正定矩阵标准型的复合矩阵的一般形式

实对称正定矩阵的复合矩阵正定性的研究已有结论,但对于一般意义下的正定矩阵的复合矩阵是否仍然是正定的研究需要利用一般的正定矩阵的标准形的复合矩阵进行讨论,给出了一般公式及具体算法,为讨论其复合矩阵的正定性提供了基础条件.     

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     

浅谈正定二次型的习题课设计

在高等代数的实二次型内容中,正定二次型占有特殊的地位.本文从概念的回顾、正定二次型与正定矩阵的判断、二次型正定及矩阵正定的性质、其它类型二次型四个方面来设计正定二次型的习题课,并通过具体例子说明例题、习题精选的原则.     

实正定矩阵的复合矩阵的正定性

本文讨论了实正定矩阵的复合矩阵的正定性，并且给出了实正定矩阵的复合矩阵仍为正定矩阵的一个充要条件.     

一类广义正定矩阵的性质

广义正定矩阵有一系列推广,研究了一类广义正定矩阵,得到了这类广义正定矩阵的一些性质,并利用这些性质得到了这类广义正定矩阵的一些行列式不等式.     

Global Existence of Positive Solutions for Volterra Integral Equations

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一类捕食-食饵-互惠模型非常数正平衡解的存在性及分歧

讨论了一类捕食-食饵-互惠反应扩散系统的非常数正平衡解.首先分析了常数正平衡解的稳定性,其次;利用最大值原理和Harnack不等式给出了正解的失验估计.在此基础上,利用积分性质进一步讨论了非常数正解的不存在性,相应地证明了当扩散系数d_2 d_3大于特定正常数且扩散系数d_1有界时此模型没有非常数正解.同时利用度理论证明了当模型的线性化算子的正特征值的代数重数是奇数且扩散系数d_3不小于给定正常数时此模型至少存在一个非常数正解,最后研究了非常数正解的分歧.     

On the Spectra of Striped Sign Patterns

Sign patterns consisting of some positive and some negative columns, with at least one of each kind, are shown to allow any self-conjugate spectrum, and thus to allow any inertia. In the case of the n ×n sign pattern with all columns positive, given any self-conjugate multiset consisting of n -1 complex numbers supplemented by a sufficiently large positive number, it is shown how to construct a positive normal matrix whose spectrum is this multiset. Thus, the positive sign pattern allows any inertia with at least one positive eigenvalue.     

On the Spectra of Striped Sign Patterns

Sign patterns consisting of some positive and some negative columns, with at least one of each kind, are shown to allow any self-conjugate spectrum, and thus to allow any inertia. In the case of the n × n sign pattern with all columns positive, given any self-conjugate multiset consisting of n m 1 complex numbers supplemented by a sufficiently large positive number, it is shown how to construct a positive normal matrix whose spectrum is this multiset. Thus, the positive sign pattern allows any inertia with at least one positive eigenvalue.     

一类具正负系数中立型方程的正解

研究一类具有正负系数的中立型微分方程,得到了这类方程存在正解的充分条件和存在有界正解的充要条件,并据此给出了其存在无界正解的充分条件。     

On the Existence of Positive Eigenvalues for Linear and Nonlinear Equations with Indefinite Weight

We prove the existence of positive eigenvalues for two types of equations having an indefinite weight. We consider the linear case, and we show the existence of a positive principal eigenvalue corresponding to a positive eigenfunction. For the nonlinear case, we show that the set of positive eigenvalues is an unbounded interval, and that the corresponding eigenfunctions are positive.     

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams. Received: June 28, 2000     

具有阶段结构且食饵有避难所的模型分析

研究了捕食者具有阶段结构且食饵有避难所的非自治捕食系统.利用Lyapunov函数方法得到了系统持续生存的条件,以及在一定条件下存在唯一全局渐进稳定的周期正解.对于更广泛的概周期现象,也得到了存在唯一全局渐进稳定的概周期正解的充分条件.     

一类中立型时滞差分方程的正解

研究一类中立型时滞差分方程面　　　　的正解,获得了方程存在最终正解以及非增正解的充要条件,建立了比较定理．     

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and Newton iterative solution

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive eigenvector is convergent. Moreover, such positive eigenvectors form a monotone increasing and continuous function of the corresponding eigenvalues.     

Generic existence and uniqueness of positive eigenvalues and eigenvectors

We consider a closed cone of positive operators on an ordered Banach space and prove that a generic element of this cone has a unique positive eigenvalue and a unique (up to a positive multiple) positive eigenvector. Moreover, the normalized iterations of such a generic element converge to its unique eigenvector.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

中立型时滞逻辑斯谛方程无界正解的振动性

关于中立型逻辑斯谛方程有界正解的研究已有了长足的发展,而无界正解的研究在已有文献中却非常少见。本文利用振动性原理,结合分析方法得到其无界正解的渐近行为,由此建立中立型时滞逻辑斯谛方程无界正解振动的充分与必要条件,所获结果推广了相关文献的结论。