Applied Mathematics Commons^™

Non-Smooth Solutions Of The Navier-Stokes Equation And Their Means, James Glimm, Jarret Petrillo

Department of Applied Mathematics & Statistics Faculty Publications

Non-smooth (finite time blowup) Leray-Hopf solutions of the incompressible Navier-Stokes equation are constructed. The initial data for blowup is characterized by nonzero energy related turbulent fluctuations. The construction occurs in a finite periodic cube T3. The mean value of a weak solution of the Navier-Stokes equation is identified as a smooth solution of the Navier-Stokes equation.

Scaling Laws, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

Scaling laws, accurate to all orders and for intermediate times, are constructed for Navier-Stokes and Euler fluids and for quantum Yang-Mills fields. A maximum entropy principle is assumed. A finite periodic spatial volume is assumed.

Conventional renormalization group scaling is accurate to all orders in a large time asymptotic limit, but it fails at intermediate times at less than leading order due to blowups and a possible resetting of the gauge group.

The extension to quantum general relativity is indicated.

The Pure Yang-Mills Field: Repeated Blowups, James Glimm, James G. Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The temporal scaling of the quantum Yang-Mills solution is characterized by analytic solutions for a limited time interval, each ended by a discontinuity (blowup).

For general initial conditions, there will be a series of blowups and a possible resetting of the gauge group..

The principle of a maximum rate of entropy production is assumed. A finite periodic spatial volume is assumed.

The Boson Yang-Mills Field: The Loop Expansion, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

This paper demonstrates convergence of the loop expansion for Yang-Mills fields.

The loop construction of perturbation theory is based on the axial gauge, ghost states, the BRST framework and the Gribov extension of the Hamiltonian, with the loop expansion perturbatively renormalizable.

The construction is established by renormalized perturbation theory convergent to all finite orders.

Two distinct Yang-Mills theories are constructed, one based on short distance asymptotics and the other based on long distance asymptotics.

The construction depends on an assumed principle of a maximum rate of entropy production.

The paper has sufficient generality to include existence for all cases that …

The Pure Yang-Mills Field: Axioms And The Mass Gap, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The infinite volume limit is constructed based on previously constructed finite volume pure Yang-Mills quantum gauge field theories. For this limit theory, Osterwalder-Schrader axioms are verified and the existence of a mass gap in the energy spectrum is proven.

Requirements for the Millennium Yang-Mills prize are fulfilled.

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

A Novel Mathematical Model Of Hiv Transmission Incorporating The Effects Of Treatment And Pre-Exposure Prophylaxis: Sensitivity Analysis And Numerical Simulations, Erick Manuel Delgado Moya

Mathematical Modelling and Numerical Simulation with Applications

Human immunodeficiency virus (HIV) continues to be a public health problem in many countries of the world, and Pre-exposure prophylaxis (PrEP) is a preventive method for HIV, which has shown great efficacy and is in use worldwide. This work presents a new mathematical model for HIV transmission incorporating PrEP use and evaluates the impact of PrEP along with its increasing use in a population. The construction of the model takes into account three forms of diagnosis: diagnosis of individuals in risky sexual contact, diagnosis after risky contact (diagnosis in the undiagnosed infected compartment), and diagnosis associated with attempting to enter …

(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 …

Construction And Data-Driven Analysis Of A Stochastic, Individual-Based Opioid Epidemiology Network Model, Leigh Bennett Pearcy, Owen Queen, Vincent Jodoin, Suzanne Lenhart, Christopher Strickland

Mathematical Modelling and Numerical Simulation with Applications

While substance use epidemiology has been an active area of mathematical research in recent years, the social and mental processes that are involved in the development of substance use disorders have presented challenges to advancing the epidemiological theory and how they differ from the contraction of pathogenic disease. Such distinction is especially pertinent in the context of the current United States opioid epidemic and its intersection with the recent COVID-19 pandemic, as both prescription drugs and social influence play major roles in the development of opioid use disorder. In this paper, we construct a stochastic network model capturing how individual …

Optimal Control Of Stochastic Systems: A Numerical Study Of The Stochastic Linear Quadratic Regulator Framework, Gülşen Orucova Büyüköz, Yaprak Güldoğan Dericioğlu, Tuğçem Partal

Mathematical Modelling and Numerical Simulation with Applications

Optimal control of stochastic linear systems is fundamental in control theory, with applications in robotics, finance, and engineering. The Stochastic Linear Quadratic Regulator (SLQR) derives optimal feedback laws via the Riccati equation but requires numerical discretization of the resulting stochastic dynamics. Despite extensive studies on numerical methods for stochastic differential equations, their performance within the SLQR framework remains insufficiently explored. This study compares two predictor–corrector schemes of different orders: the Order 1.0 Predictor-Corrector (PC) method and the Order 2.0 Weak PC method. A one-dimensional linear quadratic problem with a closed-form solution enables precise error evaluation against the analytical trajectory. Convergence …

