Optimal Bounds for the Characteristic Frequencies of Vibrating Membranes

发展中国家科学奖将 支持马克教授的研究合作 密苏里州、哥伦比亚大学和 教皇拉斐尔·本古里亚教授 智利天主教大学。 该项目旨在 解决与数学相关的问题 圆形桶的尺寸。 研究人员提出， 第一泛音与基音的比率 “翘曲”鼓，其中跨越一个封闭的曲线在三个- 维度空间 他们还将考虑其他 类型的表面，如球体的一部分 表面，这是非常重要的几何 弯曲的表面。 通过技术， 调查人员将审查早期的合作 其他音调的比率界限，以便 改进现有的界限，也许可以获得最佳的 约束，沿着给出最佳 结果. 这项研究预计将产生大量的 对特征值界的贡献 微分算子。 美国方面将获得 继续与智利专家合作， 这方面 智方将受益于 继续与美国直接合作。 对应物。

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.