Latest content added for Digital Library Partner: UNT Librarieshttps://digital.library.unt.edu/explore/partners/UNT/browse/?fq=str_degree_department:Department+of+Mathematics2017-02-19T19:42:09-06:00UNT LibrariesThis is a custom feed for browsing Digital Library Partner: UNT LibrariesA 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>Reduced Ideals and Periodic Sequences in Pure Cubic Fields2016-03-04T16:14:01-06:00https://digital.library.unt.edu/ark:/67531/metadc804842/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc804842/"><img alt="Reduced Ideals and Periodic Sequences in Pure Cubic Fields" title="Reduced Ideals and Periodic Sequences in Pure Cubic Fields" src="https://digital.library.unt.edu/ark:/67531/metadc804842/small/"/></a></p><p>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.</p>Contributions to Descriptive Set Theory2016-03-04T16:14:01-06:00https://digital.library.unt.edu/ark:/67531/metadc804953/<p><a href="https://digital.library.unt.edu/ark:/67531/metadc804953/"><img alt="Contributions to Descriptive Set Theory" title="Contributions to Descriptive Set Theory" src="https://digital.library.unt.edu/ark:/67531/metadc804953/small/"/></a></p><p>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 mice with Woodin cardinals, a counterpart to Steel’s result that the L[T2n+1] are extender models, and finally show that the generalized contiuum hypothesis holds in these models, solving a conjecture of Woodin.</p>