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+{"text":"\\section{Introduction}\nIn this paper we give a contribution to the general problem of the asymptotic analysis of systems of lattice interactions of the form \n\\begin{equation}\\label{1}\n\\sum_{i,j\\in{\\cal L}} a^\\varepsilon_{ij}|u_i-u_j|\n\\end{equation}\nwhere ${\\cal L}$ is a periodic lattice in $\\mathbb R^d$, $\\varepsilon>0$ is a parameter tending to $0$, and  $a^\\varepsilon_{ij}$ are non-negative coefficients. \nThese functionals depend on (scalar) `spin functions' with $u_i\\in\\{0,1\\}$, somehow related to ferromagnetic energies in the terminology of Statistical Mechanics (where usually $u_i\\in\\{-1,1\\}$). \n\n\nWe investigate coerciveness properties related to such energies in a discrete-to-continuum process, where\nthe values $u^\\varepsilon_i$ are identified as the values $u^\\varepsilon(\\varepsilon i)$ of a function defined on $\\varepsilon{\\cal L}$. In this way \na continuum limit of $u^\\varepsilon$ can be defined as a limit of their piecewise-constant interpolations; \ne.g., defined as $u^\\varepsilon(x)= u^\\varepsilon_i$ if the point of minimum distance of $\\varepsilon{\\cal L}$ from $x$ is $\\varepsilon i$.\nCoerciveness is established by exhibiting scales $s_\\varepsilon$ such that if $u^\\varepsilon_i$ are such that \n\\begin{equation}\\label{2}\n\\sum_{i,j\\in{\\cal L}} a^\\varepsilon_{ij}|u^\\varepsilon_i-u^\\varepsilon_j|\\le s_\\varepsilon, \n\\end{equation}\nthen the interpolations $u^\\varepsilon$ are precompact in some topology and their limit points are in general non trivial. This can be expressed by proving that the domain of the $\\Gamma$-limit of the scaled energies\n\\begin{equation}\\label{2bis}\n{1\\over s_\\varepsilon}\\sum_{i,j\\in{\\cal L}} a^\\varepsilon_{ij}|u^\\varepsilon_i-u^\\varepsilon_j| \n\\end{equation}\nin that topology\nis not trivial.\n\n\n\nThe simplest case that has been previously treated \\cite{CDL,ABC} is nearest-neighbour interactions; i.e, when $a^\\varepsilon_{ij}$ are strictly positive only when $i,j$ are nearest neighbours (n.n.~for short) in the Delaunay triangulation of ${\\cal L}$ (e.g.,\n$|i-j|=1$ if ${\\cal L}=\\mathbb Z^d$). In this case choosing $s_\\varepsilon={\\varepsilon^{1-d}}$ gives that the scaled energies \n\\begin{equation}\\label{3}\n\\sum_{i,j\\in{\\cal L}\\ i,j\n \\,\\rm n.n.} \\varepsilon^{d-1} a^\\varepsilon_{ij}|u^\\varepsilon_i-u^\\varepsilon_j|\n\\end{equation}\ncan be directly seen as a (possibly anisotropic) perimeter of the sets $\\{x: u^\\varepsilon(x)=1\\}$ defined through the piecewise-constant interpolation of $u^\\varepsilon$ from the scaled lattice $\\varepsilon{\\cal L}$. Then, the compactness properties of sets of equibounded perimeter ensure the coerciveness in $L^1_{\\rm loc}(\\mathbb R^d)$ and the limits are characteristic functions. Moreover, the $\\Gamma$-limit of the energies can be described\nby an energy defined on sets of finite perimeter $A$, which, in the simplest homogenous case, takes the form\n\\begin{equation}\\label{4}\n\\int_{\\partial A} \\varphi(\\nu)d{\\mathcal H}^{d-1}.\n\\end{equation}\nThe same scaling works for finite-range interactions; i.e., when $a^\\varepsilon_{ij}$ is $0$ if $|i-j|>R$ for some $R$, even though the energies in that case must be interpreted as a non-local perimeter \\cite{BP}.\nThe finiteness of the range of the interactions can be weakened to a decay condition that can be quantified as \n\\begin{equation}\\label{4bis}\n\\sup\\Bigl\\{\\sum_{j\\in{\\cal L}\\setminus \\{i\\}}a^\\varepsilon_{i j}|j-i|: i\\in{\\cal L},\\ \\varepsilon>0\\Bigr\\}<+\\infty,\n\\end{equation}\neven though the limit energies may have a non-local part if the `tails' of these series are not uniformly negligible \\cite{AG}. We note that such analysis is valid beyond pair potentials and generalizes to classes of many-point interactions (see \\cite{BK}).\n\nIf the decay assumptions \\eqref{4bis} do not hold then the `natural' scaling for the energies may be different from the `surface scaling' $\\varepsilon^{d-1}$, and we might exit the class of interfacial energies. An extreme case is that of `dense graphs', which is better stated in a bounded domain; i.e., when \nconsidering energies\n\\begin{equation}\\label{5}\n\\sum_{i,j\\in{\\cal L}\\cap{1\\over\\varepsilon} Q} a^\\varepsilon_{ij}|u_i-u_j|,\n\\end{equation}\nwith $Q$ a cube in $\\mathbb R^d$, and suppose that $a^\\varepsilon_{ij}\\ge c>0$ for (a positive percentage of) all interactions.\nIn that case the scaling is $s_\\varepsilon= {\\varepsilon}^{-2d}$, and the limit behaviour is described by a more abstract limit functional called a `graphon' energy \\cite{Benjamini,Borgs2012,Lovasz2006,Lovasz2012}, which can be viewed as a relaxation of a double integral on $(0,1)$ of the form\n\\begin{equation}\\label{5}\n\\int_{(0,1)\\times(0,1)}W(x,y)|v(x)-v(y)|\\,dx\\,dy\n\\end{equation}\ndefined on $BV((0,1);\\{0,1\\})$ after a complex and rather abstract relabeling procedure and identification of functions defined on $Q$ with functions defined on $(0,1)$ (see \\cite{BCD}).\nFor sparse graphs (i.e., graphs which are not dense according to the definition above) and interactions not satisfying the decay conditions \\eqref{4bis}, the correct scaling, the relative convergence and the form of the $\\Gamma$-limit is a complex open problem. In \\cite{BCS} an example is given of one-dimensional energies with range $R_\\varepsilon={1\/\\sqrt\\varepsilon}$ such that\na non-trivial $\\Gamma$-limit exists for $s_\\varepsilon={1\/\\sqrt\\varepsilon}$ with respect to the $L^\\infty$-weak$^*$ convergence, but it is defined on {\\it all} functions of bounded variation with values in $[0,1]$ (and not only those with values in $\\{0,1\\}$). In that example a crucial issue is the topology of the graph of the connections where $a^\\varepsilon_{ij}\\neq0$.\n\nIn this paper we consider an intermediate case; i.e., when the decay condition described above does not hold, and $a^\\varepsilon_{ij}\\ge c>0$ when $|i-j|\\le R_\\varepsilon$ with $R_\\varepsilon>\\!>1$ but the topology of the interactions within that range is that of a `dense' graph.\nWe further make the assumption $\\varepsilon R_\\varepsilon<\\!<\\!1$ so that the discrete-to-continuum process makes sense. We note that this latter condition is not restrictive upon a redefinition of $\\varepsilon$ in terms of $R_\\varepsilon$; e.g.~taking $R_\\varepsilon^{-1\/2}$ in the place of $\\varepsilon$. We keep the dependence of our system on $R_\\varepsilon$ and $\\varepsilon$ separate since these parameters may be defined independently in applications. Under these conditions we have \n$$\ns_\\varepsilon={R_\\varepsilon^{d+1}\\over \\varepsilon^{d-1}},\n$$\nand with this scaling functions of equi-bounded energy interpolated on the lattice $\\varepsilon{\\cal L}$ converge to a characteristic function of a set of finite perimeter. The main argument for obtaining this result is by coarse-graining. Namely, we average the values of $u^\\varepsilon$ for interaction on cubes with side length of order $R_\\varepsilon$, so that we can think of those averages as labelled on $\\varepsilon R_\\varepsilon\\mathbb Z^d$. We prove first that those labels for which averages are not essentially close to $0$ and $1$ are negligible; hence, we may regard such functions as spin functions defined on a cubic lattice. Then, we show\nthat the arguments used for nearest-neighbour interactions of \\cite{ABC} can be adapted for the interpolated functions\nof the averages. Once a limit set of finite perimeter is obtained we can prove the convergence of the interpolations of the original functions to the same set.\n\nAs an application of this scaling argument we show that for \n\\begin{equation}\\label{6}\na^\\varepsilon_{ij}= a\\Bigl({i-j\\over R_\\varepsilon}\\Bigr),\n\\end{equation}\nwhere $a$ is a positive function with $\\int a(\\xi)|\\xi|d\\xi$ finite,\nthe $\\Gamma$-limit of the energies \n\\begin{equation}\\label{7}\nF_\\varepsilon(u)={\\varepsilon^{d-1}\\over R_\\varepsilon^{d+1}}\\sum_{i,j\\in\\mathbb Z^d} a^\\varepsilon_{ij}|u_i-u_j|,\n\\end{equation}\ndefined on the cubic lattice of $\\mathbb R^d$,\nis given by an energy as in \\eqref{4} with\n\\begin{equation}\\label{8}\n\\varphi(\\nu)=\\int_{\\mathbb R^d} a(\\xi)|\\langle \\xi,\\nu\\rangle|d\\xi.\n\\end{equation}\nIn particular, if $a$ is radially symmetric then \\eqref{4} is simply a multiple of the perimeter of $A$.\nIt is interesting to note that in a sense the case $R_\\varepsilon\\to+\\infty$ can be seen as a limit of the case of $R_\\varepsilon$ finite, for which the $\\Gamma$-limit is of the form \\eqref{4} with the integrand $\\varphi(\\nu)$ given by a discretization of the integral in \\eqref{8}\n(as seen in \\cite{Ch,BG, BLB} in a slightly different context). This convergence can be re-obtained using the results in \\cite{GS}, where transportation maps are used to transform discrete energies in convolution functionals.\n\n\n\n\n\n\\section{A compactness result}\nWe denote by $Q_R=[-R\/2,R\/2)^d$ the (semi-open) coordinate cube centered in $0$ and with side length $R$ in $\\mathbb R^d$, by $B_R$ the open ball  centered in $0$ and with side length $R$ in $\\mathbb R^d$, and by $e_1,\\dots, e_d$ the vectors of the canonical basis of $\\mathbb R^d$. Moreover, $\\mathcal H^{d-1}$ denotes the $d-1$-dimensional Hausdorff measure and $|\\cdot|$ the Lebesgue $d$-dimensional measure.  \n\nLet ${\\cal L}\\subset\\mathbb R^d$ be a discrete periodic set. We can suppose without loss of generality that it is periodic in the coordinate directions with period $1$; i.e., $${\\cal L}+e_i={\\cal L}\\hbox{ for all }i\\in\\{1,\\ldots,d\\}.$$\nThe Voronoi cells of ${\\cal L}$ are defined as\n$$\nV_i=\\{x\\in\\mathbb R^d: |x-i|<|x-j|\\hbox{ for all }j\\in{\\cal L}, \\ j\\neq i\\,\\}.\n$$\nBy the periodicity of ${\\cal L}$ there exists a constant $C_{\\cal L}>0$ such that\n\\begin{equation}\\label{Voro}\n{1\\over C_{\\cal L}}\\le |V_i|\\le C_{\\cal L},\\qquad {1\\over C_{\\cal L}}\\le{\\mathcal H}^{d-1}(\\partial V_i)\\le C_{\\cal L}.\n\\end{equation}\n\nEach spin function $u\\colon\\varepsilon\\mathbb {\\cal L}\\to\\{0,1\\}$ is identified with its piecewise-constant interpolation, which is the $L^\\infty$ function  defined by\n$$\nu(x)= u(\\varepsilon i)\\ \\hbox{ if } x\\in\\varepsilon V_i\\hbox{\\,; i.e., } |x-\\varepsilon i|<|x-\\varepsilon j| \\hbox{ for all }j\\in{\\cal L}, \\ j\\neq i\\,.\n$$\nNote that by \\eqref{Voro} the $L^1$ norm of such $u$ is equivalent to\n$\\varepsilon^d\\#\\{i: u_i\\neq 0\\}$.\n\n\nIn this section we prove coerciveness properties for energies $E_\\varepsilon$ defined on\nspin functions  $u\\colon\\varepsilon{\\cal L}\\to\\{0,1\\}$  by \n\\begin{equation}\\label{fe} \nE_\\varepsilon(u)=\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i,j\\in{\\cal L},\\ i-j\\in Q_{\\eta\/\\varepsilon}} |u_i-u_j|,\n\\end{equation}\nwhere we denote $u_i=u(\\varepsilon i)$, and $\\eta=\\eta_\\varepsilon$ are such that\n\\begin{equation}\\label{eta}\n\\lim_{\\varepsilon\\to 0}\\eta_\\varepsilon=\\lim_{\\varepsilon\\to0}{\\varepsilon\\over\\eta_\\varepsilon}=0.\n\\end{equation}\n \n \n \n\\begin{lemma}[Compactness]\\label{lemma}\nLet $u^\\varepsilon$ be spin functions such that $E_\\varepsilon(u^\\varepsilon)$ is equibounded. Then, up to subsequences, the corresponding piecewise-constant interpolations, still denoted by $u^\\varepsilon$, converge in $L^1_{\\rm loc}(\\mathbb R^d)$ to $u=\\chi_A$, where $A$ is a set of finite perimeter.\n\\end{lemma}\n\n\\begin{proof} The idea of the proof is to subdivide the set of indices ${\\cal L}$ into disjoint cubes of side-length $\\eta\/4\\varepsilon$. The factor $4$ is chosen so that if we consider $i,j$ indices  belonging to two neighbouring cubes with this side-length, respectively, then $i-j\\in Q_{\\eta\/\\varepsilon}$ so that they interact in energy $E_\\varepsilon$. In such a way we can associate to each $u^\\varepsilon$ and each such smaller cube the value $0$ or $1$ of the `majority phase', if such majority phase is sufficiently close to $0$ and $1$, respectively, while we prove that the remaining cubes can be neglected. In this way we will construct coarse-grained functions for which the energy $E_\\varepsilon$ can be viewed as a standard nearest-neighbour ferromagnetic energy and the compactness then follows by interpreting spin functions as sets of finite perimeter.\n\\bigskip\n\n \nFor any $k\\in\\mathbb Z^d$ we set \n$$Q_k^\\varepsilon=\\frac{\\eta}{4\\varepsilon}k+Q_{\\frac{\\eta}{4\\varepsilon}}.$$\nFor $u:\\varepsilon{\\cal L}\\to\\{0,1\\}$ we define\n$$D(\\varepsilon,k)(u)=\\frac{\\big|\\#\\{i\\in Q_k^\\varepsilon\\cap {\\cal L}: u_i=1\\}-\\#\\{i\\in Q_k^\\varepsilon\\cap {\\cal L}: u_i=0\\}\\big|}{\\# (Q_k^\\varepsilon\\cap {\\cal L})}.$$ \nNote that $D(\\varepsilon,k)(u)$ measures how much the function $u$ is close to its majority phase; more precisely, $D(\\varepsilon,k)(u)=1$ if $u$ is constant on $Q_k^\\varepsilon\\cap {\\cal L}$, while $D(\\varepsilon,k)(u)=0$ if the values of $u$ are equally distributed between $0$ and $1$ in $Q_k^\\varepsilon\\cap {\\cal L}$.\n\nWith fixed $\\delta\\in(0,1)$, we define \n$$\n\\mathcal B^\\varepsilon(u)=\\{k\\in\\mathbb Z^d:D(\\varepsilon,k)(u)<1-\\delta\\}.\n$$\nThe  $Q_k^\\varepsilon$ corresponding to $k\\in \\mathcal B^\\varepsilon(u)$ will be considered as the cubes where $u$ is not close to a phase $1$ or $0$. We will first show that such cubes are negligible. \nIndeed, note that thanks to the first inequality in \\eqref{Voro} the number of pairs of indices $i,j$ within $Q^\\varepsilon_k$ are of order $(\\eta\/\\varepsilon)^{2d}$ and hence there exists $C_\\delta>0$ such that if $k\\in \\mathcal B^\\varepsilon(u)$, then the number of `interactions within the cube' $Q_k^\\varepsilon$ is at least\n$C_\\delta (\\frac{\\eta}{\\varepsilon})^{2d}$; namely,\n$$\\#\\{(i,j): i,j\\in Q_k^\\varepsilon\\cap{\\cal L}, u_i\\neq u_j\\}\n\\geq C_\\delta \\Big(\\frac{\\eta}{\\varepsilon}\\Big)^{2d}.$$\n\nHence, if $u^\\varepsilon$ are as in the hypotheses of the lemma; that is, $F_\\varepsilon(u^\\varepsilon)\\leq c$, we have\n\\begin{equation*}\n\\#\\mathcal B^\\varepsilon(u^\\varepsilon)\\leq\\frac{c}{C_\\delta}\\eta^{1-d}.\n\\end{equation*}\nWe can estimate the measures \n\\begin{equation}\\label{misura-diversi}\n\\Big|\\bigcup_{k\\in\\mathcal B^\\varepsilon(u^\\varepsilon)} \\varepsilon Q_k^\\varepsilon\\Big|=\n\\#\\mathcal B^\\varepsilon(u^\\varepsilon){\\eta^d\\over 4^d}\n\\leq\\frac{c}{4^dC_\\delta}\\eta,\n\\end{equation}\n\\begin{equation}\\label{bordo-diversi}\n{\\mathcal H}^{d-1}\\Bigl(\\partial\\bigcup_{k\\in\\mathcal B^\\varepsilon(u^\\varepsilon)} \\varepsilon Q_k^\\varepsilon\\Big)=\n\\#\\mathcal B^\\varepsilon(u^\\varepsilon){2d\\,\\eta^{d-1}\\over 4^{d-1}}\n\\leq\\frac{2cd}{4^{d-1}C_\\delta}.\n\\end{equation}\n\nAs for the indices such that $D(\\varepsilon,k)(u)\\geq 1-\\delta$, we subdivide them into the sets\n\\begin{eqnarray*}\n\\mathcal A^\\varepsilon_1(u)=\\{ k\\in\\mathbb Z^d: D(\\varepsilon,k)(u)\\geq 1-\\delta, \\#\\{i\\in Q_k^\\varepsilon: u_i=1\\}>\\#\\{i\\in Q_k^\\varepsilon: u_i=0\\}\\}\\\\\n\\mathcal A^\\varepsilon_0(u)=\\{ k\\in\\mathbb Z^d: D(\\varepsilon,k)(u)\\geq 1-\\delta, \\#\\{i\\in Q_k^\\varepsilon: u_i=1\\}<\\#\\{i\\in Q_k^\\varepsilon: u_i=0\\}\\}\n\\end{eqnarray*}\nand define\n\\begin{eqnarray*}\nK_j^\\varepsilon(u)=\\bigcup_{k\\in \\mathcal A^\\varepsilon_j(u)} \\varepsilon Q_k^\\varepsilon\\hbox{ for }j=0,1.\n\\end{eqnarray*}\n\nIn order to estimate the measure of the boundary of $K_1^\\varepsilon(u)$ we estimate the number of cubes $Q_k^\\varepsilon$ with  \n$k\\in \\mathcal A^\\varepsilon_1(u)$ which have a side in common with a cube $Q_{k'}^\\varepsilon$ with \n$k'\\in\\mathcal A^\\varepsilon_0(u)$, parameterized on the set\n$$\n\\mathcal A^\\varepsilon(u):=\\{k\\in\\mathcal A_1^\\varepsilon(u): k+e_j\\in\\mathcal A_0^\\varepsilon(u) \\hbox{ for some } j=1,\\dots, d\\}\n$$\nTo that end, note that if $D(\\varepsilon,k)(u)\\ge 1-\\delta$ and $k\\in \\mathcal A^\\varepsilon_1(u)$ then \n$$\n\\#\\{i\\in Q_k^\\varepsilon\\cap {\\cal L}: u_i=1\\}\\ge \\Bigl(1-{\\delta\\over 2}\\Bigr)\\#\\{i\\in Q_k^\\varepsilon\\cap {\\cal L}\\},\n$$\nso that, again recalling the first inequality in \\eqref{Voro}, each site $i\\in Q_k^\\varepsilon$ such that $u_i=1$ interacts with $C'_\\delta (\\frac{\\eta}{\\varepsilon})^{d}$\nand conversely for each site $i\\in Q_{k'}^\\varepsilon$ such that $u_i=0$. Hence, the \ninteracting pairs $(i,j)\\in Q_k^\\varepsilon\\times Q_{k'}^\\varepsilon$ are at least \n$C'_\\delta(\\frac{\\eta}{\\varepsilon})^{2d}$.\n\nHence, \n$$\nE_\\varepsilon(u)\\ge \\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\#\\mathcal A^\\varepsilon(u)C''_\\delta(\\frac{\\eta}{\\varepsilon})^{2d}= C''_\\delta\\#\\mathcal A^\\varepsilon(u)\\eta^{d-1}\n$$\nso that \n$$\n\\#\\mathcal A^\\varepsilon(u)\\le  {1\\over C''_\\delta}E_\\varepsilon(u)\\eta^{1-d}.\n$$\n\nFor the functions $u^\\varepsilon$ we then obtain \n$$\n\\#\\mathcal A^\\varepsilon(u^\\varepsilon)\\le  \\frac{c}{C''_\\delta}\\eta^{1-d},\n$$\nso that\n\\begin{eqnarray*}\n{\\mathcal H}^{d-1}(\\partial K_1^\\varepsilon(u^\\varepsilon))&\\le & 2d\\biggl(\n{\\mathcal H}^{d-1}\\Bigl(\\bigcup_{k\\in\\mathcal A^\\varepsilon(u^\\varepsilon)} \\varepsilon Q_k^\\varepsilon\\Bigr)\n+{\\mathcal H}^{d-1}\\Bigl(\\bigcup_{k\\in\\mathcal B^\\varepsilon(u^\\varepsilon)} \\varepsilon Q_k^\\varepsilon\\Bigr)\\biggr)\\\\\n&\\le& 2d\\Bigl(\\#\\mathcal A^\\varepsilon(u^\\varepsilon) {2d\\,\\eta^{d-1}\\over 4^{d-1}}+\\frac{2cd}{4^{d-1}C_\\delta}\\Bigr)\\le C^{'''}_\\delta\n\\end{eqnarray*}\nwhere $C^{'''}_\\delta$ is a positive constant depending only on $d, c$ and $\\delta$. \nBy the compactness of sets of equibounded perimeter this shows that the characteristic functions of the sets $K_1^\\varepsilon(u^\\varepsilon)$ are compact in $L^1_{\\rm loc}(\\mathbb R^d)$. The symmetric argument shows also that $K_0^\\varepsilon(u^\\varepsilon)$ are compact in $L^1_{\\rm loc}(\\mathbb R^d)$. Moreover, if we denote a limit of the sets  $K_j^\\varepsilon(u^\\varepsilon)$ by $K_j$ then we have \n\\begin{equation}\\label{tutto}|\\mathbb R^d\\setminus (K_0\\cup K_1)|=0\\end{equation} by\n\\eqref{misura-diversi}. We highlight the possible dependence of the sets obtained by this procedure on $\\delta$ by renaming them \n$K_1^\\delta$ and $K_0^\\delta$.\n\nNote that if $\\delta<\\delta'$ then \n$$\nK_1^{\\delta'}\\subset K_1^\\delta\\hbox{ and } \nK_0^{\\delta'}\\subset K_0^\\delta.\n$$\nSince in both cases \\eqref{tutto} holds, then we must have $K_1^{\\delta'}=K_1^\\delta$ \nand $K_0^{\\delta'}=K_0^\\delta$, so that these sets \nare independent of $\\delta$ and we may go back to denoting them by $K_1$ \nand $K_0$.\n\n\nWe can now prove the convergence of $u^\\varepsilon$. Fixed $\\delta<1$ as above, we write\n$$\nu^\\varepsilon= u^\\varepsilon\\chi_{K_1^\\varepsilon(u^\\varepsilon)}+ u^\\varepsilon\\chi_{K_0^\\varepsilon(u^\\varepsilon)}+ u^\\varepsilon\\chi_{\\mathbb R^d\\setminus (K_1^\\varepsilon(u^\\varepsilon)\\cup K_0^\\varepsilon(u^\\varepsilon))}.\n$$\nBy \\eqref{misura-diversi} the last term converges to $0$ in $L^1(\\mathbb R^d)$\nAs for the other two terms we localize the convergence by restricting to a cube $Q_R$.\nNote that for $k\\in \\mathcal A^\\varepsilon_1(u^\\varepsilon)$ we have\n$$\n\\|u^\\varepsilon- 1\\|_{L^1(\\varepsilon Q^\\varepsilon_k)}\\le C(1-\\delta)\\eta^d,\n$$\nso that \n$$\n\\|u^\\varepsilon\\chi_{K_1^\\varepsilon(u^\\varepsilon)\\cap Q_R}- \\chi_{K_1^\\varepsilon(u^\\varepsilon)\\cap Q_R}\\|_{L^1(\\mathbb R^d)}\\le C(1-\\delta)R^d \n$$\nwhere $C$ denotes a positive constant not depending on $\\delta.$\nAnalogously for $k\\in \\mathcal A^\\varepsilon_0(u^\\varepsilon)$ we have\n$$\n\\|u^\\varepsilon\\|_{L^1(\\varepsilon Q^\\varepsilon_k)}\\le C(1-\\delta)\\eta^d,\n$$\nand hence \n$$\n\\|u^\\varepsilon\\chi_{K_0^\\varepsilon(u^\\varepsilon)\\cap Q_R}\\|_{L^1(\\mathbb R^d)}\\le C(1-\\delta)R^d.\n$$\nWe then have, by the local convergence of $K^\\varepsilon_j(u^\\varepsilon)$, \n\\begin{eqnarray*}&&\n\\limsup_{\\varepsilon\\to0}\\|u^\\varepsilon \\chi_{K_1^\\varepsilon(u^\\varepsilon)\\cap Q_R}-\\chi_{K_1\\cap Q_R}\\|_{L^1(\\mathbb R^d)}\n\\\\\n&\\le&\n\\limsup_{\\varepsilon\\to 0}\\Bigl(\\|u^\\varepsilon \\chi_{K_1^\\varepsilon(u^\\varepsilon)\\cap Q_R}-\\chi_{K^\\varepsilon_1(u^\\varepsilon)\\cap Q_R}\\|_{L^1(\\mathbb R^d)}+\n\\| \\chi_{K_1^\\varepsilon(u^\\varepsilon)\\cap Q_R}-\\chi_{K_1\\cap Q_R}\\|_{L^1(\\mathbb R^d)}\\Bigr)\n\\\\\n&\\le&C(1-\\delta)R^d,\n\\end{eqnarray*}\nand\n$$\n\\limsup_{\\varepsilon\\to0}\\|u^\\varepsilon \\chi_{K_0^\\varepsilon(u^\\varepsilon)\\cap Q_R}\\|_{L^1(\\mathbb R^d)}\n\\le C(1-\\delta)R^d,\n$$\nso that, by the arbitrariness of $\\delta$, $u^\\varepsilon$ converge locally to $\\chi_{K_1}$.\n\\end{proof}\n\n\\begin{remark}\\rm \nThe proof of Lemma \\ref{lemma} works exactly in the same way if we suppose that `almost all' pairs of indices of ${\\cal L}$ within  $Q_{\\eta\/\\varepsilon}$ interact; namely, if in place of energy \\eqref{fe} we consider\n\\begin{equation}\\label{fe-a} \nE_\\varepsilon(u)=\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i,j\\in{\\cal L}:\\ i-j\\in Q_{\\eta\/\\varepsilon}} a^\\varepsilon_{ij} |u_i-u_j|,\n\\end{equation}\nwith the requirement that there exists $c>0$ such that\n\\begin{equation}\\label{lim-a} \n\\lim_{\\varepsilon\\to 0}{\\#\\{(i,j):i,j\\in x+Q_{\\eta\/\\varepsilon}: a^\\varepsilon_{ij}\\ge c\\}\\over \\#\\{(i,j):i,j\\in x+Q_{\\eta\/\\varepsilon}\\}}=1\n\\end{equation}\nuniformly in $x\\in\\mathbb R^d$. Condition \\eqref{lim-a} is trivially satisfied by energies \\eqref{fe} for $c=1$.\n\nNote that condition \\eqref{lim-a} cannot be relaxed to `having a proportion' of pairs of indices of ${\\cal L}$ within  $Q_{\\eta\/\\varepsilon}$ interacting, however large this proportion may be below $1$; i.e., it is not sufficient that\n\\begin{equation}\\label{lim-a2} \n\\lim_{\\varepsilon\\to 0}{\\#\\{(i,j):i,j\\in x+Q_{\\eta\/\\varepsilon}: a^\\varepsilon_{ij}\\ge c\\}\\over \\#\\{(i,j):i,j\\in x+Q_{\\eta\/\\varepsilon}\\}}\\ge \\lambda,\n\\end{equation}\nfor any $\\lambda<1$. To check this, we may consider the following example: choose ${\\cal L}=\\mathbb R^d$, fix $N\\in{\\mathbb N}$, and define\n$$\na^\\varepsilon_{ij}=\\begin{cases}\n1 &\\hbox{ if }i-j\\in Q_{\\eta\/\\varepsilon}\\hbox{ and both }i,j\\not\\in N\\mathbb Z\\\\\n0 &\\hbox{ otherwise.}\n\\end{cases}\n$$\nThen \\eqref{lim-a2} holds for $\\lambda= \\bigl(1-{1\\over N^d}\\bigr)^2$ but, if we define \n$$\nu^\\varepsilon_i=\\begin{cases}\n1 &\\hbox{ if }i\\not\\in N\\mathbb Z\\\\\n0 &\\hbox{ if }i\\in N\\mathbb Z,\n\\end{cases}\n$$\nthen $u^\\varepsilon$ converge weakly in $L^1_{\\rm loc}(\\mathbb R^d)$ to the constant $1-{1\\over N^d}$.\nSince $E_\\varepsilon(u_\\varepsilon)=0$ this shows that Lemma \\ref{lemma} does not hold.\n\nIn this example the subset $N\\mathbb Z^d$ of $\\mathbb Z^d$ can be considered as a `perforation' of the domain and can be treated as such, considering convergence only of the restriction of the functions to $\\mathbb Z^d\\setminus N\\mathbb Z^d$ (see Section 3 of \\cite{BCPS}). However, the situation can be more complicated if we take\n$$\na^\\varepsilon_{ij}=\\begin{cases}\n1 &\\hbox{ if }i-j\\in Q_{\\eta\/\\varepsilon}\\hbox{ and both }i,j\\not\\in N\\mathbb Z\\\\\nc_\\varepsilon &\\hbox{ otherwise,}\n\\end{cases}\n$$\nthat can be regarded as representing a `high-contrast medium', for which the effect of the `perforation' cannot be neglected and for some values of $c_\\varepsilon$ may give a `double porosity' effect \\cite{BCPS}.\n\\end{remark}\n\n\n\n\\section{Homogenization of long-range lattice systems}\n\nLet $a:\\mathbb R^d\\to [0,+\\infty)$ be such that $a(\\xi)|\\xi|$ is Riemann integrable on bounded sets and such that\n\\begin{equation}\\label{a1}\n\\int_{\\mathbb R^d} a(\\xi)|\\xi|\\,d\\xi<+\\infty,\n\\end{equation}\nand \n\\begin{equation}\\label{a2}\n a(\\xi)\\geq c_0 \\hbox{ if } \\ |\\xi|\\leq r_0\n\\end{equation}\nfor some $c_0,r_0>0$.\n\n\nGiven $\\varepsilon,\\eta=\\eta_\\varepsilon$ satisfying \\eqref{eta} we define the coefficients\n$$\na_{ij}^\\varepsilon=a_{i-j}^\\varepsilon=a\\Big(\\frac{\\varepsilon (i-j)}{\\eta}\\Big)\n$$ \nfor $i,j\\in\\mathbb Z^d$,\nand the energies \n\\begin{equation}\\label{fe-a} \nF_\\varepsilon(u)=\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i,j\\in\\mathbb Z^d} a^\\varepsilon_{i-j}|u_i-u_j|.\n\\end{equation}\n\n\n\\begin{definition}\\label{conv}\nA family $\\{u^\\varepsilon\\}$ of functions $u^\\varepsilon:\\varepsilon\\mathbb Z^d\\to\\{0,1\\}$ {\\rm converges} to a set $A\\subset\\mathbb R^d$ if the piecewise-constant interpolations of $u^\\varepsilon$ converge to the characteristic function $\\chi_A$ in $L^1_{\\rm loc}(\\mathbb R^d)$ as $\\varepsilon\\to0$.\n\\end{definition}\n\nBy hypothesis \\eqref{a2} we may apply Compactness Lemma \\ref{lemma}, obtaining that the family $\\{F_\\varepsilon\\}$ is coercive with respect to this convergence.\n\n\\begin{proposition}\nLet $\\{u^\\varepsilon\\}$ be such that $\\sup_\\varepsilon F_\\varepsilon(u^\\varepsilon)<+\\infty$. Then, up to subsequences, there exists a set of finite perimeter $A$ such that $u^\\varepsilon$ converge to $A$ in the sense of Definition {\\rm\\ref{conv}}.\n\\end{proposition}\n\nThis coerciveness property justifies the computation of the $\\Gamma$-limit of $F_\\varepsilon$ with respect to the convergence in \nDefinition \\ref{conv}. We use standard notation in the theory of sets of finite perimeter (see e.g.~\\cite{LN98,Maggi}).\n\n\\begin{theorem}[Homogenization]\\label{theorem}\nThe functionals defined in \\eqref{fe-a} $\\Gamma$-converge with respect to the convergence in Definition {\\rm\\ref{conv}} to the \nfunctional $F$ defined on sets of finite perimeter by\n\\begin{equation}\\label{f-a} \nF(A)=\\int_{\\partial^* A} \\varphi_a(\\nu)d{\\mathcal H}^{d-1},\n\\end{equation}\nwhere $\\partial^* A$ denotes the reduced boundary of $A$, $\\nu$ the outer normal to $A$ and $\\varphi_a$ is given by\n\\begin{equation}\\label{fi-a}\n\\varphi_a(\\nu)=\\int_{\\mathbb R^d} a(\\xi)|\\langle\\xi,\\nu\\rangle|d\\xi.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nIn order to better illustrate the proof in the general $d$-dimensional case we first deal with the one-dimensional case, in which we may highlight the coarse-graining procedure without the technical complexities of the higher-order geometry. \nIn this case\nwe may rewrite the energies as\n$$\nF_\\varepsilon(u)=\\frac{\\varepsilon^{2}}{\\eta^{2}}\\sum_{\\xi\\in\\mathbb Z}\\sum_{i\\in\\mathbb Z} a^\\varepsilon_{\\xi}|u_{i+\\xi}-u_i|.\n$$\nThe relevant \ncomputation is that of the lower bound for the target $A=[0,+\\infty)$. Let $u^\\varepsilon$ converge to $A$. For each $\\xi\\in\\mathbb Z\\setminus\\{0\\}$ and $i\\in\\{1,\\ldots,|\\xi|\\}$ we consider the function $u^\\varepsilon$ restricted to $\\varepsilon i+\\varepsilon \\xi\\mathbb Z$. By the $L^1_{\\rm loc}$ convergence, we may suppose that each such restriction changes value; i.e., there exists some $k_{i,\\xi}\\in\\mathbb Z$ such that \n$$\nu^\\varepsilon(\\varepsilon i+\\varepsilon k_{i,\\xi}\\xi)=0 \\hbox{ and } u^\\varepsilon(\\varepsilon i+\\varepsilon (k_{i,\\xi}+1)\\xi)=1.\n$$\nThe set of $\\xi$ and $i$ for which this does not hold is negligible for $\\varepsilon\\to0$; the precise proof is directly given for the $d$-dimensional functionals below. For each $\\xi$ we then have\n$$\n\\sum_{i=1}^{|\\xi|} \\sum_{k\\in\\mathbb Z}a^\\varepsilon_{\\xi}|u^\\varepsilon_{i+(k+1)\\xi}-u^\\varepsilon_{i+k\\xi}|\\ge |\\xi| a\\Bigl({\\varepsilon\\over\\eta}\\xi\\Bigr),\n$$\nso that \n\\begin{eqnarray*}\n\\liminf_{\\varepsilon\\to0} F_\\varepsilon(u^\\varepsilon)\\ge\\liminf_{\\varepsilon\\to0}{\\varepsilon^2\\over\\eta^2}\\sum_{\\xi\\in\\mathbb Z} |\\xi| a\\Bigl({\\varepsilon\\over\\eta}\\xi\\Bigr)=\\liminf_{\\varepsilon\\to0}\\sum_{\\xi\\in\\mathbb Z}  {\\varepsilon\\over\\eta} a\\Bigl({\\varepsilon\\over\\eta}\\xi\\Bigr)\\Bigl|{\\varepsilon\\over\\eta}\\xi\\Bigr|,\n\\end{eqnarray*}\nthe latter being a Riemann sum giving the integral $\\displaystyle\\int_{\\mathbb R}a(\\xi)|\\xi|\\,d\\xi$, which is $F(A)$.\n\n\\bigskip\nWe now deal with the $d$-dimensional case. The proof of the lower bound follows the argument above, but is more complex since we must take into account the direction of the interaction vectors $\\xi$.\n\nWe prove the inequality by applying the blow-up technique (see \\cite{fomu} and \\cite{BMS}, and for instance \\cite{BS,BP,NSS} for the discrete setting). \n\nWe assume that the sequence $\\{F_\\varepsilon(u^{\\varepsilon})\\}$ is equibounded and that $u^\\varepsilon$ converge in $L^1_{\\rm loc}(\\mathbb R^d)$ to $u=\\chi_A$, where $A$ is a set of finite perimeter. Up to subsequences, we can assume that $\\liminf_{\\varepsilon\\to 0} F_\\varepsilon(u^\\varepsilon)=\\lim_{\\varepsilon\\to 0}F_\\varepsilon(u^\\varepsilon)$. \nWe define the localized energy on an open set $U$ by\n$$\nF_\\varepsilon(u^{\\varepsilon};U)= \\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i\\in U} \\sum_{j\\in\\mathbb Z^d}a^\\varepsilon_{i-j}|u^\\varepsilon_i-u^\\varepsilon_j|, \n$$\nand define the measures $\\mu_\\varepsilon(U)=F_\\varepsilon(u^{\\varepsilon};U)$; since the family $\\{\\mu_\\varepsilon\\}$ is equibounded, we can assume that  $\\mu_\\varepsilon\\buildrel * \\over \\rightharpoonup \\mu$ up to subsequences. \nNow, let $\\lambda=\\mathcal H^{d-1}\\ \\rule{.4pt}{7pt}\\rule{6pt}{.4pt}\\  \\partial^\\ast A$; the lower bound inequality \nfollows if we show that for $\\mathcal H^{d-1}$-a.a.~$x\\in\\partial^\\ast A$ we have \n$$\\frac{d\\mu}{d \\lambda}(x)\\geq \\varphi_a(\\nu),$$ \nwhere $\\frac{d\\mu}{d \\lambda}$ denotes the Radon-Nikodym derivative of $\\mu$ with respect to the Hausdorff $d-1$-dimensional measure $\\lambda$. \nBy the Besicovitch Derivation Theorem, for $\\mathcal H^{d-1}$-a.a.~$x\\in\\partial^\\ast A$ we have that \n$$\\frac{d\\mu}{d \\lambda}(x)=\\lim_{\\varrho\\to 0}\\frac{\\mu(Q_\\varrho^\\nu(x))}{\\lambda(Q_\\varrho^\\nu(x))},$$\nwhere $\\lambda$ is the measure $\\mathcal H^{d-1}\\ \\rule{.4pt}{7pt}\\rule{6pt}{.4pt}\\  \\partial^\\ast A$, $\\nu$ is the normal vector to $\\partial^\\ast A$ at $x$ and \n$Q_\\varrho^\\nu(x)$ is a cube centered in $x$ with side length $\\varrho$ and a face orthogonal to $\\nu$. \nWe can fix $x=0$ and denote $Q_\\varrho^\\nu(0)$ by $Q_\\varrho^\\nu$. \nHence, the lower bound follows if we show that \n\\begin{equation}\\label{lower}\n\\lim_{\\varrho\\to 0}\\liminf_{\\varepsilon\\to 0}\\frac{1}{\\varrho^{d-1}}F_\\varepsilon(u^{\\varepsilon};Q_\\nu^\\varrho) \\ge \\varphi_a(\\nu). \n\\end{equation}\nWe may therefore assume that $\\varrho=\\varrho_\\varepsilon$ be such that $\\frac{\\varepsilon}{\\varrho}\\to 0$ and the scaled functions $u^\\varepsilon(\\frac{\\varepsilon}{\\varrho}i)$ interpolated on the lattice $\\frac{\\varepsilon}{\\varrho}\\mathbb Z^d$ converge to the characteristic function of the half space $H^\\nu=\\{x: \\langle x,\\nu\\rangle<0\\}$ on $Q_\\nu^1$. We define \n$$\nA_\\varepsilon:=\\{x\\in Q_\\nu^1: u^\\varepsilon(x)\\neq \\chi_{H^\\nu}(x)\\},\n$$\nso that $|A_\\varepsilon|\\to 0$.\n\n\nIf we define\n$$I_{\\varepsilon\/\\varrho}^\\xi=\\Bigl\\{i\\in\\mathbb Z^d: \\frac{\\varepsilon}{\\varrho} i, \\frac{\\varepsilon}{\\varrho} (i+\\xi)\\in Q_\\nu^1\\Bigr\\}$$\nthen  \n$$\nF_\\varepsilon(u^{\\varepsilon};Q_\\nu^\\varrho)\\ge \\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{\\xi\\in\\mathbb Z^d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\sum_{i\\in I_{\\varepsilon\/\\varrho}^\\xi}\n\\Bigl|u^\\varepsilon(\\frac{\\varepsilon}{\\varrho} (i+\\xi))-u^\\varepsilon(\\frac{\\varepsilon}{\\varrho}i)\\Bigr|.$$\n\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[width=0.6\\textwidth]{Fig1bis}}\n\\caption{The set $R^{\\alpha,\\xi}$}\n\\label{Fig1}\n\\end{figure} \nWe begin by estimating\n$$\n\\sum_{i\\in I_{\\varepsilon\/\\varrho}^\\xi}\n\\Bigl|u^\\varepsilon(\\frac{\\varepsilon}{\\varrho} (i+\\xi))-u^\\varepsilon(\\frac{\\varepsilon}{\\varrho}i)\\Bigr|=\\#\\Bigl\\{i\\in I_{\\varepsilon\/\\varrho}^\\xi: \nu^\\varepsilon(\\frac{\\varepsilon}{\\varrho} (i+\\xi))\\neq u^\\varepsilon(\\frac{\\varepsilon}{\\varrho}i)\\Bigr\\}.$$\nWith fixed  $\\alpha\\in (0,1)$ for each $\\xi\\in\\mathbb Z^d$ satisfying \n\\begin{equation}\\label{alfa-xi}\\Bigl|\\langle \\frac{\\xi}{|\\xi|},\\nu\\rangle\\Bigr|\\geq  \\frac{\\alpha}{\\sqrt{1+\\alpha^2}},\n\\end{equation}\nwe define \n$$P^{\\alpha,\\xi}=\\Big\\{y\\in \\Pi_\\nu\\cap Q_\\nu^1: y\\pm \\frac{\\alpha}{2|\\langle \\xi,\\nu \\rangle|} \\xi\\in Q_\\nu^1\\Big\\},$$\nwhich is not empty by \\eqref{alfa-xi}, and \n$$R^{\\alpha,\\xi}=\\Big\\{x\\in Q_\\nu^1: x=y+t \\xi, \\ y\\in P^{\\alpha,\\xi}, \\ -\\frac{\\alpha}{2|\\langle \\xi,\\nu \\rangle|}\\leq t\\leq  \\frac{\\alpha}{2|\\langle \\xi,\\nu \\rangle|}\\Big\\}$$ \n(see Fig.~\\ref{Fig1}). \nFurthermore, we fix $\\beta$ with\n\\begin{equation}\\label{beta}\\beta> \\frac{\\alpha}{\\sqrt{1+\\alpha^2}}.\n\\end{equation} \nSince we will restrict our arguments to sets $P^{\\alpha,\\xi}$ and $R^{\\alpha,\\xi}$ above with $\\xi$ satisfying\n\\begin{equation}\\label{beta-xi}\\Bigl|\\langle \\frac{\\xi}{|\\xi|},\\nu\\rangle\\Bigr|\\geq  \\beta;\n\\end{equation}\nwe omit the dependence of the sets $P^{\\alpha,\\xi}$ and $R^{\\alpha,\\xi}$ on $\\nu$, since the estimates we will obtain will be independent on $\\nu$.\n\nAs in the one-dimensional case we consider the functions restricted to the discrete lines $\\frac{\\varepsilon}{\\varrho}i+\\frac{\\varepsilon}{\\varrho}\\xi\\mathbb Z$.\nThe parameter $\\alpha$ is introduced so as to estimate the number of sites of such discrete lines inside $Q_\\nu^1$.\nWe then set\n$$B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}=\\Bigl\\{i\\in \\mathbb Z^d: \\frac{\\varepsilon}{\\varrho}i\\in R^{\\alpha,\\xi} \\hbox{ and } \nu^\\varepsilon \\hbox{ is not constant in } \\Big(\\frac{\\varepsilon}{\\varrho}i+\\frac{\\varepsilon}{\\varrho}\\xi\\mathbb Z\\Big)\\cap R^{\\alpha,\\xi}\\Bigr\\}.$$\nNote that if $i\\in B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}$ then $i+k\\xi\\in B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}$ for all $k$ with $\\frac{\\varepsilon}{\\varrho}(i+k\\xi)\\in R^{\\alpha,\\xi}$, so that, if we define the equivalence relation $i\\sim i^\\prime$ if $i-i^\\prime\\in \\xi\\mathbb Z$, \nwe may set\n$$\\widetilde B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}=B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}\/\\sim$$ \ngetting\n$$\n\\#\\Bigl\\{i\\in I_{\\varepsilon\/\\varrho}^\\xi: \nu^\\varepsilon(\\frac{\\varepsilon}{\\varrho} (i+\\xi))\\neq u^\\varepsilon(\\frac{\\varepsilon}{\\varrho}i)\\Bigr\\}\\geq \n\\#\\widetilde B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}.