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+{"text":"\\section{Introduction}\n\nThe Kronecker product of matrices is known to be ubiquitous \\cite{VL00},\nand our aim here is to investigate the $n$-fold Kronecker product of \na complex square matrix $M\\in\\C^{d\\times d}$ with itself, \n\\[\nM^{n\\otimes} \\ = \\\n\\underbrace{M \\otimes \\cdots \\otimes M}_{n\\,\\text{times}},\\qquad n\\in\\N,\n\\]\nand to apply our findings to the parametrization of semiclassical wave packets. \n\n\\subsection{The motivation}\nWe encountered a variant of the $n$-fold Kronecker product when\nstudying linear changes in the para\\-metrization of semiclassical wave packets. \nSemiclassical wave packets have first been proposed in \\cite{Hag85} as a \nmultivariate non-isotropic generalization of the Hermite functions. \nSee also \\cite{Hag98}.\nA family of semiclassical wave packets \n\\[\n\\left\\{\\varphi_{\\boldsymbol{k}}[A,\\,B; \\boldsymbol{a},\\,\\boldsymbol{\\boldsymbol{\\eta}}]: \\boldsymbol{k}\\in\\N^d\\right\\}\n\\]\nis parametrized by two invertible complex matrices $A,\\,B\\in{\\rm GL}(d,\\C)$ and \ntwo real vectors $\\boldsymbol{a},\\boldsymbol{\\boldsymbol{\\eta}}\\in\\R^d$. It \n forms an orthonormal basis of the Hilbert space of square integrable \nfunctions. Here, we focus on the more delicate dependence \non the parametrizing matrices $A$ and $B$. We therefore take $\\boldsymbol{a}=\\boldsymbol{\\eta}=\\boldsymbol{0}$ and simply write \n$\\varphi_{\\boldsymbol{k}}[A,\\,B]$ for the corresponding wave packet. \nA wave packet with $|\\boldsymbol{k}|=n$ is the product \nof a multivariate polynomial of order $n$ times a complex-valued Gaussian.\n\nIf the parameter matrix $A$ has real entries only, \nthen the polynomial can be factorized into \nunivariate Hermite polynomials. A linear change of the \nparametrization, \n\\[\nA'=AM, \\quad B' = BM\\quad\\text{for some }M\\in{\\rm GL}(d,\\C),\n\\]\nresults in a formula for the wave packet $\\varphi_{\\boldsymbol{k}}[A',\\,B']$ \ninvolving wave packets in the old parametrization weighted by coefficients \nstemming from the $n$-fold Kronecker product $M^{n\\otimes}$. \nThe following analysis will reveal the relevant symmetric subspaces \nand corresponding orthogonal projections such that the resulting \n$n$-fold symmetric Kronecker product explicitly describes the wanted change \nof the parametrization.\n\n\\subsection{Two-fold symmetric Kronecker products}\nIn semidefinite programming\n(See for example \\cite{AHO98} or \\cite[Appendix~E]{Kle02}.),\nthe two-fold Kronecker product has notably occurred in combination with \nsubspaces of a particular symmetry property. One considers the space\n\\[\nX_2 \\ =\\ \\left\\{\\boldsymbol{x}\\in\\C^{d^2}: \\boldsymbol{x} = {\\rm vec}(X),\n\\,X\\ =\\ X^t\\in\\C^{d\\times d}\\right\\},\n\\]\nthat contains those vectors that can be obtained by the row-wise vectorization\nof a complex symmetric $d\\times d$ matrix, that is, a complex matrix coinciding with its transpose matrix.\nThe dimension of the space $X_2$ is\n\\[\nL_2 \\ =\\  \\tfrac12\\,d\\,(d+1).\n\\]\nOne can prove that this space is invariant under Kronecker products,\nin the sense that for all matrices $M\\in\\C^{d\\times d}$, one has\n\\[\n(M\\otimes M)\\,\\boldsymbol{x}\\in X_2,\\quad \\text{whenever}\\;\\; \\boldsymbol{x}\\in X_2.\n\\]\nNow one uses the standard basis of $\\C^{d^2}$ for constructing an\northonormal basis of the subspace $X_2$ and\ndefines a corresponding sparse $L_2\\times d^2$ matrix~$P_2$ that has the\nbasis vectors as its rows. The symmetric Kronecker product of $M$ with itself\nis then the $L_2\\times L_2$ matrix\n\\[\nS_2(M) \\ =\\ P_2\\,\\left(M\\otimes M\\right)\\, P_2^*.\n\\]\n\n\\subsection{$\\bf n$-fold symmetric Kronecker products}\nHow does one extend this construction to symmetrizing $n$-fold Kronecker \nproducts? It is instructive to revisit the second order space in two dimensions \nand to write a vector $\\boldsymbol{x}\\in X_2$ as\n\\[\n\\boldsymbol{x}\\ =\\ (x_{(2,0)},\\,x_{(1,1)},\\,x_{(1,1)},\\,x_{(0,2)})^t.\n\\]\nThis labelling uses the multi-indices $\\boldsymbol{k}=(k_1,\\,k_2)\\in\\N^2$\nwith $k_1+k_2=2$ in the redundant enumeration\n\\[\n\\boldsymbol{\\nu_2} \\ =\\  \\left((2,0),\\,(1,1),\\,(1,1),\\,(0,2)\\right).\n\\]\nThis description allows for a straightforward extension\nto higher order $n$ and dimension $d$.  \nOne works with a redundant enumeration of the multi-indices\n$\\boldsymbol{k}=(k_1,\\,\\ldots,\\,k_d)\\in\\N^d$ with $k_1+\\cdots+k_d=n$, \ncollects them in a row vector $\\boldsymbol{\\nu_n}$, \nand defines\n\\[\nX_n \\ = \\ \\left\\{ \\boldsymbol{x}\\in\\C^{d^n}: \\ \\text{For all } j,j'\\in\\{1,\\ldots,d^n\\}, \\ x_{j} = x_{j'}\\;\\;\\text{if}\\;\\;\n\\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\nu_n}(j')\\right\\}.\n\\]\nThe dimension of $X_n$ equals the number of mult-indices in $\\N^d$\nof order $n$, that is the binomial coefficient\n\\[\nL_n\\ =\\ \\binom{n+d-1}{n}.\n\\]\nAnd again, we can prove invariance in the sense that for all\n$M\\in\\C^{d\\times d}$\n\\[\nM^{n\\otimes}\\,\\boldsymbol{x}\\in X_n,\\quad\\text{whenever}\\;\\; \\boldsymbol{x}\\in X_n.\n\\]\nSee Proposition~\\ref{prop:kron}. Then, we use the standard basis of\n$\\C^{d^n}$ to build an orthonormal basis of $X_n$ and assemble the\ncorresponding sparse $L_n\\times d^n$ matrix $P_n$.\nAll this motivates the definition of the $n$-fold\nsymmetric Kronecker product as\n\\[\nS_n(M) \\ =\\ P_n\\,M^{n\\otimes}\\,P_n^*.\n\\]\nThe matrix $S_n(M)$ is of size $L_n\\times L_n$ and inherits structural\nproperties as invertibility or unitarity from the matrix $M$.\nSee Lemma~\\ref{lem:str}.\n\n\\bigskip\nOur main result Theorem~\\ref{theo:main} provides an explicit formula for the \naction of the matrix $S_n(M)$ in terms of \nmultinomial coeffients and powers of the entries of the original matrix~$M$. \nLabelling the components of a vector $\\boldsymbol{y}\\in\\C^{L_n}$\nby multi-indices of order $n$, \nwe obtain for all $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n$ that\n\\begin{align*}\n&\\left(S_n(M)\\,\\boldsymbol{y}\\right)_{\\boldsymbol{k}}\n=\\,\\frac{1}{\\sqrt{\\boldsymbol{k}!}}\\sum_{|\\boldsymbol{\\boldsymbol{\\alpha_1}}|=k_1}\\cdots\n\\sum_{|\\boldsymbol{\\boldsymbol{\\alpha_d}}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\boldsymbol{\\alpha_1}}}\\cdots\n\\binom{k_d}{\\boldsymbol{\\boldsymbol{\\alpha_d}}}\\,\\boldsymbol{\\boldsymbol{m_1}}^{\\boldsymbol{\\boldsymbol{\\alpha_1}}}\\cdots \\boldsymbol{\\boldsymbol{m_d}}^{\\boldsymbol{\\boldsymbol{\\alpha_d}}}\\\\ \n&\\hspace*{20em} \n\\times\\ \\sqrt{(\\boldsymbol{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}})!}\n\\ y_{\\boldsymbol{\\boldsymbol{\\alpha_1}}+\\cdots+\\boldsymbol{\\boldsymbol{\\alpha_d}}},\n\\end{align*}\nwhere $\\boldsymbol{\\boldsymbol{m_1}},\\,\\ldots,\\,\\boldsymbol{\\boldsymbol{m_d}}\\in\\C^d$ denote the row vectors of $M$. \nThe summations range over multi-indices $\\boldsymbol{\\boldsymbol{\\alpha_1}},\\ldots,\\boldsymbol{\\boldsymbol{\\alpha_d}}\\in\\N^d$ with \n$|\\boldsymbol{\\boldsymbol{\\alpha_1}}|=k_1$, \\ldots, $|\\boldsymbol{\\boldsymbol{\\alpha_d}}|=k_d$. They are weighted with multinomial coefficients stemming from the $n$-fold Kronecker product \n$M^{n\\otimes}$, whereas the square roots of the factorials originate \nin the orthonormalization of the row vectors of the matrix~$P_n$.\n\n\\subsection{Application to semiclassical wave packets}\nWe consider semiclassical wave packets $\\varphi_{\\boldsymbol{k}}[A,\\,B]$, $\\boldsymbol{k}\\in\\N^d$, parametrized by two\nmatrices $A,\\,B\\in{\\rm GL}(d,\\,\\C)$. The $\\boldsymbol{k}^{\\mbox{\\scriptsize th}}$ wave packet is the  \nproduct of a multivariate polynomial $p_{\\boldsymbol{k}}[A]$ and the complex-valued Gaussian function \n\\[\n\\varphi_{\\boldsymbol{0}}[A,B](x) \\ =\\  (\\pi\\,\\hbar)^{-d\/4}\\,\\det(A)^{-1\/2}\\,\n\\exp\\!\\left(\\,-\\,\\frac{\\langle \\boldsymbol{x},\\,B\\,A^{-1}\\,\\boldsymbol{x}\\rangle}{2\\,\\hbar}\\right),\\qquad \\boldsymbol{x}\\in\\R^d,\n\\]\n where $\\hbar>0$ is the semiclassical parameter used for the overall scaling. See Definition~\\ref{def:packet}. \n The polynomial family \n \\[\n \\left\\{p_{\\boldsymbol{k}}[A]: \\boldsymbol{k}\\in\\N^d\\right\\}\n \\] \n obeys a three-term recurrence relation and a Rodrigues type representation. See \\cite[Proposition~4]{LT14} and  \\cite[Theorem~4.1]{Hag15}. It is orthogonal with respect to the Gaussian weight function \n $|\\,\\varphi_{\\boldsymbol{0}}[A,B](x)\\,|^2$, but differs from the standard Hermite polynomials on $\\R^d$ that are biorthogonal and not orthogonal. See for example \\cite[\\S6]{IZ17}. \n If the matrix $A$ is real, then the polynomials $p_{\\boldsymbol{k}}[A]$ are real and factorize into univariate scaled \n Hermite polynomials. In the complex case, we encounter a more intricate structure that we wish to explore both for theoretical and numerical reasons.\n \n \\bigskip\n We consider a change of parametrization\n\\[\nA'=A\\,M, \\qquad B'=B\\,M\n\\]\ninduced by a suitably chosen invertible matrix $M\\in{\\rm GL}(d,\\C)$.\nCollecting all wave packets $\\varphi_{\\boldsymbol{k}}[A,\\,B]$ of order  $|\\boldsymbol{k}|=n$ as the components of a \nformal vector of wave functions $\\vec\\varphi_n[A,\\,B]$, the  formula of \nTheorem~\\ref{theo:main} allows us to identify the change of parametrization explicitly as\n\\[\n\\vec\\varphi_n[A',\\,B']\\ =\\ \\det(M)^{-1\/2}\\,S_n(M)\\,\n\\vec\\varphi_n[A,\\,B],\n\\]\nsee Corollary~\\ref{MainResult} in Section \\ref{Section5.3}. That is, the $n$-fold symmetric \nKronecker product explicitly transforms one parametrization into another one.\n\n\\bigskip\nRecently, E.~Faou, V.~Gradinaru, and C.~Lubich \\cite{FGL09,L} have used semiclassical wave packets for the numerical  \ndiscretization of semiclassical quantum dynamics. See also \\cite{GH14}. \nThe computationally demanding step of this method is the assembly of the Galerkin matrix \nfor the potential function $V:\\R^d\\to\\R$ according to\n\\[\n\\left\\langle \\varphi_{\\boldsymbol{k}}[A,\\,B], V\\varphi_{\\boldsymbol{l}}[A,\\,B]\\right\\rangle\\ =\\\n\\int_{\\R^d}\\,\\overline{\\varphi_{\\boldsymbol{k}}[A,\\,B](x)}\\, V(x)\\,\\varphi_{\\boldsymbol{l}}[A,\\,B](x)\\,dx, \n\\]\nwhere the multi-indices $\\boldsymbol{k},\\boldsymbol{l}\\in\\N^d$ are bounded in modulus by some truncation value~$N\\in\\N$, \nthat determines the dimension of the Galerkin space. \nIf the wave packets are parametrized by a matrix $A$ that has only real entries, then they \nfactorize into univariate Hermite functions, and the multi-dimensional integral becomes the product \nof one-dimensional ones. \\cite[Chapter 5.9]{B17} presents a two-dimensional numerical test case, \ntransforming a linear combination of semiclassical wave packets of order $n=4$ from one parametrization to another one  \nusing a tree-based implementation of the $n$-fold symmetric Kronecker product. The \ntransformation error is in the order of machine precision. This experiment suggests \na new numerical method for semiclassical quantum dynamics using the change \nof parametrization via  $n$-fold symmetric Kronecker products. Such a method assembles\nthe Galerkin matrix in terms of univariate Hermite functions. \nThen, the known large order asymptotics of the Hermite functions should allow one to stabilize the \nnumerical evaluation of the integrands \\cite{TTO16}, such that larger values of the truncation value $N$ become feasible.\n\n\n\n\n\\subsection{Organization of the paper}\nIn the next Section\nwe start with some combinatorics for \nexplicitly relating the lexicographic enumeration of multi-indices of order $n$ \nwith the redundant enumeration $\\boldsymbol{\\nu_n}$. Then we introduce the symmetric \nsubspaces $X_n$ in Section~\\ref{sec:sym} and construct an orthonormal \nbasis together with the \ncorresponding matrix $P_n$. There we also discuss symmetric subspaces \nand our basis construction in tensor terminology.\nIn Section \\ref{sec:kron}, we define the\n$n$-fold symmetric Kronecker product and prove our main results \nProposition~\\ref{prop:kron} and Theorem~\\ref{theo:main}.\nAn introduction to semiclassical wave packets and the description of linear \nchanges in their parametrization by symmetric Kronecker products is given in \nSection~\\ref{sec:sem}.\n\n\\subsection{Notation}\nVectors and multi-indices are bold.\nOn some occasions we shall use the binomial coefficient\n\\[\n\\binom{n}{j}\\ =\\ \\frac{n!}{(n-j)!\\,j!},\\qquad\n\\text{for non-negative integers }n\\ge j.\n\\]\nWe write a multi-index $\\boldsymbol{k}\\in\\N^d$ as a row vector $\\boldsymbol{k}=(k_1,\\,\\ldots,\\,k_d)$.\nWe use the modulus $|\\boldsymbol{k}|=k_1+\\cdots+k_d$, and the multinomial coefficient\n\\[\n\\binom{|\\boldsymbol{k}|}{\\boldsymbol{k}}\\ =\\ \\frac{|\\boldsymbol{k}|!}{k_1!\\,\\cdots\\,k_d!},\\qquad\n\\text{for}\\;\\;\\boldsymbol{k}\\in\\N^d.\n\\]\nWe adopt the convention that any multinomial coefficient with any negative \nargument is defined to be $0$. We also use the $\\boldsymbol{k}^{\\mbox{\\scriptsize th}}$ power of a vector, \n\\[\n\\boldsymbol{x}^k\\ =\\ x_1^{k_1}\\,\\cdots\\,x_d^{k_d},\\qquad \\boldsymbol{x}\\in\\C^d.\n\\]\n\n\\section{Combinatorics}\\label{sec:com}\n\n\\subsection{Reverse Lexicographic ordering} \nFirst we enumerate the set multi-indices of order $n$ in $d$ dimensions,\n\\[\n\\left\\{\\boldsymbol{k}\\in\\N^d: |\\boldsymbol{k}|=n\\right\\},\\qquad n\\in\\N,\n\\]\nin reverse lexicographic ordering and collect them\nas components of a formal row vector denoted by $\\boldsymbol{\\ell_n}$.\nThe length of the vector $\\boldsymbol{\\ell_n}$ is the binomial coefficient\n\\[\nL_n\\ =\n\\ \\binom{n+d-1}{n}\n\\]\nOne can think of this in the following way \\cite{JHT}:\nThe multi-indices $\\boldsymbol{k}$ of order $n$ in $d$ dimensions are in a one-to-one\ncorrespondence with the sequences of $n$ identical balls and $d-1$\nidentical sticks. The sticks partition the line into $d$ bins into which one can\ninsert the $n$ balls.\n(The first bin is to the left of all the sticks, and contains $k_1$\nballs; the last bin is to the right of all the sticks, and contains $k_d$ balls; for\n$2\\le j\\le d-1$, the $j^{\\mbox{\\scriptsize th}}$ bin\nis between sticks $j-1$ and $j$, and it contains $k_j$ balls.)\n{\\it E.g.}, the multi-index $(3,\\,2,\\,0,\\,1)$ in four dimensions\ncorresponds to\n$$\n{\\color{red}\\bullet}\\quad{\\color{red}\\bullet}\\quad{\\color{red}\\bullet}\\quad\n{\\color{blue}|}\\quad{\\color{red}\\bullet}\\quad{\\color{red}\\bullet}\\quad\n{\\color{blue}|}\\quad{\\color{blue}|}\\quad{\\color{red}\\bullet}.\n$$\nIf all these objects were distinguishable, there would be $(n+d-1)!$ \npermutations, but since the balls are all identical, one must divide by $n!$,\nand since the sticks are all identical, one must divide by $(d-1)!$.\n\n\\subsection{A redundant enumeration}\nNext we redundantly enumerate and collect multi-indices of modulus $n$ in a \n vector $\\boldsymbol{\\nu_n}$ of length $d^n$. Each entry of the vector~$\\boldsymbol{\\nu_n}$ is a multi-index of modulus $n$. Some of these entries occur repeatedly, since our enumeration is redudant. We proceed recursively and set\n\\[\n\\boldsymbol{\\nu_0}=\\left((0,\\,\\ldots,\\,0)\\right),\\quad\n\\boldsymbol{\\nu_{\\boldsymbol{1}}}=(\\boldsymbol{\\boldsymbol{e_1}}^t,\\,\\ldots,\\,\\boldsymbol{e_d}^t),\\quad\n\\]\nand\n\\[\n\\boldsymbol{\\nu_{n+1}}\\ =\\ {\\rm vec}\n\\begin{pmatrix}\\boldsymbol{\\nu_n}(1)+\\boldsymbol{\\boldsymbol{e_1}}^t & \\ldots & \\boldsymbol{\\nu_n}(d^n)+\\boldsymbol{\\boldsymbol{e_1}}^t\\\\ \n\\vdots & & \\vdots\\\\ \\boldsymbol{\\nu_n}(1)+\\boldsymbol{e_d}^t& \\ldots & \n\\boldsymbol{\\nu_n}(d^n)+\\boldsymbol{e_d}^t\\end{pmatrix},\\qquad n\\ge0,\n\\]\nwhere $\\boldsymbol{\\boldsymbol{e_1}},\\ldots,\\boldsymbol{e_d}\\in\\C^d$ are the standard basis vectors of $\\C^d$,\nand ${\\rm vec}$ denotes the row-wise vectorization of a matrix into a row vector.\n\n\\vskip 5mm\nFor example, for $d=2$, we have\n\\begin{align*}\n\\boldsymbol{\\ell_1} \\ &= \\ \\left( (1,0),\\,(0,1)\\right),\\\\\n \\boldsymbol{\\nu_1} \\ &= \\ \\left( (1,0),\\,(0,1)\\right),\\\\\n \\boldsymbol{\\ell_2}\\ &=\\ \\left((2,0),\\,(1,1),\\,(0,2)\\right),\\\\\n\\boldsymbol{\\nu_2}\\ &=\\ \\left((2,0),\\,(1,1),\\,(1,1),\\,(0,2)\\right),\\\\\n\\boldsymbol{\\ell_3}\\ &=\\ \\left((3,0),\\,(2,1),(1,2),\\,(0,3)\\right),\\\\\n\\boldsymbol{\\nu_3}\\ &=\\ \\left((3,0),\\,(2,1),\\,(2,1),\\,(1,2),\\,(2,1),\\,(1,2),\\,(1,2),\\,\n(0,3)\\right).\n\\end{align*}\nWe observe that the multi-index $(1,1)$ appears twice in $\\boldsymbol{\\nu_2}$, since \n\\[\n(1,1) = \\boldsymbol{\\nu_{\\boldsymbol{1}}}(1)+\\boldsymbol{e_2}^t = \\boldsymbol{\\nu_{\\boldsymbol{1}}}(2)+\\boldsymbol{\\boldsymbol{e_1}}^t.\n\\]\nThe modulus three multi-index $(2,1)$ can be generated as \n\\[\n(2,1) = \\boldsymbol{\\nu_2}(1)+\\boldsymbol{e_2}^t = \\boldsymbol{\\nu_2}(2)+\\boldsymbol{\\boldsymbol{e_1}}^t = \\boldsymbol{\\nu_2}(3)+\\boldsymbol{\\boldsymbol{e_1}}^t,\n\\] \nand therefore appears three times in $\\boldsymbol{\\nu_3}$. \n\n\n\n\\subsection{A partition}\nFor relating the lexicographic and the redundant enumeration,\nwe define the mapping\n\\[\n\\sigma_n: \\{1,\\,\\ldots,\\,L_n\\}\\to {\\mathcal P}(\\{1,\\,\\ldots,\\,d^n\\}) \n\\]\nso that for all $i\\in\\{1,\\,\\ldots,\\,L_n\\}$ and $j\\in\\{1,\\,\\ldots,\\,d^n\\}$\nthe following holds:\n\\[\nj\\in\\sigma_n(i)\\ \\Longleftrightarrow\\ \\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\ell_n}(i).\n\\]\n\n\\vskip 5mm\nFor example, for $d=2$, we have\n\\[\n\\sigma_2(1)=\\{1\\},\\quad \\sigma_2(2)=\\{2,\\,3\\},\\quad\n\\sigma_2(3)=\\{4\\}\n\\]\nand\n\\[\n\\sigma_3(1)=\\{1\\},\\quad \\sigma_3(2)=\\{2,\\,3,\\,5\\},\n\\quad\\sigma_3(3)=\\{4,\\,6,\\,7\\},\\quad\\sigma_3(4)=\\{8\\}.\n\\]\n\n\\vskip 5mm\nWe observe the following partition property.\n\n\\begin{lemma}\\label{lem:sigma}\nWe have \n\\[\n\\#\\sigma_n(i)\\ =\\ \\binom{n}{\\boldsymbol{\\ell_n}(i)},\\qquad i=1,\\,\\ldots,\\,L_n,\n\\] \nand\n$\\displaystyle\\bigcup_{i=1,\\,\\ldots,\\,L_n}\\sigma_n(i) = \\{1,\\,\\ldots\\,d^n\\}$,\n~where the union is pairwise disjoint.\n\\end{lemma}\n\n\\begin{proof}\nWe first prove that we have a partition property. For any\n$j\\in\\{1,\\,\\ldots,\\,d^n\\}$ there exists $i\\in\\{1,\\,\\ldots,\\,L_n\\}$ so that \n$\\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\ell_n}(i)$. So, we clearly have\n\\[\n\\bigcup_{i=1,\\ldots,L_n} \\sigma_n(i) = \\{1,\\,\\ldots,\\,d^n\\}.\n\\]\nMoreover, since $j\\in\\sigma_n(i)\\cap\\sigma_n(i')$ is equivalent to \n$\\boldsymbol{\\ell_n}(i)=\\boldsymbol{\\ell_n}(i')$, that is, $i=i'$, the union is disjoint. \n\nFor proving the claimed cardinality,  we argue by induction.\nFor $n=0$, we have\n\\[\n\\boldsymbol{\\ell_{0}}\\ =\\ \\left((0,\\,\\ldots,\\,0)\\right)\\ =\\ \\boldsymbol{\\nu_{0}},\\qquad\n\\sigma_{0}(1) = \\{1^0\\},\\qquad\n\\#\\sigma_{0}(1) = 1.\n\\]\n\nFor the inductive step,\nwe observe that in the redundant enumeration $\\boldsymbol{\\nu_n}$, \nthe multi-index $\\boldsymbol{k}=(k_1,\\,\\ldots,\\,k_d)$ can be generated from $d$ possible\nentries in $\\boldsymbol{\\nu_{n-1}}$,  \n\\[\n(k_1-1,\\,\\,k_2,\\,\\ldots,\\,k_d),\\ \\ldots,\\ (k_1,\\,\\ldots,\\,k_{d-1},\\,k_d-1)\n\\] \nby adding $\\boldsymbol{\\boldsymbol{e_1}}^t,\\,\\ldots,\\,\\boldsymbol{e_d}^t$, respectively.\nOf course, such an entry only  belongs to $\\boldsymbol{\\nu_{n-1}}$\nif all its components are non-negative.\nFor each of these indices with all entries non-negative, there is a unique\nnumber $j\\in\\{1,\\,2,\\,\\dots,\\,L_{n-1}\\}$, such that $\\boldsymbol{\\ell_{n-1}}(j)$\nis the given index.\nIf one of these indices has a negative entry, we define $j=-1$ and\n$\\sigma_{n-1}(-1)$ to be the empty set, {\\it i.e.},\n\\[\n\\sigma_{n-1}(-1)\\ =\\ \\{\\},\\quad\\mbox{whose cardinality is } 0.\n\\]\nWe list the $d$ numbers defined this way as $i_1,\\,\\ldots,\\,i_d$, and note\nthat all the positive values in this list must be distinct.\nThen,\n\\begin{eqnarray}\\nonumber\n\\#\\sigma_n(i)&=& \n\\sum_{m=1}^d\\,\\#\\sigma_{n-1}(i_m)\n\\\\[3mm]\\nonumber\n&=&\n\\sum_{m=1}^d\\,\\binom{n-1}{k_1,\\,\\ldots,\\,k_m-1,\\,\\ldots,\\,k_d}\n\\\\[3mm]\\nonumber\n&=&\n\\sum_{m=1}^d\\,\\left.\n\\begin{cases}\\frac{(n-1)!}{k_1!\\,\\cdots(k_m-1)!\\,\\cdots\\,k_d!}&k_m>0\\\\ \n0&k_m=0\\end{cases}\n\\right\\}\n\\\\[3mm]\\nonumber\n&=& \\frac{(n-1)!\\,(k_1+\\cdots+k_d)}{k_1!\\,\\cdots\\,k_d!}\n\\\\[3mm]\\nonumber\n&=&\\binom{n}{\\boldsymbol{k}}.\n\\end{eqnarray}\n\\end{proof}\n\n\n\\begin{remark}\nConsider $i\\in\\{1,\\ldots,L_n\\}$. A number $j\\in\\{1,\\ldots,d^n\\}$ is contained in the set $\\sigma_n(i)$, if the multi-index $\\boldsymbol{\\nu_n}(j)$ coincides with the multi-index $\\boldsymbol{\\ell_n}(i)$. The previous Lemma~\\ref{lem:sigma} verifies that the cardinality $\\#\\sigma_n(i)$ is the number of unique permutations of the multi-index $\\boldsymbol{\\ell_n}(i)$.\n\\end{remark}\n\n\n\\section{Symmetric subspaces}\\label{sec:sym}\nWe next analyze the symmetric spaces\n\\[\nX_n \\ = \\ \\left\\{ \\boldsymbol{x}\\in\\C^{d^n}: \\ \\text{For all } j,j'\\in\\{1,\\ldots,d^n\\}, \\ x_{j} = x_{j'}\\;\\;\\text{if}\\;\\;\n\\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\nu_n}(j')\\right\\}\n\\]\nfor $n\\in\\N$.\nWe have $X_0=\\C$ and $X_1=\\C^d$, whereas $X_n$ is a proper subset of \n$\\C^{d^n}$ for $n\\ge2$.\n\n\\vskip 5mm\nFor example, for $d=2$,\n\\begin{align*}\nX_2\\ &=\\ \\left\\{\\boldsymbol{x}\\in\\C^4: x_2=x_3\\right\\},\\\\\nX_3\\ &=\\ \\left\\{\\boldsymbol{x}\\in\\C^8: x_2=x_3=x_5,\\ x_4=x_6=x_7\\right\\}. \n\\end{align*}\n\n\\vskip 5mm\nAny vector $\\boldsymbol{x}\\in X_n$ has $d^n$ components, but the components that\ncorrespond to the same multi-index in the redundant enumeration\n$\\boldsymbol{\\nu_n}(1),\\,\\ldots,\\,\\boldsymbol{\\nu_n}(d^n)$ have the same value. \nHence, at most $L_n$ components of $\\boldsymbol{x}\\in X_n$ are  different.\nThey may be labelled by the multi-indices $\\boldsymbol{\\ell_n}(1),\\,\\ldots,\\,\\boldsymbol{\\ell_n}(L_n)$,\nand we often refer to them by\n\\[\nx_{\\boldsymbol{\\ell_n}(i)},\\qquad i=1,\\,\\ldots,\\,L_n.\n\\]\n\n\\vskip 5mm\nThe symmetric subspaces $X_n$, $n\\in\\N$, can also be obtained as \nthe vectorization of symmetric tensor spaces, \nand we next relate this alternative point of view to ours.\n\n\\subsection{The symmetric spaces in tensor terminology}\nThe second order subspace\n\\[\nX_2\\ =\\ \\left\\{\\boldsymbol{x}\\in\\C^{d^2}: x_j = x_{j'}\\;\\;\\text{if}\\;\\;\n\\boldsymbol{\\nu_2}(j)=\\boldsymbol{\\nu_2}(j')\\right\\}\n\\]\ncan also be described in terms of matrices. Since\n\\[\n\\boldsymbol{\\nu_2} = {\\rm vec}\\begin{pmatrix}\\boldsymbol{\\boldsymbol{e_1}}^t+\\boldsymbol{\\boldsymbol{e_1}}^t & \\ldots & \\boldsymbol{e_d}^t+\\boldsymbol{\\boldsymbol{e_1}}^t\\\\\n\\vdots & & \\vdots\\\\ \\boldsymbol{\\boldsymbol{e_1}}^t+\\boldsymbol{e_d}^t & \\ldots & \\boldsymbol{e_d}^t+\\boldsymbol{e_d}^t\\end{pmatrix},\n\\]\nwe may write\n\\[\nX_2\\ =\\ \\left\\{ \\boldsymbol{x}\\in\\C^{d^2}:\n\\boldsymbol{x}={\\rm vec}(X),\\;\\; X =X^t\\in\\C^{d\\times d}\\right\\}.\n\\]\nAlternatively, as in \\cite[\\S2.3]{VLV15},  one may permute the standard basis \nvectors $\\boldsymbol{\\boldsymbol{e_1}},\\,\\ldots,\\,\\boldsymbol{e_{d^2}}\\in\\C^{d^2}$\naccording to the $d^2\\times d^2$ permutation matrix\n\\[\n\\Pi_{dd}\\ =\\\n\\left( \\boldsymbol{e_{1+0\\cdot d}},\\,\\boldsymbol{e_{1+1\\cdot d}},\\,\\ldots,\\,\\boldsymbol{e_{1+(d-1)\\cdot d}},\\,\n\\ldots,\\,\\boldsymbol{e_{d+0\\cdot d}},\\,\\boldsymbol{e_{d+1\\cdot d}},\\,\\ldots,\\,\\boldsymbol{e_{d^2}} \\right)\n\\]\nand characterize the symmetric subspace as\n\\[\nX_2\\ =\\ \\left\\{\\boldsymbol{x}\\in\\C^{d^2}: \\Pi_{dd}\\,\\boldsymbol{x} =\\boldsymbol{x}\\right\\}.\n\\]\nMore generally, the higher order symmetric spaces $X_n$ can also be described \nin terms of higher order tensors. \nA tensor $X\\in\\mathbb C^{d\\times\\cdots\\times d}$ of order $n$ is called symmetric, if\n\\[\nX_{i_1,\\ldots,i_n}  = X_{\\sigma(i_1),\\ldots,\\sigma(i_n)},\\qquad\\text{for any permutation}\\,\\sigma\\in S_d,\n\\]\nsee for example \\cite[Section 2.2]{KB09} or \\cite[Chapter 3.5]{Hack12}, and an inductive argument with respect to $n$ shows that\n\\[\nX_n\\ =\\ \\left\\{ \\boldsymbol{x}\\in\\C^{d^n}:\n\\boldsymbol{x}={\\rm vec}(X),\\;\\; X\\in\\C^{d\\times\\cdots\\times d}\\ \\text{symmetric}\\right\\}.\n\\]\n\n\\subsection{Relation between the subspaces}\nDue to the recursive definition of the redundant multi-index enumeration,\nthe symmetric subspaces of neighboring order can be easily related to each other\nas follows.\n\n\\begin{lemma}\\label{lem:dec}\nThe symmetric subspace $X_{n+1}$ is contained in the\n$d$-ary Cartesian product of the symmetric subspace $X_n$, \n\\[\nX_{n+1} \\subseteq X_n \\times \\cdots \\times X_n,\\qquad n\\in\\N.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWe decompose a vector \n\\[\n\\boldsymbol{x}\\ =\\ (\\boldsymbol{\\boldsymbol{x^{(1)}}},\\,\\ldots,\\,\\boldsymbol{\\boldsymbol{x^{(d)}}})^t\\in X_{n+1}\n\\] \ninto $d$ subvectors with $d^n$ components each.\nThe $d^{n+1}$ components of $\\boldsymbol{x}$ can be labelled by the multi-indices \n\\[\n\\boldsymbol{\\nu_n}(1)+\\boldsymbol{\\boldsymbol{e_1}}^t,\\,\\ldots,\\,\\boldsymbol{\\nu_n}(d^n)+\\boldsymbol{\\boldsymbol{e_1}}^t,\\,\\ldots,\\,\n\\boldsymbol{\\nu_n}(1)+\\boldsymbol{e_d}^t,\\,\\ldots,\\,\\boldsymbol{\\nu_n}(d^n)+\\boldsymbol{e_d}^t,\n\\] \nso that the components of the subvector $\\boldsymbol{x^{(m)}}$, $m=1,\\ldots,d$,\ncan be labelled by\n\\[\n\\boldsymbol{\\nu_n}(1)+\\boldsymbol{e_m}^t,\\,\\ldots,\\,\\boldsymbol{\\nu_n}(d^n)+\\boldsymbol{e_m}^t.\n\\]\nHence, \n\\[\nx^{(m)}_j = x^{(m)}_{j'}\\quad\\mbox{if}\\quad \\boldsymbol{\\nu_n}(j) = \\boldsymbol{\\nu_n}(j'),\n\\] \nand $\\boldsymbol{x^{(m)}}\\in X_n$ for all $m=1,\\,\\ldots,\\,d$.\n\\end{proof}\n\nThe two-dimensional examples \n\\[\nX_1=\\C^2\\quad\\mbox{and}\\quad\nX_2=\\left\\{\\boldsymbol{x}\\in\\C^4: x_2=x_3\\right\\}\n\\] \nshow that the inclusion of \nLemma~\\ref{lem:dec} is in general not an equality.\n\n\\subsection{An orthonormal basis}\nWe now use the standard basis of $\\C^{d^n}$ to construct an orthonormal \nbasis of the symmetric subspace $X_n$. \n\n\\begin{lemma} \nLet $\\boldsymbol{\\boldsymbol{e_1}},\\ldots,\\boldsymbol{e_{d^n}}$ be the standard basis vectors of $\\C^{d^n}$, and \ndefine the vectors\n\\[\n\\boldsymbol{p_i}\\ =\\ \\frac{1}{\\sqrt{\\# \\sigma_n(i)}}\\ \\sum_{j\\in\\sigma_n(i)}\\boldsymbol{e_j},\\qquad\ni=1,\\,\\ldots,\\,L_n.