Applied Mathematics Commons^™

Weak Solutions Of The Navier-Stokes Equation And Their Euler Limits, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The existence of a weak solution of the incompressible isothermal Navier-Stokes equation with given initial conditions in the Sobolev space $\mathcal{H}_{-2}$ for energy fluctuations and in $\mathcal{H}_{-3}$ for enstrophy fluctuations is assumed. The existence is uniform with respect to the Euler limit of zero viscosity $\nu$. Thus, existence of weak solutions of the Euler equation with given initial conditions is established, and these Euler solutions are the zero viscosity limit of Navier-Stokes solutions, provided the Navier-Stokes solutions exist.

The Yang-Mills Field: The Fermion Sign Problem, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A new resolution of the Fermion sign problem is accomplished based on the theory of homology groups and thimbles. The sign problem for residual states not reached by the homology groups and thimbles is resolved by a perturbative converged Bosonic expansion. The proofs depend on an assumed principle of a maximum rate of entropy production. Two Yang-Mills gauge field theories are constructed, one based on short distance asymptotics and the other based on long distance asymptotics.

Quantum General Relativity V2, James Glimm, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

We construct a quantum general relativity theory.. The construction includes specification of a cosmological model. Based on type Ia supernova data, the model has zero dark energy and zero cosmological constant.

The analysis is at the level of theoretical physics, with some mathematical details omitted.

The Pure Yang-Mills Field: I: Fixed Time Existence, James Glimm, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

The convergence of renormalized perturbation theory to all finite orders is defined and shown to be valid for the fixed time perturbation theory of a pure Yang-Mills field.

Two pure Yang-Mills quantum gauge field 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.

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.

Analytical Study Of Transient Mixed Convective Radiative Jeffrey Fluid Flow With Diffusion–Thermo And Chemical Reaction, V. Sathiya, R. Vijayaragavan, B. Rushi Kumar

Mansoura Engineering Journal

This research examines the behavior of unsteady mixed convective radiative Jeffrey fluid flow over a permeable moving plate with a diffusion thermo effect. The study incorporates multiple factors, including aligned magnetic fields, heat generation, radiation, and chemical reactions. The behavior of Jeffrey fluid under these combined conditions is particularly relevant to the design of efficient heat exchangers, MHD generators, and cooling systems for electronic components. A regular perturbation technique was employed to solve the governing equations, yielding distributions for velocity, temperature, and species concentration. These solutions enabled the derivation of expressions for skin friction, Nusselt number, and Sherwood number. Through …

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 …

All Games Have Equilibria, Arthur Paul Pedersen, M. Ali Khan, 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 …

Global Weak Solutions Of Optical Variational Wave System, Shahrazad Hamed Alnafie

Graduate Theses, Dissertations, and Problem Reports (ETD)

                                                       ABSTRACT

                   Global Weak Solutions of Optical Variational Wave System

                                        Shahrazad Hamed Mahal Alnafie

The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.

We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable …

Quantum Mechanics As A Framework For Data Assimilation And Its Application To Atmospheric Parameterization, David Freeman

Dartmouth College Ph.D Dissertations

Quantum mechanics, as a mathematical system, can be understood as a generalization of classical probability theory. Quantum Mechanical Data Assimilation (QMDA) is a method in which classical dynamical systems are embedded into a quantum mechanical setting, with an associated data assimilation scheme leveraging the operator algebraic setting. In this dissertation, the algebraic structure underlying the operator theoretic formulation of QMDA is discussed. A procedure for closure of dynamical systems based on QMDA, known as Quantum Mechanical Closure (QMCl), is then constructed, and the procedures for constructing the quantum embeddings and implementing QMCl in practice are laid out and implemented for …

A Stability Analysis Of The Phase-Lock Equations, Brian M. Sunguza

UNF Graduate Theses and Dissertations

Ginzburg and Landau have provided a set of equations that relate superconductivity to magnetic fields. Through a transformation process, Zhan has derived what are now called the phase-lock equations. A stability analysis of the spatially-independent phase-lock equations is the purpose of this presentation. This simplification is significant since it allowed for the analytical determination of equilibria, their stability, and the influence of a periodic forcing function. Through the use of an original code, numerical simulations are shown to corroborate the analytical results described above.

