Mathematics Commons^™

Bias, Structure, And Inference In Applied Network Analysis, Anna Vasenina

Dartmouth College Ph.D Dissertations

This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein–protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Some Results In Maximal Pattern Complexity, Casey Schlortt

Electronic Theses and Dissertations

For a finite alphabet 𝒜 and a sequence 𝑥 ∈ 𝒜^ℕ_⊬ , Kamae and Zamboni combined the ideas of block complexity and topological sequence entropy to define the maximal pattern complexity, 𝑝^∗_𝑛 (𝑥). They defined an aperiodic sequence 𝑥 over two letters as pattern Sturmian if it had the lowest possible maximal pattern complexity, 2𝑛. Later, Kamae and Rao extended their definition of pattern Sturmian sequences to be sequences over ℓ ≥ 2 letters which are not periodic by projection and have maximal pattern complexity ℓ𝑛.

This dissertation answers a question posed by Kamae and Zamboni …

Beyond The Best Fit: Fitting Sir-Style Models To Data While Prioritizing Biological Meaning, Meredith L. Greer

CODEE Journal

Many students find comfort in mathematics because math classes have been a place where they can find the “right answer” to exercises. Modeling courses typically emphasize more nuance, teaching students to try modeling approaches, compare back with real-world mechanisms and data, and then refine their models, in an ongoing cycle. One topic, however, can cause students to quickly revert into “right answer” mode: fitting a model to data. Phrases such as best fit and minimize the distance suggest there is one optimal solution that can be determined by an algorithm, and these algorithms typically have no relationship to the biology …

Uniform Stability Of Katyusha In Strongly-Convex Settings, Don Li

University Honors Theses

Acceleration of convergence and reduction of variance constitute a trade-off in the design of stochastic optimization machine learning algorithms. Katyusha was introduced to address this trade-off, synthesizing Nesterov Accelerated Gradient (NAG) and Stochastic Variance-Reduced Gradient (SVRG) into a single first-order optimizer with promising empirical performance. However, the generalization properties of Katyusha remain largely unexplored. We conjecture that, in the smooth quadratic regime (i.e., under assumptions of strong convexity and smoothness of the loss function, and boundedness of gradients), Katyusha is uniformly stable in the sense of Bousquet and Elisseeff. Instantiating our framework for NAG, we extend the use of Lyapunov …

The Method Of Periodic Averaging Applied To Reduced Coupled Mode Theory Models For Fiber Laser Amplifiers, Rebecca Nicole Bryant

Dissertations and Theses

Fiber laser amplifier (FLA) models are often implemented without rigorous mathematical justification or thorough numerical validation. Without a proper theoretical basis for assumptions and approximations, or a technical analysis of model performance, there is significant uncertainty about the limitations of any given reduced model and its suitability for an application. This research aims to address the lack of comprehensive assessment of FLA models by directly comparing distinct models and recommending a mathematical alternative to replace heuristic model-reduction techniques. The work in this dissertation is divided into two projects: a comparative study that uses existing FLA models to assess the validity …

Quantitatively Hyper-Positive Real Rational Functions Iii, Daniel Alpay, Izchak Lewkowicz

Mathematics, Physics, and Computer Science Faculty Articles and Research

Hyper-Positive Real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of Hyper-positive functions, related through nested inclusions, are introduced. Structurally, this family of functions turns out to be matrix-convex and closed under inversion. A state-space characterization of these functions through a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the classical Linear Matrix Inclusions, associated with passive systems, are here substituted by Quadratic Matrix Inclusions.

Course Portfolio: Elements Of Physics Phys 151, Evan A. Rich

UNL Faculty Course Portfolios

This course portfolio documents the instructional design, teaching methods, and ongoing assessment efforts for PHYS 151: Elements of Physics, an algebra-based introductory physics course at the University of Nebraska-Lincoln. The course serves a broad undergraduate population, including architecture, construction management, and life science majors. The portfolio describes the teaching framework that integrates pre-lecture video preparation, active in-class engagement through iClicker questions, and collaborative weekly recitation sections, all unified around a structured six-step problem-solving approach. A central concern of the course is building students’ self-efficacy in physics, particularly among those with math anxiety or limited preparation. Two assessments are reported: a …

Obstructions To Some Injective Oriented Colourings, Russell J. Campbell, Nancy E. Clarke, Gary Macgillivray

Theory & Applications of Graphs

Each of several possible definitions of local injectivity for a homomorphism of an oriented graph $G$ to an oriented graph $H$ leads to an injective oriented colouring problem. For each case in which such a problem is solvable in polynomial time, we identify a set $\mathcal{F}$ of oriented graphs such that an oriented graph $G$ has an injective oriented colouring with the given number of colours if and only if there is no $F \in \mathcal{F}$ for which there is a locally-injective homomorphism of $F$ to $G$.

