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Results on Non-Club Isomorphic Aronszajn Trees

Description: In this dissertation we prove some results about the existence of families of Aronszajn trees on successors of regular cardinals which are pairwise not club isomorphic. The history of this topic begins with a theorem of Gaifman and Specker in the 1960s which asserts the existence from ZFC of many pairwise not isomorphic Aronszajn trees. Since that result was proven, the focus has turned to comparing Aronszajn trees with respect to isomorphisms on a club of levels, instead of on the entire treā€¦ more
Date: August 2020
Creator: Chavez, Jose
Partner: UNT Libraries
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Determinacy of Schmidt's Game and Other Intersection Games

Description: Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our mā€¦ more
Date: May 2020
Creator: Crone, Logan
Partner: UNT Libraries
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Invariants of Polynomials Modulo Frobenius Powers

Description: Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-ā€¦ more
Date: May 2020
Creator: Drescher, Chelsea
Partner: UNT Libraries
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Winning Sets and the Banach-Mazur-McMullen Game

Description: For decades, mathematical games have been used to explore various properties of particular sets. The Banach-Mazur game is the prototypical intersection game and its modifications by e.g., W. Schmidt and C. McMullen are used in number theory and many other areas of mathematics. We give a brief survey of a few of these modifications and their properties followed by our own modification. One of our main results is proving that this modification is equivalent to an important set theoretic game, ā€¦ more
Date: May 2020
Creator: Ragland, Robin
Partner: UNT Libraries
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Applications of a Model-Theoretic Approach to Borel Equivalence Relations

Description: The study of Borel equivalence relations on Polish spaces has become a major area of focus within descriptive set theory. Primarily, work in this area has been carried out using the standard methods of descriptive set theory. In this work, however, we develop a model-theoretic framework suitable for the study of Borel equivalence relations, introducing a class of objects we call Borel structurings. We then use these structurings to examine conditions under which marker sets for Borel equivalā€¦ more
Date: August 2019
Creator: Craft, Colin N.
Partner: UNT Libraries

A Global Spatial Model for Loop Pattern Fingerprints and Its Spectral Analysis

Description: The use of fingerprints for personal identification has been around for thousands of years (first established in ancient China and India). Fingerprint identification is based on two basic premises that the fingerprint is unique to an individual and the basic characteristics such as ridge pattern do not change over time. Despite extensive research, there are still mathematical challenges in characterization of fingerprints, matching and compression. We develop a new mathematical model in the spaā€¦ more
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Date: August 2019
Creator: Wu, Di
Partner: UNT Libraries

A Novel Two-Stage Adaptive Method for Estimating Large Covariance and Precision Matrices

Description: Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample cā€¦ more
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Date: August 2019
Creator: Rajendran, Rajanikanth
Partner: UNT Libraries
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Prophet Inequalities for Multivariate Random Variables with Cost for Observations

Description: In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a bā€¦ more
Date: August 2019
Creator: Brophy, Edmond M.
Partner: UNT Libraries
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Abelian Group Actions and Hypersmooth Equivalence Relations

Description: We show that any Borel action on a standard Borel space of a group which is topologically isomorphic to the sum of a countable abelian group with a countable sum of lines and circles induces an orbit equivalence relation which is hypersmooth. We also show that any Borel action of a second countable locally compact abelian group on a standard Borel space induces an orbit equivalence relation which is essentially hyperfinite, generalizing a result of Gao and Jackson for the countable abelian grouā€¦ more
Date: May 2019
Creator: Cotton, Michael R.
Partner: UNT Libraries
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Annihilators of Bounded Indecomposable Modules of Vec(R)

Description: The Lie algebra Vec(ā„) of polynomial vector fields on the line acts naturally on ā„‚[]. This action has a one-parameter family of deformations called the tensor density modules F_Ī». The bounded indecomposable modules of Vec(ā„) of length 2 composed of tensor density modules have been classified by Feigin and Fuchs. We present progress towards describing the annihilators of the unique indecomposable extension of F_Ī» by F_(Ī»+2) in the non-resonant case Ī» ā‰  -Ā½. We give the intersection of the annihilā€¦ more
Date: May 2019
Creator: Kenefake, Tyler Christian
Partner: UNT Libraries
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Equivalence of the Rothberger and k-Rothberger Games for Hausdorff Spaces

Description: First, we show that the Rothberger and 2-Rothberger games are equivalent. Then we adjust the former proof and introduce another game, the restricted Menger game, in order to obtain a broader result. This provides an answer in the context of Hausdorff spaces for an open question posed by Aurichi, Bella, and Dias.
Date: May 2019
Creator: Hiers, Nathaniel Christopher
Partner: UNT Libraries
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Infinitary Combinatorics and the Spreading Models of Banach Spaces

Description: Spreading models have become fundamental to the study of asymptotic geometry in Banach spaces. The existence of spreading models in every Banach space, and the so-called good sequences which generate them, was one of the first applications of Ramsey theory in Banach space theory. We use Ramsey theory and other techniques from infinitary combinatorics to examine some old and new questions concerning spreading models and good sequences. First, we consider the lp spreading model problem which asksā€¦ more
Date: May 2019
Creator: Krause, Cory A.
Partner: UNT Libraries
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A Random Walk Version of Robbins' Problem

