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 Degree Discipline: Mathematics
Absolute Continuity and the Integration of Bounded Set Functions

Absolute Continuity and the Integration of Bounded Set Functions

Date: May 1975
Creator: Allen, John Houston
Description: 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.
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Abstract Measure

Abstract Measure

Date: 1957
Creator: Bridges, Robert Miller
Description: This study of abstract measure covers classes of sets, measures and outer measures, extension of measures, and planer measure.
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Abstract Vector Spaces and Certain Related Systems

Abstract Vector Spaces and Certain Related Systems

Date: August 1961
Creator: Goddard, Alton Ray
Description: The purpose of this paper is to make a detailed study of vector spaces and a certain vector-like system.
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Additive Functions

Additive Functions

Date: June 1963
Creator: McNeir, Ridge W.
Description: 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).
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Algebraic Integers

Algebraic Integers

Date: August 1969
Creator: Black, Alvin M.
Description: 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.
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Algebraic Properties of Semigroups

Algebraic Properties of Semigroups

Date: May 1971
Creator: Lumley, Robert Don
Description: 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.
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Algebraically Determined Rings of Functions

Algebraically Determined Rings of Functions

Date: August 2010
Creator: McLinden, Alexander Patrick
Description: 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.
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Algebraically Determined Semidirect Products

Algebraically Determined Semidirect Products

Date: May 2011
Creator: Jasim, We'am Muhammad
Description: 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 ...
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A*-algebras and Minimal Ideals in Topological Rings

A*-algebras and Minimal Ideals in Topological Rings

Date: May 1973
Creator: Wei, Jui-Hung
Description: The present thesis mainly concerns B*-algebras, A*-algebras, and minimal ideals in topological rings.
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Algorithms of Schensted and Hillman-Grassl and Operations on Standard Bitableaux

Algorithms of Schensted and Hillman-Grassl and Operations on Standard Bitableaux

Date: August 1983
Creator: Sutherland, David C. (David Craig)
Description: In this thesis, we describe Schensted's algorithm for finding the length of a longest increasing subsequence of a finite sequence. Schensted's algorithm also constructs a bijection between permutations of the first N natural numbers and standard bitableaux of size N. We also describe the Hillman-Grassl algorithm which constructs a bijection between reverse plane partitions and the solutions in natural numbers of a linear equation involving hook lengths. Pascal programs and sample output for both algorithms appear in the appendix. In addition, we describe the operations on standard bitableaux corresponding to the operations of inverting and reversing permutations. Finally, we show that these operations generate the dihedral group D_4
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The Analogues for t-Continuity of Certain Theorems on Ordinary Continuity

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

Date: 1941
Creator: Parrish, Herbert C.
Description: This study investigates the relationship between ordinary continuity and t-continuity.
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Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions

Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions

Date: December 2002
Creator: Valdes, LeRoy I.
Description: 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.
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The Analytical Development of the Trigonometric Functions

The Analytical Development of the Trigonometric Functions

Date: 1951
Creator: Mackey, Pearl Cherrington
Description: This thesis is a study of the analytical development of the trigonometric functions.
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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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Applications of Graph Theory and Topology to Combinatorial Designs

Applications of Graph Theory and Topology to Combinatorial Designs

Date: December 1988
Creator: Somporn Sutinuntopas
Description: 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, ...
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Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory

Applications of Rapidly Mixing Markov Chains to Problems in Graph Theory

Date: August 1993
Creator: Simmons, Dayton C. (Dayton Cooper)
Description: 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.
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An Approximate Solution to the Dirichlet Problem

An Approximate Solution to the Dirichlet Problem

Date: August 1964
Creator: Redwine, Edward William
Description: 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.
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Around the Fibonacci Numeration System

Around the Fibonacci Numeration System

Date: May 2007
Creator: Edson, Marcia Ruth
Description: 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.
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Aspects of Universality in Function Iteration

Aspects of Universality in Function Iteration

Date: December 1991
Creator: Taylor, John (John Allen)
Description: This work deals with some aspects of universal topological and metric dynamic behavior of iterated maps of the interval.
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Atmospheric Gusts and Their Effect on Aircraft

Atmospheric Gusts and Their Effect on Aircraft

Date: August 1958
Creator: Walling, Waunnetta Keene
Description: This thesis investigates atmospheric gusts and their effect on aircraft.
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Automorphism Groups

Automorphism Groups

Date: August 1965
Creator: Edwards, Donald Eugene
Description: 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.
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Automorphism Groups of Strong Bruhat Orders of Coxeter Groups

Automorphism Groups of Strong Bruhat Orders of Coxeter Groups

Date: August 1986
Creator: Sutherland, David C. (David Craig)
Description: In this dissertation, we describe the automorphism groups for the strong Bruhat orders A_n-1, B_n, and D_n. In particular, the automorphism group of A_n-1 for n ≥ 3 is isomorphic to the dihedral group of order eight, D_4; the automorphism group of B_n for n ≥ 3 is isomorphic to C_2 x C_2 where C_2 is the cyclic group of order two; the automorphism group of D_n for n > 5 and n even is isomorphic to C_2 x C_2 x C_2; and the automorphism group of D_n for n ≥ 5 and n odd is isomorphic to the dihedral group D_4.
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Axiom of Choice Equivalences and Some Applications

Axiom of Choice Equivalences and Some Applications

Date: August 1983
Creator: Race, Denise T. (Denise Tatsch)
Description: In this paper several equivalences of the axiom of choice are examined. In particular, the axiom of choice, Zorn's lemma, Tukey's lemma, the Hausdorff maximal principle, and the well-ordering theorem are shown to be equivalent. Cardinal and ordinal number theory is also studied. The Schroder-Bernstein theorem is proven and used in establishing order results for cardinal numbers. It is also demonstrated that the first uncountable ordinal space is unique up to order isomorphism. We conclude by encountering several applications of the axiom of choice. In particular, we show that every vector space must have a Hamel basis and that any two Hamel bases for the same space must have the same cardinality. We establish that the Tychonoff product theorem implies the axiom of choice and see the use of the axiom of choice in the proof of the Hahn- Banach theorem.
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Basic Fourier Transforms

Basic Fourier Transforms

Date: January 1962
Creator: Cumbie, James Randolph
Description: 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.
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