More About Geršgorin‐type theorems

In the recent book of R.S. Varga, [3], one of two main recurring themes is that a nonsingular theorem for matrices gives rise to an equivalent eigenvalue inclusion set in the complex plane, and conversely. If such nonsingularity result can be extended via irreducibility, usually this can be used for obtaining more information about the boundary of the corresponding eigenvalue inclusion set. Here we will start with one of Geršgorin‐type theorem for eigenvalue inclusion, given in [1], (for which exists corresponding equivalent statement about nonsingularity of a particular class of matrices) and use it for proving necessary conditions for an eigenvalue to lie on the boundary of localization area. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithm for computing minimal Geršgorin sets

The existing algorithms for computing the minimal Ger?gorin set are designed for small and medium size (irreducible) matrices and based on Perron root computations coupled with bisection method and sampling techniques. Here, we first discuss the drawbacks of the existing methods and present a new approach based on the modified Newton's method to find zeros of the parameter dependent left‐most eigenvalue of a Z‐matrix and a special curve tracing procedure. The advantages of the new approach are presented on several test examples that arise in practical applications. Copyright © 2015 John Wiley & Sons, Ltd.     

Ore‐type degree conditions for a graph to be H‐linked

Given a fixed multigraph H with V(H) = {h₁,…, h_m}, we say that a graph G is H‐linked if for every choice of m vertices v₁, …, _vm in G, there exists a subdivision of H in G such that for every i, v_i is the branch vertex representing h_i. This generalizes the notion of k‐linked graphs (as well as some other notions). For a family of graphs, a graph G is ‐linked if G is H‐linked for every . In this article, we estimate the minimum integer r = r(n, k, d) such that each n‐vertex graph with is ‐linked, where is the family of simple graphs with k edges and minimum degree at least . © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 14–26, 2008     

The sub‐Riemannian cut locus of H‐type groups

In the present paper we prove that the sub‐Riemannian cut locus at the origin of a wide class of nilpotent groups of step two, called H‐type groups, corresponds to the center of the group. We obtain this result by completely describing the sub‐Riemannian geodesics in the group, and using these to obtain three disjoint sets of points in the group determined by the number of geodesics joining them to the origin.     

某类双调和映射的Landau型定理

令f(z)=h(z)+g(z)为开单位圆盘U内的调和映射.本文首先建立了有界调和映射的精确系数估计,其次得到了满足有界性条件|h(z)|+|g(z)|≤M的正规化调和映射的精确系数估计.作为其应用,得到了双调和映射的一些Landau型定理.这些结果改进和推广了早期作者的相关结果.     

Coefficient Estimates for Certain Subclasses of Bi-Univalent Ma-Minda Mocanu-Convex Functions

In this paper, we introduce and investigate a new subclass of the function class Σ of bi-univalent functions of the Mocanu-convex type defined in the open unit disk, satisfy Ma and Minda subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients |a2| and |a3| for functions in the new subclass introduced here. Further Application of Hohlov operator to this class is obtained. Several(known or new) consequences of the results are also pointed out.     

广义对角占优矩阵的两定理

本文根据矩阵A的一系列三元线性不等式组的非空解集的存在性,给出了判定矩阵A为广义对角占优的两个充分条件。     

Comparison Theorems for Four Dimensional Regular Matrices

A four dimensional summability matrix is stronger than convergence in the Pringsheim sense if it sums a divergent double sequence. Presented in this paper are sufficient conditions on a four dimensional matrix transformation which ensure that the summability matrix is stronger than convergence in the Pringsheim sense. Also other sufficient conditions will be presented to ensure the failure of inclusion between two summability matrices.AMS Subject Classification (2000): Primary 40B05.     

Hamilton算子矩阵的半群生成定理

研究了Hamilton算子矩阵的半群生成问题,得到其生成C_0半群的若干充分必要条件,并给出其成为半群无穷小生成元时的谱分布.作为应用,基于Hamilton系统的半群方法,给出一类四阶微分方程混合问题的古典解.     

Tauberian Theorems for Sequences Satisfying Certain Convolution Relations

The purpose of this paper is to prove qualitative and optimal quantitative Tauberian theorems for the sequence λ defined by the relation g′ = λ∗︁g ,where ∗︁ denotes the Cauchy convolution, g is a given sequence with g(0) = 0 satisfying suitable growth conditions, and g′ is defined by g′(n) = (n + 1)g(n + 1) for n ∈ ℕ = {0, 1, 2, …}. The problem originates in the analytic theory of the distribution of prime elements in additive arithmetic semigroups developed by Knopfmacher , and in recent work of Indlekofer , Manstavičius and Warlimont , of Zhang , and of Warlimont . The new results are based on the approximation of g by recurrent sequences and do not assume λ ≥ 0; the method of proof consists in the transition to generating power series and utilizes a weighted version of a Wiener type inversion theorem due to Lucht and Reifenrath .     

Limit Theorems for Spectra of Positive Random Matrices under Dependence

We study classical ensembles of sample covariance matrices introduced by Wishart. We discuss Stein's method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure for high-dimensional matrices. The method is based on a differential equation for the density of the Marchenko-Pastur law. Bibliography: 19 titles. __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 92–123.     

New Ore‐Type Conditions for H‐Linked Graphs

For a fixed (multi)graph H, a graph G is H‐linked if any injection f: V(H)→V(G) can be extended to an H‐subdivision in G. The notion of an H ‐linked graph encompasses several familiar graph classes, including k‐linked, k‐ordered and k‐connected graphs. In this article, we give two sharp Ore‐type degree sum conditions that assure a graph G is H ‐linked for arbitrary H. These results extend and refine several previous results on H ‐linked, k‐linked, and k‐ordered graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 71:69–77, 2012     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

关于四元数矩阵之迹的几个定理

R.Rellman对两个正定实矩阵建立了与Cauchy—Schwarz不等式相类似的结果,引起人们的关注,对Rellman不等式进行深入的研究.但对四元数矩阵之迹的研究至今未见.如所熟知,四元数体的非交换性,已经给四元数代数理论的研究带来了巨大的困难,它也必然影响到四元数矩阵迹的性质.事实上,关于实(或复)矩阵的几个简单性质:     

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V W is linear and K V and D W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal X's satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X – B* XB will be studied for certain cones K of entry-wise nonnegative X's.     

The Spectral Shift Function for Certain Block Operator Matrices

Let L₀ be a 2 × 2 diagonal self‐adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L₀ move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.     

On the threshold for the Maker‐Breaker H‐game

We study the Maker‐Breaker H‐game played on the edge set of the random graph . In this game two players, Maker and Breaker, alternately claim unclaimed edges of , until all edges are claimed. Maker wins if he claims all edges of a copy of a fixed graph H; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an H‐game is given by the threshold of the corresponding Ramsey property of with respect to the graph H. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 558–578, 2016     

四元数自共轭矩阵与行列式的几个定理

本文继续使用文献[1],[2],[3],[4],[5]的符号和术语。对四元数体Q上的自共轭矩阵与行列式进行讨论得到几个重要定理。为此,先作几点说明。 2.设A为四元数体Q上的一个n阶矩阵,若A=(即,A=a_(ij),a_(ij)∈Q。恒有a_(ij)=a_(ji))。则说A是四元数体Q上的一个自共轭矩阵。自共轭四元矩阵A的行列式记为‖A‖。