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On the Cohomology of the Complement of a Toral Arrangement
The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.
Date:
August 1999
Creator:
Sawyer, Cameron Cunningham
System:
The UNT Digital Library
The Computation of Ultrapowers by Supercompactness Measures
The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.
Date:
August 1999
Creator:
Smith, John C.
System:
The UNT Digital Library
Examples and Applications of Infinite Iterated Function Systems
The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.
Date:
August 2000
Creator:
Hanus, Pawel Grzegorz
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
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
Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World
Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.
Date:
August 2002
Creator:
Arthur, Christopher
System:
The UNT Digital Library
Complemented Subspaces of Bounded Linear Operators
For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator between Banach spaces X and Y , K(X, Y ) be the space of all compact operators, and W(X, Y ) be the space of all weakly compact operators. We denote space all operator ideals by O.
Date:
August 2003
Creator:
Bahreini Esfahani, Manijeh
System:
The UNT Digital Library
The Study of Translation Equivalence on Integer Lattices
This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.
Date:
August 2003
Creator:
Boykin, Charles Martin
System:
The UNT Digital Library
Thermodynamical Formalism
Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
Date:
August 2004
Creator:
Chousionis, Vasileios
System:
The UNT Digital Library
Lyapunov Exponents, Entropy and Dimension
We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.
Date:
August 2004
Creator:
Williams, Jeremy M.
System:
The UNT Digital Library
A Computation of Partial Isomorphism Rank on Ordinal Structures
We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form.
Date:
August 2006
Creator:
Bryant, Ross
System:
The UNT Digital Library
Characterizations of Continua of Finite Degree
In this thesis, some characterizations of continua of finite degree are given. It turns out that being of finite degree (by formal definition) can be described by saying there exists an equivalent metric in which Hausdorff linear measure of the continuum is finite. I discuss this result in detail.
Date:
August 2006
Creator:
Irwin, Shana
System:
The UNT Digital Library
A Characterization of Homeomorphic Bernoulli Trial Measures.
We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space.
Date:
August 2006
Creator:
Yingst, Andrew Q.
System:
The UNT Digital Library
Hyperbolic Monge-Ampère Equation
Access:
Use of this item is restricted to the UNT Community
In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
Date:
August 2006
Creator:
Howard, Tamani M.
System:
The UNT Digital Library
A Constructive Method for Finding Critical Point of the Ginzburg-Landau Energy Functional
In this work I present a constructive method for finding critical points of the Ginzburg-Landau energy functional using the method of Sobolev gradients. I give a description of the construction of the Sobolev gradient and obtain convergence results for continuous steepest descent with this gradient. I study the Ginzburg-Landau functional with magnetic field and the Ginzburg-Landau functional without magnetic field. I then present the numerical results I obtained by using steepest descent with the discretized Sobolev gradient.
Date:
August 2008
Creator:
Kazemi, Parimah
System:
The UNT Digital Library
Spaces of operators containing co and/or l ∞ with an application of vector measures.
The Banach spaces L(X, Y), K(X, Y), Lw*(X*, Y), and Kw*(X*, Y) are studied to determine when they contain the classical Banach spaces co or l ∞. The complementation of the Banach space K(X, Y) in L(X, Y) is discussed as well as what impact this complementation has on the embedding of co or l∞ in K(X, Y) or L(X, Y). Results concerning the complementation of the Banach space Kw*(X*, Y) in Lw*(X*, Y) are also explored and how that complementation affects the embedding of co or l ∞ in Kw*(X*, Y) or Lw*(X*, Y). The l p spaces for 1 ≤ p < ∞ are studied to determine when the space of compact operators from one l p space to another contains co. The paper contains a new result which classifies these spaces of operators. Results of Kalton, Feder, and Emmanuele concerning the complementation of K(X, Y) in L(X, Y) are generalized. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis as well as a new proof of the fact that l ∞ is prime.
Date:
August 2008
Creator:
Schulle, Polly Jane
System:
The UNT Digital Library
On the density of minimal free subflows of general symbolic flows.
This paper studies symbolic dynamical systems {0, 1}G, where G is a countably infinite group, {0, 1}G has the product topology, and G acts on {0, 1}G by shifts. It is proven that for every countably infinite group G the union of the minimal free subflows of {0, 1}G is dense. In fact, a stronger result is obtained which states that if G is a countably infinite group and U is an open subset of {0, 1}G, then there is a collection of size continuum consisting of pairwise disjoint minimal free subflows intersecting U.
Date:
August 2009
Creator:
Seward, Brandon Michael
System:
The UNT Digital Library
Algebraically Determined Rings of Functions
Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.
Date:
August 2010
Creator:
McLinden, Alexander Patrick
System:
The UNT Digital Library
Uniformly σ-Finite Disintegrations of Measures
A disintegration of measure is a common tool used in ergodic theory, probability, and descriptive set theory. The primary interest in this paper is in disintegrating σ-finite measures on standard Borel spaces into families of σ-finite measures. In 1984, Dorothy Maharam asked whether every such disintegration is uniformly σ-finite meaning that there exists a countable collection of Borel sets which simultaneously witnesses that every measure in the disintegration is σ-finite. Assuming Gödel’s axiom of constructability I provide answer Maharam's question by constructing a specific disintegration which is not uniformly σ-finite.
Date:
August 2011
Creator:
Backs, Karl
System:
The UNT Digital Library
Mycielski-Regular Measures
Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.
Date:
August 2011
Creator:
Bass, Jeremiah Joseph
System:
The UNT Digital Library
Strong Choquet Topologies on the Closed Linear Subspaces of Banach Spaces
In the study of Banach spaces, the development of some key properties require studying topologies on the collection of closed convex subsets of the space. The subcollection of closed linear subspaces is studied under the relative slice topology, as well as a class of topologies similar thereto. It is shown that the collection of closed linear subspaces under the slice topology is homeomorphic to the collection of their respective intersections with the closed unit ball, under the natural mapping. It is further shown that this collection under any topology in the aforementioned class of similar topologies is a strong Choquet space. Finally, a collection of category results are developed since strong Choquet spaces are also Baire spaces.
Date:
August 2011
Creator:
Farmer, Matthew Ray
System:
The UNT Digital Library
A Relation for Point Sets in a Topological Space
The purpose of this thesis is to investigate the relation Z for point sets in a topological space. There were two original goals which caused the study.
Date:
August 1962
Creator:
Warndof, Joseph C.
System:
The UNT Digital Library
Linear Transformations in Linear Spaces
This thesis is a study of linear spaces and linear transformations in normed linear spaces. The notion of a field, in particular the complex number field, is assumed in this paper.
Date:
August 1962
Creator:
Westley, Kent N.
System:
The UNT Digital Library
Euclidean N-space
This study of the Euclidean N-space looks at some definitions and their characteristics, some comparisons, boundedness and compactness, and transformations and mappings.
Date:
August 1962
Creator:
Horner, Donald R.
System:
The UNT Digital Library