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Uniqueness of Positive Solutions for Elliptic Dirichlet Problems (open access)

Uniqueness of Positive Solutions for Elliptic Dirichlet Problems

In this paper we consider the question of uniqueness of positive solutions for Dirichlet problems of the form - ฮ” u(x)= g(ฮป,u(x)) in B, u(x) = 0 on ฯ‘B, where A is the Laplace operator, B is the unit ball in Rห†N, and A>0. We show that if g(ฮป,u)=uห†(N+2)/(N-2) + ฮป, that is g has "critical growth", then large positive solutions are unique. We also prove uniqueness of large solutions when g(ฮป,u)=A f(u) with f(0) < 0, f "superlinear" and monotone. We use a number of methods from nonlinear functional analysis such as variational identities, Sturm comparison theorems and methods of order. We also present a regularity result on linear elliptic equation where a coefficient has critical growth.
Date: December 1990
Creator: Ali, Ismail, 1961-
System: The UNT Digital Library
ฯ€-regular Rings (open access)

ฯ€-regular Rings

The dissertation focuses on the structure of ฯ€-regular (regular) rings.
Date: May 1993
Creator: Badawi, Ayman R.
System: The UNT Digital Library
Weakly Dense Subsets of Homogeneous Complete Boolean Algebras (open access)

Weakly Dense Subsets of Homogeneous Complete Boolean Algebras

The primary result from this dissertation is following inequality: d(B) โ‰ค min(2^< wd(B),sup{ฮป^c(B): ฮป < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if XโŠ‚B is ฮบโ€”weakly dense for 1 < ฮบ < sat(B), then sup{wd_ฮบ(B):ฮบ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b โ‚ฌ B\{0} such that {xโˆ—b: x โˆˆ X} is dense in B|b = {c โ‚ฌ B: c โ‰ค b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), โ€ฆ
Date: August 1990
Creator: Bozeman, Alan Kyle
System: The UNT Digital Library
The Torus Does Not Have a Hyperbolic Structure (open access)

The Torus Does Not Have a Hyperbolic Structure

Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of automorphisms on the covering space is equivalent to the original surface.
Date: August 1992
Creator: Butler, Joe R.
System: The UNT Digital Library
Multifractal Analysis of Parabolic Rational Maps (open access)

Multifractal Analysis of Parabolic Rational Maps

The investigation of the multifractal spectrum of the equilibrium measure for a parabolic rational map with a Lipschitz continuous potential, ฯ†, which satisfies sup ฯ† < P(ฯ†) xโˆˆJ(T) is conducted. More specifically, the multifractal spectrum or spectrum of singularities, f(ฮฑ) is studied.
Date: August 1998
Creator: Byrne, Jesse William
System: The UNT Digital Library
The Use of the Power Method to Find Dominant Eigenvalues of Matrices (open access)

The Use of the Power Method to Find Dominant Eigenvalues of Matrices

This paper is the result of a study of the power method to find dominant eigenvalues of square matrices. It introduces ideas basic to the study and shows the development of the power method for the most well-behaved matrices possible, and it explores exactly which other types of matrices yield to the power method. The paper also discusses a type of matrix typically considered impossible for the power method, along with a modification of the power method which works for this type of matrix. It gives an overview of common extensions of the power method. The appendices contain BASIC versions of the power method and its modification.
Date: July 1992
Creator: Cavender, Terri A.
System: The UNT Digital Library
Duals and Reflexivity of Certain Banach Spaces (open access)

Duals and Reflexivity of Certain Banach Spaces

The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.
Date: August 1991
Creator: Dahler, Cheryl L. (Cheryl Lewis)
System: The UNT Digital Library
Convergence of Conditional Expectation Operators and the Compact Range Property (open access)

Convergence of Conditional Expectation Operators and the Compact Range Property

The interplay between generalizations of Riezs' famous representation theorem and Radon-Nikodรฝm type theorems has a long history. This paper will explore certain aspects of the theory of bounded linear operators on continuous function spaces, Radon-Nikodรฝm type properties, and their connections.
Date: August 1992
Creator: Dawson, C. Bryan (Charles Bryan)
System: The UNT Digital Library
Concerning Integral Approximations of Bounded Finitely Additive Set Functions (open access)

