Mathematics Department Colloquium (20072008)
Talks in Spring 2008
April 25, Friday, 1pm in Jones 131
Speaker: Zlatko Drmac, University of Zagreb
Title: Subspace gap residuals for RayleighRitz approximations
Abstract:
Large scale eigenvalue and singular value computations are usually based
on extracting information from a compression of the matrix to suitably
chosen low dimensional subspaces. This paper introduces new a posteriori
relative error bounds based on a residual expressed using the largest
principal angle (gap) between relevant subspaces. The eigenvector
approximations are estimated using subspace gaps and relative separation
of the eigenvalues.
April 21, Monday, 3pm in Jones 131
Speaker: David Shoikhet, ORT College Braude, Israel,
Title: Old and New in Complex Dynamics
Abstract.
April 7, Monday, 3pm in Jones 131
Speaker: Rika Hagihara College of William and Mary,
Title: Complex Dynamics Lacking Period 2 Orbits
Abstract:
Most polynomials and rational functions can be easily solved for
periodic points of all periods. Periodic points play an important role in
the theory of dynamical systems. In this talk we will study quadratic
rational maps that are missing period 2 orbits. We will introduce the
parameter space of such maps, and investigate how different kinds of
dynamics are reflected in it. We will also compare the parameter space of
quadratic rational maps lacking period 2 orbits with the Mandelbrot set,
the parameter space for all quadratic polynomials, to see the similarities
and differences.
Talks in Fall 2007
November 30, Friday, 3pm in Jones 131
Speaker: Shahla Nasserasr, College of William and Mary,
Title:
Primitive Digraphs with Smallest Large Exponent
Abstract.
November 16, Friday, 3pm in Jones 131
Speaker: Evelyn Sander, George Mason University,
This is actually a
CSUMS Lecture.
4pm
Speaker: Zlatko Drmac, University of Zagreb
Title:
Computing matrix spectral decompositions in finite precision arithmetic
Abstract:
Many problems in numerical linear algebra can be considered
completely solved from the purely algebraic (or analytic)
point of view. However, in the real world applications, we
have somewhat different picture. The initial matrices are
usually given with uncertainties (as measurement errors, errors
from previous computation) and further computation of a matrix
function (such as e.g. the rank, the inverse, or the eigenvalues)
is carried out over a finite number of rationals (machine numbers)
using a finite precision machine arithmetic (in which adding three
numbers accurately represents a nontrivial challenge).
We will address these issues in numerical computation of the
spectral and the SVD decompositions, and present some results
of the recent exciting development, with contributions of several
researches (Barlow, Demmel, Drmac, Gu, Koev, Eisenstat,
Veselic). Our approach will follow the following paradigm:
i) Use perturbation theory to describe classes of matrices
together with classes of admissible perturbations for which
computation with high relative accuracy is possible. This, in
some cases, means changing the usual matrix representation.
ii) Develop an algorithm capable of achieving the theoretical
accuracy. This may require different approach for each separate
class of matrices. However, some unifying principles are developed.
We show that a new variant of the Jacobi method can be used as
a core routine for all cases.
iii)Implement the algorithms in reliable mathematical software
and prove that the implementation has the required accuracy
properties. This can be tedious and not satisfactory because
it involves technical details, hardware and compiler issues.
As an example how tricky this can be, we show how a subtle
numerical bug (dating back to LINPACK, 1971.) survived 36 years
in all numerical libraries before being detected.
November 9, Friday, 3pm in Jones 131
Speaker: Jianjun Paul Tian, College of William and Mary
Title: Spin representations of Artin's braid group
Abstract:
Motivated by distinguishing two opposite orientations of a knot, we
construct new linear representations of Artin's braid group, spin
representations and multiparameter Burau representations, by smoothing a
representation variety. We will work on unitary matrices SU(2,C) for
explicit computation purpose.
The talk is based on the following
paper.
