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Course Insight Portfolio: Math 1060 Calculus Of One Variable I, Julianne Barnhart, Rayna Elizabeth Maleki, Antsa Rakotondrafara

Publications

We presented this Course Insight Portfolio project as a final assignment for the class Teaching Undergraduate Math MATH 9700 in Summer 2025. It covers the Clemson class MATH 1060 Calculus of One Variable I and is a fruit of the collaborative work of Julianne Barnhart, Rayna Maleki, and Antsa Rakotondrafara.

Quotients, Equivalence Relations, And Normality In Non-Associative Algebra With Regards To Loops And Quasigroups, Matthew L. Mulholland

All NMU Master's Theses

This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be …

Stability Analysis Of Turbulent Fluid Flow, Adam D. Schroeder

Mathematics, Statistics, and Computer Science Honors Projects

Hydrodynamic stability refers to the study of when and how laminar flows transition to turbulence. This includes investigations of the mechanisms of transition, as well as the classification of known flow configurations as either stable or unstable and the identification of critical values of flow parameters at which this bifurcation occurs. In this thesis, we introduce the mathematical theory behind continuum mechanics and fluid dynamics as well as some tools from the study of dynamical systems. We apply these concepts to the linear stability analysis of zero pressure gradient flat plate flow via numerical simulations in OpenFOAM, discussing both the …

P-Adic Quantum Mechanics, Infinite Potential Wells, And Continuous-Time Quantum Walks, Nathaniel P. Mayes

Theses and Dissertations

In this thesis, we introduce a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. …

The Herzog-Takayama Resolution Over A Skew Polynomial Ring, Linoy Utkina

Theses and Dissertations

Let k be a field, and let I be a monomial ideal in the polynomial ring R = k[x1,..., xn]. In her thesis, Taylor introduced a complex that yields a finite free resolution of R/I as an R-module. Building on Taylor’s work, Ferraro, Martin, and Moore extended this construction to monomial ideals in skew polynomial rings. Because the Taylor resolution is typically not minimal, subsequent research efforts went into identifying specific classes of ideals whose minimal free resolutions can be constructed more simply. In 1990, Eliahou and Kervaire devised an approach for handling minimal resolutions of …

Camassa-Holm Type Equations And Multi-Peakon Solutions, Yonghong Chen

Theses and Dissertations

This thesis investigates the multi-peakon solutions of a class of Camassa-Holm type equations with quadratic nonlinearities, specifically focusing on the differential equation mt +θmu+ amxu+bmux = 0, where m = u−uxx, and θ, a, b are arbitrary constants. We derive the nonlocal form of the equation and explore its peakon dynamics, including the well-known Camassa–Holm, Degasperis–Procesi, and Holm–Staley b-family equations. By employing a practical computational approach, we analyze the N-peakon solutions and present the corresponding dynamical systems, particularly focusing on the two-peakon case. The results will …

Congruences In Arithmetic Progression For Coefficients Of Gaussian Polynomials And Crank Statistics, Joselyne Aniceto

Theses and Dissertations

The study of partition congruences, inspired by Ramanujan’s discoveries for ��(��) over a century ago, remains a central topic in this field. This dissertation examines congruence properties in two restricted partition functions: ��(��,��), which counts partitions of �� into at most �� parts, and ��(��,��,��), which further limits the size of the largest part to be at most ��. Building on Kronholm’s 2007 result, now known as the Interval Theorem, and a recent result by Eichhorn, Engle, and Kronholm, we establish new infinite families of congruences for ��(��,��,��). This dissertation extends not only the recent results of Eichhorn, Engle, …

Algebraic Properties Of Boolean Models, Harrison Fisher

All Theses

Boolean models are n-tuples of polynomial functions in n variables over the finite field of order 2. These models define finite dynamical systems which are used for modeling many different biological systems such as gene regulatory networks. These systems can be defined by updating every function synchronously, or by updating one function at a time asynchronously. In this project, we discuss a method for reverse engineering the model space of all Boolean models which fit a set of partial asynchronous data. This method is a generalization of a known method for synchronous data. In addition, we show that given the …

Relaxation Maximum-Based Iteration Method For Solving The Generalized Absolute Value Equation, Ximing Fang, Zhidong Wang, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