$$\n\nWe can estimate the number of `discrete lines' intersecting $R^{\\alpha,\\xi}$ as\n\\begin{eqnarray*}\n\\#\\Bigl(\\Bigl\\{i\\in\\mathbb Z^d: \\frac{\\varepsilon}{\\varrho}i\\in R^{\\alpha,\\xi}\\Bigr\\}\/\\sim\\Bigr)  &\\geq&  \n\\frac{|R^{\\alpha,\\xi}|}{\\bigl(\\frac{\\varepsilon}{\\varrho}\\bigr)^d}\\frac{1}{\n\\frac{\\alpha}{\\frac{\\varepsilon}{\\varrho}|\\langle\\xi,\\nu\\rangle|}\n}-C_\\alpha|\\xi|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d} \\\\\n&\\geq&  \n\\mathcal H^{d-1}(P^{\\alpha,\\xi})|\\langle\\xi,\\nu\\rangle|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}-\nC_\\alpha|\\xi|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d},\n\\end{eqnarray*}\nwhere the last term is an error term accounting for the cubes intersecting the boundary of $ R^{\\alpha,\\xi}$.\n\nNote that to every element of the complement of  $\\tilde B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}$ \nthere correspond at least\n$\\lfloor \\frac{\\alpha}{\\frac{\\varepsilon}{\\varrho} |\\langle \\xi, \\nu\\rangle|}\\rfloor$ points in ${\\varepsilon\\over\\varrho}\\mathbb Z^d\\cap A_\\varepsilon\\}$, so that\nfor $\\varepsilon$ sufficiently small we get\n\\begin{eqnarray*}\n&&\\#\\Bigl(\\Bigl\\{i\\in \\mathbb Z^d: \\frac{\\varepsilon}{\\varrho}i\\in R^{\\alpha,\\xi} \\hbox{ and } \nu^\\varepsilon \\hbox{ constant in } \\big(\\frac{\\varepsilon}{\\varrho}i+\\frac{\\varepsilon}{\\varrho}\\xi\\mathbb Z\\big)\\cap R^{\\alpha,\\xi}\\Bigr\\} \n\/\\sim \\Bigr)\\\\\n&&\\hspace{1cm}\\leq \\frac{|A_\\varepsilon|}{\\bigl(\\frac{\\varepsilon}{\\varrho}\\bigr)^d} \\frac{\\frac{\\varepsilon}{\\varrho}|\\langle \\xi, \\nu\\rangle|}{\\alpha} + C'_\\alpha|\\xi|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d}\\\\\n&&\\hspace{1cm}= \\frac{1}{\\alpha}|A_\\varepsilon| \\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}\n|\\langle \\xi, \\nu\\rangle| + C'_\\alpha|\\xi|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d},\n\\end{eqnarray*}\nwith $C'_\\alpha$ again a positive constant accounting for boundary cubes,\nand hence \\begin{equation}\\label{stima}\n\\#\\widetilde B^{\\alpha,\\xi}_{\\varepsilon\/\\varrho}\\geq\n\\mathcal H^{d-1}(P^{\\alpha,\\xi})|\\langle\\xi,\\nu\\rangle|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}-\\frac{1}{\\alpha}|A_\\varepsilon| \\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}\n|\\langle \\xi, \\nu\\rangle|\n-\n(C_\\alpha+C'_\\alpha)|\\xi|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d}.\n\\end{equation}\nBy \\eqref{beta-xi} we can estimate\n\\begin{equation}\\label{misuraP}\n\\mathcal H^{d-1}(P^{\\alpha,\\xi})\\geq \\Bigl(1-\\frac{\\alpha}{2|\\langle\\xi,\\nu\\rangle|}|\\xi-\\langle\\xi,\\nu\\rangle\\nu|\\Bigr)^{d-1}\n\\geq \\Big(1-\\frac{\\alpha}{2}\\sqrt{\\frac{1}{\\beta^2}-1}\\Big)^{d-1}\n\\end{equation}\n(see also Fig.~\\ref{Fig1}), and hence, upon fixing $R>0$ and introducing the set \n$$\\Xi_\\varepsilon^\\nu(R,\\beta)=\n\\Bigl\\{\\xi\\in \\mathbb Z^d: |\\xi|\\leq \\frac{\\eta}{\\varepsilon}R, \\Bigl|\\langle \\frac{\\xi}{|\\xi|},\\nu \\rangle\\Bigr|\\geq \\beta\\Bigr\\},$$ \nby \\eqref{stima} and \\eqref{misuraP} we have  \n\\begin{eqnarray*}\n\\frac{1}{\\varrho^{d-1}}F_\\varepsilon(u^{\\varepsilon};Q_\\nu^\\varrho) &\\geq & \n\\frac{1}{\\varrho^{d-1}}\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\mathcal H^{d-1}(P^{\\alpha,\\xi})|\\langle\\xi,\\nu\\rangle|\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}\\\\\n&&-\\frac{1}{\\varrho^{d-1}}\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\frac{1}{\\alpha}|A_\\varepsilon| \\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{1-d}\n|\\langle \\xi, \\nu\\rangle|\\\\\n&&-\n(C_\\alpha+C'_\\alpha)\\Bigl(\\frac{\\varepsilon}{\\varrho}\\Bigr)^{2-d}\n\\frac{1}{\\varrho^{d-1}}\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\n\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)|\\xi|\\\\\n&\\geq& \n\\Big(1-\\frac{\\alpha}{2}\\sqrt{\\frac{1}{\\beta^2}-1}\\Big)^{d-1}\n\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}\\Bigl(\\frac{\\varepsilon}{\\eta}\\Bigr)^{d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n|\\langle \\frac{\\varepsilon}{\\eta}\\xi,\\nu\\rangle|\\\\\n&&-\\frac{1}{\\alpha}|A_\\varepsilon| \\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}\\Bigl(\\frac{\\varepsilon}{\\eta}\\Bigr)^{d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n|\\langle \\frac{\\varepsilon}{\\eta}\\xi,\\nu\\rangle|\\\\\n&&-\n(C_\\alpha+C'_\\alpha)\n\\frac{\\varepsilon}{\\varrho}\n\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}\\Bigl(\\frac{\\varepsilon}{\\eta}\\Bigr)^{d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\\Bigl|\\frac{\\varepsilon}{\\eta}\\xi\\Bigr|.\n\\end{eqnarray*}\n\nSince $|A_\\varepsilon|\\to 0$ and \n\\begin{eqnarray*}\n \\lim_{\\varepsilon\\to 0}\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}\\Bigl(\\frac{\\varepsilon}{\\eta}\\Bigr)^{d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\Bigl|\\langle \\frac{\\varepsilon}{\\eta}\\xi,\\nu\\rangle\\Bigr|\n&=&\\int_{\\{|\\xi|\\leq R, |\\langle \\xi\/|\\xi|, \\nu\\rangle|\\ge \\beta\\}}a(\\xi)|\\langle \\xi,\\nu\\rangle|\\, d\\xi,\\\\\n \\lim_{\\varepsilon\\to 0}\\sum_{\\xi\\in\\Xi_\\varepsilon^\\nu(R,\\beta)}\\Bigl(\\frac{\\varepsilon}{\\eta}\\Bigr)^{d}a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\Bigl|\\frac{\\varepsilon}{\\eta}\\xi\\Bigr|\n&=&\\int_{\\{|\\xi|\\leq R, |\\langle \\xi\/|\\xi|, \\nu\\rangle|\\ge \\beta\\}}a(\\xi)| \\xi|\\, d\\xi,\n\\end{eqnarray*}\nwe get\\begin{eqnarray*}\\liminf_{\\varepsilon\\to0}\n\\frac{1}{\\varrho^{d-1}}F_\\varepsilon(u^{\\varepsilon};Q_\\nu^\\varrho) &\\geq & \n\\Big(1-\\frac{\\alpha}{2}\\sqrt{\\frac{1}{\\beta^2}-1}\\Big)^{d-1}\n\\int_{\\{|\\xi|\\leq R, |\\langle \\xi\/|\\xi|, \\nu\\rangle|\\ge\\beta\\}}a(\\xi)|\\langle\\xi,\\nu\\rangle|\\, d\\xi .\n\\end{eqnarray*}\nNote that by \\eqref{beta} we may let first  $\\alpha\\to 0$ and then $\\beta\\to 0$.\nWe eventually obtain\n\\begin{eqnarray*}\n\\liminf_{\\varepsilon\\to0}\\frac{1}{\\varrho^{d-1}}F_\\varepsilon(u^{\\varepsilon};Q_\\nu^\\varrho) \\geq \n\\int_{\\{|\\xi|\\le R\\}}a(\\xi)|\\langle\\xi,\\nu\\rangle|\\, d\\xi,\n\\end{eqnarray*}\nwhich, by the arbitrariness of $R$, gives \\eqref{lower}.\n\n\n\\begin{figure}[h!]\n\\centerline{\\includegraphics[width=0.8\\textwidth]{Fig2bis}}\n\\caption{Upper-bound construction}\n\\label{Fig2}\n\\end{figure} \n\\bigskip\nThe upper bound is obtained by a density argument (see \\cite{GCB} Section 1.7). Hence, it suffices to treat the case of $A$ polyhedral. In this case it suffices to take (the interpolations) $u^\\varepsilon_i=\\chi_A(\\varepsilon i)$ for $i\\in\\mathbb Z^d$. \nIndeed, \nwe write $\\partial A$ as a union of $N$ $d-1$-dimensional polytopes $\\Sigma_k$ and we denote by $\\nu_k$ the outer normal to $\\Sigma_k$ and by $K$ the $d-2$-dimensional skeleton of $A$. \n\nWe note that there exists a constant $C$ depending only on $A$ such that, for any $\\eta,R>0$,\nafter removing the closed neighborhood\n$K+\\overline B_{C \\eta R}$ from $\\partial A$,  \nwe obtain a disjoint collection $\\widetilde \\Sigma_1,\\dots, \\widetilde \\Sigma_N$ \nwith $\\widetilde \\Sigma_k\\subset \\Sigma_k$ such that \n$$\\Big(\\widetilde \\Sigma_k+B_{\\eta R} \\Big)\\cap \\Big(\\widetilde \\Sigma_{k'}+B_{\\eta R}\\Big)=\\emptyset \n\\ \\ \\hbox{ for any } k\\neq k'$$\n(see Fig.~\\ref{Fig2}). \nHence, for any $\\xi\\in\\mathbb Z^d$ with $|\\varepsilon\\xi|\\le \\eta R$, $k\\in\\{1,\\ldots,N\\}$, and \n$j\\in\\mathbb Z^d$ such that the line $\\varepsilon j+\\varepsilon\\xi\\mathbb R$ intersects $\\widetilde \\Sigma_k$, \nthe values $u^\\varepsilon_i$ change only once on the points of the discrete lines $\\varepsilon j+\\varepsilon \\xi\\mathbb Z$ which lie in a $\\eta R$ neighbourhood of \n\\widetilde\\Sigma_k$. We note that for lines intersecting $\\Sigma_k$ at a point of distance not larger than $C\\eta R$ from $K$, such changes of value are at most $N$; then, repeating the counting argument used in the lower bound, we obtain\n\\begin{eqnarray*}\n&&\\limsup_{\\varepsilon\\to 0}\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{\\substack{\\xi\\in\\mathbb Z^d\\\\ |\\varepsilon\\xi|\\leq\\eta R}} \\!\\!a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\\sum_{i\\in\\mathbb Z^d}|u^\\varepsilon_{i+\\xi}-u^\\varepsilon_i|\\\\\n&&\\hspace{1cm}\\leq\\limsup_{\\varepsilon\\to 0}\\Biggl(\\sum_{k=1}^N \\mathcal H^{d-1}(\\Sigma_k)\\frac{\\varepsilon^{d}}{\\eta^{d}}\\sum_{\\substack{\\xi\\in\\mathbb Z^d\\\\ |\\varepsilon\\xi|\\leq\\eta R}} \\!\\!a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\n\\Bigr|\\langle \\frac{\\varepsilon}{\\eta}\\xi, \\nu_k\\rangle\\Bigl| +O(\\eta R)\\Biggr)\\\\\n&&\\hspace{1cm}\\leq\\limsup_{\\varepsilon\\to 0}\\sum_{k=1}^N  \\mathcal H^{d-1}(\\Sigma_k)\\int_{ \\{|\\xi|\\leq R\\}} a(\\xi)\n|\\langle \\xi, \\nu_k\\rangle|\\, d\\xi \\\\\n&&\\hspace{1cm}\\leq \\int_{\\partial A}\\varphi_a(\\nu)d{\\mathcal H}^{d-1}.\n\\end{eqnarray*}\nSince also for $|\\varepsilon\\xi|\\ge \\eta R$  the changes of value of $u^\\varepsilon_i$ are at most $N$, we then get\n$$\\limsup_{\\varepsilon\\to 0}\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{\\substack{\\xi\\in\\mathbb Z^d\\\\ |\\varepsilon\\xi|>\\eta R}} \\!\\!a \\Bigl(\\frac{\\varepsilon}{\\eta}\\xi\\Bigr)\\sum_{i\\in\\mathbb Z^d}|u^\\varepsilon_{i+\\xi}-u^\\varepsilon_i|\\leq \nN{\\mathcal H}^{d-1}(\\partial A)\\int_{\\{|\\xi|>R\\}}a(\\xi)|\\xi|d\\xi.\n$$\nSince this term vanishes as $R\\to+\\infty$ the upper bound follows.\n\\end{proof}\n\n\\begin{example}\\label{example}\\rm\nIf $a$ is radially symmetric, then we have \n\\begin{equation}\\label{fe-symm} \nF(A)= \\sigma {\\mathcal H}^{d-1}(\\partial^*A),\n\\end{equation}\nwhere $\\sigma$ is given by\n\\begin{equation}\\label{fe-symm-s} \n\\sigma=\\int_{\\mathbb R^d}a(\\xi)|\\xi_1|d\\xi.\n\\end{equation}\n\nIn particular, we may take $a= \\chi_{B_1}$ the characteristic function of the unit ball in $\\mathbb R^d$. In this case the limit of \n\\begin{equation}\\label{fe-ball} \nF_\\varepsilon(u)=\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i,j\\in\\mathbb Z^d\\ |i-j|<\\eta\/\\varepsilon} |u_i-u_j|.\n\\end{equation}\nis given by \n\\begin{equation}\\label{fe-ball-s} \n\\sigma=\\int_{B_1}|\\xi_1|d\\xi.\n\\end{equation}\n\\end{example}\n\n\\begin{remark}[local version]\\label{remark}\\rm\nIf $\\Omega\\subset\\mathbb R^d$ is an open set with Lipschitz boundary we may define\n\\begin{equation}\\label{fe-a-omega} \nF_\\varepsilon(u)=\\frac{\\varepsilon^{2d}}{\\eta^{d+1}}\\sum_{i,j\\in\\mathbb Z^d\\cap{1\\over\\varepsilon}\\Omega} a^\\varepsilon_{i-j}|u_i-u_j|.\n\\end{equation}\nThen the $\\Gamma$-limit is \n\\begin{equation}\\label{f-a-omega} \nF(A)=\\int_{\\Omega\\cap\\partial^* A} \\varphi_a(\\nu)d{\\mathcal H}^{d-1},\n\\end{equation}\nwith minor modifications in the proof.\n\\end{remark}\n\n\n\\bigskip\n\n\\noindent{\\bf Acknowledgments.}\nAndrea Braides acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Margherita Solci acknowledges \nthe project ``Fondo di Ateneo per la ricerca 2019\", funded by the University of Sassari. We thank the anonymous referee of \\cite{BCS}, who drew our attention to the problem in this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:intro} \\IEEEPARstart{A}{ccurate} indoor localization\nis increasingly important due to the surging position based services\nsuch as position tracking, navigation and robot movement control.\nIn this research field, visible light positioning (VLP) technologies\nand computer vision based localization have the advantage of high\naccuracy and low cost. Visible light positioning technologies exploit\nvisible light signals for determining the position of the receiver.\nVisible light possesses strong directionality and low multipath interference,\nand thus VLP can achieve high accuracy localization performance \\cite{Pathak2015Visible,lim2015ubiquitous,yang2019relay}.\nAdditionally, VLP utilizes light-emitting diodes (LEDs) as transmitter.\nBenefited from the increasing market share of LEDs, VLP has relatively\nlow cost on infrastructure \\cite{Pathak2015Visible,yang2016enhanced}.\nOn the other hand, computer vision based localization relies on the\nimages of the reference features captured by cameras to estimate the\nposition and pose of cameras with high accuracy \\cite{piasco2018survey,ben2014review}.\nCameras can provide an extensive amount of information at limited\npower consumption, small size and reasonable cost \\cite{ben2014review}.\nAdditionally, nowadays cameras are essential parts of smartphones,\nwhich further corroborates the feasibility of vision based localization\ntechnologies \\cite{do2016depth}. Therefore, VLP and computer vision\nbased localization have been gained increasing attentions in recent\nyears \\cite{do2016depth,ICRA2018Manhattan}.\n\nTypical VLP algorithms include proximity \\cite{Sertthin20106}, fingerprinting\n\\cite{Qiu2016Let}, time of arrival (TOA) \\cite{wang2013TOA}, angle\nof arrival (AOA) \\cite{zhu2017ADOA}, received signal strength (RSS)\n\\cite{bai2019camera,bai2020enhanced,li2014epsilon,yasir2015rssjlt}\nand image sensing \\cite{li2018vlc}. Proximity and fingerprinting\ncannot estimate the receiver pose even though only a single luminaire\nis required. Additionally, the accuracy of proximity is insufficient\n\\cite{bai2019camera} while fingerprinting requires at least three\nluminaires to reduce the effect of ambiguity issues \\cite{vegni2012indoor}.\nAmong these VLP algorithms, RSS algorithms are most widely-used due\nto their high accuracy and low cost \\cite{bai2019camera}. However,\nRSS algorithms require multiple luminaires for localization, like\nimage sensing, TOA and AOA algorithms \\cite{do2016depth}. Moreover,\nRSS algorithms rely on accurate channel model, which is challenging\nto be achieved in practice. A popular assumption in RSS algorithms\nis that the radiation pattern of LEDs is the Lambertian model which\nmay not be true for many luminaires especially when a lampshade is\nused \\cite{miramirkhani2015channel}. Meanwhile, the estimated channel\ngain is affected by sunlight, dust and shadowing in practice \\cite{Dong2017study,cailean2017current}.\nTherefore, the feasibility of RSS algorithms is limited.\n\nOn the other hand, typical computer vision based localization methods\ninclude Perspective-n-Line (PnL) and Perspective-n-Point (PnP). The\nmethods are usually performed by analyzing $n$ correspondences between\nthree dimensional (3D) reference features and their two dimensional\n(2D) projections on the image (i.e., 3D-2D correspondences), where\nthe features are either points or lines \\cite{xu2016pose}. In particular,\nPnP methods employ the point features, while PnL methods employ the\nline features. Compared with the point features, line features can\ncarry richer information \\cite{xu2016pose}. Therefore, compared with\nPnP methods, PnL methods can achieve higher detection accuracy and\nare more robust to occlusions \\cite{pvribyl2017absolute,vakhitov2016accurate}.\nHowever, PnL methods need 3D-2D correspondences which are difficult\nto obtain. In existing PnL studies, the 3D-2D correspondences assumed\nto be perfectly known in advance \\cite{xu2016pose,xiaojian2008analytic},\nwhich is impractical in practice \\cite{ICRA2018Manhattan}. To circumvent\nthis challenge, the work in \\cite{ICRA2018Manhattan} proposed a method\nto find the 3D-2D correspondences for the scenario where the number\nof the vertical lines are more than that of horizontal lines. However,\nthis method cannot be applied to the scenario where there is no significant\ndifference between the numbers of horizontal and vertical lines, such\nas the scenario where the rectangular beacons are deployed on the\nceiling. Therefore, the feasibility of the method is constrained.\n\nThe main contribution of this paper is a novel visible light communication\n(VLC) assisted Perspective-four-Line algorithm (V-P4L), which can\nachieve feasible and accurate indoor localization. \\textit{To the\nauthors' best knowledge, this is the first localization algorithm\nthat only requires a single luminaire}\\footnote{All the LEDs in the luminaire transmit the same information for ease\nof implementation. Note that the proposed algorithm can also be implemented\nwhen LEDs in the luminaire transmit different information. In this\ncase, the 3D-2D correspondence can be obtained directly from the different\ninformation. However, this requires higher implementation complexity\nand the robustness of the link may be affected by inter-channel-interference.\nTherefore, in this work, we adopt the former strategy for higher robustness\nand lower complexity.}\\textit{ for position and pose estimation without given\n3D-2D correspondence}. The key contributions of this paper include:\n\\begin{itemize}\n\\item We propose an indoor localization algorithm termed as V-P4L, which\nuses camera to simultaneously capture the information in time and\nspace domains of LEDs to achieve high feasibility and high accuracy.\nBased on the plane and solid geometry theory, V-P4L estimates the\nluminaire's information in the camera coordinate system first using\nthe space-domain information of LEDs. Then, based on the single-view\ngeometry theory and the time-domain information of LEDs, V-P4L can\nestimate the pose and position of the camera exploiting the luminaire's\ninformation in different coordinate systems. In this way, V-P4L can\nestimate the position and pose of the receiver only using a single\nluminaire.\n\\item To avoid the requirement of the 3D-2D correspondences, the time-domain\ninformation transmitted by VLC and the linear least square (LLS) method\nare exploited in V-P4L to properly match 3D-2D correspondences. Based\non the time-domain information, V-P4L can obtain the information of\nLEDs in the world coordinate. Then, based on the LLS method and the\nLEDs' information in different coordinate systems, the 3D-2D correspondences\ncan be properly matched. In this way, V-P4L can achieve high feasibility.\n\\item To further improve the feasibility of V-P4L, we then propose a correction\nalgorithm for V-P4L to correct for the scenarios with LED height differences based on the information in both time and space domains. When LEDs have different heights, based on the single-view\ngeometry theory and the LLS method, the correction algorithm can first\nestimate the pose and 2D position of the camera, and then based on\nthe single-view geometry theory and a simple optimization method,\nthe correction algorithm can estimate the 3D position of the camera.\nIn this way, V-P4L can be used regardless of the height differences\namong LEDs.\n\\end{itemize}\nSimulation results show that for V-P4L the position error is always\nless than $15\\,\\mathrm{cm}$ and the orientation error is always less\nthan $3{^{\\circ}}$ using popular indoor luminaires.\n\nThe rest of the paper is organized as follows. Section \\ref{sec:System-Model}\nintroduces the system model. Section \\ref{sec:luminaire information}\ncalculates the luminaire information in the camera coordinate system.\nThe proposed basic algorithm of V-P4L is detailed in Section \\ref{sec:V-P4L-SH},\nand the proposed correction algorithm is detailed in Section \\ref{sec:V-P4L-DH}.\nSimulation results are presented in Section \\ref{sec:simulation}.\nFinally, the paper is concluded in Section \\ref{sec:CONCLUSION}.\n\nThe following notations are used throughout the paper: $A$ and $a$\nwith or without subscript denote scalars; $\\mathbf{v}$ denotes a\ncolumn vector and $\\mathbf{A}$ stands for a matrix; $\\left|A\\right|$\ndenotes the absolute value of $A$; $\\mathbf{A}^{\\mathrm{T}}$, $\\mathbf{A}^{\\mathrm{-1}}$,\n$\\det\\left(\\mathbf{A}\\right)$ and $\\left\\Vert \\mathbf{A}\\right\\Vert _{2}$\nindicate the transpose, inverse, determinant and Eculidean norm of\n$\\mathbf{A}$, respectively; $\\mathbf{v}\\times\\mathbf{u}$ denotes\nthe cross product of $\\mathbf{v}$ and $\\mathbf{u}$; $\\mathbf{v}\\cdot\\mathbf{u}$,\n$\\mathbf{A}\\cdot\\mathbf{v}$ and $\\mathbf{A}\\mathbf{B}$ denotes the\ndot products of $\\mathbf{v}$ and $\\mathbf{u}$, $\\mathbf{A}$ and\n$\\mathbf{v}$, and $\\mathbf{A}$ and $\\mathbf{B}$, respectively;\n$\\hat{A}$, $\\hat{\\mathbf{A}}$ and $\\hat{\\mathbf{v}}$ represent\nthe estimate of $A$, $\\mathbf{A}$ and $\\mathbf{v}$, respectively.\nIn addition, there are some special or important symbols used throughout\nin this paper, which are listed in Table \\ref{tab:Symbols} with their\nmeaning. In particular, $P$ with subscript denotes the 3D vertex\nof the LED luminaire; $p$ with subscript denotes the 2D point on\nthe image plane; $L$ with subscript represents the 3D line; $l$\nwith subscript represents the 2D line on the image plane. We use subscript\nto represent the indices or objects that the scalars, points, vectors\nand matrices corresponding to. For example, $P_{i}$ denotes the $i$th\nvertex of the luminaire, and $\\rho_{l_{ij}}$ denotes a parameter\nin the equation of $l_{ij}$. Furthermore, we use the superscript\nto represent the coordinates of points and vectors in different coordinate\nsystems. For example, the coordinates of the 2D point $p_{i}$ in\nthe world, camera, image and pixel coordinates are denoted as $p_{i}^{\\mathrm{w}}$,\n$p_{i}^{\\mathrm{c}}$, $p_{i}^{\\mathrm{i}}$ and $p_{i}^{\\mathrm{p}}$,\nrespectively.\n\n\\global\\long\\def\\arraystretch{1.1\n\\begin{table}\n\\caption{\\label{tab:Symbols}Symbols and Their Meaning.}\n\n\\centering{\n\\begin{tabular}{r|>{\\raggedright}p{12cm}}\n\\hline\nSymbol & Meaning\\tabularnewline\n\\hline\n\\hline\n$\\left(u_{0},v_{0}\\right)^{\\mathrm{T}}$ & Pixel coordinate of $o^{\\textrm{i}}$\\tabularnewline\n\\hline\n$f$ & Focal length\\tabularnewline\n\\hline\n$f_{u}$, $f_{v}$ & Focal ratios\\tabularnewline\n\\hline\n$d_{x}$, $d_{y}$ & Physical size of each pixel\\tabularnewline\n\\hline\n$P_{i}$ & The $i$th vertex of the luminaire\\tabularnewline\n\\hline\n$p_{i}$ & Projection of $P_{i}$ on the image plane\\tabularnewline\n\\hline\n$p_{i}^{\\textrm{p}}$\/$p_{i}^{\\textrm{i}}$ & Pixel\/Image coordinate of $p_{i}$\\tabularnewline\n\\hline\n$P_{i}^{\\textrm{c}}$\/$P_{i}^{\\textrm{w}}$ & Camera\/World coordinate of $P_{i}$\\tabularnewline\n\\hline\n$L_{ij}$ & 3D reference line connecting $P_{i}$ and $P_{j}$\\tabularnewline\n\\hline\n$l_{ij}$ & 2D projection of $L_{ij}$ on the image plane\\tabularnewline\n\\hline\n$\\mathbf{n}_{L_{ij}}^{\\mathrm{c}}$ & Direction vector of $L_{ij}$ in CCS\\tabularnewline\n\\hline\n$\\phi_{l_{ij}}$ & Rotate angle from $y^{\\textrm{i}}$-axis to $l_{ij}$ in anticlockwise\ndirection\\tabularnewline\n\\hline\n$\\rho_{l_{ij}}$ & Distance from $o^{\\textrm{i}}$ to $l_{ij}$\\tabularnewline\n\\hline\n$\\mathbf{n}_{\\mathrm{LED}}^{\\textrm{w}}$\/$\\mathbf{n}_{\\mathrm{LED}}^{\\textrm{c}}$ & Normal vector of the luminaire in WCS\/CCS\\tabularnewline\n\\hline\n$\\Pi_{ij}$ & Lateral face determined by the vertices $P_{i}$, $P_{j}$ and $o^{\\textrm{c}}$\\tabularnewline\n\\hline\n$\\mathbf{n}_{\\Pi_{ij}}^{\\mathrm{c}}$ & Normal vector of $\\Pi_{ij}$ in CCS\\tabularnewline\n\\hline\n$S$ & Area of the luminaire\\tabularnewline\n\\hline\n$H$ & Distance from $o^{\\textrm{c}}$ to the luminaire\\tabularnewline\n\\hline\n$V$ & Volume of rectangular pyramid $o^{\\textrm{c}}-P_{1}P_{2}P_{3}P_{4}$\\tabularnewline\n\\hline\n$V_{i}$ & Volume of triangular pyramid $o^{\\textrm{c}}-P_{i}P_{j}P_{k}$\\tabularnewline\n\\hline\n$\\varphi$, $\\theta$, $\\psi$ & Euler angles corresponding to the $x^{\\mathrm{c}}-$axis, $y^{\\mathrm{c}}-$axis\nand $z^{\\mathrm{c}}-$axis\\tabularnewline\n\\hline\n$\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}$\/$\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}}$ & Pose\/Position of the camera in WCS\\tabularnewline\n\\hline\n\\end{tabular}\n\\end{table}\n\\vspace{-0cm}\n\n\n\\section{\\label{sec:System-Model}System Model}\n\nThe system diagram is illustrated in Fig. \\ref{fig:A-system-block}.\nFour coordinate systems are utilized for localization, which are the\npixel coordinate system (PCS) $o^{\\textrm{p}}-u^{\\textrm{p}}v^{\\textrm{p}}$\non the image plane\\footnote{As shown in Fig. \\ref{fig:A-system-block}, the image plane is a virtual\nplane. In this paper, the camera is a standard pinhole camera. The\nactual image plane is behind the camera optical center (i.e., the\npinhole), $o^{\\textrm{c}}$. To show the geometric relations more\nclearly, the virtual image plane is set up in front of $o^{\\textrm{c}}$\nas done in many papers \\cite{masselli2014new,kneip2011novel}. In\nparticular, the virtual image plane and the actual image plane are\ncentrally symmetric, and $o^{\\textrm{c}}$ is the center of symmetry.}, the image coordinate system (ICS) $o^{\\textrm{i}}-x^{\\textrm{i}}y^{\\textrm{i}}$\non the image plane, the camera coordinate system (CCS) $o^{\\textrm{c}}-x^{\\textrm{c}}y^{\\textrm{c}}z^{\\textrm{c}}$\nand the world coordinate system (WCS) $o^{\\textrm{w}}-x^{\\textrm{w}}y^{\\textrm{w}}z^{\\textrm{w}}$.\nIn PCS, ICS and CCS, the axes $u^{\\textrm{p}}$, $x^{\\textrm{i}}$\nand $x^{\\textrm{c}}$ are parallel to each other and, similarly, $v^{\\textrm{p}}$,\n$y^{\\textrm{i}}$ and $y^{\\textrm{c}}$ are also parallel to each\nother. Additionally, $o^{\\textrm{p}}$ is at the upper left corner\nof the image plane and $o^{\\textrm{i}}$ is at the center of the image\nplane. Moreover, $o^{\\textrm{i}}$ is termed as the principal point,\nwhose pixel coordinate is $\\left(u_{0},v_{0}\\right)^{\\mathrm{T}}$.\nIn contrast, $o^{\\textrm{c}}$ is termed as the camera optical center.\nFurthermore, $o^{\\textrm{i}}$ and $o^{\\textrm{c}}$ are on the optical\naxis. The distance between $o^{\\textrm{c}}$ and $o^{\\textrm{i}}$\nis the focal length $f$, and thus the $z$-coordinate of the image\nplane in CCS is $z^{\\mathrm{c}}=f$.\n\nA VLC-enabled rectangular LED luminaire is constructed by four vertices\n$P_{i}$ ($i\\in\\left\\{ 1,2,3,4\\right\\} $) mounted on the ceiling.\nThe four 3D reference lines $L_{ij}$ ($i,j\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j$)\nare the edges of the luminaire. In addition, $P_{i}^{\\textrm{w}}=\\left(x_{i}^{\\textrm{w}},y_{i}^{\\textrm{w}},z_{i}^{\\textrm{w}}\\right)^{\\mathrm{T}}$\nis the world coordinate of the $i$th vertex of the luminaire, which\nis assumed to be known at the transmitter and can be transmitted by\nVLC as the time-domain information \\cite{yang2016enhanced,yang2019relay}.\nMoreover, the unit normal vector of the luminaire in WCS, $\\mathbf{n}_{\\mathrm{LED}}^{\\textrm{w}}$,\ncan be calculated by the world coordinates of the luminaire's vertices.\n\nOn the other hand, the receiver is a standard pinhole camera which\nis not coplanar with the luminaire. Therefore, the transmitter and\nthe receiver produced a rectangular pyramid $o^{\\textrm{c}}-P_{1}P_{2}P_{3}P_{4}$\nwhich contains many space-domain information. In the rectangular pyramid\n$o^{\\textrm{c}}-P_{1}P_{2}P_{3}P_{4}$, the rectangle $P_{1}P_{2}P_{3}P_{4}$\nis called the base face. Meanwhile, we define $\\Pi_{ij}$ as the lateral\nface determined by the vertices $P_{i}$, $P_{j}$ ($i,j\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j$)\nand $o^{\\textrm{c}}$. In addition, $P_{i}$ is the $i$th vertex\nand $o^{\\textrm{c}}$ is called the apex. In the camera, $p_{i}$\nis the projection of $P_{i}$ on the image plane. Moreover, $l_{ij}$\nis the 2D projection on the image plane of $L_{ij}$. Note that many\nexisting PnL algorithms assume that the 3D-2D correspondences $\\left(L_{ij}\\iff l_{ij}\\right)$\nare known in advance, which is too ideal in practice \\cite{ICRA2018Manhattan}.\nIn contrast, in this work, the 3D-2D correspondences are unknown.\nTo estimate the pose and position of the receiver without the 3D-2D\ncorrespondences, the camera is used to simultaneously capture the\ntime- and space-domain information.\n\nThe pixel coordinate of $p_{i}$ is denoted by $p_{i}^{\\textrm{p}}=\\left(u_{i}^{\\textrm{p}},v_{i}^{\\textrm{p}}\\right)^{\\mathrm{T}}$,\nand this coordinate can be obtained by the camera through image processing\n\\cite{li2018vlc}. Based on the single-view geometry theory, the image\ncoordinate of $p_{i}$, $p_{i}^{\\textrm{i}}=\\left(x_{i}^{\\textrm{i}},y_{i}^{\\textrm{i}}\\right)^{\\mathrm{T}}$,\ncan be obtained as follows:\n\\begin{equation}\np_{i}^{\\textrm{i}}=\\begin{bmatrix}d_{x}\\\\\nd_{y}\n\\end{bmatrix}p_{i}^{\\textrm{p}}-\\begin{bmatrix}u_{0}d_{x}\\\\\nv_{0}d_{y}\n\\end{bmatrix},\\label{eq:1}\n\\end{equation}\nwhere $d_{x}$ and $d_{y}$ are the physical size of each pixel in\nthe $x$ and $y$ directions on the image plane, respectively. The\ncamera's intrinsic parameters, including $\\left(u_{0},v_{0}\\right)^{\\mathrm{T}}$\nand the focal ratio $f_{u}=\\frac{f}{d_{x}}$ and $f_{v}=\\frac{f}{d_{y}}$,\ncan be calibrated in advance \\cite{bai2019camera}. The transformation\nfrom CCS to WCS can be expressed as follows \\cite{xu2016pose}:\n\\begin{equation}\nP^{\\textrm{w}}=\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}\\cdot P^{\\textrm{c}}+\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}},\\label{eq:3}\n\\end{equation}\nwhere $P^{\\textrm{w}}$ and $P^{\\textrm{c}}$ are the world and camera\ncoordinates of the same object, respectively. In addition, $\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}$\nand $\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}}\\in\\mathbb{R}^{3}$ denote\nthe pose and the position of the camera in WCS, respectively. The\ntask of the localization is to find out $\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}$\nand $\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}}$.\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[scale=0.6]{system5}\n\\par\\end{centering}\n\\caption{\\label{fig:A-system-block}The system diagram of the proposed algorithm.}\n\\end{figure}\n\n\\vspace{-0.4cm}\n\n\n\\section{\\label{sec:luminaire information}Calculating The Luminaire Information\nIn CCS}\n\nIn this section, the information of the luminaire in CCS, including\nits normal vector $\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}$ and its\nvertices' coordinates $P_{i}^{\\textrm{c}}$, $i\\in\\left\\{ 1,2,3,4\\right\\} $,\nis calculated based on the space-domain information in two steps.\nIn the first step, $\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}$ is estimated\nbased on the plane and solid geometry theory. Then, based on $\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}$,\n$P_{i}^{\\textrm{c}}$, $i\\in\\left\\{ 1,2,3,4\\right\\} $ are estimated\nby the solid geometry theory.\n\n\\vspace{-0.5cm}\n\n\n\\subsection{\\label{subsec:normal vector of LED in CCS}The Normal Vector Of The\nLuminaire In CCS}\n\nIn ICS, the point-normal form equation of a given $l_{ij}$ can be\nexpressed as \\cite{xiaojian2008analytic}:\n\\begin{equation}\nx^{\\textrm{i}}\\cos\\phi_{l_{ij}}+y^{\\textrm{i}}\\sin\\phi_{l_{ij}}=\\rho_{l_{ij}},\\label{eq:50}\n\\end{equation}\nwhere $\\left(x^{\\textrm{i}},y^{\\textrm{i}}\\right)^{\\mathrm{T}}$ is\nthe image coordinate of a point on $l_{ij}$ which can be obtained\nby the single-view geometry theory , $\\phi_{l_{ij}}$ is the rotate\nangle from $y^{\\textrm{i}}$-axis to $l_{ij}$ in anticlockwise direction,\nand $\\rho_{l_{ij}}$ is the distance from $o^{\\textrm{i}}$ to $l_{ij}$.\nSince $p_{i}$ and $p_{j}$ are on $l_{ij}$, $\\phi_{l_{ij}}$ and\n$\\rho_{l_{ij}}$ can be obtained based on the image coordinates of\n$p_{i}$ and $p_{j}$. From (\\ref{eq:3}), there are two points whose\nimage coordinates are $p_{l_{ij},1}^{\\textrm{i}}=\\left(\\frac{\\rho_{l_{ij}}}{\\cos\\phi_{l_{ij}}},0\\right)^{\\mathrm{T}}$\nand $p_{l_{ij},2}^{\\textrm{i}}=\\left(0,\\frac{\\rho_{l_{ij}}}{\\sin\\phi_{l_{ij}}}\\right)^{\\mathrm{T}}$\nare on $l_{ij}$. Since the two points are also on the image plane,\ntheir camera coordinates are $p_{l_{ij},1}^{\\textrm{c}}=\\left(\\frac{\\rho_{l_{ij}}}{\\cos\\phi_{l_{ij}}},0,f\\right)^{\\mathrm{T}}$\nand $p_{l_{ij},2}^{\\textrm{c}}=\\left(0,\\frac{\\rho_{l_{ij}}}{\\sin\\phi_{l_{ij}}},f\\right)^{\\mathrm{T}}$.\nSince the two points and $o^{\\textrm{c}}=\\left(0,0,0\\right)^{\\mathrm{T}}$\nare on $\\Pi_{ij}$, we can represent $\\Pi_{ij}$ in CCS in the general\nform as:\n\n\\begin{equation}\nA_{\\Pi_{ij}}x^{\\mathrm{c}}+B_{\\Pi_{ij}}y^{\\mathrm{c}}+C_{\\Pi_{ij}}z^{\\mathrm{c}}=0,\\label{eq:8}\n\\end{equation}\nwhere $A_{\\Pi_{ij}}=f\\cos\\phi_{l_{ij}}$, $B_{\\Pi_{ij}}=f\\sin\\phi_{l_{ij}}$\nand $C_{\\Pi_{ij}}=-\\rho_{l_{ij}}$.\n\nIn CCS, the general form equation of the rectangle $P_{1}P_{2}P_{3}P_{4}$\ncan be expressed as:\n\\begin{equation}\nA_{\\mathrm{LED}}x^{\\mathrm{c}}+B_{\\mathrm{LED}}y^{\\mathrm{c}}+C_{\\mathrm{LED}}z^{\\mathrm{c}}=1,\\label{eq:12}\n\\end{equation}\nwhere $A_{\\mathrm{LED}}$, $B_{\\mathrm{LED}}$ and $C_{\\mathrm{LED}}$\nare unknown constants. From (\\ref{eq:12}), the normal vector of the\nrectangle $P_{1}P_{2}P_{3}P_{4}$ can be expressed by $\\left(A_{\\mathrm{LED}},B_{\\mathrm{LED}},C_{\\mathrm{LED}}\\right)^{\\mathrm{T}}$.\nIn CCS, let $\\mathbf{n}_{\\Pi_{ij}}^{\\mathrm{c}}=\\left(A_{\\Pi_{ij}},B_{\\Pi_{ij}},C_{\\Pi_{ij}}\\right)^{\\mathrm{T}}$\n($i,j\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j$) denotes the normal vector\nof $\\Pi_{ij}$ and $\\mathbf{v}_{L_{ij}}^{\\mathrm{c}}\\in\\mathbb{R}^{3}$\n($i,j\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j$) denotes the direction\nvector of $L_{ij}$. Since $L_{ij}$ is the intersection line of the\nrectangle $P_{1}P_{2}P_{3}P_{4}$ and $\\Pi_{ij}$, $\\mathbf{v}_{L_{ij}}^{\\mathrm{c}}$\ncan be calculated as $\\mathbf{v}_{L_{ij}}^{\\mathrm{c}}=\\left(A_{\\mathrm{LED}},B_{\\mathrm{LED}},C_{\\mathrm{LED}}\\right)^{\\mathrm{T}}\\times\\mathbf{n}_{\\Pi_{ij}}^{\\mathrm{c}}$.\nBased on the solid geometry, we have:\n\\begin{equation}\n\\begin{cases}\n\\mathbf{v}_{L_{34}}^{\\mathrm{c}}\\cdot\\mathbf{n}_{\\Pi_{12}}^{\\mathrm{c}}=0\\\\\n\\mathbf{v}_{L_{41}}^{\\mathrm{c}}\\cdot\\mathbf{n}_{\\Pi_{23}}^{\\mathrm{c}}=0.\n\\end{cases}\\label{eq:11}\n\\end{equation}\nDefine $m=\\frac{A_{\\mathrm{LED}}}{C_{\\mathrm{LED}}}$ and $n=\\frac{B_{\\mathrm{LED}}}{C_{\\mathrm{LED}}}$,\nand we can obtain $m$ and $n$ as the functions of $A_{\\Pi_{ij}}$,\n$B_{\\Pi_{ij}}$ and $C_{\\Pi_{ij}}$ by solving (\\ref{eq:11}). Therefore,\nthe normalized normal vector of the rectangle $P_{1}P_{2}P_{3}P_{4}$\n(i.e., the orientation of the luminaire) in CCS can be expressed as:\n\\begin{equation}\n\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}=\\left(\\cos\\alpha,\\cos\\beta,\\cos\\gamma\\right)^{\\mathrm{T}},\\label{eq:13}\n\\end{equation}\nwhere:\n\\begin{equation}\n\\begin{cases}\n\\cos\\alpha=\\frac{m}{\\sqrt{m^{2}+n^{2}+1}}\\\\\n\\cos\\beta=\\frac{n}{\\sqrt{m^{2}+n^{2}+1}}\\\\\n\\cos\\gamma=\\frac{1}{\\sqrt{m^{2}+n^{2}+1}}.\n\\end{cases}\\label{eq:14}\n\\end{equation}\n\\vspace{-0.6cm}\n\n\n\\subsection{\\label{subsec:led camera coordinate}Camera Coordinates Of The Luminaire's\nVertices}\n\nSince $P_{1}$ is the intersection point of the rectangle $P_{1}P_{2}P_{3}P_{4}$,\n$\\Pi_{12}$ and $\\Pi_{41}$, its camera coordinate can be calculated\nas $P_{1}^{\\textrm{c}}=\\frac{\\mathbf{M}_{P_{1}}}{C_{\\mathrm{LED}}}$,\nwhere:\n\\begin{equation}\n\\mathbf{M}_{P_{1}}=\\begin{bmatrix}m & n & 1\\\\\nA_{\\Pi_{12}} & B_{\\Pi_{12}} & C_{\\Pi_{12}}\\\\\nA_{\\Pi_{41}} & B_{\\Pi_{41}} & C_{\\Pi_{41}}\n\\end{bmatrix}^{-1}\\cdot\\begin{bmatrix}1\\\\\n0\\\\\n0\n\\end{bmatrix}.