Schedule for JBK80
POSTERS

D.S. Ahluwalia (New
Jersey Institute of Technology)
"DIFFRACTION BY A LOSSY HALFPLANE"
ABSTRACT: The problem of diffraction of plane electromagnetic waves
by a thin lossy halfplane is formulated by the aid of Helmholtz's
equation satisfying two types of boundary conditions on the surfaces
of the halfplane, one of which holds good near the edge and the
other away from it. While the boundary condition, that applies to
points sufficiently away from the edge of the halfplane under consideration,
is of the impedancetype with a known constant impedance, the boundary
conditions for points in a small unknown neighbourhood of the edge
involves variable impedance, becoming infinite at the edge, as has
been shown by Jones [ 1 ] recently, in the cases of wedges of arbitrary
wedgeangles, the halfplane being thought of as a wedge of angle
2p.
The present mixed boundary value problem involves an extra unknown
parameter, representing the small distance e ( > 0 ) near the
edge of the halfplane in which the impedance is not a constant.
It is shown that the problem can be cast into a threepart WienerHopf
problem which is in contrast to the wellknown twopart WienerHopf
problem arising in the circumstances when the impedance is assumed
to be a known constant everywhere on the surface of the halfplane.
This threepart WienerHopf problem is handled for solution, by
using an approximate analysis of a special type , and the unknown
parameter e of the problem is determined by using the assumption
that the farfield at points lying in the plane of the scatterer
but far away from it equals the farfield at those points that correspond
to the nonlossy impedance halfplane, on the surface of which the
same constantimpedance boundary condition holds everywhere.
Reference:
1. Jones , D.S." Impedance of a lossy wedge ", IMA Journal
of Applied
Mathematics (2001 ), Vol. 66, pp. 411422.

Bedros Afeyan (Polymath
Research Inc.)
"Nonlinear Interacting Waves in Vlasov Phase Space
Using Wavelet Decomposition Based Nonuniform Adaptive Grids"
ABSTRACT: Solutions of the VlasovPoisson system of equations are
presented of ponderomotively driven electron plasma and electron
acoustic waves. Somewhat surprisingly, long lived (metastable) highly
nonlinear states persist, long after the ponderomotive forces have
been turned off, due to selfconsistent particle trapping and despite
strong nonlinear wavewave interactions. These are challenging to
capture numerically. We present an adaptive grid refinement technique,
within the context of a SemiLagrangian method, which is based on
fast wavelet transforms of the phase space trajectories. Longer
time scale and more reliable simulations may be achieved by this
technique than via traditional uniform grid interpolation methods
where unphysical entropy increase and excessive smoothing of fine
scale features can easily corrupt the solutions.

Paul Barbone (Boston
University)
"An elastic inverse problem in medical imaging"
ABSTRACT: Elastography, the imaging of soft tissue on the basis
of (shear) elastic modulus, is an emerging imaging method. The technique
relies on being able to image soft tissue while it is being deformed
by a set of externally applied forces. Through image processing,
the displacement (or sometimes velocity) field everywhere in the
region of interest is inferred. An inverse problem for the elastic
modulus results, given the measured displacement fields, an assumed
form of the tissue's constitutive equation (e.g.\ linear elastic),
and the law of conservation of momentum. We formulate, study and
solve this inverse problem. We find that the standard elastography
inverse problem is nonunique.
We describe practical formulations that are unique, discuss continuity
of the solution on the data, and existence of solution. Further,
we develop a novel stable numerical method required to solve the
resulting advective hyperbolic systems of equations, and present
examples.

Norman Bleistein (Center
for Wave Phenomena)
"Seismic Inversion with oneway wave equations."
(G. Q. Zhang, Y. Zhang and N. Bleistein)
ABSTRACT: One way wave operators are powerful tools for forward
modeling and inversion. However, their implementation involves introduction
of the square root of an operator as a pseudodifferential operator.
Exact representations of such square roots are illusive, except
in the simplest of cases. Here, singling out depth as the preferred
direction of propagation, we introduce a representation of the square
root operator as an integral in which a rational function of the
transverse Laplacian appears in the integrand. This allows us to
solve the resulting oneway wave equations with the simple device
of introducing an auxiliary function that satisfies a lower dimensional
wave equation in transverse variables only. We verify that ray theory
for these oneway wave equations leads to oneway eikonal equations
and the correct leading order transport equation for the full wave
equation. We then introduce appropriate boundary conditions at z
= 0 to generate waves at depth whose quotient leads to a reflector
map and estimate of the raytheoretical reflection coefficient on
the reflector. This method is known as "true amplitude wave
equation migration" in the geophysics literature. Computer
output confirms the
accuracy of the method for such inversion

