Keywords

Inverse eigenvalue problem, music, drum

Abstract

The central theme of this thesis deals with problems related to the question, “Can one hear the shape of a drum?” first posed formally by Mark Kac in 1966. More precisely, can one determine the shape of a membrane with fixed boundary from the spectrum of the associated differential operator? For this paper, Kac received both the Lester Ford Award and the Chauvant Prize of the Mathematical Association of America. This problem has received a great deal of attention in the past forty years and has led to similar questions in completely different contexts such as “Can one hear the shape of a graph associated with the Schrödinger operator?”, “Can you hear the shape of your throat?”, “Can you feel the shape of a manifold with Brownian motion?”, “Can one hear the crack in a beam?”, “Can one hear into the sun?”, etc. Each of these topics deals with inverse eigenvalue problems or related inverse problems. For inverse problems in general, the problem may or may not have a solution, the solution may not be unique, and the solution does not necessarily depend continuously on perturbation of the data. For example, in the case of the drum, it has been shown that the answer to Kac’s question in general is “no.” However, if we restrict the class of drums, then the answer can be yes. This is typical of inverse problems when a priori information and restriction of the class of admissible solutions and/or data are used to make the problem well-posed. This thesis provides an analysis of shapes for which the answer to Kac's question is positive and a variety of interesting questions on this problem and its variants, including cases that remain open. This thesis also provides a synopsis and perspectives of other types of “can one hear” problems mentioned above. Another part of this thesis deals with aspects of direct problems related to musical instruments.

Notes

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Graduation Date

2013

Semester

Spring

Advisor

Nashed, M

Degree

Master of Science (M.S.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematical Science; Industrial Mathematics

Format

application/pdf

Identifier

CFE0004643

URL

http://purl.fcla.edu/fcla/etd/CFE0004643

Language

English

Release Date

May 2013

Length of Campus-only Access

None

Access Status

Masters Thesis (Open Access)

Subjects

Dissertations, Academic -- Sciences, Sciences -- Dissertations, Academic

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Included in

Mathematics Commons

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