Colloquia — Spring 2002
Friday, May 3, 2002
| Title |
Anatomy of the Lorenz Attractor |
| Speaker |
Professor Divakar Viswanath
University of Michigan |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 261 |
| Sponsor |
Professors A. Mukherjea and J. S. Ratti |
Abstract
The butterfly-like Lorenz attractor is one of the best known images of chaos.
Although trajectories shuttle unpredictably between the two wings of the butterfly,
it is possible to systematically break up the attractor into periodic orbits.
There are a total of 111011 periodic orbits whose symbol sequence is of length
20 or less. The Cantor structure of the Lorenz attractor is closely related
to symbolic dynamics. Finally, there are periodic orbits arbitrarily close to
any given point on the Lorenz attractor, and I will show a method to compute
them. This method gives an algorithmic realization of one of the basic assumptions
of hyperbolicity theory.
Friday, April 26, 2002
| Title |
Noncommutative Geometry and its Applications |
| Speaker |
Dr. Jerry Kaminker
Indiana University-Purdue University at Indianapolis |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Dr. Mohamed Elhamdadi |
Abstract
Noncommutative geometry, as introduced by Alain Connes, has its origins in
the study of elliptic differential operators. It has proved to be a useful tool
in geometry, analysis and physics. In this talk we will give an introduction
to the basic ideas of this theory and describe some applications to solid state
physics.
Friday, April 12, 2002
| Title |
Nonlinear Analysis and the Magic of Degree Theory |
| Speaker |
Professor Athanassios Kartsatos |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 118 |
Abstract
We start with a continuously differentiable function on the closure cB
of an open and bounded set B in Rn. We define the degree
mapping d(f, B, p) for points p
that are not in f(bB) and are not in f(Qf),
where bB is the boundary of B and Qf is the
set of critical points of f in B. d(f,
B, p) is an integer-valued function with 4 basic properties.
Using Sard's Lemma and the Weierstrass theorem, we extend this degree mapping
to an arbitrary continuous mapping f. Using the fact that every compact
operator in a Banach space can be uniformly approximated by compact operators
of finite-dimensional range, we extend the degree mapping to a mapping d(I
- T, B, p), where T : cB
to X is a compact operator on the closure cB of an open
and bounded subset B of X with boundary bB,
and p is in X but not in (I - T)(bB).
The operator T may be nonlinear. The 4 properties of the degree above
are maintained. Three of these fundamental properties are:
- (degree of the identity) d(I, B, p)
= 1 if p is in B, and d(I, B,
p) = 0 if p is not in cB;
- (homotopy invariance) d(I - H(t,.), B,
p) = constant for all t in [0,1], where H
is compact;
- (solvability) if d(I - T, B,
0) is well-defined and d(I - T, B,
0) not = 0, then (I - T)(x) = 0, for some
x in B.
Thus, Tx = x, i.e. T has a fixed point in B.
Beautiful!
Now, let's take this to skies!
Where is the “MAGIC?” Come to the talk and find out!
Friday, April 5, 2002
| Title |
Finite and Artinian Chain Rings |
| Speaker |
Dr. Youssef Alkhamees
King Saud University
Riyadh, Saudi Arabia |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor Edwin Clark |
Abstract
We will cover some results involving the structure, the enumeration, and the
group of automorphisms of finite and Artinian chain rings using coefficient
rings and distinguished basis.
Wednesday, April 3, 2002
| Title |
A Very Asymmetric Function on the Reals |
| Speaker |
Professor Peter Komjath
Eotvos University
Budapest, Hungary |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor Vilmos Totik |
Abstract
We consider the existence of a function f : R →
R with limh → 0 max (f(x -
h), f(x + h)) = ∞ for every
x ∈ R. Under the continuum hypothesis we prove the
existence. If the negation of the continuum hypothesis is assumed, then there
are models where such functions exist and there are models where they do not
exist.
Friday, March 29, 2002
| Title |
Convolutions for Orthogonal Polynomials and Lie Algebra Representations
|
| Speaker |
Dr. Erik Koelink
Technische Universiteit Delft
The Netherlands |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor Mourad Ismail |
Abstract
A number of the classical orthogonal polynomials, such as Charlier, Hermite,
Meixner, Meixner-Pollaczek polynomials, satisfy convolution identities that
can usually be derived from a generating function. These identities are special
cases of more general convolution identities that can be obtained from tensor
product representations of the Lie algebra sl(2). The focus will be on the Meixner-Pollaczek
polynomials, but this is just a simple example of the results that can be obtained
in general.
Friday, March 22, 2002
| Title |
Monge-Kantorovich Mass Transfer and Variational Principle
for Gas Dynamics |
| Speaker |
Dr. Chaocheng Huang
Wright State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor Yuncheng You |
Abstract
The original mass transfer problem, proposed by Monge in the 1780's, asks how
to find the cheapest way to move a pile of soil or rubber into an excavation.
Mathematically, given two Radon measures (mass distributions) μ and ν
with the same total mass and the cost function (the unit cost of moving mass
at x to y) c(x,y), one
looks for the optimal mapping (shipping plan) T that minimizes the
total cost
I(T) = ∫ c(x,T(x)) dμ
over all mappings T that preserve the measures. No major progress
was made until 1940 when Kantrovich introduced a dual problem and a relaxed
variant of Monge's cost functional that remarkably transforms into a linear
problem.
