Mathematics Commons^™

An Introduction To Virtual Spatial Graph Theory, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. We state the definitions, provide some examples, and survey the known results. We hope that this paper will help lead to rapid development of the area.

Boundary Slopes Of 2-Bridge Links Determine The Crossing Number, Jim Hoste, Patrick D. Shanahan

Mathematics, Statistics and Data Science Faculty Works

A diagonal surface in a link exterior M is a properly embedded, incompressible, boundary incompressible surface which furthermore has the same number of boundary components and the same slope on each component of the boundary of M. We derive a formula for the boundary slope of a diagonal surface in the exterior of a 2-bridge link which is analogous to the formula for the boundary slope of a 2-bridge knot found by Hatcher and Thurston. Using this formula we show that the diameter of a 2-bridge link, that is, the difference between the smallest and largest finite slopes of …

Computing Boundary Slopes Of 2-Bridge Links, Jim Hoste, Patrick Shanahan

Mathematics, Statistics and Data Science Faculty Works

We describe an algorithm for computing boundary slopes of 2-bridge links. As an example, we work out the slopes of the links obtained by 1/k surgery on one component of the Borromean rings. A table of all boundary slopes of all 2-bridge links with 10 or less crossings is also included.

Virtual Spatial Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

Two natural generalizations of knot theory are t he study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spat ial graphs.

Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans

Mathematics, Statistics and Data Science Faculty Works

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can …

Intrinsic Linking And Knotting In Virtual Spatial Graphs, Thomas Fleming, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.

Cohomology Of The Adjoint Of Hopf Algebras, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Masahico Saito

Mathematics, Statistics and Data Science Faculty Works

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.

Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Ii, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field. In the present work, we develop two algebraic frameworks corresponding to two interpretations of Kubota's n-Hilbert reciprocity formalism, presented in a quasi-dualized topological form in Part I, delineating two sheaf-theoretic routes toward resolving the aforementioned (open) problem. The first approach centers on factoring sheaf morphisms eventually to yield a splitting homomorphism for Kubota's n-fold cover of the adelized special linear group …

Characterizations Of Pseudo-Codewords Of Ldpc Codes, Ralf Koetter, Wen-Cheng W. Li, Pascal O. Vontobel, Judy L. Walker

Department of Mathematics: Faculty Publications

An important property of high-performance, low complexity codes is the existence of highly efficient algorithms for their decoding. Many of the most efficient, recent graph-based algorithms, e.g. message passing algorithms and decoding based on linear programming, crucially depend on the efficient representation of a code in a graphical model. In order to understand the performance of these algorithms, we argue for the characterization of codes in terms of a so called fundamental cone in Euclidean space which is a function of a given parity check matrix of a code, rather than of the code itself. We give a number of …

Effect Of Inner Scale Atmospheric Spectrum Models On Scintillation In All Optical Turbulence Regimes, Kenneth Mayer

Electronic Theses and Dissertations

Experimental studies have shown that a "bump" occurs in the atmospheric spectrum just prior to turbulence cell dissipation.1,3,4 In weak optical turbulence, this bump affects calculated scintillation. The purpose of this thesis was to determine if a "non-bump" atmospheric power spectrum can be used to model scintillation for plane waves and spherical waves in moderate to strong optical turbulence regimes. Scintillation expressions were developed from an "effective" von Karman spectrum using an approach similar to that used by Andrews et al.8,14,15 in developing expressions from an "effective" modified (bump) spectrum. The effective spectrum extends the Rytov approximation into all optical …

Soliton Solutions Of Nonlinear Partial Differential Equations Using Variational Approximations And Inverse Scattering Techniques, Thomas Vogel

Electronic Theses and Dissertations

Throughout the last several decades many techniques have been developed in establishing solutions to nonlinear partial differential equations (NPDE). These techniques are characterized by their limited reach in solving large classes of NPDE. This body of work will study the analysis of NPDE using two of the most ubiquitous techniques developed in the last century. In this body of work, the analysis and techniques herein are applied to unsolved physical problems in both the fields of variational approximations and inverse scattering transform. Additionally, a new technique for estimating the error of a variational approximation is established. Note that the material …