\n\\]\nThen, $\\{\\boldsymbol{p_1},\\,\\ldots,\\,\\boldsymbol{p_{L_n}}\\}$ forms an orthonormal basis of the space \n$X_n$.\n\\end{lemma}\n\n\\begin{proof}\nFor all $i=0,\\,\\ldots,\\,L_n$ and $j,\\,j'=1,\\,\\ldots,\\,d^n$, we have\n\\[\n(\\boldsymbol{p_i})_j\\ =\\ \\left\\{\\begin{array}{ll} (\\#\\sigma_n(i))^{-1\/2},\n&\\,\\text{if}\\ j\\in\\sigma_n(i),\\\\ 0, &\\,\\text{otherwise.}\\end{array}\\right.\n\\]\nSince $j\\in\\sigma_n(i)$ if and only if $\\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\ell_n}(i)$, we have \n\\[\n(\\boldsymbol{p_i})_j\\ =\\ (\\boldsymbol{p_i})_{j'} \\quad\\text{if}\\quad \\boldsymbol{\\nu_n}(j)=\\boldsymbol{\\nu_n}(j'),\n\\]\nand thus $\\boldsymbol{p_i}\\in X_n$. We also observe, that for all $i,\\,i'=1,\\,\\ldots,\\,L_n$,\n\\begin{align*}\n\\langle \\boldsymbol{p_i},\\,\\boldsymbol{p_{i'}}\\rangle\\ &=\\\n\\frac{1}{\\sqrt{\\#\\sigma_n(i)\\cdot\\#\\sigma_n(i')}}\\,\n\\sum_{j\\in\\sigma_n(i)}\\,\\sum_{j'\\in\\sigma_n(i')}\\,\n\\langle \\boldsymbol{e_j},\\,\\boldsymbol{e_{j'}}\\rangle\\ \\\\\n&=\\ \\delta_{i,i'}.\n\\end{align*}\nHence, the vectors $\\boldsymbol{p_1},\\,\\ldots,\\,\\boldsymbol{p_{L_n}}$ are orthonormal.\nMoreover, for all $\\boldsymbol{x}\\in X_n$, we have\n\\begin{align*}\n\\langle \\boldsymbol{p_i},\\,\\boldsymbol{x}\\rangle\\ &=\\ \\frac{1}{\\sqrt{\\#\\sigma_n(i)}}\\,\n\\sum_{j\\in\\sigma_n(i)}\\,\\langle \\boldsymbol{e_j},\\,\\,\\boldsymbol{x}\\rangle\\\\\n&=\\\n\\sqrt{\\#\\sigma_n(i)}\\,x_{\\boldsymbol{\\ell_n}(i)},\n\\end{align*}\nand therefore\n\\begin{align*}\n\\boldsymbol{x}\\ &=\\ \\sum_{j=1}^{d^n}\\,\\langle \\boldsymbol{e_j},\\,\\boldsymbol{x}\\rangle\\,\\boldsymbol{e_j}\\ =\\ \n\\sum_{i=1}^{L_n}\\,\\sum_{j\\in\\sigma_n(i)}\\,\n\\langle \\boldsymbol{e_j},\\,\\boldsymbol{x}\\rangle\\,\\boldsymbol{e_j}\\\\\n&=\\ \n\\sum_{i=1}^{L_n}\\,x_{\\boldsymbol{\\ell_n}(i)}\\,\\sqrt{\\#\\sigma_n(i)}\\,\\boldsymbol{p_i}\n \\ =\\ \\sum_{i=1}^{L_n}\\,\\langle \\boldsymbol{p_i},\\,\\boldsymbol{x}\\rangle\\,\\boldsymbol{p_i}.\n\\end{align*}\n\\end{proof}\n\n\nThe orthonormal basis $\\{\\boldsymbol{p_1},\\ldots,\\boldsymbol{p_{L_n}}\\}$ of the symmetric subspace \n$X_n$ may be viewed as a normalized version of an orthogonal basis \n\\[\n\\big\\{X^{(1)},\\ldots,X^{(L_n)}\\big\\}\n\\] \nof the space of $d$-dimensional symmetric tensors of order $n$ constructed as follows: For each $i=1,\\ldots,L_n$ the multi-index $\\boldsymbol{\\ell_n}(i)=(k_1,\\ldots,k_d)$ defines the non-zero elements of the corresponding basis tensor $X^{(i)}\\in\\C^{d\\times\\cdots\\times d}$ according to \n\\[\nX^{(i)}_{j_1,\\ldots,j_d}\\neq 0 \\quad\\mbox{if}\\quad (j_1,\\ldots,j_d) \n\\mbox{ comprises $k_1$ times $1$, \\ldots, $k_d$ times $d$.}\n\\]\nMoreover, by symmetry, all non-vanishing entries of the tensor $X^{(i)}$ have to be the same. \nThe following Table~\\ref{tab:basis} illustrates this alternative line of thought for the third order symmetric subspace in dimension $d=2$.\n\\begin{table}[h]\n\\begin{tabular}{c|c|c|c|c}\n$i$ & $\\boldsymbol{\\ell_3}(i)$ & $\\boldsymbol{p_i}$ & non-zero elements of $X^{(i)}$ & $\\#\\sigma_3(i)$\\\\ \\hline\n$1$ & $(3,0)$ & $ \\boldsymbol{e_1}$ & $(1,1,1)$ & $1$\\\\ \n$2$ & $(2,1)$ & $ \\frac{1}{\\sqrt{3}}(\\boldsymbol{e_2}+\\boldsymbol{e_3}+\\boldsymbol{e_5})$ & \n$(2,1,1)$, $(1,2,1)$, $(1,1,2)$ & $3$\\\\ \n$3$ & $(1,2)$ & $ \\frac{1}{\\sqrt3}(\\boldsymbol{e_4}+\\boldsymbol{e_6}+\\boldsymbol{e_7})$ & $(2,2,1)$, $(2,1,2)$, $(1,2,2)$ & $3$\\\\ \n$4$ & $(0,3)$ & $ \\boldsymbol{e_8}$ & $(2,2,2)$ & $1$\\\\\n\\end{tabular}\n\\bigskip\n\\caption{\\label{tab:basis}The table lists the orthonormal basis $\\{\\boldsymbol{p_1},\\ldots,\\boldsymbol{p_4}\\}$ of the third symmetric subspace $X_3$ for dimension $d=2$. It also specifies the non-vanishing entries of a corresponding basis $\\{X^{(1)},\\ldots, X^{(4)}\\}$ of the space of symmetric tensors of size $2\\times 2\\times 2$.}\n\\end{table}\n\n\\subsection{An orthonormal matrix}\nThe orthonormal basis vectors $\\boldsymbol{p_1},\\ldots,\\boldsymbol{p_{L_n}}\\in X_n$\nallow us to define the sparse rectangular $L_n\\times d^n$ matrix\n\\[\nP_n\\ =\\ \\begin{pmatrix}\\boldsymbol{p_1}^t\\\\ \\vdots\\\\ \\boldsymbol{p_{L_n}}^t\\end{pmatrix}\n\\]\nthat has the $L_n$ basis vectors as its rows. For example, for $d=2$, we have\n\\begin{align*}\nP_2\\ &=\\ \\begin{pmatrix}\\boldsymbol{\\boldsymbol{e_1}}^t\\\\ \\frac{1}{\\sqrt{2}}(\\boldsymbol{e_2}^t+\\boldsymbol{e_3}^t)\\\\ \n\\boldsymbol{e_4}^t\\end{pmatrix}\\ \\\\\n&=\\\n\\begin{pmatrix}1&0&0&0\\\\ 0&\\frac{1}{\\sqrt2}&\\frac{1}{\\sqrt2}&0\\\\\n0&0&0&1\\end{pmatrix},\\\\*[2ex]\nP_3\\ &=\\ \\begin{pmatrix}\\boldsymbol{\\boldsymbol{e_1}}^t\\\\\n\\frac{1}{\\sqrt{3}}(\\boldsymbol{e_2}^t+\\boldsymbol{e_3}^t+\\boldsymbol{e_5}^t)\\\\\n\\frac{1}{\\sqrt3}(\\boldsymbol{e_4}^t+\\boldsymbol{e_6}^t+\\boldsymbol{e_7}^t)\\\\ \\boldsymbol{e_8}^t\\end{pmatrix} \n\\ \\\\\n&=\\ \\begin{pmatrix}1&0&0&0&0&0&0&0\\\\\n0&\\frac{1}{\\sqrt3}&\\frac{1}{\\sqrt3}&0&\\frac{1}{\\sqrt3}&0&0&0\\\\ \n0&0&0&\\frac{1}{\\sqrt3}&0&\\frac{1}{\\sqrt3}&\\frac{1}{\\sqrt3}&0\\\\\n0&0&0&0&0&0&0&1\\end{pmatrix}.\n\\end{align*}\n\n\\bigskip\nWe summarize some properties of the matrix $P_n$ and\nof its adjoint $P_n^*$ and calculate explicit formulas for their actions on\nvectors.\n\n\\begin{proposition}\\label{prop:pn}\nThe $L_n\\times d^n$ matrix $P_n$ and its adjoint $P_n^*$ satisfy\n\\[\nP_n\\,P_n^*\\ =\\ \\Id_{L_n\\times L_n},\\qquad {\\rm range}(P_n^*) \\ =\\ X_n.\n\\]\nMoreover, for all $\\boldsymbol{x}\\in X_n$, \n\\[\n(P_n\\,\\boldsymbol{x})_i \\ =\\\n\\sqrt{\\#\\sigma_n(i)}\\ x_{\\boldsymbol{\\ell_n}(i)},\\qquad i=1,\\,\\ldots,\\,L_n,\n\\]\nand for all $\\boldsymbol{y}\\in\\C^{L_n}$, \n\\[\n(P_n^*\\,\\boldsymbol{y})_{\\boldsymbol{\\ell_n}(i)}\\ =\\ 1\\left\/ \\sqrt{\\#\\sigma_n(i)}\\right.\\ y_i,\n\\qquad i=1,\\,\\ldots,\\,L_n.\n\\]\nIn particular, \n\\[\nP_n^*\\,P_n\\,\\boldsymbol{x} \\ =\\  \\boldsymbol{x},\\quad\\text{whenever}\\;\\; \\boldsymbol{x}\\in X_n.\n\\]\n\\end{proposition}\n\n\\begin{proof} \nThe two properties $P_n\\,P_n^*\\ =\\ \\Id_{L_n\\times L_n}$ and\n${\\rm range}(P_n^*) \\ =\\ X_n$  equivalently say, that \nthe row vectors of $P_n$ build an orthonormal basis of $X_n$. \n\n\\medskip\nFor any $y\\in\\C^{L_n}$, the vector $P_n^*\\,y$ is a linear combination\nof the column vectors $\\boldsymbol{p_1},\\,\\ldots,\\,\\boldsymbol{p_{L_n}}$ and therefore in $X_n$. \nLabelling its components by $\\boldsymbol{\\ell_n}(1),\\,\\ldots,\\,\\boldsymbol{\\ell_n}(L_n)$, we obtain\n\\begin{align*}\n(P_n^*\\,\\boldsymbol{y})_{\\boldsymbol{\\ell_n}(i)}\\ &=\\ \\sum_{i'=1}^{L_n}\\,(p_{i'})_{\\boldsymbol{\\ell_n}(i)}\\,y_{i'}\n\\\\ \n&=\\ \\frac{1}{\\sqrt{\\#\\sigma_n(i)}}\\ y_i,\\qquad i=1,\\,\\ldots,\\,L_n.\n\\end{align*}\nFor $\\boldsymbol{x}\\in X_n$ and $i=1,\\,\\ldots,\\,L_n$, we obtain\n\\begin{align*}\n(P_n\\,\\boldsymbol{x})_i\\ &=\\ \\sum_{j=1}^{d^n} (\\boldsymbol{p_i})_{\\boldsymbol{\\nu_n}(j)}\\,\nx_{\\boldsymbol{\\nu_n}(j)} \\\\\n&= \\frac{\\#\\sigma_n(i)}{\\sqrt{\\#\\sigma_n(i)}}\\\nx_{\\boldsymbol{\\ell_n}(i)} =\\ \\sqrt{\\#\\sigma_n(i)}\\ x_{\\boldsymbol{\\ell_n}(i)},\n\\end{align*}\nsince  $\\#\\sigma_n(i)$ components of $\\boldsymbol{p_i}$ do not vanish. In particular, \n\\[\n(P_n^*\\,P_n\\,\\boldsymbol{x})_{\\boldsymbol{\\ell_n}(i)}\\ =\\ \\frac{1}{\\sqrt{\\#\\sigma_n(i)}}\\\n(P_n \\boldsymbol{x})_i\\ =\\ x_{\\boldsymbol{\\ell_n}(i)}.\n\\]\n\\end{proof}\n\n\\section{Symmetric Kronecker products}\\label{sec:kron}\n\n\\subsection{Iterated Kronecker products}\nWe next investigate the action of an $n$-fold Kronecker product on the\nsymmetric subspace $X_n$, $n\\in\\N$. First, we prove the invariance \nof the symmetric spaces under multiplication with iterated Kronecker products.\n\\begin{lemma}\\label{lem:inv} \nFor all $M\\in\\C^{d\\times d}$ we have $M^{n\\otimes}\\boldsymbol{x} \\in X_n$ whenever $\\boldsymbol{x}\\in X_n$.\n\\end{lemma}\n\\begin{proof} Applying $M^{n\\otimes}$ to a tensor $X\\in\\C^{d\\times\\cdots\\times d}$ of order $n$, \nwe obtain the tensor\n\\[\n(M^{n\\otimes}X)_{i_1,\\ldots,i_n} = \\sum_{j_1=1}^d \\cdots \\sum_{j_n=1}^d M_{i_1,j_1}\\cdots M_{i_n,j_n} X_{j_1,\\ldots,j_n}.\n\\]\nFor a symmetric tensor $X$, we then have\n\\begin{align*}\n(M^{n\\otimes}X)_{\\sigma(i_1),\\ldots,\\sigma(i_n)} \n&= \\sum_{j_1=1}^d \\cdots \\sum_{j_n=1}^d M_{\\sigma(i_1),j_1}\\cdots M_{\\sigma(i_n),j_n} X_{j_1,\\ldots,j_n}\\\\\n&= \\sum_{k_1=1}^d \\cdots \\sum_{k_n=1}^d M_{i_1,k_1}\\cdots M_{i_n,k_n} X_{k_1,\\ldots,k_n}\\\\\n&=(M^{n\\otimes}X)_{i_1,\\ldots,i_n}.\n\\end{align*}\nThat is, $M^{n\\otimes}X$ is a symmetric tensor, too. By vectorisation we then obtain that $M^{n\\otimes}\\boldsymbol{x}\\in X_n$ whenever $\\boldsymbol{x}\\in X_n$.\n\\end{proof}\nWe now derive an explicit formula for the components of a vector $M^{n\\otimes} \\boldsymbol{x}$  in terms of the row vectors of the matrix $M$. \n\n\\begin{proposition}\\label{prop:kron} \nLet $M\\in\\C^{d\\times d}$, and denote by $\\boldsymbol{m_1},\\,\\ldots,\\,\\boldsymbol{m_d}\\in\\C^d$\nthe row vectors of the matrix $M$. Then, for all $\\boldsymbol{x}\\in X_n$, the components of the vector $M^{n\\otimes}\\boldsymbol{x}\\in X_n$ \ncan be labelled by multi-indices $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n$ and satisfy\n\\[\n\\left(M^{n\\otimes}\\boldsymbol{x}\\right)_k\\ =\\ \\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\n\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\,\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\\nx_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}},\n\\]  \nwhere the summation ranges over $\\boldsymbol{\\alpha_1},\\,\\ldots,\\,\\boldsymbol{\\alpha_d}\\in\\N^d$ with\n$|\\boldsymbol{\\alpha_1}|=k_1,\\ldots,|\\boldsymbol{\\alpha_d}| = k_d$.\n\\end{proposition}\n\n\\begin{proof}\nFor $n=1$, we have $M^{n\\otimes} = M$ and $X_n=\\C^d$,\nand our formula reduces to usual matrix-vector multiplication written as \n\\[\n(M\\boldsymbol{x})_{e_k}\\ =\\ \\sum_{j=1}^d\\,\\boldsymbol{\\boldsymbol{m_j}}^{\\boldsymbol{e_k}}\\,x_{\\boldsymbol{e_j}},\\qquad\nk=1,\\,\\ldots,\\,d.\n\\]\nFor the inductive step, we consider\n$\\boldsymbol{x}=(\\boldsymbol{x^{(1)}},\\,\\ldots,\\,\\boldsymbol{x^{(d)}})\\in X_{n+1}$ decomposed as in \nLemma~\\ref{lem:dec} with $\\boldsymbol{x^{(j)}}\\in X_n$. We compute\n\\begin{align*}\nM^{(n+1)\\otimes}\\boldsymbol{x}\\ &=\\ \\begin{pmatrix}\nm_{11}\\,M^{n\\otimes}&\\ldots &m_{1d}\\,M^{n\\otimes}\\\\ \n\\vdots &&\\vdots\\\\ m_{d1}\\,M^{n\\otimes}&\\ldots&m_{dd}\\,M^{n\\otimes}\n\\end{pmatrix}\\begin{pmatrix}\\boldsymbol{x^{(1)}}\\\\ \\vdots \\\\\\boldsymbol{x^{(d)}}\n\\end{pmatrix}\n\\\\*[2ex] \n&=\\\n\\begin{pmatrix}\nm_{11}\\,M^{n\\otimes} \\boldsymbol{x^{(1)}} +\\cdots +m_{1d}\\,M^{n\\otimes}\\boldsymbol{x^{(d)}}\\\\ \n\\vdots \\\\\nm_{d1}\\,M^{n\\otimes} \\boldsymbol{x^{(1)}}+\\cdots +m_{dd}\\,M^{n\\otimes} \\boldsymbol{x^{(d)}}\n\\end{pmatrix}.\n\\end{align*}\nBy Lemma~\\ref{lem:inv} we have for all $j=1,\\,\\ldots,\\,d$, that\n\\[\nm_{j1}\\,M^{n\\otimes} \\boldsymbol{x^{(1)}}+\\cdots+m_{jd}\\,M^{n\\otimes}\\boldsymbol{x^{(d)}}\n\\in X_n.\n\\]\nThe components of these vectors can be labelled by $\\boldsymbol{k}\\in\\N^d$ with\n$|\\boldsymbol{k}|=n$, and we have\n\\begin{align*}\n&\\left(m_{j1}M^{n\\otimes}\\boldsymbol{x^{(1)}}+\\cdots+\nm_{jd}\\,M^{n\\otimes}\\boldsymbol{x^{(d)}}\\right)_k\\\\\n&=\\\n\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\, \n\\left(m_{j1}\\, \\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\,\nx^{(1)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\\cdots\\right.\\\\\n&\\hspace*{21em} \\left.+\\ \nm_{jd}\\,\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\n\\,x^{(d)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\right).\n\\end{align*}\nThe $j$th of these sums can be rewritten as\n\\begin{align*}\n& \\sum_{|\\boldsymbol{\\alpha_j}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\alpha_j}}\\, \n\\left(\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{\\boldsymbol{e_1}}}\\,\nx^{(1)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\n\\cdots\\,+\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{e_d}}\\,\nx^{(d)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\right)\n\\\\\n&=\\ \\sum_{|\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}}}\\, \n\\boldsymbol{m_j}^{\\boldsymbol{\\beta_j}}\\,\nx^{(1)}_{\\boldsymbol{\\alpha_1}+\\cdots+(\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}})+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\\cdots\\,+\\\\\n&\\hspace{12em}\n\\sum_{|\\boldsymbol{\\beta_j}-\\boldsymbol{e_d}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{e_d}}\\,\n\\boldsymbol{m_j}^{\\boldsymbol{\\beta_j}}\\,\nx^{(d)}_{\\boldsymbol{\\alpha_1}+\\cdots+(\\boldsymbol{\\beta_j}-\\boldsymbol{e_d})+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\end{align*}\nNow we observe that for all $r=1,\\ldots,d$,\n\\[\nx^{(r)}_{\\boldsymbol{\\alpha_1}+ \\cdots+(\\boldsymbol{\\beta_j}-\\boldsymbol{e_r})+\\cdots+\\boldsymbol{\\alpha_d}} = x_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\beta_j}+\\cdots+\\boldsymbol{\\alpha_d}},\n\\]\nso that\n\\begin{align*}\n& \\sum_{|\\boldsymbol{\\alpha_j}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\alpha_j}}\\, \n\\left(\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{\\boldsymbol{e_1}}}\\,\nx^{(1)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\n\\cdots\\,+\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{e_d}}\\,\nx^{(d)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\right)\n\\\\\n&=\\ \\sum_{|\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}}}\\, \n\\boldsymbol{m_j}^{\\boldsymbol{\\beta_j}}\\,\nx_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\beta_j}+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\\cdots\\,+\\\\\n&\\hspace{12em}\n\\sum_{|\\boldsymbol{\\beta_j}-\\boldsymbol{e_d}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{e_d}}\\,\n\\boldsymbol{m_j}^{\\boldsymbol{\\beta_j}}\\,\nx_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\beta_j}+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\end{align*}\nSince all multi-indices $\\boldsymbol{\\beta_j}\\in\\N^d$ with $|\\boldsymbol{\\beta_j}|=k_j+1$ satisfy\n\\[\n\\binom{k_j+1}{\\boldsymbol{\\beta_j}} = \\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{\\boldsymbol{e_1}}}+ \\cdots + \\binom{k_j}{\\boldsymbol{\\beta_j}-\\boldsymbol{e_d}},\n\\]\nwe can write\n\\begin{align*}\n& \\sum_{|\\boldsymbol{\\alpha_j}|=k_j}\\,\\binom{k_j}{\\boldsymbol{\\alpha_j}}\\, \n\\left(\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{\\boldsymbol{e_1}}}\\,\nx^{(1)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}+\\,\n\\cdots\\,+\\boldsymbol{m_j}^{\\boldsymbol{\\alpha_j}+\\boldsymbol{e_d}}\\,\nx^{(d)}_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\right)\n\\\\\n&=\\ \\sum_{|\\boldsymbol{\\beta_j}|=k_j+1}\\,\\binom{k_j+1}{\\boldsymbol{\\beta_j}}\\, \n\\boldsymbol{m_j}^{\\boldsymbol{\\beta_j}}\\,x_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\beta_j}+\\cdots+ \\boldsymbol{\\alpha_d}}\n\\end{align*}\nand obtain\n\\begin{align*}\n&\\left(m_{j1} M^{n\\otimes}\\boldsymbol{x^{(1)}}+\\,\\cdots\\,+ m_{jd}\\, \nM^{n\\otimes}\\boldsymbol{x^{(d)}}\\right)_k\\ =\\\\\n&\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\cdots\\sum_{|\\boldsymbol{\\alpha_j}|=k_j+1}\\cdots \n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\cdots\n\\binom{k_j+1}{\\boldsymbol{\\alpha_j}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\\\\n& \\hspace*{20em}\n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}} x_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\end{align*}\nHence, $M^{(n+1)\\otimes}\\boldsymbol{x}$ has at most $L_{n+1}$ \ndistinct components that can be labelled by $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n+1$.\nThey satisfy\n\\[\n\\left(M^{(n+1)\\otimes}\\boldsymbol{x}\\right)_k\\ =\\\n\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\, \n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\ x_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\]\n\\end{proof}\n\n\\begin{remark}\nThe invariance property of Lemma~\\ref{lem:inv} can also be proved alongside \nthe inductive argument given in the proof of Proposition~\\ref{prop:kron} without \nusing tensor terminology.\n\\end{remark}\n\n\n\\subsection{Symmetric Kronecker products}\nHaving proven that $n$-fold Kronecker products leave the $n$th symmetric \nsubspace invariant, we define the $n$-fold symmetric Kronecker product as \nfollows:\n\n\\begin{definition}\nFor $M\\in\\C^{d\\times d}$ and $n\\in\\N$,\nwe define the $L_n\\times L_n$ matrix\n\\[\nS_n(M) \\ =\\ P_n \\,M^{n\\otimes}\\, P_n^*\n\\]\nand call it the {\\em $n$-fold symmetric Kronecker product} of the matrix $M$.\n\\end{definition}\n\nThe $n$-fold symmetric Kronecker product has useful structural properties.\n\n\\begin{lemma}\\label{lem:str}\nThe $n$-fold symmetric Kronecker product $S_n(M)$ of a matrix \n$M\\in\\C^{d\\times d}$ satisfies\n$S_n(M)^* = S_n(M^*)$.\nIf $M\\in{\\rm GL}(d,\\C)$, then\n\\[\nS_n(M)\\in{\\rm GL}(L_n,\\C)\\quad\\text{with}\\quad S_n(M)^{-1}\\ =\\\nS_n(M^{-1}).\n\\]\nIn particular, if $M\\in U(d)$, then $S_n(M)\\in U(L_d)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $(M\\otimes M)^*=M^*\\otimes M^*$ and\n$(M^{n\\otimes})^*=(M^*)^{n\\otimes}$, we have \n\\begin{align*}\nS_n(M^*)\\ &=\\ P_n(M^{n\\otimes})^*P_n^*\\\\\n& =\\ S_n(M^*).\n\\end{align*}\nIf $M$ is invertible, then $M\\otimes M$ is invertible with\n$(M\\otimes M)^{-1} = M^{-1}\\otimes M^{-1}$.\nBy Proposition~\\ref{prop:kron},\n\\[\n(M^{n\\otimes})^{-1}\\boldsymbol{p_j}\\in X_n,\\qquad j=1,\\,\\ldots,\\,L_n.\n\\]\nProposition~\\ref{prop:pn} yields $P_n^*\\,P_n\\,\\boldsymbol{x}=\\boldsymbol{x}$ for all $\\boldsymbol{x}\\in X_n$ and \n\\[\nP_n\\,\\boldsymbol{p_j} \\ =\\ \\boldsymbol{e_j},\\qquad j=1,\\,\\ldots,\\,L_n,\n\\] \nwhere $\\boldsymbol{\\boldsymbol{e_1}},\\,\\ldots,\\,\\boldsymbol{e_{L_n}}\\in\\C^{L_n}$ are the standard basis vectors. \nTherefore,\n\\begin{align*}\nS_n(M)\\,S_n(M^{-1})\\,\\boldsymbol{e_j} \\ &=\\\nP_n\\,M^{n\\otimes}\\,P_n^*\\,P_n\\,(M^{n\\otimes})^{-1}\\boldsymbol{p_j}\\\\\n& =\\\nP_n\\,\\boldsymbol{p_j}\\\\\n& =\\ \\boldsymbol{e_j},\n\\end{align*}\nthat is, \n\\[\nS_n(M)\\,S_n(M^{-1})\\ = \\ {\\rm Id}_{L_n\\times L_n}.\n\\]\n\\end{proof}\n\n\\subsection{The main result}\n\nThe explicit formula of Proposition~\\ref{prop:kron}\nfor the $n$-fold Kronecker product also \nallows a detailed description of the $n$-fold symmetric Kronecker product.\n\n\\begin{theorem}\\label{theo:main}\nLet $M\\in\\C^{d\\times d}$, and denote by $\\boldsymbol{m_1},\\,\\ldots,\\,\\boldsymbol{m_d}\\in\\C^d$\nthe row vectors of the matrix $M$. Then, the $n$-fold symmetric Kronecker \nproduct satisfies for all $\\boldsymbol{y}\\in\\C^{L_n}$ and $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n$,\n\\begin{align*}\n&\\left(S_n(M)\\,\\boldsymbol{y}\\right)_k\n=\\,\\frac{1}{\\sqrt{\\boldsymbol{k}!}}\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\cdots\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\cdots\n\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\,\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\\\\n&\\hspace*{20em}\n\\times \\ \\sqrt{(\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d})!}\\ y_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}},\n\\end{align*}\nwhere the summations range over $\\boldsymbol{\\alpha_1},\\ldots,\\boldsymbol{\\alpha_d}\\in\\N^d$ with \n$|\\boldsymbol{\\alpha_1}|=k_1$, \\ldots, $|\\boldsymbol{\\alpha_d}| = k_d$, and the components of $\\boldsymbol{y}\\in\\C^{L_n}$ are denoted by multi-indices of order $n$.\n\\end{theorem}\n\n\\begin{proof} \nBy Lemma~\\ref{lem:sigma}, we have\n\\[\n\\#\\sigma_n(i) = \\binom{n}{\\boldsymbol{\\ell_n}(i)},\\qquad\\mbox{for }i=1,\\,\\ldots,\\,L_n.\n\\]\nBy Proposition~\\ref{prop:pn}, we have $P_n^*\\,\\boldsymbol{y}\\in X_n$ and\n\\[\n(P_n^*\\,\\boldsymbol{y})_{\\boldsymbol{\\ell_n}(i)}\\ =\\ 1\\left\/\\sqrt{\\binom{n}{\\boldsymbol{\\ell_n}(i)}}\\ \\right.y_i,\n\\qquad\\mbox{for }i=1,\\,\\ldots,\\,L_n.\n\\]\nThis can be reformulated as\n\\[\n(P_n^*\\,\\boldsymbol{y})_\\alpha = \\sqrt{\\frac{\\alpha !}{n!}} \\ y_{\\boldsymbol{\\alpha}},\\qquad \n\\mbox{for }\\boldsymbol{\\alpha}\\in\\N^d\\ \\mbox{with }|\\boldsymbol{\\alpha}| = n.\n\\]\nBy Lemma~\\ref{lem:inv}, the $n$-fold Kronecker product leaves\n$X_n$ invariant, and by Proposition~\\ref{prop:kron} we have for all $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n$,\n\\begin{align*}\n&\\left(M^{n\\otimes}\\,P_n^*\\,\\boldsymbol{y}\\right)_k\\ \\\\\n&=\\\n\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\\sum_{|\\boldsymbol{\\alpha_d}|=k_d} \n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\,\n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\, \n(P_n^*\\,\\boldsymbol{y})_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\\\\n&=\\ \\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\, \n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\,\n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\\\\n& \\hspace*{20em}\n\\times\\ \\sqrt{\\frac{(\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d})!}{n!}}\\\ny_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\end{align*}\nBy Proposition~\\ref{prop:pn}, we have for all $\\boldsymbol{x}\\in X_n$\n\\[\n(P_n\\,\\boldsymbol{x})_i\\ =\\ \\sqrt{\\binom{n}{\\boldsymbol{\\ell_n}(i)}}\\ x_{\\boldsymbol{\\ell_n}(i)},\n\\qquad\\mbox{for } i=1,\\,\\ldots,\\,L_n,\n\\] \nso that\n\\begin{align*}\n&\\left(P_n\\,M^{n\\otimes}\\,P_n^*\\,\\boldsymbol{y}\\right)_k\\\\\n&=\\ \\sqrt{\\frac{n!}{\\boldsymbol{k}!}}\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\cdots\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d} \n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\cdots\\binom{k_d}{\\boldsymbol{\\alpha_d}}\n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\\\\n&\\hspace*{20em}\n\\times\\ \\sqrt{\\frac{(\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d})!}{n!}}\\,\ny_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\\\\n&=\\ \\frac{1}{\\sqrt{\\boldsymbol{k}!}}\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\cdots\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\cdots\n\\binom{k_d}{\\boldsymbol{\\alpha_d}}\n\\boldsymbol{m_1}^{\\boldsymbol{\\alpha_1}}\\cdots \\boldsymbol{m_d}^{\\boldsymbol{\\alpha_d}}\\\\\n& \\hspace*{20em}\n\\times\\ \\sqrt{(\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d})!}\\ y_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}.\n\\end{align*}\n\\end{proof}\n\n\\section{Application to semiclassical wavepackets}\\label{sec:sem}\n\\subsection{Parametrizing Gaussians}\n\nWe consider two complex invertible matrices $A,\\,B\\in{\\rm GL}(d,\\,\\C)$\nthat satisfy the conditions\n\\begin{align}\\label{eq:mat1}\nA^tB - B^t A\\ &=\\ 0,\n\\\\ \\label{eq:mat2}\nA^*B + B^* A\\ &=\\ 2\\,{\\rm Id}_{d\\times d}.\n\\end{align}\nThese two conditions imply that $B\\,A^{-1}$\nis a complex symmetric matrix such that its real part\n\\[\n{\\rm Re}(B\\,A^{-1})\\ =\\ (A\\,A^*)^{-1}\n\\]\nis a Hermitian and positive definite matrix, see \\cite{Hag80}.\nLet $\\hbar>0$ and define\nthe multivariate complex-valued Gaussian function\n\\begin{align}\\nonumber\n&\\varphi_{\\boldsymbol{0}}[A,\\,B]: \\R^d\\to\\C,\\\\*[-1ex] \\label{eq:gauss} \\\\*[-3ex] \\nonumber\n&\\varphi_{\\boldsymbol{0}}[A,\\,B](\\boldsymbol{x})\\ =\\ (\\pi\\,\\hbar)^{-d\/4}\\,\\det(A)^{-1\/2}\\,\n\\exp\\!\\left(\\,-\\,\\frac{\\langle \\boldsymbol{x},\\,B\\,A^{-1}\\,\\boldsymbol{x}\\rangle}{2\\,\\hbar}\\right).\n\\end{align}\nThen, $\\varphi_{\\boldsymbol{0}}[A,\\,B]$ is a square-integrable function,\nand the constant factor $\\det(A)^{-1\/2}$\nensures normalization according to\n\\[\n\\int_{\\R^d}\\,\\left|\\varphi_{\\boldsymbol{0}}[A,\\,B](\\boldsymbol{x})\\right|^2 \\,d\\boldsymbol{x}\\ =\\ 1.\n\\]\nChanging the parametrization by a unitary matrix changes the Gaussian \nfunction only by constant multiplicative factor of modulus one:\n\n\\begin{lemma}\\label{lem:gauss}\nLet $A,\\,B\\in{\\rm GL}(d,\\,\\C)$ satisfy the conditions \n(\\ref{eq:mat1}--\\ref{eq:mat2}).\nThen, for all unitary matrices $U\\in U(d)$,\nthe matrices $A'=A\\,U$ and $B'=B\\,U$ also satisfy the \nconditions (\\ref{eq:mat1}--\\ref{eq:mat2}). Moreover, \n\\[\n\\varphi_{\\boldsymbol{0}}[A',\\,B'] = \\det(U)^{-1\/2} \\,\\varphi_{\\boldsymbol{0}}[A,\\,B].\n\\]\n\\end{lemma}\n\n\\begin{proof} We observe that\n\\begin{align*}\n(A')^tB' -(B')^t A'\\ &=\\  U^t (A^t B -BA)U\\ =\\ 0\\\\\n(A')^*B' + (B')^*A'\\ &=\\ U^*(A^* B + BA)U\\ =\\ 2\\,{\\rm Id}_{d\\times d}.\n\\end{align*}\nand $B'\\,(A')^{-1} = B\\,U\\,U^*A^{-1} = BA^{-1}$. Therefore, \n\\begin{align*}\n\\varphi_{\\boldsymbol{0}}[A',\\,B']\\ &=\\ (\\pi\\,\\hbar)^{-d\/4}\\,\\det(A\\,U)^{-1\/2}\\,\n\\exp\\left(\\,-\\,\\frac{\\langle \\boldsymbol{x},\\,B\\,A^{-1}\\,\\boldsymbol{x}\\rangle}{2\\,\\hbar}\\right)\\\\\n&=\\ \\det(U)^{-1\/2}\\,\\varphi_{\\boldsymbol{0}}[A,\\,B].\n\\end{align*}\n\\end{proof}\n\n\\subsection{Semiclassical wave packets}\nFollowing the construction of \\cite{Hag98}, we consider\n$A,\\,B\\in{\\rm GL}(d,\\,\\C)$ satisfying the conditions\n(\\ref{eq:mat1}--\\ref{eq:mat2}) and introduce the vector of raising operators\n\\[\n\\Rr[A,\\,B]\\ =\\\n\\frac{1}{\\sqrt{2\\hbar}}\\left(B^*\\,\\boldsymbol{x} - iA^*(-i\\hbar\\nabla_{\\boldsymbol{x}})\\right)\n\\]\nthat consists of $d$ components,  \n\\[\n\\Rr[A,\\,B]\\ = \\\n\\begin{pmatrix}\\Rr_1[A,\\,B]\\\\ \\vdots \\\\ \\Rr_d[A,\\,B]\\end{pmatrix}.\n\\]\nThe raising operator acts on Schwartz functions $\\psi:\\R^d\\to\\C$ as\n\\[\n\\left(\\Rr[A,\\,B]\\psi\\right)(\\boldsymbol{x})\\ =\\ \\frac{1}{\\sqrt{2\\hbar}}\n\\left(B^*\\,\\boldsymbol{x}\\,\\psi(\\boldsymbol{x}) - iA^*(-i\\hbar\\nabla_{\\boldsymbol{x}}\\psi)(\\boldsymbol{x})\\right),\\qquad \\boldsymbol{x}\\in\\R^d.\n\\]\n\nPowers of the raising operator now generate the semiclassical wave packets.\n\n\\begin{definition}[Semiclassical wave packet]\\label{def:packet} \nLet $A,\\, B\\in{\\rm GL}(d,\\,\\C)$ satisfy the conditions (\\ref{eq:mat1}--\\ref{eq:mat2}) \nand $\\varphi_{\\boldsymbol{0}}[A,B]$ denote the Gaussian function of (\\ref{eq:gauss}). Then, the $\\boldsymbol{k}^{\\mbox{\\scriptsize th}}$\nsemiclassical wave packet is defined by\n\\[\n\\varphi_{\\boldsymbol{k}}[A,\\,B]\\ =\\ \\frac{1}{\\sqrt{\\boldsymbol{k}!}}\\ \\Rr[A,B]^{\\boldsymbol{k}}\\varphi_{\\boldsymbol{0}}[A,\\,B],\\qquad\n\\boldsymbol{k}\\in\\N^d\\ .\n\\]\n\\end{definition}\n\nWe note that the $\\boldsymbol{k}^{\\mbox{\\scriptsize th}}$ power of the raising operator\n\\[\n\\Rr[A,\\,B]^{\\boldsymbol{k}} = \\Rr_1[A,\\,B]^{k_1}\\,\\cdots\\,\\Rr_d[A,B]^{k_d}\n\\]\ndoes not depend on the ordering,\nsince the components commute with one another due to the compatibility \nconditions (\\ref{eq:mat1}--\\ref{eq:mat2}).