Mathematics Commons^™

The Cohomology Of The Extended Morava Stabilizer Group, With Trivial Coefficients, At Large Primes, Mohammad Behzad Kang

Wayne State University Dissertations

The goal of this thesis is to calculate the cohomology of the extended height n Morava stabilizer group, with trivial coefficients, for all heights n and all primes p>>n. To do this, we construct a family of deformations – parametrized over an affine line and smooth away from a single point – of Ravenel's Lie algebra model for the Morava stabilizer group. The singular fiber of the resulting bundle of differential graded algebras is the Chevalley-Eilenberg DGA of Ravenel's Lie algebra model, while the smooth fibers have very understandable cohomology. From here, tools such as parallel transport, connections, and …

Ookami: An A64fx Computing Resource, A. C. Calder, E. Siegmann, C. Feldman, S. Chheda, Dennis C. Smolarski Sj, F. D. Swesty, A. Curtis, J. Dey, D. Carlson, B. Michalowicz, R. J. Harrison

Mathematics and Computer Science

We present a look at Ookami, a project providing community access to a testbed supercomputer with the ARM-based A64FX processors developed by a collaboration between RIKEN and Fujitsu and deployed in the Japanese supercomputer Fugaku. We provide an overview of the project and details of the hardware, and describe the user base and education/training program. We present highlights from previous performance studies of two astrophysical simulation codes and present a strong scaling study of a full 3D supernova simulation as an example of the the machine’s capability.

Preliminary Results Of Pythagorean N-Tuples, Cara Admiraal

Science University Research Symposium (SURS)

This presentation will introduce the idea of extending the Pythagorean Theorem in higher dimensions. First, I will highlight and recognize key patterns of primitive Pythagorean Triples by examining visual and algebraic representations. I will then present key findings and questions surrounding the idea of a Pythagorean quadruple, quintuple, and n-tuple. Lastly, I will propose different branches of exploration that will be researched in the coming months.

Examining Course Achievement In An Undergraduate Psychology Statistics Course Through The Lens Of Machine Learning Techniques, Sunny Nguyet Le

CGU Theses & Dissertations

The Introductory to Psychology Statistics course stands as a notable challenge for psychology majors, often acting as a gatekeeper course. This study embarks on two primary objectives using machine learning techniques: (1) to identify the determinants of overall course achievement, specifically course grade, and (2) to investigate the influence of statistics anxiety and statistics self-efficacy, when both are present, on overall course grade. Employing a machine-learning approach, both objectives were effectively addressed. The study involved the development of a self-reported questionnaire consisting of perceptions of statistics anxiety and statistics self-efficacy, along with other demographic and academic background variables. Conducted at …

Proof Of The Toponogov Conjecture On Complete Surfaces, Brendan Guilfoyle, Wilhelm Klingenberg

Department of Mathematics Publications

We prove a conjecture of Toponogov on complete convex planes, namely that such planes must contain an umbilic point, albeit at infinity. Our proof is indirect. It uses Fredholm regularity of an associated Riemann-Hilbert boundary value problem and an existence result for holomorphic discs with Lagrangian boundary conditions, both of which apply to a putative counterexample. Corollaries of the main theorem include a Hawking-Penrose singularity-type theorem, as well as the proof of a conjecture of Milnor’s from 1965 in the convex case.

Every Feasibly Computable Reals-To-Reals Function Is Feasibly Uniformly Continuous, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

It is known that every computable function is continuous; moreover, it is computably continuous in the sense that for every ε > 0, we can compute δ > 0 such that δ-close inputs lead to ε-close outputs. It is also known that not all functions which are, in principle, computable, can actually be computed: indeed, the computation sometimes requires more time than the lifetime of the Universe. A natural question is thus: can the above known result about computable continuity of computable functions be extended to the case when we limit ourselves to feasible computations? In this paper, we prove that this …

From Normal Distribution To What? How To Best Describe Distributions With Known Skewness, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we only have partial information about the probability distribution -- e.g., all we know is its few moments. In such situations, it is desirable to select one of the possible probability distributions. A natural way to select a distribution from a given class of distributions is the maximum entropy approach. For the case when we know the first two moments, this approach selects the normal distribution. However, when we also know the third central moment -- corresponding to skewness -- a direct application of this approach does not work. Instead, practitioners use several heuristic techniques, techniques …

Solid Angle Measure Approximation Methods For Polyhedral Cones, Allison Fitisone

Theses and Dissertations--Mathematics

Polyhedral cones are of interest in many fields, like geometry and optimization. A simple, yet fundamental question we may ask about a cone is how large it is. As cones are unbounded, we consider their solid angle measure: the proportion of space that they occupy. Beyond dimension three, definitive formulas for this measure are unknown. Consequently, devising methods to estimate this quantity is imperative. In this dissertation, we endeavor to enhance our understanding of solid angle measures and provide valuable insights into the efficacy of various approximation techniques.