Stability And Preservation Properties Of Multisymplectic Integrators, Tomasz Wlodarczyk

Electronic Theses and Dissertations

This dissertation presents results of the study on symplectic and multisymplectic numerical methods for solving linear and nonlinear Hamiltonian wave equations. The emphasis is put on the second order space and time discretizations of the linear wave, the Klein-Gordon and the sine-Gordon equations. For those equations we develop two multisymplectic (MS) integrators and compare their performance to other popular symplectic and non-symplectic numerical methods. Tools used in the linear analysis are related to the Fourier transform and consist of the dispersion relationship and the power spectrum of the numerical solution. Nonlinear analysis, in turn, is closely connected to the temporal …

Pade Approximants And One Of Its Applications, Tame-Kouontcho Fowe

Electronic Theses and Dissertations

This thesis is concerned with a brief summary of the theory of Pade approximants and one of its applications to Finance. Proofs of most of the theorems are omitted and many developments could not be mentioned due to the vastness of the field of Pade approximations. We provide reference to research papers and books that contain exhaustive treatment of the subject. This thesis is mainly divided into two parts. In the first part we derive a general expression of the Pade approximants and some of the results that will be related to the work on the second part of the …

A Mathematical Study Of Two Retroviruses, Hiv And Htlv-I, Dana Ali Baxley

Electronic Theses and Dissertations

In this thesis, we examine epidemiological models of two different retroviruses, which infect the human body. The two viruses under study are HIV or the human immunodefiency virus and HTLV-I, which is the human T lymphotropic virus type I. A retrovirus is a virus, which injects its RNA into the host, rather than it's DNA. We will study each of the different mathematical models for each of the viruses separately. Then we use MATLAB-SIMULINK to analyze the models by studying the reproductive numbers in each case and the disease progression by examining the graphs. In Chapter 1, we mention basic …

Emergence Of Singular Structures In Oldroyd-B Fluids, Becca Thomases, Michael Shelley

Mathematics Sciences: Faculty Publications

Numerical simulations reveal the formation of singular structures in the polymer stress field of a viscoelastic fluid modeled by the Oldroyd-B equations driven by a simple body force. These singularities emerge exponentially in time at hyperbolic stagnation points in the flow and their algebraic structure depends critically on the Weissenberg number. Beyond a first critical Weissenberg number the stress field approaches a cusp singularity, and beyond a second critical Weissenberg number the stress becomes unbounded exponentially in time. A local approximation to the solution at the hyperbolic point is derived from a simple ansatz, and there is excellent agreement between …

Estimates Of Positive Solutions To A Boundary Value Problem For The Beam Equation, Bo Yang

Faculty Articles

We consider a two-point boundary value problem for the fourth order beam equation. New upper and lower estimates of positive solutions of the problem are obtained.

A Note On Clean Abelian Groups, Brendan Goldsmith, P. Vamos

Articles

Nicholson defined a ring to be clean if every element is the sum of a unit and an idempotent. A module is clean if its endomorphism algebra is clean. We show that torsion-complete Abelian p-groups are clean and characterize the clean groups among the class of totally projective p-groups. An example is given of a clean p-group which is neither totally projective nor torsion- complete

Optimising The Ratio Of Horseradish Peroxidase And Glucose Oxidase On A Bienzyme Electrode: Comparison Of A Theoretical And Experimental Approach, Dana Mackey, Anthony Killard, Adriano Ambrosi, Malcolm Smyth

Articles

This study compares the behaviour of an electrochemical enzyme biosensor with a theoretical analysis based on a mathematical model and numerical simulation. The biosensor is based on a bi-enzyme channelling configuration, employing the enzymes glucose oxidase and horseradish peroxidase, with direct electron transfer of horseradish peroxidase at a conducting polymer electrode. This was modelled by a system of partial differential equations and boundary conditions representing convective and diffusive transport of the substrates glucose and hydrogen peroxide, as well as reaction kinetics of the bienzyme electrode. The main parameter investigated was the ratio of the two immobilised enzymes, with the aim …

