Mathematical Sciences: Studies in Analysis

该奖项支持的数学研究集中在复域微分方程解的性质上。具体来说，我们将研究代数微分方程。也就是说，在微分方程中，因变量的导数只以多项式形式出现。这种方程的解称为微分代数。本课题所采取的观点是确定用微分代数解析函数的有限类可以导出的复微积分。例如，是否有可能对所有这样的整个函数进行分类？如果有的话，这些函数的增长限制是什么？还将努力分析这类函数的迭代，并确定在何种程度上可以用该类中的单个函数在规定的值内插任意复杂序列。微分方程在模拟物理宇宙中起着至关重要的作用。通常情况下，虽然已知解在理论上存在，但可能无法用基本方法表示。本研究的目的是开发可以作为微分方程解的广义函数的性质，并将这些性质用作综合分类方案的一部分。

The mathematical research supported by this award focuses on the nature of solutions of differential equations in the complex domain. Specifically, work will be done investigating algebraic differential equations. That is, differential equations in which the derivatives of the dependent variable enter only polynomially. A solution to such an equation is called differentially algebraic. The point of view taken in this project is one of determining the complex calculus one can derive with the restricted class of differentially algebraic analytic functions. For example, is it possible to classify all such entire functions? What, if any, are the growth restrictions on such functions? Efforts will also be made to analyze the iterates of such functions and to establish the extent to which interpolation of arbitrary complex sequences is possible at prescribed values by a single function in the class. Differential equations play an essential role in modeling the physical universe. It is often the case that solutions, although known to exist theoretically, may not be expressible by elementary means. The purpose of this research is to develop properties of broad classes of functions that can arise as solutions of differential equations and to use these properties as part of a comprehensive classification scheme.