\nThe set \n\\[\n\\left\\{\\varphi_{\\boldsymbol{k}}[A,\\,B]: \\boldsymbol{k}\\in\\N^d\\right\\}\n\\] \nforms an orthonormal basis of the space of square-integrable functions \n$L^2(\\R^d,\\,\\C)$.\n\n\\subsection{Hermite polynomials}\nBy its construction, the $\\boldsymbol{k}^{\\mbox{\\scriptsize th}}$ semiclassical wave packet is a multivariate\npolynomial of degree $|\\boldsymbol{k}|$ in $\\boldsymbol{x}$ times the complex-valued Gaussian function $\\varphi_{\\boldsymbol{0}}[A,\\,B]$, that is, \n\\[\n\\varphi_{\\boldsymbol{k}}[A,\\,B](x)\\ =\\ \\frac{1}{\\sqrt{2^{|\\boldsymbol{k}|}\\,\\boldsymbol{k}!}}\\,p_{\\boldsymbol{k}}[A](\\boldsymbol{x})\\,\\varphi_{\\boldsymbol{0}}[A,\\,B](\\boldsymbol{x}),\\qquad \\boldsymbol{x}\\in\\R^d,\n\\]\nThe polynomials $p_{\\boldsymbol{k}}[A]$ are determined by the matrix\n$A\\in{\\rm GL}(d,\\,\\C)$, see \\cite{Hag15}, and satisfy the three-term recurrence relation\n\\[\n\\left(p_{k+\\boldsymbol{e_j}}[A]\\right)_{j=1}^d\\ =\\ \\frac{2}{\\sqrt{\\hbar}}\nA^{-1}\\boldsymbol{x}\\,p_{\\boldsymbol{k}}[A] - 2A^{-1}\\overline{A}\n\\left(k_j\\,p_{\\boldsymbol{k}-\\boldsymbol{e_j}}[A]\\right)_{j=1}^d,\n\\]\nsee \\cite[Chapter V.2]{L}.\nWhenever the parameter matrix $A$ has all entries real,\n$A\\in{\\rm GL}(d,\\,\\R)$, then the polynomials factorize according to \n\\[\np_{\\boldsymbol{k}}[A](\\boldsymbol{x})\\ =\\ \\prod_{j=1}^d\\,\nH_{k_j}\\!\\left(\\tfrac{1}{\\sqrt{\\hbar}}(A^{-1}\\boldsymbol{x})_j\\right),\\qquad \\boldsymbol{x}\\in\\R^d,\n\\]\nwhere $H_n$ is the $n^{\\mbox{\\scriptsize th}}$ Hermite polynomial, $n\\in\\N$,\ndefined by the univariate three-term recurrence relation\n\\[\nH_{n+1}(y)\\ =\\ 2\\,y\\,H_n(y)\\,-\\,2\\,n\\,H_{n-1}(y),\\qquad y\\in\\R.\n\\]\n\nThe real parameter case, however, is rather exceptional when using \nsemiclassical wave packets for their key application in molecular quantum \ndynamics. \nThere, the parameter matrices $A(t)$ and $B(t)$, $t\\in\\R$,\nare time-dependent and determined by a system of ordinary differential \nequations. \nFor the particularly simple, but instructive example of\nharmonic oscillator motion,\none can even write the solution explicitly as\n\\begin{align*}\nA(t)\\ &=\\ \\cos(t)\\,A(0) + i\\,\\sin(t)\\,B(0),\\\\\nB(t)\\ &=\\ i\\,\\sin(t)\\,A(0) + \\cos(t)\\,B(0).\n\\end{align*}\nHence, the matrix $A(t)$\ncannot be expected to have only real entries, and the crucial matrix factor \n$A(t)^{-1}\\overline{A(t)}$ \nin the three term recurrence relation generates multivariate polynomials \nbeyond a tensor product representation. \n\n\\subsection{Changing the parametrization}\\label{Section5.3}\nIf $A,\\,B\\in{\\rm GL}(d,\\,\\C)$ satisfy the compatibility \nconditions (\\ref{eq:mat1}--\\ref{eq:mat2}),\nthen $|A| = \\sqrt{AA^*}$ is a real symmetric, \npositive definite matrix, \nand the singular value decomposition of $A$,\n\\[\nA\\ =\\ V\\,\\Sigma W^*\\quad\\text{with}\\quad\n\\Sigma={\\rm diag}(\\sigma_1,\\ldots,\\sigma_d)\\;\\;\\text{positive definite},\n\\]\nis given by an orthogonal matrix $V\\in O(d)$ and a unitary matrix\n$W\\in U(d)$. This provides two natural ways for transforming\n$A' = A\\,U$ with $A'\\in{\\rm GL}(d,\\,\\R)$ and $U\\in U(d)$.\nOne may work with the polar decomposition of $A$, \n\\[\nA'\\ =\\ |A|\\ =\\ V\\,\\Sigma\\,V^*\\quad\\text{and}\\quad U=W\\,V^*,\n\\]\nor alternatively with\n\\[\nA'\\ =\\ V\\,\\Sigma\\quad\\text{and}\\quad U=W.\n\\]\n\nBoth choices provide a unitary transformation to the real case,\nand we ask how to relate different families of wave packets that correspond\nto unitarily linked parametrizations.\nFor an explicit description, we collect the semiclassical wave packets\nof order $n$ in one formal vector\n\\[\n\\vec\\varphi_n[A,\\,B]\\ =\\\n\\begin{pmatrix}\\varphi_{\\boldsymbol{\\ell_n}(1)}[A,\\,B]\\\\ \\vdots\\\\\n\\varphi_{\\boldsymbol{\\ell_n}(L_n)}[A,\\,B]\\end{pmatrix},\n\\]\nwhose components are labelled by the multi-indices\n$\\boldsymbol{\\ell_n}(1),\\,\\ldots,\\,\\boldsymbol{\\ell_n}(L_n)$.\nThen, we use the $n$-fold symmetric Kronecker product in the following way:\n\n\\begin{corollary}\\label{MainResult}\nLet $A,\\,B\\in{\\rm GL}(d,\\,\\C)$ satisfy the conditions\n(\\ref{eq:mat1}--\\ref{eq:mat2}),\nand consider the matrices $A'=A\\,U$, $B'=B\\,U$ with $U\\in U(d)$.\nThen,  \n\\[\n\\vec\\varphi_n[A',\\,B']\\ =\\ \\det(U)^{-1\/2}\\,S_n(U)\\,\n\\vec\\varphi_n[A,\\,B],\\qquad n\\in\\N.\n\\]\n\\end{corollary}\n\n\\begin{proof}\nWe observe that the raising operators transform according to\n\\[\n\\Rr[A',\\,B']\\ =\\ \\Rr[A\\,U,\\,B\\,U] \\ =\\  U^*\\,\\Rr[A,\\,B],\n\\]\nwhich means for the components  that\n\\[\n\\Rr_j[A',\\,B']\\ =\\ \\sum_{m=1}^d\\,\n\\overline{u_{mj}}\\ \\Rr_m[A,\\,B],\\qquad j=1,\\,\\ldots,\\,d.\n\\]\nSince all components of the raising operators commute which each other,\nwe can use the multinomial theorem and obtain for all $n\\in\\N$ that\n\\[\n\\Rr_j[A',\\,B']^n\\ =\\ \\sum_{|\\boldsymbol{\\alpha}|=n}\\,\n\\binom{n}{\\boldsymbol{\\alpha}}\\,\\overline{\\boldsymbol{u_j}^{\\boldsymbol{\\alpha}}}\\ \n\\Rr[A,\\,B]^{\\boldsymbol{\\alpha}},\n\\]\nwhere $\\boldsymbol{u_1},\\,\\ldots,\\,\\boldsymbol{u_d}\\in\\C^d$\ndenote the column vectors of the matrix $U$. \nThis implies for any $\\boldsymbol{k}\\in\\N^d$ with $|\\boldsymbol{k}|=n$,\n\\[\n\\Rr[A',\\,B']^{\\boldsymbol{k}}\\ =\\ \\sum_{|\\boldsymbol{\\alpha_1}| = k_1}\\,\\cdots\\,\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\, \n\\overline{u_{\\boldsymbol{1}}^{\\boldsymbol{\\alpha_1}}}\\,\\cdots\\,\\overline{\\boldsymbol{u_d}^{\\boldsymbol{\\alpha_d}}}\\\n\\Rr[A,\\,B]^{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}},\n\\]\nwhere the $d$ summations run over $\\boldsymbol{\\alpha_1},\\,\\ldots,\\,\\boldsymbol{\\alpha_d}\\in\\N^d$.\nTogether with Lemma~\\ref{lem:gauss}, we therefore obtain\n\\begin{align*}\n&\\varphi_{\\boldsymbol{k}}[A',\\,B']\\ =\\\n\\frac{1}{\\sqrt{\\boldsymbol{k}!}}\\,\\Rr[A',\\,B']^{\\boldsymbol{k}} \\,\\varphi_{\\boldsymbol{0}}[A',\\,B'] \\\\\n&=\\ \\frac{1}{\\sqrt{\\det(U)\\,\\boldsymbol{k}!}}\\,\\sum_{|\\boldsymbol{\\alpha_1}| = k_1}\\,\\cdots\\,\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\n\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\\binom{k_d}{\\boldsymbol{\\alpha_d}}\\,\n\\overline{u_{\\boldsymbol{1}}^{\\boldsymbol{\\alpha_1}}}\\,\\cdots\\,\\overline{\\boldsymbol{u_d}^{\\boldsymbol{\\alpha_d}}}\\\\ \n&\\hspace*{21em}\\times\\quad\\Rr[A,\\,B]^{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}\\,\n\\varphi_{\\boldsymbol{0}}[A,B]\\\\[3mm]\n&=\\ \\frac{1}{\\sqrt{\\det(U)\\,\\boldsymbol{k}!}}\\,\\sum_{|\\boldsymbol{\\alpha_1}|=k_1}\\,\\cdots\\,\n\\sum_{|\\boldsymbol{\\alpha_d}|=k_d}\\,\\binom{k_1}{\\boldsymbol{\\alpha_1}}\\,\\cdots\\,\n\\binom{k_d}{\\boldsymbol{\\alpha_d}}\n\\;\\overline{u_{\\boldsymbol{1}}^{\\boldsymbol{\\alpha_1}}}\\,\\cdots\\,\\overline{\\boldsymbol{u_d}^{\\boldsymbol{\\alpha_d}}}\\\\\n&\\hspace*{18em}\\times\\,\\sqrt{(\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d})!}\\\n\\varphi_{\\boldsymbol{\\alpha_1}+\\cdots+\\boldsymbol{\\alpha_d}}[A,\\,B].\n\\end{align*}\nBy Theorem~\\ref{theo:main}, we then obtain\n\\[\n\\vec\\varphi_n[A',\\,B']\\ =\\ \\det(U)^{-1\/2}\\,S_n(U)\\,\\vec\\varphi_n[A,\\,B].\n\\]\n\\end{proof}\n\n\\section{Conclusion}\nWe have derived an explicit formula for the action of \n$n$-fold Kronecker products on symmetric subspaces. See Theorem~\\ref{theo:main}. \nOur findings generalize results on two-fold symmetric Kronecker products discussed \nin the literature on semidefinite programming \\cite{AHO98}, \\cite[Appendix~E]{Kle02}.\nThe new formula allows one to write a linear change of the parametrization of semiclassical wave packets \nin terms of a $n$-fold symmetric Kronecker product. \nSemiclassical wave packetshave an associated family of multivariate orthogonal \npolynomials. Our result provides an explicit transformation of these polynomials to a tensor \nproducts of scaled univariate Hermite polynomials. Moreover, semiclassical wave packets\nhave been used in \\cite{FGL09}, \\cite[Chapter~5]{L} and \\cite{GH14}\nfor a Galerkin discretization of multi-dimensional molecular quantum dynamics. \nThe explicit formula for the effect of a change of parametrization will allow to convert \nthe multi-dimensional Galerkin integrals of the method to a product of one-dimensional integrals,\nresulting in a more accurate and stable numerical method, see \\cite[Chapter~5]{B17} for \nnumerical experiments in this direction.\n\n\\subsection*{Acknowledgements} This research was partially supported by the U.S.~National Science Foundation\nGrant DMS--1210982 and the German Research Foundation (DFG), Collaborative Research Center \nSFB\/TRR 109. The authors thank the anonymous referees for their constructive help in improving \nthe clarity of presentation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{Introduction}\n\n Inflation is generally considered to be a reasonable solution to many of the\nfundamental problems within the standard cosmological model.  In\noriginal proposals \\cite{Guth}, the early universe experiences a\nperiod of accelerated expansion and essentially expands at an\nexponential rate, (i.e., $R(t)\\propto {\\rm e}^{At}$ where $R(t)$ can be\nconsidered as the size of the universe and $A$ is some positive\nconstant). Since these early stages, there have been a variety of\ninflationary models that include scalar fields that have been\nproposed \\cite{Olive90}, and scalar fields have come to play an\nimportant role in determining the dynamics of the early universe.\nIn one important class of inflationary models the condition of\nexponential expansion is relaxed, and the universe grows at a\npower-law rate, $R(t)\\propto t^p$, where $p>1$ \\cite{Luchin85}. In\nparticular, power-law inflationary models arise in models with a\nscalar field $\\phi$ having an exponential potential\n$V(\\phi)=V_0{\\rm e}^{k\\phi}$ \\cite{Wetterich}. Although power-law\ninflation is successful in solving the horizon and flatness\nproblems, inflation in these models persists into the indefinite\nfuture and a phase\n transition\nis required to bring\ninflation to an end (however, see \\cite{BC99}).\n\nSpatially homogeneous  models\ncontaining a scalar field $\\phi$ with an exponential potential have been analyzed extensively\n\\cite{Halliwell87,Coley97a}.\nIt is known that all ever-expanding scalar field models  experience power-law\ninflation when the parameter $k^2<2$; i.e., when the potential is sufficiently flat.\n  The models have also been studied when $k^2>2$\n\\cite{Coley97a}.\nRecently cosmological models containing both a scalar field with an\nexponential potential and a perfect fluid with a\nlinear barotropic equation of state\nhave been studied.  It is found that in the general\nclass of Bianchi type B models that the power-law inflationary solution is\nstill the global attractor in the physically realistic regime (i.e., when\n$\\gamma>2\/3$) if  $k^2<2$ \\cite{Billyard99}.\nInterestingly, the addition of a barotropic perfect\nfluid creates the existence of a new type of solution, appropriately called a\nmatter scaling solution \\cite{Copeland98,Billyard98,vandenHoogen99},\nin which the energy-density of the scalar field scales with that of the matter; the effective\nequation of state\nfor the scalar field is the same as that of the perfect fluid. The stability of\nthe matter scaling solution has been studied in \\cite{Copeland98,Billyard98}.\n\n\nExponential potentials arise in many theories of the fundamental\ninteractions including superstring and higher-dimensional theories\n\\cite{Olive90,Green}. Typically, `realistic' supergravity theories\npredict steep exponential potentials \\cite{Green} (i.e., $k^2>2$),\neffectively eliminating the possibility of power-law inflation.\nHowever, dimensionally reduced higher-dimensional  theories also\npredict numerous scalar fields, and so it is of interest to study\nmodels with multiple scalar fields.\n\nIn the recent work of Liddle, Mazumdar and Schunck~\\cite{Liddle98} the effect of additional scalar fields with\nindependent exponential potentials was considered. They assumed $n$ scalar fields\nin a spatially flat Friedmann-Robertson-Walker (FRW) universe. They found that\nan arbitrary number of scalar fields with\nexponential potentials evolve towards a novel inflationary scaling solution,\nwhich they termed\n{\\em assisted inflation}, in which all of the scalar fields\nscale with one another (and are hence non-negligible asymptotically)\nwith the result that inflation occurs\neven if each of the individual potentials is too steep to support\ninflation on its own.\nThe existence\nof multiple uncoupled scalar fields, each having an exponential potential, could therefore,\nthrough a combined\n(or {\\em assisted}) effort, be a source for power-law inflation. This is true\neven though\neach individual scalar field need not be a source for inflation, and might therefore\nlead to compatibility with supergravity theory.\n\n\nIn a recent dynamical analysis \\cite{Malik98} it was shown that this assisted inflationary solution is a\nlate-time attractor in the class of zero-curvature FRW models. This was done by\nchoosing a redefinition of the fields (a rotation\nin field space) which allows the effective potential\nfor field variations orthogonal to this solution to be written down; in analogy with models\nof\nhybrid inflation~\\cite{hybrid1} it was then shown that\nthis potential has a global minimum along the attractor solution.\nAlso, analytic solutions describing homogeneous and inhomogeneous\nperturbations about the attractor solution without resorting to slow-roll\napproximations were presented in \\cite{Malik98}, and curvature and isocurvature perturbation spectra\nproduced from vacuum fluctuations during assisted inflation were discussed.\n\n\n\n\n\nIn this paper we shall present a\nqualitative analysis of models with the action\n\n\\begin{equation}\n\\label{action}\nS=\\int d^4 x \\sqrt{-g} \\left[ R\n-\\frac{1}{2} \\sum_{i=1}^n (\\nabla\\phi_i^2)\n-V_0 \\sum_{i=1}^{n} e^{k_i \\phi_i}\\right] + S_m,\n\\end{equation}\nwhere $S_m$ is the matter contribution.\nAlmost all previous analyses of multiple scalar field inflationary\nmodels  have assumed zero-curvature\nFRW spacetimes with no matter; here we extend the\nanalysis to include both curvature and matter. In section \\ref{model} we shall\npresent the governing equations for $n$ scalar fields with exponential potentials\nand matter. In section \\ref{twomodel} we shall study the two-scalar field model\n with no matter, and, in particular,\ndiscuss the stability of the two-field assisted inflationary model. In section \\ref{twomodelmatter} we shall\nstudy the two-scalar field model with barotropic matter. In section \\ref{threemodel} we shall discuss three- and\nmulti-scalar field models. In section \\ref{conclusions} we present our conclusions.\n\n\n\\subsection{Generalized Assisted Inflation}\n\n\nRecently, models  with $n \\times m$ scalar fields $\\phi_{ij}$ and containing multiplicative\nexponential terms in the effective potential of the form\n\\[\nV_{eff}\\equiv\\sum_{i=1}^n \\prod_{j=1}^{m}V_0{\\rm e}^{k_{ij}\\phi_{ij}}=\\sum_{i=1}^n V_0^m{\\rm e}^{\\sum_{j=1}^m k_{ij}\\phi_{ij}},\n\\]\nwhere $1\\leq i\\leq n$ and $1\\leq j\\leq m$ and $k_{ij}$ are $n \\times m$\nreal positive constants which are not zero, have also been\nstudied. A qualitative analysis of the $m=1$ case has been given in \\cite{vandenHoogen99d},\nwhere  an analogy was made with the dynamics of\nsoft inflation \\cite{Berkin90}.\n\nIn \\cite{Copeland99} a class of spatially flat FRW\nmulti-scalar fields models with multiplicative exponential potentials were studied.\nPotentials of this form are quite common in dimensionally reduced\nsupergravity models \\cite{stelle,reall}.\nExact two-field and general $n$-field power-law scaling inflationary solutions were obtained,\nwhich were demonstrated to be late-time attractors,  generalizing the\nassisted inflationary solutions previously obtained \\cite{Liddle98};\nthis behaviour was dubbed `generalized assisted inflation'.\nIt was shown that it is more difficult to obtain assisted inflation in\nthese generalized models with cross-couplings between the scalar fields\nin the potential; the fields in any one exponential term tend to conspire to act\nagainst one another rather than assist each other (a result also noticed in\n\\cite{reall}). However,\nas with the original version of assisted inflation,\nthis inhibiting affect can be compensated for\nif there are enough exponential terms present in the potential (i.e. if $n$ is\nlarge enough) \\cite{Liddle98,Copeland99}.\n\nThe dynamics of `generalized assisted inflation' was investigated in more detail in \\cite{GL}.\nBy introducing field rotations, which results in the introduction\nof two orthogonal fields one of which is massless and the\nother posseses an exponential potential \\cite{Malik98,GL},\nthe nature of the late--time\nattractor solution in a particular class of models was determined.\nA dimensionally reduced action resulting from within the context of\na generalised toroidal compactification of higher--dimensional fields\nin Einstein gravity minimally coupled to massless scalar fields was shown\nto give rise to a model of the form under investigation, and it was shown how\nthe addition of interactions between the fields impede\ninflation in this model.\n\nSimilar  behaviour was also noted by Kanti and Olive \\cite{ko}\nin multi-field assisted inflationary models with standard chaotic polynomial (rather than exponential)\npotentials, which can arise in modern Kaluza-Klein\ntheories (and are a natural outcome of the compactification of\nhigher dimensional theories down to four dimensions). Indeed, Kanti and\nOlive \\cite{Kanti99} have recently proposed a possible realisation of assisted inflation based\non the compactification of a five-dimensional Kaluza Klein model, and have shown how\nthe additional fields of the assisted sector actually impede\ninflation (they also showed that the assisted sector, coming\nfrom a Kaluza Klein compactification,  eliminates the need for a fine-tuned quartic\ncoupling to drive chaotic inflation).  In Kaloper and Liddle \\cite{KL} the dynamics of a simple implementation of the idea in Kanti and Olive \\cite{Kanti99} was analyzed in more detail.  Since assisted inflation no longer corresponds to an asymptotic attractor, they found that as inflation proceeds the number of fields participating in the assisted behaviour decreases resulting in the interesting novel feature that the density perturbations generated retain some information about the initial conditions.\n\n\n\n\n\n\n\n\\section{The Model} \\label{model}\n\n\nWe shall assume that the spacetime is spatially homogeneous and isotropic.  The line element for such a spacetime has the form\n\\[ds^2=-dt^2+R^2(t)\\left[\\frac{dr^2}{1-kr^2}+r^2(d\\theta^2+\\sin^2(\\theta)d\\phi^2)\\right]\n\\]\nwhere $k=+1,-1,0$ determines whether the model is  closed (positive-curvature), open (negative-curvature), or flat (zero-curvature).\n\nWe shall consider $n$ scalar fields $\\phi_{i}$, where $1\\leq i\\leq n$, in which the effective potential has the form\n\\[\nV_{eff}\\equiv\\sum_{i=1}^n V_0{\\rm e}^{k_{i}\\phi_{i}},\n\\]\nwhere the $k_{i}$ are real non-zero positive constants.  We also assume that there exists a non-interacting perfect fluid\nwith density $\\rho$ and pressure\n\\begin{equation}\np=(\\gamma-1)\\rho,\\label{es}\n\\end{equation}\nand we shall assume that $1\\leq\\gamma<2$.\nThe Einstein field equations, the conservation equations, together with the Klein-Gordon equations for the scalar fields,\n yield the following autonomous system of ordinary differential equations:\n\\begin{eqnarray}\nH^2-\\frac{1}{6}\\left(\\sum_{i=1}^{n}\\dot\\phi_{i}^2\\right)-\\frac{1}{3}V_{eff}-\\frac{1}{3}\\rho=-\\frac{^3\\!R}{6}, \\nonumber\\\\\n\\dot{H}=-H^2 - \\frac{1}{3}\\left(\\sum_{i=1}^{n}\\dot\\phi_{i}^2\\right)+\\frac{1}{3} V_{eff} -\\frac{1}{6}(3\\gamma-2)\\rho,\\label{system1}\\\\\n\\dot \\rho=-3\\gamma H \\rho, \\nonumber \\\\\n\\ddot\\phi_{i}+3H\\dot\\phi_{i}+k_{i}V_0e^{k_{i}\\phi_{i}}=0,\\nonumber\n\\end{eqnarray}\n where $^3\\!R=k\/R^2$ is the curvature of the spacelike hypersurfaces, $H=\\dot R\/R$ is the Hubble expansion, and an overdot\n represents differentiation with respect to coordinate time $t$.  Units have been chosen so that $8\\pi G = c =1$.\n\n\nTo analyze the dynamical system given in equation (\\ref{system1}) we transform the system using expansion normalized variables.  Expansion normalized variables have had great success in analyzing the asymptotic behaviour of many cosmological models.  See \\cite{Coley97a,WainwrightEllis97} for arguments in support of using dimensionless expansion normalized variables.  One primary reason involves the decoupling of one of the differential equations, which effectively reduces the dimension of the system by one, and, in some cases, compactifies the phase space.  We choose expansion normalized variables of the form\n\\begin{eqnarray}\n\\Omega=\\frac{\\rho}{3H^2},\\qquad\n\\Phi_i=\\frac{\\sqrt{V_0}{\\rm e}^{k_{i}\\phi_{i}\/2}}{\\sqrt{3}H},\\qquad\n\\Psi_{i}=\\frac{\\dot\\phi_{i}}{\\sqrt{6}H},\\qquad \\frac{dt}{d\\tau}=\\frac{1}{H}.\\label{expansion}\n\\end{eqnarray}\n  The resulting dynamical system describing these perfect fluid multiple scalar field models becomes\n\\begin{eqnarray}\n\\frac{d\\Omega}{d\\tau} &=& \\Omega(2q-3\\gamma+2),\\nonumber \\\\\n\\frac{d\\Psi_{i}}{d\\tau} &=& \\Psi_{i}(q-2)-\\frac{\\sqrt{6}}{2}k_{i}\\Phi_{i}^2,\\label{system}\\\\\n\\frac{d\\Phi_i}{d\\tau} &=& \\Phi_i\\left(q + 1 + \\frac{\\sqrt{6}}{2}k_{i}\\Psi_{i} \\right),\\nonumber\n\\end{eqnarray}\nfor $(1\\leq i \\leq n)$, where the deceleration parameter has the following form\n\\[ q = \\frac{(3\\gamma-2)}{2}\\Omega +2\\sum_{i=1}^{n}\\Psi_{i}^2-\\sum_{i=1}^{n}\\Phi_i^2,\n\\]\nand\n\\[ \\frac{^3R}{6 H^2}=-1 +\\Omega +\\sum_{i=1}^{n}\\Psi_{i}^2+\\sum_{i=1}^{n}\\Phi_i^2.\n\\]\n\n\nAssuming a non-negative energy density (i.e., $\\Omega\\geq0$) and\nif $^3\\!R \\leq 0$, (i.e., in the negative and zero-curvature cases) the phase space for the dynamical system\nin the expansion normalized variables $(\\Omega,\\Phi_i,\\Psi_{i})$ is compact.  If $^3R>0$ (i.e., in the positive\ncurvature case) then the transformation given by equation (\\ref{expansion}) becomes singular when $H=0$.  Here we\nshall only make some partial comments with regards to the asymptotic behaviour of the positive curvature models.\nAll of the equilibrium points correspond to self-similar cosmological models and hence to power-law solutions\n\\cite{Billyard99}.\n\n\n\n\n\n\n\\section{Qualitative Analysis of Two-Scalar field Model} \\label{twomodel}\n\n We shall first discuss the dynamics of the model with only two minimally coupled scalar fields\n and with no matter.\n  We obtain this model by setting $n=2$ and $\\Omega=0$ in (\\ref{system}).\nIn this case we obtain the four-dimensional dynamical system\ngiven by:\n\\begin{eqnarray}\n\\frac{d\\Psi_{1}}{d \\tau}&=&\\Psi_{1}(q-2)-\\frac{\\sqrt{6}}{2}k_{1}{\\Phi_1}^2 \\nonumber\\\\\n\\frac{d\\Psi_{2}}{d \\tau}&=&\\Psi_{2}(q-2)-\\frac{\\sqrt{6}}{2}k_{2}{\\Phi_2}^2 \\nonumber\\\\\n\\frac{d\\Phi_{1}}{d \\tau}&=&\\Phi_{1}(q+1+\\frac{\\sqrt{6}}{2}k_1\\Psi_{1}) \\nonumber\\\\\n\\frac{d\\Phi_{2}}{d \\tau}&=&\\Phi_{2}(q+1+\\frac{\\sqrt{6}}{2}k_2\\Psi_{2})\\label{DS2}\n\\end{eqnarray}\nwhere\n$$q=2\\Psi_{1}^2+2\\Psi_{2}^2-\\Phi_{1}^2-\\Phi_{2}^2$$\nand\n\\[\\frac{^3R}{6 H^2}=-1+\\Psi_{1}^2+\\Psi_{2}^2+\\Phi_{1}^2+\\Phi_{2}^2.\\]\n\n\nIt is possible to choose simplified variables as in \\cite{LuPope} via a rotation\nin field space; although this would simplify the analysis of the assisted inflationary\nsolution, it would perhaps be more difficult to discribe all of the qualitative properties\nof the models and relate this analysis to previous work.\n\n\n\n\\subsection{Assisted Inflation}\n\nThe flat {\\em Assisted Inflation} model \\cite{Liddle98} corresponds to\nthe equilibrium point $A$ of the system (\\ref{DS2}) given by\n\n\\[ \\{\\Psi_1, \\Psi_2, \\Phi_1, \\Phi_2\\}^A \\equiv \\nonumber\\]\n\\begin{equation}\n       \\{-\\frac{k_1k_2^{\\ 2}}{\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})},\n     -\\frac{k_1^{\\ 2}k_2}{\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})},\n     k_2\\frac{\\sqrt{6(k_1^{\\ 2}+k_2^{\\ 2})-k_1^{\\ 2}k_2^{\\ 2}}}\n     {\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})},\n     k_1\\frac{\\sqrt{6(k_1^{\\ 2}+k_2^{\\ 2})-k_1^{\\ 2}k_2^{\\ 2}}}\n     {\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})} \\}, \\label{AAA}\n\\end{equation}\nwhich is equivalent to\n\\[ \\{-\\frac{K^2}{\\sqrt{6}k_1},-\\frac{K^2}{\\sqrt{6}k_2},\\frac{\\sqrt{K^2(6-K^2)}}{\\sqrt{6}k_1},\\frac{\\sqrt{K^2(6-K^2)}}{\\sqrt{6}k_2}\\}\\]\nwhere\n\\[K^{-2}\\equiv\\frac{1}{{k_1}^{2}}+\\frac{1}{{k_2}^{2}} .\\]\nThe deceleration parameter for this solution is given by\n\n\\begin{eqnarray}\n q_A &\\equiv& \\frac{k_1^{\\ 2}k_2^{\\ 2}-2(k_1^{\\ 2}+k_2^{\\ 2})}{2(k_1^{\\ 2}+k_2^{\\ 2})}\\nonumber \\\\\n&=& \\frac{K^2-2}{2}\\label{qA}\n\\end{eqnarray}\nand hence this solution, with\n\\[R(t) \\propto t^p\\nonumber\\]\nand\n \\[ k_1 \\phi_1 = k_2 \\phi_2,\\nonumber \\]\nis inflationary ($q_A<0$) if\n\\begin{equation}\n p \\equiv 2 \\sum_{i=1}^2 \\frac{1}{{k_i}^2} = 2K^{-2}= \\frac{1}{1+q_A} > 1; \\qquad\n 2 \\geq K^{2}. \\label{ppp}\n\\end{equation}\n\nSince a single scalar field can only give rise to an inflationary power-law solution\nif $\\frac{1}{{k_i}^2}> \\frac{1}{2}$ for $i=$ 1 or 2 \\cite{Wetterich,Billyard99}, this means that the two-scalar field model can be inflationary\neven when the each of the individual potentials is too steep for the corresponding single scalar field model to inflate\n(and hence the terminology {\\em assisted} inflation).\nThe eigenvalues corresponding to the equilibrium point $A$ are given by  (See Appendix for details.)\n\\begin{equation}\nK^2-2, \\quad \\frac{K^2-6}{2}, \\quad \\frac{1}{4}\\left((K^2-6)\\pm\\sqrt{(K^2-6)^2+8K^2(K^2-6)}\\right) \\label{eigen}\n\\end{equation}\nHence this equilibrium point is stable when (\\ref{ppp}) is satisfied, and so the\ncorresponding assisted inflationary solution is a late-time attractor \\cite{Malik98}.\n\n\n\n\n\\subsection{Stability of Equilibria}\n\nWe note that several of the equilibrium points occur in the three-dimensional invariant set\ncorresponding to\nthe zero-curvature models\ndefined by\n\n\\[1=\\Psi_{1}^2+\\Psi_{2}^2+\\Phi_{1}^2+\\Phi_{2}^2.