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**Department:**Department of Mathematics

**Collection:**UNT Theses and Dissertations

### 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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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4476/

### Determinacy-related Consequences on Limit Superiors

**Date:**May 2013

**Creator:**Walker, Daniel

**Description:**Laczkovich proved from ZF that, given a countable sequence of Borel sets on a perfect Polish space, if the limit superior along every subsequence was uncountable, then there was a particular subsequence whose intersection actually contained a perfect subset. Komjath later expanded the result to hold for analytic sets. In this paper, by adding AD and sometimes V=L(R) to our assumptions, we will extend the result further. This generalization will include the increasing of the length of the sequence to certain uncountable regular cardinals as well as removing any descriptive requirements on the sets.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc271913/

### 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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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3702/

### Development of a Geometry from a Set of Axioms

**Date:**May 1973

**Creator:**Glasscock, Anita Louise

**Description:**The purpose of this paper is to develop a geometry based on fourteen axioms and four undefined terms.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131608/

### A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions

**Date:**August 1954

**Creator:**Allen, Alfred I.

**Description:**The purpose of this paper is to define and develop a set of functions of an area in such a manner as to be analogous to the trigonometric and the hyperbolic functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130370/

### A Development of the Exponential and Logarithmic Functions

**Date:**1953

**Creator:**Mackey, Benford B.

**Description:**This thesis discusses a development of the exponential and logarithmic functions.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107841/

### The Development of the Natural Numbers by Means of the Peano Postulates

**Date:**1951

**Creator:**Baugh, Orvil Lee

**Description:**This thesis covers the development of the natural numbers by means of the peano postulates.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc96984/

### A Development of the Peano Postulates

**Date:**May 1963

**Creator:**Peek, Darwin Eugene

**Description:**The purpose of this paper is to develop the Peano postulates from a weaker axiom system than the system used by John L. Kelley in General Topology. The axiom of regularity which states "If X is a non-empty set, then there is a member Y of X such that the intersection of X and Y is empty." is not assumed in this thesis. The axiom of amalgamation which states "If X is a set, then the union of the elements of X is a set." is also not assumed. All other axioms used by Kelley relevant to the Peano postulates are assumed. The word class is never used in the thesis, though the variables can be interpreted as classes.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc108207/

### A Development of the Real Number System

**Date:**August 1961

**Creator:**Matthews, Ronald Louis

**Description:**The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130475/

### A Development of the Real Number System by Means of Nests of Rational Intervals

**Date:**1949

**Creator:**Williams, Mack Lester

**Description:**The system of rational numbers can be extended to the real number system by several methods. In this paper, we shall extend the rational number system by means of rational nests of intervals, and develop the elementary properties of the real numbers obtained by this extension.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc83636/

### Differentiable Functions

**Date:**June 1966

**Creator:**McCool, Kenneth B.

**Description:**The primary purpose of this thesis is to carefully develop and prove some of the fundamental, classical theorems of the differential calculus for functions of two real variables.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130691/

### Differentiation in Banach Spaces

**Date:**December 1972

**Creator:**Heath, James Darrell

**Description:**This thesis investigates the properties and applications of derivatives of functions whose domain and range are Banach spaces.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc131585/

### Dimension spectrum and graph directed Markov systems.

**Access:**Use of this item is restricted to the UNT Community.

**Date:**May 2006

**Creator:**Ghenciu, Eugen Andrei

**Description:**In this dissertation we study graph directed Markov systems (GDMS) and limit sets associated with these systems. Given a GDMS S, by the Hausdorff dimension spectrum of S we mean the set of all positive real numbers which are the Hausdorff dimension of the limit set generated by a subsystem of S. We say that S has full Hausdorff dimension spectrum (full HD spectrum), if the dimension spectrum is the interval [0, h], where h is the Hausdorff dimension of the limit set of S. We give necessary conditions for a finitely primitive conformal GDMS to have full HD spectrum. A GDMS is said to be regular if the Hausdorff dimension of its limit set is also the zero of the topological pressure function. We show that every number in the Hausdorff dimension spectrum is the Hausdorff dimension of a regular subsystem. In the particular case of a conformal iterated function system we show that the Hausdorff dimension spectrum is compact. We introduce several new systems: the nearest integer GDMS, the Gauss-like continued fraction system, and the Renyi-like continued fraction system. We prove that these systems have full HD spectrum. A special attention is given to the backward continued fraction ...

