote that\n$\n\\rho_3(s,\\theta)=\\kappa '_1(s)\\cos\\theta +\\kappa _1(s)\\kappa _2(s)\\sin\\theta$ and\n\\[\n\\eta _3(s,\\theta)=(2\\kappa '_1(s)\\kappa _2(s)+\\kappa _1(s)\\kappa '_2(s))\\sin\\theta +(\\kappa ''_1(s)-\\kappa _1(s)\\kappa ^2_2(s))\\cos\\theta-\\kappa _1(s)\\kappa _2(s)\\kappa _3(s),\n\\]\nWe can also show that\n$\\rho_3(s,\\theta)=\\eta _3(s,\\theta)=0$ if and only if $\\rho _3(s,\\theta)=\\sigma _3(s)=0$,\nwhere\n\\[\n\\sigma _3(s)=\\left[\\kappa _1\\kappa _2(\\kappa _1''-\\kappa _1\\kappa _2^2)\n-\\kappa _1'(2\\kappa _1'\\kappa _2+\\kappa _1\\kappa '_2)\\mp\n\\kappa _1\\kappa _2\\kappa _3\\sqrt{(\\kappa _1\\kappa _2)^2+(\\kappa '_1)^2}\\right](s).\n\\]\n\\par\nWe can unify the invariants $\\sigma _i(s)$, $(i=1,2,3)$ as follows:\n\\[\n\\sigma (s)=\\left[\\kappa _1\\kappa _2(\\kappa _1''-\\kappa _1\\kappa _2^2)\n-\\kappa _1'(2\\kappa _1'\\kappa _2+\\kappa _1\\kappa '_2)\\mp\\delta _2\n\\kappa _1\\kappa _2\\kappa _3\\sqrt{\\delta _1(\\kappa _1\\kappa _2)^2+\\delta _2(\\kappa '_1)^2}\\right](s).\n\\]\n\\par\\noindent\n\\subsection{Classifications of singularities}\nBy using the results of the three cases, we classify the singularities of the lightlike hypersurface along $\\mbox{\\boldmath $\\gamma$}$ as an application of the unfolding theory of functions. \nFor a function $f(s)$, we say that $f$\nhas {\\it $A_k$-singularity } at $s_0$ if $f^{(p)}(s_0)=0$  for all\n$1\\leq p\\leq k$ and $f^{(k+1)}(s_0)\\neq 0.$\n Let $F$ be an  $r$-parameter unfolding of $f$\nand $f$ has  $ A_k $-singularity $(k\\geq 1)$ at $s_0$. We denote\nthe $(k-1)$-jet of the partial derivative $\\partial F\/\n\\partial x_i$ at $s_0$\n as\n$$j^{(k-1)}\\left(\\frac{\\partial F}{\\partial x_i}(s,\\mbox{\\boldmath $x$}_0)\\right)(s_0)\n =\\sum\\limits _{j=1}^{k-1} \\alpha_{ji}(s-s_0)^j,~~  (i=1, \\cdots ,r).$$\n  If the rank of $k\\times r$ matrix  $(\\alpha_{0i},\\alpha_{ji}) $ is  $k~(k\\leq\nr)$, then $F$ is called a {\\it versal unfolding} of $f$,\n    where $ \\alpha_{0 i}=\\partial F\/\\partial x_i(s_0,\\mbox{\\boldmath $x$}_0)$.\n\\par\nInspired by the propositions in the previous subsections, we define the following set:\n\\[\nD^\\ell _F=\\left\\{\\mbox{\\boldmath $x$}\\in  {\\mathbb R}^r\\mid \\exists s\\in {\\mathbb R},\\ F(s,\\mbox{\\boldmath $x$})=\\frac{\\partial F}{\\partial s}(s,\\mbox{\\boldmath $x$})=\\cdots =\\frac{\\partial ^\\ell F}{\\partial s^\\ell }(s,\\mbox{\\boldmath $x$})= 0 \\right\\},\n\\]\nwhich is called a {\\it discriminant set of order $\\ell $.}\nOf course, $D^1_F=D_F$ and $D^2_F$ is the set of singular points of $D_F.$\nTherefore, we have the following proposition.\n\\begin{Pro}\nFor all the cases, we have\n\\[\nD_G=D^1_G=\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R}),\\ D^2_G=\\mathbb{LF}_C\\ \\mbox{and}\\ D^3_G\\ \n\\mbox{is\\ the\\ critical\\ value\\ set\\ of}\\ \\mathbb{LF}_C.\n\\]\n\\end{Pro}\nIn order to understand the geometric properties of the discriminant set of order $\\ell$, we introduce\nan equivalence relation among the unfoldings of functions.\nLet $F$ and $G$ be $r$-parameter unfoldings of $f(s)$ and  $g(s)$, respectively.\nWe say that $F$ and $G$ are {\\it P-$\\mathcal{R}$-equivalent} if\nthere exists a diffeomorphism germ $\\Phi :({\\mathbb R}\\times{\\mathbb R}^r,(s_0,\\mbox{\\boldmath $x$}_0))\\longrightarrow ({\\mathbb R}\\times{\\mathbb R}^r, (s'_0,\\mbox{\\boldmath $x$}'_0))$\nof the form $\\Phi (s,\\mbox{\\boldmath $x$})=(\\Phi _1(s,\\mbox{\\boldmath $x$}),\\phi (\\mbox{\\boldmath $x$}))$ such that $G\\circ\\Phi =F.$\nBy straightforward calculations, we have the following proposition.\n\\begin{Pro} Let $F$ and $G$ be $r$-parameter unfoldings of $f(s)$ and  $g(s)$, respectively. If $F$ and $G$ are P-$\\mathcal{R}$-equivalent by a diffeomorphism germ\n$\\Phi :({\\mathbb R}\\times{\\mathbb R}^r,(s_0,\\mbox{\\boldmath $x$}_0))\\longrightarrow ({\\mathbb R}\\times{\\mathbb R}^r, (s'_0,\\mbox{\\boldmath $x$}'_0))$ of the form\n $\\Phi (s,\\mbox{\\boldmath $x$})=(\\Phi _1(s,\\mbox{\\boldmath $x$}),\\phi (\\mbox{\\boldmath $x$}))$, then $\\phi (D^\\ell _F)=D^\\ell _G$ as set germs.\n\\end{Pro}\n    \nWe have the following classification theorem of versal unfoldings \\cite[Page 149, 6.6]{Bru-Gib}.\n \\begin{Th}\n Let $F:({\\mathbb R}\\times{\\mathbb R}^r,(s_0,\\mbox{\\boldmath $x$}_0))\\longrightarrow {\\mathbb R} $ be an\n$r$-parameter unfolding of $f$ which has  $A_k$-singularity at\n$s_0$. Suppose  $F$ is a versal unfolding of $f$, then $F$ is P-$\\mathcal{R}$-equivalent to\none of the following unfoldings:\n\\par\n{\\rm (a)} $k=1$ {\\rm ;} $\\pm s^2+x_1$,\n\\par\n{\\rm (b)} $k=2$ {\\rm ;} $s^3+x_1+sx_2$,\n\\par\n{\\rm (c)} $k=3$ {\\rm ;} $\\pm s^4+x_1+sx_2+s^2x_3,$\n\\par\n{\\rm (d)} $k=4$ {\\rm ;} $s^5+x_1+sx_2+s^2x_3+s^3x_4.$\n\\end{Th}\n\\par\\noindent\nWe have the following classification result as a corollary of the above theorem.\n\\begin{Co}\n Let $F:({\\mathbb R}\\times\n{\\mathbb R}^r,(s_0,\\mbox{\\boldmath $x$}_0))\\longrightarrow {\\mathbb R} $ be an\n$r$-parameter unfolding of $f$ which has  $A_k$-singularity at\n$s_0$. Suppose  $F$ is a versal unfolding of $f$, then we have the following assertions:\n\\par\\noindent\n{\\rm (a)} If $k=1$, then $ D_F $ is diffeomorphic to $\\{0\\}\\times\n{\\mathbb R}^{r-1}$ and $D^2_F=\\emptyset.$\n\\par\\noindent\n{\\rm (b)} If  $k=2$, then $D _F $ is diffeomorphic to $C(2,3)\\times {\\mathbb R}^{r-2},$\n$D^2_F$ is diffeomorphic to $\\{\\mbox{\\boldmath $0$}\\}\\times{\\mathbb R}^{r-2}$\nand $D^3_F=\\emptyset.$\n\\par\\noindent\n{\\rm (c)} If $k=3$, then $D _F $ is diffeomorphic to $SW\\times {\\mathbb R}^{r-3},$ \n $D^2_F$ is diffeomorphic to $C(2,3,4)\\times {\\mathbb R}^{r-3},$\n$D^3_F$ is diffeomorphic to $\\{\\mbox{\\boldmath $0$}\\}\\times {\\mathbb R}^{r-3}$ and $D^4_F=\\emptyset.$\n\\par\\noindent\n{\\rm (d)} If $k=4$, then $D _F $ is locally diffeomorphic to $BF\\times {\\mathbb R}^{r-4},$\n$D^2_F$ is diffeomorphic to $C(BF)\\times {\\mathbb R}^{r-4},$\n$D^3_F$ is diffeomorphic to $C(2,3,4,5)\\times {\\mathbb R}^{r-4}$,\n$D^4_F$ is diffeomorphic to $\\{\\mbox{\\boldmath $0$}\\}\\times{\\mathbb R}^{r-4}$ and $D^5_F=\\emptyset.$\n\\par\\noindent\nWe remark that all of diffeomorphisms in the above assertions are diffeomorphism germs.\n\\end{Co}\nHere, we call $BF=\\{(x_1,x_2,x_3.x_4)\\mid x_1 = 5u^4+3vu^2+2wu,x_2=4u^5+2vu^3+wu^2,x_3=u,x_4=v\\}$ a {\\it butterfly}.\nWe have the following key proposition on $H.$\n\\begin{Pro}\n If $h(s)$ has $A_k$-singularity $(k=1,2,3,4)$ at $p_0$, then $H$ is a versal unfolding of $h.$\n\\end{Pro}\n\\par\\noindent{\\it Proof. \\\/}\\ \nWe denote that $\\mbox{\\boldmath $\\gamma$}(s)=(X_{-1}(s),X_0(s),X_1(s),X_2(s),X_3(s))\\ {\\rm and}\\\n\\mbox{\\boldmath $\\lambda$} =(\\lambda_{-1},\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3).$\n\\par\\noindent\nBy definition, we have\n$$\nH(s,\\mbox{\\boldmath $\\lambda$} )=-\\lambda_{-1}X_{-1}(s)-\\lambda_0X_0(s)+\\lambda_1X_1(s)+\\lambda_2X_2(s)+\\lambda_3X_3(s)+1.\n$$\nThus we have \n$$\n\\frac{\\partial H}{\\partial \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$})=-X_i (s), {\\rm and} \\ \\frac{\\partial^2 H}{\\partial s\\partial \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$})=-X_i' (s), \\ {\\rm for} (i=-1,0)\n$$\n$$\n\\frac{\\partial H}{\\partial \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$})=X_i (s), {\\rm and} \\ \\frac{\\partial^2 H}{\\partial s\\partial \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$})=X_i' (s), \\ {\\rm for} (i=1,2,3)\n$$\nFor a fixed $\\mbox{\\boldmath $\\lambda$}_0=(\\lambda_{0-1}\\lambda_{00}, \\lambda_{01}, \\lambda_{02}, \\lambda_{03}),$ the 3-jet of $\\partial H\/\\partial\n  \\lambda_i(s,\\mbox{\\boldmath $\\lambda$}_0)(i=-1,0,1,2,3)$ at $s_0$ is\n  $$j^{(3)}\\frac{\\partial H}{\\partial\n  \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$}_0)(s_0)=-X_i'(s_0)(s-s_0)-\\frac{1}{2}X_i''(s_0)(s-s_0)^2-\\frac{1}{3}X_i'''(s_0)(s-s_0)^3, \\ (i=-1,0).$$\n$$j^{(3)}\\frac{\\partial H}{\\partial\n  \\lambda_i}(s,\\mbox{\\boldmath $\\lambda$}_0)(s_0)=X_i'(s_0)(s-s_0)+\\frac{1}{2}X_i''(s_0)(s-s_0)^2+\\frac{1}{3}X_i'''(s_0)(s-s_0)^3, \\ (i=1,2,3).$$\n\\par\\noindent\nIt is enough to show that the rank of the following matrix A is four,\n$$A=\\left(\n\\begin{array}{ccccc}\n -X_{-1}(s_0)& -X_0 (s_0)& X_1 (s_0)& X_2 (s_0)& X_3 (s_0)\\\\\n-X'_{-1}(s_0)& -X_0'(s_0)& X_1'(s_0)& X_2'(s_0)& X_3'(s_0)\\\\\n-X''_{-1}(s_0)& -X_0''(s_0)& X_1''(s_0)& X_2''(s_0)& X_3''(s_0)\\\\\n-X'''_{-1}(s_0)& -X_0'''(s_0)& X_1'''(s_0)& X_2'''(s_0)& X_3'''(s_0)\n\\end{array}\n\\right).$$\nHere we consider the following matrix B,\n$$B=\\left(\n\\begin{array}{ccccc}\n -X_{-1}(s_0)& -X_0 (s_0)& X_1 (s_0)& X_2 (s_0)& X_3 (s_0)\\\\\n-X'_{-1}(s_0)& -X_0'(s_0)& X_1'(s_0)& X_2'(s_0)& X_3'(s_0)\\\\\n-X''_{-1}(s_0)& -X_0''(s_0)& X_1''(s_0)& X_2''(s_0)& X_3''(s_0)\\\\\n-X'''_{-1}(s_0)& -X_0'''(s_0)& X_1'''(s_0)& X_2'''(s_0)& X_3'''(s_0)\\\\\n-n_{-1}(s_0)& -n_{0}(s_0)& n_{1}(s_0)& n_{2}(s_0)& n_{3}(s_0)\\\\\n\\end{array}\n\\right),$$\nwhere we denote $\\mbox{\\boldmath $n$}_1(s_0)=(n_{-1}(s_0), n_{0}(s_0), n_{1}(s_0), n_{2}(s_0), n_{3}(s_0)).$\nIn fact, \n\\begin{eqnarray*}\n{\\rm det}B&=&{\\rm det}^{t}(\\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $\\gamma$}'(s), \\mbox{\\boldmath $\\gamma$}''(s), \\mbox{\\boldmath $\\gamma$}'''(s),\\mbox{\\boldmath $n$}_{1})\\\\\n&=&{\\rm det}^{t}(\\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $\\gamma$}'(s),\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s),\\mbox{\\boldmath $b$}, \\mbox{\\boldmath $n$}_{1})\\\\\n&=&{\\rm det}^{t}(\\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $\\gamma$}'(s),\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s)-\\kappa_{1}\\mbox{\\boldmath $n$}_{1},\\mbox{\\boldmath $b$}-\\kappa_{1}\\mbox{\\boldmath $n$}_{1},\\mbox{\\boldmath $n$}_{1}),\\\\\n\\end{eqnarray*}\nwhere $\\mbox{\\boldmath $b$}=\\mbox{\\boldmath $\\gamma$}'''(s)+\\frac{\\kappa'_{1}(s)}{\\kappa_{1}(s)}\\mbox{\\boldmath $\\gamma$}(s)+(\\delta_{1}\\kappa^2_{1}(s)-1)\\mbox{\\boldmath $\\gamma$}'(s)+(1-\\frac{\\kappa_{1}'(s)}{\\kappa_{1}})\\mbox{\\boldmath $\\gamma$}''(s),$\nand $\\mbox{\\boldmath $\\gamma$}(s), \\mbox{\\boldmath $\\gamma$}'(s),\\mbox{\\boldmath $\\gamma$}''(s)-\\mbox{\\boldmath $\\gamma$}(s)-\\kappa_{1}\\mbox{\\boldmath $n$}_{1},$ $\\mbox{\\boldmath $b$}-\\kappa_{1}\\mbox{\\boldmath $n$}_{1}$ and $\\mbox{\\boldmath $n$}_{1}$  are linearly independent each other in all Case 1,2,3, respectively. Therefore we have detB$\\neq0$. This means that ${\\rm rank A}=4.$\nThis completes the proof.\n\\hfill $\\Box$\\vspace{3truemm} \\par\n\n\n\\par\\noindent\nFinally, we can apply Corollary 8.5 to our condition. Then we have the following theorem:\n\\begin{Th}\nLet $\\mbox{\\boldmath $\\gamma$}:I\\longrightarrow AdS^4$ be a spacelike curves with $\\kappa _1(s)\\not= 0.$\n\\par\\noindent\n{\\rm (A)} For the lightlike hypersurfaces $\\mathbb{LH}_C((s,\\theta),t)$ of $C=\\mbox{\\boldmath $\\gamma$} (I)$ in the Case 1,  \nwe have the following assertions:\n\\par\\noindent\n{\\rm (1)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $C(2,3)\\times {\\mathbb R}^2$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi )$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0),\n$$\nand $\\rho_1(s_0,\\theta_0)\\neq0.$ In this case, the lightlike focal set $\\mathbb{LF}_C$ is non-singular.\n\\par\\noindent\n{\\rm(2)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $SW\\times {\\mathbb R}$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi )$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0),\n$$\n$\\rho _1(s_0,\\theta_0)=0$ and $\\sigma _1(s_0)\\neq 0.$\nIn this case, the lightlike focal set $\\mathbb{LF}_C$ is locally diffeomorphic to $C(2,3,4)\\times{\\mathbb R}$ and the critical value set of $\\mathbb{LF}_C$ is a regular curve.\n\n\\par\\noindent\n{\\rm(3)} The lightlike hypersurface $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$  is locally diffeomorphic to $BF$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi )$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=-\\frac{1}{\\kappa_1(s_0)}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta _0),\n$$\n$\\rho _1(s_0,\\theta _0)=0$, $\\sigma _1(s_0)=0$ and $\\sigma '_1(s_0)\\not= 0.$\nIn this case, the lightlike focal set $\\mathbb{LF}_C$ is is locally diffeomorphic to $C(BF)\\times{\\mathbb R}$ and the critical value set is locally diffeomorphic to the $C(2,3,4,5)$-cusp.\n\n\\smallskip\n\\par\\noindent\n{\\rm (B)} For the lightlike hypersurfaces $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ of $C=\\mbox{\\boldmath $\\gamma$}(I)$ in the Case 2,  \nwe have the following assertions:\n\\par\\noindent\n{\\rm (1)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $C(2,3)\\times {\\mathbb R}^2$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0)\n$$\nand  $\\rho_2(s_0,\\theta _0)\\neq0.$ In this case, the lightlike focal set\n$\\mathbb{LF}_C$ is non-singular.\n\\par\\noindent\n{\\rm(2)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $SW\\times {\\mathbb R}$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0),\n$$\n$\\rho _2(s_0,\\theta _0)=0$ and \n$\\sigma (s_0)\\neq0.$ In this case, the lightlike focal set $\\mathbb{LF}_C$ is locally\ndiffeomorphic to $C(2,3,4)\\times{\\mathbb R}$ and the critical value set of $\\mathbb{LF}_C$ is a regular curve.\n\\par\\noindent\n{\\rm(3)} he lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $BF$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0),\n$$\n$\\rho _2(s_0,\\theta _0)=0$, \n$\\sigma _2(s_0)=0$ and $\\sigma '_2(s_0)\\neq 0.$ In this case, the lightlike focal set\n$\\mathbb{LF}_C$ is locally\ndiffeomorphic to $C(BF)\\times{\\mathbb R}$ and the critical value set of $\\mathbb{LF}_C$ is locally diffeomorphic to the $C(2,3,4,5)$-cusp.\n\n\\smallskip\n\\par\\noindent\n{\\rm (C)} For the lightlike hypersurfaces $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ of $C=\\mbox{\\boldmath $\\gamma$}(I)$ in the Case 3,  \nwe have the following assertions:\n\\par\\noindent\n{\\rm (1)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $C(2,3)\\times {\\mathbb R}^2$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0),\n$$\nand $\\rho _3(s_0,\\theta _0)\\neq0.$ In this case, the lightlike focal set\n$\\mathbb{LF}_C$ is non-singular.\n\n\\par\\noindent\n{\\rm(2)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to  $SW\\times {\\mathbb R}$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0),\n$$\n$\\rho _3(s_0,\\theta _0)=0$ and $\\sigma _3(s_0)\\neq 0.$ In this case, the lightlike focal set\n$\\mathbb{LF}_C$ is locally\ndiffeomorphic to $C(2,3,4)\\times{\\mathbb R}$ and the critical value set of $\\mathbb{LF}_C$ is a regular curve.\n\\par\\noindent\n{\\rm(3)} The lightlike hypersurface  $\\mathbb{LH}_C(N_1(C)[\\mbox{\\boldmath $n$}^T]\\times{\\mathbb R} )$ is locally diffeomorphic to $BF$ at\n$\\mbox{\\boldmath $\\lambda$}_0$ if and only if there exist $\\theta _0\\in [0,2\\pi)$ such that \n$$\np_0-\\mbox{\\boldmath $\\lambda$} _0=\\frac{1}{\\kappa_1(s_0)\\cos\\theta_0}\\mathbb{NG}(\\mbox{\\boldmath $n$}^T)(s_0,\\theta_0),\n$$\n$\\rho _3(s_0,\\theta _0)=0$, $\\sigma _3(s_0)= 0$ and $\\sigma '_3(s_0)\\not= 0.$\nIn this case, the lightlike focal set\n$\\mathbb{LF}_C$ is locally\ndiffeomorphic to $C(BF)\\times{\\mathbb R}$ and the critical value set of $\\mathbb{LF}_C$ is locally diffeomorphic to the $C(2,3,4,5)$-cusp.\n\\end{Th}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA lamb escapes from a farm and has the bad idea to roam near a precipice.\nThe shepherd wonders what is the best strategy to catch her lamb alive.\nIndeed, the lamb wanders randomly when the shepherd stays still.  However the\nlamb is skittish and moves away from the shepherd and toward the precipice\nwhenever the shepherd approaches.  Should the shepherd stay still and hope\nthe lamb will come to her, or should she walk toward the lamb and hope that\nshe reaches the lamb before it goes over the precipice?\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=300pt]{setup1.pdf}\n\\caption{Equivalent formulations of the lamb capture process.  (a) Shepherd's\n  reference frame: the lamb and the precipice drift toward the shepherd at\n  speeds $\\alpha c$ and $c$.  (b) Reference frame where the lamb only diffuses: the shepherd and\n  the precipice approach the lamb at speeds $c_1=\\alpha c$ and\n  $c_2=(1-\\alpha) c$.}\n\\label{setup}\n\\end{figure}\n\nIf the shepherd walks toward the lamb with speed $c$, we assume that the lamb\nmoves away at a slower speed $(1-\\alpha)c$, with\n$0\\leqslant \\alpha \\leqslant 1$, that is superimposed on its diffusive\nmotion.  In the reference frame of the shepherd (Fig.~\\ref{setup}(a)), the\nlamb diffuses and approaches the shepherd with speed $\\alpha c$.  However,\nthe precipice also approaches the lamb with speed $(1-\\alpha)c$.  Initially,\nthe lamb is at $x_0$, the precipice is at $L_0$, while the shepherd is fixed\nat the origin.  The probability that the lamb and shepherd meet before the\nlamb goes over the precipice coincides with the \\emph{splitting probability}\nfor the lamb to eventually reach the origin.  In turn, this latter problem is\nequivalent to the splitting probability for unbiased diffusion that starts at\n$x_0'=x_0\\!-\\!L_0\/2$ to reach the left edge of an asymmetrically contracting\ncage, whose left and right edges are at $-L_0\/2\\!+\\!c_1 t$ and\n$L_0\/2\\!-\\!c_2 t$, respectively, with $c_1=\\alpha c$ and\n$c_2=(1\\!-\\!\\alpha) c$ (Fig.~\\ref{setup}(b)).  While the contracting cage is\nthe relevant situation for the lamb capture problem, we also investigate the\ncase of the expanding cage.  Although the value of $\\alpha$ does not need to\nbe restricted, we study the range $\\alpha \\in [0,1]$, where the splitting\nprobability has the richest behavior.\n\nConditional exit from a fixed interval is a classic problem of random-walk\ntheory \\cite{Gardiner,VanKampen,Weiss,Redner}. The role of a moving boundary\nhas been considered recently, e.g., a diffusing or an oscillating trap at the\nedge of the interval \\cite{Tzou:2014,Holcman:2009,Tejedor:2011}.  For ballistically moving\nboundaries, which is our focus, the infinite-time survival probability in an\nasymmetrically expanding cage and the time-dependent survival probability in\na symmetrically expanding cage were investigated by Bray and\nSmith~\\cite{Bray:2007,Bray:2007b}. Here we derive exact expressions for the\nfirst-passage probability at any time and the splitting probabilities to one\nof the edges of the cage when each wall moves ballistically at an arbitrary\nspeed. Contracting and expanding cages are considered. The surprising consequence of our results is that the splitting\nprobabilities depend \\emph{non-monotonically} on the speed $c$ for a range of\ninitial conditions and $\\alpha$ values.\n\nIn Sec.~\\ref{sec:exp}, we derive the splitting probability in an expanding\ncage.  In Sec.~\\ref{sec:gen}, we determine the first-passage probability at\nany time and the splitting probability to one of the boundaries for both an\nexpanding and contracting cage.  Finally, in Sec.~\\ref{sec:disc}, we\ndemonstrate that the splitting probability can have a non-monotonic\ndependence on the speed $c$.  Using these results, we answer the shepherd's\nquestion of what is the best strategy to capture the lamb.\n\n\\section{ Expanding Cage}\n\\label{sec:exp}\n\nIt is convenient to treat the problem in the reference frame of the left\nboundary (Fig.~\\ref{setup}(a)), where the Brownian particle drifts to the\nright with speed $\\alpha c$ and the right boundary also drifts to the right\nwith speed $c$.  Thus the right boundary is located at $L(t)=L_0\\!+\\!ct>L_0$.\nWe focus on the splitting probability to reach the left edge of an expanding cage\n$\\mathcal{L}^{\\mathrm{e}}(x_0,L_0)$ as a function of the initial particle\nposition $x_0$ and the initial interval length $L_0$ (with\n$\\mathcal{R}^{\\mathrm{e}}$ the splitting probability to the right\nedge).  Following the approach of Ref.~\\cite{Bray:2007}, the backward\nFokker-Planck equation for the splitting probability is\n\\begin{subequations}\n\\begin{equation}\nD \\frac{\\partial^2\\mathcal{L}^{\\mathrm{e}}}{\\partial x_0^2} +\\alpha c \\frac{\\partial\n \\mathcal{L}^{\\mathrm{e}}}{\\partial x_0}+c \\frac{\\partial\\mathcal{L}^{\\mathrm{e}}}{\\partial L_0}=0\\,, \n\\end{equation}\nor, by introducing the rescaled variables $y=c x_0\/D$ and $\\lambda=cL_0\/D$, \n\\begin{equation}\\label{FPasym}\n\\frac{\\partial^2 \\mathcal{L}^{\\mathrm{e}}}{\\partial y^2} +\\alpha \\partialderiv{\\mathcal{L}^{\\mathrm{e}}}{y} +\\partialderiv{\\mathcal{L}^{\\mathrm{e}}}{\\lambda}=0\\,, \n\\end{equation}\n\\end{subequations}\nwith $0 \\leqslant y \\leqslant \\lambda$ and the boundary conditions\n$\\mathcal{L}^{\\mathrm{e}}(0,\\lambda)=1$ and $\\mathcal{L}^{\\mathrm{e}}(\\lambda,\\lambda)=0$. \n\nIn the spirit of \\cite{Bray:2007}, we seek a solution of the form\n\\begin{equation}\n\\label{gen}\n\\mathcal{L}^{\\mathrm{e}}(y,\\lambda)=\\sum\\limits_{n\\in \\mathbb{Z}} \\big[a_n e^{ny} \\!+\\! b_n e^{-(n+\\alpha)y} \\big] e^{-(n+\\alpha)n\\lambda}.\n\\end{equation}\nThe left boundary condition $\\mathcal{L}^{\\mathrm{e}}(0,\\lambda)=1$ leads to\n\\begin{subequations}\n\\label{CL}\n\\begin{equation}\\label{CLleft}\n\\mathcal{L}^{\\mathrm{e}}(0,\\lambda)=\\sum\\limits_{n\\in\\mathbb{Z}} (a_n+b_n) \\mathrm{e}^{-n(n+\\alpha)\\lambda}=1\\,,\n\\end{equation}\nwhile the right boundary condition $\\mathcal{L}^{\\mathrm{e}}(\\lambda,\\lambda)=0$ gives\n\\begin{equation*}\n\\label{bc-right}\n\\mathcal{L}^{\\mathrm{e}}(\\lambda,\\lambda)=\\sum\\limits_{n\\in\\mathbb{Z}} \n\\big[a_n e^{n\\lambda}+b_ne^{-(n+\\alpha)\\lambda}\\big] e^{-(n+\\alpha)n\\lambda}=0\\,.\n\\end{equation*}\nBy shifting the index of the second sum, $n\\to n-1$, and after some simple\nalgebra, the condition above can be written as\n\\begin{equation}\\label{CLright}\n\\mathcal{L}^{\\mathrm{e}}(\\lambda,\\lambda)=\\sum\\limits_{n\\in\\mathbb{Z}} (a_n+b_{n-1}) \\mathrm{e}^{n(1-n-\\alpha)\\lambda}=0\\,.\n\\end{equation}\n\\end{subequations}\nSince Eqs.~\\eqref{CL} hold for all $\\lambda$, we obtain the following\nrelations for the coefficients in \\eqref{gen}:\n\\begin{eqnarray}\\label{relations}\n\\begin{cases}\na_0+b_0=1\\,, \\\\\na_n+b_n=0 &\\qquad \\forall n \\neq 0\\,, \\\\\na_n+b_{n-1}=0 &\\qquad \\forall n\\,. \\\\\n\\end{cases}\n\\end{eqnarray}\n\nIf the initial cage length $L_0\\to\\infty$ and the initial position of the\nparticle is far from either boundary, then the splitting probability\n$\\mathcal{L}^{\\mathrm{e}}\\to 0$.  Using these facts in \\eqref{gen} imposes $a_0=0$.\nTogether with the relations~\\eqref{relations}, we finally obtain the\nsplitting probability to the left edge:\n\\begin{equation}\\label{piasym}\n\\mathcal{L}^{\\mathrm{e}}(y,\\lambda)=e^{-\\alpha y} +\\sum\\limits_{n=1}^{\\infty} \\big[\n  e^{-(n+\\alpha)y} - e^{ny}\\big] e^{-n(n+\\alpha)\\lambda}\\,.\n\\end{equation}\nNote that the splitting probability to the right edge can be obtained from\n\\eqref{piasym} by \n$\\mathcal{R}^{\\mathrm{e}}(y,\\lambda)\\big|_{\\alpha}=\\mathcal{L}^{\\mathrm{e}}(\\lambda-y,\\lambda)\\big|_{1-\\alpha}$.\n\nTo obtain the solution in the second formulation where an unbiased Brownian particle\nstarts at $x'_0=x_0\\!-\\!L_0\/2$ in a cage whose boundaries are located at\n$-L_0\/2\\!-\\!c_1 t$ and $L_0\/2\\!+\\!c_2 t$ at time $t$ (Fig.~\\ref{setup}(b)),\nwe replace $\\alpha$, $c$ and $x_0$ with their corresponding expressions in\nterms of $c_1$, $c_2$ and $x'_0$ to give\n\\begin{align}\n\\label{asymforward}\n\\mathcal{L}^{\\mathrm{e}}(x'_0,L_0)&=e^{-c_1(2x'_0+L_0)\/2D}\n +   \\sum\\limits_{n=1}^{\\infty} e^{-n[n(c_1+c_2)+c_1] L_0\/D} \\nonumber \\\\\n& \\qquad \\qquad  \\times \\left\\{\n  e^{[-n(c_1+c_2)+c_1](2x'_0+L_0)\/2D} - e^{[n(c_1+c_2)](2x'_0+L_0)\/2D}\\right\\}\\,.\n\\end{align}\nWhen the initial interval length $L_0\\gg 1$, the splitting probability is\nwell approximated by its first term\n\\begin{equation}\\label{approxPi1}\n\\mathcal{L}^{\\mathrm{e}}(x'_0,L_0) \\sim e^{-c_1(2x'_0+L_0)\/2D},\n\\end{equation}\nwhich is exponentially larger than all higher-order terms in the series in\n\\eqref{asymforward}.\n\nAs a useful counterpoint, we can recover this last result by applying the\n``free approximation''~\\cite{Krapivsky:1996,Redner}, in which the\nconcentration within the interval is assumed to retain the same Gaussian form\nas a diffusing particle on the infinite line with no imposed boundary\nconditions. This approximation relies on the boundaries being outside the\nrange where the probability distribution is appreciable.  Thus we assume that\nthe concentration profile is\n\\begin{equation*}\nc(x,t)=\\frac{A(t)}{\\sqrt{4\\pi Dt}}\\, e^{-(x-x'_0)^2\/4Dt}\\,,\n\\end{equation*}\nwhere the unknown amplitude $A(t)$ accounts for the loss of probability\nwithin the domain and should be determined self consistently.  In this free\napproximation, the flux at any point in space is\n\\begin{equation*}\nj= -D\\frac{\\partial c}{\\partial x} = \\frac{A(t)(x-x'_0)}{\\sqrt{16 \\pi Dt^3}}\\,\n  e^{-(x-x'_0)^2\/4Dt}.\n\\end{equation*}\nThus the total flux leaving the cage is the sum of the flux at the two\nboundaries:\n\\begin{equation*}\n  \\phi(t)=|j_1|+j_2= D \\left. \\frac{\\partial c}{\\partial x}\\right|_{x=-\\frac{L_0}{2}-c_1t} -D \\left. \\frac{\\partial c}{\\partial x}\\right|_{x=\\frac{L_0}{2}+c_2t}.\n\\end{equation*}\nFrom the exiting flux, the rate equation for the overall amplitude $A(t)$ is\n\\begin{align}\n\\label{RELxv}\n\\frac{d A}{dt} = -A\\left[ \\frac{\\frac{L_0}{2}-x'_0+c_2 t}{\\sqrt{16\\pi\n      Dt^3}}\\,e^{-(\\frac{L_0}{2}-x'_0+c_2 t)^2\/4Dt}\n +\\frac{\\frac{L_0}{2}+x'_0+c_1 t}{\\sqrt{16\\pi  Dt^3}}\\,e^{-(\\frac{L_0}{2}+x'_0+c_1 t)^2\/4Dt}\\right].\n\\end{align}\nIntegrating this equation to finite time, the  amplitude is\n\\begin{align}\n\\ln A(t)=-\\frac{e^{-c_2(L_0-2x'_0)\/2D}}{2}\\,\n\\mathrm{erfc}\\!\\left(\\frac{\\frac{L_0}{2}\\!-\\!x'_0\\!-\\!c_2t}{\\sqrt{4Dt}}\\right) \n-\\frac{e^{-c_1(L_0+2x'_0)\/2D}}{2}\\,\n\\mathrm{erfc}\\!\\left(\\frac{\\frac{L_0}{2}\\!+\\!x'_0\\!-\\!c_1  t}{\\sqrt{4Dt}}\\right).\n\\end{align}\nFor $t\\to\\infty$, this expression reduces to\n\\begin{align}\n\\label{Ainfv}\nA(t\\!=\\!\\infty)&=\\exp\\left[ -e^{-c_2(L_0-2 x'_0)\/2D}- e^{-c_1(L_0+2x'_0)\/2D}\\right]\\nonumber \\\\\n& \\sim 1-e^{-c_2(L_0-2 x'_0)\/2D}- e^{-c_1(L_0+2x'_0)\/2D}\\qquad L_0\\to\\infty\\,.\n\\end{align}\nHere $A(t\\!=\\!\\infty)$ represents the large-time limit of the survival\nprobability, namely, the probability that the particle has not touched either\nof the boundaries.  It can thus be written as\n$A(t\\!=\\!\\infty)=1-\\mathcal{L}^{\\mathrm{e}}(x'_0,L_0)-\\mathcal{R}^{\\mathrm{e}}(x'_0,L_0)$,\nwhich finally leads to\n\\begin{eqnarray}\n\\mathcal{L}^{\\mathrm{e}}(x'_0,L_0) &\\sim e^{-c_1 (L_0+2 x'_0)\/2D}\\qquad\n\\mathcal{R}^{\\mathrm{e}}(x'_0,L_0) & \\sim e^{-c_2(L_0-2 x'_0)\/2D}\\,,\n\\end{eqnarray}\nin agreement with Eq.~\\eqref{approxPi1}.\n\nUnfortunately, this backward Fokker-Planck approach does not seem to be\nadaptable to a contracting cage.  In this case, the general solution involves\na complex exponential dependence in $y$.  Therefore, the device used to\nsimplify the right boundary condition to the form given in \\eqref{bc-right}\nno longer holds.  To obtain the splitting probability in this case, we apply\na more general framework following the methods\nof~\\cite{Bray:2007b,Krapivsky:1996}.\n\n\\section{Contracting and Expanding Cage}\n\\label{sec:gen}\n\nWe now turn to the general case of a cage that can be either contracting or expanding.\nAgain, we study the problem in the reference frame of the left boundary that\nis fixed at $x=0$.  For the expanding cage, the right boundary moves to the\nright at speed $c$ and the particle drifts to the right with speed $\\alpha c$\nin addition to its diffusion.  For the contracting cage, the right boundary\nand the particle both drift to the left.  Let $x$ denote the position of the\nparticle at time $t$.  We first compute the propagator $p(x,t)$ for the\nparticle in this cage, viz., the probability for the particle to be at $x$ at\ntime $t$, by solving the forward Fokker-Planck equation (see, e.g.,\n\\cite{Redner})\n\\begin{equation}\\label{forward}\n\\partialderiv{p}{t} \\pm \\alpha c \\frac{\\partial p}{\\partial x}=D \\frac{\\partial^2 p}{\\partial x^2}\\,,\n\\end{equation}\nwith the initial and boundary conditions $p(x,0)=\\delta(x\\!-\\!x_0)$ and\n$p(0,t)=p\\big(L(t),t\\big)=0$, and with $L(t)=L_0\\pm ct$.  Here the upper sign\ncorresponds to the expanding cage and the lower sign to the contracting\ncage throughout this section.  For the contracting cage the process stops\nwhen the two boundaries meet at $t=L_0\/c$.\n\nWhen the boundaries are immobile, so that the cage length $L(t)=L_0$ is\nconstant, the elemental solutions of the Fokker-Planck equation with these\nboundary conditions are well known~\\cite{Redner}\n\\begin{equation*}\nf_n(x,t)=\\sin\\left(\\frac{n\\pi x}{L_0}\\right) \\exp\\left( \\pm\\frac{\\alpha\n    cx}{2D}-\\frac{\\alpha^2 c^2 t}{4D}  -\\frac{n^2 \\pi^2 Dt}{L_0^2} \\right)\\, \\qquad \\textrm{with $n\\in\\mathbb{N}$}.\n\\end{equation*}\nTo account for the interval length changing linearly with time, we follow the\nmethod employed in~\\cite{Krapivsky:1996} and adapted by Bray and\nSmith~\\cite{Bray:2007b} and postulate a solution to Eq.~\\eqref{forward} with\nabsorbing boundary conditions at $x=0$ and $x=L(t)$ of the form\n\\begin{equation}\n\\label{pn}\np_n(x,t)=g(x,t) \\sin\\bigg( \\frac{n\\pi x}{L(t)}  \\bigg) \n\\exp \\bigg(\\pm\\frac{\\alpha cx}{2D}-\\frac{\\alpha^2 c^2 t}{4D} \\bigg) \n\\exp\\bigg(-n^2 \\pi^2D \\int_0^t \\frac{\\diff{t'}}{L^2(t')} \\bigg). \n\\end{equation}\nSubstituting this trial function into Eq. \\eqref{forward}, we obtain\n\\begin{equation}\\label{gx}\n\\bigg( D \\frac{\\partial^2g}{\\partial x^2} -\\frac{\\partial g}{\\partial t}\\bigg) \n\\tan\\bigg(\\frac{n\\pi x}{L(t)} \\bigg)=-\n\\frac{n \\pi}{L(t)} \\bigg( 2D \\frac{\\partial g}{\\partial x} \\pm \\frac{cx}{L(t)} g \\bigg).\n\\end{equation}\n\nWe notice, as done in~\\cite{Bray:2007b}, that we can seek a form for $g(x,t)$\nthat cancels both the left- and right-hand sides of \\eqref{gx}.  Thus\n$g(x,t)$ must simultaneously solve\n\\begin{align*}\n\\begin{split}\n& D \\frac{\\partial^2g}{\\partial x^2} =\\frac{\\partial g}{\\partial t}\\,, \\\\\n& 2D \\frac{\\partial g}{\\partial x}=\\mp\\frac{cx}{L(t)} g\\,.\n\\end{split}\n\\end{align*}\nThese equations imply that the function $g(x,t)$ has the form\n\\begin{equation}\ng(x,t)=\\frac{K}{\\sqrt{L(t)}}\\,\\, e^{\\mp x^2 c\/4D L(t)} \\,, \n\\end{equation}\nwith $K$ a constant. The general solution can now be written as a\nsuperposition of the basis functions $p_n(x,t)$:\n\\begin{equation}\np(x,t)=  \\sum\\limits_{n\\in\\mathbb{N}} \\frac{a_n}{\\sqrt{L(t)}}\n           \\sin\\bigg(\\frac{n\\pi x}{L(t)} \\bigg)  \n\\exp\\bigg(\\mp\\frac{x^2c}{4DL(t)} \\pm\\frac{\\alpha c x}{2D}  \\bigg)  \n\\exp\\bigg(-\\frac{\\alpha^2 c^2 t}{4D}-\\frac{n^2\\pi^2 Dt}{L_0L(t)}\\bigg)\\,.\n\\end{equation}\nFor the initial condition $p(x,0)=\\delta(x-x_0)$, we use the identity\n\\begin{equation*}\n\\sum\\limits_{n\\in\\mathbb{N}} \\sin\\left(\\frac{n\\pi x}{L_0} \\right) \\sin\\left(\\frac{n\\pi x_0}{L_0} \\right) =\\frac{L_0}{2} \\,\\delta(x-x_0),\n\\end{equation*}\nfor $0\\leqslant x,x_0 \\leqslant L_0$ to finally obtain the solution for the\ngiven initial condition as\n\\begin{align}\np(x,t)&=  \\sum\\limits_{n\\in\\mathbb{N}} \\frac{2}{\\sqrt{L_0 L(t)}}\n   \\sin\\bigg(\\frac{n\\pi x}{L(t)} \\bigg)\\sin\\bigg(\\frac{n\\pi x_0}{L_0} \\bigg)  \\nonumber \\\\\n& \\qquad \\times \\exp\\bigg(\\mp\\frac{c(x^2-x_0^2)}{4D L(t)} \\pm\\frac{\\alpha c (x\\!-\\!x_0)}{2D} -\\frac{\\alpha^2 c^2 t}{4D}-\\frac{n^2\\pi^2 Dt}{L_0L(t)}\\bigg)\\,. \n\\end{align}\nThe first-passage probability $F$ to the left edge of the cage is therefore\n\\begin{align}\nF(0,t)&=D \\partialderiv{p}{x}\\Big|_{x=0}\\,,\\nonumber \\\\\n&= \\sum\\limits_{n\\in\\mathbb{N}} \\frac{2n\\pi D}{\\sqrt{L_0L^3(t)}}  \\sin\\!