Applied Mathematics Commons^™

Divergence-Free Smoothed Particle Hydrodynamics In A Stream Digital Twin, Austin Hartley

All Theses

Digital Twins (DT) are being explored by the South Carolina (SC) water community to simulate how SC streams will flow at various water levels. Currently, a DT called Gilligan simulates these streams utilizing weakly-incompressible Smoothed Particle Hydrodynamics (SPH). This method does not strictly enforce incompressibility, which leads to unrealistic water flows and unwanted visual artifacts that require post-processing effects to hide. To address these problems and simulate more realistic water flows, the Gilligan stream logic is updated and a state-of-the-art SPH method that enforces incompressibility—Divergence-Free SPH (DFSPH)—is implemented within the Gilligan framework. DFSPH is able to make use of two …

Numalyze: Numerical Analysis Web Application, Dev Kapupara

Electronic Theses, Projects, and Dissertations

Numalyze is an online platform that allows users to run and apply different numerical methods in real time. The application is built using Python and the Flask web framework. It provides an interface where users input mathematical functions and parameters to see the results for root-finding and integration methods, and to also perform reductions on matrices. By using a light-weight web framework and self-coded algorithms which removes dependency on massive external libraries—this application connects theoretical concepts to their practical implementation. It enables students and researchers to visualize the series of steps that each algorithm takes to compute results. Moreover, the …

Identifying And Characterizing Transition Cells In Developmental Processes From Scrna-Seq Data, Yuanxin Wang

Dissertations and Theses (Open Access)

During the development of multicellular organisms, individual cells make distinct decisions about their cell types and states. Understanding the molecular mechanisms underlying cellular state transitions at different developmental stages provides deep insights into physiology, morphology and the etiology of diseases. Single-cell RNA-sequencing (scRNA-seq), which is widely used to study complex cell states and dynamic gene expression patterns, enables us to investigate molecular mechanisms of cellular state transitions. Currently, however, computational tools available for identifying cellular states and state transitions remain limited.

Although trajectory-based methods such as Monocle and Slingshot assume that state transitions generate continuous expression profiles, they cannot distinguish …

Using Gaussian Process Regression To Learn Thermodynamic Equations Of State With Uncertainty Quantification, Austen T. Lee

Chemical Engineering Undergraduate Honors Theses

This study investigates the use of derivative-informed Gaussian Process (GP) models to estimate thermodynamic behavior across temperature and density by building a Helmholtz-based equation of state. Argon, a stable monatomic gas, was chosen as a case study within the vapor region. The GP model was trained using values of experimentally measurable properties found by taking first and second derivatives of the original potential function. Results show that while the GP model offered uncertainty quantification and informed thermodynamic behavior, it predicted values that deviated from the ground truth depending on the property. The model exhibited high confidence in regions with substantial …

Sliding Window Method For Simulating Action Potentials In Axons, Hayden Reed

Honors Theses

ABSTRACT Hodgkin and Huxley’s nonlinear partial differential equations model the excitation and propagation of action potentials in neurons, and there have been numerous attempts at finding the best numerical solution method. This thesis proposes a novel approach to solving these equations: the Sliding Window method, in which a fixed sub-interval is found through capturing the signal’s head and tail. The system is then solved on the sub-interval instead of the entire interval. Using the Sliding Window technique also involves implementing the backward and forward Euler methods and the finite difference method. It will be demonstrated that, in utilizing the Sliding …

A Profile Wald Test In M-Estimation, Reagan Kesseku

Open Access Theses & Dissertations

Despite the growing popularity of machine learning-based inference, classical statistical inference remains highly relevant in modern data science due to its interpretability and theoretical rigor. Among its core tools, the likelihood ratio test, Wald test, and score test are foundational methods for hypothesis testing within the maximum likelihood framework. Although these tests are asymptotically equivalent under regularity conditions, each offers distinct advantages depending on the context, computational demands, and the availability of parameter estimates. In this dissertation, we introduce a fourth method, the Profile Wald Test (PWT), within the broader M-estimation framework. The PWT is based on profile estimators of …

Simulations Of Richtmyer-Meshkov Instability Using High Order Weno Methods, Ryan Holley

