Mathematical Sciences: Mathematical Prediction of the Physical Properties of Materials and Media

DMS-9523292 Bruno与这一提议相关的数学研究涉及到偏微分方程组的经典理论及其在材料和介质物理性质研究中的应用；它们构成了我过去五年研究的自然延续。通过使用齐次化、积分方程组、谱分析和正则化的方法以及计算技术，我打算发展数学技术来研究和预测复合材料、形状记忆材料和等离子体的物理性质，以及一般障碍物对电磁辐射的散射。在电磁学领域，最近一些计算方法被引入新的领域，包括障碍物的遥感以及通过电磁散射手段检测受损生物组织(可能是癌症)的数学相关问题。在以前的应用中，这些方法是非常成功的；这些方法将被用于处理形状记忆合金。这些材料可用于各种应用(包括低温发电、医疗设备等)。通过利用它们的能力，在它们变形后，通过适当改变它们的温度来恢复它们的形状。

DMS-9523292 Bruno The mathematical investigations associated with this proposal relate to the classical theory of Partial Differential Equations and its applications to the study of the physical properties of materials and media; they constitute a natural continuation of my research of the last five years. By using methods of homogenization, integral equations, spectral analysis and regularity together with computational techniques I intend mathematical techniques will be developed to study and predict the physical properties of composite materials, shape memory materials and plasmas, as well as the scattering of electromagnetic radiation by general obstacles. In the area of electromagnetism some computational methods are introduced recently to new areas, including remote sensing of obstacles as well as the mathematically related problem of detection of damaged biological tissue (perhaps cancer) by means of electromagnetic scattering. In its previous applications these methods were highly successful; These methods will be used to deal with shape memory alloys. These materials can be used in a variety of applications (including generation of electricity from low temperature sources, medical devices, etc.) by exploiting their ability to recover their shape, after they have been deformed, by means of appropriate changes in their temperature.