Other Mathematics Commons^™

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 …

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 …

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 …

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. …

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 …

Exact Solution Of Steady Seepage In An Asymmetrical Domain Underneath A Cofferdam, Tan Nguyen

UNLV Theses, Dissertations, Professional Papers, and Capstones

For decades, the evaluation of steady state seepage beneath asymmetrical cofferdams has relied on numerical methods, such as Finite Element Methods (FEM), Finite Difference Method (FDM), Finite Volume Method (FVM), Mesh Reduction Method (MRM), Meshless Method (MM), Boundary Element Method (BEM), or geometric idealizations, most notably Griffiths' vertical Method of Fragments assumption. While exact closed form solutions via Schwarz-Christoffel (SC) conformal mapping have been well established for symmetrical geometries (Banerjee and Muleshkov), the generalized asymmetrical case has historically remained an intractable mathematical frontier. The primary barrier to an exact analytical solution has been the "crowding problem," a numerical phenomenon where …

The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal

Honors Theses

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Leveraging Information Theory And Ecological Network Analysis To Monitor Communication Networks In A Student Aerospace Team, Christine Sessions

Doctoral Dissertations and Master's Theses

Effective communication is a critical component of successful collaboration in group projects and team settings. However, systematically tracking and analyzing team communications can be challenging, especially in complex, multi-member teams such as those found in the aerospace industry. This paper explores an information-theoretic approach that leverages encoding techniques and graph theory to analyze communication networks within a university’s multi-year, student-led cubesat design project (Project COMET). By utilizing Shannon Entropy as a measure of information flow, encoding communication patterns, and analyzing Ecological Network Analysis parameters, this research aims to understand how an Embry-Riddle Aeronautical University student project team evolves over the …

Primitive Pythagorean Triples In Lean And Reduction Modulo Odd Prime Powers, Luke Biddle

Mathematical Sciences Undergraduate Honors Theses

Primitive Pythagorean triples (PPTs) are (a,b,c) triples that satisfy the Pythagorean theorem and share no other common factors outside of 1. This project examines these PPTs reduction modulo odd prime powers by combining proof writing and number-theoretical analysis with the process of verification and formalization in the Lean proof coding language. Using the parameterization of PPTs generated by using the unit circle with additional conditions, we investigate how these triples behave modulo  for odd primes , with emphasis on counting the number of elements in the set of PPTs (a,b,c) modulo pⁿ . By using cases based on initial …

A New Approach To Generate Combinatorial Patterns In Logical Analysis Of Data And Its Application To Predict College Retention, Salihah Ahmed E. Jaafari

Theses and Dissertations

Student retention and degree completion remain central challenges for higher-education institutions, with significant implications for student success, institutional effectiveness, and public accountability. While advances in predictive analytics have enabled earlier identification of students at risk of withdrawal, many commonly used machine learning approaches suffer from limited interpretability, constraining their practical usefulness for advising, intervention, and policy decision making. This dissertation addresses the problem of predicting student persistence by developing and evaluating optimization based, interpretable classification models within the Logical Analysis of Data (LAD) framework. Building on existing LAD formulations, this research introduces two novel pattern generation models, the Best Term …

Examining Student Practices In A Technology-Mediated Math Classroom, Bradley T. Dodd

Electronic Theses, Projects, and Dissertations

It is important for math educators to make sense of student thinking in the classroom. Without opportunities to practice this skill math educators struggle to improve, particularly when students are engaging in mathematics using technology. As such, there is a need for video artifacts of students engaging with mathematics using technology for use in professional development activities (Lovett 2020). In accordance with Lovett et al.'s (2020) design principles for examining student practices in a technology-mediated environment, I carried out a study to determine whether these artifacts of student work could be created working with college undergraduates as participants. Students engaged …

Decision Making For Large-Scale Problems Under Uncertainty And Conflict, Benjamin J. Hamlin

All Dissertations

Large-scale decision-making problems appear in many areas including long-range forecasting such as energy generation forecasting. Many such problems are subject to conflicting objectives and uncertain data, and can be modeled as linear optimization problems. We study novel theoretical results and algorithms for large-scale linear decision problems under conflict and uncertainty. First, we propose a parametric Benders decomposition algorithm for solving large-scale linear optimization problems with multiple objectives or deterministically uncertain objectives. Second, we extend the parametric Benders decomposition to a multi-stage setting, developing a parametric stochastic dual dynamic programming algorithm, which enables decision-making when conflicts and uncertainty have planning impacts …

