Applied Mathematics Colloquia Series
Archive

Spring 1999

January

01/15-11/30
Russel Caflisch
Department of Mathematics
UCLA

Atomistic, continuum and bulk models for epitaxial growth

01/22-11/30
Robert Ecke
Los Alamos National Laboratory

Pattern Formation, Amplitude Equations and Spatio-Temporal Dynamics

Patterns in systems far from equilibrium have been the subject of extensive investigation during the past two decades. I will describe experiments which quantitatively address the relevance of amplitude equations to real physical systems. Most of the examples will be drawn from Rayleigh-Benard convection. I will also discuss states with persistent chaotic dynamics and describe characterization techniques used to understand these dynamics.

February

02/05-11/30
Madan Lal Mehta
Service de Physique Theorique
Centre d'Etudes Saclay, France

Matrices Coupled in a Chain: Correlations and Spacings

The general correlation function for the eigenvalues of p hermitian matrices coupled in a chain will be expressed as a single determinant. For this a slight generalization of a theorem of Dyson will be used. Spacing functions for the same chain of matrices will also be considered.

02/12-11/30
Nancy Koppel
Center for BioDynamics and Department of Mathematics
Boston University

Rhythms of the Nervous System

The nervous system is perpetually active, in waking and in sleep. The dynamics include a large collection of rhythms in different frequency ranges, associated with different behavioral states. It is not well understood how the rhythms arise, what determines their frequencies and how synchronization can take place over long distances in the nervous system. This talk will discuss the use of mathematics to help unravel these and other mysteries, and uncover dynamical structure that suggests functional implications of the rhythmic patterns.

02/19-11/30
Harry Swinney
Physics Department and Center for Nonlinear Dynamics
University of Texas at Austin

Patterns in Oscillated Sand: Squares, Hexagons, Stripes and Oscillons

Experiments on vertically oscillated layers of particles (sand, rice, bronze balls,...) reveal that, when the peak acceleration exceeds a critical value (2.5g), standing wave patterns spontaneously form and oscillate at half the excitation frequency. Square, stripe, hexagonal, and spiral patterns can form, depending on the oscillation frequency and peak acceleration of the container. Localized standing waves--oscillons--also form for a narrow parameter range, and these oscillons can bind to form dimers and more complex molecules. Many of the observations are accurately reproduced by molecular dynamics simulations of colliding particles that conserve linear and angular momentum but dissipate energy; these simulations provide insight into the transport processes.

02/26-11/30
Randall J. LeVeque
Departments of Mathematics and Applied Mathematics
University of Washington

High-Resolution Finite-Volume Methods for Waves in Rapidly-Varying Heterogeneous Media

High resolution multi-dimensional finite volume methods for conservation laws and other hyperbolic systems have been developed and implemented in the software package CLAWPACK. These methods have recently been applied to the solution to linear acoustics and elasticity problems with discontinuous and rapidly-varying material parameters. The solution of the "Riemann problem" at cell interfaces provides an accurate resolution of waves into reflected and transmitted parts at discontinuities, and the resulting methods can give accurate numerical solutions even when the material interfaces are not aligned with the grid. For rapidly-varying media this approach can be used to solve the equations directly, without resorting to homogenization, if one can afford a fine-scale calculation. These numerical methods may also serve as a useful tool in testing and comparing homogenization theories.

March

03/05-11/30
Hans G. Kaper
Mathematics and Computer Science Division
Argonne National Laboratory

The Ginzburg-Landau Equations of Superconductivity

This talk will introduce the GL equations of superconductivity, discuss their functional formulation, and present an analysis of their dynamics. The mathematics will be complemented with some computational case studies of the dynamic behavior of vortices in type-II superconductors.