Graduate Theses and Dissertations

Turbulent mixing due to hydrodynamic instabilities occurs in a broad spectrum of engineering, astrophysical and geophysical applications. Theory, experiment, and numerical simulation help us to understand the dynamics of interface instabilities between two fluids. This thesis presents an increasingly accurate and robust front tracking method for the numerical simulations of shock-induced turbulent mixing known as Richtmyer-Meshkov Instability (RMI). Front tracking is an adaptive computational method, where the interface instability is explicitly represented as lower dimensional manifolds moving through a rectangular grid. All the cell-center states (density, velocity and pressure) are updated using higher order weighted essentially non-oscillatory (WENO) scheme. Performance …

Car Price Prediction Using Machine Learning: Analyzing The Dvm-Car Dataset, Yaman Abu Ghareebaih

Electronic Theses and Dissertations

The objective of this study is to predict car prices using machine learning models and the DVM-CAR dataset, which includes over 1.4 million images and car specifi- cations from 899 car models. Key factors such as mileage, engine power, and year of registration were analyzed for their correlation with car prices. Extensive data cleaning was performed, including filling missing values, identifying outliers, and normalizing numerical variables. Discrete variables like car make and body type were encoded using one-hot encoding. Linear relationships were analyzed with Multiple Logistic Regression, and Random Forest models were used for nonlinear patterns. Model performance was evaluated …

Investigating The Privacy-Utility Trade-O↵ In Synthetic Data Generation Using Correlated Attribute Mode, Kofi Sarfo

Electronic Theses and Dissertations

This thesis explores the privacy-utility trade-off in synthetic data generation using the Correlated Attribute Mode of DataSynthesizer, which employs Bayesian networks to model attribute dependencies. It focuses on integrating differential privacy mechanisms, particularly the Laplace mechanism, to inject controlled noise into synthetic data and enhance privacy protection. As organizations face challenges balancing data-driven decision-making with privacy regulations such as the General Data Protection Regulation and the California Consumer Privacy Act, synthetic data offers a solution by creating artificial datasets that preserve statistical properties while balancing data privacy and utility. This research investigates how different differential privacy parameters epsilon affect data …

The Little Diagram That Could: Geometric Properties And Statistical Applications Of Persistence Diagrams In Topological Data Analysis, Eugene Kler

McKelvey School of Engineering Graduate Student Theses & Dissertations

Topological Data Analysis (TDA) is a collection of techniques for data analysis that leverages topological invariants of spaces formed from data points. These methods excel at extracting useful information from noisy or sparse data, making them attractive to many mathematicians, statisticians, and scientists. In this thesis, we explore TDA on three fronts: algebraic foundations, statistical applications, and metric properties. Throughout, the central object of study is the Persistence Diagram (PD), a summary of the changes in homology that occur as one builds simplicial complexes from the data by increasing a parameter.

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 …

Towards Advancing Streamflow And Peak Flow Prediction With Machine Learning: Identifying Infrastructure At Risk, Sudan Pokharel

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Due to climate change and its impact, the need for adaptive strategies for natural disaster mitigation and resource management has never been more urgent. Central to this is water resource management, which is essential for sustainable human activities, ecological balance, and the mitigation of natural hazards like floods. Streamflow is a crucial element of water resource management and plays a vital role in planning and building water infrastructure, implementing emergency response plans, supporting flood mitigation initiatives, and regulating agricultural and industrial use. However, accurate prediction of streamflow still remains a challenge due to the complex non-linear and non-stationary interaction between …

Physics Embedded Neural Network: A Novel Data-Free Numerical Method For Solving Computational Physics Problems, Pawan Gaire

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

A novel approach for solving partial differential equations (PDEs) using neural networks for scientific computing is introduced. The proposed approach, referred to as physics-embedded neural network (PENN), features a unique architecture that incorporates the PDE and boundary conditions information directly within the final fully-connected layer of the feed-forward neural network (NN). The key aspect of PENN is the parallel numerical embedding of a differential equation associated with physical problems within the activation function of the network’s final layer. This integration leads to a new class of computational solvers competitive with classical methods like the Finite Element Method (FEM) and capable …

The Numerical Method For Finding All The Zeros Of A Function F (X) On An Interval [A,B], Franissa Simon

Honors Theses

In this study, we develop a simple mathematical method for finding all the roots of a function on a specified interval. Existing classical numerical methods, including the Bisection Method, Secant Method, and Newton Method, cannot find all the zeros of a function f (x) on an interval [a,b]. Popular numerical root solvers like Matlab’s ‘fzero’, Maple’s ‘fsolve’, and SageMath’s ‘find_root’ typically yield only a single zero. This thesis explores a proposed interval computation bisection method, a systematic approach based on the traditional Bisection Method and interval computation. Unlike traditional bisection, which relies on the intermediate value theorem, this approach uses …

