Mathematics Commons^™

Subgroups Of Coxeter Groups And Stallings Foldings, Jake A. Murphy

LSU Doctoral Dissertations

For each finitely generated subgroup of a Coxeter group, we define a cell complex called a completion. We show that these completions characterizes the index and normality of the subgroup. We construct a completion corresponding to the intersection of two subgroups and use this construction to characterize malnormality of subgroups of right-angled Coxeter groups. Finally, we show that if a completion of a subgroup is finite, then the subgroup is quasiconvex. Using this, we show that certain reflection subgroups of a Coxeter are quasiconvex.

Analytic Wavefront Sets Of Spherical Distributions On The De Sitter Space, Iswarya Sitiraju

LSU Doctoral Dissertations

In this work, we determine the wavefront set of certain eigendistributions of the Laplace-Beltrami operator on the de Sitter space. Let G′ = O_1,n(R) be the Lorentz group, and let H′ = O_1,n−1(R) ⊂ G′ be its subset. The de Sitter space dSⁿ is a one-sheeted hyperboloid in R^1,n isomorphic to G′/H′. A spherical distribution is an H′-invariant eigendistribution of the Laplace-Beltrami operator on dSⁿ. The space of spherical distributions with eigenvalue λ, denoted by D_λ^H'(dSⁿ), has dimension 2. We construct a basis for the space of …

Study Of Hybrid Nanofluid Flow In A Stationary Cone-Disk System With Temperature-Dependent Fluid Properties, A. S. John, Mahanthesh Basavarajappa, G. Lorenzini

School of Mathematical & Statistical Sciences Faculty Publications

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation …

The Modular Generalized Springer Correspondence For The Symplectic Group, Joseph Dorta

LSU Doctoral Dissertations

The Modular Generalized Springer Correspondence (MGSC), as developed by Achar, Juteau, Henderson, and Riche, stands as a significant extension of the early groundwork laid by Lusztig's Springer Correspondence in characteristic zero which provided crucial insights into the representation theory of finite groups of Lie type. Building upon Lusztig's work, a generalized version of the Springer Correspondence was later formulated to encompass broader contexts.

In the realm of modular representation theory, Juteau's efforts gave rise to the Modular Springer Correspondence, offering a framework to explore the interplay between algebraic geometry and representation theory in positive characteristic. Achar, Juteau, Henderson, and Riche …

How Generative Ai Models Such As Chatgpt Can Be (Mis)Used In Spc Practice, Education, And Research? An Exploratory Study, Fadel M. Megahed, Ying-Ju (Tessa) Chen, Joshua A. Ferris, Sven Knoth, L. Allison Jones-Farmer

Mathematics Faculty Publications

Generative Artificial Intelligence (AI) models such as OpenAI's ChatGPT have the potential to revolutionize Statistical Process Control (SPC) practice, learning, and research. However, these tools are in the early stages of development and can be easily misused or misunderstood. In this paper, we give an overview of the development of Generative AI. Specifically, we explore ChatGPT's ability to provide code, explain basic concepts, and create knowledge related to SPC practice, learning, and research. By investigating responses to structured prompts, we highlight the benefits and limitations of the results. Our study indicates that the current version of ChatGPT performs well for …

Update From Aristotle To Newton, From Sets To Fuzzy Sets, And From Sigmoid To Relu: What Do All These Transitions Have In Common?, Christian Servin, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we show that there is a -- somewhat unexpected -- common trend behind several seemingly unrelated historic transitions: from Aristotelian physics to modern (Newton's) approach, from crisp sets (such as intervals) to fuzzy sets, and from traditional neural networks, with close-to-step-function sigmoid activation functions to modern successful deep neural networks that use a completely different ReLU activation function. In all these cases, the main idea of the corresponding transition can be explained, in mathematical terms, as going from the first order to second order differential equations.

Variable-Order Fractional Laplacian And Its Accurate And Efficient Computations With Meshfree Methods, Yixuan Wu, Yanzhi Zhang

Mathematics and Statistics Faculty Research & Creative Works

The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian (-Δ) α (x) / 2 with 0 < α (x) ≤ 2, which will also be referred as the variable-order fractional Laplacian if α(x) is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian (-Δ) α (x) / 2. Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension d ≥ 1. Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen–Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives.