Analysis And Numerical Investigation Of A Breast Cancer Treatment Model Incorporating Ketogenic Diet And Immune Boosters Via Optimal Control Theory, Kunnisai Muniroh, Ummu Habibah, Wuryansari Muharini Kusumawinahyu, Nur’Izzati Hamdan

Mathematical Modelling and Numerical Simulation with Applications

This paper develops a mathematical model to investigate breast cancer dynamics by incorporating tumor–immune interactions, ketogenic diet effects, and medical treatment. The model is formulated as a system of nonlinear ordinary differential equations and analyzed within an optimal control framework. Time-dependent control variables are introduced to represent treatment strategies aimed at minimizing tumor progression while reducing therapeutic costs. The model’s well-posedness is established through positivity and boundedness analysis. The necessary conditions for optimality are derived using Pontryagin’s Minimum Principle, resulting in a coupled system of state and adjoint equations. Numerical solutions are obtained using the fourth-order Runge–Kutta method combined with …

Mathematical Analysis And Numerical Simulation Of A Fractional-Order Sir-Si Model For Malaria Transmission Dynamics, Gassan A. M. O. Farah, Abdulaziz Y. A. Mukhtar, Kailash C. Patidar

Mathematical Modelling and Numerical Simulation with Applications

This study investigates the complex transmission dynamics of malaria, a critical global health challenge, with a focus on the African continent. We introduce a novel approach that employs Fractional Differential Equations (FDEs) to advance the understanding of malaria spread and control. Specifically, we develop a new SIR-SI model using the Caputo fractional operator, which captures the memory effects and time-delay characteristics inherent in real-world epidemiological systems. A detailed analysis of the model's solvability and uniqueness is conducted using fixed-point theory. To obtain an analytical solution, the system is solved via the Laplace transform method, with solutions expressed in closed form …

Mathematical Modeling Of Awareness-Driven Interventions For Gender-Based Violence Control In A Closed Population, Furaha M. Chuma, Edward K. Ngailo, Zubeda S. Mussa

Tanzania Journal of Science

Gender-based violence (GBV) remains a critical global health issue with profound social, economic, and psychological implications. Despite concerted efforts to address this challenge, traditional intervention approaches often fall short of effectively mitigating its prevalence. In recent years, there has been growing recognition of the potential impact of awareness-based interventions in addressing various social and health crises. This paper introduces a modeling framework to explore the efficacy of awareness-based interventions in influencing the dynamics of the GBV epidemic within a closed population. A detailed sensitivity analysis, utilizing a normalized sensitivity index method, reveals that the violence transfer rate between perpetrators and …

Modelling The Impact Of Multiple Control Strategies On The Spread Of Phyllody Disease In Sesame, Augustino I. Msigwa, Furaha M. Chuma

Tanzania Journal of Science

Phyllody disease, caused by phytoplasma and transmitted by insect vectors such as leafhoppers, poses a serious threat to the productivity of sesame (Sesamum indicum) as it has caused substantial yield and oil-quality losses worldwide, with documented seed-yield reductions ranging from 38% to as high as 80% in severe outbreaks, particularly in Africa and India. Understanding phyllody dynamics is therefore critical to safeguard yields and trade. While some modeling efforts exist for sesame phyllody (for example, statistical and prediction-focused approaches), mechanistic transmission models (SEIR) assessing its full impact remain scarce. This study therefore develops and analyses impact of multiple control strategies …

Modeling And Forecasting Wholesale Gasoline Prices In Tanzania Using Arima And Neural Network Autoregressive Models, Thadei D. Sagamiko, Edward K. Ngailo

Tanzania Journal of Science

Fluctuations in wholesale gasoline prices significantly impact the economy of import-dependent countries like Tanzania, making accurate forecasting essential for policymakers and market participants. This study forecasts monthly wholesale gasoline prices in Dar es Salaam by comparing the Autoregressive Integrated Moving Average (ARIMA) model with a Neural Network Autoregressive (NNAR) model. Using data from January 2015 to December 2024 from the Energy and Water Utilities Regulatory Authority (EWURA), the models were evaluated. The ARIMA(1,1,1) model was identified as the best-fitting linear model, but the NNAR(11,7) model demonstrated superior accuracy. On the holdout test set, the NNAR model achieved a lower Mean …

Simulation Based Model For Canine Distemper Virus In Prey-Predator Dynamics, Mussa A. Stephano

Tanzania Journal of Science

This study investigates the dynamic interactions of a prey-predator ecosystem infected with Canine Distemper Virus (CDV), focusing on predation and disease transmission. A deterministic model, coupled with Monte Carlo simulations (MCS), employed to capture both stochastic variability and environmental fluctuations. The findings indicate that classical prey-predator cycles provide the baseline for population dynamics, but the introduction of CDV and environmental noise fundamentally alters system behavior. Initially, populations exhibit oscillations that gradually stabilize into dynamic equilibria. However, the disease persists within both prey and predator populations, suggesting endemicity rather than extinction. Predation plays a dual role in disease dynamics, while it …

