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**Degree Discipline:**Mathematics

**Collection:**UNT Theses and Dissertations

### Continuous Combinatorics of a Lattice Graph in the Cantor Space

**Date:**May 2016

**Creator:**Krohne, Edward William

**Description:**We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

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### The Relative Complexity of Various Classification Problems among Compact Metric Spaces

**Date:**May 2016

**Creator:**Chang, Cheng

**Description:**In this thesis, we discuss three main projects which are related to Polish groups and their actions on standard Borel spaces. In the first part, we show that the complexity of the classification problem of continua is Borel bireducible to a universal orbit equivalence relation induce by a Polish group on a standard Borel space. In the second part, we compare the relative complexity of various types of classification problems concerning subspaces of [0,1]^n for all natural number n. In the last chapter, we give a topological characterization theorem for the class of locally compact two-sided invariant non-Archimedean Polish groups. Using this theorem, we show the non-existence of a universal group and the existence of a surjectively universal group in the class.

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### An Exploration of the Word2vec Algorithm: Creating a Vector Representation of a Language Vocabulary that Encodes Meaning and Usage Patterns in the Vector Space Structure

**Date:**May 2016

**Creator:**Le, Thu Anh

**Description:**This thesis is an exloration and exposition of a highly efficient shallow neural network algorithm called word2vec, which was developed by T. Mikolov et al. in order to create vector representations of a language vocabulary such that information about the meaning and usage of the vocabulary words is encoded in the vector space structure. Chapter 1 introduces natural language processing, vector representations of language vocabularies, and the word2vec algorithm. Chapter 2 reviews the basic mathematical theory of deterministic convex optimization. Chapter 3 provides background on some concepts from computer science that are used in the word2vec algorithm: Huffman trees, neural networks, and binary cross-entropy. Chapter 4 provides a detailed discussion of the word2vec algorithm itself and includes a discussion of continuous bag of words, skip-gram, hierarchical softmax, and negative sampling. Finally, Chapter 5 explores some applications of vector representations: word categorization, analogy completion, and language translation assistance.

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### Optimal Strategies for Stopping Near the Top of a Sequence

**Date:**December 2015

**Creator:**Islas Anguiano, Jose Angel

**Description:**In Chapter 1 the classical secretary problem is introduced. Chapters 2 and 3 are variations of this problem. Chapter 2, discusses the problem of maximizing the probability of stopping with one of the two highest values in a Bernoulli random walk with arbitrary parameter p and finite time horizon n. The optimal strategy (continue or stop) depends on a sequence of threshold values (critical probabilities) which has an oscillating pattern. Several properties of this sequence have been proved by Dr. Allaart. Further properties have been recently proved. In Chapter 3, a gambler will observe a finite sequence of continuous random variables. After he observes a value he must decide to stop or continue taking observations. He can play two different games A) Win at the maximum or B) Win within a proportion of the maximum. In the first section the sequence to be observed is independent. It is shown that for each n>1, theoptimal win probability in game A is bounded below by (1-1/n)^{n-1}. It is accomplished by reducing the problem to that of choosing the maximum of a special sequence of two-valued random variables and applying the sum-the-odds theorem of Bruss (2000). Secondly, it is assumed the sequence is ...

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### Reduced Ideals and Periodic Sequences in Pure Cubic Fields

**Date:**August 2015

**Creator:**Jacobs, G. Tony

**Description:**The “infrastructure” of quadratic fields is a body of theory developed by Dan Shanks, Richard Mollin and others, in which they relate “reduced ideals” in the rings and sub-rings of integers in quadratic fields with periodicity in continued fraction expansions of quadratic numbers. In this thesis, we develop cubic analogs for several infrastructure theorems. We work in the field K=Q(), where 3=m for some square-free integer m, not congruent to ±1, modulo 9. First, we generalize the definition of a reduced ideal so that it applies to K, or to any number field. Then we show that K has only finitely many reduced ideals, and provide an algorithm for listing them. Next, we define a sequence based on the number alpha that is periodic and corresponds to the finite set of reduced principal ideals in K. Using this rudimentary infrastructure, we are able to establish results about fundamental units and reduced ideals for some classes of pure cubic fields. We also introduce an application to Diophantine approximation, in which we present a 2-dimensional analog of the Lagrange value of a badly approximable number, and calculate some examples.

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### Trees and Ordinal Indices in C(k) Spaces for K Countable Compact

**Date:**August 2015

**Creator:**Dahal, Koshal Raj

**Description:**In the dissertation we study the C(K) spaces focusing on the case when K is countable compact and more specifically, the structure of C() spaces for < ω1 via special type of trees that they contain. The dissertation is composed of three major sections. In the first section we give a detailed proof of the theorem of Bessaga and Pelczynski on the isomorphic classification of C() spaces. In due time, we describe the standard bases for C(ω) and prove that the bases are monotone. In the second section we consider the lattice-trees introduced by Bourgain, Rosenthal and Schechtman in C() spaces, and define rerooting and restriction of trees. The last section is devoted to the main results. We give some lower estimates of the ordinal-indices in C(ω). We prove that if the tree in C(ω) has large order with small constant then each function in the root must have infinitely many big coordinates. Along the way we deduce some upper estimates for c0 and C(ω), and give a simple proof of Cambern's result that the Banach-Mazur distance between c0 and c = C(ω) is equal to 3.

