Mathematics Commons^™

Quasistatic Peridynamics, Existence Of Unique Solution In The Presence Of Damage, Nuwanthi N. Samarawickrama

LSU Doctoral Dissertations

A mathematical model for damage propagation based on nonlocal potentials is developed within the framework of peridynamics. This model is applied to simulate damage evolution in cyclically loaded structures. By neglecting inertial effects, a well-posed quasistatic formulation for cyclic loading is obtained.\\ The resulting equation is expressed as a nonlocal and nonlinear integral operator that couples damage evolution to the deformation field.\\ This coupling occurs through the product of a damage factor and the derivative of a force potential. The damage factor ranges between zero and one, where one represents undamaged material and zero indicates complete damage.\\ It serves to …

How Multi-Scale Modeling Can Help Examine Social Determinants Of Health And Resulting Disparities, Kyoko Yoshida, Elsje Pienaar, Shalanda A. Bynum, Naomi Chesler, Mitchel J. Colebank, Jessie Heneghan, Nadra Tyus, Jasmine Miller-Kleinhenz, Bruce Y. Lee

Faculty Publications

Social determinants of health (SDOH) are the conditions in which people live, work, and play, and the wider set of factors (e.g., social and economic systems and policies) that shape a person’s daily life. SDOH can differ significantly across communities and populations, having positive impacts for some and negative impacts for others. Ultimately, this results in differences in health and disease distribution, that are known as health disparities. Despite the known impacts of SDOH and calls to characterize, address, reduce, and eliminate health disparities, they persist and, in some cases, have worsened. To address this challenge, a session at the …

Quantitative Boundary Doubling Estimates For Elliptic Equations, Jack Dalberg

LSU Doctoral Dissertations

We present an approach for obtaining quantitative boundary doubling inequalities for elliptic equations with Neumann boundary conditions. Carleman estimates are used to prove three-ball inequalities, which are then used to prove quantitative doubling inequalities, with bootstrapping from the interior to the boundary. This approach is illustrated by its application to the Laplace eigenvalue problem with homogeneous Neumann boundary conditions, where sharp doubling inequalities are recovered.

When then consider a equation with non homogeneous Neumann boundary conditions. By following the approach, we are able to obtain potentially sharp results. Finally, we are able to get an improvement on previously obtained results …

Gain Threshold Optimization Using Fano Resonance, Alina Oktiabrskaia

LSU Doctoral Dissertations

The study of resonances in electromagnetics plays a critical role in the design of optical systems. This dissertation investigates the interaction between resonance and gain in optical structures to establish a universal principle for achieving ultra-low-threshold lasing. Through the analysis of geometric symmetries, material properties, and coupling mechanisms, this research develops prototype structures applicable to a wide range of optical and electromagnetic systems. A range of models is considered, starting from a simple onedimensional string-resonator system (based on the model of H. Lamb), then advancing to two- and three-dimensional waveguide models, and culminating with a realistic high-contrast model in open …

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 …

Modern Procedural Terrain Generation Techniques And Their Background, Hunter A. Barton

2025 Symposium

Procedural terrain generation has become a staple in many digital environments, enabling the automated creation of large-scale and realistic landscapes for applications such as video games and movies. This paper provides an in-depth look at smooth noise functions and their use for terrain generation, as well as an overview of some more modern methods of generation. A method utilizing machine learning stlye transfer was reproduced for this paper with some alterations to improve visualization and realism.

A Theory Of Fundamental Strata For Twisted Formal Connections, Sorawit Viwanthananut

LSU Doctoral Dissertations

Let $G$ be a complex reductive group. A fundamental stratum for $G$ is a triple $(x,r,\beta)$ where $x$ is a point in the Bruhat-Tits building of $G$, $r$ is a nonnegative real number called depth of the stratum, and $\beta$ is a semistable functional on the Moy-Prasad filtration of $\fg$ associated to $x$ at level $r$. Fundamental strata were first introduced to classify admissible representations of a $p$-adic reductive group. More recently, Bremer and Sage have shown that fundamental strata play an important role in the geometric Langlands program and developed a theory of fundamental strata for $G$-connections. In this …

Exact Sampling Of The Six-Vertex Model Using Coupling From The Past, Malaeka Amir

DePaul Discoveries

This paper aims to explore the six-vertex model through simulations designed to investigate the behavior of configurations under specific domain wall boundary conditions. To generate random configurations, we employ the Markov Chain Monte Carlo method while addressing the challenge of mixing times by utilizing the Coupling from the Past (CFTP) algorithm. Implemented in Python, our approach leverages CFTP to ensure exact sampling, avoiding the uncertainty of convergence in traditional Monte Carlo methods. We explore the monotonicity property within this framework and prove that it is only maintained by the steps of this algorithm for very particular values of the parameters.

