Mathematics Commons^™

The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell

Mathematics, Statistics and Data Science Faculty Works

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbid- den number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.

A Stochastic Model For Microbial Fermentation Process Under Gaussian White Noise Environment, Yanping Ma

Mathematics, Statistics and Data Science Faculty Works

In this paper, we propose a stochastic model for the microbial fermentation process under the framework of white noise analysis, where Gaussian white noises are used to model the environmental noises and the specific growth rate is driven by Gaussian white noises. In order to keep the regularity of the terminal time, the adjustment factors are added in the volatility coefficients of the stochastic model. Then we prove some fundamental properties of the stochastic model: the regularity of the terminal time, the existence and uniqueness of a solution and the continuous dependence of the solution on the initial values.

Solving The Ko Labyrinth, Alissa Crans, Robert J. Rovetti, Jessica Vega

Mathematics, Statistics and Data Science Faculty Works

The KO Labyrinth is a colorful spherical puzzle with 26 chambers, some of which can be connected via holes through which a small ball can pass when the chambers are aligned correctly. The puzzle can be realigned by performing physical rotations of the sphere in the same way one manipulates a Rubik’s Cube, which alters the configuration of the puzzle. The goal is to navigate the ball from the entrance chamber to the exit chamber. We find the shortest path through the puzzle using Dijkstra’s algorithm and explore questions related to connectivity of puzzle with the adjacency matrix, distance matrix, …

Enhanced Lasso Recovery On Graph, Xavier Bresson, Thomas Laurent, James Von Brecht

Mathematics, Statistics and Data Science Faculty Works

This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on graphs. More precisely, we present a non-convex, non-smooth algorithm that outperforms the standard convex Lasso technique. We carry out numerical experiments on three benchmark graph datasets.

Numerical Study Of The Supercritical Solution Of The Stationary Forced Korteweg-De Vries (Sfkdv) Equation, Sumi Dey

Open Access Theses & Dissertations

The time independent surface waves on a two dimensional incompressible and inviscid fluid flow passing over a bump on a flat bottom are considered. The stationary forced Korteweg-de Vries (fKdV) equation is defined in an unbounded interval. In this project, this unbounded interval is considered as three parts - on the bump, left hand side of the bump and right hand side of the bump. Exact boundary conditions are applied to the end points of the bump. All the possible combination of boundary conditions are used. A numerical approach is proposed to find the solitary wave solutions of fKdV equation …

When Can The Primitive Element Be Written As A Sum Of Two Algebraic Elements Adjoined To The Field Of The Rational Numbers, Mohamad Moussa

Open Access Theses & Dissertations

Given a field F and elements \alpha and \beta not in F, then F(\alpha, \beta) is the smallest field containing \alpha,\beta, and F. A simple extension is a field extension which is generated by the adjunction of a single element. The Primitive Element Theorem says that if F is a field of characteristic 0, and \alpha and \beta are algebraic over F, then there is an element \gamma in F(\alpha ,\beta ) such that

F(\alpha ; \beta ) = F(\gamma). When can we say that \gamma=\alpha+\beta? We will introduce some situations where \gamma=\alpha+\beta is true and some when this is …

Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur

Turkish Journal of Mathematics

The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions.

Spherically Symmetric Finsler Metrics With Scalar Flag Curvature, Weidong Song, Fen Zhou

Turkish Journal of Mathematics

In this paper, we study spherically symmetric Finsler metrics F= y \phi( x ,\frac{}{ y }), where x \in B^n(r) \subset R^n, y \in T_xB^n(r)\{0} and \phi:[0,r)\times R \rightarrow R. By investigating a PDE equivalent to these metrics being locally projectively flat, we manufacture projectively flat spherically symmetric Finsler metrics in terms of error functions and, using Shen's result, we give its flag curvature.

A Note On Infinite Groups Whose Subgroups Are Close To Be Normal-By-Finite, Francesco De Giovanni, Federica Saccomanno

Turkish Journal of Mathematics

A group G is said to have the CF-property if the index X:X_G is finite for every subgroup X of G. Extending previous results by Buckley, Lennox, Neumann, Smith, and Wiegold, it is proven here that if G is a locally graded group whose proper subgroups have the CF-property, then G is abelian-by-finite, provided that all its periodic sections are locally finite. Groups in which all proper subgroups of infinite rank have the CF-property are also studied.

The Ext-Strongly Gorenstein Projective Modules, Jie Ren

Turkish Journal of Mathematics

In this paper, we introduce and study Ext-strongly Gorenstein projective modules. We prove that the class of Ext-strongly Gorenstein projective modules is projective resolving. Moreover, we consider Ext-strongly Gorenstein projective precovers.