\\label{eq:100}\n\\end{equation}\nThe other three $\\mathbf{M}_{P_{i}}$ ($i\\in\\left\\{ 2,3,4\\right\\} $)\ncan be calculated in the similar method of (\\ref{eq:100}). In general,\nthe camera coordinate $P_{i}^{\\textrm{c}}$ ($i\\in\\left\\{ 1,2,3,4\\right\\} $)\ncan be calculated as follows:\n\\begin{equation}\nP_{i}^{\\textrm{c}}=\\frac{\\mathbf{M}_{P_{i}}}{C_{\\mathrm{LED}}}.\\label{eq:37}\n\\end{equation}\nFrom (\\ref{eq:37}), we can observe that $P_{i}^{\\textrm{c}}$ can\nbe represented according to $C_{\\mathrm{LED}}$. Next, we will calculate\n$C_{\\mathrm{LED}}$ based on the solid geometry.\n\nThe volume of the rectangular pyramid $o^{\\textrm{c}}-P_{1}P_{2}P_{3}P_{4}$\ncan be calculates as $V=\\frac{1}{3}SH$, where $S$ is the area of\nthe luminaire and is known in advance. Additionally, $H=\\frac{1}{C_{\\mathrm{LED}}\\sqrt{m^{2}+n^{2}+1}}$\nis the distance from $o^{\\textrm{c}}$ to the rectangle $P_{1}P_{2}P_{3}P_{4}$.\nFor the triangular pyramid $o^{\\textrm{c}}-P_{1}P_{2}P_{3}$, its\nvolume can be calculated as follows:\n\\begin{equation}\nV_{1}=\\frac{1}{6}\\left|\\det(\\mathbf{M}_{V_{1}})\\right|,\\label{eq:19}\n\\end{equation}\nwhere $\\mathbf{M}_{V_{1}}=\\left[P_{1}^{\\textrm{c}},P_{2}^{\\textrm{c}},P_{3}^{\\textrm{c}}\\right]^{\\mathrm{T}}$.\nSubstituting (\\ref{eq:37}) into (\\ref{eq:19}), we have $V_{1}=\\frac{q_{1}}{C_{\\mathrm{LED}}^{3}}$,\nwhere $q_{1}=\\frac{1}{6}\\left|\\det(\\mathbf{M}_{q_{1}})\\right|$, where\n$\\mathbf{M}_{q_{1}}=\\left[\\mathbf{M}_{P_{1}},\\mathbf{M}_{P_{2}},\\mathbf{M}_{P_{3}}\\right]^{\\mathrm{T}}$.\nThe volumes of the other three triangular pyramid $o^{\\textrm{c}}-P_{2}P_{3}P_{4}$,\n$o^{\\textrm{c}}-P_{3}P_{4}P_{1}$ and $o^{\\textrm{c}}-P_{4}P_{1}P_{2}$,\ndenoted by $V_{2}$, $V_{3}$ and $V_{4}$, respectively, can be obtained\nin the same way. Since $V=\\frac{1}{2}\\sum_{i=1}^{4}V_{i}$, $C_{\\mathrm{LED}}$\ncan be calculated as follows:\n\\begin{equation}\nC_{\\mathrm{LED}}=\\sqrt{\\frac{3\\sum_{i=1}^{4}q_{i}\\cdot\\sqrt{m^{2}+n^{2}+1}}{2S}}.\\label{eq:21}\n\\end{equation}\nSubstituting (\\ref{eq:21}) into (\\ref{eq:37}), $P_{i}^{\\textrm{c}}$\n($i\\in\\left\\{ 1,2,3,4\\right\\} $) can be obtained.\n\n\\vspace{-0.3cm}\n\n\n\\section{\\label{sec:V-P4L-SH}The Basic Algorithm of V-P4L}\n\nIn this section, the basic algorithm of V-P4L is proposed for scenarios\nwhere LEDs have the same height. The basic algorithm of V-P4L contains\nthree steps. In the first step, based on the orientation information\nof the luminaire estimated in Section \\ref{sec:luminaire information},\nthe rotation angles corresponding to the $x^{\\mathrm{c}}-$axis and\n$y^{\\mathrm{c}}-$axis can be obtained by the single-view geometry\ntheory. Then, based on the LLS method and the single-view geometry\ntheory, the basic algorithm of V-P4L can properly match the 3D-2D\ncorrespondences, and obtain the rotation angles corresponding to the\n$z^{\\mathrm{c}}-$axis and the 2D coordinate of the camera. Finally,\nbased on the single-view geometry theory, V-P4L can estimate the $z$-coordinate\nof the camera.\n\n\\subsection{\\label{subsec:xy_angles}Calculate the rotation angles corresponding\nto the $x^{\\mathrm{c}}-$axis and $y^{\\mathrm{c}}-$axis}\n\nLet $\\mathbf{R}_{X}$, $\\mathbf{R}_{Y}$ and $\\mathbf{R}_{Z}$ denote\nthe rotation matrices of WCS along the $x^{\\mathrm{c}}-$axis, $y^{\\mathrm{c}}-$axis\nand $z^{\\mathrm{c}}-$axis, respectively. Given $\\mathbf{R}_{X}$,\n$\\mathbf{R}_{Y}$ and $\\mathbf{R}_{Z}$ as follows \\cite{Taylor1986rotation}:\n\\begin{equation}\n\\mathbf{R}_{X}=\\begin{bmatrix}1 & 0 & 0\\\\\n0 & \\cos\\varphi & -\\sin\\varphi\\\\\n0 & \\sin\\varphi & \\cos\\varphi\n\\end{bmatrix},\\label{eq:22}\n\\end{equation}\n\\begin{equation}\n\\mathbf{R}_{Y}=\\begin{bmatrix}\\cos\\theta & 0 & \\sin\\theta\\\\\n0 & 1 & 0\\\\\n-\\sin\\theta & 0 & \\cos\\theta\n\\end{bmatrix},\\label{eq:23}\n\\end{equation}\nand\n\\begin{equation}\n\\mathbf{R}_{Z}=\\begin{bmatrix}\\cos\\psi & -\\sin\\psi & 0\\\\\n\\sin\\psi & \\cos\\psi & 0\\\\\n0 & 0 & 1\n\\end{bmatrix},\\label{eq:24}\n\\end{equation}\nwhere $\\varphi\\in\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]$, $\\theta\\in\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]$\nand $\\psi\\in\\left(-\\pi,\\pi\\right]$ are the unknown Euler angles corresponding\nto the $x^{\\mathrm{c}}-$axis, $y^{\\mathrm{c}}-$axis and $z^{\\mathrm{c}}-$axis,\nrespectively, the rotation matrix $\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}$\nfrom CCS to WCS can be given as \\cite{Taylor1986rotation}:\n\\begin{equation}\n\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}=\\mathbf{R}_{Z}\\mathbf{R}_{Y}\\mathbf{R}_{X}.\\label{eq:25}\n\\end{equation}\nIn this section, the basic algorithm of V-P4L is proposed for scenarios\nwhere LEDs have the same height. Therefore, the normal vector of the\nluminaire can be denoted by $\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}=\\left(0,0,1\\right)^{\\mathrm{T}}$.\nBased on the single-view geometry theory, the relationship between\n$\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}=\\left(0,0,1\\right)^{\\mathrm{T}}$\nand $\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}=\\left(\\cos\\alpha,\\cos\\beta,\\cos\\gamma\\right)^{\\mathrm{T}}$\ncan be given as \\cite{xu2016pose}:\n\\begin{equation}\n\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{w}}=\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}\\cdot\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}.\\label{eq:40}\n\\end{equation}\nTherefore, we have:\n\\begin{equation}\n\\begin{cases}\n\\cos\\alpha=-\\sin\\theta\\\\\n\\cos\\beta=\\cos\\theta\\cdot\\sin\\varphi\\\\\n\\cos\\gamma=\\cos\\theta\\cdot\\cos\\varphi.\n\\end{cases}\\label{eq:29}\n\\end{equation}\nThe estimated rotation angles $\\hat{\\varphi}$ and $\\hat{\\theta}$\ncan be obtained by solving (\\ref{eq:29}).\n\n\\subsection{\\label{subsec:z_angle tx ty}Calculate the rotation angles corresponding\nto the $z^{\\mathrm{c}}-$axis and the 2D coordinate of the camera}\n\nBased on the single-view geometry theory, the relationship between\n$P_{i}^{\\textrm{c}}$ ($i\\in\\left\\{ 1,2,3,4\\right\\} $) and $P_{i}^{\\textrm{w}}=\\left(x_{i}^{\\textrm{w}},y_{i}^{\\textrm{w}},z_{i}^{\\textrm{w}}\\right)^{\\mathrm{T}}$\ncan be given as \\cite{xu2016pose}:\n\\begin{equation}\nP_{i}^{\\textrm{w}}=\\mathbf{R}_{\\mathrm{c}}^{\\mathrm{w}}\\cdot P_{i}^{\\textrm{c}}+\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}},\\label{eq:26}\n\\end{equation}\nwhere $P_{i}^{\\textrm{w}}$ is known in advance and can be obtained\nby the camera as the time-domain information. In addition, $P_{i}^{\\textrm{c}}$\nis the space-domain information that is estimated in Subsection \\ref{subsec:led camera coordinate}.\nMoreover, $\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}}=\\left(t_{x},t_{y},t_{z}\\right)^{\\mathrm{T}}$\nis the 3D world coordinate of the camera. In (\\ref{eq:26}), there\nare four unknown parameters $\\psi$, $t_{x}$, $t_{y}$ and $t_{z}$.\nIf the 3D-2D correspondences are known in advance, we can easily obtain\nthe four unknown parameters with the four vertices' world and camera\ncoordinates. However, as analyzed in Section \\ref{sec:intro}, the\n3D-2D correspondences are unknown for practical considerations. In\nthis paper, we can calculate these parameters based on the LEDs' information\nin both time and space domains. For mathematical analysis, we define:\n\\begin{equation}\n\\mathbf{R}_{Y}\\mathbf{R}_{X}=\\left[\\begin{array}{ccc}\n\\cos\\theta & \\sin\\theta\\cdot\\sin\\varphi & \\sin\\theta\\cdot\\cos\\varphi\\\\\n0 & \\cos\\varphi & -\\sin\\varphi\\\\\n-\\sin\\theta & \\cos\\theta\\cdot\\sin\\varphi & \\cos\\theta\\cdot\\cos\\varphi\n\\end{array}\\right]=\\begin{bmatrix}a_{1} & a_{2} & a_{3}\\\\\nb_{1} & b_{2} & b_{3}\\\\\nc_{1} & c_{2} & c_{3}\n\\end{bmatrix},\\label{eq:35}\n\\end{equation}\nand thus we can rewrite (\\ref{eq:26}) as follows:\n\\begin{equation}\nP_{i}^{\\textrm{w}}=\\begin{bmatrix}\\cos\\psi & -\\sin\\psi & 0\\\\\n\\sin\\psi & \\cos\\psi & 0\\\\\n0 & 0 & 1\n\\end{bmatrix}\\cdot\\begin{bmatrix}a_{1} & a_{2} & a_{3}\\\\\nb_{1} & b_{2} & b_{3}\\\\\nc_{1} & c_{2} & c_{3}\n\\end{bmatrix}\\cdot P_{i}^{\\textrm{c}}+\\mathbf{t_{\\mathrm{c}}^{\\mathrm{w}}}.\\label{eq:27}\n\\end{equation}\nThe three unknown parameters $\\psi$, $t_{x}$ and $t_{y}$ in (\\ref{eq:27})\ncan be calculated by the LLS estimator, which can be expressed in\na matrix form as follows:\n\\begin{equation}\n\\mathbf{A_{\\mathit{rs}}}\\cdot\\mathbf{x}=\\mathbf{b_{\\mathit{ij}}},\\label{eq:31}\n\\end{equation}\nwhere $\\mathbf{A}_{\\mathit{rs}}=\\left[\\mathbf{A}_{r};\\mathbf{A}_{s}\\right]$\nand $\\mathbf{b_{\\mathit{ij}}}=\\left[\\mathbf{b}_{i};\\mathbf{b}_{j}\\right]$\n($r,s,i,j\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j,r\\neq s$), where:\n\\begin{equation}\n\\mathbf{A}_{r}=\\begin{bmatrix}\\left[a_{1},a_{2},a_{3}\\right]^{\\mathrm{T}}\\cdot P_{r}^{\\textrm{c}} & -\\left[b_{1},b_{2},b_{3}\\right]^{\\mathrm{T}}\\cdot P_{r}^{\\textrm{c}} & 1 & 0\\\\\n\\left[b_{1},b_{2},b_{3}\\right]^{\\mathrm{T}}\\cdot P_{r}^{\\textrm{c}} & \\left[a_{1},a_{2},a_{3}\\right]^{\\mathrm{T}}\\cdot P_{r}^{\\textrm{c}} & 0 & 1\n\\end{bmatrix}(r\\in\\left\\{ 1,2,3,4\\right\\} ),\\label{eq:31-1}\n\\end{equation}\n\n\\begin{equation}\n\\mathbf{x}=\\left[\\cos\\psi,\\sin\\psi,t_{x},t_{y}\\right]^{\\mathrm{T}},\\label{eq:31-2}\n\\end{equation}\nand\n\\begin{equation}\n\\mathbf{b}_{i}=\\left[x_{i}^{\\textrm{w}},y_{i}^{\\textrm{w}}\\right]^{\\mathrm{T}}(i\\in\\left\\{ 1,2,3,4\\right\\} ).\\label{eq:31-3}\n\\end{equation}\nTherefore, the unknown parameters can be given by:\n\\begin{equation}\n\\mathbf{\\hat{x}=(A_{\\mathit{rs}}^{\\mathrm{T}}A_{\\mathit{rs}})^{\\mathrm{-1}}A_{\\mathit{rs}}^{\\mathrm{T}}b_{\\mathit{ij}}},\\label{eq:32}\n\\end{equation}\nwhere $\\hat{\\mathbf{x}}=\\left[\\hat{\\cos\\psi},\\hat{\\sin\\psi},\\hat{t}_{x},\\hat{t}_{y}\\right]^{\\mathrm{T}}$\nis the estimate of $\\mathbf{x}$.\n\nSince the 3D-2D correspondences are not known in advance, given a\ncertain $\\mathbf{A}_{\\mathit{rs}}$ and $\\mathbf{b_{\\mathit{ij}}}$\n($r,s,i,j\\in\\left\\{ 1,2,3,4\\right\\} ,r\\neq s,i\\neq j$), we cannot\nobtain their exact correspondence relationship. Fortunately, there\nare four $\\mathbf{A}_{r}$ ($r\\in\\left\\{ 1,2,3,4\\right\\} $) and $\\mathbf{b}_{i}$\n($i\\in\\left\\{ 1,2,3,4\\right\\} $), and that means there are only $\\mathrm{C}_{4}^{2}=6$\ndifferent $\\mathbf{A}_{\\mathit{rs}}$ and $\\mathbf{b_{\\mathit{ij}}}$\n($r,s,i,j\\in\\left\\{ 1,2,3,4\\right\\} ,r\\neq s,i\\neq j$). Therefore,\nfor each $\\mathbf{b_{\\mathit{ij}}}=\\left[\\mathbf{b}_{i},\\mathbf{b}_{j}\\right]^{\\mathrm{T}}$\n($i,j\\in\\left\\{ 1,2,3,4\\right\\} $, $i\\neq j$), we can obtain 6 candidate\nsolutions corresponding to 6 $\\mathbf{A}_{\\mathit{rs}}$ ($r,s\\in\\left\\{ 1,2,3,4\\right\\} ,r\\neq s$),\nand one of which is $\\hat{\\mathbf{x}}_{ij}$, where $\\hat{\\mathbf{x}}_{ij}$\nrepresents the exact $\\hat{\\mathbf{x}}$ corresponding to $\\mathbf{b_{\\mathit{ij}}}$.\nTherefore, we can obtain total 36 solutions which can further be separated\ninto 6 groups according to 6 different $\\mathbf{b_{\\mathit{ij}}}$.\nTo obtain a reasonable solution, here we propose a strategy that estimates\n$\\hat{\\mathbf{x}}$ by averaging the 6 closest solutions (i.e., 6\n$\\hat{\\mathbf{x}}_{ij}$) in the 6 groups of solutions, which can\nbe expressed as follows:\n\\begin{equation}\n\\hat{\\mathbf{x}}=\\frac{1}{6}\\sum_{i=1}^{4}\\sum_{j=1,j>i}^{4}\\hat{\\mathbf{x}}_{ij}.\\label{eq:39}\n\\end{equation}\nThis strategy will be verified in simulations. In this way, based\non the information in both time and space domains, the basic algorithm\nof V-P4L can properly match the 3D-2D correspondences, and obtain\nthe rotation angles corresponding to the $z^{\\mathrm{c}}-$axis $\\psi$\nand the 2D coordinate of the camera $\\left(t_{x},t_{y}\\right)^{\\mathrm{T}}$.\n\n\\subsection{\\label{subsec:tz}Calculate the $z$-coordinate of the camera}\n\nIn Subsection \\ref{subsec:z_angle tx ty}, we have obtained $\\psi$\nand $\\left(t_{x},t_{y}\\right)^{\\mathrm{T}}$. In (\\ref{eq:27}), there\nis still one unknown parameter $t_{z}$. In this section, the basic\nalgorithm of V-P4L is proposed for scenarios where LEDs have the same\nheight, i.e., $z_{1}^{\\unit{w}}=z_{2}^{\\unit{w}}=z_{3}^{\\unit{w}}=z_{4}^{\\unit{w}}$.\nBased on the single-view geometry theory, we can obtain the relationship\nbetween $z_{i}^{\\textrm{w}}$ and $t_{z}$ from (\\ref{eq:27}) as:\n\\begin{equation}\nz_{i}^{\\textrm{w}}=\\left[c_{1},c_{2},c_{3}\\right]^{\\mathrm{T}}\\cdot P_{i}^{\\textrm{c}}+t_{z}.\\label{eq:33}\n\\end{equation}\nThe estimated $z$-coordinate of the camera in WCS $\\hat{t}_{z}$\ncan be calculated as follows:\n\\begin{equation}\n\\hat{t}_{z}=\\frac{1}{4}\\left(\\sum_{i=1}^{4}z_{i}^{\\textrm{w}}-\\sum_{i=1}^{4}\\left[\\hat{c}_{1},\\hat{c}_{2},\\hat{c}_{3}\\right]^{\\mathrm{T}}\\cdot P_{i}^{\\textrm{c}}\\right),\\label{eq:30}\n\\end{equation}\nwhere $\\hat{c}_{k}$ $\\left(k\\in\\left\\{ 1,2,3\\right\\} \\right)$ is\nthe estimate of $c_{k}$.\n\nIn this way, the estimated pose $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}$\nand position of the camera $\\mathbf{\\hat{t}_{\\mathrm{c}}^{\\mathrm{w}}}$\ncan be obtained without the ideal 2D-3D correspondence assumption.\n\n\\vspace{-0cm}\n\n\n\\section{\\label{sec:V-P4L-DH}The Correction algorithm of V-P4L}\n\nMost of existing studies including the basic algorithm of V-P4L proposed\nin Subsection \\ref{sec:V-P4L-SH} assume that LEDs have the same height\n\\cite{bai2020enhanced,li2014epsilon}. However, this may not always\nbe true in practice. For instance, ceilings may be tilted due to the\nimperfect decoration or deliberate design. In these scenarios, the\nlocalization accuracy can be significantly degraded. Therefore, in\nthis subsection, based on the basic algorithm of V-P4L, we propose\na correction algorithm of V-P4L for the scenarios where LEDs have\ndifferent heights, V-P4L-DH. Based on the single-view geometry theory\nand the LLS method, V-P4L-DH can properly match the 3D-2D correspondences\nand obtain the 2D position of the camera. Based on the 2D localization,\nV-P4L-DH can achieve 3D localization by a simple optimization method.\n\n\\subsection{\\label{subsec:DH-2D}2D Localization}\n\nFor 2D-localization case where the $z$-coordinate of the camera $t_{z}$\nis known in advance, based on the single-view geometry theory, (\\ref{eq:33})\ncan be rewritten as follows:\n\\begin{equation}\n\\left[c_{1},c_{2},c_{3}\\right]^{\\mathrm{T}}\\cdot P_{i}^{\\textrm{c}}=z_{j}^{\\textrm{w}}-t_{z},\\label{eq:42}\n\\end{equation}\nwhere $i,j\\in\\left\\{ 1,2,3,4\\right\\} $. Since the 3D-2D correspondence\nare not known in advance, given a certain $P_{i}^{\\textrm{c}}$ and\n$z_{j}^{\\textrm{w}}$, we do not know their correspondence relationship.\nFortunately, there are four $P_{i}^{\\textrm{c}}$ and $z_{j}^{\\textrm{w}}$,\nand thus we can estimate $c_{k}$ $\\left(k\\in\\left\\{ 1,2,3\\right\\} \\right)$\nusing the same LLS method of solving (\\ref{eq:32}). Then, from (\\ref{eq:35}),\nwe have:\n\\begin{equation}\n\\begin{cases}\n\\hat{c}_{1}=-\\sin\\theta\\\\\n\\hat{c}_{2}=\\cos\\theta\\cdot\\sin\\varphi\\\\\n\\hat{c}_{3}=\\cos\\theta\\cdot\\cos\\varphi,\n\\end{cases}\\label{eq:43}\n\\end{equation}\nwhere $\\hat{c}_{k}$ $\\left(k\\in\\left\\{ 1,2,3\\right\\} \\right)$ is\nthe estimate of $c_{k}$. The estimated rotation angles corresponding\nto the $x^{\\mathrm{c}}-$axis $\\hat{\\varphi}$ and $y^{\\mathrm{c}}-$axis\n$\\hat{\\theta}$ can be obtained by solving (\\ref{eq:43}). Then, the\npose of the camera $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}$ and\n2D position of the camera $\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}=\\left(\\hat{t}_{x},\\hat{t}_{y}\\right)^{\\mathrm{T}}$\ncan be obtained according to (\\ref{eq:35})$-$(\\ref{eq:39}), where\n$\\hat{t}_{x}$ and $\\hat{t}_{y}$ are the estimated $x$ and $y$-coordinates\nof the camera, respectively.\n\n\\subsection{\\label{subsec:DH-3D}3D Localization}\n\nFor 3D-localization case, the $z$-coordinate of the camera $t_{z}$\nis not known in advance. For indoor scenario, the range of $t_{z}$\nmust be $[0,H_{\\mathrm{m}})$, where $H_{\\mathrm{m}}$ is the maximum\nheight of the room. Based on the above 2D-localizatoin algorithm,\nfor different $t_{z}\\in[0,H_{\\mathrm{m}})$, we can obtain different\n$\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\left(t_{z}\\right)$ and\n$\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}\\left(t_{z}\\right)$.\nBased on the estimated normal vector of the luminaire in CCS $\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}$,\nwe have \\cite{xu2016pose}:\n\\begin{equation}\n\\mathbf{\\hat{n}}_{\\mathrm{LED}}^{\\mathrm{w}}\\left(t_{z}\\right)=\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\left(t_{z}\\right)\\cdot\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}},\\label{eq:41}\n\\end{equation}\nwhere $\\mathbf{\\hat{n}}_{\\mathrm{LED}}^{\\mathrm{w}}\\left(t_{z}\\right)$\ndenotes the estimated normal vector of the luminaire in WCS when the\n$z$-coordinate of the camera is $t_{z}$. Since the world coordinates\nof the luminaire's vertices $\\mathbf{s}_{i}^{\\textrm{w}}$ ($i\\in\\left\\{ 1,2,3,4\\right\\} $)\nare known in advance, the actual normal vector of the luminaire in\nWCS $\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}$ can be calculated as follows:\n\\begin{equation}\n\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}=\\left(P_{i}^{\\textrm{w}}-P_{j}^{\\textrm{w}}\\right)\\times\\left(P_{i}^{\\textrm{w}}-P_{k}^{\\textrm{w}}\\right),\\label{eq:44}\n\\end{equation}\nwhere $i,j,k\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j\\neq k$. Therefore,\nthe difference between the estimated and actual normal vectors of\nthe luminaire in WCS can be given as:\n\\begin{equation}\n\\Delta G\\left(t_{z}\\right)=\\left\\Vert \\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}-\\mathbf{\\hat{n}}_{\\unit{LED}}^{\\textrm{w}}\\left(t_{z}\\right)\\right\\Vert _{2}.\\label{eq:45}\n\\end{equation}\nThe estimated $z$-coordinate of the camera $\\hat{t}_{z}$ can be\nobtained by the minimum $\\Delta G\\left(t_{z}\\right)$, i.e.:\n\\begin{equation}\n\\hat{t}_{z}=\\arg\\,\\min_{t_{z}}\\Delta G\\left(t_{z}\\right).\\label{eq:46}\n\\end{equation}\n\nTo reduce the complexity of V-P4L-DH, we propose a $n$-step segmentation\noptimization strategy for V-P4L-DH. In the first step, we divide the\nrange of $t_{z}$ into $N$ segments evenly, and set:\n\\begin{equation}\nt_{z}\\in\\left\\{ 0,\\frac{H_{\\mathrm{m}}}{N},2\\frac{H_{\\mathrm{m}}}{N},3\\frac{H_{\\mathrm{m}}}{N},\\ldots,H_{\\mathrm{m}}\\right\\} ,\\label{eq:61}\n\\end{equation}\ni.e., we set the interval between adjacent $t_{z}$ as $\\varepsilon_{1}\\triangleq\\frac{H_{\\mathrm{m}}}{N}$.\nSubstituting all the $t_{z}$ into (\\ref{eq:45}), we have $\\Delta G\\left(t_{z}\\right)=\\left\\{ \\Delta G\\left(t_{z,0}\\right),\\Delta G\\left(t_{z,1}\\right),\\ldots,\\Delta G\\left(t_{z,N}\\right)\\right\\} $.\nAccording to (\\ref{eq:46}), we can find the minimal $\\Delta G\\left(t_{z}\\right)$.\nWe denote the minimal $\\Delta G\\left(t_{z}\\right)$ by $\\Delta G\\left(t_{z,i}\\right)$\nwhere $i\\in\\left\\{ 0,1,\\ldots,N\\right\\} $ is the index of the $t_{z}$\nthat corresponding to $\\Delta G\\left(t_{z,i}\\right)$. In the second\nstep, we reduce the range of $t_{z}$ to $\\left(\\frac{H_{\\mathrm{m}}}{N}\\left(i-1\\right),\\frac{H_{\\mathrm{m}}}{N}\\left(i+1\\right)\\right)$.\nWe set:\n\\begin{equation}\nt_{z}\\in\\left\\{ \\frac{H_{\\mathrm{m}}}{N}\\left(i-1\\right)+\\varepsilon_{2},\\frac{H_{\\mathrm{m}}}{N}\\left(i-1\\right)+2\\varepsilon_{2},\\ldots,\\frac{H_{\\mathrm{m}}}{N}\\left(i+1\\right)-\\varepsilon_{2}\\right\\} ,\\label{eq:60}\n\\end{equation}\ni.e., the interval between adjacent $t_{z}$ is $\\varepsilon_{2}$,\nwhere $\\varepsilon_{2}<\\varepsilon_{1}$. We repeat the process of\nthe first step for the second step to the $n$th step until we can\nobtain the precise $\\hat{t}_{z}$ according to (\\ref{eq:46}), and\nobtain the optimal $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}$ and\n$\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}$. Therefore, based\non the simple optimization method, we can obtain the pose and 3D position\nof the receiver when LEDs have different heights.\n\nIn this way, when LEDs have the same height, the basic algorithm of\nV-P4L can be implemented for high accuracy at low complexity. When\nLEDs have different heights, V-P4L-DH can be implemented for high\naccuracy. In summary, V-P4L algorithm is elaborated in Algorithm 1.\nAlthough V-P4L requires the LED luminaire to be a rectangle, it is\nrobust to partial occlusion, which is meaningful due to the limitation\non the camera's field of view. For instance, if the projection of\n$P_{2}$, $p_{2}$, is blocked by barriers and not on the image plane\nas shown in Fig. \\ref{fig:occlusion}, the pixel coordinate of $p_{2}$\ncan be determined by the intersection of $l_{12}$ and $l_{23}$,\nand thus V-P4L can be still successively implemented.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.6]{occlusion}\n\\par\\end{centering}\n\\caption{\\label{fig:occlusion}A occlusion scenario where the projection of\n$P_{2}$, $p_{2}$, is blocked by barriers and not on the image plane.}\n\\end{figure}\n\n\\begin{algorithm*}\n\\caption{V-P4L Algorithm.}\n\n\\textbf{Input:}\n\n\\:\\qquad{}$P_{i}^{\\textrm{w}}=\\left(x_{i}^{\\textrm{w}},y_{i}^{\\textrm{w}},z_{i}^{\\textrm{w}}\\right)^{\\mathrm{T}}$\n($i\\in\\left\\{ 1,2,3,4\\right\\} $);\n\n\\:\\qquad{}$p_{j}^{\\textrm{p}}=\\left(u_{j}^{\\textrm{p}},v_{j}^{\\textrm{p}}\\right)^{\\mathrm{T}}$\n($j\\in\\left\\{ 1,2,3,4\\right\\} $);\n\n\\:\\qquad{}$u_{0}$, $v_{0}$, $f$, $f_{u}$, $f_{v}$ and $\\varepsilon_{1}$.\n\n\\begin{algorithmic}[1]\n\n\\STATE Calculate $p_{i}^{\\textrm{i}}=\\left(x_{i}^{\\textrm{i}},y_{i}^{\\textrm{i}}\\right)^{\\mathrm{T}}$($i\\in\\left\\{ 1,2,3,4\\right\\} $)\naccording to (\\ref{eq:1}).\n\n\\FOR {$i$ = 1 $\\to$ $4$, $j$ = 1 $\\to$ $4$ and $j\\neq i$}\n\n\\STATE Calculate $\\phi_{l_{ij}}$ and $\\rho_{l_{ij}}$ by $p_{i}^{\\textrm{i}}$\nand $p_{j}^{\\textrm{i}}$.\n\n\\STATE $A_{\\Pi_{ij}}\\leftarrow f\\cos\\phi_{l_{ij}}$, $B_{\\Pi_{ij}}\\leftarrow f\\sin\\phi_{l_{ij}}$\nand $C_{\\Pi_{ij}}\\leftarrow-\\rho_{l_{ij}}$, and then $\\mathbf{n}_{\\Pi_{ij}}^{\\mathrm{c}}\\leftarrow\\left(A_{\\Pi_{ij}},B_{\\Pi_{ij}},C_{\\Pi_{ij}}\\right)^{\\mathrm{T}}$.\n\n\\STATE $\\mathbf{v}_{L_{ij}}^{\\mathrm{c}}\\leftarrow\\left(A_{\\mathrm{LED}},B_{\\mathrm{LED}},C_{\\mathrm{LED}}\\right)^{\\mathrm{T}}\\times\\mathbf{n}_{\\Pi_{ij}}^{\\mathrm{c}}$,\nwhere $\\left(A_{\\mathrm{LED}},B_{\\mathrm{LED}},C_{\\mathrm{LED}}\\right)^{\\mathrm{T}}$\ndenotes the normal vector of the luminaire in CCS.\n\n\\ENDFOR\n\n\\STATE Define $m=\\frac{A_{\\mathrm{LED}}}{C_{\\mathrm{LED}}}$ and\n$n=\\frac{B_{\\mathrm{LED}}}{C_{\\mathrm{LED}}}$. Calculate $m$ and\n$n$ according to $\\mathbf{v}_{L_{34}}^{\\mathrm{c}}\\cdot\\mathbf{n}_{\\Pi_{12}}^{\\mathrm{c}}=0$\nand $\\mathbf{v}_{L_{41}}^{\\mathrm{c}}\\cdot\\mathbf{n}_{\\Pi_{23}}^{\\mathrm{c}}=0$.\n\n\\STATE $\\cos\\alpha\\leftarrow\\frac{m}{\\sqrt{m^{2}+n^{2}+1}}$, $\\cos\\beta\\leftarrow\\frac{n}{\\sqrt{m^{2}+n^{2}+1}}$\nand $\\cos\\gamma\\leftarrow\\frac{1}{\\sqrt{m^{2}+n^{2}+1}}$, and then\n$\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}\\leftarrow\\left(\\cos\\alpha,\\cos\\beta,\\cos\\gamma\\right)^{\\mathrm{T}}$.\n\n\\STATE Calculate $\\mathbf{M}_{P_{1}}$ according to (\\ref{eq:100}),\nand $\\mathbf{M}_{P_{i}}$ ($i\\in\\left\\{ 2,3,4\\right\\} $) can be calculated\nin the same way as $\\mathbf{M}_{P_{1}}$.\n\n\\STATE $q_{1}\\leftarrow\\frac{1}{6}\\left|\\det(\\mathbf{M}_{q_{1}})\\right|$,\nwhere $\\mathbf{M}_{q_{1}}=\\left[\\mathbf{M}_{P_{1}},\\mathbf{M}_{P_{2}},\\mathbf{M}_{P_{3}}\\right]^{\\mathrm{T}}$,\nand $q_{i}$ ($i\\in\\left\\{ 2,3,4\\right\\} $) can be calculated in\nthe same way as $q_{1}$.\n\n\\STATE $C_{\\mathrm{LED}}\\leftarrow\\sqrt{\\frac{3\\sum_{i=1}^{4}q_{i}\\cdot\\sqrt{m^{2}+n^{2}+1}}{2S}}$.\n\n\\STATE $P_{i}^{\\textrm{c}}\\leftarrow\\frac{\\mathbf{M}_{P_{i}}}{C_{\\mathrm{LED}}}$\n$\\left(i\\in\\left\\{ 1,2,3,4\\right\\} \\right)$.\n\n\\IF {$z_{1}^{\\textrm{w}}=z_{2}^{\\textrm{w}}=z_{3}^{\\textrm{w}}=z_{4}^{\\textrm{w}}$}\n\n\\STATE $\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}\\leftarrow\\left(0,0,1\\right)^{\\mathrm{T}}$.\n\n\\STATE Calculate $\\hat{\\varphi}$ and $\\hat{\\theta}$ according to\n(\\ref{eq:29}).\n\n\\STATE Calculate $\\hat{\\psi}$, $\\hat{t}_{x}$ and $\\hat{t}_{y}$\naccording to (\\ref{eq:31}) - (\\ref{eq:32}).\n\n\\STATE Calculate $\\hat{t}_{z}$ according to (\\ref{eq:30}) if $t_{z}$\nis not known in advance.\n\n\\ELSE\n\n\\IF {$t_{z}$ is known in advance}\n\n\\STATE Calculate $\\hat{c}_{i}$ $\\left(i\\in\\left\\{ 1,2,3\\right\\} \\right)$\nfollowing the same method of solving (\\ref{eq:32}).\n\n\\STATE Calculate $\\hat{\\varphi}$ and $\\hat{\\theta}$ according to\n(\\ref{eq:43}).\n\n\\STATE Calculate $\\hat{\\psi}$, $\\hat{t}_{x}$ and $\\hat{t}_{y}$\naccording to (\\ref{eq:35}) - (\\ref{eq:39}). Therefore, $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}$\nand $\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}=\\left(\\hat{t}_{x},\\hat{t}_{y}\\right)^{\\mathrm{T}}$\ncan be obtained.\n\n\\ELSE\n\n\\STATE $\\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}\\leftarrow\\left(P_{i}^{\\textrm{w}}-P_{j}^{\\textrm{w}}\\right)\\times\\left(P_{i}^{\\textrm{w}}-P_{k}^{\\textrm{w}}\\right)$,\nwhere $i,j,k\\in\\left\\{ 1,2,3,4\\right\\} ,i\\neq j\\neq k$.\n\n\\FOR {$i=0\\rightarrow\\frac{H_{\\mathrm{m}}}{\\varepsilon_{1}}$}\n\n\\STATE $\\hat{t}_{z,i}\\leftarrow i\\varepsilon_{1}$.\n\n\\STATE Calculate $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\left(t_{z,i}\\right)$\nand $\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}\\left(t_{z,i}\\right)$.\n\n\\STATE $\\mathbf{\\hat{n}}_{\\mathrm{LED}}^{\\mathrm{w}}\\left(t_{z,i}\\right)\\leftarrow\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\left(t_{z,i}\\right)\\cdot\\mathbf{n}_{\\mathrm{LED}}^{\\mathrm{c}}$.\n\n\\STATE $\\Delta G\\left(t_{z,i}\\right)\\leftarrow\\left\\Vert \\mathbf{n}_{\\unit{LED}}^{\\textrm{w}}-\\mathbf{\\hat{n}}_{\\unit{LED}}^{\\textrm{w}}\\left(t_{z,i}\\right)\\right\\Vert _{2}$.\n\n\\ENDFOR\n\n\\STATE $\\hat{t}_{z}\\leftarrow\\min_{t_{z}}\\Delta G\\left(t_{z}\\right)$.\nMeanwhile, $\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\leftarrow\\mathbf{\\hat{R}}_{\\mathrm{c}}^{\\mathrm{w}}\\left(\\hat{t}_{z}\\right)$\nand $\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}=\\left(\\hat{t}_{x},\\hat{t}_{y}\\right)^{\\mathrm{T}}\\leftarrow\\mathbf{\\hat{t}_{\\mathrm{c,2D}}^{\\mathrm{w}}}\\left(\\hat{t}_{z}\\right)$.\n\n\\ENDIF\n\n\\ENDIF\n\n\\end{algorithmic}\n\n\\textbf{Output:} $\\mathbf{\\hat{R}}_{\\mathrm{c,est}}^{\\mathrm{w}}$\nand $\\mathbf{\\hat{t}_{\\mathrm{c,est}}^{\\mathrm{w}}}=\\left(\\hat{t}_{x},\\hat{t}_{y},\\hat{t}_{z}\\right)^{\\mathrm{T}}$.\n\n\\label{algorithm1}\n\\end{algorithm*}\n\n\n\\section{\\label{sec:simulation}SIMULATION RESULTS AND ANALYSES}\n\n\\global\\long\\def\\arraystretch{0.9\n\\begin{table}[t]\n\\centering{}\\centering{}\\caption{\\label{tab:Parameters-used-for}System Parameters.}\n\\begin{tabular}{>{\\raggedright}m{5.5cm}|>{\\raggedright}m{1.6cm}|>{\\centering}m{2.5cm}}\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}Parameter}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}Value}}\\tabularnewline\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}Room size ($\\textrm{length}\\times\\textrm{width}\\times\\textrm{height}$)}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}$5\\,\\unit{m}\\times5\\,\\unit{m}\\times3\\,\\unit{m}$}}\\tabularnewline\n\\hline\n\\multirow{2}{5.5cm}{{\\footnotesize{}{}{}{}{}{}{}{}{}Length and width of LED luminaire}} & \\centering{}{\\footnotesize{}{}{}{}{}{}{}{}{}Length}  & {\\footnotesize{}{}{}{}{}{}{}{}{}Width}\\tabularnewline\n\\cline{2-3} \\cline{3-3}\n & \\multirow{1}{1.6cm}{\\centering{}{\\footnotesize{}{}{}{}{}{}{}{}{}$120\\,\\mathrm{cm}$}} & \\centering{}{\\footnotesize{}{}{}{}{}{}{}{}{}$20\\,\\mathrm{cm}-100\\,\\mathrm{cm}$}\\tabularnewline\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}LED semi-angle, $\\Phi_{\\nicefrac{1}{2}}$}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}$60{^{\\circ}}$}}\\tabularnewline\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}Principal point of camera}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}$\\left(u_{0},v_{0}\\right)=\\left(320,240\\right)$}}\\tabularnewline\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}Focal ratio of camera}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}$f_{u}=f_{v}=800$}}\\tabularnewline\n\\hline\n{\\footnotesize{}{}{}{}{}{}{}{}{}The distance between the PD\nand the camera in eCA-RSSR, $d_{\\mathrm{pc}}$}  & \\multicolumn{2}{c}{{\\footnotesize{}{}{}{}{}{}{}{}{}$1\\,\\mathrm{cm}$}}\\tabularnewline\n\\hline\n\\end{tabular}\n\\end{table}\n\nSince V-P4L combines VLC and computer vision based localization, a\nVLP algorithm named enhanced camera assisted received signal strength\nratio algorithm (eCA-RSSR) \\cite{bai2020enhanced}, and a typical\ncomputer vision algorithm termed P4L algorithm \\cite{xiaojian2008analytic}\nare conducted as the baselines schemes in this section. \\vspace{-0.5cm}\n\n\n\\subsection{Simulation Setup}\n\nThe system parameters are listed in Table \\ref{tab:Parameters-used-for}.\nThe LED luminaire is deployed in the center of the ceiling. The length\nof the luminaire which is along the $x^{\\mathrm{w}}$-axis is set\nto $120\\,\\mathrm{cm}$ \\cite{Qiu2016Let,philips}, and the widths\nof luminaire which is along the $y^{\\mathrm{w}}$-axis are varied\naccording to configurations. We set the rectangular luminaire tilt\nwith various angles along the $y^{\\mathrm{w}}$-axis to represent\nthat LEDs have different heights. All statistical results are averaged\nover $1000$ independent runs. For each simulation run, the receiver\npositions are selected in the room randomly. The pinhole camera is\ncalibrated. The image noise is modeled as a white Gaussian noise having\nan expectation of zero and a standard deviation of $2\\;\\unit{pixels}$\n\\cite{zhou2019robust}. Since the image noise affects the pixel coordinate\nof the luminaire's projection on the image plane, the pixel coordinate\nis obtained by processing 20 images for the same position. Moreover,\nwe use two-step segmentation optimization strategy, and we set $\\varepsilon_{1}=10\\:\\mathrm{cm}$\nand $\\varepsilon_{2}=1\\:\\mathrm{cm}$.\n\nNote that since eCA-RSSR requires three LEDs for localization, we\nassume that the four LEDs at the vertices of the luminaire are used\nfor eCA-RSSR and we choose the three LEDs with the highest RSSs to\nachieve best performance for eCA-RSSR. Additionally, all the LEDs\ntransmit different information in eCA-RSSR, which is not required\nin V-P4L and the P4L algorithm. Therefore, compared with V-P4L, the\nVLC link of eCA-RSSR is more complex. Furthermore, eCA-RSSR relies\non the perfect Lambertian pattern model. However, the VLC channel\nmodel can be quiet different from the Lambertian pattern model even\nusing the LED having nearly-ideal Lambertian pattern, as shown in\n\\cite{miramirkhani2015channel}, and the difference can be over 100\\%\nin certain cases. Therefore, we set a random deviation $\\delta_{\\mathrm{1}}\\leq10\\%$\nfor the Lambertian pattern model for eCA-RSSR, conservatively. On\nthe other hand, the P4L algorithm exploits a rectangle to estimate\nthe position and pose of the camera. The P4L algorithm assumes that\nthe camera knows the 3D-2D correspondences. However, the beacon in\nthe P4L algorithm cannot convey time-domain information to the camera,\nwhich make the assumption impractical. In addition, the method to\nfind the 3D-2D line correspondences given by \\cite{ICRA2018Manhattan}\nis also not practical when the camera captures the beacons on the\nceiling as stated in Section \\ref{sec:intro}. Therefore, we set a\nrandom error rate $\\delta_{\\mathrm{2}}\\leq10\\%$ for the 3D-2D correspondences,\nconservatively. Moreover, the P4L algorithm can only obtain the relative\nposition. For comparison with V-P4L, we transform the relative position\ninto the absolute position for the P4L algorithm in this section.\n\nWe evaluate the performance of V-P4L in terms of its accuracy of position\nand pose estimation. We define position error as:\n\\begin{equation}\nPE=\\left\\Vert \\mathbf{r}_{\\unit{true}}^{\\textrm{w}}-\\mathbf{r}_{\\unit{est}}^{\\textrm{w}}\\right\\Vert ,\\label{eq:48}\n\\end{equation}\nwhere $\\mathbf{r}_{\\unit{true}}^{\\textrm{w}}=\\left(x_{r,\\unit{true}}^{\\textrm{w}},y_{r,\\unit{true}}^{\\textrm{w}},z_{r,\\unit{true}}^{\\textrm{w}}\\right)$\nand $\\mathbf{r_{\\textrm{est}}^{\\textrm{w}}}=\\left(x_{r,\\mathscr{\\textrm{est}}}^{\\textrm{w}},y_{r,\\textrm{est}}^{\\textrm{w}},z_{r,\\textrm{est}}^{\\textrm{w}}\\right)$\nare the actual and estimated world coordinates of the receiver, respectively.\nAdditionally, the accuracy of pose estimation can be measured by the\norientation error which is defined as:\n\\begin{equation}\nOE=\\left|\\varTheta_{\\mathrm{true}}-\\varTheta_{\\mathrm{est}}\\right|,\\label{eq:49}\n\\end{equation}\nwhere $\\varTheta_{\\mathrm{true}}$ and $\\varTheta_{\\mathrm{est}}$\nare the actual and estimated rotation angles, respectively.\n\nIn this section, we will evaluate the performance of V-P4L under various\ntilted angles of the luminaire, various widths of the luminaire and\nvarious image noise. We compare for both scenarios where LEDs have\nthe same height and have different heights, which are denoted by SH\nand DH, respectively in the figures. Since the basic algorithm of\nV-P4L is used when LEDs have the same height, we denote the basic\nalgorithm of V-P4L by V-P4L-SH in figures.\\vspace{-0.4cm}\n\n\n\\subsection{\\label{subsec:Accuracy-Performance-1}Effect Of Luminaire's Tilted\nAngle On Accuracy Performance}\n\n\\begin{figure*}[t]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\setlength{\\belowcaptionskip}{-8pt}\n \\centering \\subfigure[Without occlusion.]{\\label{secfig1} \\includegraphics[scale=0.5]{tilt_accuracy}\n} \\subfigure[With occlusion.]{\\label{secfig2} \\includegraphics[scale=0.5]{occlusion_accuracy}\n} \\caption{\\label{fig:tilt angle PE}The comparison of position errors (PEs)\nwith varying tilted angles of the luminaire between the basic algorithm\nof V-P4L and V-P4L-DH. (a) There is no occlusion. (b) The projection\nof $P_{2}$, $p_{2}$, is blocked by barriers and not on the image\nplane as shown in Fig. \\ref{fig:occlusion}.}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\setlength{\\belowcaptionskip}{-8pt}\n \\centering \\subfigure[OEs along the x-axis.]{ \\includegraphics[width=0.31\\linewidth]{tilt_angle_x}\n} \\subfigure[OEs along the y-axis.]{ \\includegraphics[width=0.31\\linewidth]{tilt_angle_y}\n} \\subfigure[OEs along the z-axis.]