Concerning Integral Approximations of Bounded Finitely Additive Set Functions

The purpose of this paper is to generalize a theorem that characterizes absolute continuity of bounded finitely additive set functions in the form of an integral approximation. We show that his integral exists if the condition of absolute continuity is removed.
Date: August 1992
Creator: Dawson, Dan Paul
System: The UNT Digital Library
Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation (open access)

Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ยฑ1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.
Date: August 1998
Creator: Debrecht, Johanna M.
System: The UNT Digital Library
รdรกm's Conjecture and Its Generalizations (open access)

รdรกm's Conjecture and Its Generalizations

This paper examines idam's conjuecture and some of its generalizations. In terms of Adam's conjecture, we prove Alspach and Parson's results f or Zpq and ZP2. More generally, we prove Babai's characterization of the CI-property, Palfy's characterization of CI-groups, and Brand's result for Zpr for polynomial isomorphism's. We also prove for the first time a characterization of the CI-property for 1 SG, and prove that Zn is a CI-Pn-group where Pn is the group of permutation polynomials on Z,, and n is square free.
Date: August 1990
Creator: Dobson, Edward T. (Edward Tauscher)
System: The UNT Digital Library
Overrings of an Integral Domain (open access)

Overrings of an Integral Domain

This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a ฯ€-ideal, the properties of a domain which have the property that each ideal is a finite product of ฯ€-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of ฯ€-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of ฯ€-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.
Date: August 1992
Creator: Emerson, Sharon Sue
System: The UNT Digital Library
Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains (open access)

Existence of Many Sign Changing Non Radial Solutions for Semilinear Elliptic Problems on Annular Domains

The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.
Date: August 1998
Creator: Finan, Marcel Basil
System: The UNT Digital Library
Existence and Multiplicity of Solutions for Semilinear Elliptic Boundary Value Problems (open access)

Existence and Multiplicity of Solutions for Semilinear Elliptic Boundary Value Problems

This thesis studies the existence, multiplicity, bifurcation and the stability of the solutions to semilinear elliptic boundary value problems. These problems are motivated both by the mathematical structure and the numerous applications in fluid mechanics chemical reactions, nuclear reactors, Riemannian geometry and elasticity theory. This study considers the problem for different classes of nonlinearities and obtain the existence and multiplicity of positive solutions.
Date: August 1992
Creator: Gadam, Sudhasree
System: The UNT Digital Library
Using Steepest Descent to Find Energy-Minimizing Maps Satisfying Nonlinear Constraints (open access)

Using Steepest Descent to Find Energy-Minimizing Maps Satisfying Nonlinear Constraints

The method of steepest descent is applied to a nonlinearly constrained optimization problem which arises in the study of liquid crystals. Let ฮฉ denote the region bounded by two coaxial cylinders of height 1 with the outer cylinder having radius 1 and the inner having radius ฯ. The problem is to find a mapping, u, from ฮฉ into R^3 which agrees with a given function v on the surfaces of the cylinders and minimizes the energy function over the set of functions in the Sobolev space H^(1,2)(ฮฉ; R^3) having norm 1 almost everywhere. In the variational formulation, the norm 1 condition is emulated by a constraint function B. The direction of descent studied here is given by a projected gradient, called a B-gradient, which involves the projection of a Sobolev gradient onto the tangent space for B. A numerical implementation of the algorithm, the results of which agree with the theoretical results and which is independent of any strong properties of the domain, is described. In chapter 2, the Sobolev space setting and a significant projection in the theory of Sobolev gradients are discussed. The variational formulation is introduced in Chapter 3, where the issues of differentiability and existence of โ€ฆ
Date: August 1994
Creator: Garza, Javier, 1965-
System: The UNT Digital Library
Manifolds, Vector Bundles, and Stiefel-Whitney Classes (open access)

Manifolds, Vector Bundles, and Stiefel-Whitney Classes

The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.
Date: August 1990
Creator: Green, Michael Douglas, 1965-
System: The UNT Digital Library
Algebraic Number Fields (open access)

Algebraic Number Fields

This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
Date: August 1991
Creator: Hartsell, Melanie Lynne
System: The UNT Digital Library
Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems (open access)

Sufficient Conditions for Uniqueness of Positive Solutions and Non Existence of Sign Changing Solutions for Elliptic Dirichlet Problems

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.
Date: August 1995
Creator: Hassanpour, Mehran
System: The UNT Digital Library
Minimality of the Special Linear Groups (open access)

Minimality of the Special Linear Groups

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 Sโˆž.
Date: December 1997
Creator: Hayes, Diana Margaret
System: The UNT Digital Library
Properties of Bicentric Circles for Three-Sided Polygons (open access)