A Parallel Interval Modeling Framework For Nonlinear Systems: Application To A Modified Duffing Oscillator, Roman Voliansky, Nina Volianska

Northeast Journal of Complex Systems (NEJCS)

The paper presents a mathematical framework for converting nonlinear dynamical systems into parallel forms. This framework replaces the exact system motion equations with interval equations, enabling the representation of nonlinear functions over piecewise linear domains. Such representation enables the description of system motions using linear-like differential equations, which can be analyzed and manipulated using well-known control methods. One such method is eigenvalue analysis, a powerful tool in classical control theory since many techniques rely on the system’s characteristic polynomial and its eigenvalues. We apply this method to define interval system eigenvalues and track their variation during system operation. These eigenvalues …

Lipscomb University Parking Garage Analysis, Morgan Hayes, Reagan Maxwell, Daniel Diaz Tortolero, Dominick Dingus, Michael Ent

Student Scholar Symposium

Many students struggle to easily find parking during the school day at Lipscomb University, and this problem will only be emphasized with the potential removal of the nearby Stokes parking lot (containing 255 spaces). To combat this need and help provide additional room for university growth, we propose the addition of a parking garage west of the Fields engineering building, between Belmont Boulevard and Grandview Drive. This garage would be separated from the current garage behind Fields and be slightly larger, containing about 500 spaces over five levels. We evaluate the current number of parking spaces needed at Lipscomb based …

Gauge Invariance And Repeated Time Isolated Solution Discontinuities, James Glimm, James Glimm, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The main conclusion of this paper is (a) a gauge structure for the underlying physics, at or above a critical dimension, forces multiple time isolated solution discontinuities (blowups). The main conclusion (b) is that the solution is one Sobolev index more singular that the natural Hilbert space H. Proofs are given for Navier-Stokes fluids.

Using Feynman's Technique To Evaluate Non-Elementary Integrals Used In Physics, Dominick Dingus, Christopher Wengert, Madouna Barsoum

Student Scholar Symposium

In numerous physical applications, such as those in the fields of optics and thermodynamics, the quantitative description of various phenomena requires evaluating certain definite integrals whose antiderivatives cannot be expressed using elementary functions. As a result, these integrals are typically evaluated numerically using a suitable program. However, it is possible to evaluate some of these types of integrals using a strategy known as Feynman’s Technique, an approach named after Richard Feynman, the American physicist who popularized the method during the mid-20th century. This project aims to highlight the usage of Feynman’s Technique for evaluating some of these integrals while applying …

Assessing The Electrical Efficiency Of The Thompson Coil, Christopher Wengert, Dominick Dingus, Madouna Barsoum, Jacqueline Ceballos, Michael Ent, Jackson Stephens

Student Scholar Symposium

The Thompson Coil is a well-known system used in undergraduate electromagnetism courses to demonstrate electromagnetic induction. Traditionally, the system employs a “jumping ring” to visualize induced currents. This project seeks to repurpose this phenomenon for practical energy transfer by replacing the jumping ring with a wire coil, allowing the measurement of the power delivered to an electric load. Using Faraday’s Law of Induction, 𝜖 = −𝑑Φ𝐵 /𝑑𝑡 , where Φ𝐵 represents the magnetic flux, the induced electromotive force (EMF) in the receiving coil is determined. Finally, the system’s power transfer efficiency is evaluated using the formula 𝜂 =𝑃𝑜𝑢𝑡/ 𝑃𝑖𝑛 ×100% …

Complex Numbers And Phone Signals, Ava Barrett, Rebecca Brazeal, Javier Gutierrez

Student Scholar Symposium

Though widely used, complex numbers are often introduced in coursework without real-world context. Our team examines their application in electromagnetic waves and communication systems, focusing on how they help model and analyze phone signals. By connecting concepts from physics and math, we demonstrate their role in signal modulation and transmission. Through this project, we aim to highlight the practical importance of complex numbers in modern technology and provide a clearer understanding of their use in communication systems that power everyday devices like smartphones.

