Mathematics Commons^™

The Waldo Dataset, Mary E. Koone, Rosie Kallie, Vassilis Athisos, Laurel S. Stvan

Computer Science and Engineering Datasets - Archive

Distinct from the task of predicting the author of a document (authorship attribution), we focus on addressing the issue of how to estimate the similarity between the written language styles of authors. To do so, we present a dataset of metadata derived by asking human annotators, who were presented with three documents, to identify which two were written by the same author and which was written by a different author. The dataset has over 400 such annotations, creating a companion to the Amazon Web Services (AWS) customer review dataset, laying the groundwork for crowdsourcing applications to other natural language processing …

Uncertainty Quantification, Propagation & Conjunction Assessment In Orbital Mechanics Using Generalized Polynomial Chaos Expansion & 2-Dimensional Conjunction Plane Analysis Techniques, Monalisa Karim

Mechanical and Aerospace Engineering Theses

Uncertainties, that are inherent to dynamic models, can be associated with state initial conditions, force modelling errors, navigation and actuation errors. In system modelling stochastic differential equations are used to represent dynamic phenomena with uncertainties, for which the solutions are probability density functions of quantities of interest characterizing the realization of the stochastic processes. In Polynomial Chaos Expansion (PCE) propagation, these solutions are represented as weighted sums of multivariate spectral polynomials that are functions of the input random variables. Generalized polynomial chaos expansion (gPC) is an extension to the original homogenous PCE which projects the random solution onto a basis …

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

Soft-Constrained Variants Of T-Distributed Stochastic Neighbor Embedding For Global Structure Preservation, Joseph A. Balderas

Mathematics Dissertations

Dimensionality reduction (DR) is a fundamental tool in data science and machine learning that transforms high-dimensional data into a low-dimensional representation while preserving important structural properties of the original data. Among modern DR methods, t-distributed stochastic neighbor embedding (t-SNE) has become one of the most widely used techniques for visualization due to its strong ability to preserve local neighborhood structure and produce visually separated clusters. However, despite its popularity, t-SNE is well known to struggle with preserving global structure of data, often producing embeddings in which distances between clusters and neighborhoods do not accurately reflect relationships in the high-dimensional space. …

Examining Student Understanding Of Series Convergence Using Script Writing, Eduardo Torres Manzanarez

Mathematics Dissertations - Archive

This exploratory study investigates second-semester calculus students’ understanding of series convergence using both nonscripting and scripting-based tasks. Researchers recognize that students encounter difficulties with learning series, including series convergence, and that there is a need for more meaningful tasks that aid students’ learning of series. Script writing, used mainly with prospective mathematics teachers, can be used to explore mathematical understandings. Thus, this study examines how script writing, in the form of a scripting task, elicits students’ understanding of series convergence in contrast to nonscripting-based tasks to determine the potential of script writing in assessing this student understanding. We examine students’ …

Self-Tor Persistence Of Modules Over Determinantal Rings, Tatheer F. Ajani

Mathematics Dissertations - Archive

Tor-persistence is the claim that Tor of a module with itself is only zero if the module has finite projective dimension. Work done by Avramov, Iyengar, Nasseh, Sather-Wagstaff, and various other authors have proved Tor-persistence of modules over certain rings. In this work, we will prove Tor-persistence for certain modules over determinantal rings, specifically for the hypersurface defined by the determinant of a generic matrix. We will then give an explicit proof that Tor^R_2(M,M) is never zero, that Tor^R_1(M,M)=0, and due to the periodicity of the given free resolution, our result can be extended to the entire complex, showing that …

Algebra Structures For The Koszul Homology Of Minimal Intersections, Kathryn A. Grebel

Mathematics Dissertations - Archive

The Koszul homology of a local ring is a powerful tool in commutative algebra as it provides information on the structure and properties of the ring. In this research, we explore the relationship between quotients of regular local rings and their Koszul homology algebra. One such relationship is detailed by the Tate-Assmus theorem, which asserts, in part, that a ring is a complete intersection if and only if the Koszul homology is generated by its degree 1 homology elements. An objective of this research is to examine and identify the properties of a minimal intersection and its Koszul homology algebra. …