{ \\includegraphics[width=0.31\\linewidth]{tilt_angle_z}\n} \\caption{\\label{fig:tilt angle OE}The comparison of orientation errors (OEs)\nalong the $x$-axis, $y$-axis and $z$-axis between V-P4L-SH and\nV-P4L-DH.}\n\\end{figure*}\n\nWe first evaluate the effect of the tilted angles of the luminaire\non localization accuracy of V-P4L for both 2D and 3D localization.\nThis performance is represented by the means of PEs with the tilted\nangles of the luminaire varying from $0{^{\\circ}}$ to $40{^{\\circ}}$.\nThe width of the luminaire is $40\\;\\unit{cm}$. As shown in Fig. \\ref{secfig1},\nfor 2D localization, when the tilted angle of the luminaire is $0{^{\\circ}}$,\ni.e., LEDs have the same height, the basic algorithm of V-P4L can\nobtain a slight better performance than V-P4L-DH. However, as the\ntilted angle of the luminaire varying from $0{^{\\circ}}$ to $40{^{\\circ}}$,\nthe means of PEs of the basic algorithm of V-P4L increase from about\n5 cm to about 98 cm. For 3D localization, the means of PE of the basic\nalgorithm of V-P4L increase from about 10 cm to about 135 cm. In contrast,\nfor all the tilted angles of the luminaire, the means of PEs of V-P4L-DH\nare less than 18 cm for both 2D and 3D localization. Since in 3D localization\nthe estimated 2D position and pose is optimized meanwhile when search\nfor the optimal $\\hat{t}_{z}$, V-P4L can achieve better performance\nfor 3D localization than 2D localization.\n\nSince the accuracy of pose estimation is also affected by the tilted\nangle of the luminaire, we then evaluated the effect of the tilted\nangles on OEs of V-P4L for both 2D and 3D localization. As shown in\nFig. \\ref{fig:tilt angle OE}, the OEs along the $x$-axis, $y$-axis\nand $z$-axis are shown seperately. When the tilted angle of the luminaire\nis $0{^{\\circ}}$, the basic algorithm of V-P4L can obtain slight\nbetter performance than V-P4L-DH. However, as the tilted angle of\nthe luminaire varying from $0{^{\\circ}}$ to $40{^{\\circ}}$, the\nmeans of OEs of the basic algorithm of V-P4L increase from about $2{^{\\circ}}$\nto about $20{^{\\circ}}$. In contrast, for all the tilted angles of\nthe luminaire, V-P4L-DH can achieve consistent well performance, and\nthe means of OEs are always less than $6{^{\\circ}}$.\n\nWe also evaluate the effect of partial occlusion as shown in Fig.\n\\ref{fig:occlusion} on localization accuracy of V-P4L. Figure \\ref{secfig1}\nshows the accuracy performance of position estimation when there is\nno occlusion in the scenario. In contrast, Fig. \\ref{secfig2} shows\nthe accuracy performance of position estimation when $p_{2}$ is blocked\nand is not on the image plane. As shown in Fig. \\ref{secfig2}, when\nthe luminaire is partially blocked, V-P4L can still achieve high accuracy.\nIn particular, the accuracy performance of the basic algorithm of\nV-P4L is almost the same in Fig. \\ref{secfig1} and Fig. \\ref{secfig2}.\nIn addition, for V-P4L-DH, the 2D-localization accuracy reduces about\n2 cm, while the 3D-localization accuracy improves about 4 cm. Therefore,\nV-P4L is robust to partial occlusion as introduced in Section \\ref{sec:V-P4L-DH}.\n\nIn this subsection, we have verified V-P4L can achieve high accuracy\nfor both 2D and 3D localization. Since 2D localization is the special\ncase of 3D localization where the height of the receiver is known\nin advance, in the following subsections, we will only show the simulation\nresults for 3D localization.\n\n\\subsection{\\label{subsec:Accuracy-Performance}Effect Of Luminaire's Widths\nOn Accuracy Performance}\n\nWe then evaluate the effect of the luminaire's width on localization\naccuracy of V-P4L. This performance is represented by the means of\nPEs with the width varying from $20\\;\\unit{cm}$ to $100\\;\\unit{cm}$.\nFor the scenarios where LEDs have different heights, the tilted angle\nof the luminaire is $20{^{\\circ}}$. As shown in Fig. \\ref{fig:width compare1},\nfor both the scenarios where LEDs have the same height and have different\nheights, V-P4L is able to obtain the best performance among the three\nalgorithms. When LEDs have the same height, the means of PEs of the\nbasic algorithm of V-P4L are below $15\\;\\unit{cm}$. In contrast,\nfor eCA-RSSR, the means of PEs are around 70 cm as the width of the\nluminaire increases from $20\\;\\unit{cm}$ to $100\\;\\unit{cm}$. Additionally,\nfor the P4L algorithm, the means of PEs decrease from over $100\\;\\unit{cm}$\nto about $40\\;\\unit{cm}$. On the other hand, when LEDs have different\nheights, the means of PEs of V-P4L-DH decrease from about 10 cm to\n5 cm. In contrast, the means of PEs of both eCA-RSSR and the P4L algorithm\nare higher than 50 cm for all the widths of the luminaire. As shown\nin Fig. \\ref{fig:width compare1}, the localization accuracy increases\nwith the increase of the width for V-P4L. However, the PEs of V-P4L\nare always less than $15\\,\\unit{cm}$ regardless of the height differences\namong LEDs using a single LED luminaire whose width is longer than\n$20\\;\\unit{cm}$, and thus V-P4L can be applied to popular indoor\nluminaires.\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.5]{area_3D_accuracy}\n\\par\\end{centering}\n\\caption{\\label{fig:width compare1}The comparison of position errors (PEs)\nwith varying widths of the luminaire among eCA-RSSR, the P4L algorithm\nand V-P4L for both the scenarios where LEDs have the same height and\nhave different heights.}\n\\end{figure}\n\nSince the accuracy of pose estimation is also affected by the width\nof the LED luminaire, we then compare the OEs between V-P4L and the\nP4L algorithm with varying widths of the luminaire. As shown in Fig.\n\\ref{fig:width orientation 3D}, for V-P4L, all the means of OEs along\nthe $x$-axis, $y$-axis and $z$-axis are less than $3.5{^{\\circ}}$\nregardless of the height differences among the LEDs. When LEDs have\nthe same height, the means of OEs along the $x$-axis and $y$-axis\ndecrease from about $2{^{\\circ}}$ to about $0.5{^{\\circ}}$ and from\n$1{^{\\circ}}$ to about $0.3{^{\\circ}}$, respectively for both the\nbasic algorithm of V-P4L and the P4L algorithm. Additionally, the\nmeans of OEs of the basic algorithm of V-P4L along the $z$-axis decrease\nfrom about $3{^{\\circ}}$ to about $0.5{^{\\circ}}$, which is over\n$5{^{\\circ}}$ better than that of the P4L algorithm. On the other\nhand, when LEDs have different heights, for V-P4L-DH, the means of\nOEs along the $x$-axis, $y$-axis and $z$-axis decrease from $2.5{^{\\circ}}$\nto $1.5{^{\\circ}}$, from $1.5{^{\\circ}}$ to $0.5{^{\\circ}}$ and\nfrom $2.2{^{\\circ}}$ to $1.5{^{\\circ}}$, respectively. In contrast,\nfor the P4L algorithm, the means of OEs along the $x$-axis, $y$-axis\nand $z$-axis are about $9{^{\\circ}}$, $6{^{\\circ}}$, $13{^{\\circ}}$,\nrespectively. Therefore, compared with the P4L algorithm, V-P4L can\nobtain higher accuracy for pose estimation using popular indoor luminaire.\n\\begin{figure*}[t]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\setlength{\\belowcaptionskip}{-8pt}\n \\centering \\subfigure[OEs along the x-axis.]{ \\includegraphics[width=0.31\\linewidth]{area_3D_angle_x}\n} \\subfigure[OEs along the y-axis.]{ \\includegraphics[width=0.31\\linewidth]{area_3D_angle_y}\n} \\subfigure[OEs along the z-axis.]{ \\includegraphics[width=0.31\\linewidth]{area_3D_angle_z}\n} \\caption{\\label{fig:width orientation 3D}The comparison of orientation errors\n(OEs) along the $x$-axis, $y$-axis and $z$-axis between the P4L\nalgorithm and V-P4L with varying widths of the luminaire.}\n\\end{figure*}\n\n\n\\subsection{\\label{subsec:noise effect}Effect Of Image Noise On Accuracy Performance}\n\nIn this subsection, we evaluate the effect of the image noise on the\nlocalization performance of V-P4L when the width of the luminaire\nis $40\\;\\unit{cm}$. When LEDs have different heights, the tilted\nangle of the luminaire is $20{^{\\circ}}$. The image noise is modeled\nas a white Gaussian noise having an expectation of zero and a standard\ndeviation, $\\sigma_{n}$, ranging from $0$ to $4\\;\\unit{pixels}$\n\\cite{zhou2019robust}. Figure \\ref{fig:noise PE} shows the means\nof PEs versus image noises. As shown in Fig. \\ref{fig:noise PE},\nwhen LEDs have the same height, the means of PEs of the basic algorithm\nof V-P4L increase from $0\\;\\unit{cm}$ to $20\\;\\unit{cm}$ as the\nimage noise increases from $0$ to $4\\;\\unit{pixels}$. In contrast,\nfor the P4L algorithm, the means of PEs increase from 30 cm to 40\ncm. Additionally, for eCA-RSSR, the means of PEs are around 75 cm.\nWhen LEDs have different heights, the means of PEs of V-P4L-DH increase\nfrom 0 cm to 11 cm. In contrast, for the P4L algorithm, the means\nof PEs are about 85 cm. Additionally, for eCA-RSSR, the means of PEs\nare about 135 cm. Therefore, compared with the P4L algorithm and eCA-RSSR,\nV-P4L can obtain higher accuracy for position estimation.\n\nFinally, we compare the accuracy of pose estimation between V-P4L\nand the P4L algorithm under different image noises. Figure \\ref{fig:noise OE 3D}\nshow the means of orientation errors along $x$-axis, $y$-axis and\n$z$-axis with the image noise ranging from $0$ to $4\\;\\unit{pixels}$.\nWhen LEDs have the same height, the means of OEs of the basic algorithm\nof V-P4L along $x$-axis, $y$-axis and $z$-axis increase from $0{^{\\circ}}$\nto $2.5{^{\\circ}}$, from $0{^{\\circ}}$ to $1.2{^{\\circ}}$ and from\n$0{^{\\circ}}$ to $3.8{^{\\circ}}$, respectively. In contrast, for\nthe P4L algorithm, the means of OEs along the $x$-axis and $y$-axis\nincrease from $0{^{\\circ}}$ to $1.2{^{\\circ}}$ and from $0{^{\\circ}}$\nto $0.5{^{\\circ}}$, respectively, which is slight better than the\nbasic algorithm of V-P4L. However, the means of OEs of the P4L algorithm\nalong the $z$-axis increase from about $6{^{\\circ}}$ to about $8{^{\\circ}}$\nwhich is over $4{^{\\circ}}$ worse than that of the basic algorithm\nof V-P4L. On the other hand, when LEDs have different heights, for\nV-P4L-DH, the means of OEs along $x$-axis, $y$-axis and $z$-axis\nincrease from about $1{^{\\circ}}$ to less than $3.5{^{\\circ}}$.\nIn contrast, for the P4L algorithm, the means of OEs along $x$-axis,\n$y$-axis and $z$-axis are about $9{^{\\circ}}$, $6{^{\\circ}}$,\n$13{^{\\circ}}$, respectively. Therefore, compared with the P4L algorithm,\nV-P4L can obtain more stable and accurate pose estimation regardless\nof the image noise.\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.5]{noise_3D_accuracy}\n\\par\\end{centering}\n\\caption{\\label{fig:noise PE}The comparison of position errors (PEs) with\nvarying image noise among eCA-RSSR, the P4L algorithm, V-P4L for both\nthe scenarios where LEDs are at the same height and at different heights.}\n\n\\end{figure}\n\\begin{figure*}[t]\n\\setlength{\\abovecaptionskip}{0.2cm}\n\\setlength{\\belowcaptionskip}{-8pt}\n \\centering \\subfigure[OEs along the x-axis.]{ \\includegraphics[width=0.31\\linewidth]{noise_3D_angle_x}\n} \\subfigure[OEs along the y-axis.]{ \\includegraphics[width=0.31\\linewidth]{noise_3D_angle_y}\n} \\subfigure[OEs along the z-axis.]{ \\includegraphics[width=0.31\\linewidth]{noise_3D_angle_z}\n} \\caption{\\label{fig:noise OE 3D}The comparison of orientation errors (OEs)\nalong the $x$-axis, $y$-axis and $z$-axis between the P4L algorithm\nand V-P4L with varying image noise.}\n\\end{figure*}\n\n\\vspace{-0.3cm}\n\n\n\\section{CONCLUSION}\n\n\\label{sec:CONCLUSION}We have proposed a novel indoor localization\nalgorithm named V-P4L that estimates the position and pose of the\ncamera using a single, VLC-enabled LED luminaire. The camera is used\nto simultaneously capture the information in both time and space domains.\nBased on the information captured by the camera, V-P4L does not require\nthe 3D-2D correspondences. Moreover, V-P4L can be implemented regardless\nof the height differences among LEDs. Therefore, V-P4L can achieve\nhigher feasibility and higher accuracy than eCA-RSSR and the conventional\nPnL algorithms. Simulation results have shown that for V-P4L the position\nerror is always less than $15\\,\\unit{cm}$ and the orientation error\nis always less than $3{^{\\circ}}$ using popular indoor luminaires.\nIn the future, we will experimentally implement V-P4L based on a dedicated\ntest bed.\n\n\\vspace{-0.6cm}\n\n \\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRapid advances in deep learning (DL) has enabled significant remote sensing applications in automated systems for disaster management. According to the World Health Organization (WHO), 90,000 people die 160 million people are severely affected by a variety of natural disasters. Furthermore, 45 million people are displaced from their homes due to a variety of natural disasters and political violence. In the whole year of 2019, there were 409 severe natural disasters occurred worldwide. Therefore, there is clearly a challenge to quantify the magnitude of said disasters and conflicts, in order to inform a range of government and NGO systems to provide appropriate assistance and recovery plans.\n\n\\subsection{Automated Disaster Quantification and Management}\nIn post disaster scenarios, site exploration and disaster evaluation is challenged by dangerous and difficult environments for first responders. Scalable quantification, updated quantification is extremely challenging over vast areas. For example Hurricane Katrina (2005) caused damage over 230,000 km$^{2}$ grossing to over \\$160bn of damage. \n\nSatellite images have been previously used for understanding site access and automating responses, but manual feature processing is often slow and lacks scalability to large areas. Recent advances in deep learning can identify of hazard zones, provide risk assessment and insurance compensation. The data used includes a combination of aerial drone photography, satellite imaging, and social media data. In recent years, automated disaster detection system \\cite{Amit16} have used CNNs to extract data from the immediate disaster area and prioritise resource distribution and recognise viable infrastructure for aid delivery \\cite{RSOS}. The results show that the accuracy of disaster detection is 80-90\\%. Similar work using aerial photos from UAVs have been used with a VGGNet deep learning model achieving 91\\% accuracy \\cite{Kamilaris18}. Recognising the complex discontinuities in disaster images, improvements were made by adding a residual connection and extended convolution to the previous CNN frameworks \\cite{Duarte18}. When combined with the feature maps generated by aerial images and satellite image samples, the work improved the overall classification of the satellite images for building damage by nearly 4\\%. More recently, researchers have developed a new deep learning approach to analyse flood disaster images and quickly detect areas that have been flooded or destroyed to assess the extent and severity of damage \\cite{Sublime19}. However, the open challenge in this area is \\textit{scene parsing}, whereby recognising and segmenting livable objects (e.g. houses, infrastructure) from natural environment (e.g. trees and open spaces) is critical to quantifying the built livable space impact. Another open challenge is that quantification thus for in the above work has focused on singular dimensions, however we must recognise that economic damage and human fatality are both important indicators. Therefore, we would need to develop a model that combines them.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{Fig1.pdf}\n\t\\caption{Disaster scene parsing using PSPNet: (a) pre- and post-disaster images containing both built and natural environment, (b-c) feature map feeds a pyramid pooling module to achieve (d) contextual object segmentation that informs damage quantification.}\n\t\\label{1}\n\\end{figure*}\n\n\n\\subsection{Contribution}\nThis paper aims to find a way to describe the severity of the disaster and try to quantify the disaster automatically. Our contributions and novelties are as follows:\n\n(1) We use over 100 case studies from open sources, which includes before and after photos, disaster information (e.g. date, cost, fatality, etc.) and created a novel database which previously did not exist before for academia. \n\n(2) We compare two state-of-the-art approaches in deep learning, the readily available Residual Network (ResNet) and the more suitable custom configured Pyramid Scene Parsing Network (PSPNet). The former ResNet has a lower requirement for image quality, whereas the latter PSPNet is capable of scene parsing for livable space separation from natural environment. We compare the trade-off performance of image quality requirement vs. damage quantification accuracy.\n\n(3) We develop a multi-linear regression model to map the neural network outputs from (2) to the actual economic and human damage values. We keep (2) and (3) separate in order to enable the neural network models in (2) to be useful to a much wider set of humanitarian contexts beyond damage quantification. \n\n(4) Finally we train our mode on historical cases across a spectrum of global  disasters and then test our model on a more recent event.\n\nAs far as we are aware of, no such combined deep learning and disaster quantification system exists. As far as we are aware of, no such combined deep learning and disaster quantification system exists. The system fits into a wider need to automate humanitarian disaster response, allocate and prioritise global emergency resources to either preemptively address or respond to disasters \\cite{Guo19}.\n\n\n\\section{Data and Methodology}\n\n\\subsection{Database}\nWe source all our images and accompanying disaster data from open source websites. The images should have high quality (256\u00d7256 dpi) and 124 disasters are covered ranging from forest fire, earthquake, mudslide, tsunami, volcanic eruption, hurricane, typhoon, tornado and major industrial explosions, spanning several recent years around the world. Our major sources of data are: (1) EM-DAT international disaster database established by WHO and the Belgian Government \\cite{EM-DAT}, (2) CRED Centre for Research on the Epidemiology of Disasters, (3) Inria Aerial Image Dataset \\cite{Maggiori17}, and (4) xBD dataset \\cite{Gupta19}. We summarize our data in Table 1.\n\n\\begin{table}\n\\small\n\\centering\n\\caption{Data Parameters}\n\\label{Table1}\n\\begin{tabular}{cc}\n\\hline\nData or Parameter        & Value \\\\ \\hline\nCase Studies             & 124 \\\\\nDisaster Categories      & 9 \\\\\nTime Span                & 2004 to 2019 \\\\\nLocation                 & Global \\\\\nDeath Toll               & 2 to 165,000 \\\\\nEconomic Damage          & \\$0.12 to 360 billion \\\\\nLiving Space             & villages to major cities \\\\\nResolution               & 0.3m to 3m \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\small\n\\centering\n\\caption{Baseline ResNet-34 for Robust Disaster Detection: Architecture and Training\/Testing Parameters}\n\\label{Table2}\n\\begin{tabular}{ccc}\n\\hline\nLayer       & Output Size       & Details \\\\ \\hline\nConv 1      & 112$\\times$112    & 7$\\times$7, 64, stride 2 \\\\\nConv 2      & 56$\\times$56      & 3$\\times$3 max pool, stride 2 \\\\\n            &                   & $[3\\times3,64,3\\times3,64]\\times3$\\\\\nConv 3      & 28$\\times$28      & $[3\\times3,128,3\\times3,128]\\times4$\\\\ \nConv 4      & 14$\\times$14      & $[3\\times3,128,3\\times3,256]\\times6$\\\\ \nConv 5      & 7$\\times$7        & $[3\\times3,128,3\\times3,512]\\times3$\\\\  \nPool, FC  & 1$\\times$1        & Average, 1000-d FC, SoftMax \\\\ \\hline\nTraining    &                   & Value \\\\ \\hline\nBatch Size  &                   & 16 \\\\\nEpochs      &                   & 30 \\\\\nLearning Rate   &               & 0.00001 \\\\\nLoss Func.  &                   & Cross Entropy \\\\\nOptimizer   &                   & Adam \\\\\nFLOPs       &                   & $3.6\\times10^{9}$ \\\\\nTest Performance &              & 92\\% Accuracy, 0.05 Loss \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\small\n\\centering\n\\caption{PSPNet for Scene Parsing Disaster Scale Quantification through Build Environment Segmentation}\n\\label{Table3}\n\\begin{tabular}{cc}\n\\hline\nTraining    & Value \\\\ \\hline\nBackbone    & DenseNet \\cite{DenseNet} \\\\\nBatch Size  & 4 \\\\\nEpochs      & 100 \\\\\nLearning Rate   & 0.0001 \\\\\nLoss Func.  & Cross Entropy \\\\\nOptimizer   & Adam \\\\\nTest Performance & 84-88\\% Accuracy, 0.21 Loss \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{Fig2.pdf}\n\t\\caption{Baseline ResNet and Scene Parsing PSPNet: data, training, and performance. (a) ResNet produces a probability score for a disaster with high accuracy and is robust to noise, whereas (b) PSPNet can quantify the amount of built environment damage with a lower accuracy and is not robust to noise.}\n\t\\label{2}\n\\end{figure*}\n\n\\begin{table*}[t]\n\\centering\n\\caption{Case Study Results on Best and Worst Performance}\n\\label{Table4}\n\\begin{tabular}{ccccc}\n\\hline\nLocation    & Disaster          & Damage            & ResNet    & PSPNet \\\\ \\hline\nJapan       & Earthquake (2011) & \\$369bn 19k death & 100\\%     & 90\\% \\\\\nIndonesia   & Tsunami (2004)    & \\$94bn 16k death  & 99\\%     & 80\\% \\\\\nOklahoma    & Tornado (2013)    & \\$20bn 29 death   & 90\\%     & 65\\% \\\\ \nPhilippines & Typhoon (2013)    & \\$30bn 7k death   & 69\\%     & 71\\% \\\\  \nGuatemala   & Volcano (2018)    & \\$0.12bn 461 death& 69\\%     & 45\\% \\\\  \nMontecito   & Mudslide (2018)   & \\$2bn 21 death    & 63\\%     & 59\\% \\\\ \nBahamas     & Hurricane (2019)  & \\$4.7bn 370 death & 55\\%     & 40\\% \\\\  \nTexas       & Hurricane (2017)  & \\$125bn 88 death  & 52\\%     & 64\\% \\\\ \nMalibu      & Fire (2018)       & \\$6bn 2 death     & 51\\%     & 46\\% \\\\  \nFlorida     & Hurricane (2018)  & \\$25bn 74 death   & 51\\%     & 63\\% \\\\  \n\\end{tabular}\n\\end{table*}\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{Fig3.pdf}\n\t\\caption{Prediction of Severity and Damages using Unsupervised Clustering and a Multi-linear regression linking CNN outputs with Damage Data.}\n\t\\label{3}\n\\end{figure*}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{Fig4.pdf}\n\t\\caption{Case Study on Beirut Port Explosion 2020: (a) raw photos, and (b) PSPNet Output and Predicted Damage.}\n\t\\label{4}\n\\end{figure}\n\n\\subsection{ResNet for Baseline Classification}\nWe first consider the widely used ResNet model \\cite{ResNet}, which is designed to get an initial output for the severity of disasters according to the images for pre- and post-disaster. ResNet has the advantage of being relatively robust as poor image quality and different noises. Noises can often arise due to environmental conditions (e.g. smoke from conflict or fire) or compression of image due to communication bandwidth and edge processing limits. As a baseline, we first attempt a binary ResNet model with two classifications, disaster and non-disaster, and the output is the final classification with a probability based on training. A total of 271 images was used for training and 44 images used for validation. In the ResNet-34 model, 34 convolutional layers are used and the architecture used is given in Table 2. The baseline ResNet (see Figure 2a) is robust to noise (loss 0.05) and has a high accuracy of 92\\%, but it can only quantify the likelihood of a disaster present. This is sufficient for prioritising further investigation on a large scale, but is not effective in quantifying the magnitude of the damage as it cannot classify the built environment and assign a notion of contextualised value or worth.\n\n\\subsection{PSPNet for Scene Parsing}\nWhilst the baseline ResNet is robust to noise and has a high accuracy, it can only quantify the likelihood of a disaster present. PSPNet is therefore selected and configured, because it can segment all build environments and label them for quantification later in the paper \\cite{PSPNet}. The proportion of constructions area in the satellite images before and after disasters can be calculated. Through these two data, the change rate of constructions can be obtained. From the change rate of constructions, disaster situations can be reflected quantitatively. A total of 661 images was used for training and 59 images used for validation. In the PSPNet model, a backbone is based on DenseNet \\cite{DenseNet}. A feature map CNN (see Figure 1) of 3 layers pooled at 4 different sizes. After which they are convolved with $1\\times1$ filters to reduce the depth of the feature. Next, all the features are up-sampled and concatenated. The model of PSPNet also has been created and pre-trained by Pytorch and the process of training model is displayed in Figure 1 with architecture in Table 2.\n\nUsing the output segmentation (white colour indicates healthy built environment), we are able to identify the volume of change after a disaster. The scene parsing PSPNet (see Figure 2b) is not as robust to noise (loss 0.21) as ResNet, and has a lower accuracy of 84-88\\%, but it can quantify the damage to built environment from a disaster. This enables us to attempt to quantify the economic and human cost of at post disaster analysis or during an unfolding disaster. \n\n\\section{Results for Disaster Cost Quantification}\n\n\\subsection{Disaster Case Studies}\nThe outputs of ResNet-34 and PSPNet model are listed in Table 4 for a few selected case study in different areas of the world, under different type and scale disasters, with varying levels of damage. From the table, it can be seen that the difference between ResNet-34 and PSPNet has an average of 8\\%. However, we can see that both are quite accurate (80-100\\%) in predicting the disaster and level of damage for large disasters (e.g. Japanese 2011 earthquake or Indonesia 2004 tsunami), but has poor accuracy for small disasters (e.g. Malibu fire 2019 or Florida hurricane 2018). This is partly because of the lack of clear differentiation in damage in smaller disasters, as well as the lower volume of training data (e.g. training is biased towards larger data sets).\n\n\n\\subsection{Projecting the Economic and Life Loss}\nPreviously the CNNs were able to identify which features mapped to damages, but could not yet appropriate a economic and human life cost to the damages. The 2 primary damage labels for every event are economic loss ($\\$bn$) and the number of deaths. We wish to cluster them into a number of severity categories in order to reduce the resolution and dimensionality of the problem into a single \\textit{severity class} scale. \n\nWe select unsupervised $k$-means to cluster over 2000 cases of natural disasters from 2000 to 2019 using data from EM-DAT \\cite{EM-DAT}. From the Figure 3, it can be found that all data has been divided into three clusters, a different colour distinguishes every cluster of severity. This analysis is relatively reasonable because three cluster centres (three stars in Figure 3a-left) show the linear relationship, it is smooth and clear to judge red, blue and green area in the graph representing gentle, medium and severe disasters respectively. The number of cluster 1 cases is 779, which is 39\\% of all cases, and proportions of cluster 2 and cluster 3 are 32\\% and 29\\%. \n\nIn Figure 3a-right, we put every event into the corresponding cluster, and then plot outputs corresponding to every event into a graph as shown in Figure 3-2 with both ResNet-34 and PSPNet outputs, the linear relationships relate CNN outputs with the actual damage scale. The confidence interval of both models is set to 95\\% (see Figure 3b). It can be found that the R2 of ResNet-34 model is 0.48, which is much smaller than that of PSPNet at 0.76. Our multi-linear regression model relating PSPNet Loss output to predicted damages is:\n\\begin{equation}\n    \\text{Loss} = 0.1 \\times \\text{Economic Loss} + 0.038 \\times \\text{Deaths} + 0.12,\n\\end{equation} and we have checked that the residue is normal distributed.\n\n\\subsection{Case Study: Beirut Port Explosion 2020}\nIn 4th August of 2020, there was a powerful explosion in the port warehouse area of the Lebanese capital Beirut, causing widespread damage to the capital and destroying almost all buildings near the sea. At least 154 people have been killed, and nearly 5000 injured in the explosions (The Guardian, 2020). The main effect of the explosion radiated out from the point of explosion, so it is unnecessary to input wider satellite images of distant Beirut areas and only a single port area satellite image is used. First, we put these two images (Figure 4a) imported into the database with some necessary information such as location, date, and number of deaths. The economic loss is uncertain now, so the purpose is to find an approximate economic loss of this disaster. Second, using the pre-trained PSPNet model, we predict the damage in Figure 4b. The predicted loss in built livable space 72\\% under the severe category. Using Equation 1, the predicted economic loss is \\$15.6bn US dollars. This corresponds with early government estimates of \\$10-15bn. As such, despite large explosions were not part of the training data, the PSPNet model can be successfully used in disaster management to predict and assess disaster costs.\n\n\\section{Conclusions}\n\nHumanitarian disasters and political violence cause significant damage to our living space. The reparation cost to built livable space (e.g. homes, infrastructure, and the ecosystem) is often difficult to quantify in real-time. Real-time quantification is critical to both informing relief operations, but also planning ahead for rebuilding. Here, we used satellite images before and after major crisis around the world for the last 20 years to train a new Residual Network (ResNet) and Pyramid Scene Parsing Network (PSPNet) to quantify the magnitude of the damage. ResNet offers the robustness to poor image quality, whereas PSPNet offers scene parsing contextualised analysis of damage. \n\nBoth of these techniques are useful and can be cascaded: (Step 1) ResNet can identify priority areas on low resolution imagery over a large area with 90\\% accuracy, and this can be followed up with (Step 2) PSPNet to accurately identify the level of damage with 80\\% accuracy. As there are multiple damage dimensions to consider (e.g. economic loss, death-toll), we fitted a multi-linear regression model to quantify the overall damage cost to the economy and human lives. To validate our model, we successfully match our prediction to the ongoing recovery in the 2020 Beirut port explosion. These innovations provide a better quantification of overall disaster magnitude and inform intelligent humanitarian systems of unfolding disasters. \n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Derivation of the general form of $\\mathcal{F}_p[\\emph{n}]$}\n\\label{sec:GPC}\n\nIn a first step, we recall the one-body $N$-representability conditions. We remind the reader that we enumerated the $N$-particle configurations $\\bd{q}$ by $r=1,.\\ldots,R$.  Then, an $N$-particle state has the form $\\ket{\\Psi}=\\sum^{R}_{r'=1} \\alpha_{r'} \\ket{r'}$, where $\\ket{r'}$ represents a Slater determinant formed from one-particle states $\\ket{q}$, $q=1,\\ldots,d$.  $d$ is the dimension of the one-particle Hilbert space. Let us consider a single Slater determinant $\\ket{r}$, i.e., $\\alpha_{r'}=1$ for $r'=r$ and $0$ otherwise.  The corresponding  natural occupation numbers (NONs) are denoted by the vector ${\\bbf{v}^{(r)}}$. Its  $q$-th component is $v^{(r)}_q=1$\nif $\\ket{r}$ contains the one-particle state $\\ket{q}$, and otherwise zero.  ${\\bbf{v}^{(r)}}$, $r=1,\\cdots,R$ are the extremal points of the set $\\mathcal{P}^1_N$ of pure-state $N$-representable $\\bbf{n}=(n_q)$  and build the vertices of a polytope\n$P\\equiv \\mathcal{P}^1_N$ in the $d$-dimensional space of the NONs. Having determined the vertices one has to find the polytope's facets.\nThis in general is a nontrivial task.\nEach facet, $F_j$, $j=1,\\cdots,J$, of $P$  is part of a $(d-1)$-dimensional hyperplane defined by $D^{(j)}({\\bbf{n}})=0$ where $D^{(j)}({\\bbf{n}})=\\kappa^{(j)}_0 + \\sum\\limits_{k=1}^{d} \\kappa^{(j)}_k n_k$. The coefficients, $\\kappa^{(j)}_i$, $i=0,1,\\cdots,d$, are integers. The polytope is the intersection of the hyperplanes defined by $D^{(j)}({\\bbf{n}}) \\geq0$,  for all $j$. Therefore, necessary \\textit{and} sufficient conditions for the pure-state $N$-representability are the constraints $D^{(j)}({\\bbf{n}}) \\geq0$ , $j=1,\\cdots,J$.\n\nThe functions $D^{(j)}({\\bbf{n}})$ also allow us to decompose the vertices into two sets.\nFor given $j$ we decompose the index set $\\{1,\\cdots ,R\\}$ into a set $I_j=\\{r_1,\\cdots,r_j\\}$ and its complement  such that  ${\\bbf{v}}^{(r)} \\in F_j$ for $r \\in I_j$ and ${\\bbf{v}}^{(r)} \\notin F_j$\notherwise. It is $D^{(j)}({\\bbf{v}}^{(r)})=0$ for $r \\in I_j$  and $D^{(j)}({\\bbf{v}}^{(r)}) > 0$ for $r \\notin I_j$ \\cite{K09,SBV17}.\n\nWhat remains is  the derivation of the relation between $\\{D^{(j)}({\\bbf{n}})\\}$ and $\\{|\\alpha_r|^2\\}$.\nEach function $D^{(j)}({\\bbf{n}})$ determines an operator $D^{(j)}(\\hat{{\\bbf{n}}})$ with $\\hat{n}_q=c^\\dag_q c_q $. Since $\\langle \\Psi|\\hat{n}_q|\\Psi\\rangle=n_q$ we have $\\langle \\Psi|D^{(j)}(\\hat{{\\bbf{n}}}) |\\Psi\\rangle=D^{(j)}({\\bbf{n}})$. On the other hand we can substitute $|\\Psi\\rangle=\\sum^{R}_{r'=1} \\alpha_{r'}|r'\\rangle$ on its l.h.s.. With $({\\bf{A}})_{jr} \\equiv A_{jr} := D^{(j)}({\\bbf{v}}^{(r)})$\n this leads to\n\n\n\n\\begin{equation} \\label{eqA1}\nD^{(j)}({\\bbf{n}}) = \\sum^{R}_{r'=1} A_{jr'} |\\alpha_{r'}|^2 \\\\ \\ .\n\\end{equation}\nThis equation establishes a relation between the NONs and $\\{ |\\alpha_r|^2\\}$ involving the functions $\\{D^{(j)}({\\bbf{n}})\\}$ which define the domain of pure-state representability.\n\nFor the constrained minimization of $\\langle \\Psi|\\hat{V}|\\Psi\\rangle= \\sum_{r,r'} V_{rr'}\n\\alpha^{*}_r \\alpha_{r'}$ for fixed ${\\bbf{n}}$ we assume for a moment that\n $V_{rr'}=- \\tilde{\\eta}_r \\tilde{\\eta}_{r'}|V_{rr'}|$ for $r \\neq r'$, as well as,  real coefficients $\\alpha_r= \\eta_r|\\alpha_r|$) with $\\tilde{\\eta}_r =\\pm 1$ and  $\\eta_r =\\pm 1$. In that case the  minimization with respect to the phase factors  $\\{\\eta_r \\}$ is accomplished by the choice $\\tilde{\\eta}_r \\equiv \\eta_r $. Then the expectation value\n  $\\langle \\Psi|\\hat{V}|\\Psi\\rangle$ takes the form\n\n\n \\begin{equation} \\label{eqA1a}\n\\tilde{\\mathcal{F}} [\\{|\\alpha_r|\\}] = \\sum_{r} V_{rr}  |\\alpha_r|^2 - \\sum_{r \\neq r'}  |V_{rr'}|  |\\alpha_r| |\\alpha_{r'}|\\\\.\n\\end{equation}\n To derive $\\mathcal{F}[\\bd{n}]$ we have to determine $\\{|\\alpha_r|\\}$ as a function of $\\bbf{n}$.\n This can be done as follows. Introducing the symmetric and semi-definite  matrix ${\\bf{C}}={\\bf{A}}^t \\bf{A}$ and  operating with ${\\bf{A}}^t$ on Eq.