Analysis Commons^™

New Perturbation Bounds In Unitarily Invariant Norms For Subunitary Polar Factors, Lei Zhu, Wei-Wei Xu, Hao Liu, Li-Juan Ma

Electronic Journal of Linear Algebra

Let $A\in\mathbb{C}^{m \times n}$ have generalized polar decomposition $A = QH$ with $Q$ subunitary and $H$ positive semidefinite. Absolute and relative perturbation bounds are derived for the subunitary polar factor $Q$ in unitarily invariant norms and in $Q$-norms, that extend and improve existing bounds.

Iteration With Stepsize Parameter And Condition Numbers For A Nonlinear Matrix Equation, Syed M. Raza Shah Naqvi, Jie Meng, Hyun-Min Kim

Electronic Journal of Linear Algebra

In this paper, the nonlinear matrix equation $X^p+A^TXA=Q$, where $p$ is a positive integer, $A$ is an arbitrary $n\times n$ matrix, and $Q$ is a symmetric positive definite matrix, is considered. A fixed-point iteration with stepsize parameter for obtaining the symmetric positive definite solution of the matrix equation is proposed. The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Several numerical examples are presented to show the efficiency of the proposed iterative method with proper stepsize parameter and the sharpness of the three kinds of condition numbers.

United States Population Future Estimates And Long-Term Distribution, Sean P. Brogan

DePaul Discoveries

The population of the United States has always increased year over year. Even now with decreasing birth rates, the overall population continues to grow when looking at conventional models. The present study specifically examines what would happen to the U.S. population if we were to maintain the current birth and survival rates into the future. By 2050, our research shows that the U.S. population will become much older and cease to grow at all.

Analysis Of 2016-17 Major League Soccer Season Data Using Poisson Regression With R, Ian D. Campbell

Undergraduate Theses and Capstone Projects

To the outside observer, soccer is chaotic with no given pattern or scheme to follow, a random conglomeration of passes and shots that go on for 90 minutes. Yet, what if there was a pattern to the chaos, or a way to describe the events that occur in the game quantifiably. Sports statistics is a critical part of baseball and a variety of other of today’s sports, but we see very little statistics and data analysis done on soccer. Of this research, there has been looks into the effect of possession time on the outcome of a game, the ...

Properties And Convergence Of State-Based Laplacians, Kelsey Wells

Dissertations, Theses, and Student Research Papers in Mathematics

The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peridynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different ...

The Perfect Professor, Samuel Legere

Honors Projects in Mathematics

The purpose of this project is to identify the similarities and differences between student and faculty perspectives of "The Perfect Professor". Student preferences for professor characteristics may vary, but what is it about the students that causes them to think or feel this way? These predictors may be something such as a student's major, age, gender, race, or even whether or not they are an athlete. I have conducted a survey for both students and faculty to identify these correlating qualities and gather data in hopes to take the first steps that data analysts may then use to match ...

Geometry And Analysis Of Some Euler-Arnold Equations, Jae Min Lee

All Dissertations, Theses, and Capstone Projects

In 1966, Arnold showed that the Euler equation for an ideal fluid can arise as the geodesic flow on the group of volume preserving diffeomorphisms with respect to the right invariant kinetic energy metric. This geometric interpretation was rigorously established by Ebin and Marsden in 1970 using infinite dimensional Riemannian geometry and Sobolev space techniques. Many other nonlinear evolution PDEs in mathematical physics turned out to fit in this universal approach, and this opened a vast research on the geometry and analysis of the Euler-Arnold equations, i.e., geodesic equations on a Lie group endowed with one-sided invariant metrics. In ...

On Some Geometry Of Graphs, Zachary S. Mcguirk

All Dissertations, Theses, and Capstone Projects

In this thesis we study the intrinsic geometry of graphs via the constants that appear in discretized partial differential equations associated to those graphs. By studying the behavior of a discretized version of Bochner's inequality for smooth manifolds at the cone point for a cone over the set of vertices of a graph, a lower bound for the internal energy of the underlying graph is obtained. This gives a new lower bound for the size of the first non-trivial eigenvalue of the graph Laplacian in terms of the curvature constant that appears at the cone point and the size ...

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

All Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices representing ...

Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell

Brandon Russell

Determinantal Representations Of Elliptic Curves Via Weierstrass Elliptic Functions, Mao-Ting Chien, Hiroshi Nakazato

Electronic Journal of Linear Algebra

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

Under The Influence, Leonardo Cavicchio

Honors Projects in Mathematics

The purpose of this Honors Capstone entitled Under the Influence is to assess the validity of claims concerning the possible influence of roommates on one another, concerning alcohol on college campuses. This will be done by examining data collected in a prior study conducted over a two-year period. This analysis will focus on how alcohol consumption changes in correlation with the personality factors of roommates over an extended period of time. This secondary analysis of de-identified data will focus on primary and secondary subquestions. The primary question that will be addressed with the data set collected from the University of ...

Understanding Natural Keyboard Typing Using Convolutional Neural Networks On Mobile Sensor Data, Travis Siems

Computer Science and Engineering Theses and Dissertations

Mobile phones and other devices with embedded sensors are becoming increasingly ubiquitous. Audio and motion sensor data may be able to detect information that we did not think possible. Some researchers have created models that can predict computer keyboard typing from a nearby mobile device; however, certain limitations to their experiment setup and methods compelled us to be skeptical of the models’ realistic prediction capability. We investigate the possibility of understanding natural keyboard typing from mobile phones by performing a well-designed data collection experiment that encourages natural typing and interactions. This data collection helps capture realistic vulnerabilities of the security ...

Building A Better Risk Prevention Model, Steven Hornyak

National Youth-At-Risk Conference Savannah

This presentation chronicles the work of Houston County Schools in developing a risk prevention model built on more than ten years of longitudinal student data. In its second year of implementation, Houston At-Risk Profiles (HARP), has proven effective in identifying those students most in need of support and linking them to interventions and supports that lead to improved outcomes and significantly reduces the risk of failure.

Theoretical Analysis Of Nonlinear Differential Equations, Emily Jean Weymier

Electronic Theses and Dissertations

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy ...

Infinitely Many Solutions To Asymmetric, Polyharmonic Dirichlet Problems, Edger Sterjo

All Dissertations, Theses, and Capstone Projects

In this dissertation we prove new results on the existence of infinitely many solutions to nonlinear partial differential equations that are perturbed from symmetry. Our main theorems focus on polyharmonic Dirichlet problems with exponential nonlinearities, and are now published in Topol. Methods Nonlinear Anal. Vol. 50, No.1, (2017), 27-63. In chapter 1 we give an introduction to the problem, its history, and the perturbation argument itself. In chapter 2 we prove the variational principle of Bolle on the behavior of critical values under perturbation, and the variational principle of Tanaka on the existence of critical points of large augmented ...

What Makes A Theory Of Infinitesimals Useful? A View By Klein And Fraenkel, Vladimir Kanovei, Karin Katz, Mikhail Katz, Thomas Mormann

Journal of Humanistic Mathematics

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein.

Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang

Mathematics Faculty Publications

In this paper, we provide a direct and constructive proof of weak factorization of h¹ (R × R) (the predual of little BMO space bmo(R × R) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f ∈ h¹ (R × R) there exist sequences {α^k _j } ∈ and functions g_j ^k, h^k _j ∈ L² (R² ) such that ∞ ∞ f = α^k _j ^k _j H₁ H₂ g_j ^k − g_j ^kH₁ H₂ h^k k=1 j=1 in the sense of h¹ (R × R), where H₁ and H ...

The Boundedness Of The Hardy-Littlewood Maximal Function And The Strong Maximal Function On The Space Bmo, Wenhao Zhang

CMC Senior Theses

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A₁ weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps ...

On Spectral Theorem, Muyuan Zhang

Honors Theses

There are many instances where the theory of eigenvalues and eigenvectors has its applications. However, Matrix theory, which usually deals with vector spaces with finite dimensions, also has its constraints. Spectral theory, on the other hand, generalizes the ideas of eigenvalues and eigenvectors and applies them to vector spaces with arbitrary dimensions. In the following chapters, we will learn the basics of spectral theory and in particular, we will focus on one of the most important theorems in spectral theory, namely the spectral theorem. There are many different formulations of the spectral theorem and they convey the "same" idea. In ...