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Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments
Title
Can One Hear...? An Exploration Into Inverse Eigenvalue Problems Related to Musical Instruments.
Creator
Adams, Christine, Nashed, M, Mohapatra, Ram, Kaup, David, University of Central Florida
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...
Show moreThe 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.
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Date Issued
2013
Identifier
CFE0004643, ucf:49886
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004643
Nitrogen-Containing Materials for Mechanochemical Synthesis, Luminescence Analysis, and Heterogeneous Catalysis
Title
Nitrogen-Containing Materials for Mechanochemical Synthesis, Luminescence Analysis, and Heterogeneous Catalysis.
Creator
Nash, David, Zhai, Lei, Hampton, Michael, Harper, James, Rex, Matthew, Blair, Richard, University of Central Florida
Abstract / Description
Various inorganic nitrogen-containing materials have been exploited for their different properties. Several nitride materials are commercially attractive due to their mechanical properties making them suitable for ceramic industries and wide bandgaps fitting for use as semiconductor and insulator materials, as well as optoelectronics. Nitride materials can exhibit versatility in applications such as the use of gallium nitride to make blue LEDs, nitrides of titanium and silicon being utilized...
Show moreVarious inorganic nitrogen-containing materials have been exploited for their different properties. Several nitride materials are commercially attractive due to their mechanical properties making them suitable for ceramic industries and wide bandgaps fitting for use as semiconductor and insulator materials, as well as optoelectronics. Nitride materials can exhibit versatility in applications such as the use of gallium nitride to make blue LEDs, nitrides of titanium and silicon being utilized as medical implants for their chemical inertness and hardness, and the heavy use of boron nitride as a solid lubricant in the cosmetic industry. Amines have been used as nitrogen-containing organic ligands in organometallic complexes that exhibit phenomenal photophysical properties. These complexes have been heavily studied for potential applications in optoelectronics and chemical sensing. This dissertation will focus on two nitrogen-containing materials that have yet to be explored for the potential applications to be discussed. The first is hexagonal-boron nitride (h-BN), which was previously mentioned to have a substantial use in the cosmetic industry, giving products such as lipstick, foundation, and blush their slick feeling. Computational models have shown the possibility of altered electronic properties of defect sites in the h-BN sheets. These defect sites will be explored experimentally to determine any catalytic activity. Specifically, the hydrogenation reaction using defect-laden hexagonal-boron nitride will be investigated. Successful catalysis would add to the short list of non-metal catalyst, and provide an alternative catalyst that costs significantly less than the traditional metal catalysts commonly used in commercial industries. The second of the two nitrogen-containing materials is a class of metal complexes based on organometallic clusters of copper(I) iodide. Copper(I) iodide clusters formed with amine ligands have been studied for around four decades and the photophysics behind their photoluminescent properties are well understood. Much of the work has been done for use as a potential emissive material in the optoelectronics field. They have also been studied for applications in the sensing of environmental compounds. Here, research will display its use as a novel sensor for narcotic substances. This forensic application will be further explored to develop and eventually commercialize a complete field drug testing system for law enforcement and crime lab use, with the goal to equip law enforcement personnel with a presumptive drug testing method that is accurate, easy-to-use, safe, adaptable, and affordable. This system will consist of a narcotic drug-indicating test strip, a handheld fluorescence spectrometer manufactured in-house using relatively inexpensive parts, and a mobile app that will leverage photoemission data of the tested drug samples collected by multiple crime labs to provide the ability for sample-to-reference data matching. Law enforcement users would have the ability to rapidly identify an unknown substance by applying it to a test strip, testing it using the spectrometer, and capturing an image of the resulting photoemission and analyzing the spectral profile in search of a match with the support of a cloud database.
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Date Issued
2017
Identifier
CFE0007129, ucf:52297
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007129