Real meromorphic functions

实亚纯函数

• 批准号: 0555279
• 负责人: Alexandre Eremenko
• 金额: $ 33.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555279&HistoricalAwards=false
• 关键词: Real; meromorphic; functions

ABSTRACTThe proposer intends to continue his study of the distributionof roots and critical points of real meromorphic functionsusing the geometric methods developed in his previous work.The main directions of the proposed research are the following.a) The study of the Wronski map, both in the real and complex domains, and of the related pole assignment map.b) The study of the distribution of roots of successivederivatives of real entire functions.c) Further investigation of the relation between the rateof oscillation of real functions and their spectral properties.d) The study of existence and uniqueness ofmetrics of positive curvature with conic singularities on compact surfaces.One of the basic questions in mathematics and its applications iswhether a given equation or a system of equations has solutions,how many, and where are they located. In the theory of meromorphic functions one studies these questions for equations of the typef(z)=a, where a is a given complex number and f a given meromorphic function. The class of meromorphic functions includes elementary functions, such as rational, exponential and trigonometric ones, as well as the special functions, a. k. a. higher transcendental functions, such as the Gamma function, Airy functions, elliptic functions and so on. Most functions arising in applications of mathematics belong to this class. In modern mathematics, questions about resolvability of equations are usually formulated in geometric language, which makes the results appealing to our geometric intuition.The proposer plans to continue his study of geometric theory of meromorphic functions. Most of the proposed research is related to existence of real solutions, a moresubtle question than the existence of complex solutions, which are usually studied. One of the original motivations (beside intrinsic mathematical importance of these questions) was the so-called "pole placement problem", which is a major unsolved mathematical problem in control theory of linear systems. The results in this area will have implications for the design of complicated automatic control systems. These results would establish limitations on the possibility to control a system of given size by a control device of certain class. Another important area of the broader impact is the recently discovered connection of the problems considered in this proposal with physics, more precisely, with the exactly solvable models of ferromagnetism.

作者打算继续用他以前工作中发展的几何方法来研究实亚纯函数的根和临界点的分布。所提出的研究的主要方向如下：a)在实域和复域上的Wronski映射的研究，以及相关的极点配置映射。b)研究实整函数的连续导数的根的分布。c)进一步研究实函数的振动率与它们的谱性质之间的关系。d)研究紧曲面上具有二次奇性的正曲率度量的存在唯一性。数学及其应用中的一个基本问题是给定的方程或方程组是否有解，有多少解，解的位置在哪里。在亚纯函数理论中，人们研究了方程f(Z)=a的这些问题，其中a是给定的复数，f是给定的亚纯函数。亚纯函数类包括初等函数，如有理函数、指数函数、三角函数，以及特殊函数，又称高阶超越函数，如Gamma函数、Ary函数、椭圆函数等。数学应用中出现的大多数函数都属于这一类。在现代数学中，关于方程可解性的问题通常是用几何语言表示的，这使得结果更符合我们的几何直觉。大多数已提出的研究都与实解的存在性有关，与通常研究的复解的存在性相比，实解的存在性是一个更为复杂的问题。最初的动机之一(除了这些问题的内在数学重要性之外)是所谓的极点配置问题，它是线性系统控制理论中一个主要的未解决的数学问题。这方面的结果将对复杂的自动控制系统的设计具有一定的指导意义。这些结果将对通过特定类别的控制设备控制给定大小的系统的可能性建立限制。另一个更广泛影响的重要领域是最近发现的这个提议中考虑的问题与物理学，更准确地说，与铁磁性的精确可解模型之间的联系。