Mathematics Commons^™

From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville

Topology

No abstract provided.

Introduction To Reconstructability Analysis, Martin Zwick

Complex Systems Faculty Publications and Presentations

This talk will introduce Reconstructability Analysis (RA), a data modeling methodology deriving from the 1960s work of Ross Ashby and developed in the systems community in the 1980s and afterwards. RA, based on information theory and graph theory, is a member of the family of methods known as ‘graphical models,’ which also include Bayesian networks and log-linear techniques. It is designed for exploratory modeling, although it can also be used for confirmatory hypothesis testing. RA can discover high ordinality and nonlinear interactions that are not hypothesized in advance. Its conceptual framework illuminates the relationships between wholes and parts, a subject …

Reality Versus Grant Application Research “Plans”, Linda Burhansstipanov, Linda U. Krebs, Daniel Petereit, Mark Dignan, Sheikh Iqbal Ahamed, Michele Sargent, Krisin Cina, Kimberly Crawford, Doris Thibeault, Simone Bordeaux, Shalini Kanekar, Golam Mushih Tanimul Ahsan, Drew Williams, Ivor D. Addo

Mathematics, Statistics and Computer Science Faculty Research and Publications

This article describes the implementation of the American Indian mHealth Smoking Dependence Study focusing on the differences between what was written in the grant application compared to what happened in reality. The study was designed to evaluate a multicomponent intervention involving 256 participants randomly assigned to one of 15 groups. Participants received either a minimal or an intense level of four intervention components: (1) nicotine replacement therapy, (2) precessation counseling, (3) cessation counseling, and (4) mHealth text messaging. The project team met via biweekly webinars as well as one to two in-person meetings per year throughout the study. The project …

A Note On Sum, Difference, Product And Ratio Of Kumaraswamy Random Variables, Avishek Mallick, Indranil Ghosh, Gholamhossein G. Hamedani

Mathematics, Statistics and Computer Science Faculty Research and Publications

Explicit expressions for the densities of S = X₁ + X₂ , D = X₁ − X₂ , P = X₁X₂ and R= X₁/X₂ are derived when X₁ and X₂ are independent or sub-independent Kumaraswamy random variables. The expressions appear to involve the incomplete gamma functions. Some possible real life scenarios are mentioned in which such quantities might be of interest.

Green's Functions And Lyapunov Inequalities For Nabla Caputo Boundary Value Problems, Areeba Ikram

Department of Mathematics: Dissertations, Theses, and Student Research

Lyapunov inequalities have many applications for studying solutions to boundary value problems. In particular, they can be used to give existence-uniqueness results for certain nonhomogeneous boundary value problems, study the zeros of solutions, and obtain bounds on eigenvalues in certain eigenvalue problems. In this work, we will establish uniqueness of solutions to various boundary value problems involving the nabla Caputo fractional difference under a general form of two-point boundary conditions and give an explicit expression for the Green's functions for these problems. We will then investigate properties of the Green's functions for specific cases of these boundary value problems. Using …

Mathematical Models, Patty Wagner, Marnie Phipps

Mathematics Grants Collections

This Grants Collection for Mathematical Models was created under a Round Nine ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Preliminary Results Of Bayesian Networks And Reconstructability Analysis Applied To The Electric Grid, Marcus Harris, Martin Zwick

Complex Systems Faculty Publications and Presentations

Reconstructability Analysis (RA) is an analytical approach developed in the systems community that combines graph theory and information theory. Graph theory provides the structure of relations (model of the data) between variables and information theory characterizes the strength and the nature of the relations. RA has three primary approaches to model data: variable based (VB) models without loops (acyclic graphs), VB models with loops (cyclic graphs) and state-based models (nearly always cyclic, individual states specifying model constraints). These models can either be directed or neutral. Directed models focus on a single response variable whereas neutral models focus on all relations …

Finding Meaning In A Multivariable World: A Conceptual Approach To An Algebra-Based Second Course In Statistics, Karen Mcgaughey, Beth Chance, Nathan L. Tintle, Soma Roy, Todd Swanson, Jill Vander Stoep

Faculty Work Comprehensive List

Although the teaching of the first course in statistics has improved dramatically in recent years, there has been less focus on a similarly conceptual-based second course aimed at non-majors. We present a curriculum for the second course, designed to expand statistical literacy across disciplines, which focuses on conceptual understanding of multivariable relationships through data visualization, study design, the role of confounding variables, reduction of unexplained variation, and simulation-based inference, rather than the mathematically-based discourse often used in the second course. Our curriculum uses a student-centered pedagogical approach, utilizing guided discovery activities based on real-world case studies, facilitated by student-focused technology …

A Greedy Algorithm For Finding A Large 2-Matching On A Random Cubic Graph, Deepak Bal, Patrick Bennett, Tom Bohman, Alan Frieze

Department of Mathematics Faculty Scholarship and Creative Works

A 2-matching of a graph G is a spanning subgraph with maximum degree two. The size of a 2-matching U is the number of edges in U and this is at least n-k(U) where n is the number of vertices of G and κ denotes the number of components. In this article, we analyze the performance of a greedy algorithm 2greedy for finding a large 2-matching on a random 3-regular graph. We prove that with high probability, the algorithm outputs a 2-matching U with k(U)=Θ(n1/5).

