Mathematics Commons^™

Mountain Passes And Saddle Points, James Bisgard

All Faculty Scholarship for the College of the Sciences

Variational methods find solutions of equations by considering a solution as a critical point of an appropriately chosen function. Local minima and maxima are well-known types of critical points. We explore methods for finding critical points that are neither local maxima or minima, but instead are mountain passes or saddle points. Criteria for the existence of minima or maxima are well known, but those for mountain passes or saddle points are less well known. We give an accessible treatment of some criteria for the existence of such points (including the mountain pass lemma), as well as describe a method that …

On The Sum Of The Reciprocals Of Squares, Hussam Ibrahim

Celebration of Learning

The Fourier Series has always been a great tool by turning the most complicated functions into a simple approximation by several sine and cosine terms. However, given the fact that its called an approximation arouses the question of how accurate is this approximation, especially with functions that have a corner at x values. In this project, we will show how the series that is the sum of the reciprocals of squares arises in the study of Fourier series. We will discuss the convergence and the rate of convergence of this series. This result arose during exploration on mathematical software Sage. …

Differential Geometry: Curvature And Holonomy, Austin Christian

Math Theses

We develop the basic language of differential geometry, including smooth manifolds, bundles, and differential forms. Once this background is established, we explore parallelism in smooth manifolds -- in particular, in Riemannian manifolds -- and conclude by presenting a proof of the Ambrose-Singer theorem, which relates parallelism (holonomy) to curvature in principal bundles.

The P -Royden And P -Harmonic Boundaries For Metric Measure Spaces, Marcello Lucia, Michael J. Puls

Publications and Research

Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfy certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

Using Maya And Mathematica To Create Mathematical Art, Allison Carr

Honors Projects

The project consists of six pieces of art that were created in Maya using mathematical objects created in Mathematica. Accompanying the art are artist statements for each piece, the mathematics behind each object used, and the Mathematica codes used to generate the objects.

Section Abstracts: Astronomy, Mathematics And Physics With Material Science

Virginia Journal of Science

Abstracts of the Astronomy, Mathematics, and Physics with Material Science Section for the 93rd Annual Meeting of the Virginia Academy of Science, May 21-23, 2015, James Madison University, Richmond, Virginia

Book Review: How To Bake Pi: An Edible Exploration Of The Mathematics Of Mathematics, Darren B. Glass

Math Faculty Publications

If you think about it, mathematics is really just one big analogy. For one example, the very concept of the number three is an drawing an analogy between a pile with three rocks, a collection of three books, and a plate with three carrots on it. For another, the idea of a group is drawing an analogy between adding real numbers, multiplying matrices, and many other mathematical structures. So much of what we do as mathematicians involves abstracting concrete things, and what is abstraction other than a big analogy? [excerpt]

Standing On The Shoulders Of The Giants: Why Constructive Mathematics, Probability Theory, Interval Mathematics, And Fuzzy Mathematics Are Important, Vladik Kreinovich

Departmental Technical Reports (CS)

Recent death of Ray Moore, one of the fathers of interval mathematics, inspired these thoughts on why interval computations -- and several other related areas of study -- are important, and what we can learn from the successes of these areas' founders and promoters.

Ua66 Ogden College Of Science & Engineering Newsletter, Cheryl Stevens, Dean

Ogden College of Science & Engineering Publications

No abstract provided.

Counting Convex Sets On Products Of Totally Ordered Sets, Brandy Amanda Barnette

Masters Theses & Specialist Projects

The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing …

Analysis Of Discrete Fractional Operators And Discrete Fractional Rheological Models, Meltem Uyanik

Masters Theses & Specialist Projects

This thesis is comprised of two main parts: Monotonicity results on discrete fractional operators and discrete fractional rheological constitutive equations. In the first part of the thesis, we introduce and prove new monotonicity concepts in discrete fractional calculus. In the remainder, we carry previous results about fractional rheological models to the discrete fractional case. The discrete method is expected to provide a better understanding of the concept than the continuous case as this has been the case in the past. In the first chapter, we give brief information about the main results. In the second chapter, we present some fundamental …

Boundary Problems For One And Two Dimensional Random Walks, Miky Wright

Masters Theses & Specialist Projects

This thesis provides a study of various boundary problems for one and two dimensional random walks. We first consider a one-dimensional random walk that starts at integer-valued height k > 0, with a lower boundary being the x-axis, and on each step moving downward with probability q being greater than or equal to the probability of going upward p. We derive the variance and the standard deviation of the number of steps T needed for the height to reach 0 from k, by first deriving the moment generating function of T. We then study two types of two-dimensional random walks with …

Counting On R-Fibonacci Numbers, Arthur Benjamin, Curtis Heberle

All HMC Faculty Publications and Research

We prove the r-Fibonacci identities of Howard and Cooper using a combinatorial tiling approach.

