Applied Mathematics Commons^™

Problems In Chemical Graph Theory Related To The Merrifield-Simmons And Hosoya Topological Indices, William B. O'Reilly

College of Graduate Studies: Theses & Dissertations

In some sense, chemical graph theory applies graph theory to various physical sciences. This interdisciplinary field has significant applications to structure property relationships, as well as mathematical modeling. In particular, we focus on two important indices widely used in chemical graph theory, the Merrifield-Simmons index and Hosoya index. The Merrifield-Simmons index and the Hosoya index are two well-known topological indices used in mathematical chemistry for characterizing specific properties of chemical compounds. Substantial research has been done on the two indices in terms of enumerative problems and extremal questions. In this thesis, we survey known extremal results and consider the generalized …

Quantification Of Antiviral Drug Tenofovir (Tfv) By Surface-Enhanced Raman Spectroscopy (Sers) Using Cumulative Distribution Functions (Cdfs), Marguerite R. Butler, Jana Hrncirova, Meredith Clark, Sucharita Dutta, John B. Cooper

Chemistry & Biochemistry Faculty Publications

Surface-enhanced Raman spectroscopy (SERS) is an ultrasensitive spectroscopic technique that generates signal-enhanced fingerprint vibrational spectra of small molecules. However, without rigorous control of SERS substrate active sites, geometry, surface area, or surface functionality, SERS is notoriously irreproducible, complicating the consistent quantitative analysis of small molecules. While evaporatively prepared samples yield significant SERS enhancement resulting in lower detection limits, the distribution of these enhancements along the SERS surface is inherently stochastic. Acquiring spatially resolved SERS spectra of these dried surfaces, we have shown that this enhancement is governed by a power law as a function of analyte concentration. Consequently, by definition, …

Fundamental Computational Problems And Algorithms For Superhypergraphs, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Hypergraphs extend traditional graphs by allowing edges (known as hyperedges) to connect more than two vertices, rather than just pairs. This paper explores fundamental problems and algorithms in the context of SuperHypergraphs, an advanced extension of hypergraphs enabling modeling of hierarchical and complex relationships. Topics covered include constructing SuperHyperGraphs, recognizing SuperHyperTrees, and computing SuperHyperTree-width. We address a range of optimization problems, such as the SuperHy-pergraph Partition Problem, Reachability, Minimum Spanning SuperHypertree, and Single-Source Shortest Path. Furthermore, adaptations of classical problems like the Traveling Salesman Problem, Chinese Postman Problem, and Longest Simple Path Problem are presented in the SuperHypergraph framework.

Hamiltonian Models For The Propagation Of Long Gravity Waves, Higher-Order Kdv-Type Equations And Integrability, Rossen Ivanov

Book chapter/book

A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to a KdV approximation with higher order nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The HKdV is related to the known integrable PDEs with an explicit nonlinear and nonlocal transformation.

Quantitative Verification For Massive Linear Systems, Qing Liu

School of Computing: Dissertations, Theses, and Student Research

The verification of linear systems has been an active area of research for decades. Reachability analysis is a key component in verification problems. It involves computing the system’s reachable set, the set of reachable states in the state space from a given set of initial states. Most verification methods primarily focus on qualitative verification, which answers whether or not a system may violate specified safety conditions. This paper extends this qualitative verification to quantitative verification by introducing a novel approach, employing probabilistic stars (Probstars) to compute reachable sets, which augment traditional star sets by integrating Gaussian-distributed random variables with …

Deterministic And Stochastic Dynamics Of Marine Food Webs, Julian A. Hernandez S.

