Applied Mathematics Colloquia Series
Archive

Fall 2005

September

09/23
John H. Maddocks
Chair, Applied Analysis
Swiss Federal Institute of Technology

Multi-Scale Models of Sequence-Dependent DNA Mechanics

I will survey some continuum mechanics approaches to modeling the sequence-dependent physical properties of DNA fragments at the biologically pertinent length scales of a few tens to a few hundreds of base pairs.

09/30
W. B. Hubbard
Professor, Lunar & Planetary Laboratory
University of Arizona

Diagnostics of Deep Structure in Giant Planets

High-resolution, high-precision maps of surface shape and gravity areavailable for most of the terrestrial planets, but such data are stilllacking for the giant planets because of the difficulty of executinglow-periapse orbiter missions. With the approval of the Juno mission to Jupiter, a highly capable geodetic mission to a giant planet is now inprospect, and a Juno-like mission could eventually fly to Neptune. With the capability of measuring near-surface gravity anomalies well below a milligal, it becomes possible to probe the deep dynamics of giant planets. Jupiter's static (zonal) gravitational spectrum will depend on the style of interior convection. Measurements of dynamic (tidal) response and pole precession may provide independent information about the deep interior. Measurements of windspeeds and geodetic data in Saturn and Neptune already provide us with constraints on interior dynamics in those planets: a reported dramatic shift in Saturn's equatorial windspeed should be reflected in its measurable atmospheric shape, and, if deep-seated, its near-surface gravity. Available data and models show that Neptune's pronounced retrograde equatorial wind does not entrain a substantial fraction of the planet's mass.

October

10/07
Annalisa Calini
Associate Professor, Department of Mathematics
College of Charleston

Finite-Gap Solutions of the Vortex Filament Equation

For the class of quasi-periodic solutions of the vortex filament equation we study connections between the algebro-geometric data used for their explicit construction, and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such as Euler elastica, constant torsion curves, and self-intersecting filaments. We also prove generalizations of these connections to higher genus. This is joint work with Tom Ivey.

10/14
Karl P. Hadeler
Chair, Biomathematics
University of Tuebingen

Slow deposition of granular matter

Granular matter, being soft matter (sand, foam, sludge etc.) as opposed to the classical states of solid, fluid, and gas, poses a variety of unconvential mathematical problems.

We consider slow deposition of granular matter under the influence of gravity, in the absence of wind forces, neglecting inertia. Following the approach of de Gennes and the BCRE model and guided by experiment, we design a model in the form of two hyperbolic partial differential equations for the bulk of material at rest, respecting the angle of repose, and a moving surface layer.

We then consider the problem of appropriate boundary conditions depending on the geometry, e.g., deposition of granular matter in silos, on flat surfaces, on flat surfaces with elevated boundaries, on arbitrary obstacles.

The correponding stationary problems lead to a variety of boundary values problems and obstacle problems for the eikonal equation. Physically meaningful solutions can be characterized in various ways, e.g., by viscosity approaches. Here variational principles are invoked, as well as Perron approaches and finally and most effectively methods based on transport paths.

We underline the striking correspondence between the Dirichlet problems for the Laplace/Poisson equation and the eikonal equation.

An outlook can be given to other time-dependent processes such as superposition of sand heaps and sand ripple formation.

10/21
Kenneth M. Golden
Professor, Mathematics
University of Utah

Fluid transport in sea ice

The polar sea ice packs play a key role in earth's ocean-climate system, and are sensitive indicators of climatic change. As a material, sea ice is a porous composite of pure ice with brine and air inclusions, whose microstructural properties depend strongly on temperature. The transport of brine, which carries salt, heat, and nutrients through sea ice, controls a broad range of geophysical, oceanographic, and biological processes. However, measurements of the fluid permeability of sea ice are sparse and little is known theoretically. We give mathematical formulations of the two key problems of fluid transport in sea ice: bulk flow of brine, and diffusion of dissolved substances such as pollutants or bacterial enzymes. We present a comprehensive theory for the fluid permeability of sea ice, based on rigorous bounds, continuum percolation theory, hierarchical models, network simulation, and microstructural imaging. Our theoretical results closely capture laboratory and Arctic field data. The role of fluid advection in thermal transport through sea ice is also discussed. This work is joint with Hajo Eicken, Geophysical Institute, University of Alaska Fairbanks, Jingyi Zhu, Department of Mathematics, University of Utah, and four University of Utah undergraduate students in the Mathematics REU program.

10/28
Stuart S. Antman
Dept. of Mathematics & Institute for Physical Science & Technology
University of Maryland

Analytic Consequences of Incompressibility

A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. (Rubber and much living tissue (which is composed mostly of water) are examples of incompressible materials.) Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies (like worms, snakes, and eels) governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they admit some tricky a priori bounds and they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton's Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton's Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs, which have a nonstandard structure, admit, under favorable assumptions, periodic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.

November

11/04
J. D. Humphrey
Deptartment of Biomedical Engineering
Texas A&M

Stability and Growth of Intracranial Aneurysms

Intracranial aneurysms occur in 2-5% of the population; they are thin balloon-like dilatations of a weakened portion of the wall. A goal of this work is to use continuum mechanics and mathematical modeling to understand better the reasons why these aneurysms enlarge and rupture. Unfortunately, such rupture often occur without warning, and it results in high mortality and morbidity. We will show that recent studies of model lesions suggest that longstanding suspicions that these lesions are inherently unstable are probably not correct, rather enlargement and rupture likely depend on a continual turnover of structural proteins within the wall. Hence, we will also present a new model of growth and remodeling of these lesions, one based on the hypothesis that new proteins are deposited and organized within the wall in response to alterations in the local stress field.

11/18
John Lowengrub
Professor & Chair, Department of Mathematics
UC Irvine

Nonlinear simulation of angiogenesis and solid tumor growth

In this talk, I will focus on recent efforts to study solid tumor progression. Here we focus on a continuum-scale description and pose the problem in terms of conservation laws for nutrients, chemical factors and tumor cell populations. We focus first on single-phase models. We analyze the equations and develop accurate, adaptive numerical schemes. We will present simulations of the complex nonlinear coupling between the progression of the tumor and neovascularization. We demonstrate the predictive capability of the model through comparisons with in vitro and in vivo experimental studies of tumor growth. We then discuss extensions to multiphase and mixture models and discuss the effects of residual stress and cell-to-cell adhesion.

December

12/02
Andrea L. Bertozzi
Professor, Mathematics
UCLA

The analysis and design of swarms

The cohesive movement of a biological population is a commonly observed natural phenomenon. With the advent of platforms of unmanned vehicles, this occurrence is attracting renewed interest from the engineering community. This talk will review recent research results on both modeling and analysis of biological swarms and also design ideas for efficient algorithms to control groups of autonomous agents. For biological models we consider two kinds of systems: driven particle systems based on force laws and continuum models based on kinematic rules. Both models involve long-range social attraction and short range dispersal and yield patterns involving clumping, mill vortices, and surface-tension-like effects. For artificial platforms we consider the problem of boundary tracking of an environmental material and consider both computer models and demonstrations on real platforms of robotic vehicles. We also consider the motion of vehicles using artificial potentials.