AuthorSimek, Olga, 1966-
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PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractThis dissertation consists of three main results regarding heat trace asymptotics for bounded domains with cusps. First, a refined asymptotic expansion for the Dirichlet heat trace θ(t) = Treᵗ(Δ)ᴰ on a planar domain with a single cusp is presented. First three terms appeared earlier in the physics literature. We prove them, together with logarithmic remainder estimate. Second, we obtain similar results for a family of three-dimensional solids of revolution with a cusp. Third, we calculate bounds for the Neumann heat trace Ψ(t) = Treᵗ(Δ)ᴺ on a planar region a cusp, which then allow us to conclude the first two terms in the asymptotic expansion of Ψ(t). For the upper bound, we use Golden-Thompson inequality. All the results for the Dirichlet heat trace and the calculation of the lower bound for the Neumann heat trace use Brownian motion.
Degree ProgramGraduate College