对称矩阵与反对称矩阵广义特征值反问题的拓广

定义了上三角等次对角线矩阵和上三角交错次对角线矩阵;讨论了矩阵方程AX-XA=0的对称解与AX XA=0的反对称解.在此基础上考虑了以下问题的可解性:给定A∈Rn×m,D∈Rm×m,分别求X,Y∈SRn×n和X,Y∈ASRn×n,使得XA=YDA.     

Verified Computation of Eigenpairs in the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils

Consider an optimization problem arising from the generalized eigenvalue problem Ax=λBx,where A,B∈Cm×n and m>n.Ito et al.showed that the optimization problem can be solved by utilizing right singular vectors of C:=[B,A].In this paper,we focus on computing intervals containing the solution.When some singular values of C are multiple or nearly multiple,we can enclose bases of corresponding invariant subspaces of CHC,where CH denotes the conjugate transpose of C,but cannot enclose the corresponding right singular vectors.The purpose of this paper is to prove that the solution can be obtained even when we utilize the bases instead of the right singular vectors.Based on the proved result,we propose an algorithm for computing the intervals.Numerical results show property of the algorithm.     

A Christoffel–Darboux formula for multiple orthogonal polynomials

Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel–Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Riemann–Hilbert problem for multiple orthogonal polynomials, which will be useful for asymptotic analysis.     

A Generalized Rational Interpolation Problem and the Solution of the Welch–Berlekamp Key Equation

We show that an algorithm designed to solve the Welch–Berlekamp key equation may also be used to solve a more general problem, which can be regarded as a finite analogue of a generalized rational interpolation problem. As a consequence, we show that a single algorithm exists which can solve both Berlekamp's classical key equation (usually solved by the Berlekamp–Massey algorithm) and the Welch–Berlekamp key equation which arise in the decoding of Reed–Solomon codes.     

Newton's Method for a Generalized Inverse Eigenvalue Problem

A kind of generalized inverse eigenvalue problem is proposed which includes the additive, multiplicative and classical inverse eigenvalue problems as special cases. Newton's method is applied, and a local convergence analysis is given for both the distinct and the multiple eigenvalue cases. When the multiple eigenvalues are present we show how to state the problem so that it is not over-determined, and discuss a Newton-method for the modified problem. We also prove that the modified method retains quadratic convergence, and present some numerical experiments to illustrate our results. © 1997 by John Wiley & Sons, Ltd.     

Clifford分析中广义超正则函数的一个非线性边值问题

讨论了Cliffrd分析中广义超正则函数的一个非线性边值问题.首先将广义超正则函数分解为两个奇异积分算子,然后给出了广义超正则函数的Plemelj公式及相关奇异积分算子的性质,最后利用Schauder不动点原理证明了广义超正则函数的一个非线性边值问题的解的存在性及积分表达式.     

给定极点的有理函数插值序列的收敛性

设Γ∈Ｃ（１，α），α＞０．Ｇ是复平面上以Γ为边界的有界单连通区域．本文考虑了极点位于G外部，以广义Ｆａｂｅｒ-Ｄzｒｂａsｊａｎ有理函数的零点为插值结点的Ｌａｇｒａｎｇｅ插值有理函数序列对Ａ（G）和Ｅ^ｑ（Ｇ）（１＜ｑ＜＋∞）中函数的一致逼近和平均逼近阶的估计．     

广义齐次函数和二体问题

讨论了齐次函数概念的推广,并应用于求解二体问题．     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

Orthogonal Rational Functions and Frequency Analysis

One way of finding unknown frequencies in a trigonometric signal is to use Szeg theory, where under certain conditions asymptotic behavior of zeros of Szeg polynomials lead to the frequencies. Recently this was extended to generalized Szeg theory, i.e. where polynomials are replaced by certain rational functions.This note presents a brief overview of some of the Szeg theory, including also a general formula for the monic orthogonal rational functions. Moreover, for a certain measure, constructed from the observations of the signal, the moments are explicitely determined.Finally a simple example is included, indicating the connection between location of an interpolation point and the way zeros approach frequency points.     

