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Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments

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Date Issued:
2013
Abstract/Description:
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 Schr(&)#246;dinger 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.
Title: Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments.
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Name(s): Adams, Christine, Author
Nashed, M, Committee Chair
Mohapatra, Ram, Committee Member
Kaup, David, Committee Member
, Committee Member
University of Central Florida, Degree Grantor
Type of Resource: text
Date Issued: 2013
Publisher: University of Central Florida
Language(s): English
Abstract/Description: 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 Schr(&)#246;dinger 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.
Identifier: CFE0004643 (IID), ucf:49886 (fedora)
Note(s): 2013-05-01
M.S.
Sciences, Mathematics
Masters
This record was generated from author submitted information.
Subject(s): Inverse Eigenvalue Problem -- Music -- Drum
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0004643
Restrictions on Access: public 2013-05-15
Host Institution: UCF

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