Mathematics Commons^™

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 …

Towards Improving The Performance Of The Adcirc Storm Surge Modeling Software, Nick Weldner, Tim Stitt, Jijun Tang

Caravel Undergraduate Research Journal

Accurately predicting storms and hurricanes is critical to saving lives and reducing economic loss. Therefore, it is necessary to use the most efficient software and hardware technology available in order to improve the performance and fidelity of these predictive mathematical models. For over ten years, the Computational Hydraulics Lab (CHL) at the University of Notre Dame has been involved in developing the high-resolution ADvanced CIRCulation (ADCIRC) storm surge model to predict storm surges in coastal areas. The objective of the work reported here was to port a novel adaption of the parallel ADCIRC code to the state-of-the-art Intel Xeon Phi …

Proofs Of A Trigonometric Inequality, Cameron Butler, Wei-Kai Lai

Caravel Undergraduate Research Journal

A trigonometric inequality is introduced and proved using Hölder’s inequality, Cauchy-Schwarz inequality, and Chebyshev’s order inequality.

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 Low-Rank Solver For The Stokes–Darcy Model With Random Hydraulic Conductivity And Beavers–Joseph Condition, Yujun Zhu, Yulan Ning, Zhipeng Yang, Xiaoming He, Ju Ming

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes, analyzes, and demonstrates an efficient low-rank solver for the stochastic Stokes-Darcy interface model with a random hydraulic conductivity both in the porous media domain and on the interface. We consider three interface conditions with randomness, including the Beavers–Joseph interface condition with the random hydraulic conductivity, on the interface between the free flow and the porous media flow. Our solver employs a novel generalized low-rank approximation of the large-scale stiffness matrices, which can significantly cut down the computational costs and memory requirements associated with matrix inversion without losing accuracy. Therefore, by adopting a suitable data compression ratio, the …

A Computational And Experimental Study Of Energy Extraction From Vertical Axis Wind Turbines In Realistic Environments, Kevin Vargas

Theses, Dissertations and Culminating Projects

This thesis presents preliminary analyses on the performance of small scale vertical axis wind turbines (VAWTs) conducted through both experimental and computational methods. Laboratory experiments were carried out in the Complex Fluids Laboratory at Montclair State University to evaluate turbine behavior under controlled conditions. The key focus of this work was to investigate the impact of inlet blockage on turbine performance, simulating real-world environmental constraints such as flow obstruction and urban interference and measuring power generated via laboratory tools. Computational fluid dynamic (CFD) simulations were performed to further analyze flow characteristics and validate experimental observations and trends. Turbine geometries were …

Bounded Multiplication And Composition Operators On Sequence Besov Spaces, Robert Anderson

Mathematical Sciences Undergraduate Honors Theses

The sequence Besov space $b_p$, $p>1$, is the space of analytic functions $f(z)=\sum_{n=0}^{\infty} a_n\, z^n$ on the open unit disk

$\mathbb D$ with $$\sum_{n=0}^{\infty} n^{p-1}\, |a_n|^p< \infty\, .$$

We will give a brief introduction to the space $b_p$, as well as explore how certain operators behave on the space. For example, let $\varphi$ be an analytic self-map of $\mathbb D$. We study the multiplication operator $M_\varphi$ on $b_p$ and look at examples including when $\varphi$ is a polynomial. We also study the composition operator $C_\varphi$ on $b_p$, and in more depth on $b_2$ the Dirichlet space. The culmination will be an …

Numerical And Harmonic Analysis Of Simplex Number Parity, Hunter Dm Hannula

All NMU Master's Theses

The primary object of this thesis is the study of periodicity in the parity of simplex numbers by number-theoretic and harmonic methods. The regular d-simplex numbers are introduced geometrically, arithmetically, and combinatorially. We demonstrate that all sequences indexing even d-simplex numbers are defined by finitely many congruences in a single modulus, and are thus quasiperiodic. We therefore show that these "even index-sequences" are  particular elements in an affine space of functions, providing a natural decomposition result. We then introduce the discrete Fourier transform to construct the periodic parts of each index sequence, enabling the development of explicit forms for the …

Galois Action And Arithmetic In Algebraic Number Fields, Jared Kettinger

All Dissertations

This dissertation explores the arithmetic of numerous algebraic objects living within an algebraic number field from submonoids of the integers up to localizations of the ring of integers. We begin with a study of factorization in proper orders using an element-theoretic approach. In Chapter 2, by defining a natural generalization of the Davenport constant, we are able to determine the elasticity of certain orders whose integral closure is a unique factorization domain. In Chapter 3, using ideal-theoretic analogues, we are able to significantly broaden the scope of our results and the literature on factorization in orders. In particular, we give …

Mississippi Early Childhood Math Resource Guide: Supporting Mathematical Development From Birth Through Age Eight, Melissa Luckett, Systems Change Lab, Mississippi State University, Social Science Research Center, Mississippi State University

Books

The Mississippi Early Childhood Math Resource Guide is a comprehensive, research-informed digital publication designed to support the development of foundational mathematical skills in children from birth through age eight. Developed by the Systems Change Lab at Mississippi State University's Social Science Research Center, the guide translates evidence-based early mathematics concepts into practical, accessible strategies for educators, families, and caregivers. 

