Mathematics Commons^™

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( …

Assessing The Impact Of Vaccination And Behavior Change On Outbreaks Of Emerging Respiratory Diseases, Mohammed Hameed Alharbi

Mathematics Dissertations - Archive

The purpose of this dissertation is to use mathematical models to evaluate the impact of characteristics of respiratory diseases like influenza and COVID-19 either alone or co-circulating and how might influenza vaccine affect this interplay. First, we assess the effects of matching and mismatching between vaccine strains and circulating strains during the Hajj. Then we evaluate the impact of the proportion of asymptomatic COVID- 19 infections on the magnitude of an epidemic under three different behavior change scenarios. Finally, we model the co-circulation of influenza and COVID-19 to investigate whether the influenza vaccine increases the combined disease burden of influenza …

Exploring Development Of Problem Solving Strategies In Emerging Mathematicians, Andrew Charles Kercher

Mathematics Dissertations - Archive

To solve an unfamiliar mathematics problem, students of the subject must know more than the appropriate prerequisite content knowledge. They must also know how best to strategically apply their knowledge, how to monitor and gauge the effectiveness of their work, and how to respond (both cognitively and emotionally) to unanticipated results. Expanding current research on the types of experiences that foster these skills is the objective of this study. In a sequence of two task-based interviews, eight graduate and four upper-division undergraduate mathematics students solved non-traditional mathematics problems, used their work on these problems as a basis to comment on …

Current Trends In Random Walks On Random Lattices, Jewgeni H. Dshalalow

Mathematics and System Engineering Faculty Publications

In a classical random walk model, a walker moves through a deterministic d-dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset …

Mathematical Models Of Covid-19, Kate Faria

Honors Program Theses and Projects

For more than a year, the COVID-19 pandemic has been a major public health issue, affecting the lives of most people around the world. With both people’s health and the economy at great risks, governments rushed to control the spread of the virus. Containment measures were heavily enforced worldwide until a vaccine was developed and distributed. Although researchers today know more about the characteristics of the virus, a lot of work still needs to be done in order to completely remove the disease from the population. However, this is true for most of the infectious diseases in existence, including Influenza, …

Kamila Clustering For A Mixed-Type Data Analysis Of Illinois Medicare Data, Heather Elizabeth Baldacci

Undergraduate Honors Thesis Collection

The Centers for Medicare and Medicare Services (CMS) releases annual reports regarding the Market Saturation and Utilization of nationwide Medicare coverage. CMS data provide an opportunity for an in-depth analysis of Medicare usage patterns within the United States that may provide insight into socioeconomic conditions in certain regions. To discover any potential patterns, the KAMILA (KAy-means for MIxed LArge data sets) clustering algorithm has been utilized within the most recent CMS dataset from 2018. Due to the large size of the original data set, the focus of this research has been limited to Illinois Medicare data, grouped by the 102 …

Home Sweet Home: A Statistical Analysis Of Home Court Advantage In Division I College Basketball, Lauren Isabel Turnbull

Undergraduate Honors Thesis Collection

Home court advantage is a term that has been used throughout the basketball world by everyone from broadcasters and TV analysts to athletes and coaches to fans. It is a reference to the perceived benefit that the home team receives by playing in their own arena or stadium. Much of the prior research on home court advantage has centered around the reasons that it exists and how it impacts the play on the court through box score statistics. Building on this previous research, this study performs a statistical analysis of men's college basketball to determine if home court advantage is …

Secondary Teachers’ Noticing Of Students’ Mathematical Thinking As They Participate In A Professional Development Program Centered On Task-Based Student Interviews, Gurkan Kose

Theses, Dissertations and Culminating Projects

Teacher’s noticing of students’ mathematical thinking has been an important focus of research in the past two decades (e.g., Jacobs et al., 2010; Sherin et al., 2011). Noticing matters, but it is not an end in itself (Schoenfeld, 2011). It is operationalized within the context of teachers’ dispositions and knowledge which shape decisions teachers make while responding to student thinking and planning the next instructional steps. In order for teachers to adapt productive beliefs about how children learn and shift to student-centered instruction, they need to acknowledge the importance of understanding students' existing conceptions of mathematical ideas (Carpenter & Lehrer, …

2-Adic Valuations Of Square Spiral Sequences, Minh Nguyen

Honors Theses

The study of p-adic valuations is connected to the problem of factorization of integers, an essential question in number theory and computer science. Given a nonzero integer n and prime number p, the p-adic valuation of n, which is commonly denoted as νp(n), is the greatest non-negative integer ν such that p ν | n. In this paper, we analyze the properties of the 2-adic valuations of some integer sequences constructed from Ulam square spirals. Most sequences considered were diagonal sequences of the form 4n 2 + bn + c from the Ulam spiral with center value of 1. Other …

Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang

Electronic Theses and Dissertations

While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …

Constructions & Optimization In Classical Real Analysis Theorems, Abderrahim Elallam

