Related papers: Hearing the shape of a triangle

This note begins with an introduction to the inverse isospectral problem popularized by M. Kac's 1966 article in the American Mathematical Monthly, "Can one hear the shape of a drum?" Although the answer has been known for some twenty years…

We study a variation of Kac's question, "Can one hear the shape of a drum?" if we allow ourselves access to some additional information. In particular, we allow ourselves to ``hear" the local Weyl counting function at each point on the…

We answer Mark Kac's famous question, "can one hear the shape of a drum?" in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all…

Can one hear the shape of a drum? was proposed by Kac in 1966. The simple answer is NO as shown through the construction of iso-spectral domains. There already exists 17 families of planar domains which are non-isometric but display the…

In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was…

This article explores a variant of Kac's famous problem, "Can one hear the shape of a drum?", by addressing a geometric inverse problem in acoustics. Our objective is to reconstruct the shape of a cuboid room using acoustic signals measured…

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point $x$ on…

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct…

The question whether one can recover the shape of a geometric object from its Laplacian spectrum ('hear the shape of the drum') is a classical problem in spectral geometry with a broad range of implications and applications. While…

We answer Mark Kacs famous question - can one hear the shape of a drum - in the negative for orbifolds that are spherical space forms. This is done by extending the techniques developed by A. Ikeda on Lens Spaces to the orbifold setting.…

We prove that 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…

``Can one hear the shape of a drum?'' was a question posed (and made famous) by mathematician Mark Kac in the mid-1960s. It addresses whether a deeper connection exists between the resonance modes (eigenmodes) of a drum and its shape. Here…

The famous question of Mark Kac "Can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator can be legitimately extended to scattering…

In this thesis I demonstrate that isospectral domains, that is domains of differing geometric shapes that possess identical spectra, do not remain isospectral when subject to uniform rotation. One thus *can* hear the shape of a rotating…

The spectral action in noncommutative geometry naturally implements an ultraviolet cut-off, by counting the eigenvalues of a (generalized) Dirac operator lower than an energy of unification. Inverting the well known question "how to hear…

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question ``Can one hear the shape…

The inverse problem of determining the cross-sectional area of a human vocal tract during the utterance of a vowel is considered in terms of the data consisting of the absolute value of sound pressure at the lips. If the upper lip is curved…

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. This problem is related to the question "Can one hear the shape of a…

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It…

We introduce the new concept of D-geometry (or "drum geometry"), which has been recently discovered by the author in \cite{KT-DRUMS} when constructing and classifying isospectral and length equivalent drums under certain constraints. We…