The Sound of Symmetry
The Sound of Symmetry
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对称之声
DOI：
10.4169/amer.math.monthly.122.9.815
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Rowlett
中科院分区：
文献类型：
--
作者：
Zhiqin Lu;J. Rowlett
谱反问题是M. Kac在1966年的文章《一个人能听到鼓的形状吗？》虽然答案已经知道了20多年，但许多悬而未决的问题仍然存在。面向普通观众，读者面临的挑战是完成整个互动介绍逆谱理论的练习。主要技术用于逆谱问题的收集和讨论，然后用来证明，人们可以听到的形状：平行四边形，锐边形，正n边形。最后，我们表明，人们可以真实地听到的形状的规则n边形之间的所有凸n-凸，因为它是唯一确定的有限数量的特征值;对称的声音真的可以听到！
Abstract The inverse spectral problem was popularized by M. Kac's 1966 article in THIS MONTHLY “Can one hear the shape of a drum?” Although the answer has been known for over twenty years, many open problems remain. Intended for general audiences, readers are challenged to complete exercises throughout this interactive introduction to inverse spectral theory. The main techniques used in inverse spectral problems are collected and discussed, then used to prove that one can hear the shape of: parallelograms, acute trapezoids, and the regular n-gon. Finally, we show that one can realistically hear the shape of the regular n-gon amongst all convex n-gons because it is uniquely determined by a finite number of eigenvalues; the sound of symmetry can really be heard!