Strichartz Estimates For Many Particle Dispersive Equations, Tristan Reynoso

LSU Doctoral Dissertations

Dispersive equations are useful for describing a wide variety of phenomena in which solutions disperse through the space as time progresses. These equations show up frequently in physics, especially when studying quantum mechanical and fluid related systems. The single particle variants of equations such as the Schr\"{o}dinger and Wave equations have been studied at great length throughout modern history. Over the last couple decades progress has been made toward extending single particle dispersive equations to cover their many body counterparts. Space-time Strichartz estimates for the homogenous $N$-particle Schr\"{o}dinger equation with small interacting potentials was recently established on both $\mathbb{R}^d$ and …

Dynamics Of Random Graphs And Interacting Particle Systems, Aritra Mandal

Doctoral Theses

The talk will be based on the thesis which has broadly two objectives. In the first part of the thesis we study a certain random dynamics of discrete sparse graphs on n vertices(based on the Moran model) and understand the limit as n goes to infinity. In the second part of the thesis we provide a trajectorial representation for a semi linear parabolic partial differential equation. In this talk we will mainly focus on the results concerning the dynamics of sparse graphs. We will begin with the preliminaries of the sparse graphs and convergences. Following which we will define the …

Frieze Symmetry Patterns Of Pennsylvania German Fraktur, Lorelei Koss

LASER Journal

Fraktur is an American folk art developed by German-speaking communities in Pennsylvania, New Jersey, Ohio, Virginia, Maryland, and North Carolina during the 18th and 19th centuries. This art form features ornate calligraphy combined with illustrations of flowers, birds, angels, and geometric border designs. It was often used for birth and baptismal certificates, religious texts, educational materials, and public notices. Fraktur commonly incorporates frieze pattern designs. Artistic designs based on frieze patterns appear across a wide range of cultures, and research has shown that cultural groups tend to favor certain types of frieze patterns. This paper investigates which types of frieze …

An Elementary Approach To The Probability Distribution Of The Product Of Multiple Random Variables, Suwen Tian

Rose-Hulman Undergraduate Mathematics Journal

Abstract There arenindependent identically distributed (i.i.d.) uniform random variables defined as ξ1,ξ2,…,ξn, valued in interval [0,k](k>0), and there is a constantmvalued in interval (0,kn). We study the distribution of the product of these random numbers and prove a formula calculating Pr(∏i=1nξi≤m). Interestingly, we find that the result is exactly the sum of the firstnterms in the Taylor series expansion of the function exp(x) with x=nlnk-lnm. Through considering the corresponding probability density function, we make an extension of the formula calculating Pr(∏i=1nξi≤m) to any positive realn, and the extended formula can be written in a …

Exploring System Identification Of Non-Linear Dynamics Using The Weighted Composition Operator And The Liouville Operator, Chukwuebuka Amagwula

USF Tampa Graduate Theses and Dissertations

System identification is the process of determining mathematical models that describe the dynamics of a system from data. Dynamic Mode Decomposition (DMD) and Sparse Identification of Nonlinear Dynamical Systems (SINDy) are two distinct approaches used for this purpose.

DMD identifies dominant spatiotemporal modes and eigenvalues that describe the evo lution of a system. It assumes a near-linear representation of dynamics and is closely linked to the Koopman operator, making it ideal for analyzing fluid flows, oscillatory systems, and modal structures. The DMD method uses time series data where each data point is referred to as a snapshot and represents the …

On The Hamilton-Waterloo Problem With A Single Factor Of 6-Cycles, Zazil Santizo Huerta, Melissa S. Keranen

Michigan Tech Publications

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (CM, CN)decomposition of Kv into α CM-factors and β CN-factors. We denote a solution to the uniform Hamilton-Waterloo problem by HWP(v, M, N, α, β). Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the gcd(M, N) = {2, 3}, with a specific focus on the parameter M = 6. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform 2-factorizations. This innovative …

