Mathematics Commons^™

A Computational Approach To The Game Of Cycles, Jocelyn Garcia, Mike Janssen, Eliza Kautz

Faculty Work Comprehensive List

In Mathematics for Human Flourishing, Francis E. Su introduced The Game of Cycles, a game played on a finite simple planar graph. In game play, opponents alternate adding direction to the edges of the graph with the goal of creating a cycle or making the last legal move. Recent work has sought to determine winning strategies on certain classes of graphs. We introduce a tabular representation of a game state and provide a computer program that determines which player has a winning strategy on any legal game board. The program builds a directed graph of all possible game states, utilizing …

Betti Numbers Of Generic Ideals, Jason R. Howell

Electronic Theses & Dissertations (2024 - present)

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe …

Investigation Of Dynamic Adsorption And Desorption Of Polymer Nanogel In Porous Media Through Microfluidics, Junchen Liu, Fuqiao Bai, Abdulaziz A. Almakimi, Mingzhen Wei, Xiaoming He, Ibnelwaleed A. Hussein, Baojun Bai

Mathematics and Statistics Faculty Research & Creative Works

Understanding the transport and retention of elastic nanogel and microgel particles in porous media has been a significant research subject for decades, essential to the application of enhanced oil recovery (EOR). However, a lack of dynamic adsorption and desorption studies, in which the kinetics in porous media are seldom investigated, hinders the design and application of polymer nanogel in underground porous media. In this work, we visualized and quantified the transport and dynamic adsorption of polymer nanogel in 3D glass micromodels that were manufactured by packing glass beads in capillaries. Calibrating the linearity of fluorescence intensity to concentration, we calculated …

Introducing Triangular Groups And Further Results On Magic Groups, Nicholas Charles Fleece

Doctoral Dissertations

In this research, we discuss two new topics in group theory. First, we define an n-magic square in a group to be a nxn array of group elements whose rows, columns, and diagonals have the same product. This definition is akin to the idea of magic squares in the integers. Groups that have an n-magic square are said to be n-magic. We begin with some preliminary results and focus much of our attention on 3-magic groups, though we also give some results for higher n. Through a series of propositions, we ultimately prove a characterization theorem for 3-magic finitely generated …

Comparison Of Methane Emission Quantification Methods At Landfills In New York State, Alexandra M. Catena

Electronic Theses & Dissertations (2024 - present)

Greenhouse gas (GHG) emission increases are a consequence of global population rise and economic growth. It is well known that carbon dioxide is the dominant GHG by mass, but methane has received increasing attention recently due to its high warming potential and relatively short lifetime of about a decade. Methane has several different types of both natural and anthropogenic sources including wetlands, waste management, energy, agriculture and farming, and wildfires, which makes tracking and reducing methane emissions a complex problem. The emission rates of each of these sources are highly uncertain due to differing emission estimation methods, which can be …

Isochronous And Period-Doubling Diagrams For Symplectic Maps Of The Plane, T. Zolkin, S. Nagaitsev, I. Morozov, S. Kladov, Y. -K. Kim

Physics Faculty Publications

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic Hénon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more …

Reasoning Through Change: Exploring Covariational Thinking In Student Exponential Graph Interpretation, Samantha Lynn Pearson

Honors Theses and Capstones

This study explores how life science students reason about exponential functions in the context of an introductory physics course. While exponential functions are essential in both physics and biology—appearing in scenarios like population growth and temperature cooling—students often encounter challenges in interpreting them. Using a resources framework, we examined the cognitive resources these students bring to exponential reasoning, focusing on what ideas and strategies they already possess. Our methodology involved conducting think-aloud interviews with nine students from PHYS 402. Interviews were coded using a framework informed by prior research on covariational reasoning and exponential understanding. Results show that students generally …

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Multipliers Between ℓᴾ Spaces, Raymond Cheng

Mathematics & Statistics Faculty Publications

For 0 < p ⩽ ∞ and 0 < r ⩽ ∞, the space 𝔐_p,r of (coefficient) multipliers from ℓ^p and ℓ^r is completely characterized. This is elementary in most instances. The interesting case 0 < r < p < ∞ requires more effort, and it is shown that a sequence of complex numbers belongs to 𝔐_p,r if and only if the sequence of their absolute values has a non increasing rearrangement (h₀,h₁,h₂,...) satisfying

(∞

Σ (k +1)^(p-r)/p (h^r_k - h^r_k+1)^1/r) < ∞

k = 0

In that case, the expression on the left is the norm of the multiplier, and it is a compact operator. Further upper and lower bounds are given for the multiplier norm.