\\left( \\frac{n\\pi x_0}{L_0}  \\right) \n \\exp\\!\\bigg(\\pm \\frac{c x_0^2}{4DL_0} \\mp\\frac{\\alpha c x_0}{2D} -\\frac{\\alpha^2 c^2 t}{4D}-\\frac{n^2\\pi^2 Dt}{L_0L(t)}\\bigg)\\,.\n\\end{align}\n\nThe splitting probability to the left edge is the time integral of this\nfirst-passage probability.  As noted previously, the temporal integration\nrange depends on the sign of the speed.  Finally, the splitting\nprobabilities for the contracting and expanding cage, $\\mathcal{L}^{\\mathrm{c}}$\nand $\\mathcal{L}^{\\mathrm{e}}$, respectively, are\n\\begin{subequations}\n\\label{pis}\n\\begin{align}\\label{piplus}\n\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)=\\int_0^{L_0\/c} \\diff{t} F(0,t) = &\\sum\\limits_{n\\in\\mathbb{N}} \\frac{2n\\pi D}{c\\sqrt{L_0}} \\sin\\Big(\n  \\frac{n\\pi x_0}{L_0} \\Big) \ne^{-c(x_0-\\alpha L_0)^2\/4DL_0}  \\nonumber \\\\\n& \\times  e^{n^2\\pi^2 D\/cL_0}\\int_{0}^{L_0} \\frac{\\diff{L}}{L^{3\/2}} \\exp\\Big(\\frac{\\alpha^2 cL}{4D}-\\frac{n^2\\pi^2 D}{cL}\\Big), \n\\end{align}\n\\begin{align}\\label{pimoins}\n\\mathcal{L}^{\\mathrm{e}}(x_0,L_0)=\\int_0^{\\infty} \\diff{t} F(0,t) =& \\sum\\limits_{n\\in\\mathbb{N}} \\frac{2n\\pi D}{c\\sqrt{L_0}} \\sin\\Big(\n  \\frac{n\\pi x_0}{L_0} \\Big) \ne^{c (x_0-\\alpha L_0)^2\/4DL_0}  \\nonumber \\\\\n& \\times  e^{-n^2\\pi^2 D\/cL_0} \\int_{L_0}^{\\infty} \\frac{\\diff{L}}{L^{3\/2}} \\exp\\Big(-\\frac{\\alpha^2 cL}{4D}+\\frac{n^2\\pi^2 D}{cL}\\Big).\n\\end{align}\n\\end{subequations}\nExpression \\eqref{pimoins} for the splitting probability in an expanding cage\ncan be shown numerically to perfectly match the simpler form \\eqref{piasym}\nderived by the backward Fokker-Planck equation.  \n\nHowever, these splitting probabilities are not convenient for numerical\nevaluation.  Instead, it is expedient to use the Poisson summation\nformula~\\cite{Olver}\n\\begin{equation*}\n\\sum_{n\\in \\mathbb{Z}} h(n)= \\sum_{m\\in\\mathbb{Z}} \\hat{h}(2\\pi m),\n\\end{equation*}\nwith $\\hat{h}(x)=\\int_{-\\infty}^{+\\infty} \\diff{t} e^{-ixt} f(t)$, to give\nthe alternative expression \n\\begin{align}\n  &\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)= \\sum\\limits_{m\\in\\mathbb{Z}} \\sqrt{\\frac{c}{4\\pi D}}\n    e^{-c(x_0-\\alpha L_0)^2\/4DL_0}\\!\n\\int_0^{L_0} \\frac{\\diff{L}}{(L_0\\!-\\!L)^{3\/2}} \\; e^{\\alpha^2 cL\/4D} \\nonumber \\\\\n  & \\quad \\times \\!\\exp\\!\\bigg[\\!-\\frac{cL(4L_0^2m^2\\!+\\!x_0^2)}{4DL_0(L_0\\!-\\!L)}\\bigg]  \n\\left\\{x_0 \\cosh\\bigg[\\frac{cLx_0m}{D(L_0\\!-\\!L)}\\bigg] \n- 2mL_0 \\sinh\\bigg[ \\frac{cLx_0m}{D(L_0\\!-\\!L)}\\bigg] \\right\\},\n\\end{align}\nwhich is more suitable for numerical evaluation (and similarly for $\\mathcal{L}^{\\mathrm{e}}(x_0,L_0)$).\n\n\n\n\\section{Optimal Capture Criterion}\n\\label{sec:disc}\n\nWe now turn to our original question: what is the optimal strategy for the\nshepherd to catch her skittish lamb without driving it over the precipice?\nIn the shepherd's reference frame, the lamb approaches at speed $\\alpha c$\nwhile the precipice approaches at a higher speed $c$.  The probability to\ncatch the lamb in this contracting cage is the splitting probability\n$\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)$ \\eqref{piplus}.  What speed $c$ maximizes this\nsplitting probability?\n\nPartial conclusions can be drawn by studying the limits of $c\\to 0$ and\n$c\\to\\infty$.  If $c=0$, the splitting probability\n$\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)$ is a linear function of its initial position\n(see, e.g.,~\\cite{Redner})\n\\begin{equation*}\n\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)=\\frac{L_0-x_0}{L_0}.\n\\end{equation*}\nThe qualitative behavior of $\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)$ when $c\\to\\infty$ can also be\neasily understood.  In this limit, if the time $x_0\/(\\alpha c)$ for the lamb\nto reach the shepherd is smaller than the time $(L_0-x_0)\/[(1-\\alpha)c]$ for\nthe precipice to catch up to the lamb, then $\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)\\to 1$, while\n$\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)\\to 0$ otherwise.  These two times match when\n$\\alpha=x_0\/L$.  Thus in the limit of large speed, the splitting probability\nreduces to a step function, with $\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)\\approx 1$ for\n$x_0\/L<\\alpha$ and $\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)\\approx 0$ for $x_0\/L>\\alpha$ .\n\nLet us now focus on the first order of the splitting probability for small speeds.  In this\ncase, the integral over $L$ in Eq.~\\eqref{piplus}\n\\begin{equation*}\n\\mathcal{I} \\equiv \\int_{0}^{L_0} \\frac{\\diff{L}}{L^{3\/2}} \\exp\\left(\\frac{\\alpha^2 cL}{4D}-\\frac{n^2\\pi^2 D}{cL}\\right)\n\\end{equation*}\ncan be developed with respect to $c$ by expanding\n$\\exp\\left(\\alpha^2 c L\/4D\\right)$ and integrating by parts.  This yields\n\\begin{equation}\n\\mathcal{I} \\sim \\exp\\left(-\\frac{n^2\\pi^2D}{ c L_0}\\right) \\left[ \\frac{c\\sqrt{L_0}}{n^2\\pi^2D} -\\frac{c^2L_0^{3\/2}}{2n^4\\pi^4D^2} \\left(1-\\frac{n^2\\pi^2\\alpha^2}{2} \\right) \\right].\n\\end{equation}\nThen using\n\\begin{align}\n&\\sum\\limits_{n=1}^{\\infty} \\frac{(-1)^n}{n} \\sin(nz)=-\\frac{z}{2}\\,, \\nonumber\\\\\n&\\sum\\limits_{n=1}^{\\infty} \\frac{(-1)^n}{n^3} \\sin(nz)= \\frac{z^3}{12} -\\frac{\\pi^2 z}{12}\\,,\\nonumber\n\\end{align}\nwe obtain the small-speed behavior of the splitting probability:\n\\begin{equation}\n\\label{Pi+-smallc}\n\\mathcal{L}^{\\mathrm{e,c}}(x_0,L_0) = \\frac{L_0-x_0}{L_0} \\mp \\frac{cx_0}{6D}  \\frac{(L_0-x_0) \\big[(3\\alpha-1)L_0-x_0\\big]}{L_0^2}\\, +o(c).\n\\end{equation}\nHere we also quote the limiting form splitting probability for the expanding\ncage, which is found from \\eqref{pimoins} by the same steps as outlined above.\n\nAs a result of the $\\alpha$ and $x_0$ dependence of the first-order term in\n$c$ given above, $\\mathcal{L}^{\\mathrm{c}}(x_0,L_0)$ is an increasing function of\nspeed at $c=0$ when $\\alpha>\\frac{1}{3}(x_0\/L_0\\!+\\!1)$ and decreasing otherwise.\nCombining this fact with the limiting behavior for $c\\to\\infty$, we deduce\nthat the splitting probability can be a non-monotonic function of the speed,\nfor specific values of $x_0$ and $\\alpha$.  This leads to rich behaviors for\nthe splitting probability, as illustrated in Fig.~\\ref{diagphase1}(a).\n\n\\begin{figure}[ht]\n\\centerline{\\qquad\\qquad\n\\includegraphics[width=180pt]{phase-diag-cont_pspdftex.pdf}\\qquad\\qquad\n\\includegraphics[width=180pt]{phase-diag-exp_pspdftex.pdf}}\n\\caption{Splitting probability phase diagram for the (a) contracting and (b)\n  expanding cage in the speed ratio ($\\alpha$) and initial position ($x_0$)\n  plane.  In (a), $\\mathcal{L}^{\\mathrm{c}}\\approx 1$ for $c\\to\\infty$ above the\n  solid line, while $\\mathcal{L}^{\\mathrm{c}}\\approx 0$ for $c\\to\\infty$ below.\n  Above the dashed line $\\mathcal{L}^{\\mathrm{c}}$ is an increasing function of $c$\n  at $c=0$ and a decreasing function below.  These lines delineate four zones\n  of behavior as discussed in the text.  In (b), there are two zones.}\n\\label{diagphase1}\n\\end{figure}\n\nWe can now give advice to the shepherd.  There are four distinct strategies,\ncorresponding to the four zones of Fig.~\\ref{diagphase1}(a):\n\\begin{enumerate}\n\n\\item ``Dangerous zone'' (lower right).  Here, either the lamb is very\n  fearful, $\\alpha\\ll 1$, or sufficiently close to the precipice,\n  $\\alpha<\\frac{1}{3}(x_0\/L_0\\!+\\!1)$ and $\\alpha<x_0\/L_0$, so that\n  $\\mathcal{L}^{\\mathrm{c}}$ monotonically decreases with $c$.  Thus the optimum\n  strategy for the shepherd is to not move.\n\n\\item ``Safe zone'' (upper left).  Here, either the lamb is not very fearful\n  or is sufficiently close to the shepherd, $\\alpha>\\frac{1}{3}(x_0\/L_0\\!+\\!1)$\n  and $\\alpha>x_0\/L_0$, so that $\\mathcal{L}^{\\mathrm{c}}$ monotonically increases\n  with $c$.  The shepherd should run as fast as possible to maximize the\n  probability to catch the lamb.\n\n\\item ``Optimizable zone'' (upper right).  Here, the lamb is close to the\n  precipice but not too fearful, $\\alpha>\\frac{1}{3}(x_0\/L_0\\!+\\!1)$ and\n  $\\alpha<x_0\/L_0$ so that $\\mathcal{L}^{\\mathrm{c}}$ has a maximum as a function of $c$.  Thus\n  there is an optimal speed that maximizes the probability for the shepherd\n  to catch the lamb.\n\n\\item ``Dilemma zone'' (lower left).  Here the lamb is close to the shepherd\n  but also very fearful, $\\alpha<\\frac{1}{3}(x_0\/L_0\\!+\\!1)$ and\n  $\\alpha>x_0\/L_0$.  Thus $\\mathcal{L}^{\\mathrm{c}}$ initially decreases with $c$\n  before eventually increasing.  Thus if the shepherd is unfit, she should\n  stay still.  However, if she is sufficiently fit, she should run as fast as\n  possible.\n\\end{enumerate}\n\nFor the expanding cage, the splitting probability given in\nEq.~\\eqref{Pi+-smallc} is now an increasing function of speed at $c=0$ for\n$\\alpha<\\frac{1}{3}(x_0\/L_0\\!+\\!1)$.  Using this small-speed dependence, together\nwith the limiting behavior $\\mathcal{L}^{\\mathrm{e}}(x_0,L_0)\\to 0$ for large speed,\ngives the phase diagram shown Fig.~\\ref{diagphase1}(b).  There again exists a\nzone in the phase diagram where the splitting probability can be maximized\nwith respect to the speed.  Here, the maximum of the splitting probability\narises from the interplay between two competing effects from the cage\nexpansion.  Indeed, consider the complementary probability that the lamb\nnever reaches the left boundary.  The larger the speed, the lower the\nprobability for the lamb to reach the left boundary.  At the same time, the\nlarger the speed, the higher the probability for the lamb to also \\emph{not}\nreach the right boundary, which implicitly increases the probability to reach\nthe left boundary.  As a result of these competing effects, there exists an\noptimal speed of expansion that maximizes the splitting probability.\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=180pt]{splitting_max.pdf}\\qquad\\qquad\n\\includegraphics[width=180pt]{splitting_min.pdf} \n\\caption{Splitting probability to catch the lamb in the (a) ``optimizable\n  zone'' and (b) ``dilemma zone''.  In (a), the splitting probability is\n  increased substantially (by more than a factor 3 for $\\alpha=0.92$,\n  $x_0=19$, and $L_0=20$) when the shepherd moves at the optimal speed\n  $c_{\\textrm{opt}}$ instead of staying still.  In (b), the probability to\n  catch the lamb is diminished if the shepherd runs slower than speed $c^*$.\n  Here the parameter values are $\\alpha=0.08$, $x_0=1$ and $L_0=20$).}\n\\label{minmax}\n\\end{figure}\n\n\\section{Conclusion}\n\nWe analytically determined the splitting probabilities for a one-dimensional\nBrownian motion in a cage whose boundaries move at constant speeds $c_1$ and\n$c_2$.  We analyzed both the cases of contracting and expanding cages.  In\naddition, we calculated the time-dependent first-passage probabilities at\neach of the boundaries.  Intriguing behaviors of the splitting probabilities\narise as a consequence of the ballistic boundary motion.  Indeed, we found that the\nsplitting probabilities can vary non-monotonically with the relative speeds\nof the boundaries, depending on the initial position of the Brownian particle\nand the ratio between the two boundary speeds.  In the context of a fearful\nlamb near a precipice (Fig.~\\ref{setup}), this non-monotonicity defines a\nnon-trivial optimal strategy for the shepherd to catch the lamb.\n\nThis approach could be extended to determine the splitting probability to a\nsubset of a growing $d$-dimensional sphere for a particle that starts\nsomewhere within the interior.  It would also be interesting to extend the\nconditional exit problem to the case of non-linear displacements of the\nboundaries where\nthe lamb capture probability should also have a non-trivial optimization.\n\n\\bigskip\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere are several different ways of framing the results of this paper.\nOut main object of study will be CM-elliptic curves over $\\mathbb{Z}_p$ which are supersingular at $p$.\nThe results we obtain will primarily be directed towards trying to address the following three questions:\n\\begin{enumerate}\n\\item When are there elliptic curves defined over $\\mathbb{Z}_p$ with CM by an order $\\mathcal{O}$ in a quadratic imaginary field $K$ in which $p$ is inert and where $p$ does not divide the conductor of $\\mathcal{O}$?\n\\item What factors affect the possible reductions of their $j$-invariants modulo $p$ amongst the set of all supersingular $\\mathbb{F}_p$-rational $j$-invariants?\n\\item Given an $\\mathbb{F}_p$-rational supersingular $j$-invariant which admits CM by $\\mathcal{O}$, when does there exist an elliptic curve defined over $\\mathbb{Z}_p$, with CM by $\\mathcal{O}$ which reduces to it.\n\\end{enumerate}\n\nOne natural source of interest in these questions is the following observation of Ernst Kani:\n\\begin{prop*}\nEvery $\\mathbb{F}_p$ elliptic curve with CM by $\\mathcal{O}$ lifts to $\\mathbb{Z}_p$ (with a lifting of its CM to $\\overline{\\mathbb{Z}}_p$) if and only if $p$ does not divide the conductor of the ring $\\mathbb{Z}[j(E_1),\\ldots, j(E_n)]$ generated by the $j$ invariants of all elliptic curves with CM by $\\mathcal{O}$.\n\\end{prop*}\n\\begin{rmk}\nThis ring $\\mathbb{Z}[j(E_1),\\ldots, j(E_n)]$ is a very natural order in the ring class field of $\\mathcal{O}$, its structure is mysterious.\n\\end{rmk}\n\n\n\nThe results we obtain are somehow in contrast to the same question asked for elliptic curves over $\\mathbb{Z}_{p^2}$ the unramified quadratic extension of $\\mathbb{Z}_p$.\nIn particular, for the same questions asked over $\\mathbb{Z}_{p^2}$, we have the following answers:\n\\begin{enumerate}\n\\item There are always CM-elliptic curves over $\\mathbb{Z}_{p^2}$ with CM by $\\mathcal{O}$ an order in a quadratic imaginary field $K$ in which $p$ is inert, and where $p$ does not divide the conductor.\n\\item From the work of Cornut-Vatsal \\cite{CornutVatsal1,CornutVatsal2} and Jetchev-Kane \\cite{JetchevKane} we have that the reductions of the $j$-invariants of elliptic curves with CM by $\\mathcal{O}$ are equidistributed among the supersingular values in $\\mathbb{F}_{p^2}$ (as we vary the conductors $\\mathcal{O}$ subject to certain congruence conditions). Moreover, for each $p$ and all but finitely many $\\mathcal{O}$ where $p$ is inert, the map from elliptic curves with CM by $\\mathcal{O}$ to supersingular $j$-invariants in $\\mathbb{F}_{p^2}$ is surjective.\n\\item By the work of Deuring \\cite{Deuring} we know that given a supersingular elliptic curve $\\overline{E}$ with CM by $\\mathcal{O}$ there always exists a lift to an elliptic curve over $\\mathbb{Z}_{p^2}$ with CM by $\\mathcal{O}$ which reduces to $\\overline{E}$.\n\\end{enumerate}\n\n\n\nThe results we obtain are motivated by computations, some of the data from which is presented in the Appendix, which gave results which seemed contrary to the above. In particular if we consider only the elliptic curves which are defined over $\\mathbb{Z}_p$ then:\n\\begin{itemize}\n\\item \nThey are not always surjective onto supersingular $\\mathbb{F}_p$ values as we vary $\\mathcal{O}$ among \n\\begin{itemize} \n\\item maximal orders subject to certain congruence conditions on the discriminant;\n\\item orders in a certain fixed $K$ subject to certain congruence conditions on the conductor;\n\\item orders subject to certain congruence conditions on the conductor and discriminant of $K$.\n\\end{itemize}\n\\item\nThe set of possible values, and hence the overall distributions depends on congruence conditions on both the discriminant of $K$ and the conductor of $\\mathcal{O}$.\n\\item\nFor certain congruence conditions on discriminants and conductors there are irreducible factors which always appear together, in equal numbers. So the appearance of a given factor is not independent on the appearance of another.\n\\end{itemize}\n\n\nWe should emphasize before proceeding that the above does not actually conflict with the aforementioned equidistribution results. Firstly, because the $\\mathbb{Z}_p$ curves have density $0$ among all curves, but moreover, because the data does suggest the following:\n\\begin{itemize}\n\\item Varying $\\mathcal{O}$ subject to congruence conditions on the discriminants and\/or conductors; the $j$-invariants which appear are very likely to following a simple distribution.\n\\item The equidistribution results should work primarily with conditions of the form $\\ell \\not\\vert D\\mathfrak f$ rather than $\\ell \\vert D\\mathfrak f$ whereas the difference in behavior is primarily from a comparison of these cases.\n\nConsequently, it is possible or perhaps even likely that based on heuristic arguments and the precise types of families considered in the work of Cornut-Vatsal and Jetchev-Kane, some form of the equidistribution results one would have expected still hold for the $\\mathbb{Z}_p$-terms.\n\n\\item The work of \\cite{CornutVatsal2} actually describes circumstances in which there can be correlation between frequencies, however we should note that the correlations we observe are only apparent for $\\mathbb{Z}_p$ values and not the $\\mathbb{Z}_{p^2}$ values with the same reductions, so even if they have the same underlying explanation, the phenomenon is still somewhat distinct.\n\\end{itemize}\n\nThis paper is organized as follows:\n\\begin{itemize}\n\\item In Section \\ref{sec:bg} we introduce the relevant background.\n\\item In Section \\ref{sec:res} we state and prove our results\n\\item In Section \\ref{sec:cq} we discuss two natural questions our work leaves open.\n\\item In \\ref{sec:ad} we discuss the computations and data on which are work is based.\n\\end{itemize}\n\n\n\n\n\n\\section{Background}\\label{sec:bg}\n\nIn this section we will be introducing the results necessary to state and prove our theorems. \nMuch of what we are saying is very well known, and can be found in many references on the theory of complex multiplication.\nSome results which are perhaps less well known can be found in \\cite{SchertzCM}, \\cite{Deuring}, \\cite{Ibukiyama} or \\cite{Dorman1989global}.\n\nWe recall the following important facts:\n\\begin{thm}\nIf $E$ is an elliptic curve over a field of characteristic $0$ then either:\n\\begin{itemize}\n\\item $End(E) = \\mathbb{Z}$, this is the general case.\n\\item $End(E) = \\mathcal{O}$, for $\\mathcal{O} \\subset \\mathbb{Q}(\\sqrt{-D})$ an order in a quadratic imaginary field, this is the so-called CM-case.\n\\end{itemize}\n\\end{thm}\n\nWe will be interested in the CM or complex multiplication case in characteristic $0$, where we have the following classification result:\n\\begin{thm}\nThe elliptic curve $E_\\tau = \\mathbb{C} \/  (\\mathbb{Z} \\oplus \\tau\\mathbb{Z})$ has $End(E) = \\mathcal{O}$ if and only if \n\\begin{enumerate}\n\\item $\\tau \\in \\mathbb{Q}(\\sqrt{-D})$, that is $\\tau$ generates a (complex) quadratic field, and\n\\item $\\mathbb{Z} + \\tau\\mathbb{Z} \\subset \\mathbb{Q}(\\sqrt{-D})$  is a (projective) $\\mathcal{O}$-module.\n\\end{enumerate}\nMoreover, for any algebraically closed field $C$ of characteristic $0$ there is a bijective correspondence between\nelliptic curves over $C$ with $End(E) = \\mathcal{O}$ and $C\\ell(\\mathcal{O})$ the ideal class group of $\\mathcal{O}$.\n\\end{thm}\n\n\\begin{rmk}\nNote that there is an essentially equivalent bijection between $C\\ell(\\mathcal{O})$ and pairs $(E,\\rho:\\mathcal{O}\\overset{\\sim}\\rightarrow \\End(E))$ of $E$ and an isomorphism of $\\mathcal{O}$ with $\\End(E)$ with a fixed CM-type. This bijection extends to characteristic $p$ where we consider instead certain optimal embeddings $\\mathcal{O}\\rightarrow \\End(E)$.\n\nIn the definition of $P_{\\mathcal{O}}(X)$ below, it is conceptually better to be considering the bijection of the theorem.\n\\end{rmk}\n\n\\begin{thm}\nIf $E$ is an elliptic curve over a field of characteristic $p$ then either:\n\\begin{itemize}\n\\item $End(E) = \\mathbb{Z}$, this is the general case.\n\\item $End(E) = \\mathcal{O}$, for $\\mathcal{O} \\subset \\mathbb{Q}(\\sqrt{-D})$ an order in a quadratic imaginary field in which $p$ splits.\n\\item $End(E) = \\mathbb{B}$, for $\\mathbb{B}$ a maximal order in a quaternion algebra over $\\mathbb{Q}$ ramified only at $p$ and $\\infty$. This is the so-called supersingular case.\n\\end{itemize}\n\\end{thm}\n\nFrom the above we see that if ever we can reduce a CM elliptic curve $E$ at a prime inert in $\\mathcal{O}$ we will obtain a supersingular elliptic curve. In the characteristic $p$ setting it will be this case we are most interested in.\n\n\\begin{nota}\nLet $m\\in\\mathbb{Z}^+$ be square free so that $K=\\mathbb{Q}(\\sqrt{-m})$ is the quadratic imaginary field of discriminant $D$, denote by $\\mathcal{O}_K$ its maximal order and $\\mathcal{O} = \\mathcal{O}_{K,\\mathfrak f} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ an order of conductor $\\mathfrak f\\in \\mathbb{Z}$. Denote by:\n\\[ P_{\\mathcal{O}}(X) = \\prod_{\\mathfrak a \\triangleleft \\mathcal{O}} (X-j(\\mathbb{C}\/\\mathfrak a)). \\]\nDenote by $L$ the splitting field of $P_{\\mathcal{O}}(X)$ over $K$.\n\\end{nota}\n\nThe following facts are well known, for a reference see for example \\cite{SchertzCM}.\n\\begin{itemize}\n\\item $P_{\\mathcal{O}}(X) \\in \\mathbb{Z}[X]$ and is irreducible over $K$.\n\\item $L$ is abelian over $K$, with $\\Gal(L\/K) \\simeq C\\ell(\\mathcal{O})$, the action being the natural permutation action of $C\\ell(\\mathcal{O})$ on the roots.\n\\item $L$ is galois over $\\mathbb{Q}$, the action of $\\Gal(K\/\\mathbb{Q})$ on $ C\\ell(\\mathcal{O})$ being $g\\mapsto g^{-1}$ so that $\\Gal(K\/\\mathbb{Q})$ is a generalized dihedral group.\n\\item The action of complex conjugation on the ideals of $K$, agrees with the action on the set $CM(\\mathcal{O})$ which agrees with the action of $\\Gal(K\/\\mathbb{Q})$.\n\\item $L\/K$ is ramified only at primes over $\\mathfrak f$, whereas $L\/\\mathbb{Q}$ is ramified only at primes over $D\\mathfrak f$.\n\\end{itemize}\nWe shall denote by $N = \\mathbb{Q}(j) = \\mathbb{Q}[X]\/(P_{\\mathcal{O}}(X)) \\subset L$.\n\nBased on the above we can conclude the following:\n\\begin{itemize}\n\\item If $p$ is inert in $K$ and $p$ does not divide $\\mathfrak f$ (or equivalently that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$) then $p$ splits in $L\/K$.\n\\item If $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$ then $P_{\\mathcal{O}}(X) $ factors as a product of quadratic and linear terms over $\\mathbb{Z}_p$.\n\\end{itemize}\n\n\\begin{rmk}\nThe above agrees with the fact that the reductions of these elliptic curves must be defined over $\\mathbb{F}_{p^2}$, as they are known to be supersingular.\n\\end{rmk}\n\n\\begin{prop}\nIf $p$ is inert in $K$ and $E$ is an elliptic curve with CM by $\\mathcal{O}$ then the reduction of $E$ modulo $p$ is supersingular.\nIn particular, $\\End(\\overline{E}) = \\mathbb{B}$, where $\\mathbb{B}$ is a maximal order in a quaternion algebra ramified only at $p$ and infinity.\nMoreover, there is a bijection between elliptic curves with CM by $\\mathcal{O}$ and pairs:\n\\[ (\\mathcal{O} \\subset \\mathbb{B}) \\]\nof $\\mathcal{O}$ with an optimal embedding into a maximal order $\\mathbb{B}$ as above.\n\nMoreover, the natural action of the ideals of $\\mathcal{O}$ by conjugation on such pairs $(\\mathcal{O} \\subset \\mathbb{B})$ agrees with the action of the ideals on the collection of elliptic curves with CM by $\\mathcal{O}$.\n\\end{prop}\nSee \\cite{Dorman1989global}.\n\nFrom now on we shall be working in the setting where $p$ is split in $K$ and $p$ does not divide $\\mathfrak f$.\nIn particular we are assuming that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\n\n\\begin{prop}\nIf $P_{\\mathcal{O}}(X) $ has a linear factor over $\\mathbb{Z}_p$, the number of such linear factors is $\\abs{\\Gal(L\/K)[2]}$ the size of the two torsion of the class group.\n\\end{prop}\n\\begin{proof}\nBy basic algebraic number theory we must count the size of the conjugacy class of Frobenius.\nThis is then a basic property to dihedral groups.\n\\end{proof}\n\n\\begin{rmk}\nIf $\\abs{\\Gal(L\/K)[2]} = 1$ then $P_{\\mathcal{O}}(X) $ has a unique linear factor over $\\mathbb{Z}_p$.\n\\end{rmk}\n\n\\begin{thm}[Deuring]\nIf $E$ corresponds to the data $(\\mathcal{O} \\subset \\mathbb{B})$ then the reduction of $E$ modulo $p$ is defined over $\\mathbb{F}_p$ (rather than simply $\\mathbb{F}_{p^2}$) if and only if $\\mathbb{B}$ contains $\\mathbb{Z}[\\sqrt{-p}]$.\n\\end{thm}\nSee \\cite{Deuring}.\n\nIn \\cite{Ibukiyama} Ibukiyama gives a complete classification of the maximal orders $\\mathbb{B}$ which contain $\\mathbb{Z}[\\sqrt{-p}]$.\n\n\\begin{nota}\nFix $p$ and $q = 3\\pmod{8}$ such that $\\mathbb{B} = (-p,-q)$ is the quaternion algebra ramified only at $p$ and $\\infty$.\nFix $\\alpha,\\beta\\in \\mathbb{B}$ such that $\\alpha^2 = -p$, $\\beta^2=-q$ and $\\alpha\\beta = -\\beta\\alpha$.\nChoose $r \\in \\mathbb{Z}$ such that $r^2 + p = mq$ for some $m\\in \\mathbb{Z}$.\n \nDenote:\n\\[   O(p,q,r,m) = \\mathbb{Z} + \\mathbb{Z} \\frac{\\alpha(1+\\beta)}{2}+ \\mathbb{Z}\\frac{1+\\beta}{2}  + \\mathbb{Z}\\frac{(r+\\alpha)\\beta}{q} \\]\nIf $p=3\\pmod{4}$ \nchoose $r' \\in \\mathbb{Z}$ such that $(r')^2 + p = 4m'q$ for some $m'\\in \\mathbb{Z}$.\nDenote:\n\\[   O'(p,q,r',m') = \\mathbb{Z} + \\mathbb{Z} \\frac{1+\\alpha}{2} + \\mathbb{Z}\\beta + \\mathbb{Z}\\frac{(r+\\alpha)\\beta}{2q}. \\]\n\\end{nota}\n\n\\begin{thm}[Ibukiyama]\nThe sets $O(p,q,r,m)$ (and $O'(p,q,r',m')$) are maximal orders of $\\mathbb{B}$, their isomorphism classes depend only on $q$ and not on $r$ or $m$. Moreover, all pairs consisting of a maximal order in $\\mathbb{B}$ with an embedding of $\\mathbb{Z}[\\sqrt{-p}]$ are of the form $O(p,q,r,m)$ (or $O'(p,q,r',m')$) with the embedding taking $\\sqrt{-p} \\rightarrow \\pm\\alpha$.\n\nThe orders $O(p,q,r,m)$ and $O'(p,q,r',m')$ are only ever isomorphic if they correspond to the $j$-invariant $1728$.\n\\end{thm}\nSee \\cite{Ibukiyama}.\n\n\\begin{rmk}\nIn $O(p,q,r,m)$ we may write:\n\\[ \\alpha = 2\\left(\\frac{\\alpha(1+\\beta)}{2}\\right) -q \\left(\\frac{(r+\\alpha)\\beta}{q}\\right) + qr. \\]\n\\end{rmk}\n\n\\begin{rmk}\nWe can count the number of isomorphism classes of $O(p,q,r,m)$ (respectively $ O'(p,q,r',m')$) by looking at the class numbers $h_{p}$ for $\\mathbb{Z}[\\sqrt{-p}]$ (and $\\tilde{h}_p$ for $\\mathbb{Z}[(1+\\sqrt{-p})\/2]$), we have the following standard formulas (for $p\\neq 3$):\n\\begin{itemize}\n\\item The number of supersingular $j$ invariants over $\\mathbb{F}_{p^2}$ is $n=\\floor{(p-1)\/12} + e_{0} + e_{1728}$, where $e_{x}$ is $0$ or $1$ depending on if $x$ is supersingular at $p$.\n\\item If $p = 7 \\pmod{8}$ then $h_{p} = \\tilde{h}_{p}$ and there are $(h_{p}+1)\/2$ options for both $O(p,q,r,m)$ and $O'(p,q,r',m')$.\n\\item If $p = 3 \\pmod{8}$ then $h_{p} = 3\\tilde{h}_{p}$ and there are $(h_p+1)\/2$ options for $O'(p,q,r',m')$ and $(\\tilde{h}_p+1)\/2$ options for $O'(p,q,r',m')$.\n\\item If $p= 1 \\pmod{4}$ there are $h_p\/2$ options for $O(p,q,r,m)$.\n\\end{itemize}\nCombining the above allows us to compute the number of $\\mathbb{F}_p$ rational supersingular values in terms of $h_{p}$.\n\nMore generally, if we fix $K=\\mathbb{Q}(\\sqrt{-D})$ a quadratic imaginary field of discriminant $-D$ and class number $h_K$.\nFix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ and write $\\mathfrak f = \\prod q_i^{a_i}$\nThe class number of $\\mathcal{O}$ is given by:\n\\[   h_{\\mathcal{O}} = \\epsilon h_K \\prod_{i} \\left(q_i - \\left(\\frac{-D}{q_i}\\right)\\right)q_i^{a_i-1} \\]\nwhere $\\epsilon = 1$ unless $D=-3$ or $D=-4$.\n\nIf $D=-3$ and the formula above is divisible by $3$ then $\\epsilon =\\tfrac{1}{3}$.\nIf $D=-4$ and the formula above is divisible by $2$ then $\\epsilon =\\tfrac{1}{2}$.\n\\end{rmk}\n\n\\begin{thm}[Halter-Koch]\nIf $n$ is the number of prime divisors of $D\\mathfrak f$ and $2$ does not divide $\\mathfrak f$ then :\n\\[ \\abs{C\\ell(\\mathcal{O})[2]} = \\begin{cases} 2^{n-1}  & D\\mathfrak f \\text{ odd} \\\\\n                                                              2^{n-2}  & 2 || D\\mathfrak f  \\\\\n                                                              2^{n-1}  & 4 || D\\mathfrak f  \\\\\n                                                              2^{n-1}  & 8 || D\\mathfrak f  \\\\\n                                                              2^{n}  & 16 | D\\mathfrak f  \\end{cases}. \\]\n\nMore precisely, the maximal $2$-extension of the ring class field of $\\mathcal{O}$ contains:\n\\[ \\mathbb{Q}(\\sqrt{(-1)^{(q-1)\/2} q}) \\]\nwhere $q$ is an odd prime factor of $D\\mathfrak f$.\n\nIf $D = -8m$ then the maximal $2$-extension of the ring class field of $\\mathcal{O}$ contains:\n\\[ \\mathbb{Q}(\\sqrt{(-1)^{(m-1)\/2} 2}).  \\]\n\nIf $D = 4\\pmod 8$ and $4|\\mathfrak f$ then the maximal $2$-subextension of the ring class field of $\\mathcal{O}$ contains:\n\\[ \\mathbb{Q}( \\sqrt{2}) .\\]\n\nIf $D$ is odd, and $8|\\mathfrak f$ then the maximal $2$-subextension of the ring class field of $\\mathcal{O}$ contains:\n\\[ \\mathbb{Q}( \\sqrt{2}). \\]\n\nIf $D = 4\\pmod{8}$, or $2|\\mathfrak f$ and $2|D$, or $D$ is odd and $4|\\mathfrak f$ then the maximal $2$-extension of the ring class field of $\\mathcal{O}$ contains:\n\\[ \\mathbb{Q}(\\sqrt{-1}). \\]\n\nThe above fields generate the genus field $F$, that is the maximal $2$-subextension of the ring class field of $\\mathcal{O}$.\n\\end{thm}\nSee \\cite[Thm 6.1.4]{SchertzCM}.\n\n\n\\section{Results}\\label{sec:res}\n\nIn this section we will present our main theorems. These are primarily structured to address the entries in the data which we present in \\ref{sec:ad}.\nWe will begin by looking at certain conditions under which there can be no elliptic curves over $\\mathbb{Z}_p$ at all.\n\n\n\\begin{thm}\\label{thm:nolinear}\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ of conductor $\\mathfrak f \\in \\mathbb{Z}$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nThere are no elliptic curves over $\\mathbb{Z}_p$ with CM by $\\mathcal{O}$ if any of the following occur:\n\\begin{itemize}\n\\item  there is an odd prime factor $q$ of $D\\mathfrak f$ with $\\left(\\dfrac{-p}{q}\\right) = -1$\n\\item $p=1\\pmod{4}$ and $16 | D\\mathfrak f^2$.\n\\item $p=3\\pmod{8}$ and $8 | D$.\n\\item $p=3\\pmod{8}$ and $64 | D\\mathfrak f^2$\n\\end{itemize}\nOtherwise there are exactly $\\abs{C\\ell(\\mathcal{O})[2]}$ $j$-invariants for elliptic curves over $\\mathbb{Z}_p$ with CM by $\\mathcal{O}$.\n\\end{thm}\n\n\\begin{rmk}\nThe condition that there is an odd prime factor $q$ of $D$ with $\\left(\\dfrac{-p}{q}\\right) = -1$ implies in particular that the quaternion algebra\n  $(-p,-D)$ is ramified at $q$. Though this can be used to justify the condition for those $q|D$, we will not follow this strategy of proof, rather we give a proof which has a more natural connection to class field theory.\n\nThe condition on odd primes cannot be extended to even primes by use of the Kronecker symbol, the dependence on the behavior at $2$ is more subtle.\n\\end{rmk}\n\n\n\n\nWe shall use the following two lemmas.\n\\begin{lemma}\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nThe polynomial $P_{\\mathcal{O}}(X)$ has a linear factor over $\\mathbb{Z}_p$ if and only if $N = \\mathbb{Q}(j(\\mathcal{O}))$ has no quadratic subextension in which $p$ is inert.\n\\end{lemma}\n\\begin{proof}\nIf there is a quadratic subextension of $N$ which is inert, then all factors of $p$ in $N$ have inertial degree $2$, and thus there can be no linear factors.\n\nConversely, suppose every factor of $p$ in $N$ has inertial degree $2$.