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Articles 1 - 14 of 14
Full-Text Articles in Analysis
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Constructing Orthonormal Bases With The Residuals Of Successive Approximations, An Introduction To Multiresolution Analysis, Elijah J. Guptill
Master's Theses
Wavelets and wavelet analysis are used in the study of signal processing, quantum field theory, functional analysis, multifractal analysis, and various other areas of mathematics. Multiresolution analysis provides a framework for building a wavelet basis of $\mathcal{L}^{2}(\mathbb{R})$ from a scaling function $\phi$, whose dyadic dilations and translations, $\{2^{j /2}\phi(2^{j}x-k):j,k\in \mathbb{Z}\}$, approximate $\mathcal{L}^{2}(\mathbb{R})$. One of the key properties of $\phi$ is that it must satisfy $\phi(x)=\sum_{k\in \mathbb{Z}}{p_{k}2^{j /2}\phi(2^{j}x-k)}$ with respect to the norm on $\mathcal{L}^{2}(\mathbb{R})$. This equation is called a two-scale difference equation. Such equations enforce a regularity on the ordinary generating function $2^{-1 /2}\sum_{k\in \mathbb{Z}}{p_{k}z^{k}}$, known as the quadrature condition. …
On Sobolev Spaces And The Existence Of Weak Solutions To Boundary Value Problems, Skye X. Paul
On Sobolev Spaces And The Existence Of Weak Solutions To Boundary Value Problems, Skye X. Paul
Master's Theses
Many boundary value problems that arise in mathematical models have close connections to second order elliptic partial differential equations. This thesis introduces the idea of weak derivatives and Sobolev Spaces to generalize possible solutions. Using functional analysis centered around the Lax-Milgram theorem, we show the existence of these generalized solutions to boundary value problems including Laplace's Equation, 2nd order linear ODEs, and ultimately a general second order elliptic PDE. The work cumulates with recovering a number of central theorems of functional analysis in the context of Sobolev Spaces, creating a new perspective on the solvability of these boundary value problems.
Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook
Topiarism: The Kernel Embedding Of Distributions Applied To Modern Portfolio Theory, Stephen G. Cook
Master's Theses
The method of kernel embedding of distributions on a set $\Omega$ into a reproducing kernel Hilbert space is a method of studying the space of measures on a set $\Omega$ using Hilbert space geometry. Because positively weighted portfolios can be interpreted as nonnegative probability measures on the space of assets, we are able to apply this technique to portfolio theory. In this thesis, we discuss the theory of "topiarism", the study of positively weighted probability measures on compact sets under the kernel embedding of distributions. Given a specific payoff function $\psi$ on the set of assets $\Omega$, we optimize a …
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Banach Algebras And The Gelfand Theory Of Group Algebras On Locally Compact Abelian Groups, James Gabriel Bonvanie
Master's Theses
A Banach algebra is a complex algebra that is simultaneously a Banach space in which the norm is submultiplicative. Notably, $L^1(\mathbb{R})$ with the convolutional product is an Abelian, non-unital Banach algebra that admits an approximate identity. We rectify $L^1(\mathbb{R})$ lacking a unit via the unitization $L^1(\mathbb{R})\times\mathbb{C}$ with identity $(0,1)$. Unitization opens the discussion to the spectrum $\sigma(x)$ of a Banach algebra element, in which the spectrum is a nonempty, compact subset of the complex plane. The spectrum of an Abelian Banach algebra is fully characterized with multiplicative linear functionals, and we prove that the Fourier transform is the unique multiplicative …
Matrix Approximation And Image Compression, Isabella R. Padavana
Matrix Approximation And Image Compression, Isabella R. Padavana
Master's Theses
This thesis concerns the mathematics and application of various methods for approximating matrices, with a particular eye towards the role that such methods play in image compression. An image is stored as a matrix of values with each entry containing a value recording the intensity of a corresponding pixel, so image compression is essentially equivalent to matrix approximation. First, we look at the singular value decomposition, one of the central tools for analyzing a matrix. We show that, in a sense, the singular value decomposition is the best low-rank approximation of any matrix. However, the singular value decomposition has some …
Complex Dimensions Of 100 Different Sierpinski Carpet Modifications, Gregory Parker Leathrum
Complex Dimensions Of 100 Different Sierpinski Carpet Modifications, Gregory Parker Leathrum
Master's Theses
We used Dr. M. L. Lapidus's Fractal Zeta Functions to analyze the complex fractal dimensions of 100 different modifications of the Sierpinski Carpet fractal construction. We will showcase the theorems that made calculations easier, as well as Desmos tools that helped in classifying the different fractals and computing their complex dimensions. We will also showcase all 100 of the Sierpinski Carpet modifications and their complex dimensions.
