- Abstract Measure
- This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure. digital.library.unt.edu/ark:/67531/metadc107950/
- Abstract Vector Spaces and Certain Related Systems
- The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system. digital.library.unt.edu/ark:/67531/metadc130465/
- Additive Functions
- The purpose of this paper is the analysis of functions of real numbers which have a special additive property, namely, f(x+y) = f(x)+f(y). digital.library.unt.edu/ark:/67531/metadc108204/
- Algebraic Integers
- The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers. digital.library.unt.edu/ark:/67531/metadc131119/
- Algebraic Properties of Semigroups
- This paper is an algebraic study of selected properties of semigroups. Since a semigroup is a result of weakening the group axioms, all groups are semigroups. One facet of the paper is to demonstrate various semigroup properties that induce the group axioms. digital.library.unt.edu/ark:/67531/metadc131373/
- Algebraically Determined Rings of Functions
- Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring. digital.library.unt.edu/ark:/67531/metadc31543/
- Algebraically Determined Semidirect Products
- Let G be a Polish group. We say that G is an algebraically determined Polish group if given any Polish group L and any algebraic isomorphism from L to G, then the algebraic isomorphism is a topological isomorphism. We will prove a general theorem that gives useful sufficient conditions for a semidirect product of two Polish groups to be algebraically determined. This will smooth the way for the proofs for some special groups. For example, let H be a separable Hilbert space and let G be a subset of the unitary group U(H) acting transitively on the unit sphere. Assume that -I in G and G is a Polish topological group in some topology such that H x G to H, (x,U) to U(x) is continuous, then H x G is a Polish topological group. Hence H x G is an algebraically determined Polish group. In addition, we apply the above the above result on the unitary group U(A) of a separable irreducible C*-algebra A with identity acting transitively on the unit sphere in a separable Hilbert space H and proved that the natural semidirect product H x U(A) is an algebraically determined Polish group. A similar theorem is true for the natural semidirect product R^{n} x G(n), where G(n) = GL(n,R), or GL^{+}(n,R), or SL(n,R), or |SL(n,R)|={A in GL(n,R) : |det(A)|=1}. On the other hand, it is known that the Heisenberg group H_{3}(R) , (R, +), (R{0}, x), and GL^{+}(n,R) are not algebraically determined Polish groups. digital.library.unt.edu/ark:/67531/metadc67993/
- A*-algebras and Minimal Ideals in Topological Rings
- The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings. digital.library.unt.edu/ark:/67531/metadc131622/
- The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity
- This study investigates the relationship between ordinary continuity and t-continuity. digital.library.unt.edu/ark:/67531/metadc75274/
- Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions
- Hill and Monticino (1998) introduced a constructive method for generating random probability measures with a prescribed mean or distribution on the mean. The method involves sequentially generating an array of barycenters that uniquely defines a probability measure. This work analyzes statistical properties of the measures generated by sequential barycenter array constructions. Specifically, this work addresses how changing the base measures of the construction affects the statististics of measures generated by the SBA construction. A relationship between statistics associated with a finite level version of the SBA construction and the full construction is developed. Monte Carlo statistical experiments are used to simulate the effect changing base measures has on the statistics associated with the finite level construction. digital.library.unt.edu/ark:/67531/metadc3304/
- The Analytical Development of the Trigonometric Functions
- This thesis is a study of the analytical development of the trigonometric functions. digital.library.unt.edu/ark:/67531/metadc97030/
- 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. digital.library.unt.edu/ark:/67531/metadc4971/
- Applications of Graph Theory and Topology to Combinatorial Designs
