diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgbgw" "b/data_all_eng_slimpj/shuffled/split2/finalzzgbgw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgbgw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe present article addresses rank computation \nof a {\\em linear symbolic matrix}---\na matrix of the following form: \n\\begin{equation}\\label{eqn:A}\nA = A_1 x_1 + A_2 x_2 + \\cdots + A_m x_m,\n\\end{equation}\nwhere each $A_i$ is an $n \\times n$ matrix \nover a field ${\\mathbb{K}}$, $x_i$ $(i=1,2,\\ldots,m)$ are variables, \nand $A$ is viewed as a matrix over ${\\mathbb{K}}(x_1,x_2,\\ldots,x_m)$. \nThis problem, sometimes called {\\em Edmonds' problem}, \nhas fundamental importance in a wide range \nof applied mathematics and computer science; see~\\cite{Lovasz89}. \nEdmonds' problem (on large field ${\\mathbb{K}}$) is a representative problem\nthat belongs to RP---the class of problems having a randomized polynomial time algorithm---but is not known to belong to P.\nThe existence of a deterministic polynomial time algorithm \nfor Edmonds' problem is one of the major open problems in theoretical computer science.\n\n\nIn 2015, \nIvanyos, Qiao, and Subrahmanyam~\\cite{IQS15a} \nintroduced a noncommutative formulation of the Edmonds' problem, called \nthe {\\em noncommutative Edmonds' problem}.\nIn this formulation, linear symbolic matrix $A$ is regarded as \na matrix over the {\\em free skew field} ${\\mathbb{K}}(\\langle x_1,\\ldots,x_m \\rangle)$, which is the ``most generic\" skew field of fractions of noncommutative polynomial ring \n${\\mathbb{K}} \\langle x_1,\\ldots,x_m \\rangle$. \nThe rank of $A$ over the free skew field is called \nthe {\\em noncommutative rank}, or {\\em nc-rank}, which is denoted by $\\mathop{\\rm nc\\mbox{-}rank} A$.\nContrary to the commutative case, \nthe noncommutative Edmonds' problem \ncan be solved in polynomial time.\n\\begin{Thm}[\\cite{GGOW15,IQS15b}]\n\tThe nc-rank of a matrix $A$ of form (\\ref{eqn:A}) \n\tcan be computed in polynomial time.\n\\end{Thm}\nAs well as the result, the algorithms for nc-rank are stimulating\nsubsequent researches. \nThe first polynomial time algorithm is due to \nGarg, Gurvits, Oliveira, and Wigderson~\\cite{GGOW15} for the case of \n${\\mathbb{K}} = {\\mathbb{Q}}$.\nThey showed that Gurvits' {\\em operator scaling algorithm}~\\cite{Gurvits04}, \nwhich was designed for solving a special class ({\\em Edmonds-Rado class}) of Edmonds' problem, can solve nc-singularity testing (i.e., testing whether $n = \\mathop{\\rm nc\\mbox{-}rank} A$)\nin polynomial time. \nThe operator scaling algorithm has rich connections \nto various fields of mathematical sciences.\nParticularly, nc-singularity testing can \nbe formulated as a \ngeodesically-convex optimization problem on Riemannian manifold $GL_n({\\mathbb{R}})\/O_n({\\mathbb{R}})$, \nand the operator scaling can be viewed as \na minimization algorithm on it; see \\cite{AGLOW}.\nFor explosive developments after \\cite{GGOW15},\nwe refer to e.g., \\cite{BFGOWW} and the references therein. \n\nIvanyos, Qiao, and Subrahmanyam~\\cite{IQS15a,IQS15b} \ndeveloped the first polynomial time algorithm for the nc-rank \nthat works on an arbitrary field ${\\mathbb{K}}$. \nTheir algorithm is viewed as a ``vector-space \ngeneralization\" of the augmenting path algorithm \nin the bipartite matching problem.\nThis indicates a new direction \nin combinatorial optimization, since \nEdmonds' problem generalizes \nseveral important combinatorial optimization problems.\nInspired by their algorithm, \n\\cite{HI_2x2} developed a combinatorial polynomial time algorithm\nfor a certain algebraically constraint 2-matching problem in a bipartite graph, which corresponds to the (commutative) Edmonds' problem for a linear symbolic matrix in~\\cite{IwataMurota95}.\nAlso, \na noncommutative algebraic formulation that \ncaptures weighted versions\nof combinatorial optimization problems\nwas studied in \\cite{HH_degdet,HI_degdet,Oki}.\n\nThe main contribution of this paper is \na significantly different polynomial time algorithm \nfor computing the nc-rank on an arbitrary field ${\\mathbb{K}}$. \nWhile describing \nthe above algorithms and validity proofs is rather tough work, \nthe algorithm and proof presented in this paper \nare conceptually simple, elementary, \nand relatively short. \nFurther, it is also relevant to\nthe following two cutting edge issues in \ndiscrete and continuous optimization:\n\\begin{itemize}\n\t\\item submodular optimization on a modular lattice.\n\t\\item convex optimization on a CAT(0) space.\n\\end{itemize}\n\n\n A {\\em submodular function} $f$ on a lattice ${\\cal L}$\nis a function $f:{\\cal L} \\to {\\mathbb{R}}$ \nsatisfying $f(p) + f(q) \\geq f(p \\vee q) + f(p \\wedge q)$ for $p,q \\in {\\cal L}$.\nSubmodular functions on Boolean lattice $\\{0,1\\}^n$ are well-studied, and\nhave played central roles in the developments of combinatorial optimization; see \\cite{FujiBook}.\nThey are correspondents of convex functions \n({\\em discrete convex functions}) \nin discrete optimization; see \\cite{MurotaBook}.\nOptimization of submodular functions beyond Boolean lattices, particularly on modular lattices, \nis a new research area that has just started; \nsee \\cite{FKMTT14,HH16L-convex,Kuivinen11} on this subject.\n\nA {\\em CAT(0) space} is a (non-manifold) generalization \nof nonpositively curved Riemannian manifolds; see~\\cite{BrHa}. \nWhile CAT(0) spaces have been studied mainly in geometric group theory,\ntheir effective utilization in applied mathematics \nhas gained attention; see e.g.,\\cite{BHV01}.\nA CAT(0) space is a uniquely-geodesic metric space, \nand convexity concepts are defined along unique geodesics.\nTheory of algorithms and optimization on CAT(0) spaces is now being pioneered; see e.g., ~\\cite{Bacak13,Bacak14,BacakBook,Hayashi,Owen11}.\n\nOur algorithm is obtained as a combination \nof these new optimization approaches.\nWe hope that this will bring new interactions to the nc-rank literature.\nWhile it is somehow relevant to \ngeodesically-convex optimization mentioned above,\nwe deal with optimization on \ncombinatorially-defined non-manifold CAT(0) spaces.\nThe most important implication of our result is that \nconvex optimization algorithms on such spaces can be a tool of showing polynomial time complexity. \n\n\n\\paragraph{Outline.} Let us outline our algorithm.\nAs shown by Fortin and Reutenauer~\\cite{FortinReutenauer04}, \nthe nc-rank is given by the optimum value of\nan optimization problem:\n\\begin{Thm}[\\cite{FortinReutenauer04}]\\label{thm:FR}\n\tLet $A$ be a matrix of form~{\\rm (\\ref{eqn:A})}.\n\tThen $\\mathop{\\rm nc\\mbox{-}rank} A$ is equal to the optimal value of the following problem:\n\t\\begin{eqnarray*}\n\t\t{\\rm FR}: \\quad {\\rm Min.} && 2n - r - s \\\\\n\t\t{\\rm s.t.} && \\mbox{$S A T$ has an $r \\times s$ zero submatrix,} \\\\\n\t\t&& S, T \\in GL_n({\\mathbb{K}}).\n\t\\end{eqnarray*}\n\\end{Thm}\nAs in~\\cite{IQS15a,IQS15b}, our algorithm \nis designed to solve this optimization problem. \nThis problem FR can also be formulated as an optimization problem on the modular lattice of vector subspaces in ${\\mathbb{K}}^n$, as follows. \nRegard each matrix $A_i$ as a bilinear form ${\\mathbb{K}}^n \\times {\\mathbb{K}}^n \\to {\\mathbb{K}}$ by\n\\[\nA_i(x,y) := x^{\\top} A_i y \\quad (x,y \\in {\\mathbb{K}}^n).\n\\]\nThen the condition of FR says that \nthere is a pair of vector subspaces $U$ and $V$ of dimension $r$ and $s$, respectively, that\nannihilates all bilinear forms, i.e., $A_i(U,V) := \\{0\\}$.\nThe objective function is written as $2n - \\dim U - \\dim V$.\nTherefore, FR is equivalent to the following problem \n({\\em maximum vanishing subspace problem; MVSP}):\n\\begin{eqnarray*}\n\t{\\rm MVSP}: \\quad\n\t{\\rm Min.} && \n\t- \\dim X - \\dim Y \\\\\n\t{\\rm s.t.} && A_i(X,Y) =\\{0\\} \\quad (i=1,2,\\ldots,m), \\\\\n\t&& X,Y: \\mbox{vector subspaces of ${\\mathbb{K}}^n$}.\n\\end{eqnarray*}\nIt is a basic fact that\nthe family ${\\cal L}$ of all vector subspaces in $\\mathbb{K}^n$\nforms a modular lattice with respect to \nthe inclusion order.\nHence, MVSP is an optimization problem over ${\\cal L} \\times {\\cal L}$. \nFurther, by reversing the order of the second ${\\cal L}$, \nit can be viewed as a {\\em submodular function minimization (SFM)} on modular lattice ${\\cal L} \\times {\\cal L}$;\nsee Proposition~\\ref{prop:submodular} in Section~\\ref{subsec:submodular}. \n\n\n\nContrary to the Boolean case, \nit is not known generally whether a submodular function \non a modular lattice can be minimized in polynomial time.\nThe reason of polynomial-time solvability of SFM on Boolean lattice $\\{0,1\\}^n$ \nis the {\\em Lov\\'asz extension}~\\cite{Lovasz83}---\na piecewise-linear interpolation $\\bar f: [0,1]^n \\to {\\mathbb{R}}$ of function $f:\\{0,1\\}^n \\to {\\mathbb{R}}$ such that \n$\\bar f$ is convex if and only if $f$ is submodular. \nFor SFM on a modular lattice, however, \nsuch a good convex relaxation to $\\mathbb{R}^n$ is not known.\n\nA recent study~\\cite{HH16L-convex} introduced\nan approach of constructing \na convex relaxation of SFM on a modular lattice, \nwhere the domain of the relaxation is a CAT(0) space.\nThe construction is based on the concept of an {\\em orthoscheme complex}~\\cite{BM10}. \nConsider the order complex $K({\\cal L})$ of ${\\cal L}$,\nand endow each simplex with a specific Euclidean metric.\nThe resulting metric space $K({\\cal L})$ is called\nthe orthoscheme complex of ${\\cal L}$, and \nis dealt with as a continuous relaxation of ${\\cal L}$. \nThe details are given in Section~\\ref{subsub:K(L)}.\nFigure~\\ref{fig:folder} illustrates\nthe orthoscheme complex of a modular lattice with rank $2$, which is obtained by gluing Euclidean \nisosceles right triangles \nalong longer edges.\n\t\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.4]{folder.pdf}\n\t\t\\caption{An orthoscheme complex}\n\t\t\\label{fig:folder}\n\t\\end{center}\n\\end{figure}\\noindent\nThe orthoscheme complex of a modular lattice \nwas shown to be CAT(0)~\\cite{CCHO}. \nThis enables us to consider geodesically-convexity \nfor functions on $K({\\cal L})$.\nIn this setting, a submodular function $f:{\\cal L} \\to {\\mathbb{R}}$ \nis characterized by the convexity of \nits piecewise linear interpolation, i.e., Lov\\'asz extension \n$\\bar f:K({\\cal L}) \\to {\\mathbb{R}}$~\\cite{HH16L-convex}.\nAccording to this construction, we obtain \nan exact convex relaxation of MVSP in a CAT(0)-space.\n\nOur proposed algorithm is obtained by applying \nthe {\\em splitting proximal point algorithm (SPPA)} to \nthis convex relaxation.\nSPPA is a generic algorithm that minimizes a convex function of a separable form \n$\\sum_{i=1}^N f_i$, \nwhere each $f_i$ is a convex function. \nEach iteration of the algorithm updates \nthe current point $x$ to its {\\em resolvent} of $f_i$---\na minimizer of \n$y \\mapsto f_i(y) + (1\/\\lambda)d(y,x)^2$, where $i$ is chosen cyclically.\nBa\\v{c}\\'ak~\\cite{Bacak14} showed that \nSPPA generates a sequence convergent to a minimizer of $f$\n(under a mild assumption).\nSubsequently, Ohta and P\\'alfia~\\cite{OhtaPalfia15} \nproved a sublinear convergence of SPPA.\n\nThe main technical contribution is to show that \nSPPA is applicable to the convex relaxation of MVSP \nand becomes a polynomial time algorithm for MVSP: \nWe provide an equivalent convex relaxation of MVSP with\na separable objective function $\\sum_i f_i$, and show that \nthe resolvent of each $f_i$ can be computed in polynomial time. \nBy utilizing the sublinear convergence estimate, \na polynomial number of iterations for SPPA\nidentifies an optimal solution of MVSP. \n\n\nCompared with the existing algorithms, \nthis algorithm has advantages and drawbacks. \nAs mentioned above, \nour algorithm and its validity proof are relatively simple. \nParticularly, it can be uniformly written for an arbitrary field ${\\mathbb{K}}$, where\nonly the requirement for ${\\mathbb{K}}$ is that arithmetic operations is executable. \nNo care is needed for a small finite field, \nwhereas the algorithm in \\cite{IQS15a,IQS15b} needs a field extension.\nOn the other hand, our algorithm is very slow; see Theorem~\\ref{thm:prox}.\nThis is caused by using a generic and primitive algorithm (SPPA) \nfor optimization on CAT(0) spaces. \nWe believe that this will be naturally improved in future developments. \n\nThe problematic point of our algorithm \nis bit-complexity explosion for the case of ${\\mathbb{K}} = {\\mathbb{Q}}$.\nOur algorithm updates feasible vector subspaces in MVSP, \nand can cause an exponential increase of the bit-size \nrepresenting bases of those vector subspaces.\nTo resolve this problem and\nmake use of the advantage in finite fields, \nwe propose a reduction of nc-rank computation on ${\\mathbb{Q}}$ \nto that on $GF(p)$.\nThis reduction is an application of the $p$-adic valuation \non ${\\mathbb{Q}}$.\nWe consider a weighted version of the nc-rank, which \nwas introduced by~\\cite{HH_degdet} for ${\\mathbb{K}}(t)$\n and is definable \nfor an arbitrary field with a discrete valuation.\nThe corresponding optimization problem MVMP is \na discrete convex optimization on a representative CAT(0) space---the {\\em Euclidean building} for $GL_n({\\mathbb{Q}})$ (or $GL_n({\\mathbb{Q}}_p)$). \nThis may be viewed as a \n$p$-adic counterpart of the above geodesically-convex optimization approach on $GL_n({\\mathbb{R}})\/O_n({\\mathbb{R}})$ \nfor nc-singularity testing on ${\\mathbb{Q}}$. \nBy using an obvious relation of the $p$-adic valuation of \na nonzero integer and its bit-length in base $p$, \nwe show that nc-singularity testing on ${\\mathbb{Q}}$ reduces to a polynomial number of nc-rank computation over the residue field $GF(p)$, \nin which the required bit-length is polynomially bounded.\n\n\n\n\n\\paragraph{Organization.}\nThe rest of this paper is organized as follows.\nIn Section~\\ref{sec:pre}, \nwe present necessary backgrounds on convex optimization on CAT(0) space,\nmodular lattices, and submodular functions. \nIn Section~\\ref{sec:algorithm}, \nwe present our algorithm and show its validity.\nIn Section~\\ref{sec:p-adic}, \nwe present the $p$-adic reduction for\nnc-rank computation on ${\\mathbb{Q}}$.\n\n\n\n\n\n\\paragraph{Original motivation: \nBlock triangularization of a partitioned matrix.}\nThe original version~\\cite{HamadaHirai} of this paper dealt with block triangularization of a matrix with \nthe following partition structure:\n \\[\n A = \\left(\n \\begin{array}{ccccc}\n A_{11} & A_{12} &\\cdots & A_{1\\nu} \\\\\n A_{21} & A_{22} &\\cdots & A_{2\\nu} \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n A_{\\mu1}&A_{\\mu2} &\\cdots & A_{\\mu \\nu}\n \\end{array}\\right),\n \\]\n where $A_{\\alpha \\beta}$ is an $n_\\alpha \\times m_\\beta$ \n matrix over field ${\\mathbb{K}}$ for $\\alpha \\in [\\mu]$ and\n $\\beta\\in [\\nu]$. \n Consider the following block triangularization \n \\[\n A\\mapsto PEAFQ\n =\\left[\\begin{array}{ccccc}\n \\multicolumn{1}{|c}{} & & & \\\\\\cline{1-1}\n & \\multicolumn{1}{|c}{} && \\text{\\huge $*$}\\\\\\cline{2-3}\n & & & \\multicolumn{1}{|c}{} &\\\\\\cline{4-4}\n \\text{\\huge $O$} & & & & \\multicolumn{1}{|c}{} \\\\\\cline{5-5}\n \\end{array}\\right],\n \\]\n where $P$ and $Q$ are permutation matrices and \n $E$ and $F$ are regular transformations ``within blocks,\" \n i.e., $E$ and $F$ are block diagonal \n matrices with block diagonals \n $E_\\alpha \\in GL_{n_{\\alpha}}({\\mathbb{K}})$ $(\\alpha \\in [\\mu])$ and $F_\\beta \\in GL_{m_{\\beta}}({\\mathbb{K}})$ \n $(\\beta \\in [\\mu])$, respectively.\n Such a block triangularization was addressed by Ito, Iwata, and Murota~\\cite{ItoIwataMurota94} for motivating \n analysis on physical systems with (restricted) symmetry.\n %\n The most effective block triangularization \n is determined by arranging a maximal chain of \n maximum-size zero-blocks exposed in $EAF$, \n where the size of a zero block is defined as \n the sum of row and column numbers.\n %\nThis generalizes the classical {\\em Dulmage-Mendelsohn decomposition} \nfor bipartite graphs and Murota's {\\em combinatorial canonical form} for layered mixed matrices; see \\cite{HH16DM,MurotaBook}.\n\nFinding a maximum-size zero-block \nis nothing but FR (or MVSP) for\nthe linear symbolic matrix obtained by multiplying variable $x_{\\alpha\\beta}$ to $A_{\\alpha\\beta}$; \nsee \\cite[Appendix]{HH_degdet} \nfor details. \nThe original version of our algorithm was \ndesigned for this zero-block finding. \nLater, we found that this is essentially nc-rank computation. \nThis new version improves analysis (on Theorem~\\ref{thm:prox}), \nsimplifies the arguments, particularly \nthe proof of Theorem~\\ref{thm:P2}, and includes \nthe new section for the $p$-adic reduction.\n\n\\section{Preliminaries}\\label{sec:pre}\n\n\n\nLet $[n]$ denote $\\{1,2,\\ldots,n\\}$.\nLet ${\\mathbb{R}}$, ${\\mathbb{Q}}$, ${\\mathbb{Z}}$ denote the sets of real, rational, and integer numbers, respectively.\nLet $1_X$ denote the vector in ${\\mathbb{R}}^n$ \nsuch that $(1_X)_i = 1$ if $i \\in X$ and zero otherwise.\nThe $i$-unit vector $1_{\\{i\\}}$ is simply written as $1_i$.\n\n\n\n\n\\subsection{Convex optimization on CAT(0)-spaces}\\label{sec:CAT(0)}\n\n\\subsubsection{CAT(0)-spaces}\nLet $K$ be a metric space with distance function $d$. \nA {\\em path} in $K$ is a continuous map $\\gamma:[0,1] \\to L$, where\nits length is defined as $\\sup \\sum_{i=0}^{N-1} d(\\gamma(t_i), \\gamma(t_{i+1}))$\nover $0=t_0 < t_1 < t_2 < \\cdots < t_N = 1$ and $N > 0$.\nIf $\\gamma(0) = x$ and $\\gamma(1) = y$, then \nwe say that a path $\\gamma$ {\\em connects} $x,y$. \nA {\\em geodesic} is a path $\\gamma$ satisfying\n$d(\\gamma(s), \\gamma(t)) = d(\\gamma(0), \\gamma(1)) |s - t|$ \nfor every $s,t \\in [0,1]$.\nA {\\em geodesic metric space} is a metric space $K$ in which\nany pair of two points is connected by a geodesic.\nAdditionally, \nif a geodesic connecting any points is unique, then\n$K$ is called {\\em uniquely geodesic}.\n\nWe next introduce a CAT(0) space.\nInformally, it is defined as a geodesic metric space \nin which any triangle is not thicker than the corresponding triangle in Euclidean plane. \nWe here adopt the following definition.\nA geodesic metric space $K$ is said to be {\\em CAT(0)} \nif for every point $x \\in K$, every geodesic $\\gamma:[0,1] \\to K$ and $t \\in [0,1]$, it holds\n\\begin{equation}\\label{eqn:CAT(0)}\nd(x,\\gamma(t))^2 \\leq (1-t) d(x,\\gamma(0))^2 + td(x,\\gamma(1))^2 - t(1-t)d(\\gamma(0),\\gamma(1))^2.\n\\end{equation}\nThe following property of a CAT(0) space is a basis of introducing convexity.\n\\begin{Prop}[{\\cite[Proposition 1.4]{BrHa}}]\\label{prop:uniquely-geodesic}\n\tA {\\rm CAT}$(0)$-space is uniquely geodesic.\n\\end{Prop}\n\nSuppose that $K$ is a {\\rm CAT}$(0)$ space.\nFor points $x,y$ in $K$, let $[x,y]$ \ndenote the image of a unique geodesic $\\gamma$ connecting $x,y$.\nFor $t \\in [0,1]$, the point $p$ on $[x,y]$ \nwith $d(x,p)\/d(x,y) = t$ is formally written \nas $(1-t)x + t y$.\n\nA function $f: K \\to {\\mathbb{R}}$ is said to be {\\em convex}\nif for all $x,y \\in K, t \\in [0,1]$\nit satisfies \n\\begin{equation*\n f((1-t) x + t y) \\leq (1-t) f(x) + t f(y).\n\\end{equation*}\nIf it satisfies a stronger inequality\n\\begin{equation*\nf((1-t) x + t y) \\leq (1-t) f(x) + t f(y) - \\frac{\\kappa}{2} t(1-t) d(x,y)^2\n\\end{equation*}\nfor some $\\kappa > 0$, then $f$ is said to be \n{\\em strongly convex} with parameter $\\kappa > 0$.\nIn this paper, \nwe always assume that a convex function is continuous.\nA function $f:K \\to {\\mathbb{R}}$ is said to be {\\em $L$-Lipschitz}\nwith parameter $L \\geq 0$\nif for all $x,y \\in K$\nit satisfies \n\\begin{equation*}\n\t|f(x) - f(y)| \\leq L d(x,y).\n\\end{equation*}\n\n\n\\begin{Lem}\\label{lem:d^2}\n\tFor any $z \\in K$,\n\tthe function $x \\mapsto d(z,x)^2$ is strongly convex with parameter $\\kappa = 2$, and \n\tis $L$-Lipschitz with $L = 2\\mathop{\\rm diam} K$, where \n\t$\\mathop{\\rm diam} K:= \\sup_{x,y \\in K} d(x,y)$ denotes the diameter of $K$\n\t\n\\end{Lem}\nThe former follows directly from the definition (\\ref{eqn:CAT(0)}) of CAT(0)-space.\nThe latter follows from $d(z,x)^2 - d(z,y)^2 \\leq \n(d(z,x)+ d(z,y))(d(z,x) - d(z,y)) \n= (d(z,x)+ d(z,y)) d(x,y) \\leq (2 \\mathop{\\rm diam} K) d(x,y)$.\n\n\n\n\\subsubsection{Proximal point algorithm}\nLet $K$ be a complete CAT(0)-space (which \nis also called an {\\em Hadamard space}).\nFor a convex function $f: K \\to {\\mathbb{R}}$ and $\\lambda > 0$\nthe {\\em resolvent} of $f$ is a map $J_\\lambda^f: K \\to K$ defined by\n\\begin{equation*}\nJ_{\\lambda}^f (x) := \\mathop{\\rm argmin} _{y \\in K} \\left( f(y) + \\frac{1}{2\\lambda} d(x,y)^2 \\right)\n\\quad (x \\in K).\n\\end{equation*}\nSince the function $y \\mapsto f(y) + \\frac{1}{2\\lambda} d(x,y)^2$ is strongly convex with parameter $1\/\\lambda > 0$, \nthe minimizer is uniquely determined, and $J_{\\lambda}^f$ is well-defined; see \\cite[Proposition 2.2.17]{BacakBook}.\n\nThe {\\em proximal point algorithm (PPA)} is \nto iterate updates $x \\leftarrow J_{\\lambda}^f(x)$.\nThis simple algorithm generates a sequence converging to a minimizer of $f$\nunder a mild assumption; see \\cite{Bacak13,BacakBook}.\nThe {\\em splitting proximal point algorithm (SPPA)}~\\cite{Bacak14,BacakBook}, which we will use, minimizes a convex function $f: K \\to {\\mathbb{R}}$ represented as the following form\n\\begin{equation*}\n\tf := \\sum_{i=1}^m f_i,\n\\end{equation*}\nwhere each $f_i:K \\to {\\mathbb{R}}$ is a convex function.\nConsider a sequence $(\\lambda_k)_{k=1,2,\\ldots,}$ satisfying\n\\begin{equation*}\n\\sum_{k=0}^{\\infty} \\lambda_k = \\infty, \\quad \\sum_{k=0}^\\infty \\lambda_k^2 < \\infty.\n\\end{equation*}\n \n\\begin{description}\n\\item[Splitting Proximal Point Algorithm (SPPA)]\n\\item[$\\bullet$] Let $x_0 \\in K$ be an initial point.\n\\item[$\\bullet$] For $k=0,1,2,\\ldots$, repeat the following:\n\\[\nx_{km+i} := J_{\\lambda_k}^{f_i} (x_{km+i -1}) \\quad (i=1,2,\\ldots,m).\n\\]\n\\end{description}\nBa\\v{c}\\'{a}k~\\cite{Bacak14} showed that the sequence generated by\nSPPA converges to a minimizer of $f$ if \n$K$ is locally compact.\nOhta and P\\'alfia~\\cite{OhtaPalfia15} \nproved sublinear convergence of SPPA if \n$f$ is strongly convex and $K$ is not necessarily locally compact.\n\\begin{Thm}[{\\cite{OhtaPalfia15}}]\\label{thm:OhtaPalfia}\nSuppose that \n$f$ is strongly convex with parameter $\\epsilon > 0$\nand each $f_i$ is $L$-Lipschitz.\nLet $x^*$ be the unique minimizer of $f$.\nDefine the sequence $(\\lambda_k)$ by\n\\begin{equation*}\n\\lambda_k := 1\/ \\epsilon (k+1).\n\\end{equation*} \nThen the sequence $(x_\\ell)$ generated by SPPA satisfies\n\\begin{equation*}\nd(x_{km}, x^*)^2 = O\\left(\\frac{\\log k}{k} \\frac{L^2m^2}{\\epsilon^2} \\right)\n\\quad (k =1,2,\\ldots).\n\\end{equation*} \n\\end{Thm}\n\n\n\\subsection{Geometry of modular lattices}\\label{sec:modularlattice}\nWe use basic terminologies and facts in lattice theory; see e.g., \\cite{Gratzer}.\nA {\\em lattice} ${\\cal L}$ \nis a partially ordered set\nin which every pair $p,q$ of elements \nhas meet $p \\wedge q$ (greatest common lower bound) \nand join $p \\vee q$ (lowest common upper bound). \nLet $\\preceq$ denote the partial order, where \n$p \\prec q$ means $p \\preceq q$ and $p \\neq q$.\nA pairwise comparable subset of ${\\cal L}$, \narranged as $p_0 \\prec p_1 \\prec \\cdots \\prec p_k$,\nis called a {\\em chain} (from $p_0$ to $p_k$),\nwhere $k$ is called the length. \nIn this paper, we only consider lattices in which \nany chain has a finite length.\nLet ${\\bf 0}$ and ${\\bf 1}$ denote the minimum and maximum elements of ${\\cal L}$, respectively.\nThe rank $r(p)$ of element $p$ is defined \nas the maximum length of a chain from ${\\bf 0}$ to $p$. \nThe rank of lattice ${\\cal L}$ is defined as the rank of ${\\bf 1}$. \nFor elements $p,q$ with $p \\preceq q$\nthe {\\em interval} $[p, q]$ is the set of elements $u$ with $p \\preceq u \\preceq q$.\nRestricting $\\preceq$ to $[p, q]$, the interval $[p, q]$ \nis a lattice with maximum $q$ and minimum $p$. \nIf $p \\neq q$ and $[p,q] = \\{p,q\\}$, we say that $q$ {\\em covers} $p$ and denote $p \\prec: q$ or $q :\\succ p$.\nFor two lattices ${\\cal L}, {\\cal M}$, \ntheir direct product ${\\cal L} \\times {\\cal M}$ \nbecomes a lattice, where the partial order on ${\\cal L} \\times {\\cal M}$ \n is defined by $(p,p') \\preceq (q,q') \\Leftrightarrow p \\preceq q, p' \\preceq q'$.\n\n\nA lattice ${\\cal L}$ is called {\\em modular} if \nfor every triple $x,a,b$ of elements with $x \\preceq b$, \nit holds $x \\vee (a \\wedge b) = (x \\vee a) \\wedge b$.\nA modular lattice satisfies the Jordan-Dedekind chain condition. This is, the lengths of \nmaximal chains of every interval \nare the same. \nAlso, we often use the following property:\n\\begin{equation}\\label{eqn:basic}\np \\prec: p'\\ \\Rightarrow \\ p \\wedge q = p' \\wedge q \\ \\\n\\mbox{or} \\ \\ p \\wedge q \\prec: p' \\wedge q. \n\\end{equation}\nThis can be seen from the definition of modular lattices, \nand holds also when replacing $\\wedge$ by $\\vee$.\n\n\nA modular lattice ${\\cal L}$ is said to be {\\em complemented} \nif every element can be represented as a join of atoms, where\nan {\\em atom} is an element of rank $1$.\nIt is known that for a complemented modular lattice, every interval is complemented modular, \nand a lattice obtained by reversing the partial order \nis also complemented modular. \nThe product of two complemented modular lattices is also complemented modular.\n\n\nA canonical example of a complemented modular lattice is \nthe family ${\\cal L}$ of all subspaces of a vector space $U$, \nwhere the partial order is the inclusion order with \n$\\wedge = \\cap$, and $\\vee = +$.\nAnother important example is a {\\em Boolean lattice}---a lattice isomorphic to \nthe poset $2^{[n]}$ of all subsets \nof $[n]$ with respect to the inclusion order $\\subseteq$.\n\n\n\\subsubsection{Frames---Boolean sublattices in a complemented modular lattice}\nLet ${\\cal L}$ be a complemented modular lattice of rank $n$, and let\n$r$ denote the rank function of ${\\cal L}$. \nA complemented modular lattice is equivalent \nto a {\\em spherical building of type A}~\\cite{BuildingBook}. \nWe consider a lattice-theoretic counterpart of an {\\em apartment}, which is a maximal Boolean sublattice of ${\\cal L}$.\n\nA {\\em base} is a set of $n$ atoms $a_1,a_2,\\ldots,a_n$ \nwith $a_1 \\vee a_2 \\vee \\cdots \\vee a_n = {\\bf 1}$.\nThe sublattice $\\langle a_1,a_2,\\ldots, a_n \\rangle$ \ngenerated by a base $\\{a_1,a_2,\\ldots,a_n\\}$\nis called a {\\em frame}, which is isomorphic\nto a Boolean lattice $2^{[n]}$ \nby the map\n\\[\nX \\mapsto \\bigvee_{i \\in X} a_i. \n\\]\n\\begin{Lem}[{see e.g.,\\cite{Gratzer}}]\\label{lem:frame}\nLet ${\\cal L}$ be a complemented modular lattice of rank $n$.\n\\begin{itemize}\n\t\\item[(1)] For chains ${\\cal C},{\\cal D}$ in $\\cal L$, \n\tthere is a frame ${\\cal F} \\subseteq {\\cal L}$ containing $\\cal C$ and $\\cal D$.\n \\item[(2)] For a frame ${\\cal F}$ and an ordering $a_1,a_2,\\ldots, a_n$ of its basis, \n define map $\\varphi_{a_1,a_2,\\ldots, a_n}: {\\cal L} \\to {\\cal F}$ by\n \\begin{equation}\\label{eqn:retraction}\n p \\mapsto \\bigvee \\{a_i \\mid i \\in [n]: p \\wedge (a_1 \\vee a_2 \\vee \\cdots \\vee a_{i}) :\\succ p \\wedge (a_1 \\vee a_2 \\vee \\cdots \\vee a_{i-1}) \\}.\n \\end{equation}\n Then $\\varphi_{a_1,a_2,\\ldots, a_n}$ is a retraction to ${\\cal F}$ such that it is rank-preserving (i.e., $r(p) = r(\\varphi(p))$)\n and order-preserving (i.e., $p \\preceq q \\Rightarrow \\varphi(p) \\preceq \\varphi(q)$).\n\\end{itemize}\n\\end{Lem}\nThis is nothing but a part of the axiom of building, \nwhere the map in (2) is essentially\na {\\em canonical retraction} to an apartment.\n\\begin{proof}\t\n\tWe show (1) by the induction on $n$.\n\tSuppose that ${\\cal C} = ({\\bf 0} = p_0 \\prec p_1 \\prec \\cdots \\prec p_n = {\\bf 1})$\n\tand ${\\cal D} = ({\\bf 0} = q_0 \\prec q_1 \\prec \\cdots \\prec q_n = {\\bf 1})$.\n\t%\n\tConsider the maximal chains ${\\cal C}', {\\cal D}'$ \n\tfrom ${\\bf 0}$ to $p_{n-1}$, \n\twhere ${\\cal C}' := ({\\bf 0} = p_0 \\prec p_1 \\prec \\cdots \\prec p_{n-1})$ \n\tand ${\\cal D}'$ \n\tconsists of $q_i' := p_{n-1} \\wedge q_i$ $(i=0,1,\\ldots,n)$.\n\tNote that the maximality of ${\\cal D}'$ follows from (\\ref{eqn:basic}).\n\tBy induction, there is a frame $\\langle a_1,a_2,\\ldots,a_{n-1} \\rangle$ \n\tof the interval $[{\\bf 0}, p_{n-1}]$ (that is a complemented modular lattice of rank $n-1$)\n\tsuch that it contains ${\\cal C}', {\\cal D}'$. \n\tConsider the first index $j$ such that $q_j \\not \\preceq p_{n-1}$.\n\tThen $q'_i = q_i$ for $i f^*$.\n\tThen $f(p_i) \\geq f^* + 1$.\n\tHence $\\overline f(x) = \\sum_{i} \\lambda_i f(p_i) \n\t\\geq \\sum_{i}\\lambda_i (f^*+1) = f^* + 1$.\n\tHowever this contradicts $\\overline{f}(x) - f^* < 1$.\n\\end{proof}\n\n\n\n\n\\section{Algorithm}\\label{sec:algorithm}\n\n\\subsection{Nc-rank is submodular minimization}\\label{subsec:submodular}\nConsider MVSP for a linear symbolic matrix $A = \\sum_{i=1}^m A_i x_i$.\nLet us formulate MVSP as \nan unconstrained submodular function minimization \nover a complemented modular lattice.\nLet ${\\cal L}$ and ${\\cal M}$ \ndenote the lattices of all \nvector subspaces of $\\mathbb{K}^{n}$, where \nthe partial order of ${\\cal L}$ is the inclusion order \nand the partial order of ${\\cal M}$ is the reverse inclusion order. \nLet $R_{i} = R_{A_{i}}: {\\cal L} \\times {\\cal M} \\to {\\mathbb{Z}}$ be defined by\n\\begin{equation*\nR_i (X,Y) := \\mathop{\\rm rank} A_i |_{X \\times Y} \\quad ((X,Y) \\in {\\cal L} \\times {\\cal M}),\n\\end{equation*}\nwhere $A_i |_{X \\times Y}: X \\times Y \\to \\mathbb{K}$ is \nthe restriction of $A_i$ to $X \\times Y$. \nThen the condition $A_i (X,Y) = \\{0\\}$ in MVSP can be written as \n$R_{i}(X,Y) = 0$.\nBy using $R_{i}$ as a penalty term, \nconsider the following unconstrained problem:\n\\begin{eqnarray*}\n\t{\\rm MVSP}_{R}: \\quad\n\t{\\rm Min.} && \n\t- \\dim X - \\dim Y\n\t+ (2n+1) \\sum_{i=1}^mR_{i}(X,Y) \\\\\n\t{\\rm s.t.} && (X,Y) \\in {\\cal L} \\times {\\cal M}.\n\\end{eqnarray*}\nThen it is easy to see:\n\\begin{Lem}\n\tAny optimal solution of MVSP$_R$ is optimal to MVSP.\t\n\\end{Lem}\n\n\\begin{Prop}\\label{prop:submodular}\n\tThe objective function of MVSP$_R$ is submodular on ${\\cal L} \\times {\\cal M}$.\n\\end{Prop}\n\\begin{proof}\nSubmodularity of $X \\mapsto - \\dim X$\nand $Y \\mapsto - \\dim Y$ directly follows from \n$\\dim X+ \\dim X' = \\dim (X \\cap X') + \\dim (X+X')$.\t\nThus it suffices to verify \nthat $R = R_i:{\\cal L} \\times {\\cal M} \\to {\\mathbb{Z}}$ is submodular: \t\n\\begin{equation*\nR(X,Y) + R(X',Y') \\geq R(X \\cap X', Y + Y') + R(X + X', Y \\cap Y').\n\\end{equation*}\t\nNote that \nan equivalent statement appeared in \\cite[Lemma 4.2]{IwataMurota95}. \t\n\t\nBy Lemma~\\ref{lem:frame}, \nthere is a base $\\{ a_1,a_2,\\ldots,a_n\\}$ of ${\\cal L}$ \nwith $X, X', X \\cap X', X + X' \\subseteq \\langle a_1,a_2,\\ldots,a_n \\rangle$, \nand there is \na base $\\{ b_1,b_2,\\ldots,b_n\\}$ of ${\\cal M}$ \nwith $Y, Y', Y \\cap Y', Y + Y' \\subseteq \\langle b_1,b_2,\\ldots,b_n \\rangle$.\nConsider the matrix representation $A = (A(a_i,b_j))$ \nwith respect to these bases.\nFor $I,J \\subseteq [n]$, \nlet $A[I,J]$ \nbe the submatrix of $A$ with row set $I$ and column set~$J$.\nSubmodularity of $R$ follows from \nthe rank inequality\n\\begin{equation*}\n\\mathop{\\rm rank} A[I,J] + \\mathop{\\rm rank} A[I',J'] \\geq \\mathop{\\rm rank} A[I \\cap I',J \\cup J'] \n+ \\mathop{\\rm rank} A[I \\cup I',J \\cap J'].\n\\end{equation*}\nSee~\\cite[Proposition 2.1.9]{MurotaBook}.\t\n\\end{proof}\nThus,\nMVSP$_R$ has a convex relaxation on CAT(0) space \n$K({\\cal L} \\times {\\cal M}) = K({\\cal L}) \\times K({\\cal M})$ with objective function $g$ that is the Lov\\'asz extension\n\\begin{equation}\\label{eqn:g}\ng(x,y) := -\\overline{\\dim}(x) - \\overline{\\dim} (y) + (2n+1) \\sum_{i=1}^{m} \\overline{R_i}(x,y).\n\\end{equation}\n\n\n\n\n\n\n\\subsection{Splitting proximal point algorithm for nc-rank}\nWe apply SPPA\nto the following perturbed version of the convex relaxation:\n\\begin{eqnarray*}\n\t\\quad {\\rm Min.} && \n\t\t- \\overline{\\dim} (x) - \\overline{\\dim}(y) \n\t+ (2n+1) \\sum_{i=1}^m \\overline{R_{i}}(x,y) + (1\/8n) (d(\\mathbf{0}, x)^2 + d(\\mathbf{0},y)^2)\\\\\n\t{\\rm s.t.} && (x,y) \\in K({\\cal L}) \\times K({\\cal M}). \n\\end{eqnarray*}\nWe regard the objective function $\\tilde g$ as \n$\\sum_{i=1}^{m+2} f_i$, where $f_i$ is defined by\n\\begin{equation*}\nf_i(x,y) := \\left\\{\n\\begin{array}{cl}\n- \\overline{\\dim}(x) + (1\/8n) d(\\mathbf{0},x)^2 & {\\rm if}\\ \nk = m+1, \\\\\n- \\overline{\\dim}(y) + (1\/8n) d(\\mathbf{0},y)^2 & {\\rm if}\\ \nk = m+2, \\\\\n(2n+1) \\overline{R_{i}}(x, y) & \n{\\rm if}\\ 1 \\leq i \\leq m\n\\end{array}\\right.\n\\end{equation*}\n\\begin{Thm}\\label{thm:prox}\n\tLet $(z_{\\ell})$ be the sequence obtained by SPPA \n\tapplied to $\\tilde g = \\sum_{i=1}^{m+2} f_i$ with $\\epsilon := 1\/2n$.\n\tFor $\\ell = \\Omega ( n^{12}m^5 \\log nm)$, the support of $z_{\\ell} =(x_\\ell, y_\\ell)$ \ncontains a minimizer of MVSP.\n\\end{Thm}\n\\begin{proof}\n\tWe first show that $f_i$ \n\tis $L$-Lipschitz with\n\t$\n\tL = O(n^{5\/2}).\n\t$\n\tBy Lemma~\\ref{lem:d^2}, Proposition~\\ref{prop:basic_K(L)} (4-2), and Proposition~\\ref{prop:Lovasz_ext}~(2), \n\tthe Lipschitz constants of \n\t\t $\\overline{\\dim}$ and $d({\\bf 0},\\cdot)^2$ are $O(n^{3\/2})$ and $O(\\sqrt{n})$, respectively.\n\t\t %\n\t\t Therefore, if $i = m+1$ or $m+2$,\n\t\t then the Lipschitz constant of $f_i$ is $O(n^{3\/2})$.\n\t\t %\n\t\tThe Lipschitz constant of other $f_i$ \n\t\tis $O(n^{5\/2})$.\t\n\t\t\t\nThe objective function is strongly convex with parameter $1\/2n$.\nLet $\\tilde z$ denote the minimizer of $\\tilde g$.\nBy Theorem~\\ref{thm:OhtaPalfia}, we have\n\\begin{eqnarray*}\n&& \\tilde g(z_{k(m+2))}) -\\tilde g(\\tilde z) \\leq (m+2) L d(z_{k(m+2)},\\tilde z) = \nO\\left( \\sqrt{\\frac{\\log k}{k}} n^{6} m^2 \\right). \n\\end{eqnarray*}\nThus, for $k = \\Omega (n^{12} m^4 \\log nm)$, it holds\n\t$\n\t\\tilde g(z_{k(m+2)}) - \\tilde g(\\tilde z) < 1\/2$.\n\t\n\tLet $z^*$ be a minimizer of $g$ of (\\ref{eqn:g}).\n\tThen we have $g(z_{k(m+2)}) - g(z^*) \n\t= g(z_{k(m+2)}) - g(\\tilde z) + g(\\tilde z) - g(z^*) \n\t\\leq \\tilde g(z_{k(m+2)}) - \\tilde g(\\tilde z) + (1\/8n) d(\\mathbf{0},\\tilde z)^2 + \\tilde g(\\tilde z) - \\tilde g(z^*) + (1\/8n) d(\\mathbf{0},z^*)^2\n\t\\leq \\tilde g(z_{k(m+2)}) - \\tilde g(\\tilde z) + 1\/2 < 1$.\nBy Proposition~\\ref{prop:Lovasz_ext}~(3), the support of $z_{k(m+2)}$ contains a minimizer of MVSP.\n\\end{proof}\nThus, after a polynomial number of iterations, \na minimizer $(X^*,Y^*)$ of MVSP exists in the support of $z_{\\ell}$.\nOur remaining task \nis to show that the resolvent of each summand $f_i$\ncan be computed in polynomial time.\n\n\\subsubsection{Computation of the resolvent for $f_i = - \\overline{\\dim} + (1\/8n) d(\\mathbf{0}, \\cdot)^2$}\nFirst we consider the resolvent of \n$- \\overline{\\dim} + (1\/8n) d(\\mathbf{0}, \\cdot)^2$.\nThis is an optimization problem over the orthoscheme complex of a single lattice.\nIt suffices to consider the following problem.\n\\begin{eqnarray*}\t\n{\\rm P1: \\quad Min}. && - \\overline{\\dim}(x) \n+ \\epsilon d(\\mathbf{0},x)^2 + \\frac{1}{2\\lambda} d(x,x^0)^2 \\\\\n{\\rm s.t.} && x \\in K({\\cal L}),\n\\end{eqnarray*}\nwhere $\\epsilon, \\lambda > 0$, and $x^0 \\in K({\\cal L})$.\n\\begin{Lem}\n\tSuppose that $x^0$ belongs to a maximal simplex $\\sigma$. \n\tThen the minimizer $x^*$ of P1 \n\texists in $\\sigma$.\n\\end{Lem}\n\\begin{proof}\n\tLet $x^0 = \\sum_{i=0}^n \\lambda_i p_i$ for\n\tthe maximal chain $\\{p_i\\}$ of $\\sigma$.\n\tLet $x^* = \\sum_{i} \\mu_i q_i$ be the unique minimizer of P1.\n\tConsider a frame ${\\cal F} = \\langle a_1,a_2,\\ldots,a_n \\rangle$ \n\tcontaining chains $\\{p_i\\}$ and $\\{q_i\\}$.