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

AF：小：偏差分-微分方程组的计算代数方法

• 批准号: 1714425
• 负责人: Alexander Levin
• 金额: $ 18.17万
• 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: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1714425&HistoricalAwards=false
• 关键词: AF; Small; Computational; Algebraic; Methods

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.

自然过程（例如雪花的形成）可以揭示物理定律（通常被建模为数学方程系统）与对称性（可以在数学上捕获为群体行为）之间的深层联系。 在可以帮助求解方程组的计算机代数软件中，如果能够确保运算尊重群体行为并保持对称性，那就太好了。 尽管微分和差分代数算法方法已有六十多年的历史，但目前还没有有效的计算技术来分析代数偏微分方程 (PDDE) 系统和具有群作用的更一般的偏微分方程 (PDE) 系统。该项目旨在开发理论和算法来确定 PDDE 系统或具有由变换群（例如，具有对称性的 PDE）作用施加的附加条件的 PDE 系统的解的结构，用于描述物理、化学或生物过程对称性。 该项目的教育目标是让 PI 开发的符号计算应用跨部门项目不仅涉及美国天主教大学 (CUA) 的计算机科学、数学和物理专业的本科生，还涉及美国天主教大学 (CUA) 的工程专业的学生。自动控制和生物学专业领域，研究生物系统的连续和离散数学模型。该项目的主要研究方向是：（1）开发差分-微分消除以及将代数PDDE和群作用PDE系统的解集分解为可表征（？简单？）分量并集的计算方法和算法。 。将这些技术扩展到具有加权基本运算符的系统。 (2)详细阐述了评估上述系统解集的维数特征的方法和算法。特别是，PI 将获得用于计算表达 PDDE 系统的爱因斯坦强度的维数多项式和拟多项式的算法。 (3) 将所开发的技术应用于物理、自动控制、化学和生物学中出现的偏微分方程和具有群作用的偏微分方程系统。 该项目的主要方法和途径包括差分-微分多项式的特征集技术及其对多项排序和加权基本算子情况的推广、相对Groebner基方法、维数多项式和拟多项式技术以及代数 PDDE 和具有群作用的 PDE 的分解方法。 研究结果将在 CUA 的跨学科研究项目中得到展示。

项目成果

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

过滤自由模块中的相对约简和 Buchberger 算法

• DOI: 10.1007/s11786-017-0317-1
• 发表时间: 2017-12
• 期刊: Mathematics in Computer Science
• 影响因子: 0.8
• 作者: Fürst, Christoph;Levin, Alexander
• 通讯作者: Levin, Alexander

Bivariate Kolchin-type dimension polynomials of non-reflexive prime difference-differential ideals. The case of one translation

非自反素差-微分理想的二元 Kolchin 型维数多项式。

• DOI: 10.1016/j.jsc.2019.10.014
• 发表时间: 2021-01
• 期刊: Journal of Symbolic Computation
• 影响因子: 0.7
• 作者: Levin; Alexander
• 通讯作者: Alexander

Bivariate Dimension Quasi-polynomials of Difference–Differential Field Extensions with Weighted Basic Operators

双变量维数差拟多项式——带加权基本算子的微分域扩展

• DOI: 10.1007/s11786-018-0361-5
• 发表时间: 2019-06
• 期刊: Mathematics in Computer Science
• 影响因子: 0.8
• 作者: Levin; Alexander
• 通讯作者: Alexander

A New Type of Difference Dimension Polynomials

一种新型的差维多项式

• DOI: 10.1007/s11786-022-00540-9
• 发表时间: 2022-12
• 期刊: Mathematics in Computer Science
• 影响因子: 0.8
• 作者: Levin; Alexander
• 通讯作者: Alexander

Some Properties and Invariants of Multivariate Difference-Differential Dimension Polynomials.

多元差分微分维多项式的一些性质和不变量。

• 发表时间: 2018-06
• 期刊: Proceedings of the 24th Conference on Applications of Computer Algebra - ACA 2018
• 作者: Levin; A.
• 通讯作者: A.