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Monday, November 17, 2008 - 13:00 ,
Location: Skiles 255 ,
Maoan Han ,
Shanghai Normal University ,
Organizer: Haomin Zhou

Let H(m) denote the maximal number of limit cycles of polynomial systems of degree m. It is called the Hilbert number. The main part of Hilbert's 16th problem posed in 1902 is to find its value. The problem is still open even for m=2. However, there have been many interesting results on the lower bound of it for m\geq 2. In this paper, we give some new lower bounds of this number. The results obtained in this paper improve all existing results for all m\geq 7 based on some known results for m=3,4,5,6. In particular, we confirm the conjecture H(2k+1) \geq (2k+1)^2-1 and obtain that H(m) grows at least as rapidly as \frac{1}{2\ln2}(m+2)^2\ln(m+2) for all large m.

Monday, November 10, 2008 - 13:00 ,
Location: Skiles 255 ,
Guowei Wei ,
Michigan State University ,
Organizer: Haomin Zhou

Solvation process is of fundamental importance to other complex biological processes, such signal transduction, gene regulation, etc. Solvation models can be roughly divided into two classes: explicit solvent models that treat the solvent in molecular or atomic detail while implicit solvent models take a multiscale approach that generally replaces the explicit solvent with a dielectric continuum. Because of their fewer degrees of freedom, implicit solvent methods have become popular for many applications in molecular simulation with applications in the calculations of biomolecular titration states, folding energies, binding affinities, mutational effects, surface properties, and many other problems in chemical and biomedical research. In this talk, we introduce a geometric flow based multiscale solvation model that marries a microscopic discrete description of biomolecules with a macroscopic continuum treatment of the solvent. The free energy functional is minimized by coupled geometric and potential flows. The geometric flow is driven not only by intrinsic forces, such as mean curvatures, but also by extrinsic potential forces, such as those from electrostatic potentials. The potential flow is driven mainly by a Poisson-Boltzmann like operator. Efficient computational methods, namely the matched interface and boundary (MIB) method, is developed for to solve the Poisson- Boltzmann equation with discontinuous interface. A Dirichlet- to-Neumann mapping (DTN) approach is developed to regularize singular charges from biomolecules.

Monday, October 27, 2008 - 13:00 ,
Location: Skiles 255 ,
George Biros ,
CSE, Georgia Tech ,
Organizer: Haomin Zhou

Fluid membranes are area-preserving interfaces that resist bending. They are models of cell membranes, intracellular organelles, and viral particles. We are interested in developing simulation tools for dilute suspensions of deformable vesicles. These tools should be computationally efficient, that is, they should scale well as the number of vesicles increases. For very low Reynolds numbers, as it is often the case in mesoscopic length scales, the Stokes approximation can be used for the background fluid. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the nonlocal hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present two semi-implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme. We discretize the equations by using a spectral method in space, and a multistep third-order accurate scheme in time. We use the fast multipole method to efficiently compute vesicle-vesicle interaction forces in a suspension with a large number of vesicles. We report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme. Joint work with: Shravan K. Veerapaneni, Denis Gueyffier, and Denis Zorin.

Wednesday, October 22, 2008 - 10:00 ,
Location: Skiles 269 ,
Arthur Szlam ,
UCLA ,
Organizer: Haomin Zhou

SPECIAL TIME AND LOCATION FOR THIS WEEK ONLY

The k-planes method is the generalization of k-means where the representatives of each cluster are affine linear sets. In this talk I will describe some possible modifications of this method for discriminative learning problems.

Monday, October 6, 2008 - 13:00 ,
Location: Skiles 255 ,
Shengfu Deng ,
School of Mathematics, Georgia Tech ,
Organizer: Haomin Zhou

We consider the three-dimensional gravity-capillary waves on water of finite-depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamical system in stead of using Hamiltonian formulation method. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants.

Monday, September 29, 2008 - 13:00 ,
Location: Skiles 255 ,
Silas Alben ,
School of Mathematics, Georgia Tech ,
Organizer: Haomin Zhou

We discuss two problems. First: When a piece of paper is crumpled, sharp folds and creases form. These are distributed over the sheet in a complex yet fascinating pattern. We study experimentally a two-dimensional version of this problem using thin strips of paper confined within rings of shrinking radius. We find a distribution of curvatures which can be fit by a power law. We provide a physical argument for the power law using simple elasticity and geometry. The second problem considers confinement of charged polymers to the surface of a sphere. This is a generalization of the classical Thompson model of the atom and has applications in the confinement of RNA and DNA in viral shells. Using computational results and asymptotics we describe the sequence of configurations of a simple class of charged polymers.

Monday, September 22, 2008 - 13:00 ,
Location: Skiles 255 ,
Dongbin Xiu ,
Division of Applied Math, Purdue University ,
Organizer: Haomin Zhou

There has been growing interest in developing numerical methods for stochastic computations. This is motivated by the need to conduct uncertainty quantification in simulations, where uncertainty is ubiquitous and exists in parameter values, initial and boundary conditions, geometry, etc. In order to obtain simulation results with high fidelity, it is imperative to conduct stochastic computations to incorporate uncertainty from the beginning of the simulations. In this talk we review and discuss a class of fast numerical algorithms based on generalized polynomial chaos (gPC) expansion.The methods are highly efficient, compared to other traditional In addition to the forward stochastic problem solvers, we also discuss gPC-based methods for addressing "modeling uncertainty", i.e., deficiency in mathematical models, and solving inverse problems such as parameter estimation. ones, and suitable for stochastic simulations of complex systems.

Monday, September 15, 2008 - 13:00 ,
Location: Skiles 255 ,
Peijun Li ,
Department of Mathematics, Purdue University ,
Organizer: Haomin Zhou

Near-field optics has developed dramatically in recent years due to the possibility of breaking the diffraction limit and obtaining subwavelength resolution. Broadly speaking, near-field optics concerns phenomena involving evanescent electromagnetic waves, to which the super-resolving capability of near-field optics may be attributed. In order to theoretically understand the physical mechanism of this capability, it is desirable to accurately solve the underlying scattering problem in near-field optics. We propose an accurate global model for one of the important experimental modes of near-field optics, photon scanning tunneling microscopy, and develop a coupling of finite element and boundary integral method for its numerical solution. Numerical experiments will be presented to illustrate the effectiveness of the proposed method and to show the features of wave propagation in photon scanning tunneling microscope. The proposed model and developed method have no limitations on optical or geometrical parameters of probe and sample, they can be used for realistic simulations of various near-field microscope configurations.