Digital Commons Network^™

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 …

Whitney Extension Problem For Fractional Sobolev Spaces And Besov Spaces, Han Li

Dissertations, Theses, and Capstone Projects

In the dissertation, we go through the development of the Whitney extension problem and prove a type of results for the Whitney extension problem for homogeneous fractional Sobolev spaces and homogeneous Besov spaces.

This dissertation consists of four chapters:

Chapter 1: We recall the history of the Whitney extension problem and talk about some early works which have been done for the Whitney extension problem. We also mention our new results.

Chapter 2: We introduce some basic notations, definitions and preliminary results.

Chapter 3: We show the existence of a bounded linear extension operator for homogeneous fractional Sobolev space L …

Quiver Of Affine Monoid Of A Vector Space Over Finite Field, James Junie Chen Cleary

Dissertations, Theses, and Capstone Projects

In this paper, we study the quiver of the complex monoid algebra CAFF(n, q). There are n + 1 maximal subgroups of AFF(n, q), each isomorphic to AGL(k, q) for some 0 ≤ k ≤ n. Every irreducible representation of CAFF(n, q) arises from a character of CAGL(k, q) for a suitable k. Thus, we study two different approaches to classifying the characters of CAGL(k, q). Next, we compute the full quiver Q(CAFF(n, q)). Finally, we show that this quiver is a disjoint union of straight-line paths and that its basic algebra has radical square zero. Hence, it has finite …

Computational Insights Into Nucleosome Dynamics In Epigenetics Using Molecular Dynamics Simulations, Rutika Patel

Dissertations, Theses, and Capstone Projects

Nucleosome core particles (NCP) are the building blocks that form a highly organized and compact chromatin structure. Nucleosomes package DNA in the nucleus of eukaryotic cells. The NCP consists of about 147 base pairs of DNA wrapped around the histone octamer, with 1.65 superhelical turns in a left-handed manner. The histone octamer is composed of two copies of H3, H4, H2A, and H2B. Together with histone H1 and linker DNA, they further assemble into a higher-order chromatin structure. The nucleosome complex is stabilized by electrostatic interactions between positively charged histone residues and the negatively charged DNA backbone. To effectively access …

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 …

On The Fractional Laplacian Type Operator, Maysam Abdulnaser Zain

Theses

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 ��ₖ,ₐ, its kernel B_k,a (x,y), and the associated translation operator and convolution structures. Using these tools, we construct the corresponding heat …

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 …

Hyper-Bishops, Hyper-Rooks, And Hyper-Queens: Percentage Of Safe Squares On Higher Dimensional Chess Boards, Caroline Cashman, Joseph Cooper, Raul Marquez, Steven J. Miller, Jenna Shuffelton

School of Mathematical & Statistical Sciences Faculty Publications

Chess has inspired an abundance of mathematical problems, especially in combinatorics and probability. One such problem, initially studied by Miller, Sheng, and Turek, considers the proportion of safe spaces when randomly placing n rooks on an 𝑛×𝑛 chess board. They show that as n approaches infinity, the proportion of safe spaces converges to 1/𝑒2. We first generalize their results to bishops and queens. This problem is significantly more interesting and difficult; while a rook attacks the same number of spaces regardless of its position, this is not so for bishops and queens. We prove that the proportion of safe spaces …

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.

Fractional Bernstein Polynomial Approximations For Nonlinear Timefractional Partial Differential Equations, Reem Abdul Quzli

Theses

This thesis studies the numerical approximation of nonlinear time-fractional partial differential equations using fractional Bernstein polynomials. The main model considered is the nonlinear time-fractional foam drainage equation, in which the classical time derivative is replaced by the Caputo fractional derivative. This formulation introduces memory effects into the model and allows the present drainage behavior to depend on the previous evolution of the liquid fraction.

