Applied Mathematics Colloquia Series
Archive

Spring 2004

January

01/09
Robert Pego
Department of Mathematics
University of Maryland, College Park

Dynamic Scaling in Smoluchowski Ripening and Burgers Turbulence

Smoluchowski's coagulation equation is a fundamental mean-field model for the agglomeration of clusters. I'll describe comprehensive results obtained recently with Govind Menon regarding dynamic scaling of solutions for certain `solvable' rate kernels. We work in a framework for dynamic scaling analysis that ties together probability and dynamical systems theory in inspiring and profitable ways.

Our results apply in particular to Burgers' turbulence model, a basic model for understanding statistics of solutions of nonlinear PDEs. Via work of J.Bertoin, we obtain a complete classification of universality classes for dynamic scaling of shock size distributions for the inviscid Burgers equation, with initial velocity that is random with stationary, independent increments with no positive jumps.

01/16
G.M. Homsy
Department of Mechanical & Environmental Engineering
University of California, Santa Barbara

Novel Marangoni Flows

In this talk I will describe three recent studies of novel Marangoni flows, i.e. flows that are driven by tangential stresses that are produced by temperature, compositional, or electrical fields. The first two of these are flows driven or modified by the non-uniform in-situ production of surfactants by chemical reactions. Such surfactant gradients give rise to surface tension gradients which drive bulk flows. We study experimentally the effect of such reactions on viscous fingering in the tip-splitting regime, finding that Marangoni stresses result in wider fingers and a suppression of the tip-splitting instability. We then describe an amazing phenomena of spontaneous, self-sustained chemically driven oscillations at the tip of a drop suspended from the tip of a needle and connect this phenomena to the well-known tip-streaming in extensional flow near drops. Finally, we describe theory and experiment on the manipulation of tangential electrical stresses to drive chaotic advection in translating drops of dielectric liquids.

01/23
Erik Winfree
California Institute of Technology

Arthur Winfree Memorial Lecture: Clocks and Singularities in Fault-Tolerant Computing

As computers are built from ever smaller devices, with logical functions ultimately occurring statistically within individual molecules, issues of fault-tolerance become critical. Because at these scales transmission of information also becomes unreliable, local models such as cellular automata are often preferred for the formulation of fundamental results. Similarly, as global clocks cannot be relied upon, asynchronous operation must be considered. In this talk, I will informally describe the development of fault-tolerant cellular automaton models in one, two, and three dimensions. Emphasis will be on recent work, with Matt Cook and Peter Gacs, that has made use of topological concepts from the study of excitable media (which I learned as a boy from my father!) to develop new fault-tolerant asynchronous cellular automata.

February

02/20
Renu Malhotra
Lunar & Planetary Laboratory
University of Arizona

From Dust to Dust: The Evolution of Dust in Planetary Systems

Our current paradigm has planets forming within circumstellar disks by the accumulation of micron-sized solid dust grains into planetesimals, followed by the accretion of planetesimals into planets. The leftovers of planet formation are so-called debris disks, the remnant planetesimals that do not accrete into large planets; the asteroid belt and the Kuiper Belt constitute the debris disk of our Solar system. Over billions of years, the remnant planetesimals disintegrate back into small dust grains, the majority of which escape back into the interstellar medium. In this talk, I will describe our current picture as well as outstanding problems in the dynamical transport of dust and solids in the various steps of this cycle of dust-to-planets-to-dust. I will emphasize recent work on this topic with graduate student Amaya Moro-Martin, and current efforts on the astronomical detection of exo-solar debris disks and their planetary systems.

02/27
Eric Kostelich
Department of Mathematics
Arizona State University

Data Assimilation: Finding the Initial Conditions in Large Dynamical Systems

Data assimilation refers to the question of how to determine the initial conditions for a given dynamical model (such as a weather forecast model) from a collection of measurements. Dynamical weather models often exhibit local low dimensionality: in certain regions of the earth, vectors of the forecast uncertainty in the dynamical variables tend to span a subspace whose dimension is much less than that of the full atmospheric state vector in the region. This talk will outline a data assimilation scheme with comparatively modest computational costs that attempts to reduce uncertainties in the estimated current state of the atmosphere in the region by using only this lower dimensional subspace. The resulting local analyses are used to construct estimates of the global state vector, which in turn is used to generate forecasts. I will present preliminary results of its application to a 40-variable dynamical model and to the National Weather Service's global forecast model. The implementation of the method on massively parallel computer architectures, such as IBM's Blue Gene, will also be discussed.