The generalized absolute value equation (GAVE) has wide applications in scientific computing. Establishing a high performance computing method to solve the GAVE is a hot research topic in recent years. In this paper, with the aid of the maximum function, the GAVE is decomposed of two equations, and then we present the relaxation maximum-based (RM) iteration method. To see the feasibility of the method, we discuss the necessary and sufficient conditions for the GAVE to have a unique solution. Next, the convergence analysis of the RM iteration is discussed under some convergence conditions. Moreover, some numerical examples of low and …

The Improved New Intersection Theorem Revisited, Lars Winther, Luigi Ferraro

School of Mathematical & Statistical Sciences Faculty Publications

We prove a generalized version of Evans and Griffith’s improved new intersection theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least dimR−dimR/I. This improves the bound htI obtained by Avramov, Iyengar, and Neeman in 2018.

Stochastic Analysis With Operational Calculus In Reliability, Hend Hamdan Al-Jahani

Theses and Dissertations

Our dissertation models several reliability systems subject to occasional random shocks of random magnitudes W1,W2,..occurring at times tau0, tau2,..In Model 1, the underlying system becomes inoperational if it wears out due to aging specified by a monotone increasing continuous function Delta identifying system's wear at any time t>=0 . The system is deemed inoperational if delta crosses a sustainability threshold at some moment T=(delta )^-1(D). The precise time T is difficult to identify making the realistic failure time randomly delayed and thus identified at some opportune observation epoch. The system can also fail due to damages by external shocks, …

The Food Truck: A Multi-Product Newsvendor With Trans-Shipment Cost, Samuel Ajibola

Electronic Theses and Dissertations

The Newsvendor Problem is a key model in supply chain management that focuses on determining the optimal order quantity to minimize costs under uncertain demand. This thesis introduces the Food Truck Problem, an extension of the Newsvendor model that incorporates nonlinear transshipment costs for inventory transportation. In this context, a Food Truck must determine the optimal stock levels for multiple products while minimizing costs related to stock shortages, excess inventory, and transportation. Unlike traditional Newsvendor models, our approach explicitly considers a quadratic transshipment cost, which necessitates the use of Lagrangian duality and Karush-Kuhn-Tucker (KKT) conditions for analysis. Moreover, we apply …

Constructing Binary Encoding Matrices From Joined Graphs, Joshua Avalos

Electronic Theses, Projects, and Dissertations

Codes and technology are part of our daily lives and allow the modern world to function, and for us to have conveniences in our lives such as smartphones that can be used to privately call people on the other side of the planet, and for secure access to the internet. In this thesis we will explore the construction of binary codes created by vertex-edge incidence matrices of planar graphs. The Hamming (7,4) code was an incredible code that allowed the detection and correction of errors after receiving them through a transmission. We will explore the possibility of the creation of …

Decision Space Decomposition For Multiobjective Programs, Emma Soriano

All Dissertations

Being inspired by the parametric decomposition theorem for multiobjective optimization problems (MOPs) of Cuenca and Miguel (2017), and by the block- coordinate descent for single objective optimization problems, we present a decom- position theorem for computing the set of minimal elements of a partially ordered set. This set is decomposed into subsets whose minimal elements are used to retrieve the overall minimal elements. We apply this approach to strictly convex MOPs de- composing their decision space into lines. The line decomposition benefits from the fact that a multiobjective line search problem is equivalent to solving a collection of single objective …

The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper

All Dissertations

Many physical systems, whether they are ocean waves or particles moving through space, can be described using the mathematical language of “partial differ- ential equations.” In many circumstances, it is useful to study how these equations amplify an input to the equation, in which case the amplification factor is called an “eigenvalue.” The usefulness of these amplification factors is that they can be used to describe properties of the physical system. In this dissertation, I have studied this amplification factor for a related set of equations called “Toeplitz operators.” In par- ticular, I have studied eigenvalues using statistical techniques. The …

Efficient Fully Bayesian Approach To Brain Activity Mapping With Complex-Valued Fmri Data, Zhengxin Wang, Daniel B. Rowe, Xinyi Li, D. Andrew Brown

Mathematical and Statistical Science Faculty Research and Publications

Functional magnetic resonance imaging (fMRI) enables indirect detection of brain activity changes via the blood-oxygen-level-dependent (BOLD) signal. Conventional analysis methods mainly rely on the real-valued magnitude of these signals. In contrast, research suggests that analyzing both real and imaginary components of the complex-valued fMRI (cv-fMRI) signal provides a more holistic approach that can increase power to detect neuronal activation. We propose a fully Bayesian model for brain activity mapping with cv-fMRI data. Our model accommodates temporal and spatial dynamics. Additionally, we propose a computationally efficient sampling algorithm, which enhances processing speed through image partitioning. Our approach is shown to be …

