Mathematics Commons^™

Free Energy Differences In Nonequilibrium Thermodynamic Processes, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Systems which may be allowed to go out of equilibrium have been of interest recently. A quantity is formulated whose average over an ensemble of microscopic realizations of the process depends only on the initial and final states. This is so even though the system may not be in equilibrium during the process. A generalization to the case where the initial and final states are not equilibrium states is developed here. Quantum analogues of these relations are derived, and an indication of how this might be applied to study entropy increase in thermodynamics is presented.

Traveling Wave Phenomena Of Inhomogeneous Half-Wave Equation, Zhaosheng Feng, Yu Su

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0 < | v | < 1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases.

Waves In Cosmological Background With Static Schwarzschild Radius In The Expanding Universe, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe.

Enumeration Of Level-K Hypertriangulations, Patrick J. Kaylor

Theses and Dissertations

Triangulations are a classical object in discrete and computational geometry that finds it uses in many other fields, including numerous applications. In this thesis we approach the question of enumerating all hypertriangulations of a point set, the family of tilings introduced by Olarte and Santos in 2022 as induced projections of hypersimplices. Unfortunately, the usual approach to enumeration of trinagulations using flips may not work for hypertriangulations as flip-connectivity is only conjectured for that family even in the two-dimensional case. As a result, we develop a DFS-based algorithm and present its Python implementation that enumerates all hypertriangulations of small point …

Missing Data Imputation With Longitudinal Data, Joseph Omar Alanis

Theses and Dissertations

In the repeated measures longitudinal datasets where missing data is a relatively common issue, we explored different imputation methods, including machine learning (ML) methods in order to examine potential efficiencies for using traditional versus newer computational methods. In order to accomplish said comparison, we used a Monte Carlo simulation experiment of a population of size N=70000 to mimic a clinical trial, with different scenarios of missing at random (MAR) data in the response variable. To compare the behavior of each method resulting from the difference between population and sample dataset, a real dataset was used from the Boston College on …

Recognizing Triangles Using Weighted Ehrhart Polynomials, Rafael Sarabia

Theses and Dissertations

Ehrhart theory is a classical topic for polygons or polytopes with vertices with integer coordinates. One of the main results of the theory claims that for every convex polygon P with integer vertices, the number of integer points in its nth dilation is a polynomial in n, the Ehrhart polynomial of P. This property can be extended if for a polynomial weight w(x, y), we sum values of w(x, y) over all points with integer coordinates in the nth dilation of P. The resulted polynomial is called the weighted Ehrhart polynomial of P associated with the weight w. In this …

Nilpotent Global Centers Of Generalized Polynomial Kukles System With Degree Three, Hebai Chen, Zhaosheng Feng, Rui Zahg

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we study and characterize the nilpotent global centers of a generalized polynomial Kukles system with degree three. A sufficient and necessary condition of global centers is established under certain parametric conditions.

The Arbitrariness Of Symmetry In Mathematical Proofs, Melisa Vivanco

Philosophy Faculty Publications

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory power. The examples developed in this paper constitute compelling evidence supporting these claims.

Constrained Quantization For The Cantor Distribution, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

The theory of constrained quantization has been recently introduced by Pandey and Roychowdhury. In this paper, they have further generalized their previous definition of constrained quantization and studied the constrained quantization for the classical Cantor distribution. Toward this, they have calculated the optimal sets of n-points, nth constrained quantization errors, the constrained quantization dimensions, and the constrained quantization coefficients, taking different families of constraints for all n∈N. The results in this paper show that both the constrained quantization dimension and the constrained quantization coefficient for the Cantor distribution depend on the underlying constraints. It also shows that the constrained quantization …

Multi-Symplectic Method For The Two-Component Camassa–Holm (2ch) System, Xiaojian Xi, Weipeng Hu, Bo Tang, Pingwei Deng, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the …

