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Contributions To The Theory Of Neighborhoods And Its Applications, Michael Joseph Cullinane

Doctoral Dissertations

Neighborhoods have played a fundamental role in general topology since the birth of the field. This work outlines the historial evolution of the notion of neighborhood and employs neighborhood assignments, weak neighborhood assignments, and a naturally induced notion of duality in a study of non-Hausdorff topological spaces. Neighborhood characterizations of various classes of spaces, among them the developable and the pseudometrizable spaces, are obtained. A generalization of topological spaces based upon a primitive notion of neighborhood is explored and examples are supplied to motivate the investigation.

Contributions To The Theory Of Distance Functions And Its Application In General Topology, Lawrence Valery Neveu

Doctoral Dissertations

As non-Hausdorff spaces are becoming more important in topology, there is a need to consider new notions in topology to supplement the usual structures. This work uses distance functions to find useful generalizations (in a non-Hausdorff context) of the classes of spaces that are important in the Hausdorff setting. We begin, in the first part, with a historical overview that traces the evolution of the notion of distance and its role in the development of general topology.

In the second part of this work, we launch our study of distance functions. Using non-symmetric distance functions, called asymmetric, we generalize the …

Student Interactions And Mathematics Discourse: A Study Of The Development Of Discussions In A Fifth-Grade Classroom, Sharon Marie Soucy Mccrone

Doctoral Dissertations

Mathematics reform efforts are gaining attention and support in the years since the dissemination of the National Council of Teachers of Mathematics (NCTM) Standards documents. Encouraging student interactions in small and large group settings, and promoting discussion and argumentation of mathematical ideas among students are two possible implications of the vision presented in the Standards. The goal of having mathematics discussions, however, can present a variety of classroom challenges. Many factors influence classroom discourse and need to be addressed in ways that inform teachers as they work toward creating a more interactive, discussion-based mathematics classroom.

The study examines the development …

Perfect Matchings: Modified Aztec Diamonds, Covering Graphs Andn-Matchings, Adriana Badauta Cransac

Doctoral Dissertations

In the Introduction, we present the problems we are going to study and we establish the basic definitions, concepts and results that are used throughout.

We begin the first chapter with a presentation of the Aztec diamond and the behaviour of its random domino tilings. We introduce the dual-matching-problem and we explore the structure of the perfect matchings of modified Aztec diamonds. We show that some of these matchings can be extended to matchings of the dual Aztec diamond, pointing out a bijection between these types of matchings. We determine the number of perfect matchings for each of the modified …

On Some Range Inclusions, Mihaela Anca Poplicher

Doctoral Dissertations

The de Branges-Rovnyak spaces have been introduced by Louis de Branges and James Rovnyak, and have been studied by several authors, in particular by Donald Sarason. The object of this thesis is to study more extensively some properties of this class of spaces. In particular, which of these spaces are invariant under the action of which composition operators, or adjoints of composition operators.

Some known facts and a couple of new properties are included in Chapter 1. There it is shown that the norm of the kernel function for evaluation at a point in the unit disc (in the de …