Mcfadden's Discrete Choice And Softmax Under Interval (And Other) Uncertainty: Revisited, Bartlomiej Jacek Kubica, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Studies of how people actually make decisions have led to an empirical formula that predicts the probability of different decisions based on the utilities of different alternatives. This formula is known as McFadden's formula, after a Nobel prize winning economist who discovered it. A similar formula -- known as softmax -- describes the probability that the classification predicted by a deep neural network is correct, based on the neural network's degrees of confidence in the object belonging to each class. In practice, we usually do not know the exact values of the utilities -- or of the degrees of confidence. …

How To Gauge Inequality And Fairness: A Complete Description Of All Decomposable Versions Of Theil Index, Saeid Tizpaz-Niari, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In general, in statistics, the most widely used way to describe the difference between different elements of a sample if by using standard deviation. This characteristic has a nice property of being decomposable: e.g., to compute the mean and standard deviation of the income overall the whole US, it is sufficient to compute the number of people, mean, and standard deviation over each state; this state-by-state information is sufficient to uniquely reconstruct the overall standard deviation. However, e.g., for gauging income inequality, standard deviation is not very adequate: it provides too much weight to outliers like billionaires, and thus, does …

Paradox Of Causality And Paradoxes Of Set Theory, Alondra Baquier, Bradley Beltran, Gabriel Miki-Silva, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Logical paradoxes show that human reasoning is not always fully captured by the traditional 2-valued logic, that this logic's extensions -- such as multi-valued logics -- are needed. Because of this, the study of paradoxes is important for research on multi-valued logics. In this paper, we focus on paradoxes of set theory. Specifically, we show their analogy with the known paradox of causality, and we use this analogy to come up with similar set-theoretic paradoxes.

Number Representation With Varying Number Of Bits, Anuradha Choudhury, Md Ahsanul Haque, Saeefa Rubaiyet Nowmi, Ahmed Ann Noor Ryen, Sabrina Saika, Vladik Kreinovich

Departmental Technical Reports (CS)

In a computer, usually, all real numbers are stored by using the same number of bits: usually, 8 bytes, i.e., 64 bits. This amount of bits enables us to represent numbers with high accuracy -- up to 19 decimal digits. However, in most cases -- whether we process measurement results or whether we process expert-generated membership degrees -- we do not need that accuracy, so most bits are wasted. To save space, it is therefore reasonable to consider representations with varying number of bits. This would save space used for representing numbers themselves, but we would also need to store …

Data Fusion Is More Complex Than Data Processing: A Proof, Robert Alvarez, Salvador Ruiz, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

Empirical data shows that, in general, data fusion takes more computation time than data processing. In this paper, we provide a proof that data fusion is indeed more complex than data processing.

How To Fairly Allocate Safety Benefits Of Self-Driving Cars, Fernando Munoz, Christian Servin, Vladik Kreinovich

Departmental Technical Reports (CS)

In this paper, we describe how to fairly allocated safety benefits of self-driving cars between drivers and pedestrians -- so as to minimize the overall harm.

Using Known Relation Between Quantities To Make Measurements More Accurate And More Reliable, Niklas Winnewisser, Felix Mett, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Most of our knowledge comes, ultimately, from measurements and from processing measurement results. In this, metrology is very valuable: it teaches us how to gauge the accuracy of the measurement results and of the results of data processing, and how to calibrate the measuring instruments so as to reach the maximum accuracy. However, traditional metrology mostly concentrates on individual measurements. In practice, often, there are also relations between the current values of different quantities. For example, there is usually an known upper bound on the difference between the values of the same quantity at close moments of time or at …

Festival Of Research Abstracts, Spring 2024, College Of Science And Mathematics, Wright State University

Festival of Research

The collection of abstracts accepted for the Spring 2024 Festival of Research hosted by the Wright State University College of Science and Mathematics.

2-Accessibility Of The Lucas Numbers/Fibonacci Like Sequences/Wythoff Array, Cameron Lejeune

Mathematics Senior Capstone Papers

In this project we will be delving into the combinatorics side of mathematics, with basic a graph theory idea such as set coloring. It will be a continuation of a question formed by the works of Bruce M. Landman, and Aaron Robertson, ”3-Accessibility of the Fibonacci Numbers.” They were able to prove the Fibonacci Numbers to be 2-Accessible, and while they did not supply the induction proof, we were able to construct and provide a proof. This project will use their lemmas and propositions to attempt to prove 2-Accessibility of the Lucas Numbers, Fibonacci-like sequences, and the Zeckendorf-Wythoff Array.