P-Adic Quantum Mechanics, The Dirac Equation, And The Violation Of Einstein Causality, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘-adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1/℘, where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘-adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘-adic space imposes substantial restrictions on the solutions of the new …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney

Theses and Dissertations

Bucklew and Wise (1982) established that the quantization dimension of an absolutely continuous probability measure on a Euclidean space equals the Euclidean dimension of the space, and that the quantization coefficient is finite and positive. This thesis explores the variability of quantization coefficients for uniform distributions on various geometric structures, including line segments, circles, and regular polygons. We derive the conditional optimal sets of n-points and calculate the n-th conditional quantization errors for uniform distributions under different conditions. On line segments, we identify optimal point distributions as either equally spaced or geometrically spaced towards fixed end-points. For circles, we analyze …

Application And Analysis Of Machine Learning And Deep Learning Algorithms In Detection Of Ddos Cyberattacks, Dipok Deb

Theses and Dissertations

A Distributed Denial-of-Service (DDoS) attack involves overwhelming a target system's data bandwidth or computational resources, often using multiple attack systems, aiming to slow down or disable the targeted system. Detecting and mitigating DDoS attacks effectively remains challenging due to their varying characteristics. One of the promising approaches involves developing an AI based Intrusion Detection System (IDS) against cyberattacks. In this study, we aim to develop an AI based Intrusion Detection System (IDS) for DDoS threat detection using Machine Learning, Deep Learning, or hybrid techniques. Different Machine Learning (ML) and Deep Learning (DL) algorithms like Random Forest (RF), Naïve Bayes (NB), …

On Angles In Higher Order Brillouin Tessellations And Related Tilings In The Plane, Herbert Edelsbrunner, Alexey Garber, Mohadese Ghafari, Teresa Heiss, Morteza Saghafian

School of Mathematical & Statistical Sciences Faculty Publications

For a locally finite set in R 2 , the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R 2 is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in …

Using Chatgpt As A Thought Partner In Writing Relevant Proportional Reasoning Word Problems, Andrea Berryhill, Lendon Chandler, Liza Bondurant, Bima Sapkota

School of Mathematical & Statistical Sciences Faculty Publications

In this article, we, two preservice mathematics teachers (PSTs) and two mathematics teacher educators (MTEs), explore the potential of using ChatGPT, a large language model (LLM), as a thought partner to create relevant proportional reasoning word problems. We detail our iterative process of refining our inputs and critically evaluating ChatGPT's outputs. Our takeaways from this journey offer valuable insights for educators seeking to leverage the capabilities of LLMs in their lesson-planning endeavors.

On The Size Of Maximal Binary Codes With 2, 3, And 4 Distances, Alexander Barg, Alexey Glazyrin, Wei-Jiun Kao, Ching-Yi Lai, Pin-Chieh Tseng, Wei-Hsuan Yu

School of Mathematical & Statistical Sciences Faculty Publications

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths n≥6 as well as the exact size of maximal binary constant weight codes with 2, 3, and 4 distances for several values of the weight and for all but small lengths.

Enhancing Tumor Classification Through Machine Learning Algorithms For Breast Cancer Diagnosis, Lawrence Agbota, Edmund F. Agyemang, Priscilla Kissi-Appiah, Lateef Moshood, Akua Osei- Nkwantabisa, Vincent Agbenyeavu, Abraham Nsiah, Augustina Adjei

School of Mathematical & Statistical Sciences Faculty Publications

In cancer diagnosis, machine learning helps improve cancer detection by providing doctors with a second perspective and allowing for faster and more accurate determination and decisions. Numerous studies have used both classic machine learning approaches and deep learning to address cancer classification. In this study, we examine the efficacy of five commonly used machine learning algorithms; both traditional and deep learning models namely, Logistic Regression, Support Vector Machines (SVM), Random Forest (RF), Decision Tree and Deep Neural Networks (DNN). We analyze their ability to properly classify tumors as Benign or Malignant using the Wisconsin breast cancer dataset (WBCD). Random Forest …

