关于矩阵族G~n(θ,△t)一致有界性的局部(J)条件

本文第一个作者曾基于Kreiss预解条件给出稳定性问题的(J)条件([1]).本文应用[2]中的局部化技巧得到了矩阵族G~n(θ,△t)一致有界(稳定)的局部(J)条件.作为一个有趣的应用,我们在较弱的条件下改进了[2]、[3]、[4]中的主要结果.     

套网格有限差分方法求解——一阶双曲方程初边值问题的稳定性条件

本文讨论用套网格(Nesting grid)有限差分方法求解一阶双曲方程初边值问题,即在不同的区域做不同的网格剖分,选用相同或不同精度的差分格式.这种方法亦称杂交法(Hybrid Difference Method)或混合型差分方法(Mixed Difference Method),它广泛应用于数值天气预报和流体力学数值计算中,特别,对于局部区域上的解,其梯度变化激烈,而在其余区域上解的梯度变化平稳时,选用这种方法更有优越性。 [5,8,10]是套网格差分格式稳定性方面的工作,上述工作均以Kreiss定理为基础,针对两层显式耗散格式讨论,因而不便于应用,本文旨在利用GKS理论,寻求一般形式套网格差分格式稳定性的判别条件,§1针对模型问题建立套网格差分格式的一般形式,并介绍GKS理论的一种变形;§2建立套网格差分格式稳定性判别条件;§3是对一类差分格式和网格条件给出易于检验的稳定性判别准则;§4推广了Ciment匹配定理,并证明§3中的主要结果,最后§5是数值例子, 本文采用[1],[3]的符号。     

Kreiss矩阵定理的新证明

熟知,Kreiss矩阵定理在差分法稳定性理论中占有十分重要的位置.定理的证明相当复杂,[2]中收入的证明虽经Morton和Schechter作了适当处理,但是被称为证明核心的(R)?(S)的过程仍然繁琐.本文给出一种简单直观的新证明.在证明回路(A)?(R)?(S)?(H)?(A)中,(A)?(R)和(H)?(A)沿用[2]的证明,一并给出.顺便指出,新证明并不影响[2]中对另一重要定理(Buchanan准则)证明的简化,     

以酉辛群为特征边界的双曲空间的调和分析

§1.前言 本文采用四元数体的观点,讨论了以酉辛群USP(2n)为特征边界的双曲空间 {I-Z>0,ZJ=J}(1.1) 的调和函数论,从另一角度建立酉辛群上的Abel收敛定理的出发点。 §2.四元数体上的第一类典型域 华罗庚教授、万哲元教授“典型群”中建立了一般非交换体上方阵行列式理论。 四元数体的工阶方阵表示     

四阶杆振动方程的一族高稳定的十字架格式

用辛几何的观点得到了四阶杆振动方程的一族十字架辛格式，对于四阶杆振动方程的稳定条件不一定随时间方向的精度的提高而放宽，而随空间方向精度的提高稳定范围缩小．数值例子表明单辛算法具有良好的数值稳定性．     

非自治Caputo分数阶发展方程弱解的适定性与不变集

该文分析一族含有依赖时间参数t线性算子的时间分数阶非自治发展方程,利用Lions表示定理,得到了弱解适定性的充分条件;基于正交投影,建立了时间分数阶发展方程弱解的不变性准则.该文所研究方程中的算子是依赖时间的.     

带有偏差变元的广义Hamilton系统周期解的存在性

应用变分方法与Morse理论,本文讨论下面含有时滞的广义Hamilton系统的周期解,J*du-dt=g(t,u(t-r1),…,u(t-rs))其中J*是非奇异2n×2n反对称矩阵.在一定条件下,本文得到上述方程至少存在两个非平凡2π-周期解而对于一般的微分系统,本文给出其具有变分结构的判定性准则.     

