Hearing the shape of a triangle

2012/08/15 by Grieser, Daniel, Maronna, Svenja · 1 citation
#35P99 #35R30 #51N20 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1208.3163

Abstract

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^-2x.

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