Can Compiled Player War Predict Mlb Team Win Percentage?, Thomas Hertel

Mathematics Senior Capstone Papers

The baseball statistic Wins Above Replacement (WAR) is a complex yet effective metric for approximating the contribution of a player to his team through representing how many more wins his team ought to earn than with a AAAA player in his stead. The intent of this paper is to test whether the accumulation of player’s WAR can be extended to satisfy another discipline of SABRmetrics, i.e. predicting an entire team’s future performance, through modeling the sum’s correspondence (or lack thereof) to wins. If WAR is shown to extend in this way then franchise front offices could feasibly isolate it as …

Using Multiple Regression Analysis To Determine The Strength Of Certain Factors On Student Absenteeism, Emmitt Antwine

Mathematics Senior Capstone Papers

During the 2015-16 academic year, approximately 16% of the student population—exceeding 7 million students—were absent from school for 15 days or more. The escalation in chronic absenteeism is influenced by various factors, including poor health conditions, nonstandard work schedules of parents, socioeconomic disadvantages, changes in household compositions, frequent residential relocation, and substantial family responsibilities. Previous research on student absenteeism has examined the negative impacts that chronic absenteeism has had on students in diverse communities such as racial minorities, students with disabilities, and English Language Learners communities. We utilized information obtained from the U.S. Department of Education’s Civil Rights Data Collection …

An Exploration Of The Sums Of Two Squares And Pentagonal Numbers, Jacob Von Tress

Mathematics Senior Capstone Papers

In the field of number theory, square numbers are very significant, and finding the sums of square numbers is a topic of certain interest to mathematicians. The most immediate application for adding together two square numbers is to identify Pythagorean triples. However, apart from seeking sums of two squares that are squares themselves, interesting patterns emerge that have fascinated number theorists for decades. Particularly, the distribution of a number’s divisors can explicitly determine how many ways that number can be written as a sum of two squares. Furthermore, pentagonal numbers, similar to square numbers, can be visualized by drawing a …

Matroids Stemming From The Maximal Relaxation Of Graphic Matroids, Landen Nguyen

Mathematics Senior Capstone Papers

This research explores the base perspective for maximal relaxation of graphic matroids. We begin by covering the required knowledge of graph theory and the basics of matroid theory required to conduct this research such as definitions of matroids up to the definitions of relaxation, hyperplanes, and uniform matroids. We then analyze the conjecture in matroids obtained from simple connected graphs.

Four Colorful Methods For Finding The Chromatic Polynomial, Emerson Statom

Mathematics Senior Capstone Papers

Mathematicians apply algebraic graph theory to address and interpret problems arising out of data structures and optimization. For example, graph coloring problems have intrigued mathematicians and computer scientists alike. The chromatic polynomial was created to help solve such problems. This research aims to create a greater understanding of the underlying structures behind chromatic polynomials by analyzing the algebraic, deletion-contraction, and color partition methods. Each method has differing applications, but by viewing them all together, this research hopes to show the depth of this most interesting subject in a condensed manner.

Factors That Contribute To Completing Gateway Math Course At Southern University, Emma Colvin

Mathematics Senior Capstone Papers

At Southern University in Shreveport, Louisiana, there are several pathways students can take to complete the institution’s required Gateway Math Courses (GMC). The GMC consists of Math 133, Math 135, and Math 136. Currently, there is a significantly low passing rate for these courses potentially due to several factors such as the student’s math ACT score, the student’s age, whether or not the student took optional developmental math courses beforehand, and whether or not the student took these courses online or in person.