~(\\ref{eqA1}) one obtains $\\sum^{J}_{j=1} ({\\bf{A}}^t)_{rj}D^{(j)}({\\bbf{n}})= \\sum^{R}_{r'=1} C_{rr'} |\\alpha_{r'}|^2$. If $d=R$, there are as many NONs as  coefficients $\\{|\\alpha_r|\\}$. In that case $\\{ |\\alpha_r|\\}$ is uniquely determined by $\\{D^{(j)}({\\bbf{n}})\\}$, i.e. by the NONs, ${\\bbf{n}}$. This always holds if the poytope $P$ is a simplex, which was the case for the example of three fully polarized electrons on a ring of six lattice sites. Note that there are $J$ functions\n $\\{D^{(j)}({\\bbf{n}})\\}$ and only\n$d \\leq J$ NONs.  In  case $d < R$ (occurs for  $L$ and $N$ large enough),  however,  $\\{|\\alpha_r|\\}$ are not uniquely determined by the NONs.\nIn that case $\\bf{C}$ has zero-eigenvalues, i.e., its rank, $d$, is smaller  than  $R$. Let $\\{{\\bf{w}}^{(l)}\\}$ be the eigenvectors of $\\bf{C}$ and $\\{c_l\\}$ its corresponding eigenvalues. $c_l > 0$ for $l=1,\\cdots,d$ and\n$c_l = 0$ for $l=d+1,\\cdots,R$. Substituting the expansion\n\n\\begin{equation}  \\label{eqA2b}\n( |\\alpha_1|^2,\\cdots, |\\alpha_{R}|^2)= \\sum^{R}_{l=1} a^{(l)} {\\bf{w}}^{(l)}\n\\end{equation}\ninto the equation $\\sum^{J}_{j=1} ({\\bf{A}}^t)_{rj}D^{(j)}({\\bbf{n}})= \\sum^{R}_{r'=1} C_{rr'} |\\alpha_{r'}|^2$ from above and taking the orthonormality of $\\{{\\bf{w}}^{(l)}\\}$ into account allows us to determine $a^{(l)}$ for $l=1,\\ldots,d$. This yields $a^{(l)}=\\sum^{d}_{l=1} \\big(c^{-1}_l {\\bf{w}}^{(l)t} {\\bf{A}}^t {\\bf{D}}({\\bbf{n}}) \\big) w_r^{(l)}$\nfor $l=1,\\ldots,d$. Substituting these $a^{(l)}$ into the r.h.s. of  Eq.~(\\ref{eqA2b}) we arrive at\n\n\n\n\\begin{eqnarray}  \\label{eqA2}\n|\\alpha_{r}|({\\bf{n}},{\\bf{a}})\n&=&\\Big[\\sum^{J}_{j=1}b^{(j)}_rD^{(j)}({\\bf{n}}) + \\sum^{R}_{l=d+1} a^{(l)} w_r^{(l))} \\Big]^{1\/2}\\!\\!\\!.\n\\end{eqnarray}\nThe coefficients $\\{b^{(j)}_r\\}$ follow from\n$\\sum^{d}_{l=1} \\big(c^{-1}_l {\\bf{w}}^{(l)t} {\\bf{A}}^t {\\bf{D}}({\\bbf{n}}) \\big) w_r^{(l))}=\\sum^{J}_{j=1} b^{(j)}_rD^{(j)}({\\bbf{n}})$ where  ${\\bf{D}}({\\bbf{n}})=(D^{(1)}({\\bbf{n}}),\\cdots,D^{(J)}({\\bbf{n}}))^t$.\nThe absolute values $\\{|\\alpha_r|\\}$ are fixed by the NONs through $\\{D^{(j)}({\\bbf{n}})\\}$ and by the independent real variables ${\\bf{a}}=(a^{(d+1)},\\ldots, a^{(R)})$.\n\n\nTo get $\\mathcal{F}_p$ the result Eq.~(\\ref{eqA2}) has to be substituted into  Eq.~(\\ref{eqA1a}) with a subsequent minimization with respect to ${\\bf{a}}$. This is\na nontrivial problem which in general can not be performed analytically. Substituting its solution ${\\bf{a}}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V})$ into Eq.~(\\ref{eqA2}) and this expression into Eq.~(\\ref{eqA1a}) yields the functional\n\n\\begin{widetext}\n\\begin{equation} \\label{eqA3}\n\\mathcal{F}_p[{\\bbf{n}}]= \\sum^{R}_{r,r'=1}V_{rr'} \\sqrt{\\sum^{J}_{j=1}\\Big[b^{(j)}_rD^{(j)}({\\bbf{n}})+\\overline{a}_r(\\{D^{(j)}({\\bbf{n}})\\}, \\hat{V})\\Big]} \\,\\,\n\\sqrt{\\sum^{J}_{j=1}\\Big[b^{(j)}_{r'}D^{(j)}({\\bbf{n}})+\\overline{a}_{r'}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V})\\Big]}  \\\n\\end{equation}\n\\end{widetext}\nwhere $V_{rr'}=-|V_{rr'}|$ for all $r \\neq r'$ and $\\overline{a}_r(\\{D^{(j)}(\\bbf{n})\\},\\hat{V})\n=\\sum^{R}_{l=d+1} \\, a^{(l)}(\\{D^{(j)}(\\bbf{n})\\},\\hat{V}) \\, w^{(l)}_r$. Note, the dependence on $\\hat{V}$\noccurs through the matrix elements $\\{V_{rr'}\\}$ of $\\hat{V}$.\n\\\\\n\n\n\n\n\nThe result (\\ref{eqA3}) simplifies for ${\\bbf{n}}$ close to a facet $F_j$. Remember that $D^{(j)}({\\bbf{n}}) \\to 0$ for ${\\bbf{n}} \\to F_j$. As described above we can decompose the set $r=1,\\ldots,R$ of the  vertex-indices  into two subsets, $I_j$ and its complement. Then it follows from Eq.~(\\ref{eqA1})\nthat $ |\\alpha_r |= \\sqrt{D^{(j)}({\\bbf{n}})} \\, {\\beta}_{r}$ for all $r \\notin I_j$ and\n $|\\alpha_r|=|\\alpha^{(j)}_r| + \\mathcal{O}(D^{(j)}({\\bbf{n}})) $ for all $r \\in I_j$.  The real and non-negative variables $\\bd{\\beta}=(\\beta_{r})$ have to fulfil\n $\\sum_{r' \\notin I_j} A_{jr'}({\\beta}_{r'})^2 =1$, which follows from Eq.~(\\ref{eqA1}).\n  $\\{\\alpha_r^{(j)}\\}$ are the coefficients of the normalized $N$ particle state $|\\Psi^{(j)}\\rangle= \\sum_{r' \\in I_j} \\alpha_{r'}^{(j)} |r'\\rangle$\n build from Slater determinants $|r\\rangle$ corresponding  to the vertices of the facet $F_j$, only. Substituting these quantities into Eq.~(\\ref{eqA1a}) yields\n\n\n\\begin{eqnarray} \\label{eqA4}\n\\tilde{\\mathcal{F}} [\\{|\\alpha_r|\\}] &=& \\tilde{\\mathcal{F}} [\\{|\\alpha_r^{(j)|}\\}] + \\nonumber\\\\\n&+& \\delta\\tilde{\\mathcal{F}} [\\{|\\alpha^{(j)}_r|\\},\\bd{\\beta}]\\sqrt{D^{(j)}({\\bbf{n}})} + \\nonumber\\\\\n&+& \\mathcal{O}(D^{(j)}({\\bbf{n}}))  \\ ,\n\\end{eqnarray}\nwith\n\\begin{equation} \\label{eqA5}\n\\tilde{\\mathcal{F}} [\\{|\\alpha^{(j)}_r|\\}] = \\sum_{r,r' \\in I_j} V_{rr}  |\\alpha^{(j)}_r|  |\\alpha^{(j)}_{r'}|\n\\end{equation}\nand\n\\begin{equation} \\label{eqA6}\n\\delta\\tilde{\\mathcal{F}} [\\{|\\alpha^{(j)}_r|\\},\\bd{\\beta}] = 2\\sum_{r \\in I_j ,r' \\notin I_j} V_{rr'}  |\\alpha^{(j)}_r|  \\, \\beta_{r'}\\\\.\n\\end{equation}\nAgain, it is $V_{rr'}=-|V_{rr'}|$  for all $r \\neq r'$.\n\n\n$\\tilde{\\mathcal{F}} [\\{|\\alpha^{(j)}_r|\\}]$ is like   $\\tilde{\\mathcal{F}} [\\{|\\alpha_r|\\}]$ but restricted to a subspace spanned by all basis  states $|r\\rangle$ with $r \\in I_j$.\nIts minimization with respect to $\\{|\\alpha^{(j)}_r|\\}$ has to  be performed in analogy to that of  $\\mathcal{F}_p [\\{|\\alpha_r|\\}]$, but now  under the constraint   ${\\bbf{n}^{(j)}}$ fixed.  $\\bbf{n}^{(j)}$  is a chosen reference point in\n$F_j$ which is the limiting point of $\\bd{n} \\to F_j$. This minimization process yields  $\\{|\\alpha^{(j)}_r|({\\bbf{n}^{(j)}})\\}$ and finally\n $\\mathcal{F}_p [\\{|\\alpha^{(j)}_r|({\\bbf{n}^{(j)}})\\}] = \\mathcal{F}^{(j)}_p [{\\bbf{n}^{(j)}}]$. Note, the dependence of\n $\\{|\\alpha^{(j)}_r|\\}$ on $\\{V_{r,r'}\\}$, $r,r' \\in I_j$ is suppressed.\n Furthermore, $ \\{|\\alpha^{(j)}_r|\\}$\n in the second line of Eq.~(\\ref{eqA4}) has to be replaced by $\\{|\\alpha^{(j)}_r|({\\bbf{n}^{(j)}})\\}$.\n It remains the minimization of  $\\delta\\tilde{\\mathcal{F}}$ with respect to $\\bd{\\beta}$ which yields\n$\\bd{\\beta}({\\bbf{n}^{(j)}})$ where the dependence on $\\{V_{rr'}\\}$ is suppressed, as well.\nThis completes the minimization of  $\\tilde{\\mathcal{F}} [\\{|\\alpha_r|\\}]$\nfor $\\bf{n}$ approaching  ${\\bf{n}}^{(j)}$ in  $F_j$. The  functional takes the final form\n\n\n\n\\begin{widetext}\n\\begin{equation} \\label{eqA7}\n\\mathcal{F}_p [{\\bbf{n}}] = \\mathcal{F}^{(j)}_p [{\\bbf{n}^{(j)}}]\n-2\\sum_{r \\in I_j ,r' \\notin I_j} |V_{rr'}| \\,|\\alpha^{(j)}_r|({\\bbf{n}^{(j)}}) \\, \\beta_{r'}({\\bbf{n}^{(j)}}) \\sqrt{D^{(j)}({\\bbf{n}})}\n+ \\mathcal{O}(D^{(j)}({\\bbf{n}}))\n\\end{equation}\n\\end{widetext}\n\n\n\nIn case that the interaction matrix does \\emph{not} have the form $V_{rr'} =- \\tilde{\\eta}_r \\tilde{\\eta}_{r'}|V_{rr'}|$ we have to minimize the functional\n\\begin{equation}\\label{eqA2a}\n\\tilde{\\mathcal{F}}[\\{|\\alpha_r|\\},\\bd{\\eta}] = \\sum_{r,r'}  V_{rr'} \\eta_r^\\ast \\,\\eta_{r'} |\\alpha_r| |\\alpha_{r'}|  \\ ,\n\\end{equation}\nwhere $\\bd{\\eta}=\\{\\eta_r\\}$ again are the phase factors of $\\{\\alpha_r\\}$.\nSimilar as above , for fixed  $(\\bd{n},\\bd{\\eta})$  we require additional parameters   ${\\bf{a}}=(a^{(l)})$ in order to fix $\\{|\\alpha_r|\\}$. Performing the minimization on the r.h.s. of Eq.~(\\ref{eqA2a}) with respect to $\\bf{a}$\nyields $a^{(l)}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}, \\bd{\\eta})$ and $\\{|\\alpha_r|\\}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}, \\bd{\\eta})$\nfollows from Eq.~(\\ref{eqA2}) by substituting  $a^{(l)}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}, \\bd{\\eta})$. Then, substitution of\n$\\{|\\alpha_r|\\}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}, \\bd{\\eta})$ into Eq.~(\\ref{eqA2a}) yields\n$\\tilde{\\mathcal{F}}[\\bd{n},\\bd{\\eta}]$. The final step concerns the minimization with respect to the Ising-like variables\n$\\bd{\\eta}$. Let $ \\overline{\\bd{\\eta}}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V})$ denote the minimizing phase factors. The substitution of those\ninto $\\tilde{\\mathcal{F}}[\\bd{n},\\bd{\\eta}]$ yields the final result for  $\\mathcal{F}_p$ given in Eq.~(9) of the main text with\n$\\overline{a}_r\\big(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}\\big)=\\sum^{R}_{l=d+1}\\, a^{(l)}\\big(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V}, \\overline{\\bd{\\eta}}(\\{D^{(j)}({\\bbf{n}})\\},\\hat{V})\\big) w^{(l)}_r$.\n\n\n\n\n\\subsection{Derivation of the exchange force for the case of $\\mathcal{P}_N^1$ being a simplex}\nIn the case where $\\mathcal{P}_N^1$ takes the form of a simplex, the derivation of the exchange force is apparently much easier (cf.~Eq.~(6)) than for the general case of an arbitrary polytope $\\mathcal{P}_N^1= \\mathcal{E}_N^1$. We prove in the following that this exchange force is repulsive in the sense that it repels $\\bbf{n}$ from the boundary of $\\mathcal{P}_N^1$. For this, we revisit Levy's construction where we use again the ansatz $\\ket{\\Psi}=\\sum_{r=1}^R \\eta_r |\\alpha_r|$ and assume that the interaction matrix elements $V_{r r'}$ and therefore also the phases factors $\\eta_r$ are real-valued, i.e., $\\eta_r =\\pm 1$. As in the main text, we label the one-body $N$-representability constraints $D^{(r)}(\\bbf{n})\\geq 0$ such that the respective facet does not contain the vertex $\\bd{v}_r$, i.e.~we have $D^{(r)}(\\bd{v}_{r'})=0$ whenever $r\\neq r'$. For simplicity, we ``normalize'' each $D^{(r)}\\geq 0$ such that $D^{(r)}(\\bd{v}_r) =1$. Moreover, we recall Eq.~(5), i.e.\n\\begin{equation}\\label{DvsAsimplex}\nD^{(r)}(\\bbf{n}) = |\\alpha_r|^2\\,.\n\\end{equation}\nLet us now consider $\\bbf{n}$ very close, in a distance $\\varepsilon$ to the facet described by $D^{(s)}\\equiv0$ and assume that the distances $D^{(r)}(\\bbf{n})$, to all other facets are much larger, i.e., $D^{(r)}(\\bbf{n})\\gg D^{(s)}(\\bbf{n})\\equiv \\varepsilon$ for all $r\\neq s$. W.l.o.g.~we assume $s=1$. Resorting to Levy's construction and the general ansatz for $\\ket{\\Psi}$, we find\n\\begin{widetext}\n\\begin{eqnarray}\n\\mathcal{F}_p[\\bbf{n}] &=& \\min_{\\{\\eta_r\\}} \\sum_{r,r'=1}^R \\eta_r \\eta_{r'} V_{r r'} \\sqrt{D^{(r)}(\\bbf{n})D^{(r')}(\\bbf{n})} \\nonumber \\\\\n&=& \\min_{\\{\\eta_r\\}_{r>1}} \\min_{\\eta_1}\\left[\\sum_{r,r'>1} \\eta_r \\eta_{r'} V_{r r'} \\sqrt{D^{(r)}(\\bbf{n})D^{(r')}(\\bbf{n})} +\n2 \\sum_{r>1} \\eta_r \\eta_1 V_{r 1} \\sqrt{D^{(r)}(\\bbf{n})D^{(1)}(\\bbf{n})} +\nV_{1 1} D^{(1)}(\\bbf{n})\n\\right]\\nonumber \\\\\n&=& \\min_{\\{\\eta_r\\}_{r>1}}\\left[\\sum_{r,r'>1} \\eta_r \\eta_{r'} V_{r r'} \\sqrt{D^{(r)}(\\bbf{n})D^{(r')}(\\bbf{n})} -\n2 \\sqrt{D^{(1)}(\\bbf{n})}\\,\\left|\\sum_{r>1} \\eta_r  V_{r 1} \\sqrt{D^{(r)}(\\bbf{n})} \\right| +\nV_{1 1} D^{(1)}(\\bbf{n})\n\\right]\\,.\n\\end{eqnarray}\n\\end{widetext}\nWe remind the reader that the one-body $N$-representability constraints read $D^{(r)}({\\bbf{n}})=\\kappa^{(r)}_0 + \\sum^{d}_{q=1}\\kappa^{(r)}_qn_q \\geq 0$.  The gradient $\\nabla_{{\\bd{n}}} \\mathcal{F}_p[\\bbf{n}]$  contains  products of\n  $\\partial \\mathcal{F}_p \/ \\partial D^{(r)}$ and  $\\nabla_{{\\bd{n}}}D^{(r)}({\\bbf{n}})$. The latter equals the vector\n  ${\\bd{\\kappa}^{(r)}}=(\\kappa^{(r)}_1,\\ldots,\\kappa^{(r)}_d)^t$, which is anti-parallel to the normal vector of the corresponding facet.\n Taking now the gradient of $\\mathcal{F}_p[\\bbf{n}]$, only the term in the middle yields a contribution which diverges in the limit $\\varepsilon \\rightarrow 0^+$ (since we assumed $D^{(r)}(\\bbf{n})\\gg D^{(1)}(\\bbf{n})\\equiv \\varepsilon$ for all $r>1$).\n $\\partial \\mathcal{F}_p \/ \\partial D^{(1)}$  is proportional to $1\/\\sqrt{D^{(1)}(\\bbf{n})}$ which is positive,\n and its prefactor $-\\,\\left|\\sum_{r>1} \\eta_r  V_{r 1} \\sqrt{D^{(r)}(\\bbf{n})} \\right|$ is apparently negative.\n Consequently, the  exchange force ${\\bd{f}}_{ex}(\\bd{n}) = - \\nabla_{{\\bd{n}}} \\mathcal{F}_p[\\bbf{n}]$ is parallel\n to ${\\bd{\\kappa}^{(1)}}$, i.e., it points towards the interior of the polytope. Hence, the exchange force is repulsive in the sense that it repels  $\\bbf{n}$ from the polytope's boundary.\n\n\n\n\n\n\n\n\\section{Proof of $ \\mathcal{F}_p =\\mathcal{F}_e$}\n\\label{sec:Proof}\nWe assume $\\alpha_r= \\eta_r |\\alpha_r |$ in  $|\\Psi\\rangle = \\sum^{R}_{r=1}\\alpha_r |r\\rangle $ to be real and that the  interaction matrix elements are of the form $V_{rr'} \\equiv \\langle r |\\hat{V}|r'\\rangle= - \\tilde{\\eta}_r \\tilde{\\eta}_{r'}| V_{rr'}|$  for all $r \\neq r'$.  $\\eta_r= \\pm 1$ and $\\tilde{\\eta}_r= \\pm 1$.  Then the minimization of the expectation value $\\langle \\Psi |\\hat{V}|\\Psi\\rangle$ with respect to the phase factors $\\{\\eta_r\\}$ is done for $\\eta_r \\equiv \\tilde{\\eta}_r$ leading to\n\\begin{eqnarray} \\label{eqB1}\n\\tilde{\\mathcal{F}}[\\{|\\alpha_r|\\}]&=& \\min_{\\{\\eta_r\\}} \\langle \\Psi |\\hat{V}|\\Psi\\rangle  \\nonumber\\\\\n&=&\\sum^{R}_{r=1} V_{rr}  |\\alpha_r|^2 - \\sum^{R}_{r \\neq r'=1}  |V_{rr'}|  |\\alpha_r| |\\alpha_{r'}|  \\,  .\n\\end{eqnarray}\n\nChoose an $N$-particle ensemble $ \\hat{\\Gamma} = \\sum^{R}_{r,r'=1} \\Gamma_{rr'} |r \\rangle \\langle r'|$ . Then it follows $\\langle \\hat{V} \\rangle_{\\hat{\\Gamma}}=Tr_N(\\hat{V}\\hat{\\Gamma})=\\sum^{R}_{r=1} V_{rr} \\Gamma_{rr} - \\sum^{R}_{r \\neq r'=1} \\tilde{\\eta}_r \\tilde{\\eta}_{r'} |V_{rr'}| \\Gamma_{rr'}$. A necessary condition for $\\hat{\\Gamma} \\geq 0$\nis $|\\Gamma_{rr'}|^2 \\leq \\Gamma_{rr}\\Gamma_{r'r'}$ for all $r \\neq r'$. The choice $\\Gamma_{rr'} =\\tilde{\\eta}_r \\tilde{\\eta}_{r'}\\sqrt{\\Gamma_{rr}\\Gamma_{r'r'}}$ minimizes  $\\langle \\hat{V} \\rangle_{\\hat{\\Gamma}}$ for \\textit{fixed} diagonal elements $\\{\\Gamma_{rr}\\}$ and leads to $\\min_{\\{\\Gamma_{r r'}\\}_{r \\neq r'}} \\langle \\hat{V} \\rangle_{\\hat{\\Gamma}} \\equiv \\tilde{\\mathcal{F}} [\\{\\sqrt{\\Gamma_{rr}}\\}]$.\nThis choice also implies  $\\hat{\\Gamma}= |\\Phi \\rangle  \\langle \\Phi|$ with $ |\\Phi \\rangle = \\sum^{R}_{r=1} \\tilde{\\eta}_r \\sqrt{\\Gamma_{rr}} \\, |r \\rangle$, i.e. the corresponding $N$-particle density operator, $\\hat{\\Gamma}$, is positive semi-definite. Final minimization of $\\tilde{\\mathcal{F}}\\{|\\alpha|_r\\}$ and  $\\tilde{\\mathcal{F}}\\{\\sqrt{\\Gamma_{rr}}\\}$ with respect to $\\{|\\alpha|_r\\}$ and\n$\\{\\sqrt{\\Gamma_{rr}}\\}$ under  constraints $\\bd{n}=\\{n_q\\}$ fixed, leads to $\\mathcal{F}_p=\\mathcal{F}_e \\equiv \\mathcal{F}$.\n\n\n\n\n\n\n\n\n\\section{Derivation of $\\mathcal{F}_p[\\emph{n}]$ for $N$=3 fully polarized electrons in one dimension and $L$=6}\n\\label{sec:Full Polarized}\n\nThe one-particle momenta $k=(2\\pi\/6) \\, \\nu$ from the first Brillouin zone are chosen as $\\nu=0,1,2,3,4,5$.  Taking only nearest neighbor hopping into account  this leads to the one-particle energies $\\varepsilon_\\nu=-2t \\cos (2 \\pi \\nu\/6)$.\n$t >0$ is the nearest neighbor hopping parameter. The ground state for noninteracting spinless electrons ( i.e., $\\hat{V} \\equiv 0$) is $\\ket{\\nu_1,\\nu_2,\\nu_3}^{(0)}=|0,1,5 \\rangle$ for which the total momentum is $K=(2\\pi\/6))(1+5)$(mod$6$)$=0$. If $|\\langle {\\bd{q}} |\\hat{V}| {\\bd{q}'}  \\rangle| $ for\nthe $3$-particle states $| {\\bd{q}} \\rangle=  c^{\\dagger}_{q_1\\uparrow}c^{\\dagger}_{q_2\\uparrow}c^{\\dagger}_{q_3\\uparrow} |0\\rangle$ is below a critical value for all $ {\\bd{q}} $, $ {\\bd{q}'} $ with $K=0$ the ground state of the interacting system will stay in this symmetry sector. It is easy to show that the zero-momentum space is spanned by four states $\\ket{\\nu_1,\\nu_2,\\nu_3}=|0,1,5 \\rangle,$ $|0,2,4 \\rangle$, $|1,2,3 \\rangle$ and $|3,4,5 \\rangle$, denoted by $|r\\rangle$, $r=1, \\cdots, 4$. Then a general three-particle state in this symmetry sector is represented as\n$|\\Psi\\rangle=\\sum\\limits_{r=1}^4 \\alpha_r |r\\rangle$.\n Note that the number, $d\\equiv L=6$, of one-particles states $|q \\rangle$  is larger than,  $R=4$, the dimension of the three-particle subspace. This implies besides the normalization $\\sum^{5}_{\\nu=0} n_{\\nu} =3$ additional identities for the NONs ($n_{\\nu}$)  independent on $\\{\\alpha_r\\}$. Since fully polarized electrons correspond to spinless fermions, the spin variables are suppressed.\n\n\n It is straightforward to determine\n $\\{n_{\\nu}\\}$ as a function of $\\{\\alpha_r\\}$. From this relation and the normalization condition for $\\{\\alpha_r\\}$ one obtains\n\n\n  \\begin{equation} \\label{eqD1}\nn_{3}=1-n_0, \\,\\, n_ {4}=1-n_1, \\,\\, n_5=1-n_2 \\quad.\n\\end{equation}\n\nAccordingly, there are three-independent NONs, only. We choose ${\\bbf{n}}=(n_0, n_1,n_2)$.\n\nNow one could follow the general scheme described in the main text to determine the vertices of the polytope and then the facets which yields the functions $\\{D^{(j)}({\\bbf{n}})\\}$. Since for the present case the set of linear equations\nrelating  $\\{n_{\\nu}\\}$ and  $\\{\\alpha_r\\}$ is rather simple one can solve this set directly. One obtains for $r=1,\\cdots,4$\n\\begin{equation} \\label{eqD2}\n |\\alpha_r| = \\sqrt{D^{(r)}({\\bbf{n}})\/2} \\,.\n\\end{equation}\nwith\n\\begin{eqnarray} \\label{eqD3}\n&& D^{(1)}({\\bbf{n}})= n_0+n_1-n_2  \\nonumber\\\\\n&& D^{(2)}({\\bbf{n}})= n_0-n_1+n_2 \\nonumber\\\\\n&&  D^{(3)}({\\bbf{n}})=2 - n_0-n_1- n_2 \\nonumber\\\\\n&& D^{(4)}({\\bbf{n}})=-n_0+n_1+n_2\n\\end{eqnarray}\nThe validity of $\\sum\\limits_{r=1}^4 |\\alpha_r|^2=1$ is obvious. Eq.~(\\ref{eqD3}) ist identical to Eq.~(3) of the main text.\n\n$ |\\alpha_r| \\geq 0$ and Eq.~(\\ref{eqD2}) yields  the generalized  constraints, $D^{(j)} ({\\bbf{n}}) \\geq 0$,\\, $j=1,\\ldots,4$  on ${\\bbf{n}}$. These guarantee that ${\\bbf{n}}$ is pure-state $N$-representable, in the\nsector $K=0$. They define four planes building a three-dimensional polytope (a tetrahedra, which is a simplex). This polytope is identical to that of the so-called Borland-Dennis setting for three spinless fermions in a six-dimensional one-fermion Hilbert space without any symmetry conditions \\cite{BD72,R07}.\nSubstituting $\\{ |\\alpha_r|\\}$ from Eq.~(\\ref{eqD2}) into\n$\\bra{\\Psi}\\hat{V}\\ket{\\Psi}=\\sum_{r,r'} V_{rr'}\\eta_r \\eta_{r'} \\,  |\\alpha_r| \\, |\\alpha_{r'}|$ and minimizing with respect to $\\{\\eta_r\\}$ one obtains the final result which is of the form of Eq.~(6) with $\\{D^{(j)}(\\bd{n})\\}$ from Eq.~(\\ref{eqD3}).\n\n\n\n\n\n\n\n\n\n\n\n\\section{Derivation of $ \\mathcal{F}[\\emph{n}]$ for the Hubbard-square, $N=4$, $L=4$, $K=2(2\\pi\/4)$, $S=0$  and parity $p=-1$}\n\\label{sec:Derivation}\n\n\nTo determine all Slater determinants $|k_1 m_1, k_2 m_2, k_3 m_3, k_4 m_4 \\rangle$ with total momentum\n$K=\\sum\\limits_{n=1} ^4 k_n ({\\rm mod}\\, 2\\pi) =  2\\pi\/4 \\,\\sum\\limits_{n=1} ^4 \\nu_n ({\\rm mod}\\, 4) =2 \\cdot  2\\pi\/4$ and total magnetization $M_z=\\sum\\limits_{n=1}^4 m_n=0$ ($m_n = \\pm 1\/2$) is straightforward. With $\\nu_n \\in \\{0,1,2,3\\}$ skipping $2\\pi\/4$ and use of $+1\/2=\\uparrow ,\\, -1\/2=\\downarrow$ one obtains ten states\n\n\\begin{eqnarray} \\label{eqC1}\n&& |0\\uparrow, 0\\downarrow, 3 \\uparrow, 3\\downarrow \\rangle, \\ |0\\uparrow, 0\\downarrow, 1 \\uparrow, 1\\downarrow \\rangle, \\nonumber\\\\\n&&  |2 \\uparrow, 2\\downarrow,3\\uparrow, 3\\downarrow \\rangle,  |1\\uparrow, 1\\downarrow, 2 \\uparrow, 2\\downarrow \\rangle,\\nonumber\\\\\n&&  |0\\downarrow, 1 \\uparrow, 2\\downarrow, 3\\uparrow  \\rangle,  | 0\\uparrow, 1 \\downarrow, 2\\uparrow, 3\\downarrow \\rangle,\\nonumber\\\\\n&&  0\\downarrow, 1 \\downarrow, 2\\uparrow, |3\\uparrow \\rangle,  |0\\uparrow, 1 \\uparrow, 2\\downarrow, 3\\downarrow \\rangle, \\nonumber\\\\\n&&   |0\\uparrow, 1 \\downarrow, 2\\downarrow, 3\\uparrow \\rangle,  0\\downarrow, 1 \\uparrow, 2\\uparrow, |3\\downarrow \\rangle \\ .\n\\end{eqnarray}\n\nDue to the isotropy in spin space and the reflection symmetry $P : i \\to L-i+1$ implying $P : \\nu \\to - \\nu (mod L)$ all basis states can be chosen to be eigenstates of the operator of the total spin squared, $\\vec{\\hat{S}}^2$, and the parity operator $\\hat{P}$ with eigenvalues $S(S+1)$ and $p= \\pm 1$, respectively. The ground state for zero interactions is two-fold degenerate. The degeneracy is lifted in first order in $U$. The corresponding groundstate for $U=0^{+}$ is given by $\\frac{1}{\\sqrt{2}}[ |0\\uparrow, 0\\downarrow, 3 \\uparrow, 3\\downarrow \\rangle -  |0\\uparrow, 0\\downarrow, 1 \\uparrow, 1\\downarrow \\rangle]$ which is an eigenstate of $\\vec{S}^2$ and  $\\hat{P}$ with eigenvalues $0$ and $p= - 1$, respectively.\n\n\n\nThen we get for  $S=0$ and $p=-1$ the following three basis states\n\n\\begin{eqnarray} \\label{eqC2}\n&& \\frac{1}{\\sqrt{2}}\\Big[  0\\uparrow, 0\\downarrow,|3 \\uparrow, 3\\downarrow \\rangle - |0\\uparrow, 0\\downarrow, 1 \\uparrow, 1\\downarrow \\rangle \\Big] \\ ,  \\nonumber\\\\\n&& \\frac{1}{\\sqrt{2}}\\Big[ |1 \\uparrow, 1 \\downarrow, 2 \\uparrow, 2 \\downarrow \\rangle + | 2\\uparrow, 2\\downarrow,3\\uparrow, 3\\downarrow \\rangle \\Big]  , \\nonumber\\\\\n&& \\frac{1}{4\\sqrt{3}} \\Big[-2 \\, \\big( |0\\downarrow, 1 \\uparrow, 2\\downarrow,3\\uparrow \\rangle +  | 0\\uparrow, 1 \\downarrow, 2\\uparrow,3\\downarrow \\rangle \\big) \\nonumber\\\\\n&& + \\big( 0\\downarrow, 1 \\downarrow, 2\\uparrow, |3\\uparrow \\rangle + |0\\uparrow, 1 \\uparrow, 2\\downarrow, 3\\downarrow \\rangle \\big) \\nonumber\\\\\n&& + \\big( | 0\\uparrow, 1 \\downarrow, 2\\downarrow, 3\\uparrow \\rangle +  |0\\downarrow, 1 \\uparrow, 2\\uparrow, 3\\downarrow,  \\rangle \\big)\\Big]  \\ ,\n\\end{eqnarray}\nwhich will be denoted by $|r \\rangle$ , $r= 1, \\cdots, 3$. With $| \\Psi\\rangle = \\sum\\limits^3_{r=1} \\alpha_r |r\\rangle$ it is straightforward  to express the NONs  by $\\{|\\alpha_r|^2 \\}$ :\n\n\\begin{eqnarray} \\label{eqC3}\n&& n_{0 \\uparrow} = |\\alpha_1|^2 + \\frac{1}{2} |\\alpha_3|^2 \\nonumber\\\\\n&& n_{1 \\uparrow} = \\frac{1}{2} [|\\alpha_1|^2 + |\\alpha_2|^2 +|\\alpha_3|^2] = \\frac{1}{2}\\nonumber\\\\\n&& n_{2 \\uparrow} =  |\\alpha_2|^2 +  \\frac{1}{2} |\\alpha_3|^2 \\nonumber\\\\\n&& n_{3 \\uparrow} =\\frac{1}{2} [|\\alpha_1|^2 + |\\alpha_2|^2 +|\\alpha_3|^2] = \\frac{1}{2} \\ ,\n\\end{eqnarray}\nwhere the normalization $\\sum\\limits^3_{r=1} |\\alpha_r|^2 =1$ was used. Due to $S=0$ it is\n$n_{\\mu \\uparrow} =n_{\\mu \\downarrow}$.  Furthermore Eq. (\\ref{eqC3}) implies\n$n_{0 \\uparrow}+n_{2 \\uparrow}=1$.  With  $n_{\\mu} \\equiv n_{\\mu \\uparrow} =n_{\\mu \\downarrow}$ it follows from Eq.~(\\ref{eqC3})\n\n\\begin{eqnarray} \\label{eqC4}\n&& n_{0} =|\\alpha_1|^2 + \\frac{1}{2} |\\alpha_3|^2 = 1 -n_2\\nonumber\\\\\n&& n_{2} =|\\alpha_2|^2 + \\frac{1}{2} |\\alpha_3|^2\\nonumber\\\\\n&& n_{1} = n_{3} = 1\/2 \\ .\n\\end{eqnarray}\n\nAccordingly there is one independent NON, only. We choose  $n_2$ and  identify ${\\bbf{n}}$ with $n_2$ , being restricted to $0 \\leq n_2 \\leq 1$.\nTherefore the ``facets'' are defined be $D^{(1)}({\\bbf{n}})=1-n_2=0$ and $D^{(2)}({\\bbf{n}})=n_2=0$\n\nThe matrix $(A_{jr}) \\equiv (D^{(j)}({\\bbf{v}}^{(r)}))$ in Eq.~(\\ref{eqA1})  becomes\n\n\n\\begin{eqnarray} \\label{eqC5}\n(A_{jr})&=&  \\frac{1}{2}\\left(\n\\begin{array}{ccc}\n2 & 0 &1\\\\\n0 & 2&1 \\\\\n \\end{array}\n\\right)  \\  ,\n\\end{eqnarray}\nwhich leads to (see part I)\n\n\n\\begin{eqnarray} \\label{eqC6}\n(C_{rr'})&=&  \\frac{1}{4} \\left(\n\\begin{array}{ccc}\n4 & 0& 2 \\\\\n0 & 4& 2 \\\\\n2 & 2& 2\n \\end{array}\n\\right)  \\  .\n\\end{eqnarray}\nThe eigenvalues $\\{c_l\\}$and the corresponding orthonormalized eigenvectors $\\{(w^{(l)}_r)\\}$ are $c_1=1 \\ , c_2=3\/2 \\ ,c_3=0 $ and $(w^{(1)}_r)=(1\/\\sqrt{2})(1,-1,0)^t  \\ , (w^{(2)}_r)=(1\/\\sqrt{3})(1,1,1)^t  \\ , (w^{(3)}_r)=(1\/\\sqrt{6})(1,1,-2)^t$, respectively. Substituting these expressions into Eq.~(\\ref{eqA2b}) yields\n\n\n\\begin{eqnarray} \\label{eqC7}\n\\begin{pmatrix}\n|\\alpha_1|^2\\\\\n|\\alpha_2|^2\\\\\n|\\alpha_3|^2\\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\nD^{(1)}({\\bbf{n}}) +(a^{(3)}-\\frac{1}{\\sqrt{6}})w^{(3)}_1\\\\\nD^{(2)}({\\bbf{n}}) +(a^{(3)}-\\frac{1}{\\sqrt{6}})w^{(3)}_2\\\\\n(a^{(3)}-\\frac{1}{\\sqrt{6}})w^{(3)}_3\n\\end{pmatrix}  \\ .\n\\end{eqnarray}\n\\\\\nIt is straightforward to calculate the matrix elements  $V_{rr'}=\\langle r|\\hat{V} |r'  \\rangle $ of the Hubbard interaction $\\hat{V}=U \\sum^4_{i=1} \\hat{n}_{i\\uparrow}\\hat{n}_{i\\downarrow}$. As a result one obtains\n\n\n\\begin{eqnarray} \\label{eqC8}\n(V_{rr'})&=&(U\/4) \\left(\n\\begin{array}{ccc}\n3 &-1&  -\\sqrt{6}\\\\\n-1 & 3& -\\sqrt{6}\\\\\n  -\\sqrt{6}& -\\sqrt{6}&6\\\\\n\\end{array}\n\\right)  \\ .\n\\end{eqnarray}\nNote, For $U >0$ all \\textit{nondiagonal} elements are negative. Therefore, the Hubbard interaction belongs to the class of pair interactions for which Eqs.~(\\ref{eqA3}) and (\\ref{eqA7})  hold. Therefore it is $\\mathcal{F}_p[\\bbf{n}]=\\mathcal{F}_e[\\bbf{n}] \\equiv \\mathcal{F}[\\bbf{n}]$\n\n To get the functional $\\mathcal{F}[\\bbf{n}]$  we have to minimize $\\sum^3_{r,r'=1} V_{rr'}\\alpha_r\\alpha_{r'}$\nwith respect to the single independent degree of freedom, $a^{(3)}$, and have to follow the scheme described in part I . This leads to $\\mathcal{F}[\\bbf{n}]$ of the form of Eq.~(9). Here we illustrate another route where first   $(V_{rr'})$ is diagonalized. The eigenvalues of $(V_{rr'})$ are $U(1,2,0)$ with corresponding orthonormalized eigenvectors $\\vec{v}_1=(1\/\\sqrt{2})(1,-1,0)^t$, $\\vec{v}_2=1\/(2\\sqrt{2})(1,1,-\\sqrt{6})^t$ and $\\vec{v}_3=(\\sqrt{3\/8})(1,1,2\/\\sqrt{6})^t$. That one of its eigenvalues vanishes, will be crucial when we will discuss the strong coupling limit $U \\to \\infty$ below.\n\nUsing the  eigenvectors $\\{\\vec{v}_r\\}$ it follows\n$\\vec{\\alpha}=(\\alpha_1,\\alpha_2,\\alpha_3)^t= \\sum^3_{r=1} \\bar{\\alpha}_r  \\vec{v}_r$\n with $\\sum^3_{r=1} \\bar{\\alpha}_r^2=1$, where  $\\bar{a}_r=(\\vec{\\alpha} \\cdot \\vec{v}_r)$ are real. Then it is\n\n\\begin{equation} \\label{eqC9}\n\\sum^3_{r,r'=1} V_{rr'}\\alpha_r\\alpha_{r'}=U(\\bar{\\alpha}_1^2 + 2\\bar{\\alpha}_2^2) \\ .\n\\end{equation}\nThe  second line  of Eq.~(\\ref{eqC4}) (the constraint for the independent NON, $n_2$) becomes\n\n\\begin{equation} \\label{eqC10}\n1 - \\bar{a}_1[\\bar{\\alpha}_2 + \\sqrt{3}\\sqrt{1-(\\bar{\\alpha}_1^2+\\bar{\\alpha}_2^2)}]=2n_2  \\\n\\end{equation}\nfrom which it follows\n\n\\begin{equation} \\label{eqC11}\n\\bar{\\alpha}^{(\\pm )}_2(n_2;\\bar{\\alpha}_1)=\\Big[(\\frac{1}{2}-n_2) \\pm \\sqrt{3}\\sqrt{\\bar{\\alpha}_1^2(1-\\bar{\\alpha}_1^2) - (\\frac{1}{2}-n_2)^2}\\Big]\/(2\\bar{\\alpha}_1) \\ .\n\\end{equation}\n\n\n$\\bar{\\alpha}^{(\\pm )}_2(n_2;\\bar{\\alpha}_1)$ put into the r.h.s. of  Eq.~(\\ref{eqC9}) yields the functional\n\n\\begin{equation} \\label{eqC12}\n\\tilde{\\mathcal{F}}[{\\bbf{n}};\\bar{a}_1]=U[\\bar{\\alpha}_1^2 + 2\\bar{\\alpha}^{(\\pm )}_2(n_2;\\bar{\\alpha}_1)^2]  \\ ,\n\\end{equation}\n which has to be minimized with respect to $\\bar{a}_1$.\nNote that $\\bar{\\alpha}^{(+)}_2(n_2;\\bar{\\alpha}_1)  \\equiv - \\bar{\\alpha}^{(-)}_2(1-n_2;\\bar{\\alpha}_1)$.\n Since $\\tilde{\\mathcal{F}}[{\\bbf{n}};\\bar{\\alpha}_1]$ involves $\\big(\\bar{\\alpha}^{(\\pm )}_2(n_2;\\bar{\\alpha}_1\\big)^2$ the minimization with\n respect to $\\bar{\\alpha}_1$ yields a functional $\\mathcal{F}[n_2]$ exhibiting the particle-hole symmetry\n $\\mathcal{F}[n_2]=\\mathcal{F}[1-n_2]$ [55].\nThe resulting equation from that minimization is a polynomial in $\\bar{\\alpha}_1^2$ of degree \\textit{six}. Its roots can not be calculated analytically.   Therefore, we use this situation  to demonstrate the power of a perturbative approach leading in the  weak and strong coupling limit $0 \\leq U \\ll 1$ and $U \\gg 1$, respectively, to asymtotically exact results, obtained analytically.\n\n\n \\subsection{weak coupling limit}\n\n The ground state for $U \\to 0^{+}$ is twofold degenerate (both states in the first line of Eq.~(\\ref{eqC1})).\n In first order in $U$ the degeneracy is lifted and the ground state is given by the state in the first line of Eq.~(\\ref{eqC2}). Consequently the coefficients $\\alpha_r$ in $|\\Psi\\rangle$ must fulfil $\\alpha_1 \\to 1$\n and $\\alpha_r \\to 0$ for $r=2,3$ which implies $\\bar{\\alpha}_1 = 1\/\\sqrt{2} + x_1$and  $\\bar{\\alpha}_2 = 1\/(2\\sqrt{2}) + x_2$ with\n $|x_r| \\to 0 $.  Taking this small-U dependence into account it follows from Eq.~(\\ref{eqC4}) that $n_2 \\to 0$,\n i.e., $n_2$ is the \\textit{smallness parameter} for the pertubative calculation of $\\mathcal{F}$ for weak coupling.  This is intuitively clear, because it is the occupation number of the highest one-particle level $|\\nu=2\\rangle$. Eq.~(\\ref{eqC12}) becomes\n\n\n\\begin{equation} \\label{eqC13}\n\\tilde{\\mathcal{F}}[{\\bbf{n}};\\bar{\\alpha}_1]=3U\/4 + U [\\sqrt{2}(x_1+x_2(n_2;x_1)) + h.o.t.] \\ ,\n\\end{equation}\n\nwhere $h.o.t.$ stands for higher order terms.\nExpanding the r.h.s. of Eq.~(\\ref{eqC11}) with respect to $x_1$ leads to\n\n\n\\begin{eqnarray} \\label{eqC14}\n(x_1+x_2(n_2;x_1))&=&[x_1 - \\sqrt{6}\\sqrt{n_2 -2x_1^2} \\nonumber\\\\\n&+& h.o.t.]\/(2(1+\\sqrt{2}x_1)) \\ ,\n\\end{eqnarray}\nwhere the minus-sign in Eq.~(\\ref{eqC11}) has to be used. The plus-sign has to be chosen for $n_2 \\to 1$.\nPutting this result into Eq.~(\\ref{eqC13}) and minimizing with respect to $x_1$ one obtains\n\n\n\\begin{equation} \\label{eqC15}\nx^{(min)}_1 = -\\sqrt{n_2}\/\\sqrt{26} +  \\mathcal{O}(n_2)  \\ .\n\\end{equation}\nThe fact that the leading order of $x^{(min)}_1$  is proportional to $\\sqrt{n_2}$ justifies a postiori that we have not taken into account the higher order terms in Eq.~(\\ref{eqC13}).\nFinally substituting $x^{(min)}_1$ into the r.h.s. of Eq.~(\\ref{eqC14}) we get from Eq.~(\\ref{eqC13})  the functional\nin the weak coupling limit\n\n\\begin{equation} \\label{eqC16}\n\\mathcal{F}[{\\bbf{n}}] = U[3\/4 - \\sqrt{13\/4} \\sqrt{n_2} + \\mathcal{O}(n_2)] \\ .\n\\end{equation}\n\n\n\\subsection{strong coupling limit}\n\nIt is known that the ground state energy of the Hubbard model at half filling converges to zero for $U \\to \\infty$ \\cite{F91}. Therefore, it follows from Eq.~(\\ref{eqC9}) that  in the strong coupling limit it must be $\\bar{a}_1 \\to 0$ and  $\\bar{a}_2 \\to 0$, i.e. only the eigenvector $\\vec{v}_3$ of $(V_{r'r})$ with eigenvalue $0$ contributes on the l.h.s. of Eq.~(\\ref{eqC9}) . From Eq.~(\\ref{eqC10})  we obtain $n_2 \\to (1\/2)^{-}$, i.e., $\\delta=(\\frac{1}{2}-n_2) \\geq 0$ is the \\textit{smallness parameter} for the perturbative construction of the functional  $\\mathcal{F}$ in the strong coupling limit. Since $\\bar{\\alpha}_1 \\to 0$,  Eq.~(\\ref{eqC11}) implies first that we have to choose the minus-sign, and second the square root must converge to $\\delta$, in order that $\\bar{\\alpha}_2 \\to 0$. The latter condition becomes satisfied if\n\n\n\\begin{equation} \\label{eqC17}\n\\bar{\\alpha}_1=2 \\delta(1+y_1)\/\\sqrt{3}  \\ .\n\\end{equation}\nSubstituting this expression into  Eq.~(\\ref{eqC11}) (with the minus-sign) and expanding its r.h.s. with respect to $\\delta$ and $y_1$\nleads to\n\n\n\\begin{equation} \\label{eqC18}\n\\bar{\\alpha}_2(y_1;\\delta)=\\sqrt{3}\\big(\\frac{2}{3}\\delta^2 - y_1 + h.o.t. \\big) \\ .\n\\end{equation}\nNext, $\\bar{\\alpha}_1$ and $\\bar{\\alpha}_2$ from Eqs.~(\\ref{eqC17}) and (\\ref{eqC18}) are put into Eq.~(\\ref{eqC12}) which yields\n\n\n\\begin{equation} \\label{eqC19}\n\\tilde{\\mathcal{F}}[{\\bbf{n}},\\bar{\\alpha}_1] \\equiv \\tilde{\\mathcal{F}}[\\delta,y_1]=U\\Big[\\frac{4}{3}\\delta^2(1+y_1)^2 + \\frac{1}{3}(2\\delta^2 - 3y_1)^2 + h.o.t.\\Big]  \\ .\n\\end{equation}\nIts minimum is taken at\n\n\n\\begin{equation} \\label{eqC20}\ny_1(\\delta)= \\frac{4}{9}\\big(\\delta^2 - \\frac{2}{9}\\delta^4 + h.o.t.\\big)                  \\ .\n\\end{equation}\nIn a final step $y_1(\\delta)$ is substituted into the r.h.s. of Eq.~(\\ref{eqC19}) leading to the \\textit{exact} functional\n\n\n\\begin{equation} \\label{eqC21}\n\\mathcal{F}[{\\bbf{n}}]=U\\Big[\\frac{4}{3}(\\frac{1}{2}-n_2)^2  + \\frac{40}{27}(\\frac{1}{2}-n_2)^4 + \\mathcal{O}((\\frac{1}{2}-n_2)^6)\\Big]  \\ .\n\\end{equation}\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent The Grothendieck ring $K_0(\\Var_S)$ of varieties over a separated scheme $S$\nis as group spanned by isomorphism classes $[X]$ of separated schemes $X$ of finite type over $S$\nwith relations allowing to cut and paste.\nThe ring structure is given by the fiber product.\nThis ring is useful because additive invariants of varieties,\nfor example the Euler characteristic \nand the number of points over a finite field in positive characteristic,\nfactor through this ring.\nTherefore the Grothendieck ring of varieties and\nlocalizations of it are used in motivic integration\nas universal value rings.\n\nLet $S$ now be a scheme with a good action of a finite group $G$,\nand consider the category $(\\Sch_{S,G})$ of separated $S$-schemes with good $G$-action.\nA group action on $S$ is called \\emph{good} if every orbit lies in an affine subscheme of $S$,\nwhich insures that the quotient exists in the category of schemes,  see Section~\\ref{assumtions}.\nThe equivariant Grothendieck ring $K_0^G(\\Var_S)$ of varieties over $S$, see Definition \\ref{equivariant Gring},\nis generated by isomorphism classes $[X]$ of objects $X$ in this category.\nWhenever $Y$ is a $G$-invariant closed subscheme of $X$,\none asks the class of $X$ to be equal to the sum of the class of $Y$\nand the class of $X\\setminus Y$.\nMoreover one asks the classes of two affine bundles with affine $G$-action to be equal if they have the same rank and the same base.\nThe ring structure is again given by the fiber product.\n\nUsing the\nequivariant Grothendieck ring of varieties \nas value ring allows us to also encode some group action on a scheme $X$,\nas done for example with the monodromy action on the motivic Zeta function, see \\cite{DL3}.\nTo get a well defined theory of motivic integration with group actions,\none needs to make actions on affine bundles 'trivial'.\nThis is where the last relation in the definition of the equivariant Grothendieck ring of varieties actually comes from.\n\nOne can ask now how to relate the equivariant Grothendieck ring with the usual one.\nA natural thing to do is to divide \nout the action,\ni.e., to send the class $[X]\\in K_0^G(\\Var_S)$ of a scheme $X$ with good $G$-action\nto the class of its quotient $X\/G$ in $K_0(\\Var_{S\/G})$.\nBittner showed that such a quotient map\nis well defined if $S$ is a variety over a field of characteristic zero and the action of $G$ on $S$ is free,\nsee \\cite[Lemma~3.2]{MR2106970}.\nThe proof uses\nthat in this case also the action on an affine bundle in the category $(\\Sch_{S,G})$ is free,\nand thus the quotient \nis again an affine bundle.\nFor general $G$-action on $S$ this is not the case.