Colloquia -- Spring 2002

Friday, May 3, 2002

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

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

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:

1. (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;
2. (homotopy invariance) d(I - H(t,.), B, p) = constant for all t in [0,1], where H is compact;
3. (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.

Now, let's take this to skies!

Where is the “MAGIC?” Come to the talk and find out!

Friday, April 5, 2002

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

Abstract

We consider the existence of a function f : R → R with lim_{h → 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

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

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

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

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

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

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

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

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

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.

Please direct questions to mthmaster@nosferatu.cas.usf.edu.
Last updated: 29-Apr-2002.
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