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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
Topics in Fractal Geometry
In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.
Date:
August 1994
Creator:
Wang, JingLing
System:
The UNT Digital Library
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 S∞.
Date:
December 1997
Creator:
Hayes, Diana Margaret
System:
The UNT Digital Library
A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities
Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1
Date:
May 2001
Creator:
May, Russell J.
System:
The UNT Digital Library
Universal Branched Coverings
In this paper, the study of k-fold branched coverings for which the branch set is a stratified set is considered. First of all, the existence of universal k-fold branched coverings over CW-complexes with stratified branch set is proved using Brown's Representability Theorem. Next, an explicit construction of universal k-fold branched coverings over manifolds is given. Finally, some homotopy and homology groups are computed for some specific examples of Universal k-fold branched coverings.
Date:
May 1993
Creator:
Tejada, Débora
System:
The UNT Digital Library
Natural Smooth Measures on the Leaves of the Unstable Manifold of Open Billiard Dynamical Systems
In this paper, we prove, for a certain class of open billiard dynamical systems, the existence of a family of smooth probability measures on the leaves of the dynamical system's unstable manifold. These measures describe the conditional asymptotic behavior of forward trajectories of the system. Furthermore, properties of these families are proven which are germane to the PYC programme for these systems. Strong sufficient conditions for the uniqueness of such families are given which depend upon geometric properties of the system's phase space. In particular, these results hold for a fairly nonrestrictive class of triangular configurations of scatterers.
Date:
December 1998
Creator:
Richardson, Peter A. (Peter Adolph), 1955-
System:
The UNT Digital Library
π-regular Rings
The dissertation focuses on the structure of π-regular (regular) rings.
Date:
May 1993
Creator:
Badawi, Ayman R.
System:
The UNT Digital Library
Polish Spaces and Analytic Sets
A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.
Date:
August 1997
Creator:
Muller, Kimberly (Kimberly Orisja)
System:
The UNT Digital Library
The Pettis Integral and Operator Theory
Access:
Use of this item is restricted to the UNT Community
Let (Ω, Σ, µ) be a finite measure space and X, a Banach space with continuous dual X*. A scalarly measurable function f: Ω→X is Dunford integrable if for each x* X*, x*f L1(µ). Define the operator Tf. X* → L1(µ) by T(x*) = x*f. Then f is Pettis integrable if and only if this operator is weak*-to-weak continuous. This paper begins with an overview of this function. Work by Robert Huff and Gunnar Stefansson on the operator Tf motivates much of this paper. Conditions that make Tf weak*-to-weak continuous are generalized to weak*-toweak continuous operators on dual spaces. For instance, if Tf is weakly compact and if there exists a separable subspace D X such that for each x* X*, x*f = x*fχDµ-a.e, then f is Pettis integrable. This nation is generalized to bounded operators T: X* → Y. To say that T is determined by D means that if x*| D = 0, then T (x*) = 0. Determining subspaces are used to help prove certain facts about operators on dual spaces. Attention is given to finding determining subspaces far a given T: X* → Y. The kernel of T and the adjoint T* of T are used …
Date:
August 2001
Creator:
Huettenmueller, Rhonda
System:
The UNT Digital Library
Three Topics in Descriptive Set Theory
This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.
Date:
May 2010
Creator:
Kieftenbeld, Vincent
System:
The UNT Digital Library
Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank
Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.
Date:
August 2013
Creator:
Dahal, Rabin
System:
The UNT Digital Library
Level Curves of the Angle Function of a Positive Definite Symmetric Matrix
Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix.
Date:
December 2009
Creator:
Bajracharya, Neeraj
System:
The UNT Digital Library
Determinacy-related Consequences on Limit Superiors
Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets.
Date:
May 2013
Creator:
Walker, Daniel
System:
The UNT Digital Library
Graev Metrics and Isometry Groups of Polish Ultrametric Spaces
This dissertation presents results about computations of Graev metrics on free groups and characterizes isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces. In Chapter 2, computations of Graev metrics are performed on free groups. One of the related results answers an open question of Van Den Dries and Gao. In Chapter 3, isometry groups of countable noncompact Heine-Borel Polish ultrametric spaces are characterized. The notion of generalized tree is defined and a correspondence between the isomorphism group of a generalized tree and the isometry group of a Heine-Borel Polish ultrametric space is established. The concept of a weak inverse limit is introduced to capture the characterization of isomorphism groups of generalized trees. In Chapter 4, partial results of isometry groups of uncountable compact ultrametric spaces are given. It turns out that every compact ultrametric space has a unique countable orbital decomposition. An orbital space consists of disjoint orbits. An orbit subspace of an orbital space is actually a compact homogeneous ultrametric subspace.