\nlet $\\mathfrak p$ be a prime of $L$ over $p$ and let $\\sigma$ be a generator for the decomposition group of $\\mathfrak p$ and  let $\\tau$ be a generator of $\\Gal(L\/N)$.\nThen \n\\begin{itemize}\n\\item $\\sigma$ is indivisible with exact order $2$, because this is true of $\\Frob_p$.\n\\item $\\sigma$ and $\\tau$ are not conjugate, since if $\\tau$ were a conjugate of $\\Frob_p$ the field $N=L^\\tau$ would have a non-inert prime.\n\\item $\\sigma$ and $\\tau$ commute since $\\sigma$ has order $2$.\n\\item $\\sigma\\tau$ is in $\\Gal(L\/K)$ as they both act non-trivially on $K$.\n\\item It follows from the above, and the basic structure of dihedral groups, that $\\sigma\\tau$ is indivisible with exact order $2$.\n\\end{itemize}\nThus we may write:\n \\[ \\Gal(L\/K) = \\langle \\sigma\\tau \\rangle \\times H \\]\nand thus\n\\[  \\Gal(L\/\\mathbb{Q})= \\langle \\sigma \\rangle \\times (H\\rtimes \\langle \\tau \\rangle). \\]\nWe see that $G=(H\\rtimes \\langle \\tau \\rangle)$ is a normal subgroup of $\\Gal(L\/\\mathbb{Q})$, moreover, the field $L^G$ is an inert quadratic subextension of $N$.\n\\end{proof}\n\n\\begin{lemma}\nThe maximal $2$-subextension of $N$ is the totally real subfield $M$ of $F$ the genus field of $L$.\n\\end{lemma}\n\\begin{proof}\nIt suffices to show that $N$ has a real embedding since any composite of quadratic extensions is either totally complex or totally real.\n\nTo see this we use the fact that:\n\\[ \\overline{j(\\mathfrak a)} = j(\\overline{\\mathfrak a}).  \\]\nIt is thus sufficient to find $\\mathfrak a$ such that $\\overline{\\mathfrak a} = \\mathfrak a$, but indeed we may simply take $\\mathfrak a = \\mathcal{O}$.\n\\end{proof}\n\n\n\\begin{proof}[proof of Theorem \\ref{thm:nolinear}]\nThe idea of the proof is to show that $p$ is inert in a quadratic subextension of the totally real subfield $N$ of $F$ if and only if one of the conditions of the theorem holds.\n\nTo show this we must find a subextension of $N$ defined by adjoining the square root of a positive integer which is not a square modulo $p$, in each of the following cases we describe how to find such a non-square.\nNote that if $q=3\\pmod{4}$ then $\\sqrt{-Dq} \\in N$ whereas if $q=1\\pmod{4}$ then $\\sqrt{q}\\in N$.\n\\begin{itemize}\n\\item\nConsider the case where $p = 1 \\pmod{  4}$ and $4 || D$. In this case there exists odd prime factor $q'$ of $D$ with $\\left(\\dfrac{-p}{q'}\\right) = -1$.\nMoreover, $D$ has a factor $q$ such that both $\\pm q$ are not squares mod $p$.\n\n\\item Suppose there is an odd prime factor $q$ of $D\\mathfrak f$ with $\\left(\\dfrac{-p}{q}\\right) = -1$.\n\\begin{itemize}\n \\item if $q=p=3\\pmod 4$ we obtain $\\left(\\dfrac{q}{p}\\right) = -1$ and thus $-Dq$ is not a square mod $p$.\n \\item if $q=3\\pmod 4$, $p=1\\pmod{4}$ and $2 \\not\\vert D$ we obtain $\\left(\\dfrac{q}{p}\\right) = 1$ and thus $-Dq$ is not a square mod $p$.\n \\item if $q=1\\pmod 4$ we obtain $\\left(\\dfrac{q}{p}\\right) = -1$ and thus $q$ and is not a square mod $p$.\n\\end{itemize}\n\n\\item Suppose $p= 3 \\pmod{  8} $ and $8|D $ and $D\/8 = 3 \\pmod 4$ then $D$ has a factor $d$ congruent to $3\\pmod 4$ which is not a square mod $p$.\n\\item Suppose $p= 3 \\pmod{  8} $ and $8|D$ and $D\/8 = 1 \\pmod 4$ then $\\sqrt{2}$ is not a square mod $p$.\n\n\\item Suppose $p=3\\pmod{8}$ and $64 | D\\mathfrak f^2$ then $\\sqrt{2}$ is not a square mod $p$.\n\n\\item Suppose $p=1\\pmod{4}$ and $16 | D\\mathfrak f^2$ then $D$ has a factor $q$ such that both $\\pm q$ are not square mod $p$.\n\n\\end{itemize}\nThe above covers all of the cases of the theorem.\n\nTo prove the converse we remark that if $p$ is inert in $N$ it is inert in a quadratic subextension of one of the following types: \n\\begin{itemize}\n\\item $\\mathbb{Q}(\\sqrt{q})$ where $q | fD$ or \n\\item $\\mathbb{Q}(\\sqrt{q_1q_2})$ where both $q_1,q_2 = 3\\pmod{4}$ and $q_1q_2|fD$.\n\\end{itemize}\nas such fields generate the genus field of $N$.\nCompleting the proof follows a similar case analysis to the above.\n\\end{proof}\n\n\nWe now shift to discussing a phenomenon whereby certain $\\mathbb{F}_p$ reductions are disallowed based on the differing behavior of $2$.\n\\begin{rmk}\nIn the following theorem we will be distinguishing the supersingular $j$-invariants in $\\mathbb{F}_p$ by identifying them as roots of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ or $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\nTo understand the significance we recall the theorems above of Ibukiyama which asserted that this naturally divides the supersingular values into two almost disjoint sets.\nMore precisely, we have that for $p=3\\pmod{4}$ these polynomials factor as $(X-1728)\\prod_i (X-\\alpha_i)^2$ whereas for $p=1\\pmod{4}$ the factorization is $\\prod_i (X-\\alpha_i)^2$. In each case the $\\alpha_i$ are distinct in $\\mathbb{F}_p$.\nFurthermore, in the case $p=3\\pmod{4}$ the $\\alpha_i$ for $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ are distinct from those for $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\nThe polynomial for $\\sqrt{-2}$ is precisely $P_{\\mathbb{Z}[\\sqrt{-2}]}(X)=X-8000$.\n\\end{rmk}\n\n\\begin{thm}\\label{thm:tworam}\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ of conductor $\\mathfrak f\\in \\mathbb{Z}$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nLet $j$ be a $\\mathbb{Z}_p$ root of $P_{\\mathcal{O}}(X)$.\n\\begin{itemize}\n\\item If $p=7\\pmod{4}$ \n\\begin{itemize}\n\\item If $2$ is unramified in $K$ and $2 \\not\\vert \\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$.\n\\item If $2$ is unramified in $K$ and $2 | \\mathfrak f$ but $8 \\not\\vert\\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is unramified in $K$ and $8 | \\mathfrak f$ then $j$ is a root of either $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ or $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is tamely ramified in $K$ and $2 \\not\\vert \\mathfrak f$ or $4|\\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ or $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is tamely ramified in $K$ and $2 || \\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is wildly ramified in $K$ and $2 \\not\\vert \\mathfrak f$ or $4|\\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is wildly ramified in $K$ and $2 | \\mathfrak f$ or $4|\\mathfrak f$ then $j$ is a root of either $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ or  $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\end{itemize}\n\\item If $p=3\\pmod{8}$ \n\\begin{itemize}\n\\item If $2$ is unramified in $K$ and $2\\not\\vert\\mathfrak f$ or $4||\\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$.\n\\item If $2$ is unramified in $K$ and $2 || \\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is unramified in $K$ and $8 | \\mathfrak f$ then there are no linear terms.\n\\item If $2$ is tamely ramified in $K$ and $2 \\not\\vert \\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$ or $P_{\\mathbb{Z}[(1+\\sqrt{-p})\/2]}(X)$.\n\\item If $2$ is tamely ramified in $K$ and $2 || \\mathfrak f$ then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$.\n\\item If $2$ is tamely ramified in $K$ and $4 | \\mathfrak f$ then there are no linear terms.\n\\item If $2$ is wildly ramified in $K$ then there are no linear terms.\n\\end{itemize}\n\n\\item If $p=1\\pmod{4}$\n\\begin{itemize}\n\\item If $2$ is unramified in $K$ and $4 \\not\\vert \\mathfrak f$  then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$.\n\\item If $2$ is unramified in $K$ and $4 \\vert \\mathfrak f$ then there are no linear terms.\n\\item If $2$ is tamely ramified then there are no linear terms.\n\\item If $2$ is wildly ramified in $K$ and $2 \\not\\vert \\mathfrak f$  then $j$ is a root of $P_{\\mathbb{Z}[\\sqrt{-p}]}(X)$.\n\\item If $2$ is wildly ramified in $K$ and $2 \\vert \\mathfrak f$ then there are no linear terms.\n\\end{itemize}\n\\end{itemize}\n\\end{thm}\n\nTo prove this we will make use of the following lemma.\n\\begin{lemma}\\label{lem:frob}\nIf $E$ is an elliptic curve over $\\mathbb{Z}_p$ with CM by $\\mathcal{O}$ which corresponds to a datum $(\\mathcal{O} \\subset \\mathbb{B})$ then the Galois Frobenius $\\Frob_p$ acting on $E(\\overline{\\mathbb{Q}}_p)$ over $\\mathbb{Z}_p$ induces the endomorphism Frobenius $\\widetilde{\\Frob_p}$ of $\\overline{E}$. \nMoreover we have:\n\\begin{itemize}\n\\item $\\Frob_p$, the Galois action of Frobenius on $E$, acts on $\\mathcal{O}$ by $x\\mapsto \\overline{x}$.\n\\item $\\widetilde{\\Frob_p}$, the endomorphism of $\\overline{E}$,  satisfies $\\widetilde{\\Frob_p} x = \\overline{x} \\widetilde{\\Frob_p}$ for $x\\in \\mathcal{O}$.\n\\item $\\Frob_p^2$, the Galois action of Frobenius on $E$, commutes with $\\mathcal{O}$.\n\\item $\\widetilde{\\Frob_p}^2$, the endomorphism of $\\overline{E}$, satisfies $\\widetilde{\\Frob_p}^2 = -p$.\n\\end{itemize}\nIn particular $\\widetilde{\\Frob_p} \\in \\mathcal{O}^\\perp$ is an element of norm $p$.\n\\end{lemma}\nSee \\cite{SchertzCM}.\n\n\n\\begin{proof}[proof of Theorem \\ref{thm:tworam}]\nWe must show, using the classification of maximal orders containing $\\sqrt{-p}$ by Ibukiyama, that the only CM-orders in $\\alpha^\\perp$ are those satisfying the conditions of the theorem.\n\nWe note that in selecting the values of $q$, $r$ and $m$ we may assume by replacing $r$ by $r+aq$ that $8|r$.\nWith this assumption we have that $pq = m \\pmod{8}$.\nWhen selecting $q$, $r'$ and $m'$ we must have that $r'$ is odd, when $p=3\\pmod 8$ this implies that $m$ is odd.\n\nWe observe the following important facts about $\\alpha^\\perp$ in the various cases:\n\\begin{enumerate}\n\\item For the maximal orders of the form $O'(p,q,r',m')$ we have that $\\alpha^\\perp$ contains no elements with odd trace.\n\n\\item For the maximal orders of the form $O'(p,q,r',m')$ we have that all primitive elements of $\\mathbb{Z}[\\sqrt{-p}]^\\perp$ are of the form:\n \\[ y\\beta + z\\frac{(r'+\\alpha)\\beta}{2q} \\]\nfor some choice of $y$ and $z$ coprime.\n\nThe square of such an element is:\n\\[ -y^2q -z^2m - yzr'. \\]\nNotice that if $p = 3\\pmod8$ this cannot be even.\n\\item For the maximal orders of the form $O(p,q,r,m)$ we have that all primitive elements of odd trace in $\\mathbb{Z}[\\sqrt{-p}]^\\perp$ are of the form:\n \\[ y\\beta + z\\frac{(r+\\alpha)\\beta}{q} \\]\nfor some choice of $y$ and $z$ coprime, with $z$ odd.\n\nThe square of such an element is:\n\\[ -y^2q -z^2m - 2yzr \\]\nmodulo $8$ this becomes:\n\\[ -q(y^2-z^2p). \\]\nNotice that if this is odd, then $y$ is even and $-q(y^2-z^2p) = pq \\pmod{8}$.\nAlso, if it is even then $y$ and $z$ are both odd and it is divisible by $(1-p) | -q(y^2-z^2p)$.\n\\end{enumerate}\nBy considering each of the cases of the theorem, the above allows us to conclude the result.\n\\end{proof}\n\n\\begin{prop}\\label{prop:hasroots}\nSuppose there exists $\\mathbb{Z}[\\sqrt{-D}] = \\mathcal{O} \\subset \\alpha^\\perp$, then $P_\\mathcal{O}(X)$ has $\\mathbb{Z}_p$ roots.\n\\end{prop}\n\\begin{proof}\nBy the above argument we note that $\\mathcal{O} \\subset \\alpha^\\perp$ implies the existence of a solution to:\n\\[ y^2q + z^2m +2yzr = D \\qquad\\text{ or }\\qquad y^2q +z^2m + yzr' = D.  \\]\nIn the first case, multiplying by $q$ we obtain:\n\\[qD = y^2q^2 + z^2(p+r^2) + 2yzrq =  z^2p + (yq+rz)^2. \\]\nreducing modulo $8$ and modulo all the odd prime factors of $D$ we obtain the result.\nIn the second case, multiplying by $4q$ we obtain:\n\\[ 4qD = 4y^2q^2 + z^2(p+r^2) + 2yzrq =  z^2p + (2yq+rz)^2 \\]\nand the result follows similarly.\n\\end{proof}\n\n\\begin{rmk}\nNote that the above does not actually prove the converse to Lemma \\ref{lem:frob} though it would provide for an alternate proof for one direction of Theorem \\ref{thm:nolinear}.\n\\end{rmk}\n\nWe now explain the phenomenon where in specific circumstances certain $\\mathbb{F}_p$ reductions always occur with the same frequency.\nBased on \\cite{CornutVatsal2} we should expect that this is caused by systematic collections of isogenies (coming from Hecke relations), and in our case we should expect $2$-isogenies to play a role.\n\n\\begin{lemma}\\label{lem:qchoice}\nIf $\\sqrt{q_2} \\in  \\alpha^\\perp$ then $\\mathcal{O} \\simeq O(p,q_2,r,m)$ or $O(p,q_2,r',m')$ for some choice of $r,m$ or $r',m'$.\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Prop 2.1 and Rmk 2.2]{Ibukiyama} the conditions:\n\\[q_1q_2 =  z^2p + (yq_1+rz)^2 \\qquad \\text{or} \\qquad 4q_1q_2  = z^2p + (2yq_1+rz)^2 \\]\nimply that $q_1$ and $q_2$ satisfy $O(p,q_1,r_1,m_1) \\simeq O(p,q_2,r_2,m_2)$ or respectively $O'(p,q_1,r_1',m_1') \\simeq O'(p,q_2,r_2',m_2')$. The results then follow from the proof of Proposition \\ref{prop:hasroots}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:3mod4cong}\nFix $p=3\\pmod{4}$.\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nSuppose further that $2$ is tamely ramified in $K$ but $2$ does not divide $\\mathfrak f$.\n\nSuppose that $\\mathcal{O}$ is optimally embedded in $O(p,q,r,m)$ and contained in $\\alpha^\\perp$.\nLet $\\mathfrak a^2 = (2)$ in $\\mathcal{O}$.\nThen $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1} \\simeq O'(p,q,r',m')$ is a maximal order with an optimal embedding of $\\mathcal{O}$.\nConsequently, if $E$ is an elliptic curve over $\\mathbb{Z}_p$ with CM by $\\mathcal{O}$  whose reduction has endomorphism ring $O(p,q,r,m)$, then the reduction of $\\mathfrak a\\ast E$ has endomorphism ring $O'(p,q,r',m')$ with the exact same choice of $q$.\n\nConversely, if $E$ is an elliptic curve over $\\mathbb{Z}_p$ with CM by $\\mathcal{O}$ whose reduction has endomorphism ring $O'(p,q,r',m')$, then the reduction of $\\mathfrak a\\ast E$ has endomorphism ring $O(p,\\tilde{q},\\tilde{r},\\tilde{m})$ for some $\\tilde{q}$ such that $O'(p,q,r',m')\\simeq O'(p,\\tilde{q},\\tilde{r}',\\tilde{m}')$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathcal{O} = \\mathbb{Z}[\\gamma = \\sqrt{q_2}]$.\nIt suffices to show that $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1}$ contains both $\\frac{1+\\alpha}{2}$ and $\\beta$.\n\nWe note that $\\mathfrak a = (2,1+\\gamma)$ and $\\mathfrak a^{-1} = (1,\\frac{1-\\gamma}{2})$. It follows immediately that $\\beta \\in \\mathfrak a O(p,q_1,r,m) \\mathfrak a^{-1}$.\n\nNow we may write $\\gamma = y\\beta + z\\frac{r+\\alpha}{q}\\beta$ with $y$ and $r$ even and $z$ odd.\nNow, by observing that:\n\\[ \\left(\\frac{1+\\alpha}{2}\\right) = (1+\\gamma)\\left(\\tfrac{1}{2}(-zm+ry+1) + (zm+ry)\\left(\\frac{1+\\beta}{2}\\right) - \\tfrac{1}{2}(yq+zr)\\left(\\frac{r+\\alpha}{q}\\beta\\right)\\right)\\left(\\frac{1-\\gamma}{2}\\right) \\]\nand that the right hand side is in $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1}$\nwe conclude by Lemma \\ref{lem:qchoice} that $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1} \\simeq O'(p,q,r',m')$.\n\nNow suppose we start with $\\mathcal{O}$ optimal in $O'(p,q,r',m')$.\nAttempting to reverse the above calculation cannot work in general as we no longer have $r$ and $m$ but $r'$ and $m'$. However, we observe that:\n \\[\\left((1+\\gamma)\\left(\\frac{1+\\alpha}{2}\\right)\\left(\\frac{1-\\gamma}{2}\\right) - \\left(\\frac{1+\\gamma^2}{4}\\right)\\alpha\\right) \\in \\mathfrak a O'(p,q,r',m') \\mathfrak a^{-1}\\]\nis perpendicular to $\\alpha$ and has odd trace.\nHence, $\\mathfrak a O'(p,q,r',m') \\mathfrak a^{-1} \\simeq O(p,\\tilde{q},\\tilde{r},\\tilde{m})$. The result now follows.\n\\end{proof}\n\n\\begin{rmk}\nNote, that we could not simply run the first part of the above argument in the opposite direction to go from $O'(p,q,r',m')$ to $O(p,q,r,m)$, in particular this would be impossible in any case where the class groups which classify $O'(p,q,r',m')$ and $O(p,q,r,m)$ are not in bijection. \n\\end{rmk}\n\n\\begin{thm}\\label{thm:7freq}\nFix $p = 7\\pmod 4$.\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nSuppose further that $2$ is tamely ramified in $K$ but $2$ does not divide $\\mathfrak f$.\n\nIt we consider the set of supersingular values of $\\mathbb{F}_p$ except $1728$, each $j$-invariant $J$ has a partner $\\tilde{J}$ such that, the frequency of the appearance of $X-J$ and $X-\\tilde{J}$ as the reduction of irreducible linear factors of $P_{\\mathcal{O}}(X)$ modulo $p$ is the same.\n\\end{thm}\n\\begin{proof}\nWe first observe that if $E$ is defined over $\\mathbb{Z}_p$ then so too is $\\mathfrak a\\ast E$. This follows by observing that the collection of endomorphisms in $\\mathfrak a$ is Galois stable.\nMoreover, in the case $p = 7\\pmod 4$ the map from $O(p,q,r,m)$ to $O'(p,q,r',m')$ being injective implies it is bijective as the collections have the same size.\n\nBy Lemma \\ref{lem:3mod4cong} it now follows that $O(p,q,r,m)$ and $O'(p,q,r',m')$ must occur with the same frequency.\n\nWe note that $j$-invariant $1728$ is the only one that can ever be identified with itself through this process, and in fact it must, because the class group has odd order.\n\\end{proof}\n\n\\begin{rmk}\nFor $p=3\\pmod 4$ we obtain other less obvious relationships between the counts for maximal orders of type $O'$ and of type $O$ arising from the fact that the map is generically $3:1$. In particular, in general the frequency for those of type $O'$ is the sum of the frequencies of a specific collection of three of orders of type $O$.\nWe note that there will be a curve which is $2$-isogenous to the one with $j$-invariant $1728$.\n\nWe should point out that the $\\mathbb{F}_p$ points of the $2$-torsion is well understood, that there is a unique $\\mathbb{F}_p$ rational $2$-torsion point is suggestive of the above results, but does not show that the association is between $O(p,q,r,m)$ and $O'(p,q,r',m')$ and certainly not that it `respects $q$'.\n\\end{rmk}\n\n\\begin{thm}\\label{thm:1freq}\nFix $p=1\\pmod 4$.\nFix $K=\\mathbb{Q}(\\sqrt{-D})$ of discriminant $-D$. Fix an order $\\mathcal{O} = \\mathbb{Z} + \\mathfrak f\\mathcal{O}_K$ and suppose that $\\left(\\frac{-D\\mathfrak f^2}{p}\\right) = -1$.\nSuppose further that $2$ is wildly ramified in $K$ but $2$ does not divide $\\mathfrak f$.\n\nIt we consider the set of supersingular values of $\\mathbb{F}_p$, each $j$-invariant $J$ has a partner $\\tilde{J}$ such that, the frequency of the appearance of $X-J$ and $X-\\tilde{J}$ as the reduction of irreducible linear factors of $P_{\\mathcal{O}}(X)$ modulo $p$ is the same.\n\nThis partner $\\tilde{J}$ is independent of $K$ and $\\mathcal{O}$ and depends only on $p$.\n\\end{thm}\n\\begin{proof}\nSet $\\mathfrak a^2 = (2)$ in $\\mathcal{O}$. \nIn this case we have $\\mathfrak a = (2,\\gamma)$ and $\\mathfrak a^{-1} = (1,\\tfrac{1}{2}\\overline{\\gamma})$.\nAs in the previous case, we must only show that $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1}$ is independent of $\\mathcal{O}$.\n\nNow set $\\mathfrak b^2 = (2)$ in $\\mathbb{Z}[\\sqrt{-p}]$.\nWe have that $\\mathfrak b = (2,1+\\alpha)$.\n\nWe recall that we have $\\gamma = y\\beta + z\\frac{r+\\alpha}{q}\\beta = \\tfrac{1}{q}(yq +zr + z\\alpha)\\beta$ with $r$ even and both $y$ and $z$ odd.\n\nWe claim that $(1+\\alpha) \\in \\mathfrak a O(p,q,r,m)$. Indeed, as $\\beta\\in O(p,q,r,m)$ we have $yq +zr + z\\alpha = \\gamma\\beta \\in \\mathfrak a O(p,q,r,m)$. Since $2\\in \\mathfrak a$ the claim then follows immediately.\nConversely, it is clear that $q\\gamma \\in \\mathfrak b O(p,q,r,m)$. As $q$ is odd, and $2 \\in \\mathfrak b$ we also have that $\\gamma\\in \\mathfrak b O(p,q,r,m)$.\nWe thus have shown that $\\mathfrak a O(p,q,r,m) = \\mathfrak b O(p,q,r,m)$.\n\nIt follows that $\\mathfrak a O(p,q,r,m) \\mathfrak a^{-1} = \\mathfrak b O(p,q,r,m) \\mathfrak b^{-1}$ is independent of $\\mathcal{O}$.\n\\end{proof}\n\n\\begin{rmk}\nIn this case the uniqueness of the $\\mathbb{F}_p$-rational $2$-torsion points is sufficient to conclude the result.\n\\end{rmk}\n\n\n\\section{Further Questions}\\label{sec:cq}\n\nOur results suggest the following natural questions:\n\n\\begin{qu}\nIn Theorem \\ref{thm:tworam} we gave necessary conditions for a datum $(\\mathcal{O}\\subset\\mathbb{B})$ to correspond to an elliptic curve over $\\mathbb{Z}_p$. Moreover, Proposition \\ref{prop:hasroots} gives the impression that this may be sufficient.\nIt is natural to ask, if these conditions are in fact sufficient.\n\n\\begin{enumerate}\n\\item[(a)]\nMore precisely, given an elliptic curve over $\\mathbb{F}_p$, and an endomorphism (defined over some extension) when can we lift the curve to $\\mathbb{Z}_p$ such that the endomorphism lifts to some extension?\n\\item[(b)]\nIs it sufficient that the endomorphism be perpendicular to Frobenious in the endomorphism algebra over $\\overline{\\mathbb{F}_p}$? \n\\end{enumerate}\n\\end{qu}\nA thorough answer to this question would shed light on the structure of $\\mathbb{Z}[j(E_1),\\ldots,j(E_n)]$ as remarked in the introduction.\n\n\\begin{qu}\nTheorems \\ref{thm:7freq} and \\ref{thm:1freq} give situations in which there are automatic relationships between certain roots of $P_\\mathcal{O}(X)$.\nAs remarked a simalar result holds for the same reason when $p=3\\pmod 8$.\n\\begin{enumerate}\n\\item[(a)]\nIt is natural to ask if there are other situations in such relationships must exist? In particular are there situations where the role of $2$ can be replaced by some other prime?\n\\item[(b)]\nThe method of proof also suggests that we could anticipate relations between the roots of $P_\\mathcal{O}(X)$ between two different orders in the same field whose conductors differ by a factor of $2$. Can the combinatorics of this be made more precise?\n\\end{enumerate}\n\\end{qu}\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\n I would like to thank Prof. Eyal Goren for suggesting the computations which led to the discovery of these results.\n I would like to thank Prof. Ernst Kani for some useful discussions as well as recommending several references.\nI would like to thank the many developers of SAGE without which the computations through which we uncovered these results would not be possible.\nI would also like to thank the SAGE Notebook project for the use of various computing resources which they have made available through their various funding sources.\n\n\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDue to the rapidly increasing spectral resolution of galaxy surveys\n(e.g.\\ Sloan Digital Sky Survey), modelling any galaxy with spectral\nsynthesis technique requires a high spectral resolution stellar\nlibrary.\n\nOver the last years much progress has been made in synthesis models \n(e.g.\\ Pelat \\cite{Pel97}; Leitherer et al.\\ \\cite{Lei99};  Moultaka \\& Pelat \\cite{Mou00}; Bruzual \\& Charlot \\cite{Bru03};\n Le Borgne et al.\\ \\cite{LeB04}; \nCid Fernandes et al.\\ \\cite{Cid05}). Moderate to high spectral resolution observations\nof stars in order to construct reliable template libraries have also been performed\nin the visible domain (e.g.\\ STELIB of Le Borgne et al.\\ \\cite{LeB03}; UVES of\nBagnulo et al.\\ \\cite{Bag03}; CoudeFed of Valdes et al.\\ \\cite{Val04}) and in the\nIR (e.g.\\ Dallier et al.\\ \\cite{Dal96}; Meyer et al.\\ \\cite{Mey98}; Ivanov et al.\\ \\cite{Iva04}). \nHowever, the major limitation of all these libraries\nis the sampling of stellar parameters such as metallicity.\n\nOne way to avoid such a difficulty is to build libraries of\ntheoretical stellar spectra in order to choose the desired physical\nparameters.  In this sense, some extensive libraries of synthetic\nspectra have appeared recently for the visible range.\n\nMurphy \\& Meiksin (\\cite{Mur04}), based on Kurucz's ATLAS9 model\natmospheres (Kurucz \\cite{Kur93}), have built a high resolution\n($\\lambda \/ \\Delta\\lambda = 250 000$) stellar library over an extended\nvisible range (3000 to 10000 \\AA). The convection zone is treated\nusing the Mixing Length Theory (MLT) with the overshooting treatment\nof Kurucz (cf.\\ Castelli et al. \\cite{Cas97}). This library provides spectra for 54 values of effective temperature\nfrom 5250 to 50000 K, 11 values of log surface gravity from 0.0 to 5.0\nand 19 metallicities from -5.0 to 1.0. They compared their synthetic library with observed spectra (STELIB\nlibrary of Le Borgne et al. \\cite{LeB03}) for the colours and the Lick\nindices and found in general a good agreement.\n\n\nStill based on Kurucz's models, but with enhanced molecular line lists\nand the overshooting option switched off, Munari et al. (\\cite{Mun05})\nalso present a library of synthetic spectra, for a similar wavelength\nrange (2500 to 10500 \\AA). They use a new grid of ATLAS9 model\natmospheres (Castelli \\& Kurucz \\cite{Cas03}). The effective\ntemperature of these spectra is contained between 3500 and 47500 K,\nthe log surface gravity between 0.0 and 5.0 and the metallicity\nbetween -2.5 and 0.5. These spectra are computed at a resolving power\nof $\\lambda \/ \\Delta\\lambda = 500 000$ and then Gaussian convolved to\nlower resolution ($\\leq$ 20 000). In contrast with Murphy \\& Meiksin\n(\\cite{Mur04}), the predicted energy level lines are not included into\nthe line lists used to build this library, as Munari et al.\n(\\cite{Mun05}) favour the spectroscopic use rather than the\nphotometric use of the synthetic spectra. The addition of these\n``predicted lines'' permits to have a better statistical flux\ndistribution, but individual wavelengths can be wrong by up to 5\\%.\n\n\nWe would like to point out here that usually only the lower lying energy\nlevels of atoms have been determined in the laboratory, particularly for\ncomplex spectra such as those from neutral or singly ionized iron. If only\nthose transitions were taken into account, the atmospheric line blanketing\ncomputed from such data would be severly incomplete. A lot of weak lines,\npossibly unidentified even in the solar spectrum but nevertheless present, would\nbe missed. This would lead to an overestimation of the ultraviolet flux which\nin turn would be ``compensated'' by a lack of flux in the visual (Kurucz\n\\cite{Kur92}). To avoid this deficiency and to improve the temperature\nstructure of the model atmospheres, the spectrophotometric flux distribution,\nand the photometric colors requires to account for lines for which one or\nboth energy levels have to be predicted from quantum mechanical calculations.\nThis has been one of the main goals of the ATLAS9 models of Kurucz\n(\\cite{Kur92,Kur93}). As the theoretical predictions are accurate to only a few\npercent, individual wavelengths can be wrong by up to a few 100~{\\AA} in the\nvisual. Also the line oscillator strengths are sufficiently accurate merely\nin a statistical sense. This still permits to improve the total flux within\na wavelength band of a few dozen~{\\AA} in the visual, but individual features\ndo appear in the wrong part of the spectrum. For spectroscopy at higher\nresolutions, particularly if the spectrum is rectified or when working within\nsmall wavelength bands, adding the predicted energy level lines ``pollutes''\nthe theoretical spectrum with extra ``noise''. This has to be avoided in\nlibraries devoted to automatic fitting procedures or if particular line\nfeatures are essential to identify a certain spectral type (lines predicted\nfor the wrong wavelength will make this more difficult). Hence, with present\natomic data, either choice is only a compromise solution.\n\nBy combining three different model atmospheres, the high-resolution stellar library of\nMartins et al.\\ (\\cite{Mar05}) provides the largest coverage in effective temperature\n(from 3000 to 50000 K) and log surface gravity (from -0.5 to 5.5). This library, still\nin the visible wavelength range (3000 to 7000~\\AA), uses the non-LTE model atmosphere\nTLUSTY (Hubeny \\cite{Hub88}, Hubeny \\& Lanz \\cite{Hub95}, Lanz \\& Hubeny \\cite{Lan03})\nfor $T_\\mathrm{eff}\\geq 27500 K$, the Kurucz's ATLAS9 model for\n$4750\\leq T_\\mathrm{eff}\\leq 27000 K$ and Phoenix\/NextGen models\n(Allard \\& Hauschildt \\cite{All95}, Hauschildt et al.\\ \\cite{Hau99}) which use\nspherical symmetry for cooler stars with low surface gravity. A comparison with\nobserved spectra (from the STELIB library of Le Borgne et al.\\ \\cite{LeB03} and the\nIndo-US library of Valdes et al.\\ \\cite{Val04}) shows the good agreement of this\ntheoretical library with observations.\n\nThus, the visible range is now quite well covered by theoretical libraries, for\nphotometric use as well as spectroscopic use and for a wide range of physical\nparameters. All the comparisons with observations show that these theoretical spectra\ncan reasonably well mimic real stars, at least at the spectral resolution where the\ncomparisons were made.\n\nThe goal of the present work is to go one step further, exploring the near-infrared\nrange where few observed fully calibrated and no theoretical libraries are available. This research\ntakes place in a more general framework which consists in the synthesis of the stellar\npopulation of galaxies hosting active galactic nuclei by an inverse method, described\nin Pelat (\\cite{Pel97}) and Moultaka \\& Pelat (\\cite{Mou00}). The H-band provides very\ngood luminosity discriminators for stars later than K0 (cf Dallier et al.\\ \\cite{Dal96})\nand the particular region 1.57-1.64 $\\mu$m of the H-band is clear of strong emission\nlines (except Brackett lines). It allows to sample the stellar content of the very\nnucleus of Seyfert~1 galaxies, at the contrary of the visible range where the strong\nbroad emission lines of the active nucleus contaminate so much the spectra of the\ngalactic inner part that there are too few absorption lines from the stellar component\nto synthesize this region.\n\nThe lack of stellar observations at medium resolution for the\nnear-infrared range, especially for super-metallic stars, drove us to\nwork with theoretical spectra. But the behavior of model atmospheres\nand fluxes is not very well known in the infrared.\n\nDecin et al. (\\cite{Dec03}) have compared several observed stars with\ntheoretical spectra computed with the MARCS models (Gustafsson et al.\n\\cite{Gus75}, Plez et al. \\cite{Ple92}) in the range 2.38 to 12 $\\mu$m\nfor the ISO-SWS calibration, at a resolving power $R \\simeq 1000$. \nThis study points out the difficulties of modelisation due to strong\nmolecular opacities and the bad accuracy and completeness of the\natomic data in these wavelength ranges.\n\n\nIn this paper, we compute theoretical spectra using the NeMo (Vienna New Model) grid of\natmospheres (Heiter et al. \\cite{Hei02}, Nendwich et al. \\cite{Nend04}) based on the model\natmosphere code ATLAS9 by Kurucz (\\cite{Kur93}, \\cite{Kur98}) and\nCastelli et al. (\\cite{Cas97}), combined with the list of absorption lines\nVALD (Vienna Atomic Line Database, Kupka et al. \\cite{Kup99}), eventually completed by\nmolecular data collected by one of us (VT). These models are described in\nSect.~\\ref{model}. Synthetic stellar spectra are computed with the code\nSynthV (built by VT), as shown in Sect.~\\ref{spec}, using the model atmospheres\ndescribed in Sect.~\\ref{model} as input. Several tests on the input\nparameters of the spectra are done in Sect.~\\ref{test}.\n\nIn a first step of applying our synthesis calculations (Sect.~\\ref{visible}), we\ncompare a set of observed stellar spectra with their corresponding models in the\nvisible wavelength range (5000 to 9000 \\AA) to check the range of validity of the\nNeMo grid, exploring the whole range of physical parameters (effective temperature,\nsurface gravity and metallicity). In a second step, we generate synthetic spectra\nin the near-infrared range and compare them with observed ones. The results of\nthis comparison are described in Sect.~\\ref{infrared}, as are tests which\ndemonstrate that the particular choice of model atmospheres can be expected to\nbe less important than the set of line lists used for the computation of spectra.\nOur conclusions are summarized in Sect.~\\ref{conclusions}.