\n\tNotice $K({\\cal F}) \\simeq [0,1]^n$.\n\tLet $(x^0_1,x^0_2,\\ldots,x^0_n)$ and $(x^*_1,x^*_2,\\ldots,x^*_n)$ be \n\tthe ${\\cal F}$-coordinates of $x^0$ and $x^*$, respectively. \n\tBy~(\\ref{eqn:F-coordinate}), \n\t it holds $\\overline{\\dim}(x) = \\sum_i x_i$, \n\tsince $x = \\sum_{k=0}^n \\lambda_k a_{i_1} \\vee a_{i_2} \\vee \\cdots \\vee a_{i_k} \\simeq \\sum_{k} \\lambda_k 1_{\\{i_1,i_2,\\ldots,i_k\\}}$.\n\tHence\n\tthe objective function of P1 is written as\n\t\\[\n\t- \\sum_{i=1}^n x_i + \\epsilon \\sum_{i=1}^n x_i^2 + \\frac{1}{2\\lambda} \\sum_{i=1}^n (x_i - x^0_i)^2.\n\t\\] \n\tWe can assume that $p_i = a_1 \\vee a_2 \\vee \\cdots \\vee a_i$ by relabeling.\n\tThen $x^0_1 \\geq x^0_2 \\geq \\cdots \\geq x^0_n$.\n\tSuppose that $x^0_i > x^0_{i+1}$.\n\tThen $x^*_i \\geq x^*_{i+1}$ must hold. \n\tIf $x^*_i < x^*_{i+1}$, then interchanging the $i$-coordinate and $(i+1)$-coordinate of \n\t$x^*$ gives rise to another point in $K({\\cal F})$ having a smaller objective value. This is a contradiction to the optimality of $x^*$.\n\tSuppose that $x^0_i = x^0_{i+1}$. \n\tIf $x^*_i \\neq x^*_{i+1}$, \n\tthen replace both $x_i^*$ and $x_{i+1}^*$ by $(x^*_i+x^*_{i+1})\/2$ \n\tto decrease the objective value, which is a contradiction.\n\tThus $x^*_1 \\geq x^*_2 \\geq \\cdots \\geq x^*_n$.\n\tBy~(\\ref{eqn:recover}), \n\tthe original coordinate is written as \n\t$x^* = (1- x^{*}_1) {\\bf 0} + \\sum_{i=1}^n (x^*_i - x^*_{i+1}) (a_{1} \\vee a_2 \\vee \\cdots \\vee a_i) = \\sum_{i} (x^*_i - x^*_{i+1}) p_i$ \n\t(with $x_0^* = 1$ and $x^*_{n+1} = 0$). \n\tThis means that $x^*$ belongs to $\\sigma$.\n\\end{proof}\nAs in the proof, to solve P1, consider (implicitly) a frame ${\\cal F}$ \ncontaining the chain $\\{p_i\\}$ for $x^0 = \\sum_i \\lambda_i p_i$, \nand the following Euclidean convex optimization problem:\n\\begin{eqnarray*}\n{\\rm P1': \\quad Min}. && - \\sum_{i=1}^n x_i + \\epsilon \\sum_{i=1}^n x_i^2 + \\frac{1}{2\\lambda} \\sum_{i=1}^n (x_i - x^0_i)^2 \\\\\n{\\rm s.t.} && 0 \\leq x_i \\leq 1 \\quad (1 \\leq i \\leq n),\n\\end{eqnarray*}\nwhere $x$ and $x^0$ are represented in the ${\\cal F}$-coordinate.\nThen the optimal solution $x^*$ of P1$'$ is obtained coordinate-wise. \nSpecifically, $x^*_i$ is $0$, $1$, or $(x_i^0 + \\lambda)\/(1+ 2\\epsilon \\lambda)$ for each $i$. According to~(\\ref{eqn:recover}), the expression in $K({\\cal L})$ is recovered.\n\\begin{Thm}\\label{thm:P1}\n\tThe resolvent of $f_i = - \\overline{\\dim} + (1\/4n) d(\\mathbf{0}, \\cdot)^2$ \n\tis computed in polynomial time.\n\\end{Thm}\n\\subsubsection{Computation of the resolvent for $f_i = (2n+1) \\overline{R_{i}}$}\nNext we consider the computation of \nthe resolvent of~$(2n+1) \\overline{R_{i}}$. \nIt suffices to consider the following problem for $R = R_{A_i}$:\n\\begin{eqnarray*}\t\n\t{\\rm P2: \\quad Min.} && \\overline{R}(x,y) + \\frac{1}{2 \\lambda} ( d(x,x^0)^2 + d(y,y^0)^2) \\\\\n\t{\\rm s.t.} && (x,y) \\in K({\\cal L}) \\times K({\\cal M}),\n\\end{eqnarray*}\nwhere $\\lambda > 0$, $x^0 \\in K({\\cal L})$, and $y^0 \\in K({\\cal M})$.\nAs in the case of P1, we reduce P2 to a convex optimization \nover $[0,1]^{2n}$\nby choosing a special frame $\\langle e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n\\rangle$ of ${\\cal L} \\times {\\cal M}$.\n\nFor $X \\in {\\cal L}$, let $X^{\\bot}$ denote the subspace in ${\\cal M}$ defined by\n\\begin{equation*}\nX^{\\bot} := \\{ y \\in {\\mathbb{K}}^n \\mid A_i(x,y) = 0 \\ (x \\in X) \\}.\n\\end{equation*}\nNamely $X^{\\bot}$ is the orthogonal subspace of $X$ with respect to the bilinear form $A_i$.\nFor $Y \\in {\\cal M}$, let $Y^{\\bot} \\in {\\cal L}$ be defined analogously.\nLet $U_0 \\in {\\cal L}$ and $V_0 \\in {\\cal M}$ denote the left and right kernels of $A_i$, respectively:\n\\begin{eqnarray*}\n\tU_0 &:= &\\{ x \\in {\\mathbb{K}}^n \\mid A_i(x,y) = 0 \\ (y \\in {\\mathbb{K}}^n) \\}. \\\\ \n\tV_0 &:= & \\{ y \\in {\\mathbb{K}}^n \\mid A_i(x,y) = 0 \\ (x \\in {\\mathbb{K}}^n) \\}. \n\\end{eqnarray*}\n\nLet $k := \\mathop{\\rm rank} A_i$.\nAn {\\em orthogonal frame} ${\\cal F} = \\langle e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n\\rangle$\nis a frame of ${\\cal L} \\times {\\cal M}$ satisfying the following conditions:\n\\begin{itemize}\n\\item $\\langle e_1,e_2,\\ldots,e_n \\rangle$ is a frame of ${\\cal L}$.\n\\item $\\langle f_1,f_2,\\ldots,f_n\\rangle$ is a frame of ${\\cal M}$.\n\\item $e_{k+1} \\vee e_{k+2} \\vee \\cdots \\vee e_n = U_0$.\n\\item $f_1 \\vee f_2 \\vee \\cdots \\vee f_k = V_0$ ($\\Leftrightarrow$ $f_1 \\cap f_2 \\cap \\cdots \\cap f_k = V_0$ ).\n\\item $f_i = {e_i}^{\\bot}$ for $i=1,2,\\ldots,k$.\n\\end{itemize}\nFigure~\\ref{fig:frame} is an intuitive illustration of an orthogonal frame.\n\t\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.45]{frame.pdf}\n\t\t\\caption{An orthogonal frame}\n\t\t\\label{fig:frame}\n\t\\end{center}\n\\end{figure}\\noindent\n\n\\begin{Prop}\\label{prop:Lovasz}\n\tLet ${\\cal F} = \\langle e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n\\rangle$ be an orthogonal frame.\n\tThe restriction of the Lov\\'asz extension $\\overline{R}$ to \n\t$K({\\cal F}) \\simeq [0,1]^n \\times [0,1]^n$ can be written as\n\t\\begin{equation}\n\t\\overline{R} (x,y) = \\sum_{i=1}^k \\max\\{0, x_i - y_i\\},\n\t\\end{equation}\n\twhere $(x_1,x_2,\\ldots,x_n)$ is the $\\langle e_1,e_2,\\ldots,e_n \\rangle$-coordinate of $x$\n\tand $(y_1,y_2,\\ldots,y_n)$ is the $\\langle f_1,f_2,\\ldots,f_n \\rangle$-coordinate of $y$.\n\\end{Prop}\n\\begin{Prop}\\label{prop:orthogonal_frame}\n\tLet ${\\cal X}$ and ${\\cal Y}$ be maximal chains of ${\\cal L}$ and ${\\cal M}$, respectively. \n\tThen there exists an orthogonal frame ${\\cal F} = \\langle e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n\\rangle$ satisfying\n\t\\begin{equation}\\label{eqn:XuYbot}\n\t{\\cal X} \\cup {\\cal Y}^{\\bot} \\subseteq \n\t\\langle e_1,e_2,\\ldots,e_n \\rangle,\\ {\\cal X}^{\\bot} \\cup {\\cal Y} \\subseteq \n\t\\langle f_1,f_2,\\ldots,f_n \\rangle. \n\t\\end{equation}\n\tSuch a frame can be found in polynomial time.\t\n\\end{Prop}\n\n \\begin{Prop}\\label{prop:R(x,y)}\n\t Let ${\\cal X}$ and ${\\cal Y}$ \n\t be maximal chains corresponding to \n\t maximal simplices containing $x^0$ and $y^0$, respectively.\n For an orthogonal frame ${\\cal F}$ satisfying $(\\ref{eqn:XuYbot})$, \n the minimizer $(x^*,y^*)$ of P2 exists in $K({\\cal F})$.\n\\end{Prop}\n\n\nThe above three propositions are proved in Section~\\ref{subsub:proof}.\nAssuming these, we proceed the computation of the resolvent.\nFor an orthogonal frame satisfying~(\\ref{eqn:XuYbot}), the problem P2 is equivalent to\n\\begin{eqnarray*}\t\n\t{\\rm P2': \\quad Min.} && \\sum_{i=1}^k \\max\\{0,x_i - y_i \\} + \\frac{1}{2 \\lambda} \\left\\{\\sum_{i=1}^m (x_i -x^0_i)^2 + \\sum_{i=1}^n (y_i - y^0_i)^2 \\right\\} \\\\\n\t{\\rm s.t.} && 0 \\leq x_i \\leq 1, 0 \\leq y_i \\leq 1 \\quad (0 \\leq i \\leq n).\n\\end{eqnarray*}\nAgain this problem is easily solved coordinate-wise.\nObviously $x^*_i = x^0_i$ and $y^*_i = y^0_i$ for $i > k$.\nFor $i \\leq k$, \n$(x^*_i,y^*_i)$ \nis the minimizer of the $2$-dimensional problem.\nObviously this can be solved in constant time.\n\\begin{Thm}\\label{thm:P2}\n\tThe resolvent of $f_i = (2n+1)\\overline{R_i}$ is computed in polynomial time.\n\\end{Thm}\n\n\n\n\\begin{Rem}[Bit complexity]\\label{rem:bit}\n\tIn the above SPPA, \n\tthe required bit-length for coefficients of $z \\in K({\\cal L} \\times {\\cal M})$ \n\tis bounded polynomially in $n,m$. Indeed,\n\tthe transformation between the original coordinate and an ${\\cal F}$-coordinate\n\tcorresponds to multiplying a triangular matrix consisting of $0,\\pm 1$ entries; see (\\ref{eqn:recover}). \n\tIn each iteration $k$,\n\tthe optimal solution of quadratic problem P1$'$ or P2$'$ \n\tis obtained by \n\tadding (fixed) rational functions in $n,m,k$ \n\tto (current points) $x_i^0, y_i^0$ \n\tand multiplying a (fixed) $2 \\times 2$ rational matrix in $n,m,k$. \n\t%\n\tConsequently, the bit increase is polynomially bounded.\n\n\tOn the other hand, in the case of ${\\mathbb{K}} = {\\mathbb{Q}}$, we could not exclude the possibility of an exponential increase of the bit-length for the basis of a vector subspace appearing in the algorithm. \n\\end{Rem}\n\n\n\n\\subsubsection{Proofs of Propositions~\\ref{prop:Lovasz}, \\ref{prop:orthogonal_frame}, and \\ref{prop:R(x,y)}}\\label{subsub:proof}\nWe start with basic properties of $(\\cdot)^{\\bot}$, which follow from elementary linear algebra.\n\\begin{Lem}\\label{lem:bot}\n\t\\begin{itemize}\n\t\t\\item[{\\rm (1)}] If $X \\subseteq X'$, then $X^{\\bot} \\supseteq {X'}^{\\bot}$ and $\\dim {X}^{\\bot} - \\dim {X'}^{\\bot} \\leq \\dim X'-\\dim X$.\n\t\t\\item[{\\rm (2)}] $(X + X')^{\\bot} = X^{\\bot} \\cap {X'}^{\\bot}$. \n\t\t\\item[{\\rm (3)}] $X^{\\bot \\bot} \\supseteq X$.\n\t\t\\item[{\\rm (4)}] $X \\mapsto X^{\\bot}$ \n\t\tinduces an isomorphism between $[U_0, {\\mathbb{K}}^n]$ and \n\t\t$[{\\mathbb{K}}^n,V_0]$ with inverse $Y \\mapsto Y^{\\bot}$. In particular, \n\t\t$X^{\\bot \\bot \\bot} = X^{\\bot}$.\n\t\\end{itemize}\n\\end{Lem}\n\nAn alternative expression of $R$ by using $(\\cdot)^{\\bot}$ is given.\n\\begin{Lem}\\label{lem:formula_R}\n\t$R(X,Y) = \\dim Y - \\dim Y \\cap X^{\\bot} = \\dim X - \\dim X \\cap Y^{\\bot}$.\n\\end{Lem}\n\\begin{proof}\n\tConsider bases $\\{a_1,a_2,\\ldots,a_{\\ell}\\}$ of $X$ and $\\{ b_1,b_2,\\ldots,b_{\\ell'}\\}$ of $Y$.\n\tWe can assume that\n\t$\\{a_{k'+1},a_{k'+2},\\ldots,a_{\\ell'}\\}$ is a base of \n\t$Y \\cap X^{\\bot}$.\n\tConsider the matrix representation $(A_i (a_{i'},b_{j'}))$ of $A_i|_{X \\times Y}$ \n\twith respect to these bases. \n\tIts submatrix of $k'+1,k'+2,\\ldots,\\ell'$-th columns \n\tis a zero matrix.\n\tOn the other hand, \n\tthe submatrix of $1,2,\\ldots, k'$-th columns\n\tmust have the column-full rank $k'$. \n\tThus, the rank $R(X,Y)$ of $A_i|_{X \\times Y}$ is \n\t$k' = \\ell' - (\\ell'-k') = r(Y) - r(Y \\cap X^{\\bot})$.\n\tThe second expression is obtained similarly.\n\\end{proof}\n\n\n\n\\paragraph{Proof of Proposition~\\ref{prop:Lovasz}.}\nAn orthogonal frame \n$\\langle e_1,e_2,\\ldots,e_n,f_1,f_2,\\ldots,f_n \\rangle$ \nis naturally identified with Boolean lattice \n$2^{[2n]} \\simeq 2^{[n]} \\times 2^{[n]}$.\nNotice that ${e_i}^{\\bot} = f_i$ if $i \\leq k$ \nand ${e_i}^{\\bot} = {\\mathbb{K}}^n$ if $i > k$.\nThe latter fact follows from \n$e_i \\subseteq U_0 \\Rightarrow {e_i}^{\\bot} \\supseteq U_0^{\\bot} = {\\mathbb{K}}^n$.\nBy Lemma~\\ref{lem:bot}~(2), \nwe have\n$X^{\\bot} = X \\cap \\{1,2,\\ldots,k\\}$ for $X \\in 2^{[n]}$.\nBy Lemma~\\ref{lem:formula_R} \nand $\\dim Y = n - |Y|$ for \n$Y \\in 2^{[n]} \\simeq \\langle f_1,f_2,\\ldots,f_n \\rangle$ \n(with inclusion order reversed),\n we have\n\\[\n\tR(X,Y) = |Y \\cup (X \\cap [k]) | - |Y| = |(X \\setminus Y) \\cap [k]|.\n\\]\nIdentify $2^{[n]} \\times 2^{[n]}$ with $\\{0,1\\}^n \\times \\{0,1\\}^n$ \nby $(X,Y) \\mapsto (1_{X}, 1_{Y})$.\nThen $R$ is also written as\n\\begin{equation*}\nR(x,y) = \\sum_{i=1}^k \\max \\{0, x_i - y_i\\} \\quad ((x,y) \\in \\{0,1\\}^n \\times \\{0,1\\}^n).\n\\end{equation*}\nObserve that the Lov\\'asz extension of \n$(x_i,y_i) \\mapsto \\max \\{0, x_i - y_i\\}$ is \nobtained simply by extending the domain to $[0,1]^2$.\nHence, we obtain the desired expression.\n\n\n\n\n\\paragraph{Proof of Proposition~\\ref{prop:orthogonal_frame}.}\nBy Lemma~\\ref{lem:frame}, \nwe can find (in polynomial time) a frame $\\langle e_1,e_2,\\ldots,e_n \\rangle$ \ncontaining two chains ${\\cal X}$ and ${\\cal Y}^{\\bot}$.\nSuppose that ${\\cal X} = \\{X_i\\}_{i=0}^n$ and ${\\cal Y} = \\{Y_i\\}_{i=0}^n$.\nWe can assume that $e_{k+1} \\vee e_{k+2} \\vee \\cdots \\vee e_{n} = {Y_0}^{\\bot} = U_0$.\nLet $f_i := {e_i}^{\\bot}$ for $i=1,2,\\ldots,k$.\nThen $f_1 \\vee f_2 \\vee \\cdots \\vee f_n = V_0$ holds,\nsince, by Lemma~\\ref{lem:bot}~(2), we have \n$V_0 = (e_1 \\vee e_2 \\vee \\cdots \\vee e_n)^{\\bot} = {e_1}^{\\bot} \\vee {e_2}^{\\bot} \\vee \\cdots \\vee {e_n}^{\\bot} = f_1 \\vee f_2 \\vee \\cdots f_k \\vee {\\mathbb{K}}^n \\vee \\cdots \\vee {\\mathbb{K}}^n = f_1 \\vee f_2 \\vee \\cdots \\vee f_k$. \n\n\nConsider the chain ${\\cal Y}^{\\bot \\bot}$ in ${\\cal M}$.\nThen ${\\cal Y}^{\\bot \\bot} \\subseteq \\langle f_1,f_2,\\ldots,f_k \\rangle$ since\neach $Y_i^{\\bot}$ is the join of a subset of $e_1,e_2,\\ldots,e_n$.\nTaking $(\\cdot)^\\bot$ as above, \n$Y_i^{\\bot \\bot}$ is represented as the join of a subset of $f_1,f_2,\\ldots,f_k$.\nConsider a consecutive pair $Y_{i-1}, Y_{i}$ in ${\\cal Y}$.\nConsider ${Y_{i-1}}^{\\bot \\bot}$ and ${Y_{i}}^{\\bot \\bot}$.\nThen, by Lemma~\\ref{lem:bot}~(3), \n${Y_{i-1}}^{\\bot \\bot} \\preceq Y_{i-1}$ and ${Y_{i}}^{\\bot \\bot} \\preceq Y_{i}$.\nSuppose that ${Y_{i-1}}^{\\bot \\bot} \\neq {Y_{i}}^{\\bot \\bot}$.\nThen ${Y_{i-1}}^{\\bot \\bot} \\prec: {Y_{i}}^{\\bot \\bot}$ (by (\\ref{eqn:basic}) and Lemma~\\ref{lem:bot}~(1)).\nThus, for some $f_j$ $(1 \\leq j \\leq k)$, \nit holds ${Y_{i}}^{\\bot \\bot} = f_j \\vee {Y_{i-1}}^{\\bot \\bot}$.\nHere $f_j \\not \\preceq Y_{i-1}$ must hold.\nOtherwise ${Y_{i-1}}^{\\bot \\bot} \\succeq {f_j}^{\\bot \\bot} = {e_j}^{\\bot \\bot \\bot} = f_j$, \nwhich contradicts ${Y_{i-1}}^{\\bot \\bot} \\prec: {Y_{i}}^{\\bot \\bot} = f_j \\vee {Y_{i-1}}^{\\bot \\bot}$.\nAlso, $f_j \\preceq Y_i^{\\bot \\bot} \\preceq Y_i$.\nThus $Y_{i} = Y_{i-1} \\vee f_j$.\nTherefore, for each $i$ with ${Y_{i-1}}^{\\bot \\bot} = {Y_{i}}^{\\bot \\bot}$, \nwe can choose an atom $f$ with $Y_i = f \\vee Y_{i-1}$ to add to $f_1,f_2,\\ldots,f_k$, \nand obtain a required frame \n$\\langle f_1,f_2,\\ldots f_n \\rangle$ (containing ${\\cal X}^{\\bot}$ and ${\\cal Y}$).\n\n\n\n\n\n\\paragraph{Proof of Proposition~\\ref{prop:R(x,y)}.}\nConsider retractions $\\varphi \n:= \\varphi_{e_n,e_{n-1},\\ldots,e_{k+1},e_1,e_2,\\ldots,e_k}: {\\cal L} \\to \\langle e_1,e_2,\\ldots,e_n \\rangle$ \nand $\\phi := \\varphi_{f_1,f_2,\\ldots,f_n}: {\\cal M} \\to \\langle f_1,f_2,\\ldots,f_n \\rangle$; \nsee Lemma~\\ref{lem:frame}~(2) for definition.\nDefine a retraction $(\\bar \\varphi, \\bar \\phi): K({\\cal L}) \\times K({\\cal M}) \\to K(\\langle e_1,\\ldots,e_n,f_1,\\ldots,f_n \\rangle)$ by\n\\begin{equation*}\n(\\bar \\varphi, \\bar \\phi)(x,y) := (\\bar\\varphi(x),\\bar\\phi(y)) \\quad ((x,y) \\in K({\\cal L}) \\times K({\\cal M}))\n\\end{equation*} \nOur goal is to show that $(\\bar \\varphi, \\bar \\phi)$\ndoes not increase the objective value of P2.\n\nFirst we show\n\\begin{equation}\\label{eqn:phi_bot}\n(\\phi(Y))^{\\bot} = \\varphi(Y^{\\bot}) \\quad (Y \\in {\\cal M}).\n\\end{equation}\nIndeed, letting $F_i := f_1 \\vee f_2 \\vee \\cdots \\vee f_i$ and $E_i := e_1 \\vee e_2 \\vee \\cdots \\vee e_i$, we have\n\\begin{eqnarray}\n(\\phi(Y))^{\\bot} & = & \\left( \\bigvee \\{f_i \\mid i \\in [n]: Y \\wedge F_i :\\succ Y \\wedge F_{i-1} \\} \\right)^{\\bot} \\nonumber \\\\\n& = & \\left( V_0 \\wedge \\bigvee \\{f_i \\mid i \\in [n]: Y \\wedge F_i :\\succ Y \\wedge F_{i-1} \\} \\nonumber \\right)^{\\bot} \\nonumber \\\\\n& =& \\left( \\bigvee \\{f_i \\mid i \\in [k]: Y \\wedge F_i :\\succ Y \\wedge F_{i-1} \\} \\nonumber \\right)^{\\bot} \\nonumber \\\\\n& =& \\left( \\bigvee \\{f_i \\mid i \\in [k]: (Y \\wedge V_0) \\wedge F_i :\\succ (Y \\wedge V_0) \\wedge F_{i-1} \\} \\nonumber \\right)^{\\bot} \\nonumber \\\\\n& =& \\bigvee \\{U_0 \\vee e_i \\mid i \\in [k]: Y^{\\bot} \\wedge (U_0 \\vee E_i) :\\succ Y^{\\bot} \\wedge (U_0 \\vee E_{i-1}) \\} = \\varphi(Y^{\\bot}). \\nonumber\n\\end{eqnarray}\nThe second equality follows from \n$(V_0 + Z)^{\\bot} = V_0^{\\bot} \\cap Z^{\\bot} = {\\mathbb{K}}^n \\cap Z^{\\bot} = Z^\\bot$.\nThe third from the modularity:\nLet $A := \\bigvee \\{f_i \\mid i \\in [k]: Y \\wedge F_i :\\succ Y \\wedge F_{i-1} \\}$ and \n$B := \\bigvee \\{f_i \\mid i \\in [n] \\setminus [k]: Y \\wedge F_i :\\succ Y \\wedge F_{i-1} \\}$. \nThen $V_0 \\wedge B = {\\mathbb{K}}^n$ and $Y = A \\vee B$. \nThus we have $A = (V_0 \\wedge B) \\vee A = V_0 \\wedge Y$.\nThe forth follows from $f_i \\preceq V_0$ for $i \\in [k]$.\nThe fifth follows from Lemma~\\ref{lem:bot}~(4). Note \nthat by $Y^{\\bot} \\succeq U_0$ \neach atom $e_i$ with $i \\geq k+1$ is taken in the join of the definition (\\ref{eqn:retraction}) of $\\varphi = \\varphi_{e_n,e_{n-1},\\ldots,e_{k+1},e_1,e_2,\\ldots,e_k}$.\n \nNext we show\n\\begin{equation}\\label{eqn:R_phi}\nR(\\varphi(X), \\phi(Y)) \\leq R(X, Y) \\quad (X \\in {\\cal L}, Y \\in {\\cal M}).\n\\end{equation}\nIndeed, for $r = \\dim$, we have\n$\nR(\\varphi(X), \\phi(Y)) \n= r(\\varphi(X)) - r(\\varphi(X) \\wedge \\phi(Y)^{\\bot})\n= r(X) - r(\\varphi(X) \\wedge \\varphi(Y^{\\bot}))\n\\leq r(X) - r(\\varphi(X \\wedge Y^{\\bot}))\n= r(X) - r(X \\wedge Y^{\\bot}) = R(X,Y).\n$\nIn the second equality, \nwe use (\\ref{eqn:phi_bot}) and rank-preserving property of $\\varphi$. \nThe inequality follows from order-preserving property \n$\\varphi(X) \\wedge \\varphi(Y^{\\bot}) \\succeq \\varphi(X \\wedge Y^{\\bot})$.\n\nBy (\\ref{eqn:R_phi}), \nwe have $R(\\bar \\varphi(x), \\bar \\phi(y)) \\leq R(x,y)$; \nrecall the isometry between $K({\\cal L} \\times {\\cal M})$\nand $K({\\cal L}) \\times K({\\cal M})$ (Section~\\ref{subsub:K(L)}).\n Since $\\bar \\varphi$ and $\\bar \\phi$ are nonexpansive retractions (Proposition~\\ref{prop:basic_K(L)})),\nwe have $d(x^0,x) \\geq d(\\bar \\varphi(x^0), \\bar \\varphi(x)) = d(x^0, \\bar \\varphi(x))$ and $d(y^0,y) \\geq d(\\bar \\phi(y^0), \\bar \\phi(y)) = d(y^0, \\bar \\phi(y))$.\nThus, $(\\bar \\varphi, \\bar \\phi)$ has a desired property \nto prove the statement.\n\n\\section{A $p$-adic approach to nc-rank over ${\\mathbb{Q}}$}\\label{sec:p-adic}\n\nIn this section, we consider \nnc-rank computation of $A = \\sum_{i=1}^m A_i x_i$, where\neach $A_i$ is a matrix over ${\\mathbb{Q}}$. \nSpecifically, we assume that each $A_i$ is an integer matrix.\nAs remarked in Remark~\\ref{rem:bit}, \nthe algorithm in the previous section has no polynomial guarantee for the length of bits representing bases of vector subspaces.\nInstead of controlling bit sizes,\nwe consider to reduce \nnc-rank computation over ${\\mathbb{Q}}$ to\nthat over $GF(p)$ (for small $p$).\n\nFor simplicity, we deal with \nnc-singularity testing of $A$. \nHere $A$ is called {\\em nc-singular} if $\\mathop{\\rm nc\\mbox{-}rank} A < n$, \nand called {\\em nc-regular} if $\\mathop{\\rm nc\\mbox{-}rank} A = n$.\nWe utilize a relationship between nc-rank and the ordinary rank (on arbitrary field ${\\mathbb{K}}$).\nFor a positive integer $d$, the {\\em $d$-blow up} $A^{\\{d\\}}$ of $A$ is a linear symbolic matrix defined by\n\\begin{equation*}\nA^{\\{d\\}} := \\sum_{i=1}^m A_i \\otimes X_i,\n\\end{equation*}\nwhere $\\otimes$ denotes the Kronecker product and\n$X_i = (x_{i,jk})$ is a $d \\times d$ matrix with variable entries \n$x_{i,jk}$ $(i \\in [m],j,k \\in [d])$.\n\\begin{Lem}[\\cite{HrubesWigderson2015,Kaliuzhnyi-VerbovetskyiVinnikov2012}]\\label{lem:blowup}\n\tA matrix $A$ of form (\\ref{eqn:A})\n\tis nc-regular if and only if there is a positive integer $d$ such that $A^{\\{d\\}}$ is regular. \n\\end{Lem}\nThere is an upper bound of such a $d$. \nDerksen and Makam\\cite{DerksenMakam17} proved \na polynomial (linear) bound $d \\leq n-1$ \nby utilizing the {\\em regularity lemma} $d| \\mathop{\\rm rank} A^{\\{d\\}}$ in \\cite{IQS15a}.\nSuch bounds play an essential role in the validity of the algorithms of~\\cite{GGOW15,IQS15a,IQS15b}.\nInterestingly, our reduction presented below does not use any bound of $d$.\n\n\n\n\nFix an arbitrary prime number $p > 1$. \nLet $v_p: {\\mathbb{Q}} \\to {\\mathbb{Z}} \\cup \\{\\infty\\}$ denote the $p$-adic valuation:\n\\begin{equation*}\nv_p(u) := k\\ {\\rm if}\\ u = p^{k} a\/b, \n\\end{equation*}\nwhere $a, b$ are nonzero integers prime to $p$, and we let $v_p(0) := \\infty$.\nEvery rational $u \\in {\\mathbb{Q}}$ is uniquely represented as the $p$-adic expansion \n\\begin{equation}\\label{eqn:p-adic}\nu = \\sum_{i=k}^{\\infty} a_i p^i,\n\\end{equation}\nwhere $k= v_p(u)$ and $a_i \\in \\{0,1,2,\\ldots,p-1\\}$.\nThe leading (nonzero) coefficient $a_k$ is given as the solution of $a = b x \\mod p$. Then $u - a_k p^{k}$ is divided \nby $p^{k+1}$. Repeating the same procedure for $u - a_k p^{k}$, we obtain the subsequent coefficients in (\\ref{eqn:p-adic}). \n\nThe $2$-adic expansion of a nonnegative integer $z$ \nis the same as the binary expression of $z$, \nwhere $v_2(z)$ is equal to \nthe number of consecutive zeros from the first bit. \nThis interpretation holds for an arbitrary prime $p$.\nIn particular, \nthe $p$-adic valuation of a nonzero integer is bounded by \nthe bit-length in base $p$:\n\\begin{equation}\\label{eqn:vp_vs_bit}\nv_p (z) \\leq \\log_p |z| \\quad (z \\in {\\mathbb{Z}} \\setminus \\{0\\}).\n\\end{equation} \n\n\n\nThe $p$-adic valuation $v_p$ on ${\\mathbb{Q}}$ is extended to ${\\mathbb{Q}}(x_1,x_2,\\ldots,x_m)$ as follows.\nFor a polynomial $f \\in {\\mathbb{Q}}[x_1,x_2,\\ldots,x_m]$, define $v_p(f)$ by \n\\begin{equation}\\label{eqn:Gauss}\nv_p (f) := \\min \\{ v(a) \\mid \\mbox{$a$ \n\tis the coefficient of a term of $f$}\\}.\n\\end{equation}\nAccordingly, the valuation of \na rational function $f\/g$ is defined as $v_p (f)-v_p(g)$. \nThis is called the {\\em Gauss extension} of~$v_p$.\n\n\nOur algorithm for testing nc-singularity is \nbased on the following problem ({\\em maximum vanishing submodule problem; MVMP}): \n\\begin{eqnarray*}\n\t\\mbox{MVMP:} \\quad {\\rm Max}. && -v_p \\det P - v_p \\det Q \\\\\n\t{\\rm s.t.} && v_p (PA Q)_{ij} \\geq 0 \\quad (i,j \\in [n]), \\\\\n\t&& P,Q \\in GL_n ({\\mathbb{Q}}).\n\\end{eqnarray*}\nThis problem is definable for \nan arbitrary field with a discrete valuation, and the following arguments are applicable for such a field, while \n\\cite{HH_degdet} introduced MVMP\nfor the rational function field with one valuable.\n\nMVMP is also a discrete convex optimization \non a CAT(0) space. Indeed, \nits domain can be viewed as \nthe vertex set (the set of {\\em lattices}, certain submodules of ${\\mathbb{Q}}^n$) of \nthe Euclidean building for $GL_n({\\mathbb{Q}})$, and the objective function is an {\\em L-convex function}; see~\\cite{HH_degdet,HI_degdet}.\nA Euclidean building is a representative space admitting a CAT(0)-metric.\n\n\nThe optimal value of MVMP is denoted by \n$v_p \\mathop{\\rm Det} ' A\\in {\\mathbb{Z}} \\cup \\{\\infty\\}$, where we let $v_p \\mathop{\\rm Det} ' A := \\infty$\nif MVMP is unbounded. \nThe motivation behind this notation $v_p \\mathop{\\rm Det} ' A$ \nis explained in Remark~\\ref{rem:Det}. \n\n\n\n\n\n\nFor a feasible solution $(P,Q)$ of MVMP, \nconsider the $p$-adic expansion of \n$PA_i Q = \\sum_{k=0}^{\\infty} (PA_iQ)^{(k)} p^k$ for each $i$.\nThe leading matrix \n$(PA_i Q)^{(0)}$ consists of values $0,1,\\ldots,p-1$ and is considered in $GF(p)$.\nThen we can consider the linear symbolic matrix \n\\[\n(PAQ)^{(0)} := \\sum_{i=1}^m (PA_i Q)^{(0)} x_i \n\\]\nover $GF(p)$.\n\\begin{Lem}\\label{lem:Det<=det}\n\tFor a feasible solution $(P,Q)$ of MVMP, the following hold:\n\t\\begin{itemize}\n\t\t\\item[{\\rm (1)}] $- v_p \\det P - v_p \\det Q \\leq v_p \\det A$. In particular, $v_p \\mathop{\\rm Det} ' A \\leq v_p \\det A$.\n\t\t\\item [{\\rm (2)}]\tIf $(PAQ)^{(0)}$ is regular, \n\t\tthen $v_p \\det A = - v_p \\det P - v_p \\det Q = v_p \\mathop{\\rm Det} ' A$.\n\t\\end{itemize}\n\\end{Lem}\n\\begin{proof}\nThey follow from $\n0 \\leq v_p \\det P A Q = v_p \\det P + v_p \\det Q + v_p \\det A$.\nThe inequality holds in equality precisely \nwhen the leading matrix $(PAQ)^{(0)}$ \nis regular.\n\\end{proof}\nThe following algorithm for MVMP is due to \\cite{HH_degdet}, \nwhich originated from Murota's {\\em combinatorial relaxation algorithm}~\\cite{MurotaBook} \nand can be viewed as an descent algorithm on the Euclidean building. For an integer vector $z \\in {\\mathbb{Z}}$, \nlet $(p^{z})$ denote the diagonal matrix with diagonals \n$p^{z_1}, p^{z_2},\\ldots,p^{z_n}$ in order.\n\\begin{description}\n\t\\item[Algorithm: Val-Det]\n\t\\item[0:] Let $(P,Q) := (I,I)$.\n\t\\item[1:] Solve FR (or MVSP) for $(PAQ)^{(0)}$, and \n\tobtain optimal matrices $S,T \\in GL_n(GF(p))$ such that \n\t$S(PAQ)^{(0)}T$ has an $r \\times s$ zero submatrix in its upper-left corner.\n\t\\item[2:] If $(PAQ)^{(0)}$ is nc-singular, i.e.,\n\t$n < r+s$, then \n\tlet $(P,Q) \\leftarrow ((p^{- 1_{[r]}})SP, QT (p^{1_{[n] \\setminus [s]}}))$\n\tand go to step 1. Otherwise stop.\n\\end{description}\n \nThe initial $(P,Q)$ in step 0 is feasible with objection value $0$, as each $A_i$ is an integer matrix.\nIn step 2, $S, T$ are regarded as matrices in $GL_n({\\mathbb{Q}})$ with entries \nin $\\{0,1,\\ldots,p-1\\}$.\nObserve that each entry in the $r \\times s$ upper-left submatrix of $SPAQT$\nis divided by $p$.\nThus, the update in step 2 keeps the feasibility of $(P,Q)$.\nFurther, it strictly increases the objective value:\n$- v_p \\det (p^{-1_{[r]}}) S P - v_p \\det QT(p^{1_{[n] \\setminus [s]}}) \n= (r + s - n) - v_p \\det P - v_p \\det Q$.\nNote that $\\det S$ and $\\det T$ cannot be divided by $p$,\nsince $S$ and $T$ are invertible in modulo $p$. \nTherefore, nc-regularity of $(PAQ)^{(0)}$ \nis a necessary condition for optimality of~$(P,Q)$. \nIn fact, it is sufficient.\n\\begin{Prop}[{\\cite{HI_degdet}}]\\label{prop:optimality} \n\tA feasible solution $(P,Q)$ is optimal if and only if $(PAQ)^{(0)}$ is nc-regular. In this case, \n\t\tit holds $v_p \\mathop{\\rm Det} ' A \n\t\t= (1\/d) v_p \\det A^{\\{d\\}}$ for some $d > 0$.\n\\end{Prop}\n\\begin{proof}\n\tAs in \\cite[Lemma 4.2 (1)]{HI_degdet}, one can show\n\t$v_p \\mathop{\\rm Det} ' A = (1\/d) v_p \\mathop{\\rm Det} ' A^{\\{d\\}}$ for all $d$.\n\tBy Lemma~\\ref{lem:Det<=det}, $v_p \\mathop{\\rm Det} ' A \\leq (1\/d) v_p \\det A^{\\{d\\}}$ holds for all $d$. \n\t\n\tSuppose that $(PAQ)^{(0)}$ is nc-regular. \n\tIt suffices to show $v_p \\mathop{\\rm Det} ' A \\geq (1\/d) v_p \\det A^{\\{d\\}}$ for some $d$.\n\tBy Lemma~\\ref{lem:blowup}, \n\tfor some $d > 0$, \n\t$((PAQ)^{(0)})^{\\{d\\}} = ((PAQ)^{\\{d\\}})^{(0)} \n\t= ((P \\otimes I) A^{\\{d\\}} (Q \\otimes I))^{(0)}$ is regular. Observe that \n\t$(P \\otimes I, Q \\otimes I)$ is feasible \n\tto MVMP for $A^{\\{d\\}}$. By Lemma~\\ref{lem:Det<=det}~(2), \n\twe have $v_p \\det A^{\\{d\\}} =\n\t - v_p \\det P \\otimes I - v_p \\det Q \\otimes I \n\t = - d (v_p \\det P + v_p \\det Q) \\leq d v_p \\mathop{\\rm Det} ' A$. \n\\end{proof}\n\nFrom the proof and Lemma~\\ref{lem:blowup}, we have:\n\\begin{Cor}\\label{cor:regularity}\n\t$A$ is nc-regular if and only if $v_p \\mathop{\\rm Det} ' A < \\infty$.\n\\end{Cor}\n\nTherefore, {\\bf Val-Det} does not terminate if $A$ is nc-singular. \nA stopping criterion guaranteeing nc-singularity of $A$ is obtained as follows:\n\\begin{Prop}\\label{prop:bound}\n\tSuppose that each $A_i$ consists of integer entries whose absolute values are at most $D$.\n\tIf $A$ is nc-regular, then $v_p \\mathop{\\rm Det} ' A =O(n \\log_p nD)$.\n\tThus, $\\Omega(n \\log_p nD)$ iterations of \n\t{\\bf Val-Det} certify nc-singularity of $A$. \n\\end{Prop}\n\\begin{proof}\nSuppose that $A$ is nc-regular. By Proposition~\\ref{prop:optimality}, $v_p \\mathop{\\rm Det} ' A = (1\/d)v_p \\det A^{\\{d\\}}$ for some $d$.\nWe estimate $v_p \\det A^{\\{d\\}}$. The following argument is a sharpening of the proof of \\cite[Lemma 4.9]{HI_degdet}. \nRewrite $A^{\\{d\\}}$ as\n\\[\nA^{\\{d\\}} = \\sum_{i \\in[m],j,k\\in [d]} A_{i,jk} x_{i,jk},\n\\]\nwhere $A_{i,jk}$ is \nan $nd \\times nd$ block matrix with block size $n$ such that \nthe $(j,k)$-th block equals to $A_i$ \nand other blocks are zero.\nBy multilinearity of determinant, we have\n\\[\n\\det A^{\\{d\\}} = \\sum_{\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}} \\pm \\det A[\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}] x_{\\alpha_1}x_{\\alpha_2}\\cdots x_{\\alpha_{nd}},\n\\]\nwhere \n$\\alpha_\\gamma$ $(\\gamma \\in [nd])$ ranges over \n$\\{(i,jk)\\}_{i \\in [m], j\\in [d]}$ if \n$\\gamma$ belongs to the $k$-th block (i.e., $k= \\lceil \\gamma\/d \\rceil)$ and\n$A[\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}]$ is the $nd \\times nd$ matrix with the $\\gamma$-th column chosen from \n$A_{k,ij}$ with $\\alpha_\\gamma = (k,ij)$.\nA monomial in this expression is written as \n$a_z \\prod x_{k,ij}^{z_{k,ij}}$ for a nonnegative vector $z = (z_{i,jk}) \\in {\\mathbb{Z}}^{md^2}$ with $\\sum_{i,j} z_{i,jk} = n$ $(k \\in [d])$. The coefficient $a_z$ is given by \n\\[\na_z = \\sum_{\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}} \\pm \\det A[\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}],\n\\]\nwhere $\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}$ are taken so that \n$(i,jk)$ appears $z_{i,jk}$ times. \nThe total number of such indices is\n\\[\n\\prod_{k=1}^d \\frac{n!}{\\prod_{i,j} z_{i,jk} !} \n\\leq n^{nd}. \n\\]\nFrom Hadamard's inequality and the fact \nthat each column of $A[\\alpha_1,\\alpha_2,\\ldots,\\alpha_{nd}]$ has at most $n$ nonzero entries with absolute values at most $D$, \nwe have\n\\begin{equation}\n|a_z| \\leq n^{nd} (n^{1\/2}D)^{nd}\n\\leq n^{3nd\/2} D^{nd}.\n\\end{equation}\nTherefore, the bit length of $a_z$ in base $p$ is bounded by $O( nd \\log_p n D)$.\nBy (\\ref{eqn:vp_vs_bit}), we have $v_p \\det A^{\\{d\\}} = O(nd \\log n D)$. \nThus, $v_p \\mathop{\\rm Det} ' A = O(n \\log_p n D)$.\n\\end{proof}\n\nFor $p=2$, the algorithm {\\bf Val-Det} is executed as follows.\nInstead of updating $(P,Q)$, update $A$ as \n$A \\leftarrow (p^{-1_{[r]}})SAT(p^{1_{[n] \\setminus [s]}})$.\nThen, $A^{(0)}$ is computed as $(A_i)^{(0)} = A_i \\mod 2$.\nIn step 2, $S,T$ are $0,1$ matrices \nsuch that all entries of the $r \\times s$ corner of each $SA_iT$ \nare divided by $2$. \nHence, the next $A_i$ is again an integer matrix.\nThe bit-length bound of each entry in $A_i$ increases \nby $O(\\log_2 n)$ (starting from the initial bound $O(\\log_2 D)$).\nTherefore, until detecting nc-singularity of $A$, \nthe required bit-length is $O(n \\log_2 n \\log_2 n D)$. \n\n\n\n\n\n\n\n\\begin{Rem}[Valuations on the free skew field]\\label{rem:Det}\nAs shown by Cohn~\\cite[Corollary 4.6]{Cohn_valuation}, \nany valuation $v$ on a field ${\\mathbb{K}}$ is extended to \nthe free skew field ${\\mathbb{K}}(\\langle x_1,\\ldots,x_m \\rangle)$.\nThen we can consider the valuation $v \\mathop{\\rm Det} A$ \nof the Dieudonne determinant $\\mathop{\\rm Det} A$ of $A$.\nIf the extension $v$ is discrete and coincides with \nthe Gauss extension (\\ref{eqn:Gauss}) on ${\\mathbb{K}}\\langle x_1,x_2,\\ldots,x_m \\rangle$,\n then one can show by precisely the same argument in \\cite{HH_degdet} that $v \\mathop{\\rm Det} A$ is given by MVMP.\n\n Such an extension seems always exist; in this case, $v_p \\mathop{\\rm Det} ' = v_p \\mathop{\\rm Det} $. \n We verified the existence of an extension with the latter property\n (by adapting Cohn's argument in \\cite[Section 4]{Cohn_valuation}). \n However we could not prove the discreteness. \n Note that the arguments in this section is independent of the existence issue. \n\n\\end{Rem}\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe thank Kazuo Murota, Satoru Iwata, Satoru Fujishige, \nYuni Iwamasa for helpful comments, and \nthank Koyo Hayashi for careful reading.\nThe work was partially supported by JSPS KAKENHI Grant Numbers 25280004, 26330023, 26280004, 17K00029, and JST PRESTO Grant Number JPMJPR192A, Japan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{section:intro}\n\nBasic Life Support, or BLS, generally refers to the type of care that bystanders, first-responders, healthcare providers, and public safety professionals provide to anyone who is experiencing cardiac arrest, respiratory distress, or an obstructed airway \\cite{olasveengen2021european}. BLS mainly requires knowledge and skills in cardiopulmonary resuscitation (CPR), using automated external defibrillators (AED), and relieving airway obstructions in patients of every age. \n\nAccording to the American Heart Association's Heart and Stroke Statistics from the year 2020 \\cite{virani2020heart}, there are more than 356,000 out-of-hospital cardiac arrests (OHCAs) annually in the U.S., nearly 90 percent of them fatal. Similar results can be observed across Europe according to the EuReCa 2 report \\cite{grasner2020survival} from the year 2020, where a total of 37,054 OHCAs were confirmed where out of 8 percent of patients were discharged from hospital alive. \n\nIn the majority of cases, cardiac arrest is witnessed out-of-hospital, but emergency medical services may not arrive at the emergency location in a short time interval. Thus, execution of BLS and including steps such as resuscitation through bystanders becomes indispensable. However, survival rates of cardiac arrest victims could majorly increase if lay resuscitation rates would be higher. Despite the importance of becoming active in BLS and starting the foreseen steps, BLS skills are usually not trained regularly among people. Regular training, for instance in schools or at regular training courses, is missing in most cases. Thus, many people are not skilled in BLS procedures. The vital links for successful BLS are\ncommonly summarized with the chain of survival metaphor. It describes\nfour links for successful BLS: early recognition and call for help,\nearly bystander cardiopulmonary resuscitation (CPR), early defibrillation, and post-resuscitation care. As the chain of survival indicates, the need for bystanders to become active as early as possible is critical for increased survival chances. CPR, which describes the act of performing chest compressions and ventilation, and defibrillation with the aid of an Automated External Defibrillator (AED), are simple procedures that every adult person can execute.\n\nHowever, the question remains on how to teach people BLS skills and how to motivate them to refresh their skills regularly. Even though there is a large selection of training formats available, such as local Basic Life Support (BLS) training courses, online courses, or training sets including video tutorials on DVD and other materials, people may not be motivated or willing to pay for these courses or materials. Especially for younger people, who are accustomed to using modern technology gadgets, traditional\nteaching methods may feel outdated and may create a barrier concerning their self-motivation for training. In addition, these training formats are usually limited with regard to their capability to combine theoretical teaching of knowledge with practical skill training and assessment, such that they are suitable for independent self-training only to a limited extent.\n\nMotivated by the need to provide access to such BLS self-training approaches for everyone, modern training formats that utilize Augmented Reality (AR) or Virtual Reality (VR) technology were proposed in recent years. While VR and AR technologies have shown great potential to enhance BLS training in general and especially cardio-pulmonary resuscitation (CPR) performance, most of the existing approaches are in a rather prototypical state of implementation or do not leverage the full potential of an immersive learning environment for basic life support training. For instance, most of the existing approaches are not compliant with standardized medical guidelines for BLS or only focus on specific sub-steps of the BLS training such as CPR. Also, most of the existing approaches are experimenting by implementing different AR- or VR-based solutions, but still, the question remains which technology is most suitable for supporting BLS training in practice. To the best of our knowledge, there are no approaches that compare AR- and VR-based solutions with traditional BLS training approaches in terms of efficiency, effectiveness, and user satisfaction. Therefore, the following research questions can be raised:\n\n\\textbf{RQ1}: How can we exploit AR\/VR technology to support comprehensive BLS training in an efficient, effective, and user-friendly way?\n\n\\textbf{RQ2}: How does AR-\/VR-based BLS self-training perform in comparison to traditional training in terms of efficiency, effectiveness, and user satisfaction?