Numerical Analysis and Computation Commons^™

Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes, Martyna Wojcik

Discovery Day - Daytona Beach

Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations       It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions …

Pseudodifferential Absorbing Boundary Conditions For Waves, Lauren Taylor

Mechanical Engineering Theses

Absorbing boundary conditions (ABCs) are required to truncate the computational domain when performing Finite Element Analyses of exterior acoustic problems where physical domain is unbounded. ABC is applied on a fictitious boundary containing the scatterer and ideally allows outgoing waves to leave without non-physical reflections. Preventing artificial reflections is essential to benefit from the accuracy of the numerical method used. Otherwise, ABC acts as a reflective surface, and artificial reflections distort the solution in the entire domain which is not recoverable by any type of refinement. Pseudodifferential ABCs were used to describe the Dirichlet-to-Neumann map which maps known boundary values …

Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender

Mathematics Theses and Dissertations

This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …

(R2187) Analysis Of Neurological Impairments In Hospitalized Patients Using Cubic Neutrosophic Sets, B. Anitha, M. Lavanya

Applications and Applied Mathematics: An International Journal (AAM)

This study introduces an MCDM-based framework for identifying neurological diseases in hospitalized patients using symptom-based evaluations. A team of interns, guided by the chief doctor, was responsible for determining each patient’s precise condition from the presented neurological symptoms. To enhance diagnostic accuracy, the interns employed the TOPSIS and WASPAS methods to assess and rank the potential disease options. The combined analysis yielded a clear identification of the highest ranked disease for every patient, highlighting the effectiveness of these MCDM techniques in supporting clinical decision making.

Qualitative Analysis Of Solutions To A General Class Of Nonlinear Difference Equations With Applications, Osama Moaaz, Mohamed F. Abouelenein, Mona Anis

Mathematical Modelling and Numerical Simulation with Applications

This work examines the qualitative behavior of a general class of difference equations. We establish criteria guaranteeing the stability, periodicity, and boundedness of the solutions of the equation under consideration. In addition, we identify its invariant intervals. The theoretical results are subsequently applied to various special cases, among them the May--Host model. Numerical simulations are presented to demonstrate the dynamics of the solutions and to validate the theoretical analysis.

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Conditional Product Sampling For Gaussian Process Implicit Surfaces, Song Shi

Dartmouth College Master’s Theses

Gaussian Process Implicit Surfaces (GPISes) provide a powerful and unified stochastic geometry representation for rendering surfaces, volumes, and the rich continuum between them. Recent work has shown that GPISes can model a broad space of visual appearances under a unified light transport framework. However, practical rendering with GPISes remains challenging: existing estimators can become inefficient for particular correlation structures, and highly anisotropic or heightfield-like GPISes require specialized treatment to obtain robust variance reduction.

This thesis extends recent work on GPIS rendering by introducing a new next-event estimation (NEE) technique for anisotropic GPISes.We show that standard NEE provides diminishing benefits as …

Momentum Space Algorithm For Electronic Structure Of Double-Incommensurate Trilayer Graphene, Kenneth Silver Beard

LSU Doctoral Dissertations

Numerical algorithms for computing the electronic structure of incommensurate 2D-materials using ab initio models are critical for predicting material properties and guiding experiments. For bilayers, momentum space and continuum models have been introduced to approximate observables of ab initio tight-binding models using a momentum description, despite the lack of periodicity in the tight-binding model required for Bloch theory. A similar structure has been introduced for double-incommensurate trilayers using a continuum model, where the three lattices are mutually incommensurate. However, this description leads to a four-dimensional lattice space, and numerical convergence of the density of states has been observed to be …

Modeling Bitcoin Dynamics Using Differential Equations, Boone M. Fleenor

Departmental Honors & Graduate Capstone Projects

In this thesis, we develop and analyze two nonlinear systems of ordinary differential equations to model Bitcoin price dynamics. Analytical techniques are used to obtain exact or approximate solutions where possible. Then, numerical simulations using a fourth-order Runge–Kutta method are employed to explore system behavior beyond analytically tractable regimes. Finally, model outputs are compared to historical Bitcoin price data using normalized and resampled time series. These results suggest that deterministic models can provide meaningful insight into the structural behavior of Bitcoin markets, while highlighting the need for stochastic or time-dependent extensions for more realistic modeling.