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### Restricting Invariants and Arrangements of Finite Complex Reflection Groups

**Date:**August 2015

**Creator:**Berardinelli, Angela

**Description:**Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

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### Contributions to Descriptive Set Theory

**Date:**August 2015

**Creator:**Atmai, Rachid

**Description:**In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T2n], which are very natural canonical inner models of ZFC. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us in particular to reprove a strong partition property result on the ordinal of a Steel pointclass and derive a new boundedness principle which could be useful in the study of the cardinal structure of L(R). We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson’s proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Π12n+1 pointclasses using Jackson’s theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T2n]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T2n] are unique. This generalizes previous work of Hjorth. We then characterize the L[T2n] in term of inner models theory, showing that they actually are constructible models over direct limit of ...

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### Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation

**Date:**August 2014

**Creator:**Montgomery, Jason W.

**Description:**A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains ...

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### Hermitian Jacobi Forms and Congruences

**Date:**August 2014

**Creator:**Senadheera, Jayantha

**Description:**In this thesis, we introduce a new space of Hermitian Jacobi forms, and we determine its structure. As an application, we study heat cycles of Hermitian Jacobi forms, and we establish a criterion for the existence of U(p) congruences of Hermitian Jacobi forms. We demonstrate that criterion with some explicit examples. Finally, in the appendix we give tables of Fourier series coefficients of several Hermitian Jacobi forms.

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### A Comparison of Velocities Computed by Two-Dimensional Potential Theory and Velocities Measured in the Vicinity of an Airfoil

**Date:**June 1947

**Creator:**Copp, George

**Description:**In treating the motion of a fluid mathematically, it is convenient to make some simplifying assumptions. The assumptions which are made will be justifiable if they save long and laborious computations in practical problems, and if the predicted results agree closely enough with experimental results for practical use. In dealing with the flow of air about an airfoil, at subsonic speeds, the fluid will be considered as a homogeneous, incompressible, inviscid fluid.

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### Some Effects of the War Upon the Mathematics Curriculum and the Motivating Forces at Work as Reflected in the Dallas City Schools

**Date:**August 1945

**Creator:**Smith, R. N.

**Description:**"To discuss the effect all this war activity has had upon the Dallas Schools and to voice a protest against those who seek to discredit mathematics and at the same time to contribute a readable thesis upon the subject is largely the purpose of this study." --leaf 2

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### Absolute Continuity and the Integration of Bounded Set Functions

**Date:**May 1975

**Creator:**Allen, John Houston

**Description:**The first chapter gives basic definitions and theorems concerning set functions and set function integrals. The lemmas and theorems are presented without proof in this chapter. The second chapter deals with absolute continuity and Lipschitz condition. Particular emphasis is placed on the properties of max and min integrals. The third chapter deals with approximating absolutely continuous functions with bounded functions. It also deals with the existence of the integrals composed of various combinations of bounded functions and finitely additive functions. The concluding theorem states if the integral of the product of a bounded function and a non-negative finitely additive function exists, then the integral of the product of the bounded function with an absolutely continuous function exists over any element in a field of subsets of a set U.

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### A Partial Characterization of Upper Semi-Continuous Decompositions

**Date:**December 1973

**Creator:**Dennis, William Albert

**Description:**The goal of this paper is to characterize, at least partially, upper semi-continuous decompositions of topological spaces and the role that upper semi-continuity plays in preserving certain topological properties under decomposition mappings. Attention is also given to establishing what role upper semi-continuity plays in determining conditions under which decomposition spaces possess certain properties. A number of results for non-upper semi-continuous decompositions are included to help clarify the scope of the part upper semi-continuity plays in determining relationships between topological spaces and their decomposition spaces.

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### Proofs of Some Limit Theorems in Probability

**Date:**December 1974

**Creator:**Hwang, E-Bin

**Description:**This study gives detailed proofs of some limit theorems in probability which are important in theoretical and applied probability, The general introduction contains definitions and theorems that are basic tools of the later development. Included in this first chapter is material concerning normal distributions and characteristic functions, The second chapter introduces lower and upper bounds of the ratio of the binomial distribution to the normal distribution., Then these bound are used to prove the local Deioivre-Laplace limit theorem. The third chapter includes proofs of the central limit theorems for identically distributed and non-identically distributed random variables,

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### Valuations and Valuation Rings

**Date:**August 1975

**Creator:**Badt, Sig H.

**Description:**This paper is an investigation of several basic properties of ordered Abelian groups, valuations, the relationship between valuation rings, valuations, and their value groups and valuation rings. The proofs to all theorems stated without proof can be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1858. In Chapter I several basic theorems which are used in later proofs are stated without proof, and we prove several theorems on the structure of ordered Abelian groups, and the basic relationships between these groups, valuations, and their valuation rings in a field. In Chapter II we deal with valuation rings, and relate the structure of valuation rings to the structure of their value groups.

