Mathematics Commons^™

On Cheeger Constants Of Knots, Robert Lattimer

Electronic Theses, Projects, and Dissertations

In this thesis, we will look at finding bounds for the Cheeger constant of links. We will do this by analyzing an infinite family of links call two-bridge fully augmented links. In order to find a bound on the Cheeger constant, we will look for the Cheeger constant of the link’s crushtacean. We will use that Cheeger constant to give us insight on a good cut for the link itself, and use that cut to obtain a bound. This method gives us a constructive way to find an upper bound on the Cheeger constant of a two-bridge fully augmented link. …

The Mathematics Of Financial Portfolio Optimization Incorporating Environmental, Social, And Governance Score Information, Ian Driskill

Master's Theses

We numerically investigate the effects that Environmental, Social, and Governance (ESG) scores have on portfolio optimization with Modern Portfolio Theory assumptions and how ESG scores correlate with the market returns of a rated company's stock. Additionally, we review and analyze a research paper published in the Journal of Financial Economics regarding ESG investing titled “Responsible investing: The ESG-efficient frontier” by Pedersen, Fitzgibbons, and Lukasz. Our overall goal is provide insight for socially responsible inclined investors, to help them understand what ESG scores tell us and how those scores may effect their overall investment returns."

Information Based Approach For Detecting Change Points In Inverse Gaussian Model With Applications, Alexis Anne Wallace

Electronic Theses, Projects, and Dissertations

Change point analysis is a method used to estimate the time point at which a change in the mean or variance of data occurs. It is widely used as changes appear in various datasets such as the stock market, temperature, and quality control, allowing statisticians to take appropriate measures to mitigate financial losses, operational disruptions, or other adverse impacts. In this thesis, we develop a change point detection procedure in the Inverse Gaussian (IG) model using the Modified Information Criterion (MIC). The IG distribution, originating as the distribution of the first passage time of Brownian motion with positive drift, offers …

The Brahmagupta Equation And Record Numbers, Christine Patterson

Boise State University Theses and Dissertations

The Diophantine equation x² - Dy² = k has been explored since antiquity and has found significant applications in diverse contexts in mathematics. This thesis focuses on two aspects of solutions of this equation:

1. For fixed k, the sizes of the D values for which there is a solution, and the corresponding sizes of the x and y constituting solutions. This topic is previously uninvestigated and our contribution is data-based.
2. For fixed k and D for which the equation is solvable, statistical properties of the x and y occurring in these solutions. We prove that Benford's Law applies …

A Survey Of The Classification Of 1-Dimensional Shift Systems, Jacob Miller

Boise State University Theses and Dissertations

The one-dimensional full shift over a finite set A is the collection of bi-infinite sequences of symbols in A together with the left-shift map which shifts the indexing of the sequence. A shift space is a subset of a full shift defined by a collection of forbidden blocks, i.e., finite words which are not allowed to appear. Many shift spaces arise as the set of bi-infinite walks on a labeled graph, and many dynamical systems can be encoded as shift spaces where the dynamics are replaced by the left-shift map.

We introduce shift spaces and their basic properties, then …

Coarctation Duration And Severity Predict Risk Of Hypertension Precursors In A Preclinical Model And Hypertensive Status Among Patients, Arash Ghorbannia, Hilda Jurkiewicz, Lith Nasif, Abdillahi Ahmed, Jennifer Co-Vu, Mehdi Maadooliat, Ronald K. Woods, John F. Ladisa Jr.

Mathematical and Statistical Science Faculty Research and Publications

BACKGROUND:

Coarctation of the aorta (CoA) often leads to hypertension posttreatment. Evidence is lacking for the current >20 mm Hg peak-to-peak blood pressure (BP) gradient (BPGpp) guideline, which can cause aortic thickening, stiffening, and dysfunction. This study sought to find the BPGpp severity and duration that avoid persistent dysfunction in a preclinical model and test if predictors translate to hypertension status in patients with CoA.

