# Montclair State University

Title: Algebraic Applications in Discrete Dynamic Systems

Director: Aihua Li

Email: lia@mail.montclair.edu

Dates of Program: June 23 - August 1, 2008

Summary:

### Project I: Study of Discrete Time Series and Polynomial Modeling

This is the continuation of an ongoing student project at MSU. Through studying and developing algebraic methods in modeling discrete time series, this project aims to provide a different approach to the traditional modeling problem. We will consider time series data from an MSU biology lab about gene expression. The purpose is to analyze the changes of those data points and their interaction over time. Of increasing concerns are the mathematical models derived from these data, properties of, and measurements of the models, and subsequent interpretation of the points in terms of properties of the underground systems that produce the models.

### Project II: Study of Certain Diophantine Equations

Many of the great advances in number theory are due to the study of finding solutions to these equations. The focus of our research is on the following type of Diophantine equations: Ax2+Bym=Czn (1), where A, B, C are nonzero integers and m, n are positive integers. Recent developments in algebraic geometry have brought new methods and new ideas to solve Diophantine equations. From an earlier work of our REU students, an algorithm to search polynomial solutions whose integer values are solutions to Equation (1) was developed. Students working on this problem will apply this algorithm to search for new solutions. For each solution obtained, a study on the type of solutions derived from the polynomials will be conducted. Furthermore, we expect to develop methods of using existing solutions to obtain new solutions.

### Project III: Study of DS-Divisors of Positive Integers

The student will study behavior of DS-divisors of a positive integer. Here ’DSâ? stands for ’divisor-squaredâ?. A positive integer q is called a DS-divisor of a positive integer c if q2|c-q. Such a pair (c, q), is called a DS-pair. From a table generated for DS-pairs, we examine the existence and the numbers of positive DS-divisors of prime powers, product of two prime powers, and other cases. We also investigate patterns and structures of DS-divisors based on our observations of the table. In addition, we investigate the relationship between DS-divisors and Euler Numbers.

### Project IV: Searching Conductive Paths in Lattices.

This project is based on a recent collaborative research in material science which applies graph theory. The central question that we consider is that of the determination of the presence in sufficient density of a conductive nanofiller within a composite nonconductive matrix to permit the expression of some conductive property of the composite material. Students will be given a lattice with certain conditions. The points of the lattice are classified as either conductive or non-conductive. The conductive points are considered to be randomly distributed with probability p which we interchangeably denote as the probability or particle density. The question addressed by this study is for what value of the particle density that the expected value of the number of such conductive paths exceeds one. This is the expected onset of percolation. Students will work on searching and counting all possible conductive paths.

Student Researchers Supported by MAA:

• Ellizabeth Arango, Montclair State University
• Emil Demirel, Montclair State University
• Chinua Umoja, Morehouse College
• Dornell E. Wilson, Montclair State University

Program Contacts:

Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473

Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470