Functional Equivariance And Backward Error Analysis, Sanah Suri

Arts & Sciences Graduate Student Theses and Dissertations

Geometric, or structure-preserving, numerical integration has long been used as a framework for studying integrators that preserve a systems invariants. In backward error analysis, the approximate numerical flow of a system is viewed as the exact flow of a modified problem, allowing us to gain qualitative insights into the behavior of the numerical solution. The preservation of conservation laws by a numerical integrator can be generalized to $F$-functionally equivariant integrators, where $F$ represents an observable of the system in consideration. This thesis describes the behavior of geometric integrators through the lens of backward error analysis. First, we extend the idea …

Analysis For Idempotent States On Quantum Permutation Groups, J.P. Mccarthy

Department of Mathematics Publications

Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele’s proof yields an idempotent state in any non-empty weak-* compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

Dice Math And Probability, Warren Campbell

SEAS Faculty Publications

The manufacture of dice for tabletop games is a billion-dollar industry. In gaming circles and online forums, the concept of “cursed dice” is a popular topic. Dice are inherently unfair due to the difficulty of manufacturing them with perfect geometric tolerances and uniform material densities. A common method for testing dice fairness is the chi-square statistic, which is typically assumed to follow the chi-square distribution. However, this assumption is only asymptotically valid.

Exact distributions of the statistic can be computed for dice with few sides and a limited number of rolls—such as 2-sided (D2) and 4-sided (D4) dice—but the computational …

Let’S Teach More Accurate And Inclusive History! The Case Of Islamic Contributions To Mathematics And Science, Noah (Nuh) Aydin

Faculty Publications

Schools commonly teach a history of mathematics and science that is Eurocentric. This selective version of the history of science, called Classical Narrative by some, is rooted in colonial times and mindset, and has reinforced certain negative opinions about other cultures. It ignores or downplays contributions to mathematics and sciences from non-western civilizations, and falsely attributes many scientific discoveries to European scholars. Medieval Islamic Civilization, which had strong connections to Renaissance Europe, serves as a clear example of how this narrative is distorted. Based on research on primary sources since the middle of the 20th century, we now know that …

Exploring The Exceptional Extreme Rays Of The Copositive Cone, Trent Holmgren

All NMU Master's Theses

In this paper we will look at the exceptional extreme rays of COP^5 and COP^6 and identify if they are exposed or nonexposed. We start with some necessary background material on copositive matrices and cones. Then we construct two algorithms to see if a matrix is exposed or nonexposed.

Graph Convolutional Networks Enable Fast Hemorrhagic Stroke Monitoring With Electrical Impedance Tomography, J. Toivanen, V. Kolehmainen, A. Paldanius, A. Hänninen, A. Hauptmann, Sarah J. Hamilton

Mathematical and Statistical Science Faculty Research and Publications

Objective: To develop a fast image reconstruction method for stroke monitoring with electrical impedance tomography with image quality comparable to computationally expensive nonlinear model-based methods. Methods: A post-processing approach with graph convolutional networks is employed. Utilizing the flexibility of the graph setting, a graph U-net is trained on linear difference reconstructions from 2D simulated stroke data and applied to fully 3D images from realistic simulated and experimental data. An additional network, trained on 3D vs. 2D images, is also considered for comparison. Results: Post-processing the linear difference reconstructions through the graph U-net significantly improved the image quality, resulting in images …

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 …

Stability Analysis Of The Eulerian-Lagrangian Finite Volume Methods For Nonlinear Hyperbolic Equations In One Space Dimension, Yang Yang, Jiajie Chen, Jing Mei Qiu

Michigan Tech Publications

In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still …

Exploring Communication In Multi-Agent Cooperative Reinforcement Learning, Matthew Kalarickal

Math and Computer Science Honors Theses

This work focuses on communication strategies within a cooperative multi-agent reinforcement learning system. The goal is to explore how communication can be used among agents to potentially improve performance. The research operates within the scope of “learning tasks with communication,” where the primary aim is to solve domain-specific tasks through information exchange using explicit communication protocols. Three distinct communication strategies were implemented and explored: Combinatorial Ghost, Feature Sharing Ghost, and Move Sharing Ghost. Fully Centralized Training and Execution and Centralized Training with Decentralized Execution training approaches were based on the different communication strategy used. Performance metrics were recorded for these …

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, …

On A Conjecture On Covering Systems And An Irreducibility Question On Sparse 0,1-Polynomials, Alexandros Kalogirou

Theses and Dissertations

In 1952, H. Davenport posed the problem of determining a condition on the minimum modulus $m_0$ in a finite distinct covering system that would imply that the sum of the reciprocals of the moduli in the covering system is bounded away from 1. In 1973, P.~Erd\H os and J.~Selfridge indicated that they believed that $m_0$ > 4 would suffice. We provide a proof that this is the case in Chapter 2. Chapters 3 and 4 are dedicated to showing that $0,1$-polynomials of high degree and few terms are irreducible with high probability. Formally, let $k\in\mathbb{N}$ and $F(x)=1+\sum_{i=1}^kx^{n_i}$, where $ 0

On Berge Pancyclicity For Uniform Hypergraphs, Teegan Cole Bailey

Theses and Dissertations

An $n$-vertex graph $G$ is \textit{hamiltonian} if it contains a length $n$ cycle as a subgraph. A stronger notion of hamiltoncity is pancyclicity. A graph $G$ is \textit{pancyclic} if $G$ contains a cycle of length $k$ for every $3\leq k \leq n$. Dirac's classical result on hamiltonicity states that if an $n$-vertex graph $G$ has minimum degree $\delta(G)\geq \frac{n}{2}$, then $G$ is hamiltonian. Under the same minimum degree condition as Dirac, J. A. Bondy showed that $G$ must also be pancyclic or the exceptional graph $K_{\frac{n}{2}, \frac{n}{2}}$. In 1971, Bondy proposed a so called meta-conjecture claiming that \textit{``almost any nontrivial …

Explorations In Extremal Combinatorics: Graph Spectra, Codegree Squared Sum Of Hypergraphs And Lin-Lu-Yau Ricci Curvature, George Henry Brooks

Theses and Dissertations

This dissertation explores a variety of extremal problems in spectral graph theory, hypergraph Tur\'an theory in the $\ell_2$-norm, and Lin-Lu-Yau Ricci curvature. We begin by studying the maximization of linear combinations of eigenvalues in relation to the adjacency matrix of a graph. For a graph $G$ on $n$ vertices, we order its eigenvalues $\lambda_1 \geq \dots \geq \lambda_n$. First, we study a generalization of the maximum spread. Specifically, we consider the $(i,j)$-spread, defined as $\lambda_{i+1}-\lambda_{n-j}$, and aim to maximize that quantity over all simple graphs on $n$ vertices. We establish asymptotic bounds for all $i,j\geq 0$ and identify infinitely many …

Contextual Approach To Improve Problem Solving Skills In Ipas Subject At Elementary School: A Study Development Of Teaching Materials, Nur Hidayah, Sumarno Sumarno, Joko Siswanto

Jurnal Pendidikan Sains

The objective of this study is to assess the validity, practicality, and efficacy of instructional materials. We refer to this research as research and development, with a focus on the creation of teaching materials. The research design proposed by Borg and Gall consists of six steps: identification of potential and difficulties, data collecting, product design, design validation, design revision, and product trial. The utilized instruments comprise expert and practitioner validity sheets, teacher and student response surveys, and validation sheets for teaching materials and modules, as well as pretest and posttest questions. Experts performed the validity assessment, attaining a score of …

The Effort To Improve Students’ Problem-Solving Ability: Physics E-Modules For High School Students, Hidayatullah Hana Putra, Edi Supriana, Nasikhuddin Nasikhuddin, Annisa U. Yana, Suhailee Sohnui

Jurnal Pendidikan Sains

Physics teachers in Indonesia still face obstacles during learning, such as students who are less active and have difficulty in solving physics problems. The physics learning media used are also still not very interesting and are not based on authentic problem solving. Therefore, this study focuses on the development of PBL-based e-modules on work and energy to improve the physics problem-solving abilities of high school students. This research and development used the ADDIE design and was tested on 90 high school students in High School 1 Sumber Pucung Malang in the 2022/2023 academic year. The questionnaire instrument was used in …