Mathematics Commons^™

A Geometric Model For Syzygies Over 2-Calabi–Yau Tilted Algebras Ii, Ralf Schiffler, Khrystyna Serhiyenko

Mathematics Faculty Publications

In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra B, we construct a polygon S with a checkerboard pattern in its interior, which defines a category Diag(S). The indecomposable objects of Diag(S) are the 2-diagonals in S, and its morphisms are certain pivoting …

Steklov Eigenvalue Problems On Nearly Spherical And Annular Domains, Nathan Philip Schroeder

CGU Theses & Dissertations

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in d dimensions where d is any given positive integer. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we …

The Law Of The Iterated Logarithm For Lp-Norms Of Kernel Estimators Of Cumulative Distribution Functions, Fuxia Cheng

Faculty Publications – Mathematics

In this paper, we consider the strong convergence of Lp-norms (p ≥ 1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

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 …

An Approach To Multidimensional Discrete Generating Series, Svetlana S. Akhtamova, Tom Cuchta, Alexander P. Lyapin

Mathematics Faculty Research

We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case.

Solutions To The Kaluza-Klein Field Equations, Abel Eshete

All Graduate Theses, Dissertations, and Other Capstone Projects

This Alternate Paper Plan explores Kaluza-Klein theory, a multidimensional framework designed to unify Einstein’s gravitational field theory and Maxwell’s electromagnetic field theory. The objectives of this research can be summarized in two key areas: The first objective is to present a comprehensive introduction to the compactified Kaluza-Klein theory. The second aim involves the application of differential geometry, specifically E ́lie Cartan’s tetrad formalism, to derive exact solutions in two distinct scenarios: a. A Levi-Civita spacetime, b. A general spherical system. Furthermore, Lagrangian and Hamiltonian formalism are utilized to define stability conditions and describe gravitational lensing and Precession of Perihelion within …

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 …

Orthogonal Single-View And Multi-View Feature Selection Models Via Spectral Theory Based Methods, Zachary Chairez

Mathematics Dissertations - Archive

As the digital world continues to grow in the age of big data, there beckons a need for efficient and robust methods for data exploration. Under the umbrella of machine learning, feature selection positions itself as a fruitful approach that uncovers buried truths and illuminates important features of data, all while minimizing long-term storage requirements. In this work, we introduce a set of novel single-view and multi-view supervised feature selection models which are embedded with orthogonality constraints to maintain data's structural integrity while confining the optimal solution's search space.

Taking advantage of the underlying framework of these types of models, …

A Recursive Framework For Evaluating Moments Using Zero-Suppressed Binary Decision Diagrams, Brian Godwin Lim, Renzo Roel P. Tan, Jun Kawahara, Shin Ichi Minato, Kazushi Ikeda

Quantitative Methods and Information Technology Faculty Publications

The zero-suppressed binary decision diagram (ZDD) is a compact data structure widely used for the efficient representation of families of sparse subsets. Its inherent recursive structure also facilitates easy diagram manipulation and family operations. Practical applications generally fall under discrete optimization, such as combinatorial problems and graph theory. Given its utility, summarizing the subsets represented in the diagram using key metrics is of great value as this provides valuable insights into the characteristics of the family. The paper proposes a recursive algorithm to extract information on moments from families represented as a ZDD. Given a value for every element in …

Testing Informativeness Of Covariate-Induced Group Sizes In Clustered Data, Hasika K. Wickrama Senevirathne, Sandipan Duttta

Mathematics & Statistics Faculty Publications

Clustered data are a special type of correlated data where units within a cluster are correlated while units between different clusters are independent. The number of units in a cluster can be associated with that cluster’s outcome. This is called the informative cluster size (ICS), which is known to impact clustered data inference. However, when comparing the outcomes from multiple groups of units in clustered data, investigating ICS may not be enough. This is because the number of units belonging to a particular group in a cluster can be associated with the outcome from that group in that cluster, leading …

Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan

Mathematics & Statistics Faculty Publications

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …

Refracting Ufos: Solutions For Fermi Questions, May 2024, John Adam

Mathematics & Statistics Faculty Publications

This article, titled "Refracting UFOs: Solutions for Fermi Questions, May 2024," recounts the author's personal experience as a teenager observing a UFO sighting near their home in England. The author used their telescope to observe the object, which turned out to be lights from a distant airplane flying in a circular holding pattern. The article then presents two questions related to the observed phenomenon, including estimating the diameter of the aircraft's path and the banking angle of the aircraft at its speed. The article concludes with a solution to estimating the distance to the aircraft. The information is presented objectively …

Refracting Ufos, John Adam

Mathematics & Statistics Faculty Publications

Question 2: Assuming θ ≈ 2° = π/90 rad, estimate the distance R to the aircraft. Assume that R ≫ d. Question 1: Suppose that the observed “period of oscillation” is 1 min, and the speed of the aircraft is 200 km/h. Estimate the diameter d of the aircraft path (assuming that it is circular). Estimate the banking angle of the aircraft at this speed. While still a teenager (and aspiring astronomer), I became interested in the subject of UFOs (more appropriately named UAPs—see Ref. 1). The region in which I lived (50 km west of London) was in the …

Sky Smiley Face: Solutions For Fermi Questions, November 2024, John Adam

Mathematics & Statistics Faculty Publications

Question 2(a): By rotating Fig. 2 counterclockwise by 90°, explain why circumhorizontal arcs (CHAs) [Figs. 1(b) and (c)] form (under favorable conditions) only when the solar altitude exceeds 58° of arc. (In this case, the incoming ray enters a vertical face of an ice crystal and exits via the lower horizontal face.)Question 2(b): By considering the tilt of Earth’s imaginary N–S axis toward the plane of its orbit around the Sun (∼23.5°), explain why in the northern hemisphere CHAs cannot be seen above about latitude 56° N.

The History And Basics Of Fourier Series And Applications, Syafino Yunalfian

A with Honors Projects

This research paper discusses the basics of the Fourier series, how to construct the series, and a brief explanation of Fourier Transform.

Exploring Sigmoidal Bounded Confidence Models With Mean Field Methods, Tian Dong

HMC Senior Theses

Mathematicians use models of opinion dynamics to describe how opinions in a group of people change over time, which can yield insight into mechanisms behind phenomena like polarization and consensus. In these models, mathematicians represent the community as a graph, where nodes represent agents and edges represent possible interactions. Opinion updates are modeled with a system of differential equations (ODEs). Our work focuses on the sigmoidal bounded confidence model (SBCM), where agents update their opinion toward a weighted average of their neighbors' opinions by weighting similar opinions more heavily. Using tools developed in physics (mean-field theory), we derive a continuity …

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 $\mu \in \mathbb{Z}$.

A Study On The Correlation Between A Star's Rayleigh-Taylor Characteristic Timescale And Stellar Wind Activity, Fiona Klett

Undergraduate Journal of Mathematical Modeling: One + Two

In this paper, we investigate the correlations between a star's internal dynamics due to the Rayleigh-Taylor instability and episodes of stellar wind activity, using both a theoretical model and observational data from the NOAA.% \cite{NOAA}. Besides its relevance as an astrophysics problem, this study is also informative for models of climate change which include secular perturbations in the Sun's internal dynamics, as a potential source of solar activity variability.

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 …