Two-Interval Sturm–Liouville Operators in Direct Sum Spaces with Inner Product Multiples

In direct sum spaces with inner product multiples, we study two-interval Sturm–Liouville problems. For singular problems, we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive. This contrasts with the usual theory which requires the coupling matrix to have determinant one. This paper is in final form and no version of it will be submitted for publication elsewhere. Received: September 28, 2006.     

八元数矩阵的行列式及其性质

赋范的可除代数只有四种:实数R,复数C,四元数日和八元数O.由于八元数关于乘法非交换且非结合,如何对八元数矩阵定义行列式并使其具有较好的运算性质变得非常困难.最近,李兴民和黎丽根据"八元数自共轭矩阵的行列式应为实数"这一数学与物理上的需求,通过选择几个八元数乘积的次序和结合方式,首次给出了八元数行列式的定义.但是,与实数、复数以及四元数的相应的情形比较,如此定义的行列式,其所具备的运算性质较少.本文给出了一种新的八元数行列式的定义,它们具备了尽可能多的运算性质,同时使得"八元数自共轭矩阵的行列式为实数"不证自明.     

关于＂ Comments on a note on three-term recurrence for a tridiagonal matix ＂的进一步标记

在文章中,作者给出了块三对角矩阵行列式的一些关系式,本文在,=三基础之上给出了此类矩阵一些类似的关系式．     

Determinantal formulae for matrices with sparse inverses

The determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be “read off” from the graph of the inverse of the matrix. The only information used is the zero pattern of the inverse, and each zero pattern yields one or more corresponding formulae for the determinant.     

Determinant preserving maps: an infinite dimensional version of a theorem of Frobenius

In this paper we investigate the structure of maps on classes of Hilbert space operators leaving the determinant of linear combinations invariant. Our main result is an infinite dimensional version of the famous theorem of Frobenius about determinant preserving linear maps on matrix algebras. In this theorem of ours, we use the notion of (Fredholm) determinant of bounded Hilbert space operators which differ from the identity by an element of the trace class. The other result of the paper describes the structure of those transformations on sets of positive semidefinite matrices which preserve the determinant of linear combinations with fixed coefficients.     

An effective hybrid algorithm for computing symbolic determinants

A hybrid algorithm for computing the determinant of a matrix whose entries are polynomials is presented. It is based on the dimension-decreasing algorithm [22] and the parallel algorithm for computing a symbolic determinant of [19]. First, through the dimension-decreasing algorithm, a given multivariate matrix can be converted to a bivariate matrix. Then, the parallel algorithm can be applied to effectively compute the determinant of the bivariate matrix. Experimental results show that the new algorithm can not only reduce enormously the intermediate expression swell in the process of symbolic computation, but also achieve higher degree of parallelism, compared with the single parallel algorithm given in [19].     

ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES

An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.     

Fibonacci polynomials and Sylvester determinant of tridiagonal matrix

By means of left eigenvector method, we evaluate the determinant of a tridiagonal matrix, which extends the determinant due to Sylvester [5].     

A Note on Wiener-Hopf Determinants and the Borodin-Okounkov Identity

The continuous analogue of a Toeplitz determinant identity for Wiener-Hopf operators is proved. An example which arises from random matrix theory is studied and an error term for the asymptotics of the determinant is computed. Submitted: October 15, 2001? Revised: January 3, 2002.     

四元数矩阵的实值行列式

本文定义了任一个四元数方阵的实值行列式．实值行列式有一些好的性质，它是按照谢邦杰定义的可中心化四元数矩阵的行列式的推广．     

关于正交矩阵特征值与行列式的两个定理

以两种证明思路给出了正交矩阵特征值与行列式关系的定理,并在此基础之上提出了对称正交矩阵行列式的迹定理.     

Nonnegative determinant of a rectangular matrix: Its definition and applications to multivariate analysis

It is well known that the determinant of a matrix can only be defined for a square matrix. In this paper, we propose a new definition of the determinant of a rectangular matrix and examine its properties. We apply these properties to squared canonical correlation coefficients, and to squared partial canonical correlation coefficients. The proposed definition of the determinant of a rectangular matrix allows an easy and straightforward decomposition of the likelihood ratio when given sets of variables are partitioned into row block matrices. The last section describes a general theorem on redundancies among variables measured in terms of the likelihood ratio of a partitioned matrix.     

外部流动的Oseen耦合方法,I:Oseen耦合逼近

这篇文章考虑了具有非齐次边界条件的二维非定常外部Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程．通过将内部区域的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程和外部区域的Ｏｓｅｅｎ方程相耦合,得到了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ问题的逼近问题： Ｏｓｅｅｎ耦合问题,此外,我们证明了 Ｏｓｅｅｎ耦合方程弱解的存在性,唯一性和收敛精度．     

关于行列式计算的另类降阶法

由于二阶行列式的计算仅须求两对角线元素的乘积之差,所以计算非常简单．一般地,对高阶行列式求值,虽然可用Laplace展开公式或Gauss消去法,但是展开式会非常繁杂或计算量会很大．本文利用降阶原理,得到一种只需计算二阶行列式就可求出n（n≥3）阶方阵行列式值的另类方法．     

广义严格对角占优矩阵的几个判定条件

广义严格对角占优矩阵在计算数学、数学物理、控制论等众多领域有着广泛而重要的应用.但实际判断一个矩阵是否为广义严格对角占优矩阵却是困难的.本文利用α-对角占优矩阵的性质,给出了广义严格对角占优矩阵的几个判定条件,扩大了判别范围.     

The determinant of the scattering matrix and its relation to the number of eigenvalues

For a large class of scattering systems we study the behavior of the determinant of the scattering matrix as a function on the spectrum of the unperturbed operator. The variation of this determinant is related to the number of eigenvalues due to the perturbation. This relation generalizes results of Levinson and others. The range of physical systems to which these results apply is thus considerably extended.     

Global synchronization in arrays of coupled Lurie systems with both time-delay and hybrid coupling

In this paper, we propose and study an array of coupled delayed Lurie systems with hybrid coupling, which is composed of constant coupling, state delay coupling, and distributed delay coupling. Together with Lyapunov–Krasovskii functional method and Kronecker product properties, two novel synchronization criteria are presented within linear matrix inequalities based on generalized convex combination, in which these conditions are heavily dependent on the upper and lower bounds of state delay and distributed one. Through adjusting inner coupling matrix parameters in the derived results, we can realize the designing and applications of the addressed systems by referring to Matlab LMI Toolbox. The efficiency and applicability of the proposed criteria can be demonstrated by three numerical examples with simulations.     

New families of graphs determined by their generalized spectrum

We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. All graphs constructed here satisfy a certain extremal divisibility condition for the determinant of their walk matrix.     

非负交换整半环上矩阵的正/负行列式保持问题

设R为非负交换整半环,用M_n(R)表示R上所有n×n矩阵构成的矩阵半环.令T是M_n(R)到其自身的线性变换,若T满足|T(X)|~+=|X|~+,■X∈M_n(R)(或|T(X)|~-=|X|~-,(?)X∈Mn(R)),称T为M_n(R)上保持正行列式(负行列式)的线性变换.刻画了n≥4时,M_n(R)上保持正行列式/负行列式的线性满射形式.     

格矩阵的行列式与伴随矩阵

研究了格矩阵的行列式与伴随矩阵,给出了它们的一些代数性质,同时给出了由一个格矩阵构造一个传递矩阵的方法.