Hearing the shape of a triangle
Daniel Grieser,Svenja Maronna +1 more
TLDR
In this article, it was shown that the sum of the reciprocals of the angles of a triangle uniquely determines the shape of the triangle, using convexity arguments and the partial fraction expansion of the inverse spectral problem.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 $\sin^{-2}x$.read more
Citations
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Nodal portraits of quantum billiards: Domains, lines, and statistics
Sudhir R. Jain,Rhine Samajdar +1 more
TL;DR: A comprehensive review of nodal domains and lines of quantum billiards can be found in this article, where the nodal statistics distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself.
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The Robin Laplacian—Spectral conjectures, rectangular theorems
TL;DR: In this article, the shape optimization conjecture for the first two eigenvalues of the Robin Laplacian was investigated for rectangular boxes, where the spectral gap of each fixed rectangle is an increasing function of α and the second eigenvalue is concave.
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One can hear the corners of a drum
Zhiqin Lu,Julie Rowlett +1 more
TL;DR: In this article, the presence or absence of corners is spectrally determined in the following sense: any simply connected planar domain with piecewise smooth Lipschitz boundary and at least one corner cannot be isospectral to any connected domain, of any genus, that has smooth boundary.
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The sound of symmetry
Zhiqin Lu,Julie Rowlett +1 more
TL;DR: It is shown 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.
Journal ArticleDOI
One can hear the corners of a drum.
Zhiqin Lu,Julie Rowlett +1 more
TL;DR: In this paper, the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary.
References
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I and i
TL;DR: There is, I think, something ethereal about i —the square root of minus one, which seems an odd beast at that time—an intruder hovering on the edge of reality.
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Can One Hear the Shape of a Drum
TL;DR: Can one hear the shape of a drum? as discussed by the authors, 1966; The American Mathematical Monthly: Vol. 73, No. 4P2, pp. 1-23.
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Curvature and the Eigenvalues of the Laplacian
TL;DR: In this paper, the authors defined the spectrum of the problem of bounded regions of R d with a piecewise smooth boundary B and showed that if 0 > γ1 ≥ γ2 ≥ ≥ ≥ β3 ≥ etc.
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One cannot hear the shape of a drum
TL;DR: In this paper, Sunada's theorem was used to construct a pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, can one hear the shape of a drum?