THE EIGENVALUE PROBLEM EQUIVALENT TO MULTIVARIATE POLYNOMIAL SYSTEM

In this paper, the authors proved that finding all solutions of a given multivariate polynomial system is equivalent to solving a relative joint eigenvalue problem(Theorem 1) and in some cases one can find all solutions of the given system from the eigenvalues and vectors of one matrix or matrix pencil (Theorem 2). Especially the situation that the ideal generated by the given system is 0-dimensional is discussed.     

ASYMPTOTIC FORMULAE OF EIGENVALUE-COUNTING FUNCTION FOR ELLIPTIC EIGENVALUE PROBLEMS ON A BOUNDED DOMAIN WITH A FRACTAL BOUNDARY

IIntroductlonLet fi be a bounded open set In the n－dimensional Euclldean space R”withtheboundaryo0．In this paper,we shallessentiallydeal with thefollowingeigenvalue problem:,nD 一)。o…4km)螂川川J”人川川,x〔u,丸 人k＝1\“’“)llillj＝U．U b UI[。where丸 meanso/axj,and we suPPose that coeaclents aj。(x),J;k＝ 1,2;…,。satisfy the following conditions:(1)aj。(x)is real function and belong to Cm(R”);(2)There exists an M＞0;such that Iajk(x)l＜M;;;k二 1,2,…,n;148 Ann ofDiff Eqs Vol…     

二次自伴矩阵多项式阵的特征值

系统地论证了二次自伴矩阵多项式特征值,特征向量的性质.给出了二次自伴矩阵多项式特征值与任一非零向量所对应的二次多项式根之间的大小关系;精确地给出了二次自伴矩阵多项式是负定时参数的界;简化了二次自伴矩阵多项式的符号特征是正(负)的特征值对应特征向量间可以是线性无关等定理的证明.     

Brusselator方程解的存在性和稳定性

本文研究了一种Brusselator方程的。用不变区域证明了解的全局存在性，并通过构造上下解给出了非常数定态解渐进稳定的条件。最后推出吸引域。     

谱半径前六位的n阶单圈图

恰含一个圈的简单连通图称为单圈图。Cn记n个顶点的圈。△（i,j,κ）记C3的三个顶点上分别接出i,j,κ条悬挂边所得的图，其中i≥j≥κ≥0.Sl^n-l记Cl的某一顶点上接出n-l条悬挂边所得到的图。△（n-4 1,0,0)记△（n-4,0,0)的某个悬挂点上接出一条悬挂边所得到的图。本文证明了：若把所有n(n≥12）阶单圈图按其最大特征值从大到小的顺序排列，则排在前六位的依次是S3^n-3，△（n-4,1,0),△（n-4 1,0,0),S4^n-4,△（n-5,2,0),△（n-5,1,1)。     

Riemannian Foliations and Eigenvalue Comparison

Eigenvalue comparison theorems for the Laplacian on a Riemannian manifold generally give bounds for the first Dirichlet eigenvalue on balls in the manifold in terms of an eigenvalue arising from a geometrically or analytically simpler situation. Cheng's eigenvalue comparison theory assumes bounds on the curvature of the manifold and then compares this eigenvalue to the eigenvalue of a ball in a constant curvature space form. In this paper we examine the basic Laplacian – the appropriate Laplacian on functions that are constant on the leaves of the foliation. The main theorems generalize Cheng's eigenvalue comparison theorem and other eigenvalue comparison theorems to the category of Riemannian foliations by estimating the first Dirichlet eigenvalue for the basic Laplacian on a metric tubular neighborhood of a leaf closure. Several other facts about the the first eigenvalue of such foliated tubes as well as some needed facts about the tubes themselves are established. This comparison theory, like Cheng's theorem, remains valid for large tubes that are not homotopic to the middle leaf closure and that may have irregular boundaries. We apply these results to obtain upper bounds for the eigenvalues of the basic Laplacian on a closed manifold in terms of curvature bounds and the transverse diameter of the foliation.     

关于矩阵二次型极值的若干结果

本文出于统计研究的需要，系统地讨论了Ｒａｙｌｅｉｇｈ－Ｒｉｔｚ定理在矩阵形式下的推广，给出了若干新结果，完善了现有文献中这方面的研究。     

利用半群代数中良序基构造解稀疏多项式方程组特征值方法

本文利用半群代数k[A]中良序基,构造了求稀疏多项式方程组解的特征值矩阵,并给出了可以构造方阵的条件.     

Dirac类氢原子的超对称方法研究

应用超对称方法研究了Ｄｉｒａｃ类氢原子问题，不需繁琐的数学处理，得到和严格解法完全一致的能量本征值谱及相应的本征函数。     

Nonnegative realization of spectra having negative real parts

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list σ except for one (the Perron eigenvalue) have real parts smaller than or equal to zero.