Elementary Darboux Transformations and Factorization

We prove a general theorem on factorization of polynomial matrices and use it to reduce polynomial Darboux matrices to linear ones. We discuss some new examples of linear Darboux matrices.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 143–152, July, 2005.     

Multiple Cotangent and Generalized Eta Functions

In this paper we shall construct multiple analogue of the cotangent functions by using the multiple Hurwitz zeta functions and study their properties and special values. In particular, we express the double cotangent functions in terms of generalized eta functions of Berndt and Lewittes.     

Generalized Invariant Monotonicity and Invexity of Non-differentiable Functions

This paper is devoted to the study of relationships between several kinds of generalized invexity of locally Lipschitz functions and generalized monotonicity of corresponding Clarke’s subdifferentials. In particular, some necessary and sufficient conditions of being a locally Lipschitz function invex, quasiinvex or pseudoinvex are given in terms of momotonicity, quasimonotonicity and pseudomonotonicity of its Clarke’s subdifferential, respectively. As an application of our results, the existence of the solutions of the variational-like inequality problems as well as the mathematical programming problems (MP) is given. Our results extend and unify the well known earlier works of many authors.     

Rational Period Functions on the Hecke Groups

We describe rational period functions on the Hecke groups and characterize the ones whose poles satisfy a certain symmetry. This generalizes part of the characterization of rational period functions on the modular group, which is one of the Hecke groups.     

本征值问题的统一推导方法及讨论

针对一般情形的本征方程X″(x)-2bX′(x)+λX(x)=0结合第一、二、三类齐次边界条件的统一形式,给出有关本征值问题的统一结果,从而可直接利用分离变量法求解2U/t2=a22U/x2+a1u/x+a2t/u+a3u型等含有ux项的泛定方程的定解问题.     

Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

In this paper, we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian ? ? on a general domain Ω ? ?² subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator, in the spirit of the classical method of particular solutions (MPS). The main novelties of the proposed approach are the possibility of targeting each eigenvalue independently without the need for extensive scanning of the positive real axis and the use of small matrices. This is made possible by iterative inclusion of more basis functions in the expansions and a projection idea that transforms the minimization problem associated with MPS and its variants into a relatively simple zero-finding problem, even for expansions with very few basis functions.     

Boundary Asymptotics for Orthogonal Rational Functions on the Unit Circle

Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points { _k } _{k = 1} included in a compact subset of the unit disk, we consider the orthogonal rational functions _n that are obtained by orthogonalization of the sequence { 1, z / ₁, z ² / ₂, ... } where , with respect to the inner product In this paper we discuss the behaviour of _n (t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all _k = 0.     

可修复人机储备系统算子的本征值问题

讨论了可修复人机储备系统算子的本征值问题,讨论了系统算子非零本征值的存在性,并且系统算子一个本征值对应一个本征向量.     

On the Eigenvalue Problem for Self–Adjoint Operators with Singular Perturbations

We investigate the eigenvalue problem for self–adjoint operators with singular perturbations. The general results presented here include weakly as well as strongly singular cases. We illustrate these results on two models which correspond to so–called additive strongly singular perturbations.     

On the Semismoothness of Projection Mappings and Maximum Eigenvalue Functions

This paper analyses the properties of the projection mapping over a set defined by a constraint function whose image is possibly a nonpolyhedral convex set. Under some nondegeneracy assumptions, we prove the (strong) semismoothness of the projection mapping. In particular, we derive the strong semismoothness of the projection mapping when the nonpolyhedral convex set under consideration is taken to be the second-order cone or the semidefinite cone. We also derive the semismoothness of the solution to the Moreau–Yosida regularization of the maximum eigenvalue function.