Fractsynth: Exploring Glitch Timbres With Chaos Theory, Aidan Roach

The Transdisciplinary STEAM+ Journal

In this paper, I explore how chaos theory can be used to design a new kind of synthesizer with the primary focus of producing glitchy, unpredictable sounds. Glitch music embraces abstract sound design, malfunctioning electronics, and randomness as the main compositional elements. However, most synthesizers rely on stable, repetitive oscillators that often sound too controlled. To challenge this, I developed FractSynth, a real-time synthesizer that uses chaotic attractors–including the Logistic Map, Henon Map, and Lorenz System–as modulation sources for frequency, amplitude, and tone. The software also features real-time Lyapunov Exponent Tracking, which gives users a direct visual of how …

Interactions Of Dance And Mathematics: A Closer Look At Mathematical Sequences As Scores For Choreography From A Choreographic Lens, Caroline Wolfe

The Transdisciplinary STEAM+ Journal

This paper investigates the embodiment of infinite mathematical sequences through concert contemporary dance, examining how choreography can serve both as an artistic and analytical tool for exploring numerical patterns. Grounded in interdisciplinary literature on mathematical visualization in choreography, the study centers on two original choreographic works presented and performed at Missouri State University: Golden Ratio Sequence (Spring 2025), which maps the Fibonacci sequence and the natural applications of the Golden Ratio onto a spiral floor pattern, and Tetrahedral Numbers (Fall 2024), which employs an accumulation score to embody three-dimensional number growth through layered movement motifs and body created tetrahedra. The …

Unique Combinations Of Packing Integer Squares, Keith M. Dreiling, Austin Leanna, William Mooney

SACAD: Scholarly Activities

This research investigates a function, informally named WAK(x), that describes the number of ways to divide an integer square into integer subsquares counting only the list of parts. Previous research has shown values up to 28, though finding these values is computationally complex and requires a long runtime using computer algorithms. We attempt to find patterns in the values and many aspects of the values, hoping to find a general solution. We are unsure if a solution exists, but we have ideas for how to move forward in finding a solution.

Counting Hamiltonian Cycles In Quartic Circulant Graphs, Allison Hilliard

Seaver College Research And Scholarly Achievement Symposium

We consider the problem of counting Hamiltonian cycles in circulant graphs $C^K_n$ where $n$ is the number of vertices and $K$ is a set containing elements that correspond to the allowed edges in the circulant graphs. After sorting the cycles by a topological invariant called the winding number, we use a modified transfer matrix method to convert local data into global structures. The result is a generating function that counts the number of Hamiltonian cycles in a circulant graph with $n$ vertices. The results for $K=\{1,2\}$ and $K=\{1,3\}$ have been found by previous authors. We focus on the case where …

On Links Between A Theorem Of Schoenberg, Rohlin Decompositions Of Measures, The Bochner-Minlos Theorem And The Fock Space, Daniel Alpay, Paula Cerejeiras, Palle Jorgensen, Uwe Kaehler

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main goal of this paper is to gain new results in stochastics by drawing on, and combining, different areas that are normally not considered to be related. Thus, in this paper we extend the previous class of Gaussian-like functions ML which will allow for future generalized stochastic processes in infinite dimensional analysis. We show that an approach similar to the one by the classical Bochner-Minlos theorem for the white-noise case can be achieved by using Gaussian-like functions belonging to a large family -the ML_r classes (0 < r ≤∞). We show how Schoenberg’s theorem for positive definite functions on a Hilbert space allows to go beyond the classical setting of Bochner-Milnos theorem. Furthermore, we show that the application of the Rohlin’s disintegration theorem allows for a decomposition of the associated probability measures , see Theorems 3.2 and 4.3. We end this paper with several important examples of functions in these classes ML_r and provide some interesting counterexamples, e.g. Theorem 7.4, to get a …

Automating Cardiff Model Data Capture In Emergency Departments: Ambient Nlp Integration With Oracle-Cerner Fhir Systems, Simi Augustine, Marco A. Lopez, Jacquelyn Cheun, Chris Papesh

SMU Data Science Review

Violence and overdose events in Las Vegas occur at rates above the national average, with fewer than half of violent injuries reported to law enforcement [2,7]. The Cardiff Model offers a proven framework for standardized data collection and sharing between hospitals and public safety partners, yet many implementations still rely on manual entry. We propose an ambient triage pipeline integrated with Oracle-Cerner electronic health record systems to listen to nurse–patient dialogue, convert speech to text, extract Cardiff fields, and write standards-based FHIR Bundles for analytics. Using SMART on FHIR standards and Cerner Millennium APIs, the study evaluates whether ambient capture …