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Mathematics & Statistics

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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,

- it is more general than the Shannon entropy in that its definition is valid in the continuous and discrete domains,
- it possesses more general mathematical properties than the Shannon entropy and
- 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.

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Extension of Slater's List: New Generalizations of Rogers-Ramanujan Type Identities

Tina Garrett

Carleton College

Northfield, MN

3:00pm-4:00pm

PHY 118

Mourad Ismail

**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.

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An Algorithmic Approach to Rogers-Ramanujan Type Identities

Andrew Sills

Penn State University

4:15pm-5:15pm

PHY 108

Mourad Ismail

**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.

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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.

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Inversion of Bilateral Basic Hypergeometric Series

Michael Schlosser

University of Vienna

Vienna, AUSTRIA

3:00pm-4:00pm

TBA

Mourad Ismail

**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.

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The Generalized Chebyshev Polynomials

Yang Chen

Imperial College

London, UK

3:00pm-4:00pm

LIF 262

Mourad Ismail

**Abstract**

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

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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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Virtual Crossing Realization

Sam Nelson

Whittier College

Los Angeles, CA

3:00pm-4:00pm

LIF 262

Masahiko Saito

**Abstract**

Virtual isotopy moves on a classical knot diagram do not alter the fundamental quandle of the knot, and hence do not alter the knot type. Therefore, if we can change one classical knot diagram to another using virtual moves, we can do the same using only classical moves. The question we will consider is: can we change a given virtual move sequence into a classical move sequence?