Physical Sciences and Mathematics Commons^™

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

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

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 …

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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

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

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

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 …

Mechanism Of Hairpin Vortex Formation By Liutex, Yifei Yu

Mathematics Dissertations

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

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 …

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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

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 …

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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.

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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 …

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

Mathematics Dissertations

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 …

Exponential Tensor Modules, Khoa Hoang Nguyen

Mathematics Dissertations

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

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 …

Liutex Analysis By Pod And Dmd In Turbulent Flow After/ In Micro Vortex Generator, Xuan My Trieu

Mathematics Dissertations

Although vortex has been studied more than one hundred year, we still have not had universally accepted definition. A few well-known vortex identification methods are introduced ��,∆, ��2,������ criteria to identify coherent vortex structures the last three decades. A new Omega vortex identification method, which is defined as a ratio of the vorticity tensor norm squared over the sum of the vorticity tensor norm squared and deformation norm squared, was proposed in 2016. Two year later, the new vortex vector named Liutex (previously called Rotex) was proposed by Liu et al. with direction of local rotation axis (an eigenvector of …

Liutex-Based Vortex Identification Methods And Their Application In Dns Study Of Flat Plate Boundary Layer Transition, Pushpa Shrestha

Mathematics Dissertations

Vortices are intuitively known as the rotational motion of fluid particles, however, unambiguous and universally accepted methods of vortex definition and identification are not available to date in the literature. First-generation vortex identification methods, also known as vorticity-based vortex criterion, were first proposed by Helmholtz. But these methods have their own problems. These methods have a shear contamination problem, and these methods did not accurately show the direction of fluid rotation. So, to overcome these problems, eigenvalues based second-generation vortex identification methods like Q, Δ, λ_(2 ), λ_(ci ), and Ω have been proposed. Most of these second-generation methods are …

A Novel Supervised Dimensionality Reduction Method: Integrating Pca With Svm, Faezeh Soleimani

Mathematics Dissertations

Data curation and storage methods have changed over the past few decades with the use of new technologies, and gathering data on a huge number of features (dimensions) is now very common among diverse scientific and engineering fields. Prior to classification or regression, dimensionality reduction is necessary to eliminate irrelevant features and to deal with data with high dimensions. A number of numerical methods have already been proposed to reduce the dimension of data, for example, Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Supervised Principal Component Analysis (SPCA). In this dissertation, we will introduce a novel way of …

A Study On Approximations Of Totally Acyclic Complexes, Tyler Dean Anway

Mathematics Dissertations

Let $R$ be a commutative local ring to which we associate the subcategory $\Ktac(R)$ of the homotopy category of $R$-complexes, consisting of totally acyclic complexes. Further suppose there exists a surjection of Gorenstein local rings $Q \xrightarrowdbl{\varphi} R$ such that $R$ can be viewed as a $Q$-module with finite projective dimension. Under these assumptions, Bergh, Jorgensen, and Moore define the notion of approximations of totally acyclic complexes. In this dissertation we make extensive use of these approximations and define several novel applications. In particular, we extend Auslander-Reiten theory from the category of $R$-modules over a Henselian Gorenstein ring and show …

On A Cubic Nonlinear Equation Model Arising In Shallow Water Theory, Osama Salameh Alkhazaleh

Mathematics Dissertations

The shallow water waves theory produces numerous integrable equations with cubic non- linearity as asymptotic models. We began our work by formally deriving a model equation for the free surface elevation η with higher-order terms from shallow water in the Euler equation for an incompressible fluid with the simplest bottom and surface conditions. This model equation is truncated at the order O(ε3,εμ) and contains higher-order terms, which are useful for deriving a class of unidirectional wave equations including cubic nonlinear terms. Next, we derived an equation with cubic nonlinearity as the asymptotic method from the classical shallow-water theory by employing …

Liutex And Statistical Analysis For Fluid Transition, Charles Matthew Nehemiah Nottage

Mathematics Dissertations

A vortex can be intuitively recognized as the rotational swirling motion of the fluids. The fascination of this phenomenon brought about many years of research to define, classify, and identify the vortical structure. Throughout the decades, many vortex identification methods were developed and can be characterized into three generations. The generational methods are vorticity-based, eigenvalue-based such as Q, ��_ci, and ��_2, and Liutex-based. Before the development of Liutex, there was no mathematical definition for vortex. Is Liutex superior to vorticity and the eigenvalue-based methods? Is the vorticity vector the local rotational axis? Should vorticity be considered vortex? In this dissertation, …

The Natural Middle Of A Complete Resolution, Rebekah J. Aduddell

Mathematics Dissertations

It is widely known that minimal free resolutions of a module over a complete intersection ring have nice patterns that arise in their Betti sequences. In the late 1990's Avramov, Gasharov and Peeva defined a new class of R-modules that would exhibit similar patterns in their free resolutions. In doing so, they additionally defined the notion of critical degree for an R-module, which serves as a “flag” for when such patterns arise in the module’s Betti sequence. The main purpose of this thesis is to present an extension of critical degree to the category of totally acyclic complexes, Ktac(R), where …

Compressive Deconvolution Of Mri Imaging Via ℓ1 − ℓ2 Regularization, Talon Johnson

Mathematics Dissertations

The evolution of technology has drastically impacted the imaging field, particularly magnetic resonance imaging (MRI). Compared to other imaging technologies, MRI offers multiple contrasting mechanisms to distinguish tissues and fat, is radiation-free, and provides anatomical and molecular information about the tissue in question. However, data acquisition times to produce those images require a patient to lie still for a relatively long time. Consequently, it may lead to the voluntary or involuntary movement of the patient due to discomfort. Combined with the underlying issue of inherent noise, MRI is often blurry and contains artifacts. Mathematically, one can describe this behavior as …

Mathematical Approach Of Liutex Core Line And Liutex Core Tube For Vortex Structure Visualization, Dalal Khalid B Almutairi

Mathematics Dissertations

During the past decades, many vortex identification methods have been published to present a clear definition and identification of the vortex. However, all these methods are failed to offer a unique identification method, and they also cannot answer the six essential issues for vortex identification methods, which are: 1) absolute strength, 2) relative strength, 3) rotational axis, 4) vortex core center location, 5) vortex core size, and 6) vortex boundary. In this work, two vortex identification methods, which are never affected by the threshold, will be proposed. Moreover, this study will address two critical questions: 1) Where is the rotational …

On Different Computational Aspects For Box-Cox Transformation Cure Rate Model, Pei Wang

Mathematics Dissertations

Cure rate modeling is an emerging area of research not only in biomedical science but also in other disciplines such as sociology, criminal justice, economics and engineering reliability. In the first part of this thesis, use of the wider class of generalized gamma distributions is proposed as the distribution of the lifetime for a particular transformation cure rate model, known as the Box-Cox transformation cure rate model. The maximum likelihood estimation of the Box-Cox transformation cure model parameters is studied through the calculated bias, mean square error and coverage probabilities of the asymptotic confidence intervals. The flexibilities of both generalized …