一类分块反循环矩阵及其对角化问题

1引言分块反循环矩阵在数值分析、优化理论、泛函微分方程、工程力学等学科领域有十分重要的应用,当今电子计算机及计算技术的迅速发展为分块反循环矩阵的应用开辟了更为广阔的前景．本文讨论了分块反循环矩阵的交换性、特征根及对角化问题,得到任一分块反循环矩阵可用一个正交矩阵组线性表示和基本分块反循环矩阵在复数域上可以对角化且相似于对角阵的结论．     

一类特殊的分块反循环矩阵的特征值问题

利用r-循环矩阵的基本性质,探讨了一类特殊的分块反循环矩阵的特征值问题．     

一类特殊的分块反循环矩阵的特征值问题(英文)

利用r-循环矩阵的基本性质,探讨了一类特殊的分块反循环矩阵的特征值问题.     

关于K-分块循环矩阵及其对角化问题的讨论

给出了K-分块循环矩阵的概念,并探讨了K-分块循环矩阵的相似类及其对角化问题.     

关于一类分块矩阵弱相似性质的理论探讨

本文提出了分块矩阵弱相似的概念,探讨了分块矩阵弱相似的条件和性质,得到了若干有意义的结果,为解决有关实际问题提供了一定的理论依据。     

关于反循环矩阵的对角化问题

反循环矩阵是一种特殊类型的矩阵，它本身有许多重要的性质，而且与矩阵的对角化问题有联系．本文拟探讨反循环矩阵的对角化问题，以及任一ｎ阶方阵Ａ可对角化时，Ａ与反循环矩阵之间的关系．     

关于r-分块循环矩阵及其对角化问题的探讨

本文给出了r-分块循环矩阵的概念,并利用矩阵的张量积探讨了r-分块循环矩阵的相似类及其对角化问题,得出了一些重要的结论.     

一类特殊分块矩阵为循环矩阵的循环分块矩阵的几个性质

本文给出一类特殊分块矩阵为循环矩阵的循环分块矩阵的几个性质。     

两类反循环矩阵求逆的一种新算法

利用多项式的初等行变换式给出的反循环矩阵和对称反循环矩阵求逆的一种新算法.该方法不需要计算三角函数并且具有很少的计算量.     

对角因子分块循环矩阵的对角化和谱分解

导出了对角因子分块循环矩阵的概念，把循环矩阵的对角化和谱分解推广到具有对角因子循环结构的分块矩阵中去．     

粗糙集理论在属性约简及知识分类中的应用

本针对不完备信息系统属性约简的两种定义，证明了两的等价性。在此基础上结合粗糙集理论提出了相似矩阵、相似区间的概念，并将其应用于不完备信息系统知识分类的问题中。     

矩阵的分块初等变换与分块初等阵及其应用

讨论了矩阵分块初等变换和分块初等阵的定义和性质,利用这一工具研究了行列式的分块运算,分块矩阵的求逆和对称阵的分块合同变换等问题.     

A direct bijection between descending plane partitions with no special parts and permutation matrices

We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.     

Completeness of the Description of an Equilibrium Canonical Ensemble by a Two-Particle Partition Function

We show that in the equilibrium classical canonical ensemble of particles with pair interaction, the full Gibbs partition function can be uniquely expressed in terms of the two-particle partition function. This implies that for a fixed number N of particles in the equilibrium system and a fixed volume V and temperature T, the two-particle partition function fully describes the Gibbs partition as well as the N-particle system in question. The Gibbs partition can be represented as a power series in the two-particle partition function. As an example, we give the linear term of this expansion. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 123–132, October, 2005.     

Reflection positivity, rank connectivity, and homomorphism of graphs

It is shown that a graph parameter can be realized as the number of homomorphisms into a fixed (weighted) graph if and only if it satisfies two linear algebraic conditions: reflection positivity and exponential rank connectivity. In terms of statistical physics, this can be viewed as a characterization of partition functions of vertex coloring models.

     

基于属性约简的分类及其应用

约简是粗糙集用于数据分析的重要方法和手段,也是粗糙集理论的核心内容之一.论域U上的等价关系可以确定U的划分,U上的不可区分关系是等价关系.利用不可区分关系定义信息系统的划分协调集及划分约简集,而且证明了划分约简集的判定定理,并将其应用于实际分类问题.     

Image partition regularity near zero

Many of the classical results of Ramsey Theory are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over N and other subsemigroups of (R,+). We study several notions of image partition regularity near zero for both finite and infinite matrices, and establish relationships which must hold among these notions.     

Representing the space of linear programs as the Grassmann manifold

Each linear program (LP) has an optimal basis. The space of linear programs can be partitioned according to these bases, so called the basis partition. Discovering the structures of this partition is our goal. We represent the space of linear programs as the space of projection matrices, i.e., the Grassmann manifold. A dynamical system on the Grassmann manifold, first presented in Sonnevend et al. (Math Program 52:527–553), is used to characterize the basis partition as follows: From each projection matrix associated with an LP, the dynamical system defines a path and the path leads to an equilibrium projection matrix returning the optimal basis of the LP. We will present some basic properties of equilibrium points of the dynamical system and explicitly describe all eigenvalues and eigenvectors of the linearized dynamical system at equilibrium points. These properties will be used to determine the stability of equilibrium points and to investigate the basis partition. This paper is only a beginning of the research towards our goal. Research is supported in part by NUS Academic Research Grant R-146-000-084-112. The author wishes to thank Josef Stoer for his valuable comments on the paper and to thank Wingkeung To, Jie Wu, Xingwang Xu, Deqi Zhang and Chengbo Zhu for providing consultations on Differential Geometry and Grassmann manifolds and pointing out useful literature. The author is certainly responsible to all faults in the paper.