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### Dimension Theory

**Date:**August 1986

**Creator:**Frere, Scot M. (Scot Martin)

**Description:**This paper contains a discussion of topological dimension theory. Original proofs of theorems, as well as a presentation of theorems and proofs selected from Ryszard Engelking's Dimension Theory are contained within the body of this endeavor. Preliminary notation is introduced in Chapter I. Chapter II consists of the definition of and theorems relating to the small inductive dimension function Ind. Large inductive dimension is investigated in Chapter III. Chapter IV comprises the definition of covering dimension and theorems discussing the equivalence of the different dimension functions in certain topological settings. Arguments pertaining to the dimension o f Jn are also contained in Chapter IV.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc500690/

### Dimensions in Random Constructions.

**Date:**May 2002

**Creator:**Berlinkov, Artemi

**Description:**We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc3160/

### Direct Sums of Rings

**Date:**August 1966

**Creator:**Hughes, Dolin F.

**Description:**This paper consists of a study of the direct sum U of two rings S and T. Such a direct sum is defined as the set of all ordered pairs (s1, t1), where s1 is an arbitrary element in S and t1 is an arbitrary element in T.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130723/

### Divisibility in Abelian Groups

**Date:**August 1966

**Creator:**Huie, Douglas Lee

**Description:**This thesis describes properties of Abelian groups, and develops a study of the properties of divisibility in Abelian groups.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc130724/

### Dually Semimodular Consistent Lattices

**Date:**May 1988

**Creator:**Gragg, Karen E. (Karen Elizabeth)

**Description:**A lattice L is said to be dually semimodular if for all elements a and b in L, a ∨ b covers b implies that a covers a ∧ b. L is consistent if for every join-irreducible j and every element x in L, the element x ∨ j is a join-irreducible in the upper interval [x,l]. In this paper, finite dually semimodular consistent lattices are investigated. Examples of these lattices are the lattices of subnormal subgroups of a finite group. In 1954, R. P. Dilworth proved that in a finite modular lattice, the number of elements covering exactly k elements is equal to the number of elements covered by exactly k elements. Here, it is established that if a finite dually semimodular consistent lattice has the same number of join-irreducibles as meet-irreducibles, then it is modular. Hence, a converse of Dilworth's theorem, in the case when k equals 1, is obtained for finite dually semimodular consistent lattices. Several combinatorial results are shown for finite consistent lattices similar to those already established for finite geometric lattices. The reach of an element x in a lattice L is the difference between the rank of x*, the join of x and all ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc330641/

### Duals and Weak Completeness in Certain Sequence Spaces

**Date:**August 1980

**Creator:**Leavelle, Tommy L. (Tommy Lee)

**Description:**In this paper the weak completeness of certain sequence spaces is examined. In particular, we show that each of the sequence spaces c0 and 9, 1 < p < c, is a Banach space. A Riesz representation for the dual space of each of these sequence spaces is given. A Riesz representation theorem for Hilbert space is also proven. In the third chapter we conclude that any reflexive space is weakly (sequentially) complete. We give 01 as an example of a non-reflexive space that is weakly complete. Two examples, c0 and YJ, are given of spaces that fail to be weakly complete.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc504338/

### The Dyadic Operator Approach to a Study in Conics, with some Extensions to Higher Dimensions

**Date:**1940

**Creator:**Shawn, James Loyd

**Description:**The discovery of a new truth in the older fields of mathematics is a rare event. Here an investigator may hope at best to secure greater elegance in method or notation, or to extend known results by some process of generalization. It is our purpose to make a study of conic sections in the spirit of the above remark, using the symbolism developed by Josiah Williard Gibbs.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc75602/

### Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension

**Date:**May 1988

**Creator:**Brucks, Karen M. (Karen Marie), 1957-

**Description:**This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function, f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the ...

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc332102/

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

**Date:**May 2005

**Creator:**Coiculescu, Ion

**Description:**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 measure. Then we prove a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius operators for some functions naturally associated with functions in the family H and then, using fundamental properties of these operators, we shall prove the important result that the Hausdorff dimension of the subset of returning points depends analytically on the parameter taken from a small open subset of the n-dimensional parameter space.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc4783/

### Electronic Analog Computer Study of Effects of Motor Velocity and Driving Voltage Limits upon Servomechanism Performance

**Date:**1956

**Creator:**Haynes, Joe Preston

**Description:**The object of this thesis is (1) to demonstrate the value of an electronic analog computer for the solution of non-linear ordinary differential equations particularly when a large family of solutions is required; and (2) to obtain as a by-product results of practical applicability to servomechanism selection and analysis.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc107908/

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

**Date:**1940

**Creator:**Wilson, Carroll K.

**Description:**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.

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**Permallink:**digital.library.unt.edu/ark:/67531/metadc70281/