无限回路与它的一个应用

<正> 就说 X 是方程组(2)的非负数,本文的目的就是研究这种非负解的性质.在§1中,我们首先引进了“链”(无限回路)与“链丛”的概念,并用来定义了相关解(相关矩阵)与无关解(无关矩阵),证明了无关解不能表成异于它本身的两个非负解的加权平均(定理1).接着在§2中证明了相关解却能表成这种加权平均(定理3),并进一步讨论了这种加权平均表示法的一些性质,在§3中,我们将前面的讨论应用到随机矩阵上     

一类多复变全纯映照子族的增长和偏差定理

在一般复Banach空间X中的单位球B上引入一类全纯映照族M_g.考虑B上满足条件(Df(x))~(-1)f(x)∈M_g的正规化局部双全纯映照f(x)(其中x=0是f(x)-x的k+1阶零点)并得到其增长定理.作为应用,也得到了C~n中单位多圆柱D~n上映照f关于Jacobi矩阵Jf(z)的偏差定理,该结果统一和推广了星形映照许多子族的相应结论.     

四元数矩阵的惯性定理与稳定性

本文给出了四元数矩阵惯性的定义,讨论了四元数体上Lyapunov矩阵方程的唯一解,推广了一般惯性定理、Lyapunov稳定性定理、Carlson-Schneider定理、Stein稳定性定理等一些重要的结果到四元数矩阵,同时得出了四元数体上稳定矩阵的一些判别条件.     

An extension of the Kreiss stability theorem to families of matrices of unbounded order

The Kreiss matrix theorem asserts three necessary and sufficient conditions for a family of matrices of fixed finite order to be L₂-stable: a resolvent condition (R), a triangularization condition (S) and a Hermitian norm condition (H). We extend the Kreiss theorem to families of matrices of finite but unbounded order with the restriction that the degrees of the minimal polynomials of all matrices in the family are less than a fixed constant. For such matrix families, we show that (R) and (H) remain necessary and sufficient for L₂-stability, while (S) must be replaced by a somewhat stronger “block triangularization” condition (S′). This extended Kreiss theorem permits a corresponding extension of the Buchanan stability theorem.     

The equivalence of L2-stability,the resolvent condition,and strict H-stability

The Kreiss matrix theorem asserts that a family of N × N matrices is L₂-stable if and only if either a resolvent condition (R) or a Hermitian norm condition (H) is satisfied. We give a direct, considerably shorter proof of the power-boundedness of an N × N matrix satisfying (R), sharpening former results by showing that power- boundedness depends, at most, linearly on the dimension N. We also show that L₂-stability is characterized by an H-condition employing a general H-numerical radius instead of the usual H-norm, thus generalizing a sufficient stability criterion, due to Lax and Wendroff.     

关于“K类”准线性双曲型守恒律组整体解的研究

<正> 关于准线性双曲型守恒律组整体解的研究,其意义是许多人所熟知的.Diperna R.J.在文[1]中,利用Glimm J.格式证明了“K类”守恒律组之具有界变差初值的初值问题存在整体广义解.关于研究这类方程的意义,该文已经说明. 我们知道,对于方程式的情形,曾经用多种方法研究存在性问题,这不是为了改善证明方法,而是不同方法各自有不同实际意义.但是,迄今对于方程组,用Glimm J.格式几乎成了唯一的方法,其他只有对初值用阶梯函数逼近的方法有一些结果.     