Efficient Coupling For Random Walk With Redistribution, Elizabeth Tripp

Honors Scholar Theses

What can be said on the convergence to stationarity of a finite state Markov chain that behaves 'locally' like a nearest-neighbor random walk on the set of integers? In this work, we looked to obtain sharp bounds for the rate of convergence to stationarity for a particular non-symmetric Markov chain. Our Markov chain is a variant of the simple symmetric random walk on the state space {0, ..., N} obtained by allowing transitions from 0 to J₀ and from N to J_N. We first looked at the case where J₀ and J_N are fixed, deterministic …

Modeling Deformations Of Solids At The Continuum Scale, Paxton Turner

Honors Capstones

No abstract provided.

Matrix Exponentiation Without Scaling And Squaring, Bruno Gabriel Beltran

Honors Capstones

No abstract provided.

Encoding Knots, Rachael Keller

Honors Capstones

No abstract provided.

Nonlinear Gravitational-Wave Memory From Merging Binary Black Holes, Goran Dojcinoski

Theses, Dissertations and Culminating Projects

Gravitational waves are oscillations in spacetime that propagate throughout the universe at the speed of light. They are a prediction of Einstein’s theory of General Relativity. Detectable sources of gravitational waves are typically collisions of black holes or other compact objects (neutron stars, white dwarfs). While most gravitational-wave signals are expected to be oscillatory in nature, some will exhibit a phenomenon called gravitational-wave memory. This refers to a non-oscillatory component of the gravitational wave signal that can leave a permanent distortion (or “memory” ) in a gravitational-wave detector. The nonlinear memory effect is a type of memory signal that arises …

Interlace Polynomial Of A Special Eulerian Graph, Christian A. Hyra

Theses, Dissertations and Culminating Projects

In a recent paper, Arratia, Bollobas and Sorkin discussed a graph polynomial defined recursively, which they call the interlace polynomial. There have been previous results on the interlace polynomials for special graphs, such as paths, cycles, and trees. Applications have been found in biology and other areas. In this research, I focus on the interlace polynomial of a special type of Eulerian graph, built from one cycle of size n and n cycle three graphs. I developed explicit formulas by implementing the toggling process to the graph. I further investigate the coefficients and special values of the interlace polynomial. Some …

Geometry Of Hilbert Space Frames, Stephen Sorokanich Iii

Renée Crown University Honors Thesis Projects - All

In applied linear algebra, the term frame is used to refer to a redundant or linearly dependent coordinate system. The concept was introduced in the study of Fourier series and is pertinent in signal processing, where the reconstruction property for finite frames allows for redundant transmission of data to guard against losses due to noise. We give a brief introduction to the theory of finite frames in Section 1, including the major results that allow for the easy construction and description of frames. The subsequent sections relate to the theoretical importance of frames. As a natural extension of the definition …

Incorporating Relaxivities To More Accurately Reconstruct Mr Images, Muge Karaman, Iain P. Bruce, Daniel B. Rowe

Mathematics, Statistics and Computer Science Faculty Research and Publications

Purpose

To develop a mathematical model that incorporates the magnetic resonance relaxivities into the image reconstruction process in a single step.

Materials and methods

In magnetic resonance imaging, the complex-valued measurements of the acquired signal at each point in frequency space are expressed as a Fourier transformation of the proton spin density weighted by Fourier encoding anomalies: T₂^⁎, T₁, and a phase determined by magnetic field inhomogeneity (∆B) according to the MR signal equation. Such anomalies alter the expected symmetry and the signal strength of the k-space observations, resulting in images distorted …

The Minimal Zn-Symmetric Graphs That Are Not Zn-Spherical, Lowell Abrams, Dan Slilaty

Mathematics and Statistics Faculty Publications

Given a graph G equipped with faithful and fixed-point-free Γ-action (Γ a finite group) we define an orbit minor H of G to be a minor of G for which the deletion and contraction sets are closed under the Γ-action. The orbit minor H inherits a Γ-symmetry from G, and when the contraction set is acyclic the action inherited by H remains faithful and fixed-point free. When G embeds in the sphere and the Γ-action on G extends to a Γ-action on the entire sphere, we say that G is Γ-spherical. In this paper we determine for every odd value …

