Mathematics Commons^™

Long Path Lemma Concerning Connectivity And Independence Number, Shinya Fujita, Alexander Halperin, Colton Magnant

Mathematical Sciences: Faculty Publications

We show that, in a k-connected graph G of order n with α(G)=α, between any pair of vertices, there exists a path P joining them with

|P|≥min{n,(k−1)(n−k)/α +k}.

This implies that, for any edge e∈E(G), there is a cycle containing e of length at least

min{n,(k−1)(n−k)/α +k}.

Moreover, we generalize our result as follows: for any choice S of s≤k vertices in G, there exists a tree T whose set of leaves is S with

|T|≥min{n,(k−s+1)(n−k)/α +k}.

Independence Number And Disjoint Theta Graphs, Shinya Fujita, Colton Magnant

Mathematical Sciences: Faculty Publications

The goal of this paper is to find vertex disjoint even cycles in graphs. For this purpose, define a θ-graph to be a pair of vertices u,v with three internally disjoint paths joining u to v. Given an independence number α and a fixed integer k, the results contained in this paper provide sharp bounds on the order f(k,α) of a graph with independence number α(G)≤α which contains no k disjoint θ-graphs. Since every θ-graph contains an even cycle, these results provide k disjoint even cycles in graphs of order at least f(k,α)+1. We also discuss the relationship between this …

Meander Graphs And Frobenius Seaweed Lie Algebras, Colton Magnant, Vincent E. Coll, Anthony Giaquinto

Mathematical Sciences: Faculty Publications

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

On An Identity Of Gessel And Stanton And The New Little Göllnitz Identities, Carla D. Savage, Andrew Sills

Mathematical Sciences: Faculty Publications

We show that an identity of Gessel and Stanton [I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983) 197, Eq. (7.24)] can be viewed as a symmetric version of a recent analytic variation of the little Göllnitz identities. This is significant, since the Göllnitz–Gordon identities are considered the usual symmetric counterpart to little Göllnitz theorems. Is it possible, then, that the Gessel–Stanton identity is part of an infinite family of identities like those of Göllnitz–Gordon?

Toward this end, we derive partners and generalizations of the Gessel–Stanton identity. We show that …

Hybrid Proofs Of The Q-Binomial Theorem And Other Identities, Dennis Eichhorn, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We give "hybrid" proofs of the q-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version.

We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan.

Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.

Providing Preservice Teachers With Worthwhile Field-Based Experiences In Mathematics, Glenn Nelson

Faculty Work

Providing preservice teachers with worthwhile field-based experiences is recognized as an important component in their development as good teachers. Because mathematics instruction in general has moved from a teacher-directed, procedurally-focused process to a more student-centered, conceptually-oriented approach, preservice mathematics education classes – especially methods courses - should reflect this shift as well. Field-based opportunities can be instances for preservice teachers to personally experience such a shift to real-world relevance.

From Isotropic To Anisotropic Side Chain Representations: Comparison Of Three Models For Residue Contact Estimation, Weitao Sun, Jing He

Computer Science Faculty Publications

The criterion to determine residue contact is a fundamental problem in deriving knowledge-based mean-force potential energy calculations for protein structures. A frequently used criterion is to require the side chain center-to-center distance or the C-alpha-to-C-alpha atom distance to be within a pre-determined cutoff distance. However, the spatially anisotropic nature of the side chain determines that it is challenging to identify the contact pairs. This study compares three side chain contact models: the Atom Distance criteria (ADC) model, the Isotropic Sphere Side chain (ISS) model and the Anisotropic Ellipsoid Side chain (AES) model using 424 high resolution protein structures in the …

An Ethnographic Study Of In-Service And Pre-Service Teacher Collaboration In The Development And Execution Of Mathematics Lessons, Jessica Munns

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

While much is known about effective strategies for teaching mathematics, less is understood about how to lead new teachers to employ research-based practices in their work. To address this issue we followed a Methods of Teaching Secondary and Middle School Mathematics course at Utah State University where collaboration between inservice and pre-service teachers was emphasized. This report documents examples of the implementation of research-based teaching practices by collaborative groups during the course. The structure of collaboration followed a lesson study model involving research and planning, execution and observation, and reflection and revision of lessons. Preservice and in-service teachers met twice …

Extremal Problems For Independent Set Enumeration, Jonathan Cutler, A. J. Radcliffe

Department of Mathematics: Faculty Publications

The study of the number of independent sets in a graph has a rich history. Recently, Kahn proved that disjoint unions of Kr,r’s have the maximum number of independent sets amongst r-regular bipartite graphs. Zhao extended this to all r-regular graphs. If we instead restrict the class of graphs to those on a fixed number of vertices and edges, then the Kruskal-Katona theorem implies that the graph with the maximum number of independent sets is the lex graph, where edges form an initial segment of the lexicographic ordering. In this paper, we study three related questions. Firstly, we …

An Entropy Proof Of The Kahn-Lovasz Theorem, Jonathan Cutler, A. J. Radcliffe

Department of Mathematics: Faculty Publications

Bregman [2], gave a best possible upper bound for the number of perfect matchings in a balanced bipartite graph in terms of its degree sequence. Recently Kahn and Lovasz [8] extended Bregman’s theorem to general graphs. In this paper, we use entropy methods to give a new proof of the Kahn-Lovasz theorem. Our methods build on Radhakrishnan’s [9] use of entropy to prove Bregman’s theorem.