Electronic Theses and Dissertations

This thesis takes a closer look at three fundamental Classical Theorems in Real Analysis. First, for the Bolzano Weierstrass Theorem, we will be interested in constructing a convergent subsequence from a non-convergent bounded sequence. Such a subsequence is guaranteed to exist, but it is often not obvious what it is, e.g., if an = sin n. Next, the H¨older Inequality gives an upper bound, in terms of p ∈ [1,∞], for the the integral of the product of two functions. We will find the value of p that gives the best (smallest) upper-bound, focusing on the Beta and Gamma integrals. …

Constructing Non-Proxy Small Test Modules For The Complete Intersection Property, Benjamin Briggs, Eloísa Grifo, Josh Pollitz

Department of Mathematics: Faculty Publications

A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a …

Proper Sum Graphs, Austin Nicholas Beard

Graduate Theses/Dissertations

The Proper Sum Graph of a commutative ring with identity has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring.

Math And Voting: Voting Methods, Fair Representation, And The Electoral College, Sarah Nelson

Honors Scholar Theses

Voting is in integral part of any functioning democracy, but there exist more than just one way to count votes. Some voting methods use only a voter’s top-choice candidate, while others require a ranking of all candidates, most-preferred to least-preferred, from each voter. We examine some of these ranked-choice voting methods, including the anti-plurality method, Hare’s method, and Coomb’s method.

Because of the variety of voting methods, we introduce criteria, which allow for an evaluation of the advantages and disadvantages of each method. The criteria give various definitions for what “good” or “bad” voting methods look like, depending on context. …

Frobenius And Homological Dimensions Of Complexes, Taran Funk

Department of Mathematics: Dissertations, Theses, and Student Research

Much work has been done showing how one can use a commutative Noetherian local ring R of prime characteristic, viewed as algebra over itself via the Frobenius endomorphism, as a test for flatness or projectivity of a finitely generated module M over R. Work on this dates back to the famous results of Peskine and Szpiro and also that of Kunz. Here I discuss what work has been done to push this theory into modules which are not necessarily finitely generated, and display my work done to weaken the assumptions needed to obtain these results.

Adviser: Tom Marley

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 …

Symmetric Presentation Of Finite Groups, And Related Topics, Marina Michelle Duchesne

Electronic Theses, Projects, and Dissertations

We have discovered original symmetric presentations for several finite groups, including 2²:^.(2⁴:(2^.S₃)), M₁₁, 3:(PSL(3,3):2), S₈, and 2^.M₁₂. We have found homomorphic images of several progenitors, including 2^*18:((6x2):6), 2^*24:(2^.S₄), 2^*105:A₇, 3^*3:_m(2³:3), 7^*8:_m(PSL(2,7):2), 3^*4:_m(4²:2²), 7^*5:(2xA₅), and 5^*6:_mS₅. We have provided the isomorphism type of …

Zariski Geometries And Quantum Mechanics, Milan Zanussi

Boise State University Theses and Dissertations

Model theory is the study of mathematical structures in terms of the logical relationships they define between their constituent objects. The logical relationships defined by these structures can be used to define topologies on the underlying sets. These topological structures will serve as a generalization of the notion of the Zariski topology from classical algebraic geometry. We will adapt properties and theorems from classical algebraic geometry to our topological structure setting. We will isolate a specific class of structures, called Zariski geometries, and demonstrate the main classification theorem of such structures. We will construct some Zariski structures where the classification …

Visual Analysis Of Historical Lessons Learned During Exercises For The United States Air Force Europe (Usafe), Samantha O'Rourke

Theses/Capstones/Creative Projects

Within the United States Air Force, there are repeated patterns of differences observed during exercises. After an exercise is completed, forms are filled out detailing observations, successes, and recommendations seen throughout the exercise. At the most, no two reports are identical and must be analyzed by personnel and then categorized based on common themes observed. Developing a computer application will greatly reduce the time and resources used to analyze each After Action Report. This application can visually represent these observations and optimize the effectiveness of these exercises. The visualization is done through graphs displaying the frequency of observations and recommendations. …

Algebraic Structures And Variations: From Latin Squares To Lie Quasigroups, Erik Flinn

All NMU Master's Theses

In this Master's Thesis we give an overview of the algebraic structure of sets with a single binary operation. Specifically, we are interested in quasigroups and loops and their historical connection with Latin squares; considering them in both finite and continuous variations. We also consider various mappings between such algebraic objects and utilize matrix representations to give a negative conclusion to a question concerning isotopies in the case of quasigroups.

Peg Solitaire On Graphs In Which We Allow Merging And Jumping, Amanda L. Mckinney

Undergraduate Honors Theses

Peg solitaire is a game in which pegs are placed in every hole but one and the player jumps over pegs along rows or columns to remove them. Usually, the goal of the player is to leave only one peg. In a 2011 paper, this game is generalized to graphs. In this thesis, we consider a variation of peg solitaire on graphs in which pegs can be removed either by jumping them or merging them together. To motivate this, we survey some of the previous papers in the literature. We then determine the solvability of several classes of graphs including …

Interlace Polynomials Of Certain Graphs, Cheyenne Petzold

Theses, Dissertations and Culminating Projects

In this research, we investigated the interlace polynomials of a shell graph as well as other related graphs. A shell graph, Tn is constructed by adding edges to a cycle graph such that all vertices are adjacent to one vertex. The main results of this thesis include iterative and explicit formulas for the interlace polynomial of a shell graph, denoted q(Tn; x). A linear algebra application using the adjacency matrices of the chosen graphs is also explored.