Finding Torsion-Free Groups Which Do Not Have The Unique Product Property, Lindsay Jennae Soelberg

Theses and Dissertations

This thesis discusses the Kaplansky zero divisor conjecture. The conjecture states that a group ring of a torsion-free group over a field has no nonzero zero divisors. There are situations for which this conjecture is known to hold, such as linearly orderable groups, unique product groups, solvable groups, and elementary amenable groups. This paper considers the possibility that the conjecture is false and there is some counterexample in existence. The approach to searching for such a counterexample discussed here is to first find a torsion-free group that has subsets A and B such that AB has no unique product. We …

Adding Limit Points To Bass-Serre Graphs Of Groups, Alexander Jin Shumway

Theses and Dissertations

We give a brief overview of Bass-Serre theory and introduce a method of adding a limit point to graphs of groups. We explore a basic example of this method, and find that while the fundamental theorem of Bass-Serre theory no longer applies in this case we still recover a group action on a covering space of sorts with a subgroup isomorphic to the fundamental group of our new base space with added limit point. We also quantify how much larger the fundamental group of a graph of groups becomes after this construction, and discuss the effects of adding and identifying …

Euclidean Domains, Vandy Jade Tombs

Theses and Dissertations

In the usual definition of a Euclidean domain, a ring has a norm function whose codomain is the positive integers. It was noticed by Motzkin in 1949 that the codomain could be replaced by any well-ordered set. This motivated the study of transfinite Euclidean domains in which the codomain of the norm function is replaced by the class of ordinals. We prove that there exists a (transfinitely valued) Euclidean Domain with Euclidean order type for every indecomposable ordinal. Modifying the construction, we prove that there exists a Euclidean Domain with no multiplicative norm. Following a definition of Clark and Murty, …

Percent On & Off, Johannes Familton

Open Educational Resources

This PowerPoint was made for a Quantitative Reasoning, Liberal arts math course. This is an example of what one can do using only OER resources. This PowerPoint with the use of animation, animated gifs and a nice background keeps students engaged in the lesson. The contents of this lesson is taken from Math in Society, by David Lippman, Pierce College Ft Steilacoom. The text is licensed under a Creative Commons Attribution-Share Alike 3.0 United States License. PPT Background: http://www.MyFreePPT.com Animated gifs: http://www.zingerbug.com

Primitive Substitution Tilings With Rotational Symmetries, April Lynne D. Say-Awen, Ma. Louise Antonette N. De Las Peñas, Dirk Frettlöh

Mathematics Faculty Publications

This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are given of the substitutions σ6 and σ7 which give rise to aperiodic primitive substitution tilings with dense tile orientations and which are invariant under six- and sevenfold rotations, respectively; the derivation of the symmetry orders of their hulls is also presented.

Data Assimilation In The Boussinesq Approximation For Mantle Convection, Shane Alexander Mcquarrie

Theses and Dissertations

Many highly developed physical models poorly approximate actual physical systems due to natural random noise. For example, convection in the earth's mantle—a fundamental process for understanding the geochemical makeup of the earth's crust and the geologic history of the earth—exhibits chaotic behavior, so it is difficult to model accurately. In addition, it is impossible to directly measure temperature and fluid viscosity in the mantle, and any indirect measurements are not guaranteed to be highly accurate. Over the last 50 years, mathematicians have developed a rigorous framework for reconciling noisy observations with reasonable physical models, a technique called data assimilation. …

Congruences For Fourier Coefficients Of Modular Functions Of Levels 2 And 4, Eric Brandon Moss

Theses and Dissertations

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and Jenkins. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at infinity. We also demonstrate congruences for Fourier coefficients of some level 4 modular functions.

The Arithmetic Of Modular Grids, Grant Steven Molnar

Theses and Dissertations

Let M_k^(∞) (Gamma, nu) denote the space of weight k weakly holomorphic weight modular forms with poles only at the cusp (∞), and let widehat M_k^(∞) (Gamma, nu) subseteq M_k^(∞) (Gamma, nu) denote the space of weight k weakly holomorphic modular forms in M_k^(∞) (Gamma, nu) which vanish at every cusp other than (∞). We construct canonical bases for these spaces in terms of Maass--Poincaré series, and show that the coefficients of these bases satisfy Zagier duality.

Webwork Problems For Linear Algebra, Hashim Saber, Beata Hebda

Mathematics Ancillary Materials

This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.

Topics covered include:

• Linear Independence
• Linear Transformations
• Matrix of a Transformation
• Isomorphisms
• Eigenvalues and Eigenvectors
• Diagonalization
• Orthogonality

Preconditioning Methods For Thin Scattering Structures Based On Asymptotic Results, Josef A. Sifuentes, Shari Moskow

School of Mathematical & Statistical Sciences Faculty Publications

We present a method to precondition the discretized Lippmann–Schwinger integral equations to model scattering of time-harmonic acoustic waves through a thin inhomogeneous scattering medium. The preconditioner is based on asymptotic results as the thickness of the third component direction goes to zero and requires solving a two dimensional formulation of the problem at the preconditioning step.

Beyond Spatial Autocorrelation: A Novel Approach Using Reconstructability Analysis, David Percy, Martin Zwick

Complex Systems Faculty Publications and Presentations

Raster data are digital representations of spatial phenomena that are organized into rows and columns that typically have the same dimensions in each direction. They are used to represent image data at any scale. Common raster data are medical images, satellite data, and photos generated by modern smartphones.
Satellites capture reflectance data in specific bands of wavelength that correspond to red, green, blue, and often some infrared and thermal bands. These composite vectors can then be classified into actual land use categories such as forest or water using automated techniques. These classifications are verified on the ground using hand-held sensors. …