Point Modules And Line Modules Of Certain Quadratic Quantum Projective Spaces, Jose E. Lozano

Mathematics Dissertations - Archive

During the past 36 years, some research in noncommutative algebra has been driven by attempts to classify AS-regular algebras of global dimension four. Such algebras are often considered to be noncommutative analogues of polynomial rings. In the 1980s, Artin, Tate, and Van den Bergh introduced a projective scheme that parametrizes the point modules over a graded algebra generated by elements of degree one. In 2002, Shelton and Vancliff introduced the concept of line scheme, which is a projective scheme that parametrizes line modules.

This dissertation is in two parts. In the first part, we consider a 1-parameter family of quadratic …

Optimal Control Frameworks For A Class Of Epidemiological And Oncological Models, Asma Ali H Alghamdi

Mathematics Dissertations - Archive

In this thesis, we employ optimal control frameworks in two distinct contexts: Human immunodeficiency virus (HIV) and esophageal cancer. For HIV, we introduce a comprehensive data-driven nonlinear optimization framework designed for personalized therapies. This framework utilizes a deterministic in-host nonlinear ordinary differential equation (ODE) model and formulates two optimization problems using individual patient data. The first problem focuses on estimating patient-specific parameters through constrained optimization, while the second problem determines optimal combination therapies to reduce viral load to undetectable levels. Several numerical experiments suggest that our framework can provide a robust and effective optimal dosages with lower toxicity levels to …

On Weyl Representations Of Gl(N), Amairani Hernandez Garcia

Mathematics Dissertations - Archive

This thesis studies generalized Laurent polynomial representations of the general linear Lie algebra. These representations arise naturally as representations over the Weyl algebra consisting of differential operators on $\mathbb{C}^n$. Our main result is an explicit description of the socle filtration of $P_{\mu} = \Span \{x^{\bf m} \: | \: {\bf m} \in \mathbb Z^{n}, \: | \bf{m} | \: = \mu \}$ for $\mu \in \mathbb{Z}$.

Manifold Learning In Robotics: A Tutorial And Survey, Marcus Hawkins

Computer Science and Engineering Theses - Archive

In this article, we hope to represent the current state of the art of manifold learning in an understandable and approachable way. The authors will present a general overview core algorithms associated with linear and nonlinear dimensionality reduction techniques, give rudimentary definitions from differential geometry, and tenets of robotic perception, manipulation and path planning. Some of the historical applications of these algorithms will be presented, as well as conjectures about future uses, through examples from peer-reviewed journals.

Exploring The Role Of Undergraduate And Graduate Real Analysis Experiences In The Mathematical Trajectories Of Women Mathematicians From Historically Disenfranchised Groups, Te'a Riley

Mathematics Dissertations - Archive

This phenomenological study examines the role of undergraduate and graduate Real Analysis courses in shaping the mathematical trajectories of seven women Ph.D. mathematicians from groups historically disenfranchised in mathematics.Qualitative analysis of interviews explores various aspects of their development as mathematicians with a focus on their experiences in Real Analysis. This study applies Ryan & Deci’s (1985) Self-Determination Theory's Basic Psychological Need Theory and Critical Race Theory to analyze the trajectories of the participants. The research explores how the fulfillment of basic psychological needs in their Real Analysis courses may have influenced their academic and professional journeys. The basic psychological need …

Calculus Students’ Problem-Solving Strategies On Related Rates Of Change Problems Appearing In Online Versus Paper-And-Pencil Format, Tyson Bailey

Mathematics Dissertations - Archive

This study explores first-semester calculus students’ use of mathematical problem-solving strategies while working related rates of change problems in both an online homework format and a traditional pencil-paper format. We address two research questions: (1) How do students’ mathematical problem-solving strategies when working online homework on related rates of change problems compare with their problem-solving strategies when working paper-and-pencil homework related rates of change problems? (2) What influence does the ‘view an example’ feature in online homework have on a student’s problem-solving strategies when working an online RRC homework problem? Using scores on free-response midterm exam problems on related rates …