Visualizing Geometric Structures On Topological Surfaces, Andrea Clark

All NMU Master's Theses

We study an interplay between topology, geometry, and algebra. Topology is the study of properties unchanged by bending, stretching or twisting space. Geometry measures space through concepts such as length, area, and angles. In the study of two-dimensional surfaces one can go back and forth between picturing twists as either distortions of the geometric properties of the surface or as a wrinkling of the surface while leaving internal measures unchanged. The language of groups gives us a way to distinguish geometric structures. Understanding the mapping class group is an important and hard problem. This paper contributes to visualizing how the …

Sobolev Inequalities And Riemannian Manifolds, John Reever

Honors Scholar Theses

Sobolev inequalities, named after Sergei Lvovich Sobolev, relate norms in Sobolev spaces and give insight to how Sobolev spaces are embedded within each other. This thesis begins with an overview of Lebesgue and Sobolev spaces, leading into an introduction to Sobolev inequalities. Soon thereafter, we consider the behavior of Sobolev inequalities on Riemannian manifolds. We discuss how Sobolev inequalities are used to construct isoperimetric inequalities and bound volume growth, and how Sobolev inequalities imply families of other Sobolev inequalities. We then delve into the usefulness of Sobolev inequalities in determining the geometry of a manifold, such as how they can …

Factors That Influence Teachers' Utilization Of Siop Learning Strategies When Teaching English Language Learners, Courtney R. Stuart

Theses and Dissertations

Despite standardized certification tests and popular English Language Learner trainings, the strategies and methods used in two educators’ classes would not be identical. In this case-study we look to see what variables cause teachers to utilize certain strategies and how these variables affect the number of strategies used consistently when teaching ELLs. We study the effect of teachers’ language experiences, certifications, participation in SIOP trainings, education levels, perceived preparedness, and years spent in the classroom.

Using quantitative and qualitative data we conclude that dual-language fluency and professional advancements can lead to teachers using more ELL learning strategies in their classrooms, …

Blood Flow Through An Artery In The Presence Of A Stenosis, Martin Carrillo

Theses and Dissertations

We consider blood flow through an artery in the form of a cylindrical pipe in the presence of a stenosis. Here blood is treated as an incompressible, viscous and non-Newtonian Bingham plastic fluid. We derive the equation of continuity and the momentum equation which are obtained using mass conservation law and momentum conservation law, respectively. Assuming that the flow is due to the pressure drop and wall shear stress, we derive the expressions for the velocity component in the axial direction and the volumetric flow rate in an artery. Computational results for the axial velocity and flow rate are obtained …

Optimal Quantization For Mixtures Of Two Uniform Distributions, Eduardo Orozco

Theses and Dissertations

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Mixtures of probability distributions, also known as mixed distributions, are an exciting new area for optimal quantization. In this thesis, for a mixed distribution we determine the optimal sets of n-means and the nth quantization errors for all positive integers n.

Exploring The Beginnings Of Algebraic K-Theory, Sarah Schott

Boise State University Theses and Dissertations

According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K₀(R) as the free abelian group whose basis are the finitely generated projective R-modules with the added relation P ⊕ Q = P + Q. The purpose of this thesis is to study simple settings of the K-theory for rings and …

Cortical Dynamics Of Language, Kiefer Forseth

Dissertations and Theses (Open Access)

The human capability for fluent speech profoundly directs inter-personal communication and, by extension, self-expression. Language is lost in millions of people each year due to trauma, stroke, neurodegeneration, and neoplasms with devastating impact to social interaction and quality of life. The following investigations were designed to elucidate the neurobiological foundation of speech production, building towards a universal cognitive model of language in the brain. Understanding the dynamical mechanisms supporting cortical network behavior will significantly advance the understanding of how both focal and disconnection injuries yield neurological deficits, informing the development of therapeutic approaches.

Resources For Supporting Mathematics And Data Science Instructors During Covid-19, Eduardo C. Balreira, C. Hawthorne, G. Stadnyk, Z. Teymuroglu, M. Torres, J. R. Wares

Mathematics Faculty Research

In late May of 2020, a few months after the raging COVID-19 pandemic forced university faculty to quickly switch to online teaching, the Associated Colleges of the South (ACS) released a call for grant applications to support working groups "to help faculty within our consortium who will be teaching during the pandemic (e.g., from hybrid courses with some remote/online components to fully remote/online courses; socially distanced face-to-face courses)." We replied to this call and the ACS awarded the six of us (from four ACS schools) a Summer Rapid Response Grant in early June. The grant funded our efforts to create …