The proposed method approximates the solution by a finite expansion of fractional Bernstein basis functions. After substituting this approximation into the governing equation, the residual is expanded in powers of t^�� . The unknown coefficient …

Self-Adjoint Extensions Of Symmetric Operators, Malak Mousa

ETDs from 2020-2029

This thesis introduces operators in Hilbert Space, definitions and properties. And then introduce a method for constructing a self-Adjoint extensions of a symmetric operator and illuminating the method by example.

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 …

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 …

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.

Comparing The Sensitivity And Degree Of Boolean Functions Via The Hypercube, Anne-Caroline Rupp

University Honors Theses

This thesis studies three complexity measures of total Boolean functions f:{0,1}ⁿ → {0,1}: maximum sensitivity s(f), polynomial degree deg(f), and spectral sensitivity λ(f), where λ(f) is defined as the spectral norm of the adjacency matrix of the sensitivity graph. Building on the results of Aaronson et al., we examine the inequality chain √s(f) ≤ λ(f) ≤ deg(f) and investigate whether all three quantities can be simultaneously equal.

The first part of the thesis reverse engineers the equality cases of the two known inequalities to isolate necessary extremal conditions on both the Fourier structure of f and the local geometry …

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 …

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 …

Assessing The Effectiveness Of Tilt-Informed Assessments In Calculus I, Samuel Horelick

Inquiry: The Journal of the Virginia Community Colleges

Transparent Design in Learning and Teaching (TILT) is widely promoted as an evidence-based framework intended to clarify expectations, promote equity, and improve student learning. While prior research reports positive outcomes across many disciplines, less is known about how transparency functions in quantitative, problem-solving courses such as calculus, where students often value efficiency and autonomy. This study examines the effects of a TILT-informed assignment redesign in two sections of Calculus I at a Virginia Community College System institution. One section completed a traditional assignment, while the other completed an equivalent task redesigned to make the purpose, task, and evaluation criteria explicit. …

Differential-Geometric Methods For Neural Signed Distance Fields: Parameterized Surface Extraction And Curvature Regularization For Cad Models, Haotian Yin

Dissertations

Neural signed distance fields have emerged as a powerful framework for representing three-dimensional geometry through continuous and differentiable neural functions. Their flexibility, resolution independence, and compatibility with gradient-based optimization make them especially attractive for surface reconstruction and geometric learning. However, despite these advantages, two fundamental challenges remain for engineering-grade applications. First, higher-order geometric properties such as curvature are difficult to model reliably during training and often require computationally expensive second-order differentiation. Second, while neural signed distance fields provide implicit surface representations, they do not directly yield a globally consistent forward map or parameterization for downstream geometric processing.

This dissertation addresses …

A Symbolic Model Of Proof Acquisition In Act-R, Beckett Morris, Kerstin Haring

DU Undergraduate Research Journal Archive

Learning to construct mathematical proofs—formal arguments demonstrating the truth of a mathematical statement using logical deductions and previously established facts—is one of the most challenging skills in STEM education. This research aims to build the foundations for a symbolic cognitive model, using the ACT-R cognitive architecture and implementing in Python with the pyactr package, to explore how different proof strategies can be thought through with only symbols and rules. The model observes simple proofs, and its abilities are assessed based on its generalization capabilities, efficiency, and error patterns. By developing and analyzing such a model, this research provides new insights …

Measuring Stock Market Inefficiency Using A Multilayer Composite Efficiency Index: A Case Of The Egyptian Exchange, Patrick K. Owido, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

Financial markets play a critical role in resource allocation. Their performance depends on the decisions of millions of independent investors constantly reacting to one another. Their informational efficiency remains a subject of debate across economic systems. When informational efficiency is present at the weak form, historical price information should not consistently predict future returns. Several empirical tests of this hypothesis often focus on the behavior of aggregate market indices, and use individual efficiency proxies such as autocorrelation, GARCH-type volatility, or entropy-based measures to measure efficiency. This has often yielded mixed results, particularly in emerging markets. Here we show that testing …