The Direct And Inverse Scattering Problems For The Third-Order Operator, Ivan Toledo

Mathematics Dissertations - Archive

We consider the full-line direct and inverse scattering problems for the third-order ordinary differential equation containing two potentials decaying sufficiently fast at infinity. The direct scattering problem consists of the determination of the scattering data set when the two potentials are known. The scattering data set is made up of the corresponding scattering coefficients and the bound-state information. On the other hand, the inverse scattering problem involves the recovery of the two potentials when the scattering data set is available. We formulate the inverse scattering problem via a related Riemann--Hilbert problem on the complex plane. We describe the recovery of …

A Novel Regularized Orthonormalized Partial Least Squares Model For Multi-View Learning, Ce Bian

Mathematics Dissertations - Archive

Over the past few years, the size of data dimensions or features has been increasing in various fields of science and engineering, owing to the rapid pace of data collection and the development of more advanced storage methods. However, to handle high-dimensional data, dimensionality reduction is essential before performing classification or regression tasks to eliminate noisy features. There are several numerical methods available for reducing data dimensionality, such as Canonical Correlation Analysis (CCA), Principal Component Analysis (PCA), and Linear Discriminant Analysis (LDA). While these methods offer valuable approaches to data dimensionality reduction, they do come with certain limitations. CCA, for …

Optimal Control Frameworks For Modeling Dynamics And Androgen Deprivation Therapies In Prostate Cancer, Hussein Ed Duweh

Mathematics Dissertations - Archive

In this work, we present an optimal control approach for the assessment of treatments in prostate cancer. For this purpose, we use two different approaches, based on differential equations, to model the dynamics of prostate cancer. For the first approach, we use a system of ordinary differential equations (ODE) that model androgen-dependent and independent prostate cancer cell mechanisms. Given some synthetic patient data, we then performed a parameter estimation process by formulating an optimization problem to obtain the coefficients in this model. A second optimal control problem was formulated to obtain optimal androgen suppression therapies. A theoretical analysis of both …

Large Eddy Simulation By Using Wang’S Liutex-Based Subgrid Model, Vishwa Shah

Mathematics Dissertations - Archive

Turbulent flows and vortex structures in fluid dynamics have been captivating researchers for decades, owing to their intrinsic complexity and significance in various industrial and natural processes. Despite their fundamental importance, the definition and identification of vortices in turbulent flows continue to pose challenges, and to date, no universally accepted approach exists. This pursuit dates to the pioneering work of Hermann von Helmholtz in the 19th century, when the concept of vortices was first introduced. In 2019, Liu et al. introduced a novel physical quantity termed "Liutex" in scalar, vector, and tensor forms, providing a promising avenue for understanding and …

A Study In The Freeness Of Finitely Generated Anp-Modules Upon Restriction To Principal Subalgebras, Luke Manford Flattery

Mathematics Dissertations - Archive

We are interested in quantitative information on the freeness of modules over a truncated polynomial ring when restricting to subalgebras generated by a linear form. After investigating the structure of the truncated polynomial ring, subalgebras generated by a linear form, and corresponding vector spaces, we construct a generic representation and discuss its connection to a certain affine space. We quantify the abundance of freeness of modules using a certain variety called the rank variety. For any possible dimension we construct a module whose rank variety has that dimension. Finally, we define another variety, called the module variety, and show that …

Mechanism Of Hairpin Vortex Formation By Liutex, Yifei Yu

Mathematics Dissertations - Archive

Turbulence is still a mystery for human after more than one century’s development of fluid dynamics. Hairpin vortex formation is regarded as an essential process for a laminar flow transition to the turbulent flow. A new correct third generation vortex identification method, Liutex, was proposed in 2018, which can represent local rotation direction and reveal the local angular speed correctly. Using this powerful tool, the mechanism of hairpin vortex formation is re-examined. This dissertation (1) explains the mechanism of hairpin vortex formation by solving Orr-Sommerfeld equation using Chebyshev spectrum method (2) observes the DNS result of flat plate boundary layer …