This analysis includes novel Lyapunov functions that allowed for the analytical determination of the instability region. …

Ai-Enabled Digital Twins And Optimization Workflows For Accelerator Control, M. Yadav, A. Seryi, B. Terzic, J. Bird, J. Delayen, K. Makino, K. Ahmed, L. Van Riesen-Haupt, Q. Su, S. De Silva, S. Hossain, T. Griffin, T. Satogata

Physics Faculty Publications

We propose to develop advanced ML models, such as physics informed neural network (PINN) based surrogate models, to accurately represent accelerator phase space transport. These surrogate models will enable precise diagnosis and prediction of beam phase space evolution along the beamline, facilitating real-time control and optimization. The developed models will be tested using the Upgraded Injector Test Facility (UITF) at Thomas Jefferson National Accelerator Facility (JLab), providing a pathway toward ML-driven enhanced diagnostics and beamline control in operational accelerator environments. The primary aim will be to facilitate this by developing machine learning models that outperform traditional simulations in speed and …

General Relativity: Existence, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A mathematically rigorous renormalized perturbative expansion, truncated to all finite orders, establishes the existence of quantum general relativity theories with Fermion matter fields. The expansion is initialized with the selection of a vacuum state. Distinct infrared (IR) and ultraviolet (UV) vacuum states are considered. The IR vacuum state defines cosmology model. The UV vacuum state defines a black hole model. The vacuum state is initialized with a o3 gauge Lie algebra defining a quark-gluon plasma. With this choice, string theory is avoided. Finite order renormalized perturbation theory, defined using Bosons to avoid Fermion sign cancellation, is convergent to all finite …

Scaling Laws, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A mathematically rigorous renormalized perturbative expansion, truncated to all finite orders, establishes the existence of quantum general relativity theories with Fermion matter fields. The expansion is initialized with the selection of a vacuum state. Distinct infrared (IR) and ultraviolet (UV) vacuum states are considered. The IR vacuum state defines a cosmology model. The UV vacuum state defines a black hole model. The vacuum state is initialized with a o3 gauge Lie algebra defining a quark-gluon plasma. With this choice, string theory is avoided. Finite order renormalized perturbation theory, defined using Bosons to avoid Fermion sign cancellation, is convergent to all …

Performance Of Numerical Methods Applied To The Black–Scholes Model, Scott Cameron Williams

UNF Graduate Theses and Dissertations

We compare five numerical approaches for approximating solutions to the Black–Scholes partial differential equation for pricing European call options: FTCS, BTCS, Crank– Nicolson, Monte Carlo simulation, and a physics–informed neural network (PINN). These methods span finite difference techniques, probabilistic simulation, and machine learning. Performance is evaluated based on computational efficiency and accuracy relative to the analytical Black–Scholes solution.

Among the methods, Crank–Nicolson and the PINN demonstrated the strongest overall performance. Crank–Nicolson achieved the highest accuracy but exhibited increased runtime as the number of underlying stock price grid points grew. In contrast, the PINN produced slightly less accurate results but with …

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, ended by a discontinuity (blowup).

For general initial conditions, there will be a finite series of these discontinuities.

The time asymptotic limit of this series of discontinuities is given as the mean of these solutions, starting with the mean of the initial data. The time asymptote is the solution in $\mathcal{S}'$ of the heat equation as a stochastic process.

The time asymptote satisfies the cluster axiom, to complete the requirements for a solution of the Millennium Yang-Mills problem.

Lean computer verification of …

The Pure Yang-Mills Field, \, Iia: The Axioms, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A previously constructed pure Yang-Mills quantum gauge field theory is shown to satisfy all Osterwalder Schrader axioms.

