Mathematics Commons^™

The Connected Vertex Cover In Graphs, Kamran Mirasheh, Elham Mirashe, Ebrahim Vatandoost

Theory & Applications of Graphs

This paper presents tight bounds and characterizations for the vertex cover number and the connected vertex cover number of graphs. In particular, we identify all graphs for which βc(G) = |V (G)| − 1, proving that these are exactly the cycles and complete graphs. The analysis employs tools such as the degree matrix and the Rayleigh quotient to derive new and sharp upper bounds.

The $K$-Total Bondage Number Of A Graph, Jean-Pierre Appel, Gabrielle Fischberg, Kyle Kelley, Nathan B. Shank, Eliel Sosis

Theory & Applications of Graphs

Let \(G=(V,E)\) be a connected, finite undirected graph. A set \(S \subseteq V\) is said to be a total dominating set of \(G\) if every vertex in \(V\) is adjacent to some vertex in \(S\). The total domination number, \(\gamma_{t}(G)\), is the minimum cardinality of a total dominating set in \(G\). We define the \(k\)-total bondage of $G$ to be the minimum number of edges to remove from \(G\) so that the resulting graph has a total domination number at least \(k\) more than \(\gamma_{t}(G)\). In this work we establish general properties of \(k\)-total bondage and find exact values for …

Hyper Face-Magic Graphs, Ross Belgram, Donald Mcginn

Theory & Applications of Graphs

For a planar graph G of order n, let F(G) be the set of all faces of G embedded into R 2 , including the exterior face. A bijective vertex labeling f : V (G) → {1, 2, ..., n} induces a face labeling f ∗ : F(G) → N defined by setting f ∗ (F) equal to the sum of all labels of the boundary vertices of F. The graph G is said to be hyper face-magic if there exists a vertex labeling whose induced face labeling is constant. In this paper, we state properties of hyper face-magic graphs …

The Malaugh Operations And The Quadratic Invariant Gaps Of $\Si_4^C$, Forrest M. Hilton

ETDs from 2020-2029

Connected Julia sets of polynomials generally correspond to laminations, sets of chords of the unit disc that reflect the dynamics of the Julia set. If the circle is measured in revolutions and the polynomials studied are of degree $d$, then the dynamics on the lamination is given by the covering map $\sigma_d(t) := td \pmod 1$ where chords are mapped by their end points. Every lamination has at least one laminational invariant set, which is loosely an invariant complementary component of the lamination. That set has a significant impact on the shape of the Julia set. James Malaugh showed how …

On The Fractional Laplacian Type Operator, Maysam Abdulnaser Zain

Thesis/ Dissertation Defenses

In this thesis, we study analytical structures arising from Dunkl theory and their applications to harmonic analysis and fractional Laplacian operators. Dunkl operators are differential-difference operators associated with finite reflection groups, providing a natural generalization of the classical Fourier analysis through the introduction of root systems and multiplicity functions. Within this framework, several classical transforms appear as special cases of the (k,a)-generalized Fourier transform. We study the generalized Fourier transform, its kernel, and the associated translation operator and convolution structures. Using these tools, we construct the corresponding heat kernel and analyze the associated heat semigroup. Our main contribution concerns the …

The Impact Of Evolution And Strong Allee Effects On The Dynamics Of A Discrete-Time Predator-Prey System, Neerob Basak

Doctoral Dissertations

This dissertation investigates the dynamics of discrete-time predator-prey systems, focusing on both evolutionary responses and ecological interactions. The first part of this dissertation, in Chapters 2 and 3, explores how evolutionary processes, particularly the development of resistance to toxicants in predators, influence the persistence and stability of predator-prey populations. In this part, we extend the predator-prey model developed in Ackleh et al., 2019 to incorporate the evolution of a predator's resistance to toxicant effects. We consider three cases: (1) lethal effects, where the toxicant directly influences the predator's survival; (2) sublethal effects, where the toxicant impacts the predator's fecundity, and …

The Interplay Of Seasonality, Evolution, And Density-Dependence In Discrete-Time Predator-Prey Dynamics, Narendra Pant

