Optimal Bounds for the Characteristic Frequencies of Vibrating Membranes

这一发展中国家科学奖将支持哥伦比亚密苏里大学的Mark Ashbaugh教授和智利CATOLICA大学的Raphael Benguria教授的研究合作。该项目旨在解决与圆鼓的数学尺寸有关的猜想。研究人员建议，在三维空间中跨越闭合曲线的“翘曲”鼓中，为第一泛音与基音的比率开发良好的界限。他们还将考虑其他类型的曲面，例如球面的一部分，这对曲面的几何非常重要。通过早期合作中使用的技术，研究人员也将检查其他音调比率的界限，以改进现有的界限，也许还会获得最佳界限，以及给出最佳结果的形状。这项研究有望对微分算子的特征值界的一般领域做出重大贡献。美方将从与智利专家在这一领域的持续合作中获益。智利方面将受益于继续与美国同行直接参与。

This Science in Developing Countries award will support the research collaboration of Professor Mark Ashbaugh of the university of Missouri, Columbia, and Professor Raphael Benguria of the Pontificia Universidad Catolica de Chile. The project aims to solve conjectures related to the mathematical dimensions of circular drums. The researchers propose to develop good bounds for the ratio of the first overtone to the fundamental tone in "warped" drums, which span a closed curve in three- dimensional space. They will also consider other types of surfaces, such as a piece of a sphere's surface, which are of great importance to the geometry of curved surfaces. Through techniques used in earlier collaborations, the investigators will examine bounds for ratios o other tones as well, in order to improve existing bounds and perhaps obtain the optimal bound, along with the shape that gives the optimal result. The research is expected to yield substantial contributions to the general area of eigenvalue bounds for differential operators. The u.S. side will gain from continued collaboration with a Chilean expert in this area. The Chilean side will benefit from continued direct participation with a U.S. counterpart.