Mathematical Sciences: "Studies in Applied Mathematics"

这个项目涵盖了应用数学中当前感兴趣的两个领域。第一部分讨论了随机初始数据下双曲守恒定律在模式形成中的问题。从数学的角度分析解的统计性质、它们的概率分布函数和复杂的模式。这些问题出现在统计物理学、湍流模型和宇宙学的各种环境中。本课题的第二部分是关于具有压电成分的先进复合材料的数学建模。这些问题的数学方法包括偏微分方程的渐近性，包括均匀化理论、概率论和数值模拟。聚合物/压电陶瓷复合材料是构建高性能传感器和执行器的理想材料，可用于各种工业和医疗应用。

DMS-9504122 Avellaneda This project covers two areas of current interest in Applied Mathematics. The first part concerns problems in pattern formation arising from hyperbolic conservation laws with random initial data. The statistical properties of solutions, their probability distribution functions and the complex patterns will be analyzed from a mathematical standpoint. These problems arise in various settings in Statistical Physics, in turbulence modeling and in Cosmology. The second part of this project is concerned with the mathematical modeling of advanced composite materials with piezoelectric constituents. The mathematical methods that will be brought to bear on these problems include asymptotics for partial differential equations, including homogenization theory, probability and numerical modeling.Polymer/piezoceramic composites are ideal materials to construct high-performance sensors and actuators that can be used in a variety of industrial and medical applications.