Latest content added for Digital Library Collection: UNT Theses and Dissertationshttps://digital.library.unt.edu/explore/collections/UNTETD/browse/?fq=untl_institution:UNT&fq=str_degree_discipline:Mathematics2017-07-12T03:17:08-05:00UNT LibrariesThis is a custom feed for browsing Digital Library Collection: UNT Theses and DissertationsResults in Algebraic Determinedness and an Extension of the Baire Property2017-07-12T03:17:08-05:00https://digital.library.unt.edu/ark:/67531/metadc984214/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc984214/"><img alt="Results in Algebraic Determinedness and an Extension of the Baire Property" title="Results in Algebraic Determinedness and an Extension of the Baire Property" src="https://digital.library.unt.edu/ark:/67531/metadc984214/small/"/></a></p><p>In this work, we concern ourselves with particular topics in Polish space theory. We first consider the space A(U) of complex-analytic functions on an open set U endowed with the usual topology of uniform convergence on compact subsets. With the operations of point-wise addition and point-wise multiplication, A(U) is a Polish ring. Inspired by L. Bers' algebraic characterization of the relation of conformality, we show that the topology on A(U) is the only Polish topology for which A(U) is a Polish ring for a large class of U. This class of U includes simply connected regions, simply connected regions excluding a relatively discrete set of points, and other domains of usual interest. One thing that we deduce from this is that, even though C has many different Polish field topologies, as long as it sits inside another Polish ring with enough complex-analytic functions, it must have its usual topology. In a different direction, we show that the bounded complex-analytic functions on the unit disk admits no Polish topology for which it is a Polish ring.
We also study the Lie ring structure on A(U) which turns out to be a Polish Lie ring with the usual topology. In this case, we restrict our attention to those domains U that are connected. We extend a result of I. Amemiya to see that the Lie ring structure is determined by the conformal structure of U. In a similar vein to our ring considerations, we see that, again for certain domains U of usual interest, the Lie ring A(U) has a unique Polish topology for which it is a Polish Lie ring. Again, the Lie ring A(U) imposes topological restrictions on C. That is, C must have its usual topology when sitting inside any Polish Lie ring isomorphic to A(U).
In the last chapter, we introduce a new ideal of subsets of Polish spaces consisting of what we call residually null sets. From this ideal, we introduce an algebra consisting of what we call R-sets which is consistently a strict extension of the algebra of Baire property sets. We show that the algebra of R-sets is closed under the Alexandrov-Suslin operation and generalize Pettis' Theorem. From this, we provide new automatic continuity results and give a generalization of a result of D. Montgomery which shows that minimal assumptions on the continuity of group operations of an abstract group G with a Polish topology imply that G is actually a Polish group. We also see that many results pertaining to the algebra of Baire property sets generalize to the context of R-sets.</p>Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy2017-07-12T03:17:08-05:00https://digital.library.unt.edu/ark:/67531/metadc984121/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc984121/"><img alt="Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy" title="Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy" src="https://digital.library.unt.edu/ark:/67531/metadc984121/small/"/></a></p><p>In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.</p>A Decomposition of the Group Algebra of a Hyperoctahedral Group2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955102/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955102/"><img alt="A Decomposition of the Group Algebra of a Hyperoctahedral Group" title="A Decomposition of the Group Algebra of a Hyperoctahedral Group" src="https://digital.library.unt.edu/ark:/67531/metadc955102/small/"/></a></p><p>The descent algebra of a Coxeter group is a subalgebra of the group algebra with interesting representation theoretic properties. For instance, the natural map from the descent algebra of the symmetric group to the character ring is a surjective algebra homomorphism, so the descent algebra implicitly encodes information about the representations of the symmetric group. However, this property does not hold for other Coxeter groups. Moreover, a complete set of primitive idempotents in the descent algebra of the symmetric group leads to a decomposition of the group algebra as a direct sum of induced linear characters of centralizers of conjugacy class representatives. In this dissertation, I consider the hyperoctahedral group. When the descent algebra of a hyperoctahedral group is replaced with a generalization called the Mantaci-Reutenauer algebra, the natural map to the character ring is surjective. In 2008, Bonnafé asked whether a complete set of idempotents in the Mantaci-Reutenauer algebra could lead to a decomposition of the group algebra of the hyperoctahedral group as a direct sum of induced linear characters of centralizers. In this dissertation, I will answer this question positively and go through the construction of the idempotents, conjugacy class representatives, and linear characters required to do so.