Congruences Of The Sums Of The First Rp Fibonacci Numbers, Tyler Warzynak

Mathematics Senior Capstone Papers

This paper serves as an extension/application of a method detailed by Chang et. al. in two separate papers which worked with sums modulo p for combinatorial sequences, positive integers r, and prime numbers p. Using a known formula, the Fibonacci numbers were converted into a sum of binomial coefficients to apply the aforementioned method. A hypothesis was formed using computational methods, where a pattern was observed to hold for the first 50,000 prime numbers. This claim was then proven following the methodology from Chang et. al. using properties of Laurent polynomials, definitions and theorems related to the Fibonacci numbers, and …

Abstract Measures On Totally Ordered Sets, Gregory Whitehurst

Mathematics Senior Capstone Papers

Two specific notions of integration were the motivation of this paper: Riemann-Stieltjes and Lebesgue-Stieltjes integration. The question that arose from these notions was the idea of generalization. How can Stieltjes integration be generalized to an arbitrary measure? The buildup to this question requires a beautiful combination of measure theory and order theory to deliver a measure on totally ordered sets that provide structure for a potential generalization. This paper shows that the set of chains in a totally ordered set is a σ-algebra and defines cumulative cardinality measure on (X, Σ) in an attempt to give insight on the structure …

Does The ”Freshman 15” Exist?, Peyton Albritton

Mathematics Senior Capstone Papers

This experiment was performed to determine whether the environmental changes of adapting from high school to college impact a student’s weight. Using a Google form, we collected data from 58 college students in two Psychology 102 classes, with ages ranging from seventeen to twenty-two. The students were asked a series of questions regarding their age and gender, as well as their eating, sleeping, and activity habit changes. Data was analyzed using hypothesis testing and a t-test.

Injuries On Artificial Turf Vs. Natural Grass In The Nfl, Robert Emory

Mathematics Senior Capstone Papers

The purpose of this research is to determine if artificial turf causes more injuries than natural grass. By referencing different statistics from a sample of games from the 2023 National Football League season. Out of the ten games sampled, five were played on natural grass and five were played on artificial turf. We ran the data collected from these games through two different regression analysis models that output p-values to show what truly caused the injuries. Our model uses a player’s position, height, weight, age, and snap counts along with the field surface type to see how if an injury …

Comparison Of Linear Control Techniques For The Underactuated Nonlinear Quadcopter System, Ian Golsby

Mathematics Senior Capstone Papers

Uncrewed Aerial Vehicles (UAVs) are a prevalent technology in many fields. They must be lightweight, efficient, and stable in order to carry out their objectives or support a payload. The control system that maintains a UAV’s attitude directly contributes to the stability and efficiency of the UAV, and more efficient UAVs can be made more lightweight by reducing battery size. Because the UAV has only four degrees of control (one per motor) but requires twelve dimensions to describe its orientation and position over time, it is considered an under-actuated nonlinear complex system. In this study, we compare various linear control …

A Closer Look At The Structure Of Uniquely Pancyclic Graphs, Zak Kilpatrick

Mathematics Senior Capstone Papers

In this paper we will consider uniquely pancyclic graphs which are described as graphs on n vertices which have a unique cycle of size k for each integer k from 3 to n. To date, there are only seven known graphs of this type. To narrow down what a uniquely pancyclic graph could look like in order to aid the search in finding new ones, the focus of this paper is on the different ways the three and four cycles for these graphs can occur.

Somewhat Surprisingly, (Subjective) Fuzzy Technique Can Help To Better Combine Measurement Results And Expert Estimates Into A Model With Guaranteed Accuracy: Digital Twins And Beyond, Niklas Winnewisser, Michael Beer, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

To understand how different factors and different control strategies will affect a system -- be it a plant, an airplane, etc. -- it is desirable to form an accurate digital model of this system. Such models are known as digital twins. To make a digital twin as accurate as possible, it is desirable to incorporate all available knowledge of the system into this model. In many cases, a significant part of this knowledge comes in terms of expert statements, statements that are often formulated by using imprecise ("fuzzy") words from natural language such as "small", "very possible", etc. To translate …

The Impacts Of Covid-19 On The U.S. Life Insurance Industry: A Study Of Pandemic Mortality Rates, Allison Wheaton

Honors Projects in Mathematics and Economics

The COVID-19 pandemic has had substantial and widespread impacts around the world that are not yet completely understood but will likely be experienced for years to come. This research seeks to analyzed and quantify the pandemic’s effects on the life insurance industry in the United States through the use of mortality models. Because the life insurance industry relies on the ability to accurately predict an individual’s mortality, it is necessary to investigate the impacts of the pandemic on mortality predictions and accuracy. Impairment factors were developed using pandemic mortality data and applied to existing mortality tables to compare expected and …