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

小行星9410700 这个项目涉及的问题是确定哪些性质的弹性膜可以推出的知识的自然频率和节点线的膜。 节线是膜上当膜以固有频率被激励时没有激励的位置。 该项目有两个主要部分。 首先是获得固有频率和振型的摄动结果。 由于频率在高维问题中没有很好的间隔，因此必须解决小除数问题。该项目的第二个主要部分是关于节点域（由节点线定义的膜中的连接域）可以是长而薄的条带，其直径与膜的直径一样大。 然后需要定义近似节点域，当特征值的阶数趋于无穷大时，近似节点域的直径趋于零。 本研究的目的是建立从间接测量，特别是固有频率和节线的测量中发现振动系统特性的方法。 固有频率可以通过脉冲响应数据的频谱分析来确定。 当膜以固有频率被激励时，可以通过将激光引导到振动表面来测量节线。 反向散射中的多普勒频移最小的线是节线。根据这些数据，膜上外力的振幅和膜的（非恒定）密度的表达式可以从这些量的显式公式确定。 这些公式预计将是非常有用的获得有效的数值算法，以确定有关的数量。 ***

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. ***