POWRE: Applications of Recent Advances in Exponential Asymptotics

POWRE：指数渐近学最新进展的应用

• 批准号: 0074924
• 负责人: Rodica Costin
• 金额: $ 7.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2002-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074924&HistoricalAwards=false
• 关键词: POWRE; Applications; Recent; Advances; Exponential

Motivated by theoretical as well as major practical interests, new and powerful tools have been developed in the last decade for understanding local properties of differential operators near singularities. The proposed research uses recent techniques and results of exponential asymptotics to address questions in the classification of ordinary differential equations in singular regions, as well as analytic properties of partial differential operators. Among the questions addressed are: (1a) finding necessary and sufficient criteria for an ordinary differential equation whose linear part has several regular singular points and is homogeneous, to be analytically equivalent to its linear part (in a domain which contains the singular points); (1b) finding necessary criteria for a nonlinear equation to be analytically equivalent to its linear (inhomogeneous) part in a neighborhood of one irregular singular point of rank 1; (2) finding necessary andsufficient conditions for analytic hypoellipticity of partial differential operators of type ``sum of squares'', with special focus on Treves' conjecture.This POWRE project is jointly supported by the MPS Office of Multidisciplinary Activities (OMA) and the Division of Mathematical Sciences (DMS).

出于理论和主要实践利益的动机，在过去的十年中开发了新的和强大的工具，以理解近乎奇点的差异操作员的当地属性。 拟议的研究使用了指数渐近学的最新技术和结果来解决奇异区域中普通微分方程的分类中的问题，以及部分差分运算符的分析性能。 解决的问题包括：（1A）找到一个普通微分方程的必要标准，线性部分具有几个常规的奇异点，并且是均匀的，在分析上等同于其线性部分（在包含单数点的域中）； （1b）在分析上找到非线性方程的必要标准，等同于其线性（不均匀）部分，在一个不规则的等级1的邻域中； （2）寻找``正方形总和''类型的部分差异操作员的分析性低纤维化的必要条件，并特别关注Treves的猜想。该POWRE项目由MPS的多学科活动（OMA）和The MPS Office共同支持数学科学划分（DMS）。