Mathematical Sciences: Classical Complex Analysis

数学科学：经典复分析

• 批准号: 9800464
• 负责人: Donald Marshall
• 金额: $ 6.76万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800464&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Classical; Complex; Analysis

Proposal: DMS-9800464 Principal Investigator: Donald E. Marshall Abstract: Under this grant Marshall will investigate problems in four areas of analytic function theory. Estimates of harmonic measure and the growth of hyperbolic distance will be used to study the angular distribution of mass induced by analytic, area-integrable functions. This problem has applications to the characterization of extremal dilatations for quasi-conformal homeomorphisms of the unit disk. Secondly he will investigate the integrability properties of derivatives of one-to-one analytic functions. Thirdly, he will investigate finite interpolation problems for the bidisk. Marshall will seek continued fraction decompositions for a certain class of rational functions in several variables to solve this problem. In the fourth area, Marshall will investigate the accuracy of a promising technique for the numerical computation of conformal maps. Conformal maps (one-to-one analytic functions) have been used as a tool in science and engineering for many years. One way conformal maps are used is to transform a problem on a complicated region in the complex plane to a related problem on a "standard" region, such as a disk or half-plane, where known techniques can be used. The solution on the standard region is then transformed by the inverse of the conformal map to a solution of the original problem on the original region. Classically, this method was used for problems related to Laplace's equation. For example, temperature at equilibrium on a thin metallic plate satisfies Laplace's equation. More recently, conformal maps have found application to a wide range of numerical solutions of other partial differential equations arising in engineering problems. There are applications in electro-magnetics, vibrating membranes and acoustics, transverse vibrations and buckling of plates, elasticity, heat transfer, and fluid flow, for example. While early applications used explicit analytic representations for conformal maps, modern us es require conformal maps of more complicated regions which cannot be represented easily in terms of elementary functions. The only resort is to compute numerical approximations to the desired conformal maps. We will investigate the accuracy of a new technique which rapidly computes conformal maps and their inverses. It is fast enough that it can be used for experimentation on a typical workstation. The integrability questions we will work on deal with the difficult problem of estimating the growth rate of conformal maps. One of the applications of complex analysis to electrical engineering problems is the construction of electric circuits associated with a given transfer function. One method is to decompose the transfer function into simpler pieces, called a continued fraction expansion. In order to understand related problems in several complex variables, Marshall will seek similar decompositions for transfer functions depending on two variables.

建议：DMS-9800464首席研究员：唐纳德·E·马歇尔摘要：在此赠款下，马歇尔将调查分析功能理论四个领域的问题。谐波测量的估计值和双曲线距离的生长将用于研究由分析，可融合功能引起的质量的角度分布。该问题适用于对单元磁盘准符合形式同构的极端扩张的表征。其次，他将研究一对一分析函数的衍生物的集成性特性。第三，他将调查BIDISK的有限插值问题。马歇尔将在几个变量的某些合理功能中寻求持续的分数分解，以解决此问题。在第四区域，马歇尔将研究一种有前途的技术用于保形图的数值计算的准确性。 多年来，共形图（一对一的分析功能）已被用作科学和工程的工具。使用的一种方式，使用了共同地图的一种方法，将复杂平面中复杂区域的问题转换为可以使用已知技术的“标准”区域（例如磁盘或半平面）上的相关问题。然后，标准区域上的解决方案通过共形图的倒数转换为原始区域上原始问题的解决方案。从经典上讲，此方法用于与拉普拉斯方程相关的问题。例如，薄金属板上的平衡温度满足拉普拉斯方程。最近，共形图发现了在工程问题中引起的其他部分微分方程的多种数值解决方案的应用。例如，在电磁学，振动膜和声学，板横向振动和板的屈曲中有应用，例如，弹性，传热和流体流动。虽然早期应用使用明确的分析表示形式用于保形图，但现代美国需要更复杂区域的共形图，这些图不能轻松地以基本功能来表示。唯一的度假胜地是将数值近似值计算到所需的共形图。 我们将研究一种新技术的准确性，该技术可以快速计算保形图及其倒置。它足够快，可以用于典型的工作站实验。我们将致力于估计保形图的增长率的困难问题。复杂分析在电气工程问题上的应用之一是构建与给定传输功能相关的电路。一种方法是将传输函数分解为更简单的零件，称为持续的分数扩展。为了了解几个复杂变量中的相关问题，马歇尔将根据两个变量寻求类似的分解来传输功能。