\n\nTo answer the above-mentioned research questions, based on a collaboration with medical experts, we have identified the following requirements for the development of a novel VR- and AR-based BLS training environment to overcome the issues of the existing training approaches (see also Section 2 for a detailed analysis).\n\n\\begin{itemize}\n\\item \\textbf{R1 - Compliance with BLS Guidelines}: In 2021, the European Resuscitation Council (ERC) has published the latest basic life support guidelines which are based on the 2020 International Consensus on Cardiopulmonary Resuscitation Science with Treatment Recommendations \\cite{olasveengen2021european}. Procedures and techniques taught by the AR- and VR-based BLS training solution should be compliant with these ERC guidelines for BLS. All steps of the BLS procedure from \"Ensuring Scene Safety\" until \"Usage of AED\" should be covered to provide a comprehensive training approach.\n\\item \\textbf{R2 - Automated Assessment} \nThe AR- and VR-based training solution should be able to automatically assess user actions. For this purpose, the solution must be able to recognize user actions, match these actions to the predefined tasks of the BLS sequence, and finally evaluate the task execution. The assessment should cover the following aspects: correctness of task execution for each task, the correctness of task execution order for the whole BLS sequence, execution time, and CPR quality including compression rate and depth. \n\\item \\textbf{R3 - Real-Time Feedback and Debriefing}\nImplementation guidelines on BLS training emphasize the importance of debriefing and real-time feedback during training in order to increase learning effectiveness. Users should receive real-time feedback on their actions in visual or audible form\nwhen an action has been recognized or a task has been completed. In addition\nto these general feedback mechanisms that are valid for all steps, task-specific\nfeedback should be provided by the system during CPR by showing the current compression\nrate and depth. When a training session has finished, the system should present an overview of the training results (Debriefing). Results should show the list of tasks in combination with their individual completion result, i.e. whether the task has been completed correctly or not. Further, the system should provide hints for improvement where appropriate, such for instance when a task has been completed successfully, but a time constraint associated with the corresponding task has been exceeded.\n\\item \\textbf{R4 - Learning and Training Mode} There are two major purposes of the training environment: acquisition of knowledge and skills, and practical training of the acquired skills. To fulfill both, the training environment should be divided into two modes of operation. In the learning mode, each step of the training procedure, which by default is the BLS sequence, is taught to the user in a detailed manner by providing step-by-step instructions. In the training mode, users can test their BLS skills and knowledge in practice. While using this mode users can find out whether they are able to successfully complete all steps of a BLS scenario on their own and without external help. The steps and actions necessary to successfully pass the training mode are equivalent to the ones presented in the learning mode. However, in the training mode, the user will not get any explanations or additional help from the system. Instead, the system is built to automatically recognize and assess all the actions that the user performs.\n\\item \\textbf{R5 - Haptic Reality} To support an interactive and haptic learning environment, the AR- and VR-based training environment should be integrated with a haptic manikin to represent the victim who is in charge of BLS. The haptic manikin should deliver a hands-on learning experience for BLS and support the training of high-quality CPR especially with regard to compression rate and depth. \n\\end{itemize}\n\nOur contributions in this technical report are as follows: Firstly, we introduce a novel AR- and VR-based BLS training environment which tackles all the above-mentioned requirements. Our training environment consists of a generic training application kernel that can be instantiated for AR and VR usage scenarios. Furthermore, it integrates a custom build haptic manikin which supports an interactive learning mode where AR\/VR technology is combined with haptics. Apart from the constructive approach, we have conducted a usability evaluation where we analyze the efficiency, effectiveness, and user satisfaction of BLS training based on our AR and VR environment against traditional BLS training.\n\nThe rest of the technical report is structured as follows. In Section 2, we present and discuss the related work. In Section 3, we describe the conceptual solution of our AR- and VR-based BLS training environment. In Section 4, we show the details of its implementation. In Section 5, we present and discuss the main results of the usability evaluation. In Section 6, we conclude the technical report and give an outlook for future work. \n\\section{Related Work}\n\\label{section:relWork}\n\nAugmented Reality (AR) and Virtual Reality (VR) have been a topic of intense research in the last decades. In the past few years, massive advances in affordable consumer hardware and accessible software frameworks are now bringing AR and VR to the masses. While VR interfaces support the interaction in an immersive computer generated 3D world and have been used in different application domains such as training~\\cite{DBLP:conf\/vrst\/YigitbasJSE20}, robotics \\cite{DBLP:conf\/seams\/YigitbasKJE21}, education (e.g.,\\cite{DBLP:conf\/mc\/YigitbasTE20}, \\cite{DBLP:conf\/models\/YigitbasGWE21}), prototyping \\cite{DBLP:conf\/vl\/YigitbasKGE21}, or healthcare~\\cite{DBLP:conf\/mc\/YigitbasHE19}, AR enables the augmentation of real-world physical objects with virtual elements. In previous works, AR has been already applied for different aspects such as product configuration (e.g., \\cite{DBLP:conf\/hcse\/GottschalkYSE20}, \\cite{DBLP:conf\/hcse\/GottschalkYSE20a}), prototyping \\cite{DBLP:conf\/hcse\/JovanovikjY0E20}, planning and measurements (e.g., \\cite{DBLP:conf\/eics\/EnesScaffolding}, \\cite{DBLP:conf\/eics\/KringsYBE22}), robot programming \\cite{DBLP:conf\/interact\/YigitbasJE21}, or for realizing smart interfaces (e.g., \\cite{DBLP:conf\/eics\/KringsYJ0E20}, \\cite{DBLP:conf\/interact\/YigitbasJ0E19}). Besides this broad view of application domains, several approaches for AR- and VR-based training in the medical domain have been proposed in recent years \\cite{hsieh2018preliminary}. Improvements in modern AR and VR technology make researchers gain increasing interest in using this technology for medical education, including resuscitation training (\\cite{everson2021virtual}, \\cite{kuyt2021use}). Following the problem statement, research questions, and requirements overview as outlined in Section 1, we present and describe related approaches for AR- and VR-based BLS training. \n\n\\subsection{AR-based approaches}\n\nStrada et al.\\cite{strada2019holo} propose an AR-based self-training and self-evaluation tool for Basic Life Support and Defibrillation which is called Holo-BLSD. It utilizes Microsoft HoloLens to create a first aid emergency scenario where the user is confronted with a recumbent cardiac arrest victim lying on the floor. Several objects that are involved in a typical BLS scenario are modeled via holograms, such as a public phone station used for emergency calls, an AED, or other non-player characters (NPCs) that can be given instructions to. Holo-BLSD offers three modes of operation. In the learning mode, detailed instructions on the BLS procedure are given step by step via voice output from a virtual trainer. In the rehearsal mode, trainees are asked to execute the procedure by themselves, while still getting feedback on their actions from the application. Finally, in the evaluation mode, the trainees do not get any feedback, but their actions are assessed by the application. The automated assessment evaluates all steps except CPR quality. When performing CPR, only the compression frequency is automatically assessed, an automatic evaluation of compression depth and chest recoil is not included. The suitability of the proposed tool is analyzed by conducting a user study. However, there is no direct comparison with VR-based approaches or traditional training which can be used as a basis to address our stated research questions. \n\nHoloCPR by Johnson et al. \\cite{johnson2018holocpr} is a further AR-based approach that targets helping uneducated bystanders with performing timely resuscitation of a cardiac arrest victim. This approach makes use of simple visualizations, such as arrows, circles, texts, and simple animations, in order to give instructions to the user and guide the user's attention towards points of interest. There is no virtual emergency scenario created, and there are no virtual environmental objects or persons modeled that the user could interact with. To demonstrate the effectiveness of their proposed application, the authors conduct a between-subjects experiment with several participants. Experiment results indicate that, while using HoloCPR, participants show reduced initial reaction times and are faster in interpreting CPR instructions given by the AR system as compared to traditional 2D instructions, resulting in faster action. Further, HoloCPR facilitates procedural adherence, i.e. it is easier for HoloCPR users to follow individual steps of the BLS procedure. However, HoloCPR is not specifically designed as a BLS training application as it does not fully conform to the BLS guidelines. Also, it enables no further insides about a comparison to traditional training methods.\n\nJavaheri et al. \\cite{javaheri2018stayin} combine a traditional training manikin and Microsoft HoloLens with a dedicated client-server application running on a third device. The training manikin is equipped with several pressure sensors, which are able to measure data on CPR compression depth and frequency. Via the client-server application, direct data communication between the manikin's sensors and the HoloLens is possible, such that direct feedback on current CPR quality can be given to the user in real-time. The sensor-based approach allows for high-quality automated assessment of CPR performance.\nSimilar to the previous approaches, the application comprises different modes of operation. In the first phase, a virtual trainer explains the basics of how to perform CPR. In the second phase, the learned techniques have to be practically applied to the physical training manikin, and the trainee's CPR performance is automatically assessed. In both phases, the application can be controlled via voice commands. While this approach supports a detailed CPR tutorial it is not comprehensive enough to cover a whole BLS training process. Furthermore, a usability evaluation and comparison against traditional approaches are missing such that our research questions are not fully tackled by this approach. \n\nSimilar to the previous approach, Kwon et al. \\cite{kwon2014heartisense} developed a CPR training system employing a dedicated sensor kit that is inserted into a traditional training manikin. Via these sensors, the CPR quality can be measured in terms of compression location, depth, and frequency. Further, sensors can measure the strength and volume of artificial respiration, and they can detect whether the victim's airways have been cleared. The concept has been implemented in two ways: As an AR training simulator using the projection method, and as a mobile application. In both cases, visual and auditory feedback is given, for instance when the user carries out correct or incorrect activities, respectively. An automated assessment of the user's CPR performance is given, and hints for possible improvements are provided. By conducting a user study, the authors analyze the effectiveness of their proposed system. They conclude that their system is able to provide accurate training assessment and that by using their system the retention of training content can be enhanced. However, the proposed system focuses on CPR training and assessment, thus it is not suitable for comprehensive BLS training. A dedicated learning mode in which the required methods and techniques are taught is not part of the work.\n\nIn the ARLIST project conducted by Pretto et al. \\cite{pretto2009augmented}, an AR environment for general life support training has\nbeen developed and evaluated. The main goal of the project is to enhance traditional training by augmenting it with auditory feedback and visualizations in the form of AR projections onto the training manikin. For instance, facial expressions and skin injuries projected onto the manikin reflect the physical state of the virtual patient, and cardiac or pulmonary sounds played via speakers installed in the back of the manikin's neck enable the trainee to autonomously assess the patient's consciousness. Additionally, a waistcoat and an adapted stethoscope have been created that allow the trainees to execute cardiac and pulmonary auscultation during training. Actions carried out by the trainees are automatically recorded\nduring training, and the created logs can later be viewed by instructors via a companion tool that has been developed along with the AR environment. In a user study, the authors evaluate whether their proposed system is able to increase the realism of life support training. They conclude that their system improves accuracy and objectivity as well as the trainees' autonomy during training.\n\nBalian et al. \\cite{balian2019feasibility} conducted a feasibility study of an AR CPR training approach that seeks to supplement traditional training with the potential advantages AR technology offers. The system uses an AR headset (HoloLens) to show a model of the human circulatory system to the user next to a Laerdal recording CPR manikin. To provide feedback to the user the model changes the rate of blood flow displayed proportional to the rate of chest compressions applied. Further, to train consistent CPR the system uses an audible heartbeat of 110 bpm when applied compression rate is not within the tolerance of \u00b110 bpm to remind and correct the user. The authors conducted a study with 51 health care providers using their AR system and collected performance data and quantitative survey data. The survey responses showed promising results concerning the usefulness of the blood flow\nvisualization and the willingness to use the system again [34]. The authors aggregated similar responses to open-ended questions. The most liked\nfeature was the real-time audio-visual feedback. However, the two most common suggested improvements were placing the blood flow simulation above the manikin (not to the side), and providing users with real-time performance\nstatistics. While this approach addresses many requirements, it is specialized in CPR and does not cover the whole BLS training. Also, real-time performance statistics are missing and comparison against traditional training methods has not been conducted. \n\nSimilarly, Ingrassia et al. \\cite{ingrassia2020augmented} conducted a usability study of an AR system for basic life support training. The system uses the HoloLens and overlays an AR scenario onto a CPR training\nmanikin equipped with computer vision markers. This system does not only focus on CPR but demonstrates the breadth of potential for AR\nsystems. The methods of interaction enabled by the HoloLens\nare introduced individually to lead the participants through a\nseries of AR simulated basic life support actions. These include\nsafety, checking responsiveness, contacting, and interacting with\nemergency services, and AED retrieval and application. A total\nof 26 participants were guided through the multistep AR simulated\nresuscitation procedure and recorded survey responses to\nidentify the system usability and ISO 9241-400 ergonomics on\na Likert scale. Results show that the system was easy to use, comfortable,\nand required minimal cognitive effort. While this approach supports the wider application of AR training for basic life support, it is not leveraging insides about the efficiency, effectiveness, and user satisfaction of such a solution compared to traditional BLS training. \n\n\\subsection{VR-based approaches}\n\nVReanimate II \\cite{bucher2019vreanimate} (successor of VReanimate\\cite{blome2017vreanimate}) is a VR application intended for non-verbal guidance in virtual first aid scenarios. In VReanimate, no text or speech is used. Giving instructions via graphical visualizations only avoids the need for the user to read and interpret texts during training, thus allowing the creation of fast-paced, uninterrupted real-world\nscenarios. The application is based on the ERC 2015 life support guidelines \\cite{perkins2015european} and comprises the basic steps of the BLS algorithm. However, a fully automated assessment of the trainee's performance is not part of the application. The authors evaluate the usability and effectiveness of their approach by conducting interviews and an experiment.\n\nEMERGENZA \\cite{ferracani2015natural} is a serious game designed for the training of emergency medical personnel. Different virtual emergency scenarios are created, including NPCs that the user can interact with. The scenarios are designed to train the basic life support procedure. However, since this approach is completely virtual and does not make use of a dedicated training manikin, high-quality CPR training cannot be provided. The application is controlled via gestures. The authors utilize the Microsoft Kinect in order to translate the user's gestures into the virtual environment. All actions performed during training are logged and can later be used for debriefing. A usability evaluation of the proposed system is conducted in the form of a small user study. Training effectiveness is not evaluated.\n\nWhile the above-described approaches have the BLS training in scope, there are also specialized VR-based approaches focusing on CPR training. Semeraro et al. \\cite{semeraro2009virtual}, for instance, demonstrated traditional CPR training enhanced with VR in 2009 with a system using a VR headset, a CPR training manikin (Laerdal HeartSim 4000), data gloves, and tracking devices. The VR headset displays a first-person perspective cardiac arrest simulation to the user, and the data gloves track the user's hand position for representation within the virtual environment. The researchers use audio and visual detail with the intent to make the training scenario more realistic. A refined prototype VR CPR standard manikin training system with hand tracking and handsfree Kinect motion detection was presented in their follow-up work \\cite{semeraro2019virtual}. This system was trialed with 43 medical students. The test collected compression rate and compression depth simultaneously\nwith Resusci-Anne and the developed motion tracking tool. The results showed that the data collected by both systems were comparable demonstrating that it is possible to collect CPR metrics externally from the manikin and use them in a VR scenario. Furthermore, Vaughan et al. \\cite{vaughan2019cpr} demonstrated a VR CPR training system specifically for schools. Similarly, it uses a commercially available VR headset (Oculus Rift) and a hand tracking solution that does not require gloves (Leap Motion). The simulation presents\ncommon incident scenarios and utilizes a 3-D scan of the training\nmanikin used to ensure correct visual-physical alignment. The\nauthors stress the importance of CPR training at an early age and\npoint to the growing trend in medical simulation to underpin the\ndecision to develop this system. Finally, Almousa et al. \\cite{almousa2019virtual} developed a VR-CPR system that focuses on the gamification of CPR training. This approach also uses a first-person perspective simulation of a cardiac arrest patient and uses hand\ntracking for hand position measurement relative to the manikin,\ndetermining compression rate and compression depth. The novelty of this system is the exploration of gamification features, real-time feedback, and a progressive difficulty system. As the difficulty is advanced, the user experiences reduced feedback and guidance and increased distractions. Competition with other trainees is incentivized with a leader board,\nto encourage repeated training and prevent degradation of CPR\nskills over time. Evaluation results of their VR-CPR solution\ncompared with traditional CPR training methods show that the testers liked the idea of gamification and found it exciting and engaging. While the above-described approaches combine innovative ideas such as VR and gamification to enhance CPR training, their scope is restricted to CPR and comprehensive BLS training is not covered. Furthermore, a detailed comparison between VR-, AR-based training versus traditional training is not covered in a comprehensive usability evaluation such that strengths and weaknesses of each technology can be identified. \n\n\\subsection{Discussion}\n\nA summary overview of the presented related work approaches is depicted in Figure \\ref{fig:relWOrk}. As it can be seen in the overview, none of the proposed systems are fully compliant with the new ERC basic life support guidelines \\cite{olasveengen2021european} in terms of the taught methods, techniques, and procedures. While some of the existing approaches focus on specific steps of the BLS training, the necessary BLS steps are usually not included in their entirety. For instance, assessment of respiration checks and airway clearance is only included in one of the systems \\cite{kwon2014heartisense}. Since properly executed CPR is one of the most important steps in life\nsupport, the majority of the works focus on this topic. High-quality automatic\nassessment of CPR performance is non-trivial to achieve, as it usually requires the\nemployment of dedicated sensors that are attached to or inserted into the training\nmanikin. Thus, automated assessment of CPR quality is only included in those\napproaches that put their entire focus on CPR assessment, but do not create a\ntraining environment that can be used for independent self-training which covers the whole BLS process. An important feature for virtual training systems is a dedicated learning mode that teaches necessary skills. Without teaching skills, trainees are not able to learn how to perform actions properly, and they may not be able to recognize their mistakes.\nThus, training becomes ineffective, and independent self-training without\na human instructor is not possible. Those systems that are meant to be used for independent self-training usually include such a learning mode such as \\cite{strada2019holo} or \\cite{semeraro2009virtual}. However, most of the existing approaches do not separate between a training and exam mode to systematically enhance the learning outcomes of the training session. \nThe comparison shows that a complete and automatic assessment of the trainees'\nperformance is difficult. None of the considered approaches, except the VR-based approach by Almousa et al. \\cite{almousa2019virtual}, includes a fully automated assessment for all steps of the BLS procedure. Another important limitation of the considered approaches is the lack of an included debriefing feature. Some approaches state that data acquired during training are logged and those data can later be used for debriefing. However, debriefing is never part of the existing AR and VR application as such but is always carried out via separate companion tools, such as a usual desktop application. In summary, none of the existing approaches come up with a comprehensive AR\/VR BLS training environment that fulfills all stated requirements. Furthermore, there is no existing work that provides a detailed comparative usability analysis to identify benefits and limitations of AR-\/ and VR-based solutions for BLS training in terms of efficiency, effectiveness, and user satisfaction compared with traditional methods (R6 which relates to RQ2). \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\linewidth]{figures\/RelWork\/RelatedWorkOverview.pdf}\n \\caption{Related Work Overview}\n \\label{fig:relWOrk}\n\\end{figure}\n\n\n\n\\section{Solution Overview}\n\\label{section:solOverview}\n\nTo address the first research question (RQ1) motivated in Section 1, we have designed a novel AR- and VR-based BLS training environment which tackles the described requirements R1-R5. A high-level architectural overview of our training environment is shown in Figure \\ref{fig:solOverview:HighLevel}.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=.8\\linewidth]{figures\/SolutionOverview\/HighLevelOverview.pdf}\n \\caption{High-level Solution Overview}\n \\label{fig:solOverview:HighLevel}\n\\end{figure}\n\nIn the following, each component of the architectural overview will be described in more detail. In Section \\ref{section:solOverview:BLSSequence}, we first present the ERC guideline compliant process, which builds up the basis for creating our BLS training environment. With that in mind, we present in Section \\ref{section:solOverview:Software} the concept for the software part of the training environment, followed by Section \\ref{section:solOverview:Hardware}, where we present the concept of the custom hardware for realizing an interactive and haptic training environment.\n\n\n\\subsection{BLS Sequence}\\label{section:solOverview:BLSSequence}\nAs a basis for our training, we use the ERC guidelines for BLS based on \\cite{olasveengen2021european} (R1). With them, we derived the process shown as a UML activity diagram in Figure \\ref{fig:eval:blsProcess}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/SolutionOverview\/BLS_PROCESS.pdf}\n \\caption{BLS Process}\n \\label{fig:eval:blsProcess}\n\\end{figure}\n\n\nFirst, the user has to ensure that the scene is safe and nobody is in danger. After that, it needs to be checked whether the victim is responding. The user should talk to the victim and shake their shoulders to see if there is a reaction. If the victim is not reacting, the airways need to be opened, so that the breathing can be checked afterward. If there are other helpers present, the breathing status should be communicated. After that, the user should make an emergency call (or ask somebody else to do so) and send somebody to get a defibrillator. In a real accident, the victim should never be left alone. Thus, if no helpers are nearby an AED is not used. In the next step, cardiopulmonary resuscitation (CPR) has to be performed. That contains two parts, compressions and ventilations. For the compressions, the rate (compressions per minute) and the depth are evaluated. After 30 compressions, the victim needs to be ventilated. If a defibrillator (AED) is available it should be used immediately. First, the AED pads need to be placed. Then, the AED analyses the heart rhythm. During that time, nobody should touch the victim. So the user has to communicate to other helpers, that they should stand back. After analyzing, the AED will display whether a shock is necessary. If that is the case, the user needs to trigger the shock. In our training process, that is the last step to execute, in real-life, the CPR should be continued until the ambulance arrives.\n\n\n\\subsection{Software}\\label{section:solOverview:Software}\nAs in AR applications the users still see their environment, we decided to create a scenario close to real resuscitation training. The user is in a safe space (e.g. a training center or an office) and stands in front of a resuscitation manikin simulating the victim. We then use augmented reality to explain what to do and give feedback on the executed steps. Everything shown to the user should be correctly placed around the manikin. So we use the manikin as an anchor to sync the position of the virtual objects with the real world.\n\nFor VR applications we are not bound to the place the user is in. We can create whatever environment we want. To keep it simple, we decided to use a scenario where the victim is in an encapsulated garden. Here, we do not need a detailed resuscitation manikin as for AR, but we can use a simplified version. Despite the environment and the manikin, the application itself should be similar for AR and VR. Thus, in the following, we present the concept for the applications independent from the used technology.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/SolutionOverview\/FATOverview.pdf}\n \\caption{Architectural Overview}\n \\label{fig:solOverview:ApplicationOverview}\n\\end{figure}\n\nFigure \\ref{fig:solOverview:ApplicationOverview} shows the overview of the BLS Training architecture. \n\\paragraph{Task Management}\nTo allow the application to keep track of which steps the user executed, and how well it was done, we developed a \\textit{Task Management}. That consists of a \\textit{Task List} which we derive from the process described in the previous section. Every step in the process corresponds to a \\textit{Task} the user needs to execute. The training starts with the first task and guides the user sequentially through the process. After one task was finished, the next task is selected. With that, we can record how well a task was executed, and automatically assess the user's performance (R2).\n\n\\paragraph{Building Blocks}\nAs the task management only realizes the basic logic of the process, we created several \\textit{Building Blocks} which can be activated by a task and realize the interaction of the user with the application.\n\n\\begin{itemize}\n \\setlength\\itemsep{.5em}\n \\item[] \\textit{Text\/Audio.} We needed a possibility to display static information to the user. That can be done in multiple ways. For conventional training, there exist texts or images to describe a certain step, or a trainer explains and demonstrates it. For our applications, we combined all those possibilities to retrieve the highest possible learning effect. We created \\textit{Texts} describing the current step, how to execute it, and why it needs to be executed. These texts have been derived from the ERC guidelines for BLS. As it can be tiring for a user to only read text, and to simulate a trainer explaining the content, we provide \\textit{Audio} files of the text being read. In the case users missed something or have problems understanding the language, we additionally show the text itself. In addition to the texts, we show corresponding images depicting the task. \n \n \\item[] \\textit{Animations.} Until now, we adapted techniques used in real life so we can use them in our application. But as we are in a digital environment we can control, we also created \\textit{Animations} showing how the described steps are executed. With that, a user can walk around and watch the animation from the spot they like to fully understand how to execute a step properly. \n \n \\item[] \\textit{(Interactable) Objects}. In real accidents, multiple helpers are likely to be present at the accident scene. Thus it is also important, that users learn how to communicate with them and where tasks can be delegated to someone else. We provide an avatar representing other helpers. As the users cannot directly interact with it, we call it a (static) \\textit{Object}. Additionally, we needed some \\textit{Interactable Objects} the user can directly interact with. For our BLS training, we created \\textit{broken glasses, a mobile phone, and a defibrillator with its electrode pads}. \n \n \\item[] \\textit{Voice Commands.} To evaluate if a user correctly communicated with bystanding helpers, we integrated the possibility to recognize \\textit{Voice Commands} which corresponds to the information and tasks given to them. \n \n \\item[] \\textit{Manikin\/CPR simulator.} Chest compressions are a central aspect of the BLS sequence. To increase immersion and realism, we created a haptic \\textit{Manikin\/CPR Simulator}. In Section \\ref{section:solOverview:Hardware}, we explain the simulator's hardware concept in more detail. \n While performing compressions, it is substantial that the rate and depth are correct. So we display them both, so the user can easily see how they are doing. \n \n \\item[] \\textit{Position Trigger}. When interacting with the manikin, the user's head or hands need to be in a specific position. For example, when ventilating the victim, the user's head is right above the victim's head. To retrieve that information, we created \\textit{Position Triggers} which raise an event after the correct position is reached.\n \n\\end{itemize}\nTo let the users know, if they are doing well, we give them real-time feedback (R3) while they execute the training. Whenever a task was executed correctly, the user gets notified by a sound cue. In addition, during CPR, we also display the current compression depth and rate.\n\nWith that, we described our concept for the \\textit{learning} part of the application where we offer explanations, descriptions, and guidance through the training. \n\nIn addition to that, we developed a \\textit{training} mode (R4). In the \\textit{training} mode, explanations and guiding elements are omitted so the users have to do everything on their own. \n\n\\paragraph{Debriefing}\nWhen a training session has finished, the system presents an overview of the training results. Results show the list of tasks in combination with their completion result, i.e. whether the task has been completed correctly or not. Further, the system provides hints for improvement where appropriate, such as when a task has been completed successfully, or a time constraint associated with the corresponding task has been exceeded. In addition to the real-time feedback during the execution, results for CPR (average compression rate and depth and whether the chest was always fully released) are also presented during the debriefing. The application can also give specific improvement suggestions, for instance when the compression rate could be optimized in one or the other direction.\nTo not overwhelm the user, we created two levels of detail. First, an overview of all the tasks, and second, more detailed insight into a single task.\n\n\\subsection{Hardware}\\label{section:solOverview:Hardware}\nAs working on\/with the victim is an essential part of BLS, we need some kind of hardware to train that. That is especially important for performing chest compressions. Without real-life hardware, the users would perform compressions in the air or on the ground without realistic resistance. In addition to that, we could not precisely assess the rate and depth of the compressions. For conventional training, there are resuscitation manikins available to buy. Standard manikins which allow chest compressions and ventilation are mostly used in conventional training. But, as we want to evaluate the rate and depth of the compressions, we need sensors to be integrated into the manikin. \n\n\\textit{Manikins with sensors} are also commercially available, such as the Laerdal Resuci Anne\\footnote{\\url{https:\/\/laerdal.com\/us\/products\/simulation-training\/emergency-care-trauma\/resusci-anne-simulator\/}}. They are mostly used to train medical professionals and thus consist of many complex sensors. \nAs these commercially available professional manikins are usually closely tied to their own companion tools and apps, accessing their sensor data is not easily possible. Thus, we have explored an alternative solution where we created our own CPR manikin, which is equipped with only those sensors we need for our application (R5).