Luis L. Bonilla (Universidad
Carlos III de Madrid)
"Free boundary problems describing twodimensional
pulse recycling and motion in semiconductors"
ABSTRACT: Pulses of the electric field may be generated and destroyed
at metal contacts in semiconductors presenting negative differential
conductivity in their currentfield characteristics. Pulse recycling
and motion give rise to oscillations of the current used to design
oscillatory devices such as Gunn diodes or many nanostructure oscillators.
In joint work with Ramon Escobedo and Francisco Higuera, we present
an asymptotic analysis of these phenomena in twodimensional semiconductor
samples with circular contacts. A moving pulse of the electric field
far from contacts is approximated by a moving free boundary separating
regions where the electric potential solves a Laplace equation with
subsidiary boundary conditions. The dynamical condition for the
motion of the free boundary is a HamiltonJacobi equation. This
free boundary problem can be solved exactly in simple onedimensional
and axisymmetric geometries. In the general case, we solve the free
boundary problem numerically and compare with the numerical solution
of the full model.

Russel Caflisch (UCLA)
"Simultation methods for American options"
ABSTRACT: American options allow exercise at an optimal stopping
time. This optimal stopping time is difficult to include in Monte
Carlo simulation. This talk will review several approaches to overcoming
this obstacle, as well as some new results on quasiMonte Carlo
evaluation of American options.

Ana Carpio (Universidad
Complutense de Madrid)
"Depinning of dislocations in crystal structures"
ABSTRACT: The static stress to depin a 2D dislocation, the lower
dynamic stress needed to keep it moving, its velocity and displacement
vector profiles are computed from first principles. We use a simplified
discrete model whose far field distortion tensor decays algebraically
with distance, as usual in elasticity. An analytical description
of dislocation depinning in the strongly overdamped case (including
the effect of fluctuations) is given. We also show that a set of
N parallel egde dislocations can depin a given one provided N=O(L),
L being an average interaction distance, and define a limiting dislocation
density. This is joint work with Luis L.L.
Bonilla (Universidad Carlos III de Madrid).

Aloknath Chakrabarti
(New Jersey Institute of Technology)
"Role of weakly singular integral equations in surface
water wave scattering"
ABSTRACT: A Treatment of handling a class of boundary value problems
in the theory of surface water wave scattering , that uses only
weakly singular integral equations, is demonstrated.
The present treatment of the boundary value problems under consideration
is an alternative to the use of either strongly singular or hypersingular
integral equations.

Tom Chou (UCLA)
"A lengthdynamics Tonks gas theory of histone isotherms"
Abstract: The coverage of histone proteins on linear polymers such
as DNA is computed by using a onedimensional Tonks gas model.
We find a twostage adsorption process and fluctuations in coverage
that are maximal when the mean coverage is also maximal. Proteinprotein
correlations are also computed.

Diego Dominici (UIC)
"Ray solution of a singularly perturbed elliptic PDE
with applications to communications networks"
ABSTRACT: We analyze a second order, linear, elliptic PDE with mixed
boundary conditions. This problem arose as a limiting case of a
Markovmodulated queueing model for data handling switches in communications
networks. We use singular perturbation methods to analyze the problem.
In particular we use the ray method to solve the PDE in the limit
where convection dominates diffusion. We show that there are both
interior and boundary caustics, as well as a cusp point where two
caustics meet, an internal layer, boundary layers and a corner layer.
Our analysis leads to approximate formulas for the queue length
(or buffer content) distribution at the switch.

Patrick S Hagan
(Bear Stearns)
"Managing Smile Risk :)"
ABSTRACT: Local volatility (implied tree) models are the most popular
method for pricing and hedging options in the presence of market
smiles and skews. A careful analysis of these models show that they
predict that market smiles move in the opposite direction as the
price of the underlying asset, contrary to all trading experience.
This difference causes the hedges to be unstable, which can lead
to serious ëleakageí in option books. A deeper look
at the theory leads us to a stochastic volatility model, the SABR
model. We solve this model to obtain a closed form solution for
the implied volatility. This solution shows good agreement with
the observed volatility smiles, and more importantly, shows that
the SABR model predicts the correct smile dynamics, which leads
to stable hedges

Robert C. Hampshire (Princeton University)
"Asymptotic Analysis of the User Response Time for
a Webserver During Periods of Congestion"
ABSTRACT: We model the user response time for a webserver as the
sojourn time of a virtual customer in a processor sharing queue.
Moreover, we can model periods of congestion or slow response times
by assuming that the mean arrival rate for job requests exceeds
the average job processing rate. Applying asymptotic methods, we
obtain tractable fluid and diffusion approximations for these response
times. We also extend this analysis to the case of time varying
arrival rates.
This is joint work with William A. Massey of Princeton University
and Mor HarcholBalter of Carnegie Mellon
University.