\\]\nWhen matter is included, there exists a monotonic function so that in the full\ndynamical phase space there can be no periodic or recurrent orbits\nand the global dynamics can be determined.\nThis implies that the  qualitative\nfeatures  described in this\nsection can be more rigorously proven.\nAll of the equilibrium points and their corresponding eigenvalues are listed in Table \\ref{table1}.\nUsing this  this table let us discuss the local stability of these equilibrium points.\n\n\nAs noted above the equilibrium point $A$, given by (\\ref{AAA}), corresponds to the assisted\ninflationary solution. It exists for all parameter values satisfying\n\n\\begin{equation} \\frac{1}{6} < \\frac{1}{{k_1}^{2}} + \\frac{1}{{k_2}^{2}}, \\label{e1} \\end{equation}\nand is a sink (late-time attractor) for all parameter values satisfying (\\ref{ppp}) (else it is\na saddle).\n\nThere are two equilibrium points, denoted by $P_1$ and $P_2$, whose coordinate values and\nassociated eigenvalues are given in Table \\ref{table1}, which correspond to zero-curvature\npower-law solutions in which one scalar field (either $\\phi_1$ or $\\phi_2$, respectively)\nis negligible; these solutions exist if $\\frac{1}{6} < \\frac{1}{k_i^2}$ and are inflationary if,\nin addition, $\\frac{1}{2} < \\frac{1}{k_i^2}$ (for each $i=1,2$, respectively)\nand correspond to the well-known single scalar field power-law solutions \\cite{Luchin85,Wetterich}.\nFrom Table \\ref{table1} we see that each $P_i$ has two negative eigenvalues and one positive\neigenvalue for all relevent\nparameter values and an additional eigenvalue which is negative if $k_i^2<2$ (and positive for\n$2<k_i^2<6$); hence these points are saddles and have a one- or two-dimensional unstable manifold\ndepending upon whether $k_i^2<2$ or $k_i^2>2$, respectively.\n\n\n\nThere also exist equilibrium points, denoted by $CS_1$, $CS_2$ and $CS$,\n whose coordinate values and the\nassociated eigenvalues are given in Table \\ref{table1}.\nThe solutions correspond to power-law solutions in which the curvature scales with the first scalar field, the second scalar field or both,\nrespectively. The single-field curvature scaling equilibrium points $CS_1$ and $CS_2$ are both saddles.\nThe two-field curvature scaling equilibrium point $CS$ is a sink whenever\n$\\frac{1}{2} > \\frac{1}{k_1^2} + \\frac{1}{k_2^2}$ (otherwise a saddle).  Whenever the two-field curvature scaling solution is stable, it necessarily has negative curvature.\n\nThere is an equilibrium point, denoted by $M$, corresponding to the Milne form of flat spacetime,\nwhich is always a saddle.\n\nFinally, there is a one-dimensional set of equilibrium points parametrized by $\\Psi_0$, denoted by\n$MSF$, corresponding to zero-curvature massless scalar field models (in which both potentials are\nzero). There is one zero eigenvalue corresponding to the fact that there is a one-dimensional set of equilibrium points.\nThere are values for $\\Psi_0$ for which the remaining three eigenvalues are positive and hence a subset\nof $MSF$ are sources (the remainder are saddles). These correspond to well-known early-time attracting massless scalar\nfield  models \\cite{Billyard99}.\n\n\n\\subsection{Discussion}\n\n\nFrom the analysis above we conclude that the two-field assisted inflationary solution $A$ is the global attractor when\n$\\sum_{i=1}^2 k_i^{-2} > \\frac{1}{2}$ and the two-field curvature scaling solution $CS$ is the global\nattractor when\n$\\sum_{i=1}^2 k_i^{-2} < \\frac{1}{2}$. The massless scalar field solutions $MSF$ are always the\nearly-time attractors.\n\nIn all cases both scalar fields are non-negligible in generic late-time behaviour. This is contrary to the\ncommonly held belief that in multi-field models with exponential potentials the scalar field\nwith the shallowest potential (i.e., smallest value of $k$) would dominate at late times. Indeed, we have shown that the single\nfield power-law inflationary models always correspond to saddles, so that we have the rather surprising\nresult that generically a single scalar field model {\\em never} dominates at late-times!\n\nWe note that both the assisted inflationary solution and the massless scalar field early-time attractors\ncorrespond to zero-curvature models. However, the curvature is not always dynamically negligible asymptotically\nbecause the two-field curvature scaling solution has non-zero curvature.\n\nThere is a range of parameter values for which the assisted\ninflationary solution is the global late-time attractor (when the\nsolution is non-inflationary it corresponds to a saddle). For all\nof these parameter values the single field power-law solutions\n$P_1$ and $P_2$ are saddles. However, there are allowable\nparameter values for which either $P_1$ and $P_2$ are both\ninflationary, or one is inflationary while the other is not, or\nand both are non-inflationary. This might give rise to some new\ninteresting physical scenarios. For example, a model could\nasymptote towards an inflationary single field solution $P_i$,\nstay close to $P_i$ for an arbitrarily long period of time (since\n$P_i$ is an equilibrium point) inflating all the time, and then\neventually leave $P_i$ and evolve towards the stable attracting\ninflationary solution $A$. (Note that if either of $P_1$ or $P_2$\nare inflationary, then $A$ is necessarily inflationary --see\n\\ref{ppp}). This is akin to a double-inflationary model\n\\cite{doubleinflate} in which the density fluctuations on large\nand small scales decouple (i.e., the scale invariance of the\nspectrum is broken) thereby allowing the possibility of more on\nlarge scales which is in better accord with observations.\n\n\n\n\n\n\n\\section{Qualitative Analysis of Two-Scalar field Model with Matter} \\label{twomodelmatter}\n\n To understand the underlying dynamics of the model with matter (i.e., with $\\Omega\\not=0$) we shall\n shall study the model with two minimally coupled scalar fields together with matter having\n  energy density $\\rho$ with the barotropic equation of state given by (\\ref{es}).  This model\nis obtained  by setting $n=2$ in (\\ref{system}), whence we obtain the five-dimensional dynamical system  given by:\n\\begin{eqnarray}\n\\frac{d\\Omega}{ d\\tau} &=& \\Omega(2q-3\\gamma+2) \\nonumber\\\\\n\\frac{d\\Psi_{1}}{d \\tau}&=&\\Psi_{1}(q-2)-\\frac{\\sqrt{6}}{2}k_{1}{\\Phi_1}^2 \\nonumber\\\\\n\\frac{d\\Psi_{2}}{d \\tau}&=&\\Psi_{2}(q-2)-\\frac{\\sqrt{6}}{2}k_{2}{\\Phi_2}^2 \\nonumber\\\\\n\\frac{d\\Phi_{1}}{d \\tau}&=&\\Phi_{1}(q+1+\\frac{\\sqrt{6}}{2}k_1\\Psi_{1}) \\nonumber\\\\\n\\frac{d\\Phi_{2}}{d \\tau}&=&\\Phi_{2}(q+1+\\frac{\\sqrt{6}}{2}k_2\\Psi_{2})\n\\end{eqnarray}\nwhere\n$$q=\\frac{3\\gamma-2}{2}\\Omega+2\\Psi_{1}^2+2\\Psi_{2}^2-\\Phi_{1}^2-\\Phi_{2}^2$$\nand\n\\[\\frac{^3R}{6 H^2}=-1+\\Omega+\\Psi_{1}^2+\\Psi_{2}^2+\\Phi_{1}^2+\\Phi_{2}^2\\]\n\n\n\n\\subsection{Invariant Sets and Monotonic Functions}\nThe zero-curvature models\nconstitute  a four-dimensional invariant set. The models with no matter also\nconstitute a four-dimensional invariant set.\n\nThe function\n\\begin{equation}\nW=\\frac{\\Omega^2}{(\\Omega+\\Psi_{1}^2+\\Psi_{2}^2+\\Phi_{1}^2+\\Phi_{2}^2-1)^2}\n\\end{equation}\nhas derivative\n\\begin{equation}\n\\frac{d W}{d\\tau}=2(2-3\\gamma)W.\n\\end{equation}\nWe observe that this function is monotonic when $\\Omega\\not = 0$  (i.e., non-zero matter)\nand $(\\Omega+\\Psi_{1}^2+\\Psi_{2}^2+\\Phi_{1}^2+\\Phi_{2}^2-1)\\not=0$ (i.e., non-zero curvature).\nWe also observe that the sign of $3\\gamma-2$ signifigantly changes the dynamics of these models.\nFor example, in the case of interest here  $3\\gamma-2>0$, whence $W$ is a decreasing function of time $\\tau$.\n This immediately implies that\n\\begin{itemize}\n\\item There exist no periodic or recurrent orbits in the full five-dimensional phase space\n(this does not preclude the existence of closed orbits in the\ninvariant sets $\\Omega=0$ and $^3R=0$;\nhowever, we shall be primarily concerned with the dynamics of the models in the complete phase-space\nwith matter and non-zero curvature).\n\\item The future asymptotic state lies within the invariant set $\\Omega=0$.\nMatter becomes dynamically unimportant to the future.\n\\item The past asymptotic state lies within the set of zero-curvature models.\n\\end{itemize}\n\n\n\n\\subsection{Stability of Equilibria}\n\nThe equilibrium points can be classisfied into two sets; those with $\\Omega=0$ and those with $\\Omega\\not = 0$.\nAll equilibrium points listed in Table \\ref{table1} exist in this case with $\\Omega=0$, and\nTable \\ref{table2} lists the equilibrium points with $\\Omega=0$ together with the additional eigenvalue due to the addition of matter.\nUsing the function $W$ above,\nwe can further conclude that those equilibrium points in the\nset $\\Omega\\not=0$ necessarily must have zero-curvature. Table \\ref{table3}\nlists the eigenvalues found in the invariant set $\\Omega \\not =0$.\n\nLet us focus on the stability of the attractors in the full physical phase space. All late-time\nattractors (sinks) occur in the invariant set $\\Omega=0$. In the previous section we found that $A$ and $CS$\nare the only sinks in the invariant set $\\Omega=0$\n(clearly, all of the saddles remain saddles in the full five-dimensional phase space). The additional eigenvalue\nfor the equilibrium point $A$ in the full physical phase space is given in Table \\ref{table2} and is negative\nif $\\sum_{i=1}^2 k_i^{-2} > \\frac{1}{3 \\gamma}$. But this is always satisfied when\n$\\sum_{i=1}^2 k_i^{-2} > \\frac{1}{2}$ and $\\gamma > \\frac{2}{3}$,\nand hence $A$ is a sink and assisted inflation\nis a global attractor.\nSimilarly, from Table \\ref{table2} the equilibrium point $CS$ is always a sink for\n$\\sum_{i=1}^2 k_i^{-2} < \\frac{1}{2}$ and hence the two-field curvature scaling solution remains the global\nattractor in this case.\n\n\nThe early-time attractors lie in the zero-curvature invariant set and\nconsist of massless scalar field models. From Table \\ref{table2} we see that\nthe massless scalar field models corresponding to the  repelling\nequilibrium points $MSF$ are always sources (for $\\gamma<2$).\n\n\n\\subsection{Matter Scaling Solutions}\n\nIn the case of a single scalar field\nthere exist zero-curvature FRW `matter scaling' solutions when the exponential potential is too steep to drive inflation,\nin which the scalar field energy\ndensity tracks that of the perfect fluid (so that at late\ntimes neither field is neglible) \\cite{Wetterich}.  In \\cite{Copeland98}\nit was shown that whenever these matter scaling\nsolutions exist they are the unique late-time attractors\nwithin the class of flat\nFRW models .  The cosmological consequences of these scaling\nmodels have been further studied in \\cite{FJ}.  For example, in\nthese models the scalar field energy density tracks that of the\nperfect fluid and a significant fraction of the current energy\ndensity of the Universe may be contained in the homogeneous scalar field\nwhose dynamical effects mimic cold dark\nmatter;  the tightest constraint on these cosmological models comes from\nprimordial nucleosynthesis bounds on any such relic density\n\\cite{Wetterich,Copeland98,FJ}.\nThe stability of these flat, isotropic matter scaling solutions was studied within\nthe class of spatially homogeneous cosmological models with a barotropic perfect\nfluid and a scalar field with an exponential\npotential in \\cite{Billyard98}. It was found that\nwhile the matter scaling solutions are stable to shear\nperturbations, for\nrealistic matter with $\\gamma \\ge 1$ they are\nunstable to curvature perturbations.\n\n\n\nReturning to the models under investigation here, none of the equilibrium points with $\\Omega\\not=0$\ncan be late-time attractors for $\\gamma > \\frac{2}{3}$. Indeed, from Table \\ref{table3} all such equilibrium points are seen to be\nsaddles. In particular, the two-field matter scaling solution corresponding to the equilibrium point\n$MS$, which exists for\n$\\sum_{i=1}^2 k_i^{-2} < \\frac{1}{3 \\gamma}$, is a saddle. From Table \\ref{table3}\nwe see that the first eigenvalue associated with $MS$ is positive, while the real parts of the remaining\nfour eigenvlaues are all negative. This is consistent with the stability analysis of\nmatter scaling solutions in models with a single scalar field  which found that the models were unstable to curvature\nperturbations when $\\gamma> \\frac{2}{3}$ \\cite{Billyard98}. However, these two-field\nmatter scaling solutions may still be of physical import. We note that when the curvature\nis zero, the two-field matter scaling solution is an attractor (all four eigenvalues of $MS$ in the\nfour-dimensional zero-curvature invariant set have negative real parts -- so that $MS$ is a sink\nin this invariant set), as in the case for the matter scaling solution in a single field model.\nNote also from Table \\ref{table3} that both of the single-field\nmatter scaling solutions, corresponding\nto the equilibrium points $MS_1$ and $MS_2$, have two positive eigenvalues, so that again the solution\nwith multiple scalar fields is the `stronger' attractor.\n\n\n\n\n\n\n\\section{Qualitative Analysis of Three-Scalar Field Model} \\label{threemodel}\n\nLet us now consider models with more than two scalar fields. For simplicity, we\nshall exclude a matter term here. However, from the previous section we easily determine\nthe essential properties resulting from the inclusion of a matter field. In particular, in this case\na monotonic function exists and this enables us to prove the qualitative results outlined below.\nLet us begin with the three-scalar-field model,\nobtained by setting $n=3$ and $\\Omega=0$ in (\\ref{system}).\nIn this case the resulting six-dimensional dynamical system is\ngiven by:\n\\begin{eqnarray}\n\\frac{d\\Psi_{1}}{d \\tau}&=&\\Psi_{1}(q-2)-\\frac{\\sqrt{6}}{2}k_{1}{\\Phi_1}^2 \\nonumber\\\\\n\\frac{d\\Psi_{2}}{d \\tau}&=&\\Psi_{2}(q-2)-\\frac{\\sqrt{6}}{2}k_{2}{\\Phi_2}^2 \\nonumber\\\\\n\\frac{d\\Psi_{3}}{d \\tau}&=&\\Psi_{3}(q-2)-\\frac{\\sqrt{6}}{2}k_{2}{\\Phi_3}^2 \\nonumber\\\\\n\\frac{d\\Phi_{1}}{d \\tau}&=&\\Phi_{1}(q+1+\\frac{\\sqrt{6}}{2}k_1\\Psi_{1}) \\nonumber\\\\\n\\frac{d\\Phi_{2}}{d \\tau}&=&\\Phi_{2}(q+1+\\frac{\\sqrt{6}}{2}k_2\\Psi_{2})\\nonumber\\\\\n\\frac{d\\Phi_{3}}{d \\tau}&=&\\Phi_{3}(q+1+\\frac{\\sqrt{6}}{2}k_2\\Psi_{3})\\label{DS3}\n\\end{eqnarray}\nwhere\n$$q=2\\Psi_{1}^2+2\\Psi_{2}^2+2\\Psi_{3}^2-\\Phi_{1}^2-\\Phi_{2}^2-\\Phi_{3}^2$$\nand\n\\[\\frac{^3R}{6 H^2}=-1+\\Psi_{1}^2+\\Psi_{2}^2+\\Psi_{3}^2+\\Phi_{1}^2+\\Phi_{2}^2+\\Phi_{3}^2\\]\n\n\n\nAgain it would possible to choose simplified variables as in \\cite{LuPope} via a rotation\nin field space as was done in recent work \\cite{Malik98,GL}. However, we shall not do this\nhere. Indeed, we shall not present a complete qualitative analysis similar to that done in section \\ref{twomodel},\nsince the essential features are similar and the detailed analysis would be long and painful.\nRather, let us describe the main effects of including a third scalar field on the inflationary solutions.\n\nThere exists a zero-curvature assisted inflationary solution which\nnow corresponds to the equilibrium point given by\n\n\\[   \\{ \\Psi_i=-\\frac{K^{2}}{\\sqrt{6}k_i},\n \\Phi_i=\\frac{\\sqrt{K^{2}(6-K^{2})}}{\\sqrt{6}k_i} \\}  \\]\nwhere\n\\[K^{-2}\\equiv k_1^{-2} + k_2^{-2} +k_3^{-2}. \\]\nIn this solution all of the three scalar fields scale together at late times.\nThe corresponding eigenvalues are\n\n\\begin{eqnarray*}\nK^{2}-2, \\frac{K^{2}-6}{2},\n\\frac{1}{4}\\left((K^{2}-6)\\pm\\sqrt{(K^{2}-6)^2+8K^{2}(K^{2}-6)}\\right),\\\\\n\\frac{1}{4}\\left((K^{2}-6)\\pm\\sqrt{(K^{2}-6)^2+8K^{2}(K^{2}-6)}\\right).\n\\end{eqnarray*}\nIt is known \\cite{Liddle98} to be a stable late-time attractor for all parameter values\nfor which the solution is inflationary (i.e., $K^{2}<2$; recall the\npoint does not exist if $K^{2}>6$).\n\nThere are three solutions in which two scalar fields scale together asymptotically and the third is negligible.\nAssuming that the third scalar field is zero ($\\Psi_{3}=\\Phi_{3}=0$), the coordinates of the corresponding\nequilibrium point, denoted by $P_{120}$, are given by\n\n\n\n\n\\begin{equation}\n      \\{-\\frac{k_1k_2^{\\ 2}}{\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})},\n     -\\frac{k_1^{\\ 2}k_2}{\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})}, 0,\n     k_2\\frac{\\sqrt{6(k_1^{\\ 2}+k_2^{\\ 2})-k_1^{\\ 2}k_2^{\\ 2}}}\n     {\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})},\n     k_1\\frac{\\sqrt{6(k_1^{\\ 2}+k_2^{\\ 2})-k_1^{\\ 2}k_2^{\\ 2}}}\n     {\\sqrt{6}(k_1^{\\ 2}+k_2^{\\ 2})}, 0 \\}. \\label{BBB}\n\\end{equation}\nFour of the eigenvalues are given by (\\ref{eigen}), which all have negative real parts.\n\n\n\nThere are three solutions in which one scalar field scale dominates asymptotically and the remaining two are negligible.\nAssuming that the first scalar field is non-zero ($\\Psi_{1} \\ne 0 \\ne \\Phi_{1}$), the coordinates of the corresponding\nequilibrium point, denoted by $P_{100}$, are given by\n\n\n\\[ \\{-\\frac{k_1}{\\sqrt{6}}, 0 , 0, \\sqrt{1-\\frac{k_1^{\\ 2}}{6}}, 0, 0 \\}. \\]\nTwo of the eigenvalues are negative, one is positive, and there is an additional eigenvalue which\nis negative if $k_1^2<2$ and positive if $2<k_1^2<6$.\n\nIn both of these cases the additional (remaining) two eigenvlaues can be calculated and are given by\n\\[ \\{q-2<0, q+1>0 \\}, \\]\nwhere $q$ is the deceleration parameter evaluated at the equilibrium point. Hence, the point $P_{120}$ is a saddle with one eigenvalue\nwith positive real part. The equilibrium points denoted by $P_{103}$ and $P_{023}$ are also saddles with one eigenvalue\nwith positive real part. In additional, the point $P_{100}$ is a saddle with two eigenvalues with positive real\nparts (if $k_i^2<2$, and three eigenvalues with positive real parts if $k_i^2>2$). The same is true for the\nequilibrium points denoted by $P_{020}$ and $P_{003}$.\n\nConsequently there is a nested set of equilibrium points. At the top is the stable three-scalar field\nassisted inflationary solution. In the next layer there are three two-scalar field models which are saddles\nwith  one eigenvalue\nwith positive real part. In the final layer there there are three one-scalar field models which are saddles\nwith  two eigenvalues\nwith positive real parts (or three eigenvalues\nwith positive real parts). Associated with this dynamical nesting are cosmological models with very\ninteresting physical properties.\n\n\nThis will follow through in the case of $n$ scalar fields. There will be a unique stable $n$-scalar field\nassisted inflationary solution. There will then be $n$ of the ($n-1$)-scalar field models which are saddles\nwith  one eigenvalue\nwith positive real part. There will be $\\frac{1}{2} n(n-1)$ of the ($n-2$)-scalar field models which are saddles\nwith  two eigenvalues\nwith positive real parts. And so on. Finally, there will be $n$ of the ($1$)-scalar field models which are saddles\nwith  $n-1$ (or $n-2$) eigenvalues\nwith positive real parts. As one `goes up' the nested structure the equilibrium points respectively become\n`stronger attractors' (i.e., the stable manifold of the equilibrium points increases in dimension).\n\nThere is also a three-field curvature scaling solution corresponding to the\nequilibrium point given by\n\n\\[ \\{\\Psi_i=-\\frac{2}{\\sqrt{6}k_i},\\Phi_i=\\frac{2}{\\sqrt{3}k_i}\\} \\]\nwhose associated eigenvalues are given by\n\n\\[ -1\\pm\\sqrt{1+4K^{-2}(2-K^{2})} , -1\\pm\\sqrt{3}i , -1\\pm\\sqrt{3}i. \\]\nThis equilibrium point is a\nsink whenever $K^{2}>2$, in which case it represents an FRW model with negative\ncurvature $\\frac{2-K^{2}}{K^{2}}$\n (else it is a saddle and represents a positive\ncurvature model).\n\n\nFinally, there are saddle equilibrium points corresponding to the Milne model and the\none- and two-field\ncurvature scaling solutions, and a set of equilibrium points with\n$\\{\\sum_{i=1}^{3}\\Psi_i^{\\ 2}=1,\\Phi_i=0\\}$ corresponding to massless scalar\nfield\nmodels, a subset of which are sources.\n\nA complete qualitative analysis can be done for $n$ scalar field models.\nAll of these results can be proven by induction (see, for example, \\cite{Malik98}).\nThe $n$-scalar field\nassisted inflationary solution is given by \\cite{Liddle98}\n\n\\[R(t) \\propto t^p\\nonumber\\]\nand\n \\[ k_i \\phi_i = k_j \\phi_j; \\forall 1 \\le i \\ne j \\le n,\\nonumber \\]\nand\n\n\\[ p \\equiv 2 \\sum_{i=1}^n \\frac{1}{{k_i}^2} > 1. \\]\nWe note that in the two-scalar field model, although inflation can occur for potentials that are steeper\nthan in the single-field case, it cannot occur for arbitrarily steep potentials. For example,\nif $k_1 = k_2 \\equiv k$, then inflation occurs if $k^2<4$. However, for $n$-fields, if\n$k_i = k$ for all $i$, then inflation occurs if $k^2<2n$; e.g., $k^2<8$ for four scalar field models.\n\n\n\n\\section{Conclusion} \\label{conclusions}\n\nWe have studied multi-scalar field FRW cosmological models with\nexponential potentials, extending previous analysis by including\nnon-zero curvature and barotropic matter. We have used dynamical\nsystems techniques, and by establishing a monotonic function in\nthe complete dynamical phase space (which includes both matter and\ncurvature), we have been able to deduce global results.\n\n\nIn section \\ref{twomodel} a comprehensive qualitative analysis was presented in the case of two scalar fields\nwith no matter.\nWe concluded that the two-field assisted inflationary solution $A$ is the global attractor when\n$\\sum_{i=1}^2 k_i^{-2} > \\frac{1}{2}$ and the two-field curvature scaling solution $CS$ is the global\nattractor when\n$\\sum_{i=1}^2 k_i^{-2} < \\frac{1}{2}$. A subset of the massless scalar field solutions $MSF$ are always the\nearly-time attractors.\nConsequently, we found that in all cases both scalar fields are non-negligible in generic late-time behaviour; this\nis an interesting and unexpected result and is contrary to the\ncommonly held belief that in multi-field models with exponential potentials the scalar field\nwith the shallowest potential would dominate at late times (indeed, we have shown that the single\nfield power-law inflationary models always correspond to saddles).\nWe note that both the assisted inflationary solution and the massless scalar field early-time attractors\ncorrespond to zero-curvature models. However, the curvature is not always dynamically negligible asyptotically\nbecause the two-field curvature scaling solution has non-zero curvature.\n\n\n\nThe zero-curvature assisted inflationary FRW scaling solutions \\cite{Liddle98} are of particular importance\nsince, through the combined effect of multiple uncoupled scalar fields, each having an exponential potential,\npower-law inflation is possible\neven when\neach individual scalar field need not be a source for inflation.\nWe have discussed the stability of the two-field assisted inflationary model, and generalized previous results by\nincluding non-zero curvature to\nshow that for an appropriate\nrange of parameter values the assisted inflationary solution is the global late-time\nattractor. For these parameter\nvalues the single field power-law solutions $P_1$ and $P_2$ were shown to be saddles, and we showed that there are allowable parameter\nvalues for which either $P_1$ and $P_2$ are both inflationary, or one is inflationary while the other is not, or\nand both are non-inflationary, perhaps leading to\nnew interesting physical scenarios.\n\n\nIn section \\ref{twomodelmatter} we\nstudied the two-scalar field model with barotropic matter. A monotonic function was established\nin the resulting phase space. This proved that the matter must be negligible at late times and we\nfound that\n$A$ and $CS$ are the only global sinks and that consequently assisted inflation and the two-field curvature scaling solution\nare the global late-time attractors in their appropriate respective parameter ranges. This confirmed the result\nthat both scalar fields must be dynamically non-negligible in generic late-time behaviour, and\nestablishes the  stability of the two-field assisted inflationary model when matter is included.\nThe monotonic function also shows that\nthe early-time attractors lie in the zero-curvature invariant set, and we showed that they\nconsist of a subset of the massless scalar field models.\n\nFor $\\gamma > \\frac{2}{3}$, all of the equilibrium points with\n$\\Omega\\not=0$ were shown to be saddles (see Table \\ref{table3}).\nThe two-field matter scaling solution corresponding to the\nequilibrium point $MS$ was shown to have a single positive\neigenvalue. Both of the single-field matter scaling solutions,\ncorresponding to the equilibrium points $MS_1$ and $MS_2$, were\nshown to have two positive eigenvalues, so that again the solution\nwith multiple scalar fields is the `stronger' attractor. We note\nthat when the curvature is zero, the two-field matter scaling\nsolution is the late-time attractor, consistent with the stability\nanalysis in \\cite{Billyard98}. These matter scaling solutions, and\nparticularly the two-field matter scaling solutions, give rise to\nnew transient dynamical behaviour and may be of physical import.\nFor example, there are solutions which spend a period of time with\nthe scalar field mimicking the barotropic fluid in which there is\na non-negligible scalar field (dark matter) energy density\n(corresponding to a matter scaling saddle equilibrium point) and\nsubsequently evolve towards a scalar-field dominated power-law\ninflationary epoch (corresponding to a single-field saddle\nequilibrium point or a two-field assisted inflationary attractor)\nwith an accelerated expansion, perhaps explaining current high\nredshift data.\n\n In section \\ref{threemodel} we discussed three- and\nmulti-scalar field models (where, for simplicity, a matter term\nwas excluded). In the three-scalar field model we again\nestablished the assisted inflationary solution and three-field\ncurvature scaling solutions as the stable late-time attractors. We\nthen considered  $n$-scalar field models, and established a nested\nstructure for the $m$-field scaling (assisted inflationary)\nsolutions. The $n$-scalar field assisted inflationary solution is\nagain the late-time attractor. All of the $m$-field (with $m<n$)\nscaling solutions are saddles and in general the equilibrium\npoints corresponding to the $m$-field scaling solutions will have\n$n-m$ eigenvalues with positive real parts so that the equilibrium\npoints corresponding to the greater number of non-negligible\nscalar fields are, respectively,  the `stronger attractors'.