Applied Mathematics Colloquia Series
Archive

Spring 2006

February

02/03
R. Choksi
Department of Computer Science
Simon Fraser University, BC Canada

Periodic Phase Separation

Diblock copolymer melts present a physical paradigm for periodic phase separation. On the other hand, a simple model gives rise to a nonlocal functional which itself is a mathematical paradigm for energy-driven pattern formation associated with short and long-range interactions. In this talk, I will discuss several mathematical aspects associated with the functional and the underlying physical problem. These will include modeling, analysis of periodicity and scale, numerics, and the geometry on minimizing structures. I will then focus on this last issue, presenting some very recent results obtained with P. Sternberg (Indiana).

02/10
J. Moehlis
Department of Mechanical Engineering
UC Santa Barbara

Optimal Inputs for Phase Models of Spiking Neurons

Variational methods are used to determine the optimal currents that elicit spikes in various phase reductions of neural oscillator models. We show that, for a given reduced neuron model and target spike time, there is a unique current that minimizes a square-integral measure of its amplitude. For intrinsically oscillatory models, we further demonstrate that the form and scaling of this current is determined by the model's phase response curve. These results reflect the role of intrinsic neural dynamics in determining the time course of synaptic inputs to which a neuron is optimally tuned to respond, and are illustrated using phase reductions of neural models valid near typical bifurcations to periodic firing, as well as the Hodgkin-Huxley equations. This work is in collaboration with Eric Shea-Brown and Herschel Rabitz.

02/17
K. Bhattacharya
California Institute of Technology

Variational Problems in the Study of Ferroelectric Solids

Ferroelectric solids display a host of interesting electrical, mechanical and optical properties, and are thus used in a variety of mechanical, electronic and photonic devices. These solids are spontaneously electrically polarized, and the polarization distribution in a material often form intricate patterns known as domain pattern. Further, the polarization and domain patterns can be affected by mechanical stress, electrical voltage and optical illumination, giving rise to interesting properties. This talk will describe a number of variational problems motivated by the study of ferroelectric solids. It will describe how models of ferroelectric materials lead to non-convex variational problems whose solutions contain fine-scale oscillations. It will also describe some asymptotic problems and how they can be addressed using Gamma-convergence. This talk will provide the necessary physical and mathematical background, illustrate the key ideas using simple examples, describe theoretical predictions and discuss their experimental validation.

02/24
D. Levermore
Applied Mathematics and Scientific Computing Program, Inst. for Physical Science and Technology, Dept. of Mathematics
University of Maryland

Fluid Dynamics from Boltzmann Equations

We survey some recent results that establish the validity of fluid dynamical systems from the Boltzmann equation. The starting point will be the DiPerna-Lions theory of global solutions to the Boltzmann equation. The target fluid systems will be incompressible and weakly compressible. A new multiple time-scale result will be presented that unifies the acoustic and Stokes limits. Open problems will be discussed.

March

03/03
H. Qian
Department of Applied Mathematics
University of Washington, Seattle

From Singular Perturbations to Stochastic Models: Michaelis-Menten Kinetics Revisited

Michaelis-Menten equation is one of the best known mathematical theories in biochemistry. Its standard derivation is based on a simple treatment in terms of singular perturbation. Recent experimental developments in enzymology have made it possible that to study one enzyme molecule at a time. Thus a more realistic model for single-molecule enzymology is stochastic.

I shall present (1) classic singlar perturbation treatment (a la J.D. Murray and Lee & Segel), (2) single enzyme based stochastic model, (3) a semi-Markov formalism to enzyme kinetic which yields some new mathematical results, and (4) a singular perturbation treatment of systems with small number of enzymes. The relations between (1), (2) and (4) will be discussed.

03/24
J. Dold
School of Mathematics
University of Manchester, Manchester UK

Bushfires and some models of bushfires

The Bush is synonymous with Countryside in much of the Anglophone world, especially in the southern hemisphere. Australia can rightly claim the most fire-prone environment in the world and "bushfires" there are a major problem, as they can be anywhere when they get out of hand. They have been studied experimentally leading to some interesting questions about their propagation and behaviour under reasonably controllable conditions. In other cases, they can still defy all reasonable predictions. The seminar will describe some of the behaviour of bushfires and the ways in which they can be modelled. It will also touch on some of the more extreme phenomena that can arise, including sometimes disputed descriptions of survivors, and it will offer some conjectures about the underlying physics.

03/31
Michel Destrade
Laboratoire de Modélisation en Mécanique
Université Pierre et Marie Curie, Paris, France

Linear, Linearized, and Nonlinear Waves in Solids

The concept of "waves in solids" usually conjures up the image of seismic waves, the modeling of which can be dated back to the works of Lord Rayleigh (1885) and many others. Amazingly, the same solutions can be used for wavelengths ranging from the kilometer scale (earthquakes) all the way to the Angstrom scale (high-frequency signal filters used in cell phones). Of course the governing equations must be modified in the process to account for anisotropy, piezoelectricity, and other effects which complicate the analysis, but also reveal surprising possibilities such as the celebrated shear-horizontal Bleustein-Gulyaev surface wave. This talk devotes its first part to an exposition of some analytical results obtained for surface waves and edge waves, which are prime examples of "linear waves in linear solids". The second part is devoted to "linearized waves in nonlinear solids", which are infinitesimal waves superimposed onto the large, static, homogeneous deformation of a hyperelastic solid. These small-amplitude motions are used to study the influence of pre-stress on the stability of geophysical structures or loaded rubber mounts, or to measure residual stresses in soft biological tissues. Finally, the talk evokes some nonlinear waves or more precisely, some exact traveling wave solutions to the equations of motion in fully nonlinear elasticity. These include generalizations of some odd "linear waves in nonlinear solids" which are of finite amplitude and yet whose propagation ends up being governed by linear differential equations. Another development is that of a simple way to account for inherent characteristic lengths in a solid, to model the dispersion of bulk waves; then, by fine-tuning dispersive and nonlinear effects, solitary and compact traveling waves can be uncovered.

April

04/14
John Kececioglu
Department of Computer Science
University of Arizona

Inverse parametric sequence alignment

For as long as biologists have been computing alignments of sequences, the question of what values to use for scoring substitutions, insertions, and deletions has persisted. While some choices for substitution scores are now common, largely due to convention, insertion and deletion penalties are often found by trial and error. An objective way to resolve this question is to determine the appropriate values by solving the Inverse Parametric Sequence Alignment Problem: Given examples of biologically-correct alignments, find parameter values that make the examples be optimal-scoring alignments of their sequences.

We present a new polynomial-time algorithm for Inverse Sequence Alignment based on linear programming that is conceptually simple, runs fast in practice, and can handle hundreds of parameters simultaneously. Computational results show that, for the first time, we can find best-possible values for all 212 parameters of the standard protein-sequence scoring-model from hundreds of alignments in a few minutes of computation. This is joint work with Eagu Kim.

04/28
Antti Kupiainen
Department of Mathematics & Statistics
University of Helsinki

On Fourier's Law of Heat Conduction

Fourier's law states that a temperature gradient gives rise to a heat flux proportional to it, the proportionality constant being the thermal conductivity. temperature and heat flux are macroscopic quantities determined by the underlying microscopic Hamiltonian or quantum dynamics. The problem of deriving the Fourier's law is to show how such a dynamical system can give rise to the macroscopic irreversible behaviour. We review this problem in the context of a classical mechanical system consisting of coupled anharmonic oscillators. In the second part of the talk we describe a "closure" approximation to the equations determining the stationary state of the system and outline the derivation of Fourier's law from these approximative equations.