Likelihood Inference For Flexible Cure Models With Interval Censored Data, Jodi Treszoks

Mathematics Dissertations - Archive

Models for survival data with a surviving fraction, known as cure rate models, play a vital role in survival analysis. Due to the improvement of intervening methodologies, some subjects are seen to be immune permanently. While cure rate models have been studied extensively in the recent literature with a standard assumption of right-censored data, in many applied settings, such as recidivism studies or medical studies where the event of interest is not immediately harmful, continuous observation of a subject is impracticable. We call lifetime data generated with discrete follow-up times as interval-censored. In this thesis, we extend several existing cure …

Academic Integration And Self-Regulation Strategies In Precalculus And Calculus I, Kyle Russell Turner

Mathematics Dissertations - Archive

This case study examines the self-regulation strategies and academic integration of first-semester undergraduate students enrolled in precalculus and first-semester calculus at a large urban university in the southwestern United States. I use a sequence of interviews to examine the relationship of mathematics self-efficacy and mathematics identity on students’ use of these strategies. Five interviews occurred during the Fall 2021 semester with three precalculus and six calculus students, and I distributed initial surveys to thirteen first-semester calculus sections and five precalculus sections. To analyze the data, I integrated frameworks from Zimmerman and Pons (1986) and Wolters (1998) on the self-regulation strategies …

Decomposition Of Modules And Tensor Products Over Principal Subalgebras Of Truncated Polynomial Rings, Kevin Steine Jr. Harris

Mathematics Dissertations - Archive

The topic of my dissertation is to investigate the behavior of modules and tensor products over a truncated polynomial ring with prime characteristic. This investigation utilizes principal subalgebras of the truncated polynomial ring as the main tool for studying these objects. Then, we investigate if these modules and their tensor products have a similar behavior when viewed over more general truncated polynomial rings. In particular, we aim to investigate the behavior of these objects when we replace principal subalgebras over a field with prime characteristic by hypersurfaces over a field with no characteristic restriction.

Abrading The Enigma Of The Wound Healing Process: Modeling The Inflammation, Proliferation, And Maturation Stage, Amanda Patrick

Mathematics Dissertations - Archive

Wound healing encompasses a group of processes categorized into overlapping stages known as the inflammation, proliferation, and maturation/remodeling stage. The dynamics of these processes are important in studying outcomes of wound care and determining factors that contribute to certain wound outcomes. A system of ordinary differential equations is constructed for the inflammation, proliferation, and remodeling stage. Parameter sets for this model are investigated based on output dynamics according to the literature and based on experimental data. A bifurcation analysis is conducted to determine sudden changes that can occur in the inflammation system. Fourier Amplitude Sensitivity Test (FAST) is implemented to …

On Some Problems In Sparse Hybrid Imaging, Non-Standard Finite Difference Methods, And Fokker-Planck Frameworks In Esophageal Cancer, Madhu Gupta

Mathematics Dissertations - Archive

In this thesis, we first discuss nonlinear optimization frameworks for the sparsity- based on nonlinear reconstruction of parameters in hybrid imaging modalities such as current density impedance imaging (CDII) and two-photon photoacoustic computed tomography (2P-PACT). The framework comprises minimizing an objective functional involving a least square fit and some regularization terms that promote sparsity patterns and enhance the edges to facilitate high contrast and resolution. Next, we show the construction and analysis of the second-order nonstandard finite difference methods (NSFD) scheme for theta methods and explicit Runge-Kutta method. Finally, we present an application of the NSFD scheme for Fokker-Planck (FP) …

Higher-Order Nonstandard Finite Difference Methods For Autonomous Differential Equations With Applications In Mathematical Ecology, Fawaz Karhan R Alalhareth