Post-Baccalaureate Research Experiences For Students At Two Hispanic-Serving Institutions (Experience), Dessaray Monique Gorbett, Benjamin C. Flores, Cristina Villalobos, Sara E. Rodriguez, Ariana (Ari) Arciero, Josef A. Sifuentes

School of Mathematical & Statistical Sciences Faculty Publications

This study examines the implementation of a one-year program that offered post-baccalaureate research experiences to a cohort of recent STEM graduates from two large Hispanic-serving institutions. Both institutions are collaborators in an NSF Louis Stokes Alliance for Minority Participation (LSAMP) grant, which is dedicated to enhancing the academic experience of historically underrepresented minorities. Applicants were either self-nominated or were nominated by faculty to participate in the post-baccalaureate program. Following an application process that included an interview, selected participants were matched to research faculty and committed to engage in a high-impact research project for at least one semester and for up …

Bounds For The Regularity Radius Of Delone Sets, Nikolay Dolbilin, Alexey Garber, Egon Schulte, Marjorie Senechal

School of Mathematical & Statistical Sciences Faculty Publications

Delone sets are discrete point sets X in Rd characterized by parameters (r, R), where (usually) 2r is the smallest inter-point distance of X, and R is the radius of a largest “empty ball” that can be inserted into the interstices of X. The regularity radius ρ^d is defined as the smallest positive number ρ such that each Delone set with congruent clusters of radius ρ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our “Weak Conjecture” states that ρ^d=O(d2log2d)R as d→∞ , …

Oscillations In Neuronal Activity: A Neuron-Centered Spatiotemporal Model Of The Unfolded Protein Response In Prion Diseases, Elliot M. Miller, Tat Chung D. Chan, Carlos Montes-Matamoros, Omar Sharif, Laurent Pujo-Menjouet, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P …

Statistics For Anticyclotomic Iwasawa Invariants Of Elliptic Curves, Jeffrey Hatley, Debanjana Kundu, Anwesh Ray

School of Mathematical & Statistical Sciences Faculty Publications

We study the average behaviour of the Iwasawa invariants for Selmer groups of elliptic curves, considered over anticyclotomic Zp -extensions in both the definite and indefinite settings. The results in this paper lie at the intersection of arithmetic statistics and Iwasawa theory.

A Novel Consumer-Centric Metric For Evaluating Hearing Device Audio Performance, Vinaya Manchaiah, Steve Taddei, Abram Bailey, De Wet Swanepoel, Hansapani Rodrigo

School of Mathematical & Statistical Sciences Faculty Publications

Background and Aim: The emergence of direct-to-consumer hearing devices has introduced confusion in making appropriate choices, highlighting the need for users to be well-informed for optimal device selection. Currently, no established metric offers insights into the sound performance of these devices. This study aimed to introduce and assess a novel consumer-centric metric (i.e., SoundScore) for hearing device audio performance.

Method: The SoundScore metric was created based on five dimensions of hearing device audio performance (i.e., speech benefit in quiet and moderate, speech benefit in loud, own voice perception, feedback control, streamed music sound quality). Tests were conducted under lab conditions …

Studying Hilbert’S 10th Problem Via Explicit Elliptic Curves, Debanjana Kundu, Antonio Lei, Florian Sprung

School of Mathematical & Statistical Sciences Faculty Publications

N. García-Fritz and H. Pasten showed that Hilbert’s 10th problem is unsolvable in the ring of integers of number fields of the form Q(p–√3,−q−−−√) for positive proportions of primes p and q. We improve their proportions and extend their results to the case of number fields of the form Q(p–√3,Dq−−−√) , where D belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.

