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

AF：小：差分微分方程组的算法代数方法

• 批准号: 2139462
• 负责人: Alexander Levin
• 金额: $ 18.77万
• 依托单位: Catholic University of America
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-05-01 至 2025-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2139462&HistoricalAwards=false
• 关键词: AF; Small; Algorithmic; Algebraic; Methods

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.

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创建现实生活现象数学模型的主要工具。尽管取决于多个因素的连续时间和离散时间过程分别由偏微分方程组和差分方程组描述，但同时包含连续和离散分量的过程（例如由于传输质量所需的时间而导致时滞的过程，能量或信息）由偏微分方程组（PDDE）控制。 此外，物理、化学或生物过程的特征通常具有一定的对称性，可以在数学上将其捕获为变换群动作。 因此，PDDE系统和具有群作用的系统的计算方法和算法的开发在应用中至关重要。 尽管微分和差分代数的构造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。 该项目旨在开发理论、方法和算法来确定此类方程组的解的结构，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关于符号计算应用的跨部门项目，该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。本项目的重点研究方向如下。 (1) 开发用于差分-微分消除以及将代数 PDDE 系统的解集分解为简单分量并集的计算方法和算法。将所获得的技术扩展到具有群体动作和/或加权算子的系统。 (2) 开发在差分-微分模和代数中构建 Groebner 型基的算法。将这些算法应用于计算应用中出现的代数 PDDE 的维数函数。 (3)利用广义Groebner基和差分-微分特征集技术对代数微分方程的有限差分近似进行一致性分析。 (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基础作用的PDDE系统。 该项目的主要方法和途径包括广义差-微分特征集和相关Groebner基的技术、维数多项式和拟多项式的使用、代数PDDE系统以及具有群作用和加权基本的系统的分解方法。运营商。结果将在 CUA 的跨学科研究项目中得到展示。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Generalized characteristic sets and new multivariate difference dimension polynomials

广义特征集和新的多元差分维多项式

• DOI: 10.1007/s00200-023-00628-0
• 发表时间: 2024-01
• 期刊: Communication and Computing
• 作者: Levin; Alexander
• 通讯作者: Alexander

Reduction with Respect to the Effective Order and a New Type of Dimension Polynomials of Difference Modules

有效阶数的约简及一类新型差分模维数多项式

• DOI: 10.1145/3476446.3535497
• 发表时间: 2022-07
• 期刊: 2022 International Symposium on Symbolic and Algebraic Computation
• 作者: Levin; Alexander
• 通讯作者: Alexander