A Bridge Too Low, John Adam

Mathematics & Statistics Faculty Publications

The article "A bridge too low" in the Physics Teacher journal discusses a low bridge near the River Greta in Keswick, England, with an arch shaped like a semiellipse. It presents questions about the maximum height a person of a certain height can walk under the bridge without hitting their head, the cross-sectional area of the arch, its eccentricity, and perimeter. The article also mentions the approximation by Indian mathematician Srinivasa Ramanujan for the perimeter of an ellipse and invites readers to find the answers online.

A Question Of Transparency, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Question 2: (a) How "long" (in meters) might such a rain shower or fog bank be?

Hint: Suppose you are looking along a linear stack of S cubes with 1-m sides (through the rain or fog). Each cube contains N drops. If p is the visibility (i.e., the fraction of …

A Question Of Transparency: Solutions For Fermi Questions, March 2025, John Adam

Mathematics & Statistics Faculty Publications

Question 1: Why is it easier to see through rain than fog?

Start thinking about this by imagining a fixed volume (V) of water being dispersed into, say, N identical droplets of diameter d. Surface area and volume considerations should lead to the answer in terms of V and d.

Solution to Question 1: N = VI(πd³/6) = 6V/πd³ ≈ 2V/d³.

The cross-sectional area A of each drop is πd²/4 ≈ 3d²/4, so the total area blocked off (assuming no overlapping drops—so this is an upper bound) is NA ≈ 1.5V/d, so the area blocked off is inversely proportional to …

Ms-Yolo: Infrared Object Detection For Edge Deployment Via Mobilenetv4 And Slideloss, Jiali Zhang, Thomas S. White, Haoliang Zhang, Wenqing Hu, Donald C. Wunsch, Jian Liu

Mathematics and Statistics Faculty Research & Creative Works

Infrared imaging has emerged as a robust solution for urban object detection under low-light and adverse weather conditions, offering significant advantages over traditional visible-light cameras. However, challenges such as class imbalance, thermal noise, and computational constraints can significantly hinder model performance in practical settings. To address these issues, we evaluate multiple YOLO variants on the FLIR ADAS V2 dataset, ultimately selecting YOLOv8 as our baseline due to its balanced accuracy and efficiency. Building on this foundation, we present MS-YOLO (MobileNetv4 and SlideLoss based on YOLO), which replaces YOLOv8's CSPDarknet backbone with the more efficient MobileNetV4, reducing computational overhead by 1.5% …

A Bibliographic And Topic Modeling Analysis Of The P-Adic Theory Literature Using Latent Dirichlet Allocation, Humberto Llinás, Ismael Gutiérrez, Anselmo Torresblanca, Javier De La Hoz, Brian Llinás

Computer Science Faculty Publications

P-adic analysis, introduced by Kurt Hensel in the early 20th century, has developed into a fundamental area of mathematical research with broad applications in number theory, algebraic geometry, and mathematical physics. This study aims to examine the thematic evolution and scholarly impact of p-adic research through a comprehensive topic modeling and bibliometric analysis. Using classical bibliometric techniques (e.g., performance analysis, co-authorship, and co-citation networks) combined with Latent Dirichlet Allocation (LDA), we analyzed 7388 peer-reviewed documents published between 1965 and 2024. The computational workflow was conducted using R (version 4.4.1) and VOSviewer (version 1.6.20), which enabled the identification of 20 distinct …

Generalizations Of Some Q-Product Identities Of Ramanujan And Others, Timothy J. Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

By considering certain limiting cases of a WP-Bailey chain discovered by Andrews, and also limiting cases of certain classical summation formulae for basic hypergeometric series, we derive new expressions for certain Lambert series in terms of basic hypergeometric series. In some cases, the resulting series involve an arbitrary Bailey pair. This allows for the derivation of new basic hypergeometric expansions for some q-products and series that Ramanujan expressed in terms of Lambert series. Some of Ramanujan’s identities are extended to more general relations containing one or more free parameters.

Gorenstein Flat Preenvelopes Over Coherent Rings, Alec Moore

College of Graduate Studies: Theses & Dissertations

The existence of precovers and preenvelopes of Gorenstein flat modules is of great interest in the field of Gorenstein homological algebra. We give a sufficient condition in order for the class of Gorenstein flat modules to be preenveloping. More precisely, we prove that if the ring R is coherent such that every injective module has finite flat dimension, then every R-module has a Gorenstein flat preenvelope.

On The Combinatorial Invariance For Kazhdan-Lusztig Polynomials, Grover C. Harrell Iii

College of Graduate Studies: Theses & Dissertations

This thesis will be a discussion on the Combinatorial Invariance Conjecture for Kazhdan Lusztig polynomials. The conjecture is widely suspected to be true; and there is an abun dance of computational evidence which supports it. Despite this, no complete proof has been discovered for more than forty years. We will explore some known results about the CIC, particularly those by Dyer, Incitti, Brenti, Caselli, and Marietti.

