AF: Small: Symbolic Computation and Difference and Differential Equations

数学和物理中的许多函数都是由差分或微分方程定义的。理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。研究人员将开发算法来揭示这些属性。特别是，他将开发算法，以确定代数和微分关系之间发生的解决方案的一组给定的线性差分方程。此外，他将发展理论，并给出算法来衡量的代数行为的解决方案的参数化线性微分方程作为一个不同的参数。最后，他提出攻击的问题，因式分解欠定系统的偏微分方程。虽然不同的外观，这些问题将被攻击使用技术的基础上研究的基本对称性的定义方程。在过去，调查人员作出了贡献的理论和算法的发展，以解决微分和差分方程，特别是伽罗瓦理论的这些方程和算法，以解决他们的封闭形式。目前的项目部分完善和扩展这项工作，但超出了攻击更广泛的问题上面提到的。这项研究解决的基础和计算问题的线性差分和微分方程系统的代数行为。它使许多科学领域的研究人员能够理解这些方程解的定性行为。研究人员将开发算法，这将是有用的数论，组合和分析。此外，研究人员预计，这些算法将成为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人力资源开发，促进与其他科学领域的互动以及国际合作的发展。研究人员将继续不仅培养他的博士学位。学生，但进一步发展与他的同事在北卡罗来纳州州立大学的计划，以培养学生在广泛的主题在符号计算。他将赞助一位博士后学者，让这位学者参与这里提出的研究，并发展学者的教学技能，使学者融入科学界。他将继续并扩大他的工作，修订本科抽象代数课程，包括符号计算作为核心议题。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思想，特别是符号分析。他将继续与德国、法国和中国的研究人员进行合作，并将邀请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.