Mathematical Sciences: Partial Differential Equations & Analysis

数学科学：偏微分方程

• 批准号: 9213595
• 负责人: Richard Beals
• 金额: $ 60.03万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1996-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9213595&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Partial; Differential; Equations

This award supports the work of three researchers investigating several areas of broad mathematical interest and importance. Some work will be conducted jointly while other will be addressed by single investigators. One can divide the studies into three main themes: nonlinear partial differential equations, wavelet analysis and complex function theory. In the first, efforts will be made to explore how the one-dimensional quantum version of the inverse scattering methods which replace the classical action-angle variables can be applied to understanding the corresponding higher dimensional systems. Work is also progressing on the investigation of inverse geophysical problems involving surface wave phenomena. Progress on model scalar cases suggests that analysis of the full system of equations can be obtained. Studies of adapted wave form analysis, a collection of FFT- like adapted transform algorithms, continues to discover and refine new orthonormal bases which decompose functions and operators into almost diagonal form. The interplay between these theoretical advances and the application to numerical signal processing has led to highly successful decomposition techniques using entropy based stopping time searches. The applications lead, in turn, to a host of new questions in harmonic analysis. One of these will be that of creating a new concept of theoretical dimension of a function designed to measure the number of parameters necessary to describe the function with wave forms taken from a given library. Studies on the fine structure of harmonic measure also continue. Here the goals are to find the asymptotic behavior of harmonic measure in a variety of settings. Work will also be done estimating the length of level lines of the Green's function on bounded simply connected domains. It is believed that level line length varies with the fourth root of the value of constancy. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them and validate methods for expressing solutions. Harmonic analysis focuses on the decomposition of functions into component parts which best reflect their oscillatory characteristics, while complex function theory seeks to describe the geometric and analytic properties of differentiable functions of a complex variable and their generalizations to higher dimensions.

该奖项支持三名研究人员调查几个具有广泛数学兴趣和重要性的领域的工作。一些工作将联合进行，而另一些工作将由单独的调查人员处理。人们可以将研究分为三个主题：非线性偏微分方程组、小波分析和复函数理论。首先，我们将努力探索如何应用一维量子形式的逆散射方法来代替经典的作用角变量来理解相应的高维系统。涉及面波现象的地球物理反问题的研究工作也在取得进展。模型标量情形的进展表明，可以获得对整个方程组的分析。自适应波形分析是一组类似于FFT的自适应变换算法的集合，它的研究不断发现和提炼将函数和算子分解成几乎对角线形式的新的正交基。这些理论进步和在数字信号处理中的应用之间的相互作用导致了使用基于熵的停止时间搜索的分解技术的高度成功。这些应用反过来又导致了调和分析中的一系列新问题。其中之一将是创建函数的理论维度的新概念，其设计用于测量用从给定库获取的波形来描述函数所需的参数的数量。对调和测度的精细结构的研究也在继续。这里的目标是找出调和测度在各种设置下的渐近行为。此外，还将对有界单连通区域上格林函数的水平线长度进行估计。认为水平线长度随恒定值的四次根而变化。偏微分方程式是物理科学中数学建模的基础。人们知道，涉及运动、材料和能量等连续变化的现象遵循某些一般规律，这些规律可以用偏导数之间的相互作用和关系来表示。数学的关键作用不是描述这些关系，而是从这些关系中提取定性和定量的意义，并验证表达解的方法。调和分析侧重于将函数分解成最能反映其振荡特性的组成部分，而复变函数论则致力于描述复变量可微函数的几何和解析性质以及它们在高维上的推广。