03/12-11/30
John Hunter
Department of Mathematics
University of California, Davis

Nonlinear Gravitational Waves

Gravitational wave propagation is one of the most important features of Einstein's general theory of relativity. The Einstein field equations are highly nonlinear, and a question of fundamental interest is how nonlinearity affects the propagation of gravitational waves. We will describe an asymptotic solution of the vacuum Einstein field equations which describes the propagation of a thin, large amplitude gravitational wave into a slowly varing, curved space-time. The resulting asymptotic equations have the same form as well-known exact equations for colliding plane gravitational waves, without one of the usual constraint equations. The colliding plane wave equations are therefore canonical (1+1) dimensional equations for nonlinear gravitational waves.

03/26-11/30
Claudia Neuhauser
School of Mathematics
University of Minnesota

The role of explicit space in plant competition models

We present a spatial version of the Lotka-Volterra model of interspecific competition in which two species occupy sites on a square lattice. The non-spatial version of this model predicts coexistence when intraspecific competition is stronger than interspecific competition. We show that spatially local interactions can change the effective competitive interaction between the two species. Species tend to segregate as the strength of interspecific competition increases. Furthermore, the set of parameters for which coexistence is possible in the non- spatial case is reduced.

(Joint work with Stephen W. Pacala)

April

04/02-11/30
Michael J. Shelley
Courant Institute
New York University

The hydrodynamics of growing and flexing filaments

04/09-11/30
François Golse
Departement de Mathematiques et d'Informatique
Ecole Normale Supérieure

The Euler limit of the Vlasov-Poisson system, in various contexts

04/16-11/30
Catherine Garcia-Reimbert
Department de Mathematics and Mechanics
FENOMEC

Induced transparency in liquid crystals

Liquid crystals are optically active media whose properties change when they are subjected to magnetic or electric fields. This dependence on applied electric or magnetic fields can have a profound influence on the propagation of light through the crystal. In this work the phenomenon of induced transparency of a liquid crystal is studied.

Induced transparency is the dependence of the optical properties of a liquid crystal on the strength of impinging electromagnetic radiation. Light of low amplitude cannot propagate thought the crystal and the crystal remains opaque. However, when the light reaches a critical amplitude, it can propagate thought the crystal, changing the orientation of the director so that the crystal becomes transparent. In the present work a model is developed to explain induced transparency. The equations of this model are then solved numerically to predict parameter values for which induced transparency can be observer experimentally.

04/23-11/30
Graeme Milton
Department of Mathematics
University of Utah

Using analyticity to check the consistency of experimental measurements

This is joint work with David Eyre, Joe Mantese and Roderic Lakes. Experimental measurements often give the real and imaginary parts of the complex dielectric constant or magnetic permeability (or the bulk and shear moduli of viscoelastic materials) as a function of frequency. It is well known that the real and imaginary parts are Hilbert transforms of each other, with respect to frequency. These relations, known as the Kramers-Kronig dispersion relations, have a long history dating back to 1926 and have proved to be an invaluable tool for interpreting data. Such dispersion relations are prevalent throughout physics and derive from the causal nature of the response of materials, bodies or particles to electromagnetic, elastic or other fields.

In actual experiments one only makes measurements over a finite range of frequencies. This corresponds to the following mathematical problem: suppose we know the imaginary part of an analytic function is positive in the upper half plane, and is known along two disjoint intervals along the real axis. What can one say about the real part of the function? Our incomplete knowledge of the imaginary part of the function means that we can only derive inequalities on the real part. These inequalities generalize the Kramers-Kronig relations, providing bounds on the combinations of data values that are permissible over the measured frequency interval. They can provide highly accurate interpolation formulas for the real part, given its value at a few selected frequencies and given the imaginary part over a range of frequencies. We illustrate the practical utility of the bounds as applied to high frequency transmission line measurements of the complex relative magnetic permeability of a composite made with equal parts (by volume) of barium titanate and a magnesium-copper-zinc ferrite. For viscoelasticity we obtain even tighter bounds by assuming the complex bulk and shear moduli derive from a relaxation model.

04/30-11/30
Catherine Sulem
Department of Mathematics
University of Toronto

Modulation of water waves: Some rigorous results