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Topological Conjugacy Relation on the Space of Toeplitz Subshifts

Description: We proved that the topological conjugacy relation on $T_1$, a subclass of Toeplitz subshifts, is hyperfinite, extending Kaya's result that the topological conjugate relation of Toeplitz subshifts with growing blocks is hyperfinite. A close concept about the topological conjugacy is the flip conjugacy, which has been broadly studied in terms of the topological full groups. Particularly, we provided an equivalent characterization on Toeplitz subshifts with single hole structure to be flip invariā€¦ more
Date: August 2022
Creator: Yu, Ping

The D-Variant of Transfinite Hausdorff Dimension

Description: In this lecture we introduce a new transfinite dimension function for metric spaces which utilizes Henderson's topological D-dimension and ascribes to any metric space either an ordinal number or the symbol Ī©. The construction of our function is motivated by that of Urbański's transfinite Hausdorff dimension, tHD. Henderson's dimension is a topological invariant, however, like Hausdorff dimension and tHD the function presented will be invariant under bi-Lipschitz continuous maps and generally nā€¦ more
Date: May 2022
Creator: Decker, Bryce
open access

Counting Plane Tropical Curves via Lattice Paths in Polygons

Description: A projective plane tropical curve is a proper immersion of a graph into the real Cartesian plane subject to some conditions such as that the images of all the edges must be lines with rational slopes. Two important combinatorial invariants of a projective plane tropical curve are its degree, d, and genus g. First, we explore Gathmann and Markwig's approach to the study of the moduli spaces of such curves and explain their proof that the number of projective plane tropical curves, counting multā€¦ more
Date: December 2021
Creator: Zhang, Yingyu

A New Class of Stochastic Volatility Models for Pricing Options Based on Observables as Volatility Proxies

Description: One basic assumption of the celebrated Black-Scholes-Merton PDE model for pricing derivatives is that the volatility is a constant. However, the implied volatility plot based on real data is not constant, but curved exhibiting patterns of volatility skews or smiles. Since the volatility is not observable, various stochastic volatility models have been proposed to overcome the problem of non-constant volatility. Although these methods are fairly successful in modeling volatilities, they still reā€¦ more
Date: December 2021
Creator: Zhou, Jie

Optimal Pair-Trading Decision Rules for a Class of Non-linear Boundary Crossings by Ornstein-Uhlenbeck Processes

Description: The most useful feature used in finance of the Ornstein-Uhlenbeck (OU) stochastic process is its mean-reverting property: the OU process tends to drift towards its long- term mean (its equilibrium state) over time. This important feature makes the OU process arguably the most popular statistical model for developing best pair-trading strategies. However, optimal strategies depend crucially on the first passage time (FPT) of the OU process to a suitably chosen boundary and its probability densiā€¦ more
Date: December 2021
Creator: Tamakloe, Emmanuel Edem Kwaku
open access

On the Subspace Dichotomy of Lp[0; 1] for 2 < p < āˆž

Description: The structure and geometry of subspaces of a given Banach space is among the most fundamental questions in Functional Analysis. In 1961, Kadec and Pelczyński pioneered a field of study by analyzing the structures of subspaces and basic sequences in L_p[0,1] under a naturally occurring restriction of p, 2 < p <\infty. They proved that any infinite-dimensional subspace X\subset L_p[0,1] for 2<p<\infty must either be isomorphic to l_2 and complemented in L_p or must contain a complemented subspaceā€¦ more
Date: August 2021
Creator: James, Christopher W
open access

Optimal Look-Ahead Stopping Rules for Simple Random Walk

Description: In a stopping rule problem, a real-time player decides to stop or continue at stage n based on the observations up to that stage, but in a k-step look-ahead stopping rule problem, we suppose the player knows k steps ahead. The aim of this Ph.D. dissertation is to study this type of prophet problems for simple random walk, determine the optimal stopping rule and calculate the expected return for them. The optimal one-step look-ahead stopping rule for a finite simple random walk is determined in ā€¦ more
Date: August 2021
Creator: Sharif Kazemi, Zohreh

Radial Solutions of Singular Semilinear Equations on Exterior Domains

Description: We prove the existence and nonexistence of radial solutions of singular semilinear equations Ī”u + k(x)f(u)=0 with boundary condition on the exterior of the ball with radius R>0 in ā„^N such that lim r ā†’āˆž u(r)=0, where f: ā„ \ {0} ā†’ā„ is an odd and locally Lipschitz continuous nonlinear function such that there exists a Ī² >0 with f <0 on (0, Ī²), f >0 on (Ī², āˆž), and K(r) ~ r^-Ī± for some Ī± >0.
Date: May 2021
Creator: Ali, Mageed Hameed
open access

Contributions to Geometry and Graph Theory

Description: In geometry we will consider n-dimensional generalizations of the Power of a Point Theorem and of Pascal's Hexagon Theorem. In generalizing the Power of a Point Theorem, we will consider collections of cones determined by the intersections of an (n-1)-sphere and a pair of hyperplanes. We will then use these constructions to produce an n-dimensional generalization of Pascal's Hexagon Theorem, a classical plane geometry result which states that "Given a hexagon inscribed in a conic section, theā€¦ more
Date: August 2020
Creator: Schuerger, Houston S

Graded Hecke Algebras for the Symmetric Group in Positive Characteristic

Description: Graded Hecke algebras are deformations of skew group algebras which arise from a group acting on a polynomial ring. Over fields of characteristic zero, these deformations have been studied in depth and include both symplectic reflection algebras and rational Cherednik algebras as examples. In Lusztig's graded affine Hecke algebras, the action of the group is deformed, but not the commutativity of the vectors. In Drinfeld's Hecke algebras, the commutativity of the vectors is deformed, but not ā€¦ more
Date: August 2020
Creator: Krawzik, Naomi
open access

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
open access

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
open access

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
open access

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
open access

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.

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

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
Access: Restricted to UNT Community Members. Login required if off-campus.
Date: August 2019
Creator: Rajendran, Rajanikanth
open access

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.
open access

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.
open access

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
open access

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
open access

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.
open access

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
open access

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
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