Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Title: Experimental Mathematics Applied to the Study of Non-linear Recurrences
Speaker: Emilie Hogan, Rutgers University
Date: Thursday, April 7, 2011 5:00pm
Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ
In this thesis defense I will talk about two topics within the field of non-linear recurrences that appear as two chapters of my dissertation. First I will talk about global asymptotic stability in rational recurrences. A recurrence is globally asymptotically stable when the sequence it produces converges to an equilibrium solution given any initial conditions. I will explain the algorithm that I developed, in joint work with Doron Zeilberger, that takes as input a rational recurrence relation conjectured to be globally asymptotically stable, and outputs a rigorous proof of its stability. I will show how this algorithm has been applied to prove global asymptotic stability of some rational recurrences.
Secondly I will discuss a new type of generalized recurrence. Instead of producing a single sequence, these generalized recurrences produce infinitely many sequences from one set of initial conditions. I will present two families that produce rational numbers when complex numbers are expected, and observe that exponential sequences are being produced.
In addition, I will briefly mention the third topic from my thesis: a 3 parameter family of rational recurrences that produces integer sequences. I will mention the process used to prove integrality of these sequences. This topic was inspired by the Somos recurrences.
This is a Ph.D. Thesis Defense, but open to the public.