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

与这一提议相关的数学研究涉及到经典的偏微分方程理论及其在材料和介质物理性质研究中的应用；它们是我过去五年研究的自然延续。我打算利用均匀化、积分方程、谱分析和正则化方法以及计算技术，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，以及电磁辐射在一般障碍物下的散射。在电磁学领域，一些计算方法最近被引入到新的领域，包括障碍物的遥感以及利用电磁散射检测受损生物组织（可能是癌症）的数学相关问题。在以前的应用中，这些方法非常成功；这些方法将用于处理形状记忆合金。这些材料可以用于各种应用（包括从低温源发电，医疗设备等），利用它们在变形后恢复形状的能力，通过适当改变它们的温度。

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.