Brown Analysis Seminar
Abstract: The disk algebra A consists of all continuous functions f on the unit circle such that f_{n} = 0 for all n < 0, where f_{n} is the nth Fourier coefficient of f. In 1956, R. Arens and I. Singer generalized A as follows: they replaced the circle by the torus T^{2}, and they fixed a positive irrational number a. They defined the algebra A_{a} as the algebra of all continuous functions f on T^{2} such that f_{n,m} = 0 for all (n,m) in the lattice half plane n + ma < 0. It turned out (in their paper and in later work by H. Helson and D. Lowdenslager) that A_{a} shares many function-theoretic properties with A. Today, we shall show that the functions in A_{a} can be characterized as those functions which extend to a certain three-manifold in C^{2} with boundary T^{2} and which satisfy the Cauchy-Riemann equation on that manifold.
Brown University Center for Statistical Sciences Seminar
Abstract: In this talk we define a spatio-temporal model with location dependent parameters to describe temporal variation and spatial nonstationarity. We consider the prediction of observations at unknown locations using known neighbouring observations. Further we propose a local least squares based method to estimate the parameters at unobserved locations. The sampling properties of these estimators are investigated. We also develop a statistical test for spatial stationarity. In order to derive the asymptotic results we show that the spatially nonstationary process can be locally approximated by a spatially stationary process. We illustrate the methods of estimation with some simulations and a real data example.
Joint Lefschetz Center for Dynamical Systems/PDE Seminar
Brown University Graduate School
Dissertation Defense Information Seminar
Scientific Computing Seminar
Abstract: Biological cells interact through chemotaxis when cells secret diffusing chemical (chemoattractant) and move towards gradient of chemoattractant creating effective nonlocal attraction between cells. Macroscopic description of cellular density dynamics through Keller-Segel model has striking qualitative similarities with nonlinear Schrodinger equation including critical collapse in two dimensions and supercritical in three dimensions. Critical collapse has logarithmic corrections to $(t_0-t)^{1/2}$ scaling law of self-similar solution. Regularization of collapse requires taking into account finite size of cells at microsopic level of cellular dynamics description. Microscopic motion of eucaryotic cells is accompanied by random fluctuations of their shapes which creates serious challenges in microscopic level simulations. We derive a nonlinear diffusion equation coupled with chemoattractant from microscopic cellular dynamics in dimensions one and two using excluded volume approach. Nonlinear diffusion coefficient depends on cellular volume fraction and it provides regularization (prevention) of cellular density collapse. A very good agreement is shown between Monte Carlo simulations of the microscopic Cellular Potts Model and numerical solutions of the derived macroscopic equations for relatively large cellular volume fractions.
Special PDE Seminar
Brown University Graduate School
Dissertation Defense Information Seminar
Brown University Graduate School
Dissertation Defense Information Seminar
PDE Seminar
Department of Mathematics Colloquium