Invariant Basis Number And Basis Types For C*-Algebras, Philip M. Gipson

Department of Mathematics: Dissertations, Theses, and Student Research

We develop the property of Invariant Basis Number (IBN) in the context of C*-algebras and their Hilbert modules. A complete K-theoretic characterization of C*- algebras with IBN is given. A scheme for classifying C*-algebras which do not have IBN is given and we prove that all such classes are realized. We investigate the invariance of IBN, or lack thereof, under common C*-algebraic construction and perturbation techniques. Finally, applications of Invariant Basis Number to the study of C*-dynamical systems and the classification program are investigated.

Adviser: David Pitts

Bioinformatic Game Theory And Its Application To Cluster Multi-Domain Proteins, Brittney Keel

Department of Mathematics: Dissertations, Theses, and Student Research

The exact evolutionary history of any set of biological sequences is unknown, and all phylogenetic reconstructions are approximations. The problem becomes harder when one must consider a mix of vertical and lateral phylogenetic signals. In this dissertation we propose a game-theoretic approach to clustering biological sequences and analyzing their evolutionary histories. In this context we use the term evolution as a broad descriptor for the entire set of mechanisms driving the inherited characteristics of a population. The key assumption in our development is that evolution tries to accommodate the competing forces of selection, of which the conservation force seeks to …

Development Of A Tridimensional Measuring Application For Ipads, Michael Casebolt, Nicolas Kouatli, Jack Mullen

Computer Science and Software Engineering

In today’s fast-paced distribution centers workers and management alike are constantly searching for the quickest and most efficient way to package items for distribution. Even with the advancement of app-oriented solutions to a variety of problems across many industries there is a distinct unmet need in distribution environments for an application capable of increasing the efficiency and accuracy of packaging items. This senior project focused on the development and testing of an application utilizing the Structure Three Dimensional Sensor and a 4th generation iPad to scan an object or group of objects to be packaged and determine the overall dimensions …

Student Understanding Of Function And Success In Calculus, Daniel I. Drlik

Boise State University Theses and Dissertations

The purpose of this study was to determine if there is a relationship between student success in calculus and student understanding of function. Student understanding of function was measured using two questionnaires, one of which is a modification of an existing measure based on APOS theory. The other I developed with items from the concept image literature. The participants of this study were 116 high school students who were enrolled in a first-year calculus course. The results of the questionnaires were aligned to course exam scores to determine connections between function understanding and rate of success in calculus.

A major …

Eigenvectors Of Interpoint Distance Matrices, Michelle F. Craft

Departmental Honors & Graduate Capstone Projects

In this paper, the eigenvectors of interpoint distance matrices will be discussed. When plotted against each other, the eigenvectors of the distance matrix of evenly spaced points in one dimension produce some interesting patterns. An explanation and description of the patterns will be discussed. After examining many aspects of the general Euclidean interpoint distance matrix of order N, D, as well as characteristics of the eigenvectors themselves,some conclusions can be made. Furthermore, research revealed a similarity between our matrices, D, and the Discrete Cosine Transform Matrix, DCT-2. This research led to additional conclusion about our matrices D and allowed for …

Spectrally Equivalent Matrix Polynomials: Non-Standard Representations And Preservation Of Structure, Vasilije Perovic

Dissertations

Abstract attached as separate file.

On Mikhailov's Reduction Group, Tihomir Valchev

Articles

We present a generalization of the notion of reduction group which allows one to study in a uniform way certain classes of nonlocal $S$-integrable equations like Ablowitz-Musslimani's nonlocal Schr\"odinger equation. Another aspect of the proposed generalization is the possibility to derive in a systematic way solutions to S-integrable equations with prescribed symmetries.

Averaged Instrumental Variables Estimators, Yoonseok Lee, Yu Zhou

Center for Policy Research

We develop averaged instrumental variables estimators as a way to deal with many weak instruments. We propose a weighted average of the preliminary k-class estimators, where each estimator is obtained using different subsets of the available instrumental variables. The averaged estimators are shown to be consistent and to satisfy asymptotic normality. Furthermore, its approximate mean squared error reveals that using a small number of instruments for each preliminary k-class estimator reduces the finite sample bias, while averaging prevents the variance from inflating. Monte Carlo simulations find that the averaged estimators compare favorably with alternative instrumental-variable-selection approaches when the strength levels …