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Two coloring problems on matrix graphs  [PDF]
Zhe Han,Mei Lu
Mathematics , 2015,
Abstract: In this paper, we propose a new family of graphs, matrix graphs, whose vertex set $\mathbb{F}^{N\times n}_q$ is the set of all $N\times n$ matrices over a finite field $\mathbb{F}_q$ for any positive integers $N$ and $n$. And any two matrices share an edge if the rank of their difference is $1$. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let $\chi'_d(N\times n, q)$ (resp. $\chi_d(N\times n, q)$) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most $d$ (resp. exactly $d$) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of $\chi'_d(N\times n,q)$ and give some upper and lower bounds on $\chi_d(N\times n,q)$.
Problems on Polytopes, Their Groups, and Realizations  [PDF]
Egon Schulte,Asia Ivic Weiss
Mathematics , 2006,
Abstract: The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2005.
Scribability problems for polytopes  [PDF]
Hao Chen,Arnau Padrol
Mathematics , 2015,
Abstract: In this paper we study various scribability problems for polytopes. We begin with the classical $k$-scribability problem proposed by Steiner and generalized by Schulte, which asks about the existence of $d$-polytopes that can not be realized with all $k$-faces tangent to a sphere. We answer this problem for stacked and cyclic polytopes for all values of $d$ and $k$. We then continue with the weak scribability problem proposed by Gr\"unbaum and Shephard, for which we are able to complete the work by Schulte by presenting non weakly circumscribable $3$-polytopes. Finally, we propose new $(i,j)$-scribability problems, in a strong and a weak version, which generalize the classical ones. They ask about the existence of $d$-polytopes that can not be realized with all their $i$-faces "avoiding" the sphere and all their $j$-faces "cutting" the sphere. We are able to provide such examples for all the cases where $j-i \le d-3$.
A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space  [PDF]
Rui Huang,Xiaodong Wu,Ruihe Wang,Hui Li
Algebra , 2014, DOI: 10.1155/2014/392467
Abstract: Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided. 1. Introduction In recent years, a lot of matrix problems have been used widely in the fields of structural design, automatic control, physical, electrical, nonlinear programming and numerical calculation, for example, a matrix Eigen value problem was applied for mixed convection stability analysis in the Darcy media by Serebriiskii et al. [1] and some of the problems based on the nonskew symmetric orthogonal matrices were studied by Hamed and Bennacer in 2008 [2], but some of the matrix inverse problems still need further research in order to make it easier to discuss relevant issues. Therefore, in this paper, we studied the inverse problems of two kinds of special matrix equations based on the existing research achievements, moreover, the expressions and conditions of the matrix solutions are given by related matrix-calculation methods. Some definitions and assumptions of the inverse problem for two forms of special matrices are given in Section 2. In Sections 3 and 5 we discuss the existence and expressions of general solution based on the two classes of matrices, and in Sections 4 and 6 we prove the uniqueness of matrices for researching related inverse problems. 2. Definitions and Assumptions of Inverse Problems for Two Forms of Special Matrices In order to research some inverse problems of related matrices, we give the following definitions and assumptions. Definition 1. When , , , , , and , will be called the first-class special symmetric matrix and the set of these special symmetric matrices is denoted by . The corresponding problems are as follows. Problem 1. When , can be obtained, so that . Problem 2. When , can be obtained, so that , where is the solution set of the first problem. Definition 2. When , and , will be called the second-class special symmetric matrix and the set of these special symmetric matrices is denoted by . The corresponding problems are as follows. Problem 1. When , can be found, so that . Problem 2. When , can be found, so that , where is the solution set of the first problem. 3. Existence and Expression of General Solutions Based on the First-Class Special Symmetric Matrix for Problem 1 To research the structure and properties of the special symmetric matrix , first of all, we have the
Containment problems for polytopes and spectrahedra  [PDF]
Kai Kellner,Thorsten Theobald,Christian Trabandt
Mathematics , 2012,
Abstract: We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational complexity, extending results on the polytope/polytope-case by Gritzmann and Klee to the polytope/spectrahedron-case. For various restricted containment problems, NP-hardness is shown. We then study in detail semidefinite conditions to certify containment, building upon work by Ben-Tal, Nemirovski and Helton, Klep, McCullough. In particular, we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron. It is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases, including containment of a spectrahedron in a polyhedron. Moreover, in the case of bounded $S_A$ the criteria will always succeed in certifying containment of some scaled spectrahedron $\nu S_A$ in $S_B$.
Extremal edge polytopes  [PDF]
Tuan Tran,Günter M. Ziegler
Mathematics , 2013,
Abstract: The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope  [PDF]
Karola Mészáros,Alejandro H. Morales,Jessica Striker
Mathematics , 2015,
Abstract: In this paper we study an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaux of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley's triangulation of order polytopes, the Postnikov-Stanley triangulation of flow polytopes and the Danilov-Karzanov-Koshevoy triangulation of flow polytopes. We show that when a graph $G$ is a planar graph, in which case the flow polytope $F_G$ is also an order polytope, Stanley's triangulation of this order polytope is one of the Danilov-Karzanov-Koshevoy triangulations of $F_G$. Moreover, for a general graph $G$ we show that the set of Danilov-Karzanov-Koshevoy triangulations of $F_G$ is a subset of the set of Postnikov-Stanley triangulations of $F_G$. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.
A general T-matrix approach applied to two-body and three-body problems in cold atomic gases  [PDF]
Xiaoling Cui
Physics , 2010, DOI: 10.1007/s00601-011-0298-6
Abstract: We propose a systematic T-matrix approach to solve few-body problems with s-wave contact interactions in ultracold atomic gases. The problem is generally reduced to a matrix equation expanded by a set of orthogonal molecular states, describing external center-of-mass motions of pairs of interacting particles; while each matrix element is guaranteed to be finite by a proper renormalization for internal relative motions. This approach is able to incorporate various scattering problems and the calculations of related physical quantities in a single framework, and also provides a physically transparent way to understand the mechanism of resonance scattering. For applications, we study two-body effective scattering in 2D-3D mixed dimensions, where the resonance position and width are determined with high precision from only a few number of matrix elements. We also study three fermions in a (rotating) harmonic trap, where exotic scattering properties in terms of mass ratios and angular momenta are uniquely identified in the framework of T-matrix.
Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations  [PDF]
Elias Jarlebring,Michiel E. Hochstenbach
Mathematics , 2008,
Abstract: Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called matrix pencil methods, the general ideas used as well as the proofs differ considerably. Moreover, the available theory hardly reveals the relations between the different methods. In this work, a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem: the polynomial two-parameter eigenvalue problem, of which the quadratic two-parameter eigenvalue problem is a special case. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. We also recognize a few new matrix pencil variants to determine DDE stability. Finally, the recognition of the new types of eigenvalue problem opens a door to efficient computation of DDE stability.

计算数学 , 1992,
Abstract: A parallel algorithm of matrix inversion by Gauss-Jordan's method on the AP(Array-Processor) is discussed. Then the distribution problem of matrix eigenvaluesis deaet with and a parallel iterative method for determining the number of matrixeigenvalues on the AP, is presented.
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