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A Presentation of Current Research on Partitions of Lines and Space (open access)

A Presentation of Current Research on Partitions of Lines and Space

We present the results from three papers concerning partitions of vector spaces V over the set R of reals and of the set of lines in V.
Date: December 1999
Creator: Nugen, Frederick T.
System: The UNT Digital Library
Hyperspace Topologies (open access)

Hyperspace Topologies

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.
Date: August 2001
Creator: Freeman, Jeannette Broad
System: The UNT Digital Library
Borel Determinacy and Metamathematics (open access)

Borel Determinacy and Metamathematics

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.
Date: December 2001
Creator: Bryant, Ross
System: The UNT Digital Library
Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World (open access)

Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World

Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.
Date: August 2002
Creator: Arthur, Christopher
System: The UNT Digital Library
Determining Properties of Synaptic Structure in a Neural Network through Spike Train Analysis (open access)

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

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.
Date: May 2007
Creator: Brooks, Evan
System: The UNT Digital Library
A Detailed Proof of the Prime Number Theorem for Arithmetic Progressions (open access)

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
System: The UNT Digital Library
Thermodynamical Formalism (open access)

Thermodynamical Formalism

Thermodynamical formalism is a relatively recent area of pure mathematics owing a lot to some classical notions of thermodynamics. On this thesis we state and prove some of the main results in the area of thermodynamical formalism. The first chapter is an introduction to ergodic theory. Some of the main theorems are proved and there is also a quite thorough study of the topology that arises in Borel probability measure spaces. In the second chapter we introduce the notions of topological pressure and measure theoretic entropy and we state and prove two very important theorems, Shannon-McMillan-Breiman theorem and the Variational Principle. Distance expanding maps and their connection with the calculation of topological pressure cover the third chapter. The fourth chapter introduces Gibbs states and the very important Perron-Frobenius Operator. The fifth chapter establishes the connection between pressure and geometry. Topological pressure is used in the calculation of Hausdorff dimensions. Finally the sixth chapter introduces the notion of conformal measures.
Date: August 2004
Creator: Chousionis, Vasileios
System: The UNT Digital Library
Lyapunov Exponents, Entropy and Dimension (open access)

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.
System: The UNT Digital Library
Applications in Fixed Point Theory (open access)

Applications in Fixed Point Theory

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.
Date: December 2005
Creator: Farmer, Matthew Ray
System: The UNT Digital Library
Mathematical Modeling of Charged Liquid Droplets: Numerical Simulation and Stability Analysis (open access)

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

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.
Date: May 2006
Creator: Vantzos, Orestis
System: The UNT Digital Library
Characterizations of Continua of Finite Degree (open access)

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
System: The UNT Digital Library
Compact Operators and the Schrödinger Equation (open access)

Compact Operators and the Schrödinger Equation

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.
Date: December 2006
Creator: Kazemi, Parimah
System: The UNT Digital Library
On the density of minimal free subflows of general symbolic flows. (open access)

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

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.
Date: August 2009
Creator: Seward, Brandon Michael
System: The UNT Digital Library
Some Fundamental Properties of Power Series (open access)

Some Fundamental Properties of Power Series

A study to deduce some fundamental properties of power series.
Date: August 1939
Creator: Rogers, Curtis, A.
System: The UNT Digital Library
A Comparison of the Group and the Individual Techniques of Teaching Algebra (open access)

A Comparison of the Group and the Individual Techniques of Teaching Algebra

This thesis presents findings made during an examination and comparison of the individualized unit and traditional group methods used to teach ninth grade algebra in Hillsboro, Texas.
Date: August 1940
Creator: Knox, Beulah
System: The UNT Digital Library
To Develop and to Evaluate a Mathematics Curriculum for the Mentally Retarded on the Junior High School Level (open access)

To Develop and to Evaluate a Mathematics Curriculum for the Mentally Retarded on the Junior High School Level

This thesis presents the results of a study conducted to determine the math capabilities of mentally challenged students attending Reagan Junior High School in Wichita Falls, Texas. Survey results are used to determine needed curriculum changes.
Date: August 1940
Creator: Miller, Zola Catheryn
System: The UNT Digital Library
Some Properties of Certain Generalizations of the Sum of an Infinite Series (open access)

Some Properties of Certain Generalizations of the Sum of an Infinite Series

This thesis attempts to establish properties of Hölder and Cesàro summable series analogous to those of ordinary convergent series and also to establish properties that are possibly different from those of convergent series.
Date: 1941
Creator: Hill, William F.
System: The UNT Digital Library
The Cantor Ternary Set and Certain of its Generalizations and Applications (open access)

The Cantor Ternary Set and Certain of its Generalizations and Applications

This thesis covers the Cantor Ternary Set and generalizations of the Cantor Set, and gives a complete existential theory for three set properties: denumerability, exhaustibility, and zero measure.
Date: 1942
Creator: Hembree, Gwendolyn
System: The UNT Digital Library
The Elementary Transcendental Functions of a Complex Variable as Defined by Integration (open access)

The Elementary Transcendental Functions of a Complex Variable as Defined by Integration

The object of this paper is to define the elementary transcendental functions of a complex variable by means of integrals, and to discuss their properties.
Date: 1940
Creator: Wilson, Carroll K.
System: The UNT Digital Library
Number Theoretic Analogies for Certain Theorems and Processes in the Theory of Equations (open access)

Number Theoretic Analogies for Certain Theorems and Processes in the Theory of Equations

The aim of this paper is to exhibit analogs in Number Theory of certain well known theorems and methods of the Theory of Equations.
Date: 1940
Creator: Witt, F. L.
System: The UNT Digital Library
Some Properties of Transfinite Cardinal and Ordinal Numbers (open access)

Some Properties of Transfinite Cardinal and Ordinal Numbers

Explains properties of mathematical sets, algebra of sets, and set order types.
Date: 1940
Creator: Cunningham, James S.
System: The UNT Digital Library
The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity (open access)

The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity

This study investigates the relationship between ordinary continuity and t-continuity.
Date: 1941
Creator: Parrish, Herbert C.
System: The UNT Digital Library
The History of the Calculus (open access)

The History of the Calculus

The purpose of this essay is to trace the development of the concepts of the calculus from their first known appearance, through the formal invention of the method of the calculus in the second half of the seventeenth century, to our own day.
Date: 1945
Creator: Ashburn, Andrew
System: The UNT Digital Library
Some Fundamental Properties of Gamma and Beta Functions (open access)

Some Fundamental Properties of Gamma and Beta Functions

This paper consists of a discussion of the properties and applications of certain improper integrals, namely the gamma function and the beta function. There are also specific examples of application of these functions in certain fields of applied science.
Date: 1941
Creator: Nolen, Robert L.
System: The UNT Digital Library