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May 14, 2008
Dr. Isaac Harari
Tel Aviv Unversity, Visiting Professor at Duke University
Title: Direct Computation of Inverse Potential Problems with Interior Data
Abstract:The inverse problem of heat conduction provides a model for many applications governed by diffusion equations, for which interior data are available. Finding the single, unknown, thermal conductivity field by direct (i.e. non-iterative) computation requires at least two interior temperature fields. The strong form of the problem is governed by two partial differential equations of pure advective transport. For the problem to have a solution the measured temperature fields must satisfy a compatibility condition, which may become prohibitive in the presence of noise. The standard weak (variational) form does not provide a suitable basis for finite element computation. A conventional least-squares variational approach yields a well-posed problem, but does not perform well numerically.
The Adjoint Weighted Equation (AWE) is a novel variational formulation
for solving the two-field problem. In this case, the problem is well
posed when the gradients of the two given temperature fields are
linearly independent in the entire domain, a weaker condition than
the compatibility required by the strong form. The solution of the
AWE formulation is equivalent to that of the strong form when both
are well posed. The Galerkin discretization of the AWE formulation
converges with optimal rates. Computational examples confirm these
optimal rates, and demonstrate superior performance to conventional
numerical methods on problems with both smooth and rough coefficients
This work was done in collaboration with Assad Oberai of RPI and Paul
Barbone of BU. We are currently extending these ideas to problems of
April 30, 2008
Dr. Anton Gorodetski
University of California, Irvine
Title: On the stochastic layer of the standard map
Abstract: The Chirikov-Taylor standard map of the two-torus fk(x; y) = (x + y + sin(2Px); y + k sin(2Px)) (mod 1). was initially introduced in 1960s in the context of electron dy
namics in microtrons. It is the most famous example of a sym
plectic twist map, and is related to an extensive list of phys
ical problems. A celebrated long-standing conjecture claims
that fk has a stochastic layer (a transitive set of points with
non-zero Lyapunov exponents) of positive measure for non-zero values of the parameter k. At the present time it is not even known whether there exists at least one value of k such that fk maps T2 to T2 has a stochastic layer of positive measure. Using the recent results on homoclinic bifurcations of area- preserving maps, we show that stochastic layer of fk has full Hausdor® dimension for large topologically generic values of the parameter.
April 23, 2008
Dr. Xiu Ye
University of Arkansas at Little Rock
Title: Divergence Free Finite Element Methods for the Navier-Stokes Equations by H(div) Elements
Abstract: We derive and analyze a numerical formulation for the Navier-Stokes equations which make use of H(div) elements. The finite element solutions feature exact satisfaction of the continuity equation which is highly desired in practical applications. In the formulation, we seek the velocity from the exactly divergence free subspace of H(div) elements. Therefore solving a saddle point problem can be avoided. Numerical examples are provided.
April 9, 2008
Dr. Jose Carrillo
Autonomous University of Barcelona, Barcelona, Spain
Title: The Patlak-Keller-Segel model: free energies, geometric inequalities and gradient flows
Abstract: We will review some of the results known about this classical problem in mathematical biology. The long-time asymptotics of this model of cell motility due to chemotaxis will be analysed in the critical case. Its connection to free energies and the logarithmic HLS inequality will lead to the proof of infinite time aggregation in the critical case. A similar problem with nonlinear diffusion exhibiting analogous behavior will be studied in any dimensions.
April 9, 2008
Dr. Pavel Dubovski
Stevens Institute of Technology, Hoboken, NJ
Title: On the coagulation equation with sources
Abstract: Smoluchowski coagulation equation describes the evolution of gluing particles and possesses the mass conservation law. However, in many branches of physics and industry, this partial integrodifferential equation should be investigated with the external source term, which leads to the failing the mass conservation property. Such a consideration is essential under the analysis of aerosol pollution (source) in atmospheric clouds. We discuss the existence and uniqueness issues of the solution, evaluate its principal properties, give the examples of non-existence. If time allows, we will also briefly touch the sol-gel phase transition problem for this partial integrodifferential equation..
March 26, 2008
Dr. Lenny Fukshansky
Claremont McKenna College
Title: Sphere packing, lattices, and Epstein zeta function
Abstract: The sphere packing problem in dimension N asks for an arrangement of non-overlapping spheres of equal radius which occupies the largest possible proportion of the corresponding Euclidean space. This problem has a long and fascinating history. In 1611 Johannes Kepler conjectured that the best possible packing in dimension 3 is obtained by a face centered cubic and hexagonal arrangements of spheres. A proof of this legendary conjecture has finally been published in 2005 by Thomas Hales. The analogous problem in dimension 2 has been solved by Laszlo Fejes Toth in 1940, and this really is the extent of our current knowledge. If, however, one only considers lattice packings, i.e. arrangements of spheres with centers at points of a lattice, more is known.
In this talk, I will introduce the sphere packing problem, briefly surveying its history and known results. I will then restrict to lattice packings, describing a connection between the problem of finding an optimal lattice packing in a given dimension and minimization problem for Epstein zeta function on the space of unimodular lattices in this dimension. I will also introduce some important classes of lattices which are expected to solve these related problems, and will demostrate these concepts on the well understood 2-dimensional case. I will conclude with a certain approximation lemma which shows how good of a packing density one can expect from lattices with rational bases coming from one of these important classes of lattices in dimension 2.
March 25, 2008
Dr. Wallace Martindale
Department of Mathematics, University of Massachusetts
Title: One problem leads to another: some remarks on prime rings
Abstract: This is an expository talk, in part a personal account of how
attempts to solve a particular problem (in this case, Herstein's Lie
isomorphism conjectures of 1961) led to the development of some new areas
in noncommutative ring theory. In my case, I was led (in the 1960's) to
the notion of the extended centroid and the development of GPI
(generalized polynomial identity) theory to prime rings. In the case of my
colleagues (i.e. Beidar, Bresar, and Chebotar) the solution of Herstein's
conjectures was in fact the motivating factor for the very recent theory
of "functional identities". I will mostly just illustrate what is
happening with examples and will steer clear of heavy-handed
technicalities and long-winded theorems.
March 05, 2008
Dr. Bogdan Suceava
Department of Physics, California State University, Fullerton
Title: Curvature Inequalities
Abstract: John Nash's embedding theorem represented half a century ago the starting point of contemporary submanifold geometry. A natural question is what is the best possible way in which a given Riemannian space can be embedded into a given ambient space? Recently, following an idea of B.-Y.Chen, new curvature invariants have been introduced, to help us answer some questions in submanifold geometry. For example, the classical obstruction to minimal isometric immersions into Euclidean spaces is that Ricci tensor is nonnegative definite. We will present a method to construct examples of Riemannian manifolds with Ric<0 which don't admit any minimal isometric immersion into Euclidean spaces for any codimension, as a consequence of a result due to B.-Y.Chen. We will also survey new developments in the area of geometric inequalities, including Zhiqin Lu's recent proof of normal scalar conjecture.