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.     

Cramer法则在数值计算中的若干应用

用Cramer法则给出了Lagrange插值公式和Newton插值公式的简洁证明,同时得到了Vandermonde矩阵的逆矩阵的LU分解.     

Vandermonde行列式对Lagrange插值公式的应用

给出了广义Vandermonde行列式的一种求法,并运用它给出了Lagrange插值公式的几个证明.     

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.     

整系数多项式有理根的一个新求法

求整系数多项式的有理根 ,现行的高等数学书中只有一个经典的方法 ,本文给出了第二个有趣的简捷方法 ,这种方法的主要过程只需进行简单的算术运算     

整系数多项式有理根一个新求法的再探讨

设f (x)为整系数多项式,α为有理数,对n个不同的整数t1,…,tn,gα(tk) =f (tk)tk-α都是整数,那么α是f (x)的根的充要条件是f (t) =∑ni=1∏1≤j≤nj≠it-tjti-tjgα(ti) ( t∈Z) .     

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,当它们可逆时这些矩阵的逆矩阵以及逆矩阵的所有元素之和.     

Lagrange插值公式与Hermite插值公式的注记

给出一种基于商的形式的Lagrange与Hermite插值公式及其证明,同时还给出了两个相关的不等式.     

以线性代数观点看常用多项式插值方法

数学各分支之间存在很多共同的思想和方法.本文结合线性代数中关于线性空间的相关理论及观点来看数值分析中常用的多项式插值方法.一方面借助线性空间的基与坐标理论将常用多项式插值方法统一起来,并借助过渡矩阵给出了不同插值方法之间的通用转换公式.另一方面还可以通过构造特殊基函数组来产生新的多项式插值方法.     

关于多项式保形插值理论及应用

保形插值在CAGD中有着重要的应用,本综述了多项式保形插值的理论及应用。     

一类E-Vandermonde矩阵的求逆及逆的迭代公式

本文给出一类E-Vandermonde矩阵和广义E-Vandermonde矩阵可逆的条件及逆的矩阵表达式，并给出了求逆的迭代公式．     

An operator theory of linear functional differential equations

An operator theory, based on convolution rings and modules, is developed for various classes of first-order vector functional differential equations of the retarded type with finite and infinite delays. The existence and construction of complete solutions is approached in a novel manner by incorporating initial data into the operator framework. Results are then obtained on exponential and asymptotic stability. The operator framework is also applied to the study of state feedback. Constructive results on spectrum assignability and stabilizability by feedback are given.     

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.     

Lower bounds for the condition number of a real confluent Vandermonde matrix

Lower bounds on the condition number of a real confluent Vandermonde matrix are established in terms of the dimension , or and the largest absolute value among all nodes that define the confluent Vandermonde matrix and the interval that contains the nodes. In particular, it is proved that for any modest (the largest multiplicity of distinct nodes), behaves no smaller than , or than if all nodes are nonnegative. It is not clear whether those bounds are asymptotically sharp for modest .

     

An example of application of the Nielsen theory to integro-differential equations

A new nontrivial example of an application of the Nielsen fixed-point theory is presented, this time, to integro-differential equations. The emphasis is on the parameter space so that no subdomain becomes invariant under the related solution (Hammerstein) operator. Thus, at least three (harmonic) periodic solutions are established to a planar integro-differential system.

     

关于集{N_k~-(x)}_k~n=0的一个定理的新证明(英文)

设N（x）=为定义在复平面C上的函数．V（x1，…xn）为n×n矩阵，其i行，j列的元素为N（xh）.小本给出行列式detV（x1，…xb）≠0的另一种证明．     

Ergodic theory and linear differential equations

The usual notions of “elliptic” and “hyperbolic” as applied to linear ODE's are generalized to certain two-dimensional systems (including almost-periodic ones) in which the coefficient matrix depends on time. An example due to Million??ikov is used to indicate certain limits to the theory.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.