Applied Mathematics Colloquia Series
Archive

Fall 2007

September

09/14
Celeste Sagui
Department of Physics
North Carolina State University

Accurate and Efficient Electrostatics for Large-Scale Biomolecular Simulations

An accurate and numerically efficient treatment of electrostatics is essential for biomolecular simulations, in particular, when a smooth interface to quantum chemical descriptions is needed. Force field used in classical biomolecular simulation codes such as AMBER and CHARMM assign "partial charges" to every atom in a simulation in order to model the interatomic electrostatic forces. The respective charge values are obtained via least-squares fitting to the Coulombic potential produced by quantum chemical procedures Unfortunately, the fitting procedure for large, conformationally flexible molecules is under-determined, which is a major source of errors. There are two main problems associated with the treatment of classical electrostatics: (i) how does one eliminate artifacts associated with the point charges as used in force fields, and thereby improve the electrostatic potentials in a physically meaningful way?; (ii) how does one efficiently simulate the very costly long-range electrostatic interactions? Here, we present results on a recently developed distributed multipole method and discuss the importance of this method for large scale biomolecular simulations.

Refreshments at 3:30 in Math 401N
09/21
Mike Marder
Center for Nonlinear Dynamics and Department of Physics
University of Texas at Austin

Geometry and Elasticity of Leaves and Flowers

Many lovely buckling patterns can be induced in thin sheets by modifications of the metric. I will discuss experiments of Eran Sharon that explored this phenomenon first in plastic sheets and then in leaves. From a theoretical point of view, modifying the metrics of thin sheets creates a situation where it is no longer clear how much can be deduced from differential geometry, and how much must be added from elasticity. Finally I will mention an application to graphene.

09/28
Don Wang
Department of Mathematics
University of Arizona

Return Maps around Homoclinic Solutions

In the long course of the development of the modern theory of dynamical systems, ordinary differential equations have served as both a constant inspiration and a test ground. Periodically forced second order equations have been studied extensively in history. When an autonomous system with a saddle point and a homoclinic solution is subjected to periodic perturbations, the stable and unstable manifold of the perturbed saddle intersect transversally, creating a homoclinic tangle for the time-T map. Milnikov's method has been introduced for the purpose of detecting this homoclinic tangle in concrete systems of differential equations.

In this talk we follow a new route. Instead of studying the time-T map, about which our knowledge has been based mainly on geometric descriptions, we construct a Poincare section that mixes the original phase dimensions with time in the extended phase space and compute explicitly the return maps induced by the differential equations. It has turned out that, for dissipative equations, these return maps are rank one maps studied intensively by Young and myself in the last ten years. The conclusions of these studies, consequently, are applied in a rather natural fashion to the analysis of periodically forced second order equations with homoclinic solutions.

This talk is designed for a general audience, and the prerequisites are kept to a minimum.

October

10/12
C. David Levermore
Department of Mathematics and Institute for Physical Science and Technology
University of Maryland, College Park

Weakly Nonlinear-Dissipative Approximations of Hyperbolic-Parabolic Systems with Entropy

Hyperbolic-parabolic systems have spatially homogeneous equilibria. Whenathe dissipation is weak, one can derive weakly nonlinear-dissipativeaapproximations that govern perturbations of these equilibria. Theseaapproximations are quadratically nonlinear. Up to a linearatransformation, they are independent of the dependent variables used toaexpress the original system. When the original system has an entropy,athe approximation is formally dissipative in a natural Hilbert space. Weashow that under a mild structural hypothesis, this approximation hasaglobal weak solutions for all initial data in that Hilbert space. Thisatheory applies to the compressible Navier-Stokes system. The resultingaapproximate system is an incompressible Navier-Stokes system coupled toaequations that govern the acoustic modes. The solution of thisaapproximate system is unique if the incompressible modes are uniquelyadetermined.a

10/19
Kristin R. Swanson
Departments of Pathology and Applied Mathematics
University of Washington

A Rule for Mathematical Modeling in Individualizing Treatment of Brain Tumors

Malignant gliomas are brain tumors that differ from most other tumors by their aggressive diffuse invasion of the surrounding normal tissue which contributes to the dismal 6 to 12 months prognosis. The remarkable continuing development of medical imaging has increased the ability to detect gliomas, but has not been able to sufficiently define the extent of invasion of the tumor cells peripheral to the bulk tumor mass. For this reason, only the "tip of the iceberg" of the full lesion is visible by standard imaging tools (MRI, CT, PET). We use mathematical modeling to predict the true extent of the tumor given the limited information provided by standard clinical imaging. Our mathematical model can be applied to an individual patient to calculate the two factors (proliferation and infiltrative potential) precisely enough to display the past, present and future distributions of tumor cell concentrations down to the individual cell, well beyond the "edge "of the tumor defined by current imaging.

