Schedule for JBK80


D.S. Ahluwalia (New Jersey Institute of Technology)
ABSTRACT: The problem of diffraction of plane electromagnetic waves by a thin lossy half-plane is formulated by the aid of Helmholtz's equation satisfying two types of boundary conditions on the surfaces of the half-plane, 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 half-plane under consideration, is of the impedance-type 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 wedge-angles, the half-plane 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 half-plane in which the impedance is not a constant. It is shown that the problem can be cast into a three-part Wiener-Hopf problem which is in contrast to the well-known two-part Wiener-Hopf problem arising in the circumstances when the impedance is assumed to be a known constant everywhere on the surface of the half-plane. This three-part Wiener-Hopf 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 far-field at points lying in the plane of the scatterer but far away from it equals the far-field at those points that correspond to the non-lossy impedance half-plane, on the surface of which the same constant-impedance boundary condition holds everywhere.
1. Jones , D.S.-" Impedance of a lossy wedge ", IMA Journal of Applied
Mathematics (2001 ), Vol. 66, pp. 411-422.


Bedros Afeyan (Polymath Research Inc.)
"Nonlinear Interacting Waves in Vlasov Phase Space Using Wavelet Decomposition Based Nonuniform Adaptive Grids"
ABSTRACT: Solutions of the Vlasov-Poisson 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 self-consistent particle trapping and despite strong nonlinear wave-wave interactions. These are challenging to capture numerically. We present an adaptive grid refinement technique, within the context of a Semi-Lagrangian 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 one-way 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 pseudo-differential 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 one-way 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 one-way wave equations leads to one-way 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 ray-theoretical 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 two-dimensional 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 current-field 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 two-dimensional 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 Hamilton-Jacobi equation. This free boundary problem can be solved exactly in simple one-dimensional 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 quasi-Monte 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 length-dynamics Tonks gas theory of histone isotherms"
Abstract: The coverage of histone proteins on linear polymers such as DNA is computed by using a one-dimensional Tonks gas model.
We find a two-stage adsorption process and fluctuations in coverage that are maximal when the mean coverage is also maximal. Protein-protein 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 Markov-modulated 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 Harchol-Balter of Carnegie Mellon


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 cell-like 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 back-propagating 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 surfactant-producing chemical reactions on viscous fingering instabilities, and chemically driven oscillations and tip-streaming in drops.


Sam Howison (Oxford University)
"Existence, uniqueness and blow-up for the Muskat problem"
ABSTRACT: The Muskat problem is the two-phase Hele-Shaw 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 Hele-Shaw problem, it has a `forward' (stable, well-posed) direction in which a more viscous fluid displaces a less viscous one, and a correspondingly ill-posed 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 finite-time blow-up 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 Brain-Cell Microenvironment"
ABSTRACT: In the brain-cell 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 brain-cell 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 knife-edge. Parallels with Rayleigh's hypothesis in optics are drawn (cf. Keller 2000).


Jean-Marc Vanden-Broeck (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 two-layer 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 Cahn-Hilliard equation"
ABSTRACT: The convective Cahn-Hilliard 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 sharp-interface theory through a matched asymptotic analysis, resulting in a nearest-neighbors 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 anti-kink) 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 kink-ternary interaction; two kinks meet an anti-kink 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).

[1] S. J. Watson,
Crystal growth, coarsening and the convective Cahn-Hilliard 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 Cahn-Hilliard 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 long-time 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 non-uniformity in the films. Following the initial instabilities, the films breaks up into near-equilibrium droplets connected by an ultra-thin 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 long-wave 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.


Yuan-Nan 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.