Mathematics Commons^™

Approximations Of Koopman Operator Semigroups, Ibrahem Al Jabea

LSU Doctoral Dissertations

The main purpose of this dissertation is to study approximation methods for nonlinear systems using Bernhard Koopman's Global Linearization Method or Sophus Lie's method of continuous transformation groups. This approach enables the application of linear semigroup methods to a nonlinear system by focusing on the dynamics of the observables of the states, rather than directly studying the dynamics of the states. In this dissertation, we studied the pointwise semigroup and introduce the modified space $C_m(\Omega)$ and the modified Koopman-Lie semigroups. We use a splitting operator and outline a systematic approach for approximating the pointwise Koopman-Lie semigroup flows \begin{equation*} t\to T(t)g(x) …

Construction Of Stock Portfolios By Machine Learning Methods, Alfan Gehad Abulehia

Theses

We study the theory and application of two machine learning (ML) algorithms: Long-Short Term Memory Network (LSTM) and Random Forest. The study begins by providing an overview of mathematical foundation, highlighting the significance of mathematics in machine learning. We study and explain the mathematical details involved in these two algorithms. We also study closely related subjects which include but not limited to gradient descent, automatic differentiation, and recurrent neural network. The main ideas behind these topics form the foundation of any ML algorithms.

As an application of the ML algorithms, we focus on the U.S. stock market. We build stock …

Quadratic Stochastic Processes: Algebraic Structures And Their Applications, Taimun Saleh Qaisar

Dissertations

This research focuses on the algebraic structures of the Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠). In this work, we first study 𝜉𝑎- Quadratic Stochastic Operators (𝑄𝑆𝑂𝑠) linked to the partition P3. We simultaneously discuss the dynamics of the obtained 𝑄𝑆𝑂𝑠. Moreover, algebraic structure of the associated genetic algebra is studied. Further, we build Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠) using the given Markov processes. Consequently, we obtain an ordinary differential equation for the resultant Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠). Besides, we apply the solution of this ordinary differential equation for the option pricing problem. Thereafter, we construct Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠) in three-dimensional space by …

Stability Analysis In The Twist-Bend Nematic Liquid Crystal Model, Zhenqiang Li

Mathematics & Statistics Theses & Dissertations

The recently discovered twist-bend nematic liquid crystal (LC) phase is characterized by a nanoscale helical modulation of the nematic director n, forming a conical helix along the z-axis at an oblique angle θ. While many models assume a constant cone angle and equal elastic constants K11 = K22 = K33, this dissertation removes both assumptions by considering a fully anisotropic elastic energy with K11 ≠ K22 ≠ K33, and allowing θ to vary spatially. We analyze the stability of this system under frustrated and free boundary conditions using variational methods. …

The Tate Resolution Over A Complete Intersection Ring, Kolton O'Neal

Honors Program: Senior Projects (Public)

Given a Noetherian commutative ring R and an ideal I ⊆ R, Tate provided a construction in [5] to produce a DG R-algebra that is also a free resolution for R/I. In this work, we review free resolutions and DG algebras, describe Tate’s construction, and present a proof of a result from Tate’s paper about his construction when a regular sequence is involved. Specifically, this result is that it only takes two steps of Tate’s construction to resolve a characteristic 0 field k over k[[x1, . . . , xn]]/(f1, . . . , fc), where f1, . . . …

Transient Thermal Energy Harvesting At A Single Temperature Using Nonlinearity, Tamzeed B. Amin, James M. Mangum, Md R. Kabir, Syed M. Rahman, Pradeep Ashaduzzaman, Pradeep Kumar, Luis L. Bonilla, Paul M. Thibado

Physics Faculty Publications and Presentations

The authors present an in-depth theoretical study of two nonlinear circuits capable of transient thermal energy harvesting at one temperature. The first circuit has a storage capacitor and diode connected in series. The second circuit has three storage capacitors, and two diodes arranged for full wave rectification. The authors solve both Ito-Langevin and Fokker-Planck equations for both circuits using a large parameter space including capacitance values and diode quality. Surprisingly, using diodes one can harvest thermal energy at a single temperature by charging capacitors. However, this is a transient phenomenon. In equilibrium, the capacitor charge is zero, and this solution …

Navigating Geometry With Interactive Lessons: A Collection Of Resources In Geogebra That Address Frequently Held Misconceptions, Sara Vance

Honors Program: Senior Projects (Public)

Nearly every high school student in the United States is required to take a geometry class, and for many, it’s unlike any math class they’ve taken before. Additionally, there are certain topics with which past and present geometry students alike tend to struggle. This project was created to address these issues; my goal is that these materials will help students better understand these topics and clear up their frequently held misconceptions about geometry. This project was supported by interviews I conducted with veteran high school geometry teachers with over 75 years of teaching math between them.