- This dissertation is concerned with the existence and the isomorphism of designs. The first part studies the existence of designs. Chapter I shows how to obtain a design from a difference family. Chapters II to IV study the existence of an affine 3-(p^m,4,λ) design where the v-set is the Galois field GF(p^m). Associated to each prime p, this paper constructs a graph. If the graph has a 1-factor, then a difference family and hence an affine design exists. The question arises of how to determine when the graph has a 1-factor. It is not hard to see that the graph is connected and of even order. Tutte's theorem shows that if the graph is 2-connected and regular of degree three, then the graph has a 1-factor. By using the concept of quadratic reciprocity, this paper shows that if p Ξ 53 or 77 (mod 120), the graph is almost regular of degree three, i.e., every vertex has degree three, except two vertices each have degree tow. Adding an extra edge joining the two vertices with degree tow gives a regular graph of degree three. Also, Tutte proved that if A is an edge of the graph satisfying the above conditions, then it must have a 1-factor which contains A. The second part of the dissertation is concerned with determining if two designs are isomorphic. Here the v-set is any group G and translation by any element in G gives a design automorphism. Given a design B and its difference family D, two topological spaces, B and D, are constructed. We give topological conditions which imply that a design isomorphism is a group isomorphism. digital.library.unt.edu/ark:/67531/metadc331968/
- Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory
- In this dissertation the results of Jerrum and Sinclair on the conductance of Markov chains are used to prove that almost all generalized Steinhaus graphs are rapidly mixing and an algorithm for the uniform generation of 2 - (4k + 1,4,1) cyclic Mendelsohn designs is developed. digital.library.unt.edu/ark:/67531/metadc277740/
- An Approximate Solution to the Dirichlet Problem
- In the category of mathematics called partial differential equations there is a particular type of problem called the Dirichlet problem. Proof is given in many partial differential equation books that every Dirichlet problem has one and only one solution. The explicit solution is very often not easily determined, so that a method for approximating the solution at certain points becomes desirable. The purpose of this paper is to present and investigate one such method. digital.library.unt.edu/ark:/67531/metadc130550/
- 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 give a formula for R(n) in terms of binomial coefficients modulo two. digital.library.unt.edu/ark:/67531/metadc3676/
- Aspects of Universality in Function Iteration
- This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval. digital.library.unt.edu/ark:/67531/metadc278799/
- Atmospheric Gusts and Their Effect on Aircraft
- This thesis investigates atmospheric gusts and their effect on aircraft. digital.library.unt.edu/ark:/67531/metadc108016/
- Automorphism Groups
- This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper. digital.library.unt.edu/ark:/67531/metadc130613/
- Basic Fourier Transforms
- The purpose of this paper is to develop some of the more basic Fourier transforms which are the outgrowth of the Fourier theorem. Although often approached from the stand-point of the series, this paper will approach the theorem from the standpoint of the integral. digital.library.unt.edu/ark:/67531/metadc130482/
- 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. digital.library.unt.edu/ark:/67531/metadc3061/
- Borel Sets and Baire Functions
- This paper examines the relationship between Borel sets and Baire functions. digital.library.unt.edu/ark:/67531/metadc163964/
- Bounded, Finitely Additive, but Not Absolutely Continuous Set Functions
- In leading up to the proof, methods for constructing fields and finitely additive set functions are introduced with an application involving the Tagaki function given as an example. Also, non-absolutely continuous set functions are constructed using Banach limits and maximal filters. digital.library.unt.edu/ark:/67531/metadc332375/
- The Buckling of a Uniformly Compressed Plate with Intermediate Supports
- This problem has been selected from the mathematical theory of elasticity. We consider a rectangular plate of thickness h, length a, and width b. The plate is subjected to compressive forces. These forces act in the neutral plane and give the plate a tendency to buckle. However, this problem differs from other plate problems in that it is assumed that there are two intermediate supports located on the edges of the plate parallel to the compressive forces. digital.library.unt.edu/ark:/67531/metadc83479/
- 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. digital.library.unt.edu/ark:/67531/metadc70344/
- 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 extension of work of Bringmann, Conley, and Richter in the rank 1case. digital.library.unt.edu/ark:/67531/metadc283833/
- Certain Properties of Functions Related to Exhaustibility
- In this thesis, we shall attempt to present a study of certain properties of real functions related to the set property exhaustible. digital.library.unt.edu/ark:/67531/metadc107821/
- A Characterization of Homeomorphic Bernoulli Trial Measures.