In this talk, I shall briefly introduce recent developments on the mass transfer
problem for the distance function c(x,y) =
|x-y|p and its applications to kinetic equations
raised in the gas dynamics, for instance, the Kramers system and Vlasov-Poisson-Fokker-Planck
(VPFP) system. In particular, I shall show how to use the Monge-Kantorovich
cost functional to establish a semi-discrete variational principle. The variational
principle demonstrates an interesting phenomenon: VPFP dynamics may be viewed
as the steepest descent of the total energy with respect to the Monge-Kantorovich
functional.
Friday, March 8, 2002
| Title |
Crystallographic Invariants |
| Speaker |
Dr. David Rabson
Department of Physics, USF |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Dr. Mohamed Elhamdadi |
Abstract
We describe three invariants that come out of the homological description of
crystallography. The first corresponds simply to the systematic extinctions
present in all but two of the 157 periodic non-symmorphic space groups. The
second, while perhaps less familiar, has been noted before: the two exceptional
non-symmorphic periodic space groups (as well as others) exhibit a necessary
electronic degeneracy, or “band sticking,” at defined points in the
Brillouin zone. The third invariant, present for example in a rank-five, tetragonal
modulated crystal, is new; we will discuss its physical implications.
Friday, March 1, 2002
| Title |
Groups of Automata and Their Geometry |
| Speaker |
Dr. Zoran Sunik
Research Assistant Professor
University of Nebraska, Lincoln |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Dr. Natasha Jonoska |
Abstract
We introduce several examples of groups that can be realized by automata and
explore their connections to geometric group theory, dynamical systems, random
walks, spectra and other areas of mathematics, thus demonstrating once again
the unity and the beauty of the subject.
Friday, February 22, 2002
| Title |
Generators and Relations for the Mapping Class Groups of Surfaces |
| Speaker |
Dr. Susumu Hirose
Michigan State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Dr. Masahiko Saito |
Abstract
Two homeomorphisms are called isotopic if we can deform one to the other. The
set of isotopy classes of homeomorphisms on a surface has a group structure
defined by composition of homeomorphisms. We call this group “the mapping
class group of the surface.” This group is one of the central subject of
low dimensional topology, because this group has a deep relationship with the
classification of 3-manifolds. In this talk, I will introduce a set of generators
of the mapping class group, a presentation of this group, and a method to obtain
this presentation.
Friday, February 8, 2002
| Title |
Optimization of Linear Error Correcting Codes |
| Speaker |
Dr. Larry Dunning
Professor of Computer Science
Bowling Green State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor Edwin Clark |
Abstract
Suppose that a given linear block code is to be used for error-correction /error-detection.
It is well known that such a code can be placed in systematic form where the
message bits appear in the codeword. However, other encodings may provide better
performance when the error rates for the message bits are considered individually.
The greedy algorithm of matroid theory can be applied in this situation to obtain
encodings that are optimal with respect to a number of different evaluation
measures. In particular, the probability of message bit error and a generalization
of the Hamming metric will be considered in detail.
Friday, February 1, 2002
| Title |
Best Approximation in Sobolev Spaces |
| Speaker |
Dr. Xin Li
University of Central Florida |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor Vilmos Totik |
Abstract
For numerical solutions of differential equations, approximation with respect
to a Sobolev norm (a norm involving both the function and its derivatives) is
more appropriate. I will illustrate the use of best polynomial and rational
approximation in Sobolev spaces, demostrate some basic properties of polynomials
orthogonal in Sobolev-Laguerre and Sobolev-Legendre spaces, and discuss a general
framework on orthogonal rational functions in Sobolev spaces.
Wednesday, January 23, 2002
| Title |
Asymptotic Properties of a Simple TCP Model |
| Speaker |
Professor Goran Högnäs
Åbo Akademi University
Finland |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 120 |
| Sponsor |
Professor Arunava Mukherjea |
Abstract
We examine a simple discrete time Markov chain model of TCP congestion control
and prove that it has a unique invariant measure. We also show that if the process
is scaled by a factor sqrt(p) (where p is an error probability),
then the invariant measures converge to a limit as p tends to 0.
If the scaled process is transformed in a suitable way we show that it converges
to a piecewise linear limit process. The unique invariant measure of the limit
process coincides with the limit of the invariant measures above and can be
easily computed.
Joint work with graduate student Niclas Carlsson.
Friday, January 11, 2002
| Title |
Finding Meromorphic Solutions by Nevanlinna Theory |
| Speaker |
Dr. Yik-Man Chiang
Department of Mathematics
Hong Kong University of Science & Technology |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor Mourad Ismail |
Abstract
We discuss how to use the classical Nevanlinna theory of meromorphic functions
in the complex plane to find meromorphic solutiions of certain ordinary algebraic
differential equations with constant or polynomial coefficients. The method
will combine with local series analysis to solve explicitly a subclass of certain
ODEs. The idea behind is connected with Kowalevskaya's solution (1880's) to
describing the mass, centre of mass, and moment of inertia of a spinning top
that finding meromorphic solutions can be a useful tool of integrability.