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We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in the plane, to develop a singularity at some finite time or converge to an m-fold circle as time goes to infinity. For the area preserving flow, the positivity of the enclosed algebraic area determines whether the curvature blows up in a finite time or not, while for the length preserving curvature flow, it is the positivity of an energy associated with initial curve that plays such a rule.

题目：从数学的角度看计算的本质与未来主讲人：姚鹏晖 副教授（南京大学 计算机科学与技术系）地 点：致远楼101室时 间：2019年5月17日15：30-16：30内容简介：理论计算机科学是计算机科学的一个分支，主要研究有关计算的逻辑化、抽象化和数学化的问题，从理论的角度回答 “什么是计算” 和 “怎样更高效地计算”这两个问题。这个报告中，将简要回顾下“计算”的历史，介绍当今计算机学家们对“计算”的理解，包括进入量子信息...

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the L^1 in time Lipschitz estimate of the velocity field can not be obtained by energy method. Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid, namely, striated regularity can help to get the L^/infty boundedness of the double Riesz transform, we derive the a priori L^1 in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space.

In his famous work on resolution of singularities, Slodowy pointed out that the McKay correspondence could be generalized to all affine Dynkin diagrams and showed this for most of the cases. We will first thoroughly explain the McKay-Slodowy correspondence using finite group theory, including the missing cases of A_{2n}^{(2)} and A_2^{(2)} in the literature. After the McKay-Slodowy correspondence is firmly established, we then show that the Poincare series for the tensor algebra of the induction and restriction of the fundamental modules realize all exponents of affine Lie algebras except A_{2n}^{(1)}. This is joint work with Danxia Wang and Honglian Zhang.

We are concerned with the three-dimensional time-dependent incompressible magnetohydrodynamic (MHD) equations with magnetic vector potential formulations. Compared with the traditional B formulations, the new MHD system has the advantage that it can ensure a direct exact discrete divergence-free magnetic induction in practical numerical computations. Using a mixed finite element approach, we discretize the velocity and pressure by stable finite elements, and the mag

Exciton diffusion length plays a vital role in the function of opto-electronic devices. Oftentimes, the domain occupied by a organic semiconductor is subject to surface measurement error. In many experiments, photoluminescence over the domain is measured and used as the observation data to estimate this length parameter in an inverse manner based on the least square method. However, the result is sometimes found to be sensitive to the surface geometry of the domain. We propose an asymptotic-based method as an approximate forward solver whose accuracy is justified both theoretically and numerically.

This talk is concerned with a type of nonlinear Schrodinger equation with potential possessing non-isolated critical points. we obtain the necessary condition, existence and local uniqueness of the positive single peak solution with concentrating at this kind of points. These types of results will also be mentioned for the BEC model. Here the main difficulty is the degeneracy and inhomogeneity of the potantial at the concentrating point.