Applied Math Seminar

Abstracts

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Gerry Kaiser

Electromagnetic wavelets were introduced in 1994 as a family of "small" EM waves related by conformal spacetime operations (generalizing translation and scaling for 1D wavelets) that can be used to analyze and synthesize solutions of the homogeneous Maxwell equations. They are closely related to the complex-source pulsed beams (CSPB) used extensively in engineering, whose construction usually appears as an ad-hoc "analytic extension" of the wave propagator associated with a formal displacement of a point source to complex spacetime. Perhaps for this reason, CSPB have been regarded as a mathematical "trick" useful mainly to model the behavior of sharp-beam solutions of the wave equation. Little attention has been devoted to investigating the possibility of their physical realization, perhaps because their singular and convoluted behavior in the near zone appeared to make the requisite analysis very difficult. Recently I have e xtended earlier work on "complex-distance potential theory" fully to the hyperbolic regime and computed the source distributions responsible for EM wavelets and their 4D Fourier transforms. Surprisingly, all the complications drop away in the Fourier domain. Difficult integrals lead to "miraculous" cancellations and the sources take on a simple and clear form. This suggests that practical computations can be performed with EM wavelets using FFT-based algorithms. It also opens the door to attempting the experimental realization of antenna sources to emit and detect the EM wavelets. The accomplishment of both objectives would make it possible to implement a generalized radar concept based on EM wavelets proposed earlier.

Reference:
Physical wavelets and their sources: Real physics in complex space-time. Topical Review, J. Physics A 36 #30, R291--R338, 2003.

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Aaron Masino

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Phil Broadbridge

When we mention the "Navier-Stokes equations", most people immediately think of the momentum transport equations for an incompressible Newtonian fluid with constant viscosity. When we allow the fluid to be compressible, with temperature-dependent viscosity, the momentum equations are coupled to the heat and mass transport equations. This system of equations is quite daunting, but the steady axisymmetric flows are surprisingly tractable.

We use Lie symmetry analysis to determine those temperature-dependent viscosity and thermal conductivity functions that admit explicit solution of the steady axisymmetric Navier-Stokes system. In this system, the only heat source is that due to viscous dissipation. Exact solutions are constructed for steady Couette flow or vortex flow of a compressible Boltzmann hard-sphere gas with viscosity and conductivity increasing with temperature, as predicted by kinetic theory. The temperature of a vortex is predicted to be positive only in the exterior of a thin cylindrical core whose small radius is determined uniquely in terms of atomic parameters. A core of exclusion, of the same magnitude as in the Boltzmann gas, exists also in the vortex of an ideal compressible gas with temperature-independent properties. The two-dimensional steady flow solution cannot exist inside the core , wherein there must be more general transport mechanisms such as upwelling, or more general non-frictional heat sources such as latent heat release. A similar result is obtained for an incompressible liquid, with viscosity decreasing with temperature.

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Rich Braun

A brief summary of the 2003 Mathematial Problems in Industry workshop, held at WPI in Worcester, MA, will be given. Two problems will be discussed in a little more detail. One is about strategy for optimal routing of elevators, and the other is about geometrical design of a "laying head", which coils wire entering at about 100 m/s.

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Bruce Lipphardt

HF radars measure synoptic surface currents near the coast at time intervals like one hour with spatial resolution of a few kilometers. Although increasing numbers of HF radars make synoptic surface current maps routinely available, these maps are not useful for environmental or ecological studies, which require quantitative descriptions of near surface transport to study the impacts of upwelling, the ecology of nearshore habitats, or the effects of harmful algal blooms or oil spills. Particle studies provide a natural way to reveal synoptic surface transport properties embedded in a synoptic Eulerian velocity archive.

A six month archive of hourly HF radar maps of surface currents in Monterey Bay is used to compute hourly synoptic Lagrangian maps (SLMs) which describe the residence time and escape fate for a regular grid of simulated particles at the bay's surface. The flow is assumed to be two-dimensional, and space and time gaps in the measurements are filled using an objective mapping method. Results thus far show that there is no apparent "climatology" for surface transport: large spatial scale variations are seen over daily, weekly and monthly scales. The fraction of particles that leave the bay also varies significantly over time scales of days to months.

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Richard Kleeman

An important question for practical prediction of both the weather and climate is how it may vary from one set of initial conditions to another. Some predictions are much more reliable than others. In this talk we introduce a new theoretical framework from information theory for addressing this practical concern. This involves entropic functionals on the space of probability density functions. There are interesting links between this work and non-equilibrium statistical mechanics which are explained. The framework is then applied to practical prediction problems and results are shown from both climate (El Nino) and weather prediction models. Conventional wisdom asserts that variations in prediction ensemble spread (or probability distribution dispersion) controls utility variations for weather prediction. Contrary to this we find that it is the "signal" amplitude in the initial conditions that controls this and variations in the "noise" or "spread" are often not important.

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Robert Wilson

In 1992 the World Bank released a ground-breaking study: The Global Burden of Disease. The project introduced a measure that combines life expectancy and time lived with disabilities, (the disability-adjusted life-year, DALY). This presentation will chronicle the development of the DALY, some controversial applications, and an application of the methodology to measure the heath of small geographic areas in the State of Delaware.

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Luis Melara

Developing numerical methods for predicting microstructure in materials is an extremely large and important research area. Two examples of material microstructures are Austenite and Martensite. Austenite is a microscopic phase with simple crystallographic structure while Martensite is one with a more complex structure. One important task in materials science is the development of numerical procedures which accurately predict microstructures in Martensite near an Austenite-twinned-Martensite interface. A family of techniques is presented in this talk. The method involves the solution of an unconstrained and constrained optimization problem using a quasi-Newton optimization algorithm combined with two different finite element approximations of a total energy functional. Preliminary results suggest that the constrained minimizers of this energy functional that are located by the presented numerical algorithm display the desired characteristics.

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Harris Wong

A thin liquid film experiences additional intermolecular forces when the film thickness h is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by disjoining pressure P. Since P dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, P = P(h) because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope hx leads to a P = P(h, hx) that allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. We find P = P(h, hx, hxx). The curvature term hxx prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. We also simulate numerically the evolution of a film step with the new disjoining pressure included.

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John McCuan

There is a well developed theory concerning the regularity of a capillary meniscus in the neighborhood of a corner point, i.e., a point in the wall of a cylindrical tube with a corner; a square tube has four such points. Special interest derives from the fact that regularity (even at the level of L^infty estimates) depends discontinuously on the wetting properties of the liquid. Nevertheless, there are some fundamental open questions concerning such configurations. I will describe the known theory, some open problems, and some recently constructed examples which shed light on possible behavior. The examples were constructed in joint work with R. Huff.