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  Partner: UNT Libraries
 Decade: 2000-2009
 Degree Discipline: Mathematics
 Degree Level: Master's
 Collection: UNT Theses and Dissertations
Applications in Fixed Point Theory

Applications in Fixed Point Theory

Date: December 2005
Creator: Farmer, Matthew Ray
Description: Banach's contraction principle is probably one of the most important theorems in fixed point theory. It has been used to develop much of the rest of fixed point theory. Another key result in the field is a theorem due to Browder, Göhde, and Kirk involving Hilbert spaces and nonexpansive mappings. Several applications of Banach's contraction principle are made. Some of these applications involve obtaining new metrics on a space, forcing a continuous map to have a fixed point, and using conditions on the boundary of a closed ball in a Banach space to obtain a fixed point. Finally, a development of the theorem due to Browder et al. is given with Hilbert spaces replaced by uniformly convex Banach spaces.
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Borel Determinacy and Metamathematics

Borel Determinacy and Metamathematics

Date: December 2001
Creator: Bryant, Ross
Description: Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.
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Characterizations of Continua of Finite Degree

Characterizations of Continua of Finite Degree

Date: August 2006
Creator: Irwin, Shana
Description: 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.
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Compact Operators and the Schrödinger Equation

Compact Operators and the Schrödinger Equation

Date: December 2006
Creator: Kazemi, Parimah
Description: In this thesis I look at the theory of compact operators in a general Hilbert space, as well as the inverse of the Hamiltonian operator in the specific case of L2[a,b]. I show that this inverse is a compact, positive, and bounded linear operator. Also the eigenfunctions of this operator form a basis for the space of continuous functions as a subspace of L2[a,b]. A numerical method is proposed to solve for these eigenfunctions when the Hamiltonian is considered as an operator on Rn. The paper finishes with a discussion of examples of Schrödinger equations and the solutions.
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A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions

A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions

Date: May 2004
Creator: Vlasic, Andrew
Description: 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.
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Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis

Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis

Date: May 2007
Creator: Brooks, Evan
Description: A "complex" system typically has a relatively large number of dynamically interacting components and tends to exhibit emergent behavior that cannot be explained by analyzing each component separately. A biological neural network is one example of such a system. A multi-agent model of such a network is developed to study the relationships between a network's structure and its spike train output. Using this model, inferences are made about the synaptic structure of networks through cluster analysis of spike train summary statistics A complexity measure for the network structure is also presented which has a one-to-one correspondence with the standard time series complexity measure sample entropy.
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Hyperspace Topologies

Hyperspace Topologies

Date: August 2001
Creator: Freeman, Jeannette Broad
Description: In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.
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Lyapunov Exponents, Entropy and Dimension

Lyapunov Exponents, Entropy and Dimension

Date: August 2004
Creator: Williams, Jeremy M.
Description: 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.
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Mathematical Modeling of Charged Liquid Droplets: Numerical Simulation and Stability Analysis

Mathematical Modeling of Charged Liquid Droplets: Numerical Simulation and Stability Analysis

Date: May 2006
Creator: Vantzos, Orestis
Description: The goal of this thesis is to study of the evolution of 3D electrically charged liquid droplets of fluid evolving under the influence of surface tension and electrostatic forces. In the first part of the thesis, an appropriate mathematical model of the problem is introduced and the linear stability analysis is developed by perturbing a sphere with spherical harmonics. In the second part, the numerical solution of the problem is described with the use of the boundary elements method (BEM) on an adaptive mesh of triangular elements. The numerical method is validated by comparison with exact solutions. Finally, various numerical results are presented. These include neck formation in droplets, the evolution of surfaces with holes, singularity formation on droplets with various symmetries and numerical evidence that oblate spheroids are unstable.
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On the density of minimal free subflows of general symbolic flows.

On the density of minimal free subflows of general symbolic flows.

Date: August 2009
Creator: Seward, Brandon Michael
Description: This paper studies symbolic dynamical systems {0, 1}G, where G is a countably infinite group, {0, 1}G has the product topology, and G acts on {0, 1}G by shifts. It is proven that for every countably infinite group G the union of the minimal free subflows of {0, 1}G is dense. In fact, a stronger result is obtained which states that if G is a countably infinite group and U is an open subset of {0, 1}G, then there is a collection of size continuum consisting of pairwise disjoint minimal free subflows intersecting U.
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