This collection of interactive …

Unavoidable Immersions And Topological Minors Of K-Edge-Connected Graphs, Brittian Qualls

LSU Doctoral Dissertations

In this work, we present results concerning the unavoidable structures in large and infinite k-edge-connected graphs. These results are inspired by the classical result of Ramsey, who proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either K_r or $\overline{K_r}$. We consider different graph containment relations, focusing primarily on the immersion relation. In the case of finite graphs, we provide the unavoidable immersions of 4-edge-connected graphs and prove that linear edge-connectivity suffices to immerse the graph C_t,r.

This dissertation also considers infinite graphs. We present the …

An In-Depth Analysis Of The Bellarmine University Student Athlete Experience, Mattingly E. Spalding

Undergraduate Theses

The goal of this thesis is to improve the student athlete experience at Bellarmine University through direct feedback from our current student athletes. By defining what parts of the student athlete experience they value most, Bellarmine can reflect on the support currently provided in these areas. Additionally, if it is concluded through the response that high valued areas are not being satisfied, Bellarmine can work to improve the support in these areas. Or, if there is abundant support in a low-valued area, Bellarmine could think to shift resources into a more needed concentration. Through the design sample, data was accumulated …

Circles Are So Euclidean: Exploring Convex Bodies, Norms, Symmetries, And Isometries Via The Minkowski Functional, Fanni Kertesz

Undergraduate Theses

In a real vector space, a symmetric convex body containing the origin and compact under the Euclidean norm uniquely defines a norm via the Minkowski functional, where the closed unit ball of this norm is the convex body. This establishes a one-to-one correspondence between convex bodies and norms. The norm derived from the convex body allows for the definition of a metric, enabling the study of isometries within the space. However, the symmetries of the convex body itself do not completely determine the isometries of the associated Minkowski space. Instead, the symmetries of the most symmetric shape a convex body …

Robust Spacecraft Autonomy For Deep Space Exploration In Special Euclidean Group Se(3), Matthew Wittal

Doctoral Dissertations and Master's Theses

Over the past half-century, humanity has gained extensive experience conducting manned spaceflight near Earth. Arguably, "near Earth" could even include the Moon — the most distant destination humans have reached. However, "near" in this work primarily refers low Earth orbit (LEO). One could argue that we have not truly left Earth since the Apollo, as spacecraft in some LEOs remain subject to atmospheric drag thus emphasizing their continued connection to Earth's immediate environment. Reflecting on this, it becomes clear that humanity has largely remained bound to Earth’s immediate vicinity since the Apollo missions reached the Moon. However, that is set …

On Unavoidable Infinite Hypergraphs, Samuel Weiner

LSU Doctoral Dissertations

Ramsey's Theorem states that every infinite graph contains either K_∞ or $\overline{K_∞}$ as an induced subgraph. For this reason, K_∞ and $\overline{K_∞}$ are often referred to as the unavoidable infinite graphs. Many similar results have characterized the unavoidable members for various classes of infinite graphs; perhaps the most notable of these is König's Infinity Lemma, which states that every infinite, connected, locally finite graph contains a ray as an induced subgraph. From these findings, one can easily deduce that every infinite, connected graph contains an infinite clique, star, or ray as an induced subgraph; …

On Regularity And Convergence Of Solutions To The Boltzmann-Enskog Equations, Christian Ennis

LSU Doctoral Dissertations

The Boltzmann equation describes the time evolution of the density function in position-velocity space for a classical particle subjected to possible collisions by other particles in a diluted gas that expands in vacuum for a given initial distribution. While many authors have studied the probabilistic interpretation of the spatially homogeneous Boltzmann equation, there is a dearth of articles on the stochastic framework of the full (that is, spatially inhomogeneous) Boltzmann equation. In this thesis, we examine a stochastic process, developed by S. Albevario, B. Ruediger, and P. Sundar, whose law is a weak solution to a mollified Boltzmann equation. This …

