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

微分式、差分式和差分式--微分方程式是科学家和工程师用来为现实生活现象建立数学模型的主要工具。依赖于多种因素的连续时间过程和离散时间过程分别由偏微分方程组和差分方程组来描述，而同时包括连续和离散分量的过程(例如，由于传输质量、能量或信息所需的时间而导致时滞的过程)则由偏差分方程组(PDDEs)来描述。此外，物理、化学或生物过程的特征通常具有一定的对称性，这些对称性可以在数学上被捕获为变换群作用。因此，发展PDDES系统和这类具有群体作用的系统的计算方法和算法在应用中是非常重要的。尽管微分和差分代数中的构造方法已经有60多年的历史，但目前还没有有效的计算技术来计算代数偏微分方程。这个项目的目的是发展理论、方法和算法来确定这类方程系统的解的结构，包括具有对称群作用的代数PDE。研究成果将应用于描述物理、化学和生物数学模型的系统。该项目的教育目标是创建一个关于符号计算应用的跨部门项目，将涉及美国天主教大学(CUA)计算机科学、数学、物理和生物学专业的本科生和研究生。本课题的主要研究方向如下。(1)发展了差-微分消去法和将代数偏微分方程组的解集分解成简单分量并的计算方法和算法。将所获得的技术扩展到具有群操作和/或加权算子的系统。(2)差分模和代数中Groebner型基的构造算法的发展。这些算法在应用中出现的代数PDDEs的维度函数的计算中的应用。(3)基于广义Groebner基和差-微分特征集的代数微分方程差分逼近的相合性分析。(4)将所获得的方法和算法应用于在物理、工程、化学和生物模拟中起基础作用的PDDEs系统。该项目的主要方法和途径包括广义差-微分特征集和相对Groebner基的技术，维多项式和拟多项式的使用，以及代数偏微分方程组和这类具有群作用和加权基本算子的系统的分解方法。结果将在加州大学的跨学科研究项目中得到展示。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Differential, difference and difference-differential equations constitute main tools that scientists and engineers use to create mathematical models of real-life phenomena. Whereas continuous-time and discrete-time processes depending on several factors are described by systems of partial differential and difference equations, respectively, processes that include both continuous and discrete components (such as processes with time delay caused by the time required to transport mass, energy or information) are governed by systems of partial difference-differential equations (PDDEs). Furthermore, very often characteristics of physical, chemical or biological processes have certain symmetries, which can be captured mathematically as transformation group actions. Thus, the development of computational methods and algorithms for systems of PDDEs and such systems with group action is of primary importance in applications. Despite the over sixty-year history of constructive methods in differential and difference algebra, there are currently no efficient computational techniques for algebraic PDDEs. This project aims to develop the theory, methods and algorithms to determine the structure of solutions of systems of such equations including algebraic PDDEs with symmetry group actions. The research results will be applied to systems that describe mathematical models in physics, chemistry and biology. The educational goal of the project is to create an interdepartmental program on applications of symbolic computation that will involve undergraduate and graduate majors in computer science, mathematics, physics and biology at the Catholic University of America (CUA). The key research directions of this project are as follows. (1) Development of computational methods and algorithms for difference-differential elimination and for decomposition of solution sets of systems of algebraic PDDEs into unions of simple components. Extension of the obtained techniques to systems with group actions and/or weighted operators. (2) Development of algorithms for building Groebner-type bases in difference-differential modules and algebras. Applications of these algorithms to the computation of dimension functions of algebraic PDDEs that arise in applications. (3) Consistency analysis of finite difference approximations of algebraic differential equations via the techniques of generalized Groebner bases and difference-differential characteristic sets. (4) Application of the obtained methods and algorithms to systems of PDDEs that play fundamental roles in physics, engineering, chemical and biological modeling. The main methods and approaches of the project include the techniques of generalized difference-differential characteristic sets and relative Groebner bases, the use of dimension polynomials and quasi-polynomials, and decomposition methods for systems of algebraic PDDEs and such systems with group action and weighted basic operators. The results will be demonstrated in interdisciplinary research projects at CUA.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.