Applied Mathematics Commons^™

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.

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 mathematical resolution of the Fermion sign problem is accomplished based on the theory of homology groups and thimbles. The original contribution of this paper is the mathematicly rigorous convergence of residual states. The residual states remain after use of a rigorous Picard-Lefschetz saddle point optimization.

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.

The Pure Yang-Mills Field. I: Eistence, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

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 is based on the axial gauge, ghost states, the BRST framework and 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 to all finite orders.

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

The Pure Yang-Mills Field: The Mass Gap; Axioms Minus One, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

A strictly positive gap at low temperatures is demonstrated in the energy spectrum of the previously constructed pure Yang-Mills quantum gauge field theories. Axioms other than the cluster axiom are verified.

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.

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.

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 a cosmology model. The UV vacuum state defines a black hole model.

The vacuum state is initialized with a $\mathfrak{o}^3$ 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 …

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.

The principle of a …

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.

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.

Brst Conserved Moments For Fluids, James Glimm

Department of Applied Mathematics & Statistics Faculty Publications

This note adds a missing proof for an assertion in the original paper Gauge Invariance and Repeated Time Isolated Solution Discontinuities.

The assertion is that under weak limits and within a constant energy isosurface, enstrophy is conserved up to losses due to viscous dissipation.

According to the BRST theory for quantum Yang-Mills fields, the conserved moments are exactly those with infinite vacuum expectation values. Application of the BRST theory to fluids is explored. The BRST restriction to the loop expansion (fixed space) is justified by the reqirement that a perturbative expansion can be constructed. In this case the theory predicts …

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.

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 a finite sequence of blowups.

The extension to quantum general relativity is indicated.

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

Department of Applied Mathematics & Statistics Faculty Publications

A pure quantum general relativity field is one lacking in matter. Such a field has a Lorentzian space-time geometry. A renormalized perturbation expansion, truncated to all finite orders, establishes  the existence of pure quantum general relativity with full mathematical rigor

Entropic Foundation Of Finance And Physics: Securities Price Dynamics And Quantum Theory, Mohammad Abedi

Electronic Theses & Dissertations (2024 - present)

In many scientific and financial contexts, we must reason and make predictions under conditions of incomplete information. This dissertation develops Entropic Dynamics (ED) as a unified framework for deriving dynamical laws directly from principles of inference. Within this approach, probability distributions represent states of knowledge, and their evolution is determined through entropy maximization subject to relevant constraints. This leads to a novel concept of entropic time and a formulation of dynamics as an inferential process. In this talk, I will present how ED provides a common foundation across multiple domains. In physics, quantum dynamics for particles and scalar fields in …

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 …

The Ellipsoidal Principal Semi-Axis Geometry Of The Solution To An Ivp For A Matrix Diffusion Pde, Brandon M. Fox

Electronic Theses & Dissertations (2024 - present)

We analyze the underlying geometry of the solution to an IVP for a matrix diffusion PDE. We first derive the fundamental solution to the PDE. We then determine the unique solution to the IVP. From there, we begin analyzing its underlying geometry. We first observe that the geometry exhibits an ellipsoidal nature. Furthermore, we observe that it is described by the principal semi-axis geometry of the ellipsoids associated with the solution. This conclusion follows from applying the Principal Axis Theorem to the associated ellipsoids to establish their principal semi-axis geometry, as governed by the eigenstructure of the matrix. This thesis …

Stochastic Fractional-Order Memristive Fuzzy Bam Neural Networks With Time Delays And Leakage Term For Finite-Time Stability Analysis, J. Kumar, M. Syed Ali, Sumaya Sanober, Mohammad Yarish, Abeer M. Alotaibi, Tarek F. Ibrahim

Computer Science Faculty Publications

In this study, a finite-time stability analysis with time delays and a leakage term is conducted on stochastic fractional-order memristive fuzzy BAM neural networks. FOMFBAMNNs are developed using set-valued map theories as well as differential inclusion. We obtained several significant adequate criteria of uniform stability in the mean square of such networks by using analytical methods and inequality approaches, such as Cauchy–Schwarz inequality and Burkholder–Davis–Gundy inequality. In addition to examining two different fractional-order derivatives between the U-layer and V-layer synchronously with fractional order, the existence, uniqueness, and stability of its equilibrium point are also shown ½ ≤ α ≤ 1. …

Design And Analysis Of Modern Quantum Neural Network Architectures For Intelligent Systems, Lakshmi Chandrakanth Kasireddy, Prabhakara Rao Kapula, Dineshkumar Rajendran, Neha Bharani, Srikanth Pulipeti, Islombek Khushvaktov

