Mathematics Commons^™

Standing Waves Of Schrödinger Equation With Constant Magnetic Field And Combined Power Nonlinearities, Zhaosheng Feng, Yu Su

School of Mathematical & Statistical Sciences Faculty Publications

We study standing waves of the Schrödinger equation with constant magnetic field and combined power nonlinearities, which describes a single non-relativistic quantum particle in the presence of an electromagnetic field. We develop the local minima geometry method to establish the existence, estimates and mass collapse results for this equation, which generalize exiting results in the literature.

Existence And Stability Theory Of A Neurologically Inspired Parabolic Pde Model With A Nonlinear Time-Delayed Boundary Condition, Gangadhara Boregowda, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we establish the existence of a positive, bounded solution for a class of parabolic partial differential equations with nonlinear boundary conditions, where the boundary conditions depend on the solution on the boundary at a time 𝜏≥0 in the past. These equations model the production dynamics of a protein species by a single cell, where a feedback mechanism downregulates the protein’s production. Furthermore, we analyze the stability of a nontrivial steady-state solution and provide sufficient conditions on the nonlinearity parameter, boundary flux, and time-delay that ensure the occurrence of a Hopf bifurcation.

Long-Term Behavior Of Subordinated Branching Processes With Prevailing Emigration, George Yanev

School of Mathematical & Statistical Sciences Faculty Publications

This paper deals into the long-term behavior of subordinated critical branching processes with migration. We focus on scenarios where emi­gration is the dominant factor and introduce additional randomness in timing through a subordination mechanism, involving renewal processes. The key findings highlight how the initial population size and the interar­rival mean time influence both asymptotic behavior of the non-extinction probability and corresponding Yaglom type limit theorems. We also study an alternating regenerative process, when the population cycles between zero and positive states. This research complements previous studies for processes when immigration prevails over emigration.

Quantization For The Mixtures Of Uniform Distributions On Connected And Disconnected Line Segments, Asha Barua, Gustavo Fernandez, Ashley Gomez, Olga Lopez, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we have studied various mixed distributions generated by two uniform distributions: first, where the supports are two connected line segments, and second, where the supports are two disconnected line segments. For these mixed distributions, we have determined the optimal sets of n-means and the corresponding nth quantization errors for all positive integers n. The methods developed in this paper can be applied more generally to investigate optimal quantization for any mixed distribution P:=pP1+(1−p)P2, where P1 and P2 are arbitrary probability distributions supported on either connected or disconnected line segments, and (p,1−p) is any probability …

Murmurations Of Modular Forms And P-Power Coefficients, Debanjana Kundu, Katharina Müller

School of Mathematical & Statistical Sciences Faculty Publications

We extend the work of N. Zubrilina on murmuration of modular forms to the case when prime-indexed coefficients are replaced by squares of primes. Our key observation is that the shape of the murmuration density is the same.

2-Adic Quantum Mechanics, Continuous-Time Quantum Walks, And The Space Discreteness, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The authors show that a large class of 2-adic Schrödinger equations is the scaling limit of certain continuous-time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, new types of continuous-time quantum walks (CTQWs) on graphs using two symmetric matrices are constructed. The transport between nodes in one direction is described by one matrix, while the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations …

Congruences For Quotients Of Klein Forms, Jeffery Opoku

Theses and Dissertations

This dissertation investigates the arithmetic properties of some modular forms and eta quotients, focusing on the divisibility properties and congruence relations satisfied by these. The first part examines quotients of the Rogers-Ramanujan and Rogers-Selberg functions, defined by \[ f(\tau) = q^{r} (q^5; q^5)^{a_0} (q, q^4; q^5)^{a_1} (q^2, q^3; q^5)^{a_2}= \sum_{n=r}^{\infty} P_{a_0,a_1,a_2}(n-r) q^n, \] and \[ g(\tau) = q^{s} (q^7; q^7)^{a_0} (q, q^6; q^7)^{a_1} (q^2, q^5; q^7)^{a_2} (q^3, q^4; q^7)^{a_3}= \sum_{n=s}^{\infty} P_{a_0,a_1,a_2,a_3}(n-s) q^n, \] respectively. We establish conditions on the exponents \(a_0, a_1, a_2, a_3\) and residue classes \(r\) and $s$ modulo $p$ such that \(P_{a_0, a_1, a_2}(pn - r) \equiv …

Optimal Quantization For Nonuniform Discrete Distributions, Russel Cabasag, Samir Huq, Eric Mendoza, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