Clinicians routinely question the efficacy of treatment and, to date, have very few quantitative tools for assessing treatment response in real time. Further, as new experimental therapies (radio and chemo) are introduced, an early and robust measure of effect is necessary in individual patients. These problems are compounded for those therapies that target molecular changes that only a subset of tumors will have and others not have, since tissue sampling may not be available in all. It is clear to us that truly revolutionary research to improve the humbling prognosis of gliomas cannot proceed without knowledge of the in vivo interactions of the two most salient features of their progression - dispersal and net proliferation - provided through mathematical modeling.

10/26
Garry Webb
Institute of Geophysics and Planetary Physics
University of California Riverside

Hamiltonian Theory of Nonlinear Traveling Waves in Multi-Fluid, Charge-Neutral Space Plasmas

Fully nonlinear traveling waves in multi-fluid plasmas have recently received considerable theoretical and observational attention in space plasma physics, as a consequence of progress in high time resolution satellite observations in the near Earth plasma environment by the FAST, Polar, Geotail and Cluster spacecraft. In this talk we present a dual (multi-symplectic) Hamiltonian description of traveling waves in a charge-neutral, electron-proton, non-relativistically moving plasma. The analysis is facilitated by using the de Hoffman-Teller (dHT) frame of MHD shock theory to simplify the transverse electron and proton momentum equations. We show that the governing equations are exactly integrable in the case where the total transverse momentum fluxes of the system are zero in the dHT frame. Numerical examples of integrable, oblique, traveling waves in a cold plasma are used to illustrate the physics. The transverse electron and proton velocity components exhibit complex rosette type patterns. The role of separatrices in the phase space, the rotational integral, and the longitudinal structure equation on the different wave forms are discussed.

November

11/02
Miroslav Kolesik
College of Optical Sciences
University of Arizona

Nonlinear Wave Mixing in Femtosecond Optics

More than a decade ago, the phenomena of long distance propagation and filamentation were discovered in high-power, ultrashort light pulses propagating in gaseous and condensed bulk media. This instigated keen interest and earnest research, motivated by both the basic physics and many practical applications. A great deal of our knowledge concerning these phenomena originates in modeling and numerical simulations. Realistic numerical modeling continues to provide new insights into the underlying physics that would be extremely difficult to understand from experiments alone. Here, I present an overview of numerical methods capable to study the highly nonlinear processes that accompany optical filamentation, such as explosive broadening of spectrum, plasma generation and harmonic generation. I will also discuss an effective nonlinear weave mixing paradigm that provides us with an intuitive yet quantitative way to understand complex experimental data.

11/09
Annette Peko Hosoi
Department of Mechanical Engineering
MIT

Optimizing Low Reynolds Number Locomotion

In this talk I will discuss two optimization topics related to low Reynolds number locomotion: optimal stroke patterns in linked swimmers and optimal fluid material properties in adhesive locomotion. In contrast to many optimization problems, we do not consider geometry; rather, we optimize the swimming kinematics or fluid material properties for a given geometrical configuration. In the first case, we begin by optimizing stroke patterns for Purcell's 3-link swimmer. We model the swimmer as a jointed chain of three slender links moving in an inertialess flow. The swimmer is optimized for both efficiency and speed. In the second case, we analyze the adhesive locomotion used by common gastropods such as snails and slugs. Such organisms crawl on a solid substrate by propagating muscular waves of shear stress on a viscoelastic mucus. Using a simple mechanical model, we derive criteria for favorable fluid material properties to lower the energetic cost of locomotion.

11/16
Emina Soljanin
Bell Laboratories, Murray Hill, NJ

What Are the Benefits of Coding in Content Distribution Networks?

Network coding is an elegant and novel technique introduced at the turn of the millennium to improve network throughput and performance. Its power comes from allowing network nodes to combine and process different incoming information streams. It is widely believed that potential benefits of network coding are huge, in particular in terms of throughput and security. By using techniques of combinatorial optimization and occupancy models, this talk will examine whether such beliefs are generally justified, and quantify coding benefits for certain network topologies and traffic scenarios.

11/30
Roberto Camassa
University of North Carolina

Spinning Rods, Microfluidics, and Mucus Propulsion by Cilia in the Lung

Understanding and modeling how human lungs function is in large part based on the hydrodynamics of the mucus fluid layers that coat lung airways. In healthy subjects, the beating of cilia is thought to be the primary method of moving mucus. With the aim of establishing a quantitative benchmark of how cilia motion propels the surrounding fluid, we study the idealized situation of one rod spinning in a fluid obeying the Stokes approximation, the appropriate limit for a Newtonian fluid with typical dimensions, and time scales of cilia dynamics.

New approximate-for cylindrical rods pinned to a flat plane boundary-and exact-for ellipsoidal rods freely spinning around their center-solutions for the fluid motion will be presented and compared with the experimental data collected with spinning magnetic nano-rods in water. In order to assess the influence of Brownian perturbations in this micro-scale experiment, data from an experimental set-up scaled by dynamical similarity to macroscopic (table-top) dimensions will also be presented and compared to the theoretical predictions.