Decomposition And Coordination For Multiobjective Optimization: A Framework And Methodology, Philip J. De Castro

All Dissertations

In this work, we consider finding Pareto efficient solutions for complex multiobjective optimization problems (MOPs). Complex MOPs are unique in the literature because they have many more objective functions than is typically considered. In fact, such complex MOPs will have 30+ objective functions. This large problem size presents computational and coginitive difficulties. Computationally, standard techniques for solving MOPs are often ineffective and cognitively it is difficult for a decision maker (DM) to handle all of the information provided in such a large problem. To address these challenges, we develop a decomposition and coordination framework. This framework will allow us to …

Properties Of Eigenvalues Of The Fractal Laplacian, Eric Stachura, Andrew Chincea

Symposium of Student Scholars

We investigate the properties of the eigenvalues of the fractal Laplacian. We begin by defining the fractal Laplacian operator in one dimension and formulate the corresponding Dirichlet eigenvalue problem. Analytical solutions are obtained for specific fractal parameters, and computational results illustrate the structure of eigenvalues and their associated eigenfunctions. We extend our analysis to two dimensions using separation of variables. Our findings contribute to a deeper understanding of how fractal geometry affects the spectral characteristics of differential operators.

Developing Predictive Mathematical Model For Optimizing Coating Weight Variation In Galvalume Production: A Case Study Of A Metal Industry, Victoria Mahabi

Tanzania Journal of Engineering and Technology (TJET)

Variations in coating weight for galvanized steel sheets can result in notable differences between batches. Such variations may cause various issues, such as diminished corrosion resistance, lower mechanical strength, and visual defects, which can ultimately drive-up costs, lead to customer dissatisfaction, and pose safety risks. Even with attempts to manage elements like air knife pressure and line speed, coating weight inconsistencies remain challenging. The research focuses on developing a predictive mathematical model designed to optimize variations in coating weight during Galvalume production. The critical parameters influencing coating weight variation were identified and analysed using a systematic literature review, primary data …

A Fuzzy Based Framework For Sustainable Technology Selection In Small-Scale Gold Mining Operations, John M. Kafuku

Tanzania Journal of Engineering and Technology (TJET)

Small-scale gold mining (SSGM) operations in Tanzania has been operating inefficiently due to inadequate mining processing technologies, poor working tools, lack of enough capital, and insufficient electricity. Despite the efforts made by different stakeholders in boosting the sustainability of SSGM yet the sector has not reached the expected goal. This paper proposes a framework for appropriate technology selection to help small scale gold miners in evaluating various gold mineral processing technologies. The framework utilizes the fuzzy logic set theory for technology evaluation and selection. The developed framework for technology selection upon validation provided results that technology adequacy of more than …

Improving Image Quality In Electrical Capacitance Tomography Using Otsu Thresholding, Josiah Nombo

Tanzania Journal of Engineering and Technology (TJET)

Electrical Capacitance Tomography (ECT) is an imaging technique used in industrial process monitoring, particularly for monitoring and measuring the composition of multiphase flows. Despite its widespread application, the commonly used Linear Back Projection (LBP) algorithm often produces low-quality images due to its limited ability to handle high permittivity contrasts and nonlinearities. This study investigates the use of Otsu thresholding as a post-processing technique to enhance ECT image quality. By maximizing inter-class variance in the image histogram, Otsu thresholding improves contrast, clarity, and structural definition, enabling more effective segmentation of oil and gas components in multiphase flows. The proposed Otsu-based reconstruction …

Development Of A Microcontroller-Based Intelligent Traffic Light Control System For Vehicular Movement In T-Junctions, Frederick O. Ehiagwina

Tanzania Journal of Engineering and Technology (TJET)

This research is devoted to the issue of regulating traffic congestion in major cities using light-dependent resistors coupled with the PIC16F877A microcontroller. This study proposes an intelligent traffic control system for T-Junctions, utilizing sensing and control to optimize traffic flow through dynamic phase adjustments and congestion reduction, enabled by a microcontroller-based decision-making system. The proposed system reduces traffic congestion, automates control, and enhances safety, minimizing accidents and lowering infrastructure costs. Under simulated environment, it demonstrates an average response time of 50 ms and achieves 99% accuracy in displaying the correct countdown. Finally, the number of state transitions handled per minute …