Date of Thesis

Spring 2018

Thesis Type

Honors Thesis (Bucknell Access Only)

Degree Type

Bachelor of Science

Major

Mathematics

Minor, Emphasis, or Concentration

Dance

First Advisor

Emily B. Dryden

Second Advisor

Jeffrey J. Langford

Keywords

Heat Content, Polygons, Asymptotics

Abstract

Mark Kac originally posed the question "Can you hear the shape of a drum?" in 1966. That is, does the set of frequencies at which a drum oscillates uniquely determine the shape of the drumhead? This question was partially answered by Gordon, Webb and Wolpert in 1992, who found two different drums that make the same sounds. Therefore, in general, you cannot hear the shape of a drum. Later, Greiser and Maronna showed that amongst triangular drums, "you can hear the shape" of a triangular drum. We tackle a similar question, but instead of determining a drum's shape based on the frequencies at which it oscillates, we ask if you can determine the shape of a drum from its heat content, or thermal energy. van den Berg and Srisatkunarajah showed that the heat content $q(t)$ for a polygon $\Omega$ has a small time asymptotic expansion \[q(t)=|\Omega|-\frac{2|\partial \Omega|}{\sqrt{\pi}}t^{\frac{1}{2}}+4t \Phi_\Omega+O(e^{-k/t}) \] where $|\Omega|$ is the area, $|\partial\Omega|$ is the perimeter, $\Phi_\Omega$ is a function of the interior angles of $\Omega$, and $k$ is a constant related to the length of the shortest side. We show that $|\Omega|,|\partial\Omega|,$ and $\Phi_\Omega$ uniquely determine parallelograms amongst parallelograms, and uniquely determine acute trapezoids amongst acute trapezoids. However, we provide examples to show that in general, these three quantities do not uniquely determine quadrilaterals within the class of quadrilaterals, nor do they determine trapezoids within the class of trapezoids. We also show that the equiangular $n$-gons minimize the value of the function $\Phi_\Omega$ and we establish the explicit lower bound $\frac{\pi}{4}\leq \Phi_\Omega$.

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