METHODS:

Rabbits (n=75; 5–12/group) were exposed to mild, intermediate, or severe CoA (≤12, 13–19, ≥20 mm Hg BPGpp) for ≈1, 3, or 22 weeks using dissolvable and permanent sutures with thickening, stiffening, contraction, and endothelial …

Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, Thomas Rolando Mellema

Electronic Theses and Dissertations

Security system design flaws will create greater risks and repercussions as the systems being secured further integrate into our daily life. One such application example is incorporating the powerful potential of the concept of the Internet of Things (IoT) into software services engineered for improving the practices of monitoring and prescribing effective healthcare to patients. A study was performed in this application area in order to specify a security system design for a Health Prescription Assistant (HPA) that operated with medical IoT (mIoT) devices in a healthcare environment. Although the efficiency of this system was measured, little was presented to …

Standard And Non-Standard Log-Linear Models For 2 × 2 Contingency Tables, G M Toufiqul Hoque

UNLV Theses, Dissertations, Professional Papers, and Capstones

Log-linear models can be used to model the joint relationship of two or more categorical variables in a multiway contingency table. In a log-linear model, the logarithm of the expected joint counts (or the logarithm of the joint probabilities) in a contingency table can be written as a linear model.

Most log-linear models used in practice are standard. Standard log-linear models include the traditional parameter terms we see in ANOVA models: an overall effect, main effects, and various kinds of interaction terms.

Standard log-linear models are divided into hierarchical and non-hierarchical. Hierarchical models satisfy the hierarchy principle: if a higher-order …

Ramanujan Type Congruences For Quotients Of Klein Forms, Timothy Huber, Nathaniel Mayes, Jeffery Opoku, Dongxi Ye

School of Mathematical & Statistical Sciences Faculty Publications

In this work, Ramanujan type congruences modulo powers of primes p≥5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t-core partitions known to satisfy Ramanujan type congruences for p=5,7,11. The vectors of exponents corresponding to products that are modular forms for Γ1(p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ1(p) of weights 1≤k≤5 for primes 5≤p≤19 and whose Fourier coefficients …

Local Existence Of Solutions To A Nonlinear Autonomous Pde Model For Population Dynamics With Nonlocal Transport And Competition, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Highlights

• Partial differential equation models are ubiquitous in applied sciences.

• A partial differential equation based in ecology is studied for solution existence.

• Energy methods and convergence analysis lead to local classical solutions.

Abstract

In this paper, we prove that a particular nondegenerate, nonlinear, autonomous parabolic partial differential equation with nonlocal mass transfer admits the local existence of classical solutions. The equation was developed to qualitatively describe temporal changes in population densities over space through accounting for location desirability and fast, long-range travel. Beginning with sufficiently regular initial conditions, through smoothing the PDE and employing energy arguments, we obtain a sequence …

Bernstein Polynomials Method For Solving Multi-Order Fractional Neutral Pantograph Equations With Error And Stability Analysis, M. H. T. Alshbool

All Works

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series …

Identifying Disease-Related Gene-Environment Interactions Based On Method Of Moments, Linchuan Shen

UNLV Theses, Dissertations, Professional Papers, and Capstones

Human diseases are often caused by a complex interplay of multiple factors, including genetics and environmental factors. These factors can play critical roles in the development and progression of diseases. Although genome-wide association studies (GWAS) have successfully identified many genetic variants associated with human diseases, the estimated effects of these variants are small and can explain only a relatively small portion of the heritability of the underlying diseases.

Detecting gene-environment interactions (G × E) can shed light on the biological mechanisms of diseases. However, most existing methods that investigate G × E only look at how one environmental …

Conditional Quantization For Uniform Distributions On Line Segments And Regular Polygons, Tsianna Danielle Dominguez

Theses and Dissertations

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the support are preselected, then the quantization is called a conditional quantization. In this thesis, we have investigated the conditional quantization for the uniform distributions defined on the unit line segments and m-sided regular polygons, where m ≥ 3, inscribed in a unit circle.

A Study Of Quantitative Reasoning Instructors’ Choices And Motivations When Teaching Quantitative Reasoning For The First Time, Trish Ann Harding

Theses and Dissertations

This qualitative study delves into the instructional decision-making processes of post-secondary instructors teaching quantitative reasoning (QR) courses for the first time. The study aims to address the gap in understanding how first-time QR instructors navigate the complexities of curriculum design and pedagogical strategies, and how these experiences contribute to their professional development. The research questions center on identifying the instructional decisions made by these instructors, exploring the factors influencing their decision-making, and understanding the impact of exercising agency has on their professional development. Through in-depth exploration, this study seeks to shed light on the challenges and opportunities faced by first-time …

Approximation Via Degree Reduction Of Nonlinearities With Applications To Turbulent Flows, Flame Fronts, And Magnetohydrodynamics, Matthew Enlow

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

We perform an analytical and computational investigation on the effectiveness of a locally bounded truncation function, which we call a calming function, when applied to the nonlinear terms of several dissipative partial differential equations. In particular, the 3D Navier-Stokes equations of incompressible fluid flow, the 2D Kuramoto-Sivashinsky equations of laminar flame fronts, and the 2D MHD-Boussinesq equations of magnetohydrodynamics. Each of these equations have open questions about the global existence and uniqueness of their solutions. These calming functions effectively reduce the algebraic degree of select nonlinear terms, thus one can verify global wellposedness for these "calmed systems." More specifically, in …

