Upcoming Events

Wednesday, September 16, 2009 at 3:00 PM in SAS 4201
Irina Kogan, NC State Mathematics
Conservation Laws with Prescribed Eigencurves
We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan-Kähler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class. This is a joint work with Kris Jenssen from Penn State University.

Wednesday, September 23, 2009 at 3:00 PM in SAS 4201
Mark Hoefer, NC State Mathematics
Two-Dimensional Supersonic, Superfluidic Flows
Dispersive shock waves (DSWs) are studied theoretically in the context
of two-dimensional (2D) supersonic flow of a superfluid. Employing
Whitham averaging theory for the repulsive Gross-Pitaevskii (GP)
equation, modeling the superfluidic Bose-Einstein condensate, suitable
jump and entropy conditions are obtained for an oblique DSW, a
fundamental building block for 2D flows with boundaries. In analogy
to oblique viscous shock waves (VSWs), these conditions yield analytic
relations between Mach number ($M$), velocity deflection angle
($theta$) and wave angle ($beta$). Unlike VSWs, the
$M$-$theta$-$beta$ phase diagram for DSWs displays four distinct
regions associated with phase transitions in supersonic flow over a
corner which are predicted and verified by numerical computations of
the GP equation. Quasi-stationary DSWs, shock detachment due to
transonic flow, spontaneous excitation of vortices and the onset of
turbulent dynamics associated with cavitation of the superfluid are
observed.

Wednesday, September 30, 2009 at 3:00 PM in SAS 4201
Kevin Lin, University of Arizona Mathematics Department
Spike-Time Reliability of Layered Neural Oscillator Networks
This talk concerns the reliability of large networks of coupled neural
oscillators driven by fluctuating stimuli. Reliability means that
upon repeated presentations of a given stimulus, the network gives
essentially the same response each time; whether a network is reliable
can impact its ability to encode information via the precise timing of
spikes. Focusing on certain layered network architectures commonly
used in neuroscience, I will explain -- via a combination of
qualitative theory and numerical simulations -- how factors like
network architecture affect reliability. I will also discuss the
effects of noise, and show that some types of noise affect reliability
more seriously than others.

This is joint work with Eric Shea-Brown and Lai-Sang Young.

Wednesday, October 14, 2009 at 3:00 PM in SAS 4201
Xiao-Biao Lin, NC State Mathematics
Spherically Symmetric Waves for a Liquid/vapor Phase Transition Model
Dynamic flow involving liquid/vapor phase transition is an important phenomenon
in many engineering processes. For example, gasoline in internal combustion engine must be injected into the cylinder, vaporized and well mixed with oxygen before ignited by sparks. Understanding the evaporation process and the relation among the parameters of the system may help improve the fuel efficiency of the engine. In this talk, we discuss fluid flow involving liquid/vapor phase transition in a cone shaped section, simulating the flow in fuel injection nozzles. Assuming that the flow is spherically symmetric, and the fluid has high specific heat, we look for standing wave solutions inside the nozzle.

The model consists of a system of viscous conservation laws coupled with a reaction-diffusion equation.
We look for two types of standing waves:
(1) Explosion
(2) Evaporation

If the diffusion coefficient, viscosity and typical reaction time are small, the system is singularly perturbed.
Transition from liquid mixture to vapor occurs in an internal layer inside the nozzle.
First, a matched formal asymptotic solution is constructed. Then we look for a real solution near the approximation.

Wednesday, October 21, 2009 at 3:00 PM in SAS 4201
Michael Shearer, NC State Mathematics
Shock waves in particle size segregation
Shock waves in continuum models of particle size segregation in avalanches are sharp interfaces separating
regions of different concentrations of large and small particles. Under the typical shearing present in avalanche
flow, shocks form from initially smooth conditions, and then break to form mixing zones that evolve and
propagate with the flow. The analysis and simulation of these unexpected patterns are based on the recent
model of Gray and Thornton, which combines a linear bulk shearing motion in the direction of the avalanche,
with a nonlinear segregation motion normal to the flow. Using techniques from the theory of hyperbolic
conservation laws, we are able to describe shock formation completely, and to characterize most of what
happens after a shock breaks.
In related work, we describe experiments and simulations designed to test the Gray-Thornton model in
a Couette cell. Here, a layer of small glass spheres lies initially over a layer of larger spheres of the same
density. Upon rotating the lower confining circular plate, the spheres mix and then resegregate. The Gray
Thornton model is modified to account for the non-uniform shear present in this configuration. The modified
model captures the main features of the evolution, but there are interesting additional effects that are not
well described by the model.
This is joint work with graduate students Lindsay May, Nick Giffen, and physicist Karen Daniels.

Wednesday, October 28, 2009 at 3:00 PM in SAS 4201
John Harlim, NC State Mathematics
Filtering Turbulent Sparsely Observed Geophysical Flows
Filtering sparsely turbulent signals from nature is a central problem of contemporary data assimilation. In this talk, I will discuss filtering sparsely observed turbulent signals in which we simulate observations by adding white noise to solutions of the quasi-geostrophic (QG) model with turbulent cascades from baroclinic instability. In particular, we consider two separate regimes with varying Rossby radius mimicking the ``atmosphere'' and the ``ocean''. In the ``atmospheric'' case, large scale turbulent fluctuations are dominated by barotropic zonal jets with non-Gaussian statistics while the ``oceanic'' case has large scale blocking regime transitions with barotropic zonal jets and large scale Rossby waves.
I will discuss how to use tools from applied mathematics to develop cheap reduced stochastic filters, including the radical linear stochastic filters with model errors and the very recently developed exactly solvable Stochastic Parameterization Extended Kalman Filter (SPEKF) with additive and multiplicative bias corrections ``on the fly''. We will compare these cheap reduced filtering strategies with the state-of-the-art Local-Least-Squares Ensemble Adjustment Kalman Filter (LLS-EAKF) and we shall see that the cheap filters supersede the sophisticated LLS-EAKF in the numerically stiff ``oceanic" regime.

Wednesday, November 4, 2009 at 3:00 PM in SAS 4201
Rich McLaughlin, UNC Applied Math
Darwin's theorem, Taylor Diffusion, and falling spheres in stratified fluids
The motion of bodies falling through stratified fluids arises naturally in the context of carbon (marine snow) settling in the ocean, and the details of the settling rates may play a role in assessing the role of the ocean in the earth's carbon cycle. In this lecture, we look at phenomena associated with falling spheres in stratified fluids, and in particular focus upon the critical parameters setting when transient sphere levitation is possible. Some of the mechanisms responsible for this behavior are related to fluid entrainment (Darwin drift), and anomalous mixing of the entrained fluid (Taylor diffusion). We review the results of Darwin and Taylor, and present new results regarding anomalous short time mixing, and asymptotic corrections to Darwin's drift volume. New arrestment criteria derived from the energetics of the fluid+body system will be presented and compared to experimental observations.

this is joint work with Roberto Camassa, Joyce Lin, Zhi Lin, Keith Mertens, Matthew Moore, Ashwin Vaidya, and Claudio Viotti

Thursday, November 19, 2009 at 2:30 PM in SAS 4201
Matthew Nicol, University of Houston Mathematics Department
Statistical properties of dynamical systems
We discuss recent progress in understanding certain key statistical properties of nonuniformly hyperbolic dynamical systems, in particular large deviations, extreme value theory and approximation by Brownian motion.

(This Differential Equations seminar is joint with the Probability and Random Systems seminar)

Wednesday, December 2, 2009 at 3:00 PM in SAS 4201
James Nolan, Duke Mathematics
TBA
TBA

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Seminar Organizer: Mark Hoefer