Applied Mathematics Colloquia Series
Archive

Fall 2006

September

09/15
Wendy Zhang

Singularity formation on a liquid interface: breaking a drop, pulling a neck & making a splash

The surface of a liquid is easily deformed, stretched, and broken. Some examples are surface-tension driven break-up of a bubble, withdrawal and entrainment of stratified liquid layers and the splash of a liquid drop upon impact. Here we develop theoretical and simulation tools to analyze the dynamics and compare our results against experiments. Taken together, the results indicate that singularity formation can organize the evolution of the interface near a topological transition in distinctively different, and unexpected, ways. The asymptotic dynamics associated with the singularity needs not be universal, but can instead retain a memory of initial and boundary conditions.

09/22
Russel Caflisch

Complex singularities for imcompressible Euler equations

This talk will describe the possibility of singularity development for the Euler equations for incompressible inviscid flow. Our approach to this problem is to study the structure and dynamics of singularities in the complex plane. This includes a numerical study of complex traveling wave solutions to the Euler equations and an unfolding for singularities that have special form.

09/29
Hal Smith

The Dynamics of Bacterial Infection, Innate Immune Response, and Antibiotic Treatment

We develop a simple mathematical model of a bacterial colonization of host tissue which takes account of nutrient availability and innate immune response. The model features an infection-free state which is locally but not globally attracting implying that a super-threshold bacterial inoculum is required for successful colonization and tissue infection. A subset $B$ of the domain of attraction of the disease-free state is explicitly identified. The dynamics of antibiotic treatment of the infection is also considered. Successful treatment results if the antibiotic dosing regime drives the state of the system into B.

October

10/06
Roxana Smarandache

Algebraic Analysis of the Performance of Low-Density Parity-Check Convolutional Codes

Abstract: Message-passing iterative decoders for low-density parity-check (LDPC) block codes (linear binary codes in the nullspace of a sparse matrix or, equivalently, with an associated bipartite graph with very few edges) are known to be subject to decoding failures due to so-called pseudo-codewords. These are real-valued vectors that can be loosely described as error patterns that cause non-convergence in iterative decoding due to the fact that the algorithm works locally and can give priority to a vector that fulfills the equations of a graph cover rather than the graph itself. The function that measures the effect that decoding failures have on the performance of the code is given not by the minimum Hamming weight but by the minimum pseudo-weight. This is defined, for the case of an additive white Gaussian noise channel (AWGNC), as where P is the set of all pseudo-codewords of the code.

In this talk we address the pseudo-codeword problem from the convolutional-code perspective. In particular, we compare the performance of LDPC convolutional codes, (linear codes over the rational field A = F (D)), with that of their “wrapped" quasicyclic block versions (linear codes over some ring of the form F [Y ]/(Y r – 1)), and we show that the minimum pseudo-weight of an LDPC convolutional code is at least as large as the minimum pseudo-weight of an underlying quasi-cyclic code. This result, which parallels a well-known relationship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudo-codeword in the convolutional code induces a pseudo-codeword in the block code with pseudo-weight no larger than that of the convolutional code's pseudo-codeword. This difference in the weight spectra leads to improved performance at low-to-moderate signal-to-noise ratios for the convolutional code.

10/13
Dionisios Margetis

Evolution of crystal surfaces: From discrete schemes to continuum laws

In traditional settings such as fluids and classical elasticity the starting point (truth'') is often identified with continuum equations for macroscopic variables of interest (densities etc). But in many cases of mathematical modeling this perspective is changed: The truth is atomistic, or takes the form of discrete schemes, by which evolution laws must be determined and analyzed at the macroscale. In this talk I focus on the evolution of crystal surfaces as a prototypical case of two-scale modeling and analysis, with implications in the design of novel devices. The governing, discrete equations represent the motion of interacting line defects, steps'' with atomic height. In the appropriate limit a nonlinear PDE is derived for the surface height, and free-boundary problems are formulated for the surface motion. I show analytically how microscopic details of the crystal enter the requisite boundary conditions, and thus affect evolution at the macroscale.

10/20
Jerry Moloney

Nonlinear X’s and O’s in ultrashort laser pulse propagation

The newly emerging field of Extreme Nonlinear Optics is pushing pulsed laser technology to levels never imagined just a decade ago. Record peak intensities of 10 22 Watts cm 2 have been demonstrated in the laboratory (10 27 Watts cm 2is enough to break down the vacuum and create an electron/positron pair!). In parallel, ultrashort pulse durations have recently been pushed into the attosecond (10 -18 of a second!) regime. More modest femtosecond duration pulses (10 -18 of a second) with peak intensities reaching 10 14 Watts cm 2, are capable of breaking down the main molecular constituents of air (i.e N 2 and O 2) forming critically self-focused light strings that appear to propagate anomalously long distances in the atmosphere. These leave electron-ion plasma channels in their wake that can potentially act as lightning rods and the plasma channels themselves emit burst of THz radiation.