This paper explores the process of optimal quantization for several types of discrete probability distributions. Quantization is a technique used to approximate a complex distribution with a smaller set of representative points, which is important in fields such as data compression and signal processing. We begin by examining two specific nonuniform distributions over a finite set of values and identify the best representative points for different levels of approximation. We then extend our analysis to two infinite discrete distributions: one supported on the reciprocals of natural numbers and another on the natural numbers themselves. For these distributions, we compute the …

Conditional Optimal Sets And The Quantization Coefficients For Some Uniform Distributions, Evans Nyanney, Megha Pandey, Mrinal Kanti Roychowdhury

School of Mathematical & Statistical Sciences Faculty Publications

Bucklew and Wise (1982) showed that the quantization dimension of an absolutely continuous probability measure on a given Euclidean space is constant and equals the Euclidean dimension of the space, and the quantization coefficient exists as a finite positive number. By giving different examples, in this paper, we have shown that the quantization coefficients for absolutely continuous probability measures defined on the same Euclidean space can be different. We have taken uniform distribution as a prototype of an absolutely continuous probability measure. In addition, we have also calculated the conditional optimal sets of n-points and the nth conditional quantization errors …

A Generalization Of Franklin’S Partition Identity And A Beck-Type Companion Identity, Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, Ren Watson

School of Mathematical & Statistical Sciences Faculty Publications

Euler’s classic partition identity states that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.

Spike Timing-Dependent Plasticity And Random Inputs Shape Interspike Interval Regularity Of Model Stn Neurons, Thoa Thieu, Roderick Melnik

School of Mathematical & Statistical Sciences Faculty Publications

Background/Objectives: Neuronal oscillations play a key role in the symptoms of Parkinson’s disease (PD). This study investigates the effects of random synaptic inputs, their correlations, and the interaction with synaptic dynamics and spike timing-dependent plasticity (STDP) on the membrane potential and firing patterns of subthalamic nucleus (STN) neurons, both in healthy and PD-affected states. Methods: We used a modified Hodgkin–Huxley model with a Langevin stochastic framework to study how synaptic conductance, random input fluctuations, and STDP affect STN neuron firing and membrane potential, including sensitivity to refractory period and synaptic depression variability. Results: Our results show that random inputs significantly …

Dissections Of Lacunary Eta Quotients And Identically Vanishing Coefficients, Timothy Huber, James Mclaughlin, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

For any function A(q)=∑∞n=0anqn defineA(0):={n∈N:an=0}.Now suppose C(q) and D(q) are two functions whose m-dissections are given byC(q)=c0G0(qm)+c1qG1(qm)+…+cm−1qm−1Gm−1(qm),D(q)=d0G0(qm)+d1qG1(qm)+…+dm−1qm−1Gm−1(qm).If it is the case that ci=0⟺di=0, i=0,1,…,m−1, then we say that C(q) and D(q) have similar m-dissections, and then it is also clear that C(0)=D(0), in which case we say that C(q) and D(q) have identically vanishing coefficients. In the present paper some new 4-dissections of particular eta quotients are developed. These are used in conjunction with known 2- and 3-dissections to prove many results on the identical vanishing of coefficients of various eta quotients, results which were found experimentally …

P-Adic Cellular Neural Networks With Delay, Baboucarr Dibba

Theses and Dissertations

This dissertation presents a novel framework for p-adic reaction-diffusion cellular neural networks (CNNs) with delay, offering new insights into the stability and dynamic behavior of these networks. Through numerical simulations, we explore their response to various conditions, highlighting their capability to model complex systems. Additionally, this work reviews cutting-edge developments in p-adic CNNs, particularly their application to advanced image processing tasks such as edge detection and noise filtering, demonstrating their effectiveness in preserving critical image features while filtering out noise. This dissertation is written in collaboration with my Ph.D supervisor and Dr. Zambrano-Luna, Brian.

A Study Of Quasigroups, A Computational Approach On Isotopism And Isomorphism Hierarchical Structure, Runaldo Montrose

Theses and Dissertations

Quasigroups morphism classification can present a real computational challenge. As of today, the number of quasigroups that can be generated from a finite set S of cardinality n is known for very small value of n. Though we know the number of quasigroups for small n, generating them in real time is a whole other issue that leads to a bigger challenge of building their isomorphy classes, because the algorithm used required large computing memory resources. In this paper we will expose the computational complexity of constructing their hierarchical morphism structure from isotopism to isomorphism. As an improvement, …

Class Groups And Selmer Groups In Special Families, Debanjana Kundu, Abhishek _

School of Mathematical & Statistical Sciences Faculty Publications

We explore the relationship between (3-isogeny induced) Selmer group of an elliptic curve and the (3 part of) the ideal class group, over certain non-abelian number fields.

