PARTIALLY ORDERED LINEAR ALGEBRA AND ITS APPLICATIONS

这项研究将集中在两个方向：(1)利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性质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，例如数学人口学、种群和遗传学。我们的工作将集中在Dedekin sigma完全偏序线性代数(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会提供一种新的方法来看待数学的一些分支，如有序群论、算子代数和矩阵理论；参见B.Eck mann的《Topology，Algebra，Analysis-Relationments and Missing Links》，Amer的通告。数学课。SoC。46(1999)，P520-527。具体地说，我们的项目在POLAS的研究中有以下目标：(1)刻画包括全序群在内的不同有序群，(2)刻画包括三角算子和谱算子的三角型POLAS，(3)研究列模型中与马尔可夫链有关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，以及(6)寻求正线性算子在数学人口学中的应用。关于上述目标，人们提出了许多猜想，例如，我们猜想Pola中的一个全序群一定是可交换的。我们研究的全有序基团比文献中出现的要普遍得多。另一个猜想与马尔可夫链上的一个极限定理有关：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&lt；2且u=u/2，则存在Lim x/5。这里，x可以被解释为马尔可夫转变。我们在这里使用的极限是关于顺序收敛的。但范数和拓扑学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.