Classes Of Analytic Functions Defined By Salagean Derivative Operator Associated With Neutrosophic Generalized Poisson Distribution, Soliu O. Opeyemi Okunola, Olushola Adeyemo, Sayo A. Abidemi Gbangbala, Folorunso I. Isola Akinwale

Neutrosophic Systems with Applications

This study introduces and analyses new subclasses of analytic functions by applying the Salagean derivative operator to the Neutrosophic Generalized Poisson Distribution (NGPD) series. We develop a model where the mean parameter is treated as an interval or set to account for indeterminacy in complex systems. By employing Stirling numbers of the second kind and decreasing factorials, we derive necessary and sufficient coefficient inequalities and inclusion relations for these new subclasses. Numerical results and graphical illustrations demonstrate the sensitivity of these functions to orientation and the neutrosophic parameter, providing a framework for applications in fields like medical imaging and network …

1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, ..., Annemarie Torresen

Masters Theses

1

I invite you to see it and feel it and sit in it and breathe it in and

hold it in your lap. Art isn’t too scary, and neither is math.

2

bouncing between bounds of a binary

bippity boppity! let’s break brains and bread

balance or belly flop, what’s mine is yours:

bumptious and bumbling and barely able

i offer it broken and let you like it that way

Goodness-Of-Fit Test For The Kumaraswamy Distribution Via Energy Distance Approach With Applications To Real Data, Joseph Njuki, Thomas Gilbert

Mathematics and Statistics

In this article, we develop a goodness-of-fit test for the Kumaraswamy distribution based on energy statistics. Due to the availability of its quantile (inverse) function, the Kumaraswamy distribution has been shown to be the preferred alternative to the Beta distribution, since both have bounded support in the (0,1) interval. The proposed test procedure is simple and more powerful against general alternatives. Under different settings, simulations show that the proposed test is capable of being well controlled for any given significance (nominal) levels. In terms of power comparisons, the proposed test outperforms other existing methods in different settings. We then apply …

Conditional Product Sampling For Gaussian Process Implicit Surfaces, Song Shi

Dartmouth College Master’s Theses

Gaussian Process Implicit Surfaces (GPISes) provide a powerful and unified stochastic geometry representation for rendering surfaces, volumes, and the rich continuum between them. Recent work has shown that GPISes can model a broad space of visual appearances under a unified light transport framework. However, practical rendering with GPISes remains challenging: existing estimators can become inefficient for particular correlation structures, and highly anisotropic or heightfield-like GPISes require specialized treatment to obtain robust variance reduction.

This thesis extends recent work on GPIS rendering by introducing a new next-event estimation (NEE) technique for anisotropic GPISes.We show that standard NEE provides diminishing benefits as …

Design Considerations For Hypertension Chronotherapy Trials: Insights From Experience And Modelling, Olivia Walch, Amy Rogers, Yitong P. Huang, Marc D. Ruben, Kenneth A. Dyar, Robert W. V. Flynn, Isla S. Mackenzie, Roberto Manfredini, Francesco P. Cappuccio, Filippo Pigazzani

Mathematics Sciences: Faculty Publications

Chronotherapy aims to maximise treatment efficacy while minimising side effects by scheduling treatment according to personal biological rhythms. In recent years, randomised clinical trials (RCTs) have been conducted to evaluate whether scheduled blood pressure interventions can improve patient outcomes. However, reports of time-of-day effects have attracted rebuttals and engendered methodological debate. A perfectly controlled chronotherapy trial (i.e., a trial that assesses the effect of assigning time of intervention) will never be feasible in the real world; yet some factors may be more critical to consider and control for than others. To advance the conversation about how best to evaluate the …

Parameter Estimation In Ode Models Using Least-Squares Regression, Ulrich A. Hoensch

CODEE Journal

We present a method of estimating model parameters for non-linear ODEs using least-squares regression. The coefficient of determination can be used as a measure of model fit. The method is demonstrated using US population data to fit a logistic growth model. Also, a competing species model is used to describe the interaction of two different species of yeast.