\n\n\n\\section{Description of the model atmospheres}\\label{model}\n\n\nNeMo differs from the original grids of model atmospheres based on\nATLAS9 in the treatment of the convective energy transport. It\nprovides also a higher vertical resolution of the atmospheres and a\nfiner grid in effective temperature and surface gravity.\n\n\nThis model grid of stellar atmosphere uses convection treatment\nwithout overshooting. The overshooting prescription has been\nintroduced by Kurucz (\\cite{Kur93}, \\cite{Kur98}) and modified by\nCastelli et al. (\\cite{Cas97}). It was supposed to take into account\nthe change in the temperature gradient of the stable atmosphere layers\nnear a convective zone due to the overshooting of gas from that zone\ninto the stellar atmosphere. But this prescription is left aside in\nthe present work, because, even if the properties of various numerical\nsimulations are well described and in good agreement when compared to\nobservations of the Sun, models with overshooting are worse than\nmodels without for other stellar types (see Heiter et al. \\cite{Hei02}\nfor a detailed discussion).\n\nThe NeMo grids offer a choice among different convection models. One of them is the\nmixing length theory (MLT), with $\\alpha = 0.5$. The parameter $\\alpha\\ $ represents\nthe ratio between the characteristic length (distance traveled by an element of fluid\nbefore its dissolution) and the scale height of the local pressure. This parameter is\nsubject to discussion: according to comparisons between observed and computed energy\ndistributions for the Sun done by Castelli et al. (\\cite{Cas97}), $\\alpha$ should be\nset at least to 1.25, but Van't Veer \\& M\\'egessier (\\cite{vVe96}), using the same\ncodes and input data as Castelli et al. (\\cite{Cas97}), but different observations for\nthe Sun, found that $\\alpha = 0.5$ is required to fit both H$_\\alpha$ and H$_\\beta$\nprofiles. Fuhrmann et al. (\\cite{Fur93}) were the first to notice that a value of 0.5\nfor the parameter $\\alpha$ is needed to reproduce the Balmer line profiles of cool\ndwarf stars. In addition, this parameter has to span a large domain (from 1 to 3) to\nreproduce the red giants (Stothers \\& Chin \\cite{Sto97}). The alternative convective\nmodels available in the NeMo grids are of \"Full Spectrum Turbulence\" (FST) type.\nIntroduced by Canuto \\& Mazzitelli (\\cite{Can91}, \\cite{Can92}; thereafter\nCM model) and Canuto, Goldman \\& Mazzitelli (\\cite{Can96}; thereafter CGM model),\nthese models avoid the one-eddy approximation of MLT. In addition, both models were\nsuggested to be used with a scale length different from the usual multiple $\\alpha$\nof the local pressure scale height (see Heiter et al. \\cite{Hei02} for further details).\n\nThe latter models were introduced in NeMo to allow a choice among different\ntreatments of the internal structure of the stars, depending on the aim of the\nmodel computation and its underlying assumption of how to describe the\nconvective energy transport within the limitations of a simple convection model\n(using only algebraic rather than differential equations). \n\nTwo levels of vertical resolution are also offered and hence we can\neither work with 72 or 288 layers. The MLT models are computed with 72\nlayers, CM models with 288 and CGM ones are computed for both values.\n\nThe metallicity of the model atmopheres covers a large range between\n-2.0 and +1.0 dex and have 13 different values. This range of\nmetallicity is enough for our purpose. The super metal rich stars, in\nparticular, are represented with five different levels of metallicity\n(+0.1, +0.2, +0.3, +0.5 and +1.0 dex) reaching the highest possible\nvalue for a real star.\n\nNeMo provides model atmospheres for effective temperatures between\n4000K and 10000K, by successive steps of 200K; for lowest\ntemperatures, the model atmospheres computed with ATLAS9 become\ninadequate, mainly because of the molecular opacities which become\nvery important for cool stars. The MARCS6 models (Gustafsson et al.\n\\cite{Gus75}, Plez et al. \\cite{Ple92}), more dedicated to the cool\nstars, handle this problem with a more complete treatment of molecular\nopacity.\n\nThe available values for the surface gravity (log g) of the stellar\natmospheres in the NeMo grid span a range from 2.0 to 5.0 with steps\nof 0.2. It is bounded at 2.0 owing to the plane-parallel approximation\nused in ATLAS9; for lower values of log g, spherically symmetric\ngeometry should be used\ninstead (cf.\\ Hauschildt et al.\\ \\cite{Hau99} and Baraffe et al.\\ \\cite{Bar02}).\n\n\nOther models, working with the appropriate geometry like MARCS6 or\nPhoenix\/NextGen (Allard \\& Hauschildt \\cite{All95}, Hauschildt et al.\n\\cite{Hau99}) are necessary for these small values. Indeed, MARCS6,\nwhose main purpose is to model cool stars, uses also the approximation\nof spherically symmetric geometry to reproduce supergiants and the\ncool giants stars, which have a low surface gravity (Plez et al.\\\n\\cite{Ple92}).  NextGen models, like ATLAS9, assume LTE and\nplane-parallel geometry for dwarf stars, but a spherical symmetry is\nused for low-gravity giant and pre-main sequence stars (log g $<$ 3.5,\nsee Hauschildt et al.\\ \\cite{Hau99}).\n\nContrary to NeMo and MARCS6, NextGen can use a non-LTE model for high\ntemperature stars. But using NLTE does not improve significantly the\nmodelisation of our observed stars, as NLTE effects begin to occur\nonly from 7000 K to higher effective temperature (Hauschildt et al.\n\\cite{Hau99}), but are still small to at least 10000 K. Moreover,\nNextGen does not reproduce well enough the individual lines owing to\nthe treatment of atomic and molecular lines with a direct opacity\nsampling method. Indeed, working with opacity distribution functions,\nlike in ATLAS9, would ask too much computer resources when using NLTE\ncalculations (Hauschildt et al. \\cite{Hau99}). In addition to that,\ntoo few layers are used in the published models to describe the bottom\npart of the photosphere.\n\nHowever, NextGen could be an alternative to ATLAS9 type model atmospheres in\na next step of our project, for generating spectra of stars with a log g below\n2.0 (for which spherical symmetry is needed) and\/or an effective temperature\nabove 10000 K.\n\nBertone et al. (\\cite{Ber04}) have compared both ATLAS9 and NextGen models to\nobservations in the visible range along the whole spectral-type sequence. The\nconclusions of this work are that both models reproduce very well the spectral energy\ndistribution of F type stars and earlier but this good agreement decreases at lower\ntemperature, especially for K stars, owing to the lack of molecular treatment in those\nmodels. ATLAS9 provides a better fit, in general, from B to K type stars but as said\npreviously NextGen is more suitable for M stars, due to the use of spherical geommetry\nfor the giants and a more complete molecular line opacity. However, Martins et\nal.\\ (\\cite{Mar05}) note that this comparison is made with a previous generation\nof NextGen models, using for example a mixing length parameter of 1 instead of\n2, preferred by hydrodynamic models. This is also true for the ATLAS9 models, as\nBertone et al.\\ (\\cite{Ber04}) did not use the latest versions of ATLAS9, including\nnew opacity distribution functions (Castelli \\& Kurucz \\cite{Cas03}), computed with\nmore up-to-date solar abundances and molecular contributions than the previous one.\n\nThus, a new comparison with observations in the visible range is not\nunnecessary. Moreover, an extensive comparison of spectra based on the\nNeMo grid of model atmospheres for the entire range of A to early M stars\nincluding both dwarfs and giants has not been done before. We hence begin\nour comparison with observations in the visual before proceeding to the\ninfrared. The implications of changing abundances or the description of\nconvection at spectral resolutions relevant for studies of galaxies\nare included as part of the discussion of our comparisons.  \n\n\n\\section{Obtaining a theoretical spectrum}\\label{spec}\n\n\nFirst of all, we downloaded the model atmospheres corresponding to the stellar types\nwanted from the NeMo website (http:\/\/ams.astro.univie.ac.at\/nemo\/). The models are\nclassified according to the convection model (CM, CGM or MLT) and to the number of\nlayers representing the atmosphere. The model CGM with 72 layers is detailed enough\nfor our purpose. Models with 288 layers are used only for specific applications like\nthe calculation of the convective scale length in stellar interior models\n(Heiter et al.\\ \\cite{Hei02}). Reduced to the medium resolution of our observations,\nboth computations of a model with 72 and 288 layers respectively give similar spectra.\n\nThe next parameter to determine is the microturbulence velocity. For cool dwarf\nstars, this velocity is low: about 0--1~km\/s, but the value is increasing\ntowards higher luminosities, reaching values as high as 5~km\/s (Gray \\cite{Gra92}).\nA few stars do not follow this rule: hot stars, like B and O type, have\na negligible microturbulence velocity and some specific types of A stars can\neither have a null velocity (Ap type stars, for them magnetic field effects are\nimportant instead) or a velocity of 4~km\/s (Am type stars). As the\nmicroturbulence velocity has only a small influence on the overall shape of the\nspectra and on the line profiles at our spectral resolution, we can use a common\nvalue of 2~km\/s for comparison with all our stellar spectra, composed by A to\nearly-M type dwarf and F to K type giant stars, as 2~km\/s is a good compromise\nfor these stars (Gray \\cite{Gra92}).\n\nThen, the three main physical parameters of the star have to be\nchosen. The metallicity, the effective temperature and the surface\ngravity of the theoretical stellar spectrum should correspond as good\nas possible to the observed star to be compared. Therefore, once the\nstellar characteristics are determined, the nearest set of parameters\n(T, log g, Z) in the NeMo grid is taken.\n\nThe metallicity is taken from Nordstroem et al. (\\cite{Nor04}), Cayrel\nde Strobel et al. (\\cite{Cay01}) and Barbuy \\& Grenon (\\cite{Bar90})\nwhen available for individual stars of the sample, otherwise assumed\nto be solar.\n\nNordstroem et al. (\\cite{Nor04}) have determined the effective\ntemperature of most of the dwarf stars in our sample, for other stars\nthe effective temperature is assumed, as well as the surface gravity,\naccording to their spectral type from the corresponding values of\ntemperature and gravity as given in Schmidt-Kaler (\\cite{Sch82}) and\nGray (\\cite{Gra92}).\n\nOnce the most suitable model atmosphere is determined, we can generate\na theoretical flux calibrated spectrum with the code SynthV (by VT).\nThis code requires several input parameters such as the wavelength\nrange for which the spectrum will be computed, as well as the\nwavelength step. This step has to be small enough compared to 2.5~\\AA\\\nas the desired opacity in each wavelength point includes absorption\nfrom all nearest lines within 2.5 \\AA, so large wavelength steps would\ngive wrong results. We take 0.1~\\AA\\ in the visible and in the\ninfrared range. Then, we can enter a rotation profile for the star, if\nneeded, and indicate a list of absorption lines to be used.  For our\nwork, we take the Vienna Atomic Linelist Database (VALD), completed by\nseveral molecular line lists (C$_2$, CN, CO, H$_2$, CH, NH, OH, MgH,\nSiH, SiO, TiO, H$_2$O \\footnote{file VColl\\_molec.lns built by VT\n    from Kurucz' CDROMs 15, 24, 25, and 26, see Kurucz \\cite{Kur93b}\n    and \\cite{Kur99}; the file is available from V. Tsymbal upon\n    request}).  The line profiles are approximated by a Voigt\nfunction. SynthV also provides the possibility to change individual\nabundances.\n\nThe final theoretical spectrum has to be reduced to the same\nresolution and the same sampling as the observed spectrum for further\ncomparisons. So, the calculated spectrum is Gaussian smoothed and\nresampled by Fourier interpolation to the same step as the observed\nspectrum.\n\n\n\\section{Testing parameters}\\label{test}\n\n\nThe determination of the physical parameters of the observed stars is\nnot as acurate as we would like.  So it is necessary to investigate\nthe nearest values of T$_\\mathrm{eff}$ \/ log g \/ Z of the grid. The\nchemical abundances can also be changed; as these abundances are not\nvery well determined, it is crucial to notice how a variation of the\nabundance of one element modifies the spectrum.\n\nThe most important change is caused by the temperature. Indeed, the step of\n{200 K} as was chosen for the grid computation is still quite large for our \npurpose and a deviation of this range can be dramatic for the slope of the\nspectrum. The coldest stars (M, K and even G type) are the ones most affected\nby a change of 200~K.\n\nFig. \\ref{vis-dw} and \\ref{vis-gi} show the evolution of the spectra\nwith temperature in the visible range for the dwarfs and the giants,\nrespectively, and Fig. \\ref{ir-kgi} shows this evolution for dwarf\nstars in the infrared range. We can see that for intermediate\ntemperature, there is mainly a difference in continuum. But for the\nextreme values, the modification of the spectrum is more dramatic, as\nit affects also the absorption line features.\n\n\\begin{figure*}\n\\centering\n  \\includegraphics[angle=-90,width=12cm]{temp_dw_up.ps}\n  \\caption{Results of a variation of temperature in the visible range\n    for dwarf stars. From the top to the bottom: T=6000K to 4600K with\n    a step of 200K between two spectra, arbitrarily shifted by a\n    constant value for the purpose of clarity.}\n  \\label{vis-dw}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n  \\includegraphics[angle=-90,width=12cm]{temp_gi_up.ps}\n  \\caption{Results of a variation of temperature in the visible range\n    for giant stars. From the top to the bottom: T=5000K to 4000K with\n    a step of 200K between two spectra, arbitrarily shifted by a\n    constant value for the purpose of clarity.}\n  \\label{vis-gi}\n\\end{figure*}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[angle=-90,width=12cm]{ir_temp_dw_up.ps}\n\n  \\caption{Results of a variation of temperature in the infrared range\n    for dwarf stars. From the top to the bottom: T=7200K, T=6400K and\n    T=6000K to 5200K with a step of 200K, arbitrarily shifted by a\n    constant value for the purpose of clarity.}\n  \\label{ir-kgi}\n\\end{figure*}\n\nA change in log g causes a variation of the line profiles. This\nparameter is not very well known for observed stars. So it is\nimportant to test different values around a first guess. Even if the\nvariation caused by a gap of 0.2 in log g is not very important, it\ncan improve the comparison.\n\nThe metallicity has an influence on the slope of the continuum:\nincreasing the metallicity of a theoretical spectrum has a similar\ninfluence on the continuum as decreasing the temperature (see e.g.\nRam\\'irez \\& Mel\\'endez, 2005, for a detailled discussion). A\nvariation of metallicity has also a clear influence on the strength of\nthe absorption lines.\n\nThe variation of individual abundances can also cause some changes in the\nspectra. SynthV uses by default solar abundances of Anders \\& Grevesse\n(\\cite{And89}). But some studies more recent like Kurucz (\\cite{Kur93}) or Holweger (\\cite{Hol01}) give different\nvalues for various elements (like He, Fe, O, C, N...). These changes can be\nquite important (up to 0.2 dex for Fe). A simple test to probe the influence of\ndifferent abundances was made comparing two spectra assuming the same physical\nparameters, except for the solar abundance of Holweger (\\cite{Hol01}) in one\ncase and the one used by Kurucz (\\cite{Kur93}) in the other. The result shows only slight differences,\nand the comparison with our observations cannot determine whether one is better\nthan the other. For our work, we chose the values given by Holweger (\\cite{Hol01}).\n\nIn order to fit metallic stars, it is also important to check the\ninfluence of the modification of an individual element abundance on\nthe synthetic spectrum. A change of the individual abundances of O and\nC leads to a modification of OH and CO line strengths, respectively.\nIndeed, when the ratio C\/O increases, more CO molecules will be\nformed; on the other hand, when it decreases, more oxygen will be left\nto form OH molecules (see Decin et al. \\cite{Dec00}).\n\nFor the hottest stars, it is important to take into account rotational\nvelocity.  At the medium resolution of our observations, a convolution\nwith a Gaussian profile is good enough to reproduce this effect.\nHence, we do not need to include a more accurate description of the\nchange in the profile of the lines caused by the rotational velocity.\nConsequently, even though SynthV allows to compute spectra with a\nrotational velocity profile for the lines, we compute the spectra\nwithout rotational velocity in order to save computation time and\nconvolve them afterwards with a gaussian profile.\n\nAdditional tests have been made with a different microturbulence\nvelocity for a very cool star (4 km\/s for a M0V-type spectrum in the\nvisible range) and a different number of layers for the convection\nmodel (288 instead of 72 for the same CGM model and the same physical\nparameters). At the resolution of the observed samples, these\nmodifications do not lead to any difference.\n\n\n\\section{Results in the visible range}\\label{visible}\n\n\nAlthough our goal is to explore spectra in the infrared wavelength range, a study of\nthe behavior of the NeMo model atmospheres and spectra in the visible range provides\nus a good indication of their reliability as a function of the physical parameters of\nthe stars.\n\n\n\\subsection{Observations in the visible}\n\n\n18 spectra of observed stars (A to M dwarfs and G to K giants)\ncorresponding to the range of the parameters in the NeMo grid have\nbeen compared to theoretical stellar spectra. This sample of\nobserved spectra at a resolving power of R $\\simeq$ 600 is taken from\nthe stellar library used by Boisson et al. (\\cite{Boi00}). One part\nof these observations comes from the stellar library of Silva \\&\nCornell (\\cite{Sil92}), made at the KPNO with the MARK III spectrograph,\nthey cover the wavelength range 3500-9000\\AA. These spectra\nare, for most of them, a mean-value of several stars of nearby\nspectral type. The name of these stars and the mean associated\nspectral types as given by Silva \\& Cornell are listed in Table\n\\ref{list_vis}.  The remaining of the library, mainly supermetallic\nstars, were observed by Serote Roos et al. (\\cite{Ser96}) at the CFHT\nwith the Herzberg spectrograph and at OHP with the Aurelie\nspectrograph. The spectral range is limited to 5000-9000\\AA.\nThe name, spectral type (or associated mean-spectral type) and\nparameters of these stars are listed in Table \\ref{list_vis}.\n\n\\begin{table*}\n\\caption{List of observed stars in the visible, with the values of the parameters\ntaken from the mean-values listed by Gray (\\cite{Gra92}) and Schmidt-Kaler\n(\\cite{Sch82}) or from (1) Nordstroem et al.\\ (\\cite{Nor04}),\n(2) Cayrel de Strobel et al.\\ (\\cite{Cay01}) and (3) Barbuy \\& Grenon (\\cite{Bar90}).\nIn the last column, when no information on metallicity is available, a tick mark\nreplaces it. The quantity $<v\\sin i>$ is given in km\/s.}\n\\label{list_vis}\n\\centering\n\\begin{tabular}{|p{4.0cm} c c p{2.8cm} c p{2.5cm}|}\n\\hline\nName & Spectral Type & $<v\\sin i>$ & $T_\\mathrm{eff}$(K) & log g & [Fe\/H] \\\\\n\\hline\nHD116608,HD190785,HD124320 & A1-3 V & $145$ & $8900$ & $4.2$ & - \\\\\n\\& HD221741& & & & & \\\\\nHD88815 & F2 V & $90$ & $7244^1$ & $4.3$ & $-0.13^1$ \\\\\nHD187691, HD149890 & F8-9 V & $7$ & $6026^1$, $5902^1$ & $4.4$ & $0.07^2$, $-0.44^1$ \\\\\nHD121370 & rG0 IV & $5$ & $5957^1$ & $4.4$ & $+0.27^2$ \\\\\nHD38858 & G4 V & $3$ & $5636^1$ & $4.5$ & $-0.26^1$ \\\\\nHD161797 & rG5 IV & $3$ & $5700$ & $4.5$ & $+0.23^2$ \\\\\nHD149661,HD151541,HD33278, & G9K0 V & $2$ & $5176^1$, $5236^1$,\n   $5300$, & $4.5$ & $0.01^1$, $-0.36^1$, - \\\\\nHD23524,SAO66004,SAO84725 & & & $5200^1$, $5300$, $5300$& &  $-0.49^1$, -, -\\\\\nHD93800 & rK0 V & $2$ & $5250$ & $4.5$ & $+0.43^3$ \\\\\nHD39715 & rK3 V & $1$ & $4850$ & $4.6$ & $+0.33^3$ \\\\\nHD36395 & rM1 V & $1$ & $3850$ & $4.6$ & $+0.6^2$ \\\\\n\\hline\nHD15866, HD25894, HD2506 & G0-4 III & $10$ & $5500$ & $3.0$ & - \\\\\nHD163993 & wG8 III & $3$ & $4950$ & $2.7$ & $-0.1^2$ \\\\\nHD72324 & G9 III & $3$ & $4900$ & $2.7$ & - \\\\\nHD33506, HD112989 & rG9K2 III & $2$ & $4700$ & $2.6$ & $+0.14^2$ \\\\\nSAO76803 & K2 III & $2$ & $4500$ & $2.5$ & - \\\\\nHD181984 & rK2 III & $2$ & $4500$ & $2.5$ & $+0.1^2$\\\\\nHD176670 & rK3 III & $1$ & $4300$ & $2.2$ & $-0.03^2$ \\\\\nHD154733, HD21110 & K4 III & $1$ & $4000$ & $2.0$ & $-0.14^2$,- \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nThe atmospheric bands are removed from the observed spectra to compare with the theoretical spectra.\n\n\n\\subsection{Comparisons}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{f2v_up2.ps}\n\\caption{The observed spectrum (in black) is from a F2V type star (HD 88815), the\n         theoretical one (in grey) is computed with the following parameters:\n\t\t T=7200K, log g=4.0, [M\/H]=-0.1. Atmospheric bands are removed from\n\t\t the observed spectrum. The residual between the two spectra is \n\t\t (theoretical~flux~-~observed~flux)\/theoretical~flux.}\n\\label{v-f2v}\n\\end{figure*}\n\nWhen the physical parameters (see Table \\ref{list_vis})\nare known, we took the model with the nearest values, otherwise we used the mean values\naccording to the spectral type as starting points and investigated the nearby values to\nfind the best agreement between the observed and the computed spectra. The theoretical\nspectra are computed with a wavelength step of 0.1~\\AA, corresponding to a resolution\nof 60000 at 6000~\\AA, then Gaussian smoothed to the resolution of the observed spectra.\nBy Fourier interpolation, we reduce the computed spectra to the same wavelength step\nas the observations. Spectra are normalized to 1 in the range 5440-5460\\AA. When the star\nhas a rotational velocity, we convolve the corresponding computed spectrum with\na Gaussian of the same velocity.\n\nThe agreement between the observed and computed spectra is\nsatisfactory for effective temperatures ranging 4600 to 9000 K, 9000 K\ncorresponding to the highest temperature for the stars composing our\nsample. For these spectra, the main discrepancies, which consist in\ndifferences in the slope of the blue extremity of the continuum, can\nbe explained by the difficulty to have a good flux calibration at the\nwavelength ends of the observational data (in particular at 5000\\AA\\\nwhere strong MgI, MgH and FeI absorptions are present). Various\nexamples of comparisons for these stars are shown in\nFig.~\\ref{v-f2v}-\\ref{v-k0v}.  The spectral type of the observed stars\nand the physical parameters used to compute the theoretical spectra\nare noticed for each figure.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{g0iv_up2.ps}\n\\caption{The observed spectrum (in black) is from a G0IV type star (HD\n  121370), the theoretical one (in grey) is computed with the\n  following parameters: T=6000K, log g=4.4, [M\/H]=+0.3.}\n\\label{v-g0iv}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{g8iii_up.ps}\n\\caption{The observed spectrum (in black) is from a G8III type star\n  (HD 163993), the theoretical one (in grey) is computed with the\n  following parameters: T=5000K, log g=2.8, [M\/H]=-0.1.}\n\\label{v-g8iii}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{k0v_up2.ps}\n\\caption{The observed spectrum (in black) is from a K0V type star (HD\n  93800), the theoretical one (in grey) is computed with the following\n  parameters: T=5200K, log g=4.4, [M\/H]=+0.5.}\n\\label{v-k0v}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{m1v_p10_up2.ps}\n\\caption{The observed spectrum (in black) is from a M1V type star (HD\n  36395), the theoretical one (in grey) is computed with the following\n  parameters : T=4000K, log g=4.6, [M\/H]=+1.0. The scale for the\n  residual is twice the scale of the previous figures.}\n\\label{v-m1v}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{m1v_p05_up2.ps}\n\\caption{Same as Fig. \\ref{v-m1v}, with [M\/H]=+0.5 for the theoretical spectrum.}\n\\label{v-m1v_5}\n\\end{figure*}\n\n\nThree stars have a temperature below 4400 K, one dwarf and two giants. The M dwarf\nstar (HD 36395) is particular as it has a very high metallicity. It is the most\nmetallic star of Cayrel de Strobel's catalog.\\footnote{[Fe\/H]=+0.6 dex} Indeed, to\nfit correctly the observations, we need to set the metallicity of the theoretical star\nas high as possible (Fig.~\\ref{v-m1v}), but a metallicity of [M\/H]=+1.0 dex is not\nrealistic. Fig.~\\ref{v-m1v_5} shows that [M\/H]=+0.5 dex is not sufficient. However,\nrecently Woolf \\& Wallerstein (\\cite{Woo05}) have found a temperature of 3760~K (instead of 3850~K) and\na metallicity of [Fe\/H]=+0.2 dex for this star. So the discrepancies may be simply\ndue to the difference in temperature, out of reach for NeMo.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{k4iii_up2.ps}\n\\caption{The observed spectrum (in black) is the mean-value of two K4III type stars\n         (HD 154733 and HD 21110), the theoretical one (in grey) is computed with the\n\t\t following parameters: T=4000K, log g=2.0, solar metallicity. The scale of\n\t\t this plot is not the same as for the previous figures.}\n\\label{v-k4iii}\n\\end{figure*}\n\n\nThe two others are K giant stars. First it is worth to say that at\nthis low temperature, a difference of 200 K causes a variation of the\nslope of the continuum more important than for the highest\ntemperatures. Thus, the continuum of the K3III observed star can\nneither be correctly fitted by a computed spectrum with a temperature\nof 4400 K nor with 4200 K. The first one is too blue and the second\none is too red.  Another difficulty comes from the Na line at 5894\n\\AA\\ and the MgH band computed too strong compared to the\nobservations. These absorption features are very sensitive to a\nvariation of temperature and a too strong value means that the\ntemperature is too low, but increasing the temperature would lead to a\ncontinuum too blue. This is particularly clear for the K4III star,\nwhich lies on the boundaries of the NeMo grid for temperature and\ngravity (T=4000 K, log g=2.0), as we can see on Fig. \\ref{v-k4iii}.\nThis star comes from the stellar library of Silva \\& Cornell (1992)\nwhose original spectra extend down to 3500\\AA.  The agreement for the\nslope of the continuum is satisfactory, but the computed NaI line,\n  the MgH band as well as other lines and molecular bands such as e.g.\n  CaII, CN, Gband are by far too strong.  If we set the temperature at\n  4200~K, which is still reasonable for this kind of star, the\n  accordance for the lines and molecular bands would be better, but\n  the slope of the observed continuum would be too red.\n\nThe hypothesis of plane-parallel geometry of the model begins to\nbecome unrealistic (low log gravity) and the molecular opacities,\nwhich are not taken sufficiently into account in ATLAS9, become\nimportant for these cool stars.\n\n\n\\section{Results in the infrared range}\\label{infrared}\n\n\nWe can immediatly say that there are, by far, more discrepancies\nbetween the computed and the observed spectra in the infrared than in\nthe visible range, even at the medium resolution we have.\n\n\n\n\\subsection{Observations in the infrared}\n\n\nThe observed spectra for this wavelength range come from Meyer et al. (\\cite{Mey98})\nand Boisson et al. (\\cite{Boi02}). The Meyer's ones are observations at a resolving\npower of R $\\simeq$ 3000 at 1.6 $\\mu$m with the KPNO Mayall 4 m Fourier Transform\nSpectrometer; these spectra have to be calibrated in flux. The stars from Boisson\net al. (\\cite{Boi02}) come from the ISAAC spectrograph, mounted on the VLT telescope,\nat a resolving power of R $\\simeq$ 3300 at 1.6 $\\mu$m. From these samples, we selected\n23 stars matching the available parameters of NeMo (A to M dwarfs and F to K giants),\nlisted in Table \\ref{list_ir}.\n\n\\begin{table*}\n\\caption{List of observed stars in the infrared, with the values of the parameters\n         taken from the mean-values listed by Gray (\\cite{Gra92}) and Schmidt-Kaler\n\t\t (\\cite{Sch82}) or from (1) Nordstroem et al. (\\cite{Nor04}),\n\t\t (2) Cayrel de Strobel et al. (\\cite{Cay01}) and (3) Barbuy \\& Grenon\n\t\t (\\cite{Bar90}). In the last column, when no information on metallicity\n\t\t is available, a tick mark replaces it.}\n\\label{list_ir}\n\\begin{tabular}{|l c c c c c|}\n\\hline\nName & Spectral Type & $<v\\sin i>$(km\/s) & $T_\\mathrm{eff}$(K) & log g & [Fe\/H] \\\\\n\\hline\nHD159217 & A0 V & $150$ & $9700$ & $4.3$ & $-$ \\\\\nHD27397 & F0 IV & $120$ & $7100$ & $4.3$ & $-$ \\\\\nHD48501 & F2 V & $9$0 & $6850$ & $4.3$ & $+0.01$ \\\\\nHD26015 & F3 V & 65 & $6776^1$ & $4.3$ & $+0.11^1$\\\\\nHD30606 & F6 V & 10 & $6152^1$ & $4.4$ & $-0.01^2$ \\\\\nHD98231 & F8.5 V & 7 & $5794^1$ & $4.4$ & $-0.35^2$ \\\\\nHD112164 & rG1 V & 5 & $5768^1$ & $4.4$ & $+0.24^2$ \\\\\nHD10307 & G1.5 V & 4 & $5781^1$ & $4.4$ & $-0.04^2$ \\\\\nHD98230 & G2 V & 4 & $5794^1$ & $4.5$ & $-0.34^2$ \\\\\nHD106116 & rG4 V & 3 & $5572^1$ & $4.5$ & $+0.15^2$ \\\\\nHD20618 & G6 IV & 2 & $5600$ & $4.5$ & $-$ \\\\\nHD101501 & G8 V & 2 & $5408^1$ & $4.5$ & $+0.03^2$ \\\\\nHD185144 & K0 V & 2 & $5212^1$ & $4.5$ & $-0.29^1$ \\\\\nHD22049 & K2 V & 2 & $5117^1$ & $4.6$ & $-0.14^2$ \\\\\nHD39715 & rK3 V & 1 & $4850$ & $4.6$ & $+0.33^3$ \\\\\nHD131977 & K4 V & 1 & $4700$ & $4.6$ & $+0.03^2$ \\\\\nHD201902 & K7 V & 1 & $4100$ & $4.6$ & $-0.63^2$ \\\\\nGL338 & M0 V & 1 & $3900$ & $4.6$ & $-$ \\\\\n\\hline\nHD89025 & F0 III & $80$ & $7100$ & $3.4$ & $-$ \\\\\nHD432 & F2 III & $75$ & $7278^1$ & $3.2$ & $+0.18^1$ \\\\\nHD107950 & G6 III & $4$ & $5050$ & $2.8$ & $-0.16^2$ \\\\\nHD197989 & K0 III & $2$ & $4800$ & $2.7$ & $-0.18^2$ \\\\\nHD3627 & K3 III & $2$ & $4300$ & $2.0$ & $+0.04^2$ \\\\\n\\hline\n\\end{tabular}\n\n\\end{table*}\n\n\n\\subsection{Comparisons}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_f85v_up.ps}\n\\caption{The observed spectrum (in black) is from a F8.5V type star (HD 98231), the\n         theoretical one (in grey) is computed with the following parameters:\n\t\t T=5800K, log g=4.6, [M\/H]=-0.3dex.}\n\\label{ir-f85v}\n\\end{figure*}\n\nAround 1.6 $\\mu$m hot stars are dominated by the Brackett lines, and a\ngood determination of the rotational velocity of these stars, which\nbroadens the lines, is very important to have the best possible match\nbetween the computation and the observation. In our wavelength range,\nthe Brackett lines at 1.588, 1.611 and 1.641 $\\mu$m are nearly the\nonly features of the observed spectra. They are well fitted by the\ntheoretical spectra.\n\nWhen the temperature decreases, some atomic features appear and the\ncomparison between observed and computed spectra deteriorates.\nIndeed, for the F6V, a quite large amount of metallic lines, visible\nin the observed spectra, are not present or too weak in the computed\nspectra and this trend continues with the F8.5V (Fig. \\ref{ir-f85v}).\nThe continuum of these observed spectra is very well reproduced, but\nthis is not the case for the lines: most of the metallic lines are\ncomputed too weak.\n\nIn addition, for the F8.5V star, the Brackett lines at 1.611 and 1.641\n$\\mu$m, are computed too strong. Indeed, the Brackett lines have\nalmost disappeared in the observed spectrum but are still strong in\nthe computation.\n\nThese Brackett lines are also present in all the theoretical G stars,\nwhich is not always the case for observed stars, as seen in\nFig.~\\ref{ir-g4v}. The behaviour of the computed Brackett lines\ntowards the temperature is shown on Fig.