Mathematics Dissertations - Archive

Nonstandard finite difference (NSFD) methods have been widely used to numerically solve various problems in biology. In recent years, NSFD methods have been proposed that preserve essential properties of the solutions of general autonomous differential equations, such as positivity and elementary stability, among others. However, those methods are only of first-order accuracy. In the first part of this dissertation, we construct and analyze two second-order modified positive and elementary stable nonstandard (PESN) numerical methods for n-dimensional autonomous differential equations. The new PESN methods are generalized versions of the explicit Euler's method and second-order accurate, thereby improving the order of accuracy …

Performance Of Density Estimators In Additive Measurement Error Models Based On Right Censored Data, Hrishabh Khakurel

Mathematics Dissertations - Archive

In the deconvolution problem for right censored data, one is interested in estimating the density of a contaminated variable X when X satisfies Z= X+ E, where E is a measurement error with a known distribution, and the observable variable Z is right-censored. Zhu, Sun, Khakurel, and Wang (2021) applied the Inverse Probability of Censoring Weighted Average method and derived the estimators of the unknown density of X. In this study, we evaluate the performance of the density estimators both in theory and in simulation. We derive the theoretical upper bounds for Mean Squared Error (MSE) of the estimator and …

A Necessary And Sufficient Condition For The Asymptotic Normality Of The Quantile Estimator In The Deconvolution Problem, Jeremy R. Valdez

Mathematics Dissertations - Archive

In this study, we examine the estimation of a quantile function when we have n observations coming from the convolution model contaminated by additive measurement errors. Under certain assumptions, a kernel type deconvolution quantile estimator of the unknown quantile function is proposed. Moreover, we discuss the necessary and sufficient condition on the bandwidth in order to investigate the limiting distribution of the deconvolution kernel quantile estimator when the error terms follow either an ordinary smooth or super smooth distribution. A bootstrap approach is used to select the optimal bandwidth to construct approximate distribution free confidence bands for the quantile function …

Twisting Systems And Some Quantum P³S With Point Scheme A Rank-2 Quadric, Hung Viet Tran

Mathematics Dissertations - Archive

In 1996, J. J. Zhang introduced the concept of twisting a graded algebra by a twisting system, which generalizes the concept of twisting a graded algebra by an automorphism (the latter concept having been introduced in an article by M. Artin, J. Tate and M. Van den Bergh in 1991). Twisting using a twisting system is an equivalence relation and certain important algebraic properties of the original algebra are carried over to the twisted algebra. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few known examples of nontrivial twisting systems …

Exponential Tensor Modules, Khoa Hoang Nguyen

Mathematics Dissertations - Archive

Representation theory of Lie algebra of a finite dimensional reductive Lie algebra g is a long-standing problem. The ultimate goal is to classify all representations of g. However. the only case only case when a complete classification is obtained is the case of g = sl(2). Hence, it is natural to study certain categories of representations of g for which some finiteness conditions on the action of certain elements of g is enforced. In this thesis, we introduce a class of representations T (g, V, S) of sl(n + 1) of mixed tensor type. By varying the polynomial g, the …

Quantized Enveloping Superalgebra Of Type P, Saber Murad Ahmed

Mathematics Dissertations - Archive

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ attached to the Lie superalgebra $\mathfrak{p}_n$ of type P. The superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ is a quantization of a Lie bisuperalgebra structure on $\mathfrak{p}_n$ and we study some of its basic properties. We determine representations of the superalgebra $\mathfrak{U}_q\mathfrak{p}_n$ and derive its Drinfeld-Jimbo relations. We prove the triangular decomposition of $\mathfrak{U}_q\mathfrak{p}_n$ and introduce some preliminary results concerning the highest weight representation of $\mathfrak{U}_q\mathfrak{p}_n$. We also introduce the periplectic q-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q\mathfrak{p}_n$-module structure on $\mathbb{C}(n|n)^{\otimes \ell}$. Finally, we propose a definition for a new periplectic q-Schur …