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Polish Spaces and Analytic Sets

Description: A Polish space is a separable topological space that can be metrized by means of a complete metric. A subset A of a Polish space X is analytic if there is a Polish space Z and a continuous function f : Z —> X such that f(Z)= A. After proving that each uncountable Polish space contains a non-Borel analytic subset we conclude that there exists a universally measurable non-Borel set.
Date: August 1997
Creator: Muller, Kimberly (Kimberly Orisja)
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Partner: UNT Libraries
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Algebraic Number Fields

Description: This thesis investigates various theorems on polynomials over the rationals, algebraic numbers, algebraic integers, and quadratic fields. The material selected in this study is more of a number theoretical aspect than that of an algebraic structural aspect. Therefore, the topics of divisibility, unique factorization, prime numbers, and the roots of certain polynomials have been chosen for primary consideration.
Date: August 1991
Creator: Hartsell, Melanie Lynne
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Conway's Link Polynomial: a Generalization of the Classic Alexander's Knot Polynomial

Description: The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define … more
Date: December 1986
Creator: Woodard, Mary Kay
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Hyperspaces

Description: This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are disc… more
Date: December 1976
Creator: Voas, Charles H.
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Continuous, Nowhere-Differentiable Functions with no Finite or Infinite One-Sided Derivative Anywhere

Description: In this paper, we study continuous functions with no finite or infinite one-sided derivative anywhere. In 1925, A. S. Beskovitch published an example of such a function. Since then we call them Beskovitch functions. This construction is presented in chapter 2, The example was simple enough to clear the doubts about the existence of Besicovitch functions. In 1932, S. Saks showed that the set of Besicovitch functions is only a meager set in C[0,1]. Thus the Baire category method for showing the e… more
Date: December 1994
Creator: Lee, Jae S. (Jae Seung)
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Primitive Substitutive Numbers are Closed under Rational Multiplication

Description: Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is … more
Date: August 1998
Creator: Ketkar, Pallavi S. (Pallavi Subhash)
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The History of the Calculus

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

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.
Date: 1940
Creator: Shawn, James Loyd
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Fourier Transforms of Functions on a Finite Abelian Group

Description: This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform… more
Date: August 1982
Creator: Currey, Bradley Norton
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Interpolation and Approximation

Description: In this paper, there are three chapters. The first chapter discusses interpolation. Here a theorem about the uniqueness of the solution to the general interpolation problem is proven. Then the problem of how to represent this unique solution is discussed. Finally, the error involved in the interpolation and the convergence of the interpolation process is developed. In the second chapter a theorem about the uniform approximation to continuous functions is proven. Then the best approximation and… more
Date: May 1977
Creator: Lal, Ram
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Explicit Multidimensional Solitary Waves

Description: In this paper we construct explicit examples of solutions to certain nonlinear wave equations. These semilinear equations are the simplest equations known to possess localized solitary waves in more that one spatial dimension. We construct explicit localized standing wave solutions, which generate multidimensional localized traveling solitary waves under the action of velocity boosts. We study the case of two spatial dimensions and a piecewise-linear nonlinearity. We obtain a large subset of th… more
Date: August 1990
Creator: King, Gregory B. (Gregory Blaine)
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Finite Difference Methods for Approximating Solutions to the Heat Equation

Description: This paper is concerned with finite difference methods for approximating solutions to the partial differential heat equation. The first chapel gives some introductory background into the physical problem, then motivates three finite difference methods. Chapters II through IV provide statements and proofs for the theorems used in the methods of Chapter I. The final Chapter, V, provides conclusions and an indication of future work. An appendix includes the computer codes written by the author wit… more
Date: August 1982
Creator: Neuberger, Barbara O. (Barbara Osher)
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Some Properties of Derivatives

Description: This paper is concerned with certain properties of derivatives and some characterizations of linear point sets with derivatives. In 1946, Zygmunt Zahorski published a letter on this topic listing a number of theorems without proof, and no proof of these assertions has been published. Some of the theorems presented here are paraphrases of Zahorski's statements, developed in a slightly different order.
Date: 1951
Creator: Dibben, Philip W.
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