Colloquia — Fall 2006
Friday, December 8, 2006
| Title |
Multiwavelets on the Interval |
| Speaker |
Patrick Van Fleet
University of St. Thomas |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Dr. Catherine Bénéteau |
Abstract
In this talk, we present a method for constructing scaling vectors and multiwavelets
on the interval. While other constructions exist (for example, Dahmen/Micchelli
1997, Goh/Jiang/Xia 2000, and Lakey/Pereya 2000), ours is different in that
we take an existing scaling vector for L2(R) and modify it to construct
a scaling vector on L2([a,b]). The process is similar to that employed
by Daubechies 1992 and Meyer 1991. Moreover, our construction allows us to
build nonnegative scaling functions if so desired. The talk will conclude with
some examples of our method.
Friday, December 1, 2006
| Title |
Inverse scattering transform for the vector NLS equation
with non-vanishing boundary conditions |
| Speaker |
Barbara Prinari
Dipartimento di Fisica and Sezione INFN
Universita di Lecce, Italy |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 130 |
| Sponsor |
Dr. Wen-Xiu Ma |
Abstract
The inverse scattering transform for the vector defocusing vector nonlinear
Schrodinger NLS equation with non-vanishing boundary values at infinity is
constructed. The direct scattering problem is formulated on a two-sheeted covering
of the complex plane. On the direct side, two out of the six scattering eigenfunctions
do not admit an analytic extension on either sheet of the surface. Two additional
analytic solutions are constructed by considering *adjoint* eigenfunctions.
The discrete spectrum, bound states and symmetries of the direct problem are
discussed. In general a discrete eigenvalue corresponds to a quartet of zeros
(poles) of certain scattering data. The inverse scattering problem is formulated
in terms of a Riemann-Hilbert (RH) problem in the upper/lower half planes of
a suitable uniformization variable. Special soliton solutions, which have dark
solitonic behavior in both components, and ones which have one dark and one
bright component, are constructed from the poles in the RH problem. The linear
limit is obtained from the RH problem and is shown to correspond to the Fourier
solution obtained from the linearized vector NLS system.
Friday, November 17, 2006
| Title |
Applications of Potential Theory to Problems in Numerical
Linear Algebra |
| Speaker |
John Rossi
Virginia Tech |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Professor Dmitry Khavinson |
Abstract
It is shown that convergence rates of iterative methods based on Krylov subspace
techniques can be quantified via elementary results in 2-dimensional potential
theory involving Green's functions and capacity. We will demonstrate our results
by looking at three different problems involving a very large matrix A:
- Solving Ax = b
- Finding the eigenspaces of A
- Computing f(A) where f is analytic in a domain containing the eigenvalues
of A.
Wednesday, November 15, 2006
| Title |
On Braid Index of Connected Sum of 2-Knots |
| Speaker |
Shin Satoh
Kobe University
Kobe, Japan |
| Time |
2:00-3:00 p.m. |
| Place |
NES 103 |
| Sponsor |
Dr. Masahiko Saito |
Abstract
Every 1-dimensional knot can be deformed into the closure of a 1-dimensional
braid (Alexander's theorem). The braid index of a 1-knot K, denoted by Braid(K),
is defined to be the minimal number of the strings for all such 1-braids. For
the connected sum K#L of two 1-knots K and L, Birman and Menasco prove the
equality Braid(K#L) = Braid(K) + Braid(L) - 1.
For a 2-dimensional knot (a knotted 2-sphere in 4-space), the braid index
is similarly defined by introducing the notion of a 2-dimensional braid. However,
in the study with Kamada and Takabayashi, we prove the inequality Braid(K#L)
< Braid(K) + Braid(L) - 1 for any non-trivial 2-knots K and L.
The aim of this talk is to give a new proof which is simpler than the original
one.
Friday, October 27, 2006
| Title |
Symmetric Groups, General Linear Groups, and a Greek Island |
| Speaker |
Bhama Srinivasan
University of Illinois at Chicago |
| Time |
2:00-3:00 p.m. |
| Place |
TBA |
| Sponsor |
Dr. Xiang-dong Hou |
Abstract
The classical representation theory of symmetric groups involves rich combinatorics
such as partitions and Young diagrams. Similar concepts arise also in the representation
theory of general linear groups over finite fields. The talk will first give
a description of these two theories, and then mention more recent work on classical
groups such as orthogonal groups. Finally the talk will mention some mysterious
objects which arose from the theory and have been named after a Greek island.
Friday, October 6, 2006
| Title |
Homological Symbols |
| Speaker |
Marian Anton
University of Kentucky |
| Time |
3:00-4:00 p.m. |
| Place |
TBA |
| Sponsor |
Dr. Mohamed Elhamdadi |
Abstract
We define some symbols called “homological symbols” and explain
their relationship with group theory and number theory. The talk will be accessible
to graduate students.