Mathematical Sciences: Classical Complex Analysis

数学科学：经典复分析

• 批准号: 9302823
• 负责人: Donald Marshall
• 金额: $ 6万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-04-15 至 1996-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302823&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Classical; Complex; Analysis

Work supported by this award focuses on problems arising in the mathematical theory of complex function theory. Three primary research themes will be followed. The first concerns angular derivatives of conformal maps, that is, mappings of domains in the complex plane which are univalent and analytic. It has been a long standing problem to determine whether or not a conformal mapping of a simply connected region onto the unit disk has an angular derivative. Although many special cases are known to be true, only recently has the techniques of extremal length been developed to the point where a general existence result is now possible. It has been possible to obtain a precise statement about the existence for the heretofore unknown case of strip domains. More work remains. A second area of work will focus on the continued refinement of numerical algorithms for conformal maps. A fast, reliable program has been developed and used primarily for finding experimental properties of maps. Work will now be done to understand convergence properties and use them to study the classical corona problem for triply-connected domains. Finally, efforts will be made to determine whether the class of interpolating Blaschke products generate the entire space of bounded holomorphic functions on the disk. It was shown in 1976 that if one uses all Blaschke products, the statement is true. The interpolating products are much less likely to occur, yet have more regularity and therefore are a better class to use as approximates. Complex analysis, the study of differentiable functions of a complex variable, lies at the heart of vast areas of mathematics stretching from number theory through potential theory and on to linear algebra and numerical analysis. This particular project combines some long-standing problems and methods with new applications of computational geometry and graph theory.

该奖项支持的工作集中在复杂功能理论的数学理论中引起的问题。 将遵循三个主要研究主题。 第一个关注的是共形图的角衍生物，即，在复杂平面中的域映射是无数和分析的。 确定简单连接区域到单位磁盘上是否具有角度衍生物的共同映射是否具有长期问题。 尽管已知许多特殊案例是正确的，但直到最近才开发出最大长度的技术，以至于现在可能存在一般存在的结果。 可以获得有关迄今未知域案例的存在的精确陈述。 还有更多的工作。 第二个工作区域将集中于保形图的数值算法的持续改进。 已经开发了一个快速，可靠的程序，主要用于查找地图的实验性能。 现在将完成工作以了解收敛属性，并使用它们来研究三重连接域的经典电晕问题。最后，将努力确定插值类产品是否会在磁盘上产生有界全态函数的整个空间。 1976年，人们表明，如果人们使用所有Blaschke产品，则该陈述是正确的。插值产品的发生可能性要小得多，但具有更多的规律性，因此是更好的类别作为近似值。 复杂的分析是对复杂变量的可区分函数的研究，位于数学领域的核心，从数字理论到潜在理论，再到线性代数和数值分析。 这个特定的项目将一些长期存在的问题和方法与计算几何学和图理论的新应用结合在一起。