he focus of this talk will be on describing ultrashort pulse propagation models that will allow us to access current and future extreme intensity and ultrashort time interactions with materials. From these models we can seamlessly derive the Nonlinear Schrödinger equation (NLS) and the so-called Nonlinear Envelope (NEE) model of Brabec and Krausz. We will also see that the critical collapse singularity of the 2D NLS equation plays a central role in initiating many of the experimentally observed phenomena. A dramatic manifestation of critical collapse (self-focusing) is the generation of a white light supercontinuum spectrum that spans the entire ultraviolet through visible to far-infrared spectral region. We will see that the generation of this spectrum can be ascribed to a classical 3-wave interaction and its shape is dependent on the dispersion properties of the interacting medium. When a femtosecond duration pulse critically self-focuses in a water cell, the shape of the latter’s spectrum will depend on whether the pulse central wavelength is in the normal (X-Wave) or anomalous (O-Wave) dispersion region of water.

10/27
Constance Schober

The Formation of Rogue Waves in NLS Models: Modelling and Phase Singularities

Rogue waves in deep water are investigated in the framework of the nonlinear Schroedinger (NLS) equation and the modified Dysthe (MD) equation. Phase modulation of higher order homoclinic solutions of the NLS equation can produce coalescence of unstable spatial modes, creating waves of maximal amplitude. Numerical simulations of the MD equation indicate that a chaotic regime increases the likelihood of rogue wave formation, and that enhanced focusing occurs due to chaotic phase evolution. In this talk, we will discuss the models, higher order homoclinic solutions of the NLS equation, and show that the formation of extreme waves in random oceanic sea states characterized by JONSWAP power spectra is well predicted by the proximity to homoclinic solutions of the NLS equation. Finally, we will discuss recent work on the relation between wave amplification and phase singularities.

November

11/03
Stephen Morris

Order and disorder in columnar joints

Columnar joints are three-dimensional fracture networks that form in cooling basaltic lava flows. The network organizes the solid flow into ordered, mostly hexagonal columns. Famous examples include the Giant's Causeway in Northern Ireland, Fingal's cave in Scotland and The Devil's Postpile in California. The same pattern can be observed on a smaller scale in desiccating corn starch, and in some other materials. We have made the first three dimensional study of the evolution of the network in corn starch and relate these observations to the mature patterns observed in basalt. The starch patterns are statistically similar to those found in the Giant's Causeway, suggesting that mature columnar joint patterns contain inherent residual disorder. We find that the starch patterns can be made more similar to the basaltic ones using controlled drying rate conditions. Discontinuous transitions in pattern scale can be observed under constant external drying conditions, which may prompt a reinterpretation of similar transitions found in basalt. Experiments with starch suggest that the scale of the regular column pattern is not unique, but rather that a given scale can be stable over a range of different fracture speeds.

11/10
Peter Glynn

Numerical schemes for simulating SDEs under the total variation norm

Discretization error for simulation of SDE's has been widely studied when the error is measured in the "weak" or "strong" sense. The weak error describes the error associated with computing expectations of smooth functions of the state at time t, whereas the strong error is intended to measure the quality of the discretization for path functionals of the SDE. However, the strong error only guarantees rates of convergence for the error associated with path functionals that are suitably continuous. Furthermore, the strong error has the theoretically undesirable property that it is computed in terms of a very specific "coupling" between the solution to the SDE and the approximating process. To overcome these difficulties, we consider instead the total variation distance between the probability distribution of the solution to the SDE and that of the approximation (restricted to the filtration associated with the processes observed at the discretization epochs). In this norm, it turns out that the Euler method does not converge. We therefore propose a new approximation scheme that computes solutions that are accurate in the total variation distance. This represents joint work with Jose Antonio Perez and Jonathan Goodman.

11/17
Graeme Walter Milton

Cloaking and Opaque Perfect Lenses

We show how a slightly lossy superlens of thickness d cloaks collections of polarizable line dipoles or point dipoles or finite energy dipole sources that lie within a distance of d/2 of the lens. In the limit as the loss in the lens tends to zero, these become essentially invisible from the outside through the cancelling effects of localized resonances generated by the interaction with the superlens. The lossless perfect Veselago lens has attracted a lot of debate. It is shown that as time progresses the lens becomes increasingly opaque to any physical dipole source located within a distance d/2 from the lens and which has been turned on at time $t=0$. Here a physical source is defined as one which supplies a bounded amount of energy per unit time. In fact the lens cloaks the source so that it is not visible from behind the lens either. For sources which are turned on exponentially slowly there is an exact correspondence between the response of the perfect lens in the long time constant limit and the response of lossy lenses in the low loss limit. This is joint work with Nicolae Nicorovici and Ross McPhedran.

December

12/01
Jack Xin

Auditory Transforms and Applications

The human ear is a sound analyzer with uniform and fine frequency resolution at low frequencies, yet nonuniform and coarse frequency resolution towards higher frequencies. We construct discrete invertible transforms mimicing the ear by building auditory characteristics into the discrete Fourier transform, so called auditory transforms. Such transforms offer new tools for signal processing, with applications in sound analysis and enhancment.