Conformal And Geometric Properties Of The Camassa-Holm Hierarchy, Rossen Ivanov

Articles

Integrable equations with second order Lax pair like KdV and Camassa-Holm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this ontribution. The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of functions, which helps to describe the Inverse Scattering Transform (IST) for the Camassa-Holm hierarchy as a Generalised Fourier Transform (GFT). Using GFT we describe explicitly some members of the CH hierarchy, including integrable deformations for the CH equation. Also we show that solutions …

[Introduction To] Schaum's Outline Of Data Structures With Java, John R. Hubbard

Bookshelf

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Bounds And Comparisons For Weighted Renewal-Type Integral Equations, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this note, inequalities and bounds for weighted renewal-type integral equations are presented. Some upper and lower bounds for the weighted renewal-type integral equations with monotone weight functions are derived. Some upper and lower bounds for the weighted renewal-type equations with monotone weight functions are derived. Bounds for the difference between two weighted renewal functions as well between the parent and weighted renewal functions are obtained in terms of the parent renewal reliability functions and their first and second moments. Relations for renewal-type integrals of the ruin probability are presented. Some inequalities, bounds and convergence results are also established.

Solving Ramanujan's Differential Equations For Eisenstein Series Via A First Order Riccati Equation, James M. Hill, Bruce C. Berndt, Timothy Huber

School of Mathematical & Statistical Sciences Faculty Publications

In this paper we prove that Ramanujan's differential equations for the Eisenstein series P, Q, and R are invariant under a simple one-parameter stretching group of transformations. Using this, we show that the three differential equations may be reduced to a first order Riccati differential equation, the solution of which may be represented in terms of hypergeometric functions. The resulting formulas allow for the derivation of parametric representations of P, Q, and R, analogous to representations in Ramanujan's second notebook. In contrast, in the classical approach, one first needs to derive the fundamental formula connecting theta functions with elliptic integrals. …

The Sheffer Group And The Riordan Group, Tian-Xiao He, Peter Shiue, Leetsch Hsu

Scholarship

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs.

Symbolization Of Generating Functions; An Application Of The Mullin–Rota Theory Of Binomial Enumeration, Tian-Xiao He, Peter S, Leetsch Hsu

Scholarship

We have found that there are more than a dozen classical generating functions that could be suitably symbolized to yield various symbolic sum formulas by employing the Mullin–Rota theory of binomial enumeration. Various special formulas and identities involving well-known number sequences or polynomial sequences are presented as illustrative examples. The convergence of the symbolic summations is discussed.

Construction Of Biorthogonal B-Spline Type Wavelet Sequences With Certain Regularities, Tian-Xiao He

Scholarship

No abstract provided.

Two Number-Theoretic Problems That Illustrate The Power And Limitations Of Randomness, Andrew Shallue

Scholarship

This thesis contains work on two problems in algorithmic number theory. The first problem is to give an algorithm that constructs a rational point on an elliptic curve over a finite field. A fast and easy randomized algorithm has existed for some time. We prove that in the case where the finite field has characteristic 2, there is a deterministic algorithm with the same asymptotic running time as the existing randomized algorithm.

Torsion-Free Groups And Modules With The Involution Property, Brendan Goldsmith, C. Meehan, S.L. Wallutis

Articles

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of -adic integers.

Closure Under Transfinite Extensions, Edgar Enochs, Alina Iacob, Overtoun Jenda

Mathematical Sciences: Faculty Publications

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen's theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions.

On Energy And Expected Uncertainty Measures In Weighted Distributions, Broderick O. Oluyede, Mekki Terbeche

Mathematical Sciences: Faculty Publications

In this note, bounds and inequalities for the comparisons of weighted energy functions, entropy, and discrimination information measures and their unweighted counterparts are presented. Inequalities for weighted expected uncertainty, cross-entropy or discrimination information measures are also presented. A useful result on the convergence of the weighted kernel density informational energy estimates is given and some informational energy applications presented.

A Note On Relations For Reliability Measures In Zero-Adjusted Models, Broderick O. Oluyede, Mavis Pararai

Mathematical Sciences: Faculty Publications

In this note we examine and study relations in zero-adjusted models. Relations for reliability measures in the adjusted and unadjusted models are established and appropriate comparisons including the relative error are presented. The relative error is shown to be a decreasing function of the counts. Some inequalities and comparisons for weighted zero-adjusted models are established.