AF: Small: Symbolic Computation and Difference and Differential Equations

AF：小：符号计算以及差分和微分方程

• 批准号: 1017217
• 负责人: Michael Singer
• 金额: $ 47万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1017217&HistoricalAwards=false
• 关键词: AF; Small; Symbolic; Computation; Difference

Many of the functions of interest in mathematics and physics are defined by difference or differential equations. Understanding their algebraic properties is key to using these functions to describe physical and mathematical phenomena. The investigator will develop algorithms that will reveal these properties. In particular, he will develop algorithms to determine the algebraic and differential relations that occur among solutions of a given set of linear difference equations. In addition, he will develop theory and give algorithms to measure the algebraic behavior of solutions of parameterized linear differential equations as one varies the parameter. Finally, he proposes to attack the problem of factoring underdetermined systems of partial differential equations.Although disparate in appearance, these problems will be attacked using techniques based on studying the underlying symmetries of the defining equations. In the past, the investigator has contributed to the development of theory and algorithms to solve differential and difference equations, in particular to the Galois theories of these equations and algorithms to solve them in closed form. The present project in part refines and extends this work but goes beyond to attack the broader problems mentioned above.This research addresses foundational and computational issues concerning the algebraic behavior of systems of linear difference and differential equations. It allows researchers in many scientific fields to understand aspects of the qualitative behavior of solutions of these equations. The researcher will develop algorithms that that will be useful to number theorists, combinatorists and analysts. In addition, the investigator anticipates that these algorithms will form the foundation on which Maple and Mathematica code are based and so have an impact on the education and day-to-day work of engineers and other scientists.Key components of this project are the development of human resources, the fostering of interactions with other scientific fields and the advancement of international collaborations. The investigator will continue to not only train his Ph.D. students but to further develop with his colleagues a program at NC State University to train students in a broad range of topics in Symbolic Computation. He will sponsor a postdoctoral scholar, involving this scholar in the research proposed here as well as develop the scholar's teaching skills and integrate the scholar into the scientific community. He will continue and expand his work on revising the undergraduate Abstract Algebra curriculum to include Symbolic Computation as a core topic. In addition he will continue to organize workshops aimed at students and colleagues in diverse fields to disseminate to a broad scientific community the ideas of Symbolic Computation in general and Symbolic Analysis in particular. He will continue his recent collaborations with researchers in Germany, France and China and will involve graduate students from NC State University in these projects, allowing them to integrate themselves in the international research community.

数学和物理学中许多有趣的函数都是由差分方程或微分方程定义的。了解它们的代数性质是使用这些函数描述物理和数学现象的关键。研究人员将开发能够揭示这些特性的算法。特别是，他将开发算法来确定给定线性差分方程组的解之间出现的代数和微分关系。此外，他还将发展理论并给出算法来测量参数化线性微分方程解在改变参数时的代数行为。最后，他提出解决偏微分方程欠定系统的因式分解问题。虽然表面上各不相同，但这些问题将使用基于研究定义方程的基本对称性的技术来解决。过去，研究者为求解微分方程和差分方程的理论和算法的发展做出了贡献，特别是这些方程的伽罗瓦理论和以封闭形式求解它们的算法。目前的项目部分地完善和扩展了这项工作，但超越了解决上述更广泛的问题。这项研究解决了有关线性差分和微分方程组的代数行为的基础和计算问题。它使许多科学领域的研究人员能够了解这些方程解的定性行为的各个方面。研究人员将开发对数论学家、组合学家和分析师有用的算法。此外，研究人员预计这些算法将构成 Maple 和 Mathematica 代码的基础，从而对工程师和其他科学家的教育和日常工作产生影响。该项目的关键组成部分是开发人力资源、促进与其他科学领域的互动以及推进国际合作。研究人员不仅将继续培养他的博士学位。学生，但与他的同事进一步开发北卡罗来纳州立大学的一个项目，以培训学生符号计算的广泛主题。他将赞助一名博士后学者，让该学者参与此处提出的研究，并发展该学者的教学技能并将该学者融入科学界。他将继续并扩展他的本科抽象代数课程修订工作，将符号计算作为核心主题。此外，他将继续组织针对不同领域的学生和同事的研讨会，向广泛的科学界传播符号计算的一般思想，特别是符号分析的思想。他将继续与德国、法国和中国的研究人员最近的合作，并将让北卡罗来纳州立大学的研究生参与这些项目，使他们能够融入国际研究界。