Physical Sciences and Mathematics Commons^™

Variational And Partial Differential Equation Models For Color Image Denoising And Their Numerical Approximations Using Finite Element Methods, Miun Yoon

Masters Theses

Image processing has been a traditional engineering field, which has a broad range of applications in science, engineering and industry. Not long ago, statistical and ad hoc methods had been main tools for studying and analyzing image processing problems. In the past decade, a new approach based on variational and partial differential equation (PDE) methods has emerged as a more powerful approach. Compared with old approaches, variational and PDE methods have remarkable advantages in both theory and computation. It allows to directly handle and process visually important geometric features such as gradients, tangents and curvatures, and to model visually meaningful …

Nikolski's Approach To The Theorems Of Beurling And Nyman Regarding Zeros Of The Riemann Ζ-Function, Jared R. Bunn

Masters Theses

In this thesis we present the proof of a theorem by Nikolai Nikolski. This theorem leads to a more general theorem by Nikolski regarding zero free regions of the Riemann ζ-function. This theorem is an improvement on the theorems that Nyman and Beurling proved in the nineteen fifties. Nikolski’s approach uses, in addition to step function approximations introduced by Nyman, distance functions to give more flexibility, including possible numerical experiments. The introduction discusses the Riemann Hypothesis, which always surrounds any study of the Riemann ζ-function.

The background material discussed in this thesis gives all the necessary prerequi- sites …

Coarse Structures And Higson Compactification, Christian Stuart Hoffland

Masters Theses

FOLLOWING John Roe in his Lectures on Coarse Geometry, we begin by describing the large-scale structure of metric spaces by means of coarse maps between them, those being maps which preserve distances at large scales. Using these techniques, we demonstrate that the real numbers and the integers have the same large scale structure--or are coarsely equivalent--but that the real line is coarsely equivalent to neither the Euclidean plane nor the set of positive real numbers. Following a generalization of these concepts for general topological spaces with the introduction of an abstract coarse structure on the space, we show, among …