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Past talks for Spring 2011

**January 14, 2011 **

Speaker: Liping Liu, Department of Mechanical Engineering
University of Houston

**Title:
**A differential approach to energy bounds of multiphase composites

**Abstract: ** In this paper we present a new method of deriving microstructure-dependent
bounds on the effective properties of general heterogeneous media. We first
define and calculate the higher-order polarization tensors for multiphase
heterogeneous media, and next derive a differential inequality on the energy
with the initial condition given by the polarization tensors. Using the
comparison theorem we obtain bounds on the energy induced by the
inhomogeneities. These new bounds, taking into account of the average
Eshelby tensors for homogeneous problems, are much tighter than the
microstructure-independent bounds such as the classic Hashin-Shtrikman
bounds on one hand, and on the other hand, recover the classic bounds by
minimizing or maximizing the bounds over all possible average Eshelby
tensors. Also, these bounds are applicable to non-well-ordered composites
and multifunctional composites. It is anticipated that this new approach
will be useful for the modeling and optimal design of a variety of
heterogeneous media.

**January 21, 2011 **

Speaker: Grady B. Wright
, Department of Mathematics
Boise State University

**Title:
**Reconstruction and Decomposition of Vector Fields on the Sphere
with Applications

**Abstract: ** Vector fields tangent to the surface of the sphere appear in many
applications from the geophysical sciences. We present a new
numerical technique based on radial basis functions (RBFs) for
reconstructing tangent vector fields from samples of the field at
“scattered” locations on the surface of the sphere. The method is
entirely free of any coordinate singularities, naturally provides a
way to decompose the reconstructed field into its individual
Helmholtz/Hodge components (i.e. its divergence- and curl-free parts),
and can be used to approximate surface derivatives of the field such
as the surface divergence and curl. Applications for fitting and
decomposing vector fields are discussed in addition to a new method
for making the the computations efficient. If time permits, we also
discuss the generalization of the reconstruction method to more
general manifolds.

**January 24, 2011**

Speaker: Hyeonbae Kang, Department of Mathematics
Inha University, S. Korea

**Title:
**Generalized Polarization Tensors for Imaging

**Abstract: ** I will show that Generalized Polarization
tensors can be used to recover fine details of shape. I will
also briefly discuss about their connection to cloaking.

**Febryary 4, 2011**

Speaker: Tamar Shinar, Courant Institute

**Title:
**Numerical studies of microtubule-based motion in the single-celled C. elegans embryo

**Abstract: ** We develop a simple model of microtubule-based pronuclear motion in a
single-celled C. elegans embryo. The model consists of a model for
microtubule dynamic instability, a Newtonian, viscous fluid contained within
an enclosing geometry for the cytoplasm, a rigid body for the pronucleus,
and a motor protein load-velocity relationship. Motor proteins distributed
throughout the cytoplasm interact with microtubule filaments by sliding
along them with a velocity that depends on their load. They in turn pull on
the filaments, resulting in translation of the microtubule-bound
pronucleus. Our simulations show pronuclear migration, and moreover, a
robust pronuclear centration and rotation very similar to that observed in
vivo. I will also briefly describe the numerical method for the coupled
simulation of the Stokes fluid and rigid structures.

**February 7, 2011**

Speaker: Niklas Wellander
, (Swedish Defence Research Agency and Lund University)

**Title:
**Cloaking by change of variables - The fixed frequency case

**Abstract: ** We present the fundamentals for electromagnetic cloaking by
means of change of variables. The method relies on the non-uniqueness of
the inverse scattering problem. The scattering of electromagnetic energy
is put into a variational form. The domain containing the cloak and the
cloaked object is initially filled with the surrounding material (in
general vacuum for the most interesting applications). The cloak is the
effect of a singular transformation, which when interpreted as an active
transform defines the properties of the cloak explicitly. Greenleaf,
Lassas and Uhlmann (2003) used a coordinate transform to define a
surrounding heterogeneous medium for a cloak in the electrical impedance
tomography problem. Kohn, Shen, Vogelius and Weinstein (2008) used a
nonsingular transform to produce a near-cloak in a variational setting
of electrical impedance tomography.
We extend the previous results and show some new estimates.
This work is in colaboration with Gerhard Kristensson.

**February 11, 2011**

Speaker: Inbo Sim, Ulsan University, Korea

**Title:
**Discrete spectrum for a p-Laplace eigenvalue problems with singular
sign-changing weights

**Abstract: ** This talk is based on joint work with R. Kajikiya and Y. Lee.
We study the existence of spectrum for one-dimensional p-Laplace eigenvalue
problems with singular weights subject to Dirichlet boundary condition.
For a certain class
of singular sign-changing weights, we prove the existence of a discrete
spectrum. Proofs are
based on the $C^1[0,1]$-regularity of solutions and construction of the
first eigenvalue in
a variational set-up. We also present an example of a weight for which the
spectrum is
continuous and any corresponding eigenfunction does not belong to $C^1[0,1].$

**February 18, 2011**

Speaker: Martin Short, UCLA, Department of Mathematics

**Title:
**Suppressing and solving crimes with math

**Abstract: ** Law enforcement officials are always looking for ways to
decrease crime further, and to solve those crimes that do occur
Increasingly, the police are turning to ever more technical approaches
to accomplish these goals. In this talk, we will cover a number of
mathematical models and techniques that speak to these issues. Using
evolutionary game theory, we will show that criminal informants are
essential to creating a low-crime society, and provide estimates for
the optimal number of such individuals. We will address the problem
of "geographic profiling" - estimating the location of a criminal's
residence given the locations of his criminal acts - using an inverse
Fokker-Planck model. And (time permitting) we will discuss how one
might determine which gangs were involved in a particular violent
incident by solving a constrained maximization problem.