~\\ref{ir-kgi}; the Brackett\nlines are still present in theoretical spectra for temperatures as low\nas 5200K.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_g4v_up.ps}\n\\caption{The observed spectrum (in black) is from a G4V type star (HD 106116), the\n         theoretical one (in grey) is computed with the following parameters:\n\t\t T=5800K, log g=4.4, [M\/H]=+0.3dex.}\n\\label{ir-g4v}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_k2v_up.ps}\n\\caption{The observed spectrum (in black) is from a K2V type star (HD 22049), the\n         theoretical one (in grey) is computed with the following parameters: T=4800K,\n         log g=4.6, [M\/H]=-0.1dex.}\n\\label{ir-k2v}\n\\end{figure*}\n\nThen, when the temperature decreases, as expected, the Brackett lines\nare fainter and atomic lines (FeI, SiI, MgI and CaI) become stronger\nand stronger for the observed spectra, as well as the theoretical\nspectra.  But the agreement between the computed and the observed\nstars becomes worse.  From Fig.~\\ref{ir-g4v} to \\ref{ir-k3iii}, the\nresiduals between observed and theoretical spectra show that several\nabsorption lines are missing in the theoretical stars, iron lines for\nthe most part. The model atmosphere is not the reason because the\ncontinuum shape is very good and several lines match perfectly, but\nthe line list needs to be improved.\n\nIn the infrared range, the lack of several metallic and molecular\nlines causes the discrepancies, with enhanced differences at low\ntemperature due to the greatest strength of the lines for the coolest\nstar (Fig. \\ref{ir-k3iii}).  Fig.~\\ref{ir-k2v} and \\ref{ir-k4v}\npresent two similar stars (K2V and K4V, respectively), the first one\nis from Meyer and the second one is a VLT observation at higher\nresolution. Both comparisons show this lack of absorption lines, with\nmore details visible for Fig. \\ref{ir-k4v}.\n\nThe comparison for the coolest dwarf star of this sample, a M0V, is\nnot so bad for such a low temperature. The continuum is good in spite\nof the limitations of the model (Fig.~\\ref{ir-m0v}).  We notice,\nhowever, that at the contrary of the previous spectra, the absorption\nlines are computed too strong for the theoretical spectrum, as seen\nthanks to the residual.  This is probably due to the limit of validity\nof the model atmospheres, as already seen in the visible range.\n\n\\bigskip\n\nIn order to investigate further which lines are missing in the\ncomputations, we have compared the high-resolution spectrum of the\nwell-known K1III star Arcturus (Hinkle et al. 1995) to a theoretical\nspectrum computed with the parameters of this star ($T_\\mathrm{eff}$ =\n4400 K, log g = 2.0, [M\/H]=-0.2, $v \\sin i$ = 3.5 km\/s) and point out\nthe discrepancies: Fig. \\ref{ir_arct_zoom} shows a detail of this\ncomparison, and Table \\ref{arct_lines} lists the missing lines in the\nwhole range. \nIn addition to the lines quoted in Table \\ref{arct_lines}, several\nother features are computed too weak, in particular OH and CO\nmolecular bands, certainly due to an inaccurate determination of the\noscillator strengths, as discussed in Lyubchik et al. (\\cite{Lyu04}).\n\nThis study, based on the NeMo grids of atmospheres, remains valid for the entire\nfamilly of the ATLAS models. Indeed, as shown in Fig.~\\ref{nemo_kur}, two theoretical\nspectra computed for the same physical parameters, with the NeMo grid and the ATLAS9\nmodels with the overshooting prescription (Kurucz \\cite{Kur93}, \\cite{Kur98}; Castelli\net al.~\\cite{Cas97}, respectively), are very similar at our spectral resolution. The\ndiscrepancies between the two different theoretical spectra are very faint compared\nto the discrepancies between the models and the observed spectra.\n\nThe same ATLAS9 models, but without overshooting (NOVER models, Castelli et\nal.~\\cite{Cas97}), present even less differences with the NeMo spectra, in particular\nthe slight discrepancy found for the Brackett lines disappear. They are more sensitive\nthan other lines to the fact that the Kurucz overshooting prescription changes the\ntemperatures at Rosseland optical depths of 0.1 to 0.5.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_k4v_up.ps}\n\\caption{The observed spectrum (in black) is from a K4V type star (HD 131977), the\n         theoretical one (in grey) is computed with the following parameters: T=4800K,\n\t\t log g=4.6, solar metallicity.}\n\\label{ir-k4v}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_k3iii_up.ps}\n\\caption{The observed spectrum (in black) is from a K3III type star (HD 3627), the\n         theoretical one (in grey) is computed with the following parameters: T=4400K,\n\t\t log g=2.0, solar metallicity.}\n\\label{ir-k3iii}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_m0v_up.ps}\n\\caption{The observed spectrum (in black) is from a M0V type star (GL 338), the\n         theoretical one (in grey) is computed with the following parameters: T=4000K,\n\t\t log g=4.6, solar metallicity.}\n\\label{ir-m0v}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{ir_k3iii_theo_up.ps}\n\\caption{Comparison between two theoretical models (ATLAS9 and NeMo)\n  with the following parameters: T=4400K, log g=2.0, solar\n  metallicity. Note that the scale for the residual is enhanced\n  compared to all other figures.}\n\\label{nemo_kur}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{arct_up.ps}\n\\caption{Comparison between a high resolution observation\n($\\frac{\\lambda}{\\Delta\\lambda} \\simeq 100000$) of Arcturus (in\n  black) and the corresponding theoretical star (in grey). The flux is\n  given with continuum normalized to 1 in order to better show the\n  missing lines.}\n\\label{ir_arct}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90,width=12cm]{arct_zoom_up.ps}\n\\caption{The same as Fig. \\ref{ir_arct} but zoomed in into a limited\n         wavelength range.}\n\\label{ir_arct_zoom}\n\\end{figure*}\n\n\n\\begin{table}\n\\caption{List of the missing lines, according to the comparison between Arcturus and\n          a computed spectrum. The Ni line is not missing as such but shifted by\n\t\t  2~\\AA\\ in the atomic database.}\n\\label{arct_lines}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nwavelength ($\\mu$m)& element & & wavelength ($\\mu$m)& element \\\\\n\\hline\n\n1.5764 & Fe & & 1.6208 & Fe \\\\\n1.5893 & Fe & & 1.6214 & Fe \\\\\n1.5895 & Fe & & 1.6231 & Fe \\\\\n1.5913 & Fe & & 1.6285 & Fe \\\\\n1.5939 & Fe & & 1.6316 & Fe \\\\\n1.5954 & Fe & & 1.6319 & Fe \\\\\n1.5968 & Fe & & 1.6362 & Ni \\\\\n1.6007 & Fe & & 1.6394 & Fe \\\\\n1.6008 & Fe & & 1.6440 & Fe \\\\\n1.6041 & Fe & & 1.6450 & OH \\\\\n1.6071 & Fe & & 1.6517 & Fe \\\\\n1.6076 & Fe & & 1.6524 & Fe \\\\\n1.6088 & Fe & & 1.6532 & Fe \\\\\n1.6116 & Fe & & 1.6569 & Fe \\\\\n1.6126 & Fe & & & \\\\\n1.6175 & Fe & & & \\\\\n1.6195 & Fe & & & \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}  \\label{conclusions}\n\n\nIn spite of some discrepancies, the comparisons between observed and\ntheoretical spectra in the visible range suggests a reasonable\nagreement, even at the limits of the parameter range of NeMo.\nHowever, we have to be careful when we compute a model near the lower\nlimit in temperature.\n\nThe spectra modelized with NeMo can be used to build a theoretical\nspectral library for A to K dwarf and giant stars in the visible\nrange, but this is not the case for the near-infrared range.\n\nIndeed, in the range 1.57 to 1.67 $\\mu$m, the spectra computed do not reproduce\nvery well the observations. Albeit the good agreement for the overall flux\ndistribution shape, we can see that there are many differences for the line\nfeatures when focusing on details of the spectra. The strength of the infrared\nabsorption lines is usually underestimated in calculations, and some lines are\nsimply missing (Fe, OH and CO lines are the most problematic ones). As pointed\nout by Decin et al.\\ (\\cite{Dec03}) for the MARCS6 models and\nLyubchik et al.\\ (\\cite{Lyu04}) for NextGen models of ultracool dwarfs and\na Kurucz model for Arcturus, it was not possible to generate synthetic spectra\nwhich can reproduce observed spectra in the infrared with the line lists that\nhave been used in constructing the model atmospheres, even at a medium\nresolution. In particular, the oscillator strengths are still not known\nsufficiently well.\n\nWe have also performed a comparison of spectra that include the lines\nfrom iron peak elements with predicted energy levels, as published by\nKurucz (\\cite{Kur98}), with the observations of Arcturus discussed\nabove.  As to be expected, such spectra contain more lines, but the\ninaccuracy of their energy levels frequently places them at the wrong\nwavelengths and the overall match hardly improves (the total flux\ndistribution over all wavelength ranges, particularly in the\nultraviolet, is closer to observations when including this set of\nlines, but for the limited wavelength range around the H band the\neffects are small and sufficiently compensated when setting the zero\npoint of the flux distribution). For the case of Arcturus we also\nperformed a comparison with spectra computed with PHOENIX in LTE (P.\\\nHauschildt, priv.\\ comm.\\ 2005). The resulting spectra for the 1.57 to\n1.67~$\\mu$m range were found to be rather similar to those from\nNeMo\/VALD\/SynthV when including the predicted level lines.  One\nimportant reason for this is certainly the fact that the atomic line\nlists for PHOENIX are essentially those of Kurucz (\\cite{Kur98}).\nBecause the flux distribution of the PHOENIX spectra is similar to the\nobservations as well, at least for the K giants the detailed choice of\nthe model atmosphere code appears to be clearly less important than\nthe choice of atomic line lists (note that PHOENIX uses its own\ncollection of molecular line lists, different from the one we have\nused here). Considering the uniformity of the deterioration of the\nmatch of spectra in the 1.57 to 1.67~$\\mu$m range when looking at the\nsequence from F to K stars we conclude that the insufficient line\nlists, and in particular lists of atomic lines, are the main obstacle\nfor a more satisfactory match of observed spectra of these groups of\nstars. The modelizing in the infrared range needs some further\nimprovements, in particular for the absorption lines database, before\nto build a theoretical spectral library which can be used with high\nbenefit instead of an observed star library.\n\nThe lack of M stars in spectral library would be very much prejudicial\nto the study of stellar populations as the variations of their strong\natomic lines and molecular bands along their evolution from dwarf to\nsupergiant to giant provides very good age discriminators (from 10$^6$\nto 10$^{10}$yrs). M stars peak in a wavelength range which is not much\nabsorbed even in heavily reddened region, as young stellar clusters,\nmaking them easily detectable. Moreover they are known to be very\nimportant contributors to the stellar populations of galaxies as well\nfor the mass as for the luminosity following the age of the\npopulation. All this make a good theoretical library of M stars very\ncritical in order to extend incomplete observed library. \n\nThe prospects of constructing such a library from the upcoming generation\nof model atmospheres (MARCS, PHOENIX, perhaps future versions of ATLAS)\nare indeed improving because of the enormous efforts spent in extending\nthe molecular line data and equation of state. To match\nthe spectra of the hot end of M stars in the H band will nevertheless\nrequire more complete atomic data, although this is less crucial as for\nK stars. Efforts along this direction are currently made.\n\n\\begin{acknowledgements}\nFK gratefully acknowledges the hospitality of the Observatoire de Paris-Meudon\nduring his stays as an invited visitor. \nJF also thanks the nice people of the AMS group at Vienna Observatory for \ntheir hospitality and help during part of this work. \nVT acknowledges the Austrian Fonds zur F\\\"orderung der wissenschaftlichen\nForschung FwF (P17580) and by the BM:BWK (project COROT).\nThis research has made use of the model atmosphere grid NeMo, provided by\nthe Department of Astronomy of the University of Vienna, Austria, and\nfunded by the Austrian FwF (P14984).\nWe are thankful to Peter Hauschildt who has computed comparison spectra for\nArcturus for us which helped to demonstrate the importance of the line lists\nused relative to the particular choice of model atmosphere codes.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{APPENDIX}\n\\section*{Proof of Lemma \\ref{lemma:concave}}\nLet $\\mathbf{s} = \\{s_1,..s_m\\}$, since $\\mathbf{s}$ is a unit vector, there is only one value among $s_1,..s_m$ is $1$ and the others are all $0$. By extending the Equ. \\ref{equ:delta_g}, we have\n\\begin{align*}\n\\Delta_\\mathbf{s} g(\\mathcal{P}, \\mathbf{x}) = \\sum_{p \\in \\mathcal{P}} \\beta_p \\Big( \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e)) \\Big)\n\\end{align*}\nFor each path $p$, we will prove that:\n\\begin{align*}\n& \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e)) \\\\\n& \\quad \\geq \\gamma \\cdot (\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e))) \n\\end{align*}\nWe consider three cases:\n\\begin{itemize}\n\\item $\\mathtt{T} \\geq  \\sum_{e \\in p} f_e(x_e+s_e) \\geq  \\sum_{e \\in p} f_e(x_e)$. Then we have\n\\begin{align*}\n&\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e)) \\\\\n& \\quad \\quad = \\sum_{e \\in p} f_e(x_e+s_e) - \\sum_{e \\in p} f_e(x_e) \\\\ \n&\\quad \\quad \\geq \\gamma \\cdot \\Big( \\sum_{e \\in p} f_e(y_e+s_e) - \\sum_{e \\in p} f_e(y_e) \\Big) \\\\\n& \\quad \\quad \\geq \\gamma \\cdot \\Big(\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e)) \\Big) \n\\end{align*}\n\\item $\\sum_{e \\in p} f_e(x_e+s_e) \\geq \\mathtt{T} \\geq  \\sum_{e \\in p} f_e(x_e)$. In this case,\n\\begin{align*}\n\\sum_{e \\in p} f_e(y_e + s_e) \\geq \\sum_{e \\in p} f_e(x_e + s_e) \\geq \\mathtt{T}\n\\end{align*}\nalso\n\\begin{align*}\nmin(\\mathtt{T}, \\sum_{e\\in p}f_e(y_e)) \\in [\\sum_{e \\in p} f_e(x_e), \\mathtt{T}]\n\\end{align*}\nTherefore:\n\\begin{align*}\n&\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e)) \\\\\n& \\quad \\quad = \\mathtt{T} - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e)) \\\\\n& \\quad \\quad \\leq \\mathtt{T} - \\sum_{e \\in p} f_e(x_e) \\\\\n& \\quad \\quad = \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e))\n\\end{align*}\n\\item $\\sum_{e \\in p} f_e(x_e+s_e) \\geq  \\sum_{e \\in p} f_e(x_e) \\geq \\mathtt{T}$. This case is trivial because both $\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(y_e))$ and $\\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e+s_e)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e))$ are $0$.\n\\end{itemize}\nHence, $\\Delta_\\mathbf{s} g(\\mathcal{P}, \\mathbf{x}) \\geq \\gamma \\Delta_\\mathbf{s} g(\\mathcal{P}, \\mathbf{y})$, which completes the proof.\n\n\\section*{Proof of Theorem \\ref{theorem:greedy_approx}}\nDenote $\\mathbf{x}^*$ is optimal solution to the \\texttt{QoSD} instance. Define $\\mathbf{x}_i$ as our obtained solution before the $i^\\textnormal{th}$ iteration in Alg. \\ref{alg:greedy_blocking}. Denote $\\mathbf{x}^o_i$ as an optimal solution that is in additional to $\\mathbf{x}_i$ to block all paths in $\\mathcal{P}$. We have:\n\n\\begin{align} \\label{equ:x0x}\n||\\mathbf{x}^*|| \\geq ||\\mathbf{x}^* \/ \\mathbf{x}_i|| \\geq ||\\mathbf{x}^o_i||\n\\end{align}\n\nAssume $\\mathbf{x}^o_i = \\sum_{i=1}^l \\mathbf{u}_i$ where $\\mathbf{u}_i$ is a unit vector. We have:\n\\begin{align}\n&\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i + \\mathbf{x}^o_i) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) = \\sum_{j=1}^l \\Delta_{\\mathbf{u}_j} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i + \\sum_{z=1}^{j-1} \\mathbf{u}_z) \\\\\n& \\quad \\quad \\leq \\frac{1}{\\gamma} \\sum_{j=1}^l \\Delta_{\\mathbf{u}_j} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) \\quad \\quad \\quad (\\textnormal{Lemma \\ref{lemma:concave}})\\\\\n& \\quad \\quad \\leq \\frac{||\\mathbf{x}^o_i||}{\\gamma} \\max_\\mathbf{s} \\Delta_\\mathbf{s} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) \\\\\n& \\quad \\quad \\leq \\frac{\\mathtt{OPT}}{\\gamma} (\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i)) \\label{equ:hmm}\\\\\n& \\quad \\quad = \\frac{\\mathtt{OPT}}{\\gamma} (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) - (|\\mathcal{P}| \\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1})))\n\\end{align}\n\nEqu. \\ref{equ:hmm} follows by greedy selection. Since $\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_1 + \\mathbf{x}^o_i) = |\\mathcal{P}| \\mathtt{T}$,\n\\begin{align*}\n|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) \\leq (1-\\frac{\\gamma}{\\mathtt{OPT}}) (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i))\n\\end{align*}\n\nNote that the Alg. \\ref{alg:greedy_blocking} will terminate after $||\\mathbf{x}||$ iterations. Therefore: \n\n\n\\begin{align*}\n&|\\mathcal{P}| \\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{||\\mathbf{x}||}) \\leq (1-\\frac{\\gamma}{\\mathtt{OPT}}) (|\\mathcal{P}| \\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{||\\mathbf{x}||-1})) \\leq ... \\\\\n& \\quad \\quad \\leq (1-\\frac{\\gamma}{\\mathtt{OPT}})^{||\\mathbf{x}||} (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\{0\\}^{|E|})) \\\\\n& \\quad \\quad \\leq (1-\\frac{\\gamma}{\\mathtt{OPT}})^{||\\mathbf{x}||} |\\mathcal{P}| \\mathtt{T}\n\\end{align*}\n\nSince there should be at least a path $p \\in \\mathcal{P}$ whose overall delay is at most $\\mathtt{T} - 1$ in final round, we have $|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_l) \\geq 1$. Therefore:\n\n\\begin{align*}\n||\\mathbf{x}|| \\leq \\frac{\\ln |\\mathcal{P}| \\mathtt{T}}{\\ln \\frac{1}{1-\\frac{\\gamma}{\\mathtt{OPT}}}} = \\frac{\\ln |\\mathcal{P}| \\mathtt{T}}{\\ln (1 + \\frac{\\gamma\/\\mathtt{OPT}}{1 - \\gamma\/\\mathtt{OPT}})}\n\\end{align*}\n\nWe have $\\ln(1+x) \\geq x - \\frac{x^2}{2}$ for $x \\in (0,1)$. So\n\n\\begin{align*}\n||\\mathbf{x}|| \\leq \\frac{\\ln |\\mathcal{P}| \\mathtt{T}}{\\frac{\\gamma}{\\mathtt{OPT}} (1 - \\frac{\\gamma}{2\\mathtt{OPT}})} \\leq \\mathtt{OPT} \\cdot O(\\frac{\\ln |\\mathcal{P}| \\mathtt{T}}{\\gamma})\n\\end{align*}\n\nAnd since $|\\mathcal{P}| \\leq n^\\mathtt{h}$, \\texttt{IG} obtains $O(\\frac{1}{\\gamma} (\\mathtt{h}\\ln n + \\ln \\mathtt{T}))$ approximation guarantee, which completes the proof.\n\n\\section*{Proof of Theorem \\ref{theorem:trunk_approx}}\nDenote $\\mathbf{x}^* = \\{x_1^*, ... x_m^*\\}$ as optimal solution to the \\texttt{QoSD} problem. Define $\\mathbf{x}_i = \\{x_1,...x_m\\}$ is our obtained solution before the $i^\\textnormal{th}$ iteration in Alg. \\ref{alg:trunk_adding}. Denote $\\mathbf{x}^o_i = \\{x_1^o, ... x_m^o\\}$ as an optimal solution in additional to $\\mathbf{x}_i$ to block all paths in $\\mathcal{P}$. We have:\n\n\\begin{align*}\n||\\mathbf{x}^*|| \\geq ||\\mathbf{x}^* \/ \\mathbf{x}_i|| \\geq ||\\mathbf{x}^o_i||\n\\end{align*}\n\nDenote $\\mathbf{v}(e) = \\{x_1,..x_{e-1},x_e + x_e^o,...x_m + x_m^o\\}$. Trivially, $\\mathbf{v}(1) = \\mathbf{x}_i + \\mathbf{x}^o$ and $\\mathbf{v}(m+1) = \\mathbf{x}_i$. Assume $\\mathbf{u}(e_i,j_i)$ is the vector we would add into solution $\\mathbf{x}_i$ in iteration $i^\\textnormal{th}$. We have following lemma.\n\n\\begin{lemma}\nFor all $e \\in E$, we have:\n\\begin{align*}\n\\frac{\\Delta_{\\mathbf{u}(e_i,j_i)} \\mathtt{D}(\\mathcal{P},\\mathbf{x}_i)}{j_i} \\geq \\frac{\\mathtt{D}(\\mathcal{P},\\mathbf{v}(e)) - \\mathtt{D}(\\mathcal{P},\\mathbf{v}(e+1))}{x_e^o}\n\\end{align*} \n\\end{lemma}\n\n\\begin{proof}\nDenote $\\mathbf{w}(e) = \\{x_1,...x_{e-1}, x_e + x_e^o, x_{e+1}, ... x_m\\}$. Consider a single path $p \\in \\mathcal{P}$, denote\n\\begin{align*}\n&h(p,s) =  \\sum_{e \\in p \\& e < s} f_e(x_e) + \\sum_{e \\in p \\& e \\geq s} f_e(x_e + x_e^o) \\\\\n&g(p,s) = \\sum_{e \\in p \\& e \\neq s} f_e(x_e) + f_e(x_s + x_s^o)\n\\end{align*}\nthen we have:\n\\begin{align*}\n&\\mathtt{r}(p,\\mathbf{v}(s)) - \\mathtt{r}(p,\\mathbf{v}(s+1)) = \\min(\\mathtt{T}, h(p,s)) - \\min(\\mathtt{T}, h(p,s+1)) \\\\\n&\\mathtt{r}(p,\\mathbf{w}(s)) - \\mathtt{r}(p,\\mathbf{x}_i) = \\min(\\mathtt{T}, g(p,s)) - \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e))\n\\end{align*}\n\nTrivially, we have that:\n\\begin{align*}\n&h(p,s) - h(p, s+1) = g(p,s) - \\sum_{e\\in p} f_e(x_e) = f_{s}(x_{s} + x_{s}^o) - f_{s}(x_{s})\n\\end{align*}\nand due to monotonicity of $\\mathtt{r}(p, \\mathbf{x})$\n\\begin{align*}\n&h(p,s) \\geq g(p,s) \\\\\n&h(p,s+1) \\geq \\sum_{e \\in p} f_e(x_e)\n\\end{align*}\nTherefore, using the similar proof as lemma \\ref{lemma:concave}, we have:\n\\begin{align*}\n&\\mathtt{D}(\\mathcal{P},\\mathbf{v}(e)) - \\mathtt{D}(\\mathcal{P},\\mathbf{v}(e+1)) \\leq \\mathtt{D}(\\mathcal{P},\\mathbf{w}(e)) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) \\\\\n& \\quad \\quad \\quad = \\Delta_{\\mathbf{u}(e, x_{e}^o)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i)\n\\end{align*}\nDue to \\texttt{AT} selection, we have that:\n\\begin{align*}\n\\frac{\\Delta_{\\mathbf{u}(e, x_{e}^o)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i)}{x_{e}^o} \\leq \\frac{\\Delta_{\\mathbf{u}(e_i,j_i)} \\mathtt{D}(\\mathcal{P},\\mathbf{x}_i)}{j_i}\n\\end{align*}\nin which the lemma follows.\n\\end{proof}\n\nNow, we will find the approximation guarantee of \\texttt{AT} solution. We have:\n\\begin{align*}\n&\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i + \\mathbf{x}^o_i) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) = \\sum_{e} (\\mathtt{D}(\\mathcal{P}, \\mathbf{v}(e)) - \\mathtt{D}(\\mathcal{P}, \\mathbf{v}(e+1))) \\\\\n& \\quad \\quad \\quad \\leq \\sum_e \\frac{x_e^o}{j_i} \\Delta_{\\mathbf{u}(e_i,j_i)} \\mathtt{D}(\\mathcal{P},\\mathbf{x}_i) \\\\\n& \\quad \\quad \\quad \\leq \\frac{\\mathtt{OPT}}{j_i} (\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i))\n\\end{align*}\nSince $\\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i + \\mathbf{x}_i^o) = |\\mathcal{P}| \\mathtt{T}$, we have\n\\begin{align*}\n|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) \\leq (1 - \\frac{j_i}{\\mathtt{OPT}}) (|\\mathcal{P}| \\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i))\n\\end{align*}\nAssume \\texttt{AT} stops after $l$ iterations, we have\n\\begin{align}\n&|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{l}) \\leq \\prod_{i=1}^l (1-\\frac{j_i}{\\mathtt{OPT}}) (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\{0\\})) \\\\\n&\\quad \\quad \\leq \\Big( 1 - \\frac{\\sum_{i=1}^l j_i}{l \\cdot \\mathtt{OPT} } \\Big)^l (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\{0\\})) \\label{equ:cauchy_ta} \\\\\n& \\quad \\quad \\leq e^{-\\frac{||\\mathbf{x}||}{\\mathtt{OPT}}} (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\{0\\})) \\label{equ:e_ta}\n\\end{align}\nEqu. \\ref{equ:cauchy_ta} comes from the following Cauchy theorem\n\\begin{theorem} \\label{theorem:cauchy}\n(\\textnormal{Cauchy Theorem} \\cite{cauchyInequality}) Given $n$ non-negative numbers $x_1,...x_n$, we have\n\\begin{align*}\n\\prod_{i=1}^n x_i \\leq (\\frac{\\sum_{i=1}^n x_i}{n})^n\n\\end{align*}\n\\end{theorem}\nEqu. \\ref{equ:e_ta} comes from observation that $(1-\\frac{x}{n})^n \\leq e^{-x}$.\n\nTherefore, $||\\mathbf{x}||_1 \\leq \\mathtt{OPT} \\ln |\\mathcal{P}|\\mathtt{T}$. Since $|\\mathcal{P}|$ is bounded by $n^\\mathtt{h}$, \\texttt{AT} obtains $O(\\mathtt{h} \\log n + \\log \\mathtt{T})$ approximation guarantee.\n\n\\section*{Proof of Lemma \\ref{lem:greedy_sampling}}\nDenote $\\mathbf{v}_i$ as the budget vector $\\mathbf{v}$ after greedily selecting first $i$ unit vectors, then by monotonicity $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v}^o) \\leq \\hat{D}(\\mathcal{P}, \\mathbf{v} + \\mathbf{v}^o + \\mathbf{v}_i)$. We have\n\\begin{align}\n&\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v}^o) \\leq \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o + \\mathbf{v}_i) \\\\\n& \\quad = \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i) + \\sum_{j=1}^q \\Delta_{\\mathbf{u}_j} \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i + \\sum_{i=1}^{j-1} \\mathbf{u}_i) \\\\ \n& \\quad \\leq \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i) + \\frac{1}{\\gamma} \\sum_{j=1}^q \\Delta_{\\mathbf{u}_j}\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i) \\\\\n& \\quad \\leq \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i) + \\frac{q}{\\gamma} (\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i+1}) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i)) \\label{equ:greedy_select}\n\\end{align}\nThe inequality (\\ref{equ:greedy_select}) is due to greedy selection. Therefore,\n\\begin{align*}\n&\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i+1}) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i) \\geq \\frac{\\gamma}{q}(\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_i)) \n\\end{align*}\nWhich also means\n\\begin{align*}\n&\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i+1}) \\\\\n& \\quad \\leq (1-\\frac{\\gamma}{q}) (\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i}))\n\\end{align*}\nTherefore\n\\begin{align*}\n\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}) \\leq (1-\\frac{\\gamma}{q})^q (\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x}))\n\\end{align*}\nSo\n\\begin{align*}\n&\\Delta_{\\mathbf{v}}\\hat{B}(\\mathcal{P},\\mathbf{x}) \\geq (1-(1-\\frac{\\gamma}{q})^q) \\Delta_{\\mathbf{v}^o}\\hat{B}(\\mathcal{P},\\mathbf{x}) \\\\\n&\\quad \\quad \\geq (1-e^{-\\gamma}) \\Delta_{\\mathbf{v}^o}\\hat{B}(\\mathcal{P},\\mathbf{x})\n\\end{align*}\nwhich completes the proof.\n\n\\section*{Proof of Theorem \\ref{theorem:sampling_approx}}\nFirst, considering the greedy selection $\\mathbf{v}$ in each sampling iteration, from lemma \\ref{cor:greedy_sa}, we have\n\\begin{align*}\n\\Delta_\\mathbf{v} B(\\mathbf{x}) \\geq (1-e^{-\\gamma})(1-\\epsilon) \\Delta_{\\mathbf{v}^*} B(\\mathbf{x})\n\\end{align*}\nwith probability at least $1-\\delta\/||\\mathtt{b}||$.\n\nDenote $\\mathbf{x}^o = \\sum_i^s \\mathbf{u}_i$ as an optimal solution, which is additional to $\\mathbf{x}$, can block all paths in $\\mathcal{F}$ ($\\mathbf{u}_i$ is a unit vector). Let split $\\mathbf{x}^o$ into $l = \\ceil{\\frac{||\\mathbf{x}^o||}{q}}$ parts $L_1,...L_l$ where $L_i = \\sum_{j=(i-1)q+1}^{iq} \\mathbf{u}_j$, we have: \n\n\\begin{align*}\n&\\Delta_{\\mathbf{x}^o} B(\\mathbf{x}) = \\sum_{j=1}^{l} \\Delta_{L_j} B(\\mathbf{x} + L_1 + ... + L_{j-1}) \\leq \\frac{1}{\\gamma} \\sum_{j=1}^{l} \\Delta_{L_j} B(\\mathbf{x})\n\\end{align*}\nTherefore, there should be at least a value $\\Delta_{L_j} B(\\mathbf{x}) \\geq \\frac{\\gamma}{l} \\Delta_{\\mathbf{x}^o} B(\\mathbf{x})$. And since $||L_j|| \\leq q$, we have:\n\n\\begin{align*}\n\\Delta_{\\mathbf{v}} B(\\mathbf{x}) \\geq \\frac{\\gamma}{l} (1-e^{-\\gamma}) (1-\\epsilon) \\Delta_{\\mathbf{x}^o} B(\\mathbf{x}) \n\\end{align*}\nwhich also means\n\\begin{align*}\n|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}+\\mathbf{v}) \\leq (1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon)) (|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}))\n\\end{align*}\n\nNow, denote $\\mathbf{x}_i$ as our solution after the $i^\\textnormal{th}$ iteration of Alg. \\ref{alg:sampling_solution}. We have\n\n\\begin{align*}\n|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}_{i+1}) \\leq (1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon)) (|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}_i))\n\\end{align*}\n\nAssume the algorithm terminates after $g$ iterations, we have:\n\\begin{align*}\n|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}_g) &\\leq (1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon)) (|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}_{g-1})) \\leq ... \\\\\n& \\leq \\Big(1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon)\\Big)^g (|\\mathcal{F}|\\mathtt{T} - B(\\{0\\}^m)) \n\\end{align*}\nEach inequality happens with probability at least $1- \\frac{\\delta}{||\\mathtt{b}||}$. So the probability such that $|\\mathcal{F}|\\mathtt{T} - B(\\mathbf{x}_g) \\leq \\Big(1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon)\\Big)^g (|\\mathcal{F}|\\mathtt{T} - B(\\{0\\}^m))$ is at least $1 - \\frac{\\delta g}{||\\mathtt{b}||} \\geq 1 - \\delta$. Moreover, since in the $g^\\textnormal{th}$ iteration, there should exist a path $p \\in \\mathcal{F}$, whose length smaller than $\\mathtt{T}$. So, the maximum length of $p$ is $\\mathtt{T}-1$. Therefore\n\n\\begin{align*}\n1 \\leq (1 - \\frac{q\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon))^g |\\mathcal{F}|\\mathtt{T}\n\\end{align*}\n\nSo $g \\leq O(\\frac{\\ln \\mathtt{T} + h \\ln d}{q \\gamma (1-e^{-\\gamma})(1-\\epsilon)}) \\mathtt{OPT}$. Since in each iteration, a budget vector $\\mathbf{v}$, $||\\mathbf{v}|| \\leq q$ is added into solution, out final solution guarantees $O(\\frac{\\ln \\mathtt{T} + h \\ln d}{\\gamma (1-e^{-\\gamma})(1-\\epsilon)})$ approximation ratio with probability at least $1-\\delta$.\n\n\\section{Introduction}\n\nGraph connectivity is considered as an important metric on measuring the functionality of a network. Typically, the connectivity-related problems usually ask for the minimum-size set of components (nodes or edges) whose removal disconnects the target set of nodes. This consideration has led to the investigation of many forms of cutting problems in a network: \\textit{e.g} the minimum cut problem, the minimum multicut problem, the sparest cut problem \\cite{vazirani2013approximation} and the most recent work, the Length-Bounded Multicut (\\texttt{LB-MULTICUT}) problem \\cite{kuhnle2018network}. In addition, various measures based on connectivity have formed the framework for assessment of network resilience to external attacks \\cite{grubesic2008comparative,sen2009region,shen2013discovery,shen2012adaptive,nguyen2013detecting,dinh2014bound,dinh2015network,dinh2015assessing, pan2018vulnerability, dinh2010approximation, mishra2014cascading}. \n\nHowever, many network applications now consider other factors when determining a network functionality in addition to connectivity. For example, in Bitcoin network, to guarantee synchronization, not only the network connectivity is required but a network is also configured in order to ensure the broadcasting time of transaction messages under several seconds \\cite{apostolaki2017hijacking}. As another example, consider a time-sensitive delivery on a road network, where edge weights represents the travel time between destinations. Connectivity between a source and a destination is insufficient when a guarantee on the delivery time is required.\n\nTherefore, a natural question is whether a tech-savvy attacker can damage the network functionality without impacting the connectivity? Under various forms, this kind of attacks actually is common, yet stealthy. For example, in the I-SIG system, real-time vehicle trajectory data transmitted using the CV technology are used to intelligently control the duration and sequence of traffic signals \\cite{CvAttack,CVpilot,CVpilot2,checkoway2011comprehensive,koscher2010experimental,mazloom2016security,chen2018exposing}. An adversary, therefore, can compromise multiple vehicles and send malicious messages with false data (e.g., speed and location) to the I-SIG system to impact the traffic control decisions. As reported by previous works, it has been shown that even one single attack vehicle can manipulate the intelligent traffic control algorithm in the I-SIG system and cause severe traffic jams \\cite{CvAttack,chen2018exposing}. To understand the severity of such attack, it is necessary to study on which roads the attackers can target to and what is the minimum number of vehicles the attackers have to compromise to cause large-scale congestions, e.g. traveling from two certain locations takes several hours longer than usual. Such attack can be for political or financial purposes, e.g. blocking traffics of business competitors \\cite{CvAttack}.  \n\n\n\nAs another example, in Bitcoin network or any Blockchain-based applications, an attacker can target to damage the consensus between copies of public ledger of major miners by delaying block propagation between them. Recent works \\cite{apostolaki2017hijacking} have shown that after receiving request for a block information from another node, a Bitcoin node can have up to 20 minutes to respond. An attacker, therefore, can flood the Bitcoin nodes with too many requests or ``dust'' messages to handle, thus delay their block delivery. By flooding multiple nodes, the attacker can disrupt miners to reach consensus on a certain state of Blockchain. The impact of this attack varies relying upon the victims. If the victim is a merchant, it is vulnerable to double spending attacks \\cite{DoubleSpend}. If the victim is a miner, the attack wastes its computational power \\cite{pinzon2016double}. If the victim is a regular node, it will have an outdated view of the Blockchain, and thus more vulnerable to the temporal attacks which exploit the lagging in Blockchain synchronization \\cite{pinzon2016double,dennis2016temporal}. Therefore, it is necessary to study which nodes are critical and how the attacker should attack such nodes (e.g. how much bandwidth consumption) to impact the Bitcoin network functionality, e.g. causing major miners several hours to reach consensus.\n\nWith this motivation, we consider the \\textit{Quality of Service Degradation} (\\texttt{QoSD}) problem. Given a directed graph $G$ representing a network, threshold $\\mathtt{T}$ and set of pairs $S$ in $G$, the objective is to identify a minimum budget to increase the edge (or node) weights to ensure the weighted, shortest-path distance between each pair in $S$ is no smaller than $\\mathtt{T}$. Intuitively, the goal of this problem is to assess how robust the network is; the greater budget to increase edge weights found, the more resilient the network is to the perturbation in terms of edge weights. In addition, the budget to increase weight of a edge in the solution provides an indication of the importance of this edge to the desired functionality. \n\nIn the context of network reliability, Kuhnle et al. \\cite{kuhnle2018network} have recently studied a special case of our problem  under the name \\texttt{LB-MULTICUT}. Different to our problem, the objective of this problem is to identify a minimum set of edges whose {\\bf removal} ensures the distance between each pair of nodes is no smaller than $\\mathtt{T}$. Directly adopting the \\texttt{LB-MULTICUT} solutions to our \\texttt{QoSD} problem is not feasible since most of those solutions exploited a trait that their problems can be formulated by Integer Programming and exhibit submodular behaviors. \\texttt{QoSD} problem, on the other hand, is shown to be neither submodular nor supermodular, making \\texttt{QoSD} more challenging to devise an efficient algorithm. Also, modern networked systems are increasingly massive in scale, often with size of millions of vertices and edges. The need for a scalable algorithm on large-scale networks poses another challenge for our problem. Motivated by these observations, the main contribution of this work are as follows.\n\n\\begin{itemize}\n\\item We provide three highly scalable algorithms for our problem: Two iterative algorithms, $\\mathtt{IG}$ and $\\mathtt{AT}$, with approximation ratio $O(\\gamma^{-1}(\\ln \\mathtt{T} + \\mathtt{h} \\ln n))$ and $O(\\ln \\mathtt{T} + \\mathtt{h} \\ln n)$ respectively, where $\\gamma$ is a metric measuring the concave property of edge weight functions w.r.t a budget to increase edge weights, \\texttt{h} is the maximum number of edges of a path connecting between a pair in $S$, and $n$ is the number of nodes in $G$; and $\\mathtt{SA}$, a probabilistic approximation algorithm returning $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln \\mathtt{d}}{\\gamma (1-e^{-\\gamma}) (1-\\epsilon)})$ approximation result with high probability, where $\\mathtt{d}$ is the maximum degree of $G$. \n\\item When the edge weight functions are linear w.r.t the cost to increase edge weight, we propose $\\mathtt{LR}$, a randomized rounding algorithm based on LP relaxation of the problem. $\\mathtt{LR}$ provides $O(\\mathtt{h} \\ln n)$ approximation guarantee\n\\item We extensively evaluate our algorithms on both synthetic networks and large-scale, real-world networks. All of our four algorithms are demonstrated to scale to networks with millions of nodes and edges in under a few hours and return nearly optimal solutions. Also, the experiments show the trade-off between our proposed algorithms in terms of runtime and quality of solution.\n\\end{itemize}\n\n\n\n\\textit{Organization.} The rest of this paper is organized as follows. Section \\ref{sec:related} reviews literatures related to our problem. In Section \\ref{sec:problem}, we formally define the problem and discuss its challenges. The four solutions, \\texttt{IG}, \\texttt{AT}, \\texttt{SA} and \\texttt{LR}, are presented in Section \\ref{sec:igta}, \\ref{sec:sa}, \\ref{sec:lr} and \\ref{sec:discussion}, respectively. In Section \\ref{sec:experiment}, we evaluate our algorithms, comparing to heuristic methods for the general case and to algorithms in \\cite{kuhnle2018network} for the special case. Finally, Section \\ref{sec:conclusion} concludes the paper.\n\n\\section{Related works} \\label{sec:related}\n\n\\textbf{Relationship with Kuhnle et al.} \\cite{kuhnle2018network} Kuhnle et al. has studied the Length-Bounded Multicut Problem (\\texttt{LB-MULTICUT}). The objective of this problem is to identify a minimum set of edges whose {\\bf removal} ensures the distance between each pair of nodes of a given set $S$ is no smaller than $\\mathtt{T}$. \\texttt{LB-MULTICUT} is a special case of \\texttt{QoSD} where we restrict to two conditions: 1) the only way to increase an edge weight is making the weight greater than \\texttt{T} and 2) the cost of doing so is uniform among edges.\n\n\nOur \\texttt{QoSD} problem is more general and realistic than \\texttt{LB-MULTICUT}, as briefly discussed earlier. In the adversarial perspective, it is impractical to remove edges out of a network structure. Taking the I-SIG system as an example, the attacker can only damage the network functionality by compromising multiple vehicles, causing severe traffic jams on road network rather than physically damaging road lines. Furthermore, on the Bitcoin-based applications, the Bitcoin protocol only allows a maximum delay of 20 minutes for any packet delivery. For any damage of a P2P connection, the protocol creates another connection to guarantee the connectivity of Bitcoin network. Thus, the \\texttt{LB-MULTICUT} cannot be applied on those two applications.\n\n\nOther than the special case, \\texttt{LB-MULTICUT} and \\texttt{QoSD} are fundamentally different, thus solutions to \\texttt{LB-MULTICUT} are not readily applied to \\texttt{QoSD}. More specifically, Kuhnle {\\em et al.} proposed three approximation algorithms for \\texttt{LB-MULTICUT}, which are $\\mathtt{MIA}$, $\\mathtt{TAG}$, $\\mathtt{SAP}$ \\cite{kuhnle2018network}. We are going to discuss the limits of these algorithms w.r.t solving \\texttt{QoSD}.\n\nThe general idea of \\texttt{MIA} is to find the multicut of sub-graphs of the input network such that each optimal multicut is a lower bound of the optimal solution of \\texttt{LB-MULTICUT} instance. In this solution, the authors exploit the similarity between \\texttt{LB-MULTICUT} and the multicut problem where cutting an edge in a single path is sufficient to disconnect this path. With the multicut solution, \\texttt{MIA} utilizes the $O(n^{11\/23})$ approximation algorithm proposed by Agarwal et al. \\cite{agarwal2007improved}. Thus \\texttt{MIA}'s performance guarantee is bounded by $O(Mn^{11\/23})$ where $M$ is the number of considered subgraphs. Our problem does not require edge removals, so there is not clear connection with multicut. Therefore, we find it infeasible to apply \\texttt{MIA}, even with modification, to solve our problem.\n \nThe next algorithm of \\texttt{LB-MULTICUT} is \\texttt{TAG}. In general, \\texttt{TAG} is a dynamic algorithm, which uses a primal-dual solution to bound the worst-case performance under incremental graph changes and improves the solution in practice by periodic pruning. \\texttt{TAG} utilizes the trait that cutting all edges, which are in the maximal set of disjoint paths connecting target pairs of nodes, is sufficient to disconnect those pairs. However, this solution may not be practical in our problem. Increasing weights of those edges to maximum does not guarantee the shortest paths, which connect target pairs of nodes, no smaller than the threshold $\\mathtt{T}$.\n \n\n\n\n\n\nThe $\\mathtt{SAP}$ algorithm is a greedy, sampling-based solution with an $O(\\mathtt{h} \\log n)$ approximation guarantee ($\\mathtt{h}$ is the maximum number of edges of a single path connecting a pair in $S$), which holds with the probability of at least $1-1\/m$. Our algorithm $\\mathtt{SA}$ is inspired by $\\mathtt{SAP}$ in that we also use a greedy approach based on path samples, generated by using probabilistic hints based upon shortest path computations to guide the sampling. However, since our objective function is non-submodular, we prove that an approximation guarantee of $\\mathtt{SA}$ depends on $\\gamma$, where $\\gamma$ measures the concave property of edge weight functions. Moreover, we boost the process of obtaining a feasible solution by allowing a finite budget of at most $q$ to be added on each step of sampling, where $q$ can be any number. We prove that $q$ does not impact the performance guarantee of $\\mathtt{SA}$.\n\n\\textbf{Optimization on Integer Lattice.} As there is a finite budget to increase the edge weight, we model our problem in a form of minimization problem on Integer Lattice: given a set of functions $\\{f_i | f_i : (\\mathbb{Z}^+ \\cup \\{0\\})^n \\rightarrow \\mathbb{R}^+\\}$ on the Integer Lattice, the objective is to minimize the cardinality of $\\mathbf{x}$ that $f_i(\\mathbf{x}) \\geq \\theta_i$ for all $i$. The optimization on the Integer Lattice has received much attention recently. However, most of those works focus on the maximization version, which asks for maximizing $f(\\mathbf{x})$ under a cardinality constraint $||\\mathbf{x}|| \\leq k$. When $f$ is non-submodular, those works exploits either the submodularity ratio $\\gamma_s$ \\cite{das2011submodular}, generalized curvature $\\alpha$ \\cite{bian2017guarantees} or the diminishing-return ratio $\\gamma_d$ \\cite{kuhnle2018fast,lehmann2006combinatorial} to devise approximation solutions with performance guarantee in terms of those parameters. However, the fact that those parameters can be small and computationally hard to obtain on several real-world objectives raises a concern on those theoretical approximation ratios. For example, Kuhnle et al. \\cite{kuhnle2018fast} proposed a fast maximization of Non-Submodular, Monotonic Functions on the Integer Lattice with approximation ratio $(1-e^{-\\gamma_d \\gamma_s} -\\eta)$ for any $\\eta > 0$. If $\\gamma_d$ or $\\gamma_s$ is $0$, this ratio will be smaller than $0$. In our work, we utilize the concave property of edge weight functions to introduce the \\textit{concave ratio} $\\gamma$, which we use to prove the theoretical guarantee of $\\mathtt{IG}$ and $\\mathtt{SA}$, and bound the sampling size of $\\mathtt{SA}$. $\\gamma$ can be found easily from the derivative of edge weight functions or scanning through all edge weight functions with $O(m)$ time complexity. $\\gamma$ can be small in some cases, so we devise the $\\mathtt{AT}$ solution from an improved $\\mathtt{IG}$ algorithm, which discards the dependence on $\\gamma$ value to obtain better theoretical performance guarantee but a worse runtime in trade-off. \n\n\\textbf{Classical Multicut Problem.} The Multicut problem asks for the minimum number of edges (or nodes) whose removal ensures each pair in $S$ is topologically disconnected. For the edge version in an undirected graph, an $O(\\log k)$ approximation was developed by Garg et al. \\cite{garg1996approximate} by considering multicommodity flow. In directed graphs, Gupta \\cite{gupta2003improved} developed an $O(\\sqrt[]{n})$ approximation algorithm, which was later improved to $O(n^{11\/23})$ by Agarwal et al. \\cite{agarwal2007improved}. These solutions were based on the optimal solution of the linear relaxation modeling the problem instance. Our \\texttt{LR} algorithm was inspired by this approach but we have to deal with the challenge that a LP-optimal value of each edge could be larger than 1. Therefore, any discretization technique of the Multicut problem cannot be directly applied to our problem. We have devised a randomized rounding technique on which we can obtain a feasible solution with high probability while ensuring an $O(\\mathtt{h} \\log n)$ performance ratio.\n\n\n\n\n\\section{Problem Formulation} \\label{sec:problem}\n\n\nIn this section, we formally define the \\textit{Quality of Service Degradation} ($\\mathtt{QoSD}$) problem in the format of cardinality minimization on the Integer Lattice and present challenges on solving $\\mathtt{QoSD}$.  \n\nWe abstract the network using a weighted directed graph $G=(V,E)$ with $|V|=n$ nodes and $|E| = m$ directed edges. Each edge $e$ is associated with a function $f_e: \\mathbb{Z}^\\geq \\rightarrow \\mathbb{Z}^+$ which indicates the weight of $e$ w.r.t a budget to increase weight of $e$. In another word, if we spend $x$ on edge $e$, the weight of edge $e$ will be $f_e(x)$. $f_e$ is monotonically increasing.\n\n\nLet $b_e$ be the maximum possible budget to increase the weight of edge $e$. Denote $\\mathbf{x} = \\{x_1,...x_m\\}$ is a vector where $x_i$ is the budget to increase weight of the $i^\\textnormal{th}$ edge and similarly $\\mathtt{b} = \\{b_1,...b_m\\}$, we have $x_i \\leq b_i$ $\\forall i\\in [1,m]$. $\\mathtt{b}$ is called \\textit{the box}. The overall budget to increase weight of all edges is denoted by $||\\mathbf{x}|| = \\sum_e x_e$. Let $f=\\{f_1,f_2,...f_m\\}$ be a set of edge weight functions. Note that, for simplicity, the notation $e$ is used to present an edge in $E$ and also the index of this edge, i.e. if we write $x_e$, we mean the budget to increase the weight of edge $e$ (to $f_e(x)$) and also the element in $\\mathbf{x}$ that is corresponding to $e$. The same rule is applied with $b_e, f_e$. Also, if we write $e-1$ (or $e+1$), we indicate the edge right next to $e$ on the left (right) in $\\mathbf{x}$.\n\nA path $p = p_0,p_1,...p_l \\in G$ is a sequence of vertices such that $(p_{i-1},p_i) \\in E$ for $i=1,..,l$. A path can also be understood as the sequence of edges $\\{(p_0,p_1), (p_1,p_2),... (p_{k-1}, p_k)\\}$. In this work, a path is used interchangeably as a sequence of edges or a sequence of nodes.\nA \\textit{single path} is a path containing no cycles (i.e repeated vertices). Under a budget vector $\\mathbf{x}$, the length of a path $p$ is defined as $\\sum_{e \\in p} f_e(x_e)$. We now formally define $\\mathtt{QoSD}$ as follows:\n\n\\begin{mydef} \\textnormal{Quality of Service Degradation ($\\mathtt{QoSD}$).} Given a directed graph $G=(V,E)$, a set $f = \\{f_e: \\mathbb{Z}^\\geq \\rightarrow \\mathbb{Z}^+ \\}$ of edge weight functions, a box $\\mathtt{b}$ and a target set $S = \\{(s_1,t_1),...(s_k,t_k)\\}$, determine a minimum budget $||\\mathbf{x}||$ such that under $\\mathbf{x}$, the weighted, shortest-path between each pair in $S$ exceeds a threshold $\\mathtt{T}$. A problem instance may be represented by the tuple $(G,f,\\mathtt{b},S,\\mathtt{T})$\n\\end{mydef}\n\nFor each edge $e \\in E$, let $w_e = f_e(0)$ denote the initial weight of $e$. In this work, we assume $w_e > 0$ for all $e \\in E$, which can be justified by the fact that most networks have positive costs associated with their edges, even when there is no interference from external sources (i.e., propagation delay in communication networks,  processing delay in Blockchains). \n\nLet $\\mathcal{P}_i$ denote a set of simple paths connecting the pair $(s_i,t_i) \\in S$ and $\\sum_{e \\in p} w_e < \\mathtt{T}$ for all $p \\in \\mathcal{P}_i$. Let $\\mathcal{F} = \\cup^k_{i=1} \\mathcal{P}_i$, we call a path $p \\in \\mathcal{F}$ a \\textit{feasible path} and $\\mathcal{F}$ is a set of all feasible paths in $G$. Let $\\mathtt{w} = \\min_e w_e$, it is trivial that the number of edges of a feasible path is upper-bounded by $\\ceil{\\frac{\\mathtt{T}}{\\mathtt{w}}}$. Denote $\\mathtt{h} = \\ceil{\\frac{\\mathtt{T}}{\\mathtt{w}}}$.\n\nUnder $\\mathbf{x}$, given a pair of nodes $(s,t)$, if there exists no single path $p$ from $s$ to $t$ which satisfies $\\sum_{e \\in p} f_e(x_e) < \\mathtt{T}$, we call $s$ is separated from $t$ or the pair $(s,t)$ is \\textit{separated} by $\\mathbf{x}$. Also, given a feasible path $p \\in \\mathcal{F}$, if $\\sum_{e \\in p} f_e(x_e) \\geq \\mathtt{T}$, we call $p$ is \\textit{blocked} by $\\mathbf{x}$ or $\\mathbf{x}$ blocks $p$.\n\nThe $\\mathtt{QoSD}$ problem can be formulated as the follows:\n\n\\begin{align}\n\\min & \\quad ||\\mathbf{x}|| \\\\\n\\text{s.t. } & \\sum_{e \\in p} f_e(x_e) \\geq \\mathtt{T} && \\forall  p \\in \\mathcal{F} \\label{con:path} \\\\\n& x_e \\leq b_e && \\forall e \\in E \\\\\n& x_e \\in \\mathbb{Z}^+ \\cup \\{0\\} && \\forall e \\in E\n\\end{align}\nNote that even $x_e \\in \\mathbb{Z}^+ \\cup \\{0\\}$, this is not an Integer Program because $f_e(x)$ may not be a linear function. \n\nWe can see this formulation as the cardinality minimization on the Integer lattice to satisfy multiple constraints. Before going further, we will look at several notations, mathematical operators on Integer lattice, which will be used along the theoretical proofs of our algorithms. Given $\\mathbf{x} = \\{x_1,...x_m\\}, \\mathbf{y} = \\{y_1,...y_m\\} \\in \\mathbb{Z}^m$, we have:\n\\begin{align*}\n\\mathbf{x} + \\mathbf{y} & = \\{x_1+y_1,...x_m+y_m\\} \\\\\n\\mathbf{x} - \\mathbf{y} & = \\{x_1 - y_1,... x_m - y_m\\} \\\\\n\\mathbf{x} \\wedge \\mathbf{y} & = \\{ \\min(x_1,y_1),...\\min(x_m,y_m)\\} \\\\\n\\mathbf{x} \\vee \\mathbf{y} &= \\{\\max(x_1,y_1),... \\max(x_m,y_m)\\} \\\\\n\\mathbf{x} \/ \\mathbf{y} &= \\{\\max(x_1 - y_1,0),... \\max(x_n - y_n, 0)\\}\\\\\nc \\mathbf{x} & = \\{cx_1,...cx_m\\} \\quad \\forall c \\in \\mathbb{Z}\n\\end{align*}\n\nMoreover, we say $\\mathbf{x} \\leq \\mathbf{y}$ if $x_i \\leq y_i$ for all $i \\in [1,m]$, the similar rule is applied to $<,\\geq,>$. \n\nLet $\\mathbf{s}_i$ be a unit vector with the same dimension with $\\mathbf{x}$, $\\mathbf{s}_i$ has value $1$ in the $i^{th}$ element and $0$ elsewhere. Therefore, we could also write $\\mathbf{x} = \\sum_{i=1}^m x_i \\mathbf{s}_i$. Table \\ref{table:notation} summarizes all the notations we have so far.\n\n\n\\textit{Discussion.} Given an instance of $\\mathtt{QoSD}$ $(G,f,\\mathtt{b},S,\\mathtt{T})$, the optimal solution can be obtained by formulating the problem as the following Integer Programming (IP):\n\\vspace*{-5px}\n\\begin{align}\n\\min & \\quad \\sum_e \\sum^{b_e}_{i=0} i \\cdot y_{e,i} \\\\\n\\text{s.t. } & \\sum_{i=0}^{b_e} y_{e,i} = 1 && \\forall e \\in E  \\label{con:cost}\\\\\n&\\sum_{e \\in p} \\sum_{i=0}^{b_e} f_e(i) \\cdot y_{e,i} \\geq \\mathtt{T} && \\forall  p \\in \\mathcal{F} \\label{con:costpath} \\\\\n& y_{e,i} \\in \\{0,1\\} && \\forall e \\in E, i \\in [0,b_e]\n\\end{align}\nwhere $y_{e,i}$ is an indicator variable which is $1$ if $x_e = i$ and $0$ otherwise. The first constraint (Eq. \\ref{con:cost}) is to guarantee the budget to increase weight of edge $e$ is a value in range $[0,b_e]$ and the second constraint (Eq. \\ref{con:costpath}) is to ensure the length of each feasible path is at least $\\mathtt{T}$. However, solving this IP is extremely expensive. Not only because solving IP is NP-hard (the performance is strongly dependent on which solver is used) but also listing all the paths for the second constraint is very expensive in practice since it requires $O(m^\\mathtt{h})$ in the worst case. Our algorithms are designed to be efficient even when $G$ is large and hence do not require a listing of $\\mathcal{F}$ or an optimal solution of the linear relaxation of this IP formulation.\n\n\\textit{Hardness and Inapproximability.} Since \\texttt{LB-MULTICUT} is a special case of $\\mathtt{QoSD}$, $\\mathtt{QoSD}$ is NP-hard. Furthermore, any inapproximability result of \\texttt{LB-MULTICUT} or the Multicut problem is also the inapproximability of \\texttt{QoSD}. We summarize those results as follows:\n\\begin{itemize}\n\\item Kuhnle et al. \\cite{kuhnle2018network} Let $\\mathtt{T} \\geq 16$. Unless $NP \\subseteq BPP$, there is no polynomial-time algorithm to approximate $\\mathtt{QoSD}$ within a factor of $\\floor{\\frac{\\mathtt{T}}{6}} - 1 - \\epsilon$ for any $\\epsilon > 0$.\n\\item Lee et al. \\cite{lee2016improved}: When $\\mathtt{T}$ is fixed and initial edge weights are uniform, $\\mathtt{QoSD}$ is inapproximable within a factor of $\\Omega(\\sqrt[]{T})$ assuming the Unique Games Conjecture.\n\\item Chawla et al. \\cite{chawla2006hardness}: There exist no $O(\\log \\log n)$-approximation algorithm for $\\mathtt{QoSD}$ unless $P=NP$. \n\\end{itemize}\n\n\\textit{Node version of the problem.} The node version of the $\\mathtt{QoSD}$ problem asks for the minimum budget to increase node weights rather than edge weights in the problem definition above. All our four algorithms can be easily adapted for the node version and keep the same theoretical performance guarantees.\n\n\\begin{table}[t]\n\\centering\n\\caption{Notation} \\label{table:notation}\n\\begin{tabular}{c|l}\n\\toprule\n     Notation & Definition   \\\\ \n \\midrule\n $G=(V,E)$ & Input directed graph \\\\\n $V,E$\t& Vertex and edge sets of $G$, respectively \\\\\n $n,m$\t& Number of vertices, edges in $G$, respectively \\\\\n \\texttt{d}\t& The maximum degree of $G$ \\\\\n $S$\t& The set of target pairs of nodes \\\\\n $k$\t& The number of pairs in target set $S$ \\\\\n $\\mathtt{T}$\t& The threshold on the path length \\\\\n $f_e(x)$\t\t& The weight function of edge $e$ w.r.t a budget $x$ \\\\\n $f$\t\t\t& The set of all weight functions of edges in $G$ \\\\\n $\\mathcal{F}$\t& The set of all feasible paths \\\\\n $\\mathtt{h}$\t& The maximum number of edges of a path in $\\mathcal{F}$ \\\\\n $q$\t& The maximum added cost in each iteration of \\texttt{SA} \\\\\n $\\mathbf{x} = \\{x_1,..x_m\\}$ & The budget vector, $x_i$ is the budget on edge $i$\\\\\n $\\mathbf{s}_i$ & Unit vector, 1 in the $i^{\\textnormal{th}}$ element and $0$ elsewhere \\\\\n $\\gamma$\t\t& The concave ratio of the function set $f$ \\\\\n $\\alpha$\t\t& Bias parameter in the sampling of $\\mathtt{SA}$\\\\\n $\\mathtt{x}^*$\t& Optimal solution to the problem instance\\\\\n $\\mathtt{OPT} = ||\\mathbf{x}^*||$\t& Size of optimal solution\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\\section{Iterative solution} \\label{sec:igta}\n\n\nThere are two challenging tasks to solve the \\texttt{QoSD} problem. The first one is the number of feasible paths could be extremely large, thus we need to avoid listing all the feasible paths as discussed earlier. The second challenge is that the objective function of \\texttt{QoSD} can be non-submodular, depending on the edge weight functions. We handle the challenges via two different algorithms: \\textit{Iterative Greedy} (\\texttt{IG}) and \\textit{Adaptive Trading} (\\texttt{AT}). After the discussion of \\texttt{IG} and \\texttt{AT}, we provide the theoretical analysis and approximation guarantee of both algorithms.\n\n\n\n\nTo tackle the first challenge, instead of listing all feasible paths of the network, we build a set $\\mathcal{P}$ of candidate paths which is a subset of $\\mathcal{F}$ but blocking all paths in $\\mathcal{P}$ is sufficient to separate all pairs in $S$. $\\mathcal{P}$ is built incrementally and iteratively. For each iteration, we find a budget vector $\\mathbf{x} = \\{x_1,..x_m\\}$ to block all paths in $\\mathcal{P}$. Then, we set the length of an edge $e$ to be $f_e(x_e)$. Next, we check whether $\\mathbf{x}$ is sufficient to separate all pairs in $S$ by checking whether there exists the shortest path of a certain pair in $S$ whose length is smaller than $\\mathtt{T}$. If yes, then blocking all paths in $\\mathcal{P}$ is not sufficient to separate all pairs in $S$; we add all the shortest paths of pairs whose length has not exceeded $\\mathtt{T}$ into $\\mathcal{P}$ and continue to the next iteration. If no, then $\\mathbf{x}$ is sufficient to separate all pairs in $S$; we terminate the algorithm and return $\\mathbf{x}$. The full algorithms is represented by Alg. \\ref{alg:iterative}.  \n\n\n\n\\begin{algorithm}[t]\n\t\\caption{Iterative Solution}\n    \\label{alg:iterative}\n\t\\begin{flushleft}\n\t\t\\textbf{Input} $G, f, \\mathtt{b}, \\mathtt{T},S$\\\\\n\t\t\\textbf{Output} QoS adjustment vector $\\mathbf{x}$\n\t\\end{flushleft}\n    \\begin{algorithmic}[1]\n    \t\\State $\\mathcal{P} = \\emptyset$\n\t\t\\While{There exists path $p \\in \\mathcal{F}$ whose length $< \\mathtt{T}$} \\label{line:outer_it}\n        \t\\State $\\mathcal{P} \\leftarrow \\mathcal{P} \\cup $ potentialPaths($G,S,\\mathbf{x}$). \\label{line:add_path}\n            \\State $\\mathbf{x} = \\{0\\}^m$\n            \\State Find $\\mathbf{x}$ to block all paths in $\\mathcal{P}$ \\label{line:block_path}\n        \\EndWhile\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return $\\mathbf{x}$}\n     \\end{flushleft}\n\\end{algorithm}\n\n\n\nSince the maximum number of edges of a feasible path could reach up to $\\mathtt{h} = \\floor{\\frac{\\mathtt{T}}{\\mathtt{w}}}$, the number of feasible paths of the network $G$ is upper bounded by $O(n^\\mathtt{h})$. Because we guarantee there should be at least a feasible path is added into $\\mathcal{P}$ in each iteration (line \\ref{line:add_path} Alg. \\ref{alg:iterative}), the number of iterations in Alg. \\ref{alg:iterative} is at most $O(n^\\mathtt{h})$. This is a large number and comparable to the case if we tried to enumerate all feasible paths. However in experiment, we found that the number of iterations is much smaller even on large and highly dense networks. \n\n\n\\begin{algorithm}[H]\n\t\\caption{\\textsc{potentialPaths}$(G,S,\\mathbf{x})$}\n\t\\label{alg:shortest_path}\n    \\begin{flushleft}\n    \t\\textbf{Input} $G,S, \\mathbf{x}$\\\\\n\t\\textbf{Output} Set $\\mathcal{P}$ of paths whose lengths is smaller than $\\mathtt{T}$\n    \\end{flushleft}\n    \\begin{algorithmic}[1]\n\t\t\\State Assign edge $e$ length is $f_e(x_e)$ $\\forall$  $e \\in E$\n        \\For{each pair $(s,t) \\in S$}\n        \t\\State $p \\leftarrow $ shortest path between $s$ and $t$\n            \\If{length of $p$ is smaller than $\\mathtt{T}$}\n            \t\\State $\\mathcal{P} = \\mathcal{P} \\cup p$\n            \\EndIf\n        \\EndFor\n    \\end{algorithmic}\n    \\begin{flushleft}\n    \t\\textbf{Return } $\\mathcal{P}$\n    \\end{flushleft}\n\\end{algorithm}\n\n\\begin{lemma}\nThe approximation guarantee of Alg. \\ref{alg:iterative} equals to the approximation guarantee of the algorithm that finds $\\mathbf{x}$ to block all paths in $\\mathcal{P}$\n\\end{lemma}\n\\begin{proof}\nSince $\\mathcal{P}$ is a subset of all feasible paths in $G$, the optimal solution to block all feasible paths is also a feasible solution to block all paths in $\\mathcal{P}$. Therefore, the optimal solution to block all paths in $\\mathcal{P}$ is at most the size of the optimal solution of \\texttt{QoSD}. Denote $\\mathbf{x}^o$ and $\\mathbf{x}^*$ as the optimal solutions to block paths in $\\mathcal{P}$ and $\\mathcal{F}$ respectively. Assume the algorithm in line \\ref{line:block_path} of Alg. \\ref{alg:iterative} returns $\\alpha$-approximation result. We have $||\\mathbf{x}|| \\leq \\alpha \\cdot ||\\mathbf{x}^o|| \\leq \\alpha \\cdot ||\\mathbf{x}^*||$. And since finally $\\mathbf{x}$ is a feasible solution to our problem, then the output $\\mathbf{x}$ of Alg. \\ref{alg:iterative} is within $\\alpha$  factor to optimal solution $\\mathbf{x}^*$.   \n\\end{proof}\n\nNow let us discuss the the second challenge: how to block all paths in $\\mathcal{P}$, line \\ref{line:block_path} of Alg. \\ref{alg:iterative}. To address this, we propose two algorithms, \\textit{Greedy} and \\textit{Adaptive Trading}. Before delving into the details of each algorithm, we introduce the parameter $\\gamma$, which is used to measure the concave property of weight functions. $\\gamma$ would be utilized on performance analysis for our algorithms. \n\n\n\\subsection{Concave property of weight functions}\n\nThe concave ratio of a set of functions is defined as follows:\n\n\\begin{mydef}\n(\\textnormal{Concave ratio}) The concave ratio of a set $F$ of non-negative functions is the largest scalar $\\gamma \\in [0,1]$ such that:\n\\begin{align}\nf(x + 1) - f(x) \\geq \\gamma \\cdot \\big(f(y + 1) - f(y) \\big)  \n\\end{align}\nFor all $f \\in F$ and $0 \\leq x \\leq y$\n\\end{mydef}\n\nIn our problem, the set of non-negative functions contains all weight functions of edges in $G$. Therefore, for simplicity, we denote $\\gamma$ as the \\textit{concave ratio} of these set of weight functions. Now, we will utilize $\\gamma$ to get several useful exploration for our solutions. First, given a path $p$ and a vector $\\mathbf{x}$, define: \n\\begin{align}\n\\mathtt{r}(p, \\mathbf{x}) = \\min(\\mathtt{T}, \\sum_{e \\in p} f_e(x_e)) \\label{equ:define_r}\n\\end{align}\n\nLet $g(\\mathcal{P},\\mathbf{x})$ be an arbitrary linear combination of $\\mathtt{r}(p,\\mathbf{x})$ for all $p \\in \\mathcal{P}$. $g(\\mathcal{P},\\mathbf{x})$ could be presented as follows:\n\\begin{align}\ng(\\mathcal{P}, \\mathbf{x}) = \\sum_{p \\in \\mathcal{P}} \\beta_p \\mathtt{r}(p, \\mathbf{x}) \\quad \\quad \\beta_p \\in \\mathbb{R}^+ \\textnormal{ } \\forall \\textnormal{ } p \\in \\mathcal{P}\n\\end{align}\nGiven a vector $\\mathbf{z}$, define:\n\\begin{align}\n\\Delta_{\\mathbf{z}} g(\\mathcal{P},\\mathbf{x}) = g(\\mathcal{P}, \\mathbf{x} + \\mathbf{z}) - g(\\mathcal{P}, \\mathbf{x}) \\label{equ:delta_g}\n\\end{align}\nWe have the following lemma.\n\n\\begin{lemma} \\label{lemma:concave}\nGiven two budget vectors $\\mathbf{x}, \\mathbf{y}$ where $\\mathbf{x} \\leq \\mathbf{y}$ and a unit vector $\\mathbf{s}$, we have:\n\\begin{align*}\n\\Delta_\\mathbf{s} g(\\mathcal{P}, \\mathbf{x}) \\geq \\gamma \\Delta_\\mathbf{s} g(\\mathcal{P}, \\mathbf{y})\n\\end{align*}\n\\end{lemma}\n\n\\begin{skproof}\nWithout lost of generality, we assume $\\mathbf{s} = \\mathbf{s}_i$, a unit vector which has value $1$ at the $i^\\textnormal{th}$ element and $0$ elsewhere. We prove that: given a feasible path $p$, the marginal gain of $\\mathtt{r}(p,\\mathbf{x})$ by $\\mathbf{s}_i$ is at least $\\gamma$ times the marginal gain of $\\mathtt{r}(p, \\mathbf{y})$ by $\\mathbf{s}_i$. \n\nBy definition, the $\\mathtt{r}(p,\\cdot)$ value of any budget vector cannot exceed $\\mathtt{T}$. Also, $\\mathtt{r}(p,\\mathbf{u}) \\leq \\mathtt{r}(p, \\mathbf{v})$ if $\\mathbf{u} \\leq \\mathbf{v}$. Therefore, we consider three different cases: (1) $\\mathtt{r}(p,\\mathbf{x}) < \\mathtt{r}(p, \\mathbf{x} + \\mathbf{s}_i) < \\mathtt{T}$; (2) $\\mathtt{r}(p, \\mathbf{x}) < \\mathtt{r}(p, \\mathbf{x} + \\mathbf{s}_i) = \\mathtt{T}$; and (3) $\\mathtt{r}(p, \\mathbf{x}) = \\mathtt{r}(p, \\mathbf{x} + \\mathbf{s}_i) = \\mathtt{T}$. All three cases guarantee $\\mathtt{r}(p,\\mathbf{x} + \\mathbf{s}_i) - \\mathtt{r}(p, \\mathbf{x}) \\geq \\gamma \\big( \\mathtt{r}(p,\\mathbf{y} + \\mathbf{s}_i) - \\mathtt{r}(p, \\mathbf{y}) \\big)$. Since $g(\\mathcal{P},\\mathbf{x})$ is a linear combination of $\\mathtt{r}(p, \\mathbf{x})$, the lemma follows.