Strong Neighborhood-Prime Labelings Under Ordinary & Gaussian Integers, Micheal Arnal-Brown

Murray State Theses and Dissertations

This thesis introduces and studies the notion of a strong neighborhood-prime labeling, a strengthening of the neighborhood-prime labeling where the label 1 can be assigned to any vertex in a graph. We prove that several graph families—including paths, cycles (excluding those congruent to 2 modulo 4), caterpillars, helm graphs, closed helm graphs, gear graphs, and graphs with universal vertices—admit such labelings, and also provide results to more general classes of graphs. We extend this new labeling concept to the Gaussian integers using a spiral order- ing on Z[i] and define a Gaussian analogue of strongly neighborhood-primeness. To support this extension, …

Moore Graphs, Trevor Saxton

Williams Honors College, Honors Research Projects

A Moore graph is a simple regular graph, with n vertices, degree d, and diameter k, that satisfies the Moore bound: n = 1 + d (d − 1)k − 1 d − 2 . There are graphs for which the bound is met and in which existence and uniqueness are known. For k = 2 it is known that the Moore bound is achieved for d = 2, 3, 7, with the case of d = 57 conjectured to exist. For k = 3 the bound is achieved for only d = 3 [5]. Due to the construction of …

Analyzing Car Theft Trends In Central Texas: A Comparative Study Of Waco, College Station, And Killeen, Daniel Njogu

Williams Honors College, Honors Research Projects

This study examines motor vehicle theft (MVT) trends from 2019 to 2023 in three Central Texas cities—Waco, College Station, and Killeen—using temporal analysis, geospatial hotspot mapping, and make/model data. In Killeen, thefts generally rose over the period, with notable peaks in October and on Mondays. College Station saw an overall decline in thefts but experienced a seasonal spike each March, and Waco’s thefts increased until around 2021 before beginning to fall. Local festivals—such as the Spirit of Texas in College Station and the Heart O’ Texas Fair in Waco—appear to coincide with these seasonal upticks. Hyundais and Kias were most …

The Effect Of Electrode Geometry On Excited Species Production In Atmospheric Pressure Air-Hydrogen Streamer Discharge, Shirshak Kumar Dhali, Stuart Reyes

Electrical & Computer Engineering Faculty Publications

When a gas is overvolted at or near atmospheric pressure, it results in a streamer discharge formation. Electrode geometries exert significant impact on the electrical breakdown of gases by altering the spatial profile of the electric field. In many applications the efficient generation of radicals is critical and is determined by the characteristics of the streamer discharge. We examine the effect of electrode geometry on the streamer characteristics and the production of radicals. This is performed for three different electrode geometries: plane–plane, pin–plane, and pin–pin. A two-dimensional rotationally symmetric fluid model is used for the streamer discharge simulation in the …

On Higher Level Zhu Algebras Of N-Graded Vertex Algebras Associated With Simple Leibniz Algebras That Contain Sl2, Christian Soltermann

Theses and Dissertations

In this thesis, we study how the higher-level Zhu algebras of a vertex algebra reflect the structure of associated simple Leibniz algebras. In particular, we construct a vertex algebra from a vertex algebroid containing the simple Lie algebra sl₂ and analyze its higher level Zhu algebras. The irreducible modules of this vertex algebra were completely classified in [JY20b], but the structure of its indecomposable modules remains an open problem. Since modules for higher level Zhu algebras correspond to modules of vertex algebras, studying these algebras provides a method for understanding their broader representation theory.

The Impact Of Loss Function Topology On Gradient Descent, Robert B. Skudnig Jr.

Theses and Dissertations

Gradient descent is a popular optimization method that utilizes a model’s prediction error to iteratively improve its parameters for a given task. The functions that measure this error can be defined to align with the user’s goals and sometimes satisfy metric or norm properties. It is common for these functions to measure over Rn, but any differentiable space allows for gradient descent to occur. There has been some research investigating the influence of topological spaces on optimization methods, but it is a limited field of study. This thesis further explores this phenomenon by applying a transformation prediction model to multiple …

Uncertain Labeling Graphs And Uncertain Graph Classes (With Survey For Various Uncertain Sets), Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Graph theory, a branch of mathematics, studies the relationships between entities using vertices and edges. Uncertain Graph Theory has emerged within this field to model the uncertainties present in real-world networks. Graph labeling involves assigning labels, typically integers, to the vertices or edges of a graph according to specific rules or constraints. This paper introduces the concept of the Turiyam Neutrosophic Labeling Graph, which extends the traditional graph framework by incorporating four membership values—truth, indeterminacy, falsity, and a liberal state—at each vertex and edge. This approach enables a more nuanced representation of complex relationships. Additionally, we discuss the Single-Valued Pentapartitioned …

