Colloquia — Spring 2001
Friday, May 4, 2001
| Title |
Asymptotics for Stieltjes and Van Vleck polynomials |
| Speaker |
Professor André Martinez-Finkelshtein
University of Almeria, Spain |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor E.B. Saff |
Abstract
We are interested in the zeros of the polynomial solutions of a class of second-order
linear differential equations with polynomial coefficients. The best-known examples
are the hypergeometric and the Heun equations. Heine, Stieltjes, Van Vleck and
others, have studied these equations, producing some beautiful results concerning
the location of the zeros. In particular, their electrostatic interpretation,
due to Stieltjes, is the key to the potential theory approach which allows to
answer the question about the asymptotic distribution of the zeros as their
number goes to infinity.
Friday, April 27, 2001
| Title |
The Poincaré Conjecture; a Million Dollar Problem |
| Speaker |
Professor Cameron Gordon
University of Texas at Austin |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor Masahiko Saito |
Abstract
It has been known since the 19th century that the 2-sphere is the only closed
2-manifold in which every loop contracts, i.e., can be shrunk to a point. In
1904, Henri Poincaré asked whether the analogous statement holds in dimension
3: Is the 3-sphere the only closed 3-manifold in which every loop contracts?
The assertion that the answer is “yes” has become known as the Poincaré
Conjecture. Recently, the Clay Mathematics Institute offered $1 million for
the solution of each of seven unsolved mathematical problems; the Poincare Conjecture
is one of them. It is the most famous unsolved problem in the field of topology,
and has resisted solution for almost a century. We will explain the terms n-manifold,
n-sphere, etc, and give some history and background to the conjecture,
as well as explaining how non-Euclidean geometry enters into the theory of manifolds
in dimensions 2 and 3.
Wednesday, April 25, 2001
| Title |
On Best Meromorphic Approximation of Markov Functions and
n-widths |
| Speaker |
Professor Vasiliy Prokhorov
University of South Alabama |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Professor E. B. Saff |
Abstract
The talk is devoted to some results concerning the best meromorphic approximation
on the unit circle in the space Lp of Markov functions (Cauchy transforms
of positive measure w with support supp w in (-1,1)).
Connections with n-widths, the problem of minimal Blaschke products,
the theory of Hankel operators will be discussed.
Friday, April 20, 2001
| Title |
3-manifolds that are Seifert union of solid tori |
| Speaker |
Professor Wolfgang Heil
Florida State University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor Mohamed Elhamdadi |
Abstract
Questions about the Lusternik-Schnirelman category of a 3-manifold M
lead to problems about decompositions of M into (orientable) handlebodies.
We study the class of 3-manifolds that are Seifert unions of solid tori
V1,V2; i.e. that are obtained from
V1,V2 by identifying a collection
of disjoint surfaces {F'i} in ∂ V1
with a collection of disjoint surfaces {F″i}
in ∂ V2 under homeomorphisms. (If the collection
{F'i} and {F″i}
consist of essential annuli then such a Seifert union is a Seifert fiber space).
In obtaining the classification of Seifert unions of two tori we are lead to
consider Seifert unions of any collection of punctured balls and punctured tori.
At first one might think that this would include all compact 3-manifolds with
boundary; however, it turns out that this collection of Seifert unions yields
a very restricted class.
Wednesday, April 11, 2001
| Title |
Frequency Domain Method for Linear Evolutions Equations |
| Speaker |
Professor Zhuangyi Liu
Univ. of Minnesota at Duluth |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor Yuncheng You |
Abstract
Consider a linear control system
dz(t)/dt = Az(t)
+ Bu(t), z(0) = z0,
y(t) = Cz(t) + Du(t)
where z(t), u(t), y(t)
are functions in Hilbert spaces Z, U, Y, respectively;
A generates a c0-semigroup of contractions
on Z; the operators B, C, D are
in L(U,Z), L(Z,Y),
L(U,Y), respectively.
It is known that the properties of the uncontrolled system (u(t)
= 0), such as the asymptotic behavior and smoothness of z(t),
provide crutial information to the control design. In this talk, we introduce
a Frequency domain method for the study of these properties. This method is
based on the relation between the properties of z(t) and
the norm of the resolvent operator,
||(s I-A)-1||, on the imaginary
axis. We apply this method to several systems governed by partial differential
equations with global, local, boundary, or dynamical boundary dissipative terms.
Friday, March 23, 2001
| Title |
Fluid Fingering Problems in Hele-Shaw Cells |
| Speaker |
Professor Jianzhong Su
University of Texas at Arlington |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor Yuncheng You |
Abstract
Fluid fingering phenomena arise from various physical problems such as oil
recovery and phase transition. The interface between the two different fluids
evolves according to the physical laws and generates finger like patterns. Its
motion is governed by a partial differential equation with free boundaries.
In this talk, we will first provide some physical background of fluid fingering
problems, then we will discuss the finger solutions of Hele-Shaw equations.
