Mathematical Sciences: Harmonic Analysis Algorithms

一个可再生能源股现场项目将侧重于使用先进的 计算机实验，让本科研究人员 各种当前的数学研究前沿。 本文编制了一个程序，对该问题进行了数值试验。 非线性发展方程解的定性行为 方程和椭圆系统将是可用的。 研究 多孔介质方程解的渐近性态 代表一个特定的领域。 获得的数值数据将 扩展了最近关于极值唯一性的理论结果 Sobolev不等式中的函数 其他工作将集中在φ变换，一个新的 开发了将一般函数分解为和的技术 简单的术语。 它是基于谐波的重要进展 近二十年来的分析。 除了是 相对容易理解，这种转换表现出一定的 快速傅立叶变换的速度特性。 性能 使其在数值计算中很有用;理解 这些性质是一个非常适合本科生的任务 research. 学生实习生也将参与创建 自同构群计算的数值模型 多复变量空间中的域。 这种高度 理论领域有助于数值解释 通过某些边界微分量， Chern-Moser-Tanaka不变量 繁琐的计算， 将域的边界置于标准形式， 出去 学生将发展基本的代数和符号 操纵算法，这将是必要的，学习有关 复杂的数学结构及其分析 同步

An REU site project will focus on the use of advanced computational experimentation to expose undergraduate researchers to a variety of current mathematical research frontiers. A program to carry out numerical experiments on the qualitative behavior of solutions of nonlinear evolution equations and elliptic systems will be available. Studies of asymptotic behavior of solutions to the porous medium equation represents one specific area. Numerical data obtained will amplify recent theoretical results on uniqueness for extremal functions in the Sobolev inequality. Other work will concentrate on the phi-transform, a new technique developed for decomposing general functions into sums of simple terms. It is based on important advances in harmonic analysis in the last twenty years. In addition to being relatively easy to understand, this transform exhibits certain speed characteristics of the fast Fourier transform. Properties of the transform make it useful in numerical work; understanding these properties is a task well suited for undergraduate research. Student interns will also be involved in the creation of numerical models for calculation of automorphism groups for domains in the space of several complex variables. This highly theoretical field lends itself to numerical interpretation through certain boundary differential quantities known as the Chern-Moser-Tanaka invariants. Tedious calculations necessary to put the boundary of a domain in normal form can now be carried out. Students will develop the basic algebra and symbol manipulation algorithms which will be needed, learning about the complex mathematical structures and their analysis simultaneously.