\nAgain we should emphasize that Malik and Wands \\cite{Malik98} showed that the multi-scalar field assisted inflationary solution is a late time attractor by utilizing a rotation in field space; indeed, the stable modes in a general stability analysis of this solution are presumably related to the isocurvature  perturbations orthogonal to the attractor trajectory in field space obtained in their analysis.\n\nFinally, from previous investigations \\cite{Billyard99} of\nspatially homogeneous scalar field cosmological models with an\nexponential potential and barotropic matter and from the above\nanalysis, we can conclude that the assisted inflationary solution\nis a global attractor for all ever-expanding spatially homogeneous\nmulti-field cosmological models with exponential potentials\nprovided $\\sum_{i=1}^{n}k_{i}^{-2}>1\/2$.\n We can also conclude that the multi-field\ncurvature scaling solution is a global attractor for models of\nBianchi types $V$ and $VII_h$ provided\n$\\sum_{i=1}^{n}k_{i}^{-2}<1\/2$ \\cite{vandenHoogen97a} and a\nmulti-field generalization of the Feinstein-Ibanez anisotropic\nsingle-field solution \\cite{Feinstein93} is the global attractor for models of\nBianchi types $III$ and $VI_h$ if $\\sum_{i=1}^{n}k_{i}^{-2}<1\/2$.\nIndeed, there will be $n$-field generalizations corresponding to\nall equilibrium points of the single-field Bianchi Type B models\n(cf. \\cite{Billyard99}).\n\n\nFinally, we note that spatially flat FRW matter scaling solutions\nalso exist in the context of generalized assisted inflation. In\n\\cite{GL} it was shown that in the higher-dimensional  context, in\nthe six--dimensional model the assisted dynamics between the\nscalar fields mimics the behaviour of a relativistic fluid\n$(\\gamma =4\/3$), while for higher dimensions the scalar fields\ndominate the radiation component, perhaps leading to a `moduli'\nproblem for the early universe.\n\n\n\n\n\n\n\\acknowledgements\n\nWe would like to thank Nemanja Kaloper, Jim Lidsey and David Wands for interesting discussions, and\nLaura Filion for help in checking the calculations.\nAAC and\nRJvdH are supported by grants from the Natural Sciences and Engineering Research Council of Canada.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}   \n    \nIn 1983 we began a ground-based project to provide a more secure\ncalibration of the zero-point for secondary distance indicators (such\nas the Tully-Fisher relation) by building up a database of accurate\nCepheid distances to nearby spiral galaxies.  In due course the Hubble\nSpace Telescope was launched and other activities took precedence over\nthe ground-based effort. Preliminary mention of work on Cepheids\ndiscovered in NGC~0247 was given in Freedman et al. (1988) and again\nin Catanzarite, Freedman, Horowitz \\& Madore (1994) but details were\nnever published, until now.  Here we present the photometric data and\nprovide a brief analysis leading to a distance determination to\nNGC~0247 based on nine Cepheids discovered in NGC~0247 some\ntwenty-five years ago. We then go on to compare it with new\nobservations published by Garcia-Varela et al. (2008, hereafter\n[GV08]).\n\n\\section{Observations}\nObservations for this program were carried out over a span of eight\nyears (giving a time baseline of almost 3,000 days). Oberving began\nfirst at the Cerro Tololo 4m, and was completed using the 2.5m duPont\ntelescope at Las Campanas, Chile.  Three fields of NGC~0247 were\nsurveyed in $B$, $V$, $R$, and $I$ filters at the Cerro~ Tololo\n4m~telescope during November~1984 to November~1988.  Figure~1 shows a\nphotograph of NGC~0247 with the three fields delineated. The\ncoordinates of the CCD field centers are given in Table 1.  From the\nsecond through the fourth year of the program data were obtained in\nthe service-observing mode offered at CTIO; those data were \ntaken by M. Navarrete.  For most of the runs the $512\\times320$\nRCA chip \\#5 (having a scale of 0.60~arcsec\/pxl and a total\nfield of view of ~ 3$'$ by 5$'$ at the prime focus) was used.  In 1988\na different RCA chip (\\#4) with similar characteristics was\nsubstituted.  Exposure times for these frames were typically 400~sec\nin $B$ and 300~sec in $V$, $R$ and $I$.  The frames were\nbias-subtracted, flat-fielded, and defringed using standard\ndata-reduction packages available at Cerro Tololo.  Beginning in 1990,\nthe observing program for the Sculptor galaxies shifted to the duPont\n2.5m telescope at the Las~Campanas~Observatory.  $BVRI$ CCD\nobservations covering the same three selected fields were obtained in\nDecember 1990, in September 1991, and in October, November, and\nDecember 1992.  Exposure times at this telescope were generally\n900~sec in $B$ and 600~sec in $V$, $R$ and $I$\\@.  Most of these\nobservations were electronically binned ($2 \\times 2$) at the\ntelescope.  For the 1990 and 1991 runs, the FORD1 CCD chip was used.\nFor the October and November 1992 runs a Tektronix~CCD~chip~(TEK4) was\nused; for the December 1992 runs, Tektronix CCD chips were also used\n(TEK3 and TEK4)\\@.  These chips each had dimensions of $2048 \\times\n2048$ pixels; the image scales obtained were: FORD1: 0.16~arcsec\/pxl;\nTEK3: 0.23~arcsec\/pxl; and TEK4: 0.26~arcsec\/pxl.  The survey totalled\nabout 250 exposures on 29 different nights over a span of eight years.\nTable 2 gives a journal of the observations.\n\n\n\n\n\\section{Photometry Reduction and Calibration}\nPhotometric calibration of the CTIO frames was accomplished using E-region\nstandards in E1, E2, E3, E7, E8, and E9 (Graham 1984) and in SA 98\n\\cite{lan83}.  $BVRI$ standards were taken on 20 independent\nphotometric nights.  As described in Freedman et al. (1992), a check\non the external accuracy of the photometric calibration for these runs\nwas made by individually calibrating the frames for NGC~0300.  The\nmagnitudes for the brightest stars were in agreement to within\n0.01-0.03$\\pm$0.03 mag of the average for all filter\/field\ncombinations. \n\nThe CCD frames from CTIO were reduced using both DoPHOT (Schechter,\nMateo \\& Saha 1993) and DAOPHOT (Stetson 1987), and cross-checked.\nThe unresolved background level in NGC~0247 is highly non-uniform, and\nis characterized both by regions in which there are strong spiral arms\nas well as relatively blank, interarm regions. In order to maximize\nthe detection limits of the algorithm FIND in DAOPHOT, the frames were\nfirst median-smoothed using a 7$\\times$ 7 pixel boxcar averaging\nscheme, and subtracted from the original frames.\n\nDetails of the calibration process are discussed in~\\cite{fre92}.  The\nLCO data were reduced using a variant of the DoPHOT package\n\\cite{mat89}.  This version of DoPHOT uses median smoothing to\nconstruct an initial model of the background sky before searching for\nobjects.  The sky model is refined after objects at the\nnext-to-the-lowest threshold have been found and subtracted.  The\nrefined sky model is then adopted as the baseline, and objects are\nagain found down to the lowest threshold and their PSF parameters\nre-measured.  LCO frames were brought onto the CTIO calibrated\nmagnitude system by the following process: The $(B-V)$ color\nterm for the CCD chips used at LCO relative to RCA chip used in the\nCTIO observations was measured and a correction was applied to the LCO\n$B$ photometry (the LCO $V$ photometry had no significant $(B-V)$\ncolor term.)  Next, the magnitude zero-point of each LCO frame was\noffset to the instrumental zero-point of a fiducial CTIO frame, for\nthe corresponding field and filter.  The calibration transformation\nderived for the fiducial CTIO frame was then applied to the LCO data.\n\n\n\\section{The Cepheids}\nFor Fields 1 and 3 all of the observations were tied to the\nphotometric zero point for October 7, 1988.  Observations on this\nnight were taken under excellent seeing conditions and had the best\nphotometric calibration available to us.  For Field 2, October 13,\n1988 was used to calibrate the data.  To put all of the stars in each\nfield on the same coordinate system, all frames from each field were\nspatially registered to the October 7, 1988 $V$ frame for that field,\n(since that was the best or close to best $V$ night for all three\nfields).  Coordinate transformations produced matches with $rms$\nscatter of $\\pm$0.30 pixels or better.  Calibrated, matched photometry\nfiles containing the entire set of observations for each\nfield\/bandpass combination were produced.  Stars with high internal\n$V$ magnitude dispersion were then identified as described in\n\\cite{fre94}.  These variable candidates were then subjected to a\nfurther test: a star was flagged as a Cepheid candidate only if the\nhistogram of its magnitudes was consistent with a uniform magnitude\nhistogram, as expected for Cepheids.  All three fields were searched\nfor variables down to a signal-to-noise level of 1.5$\\sigma$.  The $V$\nphotometric data for each candidate was then phased to the twelve\nperiods (in the range of 1 to 100 days) with the lowest phase\ndispersions, using a routine based on the Lafler-Kinman algorithm\n(\\cite{laf65}).  The $V$ light curves were then visually inspected and\nthe best period was selected.  Calibrated $B$, $R$, and $I$\nobservations were then phased to this adopted period and the\nmulti-wavelength light curves were inspected for consistency.\nCandidates with strong correlation of phase and amplitude between\ntheir $BVRI$ light curves, having well-determined periods, mean colors\nand well-sampled light curves characteristic of known Cepheids were\nthen identified as Cepheids.  Each of these stars was then visually\ninspected in the best image frame to check for nearby companions.\nNine Cepheids in total made it through the selection procedure.  All\nwere found in $BVRI$, with the exception of NGC~0247:[MF09]~C9, which\nwas was too faint to be recovered on the $I$ frames.  The positions\nfor the nine Cepheids in our sample are given in Table~3, the first 6\nof which are mapped over from [GV08], with the positions for C7, C8\nand C9 being on that system but having lower precision.  The\nindividual Cepheid observations are presented in Tables~4 through\n12. The light curves are shown in Figure 2.  The time-averaged\nproperties of the individual Cepheids are listed in Table~4.\n\n\n\\begin{figure}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[width=0.4\\textwidth, angle=-90]{fig2.eps}}\\hspace{0.02cm}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig3.eps}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig4.eps}}\\\\\n\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[width=0.4\\textwidth, angle=-90]{fig5.eps}}\\hspace{0.02cm}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig6.eps}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig7.eps}}\\\\\n\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[width=0.4\\textwidth, angle=-90]{fig8.eps}}\\hspace{0.02cm}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig9.eps}}\n\\resizebox{0.32\\hsize}{!}{\\includegraphics[angle=-90]{fig10.eps}}\\\\\n\\caption{$BVRI$ lightcurves for the individual Cepheids.  The plotted\nmagnitude range is 5~mag in all cases. Magnitude offsets, applied, to\nmake the lightcurves individually more visible, are given in the\nvertical-axis labels. In order from top to bottom the lightcurves are\n$I$, $R$, $V$ and $B$.}\n\\end{figure}\n\n\n\n\n\\subsection{Other Variable Stars Found in NGC~0247}\nFive variable stars which could not be classified as Cepheids were\nalso discovered.  These stars are well-isolated, their photometry is\nwell-measured by DoPHOT, and they have extremely strong $BVRI$\ncorrelation.  Three of them are very red, and have light curves with a\n``square wave'' shape.  If they are eclipsing variables then we have\nprobably been unable to determine the periods correctly.  A fourth\nobject has a light curve with the right shape to be a Cepheid, but is\nextremely red.  The fifth may be a Cepheid with an uncharacteristic\nlight curve.  The properties of these stars are summarized in Table\n13.\n\n\\section{The Distance to NGC~0247}\n\nA comprehensive review of previously published distance estimates to\nNGC~0247 is given in [GV08].\\footnote{In addition, an updated, on-line\ncompilation of distances to nearby galaxies, including NGC~0247, is\navailable through the {\\it NASA\/IPAC Extragalactic Database} at the\nfollowing URL: http:\/\/nedwww.ipac.caltech.edu\/level5\/NED1D\/intro.html.}\nIn that paper the authors also present their new $VI$ observations of\n23 Cepheids in the period range 17 to 131 days.  Based on those two\ncolors they derive a true distance modulus of 27.80$\\pm$0.09~mag\n(3.6~Mpc) tied to an LMC true distance modulus of 18.50~mag, as also\nadopted in this paper. Another important independent distance\nmeasurement to NGC~0247 worth noting here, because of its comparably\nhigh precision, is the tip of the red giant branch (TRGB) distance\nmodulus ($\\mu_o$ = 27.81~mag or 3.65~Mpc) published by Karachentsev et\nal. (2006).\n\n\\subsection{Discussion of Data}\n\nThe detected Cepheids at $B$ lie closer to the photometry limits than\nat $V$, $R$, or $I$; furthermore deriving a stable zero-point for that\nbandpass was found to be problematic.  As such, the $B$ data were used\nto confirm the periods adopted here, but because of signal-to-noise\nand other calibration problems we do not use the $B$-band data further\nin this paper. The $B$-band data are listed in this paper, but readers\nare strongly warned against using it for anything quantitative until a\nproper calibration is found.  The time-averaged data for our 9\nCepheids are given in Table 14.  The periods cited there were derived\nfrom these data alone, but will be updated later in the paper when we\nconsider  a merger with the [GV08] sample.\n\nAs can be seen in Figure 12, the NGC~0247 PL relations in $V$, $R$,\nand $I$, have smaller observed dispersions than the fiducial LMC PL\nrelations whose 2-sigma widths are shown by the dashed lines.  The\nsmall observed dispersion is presumably due to small number\nstatistics, but it could also be signalling a slight bias in the\nsample.  If the instability strip is not being fully sampled we cannot\nbe sure that these Cepheids properly reflect the mean. An external\ncheck with the results of [GV08] (Section 6 below) would suggest that\nthat bias (between samples) is at or below the 0.1 mag level.\n  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90, scale=0.65]{fig11.eps}\n\\caption{Fits of the NGC~0247 Cepheid PL-relations in $V$, $R$, and\n$I$ to the LMC PL-relation. Solid lines show the least-squares fit,\nflanked by $\\pm$2-sigma boundaries to the instability strip as derived from\nLMC calibrators.}\n\\end{center}\n\\end{figure}\n\n \n\\subsection{PL Relations and Apparent Distance Moduli}\n\nTo determine apparent $VRI$ distance moduli, residuals about the PL\nrelations for NGC~0247 Cepheids were minimized relative to the mean\nLMC PL relations given in \\cite{mad91}, and updated to the VI\ncalibration of Udalski (2000).  For a given bandpass, the LMC~PL\nrelation was iteratively shifted relative to the NGC~0247 PL relation\nuntil the $\\chi^2$ of the fit was minimized.  The off-set determined\nin this way is then the apparent distance modulus (for that bandpass)\nof NGC~0247 with respect to the LMC.  The results of the PL-fits are\nshown in Figure 3.  In the absence of other physical effects,\ndetermination of the true distance modulus and reddening can obtained\nby fitting the apparent moduli in different filters to an interstellar\nextinction law (e.g., Cardelli et al. 1989)\\footnote{Here we use A$_V$\n= 3.2$\\times$E(B-V) and A$_V$ = 2.45$\\times$E(V-I). [GV08] choose to use\na slightly different reddening law, taken from Schlegel et al. (1998), giving\nA$_V$ = 2.50$\\times$E(V-I) which is only 2\\% different from our\nadopted value.} originally discussed in Freedman (1998).\n\n\n\n\n\nIn Figure 4, the apparent distance moduli at $VRI$ for the Cepheid\nsample in NGC~0247 are plotted with respect to inverse wavelength.\nThe solid line gives a fit to a standard \\cite{car89} Galactic\nextinction law flanked by one-sigma error curves (dashed lines).  The\n$VRI$ data are very well-fitted by an extinction curve (with a small\npositive reddening equivalent to E(V-I) = 0.07~mag)\\footnote{All\nreddenings in this paper are given in terms of E(V-I). For those\nwishing the E(B-V) equivalent the appropriate conversion factor is\nE(B-V)\/E(V-I) = 2.45\/3.20 = 0.77} having an intercept corresponding to\na true distance modulus of $\\mu_o$ = 27.81$\\pm$0.10~mag\n(3.65$\\pm$0.16~Mpc). The solution using only $V$ and $I$ gives\nessentially the same numbers ($\\mu_o$ = 27.79$\\pm$0.13~mag;\n3.61$\\pm$0.23~Mpc).\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90, scale=0.60]{fig12.eps}\n\\caption{Fit of the apparent distance moduli in $V$, $R$, and $I$ to a\nGalactic extinction law (solid line). One-sigma errors on the fit are\nshown with broken lines. Plotted contours are 2, 4 and 6 sigma.}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Comparision with Garcia-Varela et al. (2008)}\n\nWe have made a positional cross-correlation of our Cepheids with those\ndiscovered by [GV08]. Six of our nine variables were recovered by the\nAraucania Project, and the correct identification of these stars\nacross the two studies is reinforced by the independently derived\nperiods which agree to better than 10\\% in most cases. We now combine\nthe two datasets, revise the periods when posible and update the $V$\nand $I$ intensity-mean magnitudes. The results of that update are\ngiven in Table 15. The combined lightcurves are shown in Figure 5.\n\n\n\\begin{figure}\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[width=0.4\\textwidth, angle=-90]{fig13.eps}}\\hspace{0.02cm}\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[angle=-90]{fig14.eps}}\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[angle=-90]{fig15.eps}}\\\\\n\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[width=0.4\\textwidth, angle=-90]{fig16.eps}}\\hspace{0.02cm}\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[angle=-90]{fig17.eps}}\n\\resizebox{0.31\\hsize}{!}{\\includegraphics[angle=-90]{fig18.eps}}\\\\\n\n\\caption{Combined $BVRI$ lightcurves for the individual Cepheids in\nNGC~0247.  The plotted magnitude range is 5~mag in all\ncases. Magnitude offsets, applied, to make the lightcurves\nindividually more visible, are given in the vertical-axis labels. In\norder from top to bottom the lightcurves are $I$, $R$, $V$ and $B$. \nSolid points are from this paper; open circles are from [GV08].}\n\\end{figure}\n\n\nThe updated VRI [MF09] sample alone gives $\\mu_V$ =\n27.97$\\pm$0.05~mag, $\\mu_R$ = 27.98$\\pm$0.06~mag, $\\mu_I$ =\n27.90$\\pm$0.06~mag, E(V-I) = 0.11$\\pm$0.03~mag resulting in $\\mu_o$ =\n27.81$\\pm$0.05~mag or 3.65$\\pm$0.08~Mpc for the 3-band fit, and\n$\\mu_o$ = 27.79$\\pm$0.13~mag (3.61$\\pm$0.23~Mpc) with E(V-I) =\n0.07$\\pm$0.04~mag for the VI fit alone.\n\n\nWe consider a progressive merger of the two datasets.  We first apply\nour standard fitting techniques to the [GV08] preferred subset of 17\nCepheids, omitting as they did, the longest and shortest-period\nCepheids in their sample. We get $\\mu_V$ = 28.21$\\pm$0.05~mag,\n$\\mu_I$ = 28.05$\\pm$0.06~mag, E(V-I) = 0.15$\\pm$0.03~mag\nresulting in $\\mu_o$ = 27.82$\\pm$0.08~mag or 3.66$\\pm$0.14~Mpc.\nThis differs from the [GV08] solution by  +0.025~mag in the true\nmodulus.  \n\nIf we now update the [GV08] sample with the revised periods and\nmagnitudes for NGC~0247:[MF09]~C2 through C6 we get $\\mu_V$ =\n28.15$\\pm$0.06~mag, $\\mu_I$ = 28.03$\\pm$0.06~mag, E(V-I) = 0.11$\\pm$\n0.03~mag resulting in $\\mu_o$ = 27.87$\\pm$0.09~mag or\n3.75$\\pm$0.15~Mpc.  Augmenting the [GV08] sample with\nNGC~0247:[MF09]~C7 \\& NGC~0247:[MF09]~C8, plus reintroducing\nNGC~0247:[MF09]~C1 with its first-epoch period and magnitude, in\naddition to its evolved values from [GV08] as described in Section 7\n(below), we get $\\mu_V$ = 28.13$\\pm$0.05~mag, $\\mu_I$ = 28.01$\\pm$\n0.06~mag, $E(V-I)$ = 0.11$\\pm$0.03~mag resulting in $\\mu_o$ =\n27.85$\\pm$ 0.09~mag or 3.72$\\pm$0.15~Mpc.  The above results are\nsummarized in Table 16.\n\n\n\n\n\n\\section{The 70-Day Cepheid NGC~0247:[MF09]~C1}\n\nIn an attempt to update the period and combine the photometry for the\nlongest-period Cepheid in our sample, NGC~0247:[MF09]~C1, we quickly\nfound that the mean magnitudes and colors derived from our data did\nnot correspond to data published for it in the [GV08] study. Figure 6\nshows the differences. In that plot our data are shown as circled\nsolid symbols, phased to our adopted period of 69.9 days.  Below those\nlight curves are the data from [GV08], shown as open cicles, phased to\ntheir period of 65.862 days (with an arbitrarily added phase shift of\n0.6 to align the lightcurves for ease of visual comparision). The\ntime-averaged V magnitudes differ by 0.8~mag, with the most recent\nepoch being fainter; while the (V-I) colors differ by 0.23~mag, with\nthe most recent data being bluer. The sense of the change eliminates a\nself-shrouding event as the possible cause. A remaining explanation\nis that the structure of the star itself may have systematically\nchanged in the intervening quarter century: in the face of a rising\nsurface temperature (indicated by the decrease in the (V-I) color) and\nthe resulting increased surface brightness, the overall radius of this\nstar may have decreased significantly.  In the process the period\ndropped by 6\\%, from 70 to 66 days. \n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90, scale=0.85]{fig19.eps}\n\\caption{The V and I lightcurves for the Cepheid C1 at the two epochs\nsurveyed by [MF09] (circled solid symbols) and [GV08] (open\ncircles). The I-band light curves are both displaced by one magnitude\nupward in the figure for ease of viewing. In addition the earlier data\nare displaced in phase by 0.6 cycles so as to align the two datasets\naround maximum light.  The data are phased to a period of 69.9 days\nfor the [MF09] observations, and to a period of 65.862 days for the\n[GV08] observations. The vertical displacement of the pairs of light\ncurves in V and in I is real, indicating that the star faded by nearly\n0.8~mag between the times of the two studies.}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nA simple linear decrease of the period with time ($\\Delta P\/P$ =\n0.0075 day\/day) fails to phase the lightcurves over the total baseline\n(and, in fact, destroys coherence within the individual observing\ncampaigns). Without undertaking more sophisticated modelling we\ndefault to the next simplest conclusion that the period change was a\ndiscontinuous event. Further monitoring of this star could be reveal\ninteresting aspects of the structure of Cepheids in general if this\nbehavior persists.\n\n\n\\section{Summary and Conclusions}\n\n\nNine Cepheids have been identified in the Galaxy NGC~0247. Six of\nthese variable stars have been independently found by [GV08].\n\nThe period and magnitude-updated VRI [MF09] sample alone gives\napparent moduli of $\\mu_V$ = 27.97$\\pm$0.05~mag, $\\mu_R$ =\n27.98$\\pm$0.06~mag, $\\mu_I$ = 27.90$\\pm$0.06~mag, and $E(V-I)$ =\n0.11$\\pm$0.03~mag resulting in $\\mu_o$ = 27.81$\\pm$0.05~mag or\n3.65$\\pm$0.08~Mpc for the 3-band fit. These data yield a true distance\nmodulus of $\\mu_{0} = 27.70\\pm0.11$~mag corresponding to a metric\ndistance of 3.47$\\pm$0.18~Mpc.\n\n\nCombining our observations with newly published data from [GV08] in the V \\& I\nbands, and updating the periods accordingly, results in a reddening of\nE(V-I) = 0.06$\\pm$0.04~mag and a (preferred) true modulus of $\\mu_{0}\n= 27.81\\pm0.05$~mag (3.65$\\pm$0.08~Mpc). This is (fortuitously)\nidentical to the TRGB distance modulus recently published by\nKarachentsev et al. (2006) further re-inforcing the consistency of\nthese two distance scales, which are based on largely independent\nassumptions, and have very different systematics.\n\n\n\nThe 70-day Cepheid NGC~0247:[MF09]~C1 deserves follow-up observations\nto see if the extraordinary changes in its magnitude, period and color\nfound between these epochs (first 1984-1992 and then 2002-2005) is an\non-going phenomenon.\n\n\\acknowledgements During the initial period in which these\nobservations were made WLF's research was supported in part by NSF\nGrants AST 87-13889 and 9116496 on the extragalactic distance scale.\nWe thank Bob Williams who provided us with Director's Discretion Time\nin 1988, and Irwin Horowitz who particiated in the early stages of\nreducing the Las Campanas data.  We also thank Jose Garcia-Varela,\nGrzegorz Pietrzynski and Wolfgang Gieren for providing their more\nrecently acquired Cepheid data in advance of publication. This\nresearch has made use of the NASA\/IPAC Extragalactic Database (NED)\nwhich is operated by the Jet Propulsion Laboratory, Caltech, under\ncontract with the National Aeronautics and Space Administration.\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section1}\n\nLet $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. Then\n\\begin{align*} \n\\mathrm{tr}_R(M)=\\sum_{f\\in \\Hom_R(M, R)} \\Im f = \\Im (\\mathrm{ev})\n\\end{align*}\nwhere $\\mathrm{ev}:\\Hom_R(M, R) \\otimes_R M \\to R$; $\\ f\\otimes x\\mapsto f(x)$ ($f\\in \\Hom_R(M, R)$ and $x\\in M$) denotes the evaluation map, is called the {\\it trace ideal} of $M$. An ideal $I$ of $R$ is called a {\\it trace ideal} if $I=\\mathrm{tr}_R(M)$ for some $R$-module $M$. \n\nThe notion of trace ideals was recently studied by several papers (\\cite{DMP, GIK2, HHS, K, Lin, LP}). In particular, Herzog, Hibi, Stamate deeply studied the trace ideal of the canonical module, and they introduced a new notion, namely, {\\it nearly Gorenstein rings} as a class of non-Gorenstein Cohen-Macaulay rings (\\cite{HHS}). Here, recall that the trace ideal of the canonical module defines the non-Gorenstein locus of the ring (\\cite[after 11.41. Lemma]{LW} and \\cite[Lemma 2.1]{HHS}). Thus, we may suppose that the ring is close to being Gorenstein if the trace ideal of the canonical module is large. Indeed, the notion of nearly Gorenstein rings is defined by the inclusion $\\mathrm{tr}_R(\\omega_R)\\supseteq \\mathfrak{m}$ for a Cohen-Macaulay local ring $(R, \\mathfrak{m})$ possessing the canonical module $\\omega_R$ of $R$.