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

微分方程、差分方程和差分-微分方程是科学家和工程师用来建立现实生活中现象的数学模型的主要工具。而依赖于几个因素的连续时间和离散时间过程分别由偏微分方程和差分方程系统描述，包括连续和离散分量的过程（例如具有由传输质量、能量或信息所需的时间引起的时间延迟的过程）由偏差分-微分方程系统（PDDE）控制。 此外，物理、化学或生物过程的特征通常具有某些对称性，这些对称性可以在数学上被捕获为变换群作用。 因此，PDDE系统和具有基团作用的系统的计算方法和算法的发展在应用中具有首要的重要性。 尽管微分和差分代数的构造性方法已有六十多年的历史，但目前还没有有效的代数PDDE计算技术。 该项目旨在发展理论、方法和算法，以确定此类方程系统的解的结构，包括具有对称群作用的代数PDDE。研究成果将应用于描述物理，化学和生物学数学模型的系统。该项目的教育目标是创建一个跨部门的程序，将涉及计算机科学，数学，物理和生物学在美国天主教大学（CUA）的本科生和研究生专业的符号计算的应用。本项目的重点研究方向如下。(1)发展计算方法和算法，用于差分-微分消除和代数PDDE系统的解集分解为简单分量的并集。扩展所获得的技术与群作用和/或加权算子的系统。(2)开发在差分微分模和代数中建立Groebner型基的算法。将这些算法应用于实际应用中出现的代数偏微分方程维数函数的计算。(3)基于广义Groebner基和差分-微分特征集的代数微分方程有限差分逼近的相容性分析。(4)将所获得的方法和算法应用于在物理，工程，化学和生物建模中发挥重要作用的PDDEs系统。 该项目的主要方法和途径包括广义差分-微分特征集和相关Groebner基技术，维数多项式和拟多项式的使用，以及代数PDDE系统和具有群作用和加权基本算子的系统的分解方法。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估来支持。

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.