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 …

A Copula-Based Framework For Multivariate Count Time Series With Mixed Marginal Distributions, Dimuthu Fernando, Yuxin Wen, Wimarsha Jayanetti

Engineering Faculty Articles and Research

We developed a class of multivariate integer-valued time series models using copula theory. Each count time series is modeled as a Markov chain, with serial dependence characterized through copula-based transition probabilities for Poisson and negative binomial marginals. Cross-sectional dependence is modeled via a trivariate Gaussian or a “t-copula”, allowing for both positive and negative correlations and providing a flexible dependence structure. Model parameters are estimated using likelihood-based inference, where the trivariate Gaussian or t-copula integrals are evaluated through standard randomized Monte Carlo methods. Simulation results, along with an analysis of annual counts of major hurricanes (Category 3+) across the North …

C*-Extreme Quantum Instruments; Completion And Disintegration Of Completely Positive Maps, Arghya Chongdar

Doctoral Theses

This thesis develops a comprehensive operator-algebraic framework for the study of completely positive (CP) instruments, with contributions spanning convexity theory, integration theory, completion problems, and disintegration theory. We begin by laying the foundational groundwork in the theory of C*-algebras and von Neumann algebras, introducing key structures such as CP maps, positive operator-valued measures (POVMs), and CP instruments, along with their dilation-theoretic properties. Pure and decomposable instruments are characterized via minimal bi-dilations, and CP instruments are realized as bivariate maps, providing a rigorous quantum analogue of classical joint measures. The thesis then investigates convexity-theoretic aspects of instruments. A structural characterization of …

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.

On The Number Of Ways To Express A Set As A Union Of Individually Interesting Sets, Alison Watson

Master's Theses

In a dataset that contains what ought rightly to be several distinct datasets placed in juxtaposition with each other, grouping datapoints based on observed similarities can be done in many ways. Topological Data Analysis (TDA) is a field of math that seeks to impose geometric structure onto datasets, thereby translating problems in statistics to problems in geometry or topology. A common truism in this field is “Data has shape and shape has meaning.” In this thesis, we use combinatorial and categorical arguments to demonstrate some shortcomings of a TDA approach on a class of inverse problems inspired by marine wildlife …

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.

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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 …

Mathematical Models Of Multisensory Detection And Decision-Making, Rebecca M. Brady

Doctoral

The brain seamlessly integrates signals from multiple sensory modalities to interpret the world efficiently. By using information from various senses, the brain can enhance its ability to detect and respond to stimuli more quickly and accurately. However, combining sensory cues from multiple modalities is only sometimes beneficial as it may lead to illusions and reduced behavioural performance. Behavioural and electrophysiological experiments have revealed that detection and decision-making strategies for multisensory cues evolve throughout human development and ageing. Additionally, studies have demonstrated that maladaptive multisensory processing is a key indicator of a proclivity to falls in older adults and individuals with …

Advanced Mathematical Modeling And Data-Driven Techniques For The Diagnosis Of Diabetes Using Continuous Glucose Monitoring (Cgm) Data, Farah Morsi

Theses

Diabetes mellitus is a major and growing health challenge, particularly in the Middle East and North Africa (MENA) region. Continuous Glucose Monitoring (CGM) provides high-resolution time-series data that capture detailed glucose fluctuations over time. However, conventional CGM summary measures, such as mean glucose, standard deviation, and time-in-range, may not fully describe the nonlinear temporal structure of glucose dynamics.

This thesis investigates nonlinear dynamical approaches for analyzing CGM time series, with a focus on recurrence-based analysis and ordinal-network analysis. Recurrence-based methods, including recurrence quantification analysis (RQA), are used to characterize geometric and temporal patterns in reconstructed phase space, while ordinal networks …

Control And Entrainment Of Oscillatory Dynamics In An Oncolytic Virus–Tumor Model Under Periodic Therapy Forcing, Aya Salaheddin Shujrawi

Theses

This thesis investigates the dynamics of a tumour–virus interaction model under sinusoidal periodic viral injection, using the three-compartment ordinary differential equation framework of Baabdulla and Hillen (2024). The aim is to characterise how the frequency and amplitude of periodic injection interact with the system's intrinsic oscillatory dynamics, and to identify conditions for stable frequency entrainment. Equilibrium and Hopf bifurcation analysis of the autonomous system yields a supercritical bifurcation at θ_H^auto ≈ 338.45 with intrinsic frequency Ω₀^auto ≈ 0.7552. Introducing a constant baseline injection u₀ = 0.05 raises the threshold to θ_H^forced ≈ 364.85 and shifts …

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 …

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. …

Fostering Innovation At The Intersection Of Maker Education And Extended Reality (Xr), Jewoong Moon, Yong Ju Jung, Soo Hyeon Kim, Younggon Bae, Bertrand Schneider

School of Mathematical & Statistical Sciences Faculty Publications

No abstract provided.

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.