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; 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，可以揭示物理定律之间的深厚连接，物理定律通常被建模为数学方程系统和对称性系统，可以用数学捕获作为小组动作。 在可以帮助求解方程式系统的计算机代数软件中，能够确保操作尊重小组动作并保留对称性将是一件好事。 尽管在差异和差异代数中具有六十年的算法方法历史，但目前尚无有效的计算技术来分析代数部分差异方程（PDDES）的系统系统，以及具有组作用的偏微分方程（PDES）的更通用系统。该项目旨在开发理论和算法，以确定PDDES或PDES系统的解决方案的结构，该系统具有转化组的作用（例如，具有对称性的PDES），用于描述与符号对象的物理，化学或生物学过程。 该项目的教育目标是在符号计算的应用中参与部门间计划，不仅在美国天主教大学（CUA）的计算机科学，数学和物理学的本科专业的专业中，还将开发专业的本科专业，还将在自动控制和生物学专业的生物学专业领域中进行工程专业的生物学和离散数学模型。该项目的关键研究方向是：（1）开发用于差异差消除的计算方法和算法，以及分解代数PDDES和PDES系统的解决方案集，并将群体作用分为具有特征性的（？简单??）组件的工会。将这些技术扩展到具有加权基本操作员的系统。 （2）详细阐述用于评估上述系统解决方案集的维度特征的方法和算法。特别是，PI将获得用于计算尺寸多项式的算法和表达Einstein pddes系统强度的准多项式。 （3）将开发的技术应用于PDDES和PDES系统，具有物理，自动控制，化学和生物学中出现的群体作用。 该项目的主要方法和方法包括针对差异定义多项式的特征设置技术及其对几个定期订单和加权基本运算符的情况的概括，相对的groebner基依基，尺寸多项式和准polynomials的技术以及分解方法，用于algebraic pddes pddes and pddes and pddes and pds and pds and pdes and pds and pdes and。 结果将在CUA的跨学科研究项目中证明。