Mathematics Commons^™

Underclosed Posets, Richard Ngo

McNair Summer Research Program

Underclosed complexes are a recent generalization of interval graphs to higher dimensions. Motivated by underclosed complexes, we define and study underclosed posets. Order ideals of these posets correspond to pure underclosed complexes. We classify which principal order ideals are rank-symmetric (and in fact are self-dual).

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …

Hidden Symmetries Of The Classical 3×3 Magic Square, Vijender Nallapu, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The classical 3×3 magic square is one of the most famous objects in recreational mathematics. Its defining feature is that every row, every column, and the two main diagonals have the same sum. However, beyond this well-known property, the square contains several hidden symmetries involving rotations, reflections, complementary pairs, parity, and balance around the central entry. In this article, we present an elementary exploration of these symmetries and show how the classical magic square provides a beautiful introduction to mathematical structure and pattern recognition.

The Mathematics Of Calendar Patterns: Day-Of-Week Phenomena, Number Patterns, And Recreational Investigations, Vijender Nallapu, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Calendars are among the most familiar mathematical structures encountered in everyday life. Beyond their practical use in organizing dates and measuring time, calendars contain a rich collection of numerical patterns, symmetries, and arithmetic relationships. Many of these patterns can be explored using only elementary mathematics, making calendars an attractive subject for recreational investigation and mathematical enrichment.

In this article we examine several mathematical properties arising from the structure of monthly calendars. We investigate arithmetic patterns associated with rows, columns, and rectangular blocks of dates, establish properties of calendar squares, and explore connections with modular arithmetic through day-of-week calculations. We also …

Vedic Mathematics And Mental Computation: Fast Multiplication Techniques, Algebraic Foundations, And Historical Perspectives, Rakesh Bhatia, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Mental computation has fascinated mathematicians, educators, and students for centuries. Various computational techniques allow arithmetic calculations to be performed rapidly without the use of calculators or computers. Among these methods, several popular techniques are associated with what is commonly known as Vedic Mathematics. In this article, we examine a collection of fast multiplication techniques, including multiplication near powers of ten, multiplication by eleven, squaring numbers ending in five, and the vertical-and-crosswise method. For each technique, we provide elementary mathematical explanations based on algebraic identities and place-value representations.

Curious Patterns And Identities Involving Consecutive Integers, Carlo Cattani, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Consecutive integers exhibit numerous surprising patterns and elegant identities. In this article, we investigate several classical properties involving products, sums, divisibility, perfect squares, cubes, pronic numbers, triangular numbers, sums of squares, and Pythagorean triples. Using only elementary algebra, symmetry arguments, and mathematical induction, we establish a collection of identities that reveal rich connections among these classical topics. The exposition is intended for students, teachers, and recreational mathematicians, illustrating how simple observations about consecutive integers can lead to beautiful mathematical patterns and deeper insights into elementary number theory.

Rockin’ Rover On The Rainbow Road, Michael Kolta, Lawrence Burgee, Ying Yuan

Transformations

This paper presents a progressive series of age-appropriate lesson plans for grades K-12 that all use the same interdisciplinary activity to educate students about Science, Technology, Engineering, Art, and Mathematics (STEAM) simultaneously. Technology from Texas Instruments (TI) was employed including a TI Nspire graphing calculator that can run Python programs, a TI Innovator Hub, and a TI Rover. The TI Rover is a small, robotic car that has sensors and is controlled by the calculator via the Hub hardware interface. A Python program was developed that uses the color sensor in the Rover to detect the color on colored paper …

The 2025 Measles Outbreak In Texas, Tamer Oraby, Martial L. Ndeffo-Mbah

School of Mathematical & Statistical Sciences Faculty Publications

Background: In 2025, Texas experienced its largest measles outbreak in decades, reporting 762 cases by mid-August. Measles is a highly contagious but vaccine-preventable infection transmitted mainly among unvaccinated individuals and capable of causing severe outcomes.

