Mathematical Sciences: Inverse Nodal Problems

在这个奖项中，首席研究员将研究一类新的反问题，称为反节点问题。这项研究的动机是关于如果一个人知道节线，就可以对振动膜说多少的问题。节线是膜的本征函数沿其消失的曲线。在一个维度中，节线是点(节点)。节点在工程应用中很有意义，因为它们代表了支撑桅杆或将轴承放置在轴上的最佳位置，而节点线则用于分析全息板中的缺陷。如果你知道节点，你能对振动弦说些什么？如果你知道节线，你能对振动膜说些什么？节线是在振动过程中产生的驻波上的固定曲线或点。这是一个逆问题的例子：一个人试图从它产生的数据中确定一个对象或过程。这类反问题具有重要的工程应用，主要研究人员将使用解析和数值方法来研究不连续问题和密度的唯一性。

In this award the principal investigator will study a new class of inverse problems called inverse nodal problems. The motivation for this study is the question concerning how much one can say about a vibrating membrane if one knows the nodal lines. The nodal lines are the curves along which the eigenfunctions of the membrane vanish. In one dimension the nodal lines are points (nodes). Nodes are of interest in engineering applications because they represent the optimal locations for the support of a mast or placement of a bearing on a shaft, while nodal lines enter into the analysis of flaws in holographic plates. What can you say about a vibrating string if you know the nodes or a vibrating membrane if you know the nodal lines? Nodal lines are the fixed curves or points on the standing waves generated during the vibration. This is an example of an inverse problem: one attempts to determine an object or process from the data it produces. Such inverse problems have important engineering applications and the principal investigator will employ analytical and numerical methods to study the discontinuous problems and the uniqueness of the density.