Mathematics Commons^™

On The Existence Of Non-Free Totally Reflexive Modules, J. Cameron Atkins

Theses and Dissertations

For a standard graded Cohen-Macaulay ring R, if the quotient R/(x) admits nonfree totally reflexive modules, where x is a system of parameters consisting of elements of degree one, then so does the ring R. A non-constructive proof of this statement was given in [16]. We give an explicit construction of the totally reflexive modules over R obtained from those over R/(x).

We consider the question of which Stanley-Reisner rings of graphs admit nonfree totally reflexive modules and discuss some examples. For an Artinian local ring (R,m) with m3 = 0 and containing the complex numbers, we describe an explicit …

A Family Of Simple Codimension Two Singularities With Infinite Cohen-Macaulay Representation Type, Tyler Lewis

Theses and Dissertations

A celebrated theorem of Buchweitz, Greuel, Knörrer, and Schreyer is that the hypersurface singularities of finite representation type, i.e. the hypersurface singularities admitting only finitely many indecomposable maximal Cohen-Macaulay modules, are exactly the ADE singularities. The codimension 2 singularities that are the analogs of the ADE singularities have been classified by Frühbis-Krühger and Neumer, and it is natural to expect an analogous result holds for these singularities. In this paper, I will present a proof that, in contrast to hypersurfaces, Frühbis-Krühger and Neumer’s singularities include a subset of singularities of infinite representation type.

Subdivision Of Measures Of Squares, Dylan Bates

Theses and Dissertations

The primary goal of our work is to establish a method to relate simple measures to a given set of moments. We calculate the moments of squares via linear polynomial weight measures and straight line cuts and use this to calculate the centre of mass of the square. The one-to-one correspondence that is found is needed to represent surfaces with gaps, which can estimate arbitrary measures on squares. From this, a subdivision scheme is developed, which successively quadrisects squares and uses the relation to estimate the new measures in order to provide a good representation of the original surface. One …

Convergence And Rate Of Convergence Of Approximate Greedy-Type Algorithms, Anton Dereventsov

Theses and Dissertations

In this dissertation we study the questions of convergence and rate of convergence of greedy-type algorithms under imprecise step evaluations. Such algorithms are in demand as the issue of calculation errors appears naturally in applications.

We address the question of strong convergence of the Chebyshev Greedy Algorithm (CGA), which is a generalization of the Orthogonal Greedy Algorithm (also known as the Orthogonal Matching Pursuit), and show that the class of Banach spaces for which the CGA converges for all dictionaries and objective elements is strictly between smooth and uniformly smooth Banach spaces.

We analyze an application-oriented modification of the CGA, …

Deep Learning: An Exposition, Ryan Kingery

Theses and Dissertations

In this paper we describe and survey the field of deep learning, a type of machine learning that has seen tremendous growth and popularity over the past decade for its ability to substantially outperform other learning methods at important tasks. We focus on the problem of supervised learning with feedforward neural networks. After describing what these are we give an overview of the essential algorithms of deep learning, backpropagation and stochastic gradient descent. We then survey some of the issues that occur when applying deep learning in practice. Last, we conclude with an important application of deep learning to the …

Polynomials Of Small Mahler Measure With No Newman Multiples, Spencer Victoria Saunders

Theses and Dissertations

A Newman polynomial is a polynomial with coefficients in f0;1g and with constant term 1. It is known that the roots of a Newman polynomial must lie in the slit annulus fz 2C: f��1 1 such that if a polynomial f (z) 2 Z[z] has Mahler measure less than s and has no nonnegative real roots, then it must divide a Newman polynomial. In this thesis, we present a new upper bound on such a s if it exists. We also show that there are infinitely many monic polynomials that have distinct Mahler measures which all lie below f, have …

Unconditionally Energy Stable Numerical Schemes For Hydrodynamics Coupled Fluids Systems, Alexander Yuryevich Brylev

Theses and Dissertations

The thesis consists of two parts. In the first part we propose several second order in time, fully discrete, linear and nonlinear numerical schemes to solve the phase-field model of two-phase incompressible flows in the framework of finite element method. The schemes are based on the second order Crank-Nicolson method for time disretizations, projection method for Navier-Stokes equations, as well as several implicit-explicit treatments for phase-field equations. The energy stability, solvability, and uniqueness for numerical solutions of proposed schemes are further proved. Ample numerical experiments are performed to validate the accuracy and efficiency of the proposed schemes thereafter.

In the …

Covering Subsets Of The Integers And A Result On Digits Of Fibonacci Numbers, Wilson Andrew Harvey

Theses and Dissertations

A covering system of the integers is a finite system of congruences where each integer satisfies at least one of the congruences. Two questions in covering systems have been of particular interest in the mathematical literature. First is the minimum modulus problem, whether the minimum modulus of a covering system of the integers with distinct moduli can be arbitrarily large, and the second is the odd covering problem, whether a covering system of the integers with distinct moduli can be constructed with all moduli odd. We consider these and similar questions for subsets of the integers, such as the set …

Nonequispaced Fast Fourier Transform, David Hughey

Theses and Dissertations

Two algorithms for fast and accurate evaluation of high degree trigonometric polynomials at many scattered points are presented. Both methods rely on highly localized kernels and the Fast Fourier Transform. The first algorithm uses the function values at uniformly distributed grid points and kernels that reproduce trigonometric polynomials, while the second method uses kernels that approximate well the function on the frequency side. Both algorithm are termed Nonequispaced Fast Fourier Transform. The first algorithm is coded in MATLAB and shown to approximate well the function to be evaluated.