# Undergraduate Research | Mathematics | SIU

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## Research

In SIU Mathematics department, we encourage undergraduate students to participate in mathematical research and supervised readings of special topics under guidance of our faculty. Annually, we organize a research conference in undergraduate mathematics research jointly with the Southeast Missouri State University, see IL-MO Conference for details.

Below are names of students and their projects in recent past.

Title:  Use reinforcement learning to optimize a game
agent to compute with scripted agents and novice human players

Details: (1) iterative algorithms for solving Riccati equations
(2) characterizing Lyapunov types of matrix stability

Term: Spring 2018
Student: Preston R. Yun
Professor: Xu, Jianhong

Title: A Solution to Pell's Equation

Details: A classical problem in the history of mathematics was calculating all the integral solutions (m, n) for a fixed D > 0 of m^2 – D·n^2 = +-1. In fact, as long as D is not a perfect square, there are infinitely many distinct pairs (m, n) satisfying the equation.  This project aims to provide a solution to Pell's Equation by introducing some elementary tools from Algebra and Number Theory, and giving similar solutions to other special higher degree cases like that of Pell's equations.

Term: Spring 2017
Student: Zeid Ghalyoun
Professor: Calvert

Title: Improving Numerical Integration Estimates with Inverse Functions

Details: Estimating the integral for power functions with powers 0<p<1 takes much more computational power via standard numerical methods than functions with powers p>1. This is, at least partially due to the fact that such power functions are not Lipschitz continuous. However, as was first shown by C.A. Laisant in 1905, the integrals of certain functions can be expressed as integrals of their inverses. This paper hopes to show that by expressing hard to approximate integrals as integrals of their inverses, methods such as Newton-Cotes formulas and Monte Carlo methods become much more useful and effective than by approximating the integral of the original function alone.

Term: Spring 2017
Student: Thomas Campbell
Professor: Schurz

Title: Natural Computation in Gene Regulatory Networks

Details: The purpose of this research is to implement computability theory in gene regulatory networks. A gene regulatory network is a collection of molecular regulators that interact with each other and other elements in the cell to output gene expression levels. These networks exhibit natural computation. The class of partial recursive functions is the smallest class satisfying the following five axioms: the successor function, the constant function, projection functions, composition, and recursion. This seeks to realize these axioms in gene regulatory networks.

Term: Fall 2016--Spring 2017
Student: Brianna Martin
Professor: Calvert

Title: Sports Data Analysis Project

Details: The group analyzed the Missouri Valley conference Men and Women's basketball data using R software, clustering, and discriminant analysis.

Term:  Fall 15
Students:  Nicole Staples, Philip Kains
Professors:  Budzban, Olive

Title: Research in Visualization in Mathematics

Details: Research in geometry, computing, and visualization.  The work started in Fall 2014 and continued through Fall 2015. Our weekly meetings (Thursday 2pm) were frequented also by a number of other interested students.

Term:  Fall 2014
Students:  Bradley Dragun, Thomas Finkenkelle, Aaron Zolotor
Professor:  Kocik

Term:  Spring 2015
Students:  Jeffrey Lawrence, Aaron Zolotor
Professor:  Kocik

Term:  Fall 2015