AF: Small: Solving and Simplifying Algebraic, Differential, and Difference Equations.

科学中的问题通常归结为数学方程。有许多计算机程序可以解各种类型的方程。方程本身通常来自计算机计算，所以它们可以很大。这个项目的目标是简化大型方程，尽可能将它们简化为更容易求解的较小方程。当计算机构造公式和方程来解决科学问题时，它们的第一个表达式不必是最简单的。可能存在更简单的公式或方程。找到它们使后续的计算更容易和更快。这个项目旨在简化的方程类型是微分方程组、递推关系和两个变量的代数方程。这些类型的方程的一个共同主题是，为了简化它们，关键的一步是对它们的奇点进行技术分析。计算机代数系统可以自动找到各种类型的方程的精确解。它们是研究和教育基础设施的重要组成部分。然而，他们经常产生不必要的复杂答案。在这个项目中开发的技术不仅有助于简化方程，而且有助于简化解，使其对研究和教学更有用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Problems in science are often reduced to mathematical equations.There are many computer programs for solving various types of equations.The equations themselves often come from computer computation, so theycan be large. This project aims to simplify large equations, reducing themwhen possible to smaller ones that can be solved more easily.When computers construct formulas and equations to tackle problems in science,their first expressions need not be the simplest ones. It is likely that simpler formulasor equations exist. Finding them makes subsequent computations easier and faster.The types of equations that this project aims to simplify are differential equations,recurrence relations, and algebraic equations in two variables. A common theme foreach of these types of equations is that in order to simplify them, a key step is atechnical analysis of their singularities.Computer algebra systems can automatically find exact solutions for various typesof equations. They are an important part of the infrastructure for researchand education. However, they often produce unnecessarily complicated answers.The techniques developed in this project will help to simplify not only equations,but solutions as well, making them more useful for research and teaching.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.