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Department: Department of Mathematics
Collection: UNT Theses and Dissertations
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### Minimality of the Special Linear Groups

Date: December 1997
Creator: Hayes, Diana Margaret
Description: Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.
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### Topics in Fractal Geometry

Date: August 1994
Creator: Wang, JingLing
Description: In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.
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### Multifractal Measures

Date: May 1994
Creator: Olsen, Lars
Description: The purpose of this dissertation is to introduce a natural and unifying multifractal formalism which contains the above mentioned multifractal parameters, and gives interesting results for a large class of natural measures. In Part 2 we introduce the proposed multifractal formalism and study it properties. We also show that this multifractal formalism gives natural and interesting results when applied to (nonrandom) graph directed self-similar measures in Rd and "cookie-cutter" measures in R. In Part 3 we use the multifractal formalism introduced in Part 2 to give a detailed discussion of the multifractal structure of random (and hence, as a special case, non-random) graph directed self-similar measures in R^d.
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### Properties of Bicentric Circles for Three-Sided Polygons

Date: August 1998
Creator: Heinlein, David J. (David John)
Description: We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.
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### Aspects of Universality in Function Iteration

Date: December 1991
Creator: Taylor, John (John Allen)
Description: This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval.
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### π-regular Rings

Date: May 1993
Description: The dissertation focuses on the structure of π-regular (regular) rings.
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### Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems

Date: August 1995
Creator: Hassanpour, Mehran
Description: In this paper we study the uniqueness of positive solutions as well as the non existence of sign changing solutions for Dirichlet problems of the form \eqalign{\Delta u + g(\lambda,\ u) &= 0\quad\rm in\ \Omega,\cr u &= 0\quad\rm on\ \partial\Omega,}where $\Delta$ is the Laplace operator, $\Omega$ is a region in $\IR\sp{N}$, and $\lambda>0$ is a real parameter. For the particular function $g(\lambda,\ u)=\vert u\vert\sp{p}u+\lambda$, where $p={4\over N-2}$, and $\Omega$ is the unit ball in $\IR\sp{N}$ for $N\ge3$, we show that there are no sign changing solutions for small $\lambda$ and also we show that there are no large sign changing solutions for $\lambda$ in a compact set. We also prove uniqueness of positive solutions for $\lambda$ large when $g(\lambda,\ u)=\lambda f(u)$, where f is an increasing, sublinear, concave function with f(0) $<$ 0, and the exterior boundary of $\Omega$ is convex. In establishing our results we use a number of methods from non-linear functional analysis such as rescaling arguments, methods of order, estimation near the boundary, and moving plane arguments.
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### Generalized Function Solutions to Nonlinear Wave Equations with Distribution Initial Data

Date: August 1996
Creator: Kim, Jongchul
Description: In this study, we consider the generalized function solutions to nonlinear wave equation with distribution initial data. J. F. Colombeau shows that the initial value problem u_tt - Δu = F(u); m(x,0) = U_0; u_t (x,0) = i_1 where the initial data u_0 and u_1 are generalized functions, has a unique generalized function solution u. Here we take a specific F and specific distributions u_0, u_1 then inspect the generalized function representatives for the initial value problem solution to see if the generalized function solution is a distribution or is more singular. Using the numerical technics, we show for specific F and specific distribution initial data u_0, u_1, there is no distribution solution.
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Date: May 1993