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Asymptotic Formula for Counting in Deterministic and Random Dynamical Systems

The lattice point problem in dynamical systems investigates the distribution of certain objects with some length property in the space that the dynamics is defined. This problem in different contexts can be interpreted differently. In the context of symbolic dynamical systems, we are trying to investigate the growth of N(T), the number of finite words subject to a specific ergodic length T, as T tends to infinity. This problem has been investigated by Pollicott and Urbański to a great extent. W… more

Date: May 2023

Creator: Naderiyan, Hamid

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Hochschild Cohomology of Finite Cyclic Groups Acting on Polynomial Rings

The Hochschild cohomology of an associative algebra records information about the deformations of that algebra, and hence the first step toward understanding its deformations is an examination of the Hochschild cohomology. In this dissertation, we use techniques from homological algebra, invariant theory, and combinatorics to analyze the Hochschild cohomology of skew group algebras arising from finite cyclic groups acting on polynomial rings over fields of arbitrary characteristic. These algebr… more

Date: May 2023

Creator: Lawson, Colin M.

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Continuity of Hausdorff Dimension of Julia Sets of Expansive Polynomials

This dissertation is in the area of complex dynamics, more specifically focused on the iteration of rational functions. Given a well-chosen family of rational functions, parameterized by a complex parameter, we are especially interested in regularity properties of the Hausdorff dimension of Julia sets of these polynomials considered as a function of the parameters. In this dissertation I deal with a family of polynomials of degree at least 3 depending in a holomorphic way on a parameter, focusi… more

Date: August 2022

Creator: Wilson, Timothy Charles

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Counting Plane Tropical Curves via Lattice Paths in Polygons

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 mult… more

Date: December 2021

Creator: Zhang, Yingyu

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Annihilators of Bounded Indecomposable Modules of Vec(R)

The Lie algebra Vec(ℝ) of polynomial vector fields on the line acts naturally on ℂ[]. This action has a one-parameter family of deformations called the tensor density modules F_λ. The bounded indecomposable modules of Vec(ℝ) 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_λ by F_(λ+2) in the non-resonant case λ ≠ -½. We give the intersection of the annihil… more

Date: May 2019

Creator: Kenefake, Tyler Christian

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Conformal and Stochastic Non-Autonomous Dynamical Systems

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 of… more

Date: August 2018

Creator: Atnip, Jason

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Hausdorff Dimension of Shrinking-Target Sets Under Non-Autonomous Systems

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 iterate… more

Date: August 2018

Creator: Lopez, Marco Antonio

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Rankin-Cohen Brackets for Hermitian Jacobi Forms and Hermitian Modular Forms

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 Her… more

Date: December 2016

Creator: Martin, James D. (James Dudley)

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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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

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 math… more

Date: May 2016

Creator: Le, Thu Anh

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Hermitian Jacobi Forms and Congruences

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.

Date: August 2014

Creator: Senadheera, Jayantha

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Centers of Invariant Differential Operator Algebras for Jacobi Groups of Higher Rank

Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extensio… more

Date: August 2013

Creator: Dahal, Rabin

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Random Iteration of Rational Functions

It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational map of degree at least two and if ϕ:ℂ→ℝ is Hölder continuous and satisfies the “thermodynamic expanding” condition P(T,ϕ) > sup(ϕ), then there exists exactly one equilibrium state μ for T and ϕ, and furthermore (ℂ,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on ℂ, using the concepts of relative pressure and relative entropy of such a system, and the variational principle … more

Date: May 2012

Creator: Simmons, David

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following quest… more

Date: December 2009

Creator: Bajracharya, Neeraj

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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The Global Structure of Iterated Function Systems

I study sets of attractors and non-attractors of finite iterated function systems. I provide examples of compact sets which are attractors of iterated function systems as well as compact sets which are not attractors of any iterated function system. I show that the set of all attractors is a dense Fs set and the space of all non-attractors is a dense Gd set it the space of all non-empty compact subsets of a space X. I also investigate the small trans-finite inductive dimension of the space of… more

Date: May 2009

Creator: Snyder, Jason Edward

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Around the Fibonacci Numeration System

Let 1, 2, 3, 5, 8, … denote the Fibonacci sequence beginning with 1 and 2, and then setting each subsequent number to the sum of the two previous ones. Every positive integer n can be expressed as a sum of distinct Fibonacci numbers in one or more ways. Setting R(n) to be the number of ways n can be written as a sum of distinct Fibonacci numbers, we exhibit certain regularity properties of R(n), one of which is connected to the Euler φ-function. In addition, using a theorem of Fine and Wilf, we… more

Date: May 2007

Creator: Edson, Marcia Ruth

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Characterizations of Continua of Finite Degree

In this thesis, some characterizations of continua of finite degree are given. It turns out that being of finite degree (by formal definition) can be described by saying there exists an equivalent metric in which Hausdorff linear measure of the continuum is finite. I discuss this result in detail.

Date: August 2006

Creator: Irwin, Shana

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Dynamics, Thermodynamic formalism and Perturbations of Transcendental Entire Functions of Finite Singular Type

In this dissertation, we study the dynamics, fractal geometry and the topology of the Julia set of functions in the family H which is a set in the class S, the Speiser class of entire transcendental functions which have only finitely many singular values. One can think of a function from H as a generalized expanding function from the cosh family. We shall build a version of thermodynamic formalism for functions in H and we shall show among others, the existence and uniqueness of a conformal mea… more

Date: May 2005

Creator: Coiculescu, Ion

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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Lyapunov Exponents, Entropy and Dimension

We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.

Date: August 2004

Creator: Williams, Jeremy M.

Item Type: Thesis or Dissertation

Partner: UNT Libraries

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A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions

We follow a research paper that J. Elstrodt published in 1998 to prove the Prime Number Theorem for arithmetic progressions. We will review basic results from Dirichlet characters and L-functions. Furthermore, we establish a weak version of the Wiener-Ikehara Tauberian Theorem, which is an essential tool for the proof of our main result.

Date: May 2004

Creator: Vlasic, Andrew

Item Type: Thesis or Dissertation

Partner: UNT Libraries