张量对称类上诱导算子的y-数值半径(英文)

本文研究了诱导矩阵K(A)的y-数值半径ry(K(A))、y-可分数值半径ryχ(K(A))与范数A2、广义矩阵函数dχG(A)之间的关系问题.利用ry(K(A))及ryχ(K(A))的概念,得到了ry(K(A))、ryχ(K(A))、‖A‖2、dGχ(A)它们之间的两个不等式.     

ON k-DIAMETER OF k-CONNECTED GRAPHS

Abstract. Let G be a k-connected simple graph with order n. The k-diameter, combining con-nectivity with diameter, of G is the minimum integer     

Computing matrix symmetrizers,finally possible via the Huang and Nong algorithm

By a theorem of Frobenius (F.G. Frobenius, Über die mit einer Matrix vertauschbaren Matrizen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1910), pp. 3–15 (also in Gesammelte Abhandlungen, Band 3, Springer 1968. pp. 415–427)), every matrix A _n,n over any field 𝔽 is the product of two symmetric ones. Using the algorithm of Huang and Nong (J. Huang and L. Nong, An iterative algorithm for solving finite-dimensional linear operator equations T(x)?=?f with applications, Linear Algebra Appl. 432 (2010), pp. 1176–1188) for linear systems, we develop an algorithm to compute a symmetric matrix S?=?S ^T ?∈?𝔽 ^n,n for which SA is symmetric for any given square matrix A?∈?𝔽 ^n,n where 𝔽?=?? or ?. The algorithm is implemented and tested in MATLAB.     

2n名选手循环赛安排问题

为解决2n名选手循环赛安排问题,给出了边矩阵及循环赛图的定义.提出了求K2n的Δ(G)个完备匹配M(i)的一种算法.介绍了8名选手循环赛图K(81)及16名选手循环赛图K(161)的形成过程.讨论了完备匹配不交的循环赛图K(2 in)的个数问题.     

k-树的补图的最小填充和树宽

一个图的最小填充问题是寻求边数最少的弦母图,一个图的树宽问题是寻求团数最小的弦母图,这两个问题分别在稀疏矩阵计算及图的算法设计中有非常重要的作用．一个k-树G的补图G称为k-补树．本文给出了k-补树G的最小填充数f(G) 及树宽TW(G)．     

3-连通、高次和坚韧图周长的估计(Ⅰ)

设G是一个n阶3-连通图,周长为C(G),独立数为,若G是1-坚韧的,且,则G的每一个最长圈是控制圈且;又若G是5/3-坚韧的或,则G是Hamilton图。     

完全多部图的无符号Laplacian特征多项式(英文)

For a simple graph G,let matrix Q（G）=D（G） + A（G） be it’s signless Laplacian matrix and Q G （λ）=det（λI Q） it’s signless Laplacian characteristic polynomial,where D（G） denotes the diagonal matrix of vertex degrees of G,A（G） denotes its adjacency matrix of G.If all eigenvalues of Q G （λ） are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K（n₁,n₂,···,n_t）.We prove that the complete t-partite graphs K（n,n,···,n）t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K（m,···,m）s（n,···,n）t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K（m,···,m,）s1（n,···,n,）s2（l,···,l）s3.     

A Newton-based method for the calculation of the distance to instability

In this paper, a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided. The method uses Newton’s method to find a two-dimensional Jordan block corresponding to a pure imaginary eigenvalue in a certain two-parameter Hamiltonian eigenvalue problem introduced by Byers [R. Byers, A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J. Sci. Statist. Comput. 9 (1988) 875-881]. This local method is augmented by a test step, previously used by other authors, to produce a global method. Numerical results are presented for several examples and comparison is made with the methods of Boyd and Balakrishnan [S. Boyd, V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L_∞-norm, Systems Control Lett. 15 (1990) 1-7] and He and Watson [C. He, G.A. Watson, An algorithm for computing the distance to instability, SIAM J. Matrix Anal. Appl. 20 (1999) 101-116].     

Quasi-classical asymptotics of solutions to the matrix factorization problem with quadratically oscillating off-diagonal elements

The paper investigates the asymptotic behavior of solutions to the 2 × 2 matrix factorization (Riemann-Hilbert) problem with rapidly oscillating off-diagonal elements and quadratic phase function. A new approach to study such problems based on the ideas of the stationary phase method and M. G. Krein’s theory is proposed. The problem is model for investigating the asymptotic behavior of solutions to factorization problems with several turning points. Power-order complete asymptotic expansions for solutions to the problem under consideration are found. These asymptotics are used to construct asymptotics for solutions to the Cauchy problem for the nonlinear Schrödinger equation at large times.     

