Digital Commons Network^™

Zonality In Graphs, Andrew Bowling

Dissertations

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

Using Visual Imagery To Develop Multiplication Fact Strategies, Gina Kling

Dissertations

The learning of basic facts, or the sums and products of numbers 0–10 and their related differences and quotients, has always been a high priority for elementary school teachers. While memorization of basic facts has been a hallmark of elementary school, current recommendations focus on a more nuanced development of fluency with these facts. Fluency is characterized by the ability to demonstrate flexibility, accuracy, efficiency, and appropriate strategy use. Despite recommendations to focus on strategy use, there is insufficient information on instructional approaches that are effective for developing strategies, particularly for multiplication facts. Using visual imagery with dot patterns has …

Irregular Orbital Domination In Graphs, Peter E. Broe

Dissertations

In recent decades, domination in graphs has become a popular area of study due in large degree to its applications to modern society and the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and of Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are …

Characterizing Undergraduate Students’ Proving Processes Around “Stuck Points”, Yaomingxin Lu

Dissertations

Learning to prove mathematical propositions is a cornerstone of mathematics as a discipline (de Villiers, 1990). However, since proving is a different mathematical activity as compared to students’ prior experience, research has also shown that many undergraduate students struggle to learn to prove, including those who major in mathematics (Moore, 1994; Selden, 2012). While the field has generated research that has analyzed the final products of proof (Selden & Selden, 2009) and there are frameworks for analyzing problem-solving processes (e.g., Carlson & Bloom, 2005; Schoenfeld, 1985, 2010), much remains to be known about analyzing undergraduate students’ proving processes. With a …

Dominating Functions In Graphs, Maria Talanda-Fisher

Dissertations

Domination in graphs has become one of the most popular areas of graph the- ory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. While this area evidently began with the work by the French mathematician Claude Berge in 1958 and the Norwegian-American mathematician Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of the survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous …

From Multi-Prime To Subset Labelings Of Graphs, Bethel I. Mcgrew

Dissertations

A graph labeling is an assignment of labels (elements of some set) to the vertices or edges (or both) of a graph G. If only the vertices of G are labeled, then the resulting graph is a vertex-labeled graph. If only the edges are labeled, the resulting graph is an edge-labeled graph. The concept was first introduced in the 19th century when Arthur Cayley established Cayley’s Tree Formula, which proved that there are n^n-2 distinct labeled trees of order n. Since then, it has grown into a popular research area.

In this study, we first review several types …

On Problems In Random Structures, Ryan Cushman

Dissertations

This work addresses two problems in optimizing substructures within larger random structures. In the first, we study the triangle-packing number v(G), which is the maximum size of a set of edge-disjoint triangles in a graph. In particular we study this parameter for the random graph G(n,m). We analyze a random process called the online triangle packing process in order to bound v(G). The lower bound on v(G(n,m)) that this produces allows for the verification of a conjecture of Tuza for G( …

Saudi Elementary Mathematics Teachers’ Knowledge For Teaching Fractions, Mona Khalifah A Aladil

Dissertations

Recent reform efforts in Saudi Arabia attend to mathematics instruction with a great deal of emphasis to improve Saudi mathematics education. Studies in different countries have confirmed that teachers’ mathematical knowledge for teaching plays an important role in mathematical quality of instruction and students’ achievement (e.g., Ball, 1990; Baumert et al., 2010; Hill, Rowan, & Ball, 2005). Yet few studies about mathematics teachers’ knowledge for teaching have been conducted in the Saudi context. This study investigates Saudi elementary mathematics teachers’ knowledge for teaching in the content strand of rational numbers with an emphasis on fractions, which is an important step …

On The Local Theory Of Profinite Groups, Mohammad Shatnawi

Dissertations

Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H^' an abelian group where H is a subgroup of G and H' is the derived group of H. One important first application is Burnside’s normal p-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. …

On Codes Over Rings: The Macwilliams Extension Theorem And The Macwilliams Identities, Noha Abdelghany

Dissertations

The MacWilliams extension theorem for code equivalence and the MacWilliams identities for weight enumerators of a code and its dual code are two of the most important results in classical coding theory. In this thesis, we study how much these two results could be extended to codes over more general alphabets, beyond finite fields. In particular, we study the MacWilliams extension theorem and the MacWilliams identities for codes over rings and modules equipped with general weight functions.

