关于Ostrowski圆盘定理的一个注记

本文对Ostrowski给出的关于矩阵的特征值估计作一些讨论，特征值分布特性被揭示，进而得到了一个判定矩阵非奇异性的充分条件．     

Heisenberg群上Folland-Stein算子的Dirichlet特征值问题

该文研究Heisenberg群上Folland－Stein算子■（α为复数）的特征值问题，证明了当|Reα|＜n时，■具离散分布的特征值．当α∈（-n,n）时，特征值均为正的．然后给出了相邻特征值之差的估计．     

利用三角分解估计实对称矩阵的特征值

鉴于直接计算矩阵特征值的工作量很大,因此在实问题中,我们有时得借助于对这些特征值的某种估计。但通常基于Gerschgorin定理的估计方法往往不能对各特征值给出足够精确的界。本文则利用半正定矩阵伴随选主元的LDL~T分解提出一种估计实对称矩阵特征值的方法,所耗费的计算量是有限的,但在大多数情况下估计的精度可以得到很大的改进。本方法特别适用于半正定矩阵非零小特征值的估计,从而可用于在计算机上确定具体数值矩阵的秩。     

关于几类矩阵的特征值分布

<正> 在矩阵论中以及应用矩阵工具的各类问题中,估计矩阵的特征值大小与分布十分重要.在[1]中,我们给出了非负矩阵(元素全非负的矩阵)最大特征值的计算与估计方法,并将此结果推广到更广的一类矩阵.在本文中,我们将对实用中几类重要矩阵给出它们特征值分布的估计.     

平稳遍历马氏链部分和序列最小值分布的渐近估计及其应用

本文给出了有限状态平稳遍历Markov链部分和序列最小值分布的一个渐近估计式并利用它对一类Athreya-KarlinBPRE．灭种概率的渐近行为作出估计．     

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.     

圆盘定理的改进与弱连对角占优矩阵

本文对圆盘定理进行了改进，给出了特征值分布新的估计，由此引出了弱连对角占优矩阵，讨论了其基本性质，重点分析了该类矩阵的逆与分裂特征，证明了在该类矩阵条件下Ｈ－相容分裂是收敛分裂，并给出迭代矩阵谱半径的上界及ＳＯＲ算法中参数ω的选取范围。     

一类二阶退化方程组的特征值估计

本文运用关于Ｈｅｉｓｅｎｂｅｒｇ群的知识讨论了一类二阶退化方程组的特征值问题．对Ｒ２ｎ＋１中的有界域上定义的特征值问题，给出了相邻特征值之差的估计，并对一类特殊方程组得到了更精确的估计．     

负压激励下含椭圆孔高弹体的屈曲分析

基于数值模拟与理论分析,研究了含周期性椭圆孔二维结构的屈曲行为。针对不同的屈曲模态,建立理论模型进行模态分析。结果表明,改变孔的几何参数,椭圆孔结构的屈曲模态会随之发生转换,理论分析与数值结果吻合良好。此外,在数值模拟中,与位移加载不同,负压激励下的单胞需要考虑力边界条件的修正,以确保其满足完备性条件。已有工作在单胞选择中常存在问题,导致错误结果。针对上述问题研究了不同单胞所对应的边界条件,并结合有限结构进行了分析与讨论。     

p-Laplace算子基本特征值率的估计

通过改进已有方法。给出了Euclid空间R^N中p-Laplace算子Dirichlet特征值问题中基本特征值率的两个估计．其中一个估计与区域有关，另一个则与区域无关．这里的结论是对已有文献中结果的改进．     

Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.     

The extrapolation of numerical eigenvalues by finite elements for differential operators

This paper discusses the extrapolation of numerical eigenvalues by finite elements for differential operators and obtains the following new results: (a) By extending a theorem of eigenvalue error estimate, which was established by Osborn, a new expansion of eigenvalue error is obtained. Many achievements, which are about the asymptotic expansions of finite element methods of differential operator eigenvalue problems, are brought into the framework of functional analysis. (b) The Richardson extrapolation of nonconforming finite elements for multiple eigenvalues and splitting extrapolation of finite elements based on domain decomposition of non-selfadjoint differential operators for multiple eigenvalues are achieved. In addition, numerical examples are provided to support the theoretical analysis.     

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported. The first author was supported by China Postdoctoral Sciences Foundation.     

Algorithm for min-range multiplication of affine forms

We consider an approximate method based on the alternate trapezoidal quadrature for the eigenvalue problem given by a periodic singular Fredholm integral equation of second kind. For some convolution-type integral kernels, the eigenvalues of the discrete eigenvalue problem provided by the alternate trapezoidal quadrature method have multiplicity at least two, except up to two eigenvalues of multiplicity one. In general, these eigenvalues exhibit some symmetry properties that are not necessarily observed in the eigenvalues of the continuous problem. For a class of Hilbert-type kernels, we provide error estimates that are valid for a subset of the discrete spectrum. This subset is further enlarged in an improved quadrature method presented herein. The results are illustrated through numerical examples.     

Finite elements for the eigenvalue problem of differential operators in unbounded intervals

The eigenvalue problem for one-dimensional differential operators with a possible essential spectrum is discretized with finite elements defined on a bounded interval together with a fundamental system of the differential equation outside of the interval. A non-pollution property of the discrete spectra is proved and the error in the approximation of isolated eigenvalues and corresponding eigenvectors is estimated. The convergence of some numerical algorithms for the solution of the subsequent discrete nonlinear eigenvalue problem is proved. The method is tested in some numerical examples.     

Trimmed linearizations for structured matrix polynomials

We discuss the eigenvalue problem for general and structured matrix polynomials which may be singular and may have eigenvalues at infinity. We derive condensed forms that allow (partial) deflation of the infinite eigenvalue and singular structure of the matrix polynomial. The remaining reduced order staircase form leads to new types of linearizations which determine the finite eigenvalues and corresponding eigenvectors. The new linearizations also simplify the construction of structure preserving linearizations.     

On Eigenvalues of the Transition Matrix of Some Count-Data Markov Chains

We analyze the eigenstructure of count-data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models.     

Recursive estimation for ordered eigenvectors of symmetric matrix with observation noise

The principal component analysis is to recursively estimate the eigenvectors and the corresponding eigenvalues of a symmetric matrix A based on its noisy observations Ak=A+Nk, where A is allowed to have arbitrary eigenvalues with multiplicity possibly bigger than one. In the paper the recursive algorithms are proposed and their ordered convergence is established: It is shown that the first algorithm a.s. converges to a unit eigenvector corresponding to the largest eigenvalue, the second algorithm a.s. converges to a unit eigenvector corresponding to either the second largest eigenvalue in the case the largest eigenvalue is of single multiplicity or the largest eigenvalue if the multiplicity of the largest eigenvalue is bigger than one, and so on. The convergence rate is also derived.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Approximation of the eigenvalues of a fourth order differential equation with non-smooth coefficients

The eigenvalues of a fourth order, generalized eigenvalue problem in one dimension, with non-smooth coefficients are approximated by a finite element method, introduced in an earlier work by the author and A. Lutoborski, in the context of a similar source problem with non-smooth coefficients. Error estimates for the approximate eigenvalues and eigenvectors are obtained, showing a better performance of this method, when applied to eigenvalue approximation, compared to a standard finite element method with arbitrary mesh.