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/SolutionOverview\/SensorCommunication.pdf}\n \\caption{Sensor Communication}\n \\label{fig:solOverview:Hardware:SensorCommunication}\n\\end{figure}\n\nFigure \\ref{fig:solOverview:Hardware:SensorCommunication} shows an overview of the connection between the CPR simulator and the application. The application can connect to a wifi chip and send commands e.g. to start\/stop sending data from a sensor. The chip then reads the sensor's value and sends it back to the application. Here, the user should be provided with real-time feedback on the compression rate and depth. Additionally, for later analysis, the received values are logged. \n\n\\paragraph{AR}\\label{section:solOverview:Hardware:AR}\nSince in AR applications the users' real environment is perceivable, we utilize a real manikin for the training. As a starting point, we employed a plastic manikin and inserted springs into the chest to allow compressions. We installed sensors in both the mannequin's chest and head. These are used for measuring compression depth during CPR and for recognizing a head tilt during airway clearance.\n\n\\paragraph{VR}\\label{section:solOverview:Hardware:VR}\nFor VR, the compression simulation can be simpler. As the users do not see the real world, we do not need a correct representation of a resuscitation manikin. In addition, every user interaction happens in the virtual environment, so we do not need a realistic manikin. Instead, for the VR hardware, we only need a device the user can push and which can measure the compressions similar to the manikin for the AR application, omitting all components used to increase the immersion.\n\\section{Implementation}\n\\label{section:impl}\nThis section describes the implementation of our AR- and VR-based BLS training environment. First, in Section \\ref{section:impl:Hardware}, we describe how we built the hardware components with which the user interacts, followed by Section \\ref{section:impl:Software}, where we present the software implementation of the two applications. In Section \\ref{section:impl:Software:Walkthrough}, we present a walkthrough of both applications and explain which of the developed building blocks were used to create the training applications. Finally, in Section \\ref{section:impl:Debriefing} we present how we realized the debriefing mode for our training.\n\n\n\\subsection{Hardware}\\label{section:impl:Hardware}\n\n\\subsubsection{AR}\\label{section:impl:Hardware:AR}\nThe resuscitation manikin is a customized display dummy. Figure \\ref{fig:implementation:Hardware:DollSchematic} shows the main components, Figures \\ref{fig:implementation:Hardware:Mesut1}-\\ref{fig:implementation:Hardware:Mesut3} show the final result of our custom manikin. \n\n\nFirst, we cut out a part of the chest and filled the space with springs, so that the chest can be pushed down by the user. To measure the depth, we use an ultrasonic sensor which measures the distance between the ground and the chest top. To retrieve the tilt angle of the head (a user needs to overextend the neck to open the victim's airways), we attached a gyro sensor to it. Both sensors are connected to an \\texttt{ESP-8266} wifi chip. The chip hosts a simple server, to which the application can connect. It can then send simple commands to control the chip. For example, the application can request data for both sensors or tell the chip to stop sending data. When data is requested, the chip checks the corresponding sensor and sends its value together with a timestamp to the application. \n\\begin{figure}\n \\begin{center} \n \\subfloat[Resuscitation Manikin Schematic\\label{fig:implementation:Hardware:DollSchematic}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/MesuSchematic.pdf}\n }\\hspace*{3pt}\n \\subfloat[Top View\\label{fig:implementation:Hardware:Mesut1}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/mesut1.pdf}\n }\n \\end{center}\n \n \\begin{center}\n \\subfloat[Bottom View\\label{fig:implementation:Hardware:Mesut2}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/mesut2.pdf}\n }\\hspace*{3pt}\n \\subfloat[Manikin as Seen by User\\label{fig:implementation:Hardware:Mesut3}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/mesut3.pdf}\n }\n \\end{center}\n \\caption{Custom Resuscitation Manikin}\n \\label{fig:implementation:Software:AR:Objects}\n\\end{figure}\n\n\n\\subsubsection{VR}\\label{section:impl:Hardware:VR}\nFor the VR application, we do not need a realistic resuscitation manikin, as the user only sees the virtual world. Taking this into account and additionally to provide an independent solution for the VR side, we created a simpler CPR simulator dedicated to the VR-based BLS training environment. Figure \\ref{fig:implementation:Hardware:VR:Sim} shows the simulator used for the VR application. Also, the sensor for the head's angle is not needed anymore as that is also tracked within the application. So we only need an ultrasonic sensor for the compressions and the \\texttt{ESP-8266} wifi chip to communicate with the application. As the applications can control when and which sensor data is sent, we could use the same code as for the AR application. The VR application will not request data from the gyro sensor, so it does not matter if there is no sensor connected.\n\\begin{figure}\n \\begin{center} \n \\subfloat[Top View\\label{fig:implementation:Hardware:VR:SimTop}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRCPRSimTop.pdf}\n }\\hspace*{3pt}\n \\subfloat[Bottom View\\label{fig:implementation:Hardware:VR:SimBottom}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRCPRSimBottom.pdf}\n }\\hspace*{3pt}\n \\subfloat[Side View\\label{fig:implementation:Hardware:VR:SimSide}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRCPRSimSide.pdf}\n }\n \\end{center}\n \\caption{Custom Resuscitation Manikin}\n \\label{fig:implementation:Hardware:VR:Sim}\n\\end{figure}\n\n\\subsection{Software}\\label{section:impl:Software}\nTo develop our applications, we used the Unity\\footnote{\\url{https:\/\/unity.com\/}} game engine. We developed two independent applications, one in AR and one in VR, which have basically the same functions but use different technologies to create them.\n\n\\subsubsection{AR}\\label{section:impl:Software:AR}\nThe AR application was developed for the \\textit{Microsoft Hololens 2}\\footnote{\\url{https:\/\/www.microsoft.com\/en-us\/hololens}}. \nWe were using Unity version 2019.4.3f1 together with the Mixed Reality Toolkit (MRTK)\\footnote{\\url{https:\/\/github.com\/Microsoft\/MixedRealityToolkit-Unity}} version 2.4.0.\nThe MRTK offers a set of components and features that facilitate the development of mixed reality applications. For instance, via the \\textit{InputSystem}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/windows\/mixed-reality\/mrtk-unity\/features\/input\/overview?view=mrtkunity-2021-05}} provided by the MRTK, head, gaze, and gesture inputs that a user executes on HoloLens can be easily retrieved. This is necessary to facilitate the recognition of user actions which is required for the automated assessment capabilities of the training system. To achieve advanced tracking techniques beyond the scope of features offered by the MRTK and the HoloLens as such, Vuforia Engine\\footnote{\\url{https:\/\/developer.vuforia.com\/}} version 9.4.6 has been utilized. Vuforia has been chosen as a technology because it offers great compatibility with Unity and provides the right amount of functionality even in its free version. Vuforia provides a set of advanced recognition features that allow recognition and tracking of real-world objects. For the training system developed\nhere, the most important feature offered by Vuforia is the recognition of image targets. Image targets are based on flat images, for instance, a simple QR code, that Vuforia can detect in the real environment. Image targets are created via a web interface in the Vuforia Engine Developer Portal and are stored locally within a device target database that can be imported into a Unity project. The image targets can then be added to a scene within Unity. We used image tracking to synchronize the virtual with the real world. To achieve this, an image marker is placed on the belly of the manikin. When the application starts, that image needs to be scanned by looking at it from a short distance while wearing the HoloLens. After the position is set, we can remove the image marker from the manikin so it does not distract the users. \n\nWe developed the application based on the concept described in Section \\ref{section:solOverview:Software} and by realizing the architecture depicted in Figure \\ref{fig:solOverview:ApplicationOverview}.\n\n\\paragraph{User}\nThe User wears the Hololens 2 with which they interact with the training application. The application itself consists of several components. First, there are the \\textit{Building Blocks} that provide the functionality to interact with the user.\n\n\\paragraph{Text\/Audio}\nTo convey content to the user, we implemented different elements.\nTo provide a \\textit{Text} to the user, we created simple description panels. In this dashboard, we can display arbitrary text using the \\textit{TextMeshPro}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/com.unity.textmeshpro.html}} package. Next to the text, there is a place to additionally integrate images as 2D \\textit{sprites}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/Sprites.html}}. The text content and the image can be changed by a script. Additionally, we provide a checklist that was created based on the tasks we defined for the user. It shows which tasks to do and which tasks are already done. \nTo provide the user a better overview of the current, previous, and next steps in the BLS sequence, we show a breadcrumb-like overview of the tasks during the whole procedure.\nThe dashboards and breadcrumbs can be seen in the top right corner of the scene displayed in Figure \\ref{fig:implementation:Software:AR:AROverview}.\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\linewidth]{figures\/Implementation\/AROverview.pdf}\n \\caption{AR Scene as Displayed to the User}\n \\label{fig:implementation:Software:AR:AROverview}\n\\end{figure}\n\nAs a next step, we developed a system to read the displayed text as \\textit{audio} to the user. We decided to record the texts in high quality rather than using a text-to-speech system to increase the immersion and better simulate real-life training. Inside Unity, audio files are used as \\textit{AudioClips}\\footnote{\\url{https:\/\/docs.unity3d.com\/ScriptReference\/AudioClip.html}}. The audio tracks are played when the corresponding task starts and the displayed text is updated so that it fits with the recording.\n\n\\paragraph{Animations}\nTo show the user \\textit{Animations} about what to do, we first needed a possibility to create such animations. As it is very time-consuming to create all animations by hand, we exploited the HoloLens' head, hand, and finger tracking capabilities in order to record the movement of the head, the hands, and the fingers by saving their position and rotation in fixed time intervals. We then converted these recordings into Unity \\textit{animations}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/AnimationSection.html}} where we used the recorded keyframes to create smooth animations. The animations are then mapped on head and hand meshes. In the bottom right corner of Figure \\ref{fig:implementation:Software:AR:AROverview}, a screenshot of an animation for shaking the victim's shoulders is shown.\n\n\\paragraph{(Interactable) Objects}\n\nTo simulate by-standing helpers, we used a simple human model, as can be seen on the left side of Figure \\ref{fig:implementation:Software:AR:AROverview}. The bystander is used to encourage the use of voice commands.\nAdditionally, there are multiple virtual objects, with which a user can interact directly. They are shown in Figure \\ref{fig:Implementation:Software:AR:InteractableObjectsOverview}.\nFirst, we added a broken glass that needs to be removed at the start of the training. The glass needs to be placed in a nearby dustbin and disappears when being dropped there. To allow the user to grab a 3D mesh, we use MRTK's \\textit{NearInteractionGrabbable}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/dotnet\/api\/microsoft.mixedreality.toolkit.input.nearinteractiongrabbable?view=mixed-reality-toolkit-unity-2020-dotnet-2.7.0}} component by adding the corresponding script to the mesh. \nNext, we needed a phone to call the emergency service. The phone is grabbable and interactive. When pressing the message button, a number pad shows up, where the user needs to enter the emergency number. For that, we used and modified the \\textit{pressable buttons}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/windows\/mixed-reality\/mrtk-unity\/features\/ux-building-blocks\/button?view=mrtkunity-2021-05}} provided by the MRTK.\nFinally, we created a defibrillator (AED) with its electrode pads. The pads can be grabbed and moved. When the pads are moved closer to their correct location on the manikin's chest, they snap to the right position. The button on the AED can be pressed to confirm delivering the shock.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.9\\linewidth]{figures\/Implementation\/ARObjectsOverview.pdf}\n \\caption{Interactable Objects}\n \\label{fig:Implementation:Software:AR:InteractableObjectsOverview}\n\\end{figure}\n\n\n\\paragraph{Voice Commands}\nAs we want to train the communication with real bystanders, we needed to implement voice commands. For that, we used MRTK's \\textit{SpeechInputHandler}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/dotnet\/api\/microsoft.mixedreality.toolkit.input.speechinputhandler?view=mixed-reality-toolkit-unity-2020-dotnet-2.7.0}} component, where we can define which keyphrases can be recognized, and what happens as a response when they are recognized. By utilizing the \\textit{SpeechInputHandler} component, voice commands are implemented as static keyphrases, which turned out to be the most reliable solution for voice recognition in the current version of the HoloLens. To guide the user to which keyphrases are available, we added hints to the AR environment using MRTK's \\textit{tooltip}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/windows\/mixed-reality\/mrtk-unity\/features\/ux-building-blocks\/tooltip?view=mrtkunity-2021-05}} component (see Fig \\ref{fig:implementation:Software:AR:AROverview}).\n\n\\paragraph{Manikin}\nIn Section \\ref{section:impl:Hardware:AR}, we already described how we created the resuscitation manikin. To give real-time feedback for the chest compressions in the application, there are two elements to consider. First, we have the compression rate, and second the compression depth. The manikin measures how far the chest is pushed multiple times every second and sends those values with a timestamp to the application.\n\n\nFigure \\ref{fig:implementation:Software:AR:UserPushData} shows how we use the incoming data. The black line is the measured distance from the top of the chest to the ground. When the application starts, we measure the default distance when the chest is not pushed. That value is set as the zero level. To find out whether the chest was pushed, we check if the distance is lower than a fixed threshold (in our case 3cm). To calculate the compression rate, we consider the moment when the threshold is reached. A push is finished, when the depth gets again higher than the threshold. The depth for a push is calculated from the lowest distance measured during a push. Therefore, we subtract the lowest distance from the zero level to get how deep the user pushed the chest. As a measure for the compression rate, we calculate the compressions per minute. Therefore, we measure the time between two pushes and calculate how many pushes per minute that would be.\nTo give feedback to the user, we provide three types of information. Figure \\ref{fig:implementation:Software:AR:CompressionDisplay} shows the display we provide to the user. On the left side, we show the current compression rate (averaged over four pushes so the number changes smoothly). In the middle, we show the number of compressions already performed. On the right, we display the current compression depth by using Unity's \\textit{slider}\\footnote{\\url{https:\/\/docs.unity3d.com\/2018.3\/Documentation\/ScriptReference\/UI.Slider.html}}. The red bar indicates how deep the chest is pushed. The lower blue bar indicates the desired depth and the top blue bar indicates the zero level. With that, we can provide real-time feedback for the compression. We also log the compression rate and depth values to analyze the user's performance.\n\n\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Sensor Data of User Push\\label{fig:implementation:Software:AR:UserPushData}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/FATPush.pdf}\n }\\hspace*{3pt}\n \\subfloat[Compression Display\\label{fig:implementation:Software:AR:CompressionDisplay}]{\n \\includegraphics[width=.45\\linewidth]{figures\/Implementation\/CPRDisplay.pdf}\n }\n \\end{center}\n \\caption{Using Sensor Data to Evaluate Compressions}\n \\label{fig:implementation:Software:AR:Compressions}\n\\end{figure}\n\n\n\n\\paragraph{Position Trigger}\nWe needed to track the position of the user's head and hands. For that, we utilize the GazeProvider component and the hand joint utilities provided by the MRTK's Input System\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/windows\/mixed-reality\/mrtk-unity\/features\/input\/gaze?view=mrtkunity-2021-05}}. We create a 3D volume like the one shown in Figure \\ref{fig:implementation:Software:AR:3DVolume}, allowing us to respond to events such as when users place their hands on the manikin's head during airway clearance.\n\\begin{figure}\n \\centering\n \\includegraphics[width=.5\\linewidth]{figures\/Implementation\/PositionTrigger.pdf}\n \\caption{3D Volume for Position Tracking}\n \\label{fig:implementation:Software:AR:3DVolume}\n\\end{figure}\n\n\\paragraph{Task Management}\n\nThe core workflow of a training session is defined by the task management. Every step of the training procedure is defined as a \\textit{Task}. \\textit{BaseTasks} are complex tasks, i.e. they are composited of one or more \\textit{SubTasks} (Figure \\ref{fig:implementation:Software:AR:TaskSystem}). A subtask, in contrast, is an atomic action that the user of the system performs, i.e. it is not further composited of more refined subtasks. As an example, utilization of the AED is one step of the BLS\/AED sequence, thus it is defined as a base task within the system. This step consists of several subtasks, such as placing the electrode pads of the AED in the correct locations on the victim's body, ensuring that nobody touches the victim, and delivering a shock.\n\n\n\nEach base task holds a list with its associated subtasks. The base tasks, in turn, are arranged within a \\textit{Tasklist}. Hence, the tasklist describes the sequence of steps of the final training procedure.\nTo connect the tasks to elements of the application, each base task is assigned to an individual \\textit{TaskModule}. A task module is an encapsulated entity responsible for the assessment of exactly one base task. It gets informed when the user did everything to complete the (sub) task and triggers the loading of the next task.\nAs the requirements for each task module can vary, we created a generic task module implementation such that any concrete implementations can inherit and, if necessary, override its basic functionality (Figure \\ref{fig:implementation:Software:AR:TaskSystem}).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.8\\linewidth]{figures\/Implementation\/TaskModuleClasses.pdf}\n \\caption{Task Management Classes}\n \\label{fig:implementation:Software:AR:TaskSystem}\n\\end{figure}\n\n\\subsubsection{VR}\\label{section:impl:Software:VR}\nThe VR application was built with Unity 2019.1.7f1 in combination with SteamVR\\footnote{\\url{https:\/\/valvesoftware.github.io\/steamvr_unity_plugin\/}} version 2.3.2. SteamVR offers a variety of functionality to support VR development. It, for example, provides some basic components like a preconfigured VR camera or objects representing the controller movement. Additionally, applications build with SteamVR work on different VR Headsets. For our development, we used the Valve Index\\footnote{\\url{https:\/\/www.valvesoftware.com\/en\/index}} VR headset. Whereas we use a real-life manikin as the victim in the AR application, we need a virtual victim for the VR variant. For that, we use a 3D model of a human lying on the ground which can be seen at the bottom in Figure \\ref{fig:implementation:Software:VR:VRSceneOverview}. To increase realism, the model uses physics and can be grabbed and moved by the user. As, during CPR, the user has to push the victim's chest, the model needs to be aligned with the real world so when the user pushes on the simulator, it looks for them as they push the chest. So we decided to attach the model to specific points on the ground so that the user can interact with the victim, but it stays at roughly the same position. \n\n\\paragraph{User}\nTo interact with the application, the user wears the Valve Index VR headset and its corresponding controllers.\nFurther, we also created all needed components as described in Section \\ref{section:solOverview:Software} and depicted in Figure \\ref{fig:solOverview:ApplicationOverview}.\n\n\\paragraph{Text\/Audio}\n\nTo display \\textit{Text} in the VR application, a description panel is placed near the ground behind the victim (Figure \\ref{fig:implementation:Software:VR:VRSceneOverview}), in the middle. We use the \\textit{TextMeshPro}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/com.unity.textmeshpro.html}} package, where arbitrary text can be inserted via script. The checklist in the background shows which big tasks are already completed. Next to the checklist, images as 2D \\textit{sprites}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/Sprites.html}} can be displayed.\nThe \\textit{Audio} descriptions are also integrated into the VR application. For that, we could use the same audio files we recorded for the AR version.\n\n\\paragraph{Animations}\nSimilarly, for the \\textit{Animations} within the VR application, we reused the Unity animations we recorded for the AR variant. At the bottom of Figure \\ref{fig:implementation:Software:VR:VRSceneOverview}, there is a screenshot of the shoulder shaking animation.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.8\\linewidth]{figures\/Implementation\/VROverview.pdf}\n \\caption{VR Scene as Displayed to the User}\n \\label{fig:implementation:Software:VR:VRSceneOverview}\n\\end{figure}\n\n\n\\paragraph{(Interactable) Objects}\nThe bystander in the VR application can be seen on the left side in Figure \\ref{fig:implementation:Software:VR:VRSceneOverview}. It is a simple human model that can be interacted with via voice commands.\nThe objects with which the user can directly interact are shown in Figure \\ref{fig:Software:VR:InteractableObejctsOverview}.\nWe created broken glasses the users need to put away at the beginning. They are shown on the left. We used the same base model as for the AR application but needed to make small adjustments to the model and create a new material to make it look good in the VR application. To allow the user to grab and move the glass, we use SteamVR's \\textit{Throwable}\\footnote{\\url{https:\/\/valvesoftware.github.io\/steamvr_unity_plugin\/api\/Valve.VR.InteractionSystem.Throwable.html}} script and added physics to the models. \nThe phone's model is the same as in the AR application, but we had to recreate the number pad. The phone itself is also a \\textit{Throwable}, but when the user grabs it, we display a number pad to dial the number. Using the \\textit{HoverButton}\\footnote{\\url{https:\/\/valvesoftware.github.io\/steamvr_unity_plugin\/api\/Valve.VR.InteractionSystem.HoverButton.html}} from SteamVR's input system for each number, the user can input the number by pushing them. The phone is shown in the middle of Figure \\ref{fig:Software:VR:InteractableObejctsOverview}.\nFinally, we needed the AED. Here, we could use the same models as we did for the AR application. We only needed to change the interaction. The pads became \\textit{Throwables} with a physics simulation and the button became a \\textit{hover button}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.8\\linewidth]{figures\/Implementation\/VRObjectsOverview.pdf}\n \\caption{Interactable Objects}\n \\label{fig:Software:VR:InteractableObejctsOverview}\n\\end{figure}\n\n\\paragraph{Voice Commands}\nTo enable voice commands, we used Unity's \\textit{PhraseRecognizer}\\footnote{\\url{https:\/\/docs.unity3d.com\/ScriptReference\/Windows.Speech.PhraseRecognizer.html}} class. With that, we can define which key phrases should be recognized and get notified when they were said. To show the user the keyphrase to say, we add a hint like the one shown in Figure \\ref{fig:implementation:Software:VR:VoiceCommandHint} when needed.\n\n\\begin{figure}\n\\begin{minipage}{.4\\linewidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{figures\/Implementation\/VRVoiceCommandHint.pdf}\n \\caption{Voice Command Hint}\n \\label{fig:implementation:Software:VR:VoiceCommandHint}\n\\end{minipage\n\\begin{minipage}{.4\\linewidth}\n \\centering\n \\includegraphics[width=.9\\linewidth]{figures\/Implementation\/VRHardwareIntegration.pdf}\n \\caption{Sensor Data of User Push}\n \\label{fig:implementation:Software:VR:UserPushData}\n\\end{minipage}\n\\end{figure}\n\n\n\\paragraph{Manikin\/CPR Simulator}\nWe already presented the CPR simulator we used in the VR application in Section \\ref{section:impl:Hardware:VR}.\nTo give real-time feedback to the user while performing compressions, we could reuse the components of the AR application (Figure \\ref{fig:implementation:Software:VR:UserPushData}). Two text fields show the rate and how many compressions were done, and the indicator on the right shows the depth.\n\n\n\\paragraph{Position Trigger}\nTo track the user's head position, we created simple 3D volumes (Figure \\ref{fig:implementation:Software:VR:3DVolume}) which can detect whenever an object enters it. Then, we can check whether the object was the user's head and trigger the next steps. That is similar to how the position trigger worked in AR. But for the hands, we used a different concept. In AR, we could only track a hand's position, but we could not track whether the user grabs e.g. the victim's shoulder. However, in VR, based on the Valve Index hand controllers we can track all interactions being made with the environment. So when the user needs to interact with the victim (e.g. shake shoulder, see Figure \\ref{fig:implementation:Software:VR:HandTracking}) we can place small 3D volumes there and add SteamVR's \\textit{Throwable} script which allows us to get notified when the user grabs\/releases it. \n\\begin{figure}\n\\begin{center}\n \\subfloat[3D Volume for Head Position Tracking\\label{fig:implementation:Software:VR:3DVolume}]{\\includegraphics[height=120pt]{figures\/Implementation\/VRHeadTracking.pdf}}\n \\hspace*{30pt}\n \\subfloat[3D Volume for Hand Position Tracking\\label{fig:implementation:Software:VR:HandTracking}]{\\includegraphics[height=120pt]{figures\/Implementation\/VRShouldersGrab.pdf}}\n \\end{center}\n \\caption{Position Tracking}\n \\label{fig:implementation:Software:VR:PositionTracking}\n\\end{figure}\n\n\\paragraph{Task Management} \nAs the AR and VR applications only differ in the way they interact with the user while the underlying application stays the same, we used the same task system for the AR as well as the VR application.\n\n\\subsection{Training Walkthrough}\\label{section:impl:Software:Walkthrough}\n\nUsing those building blocks, we created the whole training application which guides the users through the BLS sequence. Figure \\ref{fig:implementation:Software:ARWalktrough} shows a full walkthrough of all tasks in the AR as well as in the VR training. Note that the 3D model of the manikin in the AR training is not shown to the user. During the training, all virtual objects are placed on and around the real manikin. The model is just shown here for illustration purposes.\nFigures \\ref{fig:implementation:Software:ARWalktrough:0} - \\ref{fig:implementation:Software:ARWalktrough:4} are taken from the AR application, whereas Figures \\ref{fig:implementation:Software:VRWalktrough:5}-\\ref{fig:implementation:Software:VRWalktrough:7} are taken from the VR application.\nChanging the applications is only done here to illustrate the walkthrough of both applications. The users did not change the training in the middle, they stayed in their application until the end.\nIn all steps, description panels, images, and audio files are involved as we use them to explain the basics of every step. \n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Scene Safety\\label{fig:implementation:Software:ARWalktrough:0}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/ARWalkthrough\/0.pdf}\n }\\hspace*{3pt}\n \\subfloat[Check Response\\label{fig:implementation:Software:ARWalktrough:1}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/ARWalkthrough\/1.pdf}\n }\\hspace*{3pt}\n \\subfloat[Open Airways\\label{fig:implementation:Software:ARWalktrough:2}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/ARWalkthrough\/2.pdf}\n }\n \\end{center}\n \n \\begin{center} \n \\subfloat[Check Breathing\\label{fig:implementation:Software:ARWalktrough:3}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/ARWalkthrough\/3.pdf}\n }\\hspace*{3pt}\n \\subfloat[Call Ambulance\\label{fig:implementation:Software:ARWalktrough:4}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/ARWalkthrough\/4.pdf}\n }\\hspace*{3pt}\n \\subfloat[Get AED\\label{fig:implementation:Software:VRWalktrough:5}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRWalkthrough\/5.pdf}\n }\n \\end{center}\n \n \\begin{center} \n \\subfloat[Chest Compressions\\label{fig:implementation:Software:VRWalktrough:6.1}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRWalkthrough\/6.1.pdf}\n }\\hspace*{3pt}\n \\subfloat[Ventilation\\label{fig:implementation:Software:VRWalktrough:6.2}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRWalkthrough\/6.2.pdf}\n }\\hspace*{3pt}\n \\subfloat[Use AED\\label{fig:implementation:Software:VRWalktrough:7}]{\n \\includegraphics[width=.3\\linewidth]{figures\/Implementation\/VRWalkthrough\/7.pdf}\n }\n \\end{center}\n \n \\caption{AR\/VR Walkthrough}\n \\label{fig:implementation:Software:ARWalktrough}\n\\end{figure}\n\n\n\nIn the beginning, the scene safety needs to be ensured (Figure \\ref{fig:implementation:Software:ARWalktrough:0}). Here we placed the broken glasses on the victim's body. They are interactable objects and need to be dragged into the dustbin on the right side. Animations of the hands give an additional hint of what the user needs to do. \nTo check the victim's response (Figure \\ref{fig:implementation:Software:ARWalktrough:1}), we check if the user's hands are at the victim's shoulders with our position trigger. Animated hands show the movements to be done. In addition, the user needs to ask the victim if everything is fine. So we also use voice commands in this step. \nWhen the airways need to be opened (Figure \\ref{fig:implementation:Software:ARWalktrough:2}) we use a position trigger to recognize if the hands are placed on the victim's head. For the head tilt, we use the measurements of the manikin's gyro sensor to calculate the head's angle. We also show animations for the placement as well as for the tilting. In the VR application, the user is moving the head of the virtual manikin, so we can measure the tilt here.\nWhile checking the breathing (Figure \\ref{fig:implementation:Software:ARWalktrough:3}) we use a position trigger to check if the user's head is above the victim's mouth and the user is looking towards the victim's chest. Here, an animated head also shows the movement. Next, the breathing status needs to be communicated. So we use voice commands to recognize if the user correctly announces the status. \nWhen the ambulance needs to be called (Figure \\ref{fig:implementation:Software:ARWalktrough:4}), the user can grab the phone (interactable object) to dial the number. Alternatively, the bystander can be asked to call the ambulance. In this case, we also use voice commands.\nIn the next step, sending someone to get an AED (Figure \\ref{fig:implementation:Software:VRWalktrough:5}), we only use voice commands.\nWhen performing chest compressions (Figure \\ref{fig:implementation:Software:VRWalktrough:6.1}), we use the manikin's ultrasonic sensor to measure the compression depth and to display the rate and depth to the user. Animations show how the compressions need to be performed. \nAfter that, the victim needs to be ventilated (Figure \\ref{fig:implementation:Software:VRWalktrough:6.2}). Here we check if the user's head is correctly placed over the victim's head as is shown by an animation. \nFor the final step, using the AED (Figure \\ref{fig:implementation:Software:VRWalktrough:7}), we have the AED pads as interactable objects which the user needs to place and the AED's button which needs to be activated at the end. The pad placement is also shown by an animation. Before triggering the shock, it needs to be ensured that nobody touches the victim. As the user needs to communicate that, we use voice commands.\n\n\n\\subsection{Debriefing}\\label{section:impl:Debriefing}\nWhen the users perform the training, we log their performance. With that, we get the duration of how long it took to complete a task, and how (well) the task was completed. During the debriefing, the user sees the scene depicted in Figure \\ref{fig:implementation:Debriefing:Overview}.\n\\begin{figure}\n \\begin{center} \n \\subfloat[Debriefing Overview\\label{fig:implementation:Debriefing:Overview}]{\n \\includegraphics[width=.7\\linewidth]{figures\/Implementation\/DebriefingOverviewZoom.pdf}\n }\n \\end{center}\n \n \\begin{center}\n \\subfloat[Debriefing CPR\\label{fig:implementation:Debriefing:CPR}]{\n \\includegraphics[width=.7\\linewidth]{figures\/Implementation\/DebriefingCPR.jpg}\n \n \n \n \n \\end{center}\n \\caption{Debriefing}\n \\label{fig:implementation:Debriefing}\n\\end{figure}\n\nHere, we present all the tasks with some basic information about the user's performance. \nFirst, we give an overview of the overall performance compared to the last training. We show how many percent of the tasks were executed correctly and how long the user needed to complete the training. Beneath that, we show the same information but on task-level. At the top, we show how many percent of the task was executed, i.e. how many subtasks were executed correctly. That is also summarized at the bottom. In between them, we show the duration the user needed to complete the task. In addition to that, the user can zoom into the results of a single task, either by gazing at the panel of the corresponding task in the overview or by gazing at the location where the task has been executed on the manikin. To enable that, we use MRTK's \\textit{Interactbable}\\footnote{\\url{https:\/\/docs.microsoft.com\/en-us\/windows\/mixed-reality\/mrtk-unity\/features\/ux-building-blocks\/interactable?view=mrtkunity-2021-05}} script, which triggers loading the details using the \\textit{InteractableOnFocusReceiver}\\footnote{\\url{https:\/\/hololenscndev.github.io\/MRTKDoc\/api\/Microsoft.MixedReality.Toolkit.UI.InteractableOnFocusReceiver.html}}. In the VR version, users aim at the spots using a laser pointer. When the user activates the controller's trigger, we cast a ray\\footnote{\\url{https:\/\/docs.unity3d.com\/ScriptReference\/Physics.Raycast.html}} along the laser pointer and load the details for the task where the ray hits. All other components are the same for the AR and VR versions. The zones mentioned above can be seen at the bottom of the screenshot. To facilitate re-experiencing of the performed tasks, we decided to also include the animations used in the learning mode into the debriefing. However, if desired, the animations can be hidden. For the CPR, we have got additional values for the compressions (see Figure \\ref{fig:implementation:Debriefing}). We show them in two different ways. On the right side, we show an average of the compression depth and rate whereas, in the middle, we display graphs showing how the values changed over time. Both graphs can be enlarged for better readability by focusing them or selecting them with the laser pointer. The graphs are calculated dynamically. We display the user's performance (red line) together with the upper and lower threshold (green lines). The graphs themselves are generated using Unity's \\textit{LineRenderer}\\footnote{\\url{https:\/\/docs.unity3d.com\/Manual\/class-LineRenderer.html}}.