Mathematics Commons^™

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 …

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 …

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 …

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.

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 …

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 …

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 …

Heat Transfer And Micropolar Characteristic Analysis For Fluid Flow Over A Contracting Permeable Infinite-Length Cylinder: A Fractional Derivative Study, Khalid Al Homran

Theses

This study presents a comprehensive analysis of heat transfer and micropolar fluid characteristics for flow over a contracting permeable infinite-length cylinder using fractional derivative modeling. The micropolar fluid model accounts for microstructure effects that are prevalent in many practical applications, such as polymeric fluids, colloidal suspensions, and biological fluids. The inclusion of fractional derivatives allows the analysis to incorporate memory effects, which provide a more accurate depiction of the physical phenomena compared to classical integer-order models. The governing partial differential equations, representing the momentum, micropolar momentum and energy transport processes, are transformed into a set of nonlinear ordinary differential equations …

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 …

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 …

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 …

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 …

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 …

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 …

Analytical And Numerical Methods For Solving Fractional Integro-Differential Equations Using The Modified Operational Matrix Method, Nour Alzoubi

Theses

Fractional calculus has attracted considerable attention in recent years because of its wide applicability to model a variety of linear and nonlinear physical phenomena across different scientific disciplines. In particular, fractional diferential equation systems have proven to be effective in describing processes with memory and hereditary properties. However, the solvability and analysis of such systems strongly depend on the type of fractional operator employed, especially in the presence of nonlocal fractional derivatives with singular kernels, which remain an open and challenging area of research.

The main objective of this thesis is to develop efficient analytical and numerical techniques for solving …

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.

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. …