\n\nIn this paper, we study the opposite case in some sense, that is, the case where the trace ideal of the canonical module is as small as possible. It is known that for a one-dimensional generically Gorenstein Cohen-Macaulay local ring $(R, \\mathfrak{m})$ possessing the canonical module $\\omega_R$, $\\mathrm{tr}_R(\\omega_R)\\supseteq R:\\overline{R}$, where $\\overline{R}$ denotes the integral closure of $R$. Thus, we call $R$ a {\\it far-flung Gorenstein ring} if $\\mathrm{tr}_R(\\omega_R)=R:\\overline{R}$. From the definition, it seems that there is only little that we can expect for far-flung Gorenstein rings. However, we will obtain that far-flung Gorenstein rings enjoy interesting properties. \n\nTo illustrate our results, let $(R, \\mathfrak{m})$ be a (one-dimensional) far-flung Gorenstein ring. For simplicity, suppose that $R\/\\mathfrak{m}$ is infinite and $\\overline{R}$ is a local ring. Then, we obtain the bounds \n\\[\n\\mathrm{r}(R)+1\\le \\mathrm{e}(R) \\le \\binom{r+1}{2}\n\\]\nfor the multiplicity of $R$, where $\\mathrm{e}(R)$ denotes the multiplicity and $r=\\mathrm{r}(R)$ denotes the Cohen-Macaulay type of $R$ (Corollary \\ref{0.11}).  Furthermore, the upper bound can be improved if $R$ is a numerical semigroup ring. In this case, we will see that $\\mathrm{e}(R)\\le \\overline{n}(r)$, where the integer $\\overline{n}(r)$ denotes the solution of the Rohrbach problem for $r$ (Corollary \\ref{cor5.3}). Note that the Rohrbach problem is a long-standing problem in additive number theory, see \\cite{Ro, Slo}.\n\nWe also obtain that if $(R, \\mathfrak{m})$ is a far-flung Gorenstein ring,  the endomorphism algebra $\\Hom_R(\\mathfrak{m}, \\mathfrak{m})$ of the maximal ideal is again far-flung Gorenstein  (Theorem \\ref{b3.2}) and $\\Hom_R(\\omega_R, M)$ is $\\overline{R}$-free for all reflexive modules $M$ of positive rank (Theorem \\ref{thm4.1}). Examples arising from numerical semigroup rings are also explored.\n \n\n\nThe remainder of this paper is organized as follows. In Section \\ref{section2} we give a characterization of far-flung Gorenstein rings and prove the bounds $\\mathrm{r}(R)+1\\le \\mathrm{e}(R) \\le \\binom{r+1}{2}$. In Section 3 we prove Theorem 3.2. \nIn Section \\ref{section5} we study reflexive modules over far-flung Gorenstein rings. In Section \\ref{section5.5} we revisit the bound for the multiplicity of far-flung Gorenstein numerical semigroup rings in relation with the Rohrbach problem. In Section \\ref{section6} we study in more details numerical semigroup rings.\n\n\nLet us fix our notation throughout this paper. In what follows, $(R, \\mathfrak{m})$ is a Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module. Then \n$\\ell_R(M)$, $\\mu_R(M)$, and $\\mathrm{e}(M)$ denote the length, multiplicity, and the number of minimal generators of $M$, respectively. $\\mathrm{r}(R)$ and $v(R)$ denote the Cohen-Macaulay type and embedding dimension of $R$, respectively.\n\nLet $\\mathrm{Q}(R)$ denote the total ring of fractions of $R$, and let $\\overline{R}$ denote the integral closure of $R$. Then a finitely generated $R$-submodule of $\\mathrm{Q}(R)$ containing a non-zerodivisor of $R$ is called a {\\it fractional ideal}. For two fractional ideals $I$ and $J$, $I:J$ denotes the colon ideal of $I$ and $J$ which is given by the set $\\{\\alpha\\in \\mathrm{Q}(R) \\mid \\alpha J\\subseteq I\\}$  (see \\cite{HK}).\n\n\n\n\n\\begin{acknowledgments}\nWe would like to thank Mihai Cipu for telling us about the Rohrbach problem.\n\\end{acknowledgments}\n\n\n\n\n\n\n\n\n\n\\section{Far-flung Gorenstein rings and bounds of the multiplicity}\\label{section2}\n\nThroughout this paper, unless otherwise noted, let $(R, \\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension one, possessing the canonical module $\\omega_R$. \nWe set the conditions on $R$ as follows.\n\\begin{condition} \n\\begin{enumerate}[{\\rm (a)}] \n\\item There exists an $R$-submodule $C$ of $\\mathrm{Q}(R)$ such that $R\\subseteq C\\subseteq \\overline{R}$ and $C\\cong \\omega_R$.\n\\item $\\overline{R}$ is finitely generated as an $R$-module.\n\\item $\\overline{R}$ is a local ring with the maximal ideal $\\mathfrak{n}$.\n\\item There exists a non-zerodivisor $a\\in \\mathfrak{m}$ such that $(a)$ is a reduction of $\\mathfrak{m}$.\n\\end{enumerate}\n\\end{condition}\n\n\\begin{rem} \\label{a2.1}\n\\begin{enumerate}[{\\rm (i)}] \n\\item If the ring $R$ is generically Gorenstein (i.e., $R_\\mathfrak{p}$ is Gorenstein for all $\\mathfrak{p} \\in \\Ass R$) and the residue field $R\/\\mathfrak{m}$ is infinite, then $R$ satisfies the conditions (a) and (d). Indeed, there exist a canonical ideal $\\omega\\subsetneq R$ and its reduction $(a)\\subseteq \\omega$. Hence $C=\\frac{\\omega}{a}$ is the module satisfying the condition (a).\n\\item If $R$ is a numerical semigroup ring $K[|H|]=K[|t^h : h\\in H|]$, where $K[|t|]$ is formal power series ring over a field $K$ and $H$ is a subsemigroup of $\\mathbb{N}_0=\\{0, 1, 2, \\dots\\}$, then $R$ satisfies the conditions (a)-(d) (see Section \\ref{section6}). \n\\item The condition (d) implies that $\\mathfrak{m} \\overline{R}=a\\overline{R}$. Indeed, we have $(a)\\subseteq \\mathfrak{m} \\subseteq \\overline{(a)}=a\\overline{R}\\cap R\\subseteq a\\overline{R}$, where $\\overline{(a)}$ denotes the integral closure of ideal $(a)$ in the sense of \\cite[page 2]{SH}. It follows that $R\\subseteq \\frac{\\mathfrak{m}}{a}\\subseteq \\overline{R}$; hence, $\\frac{\\mathfrak{m}}{a}\\overline{R}=\\overline{R}$. \n\\item For all fractional ideals $I$ and $J$, \n\\[\nI:J\\cong \\Hom_R(J, I)\n\\] \n(\\cite[Lemma 2.1]{HK}). In particular, with the condition (a), we have $C:(C:I)=I$ (see \\cite[Definition 2.4]{HK}).\n\\end{enumerate} \n\\end{rem}\n\nUnder condition (a), we can regard the evaluation map\n\\begin{align*} \n\\mathrm{ev}: \\Hom_R(\\omega_R, R)\\otimes_R \\omega_R \\to R \n\\end{align*}\nas the multiplication map $(R:C) \\otimes_R C \\to R$; $f\\otimes x \\to fx$ ($f\\in R:C$ and $x\\in C$). It follows that $\\mathrm{tr}_R(\\omega_R)=(R:C)C$.\nThe following is a starting point of this paper.\n\n\\begin{Lemma}\\label{a2.2} {\\rm (cf. \\cite[Proposition A.1]{HHS2})}\nWith the conditions {\\rm (a)} and {\\rm (b)}, we have $R:\\overline{R}\\subseteq \\mathrm{tr}_R(\\omega_R)$.\n\\end{Lemma}\n\n\\begin{proof}\nThis follows from the fact that $\\mathrm{tr}_R(\\omega_R)=(R:C)C\\supseteq R:C\\supseteq R:\\overline{R}$.\n\\end{proof}\n\nSince the trace ideal of the canonical module defines the non-Gorenstein locus of the ring (\\cite[after 11.41. Lemma]{LW}, \\cite[Lemma 2.1]{HHS}), we may suppose that the ring is close to being Gorenstein if the trace ideal of the canonical module is large. In this paper, we study the opposite case, that is, the case where the trace ideal of the canonical module is as small as possible:\n\n\\begin{Definition}\\label{defffg}\nSuppose that $R$ satisfies the conditions {\\rm (a)} and {\\rm (b)}.\nWe say that $R$ is a {\\it far-flung Gorenstein ring} if $\\mathrm{tr}_R(\\omega_R)=R:\\overline{R}$.\n\\end{Definition}\n\n\n\\begin{rem} \nRecall that for an arbitrary Cohen-Macaulay local ring $(R, \\mathfrak{m})$ possessing the canonical module $\\omega_R$, we say that $R$ is a {\\it nearly Gorenstein ring} if $\\mathrm{tr}_R(\\omega_R)\\supseteq \\mathfrak{m}$ (\\cite{HHS}). Hence, if a one-dimensional Cohen-Macaulay local ring $R$ satisfies the conditions {\\rm (a)} and {\\rm (b)}, $R$ is nearly Gorenstein and far-flung Gorenstein if and only if $R:\\overline{R}\\supseteq \\mathfrak{m}$. This is equivalent to saying that $\\overline{R}= \\mathfrak{m}:\\mathfrak{m} (\\cong \\Hom_R(\\mathfrak{m}, \\mathfrak{m}))$. Indeed, we may assume that $R$ is not a discrete valuation ring. We then obtain that \n\\[\nR:\\overline{R}\\supseteq \\mathfrak{m} \\Leftrightarrow \\mathfrak{m} \\overline{R}\\subseteq R \\Leftrightarrow \\overline{R}\\subseteq R:\\mathfrak{m}=\\mathfrak{m} :\\mathfrak{m},\n\\]\nwhere the equality $R:\\mathfrak{m}=\\mathfrak{m} :\\mathfrak{m}$ is well understood and described in Lemma \\ref{b3.2}.\n\nIn particular, all nearly Gorenstein far-flung Gorenstein rings are almost Gorenstein rings in the sense of \\cite[Definition 3.1]{GMP} and have minimal multiplicity (see \\cite[Theorem 5.1]{GMP}).\nFurthermore, if $R=K[|H|]$ is a numerical semigroup ring, then the condition $R:\\overline{R}\\supseteq \\mathfrak{m}$ is equivalent to saying that $H=\\left<n, n+1, \\dots, 2n-1\\right>$ for some $n>0$. \n\\end{rem}\n\n\n\nThe following is a characterization of far-flung Gorenstein rings.\n\n\\begin{thm} \\label{0.8}\nSuppose that $R$ satisfies the conditions {\\rm (a)-(c)}.\nThen the following are equivalent:\n\\begin{enumerate}[{\\rm (i)}] \n\\item $R$ is a far-flung Gorenstein ring;\n\\item $\\mathrm{tr}_R(\\omega_R)\\cong \\overline{R}$;\n\\item $C^2=\\overline{R}$.\n\\end{enumerate} \n\\end{thm}\n\n\n\n\n\\begin{proof}\n(i) $\\Rightarrow$ (ii): Note that $R:\\overline{R}$ is a nonzero ideal of $\\overline{R}$. Since $\\overline{R}$ is a discrete valuation ring (for example, see \\cite[Theorem 2.2.22]{BH}), it follows that $\\mathrm{tr}_R(\\omega_R)=R:~\\overline{R}\\cong \\overline{R}$.\n\n(ii) $\\Rightarrow$ (i): $\\mathrm{tr}_R(\\omega_R)=\\alpha \\overline{R}$ for some $\\alpha\\in \\mathrm{Q}(R)$. We then have \n\\[\n\\alpha\\in \\alpha \\overline{R}=\\mathrm{tr}_R(\\omega_R)\\subseteq R,\n\\]\nthus $\\alpha \\overline{R}$ is an ideal of $\\overline{R}$ in $R$. Hence, $\\mathrm{tr}_R(\\omega_R)=\\alpha \\overline{R}\\subseteq R:\\overline{R}$. The reverse inclusion follows from Lemma \\ref{a2.2}.\n\n(i) $\\Rightarrow$ (iii): This follows from the following implications.\n\\begin{align*}\n\\text{$R$ is a far-flung Gorenstein ring} &\\Leftrightarrow R:\\overline{R} =\\mathrm{tr}_R(\\omega_R) &&\\Leftrightarrow R:\\overline{R} =(R:C)C \\\\\n&\\Rightarrow R:\\overline{R}=R:C &&\\Leftrightarrow (C:C):\\overline{R}=(C:C):C \\\\\n&\\Leftrightarrow C:C\\overline{R}=C:C^2 &&\\Leftrightarrow C^2=C\\overline{R}=\\overline{R},\n\\end{align*}\nwhere the third implication follows from the inclusions $R:\\overline{R}  \\subseteq R:C \\subseteq (R:C)C$ and the sixth equivalence follows by applying the $C$-dual $\\Hom_R(-, C)=C:-$.\n\n(iii) $\\Rightarrow$ (i): By the argument of (i) $\\Rightarrow$ (iii), $C^2=\\overline{R}$ implies that $R:\\overline{R}=R:C$. Hence, \n\\[\n\\mathrm{tr}_R(\\omega_R)=(R:C)C=(R:\\overline{R})C=R:\\overline{R},\n\\]\nwhere the third equality follows from $R:\\overline{R}\\subseteq (R:\\overline{R})C\\subseteq (R:\\overline{R})\\overline{R}=R:\\overline{R}$.\n\\end{proof}\n\n\n\nBy using Theorem \\ref{0.8} we find an upper bound of the multiplicity for far-flung Gorenstein rings. \nRecall that there is a lower bound of the multiplicity of Cohen-Macaulay local rings.\n\n\\begin{Fact} {\\rm (cf. \\cite[3.1 Proposition]{S3})}\\label{b2.5}\nIf $(R, \\mathfrak{m})$ is a Cohen-Macaulay local ring with $\\mathrm{e}(R)>1$ (not necessarily of dimension one), then $\\mathrm{r}(R)+1 \\le \\mathrm{e}(R)$. Furthermore, $\\mathrm{r}(R)+1 = \\mathrm{e}(R)$ holds if and only if $R$ has minimal multiplicity.\n\\end{Fact}\n\n\n\\begin{proof}\nWe may assume that $R\/\\mathfrak{m}$ is infinite by passing to $R \\to R[X]_{\\mathfrak{m} R[X]}$. Then we can choose a parameter ideal $Q$ as a reduction of $\\mathfrak{m}$. Hence we obtain that\n\\[\n\\mathrm{e}(R)-1=\\ell_R (R\/Q)-\\ell_R (R\/\\mathfrak{m})=\\ell_R (\\mathfrak{m}\/Q) \\ge \\ell_R ((Q:_R\\mathfrak{m})\/Q)=\\mathrm{r} (R).\n\\] \nThe equality holds true if and only if $\\mathfrak{m} = Q:_R\\mathfrak{m}$, which is equivalent to saying that $\\mathfrak{m}^2 =Q\\mathfrak{m} $ by \\cite[Theorem 2.2.]{CP}.\n\\end{proof}\n\n\n\\begin{Corollary}\\label{0.11}\nSuppose that $R$ satisfies the conditions {\\rm (a)-(d)}. If $R$ is a far-flung Gorenstein ring, then the inequalities\n\\[\n\\mathrm{r}(R)+1 \\le \\mathrm{e}(R) \\le \\binom{\\mathrm{r}(R)+1}{2}\n\\] \nhold.\n\\end{Corollary}\n\n\\begin{proof}\nBy Theorem \\ref{0.8}, $\\mu_R(\\overline{R})=\\mu_R(C^2)\\le \\binom{\\mathrm{r}(R)+1}{2}$. Meanwhile, we have $\\mu_R(\\overline{R})=\\ell_R(\\overline{R}\/\\mathfrak{m} \\overline{R})=\\ell_R(\\overline{R}\/a\\overline{R})$ by Remark \\ref{a2.1}(iii). Since $(a)$ is a reduction of $\\mathfrak{m}$, $\\ell_R(\\overline{R}\/a\\overline{R})=\\mathrm{e}(\\overline{R})$. Therefore, since $\\mathrm{e}(\\overline{R})=\\mathrm{e}(R)+ \\mathrm{e}(\\overline{R}\/R)=\\mathrm{e}(R)$ by the additivity of the multiplicity, we obtain that $\\mu_R(\\overline{R})=\\mathrm{e}(R)$.\n\\end{proof}\n\n\\begin{Corollary}\\label{cor:type2}\nSuppose that $R$ satisfies the conditions {\\rm (a)-(d)}. If $R$ is a far-flung Gorenstein ring of type $2$, then $R$ has minimal multiplicity. Moreover, under these hypotheses, the multiplicity of $R$ is $3$. \n\\end{Corollary}\n\n\n\\begin{proof}\nLet $R$ be a far-flung Gorenstein ring with  $\\mathrm{r}(R)=2$.  Then we have $3=\\mathrm{r}(R)+1 \\le \\mathrm{e}(R) \\le \\binom{\\mathrm{r}(R)+1}{2}=3$ by Corollary \\ref{0.11}. Hence $R$ has minimal multiplicity of $3$ by Fact \\ref{b2.5}.  \n\\end{proof}\n\n\\begin{ex}\\label{b2.8}\nLet $R=K[|H|]$ be a numerical semigroup ring with $\\mathrm{r}(R)=2$. Then $R$ is a far-flung Gorenstein ring if and only if $H=\\langle 3, 3n+1, 3n+2\\rangle$ for some integer $n>0$.\n\\end{ex}\n\n\\begin{proof}\nIf $R$ is a far-flung Gorenstein ring of type $2$, Corollary~\\ref{cor:type2} implies $v(R)=\\mathrm{e}(R)=3$. \nIt is known from  \\cite[Proposition 2.5]{HHS2} that when $H$ is $3$-generated and not symmetric, the ring $K[|H|]$ is far-flung Gorenstein if and only if $H=\\langle 3, 3n+1, 3n+2\\rangle$ for some integer $n >0$.\n\\end{proof}\n\nIn contrast to Corollary~\\ref{cor:type2}, far-flung Gorenstein rings need not have minimal multiplicity in general.\n\n\\begin{ex}\\label{b2.9}\nLet $R=K[|t^7, t^8, t^{11}, t^{17}, t^{20}|]$. Then $\\mathrm{tr}_R(\\omega_R)=R:\\overline{R}=t^{14}\\overline{R}$. Thus $R$ is far-flung Gorenstein, but $R$ does not have minimal multiplicity: $\\mathrm{e}(R)=7>v(R)=5$.\n\\end{ex}\n\n\n\n\nAlthough Corollary \\ref{0.11} gives an upper bound for the multiplicity, this is not sharp for numerical semigroup rings. We will return to this topic in Section \\ref{section5.5}. \n\n\n\n\\section{The endomorphism algebra $\\Hom_R(\\mathfrak{m}, \\mathfrak{m})$}\\label{section4}\n\n\nThroughout this section, let $(R, \\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension one possessing the canonical module $\\omega_R$, and suppose that $R$ satisfies the conditions {\\rm (a)-(d)}. Here, we study the far-flung Gorenstein property of the endomorphism algebra $\\Hom_R(\\mathfrak{m}, \\mathfrak{m})$ in relation with that of $R$. Set $B=\\mathfrak{m}:\\mathfrak{m}\\cong \\Hom_R(\\mathfrak{m}, \\mathfrak{m})$. The following lemmas are known, but we include a proof for the convenience of readers.\n\n\\begin{Lemma}\\label{b3.1}\n$B$ is a Cohen-Macaulay local ring of dimension one and $\\overline{B}=\\overline{R}$.\n\\end{Lemma}\n\n\\begin{proof}\n$B$ is a subring of $\\overline{R}$ and finitely generated as an $R$-module. Hence $\\overline{B}=\\overline{R}$ holds. \nFurthermore, any maximal ideal of $B$ is forced to be $\\mathfrak{n}\\cap B$. It follows that $B$ is a local ring with the maximal ideal $\\mathfrak{n}\\cap B$. \n\\end{proof}\n\n\n\n\\begin{Lemma}\\label{b3.2}\nIf $R$ is not a discrete valuation ring, then $R\\subsetneq B=R:\\mathfrak{m}$.\n\\end{Lemma}\n\n\\begin{proof}\nSuppose that there exists an element $f\\in (R:\\mathfrak{m})\\setminus (\\mathfrak{m}:\\mathfrak{m})$. Then $\\mathfrak{m} f\\subseteq R$ but $\\mathfrak{m} f\\not\\subseteq \\mathfrak{m}$. It follows that $\\mathfrak{m} f =R$ and hence $\\mathfrak{m}$ is cyclic. This shows that $R$ is a discrete valuation ring, which is a contradiction. Thus $R:\\mathfrak{m}=B$. To prove $R\\subsetneq B$, consider the short exact sequence $0\\to \\mathfrak{m} \\to R \\to R\/\\mathfrak{m} \\to 0$. By applying the functor $\\Hom_R(-, R)\\cong R:-$, we obtain that \n\\[\n0 \\to R=R:R \\to B=R:\\mathfrak{m} \\to \\Ext_R^1(R\/\\mathfrak{m}, R) \\to 0. \n\\]\nTherefore, we have $\\ell_R(B\/R)=\\mathrm{r}(R)>0$.\n\\end{proof}\n\n\\begin{Theorem}\\label{b3.3}\nSuppose that $R$ satisfies {\\rm (a)-(d)}. If $R$ is a far-flung Gorenstein ring, then $B\\cong \\Hom_R(\\mathfrak{m}, \\mathfrak{m})$ is also a far-flung Gorenstein ring.\n\\end{Theorem}\n\n\\begin{proof}\nIf $R$ is a discrete valuation ring, then the assertion of the theorem is obvious because $R=B$.  Hence we may assume that $R$ is not a discrete valuation ring. Then the assertion follows from the following claim.\n\\end{proof}\n\n\\begin{claim}\\label{claim1}\nSuppose that $R$ satisfies {\\rm (a)-(d)} and $R$ is a far-flung Gorenstein ring. Choose  an element $a \\in \\mathfrak{m}$ such that $(a)$ is a reduction of $\\mathfrak{m}$ {\\rm (}recall the condition {\\rm (d)}{\\rm )}. \nThen the following assertions hold true.\n\\begin{enumerate}[{\\rm (i)}] \n\\item $B:\\overline{B}=\\frac{1}{a} (R:\\overline{R})$.\n\\item $\\mathrm{tr}_B(\\omega_B)=\\frac{1}{a} \\mathrm{tr}_R(\\omega_R)$.\n\\end{enumerate} \t\n\\end{claim}\n\n\n\\begin{proof}[Proof of Claim \\ref{claim1}]\n(i): This follows from \n\\begin{align*}\nB:\\overline{B}=B:\\overline{R}=(R:\\mathfrak{m}):\\overline{R}=R:\\mathfrak{m}\\overline{R}=R:a\\overline{R}=\\frac{1}{a} (R:\\overline{R})\n\\end{align*}\nby Remark \\ref{a2.1}(iii) and Lemmas \\ref{b3.1} and \\ref{b3.2}.\n\n(ii): Since we have $\\omega_B\\cong \\Hom_R(B, C)\\cong C:B$, we obtain that \n\\[\n\\mathrm{tr}_B(\\omega_B)=(B:(C:B))(C:B).\n\\] \nBy noting that $B=R:\\mathfrak{m}=(C:C):\\mathfrak{m}=C:\\mathfrak{m} C$, we obtain that $C:B=C:~(C:\\mathfrak{m} C)=\\mathfrak{m} C$. Hence we have $\\mathrm{tr}_B(\\omega_B)=(B:\\mathfrak{m} C)\\mathfrak{m} C$.\n\nFurthermore, we can compute $B:\\mathfrak{m} C$ as follows:\n\\begin{align*}\nB:\\mathfrak{m} C&=(R:\\mathfrak{m}):\\mathfrak{m} C=R:\\mathfrak{m}^2 C = (R:C) : \\mathfrak{m}^2\\\\\n&=(R: \\overline{R}):\\mathfrak{m}^2= R:\\mathfrak{m}^2 \\overline{R}=R:a^2\\overline{R}=\\frac{1}{a^2} (R:\\overline{R})\\\\\n&=\\frac{1}{a^2} (R:C),\n\\end{align*}\nwhere the fourth and the eighth equalities follow from the fact $R:C=R:\\overline{R}$ (see the proof of Theorem \\ref{0.8} (i)\\implies (iii)). \nIt follows that \n\\begin{align*}\n\\mathrm{tr}_B(\\omega_B)=\\frac{1}{a^2}(R:C)\\mathfrak{m} C=\\frac{\\mathfrak{m}}{a^2}\\mathrm{tr}_R(\\omega_R)=\\frac{\\mathfrak{m}}{a^2}(R:\\overline{R})=\\frac{a}{a^2}(R:\\overline{R})=\\frac{1}{a}\\mathrm{tr}_R(\\omega_R).\n\\end{align*}\n\\end{proof}\n\nThe converse of Theorem \\ref{b3.3} is not true.\n\n\\begin{ex} \nLet $R=K[|t^4, t^5, t^6|]$ be a numerical semigroup ring, where $K$ is a field. Let $\\mathfrak{m}$ be the maximal ideal of $R$. Then $R$ is not far-flung Gorenstein by Example \\ref{b2.8}. But \n\\[\n\\mathfrak{m}:\\mathfrak{m}=K[|t^4, t^5, t^6, t^7|]\n\\]\nis a far-flung Gorenstein ring by Corollary \\ref{ffg-minmult-aseq}.\n\\end{ex}\n\n\n\n\n\n\\section{Reflexive modules over far-flung Gorenstein rings}\\label{section5}\n\nLet $(R, \\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension one possessing the canonical module $\\omega_R$. \nSuppose that $R$ satisfies the conditions {\\rm (a)-(c)}. If $R$ is a far-flung Gorenstein ring, then we observe that by applying the functor $\\Hom_R(\\omega_R, -)$ to $R$\n\\[\n\\Hom_R(\\omega_R, R) \\cong R:C = R:\\overline{R} \\cong \\overline{R}.\n\\]\nThis observation can be generalized in the following way. Let $(-)^*$ and $(-)^\\vee$ denote the $R$-dual $\\Hom_R(-, R)$ and the canonical dual $\\Hom_R(-, \\omega_R)$, respectively.\n\n\\begin{thm} \\label{thm4.1}\nLet $R$ be a far-flung Gorenstein ring, and let $M$ be a reflexive module of rank $r>0$.  Then $\\Hom_R(\\omega_R, M) \\cong \\overline{R}^r$.\n\\end{thm}\n\n\\begin{proof} \nSince $M$ is reflexive, there exists an exact sequence \n$0 \\to M \\to F_1 \\to F_2,$\nwhere $F_1$ and $F_2$ are finitely generated free $R$-modules (see, for example, \\cite[Proposition~4.1]{HKS}). Hence, by applying the canonical dual $(-)^\\vee$, we obtain a surjection \n\\[\nF_1^\\vee \\cong \\omega_R^{\\rank_R F_1} \\to M^\\vee\n\\]\nbecause $X$ is a maximal Cohen-Macaulay $R$-module, where $X$ is the image of the map $F_1 \\to F_2$.  It follows that $\\mathrm{tr}_R(\\omega_R)\\supseteq \\mathrm{tr}_R(M^\\vee)$ by \\cite[Proposition 2.8(i)]{Lin}. Note that $\\mathrm{tr}_R(M^\\vee)$ contains a non-zerodivisor. Indeed, $\\mathrm{tr}_R(M^\\vee)_\\mathfrak{p}=\\mathrm{tr}_{R_\\mathfrak{p}} ((M^\\vee)_\\mathfrak{p})=R_\\mathfrak{p}$ for all associated prime ideals  $\\mathfrak{p}$ by \\cite[Proposition 2.8(viii)]{Lin} since $M^\\vee$ has a positive rank. It follows that $\\mathrm{tr}_R(M^\\vee)\\not\\subseteq \\mathfrak{p}$ for all associated prime ideals  $\\mathfrak{p}$.\n\nTherefore, we obtain that \n\\[\n\\overline{R} \\subseteq R:(R:\\overline{R})=R:\\mathrm{tr}_R(\\omega_R) \\subseteq R:\\mathrm{tr}_R(M^\\vee) =\\mathrm{tr}_R(M^\\vee) : \\mathrm{tr}_R(M^\\vee) \\subseteq \\overline{R},\n\\]\nwhere the fourth equality follows from \\cite[Corollary 2.2]{GIK2}.\nOn the other hand, we have a surjection\n\\[\n(M^\\vee)^* \\otimes_R M^\\vee \\xrightarrow{\\mathrm{ev}} \\mathrm{tr}_R(M^\\vee) \\to 0, \n\\]\nwhere $\\mathrm{ev}: f \\otimes x \\mapsto f(x)$ ($f\\in (M^\\vee)^*$ and $x\\in M^\\vee$), by definition of the trace ideal of $M^\\vee$. Therefore, by applying the $R$-dual to the above surjection, we obtain that \n\\[\n0 \\to R: \\mathrm{tr}_R(M^\\vee) = \\overline{R} \\to \\Hom_R((M^\\vee)^*, (M^\\vee)^*).\n\\]\nSince $(M^\\vee)^*$ is a reflexive module, we can regard $(M^\\vee)^*$ as an $\\overline{R}$-module (see \\cite[Proposition 2.4]{IK} or \\cite[(7.2) Proposition]{Ba}). Since $\\overline{R}$ is a discrete valuation ring, it follows that $(M^\\vee)^* \\cong (\\overline{R})^r$.\nThis completes the proof since we have the isomorphisms\n\\[\n(M^\\vee)^* \\cong \\Hom_R(M^\\vee \\otimes_R \\omega_R, \\omega_R) \\cong \\Hom_R(\\omega_R, M^{\\vee \\vee})\\cong \\Hom_R(\\omega_R, M).\n\\] \n\\end{proof}\n\n\n\n\\section{Revisiting the upper bound of the multiplicity}\\label{section5.5}\n\n\nAlthough Corollary \\ref{0.11} gives an upper bound for the multiplicity, it is not sharp. In what follows, we investigate a sharp upper bound of the multiplicity of far-flung Gorenstein numerical semigroup rings.\nLet $(R, \\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension one possessing the canonical module $\\omega_R$. \nSuppose that $R$ satisfies the conditions (a)-(d).\nIn this section, we further assume that the canonical ring homomorphism $R\/\\mathfrak{m} \\to \\overline{R}\/\\mathfrak{n}$ is bijective. \nFor an element $x\\in \\overline{R}$, let \n\\[\nv(x) = \\ell_{\\overline{R}}(\\overline{R}\/x\\overline{R})\n\\]\ndenote the {\\it discrete valuation} of $x$.\nChoose $f_1, f_2, \\dots, f_r\\in C$ such that \n\\[\nC=\\langle f_1, f_2, \\dots, f_r\\rangle,\n\\] \nwhere $r=\\mathrm{r}(R)$. \nSet $n_i=v(f_i)$ for $1\\le i \\le r$. We may assume that $n_1 \\le n_2 \\le \\cdots \\le n_r$ after replacing in the appropriate order. \n\nAssume that $n_i=n_j$ for some $1\\le i < j \\le r$. Then we have $f_i \\overline{R}=f_j \\overline{R}=\\mathfrak{n}^{n_i}$. Hence $f_j=f_i x$ for some $x\\in \\overline{R}\\setminus \\mathfrak{n}$. Since we have the isomorphism $\\varphi: R\/\\mathfrak{m} \\to \\overline{R}\/\\mathfrak{n}$, there exists $y\\in R\\setminus \\mathfrak{m}$ such that $x-y\\in \\mathfrak{n}$. Hence \n\\[\nf_j=(x-y+y)f_i=(x-y)f_i+yf_i, \\quad \\text{i.e., $f_j-yf_i=(x-y)f_i$}.\n\\]\nTherefore, by replacing $f_j$ with $f_j-yf_i$ among the generators of $C$, we may assume that $n_i<n_j$. It follows that we can choose a minimal system of generators $f_1, f_2, \\dots, f_r$ of $C$ such that \n\\begin{align} \\label{eq1}\nn_1 < n_2 < \\cdots < n_r.\n\\end{align}\n\nWith this notation introduced we have the following.\n\n\n\n\n\\begin{thm} \\label{b5.1}\nConsider the following assertions:\n\\begin{enumerate}[{\\rm (i)}] \n\\item $R$ is a far-flung Gorenstein ring.\n\\item There exists a fractional canonical module $C=\\left<f_1, f_2, \\dots, f_r\\right>$, where $n_i=v(f_i)$ satisfies the inequalities {\\rm (\\ref{eq1})}, such that the sum-set $\\{n_i+n_j : 1\\le i\\le j \\le r\\}$ contains the integers $0, 1, \\dots, \\mathrm{e}(R)-1$.\n\\end{enumerate} \nThen {\\rm (ii) $\\Rightarrow$ (i)} holds. The implication {\\rm (i) $\\Rightarrow$ (ii)} also holds if $R$ is a numerical semigroup ring $K[|H|]$. When the condition {\\rm (ii)} holds, then $n_1=0$ and $n_2=1$.\n\\end{thm}\n\n\\begin{proof} \n(ii) $\\Rightarrow$ (i): Let $m$ be a non-negative integer, and write \n\\[\nm=s{\\cdot}\\mathrm{e}(R) + n\n\\] \nfor some integers $s\\ge 0$ and $0\\le n < \\mathrm{e}(R)$. Let $f\\in C^2$ be an element with $v(f)=n$. \nBy using Remark \\ref{a2.1}(iii) and the proof of Corollary \\ref{0.11}, we have $v(a)=\\mu_R(\\overline{R})=\\mathrm{e}(R)$, where $(a)$ is a reduction of $\\mathfrak{m}$. Hence $a^s f\\in C^2$ is an element with $v(a^s f)=s{\\cdot}\\mathrm{e}(R) + n=m$. It follows that $C^2=\\overline{R}$ by \\cite{HK2} (see also \\cite[Proposition 1]{Matsu}).\n\n(i) $\\Rightarrow$ (ii): Suppose that $R=K[|H|]$ is a numerical semigroup ring. Then a fractional canonical module of $R$ is given by \n\\[\nC=\\left<t^{{\\rm F}(H)-\\alpha} : \\alpha \\in {\\rm PF}(H)\\right>\n\\]\nwhere ${\\rm PF}(H)=\\{x\\in  {\\NZQ Z} \\setminus H: x+h\\in H \\text{ for all } 0\\neq h \\in H\\}$ is the set of pseudo-Frobenius numbers of $H$ and ${\\rm F}(H)=\\max\\{n\\in \\mathbb{Z} : n\\not\\in H\\}$ denotes the Frobenius number of $H$ (see \\cite[Example (2.1.9)]{GW} and Section \\ref{section6}). Note that $C$ is a module satisfying the condition (a). Therefore, if $R$ is a far-flung Gorenstein ring, then $C^2=\\overline{R}$ by Theorem \\ref{0.8}. \n\nOn the other hand, $v(x)\\ge \\mathrm{e}(R)$ for all $x\\in \\mathfrak{m}$ since \n\\begin{align*}\n\\ell_{\\overline{R}} (\\overline{R}\/x \\overline{R}) \\ge \\ell_{\\overline{R}} (\\overline{R}\/\\mathfrak{m} \\overline{R})=\\ell_R (\\overline{R}\/\\mathfrak{m} \\overline{R})= \\mu_R(\\overline{R})=\\mathrm{e}(R),\n\\end{align*}\nwhere the second equality follows from the assumption that $R\/\\mathfrak{m} \\cong \\overline{R}\/\\mathfrak{n}$, the third equality follows by Remark \\ref{a2.1}(iii), the fourth equality follows from the proof of Corollary \\ref{0.11}. \n\nHence, by noting that $C^2$ is generated by monomials, for each $0\\le n <\\mathrm{e}(R)$, an element $f$ with $v(f)=n$ can be chosen as $t^{\\mathrm{F}(H)-\\alpha}{\\cdot}t^{\\mathrm{F}(H)-\\beta}$, where $\\alpha, \\beta\\in \\mathrm{PF}(H)$. It follows that by setting $f_i=t^{\\mathrm{F}(H)-\\alpha_{r-i}}$, where ${\\rm PF}(H)=\\{\\alpha_1<\\alpha_2<\\cdots <\\alpha_r=\\mathrm{F}(H)\\}$, we obtain the assertion (ii).