\n\n\n\n\n\n\n\n\n\n\\section{Evaluation}\n\\label{section:Evalution}\n\nTo answer the second research question (RQ2) motivated in the introduction, we have conducted a usability evaluation to analyze the efficiency, effectiveness, and user satisfaction of our AR- and VR-based BLS training environment compared to traditional training methods. In Section \\ref{section:Evalution:setupProcedure}, we describe the used setup and procedure. Following that, we present our participants and their background in Section \\ref{section:Evaluation:Participants}. We display the results we gathered during our usability experiment in Section \\ref{section:Evalution:results}. Those results are discussed in Section \\ref{section:Evalution:discussion}. Finally, in Section \\ref{section:Evalution:threatsToValidity} we describe the threats to the validity of our study.\n\n\n\\subsection{Setup and Procedure}\\label{section:Evalution:setupProcedure}\n\nTo answer RQ2, we conducted a user study that targets comparing our AR and VR training with real-life (R) training. For the R training, the trainer explained the tasks on the manikin and the trainee performed them afterward. In this context, please note that we had no certified trainer, but rather an expert familiar with the BLS training process. The content was the same as in the digital trainings, i.e. the trainer described the same steps and used similar descriptions as the AR and the VR application. \nFor the usability experiment, we followed the process depicted in Figure \\ref{fig:eval:userStudyProcess}. \n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{figures\/Evaluation\/UserStudieProcess.pdf}\n \\caption{User Study Procedure}\n \\label{fig:eval:userStudyProcess}\n\\end{figure}\nFirst, the users completed their assigned training (Figure \\ref{fig:experiment:participantsOverview}). Next, they were asked to answer a questionnaire. We used the SUS questionnaire \\cite{brooke1996sus} to get insight into the systems' overall usability, the NASA TLX score \\cite{hart1988development} to evaluate the workload, and the presence questionnaire by Usoh et al. \\cite{DBLP:journals\/presence\/UsohCAS00}. Here, we adjusted the questions so that they fit our scenario. \n\nFinally, the users performed an exam. That means they have to perform the whole BLS sequence without any help. We measured the time and noted whenever a user completed a step, so we know in which order the tasks were executed and how long it took. Additionally, we recorded the sensor values of the manikin to analyze the compression rate and depth. We then calculate a score that indicates how well the sequence was executed. Note that we do not consider the duration here.\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[AR\\label{fig:experiment:participantsOverview:AR}]{\n \\includegraphics[width=.23\\linewidth]{figures\/Experiment\/ARSetup.pdf}\n }\\hspace*{1pt}\n \\subfloat[VR\\label{fig:experiment:participantsOverview:VR}]{\n \\includegraphics[width=.23\\linewidth]{figures\/Experiment\/VRSetup.pdf}\n }\\hspace*{1pt}\n \\subfloat[R\\label{fig:experiment:participantsOverview:R}]{\n \\includegraphics[width=.23\\linewidth]{figures\/Experiment\/RSetup.pdf}\n }\\hspace*{1pt}\n \\subfloat[Exam setup\\label{fig:experiment:participantsOverview:exam}]{\n \\includegraphics[width=.23\\linewidth]{figures\/Experiment\/Exam.pdf}\n } \n \\end{center}\n \\caption{Usability Experiment}\n \\label{fig:experiment:participantsOverview}\n\\end{figure}\n\nTo compare the execution of the BLS sequence, we introduced a simple scoring system. Every time the user executes an activity correctly, they get the corresponding score (Figure \\ref{table:evaluation:scores}). The score distribution was implemented in a similar way to Strada et al. \\cite{strada2019holo}.\n\n\\begin{figure}\n\\centering\n \\begin{tabular}{ | l | r | }\n \\hline\n \\textbf{Task name} &\\textbf{Score}\\\\\\hline\n Ensure Safety & 2 \\\\ \\hline\n Check Response & 1 \\\\ \\hline\n Open Airways & 1 \\\\ \\hline\n Make an emergency call & 2 \\\\ \\hline\n Send somebody to get an AED & 2 \\\\ \\hline\n Perform compressions & 0-4 \\\\ \\hline\n Ventilate & 2 \\\\ \\hline\n Place AED pads & 1 for each pad \\\\ \\hline\n Make people stand back & 1 \\\\ \\hline\n Trigger shock & 1 \\\\\n \\hline\n \\end{tabular}\n \\caption{Scores}\\label{table:evaluation:scores}\n\\end{figure}\nFor the compressions, the average rate (compressions per minute) and the average depth are evaluated. The average rate points have the following intervals. 95$<$avg. rate$<$125: 2 points, 80$<$avg. rate$<$95: 1 point, 125$<$avg. rate$<$140: 1 point, all other values: 0 points. For the average depth, 5$<$avg. depth$<$6 results in 2 points, all other values result in 0 points.\n\nBy adding the points for executing the tasks, we get an intermediate score. As the correct execution order is important for most tasks, we weigh the score. For that, we count the number of tasks that were executed at the correct time (i.e. the task the user executed before the current task is also its predecessor in the activity diagram). If the order is 100\\% correct, the user gets the full score, if the order is 0\\% correct, the user gets 50\\% of the intermediate score. All values in between are mapped accordingly.\n\n\\subsection{Participants}\\label{section:Evaluation:Participants}\nFor our user study, we had 21 participants, with ages ranging from 22 to 54. We had 5 female and 16 male participants. The participants were distributed randomly on the training types so that 7 participants performed the AR, VR, and R training. 19 participants had done first aid training before, but for most of them, the last training was 5 or more years ago. Only 2 persons regularly participate in first aid training every two years. Most AR participants had no to little experience with AR, whereas most VR participants had at least a little experience with VR applications.\n\n\n\n\n\n\n\\subsection{Results}\\label{section:Evalution:results}\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[SUS Score\\label{fig:eval:susScoreAll}]{\\includegraphics[height=150pt]{figures\/Evaluation\/SUSScoreARVR.pdf}}\\hspace*{5pt}\n \\subfloat[TLX Score\\label{fig:eval:TLXScore}]{\\includegraphics[height=150pt]{figures\/Evaluation\/TLXAll.pdf}}\\hspace*{5pt}\n \\subfloat[Presence Score\\label{fig:eval:PresenceScore}]{\\includegraphics[height=150pt]{figures\/Evaluation\/PresenceScoreAll.pdf}} \n \\end{center}\n \\caption{Questionnaire Scores}\n \\label{fig:eval:questionnaireScores}\n\\end{figure}\nFigure \\ref{fig:eval:susScoreAll} shows the SUS scores for the AR and VR training applications. The mean scores are $80.36$ for the AR and with $83.57$ slightly higher for the VR training. But considering the medians, the AR application is rated higher ($87.5$) than the VR application ($82.5$). It is also worth noting that the scores for the AR training are broader distributed on the scale, whereas the scores for the VR training are close together.\n\nFigure \\ref{fig:eval:TLXScore} shows the results of the NASA-TLX \\cite{hart1988development} questionnaire. Here, we can see that, on average, the workloads of the AR ($29.88$) and the VR training ($30.12$) are almost the same. A small difference can be seen at the median which is $30$ for the AR and $26.67$ for the VR application. Again, the values for the AR application are more distributed than the VR ones. A bigger difference can be seen when compared to the R training, where the mean ($42.74$), as well as the median ($38.33$), are higher. \n\nFinally, we consider the presence score proposed in \\cite{DBLP:journals\/presence\/UsohCAS00}. Here, when using the VR application users had a stronger feeling to be present at an accident scene than the users of the AR application.\n\n\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Training Score\\label{fig:eval:trainingScore}]{\\includegraphics[height=150pt]{figures\/Evaluation\/TrainingScoreAll.pdf}}\\hspace*{5pt}\n \\subfloat[Exam Score\\label{fig:eval:examScore}]{\\includegraphics[height=150pt]{figures\/Evaluation\/ExamScoreAll.pdf}}\n \\end{center}\n \\caption{Scores}\n \\label{fig:eval:scores}\n\\end{figure}\nFigure \\ref{fig:eval:scores} shows the scores the users got when executing the training and the exam. Figure \\ref{fig:eval:trainingScore} shows the scores in the training. Here, the average score differs slightly for the AR\/VR trainings (VR: $14.14$, AR: $13.86$) but they have the same median ($14$). For the real-life training, the mean ($12.43$) and the median ($12$) are lower. The scores for the VR training are less distributed than the scores for the other trainings.\n\nConsidering the exam scores of Figure \\ref{fig:eval:examScore}, we see that the score of participants of the AR (mean: $8.01$, median $7.56$) and VR training (mean: $7.88$, median: $8.25$) are close to each other, while the participants of R trainings have higher scores (mean: $10.6$, median: $9.75$). Again, the values for the VR training participants are less distributed than the others.\n\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Training Duration\\label{fig:eval:trainingDuration}]{\\includegraphics[height=150pt]{figures\/Evaluation\/TrainingDurationAll.pdf}}\\hspace*{5pt}\n \\subfloat[Exam Duration\\label{fig:eval:examDuration}]{\\includegraphics[height=150pt]{figures\/Evaluation\/ExamDurationAll.pdf}}\n \\end{center}\n \\caption{BLS Durations}\n \\label{fig:eval:blsDurations}\n\\end{figure}\nFigure \\ref{fig:eval:blsDurations} shows the durations for the trainings and exams. Considering the trainings (Figure \\ref{fig:eval:trainingDuration}) it can be seen that the R training was the fastest (mean: 6:15 mins, median: 6:09 mins). With on outlier (14:51 mins), the AR training's duration was in the middle (mean: 8:09, median: 6:42) and the VR training took slightly longer (mean: 8:41, median: 9:01).\n\nFor the exam, the participants of the AR and R training were almost equally fast. AR participants needed 1:47 mins on average, the median is 1:42. Participants of the R training needed 1:47 mins on average and had a median of 1:40 mins. Both had one outlier with 2:58 mins (AR) and 2:42 mins (R). Participants of the VR training needed more time, 2:21 mins on average with a median of 2:37 mins.\n\n\n\n\n\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Training CPR Rate Average\\label{fig:eval:trainingRate}]{\\includegraphics[height=150pt]{figures\/Evaluation\/TrainingCPRRateAVGAll.pdf}}\\hspace*{5pt}\n \\subfloat[Exam CPR Rate Average\\label{fig:eval:examRate}]{\\includegraphics[height=150pt]{figures\/Evaluation\/ExamCPRRateAVGAll.pdf}}\n \\end{center}\n \\caption{CPR Rate Averages}\n \\label{fig:eval:rates}\n\\end{figure}\nAs chest compressions are an essential part of basic life support, we also measure how well they are executed. To do so, we recorded the compression rate and depth for the trainings and the exams.\n\n\nFigure \\ref{fig:eval:trainingRate} shows the compression rate in the trainings. In our trainings, the users should perform 105 compressions per minute. Here the medians were close to each other (AR:$105$, R:$106$, VR:$101$) whereas the mean values differ more. Users of the AR training tend to be much slower than trained and R training participants tend to be too fast. That can also be observed on the average rate, AR participants had an average rate of $97.29$ and R participants $112.42$. VR participants are on average the closest to the trained rate of $105$ with an average rate of $102.26$.\n\nDuring the exam (Figure \\ref{fig:eval:examRate}), participants of the AR training still tend to be too slow. Their average rate was $105.43$ with a median of $98$. VR and R training participants tend to be a bit too fast with an average and median of $113$ for VR training participants, and an average of $114.23$ and a median of $119$.\n\n\\begin{figure}\n \\begin{center} \n \\subfloat[Training CPR Depth Average\\label{fig:eval:trainingDepth}]{\\includegraphics[height=150pt]{figures\/Evaluation\/TrainingCPRDepthAVGAll.pdf}}\\hspace*{5pt}\n \\subfloat[Exam CPR Depth Average\\label{fig:eval:examDepth}]{\\includegraphics[height=150pt]{figures\/Evaluation\/ExamCPRDepthAVGAll.pdf}}\n \\end{center}\n \\caption{CPR Depth Averages}\n \\label{fig:eval:depth}\n\\end{figure}\nThe compression depth can be seen in Figure \\ref{fig:eval:trainingDepth}. In our trainings, we aimed to train the recommended depth of 5-6cm. For the trainings (Figure \\ref{fig:eval:trainingDepth}), the R training participants were the best with an average of $5.49$ cm and a median of $5.85$ cm. The participants of the AR training (average: $4.57$ cm, median: $4.7$ cm) as well as the participants of the VR training (average: $4.03$ cm, median: $4.02$ cm did not push far enough.\nDuring the exam, the AR training participants pushed on average $5.7$ cm with a median of $5.8$ cm, but some participants pushed too far or not far enough. VR training participants with an average of $5.4$ cm and a median of $5.31$ cm pushed mostly correct. Some participants pushed a bit too far, and one outlier only pushed $3.15$ cm deep. Some R training participants pushed too far, although the average with $5.87$ cm and the median with $5.26$ cm is still in the desired range.\n\n\\subsection{Discussion}\\label{section:Evalution:discussion}\nA major aspect when comparing the results of the trainings to each other is how well the users could interact with the technology. Considering the SUS score (Figure \\ref{fig:eval:susScoreAll}), we can see a wide range of scores. But during the study, we noticed that the participants who gave the application a low SUS score had problems understanding how the technology works and how to use it properly. Whereas the participants who could work with it well gave a high rating. The same aspect can be seen for the TLX score (Figure \\ref{fig:eval:TLXScore}). The participants who could work well with the technology felt a smaller workload than users who had problems. Concerning the workload, we can see a great potential of utilizing AR and VR for BLS training. On average, the TLX score for the AR- and VR-based BLS training environment was around 30 and much better than the real-life training that received a TLX score over 40. This shows that independent self-training of the BLS procedure can be eased through an interactive and immersive training environment. A big difference between AR and VR can be seen for the presence score (Figure \\ref{fig:eval:PresenceScore}), here the VR application got a way better score. The reason is that using Virtual Reality, we could immerse the users in another place so that it felt more like being at an accident. In AR, they could see the office room they were in the whole time. As the field of view for virtual objects is quite small on the Hololens 2 and we cannot access high computing power, it is hard to immerse the users in another environment. \nWhen comparing the scores the users reached in the trainings and exams (Figures \\ref{fig:eval:trainingScore} and \\ref{fig:eval:examScore}), we can see that the scores for AR and VR are always close to each other. For the training, their scores are very high. That is because the application only continues with the next step if the previous step was executed correctly. Only the performance of the CPR can change the results. If a user executes a step wrong in the real-life training, the trainer can correct it afterward, but it was executed wrong in the first place. But considering the exam scores, users with the real-life training had better results than the others. So the 1:1 interaction with the trainer and the corrections still yield a better remembering of the tasks.\nThe results of the CPR (Figures \\ref{fig:eval:rates} and \\ref{fig:eval:depth}) are similar for all trainings in the exam. In contrast to the overall score, the execution was learned in an equal quality for all training types.\n\n\n\n\\subsection{Threats to Validity}\\label{section:Evalution:threatsToValidity}\nFor our study, we had 21 participants and, accordingly, 7 participants in each training type. Especially when comparing the AR and VR versions the results are close to each other and we could not see whether one is significantly better than the other. With that in mind, a bigger user study could yield a more convincing result.\n\nDue to the limited possibility and reliability in recognizing gestures or movement of e.g. the hands, the applications could not verify that all hand and finger movements were 100\\% accurate. Thus, it may happen that the application recognizes a step as executed correctly whereas a real trainer could correct even small mistakes. \n\nIn regular first aid training, the trainers are mostly experienced paramedics who not only explain the steps of the BLS sequence but give additional information when needed and enrich the training with personal stories and experiences. We did not integrate such additional content into our trainings, but the user's attention and thus their performance could be increased if they get some context about how the learned techniques really save lives. \n\\section{Conclusion and Outlook}\n\\label{section:conclusion}\n\nIn the event of cardiac arrest, early bystander basic life support (BLS) is vital for increased survival chances. Regular training of BLS procedures among people\nis required. While AR- and VR-based approaches have been promoted to enhance BLS training in an interactive and practical way, current existing approaches are not fully compliant with the medical BLS guidelines or focus only on specific steps of BLS training such as resuscitation. Furthermore, most of the existing training approaches do not focus on automated assessment to enhance efficiency and effectiveness through fine-grained real-time feedback. To overcome these issues, we have designed and implemented an immersive BLS training environment that supports AR as well as VR training with an interactive and haptic manikin. The main results of our usability evaluation show that AR and VR technology have the potential to increase engagement in BLS training, improve high-quality resuscitation training, and reduce the cognitive workload compared to traditional training with a personal instructor.\n\nIn future work, we plan to extend our BLS training environment to also support Advanced Life Support (ALS). While the scope of this work considers independent self-training only, multi-user training becomes more and more important, particularly for Advanced Life Support. As training is not only about learning specific skills such as resuscitation and is also about learning to work together in a team of multiple people, we plan to extend our training environment by collaboration features.\n\n\\section*{Acknowledgement}\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Key points}\n\n\\section{Introduction}\nThe physical process of wave breaking remains one of the classical unresolved problems of fluid dynamics. Considerable research effort has been devoted to this topic, but the highly nonlinear nature of the breaking process makes both observational and numerical efforts challenging. Over the years a number of diagnostic parameters have been proposed to characterise the breaking onset process, with \\citet{perlin2013breaking-waves-} providing the most recent review of progress in this field. While many parameters have been successful in characterising breaking onset for a specific subset of surface gravity waves, until recently none have proven to apply generically across a range of water depths, generation or instability mechanisms. \n\nAn approach that has shown considerable promise is based upon the evolution of the intragroup energy flux \\citep{tulin2001breaking}. The key physical concept is that breaking onset in an unsteady wave group is triggered when the energy flux convergence rate, as measured in a frame co-moving with the tallest crest in the group, exceeds a local stability level \\citep{banner_peirson_2007, derakhti2016breaking-onset-}. Recently, \\citet[hereafter B18]{barthelemy2018on-a-unified-br} proposed a breaking inception parameter $B$ that links the local energy flux\n\\begin{equation}\n \\mathbf{F} = \\mathbf{u}\\left(\\left(p-p_0\\right) + \\rho g z + 1\/2\\rho \\lVert\\mathbf{u}\\rVert^2 + E_0\\right)\n\\end{equation}\nto the local energy density\n\\begin{equation}\n E = \\rho g z + 1\/2\\rho\\lVert\\mathbf{u}\\rVert^2 + E_0.\n\\end{equation}\nHere $p$ and $p_0$ are the pressure within the fluid and at the interface, $z$ is the vertical coordinate, $\\rho$ is the density, $g$ the gravitational acceleration and $\\mathbf{u}$ the fluid velocity. As $z=0$ at the still water level, a reference energy term $E_0 = -\\rho g z_0$ ensures that $E$ is always positive. The constant $z_0$ is set to twice the water depth, the choice of which has been previously shown to have negligible impact on results \\citepalias{barthelemy2018on-a-unified-br}. \n\nThe normalised local energy flux within the crest is derived by dividing these quantities by the wave crest speed $\\mathbf{c}$,\n\\begin{equation}\\label{eq:B}\n \\mathbf{B} = \\frac{\\mathbf{F}}{E\\lVert\\mathbf{c}\\rVert}\n\\end{equation}\nand the breaking inception parameter $B$ is defined as $B=\\lVert\\mathbf{B}\\rVert$. \\citetalias{barthelemy2018on-a-unified-br} found that, when tracking $B$ for any particular crest, the transition of $B$ through the generic breaking inception threshold level of $B_\\mathrm{th}\\approx 0.85$ separates breaking and non-breaking wave crests. For those crests that exceed the threshold $B_\\mathrm{th}$, breaking onset -- defined as the instant that the crest interface height becomes multi-valued -- is observed when $B > 1.0$. Thus the breaking inception threshold of $B = B_\\mathrm{th}$ represents a point of no return beyond which breaking will occur. This picture of wave breaking has been subsequently validated in laboratory studies \\citep{saket2017on-the-threshol,SAKET2018159}, with a variety of wave packet types \\citep{DerakhtiMorteza2018Ptbs}, for deep water waves \\citep{seiffert2018simulation-of-b} and for waves shoaling on topography \\citep{Derakhti_2020}. \n\nAs a diagnostic parameter, $B$ has practical advantages. At the water surface $p-p_0=0$ and the breaking inception parameter reduces to $B=\\lVert\\mathbf{u}\\rVert\/\\lVert\\mathbf{c}\\rVert$, which can be measured in a laboratory or field setting. In addition, the horizontal components of $\\mathbf{B}$ and $\\mathbf{c}$ are much larger than the vertical components such that $B_x=F_x\/(Ec_x)$ can be used in place of $B$ to a close approximation. $B$ has also demonstrated potential as a forecasting parameter, with the breaking inception threshold $B_\\mathrm{th}$ exceeded up to half a wave period prior to breaking onset \\citepalias{barthelemy2018on-a-unified-br}. \n\nWhile the formulation of $B$ is a dynamic threshold, the underlying reason that breaking occurs only for wave crests in which $B$ exceeds $B_\\mathrm{th}$ is yet to be determined. To advance towards resolving this knowledge gap, we consider the main component of (\\ref{eq:B}), the local mechanical energy density $E$, in isolation. We investigate the hypothesis that the breaking inception threshold mimics the local energetics at the crest tip. We track the evolution of both $B$ and $E$ as the crest either relaxes from its maximum steepness without breaking, or transitions to breaking onset. Our experiment utilises direct numerical simulation with a two-phase volume-of-fluid Navier-Stokes solver \\citep{popinet2003gerris:-a-tree-,popinet2009an-accurate-ada} to examine fully nonlinear wave packets in the presence of viscosity and surface tension. Using these results we study the temporal evolution of the energy density and compare this to the evolution of $B$. \n\n\\section{Theoretical background}\n\nThe determination of breaking inception using $B$ is achieved by tracking the maximum value of $B$ within the crest, which occurs at or near the crest tip. As the crest is propagating at velocity $\\mathbf{c}$ the temporal evolution of $B$ has both a local component and a component in the frame of reference of the moving crest \\citep[equation (1.2)]{tulin-2007},\n\\begin{equation}\\label{eq:DBDt}\n \\frac{D_{c}B}{Dt} \n = \\frac{\\partial B}{\\partial t} + \\mathbf{c} \\cdot \\nabla B \n = - \\nabla \\cdot \\left( \\left[\\mathbf{u} - \\mathbf{c}\\right]B\\right),\n\\end{equation}\nwhere $D_c \/ Dt$ denotes the rate of change in the unsteady crest-following frame of reference. \n\n\\cite{DerakhtiMorteza2018Ptbs} found that the strength of breaking is proportional to the rate of change of $B$ at breaking inception ($B = B_{\\mathrm{th}}$). They defined the parameter\n\\begin{equation}\\label{eq: GammaB_def}\n\\Gamma_B = T_0 \\left. \\frac{D_c B}{Dt}\\right|_{B_\\mathrm{th}}\n\\end{equation}\nwhere the rate of change of $B$ is normalised by the local crest period $T_0$. (Note that $B$ and $\\Gamma_B$ are both dimensionless quantities.) (\\ref{eq: GammaB_def}) is a significant finding as it shows that not only does $B$ provide advance warning of breaking, it is also indicative of the strength of the breaking and the energy dissipation thereafter. \n\nIn the same manner as (\\ref{eq:DBDt}), in the crest-following frame, the local energy balance \\citep[equation (2.3.2)]{phil1977} can be expressed as\n\\begin{equation}\\label{eq:energy-balance}\n \\frac{D_{c}E}{Dt} \n = \\frac{\\partial E}{\\partial t} + \\mathbf{c \\cdot \\nabla} E \n = \\mathbf{u \\cdot f} - \\mathbf{\\nabla \\cdot F_c},\n\\end{equation}where $\\mathbf{u \\cdot f}$ is a sink term representing the work done against friction and $\\mathbf{F_c}=\\left(\\mathbf{u}-\\mathbf{c}\\right)E + \\mathbf{u}\\left(p-p_0\\right)$ is the divergence of the energy flux in the crest-following frame. \n\nOur aim is to relate the behaviour of $B$, particularly its rate of change following the crest tip $D_cB\/Dt$, to the more familiar wave energy growth rate $D_cE\/Dt$ following the crest tip. We introduce the normalised growth rate:\n\\begin{equation}\\label{eq:Gamma-E}\n\\Gamma_E=\\frac{T_0}{E - E_0} \\left.\\frac{D_cE}{Dt}\\right|_{B_{\\mathrm{th}}}.\n\\end{equation}\nIn (\\ref{eq:Gamma-E}), the local rate of change of $E$ following the crest tip is normalised by the dynamic local energy density ($E - E_0$), divided by the local crest period $T_0$. The arbitrary reference energy level $E_0$, which does not affect $D_cE\/Dt$, is suppressed in the denominator to allow a generic comparison of deep and shallow water cases. \n\n\\section{Experiment description}\n\nTo elucidate the relationship between $\\Gamma_E$ and $\\Gamma_B$, we conducted a suite of numerical simulations of breaking and non-breaking waves across a range of wave packet configurations and grid refinements (table \\ref{tab:experiments}). We used the Gerris software package \\citep{popinet2003gerris:-a-tree-} to numerically solve the two-dimensional, incompressible, variable density Navier-Stokes equations, including the effects of viscosity and surface tension. Gerris uses the Volume-Of-Fluid (VOF) method to simulate two-phase flows, with surface tension modelled through an improved implementation of the continuum-surface-force approach \\citep{popinet2009an-accurate-ada}. Gerris has been extensively validated for simulations of surface gravity waves \\citep{WRONISZEWSKI20141}, wave breaking kinematics \\citep{deike2017lagrangian-tran,pizzo2016current-generat} and energy dissipation \\citep{de2018breaking}.\n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\textwidth]{figures\/Gerris_generic_domain.pdf}}\n \\caption{The numerical wave tank. Waves are generated at the paddle boundary and travel down the tank in the positive $x$ direction. The depth of water $d$ is varied to achieve the desired depth\/wavelength ratio. A numerical sponge layer absorbs waves at the far end of the tank, while an outflow boundary minimises pressure gradients in the air phase due to paddle movement. A typical chirped wave packet ($N=5$, enlarged for clarity) is shown.}\n\\label{fig:tank}\n\\end{figure}\n\nThe model is set up as a two-dimensional numerical wave tank of length $23.5\\lambda_p$ and height $1.18\\lambda_p$, where $\\lambda_p$ is the deep-water wavelength of the wave paddle forcing (figure \\ref{fig:tank}). While computational constraints limit us to two-dimensional simulations, previous studies have shown that there is negligible difference in $B$ between two- and three-dimensional cases \\citep{barthelemy2018on-a-unified-br,DerakhtiMorteza2018Ptbs}.\n\nWaves are generated at the left-hand side of the tank. We simulate a bottom-mounted flexible flap paddle by deriving the exact solutions for velocity and pressure gradient from wavemaker theory \\citep{dean1991water-wave-mech} and apply these at the fixed boundary. This method greatly increases the computational efficiency of the model while still generating a fully nonlinear wave packet. The lateral movement $A_p$ of the simulated paddle is $<5 \\% $ of the wavelength in most cases (table \\ref{tab:experiments}) so the approximation of a fixed boundary has little effect on the results. \n\nThe motion of the paddle $x_p$ with time $t$ follows the chirped packet function \\citep{song2002on-determining-},\n\\begin{multline}\nx_p(t)=-0.25A_p\\left(1+\\tanh\\left[\\frac{4\\omega_pt}{N\\pi}\\right]\\right)\\left(1-\\tanh\\left[\\frac{4\\left(\\omega_pt-2N\\pi\\right)}{N\\pi}\\right]\\right)\\\\ \n\\times\\sin\\left(\\omega_p t\\left[1-\\frac{\\omega_pC_{ch}t}{2}\\right]\\right)\\, \n\\end{multline}\nwhere $x_p$ is a function of the paddle forcing amplitude $A_p$, the forcing frequency $\\omega_p$, the number of waves in the packet $N$ and the packet linear chirp rate $C_{ch}=1.0112 \\times 10^{-2}$. We vary $A_p$ and $N$ to generate an ensemble of non-breaking and breaking waves (table \\ref{tab:experiments}) of varying amplitude and breaking strength. \n\nEnergy absorption at the far end of the tank is achieved through a number of complementary approaches. The final $4.7\\lambda_p$ of the tank consists of a numerical sponge layer based on that derived by \\cite{clement1996coupling-of-two}, which effectively absorbs high frequency waves. The reflection of low frequency waves is minimised by gradually increasing the grid spacing within the sponge layer to enhance numerical dissipation. An outflow boundary condition is also applied to the dry portion of the lateral boundary to minimise compression of the air phase caused by the paddle motion, which further improves the performance of the model's Poisson solver. \n\nGerris uses a quadtree mesh structure which enables efficient adaptive mesh refinement \\citep{popinet2003gerris:-a-tree-}. Each level of refinement divides the parent cell into four, resulting in a maximum resolution equivalent to an uniform mesh size of of $2^j \\times 2^j$, for $j$ refinement levels. As our primary interest in this study is focused on the air-water interface and the water boundary layer, we determine the maximum required resolution based on the boundary layer thickness $\\delta={\\lambda_p}\/{\\sqrt{Re}}$ \\citep{phil1977} where $Re=\\rho U\\lambda_p \/ \\sigma$ is the wave Reynolds number. To reduce computational cost we set $Re = 4 \\times 10^4$ which allows us to resolve the boundary layer with four cells at a refinement level of $2^{10}$ and equates to a resolution of $\\lambda_p\/870$ with the scaling used. While this is smaller than our physical $Re = 1.25 \\times 10^6$, previous studies \\citep{deike2017lagrangian-tran, mostert_deike_2020} have shown that $Re = 4 \\times 10^4$ is large enough that viscous effects are not dominant and all energy within the boundary layer is adequately resolved. A limited number of experiments with a maximum refinement level of $2^{11}$ (eight cells within the boundary layer, equivalent to $\\lambda_p\/1750$) are also reported on in the following section. For all experiments, mesh refinement criteria are configured to ensure maximum resolution at the air-water interface and in regions of large vorticity.\n\nA total of 74 experiments were completed with a range of resolution, wave packet size, water depth, and paddle amplitudes (table \\ref{tab:experiments}) generating an ensemble of 285 non-breaking and 52 breaking crests for analysis. All parameters are presented as non-dimensional quantities.\n\\begin{table}\n \\begin{center}\n\\def~{\\hphantom{0}}\n \\begin{tabular}{ c c c c c }\n Refinement level & $N$ & $d\/\\lambda_p$\t& No. of cases & $A_p\/\\lambda_p$ \\\\[3pt]\n $2^{10}$ \t & 5 & 0.59 & 27 \t\t & $0.0250-0.0500$\\\\\n $2^{10}$ & 9 & 0.59 & 32 \t & $0.0250-0.0450$\\\\\n $2^{10}$ & 5 & 0.20 & 9 & $0.0800 - 0.0920 $\\\\\n $2^{11}$ & 5 & 0.59 & 3\t\t & $0.0370-0.0460$ \\\\\n $2^{11}$ & 9 & 0.59 & 3\t\t & $0.0370-0.0389$ \n \\end{tabular}\n \\caption{Summary of experiments included in this study. The model was configured using a range of mesh refinement levels, wave packets $N$ and water depth $d\/\\lambda_p$. For each configuration the amplitude of the paddle $A_p\/\\lambda_p$ was varied to generate an ensemble of breaking and non-breaking crests.}\n \\label{tab:experiments} \n \\end{center}\n\\end{table}\n\n\\section{Results}\nWe first examine the evolution of the critical parameters $B_x$, $F_x$ and $E$ for a maximally recurrent non-breaking wave (figure \\ref{fig:nb-slice}). Snapshots of the wave evolution before, at, and after the time of maximum $B$ are shown. For each parameter, a local maximum is visible at the crest of the wave. In studies utilising an inviscid solver \\citep{SeiffertBr2017Sobw, barthelemy2018on-a-unified-br} these maxima are located at the crest surface. In our simulations, where the impacts of viscosity and surface tension are included, we find that the maxima occur at the edge of the interfacial boundary layer. A consequence of limiting the Reynolds number to $4 \\times 10^4$ and effectively increasing the thickness of the turbulent boundary layer is that the depth of the maxima below the interface is amplified. However, in other aspects, such as the magnitude of $B$, our results are consistent with those previous studies.\n\nAt each time, the position and magnitude of the maxima is located with a two-dimensional spline to derive the temporal evolution of these crest values (figure \\ref{fig:nb-slice}d). Times are normalised by the local crest period $T_0$ and referenced to the time of maximum $B$ (which we set to be $t = 0$). While the absolute values of $F_x$ and $E$ differ, their evolution in time are very similar. As would be expected, the evolution of $B_x$ is closely related; however, the time of the peak value occurs slightly later than $F_x$ and $E$ due to the dependence on the crest speed $c$ (equation (\\ref{eq:B})), which undergoes a regime of deceleration and acceleration as the crest evolves \\citep{banner2014linking-reduced, fedele_banner_barthelemy_2020}.\n\nThe crest speed $c$ is a critical parameter in the calculation of $B$ but it is difficult to calculate accurately \\citep{Derakhti_2020}. We achieve this by firstly applying a smoothing filter to the interface, which removes small-scale ripples. A low-pass filter is then applied to the resultant crest positions, and a smooth cubic spline used to interpolate between data points. Comparison of the smoothed crest position with the evolution of the interface confirms that this is a robust method for calculating the crest speed. \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\textwidth]{figures\/CrestNum2_slicePlot}}\n \\caption{The evolution of $B_x$ (a), $F_x$ (b) and $E$ (c) for a non-breaking wave progressing through the growing and decaying phase. The horizontal axis is normalised by the deep-water wave length $L_0$ and the vertical axis is exaggerated by a factor of $10:1$. The temporal evolution of the maximum crest value of each parameter is shown in panel d. The time of each snapshot (A-D) is indicated by the dashed lines. Times are normalised by the instantaneous deep-water crest period $T_0$ and referenced to the time of maximum $B$. The value of $B_\\mathrm{th}$ is also indicated by the blue dashed line in panel d.