Symbolic Computation and Differential and Difference Equations

这项研究解决了有关线性差分和微分方程系统的代数行为的基础和计算问题，并使研究人员能够了解这些方程的解的定性行为的方面。这项研究开发的算法不仅对数学家有用，而且对许多领域的工程师和科学家也有用。这些算法将是基础的代码出现在符号计算的计算机软件包中使用的教育以及日常工作的工程师和其他scientists. In，特别是，这项研究探讨新的和更有效的算法的经典Picard-Vessiot理论的超定系统的微分和差分方程，并发展了新的算法来确定线性偏微分和差分方程的下定系统的初等代数性质。它还将经典的Picard-Vessiot理论及其算法扩展到具有无限维群的伽罗瓦理论，允许应用于参数化线性方程以及某些类非线性方程。

This research addresses foundational and computational issues concerning the algebraic behavior of systems of linear difference and differential equations and allows researchers to understand aspects of the qualitative behavior of solutions of these equations. This research develops algorithms that will be of use not only to mathematicians but engineers and scientists in many fields as well. These algorithms will be the basis of code appearing in symbolic computation computer packages used in the education as well as the day-to-day work of engineers and other scientists.In particular, this research investigates new and more efficient algorithms for the classical Picard-Vessiot theories of over determined systems of differential and difference equations, and develops new algorithms to determine elementary algebraic properties of under determined systems of linear partial differential and difference equations. It also extends the classical Picard-Vessiot theory and its algorithms to Galois theories with infinite dimensional groups, allowing applications to parameterized linear equations as well as some classes of nonlinear equations.