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.

In combustion theory, the propagation of premixed flames is usually described by the conventional thermal-diffusional model with standard Arrhenius kinetics. Formal asymptotic methods based on large activation energy have allowed simpler descriptions, especially when the thin flame zone is replaced by a free interface, called the flame front, which separates burned and unburned gases. At the flame front, the temperature and mass fraction gradients are discontinuous.

We get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace–Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result (Alexandrov, 2007) of Alexandrov.

Roughly speaking, the geometric group theory studies groups from the geometric point of view. We will talk about lengths, curvatures, areas on groups. In particular, we will introduce the notions of CAT(0) groups, Gromov hyperbolic groups and some problems.

We will talk about two recent work: 1. A reaction-diffusion model with nonlocal interactions between ribbed mussels, marsh grass and sediment is proposed. Mussel-grass aggregating behavior is suggested to be a result of nonlocal density-dependent interactions between mussels. The model is a reaction-diffusion system with cooperative kinetic dynamics which is outside of the regime of classical Turing instability mechanism, but the nonlocal interaction causes the generation of spatial patterns. The nonlocal model can be approximated by a bi-harmonic PDE, which produces analytic results marching the numerical simulation of full nonlocal model. 2. A reaction-diffusion model is proposed to describe the evolution of spatial distributions of ROP1 and calcium on the pollen tube tip. The cytoplasmic ROP1 activate ROP1 on the membrane and the calcium ions inhibit ROP1, while ROP1 controls calcium influx with a time delay.

In this talk, we will present our work on open Lie group actions on open manifolds. Witten genus and elliptic genera are modular topological invariants for manifolds, which are closely related to representation of loop groups and the hypothetical index theory on free loop space as well as the elliptic cohomology theory in algebraic topology. They find applications in problems of positive curvature and group actions on manifolds. In this talk, we will briefly introduce these invariants and present our joint work with Varghese Mathai on generalizing them to open manifolds with proper actions of open Lie groups.

In this talk, we shall present our recent works on primes in arithmetic progressions with friable indices, joint with Jianya Liu and Ping Xi. Denote by $/pi(x,y;q,a)$ the number of primes $p/leqslant x$ such that $p/equiv a/bmod q$ and $(p-a)/q$ is free of prime factors larger than $y$. Assume a suitable form of Elliott--Halberstam conjecture, it is proved that $/pi(x,y;q,a)$ is asymptotic to $/rho(/log(x/q)//log y)/pi(x)//varphi(q)$ on average, subject to certain ranges of $y$ and $q$, where $/rho$ is the Dickman function. Moreover, unconditional upper bounds are also obtained via sieve methods.

著名的Green-Tao定理证明了素数中存在任意长度的算术数列。Shapiro型素数是数论中的一类特殊的、重要的素数数列，本报告将介绍Shapiro型素数中的Green-Tao定理，并给出相关的定量结果。