Colloquia — Fall 1998
Friday, November 13, 1998
Domination Numbers of Small Graphs
W. Edwin Clark
This will be a self-contained talk on the tightest upper bound for the domination number of graphs with \(n\) vertices and minimum degree delta. In this talk I will concentrate on graphs with a small number of vertices. The smallest unknown value is for \(n=15\) and \(d=6\). This work apparently requires neither Banach spaces nor the Axiom of Determinancy. I will define all the necessary graph theoretic terms.
Friday, November 20, 1998
The Wigner-Ville Distribution and Time-Frequency Signal Analysis
University of Central Florida
Although the Fourier transform analysis is one of the great achievements in mathematics and has widespread applications in science and engineering, it cannot be used effectively to analyze non-stationary signals. In order to overcome the inherent difficulty of the Fourier transform, three or four different methods including the Gabor transform, the Zak transform, wavelet transform and the Wigner distribution have been developed for time-frequency signal analysis. In this talk, recent developments of the Wigner-Ville distribution, and \(M\)-band wavelet analysis will be discussed. Special attention will be given to applications to a wide variety of signals encountered in physical, engineering and biomedical sciences.
Friday, December 4, 1998
\(N\) Charges on the Sphere
Monday, December 7, 1998
New observations on the Gollnitz-Gordon and Rogers-Ramanujan identities
University of Florida
In the entire theory of partitions and \(q\)-series, the Rogers-Ramanujan identities are unmatched in simplicity, elegance, and depth. The Gollnitz-Gordon identities are the perfect analogues to the modulus \(8\) for what the Rogers-Ramanujan identities are to the modulus \(5\). In this talk two new and simple proofs of the Gollnitz-Gordon identities will be given by bisection of well known and fundamental theta function identities. Similar methods applied to the Rogers-Ramanujan identities lead to new product representations modulo \(80\). Other implications of this method of bisection include new shifted partition identities, a new proof of the quituple product identity, and modular relations for various theta series arising as generating functions of partition functions. The talk will be accessible to non-experts and graduate students.
Tuesday, December 8, 1998
The WZ Algorithms and Some Applications to Problems in Analysis and Combinatorics
Most mathematicians have heard about the large body of algorithms known collectively as the WZ algorithms, which recently earned for their creators the Steele prize. However for many of us, the processes of accessing and using these algorithms remain mysterious.
In this lecture, I will explain how two of the most powerful algorithms in these packages, zeil and hyper, can be accessed from the internet and I illustrate their use in three typical problems in applied mathematics: finding explicit formulas for the associated Legendre polynomials, calculating Hankel determinants of combinatorial polynomials, and determining connecting coefficients that arise in certain quantum chemistry problems.