关于Vandermonde矩阵及其行列式的注记

给出Vandermonde矩阵及其行列式的若干应用,揭示它在高等代数和矩阵分析等方面的重要地位.具体来说,运用Vandermonde行列式来计算几个与之相关的行列式,运用线性方程组来证明组合恒等式,给出两个特殊的Vandermonde矩阵的应用,特别是用Schur矩阵给出了樊畿不等式的一个证明,给出了Vandermonde矩阵与Cauchy矩阵的一个恒等式.     

一般广义Vandermonde行列式的直接计算公式

Vandermonde行列式构造独特,是高等代数中一个典型的行列式,在数值计算,数值逼近等领域有着广泛的应用.通过对已得到的几类广义Vandermonde行列式的结果进行变形分析,推广得到了最一般的广义Vandermonde行列式的直接计算公式.     

两类Stirling数的行列式表示

通过计算两个广义的范德蒙(Vandermonde)行列式,得到了第一类无符号Stirling数和第二类Stirling数的一种新的表示方法:用行列式来表示.     

广义Vandermonde行列式的计算公式

给出两类广义Vandermonde行列式的计算公式以及各类广义Vandermonde行列式的计算方法.     

Lagrange插值公式的几种构造性证明

利用中国剩余定理、行列式以及线性方程组理论给出了Lagrange插值公式的几种构造性证明,得到了Vandermonde矩阵的逆矩阵的一种算法.     

一类多元有理插值公式

应用作者新近建立的Gould-Hsu反演的多变量形式,本文研究一类多元有理插值公式的构造,确定了该类插值级数所表现的函数类,并给出了差分表计算的递归公式。最后作者提出了与适定性相联系的广义Vandermonde行列式的计算问题。     

关于几类矩阵的注记

特殊矩阵在矩阵分析里起着核心的作用.运用Cramer法则和Lagrange插值公式,处理循环矩阵,Vandermonde矩阵,Hilbert矩阵,Cauchy矩阵的一些基本问题:给出Ramakrishnan的矩阵分解定理的一种推广,计算Vandermonde矩阵,Hilbert矩阵,Cauchy矩阵的行列式,当它们可逆...     

二类广义Vandermonde行列式的计算

给出了二类广义Vandermonde行列式计算的显式表示式.     

杨辉与数学教育

杨辉与数学教育孙宏安（辽宁师范大学数学系）众所周知，中国古代数学取得过许多具有世界历史意义的成就，例如在线性方程组求解、一元高次方程求解、一次同余式组求解、高阶等差数列求和、多元高次方程组求解、插值法、圆周率计算等方面都有举世闻名的创造性工作．考察中...     

第三类广义Vandermonde行列式的计算

给出了第三类广义Vandermonde行列式的计算结果的显式表达式.     

一类广义Vandermonde矩阵的可逆条件及求逆公式

利用广义Vandermonde行列式的显式表示式,给出了广义Vandermonde矩阵可逆的充要条件及求逆公式.     

Divided differences and a unified approach to evaluating determinants involving generalized factorials

By employing divided differences, a unified approach to the evaluation of some determinant involving generalized factorials is proposed. Previous generalizations of the Vandermonde determinant and the Cauchy determinant due to Chu-Claudio, Chu-Wang-Zhang and Johnson are included as special cases of our unified treatment.     

On some generalizations of the Vandermonde matrix and their relations with the Euler beta-function

A multiple Vandermonde matrix which, besides the powers of variable, also contains their derivatives is introduced and an explicit expression of its determinant is obtained. for the case of arbitrary real powers, when the variables are positive, it is proved that such generalized multiple Vandermonde matrix is positive definite for appropriate enumerations of rows and columns. As an application of these results, some relations are obtained which in the one-dimensional case give the well-known formula for the Euler betafunction.     

一元切触有理插值存在性的判别方法

在用广义Vandermonde行列式给出Hermite插值多项式的表达式的基础上,分别针对iα=2,iα=3(i=1,2,…,s)的情形给出切触有理插值问题有解的条件及解的表达式.     

Optimization of the Determinant of the Vandermonde Matrix and Related Matrices

The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Gröbner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.     

Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach different from that of other recent papers on this subject, the values of these determinants are computed, recovering and extending the results of those papers.Partially supported by Research Grant PS900121 DGICYT.     

Determinants of Random Matrices and Jack Polynomials of Rectangular Shape

We consider an N -dimensional real integral, indexed by a parameter that specifies the power of a Vandermonde determinant. For two particular values of the parameter, this integral arises from matrix integrals, over real symmetric and complex Hermitian   N × N   matrices. When it is normalized, it gives the expectation of an arbitrary power of the determinant. The results are given as finite summations, using terminating hypergeometric series. We relate the integral to a specific coefficient in the Jack polynomial indexed by a partition of rectangular shape, and present data for this coefficient in terms of the parameter α.