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.     

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.     

含参数线性方程组解的判别方法注记

当线性方程组中含有未知参数时,线性方程组解的情况往往需要进行讨论.本文给出了在非齐次线性方程组系数矩阵中含有未知参数且系数行列式等于零的情况下,判定对应参数值下方程组的解是无解还是有无穷多解的两个判定定理.和以前的方法比较,本文提出的讨论方法更直接.     

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。     

随机扰动下的非自治种群的随机持久性(英文)

本文考虑非线性随机扰动下的生态种群的问题.首先给出全局正解的存在性.其次,在合理的条件下讨论随机最终有界和随机持久问题,同时也给出解的渐近估计.     

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.     

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.     

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     

具有变系数和时间延迟的随机HIV-1感染模型[英文]

本文提出了具有变系数和时间延迟的随机HIV-1感染模型.首先证明模型存在唯一全局正解,然后给出无感染均衡解渐近稳定的充分条件.     

具有脉冲效应的周期三种群捕食-食饵系统

本文讨论一类具有脉冲效应和周期系数的两个食饵一个捕食者的捕食-食饵系统的动力学行为.利用脉冲微分方程比较定理和乘子理论,证明了系统的有界性,讨论了平凡周期解和半平凡周期解的稳定性,利用重合度的理论给出了系统存在周期正解的充分条件.     

通解矩阵(英文)

用消元法程序在计算机上求含ｍ个方程ｎ个未知量的线性代数方程组的通解的问题现在仍然没有得到解决．本文首先给出一个称为“通解矩阵”的新定义，然后证明两个有关定理．以此为基础，使解决上述问题成为一件容易的事．     

A class of discrete time generalized Riccati equations

In this paper, a class of discrete-time backward non-linear equations defined on some ordered Hilbert spaces of symmetric matrices is considered. The problem of the existence of some global solutions is investigated. The class of considered discrete-time non-linear equations contains, as special cases, a great number of difference Riccati equations both from the deterministic and the stochastic framework. The results proved in the paper provide the sets of necessary and sufficient conditions that guarantee the existence of some special solutions of the considered equations as: the maximal solution, the stabilizing solution and the minimal positive semi-definite solution. These conditions are expressed in terms of the feasibility of some suitable systems of linear matrix inequalities (LMI). One shows that in the case of the equations with periodic coefficients to verify the conditions that guarantee the existence of the maximal or the stabilizing solution, we have to check the solvability of some systems of LMI with a finite number of inequations. The proofs are based on some suitable properties of discrete-time linear equations defined by the positive operators on some ordered Hilbert spaces chosen adequately. The results derived in this paper provide useful conditions that guarantee the existence of the maximal solution or the stabilizing solution for different classes of difference matrix Riccati equations involved in many problems of robust control both in the deterministic and the stochastic framework. The proofs are deterministic and are accessible to the readers less familiarized with the stochastic reasonings.     

Divergence-free wavelet solution to the Stokes problem

In this paper, we use divergence-free wavelets to give an adaptive solution to the velocity field of the Stokes problem. We first use divergence-free wavelets to discretize the divergence-free weak formulation of the Stokes problem and obtain a discrete positive definite linear system of equations whose coefficient matrix is quasi-sparse; Secondly, an adaptive scheme is used to solve the discrete linear system of equations and the error estimation and complexity analysis are given.     

On the existence of the stabilizing solution of generalized Riccati equations arising in zero-sum stochastic difference games: the time-varying case

In this paper, a large class of time-varying Riccati equations arising in stochastic dynamic games is considered. The problem of the existence and uniqueness of some globally defined solution, namely the bounded and stabilizing solution, is investigated. As an application of the obtained existence results, we address in a second step the problem of infinite-horizon zero-sum two players linear quadratic (LQ) dynamic game for a stochastic discrete-time dynamical system subject to both random switching of its coefficients and multiplicative noise. We show that in the solution of such an optimal control problem, a crucial role is played by the unique bounded and stabilizing solution of the considered class of generalized Riccati equations.     

3阶齐次线性常微分方程的球面解曲线

首先给出3阶齐次线性常微分方程的球面解曲线的定义,然后讨论其球面解曲线与系数函数的关系,最后给出了球面解曲线的几何性质.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

变系数线性微分方程初值问题数值解的小波方法

通过利用小波尺度函数的正交性并结合配点法 ,本文给出了一种求解变系数线性微分方程初值问题数值解的小波算法 .在一定的假设条件下 ,对算法的收敛性进行了理论分析 .最后 ,我们还给出了一个具体的数值计算例子 .     

线性方程组反问题的次对称正定解

给出了线性方程组Ax=b的反问题有次对称正定解的充要条件.     

相同装卸工情况下装卸工问题的最优解

装卸工问题是一个新的NP困难的组合最优化问题,寻找其性能优良的近似算法是有重要的理论意义和实用价值的.相同装卸工情况下装卸工问题的系数矩阵是全么模矩阵,利用全么模矩阵的性质可以证明这种情况下的装卸工问题是多项式可解的.然而用全么模阵的性质还不能得到解的表达式.对这种情况下一辆货车的装卸工问题,用对偶单纯形法可得到最优解和最优值的解析表达式,从而可以把这个可解问题的最优值作为一般装卸工问题的近似值.这对于分析近似算法的性态是非常重要的.