Fixed Hooks In Arbitrary Columns Of Partitions, Philip Cuthbertson

Michigan Tech Publications

In a paper by the author, Hemmer, Hopkins, and Keith, the concept of a fixed point in a sequence was applied to the sequence of first column hook lengths of a partition. In this paper, we generalize this notion to fixed hook lengths in an arbitrary column of a partition. We establish combinatorial connections between these fixed hooks and colored partitions that have interesting gap and mex-like conditions. Additionally, we obtain several generating functions for hook lengths of a given fixedness by hook length or part size in unrestricted partitions, as well as some classical restrictions such as odd and …

Explicit Modularity And Moments For Certain Hypergeometric Character Sums, Brian Grove

LSU Doctoral Dissertations

A great deal of progress in number theory throughout history has been motivated by trying to solve equations. One of the most famous challenges is to show there are no positive integer solutions to $x^{n}+y^{n} = z^{n}$ for $n > 2$, posed by Fermat around 1637. Special cases, such as the $n = 3$ and $n = 4$ cases, can be established using various algebraic manipulations. However, a general solution was elusive until the late 1990s when the combined work of Wiles \cite{Wiles} and Taylor--Wiles \cite{TaylorWiles} give a full proof.

One of the key insights used in proving Fermat's conjecture involves …

Stability Criteria For The Generalized El Borhamy-Rashad-Sobhy Equation, Mohamed El-Borhamy Assoc.Prof, Essam Eddin Rashad Prof., Fathi Mousa Dr., Mai Hamouda

Journal of Engineering Research

This article is concerned with the study of stability criteria for one of the generalization form of El Borhamy-Rashad Sobhy equation, which is a linear second-order ordinary differential equation with periodically time varying coefficients. Many engineering applications can be represented by this generalization, for instance, including the modeling of RLC circuit with time varying inductance, resistance and capacitance, and the vibration of a stretched string, whose mass per unit length is periodic, under a periodic motion. An approximate solution is derived by using the Wenhl-Kramers-Brillonin (WKB) approach. A method of constructing Liapunov function is employed to derive extra conditions for …

New View Of Some Propositions On Hesitant Fuzzy Set And Its Application In Selecting The Best Person In Any Job, Manar Mohamed Omran Dr., Arafa A. Nasef A.Dr, Reham Abd El-Aziz Abo Khadra Dr., Mahmoud Arafa Nasef Dr.

Journal of Engineering Research

An essential part of uncertainty is played by the hesitant fuzzy set (HFS). So, it can be utilized when making decisions. The suggested use of HFS to choose the best candidate for any post is thoroughly discussed and introduces new HFS concepts

Stochastic Analysis And White Noise Calculus Of Nonlinear Wave Equations With Application To Laser Generation And Propagation, Sivaguru S. Sritharan, Saba Mudaliar

Journal of Stochastic Analysis

No abstract provided.

Fractional Calculus Approach For Learning Unknown Dynamic System, Haowei(Alice) Chen

USF Tampa Graduate Theses and Dissertations

The rapidly growing interest in data-driven modeling and fractional-order systems highlights a keychallenge in modern science and engineering: capturing long-memory effects and nonlinear behaviors in real-world dynamical processes. Conventional integer-order methods, while powerful, often overlook the history-dependent nature of many phenomena—ranging from viscoelastic materials to anomalous diffusion and hereditary feedback systems. This dissertation addresses that gap by blending fractional calculus with cutting-edge machine learning to create robust, memory-aware modeling and control frameworks.

Beginning with an extension of Dynamic Mode Decomposition (DMD) to fractional-order systems, we introduce Fractional Dynamic Mode Decomposition (F-DMD) as a data-driven tool that leverages Mittag- Leffler functions …

Critical Appraisal Of Multidrug Therapy In The Ambulatory Management Of Patients With Covid-19 And Hypoxemia. Part Ii: Causal Inference Using The Bradford Hill Criteria, Eleftherios Gkioulekas, Peter A. Mccullough, Colleen Aldous

School of Mathematical & Statistical Sciences Faculty Publications

We continue the critical appraisal of three published case series of 119 COVID-19 patients with hypoxemia, treated in the United States, Zimbabwe, and Nigeria with similar ivermectin-based multidrug treatments, to assess the available evidence supporting a causal relationship between treatment and reduction in hospitalizations and mortality. A narrative review was conducted to assess the Bradford Hill criteria for a causal association. We used a previously proposed refinement of the Bradford Hill criteria that reorganized them into three categories of direct, mechanistic, and parallel evidence. The efficacy of the two most aggressive ivermectin-based multidrug protocols is supported by the Bradford Hill …