Initial-Boundary Value Problem for a Class of Linear Relaxation Systems in Arbitrary Space Dimensions

In this paper, we consider the characteristic initial-boundary value problem (IBVP) for the multi-dimensional Jin-Xin relaxation model in a half-space with arbitrary space dimension n?2. As in the one-dimensional case (n=1, see (J. Differential Equations, 167 (2000), 388-437), our main interest is on the precise structural stability conditions on the relaxation system, particularly the formulation of boundary conditions, such that the relaxation IBVP is stiffly well posed, that is, uniformly well posed independent of the relaxation parameter ε>0, and the solution of the relaxation IBVP converges, as ε→0, to that of the corresponding limiting equilibrium system, except for a sharp transition layer near the boundary. Our main result can be roughly stated as Stiff Kreiss Condition=Uniform Kreiss Condition for the relaxation IBVP we consider in this paper, which is in sharp contrast to the one-dimensional case (Z. Xin and W.-Q. Xu, J. Differential Equations, 167 (2000), 388-437). More precisely, we show that the Uniform Kreiss Condition (which is necessary and sufficient for the well posedness of the relaxation IBVP for each fixed ε), together with the subcharacteristic condition (which is necessary and sufficient for the stiff well posedness of the corresponding Cauchy problem), also guarantees the stiff well posedness of our relaxation IBVP and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate. Optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified.     

3×3上三角算子矩阵的Weyl型定理

设A∈B(H1),B∈B(H2),C∈B(H3)为给定的三个算子,用M(D,E,F)= 表示一个作用在H1(?)H2(?)H3上的3×3算子矩阵．本文首先给出存在算子D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2),使得M(D,E,F)为上半Fredholm算子(下半Fredholm算子)的充要条件．同时研究了3×3算子矩阵 M(D,E,F)的Weyl定理,α-Weyl定理,Browder定理和α-Browder定理．     

On a conjecture by le Veque and Trefethen related to the kreiss matrix theorem

In the Kreiss matrix theorem the power boundedness ofN ×N matrices is related to a resolvent condition on these matrices. LeVeque and Trefethen proved that the ratio of the constants in these two conditions can be bounded by 2eN. They conjectured that this bound can be improved toeN.In this note the conjecture is proved to be true. The proof relies on a lemma which provides an upper bound for the arc length of the image of the unit circle in the complex plane under a rational function. This lemma may be of independent interest.     

上三角算子矩阵Browder定理稳定性的判定

用σ(T)和σ_w)分别表示算子T的谱与weyl谱,π_(00)(T)={λ∈isoσ(T),0dimN(T-λI)∞},若σ(T)\σ_w(T)■π_(00)(T)成立,则就认为T满足Browder定理.主要研究了2×2上三角算子矩阵的Browder定理在紧摄动下的稳定性,并给出了判定稳定性的等价条件.     

Boundary conditions and boundary layers for a multi-dimensional relaxation model

In this paper we study the linearized relaxation model of Katsoulakis and Tzavaras in a half-space with arbitrary space dimension n?1. Our main interest is to establish the asymptotic equivalence of the relaxation system and its corresponding multi-dimensional equilibrium conservation law. We identify and rigorously justify a necessary and sufficient condition (which we refer to as stiff Kreiss condition, or SKC in short) on the boundary condition to guarantee the uniform stability of the initial-boundary value problem of the relaxation system independent of the relaxation rate. The asymptotic convergence and the corresponding boundary layer behavior are studied by Fourier-Laplace transform and a detailed asymptotic analysis. The SKC is shown to be more restrictive than the classical uniform Kreiss condition for all n?1.     

Alpha well-posedness,alpha stability,and the matrix theorems of H.-O. Kreiss

Summary For systems of partial differential equations with constant coefficients and for the corresponding difference equations the concepts of well-posedness and stability are introduced. These concepts are more general than strong well-posedness and stability on the one hand, and more restrictive than weak well-posedness (Petrovskii condition) and weak stability (von Neumann condition) on the other. Characterizations of these properties are established which partly extend the matrix theorems of H.-O. Kreiss. Also a Lax type theorem is valid in this setting.This author was supported by the Deutsche Forschungsgemeinschaft (Grant Go 261/4)     

Levinson定理证明的一些简化

<正> (一)1974年,Levinson N.证明了他的关于Riemann ζ-函数在直线σ=1/2上的零点个数的著名定理:对充分大的T有