These finger solutions are traveling wave solutions whose finger shaped interfaces
are moving along a certain direction at a constant speed. The existence of finger
solutions is shown through a fixed point argument of the Hilbert Transformation.
Wednesday, March 21, 2001
| Speaker |
Professor Laurent Barachart
INRIA — Sophia-Antipolis
France |
| Topic |
Meromorphic approximation and inverse boundary problem for
the 2-D Laplacian |
| Time |
4:00-5:00 p.m. |
| Place |
PHY 013 |
Abstract
We present a new approach to crack detection based on the meromorphic approximation
of the complex solution to a Dirichlet Neuman problem.
Such methods are computationally attractive to detect the edges of a sufficiently
regular crack. They also raise some conjectures about the behaviour of non-classical
meromorphic approximants with real residues in connection with Sobolev-type
discrete approximation to Green potentials.
Tuesday, March 20, 2001
| Title |
Traveling Wavefronts of Reaction-Diffusion Equations |
| Speaker |
Dr. Kunquan Lan
York University |
| Time |
3:00-4:00 |
| Place |
PHY 108 |
| Sponsor |
Professor A. G. Kartsatos |
| Note |
Speaker is a candidate for Asst. Prof. in PDEs. |
Abstract
Traveling waves have been studied extensively for continuous and lattice partial
differential equations. These equations arise in a wide variety of models. For
example, the reaction-diffusion models of Fisher type were proposed to describe
the spread of a favoured gene in a population. Models involving lattice differential
equations occur in biology, chemical reaction theory, image processing and pattern
recognition, material science and cellular neural networks.
When one considers traveling wave solutions, these partial differential equations
are reduced to eigenvalue problems for first or second order ordinary differential
equations.
Traveling waves for reaction-diffusion equations correspond to eigenvalue problems
for second order ordinary differential equations. They have been widely studied
using several theories. For example, the theory of monotone operators together
with degree theory and the theory of the phase plane trajectories have been
used.
In this talk, I shall review some simple continuous population models for single
species and then present our new iterative techniques and show how to apply
our theory to obtain the traveling wave solutions for reaction-diffusion equations.
This new theory is much simpler than and superior to previous ones. It can be
applied not only to obtain the ranges of wave speeds but also to provide powerful
numerical schemes to compute the waves.
Friday, March 9, 2001
| Title |
Symmetry Constraints of Zero Curvature Equations |
| Speaker |
Dr. Wen-Xiu Ma
Assistant Professor
City University of Hong Kong |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Prof. Yuncheng You |
| Note |
Speaker is a candidate for Asst. Prof. in PDEs. |
Abstract
Symmetry constraints are proposed to decompose zero curvature equations irrespective
of dimensions into specific systems of ODEs, called constrained flows. Functionally
independent and involutive systems of functions are generated from stationary
zero curvature equations, and used to show the Liouville integrability for the
constrained flows. The constraints on the potentials resulting from the symmetry
constraints give rise to involutive solutions to zero curvature equations. The
resulting constrained flows can be solved by separation of variables, and Jacobi
inversion problems of the constrained flows exhibit the integrability by quadratures
for zero curvature equations. The theory is illustrated by examples of soliton
equations.
Thursday, March 8, 2001
| Title |
On Nonlinear Wave Propagation in Media With Dispersion and Dissipation |
| Speaker |
Dr. Vladimir Varlamov
Visiting Associate Professor
University of Texas at Austin |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Prof. A. Kartsatos |
| Note |
Speaker is a candidate for Asst. Prof. in PDEs. |
Abstract
A brief history of the discovery of solitons is given, and the basic semilinear
evolution equations describing wave propagation in dispersive media are presented.
Typical examples are the Boussinesq equation, the Korteweg-de Vries equation,
the Benjamin-Bona-Mahony equation and their dissipative counterparts. The issues
of wave generation by a moving boundary are discussed. For the damped Boussinesq
equation a Cauchy problem is examined and the long-time is asymptotics is presented.
As an example of studying multidimensional problems the damped Boussinesq equation
in a disc is considered, and the long-time asymptotics is calculated. Series
in special functions are used for obtaining long-time asymptotic expansions
in bounded domains.
Tuesday, March 6, 2001
| Title |
Enumeration of Isomorphism Classes of Extensions of p-adic
Fields and Isomorphism Classes of Finite Commutative Chain Rings |
| Speaker |
Dr. Xiang-Dong Hou
Associate Professor
Wright State University |
| Time |
3:00-4:00 p.m. |
| Place |
LIF 260 |
| Sponsor |
Professor Edwin Clark |
| Note |
Speaker is a candidate for Asst. Prof. in Alg./Comb. |
Abstract
Let F be a finite extension of Qp.
Given positive integers f and e, the number of extensions of
F with residue degree f and ramification index e
in a fixed algebraic closure of F is finite; Krasner's formulas allow
one to compute this number. Our concern is the number I(F,
f,e) of F-isomorphism classes of the extension of
F with residue degree f and ramification index e.