Computer Science Faculty Publications

Quantum neural networks (QNNs) offer a principled pathway for integrating quantum computation with machine learning through superposition- and entanglement-based representations. This chapter proposes an architecture-aware design and evaluation framework for modern QNNs, emphasizing robustness and system feasibility alongside predictive performance. Multiple architectures variational QNNs, quantum convolutional neural networks, tensor-network hybrids, and fully quantum models—are assessed under a unified protocol. Experimental analysis shows that the proposed architecture-search–guided QNN achieves 91.8% classification accuracy and an F1-score of 0.914, outperforming fixed-template variational QNNs by approximately 5.6 percentage points. Under depolarizing noise with probability p = 0.10, the proposed model retains 85.3% accuracy, whereas …

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 …

Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight

Williams Honors College, Honors Research Projects

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration …

Mathematical Model Of Graphene, Douglas M. Sanor

Williams Honors College, Honors Research Projects

Graphene, a single-atom-thick layer of carbon arranged in a hexagonal lattice, exhibits exceptional mechanical, electrical, and thermal properties that make it a promising material for a wide range of engineering applications. This paper presents a mathematical framework for modeling the mechanical behavior of graphene, with a focus on atomistic-to-continuum approaches. We begin with a onedimensional Frenkel-Kontorova model that represents graphene as a discrete chain of particles interacting with both their nearest neighbors through harmonic spring potentials and an underlying substrate through van der Waals forces. Numerical simulations of this discrete model demonstrate the commensurate-toincommensurate phase transition, revealing how geometric mismatch …

Managing Multi-Drug Resistance: An Evolutionary Game Theory And Optimal Control Approach, Shukhrat Nasrulloev

Theses and Dissertations

Multi-drug resistance is an evolutionary process in which treatment eliminates sensitive cells, allowing resistant clones to dominate. This thesis investigates this process using a framework integrating population dynamics, evolutionary game theory, and optimal control theory. We develop a two-population logistic growth model describing competition between drug-sensitive and drug-resistant cells under treatment, construct dose-dependent payoff matrices and replicator dynamics to characterize evolutionary competition, and derive a critical drug level D_crit = (r_S - rR)/(dS - d_R) at which resistant cells gain a fitness advantage. An optimal control problem is formulated via Pontryagin's Maximum Principle to identify schedules …

Data-Driven Partitioning In Distributed Optimization For Networked Systems, Prosper Azameti

Theses and Dissertations

The convergence behavior of distributed optimal power flow (OPF) depends strongly on how the power network is partitioned into regions. Classical graph-based methods such as METIS are widely used, but they rely mainly on static topological criteria and do not explicitly incorporate operating-point-dependent information that may affect distributed optimization performance. This thesis develops a data-driven partitioning framework for distributed OPF using graph neural networks (GNNs). Each OPF scenario is represented as a graph in which buses are nodes and transmission lines are edges. Node and edge features capture both structural and operational characteristics of the network. Partition prediction is formulated …

Theoretical Analysis Of Filtering And Smoothing Techniques In Stochastic Time Series, Kaosarat Bola Azeez

Theses and Dissertations

Polynomial-preserving linear filters are fundamental tools for smoothing and trend extraction in noisy time series, particularly when the objective is to reduce short-run variation without distorting underlying low-degree structure. This thesis develops a general framework for such filters based on local polynomial approximation and least-squares estimation. In particular, we establish that the resulting estimators can be represented as symmetric linear filters whose weights satisfy moment conditions that ensure exact polynomial reproduction up to a specified degree. Building on this framework, we derive explicit coefficient vectors for several classical smoothing filters, including the Savitzky-Golay filters SG(5,2) and SG(7,3), as well as …

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 …

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. …

Robust Deep Learning One-Class Classification, Shahd Alnofaie

Graduate Studies Theses and Dissertations 2026

One-Class Classification (OCC) focuses on learning the characteristics of normal data and identifying observations that deviate from this learned pattern as anomalies. It is commonly used in applications such as medical diagnosis, cybersecurity, industrial monitoring, and fraud detection, where abnormal examples are often rare or unavailable during training. Classical approaches such as SVDD and LS-SVDD describe normal data using a hypersphere. While effective in some settings, these methods rely on shallow representations and can be sensitive to noise and contaminated observations. To address these limitations, this dissertation introduces a Deep LS-SVDD framework that combines hypersphere-based data description with deep neural …

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 …