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Plane Curves, Convex Curves, and Their Deformation Via the Heat Equation

Description: We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is … more
Date: August 1998
Creator: Debrecht, Johanna M.
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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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Polynomial Curve and Surface Fitting

Description: The main problems of numerical analysis involve performing analytical operations, such as integration, differentiation, finding zeroes, interpolation, and so forth, of a function when all the data available are some samples of the function. Therefore, the purpose of this paper is to investigate the following problem: given a set of data points (x[sub i], y[sub i]) which are samples of some function, determine an approximating function. Further, extend the problem to that of determining an appro… more
Date: January 1968
Creator: Capps, Ann Dowdy
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Polynomial Isomorphisms of Cayley Objects Over a Finite Field

Description: In this dissertation the Bays-Lambossy theorem is generalized to GF(pn). The Bays-Lambossy theorem states that if two Cayley objects each based on GF(p) are isomorphic then they are isomorphic by a multiplier map. We use this characterization to show that under certain conditions two isomorphic Cayley objects over GF(pn) must be isomorphic by a function on GF(pn) of a particular type.
Date: December 1989
Creator: Park, Hong Goo
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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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Product and Function Spaces

Description: 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
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Product Measure

Description: In this paper we will present two different approaches to the development of product measures. In the second chapter we follow the lead of H. L. Royden in his book Real Analysis and develop product measure in the context of outer measure. The approach in the third and fourth chapters will be the one taken by N. Dunford and J. Schwartz in their book Linear Operators Part I. Specifically, in the fourth chapter, product measures arise almost entirely as a consequence of integration theory. Both de… more
Date: August 1983
Creator: Race, David M. (David Michael)
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Proofs of Some Limit Theorems in Probability

Description: This study gives detailed proofs of some limit theorems in probability which are important in theoretical and applied probability, The general introduction contains definitions and theorems that are basic tools of the later development. Included in this first chapter is material concerning normal distributions and characteristic functions, The second chapter introduces lower and upper bounds of the ratio of the binomial distribution to the normal distribution., Then these bound are used to prov… more
Date: December 1974
Creator: Hwang, E-Bin
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Properties of Limit Functions

Description: The purpose of this paper is to investigate properties of functions which are limits of functions with prescribed properties. Chapter II asks the question "Does a function which is the limit of a sequence of functions each of which is endowed with a certain property necessarily have that property?"
Date: December 1972
Creator: Coppin, Anthony M.
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Properties of Power Series Rings

Description: This thesis investigates some of the properties of power series rings. The material is divided into three chapters. In Chapter I, some of the basic concepts of rings which are a prerequisite to an understanding of the definitions and theorems which follow are stated. Simple properties of power series rings are developed in Chapter II. Many properties of a ring R are preserved when we attach the indeterminant x to form the power series ring R[[x]]. Further results of power series rings are exami… more
Date: August 1990
Creator: O'Brien, Rita Marie
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Properties of R-Modules

Description: This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include exis… more
Date: August 1989
Creator: Granger, Ginger Thibodeaux
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Properties of Some Classical Integral Domains

Description: Greatest common divisor domains, Bezout domains, valuation rings, and Prüfer domains are studied. Chapter One gives a brief introduction, statements of definitions, and statements of theorems without proof. In Chapter Two theorems about greatest common divisor domains and characterizations of Bezout domains, valuation rings, and Prüfer domains are proved. Also included are characterizations of a flat overring. Some of the results are that an integral domain is a Prüfer domain if and only if eve… more
Date: May 1975
Creator: Crawford, Timothy B.
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