Deterministic Global 3d Fractal Cloud Model For Synthetic Scene Generation, Aaron M. Schinder, Shannon R. Young, Bryan J. Steward, Michael L. Dexter, Andrew Kondrath, Stephen Hinton, Ricardo Davila

Faculty Publications

This paper describes the creation of a fast, deterministic, 3D fractal cloud renderer for the AFIT Sensor and Scene Emulation Tool (ASSET). The renderer generates 3D clouds by ray marching through a volume and sampling the level-set of a fractal function. The fractal function is distorted by a displacement map, which is generated using horizontal wind data from a Global Forecast System (GFS) weather file. The vertical windspeed and relative humidity are used to mask the creation of clouds to match realistic large-scale weather patterns over the Earth. Small-scale detail is provided by the fractal functions which are tuned to …

Influence Of Parsimony And Work-Related Psychological Constructs In Predicting Turnover Intention When Using Machine Learning Vs Regression, Diego Figueiras

Theses, Dissertations and Culminating Projects

This dissertation explores the ongoing debate between traditional statistical regression models and machine learning (ML) algorithms in predictive modeling, focusing on the impact of sample size and the number of variables. Study 1 investigates the relationship between sample size and predictive accuracy, proposing hypotheses regarding the advantages of ML over regression as sample size increases. Additionally, the study examines the influence of the number of variables on predictive accuracy, emphasizing the trade-off between ML and regression models. Using data from the Federal Employee Viewpoint Survey, the research aims to contribute insights into the conditions favoring each modeling approach. Study 2 …

Analysis And Construction Of Artificial Neural Networks For The Heat Equations, And Their Associated Parameters, Depths, And Accuracies., Shakil Ahmed Rafi

Graduate Theses and Dissertations

This dissertation seeks to explore a certain calculus for artificial neural networks. Specifically we will be looking at versions of the heat equation, and exploring strategies on how to approximate them.

Our strategy towards the beginning will be to take a technique called Multi-Level Picard (MLP), and present a simplified version of it showing that it converges to a solution of the equation (^∂/_∂t u_d ) (t, x) = (∇² _x u_d)(t, x).

We will then take a small detour exploring the viscosity super-solution properties of solutions to such …

Derived Jet Schemes And Arc Spaces, And Arithmetic Arc Space Representability, C. Eric Overton-Walker

Graduate Theses and Dissertations

Associated to a given scheme X one can define geometric and arithmetic notions of jet schemes and arc spaces. We develop a construction of the geometric jets and arcs in the setting of derived schemes and explore consequences thereof. In particular, we prove an analogous theorem to that of one by Tommaso de Fernex and Roi Docampo concerning the cotangent sheaves of geometric jets and arcs. Our version then produces many subsequent results which allow us to prove stronger versions of results concerning geometric jets and arcs by removing unnecessary hypotheses. Separately, we explore evidence as to why, in contrast …

Flipped Classroom For Linear Algebra At Undergraduate Level, M. Thulasidas

Research Collection School Of Computing and Information Systems

In this article, we describe our experience in developing an undergraduate Linear Algebra course tailored to highlight its relevance and applicability in Computer Science. Over the course of three years, the course transitioned from a traditional direct-instruction format to a flipped-classroom design, resulting in positive student learning outcomes. This article covers the course design philosophy, its syllabus, learning objectives, and the incorporation of both quantitative and qualitative student feedback in shaping the course. Furthermore, the article shares the insights gleaned from our experience, which can serve as best practices for instructors aiming to deliver a successful Linear Algebra course for …

A Post-Quantum Mercurial Signature Scheme, Madison Mabe

All Theses

This paper introduces the first post-quantum mercurial signature scheme. We also discuss how this can be used to construct a credential scheme, as well as some practical applications for the constructions.

Weighted Ehrhart Theory: Extending Stanley's Nonnegativity Theorem, Esme Bajo, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, Josephine Yu

School of Mathematical & Statistical Sciences Faculty Publications

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of …

Representation Learning For Generative Models With Applications To Healthcare, Astronautics, And Aviation, Van Minh Nguyen

Theses and Dissertations

This dissertation explores applications of representation learning and generative models to challenges in healthcare, astronautics, and aviation.