A Dg-Algebra Structure With Divided Powers On The Generalized Taylor Resolution, Raul F, Alvarez

Theses and Dissertations

This thesis investigates the construction of a DG Γ-algebra structure on the Generalized Taylor Resolution (GTR) associated with monomial ideals. The classical Taylor resolution is known for providing a free but generally non-minimal resolution, leading to computational challenges and inefficiencies in algebraic analysis. In contrast, the GTR preserves essential algebraic structures while optimizing the resolution process, offering a more efficient and comprehensive framework for studying monomial ideals.

We introduce a novel DG Γ-structure that incorporates divided powers into the GTR, enhancing its multiplicative and homological properties. This structure preserves strict graded commutativity and is fully compatible with the differential graded …

Reuben Hersh's What Is Mathematics, Really? (Book Review), Calvin Jongsma

Faculty Work Comprehensive List

Reuben Hersh’s 1997 book What is Mathematics, Really? popularized a trend in the philosophy of mathematics that was gaining some traction at the time it was written. This essay examines Hersh’s work within the broader historical context of 19th- and 20th-century developments in mathematics and philosophy of mathematics. It also focuses on Hersh’s antagonism toward the influence of religion on (philosophy of) mathematics, concluding by briefly considering two positions taken by Christian mathematicians associated with ACMS in defense of such a connection.

Equity Vs. Excellence: Dichotomy Or Duo?, Nadjia I. Logans

Honors Program: Senior Projects (Public)

In this thesis, I investigate recent changes in mathematics education and how these developments connect to the principles of equity and excellence. I place a special focus on “active learning” methods. My methodology primarily consisted of interviews with educators, along with analyzing literature from math education researchers. I first define and explain the motivations behind active learning instruction methods. Next, I explore not only how instructors define equity and excellence but how educators perceive the role of the two principles within education. Finally, I challenge the pervasive media portrayal of equity and excellence as being mutually exclusive. Instead, I find …

Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict Arroyo Cristobal

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.

One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …

Countering Deficit Narratives Through A Blackcrit Lens By Examining Case Studies Of Black College Mathematics Students’ Classroom Experiences, Akhenaton Wilbourn

All Dissertations

This dissertation examines the lived experiences of 11 Black college mathematics students at their predominantly white institution (PWI) through the lens of Black Critical Theory (BlackCrit) to counter deficit narratives about the racial achievement gap (RAG) in mathematics education courses. Deficit narratives often attribute disparities between Black and white students’ mathematical performance to assumed deficiencies in Black students’ abilities rather than systemic inequities, historical marginalization, and ongoing anti-Blackness in educational settings (Cobb & Russell, 2015; Kress, 2021). Using multiple case studies, this study explores how anti-Blackness manifests in mathematics classrooms, how students navigate tensions between multiculturalism, neoliberal educational policies, and …

Some Interpolation Problems In The Projective Plane, Lilah Estes

Mathematical Sciences Undergraduate Honors Theses

Given some set of r general points in the projective plane, we want to better understand: what is the smallest degree of any polynomial passing through the points m times? How many linearly independent equations of this degree pass through the points m times? The investigation of these questions, particularly for the case of m=3 and r< 16, motivates the development of several results. We translate Terracini's inductive argument, a tool for evaluating the expectedness of certain sets of double points, into a version which can be used for triple points, and prove that the argument holds. We compute the minimal graded free resolutions for the ideals corresponding to up to 15 points, for m up to 6, and we conjecture a connection between the expectedness of these ideals and what their resolutions look like. Further, we prove that this conjecture holds when m=1, and we either fully or partially prove that these ideals are …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Data Science For Engineers, Heidi Moulton

Undergraduate Honors Capstone Projects

Undergraduate research is a core pillar of Utah State University’s College of Engineering. Many students become involved with research during their Junior and Senior years and begin to generate various forms of data. Most students, however, have received little formal education on how to process data, and there are currently no readily available resources within the College of Engineering. As a Mechanical Engineering and Data Science double major, I found the data processing techniques I learned in my Data Science courses invaluable as an undergraduate researcher, and now as a Mechanical Engineer at Apogee Instruments, I frequently draw upon these …

The Stem Gender Gap: Can It Be Closed?, Makenna Roberts

Undergraduate Honors Capstone Projects

Despite increasing gender parity in many professional fields, women remain underrepresented in STEM industries, particularly in math-intensive areas such as engineering, computer science, and physics. This report explores the persistent gender gap in STEM through analysis of two data sources: gendered differences in the US national Scholastic Aptitude Test (SAT) performance and post-graduate employment patterns among scientists and engineers.

