Contributed presentations

Contributed Paper Session

Saturday, 2:10-3:30 PM

Session 1 - Moderator: Shannon Lockard

2:10-2:25 Implications of the difference in math education between the US and abroad

William Velez*, Tucson

Alicia Prieto-Langarica, Youngstown State University

The bachelor's degree educational requirements between the US and most other countries are very different. We will describe these differences in this talk and discuss the implications for opportunities for study abroad for US students.

2:30-2:45 Excel Projects for a Probability Class

Jason Molitierno, Sacred Heart University

In a probability class that is often taken by math majors, the major topics are discrete distributions, continuous distributions, and joint distributions. In this talk, I will discuss projects that I give in order to reinforce the concepts of these distributions. I will discuss using Excel to visualize various distributions such as the binomial, negative binomial, Poisson, normal, exponential, and general joint distributions. I will discuss how I have students create graphs of these distributions using Excel and how I ask students to analyze these distributions.

2:50-3:05 Skolem Number of Subgraphs of the Triangular Lattice

Braxton Carrigan*, Southern Connecticut State University

G. Green, Southern Connecticut State University

A Skolem sequence can be thought of as a labelled path where two vertices with the same label are that distance apart. This concept has naturally been generalized to graph labelling. This brings rise to the question; ``what is the smallest set of consecutive positive integers we can use to proper Skolem label a graph?'' This is known as the Skolem number of the graph. In this paper we give the Skolem number for three natural vertex induced subgraphs of the triangular lattice graph.

3:10-3:25 Delayed wins in Tic-Tac-Toe type games

Klay Kruczek, Southern Connecticut State University

A positional game is a 2-player game played on a hyper-graph, where the winning lines are the hyper-edges, and each player alternates taking a vertex. In a strong game, the first player to occupy a winning line wins. The Erd\H{o}s-Selfridge Theorem states if a positional game is played on an $n$-uniform hypergraph with less than $2n-1$ hyper-edges, then Player 2 can force a draw. Erd\H{o}s and Beck created $n$-uniform hypergraphs with exactly $2n-1$ hyper-edges on which Player 1 wins in n moves. We modify these games so Player 1 is forced to use at least $2n – 3$ moves.

Session 2

2:10-2:25 Enumeration of Discrete Gradient Vector Fields on Simplicial Complexes

Andrew Tawfeek*, Amherst College

Ivan Contreras, Amherst College

Discrete Morse theory has been developed over the past few decades, since its original formulation by Forman in 1998, and has wide-ranging application today. We then provide a novel approach to enumerating the discrete gradient vector fields on finite simplicial complexes and show that the characteristic polynomial of the Laplacian serves as a special generating function for gradients. We prove this result completely for $1$-dimensional complexes, then discuss our current research on generalizing our results to higher-dimensional simplicial complexes.

2:30-2:45 Radius of convergence for Maclaurin series of $p$-trigonometric functions

Rob Poodiack*, Norwich University

Bill Wood, University of Northern Iowa

The $p$-trigonometric functions $\sin_p$ and $\cos_p$ parameterize the closed curve $|x|^p + |y|^p = 1$. To approximate the values of these functions, we would like to be able to turn on occasion to Maclaurin series. Unlike the $p=2$ case, these series for $p>2$ appear to have a finite radius of convergence. We will explore these series and why the $p>2$ case is different from our usual experiences with sine and cosine, with a side trip to generalized hyperbolic, exponential, and logarithmic functions in the process.

2:50-3:05 A Second Order Rational Difference Equation with Quadratic Terms

Zachary Kudlak*, US Coast Guard Academy

Yevgeniy Kostrov, Manhattanville College

We give the character of solutions of the following second-order rational difference equation with quadratic denominator

\[

x_{n+1}=\dfrac{\alpha + \beta x_{n}}{Bx_n + Dx_nx_{n-1} + x_{n-1}},

\]

where the coefficients are positive numbers, and the initial conditions $x_{-1}$ and $x_0$ are nonnegative such that the denominator is nonzero. In particular, we show that the unique positive equilibrium is locally asymptotically stable, and we give conditions on the coefficients for which the unique positive equilibrium is globally stable.

3:10-3:25 Predator-Prey Linear Coupling with Hybrid Species

Jean-Luc Boulnois, Babson College

The classical two-species non-linear Predator-Prey system, often used in population and epidemic modeling, is expressed in terms of a single positive coupling parameter λ. Based on standard transformations of prey and predator populations, we derive a novel λ-invariant Hamiltonian resulting in two coupled 1rst-order ODEs for “hybrid-species", albeit with one being linear; this enables the derivation of a new exact, closed-form, single quadrature solution valid for any value of λ and the system's energy. In the particular case λ = 1 the ODE system completely uncouples and a new, exact, energy-only dependent simple quadrature solution is derived. In the case λ - 1 an accurate practical approximation uncoupling the non-linear system is proposed and solutions are provided in terms of explicit quadrature together with high energy asymptotic solutions. A novel, exact, closed-form expression of the system's oscillation period valid for any value of λ and orbital energy is also derived; two fundamental properties of the period are established; for λ = 1 the period is shown to be shortest and expressed in terms of a unique energy function.