Mathematical Sciences: Applications of Numerical Methods to Problems in Geometric Function Theory

数学科学：数值方法在几何函数理论问题中的应用

• 批准号: 8800584
• 负责人: James Hummel
• 金额: $ 4万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8800584&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applications; Numerical; Methods

This project seeks to develop further the methods and theory of numerical techniques applied to the study of problems arising in geometric function theory. While the basic numerical methods are generally available, their application to specific problems such as seeking global trajectory structures for quadratic differentials are far from perfected. Many fundamental questions can be reduced to those of solving for properties of a particular conformal map which is known to satisfy a particular differential equation. These equations, the result of applying variational techniques to single out extremal functions, are of apparent simple type. But the parameters they contain are defined by awkward side conditions. This makes the equations into functional differential equations - equations which first appeared in the theory of automorphic functions and now occur in the study of Riemann surface moduli and in various questions concerning ordinary differential equations of complex arguments. In addition to the development of general methods and, eventually, programs, work will be done on specific problem areas where new approaches should flow from the computational development. These areas include studies of the space of bounded nonvanishing univalent functions defined in a disc. This subject has been the focus of intense research in recent years but now is short of fresh new approaches to the resolution of many important questions. Work will also be done analyzing the class of univalent functions whose range covers a fixed disc. Finally efforts will be made to sharpen upper and lower estimates for the universal Bloch constants. There is known to be considerable room for progress here.

该项目旨在进一步发展应用于研究几何函数理论中出现的问题的数值技术的方法和理论。 虽然基本的数值方法普遍可用，但它们在特定问题（例如寻求二次微分的全局轨迹结构）中的应用还远未完善。 许多基本问题可以简化为求解已知满足特定微分方程的特定共形映射的性质。 这些方程是应用变分技术挑选出极值函数的结果，显然是简单类型的。 但它们包含的参数是由尴尬的附带条件定义的。 这使得方程变成了泛函微分方程——这些方程首先出现在自守函数理论中，现在出现在黎曼曲面模的研究以及有关复变元常微分方程的各种问题中。 除了开发一般方法以及最终的程序之外，还将针对特定问题领域开展工作，在这些领域中，应从计算开发中得出新方法。 这些领域包括对光盘中定义的有界非零单价函数空间的研究。 这一主题近年来一直是深入研究的焦点，但现在缺乏解决许多重要问题的新方法。 还将完成分析范围覆盖固定圆盘的一类单价函数的工作。 最后，我们将努力提高通用布洛赫常数的上限和下限估计值。 众所周知，这里还有相当大的进步空间。