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### Linear Operators

**Date:**December 1975

**Creator:**Malhotra, Vijay Kumar

**Description:**This paper is a study of linear operators defined on normed linear spaces. A basic knowledge of set theory and vector spaces is assumed, and all spaces considered have real vector spaces. The first chapter is a general introduction that contains assumed definitions and theorems. Included in this chapter is material concerning linear functionals, continuity, and boundedness. The second chapter contains the proofs of three fundamental theorems of linear analysis: the Open Mapping Theorem, the Hahn-Banach Theorem, and the Uniform Boundedness Principle. The third chapter is concerned with applying some of the results established in earlier chapters. In particular, the concepts of compact operators and Schauder bases are introduced, and a proof that an operator is compact if and only if its adjoint is compact is included. This chapter concludes with a proof of an important application of the Open Mapping Theorem, namely, the Closed Graph Theorem.

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### Equivalent Sets and Cardinal Numbers

**Date:**December 1975

**Creator:**Hsueh, Shawing

**Description:**The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved.

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### The Use of Chebyshev Polynomials in Numerical Analysis

**Date:**December 1975

**Creator:**Forisha, Donnie R.

**Description:**The purpose of this paper is to investigate the nature and practical uses of Chebyshev polynomials. Chapter I gives recognition to mathematicians responsible for studies in this area. Chapter II enumerates several mathematical situations in which the polynomials naturally arise and suggests reasons for the pursuance of their study. Chapter III includes: Chebyshev polynomials as related to "best" polynomial approximation, Chebyshev series, and methods of producing polynomial approximations to continuous functions. Chapter IV discusses the use of Chebyshev polynomials to solve certain differential equations and Chebyshev-Gauss quadrature.

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### Inverse Limit Spaces

**Date:**December 1974

**Creator:**Williams, Stephen Boyd

**Description:**Inverse systems, inverse limit spaces, and bonding maps are defined. An investigation of the properties that an inverse limit space inherits, depending on the conditions placed on the factor spaces and bonding maps is made. Conditions necessary to ensure that the inverse limit space is compact, connected, locally connected, and semi-locally connected are examined. A mapping from one inverse system to another is defined and the nature of the function between the respective inverse limits, induced by this mapping, is investigated. Certain restrictions guarantee that the induced function is continuous, onto, monotone, periodic, or open. It is also shown that any compact metric space is the continuous image of the cantor set. Finally, any compact Hausdorff space is characterized as the inverse limit of an inverse system of polyhedra.

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### Properties of Some Classical Integral Domains

**Date:**May 1975

**Creator:**Crawford, Timothy B.

**Description:**Greatest common divisor domains, Bezout domains, valuation rings, and Prüfer domains are studied. Chapter One gives a brief introduction, statements of definitions, and statements of theorems without proof. In Chapter Two theorems about greatest common divisor domains and characterizations of Bezout domains, valuation rings, and Prüfer domains are proved. Also included are characterizations of a flat overring. Some of the results are that an integral domain is a Prüfer domain if and only if every overring is flat and that every overring of a Prüfer domain is a Prüfer domain.

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### Chebyshev Subsets in Smooth Normed Linear Spaces

**Date:**December 1974

**Creator:**Svrcek, Frank J.

**Description:**This paper is a study of the relation between smoothness of the norm on a normed linear space and the property that every Chebyshev subset is convex. Every normed linear space of finite dimension, having a smooth norm, has the property that every Chebyshev subset is convex. In the second chapter two properties of the norm, uniform Gateaux differentiability and uniform Frechet differentiability where the latter implies the former, are given and are shown to be equivalent to smoothness of the norm in spaces of finite dimension. In the third chapter it is shown that every reflexive normed linear space having a uniformly Gateaux differentiable norm has the property that every weakly closed Chebyshev subset, with non-empty weak interior that is norm-wise dense in the subset, is convex.

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### Topics in Category Theory

**Date:**August 1974

**Creator:**Miller, Robert Patrick

**Description:**The purpose of this paper is to examine some basic topics in category theory. A category consists of a class of mathematical objects along with a morphism class having an associative composition. The paper is divided into two chapters. Chapter I deals with intrinsic properties of categories. Various "sub-objects" and properties of morphisms are defined and examples are given. Chapter II deals with morphisms between categories called functors and the natural transformations between functors. Special types of functors are defined and examples are given.

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### Spaces of Closed Subsets of a Topological Space

**Date:**August 1974

**Creator:**Leslie, Patricia J.

**Description:**The purpose of this paper is to examine selected topologies, the Vietoris topology in particular, on S(X), the collection of nonempty, closed subsets of a topological space X. Characteristics of open and closed subsets of S(X), with the Vietoris topology, are noted. The relationships between the space X and the space S(X), with the Vietoris topology, concerning the properties of countability, compactness, and connectedness and the separation properties are investigated. Additional topologies are defined on S(X), and each is compared to the Vietoris topology on S(X). Finally, topological convergence of nets of subsets of X is considered. It is found that topological convergence induces a topology on S(X), and that this topology is the Vietoris topology on S(X) when X is a compact, Hausdorff space.

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