\n\\end{skproof}\n\n\n\\begin{lemma} \\label{lemma:concave_3}\nGiven three budget vectors $\\mathbf{x},\\mathbf{y}, \\mathbf{z}$ where $\\mathbf{x} \\leq \\mathbf{y}$ we have:\n\\begin{align*}\n\\Delta_{\\mathbf{z}} g(\\mathcal{P}, \\mathbf{x}) \\geq \\gamma \\Delta_{\\mathbf{z}} g(\\mathcal{P}, \\mathbf{y})\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathbf{z} = \\sum_{i=1}^{||\\mathbf{z}||} \\mathbf{s}_i$ where $\\mathbf{s}_i$ is a unit vector, we have:\n\\begin{align*}\n&\\Delta_\\mathbf{z} g(\\mathcal{P}, \\mathbf{y}) = \\sum_{j=1}^{||\\mathbf{z}||} \\Delta_{\\mathbf{s}_j} g(\\mathcal{P},\\mathbf{y} + \\mathbf{s}_1 + ... + \\mathbf{s}_{j-1}) \\leq \\frac{1}{\\gamma} \\Big( \\sum_{j=1}^{||\\mathbf{z}||} \\Delta_{\\mathbf{s}_j} g(\\mathcal{P},\\mathbf{x} + \\mathbf{s}_1 + ... + \\mathbf{s}_{j-1}) \\Big) \\leq \\frac{1}{\\gamma} \\Delta_{\\mathbf{z}} g(\\mathcal{P}, \\mathbf{x})\n\\end{align*}\nwhich completes the proof.\n\\end{proof}\n\nNote that $\\mathtt{r}(p, \\mathbf{x}) \\leq \\mathtt{T}$. A budget vector $\\mathbf{x}$ is sufficient to block all paths in $\\mathcal{P}$ iff $\\mathtt{r}(p, \\mathbf{x}) = \\mathtt{T}$ for all $p \\in \\mathcal{P}$. Therefore, to block all paths in $\\mathcal{P}$, we find the minimum $||\\mathbf{x}||$ such that:\n\\begin{align}\n\\mathtt{D}(\\mathcal{P}, \\mathbf{x}) = \\sum_{p \\in \\mathcal{P}} \\mathtt{r}(p, \\mathbf{x}) = |\\mathcal{P}| \\cdot \\mathtt{T}\n\\end{align}\nIn the next subsections, we devise two approximation algorithms to find such $\\mathbf{x}$ and provide their performance guarantees.\n\n\\subsection{Iterative Greedy algorithm} \nThe first algorithm to block all paths in $\\mathcal{P}$ is the \\textit{iterative greedy} algorithm (\\texttt{IG}). The general idea is that: we iteratively add a unit vector $\\mathbf{s}$ into $\\mathbf{x}$, which maximizes the marginal gain $\\Delta_\\mathbf{s} \\mathtt{D}(\\mathcal{P}, \\mathbf{x})$, until $\\mathbf{x}$ is sufficient to block all paths in $\\mathcal{P}$. Hence, the final overall budget ($||\\mathbf{x}||$) is equal to the number of iterations of the algorithm. \\texttt{IG} is fully presented by Alg. \\ref{alg:greedy_blocking}. \n\nHowever, the objective function $\\mathtt{D}(\\mathcal{P}, \\mathbf{x})$ is neither submodular nor supermodular w.r.t $\\mathbf{x}$. If each edge weight function is concave, $\\mathtt{D}(\\mathcal{P},\\cdot)$ exhibits a \\textit{submodular behavior}. On the other hand, if each weight function is convex, then $\\mathtt{D}(\\mathcal{P}, \\mathbf{x})$ can be much more than the sum of $\\mathtt{D}(\\mathcal{P}, \\mathbf{s})$ values of unit vectors $\\mathbf{s}$ constituting $\\mathbf{x}$, which is a \\textit{supermodular behavior}. The non-submodularity of $\\mathtt{D}(\\mathcal{P}, \\cdot)$ means that the $\\mathbf{x}$ returned by \\texttt{IG} may not have an $O(\\log n)$ approximation ratio. Actually the concave ratio $\\gamma$ plays an important role on the performance guarantee of \\texttt{IG}, which is proved theoretically by Theorem \\ref{theorem:greedy_approx} and would be further illustrated in the experimental evaluation.\n\n\n\n\n\n \n\n\n\\begin{algorithm}[h]\n\t\\caption{Greedy blocking paths (\\texttt{IG})}\n    \\label{alg:greedy_blocking}\n    \\begin{flushleft}\n    \\textbf{Input} $G, f, \\mathtt{b}, \\mathtt{T}, \\mathcal{P}$\\\\\n\t\\textbf{Output} a cost vector $\\mathbf{x}$\n    \\end{flushleft}\n\t\\begin{algorithmic}[1]\n    \t\\State $\\mathbf{x} = \\{0\\}^m$\n\t\t\\While{$\\mathtt{D}(\\mathcal{P},\\mathbf{x}) \\leq |\\mathcal{P}| \\mathtt{T}$} \\label{line:inner_it_greedy}\n        \t\\For{each unit vector $\\mathbf{s}$}\n            \t\\State $\\Delta_\\mathbf{s} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}) = \\mathtt{D}(\\mathcal{P}, \\mathbf{x} + \\mathbf{s}) - \\mathtt{D}(\\mathcal{P}, \\mathbf{x})$\n            \\EndFor\n            \\State $\\mathbf{x} = \\mathbf{x} + \\textnormal{argmax}_\\mathbf{s} \\Delta_\\mathbf{s} \\mathtt{D}(\\mathcal{P}, \\mathbf{x})$\n        \\EndWhile\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return $\\mathbf{x}$}\n     \\end{flushleft}\n\\end{algorithm}\n\n\n\\begin{theorem} \\label{theorem:greedy_approx}\n\\texttt{IG} returns a solution within $O(\\gamma^{-1}(\\mathtt{h} \\ln n + \\ln \\mathtt{T}))$ factor of the optimal solution for blocking all paths in $\\mathcal{P}$. \n\\end{theorem}\n\n\\begin{skproof}\nDenote $\\mathbf{x}^*$ as an optimal solution to the \\texttt{QoSD} instance ($||\\mathbf{x}^*|| = \\mathtt{OPT}$). Denote $\\mathbf{x}_i$ as our obtained solution before the $i^\\textnormal{th}$ iteration in Alg. \\ref{alg:greedy_blocking}. The key of our proof is that: the gap between $|\\mathcal{P}| \\mathtt{T}$ and $\\mathtt{D}(\\mathcal{P},\\mathbf{x})$ will be reduced after each iteration by a factor at least $1-\\frac{\\gamma}{\\mathtt{OPT}}$. To be specific:\n\\begin{align*}\n|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) \\leq (1-\\frac{\\gamma}{\\mathtt{OPT}}) (|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i))\n\\end{align*}\nThis was proved by using the property of concave ratio from lemma \\ref{lemma:concave} and the greedy selection. \n\nFurthermore, since there should exist at least a feasible path $p \\in \\mathcal{P}$ such that $\\mathtt{r}(p, \\mathbf{x}) \\leq \\mathtt{T} - 1$ before the final iteration of the algorithm, we prove that the number of iterations is upper bounded by $O(\\frac{\\ln |\\mathcal{P}| \\mathtt{T}}{\\gamma\/\\mathtt{OPT}})$. The theorem follows as the number of iterations is equal to $||\\mathbf{x}||$.\n\\end{skproof}\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Adaptive Trading algorithm} \nThe concave ratio of the edge weight functions could be very small if the weight functions are convex, which makes the approximation guarantee of \\texttt{IG} undesirable. Therefore, in this section, we propose a solution whose performance guarantee does not depend on the concave ratio $\\gamma$. We name this algorithm \\textit{Adaptive Trading} (\\texttt{AT}). \n\nThe algorithm still works in the iterative manner and terminates only when the desired $\\mathbf{x}$ is found, but different from \\texttt{IG} on how the solution $\\mathbf{x}$ is improved in each iteration. To be specific, in each iteration, the algorithm finds an amount of additional budget to increase the weight of an edge such that maximize the ratio between the increasing amount of $\\mathtt{D}(\\mathcal{P}, \\mathbf{x})$ and the additional budget. Therefore, in each iteration, the additional budget could be bigger than $1$. To find such amount, the simplest way is to scan through all possible amounts of additional budget of each edge. Note that the maximum budget which can be added to increase weight of edge $e$ is upper bounded by $b_e$. Therefore, the computation complexity in each iteration of $\\mathtt{AT}$ is upper bounded by $O(||\\mathtt{b}||)$. Denote $\\mathbf{u}(e,i) \\in \\mathbb{R}^m$ as a vector where the element corresponding to edge $e$ has value $i$ and other elements are $0$. \\texttt{AT} is fully presented in Alg. \\ref{alg:trunk_adding} and its approximation guarantee is provided by Theorem \\ref{theorem:trunk_approx}.    \n\n\n\n\\begin{algorithm}[h]\n\t\\caption{Adaptive Trading solution (\\texttt{AT})}\n    \\label{alg:trunk_adding}\n    \\begin{flushleft}\n    \\textbf{Input}  $G, f, \\mathtt{b}, \\mathtt{T}, \\mathcal{P}$\\\\\n\t\\textbf{Output} QoS adjustment vector $\\mathbf{x}$\n    \\end{flushleft}\n\t\\begin{algorithmic}[1]\n    \t\\State $\\mathcal{P} = \\emptyset$\n\t\t\\While{$\\mathtt{D}(\\mathcal{P},\\mathbf{x}) \\leq |\\mathcal{P}| \\mathtt{T}$} \\label{line:inner_it_trunk}\n        \t\\For{each edge $e \\in E$}\n            \t\\State $z_e = \\textnormal{argmax}_{z} \\frac{\\Delta_{\\mathbf{u}(e,z)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x})}{z}$\n            \\EndFor\n            \\State $\\mathbf{x} = \\mathbf{x} + \\textnormal{argmax}_{\\mathbf{u}(e,z_e)} \\frac{\\Delta_{\\mathbf{u}(e,z_e)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x})}{z_e} $\n        \\EndWhile\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return $\\mathbf{x}$}\n     \\end{flushleft}\n\\end{algorithm}\n\n\n\\begin{theorem} \\label{theorem:trunk_approx}\n\\texttt{AT} returns a solution within $O(\\mathtt{h} \\ln n + \\ln \\mathtt{T})$ factor of the optimal solution for blocking all paths in $\\mathcal{P}$. \n\\end{theorem}\n\n\\begin{skproof}\nDenote $\\mathbf{x}_i = \\{x_1,...x_m\\}$ as our obtained solution before the $i^\\textnormal{th}$ iteration in Alg. \\ref{alg:trunk_adding}. Let $\\mathbf{x}^o_i = \\{x_1^o, ... x_m^o\\}$ be an optimal solution which is in addition to $\\mathbf{x}_i$ to block all paths in $\\mathcal{P}$. Denote $\\mathbf{v}(e) = \\{x_1,..x_{e-1},x_e + x_e^o,...x_m + x_m^o\\}$. Trivially, $\\mathbf{v}(1) = \\mathbf{x}_i + \\mathbf{x}^o$ and $\\mathbf{v}(m+1) = \\mathbf{x}_i$. Let $\\mathbf{u}(e_i,j_i)$ be a vector we add into solution $\\mathbf{x}_i$ in the $i^\\textnormal{th}$ iteration. The key of our proof is that the following inequality is always guaranteed after each iteration.\n\\begin{align}\n\\frac{\\Delta_{\\mathbf{u}_i(e_i,j_i)} \\mathtt{D}(\\mathcal{P},\\mathbf{x}_i)}{j_i} \\geq \\frac{\\mathtt{D}(\\mathcal{P},\\mathbf{v}(e)) - \\mathtt{D}(\\mathcal{P},\\mathbf{v}(e+1))}{x_e^o} \\label{equ:key_trunk}\n\\end{align}\nfor any $e \\in E$. This is proved by utilizing the monotonicity of $\\mathtt{r}(p,\\mathbf{x})$ w.r.t $\\mathbf{x}$ and the trait that the selection of our algorithm ensures $\\Delta_{\\mathbf{u}(e, q)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) \\leq \\frac{q}{j_i} \\Delta_{\\mathbf{u}(e_i,j_i)} \\mathtt{D}(\\mathcal{P},\\mathbf{x}_i)$ for any $e \\in E$ and $q \\in \\mathbb{Z}^+$.\n\nFurthermore, the Eq. \\ref{equ:key_trunk} helps us to prove that: the gap between $|\\mathcal{P}|\\mathtt{T}$ and $\\mathtt{D}(\\mathcal{P}, \\mathbf{x})$ will be reduced after each iteration by a factor at least $1-\\frac{j_i}{\\mathtt{OPT}}$. To be specific:\n\\begin{align*}\n|\\mathcal{P}|\\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_{i+1}) \\leq (1 - \\frac{j_i}{\\mathtt{OPT}}) \\big(|\\mathcal{P}| \\mathtt{T} - \\mathtt{D}(\\mathcal{P}, \\mathbf{x}_i) \\big)\n\\end{align*}\nsince there should exist at least a feasible path $p \\in \\mathcal{P}$ such that $\\mathtt{r}(p, \\mathbf{x}) \\leq \\mathtt{T} - 1$ before the final iteration of the algorithm, utilizing Cauchy theorem \\cite{cauchyInequality}, we bound the budget $||\\mathbf{x}||$ by $\\mathtt{OPT} \\cdot O(\\ln |\\mathcal{P}| \\mathtt{T})$. Since \n$|\\mathcal{P}| \\leq n^\\mathtt{h}$, the theorem follows.\n\\end{skproof}\n\n\n\n\n\n\n\n\n\n\\section{Sampling Approach} \\label{sec:sa}\n\nIn this section, we introduce a sampling solution \\texttt{SA} to \\texttt{QoSD} which has $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln d}{\\gamma (1-e^-{\\gamma})(1-\\epsilon)})$ approximation guarantee with probability at least $1-\\delta$ where $\\epsilon, \\delta > 0$ are arbitrarily small numbers. \\texttt{SA} runs in polynomial time when the parameter $\\mathtt{T}$ is fixed.\n\nWe define a \\textit{blocking metric} of a budget vector $\\mathbf{x}$ as follows\n\\begin{align*}\nB(\\mathbf{x}) = \\sum_{p \\in \\mathcal{F}} \\mathtt{r}(p, \\mathbf{x})\n\\end{align*}\nIt is trivial that $\\mathbf{x}$ blocks all pairs in $\\mathcal{F}$ iff $B(\\mathbf{x}) = |\\mathcal{F}| \\cdot \\mathtt{T}$. \n\nIn essence, \\texttt{SA} attempts to minimize $||\\mathbf{x}||$ while ensuring $B(\\mathbf{x}) = |\\mathcal{F}| \\cdot \\mathtt{T}$. To do so, \\texttt{SA} works in the greedy manner as follows: in each iteration, \\texttt{SA} finds a budget vector $\\mathbf{v} = \\{v_1,...v_m\\}$, $||\\mathbf{v}|| \\leq q$, to add into $\\mathbf{x}$ which maximizes $B(\\mathbf{x} + \\mathbf{v})$. Rather than an expensive listing of $\\mathcal{F}$, an estimator is employed by path sampling procedure to find the vector $\\mathbf{v}$. This process is repeated until the budget vector $\\mathbf{x}$ is sufficient to block all paths in $\\mathcal{F}$. \\texttt{SA} is fully presented in Alg. \\ref{alg:sampling_solution}.\n\n\n\n\n\\begin{algorithm}[h]\n\t\\caption{Sampling Algorithm (\\texttt{SA})}\n    \\label{alg:sampling_solution}\n\t\\begin{flushleft}\n\t\\textbf{Input} $G, S, \\mathtt{T}, f, \\mathtt{b}$ and $q, \\epsilon, \\delta$\\\\\n\t\\textbf{Output} cost vector $\\mathbf{x}$\n\t\\end{flushleft}\n    \\begin{algorithmic}[1]\n        \\State Initiate $\\mathbf{x} = \\{0\\}^m$\n        \\While{There exists a path $p \\in \\mathcal{F}$ whose length $< \\mathtt{T}$}\n            \t\\State Generate $\\mathcal{P} = $ $\\mathcal{N}(q,\\epsilon, \\delta\/||\\mathtt{b}||)$ sample paths $p_1,p_2,...$\n            \t\\State Greedily select $\\mathbf{v}$ ($||\\mathbf{v}|| \\leq q$) that maximizes $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v})$\n                \\State $\\mathbf{x} = \\mathbf{x} + \\mathbf{v}$\n        \\EndWhile\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return } $\\mathbf{x}$\n     \\end{flushleft}\n\\end{algorithm}\n\nSince we will not list $\\mathcal{F}$, the questions now are (1) how to estimate $B(\\cdot)$; and (2) how many sample paths should be generated to bound the error between the estimator of $B(\\cdot)$ and its actual value. In sub-section \\ref{subsec:estimate}, we define the estimator $\\hat{B}(\\cdot)$ employed in each iteration of Alg. \\ref{alg:sampling_solution}. We provide the approximation guarantee of greedily selection on sub-section \\ref{subsec:greedy_sa}. Sub-section \\ref{subsec:no_sampling} provides the lower bound on the number of sampling paths to bound the error. We then put all the results together to obtain the performance guarantee of \\texttt{SA}.\n\n\n\n\\subsection{Estimator} \\label{subsec:estimate}\n\nLet an instance $(G,f,\\mathtt{b},\\mathtt{T})$ of \\texttt{QoSD} be given. Denote $\\mathcal{J}$ as a set of all single paths in $G$. For each $p \\in \\mathcal{J}$, define:\n\n\n\n\\begin{align*}\n\\mathtt{R}(p,\\mathbf{x}) =\n\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\mathtt{r}(p,\\mathbf{x})  & \\mbox{if } p \\in \\mathcal{F} \\\\\n\t\t\t0 & \\mbox{otherwise} \n\t\t\\end{array}\n\t\\right. \\label{pf1}\n\\end{align*}\n\nIt is trivial that $\\sum_{p \\in \\mathcal{J}} \\mathtt{R}(p,\\mathbf{x}) = \\sum_{p \\in \\mathcal{F}} \\mathtt{r}(p, \\mathbf{x})$. Inspired by the estimation on the number of paths in a graph \\cite{roberts2007estimating}, we define the estimator of $B(\\mathbf{x})$ in the following way: Given a probability distribution $\\rho$ on $\\mathcal{J}$ such that $\\rho(p) > 0$ for all $p \\in \\mathcal{F}$. Let $\\mathcal{P} = \\{p_1,p_2, ... p_l\\}$ be a set of $l$ paths samples from $\\rho$, $B(\\mathbf{x})$ could be estimated by\n\n\\begin{align*}\n\\hat{B}(\\mathcal{P},\\mathbf{x}) = \\frac{1}{l} \\sum_{i=1}^l \\frac{\\mathtt{R}(p_i,\\mathbf{x})}{\\rho(p_i)}\n\\end{align*}\n\n\n\n\n\\begin{lemma}\n$\\hat{B}(\\mathcal{P}, \\mathbf{x})$ is an unbiased estimator of $B(\\mathbf{x})$\n\\end{lemma}\n\n\\begin{proof}\n\\begin{align*}\n&\\mathbb{E}[\\hat{B}(\\mathcal{P}, \\mathbf{x})] = \\mathbb{E}\\Big[ \\frac{\\mathtt{R}(p,\\mathbf{x})}{\\rho(p)} \\Big] = \\sum_{p \\in \\mathcal{J}} \\frac{\\mathtt{R}(p,\\mathbf{x})}{\\rho(p)} \\cdot \\rho(p) = \\sum_{p \\in \\mathcal{J}} \\mathtt{R}(p,\\mathbf{x}) = \\sum_{p \\in \\mathcal{F}} \\mathtt{r}(p, \\mathbf{x})\n\\end{align*}\n\\end{proof}\n\n\n\nTo sampling paths, we utilize the following biased, self-avoiding random walk sampling technique, which was once proposed by Kuhnle \\cite{kuhnle2018network}. First, we randomly select a pair $(s,t)$ from $S$ and put $s$ into the sample path $p$. Considering in a certain moment, $p=\\{s,..u\\}$ ($u$ is called a tail node of $p$ at this time). The NeighborSelection procedure would select a node among the out-going neighbors of $u$ to add into $p$. The selection is as follows:  Let $\\mathcal{T}$ be the shortest-path tree directed towards $t$. Let $v$ be the parent of $u$ in $\\mathcal{T}$. If $N(u)\/p = \\{v\\}$, then the next node we add into $p$ is $v$. If $v \\in N(u)\/p$, we select $v$ with probability $\\alpha$ and the other nodes in $N(u)\/p$ with probability of $\\frac{1-\\alpha}{|N(u)\/p|-1}$. If $v \\not\\in N(u)\/p$, we select the next node uniform randomly among $N(u)\/p$. The sampling procedure ends when we meet the node $t$ or the length of $p$ exceeds $\\mathtt{T}$. With the path-sampling procedure defined, given a path $p$, we could easily find $\\rho(p)$. Also, $\\rho(p) > 0$ for all $p \\in \\mathcal{F}$. The sampling technique is fully presented in Alg. \\ref{alg:sampling_path}.\n\n\n\n\n\n\\begin{algorithm}[t]\n\t\\caption{Sampling path}\n    \\label{alg:sampling_path}\n\t\\begin{flushleft}\n\t\t\\textbf{Input} $G, S, \\mathtt{T},\\mathbf{x}$ \\\\\n\t\t\\textbf{Output} Sample path $p$\n\t\\end{flushleft}\n   \t\\begin{algorithmic}[1]\n\t\t\\State $(s,t) \\leftarrow $ randomly select a transaction. \\label{line:select_pair}\n        \\State $p \\leftarrow \\{s\\}$\n        \\Do\n        \t\\State $u = $ tail$(p)$\n            \\State Let $N(u)$ be the set of outcoming neighbors of $u$\n            \\State $p \\leftarrow p \\textnormal{ } \\cup $ NeighborSelection$(N(u),p, (s,t))$ \\label{line:neighbor_select}\n        \\doWhile{$u \\neq t$ and $\\sum_{e \\in p} f_e(x_e) < \\mathtt{T}$}\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return $p$}\n     \\end{flushleft}\n\\end{algorithm}\n\n\n\n\\subsection{Greedy selection on the estimator} \\label{subsec:greedy_sa}\nHaving defined the estimator $\\hat{B}(\\mathcal{P},\\mathbf{x})$ and the path sampling procedure, we now find the budget vector $\\mathbf{v}$, $||\\mathbf{v}|| \\leq q$, to maximize $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v})$. $\\mathbf{v}$ is found in the greedy manner as follows: we run in $q$ iterations and in each iteration, selecting the unit vector that maximizes the marginal gain of $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v})$. Since it is trivial, we will not write down the pseudo-code on how we find $\\mathbf{v}$. \n\nThe question now is what approximation guarantee $\\mathbf{v}$ can provide? Note that $\\hat{B}(\\mathcal{P}, \\mathbf{x})$ is a finite combination of functions $\\mathtt{r}(p,\\mathbf{x})$  with $p \\in \\mathcal{P}$. Hence, $\\hat{B}(\\mathcal{P}, \\mathbf{x})$ is submodular if all weight functions are concave and supermodular if they are convex. So maximizing $\\hat{B}(\\mathcal{P}, \\mathbf{x + v})$ using greedy algorithm may not return $1-1\/e$ approximation result. Therefore, similar to \\texttt{IG}, we use the concave ratio $\\gamma$ to obtain the performance guarantee of the greedy selection to maximize $\\hat{B}(\\mathcal{P},\\mathbf{x})$. \n\nDenote $\\mathbf{v}^o = \\sum_{i=1}^q \\mathbf{u}_i$ as an optimal solution that maximizes $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v})$, where $\\mathbf{u}_i$ is a unit vector ($i \\in [0,m]$). Lemma \\ref{lem:greedy_sampling} provides approximation guarantee of the greedy selection. \n\n\n\n\n\n\n\\begin{lemma} \\label{lem:greedy_sampling}\n\\begin{align*}\n\\Delta_{\\mathbf{v}}\\hat{B}(\\mathcal{P}, \\mathbf{x}) \\geq (1-e^{-\\gamma}) \\Delta_{\\mathbf{v}^o} \\hat{B}(\\mathcal{P}, \\mathbf{x}^o)\n\\end{align*}\n\\end{lemma}\n\n\\begin{skproof}\nDenote $\\mathbf{v}_i$ as the budget vector $\\mathbf{v}$ after greedily selecting first $i$ unit vectors. The key of the proof comes from the following inequality:\n\\begin{align*}\n&\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i+1}) \\leq (1-\\frac{\\gamma}{q}) \\Big(\\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}^o) - \\hat{B}(\\mathcal{P},\\mathbf{x} + \\mathbf{v}_{i})\\Big)\n\\end{align*}\nThis inequality is proved by using the property of $\\gamma$ from lemma \\ref{lemma:concave} and the trait that $\\hat{B}(\\mathcal{P}, \\mathbf{x})$ is monotone w.r.t $\\mathbf{x}$. Using this inequality, we prove that\n\\begin{align*}\n&\\Delta_{\\mathbf{v}}\\hat{B}(\\mathcal{P},\\mathbf{x}) \\geq (1-(1-\\frac{\\gamma}{q})^q) \\Delta_{\\mathbf{v}^o}\\hat{B}(\\mathcal{P},\\mathbf{x}) \\geq (1-e^{-\\gamma}) \\Delta_{\\mathbf{v}^o}\\hat{B}(\\mathcal{P},\\mathbf{x})\n\\end{align*}\nin which the lemma follows.\n\\end{skproof}\n\n\n\\subsection{Sample size and Performance guarantee} \\label{subsec:no_sampling}\nWe have proved the performance guarantee of the additional budget vector $\\mathbf{v}$ to maximize $\\hat{B}(\\mathcal{P}, \\mathbf{x} + \\mathbf{v})$. The question now is: what is the size of $\\mathcal{P}$ to bound the error between $\\Delta_\\mathbf{v} \\hat{B}(\\mathcal{P}, \\mathbf{x})$ and $\\Delta_\\mathbf{v} B(\\mathbf{x})$? In this part, we will answer this question. Then, putting together with the performance guarantee of selecting $\\mathbf{v}$ on $\\mathcal{P}$, we provide the performance guarantee of \\texttt{SA}.\n\nTo find the minimum number of samples, we utilize the following Chernoff Bound theory. \n\n\n\\begin{theorem}\n(\\textnormal{Chernoff Bound theorem} \\cite{hoeffding1963probability}) Let $X_1,X_2,...X_n$ be random variables such that $a \\leq X_i \\leq b$ for all $i$. Let $X = \\sum_{i=1}^n X_i$ and set $\\mu = \\mathbb{E}(X)$. Then for all $\\epsilon > 0$, we have:\n\\begin{align}\n\\textnormal{Pr}[X \\geq (1+\\epsilon) \\mu] \\leq \\exp(-\\frac{2\\epsilon^2 \\mu^2}{n(b-a)^2})\\\\\n\\textnormal{Pr}[X \\leq (1-\\epsilon) \\mu] \\leq \\exp(-\\frac{\\epsilon^2 \\mu^2}{n(b-a)^2}) \\label{equ:chernoff_minus} \n\\end{align}\n\\end{theorem}\n\nConsidering a path $p \\in \\mathcal{F}$, we have:\n\\begin{align*}\n\\rho(p) \\geq \\frac{1}{|S|} (\\frac{1-\\alpha}{\\mathtt{d} - 1})^\\mathtt{h} = \\Omega(\\mathtt{d}^{-\\mathtt{h}} |S|^{-1})\n\\end{align*}\nwhere $\\mathtt{d}$ is the maximum out-going degree of a node in $G$. Therefore, for any single path $p$, $0 \\leq \\frac{\\mathtt{R}(p,\\mathbf{x})}{\\rho(p)} \\leq O(\\mathtt{T} |S| \\mathtt{d}^{\\mathtt{h}})$\n\nDenote $\\mathbf{v}^*$ as an optimal solution that maximizes $\\Delta_\\mathbf{v} B(\\mathbf{x})$. \n\\begin{lemma}\nGiven $0<\\epsilon_1,\\delta_1 < 1$, with the number of sampling paths satisfies\n\\begin{align}\n|\\mathcal{P}| \\geq \\frac{\\ln(1\/\\delta_1) \\mathtt{T}^2 |S|^2 \\mathtt{d}^{2\\mathtt{h}}}{\\epsilon_1^2 \\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})} \n\\end{align}\nthe following condition is guaranteed:\n\\begin{align}\n\\textnormal{Pr}[\\Delta_{\\mathbf{v}^*} \\hat{B}(\\mathcal{P},\\mathbf{x}) \\geq (1-\\epsilon_1) \\Delta_{\\mathbf{v}^*} B(\\mathbf{x})] \\geq 1 - \\delta_1\n\\end{align}\n\\end{lemma}\n\nThis lemma is trivially derived from Eq. \\ref{equ:chernoff_minus}.\n\n\\begin{lemma}\nGiven $0 < \\epsilon_2, \\delta_2 < 1$, with the number of sampling paths satisfies\n\\begin{align*}\n|\\mathcal{P}| \\geq \\frac{\\ln(\\binom{n+q}{q}\/\\delta_2) \\mathtt{T}^2 |S|^2 \\mathtt{d}^{2\\mathtt{h}}}{2(1-e^{-\\gamma})^2 \\epsilon_2^2 \\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})}\n\\end{align*}\nwe have $\\Delta_{\\mathbf{v}_q} \\hat{D}(x) \\leq \\Delta_{\\mathbf{v}_q} D(\\mathbf{x}) + (1-e^{-\\gamma})\\epsilon_2 \\Delta_{\\mathbf{v}^*} D(\\mathbf{x})$ for all budget vectors $\\mathbf{v}_k$, which satisfy $||\\mathbf{v}_k|| = q$, with probability at least $1-\\delta_2$\n\\end{lemma}\n\n\\begin{proof}\nLet us consider an arbitrary budget vector $\\mathbf{v}_q$, $||\\mathbf{v}_q|| = q$\n\\begin{align*}\n&\\textnormal{Pr}\\Big[\\Delta_{\\mathbf{v}_q} \\hat{B}(\\mathcal{P}, \\mathbf{x}) \\geq \\Delta_{\\mathbf{v}_q} B(\\mathbf{x}) + (1-e^{-\\gamma})\\epsilon_2 \\Delta_{\\mathbf{v}^*} B(\\mathbf{x})\\Big] \\\\\n& \\quad = \\textnormal{Pr}\\Big[\\Delta_{\\mathbf{v}_q} \\hat{B}(\\mathcal{P},x) \\geq \\Delta_{\\mathbf{v}_q} B(\\mathbf{x}) \\Big(1 + (1-e^{-\\gamma})\\epsilon_2 \\frac{\\Delta_{\\mathbf{v}^*} B(\\mathbf{x})}{\\Delta_{\\mathbf{v}_q} B(\\mathbf{x})}\\Big)\\Big] \\\\\n& \\quad \\leq \\exp \\Big(\\frac{2(1-e^{-\\gamma})^2 \\epsilon_2^2 |\\mathcal{P}| \\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})}{\\mathtt{T}^2 \\mathtt{d}^{2\\mathtt{h}}}\\Big)\n\\end{align*}\nUsing the union bound theory, to let $\\Delta_{\\mathbf{v}_q} \\hat{B}(\\mathcal{P},x) \\leq \\Delta_{\\mathbf{v}_q} B(\\mathbf{x}) + (1-e^{-\\gamma})\\epsilon_2 \\Delta_{\\mathbf{x}^*} B(\\mathbf{x})$  satisfy for any budget vector $\\mathbf{v}_q$, $||\\mathbf{v}_q|| = k$, we have\n\\begin{align*}\n&\\textnormal{Pr}\\Big[\\Delta_{\\mathbf{v}_q} \\hat{B}(\\mathcal{P}, \\mathbf{x}) \\geq \\Delta_{\\mathbf{v}_q} B(\\mathbf{x}) + (1-e^{-\\gamma})\\epsilon_2 \\Delta_{\\mathbf{v}^*} B(\\mathbf{x}) \\Big] \\leq \\binom{n+q}{q} \\exp \\Big(\\frac{2(1-e^{-\\gamma})^2 \\epsilon_2^2 |\\mathcal{P}| \\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})}{\\mathtt{T}^2 |S|^2 \\mathtt{d}^{2\\mathtt{h}}} \\Big)\n\\end{align*}\nThe lemma follows by letting $\\binom{n+q}{q} \\exp \\Big(\\frac{2(1-e^{-\\gamma})^2 \\epsilon_2^2 |\\mathcal{P}| \\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})}{\\mathtt{T}^2 |S|^2 \\mathtt{d}^{2\\mathtt{h}}} \\Big) \\leq \\delta_2$\n\\end{proof}\n\n\\begin{lemma} \\label{cor:greedy_sa}\nGiven $0 \\leq \\epsilon_1,\\epsilon_2,\\delta_1,\\delta_2 \\leq 1$, let $\\epsilon \\geq \\epsilon_1 + \\epsilon_2$ and $\\delta \\geq \\delta_1 + \\delta_2$. If the number of sampling paths is at least\n\\begin{align}\n\\frac{\\mathtt{T}^2 |S|^2 \\mathtt{d}^{\\mathtt{2\\mathtt{h}}}}{\\Delta_{\\mathbf{v}^*}^2 B(\\mathbf{x})} \\max\\Big( \\frac{\\ln(1\/\\delta_1)}{\\epsilon_1^2}, \\frac{\\ln(\\binom{n+q}{q}\/\\delta_2)}{2(1-e^{-\\gamma})^2\\epsilon_2^2} \\Big) \\label{equ:first_thres}\n\\end{align}\nthe greedy algorithm on $\\mathcal{P}$ returns a budget vector $\\mathbf{v}$ that guarantees\n\\begin{align*}\n\\textnormal{Pr}[\\Delta_{\\mathbf{v}} B(\\mathbf{x}) \\geq (1-e^{-\\gamma})(1-\\epsilon) \\Delta_{\\mathbf{v}^*}B(\\mathbf{x})] \\geq 1 - \\delta\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nFor the given number of sample paths, we have\n\\begin{align}\n\\Delta_{\\mathbf{v}} B(\\mathbf{x}) & \\geq \\Delta_{\\mathbf{v}} \\hat{B}(\\mathcal{P},\\mathbf{x}) - (1-e^{-\\gamma}) \\epsilon_2 \\Delta_{\\mathbf{v}^*} B(\\mathbf{x}) \\label{equ:first_equ} \\\\\n& \\geq (1-e^{-\\gamma}) \\Delta_{\\mathbf{v}^*} \\hat{B}(\\mathcal{P},\\mathbf{x}) - (1-e^{-\\gamma}) \\epsilon_2 \\Delta_{\\mathbf{x}^*} B(\\mathbf{x}) \\\\\n& \\geq (1-e^{-\\gamma})(1-\\epsilon_1) \\Delta_{\\mathbf{v}^*} B(\\mathbf{x}) - (1-e^{-\\gamma}) \\epsilon_2 \\Delta_{\\mathbf{v}^*} B(\\mathbf{x}) \\label{equ:second_equ} \\\\\n& \\geq (1-e^{-\\gamma})(1-\\epsilon) \\Delta_{\\mathbf{v}^*} B(\\mathbf{x})\n\\end{align}\nThe inequality (\\ref{equ:first_equ}) happens with probability $1-\\delta_1$ while the inequality (\\ref{equ:second_equ}) happens with probability $1-\\delta_1$. Overall $\\Delta_{\\mathbf{v}} B(\\mathbf{x}) \\geq (1-e^{-\\gamma})(1-\\epsilon) \\Delta_{\\mathbf{v}^*}B(\\mathbf{x})$ with probability at least $(1-\\delta_1)(1-\\delta_2) \\geq 1-\\delta$.\n\\end{proof}\n\n\nThere is a drawback of the threshold (\\ref{equ:first_thres}): it depends on $\\Delta_{\\mathbf{v}^*} B(\\mathbf{x})$, which is untraceable. However, we can use the simple lower bound of $\\Delta_{\\mathbf{v}^*} B(\\mathbf{x})$ as follows: As long as the algorithm has not terminated, there should be at least a path $p \\in \\mathcal{F}$ such that the length of $p$ is at most $\\mathtt{T} - 1$. So the marginal gain of the optimal solution should be at least $1$. Therefore, we have the following threshold, which is the sufficient number of sample paths to bound the error between approximation ratio of $\\mathbf{v}$ on $\\Delta_\\mathbf{v} \\hat{B}(\\mathcal{P}, \\mathbf{x})$ and $\\Delta_\\mathbf{v} B(\\mathbf{x})$.\n\\begin{align*}\n\\mathcal{N}(q,\\epsilon,\\delta) = \\min_{\\epsilon_1; \\delta_1 } \\Bigg(\\mathtt{T}^2 |S|^2 \\mathtt{d}^{2\\mathtt{h}} \\max\\Big( \\frac{\\ln(1\/\\delta_1)}{\\epsilon_1^2}, \\frac{\\ln(\\binom{n+q}{q}\/(\\delta - \\delta_1))}{2(1-e^{-\\gamma})^2(\\epsilon - \\epsilon_1)^2} \\Big)\\Bigg)\n\\end{align*}\n\n\n\n\n\n\n\\begin{theorem} \\label{theorem:sampling_approx}\nGiven $0 < \\epsilon, \\delta < 1$, by generating $\\mathcal{N}(k,\\epsilon,\\frac{\\delta}{||\\mathtt{b}||})$ of sample paths in each sampling iteration, \\texttt{SA} returns a solution within $O(\\frac{\\mathtt{h} \\ln \\mathtt{d} + \\ln \\mathtt{T}}{\\gamma (1-e^{-\\gamma})(1-\\epsilon)})$ factor of optimum to the \\texttt{QoSD} instance with probability at least $1-\\delta$. \n\\end{theorem}\n\n\\begin{skproof}\nDenote $\\mathbf{x}^o = \\sum_i^s \\mathbf{u}_i$ as an optimal solution, which is in addition to $\\mathbf{x}$ to block all paths in $\\mathcal{F}$ ($\\mathbf{u}_i$ is a unit vector). Let $\\mathbf{v}$ be a budget vector we get from greedy selection on the sample set $\\mathcal{P}$. The key of our proof is that\n\n\\begin{align*}\n\\Delta_{\\mathbf{v}} B(\\mathbf{x}) \\geq \\frac{\\gamma q}{||\\mathbf{x}^o||} (1-e^{-\\gamma}) (1-\\epsilon) \\Delta_{\\mathbf{x}^o} B(\\mathbf{x}) \n\\end{align*}\nThis is proved by the finding that there exists a budget vector $\\mathbf{w}$ such that $||\\mathbf{w}|| \\leq q$ and $\\Delta_{\\mathbf{w}} B(\\mathbf{x}) \\geq \\frac{\\gamma q}{||\\mathbf{x}^o||} \\Delta_{\\mathbf{x}^o} B(\\mathbf{x})$.\n\nTherefore, we observe that: after each sampling iteration, the gap between $|\\mathcal{F}|\\mathtt{T}$ and $B(\\mathbf{x})$ shrinks by a factor at least $(1 - \\frac{k\\gamma}{\\mathtt{OPT}} (1-e^{-\\gamma})(1-\\epsilon))$ with probability at least $1 - \\frac{\\delta}{||\\mathtt{b}||}$.\n\nFurthermore, since there should exist at least a feasible path $p \\in \\mathcal{P}$ such that $\\mathtt{r}(p, \\mathbf{x}) \\leq \\mathtt{T} - 1$ before the final sampling iteration, we prove that the number of iterations is upper bounded by $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln \\mathtt{d}}{q \\gamma (1-e^{-\\gamma})(1-\\epsilon)}) \\mathtt{OPT}$. Since in each iteration, a budget vector $\\mathbf{v}$, $||\\mathbf{v}|| \\leq q$, is added into solution, out final solution guarantees $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln \\mathtt{d}}{\\gamma (1-e^{-\\gamma})(1-\\epsilon)})$ approximation ratio with probability at least $1-\\delta$.\n\\end{skproof}\n\n\n\n\n\n\n\n\n\n\n\n\nInterestingly, the approximation ratio of \\texttt{SA} does not depend on $q$. So whatever the value of $q$ is, the result of \\texttt{SA} always has the same upper bound, which means a large value of $q$ could reduce the number of sampling iterations but the number of sample paths in each iteration would increase as the trade-off.  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Linear Weight Functions} \\label{sec:lr}\n\nHaving considered approximation algorithms to \\texttt{QoSD}, we now propose a solution, called \\textit{Linear Rounding} (\\texttt{LR}), for the case where the edge weight functions are linear. \\texttt{LR} obtains $O(\\mathtt{h} \\log n)$ approximation guarantee, which is the best ratio compared among all the proposed solutions.\n\nFor each $e \\in E$, the weight function of $e$ is represented as $f_e(x) = \\beta_e x + \\alpha_e$, where $\\beta_e, \\alpha_e \\in \\mathbb{Z}^+$. Denote $\\beta = \\max_e \\beta_e$. The \\texttt{QoSD} instance can be solved by the following Integer Programming.