Some Types Of Hyperneutrosophic Set (1): Bipolar, Pythagorean, Double-Valued, Interval-Valued Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

The Neutrosophic Set is a mathematical framework designed to manage uncertainty, characterized by three membership functions: truth (T), indeterminacy (I), and falsity (F). In recent years, extensions such as the Hyperneutrosophic Set and SuperHyperneutrosophic Set have been introduced to address more complex scenarios. This paper proposes new concepts by extending Bipolar Neutrosophic Sets, Interval-Valued Neutrosophic Sets, Pythagorean Neutrosophic Sets, and Double-Valued Neutrosophic Sets using the frameworks of Hyperneutrosophic and SuperHyperneutrosophic Sets. Additionally, a brief analysis of these extended concepts is presented.

Exploring Concepts Of Hyperfuzzy, Hyperneutrosophic, And Hyperplithogenic Sets (I), Florentin Smarandache, Takaaki Fujita

Branch Mathematics and Statistics Faculty and Staff Publications

This work investigates the evolution of traditional set theory to address complex and ambiguous real-world phenomena. It introduces hierarchical hyperstructures and superhyperstructures, where superhyperstructures are formed by iteratively applying power sets to create nested abstractions. The focus is placed on three foundational set-based frameworks—Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets and their extensions into Hyperfuzzy Sets, HyperNeutrosophic Sets, and Hyperplithogenic Sets. These extensions are applied to various domains, including Statistics, TOPSIS, K-means Clustering, Evolutionary Theory, Topological Spaces, Decision Making, Probability, and Language Theory. By exploring these generalized forms, this paper seeks to guide and inspire further research and development in …

Symbolic Hyperplithogenic Set, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Concepts such as Fuzzy Sets, Neutrosophic Sets, and Plithogenic Sets have been widely investigated for tackling uncertainty, with numerous applications explored across various domains. As extensions of the Plithogenic Set, the HyperPlithogenic Set and the SuperHyperPlithogenic Set are also recognized. A Symbolic Plithogenic Set (SPS) is a structured set defined by symbolic components 𝑃𝑖 and coefficients 𝑎𝑖 , enabling flexible algebraic operations under a specified prevalence order. In this paper, we examine concepts including the Symbolic HyperPlithogenic Set and the Symbolic 𝑛-SuperhyperPlithogenic Set.

Some Types Of Hyperneutrosophic Set (7): Type-M, Nonstationary, Subset-Valued, And Complex Refined, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This paper builds upon the foundational advancements introduced in [26,39–43]. TheNeutrosophic Set provides a versatile mathematical framework for addressing uncertainty through its three membership functions: truth, indeterminacy, and falsity [84]. Extensions such as the Hyperneutrosophic Set and the SuperHyperneutrosophic Set have been recently proposed to address increasingly complex and multidimensional problems. Detailed formal definitions of these concepts can be found in [33]. In this paper, we extend the Type-𝑚, Nonstationary, Subset-Valued, and Complex Refined Neutrosophic Sets using the Hyperneutrosophic Set and the SuperHyperneutrosophic Set frameworks.

Significado Neutrosófico: Partes Comunes De Cosas Poco Comunes Y Partes Poco Comunes De Cosas Comunes, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Esta investigación explora la Neutrosofía, un enfoque filosófico que se centra en la identificación de elementos comunes entre conceptos opuestos y en el análisis de las diferencias entre conceptos semejantes. En este contexto, se estudian las Partes Comunes a Cosas No Comunes, que se manifiestan cuando elementos como y < antiA > comparten aspectos en su intersección, y las Partes No Comunes a Cosas Comunes, donde conceptos iguales como y difieren al exhibir elementos únicos. Este análisis permite comprender mejor la neutralidad e indeterminación representada por < neutA > y < neutB >, situados entre sus respectivos opuestos. La investigación abarca diversas áreas como la Dialéctica, el Yin …

Superhypergraph Neural Networks And Plithogenic Graph Neural Networks: Theoretical Foundations, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Hypergraphs extend traditional graphs by allowing edges to connect multiple nodes, while superhypergraphs further generalize this concept to represent even more complex relationships. Neural networks, inspired by biological systems, are widely used for tasks such as pattern recognition, data classification, and prediction. Graph Neural Networks (GNNs), a well-established framework, have recently been extended to Hypergraph Neural Networks (HGNNs), with their properties and applications being actively studied. The Plithogenic Graph framework enhances graph representations by integrating multi-valued attributes, as well as membership and contradiction functions, enabling the detailed modeling of complex relationships. In the context of handling uncertainty, concepts such as …