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December 11,
2003 11:00 -
12:00
Pattern Formation and
Dynamics in Nematic Electroconvection
Juliana Oprea
Department of Mathematics
Colorado State University, Fort Collins
The subject of the spontaneous formation of patterns in systems far from equilibrium is an exciting and fast-growing branch of
applied mathematics and physics, with strong impact on fields as diverse as ecology, chemistry, and new technologies and processes.
Dissipative systems, in which energy must be supplied through a gradient such in temperature, velocity, concentration,
etc, to maintain the physical instability, are a good step towards a description of this variety of phenomena.
In this talk, the basic mathematical tools in analyzing the selection mechanisms and the dynamics of patterns
in extended dissipative systems are summarized and exemplified on a paradigm system, the electroconvection in nematic liquid crystals.
December 4,
2003 11:00 - 12:00
On the Rate of Convergence of Series of
Random Variables
Eunwoo Nam
US Air Force Academy
Colorado Springs, CO
The
rate of convergence for an almost certainly convergent series  of
random variables is studied in this presentation. When the series  converges
to a random variable  almost
surely, the tail series  is
a well-defined sequence of random variables with  almost
surely. We shall be concerned with the rate in which converges
to almost
surely (or in probability), or equivalently, in which the tail series  converges
to 0 almost surely (or in probability). More specifically, tail series
strong laws of large number and tail series weak laws of large
numbers will be introduced, and their relationship will also be
investigated.
November 13, 2003
11:00 - 12:00
Mass
extinctions: living in a big flock can be bad for you!
Rinaldo Schinazi
Department of Mathematics
University of Colorado at Colorado Springs
We introduce a spatial
stochastic process to model mass extinctions. We assume that a given
species lives in flocks and that a new flock is started only when the
parent flock reaches a given maximum size. Our model shows that, at least
in theory, there is a critical maximum flock size above which a species is
certain to disappear and below which it may survive.
October 30, 2003
11:00 - 12:00
Soliton
solutions relative to arbitrary backgrounds for the NLS
equation and the Manakov system
Radu Cascaval
Department of Mathematics
University of Colorado at Colorado Springs
The construction of soliton
solutions relative to arbitrary backgrounds of the NLS hierarchy is
related to the addition of eigenvalues to the spectrum of the background
Lax operator via Darboux transformations. We will discuss the
generalization of this construction for the Manakov system. In nonlinear
optics the Manakov system (coupled NLS) appears as a model for wave
propagation in optical fibers in the presence of birefringence effects.
The linear and nonlinear stability of these solutions will be addressed.
October 9,
2003 11:00 to 12:00
Dispersion Management for
Randomly Varying Optical Fibers
Jamison T. Moeser
Department of Applied Math
University of Colorado at Boulder
In modern optical fiber
links, random variations in various fiber characteristics can become a
major limitation to long-scale pulse propagation. We present an
asymptotic theory for optical pulse propagation in dispersion managed
(DM) fibers whose accumulated dispersions profiles are piecewise linear
with random slope. The validity of this theory can be shown
analytically and is verified with direct numerical simulation. The
equation that describes the slow evolution of initial pulses possesses a
stationary solution which is a minimizer for a related variational
problem. These special solutions have potential for use as
bit-carriers in an optical fiber link, and for fibers with moderate
noise in the dispersion profile, perform much better than ideal DM
solitons optimized for the same fiber.
September 30, 2003 11:00 -
12:00
On Tilting and Cotilting Modules Over Artin
Algebras
Francesca
Mantese
Dipartimento di
Matematics Pura ed Applicata
Universita'
degli Studi di Padova, Italy
In this
talk we present some preliminary and basic aspects of the theory of
tilting and cotilting modules over Artin algebras. In particular we
focus on the connections between tilting theory and some classical
homological conjectures for finite dimensional algebras, such as the
Finitistic Dimension Conjecture and the Generalized Nakayama
Conjecture.
September 29, 2003
Endoprimal Abelian
Groups
Laszlo
Marki
Alfred Renyi
Institute of Mathematics
Hungarian
Academy of Sciences, Budapest
An
algebra is said to be endoprimal if all those finitary functions in it
which permute with every endomorphism of the algebra are term functions
- in the case of abelian groups the latter means the functions of the
form with
integers .
First results on endoprimality of abelian groups have shown a close
connection of endoprimality to decomposition properties including, among
others, a complete description of endoprimal torsion groups. Here we
present recent results by K. Kaarli, L. Marki and further coauthors
which solve the problem of endoprimality for separable torsion-free
abelian groups, torsion-free abelian groups of rank 3, and mixed abelian
groups of torsion-free rank 1.
September 15,
2003
Domino Chain
Reactions
Bill Briggs
Mathematics
Department
University of Colorado at Denver
Imagine standing, say, 10 meters from
a wall with a bag full of dominoes. As everyone knows, if you build a
line of dominoes, all standing on edge, between you and the wall, it
will undergo a chain reaction when the first domino is tipped. How
should you arrange the dominoes in order to minimize the travel time of
the chain reaction (a) when the dominoes have a uniform spacing and (b)
when the dominoes may have arbitrary spacing? While these problems may
sound frivolous, their solutions involve a delightful combination of
mathematical modeling, numerical calculations, and symbolic analysis -
and they lead to some surprising results.
September 15,
2003
The Mathematics of Population
Genetics
Bill Briggs
Mathematics
Department
University of Colorado at Denver
Population
genetics is the first area of biology in which rigorous mathematical
models were developed. This work was done in the 1920s by Fisher,
Sewell, and Wright. In this sense, population genetics is to biology
what Newtonian mechanics is to physics. Compared to mathematics models
in mechanics, population genetics is sadly neglected in much of our
teaching. In this talk, we will survey some of the early population
genetics models, which involve difference and differential equations. We
will see how they have been extended to present day models that are
still relevant and not fully understood.
September 11,
2003
Dynamical features of soliton
interactions in
continuous and discrete vector nonlinear Schrodinger
system
Barbara Prinari
INFN, Lecce,
Italy
We analyze the dynamical features associated
with soliton interactions in the continuous and discrete vector NLS
equation. We give general formulas for the polarization shifts in the
multisoliton interactions and show that in both cases the
interactions are pairwise and independent of the order in which the
collisions take place. We identify the various possibilities of energy
redistributions among the modes of the solitons and express them as
(generalized) linear fractional transformations. Finally, following
Steiglitz et al, we interpret the multisoliton solutions as (optical)
logic gates.
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