A Proposal To Explore Geometry With Geogebra: Graphical Exploration And Formal Demonstration Of The Collinearity Of The Barycenters Of A Polygon, Saulo Mosquera Lopez, Marlio Paredes, Walter Castro

School of Mathematical & Statistical Sciences Faculty Publications

This paper illustrates an example of a mathematical activity that teachers and students can replicate to create an experience that resembles professional mathematical activity. We extend the property “Consider a triangle ABC, any straight line and let A’, B’, C’ be the reflections of the points A, B, C on the straight line then the barycenters of the triangles ABC, A’BC, AB’C and ABC’ are collinear and the line of collinearity is perpendicular to the straight line” for any quadrilateral. It is proved that there are four additional triangles, for a total of eight, whose barycenters are collinear and that …

High Moment And Pathwise Error Estimates For Fully Discrete Mixed Finite Element Approximations Of The Stochastic Stokes Equations With Multiplicative Noise, Liet Vo

School of Mathematical & Statistical Sciences Faculty Publications

This paper is concerned with high moment and pathwise error estimates for both velocity and pressure approximations of the Euler–Maruyama scheme for time discretization and its fully discrete mixed finite element discretization. Optimal rates of convergence are established for all pth moment errors for p ≥ 2 using a novel doubling of moments technique. The almost optimal rates of convergence are then obtained using Kolmogorov’s theorem based on the high moment error estimates. Unlike for the velocity error estimate, the high moment and pathwise error estimates for the pressure approximation are proved in a time-averaged norm. In addition, the …

Circle Actions On Oriented 4-Manifolds, Donghoon Jang, Oleg R. Musin

School of Mathematical & Statistical Sciences Faculty Publications

In this present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah–Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case of isolated fixed points that if an abstract graph satisfies the Atiyah–Hirzebruch formula, then there exists a corresponding 4-dimensional oriented S1-manifold.

Predictors Of Nursing Students' Stress, Anxiety, And Depression During The Covid-19 Pandemic In A Hispanic-Serving University In South Texas: A Cross-Sectional Study, Maria I. Diaz, Eleftherios Gkioulekas, Nancy Nadeau

School of Mathematical & Statistical Sciences Faculty Publications

Background: In nursing education, there have been several studies on the impact of the COVID-19 pandemic on the ability of nursing students to cope while in nursing school.

Purpose statement: The goal of this study is to assess undergraduate nursing students' support mechanisms as predictors of stress, anxiety, and depression during the COVID-19 pandemic within a Hispanic-serving institution in South Texas.

Methods: Across-sectional design was used in this study. An online survey using self-reported questionnaires was used to gather data from an undergraduate nursing student cohort during the Fall 2021 semester. Linear regression was used to identify the predictors of …

“What Makes It Eigen-Esque-Ish?”: A Form-Function Analysis Of The Development Of Eigentheory Concepts In A Quantum Mechanics Course, Megan Wawro, Kaitlyn Stephens Serbin

School of Mathematical & Statistical Sciences Faculty Publications

Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. Words and meanings associated with eigentheory develop and vary over time, as do their associated symbols. In this study, we investigate how “eigen” develops over time in one quantum mechanics course by analyzing form-function relations (Saxe, 1999) for eigentheory concepts over 22 class sessions. We share results concerning our microgenetic and ontogenetic analyses of the creation of form-function relations and their shifts over time by characterizing the continuity and discontinuity of the various functions and forms associated with concepts …

Conditional Quantization For Some Discrete Distributions, Edgar A. Gonzalez, Mrinal Kanti Roychowdhury, David A. Salinas, Vishal Veeramachaneni

School of Mathematical & Statistical Sciences Faculty Publications

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support {12𝑛:𝑛∈ℕ}. We have …

Derived $P$-Adic Heights And The Leading Coefficient Of The Bertolini–Darmon–Prasanna $P$-Adic $L$-Function, Francesc Castella, Chi-Yun Hsu, Debanjana Kundu, Yu-Sheng Lee, Zheng Liu

School of Mathematical & Statistical Sciences Faculty Publications

Let E/Q be an elliptic curve and let p be an odd prime of good reduction for E. Let K be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which p splits. The goal of this paper is two-fold: (1) we formulate a p-adic BSD conjecture for the p-adic L-function LBDP p introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue F BDP p of LBDP p , we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected …

How Much Should Consumers With Mild To Moderate Hearing Loss Spend On Hearing Devices?, Vinaya Manchaiah, Steve Taddei, Abram Bailey, De Wet Swanepoel, Hansapani Rodrigo, Andrew Sabin

