随机图的谱矩

简单图G的k阶谱矩定义为G的特征值的k阶幂之和,记为Mk(G).应用概率和代数的方法,对于几乎所有的图G,本文给出Mk(G)的一个精确估计.此外,对于几乎所有的多部图G,本文给出了Mk(G)的上界和下界.     

加权流形上加权p-Laplace特征值问题的第一特征值下界估计（英文）

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.     

加权流形上加权p-Laplace特征值问题的第一特征值下界估计

本文研究了加权流形上加权p-Laplacian特征值问题的第一特征值下界估计的问题.利用余面积公式、Cavalieri原理以及Federer-Fleming定理,获得了由Cheeger常数或等周常数确定的第一特征值的下界估计.     

四阶不定微分算子非实特征值的界

给出了四阶正则不定微分算子仅在可积条件下的非实特征值上界和下界的估计.更一般地,非实特征值的下界可以利用Krein空间的自共轭算子得到.     

用有限元方法求解界面特征值问题

用有限元方法计算椭圆型界面特征值问题,实验数据显示近似特征值的变化规律:界面特征值问题中系数的间断性对协调和非协调Crouzeix-Raviart有限元特征值的收敛性并无影响,而且对协调有限元特征值外推以后得到高精度的解,相应的外推值还提供特征值下界;Crouzeix-Raviart元特征值提供特征值下界,这对一般有界区域如"镂空"型区域也成立.另外,还展示近似特征函数的图形.     

矩阵具有正实部特征值的判定

利用根与系数的关系.证明了特征方程没有零实部根的充要条件,给出了矩阵特征值至少具有一个正实部的充分条件,最后通过示例计算验证了该方法的有效性.     

非协调元特征值渐近下界

利用有限元收敛速度下界的结果获得某些非协调元方法新的Aubin-Nitsche估计形式,然后再结合非协调元特征值的展开式获得不需要额外条件下非协调元特征值渐近下界的结果.     

关于图的规范拉普拉斯特征值和的若干结果（英文）

对任意一个连通图G,记L(G)和L(G)分别为G的拉普拉斯矩阵和规范拉普拉斯矩阵.令μ_1≥μ_2≥…≥μ_n=0和λ_1≥λ_2≥…≥λ_n=0分别为G的拉普拉斯特征值和规范拉普拉斯特征值.本文给出了λ_1的三个新的下界.前两个下界优于Das等在[Ars Cormbin.,2015,118:143-154]中给出的下界,第三个下界优于张晓东在[Ars Combin.,2004,72:191-198]中给出的下界.另一方面讨论了规范拉普拉斯特征值与G的度序列之间的关系.同时也讨论了图的拉普拉斯特征值和规范拉普拉斯特征值之间的关系.     

二部图的单特征值

一个图的特征值通常指的是它的邻接矩阵的特征值,在图的所有特征值中,重数为1的特征值即所谓的单特征值具有特殊的重要性.确定一个图的单特征值是一个比较困难的问题,主要是没有一个通用的方法.1969年,Petersdorf和Sachs给出了点传递图单特征值的取值范围,但是对于具体的点传递图还需要根据图本身的特性来确定它的单特征值.给出一类正则二部图,它们是二面体群的凯莱图,这类图的单特征值中除了它的正、负度数之外还有0或者±1,而它们恰好是Petersdorf和Sachs所给出的单特征值范围内的中间取值.     

Laplacian第一特征值整体曲率Pinching的一个结果

紧致流形上Laplacian的第一特征值的下界估计一直以来是人们非常感兴趣的问题之一.本文在整体曲率Pinching较小的条件之下考虑这个问题,得到了相应几何条件之下的Laplacian第一特征值的一个下界估计.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

Error bounds in the approximation of eigenvalues of differential and integral operators

Various methods of approximating the eigenvalues and invariant subspaces of nonself-adjoint differential and integral operators are unified in a general theory. Error bounds are given, from which most of the error bounds in the literature can be derived. Computable error bounds are given for simple eigenvalues, and trace formulae are used to improve the accuracy of the computed eigenvalues.     

Sub-Laplacian Eigenvalue Bounds on CR Manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjöstrand asymptotic law, and can be viewed as a CR version of Korevaar's bounds for Laplace eigenvalues of conformal metrics.     

关于矩阵多项式特征值界的注记

本文讨论矩阵多项式特征值定域问题.首先对Higham和Tisseur[Linear Algebra Appl.,358(2003),5-22]得到的结果给出较详细的比较.然后利用分块矩阵谱半径的估计给出了获取特征值界的一种新办法.利用这种新办法,不但可以简明地得出很多已有的界,且对椭圆及双曲矩阵多项式得出了特征值的新的界.     

Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods

In this study, the bounds for eigenvalues of the Laplacian operator on an L-shaped domain are determined. By adopting some special functions in Goerisch method for lower bounds and in traditional Rayleigh–Ritz method for upper bounds, very accurate bounds to eigenvalues for the problem are obtained. Numerical results show that these functions can also be successfully used to solve the problem on the region with other reentrant angle.     

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

Estimates for eigenvalues of quasilinear elliptic systems. Part II

In this paper we find explicit lower bounds for Dirichlet eigenvalues of a weighted quasilinear elliptic system of resonant type in terms of the eigenvalues of a single p-Laplace equation. Also we obtain asymptotic bounds by studying the spectral counting function which is defined as the number of eigenvalues smaller than a given value.     

The spread of matrices and polynomials

The spread of a matrix (or polynomial) is the maximum distance between any two of its eigenvalues (or its zeros). E. Deutsch has recently given upper bounds for the spread of matrices and polynomials. We obtain sharper, simpler upper bounds and observe that they are also upper bounds for the sum of the absolute values of the two largest eigenvalues (or zeros).     

Upper (lower) bounds of the eigenvalues,spread and the open problems for the real symmetric interval matrices

This work is concerned with exploring the upper bounds and lower bounds of the eigenvalues of real symmetric matrices of order n whose entries are in a given interval. It gives the maximum and minimum of the eigenvalues and the upper bounds of spread of real symmetric interval matrices in all cases. It also gives the answers of the open problems for the maximum and minimum of the eigenvalues of real symmetric interval matrices. Copyright © 2012 John Wiley & Sons, Ltd.     

On the discrete spectrum of non-selfadjoint operators

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schrödinger operators with complex potentials.