Mathematical Sciences: Dimension, Smoothness and Functional Equations

在这个项目中，主要研究人员将在由某些函数方程支配的并元迭代格式的背景下研究迭代函数系统(IFS)和细分算法(SDA)。攻击方法将涉及基于小波理论的分析和数值技术。整个研究计划与佐治亚理工学院目前正在进行的分形几何学工作有联系。特别是，主要研究人员将关注函数方程解的以下三个相互关联的方面：维度和光滑性、奇点谱和反问题。对分维集合的几何分析涉及到许多不同的数学领域，可以肯定地说，这门学科为经典分析的几个部分注入了新的生命。在这个项目中，主要的研究人员将应用小波理论中的技术来研究函数方程的解集的性质，这些函数方程的图形是分形图。她的方法将结合分析和数值方法，她将在这项将分析与几何相结合的工作中与领先的人合作。

In this project the principal investigator will study iterated function systems (IFS's) and subdivision algorithms (SDA's) in the context of dyadic iteration schemes governed by certain functional equations. The method of attack will involve analytical and numerical techniques that are based upon wavelet theory. The overall research plan has points of contact with the work on fractal geometry that is being conducted currently at Georgia Tech. In particular, the principal investigator will look at the following three interrelated aspects of solutions of functional equations: dimension and smoothness, the singularity spectrum and the inverse problem. The analysis of the geometry of sets of fractal dimension involves a number of different areas of mathematics, and it is safe to say that this subject has breathed new life into several parts of classical analysis. In this project the principal investigator will apply techniques from the theory of wavelets to study the properties of solution sets of functional equations whose graphs are fractals. Her approach will combine analytical and numerical methods, and she will be collaborating with the leading people in this work that blends analysis with geometry.