On Ampleness And Pseudo-Anosov Homeomorphisms In The Free Group, Rizos Sklinos

Turkish Journal of Mathematics

We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first-order theory of non-Abelian free groups, T_{fg}, is n-ample for any n \in \omega. This result adds to the work of Pillay, which proved that T_{fg} is non-CM-trivial. The sequence witnessing ampleness is a sequence of primitive elements in F_{\omega}. Our result provides an alternative proof to the main result of a recent preprint by Ould Houcine and Tent.

(X, Y)-Gorenstein Projective And Injective Modules, Qunxing Pan, Faqun Cai

Turkish Journal of Mathematics

This paper introduces and studies (X,Y)-Gorenstein projective and injective modules, which are a generalization of Enochs' Gorenstein projective and injective modules, respectively. Our main aim is to investigate the relations among various (X,Y)-Gorenstein projective modules.

Balanced Pair Algorithm For A Class Of Cubic Substitutions, Tarek Sellami

Turkish Journal of Mathematics

In this article we introduce the balanced pair algorithm associated with 2 unimodular Pisot substitutions having the same incidence matrix. We are interested in beta-substitution related to the polynomial x^3 - ax^2 - bx-1 for a \geq b \geq 1. Applying the balanced pair algorithm to these substitutions, we obtain a general formula for the associated intersection substitution.

On P-Schemes With The Same Degrees Of Thin Radical And Thin Residue, Fatemeh Raei Barandagh, Amir Rahnamai Barghi

Turkish Journal of Mathematics

Let p and n>1 be a prime number and an integer, respectively. In this paper, first we show that any p-scheme whose thin radical and thin residue are equal is isomorphic to a fission of the wreath product of 2 thin schemes. In addition, we characterize association p-schemes whose thin radical and thin residue each have degree equal to p. We also characterize association p-schemes on p^n points whose thin radical and thin residue each have degree equal to p^{n-1}, and whose basis relations each have valency 1 or p^{n-1}. Moreover, we show that such schemes are Schurian.

Arithmetical Rank Of The Edge Ideals Of Some N-Cyclic Graphs With A Common Edge, Guangjun Zhu, Feng Shi, Yan Gu

Turkish Journal of Mathematics

In this paper, we present some lower bounds and upper bounds on the arithmetical rank of the edge ideals of some n-cyclic graphs with a common edge. For some special n-cyclic graphs with a common edge, we prove that the arithmetical rank equals the projective dimension of the corresponding quotient ring.

Good Modulating Sequences For The Ergodic Hilbert Transform, Azer Akhmedov, Doğan Çömez

Turkish Journal of Mathematics

This article investigates classes of bounded sequences of complex numbers that are universally good for the ergodic Hilbert transform in L_p-spaces, 2 \leq p \leq \infty. The class of bounded Besicovitch sequences satisfying a rate condition is among such sequence classes.

Predicting Scenarios For Successful Autodissemination Of Pyriproxyfen By Malaria Vectors From Their Resting Sites To Aquatic Habitats; Description And Simulation Analysis Of A Field-Parameterizable Model, Samson Sifael Kiware, George F. Corliss, Stephen Merrill, Dickson W. Lwetoijera, Gregor J. Devine, Silas Majambere, Gerry F. Killeen

Electrical and Computer Engineering Faculty Research and Publications

Background

Large-cage experiments indicate pyriproxifen (PPF) can be transferred from resting sites to aquatic habitats by Anopheles arabiensis - malaria vector mosquitoes to inhibit emergence of their own offspring. PPF coverage is amplified twice: (1) partial coverage of resting sites with PPF contamination results in far higher contamination coverage of adult mosquitoes because they are mobile and use numerous resting sites per gonotrophic cycle, and (2) even greater contamination coverage of aquatic habitats results from accumulation of PPF from multiple oviposition events.

Methods and Findings

Deterministic mathematical models are described that use only field-measurable input parameters and capture the biological …

Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

• GAMTE 2015 Officers
• GAMTE 2015 Conference Committee
• GAMTE 2015 Proceeding Committee
• Purposes and Goals of GAMTE
• Letter from the President
• Table of Contents

Engaging In Lesson Study At Georgia College, Angel R. Abney, Brandon Samples, Doris Santarone

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught.

I Don't Play Chess: A Study Of Chess Piece Generating Polynomials, Stephen R. Skoch

Senior Independent Study Theses

This independent study examines counting problems of non-attacking rook, and non-attacking bishop placements. We examine boards for rook and bishop placement with restricted positions and varied dimensions. In this investigation, we discuss the general formula of a generating function for unrestricted, square bishop boards that relies on the Stirling numbers of the second kind. We discuss the maximum number of bishops we can place on a rectangular board, as well as a brief investigation of non-attacking rook placements on three-dimensional boards, drawing a connection to latin squares.

