Mathematics Commons^™

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 …

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 …

Identifying Relevant Covariates In Rna-Seq Analysis By Pseudo-Variable Augmentation, Yet Nguyen, Dan Nettleton

Mathematics & Statistics Faculty Publications

RNA-sequencing (RNA-seq) technology allows for the identification of differentially expressed genes, which are genes whose mean transcript abundance levels vary across conditions. In practice, RNA-seq datasets often include covariates that are of primary interest in addition to a set of covariates that are subject to selection. Some of these covariates may be relevant to gene expression levels, while others may be irrelevant. Ignoring relevant covariates or attempting to adjust for the effect of irrelevant covariates can compromise the identification of differentially expressed genes. To address this issue, we propose a variable selection method that uses pseudo-variables to control the expected …

Adaptive Multi-Grade Deep Learning For Highly Oscillatory Fredholm Integral Equations Of The Second Kind, Jie Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

This paper studies the use of Multi-Grade Deep Learning (MGDL) for solving highly oscillatory Fredholm integral equations of the second kind. We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.

An Association Test For Ordinal Outcomes In Clustered Data With Informative Cluster Size, Hasika K. Wickrama Senevirathne, Sandipan Dutta

Mathematics & Statistics Faculty Publications

In cluster-correlated data, the number of observations in a cluster can be associated with the outcome from that cluster. This phenomenon is known as informative cluster size which can occur in cluster-randomized clinical trial data. Several studies have found that ignoring the issue of informative cluster size can produce biased results in the analysis of clustered data. Most of the existing methods for addressing informative cluster size are suited to continuous outcomes. However, ordinal outcomes and covariates are often encountered in clustered data obtained from large clinical studies. The existing methods for ordinal association testing in clustered data can produce …

Logistic-T Multinomial Mixture Model For Clustering For Microbiome Data, Wenshu Dai, Yuan Fang, Sanjeena Subedi

Mathematics & Statistics Faculty Publications

The logistic-normal multinomial distribution has been used for modelling microbiome data obtained from high-throughput sequencing technologies, which are compositional in nature. A logistic-normal multinomial distribution is a hierarchical multinomial distribution that assumes the latent variable which are the additive log-ratio (ALR) transformed proportions in a multinomial distribution follows a Gaussian distribution. Model-based clustering algorithms have also been developed for clustering microbiome data based on the logistic-normal models. However, the Gaussian assumption may violated when the ALR transformed variable exhibit heavy-tailed distributions or has outliers. Our study introduces a novel mixture of logistic-t multinomial models that effectively address these challenges. Utilizing …

A Composite Narxnn Approach To Photovoltaic Power Forecasting With Integrated Weather Inputs And Uncertainty Quantification, Denisse Urenda Castañeda, Sharmin Abdullah, Jackson Morgan, Honglun Xu, Michael Pokojovy, Tzu-Liang Tseng

Mathematics & Statistics Faculty Publications

Solar photovoltaics (PV) are a major source of sustainable energy. Yet, their power output is highly sensitive to environmental variability, particularly solar irradiance, cloud cover, wind, and temperature. Accurate forecasting of PV power is essential for efficient grid integration and energy planning, especially in applications requiring reliable longer-term forecasting rather than one-step-ahead predictions. This study presents a PV power forecasting approach using Nonlinear Autoregressive models with Exogenous Inputs (NARX), integrating large-scale numerical weather historical data as exogenous variables. Although NARX models effectively capture temporal dependencies, they can become overly dependent on historical power values, reducing responsiveness to real-time weather changes. …

A New Parallel-In-Time Direct Inverse Method For Nonlinear Differential Equations, Nail K. Yamaleev, Subhash Paudel

Mathematics & Statistics Faculty Publications

We propose a new method for parallelization of the first-order backward difference discretization (BDF1) of the first-order time derivative in nonlinear partial differential equations, such as conservation law equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at …

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 …

Studies On The Depth Formula And On Reducing Dimensions, Brian Mccourt Laverty

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation presents the author’s recent research, conducted under the supervision of Professor Olgur Celikbas, and based on two articles—one published and one in progress. These works develop two closely related research directions in commutative algebra. Together, they contribute to the subject by addressing aspects of existing conjectures, establishing new results, and introducing methods for studying homological invariants.

The first research direction concerns the depth formula, namely the equality \[ \depth_R(M)+\depth_R(N)=\depth(R)+\depth_R(M\otimes_RN) \] where $M$ and $N$ are finitely generated $R$-modules. A classical result of Auslander \cite{Aus} shows that the depth formula holds provided that either $M$ or $N$ has finite …

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 …

Categories, Homology And Sheaves For Hypergraphs, Robert Green

Electronic Theses & Dissertations (2024 - present)

Hypergraphs are a prominent tool for representing networks with connections among three or more entities. There is an inherent flexibility that allows hypergraphs to more naturally represent certain types of networks than graphs or simplicial complexes can on their own. This flexibility, however, comes at a cost, as there is a zoo of various categories and homology theories that are applicable to hypergraphs. The first chapter of this dissertation explores various categorical perspectives on hypergraphs, focusing on what the natural notion of morphism between hypergraphs should be. It also contains an exploration of the functoriality of vertex-edge duality in these …

Computational Insights Into Orthotropic Fracture: Crack-Tip Fields In Strain-Limiting Materials Under Non-Uniform Loads, Saugata Ghosh, Dambaru Bhatta, S. M. Mallikarjunaiah

