Hearing the shape of a triangle
Hearing the shape of a triangle
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听到三角形的形状
DOI：
10.1090/noti1063
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Svenja Maronna
中科院分区：
文献类型：
--
作者：
D. Grieser;Svenja Maronna
1966年，马克·卡茨提出了一个著名的问题：“人们能听到鼓的形状吗？”'.虽然这后来被证明是错误的，但它被C。所以人们可以听到三角形的形状。在介绍了一般的逆谱问题之后，我们将给出这一事实的一个新的证明。论证的中心点是要表明面积、周长和角的倒数之和唯一地决定一个三角形。这是证明使用凸性参数和部分分数扩展的$\sin^{-2}x$。
In 1966 Mark Kac asked the famous question 'Can one hear the shape of a drum?'. While this was later shown to be false in general, it was proved by C. Durso that one can hear the shape of a triangle. After an introduction to the general inverse spectral problem we will give a new proof of this fact. The central point of the argument is to show that area, perimeter and the sum of the reciprocals of the angles determine a triangle uniquely. This is proved using convexity arguments and the partial fraction expansion of $\sin^{-2}x$.