- We give conditions which, given two Bernoulli trial measures, determine whether there exists a homeomorphism of Cantor space which sends one measure to the other, answering a question of Oxtoby. We then provide examples, relating these results to the notions of good and refinable measures on Cantor space. digital.library.unt.edu/ark:/67531/metadc5331/
- 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. digital.library.unt.edu/ark:/67531/metadc5367/
- Characterizations of Some Combinatorial Geometries
- We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic. digital.library.unt.edu/ark:/67531/metadc277894/
- A Classification of Regular Planar Graphs
- The purpose of this paper is the investigation and classification of regular planar graphs. The motive behind this investigation was a desire to better understand those properties which allow a graph to be represented in the plane in such a manner that no two edges cross except perhaps at vertices. digital.library.unt.edu/ark:/67531/metadc164029/
- A Collapsing Result Using the Axiom of Determinancy and the Theory of Possible Cofinalities
- Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1 digital.library.unt.edu/ark:/67531/metadc2789/
- Compact Convex Sets in Linear Topological Spaces
- The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets. digital.library.unt.edu/ark:/67531/metadc130516/
- 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. digital.library.unt.edu/ark:/67531/metadc5453/
- Compact Topological Spaces
- The purpose of this paper is to investigate some properties of compact topological spaces and to relate these concepts to the separation properties. digital.library.unt.edu/ark:/67531/metadc130506/
- Compactness and Equivalent Notions
- One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third. digital.library.unt.edu/ark:/67531/metadc130821/
- The Comparability of Cardinals
- The purpose of this composition is to develop a rigorous, axiomatic proof of the comparability of the cardinals of infinite sets. digital.library.unt.edu/ark:/67531/metadc130514/
- A Comparative Study of Non Linear Conjugate Gradient Methods
- We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=b-Axk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized. digital.library.unt.edu/ark:/67531/metadc283864/
- Comparison of Some Mappings in Topology
- The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations. digital.library.unt.edu/ark:/67531/metadc108253/
- A Comparison of Velocities Computed by Two-Dimensional Potential Theory and Velocities Measured in the Vicinity of an Airfoil
- In treating the motion of a fluid mathematically, it is convenient to make some simplifying assumptions. The assumptions which are made will be justifiable if they save long and laborious computations in practical problems, and if the predicted results agree closely enough with experimental results for practical use. In dealing with the flow of air about an airfoil, at subsonic speeds, the fluid will be considered as a homogeneous, incompressible, inviscid fluid. digital.library.unt.edu/ark:/67531/metadc699611/
- Complemented Subspaces of Bounded Linear Operators
- For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator between Banach spaces X and Y , K(X, Y ) be the space of all compact operators, and W(X, Y ) be the space of all weakly compact operators. We denote space all operator ideals by O. digital.library.unt.edu/ark:/67531/metadc4349/
- Completely Simple Semigroups
- The purpose of this thesis is to explore some of the characteristics of 0-simple semigroups and completely 0-simple semigroups. digital.library.unt.edu/ark:/67531/metadc130974/
- Completeness Axioms in an Ordered Field
- The purpose of this paper was to prove the equivalence of the following completeness axioms. This purpose was carried out by first defining an ordered field and developing some basic theorems relative to it, then proving that lim [(u+u)*]^n = z (where u is the multiplicative identity, z is the additive identity, and * indicates the multiplicative inverse of an element), and finally proving the equivalence of the five axioms. digital.library.unt.edu/ark:/67531/metadc131462/
- Completing the Space of Step Functions
- In this thesis a study is made of the space X of all step functions on [0,1]. This investigation includes determining a completion space, X*, for the incomplete space X, defining integration for X*, and proving some theorems about integration in X*. digital.library.unt.edu/ark:/67531/metadc164014/
- A Computation of Partial Isomorphism Rank on Ordinal Structures
- We compute the partial isomorphism rank, in the sense Scott and Karp, of a pair of ordinal structures using an Ehrenfeucht-Fraisse game. A complete formula is proven by induction given any two arbitrary ordinals written in Cantor normal form. digital.library.unt.edu/ark:/67531/metadc5387/
- The Computation of Ultrapowers by Supercompactness Measures
- The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI. digital.library.unt.edu/ark:/67531/metadc2201/
- Concerning linear spaces
- The basis for this thesis is H. S. Wall's book, Creative Mathematics, with particular emphasis on the chapter in that book entitled "More About Linear Spaces." digital.library.unt.edu/ark:/67531/metadc130586/
- Concerning Measure Theory
- The purpose of this thesis is to study the concept of measure and associated concepts. The study is general in nature; that is, no particular examples of a measure are given. digital.library.unt.edu/ark:/67531/metadc131547/
- Concerning the Convergence of Some Nets
- This thesis discusses the convergence of nets through a series of theorems and proofs. digital.library.unt.edu/ark:/67531/metadc130552/
- Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation
- A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented. digital.library.unt.edu/ark:/67531/metadc699977/