Organized by developmental stages, the guide integrates learning domains, age-appropriate math skills, and everyday instructional practices to promote the use of "math talk" and embedded learning opportunities across settings. It incorporates activities, instructional strategies, and resources aligned to support both formal …

Exploring The Evolution Of Preservice Elementary Teachers' Mathematics Identity And Possible Selves: A Multi-Case Study Approach, Christa R. Mawn

Theses, Dissertations and Culminating Projects

This study explores the nature of preservice elementary teachers’ mathematics identity and possible selves and identifies shifts in their mathematics identity or possible selves over the course of a place value unit during the semester during a course on mathematics content for elementary teachers. Drawing on narrative identity and possible selves theory, this qualitative multi-case study examined the mathematics identity and possible selves of preservice elementary teachers enrolled in a mathematics content course. Course assignments were used as data sources and included written narratives, future-oriented reflections, drawings, and course artifacts. Individual cases were analyzed, and were followed by a cross-cases …

Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi

All Dissertations

Duke and Ghate independently studied the question of when it is possible for the product of two eigenforms to be an eigenform. In this dissertation, we take up a generalization of that question, namely is it possible for the product of two eigenforms to be equal to a different product of two eigenforms? Under this formulation, the question becomes closer to one about unique factorization, i.e., how closely do eigenforms work like irreducible elements? Our conjecture is that there are only finitely many cases where the product of two eigenforms is equal to a different product of two eigenforms, and …

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 …

Parameterized Polynomial Systems: Monodromy, Sparse Polynomials, And Solutions, Julianne Barnhart

All Dissertations

The lift of a loop in the base space of a branched cover to the cover induces a permutation of points in a fibre. The monodromy group of the branched cover is the permutation group generated by all such permutations. When loops are restricted to a particular subset of the base space, the corresponding permutation group induced by these loops is the restricted monodromy group. Monodromy groups encode structure and symmetries of many enumerative problems. We describe the relationship between the restricted monodromy group and the monodromy group of the original branched cover. Our main result is a local-to-global property: …

An Exploration Of The Autorotating Pendulum Model, Vlad Nita

Theses, Dissertations and Culminating Projects

Autorotation is the spontaneous rotation of an object, usually caused by an external fluid flow. The study of autorotation has many physical applications, such as in the design of wind/water turbines. In this thesis, we explore a nonlinear pendulum ordinary differential equation (ODE) which is used to model rotating plates in a fluid and has the capacity to reveal autorotation. In the context of an ODE, autorotation emerges as a bifurcation past oscillations, when the initial velocity of the system crosses a particular threshold. In his classic study from 1983, Lugt [14] utilizes this equation to capture experimental autorotation. Copeland’s …

Conceptualizing A Responsive Design Pedagogy To Support Teachers' Responsiveness To Students' Mathematical Ideas Through Making, Denish Ogweno Akuom

Theses, Dissertations and Culminating Projects

This study develops Responsive Design Pedagogy (RDP), a conceptual and analytic model of teacher responsiveness in design-based environments, to account for how responsiveness emerges through interactions among students, tools, and tasks. Although responsiveness has been widely studied in mathematics education, limited attention has been given to responsiveness within mathematical design activity and to how teachers attend to the multiple modes through which students express their mathematical ideas. Addressing these gaps, this qualitative study examined one teacher’s enactment of responsiveness in a Learning-by-Design classroom where students designed and used 3D-printed mathematical tools. Data sources included classroom video, student design artifacts, teacher …

Data Driven Monitoring And Control Of Laser Powder Bed Fusion Process, Jose De Jesus Galarza

Theses and Dissertations

The Laser Powder Bed Fusion Process (LPBF) has been one of the main processes of additive manufacturing, enabling the manufacturing of complex geometries, customization, and lightweight parts. Modern LPBF processes have integrated monitoring systems that capture the light emissions per layer for quality assurance. However, standard defect detection algorithms have not yet achieved the high precision required due to the inherently variable nature of the signal, insufficient data for model training, and the confounding effects of the print.

The processes still have some challenges, such as characterizing the roughness from the build parameters alone, improving the pore detection using the …

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 …

I Am Still Learning, Too!: Pre-Service Teachers' Experiences Of Mathematics Anxiety In An Inquiry-Based Mathematics Education Course At Utrgv, Jeremiah James Adriano Dela Rosa

Theses and Dissertations

Mathematics anxiety is a prevalent issue in mathematics education that negatively impacts students’ learning, performance, and engagement in mathematics. Prior research suggests that mathematics anxiety is often shaped by students’ experiences and emotional responses within the classroom environment.

The purpose of this study is to explore how Pre-Service Teachers experience mathematics anxiety in an Inquiry-Based Mathematics Education (IBME) classroom. This study employed a qualitative research design supported by descriptive survey data collected through the Abbreviated Mathematics Anxiety Scale (AMAS), selected components of the Fennema-Sherman Mathematics Anxiety Scale (FSMAS), and semi-structured interviews. The survey instruments were used to provide descriptive background …