}\n\\label{fig:nb-slice}\n\\end{figure}\n\nIn the breaking case (figure \\ref{fig:b-slice}) the local maxima of each parameter are more clearly defined and are located on the forward crest face at the instant of breaking. Breaking onset, defined as the time when the interface height first becomes multi-valued, occurs approximately $0.1 - 0.2$ deep-water wave periods after breaking inception ($B=B_\\mathrm{th}$). The rates of change of both $B$ and $E$ at breaking inception (i.e. $\\Gamma_B$ and $\\Gamma_E$) are smoothly varying and approximately linear (figure \\ref{fig:b-slice}d). \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\textwidth]{figures\/CrestNum1_slicePlot}}\n \\caption{As for figure \\ref{fig:nb-slice} but for a breaking wave. The vertical axis is exaggerated by a factor of $8:1$. Times are referenced to the time that breaking is first detected.}\n\\label{fig:b-slice}\n\\end{figure}\n\nWe examine the evolution of $E$ in more detail in figure \\ref{fig:E-time-series}. For non-breaking waves, the magnitude of $E$ plateaus as the maximum $B$ value is reached ($t\/T_0=0$), with the peak value of $E$ increasing as a function of the maximum wave amplitude. Conversely, in a breaking crest $E$ continues to increase through breaking inception and past breaking onset. While the absolute range of $E$ is small, there is a distinct separation in values at $t\/T_0 =0$ between the non-breaking and breaking crests.\n\nFor breaking waves the energy density convergence rate $\\Gamma_E$ (equation (\\ref{eq:Gamma-E})) is calculated by first fitting a local smooth spline to the $E$ time-series (figure \\ref{fig:E-time-series} inset). The spline is fit over the time interval for which $0.7 < B < 1.0$, chosen to optimise the spline fit for the period of interest while also capturing any variability in $E$. The first derivative of the spline yields $D_cE \/ Dt$ (equation (\\ref{eq:energy-balance})); $\\Gamma_E$ is then taken as the normalised value of $D_cE \/ Dt$ as the crest passes through $B_\\mathrm{th}$. To account for the uncertainty in the absolute value of $B_\\mathrm{th}$, $D_cE \/ Dt$ is averaged over the interval $0.85 < B < 0.86$ (shaded regions in figure \\ref{fig:E-time-series} inset). $D_cE \/ Dt$ is nearly constant at this time and we find that $\\Gamma_E$ is relatively insensitive to the choice of averaging interval. \n\nThe three breaking examples shown in figure \\ref{fig:E-time-series} are characterised as weak, moderate and strong breaking crests based on the magnitude of $\\Gamma_B$ (here calculated using an equivalent method to $\\Gamma_E$). It can be seen that the magnitude of $\\Gamma_E$ correspondingly increases with increasing $\\Gamma_B$. This is an interesting result as nothing else appears to distinguish the evolution of $E$ between these cases; there is no trend in the value of $E$ at breaking inception and $E$ is nearly identical in all three cases at breaking onset. \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\textwidth]{figures\/EnergyDensity_comparison}}\n \\caption{Evolution of the local energy density $E$ for non-breaking (black), weak (blue), moderate (red) and strong (green) breaking crests. Time is relative to the maximum $B_x$ value (non-breaking crests), or to the time of breaking onset (breaking crests). The calculation of $\\Gamma_E$ (inset) is done by fitting a smooth spline over the time period that $0.75 < B < 1.0$. $\\Gamma_E$ is taken as the average slope of the spline for the time period that $0.85 < B < 0.86$ (coloured shaded region) and is reported to $95\\%$ confidence.}\n\\label{fig:E-time-series}\n\\end{figure}\nThe strong link between $\\Gamma_E$ and $\\Gamma_B$ is seen across all breaking crests in our ensemble, regardless of wave packet configuration, water depth or model resolution (figure \\ref{fig:Gamma-comparison}). The robustness of the relationship was further tested by varying the averaging period used in the calculation of $\\Gamma$ between $0.84 < B < 0.85$ and $0.86 < B < 0.87$, with no significant impact on the results. \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=1\\textwidth]{figures\/Bx_gamma_085-086_vs_E_gamma_085-086_085}}\n \\caption{Relationship between $\\Gamma_B$ and $\\Gamma_E$ for all breaking crests from the experiments listed in table \\ref{tab:experiments}. Error bars indicate the sensitivity to varying the averaging interval when calculating $\\Gamma$. A linear regression has been applied (black dashed line) with the grey dashed lines indicating the $95\\%$ confidence (inner) and prediction (outer) intervals.}\n\\label{fig:Gamma-comparison}\n\\end{figure}\n\n\\section{Discussion and conclusions}\nThe aim of this study has been to make progress towards a physical explanation as to why the breaking inception parameter $B$ is a reliable predictor of breaking. The threshold of $B_\\mathrm{th}\\approx 0.85$ separating breaking and non-breaking waves first reported by \\citet{barthelemy2018on-a-unified-br} has since been confirmed in further independent studies. A significant feature of $B_\\mathrm{th}$ is that it provides advanced warning of breaking onset --- up to $0.2$ deep-water wave periods in our results. \\cite{DerakhtiMorteza2018Ptbs} shed further light on the subject by showing that the normalised rate of change of $B$ at breaking inception, $\\Gamma_B$, is strongly correlated to the strength of the eventual breaking event. \n\nWe have used direct numerical simulation to investigate the links between $B$ and the local crest energy density $E$. In an ensemble of experiments spanning a range of wave packets, water depths and model resolutions, we have shown that the crest energy growth rate, $\\Gamma_E$, is strongly correlated to $\\Gamma_B$ and is therefore also an indicator of the breaking strength. \n\nWe now move to a discussion on the physical interpretation of these results. Equation (\\ref{eq:energy-balance}) links the divergence of the energy flux to the rate of change of the energy density. As the work done against friction is small compared to the energy flux divergence (here $\\lVert \\mathbf{u} \\cdot \\mathbf{f} \\rVert \/ \\lVert \\nabla \\cdot \\mathbf{F_c} \\rVert =O\\left(10^{-3}\\right)$) then (\\ref{eq:Gamma-E}) can also be expressed as\n\\begin{equation}\\label{eq:Gamma-E-as-div}\n\\Gamma_E\\approx - \\frac{T_0}{E - E_0} \\left.\\nabla \\cdot \\mathbf{F_c}\\right|_{B_\\mathrm{th}}.\n\\end{equation}\nThus, $\\Gamma_E$ represents the energy flux convergence within the crest and is closely related to the mechanism that leads to breaking: an excessive flow of energy into the crest triggers a local instability which can only be dissipated through the process of breaking. However, while $\\Gamma_E$ provides a physical explanation for the process of breaking inception, the highly nonlinear nature of the breaking process makes $\\Gamma_E$ difficult to quantify except via a detailed numerical simulation. \n\nIn this study, we have re-examined the energetics of wave breaking onset through the lens of the breaking inception parameter, $B$, which is related to the normalised energy flux near the crest tip. We have shown that $\\Gamma_B$ (the rate of change of $B$) is an effective proxy for the energy growth rate, $\\Gamma_E$. Since $B = \\lVert \\mathbf{u} \\rVert \/ \\lVert \\mathbf{c} \\rVert$ at the crest surface, both $B$ and $\\Gamma_B$ can be readily measured in a laboratory or field experiment. We therefore see that the utility of the inception parameter $B$ as a predictor of wave breaking derives from its close relation to the energy flux convergence near the wave crest, which is the underlying physical process leading to breaking onset. However, an explanation for the existence of the generic breaking inception threshold $B_\\mathrm{th}\\approx 0.85$ remains to be determined.\n\n\\section*{Acknowledgements}\nThis research was supported by resource grants under the National Computational Merit Allocation Scheme (NCMAS) and the Intersect Compute Merit Allocation Scheme (ICMAS). DB is supported by an Australian Government Research Training Program (RTP) Scholarship.\n\n\\section*{Declaration of Interests} The authors report no conflict of interest.\n\\bibliographystyle{jfm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nJust after the synthesis of quasi 1D tubular\\cite{iijima1,iijima2} and 2D single layer graphene\\cite{novo1,novo2} structures, which have exhibited exceptional properties, answers to the critical question of whether Si can form similar structures have been desperately sought.\\cite{takeda,zhang,engin} This question was rather logical, since C and Si are group IV elements of the periodic table and hence they are isovalent. Also it has been attempting to adopt the technology developed behind the Si crystal to Si nanostructures. Unfortunately, the attempts of searching single layer Si in honeycomb structure have been discouraged usually by the arguments that Si does not have layered allotrope like graphite. Surprisingly, free standing silicene and germanene, namely graphene like single layer honeycomb structures of silicon and germanium, have been shown to be stable.\\cite{seymur2009} Also it has been shown that silicene shares several of the exceptional properties of graphene, such as the linearly crossing of $\\pi$- and $\\pi^*$-bands at the Fermi level (if a very small gap opening of $\\sim$ 1 meV due to the spin-orbit coupling is neglected),\\cite{liu} the ambipolar effect and the family behavior observed in nanoribbons.\\cite{seymur2010} Although the strong $\\pi - \\pi$ coupling ensures planar geometry of graphene, this coupling weakens in silicene. However, the endangered stability is regained by the rehybridization of $3s$ and $3p$ valence orbitals to four-fold $sp^3$-like bonds through the dehybridization of three-fold planar $sp^2$ bonds. This leads to the buckling of the planar honeycomb structure. Accordingly, single atomic plane of graphene is replaced by two atomic planes, which are split by a buckling of $\\Delta$=0.44\\AA~ and the alternating Si atoms at the corners of the hexagons are located in different atomic planes. Owing to the buckling (or puckering) of the structure, the vertical symmetry can be broken under perpendicular electric field which polarizes the Si atoms and hence opens a band gap.\\cite{ongunvacuum, falko, liang2012, ezawa2012} Presently, silicene has been an active field of research with several challenges.\\cite{lebeg,garcia,bai,pan,wang,angel} Search on the similar single layer, honeycomb topology have extended to group IV-IV, III-V and II-VI compounds\\cite{engin,jiang,akturk,hasan1} including SiC\\cite{sic} and ZnO,\\cite{kulkarni,zno} as well as transition metal oxides and dichalcogenides.\\cite{can1,can2} Each of these structures has been predicted to display interesting properties with potential applications in nanotechnology. Nowadays, great expectations for fundamental properties and technological application have been directed towards various single layer nanostructures and their composite, van der Waals thin films.\\cite{search}\n\nOur earlier predictions\\cite{engin,seymur2009,seymur2010} on the stability of silicene and its properties have been confirmed recently by realizing the growth and characterization of single layer silicene on Ag substrate.\\cite{lay2010, lay2012} These studies, at the same time, initiated a growing interest in silicene. Even though two dimensional semi-metallic silicene has limited applications for nanoelectronics, several new functionalities can be achieved by chemisorption of a number of foreign atoms or molecules.\\cite{peter1,peter2, lin2012}\n\nMotivated by the self-organized behavior of carbon host atom on graphene,\\cite{cac1,cac2,canethem} here we investigate the effects of silicon and carbon adatom on silicene. Silicon adatom is a host atom and its interaction with silicene may be crucial for the study of the growth of the multilayer silicene. Additionally, the decoration through uniform Si coverage may attribute useful functionalities to silicene. Being a group IV element, carbon adatom is isovalent to Si and its interaction with silicene is also important for future graphene-silicene composite materials like silicene including SiC quantum dots. Apart from Si and C adatoms, we also study the interactions of silicene with of H, O, Ti adatoms and H$_2$, H$_2$O and O$_2$ molecules. Oxygen and hydrogen atoms are known to form strong bonds on the surface of bulk silicon and are crucial elements in Si based microelectronics. In particular, it is important to know whether silicene is oxidized as easily as bulk Si does. Finally, Ti is a $3d$-transition metal atom and may form strong bonds with silicene for metallic contacts like Ti on graphene. Therefore, the interaction of these atoms with silicene and the selected functionalities attained thereof have been the focus of this work. Our study is based on free standing (suspended) silicene, since it has been shown to be stable\\cite{seymur2009} like graphene. Even if silicene is grown on Ag(111) and hence its states engage in hybridization with those of Ag(111) inducing some changes in specific states, the effects of hybridization cease at the top surface of multiple silicene layers. In this respect, our study is relevant for the strong interaction between Si, C, O, H and Ti adatoms and silicene surface.\n\nWe show that the interactions between Si adatom and silicene are complex and lead to amazing results, which are of crucial importance in the rapidly developing research on silicene: Silicon adatom on silicene pushes the Si atom underneath and readily forms a dumbbell (D) structure by donating significant electronic charge to nearest Si atoms. By engaging in a $3+1$ coordination, D decorated silicene is a structure between the fourfold coordinated diamond structure and the single layer, buckled silicene. Hence, silicene with dumbbells is slightly more favorable than pristine silicene. The cohesion of uniform D+silicene structure becomes even more superior to that of silicene when the smallest D-D distance, $d_{D-D}$, is less than $\\sqrt{3} a$ ($a$ being the lattice constant of pristine silicene). Dumbbells also display interesting dynamics and structural transformations, which are crucial for the understanding of the growth of multilayer silicene. Additionally, we also present the coverage dependent features of the D+silicene structure.\n\nCarbon adatom also creates unique reconstructions in silicene. Carbon initially forms a bond on the bridge site; however if a small barrier is passed, the C adatom substitutes one of the host silicene atoms to form a substitutional impurity. On the other hand, oxygen molecule can dissociate on silicene, where constituent oxygen atoms form strong bonds to oxidize silicene. A bridge bonded O adatom can pass from the top side to the bottom side of silicene once a small energy barrier is overcame and thus it can easily penetrate across silicene layers. H and Ti adatoms attribute magnetic properties. Silicene acquires integer magnetic moment and half-metallic character due to a specific uniform coverage of Ti.\n\n\\begin{table*}\n\\caption{Characterization of the case, where a single adatom (Si, C, H, O and Ti) adsorbed uniformly to each (4x4) supercell of silicene (corresponding to the coverage $\\Theta$=1\/32). $BS$: Binding site, where D, S, T, B and TH represents dumbbell structure, substitution of adatom, top site, bridge site and between top and hollow sites, respectively; $d$: The smallest distance between the adatom and the nearest Si atom; $E_b$: Binding energy in eV per unit cell and in kJ per mol; $E_f$: Formation energy; $E_B$: The minimum energy barrier in the migration of adatom; $E_g$: The smallest band gap between spin-unpolarized conduction and valance bands. For spin-polarized systems the gap between spin up and spin down\/spin up and spin up \/spin down and spin down conduction and valence bands are given; $Q^*$: Effective charge on the adatom; $\\mu$: the total magnetic moment per supercell. *Note that when 4 Ti adatoms are adsorbed on (4x4) supercell of silicene uniformly, the resulting structure is a half-metal (HM) with ferromagnetic order.}\n\\label{table1}\n\\begin{center}\n\\begin{tabular}{ccccccp{4cm}cc}\n\\hline \\hline\nAdatom & $BS$ & $d$ (\\AA) & $E_b$ (eV, kJ\/mol) & $E_f$ (eV) & $E_B$ (eV) & $E_g$ ($\\uparrow \\downarrow \/ \\uparrow \\uparrow \/ \\downarrow \\downarrow$)(eV) & $Q^*$ (e) & $\\mu (\\mu_B$) \\\\\n\\hline\nSi & D & 2.39 & 3.96 (380) & -0.75 & 0.92 & 0.08\/0.43\/0.43 & 0.22 & 2.0 \\\\\nC & S & 1.85 & 5.88 (564) & -1.77 & 1.52 & 0.19 & -0.42 & 0 \\\\\nH & T & 1.51 & 2.12 (203) & -0.13 & 0.26 & 0.16\/0.29\/0.24 & -0.20 & 1.0 \\\\\nO & B & 1.70 & 6.16 (591) & 2.83 & 0.65 & 0.21 & -0.34 & 0 \\\\\nTi & TH & 2.50 & 4.14 (397) & -3.59 & 0.22 & 0.09\/0.09\/0.15 (HM*) & 0.16 & 2.0 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Method}\nWe have performed spin polarized density functional theory calculations within generalized gradient approximation(GGA) including van der Waals corrections.\\cite{grimme06} We used projector-augmented wave potentials PAW,\\cite{blochl94} and the exchange-correlation potential is approximated with Perdew-Burke-Ernzerhof, PBE functional.\\cite{pbe}\n\nUsing the supercell geometry within the periodic boundary conditions, we considered the adatoms as isolated dopants. In large supercells the adatom-adatom coupling is reduced significantly and conditions of isolated adatom is approximately met. The localized states of dopants appear as flat bands. Very large supercells are not convenient from the computation point of view. Therefore, one has to optimize the supercell size. In the present study, we used $4 \\times 4$ supercells of silicene, which corresponds to the uniform coverage of one Si adatom per 32 Si host atom, namely $\\Theta$=1\/32. The size of this supercell is tested to be sufficient to minimize the adatom-adatom coupling for the purpose of the present study. On the other hand, results obtained from relatively smaller supercells are taken for the uniform coverage or decoration of adatoms. Hence, in specific cases, we also treated a uniform coverage of $\\Theta >$ 1\/32 to examine the effects of significant couplings between adatoms.\n\nThe Brillouin zone was sampled by (11x11x1) \\textbf{k}-points in the Monkhorst-Pack scheme where the convergence in energy as a function of the number of \\textbf{k}-points was tested. The number of \\textbf{k}-points were further increased to (17x17x1) in small supercell calculations. Atomic positions were optimized using the conjugate gradient method, where the total energy and atomic forces were minimized. The energy convergence value between two consecutive steps was chosen as $10^{-5}$ eV. A maximum force of 0.002 eV\/\\AA~ was allowed on each atom. Numerical calculations were carried out using the VASP software.\\cite{vasp}\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{fig1.jpg}\n\\caption{(Color online) One Si adatom adsorbed to each (4x4) supercell of silicene, which corresponds to the uniform coverage of $\\Theta$=1\/32. (a) Top and side views of the atomic configuration of the dumbbell (D) structure. Blue balls represent Si atoms. (b) Magnified view of the D structure together with the isosurface charge density. D$_1$ and D$_2$ denote Si atoms at both ends of the dumbbell; and A, E and F are silicene atoms nearest to D$_1$ and D$_2$. Excess charges on the Si atoms of the dumbbell structure are shown by numerals. (c) Energy landscape for the Si adatom on silicene calculated on a hexagon. The migration path of the Si adatom with minimum energy barrier $E_B$ is indicated by stars. (d) Contour plot of the total charge density $\\rho_{T}(\\textbf{r})$, on the horizontal plane passing through A, E and F atoms, and on the planes passing through A-D$_1$, A-D$_2$ and D$_1$ -D$_2$ bonds. (e) Energy band structure of the D+silicene structure with the dash-dotted line indicating the Fermi level. Blue(dark) and green(light) lines represent spin up and spin down states, respectively. The inset shows that the isosurface charge density of spin up states making the flat band just below the Fermi level is localized mainly at the D-structure. (f) Spin projected total density of states TDOS. Up-arrow and down-arrow stand for spin up and spin down states, respectively. The density of states DOS projected to $D_1$ is augmented four times and plotted in panel (f).}\n\\label{fig1}\n\\end{figure*}\n\nThe binding energy $E_b$, is calculated from the expression, $E_b=E_{T}[silicene]+E_{T}[A\/M]-E_{T}[A\/M+silicene]$, in terms of the total energies of bare silicene supercell and of free adatom A, (=Si, C, H, O or Ti) or molecule M, (M=H$_2$, O$_2$, H$_2$ or H$_2$O), and the structure optimized total energy of one A or one M adsorbed to each silicene supercell, respectively. All total energies are calculated in the same supercell. In our notation, $E_{b} >$0 indicates a binding structure. The formation energy, $E_f$, takes into account the binding relative to the ground state of A either in bulk crystal or in a molecule. Accordingly, for the case of hydrogen, for example, $E_{f} =-E^{'}_{b}\/2+E_b$, where $E^{'}_{b}$ is the binding energy of H$_2$ relative to hydrogen atom. Normally, while a process with $E_{f}>$0 is favored, $E_{f} <$0 may give rise to clustering or desorption under specific circumstances.\n\nThe bonding and effective charge of adatoms are characterized by calculating total charge density of adatom+silicene system, $\\rho_{T}(\\textbf{r})$. We presented the charge distribution in terms of isosurfaces and contour plots. We also carried out Mullliken analysis\\cite{mulliken} in terms of atomic orbitals of constituent atoms to obtain their effective charge, $Q^*$, as implemented in the SIESTA package.\\cite{siesta} The energy landscape of an adatom on silicene is calculated by placing the adatom to 500 different grid points in the hexagon of silicene and performing self consistent energy minimization calculation for each point. At each grid point, the $x$ and $y$ coordinates of the adatom were kept fixed while its $z$ coordinate was relaxed to minimize the total energy. The energy at each grid point is designated by a color code. The fully relaxed, spin-polarized calculations of Ti adatoms are also repeated using local basis set in SIESTA package resulting in the same magnetic moment obtained by using plane wave basis set.\\cite{vasp}\n\n\\section{Results and Discussions}\nThe interactions of silicene with the adatoms (Si, C, H, O and Ti) and molecules (H$_2$O, H$_2$ and O$_2$) are characterized by the calculation of optimum binding geometries and the corresponding atomic structures at the proximity of the adatom. The energy landscapes and the path of migrations with lowest energy barriers are calculated; relevant electronic and magnetic properties together with the spin-projected and adatom projected densities states are presented. We present a summary of our results in Table~\\ref{table1}. Here the top (T), bridge (B), hollow (H), between top and hollow (TH), and between bridge and hollow (BH) are special sites where adatoms are bound to silicene. In specific cases, the adatom can substitute the host Si atom (S) or can push it down to form a dumbbell (D) structure.\n\n\\begin{table*}\n\\caption{Variation of the properties of D+silicene (i.e. silicene with uniform coverage of dumbbell structure in different supercells). ($n$x$n$): Supercell size; $\\Theta =1\/2n^2$: uniform coverage of D structure per the number of Si atoms in silicene supercell; $(2n^2+1)$: Number of Si atoms of D+silicene in the ($n$x$n$) supercell of silicene; $d_D$: distance between two nearest neighbor dumbbell structures; $\\mu$:magnetic moment per supercell; $ES$: Electronic structure specified as metal M, or semiconductor with the band gap between valance and conduction bands, $E_g$; $E_b$: binding energy; $E_{C}[D]$: cohesive energy of Si atom in D+silicene structure; $\\Delta E_{C}$: difference between the cohesive energies of Si atom in D+silicene and pristine silicene, where positive values indicates that D+silicene structures are favorable. The cohesive energy of pristine silicene is $E_C$=3.936 eV.}\n\\label{table2}\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n\\hline \\hline\nSupercell & $\\Theta$ & $(2n^2+1)$ & $d_D$ (\\AA) & $\\mu$ ($\\mu_B$) & ES & $E_b$ (eV) & $E_{C}[D]$ (eV) & $\\Delta E_{C}$ (eV) \\\\\n\\hline\n$1 \\times 1$ & 1\/2 & 3 & 3.58 & 0 & M & 4.13 & 4.002 & 0.066 \\\\\n$ \\sqrt{3} \\times \\sqrt{3}$ & 1\/6 & 7 & 6.52 & 0 & M & 4.35 & 4.001 & 0.065 \\\\\n$2 \\times 2$ & 1\/8 & 9 & 7.70 & 0 & M & 3.89 & 3.956 & 0.020 \\\\\n$3 \\times 3$ & 1\/18 & 19 & 11.50 & 1.8 & M & 3.92 & 3.945 & 0.009 \\\\\n$4 \\times 4$ & 1\/32 & 33 & 15.40 & 2.0 & 0.083 & 3.96 & 3.939 & 0.003 \\\\\n$5 \\times 5$ & 1\/50 & 51 & 19.20 & 2.0 & 0.078 & 4.02 & 3.937 & 0.001 \\\\\n$7 \\times 7$ & 1\/98 & 99 & 23.0 & 2.0 & 0.075 & 4.03 & 3.938 & 0.002 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{fig2.jpg}\n\\caption{(Color online) One carbon adatom adsorbed to each (4x4) supercell of silicene amounting to a uniform coverage of $\\Theta$=1\/32. Blue and brown balls indicate Si and C atoms, respectively.(a) Top view of atomic configuration of C adatom adsorbed above a Si-Si bond (BH-site) with a binding energy of $E_b$=4.41 eV. This is a precursor state for the substitutional carbon atom described in (b). (b) Top and side views of the carbon atom substituting one of the Si atoms and pushing it to point M to form three fourfold coordinated Si atoms at (G,E,N). The binding energy of the C adatom at this configuration is $E_b$=5.88 eV (564kJ\/mol). (c) The energy landscape of the C adatom on silicene calculated on a hexagon. The migration path of the Si adatom with minimum energy barrier $E_B$=1.52 eV is indicated by stars. (d) Isosurfaces of the total charge density, $\\rho_{T}(\\textbf{r})$ describing the bond formation around the substitutional carbon atom. Contour plots of the total charge density in the plane passing through A, E and F atoms show the $sp^2$ hybrid bonds of the substitutional carbon atom. Contour plots in the plane of G,E,M and G,M,N atoms show four-fold bond formation. (e) Energy band structure of silicene including a single substitutional carbon atom in each (4x4) supercell. (f) The corresponding total density of states TDOS and the density of states DOS projected on substitutional carbon atom.}\n\\label{fig2}\n\\end{figure*}\n\n\\subsection{Si adatoms}\nA Si adatom, which is initially bonded to the T site, pushes down the Si atom underneath to form a dumbbell D structure. This configuration occurs without any barrier and has the binding energy of $E_b$=3.96 eV for a single Si adatom forming a D-structure in each $4 \\times 4$ supercell, which is equivalent to 380kJ per one mole of Si atoms. Therefore, a single D structure is not a predetermined configuration; it can occur while Si adatom migrates on silicene. In ~\\ref{fig1}(a-b) the atomic configuration of D structure is presented. Two Si atoms positioned at two ends of the dumbbell are specified as D$_1$ and D$_2$. The distance between the dumbbell atoms D$_1$ or D$_2$ and nearest silicene atoms (A, E and F), which are located in a horizontal plane is 2.41 \\AA. This is larger than the nearest Si-Si distance 2.28 \\AA~ in pristine silicene. The distance between D$_1$ and D$_2$ is relatively large and is 2.69 \\AA. We note that in our earlier work, it was found that carbon atom migrating on graphene can form also similar dumbbell structure, even if it is slightly less energetic relative to its B-site binding.\\cite{canethem} Recently, formation of the dumbbell structure in silicene layers has been deduced during the course of vacancy healing. \\cite{healing} \n\n\nOur recent calculations have also demonstrated that the formation process of the dumbbell structure on silicene multilayers grown on Ag substrate is practically identical to that on free-standing silicene reported in the present paper. The reconstructions such as 3$\\times$3 and $\\sqrt{3} \\times \\sqrt{3}$ that took place in the course of multilayer growth on Ag are induced by the formation of dumbbell structure. Therefore, the predictions in the present paper are important for silicene research and reveal new physical phenomena related with the formation of multilayer Si, which is apparently the precursor state for missing layered structure of silicon.\n\nThe Mulliken analysis indicates that the depletion of electronic charge from each of D$_1$ and D$_2$ atoms is +0.22 electrons, which is transferred to nearest three Si atoms of silicene. This situation suggests that strong bonds with mixed covalent-ionic character\\cite{harrison} form between nearest silicene atoms (A, E and F) and each of the dumbbell atoms, D$_1$ or D$_2$. On the other hand the D$_1$ - D$_2$ bond is relatively weak. These arguments can be depicted from isosurfaces of the total charge density and the charge density contour plots presented in ~\\ref{fig1}(d). Accordingly, each of A, E and F atoms are four-fold coordinated, and hence they mimic the bulk Si crystal by making four bonds with their nearest neighbors. Whereas D$_1$ and D$_2$ atoms are 3+1 coordinated, each of them makes three strong bonds with A, E and F, but are weakly bonded to each other. We note that having positively charged two Si atoms located above and below the Si planes of buckled silicene may attribute interesting functionalities, which may be monitored by the electric field applied perpendicular to silicene. For example, positively charged surface of D+silicene is attracted by negatively charged surfaces or vice versa for positively charged surfaces. Additionally, the work function (or photoelectric threshold) of silicene increases upon its decoration with D.\n\nIt should be noted that the formation energies of both the pristine silicene and D+silicene are negative with respect to bulk Si in diamond structure. In spite of that, one free Si adatom at the close proximity of a D structure does not form any bond with D$_1$ or D$_2$ to nucleate a cluster or an atomic chain as carbon adatom does on graphene or boron nitride\\cite{cac1, cac2} It rather moves to the third nearest neighbor and form another D structure. It appears that the D structure display a self-organizing character. The D structure occurs at the T sites of silicene; H-sites are unfavorable since Si adatom cannot form sixfold long bonds with Si atoms at the corners of the hexagon. The calculated energy landscape of the Si adatom is shown in ~\\ref{fig1}(c). The minimum energy barrier for the migration of Si adatom is estimated to be 0.92 eV. Although this barrier is significant to hinder diffusion at room temperature, at high temperatures the D structure may display interesting dynamics in the course of the growth of silicene.\n\nThe fact that the binding energy of the D structure ($E_b$=3.96 eV) is slightly higher than the cohesive energy of a Si atom forming a pristine silicene ($E_C$=3.94 eV) brings about the question whether the D+silicene with diverse decoration of D can be energetically more favorable than bare silicene and may constitute its complex derivatives. To this end, we compare the cohesive energies of Si atoms in the D+silicene structures with diverse coverage values with that in the pristine silicene. The cohesive energy per Si atom in an $n \\times n$ supercell comprising one single D structure is obtained from the energy difference between the energy of free Si atom $E_{T}[Si]$ and the total energy of the structure of one D per supercell divided by 2$n^2$+1, namely $E_{C}[D]=E_{T}[Si]-E_{T}[D+silicene]\/(2n^2+1)$. Similarly, the cohesive energy of Si atom in a $n \\times n$ silicene supercell is $E^{o}_{C}=E_{T}[Si]-E_{T}[silicene]\/2n^2$. Then, the positive values of the energy difference, $\\Delta E_{C} = E_{C} - E^{o}_{C}$ indicates that D+silicene structure is more favorable. For the sake of comparison, the cohesive energy of single Si atom in bulk silicon is calculated with the same parameters to be 4.71 eV. The cohesive energy and relevant properties of D+silicene are calculated as a function of coverage and presented in Table~\\ref{table2}.\n\nThe cohesive energy $E_C$ of the D+silicene structure decreases with decreasing coverage. It is larger than the cohesive energy of Si atom in silicene and hence is slightly more favorable energetically than pristine silicene. For $n$=1 ($\\Theta$=1\/2), the D+silicene structure has nonmagnetic ground state; it is metal and has high cohesive energy. Similarly, for a single D adsorbed to $\\sqrt{3} \\times \\sqrt{3}$ supercell, which is predicted to be a nonmagnetic metal, $\\Delta E_{C}$=65 meV per Si atom is significant. Present results confirm the recent study,\\cite{kaltsas} which found that $\\sqrt{3} \\times \\sqrt{3}$ coverage stable and has cohesive energy 48 meV per atom higher than that of pristine silicene. We believe that the difference between the calculated cohesive energies occurs from the van der Waals correction taken into account in the present study. For $n$=2 and $n$=3, $\\Delta E_C$ decreases and continues to be nonmagnetic metal. However, for $n$=4, 5 and 7, D+silicene attains spin polarized ground state and achieve $\\sim$2 $\\mu_{B}$ magnetic moment per supercell. Hence, three of them are spin polarized semiconductor with a band gap $E^{\\uparrow \\downarrow}_{g} \\sim$ 80 meV. For the case of $n$=4, the flat bands at the edges of valence and conduction bands in ~ \\ref{fig1}(e) are derived from orbital states, which are localized at the D structure with also minor contributions from other Si atoms. Similar flat bands due to D structure also occur at -2 eV in the valence band as shown in the spin polarized DOS projected to D atoms presented in ~\\ref{fig1}(f).\n\nFor $\\Theta \\leq $ 1\/32 the spins are polarized and metallic states change into semiconductors. Also, $\\Delta E_{C}$ is reduced and becomes smaller than the accuracy limits of present calculations. Apparently, various structures of D+silicene can be considered as the allotropes of the pristine silicene and display variations in the physical properties as a function of the coverage. The D structures forming uniform (1x1), ($\\sqrt{3}$x$\\sqrt{3}$), (2x2), (4x4), (5x5) and (7x7) supercells form centered hexagons of different sizes on silicene. On the other hand, two D structures contained in the ($\\sqrt{3}$x$\\sqrt{3}$) and ($n$x$n$) supercells with $n$=3,6,9,.. can form regular honeycomb structure and yield linearly crossing bands.\n\nFinally, the question of whether the dumbbell Si atoms are active sites of silicene or not is investigated through the adsorption of Ti and H$_2$O to the dumbbell Si atoms. We found that similar to bare silicene H$_2$O did not form bonds with D$_1$ adatom. The increase of the binding energy relative to that on bare silicene was only 130 meV. The binding energy of Ti atom to D$_1$ was almost half of the binding energy of Ti atom to bare silicene. We therefore arrive at the conclusion that the D structure of Si adatom gives rise to interesting electronic and magnetic properties, but it does not involve in active chemical reactions that are significantly different from bare silicene.