November 2, Friday, 3pm in Jones 131
Speaker: Roberto Costas, College of William and Mary
Title:
Extensions of discrete classical
orthogonal polynomials beyond the orthogonality
Abstract:
It is well known that the family of Hahn polynomials (Hn(x;N)) is
orthogonal with respect to a certain weight function up to N. In this
talk we present a factorization for Hahn polynomials
Abstract:
for a degree higher
than N and we prove that these polynomials can be characterized by a
discrete Sobolev orthogonality.
October 26, Friday, 3pm in Jones 131
Speaker: Carsten Collon, TUDresden
Title: Invariant tracking control of the kinematic car
Abstract:
Influencing the behavior of a dynamical system by determing inputs from
system
outputs via a feedback law is an important subject in control theory. This
talk
considers an invariant parameterization of control laws w.r.t. Lie
transformation groups. The wellknown example of the kinematic car serves
as
motivation for reviewing the choice of coordinates used to design the
control
law. The example is followed by a presentation of a "normalization
procedure"
as an approach to compute a complete set of invariants for the action of a
given Lie transformation group. These invariants can be used to obtain an
invariant parameterization of the control law.
October 19, Friday, 3pm in Jones 131
Speaker: Bill Kalies, Florida Atlantic University
Title: Computational dynamics from a topological point of view
Abstract:
In this talk, we will review some recently developed techniques
for computing global dynamics, including changes with respect
to parameters, and some applications. The talk will focus on
computing recurrence in the iteration of a continuous map
by reducing the dynamics to a finite directed graph. Then we
give results that allow the information from the reduced system
to be lifted back to information about the original map.
These methods can identify regions of recurrent an nonrecurrent
behavior and also provide rigorous computerassisted proofs
of the existence of various types of dynamical structures such
as periodic points, connecting orbits, and chaotic behavior.
October 5, Friday, 3pm in Jones 131
Speaker: J. Brown, College of William and Mary
Title: Pseudotrees, Suslin Trees, and Cardinal Functions
Abstract
September 28, Friday, 3pm in Jones 131
Speaker: C.K. Li, College of William and Mary
Title: Numerical ranges and dilations of operators
Abstract:
The numerical range of an operator A acting on a Hilbert space H
is the set
W(A) = {(Ax,x): x in H, (x,x) = 1}.
We say that D is a dilation of A if A can be viewed as a compression
of the operator D. In this talk, we will discuss how one can use the
numerical range of A to find a dilation of A with simple structure.
Extension of the result to the joint numerical range and higher
rank numerical range will also be mentioned.
September 21, Friday, 3pm in Jones 131
Speaker: Roberto S. Costas Santos, College of William and Mary
Title: Classical orthogonal polynomial. A general difference calculus
approach
Abstract:
It is well known that the classical families of orthogonal polynomials are
characterized as eigenfunctions of a second order linear
differential/difference operator. In this talk we present the essential
part of the study of classical orthogonal polynomials in a more general
context by using the differential (or difference) calculus and Operator
Theory, and, in such a way, we obtain a unified representation of them.
September 14, Friday, 3pm in Jones 131
Speaker: Katarzyna Filipiak, Agricultural University of Poznan, Poland
Title: Connectedness and optimality of block deigns under an interference
model
Abstract:
We consider experiments in which interplot interference may occur. Our aim is to
characterize connected and optimal designs under an interference model with neighbor
effects. The conditions of connectedness and optimality of designs can be formulated using
the properties of information matrices. The information matrix can be expressed as the Schur
complement of some matrices and it has such properties as symmetry, nonnegativedefiniteness
and zero row and column sums. We study such properties of information
matrices as maximality of rank, complete symmetry and maximality of the trace. We are
interested in determining designs in which:
 the sum of eigenvalues of the inverse of information matrix is minimal,
 the product of nonzero eigenvalues of the information matrix is maximal, and
 the minimal nonzero eigenvalue of information matrix is maximal.