School of Mathematical & Statistical Sciences Faculty Publications

A finite element framework is presented for analyzing crack-tip phenomena in transversely isotropic, strain-limiting elastic materials. Mechanical response is characterized by an algebraically nonlinear constitutive model, relating stress to linearized strain. Non-physical strain singularities at the crack apex are mitigated, ensuring bounded strain magnitudes. This methodology significantly advances boundary value problem (BVP) formulation, especially for first-order approximate theories. For a transversely isotropic elastic solid with a crack, the governing equilibrium equation, derived from linear momentum balance and the nonlinear constitutive model, is reduced to a second-order, vector-valued, quasi-linear elliptic BVP. This BVP is solved using a robust numerical scheme combining …

Strong Q-Analogues For Values Of The Dirichlet Beta Function, Ankush Goswami, Tim Huber

School of Mathematical & Statistical Sciences Faculty Publications

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as 𝑞→1− and explicit evaluations at CM points for |𝑞|< 1. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

Asymptotics Of Discrete Convolution Powers And Applications To Difference Schemes, Pedro Henrique Alves Silva Dos Santos

Honors Theses

In this thesis we provide Gaussian Estimates and Local Limit Theorems describing the asymptotic behavior of convolution powers of a class of complex-valued functions on $\mathbb{Z}^d$. Convolution powers arise naturally in the study of partial differential equations, as well as in random walks in probability theory. In particular, they are connected to the stability theory of difference schemes used to approximate solutions to partial differential equations. We take inspiration from the work of Vidar Thomée on stability theory to restrict our attention to convolution powers of functions whose Fourier Transforms satisfy certain local expansions. We then combine the Cauchy Integral …

Enumerating Matrices Of A Given Order Over Finite Fields, Thor Richard Gabrielsen

Honors Theses

This thesis derives a generating function that describes the matrices of a given multiplicative order over finite fields of a given order assuming that the order of a field is not a divisor of the desired order of the matrix. This is done by using the power series known as the cycle index derived from the rational canonical form. This index can be factored, and using the fact that the minimal polynomial divides some polynomial of the form x^k − 1 we can show that the minimal polynomial is squarefree. This sufficiently restricts the form of the rational canonical form …

On Values Taken By Characters Of Finite Groups, Christopher William Herbig

Graduate Research Theses & Dissertations

The characters of finite groups are a fundamental tool for analyzing the structure of finite groups and are interesting in their own right. In particular, we are concerned with how certain assumptions on the values taken by characters impact the structure of finite groups and vice versa. After establishing notation and stating some fundamental results in Chapter 2, this work begins with a presentation of results in relation to a conjecture of N. N. Hung and P. H. Tiep on fields generated by character values. In particular, we have found a large family of counterexamples to the conjecture by taking …

The Effects Of Problem-Based Learning On Mathematical Creativity And Self-Efficacy Of High School Students, Margaret Remus

Graduate Research Theses & Dissertations

Mathematical creativity is recognized as an important component of mathematics education; however, there is limited understanding of how instructional approaches can effectively support this skill among high school students, particularly through problem-based learning (PBL). This study examined the effects of problem-based learning on mathematical creativity among high school students.

This study employed a mixed-methods approach using an alternating treatment design with two groups of participants. Group 1 received the PBL intervention followed by a control condition, while Group 2 received the control condition followed by the PBL intervention. Quantitative data were collected through a mathematical creativity test measuring flexibility, fluency, …

The Spacetime Finite Element Method For Investigations Into Physics Ghost Systems And Time Parallel Preconditioning, Jax Wysong

Electronic Theses and Dissertations

This work operates on two fronts, focusing on interesting physical phenomena before turning our attention to an interesting numerical math problem. First, using the spacetime finite element method (FEM), we investigate a PDE system consisting of two Klein Gordon equations, which are coupled nonlinearly through the potential energy. The system contains a ghost (negative kinetic energy term). Systems such as these are generally deemed physically unstable, resulting in infinite energy in finite time. However, recent work has shown that this is not always the case. We investigate multiple scenarios arising from different initial conditions to characterize if/when a ghost system …

Entropic Dynamics Approach To The Classical Limit Of Quantum Mechanics: Decoupling Of The Center Of Mass Motion For A Mesoscopic Particle, Fatimah Judayba

Electronic Theses & Dissertations (2024 - present)

In the Entropic Dynamics (ED) approach, quantum mechanics is derived from the principles of entropic inference and information geometry. The ED approach differs from other interpretations by making a clear commitment to distinguishing which variables are ontic (real) and which are epistemic. The classical limit for the center of mass (CM) coordinate is achieved for a large number of particles, M →∞, while Planck’s constant ℏ remains finite. Typically, the emergence of the classical limit requires decoherence through interactions with the external environment. In this work, we investigate whether the classical behavior of the CM coordinate in a mesoscopic system …

On Quantum Processes And The Epistemic Constraints, Varun Immanuel Premkumar Immanuel

Electronic Theses & Dissertations (2024 - present)

This doctoral dissertation on the foundations of quantum theory tells the story of a conceptual protagonist I have called “Epistemic Constraint.” Here, epistemic constraints are the definite, intersubjectively agreeable, ordinary-language conditions under which experiments are described.

The usual formulation of the quantum measurement problem, which I call the Schrodingerian measurement problem, has the structure of an anomaly: if we take quantum theory at face value, we expect no definite values, and yet we see definite values in experiments. The responses to this problem have been either to solve it or to dissolve it. These responses, which have taken the form …

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 …