Theses, Dissertations and Culminating Projects

Ecologists have long been concerned with understanding the behavior and evolutionary patterns exhibited within complex ecological communities. Under- standing the delicate balance that sustains ecosystems is crucial in determining how these communities evolve over time. Recently, researchers have combined deterministic Lotka-Volterra dynamics with different types of synthetic food webs (cascade, niche and generalized cascade models), and have analyzed the mechanisms behind primary extinction events and the ensuing secondary extinction cascade. These studies also enabled the exploration of the complex interplay of species loss to explain how food web structure influences primary and secondary extinction. We have extended these ideas to …

Using A Degree-Based Network Model To Understand And Control Traffic Jams In The Atlanta Metropolitan Area, Ian Salamone-Lent, Theresa Washington, Asma Azizi

The Kennesaw Journal of Undergraduate Research

Traffic congestion is an enduring problem for major metropolitan areas, such as Atlanta, GA. Our goal is to understand the nature of traffic congestion patterns in the highway system of Cobb County in Atlanta, GA. We created a road network representative of the Cobb County highway system and then superimposed a degree-based SIR model to simulate traffic congestion on that network. The model’s parameters, propagation and dissipation rates, were estimated using empirical traffic data, which are vehicles’ speed time series and speed limit of each road in the network. We then conducted a local sensitivity analysis of the model’s key …

Structured Invariant Subspace And Decomposition Of Systems With Time Delays And Uncertainties, Huan Phan-Van, Keqin Gu

SIUE Faculty Research, Scholarship, and Creative Activity

This article discusses invariant subspaces of a matrix with a given partition structure. The existence of a nontrivial structured invariant subspace is equivalent to the possibility of decomposing the associated system with multiple feedback blocks such that the feedback operators are subject to a given constraint. The formulation is especially useful in the stability analysis of time-delay systems using the Lyapunov-Krasovskii functional approach where computational efficiency is essential in order to achieve accuracy for large scale systems. The set of all structured invariant subspaces are obtained (thus all possible decompositions are obtained as a result) for the coupled differential-difference equations …

Weak-Strong Beam-Beam Simulation With Crab Cavity Noises For The Hadron Storage Ring Of The Electron-Ion Collider, Y. Luo, B. Gamage, C. Montag, D. Marx, D. Xu, F. Willeke, H. Huang, H. Lovelace Iii, J. Berg, M. Blaskiewicz, S. Peggs, T. Satogata, V. Ptitsyn, V. Morozov, Y. Hao

Mathematics & Statistics Faculty Publications

The Electron Ion Collider (EIC), to be constructed at Brookhaven National Laboratory, will collide polarized high-energy electron beams with hadron beams, achieving luminosities of up to 1 X 10³⁴cm⁻²s⁻¹ in the center-mass energy range of 20-140 GeV. Crab cavities are employed to compensate for the geometric luminosity loss caused by a large crossing angle of 25 mrad in the interaction region. The phase noise in crab cavities will induce a significant emittance growth for the hadron beams in the Hadron Storage Ring (HSR). Various models have been utilized to study the effects of crab cavity …

A Novel Computational Analysis Of Boundary-Driven Two-Dimensional Heat Flow With Internal Heat Generation, Muhammad Abid, Madiha Bibi, Nasir Yasin, Muhammad Shahid

Mathematics & Statistics Faculty Publications

Accurate numerical solution of parabolic and elliptic partial differential equations governing two-dimensional heat transfer is critical for engineering simulations but computationally challenging. This work employs key numerical techniques finite differences, conjugate gradients, and Crank-Nicolson time stepping to solve the heat diffusion equation and analyze method performance. The Poisson equation is discretized using second-order central finite differences and solved with the conjugate gradient approach to determine the steady state solution. The transient heat equation is integrated in time via the Crank-Nicolson implicit scheme, also utilizing conjugate gradients. The methods effectively compute solutions matching analytical and boundary conditions. Convergence and stability are …

Enhancing Set-Theoretic Research Methods With Neutrosophic Sets, Maikel Yelandi Leyva Vázquez, Jesús Estupiñán Ricardo, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This study employed the integration of neutrosophic set theory with set-theoretic methods and Qualitative Comparative Analysis (QCA) to examine intricate social phenomena. Participants' opinions and attitudes were captured using neutrosophic Likert scales, which reflected elements of truth, indeterminacy, and falsity. This proposal is illustrated in a case study facilitated a more comprehensive and subtle examination, emphasizing the significance of variables such as motivation, instructional excellence, and educational resources in achieving academic accomplishment. The results of the necessary condition analysis and set coincidence analysis indicated that motivation and teaching quality have the highest individual impact. Nevertheless, the convergence of scholarly resources, …