Ribando and Aomoto independently discovered a Taylor series formula for solid angle …

Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, Danzhe Chen

CMC Senior Theses

This thesis provides a comprehensive exploration of lattice theory, emphasizing its dual significance in both theoretical mathematics and practical applications, particularly within computational complexity and cryptography. The study begins with an in-depth examination of the fundamental properties of lattices and progresses to intricate lattice-based problems such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems are analyzed for their computational depth and linked to the Subset Sum Problem (SSP) to highlight their critical roles in understanding computational hardness. The narrative then transitions to the practical applications of these theories in cryptography, evaluating the shift from …

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 …

Exploring Loss Functions In Machine Learning, Yujie Wang

CGU Theses & Dissertations

The loss function plays a critical role in machine learning. It is fundamental in training, evaluating, and optimizing machine learning models, directly impacting their effectiveness and efficiency in solving specific tasks. We explore three new loss functions and their applications. Softmax Cross-Entropy Loss, stands as a prevalent choice in neural network classification tasks. It treats all misclassifications uniformly. However, multi-class classification problems often have many semantically similar classes. We should expect that these semantically similar classes will have similar parameter vectors. We introduce a weighted loss function, the tree loss as a drop-in replacement for the cross entropy loss. The …

Modeling The Opioid Crisis In Virginia: A Differential Equations Model Assessing The Impact Of Medication-Assisted Treatment On The Addicted Population, Maniha Zehra Akram

Honors Theses

The opioid epidemic is prevalent in countless communities throughout the United States and has yet to be mitigated. Treatments for OUD (opioid use disorder) include Medication-Assisted Treatment (MAT) and treatment without medication (non-MAT), with the former being judged as more effective in terms of lower relapse rates, death rates, and criminal activity (U.S. Food & Drug Administration, 2023; SAMHSA, 2024). Motivated by the promising research on MAT, this paper models the relationship

between the treatment and addicted populations using a system of ordinary differential equations. In addition to producing closed-form equilibrium solutions, the model leads to the conclusion that expanding …

Platform-Independent Estimation Of Human Physiological Time From Single Blood Samples, Yitong Huang, Rosemary Braun

Mathematics Sciences: Faculty Publications

Abundant epidemiological evidence links circadian rhythms to human health, from heart disease to neurodegeneration. Accurate determination of an individual's circadian phase is critical for precision diagnostics and personalized timing of therapeutic interventions. To date, however, we still lack an assay for physiological time that is accurate, minimally burdensome to the patient, and readily generalizable to new data. Here, we present TimeMachine, an algorithm to predict the human circadian phase using gene expression in peripheral blood mononuclear cells from a single blood draw. Once trained on data from a single study, we validated the trained predictor against four independent datasets with …

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 …

Conventions, Definitions, Identities, And Other Useful Formulae, Robert A. Mcnees Iv

Physics: Faculty Publications and Other Works

As the name suggests, these notes contain a summary of important conventions, definitions, identities, and various formulas that I often refer to. They may prove useful for researchers working in General Relativity, Supergravity, String Theory, Cosmology, and related areas.

Estimated Glomerular Filtration Rate Slope And Risk Of Primary And Secondary Major Adverse Cardiovascular Events And Heart Failure Hospitalization In People With Type 2 Diabetes: An Analysis Of The Exscel Trial, Abderrahim Oulhaj, Faisal Aziz, Abubaker Suliman, Kathrin Eller, Rachid Bentoumi, John B. Buse, Wael Al Mahmeed, Dirk Von Lewinski, Ruth L. Coleman, Rury R. Holman, Harald Sourij

All Works

Aim: The decline in estimated glomerular filtration rate (eGFR), a significant predictor of cardiovascular disease (CVD), occurs heterogeneously in people with diabetes because of various risk factors. We investigated the role of eGFR decline in predicting CVD events in people with type 2 diabetes in both primary and secondary CVD prevention settings. Materials and Methods: Bayesian joint modelling of repeated measures of eGFR and time to CVD event was applied to the Exenatide Study of Cardiovascular Event Lowering (EXSCEL) trial to examine the association between the eGFR slope and the incidence of major adverse CV event/hospitalization for heart failure (MACE/hHF) …

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 …

Certain Incident Matrices For Lines Intersecting On Plane, Anawin Chongaumklang

Chulalongkorn University Theses and Dissertations (Chula ETD)

This work studies arrangements of lines on the plane, focusing on Fourier’s 17 lines problem, which involves constructing 101 intersection points with 17 lines. We use code notation and incident matrices to classify possible configurations and develop algorithms to determine which codes are drawable. Special attention is given to the role of multiple points and parallel families in these arrangements. Out of 924 possible codes, 658 are confirmed drawable using a combination of theoretical analysis and computer verification. For the remaining cases, new lemmas are proposed to construct additional intersection points. Our findings offer both theoretical insights and practical algorithms …

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 …

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 …

Quantifying Non-Primary Dna Formations Through Mechanical And Geometric Models, Sonia E. Teodorescu

Undergraduate Journal of Mathematical Modeling: One + Two

In this article, mechanical and geometric models for DNA strains (regarded as a helical structure in 3 dimensions, embedded into surfaces of various shapes (straight or curved cylinders, spheres, or projected into planes), are analyzed in order to obtain parameter estimates for DNA characteristics which can be used to detect the formation of secondary and tertiary formations in the presence of disorder. The models allow for the explicit representation of the DNA shape on constrained geometries and can therefore be implemented directly into {\it{ab initio}} or synthetic simulation studies.

Meromorphic Continuations Of Euler Products With Polynomial Growth Coefficients, Kledis Ahmetaj

Senior Honors Theses and Projects

In this paper, we investigate the meromorphic continuation of Euler products, with a particular focus on those with polynomial growth coefficients. Building upon the classical example of the Riemann zeta function, we derive conditions under which Euler products can be extended beyond their regions of convergence to broader domains of the complex plane, revealing singularities and deepening our understanding of their analytic properties. We begin by establishing a framework for meromorphic continuation and then introduce a general theorem that applies to Euler products of the form

where the coefficients and exponents satisfy specific constraints. We prove that such Euler products …