PARTIALLY ORDERED LINEAR ALGEBRA AND ITS APPLICATIONS

York College, New York NY

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Abstract

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.