Mathematical Sciences: Inverse Nodal Problems and Perturbation Theory in Higher Dimensions

9410700麦克劳克林这个项目涉及根据弹性膜的固有频率和节线的知识来确定弹性膜的哪些属性的问题。节线是指当膜以固有频率激励时，膜上没有激励的地方。该项目包括两个主要部分。第一种方法是得到固有频率和振型的摄动结果。由于在高维问题中频率分布不是很好，所以必须解决小因子问题。该项目的第二个主要部分涉及这样一个事实，即节域(由节线定义的膜中的连通域)可以是直径与膜的直径一样大的细长条带。那么所需要的是定义近似的节域，其直径随着本征值的阶数趋于无穷大而趋于零。这项研究的目标是建立从间接测量，特别是固有频率和节线的测量中寻找振动系统特性的方法。通过对脉冲响应数据的频谱分析，可以确定其固有频率。当薄膜以固有频率激励时，通过在振动表面引导激光来测量节线。后向散射中的多普勒频移最小的线是节线。根据该数据，可以从这些量的显式公式中确定膜上的外力的幅度和膜的(非恒定)密度的表达式。这些公式有望对获得有效的数值算法来识别所讨论的量非常有用。***

9410700 McLaughlin This project is concerned the problem of determining which properties of an elastic membrane can be deduced from knowledge of the natural frequencies and the nodal lines of the membrane. The nodal lines are places on the membrane where there is no excitation when the membrane is excited at a natural frequency. There are two major parts to the project. The first is to obtain perturbation results for the natural frequencies and mode shapes. Since the frequencies are not well spaced in higher dimensional problems, small divisor problems must be solved. The second major part of the project is related to the fact that nodal domains (connected domains in the membrane defined by the nodal lines) can be long, thin strips whose diameter is as large as the diameter of the membrane. What is required then is to define approximate nodal domains whose diameter goes to zero as the order of the eigenvalue goes to infinity. The goal of this research is to establish methods for finding properties of vibrating systems from indirect measurements, in particular, measurements of natural frequencies and nodal lines. The natural frequencies can be determined by spectral analysis of impulse response data. The nodal lines can be measured by directing a laser at the vibrating surface when the membrane is excited at a natural frequency. The lines where the Doppler shift in the backscatter is minimized are the nodal lines. From this data the amplitudes of external forces on the membrane and an expression for a (nonconstant) density of the membrane may be determined from explicit formulas for these quantities. These formulas are expected to be very useful for obtaining efficient numerical algorithms to identify the quantities in question. ***