非齐次线性方程组的半正解

线性方程组的不带负分量的非零解向量称为半正解本给出非齐次线性方程组Ax=B (B≠0)的半正解结构,进而得到该类线性方程组有半正解的充分条件和必要条件以及唯一半正解的充要条件。     

非齐次线性方程组的半正解

线性方程组的不带负分量的非零解向量称为半正解.本文给出非齐次线性方程组AX= b (b≠0)的半正解结构,进而得到该类线性方程组有半正解的充分条件和必要条件以及唯一半正解的充要条件.     

线性方程组解的有关问题

首先讨论了两个齐次线性方程组有非零公共解的充分必要条件并给出了非零公共解的一般形式,然后讨论了两个线性方程组同解的一个充分必要条件和非齐次线性方程组的线性无关解向量的个数以及非齐次线性方程组通解的表达式,最后证明了非齐次线性方程组有解的一个充分必要条件.     

关于正线性方程组正解的判定定理

给出正线性方程组正解的概念,给出并证明了正线性方程组正解的若干判别方法,对教学有一定的指导意义.     

正线性方程组正解的判别

本文给出了正线性方程组正解的概念，给出了正线性方程组正解的若干判别方法     

一类带奇异性的两点边值问题

对一类带有奇异性的两点边值问题讨论正解的存在性,在很一般的条件下,建立了摄动问题的可解性与原问题的可解性之间的关系,做为此结论的应用,对某些特殊情形,给出正解存在的充分必要条件。     

线性方程组的正解

讨论了线性方程组正解的若干性质,给出了线性方程组有正解的一个充要条件,以及由此得到的求正解的一般方法,还介绍了正解问题的若干应用.     

一类核反应堆数学模型正解的全局分歧北大核心CSCD

本文研究一类源于核反应堆的数学模型正解的存在性.该模型旨在描述与快中子流密度、反应堆温度紧密相关的核反应过程.本文主要讨论反应堆与外界有热交换的情形.从数学的角度来看,模型自身的非合作特性导致对正解存在性及相关性质的研究较为困难,适用于研究合作系统的比较原理等方法将不再有效.运用分歧理论,我们获得了该模型存在正解的充分必要条件,建立了正解的全局分歧结果,同时对正解的渐近行为进行了仔细分析.所得结果丰富并补充了核反应堆数学模型的相关理论.     

关于线性方程组同解的条件

讨论线性方程组同解的条件,得到了两个线性方程组同解的充分必要条件.     

一类核反应堆数学模型正解的全局分歧

本文研究一类源于核反应堆的数学模型正解的存在性.该模型旨在描述与快中子流密度、反应堆温度紧密相关的核反应过程.本文主要讨论反应堆与外界有热交换的情形.从数学的角度来看,模型自身的非合作特性导致对正解存在性及相关性质的研究较为困难,适用于研究合作系统的比较原理等方法将不再有效.运用分歧理论,我们获得了该模型存在正解的充分必要条件,建立了正解的全局分歧结果,同时对正解的渐近行为进行了仔细分析.所得结果丰富并补充了核反应堆数学模型的相关理论.     

Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli

The existence of multiple positive solutions of systems of singular Hammerstein integral equations is studied, where the nonlinearities involved are allowed to have singularities in their second variables and satisfy weaker conditions involving the first eigenvalues of the corresponding linear Hammerstein integral operators. Such systems contain some mathematical models arising in science and engineering. Applications are given to the existence of multiple positive radial solutions of systems of semilinear singular elliptic equations in annuli on which, to the best of our knowledge, there has been little study.     

On partial differential equations of mixed type in N independent variables

It is proved that a large class of 2-nd order linear partial differential equations of mixed type in n independent variables can be solved by using the theory of positive symmetric systems of 1-st order partial differential equations. Some quasilinear equations of mixed type can be treated in the same way.     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.     

Smoothing factor,order of prolongation and actual multigrid convergence

We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V-cycle multigrid. We derive a two-sided bound that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow and away from 1 when both the smoothing factor and an additional parameter are small enough. Besides the smoothing factor, the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix associated with small eigenvalues. Nice V-cycle convergence is guaranteed if the tangent of this angle has an upper bound proportional to the eigenvalue, whereas nice two-grid convergence requires a bound proportional to the square root of the eigenvalue. We also discuss the well-known rule which relates the order of the prolongation to that of the differential operator associated to the problem. We first define a frequency based order which in most cases amounts to the so-called high frequency order as defined in Hemker (J Comput Appl Math 32:423–429, 1990). We give a firmer basis to the related order rule by showing that, together with the requirement of having the smoothing factor away from one, it provides necessary and sufficient conditions for having the two-grid convergence rate away from 1. A stronger condition is further shown to be sufficient for optimal convergence with the V-cycle. The presented results apply to rigorous Fourier analysis for regular discrete PDEs, and also to local Fourier analysis via the discussion of semi-positive systems as may arise from the discretization of PDEs with periodic boundary conditions.     

拟线性正对称方程组的边值问题及其对混合型方程的应用

<正> 1.引言 本文有两个目的.第一个目的是讨论拟线性正对称方程组的边值问题.如所知,线性的正对称方程组是一类相当广泛的偏微分方程组,许多常见的偏微分方程都可以化为正对称方程组去.对于这个方程组,可以利用能量不等式来证明许多边值问题的适定性.本文充分利用线性正对称方程组的结果,经过适当的估计,用压缩映照原理证明了拟线性正对称方程组的边值问题的解的存在性.对求解的区域而言,问题是大范围的,但     

一类特殊微分代数方程组的性质及应用

一类特殊微分代数方程组的性质及应用顾金生，胡显承（清华大学应用数学系）ＴＨＥＰＲＯＰＥＲＴＩＥＳＯＦＡＫＩＮＤＯＦＤＩＦＦＥＲＥＮＴＩＡＬ／ＡＬＧＥＢＲＡＩＣＥＱＵＡＴＩＯＮＳＡＮＤＴＨＥＩＲＡＰＰＬＩＣＡＴＩＯＮＳ￥ＧｕＪｉｎ－ｓｈｅｎｇ；ＨｕＸｉ...     

Formulae of partial reduction for linear systems of first order operator equations

This paper deals with the reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form.     

Strictly semi-positive tensors and the boundedness of tensor complementarity problems

In this paper, we present the boundedness of solution set of tensor complementarity problem defined by a strictly semi-positive tensor. For strictly semi-positive tensor, we prove that all \(H^+(Z^+)\)-eigenvalues of each principal sub-tensor are positive. We define two new constants associated with \(H^+(Z^+)\)-eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too.     

Autonomous Planar Systems of Riccati Type

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non--Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations.     

Outputs and time lags for linear bioreactors

Linear systems of convection reaction-diffusion equations for bioreactors are shown to have a structure which allows a geometric factorization of steady state problems giving a significant reduction in their dimensionality. Moreover, convection dominated linear systems with quasisymmetric reaction terms may be further simplified by matrix transformations, which uncouple the differential equations. The boundary conditions are also uncoupled when the diagonal diffusivity matrix D governing diffusion in the bioparticle is a scalar multiple of the corresponding matrix H describing the diffusivity characteristic of the fluid boundary layers around the bioparticles. The dominant transient behaviour of the systems may be handled by establishing an analogous system of time independent equations for mean action time variables and higher moments. These equations have the same amenable structure. Outputs, time lags and various mean residence and first passage times, associated with establishing steady outputs from a concentration free initial state, can be expressed in terms of the steady state solutions and mean action time variables.