School of Mathematical & Statistical Sciences Faculty Publications

Background: This study examined the relationship between hearing device price and sound quality. Method: A novel consumer-centric metric of sound quality (“SoundScore”) was used to assess hearing devices’ audio performance. Each hearing device is tested with two fittings. The “Initial Fit” is designed to approximate the most likely fitting for an individual with a mild-to-moderate sloping sensorineural hearing loss. The “Tuned Fit” includes adjusting parameters optimized to hit prescriptive fitting targets (NAL NL2) on an acoustic manikin. Each fitting is evaluated across five dimensions. Both fittings are combined using a weighted average to create a single number from 0 to …

The Improved New Intersection Theorem Revisited, Lars Winther, Luigi Ferraro

School of Mathematical & Statistical Sciences Faculty Publications

We prove a generalized version of Evans and Griffith’s improved new intersection theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least dimR−dimR/I. This improves the bound htI obtained by Avramov, Iyengar, and Neeman in 2018.

P-Adic Quantum Mechanics, Infinite Potential Wells, And Continuous-Time Quantum Walks, Nathaniel P. Mayes

Theses and Dissertations

In this thesis, we introduce a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. …

The Herzog-Takayama Resolution Over A Skew Polynomial Ring, Linoy Utkina

Theses and Dissertations

Let k be a field, and let I be a monomial ideal in the polynomial ring R = k[x1,..., xn]. In her thesis, Taylor introduced a complex that yields a finite free resolution of R/I as an R-module. Building on Taylor’s work, Ferraro, Martin, and Moore extended this construction to monomial ideals in skew polynomial rings. Because the Taylor resolution is typically not minimal, subsequent research efforts went into identifying specific classes of ideals whose minimal free resolutions can be constructed more simply. In 1990, Eliahou and Kervaire devised an approach for handling minimal resolutions of …

A Dg-Algebra Structure With Divided Powers On The Generalized Taylor Resolution, Raul F, Alvarez

Theses and Dissertations

This thesis investigates the construction of a DG Γ-algebra structure on the Generalized Taylor Resolution (GTR) associated with monomial ideals. The classical Taylor resolution is known for providing a free but generally non-minimal resolution, leading to computational challenges and inefficiencies in algebraic analysis. In contrast, the GTR preserves essential algebraic structures while optimizing the resolution process, offering a more efficient and comprehensive framework for studying monomial ideals.

We introduce a novel DG Γ-structure that incorporates divided powers into the GTR, enhancing its multiplicative and homological properties. This structure preserves strict graded commutativity and is fully compatible with the differential graded …

Camassa-Holm Type Equations And Multi-Peakon Solutions, Yonghong Chen

Theses and Dissertations

This thesis investigates the multi-peakon solutions of a class of Camassa-Holm type equations with quadratic nonlinearities, specifically focusing on the differential equation mt +θmu+ amxu+bmux = 0, where m = u−uxx, and θ, a, b are arbitrary constants. We derive the nonlocal form of the equation and explore its peakon dynamics, including the well-known Camassa–Holm, Degasperis–Procesi, and Holm–Staley b-family equations. By employing a practical computational approach, we analyze the N-peakon solutions and present the corresponding dynamical systems, particularly focusing on the two-peakon case. The results will …

Congruences In Arithmetic Progression For Coefficients Of Gaussian Polynomials And Crank Statistics, Joselyne Aniceto

Theses and Dissertations

The study of partition congruences, inspired by Ramanujan’s discoveries for ��(��) over a century ago, remains a central topic in this field. This dissertation examines congruence properties in two restricted partition functions: ��(��,��), which counts partitions of �� into at most �� parts, and ��(��,��,��), which further limits the size of the largest part to be at most ��. Building on Kronholm’s 2007 result, now known as the Interval Theorem, and a recent result by Eichhorn, Engle, and Kronholm, we establish new infinite families of congruences for ��(��,��,��). This dissertation extends not only the recent results of Eichhorn, Engle, …

Relaxation Maximum-Based Iteration Method For Solving The Generalized Absolute Value Equation, Ximing Fang, Zhidong Wang, Zhijun Qiao

School of Mathematical & Statistical Sciences Faculty Publications

The generalized absolute value equation (GAVE) has wide applications in scientific computing. Establishing a high performance computing method to solve the GAVE is a hot research topic in recent years. In this paper, with the aid of the maximum function, the GAVE is decomposed of two equations, and then we present the relaxation maximum-based (RM) iteration method. To see the feasibility of the method, we discuss the necessary and sufficient conditions for the GAVE to have a unique solution. Next, the convergence analysis of the RM iteration is discussed under some convergence conditions. Moreover, some numerical examples of low and …