Numerical Solution Of Hybrid Nanofluid And Its Stability Over Permeable Wedge Sheet With Heat Transfer Analysis, Aisha M. Alqahtani, Zeeshan, Waris Khan, Florentin Smarandache, Nidhal Becheikh, Roobaea Alroobaea, Taseer Muhammad

Branch Mathematics and Statistics Faculty and Staff Publications

The inclusion of nanoparticles has the potential to improve the thermal efficiency of the base fluid. The field of nanofluid (NF) dynamics has attracted important attention due to its extensive range of practical uses like fuel cells, solar energy, medication administration, heat transfer, microfabrication, coolant applications, and other related domains. The aim of this study is to scrutinize the impact of Lorentz force, thermal energy, joule heating, heat source and injection parameters, and Brownian and thermoporetic diffusions on the hybrid nanofluid over the moving wedge. The stability inquiry is reported for the existing work in order to confirm the stable …

A Review Of The Hierarchy Of Plithogenic, Neutrosophic, And Fuzzy Graphs: Survey And Applications, Takaaki Fujita, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

As many readers may know, graph theory is a fundamental branch of mathematics that examines networks consisting of nodes and edges, with a focus on their paths, structures, and properties [157]. A Fuzzy Graph extends this concept by assigning a membership degree between 0 and 1 to each edge and vertex, capturing the level of uncertainty. Expanding on this idea, the Turiyam Neutrosophic Graph was introduced as an extension of both Neutrosophic and Fuzzy Graphs. Plithogenic graphs, in turn, offer a powerful approach for managing uncertainty. In this paper, we explore the relationships among various graph classes, including Plithogenic graphs, …

Applications Of Physics-Informed Graph Neural Networks In Wind Farm Layout Optimization, Emma R. Martin

Theses and Dissertations

Rapid advancements in data-driven deep learning models have led to their increased usage in recent years. Highly-effective models for conducting complex decision-making tasks have been developed. Given their success, data-driven deep learning models are being developed and improved in a variety of areas, one of which is renewable energy. In this thesis, we discuss the development of data-driven, deep learning models for optimizing wind farm layouts for maximum power output. We include a background for wind energy, the physics of fluid flow and power production, and two types of neural networks: graph neural networks and physics-informed neural networks. Additionally, we …

Analysis And Detection Of Cyber Attacks In Multi Vehicle Systems Using Macroscopic Models, Abhishek Kashyap

Mechanical and Aerospace Engineering Dissertations - Archive

The study of potential cyber-attacks in different domains is an active area of research. Given that systems are becoming more and more interconnected, cyber physical systems that operate infrastructure and/or plants can make these assets more vulnerable and open to different attack vectors. The primary focus of this research is the modeling, analysis and detection of cyber-attacks on platoons of autonomous cars and swarms of UAVs. In this work, we consider scenarios wherein an attacker may hack into a subset of vehicles in a multi-vehicle system and make subtle modifications in their parameters. Due to the interconnected nature of the …

A New Mechanistic Model Of Brain Metabolism With Optimal Parametrization, Alice Lubbe

Mathematics Dissertations - Archive

Models of glucose metabolism in the brain often focus on chemical exchanges and reactions that occur as part of the tricarboxylic acid cycle (TCA cycle). Experiments involving nuclear magnetic resonance (NMR) spectroscopy to detect and measure carbon-labeled isotopomers of metabolites such as glutamate in vivo inform kinetic, mechanistic models used to study metabolic pathways. In the present work, a new model with two compartments, astrocytic and neuronal, is developed using known biochemical processes and fit to experimental data coming from fully labeled glucose infusions. A gradient descent method is introduced and employed to obtain optimal flux parameter values involved in …

Uniform Regularity Estimates For The Stokes System In Perforated Domains, Jamison R. Wallace

Theses and Dissertations--Mathematics

We consider the Stokes equations in an unbounded domain $\omega_{\epsilon,\eta}$ perforated by small obstacles, where $\epsilon$ represents the minimal distance between obstacles and $\eta$ is the ratio between the obstacle size and $\epsilon$. We are able to obtain uniform $W^{1,q}$ estimates for solutions to the Stokes equations in such domains with bounding constants depending explicitly on $\epsilon$ and $\eta$.

