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Mathematics & Statistics

# Colloquia — Fall 2002

## Tuesday, December 17, 2002

Title
Speaker

Time
Place

Another Measure of Uncertainty
M. Rao
University of Florida
3:00pm-4:00pm
PHY 130
Arunava Mukherjea

Abstract

Information Theory originates from two path breaking papers of Claude E. Shannon (1948) in which he proposed a quantitative measure of information and uncertainty in a random phenomenon based on the classical Boltzmann Entropy of Statistical Physics. Shannon proposed that a measure of information in the occurance of an uncertain event of probability $$p$$ is $$-\log p$$. It is difficult to overestimate the influence of this work on modern information theory.

In this paper we use the cumulative distribution of a random variable to define the information content in it and thereby develop a novel measure of information that parallels Shannon entropy, which we call cumulative residual entropy (CRE). The salient features of CRE are,

1. it is more general than the Shannon entropy in that its definition is valid in the continuous and discrete domains,
2. it possesses more general mathematical properties than the Shannon entropy and
3. it can be easily computed from sample data and these computations aymptotically converge to the true values.

We discuss some applications of our new information measure to image processing and point out its advantages over the use of traditional Shannon entropy. We also give a precise formula relating CRE and Shannon entropy.

## Wednesday, December 4, 2002

Title
Speaker

Time
Place

Extension of Slater's List: New Generalizations of Rogers-Ramanujan Type Identities
Tina Garrett
Carleton College
Northfield, MN
3:00pm-4:00pm
PHY 118

Abstract

In this talk we will review several known identities of Rogers-Ramanujan Type. We will outline a method for extending these identities to give more general theorems and give several examples from L. J. Slater's 1950 list of theorems that can be improved.

## Friday, November 22, 2002

Title
Speaker

Time
Place

An Algorithmic Approach to Rogers-Ramanujan Type Identities
Andrew Sills
Penn State University
4:15pm-5:15pm
PHY 108

Abstract

108 years after their initial discovery by L. J. Rogers, the Rogers-Ramanujan identities continue to stimulate research in numerous areas of the mathematical sciences including the theory of partitions, Lie algebras, statistical physics, and symbolic computation.

I will discuss a method for producing polynomial generalizations of Rogers-Ramanujan type identities via an algorithmic method using nonhomogeneous $$q$$-difference equations. Next, I will discuss some of the implications of this method for algorithmic proof theory and statistical mechanics.

## Friday, November 22, 2002

Title

Speaker

Time
Place

Integrability Characteristics and Coherent Structures of Integrable Two-Dimensional Generalizations of NLS Type Equations
S. R. Choudhury
University of Central Florida
3:00pm-4:00pm
LIF 262
Wen-Xiu Ma

Abstract

A recent algorithmic procedure based on truncated Painlevè expansions is used to derive Lax Pairs, Darboux Transformations, Hirota Tau Functions, and various soliton solutions for integrable $$(2+1)$$ generalizations of NLS type equations [1]. In particular, diverse classes of solutions are found analogous to the dromion, instanton, lump, and ring soliton solutions derived recently for $$(2+1)$$ KdV Type Equations, the Nizhnik-Novikov-Veselov Equation, and the Broer-Kaup system. If time permits and results are available, possible applications of these solutions, as well as possible applications of the same techniques to non-isospectral integrable hierarchies, may be considered.

[1] A. V. Mikhailov and R. I. Yamilov, On Integrable Two-Dimensional Generalizations of NLS Type Equations, Phys. Letters A 230 (1997), no. 5-6, 295-300.

## Wednesday, November 20, 2002

Title
Speaker

Time
Place

Inversion of Bilateral Basic Hypergeometric Series
Michael Schlosser
University of Vienna
Vienna, AUSTRIA
3:00pm-4:00pm
TBA

Abstract

We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised $${}_6\psi_6$$ summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

## Friday, November 15, 2002

Title
Speaker

Time
Place

The Generalized Chebyshev Polynomials
Yang Chen
Imperial College
London, UK
3:00pm-4:00pm
LIF 262

Abstract

We construct explicitly using theta functions on a particular Riemann surface polynomials orthogonal over a union of several disjoint intervals.

## Tuesday, November 12, 2002

Title
Speaker

Time
Place

A Generalization of Polynomials: Nichols Algebras
Matias Grana
M.I.T.
Cambridge, MA
3:30pm-4:30pm
PHY 118
Masahiko Saito

Abstract

In the 70's two physicists, Yang (doing quantum mechanics) and Baxter (doing statistical mechanics) arrived to the same equation, the now famous Yang-Baxter equation. In the 80's two mathematicians, Drinfeld and Jimbo, constructed an enormous family of solutions to that equation, opening a branch of algebra (and physics) now known as Quantum Groups.

In a completely unrelated fashion, in his Ph.D. thesis in 1978, Warren Nichols defined a family of algebras which he called Bialgebras of Type One. By results proved in the 90's, we now know how to produce one of these algebras from any solution of the Yang-Baxter equation. We call then these algebras by “Nichols algebras”. Examples of them are polynomial algebras, exterior algebras and “quantum Borel algebras”, which are a main part of the Quantum Groups defined by Drinfeld and Jimbo.

I will define Nichols algebras and give some examples based on quandles. (Quandles are a nice tool for topologists, but this is another story). The talk is intended for a general audience; no knowledge of physics, Yang-Baxter equations, Quantum Groups, Quandles or any other “mysterious” thing is required, as I'll define every used object.

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