Latest content added for UNT Digital Library Partner: UNT Librarieshttp://digital.library.unt.edu/explore/partners/UNT/browse/?fq=str_degree_discipline:Mathematics&fq=dc_type:text_etd2016-06-28T16:28:55-05:00UNT LibrariesThis is a custom feed for browsing UNT Digital Library Partner: UNT LibrariesContinuous Combinatorics of a Lattice Graph in the Cantor Space2016-06-28T16:28:55-05:00http://digital.library.unt.edu/ark:/67531/metadc849680/<p><a href="http://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="http://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 Spaces2016-06-28T16:28:55-05:00http://digital.library.unt.edu/ark:/67531/metadc849626/<p><a href="http://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="http://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 Structure2016-06-28T16:28:55-05:00http://digital.library.unt.edu/ark:/67531/metadc849728/<p><a href="http://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="http://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>Optimal Strategies for Stopping Near the Top of a Sequence2016-03-20T10:34:12-05:00http://digital.library.unt.edu/ark:/67531/metadc822812/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc822812/"><img alt="Optimal Strategies for Stopping Near the Top of a Sequence" title="Optimal Strategies for Stopping Near the Top of a Sequence" src="http://digital.library.unt.edu/ark:/67531/metadc822812/small/"/></a></p><p>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 i.i.d. The best lower bounds are provided for the winning probabilities in game B given any continuous distribution. These bounds are the optimal win probabilities of a game A which was examined by Gilbert and Mosteller (1966).</p>Reduced Ideals and Periodic Sequences in Pure Cubic Fields2016-03-04T16:14:01-06:00http://digital.library.unt.edu/ark:/67531/metadc804842/<p><a href="http://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="http://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>Trees and Ordinal Indices in C(k) Spaces for K Countable Compact2016-03-04T16:14:01-06:00http://digital.library.unt.edu/ark:/67531/metadc804883/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc804883/"><img alt="Trees and Ordinal Indices in C(k) Spaces for K Countable Compact" title="Trees and Ordinal Indices in C(k) Spaces for K Countable Compact" src="http://digital.library.unt.edu/ark:/67531/metadc804883/small/"/></a></p><p>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.</p>Restricting Invariants and Arrangements of Finite Complex Reflection Groups2016-03-04T16:14:01-06:00http://digital.library.unt.edu/ark:/67531/metadc804919/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc804919/"><img alt="Restricting Invariants and Arrangements of Finite Complex Reflection Groups" title="Restricting Invariants and Arrangements of Finite Complex Reflection Groups" src="http://digital.library.unt.edu/ark:/67531/metadc804919/small/"/></a></p><p>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.</p>Contributions to Descriptive Set Theory2016-03-04T16:14:01-06:00http://digital.library.unt.edu/ark:/67531/metadc804953/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc804953/"><img alt="Contributions to Descriptive Set Theory" title="Contributions to Descriptive Set Theory" src="http://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>Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation2015-08-21T05:42:39-05:00http://digital.library.unt.edu/ark:/67531/metadc699977/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc699977/"><img alt="Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation" title="Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation" src="http://digital.library.unt.edu/ark:/67531/metadc699977/small/"/></a></p><p>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 of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.</p>Hermitian Jacobi Forms and Congruences2015-08-21T05:42:39-05:00http://digital.library.unt.edu/ark:/67531/metadc700083/<p><a href="http://digital.library.unt.edu/ark:/67531/metadc700083/"><img alt="Hermitian Jacobi Forms and Congruences" title="Hermitian Jacobi Forms and Congruences" src="http://digital.library.unt.edu/ark:/67531/metadc700083/small/"/></a></p><p>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.</p>