</p>Contributions to Descriptive Set Theory2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955115/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955115/"><img alt="Contributions to Descriptive Set Theory" title="Contributions to Descriptive Set Theory" src="https://digital.library.unt.edu/ark:/67531/metadc955115/small/"/></a></p><p>Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}</p>Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms2017-02-19T19:42:09-06:00https://digital.library.unt.edu/ark:/67531/metadc955117/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc955117/"><img alt="Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms" title="Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms" src="https://digital.library.unt.edu/ark:/67531/metadc955117/small/"/></a></p><p>In this thesis, we define differential operators for Hermitian Jacobi forms and Hermitian modular forms over the Gaussian number field Q(i). In particular, we construct Rankin-Cohen brackets for such spaces of Hermitian Jacobi forms and Hermitian modular forms. As an application, we extend Rankin's method to the case of Hermitian Jacobi forms. Finally we compute Fourier series coefficients of Hermitian modular forms, which allow us to give an example of the first Rankin-Cohen bracket of two Hermitian modular forms. In the appendix, we provide tables of Fourier series coefficients of Hermitian modular forms and also the computer source code that we used to compute such Fourier coefficients.</p>Quantum Drinfeld Hecke Algebras2016-08-31T22:41:47-05:00https://digital.library.unt.edu/ark:/67531/metadc862764/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc862764/"><img alt="Quantum Drinfeld Hecke Algebras" title="Quantum Drinfeld Hecke Algebras" src="https://digital.library.unt.edu/ark:/67531/metadc862764/small/"/></a></p><p>Quantum Drinfeld Hecke algebras extend both Lusztig's graded Hecke algebras and the symplectic reflection algebras of Etingof and Ginzburg to the quantum setting. A quantum (or skew) polynomial ring is generated by variables which commute only up to a set of quantum parameters. Certain finite groups may act by graded automorphisms on a quantum polynomial ring and quantum Drinfeld Hecke algebras deform the natural semi-direct product. We classify these algebras for the infinite family of complex reflection groups acting in arbitrary dimension. We also classify quantum Drinfeld Hecke algebras in arbitrary dimension for the infinite family of mystic reflection groups of Kirkman, Kuzmanovich, and Zhang, who showed they satisfy a Shephard-Todd-Chevalley theorem in the quantum setting. Using a classification of automorphisms of quantum polynomial rings in low dimension, we develop tools for studying quantum Drinfeld Hecke algebras in 3 dimensions. We describe the parameter space of such algebras using special properties of the quantum determinant in low dimension; although the quantum determinant is not a homomorphism in general, it is a homomorphism on the finite linear groups acting in dimension 3.</p>Irreducible Modules for Yokonuma-Type Hecke Algebras2016-08-31T22:41:47-05:00https://digital.library.unt.edu/ark:/67531/metadc862800/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc862800/"><img alt="Irreducible Modules for Yokonuma-Type Hecke Algebras" title="Irreducible Modules for Yokonuma-Type Hecke Algebras" src="https://digital.library.unt.edu/ark:/67531/metadc862800/small/"/></a></p><p>Yokonuma-type Hecke algebras are a class of Hecke algebras built from a Type A construction. In this thesis, I construct the irreducible representations for a class of generic Yokonuma-type Hecke algebras which specialize to group algebras of the complex reflection groups and to endomorphism rings of certain permutation characters of finite general linear groups.</p>Continuous Combinatorics of a Lattice Graph in the Cantor Space2016-06-28T16:28:55-05:00https://digital.library.unt.edu/ark:/67531/metadc849680/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc849680/"><img alt="Continuous Combinatorics of a Lattice Graph in the Cantor Space" title="Continuous Combinatorics of a Lattice Graph in the Cantor Space" src="https://digital.library.unt.edu/ark:/67531/metadc849680/small/"/></a></p><p>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.</p>The Relative Complexity of Various Classification Problems among Compact Metric Spaces2016-06-28T16:28:55-05:00https://digital.library.unt.edu/ark:/67531/metadc849626/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc849626/"><img alt="The Relative Complexity of Various Classification Problems among Compact Metric Spaces" title="The Relative Complexity of Various Classification Problems among Compact Metric Spaces" src="https://digital.library.unt.edu/ark:/67531/metadc849626/small/"/></a></p><p>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.</p>An Exploration of the Word2vec Algorithm: Creating a Vector Representation of a Language Vocabulary that Encodes Meaning and Usage Patterns in the Vector Space Structure2016-06-28T16:28:55-05:00https://digital.library.unt.edu/ark:/67531/metadc849728/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc849728/"><img alt="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" title="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" src="https://digital.library.unt.edu/ark:/67531/metadc849728/small/"/></a></p><p>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.</p>