\n\n\\medskip\n\nIn this paper,\nwe show that for an abelian group $G$, the quotient map on the $G$-equivariant Grothendieck ring is well defined in general\nif we put some extra assumptions on the stabilizers of the points of $S$.\n For $s\\in S\/G$, denote by $F_s$ the residue field of $s$,\n and by $G_s\\subset G$ the stabilizer of a point $s'\\in S$ in the inverse image of $s$ under the quotient map $S\\to S\/G$.\n With this notation, we show the following theorem:\n\\begin{thm*}[Theorem \\ref{quotientmap}]\n Let $G$ be a finite abelian group.\n Assume for all $s\\in S\/G$ that\n $F_s$  contains all $\\lvert G_s \\rvert$-th roots of unity.\n Then there is a well defined group homomorphism\n  \\[\n K^G_0(\\Var_S)\\to K_0^*(\\Var_{S\/G})\n \\]\n sending $[X]\\in K_0^G(\\Var_S)$ to $[X\/G]\\in K_0(\\Var_{S\/G})$ for every $X\\in (\\Sch_{S,G})$.\n\\end{thm*}\n\\noindent\nHere $K_0^*(\\Var_{S\/G})=K_0(\\Var_{S\/G})$, if the $G$-actions on $S$ is \\emph{tame}, \ni.e., if the characteristic of the residue field of every point in $S$ is prime to the order of its stabilizer, see Section \\ref{assumtions}.\nIf the action of $G$ on $S$ is \\emph{wild}, i.e., not tame,\n$K_0^{*}(\\Var_{S\/G})$ is equal to the modified Grothendieck ring $K_0^{\\text{mod}}(\\Var_{S\/G})$,\nin which classes of varieties connected by universal homeomorphisms are equal, see Definition \\ref{dfn modified}.\nThis is due to the fact that\nif $G$ acts wildly on a scheme $X$\nthe quotient of a closed invariant subscheme of $X$ only has a universal \nhomeomorphism onto its image in the quotient $X\/G$,\nwhich is in general not a piecewise\nisomorphism on the underlying reduced schemes, see Example \\ref{exinsep3}.\nIt is not known whether two such schemes have the same class in the usual Grothendieck ring of varieties.\n\n\n\\noindent\nIn order to prove that the quotient map is well defined,\nwe need to control in particular quotients of affine bundles by affine actions.\nTo do so, we show that the class of the quotient $V\/G$ of an affine bundle $\\varphi:V\\to B$\nin $K_0^*(\\Var_{S\/G})$\nonly depends on the rank $d$ of the bundle and its base $B$, see Lemma~\\ref{lemma}.\nWe prove the lemma by showing that all fibers of the induced map $\\varphi_G:V\/G\\to B\/G$\nhave the class of an affine space of dimension $d$\nin the Grothendieck ring of varieties,\nand conclude then by spreading out.\nTo compute the fibers of $\\varphi_G$\nwe use the following proposition:\n\\begin{prop*}[Proposition \\ref{lemwildandtame}]\n  Let $G$ be a finite abelian group with quotient $G\\to \\Gamma$.\n  Let $k$ be a field of characteristic $p>0$,\n  let $q$ be the greatest divisor of $\\lvert G\\rvert$ prime to $p$, \n  and let $K\/k$ be a Galois extension with Galois group $\\Gamma$.\n  Assume that the Galois action on $K$ lifts to a $k$-linear action of $G$\n  on a finite dimensional $K$-vector space $V$.\n  If $k$ contains all $q$-th roots of unity, then\n  \\[\n   [V\/G]=\\mathbb{L}_k^{\\Dim_KV}\\in K_0^{*}(Var_k)\n  \\]\n  with $\\mathbb{L}_k:=[\\mathbb{A}_k^1]\\in K_0^{*}(Var_k)$.\n\\end{prop*}\n\\noindent\nThis proposition was already shown in the tame case in\n\\cite[Lemma~1.1]{MR2642161}\nby decomposing $V$ into eigenspaces.\nThis method does not work in the case of wild actions.\nInstead we construct a $G$-equivariant map from $V$ to a vector space $W$ of dimension one over $K$,\nuse an induction argument to compute the fibers of the induce map between the quotients,\nand use again spreading out to conclude.\n\n\n\n\n\\medskip\n\nAs an application of our main theorem, \nwe get that \nthe quotient of the motivic nearby fiber is a well defined invariant\nwith values in $\\mathcal{M}_k$, the localization of $K_0(\\Var_k)$ with respect to $\\mathbb{L}:=[\\mathbb{A}_k^1]$,\nwith $k$ a field of characteristic zero containing all roots of unity.\nThe motivic nearby fiber, see Definition \\ref{dfn mnf},\nis an invariant of  a non-constant \nmorphism $f:X\\to \\mathbb{A}_k^1$,\nwith $X$ an irreducible smooth $k$-variety,\nand was constructed in \\cite{DL3}\nas a limit of the motivic Zeta function.\n\nWe show moreover that modulo $\\mathbb{L}$, the quotient of the motivic nearby fiber is equal to the motivic reduction $R(f)$ of $f$\nin the image of $K_{0}(\\Var_k)$ in $\\mathcal{M}_{k}$,\nsee Proposition~\\ref{application}.\nThe \\emph{motivic reduction} of $f$, see Definition \\ref{dfn mr}, \nis defined as the class of $h^ {-1}(X_0)$ in $K_0(\\Var_{k})$ modulo $\\mathbb{L}$,\nwhere $h:Y\\to X$ is any smooth modification of $f$, i.e., $Y$ is a smooth $k$-variety and $h$ is a proper morphism\ninducing an isomorphism $Y\\setminus h^ {-1}(X_0)\\to X\\setminus X_0$.\nThe definition of $R(f)$ does not depend on the choice of such an $h$ due to weak factorization, see Proposition \\ref{welldef R(f)}.\n\nFrom this result we deduce that,\nif $X$ is a smooth variety with a proper, non-constant morphism $f:X\\to \\mathbb{A}_k^1$,\nand the generic fiber $X_\\eta:=X\\times_{\\ensuremath{\\mathbb{A}}^1_k}\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}=X\\setminus X_0$ of $f$ is equal to $1$ modulo $\\mathbb{L}$\nin $K_0(\\Var_{\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}})$,\nthen the same holds for the special fiber $X_0$ of $f$ in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$.\nThis can be seen as a motivic analog of the main theorem in \\cite[Theorem~1.1]{MR2247971},\nwhich says the following:\nif $V$ is an absolutely irreducible smooth projective variety\nover a local field $K$ with finite residue field $F$\nwhich has a certain cohomological property,\nnamely that the \\'etale cohomology of $V\\times_K \\bar{K}$ has coniveau~$1$,\nthen the amount of points of the special fiber of every projective regular model of $V$ is equal to $1$ modulo $\\lvert F \\rvert$.\nWe will explain this analogy in more details in Section~\\ref{an application}.\n\n\n\n\n\\section{Preliminaries}\n\\label{assumtions}\n\n\\noindent\nFix a finite group $G$. \nLet $S$ be a separated scheme endowed with\na left action of $G$. \nIf not mentioned otherwise, all group actions will be left actions.\nWe say the action of $G$ on $S$ is \\emph{good} if every\norbit of this action is contained in an affine open subscheme of $S$.\nBy requiring the action to be good, one makes sure that the quotient exists in the category of schemes, see \\cite[Expos\\'e V.1]{MR0238860}.\nWe call an action \\emph{tame} if the characteristic\nof the residue field of every point $s\\in S$\nis zero or positive and prime to the order of the stabilizer $G_s\\subset G$ of $s$. \nWe call an action \\emph{wild} if it is not tame.\n\nIf not mentioned otherwise,\nwe assume for the rest of the text that \n$S$ is a separated scheme with a good $G$-action,\nthat the quotient $S\/G$ is locally Noetherian and separated, and that the quotient map $S\\to S\/G$ is finite.\nThis is for example true if $S$ is a separated scheme of finite type over a field $k$,\nand $G$ acts on $S$ by a group of $k$-morphisms, see \\cite[Expos\\'e V.1, Corollaire 1.5]{MR0238860}.\n\nWe denote by $(\\Sch_{S,G})$ the category whose objects are separated\nschemes of finite type over $S$ with a good $G$-action such that the structure map is $G$-equivariant, and whose morphisms are\n$G$-equivariant morphisms of $S$-schemes.\nNote that if $G$ acts tamely on $S$, the same is true for every $X\\in (\\Sch_{S,G})$,\nbecause the stabilizer of a point $x\\in X$ is \na subset of the stabilizer of the the image of $x$ in $S$.\n\nOne can check that the fiber product exists in this category:\ntake any $X,Y$ in $(\\Sch_{S,G})$. \nFor $g\\in G$ let $g_X\\in \\Aut(X)$ and $g_Y\\in \\Aut(Y)$ be the corresponding automorphisms.\nThen $g_X \\otimes g_Y$ is an automorphism of $X\\times_SY$.\nDoing the same for every $g\\in G$ we get an action of $G$ on $X\\times_SY$ with $G$-equivariant projection maps.\nThis action is good, because the fiber product is constructed using affine covers.\nIt is easy to see that $X\\times_SY$ together with the projection maps to $X$ and $Y$ is in fact the categorical fiber product in $(\\Sch_{S,G})$.\n\n\n\n\n\n\n\\section{Equivariant affine bundles}\n\\label{equvariant affine bundles}\n\n\n\n\n\\begin{defn}\\label{affine bundle}\nLet $B$ be an $S$-scheme.\nAn \\emph{affine bundle over $B$ of rank $d$} is a $B$-scheme $V$\nwith a vector bundle $E\\to B$ of rank $d$ and a $B$-morphism $\\varphi: E\\times_BV\\to V$ such that \n$\\varphi\\times p_V: E\\times _BV \\to V\\times_B V$, where $p_V$ denotes the projection to $V$, is an isomorphism of $B$-schemes.\nWe call $E$ the \\emph{translation space} of $V$.\n\nAn affine bundle $V$ over $B$ is called \\emph{$G$-equivariant}, if $V$ and $B$ are in $(\\Sch_{S,G})$, and $V\\to B$ is $G$-equivariant.\nThe $G$-action on $V\\to B$ is called \\emph{affine} if there is a $G$-action on $E$, linear over the action on $B$,\nsuch that $\\varphi$ is $G$-equivariant.\nAn action on $E$ is \\emph{linear over the action on $B$}\nif for all $g\\in G$ the\nmap $g':E \\to g_B^*E$ induced by the following Cartesian diagram\n\\begin{equation*}\\label{diagram1}\n  \\xymatrix{\n E\\ar@\/^\/[drr]^{g_E}\\ar@\/_\/[ddr]\\ar@{-->}[rd]^{g'}\\\\\n &g_B^*(E)\\ar[r]\\ar[d]& E\\ar[d]\\\\\n & B\\ar[r]^{g_B}& B\n }\n\\end{equation*}\n is a morphism of vector bundles.\n Here $g_B\\in \\Aut(B)$ and $g_E\\in \\Aut(E)$ are the automorphisms of $B$ and $E$ induced by $g$,\n and $g_B^*E:=E\\times_B B$, where $B$ is a $B$-scheme via $g_B$.\n\\end{defn}\n\n\\begin{ex}\\label{exe=v}\n Let $E$ be a vector bundle over some $B\\in (\\Sch_{S,G})$ with an action on $G$ which is linear over the action on the base $B$,\n then $E$ can also be viewed as a $G$-equivariant affine bundle with affine $G$-action.\n We call the $G$-action on $E$ \\emph{quasi-linear}.\n\\end{ex}\n\n\n\n\\begin{ex}\\label{exwildtranslation}\nLet $k$ be a field of characteristic $p>0$, and let $G=\\mathbb{Z}\/p\\mathbb{Z}$.\nLet $B=\\Spec(k)$ with trivial $G$-action, and consider $V= \\Spec(k[x])$ with the $G$-action given by sending $x$ to $x+1$.\nAs this action has no fixed point, there is no way of changing coordinates to achieve that this action is linear.\nHence in particular $V$ is not isomorphic to a $k$-vector space with linear action.\n\nLet $E=\\Spec(k[y])$ be the trivial vector bundle of dimension $1$ over $B$\nwith trivial action of $G$.\nConsider the map\n given by sending $(e,v)\\in E\\times_BV$ to $e+v\\in V$.\n As $(e,v+1)$ is mapped to $e+v+1$ this map is clearly $G$-equivariant.\nOne can check that it induces an isomorphism $V\\times_BE\\to V\\times_B V$.\nSo $V\\to B$ is a $G$-equivariant affine bundle\nwith affine $G$-action, because the action on $E$ is trivial.\n \\end{ex}\n \n \\begin{rem}\\label{trivialtorsor}\nDefinition \\ref{affine bundle} implies that a $G$-equivariant affine bundle $V\\to B$ of rank $d$ is in particular a \\emph{principal homogenous space} or \\emph{torsor}.\n Hence \\cite[Proposition 4.1]{MR559531} implies that it is locally in the \\'etale topology a \\emph{trivial torsor},\n i.e., there is a cover $\\{U_i\\}_{i\\in I}$ of $B$ in the \\'etale topology,\n such that $V_{U_i}:=V\\times_BU_i\\cong E_{U_i}:=E\\times_B U_i$, and $E_{U_i}\\cong \\mathbb{A}^d_{U_i}$ acts on $V_{U_i}$ by translation.\n  \n  By \\cite[Propostion 14]{Serre} the algebraic group $\\mathbb{G}_a^d$ is special.\n  This means by definition, see \\cite[4.1]{Serre}, that a locally trivial $\\mathbb{G}_a^d$-torsor in the \\'etale topology is already locally trivial in the Zariski topology.\n  But if we restrict $B$ to an open over which $E$ is trivial, $V$ is a locally trivial $\\mathbb{G}_a^d$-torsor in the \\'etale topology.\n  Hence we may assume that $\\{U_i\\}_{i\\in I}$ is a cover of $B$ in the Zariski topology.\n \\end{rem}  \n\n\n\\begin{rem}\\label{Vb} \nLet $B\\in (\\Sch_{S,G})$, and let $E$ be a vector bundle of rank $d$ with a $G$-action which is linear over that on $B$.\nLet $b\\in B$ be a \\emph{fixed point}, i.e., the orbit of $b$ under the action of $G$ on $B$ contains only $b$,\nand let $K$ be its residue field.\nThen $E_b:=E\\times_Bb\\cong\\Spec(K[x_1,\\dots,x_d])$, and the $G$-action on $E$ restricts to $E_b$.\nTake any $g\\in G$, and let $\\alpha\\in \\Aut(K[x_1,\\dots,x_d])$ be the corresponding automorphism of rings.\nThen we have the following commutative diagram.\n\n\\begin{equation*}\\label{diagram affine}\n  \\xymatrix{\n K[x_1,\\dots,x_d]\\\\\n &K\\otimes_KK[x_1,\\dots,x_d]\\ar@{-->}[lu]^{\\alpha'}& K[x_1,\\dots,x_d]\\ar[l]\\ar@\/_\/[ull]^{\\alpha}\\\\\n & K\\ar[u]\\ar@\/^\/[uul]& K\\ar[l]^{\\alpha\\rvert_K}\\ar[u]\n }\n\\end{equation*}\nNote that $K\\otimes_KK[x_1,\\dots,x_d] \\cong K[x_1,\\dots,x_d]$, but the $K$-structure on the first\nis given by sending $s\\in K$ to $\\alpha^{-1}(s)$.\nBy the definition $\\alpha'$ is $K$-linear.\nHence we have\n\\begin{equation}\\label{linear}\n\\alpha(x_i)=\\alpha'(x_i)=\\sum_{j=1}^d a_{ij}x_j\n\\end{equation}\nfor some $a_{ij}\\in K$.\nUsing that $\\alpha$ is a ring morphism,\nwe get that\n\\begin{equation}\\label{qlin}\n \\alpha(v+sw)=\\alpha(v)+\\alpha\\rvert_K(s)\\alpha(w)\n\\end{equation}\nfor all $v,w\\in K[x_1,\\dots,x_d]$ and $s\\in K\\subset K[x_1,\\dots,x_d]$.\nIf $s\\in k:=K^G$,\nthen $\\alpha(v+sw)=\\alpha(v)+s\\alpha(w)$,\nbecause $\\alpha\\rvert _k=\\Id$ by definition.\nNote that $K$ is a Galois extension of $k$,\nand we have a surjective map \n\\[\nG\\to \\Gal(K,k)=:\\Gamma.\n\\]\nSo $K$ is a $k$-vector space of dimension $r:=\\lvert\\Gamma\\rvert$.\nNow we can view $E_b$ as a $K$-vector space of dimension $d$, and\nhence also as a $k$-vector space of dimension $rd$.\nWe have seen that the $G$-action on $E_b$ defines a $k$-linear action on $E_b$\nwhich lifts the Galois action of $\\Gamma$ on $K$.\nThis follows from Equation (\\ref{linear}) and Equation (\\ref{qlin}).\nHence in particular the $G$-action on $E_b$ is quasi-linear.\n\\end{rem}\n\n\\begin{rem}\\label{Vb translation}\nLet $V\\to B$ be a  $G$-equivariant affine bundle of rank $d$ with affine $G$-action with translation space $E\\to B$,\nand let $b\\in B$ be a fixed point.\n Remark~\\ref{trivialtorsor}\n implies that $\\varphi_b: E_b\\times V_b \\to V_b$, with $E_b=E\\times_B b$ and $V_b=V\\times_B b$,\n is the trivial torsor,\n hence $V_b\\cong E_b \\cong \\mathbb{A}^d_K$,\n and $\\varphi_b$ sends $(v,w)\\in E_b\\times V_b$ to $v+w\\in V_b$.\n \n As $b$ is fixed under the action of $G$, the $G$-action on $E$ and $V$ restrict to $E_b$ and $V_b$.\n Moreover $\\varphi_b$ is $G$-equivariant.\n Take any $g\\in G$, and let $g_E\\in \\Aut(E_b)$ and $g_V\\in \\Aut(V_b)$ be the corresponding automorphisms.\n Fix a $0\\in V_b$.\nFor all $v\\in V_b$ we have that\n \\begin{equation}\\label{formel translation}\n   g_V(v)=g_V(v+0)=g_V(\\varphi_b ( v,0 ))=\\varphi_b (g_E(v), g_V(0))=g_E(v)+g_V(0).\n \\end{equation}\nNote that Remark \\ref{Vb} implies that\n$g_E$ is quasi-linear.\nMoreover $g_V(0)$ does only depend on $g$ and the choice of $0$, but not on $v$.\n\\end{rem}\n\n\n\\begin{rem} \\label{tame translation}\n Assumption and notation as in Remark \\ref{Vb translation}.\nLet $H\\subset G$ \nbe the subgroup consisting of all elements of order prime to the characteristic of $K$.\nAssume that $H$ is abelian.\n\nView $V_b$ as a vector space over $k=K^G$, and\nconsider the action of $H$ on $V_b$.\nBy Remark~\\ref{Vb translation} we know that\nfor every $h\\in H$ the corresponding automorphism sends $v\\in V_b$ to $A_h(v)+b_h$,\nwhere $A_h$ is a $k$-linear map and $b_h\\in V_b$.\nWe are now going to show that the action of $H$ on $V_b$ has a fixed point.\nTherefore we view $V_b$ as a scheme over $k$,\nhence $V_b\\cong \\mathbb{A}^{rd}_k=\\Spec(k[x_1,\\dots,x_{rd}])$,\nand show by induction on $n:=\\lvert H\\rvert$ that\nthe fixed point locus $V_b^H\\subset V_b$ is isomorphic to $\\mathbb{A}_k^N$ for some $N\\geq 0$.\nFor $n=1$, the statement is trivial.\n\nSo let $n>1$.\nThen there exists a nontrivial cyclic $q$-subgroup $H'$ of $H$ for some prime $q$, prime to the characteristic of $k$.\nConsider the induced action of $H'$ on $V_b$.\nAs $q\\neq p$, we can use \\cite[Corollary 5.5]{1009.1281}, which follows from a theorem of Serre in \\cite{MR2555994},\nto get that $V_b^{H'}(\\bar{k})\\neq \\emptyset$.\nHere $\\bar{k}$ is the algebraic closure of $k$.\nIn particular $V_b^{H'}$ is not the empty scheme.\nLet $h\\in H$ be a generator of $H'$.\nThen the corresponding automorphism of $k[x_1,\\dots,x_{rd}]$ sends $x_i$\nto $\\sum h_{ij}x_j +h_i$ for some $h_{ij},h_i\\in k$.\nHence $V_b^{H'}\\subset V_b$ is given by equation of the form\n$h_{ij}x_j + h_i-x_i$, hence $V_b^{H'}$ is a nonempty linear subspace of $V_b$,\nso in particular isomorphic to $\\mathbb{A}_k^N$ for some $N\\geq 0$.\n\nAs $H$ is abelian, it maps every point fixed by $H'$\nto a point fixed by  $H'$.\nHence $V_b^{H'}$ is $H$-invariant.\nAs $H'$ acts trivially on $V_b^{H'}$,\nwe get in fact an action of $H\/H'$ on $V_b^{H'}$.\nThis action is still given by some $k$-linear maps composed with some translation.\nAs the order of $H\/H'$ is smaller than $n$,\nwe can now use the induction assumption to get that\n$V_b^H=(V_b^{H'})^{H\/H'}\\cong \\mathbb{A}_k^N$ for some $N\\geq 0$.\nIn particular $V_b^H$ has a point $v_0$ over $k$.\n\nLet $g\\in H\\subset G$, and let $g_V$ be the corresponding automorphism of $V_b$.\nThen $g_V(v_0)=v_0$.\nIf we now chose $0$ in Remark~\\ref{Vb translation} to be $v_0$,\nwe get from Equation (\\ref{formel translation}) that for all $g\\in H\\subset G$ we have\n\\begin{equation*}\\label{eq tame translation}\n g_V(v)=g_E(v)+g_V(0)=g_E(v)+0=g_E(v)\n\\end{equation*}\nfor all $v\\in V_b$.\n\\end{rem}\n\n\\noindent\nNote that we can also find a fixed point in Remark \\ref{tame translation} using elementary calculations\ninstead of \\cite[Corollary 5.5]{1009.1281}.\nIn both cases\nwe need to assume that $H\\subset G$ is an abelian subgroup.\nMoreover it is crucial that\nthe order of $H$ is prime to the characteristic of $K$.\nIn case of wild actions there exist $G$-equivariant affine bundles with affine $G$-action\nsuch that there is no change of coordinates making the action quasi-linear, even if $G$ is cyclic.\nSuch an $G$-equivariant affine bundle is given in Example~\\ref{exwildtranslation}.\n\n\n\n\n\n\\section{The equivariant Grothendieck ring of varieties}\n\\label{equivariant Grothendieck ring}\n\n\\begin{defn}\\label{equivariant Gring}\nThe \\emph{equivariant Grothendieck ring of $S$-varieties}\n$K_0^G(\\Var_S)$ is defined as follows: as an abelian group, it is\ngenerated by isomorphism classes $[X]$ of elements $X\\in(\\Sch_{S,G})$. These generators are subject to the following\nrelations:\n\n\\begin{enumerate}\n\\item $[X]=[Y]+[X\\setminus Y]$, whenever $Y$ is a closed $G$-equivariant \nsub scheme of $X$ (scissors relation).\n\\item $[V]=[W]$, whenever $B\\in (\\Sch_{S,G})$, and\n$V\\rightarrow B$ and $W\\to B$ are two\n $G$-equivariant affine bundles of rank $d$ over $B$ with affine $G$-action, see Definition \\ref{affine bundle}.\n\\end{enumerate}\nFor all $X,Y\\in (\\Sch_{S,G})$, set \n $[X]\\cdot[Y]:=[X\\times_S Y]$, where\n the fiber product is taken in $(\\Sch_{S,G})$.\n This product extends bilinearly to\n $K_0^G(\\Var_S)$ and makes it into a ring.\n \nWe denote by $\\ensuremath{\\mathbb{L}}_S$\n the class of the affine line $\\ensuremath{\\mathbb{A}}^1_S$ with $G$-action induced by the action on $S$ as above.\n  If the base scheme $S$ is clear from the\n context, we write $\\ensuremath{\\mathbb{L}}$ instead of $\\ensuremath{\\mathbb{L}}_S$.\n We define $\\mathcal{M}^G_S$ as the localization\n $K_0^G(\\Var_S)[\\ensuremath{\\mathbb{L}}_S^{-1}]$.\n  \\end{defn}\n \n \\begin{nota}\n  If $G$ is the trivial group $\\{e\\}$, we write\n $K_0(\\Var_S)$ and $\\mathcal{M}_S$ instead of $K_0^G(\\Var_S)$ and\n $\\mathcal{M}^G_S$, receptively.\n Note that in this case Relation (2) becomes trivial.\n If $S=\\Spec(A)$,\n we write $K_0^G(\\Var_A)$ for $K_0^G(\\Var_S)$, $\\mathbb{L}_A$ for $\\mathbb{L}_S$, and $\\mathcal{M}_A^G$ for $\\mathcal{M}_S^G$.\n \\end{nota}\n\n \\begin{rem} \\label{rem 0div neg}\nIn \\cite{lzero} it was shown that $\\mathbb{L}_{\\mathbb{C}}$ is a zero divisor in $K_0(\\Var_{\\mathbb{C}})$.\n This means in particular the canonical map $K_0^G(\\Var_{S})\\to \\mathcal{M}_S^G$ is not an injective map in general.\n \\end{rem}\n\n\\begin{rem}\\label{remark}\nA morphism of finite groups $G'\\rightarrow G$ induces forgetful\nring morphisms\n\\[\nK^{G}_0(\\Var_S)\\rightarrow K^{G'}_0(\\Var_{S})\\mbox{\nand }\n \\mathcal{M}^{G}_S\\rightarrow \\mathcal{M}^{G'}_{S}.\n \\]\n If\n $G'\\rightarrow G$ is surjective, then these morphisms are\n injections.\n\\end{rem}\n\n\\begin{defn}\n Let $S$ be a separated scheme with an action of a profinite group\n \\[\n \\widehat{G}=\\lim_{\\stackrel{\\longleftarrow}{i\\in I}} G_i\n \\]\n factorizing through a good action of some finite quotient $G_i$.\n Then we define\n $$K_0^{\\widehat{G}}(\\Var_S):=\\lim_{\\stackrel{\\longrightarrow}{i\\in\n I}}K_0^{G_i}(\\Var_S)\\ \\mathrm{and}\\ \\mathcal{M}^{\\widehat{G}}_{S}:= \\lim_{\\stackrel{\\longrightarrow}{i\\in\n I}}\\mathcal{M}^{G_i}_{S}.$$\n \\end{defn}\n\n \\medskip\n \\noindent\n Note that in the literature on can find several different definitions of the equivariant Grothendieck ring of varieties.\n This difference always lies in relation~(2),\n which is needed to compute formulas in motivic integration.\n Our definition can be found for example in \\cite[Section 3.4]{MR2483954}.\n We are now going to discuss two alternative definitions:\n \n \\begin{rem}\n In \\cite[2.9]{DLLefschetz} and \\cite[2.4]{DL3}, instead of relation~(2) one divides out the following relation:\n   \\begin{itemize}\n  \\item[(2a)]\n  $[V]=[W]$,\n  where both $V$ and $W$ are affine spaces of degree $r$ over $B$ in $(\\Sch_{S,G})$\n  with good $G$-actions lifting the $G$-action on $B$.\n \\end{itemize}\n The problem here is that it is very hard to say how these lifts look like. For example it is not even known\n whether or not all actions of a finite cyclic group on $\\mathbb{A}_\\mathbb{C}^3$ are linearizable, i.e., whether on can find coordinates for which the action becomes linear,\n see \\cite[Section 6]{MR1423629}.\n In later definitions, there is always a restriction on the actions which one wants to consider.\n\\end{rem}\n \n \\begin{rem}\n In \\cite[Section 2.2]{MR2106970} one divides out the following relation instead of relation~(2):\n \\begin{itemize}\n\n  \\item[(2b)] $[G\\circlearrowright\\mathbb{P}(V)]=[\\mathbb{P}^n\\times(G\\circlearrowright B)]$, whenever\n  $B\\in (\\Sch_{S,G})$ and $V\\to B$\n  is a vector bundle of rank $d+1$ with a $G$-action on $V$ which is linear over the action on $B$.\n  Here $G\\circlearrowright\\mathbb{P}(V)$ denotes the projectivization of this action, whereas $\\mathbb{P}^n\\times(G\\circlearrowright B)$\n  denotes the action on $V$ on the right vector only.\n  \\end{itemize}\n Bittner uses this formulation, because she is working with projective varieties as generators for the Grothendieck ring,\nsee for example \\cite[Corollary 3.6]{MR2106970}.\nIn this context it is of course important to work with projective varieties as generators for the relations. \n  \nAs already remarked in \\cite{MR2106970},\nrelation~(2b) implies in particular that the class of two affine bundles of rank $d$ over $B$ with an affine action over the action on $B$\nhave the same class.\n\nOn the other hand,\nlet $V$ be a vector space of dimension $d+1$ over a field $k$ containing all $\\lvert G \\rvert$-th roots of unity and let $G$\nbe a finite abelian group acting linearly on $V$. These assumptions imply that\nwe can find a common eigenvector for the linear maps on $V$,\nhence there are coordinates such that the induced action on $\\mathbb{P}(V)$ sends\n\\[\n [x_0:\\dots :x_d]\\mapsto [x_0:\\sum_{i=0}^da_{1i}x_i:\\dots:\\sum_{i=0}^da_{di}x_i]\n\\]\nfor all $g\\in G$.\nHence we can decompose $\\mathbb{P}(V)$ in the $G$-invariant subschemes given by $x_0=0$ and $x_0\\neq 0$.\nThe first is of the the form $\\mathbb{P}(V')$, where $V'$ is a $k$-vector space of dimension $d$ with a linear action and\nthe action on $\\mathbb{P}(V')$ is induced by this action.\nThe second is isomorphic to $\\mathbb{A}_k^d=\\Spec(k[x_1,\\dots,x_d])$\nwith affine $G$-action sending for every $g\\in G$\n\\[\n (x_1,\\dots ,x_d)\\mapsto (\\sum_{i=1}^da_{1i}x_i+a_{10},\\dots,\\sum_{i=1}^da_{di}x_i+a_{d0}).\n\\]\nUsing an inductive argument, we can decompose $\\mathbb{P}(V)$ into $k$-vector spaces with affine $G$-action.\nAnalogously we can decompose $\\mathbb{P}(V)$ if $V$ is a vector space over $K$ with a quasi-linear action of $G$ over a $G$-action on $K$.\nTherefore we need to assume that $k:=K^G$ contains all $\\lvert G\\rvert$-th roots of unity.\nWith this decomposition one can use Proposition \\ref{lemeh} to show as in Lemma \\ref{lemma} (with the same assumptions on $S$ and $G$ as there)\nthat the class of the quotient of any $\\mathbb{P}(V)$ as in relation~(2b)\nonly depends on the rank and the base of the vector bundle $V$, hence Theorem \\ref{quotientmap} holds also for Bittner's definition.\nTo avoid the decomposition step it is more reasonable for us to work with our definition.\n\\end{rem}\n\n \n \n \\section{The modified Grothendieck ring of varieties}\n \\label{modified Grothendieck ring}\n \n \\noindent\n Due to the nature of wild actions,\n we are not able to compute quotients of such actions in the usual Grothendieck ring\n by decomposing a scheme into $G$-invariant subschemes and computing the quotient separately on these subschemes.\n The quotient of a closed subscheme has in general\n a purely inseparable map to the image of this subscheme under the quotient map.\n But in the wild case this map\n might not be a piecewise isomorphism,\n as we will see in Example~\\ref{exinsep3}.\n We do not know whether the classes of two schemes connected with such a\n morphism  have the same class in the Grothendieck ring of varieties.\nTherefore we now introduce the modified Grothendieck ring of varieties, in which their classes are the same.\n \n \\begin{defn}\nA morphism of schemes $f:Y\\to X$ is called a \\emph{universal homeomorphism},\nif for every morphism of schemes $X'\\to X$ the morphism\nof schemes $f': Y\\times_X X'\\to X'$ induced by base change is a homeomorphism.\n \\end{defn}\n\n\n \\begin{defn}\\label{dfn modified}\n Let $\\mathcal{I}_S\\subset K_0(\\Var_S)$ be the ideal\n generated by elements of the form $[X]-[Y]$\n such that there exists a universal homeomorphism\n $f:X\\to Y$. The \\emph{modified Grothendieck ring of $S$-varieties} is defined as the quotient\n  \\[\n   K_0^{\\text{mod}}(\\Var_S):=K_0(\\Var_S)\/\\mathcal{I}_S.\n  \\]\nDenote by $\\ensuremath{\\mathbb{L}}_S$\n the class of the affine line $\\ensuremath{\\mathbb{A}}^1_S$.\n  If the base scheme $S$ is clear from the\n context, we write $\\ensuremath{\\mathbb{L}}$ instead of $\\ensuremath{\\mathbb{L}}_S$.\n We define $\\mathcal{M}^{\\text{mod}}_S$ as the localization\n $K_0^{\\text{mod}}(\\Var_S)[\\ensuremath{\\mathbb{L}}_S^{-1}]$.\n \\end{defn}\n \n \\begin{nota}\n  If $S=\\Spec(A)$ is an affine scheme,\n  we write $K_0^{\\text{mod}}(\\Var_A)$ for $K_0^{\\text{mod}}(\\Var_{S})$,\n  $\\mathbb{L}_A$ for $\\mathbb{L}_S$,\n  and $\\mathcal{M}_A^{\\text{mod}}$ for $\\mathcal{M}_S^{\\text{mod}}$.\n \\end{nota}\n\n\n  \\begin{rem}\\label{mod0}\n  If $S$ is a Noetherian $\\mathbb{Q}$-scheme, then the quotient map\n  \\[\n   K_0(\\Var_S)\\to K_0^{\\text{mod}}(\\Var_S)\n  \\]\nis an isomorphism, see \\cite[Corollary 3.8.3]{MR2885336}.\nIn particular this holds if $S$ is a scheme of finite type over any field of characteristic $0$.\n\nIt is not known whether it is an isomorphism in positive characteristic.\nThe problem is that the standard specializing  morphisms used to distinguish elements in the Grothendieck ring factor through \nthe modified Grothendieck ring, see \\cite[Proposition 4.1]{k0mod}.\n \\end{rem}\n\n \\noindent\n We will now prove some technical lemmas which will be used later to compute\n quotients in the (modified) Grothendieck ring of varieties.\n \n \\begin{lem}[Spreading out for the modified Grothendieck ring]\\label{spreading out}\n  Take a directed system of Noetherian commutative rings\n  ${(A_i,\\varphi_{ij}:A_i\\to A_j)}$,\n  and denote by $A$ the direct limit of this system in the category of rings.\n  Then there exists an isomorphism of rings\n  \\[\n\\varphi^{\\text{mod}}: \\lim_{\\stackrel{\\longleftarrow}{i\\in I}}K_0^{\\text{mod}}(\\Var_{A_i})\\to K^{\\text{mod}}_0(\\Var_{A}).\n  \\]\n \\end{lem}\n\n \\begin{proof}\n  Consider the ring morphism\n    \\[\n\\varphi:  \\lim_{\\stackrel{\\longleftarrow}{i\\in I}}K_0(\\Var_{A_i})\\to K_0(\\Var_{A}).\n  \\]\n  induced by the ring morphism $\\varphi_i:K_0(\\Var_{A_i})\\to K_0(\\Var_{A})$\n  given by\n  sending the class of an $A_i$-scheme $U$ to the class of $U\\times_{\\Spec(A_i)}\\Spec(A)$.\n   By \\cite[Proposition~2.9]{MR2770561}, $\\varphi$ is an isomorphism.\n  As a universal homeomorphism is stable under base change,\n  for every universal homeomorphism $f:X\\to Y$ between two $A_i$-schemes,\n  the base change of $f$ to $\\Spec(A)$\n is also a universal homeomorphism.\n  Hence we get well defined maps ${\\varphi_i^{\\text{mod}}:K_0^{\\text{mod}}(\\Var_{A_i})\\to K_0^{\\text{mod}}(\\Var_A)}$,\n  which induce a well defined surjective map $\\varphi^{\\text{mod}}$ as in the claim.\n \n We still need to show that $\\varphi^{\\text{mod}}$ is injective.\n So let $f: X\\to Y$ be a universal homeomorphism between $A$-schemes.\n  By \\cite[Theorem 8.8.2]{MR0217086}\n  there exist an $i$ and a morphism of $A_i$-schemes $f_i : X_i \\to Y_i$\n  such that the base change of $f_i$ to $\\Spec(A)$ is $f$.\n  By \\cite[Theorem 8.10.5]{MR0217086} $f$ is a universal homeomorphism\n  if and only if there is a $j\\geq i$ \n  such that the base change of $f_i$ induced by ${\\Spec(A_j)\\to \\Spec(A_i)}$ is a universal homeomorphism.\n  Hence $\\varphi^{\\text{mod}}$ is injective.\n  \\end{proof}\n \n \\noindent\nRecall that we assume that $S$ is a separated scheme with good action\nof a finite group $G$,\nsuch that the quotient map $S\\to S\/G$ is finite\nand $S\/G$ is separated and locally Noetherian.\nThe next lemmas will enable us to decompose the quotient of schemes\nin the category $(\\Sch_{G,S})$ in the (modified) Grothendieck ring of varieties.\n\n  \\begin{lem}\\label{closed mod} \n Let $X\\in (\\Sch_{G,S})$, and denote\n  by $\\pi: X\\to X\/G$ the quotient.\n   Let $Y\\subset X$ be a closed $G$-invariant subscheme,\n   and let $Z$ be the image of $Y$ under $\\pi$.\n   Then the $G$-action on $X$ restricts to a good $G$-action on $Y$,\n   and there exists a universal homeomorphism \n   $f: Y\/G\\to Z$.\n   Hence in particular\n   \\[\n    [Y\/G]=[Z]\\in K_0^{\\text{mod}}(\\Var_{S\/G}).\n   \\]\n\\end{lem}\n\n  \\begin{proof}\nLet $i:Y\\hookrightarrow X$ be the inclusion map.\nAs $Y\\subset X$ is a $G$-invariant closed subscheme, \nthe $G$-action on $X$ restricts to $Y$. \nAs every affine subscheme of $X$\nwill restrict to an affine subscheme of $Y$,\nthis action is good.\nDenote by $\\pi_Y:Y\\to Y\/G$ the quotient map.\nAs $i$ is $G$-equivariant, we get an induced map $i_G:Y\/G\\to X\/G$ with $\\pi \\circ i=i_G\\circ \\pi_Y$.\nAs $\\pi$ maps $Y$ to $Z$, $i_G$ factors through $Z$.\nWe are going to show that ${i_G: Y\/G \\to Z}$ is a universal homeomorphism.\nBy \\cite[2.4.5.]{MR0199181} it suffices to show that $i_G$ is finite, surjective and purely inseparable.\n\nAs both the points of $X\/G$ and $Y\/G$ are just orbits of the action of $G$,\nthe map $i_G: Y\/G \\to Z$ is a bijection on points.\n\nAs $X$ is of finite type over $S$ and hence over $S\/G$ using that $S\\to S\/G$ is finite,\n$\\pi$ is finite\nby \\cite[Expos\\'e V, Corollaire 1.5]{MR0238860}.\nAs $i$ is proper, the same holds for $\\pi \\circ i$.\nAs moreover $\\pi_Y$ is surjective, $i_G$ is proper by \\cite[Proposition 12.59]{MR2675155}.\nWe have already seen that $i_G$ is quasi-finite, hence it is finite.\nIt remains to show that $i_G$ is purely inseparable, i.e., that for all $y\\in Y\/G$\nthe residue field $L$ of $y$ is purely inseparable over the residue field $K$ of $z:=i_G(y)$.\n\nUsing \\cite[Capitre V.2, Th\\'eor\\`eme 2]{MR782297}\nwe get the following:\nlet $G_x$ be the stabilizer of a point $x\\in Y\\subset X$ of the orbit of $G$ over $y$ and $z$, respectively,\nand let $M$ be the residue field of $x$.\nThen $M$ is normal over $L$ and over $K$, and $G_x$ surjects on $\\Gal(M,L)$ and $\\Gal(K,L)$.\nHence we get the following inclusions of fields\n\\[\n \\xymatrix{\n L \\ar@{^(->}[r] &M^{G_x}\\ar@{^(->}[r]& M\\\\\n K.\\ar@{^(->}[ru] \\ar@{^(->}[u]&\n }\n\\]\nAs $M$ is normal over $K$,\n $M^{G_x}$ is normal over $K$, too.\nWe now can split this extension in a separable extension $K'$ over $K$,\nand a purely inseparable extension $M^{G_x}$ over $K'$.\nObserve that $K'$ is normal over $K$,\nand therefore $K=K'^{\\Gal(K',K)}$.\nBut $\\Gal(K',K)$ is a quotient of $\\Gal (M^{G_x},K)$, and the latter is trivial. Hence $K=K'$.\nTherefore $M^{G_x}$ is purely inseparable over $K$, and hence the same holds for $L$.\nThis finishes the proof.\n  \\end{proof}\n\n\n \\begin{lem}\\label{rem tame decomposition}\n Assumptions and notation as in Lemma \\ref{closed mod}.\n Assume moreover that the action of $G$ on $X$ is tame.\nThen there exist a map $f:Y\/G\\to Z$ which is a piecewise isomorphism,\nhence in particular\n\\[\n    [Y\/G]=[Z]\\in K_0(\\Var_{S\/G}).\n   \\]\n \\end{lem}\n \n \\begin{proof}\nUse the same notation as in the proof of Lemma \\ref{closed mod}.\nAs $G$ acts tamely on $X$,\nit follows from \\cite[Capitre~V.2, Proposition 5 and Corollaire]{MR782297} that\n $M^{G_x}$ is actually equal to $L$ and to $K$, hence $L=K$.\n Hence we have a finite bijective map\n $i_G: Y\/G \\to Z$\n such that for every point $y\\in Y\/G$\n the residue field of $y$ is isomorphic with the residue field of  $i_G(y)$.\n As an $S\/G$-variety has the same class in $K_0(\\Var_{S\/G})$ as the reduced underlying scheme,\n we can assume that both $Y\/G$ and $Z$ are reduced.\n  Take a generic point $\\eta \\in Y\/G$ with function field $\\kappa_\\eta$.\n  The image $i_G(\\eta)\\in Z$ will be a generic point with function field isomorphic to $\\kappa_\\eta$.\n  Hence we can find open subschemes $U\\subset Y\/G$ and $V\\subset Z$,\n  such that $i_G:U\\to V$ is an isomorphism.\n  Now we can proceed with $i_G:Y\/G\\setminus U\\to Z\\setminus V$, and use Noetherian induction\n  to get that $i_G$ is a piecewise isomorphism.\n  The claim now follows using the scissors relation in $K_0(\\Var_{S\/G})$.\n   \\end{proof}\n\n   \\noindent\nIf the action of $G$ on $X$ is wild, we will really need\nto work in the modified Grothendieck ring, as\nthe following examples show.\n\n \\begin{ex}\\label{exinsep}\n Let $p$ be a prime, $G=\\mathbb{Z}\/p\\mathbb{Z}$ and let $k$ be a field of characteristic $p$.\n  Let $X=\\mathbb{A}^2_k=\\Spec(k[x,y])$,\n  and consider the action of $G$ on $X$ given by sending $x$ to $x+y$ and $y$ to $y$.