AF: Small: Symbolic Computation and Difference and Differential Equations

数学和物理学中的许多感兴趣的函数都是由差分式或微分方程式定义的。了解它们的代数性质是使用这些函数来描述物理和数学现象的关键。研究人员将开发揭示这些特性的算法。特别是，他将开发算法来确定给定的一组线性差分方程解之间的代数关系和微分关系。此外，他还将发展理论并给出算法，以测量参数变化时参数化线性微分方程解的代数行为。最后，他建议解决欠定偏微分方程组的因式分解问题。尽管这些问题在外观上是不同的，但这些问题将通过研究定义方程的基本对称性来解决。在过去，这位研究人员致力于微分方程组和差分方程组的理论和算法的发展，特别是这些方程的伽罗瓦理论和闭合形式的求解算法。本项目部分地改进和扩展了这项工作，但超越了上述更广泛的问题。这项研究解决了与线性差分方程组和微分方程组的代数行为有关的基本和计算问题。它允许许多科学领域的研究人员了解这些方程的解的定性行为的各个方面。研究人员将开发对数字理论家、组合学家和分析师有用的算法。此外，研究人员预计，这些算法将成为Maple和数学代码所基于的基础，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，以及促进国际合作。这位研究人员不仅将继续培训他的博士生，还将与他的同事们一起在北卡罗来纳州州立大学进一步开发一个项目，以培训学生在符号计算方面的广泛主题。他将赞助一名博士后学者，让这名学者参与这里提出的研究，并发展该学者的教学技能，使该学者融入科学界。他将继续并扩大他的工作，修订本科生的抽象代数课程，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讲习班，向广大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。

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.