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Minimal Nodal Domains For Strictly Elliptic Partial Differential Equations With Homogeneous Boundary Conditions, Charles E. Miller

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This work presents a proof of the dependence of the first eigenvalue for uniformly elliptic partial differential equations on the domain in a less abstract setting than that of Ivo Babušhka and Rudolf Výborný in 1965. The proof contained here, under rather mild conditions on the boundary of the domain, ∂Ω, demonstrates that the first eigenvalue of elliptic partial differential equation

{Lu + λu = 0 in Ω

{u = 0 on ∂Ω

depends continuously on the domain in the following sense. If a sequence of domains is such that Ω_i → Ω in …

Geodesic On Surfaces Of Constant Gaussian Curvature, Veasna Chiek

Theses Digitization Project

The goal of the thesis is to study geodesics on surfaces of constant Gaussian curvature. The first three sections of the thesis is dedicated to the definitions and theorems necessary to study surfaces of constant Gaussian curvature. The fourth section contains examples of geodesics on these types of surfaces and discusses their properties. The thesis incorporates the use of Maple, a mathematics software package, in some of its calculations and graphs. The thesis' conclusion is that the Gaussian curvature is a surface invariant and the geodesics of these surfaces will be the so-called best paths.