\n  We have that $X\/G=\\Spec(k[x^p+(p-1)xy^{p-1}, y])\\cong \\Spec(k[u,y])$.\n  Denote by $\\pi:X\\to X\/G$ the quotient map.\n  \n  Consider the $G$-invariant closed subscheme\n  $Y=\\Spec(k[x,y]\/(y))\\cong \\ensuremath{\\mathbb{A}}^1_k\\subset X$.\nThen the induced action on $Y$ is trivial,\n and the induced map $i_G$ from $Y=Y\/G$ to $Z=\\pi(Y)=\\Spec(k[u,y]\/(y))\\cong \\Spec(k[u])$ is given by sending $u$ to $x^p$.\n Note that $k(u)$ is the function field of $Z$\n and $k(x)$ is the function field of $Y\/G=Y$.\n The field extension $k(u)\\subset k(x)$ induced by $i_G$ is radical of degree $p$.\n As the characteristic of $k$ is equal to $p$, it is purely inseparable.\n \n Nevertheless, $Y$ and $Z$ are isomorphic over $k$, but this isomorphism is not given by $i_G$.\n \\end{ex}\n\n\n\n \\begin{ex}\\label{exinsep3}\n   Let $p>3$ be a prime, let  $G=\\mathbb{Z}\/p\\mathbb{Z}$, and let $k$ be a field of characteristic $p$.\n  Let $X=\\mathbb{A}^6_k=\\Spec(k[x,y,a,a',b,b'])$,\n  and consider the action of $G$ on $X$ given by sending $P(x,y,a,a',b,b')\\in k[x,y,a,a',b,b']$\n  to ${P(x,y,a+a',a',b+b',b')}$.\n  One can check that\n  \\begin{align*}\n  X\/G&=\\Spec(k[x,y,a^p+(p-1)aa'^{p-1},a',b^p+(p-1)bb'^{p-1},b'])\\\\\n  &\\cong \\Spec(k[x,y,u_a,a',u_b,b']).\n \\intertext{ Denote by $\\pi:X\\to X\/G$ the quotient map.\n  Consider}\n   Y&=\\Spec(k[x,y,a,a',b,b']\/(a',b',x^3+a^px-y^2+b^p))\\\\\n  &=\\Spec(k[x,y,a,b]\/(x^3+a^px-y^2+b^p))\\subset X.\n  \\end{align*}\n  Note that $Y$ is $G$-invariant, and\n the induced action on $Y$ is trivial.\n As \n \\[\n  x^3+u_ax-y^2+u_b=x^3+a^px-y^2+b^p+(p-1)axa'^{p-1}+(p-1)bb'^{p-1},\n \\]\n we get that\n \\[\n k[x,y,u_a,a',u_b,b']\\cap (a',b',x^3+a^px-y^2+b^p)=(a',b',x^3+u_ax-y^2+u_b),\n \\]\nand hence\n$Z=\\pi(Y)=\\Spec(k[x,y,u_a,u_b]\/(x^3+u_ax-y^2+u_b))$.\nNote that the residue field of the generic point of $Z$\nis isomorphic to the function field of the elliptic curve\n$E=\\Spec(K[x,y]\/(x^3+ax-y^2+b))$\nwith $K=K(a,b)\\cong K(u_a,u_b)$.\nLet $\\varphi: K\\to K$ be the Frobenius map.\nThen \n\\[\nE^{(p)}:=E\\times_{\\varphi}\\Spec(K)\\cong \\Spec(K[x,y]\/(x^3+a^px-y^2+b^p)).\n\\]\nNote that the residue field of the generic point of $Y=Y\/G$\n is isomorphic to the function field of $E^{(p)}$.\n Now we compute the $j$-invariant of $E$  and $E^{(p)}$:\n \\[\n  j(E)=1728\\frac{4a^3}{4a^3+27b^2} \\text{ and } j(E^{(p)})=1728\\frac{4(a^p)^3}{4(a^p)^3+27(b^p)^2} \n \\]\nHence $E$ and $E^{(p)}$ will have the same $j$-invariant\nif and only if\n ${b^2}\/{a^3}=({b^2}\/{a^3})^p$.\nAs this was only true if ${b^2}\/{a^3}$ was in $\\mathbb{F}_p$,\nthe $j$-invariants\n of $E$ and $E^{(p)}$ are different.\n Therefore $E$ and $E^{(p)}$ are not isomorphic, and \n hence $Z$ and $Y$ cannot be piecewise isomorphic.\n But by \\cite[Theorem 2.13]{lzero} this does not imply that $Z$ an $Y$ cannot have the same class in the Grothendieck ring.\n In fact it is not known whether $E$ and $E^{(p)}$,\n and thus $Z$ and $Y$, have the same class in the Grothendieck ring.\n \\end{ex}\n\n \n \\begin{rem} \\label{rem decomposing}\n Take $X\\in (\\Sch_{S,G})$, \n let $Y\\subset X$ be a closed $G$-invariant subscheme and let $U=X\\setminus Y$ be the compliment.\n Let $\\pi:X\\to X\/G$ be the quotient map.\n By\n \\cite[Expos\\'e V, Corollaire 1.5]{MR0238860}\n we have that\n$U\/G \\cong \\pi(U)$.\nHence\n\\begin{align*}\n  [X\/G]&=[U\/G]+[Y\/G]\\in K_0(\\Var_{S\/G})\\\\\n\\shortintertext{if and only if}\n [Y\/G]&=[\\pi(Y)]\\in K_0(\\Var_{S\/G}).\n\\end{align*}\nWe do not know whether this is true or not,\neven if $S=\\Spec(k)$ with $k$ a finite field, see Example \\ref{exinsep3}.\n \\end{rem}\n \n  \n\\begin{lem} \\label{lemma vezel}\nTake two schemes $V,B\\in (\\Sch_{G,S})$, and let\n${\\varphi:V\\to B}$ be a $G$-equivariant morphism of finite type.\n Denote the induced map between the quotients with ${\\varphi_G:V\/G\\to B\/G}$.\n Let $x\\in B\/G$ be a point with residue field $k$,\n let $b$ be a point in $B$ mapped to $x$ under the quotient map,\n and let $G_b$ be the stabilizer of $b$.\n Assume that $G_b$ is a normal subgroup of $G$.\n Then $G_b$ acts on $\\varphi^{-1}(b)$, this action is good, and\n \\[\n  [\\varphi_G^{-1}(x)]=[\\varphi^{-1}(b)\/G_b]\\in K_0^{\\text{mod}}(\\Var_k).\n \\]\n\\end{lem}\n\n\\begin{proof}\n Let $\\pi:B\\to B\/G$ and $\\pi_V:V\\to V\/G$ be the quotient maps.\n Note that $\\pi\\circ \\varphi=\\varphi_G\\circ \\pi_V$.\n Let $X\\subset B\/G$ be the closure of $x$ in $B\/G$.\n By construction $\\varphi^{-1}(\\pi^{-1}(X))=\\pi_V^{-1}(\\varphi_G^{-1}(X))$\n is a $G$-invariant closed subscheme mapped surjectively to $\\varphi_G^{-1}(X)$ under\n the quotient map $\\pi_V$.\n Thus\n by Lemma \\ref{closed mod}\nthere is a universal homeomorphism\n\\[\nf: \\varphi^{-1}(\\pi^{-1}(X))\/G\\to \\varphi_G^{-1}(X).\n\\]\nHence we get a universal homeomorphism\n\\[\nf_k: \\varphi^{-1}(\\pi^{-1}(X))\/G\\times_{X} \\Spec(k)\\to \\varphi_G^{-1}(X)\\times_{X} \\Spec(k)= \\varphi_G^{-1}(x), \n\\]\nbecause universal homeomorphisms are stable under base change.\n As $x$ is the generic point of $X$,\n $\\Spec(k)\\to X$ is flat.\n Hence by \\cite[Expos\\'e V, Proposition~1.9]{MR0238860},\n \\[\n  \\varphi^{-1}(\\pi^{-1}(x))\/G=(\\varphi^{-1}(\\pi^{-1}(X))\\times_{X} \\Spec(k))\/G\n  \\cong \\varphi^{-1}(\\pi ^{-1}(X))\/G\\times _{X} \\Spec(k).\n \\]\nThus in the modified Grothendieck ring we get\n\\[\n [\\varphi_G^{-1}(x)]=[\\varphi^{-1}(\\pi^{-1}(X))\/G\\times_{X}\\Spec(k)] =[\\varphi^{-1}(\\pi^{-1}(x))\/G]\\in K_0^{\\text{mod}}(\\Var_k).\n\\]\nNow consider the stabilizer $G_b$ of $b\\in \\pi^{-1}(x)$.\nAs $G_b$ is a subgroup of $G$, it acts on $V$ and $B$.\nBy construction, $\\pi^{-1}(x)$ is $G$-invariant and $\\varphi$ is $G$-equivariant,\nhence we get induced actions of $G_b$ on $\\varphi^{-1}(\\pi^{-1}(x))$ and $\\pi^{-1}(x)$,\nand an induced map \n\\[\n\\psi:\\varphi^{-1}(\\pi^{-1}(x))\/G_b\\to \\pi^{-1}(x)\/G_b.\n\\]\nAs $G_b\\subset G$ is a normal subgroup, we may consider $H:=G\/G_b$.\n$H$ acts on $\\varphi^{-1}(\\pi^{-1}(x))\/G_b$ and $\\pi^{-1}(x)\/G_b$,\nand $\\psi$ is $H$-equivariant.\nAs $\\pi^{-1}(x)$ is the inverse image of the point $x$ of the finite quotient map $\\pi$,\n$\\pi^{-1}(x)$ is a finite union of points.\nThus also $\\pi^{-1}(x)\/G_b$ is a finite union of points $P_1,\\dots,P_n$,\nand hence $\\varphi^{-1}(\\pi^{-1}(x))\/G_b\\cong \\bigcup \\psi^{-1}(P_i)$.\nAs $G_b$ is the stabilizer of $b$ and $\\pi^{-1}(x)$ is the orbit of $b$,\nthe action of $H$ on $\\pi^{-1}(x)\/G_b$ is free and transitive,\ni.e., for every pair $i,j$ there is a unique $h_{ij}\\in H$ \nwith $h_{ij}(P_i)=(P_j)$.\nHence $h_{ij}$ also maps $\\psi^{-1}(P_i)$ isomorphically to $\\psi^{-1}(P_j)$.\n\nLet $W$ be the disjoint union of $n$ copies of $\\psi^{-1}(P_1)$.\nLet $H$ act on $W$ as follows:\nfor $h\\in H$ with $h(P_i)=P_j$,\nlet the corresponding automorphism of $W$ map the $i$-th copy of $\\psi^{-1}(P_1)$ identically to the $j$-th copy.\nIt is obvious that $W\/H\\cong \\psi^{-1}(P_1)$.\nConsider the map $\\varphi: W\\to \\varphi^{-1}(\\pi^{-1}(x))\/G_b$ given on the $i$-th copy of $\\psi^{-1}(P_1)$\nby $h_{1i}\\rvert_{\\psi^{-1}(P_1)}$.\nOne can check that $\\varphi$ is a $H$-equivariant isomorphism with $H$-equivariant inverse,\nhence we get that\n\\[\n \\varphi^{-1}(\\pi^{-1}(x))\/G=\\varphi^{-1}(\\pi^{-1}(x))\/G_b\/H\\cong W\/H\\cong \\psi^{-1}(P_1).\n\\]\nWithout loss of generality we may assume that $P_1$ is the image of $b$ under the quotient map.\nAs $b$ is a component of $\\pi^{-1}(x)$, $\\varphi^{-1}(b)$ is open in $\\varphi^{-1}(\\pi^{-1}(x))$.\nMoreover  it is $G_b$-invariant, because $G_b$ is the stabilizer of $b$ and $\\varphi$ is $G_b$-equivariant.\nSo by \\cite[Expos\\'e V, Corollaire 1.4.]{MR0238860} we get that\n\\[\n \\psi^ {-1}(P_1)\\cong \\varphi^ {-1}(b)\/G_b.\n\\]\nAll together we have \n\\[\n [\\varphi_G^{-1}(x)] =[\\varphi^{-1}(\\pi^{-1}(x))\/G]=[\\psi^{-1}(P_1)]=[\\varphi^{-1}(b)\/G_b]\\in K_0^{\\text{mod}}(\\Var_k).\n\\]\n\n\\end{proof}\n\n \n\\begin{rem}\\label{rem fiber tame}\nAssumption and notation as in Lemma \\ref{lemma vezel}.\nIf $G$ acts tamely on $S$, and hence also on $V$ and $B$,\n we get that \n \\[\n  [\\varphi_G^{-1}(x)]=[\\varphi^{-1}(b)\/G_b]\\in K_0(\\Var_k).\n \\]\nThis holds, because in this case we get, as in the proof of Lemma \\ref{rem tame decomposition}, that\n$f$\nis a finite, bijective map such that the map between the residue fields of the points are isomorphic.\n Hence the same holds for $f_k$,\n and we can show, as done in Lemma \\ref{rem tame decomposition},\nthat $\\varphi_G^{-1}(x)$\n and $\\varphi^{-1}(\\pi^{-1}(X))\/G\\times \\Spec(k)\\cong \\varphi^{-1}(b)\/G_b$\n have the same class in $K_0(\\Var_k)$.\n \\end{rem}\n\n\n\n \n\\section{Quotients of vector spaces by quasi-linear actions}\n\\label{quotients of vector spaces}\n \n \\noindent\nThe aim of this section is to show a version of the following proposition in the case of wild group actions.\nThis proposition was proved in \\cite[Lemma~1.1]{MR2642161}\nas a generalization of \\cite[Lemma 5.1]{Loo}.\n\n\n \\begin{prop}\\cite[Lemma~1.1]{MR2642161}\\label{lemeh}\n  Let $G$ be a finite abelian group with quotient $G\\to \\Gamma$.\n  Let $k$ be a field of characteristic zero, or positive characteristic prime to $\\lvert G\\rvert$,\n  and let $K\/k$ be a Galois extension with Galois group $\\Gamma$.\n  Assume that the Galois action of $\\Gamma$ on $K$ lifts to a $k$-linear action of $G$\n  on a finite dimensional $K$-vector space $V$.\n  If all $\\lvert G\\rvert$-th roots of unity lie in $k$, then\n  \\[\n   [V\/G]=\\mathbb{L}_k^{\\Dim_KV}\\in K_0(Var_k).\n  \\]\n \\end{prop}\n\n \\noindent\n In \\cite[Lemma~1.1]{MR2642161} this proposition was only stated for characteristic zero.\n Going through the proof, one recognizes\n that the only assumptions on $k$ which are used are that the characters of $G$\n are $k$-rational\n and that $\\lvert G\\rvert$ is prime to the characteristic of $k$,\n and hence for every representation of $G$ on a $k$-vector space $V$, there is a decomposition of $V$ into eigenspaces over $k$.\n This is not true if the characteristic of $k$ divides $\\lvert G \\rvert$,\n even if $k$ is algebraically closed.\nNote furthermore that if the action of $G$ is tame,\nwe can decompose in the usual Grothendieck ring of varieties,\nsee Lemma~\\ref{rem tame decomposition}.\nHence the proposition really holds in $K_0 (\\Var_k)$, also if the characteristic of $k$ is positive but prime to the order of $G$.\n \n Working with wild actions,\n we cannot decompose $V$ into eigenspaces, so\n we will not be able to use the stratification of $V$ from \\cite[Lemma~5.1]{Loo} as done in \\cite[Lemma~1.1]{MR2642161}.\n Instead we will first show the claim for the case $\\Dim_K V=1$ with elementary methods,\n and then use a $G$-equivariant fibration $\\varphi: V\\to W$ to a vector space $W$ of dimension $1$ over $K$\n to conclude by induction.\n More precisely,\n we compute the classes of the fibers of the induced map $\\varphi_G: V\/G\\to W\/G$ separately\n using the induction assumption and Lemma \\ref{lemma vezel},\n and then we use spreading out, see Lemma~\\ref{spreading out}.\n To be able to use Lemma \\ref{lemma vezel} we need to work in the \n modified Grothendieck ring of varieties.\n \n\\begin{prop}\\label{lemwildandtame}\n  Let $G$ be a finite abelian group with quotient $G\\to \\Gamma$.\n  Let $k$ be a field of characteristic $p$,\n  let $q$ be the greatest divisor of $\\lvert G\\rvert$ prime to $p$, \n  and let $K\/k$ be a Galois extension with Galois group $\\Gamma$.\n  Assume that the Galois action on $K$ lifts to a $k$-linear action of $G$\n  on a finite dimensional $K$-vector space $V$.\n  If $k$ contains all $q$-th roots of unity, then\n  \\[\n   [V\/G]=\\mathbb{L}_k^{\\Dim_KV}\\in K_0^{\\text{mod}}(Var_k).\n  \\]\n \\end{prop}\n\n\\begin{proof}\nAs $G$ is a finite abelian group, we have that\n\\[\n G\\cong \\mathbb{Z}\/p^{r_1}\\mathbb{Z}\\times ... \\times \\mathbb{Z}\/p^{r_s}\\mathbb{Z}\\times \\mathbb{Z}\/q_1\\mathbb{Z}\\times \\dots \\times \\mathbb{Z}\/q_t\\mathbb{Z}\n\\]\nwith $p$ the characteristic of $k$, and $q_i$ prime to $p$.\nSet $r:=\\sum r_i$ and $q:=\\prod q_i$.\nNote that $\\lvert G\\rvert=p^rq$, and $q$ is prime to $p$.\n Set $d:=\\Dim_KV$.\nThen we have \n $V\\cong \\Spec(K[x_1,\\dots,x_d])$ as schemes.\n  The $G$-action on $V$ is given by $\\alpha_1,\\dots, \\alpha_s$ in $\\Aut_k(K[x_1,\\dots,x_d])$ with $\\alpha_l^{p^{r_l}}=\\Id$,\nand $\\beta_1,\\dots, \\beta_t$ in $\\Aut_k(K[x_1,\\dots,x_d])$ with $\\beta_l^{q_l}=\\Id$,\n  such that\n  $\\alpha_l\\rvert _K$ and $\\beta_l\\rvert_K$ generate the Galois action on $K$.\n  As $G$ is abelian, the $\\alpha_l$ and $\\beta_l$ commute.\n  \n  View $V$ as a vector space over $k$. \n  The $G$-action is given by $A_l,B_l\\in GL_k(V)$ by assumption.\n  Moreover,\n  $A_l ^{p^{r_l}}=\\Id$ and $B_l^{q_l}=\\Id$,\n  and the $A_l$ and $B_l$ commute.\n  Note that, as the order of the $B_l$ is finite of rang prime to $p$, the $B_l$ are diagonalizable over $\\bar{k}$.\n  As $B_l^{q_l}=\\Id$, all eigenvalues are $q_l$-th roots of unity, and as $q_l$ divides $q$,\n  all those eigenvalues are already in $k$ by assumption.\n  Hence $B_l$ is already diagonalizable over $k$.\n  As the $B_l$ commute, we find a basis of $V$ of common eigenvectors of all the $B_l$.\n \n Now consider the $A_l$.\n Let $E$ be any intersection of eigenspaces of the $B_l$.\n As the $A_l$ commute with the $B_l$,\n $A_l(E)=E$ for all $l$.\n Recall that $A_l^{p^{r_l}}=\\Id$,\n hence all eigenvalues of $A_l$ are $p^{r_l}$-th roots of unity,\n   and as $\\Char(k)=p$, all the eigenvalues are $1$, i.e., in particular in $k$.\n  So we find a $k$-basis of $E$\n  such that $A_l$ has upper triangle form with only $1$ on the diagonal.\n  As the $A_l$ commute, we can even find a $k$-basis of $E$ such that all the $A_l$ have upper triangle form.\n  We can do this for all intersections of eigenspaces of the $B_l$, hence we \n get a $k$-basis $B:=\\{v_1,\\dots, v_s\\}$ of $V$ such that all $A_l$ have upper triangle form with only $1$ on the diagonal,\n and $B$ consist only of eigenvectors of the $B_l$.\n \n  Consider the subset of $B$ containing\n  those $v_i$ which do not lie in the \n  sub-$K$-vector space of $V$ spanned by the $v_j$ with $j<i$.\n  This way we get a basis $B'=\\{w_1,\\dots,w_d\\}$ of $V$ as $K$-vector space\n  such that $A_l(w_i)=w_i + \\sum_{j<i}a_{lij}w_i$ for some $a_{lij}\\in K$,\n  and $B_l(w_i)=\\mu_{li}w_i$ for some $q_l$-th roots of unity $\\mu_{li}$.\nHence we may assume - after a change of coordinates - that\n  \\[\n  \\alpha_l(x_i)=x_i+\\sum_{j<i}a_{lij}x_j\\text{ and }   \\beta_l(x_i)=\\mu_{li}x_i\n  \\]\n  for some $a_{lij}\\in K$, and\n  some $q_l$-th roots of unity $\\mu_{li}$.\n  We need to show that\n    \\begin{equation}\\label{claim}    \n   [V\/G]=[\\Spec(K[x_1,\\dots,x_d]^G)]=\\mathbb{L}_k^d\\in K_0^{\\text{mod}}(Var_k).\n    \\end{equation}\nWe will show a slightly more general statement by induction on $d$,\nwe namely only ask that\n$\\alpha_l\\rvert_K$ and $\\beta_l\\lvert_K$ generate the action of $\\Gamma =\\Gal(K,k)$,\nand that\n  \\begin{equation}\\label{translation}\n  \\alpha_l(x_i)=x_i+\\sum_{j<i}a_{lij}x_j+a_{li} \\text { and } \\beta_l(x_i)=\\mu_{li}x_i\n  \\end{equation}\n  for some $a_{lij}\\in K$ and $a_{li}\\in K$, and some $q_l$-th roots of unity.\n  \n  \\medskip\n  \n  Start with $d=1$. Hence $V\\cong\\Spec(K[x_1])$, $\\alpha_l(x_1)=x_1+a_l$ for some $a_l\\in K$,\n  and $\\beta_l(x_1)=\\mu_lx_1$ for some $q_l$-th root of unity $\\mu_l$.\n  We will show the claim  for $d=1$ by jet another induction, this time on $r$.\n  If $r=0$, $\\lvert G\\rvert$ is prime to $p$, so by \\cite[Lemma~1.1]{MR2642161}  the claim is true.\n  Assume hence that the claim holds for $r-1$. \n  \n  Let ${G}':=\\mathbb{Z}\/p\\mathbb{Z}\\times \\{0\\} \\times \\dots \\times \\{0\\}\\subset G$.\n  Note that $G'\\subset G$ is a normal subgroup of order $p$ and acts on $V$.\n  The action is generated by $\\gamma=\\alpha_1^{p^{r_1-1}},$\n  and hence $\\gamma(x_1)=x_1+b$ for some $b\\in K$.\n  Note that $K$ is a Galois extension of ${K}':=K^{{G}'}$ with Galois group generated by $\\gamma\\rvert_K$.\n  \n  If $b=0$, $K[x_1]^{G'}=K^{G'}[x_1]=K'[x_1]$.\n  If $b\\neq 0$ and $\\alpha\\rvert_K=\\Id$, i.e., $K'=K$, then\n  \\[\n  K[x_1]^{G'}= K[x_1 +(p-1)b^{1-p}x_1^p]\\cong K'[y].\n  \\]\n  We now may now assume that $K\/K'$ is a field extension of degree $p$.\n  As the characteristic of $K$ is equal to $p$,\n  $K\/K'$ is an Arthin-Schreier extension,\n thus $K=K'(\\omega)$ for some $\\omega$ in $K$,\n  and $\\gamma(\\omega)=\\omega+1$.\n  Hence $\\omega^2$ is mapped to $\\omega^2 + 2\\omega + 1$ and similarly for higher powers of $\\omega$.\n  As $\\gamma$ is a linear map on the $K'$-vector space $K$ of degree $p$,\n  we find a basis $\\{1,\\omega,v_3,\\dots, v_{p} \\}$ of $K$ over $K'$\n  such that $\\gamma(v_i)=v_i+v_{i-1}$ for $i>3$, and $\\gamma(v_3)=v_3+\\omega$.\n  We can write $b$ in this basis, i.e., we have\n  \\begin{align*}\n  b&=b_1 +b_2\\omega + \\sum_{i=3}^{p} b_iv_i\\\\\n \\shortintertext\n  {for some $b_i\\in K'$.\n  Set} \n  y&:=x_1 - b' \\text{ with } b':=b_1 \\omega + \\sum _{i=2}^{p-1} b_i v_{i+1}.\n  \\end{align*}\n  We have $\\gamma(y)= y + b_{p}v_{p}$, and $K[x_1]\\cong K[y]$.\n  Using that the characteristic of $K$ is $p$, we get that $y =\\gamma^p(y)=y+b_p$,\n  hence $b_p=0$. \n  Therefore \n  \\[\n  K[x_1]^{G'}\\cong K[y]^{G'}= K^{G'}[y]= K'[y].\n  \\]\n  Now $H:=G\/{G'}$ acts on $K[x_1]^{{G'}}$, which is isomorphic to ${K'}[y]$ for some $y$ as we have seen,\n  and $(K[x_1]^{{G}'})^{H}=K[x_1]^G$.\n  The action is given by $\\alpha_l'=\\alpha_l\\rvert_{K'[y]}$,\n  and $\\beta_l'=\\beta_l\\rvert_{K'[y]}$.\n  For simplicity we write $\\alpha_l$ and $\\beta_l$ also for $\\alpha_l'$ and $\\beta_l'$.\n  Note that ${K}'$ is a Galois extension of $k$ and the Galois action is generated by $\\alpha_l\\rvert_{{K}'}$ and $\\beta_l\\rvert_{K'}$.\n  In order to use the induction assumption, we still have to show\n  that, maybe after some coordinate change, the $\\alpha_l$ and $\\beta_l$\n  are given as in Equation (\\ref{translation}).\n  \n\nIf $y=x_1$, there is nothing to show.\nLet $y=x_1 +\\tilde{b}x_1^p$ with $\\tilde{b}:=b^{1-p}(p-1)\\in K$.\nRecall that $\\alpha_l(x_1)=x_1+a_l$ for some $a_l\\in K$.\nAs $\\alpha_l$ commutes with $\\gamma$,\nwe get that $\\alpha_l(b)+a_l=b+ \\gamma(a_l)$.\nAs $\\gamma\\rvert_K=\\Id$ in this particular case, it follows that $\\alpha_l(b)=b$.\nAs $p-1\\in k$,\nwe get that\n$\\alpha_l(\\tilde{b})=\\alpha_l(b)^{p-1}\\alpha_l(p-1)=\\tilde{b}$,\nand hence\n\\[\n \\alpha_l(y)=\\alpha_l(x_1+\\tilde{b}x_1 ^p)=x_1+\\tilde{b}x_1^p+ a_l+\\tilde{b}a_l^p=y+{a}_l'\n\\]\nwith ${a}_l'=a_l+\\tilde{b}a_l^p\\in K$.\nNow recall that $\\beta_l(x_1)=\\mu_lx_1$ for some $q_l$-th root of unity $\\mu_l$.\nAs $\\beta_l$ and $\\gamma$ commute,\n$\\beta_l(b)=\\mu_lb$,\nthus $\\beta_l(\\tilde{b})=\\mu_l^{1-p}b^{1-p}(p-1)=\\mu_l^{1-p}\\tilde{b}$.\nHence\n\\[\n \\beta_l(y)=\\beta_l(x_1+\\tilde{b}x_1^p)=\\mu_lx_1+\\mu_l^{1-p}\\tilde{b}\\mu_l^px_1^p=\\mu_lx_1+\\mu_l\\tilde{b}x_1^p=\\mu_ly.\n\\]\nConsider now the case $y=x_1 -b'$ with $b'$ as above.\nView $\\beta_l$ and $\\gamma=\\alpha_1^{p^{r_1-1}}$ again as morphism of $K[x_1]$.\nRecall that $\\beta_l(x_1)=\\mu_lx_1$ and $\\gamma(x_1)=x_1+b$.\nBy assumption all the $\\beta_l$ and $\\gamma$ commute pairwise.\nHence\n\\[\n \\mu_lx_1+\\beta_l(b)=\\beta_l(x_1+b)=\\beta_l ( \\gamma(x_1)) =\\gamma( \\beta_l(x_1))=\\gamma (\\mu_lx_1)=\\mu_lx_1+\\mu_lb,\n\\]\nso $\\beta_l(b)=\\mu_lb$.\nNote that the $\\beta_l\\rvert_K$ are $k$-linear maps of the $k$-vector space $K$.\nAs $\\beta_l ^{q_l}=\\Id$ and all the $q_l$-th roots of unity lie in $k$ by assumption,\nthere is a basis of eigenvectors of $\\beta_l$ of $K$ over $k$.\nAs the $\\beta_l$ commute with each other,\nwe can even find a common basis of eigenvectors of all the $\\beta_l$.\nHence,\nusing that $b'\\in K$,\nwe can write $b'=\\sum b_{i}'$\nwith $b_i' \\in K$, and\n$\\beta_l(b_{i}')=\\mu_{li}b_{i}'$ for some $q_l$-th root of unity $\\mu_{li}$.\nMoreover we assume that if for all $l$ we have that $\\mu_{li}=\\mu_{lj}$, then $i=j$.\nWithout loss of generality we may assume that \n$\\mu_{l1}=\\mu_l$ for all $l$.\nAs $\\gamma(K)\\subset K$, and $b_i'\\in K$,\n$\\gamma(b_{i}')=b_{i}'+\\bar{b}_{i}$ for some $\\bar{b}_{i}\\in K$.\nUsing again that the $\\beta_l$ and $\\gamma$ commute,\nwe get that $\\mu_{li}b_{i}' +\\mu_{li}\\bar{b}_{i}=\\mu_{li}\\bar{b}_{i}'+\\beta_l(\\bar{b}_{i})$,\ni.e., $\\beta_l(\\bar{b}_{i})=\\mu_{li}\\bar{b}_{i}$ for all $l$.\nIn particular the $b_i$ which are not zero are linear independent.\nAs $\\gamma (b')=b' +b$, we get that $b=\\sum \\bar{b}_i$.\nHence for all $l$ we have\n\\[\n0 = \\beta_l(b)-\\mu_lb=\\sum \\beta_l(\\bar{b}_i) -\\sum \\mu_{l}\\bar{b}_{i}= \\sum  (\\mu_{li}\\bar{b}_{i}-\\mu_l \\bar{b}_i)=\\sum(\\mu_{li}-\\mu_{l})\\bar{b}_i\n\\]\nHence $\\bar{b}_{i}=0$ if $\\mu_{li}\\neq \\mu_l$ for at least one $l$.\nIn particular this implies that $\\tilde{b}:=\\sum_{i\\neq 1}b_i'$\nlies in $K ^{G'}=K'$.\nSet $\\tilde{y}:=y+\\tilde{b}=x_1-b_1'$.\nIt follows that\n$K'[y]=K'[\\tilde{y}]$, and\nthat $\\beta_l(\\tilde{y})=\\beta_l(x_1-b_1')=\\mu_lx_1-\\mu_lb_1'=\\mu_l\\tilde{y}$.\nMoreover $\\alpha_l(\\tilde{y})=\\tilde{y}+(a_l - \\alpha_l(b_1')+b_1')$,\nand $a_l':=a_l - \\alpha_l(b_1')+b_1'\\in K'$.\n\n So all together we may assume that\n $K[x_1]^{G'}=K'[y]$ and\n $\\alpha_l(y)=y+a_l'$ for some $a_l'\\in K'$,\n and $\\beta_l(y)=\\mu_ly$.\n Hence we can use the induction assumption for the $H$-action on $K'[y]$.\nThis proves Equation~(\\ref{claim}) for $d=1$.\n \n \\medskip\n \n Now assume that the claim holds for $d-1$.\n Look at $V=\\Spec(K[x_1,\\dots,x_d])$ with a $G$-action as in Equation (\\ref{translation}).\n Note that the inclusion map $K[x_1]\\hookrightarrow K[x_1,\\dots, x_d]$ is $G$-equivariant,\n if $G$ acts on $K[x_1]$ generated by $\\alpha_l'$ and $\\beta_l' $ such that the $\\alpha_l'\\rvert_K$ and $\\beta_l' \\rvert_K$ generate the Galois action on $K$,\n and $\\alpha_l'(x_1)=x_1+a_{l1}$ and $\\beta_l'(x_1)=\\mu_{l1}x_1$.\n To simplify notation we will use $\\alpha_l$ and $\\beta_l$ also for $\\alpha_l'$ and $\\beta_l'$.\n Set $W:=\\Spec(K[x_1])$,\ndenote by $\\varphi$ the $G$-equivariant map from $V$ to $W$,\nand let $\\varphi_G: V\/G\\to W\/G$ be the induced map between the quotients.\n\nLet $x \\in W\/G$ be any point with residue field $\\kappa_x$, and\nlet $w\\in W$ be a point with residue field $\\kappa_{w}$ in the inverse image of $x$ under the quotient map ${\\pi: W\\to W\/G}$.\nLet $G_w\\subset G$ be the stabilizer of $w$.\nWe get an induced action of $G_w$ on $V$ and $W$.\nNote that the action of $G_w$ on $V$ is generated by \n$\\tilde{\\alpha}_l =\\alpha_l^{s_l}$ and $\\tilde{\\beta}_l=\\beta_l^ {t_l}$ for some $s_l>0$ and some $t_l>0$, hence we have\n  \\[\n  \\tilde{\\alpha}_l(x_i)=x_i+\\sum_{j<i}\\tilde{a}_{lij}x_j+\\tilde{a}_{li} \\text{ and } \\tilde{\\beta}_l(x_i)=\\tilde{\\mu}_{li}x_i\n  \\]\n  for some $\\tilde{a}_{lij}\\in K$ and $\\tilde{a}_{li}\\in K$, and\nwith $\\tilde{\\mu}_{li}=\\mu_{li}^ {t_i}$.\n\nBy construction, $w$ is fixed under this action of $G_w$,\nand therefore we have that $\\varphi^{-1}(w)\\cong \\Spec(\\kappa_{w}[x_2,\\dots,x_d])\\subset V$\nis $G_w$-invariant.\nThe action is given by $\\gamma_l,\\delta_l\\in \\Aut_k(\\kappa_{w}[x_2,\\dots,x_d])$ with\n  \\[\n   \\gamma_l(x_i)=x_i+\\sum_{j<i, i\\neq 1}\\tilde{a}_{lij}x_j + (\\tilde{a}_{li1}\\bar{x}_1 + \\tilde{a}_{li})\\text{ and }\n  \\delta_l(x_i)=\\tilde{\\mu}_{li}x_i .\n  \\]\n  Here $\\bar{x}_1$ denotes the image of $x_1$ in $\\kappa_{w}$.\n  Note that $\\tilde{a}_{li1}\\bar{x}_1 + \\tilde{a}_{li}\\in \\kappa_{w}$.\n  Moreover $\\kappa_{w}$ is a Galois extension of $\\tilde{\\kappa}_x :=\\kappa_{w}^{G_w}$,\n  and $\\gamma_l\\rvert_{\\kappa_{w}}$ and $\\delta_l\\rvert_{\\kappa_{w}}$ generate the Galois action.\n  All together we can use the induction assumption and get that\n  \\[\n  [\\varphi^{-1}(w)\/G_w]=[\\Spec(\\kappa_{w}[x_2,\\dots,x_d]^{G_w})]= \\mathbb{L}^{d-1}_{\\tilde{\\kappa}_x}\\in K_0^{\\text{mod}}(\\Var_{\\tilde{\\kappa}_x}).\n  \\]\n  As $\\tilde{\\kappa}_x$ is purely inseparable over $\\kappa_{x}$ by \\cite[Capitre V.2, Th\\'eor\\`eme 2]{MR782297}, and hence there is a universal homeomorphism $f: \\mathbb{A}^{d-1}_{\\tilde{\\kappa}_x}\\to \\mathbb{A}^{d-1}_{\\kappa_x}$,\n  we get that\n$\\mathbb{L}^{d-1}_{\\tilde{\\kappa}_x}=\\mathbb{L}^{d-1}_{\\kappa_x}$ in $K_0^{\\text{mod}}(\\Var_{\\kappa_x})$.\nAs $G$ is abelian and thus $G_w \\subset G$ is a normal subgroup, we can use Lemma \\ref{lemma vezel} to get that\n \\begin{align}\\label{punten}\n[\\varphi_G^{-1}(x)]=[\\varphi^{-1}(w)\/G_w]=\\mathbb{L}_{\\kappa_x}^{d-1}\\in K_0^{\\text{mod}}(\\Var_{\\kappa_\\eta}).\n \\end{align}\nNow let $\\eta\\in W\/G$ be the generic point with residue field $\\kappa_\\eta$.\nBy Lemma \\ref{spreading out} there is an isomorphism\n\\[\n\\lim_{\\stackrel{\\longleftarrow}{\\kappa_\\eta \\in U \\subset W\/G}}\\!\\!\\!\\!\nK_0^{\\text{mod}}(\\Var_U)\\to K_0^{\\text{mod}}(\\Var_{\\kappa_\\eta}).\n\\]\nHence using Equation (\\ref{punten}) for $\\eta$ there is a nonempty open $U\\subset W\/G$ such that\n\\[\n[ \\varphi_G^ {-1}(U)]=\\mathbb{L}^ {d-1}_{U}\\in K_0^{\\text{mod}}(\\Var_{U}).\n\\]\nAs the separated map $U\\to \\Spec(k)$ induces a forgetful map from $K_0^{\\text{mod}}(\\Var_U)$ to $K_0^{\\text{mod}}(\\Var_{k})$,\nwe have that $[\\varphi_g^ {-1}(U)]=\\mathbb{L}^ {d-1}_{k}[U]\\in K_0^{\\text{mod}}(\\Var_{k})$.\nNote that $W\/G\\setminus U$ consist of finitely many points $P_i$.\nWe already know that $[\\varphi_G^{-1}(P_i)]=\\mathbb{L}^d_k[P_i]$ in $K_0^{\\text{mod}}(\\Var_k)$,\nsee Equation (\\ref{punten}).\nUsing the scissors relation in $K_0^{\\text{mod}}(\\Var_{S\/G})$,\nwe get that \n\\begin{align*}\nV\/G& =[\\varphi_G^{-1}(W\/G)]=[\\varphi_G^{-1}(U)]+\\sum [\\varphi_G^{-1}(P_i)]\\\\\n& =\\mathbb{L}_k^{d-1}[U]+\\sum \\mathbb{L}_k^{d-1}[P_i]\n=\\mathbb{L}^ {d-1}_{k}[W\/G]=\\mathbb{L}_k^{d-1}\n\\mathbb{L}^{}_k=\\mathbb{L}^d_k\\in K_0^{\\text{mod}}(\\Var_{k}). \n\\end{align*}\nHere we used the induction assumption again to get that $[W\/G]=\\mathbb{L}_k\\in K_0^{\\text{mod}}(\\Var_k)$.\n Claim (\\ref{claim}) now follows by induction.\n \\end{proof}\n \n \\begin{rem}\n  Consider $V=X=\\Spec(k[x,y])$ from Example \\ref{exinsep}.\n  Then we get a $G$-equivariant map $\\varphi: V\\to W=\\Spec(k[y])$,\n  where we consider the trivial action on $W$,\n  and hence a map $\\varphi_g: V\/G\\to W\/G=W$.\n  We have already shown that the induced map between\n  \\[\n  \\varphi^{-1}(0)\/G=\\Spec(k[x])\\to \\varphi_G^{-1}(0)=\\Spec(k[x^p+(p-1)xy^{p-1}])\n  \\]\n  is purely inseparable,\n  but $\\varphi^{-1}(0)\/G$ and $ \\varphi_G^{-1}(0)$ are isomorphic over $k$,\n  hence they have the same class in $K_0(\\Var_k)$.\n  This suggests that Proposition \\ref{lemwildandtame} maybe also holds in the usual Grothendieck ring,\n  at least if we assume that $k$ is a finite field.\n  But as the isomorphism between the fibers is not given in a canonical way,\n  I do not know how to do this in general, and whether it is possible at all.\n  In the case that $\\Dim_KV=1$, it follows from the proof of Proposition \\ref{lemwildandtame}.\n \\end{rem}\n \n \\begin{rem}\\label{exab}\n In the proof of Proposition \\ref{lemwildandtame} we used the fact that $G$ is abelian\n to get a $G$-invariant sub-vector space $W$ of $V$.\n In fact this assumption is necessary.\nAs was already observed in \\cite{MR2642161},\nit follows from \\cite[Proposition 3.1, ii]{Eke} and \\cite[Corollary 5.2]{Eke}\nthat there exists an $n$-dimensional $\\mathbb{C}$-vector space $V$ and a finite group $G\\subset GL(V)$\nsuch that \n\\[\n\\lim_{m\\to\\infty}\\ \\ [V^m\/G]\/\\mathbb{L}^{mn}\\neq 1 \\in \\hat{\\mathcal{M}}_{\\mathbb{C}},\n\\]\nwhere $\\hat{\\mathcal{M}}_{\\mathbb{C}}$ is the completion of $\\mathcal{M}_{\\mathbb{C}}$ by the dimension fibration.\nSo in particular there exists an $m$ large enough such that $\\mathbb{L}^{mn}\\neq [V^ m\/G]\\in K_0(\\Var_{\\mathbb{C}})$.\n\nIn fact there are explicit $V$ and $G$ for which this holds, which can be found in \\cite{MR751131}. \nThe group in Saltman's example is a $p$-group, hence in particular solvable.\nHence Proposition \\ref{lemeh} will not hold in the case of solvable groups. \n \\end{rem}\n \n \\begin{rem}\\label{exHE} \n If one does not assume that $k$ has enough roots of unity,\n  Proposition~\\ref{lemeh} does not hold.\n There is namely\n a concrete counterexample, see \\cite[Example~1.2]{MR2642161},\n of a $K=\\mathbb{Q}(\\sqrt{-1})$-vector space $V$ with $\\mathbb{Q}$-linear\n $\\mathbb{Z}\/4\\mathbb{Z}$-action lifting the action of the Galois group of $K$ over $\\mathbb{Q}$,\n such that $[V\/G]$ is not equal to $\\mathbb{L}^{\\Dim_KV}$ in\n $K_0(\\Var_{\\mathbb{Q}})$.\n \\end{rem}\n \n\n \n \n \\section{Quotients of equivariant affine bundles}\n \\label{quotients of affine bundles}\n \\noindent\nNow we use the result from the previous section to compute the class of the quotient of an\nequivariant affine bundle by an affine action\nin a (modified) Grothendieck ring of varieties.\nTake $S$ with an action of $G$ as before.\nFor a point $s\\in S\/G$,\ndenote by $F_s$ its residue field,\nand by $G_s\\subset G$ the stabilizer of a\npoint $s'\\in S$\nlying in the inverse image of $s$ under the quotient map $S\\to S\/G$.\nA priory, $G_s$ depends on the choice of $s'$.\nBut as all stabilizers of an orbit are conjugated,\n$\\lvert G_s\\rvert$ does not depend on $s'$.\n \n \\begin{lem}\\label{lemma}\n Let $G$ be a finite abelian group,\nand let $\\varphi:V\\rightarrow B$ be\na $G$-equivariant affine bundle of rank $d$ with affine $G$-action in the category $(\\Sch_{S,G})$.\nAssume that the residue field $F_s$ of every point $s \\in  S\/G$\ncontains all $\\lvert G_s \\rvert$-th roots of unity.\nThen\n\\[\n[V\/G]=\\mathbb{L}_{S\/G}^d[B\/G]\\in K_0^{\\text{mod}}(\\Var_{S\/G}).\n\\]\nIf the action of $G$ on $S$ is tame, we get\n\\[\n[V\/G]=\\mathbb{L}_{S\/G}^d[B\/G]\\in K_0(\\Var_{S\/G}).\n\\]\n \\end{lem}\n \n \\begin{proof}\nLet $\\varphi:V\\to B$  be as in the claim, and let $\\varphi_G:V\/G\\to B\/G$ be the induced map between the quotients.\n Let $x\\in B\/G$ be a point with residue field $k$.\n As $B\/G$ is an $S\/G$-scheme, $x$ lies over a point $s\\in S\/G$,\n and $k$ contains the residue field\n$F_s$ of $s$.\n Let $b\\in B$ be a point mapped to $x$ under the quotient map,\n let $K$ be the residue field of $b$ and let $G_b$ be the stabilizer of $b$ under the action of $G$.\n Without loss of generality we may assume that $b$ is mapped to $s'$ under\nthe structure map $B\\to S$.\nHence as this map is $G$-equivariant,\n$G_s$ is a subgroup of $G_b$.\nAs $G$ is abelian, $G_b\\subset G$ is a normal subgroup, and hence\n Lemma~\\ref{lemma vezel} implies that\n\\[\n [\\varphi_G^{-1}(x)]=[\\varphi^{-1}(b)\/G_b]\\in K_0^{\\text{mod}}(\\Var_k).\n\\]\nIf $G$ acts tamely on $S$,\nwe get this equation also in $K_0(\\Var_k)$, see Remark \\ref{rem fiber tame}.\n\nAs $\\varphi:V\\to B$ is a $G$-equivariant affine bundle of rank $d$ with affine $G$-action,\nand $G_b$ is a subgroup of $G$,\nthe induced action of $G_b$ makes it into a $G_b$-equivariant affine bundle of rank $d$ with affine $G_b$-action.\nAs $G_b$ is the stabilizer of $b$, $b$ is fixed under the action of $G_b$.\nHence\nby Remark \\ref{Vb translation},\n$V_b:=\\varphi^{-1}(b)$ is a $K$-vector space of rank $d$,\nand the $G_b$-action on $V$ restricts to $V_b$.\nIf the characteristic of $k$ and hence of $F_s$ is $p$,\nthis action is generated by automorphisms\n$A_l,B_l\\in \\Aut(V_b)$ such that the order of the $A_l$ is a power of $p$\nand the order of the $B_l$ divides $\\lvert G_b\\rvert$ and is prime to $p$.\nAs $G_b$ is a subgroup of $G_s$,\nthis means in particular that the order of $B_l$ divides $\\lvert G_s\\rvert$.\nIf the characteristic of $k$ is zero,\nwe may assume that the action is only generated by $B_l$s as above.\nAlso from Remark~\\ref{Vb translation}\nwe get that for all $v\\in V_b$ we have that\n$A_l(v)=A_l'(v)+ u_l$  and $B_l(v)=B_l'(v)+w_l$ for some $u_l,w_l \\in V_b$, \nsuch that $A_l',B_l'$ are quasi-linear maps of $V_b$.\nAs $G_b\\subset G$ is abelian, we may assume by Remark \\ref{tame translation} that $w_l=0$.\n\nNote that $\\tilde{k}:=K^{G_b}$\nis a field extension of $k$ and hence of $F_s$.\nBy Remark \\ref{Vb},\nthe $A_l'$ and the $B_l'$ define a $\\tilde{k}$-linear action on a $K$-vector space\n$V_b$\nlifting the Galois action of $\\Gal(K,\\tilde{k})$ on K.\nBy assumption $F_s$ contains all $\\lvert G_s\\rvert$-th roots of unity, hence the same holds for $\\tilde{k}$.\nSo using this and that $G_b$ is abelian, we can find as in the proof of Proposition \\ref{lemwildandtame} a $K$-basis\nsuch that the $A_l'$ have upper triangle form with only $1$ on the diagonal,\nand the $B_l'$ are diagonal with $q_s$-th roots of unity as eigenvalues.\nAll together we may assume that $V_b\\cong \\Spec(K[x_1, \\dots, x_d])$ as schemes, and that\nthe $G$-action on $V_b$ is given by\n$\\alpha_l,\\beta_l \\in \\Aut(K[x_1,\\dots,x_d])$ \nsuch that the $\\alpha_l\\rvert_K$ and $\\beta_l\\rvert_K$ generate the $\\Gal(K,\\tilde{k})$-action on $K$, and\n\\[\n  \\alpha_l(x_i)=x_i+\\sum_{j<i}a_{lij}x_j+a_{li} \\text { and } \\beta_l(x_i)=\\mu_{li}x_i\n\\]\nfor some $a_{lij},a_{l_i}\\in K $ and some $q$-th roots of unity $\\mu_{li}$.\nBut this is exactly the setting\nfor which we showed Proposition \\ref{lemwildandtame}, see Equation (\\ref{translation}).\nHence we get that\n\\begin{equation*}\n [\\varphi^{-1}(b)\/G_b]=[V_b\/G_b]=\\mathbb{L}_{\\tilde{k}}^d\\in K_0^{\\text{mod}}(\\Var_{\\tilde{k}}).\n\\end{equation*}\nAs $G_b$ is the stabilizer of $b\\in B$ under the action of $G$,\nand $x$ is the image of $b$ under $\\pi$,\nby \\cite[Capitre V.2, Th\\'eor\\`eme 2]{MR782297}\n$\\tilde{k}$ is a purely inseparable extension of $k$.\nHence\n$\\mathbb{L}_{\\tilde{k}}=\\mathbb{L}_k\\in K_0^{\\text{mod}}(\\Var_k)$.\nPutting everything together we get for every point $x\\in B\/G$ with residue field $k$ that\n\\begin{align}\\label{punten2}\n[\\varphi_G^ {-1}(x)]=\\mathbb{L}_k^d\\in K_0^{\\text{mod}}(\\Var_{k}).\n\\end{align}\nIf the action of $G$ on $S$ is tame, we have that the characteristic of $F_s$ is prime to the order of $G_s$, and hence the characteristic of $k$\nis prime to the order of $G_b$.\nThus by \\cite[Capitre V.2, Proposition 5 and Corollaire]{MR782297}\n$\\tilde{k}=k$.\nAs the $A_l$ are trivial in this case, the $G$-action on $V_b$ is actually quasi-linear,\nand hence by Proposition~\\ref{lemeh}\nwe get that\n\\begin{align} \\label{punten2a}\n[\\varphi_G^ {-1}(x)]=[\\varphi^{-1}(b)\/G_b]=\\mathbb{L}_k^d\\in K_0(\\Var_{k}).\n\\end{align}\nNote that, without loss of generality, we may assume that $B\/G$ is integral in the claim.\nThis is true,\nbecause we can decompose $B\/G$ in irreducible schemes using the scissors relation.