The Pure Yang-Mills Field, \, Iia: The Mass Gap, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

No abstract provided.

Mathematicly Rigorous Quantum General Relativity, I: Pure, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A pure general relativity (quantum or classical) is one lacking in matter. Such a field has a Lorentzian space time geometry. A renormalization perturbative expansion, truncated to all finite orders, establishes both the quantum and the general relativity theory with full mathematical rigor.

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 a loop expansion cutoff to all finite orders for the dynamics.

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 …

The Pure Yang-Mills Field, Repeated Blowups, James 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, ended by a discontinuity (blowup).

For general initial conditions, there will be a finite series of these discontinuities.

The time asymptotic limit of this series of discontinuities is given as the mean of these solutions, starting with the mean of the initial data. The time asymptote is the solution in $\mathcal{S}'$ of the heat equation as a stochastic process.

The time asymptote satisfies the cluster axiom, to complete the requirements for a solution of the Millennium Yang-Mills problem.

The principle of a …

Applications Of Machine Learning For Evaluating Downward-Coupled Stratosphere-Troposphere Interactions And Subseasonal Forecasts Of Opportunity, Elena M. Fernandez

Electronic Theses & Dissertations (2024 - present)

Wintertime stratospheric dynamics provide key information for understanding atmospheric teleconnections and improving subseasonal-to-seasonal (S2S) predictions on timescales of two weeks to two months. Periods of enhanced predictability, often referred to as forecasts of opportunity, arise from large-scale teleconnected variability, within which the stratosphere serves as an important precursor for tropospheric states, such as near-surface temperatures. While traditional diagnostics of downward coupled stratosphere-troposphere interactions typically rely on zonal-mean representations of wind and geopotential height, this dissertation presents an alternative vortex-centric framework through metrics that capture the daily geometric and dynamical evolution of the stratospheric polar vortex. The proposed stratospheric …

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 …

An Empirical Comparison Of K-Nearest-Neighbors And Logistic Regression Classification Models, Jackson Cushing

Graduate Studies Theses and Dissertations 2026

This thesis presents an empirical comparison of two classification methods: Logistic Regression and K Nearest Neighbors (KNN). The primary objective of this research is to evaluate the strengths and limitations of each method when applied to real-world datasets. Several publicly available datasets on diabetes, breast cancer, heart attack risk, and cardiovascular disease, were analyzed. For each dataset, K Nearest Neighbors models were implemented in the same way logistic regression had already been applied. The results demonstrate that while logistic regression offers interpretable parameter estimates and performs well when the underlying predictor and outcome relationship is approximately linear, however KNN can …

Multi-Grade Deep Learning, Yuesheng Xu

Mathematics & Statistics Faculty Publications

Deep learning requires solving a nonconvex optimization problem of a large size to learn a deep neural network (DNN). The current deep learning model is of a single-grade, that is, it trains a DNN end-to-end, by solving a single nonconvex optimization problem. When the layer number of the neural network is large, it is computationally challenging to carry out such a task efficiently. The complexity of the task comes from learning all weight matrices and bias vectors from one single nonconvex optimization problem of a large size. Inspired by the human education process which arranges learning in grades, we …

Machine Learning-Based Intrusion Detection System For Iot Networks Using The Rt-Iot 2022 Dataset, Bukunmi Ebenezer Afolabi

Theses, Dissertations and Capstones

The rapid expansion of the Internet of Things (IoT) has transformed modern computing by enabling seamless connectivity among heterogeneous devices across diverse application domains. However, this increased interconnectivity has significantly enlarged the attack surface of IoT networks, exposing them to a wide range of sophisticated cyber threats. Conventional security mechanisms often lack the capability to detect emerging attacks in real time, thereby necessitating the development of intelligent Intrusion Detection Systems (IDS) capable of accurately identifying malicious network activities. This study developed and evaluated a machine learning-based intrusion detection framework for multiclass IoT attack detection using the RT-IoT2022 dataset. The dataset …

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 …