Date:
May 2013
Creator:
Shi, Xiaohui
System:
The UNT Digital Library
Traveling Wave Solutions of the Porous Medium Equation
We prove the existence of a one-parameter family of solutions of the porous medium equation, a nonlinear heat equation. In our work, with space dimension 3, the interface is a half line whose end point advances at constant speed. We prove, by using maximum principle, that the solutions are stable under a suitable class of perturbations. We discuss the relevance of our solutions, when restricted to two dimensions, to gravity driven flows of thin films. Here we extend the results of J. Iaia and S. Betelu in the paper "Solutions of the porous medium equation with degenerate interfaces" to a higher dimension.
Date:
May 2013
Creator:
Paudel, Laxmi P.
System:
The UNT Digital Library
Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials
Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in …
Date:
August 2012
Creator:
Akter, Hasina
System:
The UNT Digital Library
Descriptive Set Theory and Measure Theory in Locally Compact and Non-locally Compact Groups
In this thesis we study descriptive-set-theoretic and measure-theoretic properties of Polish groups, with a thematic emphasis on the contrast between groups which are locally compact and those which are not. The work is divided into three major sections. In the first, working jointly with Robert Kallman, we resolve a conjecture of Gleason regarding the Polish topologization of abstract groups of homeomorphisms. We show that Gleason's conjecture is false, and its conclusion is only true when the hypotheses are considerably strengthened. Along the way we discover a new automatic continuity result for a class of functions which behave like but are distinct from functions of Baire class 1. In the second section we consider the descriptive complexity of those subsets of the permutation group S? which arise naturally from the classical Levy-Steinitz series rearrangement theorem. We show that for any conditionally convergent series of vectors in Euclidean space, the sets of permutations which make the series diverge, and diverge properly, are ?03-complete. In the last section we study the phenomenon of Haar null sets a la Christensen, and the closely related notion of openly Haar null sets. We identify and correct a minor error in the proof of Mycielski that a …
Date:
May 2013
Creator:
Cohen, Michael Patrick
System:
The UNT Digital Library
Topological uniqueness results for the special linear and other classical Lie Algebras.
Access:
Use of this item is restricted to the UNT Community
Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with non-discrete topology. The only local fields (up to isomorphism) are the real, complex, and p-adic numbers, finite extensions of the p-adic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, …
Date:
December 2001
Creator:
Rees, Michael K.
System:
The UNT Digital Library
Quantization Dimension for Probability Definitions
Access:
Use of this item is restricted to the UNT Community
The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well as some basic facts. We develop a generalized framework for the quantization dimension which extends the current theory to include a wider range of probability measures. This framework uses the theory of thermodynamic formalism and the multifractal spectrum. It is shown that at least in certain cases the quantization dimension function D(r)=Dr is a transform of the temperature function b(q), which is already known to be the Legendre transform of the multifractal spectrum f(a). Hence, these ideas are all closely related and it would be expected that progress in one area could lead to new results in another. It would …
Date:
December 2001
Creator:
Lindsay, Larry J.
System:
The UNT Digital Library
Hyperspace Topologies
In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.
Date:
August 2001
Creator:
Freeman, Jeannette Broad
System:
The UNT Digital Library
Dimensions in Random Constructions.
We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.
Date:
May 2002
Creator:
Berlinkov, Artemi
System:
The UNT Digital Library
Around the Fibonacci Numeration System
Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we give a formula for R(n) in terms of binomial coefficients modulo two.
Date:
May 2007
Creator:
Edson, Marcia Ruth
System:
The UNT Digital Library
Borel Determinacy and Metamathematics
Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.
Date:
December 2001
Creator:
Bryant, Ross
System:
The UNT Digital Library
Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis
A "complex" system typically has a relatively large number of dynamically interacting components and tends to exhibit emergent behavior that cannot be explained by analyzing each component separately. A biological neural network is one example of such a system. A multi-agent model of such a network is developed to study the relationships between a network's structure and its spike train output. Using this model, inferences are made about the synaptic structure of networks through cluster analysis of spike train summary statistics A complexity measure for the network structure is also presented which has a one-to-one correspondence with the standard time series complexity measure sample entropy.
Date:
May 2007
Creator:
Brooks, Evan
System:
The UNT Digital Library