When p2 does not divide e, we determine
I(F,f,e) completely; when
p2 divides e exactly, we determine I
(F,f,e) under some additional assumptions. Our
approach is based on results from class field theory and computations with the
Galois groups.
A topic closely related to the problem considered above is finite commutative
chain rings. A finite commutative chain ring is a finite commutative ring whose
ideals form a chain under inclusion. Such rings are useful in finite geometry
and combinatorics. Each finite commutative chain ring has a set of invariants
(p,n,f,e,t). The number
I(Qp,f,e)
is essentially the isomorphism classes of finite commutative chain rings with
invariants (p,n,f,e,t).
When p does not divide e, the number of isomorphism classes
of finite commutative chain rings with invariants (p,n,f,e,t)
has been determined by Clark and Liang. Our results on I(Qp,f,e)
settle some open cases about the number of isomorphism classes of finite commutative
chain rings.
Friday, March 2, 2001
| Title |
Bose-Mesner Algebras |
| Speaker |
Dr. Brian Curtin
NSF Post-Doctoral Fellow
University of California, Berkeley |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
TBA |
| Note |
Speaker is a candidate for Asst. Prof. in Alg./Comb. |
Abstract
We survey some of the rich algebraic combinatorics of Bose-Mesner algebras.
A Bose-Mesner algebra is a commutative complex matrix subalgebra which, in addition
to ordinary matrix product, is closed under entry-wise product and transposition
and which contains the identity and all-ones matrices.
We shall recall the origins of Bose-Mesner algebras in permutation groups and
design theory. We then discuss the recent influence of quantum algebras and
knot theory on the subject of Bose-Mesner algebras. Indeed, a certain quantum
algebra arises in connection with an important family of Bose-Mesner algebras,
and it was recently shown that spin models (the basic data for a statistical
mechanical construction of link invariants) lies in some Bose-Mesner algebra.
We have been working to elaborate on these connections.
Monday, February 26, 2001
| Title |
A Statistical Model for Latitudinal Correlations of Satellite Data |
| Speaker |
Dr. Dongseok Choi |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Professor A. N. V. Rao |
| Note |
Speaker is a candidate for Asst. Prof. in Statistics. |
Abstract
We develop a new class of models for the latitudinal correlation patterns of
satellite ozone and temperature data. We employ the monthly average series for
each 5-degree latitude zone of the TOMS data from the Nimbus 7 satellite over
14 years from November 1978 to November 1992. To each latitude zone, a temporal
regression model that includes all known physical effects and time dependency
is fitted separately by maximizing likelihood function. From the residuals of
all latitude zones, we observe strong contemporaneous spatial correlations,
which decay at different rates depending on latitudes and become negative at
moderate distances. A three-component model with two associated weight functions,
which can accommodate the main features of the concurrent spatial correlations
of the residuals is developed. We use low order ARMA models as components, which
are widely used in time series analysis. A similar analysis is done on the latitudinal
correlations of the residuals from the monthly average series of CPC temperature
data at 30mb from January 1979 to December 1993. The new class of models can
be applied to study covariance structures that consist of a few dominant factors
with varying effectiveness.
Friday, February 16, 2001
| Title |
Invariant Subspaces of Finite-Dimensional Vector Spaces |
| Speaker |
Dr. Markus Schmidmeier
Post-Doctoral Fellow
Florida Atlantic University |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor Edwin Clark |
| Note |
Speaker is a candidate for Asst. Prof. in Alg./Comb. |
Abstract
Consider a linear operator T which acts nilpotently on a
finite-dimensional vector space V and a subspace U of
V that is invariant under the action of T. As a
classification of all such systems (T,V,U) is
not feasible, we focus on those systems for which T has nilpotency
index n on V, and nilpotency index m on
U. In joint work with C. M. Ringel, we showed the following:
- There are only finitely many indecomposable systems (T,
V, U), up to equivalence, if m < 3 , or if
n < 6 , or if (m,n) is the pair (3,6).
- There are infinitely many systems, which can be classified, in case
(m,n)
is one of the pairs (6,6), (5,6), (4,6), or (3,7).
- Otherwise, a classification is not feasible.
Our study of invariant subspaces is motivated by — and related to —
the problem of classifying the pm-bounded
subgroups of a pn-bounded finite abelian group.
Friday, February 2, 2001
| Title |
Baker-Hausdorff Theorem |
| Speaker |
Professor Sam Sakmar
Department of Physics, USF |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Professor Masahiko Saito |
Abstract
Lie groups play an important role both in mathematics and physics. Because
of the non-commutativity of the multiplication rule of the Lie Algebras the
combination of the group elements of the Lie Groups is non-trivial and is given
by the Baker-Hausdorff theorem. We give the proofs of all the theorems needed
for the proof of the Baker-Hausdorff theorem.