Bifurcation Exploration Of Ion Acoustic Solitons Formation Of A Nonlinear Beta Fractional Kadomtsev-Petviashvili Burger Model In Plasma State, Mst. Razia Pervin, Alrazi Abdeljabbar, Fahad Sameer Alshammari, Mst. Shekha Khatun, Harun Or-Roshid

Mathematical Modelling and Numerical Simulation with Applications

This research presents an extensive investigation of Ion acoustic soliton dynamics governed by a Beta-fractional Kadomtsev-Petviashvili-Burgurs (KPB) model. By engaging the planar dynamical system scheme in aggregation with the extended $(\phi, \psi)$ expansion, Kudryashov expansion, and the NMKM analytic schemes, we create a broad class of exact nonlinear pattern wave solutions. The local stability edifice of the fractional plasma model is explored through bifurcation theory, enabling the far-reaching classification of all admissible phase diagrams. Conforming Ion acoustic wave structures allied with every detour alignment are systematically assembled. Owing to the fractional and dissipative appearances of the model, an all-embracing assortment …

Employing Effective Solution Methods For Caputo-Based Sequential Fractional Models, Eman A. A. Ziada, Mohamed F. Abouelenein, Hijaz Ahmad, Monica Botros

Mathematical Modelling and Numerical Simulation with Applications

This paper investigates a class of nonlinear sequential singular fractional differential equations (FDEs) involving Caputo derivatives. This type of equation has several key advantages that enhance its value, such as capturing memory and hereditary effects. Viscoelastic materials and anomalous diffusion, as well as biological systems, can take advantage of this feature. In addition, fractional derivatives possess a sequential structure that enables the implementation of multiscale processes and hierarchical memory responses. Moreover, it provides an effective and flexible framework for solving differential equations compared to classical differential equations. It can therefore be used to model complex systems in physics, biology, and …

(Si16-04) Some Fixed Point Theorems On Chatterjea Type Contractions, Irom Shashikanta Singh, Y. Mahendra Singh

Applications and Applied Mathematics: An International Journal (AAM)

This paper establishes the existence of fixed points related to strict Chatterjee contractive mappings by relaxing the compactness of the underlying spaces and the continuity of the mapping involved, using altering distance functions and comparison functions in the general setting of metric spaces. Several non-trivial and illustrative examples are provided to demonstrate, support, and validate the obtained theoretical results. In addition, a theorem that can characterize the completeness of metric spaces through the existence of fixed points is rigorously proven and discussed. Furthermore, a theorem on strict Chatterjea-type modulus contractive mappings without continuity assumptions and with relaxed compactness conditions is …

The Butterfly Effect In Economics: Exploring Chaos With A Simple Financial Model, Sandy Hs Herho

CODEE Journal

Wild swings in financial markets need not result from external shocks like earthquakes or wars—they can emerge from deterministic chaos. This article introduces kalimusada, an open-source Python library that lets students and instructors explore this phenomenon through a simple three- equation model of financial dynamics. The model couples interest rates, investment, and prices through nonlinear feedback, generating bounded but unpredictable oscillations characteristic of chaos. Tiny differences in starting conditions—smaller than any measurement could detect—grow exponentially until two initially identical economies follow completely different paths. The library provides ready-to-use tools for visualizing this “butterfly effect” in economics, computing divergence metrics, and …

A Mathematical Frameworks For Singular, Nonlinear Phenomena: Applications To Nematocyst Firing And Inhomogeneous Nls With Coulomb Potential, Abdulrahman Alharbi

Theses and Dissertations

Nematocysts are specialized cellular organelles found in all cnidarians, including corals and jellyfish, as well as in some single-celled protists such as dinoflagellates. These organelles display remarkable diversity in morphology and function, playing roles in prey capture and defense. The firing of a nematocyst is one of the fastest accelerations in nature, yet the underlying physical mechanisms remain not fully understood. In this work, we address key questions: how sufficient force is generated to overcome the fluid boundary layer, whether fluid–structure interaction models can reproduce observed dynamics, and what mechanisms trigger discharge.