Properties of Bicentric Circles for Three-Sided Polygons

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.
Date: August 1998
Creator: Heinlein, David J. (David John)
System: The UNT Digital Library
Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices (open access)

Countable Additivity, Exhaustivity, and the Structure of Certain Banach Lattices

The notion of uniform countable additivity or uniform absolute continuity is present implicitly in the Lebesgue Dominated Convergence Theorem and explicitly in the Vitali-Hahn-Saks and Nikodym Theorems, respectively. V. M. Dubrovsky studied the connection between uniform countable additivity and uniform absolute continuity in a series of papers, and Bartle, Dunford, and Schwartz established a close relationship between uniform countable additivity in ca(ฮฃ) and operator theory for the classical continuous function spaces C(K). Numerous authors have worked extensively on extending and generalizing the theorems of the preceding authors. Specifically, we mention Bilyeu and Lewis as well as Brooks and Drewnowski, whose efforts molded the direction and focus of this paper. This paper is a study of the techniques used by Bell, Bilyeu, and Lewis in their paper on uniform exhaustivity and Banach lattices to present a Banach lattice version of two important and powerful results in measure theory by Brooks and Drewnowski. In showing that the notions of exhaustivity and continuity take on familiar forms in certain Banach lattices of measures they show that these important measure theory results follow as corollaries of the generalized Banach lattice versions. This work uses their template to generalize results established by Bator, Bilyeu, and โ€ฆ
Date: August 1999
Creator: Huff, Cheryl Rae
System: The UNT Digital Library
A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema (open access)

A Continuous, Nowhere-Differentiable Function with a Dense Set of Proper Local Extrema

In this paper, we use the following scheme to construct a continuous, nowhere-differentiable function ๐‘“ which is the uniform limit of a sequence of sawtooth functions ๐‘“โ‚™ : [0, 1] โ†’ [0, 1] with increasingly sharp teeth. Let ๐‘‹ = [0, 1] x [0, 1] and ๐น(๐‘‹) be the Hausdorff metric space determined by ๐‘‹. We define contraction maps ๐‘คโ‚ , ๐‘คโ‚‚ , ๐‘คโ‚ƒ on ๐‘‹. These maps define a contraction map ๐‘ค on ๐น(๐‘‹) via ๐‘ค(๐ด) = ๐‘คโ‚(๐ด) โ‹ƒ ๐‘คโ‚‚(๐ด) โ‹ƒ ๐‘คโ‚ƒ(๐ด). The iteration under ๐‘ค of the diagonal in ๐‘‹ defines a sequence of graphs of continuous functions ๐‘“โ‚™. Since ๐‘ค is a contraction map in the compact metric space ๐น(๐‘‹), ๐‘ค has a unique fixed point. Hence, these iterations converge to the fixed point-which turns out to be the graph of our continuous, nowhere-differentiable function ๐‘“. Chapter 2 contains the background we will need to engage our task. Chapter 3 includes two results from the Baire Category Theorem. The first is the well known fact that the set of continuous, nowhere-differentiable functions on [0,1] is a residual set in ๐ถ[0,1]. The second fact is that the set of continuous functions on [0,1] which have a dense set โ€ฆ
Date: December 1993
Creator: Huggins, Mark C. (Mark Christopher)
System: The UNT Digital Library
Weak and Norm Convergence of Sequences in Banach Spaces (open access)

Weak and Norm Convergence of Sequences in Banach Spaces

We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in โ„“ยน they coincide. We then show a theorem of Rosenthal's which states that if {๐“โ‚™} is a bounded sequence in a Banach space, then {๐“โ‚™} has a subsequence {๐“'โ‚™} satisfying one of the following two mutually exclusive alternatives; (i) {๐“'โ‚™} is weakly Cauchy, or (ii) {๐“'โ‚™} is equivalent to the unit vector basis of โ„“ยน.
Date: December 1993
Creator: Hymel, Arthur J. (Arthur Joseph)
System: The UNT Digital Library
Primitive Substitutive Numbers are Closed under Rational Multiplication (open access)

Primitive Substitutive Numbers are Closed under Rational Multiplication

Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, ฮด, ฯ†) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers ฮฑ whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
Date: August 1998
Creator: Ketkar, Pallavi S. (Pallavi Subhash)
System: The UNT Digital Library