Abstracts for the Student Paper Session, Fall 2008 MAA Seaway Section Meeting

Matching Problems with Utility Functions

Lhakpa Sherpa

Elmira College

The Gale Shapley algorithm gives a solution to stable matching problems, such as the marriage problem, using a ranking matrix that provides information on how the men rank the women and vice versa. It is restrictive in the sense that once a stable matching is attained there are limited strategic moves that players (men and women) can make without hurting someone else.

This paper extends the Gale-Shapley algorithm with a utility matrix, providing the amount of utility that the men assign to the women and the women assign to men. Unlike the ranking matrix, it provides more information because it quantifies the difference between the different rankings. We show that stable matching as defined by Gale and Shapley is still attainable using the Gale Shapley algorithm. But in addition, after the stable matching is attained, it is possible for the players to make moves to better their choices.

Medians of the Degrees of Vertices in a Bipartite Graph

Hidefumi Kakizoe

Elmira College

In this research, we examine the distribution of the degrees of vertices in bipartite graph.  The "High School Prom Theorem" is well known as one theorem arguing the relationship between the means of the two groups of vertices in a bipartite graph, but there is little known about the medians of the degrees of the vertices.  We study how the medians of the two groups of vertices are likely to be different, concentrating on the case where the bipartite graph is conjugate.  For this case, we establish limit theorems on the probability of the medians and on their expected values, and a bound on the difference between the medians.

An Overview and Motivation into Higher-Level Mathematics

Ryan M. Northrup


Clarkson University

Interval mathematics is an exciting branch of applied mathematics that is practical—useful to the engineers and scientists of tomorrow.  Given the limitations of today’s measuring devices and computing equipment, interval mathematics provides an intuitive means of collecting and propagating measurement errors.  This talk will begin with an introduction to the main theories of Interval Mathematics, starting with simple operations and some examples. The talk will conclude with a glimpse into a more advanced, multi-branched application involving interval mathematics, differential equations and numerical methods.

What is a complete group?

Elizabeth Wilcox

Binghamton University

In this talk we will review the definition of a group automorphism and discuss different types of automorphisms, in particular those automorphisms known as inner automorphisms.  We will discuss complete groups and give examples of familiar groups which are complete.  This talk should be appropriate for those who have taken only an introductory class in group theory.

Linear Programming and Combinatorics

Garry Bowlin

SUNY Binghamton

Linear programming is the problem of maximizing a linear functional over a convex region given by linear constraints, and there are fast algorithms to solve problems of this type. Another use of linear programming in combinatorics is finding an upper bound to integer programming problems, where solutions are required to be integer valued. The difficulty in solving these problems lies in finding a set of valid constraints. This talk will focus on one example of this method, the Gale-Berlekamp Switching game, and how it could be used in an undergraduate combinatorics course.

Recurrence Linear Relations with Modulo Arithmetic

Manuel Lopez

Rochester Institute of Technology

We study the recurrence relation with modular arithmetic, i.e over the ring . The values generated by iterating the equation partition into equivalence classes. Such equivalence classes have two different internal structures as follows:

If then each equivalence class is a cycle.

If then each equivalence class is eventually periodic

Restricting ourselves to the elements which are periodic we find that the set of cycles they give rise to have an abelian group structure with the appropriate cycle addition. Our goal is to investigate how the number theoretic relationship between different values a and M affect the group structures. This is investigation forms the basis of the project on the discrete logarithm presented by Mark Lemay.

Extending the Logarithm in Modulo Arithmetic

Mark Lemay

Rochester Institute of Technology

Studying the recurrence relation we propose a way to expand the traditional discrete logarithm modulo M with base c. Our proposed function has all element of as its domain. We explore what is in this context the appropriate counterpart to the standard laws of logarithms. We also explore how change of base and change of modulus affects our proposed logarithm function.

Systole lengths in hyperbolic knot or link complements in S^3

Sree-Krishna Palaparthi

SUNY Buffalo

A systole in a hyperbolic 3-manifold is a shortest closed geodesic in it. Colin Adams and Alan Reid showed that the length of a systole in a hyperbolic link complement in S^3 is at most 7.35534.. In this talk I will present their result for hyperbolic knot complements in S^3 and observe that there is a better upper bound of 7.171646.. due to the improvement of the Thurston's and Gromov's 2 pi theorem by Ian Agol and Marc Lackenby.

Palindromic congruence properties of p(n,m)

Brandt Kronholm

SUNY Albany

p(n,m) is the function which enumerates the number of partitions of the non-negative integer n into exactly m parts.  In a previous publication [PAMS,133 (2005), 2891-2895] the speaker established and gave an explicit formula for an infinite family of Ramanujan-like congruences for p(n,m).  The goal of this talk is to discuss a new result showing that almost all of these congruences occur in pairs and those that do not are easily identified.

The Generalized MLE With Interval-Censored And Masked Competing Risks Data

Jiaping Wang

SUNY Binghamton

We consider the estimation problem of the joint cumulative distribution function (cdf) of the failure time T and the failure cause C of a J-component series system. The study is motivated by a cancer research data with interval-censored (IC) T and masked C. This type of data is called the interval-censored and masked competing risks (ICMCR) data. We propose to estimate the cdf by the generalized maximum likelihood estimator (GMLE). In general, there is no explicit solution for the  GMLE based on the ICMCR data. We discuss the algorithm for the GMLE. We show that with the continuous right-censored and masked competing risks data the standard GMLE is inconsistent. However, our simulation results suggest that with ICMCR data the GMLE is consistent. Moreover, we study the empirical convergent rates of the GMLE through simulation.

An Application of Cavalieri’s Principle: Calculating the Volume of the Steinmetz Solid

Yeyao Xiao

Plattsburgh State University

The solid common to two right circular cylinders with equal radii intersecting at right angle is called Steinmetz Solid, or bicylinder. Using integral calculus, one can snow that the volume V of the Steinmetz Solid with radius r is given by V=16r³/3. But the formula for the volume of Steinmetz Solid was known to Archimedes and to the Chinese mathematician Tsu Ch’ung-Chih, long before the development of calculus. Tsu Ch’ung-Chih’s calculation of the volume is an application of what is now called Cavalieri’s Principle.

Do Graphing Calculators Help Students Perform Better on the New York Mathematics A Regents?

Benjamin Norris

Niangara University

The usage of graphing calculators in today’s math classroom has been a much heated debate amongst educators. While some have hailed it as a marvelous tool that all students should have access to when performing complex math problems, others have felt the calculator does not actually help students comprehend the math they are doing. To see if graphing calculators statistically impact how well students perform in math, 15 tenth-graders were asked to complete two New York State Mathematics A Regents. During the first test scenario, students were allowed to use a graphing calculator to complete the exam. Likewise, when the students took the second Regents, they were not allowed to use calculators of any sort. After the Regents were collected and scored, the data was then analyzed to see if graphing calculators statistically had an impact on the student’s Regents score. The students also completed an informal survey as to how they felt about completing the Regents without any calculator whatsoever.

Did the students do better on their Math A Regents by having a calculator? Regardless of your personal opinion on calculators, I implore you to come and listen to this talk.