PARTIALLY ORDERED LINEAR ALGEBRA AND ITS APPLICATIONS

偏序线性代数及其应用

• 批准号: 6335951
• 负责人: TAEN-YU DAL
• 金额: $ 3.15万
• 依托单位: YORK COLLEGE
• 依托单位国家: 美国
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://reporter.nih.gov/project-details/6335951
• 关键词: biology; human population study; mathematical model; mathematics; model design /development

This research will focus in two directions: (1) a study of ordered algebraic models using linear algebra methods, (2) applications of the properties identified in the abstract models to various branches in the biology sciences. There are many examples of using positivity (order) concept to examine mathematical models related to biological studies such as mathematical demography, population, and genetics. Our work will center around a study of groups, abstract matrix models, linear operators, and probability (limit theorems) in a Dedekind sigma complete partially ordered linear algebra (pola). This idea of studying algebraic structures by using linear algebra techniques may provide a new way to look at some branches of mathematics such as ordered group theory, operator algebra, and matrix theory; see B. Eckmann, "Topology, Algebra, Analysis-Relations and Missing Links", Notices of Amer. Math. Soc. 46 (1999), p520-527. Specifically, our project has the following goals in a study of Polas: (1) to characterize different ordered groups including fully ordered groups, (2) to characterize triangular type polas including triangular operators, spectral operators, (3) to examine limit theorems related to Markov chains in a Column Model with applications on epidemic models, (5) to construct and study an abstract matrix model for column-finite matrices, and (6) to seek applications of positive linear operators in mathematical demography. Many conjectures are made with respect to the above goals, e.g., we conjecture a fully ordered group in a pola must be commutative. Fully ordered groups in our study are much more general than the ones appearing in the literatures. Another conjecture is concerned with a limit theorem on Markov chains: if 0 is less than or equal to x is less than or equal to gamma/u, where x, u are elements in a pola, gamma < 2 and u = u/2, than lim x/5 exists. Here x can be interpreted as a Markov transition. The limit we use here is with respect to the order convergence. But norm and topology play no role in this study.

本研究将集中在两个方向：（1）使用线性代数方法研究有序代数模型，（2）将抽象模型中确定的属性应用于生物科学的各个分支。有许多使用正性（顺序）概念来检查与生物学研究相关的数学模型的例子，如数学人口学，人口和遗传学。我们的工作将围绕研究组，抽象矩阵模型，线性算子，概率（极限定理）在一个Dedekind西格玛完全偏序线性代数（pola）。这种用线性代数技术研究代数结构的想法可能为研究某些数学分支提供一种新的方法，如有序群理论、算子代数和矩阵理论;见B。陈文辉，“拓扑、代数、分析关系与缺失环节”，国立台湾大学数学研究所硕士论文（1999），页520 -527。具体来说，我们的项目在Polas的研究中有以下目标：（1）刻画了不同序群包括全序群，（2）刻画了三角型极点包括三角算子，谱算子，（3）研究了列模型中马氏链的极限定理及其在流行病模型中的应用，（5）构造和研究列有限矩阵的抽象矩阵模型;（6）寻求正线性算子在数学人口学中的应用。关于上述目标作出了许多说明，例如，我们猜想pola中的全序群一定是交换群。我们所研究的全序群比文献中出现的全序群更一般。另一个猜想与马氏链的极限定理有关：如果0小于或等于x小于或等于gamma/u，其中x，u是pola中的元素，gamma < 2且u = u/2，则存在lim x/5。这里x可以解释为马尔可夫转移。这里我们使用的极限是关于阶收敛的。但范数和拓扑在本研究中没有起作用。