The Packing Chromatic Number Of Random D-Regular Graphs, Ann Wells Clifton

Theses and Dissertations

Let G = (V (G),E(G)) be a simple graph of order n and let i be a positive integer. X_i superset V (G) is called an i-packing if vertices in X_i are pairwise distance more than i apart. A packing coloring of G is a partition X = {X₁,X₂,X₃, . . . ,X_k} of V (G) such that each color class X_i is an i-packing. The minimum order k of a packing coloring is called the packing chromatic number of G, denoted by X_p(G). Let G_{n,d …}

A Family Of Multiple Harmonic Sum And Multiple Zeta Star Value Identities, Erin Linebarger, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al [Trans. Amer. Math. Soc. (to appear)]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.

Log-Concavity And Symplectic Rows, Yi Lin, Álvaro Pelayo

Mathematical Sciences: Faculty Publications

The Duistermaat-Heckman measure of a Hamiltonian torus action on a symplectic manifold (M,ω) is the push forward of the Liouville measure on M by the momentum map of the action. In this paper we prove the logarithmic concavity of the Duistermaat-Heckman measure of a complexity two Hamiltonian torus action, for which there exists an effective commuting symplectic action of a 2-torus with symplectic orbits. Using this, we show that given a complexity two symplectic torus action satisfying the additional 2-torus action condition, if the fixed point set is non-empty, then it has to be Hamiltonian. This implies a classical result …

New Families Of Weighted Sum Formulas For Multiple Zeta Values, Yuan Zhao, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

In this paper we use the generating functions and the double shuffle relations satisfied by the multiple zeta values to derive some new families of identities.

Primary Spaces, Mackey's Obstruction, And The Generalized Barycentric Decomposition, Patrick Iglesias-Zemmour, Francois Ziegler

Mathematical Sciences: Faculty Publications

We call a hamiltonian N-space primary if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) x (trivial), as an analogy with representation theory might suggest. For instance, Souriau's barycentric decomposition theorem asserts just this when N is a Heisenberg group. For general N, we give explicit examples which do not split, and show instead that primary spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full "Mackey theory" of hamiltonian G-spaces, where G is an overgroup in which N …

Properly Colored Notions Of Connectivity - A Dynamic Survey, Xueliang Li, Colton Magnant

Mathematical Sciences: Faculty Publications

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs.

Second Hamiltonian Cycles In Claw-Free Graphs, Hossein Esfandiari, Colton Magnant, Pouria Salehi Nowbandegani, Shirdareh Haghighi

Mathematical Sciences: Faculty Publications

Sheehan conjectured in 1975 that every Hamiltonian regular simple graph of even degree at least four contains a second Hamiltonian cycle. We prove that most claw-free Hamiltonian graphs with minimum degree at least 3 have a second Hamiltonian cycle and describe the structure of those graphs not covered by our result. By this result, we show that Sheehan’s conjecture holds for claw-free graphs whose order is not divisible by 6. In addition, we believe that the structure that we introduce can be useful for further studies on claw-free graphs.

Mathematics Boot Camps: A Strategy For Helping Students To Bypass Remedial Courses, Marilyn Ann Louise Hamilton

Walden Dissertations and Doctoral Studies

Many community colleges struggle to find the best strategy to help incoming at-risk students prepare for the placement test. The purpose of this quantitative quasi-experimental study, was to answer the question as to which of 2 programs, a 2-week, face-to-face mathematics refresher program, Math Boost-Up, or an online-only program, might increase the ACCUPLACER posttest scores of incoming community college students. The study used archival data for 136 students who self-selected to either participate in the Math Boost-Up program (the experiment group), or in the online-only program (the comparison group). Knowles's theory of adult learning, andragogy, served as the theoretical framework. …

Commutator Studies In Pursuit Of Finite Basis Results, Nathan E. Faulkner

Theses and Dissertations

Several new results of a general algebraic scope are developed in an effort to build tools for use in finite basis proofs. Many recent finite basis theorems have involved assumption of a finite residual bound, with the broadest result concerning varieties with a difference term (Kearnes, Szendrei, and Willard (2013+)). However, in varieties with a difference term, the finite residual bound hypothesis is known to strongly limit the degree of nilpotence observable in a variety, while, on the other hand, there is another, older series of results in which nilpotence plays a key role (beginning with those of Lyndon (1952) …

Graphs Of Classroom Networks, Rebecca Holliday

College of Graduate Studies: Theses & Dissertations

In this work, we use the Havel-Hakimi algorithm to visualize data collected from students to investigate classroom networks. The Havel-Hakimi algorithm uses a recursive method to create a simple graph from a graphical degree sequence. In this case, the degree sequence is a representation of the students in a classroom, and we use the number of peers with whom a student studied or collaborated to determine the degree of each. We expand upon the Havel-Hakimi algorithm by coding a program in MATLAB that generates random graphs with the same degree sequence. Then, we run another algorithm to find the isomorphism …