Methods: We investigate counterfactual measles control scenarios based on daycare and school closures and reactive vaccination of infants and children, including mixed interventions. We analyze the 2025 Texas outbreak using an age-structured multi-stage SEIR model formulated as a system of ordinary differential equations. The model is fit to case data using Bayesian inference to estimate the effective reproduction number and generate posterior predictive trajectories under …

From Reluctance To Resilience: The Link Between Attitude And Growth Mindset In Calculus Students, Olivia Valente, Kathryn E. Pedings-Behling, Amy N. Langville

Journal of Educational Research and Practice

Many postsecondary students enter general education mathematics courses with negative attitudes, shaping their willingness to engage with the content. This study explores the relationship between students’ attitudes towards mathematics and their growth mindset, revealing effective strategies to enhance learning experiences, particularly for students who are reluctant learners. The research investigates two questions: (1) How do students’ Attitude Toward Mathematics Inventory (ATMI) scores and Growth Mindset Scale (GMS) scores relate? and (2) Do gender or course modality affect this relationship? This study addresses a gap in literature by examining how enjoyment, motivation, self-confidence, and perceived value connect with growth mindset. The …

Algebraic And Topological Methods In Computational Neuroscience, Trong-Thuc Trang

Electronic Theses and Dissertations 2020 - Present

Neural data is incredibly rich in combinatorial, topological, and geometrical information, reflecting the intricate shape and connectivity of neural firing patterns. To decipher these structures, neuroscience increasingly relies on advanced mathematical tools to analyze neural activity. Here we study (1) neural codes within the poset P_Code of neural codes and (2) the connectivity of neural population activity within the insular cortex when responding to interoceptive information. In (1), we establish combinatorial constructions for all upward covering relations based on what we call “isolated subsets” with supporting theorems and give a slight modification of the existing downward covering relations. We …

Positive Curvature And Discrete Abelian Symmetry, Lee Kennard, Elahe Khalili Samani, Catherine Searle

Mathematics

By replacing the torus with an elementary abelian two-group, we generalize Grove and Searle’s maximal symmetry result and Wilking’s half-maximal symmetry result for positively curved manifolds with an isometric torus action. © The Author(s) 2026.

A Stabilized Weighted Interior Penalty Method For Thermal Convection Model In Heterogeneous Porous Media, Yuanyuan Hou, Qianqian Ding, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

In this article, we propose and analyze a stabilized weighted interior penalty method for solving the thermal convection problems in heterogeneous porous media. We first transform the thermal convection model into the pressure primal form with homogeneous Neumann boundary condition and develop a symmetric weighted interior penalty method to handle the discontinuous Darcy number and automatically adjust the penalty coefficient corresponding to the varying permeability. Then we recover the velocity by a stabilized method and incorporate it into the energy equation to obtain the temperature. The stability and convergence rates of the numerical solutions are rigorously proved and verified by …

Ramsey Chains In Graphs, Ritabrato Chatterjee

Dissertations

The research in this work deals with decomposition of graphs and Ramsey theory, two popular areas of research in graph theory due to numerous fascinating problems and the sheer mathematical beauty of the subjects. A decomposition {G₁, G₂, ..., G_k} of a graph G into subgraphs of G is ascending if the subgraph G_i is isomorphic to a proper subgraph of G_i+1 for i = 1, 2, ..., k − 1. The well-known and long-standing of Ascending Subgraph Decomposition Conjecture states that every graph has an ascending subgraph …

Aqqd: Annotated Quranic Qira’At Dataset, Linda Smail, Mohammed Lataifeh, Md Sohazur Islam Sozib, Arthur Diniz De Souza

All Works

AQQD (Annotated Quranic Qira'at Dataset) is an open audio dataset of Quranic recitations annotated across canonical Qira'at styles. The dataset is designed to support research in machine learning, speech and audio processing, computational linguistics, and Quranic studies. The current release contains 24,183 WAV audio files from 309 reciters and covers 70 selected Quranic Surahs segmented into representative verses and phonetic variation points. Of these, 23,111 recordings were collected from publicly available sources, including official reciter websites, the Midad repository, MP3Quran, and verified YouTube channels, while an additional controlled subset of 1,072 recordings was obtained from a single reciter recorded as …

