AF: Small: Computational Algebraic Methods for Systems of Partial Difference-Differential Equations

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Natural processes, like the formation of a snowflake, can reveal deep connections between physical laws, which are often modeled as systems of mathematical equations, and symmetries, which can be captured mathematically as group actions. In computer algebra software, which can help solve systems of equations, it would be good to be able to ensure that operations respect group actions and preserve symmetries. Despite the over sixty-year history of algorithmic approaches in differential and difference algebra, there are currently no efficient computational techniques to analyze systems of algebraic partial difference-differential equations (PDDEs) and more general systems of partial differential equations (PDEs) with group action. This project aims to develop the theory and algorithms to determine the structure of solutions of a system of PDDEs or PDEs with additional conditions imposed by the action of transformation groups (e. g., PDEs with symmetries), for describing physical, chemical, or biological processes with symmetries. The educational goal of the project is to involve into an interdepartmental program on applications of symbolic computation the PI will develop, not only undergraduate majors in computer science, mathematics, and physics at the Catholic University of America (CUA), but also engineering majors in the field of automatic control and biology majors who work with continuous and discrete mathematical models of biology systems. The key research directions of this project are: (1) Development of computational methods and algorithms for difference-differential elimination and for decomposition of solution sets of systems of algebraic PDDEs and PDEs with group action into unions of characterizable (?simple??) components. Extension of these techniques to systems with weighted basic operators. (2) Elaboration of methods and algorithms for the evaluation of dimension characteristics of the solution sets of the above-mentioned systems. In particular, the PI will obtain algorithms for computing dimension polynomials and quasi-polynomials that express the Einstein?s strength of a system of PDDEs. (3) Application of the developed techniques to systems of PDDEs and PDEs with group action that arise in physics, automatic control, chemistry and biology. The main methods and approaches of the project include the characteristic set technique for difference-differential polynomials and its generalizations to the cases of several term orderings and weighted basic operators, the relative Groebner basis method, the techniques of dimension polynomials and quasi-polynomials, and decomposition methods for algebraic PDDEs and PDEs with group action. The results will be demonstrated in interdisciplinary research projects at CUA.