实现同一度序列的图的最大团数

图G中最大完全子图的阶数称为G的团效．ω(π)和γ(π)分别表示实现度序列π=(d_1,d_2,…,d_n)的图的最大团数和最小团数．Erds,Jacobson和Lehel开始考虑确定具有相同度序列π的图的可能的团数问题．他们证明了对于充分大的n,有ω(π)-γ(π)-n一2n~(2/3)．在本文中,我们首先估计了一类特殊可图序列的ω(π)之值,其次我们建立了一个估计任意可图序列π的ω(π)之值的算法．     

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

哈密顿图的无符号拉普拉斯谱半径条件

令A(G)=(a_(ij))_(n×n)是简单图G的邻接矩阵,其中若v_i-v_j,则a_(ij)=1,否则a_(ij)=0.设D(G)是度对角矩阵,其(i,i)位置是图G的顶点v_i的度.矩阵Q(G)=D(G)+A(G)表示无符号拉普拉斯矩阵.Q(G)的最大特征根称作图G的无符号拉普拉斯谱半径,用q(G)表示.Liu,Shiu and Xue[R.Liu,W.Shui,J.Xue,Sufficient spectral conditions on Hamiltonian and traceable graphs,Linear Algebra Appl.467(2015)254-255]指出:可以通过复杂的结构分析和排除更多的例外图,当q(G)≥2n-6+4/(n-1)时,则G是哈密顿的.作为论断的有力补充,给出了图是哈密顿图的一个稍弱的充分谱条件,并给出了详细的证明和例外图.     

L(d1, d2,..., dt)-Number λ(Cn; d1, d2,...,dt) of Cycles

An L(d1,d2,...,dt)-labeling of a graph G is a function f from its vertex set V(G) to the set {0, 1,..., k} for some positive integer k such that {f(x) - f(y)| ≥ di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L(d1,d2,...,dt)-number λ(G;d1,d2,... ,dt) of G is the smallest integer number k such that G has an L(d1,d2,... ,dt)labeling with max{f(x)|x ∈ V(G)} = k. In this paper, we obtain the exact values for λ(Cn; 2, 2,1) and λ(Cn; 3, 2, 1), and present lower and upper bounds for λ(Cn; 2,..., 2,1,..., 1)     

给定顶点数和边数的连通图的Q-谱半径的界

图的无符号拉普拉斯矩阵是图的邻接矩阵和度对角矩阵的和,其特征值记为q1≥q2≥…≥qn.设C(n,m)是由n个顶点m条边的连通图构成的集合,这里1≤n-1≤m≤(n2).如果对于任意的G∈C(n,m)都有q1(G*)≥q1(G)成立,图G*∈C(n,m)叫做最大图.这篇文章证明了对任意给定的正整数a=m-n+1,如果n...     

三圈图的无符号拉普拉斯谱半径

假设图G的点集是V(G)={v_1,v_2,…,v_n},用d_(v_i)(G)表示图G中点v_i的度,令A(G)表示G的邻接矩阵,D(G)是对角线上元素等于d_(v_i)(G)的n×n对角矩阵,Q(G)=D(G)+A(G)是G的无符号拉普拉斯矩阵,Q(G)的最大特征值是G的无符号拉普拉斯谱半径.现确定了所有点数为n的三圈图中无符号拉普拉斯谱半径最大的图的结构.     

与直径和围长有关的最大亏格的下界

本文证明了如下结果:设G为直径为d的简单图,若G的围长不小于d,则当d为不小于4的偶数时,有ξ(G)≤1,即G是上可嵌入的;当d为不小于3的奇数时,有ξ(G)≤2,即γM(G)≥1/2β(G)-1.     

点度与图的上可嵌入性

本文证明了:(1) 设G是2-连通简单图,且不含K_3,若对任意一对距离为2的点u,u,有max{d(u),d(u)}>n/3-1,其中n=|V(G)|,则G是上可嵌入的,且条件中不等式的界"n/3-1"是不可达的;(2) 设G是3-连通简单图,若对任意依次相邻的三点u,u,W,有max{d(u),d(u),d(w)}≥n/6+1,其中n=|V(G)|,则G是上可嵌入的,且条件中不等式的界"n/6+1"是最好的.     

定向图的斜Randic能量

图G是一个简单无向图,G~σ是图G在定向σ下的定向图,G被称作G~σ的基础图.定向图G~σ的斜Randi6矩阵是实对称n×n矩阵R_s(G~σ)=[(r_s)_(ij)].如果(v_i,v_j)是G~σ的弧,那么(r_s)_(ij)=(d_id_j)~(-1/2)且(r_s)_(ji)=(d_id_j)~(-1/2),否则(r_s)_(ij)=(r_s)_(ji)=0.定向图G~σ的斜Randi能量RE_s(G~σ)是指R_s(G~σ)的所有特征值的绝对值的和.首先刻画了定向图G~σ的斜Randi矩阵R_s(G~σ)的特征多项式的系数.然后给出了定向图G~σ的斜Randi能量RE_s(G~σ)的积分表达式.之后给出了RE_s(G~σ)的上界.最后计算了定向圈的斜Randi能量RE_s(G~σ).     

A nested dissection approach for sparse matrix partitioning

We consider how to partition and distribute sparse matrices among processors to reduce communication cost in sparse matrix computations, in particular, sparse matrix-vector multiplication. We consider 2d distributions, where the distribution is not constrained to just rows or columns. We present a new model and an algorithm based on vertex separators and nested dissection. Preliminary numerical results for sparse matrices from real applications indicate the new method performs consistently better than traditional 1d partitioning and is often also better than current 2d methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modified Newton’s method for best rank-one approximation to tensors

In this paper, a modified Newton’s method for the best rank-one approximation problem to tensor is proposed. We combine the iterative matrix of Jacobi-Gauss-Newton (JGN) algorithm or Alternating Least Squares (ALS) algorithm with the iterative matrix of GRQ-Newton method, and present a modified version of GRQ-Newton algorithm. A line search along the projective direction is employed to obtain the global convergence. Preliminary numerical experiments and numerical comparison show that our algorithm is efficient.