Mathematical Sciences: Studies in Analysis

该奖项支持的数学研究侧重于复数域中微分方程解的性质。具体地说，将对代数微分方程式进行研究。也就是说，因变量的导数只以多项式进入的微分方程式。这种方程的解称为微分代数。在这个项目中采取的观点是确定一个人可以用受限制的微分代数解析函数类来导出的复微积分。例如，有没有可能对所有这些完整的功能进行分类？这些功能的增长限制是什么(如果有的话)？还将努力分析这类函数的迭代，并确定通过类中的单个函数以规定值对任意复序列进行内插的程度。微分方程式在模拟物理宇宙中起着至关重要的作用。通常的情况是，解虽然在理论上是已知的，但可能不能用基本的方法来表示。这项研究的目的是发展可作为微分方程解出现的广泛函数类的性质，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.