David Holcman (Keck
Center, UCSF San Francisco/Berkeley, Math dept)
"Calcium dynamic in dendritic spine and spine motility"
ABSTRACT: I will present a work in collaboration with Z. Schuss(TAU),
E. Korkotian(WIS), about modeling some neurobiological microsystems.
A dendritic spine is a celllike structure located on a dendrite
of a neuron. It conducts calcium ions from the synapse to the dendrite.
A dendritic spine can contain anywhere between a few and up to thousands
of calcium ions at a time. Internal calcium is known to bring about
fast contractions of dendritic spines (twitching) after a burst,
an action potential, or a backpropagating action potential. In
this paper, we propose an explanation of the cause and effect of
the twitching and its role in the functioning of the spine as a
conductor of calcium. We model the spine as a machine powered by
the calcium it conducts and we describe its moving parts. The latter
are proteins that are involved in the conduction process. These
proteins are found inside the dendritic spine and their spatial
distribution can be measured. We propose a molecular model of calcium
dynamics in a dendritic spine, which shows that the rapid calcium
motility in the spine is due to the concerted contraction of certain
proteins that bind calcium. The contraction induces a stream of
the cytoplasmic fluid in the direction of the dendritic shaft, thus
speeding up the time course of spinal calcium dynamics, relative
to pure diffusion. According to the proposed model, the diffusive
motion of the calcium ions is described by a system of Langevin
equations, coupled to the hydrodynamical fluid flow field induced
by contraction of proteins. These contractions occur when enough
calcium binds to specific protein molecules inside the spine. By
following the random ionic trajectories, we compute the distribution
of calcium exit time from the spine, the evolution of concentration
of calcium bound to specific proteins, the relative number of ions
pumped out, compared to the number of ions that leave at the dendritic
shaft, and so on. A computer simulation of this model of calcium
dynamics in a dendritic spine was run with any the number of calcium
ions varying from one or two, up to the hundreds. The simulation
indicates that spine motility can be explained by the basic rules
of chemical reaction rate theory at the molecular level . Analysis
of the simulation data reveals two time periods in the calcium dynamics.
In the first period calcium motion is driven by a hydrodynamical
push, while there are no push effects in the second, when ionic
motion is mainly diffusion in a domain with obstacles. A biological
conclusion is that the role of rapid motility in dendritic spines
is to increase the efficiency of calcium conduction to the dendrite
and to speed up the emptying of the spine.

George M. Homsy (UCSB)
"Some Novel Interfacial Flows"
ABSTRACT: Interfacial flows driven by applied stresses and/or surface
tension gradients occur in many applications in Chemical and Mechanical
Engineering. They involve interfacial fluid mechanics and free boundary
problems, areas to which Joe Keller has contributed. This presentation
will consist of movies (the available technology permitting) of
some new and unexpected interfacial flows, including using electrical
stresses to drive chaotic advection in drops, the effect of surfactantproducing
chemical reactions on viscous fingering instabilities, and chemically
driven oscillations and tipstreaming in drops.

Sam Howison (Oxford
University)
"Existence, uniqueness and blowup for the Muskat problem"
ABSTRACT: The Muskat problem is the twophase HeleShaw free boundary
problem; it is a model for flow in porous media and is a close relative
of the Stefan problem (both areas on which Joe Keller has published
papers). Like the HeleShaw problem, it has a `forward' (stable,
wellposed) direction in which a more viscous fluid displaces a
less viscous one, and a correspondingly illposed backward direction.
In joint work with Russel Caflisch and Mike Siegel, we show existence
and uniqueness of a smooth solution for the forward problem, and
we demonstrate by construction that it is possible to have finitetime
blowup for the backward problem.

Steven D. London (University
of Houston)
"Resistive Instability in the Earth's Outer Core: A
Thin Spherical Shell Model"
ABSTRACT: The Earth's outer core is modeled as a thin, rotating,
electrically conducting spherical shell containing a conducting
fluid. The fluid is not a perfect conductor and is therefore subject
to the possibility of resistive instability. Such instability has
been detected in numerical work. This poster discusses an attempt
to study an analytic asymptotic model for resitive instability.
We assume a thin shell with large Elsasser number and look for a
geometric optics type solution. Application of the boundary conditions
determines the complex frequency. Results of this analysis are discussed.

