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Gheorghe Craciun, UW-Mathematics

Mathematical results arising from systems biology

We describe new sufficient conditions for global injectivity of general nonlinear functions, necessary and sufficient conditions for global injectivity of polynomial functions, and related criteria for uniqueness of equilibria in nonlinear dynamical systems. Some of these criteria are graph-theoretical, others are checked using symbolic computation. We also mention some applications of these methods in the study of Bezier curves and patches, and other types of manifolds used in geometric modeling. Also, we discuss some criteria for persistence and boundedness of trajectories in polynomial or power-law dynamical systems. All these seemingly unrelated results have been inspired by the study of mathematical models in systems biology.

Jean-Marc Vanden-Broeck, University College London

The effects of electrical fields on nonlinear free surface flows


Thierry Goudon, INRIA-Lille, France

Fluid-particle flows

We are interested in flows where a disperse phase (particles) is coupled to a dense phase (fluid). The evolution of the mixture is described by a kinetic equation coupled to a hydrodynamic system (Euler or Navier-Stokes). We will discuss several mathematical questions, with a particular attention paid to asymptotic issues. We will also present relevant numerical schemes specifically adapted to the asymptotic regime.

Sang Dong Kim, Kyungpook National University, Korea

A non-standard explicit method for solving stiff initial value problems

In this talk, we present a non-standard type of an explicit numerical method for solving stiff initial value problems which not only avoids unnecessary iteration process that may be required in most implicit methods but also has such a good stability as implicit methods possess.

The proposed methods use both a Chebyshev collocation technique and an asymptotical linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler's polygon. These methods with or without usages of the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian. Numerical tests are given to support the theoretical analysis as evidences.

Jean-Luc Thiffeault, UW-Mathematics

Velocity fluctuations in suspensions of swimming microorganisms


Nick Tanushev, University of Texas

Gaussian beam methods

Gaussian beams are asymptotic high frequency solutions to hyperbolic partial differential equations that are concentrated on a single curve through space-time. Their superpositions can be used to model more general high frequency wave propagation. In this talk, I will give a brief review of Gaussian beams and discuss some recently obtained results on the asymptotic convergence rate of Gaussian beam superpositions when the initial data is of the WKB form, $a(x) exp[i k \phi(x)]$. In numerical simulations involving Gaussian beams, one of the main challenges is to represent the initial data in terms of Gaussian beams. I will present a numerical method for decomposing general high frequency initial data into a sum of Gaussian beams. Finally, I will describe some open problems in Gaussian beam methods.

Bin Dong, University of California San Diego

Mathematics in medical imaging and image analysis

Variational models and tight wavelet frame based models for image processing have been extensively studied for the past 15 years. However, it was only in recent years did people start to apply these models to medical imaging and related problems. In my talk, I will explain how did we fashion these known models in image processing properly to solve problems in medical imaging and image analysis. Furthermore, I will draw connections between variational models and one of the frame based model. Such connections not only grant geometric insights to the frame based model, but also provide us a new viewpoint of frame transform that leads to frame based models for medical image segmentations and surface reconstruction from scattered data. In addition, we also combined the idea of multiresolution analysis with that of level set method, and developed a new multiscale representation for surfaces, and then applied it to surface inpainting problems that help doctors to quantify plaque formation of blood vessels.

Hoai Minh Nguyen, NYU-Courant Institute

Cloaking via change of variables for the Helmholtz equation

A region of space is cloaked for a class of measurements if observers are not only unaware of its contents, but also unaware of the presence of the cloak using such measurements. One approach to cloaking is the change of variables scheme introduced by Greenleaf, Lassas, and Uhlmann for electrical impedance tomography and by Pendry, Schurig, and Smith for the Maxwell equation. They used a singular change of variables which blows up a point into the cloaked region. To avoid this singularity, various regularized schemes have been proposed. In this talk I present results related to cloaking via change of variables for the Helmholtz equation using the natural regularized scheme introduced by Kohn, Shen, Vogelius, and Weintein, where the authors used a transformation which blows up a small ball instead of a point into the cloaked region. I will discuss the degree of invisibility for a finite range or the full range of frequencies, and the possibility of achieving perfect cloaking. If time permits, I will mention some results related to the wave equation.

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