Advancements In Elliptic Mesh Redistribution And Automated Overset Grid Generation For Computational Fluid Dynamics, Chase Ashby

Theses and Dissertations--Mathematics

Automated and robust structured curvilinear overset grid generation remains a significant challenge for the application of computational fluid dynamics. This thesis intro- duces a specialized anisotropic Mach cone aligned adaptation algorithm for low-boom simulations and an automated procedure for generating structured curvilinear overset grids for immersed boundary simulations. Results demonstrate that anisotropic mesh adaptation can reduce computational resource usage by over 50% while maintaining sonic boom prediction accuracy across the entire sonic boom domain compared to user-constructed Mach cone aligned grids. The proposed immersed overset grid gen- eration procedure eliminates the need for complex surface generation algorithms, producing grids suitable …

Last Passage Time And Excursion Theory For Solvable Diffusions With Applications In Mathematical Finance, Yaode Sui

Theses and Dissertations (Comprehensive)

In this dissertation, we investigate the properties of last passage times and excursion theory in one-dimensional solvable diffusions, emphasizing their applications in financial modeling, particularly in option pricing. We derive closed-form formulas for the marginal distribution of last passage times and their joint distribution with process values, including the maximum and minimum of the process value. The focus is on time-homogeneous diffusions with various boundaries and imposed killing. Employing spectral expansion theory, we derive explicit formulas for distributions of last passage times in common processes such as Drifted Brownian Motion (BM), Squared Bessel (SQB), Ornstein-Uhlenbeck (OU), and Cox-Ingersoll-Ross (CIR) models. …

Investigation Of Space Charge Effects On Co2 Electrocatalytic Reduction On Gd-Doped Ceria Via Scanning Kelvin Probe And Model-Based Bayesian Analysis, Alejandro Mejia

Graduate Theses, Dissertations, and Problem Reports (ETD)

In studying novel energy conversion and storage systems, such as high-temperature electrolysis, numerous underlying fundamental physical processes remain unclear or inadequately understood. Among these, the modeling and comprehension of surface reaction mechanisms, coupled with the intricate effects of space‑charge interfaces, remains an unclear and challenging area of research.

The work of this dissertation involves the development of a 2D finite element analysis model, leveraging the robust MOOSE framework from INL. This model, featuring inhomogeneous defect thermodynamics for near-surface chemistry, formulated through Poisson‑Cahn variational theory, has been exploited for studying the electrocatalytic reduction of CO₂ on gadolinia doped ceria. The …

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

Theses and Dissertations (Comprehensive)

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

Reducing Generalization Error In Multiclass Classification Through Factorized Cross Entropy Loss, Oleksandr Horban

CMC Senior Theses

This paper introduces Factorized Cross Entropy Loss, a novel approach to multiclass classification which modifies the standard cross entropy loss by decomposing its weight matrix W into two smaller matrices, U and V, where UV is a low rank approximation of W. Factorized Cross Entropy Loss reduces generalization error from the conventional O( sqrt(k / n) ) to O( sqrt(r / n) ), where k is the number of classes, n is the sample size, and r is the reduced inner dimension of U and V.

A Class Of Game-Theoretic And Fokker-Planck Optimal Control Frameworks In Colon And Esophageal Cancer, Mesfer Alajmi Phd

Mathematics Dissertations - Archive

In this dissertation, we first present a new stochastic framework for parameter estimation and uncertainty quantification in colon cancer-induced immune responses. A stochastic process that captures the system's inherent randomness determines the dynamics of colon cancer. The stochastic framework is based on the Fokker-Planck equation, which represents the evolution of the probability density function corresponding to the stochastic process. We formulate an optimization problem that takes individual patient data with randomness present and solves it to obtain the unknown parameters corresponding to the individual tumor characteristics. Furthermore, we perform a sensitivity analysis of the optimal parameter set to identify the …

Batch Culture Models Of The Murine Gut Microbiome & The Impact Of Simple Dormancy On Dormancy-Capable Microorganisms Models, Ana C. Mendez