Our research investigates models based on osmotic pressure …

(R2151) Error Estimates Of Barycentric Lagrange Interpolation, Alvira Yawar, Swarnima Bahadur

Applications and Applied Mathematics: An International Journal (AAM)

Barycentric interpolation, which comes from Lagrange interpolation, is a useful method in numerical analysis. In this research paper, we explain how the barycentric interpolation formula is derived and discuss its features. We compare its stability and performance with the traditional Lagrange formula. First, we show how to get the barycentric formula from the Lagrange polynomial and present it as a rational function. We also provide an estimate of the error. Then, we use numerical examples to show that the barycentric formula is more stable and works better, especially when the degree of interpolation is high. Our results show that the …

Universal Systems Simulation Via Constraint Hypergraphs With Applications To Digital Twins, John Morris

All Dissertations

The characterization of systems encompasses a variety of modeling frameworks designed to capture specific behaviors and components of various system domains. Whatever the framework, the core elements of a system representation are the information of the system and a description of how that information is related. The relations in deterministic systems are functions, which, when composed to form executable processes, can be used to simulate system data. A declarative modeling framework is one that encodes mechanisms for preparing these simulations within the model structure, allowing an external agent to form the execution processes required for a given context. To date, …

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Algebraic Multigrid Methods For Nonsymmetric And Indefinite Problems: Theory And Applications, Ahsan Ali

Mathematics & Statistics ETDs

Algebraic multigrid (AMG) is a well-established and highly efficient solver for symmetric positive definite (SPD) systems arising from elliptic and parabolic PDEs, while nonsymmetric systems from hyperbolic PDEs remain a significant challenge. This dissertation develops AMG methods and theory for nonsymmetric problems. First, we develop a novel approach combining mode constraints from energy-minimization AMG with local approximations of ideal restriction in $\ell$AIR, resulting in constrained $\ell$AIR (C$\ell$AIR), which demonstrates scalable convergence across advective and diffusive problems. Second, we extend optimal AMG theory by deriving spectral radius estimates for the two-grid error transfer operator using matrix-induced orthogonality, enabling convergence predictions for …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

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 …

Property Testing Ai: An Efficient Frontier, Paul Sopher Lintilhac

Dartmouth College Ph.D Dissertations

In this dissertation, we take a step towards addressing the major problem of a lack of standardized and rigorous approaches to testing and evaluation of AI systems. Taking inspiration from both the fields of Property Testing and Property Based Testing (for programs), we develop a novel taxonomy of partially overlapping classes of properties of AI systems, including simple properties, compound properties, higher order properties, data relation properties, and architecture-utility properties. We argue that this taxonomy categorizes a diverse set of AI traits -- including accuracy, fairness, robustness, monotonicity, point-wise and global privacy properties, sensitivity, and more -- according to the …

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

From Neural Networks To Large Language Models: Innovations In Financial Ai, Mathematical Reasoning, And Structured Data Representation, Junyi Ye

Dissertations

This dissertation explores the evolution and application of artificial intelligence techniques across three critical domains: financial modeling, mathematical reasoning, and structured data analysis. The dissertation presents seven research projects that chart a progression from specialized neural architectures to sophisticated large language models (LLMs), contributing novel methodologies and frameworks at each stage.

In the financial domain, the research first introduces TS-Mixer, a MLP-based architecture for time-series forecasting that captures both feature relationships and temporal dependencies through a simple yet effective design, outperforming more complex models in S&P500 index prediction. The dissertation then presents DySTAGE, a dynamic graph representation learning framework that …

Using Permutation Groups To Identify Families Of Capacity Achieving Codes, Daniel Welchons

Department of Mathematics: Dissertations, Theses, and Student Research

When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory, but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.