Doctoral Dissertations

We extend the discrete-time mathematical models developed in (Ackleh et al., 2019) and (Ackleh et al., 2024) to account for seasonal prey reproduction and build a class of discrete-time predator-prey seasonal models. Each model distinguishes between breeding and non-breeding seasons, representing prey reproduction as a periodic function of period 2. Altogether, three different models are analyzed. In the first part, we extend the predator-prey model from (Ackleh et al., 2019) to incorporate seasonality. We study the resulting dynamics and show that when the inherent reproduction number of the prey and the invasion reproduction number of the predator are larger than …

Generating Live Heatmaps Of Edr Data Through A Spatiotemporal Weighting, Joel T. Williams, C. Sean Bohun, Alberto Fornaci, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

This paper introduces a novel method for generating live heatmaps of eddy dissipation rate (EDR) data through a spatiotemporal weighting designed to enhance turbulence visualization in aviation. As more flight data become available, approaches relying solely on in-flight EDR measurements have the potential to accurately nowcast and visualize turbulence with low computational cost. The proposed method significantly improves the turbulence visualization capabilities of common commercial aircraft. This is particularly valuable for pilot decision-making and trip planning, enhancing flight safety and operational efficiency. This approach also incorporates an innovative uncertainty threshold, which refrains from predicting when there are insufficient data, thereby …

Social Media Is A Juggernaut: Lagged Correlation Analysis Using Ngram Data On “Internet” And “Social Media” With Amplification By The Advent Of The “Iphone”, William Zywiak, Gao Niu, Nicolas Petrell, Victoria Nichele, Kirsten Hokeness

Mathematics and Economics Faculty Journal Articles

The amount of new information transmitted per day can be overwhelming. The data available through Ngram Viewer allows statistical examination to determine the most salient topics, as well as lagged correlation and lagged variance to support possible causal connections between concepts. Using this database, we determined that COVID, crypto, microplastics, and especially social media are important topics of the last few years. We also present results that suggest the development of the internet and the iPhone fueled the prominence of social media, and the access of the iPhone also increased access to the internet.

Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill

Master's Theses

Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of  $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. …

Algorithmic Problems In Automorphic Orbits Of Free Groups, Siobhan B. O'Connor

Dissertations, Theses, and Capstone Projects

One of the fundamental problems in the field of combinatorial group theory is telling when two group presentations represent isomorphic groups. Since applying a free group automorphism to the set of relators of a presentation gives an isomorphic group, we want to be able to quickly decide when looking at a relator whether a given word can be sent to it via an automorphism. We give a hands-on introduction to the automorphisms of free groups using patterns of colored beads. We show that you can make this decision correctly in constant time on average by looking for "orbit-blocking" words that …

Variational Methods For Semilinear Pdes With Dirac Singularities, Samuel J. Magill

Dissertations, Theses, and Capstone Projects

This dissertation utilizes a variational framework for semilinear elliptic equations in two dimensions with Dirac measure data. The central objects of study are equations of the form −ΔU = f(U) + Σ_j=1^N α_jδ_pj on bounded Lipschitz domains Ω ⊂ ℝ² with homogeneous Dirichlet boundary condition, and on a flat torus 𝕋², where α_j is positive for the Dirichlet setting and α_j is negative on the torus. Solutions are obtained by minimizing the restriction of an energy functional to an order interval determined by explicit sub- and supersolutions; the Euler–Lagrange equation is recovered …

Identifying Textual Predictors Of Early Termination In Clinical Trials In Medicine: An Explainable Machine-Learning Study, Rohan Ramnarain

Dissertations, Theses, and Capstone Projects

About one in five clinical trials in medicine ends early, wasting valuable resources and reducing the evidence available for developing life-saving medical treatments. This project uses a method called Trial2Vec, which is a self-supervised machine-learning method that converts clinical trial documents into dense numerical representations that capture their key design and clinical characteristics, to turn each proposed clinical trial’s written protocol into a compact numerical profile (a process referred to as embedding). These profiles are then paired with a predictive machine learning models to identify the words and phrases in the trial documents that can signal a higher risk of …

Graded Contact Geometry And The Aksz Formalism, Ivan Contreras, Nicolas Martinez Alba, Rajan Amit Mehta

Mathematics Sciences: Faculty Publications

The AKSZ formalism is a construction of topological field theories where the target spaces are differential graded symplectic manifolds. In this paper, we describe an analogue of the AKSZ formalism where the target spaces are differential graded contact manifolds. We show that the space of fields inherits a weak contact structure, and we construct a solution to the analogue of the classical master equation, defined via the Jacobi bracket. In the n =1 case, we recover the Jacobi sigma model, and in the n = 2 case, we obtain three-dimensional topological field theories associated to Courant-Jacobi algebroids.

