Date of Award

2014

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

McDonald, Patrick

Keywords

Torsional Rigidity, Quantum Graphs, Mathematics

Area of Concentration

Mathematics

Abstract

The research of Gordon, Wolpert and Webb showed that the Dirichlet eigenspectrum of a planar domain is not sufficient to identify the domain uniquely, sparking a search for a new, stronger, natural geometric invariant. Colladay, Kaganovskiy and McDonald performed calculations indicating that the torsional rigidity--the first moment of heat content may be such an invariant. We investigate this possibility by calculating the torsional rigidities of two isospectral domain pairs. In the first--a pair of disconnected planar domains constructed by S.J. Chapman--we find that the torsional rigidities are given sums of functions in two variables and, by constructing upper and lower bounds for such sums, show that the torsional rigidities do indeed differ. Our second investigation is of a quantum graph analog of the original isospectral pair example of Gordon, Wolpert and Webb. The quantum graph construction follows that of Colladay, et al. We find, unlike Colladay, et al. that the pair is not distinguished by torsional rigidity for all sets of edge-lengths.

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