Decidability Of The Orbit Problem Over Q(X), Joshua Holden, Alexa Renner

Mathematical Sciences Technical Reports (MSTR)

The orbit problem is the problem of whether, given x, y ∈ Qn and an n × n matrix A, there exists an i ∈ N such that Aix = y. In 1980, Kannan and Lipton proved that the orbit problem is decidable. We show that a generalization of the orbit problem, where the field is Q(X) for X a countable set of transcendentals, is also decidable. We define the orbit power problem for an arbitrary group G to be the problem of when, given x, y ∈ G, there exists an n ∈ Z such that xn = y. …

Advantages Of Dynamic Representation For Related Rates Problems In Calculus, Eri Osuna

Electronic Theses, Projects, and Dissertations

Related-rates problems are a standard yet persistently difficult topic in first-semester calculus. Research increasingly recommends dynamic visualization tools such as GeoGebra, but direct comparisons of static and dynamic representations in related-rates settings remain scarce. This qualitative study examines how representation type shapes the quality of students' reasoning and their perceived experience during related-rates problem solving. Six mathematics students who had completed Calculus —a group of four undergraduates and a pair of graduate students—completed a static sliding-ladder task and a dynamic airplane-and-camera task supported by an interactive GeoGebra applet, followed by an interview. Findings indicate that the two representations supported reasoning …

Geodesic Completeness And The Hopf-Rinow Theorem, Christopher Farias

Electronic Theses, Projects, and Dissertations

Differential geometry is concerned with the properties of calculus and geometry on curved n-dimensional manifolds. As a result, thinking about such a space often runs counter to the Euclidean geometer's intuition of distances, angles, and transformations. This thesis aims to build up to proving an important result in the study of Riemannian manifolds: the Hopf-Rinow theorem.

In Chapter 2, we begin by defining what a manifold is and showing that the collection of directional derivatives at a point on the manifold spans a tangent vector space. After defining a basis and a metric for this space, in Chapter 3, we …

Student Attitudes And Perceptions Of Proof In Mathematics, Emelin G. Sibrian Marquez

Electronic Theses, Projects, and Dissertations

Traditionally, mathematical proof is viewed primarily as a tool for validation or verification. However, proof holds many other important roles such as discovery, reasoning, explanation, and justification. For many students, these other roles are not always obvious. Those encountering rigorous proof for the first time often find the process abstract, intimidating or disconnected from their previous learning. This disconnect can lead to negative attitudes as students transition from computational mathematics to advanced proof-based mathematics. Utilizing a mixed-methods approach, this study examined undergraduate and graduate mathematics students at a Hispanic-Serving Institution (HSI) in Southern California. We investigated what students perceive the …

Comparing 3-Connectedness And Roundness In Matroid Theory, Blanca Delia Larios

Electronic Theses, Projects, and Dissertations

A matroid is a discrete mathematical object that abstracts and connects the various notions of independence found throughout mathematics. Such notions of independence include linear independence, algebraic independence, as well as notions of independence that arise in graph theory. There are many broad classes of matroids. Important examples include binary matroids, graphic matroids, regular matroids, uniform matroids, and various levels of connected matroids. Some of the most important problems in matroid theory involve characterizing classes of matroids so that such characterizations can be used to prove results concerning these matroid classes. This thesis is a study of two important classes …

Stabilized Continuous Galerkin Methods For Heat Transport In Fully Coupled Nonlinear Thermo-Poroelasticity In Heterogeneous Porous Media, Payel Dey

Open Access Theses & Dissertations

This thesis focuses on the numerical approximation of a coupled Thermo-Hydro-Mechanical (THM) model with advective heat transport in heterogeneous media. The governing equations consist of three coupled equations: the momentum balance, the fluid mass balance, and the energy balance. The primary unknowns are displacement, pressure, and temperature. In this system, the fluid pressure determines the Darcy flux, and this flux transports heat through the advective term in the energy equation. Since the Darcy flux is computed numerically from the flow equation, the conservation properties of the flow approximation can affect the temperature solution. This effect is especially important when thermal …