Can you hear the fractal dimension of a drum?

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dc.creatorArrighetti, W.
dc.creatorGerosa, G.
dc.date2005-03-31
dc.date.accessioned2026-07-07T06:19:39Z
dc.date.available2026-07-07T06:19:39Z
dc.descriptionElectromagnetics and Acoustics on a bounded domain are governed by the Helmholtz's equation; when such a domain is a [pre-]fractal described by means of a `just-touching' Iterated Function System (IFS) spectral decomposition of the Helmholtz's operator is self-similar as well. Renormalization of the Green's function proves this feature and isolates a subclass of eigenmodes, called ``diaperiodic'', whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator's eigenspaces. The definition of ``spectral dimension'' is given and proven to depend on diaperiodic modes only for a wide class of IFSs. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the `self-similar' spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question.
dc.description[11 pages, 2 figures] To appear in ``Applied and Industrial Mathematics in Italy'', World Scientific, 2005. Authors are with the Electronic Engineering Department, ``LaSapienza'' University of Rome, at http://www.die.uniroma1.it/strutture/labcem/
dc.identifierhttps://arxiv.org/abs/math/0503748
dc.identifierhttp://arxiv.org/abs/math/0503748
dc.identifierin ``Applied and Industrial Mathematics in Italy'' (ISBN 981-256-368-7; 600 pages), M.Primicerio, R.Spigler, V.Valente editors, pp. 65-75, World Scientific, 2005; presented at the 7th SIMAI congress (Venice, 20-24 september 2004)
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/95101
dc.subjectSpectral Theory
dc.subjectMathematical Physics
dc.subjectMetric Geometry
dc.subject28A80
dc.titleCan you hear the fractal dimension of a drum?
dc.typetext

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