\n\n\\begin{align}\n\\min & \\quad \\sum_{e \\in E} x_e & \\label{eqn:IP} \\\\\n\\text{s.t. } & \\quad \\sum_{e \\in p} (\\beta_e x_e + \\alpha_e) \\geq \\mathtt{T} && \\forall  p \\in \\mathcal{F} \\label{con:path} \\\\\n& \\quad x_e \\leq b_e && \\forall e \\in E \\label{con:max_cost}\\\\\n& \\quad x_e \\in \\mathbb{Z}^+ \\cup \\{0\\} && \\forall e \\in E \\label{con:qos_integer}\n\\end{align}\nThis IP has a simple linear relaxation by replacing constraint (\\ref{con:qos_integer}) with:\n\\begin{align}\nx_e \\in \\mathbb{R}^+ \\cup \\{0\\} && \\forall e \\in E\n\\end{align}\n\n\n\n\nAlthough constructing this relaxation maybe intractable due to the extremely large size of $\\mathcal{F}$, this LP still can be solved in polynomial time using \\textit{ellipsoid method} with a simple separation oracle similar to Multicut problem \\cite{vazirani2013approximation}. \n\nDenote the vector $\\mathbf{x}^\\prime = \\{x_1^\\prime, ...x_m^\\prime\\}$ as the optimal solution to the LP relaxation, $x_e^\\prime$ can be a real number. The problem now is how to obtain a discrete solution $\\mathbf{x}$ from $\\mathbf{x}^\\prime$ and what approximation guarantee $\\mathbf{x}$ provides? To do so, we applied the randomized rounding technique as follows: Given an edge $e$, if $x_e^\\prime$ is an integer, let $x_e = x_e^\\prime$. Otherwise, denote $\\rho_e = x_e^\\prime - \\floor{x_e^\\prime}$ and given $\\eta$, which would be defined later, then:\n\\begin{itemize}\n\\item If $\\eta \\rho_e \\geq 1$, $x_e = \\ceil{x_e^\\prime}$. Let $y_e = x_e$\n\\item If $\\eta \\rho_e < 1$, $x_e = \\ceil{x_e^\\prime}$ with probability $\\eta \\rho_e$ and $\\floor{x_e^\\prime}$ otherwise. Let $y_e = \\floor{x_e^\\prime}$\n\\end{itemize}\n\\texttt{LR} is fully presented in Alg. \\ref{alg:linear_rounding}.\n\\begin{algorithm}[t]\n\t\\caption{Linear Rounding algorithm (\\texttt{LR})}\n    \\label{alg:linear_rounding}\n\t\\begin{flushleft}\n\t\\textbf{Input} $G, S, \\mathtt{T}, \\mathtt{b}, f, \\delta$\\\\\n\t\\textbf{Output} cost vector $\\mathbf{x} = \\{x_1,..x_m\\}$\n\t\\end{flushleft}\n    \\begin{algorithmic}[1]\n        \\State $\\mathbf{x}^\\prime \\leftarrow $ optimal solution of LP-relaxation.\n        \\State $\\beta = \\max_e \\beta_e$; $\\eta = \\frac{\\beta}{1-\\exp(-\\beta)} (\\ln \\frac{n^\\mathtt{h}}{\\delta} + 1)$\n        \\For{each $e \\in E$}\n        \t\\If{$x_e^\\prime$ is a integer}\n            \t\\State $x_e = x_e^\\prime$\n            \\Else\n            \t\\State $\\rho_e = x_e^\\prime - \\floor{x_e^\\prime}$\n                \\State $x_e = \\ceil{x_e^\\prime}$ with probability $\\eta \\rho_e$; $\\floor{x_e^\\prime}$ otherwise.\n            \\EndIf\n        \\EndFor\n     \\end{algorithmic}\n     \\begin{flushleft}\n     \t\\textbf{Return } $\\mathbf{x}$\n     \\end{flushleft}\n\\end{algorithm}\n\nConsider a path $p \\in \\mathcal{F}$, it is trivial that $\\mathbf{x}$ will block $p$ if $\\sum_{e \\in p} f_e(y_e) \\geq \\mathtt{T}$. The question is whether $\\mathbf{x}$ can block $p$ if $\\sum_{e \\in p} f_e(y_e) < \\mathtt{T}$? Denote:\n\\begin{align*}\nT_p = \\mathtt{T} - \\sum_{e \\in p} f_e(y_e) \\\\\n\\mathcal{E}_p = \\{e \\in p; y_e < x_e\\}\n\\end{align*}\nSo:\n\\begin{align*}\nT_p \\leq \\sum_{e \\in \\mathcal{E}_p} \\beta_e (x^\\prime_e - y_e) = \\sum_{e \\in \\mathcal{E}_p} \\beta_e \\rho_e\n\\end{align*}\nThen the probability that $\\mathbf{x}$ does not block $p$ is given as follows:\n\n\n\n\\begin{align}\n&\\textnormal{Pr}[p \\textnormal{ is not blocked by } \\mathbf{x}] = \\textnormal{Pr}\\Big[\\sum_{e \\in p} (\\beta_e x_e + \\alpha_e) < \\mathtt{T} \\Big] = \\textnormal{Pr}\\Big[ \\sum_{e \\in \\mathcal{E}_p} \\beta_e (x_e - \\floor{x_e^\\prime}) < T_p \\Big] \\\\\n& \\quad \\quad = \\textnormal{Pr}\\Big[\\exp{\\Big(-\\sum_{e \\in \\mathcal{E}_p} \\beta_e (x_e - \\floor{x_e^\\prime}) \\Big) } > \\exp{(-T_p)}\\Big] \\\\\n& \\quad \\quad \\leq \\exp(T_p) \\cdot \\mathbb{E}\\Big[\\exp\\Big(-\\sum_{e \\in \\mathcal{E}_p} \\beta_e (x_e - \\floor{x_e^\\prime}) \\Big) \\Big] \\label{equ:markov} \\\\\n& \\quad \\quad = \\exp(T_p) \\cdot \\prod_{e \\in \\mathcal{E}_p} \\Big( \\exp(-\\beta_e) \\cdot \\eta\\rho_e + (1 - \\eta\\rho_e) \\Big) \\\\\n& \\quad \\quad \\leq \\exp(T_p) \\cdot \\Bigg( 1 - \\frac{\\sum_{e \\in \\mathcal{E}_p} \\eta \\rho_e (1 - \\exp(-\\beta_e))}{|\\mathcal{E}_p|} \\Bigg)^{|\\mathcal{E}_p|} \\label{equ:cauchy} \\\\\n& \\quad \\quad \\leq \\exp(T_p) \\cdot \\Bigg( 1 - \\frac{1-\\exp(-\\beta)}{\\beta} \\cdot \\frac{\\eta T_p}{|\\mathcal{E}_p|} \\Bigg)^{|\\mathcal{E}_p|} \\\\\n& \\quad \\quad \\leq \\exp(T_p) \\cdot \\exp\\Bigg(- \\eta T_p \\frac{1 - \\exp(-\\beta)}{\\beta} \\Bigg) \\leq \\exp\\Bigg( -\\bigg( \\eta \\frac{1-\\exp(-\\beta)}{\\beta} - 1 \\bigg) \\Bigg)\n\\end{align}\nEq. \\ref{equ:markov} comes from Markov inequality \\cite{markovInequality} while Eq. \\ref{equ:cauchy} is from Cauchy Theorem \\cite{cauchyInequality}. Since there are at most $n^\\mathtt{h}$ feasible paths in $\\mathcal{F}$, using Union Bound theory \\cite{UnionBound}, the probability that $\\mathbf{x}$ cannot block all paths in $\\mathcal{F}$ is at most\n\\begin{align} \\label{equ:prob_x}\nn^\\mathtt{h} \\cdot \\exp\\Bigg( -\\bigg( \\eta \\frac{1-\\exp(-\\beta)}{\\beta} - 1 \\bigg) \\Bigg)\n\\end{align}\n\\begin{theorem}\nGiven fixed $0 < \\delta < 1$ and $\\eta = \\frac{\\beta}{1-\\exp(-\\beta)} (\\ln \\frac{n^\\mathtt{h}}{\\delta} + 1)$, \\texttt{LR} returns a solution within $O(\\mathtt{h} \\ln n)$ factor of optimum to the \\texttt{QoSD} instance with probability at least $1-\\delta$.\n\\end{theorem}\n\\begin{proof}\nFrom Eq. \\ref{equ:prob_x} and the given $\\eta$, the probability that $\\mathbf{x}$ blocks all paths in $\\mathcal{F}$ is at least $1-\\delta$. Also\n\\begin{align*}\n&\\mathbb{E}[||\\mathbf{x}||] = \\sum_e \\Big( \\ceil{x_e^\\prime} \\eta \\rho_e + \\floor{x_e^\\prime} (1 - \\eta \\rho_e) \\Big) \\\\\n& \\quad \\leq \\sum_e \\Big( \\floor{x_e^\\prime} + \\eta \\rho_e \\Big) \\leq \\sum_e \\eta x_e^\\prime \\\\\n& \\quad = \\eta \\cdot ||\\mathbf{x}^\\prime|| \\leq \\eta \\cdot ||\\mathbf{x}^*||\n\\end{align*}\nwhich completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion} \\label{sec:discussion}\nIn this section, we discuss the trade-off between the performance guarantee and the runtime complexity of the four proposed algorithms, summarized in Table. \\ref{table:performance}.\n\nFirst, we consider the performance guarantee of the \\texttt{IG} and \\texttt{AT} algorithm. The approximation ratio of \\texttt{IG} and \\texttt{AT} are $O(\\frac{1}{\\gamma} (\\mathtt{h} \\ln n + \\ln \\mathtt{T}) )$ and $O(\\mathtt{h} \\ln n + \\ln \\mathtt{T})$ respectively, where $\\gamma$ is the \\textit{concave ratio} of edge weight functions. $\\gamma$ plays an important role in the differences between \\texttt{IG} and \\texttt{AT} solutions. The smaller $\\gamma$ is - which signifies a more convex of edge weight functions - the worse \\texttt{IG} performs. But if all edge weight functions are concave - or at least linear - $\\gamma$ equals to $1$, then \\texttt{IG} and \\texttt{AT} obtain the same approximation guarantee. Not only achieve the same ratio, the two algorithms also return the same solution because in \\texttt{AT}, $\\frac{\\Delta_{u(e,x)} \\mathtt{D}(\\mathcal{P}, \\mathbf{x})}{x}$ reaches maximum at $x = 1$. So in each iteration, the budget increases at most by $1$, and it is also the selection of \\texttt{IG}. Overall, \\texttt{AT} theoretically returns better solutions than \\texttt{IG}. \n\n\n\nHowever, in trade-off, \\texttt{AT} has higher computational complexity than \\texttt{IG}. Both algorithms use the same framework as in Alg. \\ref{alg:iterative}. The maximum number of  iterations in this framework (line \\ref{line:outer_it} of Alg. \\ref{alg:iterative}) is upper bounded by $n^\\mathtt{h}$, which is theoretically a large number. However, from our experiments on both random graphs and real-but-dense networks, the number of iterations never reach this amount. Considering the strategy of blocking paths, the number of computation in each inner iteration (line \\ref{line:inner_it_trunk} of Alg. \\ref{alg:trunk_adding}) of \\texttt{AT} is $O(||\\mathtt{b}||)$, while this number (line \\ref{line:inner_it_greedy} of Alg. \\ref{alg:greedy_blocking}) in \\texttt{IG} is $O(m)$. In the worst-case scenario, the number of inner iterations of both \\texttt{IG} and \\texttt{AT} can reach up to $O(||\\mathtt{b}||)$. Therefore, the worst-case runtime complexity of \\texttt{IG} and \\texttt{AT} is $O(n^\\mathtt{h} m ||\\mathtt{b}||)$ and $O(n^\\mathtt{h} ||\\mathtt{b}||^2)$ respectively.\n\nWith \\texttt{SA}, to obtain the $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln d}{\\gamma (1-e^{-\\gamma}) (1-\\epsilon)})$ ratio, we have to generate $\\mathcal{N}(k,\\epsilon,\\delta\/||\\mathtt{b}||)$ paths with $O(m)$ time complexity for each path in each sampling steps. Also, after sampling, a budget vector $\\mathbf{v}$ ($||\\mathbf{v}|| \\leq q$) is added into $\\mathbf{x}$, which makes the number of sampling steps at most $O(\\frac{||\\mathtt{b}||}{q})$. Moreover, the greedy selection on a sample set costs $O(qm)$ runtime complexity. Therefore, the worse-case runtime complexity of \\texttt{SA} is bounded by $O(\\mathcal{N}(k,\\epsilon, \\frac{\\delta}{||\\mathtt{b}||}) \\cdot ||\\mathtt{b}|| \\cdot m^2)$. However, if $\\mathtt{h}$ is large, the number of samples required by \\texttt{SA} becomes large and its sampling procedure dominates its runtime; this is ameliorated by trivially parallelizing the sampling process, which is possible since each sample is independent. In practice, the parameter $\\alpha$ greatly reduces the required number of samples; with $\\alpha = 0.8$, we found that $O(|S|)$ samples were sufficient to provide feasible solutions within reasonable runtime. \n\n\n\\begin{table}[t]\n\\centering\n\\caption{Algorithm performance ratio and time complexity} \\label{table:ratio}\n\\label{table:performance}\n\\begin{tabular}{c | l | l }\n\\toprule\nAlgorithm     & Approximation Ratio    & Worst-case Runtime \\\\ \n \\midrule\n\\texttt{IG}\t \t\t\t& $O(\\gamma^{-1} (\\ln \\mathtt{T} + \\mathtt{h} \\ln n))$ \t& $O(n^\\mathtt{h} m ||\\mathtt{b}||)$ \\\\\n\\texttt{AT}  \t\t& $O(\\ln \\mathtt{T} + \\mathtt{h} \\ln n)$ \t\t& $O(n^\\mathtt{h} ||\\mathtt{b}||^2)$\\\\\n\\texttt{SA}\t  \t\t& $O(\\frac{\\ln \\mathtt{T} + \\mathtt{h} \\ln \\mathtt{d}}{\\gamma (1-e^{-\\gamma}) (1-\\epsilon)})$ \t\t& $\\mathcal{N}(q,\\epsilon,\\frac{\\delta}{||\\mathtt{b}||}) \\cdot O(m^2||\\mathtt{b}||)$  \\\\\n\\texttt{LR}\t\t& $O(\\mathtt{h} \\ln n)$ \t& LP-solver$() + m$ \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nNext, consider the \\texttt{LR} solution, which is only used if all the weight functions are linear. The runtime of \\texttt{LR} strongly depends on the linear programming solver (LP-solver$()$). In the experimental evaluation, we observe that in most cases, the number of edges - whose $x_e^\\prime$ is real - is inconsiderably small. Therefore, after randomized rounding, the size of the discrete solution $\\mathbf{x}$ has a diminutive gap comparing with $\\mathbf{x}^\\prime$'s. Hence, although \\texttt{IG} and \\texttt{AT} perform fairly well in general cases, \\texttt{LR} usually returns the best solution if the edge weight functions are linear. \n\n\n\n\\section{Experimental Evaluation} \\label{sec:experiment}\nIn this section, we evaluate our proposed approximation algorithms by 1) comparing their performance to an intuitive heuristic as there is no other solution to \\texttt{QoSD}, in a general case; and 2) comparing our algorithms to \\cite{kuhnle2018network} as a special case of \\texttt{QoSD}. The experiments were conducted on a Linux machine with 2.3Ghz Xeon 18 core processor and 256GB of RAM. The programming language we used is C++. Several steps in our algorithm are parallelized by using OpenMP with 64 threads. The reported running time is real-world time, not CPU time. The source code is available at \\cite{code}.\n\n\\subsection{Experiment Settings}\nWe evaluated the following algorithms; the source code of all of our implementation is written in C++.\n\\begin{itemize}\n\\item \\texttt{AT}: In this solution, to find the shortest paths between a pair of nodes, we utilized the Dijkstra algorithm and computed each path separately. The reason for this implementation is that by doing so, we can parallelize the process by dividing it into independent tasks. Therefore, even the theoretical time complexity of the Dijkstra algorithm for all-pair shortest paths is worse than Floyd-Warshall methods, the parallelization helps to boost the performance of the Dijkstra algorithm while it is impossible to do so with Floyd-Warshall. \n\\item \\texttt{IG}: this algorithm used the same settings as \\texttt{AT}.\n\\item \\texttt{SA}: We set the bias ratio $\\alpha = 0.8$ and the number of sample paths is $O(|S|)$ for all experiments. We found this value of $\\alpha$ and the number of samples are sufficient to obtain feasible solutions within reasonable runtime in most cases. \n\\item \\texttt{LR}: We used CPLEX \\cite{cplex2009v12} to solve the linear programming. Implementing the \\textit{ellipsoid method} could result in impractical performance. So we used the same concept of \\texttt{IG} and \\texttt{AT} to solve the LP relaxation as follows: rather than listing all feasible paths in $\\mathcal{F}$, we iteratively listed the shortest LP-weighted paths as constraints until the length of the shortest paths between each pair exceeded $\\mathtt{T}$.  \n\\item Centrality Cutting ($\\mathtt{CC}$) heuristic: Centrality has been commonly used as a metric to identify critical components of a network in the literature. \\texttt{CC} works in iterative manner as follows: First, we set $\\mathbf{x} = \\{0\\}^m$. In each iteration, we found the shortest paths between a pair of nodes under the current budget vector $\\mathbf{x}$ and computed the number of appearances of each edge in those paths. The algorithm then raised the weight of the edge that appears the most to maximum. All those steps are repeated until there were no shortest paths whose length was smaller than $\\mathtt{T}$. When finding shortest paths of each pair, we also used parallelization to boost $\\mathtt{CC}$ performance.\n\\item \\texttt{SAP}, \\texttt{MIA}, \\texttt{TAG} \\cite{kuhnle2018network}: These algorithms were only implemented in comparison on the special case of \\texttt{QoSD} (the \\texttt{LB-MULTICUT} problem). The source code of those algorithms was taken from \\cite{lbcode} and it was only available for undirected networks.\n\\end{itemize}\n\nTo obtain $S$, we sampled uniformly random sets of pairs of nodes on each network. All results were averaged over 5 independent repetitions of each experiment. The weight function of each edge was selected from following functions\n\n\\begin{itemize}\n\\item A linear function $f_e(x) = \\Theta(x)$. \n\\item A convex function $f_e(x) = \\Theta(x^2)$. This function was inspired by the average delay calculation on computer networks w.r.t packet arrival rate.\n\\item A concave function $f_e(x) = \\Theta(\\ln x)$. This function was inspired by the additive metric on IoT network w.r.t packet error rate. \n\\item A cutting function: $f_e$ only received two values, $f_e(0) = 1$ and $f_e(1) = \\mathtt{T}$. This function was used when we compared our solution with the algorithms of the \\texttt{LB-MULTICUT} problem.\n\\end{itemize}\nEach function was set such that the initial weight $f_e(0)=1$ and the maximum weight was $\\mathtt{T}$. Since there exists heterogeneous coupling delays in modern networks, in our experiment, the weight function of each edge was randomly selected from the linear, convex or concave functions as mentioned above. In the experiments with the presence of \\texttt{LR}, all weight functions were linear. On the other hand, all weight functions were cutting function if compared with the algorithms of \\texttt{LB-MULTICUT}. \n\n\nThe algorithms were implemented on both synthesized networks and real-world networks. The synthesized networks we used were the Erdos-Renyi (ER) \\cite{erdos1960evolution} graphs with $240$ nodes and varied the edge density parameter $\\rho$. For the real-world networks, we used the datasets from Stanford Network Analysis Project \\cite{snapnets}, including Gnutella, Skitter and Roadnet. Skitter is highly dense IPv4 Internet topology graph, which were collected by traceroutes run daily in 2005; Gnutella is the snapshots of peer-to-peer file sharing; and RoadCA is a road network of California where intersections and endpoints are represented by nodes, and the roads connecting these intersection or endpoints are represented by undirected edges. Information of real-world datasets are summarized in Table \\ref{table:dataset}. \n\n\\begin{table}[t]\n\\centering\n\\caption{Statistics of datasets} \\label{table:dataset}\n\\begin{tabular}{l | l | r | r | r}\n\\toprule\n     Data & Type    & Nodes & Edges & Diameter \\\\ \n \\midrule\nGnutella & Directed \t& $10.9$ K & $40.0$ K  & 9 \\\\\nRoadCA & Undirected \t& $2.0$ M \t& $2.8$ M & 786\\\\\nSkitter  & Directed \t& $1.7$ M \t& $11.1$ M & 25 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Performance comparison}\n\n\\subsubsection{Small size random graph}\n\\begin{figure}[t]\n\\subfloat[Linear]{\n\t\\includegraphics[width=0.35\\textwidth]{image\/random_directed_delay1_changeP_result.png}\n\t\\label{fig:randDelay1_size}}\n~\n\\hspace{-20px}\n \\subfloat[Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/random_directed_delay2_changeP_result.png}\n\\label{fig:randDelay2_size}}\n~\n\\hspace{-20px}\n\\subfloat[Cutting]{\\includegraphics[width=0.35\\textwidth]{image\/random_undirected_delay5_changeP_result.png}\n\\label{fig:randDelay5_size}}\n    \n    \\caption{Solution quality of algorithms on Random graph}\n \t\\label{fig:random_size}\n\\end{figure}\n\n\\begin{figure}[t]\n\\subfloat[Linear]{\n\t\\includegraphics[width=0.35\\textwidth]{image\/random_directed_delay1_changeP_time.png}\n\t\\label{fig:randDelay1_time}}\n~\n\\hspace{-20px}\n \\subfloat[Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/random_directed_delay2_changeP_time.png}\n\\label{fig:randDelay2_time}}\n~\n\\hspace{-20px}\n\\subfloat[Cutting]{\\includegraphics[width=0.35\\textwidth]{image\/random_undirected_delay5_changeP_time.png}\n\\label{fig:randDelay5_time}}\n    \n    \\caption{Runtime of algorithms on Random graph}\n \t\\label{fig:random_time}\n\\end{figure}\n\n    \n\n    \n\n    \n\nIn these experiments, we compared our algorithms with the $\\mathtt{CC}$ solution on directed ER networks with $n=240$ and we varied the edges density $\\rho$. The threshold \\texttt{T} was set to be 3 and the size of $S$ was 10.\n\n\nFig. (\\ref{fig:randDelay1_size}) and Fig. (\\ref{fig:randDelay1_time}) show the results and runtimes of the algorithms when edge weight functions are linear. We notice that our four algorithms performed almost similarly in terms of quality of solution and very close to the optimal solution of LP relaxation. Meanwhile, $\\mathtt{CC}$ was far from being optimal when its solutions were always at least double to the solution of other algorithms. In terms of runtime, the ranking from best to worst was $\\mathtt{IG}$, $\\mathtt{AT}$, $\\mathtt{LR}$, $\\mathtt{CC}$ and $\\mathtt{SA}$. $\\mathtt{SA}$ performed worse especially when the edge density increased and approached to 1. This can be explained by the following: with high value of the bias parameter $\\alpha$, most paths of the sample set $\\mathcal{P}$ were the shortest paths of pairs in $S$. But because the edge density is high, there would be multiple paths between a pair of nodes whose length is smaller than $\\mathtt{T}$, which makes the number of sampling iterations on $\\mathtt{SA}$ increases. Therefore, $\\mathtt{SA}$ had a high runtime on finding shortest paths and then sampling, which was the main factor degrading its runtime. We observe that at the smallest edge density ($0.1$), all algorithms performed the best on both quality of solution and runtime, which is promising since most of real-world networks are sparse \\cite{kuhnle2018network}.\n\n\nNext, we compared our algorithms in the scenario with heterogeneous weight functions. $\\mathtt{LR}$ was no longer applicable, which explains why we did not plot $\\mathtt{LR}$ in Fig. (\\ref{fig:randDelay2_size}) and Fig. (\\ref{fig:randDelay2_time}). Although having the same quality of solution in linear delay, $\\mathtt{IG}$ performed much worse than $\\mathtt{AT}$ when its sizes of solutions were always at least 20 times of $\\mathtt{AT}$'s. The concave ratio $\\gamma$ was 0 in this scenario because there existed a weight function whose value did not change by adding several cost units. This experiment clearly illustrated the impact of concave ratio $\\gamma$ on the performance guarantee of $\\mathtt{IG}$. Moreover, $\\gamma$ also impacted on $\\mathtt{IG}$'s runtime because the \\texttt{IG} runtime is proportional to the solution size. From density $0.4$, the gap between $\\mathtt{IG}$ and $\\mathtt{AT}$'s runtime became distinguishable. \n\n\nFinally, we compared our algorithms with three methods proposed by Kuhnle et al. \\cite{kuhnle2018network} for the \\texttt{LB-MULTICUT} problem. The random graph was undirected and had 240 nodes. The size of solution and runtimes are reported in Fig. (\\ref{fig:randDelay5_size}) and Fig. (\\ref{fig:randDelay5_time}). All of our four algorithms returned the best results while \\texttt{SAP}, \\texttt{MIA} and \\texttt{TAG} were even worse than \\texttt{CC} in term of quality of solution. The gap between the final budgets of those three algorithms and our algorithms was significant with small $\\rho$ and became smaller when $\\rho$ increased. In terms of runtime, \\texttt{SAP} and \\texttt{TAG} performed the worst while \\texttt{MIA} bypassed \\texttt{SA} after $\\rho = 0.3$ and \\texttt{LR} after $\\rho = 0.5$. \\texttt{IG} and \\texttt{AT} were the fastest by far. It took less than one second for these two algorithms to finish no matter the edge density.\n\n\\subsubsection{Results on real networks}\n\nIn this subsection, we evaluate our algorithms on the real-world networks. We mainly examined the effect of varying the threshold $\\mathtt{T}$ on the algorithm performances. The number of pairs is set to be 100. We limited the runtime by a day (24 hours); any experiments, which ran longer than a day, were terminated.\n\n\\begin{figure}[t] \n\\subfloat[Gnutella - Linear]{\\includegraphics[width=0.35\\textwidth]{image\/p2p-Gnutella04_directed_delay1_changeT_result.png}\n\\label{fig:gnutellaDelay1_size}}\n~\n\\hspace{-20px}\n\\subfloat[Gnutella - Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/p2p-Gnutella04_directed_delay4_changeT_result.png}\n\\label{fig:gnutellaDelay4_size}}\n~\n\\hspace{-20px}\n\\subfloat[Skitter - Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/as-skitter_undirected_delay4_changeT_result.png}\n\\label{fig:skitterDelay4_size}}\n    \n    \\caption{Solution quality of algorithms on Gnutella and Skitter}\n \t\\label{fig:gnuski_size}\n\\end{figure}\n\n\n\\begin{figure}[t] \n\\subfloat[Gnutella - Linear]{\\includegraphics[width=0.35\\textwidth]{image\/p2p-Gnutella04_directed_delay1_changeT_time.png}\n\\label{fig:gnutellaDelay1_time}}\n~\n\\hspace{-20px}\n\\subfloat[Gnutella - Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/p2p-Gnutella04_directed_delay4_changeT_time.png}\n\\label{fig:gnutellaDelay4_time}}\n~\n\\hspace{-20px}\n\\subfloat[Skitter - Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/as-skitter_undirected_delay4_changeT_time.png}\n\\label{fig:skitterDelay4_time}}\n    \n    \\caption{Runtime of algorithms on Gnutella and Skitter}\n \t\\label{fig:gnuski_time}\n\\end{figure}\n\n    \n\n    \n\n\nFirst, we discuss the results on the smallest network, Gnutella, in which we let $\\mathtt{T}$ vary from $10$ to $50$. The result and the runtime of each algorithm are shown in Fig. (\\ref{fig:gnutellaDelay1_size}), (\\ref{fig:gnutellaDelay4_size}), (\\ref{fig:gnutellaDelay1_time}) and (\\ref{fig:gnutellaDelay4_time}). When the weight functions were all linear, we observed the same pattern as in the random graph, where the quality of solution of $\\mathtt{IG}$, $\\mathtt{AT}$ and $\\mathtt{LR}$ almost overlapped. Actually, $\\mathtt{LR}$ always returned the best solution but the gap between \\texttt{LR} and $\\mathtt{AT}$ and $\\mathtt{IG}$ was insignificant. Meanwhile, the sizes of the $\\mathtt{SA}$'s solutions were always within 1.3 factor from $\\mathtt{LR}$. However, in terms of runtime, $\\mathtt{AT}$ performed the worst among our proposed algorithms while $\\mathtt{LR}$ again was the best. The next best algorithm in terms of runtime was $\\mathtt{SA}$, which stayed within 1.4 factor from $\\mathtt{LR}$. Starting from $\\mathtt{T} = 35$, the runtime of $\\mathtt{LR}, \\mathtt{IG}$ and $\\mathtt{SA}$ almost stayed the same while the runtime of $\\mathtt{AT}$ and $\\mathtt{CC}$ kept increasing. \n\nHowever, it was a different matter in the experiments with heterogeneous weight functions, shown in Fig. (\\ref{fig:gnutellaDelay4_size}) and Fig. (\\ref{fig:gnutellaDelay4_time}). In terms of the quality of solution, the ranking from best to worst was $\\mathtt{AT}$, $\\mathtt{IG}$, $\\mathtt{SA}$ and $\\mathtt{CC}$. These algorithms were now more virtually distinguishable in solution quality. The sizes of solutions of $\\mathtt{IG}$ could be up to 1.25 factor from $\\mathtt{AT}$'s while this number of $\\mathtt{SA}$ was 4. In terms of runtime, $\\mathtt{CC}$ was no longer the worst algorithm. Starting from $\\mathtt{T} = 45$, $\\mathtt{AT}$ ran slower than $\\mathtt{SA}$ and $\\mathtt{CC}$. $\\mathtt{IG}$ was by far the fastest among the algorithms. \n\n\n    \n\n    \n\n    \n\nNext, we experimented our algorithms on large scale networks. The Roadnet network contains 2 millions of nodes but only 2.8 millions of edges, which made it the sparest network among the datasets we used for experiment. First, we varied the value of $\\mathtt{T}$ from 100 to 120 and plotted the results as in Fig. (\\ref{fig:roadDelay1_size}) and Fig. (\\ref{fig:roadDelay1_time}). $\\mathtt{SA}$ performed much worse than $\\mathtt{LR}$, $\\mathtt{AT}$ and $\\mathtt{IG}$. Its sizes of solutions were always at least 3 times greater than the others' and roughly near $\\mathtt{CC}$ when $\\mathtt{T}$ increased. Although $\\mathtt{CC}$ always returned the worst solution, it was by far the fastest. The second best in terms of runtime was $\\mathtt{IG}$ but it was always at least 15 times slower than $\\mathtt{CC}$. This number in $\\mathtt{LR}$ and $\\mathtt{AT}$ were 30 and 60 respectively. With $\\mathtt{T}=100$, $\\mathtt{SA}$ was slightly faster than $\\mathtt{LR}$ and $\\mathtt{AT}$, therefore, we reduced the experimental range of $\\mathtt{T}$ to $[70,100]$ on next experiment to observe the behaviors of $\\mathtt{SA}$. Interestingly, $\\mathtt{SA}$ performed much more faster than $\\mathtt{AT}$ and $\\mathtt{IG}$ in this range. \n\nIn the experiment with heterogeneous weight functions, as can be seen in Fig. (\\ref{fig:roadDelay4_size}) and Fig. (\\ref{fig:roadDelay4_time}), $\\mathtt{SA}$ ran up to 8 and 16 times faster than $\\mathtt{IG}$ and $\\mathtt{AT}$ respectively. However, $\\mathtt{SA}$'s solutions were still worse than those two algorithms. From our observation, we found it hard to predict the behaviors of $\\mathtt{SA}$ especially when the set $\\mathcal{F}$ becomes larger. $\\mathtt{SA}$ can perform well when this set is small and is very stable in a certain range of this set's size. But when it exceeds this range, $\\mathtt{SA}$'s runtime increases at a higher rate than any other algorithms we have considered. \n\nFinally, we evaluated the algorithms on the cutting scenario and reported the results in Fig. (\\ref{fig:roadDelay5_size}) and Fig. (\\ref{fig:roadDelay5_time}). Up to $\\mathtt{T} = 91$, \\texttt{IG} and \\texttt{AT} were the best in term of quality of solution but then were bypassed by \\texttt{TAG}. In term of runtime, in most cases, our algorithms were 30 times slower than the fastest one, \\texttt{SAP}. \n\n\n\\begin{figure}[t]\n\\subfloat[Linear]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay1_changeT_result.png}\n\\label{fig:roadDelay1_size}}\n~\n\\hspace{-20px}\n\\subfloat[Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay4_changeT_result.png}\n\\label{fig:roadDelay4_size}}\n~\n\\hspace{-20px}\n\\subfloat[Cutting]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay5_changeT_result.png}\n\\label{fig:roadDelay5_size}}\n    \n    \\caption{Solution quality of algorithms on RoadnetCA}\n \t\\label{fig:road_size}\n\\end{figure}\n\n\\begin{figure}[t]\n\\subfloat[Linear]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay1_changeT_time.png}\n\\label{fig:roadDelay1_time}}\n~\n\\hspace{-20px}\n\\subfloat[Heterogeneous]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay4_changeT_time.png}\n\\label{fig:roadDelay4_time}}\n~\n\\hspace{-20px}\n\\subfloat[Cutting]{\\includegraphics[width=0.35\\textwidth]{image\/roadNet-CA_undirected_delay5_changeT_time.png}\n\\label{fig:roadDelay5_time}}\n    \n    \\caption{Runtime of algorithms on RoadnetCA}\n \t\\label{fig:road_time}\n\\end{figure}\n\n\n    \n\nThe last network we did experiments on was Skitter, which is a dense graph where the average degree of a node is 6.5. In this experiment, we varied $\\mathtt{T}$ in the range from $5$ to $12$. Fig. (\\ref{fig:skitterDelay4_size}) and Fig. (\\ref{fig:skitterDelay4_time}) show the performance of our algorithms. $\\mathtt{IG}$, $\\mathtt{AT}$ and $\\mathtt{SA}$ could finish within the limited runtime while $\\mathtt{CC}$ was unable to run even at $\\mathtt{T} = 7$, which is why we did not show $\\mathtt{CC}$'s results from $\\mathtt{T} = 7$ in those figures. Also, this experiment clearly shows the trade-off between $\\mathtt{IG}$ and $\\mathtt{AT}$. The solution of $\\mathtt{IG}$ was up to 1.25 times of $\\mathtt{AT}$ while running faster with almost the same factor.\n\n\\subsubsection{Summary of results} The experimental results can be summarized as follows.\n\\begin{itemize}\n\\item In case of linear weight functions, $\\mathtt{LR}$ always returned the best solution. The solution quality of $\\mathtt{IG}$ and $\\mathtt{AT}$ were worse than $\\mathtt{LR}$ but usually by only a small factor. In addition, $\\mathtt{IG}$ usually ran the fastest while the runtime of $\\mathtt{AT}$ was more impacted by the varying of $\\mathtt{T}$ than the other two.\n\\item In general cases where $\\mathtt{LR}$ is no more applicable, $\\mathtt{AT}$ was always the algorithm that returned the best quality of solution. $\\mathtt{IG}$ was competitive to $\\mathtt{AT}$ only if the weight functions tent to be more concave. In trade-off, $\\mathtt{IG}$ performed much more faster than $\\mathtt{AT}$ in most experiments.\n\\item In most experiments, $\\mathtt{SA}$ was the worst among our algorithms in term of both the quality of solution and runtime. However, in several cases when the set of feasible paths was small or the input network was sparse, $\\mathtt{SA}$ outperformed our other algorithms in runtime and the intuitive heuristics in solution quality. \n\\end{itemize}\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this work, we have introduced a new \\texttt{QoSD} problem together with four solutions \\texttt{IG, AT, SA} and \\texttt{LR}, each of which scales to networks with millions of edges and nodes in under several hours and has a proven performance guarantee. Future work would include lowering the number of samples required by \\texttt{SA}, making it more scalable. In addition, bounding the size of a set of candidate paths on \\texttt{IG} and \\texttt{AT} is necessary to reduce the burden on memory and waste of works when the candidate set is undesirable, and considering the correlation in increasing the edges' weights. Following that, we will investigate more on QoS degradation assessment on interdependent networks where networks are intertwined and interdependent, making the task of devising efficient algorithms much more challenging.\n\n\\begin{acks}\nThe authors would like to thank the anonymous reviewers for their valuable comments and helpful suggestions. We would also like to show our gratitude to Dr. Figueiredo (UFRJ) for shepherding our paper. This work is supported in part by NSF EFRI-1441231, NSF CNS-1814614, and DTRA HDTRA1-14-1-0055.\n\\end{acks}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}