Short-Term Facilitation May Stabilize Parametric Working Memory Trace, Vladimir Itskov, David Hansel, Misha Tsodyks

Department of Mathematics: Faculty Publications

Networks with continuous set of attractors are considered to be a paradigmatic model for parametric working memory (WM), but require fine tuning of connections and are thus structurally unstable. Here we analyzed the network with ring attractor, where connections are not perfectly tuned and the activity state therefore drifts in the absence of the stabilizing stimulus. We derive an analytical expression for the drift dynamics and conclude that the network cannot function as WM for a period of several seconds, a typical delay time in monkey memory experiments. We propose that short-term synaptic facilitation in recurrent connections significantly improves the …

[Introduction To] Basic Statistical Tools For Improving Quality, Paul Kvam, Chang W. Kang

Bookshelf

This book is an introductory book on improving the quality of a process or a system, primarily through the technique of statistical process control (SPC). There are numerous technical manuals available for SPC, but this book differs in two ways: (1) the basic tools of SPC are introduced in a no-nonsense, simple, non-math manner, and (2) the methods can be learned and practiced in an uncomplicated fashion using free software (eZ SPC 2.0), which is available to all readers online as a downloadable product. The book explains QC7 Tools, control charts, and statistical analysis including basic design of experiments. Theoretical …

Combinatorial Aspects Of Excedances And The Frobenius Complex, Eric Logan Clark

University of Kentucky Doctoral Dissertations

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that …

Review: The Mathematics Of Sex: How Biology And Society Conspire To Limit Talented Women And Girls, Susan Jane Colley

Faculty & Staff Scholarship

No abstract provided.

Probability Theory On Time Scales And Applications To Finance And Inequalities, Thomas Matthews

Doctoral Dissertations

"In this dissertation, the recently discovered concept of time scales is applied to probability theory, thus unifying discrete, continuous and many other cases. A short introduction to the theory of time scales is provided. Following this preliminary overview, the moment generating function is derived using a Laplace transformation on time scales. Various unifications of statements and new theorems in statistics are shown. Next, distributions on time scales are defined and their properties are studied. Most of the derived formulas and statements correspond exactly to those from discrete and continuous calculus and extend the applicability to many other cases. Some theorems …

Differential Geometry Of Moving Surfaces And Its Relation To Solitons, Andrei Ludu

Publications

In this article we present an introduction in the geometrical theory of motion of curves and surfaces in R 3 , and its relations with the nonlinear integrable systems. The working frame is the Cartan’s theory of moving frames together with Cartan connection. The formalism for the motion of curves is constructed in the Serret-Frenet frames as elements of the bundle of adapted frames. The motion of surfaces is investigated in the Gauss-Weingarten frame. We present the relations between types of motions and nonlinear equations and their soliton solutions.

Solitary Waves, Periodic And Elliptic Solutions To The Benjamin, Bona & Mahony (Bbm) Equation Modified By Viscosity, S.C. Mancas, Harihar Khanal, Shahrdad G. Sajjadi

Publications

We introduce spatiotemporal solitons of the two-dimensional complex Ginzburg-Landau equation (2D CCQGLE) with cubic and quintic nonlinearities in which asymmetry between space-time variables is included. The 2D CCQGLE is solved by a powerful Fourier spectral method, i.e., a Fourier spatial discretization and an explicit scheme for time differencing. Varying the system’s parameters, and using different initial conditions, numerical simulations reveal 2D solitons in the form of stationary, pulsating and exploding solitons which possess very distinctive properties. For certain region of parameters, we have also found stable coherent structures in the form of spinning (vortex) solitons which exist as a result …

Minimal And Near Minimal Congruence Lattice Representations Of Finite Lattices By Finite Algebras On Sets Of Integers, Roger Lee Bunn

Doctoral Dissertations

"We give finite congruence lattice representations of some finite distributive, modular and nonmodular lattices by means of finite algebras on sets of integers. These representations are minimal or near minimal as determined by ρ, a mapping from the class R of finitely representable lattices into the natural numbers N"--Abstract, page iii.

Numerical Calculation Of Lyapunov Exponents For Three-Dimensional Systems Of Ordinary Differential Equations, Clyde-Emmanuel Estorninho Meador

Theses, Dissertations and Capstones

We consider two algorithms for the computation of Lyapunov exponents for systems of ordinary dierential equations: orbit separation and continuous Gram-Schmidt orthonormal-ization. We also consider two Runge-Kutta methods for the solution of ordinary dierential equations. These algorithms and methods are applied to four three-dimensional systems of ordinary dierential equations, and the results are discussed.

Middle School Mathematics Students' Perspectives On The Study Of Mathematics, Christy H. Vaughn

Walden Dissertations and Doctoral Studies

This qualitative study addressed the perceptions toward the study of mathematics by middle school students who had formerly been in a remedial mathematics program. The purpose of the study was to explore the past experiences of nine students in order to determine what is needed for them to feel successful in mathematics. The conceptual framework of the study was grounded in philosophies of motivation, including achievement goal theory, self-worth theory, self-efficacy theory, expectancy-value theory, and attribution theory. The study used a phenomenological research design to answer the key research question, which focused upon the experiences of students and the meaning …