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Convergence of Infinite Series
The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.
Date:
August 1983
Creator:
Abbott, Catherine Ann
System:
The UNT Digital Library
The Moore-Smith Limit
It is the purpose of this thesis to indicate in more detail how various limits defined in analysis, as well as other concepts not ordinarily defined as limits, may be obtained as special cases of the Moore-Smith limit.
Date:
1952
Creator:
Alexander, Donnie B.
System:
The UNT Digital Library
A Development of a Set of Functions Analogous to the Trigonometric and the Hyperbolic Functions
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.
Date:
August 1954
Creator:
Allen, Alfred I.
System:
The UNT Digital Library
Absolute Continuity and the Integration of Bounded Set Functions
The first chapter gives basic definitions and theorems concerning set functions and set function integrals. The lemmas and theorems are presented without proof in this chapter. The second chapter deals with absolute continuity and Lipschitz condition. Particular emphasis is placed on the properties of max and min integrals. The third chapter deals with approximating absolutely continuous functions with bounded functions. It also deals with the existence of the integrals composed of various combinations of bounded functions and finitely additive functions. The concluding theorem states if the integral of the product of a bounded function and a non-negative finitely additive function exists, then the integral of the product of the bounded function with an absolutely continuous function exists over any element in a field of subsets of a set U.
Date:
May 1975
Creator:
Allen, John Houston
System:
The UNT Digital Library
Integration of Vector Valued Functions
This paper develops an integral for Lebesgue measurable functions mapping from the interval [0, 1] into a Banach space.
Date:
August 1972
Creator:
Anderson, Edmond Cardell, III
System:
The UNT Digital Library
R-Modules for the Alexander Cohomology Theory
The Alexander Wallace Spanier cohomology theory associates with an arbitrary topological space an abelian group. In this paper, an arbitrary topological space is associated with an R-module. The construction of the R-module is similar to the Alexander Wallace Spanier construction of the abelian group.
Date:
May 1973
Creator:
Anderson, Stuart Neal
System:
The UNT Digital Library
Complete Ordered Fields
The purpose of this thesis is to study the concept of completeness in an ordered field. Several conditions which are necessary and sufficient for completeness in an ordered field are examined. In Chapter I the definitions of a field and an ordered field are presented and several properties of fields and ordered fields are noted. Chapter II defines an Archimedean field and presents several conditions equivalent to the Archimedean property. Definitions of a complete ordered field (in terms of a least upper bound) and the set of real numbers are also stated. Chapter III presents eight conditions which are equivalent to completeness in an ordered field. These conditions include the concepts of nested intervals, Dedekind cuts, bounded monotonic sequences, convergent subsequences, open coverings, cluster points, Cauchy sequences, and continuous functions.
Date:
August 1977
Creator:
Arnold, Thompson Sharon
System:
The UNT Digital Library
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
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
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.
Date:
January 1964
Creator:
Aslan, Farhad
System:
The UNT Digital Library
A Method for Approximating the Distributed Loads of an Airplane by Sets of Point Loads
This paper gives the derivation of a method for determining the forces to be applied to these points which will simulate the load distributed over all the airplane.
Date:
1957
Creator:
Austin, Charles Wayne
System:
The UNT Digital Library
Valuations and Valuation Rings
This paper is an investigation of several basic properties of ordered Abelian groups, valuations, the relationship between valuation rings, valuations, and their value groups and valuation rings. The proofs to all theorems stated without proof can be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1858. In Chapter I several basic theorems which are used in later proofs are stated without proof, and we prove several theorems on the structure of ordered Abelian groups, and the basic relationships between these groups, valuations, and their valuation rings in a field. In Chapter II we deal with valuation rings, and relate the structure of valuation rings to the structure of their value groups.
Date:
August 1975
Creator:
Badt, Sig H.
System:
The UNT Digital Library
Near-Rings
The primary objective of this work is to discuss some of the elementary properties of near-rings as they are related to rings. This study is divided into three subdivisions: (1) Basic Properties and Concepts of Near-Rings; (2) The Ideal Structure of Near-Rings; and (3) Homomorphism and Isomorphism of Near-Rings.
Date:
May 1972
Creator:
Baker, Edmond L.
System:
The UNT Digital Library
Completely Simple Semigroups
The purpose of this thesis is to explore some of the characteristics of 0-simple semigroups and completely 0-simple semigroups.
Date:
August 1968
Creator:
Barker, Bruce W.
System:
The UNT Digital Library
T-Functions
The main purpose of this paper is to make a detailed study of a certain class T of complex functions. The functions of the class T have a special mapping property and are meromorphic in every region. As an application of this study, certain elementary functions are defined and studied in terms of a special T-function.
Date:
June 1960
Creator:
Barlow, John Rice
System:
The UNT Digital Library
Product and Function Spaces
In this paper the Cartesian product topology for an arbitrary family of topological spaces and some of its basic properties are defined. The space is investigated to determine which of the separation properties of the component spaces are invariant.
Date:
August 1971
Creator:
Barrett, Lewis Elder
System:
The UNT Digital Library
Properties of Order Relations and Certain Partly Ordered Systems
The purpose of this paper is to present a study of partly ordered sets. It includes a rigorous development of relations based on the notion of a relation as a set, lattices, and theorems concerning the lattice of subgroups of a group.
Date:
June 1961
Creator:
Barros, David Nicholas
System:
The UNT Digital Library
A Set of Axioms for a Topological Space
Axioms for a topological space are generally based on neighborhoods where "neighborhood" is an undefined term. Then, limit points are defined in terms of neighborhoods. However, limit points seem to be the basic concept of a topological space, rather than neighborhoods. For this reason, it will be attempted to state a set of axioms for a topological space, using limit point as the undefined concept, and to delete the idea of neighborhoods from the theory.
Date:
August 1960
Creator:
Batcha, Joseph Patrick
System:
The UNT Digital Library
The Development of the Natural Numbers by Means of the Peano Postulates
This thesis covers the development of the natural numbers by means of the peano postulates.
Date:
1951
Creator:
Baugh, Orvil Lee
System:
The UNT Digital Library
On Sets and Functions in a Metric Space
The purpose of this thesis is to study some of the properties of metric spaces. An effort is made to show that many of the properties of a metric space are generalized properties of R, the set of real numbers, or Euclidean n--space, and are specific cases of the properties of a general topological space.
Date:
December 1971
Creator:
Beeman, Anne L.
System:
The UNT Digital Library
Lebesgue Linear Measure
This paper discusses the concept of a general definition of measure, and shows that the Lebesgue measure satisfies the requirements set forth for the ideal definition.
Date:
1940
Creator:
Beeman, William Edwin
System:
The UNT Digital Library
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.
Date:
August 1967
Creator:
Bell, Wayne Charles
System:
The UNT Digital Library
Some Theorems and Product Spaces
This thesis is a study of some axioms and theorems, and product spaces.
Date:
1957
Creator:
Bethel, Edward Lee
System:
The UNT Digital Library
Metric Spaces
This thesis covers fundamental properties of metric spaces, as well as completeness, compactness, and separability of metric spaces.
Date:
1957
Creator:
Bilyeu, Russell Gene
System:
The UNT Digital Library