Advanced Soliton Structures And Elliptic Wave Patterns In A Sixth-Order Nonlinear Schrödinger Equation Using Improved Modified Extended Tanh Function Method, Mina Fahim, Hamdy Mohamed Ahmed, Islam Samir, Kamal Dib

Basic Science Engineering

In this work, a sixth–order extension of the nonlinear Schrödinger equation (NLSE) within its integrable hierarchy is investigated to model higher–order nonlinear and dispersive effects relevant to optical fiber systems and nonlinear wave propagation. By employing the Improved Modified Extended Tanh Function Method, a comprehensive family of exact analytical solutions is derived, encompassing bright and dark solitons, singular soliton structures, and singular periodic solutions. In addition, solution families expressed in terms of Jacobi elliptic functions, Weierstrass doubly periodic elliptic functions, and exponential profiles are obtained. The novelty of this study lies in extending the analytical framework of the NLSE hierarchy …

Stability Analysis Of Thermohaline Convection With A Time-Varying Shear Flow Using The Lyapunov Method, Kalin Kochnev

Honors Scholar Theses

This work applies the Lyapunov method to identify instabilities and compute the growth rate of a linear time-varying system. The linear system studied describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. A time-dependent weighting matrix is employed to construct a Lyapunov function candidate. The resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted by the Lyapunov method or Floquet theory, used for comparison, will converge to the same value obtained from numerical simulations. Furthermore, the …

On Sharpest Tail Bounds For Functions Of Tail Bounded Random Variables, Stephen Harrison

Mathematics & Statistics ETDs

Consider n real/complex, independent/dependent random variables with respective tail bounds and g a measurable function of the r.v.’s. Consider f the “sharpest” tail bound of g (sharpest in the sense, if f were any less, then for some X1, ..., Xn satisfying the conditions, g(X1, ..., Xn) would not satisfy the tail f). Significant research has been done to approximate f often with high accuracy. These results are often of the form, for g in this family, and tail bounds of Xk in this family, f is bounded by some f′ with high accuracy. However, the question “what would it …

Spectral Methods And Wavelets In Quantitative Finance Problems, Davood Damircheli

Theses and Dissertations

This dissertation leverages advanced spectral methods and wavelet techniques to address complex quantitative finance models, enhancing the computation of financial derivatives and risk assessments. Building on foundational studies, this research extends these methods to broader, intricate financial contexts. The first section explores fractional-order generalized Chebyshev wavelets (FOCW) applied to fractional advection equations, relevant in both mathematics and physics. Using a regularized beta function to compute the Riemann-Liouville fractional integral operator, this study introduces a novel numerical scheme with robust accuracy, confirmed through error analysis and empirical tests. The second part examines the fractional Black-Scholes equations for option pricing under subdiffusive …

Modeling Private Debt Using U.S. Consumer Expenditure Data, Stsiapan Dziamentsyeu

Honors Capstones

This project models private household debt among U.S. consumers using data from the Consumer Expenditure Survey (CES) between 2013 and 2023. The analysis focuses on identifying how demographic and economic characteristics, such as income, housing expenditures, education, and occupation, relate to non-mortgage “other” loan balances. After initial model development produced poor residual behavior due to zero-inflation from imputed debt values, the analysis was refined to include only households reporting verifiable debt. Multiple modeling techniques, including AIC-based variable selection and Lasso regularization, were compared under a five-fold cross-validation framework. The Lasso model achieved superior predictive accuracy (RMSE = 1.55, MAE = …

Math Meets Climate: The Energy Balance Model, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to the mathematics of Earth’s climate through the classical energy balance model. Students analyze how incoming solar radiation, outgoing thermal radiation, and temperature-dependent albedo interact to determine Earth’s equilibrium temperature. Using analytical calculations and computational tools, students identify equilibrium states, assess their stability, and interpret the results through the lens of dynamical systems and bifurcation theory. The activity builds conceptual understanding of climate feedbacks, greenhouse effects, and tipping behavior using a transparent, one-variable model. Designed for applied mathematics and interdisciplinary STEM courses, this assignment emphasizes computation, physical interpretation, and real-world relevance. It is released as a …

Understanding Enso Through Mathematical Models, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to conceptual models of the El Niño–Southern Oscillation (ENSO) and guides them through a structured investigation of their physical and mathematical foundations. Students analyze the recharge–oscillator and delayed–oscillator frameworks, explore how differential equations capture ocean–atmosphere interactions, and evaluate parameter-driven changes in oscillatory behavior. A key component of the work is the guided use of generative AI as a research tool: students employ AI models to locate peer-reviewed literature, interrogate model extensions, and refine their understanding of complex mechanisms, while synthesizing all final explanations in their own words. By blending classical climate modeling with modern AI-supported inquiry, …

A Harmonious Equation: How To Facilitate The Interdisciplinary Nature Of The Arts In A High School Math Classroom, Myah Medendorp

Honors Theses

The education system is constantly evolving and advancing, and the teaching methods should advance along with it. Musical involvement has long been cited as having a positive effect on mathematical proficiency and being an easy way to boost mathematical performance but is most often used in an elementary school setting and the early learning years. Are there ways to use this scientific relationship to boost performance and engagement in a secondary school environment?

This literature review functionalization takes a closer look at this relationship and how it can be implemented effectively in a high school math classroom beyond the young …