PARTIALLY ORDERED LINEAR ALGEBRA AND ITS APPLICATIONS

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)将抽象模型中识别的属性应用于生物科学的各个分支。有许多例子使用积极性（顺序）概念来检查与生物学研究相关的数学模型，如数学人口学，人口和遗传学。我们的工作将围绕着Dedekind sigma完全部分有序线性代数（pola）中的群、抽象矩阵模型、线性算子和概率（极限定理）的研究。这种利用线性代数技术研究代数结构的想法可能为研究一些数学分支，如有序群论、算子代数和矩阵理论提供了一种新的方法；参见B. Eckmann，“拓扑、代数、分析关系和缺失环节”，美国学报。数学。社会科学46 (1999),p520-527。具体来说，我们的项目在Polas研究中有以下目标：(1)刻画包括全有序群在内的不同有序群，(2)刻画包括三角算子、谱算子在内的三角型极点，(3)研究列模型中马尔可夫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理人口学中的应用。关于上述目标，我们做了许多猜想，例如，我们猜想极点上的全序群一定是可交换的。我们研究的全序群比文献中出现的全序群要普遍得多。另一个猜想与马尔可夫链上的极限定理有关：如果0小于等于x小于等于γ /u，其中x， u是极坐标γ < 2和u = u/2中的元素，则存在lim x/5。这里x可以被解释为马尔可夫跃迁。这里的极限是关于收敛阶的。但范数和拓扑在本研究中不起作用。

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.