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One of the fifty problems of dynamical systems formulated by J. Palis and C. Pugh is the problem of the existence of a simple arc joining two structural stable systems on a smooth closed manifold [1]. Simplicity means that the arc consists of structural stable systems with the exception of a finite set of points where the system has the least deviation from a rough system, that is simple bifurcations: a saddle-node, a doubling period or a bifurcation of a heteroclinic tangency. We show that by means of such arc it is possible to change the dimension of a basic set. Namely we joint arbitrary structural stable 3-diffeomorphism with an expanding two-dimensional attractor with an Anosov diffeomorphism.

We consider astrophysical systems that are modeled by the multidimensional Euler equations with gravity. First for the homogeneous Euler equations we look at flow in the low Mach number regime. Here for conventional finite volume discretization one has excessive dissipation in this regime. We identify inconsistent scaling for low Mach numbers of the numerical flux function as the origin of this problem. Based on the Roe solver a technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations is proposed. We analyze properties of this scheme and demonstrate that its limit yields a discretization of the incompressible limit system.

We consider the dynamic vector-host-pathogen interaction motivated by such tick borne diseases as tick-borne encephalitis and Lyme disease. We stratify the vector population in terms of the stage before and after the contact with the host when co-feeding transmission may take place, and we consider the case where vector development may involve two time lags due to diapause. We derive and calculate the critical rate for the model to exhibit nonlinear oscillations.

As we know, there are differences in travel frequency among people by age, ethnicity, region, gender, income, etc. Usually only a small fraction of people are frequent travelers while most travel occasionally or never. For instance, the percentages of frequent air traveler, occasional air traveler, and never flown in the U.S. in 2015 are 7%, 71%, and 22%, respectively. In this talk, we propose a multipatch epidemic model to investigate the effect of travel behavior on the spread of infectious diseases.

在本次报告中,我们将首先回顾经典Navier-Stokes方程的正则性结果，而后介绍我们在3维各向异性的navier-Stokes方程方面的适定性结果；最后介绍我们最近有关3维经典Navier-Stokes方程的一个分量的临界正则性准则。

In this paper, two decoupled finite element methods are proposed for solving the 3D primitive equations of ocean. Based on the finite element approximation, optimal error estimates are given under the convergence condition. And the detailed algorithms are given in the section of numerical tests. Further, numerical calculations are implemented to validate the theoretical analysis and more calculations are implemented for a more meaningful problem. For both theoretical and numerical points of view, the proposed decoupled finite element methods are the effective strategies to solve the 3D primitive equations of ocean.

The endoplasmic reticulum (ER) in plant cells forms a highly dynamic network of complex geometry. ER network morphology and dynamics are influenced by a number of biophysical processes, including filament/tubule tension, viscous forces, Brownian diffusion and interactions with many other organelles and cytoskeletal elements. Previous studies have indicated that ER networks can be thought of as constrained minimal-length networks acted on by a variety of forces that perturb and/or remodel the network. Here, we study two specific biophysical processes involved in remodelling.

Dirichlet character plays an important role in number theory. Almost 100 years have passed since the fundamental contribution of G. Polya and I.M. Vinogradov (1918), there were very few substantial progress like D.A. Burgess's work in late 1950s. The past decade has seen many promising changes due to the success of new ideas and tools. In this talk, some old and new results and several open problems in the field of character sums will be presented. Also I will introduce the joint works with Profs. C.H. Jia and M.A. Korolev.