Exploring The Numerical Range Of Block Toeplitz Operators, Brooke Randell
Exploring The Numerical Range Of Block Toeplitz Operators, Brooke Randell
Master's Theses
We will explore the numerical range of the block Toeplitz operator with symbol function \(\phi(z)=A_0+zA_1\), where \(A_0, A_1 \in M_2(\mathbb{C})\). A full characterization of the numerical range of this operator proves to be quite difficult and so we will focus on characterizing the boundary of the related set, \(\{W(A_0+zA_1) : z \in \partial \mathbb{D}\}\), in a specific case. We will use the theory of envelopes to explore what the boundary looks like and we will use geometric arguments to explore the number of flat portions on the boundary. We will then make a conjecture as to the number of flat …
On The Numerical Range Of Compact Operators, Montserrat Dabkowski
On The Numerical Range Of Compact Operators, Montserrat Dabkowski
Master's Theses
One of the many characterizations of compact operators is as linear operators which
can be closely approximated by bounded finite rank operators (theorem 25). It is
well known that the numerical range of a bounded operator on a finite dimensional
Hilbert space is closed (theorem 54). In this thesis we explore how close to being
closed the numerical range of a compact operator is (theorem 56). We also describe
how limited the difference between the closure and the numerical range of a compact
operator can be (theorem 58). To aid in our exploration of the numerical range of
a compact …
An Investigation Into Crouzeix's Conjecture, Timothy T. Royston
An Investigation Into Crouzeix's Conjecture, Timothy T. Royston
Master's Theses
We will explore Crouzeix’s Conjecture, an upper bound on the norm of a matrix after the application of a polynomial involving the numerical range. More formally, Crouzeix’s Conjecture states that for any n × n matrix A and any polynomial p from C → C,
∥p(A)∥ ≤ 2 supz∈W (A) |p(z)|.
Where W (A) is a set in C related to A, and ∥·∥ is the matrix norm. We first discuss the conjecture, and prove the simple case when the matrix is normal. We then explore a proof for a class of matrices given by Daeshik Choi. We expand …
A Study Of The Design Of Adaptive Camber Winglets, Justin J. Rosescu
A Study Of The Design Of Adaptive Camber Winglets, Justin J. Rosescu
Master's Theses
A numerical study was conducted to determine the effect of changing the camber of a winglet on the efficiency of a wing in two distinct flight conditions. Camber was altered via a simple plain flap deflection in the winglet, which produced a constant camber change over the winglet span. Hinge points were located at 20%, 50% and 80% of the chord and the trailing edge was deflected between -5° and +5°. Analysis was performed using a combination of three-dimensional vortex lattice method and two-dimensional panel method to obtain aerodynamic forces for the entire wing, based on different winglet camber configurations. …
Computing Homology Of Hypergraphs, Jackson Earl
Computing Homology Of Hypergraphs, Jackson Earl
STAR Program Research Presentations
In the modern age of data science, the necessity for efficient and insightful analytical tools that enable us to interpret large data structures inherently presents itself. With the increasing utility of metrics offered by the mathematics of hypergraph theory and algebraic topology, we are able to explore multi-way relational datasets and actively develop such tools. Throughout this research endeavor, one of the primary goals has been to contribute to the development of computational algorithms pertaining to the homology of hypergraphs. More specifically, coding in python to compute the homology groups of a given hypergraph, as well as their Betti numbers …
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Invariant Subspaces Of Compact Operators And Related Topics, Weston Mckay Grewe
Mathematics
The invariant subspace problem asks if every bounded linear operator on a Banach space has a nontrivial closed invariant subspace. Per Enflo has shown this is false in general, however it is known that every compact operator has an invariant subspace. The purpose of this project is to explore introductory results in functional analysis. Specifically we are interested in understanding compact operators and the proof that all compact operators on a Hilbert space have an invariant subspace. In the process of doing this we build up many examples and theorems relating to operators on a Hilbert or Banach space. Continuing …
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Hilbert Space Theory And Applications In Basic Quantum Mechanics, Matthew Gagne
Mathematics
We explore the basic mathematical physics of quantum mechanics. Our primary focus will be on Hilbert space theory and applications as well as the theory of linear operators on Hilbert space. We show how Hermitian operators are used to represent quantum observables and investigate the spectrum of various linear operators. We discuss deviation and uncertainty and briefly suggest how symmetry and representations are involved in quantum theory.
Completeness Of Ordered Fields, James Forsythe Hall
Completeness Of Ordered Fields, James Forsythe Hall
Mathematics
The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other areas of mathematics, such as nonstandard analysis. Continuing, we study the completeness of non-Archimedean fields, and provide several examples of such fields with varying degrees of properties, using nonstandard analysis to produce some relatively "nice" (in particular, they are Cantor complete) final examples. As a small detour, …