Study Of Nanofluid Flow In A Stationary Cone–Disk System With Temperature-Dependent Viscosity And Thermal Conductivity, Anagha Susan John, Mahanthesh Basavarajappa, Igor V. Shevchuk

School of Mathematical & Statistical Sciences Faculty Publications

The substantial temperature gradient experienced by systems operating at relatively high temperatures significantly impacts the transport characteristics of fluids. Hence, considering temperature-dependent fluid properties is critical for obtaining realistic prediction of fluid behavior and optimizing system performance. The current study focuses on the flow of nanofluids in a stationary cone–disk system (SCDS), taking into account temperature-dependent thermal conductivity and viscosity. The influence of Brownian motion, thermophoresis, and Rosseland radiative flux on the heat transport features are also examined. The Reynolds model for viscosity and Chiam's model for thermal conductivity are employed. The Navier–Stokes equation, the energy equation, the incompressibility condition, …

Conditional Constrained And Unconstrained Quantization For A Uniform Distribution On A Hexagon, Christina Hamilton

Theses and Dissertations

In this thesis, we have considered a uniform distribution on a regular hexagon and the set of all its six vertices as a conditional set. For the uniform distribution under the conditional set first, for all positive integers n ≥ 6, we obtain the conditional optimal sets of n-points and the nth conditional quantization errors, and then we calculate the conditional quantization dimension and the conditional quantization coefficient in the unconstrained scenario. Then, for the uniform distribution on the hexagon taking the same conditional set, we investigate the conditional constrained optimal sets of n-points and the conditional constrained quantization errors …

How Mathematics Instructors Foster The Development Of Black Students' Mathematics Identity In Undergraduate Active Learning Mathematics Courses, Ashly J. Olusanya

Theses and Dissertations

Black students must overcome unique challenges to succeed in mathematics. Educators are tasked with identifying equitable teaching practices to support these students. Active learning (AL) is a teaching pedagogy that engages students in rigorous mathematical activities and encourages student participation. This research study will explore the professors’ beliefs about how students learn mathematics and why they use active learning in their collegiate mathematics courses. The study explores the connections between these beliefs and their reported use of instructional practices. The study also identifies the instructors’ beliefs about developing students’ mathematics identities, particularly their Black …

A Study On A Vector Complex Modified Korteweg-De Vries Equation, Changyan Shi

Theses and Dissertations

In this thesis, we systematically study a vector complex modified Kordeweg-de Vries equation by combining Hirota's bilinear method and the the Kadomtsev–Petviashvili (KP) reduction method. This vector nonlinear equation is a multi-component generalization of the well-known modified Kordeweg-de Vries (mKdV) equation and can be reduced to the known Hirota equation, Sasa-Satsuma (SS) equation, Sasa-Satsuma-mKdV equation as well as coupled Sasa-Satsuma equation. First, we bilinearize the vector complex mKdV equation under both the zero and nonzero boundary conditions by introducing auxiliary tau functions. Then, starting from two sets of bilinear equations of multi-component KP hierarchy and single-component KP-Toda …

The Perspectives Of Using Desmos For Students’ Conceptual Understanding And Procedural Fluency To Solve Linear Equations, Larmel Dimatulac Madrilejos

Theses and Dissertations

The study examines the perspectives of using the Desmos calculator of Algebra I students' conceptual understanding and procedural fluency to write, graph, and solve linear equations in Algebra I STAAR. While the students have continuously used technology for mathematics assessment, emergent bilingual students in South Texas still need help passing high-stakes testing. The framework of the study is grounded in the theory of mathematical education (knowledge of mathematics educators to teach), the theory of mathematical learning (understanding how students learn mathematics), and social constructivism. The study seeks ways to teach all students, mainly the minority, to learn …

Local Existence Of Solutions To A Nonlinear Autonomous Pde Model For Population Dynamics With Nonlocal Transport And Competition, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Highlights

• Partial differential equation models are ubiquitous in applied sciences.

• A partial differential equation based in ecology is studied for solution existence.

• Energy methods and convergence analysis lead to local classical solutions.

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …