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

自然过程，比如雪花的形成，可以揭示物理定律和对称性之间的深层联系，物理定律通常被建模为数学方程系统，而对称性可以被数学捕获为群体行为。在计算机代数软件中，它可以帮助解决方程组，能够确保操作尊重群体行为并保持对称性将是很好的。尽管微分和差分代数的算法方法已有60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系统和更一般的具有群作用的偏微分方程系统。该项目旨在发展理论和算法，以确定PDDEs系统的解的结构或由转换群（例如，具有对称性的PDEs）的作用施加的附加条件的PDEs系统的解的结构，用于描述具有对称性的物理，化学或生物过程。该项目的教育目标是将PI开发的符号计算应用纳入一个跨部门项目，不仅包括美国天主教大学（CUA）计算机科学，数学和物理学的本科专业，还包括自动控制领域的工程专业和研究生物系统连续和离散数学模型的生物学专业。本项目重点研究方向为：(1)发展微分-微分消元的计算方法和算法，以及将代数偏微分微分方程和群作用偏微分微分方程解集分解为可表征（简单）分量并的计算方法和算法。将这些技术推广到具有加权基本算子的系统。(2)阐述了上述系统解集维度特征评价的方法和算法。特别是，PI将获得计算维度多项式和准多项式的算法，这些多项式表示爱因斯坦？PDDEs系统的强度。(3)将开发的技术应用于物理、自动控制、化学和生物学中出现的PDDEs和具有群作用的PDEs系统。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在若干项排序和加权基本算子情况下的推广、相对Groebner基方法、维数多项式和拟多项式技术、代数偏微分方程和群作用偏微分方程的分解方法。研究结果将在中国农业大学的跨学科研究项目中得到展示。

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.