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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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