**February 28, 2011**

Speaker: Songming Hou
, Louisiana Tech University

**Title:
**A numerical method for solving the elliptic and elasticity interface problems

**Abstract: ** Solving elliptic and elasticity equations with interface jump
conditionsare challenging problems for most existing methods, especially with
non-smooth interfaces. Nonetheless, they have wide applications in engineering
and science. I will present a non-traditional finite element method for solving
matrix coefficient elliptic and elasticity equations with sharp-edged
interfaces in two dimensions. Some recent advances on the three dimensional
elliptic interface problems and the three-domain elliptic interface problems
will also be discussed.

**March 7, 2011**

Speaker: Igor Tsukerman
, Department of Electrical and Computer Engineering, The University of Akron

**Title:
**From Analytical to Numerical Methods and Back:
Trefftz Schemes, Whitney Forms, and Metamaterials

**Abstract: ** Several examples of a curious interplay between analytical and numerical methods
are presented. First, local Trefftz approximations lead to high-order difference
schemes and to singularity-free boundary-difference methods, with applications
in electromagnetics, photonics. and other areas. Second, Whitney-like
interpolation, well established in finite element analysis, is put to a new use
in homogenization of metamaterials. All coarse-grained fields are unambiguously
defined and satisfy Maxwell’s equations exactly; effective parameters are then
derived without any heuristic assumptions. This approach should be applicable
in areas beyond metamaterials and electromagnetic waves.

**April 4, 2011**

Speaker: Alexander Kurganov
, Department of Mathematics, Tulane University

**Title:
**Central Schemes: a Powerful Black-Box-Solver for Nonlinear Hyperbolic PDEs

**Abstract: ** I will first give a brief description of finite-volume, Godunov-type
methods for hyperbolic systems of conservation laws. These methods consist of two types of
schemes: upwind and central. My lecture will focus on the second type -- non-oscillatory
central schemes. Godunov-type schemes are projection-evolution methods. In these methods,
the solution, at each time step, is interpolated by a (discontinuous) piecewise
polynomial interpolant, which is then evolved to the next time level using the integral form of
conservation laws. Therefore, in order to design an upwind scheme, (generalized) Riemann problems have
to be (approximately) solved at each cell interface. This however may be hard
or even impossible. The main idea in the derivation of central schemes is to avoid solving
Riemann problems by averaging over the wave fans generated at cell interfaces. This strategy
leads to a family of universal numerical methods that can be applied as a black-box-solver to
a wide variety of hyperbolic PDEs and related problems. At the same time, central schemes
suffer from (relatively) high numerical viscosity, which can be reduced by
incorporating of some upwinding information into the scheme derivation -- this leads to
central-upwind schemes, which will be presented in the lecture.
During the talk, I will show a number of recent applications of the central schemes.

**April 11, 2011**

Speaker: Alexander Balk
, Department of Mathematics, University of Utah

**Title:
**Extra Invariant and Zonal Jets

**Abstract: ** I will paint a picture that has emerged due to several research papers
in the last 20 years, including the one that will be published in May.
If you look at Jupiter, you see several stripes parallel to its equator;
these are fluid jets flowing alternatively east and west. I will show
that these jets are related to the extra conservation.
Geophysical fluids (like Jupiter's atmosphere) are mathematically
similar to fusion plasmas with magnetic confinement. Here, zonal jets
also appear and play an important part: They serve as transport
barriers, limiting the flux of heat and particles from the center of the
reactor to its walls.

**April 18, 2011**

Speaker: Marc Mueller-Stoffels
, Physics Department University of Alaska Fairbanks

**Title:
**Arctic Ocean stable state dynamics in a thermodynamic coupled cell model

**Abstract: ** The Arctic ocean is mostly covered with sea ice. However, recently
this sea ice cover underwent significant changes in summer ice extent
and a transition from thick multi-year to thinner first year sea ice.
In popular and scientific literature the possibility of tipping
point-like behavior is discussed, i.e. at its extreme a rapid loss of
the remaining sea ice cover and transition to an ice-free Arctic
ocean. I will not add to the lively discussion about what kind of sea
ice cover we will have in 2100. But will introduce several
thermodynamic modeling approaches that do or do not yield a relatively
sudden transition from a perennial ice cover to a seasonal ice cover
or perennially open ocean. Based on a coupled cell model (slightly
modified from Mueller-Stoffels and Wackerbauer, Chaos, 2011) I will
demonstrate that the number and location of stable fixed points for
the system varies over the course of a year. Furthermore, I will show
some results on the introduction of surface heterogeneities, i.e.
spatially varying ice albedo.

**May 9, 2011**

Speaker: Robert Lipton
, Department of Mathematics, Louisiana State University

**Title:
**TBA

**Abstract: ** TBA

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