Description: Robbins' problem is an optimal stopping problem where one seeks to minimize the expected rank of their observations among all observations. We examine random walk analogs to Robbins' problem in both discrete and continuous time. In discrete time, we consider full information and relative ranks versions of this problem. For three step walks, we give the optimal stopping rule and the expected rank for both versions. We also give asymptotic upper bounds for the expected rank in discrete time. Finaā€¦ more
Date: December 2018
Creator: Allen, Andrew
Partner: UNT Libraries
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Conformal and Stochastic Non-Autonomous Dynamical Systems

Description: In this dissertation we focus on the application of thermodynamic formalism to non-autonomous and random dynamical systems. Specifically we use the thermodynamic formalism to investigate the dimension of various fractal constructions via the, now standard, technique of Bowen which he developed in his 1979 paper on quasi-Fuchsian groups. Bowen showed, roughly speaking, that the dimension of a fractal is equal to the zero of the relevant topological pressure function. We generalize the results ofā€¦ more
Date: August 2018
Creator: Atnip, Jason
Partner: UNT Libraries
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Hausdorff Dimension of Shrinking-Target Sets Under Non-Autonomous Systems

Description: For a dynamical system on a metric space a shrinking-target set consists of those points whose orbit hit a given ball of shrinking radius infinitely often. Historically such sets originate in Diophantine approximation, in which case they describe the set of well-approximable numbers. One aspect of such sets that is often studied is their Hausdorff dimension. We will show that an analogue of Bowen's dimension formula holds for such sets when they are generated by conformal non-autonomous iterateā€¦ more
Date: August 2018
Creator: Lopez, Marco Antonio
Partner: UNT Libraries
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Infinitely Many Solutions of Semilinear Equations on Exterior Domains

Description: We prove the existence and nonexistence of solutions for the semilinear problem āˆ†u + K(r)f(u) = 0 with various boundary conditions on the exterior of the ball in R^N such that lim rā†’āˆžu(r) = 0. Here f : R ā†’ R is an odd locally lipschitz non-linear function such that there exists a Ī² > 0 with f < 0 on (0, Ī²), f > 0 on (Ī², āˆž), and K(r) \equiv r^āˆ’Ī± for some Ī± > 0.
Date: August 2018
Creator: Joshi, Janak R
Partner: UNT Libraries
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Non-Resonant Uniserial Representations of Vec(R)

Description: The non-resonant bounded uniserial representations of Vec(R) form a certain class of extensions composed of tensor density modules, all of whose subquotients are indecomposable. The problem of classifying the extensions with a given composition series is reduced via cohomological methods to computing the solution of a certain system of polynomial equations in several variables derived from the cup equations for the extension. Using this method, we classify all non-resonant bounded uniserial extā€¦ more
Date: May 2018
Creator: O'Dell, Connor
Partner: UNT Libraries
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On Factors of Rank One Subshifts

Description: Rank one subshifts are dynamical systems generated by a regular combinatorial process based on sequences of positive integers called the cut and spacer parameters. Despite the simple process that generates them, rank one subshifts comprise a generic set and are the source of many counterexamples. As a result, measure theoretic rank one subshifts, called rank one transformations, have been extensively studied and investigations into rank one subshifts been the basis of much recent work. We will ā€¦ more
Date: May 2018
Creator: Ziegler, Caleb
Partner: UNT Libraries
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Uniserial Representations of Vec(R) with a Single Casimir Eigenvalue

Description: In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensiā€¦ more
Date: May 2018
Creator: Kuhns, Nehemiah
Partner: UNT Libraries
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A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers

Description: Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And fā€¦ more
Date: August 2017
Creator: Allen, Cristian Gerardo
Partner: UNT Libraries
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Crystallographic Complex Reflection Groups and the Braid Conjecture

Description: Crystallographic complex reflection groups are generated by reflections about affine hyperplanes in complex space and stabilize a full rank lattice. These analogs of affine Weyl groups have infinite order and were classified by V.L. Popov in 1982. The classical Braid theorem (first established by E. Artin and E. Brieskorn) asserts that the Artin group of a reflection group (finite or affine Weyl) gives the fundamental group of regular orbits. In other words, the fundamental group of the spacā€¦ more
Date: August 2017
Creator: Puente, Philip C
Partner: UNT Libraries
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A General Approach to Buhlmann Credibility Theory

Description: Credibility theory is widely used in insurance. It is included in the examination of the Society of Actuaries and in the construction and evaluation of actuarial models. In particular, the Buhlmann credibility model has played a fundamental role in both actuarial theory and practice. It provides a mathematical rigorous procedure for deciding how much credibility should be given to the actual experience rating of an individual risk relative to the manual rating common to a particular class of riā€¦ more
Date: August 2017
Creator: Yan, Yujie yy
Partner: UNT Libraries
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Numerical Values of the Hausdorff and Packing Measures for Limit Sets of Iterated Function Systems

Description: In the context of fractal geometry, the natural extension of volume in Euclidean space is given by Hausdorff and packing measures. These measures arise naturally in the context of iterated function systems (IFS). For example, if the IFS is finite and conformal, then the Hausdorff and packing dimensions of the limit sets agree and the corresponding Hausdorff and packing measures are positive and finite. Moreover, the map which takes the IFS to its dimension is continuous. Developing on previous ā€¦ more
Date: August 2017
Creator: Reid, James Edward
Partner: UNT Libraries
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Partition Properties for Non-Ordinal Sets under the Axiom of Determinacy

Description: 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.
Date: May 2017
Creator: Holshouser, Jared
Partner: UNT Libraries
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