The use of the Zak transform to obtain a general setting for Gabor Systems

Speaker: 

Professor Guido Weiss

Institution: 

Washington University

Time: 

Thursday, March 12, 2009 - 4:00pm

Location: 

RH 306

Suppose g is a square integrable function on the real line. The principal shift invariant space, , generated by g is the closure of the span of the system
B ={g(.-k): k an integer}. These spaces are most important in many areas of Analysis. This is particulrly true in the theory of Wavelets. We begin by describing a very simple method for obtaining the basic properties of and the systems B.
The systems obtained by applying, in addition to the integral translations, also the integral modulations (these are the multiplication of a function by exp(-2pinx)) are known as the Gabor systems. By using the Zak transform we show how the same methods can be used to study the basic properties of the Gabor systems and their span.
We will define the Zak transform and explain all this
in a very simple way that will be easily understood by all who know only a "smidgeon" of mathematics. A bit more challenging will be the explanation how all this can be extended to general locally compact abelian groups and their duals.
This is joint work with E. Hernandez, H. Sikic and E. N.
Wilson.

Parallel Adaptive Methods and Domain Decomposition

Speaker: 

Professor Randolph Bank

Institution: 

University of California, San Diego

Time: 

Thursday, November 6, 2008 - 4:00pm

Location: 

RH 306

We discuss a parallel adaptive meshing strategy due to Bank
and Holst. The main features are low communication costs,
a simple load balancing procedure, and the ability to
develop parallel solvers from sequential adaptive
solvers with little additional coding.
In this talk we will discuss some recent developments,
including variants of the basic adaptive paradigm,
improvements in the adaptive refinement algorithm itself,
and a domain decomposition linear equations solver
based on the same principles.

Algebraic Analysis of Dirac Operators

Speaker: 

Professor Daniele Struppa

Institution: 

Chapman University

Time: 

Thursday, October 23, 2008 - 4:15pm

Location: 

RH 306

There are several analogues of the theory of one complex variable, when the values of the functions are taken in the division algebra H of quaternions, or in a suitable Clifford algebra. These theories rely on the construction of operators which somehow imitate the Cauchy-Riemann operator; in the quaternionic case one uses the Cauchy-Fueter operator, and in the Clifford case one uses the Dirac operator. The extension to several variables has remained elusive for a long time, but it can in fact be achieved if one considers these systems from the point of view of their algebraic properties. The analysis of such operators from the point of view of the Palamodov-Ehrenpreis Fundamental Principle allows the construction of a non-trivial theory in several variables. This talk will discuss the strength of this approach, as well as some of the questions which remain open, and will be concluded with a new twist on these theories.

Indifference pricing with general semimartingales

Speaker: 

Professor Marco Frittelli

Institution: 

University of Milano

Time: 

Friday, May 9, 2008 - 3:00pm

Location: 

MSTB 254

We consider a financial market where the discounted prices of the assets available for trading are modeled by semimartingales that are not assumed to be locally bounded. In this case the appropriate class of admissible integrands is defined through a random variable W that controls the losses incurred in trading. Applying the theory of Orlicz spaces, and convex analysis we study the utility maximization problem with an unbounded random endowment.

We then apply the duality relation to compute the indifference price of a claim satisfying weak integrability conditions. The indifference price leads to a convex risk measure defined on the Orlicz space associated to the utility function.

The talk is based on joint works with S. Biagini and with S. Biagini, M. Grasselli.

Existence of rough solutions to the Einstein constraint equations without CMC or near-CMC conditions

Speaker: 

Professor Mike Holst

Institution: 

UCSD

Time: 

Wednesday, June 4, 2008 - 3:00pm

Location: 

MSTB 254

There is currently tremendous interest in geometric PDE, due in
part to the geometric flow program used successfully to attack the
Poincare and Geometrization Conjectures. Geometric PDE also play
a primary role in general relativity, where the (constrained) Einstein
evolution equations describe the propagation of gravitational waves
generated by collisions of massive objects such as black holes.
The need to validate this geometric PDE model of gravity has led to
the recent construction of (very expensive) gravitational wave
detectors, such as the NSF-funded LIGO project. In this lecture, we
consider the non-dynamical subset of the Einstein equations called
the Einstein constraints; this coupled nonlinear elliptic system must
be solved numerically to produce initial data for gravitational wave
simulations, and to enforce the constraints during dynamical simulations,
as needed for LIGO and other gravitational wave modeling efforts.

The Einstein constraint equations have been studied intensively for
half a century; our focus in this lecture is on a thirty-year-old open
question involving existence of solutions to the constraint equations
on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic
curvature. All known existence results have involved assuming either
constant (CMC) or nearly-constant (near-CMC) mean extrinsic curvature.
After giving a survey of known CMC and near-CMC results through 2007,
we outline a new topological fixed-point framework that is fundamentally
free of both CMC and near-CMC conditions, resting on the construction of
"global barriers" for the Hamiltonian constraint. We then present
such a barrier construction for case of closed manifolds with positive
Yamabe metrics, giving the first known existence results for arbitrarily
prescribed mean extrinsic curvature. Our results are developed in the
setting of a ``weak'' background metric, which requires building up a
set of preliminary results on general Sobolev classes and elliptic
operators on manifold with weak metrics. However, this allows us
to recover the recent ``rough'' CMC existence results of Choquet-Bruhat
(2004) and of Maxwell (2004-2006) as two distinct limiting cases of our
non-CMC results. Our non-CMC results also extend to other cases such
as compact manifolds with boundary.

Time permitting, we also outline some new abstract approximation theory
results using the weak solution theory framework for the constraints; an
application of which gives a convergence proof for adaptive finite
element methods applied to the Hamiltonian constraint.

This is joint work with Gabriel Nagy and Gantumur Tsogtgerel.

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