\n\nIn particular, there exist $1\\le i \\le j \\le r$ such that $v(f_i{\\cdot}f_j)=1$, i.e., $n_i+n_j=1$. It follows that $n_1=0$ and $n_2=1$.\n\\end{proof}\n\nThe assertion (ii) of Theorem \\ref{b5.1} is tightly related to the Rohrbach problem:\n\n\\begin{prob} (The Rohrbach problem, \\cite{Ro, Slo})\nLet $A$ be a set of non-negative integers with $r$ elements. Let $n(A)$ denote the integer such that \nthe sum-set \n\\[\nA+A=\\{a+b:a,b\\in A\\}\n\\] contains the integers $0, 1, \\dots, n(A)-1$ but not $n(A)$. If $0\\notin A$ then let $n(A)=-1$.\nFor $r>0$ the Rohrbach problem asks to find the integer \n\\[\n\\overline{n}(r)=\\max\\{ n(A):  |A|=r\\}. \n\\] \n\\end{prob}\n\nWith this notation we have the following result, which improves Corollary \\ref{0.11}.\n\n\\begin{cor} \\label{cor5.3}\nSuppose that $R$ is a numerical semigroup ring. If $R$ is a far-flung Gorenstein ring, then the inequality\n\\[\n\\mathrm{e}(R) \\le \\overline{n}(r)\n\\] \nholds, where $r=\\mathrm{r}(R)$.\n\\end{cor}\n\n\nThe solution $\\overline{n}(r)$ of the Rohrbach problem is known for $r\\le 25$ (see \\cite{KoCo, Slo}):\n\n\\begin{table}[htb]\n  \\begin{tabular}{|c||rrrrrrrrrrrrr|} \\hline\n$r$            & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13\\\\ \n$\\overline{n}(r)$ &1 & 3 & 5 & 9 & 13 & 17 & 21 & 27 & 33 & 41 & 47 & 55 & 65 \\\\ \\hline\n$r$            & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & $\\cdots$\\\\ \n$\\overline{n}(r)$ &73 & 81 & 93 & 105 & 117 & 129 & 141 & 153 & 165 & 181 & 197 & 213 & $\\cdots$\\\\ \\hline\n  \\end{tabular}\n\\end{table}\n\n\n\n\n The following example ensures that $\\overline{n}(r)$ provides the sharp upper bounds of the multiplicity of far-flung Gorenstein numerical semigroup rings, if $r=\\mathrm{r}(R)\\le 5$.\n\n\\begin{ex} \nLet $K$ be a field and $K[|t|]$ denote the formal power series ring. Then by using Proposition \\ref{b6.1} it is easy to check that the following hold true.\n\\begin{enumerate}[{\\rm (i)}] \n\\item Let $R_1=K[|t^5, t^6, t^{13}, t^{14}|]$. Then $R_1$ is a far-flung Gorenstein ring with $\\mathrm{r}(R_1)=3$ and $\\mathrm{e}(R_1)=5$.\n\\item Let $R_2=K[|t^9, t^{10}, t^{11}, t^{12}, t^{15}|]$. Then $R_2$ is a far-flung Gorenstein ring with $\\mathrm{r}(R_2)=4$ and $\\mathrm{e}(R_1)=9$.\n\\item Let $R_3=K[|t^{13}, t^{14}, t^{15}, t^{16}, t^{17}, t^{18}, t^{21}, t^{23}|]$. Then $R_3$ is a far-flung Gorenstein ring with $\\mathrm{r}(R_3)=5$ and $\\mathrm{e}(R_1)=13$.\n\\end{enumerate} \n\\end{ex}\n\n\\begin{quest}\n\\begin{enumerate}[{\\rm (i)}] \n\\item Is it true that for any $r\\geq 2$ there exists a far flung Gorenstein numerical semigroup ring $R$ of type $r$ with $\\mathrm{e}(R)=\\overline{n}(r)$?\n\\item For any $r\\geq 2$, what is the exact range of values of $\\mathrm{e}(R)$ when $R$ runs over all far-flung Gorenstein numerical semigroup rings of type $r$? \n\\end{enumerate}\n\\end{quest}\n\n\n\n\\section{far-flung Gorenstein numerical semigroup rings}\\label{section6}\n\n\nIn this section we investigate numerical semigroup rings. When a ring is a numerical semigroup ring, we can check the far-flung Gorenstein property by using only the data of the numerical semigroup (Proposition \\ref{b6.1}). First, let us recall some basic notation of numerical semigroup rings. \n\nA numerical semigroup $H$ is a submonoid of ${\\NZQ N}_0$ such that ${\\NZQ N}_0\\setminus H$ is finite.\nThe set \n\\[\n{\\rm PF}(H)=\\{x\\in  {\\NZQ Z} \\setminus H: x+h\\in H \\text{ for all } 0\\neq h \\in H\\}\n\\] \nis called the set of the {\\it pseudo-Frobenius numbers} of $H$. \nThe largest value in ${\\rm PF}( H)$ is called the {\\it Frobenius number} of $H$, denoted ${\\rm F}(H)$. The smallest nonzero element in $H$ is called the {\\it multiplicity} of $H$ and we denote it by ${\\rm e}(H)$.\n\nLet $K$ be a field. Then, the pseudo-Frobenius numbers of $H$ define the canonical module of the numerical semigroup ring $R=K[|H|]$. Indeed, it is known that a graded canonical module of $S=K[H]$ is $\\omega_S=\\sum_{\\alpha \\in {\\rm PF}(H)} S t^{-\\alpha}$, see \\cite[Example (2.1.9)]{GW}. After multiplication by $t^{{\\rm F}(H)}$ we obtain the canonical module\n\\[\nD=\\sum_{\\alpha \\in {\\rm PF}(H)}  St^{{\\rm F}(H)-\\alpha}\n\\] \nsuch that $S\\subseteq D\\subseteq \\overline{S}=K[t]$. By noting that $R=K[|H|]$ is the completion of $S_\\mathfrak{M}$, where $\\mathfrak{M}$ is the graded maximal ideal of $S$, we obtain that\n\\[\nC=\\sum_{\\alpha \\in {\\rm PF}(H)}  Rt^{{\\rm F}(H)-\\alpha}\n\\]\nis a module appearing in the condition (a). It is known that $\\mathrm{e}(R)=\\mathrm{e}(H)$. With this notation we have the following.\n\n\\begin{Proposition}\\label{b6.1}\n\\label{prop:ffg-h}\nLet $R=K[|H|]$ be the semigroup ring for the numerical semigroup $H$. The following statements are equivalent:\n\\begin{enumerate}[{\\rm (i)}] \n\\item  $R$ is a far-flung Gorenstein ring.\n\\item $\\{0, \\dots, {\\rm e}(H)-1\\} \\subseteq  \\{ 2 {\\rm{F}(H)}-\\alpha-\\beta: \\alpha, \\beta \\in {\\rm PF}(H)\\}$.\n\\item $\\{2{\\rm F}(H)- {\\rm e}(H)+1,\\dots, 2{\\rm F}(H)\\} \\subseteq  \\{\\alpha+\\beta: \\alpha, \\beta\\in {\\rm PF}(H)\\}$.\n\\end{enumerate}\n\\end{Proposition}\n\n\\begin{proof}\nAccording to Theorem \\ref{0.8}, $R$ is far-flung Gorenstein if and only if $C^2=K[|t|]$.  \nThis proves that (i) $\\Leftrightarrow$ (ii) since $C^2$ is generated by monomials. \nSimple algebraic manipulations show that (ii) $\\Leftrightarrow$ (iii). \n\\end{proof}\n\n\\begin{Corollary}\\label{ffg-minimal}\nLet $H$ be a numerical semigroup minimally generated by \n$a_1<\\dots< a_v$ which is of minimal multiplicity, i.e. $v=a_1$.  Then $K[|H|]$ is a far-flung Gorenstein ring if and only if \n\\begin{equation} \\label{inclusion}\n\\{2 a_v-a_1+1,\\dots, 2 a_v\\}\\subseteq \\{a_i+a_j:2\\leq i, j\\leq v\\}.\n\\end{equation}\n\\end{Corollary}\n\n\\begin{proof}\nSince $H$ has minimal multiplicity we have \n${\\rm PF}(H)=$ $\\{ a_2-a_1, \\dots, a_v-a_1\\}$ and $a_1=v$\n (see \\cite{RS}).  The conclusion now follows from Proposition \\ref{prop:ffg-h}(iii).\n\\end{proof}\n\nThis corollary has an immediate application, where we find far-flung numerical semigroup rings with minimal multiplicity   of arbitrary embedding dimension.\n\n\\begin{Corollary}\\label{ffg-minmult-aseq}\nLet $a\\geq 3$ and $d$ be coprime nonnegative integers and $H=\\langle a, a+d, \\dots, a+(a-1)d \\rangle$. Then $R=K[|H|]$ is a far-flung Gorenstein ring if and only if $d=1$.\n\\end{Corollary} \n\\begin{proof}\nAssume that $R$ is far-flung Gorenstein. Then the sum of any two generators of $H$ larger then $a$ is of the form $2a+ jd$, hence we can not generate with them all the integers in any interval of length $a$ unless $d=1$. Conversely, when $d=1$  we verify that \\eqref{inclusion} holds: on the right hand side  we have the set $\\{2a+2, \\dots, 4a-2\\}$, which clearly contains the set $\\{3a-1,\\dots, 4a-2\\}$. Thus, $R$ is a far-flung Gorenstein ring. \n\\end{proof}\n\n\nWe shall now describe the far-flung Gorenstein numerical semigroup rings of type $3$, not of minimal multiplicity. Recall that the classification of the far-flung Gorenstein numerical semigroup rings of type $2$ is given in Example \\ref{b2.8}. \n\n\\begin{Theorem}\nLet $R=K[|H|]$ be a numerical semigroup ring. Suppose that $R$ is of type 3 and not of minimal multiplicity. Then the following conditions are equivalent:\n\\begin{enumerate}[{\\rm (i)}] \n\\item $R$ is a far-flung Gorenstein ring.\n\\item \\begin{enumerate}\n\\item[{\\rm (1-1)}] $H=\\langle 5, 5m+4, 10m+6, 10m+7 \\rangle$,  where $m\\geq 1$;\n\\item[{\\rm (1-2)}] $H=\\langle 5, 5m+1, 10m+3, 10m+4 \\rangle$,  where $m\\geq 1$; \n\\item[{\\rm (2-1)}] $H=\\langle 5, 5m+2, 10m+1, 10m+3 \\rangle$,  where $m\\geq 1$;\n\\item[{\\rm (2-2)}] $H=\\langle 5, 5m+3, 10m+4, 10m+7\\rangle$,  where $m\\geq 1$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{Theorem}\n\n\\begin{proof}\n(i) $\\Rightarrow$ (ii): By Corollaries \\ref{0.11} and \\ref{cor5.3}, we have $\\mathrm{e}(R)=5$. Hence $\\embdim R=4$ because  $\\embdim R<\\mathrm{e}(R)$ and $\\embdim R>3$ by \\cite{H}. Furthermore, $C =\\sum_{\\alpha\\in \\mathrm{PF}(H)} Rt^{\\mathrm{F}(H)-\\alpha}$ is generated by either $1, t, t^2$ or $1, t, t^3$.\n\nAssume that $C= \\langle 1, t, t^2 \\rangle$. Then, since $\\mathrm{e}(R)=5$, we have either the case  \n\\begin{align*}\n\\text{{\\rm (1-1):}}& \\quad \\mathrm{PF}(H)=\\{ 5n+1, 5n+2, 5n+3\\} \\quad \\text{or}\\\\\n\\text{{\\rm (1-2):}}& \\quad \\mathrm{PF}(H)=\\{ 5n+2, 5n+3, 5n+4\\}\n\\end{align*}\nfor some $n\\ge 0$. \n\n{\\rm (1-1):} Considering the smallest nonzero element in $H$ in each congruence class modulo $5$,  there exists $m>0$ so that $H=\\langle 5, 5n+6, 5n+7, 5n+8, 5m+4 \\rangle$  and $5m+4-5=5m-1\\not\\in H$. \nThe pseudo-Frobenius numbers of $H$ are also the elements in ${\\NZQ Z} \\setminus H$ which are maximal with respect to the partial order $\\leq_H$ induced by $H$, where $a\\leq_H b$ if and only if $b-a \\in H$. \nSince $5m-1\\not\\in \\mathrm{PF}(H)$, we get  $5m-1 \\leq_H 5n+1$, or $5m-1\\leq_H 5n+2$, or $5m-1\\leq_H 5n+3$, i.e. \n\\begin{align*}\n\\begin{cases}\n5n+3-(5m-1)\\in H \\quad \\text{or} \\\\\n5n+2-(5m-1)\\in H \\quad \\text{or} \\\\\n5n+1-(5m-1)\\in H. \n\\end{cases}\n\\end{align*}\nConsidering the smallest elements in $H$  modulo 5, it is equivalent to saying that\n\\begin{align*}\n\\begin{cases}\n5n+3-(5m-1)\\ge 5m+4 \\quad \\text{or} \\\\\n5n+2-(5m-1)\\ge 5n+8 \\quad \\text{or} \\\\\n5n+1-(5m-1)\\ge 5n+7.\n\\end{cases}\n\\end{align*}\n  Of these, only $5n+3-(5m-1)\\ge 5m+4$ can happen. Hence, we get $2m\\le n$.  Thus $5n+8= 2(5m+4)+5(n-2m)$, which gives \n\\[\nH= \\langle 5,  5m+4 , 5n+6, 5n+7\\rangle.\n\\]\n\nThe fact that $5n+3 \\notin H$ is equivalent to $5n+3\\notin \\langle 5, 5m+4\\rangle$. On the other hand, for any positive integer $p$ one has $5p+3\\notin \\langle 5, 5m+4\\rangle$ if and only if there exist $u,v$ nonnegative integers such that $5p+3=5u+(5m+4)v$. Arguing modulo $5$, the previous equation has nonnegative integer solutions if and only if $5p+3\\geq 2(5m+4)$, equivalently $p\\geq 2m+1$.\nTherefore, $n\\leq 2m$, which gives that $n=2m$ and \n$$\nH=\\langle 5, 5m+4, 10m+6, 10m+7 \\rangle, \\text{ for }m\\geq 1.\n$$\nConversely, it is routine to check that such a semigroup $H$ is minimally generated by these four numbers and that ${\\rm PF}(H)\\supseteq\\{10m+1, 10m+2, 10m+3\\}$. Since $H$ is not of minimal multiplicity, the previous inclusion is an equality. This completes the analysis of the case (1-1).\n\nThe rest of the proof proceeds in a similar  way.\n\n{\\rm (1-2):} Arguing modulo $5$ we find $m>0$ so that $H=\\langle 5, 5m+1, 5n+7, 5n+8, 5n+9 \\rangle$ and $5m+1-5\\not\\in H$. Since $5m-4\\not\\in \\mathrm{PF}(H)$, we get that\n\\begin{align*}\n\\begin{cases}\n5n+2-(5m-4)\\in H \\quad \\text{or} \\\\\n5n+3-(5m-4)\\in H \\quad \\text{or} \\\\\n5n+4-(5m-4)\\in H. \n\\end{cases}\n\\end{align*}\nConsidering the smallest  element in $H$ in the same congruence class modulol $5$, it is equivalent to saying that \n\\begin{align*}\n\\begin{cases}\n5n+2-(5m-4)\\ge 5m+1 \\quad \\text{or} \\\\\n5n+3-(5m-4)\\ge 5n+7 \\quad \\text{or} \\\\\n5n+4-(5m-4)\\ge 5n+8\n\\end{cases}\n\\end{align*}\n Of these three inequalities, only  the first can happen. Hence, we get $2m-1\\le n$. Since $5n+7= 2(5m+1)+ 5(n-2m+1)$ we obtain that  \n\\[\nH= \\langle 5, 5m+1, 5n+8, 5n+9 \\rangle.\n\\]\n\nClearly, $5n+2 \\notin H$ if and only if $5n+2\\notin \\langle 5, 5m+1\\rangle$. On the other hand, for any integer $p$, $5p+2 \\in \\langle 5, 5m+1\\rangle$ if and only if $5p+2=5u+(5m+1)v$ for some nonnegative integers $u,v$, equivalently, $5p+2 \\geq  2(5m+2)$, i.e. $p\\geq 2m$. Thus $n\\leq 2m-1$, which forces $n=2m-1$ and \n$$\nH=\\langle 5, 5m+1,  10m+3, 10m+4\\rangle \\text{ with } m \\geq 1.\n$$ \nConversely, when $H$ is of this form, it is easy to check that ${\\rm PF}(H)=\\{10m-3, 10m-2, 10m-1\\}$, hence $R$ is far-flung Gorenstein.\n\nNext let us assume that $C=\\langle 1, t, t^3 \\rangle$. Then, since $\\mathrm{e}(R)=5$, we have  either\n\\begin{align*}\n\\text{{\\rm (2-1):}}& \\quad \\mathrm{PF}(H)=\\{ 5n+1, 5n+3, 5n+4\\} \\quad \\text{or}\\\\\n\\text{{\\rm (2-2):}}& \\quad \\mathrm{PF}(H)=\\{ 5n+4, 5n+6, 5n+7\\}\n\\end{align*}\nfor $n\\ge 0$. \n\n{\\rm (2-1):} In this case we can choose $m>0$ so that $H=\\langle 5, 5n+6, 5m+2, 5n+8, 5n+9 \\rangle$ and $5m+2-5\\not\\in H$. Since $5m-3\\not\\in \\mathrm{PF}(H)$, we get \n\\begin{align*}\n\\begin{cases}\n5n+1-(5m-3)\\in H \\quad \\text{or} \\\\\n5n+3-(5m-3)\\in H \\quad \\text{or} \\\\\n5n+4-(5m-3)\\in H. \n\\end{cases}\n\\end{align*}\nConsidering the smallest element in $H$  with the same residue modulo $5$, it is equivalent to saying that \n\\begin{align*}\n\\begin{cases}\n5(n-m)+4\\ge 5n+9 \\quad \\text{or} \\\\\n5(n-m)+6\\ge 5n+6 \\quad \\text{or} \\\\\n5(n-m)+7\\ge 5m+2\n\\end{cases}\n\\end{align*}\n Of these, only the latter inequality  can happen.  Hence, we get $2m-1\\le n$.  Then $5n+9=5(n-2m+1)+2(5m+2)$ and  \n\\[\nH=\\langle 5, 5m+2, 5n+6,  5n+8\\rangle.\n\\]\n\nThe fact that $5n+4\\notin H$ is equivalent to $5n+4\\notin \\langle 5, 5m+2\\rangle$. For any integer $p$, $5p+4 \\in \\langle 5, 5m+2\\rangle$ if and only if $5p+4 \\geq 2(5m+2)$, i.e. $p\\geq 2m$. We derive $n\\leq 2m-1$, hence $n=2m-1$ and \n$$\nH=\\langle 5, 5m+2, 10m+1, 10m+3\\rangle, \\text{ where } m\\geq 1.\n$$\nIt is routine to check that such a semigroup has the   pseudo-Frobenius numbers as in (2-1).\n\n{\\rm (2-2):} In this case we find  $m>0$ so that  $H=\\langle 5, 5n+11, 5n+12, 5m+3, 5n+9 \\rangle$ and $(5m+3)-5\\not\\in H$. Since $5m-2\\not\\in \\mathrm{PF}(H)$, we get \n\\begin{align*}\n\\begin{cases}\n5n+4-(5m-2)\\in H \\quad \\text{or} \\\\\n5n+6-(5m-2)\\in H \\quad \\text{or} \\\\\n5n+7-(5m-2)\\in H. \n\\end{cases}\n\\end{align*}\nArguing as before, it is equivalent to saying that \n\\begin{align*}\n\\begin{cases}\n5(n-m)+6\\ge 5n+11 \\quad \\text{or} \\\\\n5(n-m)+8\\ge 5m+3 \\quad \\text{or} \\\\\n5(n-m)+9\\ge 5n+9.\n\\end{cases}\n\\end{align*}\nOf these three inequalities, only  the middle one can occur, hence  $2m-1\\le n$. We write $5n+11=2(5m+3)+5(n-2m+1)$ and thus \n\\[\nH= \\langle 5,5m+3,  5n+9, 5n+12 \\rangle.\n\\]\nSince $5n+6\\notin H$, arguing as before we see that $n\\leq 2m-1$, and in fact $n=2m-1$. Hence\n$$\nH=\\langle 5, 5m+3, 10m+4, 10m+7\\rangle, \\text{ where } m\\geq 1.\n$$\nConversely, it is easy to check that $10m-1, 10m+1, 10m+2$ are indeed the pseudo-Frobenius numbers of  such an $H$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Circular Electron Positron Collider (CEPC) is a proposed electron-positron collider with a total circumference of 100 km and two interaction points.\nIt will be operated at center-of-mass energies from 91 GeV to 240 GeV and produces large samples of the W, the Z, and the Higgs bosons.\nIts nominal luminosity and massive boson yields are listed in Table~\\ref{plain} \\cite{r1}.\nThe CEPC can measure most of the Higgs boson properties with accuracies that exceed the ultimate precision of the HL-LHC by one order of magnitude, and also boost current precision of the Electroweak (EW) measurements by one order of magnitude. \nThe CEPC can also be upgraded to a proton-proton collider with a center-of-mass energy around 100 TeV. \n\n\\begin{table}\n\\centering\n\\caption{Running time, instantaneous and integrated luminosities at different values of the center-of-mass energy and anticipated corresponding boson yields at the CEPC. The Z boson yields of the Higgs factory and WW threshold scan operation are from the initial-state radiative return $e^{+}e^{-} \\to \\gamma Z$ process. The ranges of luminosities for the Z factory correspond to the two possible solenoidal magnetic fields, 3 or 2 Tesla.}\n\\label{plain}\n\\begin{tabular*}{\\columnwidth}{@{\\extracolsep{\\fill}}cccc@{}}\n\\hline\n \\multirow{2}{*}{Operation mode}   & \\multirow{2}{*}{ Z factory}                &  WW                                                  &\\multirow{2}{*}{Higgs factory}        \\\\\n                                                       &                                                         & threshold scan                                  &                                                       \\\\\n\\hline \n$\\sqrt{s}$ (GeV)                              & \\multirow{1}{*}{91.2}                            &  \\multirow{1}{*}{158 - 172}                  &  \\multirow{1}{*}{ 240}                        \\\\\n\\hline\nRunning time                             &   \\multirow{2}{*}{2 }                               &   \\multirow{2}{*}{1}                               & \\multirow{2}{*}{7}                            \\\\\n(years)                                       &  &  &  \\\\\n\\hline\nInstantaneous                           & \\multirow{3}{*}{17 - 32}                        &   \\multirow{3}{*}{10}                             &  \\multirow{3}{*}{3}                           \\\\\nLuminosity                                 &   &     &     \\\\\n($10^{34} cm^{-2}s^{-1}$)          &   &     &     \\\\\n\\hline\nIntegrated Luminosity                & \\multirow{2}{*}{8 - 16}                           &    \\multirow{2}{*}{2.6}                            & \\multirow{2}{*}{5.6}                       \\\\\n($ab^{-1}$)                                &     &      & \\\\\n\\hline\nHiggs yield                                &   -                                                           &     -                                                        & $10^{6}$                  \\\\\n\\hline\nW yield                                     &   -                                                            &   $10^{7} $                                            & $10^{8}$                   \\\\\n\\hline\nZ yield                                      & $10^{11 - 12}$                                        &    $10^{8}$                                             & $10^{8}$                  \\\\      \n   \\hline\n\\end{tabular*}\n\\end{table}\n\nAt 240 GeV center of mass energy, the Higgs boson is mainly produced through the ZH process at the CEPC. \nThe leading di-boson Standard Model backgrounds for the CEPC Higgs measurements are the WW and ZZ processes, see Figure~\\ref{crosssection}.\nA successful separation between the Higgs signal and the di-boson backgrounds is essential for the precise Higgs measurements. \nIn addition, the separation of the WW and ZZ events is important for the QCD measurement, the Triplet Gauge Boson Coupling measurement, and the W boson mass measurement at continuum.\n\nHalf of these di-boson events decay into 4-jet final states.\nThe separation between those 4-jet events is determined by the intrinsic boson mass distribution, the detector performance, and the jet confusion.\nThe latter refers to the uncertainties induced by the jet clustering and pairing algorithm.\nGiving the relatively small mass difference between the W boson and the Z boson, the separation between the WW and the ZZ events in the full hadronic final states is extremely demanding in the detector performance and the jet confusion control. \nTherefore it serves as a stringent benchmark for the detector design and reconstruction algorithm development. \nUsing the CEPC baseline detector geometry and software, we investigate the separation performance of the full hadronic WW and ZZ events at full simulation level.\nWe confirm that these events can be clearly separated with the CEPC baseline detector. \nThrough comparative analyses, we quantify the impacts of each component and conclude the jet confusion dominates the separation performance. \n\n\n\\begin{figure}\n\\begin{minipage}{\\columnwidth}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{crossection.png}\n\\end{minipage}\n\\caption{The cross section for unpolarized $e^{+}e^{-}$ collision, the right side shows the expected number of events at the nominal parameters of the CEPC Higgs runs at 240 GeV center-of-mass energy.}\n\\label{crosssection}\n\\end{figure}\n\nThis paper is organized as follows. \nSection 2 introduces the CEPC baseline detector geometry and the software. \nThe analysis method and the separation performance at various conditions are quantified and compared in section 3. \nUsing the Monte Carlo (MC) truth information, section 4 further analyzes the jet confusion. \nThe conclusion is summarized in section \\ref{secConclusion}. \n\n\\section{Detector geometry, software, sample and analysis method}\n\nA Particle Flow oriented detector design is the baseline detector as described the CEPC CDR~\\cite{r1}. \nThis baseline reconstructs all the visible final state particles in the most-suited detector subsystems.\nFor the CEPC physics measurements, this baseline reconstructs all the core physics objects with high efficiency, high purity, and high precision~\\cite{r1}\\cite{r6}. \nFrom inner to outer, the detector is composed of a silicon pixel vertex detector, a silicon inner tracker, a Time Projection Chamber (TPC) surrounded by a silicon external tracker, a silicon-tungsten sampling Electromagnetic Calorimeter (ECAL), a steel-Glass Resistive Plate Chambers (GRPC) sampling Hadronic Calorimeter (HCAL), a 3 Tesla superconducting solenoid, and a flux return yoke embedded with a muon detector. \nThe structure of the CEPC detector is shown in Figure~\\ref{APODISstructure}.\nIn fact, the separation of vector bosons scattering processes (with $\\nu\\nu$WW and $\\nu\\nu$ZZ final states) provides a strong motivation for the Particle Flow oriented detector design~\\cite{ILCTDR}\\cite{CLICCDR}.\n\n\\begin{figure}\n\\begin{minipage}{\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth]{APODISstructure.png}\n\\end{minipage}\n\\caption{The CEPC baseline detector. From inner to outer, the detector is composed of a silicon pixel vertex detector, a silicon inner tracker, a TPC, a silicon external tracker, an ECAL, an HCAL, a solenoid of 3 Tesla and a return yoke embedded with a muon detector. In the forward regions, five pairs of silicon tracking disks are installed to enlarge the tracking acceptance.}\n\\label{APODISstructure}\n\\end{figure}\n\n\\begin{figure}\n\\begin{minipage}{\\columnwidth}\n\\centering\n\\includegraphics[width=\\columnwidth]{softwarechain.png}\n\\end{minipage}\n\\caption{The information flow of the CEPC software chain.}\n\\label{softwarechain}\n\\end{figure}\n\n\\begin{figure}\n\\begin{minipage}{\\columnwidth}\n\\centering\n\\includegraphics[width=7.1cm, height = 6.8cm]{WW-event.png}\n\\end{minipage}\n\\caption{The display of a reconstructed WW event. This event has 82 final state particles whose energy exceed 0.5 GeV, reconstructed by Arbor. The charged particles are represented by the curves (color represent their charge) associated with calorimeter clusters. The photons are displayed as cyan straight lines associated with calorimeter clusters.}\n\\label{Reco}\n\\end{figure}\n\nThe CEPC baseline software is demonstrated in Figure~\\ref{softwarechain}. \nIt uses the Whizard~\\cite{r2}\\cite{r3} and the Pythia~\\cite{r4} generators as the starting point. \nThe detector geometry is implemented into the MokkaPlus~\\cite{r8}, a GEANT4~\\cite{r5} based full simulation module.\nThe MokkaPlus calculates the energy deposition in the detector sensitive volumes and creates simulated hits.\nFor each sub-detector, the digitization module converts the simulated hits into digitized hits by convoluting the corresponding sub-detector responses. \nThe reconstruction modules include the tracking, the Particle Flow, and the high-level reconstruction algorithms. \nThe digitized tracker hits are reconstructed into tracks via the tracking modules. \nThe particle flow algorithm, Arbor~\\cite{r6}, reads the reconstructed tracks and the calorimeter hits to build reconstructed particles. \nHigh-level reconstruction algorithms reconstruct composite physics objects such as the converted photons, the $\\tau$s, the jets, et al., and identify the flavor of the jets.  \n\n\nUsing the CEPC baseline simulation, \nwe produce inclusive samples of 38k WW and 38k ZZ events.\nThese samples include all the different quark flavors according to the SM decay branching ratios.\nTo simplify the analysis, the interference between WW and ZZ is ignored.  \nTo analyze the impact of heavy flavors, we also produce light flavor samples for comparison. \nThese light flavor samples are 30k $WW\\to u\\bar{d}\\bar{u}s$ or $u\\bar{s}\\bar{u}d$ and 27k $ZZ\\to u\\bar{u}u\\bar{u}$ events. \nFigure~\\ref{Reco} displays a reconstructed $e^{+}e^{-}\\to WW \\to u\\bar{u}s\\bar{d}$ event using Druid~\\cite{r12}.\nAll the samples are generated at the center-of-mass energy of 240 GeV. \n\n\nStarting with the fully reconstructed WW\/ZZ events, our analysis employs the jet clustering and pairing algorithm.  \nThe reconstructed particles are clustered into four RecoJets using the $\\it{k_t}$ algorithm for the $e^{+}e^{-}$ collisions ($\\it{e^+e^- k_t}$) with the FastJet package~\\cite{r7}. \nA minimal $\\chi^{2}$ method is used for the jet pairing. \nThese four RecoJets are paired into two di-jet systems. \nTheir masses are compared with the hypothesis of a WW or a ZZ event via the $\\chi^{2}$ defined as:\n\n $$\\chi^{2} = \\frac{(M_{12} - M_{B})^2 + (M_{34} - M_{B})^2}{\\sigma_{B}^{2}}. $$ \n \n\\begin{table}\n\\centering\n\\caption{The values of $\\sigma_B$ for different cases.}\n\\label{sigma}\n\\begin{tabular*}{0.9\\columnwidth}{@{\\extracolsep{\\fill}}ccc@{}}\n\\hline\n$\\sigma_B$\/GeV       & $\\sigma_W$     &  $\\sigma_Z$          \\\\\n\\hline \nGenJet                      &   2.0                     &  2.5                         \\\\\n\\hline\nRecoJet                    &   3.8                  &  4.4                          \\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\nThe quantity $M_{12}$ and $M_{34}$ refer to the masses of di-jet systems, and $M_{B}$ is the reference mass of the Z or the W boson~\\cite{r9}. \nThe $\\sigma_{B}$ is the convolution of the boson width and the detector resolution. \nAccording to~\\cite{r1}, the detector resolution is set to be 4\\% of the boson mass. \nThe values of the $\\sigma_{B}$ for different cases are listed in Table~\\ref{sigma}.