Mathematical Sciences: Discrete Analytic Function Theory Via Circle Packing

数学科学：通过圆堆积的离散解析函数理论

• 批准号: 9303135
• 负责人: Kenneth Stephenson
• 金额: $ 9.75万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303135&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Discrete; Analytic; Function

This project continues investigations into the connections between circle packing and the theory of analytic functions of a complex variable. Circle packings have long been an intriguing concept in mathematical research. In 1985 William Thurston proposed that circle packings - packings of plane domains by circles of various sizes - could lead to a completely new point of view in conformal mapping. This turned out to be the case and this project is one of several by-products of that now famous lecture. The idea has grown and prospered. Circle packings are exhibiting faithful discrete analogues to the classical results of geometric function theory such as the Schwarz-Pick lemma, Dirichlet problem, uniformization theorem, Brownian motion, etc. The current project concentrates on the discrete version of the length-area method, a powerful, but unwieldy method exercised by a few experts and not easily understood by consumers of function theory. The second part concerns infinite packings, possibly the most speculative part of the project. Most research to date considers only finite packings. The final component of the research involves packings using circle with overlap. There are fundamental results which allow for overlap, but little work has been done on this generalization. Past research led to the development of software for manipulating, displaying and animating circle packings. This is both a research tool for performing experiments and a practical method for creating the pictures needed for presentation. The software is freely available. Complex function theory is the mathematical basis for computing with and visualizing the behavior of complex functions of a complex variable. The subject has an extensive history and is normally studied concurrently as analysis and geometry. This project adds a new dimension: computation. Circle packing leads naturally to computer simulation. This in turn has led to new insights and proofs in the classical theory. To prove its worth, the new point of view will have to lead to new , unexpected discoveries.

该项目继续调查 圆填充与解析理论的联系 复变量的函数。 圆形填料长期以来一直是 数学研究中一个有趣的概念 1985年，威廉 Thurston提出圆填充-平面填充 不同大小的圆圈-可能会导致一个完全 保角映射的新观点。 这竟然是 这个案子和这个项目是其中的一个副产品， 著名的演讲。 这个想法已经成长和繁荣。 圈 填料表现出忠实的离散类似物， 几何函数论的经典结果，如 Schwarz-Pick引理，Dirichlet问题，单值化定理， 布朗运动等。目前的项目集中在 离散版本的长度面积法，一个强大的，但 一些专家使用的笨拙的方法， 消费者对功能理论的理解。 第二部分 关于无限包装，可能是最投机的一部分， 该项目 到目前为止，大多数研究只考虑有限的 填料。 研究的最后一部分涉及包装 使用带重叠的圆。 有一些基本结果， 允许重叠，但在这方面做的工作很少 一般化 过去的研究导致了软件的发展 用于操作、显示和动画化圆包装。 这 它既是一种进行实验的研究工具， 创建所需图片的实用方法 演示文稿. 该软件免费提供。 复变函数理论是数学基础， 计算和可视化复杂函数的行为 一个复杂的变量。 这门学科有着悠久的历史 并且通常同时作为分析和几何学来研究。 这个项目增加了一个新的维度：计算。 圆packing 这自然会导致计算机模拟。 这反过来又导致了 对经典理论的新认识和新证明。证明其 价值，新的观点将导致新的， 意想不到的发现