This thesis seeks to apply the …

Using Permutation Groups To Identify Family Of Capacity Achieving Codes, Daniel Joseph Welchons

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

When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.

This this thesis seeks to …

Analysis Of Systematic Trade-Offs Between Military And Healthcare Expenditure Alongside Gdp Growth Of Select Asian And Western Exporting Economies In The 21st Century, Rahul Balamurugan, Carlos Gershenson, Preethi Nanjundan, Hiroki Sayama

Northeast Journal of Complex Systems (NEJCS)

This study explores the complexity in the trade-offs between military expenditure, healthcare expenditure, and GDP growth across select Asian nations and major weapon-exporting countries, examining how nations allocate finite resources between national security and human well-being over the past two decades. Using a systems science approach, the research integrates Granger causality testing to analyze temporal and directional relationships among GDP growth, military expenditure, and healthcare expenditure, uncovering their dynamic interdependencies. The methodology includes trend and slope analysis, Granger causality testing, outlier detection, and clustering to identify heterogeneity in resource allocation strategies. Developed, weapon-exporting nations exhibit complementary trends, with strong causality …

Project Title: Maximizing The Volume Of A Cardboard Box To Save Trees– An Application Of Polynomial Functions To Address Global Issues [Mathematics], Lucie Mingla

Open Educational Resources

MAT 115 College Algebra & Trigonometry/Precalculus

Project Title: Maximizing the Volume of a Cardboard Box to Save Trees– An Application of Polynomial Functions to Address Global Issues

Reflective Narrative:

This project was inspired by my participation in the "Designing and Implementation of STEM Co-Curricular Activities" CTL seminar in Spring 2023. I am grateful to Drs. Bukurie Gjoci, Daniel Gertner, Ingrid Veras, and Midas Tsai, along with fellow participants, for their invaluable feedback that helped shape its development. The project was implemented in two College Algebra and Trigonometry courses. I participated in two seminars to further develop this project. The Community …

Integrating Sentiment Analysis In Predictive Models: A Comparative Study On Game Popularity On Steam, Khaleefa Alhemeiri

CMC Senior Theses

Over the past decades, the gaming industry has managed to evolve into a multi-billion-dollar enterprise. Gaming platforms such as Steam foster unprecedented amounts of engagement among players worldwide daily. In this thesis, we investigate the effect of incorporating sentiment-driven metrics, specifically YouTube view counts and positive reviews, into predictive models for game popularity. In addition, by comparing our linear regression sentiment-based approach to the Bayesian hierarchical folded normal model used by De Luisa et al. (2021), we can understand the many differences, strengths, and limitations of each methodology. In our thesis, we focus on three games. Each is of varying …

Mathematical Modelling Of Disease Outbreak, Favour Christian, Matthew Molloy

SURE Journal: Science Undergraduate Research Experience Journal

Establishing a model framework for more research necessitates a thorough understanding of the causes, distribution, prevalence, and evolution of infectious illnesses. The main mathematical concept used in this modelling simulation is ordinary differential equations (ODEs). The purpose of this study was to investigate the significance of the many criteria linked to a zombie virus spread. The zombie framework provides an accessible and relatively simple representation of the nature of infectious disease spread, allowing for tractable assumptions and the development of more complex situations.

The models are designed around a zombie outbreak in which the zombie virus is spread through a …

(R2100) Optimality Conditions Of A Topsis Optimization Model And Its Application On Interval-Valued Data, Sudipta Roy, Sandip Chatterjee

Applications and Applied Mathematics: An International Journal (AAM)

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is widely used in the field of multi-criteria decision analysis. Despite its popularity and widespread application, little attention has been given to the mathematical foundation that underlies the TOPSIS algorithm. The existing literature on this subject is far from comprehensive, leaving many aspects of the algorithm unexplored. This paper aims to address this gap in the literature by delving into the optimization problem associated with TOPSIS. Unlike traditional interval analysis theory, which only covers a limited scope, our approach extends to a broader range of scenarios and offers …