Specifying attracting cycles for Newton maps of polynomials

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result in Plaza and Romero [6], which shows how to find such a polynomial of degree 2n. Moreover, we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n ≥ 2 and degree d ≥ 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.     

二维自治g-Navier-Stokes系统的全局渐近行为研究

研究了有界区域上二维自治g-Navier-Stokes系统的双全局吸引子,利用非紧性测度方法,给出了一种验证其存在性的新方法.得出二维自治g-Navier-Stokes方程在有界区域上有一个非空、紧可逆H_g一V_g全局吸引子这一结论.     

On Global Attractor for 2D Kirchhoff–Boussinesq Model with Supercritical Nonlinearity

Dynamics for a class of nonlinear 2D Kirchhoff–Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to “strong” initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolev's embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Ball's method [2 Ball , J. ( 2004 ). Global attractors for semilinear wave equations . Discr. Cont. Dyn. Sys. 10 : 31 – 52 .[Crossref], [Web of Science ®] , [Google Scholar]] and on recent asymptotic quasi-stability inequalities established in [11 Chueshov , I. , Lasiecka , I. ( 2008 ). Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping . Memoirs of the American Mathematical Society, Vol. 912 . Providence , RI : American Mathematical Society .[Crossref] , [Google Scholar]]. These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C ^∞ smoothness.     

带加法白噪声的随机Boussinesq方程组的解的渐近行为

考虑速度和温度同时在加法白噪声扰动下的随机Boussinesq方程组的解的渐近特征.可以接轨道得到该随机方程组的唯一解,并可以验证该解生成随机动力系统,进而证明了该随机动力系统存在随机吸引子.     

Global existence,asymptotic behavior,and uniform attractor for a nonautonomous equation

In this paper, we establish the global existence, asymptotic behavior, and uniform attractor for a nonautonomous viscoelastic equation with a delay term. Under some suitable assumptions, we firstly prove the global well‐posedness of the problem by using the Faedo–Galerkin approximations together with some energy estimates and then obtain the general decay results of the energy via suitable Lyapunov functionals. Finally, we prove the existence of uniform attractors. Copyright © 2013 John Wiley & Sons, Ltd.     

Hypernuclear physics for neutron stars

The role of hypernuclear physics for the physics of neutron stars is delineated. Hypernuclear potentials in dense matter control the hyperon composition of dense neutron star matter. The three-body interactions of nucleons and hyperons determine the stiffness of the neutron star equation of state and thereby the maximum neutron star mass. Two-body hyperon–nucleon and hyperon–hyperon interactions give rise to hyperon pairing which exponentially suppresses cooling of neutron stars via the direct hyperon URCA processes. Nonmesonic weak reactions with hyperons in dense neutron star matter govern the gravitational wave emissions due to the r-mode instability of rotating neutron stars.