The first part investigates the use of Generative Adversarial Networks (GANs) to synthesize realistic electronic health record (EHR) data. An initial attempt at training a GAN on the MIMIC-IV dataset encountered stability and convergence issues, motivating a deeper study of 1-Lipschitz regularization techniques for Auxiliary Classifier GANs (AC-GANs). An extensive ablation study on the CIFAR-10 dataset found that Spectral Normalization is key for AC-GAN stability and performance, while Weight Clipping fails to converge without Spectral Normalization. Analysis of the training dynamics provided further …

Error Estimates For A Mixed Finite Element Method For The Maxwell's Transmission Eigenvalue Problem, Chao Wang, Jintao Cui, Jiguang Sun

Michigan Tech Publications

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell's transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the …

Inference For Time Series In Change Points And Statistical Learning, Jiaqi Li

Arts & Sciences Graduate Student Theses and Dissertations

This study aims to develop a comprehensive framework for inference theory in time series, with a focus on change-point detection and online learning. First, we introduce an $\ell^2$-based inference approach for multiple change-point detection in high-dimensional time series, which targets dense or spatially clustered signals by a novel Two-Way MOSUM (moving sum) test statistic. We derive the limiting distribution of this $\ell^2$-aggregated statistic by extending the high-dimensional Gaussian approximation to non-stationary spatial-temporal processes. Simulation exhibits promising performance of our test in detecting non-sparse weak signals, and the application to COVID-19 analysis shows the real-world relevance of our method. Next, we …

Art And Math Via Cubic Polynomials, Polynomiography And Modulus Visualization, Bahman Kalantari

LASER Journal

Throughout history, both quadratic and cubic polynomials have been rich sources for the discovery and development of deep mathematical properties, concepts, and algorithms. In this article, we explore both classical and modern findings concerning three key attributes of polynomials: roots, fixed points, and modulus. Not only do these concepts lead to fertile ground for exploring sophisticated mathematics and engaging educational tools, but they also serve as artistic activities. By utilizing innovative practices like polynomiography—visualizations associated with polynomial root finding methods—as well as visualizations based on polynomial modulus properties, we argue that individuals can unlock their creative potential. From crafting captivating …

A Discussion On Estimation Of The Best Constant For Spherical Restriction Inequalities, Hongyi Liu

Mathematics, Statistics, and Computer Science Honors Projects

The restriction conjecture asks for a meaningful restriction of the Fourier transform of a function to a sufficiently curved lower dimensional manifold. It then conjectures certain size estimates for this restriction in terms of the size of the original function. It has been proven in 2 dimensions, but it is open in dimensions 3 and larger, and is an area of much recent active effort. In our study, instead of aiming to prove the restriction conjecture, we target understanding its worst-case scenarios within known estimates. Specifically, we investigate the extension operator applied to antipodally concentrating profiles, examining the ratio of …

Classification Of Topological Defects In Cosmological Models, Abigail Swanson

Departmental Honors & Graduate Capstone Projects

In nature, symmetries play an extremely significant role. Understanding the symmetries of a system can tell us important information and help us make predictions. However, these symmetries can break and form a new type of symmetry in the system. Most notably, this occurs when the system goes through a phase transition. Sometimes, a symmetry can break and produce a tear, known as a topological defect, in the system. These defects cannot be removed through a continuous transformation and can have major consequences on the system as a whole. It is helpful to know what type of defect is produced when …

An Investigation Into The Causes Of Home Field Advantage In Professional Soccer, Paige E. Tomer

Mathematics, Statistics, and Computer Science Honors Projects

Home-field advantage is the sporting phenomenon in which the home team outperforms the away team. Despite its widespread occurrence across sports, the underlying reasons for home-field advantage remain uncertain. In this paper, we employ a range of statistical methods to explore the causal relationships of potential determinants of home-field advantage. We measure home-field advantage using match outcomes and differential metrics (e.g., differences in yellow cards received). In an attempt to narrow the research disparity between men’s and women’s sports, we utilize data from the National Women’s Soccer League (NWSL) and the English Premier League (EPL) to investigate potential causes of …

Representation Theory And Burnside's Theorem, Nathan Fronk

Senior Seminars and Capstones

In this paper we give a brief introduction to the representation theory of finite groups, and by extension character theory. These tools are extensions of group theory into linear algebra, that can then be applied back to group theory to prove propositions that are based entirely in group theory. We discuss the importance of simple groups and the Jordan-Hölder theorem in order to prepare for the statement of Burnside’s pq theorem. Lastly, we provide a proof of Burnside’s theorem that utilizes the character theory we covered earlier in the paper.