\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{fig3.jpg}\n\\caption{(Color online) One hydrogen adatom adsorbed to each (4x4) supercell of silicene. (a) Top and side views of the H adatom adsorbed at the T-site above the protruded Si atom. Blue and red balls represent Si and H atoms, respectively. (b) The local and magnified view of the adsorption geometry with the isosurfaces of the charge density showing bonding configuration. (c) Energy landscape. The path of migration of H with minimum energy barrier of $E_B$=0.26 eV is shown by stars. (d) The contour plots of the total charge density $\\rho_{T}(\\textbf{r})$ in a lateral plane passing through three Si atoms (A, E, F) nearest to H adatom. Similar contour plots passing through the plane of atoms A, G, H and A, H, E describe the bonding with the H adatom. (e) Energy band structure of silicene including a single H adatom adsorbed to each (4x4) supercell. Blue(dark) and green(light) lines indicate the spin up and spin down split bands. The flat band of H atom is indicated. (f) The corresponding spin polarized total density states.}\n\\label{fig3}\n\\end{figure*}\n\n\\subsection{C adatoms}\nFree carbon atom is first adsorbed to the BH site with a binding energy $E_b$=4.41 eV. This is only a local minimum and a precursor of another configuration with a higher binding energy of $E_b$=5.88 eV (588kJ\/mol), where C adatom substitutes one of the Si atoms of silicene and displaces it to a nearby site below the silicene layer. The latter substitutional site is 1.47 eV more energetic relative to the BH site adsorption, but needs to overcome a small energy barrier. These two configurations are shown in ~\\ref{fig2}(a-b). The contour plots of the charge density show dramatic changes in bonding. In particular, substitutional carbon atom acquires 0.42 electrons from three nearest Si atoms (A, E and F), which are not buckled anymore, but locally flattened. Under these circumstances, $sp^3$ like hybrid orbitals are dehybridized upon substitution of carbon atoms, which attains the $sp^2$ bonding with nearest three Si atoms as shown in ~\\ref{fig2}(d). We note also that silicene substituted by C atoms can be the precursor of the single layer SiC in honeycomb structure, which is planar and has in-plane stiffness of 166 J\/m$^2$.\\cite{sic} Additionally, single layer SiC is a wide band gap semiconductor with $E_g$=3.8 eV. Also the Si atom, which is displaced by by substitutional carbon is a potential candidate to form a D structure. It is therefore anticipated that carbon adatom can initiate the D structures since they at the same time grow the planar domains of Si-C compounds in silicene.\n\nThe energy landscape presented in ~\\ref{fig2}(c) shows that the substitution of C is the most energetic configuration. The minimum energy barrier $E_B$=1.52 eV revealed from this plot is rather high and blocks the migration of C adatom. The migration path of C with lowest energy barrier is marked by stars.\n\nWhile single substitutional C atom in each (4x4) supercell opens a gap of 0.19 eV as shown in ~\\ref{fig2}(e-f), the band gap shall increase with increasing concentration of substitutional C and eventually saturate at $E_g$=3.8 eV. Present findings implies that one can make a mesh of composite material from bare semimetallic silicene by forming C-doped semiconducting domains with tunable band gap.\\cite{hasan1} Upon increased C coverage one can also expect to fabricate a core-shell structure consisting of the planar SiC domains and hence quantum dots in semimetallic silicene lattice.\\cite{coreshell}\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig4.jpg}\n\\caption{(Color online) Dissociation of oxygen molecule on silicene. Dissociated oxygen atoms are adsorbed to different sites on silicene. Blue and red balls represent Si and O atoms, respectively. }\n\\label{fig4}\n\\end{figure}\n\n\\subsection{Binding of H$_2$ molecule and H atom}\nAdsorption of H atom to the surfaces of bulk Si crystal forming monohydride, dihydride and trihydride have been the subject of extensive studies earlier.\\cite{sih} The interaction of H$_2$ molecule and H atom is also essential for silicene. Similar to graphane CH\\cite{chribbon}, H atom is adsorbed to Si atoms at the corners of hexagons alternatingly from top and bottom surfaces forming silicane i.e. fully hydrogenated silicene, SiH.\\cite{sihsil1,sihsil2} Whereas, the interaction between H$_2$ and silicene is rather weak. The dissociation energy of free H$_2$ molecule is 4.5 eV and hence the formation energy for the adsorption of H atom through the dissociation of H$_2$ is negative. Therefore, molecular hydrogen neither forms strong chemical bonds with silicene, nor it dissociates. However, once the atomic hydrogen positioned on the surface of silicene, it can readily form strong bonds with silicene at T-site above the protruded of Si atoms. The binding energy is calculated to be $E_b$=2.12 eV (203kJ\/mol). The Si-H bond distance is 1.51 \\AA. In ~\\ref{fig3} we present all the relevant results related to the interaction and binding of H with silicene.\n\nThe isosurfaces of the total charge density shows the bond formation among H adatom and host Si atoms in ~\\ref{fig3}(b). As depicted by the contour plots of $\\rho_{T}(\\textbf{r})$ in ~\\ref{fig3}(d), the Mulliken analysis\\cite{mulliken} yields an excess charge of $Q^*$=-0.20 electrons on H atom. This situation is in contrast with H adatom on graphene, where charge is transferred from H to graphene, but is in compliance with the ordering of electronegativities\\cite{electrone} of Si, H and C atoms as 1.8, 2.1 and 2.5, respectively.\n\nThe energy landscape of H adatom shown in ~\\ref{fig3}(c) confirms that the T-site is really the energetically most favorable site. We also predict the minimum energy barrier is $E_B$=0.26 eV for the migration of H adatom on silicene. It appears that H adatoms are rather mobile on silicene. In this context, the possibility that the formation of H$_2$ molecule from adsorbed H atoms of SiH leading to the dissociation of H requires serious investigations in studies dealing with silicane.\n\nHydrogen adatom has dramatic effects on the electronic structure of silicene as shown in ~\\ref{fig3}(e-f). First of all, H adsorbed silicene has spin-polarized state, where the spin degeneracy is broken and spin up and spin down bands split. Spin polarization is depicted by the energy band structure of one H atom adsorbed to each (4x4) supercell and corresponding spin projected TDOS. The flat spin up and spin down bands below and above the Fermi level mainly derive from H orbitals. The dispersion of this band decreases with decreasing coverage of H and eventually appears as two localized impurity levels. While the band gap between the highest occupied spin up band and the lowest unoccupied spin down band is only $E_{g}^{\\uparrow \\downarrow}$=0.16 eV, the gap between spin up bands is $E_{g}^{\\uparrow \\uparrow}$=0.29 eV. For this spin polarized ground state of silicene+H adatom system, each H adatom contributes a magnetic moment of $\\mu$=1.0 $\\mu_B$ per supercell. On the other hand, two H adatom adsorbed to two adjacent Si atoms from different sides of silicene plane has $\\mu$=0 $\\mu_B$. We conclude this section by noting that the holes created on silicane can have magnetic moments which depend on their size and geometry.\\cite{chribbon}\n\n\\begin{figure*}\n\\includegraphics[width=15cm]{fig5.jpg}\n\\caption{(Color online) A single oxygen adatom adsorbed to (4x4) supercell of silicene. (a) Top and side views of the atomic configurations of the O adatom adsorbed at the B-site of silicene above the Si-Si bond. Blue and red balls represent Si and O atoms, respectively. (b) The local and magnified view of the adsorption site with the isosurfaces of the total charge density showing the bonding configuration. (c) Energy landscape. The migration path of O with a minimum energy barrier of $E_B$=0.65 eV is shown by stars. (d) The contour plot of the total charge density $\\rho_{T}(\\textbf{r})$ in a plane passing through through Si-Si bond and O atom. Weakening of Si-Si bond underneath and charge accumulation on O atoms is clearly seen. (e) Energy variation of the O adatom while it penetrates from upper side to the lower side of silicene. The energy barrier for this process is only 0.28 eV. In the graph, zero in the x-axis indicates the position of the silicene layer. (f) Energy band structure of silicene including the O adatom adsorbed to each (4x4) supercell. The direct band gap at $K$-symmetry point is $E_g$=0.21 eV. (g) The corresponding spin polarized total density states and the density of states DOS projected to O atom.}\n\\label{fig5}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=16cm]{fig6.jpg}\n\\caption{(Color online) One Ti adatom adsorbed to each (4x4) supercell of silicene. (a) Top and side view of atomic configuration of Ti adatom adsorbed at the TH site. Dark blue and light blue balls represent Si and Ti atoms, respectively. (b) The local and magnified view of the adsorption site with the isosurfaces of the total charge density showing bonding configuration. (c) Energy landscape. The path of migration of Ti adatom with minimum energy barrier of $E_B$=0.22 eV is shown by stars. (d) Spin polarized energy band structure of silicene including a single Ti adatom adsorbed to each (4x4) supercell. Spin up and spin down bands are shown by blue(dark) and green(light) lines. (e) The corresponding spin polarized total density states and the density of states DOS projected to the Ti atom. (f) Spin polarized energy band structure for a single Ti adsorbed to each (2x2) silicene supercell showing the half-metallic state with a metallic spin up band and a gap between spin down bands.}\n\\label{fig6}\n\\end{figure*}\n\n\\subsection{Binding of O$_2$ molecule and O adatom}\nThe interaction of O atom with graphene has been an active field of study. Both experimental and theoretical studies have shown that semi-metallic graphene changes to a semiconductor owing to a gap opening upon oxidation. Reversible oxidation-reduction process and the atomic processes thereof have been treated in several studies. \\cite{gox1,gox2,gox3,gox4} It is well-known that the oxidation of silicon surfaces and formation of SiO$_2$ film is one of the crucial processes in microelectronics. Therefore, we expect that O-silicene interaction is even more interesting. In fact, the interaction between O$_2$ and silicene surface is rather strong and leads to the dissociation of molecule into two O atoms provided that the O$_2$ molecule is in a close proximity of the active bridge sites of the silicene. The disassociated oxygen atoms are then adsorbed at different sites of silicene. In ~\\ref{fig4} the atomic configuration related with dissociation is presented.\n\nSingle O atom binds to silicene above the Si-Si bond at B site. The binding energy is calculated as $E_b$=6.16 eV (591kJ\/mol). The bond distance between O and nearest Si atom is 1.70 \\AA. The Si-Si bond underneath the O adatom is weakened and is elongated from 2.28 \\AA~ to 2.33 \\AA. The charge is transferred from silicene to O. The Mulliken analysis estimates the charge transfer from silicene to O adatom to be $Q^*$=-0.34 electrons. The bonding configuration of O adatom and charge transfer to O is presented in ~\\ref{fig5}(a-b).\n\nBoth sides of silicene are equally reactive and can easily be oxidized by O adatoms. However, the penetration of oxygen adsorbed to one side to the other side is crucial. For example, metal surfaces are protected from oxidation efficiently by graphene coating, since the penetrating oxygen adatom from one side to the other side above the metal surface has to overcome a barrier of $\\sim$6 eV.\\cite{coating1,coating2,coating3} We did the same test for silicene and calculated the energy barrier necessary for the O adatom to pass from the upper to the bottom side. In ~\\ref{fig5}(e) the variation of the total energy is shown as the O adatom is forced to penetrate from one side of the silicene to the other side. In the course of penetration, the coordinates of the atoms are fully relaxed. During penetration, one of the Si-Si bonds expands and the oxygen adatom passes through the center of this expanded Si-Si bond. The energy barrier for this penetration process is calculated as 0.28 eV, which is rather small as compared to the barrier in graphene. In the course of penetration, the Si-Si bond underneath O is slightly elongated and O itself is slightly displaced towards the center of hexagon before it arrives at the equilibrium position below silicene. Low energy barrier implies that O adatoms can easily penetrate into bilayer or multilayer silicene to oxidize them.\n\nThe ground state of the system consisting of one O atom adsorbed to the $4 \\times 4$ supercell is nonmagnetic. Energy bands of this supercell open a direct band gap of 0.21 eV at the $K$-point of the Brillouin zone, as shown in ~\\ref{fig5}(f-g). In this respect, gap opening through oxidation, hence transition from semimetallic silicene to semiconductor is reminiscent of graphene, where controlled reduction\/oxidation process by external agents, such as charging or perpendicular electric field was exploited for device applications.\\cite{gox1,gox2,gox3,gox4}\n\n\\subsection{Ti adatom}\nFinally, we consider the coverage consisting of a single Ti atom adsorbed to each (4x4) supercell of silicene. Ti atoms is adsorbed above silicene between T- and H-sites which is identified as the TH site. However, the energetically favorable binding site appears to depend on coverage. For example, the binding site switches to the B site for single Ti atom adsorbed uniformly to each (6x6) supercell of silicene. The binding energy is rather strong and is $E_b$=4.14 eV (397kJ\/mol). The minimum energy barrier to the migration of Ti adatom is revealed from the calculated energy landscape in ~\\ref{fig6}(c) to be $E_B$=0.22 eV. Hence, Ti adatom is mobile at elevated temperatures. However, the formation energy is negative and consequently Ti clustering on silicene can occur at certain circumstances. Upon binding to TH site, 0.16 electrons are transferred from the Ti adatom to the nearest Si atoms. The binding configuration and other relevant properties of Ti+silicene system are presented in ~\\ref{fig6}(a-f).\n\nTi atom is a light transition metal with an open $3d$-shell and attributes spin polarized ground state to Ti+silicene system. For the system under study, where one Ti adatom is adsorbed to each (4x4) supercell, the magnetic moment per supercell is $\\mu$=2.0 $\\mu_B$. Accordingly, spin up and spin down bands split. The linearly crossing bands open a band gap and flat $3d$-bands occur in the gap and around the Fermi level. The minimum gap between spin up bands, $E_{g}^{\\uparrow \\uparrow}$=0.09 eV occurs at $K$-point. The minimum band gap between spin down bands is indirect and $E^{\\downarrow \\downarrow}_g$=0.15 eV. However, these electronic and magnetic properties depend on the coverage. Further to the coverage of $\\Theta$=1\/32, we investigated higher Ti coverage by adsorbing four Ti atoms uniformly on each (4x4) supercell actually leading to the uniform coverage of one Ti atom for every (2x2) silicene supercell (i.e. $\\Theta$=1\/8). This way we were able to treated also the antiferromagnetic interaction among Ti adatoms. We found that the ground state for the coverage $\\Theta$=1\/8 is ferromagnetic. Moreover, the gap between spin up bands diminish to make the system metallic for spin up electrons, while the gap between the spin-down bands is reduced. Accordingly, the $\\Theta$=1\/8 coverage attained a half-metallic behavior.\n\n\\section{Conclusions}\nThe Interaction of Si, C, H, O and Ti adatoms and H$_2$, H$_2$O and O$_2$ molecules are crucial for silicene. Using density functional theory, we examined the energetics of binding and atomic configuration related with these adatoms and molecules. It is predicted that while H$_2$O is non-bonding and H$_2$ is very weakly bound, O$_2$ molecule is dissociated on silicene leading to its oxidation. Owing to small energy barrier, an oxygen atom bound to the surface of multilayer silicene can easily diffuse to oxidize other layers. We found that oxidized silicene is a semiconductor; the band gap can be tuned by oxygen coverage. Silicene changes to a spin-polarized state upon the adsorption of hydrogen atom. Similar to graphane, the magnetization and the band gap can be tuned by H adatom concentration. Titanium adatoms also attribute coverage dependent magnetic ground state and band gap, which is tuned by Ti concentration. The nonmagnetic silicene becomes a half-metal upon the uniform Ti coverage at $\\Theta$=1\/8. Even if C adatom forms strong bridge bonding with Si-Si bonds, the substitution of host silicene with carbon adatom is favored. Finally, we revealed that the formation of dumbbell structure by Si adatom can lead to stable structures with interesting coverage dependent physical properties. We believe that the allotropes of silicene consisting of uniform coverage of Si dumbbells will attract interest. It is also shown that silicene acquire diverse and important functionalities owing to its decoration with all these adatoms.\n\n\\section{Acknowledgement}\nThe computational resources have been provided by TUBITAK ULAKBIM, High Performance and Grid Computing Center (TR-Grid e-Infrastructure) and UYBHM at Istanbul Technical University through Grant No. 2-024-2007. This work was supported partially by the Academy of Sciences of Turkey(TUBA) and TUBITAK. The authors acknowledge the financial support of TUBA.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe evolution of galaxies is one of the most important topics in extragalactic astronomy. Galaxies evolve in different ways under the influence of internal or external processes, leading to different morphologies \\citepads{2004ARA&A..42..603K}. Internal secular processes are slow, with timescales equivalent to several times the rotation period of the galaxy \\citepads[e.g.][]{1979ApJ...227..714K, 1981A&A....96..164C, 1998gaas.book.....B, 2004ARA&A..42..603K, 2013seg..book....1K}, but they are particularly relevant in the low-density regions of the Universe preferred by late-type galaxies.\n\nMassive discs and bars play an important role in secular evolution. Galactic bars are formed in the early stages of a galaxy lifetime \\citepads{2012ApJ...758..136S}, shortly after the disc is settled and becomes dynamically cold. The bar behaves like a true engine of evolution of the galaxy, driving the distribution of angular momentum in the disc and causing gas inflow. Disc gas can be funnelled towards the inner regions of a galaxy, where it can lead to new structures or even help fuel AGN activity \\citepads[e.g.][]{1985A&A...150..327C, 1989Natur.338...45S,1999ApJ...525..691S,2004ApJ...600..595R,2005ApJ...632..217S,2015MNRAS.449.2421S}. Nuclear rings are stimulated by the gas inflow driven by the bar to the central parts of the galaxy. The gas inflow stagnates near inner Lindblad dynamical resonances where the gas piles up, shocks, and leads to star formation \\citepads[e.g.][]{1995ApJ...454..623K, 1995ApJ...449..508P, 1996FCPh...17...95B, 2000A&A...362..465R, 2001AJ....121.3048M, 2006MNRAS.369..529F, 2010MNRAS.402.2462C}. Studying the kinematics of older and younger stellar populations in nuclear rings can provide valuable information on when and how rings and bars were built and thus also on when the disc settled \\citepads[][]{2015A&A...584A..90G, 2019MNRAS.482..506G}.\\\n\nThe evolution of the inner structures of a galaxy can be studied using both theoretical and observational approaches. In the last few decades the theoretical approach has advanced significantly, mainly based on hydrodynamical simulations. These can reproduce the kinematics of a wide variety of galaxies at different redshifts and thus contribute to the development of a general understanding of galaxy evolution. Models like EAGLE \\citepads{2015MNRAS.446..521S} and IllustrisTNG \\citepads{2018MNRAS.473.4077P} have become very popular and successfully replicate the results of key observations.\\looseness-2\n \nOn the observational side, technical advances have resulted in tremendous progress. One of these is the introduction of integral-field spectroscopy (IFS), from which spatially resolved maps of gas and stellar kinematics in galaxies can be derived. Combining IFS data with stellar population models has allowed the study of the distribution of kinematics, age, metallicity, extinction, and colour of the stellar populations in different parts of galaxies at unprecedented resolution. From the first implementations with instruments such as OASIS or SAURON \\citepads[e.g.][]{2001MNRAS.326...23B}, IFS has evolved into revolutionary instruments like VIMOS \\citepads[][]{2003SPIE.4841.1670L}, SINFONI \\citepads[][]{2003SPIE.4841.1548E, 2004Msngr.117...17B}, PMAS\/PPaK \\citepads[][]{2005PASP..117..620R, 2004AN....325..151V, 2006PASP..118..129K}, MUSE \\citepads[][]{2010SPIE.7735E..08B}, or MEGARA \\citepads[][]{2014SPIE.9147E..0OG, 2018SPIE10702E..17G, 2018SPIE10702E..16C}, and into surveys like ATLAS3D \\citepads[][]{2011MNRAS.413..813C}, SAMI \\citepads[][]{2012MNRAS.421..872C}, MaNGA \\citepads{2015ApJ...798....7B}, CALIFA \\citepads{2016A&A...594A..36S}, or TIMER \\citepads{2015A&A...584A..90G, 2019MNRAS.482..506G}.\\looseness-2\n\nHere we use IFS data to study the secular evolution of galaxies by analysing the kinematics of different stellar populations, as well as their relationship with parameters like dust content, stellar age and metallicity. In past studies, \\citetads{1990MNRAS.245..350D} and \\citetads{1992ApJ...400L..21B} quantified how absorption affects the rotation curves in the inner regions of late-type galaxies. These works were later supplemented with mainly Monte Carlo simulations for low surface brightness \\citepads{2001ApJ...548..150M}, elliptical \\citepads{2000MNRAS.313..153B,2000MNRAS.318..798B}, and disc galaxies \\citepads{2003MNRAS.343.1081B}, and with studies using IFS data cubes leading to analytical methods for extracting the kinematics and the stellar population parameters \\citepads[e.g.][]{2006MNRAS.365...74O, 2007IAUS..241..175C}. These methods, based on the fitting of IFS data in their full spectral range to single-age and single-metallicity stellar population (SSP) models \\citepads{2003MNRAS.344.1000B, 1999ApJ...513..224V, 2010MNRAS.404.1639V}, have been successfully applied to several scenarios: the separation of counter-rotating populations in discs \\citepads[][]{2013A&A...549A...3C, 2013MNRAS.428.1296J}, bulge-disc decomposition \\citepads{2017MNRAS.465.2317J, 2017MNRAS.466.2024T, 2019MNRAS.485.1546T, 2019MNRAS.484.4298M, 2019MNRAS.488L..80M}, galaxy-halo separation \\citepads{2018MNRAS.478.4255J}, or kinematically decoupled cores \\citepads[KDC,][]{2018MNRAS.480.3215J}.\\\n\n\\citet{etheses4179}\\footnote{Available at Durham E-Theses \\url{http:\/\/etheses.dur.ac.uk\/4179\/}} studied the kinematics of different stellar populations by assuming that different spectral regions are sensitive to stars of different ages. He extracted the line-of-sight velocity distribution parameters (\\textit{v$_\\star$}, \\textit{$\\sigma_\\star$}, \\textit{$h_3$}, and \\textit{$h_4$}) for a young stellar population, traced by the H$\\beta$ line, and an old one, traced by the Ca\\,{\\sc II} Triplet, and showed that the enhanced H$\\beta$ absorption exhibited by elliptical galaxies may be produced by the presence of a disc formed by young stars.\\\n\nThe aim of the present work is to quantify how the kinematics of young ($\\lesssim 2$\\,Gyr) and old stellar populations differ in the nuclear rings of a small sample of disc galaxies, and to explore relations with age, stellar metallicity, and extinction. We extract the line-of-sight velocity and velocity dispersion to characterise the kinematics of the young and old stellar populations, using the H$\\beta$ line and the Ca\\,{\\sc II} Triplet, respectively. We compare our results with ages and metallicities obtained from the analysis of the data using the Galaxy IFU Spectroscopy Tool (GIST) pipeline \\citepads[][]{2020arXiv200901856B}, and the extinction as obtained from the fitting of the full spectra. \n\nThe paper is structured as follows. In Section~\\ref{sec:sample} we present the galaxies we used for our study. Section~\\ref{sec:methods} provides a detailed explanation of the methods and tools employed for the analysis of the data, and Section~\\ref{sec:results} summarises the results obtained from our analysis. We discuss our results in Section~\\ref{sec:discussion} and present a brief summary in Section~\\ref{sec:conclusions}.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Figures\/fluvelage_err.png}\n\\caption{Maps of kinematic and stellar population parameters for the four galaxies in our sample. (Top row) H$\\beta$ emission-line fluxes in log scale. For NGC\\,1097, NGC\\,5248, and NGC\\,1300 we overlay the flux level that defines the nuclear ring. These isophotes are also indicated in the other panels for reference, but not in NGC\\,4643 since there is no evidence for a ring in this galaxy and it is used as a control galaxy. (Second and third rows) Maps of the difference in line-of-sight velocity and velocity dispersion between the two spectral ranges used in our study (i.e. H$\\beta-$Ca\\,{\\sc II}). (Bottom row) Maps for the mean luminosity-weighted age from the analysis performed by \\citetads[][]{2020arXiv200901856B}.}\n\\label{fig:fluvelage}\n\\end{figure*}\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Figures\/fraextmetcol.png}\n\\caption{Maps of dust extinction and stellar population parameters for the four galaxies in our sample. Each row (from top to bottom) presents: the fraction of young stars, colour excess $E(B-V)$, and metallicity [M\/H] (see text for details). Isophotes as indicated in Fig.~\\ref{fig:fluvelage}.}\n\\label{fig:frametcol}\n\\end{figure*}\n\n\\section{Sample}\n\\label{sec:sample}\n\nWe use data from the Multi-Unit Spectroscopic Explorer \\citepads[MUSE;][]{2010SPIE.7735E..08B}, an integral field spectrograph installed on the Very Large Telescope (VLT) located at the Paranal Observatory in Chile. The spectral range is 4750\\,\\r{A} to 9300\\,\\r{A}, with a spectral resolution that ranges from 1700 for the blue wavelengths to 3400 for the red ones \\citepads[][]{2017A&A...608A...1B, 2017A&A...608A...5G}. This instrument offers a $1\\arcmin\\,\\times\\,1\\arcmin$ field of view with $0.2\\arcsec$ per pixel in the Wide Field Mode. These capabilities allow the production of detailed maps in terms of spatial and velocity resolution, both crucial for the study of the kinematics of the inner structures of galaxies.\\\n\nWe selected four galaxies from the Time Inference with MUSE in Extragalactic Rings (TIMER) sample \\citepads{2019MNRAS.482..506G} of 24 barred nearby galaxies which present a wide range of star-forming rings. All TIMER galaxies are selected from the {\\textit Spitzer} Survey of Stellar Structure in Galaxies \\citepads[S$^4$G;][]{2010PASP..122.1397S} and are classified as strongly barred galaxies \\citepads[SB,][]{2015ApJS..217...32B}. They also present high stellar masses, with values between $2.0\\times10^{10}$\\,M$_{\\odot}$ and $1.7\\times10^{11}$\\,M$_{\\odot}$ (as estimated in \\citealtads{2013ApJ...771...59M}, \\citealtads{2015ApJS..219....3M}, and \\citealtads{2015ApJS..219....5Q}) and low to intermediate inclinations (below $60\\degree$). The four selected galaxies were chosen to have a wide range of circumnuclear properties (e.g. the presence and size of a nuclear ring and the level of star formation).\\\n\nNGC\\,1300 is an SB(rs)bc galaxy without nuclear activity \\citepads{2004AJ....128.1124S}. It is an archetypal example of a strongly barred galaxy and has been used in many studies related to the dynamical modelling of galaxies \\citepads[e.g.][]{1996A&A...313..733L, 1997A&A...317...36L, 2000A&A...361..841A}. While the presence of a nuclear ring is not evident given the smooth appearance of the stellar light in the inner regions \\citepads[e.g.][]{1989ApJS...71..433P, 2000MNRAS.317..234P}, \\citetads{2008AJ....135..479B} found evidence of a nuclear ring of 400\\,pc radius \\citepads[][]{2010MNRAS.402.2462C} traced by hot spots of Br$\\gamma$ and [Fe\\,{\\sc ii}] emission lines. These lines are related to the photo-ionised gas around OB stars and supernova remnants respectively, thus tracing the regions where star formation activity may be taking place. \n\nNGC\\,5248 is classified as an SAB(rs)bc galaxy in the NASA\/IPAC Extragalactic Database (NED)\\footnote{\\url{http:\/\/ned.ipac.caltech.edu\/}}, and it is a prime example of a grand-design spiral. There is a very active star-forming circumnuclear ring of 650\\,pc radius \\citepads[][]{2010MNRAS.402.2462C} populated by numerous young star clusters \\citepads{1996AJ....111.2248M, 2001AJ....121.3048M}. \\citetads{2001MNRAS.324..891L} and \\citetads{2001AJ....121.3048M} detected a second small ring \\citepads[of 150\\,pc radius;][]{2010MNRAS.402.2462C} inside the first one using H$\\alpha$ maps.\\looseness-2\n\nIn addition to these two galaxies, we selected NGC\\,1097 and NGC\\,4643 as cases that illustrate better the extremes of high and low star formation activity, respectively, in the sample. \\citetads{2019MNRAS.482..506G} also employed these two galaxies as examples of the wide range of properties that can be found in the TIMER sample, performing an analysis of the stellar kinematics, age, metallicity and star formation history in the wavelength range between $4750$\\,\\r{A} and $5500$\\,\\r{A}. NGC\\,1097 is an SB(s)b galaxy and like NGC\\,5248 is a very well-studied grand-design spiral. It also harbours a 1\\,kpc-radius \\citepads[][]{2010MNRAS.402.2462C} nuclear ring which has been observed in several ranges of the electromagnetic spectrum revealing the presence of numerous stellar clusters and a considerable amount of molecular gas \\citepads[e.g.][]{2019MNRAS.485.3264P}. NGC\\,4643 was added to our sample as a control galaxy. It is an SB(rs)0\/a galaxy without a nuclear ring, which hosts little gas and in which the old component of the stellar population dominates the disc \\citepads[][]{2019MNRAS.482..506G}.\n\n\\section{Analysis methods}\n\\label{sec:methods}\n\nWe aim to derive the stellar kinematics from different spectral regions and explore any differences observed in velocity and velocity dispersion, as well as possible correlations with key parameters. The first step in this analysis is to select the appropriate wavelength regions which may be sensitive to, at least, extreme stellar populations (e.g. young and old). Following \\citetads{etheses4179}, we used the H$\\beta$ and Mg$b$ region ($4750-5500$\\,\\r{A}), and the Ca\\,{\\sc II} Triplet region ($8498-8950$\\,\\r{A}) as proxies for young and old populations, respectively.\n\nH$\\beta$ is particularly strong in absorption for stars with spectral type A. We thus assume that the kinematics derived from the spectral fitting around this line may be a good tracer of the kinematics of a relatively young stellar population. On the other hand, the Ca\\,II Triplet is most prominent in absorption in G, K, and M type stars, tracing the old stellar population. \n\nIn our analysis we make use of the stellar population analysis from \\citetads[][]{2020arXiv200901856B} obtained with the GIST pipeline\\footnote{\\url{http:\/\/ascl.net\/1907.025}} \\citepads{2019A&A...628A.117B} that was applied to the entire TIMER sample. This analysis was performed imposing a level of Voronoi binning with a target signal-to-noise ratio of 100 per spectral pixel. The stellar kinematics ($v_\\star$, $\\sigma_\\star$, h$_3$, h$_4$) and the stellar population parameters (e.g. luminosity-weighted age, metallicity [M\/H], and [$\\alpha$\/Fe]) were extracted from a specific wavelength range (from $4800-5800$\\,\\r{A}). The MILES stellar population models \\citepads{2011A&A...532A..95F, 2015MNRAS.449.1177V} were used as templates for the fitting.\n\nSince we are interested in the kinematics of the two distinct spectral regions, we re-run the GIST pipeline for the kinematics extraction for each of them separately. In this second analysis we masked the emission lines that can affect the fitting process, like the H$\\beta$ line itself. In order to make our results easily comparable with those computed in the $4800-5800$\\,\\r{A}\\ wavelength range, we imposed the same Voronoi binning. While ideally it would be desirable to also extract the stellar population parameters from each spectral range, the Ca\\,{\\sc II} Triplet region by itself is not sensitive enough to obtain reliable ages, metallicities, and [$\\alpha$\/Fe] values. This is why we rely on the average values presented in \\citetads[][]{2020arXiv200901856B}, which are sensitive to the presence of young stellar populations. Besides the fitting in each spectral region, we performed an additional fitting for the whole spectral range in order to extract the colour excess $E(B-V)$ from the binned spectra. In this second run we used the E-MILES models \\citepads[][]{2016MNRAS.463.3409V} as templates, which are identical to the MILES models in the $4800-5800$\\,\\r{A}\\ region, but allow us to extend the analysis to the Ca\\,{\\sc II} Triplet.\n\nThree of the galaxies in our sample present well-defined nuclear rings characterised by the presence of young stars. NGC\\,4643 does not, is dominated by old stellar populations, and is used here as a control galaxy. Since we want to study specific relations between the kinematic and the stellar population parameters in different regimes, we have identified different regions of the galaxies with distinct properties. We use H$\\beta$ flux maps to help us define the nuclear ring-dominated areas. Regions outside those limits are denoted as the {\\it disc}, while the area inside the ring is denoted as the {\\it nucleus} (see Fig.~\\ref{fig:fluvelage} and \\ref{fig:frametcol}). We note that we use only a portion of the MUSE field of view (FoV) for this analysis. This selection is done with the purpose of avoiding the noisy spaxels at the edges of the cubes and possible artifacts. The selected portion of the MUSE FoV covers an area of $17\\arcsec\\times17\\arcsec$ in all the galaxies in the sample, with the exception of NGC\\,5248, where the area is slightly smaller ($14\\arcsec\\times14\\arcsec$).\n\n\\section{Results}\n\\label{sec:results}\n\nWe present our results for the three regions identified for the galaxies of our sample: disc, ring, and nucleus. The results are summarised in Table~\\ref{tab:results} and include difference in line-of-sight velocity, difference in velocity dispersion, age, fraction of young stars, colour excess, metallicity, and colour.\n\n\\subsection{Kinematic differences}\n\\label{sec:kinematics}\n\nThe main result of our study is the kinematic difference between the two selected spectral ranges. The four galaxies selected from the TIMER sample illustrate scenarios where a young stellar population is present to different extents in the inner regions of the galaxies: from the extremely young inner disc of NGC\\,5248 to the lack of evidence of young stellar populations in the inner regions of NGC\\,4643. We now seek to confirm whether the young stars leave an imprint in the stellar kinematics, especially in the rings. Throughout the paper we refer to the differences in line-of-sight velocity between the two spectral ranges (young minus old) as $\\Delta v_\\star$, calculated as the absolute difference of the values of the estimated velocities for the H$\\beta$ and Ca\\,{\\sc II} Triplet lines.\n\nThe maps in Fig.~\\ref{fig:fluvelage} present a fairly homogeneous and smooth appearance with the exception of NGC\\,1097 and some small regions in NGC\\,5248. In NGC\\,1097 we can clearly identify the shape of the ring, traced mostly by regions where the \\deltaV\\ values are high. This ring spans a wide range of \\deltaV\\ values, up to $70\\,{\\rm km\\,s^{-1}}$, while in the disc and nucleus the maximum variation is about $25\\,{\\rm km\\,s^{-1}}$. These differences are somewhat milder in the other three galaxies in our sample with mean values close to zero and a typical standard deviation of $5\\,{\\rm km\\,s^{-1}}$. There are a few exceptions to this behaviour in specific locations in the disc of NGC\\,1300 and the ring of NGC\\,5248.