Fractal Geometry and Applications

分形几何及其应用

• 批准号: 0707524
• 负责人: Michel Lapidus
• 金额: $ 13万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707524&HistoricalAwards=false
• 关键词: Fractal; Geometry; Applications

This project pursues and amplifies the PI's earlier investigations of the relationships between fractal geometry, spectral geometry, and dynamical systems. We plan to study the vibrations of "fractal drums," both "drums with fractal boundary" (Laplacians on open sets with very irregular boundary) and "drums with fractal membrane" (Laplacians on fractals themselves), "fractal billiards," as well as the associated "complex fractal dimensions" (a suitable measure of the oscillations intrinsic to nonsmooth geometries and their spectra) and "fractal curvatures" (a suitable measure of the way space warps in a fractal space). Although the proposed theory is mathematically rigorous, it is also naturally physically motivated (with, e.g., applications to the scattering of waves by fractal surfaces and to the study of porous media), and has recently drawn (and will continue to draw) its impetus from the use of computer graphics and computer experiments. We also intend to pursue our mathematical and computer graphics aided study of partial differential equations such as the Laplace, heat, and wave equations on regions with fractal boundary or on fractals themselves.The proposed problems are closely connected to Mark Kac's question "Can one hear the shape of a drum?" and to its beautiful extensions from the "smooth" to the "fractal" domain by Michael Berry. The expected results should be of physical significance in condensed matter and solid state physics; for example, in the study of mechanical or electrical transport in porous or in random media, as well as of heat diffusions on fractals and in disordered systems. They should also be relevant to subterranean imaging and the study of the way information and electromagnetic signals propagate across rough terrain. (Recent engineering and physical applications include new types of cell phones, fractal antennas, loudspeakers, heat insulators, nearly optimal soundproof walls, radar detection, catalytic chemical reactions, and computer microchips.) This work may help understand the formation of fractal structures (e.g. coastlines, trees and blood vessels) in nature as well as the reason why certain biological structures, such as lungs, are both fractal and nearly optimal to fulfill their biological functions.

这个项目是对PI早期关于分形学、谱几何学和动力系统之间关系的研究的继续和深化。我们计划研究“分形鼓”的振动，包括“带分形边界的鼓”(具有非常不规则边界的开集上的拉普拉斯)和“带分形膜的鼓”(分形上的拉普拉斯人本身)，“分形台球”，以及与之相关的“复分形维”(一种适当的测量非光滑几何及其频谱固有的振荡的量度)和“分形曲率”(一种适当的量度空间在分维空间中扭曲的方式)。虽然提出的理论在数学上是严格的，但它也是自然的物理动机(例如，应用于分形表面的波散射和多孔介质的研究)，最近从计算机图形学和计算机实验的使用中获得了(并将继续获得)它的推动力。我们还打算继续我们的数学和计算机图形辅助研究偏微分方程组，如拉普拉斯方程，热方程和波动方程，在具有分形边界的区域上或在分形本身上。所提出的问题与Mark Kac的问题密切相关：人们能听到鼓的形状吗？以及迈克尔·贝里从“平滑”领域到“分形”领域的美丽延伸。预期的结果应该在凝聚态和固态物理学中具有物理意义；例如，在研究多孔或随机介质中的机械或电子传输，以及在分形学和无序系统中的热扩散。它们还应该与地下成像和研究信息和电磁信号在崎岖地形上传播的方式有关。(最近的工程和物理应用包括新型手机、分形天线、扬声器、隔热材料、近乎最佳的隔音墙、雷达探测、催化化学反应和计算机微芯片。)这项工作可能有助于理解自然界中分形结构(如海岸线、树木和血管)的形成，以及为什么某些生物结构，如肺，既是分形的，又是履行其生物功能的近乎最佳的结构。