Reduced Product Type Monoid-Module Extensions, Darryl Jent

Dissertations

In 1955, I. M. James introduced the James Construction, a free topological monoid that models the loops on the suspension of a given space. In 1969, S. Y. Husseini generalized this idea to RPT monoids: topological monoids with a free-like monoid structure that can be used to model a broader class of loop spaces. In order to prove that these topological monoids are models of loop spaces, both I. M. James and S. Y. Husseini constructed contractible spaces on which these topological monoids act. We define a topological module as a space equipped with an action by a topological monoid. …

From Total Domination To Graph Coloring, Sawyer Isaac Osborn

Dissertations

A question involving a chess piece called a prince on the 8×8 chessboard leads to a concept in graph theory involving total domination. We say a vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A subset S of the vertex set of a graph G is a total dominating set for G if every vertex in G is totally dominated by at least one vertex of S. If S is a total dominating set of G, then σ_S(v) denotes the number of …

Introduction To Mathematical Analysis Ii, Beatriz Lafferriere, Gerardo Lafferriere, Mau Nam Nguyen

PDXOpen: Open Educational Resources

These lecture notes complement the book Introduction to Mathematical Analysis I, Third Edition, with sections on integration, numerical series, series of functions, and selected advanced topics. Together, both volumes support a one semester or two-quarter course on introductory mathematical analysis.

Adopt/Adapt

If you are an instructor adopting or adapting this PDXOpen textbook, please help us understand your use by filling out this form.

Behind The Screen: A Comprehensive Framework For Digital Work Metrics And Data Integration, F. Demaria, A. Papana Dagiasis, M. Cavicchioli, T. Fabbri

Mathematics and Statistics Faculty Publications

The digital transformation and the subsequent datafication of work generate rich electronic traces that can be transformed into actionable insights through modern analytics. In this context, this study proposes a data-driven framework that leverages sidClustering and unsupervised Random Forest (RF) for clustering and feature selection, constructs composite indicators via multiple aggregation strategies, and employs visual tools to enhance the interpretability of results. A key feature of the framework is the integration of two data sources: click metadata from Microsoft 365 and employee attitudes measured through a questionnaire. This integration enables the analysis of digital work behavior (DWB) in relation to …

Equitable Decompositions: A Gateway To Spectral Theory Through Graph Automorphisms, Daniel Ford

Master's Theses

Graphs with symmetry appear throughout mathematics and its applications, from the structure of molecules and network design to combinatorial game theory. A central question in spectral graph theory is how to compute or characterise the eigenvalues of the matrices associated with such graphs. Classical decomposition methods, such as diagonalisation or Jordan normal form, accomplish this but only once some spectral information is already known. A different approach, introduced by Barrett et al. (2015), uses the automorphisms of a graph to block-diagonalise its adjacency matrix without any prior spectral information. Because one of the resulting summands is always the quotient matrix …

Gauss Composition And Orthogonal Modular Forms On Binary Lattices, Haochen Wu

Dartmouth College Ph.D Dissertations

We revisit Gauss composition over a general base scheme, with a focus on orthogonal groups. We show that the Clifford and norm functors provide a discriminant-preserving equivalence of categories between binary quadratic modules and pseudoregular modules over quadratic algebras. This perspective synthesizes the constructions of Kneser and Wood, reconciling algebraic and geometric approaches and clarifying the role of orientations and the natural emergence of narrow class groups.

As an application, we restrict to lattices and show that binary orthogonal eigenforms correspond to Hecke characters. Using theta series, we show the explicit connection between Hilbert modular forms and orthogonal modular forms …