\n\nThe presence of the rings stands out more when inspecting the differences in velocity dispersion (\\deltaS\\ , also computed as young minus old). This is most notable in NGC\\,1097 with mean \\deltaS\\ \\,values of $24\\,{\\rm km\\,s^{-1}}$, well above the mean values observed in its disc and nucleus ($\\sim\\,10\\,{\\rm km\\,s^{-1}}$ and $\\sim\\,5\\,{\\rm km\\,s^{-1}}$, respectively). For the other three galaxies, the scenario is analogous to what we observe for \\deltaV\\ . It is worth noting the slightly positive mean overall value of \\deltaS\\ observed in NGC\\,4643 ($8\\,{\\rm km\\,s^{-1}}$). Given the lack of young stellar populations the expectation is to find \\deltaS\\ \\,differences close to zero. If this was a problem due to the adopted spectral resolution as a function of wavelength in our analysis, one would expect a similar offset in equally old regions of other galaxies. This is however not observed when inspecting the disc dominated regions of NGC\\,1300, with populations similar to those in NGC\\,4346 but mean \\deltaS\\ \\,of $1\\,{\\rm km\\,s^{-1}}$. With uncertainty levels for \\deltaS\\ \\,in our measurements of $9\\,{\\rm km\\,s^{-1}}$ (see \\citealtads{2019A&A...623A..19P}) it is difficult to draw more firm conclusions.\n\n\\subsection{Age}\n\\label{sec:age}\n\nAge is an excellent tracer of the presence of the rings in our sample. As expected, NGC\\,4643 is very different from the other three galaxies in the sample. In this galaxy there is no ring structure in the age map, which appears rather homogeneous. In NGC\\,4643 we find a rather old stellar population with a mean age around $11$\\,Gyr with a small standard deviation of $0.7$\\,Gyr. \n\nIn NGC\\,1097 and NGC\\,5248 we do not only find a well defined ring-like structure, but also relatively young stellar populations within them. This is especially noticeable in the ring of NGC\\,5248. In the two rings we estimate an average age below $4$\\,Gyr ($3.5$\\,Gyr for NGC\\,1097 and $1.8$\\,Gyr for NGC\\,5248). It is remarkable that in NGC\\,5248 we find a relatively small range of ages (from $0.5$\\,Gyr to $5.0$\\,Gyr) while in the ring of NGC\\,1097 it is much larger (from $0.8$\\,Gyr to $10.7$\\,Gyr), which clearly affects the obtained mean value. On the other hand, the ring of NGC\\,1300 seems to be dominated by an intermediate-age stellar population, with a mean age of $4.6$\\,Gyr. The ages in all these rings correspond reasonably well with the ellipses we used to define them (see Sect.~\\ref{sec:methods} for details). \n\nThe other two regions defined in these galaxies display different behaviours. NGC\\,1097 has a disc with a stellar population of intermediate age ($\\simeq 6$\\,Gyr) and a central region dominated by older stars (with mean age over $8$\\,Gyr). In NGC\\,1300 the disc has an average age of $8.3$\\,Gyr, while the region inside the ring has a slightly younger mean age of $5.9$\\,Gyr with a smaller dispersion ($0.7$\\,Gyr for the central region and $1.6$\\,Gyr for the disc). The whole disc of NGC\\,5248 is populated by young stars. The three defined regions are dominated by young stars with mean ages below $3$\\,Gyr and a small dispersion which never exceeds $0.7$\\,Gyr. All the age measurements agree well with the typical age uncertainties: around $0.5$\\,Gyr and $3$\\,Gyr for the young and old stellar population, respectively \\citepads[][]{2019A&A...623A..19P}.\n\n\\subsection{Fraction of young stars}\n\\label{sec:ysf}\n\nThe fraction of young stars is recovered from the light-weighted results obtained by \\citetads[][]{2020arXiv200901856B}. We sum the weights applied during the fitting process to all the templates corresponding to stellar populations below a certain age, considering the models with ages below $2$\\,Gyr as those tracing the young stellar populations. We checked that lowering this value to $1$\\,Gyr gave consistent results. \n\nThe maps for the fraction of young stars (top row in Fig.~\\ref{fig:frametcol}) complete the information given by the age maps. NGC\\,4643 is characterised by a very homogeneous distribution of low values of the fraction of young stars, with a mean value close to zero and never exceeding $0.13$. The other three galaxies illustrate different scenarios for the presence of a well defined ring. In the case of NGC\\,1097 the ring exhibits large regions dominated by high fractions of young stars, with a mean value of $0.44$ with a dispersion of $0.22$. In the disc we find similar values (mean value of $0.33$ and a standard deviation of $0.18$), while in the nucleus we find extremely low fractions with a mean value of $0.10$ and a standard deviation of $0.10$.\n\nNGC\\,5248 shows high fractions of young stars in all the regions. The ring is recognisable by a greater presence of bins with high fractions of young stars, resulting in a slightly higher mean value, $0.63$, when compared with the other two regions ($0.53$ and $0.55$ for the disc and nucleus, respectively). In all the three regions the dispersion of values is small, around $0.15$. In contrast, NGC\\,1300 exhibits rather low fractions in all the regions, with mean values of $0.09$, $0.23$, and $0.019$ in disc, ring, and nucleus, respectively, and dispersion around $0.10$. The typical uncertainty in our measurements of the fraction of young stars is $0.13$.\n\n\\subsection{Extinction}\n\\label{sec:ebv}\n\nThe maps of extinction, characterised by the colour excess $E(B-V)$ (second row in Fig.~\\ref{fig:frametcol}), reveal a little more of the structure of the galaxies. These maps replicate and emphasise the information given by the colour maps in \\citetads{2019MNRAS.482..506G}. For NGC\\,1097 the $E(B-V)$ map shows not only the presence of dust lanes in the ring, but also along the leading edge of the bar. In this galaxy we find a wide range of values of the colour excess in the three regions, resulting in discrete mean values in each one of them ($0.08$ in the disc, $0.16$ in the ring, and $0.14$ in the nucleus, all with low standard deviations around $0.05$). In the map for NGC\\,1300 we also recognise these structures traced by higher values of $E(B-V)$ when compared with those for the rest of the galaxy. Regarding the mean values in the three regions, NGC\\,1300 exhibits higher values compared to those in NGC\\,1097: $0.14$ for the disc, $0.21$ for the ring, and $0.20$ for the nucleus, all with dispersion below $0.07$. NGC\\,5248 is slightly different, with in general higher values of the colour excess. This gives us the image of a dusty galaxy, with plenty of star-forming regions. The ring is traced by alternating hot-spots of low and high values of the extinction, like in NGC\\,1097. For this galaxy the values in the three regions are very similar: $0.23$ for the disc, $0.20$ for the ring, and $0.218$ for the nucleus. The $E(B-V)$ map for NGC\\,4643 follows a similar pattern to the colour map obtained for this galaxy, presenting a patchy distribution of very low values (with a mean value around zero). All our measurements of the colour excess are expressed with an uncertainty of $0.05$.\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Figures\/scatter_dsig-ebv.png}\n\\caption{Observed values of $\\Delta\\sigma_\\star$ against colour excess $E(B-V)$ in the three selected regions for the four galaxies in the sample. We represent the mean values of $\\Delta\\sigma_\\star$ estimated in equally spaced age bins. We also represent the $1\\,\\sigma$ uncertainty interval for these mean values as colour-shaded regions. The typical error in the $\\Delta\\sigma_\\star$ estimate is represented as the grey-shaded area. In general we find vertical distributions of the mean values of $\\Delta\\sigma_\\star$. The only exception is the ring of NGC\\,1097 where we find increasing values of $\\Delta\\sigma_\\star$ when $E(B-V)$ increases.}\n\\label{fig:scat_dsig-ebv}\n\\end{figure*}\n\n\n\\subsection{Metallicity}\n\\label{sec:metallicity}\n\nThe metallicity also reproduces quite well the ring-like structure in all the galaxies of our sample (see the third row in Fig.~\\ref{fig:frametcol}). The only exception is NGC\\,4643 in which we find some structure, corresponding to the presence of a nuclear disc, but not a ring, with higher values in the central few hundred parsecs of the galaxy. We also see a band of intermediate and high values that goes from the south-west to the north-east part of the disc and which corresponds to the bar. In general we observe a rather homogeneous distribution around the mean value of $-0.25$\\,dex. \n\nIn NGC\\,1097 there is a clear match between the H$\\beta$ amplitude line contours used to define the ring and the regions with lower metallicities (ranging from $-1.7$ to $0.24$\\,dex with an average value of $-0.33$). In the central region and the rest of the disc we find similar mean values of $0.15$ and $0.04$, respectively. \n\nIn NGC\\,1300 we report a similar match between regions with low metallicity and the isophotes obtained from the H$\\beta$ line amplitude map. For this galaxy we observe lower metallicity throughout the disc compared to the other galaxies in the sample. It is noticeable that for this galaxy the H$\\beta$ line isophotes used do not cover an important part of the ring. In these areas the metallicity reaches values similar to the central part of the galaxy (around $0$). The mean value of the metallicity in the three regions is similar: $-0.22$\\,dex for the disc, $-0.18$ for the ring, and zero for the nucleus. \n\nNGC\\,5248 also shows a correspondence between the peaks in the H$\\beta$ line amplitude map and the low metallicity regions, and, like in the case of NGC\\,1300, in the areas where the isophote is disrupted the metallicity reaches higher values, similar to the disc and the nuclear region. The mean values in the three regions defined ($-0.09$ in the disc, $-0.21$ in the ring, and $0.15$ in the nucleus) are quite similar and also to the standard deviations in each region ($0.15$, $0.30$, and $0.12$\\,dex, respectively). The typical uncertainty in the measurements of the metallicity in all the galaxies in our sample is around $0.10$\\,dex \\citepads[][]{2019A&A...623A..19P}.\\\n\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Figures\/scatter_dsig-ebv_SNR.png}\n\\caption{Changes in $\\Delta\\sigma_\\star$ with the $E(B-V)$ distribution in the ring in NGC\\,1097. We plot the results for three different target SNR for the GIST pipeline analysis (the blue series): $100$ (left panel), $250$ (central panel), and $350$ (right panel). We can observe how the highest values of the colour excess disappear when we increase the SNR. At the same time the highest values of $\\Delta\\sigma_\\star$ start to decrease. At SNR=350 we cannot clearly discern the trend we observe for SNR=100 and the distribution for the ring of NGC\\,1097 is similar to that for that of NGC\\,5248.}\n\\label{fig:scat_dsig-ebv_snr}\n\\end{figure*}\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn light of the results obtained, we now wish to confirm which parameters depend on or influence the kinematics estimated in the spectral regions around the H$\\beta$ and Ca\\,{\\sc II} Triplet lines. As mentioned above, we operate under the hypothesis that the kinematics derived using these spectral regions are sensitive to different stellar populations. We study the possible trends of the difference between the kinematics of young and old stellar populations with the various parameters extracted. We pay special attention to the behaviour of the rings, but also inspect trends in the disc and nucleus components.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{Figures\/ppxf_4643.png}\n\\caption{$\\Delta\\sigma_\\star$ against $\\Delta v_\\star$ with colour-coded values of the stellar population parameters (from top to bottom: fraction of young stars, mean age, colour excess, and metallicity) for NGC\\,4643. On the left side we represent the data with no smoothing and on the right side the \\textsc{loess}-smoothed data. The grey rectangle represents the typical error in $\\Delta v_\\star$ ($7$\\,km\\,s$^{-1}$) and $\\Delta\\sigma_\\star$ (9\\,km\\,s$^{-1}$). The uncertainties introduced by the smoothing are: $\\pm 0.019$ for the YSF, $\\pm 0.7$\\,Gyr for the mean age, $\\pm 0.016$ for $E(B-V)$, and $\\pm 0.08$\\,dex for the metallicity.}\n\\label{fig:ppxf_4643}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{Figures\/ppxf_nuc_1097.png}\n\\caption{As Fig.~5, now for the nucleus of NGC\\,1097. The uncertainties introduced by the smoothing are: $\\pm 0.10$ for the YSF, $\\pm 1.7$\\,Gyr for the mean age, $\\pm 0.06$ for E(B$-$V), and $\\pm 0.10$\\,dex for the metallicity.}\n\\label{fig:ppxf_nuc_1097}\n\\end{figure}\n\n\\subsection{Trends of the kinematic differences with stellar population properties}\n\nWe start to explore the relation between the kinematic (velocity and velocity dispersion) differences of the two spectral regions studied and the stellar population properties extracted from our analysis. We first consider the nuclear ring regions.\n\nFigure~\\ref{fig:scat_dsig-ebv} and the corresponding figures in Appendix~\\ref{append: kin_vs_prop} show the differences in velocity (\\deltaV\\ ) and velocity dispersion (\\deltaS\\ ) plotted against the values of mean age, fraction of young stars, metallicity, and colour excess. In order to make our analysis of the possible trends easier, we do not directly represent the whole clouds of points. We separate the results in each region of the galaxies in equally spaced bins for each parameter and estimate the mean value of the kinematical differences in each one of them. Additionally, for robustness, we ignore any bins with less than three elements. We study the significance of the mean value in each bin by comparing the $1\\,\\sigma$ confidence interval with the typical error in our estimations (calculated as the mean standard deviation of the errors of the difference in velocity and velocity dispersion, giving values of $7\\,{\\rm km\\,s^{-1}}$ and $9\\,{\\rm km\\,s^{-1}}$, respectively).\n\nExamining the plots for \\deltaV\\ (see Fig.~\\ref{fig:scat_dv-age}, Fig.~\\ref{fig:scat_dv-ysf}, Fig.~\\ref{fig:scat_dv-metal}, Fig.~\\ref{fig:scat_dv-ebv}), we find no significant trends in any of the regions, with the exception of \\deltaV\\ \\,vs [M\/H] where there is a tentative trend with metallicity in NGC\\,1097 and NGC\\,5248, the galaxies with the largest rings. The observed trend is such that high \\deltaV\\ \\,values are found for lower metallicities. There is some evidence that the metal-poor stars in the ring of NGC\\,1097 are due to an interaction event with a nearby galaxy which has rejuvenated the ring \\citep[see][for details]{2020arXiv200901856B}, a phenomenon which could explain this trend.\n\nRegarding \\deltaS\\,, contrary to what one may expect, we do not observe clear trends with age or with the fraction of young stars (Fig.~\\ref{fig:scat_dsig-age} and Fig.~\\ref{fig:scat_dsig-ysf}, respectively). The strongest relation seems to be with E(B$-$V). In Fig.~\\ref{fig:scat_dsig-ebv} we show that there is a trend of $\\Delta\\sigma_\\star$ with the colour excess, at least in the case of the ring of NGC\\,1097. There we find a clear trend of increasing values of $\\Delta\\sigma_\\star$ with increasing colour excess. It is especially remarkable for the highest values of E(B$-$V), where we observe high values of the mean difference of velocity dispersion. These values are equivalent to several times the typical error. In the other galaxies and the other regions of NGC\\,1097 the mean values of $\\Delta\\sigma_\\star$ in the colour excess bins are similar to the error. It is noticeable that the $1\\,\\sigma$ confidence intervals for the values of $\\Delta\\sigma_\\star$ in the ring of NGC\\,1097 are slightly higher than the typical error. Nevertheless, we consider the \\deltaS\\ \\,measurements in the ring as robust as those in the other regions of NGC\\,1097 and the other galaxies in the sample. For the remaining parameters, we do not find clear trends (the relevant plots can be found in the appendix).\\looseness-2\n\nWe do not find any significant relation of the kinematic differences with the extracted stellar population parameters in the cases of NGC\\,1300 and NGC\\,5248. We therefore will focus our attention on NGC\\,1097 and NGC\\,4643 only during the discussion, as they represent the two extreme cases of a very active ring on the one hand and a quiescent galaxy that does not exhibit any relevant kinematic difference on the other. The cases of NGC\\,1300 and NGC\\,5248 are presented in the appendix section instead.\n\nWhy do we see the trend in the ring in NGC\\,1097 and not in those in the other galaxies? We suggest that this is due to a beam-smearing effect, more pronounced in NGC\\,1300 and NGC\\,5248 than in NGC\\,1097. To confirm this, we re-ran the analysis of the data of NGC\\,1097 using the GIST pipeline at two different signal-to-noise ratios (SNR): $250$ and $350$. This change in the target SNR produces bigger Voronoi bin sizes and thus poorer spatial resolution. In the case of NGC\\,1097, for the considered field of view, at a SNR=$100$ we obtain $3609$ bins, at a SNR=$250$, $540$ bins, and at a SNR=$350$, $264$ bins. In contrast, for NGC\\,1300 at SNR=$100$ we have $537$ bins. Considering the distances for these two galaxies and a target SNR=$100$, the average size of a bin located in the ring of NGC\\,1300 is about $39{\\rm\\,pc}$, compared to around $22{\\rm\\,pc}$ in NGC\\,1097 ($74{\\rm\\,pc}$ at SNR$=350$). The effect of the change of the bin size on the ring of NGC\\,1097 can be seen in Fig~\\ref{fig:scat_dsig-ebv_snr}. There we show that increasing the bin size is translated into a loss of the higher values of the colour excess and a decrease in the higher values of $\\Delta\\sigma_\\star$ of about $10\\,{\\rm km\\,s^{-1}}$. We further discuss this in Appendix~\\ref{append: kin_1300_5248}.\\looseness-2\n\nThe main inconvenience with the way we present the trends in Figures~\\ref{fig:scat_dsig-ebv} and \\ref{fig:scat_dsig-ebv_snr} is that the most extreme values get diluted when averaging in the bins in which we separate our results. An alternative method to present our results is to confront \\deltaV\\ \\,and \\deltaS\\ \\,and colour-code the individual Voronoi bin values with different properties. The problem with this representation of the complete set of data is that for each range of values of the stellar population parameters we have a relatively large dispersion of the values of the kinematic differences. To aid the eye in identifying the main trends we have applied some smoothing using the \\textsc{loess}\\footnote{http:\/\/purl.org\/cappellari\/idl} package. This software is able to recover the trends in noisy data by applying a two-dimensional Locally Weighted Regression method \\citepads{Clev:Devl:1988}. Results are presented in Figs.~\\ref{fig:ppxf_4643}, \\ref{fig:ppxf_nuc_1097}, \\ref{fig:ppxf_out_1097}, and \\ref{fig:ppxf_ring_1097} for the whole data set of NGC\\,4643 and the three regions of NGC\\,1097. In order to illustrate the changes introduced by the smoothing process, in Appendix~\\ref{append: kin_1300_5248} we represent the original data in the left panel of each plot. We also account for the uncertainties introduced by the smoothing, estimating them as the standard deviation of the difference between the original and the smoothed data (as indicated in the caption of the figures).\n\nNGC\\,4643 (see Fig.~\\ref{fig:ppxf_4643}) does not show any trend in the plots. The kinematic differences are clustered around low values, just a few times larger than the typical uncertainties. Regarding the values of the stellar population parameters, the dynamical ranges are small, with differences between the maximum and minimum values in general smaller than the estimated uncertainty.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{Figures\/ppxf_outring_1097.png}\n\\caption{As Fig.~5, now for the disk of NGC\\,1097. The uncertainties introduced by the smoothing are: $\\pm 0.17$ for the YSF, $\\pm 1.4$\\,Gyr for the mean age, $\\pm 0.04$ for E(B$-$V), and $\\pm 0.010$\\,dex for the metallicity.}\n\\label{fig:ppxf_out_1097}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{Figures\/ppxf_ring_1097.png}\n\\caption{As Fig.~5, now for the nuclear ring of NGC\\,1097. The uncertainties introduced by the smoothing are: $\\pm 0.20$ for the YSF, $\\pm 1.6$\\,Gyr for the mean age, $\\pm 0.05$ for E(B$-$V), and $\\pm 0.36$\\,dex for the metallicity.}\n\\label{fig:ppxf_ring_1097}\n\\end{figure}\n\nThe nucleus of NGC\\,1097 (see Fig.~\\ref{fig:ppxf_nuc_1097}) presents a similar case to that of NGC\\,4643, spanning similar ranges of values. Its stellar population parameters show also relatively small dynamical ranges. There are some indications of weak trends, particularly of \\deltaV\\ \\,with mean age. In the case of the disc (see Fig.~\\ref{fig:ppxf_out_1097}) we find higher values of \\deltaS\\ \\,, but low dynamical ranges of the parameters. There is a weak trend of \\deltaV\\ with colour excess, but the range of values of E(B$-$V) ($0.04-0.2$) is not large enough to be formally significant.\n\nThe ring of NGC\\,1097 (Fig.~\\ref{fig:ppxf_ring_1097}) exhibits larger kinematic differences between the two stellar populations. In general, all the stellar population parameters span wider ranges of values and trends of \\deltaS\\ \\,with mean age and $E(B-V)$ are found. In the case of the trend with age there are indications of increasing \\deltaS\\ \\,with younger ages, but the range of mean ages spanned in this region ($2-7$\\,Gyr) is not large enough to confirm the trend. The picture with colour excess is quite different. We see a clear increase of \\deltaS\\ \\,with $E(B-V)$, which spans a wide dynamic range ($0.06-0.27$). This results suggests that we may be probing different set of stars along the line-of-sight in the two spectral ranges, and this will contribute to the elevated \\deltaS\\ \\,values.\\newline\n\n\n\\subsection{Kinematic simulations}\n\\label{sec:sims}\n\nWhen interpreting the results above, it is important to establish the expected level of kinematic differences caused by distinct populations alone (i.e. not influenced by the presence of dust). For that purpose, we have created a set of mock spectra made of young and old stellar populations with a wide range of kinematic properties. We designed the experiment to reproduce as closely as possible all the relevant details of the MUSE data, from the ranges used to the spectral resolution and sampling. We generated $\\sim$48000 test cases using the E-MILES library as the reference with the following input parameters:\n\n\\begin{itemize}\n \\item Fraction of young stars: [10$^{-2}$,10$^0$] (21 steps log-spaced)\n \\item Age$_\\mathrm{young}$: [0.1,0.4,1.0] (Gyr)\n \\item Age$_\\mathrm{old}$: 12.0 (Gyr)\n \\item $\\mathrm{[M\/H]}_\\mathrm{young}$: 0.06 dex\n \\item $\\mathrm{[M\/H]}_\\mathrm{old}$: 0.06 dex\n \\item V$_\\mathrm{young}$: [-100.0,100.0] (in steps of 25\\,km\\,s$^{-1}$)\n \\item V$_\\mathrm{old}$: [-100.0,100.0] (in steps of 25\\,km\\,s$^{-1}$)\n \\item $\\sigma_\\mathrm{young}$: [50.0,150.0] (in steps of 50\\,km\\,s$^{-1}$)\n \\item $\\sigma_\\mathrm{old}$: [150.0,250.0] (in steps of 50\\,km\\,s$^{-1}$)\n\\end{itemize}\n\nFor consistency, we fitted the data using the GIST pipeline with the same configurations as used with the real data. This included the masking of emission line-dominated regions, even though emission lines were not actually included in the mock spectra. Given the prominence of the H$\\beta$ line in the 4750$-$5500\\,\\r{A} region, we also performed a separate run without masking any parts of the spectra. The most relevant results of this experiment are presented in Fig.~\\ref{fig:sims}. Rather than presenting all the individual measurements, which are heavily dependent on our choice of input parameters for the simulations, we present the areas covering the 1\\%$-$99\\%\\ percentiles of the output distributions as a function of the input parameter. They provide a good estimate of the range of possible solutions. These regions are denoted by the shaded areas in the different panels. The red areas correspond to the fits masking the H$\\beta$ line, while the blue ones are for fits where no pixels have been excluded.\n\nThe first striking, but perhaps not surprising result of our simulations, is the lower sensitivity to differentiate between populations when the H$\\beta$ line is masked (i.e. as we find with real data). These differences are most notable in the velocity, and have a smaller effect in the velocity dispersion. This effect may explain the lack of trends observed with fraction of young stars in our MUSE data for all galaxies. The spread of most of the (red) solutions are also not much bigger than the typical uncertainty of our measurements (7\\,km\\,s$^{-1}$). Furthermore, the most extreme differences in velocity of $\\sim$17\\,km\\,s$^{-1}$\\ are for situations where the differences in velocity between the two populations is of the order of 200\\,km\\,s$^{-1}$, which is highly unlikely to be the case in our galaxies.\n\nAnother interesting feature in Fig.~\\ref{fig:sims} is the dependence of the kinematic differences on the fraction of young stars. When this fraction is either too small or too large, the resulting spectra are so dominated by old or young stars respectively that the differences are within the uncertainties of our measurements, and one effectively has a spectrum of a single stellar population. In our specific tests the largest differences are observed for intermediate fractions of young stars of around $5$\\%, and again only for extreme differences in velocity of the input populations.\n\nOur experiments confirm that with our particular setup for the kinematic extraction (i.e. masking the H$\\beta$ line), we should not observe strong trends with the fraction of young stars in the galaxies. The observed differences must thus be caused by dust extinction. This is an aspect that we cannot simulate without performing proper radiative transfer propagation to generate the input model spectra, which is well beyond the scope of this paper. \n\n\\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=\\linewidth]{Figures\/figure.png}\n \\caption{Kinematic differences between the H$\\beta$ and Ca\\,{\\sc II} triplet regions extracted from our simulations. Top panel shows the differences in velocity and velocity dispersion as a function of: (1) fraction of young stars and (2) input velocity difference between populations. Shaded areas mark the $1\\%-99\\%$ percentiles sampled by our simulations when the H$\\beta$ line is included (blue) or not (red) in the spectral fit.}\n \\label{fig:sims}\n\\end{figure}\n\nThe most striking fact about our results is that we have positive values of the difference of velocity dispersion $\\Delta \\sigma_\\star = \\sigma_{\\rm H\\beta} - \\sigma_{\\rm Ca\\,II}$. This would mean, if we assume that the H$\\beta$ line is dominated by the young stars and the Ca Triplet by older ones, that we are observing a young stellar population with a higher velocity dispersion. This seems to contradict the intuitive idea of young stellar populations having a more ordered motion than the old stellar populations. The simulations above indicate that, at the spectral resolution of our data, differences in stellar populations is not the cause of the observed kinematic differences.\n\n\\subsection{Asymmetric drift}\n\\label{sec:ad}\n\nThe asymmetric drift $v_a$ is one of the possible sources of an enhanced velocity dispersion in the kinematics traced by the H$\\beta$ line spectral region. It can be defined as the difference between the velocity of a hypothetical ensemble of stars $v_c$, which moves around the centre of a galaxy in a perfectly circular orbit, and the mean rotation speed of the stellar population under study $\\overline{v_\\phi}$. The asymmetric drift gives information about the tendency of a stellar population to lag behind other populations. The main reason for this lag can be found in the collisionless nature of the stars within a galaxy, which makes them sensitive to dynamical heating. In this kind of process the stars tend to migrate to orbits with larger radii, which leads to a decrease of the rotational velocity, as expected from angular momentum conservation. These changes are translated into an increase of the random motions of the stars and consequently an increase of the velocity dispersion of the stellar populations \\citepads{2002MNRAS.336..785S}. The more time passes, the more heating events will take place, thus it is expected to find higher velocity dispersions in old stellar populations \\citepads{1996ApJ...460..121W}.\n\n\\citetads{2016MNRAS.458.1199E} studied the slope of the inner rotation curves of a sample of spiral galaxies from the S$^4$G survey, using H$\\alpha$ Fabry-Perot data to derive the rotation velocity of the ionised-gas component. They corrected these velocities for the asymmetric drift effect to obtain the circular velocity. While their results were estimated for ionised gas, they can nevertheless provide us with a reasonable approximation of the values we could expect for the stars. \n\nWhen we evaluate the results of \\citetads{2016MNRAS.458.1199E} at a radius of $650$\\,pc, which is approximately the average radius of the rings of the galaxies in our sample, we obtain an asymmetric drift of around $2$\\,km\\,s$^{-1}$. This value is very close to the average velocity differences estimated in the rings of our galaxies, but significantly smaller than the most extreme velocity differences estimated from our data. The asymmetric drift may thus have some relevance in the ring kinematics, especially concerning the velocity dispersion distribution, but it is not the main driver in those regions where we find the higher kinematic differences between the young and old stellar populations.\n\n\\subsection{The effect of Paschen lines}\n\\label{sec:paschen}\n\nAnother possible effect that may be altering the kinematics estimated from the MUSE data is the presence of the Paschen line series in absorption in the Ca\\,II Triplet region. The Paschen series fall in the near-infrared spectral range and some of the transitions overlap with the Ca\\,II lines. We could expect to observe some dependence of this effect with age since the Paschen lines Pa\\,16, Pa\\,15 and Pa\\,13 are strong in young stars. This superposition could result in a change in the absorption lines shape, which will tend to increase the measured velocity dispersion \\citep[see, e.g., Fig.~10 in][]{2003MNRAS.340.1317V}. Given that \\deltaS\\ in the ring-dominated regions is such that velocity dispersions are larger in the H$\\beta$ region than around the Ca\\,II Triplet, it is unlikely that the effect of the Paschen lines plays any significant role in our measurements. \n\n\\section{Summary and conclusions}\n\\label{sec:conclusions}\n\nThe goal of the present work is to explore the relation between the kinematics derived from the blue and red spectral ranges and the properties linked to different stellar populations harboured in the inner regions of four nearby galaxies in the TIMER sample. We derived the velocity and velocity dispersion around the H$\\beta$ and the Ca\\,{\\sc II} Triplet spectral lines in these regions using the \\textsc{GIST} pipeline. Additionally, we estimated the colour excess E(B$-$V). Assuming that the H$\\beta$ line is sensitive to the young stellar population and that the Ca\\,{\\sc II} Triplet is dominated by the old stellar population, we analysed the correlation of the difference of velocity and velocity dispersion between both populations and the various other derived parameters. Our results can be summarised as follows:\n\n\\begin{itemize}\n \\item The young stellar population presents not only a higher velocity, but also a higher velocity dispersion than the old stellar population. This is especially remarkable in the nuclear rings, where differences between both populations reach up to $70\\pm7\\,{\\rm km\\, s^{-1}}$ in velocity and $88\\pm9\\,{\\rm km\\, s^{-1}}$ velocity dispersion in the case of NGC\\,1097. Nevertheless, the mean values of the difference in velocity in the rings are low (below $10\\,{\\rm km\\, s^{-1}}$), while the mean differences in velocity dispersion are more significant, reaching up to $24\\pm9\\,{\\rm km\\, s^{-1}}$ in the most extreme case.\n\n \\item We observe a clear trend of the difference of velocity dispersion increasing with extinction in the ring of NGC\\,1097. This trend is also observed in the other rings, but is weaker there.\n \n \\item We demonstrate with the ring of NGC\\,1097 that the physical size of the bins plays a limited role in the change of the trend of \\deltaS\\ with colour excess. Our study suggests that beam-smearing effects could be the reason for weaker trends in the rings of the other galaxies in our sample.\n\n \\item From a simulation exercise in which we mix a set of mock spectra of young and old stellar populations, we demonstrate that masking the H$\\beta$ line produces a lower sensitivity when determining differences in velocity and velocity dispersion. This explains the lack of trends with age or fraction of young stars.\n \n \\item We can rule out asymmetric drift or overlapping Paschen lines on top of the Ca\\,{\\sc II} ones as drivers of the observed kinematic differences.\n\\end{itemize}\n\nOur analysis indicates that the kinematic differences observed are not strongly related to the presence of different populations. However, the presence of dust and the physical size of the bins in our maps play important roles in the difference of velocity dispersion. We demonstrate that, in extreme situations, these differences are observable at the intermediate resolution offered by the current generation of IFUs, such as MUSE. In the near future, with the next generation of high-spectral resolution spectrographs \\citep[e.g. WEAVE;][]{weave}, it will be possible to perform even more detailed chemo-kinematic decompositions of nearby galaxies.\n\n\\begin{acknowledgements}\nBased on observations collected at the European Southern Observatory under ESO programmes 097.B-0640(A). We acknowledge financial support from the European Union's Horizon 2020 research and innovation programme under Marie Sk\\l odowska-Curie grant agreement No 721463 to the SUNDIAL ITN network, from the State Research Agency (AEI) of the Spanish Ministry of Science and Innovation and the European Regional Development Fund (FEDER) under the grant with reference PID2019-105602GB-I00, and from IAC project P\/300724, financed by the Ministry of Science and Innovation, through the State Budget and by the Canary Islands Department of Economy, Knowledge and Employment, through the Regional Budget of the Autonomous Community. JF-B acknowledges support from grants AYA2016-77237-C3-1-P and PID2019-107427GB-C32 of the Spanish Ministry of Science, Innovation and Universities (MCIU) and through the IAC project TRACES, which is partially supported by the state budget and the regional budget of the Consejer\\'ia de Econom\\'ia, Industria, Comercio y Conocimiento of the Canary Islands Autonomous Community. The Science, Technology and Facilities Council is acknowledged by JN for support through the Consolidated Grant Cosmology and Astrophysics at Portsmouth, ST\/S000550\/1. JMA acknowledge support from the MCIU by the grant AYA2017-83204-P and the Programa Operativo FEDER Andaluc\\'{i}a 2014-2020 in collaboration with the Andalucian Office for Economy and Knowledge. PC acknowledges financial support from Funda\\c{c}\\~{a}o de Amparo \\`{a} Pesquisa do Estado de S\\~{a}o Paulo (FAPESP) process number 2018\/05392-8 and Conselho Nacional de Desenvolvimento Cient\\'ifico e Tecnol\\'ogico (CNPq) process number 310041\/2018-0. GvdV acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 724857 (Consolidator Grant ArcheoDyn). TK was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(No. 2019R1A6A3A01092024).\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}