Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of automorphisms on the covering space is equivalent to the original surface.

The purpose of this paper is to explore certain properties of Banach spaces. The first chapter begins with basic definitions, includes examples of Banach spaces, and concludes with some properties of continuous linear functionals. In the second chapter, dimension is discussed; then one version of the Hahn-Banach Theorem is presented. The third chapter focuses on dual spaces and includes an example using co, RI, and e'. The role of locally convex spaces is also explored in this chapter. In the fourth chapter, several more theorems concerning dual spaces and related topologies are presented. The final chapter focuses on reflexive spaces. In the main theorem, the relation between compactness and reflexivity is examined. The paper concludes with an example of a non-reflexive space.

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 convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.

This paper examines idam's conjuecture and some of its generalizations. In terms of Adam's conjecture, we prove Alspach and Parson's results f or Zpq and ZP2. More generally, we prove Babai's characterization of the CI-property, Palfy's characterization of CI-groups, and Brand's result for Zpr for polynomial isomorphism's. We also prove for the first time a characterization of the CI-property for 1 SG, and prove that Zn is a CI-Pn-group where Pn is the group of permutation polynomials on Z,, and n is square free.

The problem of embedding a manifold in Euclidean space is considered. Manifolds are introduced in Chapter I along with other basic definitions and examples. Chapter II contains a proof of the Regular Value Theorem along with the "Easy" Whitney Embedding Theorem. In Chapter III, vector bundles are introduced and some of their properties are discussed. Chapter IV introduces the Stiefel-Whitney classes and the four properties that characterize them. Finally, in Chapter V, the Stiefel-Whitney classes are used to produce a lower bound on the dimension of Euclidean space that is needed to embed real projective space.

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.

We define and construct bicentric circles with respect to three-sided polygons. Then using inherent properties of these circles, we explore both tangent properties, and areas generated from bicentric circles.

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 the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.

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 the infinite family of standing waves, and we exhibit several interesting features of the family. Our solutions include solitary waves that carry nonzero angular momenta in their rest frames. The spatial profiles of these solutions also furnish examples of symmetry breaking for nonlinear elliptic equations.

This paper describes the tension between intuition about number theory and attempts to formalize it. I will first examine the root of the dilemma, Godel's First Incompleteness Theorem, which demonstrates that in any reasonable formalization of number theory, there will be independent statements. After proving the theorem, I consider some of its consequences on intuition, focusing on Freiling's "Dart Experiment" which is based on our usual notion of the real numbers as a line. This experiment gives an apparent refutation of the Axiom of Choice and the Continuum Hypothesis; however, it also leads to an equally apparent paradox. I conclude that such paradoxes are inevitable as the formalization of mathematics takes us further from our initial intuitions.

We describe groups of positive type and prove that a group G is of positive type if and only if G admits a non-trivial partition. We completely classify groups of type 2, and present examples of other groups of positive type as well as groups of type zero.

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.

The purpose of this thesis is to examine certain questions concerning the Cantor ternary set. The second chapter deals with proving that the Cantor ternary set is equivalent to the middle thirds set of [0,1], closed, compact, and has Lebesgue measure zero. Further a proof that the Cantor ternary set is a locally compact, Hausdorff topological group is given. The third chapter is concerned with establishing the existence of a Haar integral on certain topological groups. In particular if G is a locally compact and Hausdorff topological group, then there is a non-zero translation invariant positive linear form on G. The fourth chapter deals with proving that for any Haar integral I on G there exists a unique Haar measure on G that represents I.

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 examined in Chapter III. An important result illustrated in this chapter is that power series rings possess some of the properties of rings of polynomials.

We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.

In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups.

Several properties of Hausdorff continua are considered in this paper. However, the major emphasis is on developing the properties of the hyperspaces 2x and C(X) of a Hausdorff continuum X. Preliminary definitions and notation are introduced in Chapter I. Chapters II and III deal with the topological structure of the hyperspaces and the concept of topological convergence. Properties of 2x and C(X) are investigated in Chapter IV, while Chapters V and VI are devoted to the Hausdorff continuum X. Chapter VII consists of theorems pertaining to Whitney maps and order arcs in 2x. Examples of C(X) are provided in Chapter VIII. Inverse sequences of Hausdorff continua and of their hyperspaces are considered in Chapter IX.

We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.