Extremal Problems On Induced Graph Colorings, James Hallas

Dissertations

Graph coloring is one of the most popular areas of graph theory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. As far back as 1880, in an attempt to solve the famous Four Color Problem, there have been numerous examples of certain types of graph colorings that have generated other graph colorings of interest. These types of colorings only gained momentum a century later, however, when in the 1980s, edge colorings were studied that led to vertex colorings of various types, led by the introduction …

Variations In Ramsey Theory, Drake Olejniczak

Dissertations

The Ramsey number R(F,H) of two graphs F and H is the smallest positive integer n for which every red-blue coloring of the (edges of a) complete graph of order n results in a graph isomorphic to F all of whose edges are colored red (a red F) or a blue H. Beineke and Schwenk extended this concept to a bipartite version of Ramsey numbers, namely the bipartite Ramsey number BR(F,H) of two bipartite graphs F and H is the smallest positive integer rsuch that every red-blue coloring of the r-regular complete bipartite graph results in either …

The Role Of Sampling Variability In Developing K-8 Preservice Teachers’ Informal Inferential Reasoning, Omar Abu-Ghalyoun

Dissertations

Recent influential policy reports, such as the Common Core State Standards (CCSS-M, 2010) and Guidelines for Assessment and Instruction in Statistics Education Report, (GAISE, 2007), have called for dramatic changes in the statistics content included in the K-8 curriculum. In particular, students in these grades are now expected to develop Informal Inferential Reasoning (IIR) as a way of preparing them for formal concepts of inferential statistics such as confidence intervals and testing hypotheses. Ben-Zvi, Gil, & Apel, (2007) describe IIR as the cognitive activities involved in informally making statistical inferences. Over this path from informal to formal inference, many important …

Uniformly Connected Graphs, Nasreen Almohanna

Dissertations

Perhaps the most fundamental property that a graph can possess is that of being connected. Two vertices u and v of a graph G are connected if G contains a u-v path. The graph G itself is connected if every two vertices of G are connected. The well-studied concept of connectivity provides a measure on how strongly connected a graph may be. There are many other degrees of connectedness for a graph. A Hamiltonian path in a graph G is a path containing every vertex of G. Among the best-known classes of highly connected graph are the Hamiltonian-connected graphs, …

Generalized Line Graphs, Mohra Abdullah Z. Alqahtani

Dissertations

With every nonempty graph, there are associated many graphs. One of the best known and most studied of these is the line graph L (G) of a graph G, whose vertices are the edges of G and where two vertices of L (G) are adjacent if the corresponding edges of G are adjacent. This concept was implicitly introduced by Whitney in 1932. Over the years, characterizations of graphs that are line graphs have been given, as well as graphs whose line graphs have some specified property. For example, Beineke characterized graphs that are line graphs by forbidding certain graphs …

Probabilistic And Extremal Problems In Combinatorics, Sean English

Dissertations

Graph theory as a mathematical branch has been studied rigorously for almost three centuries. In the past century, many new branches of graph theory have been proposed. One important branch of graph theory involves the study of extremal graph theory. In 1941, Turán studied one of the first extremal problems, namely trying to maximize the number of edges over all graphs which avoid having certain structures. Since then, a large body of work has been created in the study of similar problems. In this dissertation, a few different extremal problems are studied, but for hypergraphs rather than graphs. In particular, …

The Bellringer Sequence: Investigating What And How Preservice Mathematics Teachers Learn Through Pedagogies Of Enactment, Mary A. Ochieng

Dissertations

This study examines preservice teacher learning through pedagogies of enactment—approaches to teacher education that allow preservice teachers to learn by doing what teachers do. Preservice teacher (PST) learning is examined through the implementation of the Bellringer Sequence (BRS), a pedagogy of enactment conceptualized in the study. The BRS is centered around bellringers—brief mathematical tasks implemented as students arrive for class. The BRS is a sequence of four activities centered on a bellringer: preparation (for teaching a bellringer) implementation (of the bellringer with peers), debriefing (discussing the implementation as colleagues), and written reflection (about the effectiveness of the bellringer).

Practice-based approaches …

Induced Graph Colorings, Ian Hart

Dissertations

An edge coloring of a nonempty graph G is an assignment of colors to the edges of G. In an unrestricted edge coloring, adjacent edges of G may be colored the same. If every two adjacent edges of G are colored differently, then this edge coloring is proper and the minimum number of colors in a proper edge coloring of G is the chromatic index χ^/(G) of G. A proper vertex coloring of a nontrivial graph G is an assignment of colors to the vertices of G such that every two adjacent vertices of …

Graceful Colorings And Connection In Graphs, Alexis D. Byers

Dissertations

For a graph G of size m, a graceful labeling of G is an injective function f : V (G) → {0, 1, . . . , m} that gives rise to a bijective function f ¹ : E(G) → {1, 2, . . . , m} defined by f ¹(uv) = |f (u) − f (v)|. A graph is graceful if it has a graceful labeling. Over the years, a number of variations of graceful …

Edge Induced Weightings Of Uniform Hypergraphs And Related Problems, Laars C. Helenius

Dissertations

The starting point of the research is the so called 1-2-3 Conjecture formulated in 2004 by Karoński, Luczak, and Thomason. Roughly speaking it says that the edges of any graph can be weighted from {1, 2, 3} so that the induced vertex coloring (as the sum of weights adjacent to a given vertex) is proper. The conjecture has attracted a lot of interest from researchers over the last decade but is still unanswered. More recently, the conjecture has been studied for hypergraphs.

