Mathematics Commons^™

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 …

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 …

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 …

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 …

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.

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 …

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 …

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 …

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 …

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 …

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 …

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.

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 …

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 …