\nAmong all six possible combinations (corresponding to three different jet pairings and two values of $M_{B}$), \nthe one with the minimal value of the $\\chi^{2}$ determines the event type and corresponding di-jet masses. \n\nUsing the same jet clustering and pairing setup for the RecoJets analysis, the visible particles at the MC truth level can be clustered into the GenJets and paired into di-jet systems. \nThese GenJets are corresponding to the perfect detector, \nand the separation performance using the GenJets describes the impacts of the intrinsic boson mass distribution and the jet confusion.\nIn this paper, the analyses are performed using both the RecoJets and the GenJets.  \n\n\n\\section{Separation Performance with Overlapping Ratio } \n\nUsing the method introduced above, the masses of the di-jet systems ($M_{12}$ and $M_{34}$) are calculated.  \nFigure~\\ref{czwHerror} shows the average reconstructed di-jet mass distributions of the inclusive WW and ZZ samples using the RecoJets, each normalized to unit area. \nEach distribution exhibits a clear peak at the anticipated boson mass and an artificial tail towards the other peak.\nThese tails are induced by the jet pairing algorithm, the neutrinos generated in heavy flavor quark fragmentation, and the ISR photons. \nThe peaks are clearly separated, however, the tails lead to significant confusion between the WW and ZZ events. \n \nThe confusion can be evaluated by the overlapping ratio between two distributions: \n\n$$ Overlapping\\ Ratio = \\sum_{bins}min({a_{i}, b_{i}}),$$\n\n$a_{i}$ and $b_{i}$ are the bin contents of both distributions at a same bin.  \nTo the first order, the overlapping ratio is equal to the sum of misidentification probabilities ($P_{WW\\to ZZ} + P_{ZZ \\to WW}$ in this manuscript).\nAn overlapping ratio of zero means no mis-identification. \n\nThrough a parameter scan of the generalised $\\it{k_t}$ algorithm for the ${e^+e^-}$ collision, \nthe $\\it{e^+e^- k_t}$ algorithm is chosen for this analysis as it has the minimum overlapping ratio on the inclusive sample.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{czwHerror.png}\n\\caption{The RecoJet level distributions of $0.5\\times(M_{12} + M_{34})$ of the WW and ZZ events. The overlapping ratio is $57.8\\% \\pm 0.23\\%$.}\n\\label{czwHerror}\n\\end{figure}\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{czwDiserror.png}\n\\caption{The RecoJet level distribution of $M_{12}$ versus $M_{34}$. }\n\\label{czwDiserror}\n\\end{figure}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cGzwHerror.png}\n\\caption{The GenJet level distributions of $0.5\\times(M_{12} + M_{34})$ of the WW and ZZ events. The overlapping ratio is $52.6\\% \\pm 0.25\\%$.}\n\\label{cGzwHerror}\n\\end{figure}\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cGzwDiserror.png}\n\\caption{The GenJet level distribution of $M_{12}$ versus $M_{34}$.}\n\\label{cGzwDiserror}\n\\end{figure}\n\nFigure~\\ref{czwHerror} has an overlapping ratio of $57.8\\%\\pm 0.23\\%$.\nThe correlation of $M_{12}$ versus $M_{34}$ using the RecoJet is shown in Figure~\\ref{czwDiserror}, the distributions of the WW and ZZ events are overlapped. \nFigure~\\ref{czwDiserror} has two separable peaks located on a large area of a flat plateau. \nThe latter contributes significantly to the overlapping ratio. \n\nThe separation performance at the GenJet level is also analyzed. \nFigure~\\ref{cGzwHerror} shows the distributions of average di-jet mass and has an overlapping ratio of $52.6\\% \\pm 0.25\\%$.   \nComparing to the RecoJet distributions, Figure~\\ref{cGzwHerror} exhibits much narrow peaks and similar tails.    \nThat's to say, the peak width of the RecoJet distributions are mainly dominated by the detector performance.             \nThe correlation between $M_{12}$ versus $M_{34}$ with the GenJets is shown in Figure~\\ref{cGzwDiserror}.\nAside from two clearly separable peaks, Figure~\\ref{cGzwDiserror} also has a plateau with similar contour and area comparing to Figure~\\ref{czwDiserror}, the distribution at RecoJet level.\nClearly, the common patterns of the GenJet and the RecoJet level distributions are induced by the intrinsic boson mass and the jet confusion.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{czwDissDmass.png}\n\\caption{ The RecoJet level distribution of $M_{12}$ versus $M_{34}$ with the equal mass condition. The selection efficiency for WW\/ZZ is $54\\%$\/$44\\%$.}\n\\label{czwDissDmass}\n\\end{figure}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{czwRdMass1.png}\n\\caption{ The RecoJet level distributions of $0.5\\times(M_{12} + M_{34})$ with the equal mass condition, the overlapping ratio is $39.9\\% \\pm 0.40\\%$.}\n\\label{czwRdMass1}\n\\end{figure}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cGzwDissDmass.png}\n\\caption{ The GenJet level distribution of $M_{12}$ versus $M_{34}$ with the equal mass condition. The selection efficiency for WW\/ZZ is $59\\%$\/$47\\%$.}\n\\label{cGzwDissDmass}\n\\end{figure}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cGzwRdMass1.png}\n\\caption{ The GenJet level distributions of $0.5\\times(M_{12} + M_{34})$ with the equal mass condition, the overlapping ratio is $27.1\\% \\pm 0.42\\%$.}\n\\label{cGzwRdMass1}\n\\end{figure}\n\n\nThe area of the plateau can be significantly reduced using the fact that WW and ZZ processes produce two equal mass bosons.\nWe define an equal mass condition that requires the mass difference between the two di-jet systems to be smaller than 10 GeV ($|M_{12} - M_{34}| < 10$).\nThis condition keeps roughly half of the events.\nAfter applying this equal mass condition, the overlapping ratios are improved to $39.9\\% \\pm 0.40\\%$ and $27.1\\% \\pm 0.42\\%$, corresponding to the RecoJet and the GenJet plots, \nsee Figure~\\ref{czwDissDmass} to Figure~\\ref{cGzwRdMass1}.\n\n\nThe overlapping ratios of the full hadronic WW and ZZ events can be compared with two reference values. \nThe first one is the overlapping ratio at the semi-leptonic di-boson events,\nwhere the invariant mass of the hadronic decayed W and Z bosons can be reconstructed without any jet confusion. \nThe second one is the overlapping ratio of the MC truth boson masses, which follow approximately the Breit-Wigner distributions. \nThe first value provides a reference to the jet confusion evaluation, \nand the second one describes the impact of intrinsic boson mass distributions and is the lower limit of the overlapping ratio. \n\n\n \\begin{figure} \n\\centering\n\\includegraphics[width=7.5cm, height = 7.2cm]{peizhu.png}\n\\caption{The distribution of total invariant mass of hadronic system of $ZZ\\to \\nu\\nu q\\bar{q}$, $ZH\\to\\nu\\nu(Z)q\\bar{q}(H)$, and $WW\\to\\mu\\nu q\\bar{q}$~\\cite{r1}. The overlapping ratio of WW and ZZ is $47.32\\% \\pm 0.26\\%$.}\n\\label{peizhu}\n\\end{figure}\n\nThe invariant hadronic mass distributions of semi-leptonic di-boson events ($ZZ\\to \\nu\\nu qq$, $ZH\\to\\nu\\nu(Z)qq(H)$, and $WW\\to\\mu\\nu qq$, inclusive sample) are shown in Figure~\\ref{peizhu}~\\cite{r1}.\nIt has clearly separated peaks at anticipated masses. \nThis semi-leptonic overlapping ratio is $47.3\\% \\pm 0.26\\%$.\nIt is significantly better than that of inclusive full hadronic WW and ZZ events using the RecoJets ($57.8\\% \\pm 0.23\\%$),\nbut worse than that with equal mass constraint ($39.9\\% \\pm 0.40\\%$). \n\n\nThe overlapping ratios of MC truth boson mass of WW and ZZ events are extracted. \nFor the full hadronic events, we calculate the average mass of two MC truth bosons and the overlapping ratio is $13.3\\% \\pm 0.34\\%$. \nFor the semi-leptonic event, we extract the truth level value of the mass of the hadronic decay boson, and the overlapping ratio is 12.5\\%.\nIn fact, those two values are close to the integration of two ideal Breit-Wigner distribution overlapping area according to the W and the Z boson masses and widths (12\\%).\nFor simplicity, the average value at full hadronic and semi-leptonic events (12.9\\%) is used in later discussion. \n\n\nEnergetic neutrinos can be generated via the semi-leptonic decays at the heavy flavor jet fragmentation, leading to significant missing energy and momentum. \nAt the full hadronic WW and ZZ samples, these energetic neutrinos can disturb the jet clustering and pairing performance and increase the jet confusion. \nIts impact is quantified using comparative analysis of the light jet sample. \nComparing to the inclusive sample, the overlapping ratio at light jet sample is reduced by 7.1\\% (from 39.9\\% to 32.8\\%) and 4.6\\% (from 57.8\\% to 53.2\\%), with and without the equal mass condition respectively.\n\n\nAt 240 GeV center of mass energy, a significant fraction of the WW and ZZ events have energetic ISR photons in their final states.\nThis ISR effect is included in the Whizard generator. \nThese ISR photons, once incident into the ECAL (|cos($\\theta$)| < 0.995 at the CEPC baseline), can be recorded as isolated energetic clusters.\nThose clusters may also increase of the jet confusion. \nWe define an ISR veto condition that excludes events with ISR photons whose energy exceeds 0.1 GeV. \nOnce applied on the light jet samples, the overlapping ratio can be further reduced by 3.4\\% (from 32.8\\% to 29.4\\%) and 3.6\\% (from 53.2\\% to 49.6\\%), with and without equal mass condition respectively.\n\n\nThe same analysis is performed also with GenJets and the overlapping ratio is summarized in Table~\\ref{overlappingTable} and Figure~\\ref{cCompare1}.\nFour lines, corresponding to the cases of the GenJet level or the RecoJet level, with or without the equal mass condition, are identified in Figure~\\ref{cCompare1}.\nTo be compared with two horizontal lines corresponding to the overlapping ratio of truth level boson mass distribution (12.9\\%) and that of the semi-leptonic sample (47\\%).\nSeveral interesting conclusions can be drawn: \n\n\\begin{itemize}\n\n\\item []1, For the full reconstructed samples, the WW and ZZ events could be efficiently separated. \nThe separation performance is slightly worse than the semi-leptonic events.\nHowever, the separation performance of the full hadronic events can exceed that of the semi-leptonic events, once the equal mass condition is applied. \n\n\\vspace*{0.3cm}\n\n\\item []2, It's actually the jet confusion that dominants the separation performance of the inclusive samples, as the GenJet level samples have already significant overlapping ratio. \nThe detector performance is significant on the boson peak width, but contribute only marginally to the overall separation performance. \nFor the inclusive samples without equal mass condition, the overlapping ratio only increases by 5\\% at the RecoJet level comparing to that at the GenJet level.\nMeanwhile, their relative difference becomes more significant once the equal mass condition and other restrictive conditions are applied.\n\n\\vspace*{0.3cm}\n\n\\item []3, The equal mass condition can efficiently veto events contaminated by large jet confusion. \nAt the cost of lost roughly half of the statistic, the separation ratio can be improved by roughly 20\\% for both the RecoJets and the Genjets. \nFor the GenJets with the light jet samples and ISR photons veto, the overlapping ratio is approaching to the physics lower limit of 12.9\\%. \n\n\\vspace*{0.3cm}\n\n\\item []4, The neutrinos generated in the heavy flavor jets and the ISR photons contribute approximately a constant amount of overlapping ratio for all four different cases.\nIn fact, the accumulated impact of neutrinos and ISR photons are larger than that of the detector performance: for the light jet sample with the ISR veto, the RecoJet distribution overlapping ratio ($49.6\\% \\pm 0.30\\%$) is smaller than that of the inclusive sample at the GenJet level ($52.6\\% \\pm 0.25\\%$).\nCollectively, they contribute up to 10\\% of the overall overlapping ratio on the inclusive sample. \nTherefore, adequate jet flavor tagging and ISR photon finding algorithm can be used, to significantly improve the separation performance. \n\n\\end{itemize}\n\n\n\\begin{table*}\n\\centering\n\\caption{The overlapping ratios with different conditions.}\n\\label{overlappingTable}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}cccc@{}}\n\\hline\n                                             &  Light sample                                           &  Light sample                                                   & Inclusive sample                                          \\\\\n                                             &   non energetic ISR                                  &\t\t&\t                                   \\\\\n\\hline \nRecoJet\t       &   $49.6\\% \\pm 0.30\\%$                            &  $53.2\\% \\pm 0.29\\%$                                     &   $57.8\\% \\pm 0.23\\%$                                 \\\\\n\\hline\nGenJet                                 &   $39.1\\% \\pm 0.33\\%$                             &  $48.9\\% \\pm 0.30\\%$                                      &   $52.6\\% \\pm 0.25\\%$                                 \\\\\n\\hline\nRecoJet                                &  \\multirow{2}{*}{$29.4\\% \\pm 0.71\\%$ }    & \\multirow{2}{*}{$32.8\\% \\pm 0.49\\%$}             &   \\multirow{2}{*}{$39.9\\% \\pm 0.40\\%$}         \\\\\n with equal mass condition   & \t\t&   \t\t&  \t                                     \\\\\n\\hline\nGenJet                                  &   \\multirow{2}{*}{$16.0\\%\\pm 0.72\\%$ }    &  \\multirow{2}{*}{$23.0\\% \\pm 0.51\\%$ }            &  \\multirow{2}{*}{$27.1\\% \\pm 0.42\\%$}          \\\\\nwith equal mass condition    &                                                                   &                                                                          & \\\\\n\\hline\n\\hline\nReference Values\t       &\t\t\t\t\t\t\\\\\n\\hline\nSemi-leptonic, RecoJet        &                                                                  & $47.3\\% \\pm 0.26\\%$   \t &\t                                      \\\\\n\\hline\nIntrinsic Boson Mass            &                                                                  &  $13.3\\% \\pm 0.34\\%$\t &\t\t\\\\\n\\hline\n\\end{tabular*}\n\\end{table*}\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cCompare1.png}\n\\caption{ The overlapping ratios for different cases. The X-axis indicates the different sample restrictive conditions: the light flavor samples with ISR veto condition, the light flavor samples, and the inclusive samples. }\n\\label{cCompare1}\n\\end{figure}\n\n\n\n\\section{Quantification of the jet confusion}\n\nIn this section, we analyze the correlation between the jet confusion and the overlapping ratio using the angles between the di-jet systems and the MC truth bosons. \nEach event has two di-jet systems and two MC truth level bosons.\nThe mapping with the minimal value of angle sum is selected. \n\nFigure~\\ref{cwdR1_dR2} shows the correlation of two angles between the RecoJets and the MC truth bosons of the inclusive WW events. \nFor $\\alpha_{1}$ and $\\alpha_{2}$ smaller than 0.1 radians, these two quantities are not correlated.\nThe distribution actually reflects the jet angle resolution of the CEPC baseline detector.  \nFor $\\alpha_{1}$ and $\\alpha_{2}$ larger than 0.1 radians, a strong correlation is observed between these two quantities, corresponding to significant jet confusion. \n\nWe quantify the jet confusion using the product $\\alpha = \\alpha_{1}\\times \\alpha_{2}$ as the order parameter, which increases with the jet confusion. \nFigure~\\ref{cwdR} shows the distribution of $Log_{10}(\\alpha)$ at the RecoJet level, which exhibits a gaussian-like distribution up to $Log_{10}(\\alpha) = -2$ and a flat plateau up to $Log_{10}(\\alpha) = 0.4$.\nThe plateau corresponds to the physics events with large jet confusion. \n\nTo quantify the impact of jet clustering performance, the reconstructed WW sample is divided into five subsamples with the equal statistics, see Figure~\\ref{cwdR}. \nA set of thresholds on $\\alpha$ are extracted. \nThe ZZ samples are divided also into five subsamples using the same thresholds, \nand the overlapping ratios of the same set of subsamples are calculated. \n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cwdR1_dR2.png}\n\\caption{The correlation of $\\alpha_{1}$ versus $\\alpha_{2}$ (unit in radians), the angular difference between reconstructed di-jet systems and the MC truth bosons of the inclusive WW samples.}\n\\label{cwdR1_dR2}\n\\end{figure}\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cwdR.png}\n\\caption{The distribution of $\\alpha$ ($\\alpha = \\alpha_{1}\\times \\alpha_{2}$) of the WW sample using the RecoJets. There are four vertical lines to characterize $\\alpha$ into five subsamples, each contains $20\\%$ of the statistics.}\n\\label{cwdR}\n\\end{figure}\n\nFigure~\\ref{overlappingRatio} shows the average di-jet mass distributions of each set at the RecoJet and the GenJet level.\nTheir overlapping ratios increase monotonically with the jet confusion, see Figure~\\ref{cGoverlap}.  \nThe relative difference between that of the GenJets and the RecoJets, which reflects the detector performance, became less significant. \nIn the first set - corresponding to 20\\% of the total statistics with the minimal jet confusion, the overlapping ratio of the GenJets is close to the lower limit, and that of the the RecoJets is relatively 76\\% larger (14.1\\% to 24.8\\%). \nIn the last set, for both GenJets and RecoJets, the distributions of the WW and ZZ events are similar. \nThat's to say, the jet confusion eliminates almost completely the separation power for the last 20\\% of statistics with the worst jet confusion. \n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\columnwidth]{cGoverlap.png}\n\\caption{ The overlapping ratios of different sets sorted according to the jet confusion order parameter $\\alpha$. The red\/blue lines is corresponding to the GenJet\/RecoJet. The red\/brown dashed horizontal line indicates the overlapping ratio of the semi-leptonic sample\/intrinsic boson mass distributions, respectively.\n}\n\\label{cGoverlap}\n\\end{figure}\n\n\n\\begin{figure*}\n\\begin{minipage}{\\textwidth}\n\\centering\n\\includegraphics[width=0.19\\textwidth]{czwH1.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{czwH2.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{czwH3.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{czwH4.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{czwH5.png}\n\\\\\n\\includegraphics[width=0.19\\textwidth]{cGzwH1.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{cGzwH2.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{cGzwH3.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{cGzwH4.png}\n\\hfill\n\\includegraphics[width=0.19\\textwidth]{cGzwH5.png}\n\\end{minipage}\n\\caption{The average dijet mass distributions after dividing the inclusive sample into five subsamples. From left to right, the $\\alpha$ is degrading. The distributions in the top row are using the RecoJets, the overlapping ratio is $24.8\\% \\pm 0.81\\%$, $27.6\\% \\pm 0.77\\%$, $39.1\\% \\pm 0.63\\%$, $74.1\\% \\pm 0.37\\%$ and $91.1\\% \\pm 0.22\\%$, respectively. The bottom distributions are corresponding to the GenJets, the overlapping ratio is $14.1\\% \\pm 0.89\\%$, $15.0\\% \\pm 0.83\\%$, $34.0\\% \\pm 0.65\\%$, $74.4\\% \\pm 0.37\\%$ and $91.9\\% \\pm 0.21\\%$, respectively.}\n\\label{overlappingRatio}\n\\end{figure*}\n \n It's interesting that the jet confusion takes polarized pattern in this analysis. \n Sorting the inclusive samples with the jet confusion, \n the first 40\\% of the samples have only marginal jet confusion (as the overlapping ratio is close to the lower limit).\n However, the jet confusion soon grows to be the leading impact factor of WW\/ZZ separation, and dominate the overlapping ratio for the last 40\\% of the samples. \n The critical point occurs at roughly half of the statistics. \n This S-curve in Figure~\\ref{cGoverlap} may characterize profoundly the jet clustering and pairing performance, and can be used as a reference for corresponding performance evaluation and algorithm development. \n\n\n\\section{Conclusion}\n\\label{secConclusion}\n\nThe separation of the full hadronic WW and ZZ events is an important benchmark for the CEPC detector design and performance evaluation. \nThis separation performance is determined by the intrinsic boson mass distribution, the detector performance, and the jet confusion. \nUsing the CEPC baseline simulation tool, we analyze this benchmark performance using full simulated samples. \nThe $\\it{e^+e^- k_t}$ and the minimal $\\chi^{2}$ methods are used as the jet clustering and pairing algorithms, respectively. \n\nWe quantify the separation performance using the overlapping ratio. \nComparative analyses are performed to disentangle the impacts of three components. \nThe impact of the intrinsic boson mass distribution is characterized by the overlapping ratio of the MC truth boson mass distributions, which is found to be 12.9\\%.\nThe overlapping ratio using the GenJets only includes the intrinsic boson mass and the jet confusion. \nTherefore, the relative difference between the overlapping ratios of the GenJets and the RecoJets describes the impact of detector performance. \nThe reconstructed boson masses with hadronic decay final states of the semi-leptonic events are free of the jet confusion. \nThese semi-leptonic distributions have an overlapping ratio of $47.3\\% \\pm 0.26\\%$, providing another reference. \n\nWe confirm that the full hadronic WW and ZZ events can be clearly separated at the full reconstruction level.\nUsing the RecoJets, the overlapping ratio for the inclusive full hadronic WW and ZZ event samples at the CEPC is $57.8\\% \\pm 0.23\\%$. \nAn equal mass condition can reduce the overlapping ratio to $39.9\\% \\pm 0.40\\%$, at the cost of vetoing half of the statistics. \nThe overlapping ratios of the GenJet level distributions are $52.6\\% \\pm 0.25\\%$ and $27.1\\% \\pm 0.42\\%$, with and without the equal mass condition respectively. \nComparing to the separation performance with the RecoJets, the GenJets separation performance are significantly improved - especially with the equal mass condition, but its overlapping ratio is still two times larger the lower limit of 12.9\\%.\nTherefore, we conclude that the jet confusion plays a dominant role in the WW-ZZ separation with full hadronic final states. \n\nThe overlapping ratio for WW and ZZ events with the semi-leptonic final state is estimated to be $47.3\\% \\pm 0.26\\%$, which is between that of the inclusive full hadronic samples with and without equal mass condition ($57.8\\% \\pm 0.23\\%$ and $39.9\\% \\pm 0.40\\%$).\nOnce the jet confusion is under control, the separation performance of the full hadronic events is better than that of semi-leptonic events, since the former can use mass information from both reconstructed bosons with indepentdent detector response. \n\nThe neutrinos and ISR photons play an important role in the separation performance. \nCollectively, they contribute to roughly 10\\% of the overall overlapping ratio. \nTherefore, the jet flavor tagging algorithm and ISR photon identification algorithm are crucial for the full hadronic WW and ZZ event separation. \n\nThe jet confusion is further characterized by the reconstructed angle of bosons. \nThe full hadronic WW and ZZ samples are divided into subsamples and sorted accordingly. \nFor those subsamples, the jet confusion takes a polarized pattern. \nFor the best 40\\% of the events, the difference between the reconstructed boson angle and the truth value is smaller than 0.1 radians, \nand the jet confusion is minimum.  \nThe overlapping ratio of the GenJet level distributions is close to the lower limit of 12.9\\%. \nThe separation of those events are mainly dominated by the detector performance. \nFor the last 40\\% of events, the jet confusion dominates the separation performance. \n\n\nTo conclude, our analysis confirms that the baseline CEPC detector and reconstruction software could efficiently separate the full hadronic WW and ZZ events at full reconstruction level. \nThe overall separation performance is dominated by the jet confusion. \nDedicated studies and developments on the jet clustering and pairing algorithms are required, to significantly improve the separation performance.\nAdequate ISR photon finding and jet flavor tagging could significantly improve this separation performance. \nThrough optimization of the jet clustering and pairing algorithms, for example using differential jet energy resolutions for the jet pairing $chi_2$ calculation, the iterative jet clustering, and the Multiple Variable Analyses, this performance is expected to be improved significantly. \n\n\nThe reconstruction of multi-jets events at the electron positron Higgs factories is critical for the physics reach.\nOn top of the particle flow reconstruction that produces all the final state particle, \nthe critical requirement is to identify precisely all the decay products from each color -singlet. \nIn our analysis, the identification is implemented with a straightforward jet clustering and jet pairing algorithm. \nBecause the jets at the CEPC can have low energies and large opening angles, these algorithms can lead to large jet confusions that dominate the final measurement accuracy. \nDedicated studies to control the jet confusion, or equivalently, the development of color-singlet reconstruct algorithms, are critical.\nThe WW\/ZZ separation analysis presented in this paper is an early step of these studies. \nIt not only demonstrates the physics performance of the CEPC baseline but also provides the reference and a simple quantification method to evaluate different color-singlet reconstruction algorithms. \n\n\n\n\\begin{acknowledgements}\nWe are in debt to Jianming Qian, Liantao Wang, Huaxing Zhu, and Haibo Li for their constructive suggestions.\nWe are grateful to Dan Yu, Xianghu Zhao, Hao Liang, and Yuxuan Zhang for their supports and helps.\nWe thank Gang Li and Chengdong Fu for producing the samples. \n\nThis work was supported by National Key Program for S\\&T Research and Development (Grant No.: 2016YFA0400400), the National Natural Science Foundation of China (Grant No.: 11675202), the Hundred Talent Programs of Chinese Academy of Science (Grant No.: Y3515540U1).\n\\end{acknowledgements}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}