courses title

November 14, 2007
14:00-15:00, JR243

Dr. Susana Serna
Department of Mathematics, UCLA

Title: High order accuracy in space for the numerical approximation of hyperbolic conservation laws and Hamilton-Jacobi equations

Abstract: High order accurate shock capturing schemes are designed to resolve complex shock dynamics with high accuracy in space and time avoiding spurious oscillations near discontinuities. We analyze the use of hyperbolas and parabolas as basic functions for the design of piecewise smooth reconstruction procedures and discuss the advantage of applying slope limiters to control the total variation growth. We propose the use of "power" limiters to improve the behavior of the piecewise hyperbolic method and the weighted essentially non-oscillatory (WENO) parabolic method. We present a set of numerical experiments showing the different approaches for Euler equations of gas dynamics. We extend the discussion to the approximation of the viscosity solution of Hamilton-Jacobi equations with applications to the level set reinitialization.

 

October 31, 2007
14:00-15:00, JR243

Dr. Martin B. Short
Department of Mathematics, UCLA

Title: Fluids, Form, and Function: The role of fluid dynamics in the evolution of stalactites, icicles, and aquatic microorganisms

Abstract: ABSTRACT: Engineers designing such diverse objects as bicycle helmets, automobile bodies, and ship hulls must carefully consider how their product will interact with the fluids around it. In fact, this unavoidable interaction often dictates the object's form and how it will eventually function. Of course, mother nature is not immune to this selective effect, and the shapes and actions of many natural, everyday objects are similarly determined by fluid dynamics. In this talk, I will explore this phenomenon through the offbeat examples of stalactites, icicles, and the family of Volvocalean green algae. We shall discover how, through a synthesis of thin-film fluid dynamics and calcium carbonate chemistry, stalactites are all drawn toward a universal shape; how, because of thermally buoyant boundary layers, icicles are drawn toward this same shape; and how, through the use of their flagella, species of Volvocalean algae were able to evolve to ever larger sizes.

 

October 10, 2007
14:00-15:00, JR243

Dr. Tom Chou
Department of Biomathematics and Departmentt of Mathematics, UCLA

Title: Simple Stochastic Problems in Biophysics

Abstract: In this talk, two different applications of stochastic
processes in biophysics will be presented. The first will be the
development, analysis, and simulation of a simple one-dimensional
model for track-induced molecular motion. In contrast to molecular
motors that invoke a molecular conformational change to power their
biased motion, examples of track-propelled motion arise in biophysics.
Our simple model describes track-propelled motion in terms of
asymmetric nucleation of hydrolysis waves by coupling motion of a load
particle to moving domain walls. We use asymptotic analysis in a
moving frame to compute the dependence of the motor velocity as a
function of local hydrolysis rates and find a maximum velocity at
intermediate nucleation asymmetries. The second part of this talk will
a model for receptor-mediated virus adsorption and infection of cells.
Viruses are typically classified as being endocytotic or fusogenic,
with viruses that infect mammalian cells employing both entry
mechanisms. We develop the mathematics that model the stochastic
entry process as a competition between fusion and endocytosis. We find
the parameter regimes that determine which mechanism dominates.


October 3, 2007
14:00-15:00, JR215

Dr. Vladislav Panferov
Department of Mathematics, CSUN

Title: On the Boltzmann equation in 1D in space

Abstract: Solutions of the Boltzmann equation can be defined globally in
time using the concept of weak (renormalized) solutions. The problem of
regularity of such solutions is to a large extent still open. I will
discuss the case of solutions with one-dimensional spatial dependence
(plane waves) for which new apriori estimates are available. Such
estimates allow us to obtain strong solutions globally in time and to
describe propagation of regularity properties (derivatives, moments) in
the case of cutoff collision kernels.

 

September 19, 2007
14:00-15:00, JR215

Dr. Jorge Balbas
Department of Mathematics, CSUN

Title: On the Steady State Solutions of Shallow Water Flows

Abstract: The Saint-Venant equations, a system of PDEs commonly employed
for modeling geophysical flows, enjoys the existence of (physically
relevant) steady state solutions. A good understanding of these flows and
creating the tools for solving them will allow us to further investigate
the applications of the model and to validate existing numerical schemes
for more complex flows. In this talk I will present a summary of the
conditions that lead this type of flows, the tools we developed to compute
them (exactly), and discuss some applications. This work was conducted in
collaboration with S. Karni (U. of Michigan) and A. Wetzel (U. of
Michigan).

 

September 12, 2007
14:00-15:00, JR215

Dr. Maria R. D'Orsogna
Department of Mathematics, CSUN

Title: PATTERNS, STABILITY AND COLLAPSE FOR TWO-DIMENSIONAL BIOLOGICAL
SWARMS

Abstract: One of the most fascinating biological phenomena is the
self-organization of individual members of a species moving in unison with
one another, forming elegant and coherent aggregation patterns. Schools of
fish, flocks of birds and swarms of insects arise in response to external
stimuli or by direct interaction, and are able to fulfill tasks much more
efficiently than single agents. How do these patterns arise? What are
their properties? How are individual characteristics linked to collective
behaviors? In this talk we model a swarm as a non-linear system of self
propelled agents that interact via pairwise attractive and repulsive
potentials. We are able to predict distinct aggregation morphologies,
such as flocks and vortices, and by utilizing statistical mechanics tools,
to relate the interaction potential to the collapsing or dispersing
behavior of aggregates as the number of constituents increases. We also
discuss passage to the continuum and possible applications of this work to
the development of artificial swarming teams.