\nThis relation also implies that we can consider the underlying reduced scheme structure.\nLet $\\eta\\in B\/G$ be the generic point with residue field $\\kappa_\\eta$.\nBy Lemma \\ref{spreading out} there is an isomorphism\n\\[\n\\lim_{\\stackrel{\\longleftarrow}{\\kappa_\\eta \\in U \\subset B\/G}}\\!\\!\\!\\!\nK_0^{\\text{mod}}(\\Var_U)\\to K_0^{\\text{mod}}(\\Var_{\\kappa_\\eta}).\n\\]\nThis implies using Equation (\\ref{punten2}) for $\\eta$ that we can find an open $U\\subset B\/G$ such that\n\\[\n[ \\varphi_G^ {-1}(U)]=\\mathbb{L}^ d_{U}\\in K_0^{\\text{mod}}(\\Var_{U}).\n\\]\nAs the separated map $U\\to S\/G$ induces a forgetful map from $K_0^{\\text{mod}}(\\Var_U)$ to $K_0^{\\text{mod}}(\\Var_{S\/G})$,\nwe get that $[\\varphi_G^ {-1}(U)]=\\mathbb{L}^ d_{S\/G}[U]\\in K_0^{\\text{mod}}(\\Var_{S\/G})$.\nProceed with a generic point of $B\/G\\setminus U$.\nBy Noetherian induction we get, using the scissors relation in the modified Grothendieck ring,\nthat \n\\[\n[V\/G]=\\mathbb{L}^ d_{S\/G}[{B\/G}]\\in K_0^{\\text{mod}}(\\Var_{S\/G}).\n\\]\nIf the action of $G$ on $S$ is tame,\nwe can use that by \\cite[Proposition 2.9]{MR2770561}\n\\[\n\\lim_{\\stackrel{\\longleftarrow}{\\kappa_\\eta \\in U \\subset B\/G}}\\!\\!\\!\\!\nK_0(\\Var_U)\\to K_0(\\Var_{\\kappa_\\eta}).\n\\]\nis an isomorphism, and Equation (\\ref{punten2a})\nto conclude with an analog argument as above that\n\\[\n[V\/G]=\\mathbb{L}^ d_{S\/G}[{B\/G}]\\in K_0(\\Var_{S\/G}).\n\\]\n\\end{proof}\n\n\\begin{ex}\\label{extrivial}\n Take a $B\\in (\\Sch_{S,G})$,\n and let $\\mathbb{A}^d_B\\to B$ be\nthe trivial vector bundle with $G$-action induced  by the action on $B$.\n\nAs $\\mathbb{A}_{S\/G}^d$ is flat over $S\/G$,\nit also follows directly from  \\cite[Expos\\'e V, Proposition~1.9]{MR0238860} that\n$\\mathbb{A}^d_B\/G\\cong (\\ensuremath{\\mathbb{A}}^d_{S\/G}\\times_{S\/G}B)\/G \\cong \\ensuremath{\\mathbb{A}}^d_{S\/G}\\times_{S\/G} B\/G$,\nthus\n\\[\n[\\mathbb{A}^d_B\/G]=\\mathbb{L}^d_{S\/G}[{B\/G}]\\in K_0(\\Var_{S\/G}).\n\\]\n\\end{ex}\n\n\n \n\\section{The quotient map on the equivariant Grothendieck ring of varieties}\n\\label{quotient map}\n\n\\noindent\nIn \\cite[Lemma 3.2]{MR2106970}\nit was shown that, if $G$ acts freely on a\nvariety $S$ of characteristic zero,\ntaking the quotient defines a map from $K_0^G(\\Var_S)$ to $K_0(\\Var_{S\/G})$.\nWe will now show that such a map exists for more general $S$ and $G$.\nTo make sure that the quotient $X\/G$ of a separated $S$-scheme of finite type $X$\nwith good $G$-action is a separated $S\/G$-scheme of finite type,\nwe assume, as before, that $S\/G$ is locally Noetherian and separated,\nand that the quotient map $\\pi_S:S \\to S\/G$ is finite.\nFor a point $s\\in S\/G$, we denote again by $F_s$ the residue field of $s$ and by $ G_s\\subset G$\nthe stabilizer of a point $s'\\in \\pi ^{-1}(s)\\subset S$.\n\nProving that the quotient map is well defined,\nthe main problem is to show that the second relation in Definition \\ref{equivariant Gring} does not cause troubles,\nhence we have to control quotients of $G$-equivariant affine bundles by affine $G$-actions.\nIn the proof of \\cite[Lemma 3.2]{MR2106970} it is shown that, if the action of $G$ on $S$ is free,\nthe quotient of such a bundle is again an affine bundle.\nWithout the freeness assumption the quotient can be even singular,\nhence in particular it is no an affine bundle in general.\nBut we have seen in Lemma \\ref{lemma}  that in the modified Grothendieck ring,\nor in the usual Grothendieck ring in the case of tame actions,\nthe class of the quotient of a\n$G$-equivariant affine bundle only depends on its rank and its base.\nThis enables us to show the following theorem.\n \n \\begin{thm}\\label{quotientmap}\n Let $G$ be a finite abelian group.\n Assume that the residue field $F_s$ of any point $s\\in S\/G$ contains all $\\lvert G_s\\rvert $-th roots of unity.\n Then there is a well defined group homomorphism\n \\[\n K^G_0(\\Var_S)\\to K_0^{\\text{mod}}(\\Var_{S\/G})\n \\]\n sending $[X]\\in K_0^G(\\Var_S)$ to $[X\/G]\\in K_0^{\\text{mod}}(\\Var_{S\/G})$ for every $X\\in (\\Sch_{S,G})$.\n If the $G$-action on $S$ is tame,\n it factors through a group homomorphism\n  \\[\n K^G_0(\\Var_S)\\to K_0(\\Var_{S\/G}).\n \\]\n \\end{thm}\n\n \\begin{proof} \n For all $X\\in (\\Sch_{S,G})$,\n the quotient $X\/G$ exists by \\cite[Expos\\'e V, Corollaire 1.4]{MR0238860},\n because the $G$-action on $X$ is assumed to be good.\n As the structure map $X\\to S$ is $G$-equivariant,\n $X\/G$ is an $S\/G$-scheme.\n As $S\/G$ is locally Noetherian and $S\\to S\/G$ is finite and hence of finite type,\n by \\cite[Corollaire 1.5]{MR0238860},\n $X\/G$ is of finite type over $S\/G$.\n Moreover, the fact that $X$ is separated over $S\/G$ implies\n the same for $X\/G$.\n Hence we get a well defined map from $(\\Sch_{S,G})$ to $K_0^{\\text{mod}}(\\Var_{S\/G})$ and $K_0(\\Var_{S\/G})$, respectively.\nSo in order to show the proposition, we need to show that this map factors through $K^ G_0(\\Var_S)$.\nHence we need to show that Relation (1) and Relation (2) from Definition \\ref{equivariant Gring} still hold after taking the quotient.\n\nTake any $X\\in (\\Sch_{S,G})$ and a closed subscheme $Y$ of $X$,\nclosed with respect to the action of $G$.\nSet $U:=X\\setminus Y$.\nDenote by $\\pi:X\\to X\/G$ the quotient map.\nAs $U \\subset X$ is open and $G$-invariant,\nby \\cite[Expos\\'e V, Corollaire 1.4]{MR0238860}, $\\pi(U)\\cong U\/G$ and open in $X\/G$.\nBy Lemma \\ref{closed mod}, $[\\pi(Y)]=[Y\/G]\\in K_0^{\\text{mod}}(\\Var_{S\/G})$.\nHence we get, using the scissors relation in $K_0^{\\text{mod}}(\\Var_{S\/G})$, that\n\\begin{equation}\\label{eqrel1}\n  [X\/G]  \n=[\\pi(Y)]+[\\pi(U)]\n =[Y\/G]+[U\/G] \\in K_0^{\\text{mod}}(\\Var_{S\/G}).\n\\end{equation}\nIf the action of $G$ on $S$, and hence also the action of $G$ on $X$ is tame,\nthen by Lemma \\ref{rem tame decomposition} we get Equation~(\\ref{eqrel1}) also in $K_0(\\Var_{S\/G})$.\n\nIt remains to show that Relation (2) still holds after taking the quotient.\nTake any $B\\in (\\Sch_{S,G})$, and\nlet $V\\rightarrow B$ and $W\\to B$\nbe $G$-equivariant affine bundle of rank $d$ with affine $G$-action.\nBy Lemma \\ref{lemma} we have that\n\\begin{equation}\\label{eqrel2}\n [V\/G]=\\mathbb{L}^d_{S\/G}[{B\/G}]=[W\/G]\\in K_0^{\\text{mod}}(\\Var_{S\/G}),\n\\end{equation}\n and if the action of $G$ on $S$ is tame,\n Equation (\\ref{eqrel2}) holds in $K_0(\\Var_S)$.\nAltogether, taking the quotients gives us well defined maps as in the claim.\n\\end{proof}\n \n \\begin{rem}\n By Remark \\ref{rem decomposing}\n we only get a well defined quotient map with values in the modified Grothendieck\n ring of varieties in the case of a wild action on $S$,\neven if we can show Lemma \\ref{lemma}\n in the usual Grothendieck ring of varieties.\n\\end{rem}\n\n \n\\begin{rem} \\label{necessary}\nLet $S=\\Spec(k)$ with trivial $G$-action for some group $G$.\nAssume that there exists a finite Galois extension $K\/k$ and a $d$-dimensional $K$-vector space\n$V$ with a $k$-linear $G$-action lifting the Galois action,\nsuch that $[V\/G]\\neq \\mathbb{L}_k^{d}$ in $K_0^{\\text{mod}}(\\Var_k)$.\nNote that $V\\to \\Spec(K)$ is an affine bundle of rank $d$ with affine $G$-action in the category $(\\Sch_{S,G})$,\nand the same holds for $\\mathbb{A}^ {d}_K\\to \\Spec(K)$\nwith a $G$-action induced by the Galois action on $K$.\nWe have $\\mathbb{A}^d_K\/G\\cong\\mathbb{A}^d_k$, see Example \\ref{extrivial}.\nBy definition $[V]=[\\mathbb{A}^{d}_K]\\in K_0^ G(\\Var_k)$,\nbut $[V\/G]\\neq[\\mathbb{A}^{d}_K\/G]\\in K_0^ G(\\Var_k)$,\nso there cannot be a well defined map from $K_0^ G(\\Var_k)$ to $K_0^{\\text{mod}}(\\Var_k)$ as in\nTheorem~\\ref{quotientmap}.\nHence Remark \\ref{exab} and Remark \\ref{exHE} show that Theorem \\ref{quotientmap} in general\ndoes not hold without the assumption on the residue field $F_s$ for all $s\\in S\/G$ or for a non-abelian group $G$.\n\\end{rem}\n\n\n \n  \\begin{cor} \\label{corM}\nNotation and assumptions as in Theorem \\ref{quotientmap}.\n Then there is a well defined group homomorphism\n \\[\n  \\mathcal{M}_S^G\\to \\mathcal{M}^{\\text{mod}}_{S\/G}\n \\]\nsending $\\mathbb{L}_S^{-s}[X]$ to  $\\mathbb{L}_{S\/G}^{-s}[X\/{G}]$\nfor all $X\\in (\\Sch_{S,G})$.\nIf the action of $G$ on $S$ is tame,\nwe also get a well defined group homomorphism\n$ \\mathcal{M}_S^G\\to \\mathcal{M}_{S\/G}$\n with this property.\n \\end{cor}\n \n \\begin{proof}\nUsing Theorem \\ref{quotientmap} it suffices to show that for all $j\\in \\mathbb{Z}$ \n\\begin{equation}\\label{xmall}\n \\mathbb{L}^{-j}_{S\/G}[X\/G]=\\mathbb{L}_{S\/G}^{-(j+1)}[(\\mathbb{A}^1_{S}\\times_S X)\/G] \\in \\mathcal{M}_{S\/G}\n\\end{equation}\nand hence in $\\mathcal{M}^{\\text{mod}}_{S\/G}$\nfor all $X\\in (\\Sch_{S,G})$.\nNote that the fiber product of $X$ and $\\mathbb{A}^1_S$ is taken in the category $(\\Sch_{S,G})$.\nNote that   \n  $\\mathbb{A}^1_{S}\\times_SX =\\mathbb{A}^1_{S\/G}\\times_{S\/G}X $,\n  and the action on $\\mathbb{A}^ 1_{S\/G}$ is trivial.\n   As $\\mathbb{A}^1_{S\/G}$ is flat over $S\/G$,\n   \\cite[Proposition 1.9]{MR0238860} implies that \n   $(\\mathbb{A}^1_{S\/G}\\times_{S\/G}X)\/G=\\mathbb{A}^1_{S\/G}\\times_{S\/G}X\/G$.\n   Hence Equation (\\ref{xmall}) holds.\n\\end{proof}\n\n\\noindent\n  Consider now a profinite group\n  \\[\n  \\hat{G}=\\lim_{\\stackrel{\\longleftarrow}{i\\in I}} G_i.\n  \\]\n Let $S$ be a separated scheme with a $\\hat{G}$-action factorizing through a good action of a finite quotient $G_j$ of $\\hat{G}$.\n Assume that the quotient $S\/\\hat{G}=S\/G_j$ is a separated, locally Noetherian scheme,\n and that the quotient map $\\pi_S: S\\to S\/\\hat{G}$ is finite.\n For all $s\\in S$ and $i\\geq j$, set $q_{si}:=\\lvert G_{si}\\rvert $,\n with $G_{si}\\subset G_i$ the stabilizer of a point $s'\\in \\pi_s^{-1}(s)\\subset S$\n under the action of the $G_i$.\n \n \n \\begin{cor}\\label{corprofinite}\n Let $\\hat{G}$ be a profinite abelian group.\n Assume that the residue field $F_S$ of any point $s\\in S\/\\hat{G}$ contains all $q_{si}$-th roots of unity for all $i\\geq j$.\n Then there are well defined group homomorphisms\n \\[\n  K_0^{\\hat{G}}(\\Var_S)\\to K_0^{\\text{mod}}(\\Var_{S\/\\hat{G}})\\text{ and } \\mathcal{M}_S^{\\hat{G}} \\to \\mathcal{M}^{\\text{mod}}_{S\/\\hat{G}}\n \\]\nsending the class of a separated scheme $X$\nto the class of its quotient $X\/\\hat{G}$.\nIf the action of $\\hat{G}$ on $S$ is tame, we get well defined group homomorphisms\n \\[\n  K_0^{\\hat{G}}(\\Var_S)\\to K_0(\\Var_{S\/\\hat{G}})\\text{ and } \\mathcal{M}_S^{\\hat{G}} \\to \\mathcal{M}_{S\/\\hat{G}}\n \\]\n with this property.\n \\end{cor}\n \n \\begin{proof}\n By assumption $S\/\\hat{G}=S\/G_j=S\/G_i$ for all $i\\geq j$.\n Hence\nby Theorem~\\ref{quotientmap}\nthere is a well defined map $K_0^ {G_i}(\\Var_S)\\to K_0^{\\text{mod}}(\\Var_{S\/G_i})=K_0^{\\text{mod}}(\\Var_{S\/\\hat{G}})$ sending $X\\in (\\Sch_{S,G_i})$ to its quotient\n$X\/G_i$ for all $i\\geq j$.\nUsing this map we get the following commutative diagram:\n\\[\n \\xymatrix{\n\\dots \\ar[r] & K_0^ {G_i}(\\Var_S) \\ar[r] \\ar[rd] & K_0^ {G_{i+1}}(\\Var_S)\\ar[r] \\ar[d] & \\dots  \\\\\n&  & K_0^{\\text{mod}}(\\Var_{S\/\\hat{G}})  & \n }\n\\]\nHere the maps in the first line are given as in Remark \\ref{remark}.\nHence we get an induced map\n\\[ \\lim_{\\stackrel{\\longrightarrow}{i\\in I}}K_0^{G_i}(\\Var_S) \\to\n   K_0^{\\text{mod}}(\\Var_{S\/\\hat{G}})\n\\]\nwith the required property.\nThe tame case works analog.\nThe statement for $\\mathcal{M}_S^{\\hat{G}}$\nfollow with a similar argument from Corollary \\ref{corM}.\n \\end{proof}\n \n  \n\\begin{rem}\\label{K0-linear}\n Let $k$ be a field with trivial tame $G$-action.\n Then we can view $K_0^G(\\Var_k)$ as a module over $K_0(\\Var_k)$\n by mapping the class of a $k$-scheme of finite type $X$ in $K_0(\\Var_k)$ to the class of $X$ with trivial $G$-action.\n It is clear that the quotient map is trivial on the image of $K_0(\\Var_k)$ in $K_0^G(\\Var_k)$.\n As every $k$-scheme of finite type is flat over the field $k$,\n it follows from \\cite[Proposition 1.9]{MR0238860}\n that the quotient map maps the class of $X\\times_k V$ in $K_0^G(\\Var_k)$ to $[X]\\cdot[V\/G]\\in K_0(\\Var_k)$.\n Hence it follows that the quotient map\n \\[\n  K_0^G(\\Var_k)\\to K_0(\\Var_k)\n \\]\nis $K_0(\\Var_k)$-linear.\nSimilarly we get that the quotient map $\\mathcal{M}_k^G\\to \\mathcal{M}_k$ is $\\mathcal{M}_k$-linear.\nMoreover we get the analog statements for a profinite group $\\hat{G}$.\n\\end{rem}\n\n \n \n \\section{The quotient of the motivic nearby fiber}\n \\label{an application}\n\n \\noindent\nThroughout this section, \nif not mentioned otherwise,\nlet $k$ be a field of characteristic zero\ncontaining all roots of unity,\nlet $X$ be an irreducible algebraic variety over $k$, and\nlet $f:X\\to \\mathbb{A}_k^1$ be a non-constant morphism.\nWe denote by $X_0\\subset X$ the zero locus of $f$ in $X$,\nand assume that $X_\\eta:=X \\times_{\\mathbb{A}_k^1}\\mathbb{A}_k^1\\setminus \\{0\\}=X\\setminus X_0$ is smooth.\n\n\\medskip\n\nWe are now going to use the results from the previous section\nto show that the quotient of the motivic nearby fiber is a well defined invariant with values in $\\mathcal{M}_{X_0}$.\nThe motivic nearby fiber can be attached to a map $f: X\\to \\ensuremath{\\mathbb{A}}^1_k$ as above.\nIt was constructed in \\cite{DL3}\nas a limit of the motivic Zeta function,\nand was investigated in more details in \\cite{MR2106970}.\nMoreover we show that modulo $\\mathbb{L}$, this quotient is equal to motivic reduction $R(f)$ in the image of $K_0(\\Var_{X_0})$ in $\\mathcal{M}_k$,\nsee Proposition~\\ref{application}.\nHere $R(f)$ is the class of the special fiber of a smooth modification of $f$\nin  $K_{0}(\\Var_{X_0})\/\\mathbb{L}$.\nA \\emph{smooth modification} of $f: X\\to \\mathbb{A}^1_k$ is \na smooth irreducible algebraic variety $Y$ over $k$,\ntogether with a proper morphism \n$h:Y\\to X$ such that the restriction \n$h: Y\\setminus h^ {-1}(X_0)\\to X\\setminus X_0$ is an isomorphism.\nDue to weak factorization, the definition of the motivic reduction\ndoes not depend on the choice of such a modification,\nsee Lemma \\ref{welldef R(f)}.\n\n\\medskip\n\nThis result implies the following: if $f: X\\to \\mathbb{A}_k^1$ is proper\nand $X$ is smooth, and the generic fiber $X_\\eta$ of $f$ is equal to $1$ modulo $\\mathbb{L}$ in $K_0(\\Var_{\\ensuremath{\\mathbb{A}}_k^1\\setminus \\{0\\}})$,\nthen the same holds for the special fiber of $f$ in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$,\nsee Corollary \\ref{1modL}.\nThis can be seen as a motivic analog of the main theorem in \\cite[Theorem 1.1]{MR2247971},\nwhich says that\nif $V$ is an absolutely irreducible smooth projective variety\nover a local field $K$ with finite residue field $F$ such that the $m$-th \\'etale cohomology\nof $V\\times_K \\bar{K}$, $\\bar{K}$ the algebraic closure of $K$, has\nconiveau $1$\nfor all $m\\geq1$,\nthen the number of rational points of the special fiber of any\nregular projective model of $V$ is congruent $1$ modulo $\\lvert F \\rvert$.\n\nHow does this analogy work?\nIn both cases one deduces a property of the special fiber from a property of the generic fiber of some proper and smooth scheme.\nWe are now going to outline the connection between the properties on the generic fibers and the properties on the special fibers, respectively.\n\nAs remarked in \\cite{MR2247971}, if the characteristic of $K$\nis equal to zero,\nthen the fact that the \\'etale cohomology of $V$ has coniveau $1$\nimplies that the Hodge type of the de Rham cohomology\nis $\\geq 1$,\nor equivalently that $H^q(V,\\mathcal{O}_V)=0$ for all $q\\geq 1$.\nNow\nconsider the Hodge-Deligne polynomial $HD: K_0(\\Var_K)\\to \\mathbb{Z}[u,v]$, see \\cite[Example~4.1.6]{MR2885336}.\nThis is a ring morphism sending\nthe class of a projective and smooth $K$-variety $X$ to $\\sum_{p,q} (-1)^{p+q} \\Dim_K(H^q(X,\\Omega_X^p))u^pv^q$.\nWe have that \n\\[\nHD(\\mathbb{L})=HD([\\mathbb{P}_K^1])-HD(1)=1+uv-1=uv.\n\\]\nHence, if the class of $V$ is equal to $1$ modulo $\\mathbb{L}$ in $K_0(\\Var_K)$,\n $HD(V)=1$ modulo $uv$, hence in particular $H^q(V,\\mathcal{O}_V)=0$ for $q\\geq 1$, so\nthe de Rham cohomology of $V$ has Hodge type $\\geq 1$.\n\nNote furthermore that if we have a finite field $F$,\nand the class of a variety $V_F$ over $F$ in $K_0(\\Var_F)$\nis equal to $1$ modulo $\\mathbb{L}$,\nthen $\\#(V_F(F))=1$ modulo $\\lvert F \\rvert$.\n\n\\subsubsection*{The Motivic nearby fiber}\nRecall the following notation from \\cite{DL3}.\nLet ${h: Y \\to X}$ be an \\emph{embedded resolution} of $f$, \ni.e., $h$ is a proper morphism inducing an isomorphism $Y\\setminus h^{-1}(X_0)\\to X\\setminus X_0$, $Y$ is smooth, and $h^ {-1}(X_0)$ is a simple normal crossing divisor.\nSuch an embedded resolution always exists due to resolution of singularities in characteristic zero.\nDenote by $E_i$, $i\\in J$, the irreducible components of $h^{-1}(X_0)$.\nFor each $i\\in J$, denote by $N_i$ the multiplicity of $E_i$ in the divisor $f\\circ h$\non $Y$.\nFor $I\\subset J$,\nwe consider non-singular varieties \n\\[\nE_I=\\bigcap _{i\\in I}E_i, \\text{ and } E_I^o:=E_I\\setminus \\bigcup_{j\\in J\\setminus I} E_j.\n\\]\nNote that $\\bigcup_{\\emptyset \\neq I\\subset J}E_I^o=h^{-1}(X_0)$.\n\nLet $\\mu_n\\subset \\mathbb{C}$ be the group of $n$-th roots of unity for all $n\\in \\mathbb{N}.$ \nSet ${m_I:=\\text{gcd}(N_i)_{i\\in I}}$.\nWe introduce an unramified Galois cover $\\tilde{E}_I^o$ of $E_i^o$ with Galois group \n$\\mu_{m_I}$ as follows.\nLet $U$ be an affine Zariski open subset of $Y$,\nsuch that, on $U$, $f\\circ h=uv^{m_i}$,\nwith $u$ a unit on $U$ and $v$ a morphism from $U$ to $\\mathbb{A}_k^1$.\nThen the restriction of $\\tilde{E}_I^o$ above $E_I^o\\cap U$,\ndenoted by $\\tilde{E}_I^o\\cap U$, is defined as \n\\[\n \\{(z,y)\\in \\mathbb{A}_k^1\\times(E_I^o\\cap U)\\mid z^{m_I}=u^{-1}\\}.\n\\]\nNote that $E_I^o$ can be covered by such affine open subset $U$ of $Y$.\nGluing together the $\\tilde{E}_I^o\\cap U$, in the obvious way,\nwe obtain the cover $\\tilde{E}_I^o$ of $E_I^o$ which has a natural $\\mu_{m_I}$-action\n(obtained by multiplying the $z$-coordinate with the elements of $\\mu_{m_I}$).\nThis $\\mu_{m_I}$-action on $\\tilde{E}_I^o$ induces a $\\hat{\\mu}=\\lim \\mu_i$-action on \n$\\tilde{E}_I^o$ in the obvious way.\nNote that by construction $\\tilde{E}_I^o\/\\hat{\\mu}\\cong E_I^o$.\n\n\\begin{defn}\\label{dfn mnf}\n\\cite[Definition 3.5.3]{DL3}\n With this notation the \\emph{motivic nearby fiber} is given by\n\\[\n \\mathcal{S}_f:=\\sum_{\\emptyset \\neq I\\subset J} (1- \\mathbb{L})^{\\lvert I \\rvert-1}[\\tilde{E}^o_{I}]\\in \\mathcal{M}_{X_0}^{\\hat{\\mu}}.\n\\]\nThis definition does not depend on the choice of $h:Y\\to X$, see \\cite[3.3.1]{DLLefschetz}.\n\\end{defn}\n\\noindent\nAs by Corollary~\\ref{corprofinite},\nthere is a well defined map $\\mathcal{M}_{k}^ {\\hat{\\mu}}\\to \\mathcal{M}_k$\nsending the class of a variety with $\\hat{\\mu}$-action to its quotient,\nthe quotient of the motivic nearby fiber\n \\begin{equation}\\label{Sf\/G}\n   \\mathcal{S}_f\/{\\hat{\\mu}}:=\\sum_{\\emptyset \\neq I\\subset J} (1- \\mathbb{L})^{\\lvert I \\rvert-1}[\\tilde{E}^o_{I}\/\\hat{\\mu}]=\\sum_{\\emptyset \\neq I\\subset J} (1- \\mathbb{L})^{\\lvert I \\rvert-1}[{E}^o_{I}]\n \\in \\mathcal{M}_{X_0} \n \\end{equation} \n is a well defined invariant of $f:X\\to \\mathbb{A}_k^1$.\n In particular it does not depend on the choice of $h: Y\\to X$.\n\nBy \\cite[Definition 8.1]{MR2106970},\nthere is a well defined \\emph{nearby cycle morphism}\n\\[\n \\psi : \\mathcal{M}_{\\ensuremath{\\mathbb{A}}_k^1}\\to \\mathcal{M}_k^{\\hat{\\mu}}\n\\]\nsending the class of $X\\in \\mathcal{M}_{\\mathbb{A}_k^1}$ to $\\mathcal{S}_f$ for every smooth $k$-variety $X$ with a proper map $f: X\\to \\ensuremath{\\mathbb{A}}^1_k$.\nHere $\\mathcal{S}_f$ is the image of $\\mathcal{S}_f$\nunder the map $\\mathcal{M}_{X_0}^{\\hat{\\mu}}\\to \\mathcal{M}_k^{\\hat{\\mu}}$\ninduced  by the structure map $X_0\\to \\Spec(k)$.\nThis morphism is $\\mathcal{M}_k$-linear,\nand $1=[\\ensuremath{\\mathbb{A}}^1_k]\\in \\mathcal{M}_{\\ensuremath{\\mathbb{A}}_k^1}$ is mapped to $1=[\\Spec(k)]\\in \\mathcal{M}_k^{\\hat{\\mu}}$.\nUsing Corollary \\ref{corprofinite} we get a well defined group morphism\n\\[\n \\bar{\\psi}: \\mathcal{M}_{\\ensuremath{\\mathbb{A}}^1_k}\\to \\mathcal{M}_k\n\\]\nsending the class of $X$ to the class of $\\mathcal{S}_f\/\\bar{\\mu}$ for every smooth $k$-variety $X$ with a proper map $f: X\\to \\ensuremath{\\mathbb{A}}^1_k$.\nAgain $1\\in \\mathcal{M}_{\\ensuremath{\\mathbb{A}}_k^1}$ is mapped to $1\\in \\mathcal{M}_k$, and using Remark~\\ref{K0-linear}\nwe see that $\\bar{\\psi}$ is $\\mathcal{M}_k$-linear, too.\n\nBy \\cite[List of properties 8.4]{MR2106970} the image of every $\\mathbb{A}_k^1$-scheme\nsupported in the point $0$ is trivial.\nSo we get in fact maps\n\\begin{equation}\\label{PSI}\n  \\psi: \\mathcal{M}_{\\ensuremath{\\mathbb{A}}_k^1\\setminus \\{0\\}}\\to \\mathcal{M}_k^{\\hat{\\mu}}\n \\text{ and }\n  \\bar{\\psi}: \\mathcal{M}_{\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}}\\to \\mathcal{M}_k\n\\end{equation}\nsending a smooth variety $X_\\eta$ over $\\mathbb{A}_k^1\\setminus \\{0\\}$ to $\\mathcal{S}_f$ and $\\mathcal{S}_f\/\\hat{\\mu}$, respectively,\nfor some proper map $f:X\\to \\mathbb{A}_k^1$\nwith $X$ smooth and irreducible, and $X_\\eta \\cong X\\times _{\\mathbb{A}_k^1}\\mathbb{A}_k^1\\setminus \\{0\\}$.\nThis maps are also $\\mathcal{M}_k$-linear and map $1$ to $1$.\n\n\n\\begin{rem}\nLet $X$ be a smooth and proper variety over $K=k(\\!(t)\\!)$.\nLet $\\mathcal{X}$ be proper sncd-model of $X$,\ni.e., $\\mathcal{X}$ is a proper and flat $k[\\![t]\\!]$-scheme with generic fiber isomorphic to $X$\nsuch that its special fiber $X_0$\nis a simple normal crossing divisor.\nBy \\cite[Definition~8.3]{NiSe}\nthe motivic volume $S(\\hat{X},\\hat{K})$\nwith values in $\\mathcal{M}_k$,\nwhich we can associate with the $t$-adic completion $\\hat{X}$ of $\\mathcal{X}$,\ndoes only depend on the generic fiber of $\\hat{X}$,\nand thus only on $X$ and not on the model $\\mathcal{X}$.\nBy \\cite[Proposition~8.2]{NiSe}\nwe have that\n\\[\n S(\\hat{X},\\hat{K})=\\mathbb{L}^{-m}\\sum_{\\emptyset \\neq I\\subset J} (1- \\mathbb{L})^{\\lvert I \\rvert-1}[\\tilde{E}^o_{I}]\\in \\mathcal{M}_k,\n\\]\nwhere the $\\tilde{E}_I^o$ are constructed from $X_0$\nas done above.\nIf $\\mathcal{X}$ is actually coming from some $f: X\\to \\mathbb{A}^1_k$\nas studied above, by \\cite[Theorem 9.13]{NiSe},\nwe have that\n\\[\n S(\\hat{X},\\hat{K})=\\mathbb{L}^{-(m-1)}\\mathcal{S}_f\\in \\mathcal{M}_k.\n\\]\nIf we could show that\nactually $S(\\hat{X},\\hat{K})$ was well defined in $\\mathcal{M}_k^{\\hat{\\mu}}$ (this is work in progress),\nwhere we consider the $\\tilde{E}^o_I$ with $\\hat{\\mu}$-action as done above,\nthen also\n\\[\n S(\\hat{X},\\hat{K})\/\\hat{\\mu}=\\mathbb{L}^{-m}\\sum_{\\emptyset \\neq I\\subset J} (1- \\mathbb{L})^{\\lvert I \\rvert-1}[{E}^o_{I}]\\in\\mathcal{M}_k\n\\]\nwould hold by Corollary \\ref{corprofinite},\nand we could proof results analogous to Proposition~\\ref{application} and Corollary \\ref{1modL} also in this context.\n\\end{rem}\n\n\n\\subsubsection*{Connection with the motivic reduction}\nTo be able to define the motivic reduction of $f$, we first need to proof the following lemma.\n\n\\begin{lem}\\label{welldef R(f)}\nLet $h: Y \\to X$ be any smooth modification of $f:X\\to \\mathbb{A}_k^1$.\nThen  the class of $h^ {-1}(X_0)$ in $K_0(\\Var_{X_0})\/\\mathbb{L}$ does not depend on the choice of $h$.\n\\end{lem}\n\n\\begin{proof}\nLet $h_1: Y_1\\to X$, $h_2: Y_2\\to X$ be two smooth modifications of $f$.\nWe have shown that \n\\begin{equation}\\label{hi}\n [h_1^{-1}(X_0)]=[h_2^{-1}(X_0)]\\in K_0(\\Var_{X_0})\/\\mathbb{L}.\n\\end{equation}\nConsider the fiber product $Y_{12}:=Y_1\\times_XY_2$.\nNote that the projection maps ${p_i: Y_{12}\\to Y_i}$ are proper,\nand induce isomorphisms between $ Y_{12}\\setminus p_i^{-1}(h_i^{-1}(X_0))$ and $Y_i\\setminus h_i^{-1}(X_0)$.\nAs the $h_i$ are smooth modifications of $f$,\nthe $Y_i\\setminus h_i^{-1}(X_0)$ are isomorphic to $X_\\eta =X\\setminus X_0$, which is smooth by assumption.\n\nLet $b: \\tilde{Y} \\to Y_{12}$ be an embedded resolution of \n$f\\circ h_1 \\circ p_1=f\\circ h_2\\circ p_2: Y_{12}\\to \\mathbb{A}_k^1$.\nSet $g_i:=p_i\\circ b$.\nBy construction $g_1^{-1}(h_1^{-1}(X_0))=g_2^{-1}(h_2^{-1}(X_0))$.\nHence in order to show Equation (\\ref{hi}),\nit suffices to show that $[h_i^{-1}(X_0)]=[g_i^{-1}(h_i^{-1}(X_0))]$ in $ K_0(\\Var_k)\/\\mathbb{L}$.\nBy the symmetry of the construction it suffices to show it for $i=1$.\n\nNote that $g_1: \\tilde{Y}\\to Y_1$ is a proper birational morphism over $X$ between two smooth varieties.\nBy \\cite[Remark 2.3]{MR2059227}, the Weak Factorization \nTheorem, see \\cite[Theorem 0.1.1]{MR1896232},\nholds for $g_1$, i.e., there exist a sequence of birational maps as follows\n\\[\n  \\xymatrix{\nX_1=V_0\\ar@{-->}[r]^{\\phi_1}\\ar@\/_5mm\/[rrrrr]_{g_1}& V_1\\ar@{-->}[r]^{\\phi_2}& V_2 \\ar@{-->}[r]^{\\phi_2}&\\dots \\ar@{-->}[r]^{\\phi_{l-1}} &  V_{l-1}\\ar@{-->}[r]^{\\phi_{l}}& V_l=Y_i,\n }\n\\]\nand for all $i$, either $\\phi_i:V_{i-1}\\dashrightarrow V_{i}$ or\n$\\phi_i^{-1}: V_i\\dashrightarrow V_{i-1}$\nis a morphism obtained by blowing up a smooth center.\nBy \\cite[Remark 2.4]{MR2059227}, the factorization is a factorization over $X$,\ni.e., there are structure maps $\\varphi_i: V_i\\to X$, with $\\varphi_0=h_1\\circ g_1$ and $\\varphi_l=h_1$,\nand the $\\phi_i$\nare maps over $X$.\n\nTake any $i\\in \\{1,\\dots,l\\}$.\nIf $\\phi_i: V_{i-1}\\to V_i$ is a blowup in the smooth center $C_i\\subset V_i$,\nthen we get using the scissors relation that\n\\begin{align*}\n [\\varphi_{i-1}^ {-1}(X_0)]&=[\\phi_i^{-1}(\\varphi_i^{-1}(X_0))]=[\\phi_i^{-1}(\\varphi_i^ {-1}(X_0)\\setminus C_i)]+[\\phi_i^{-1}(\\varphi_i^{-1}(X_0)\\cap C_i)]\\\\\n &=[\\varphi_i^ {-1}(X_0)\\setminus C_i]+[\\mathbb{P}_{X_0}^{\\Codim_{V_i}(C_i)}][C_i\\cap \\varphi_i^{-1}(X_0)]\\\\\n &=[\\varphi_i^ {-1}(X_0)\\setminus C_i]+[C_i\\cap \\varphi_i^{-1}(X_0)]=[\\varphi_i^ {-1}(X_0)] \\in K_0(\\Var_{X_0})\/(\\mathbb{L}).\n\\end{align*}\nAnalogously, we get the same statement if $\\phi_i^{-1}$ is a blowup. \nHence it follows that the classes of $g_1^ {-1}(h_1^{-1}(X_0))$ and $h_1^ {-1}(X_0)$ coincide in $K_0(\\Var_k)\/(\\mathbb{L})$,\nand hence the claim follows as observed above.\n\\end{proof}\n\n\\begin{defn}\\label{dfn mr}\n The \\emph{motivic reduction} $R(f)$ of $f: X\\to \\mathbb{A}_k^1$ is defined as the class of $h^{-1}(X_0)$\n  in $K_0(\\Var_{X_0})\/\\mathbb{L}$ of any smooth modification $h:Y\\to X$ of $f$.\n\\end{defn}\n\n\\begin{nota}\nLet $R\\in K_0(\\Var_{X_0})$ any  element in the inverse image of $R(f)$ under the quotient map.\nWe denote with $R(f)$ also the class of the image of $R$ in $\\mathcal{M}_{X_0}$ modulo $\\mathbb{L}$.\n\nUsing the forgetful map $K_0(\\Var_{X_0})\\to K_0(\\Var_k)$\ninduced by the structure morphism $X_0\\to \\Spec(k)$,\nwe can view $R(f)$ also as an element in $K_0(\\Var_k)\/\\mathbb{L}$.\nWe denote with $R(f)$ also its image in $K_0(\\Var_k)\/\\mathbb{L}$.\n\\end{nota}\n\n\\noindent\nNow we can combine the definition of the motivic reduction with the motivic nearby fiber,\nand get the following proposition.\n\n\\begin{prop}\\label{application} \nThe class of $R(f)$ and $S_f\/\\hat{\\mu}$ in the image of $K_0(\\Var_{X_0})$ in $\\mathcal{M}_{X_0}$ modulo $\\mathbb{L}$ coincide.\n \\end{prop}\n\n\\begin{proof}\nLet $h: Y\\to X$ be a embedded resolution of $f$, and\nlet $E_I^o$ be constructed from $h: Y\\to X$ as done above.\nThen by Equation (\\ref{Sf\/G}) we have\n\\begin{align*}\n \\mathcal{S}_f\/{\\hat{\\mu}}\n& =\\sum_{\\emptyset \\neq I\\subset J}[{E}^o_{I}] + \\mathbb{L} \\Big( \\sum_{\\emptyset \\neq I\\subset J} \\sum_{k=1}^{\\lvert I \\rvert-1} {\\lvert I \\rvert-1 \\choose k}\\mathbb{L}^{k-1}[E_I^o]\\Big)\\\\\n& =[h^{-1}({X}_0)]+\\mathbb{L} \\Big(\\sum_{\\emptyset \\neq I\\subset J} \\sum_{k=1}^{\\lvert I \\rvert-1} {\\lvert I \\rvert-1 \\choose k}\\mathbb{L}^{k-1}[E_I^o]\\Big) \\in \\mathcal{M}_{X_0}.\n\\end{align*}\nOne observes that $\\mathcal{S}_f\/{\\hat{\\mu}}$ lies in the image of $K_0(\\Var_{X_0})$.\nHence \n $\\mathcal{S}_f\/{\\hat{\\mu}}$ is equal to $[h^{-1}({X}_0)]$ modulo $\\mathbb{L}$\n in the image of $K_0(\\Var_{X_0})$ in $\\mathcal{M}_{X_0}$.\nThis is equal to $R(f)$, because $h:Y\\to X$ is an embedded resolution and hence a smooth modification \nof $f: X\\to \\mathbb{A}_k^1$.\n\\end{proof}\n\n\n \\begin{cor}\\label{1modL}\n Let $X$ be a smooth variety over $k$, and let ${f: X\\to \\mathbb{A}_k^1}$ be a proper morphism.\n If the class of $X_\\eta$ is equal to $1$ modulo $\\mathbb{L}$\n in $K_0(\\Var_{\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}})$,\n then the class of $f^{-1}(X_0)$ is equal to $1$ modulo $\\mathbb{L}$\n in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$.\n \\end{cor}\n \n\\begin{proof}\n If $[X_\\eta]=1 \\mod \\mathbb{L}\\in K_0(\\Var_{\\ensuremath{\\mathbb{A}}^1_k\\setminus\\{0\\}})$,\n we can write $[X_\\eta]=1+\\mathbb{L}[V]$ with $[V]\\in  K_0(\\Var_{\\ensuremath{\\mathbb{A}}^1_k\\setminus\\{0\\}})$,\n also for the class of $X_\\eta$ in $\\mathcal{M}_{\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}}$.\n Consider the map $\\bar{\\psi}: \\mathcal{M}_{\\ensuremath{\\mathbb{A}}^1_k\\setminus \\{0\\}}\\to \\mathcal{M}_k$, see Equation (\\ref{PSI}).\n \n On the one hand side,\n  $\\bar{\\psi}([X_\\eta])=1+\\mathbb{L}\\bar{\\psi}(V)\\in \\mathcal{M}_k$, because\n $\\bar{\\psi}$ is $\\mathcal{M}_k$-linear and maps $1$ to $1$.\n On the other hand,\n$\\bar{\\psi}([X_\\eta])=\\mathcal{S}_f\/\\hat{\\mu}$, and we have already seen that this is equal to $R(f)$\n in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$.\nHere $\\mathcal{S}_f\/\\hat{\\mu}$ and $R(f)$ \n are elements in $K_0(\\Var_k)$ via the map $K_0(\\Var_{X_0})\\to K_0(\\Var_k)$ induced by the structure map $X_0\\to \\Spec(k)$.\n Moreover $R(f)=[f^{-1}(X_0)]$, because $X$ is smooth, and hence $\\Id$ is a smooth modification of $f$.\n\nAll together $[f^{-1}(X_0)]$ is equal to $1$ modulo $\\mathbb{L}$ in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$.\n\\end{proof}\n\n\\begin{rem}\n It should be possible to proof Corollary \\ref{1modL} without using $S_f$,\n by showing that there exists a\n well defined map\n $K_0(\\Var_{\\mathbb{A}_k^1})\\to K_0(\\Var_{k})\/\\mathbb{L}$, which is $K_0(\\Var_k)$-linear, \nand sends $[X]$ to $R(f)$ if the structure map $f:X\\to \\mathbb{A}_k^1$ is proper and non-constant,\n and to $0$ if $f$ is proper and constant.\n To do this, one would need to consider the blowup relations\n in the Grothendieck ring of varieties,\n see \\cite[Theorem 5.1]{MR2059227}.\n\\end{rem}\n\n\n\n \\begin{rem}\n By Remark \\ref{rem 0div neg},\n $K_0(\\Var_{X_0})$ is not a subgroup of $\\mathcal{M}_{X_0}$, hence\nwe cannot show Proposition \\ref{application} in $K_0(\\Var_k)\/\\mathbb{L}$.\n If we could show that the motivic nearby fiber was well defined\n in $K_0^{\\hat{\\mu}}(\\Var_{X_0})\/\\mathbb{L}$, it would follow from\nthe fact that taking the quotient also\ngives a well defined map from $K_0^{\\hat{\\mu}}(\\Var_{X_0})\/(\\mathbb{L})$ to $K_0(\\Var_{X_0})\/(\\mathbb{L})$,\nthat $\\mathcal{S}_f\/\\hat{\\mu}$ would be well defined in $K_0(\\Var_{X_0})\/(\\mathbb{L})$.\nLike this we could also show that $R(f)$ is well define without using the Weak Factorization Theorem.\n\\end{rem}\n\n\n\\noindent\nIn order to avoid the problem that $K_0(\\Var_{X_0})$ is maybe not a subgroup of $\\mathcal{M}_{X_0}$,\nwe can work in the Grothendieck ring of effective motives.\nTo make things easier we work over $k$ right away.\n\nLet $\\text{Mot}^{\\text{eff}}_{k}$ be the additive category of effective motives with rational coefficients,\nlet $K_0(\\text{Mot}^{\\text{eff}}_{k})$ its Grothendieck ring, and let\n$\\mathbb{L}_{\\text{mot}}$ be the class of the Lefschetz motive in this ring.\nLet $\\text{Mot}_{k}$ be the category of motives with rational coefficients,\nlet $K_0(\\text{Mot}_{k})$ its Grothendieck ring, and let\n$\\mathbb{L}_{\\text{mot}}$ be the image of the Lefschetz motive.\nFor the precise definitions of these objects we refer to \\cite[Chapter~4]{MR2115000}.\nThe notation used here can be found for example in \\cite[Example 2.3]{MR2885336}.\nBy \\cite[Theorem 4.11]{MR2885336}\nwe get commuting maps as follows:\n\\[\n \\xymatrix{\n K_0(\\Var_k)\\ar[rr]^{\\chi^{\\text{eff}}_{\\text{mot}}} \\ar[d] & & K_0(\\text{Mot}^{\\text{eff}}_{k})\\ar[d]^{\\rho}\\\\\n\\mathcal{M}_k \\ar[rr]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\chi_{\\text{mot}}} & & K_0(\\text{Mot}^{\\text{eff}}_{k})[\\mathbb{L}_{\\text{mot}}^{-1}]\\cong K_0(\\text{Mot}_{k})\n}\n\\]\nHere $\\chi^{\\text{eff}}_{\\text{mot}}$\nmaps the class of a projective $k$-variety $X$\nto the class of the effective motive $(X,\\Id)$.\n$\\mathbb{L}$ is mapped to $\\mathbb{L}_{\\text{mot}}$.\nBy \\cite[Proposition 2.7]{MR2770561}\n$\\rho$ is injective if one assumes the following standard conjecture,\nwhich can be found for example in \\cite[Conjecture 2.5]{MR1879805}:\n\n\\begin{conj}\\label{con}\nIf $M$ and $N$\nare objects in $\\mathrm{Mot}^ {\\mathrm{eff}}_k$,\nthen $[M]=[N]\\in K_0(\\mathrm{Mot}^ {\\mathrm{eff}}_k)$\nif and only if $M$ and $N$\nare isomorphic.\n\\end{conj}\n\n\\noindent\nAs shown in \\cite[Proposition 4.4]{MR3058610}, Conjecture \\ref{con} holds if M an N are supposed to be finite dimensional.\nAn important conjecture by Kimura and O'Sullivan predicts that all the motives\n$M\\in \\text{Mot}^{\\text{eff}}_k$ are finite dimensional. See \\cite[Conjecture 2.7]{MR2167204}, or \\cite[Conjecture KS, page 390]{MR3058610} for this very precise formulation.\n\nAssume now that Conjecture \\ref{con} is true.\nHence $\\rho$ is injective,\nand as $\\mathcal{S}_f\/\\hat{\\mu}$ lies in the image of $K_0(\\Var_k)$ in $\\mathcal{M}_k$,\nthe inverse image of $\\chi_{\\text{mot}}(\\mathcal{S}_f\/\\hat{\\mu})$ under $\\rho$\nhas precisely one element,\nwhich we denote by $\\mathcal{S}_f\/\\hat{\\mu}_{\\text{mot}}$.\nSet $R(f)_{\\text{mot}}:=\\chi_{\\text{mot}}^{\\text{eff}}(R(f))$.\nProposition \\ref{application} then implies the following:\n\n\n\\begin{cor}\\label{application motives}\n$\\mathcal{S}_f\/\\hat{\\mu}_{\\text{mot}}$ and $R(f)_{\\text{mot}}$\ncoincide in $ K_0(\\text{Mot}^{\\text{eff}}_{k})$.\n\\end{cor}\n\n\n\n\n\\section*{Acknowledgments}\n\\noindent\nDuring the research for this article, I was supported by a research fellowship\n of the \\textbf{DFG} (Aktenzeichen HA 7122\/1-1).\nI thank H\\'el\\`ene Esnault for discussing her result with me.\nMoreover, I am very thankful to Johannes Nicaise for the numerous discussions we had, for the ideas he gave me and for the suggestions he made. \n\n \\bibliographystyle{babalpha}\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}