The main result of this dissertation shows in particular that an analogous conjecture holds for almost all uniform …

Highly Hamiltonian Graphs And Digraphs, Zhenming Bi

Dissertations

A cycle that contains every vertex of a graph or digraph is a Hamiltonian cycle. A graph or digraph containing such a cycle is itself called Hamiltonian. This concept is named for the famous Irish physicist and mathematician Sir William Rowan Hamilton. These graphs and digraphs have been the subject of study for over six decades. In this dissertation, we study graphs and digraphs with even stronger Hamiltonian properties, namely highly Hamiltonian graphs and digraphs.

Pedagogical Moves As Characteristics Of One Instructor’S Instrumental Orchestrations With Tinkerplots And The Ti-73 Explorer: A Case Study, James L. Kratky

Dissertations

Those supporting contemporary reform efforts for mathematics education in the United States have called for increased use of technologies to support student-centered learning of mathematical concepts and skills. There is a need for more research and professional development to support teachers in transitioning their instruction to better meet the goals of such reform efforts.

Instrumental approaches to conceptualizing technology use in mathematics education, arising out of the theoretical and empirical work in France and other European nations, show promise for use to frame studies on school mathematics in the United States. Instrumental genesis is used to describe the bidirectional and …

Chromatic Connectivity Of Graphs, Elliot Laforge

Dissertations

No abstract provided.

Resolving Classes And Resolvable Spaces In Rational Homotopy Theory, Timothy L. Clark

Dissertations

A class of topological spaces is called a resolving class if it is closed under weak equivalences and homotopy limits. Letting R(A) denote the smallest resolving class containing a space A, we say X is A-resolvable if X is in R(A), which induces a partial order on spaces. These concepts are dual to the well-studied notions of closed class and cellular space, where the induced partial order is known as the Dror Farjoun Cellular Lattice. Progress has been made toward illuminating the structure of the Cellular Lattice. For example: Chachólski, Parent, and Stanley have shown that it …

Edge Colorings Of Graphs And Their Applications, Daniel Johnston

Dissertations

Edge colorings have appeared in a variety of contexts in graph theory. In this work, we study problems occurring in three separate settings of edge colorings.

For more than a quarter century, edge colorings have been studied that induce vertex colorings in some manner. One research topic we investigate concerns edge colorings belonging to this class of problems. By a twin edge coloring of a graph G is meant a proper edge coloring of G whose colors come from the integers modulo k that induce a proper vertex coloring in which the color of a vertex is the sum of …

Phantom Maps, Decomposability, And Spaces Meeting Particular Finiteness Conditions, James Schwass

Dissertations

The purpose of this dissertation is to extend principles for detecting the existence of essential phantom maps into spaces meeting particular finiteness conditions. Zabrodsky shows that a space Y having the homotopy type of a finite CW complex is the target of essential phantom maps if and only if Y has a nontrivial rational homology group. We show this observation holds on the collection of finite generalized CW complexes. Similarly, Iriye shows a finite-type, simply connected suspension space is the target of essential phantom maps if and only if it has a nontrivial rational homology group. We show this observation …

Lnference On Differences In K Means For Data With Excess Zeros And Detection Limits, Haolai Jiang

Dissertations

Many data have excess zeros or unobservable values falling below detection limit. For example, data on hospitalization costs incurred by members of a health insurance plan will have zeros for the percentage who did not get sick. Benzene exposure measurements on petroleum re nery workers have some exposures fall below the limit of detection. Traditional methods of inference like one-way ANOVA are not appropriate to analyze such data since the point mass at zero violates typical distribution assumptions.

For testing for equality of means of k distributions, we will propose a likelihood ratio test that accounts for excess zeros or …

Modular Monochromatic Colorings, Spectra And Frames In Graphs, Chira Lumduanhom

Dissertations

Abstract attached as separate document.

Mathematical Methods Of Analysis For Control And Dynamic Optimization Problems On Manifolds, Robert J. Kipka

Dissertations

Mathematical Methods Of Analysis For Control And Dynamic Optimization Problems On Manifolds Driven by applications in fields such as robotics and satellite attitude control, as well as by a need for the theoretical development of appropriate tools for the analysis of geometric systems, problems of control of dynamical systems on manifolds have been studied intensively during the past three decades. In this dissertation we suggest new mathematical techniques for the study of control and dynamic optimization problems on manifolds. This work has several components including: an extension of the classical Chronological Calculus to control and dynamical systems which are merely …