Mathematical Systems Biology - Spatial Dynamics and Growth and Signaling

We will describe some recent applications of resolution of singularities methods to questions of interest in analysis. In particular, we will describe a recent local resolution of singularities algorithm of the speaker for real-analytic functions. This algorithm is elementary and self-contained, and makes extensive use of Newton polyhedra and local coordinate systems. Some applications will be given. These include applications to oscillatory integrals, asymptotic expansions for sublevel set volumes, and the determination of the supremum of the positive e for which |f|^{- e} is locally integrable. Here f denotes a real-analytic function.

This talk investigates the upscaling methods to the following
parabolic equation
$$
\partial_t c+\nabla\cdot(\mathbf{u}
c)-\nabla\cdot(\mathbf{D}\nabla c)=f(x,t)
$$
which stems from the application of solute transport in porous
media. Because of the high oscillating permeability of the porous
media, the Darcy velocity $\mathbf{u}$ hence the dispersion tensor
$\mathbf{D}$ has many scales with high contrast. Thus, how to
calculate the macro-scale equivalent coefficients of the above
equation becomes the target of this talk. Two kinds of upscaling
formulations are discussed in this work. The two different
equivalent coefficients computing formulations are based on the
solutions of two different cell (local) problems, which one utilizes
the elliptic operator with terms of all orders while the other only
uses the second order term. Error estimates between the equivalent
coefficients and the homogenized coefficients are given under the
assumption that the oscillating coefficients are periodic (which is
not required by our methods). Numerical experiments are carried out
for the periodic coefficients to demonstrate the accuracy of the
proposed methods. Moreover, we apply the two upscaling methods to
solve the solute transport in a porous medium with a random
log-normal relative permeability. The results show the efficiency
and accuracy of the proposed methods.

We study particle systems corresponding to highly connected queuing
networks, like Internet. We examine the validity of the so called Poisson
Hypothesis (PH), which predicts that such particle system, if started
from a reasonable initial state, relaxes to its equilibrium in time
independent of the size of the network. We show that this is indeed the
case in many situations.

However, there are networks for which the relaxation process slows down.
This behavior reflects the fact that the corresponding infinite system
undergoes a phase transition. Such transition can happen only when the
load per server exceeds some critical value, while in the low load
situation the PH behavior holds. Thus, the load plays here the same role
as the inverse temperature in statistical mechanics.

The control of the density and location of dislocations (line defects)
in heteroepitaxial thin film is very important in designing
semiconductor-based electronic devices. We have developed a level set
method based, three dimensional dislocation dynamics simulation method
to describe the motion of dislocations in thin films. The dislocation
location is given by the intersection of the zero level sets of a pair
of level set functions. This representation does not require
discretization and tracking of the dislocation, and handles topological
changes automatically. The simulation method incorporates the elastic
interactions of the dislocations and the stress fields throughout the
film and substrate. Using the above approach, various dislocation
motion and interactions within a heteroepitaxial thin film are simulated
and analyzed.