The first portion of the analysis examines SAT score data from 2018 to 2024, with a focus on high-performing test-takers. Using Python and R, data was extracted and visualized to compare male and female performance on the Math and Evidence-Based Reading …

Development Of Saddlepoint Methodologies For Sparse Sample Multiple Parameter Generalized Linear Models And Correlated Data Scenarios, Christopher Johnson

All Graduate Theses and Dissertations, Fall 2023 to Present

Parameter estimation using maximum likelihood techniques may be biased when sample sizes are small, event rates are low, or otherwise sparse counts exist in a parametric model. This in turn may lead researchers to draw invalid statistical conclusions when conventional methods are utilized. The saddlepoint approximation has potential to lessen the degree of bias in sparse data conditions through its use of moments beyond the mean and variance, which allows for more accurate approximations using a smaller number of observations. We propose two novel saddlepoint methods for use in practical analysis scenarios, as an alternative to maximum likelihood estimation. First, …

A Survey Of Master's Qualifying Exam Practices And Content In Real Analysis, Zachary M. Coverstone

All Graduate Theses and Dissertations, Fall 2023 to Present

Graduate programs in mathematics are intended to develop experts in mathematics. As part of a graduate program, many students are expected to engage in a qualifying examination that can serve myriad purposes. This dissertation investigates real analysis qualifying examinations at the master's level through a nation-wide survey of institutions in the Association of Public and Land-Grant Universities (APLU). This study reveals the practices in administration of real analysis qualifying exams, institutions reported offering traditional, pencil-and-paper exams and generally asked students to prepare individually using previously administered exams. The survey results show that approximately two in three APLU institutions offering master's …

Multiscale Integration Of Receptor-Ligand Dynamics Into Discrete And Continuous Tumor Growth Models With Application To Tyrosine Kinase Inhibitor Treatment, Romasa Qasim

Open Access Theses & Dissertations

The epidermal growth factor (EGF) receptor cascade plays a crucial role in the survival and proliferation of tumor cells. Tyrosine kinase inhibitors (TKIs) are a class of drugs that inhibit epidermal growth factor receptors (EGFRs), thereby preventing the downstream signal transduction. Despite their importance, models that link spatial receptor dynamics to tumor growth remain scarce. Further, TKIs act through selective mechanisms, inhibiting active, inactive, or all receptor states, which poses a challenge to traditional modeling approaches.

We propose to numerically study two mathematical models incorporating receptor-dynamics into cancer models to describe the impact of EGFR overexpression and TKIs. The first …

A Numerical Study Of Self-Assembling Amphiphilic Systems, Joshua Albert Sackey

Open Access Theses & Dissertations

Mixing processes in ternary mixtures involve immiscible fluids such as oil and water, and a surface-active molecule called surfactant. These physical processes find applications in various fields, including enhanced oil recovery, drug delivery design systems, and the formulation of cleaning products. Even though this process has several applications, the mathematical models describing it and the numerical methods solving it are not well understood. The underlying mathematical model is a nonlinear initial-boundary value problem involving sixth-order derivatives and belongs to the class of sixth-order Cahn-Hilliard equations. Authors Sharma and Tierra recently proposed a numerical method to approximate its solutions in two …

Yield Prediction Of Pv Solar Energy Systems And Its Application In The Energy Grid For Operational Efficiency, Pablo Bustamante

Open Access Theses & Dissertations

The generation of power from photovoltaic (PV) solar panels is influenced by a multitude of factors. These include the tilt and orientation of the solar panels, the latitude of their location, and the prevailing climate and weather conditions. Additionally, shading at specific locations, particularly if the panels are not part of a solar facility, can significantly impact their efficiency. The quality and efficiency of the panels themselves, along with the preventive maintenance of both the solar panels and associated components such as inverters and trackers, are also critical. Finally, the overall system design and installation play a vital role in …