Mathematics Dissertations - Archive

The proposed mathematical biology research utilizes mathematical models to gain insight into biological systems. These systems of ordinary differential equations model diverse topics, ranging from gut microbiomes to harmful algal blooms. A complete stability analysis, supporting phase plane portraits, bifurcation diagrams, and numerical simulations will accompany the models presented. In Chapter 2, the murine gut microbiome is modeled to match laboratory experiments in the literature. In these experiments, mice eat plasmid-carrying “donor” bacteria and naturally carry plasmid-free “resident” bacteria in their gut. The models aim to capture the behavior of plasmids, donor bacteria, and resident bacteria. Chapter 3 explores dormancy …

Unveiling The Power Of Shor's Algorithm: Cryptography In A Post Quantum World, Dylan Phares

CMC Senior Theses

Shor's Algorithm is an extremely powerful tool, in utilizing this tool it is important to understand how it works and why it works. As well as the vast implications it could have for cryptography

The Computational Search For Unidentified Central Configurations Of The Newtonian N-Body Problem, Hannah G. Havel

CURE Proceedings

The N-body problem is a field of study in mathematics and physics that involves predicting the motion of particles moving under their mutual gravitational attraction. It is vital in celestial mechanics, such as planning collision-free satellite orbit trajectories. When beginning to understand the N-body problem, we can start by looking at equal masses of these particles or celestial bodies. As particles move, their position and velocity change, both energy and angular momentum are conserved. Sets of constant energy and angular momentum, known as integral manifolds, are higher-dimensional figures that represent constraints of movement to a system. Integral manifolds are described …

A Novel Scheme Based On Bessel Operational Matrices For Solving A Class Of Nonlinear Systems Of Differential Equations, Atallah El-Shenawy, Mohamed El-Gamel, Muhammad E. Anany

Mansoura Engineering Journal

The system of ordinary differential equations arises in many natural phenomena, especially in the field of disease spread. In this paper, a perfect spectral technique is introduced to solve systems of nonlinear differential equations. The technique enhanced the Bessel collocation technique by converting the series notation of unknown variables and their derivatives to matrix relations. The Newton algorithm is developed to solve the resulting nonlinear system of algebraic equations. The effectiveness of the scheme is proved by the convergence analysis and error bound as demonstrated in Theorem 1. The scheme of solution is tested to clarify the efficiency and the …

Echolocation On Manifolds, Kerong Wang

Honors Theses

We consider the question asked by Wyman and Xi [WX23]: ``Can you hear your location on a manifold?” In other words, can you locate a unique point x on a manifold, up to symmetry, if you know the Laplacian eigenvalues and eigenfunctions of the manifold? In [WX23], Wyman and Xi showed that echolocation holds on one- and two-dimensional rectangles with Dirichlet boundary conditions using the pointwise Weyl counting function. They also showed echolocation holds on ellipsoids using Gaussian curvature.

In this thesis, we provide full details for Wyman and Xi's proof for one- and two-dimensional rectangles and we show that …

The Precedence-Constrained Quadratic Knapsack Problem, Changkun Guan

Honors Theses

This thesis investigates the previously unstudied Precedence-Constrained Quadratic Knapsack Problem (PC-QKP), an NP-hard nonlinear combinatorial optimization problem. The PC-QKP is a variation of the traditional Knapsack Problem (KP) that introduces several additional complexities. By developing custom exact and approximate solution methods, and testing these on a wide range of carefully structured PC-QKP problem instances, we seek to identify and understand patterns that make some cases easier or harder to solve than others. The findings aim to help develop better strategies for solving this and similar problems in the future.

Adaptive Neh With Constrained Nearest Neighbor Subtours For The Electric Vehicle Routing Problem With Time Windows, Andrew Struthers

All Master's Theses

The development of electric vehicles is currently considered one of the most innovative areas in manufacturing. Largely driven by the desire to reduce greenhouse emissions, electric vehicles are seen as a viable alternative to internal combustion engine cars. Starting from consumer cars, a dedicated effort is being made to translate this into commercial vehicles for freight and delivery. This research introduces a novel adaptive Nawaz, Enscore, Ham (NEH) algorithm with constrained nearest neighbor subtour (NEH-NN). This algorithm is tested on the standard benchmark problems in literature and used as a seed solution for the Genetic Algorithm (GA). The performance and …