March

03/05
Edward Kerschen
Aerospace & Mechanical Engineering
University of Arizona

Resonances Caused by Flow Past a Cavity in the Wall

Acoustic resonances leading to high unsteady pressure levels may occur in flow past cavities. The resonance loop involves a coupling between the downstream-propagating instability wave on the shear layer spanning the open face of the cavity, and acoustic waves propagating within the cavity.The elements of the disturbance field are coupled by the scattering processes that occur at the upstream and downstream ends of the cavity. A theoretical prediction method is developed which combines propagation models in the central region of the cavity with scattering models for the end regions. In the analyses of the scattering problems at the cavity ends, the square-corner geometry is treated exactly, by a method employing the Wiener-Hopf technique. The global analysis provides the frequency and growth (or decay) rate of each mode. The predicted frequencies are in generally good agreement with existing.

03/12
Tracy Jackson
University of Michigan

Multiphase Mechanics of Tumor Encapsulation and Invasion

Encapsulation responses are a widespread and primitive biological phenomenon. The mechanisms responsible for encapsulation of non-neoplastic lesions such as parasites and implanted foreign materials have been extensively researched. However, while it is well-known that many benign tumors are surrounded by a well-defined capsule and that the presence of this capsule is an integral determinant of prognosis, the mechanisms by which the capsule forms have yet to be explained fully. Two complementary theories have been postulated to explain this phenomenon. Since it is difficult to discriminate between the hypotheses using experimental techniques, mathematical modeling provides a natural approach for testing and comparing the assumptions and the consequences associated with each of them. In this talk, I will present a mathematical model that describes the processes of tumor growth and encapsulation. By performing a range of numerical simulations we are able to compare the two hypotheses for capsule formation and to predict which we believe is more likely to explain the experimental observations. Further, using a modified version of the model, it is also possible to show that transcapsular spread or invasion of the tumor may be due to the production by the tumor cells of proteases and their subsequent action.

03/26
Peter Olver
School of Mathematics
University of Minnesota

Geometric foundations of numerical algorithms and symmetry

In this talk, I will introduce a new geometric foundation for the numerical analysis of differential equations. Implementation of the general method of moving frames in this context leads to a general framework for constructing symmetry-preserving numerical approximations to differential invariants and invariant differential equations. Applications in computer vision and numerical methods for ordinary differential equations will be presented.

April

04/02
L. Gary Leal
Department of Chemical Engineering
University of California at Santa Barbara

Computational Studies of the Shear Flow Behavior of Models for Liquid Crystal Polymers

The use of liquid crystalline polymers (LCPs) as high strength materials has been limited almost exclusively to the formation of fibers such as Kevlar. This is because processing flows, other than fiber spinning, disrupt the long-range orientational order of these rather expensive materials. The mechanisms by which this occurs have remained elusive. In this study, we seek to explore this question in the context of theoretical models that have been proposed to describe LCPs. We consider the simplest of all possible flow geometries, namely the flow between parallel, flat boundaries, one of which is translating in its own plane. One reason for studying this flow is that there is a large body of experimental observations available for qualitative comparison. Although we are still at an early state of realizing our overall objectives, the work that I will discuss here seems to point in a positive direction for future studies.

04/09
Richard A. Tapia
Department of Computational and Applied Mathematics
Rice University

Inverse, Shifted Inverse, and Rayleigh Quotient Iteration as Newton's Method

The inverse, shifted inverse, and Rayleigh quotient iterations are well-known algorithms for computing an eigenvector of a symmetric matrix. In this talk we demonstrate that each one of these three algorithms can be viewed as a standard form of NewtonÂ’s method from the nonlinear programming literature, involving an norm projection. This provides an explanation for their good behavior despite the need to solve systems with nearly singular coefficient matrices. Our equivalence result also leads us naturally to a new proof that the convergence of the Rayleigh quotient iteration is q-cubic with rate constant at worst 1.

04/16
Alexander Figotin
Department of Mathematics
University of California, Irvine

Nonlinear Maxwell Equations in Inhomogeneous Media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity.

04/23
Carlos Castillo-Chavez
Department of Mathematics & Statistics
Arizona State University

Use of epidemiological models on problems associated with homeland security

Globalization and the possibility of bioterrorist acts have highlighted the pressing need for the development of theoretical and practical mathematical frameworks that may be useful in our systemic efforts to anticipate, prevent and respond to acts of destabilization. In this talk, I will highlight problems that can be addressed or formulated (at least initially) via epidemiological approaches. I will illustrate some of "new" issues with examples that include the pathogen releases in mass transportation systems, role of transient populations on small-pox outbreaks and control, time-response required to halt a foot and mouth disease outbreaks, etc.