Robert M. Miura (New
Jersey Institute of Technology)
"Dispersal of Ions in the BrainCell Microenvironment"
ABSTRACT: In the braincell microenvironment, an increase in the
extracellular potassium concentration can depolarize neurons and
affect their excitability, as well as affect glial cells. Above
a pathological level, there may result a slow chemical wave called
"spreading cortical depression". The K+ dynamics result
from diffusion in the extracellular and intracellular spaces, passive
and active ion transport across the membranes, and a spatial buffering
mechanism. From a realistic tissue structure, we build a theoretical
model and study the migration of K+ due to the injection of KCl,
as well as the induced migrations of Na+ and Cl. A square lattice
is used on which the K+, Na+, and Cl particles move with discrete
temporal and spatial steps. Different rules for each ion determine
their movements according to the lattice Boltzmann equations and
membrane current equations. We show several important effects due
to the microscopic structure of the braincell environment. A new
mechanism of buffering potassium, namely, temporal buffering, is
proposed and demonstrated.

Stefan Llewellyn Smith
(MAE, UCSD)
"Mathematical models of tidal conversion"
ABSTRACT: Tidal conversion is the process by which energy
is converted from the barotropic tide to internal gravity waves
via flow over ocean bathymetry. These internal gravity waves are
known as the "internal tide". The internal tide propagates
at a fixed angle to the vertical determined by the three fundamental
frequencies: (1) the tidal frequency, (2) the Coriolis frequency
and (3) the buoyancy frequency. We review models of tidal conversion
by topography ranging from shallow slopes to a vertical knifeedge.
Parallels with Rayleigh's hypothesis in optics are drawn (cf. Keller
2000).

JeanMarc VandenBroeck
(The University of East Anglia)
"Nonlinear waves and fronts at the interface between
two fluids"
ABSTRACT: Nonlinear waves in a forced channel flow are considered.
The forcing is due to a bottom obstruction. A weakly nonlinear analysis
is performed. The weakly nonlinear results are validated by comparison
with numerical results based on the full governing equations. Although
the problem of twolayer flows over an obstacle is a classical problem,
several branches of solutions which have never been computed before
are obtained. Experimental results are presented.

Stephen J. Watson
(Northwestern University)
"Coarsening dynamics for the convective CahnHilliard
equation"
ABSTRACT: The convective CahnHilliard equation (cCH) models the
coarsening dynamics of growing faceted crystal surfaces; the local
surface slope is the order parameter and attachment kinetics is
the dominant mass transport mechanism.
We characterize the coarsening dynamics of the (cCH) in one space
dimension [1,2].
First, we derive a sharpinterface theory through a matched asymptotic
analysis, resulting in a nearestneighbors interaction coarsening
dynamical system (CDS). From this, scaling laws for the entire coarsening
path are derived, and the relevant crossover regime is identified.
Further, two types of phase boundaries (kink and antikink) emerge
in our sharp interface theory due to the presence of convection.
Our theoretical predictions on (CDS) include the following novel
coarsening mechanisms:
i) Binary coalescence of phase boundaries is impossible;
ii) Ternary coalescence may only occur through the
kinkternary interaction; two kinks meet an antikink resulting
in a kink.
We also present direct numerical simulations on
(cCH) which confirm our predicted scaling laws as well as the above
coarsening features. In part, joint work with Felix Otto (Bonn)
and Stephen H. Davis (Northwestern).
References:
[1] S. J. Watson,
Crystal growth, coarsening and the convective CahnHilliard equation,
in Free Boundary Problems (Trento, 2002) Birkhaeuser, (P. Colli,
C. Verdi, A. Visintin, ed.)
[2] S. J. Watson et. al, ``Coarsening dynamics for
the convective CahnHilliard equation'',
to appear in Physica D (2002). [Preprint: Bonn (2001), Max Plank
Institute (April 2002)].

Thomas Witelski
(Duke University)
"Coarsening Dynamics of Dewetting Films"
ABSTRACT: Lubrication theory for unstable thin liquid films on solid
substrates is used to model the coarsening dynamics in the longtime
behavior of dewetting films. The dominant physical effects that
drive the fluid dynamics in dewetting films are surface tension
and intermolecular interactions with the solid substrate. Instabilities
in these films lead to rupture and other morphological changes that
promote nonuniformity in the films. Following the initial instabilities,
the films breaks up into nearequilibrium droplets connected by
an ultrathin film. For longer times, the fluid will undergo a coarsening
process in which droplets both move and exchange mass on slow timescales.
The dynamics of this coarsening process will be obtained through
the asymptotic reduction of the longwave PDE governing the thin
film to a set of ODEs for the evolution of the droplets. From this,
a scaling law which governs the coarsening rate is derived. This
is joint work with Karl Glasner, Dept of Math, University of Arizona.

YuanNan Young (Stanford
University)
"Improved Particle level set method"
ABSTRACT: Level set method has been used extensively to capture
interfacial phenomena. A hybrid method of particle level set method
is recently developed by Enright and Fedkiw. We improve this algorithm
by incorporating curvature into each Lagrangian particle and correct
the level set using such geometric information carried by each particle.
We also demonstrate how this improvement can be useful in capturing
levels in mixing and reacting flows.


