Mathematical Sciences: Dimension, Smoothness and Functional Equations

在这个项目中，首席研究员将研究迭代 函数系统（IFS）和细分算法（SDA） 并矢迭代格式的上下文由某些泛函 方程 攻击方法将涉及分析和 基于小波理论的数值技术。 的 总体研究计划与工作的联系点 分形几何学目前正在格鲁吉亚理工学院进行。 特别是，首席研究员将研究 以下三个相互关联的方面的解决方案的功能 方程：维数和光滑性，奇异谱和 逆问题。 分形维数集的几何分析 涉及到许多不同的数学领域，它是安全的， 我想说这个主题为几个部分注入了新的活力 经典分析。 在这个项目中， 将应用小波理论的技术来研究 函数方程解集的性质 是分形 她的方法将结合联合收割机的分析和数值 方法，她将与领先的人合作， 这项工作将分析与几何学结合起来。

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.