Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of R_II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.     

主特征值估计的新故事

本文介绍主特征值估计的两种通用方法，着重于两个侧面：来自黎曼几何的第一种方法如何应用于概率论；来自概率论的第二种方法如何应用于黎曼几何。此外，还将概述若干基本结果。     

A Note on the Difference of Gaussian Measure of Two Balls in Hilbert Spaces

§１．　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＬｅｔＨｂｅａｓｅｐａｒａｂｌｅＨｉｌｂｅｒｔｓｐａｃｅ，μｂｅａｓｙｍｍｅｔｒｉｃＧａｕｓｓｉａｎｍｅａｓｕｒｅｏｎＨａｎｄλ１≥λ２≥…０ｂｅｔｈｅｅｉｇｅｎｖａｌｕｅｓｏｆｔｈｅｃｏｖａｒｉａｎｃｅμＴ．ＺａｋｇｉｖｅｓｔｈｅｅｓｔｉｍａｔｅｏｆｔｈｅｄｉｆｆｅｒｅｎｃｅｏｆＧａｕｓｓｉａｎｍｅａｓｕｒｅｏｆｔｗｏｂａｌｌ．μ｛ｘ∈Ｈ：ｙｘｙ≤ｔ｝－μ｛ｘ∈Ｈ：ｙｘ＋ｒｙ≤ｔ｝ｉｎ［１，２］．Ｉｎｔｈｉｓｎｏｔｅｗｅｏｂｔａｉｎａｆｉｎｅｒｅｓｔｉｍａｔ…     

EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

1IntroductionInrecentyears,theexistenceandmultiplicityofsolutionsoftheproblemwerestudiedbynumerousauthors.WhereA>0,fiisasmoothdomaininR",N22.Thefundamentalquestionsaboveproblem(1.1)-(1.2):have.receivedrathersimpleanswers.Todoso,theauthorsalwaysusethenicepropertiesoftheoperatorA,forexample,thelinearity,maximumprinciple,strongmaximumprinciple,comparisonprincipleandsoon.Meanwhile,theexistenceandmultiplicityofpositiveradialsolutionsof(1.1)-(1.2)attractedmuchinterest,see[l,5-6,10,13-18]andtherefe…     

对称双随机矩阵逆特征值问题

主要研究随机矩阵逆特征值问题．特别是对称双随机矩阵和列随机矩阵逆特征值问题．对参考文献[1]与[2]的结论作了一些推广．并给出了—个数值例子．     

不定Sturm—Liouville算子的特征函数的振荡问题

研究了定义在有限区间（0，l）上的具有一般分离型边条件的不定Sturm—Liouville算子的特征函数的振荡问题．利用Prufer变换，给出了上述Sturm-Liouville算子特征值的符号指标的具体形式；得到了特征值的符号指标与Weyl函数以及Prufer角在该特征值处的罗朗展式（泰勒展式）的首项系数的符号之间的关系；最后，在上述两个结果的基础上给出了上述Sturm—Liouville算子的第n个正特征值所对应的特征函数在[0，l]内的零点个数的计算公式．     

广义边界梁的动力学建模与动态特征

对于广义边界条件Euler-Bernoulli梁,采用相对描述方式建立了可描述梁整体运动和相对变形的几何非线性及其线性化动力学模型,应用线性变换得到了该类梁的线性经典动力学方程,得到了广义边界条件下梁的横向振动代数特征方程、特征函数及特征值的退化表达式.算例分析了边界小扰动对固支-固支梁横向振动特征的影响规律.     

Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength S _j for j=1,…,K at unknown spot locations x _j ∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S _j and x _j for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths S _j , for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.      

积分方程特征值的一个下界估计式及其应用

导出了具有正定对称核的积分方程最小特征值的一个下界估计式，为此为基础，获得了相应特征值的一种算法。     

The Principal Eigenvalue for Jump Processes

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented. The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions. Received November 15, 1999, Accepted March 9, 2000