DARBOUX TRANSFORMATION OF A NONLINEAR EVOLUTION EQUATION AND ITS EXPLICIT SOLUTIONS

In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-difference equation is given. In order to solve the differential-difference equation, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equation are given.     

The ergodicity of stochastic generalized porous media equations with lévy jump

In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.     

Global existence and blow-up of solutions to a nonlocal evolution p-laplace system with nonlinear boundary conditions

In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.     

STABILITY OF EQUILIBRIA OF NEMATIC LIQUID CRYSTALLINE POLYMERS

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b*(6.7314863965), the only equilibrium state is isotropic and it is stable; when b*< b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.     

非线性不等式组的光滑近似方法及其收敛性

将非线性不等式组的求解转化成非线性最小二乘问题,利用引入的光滑辅助函数,构造新的极小化问题来逐次逼近最小二乘问题.在一定的条件下,文中所提出的光滑高斯-牛顿算法的全局收敛性得到保证.适当条件下,算法的局部二阶收敛性得到了证明.文后的数值试验表明本文算法有效.     

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order CrouzeixRaviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.     

丢翻图方x2 + a2=2yn解法的计算研究

本文研究了一类丢番图方程的解.利用对Thue方程解的估计和方程解的连分式展开,获得了所求丢番图方程解的个数、解上界的估计和一般的求解算法.最后利用该算法给出了1≤a≤108的所有解.     

几乎完全非线性多项式函数间的EA等价性

简述了几乎完全非线性(APN)多项式函数的研究现状,讨论了两类几乎完全非线性多项式函数间的扩张仿射(EA)等价性,给出了验证EA等价的一般方法.     

精心筹划 突出重点 疏通经络 整体提升——对高三解析几何复习的一些感想

笔者任教高三年级多年,每一届在指导高三学生复习备考的过程中,都会听到学生反映最惧怕“解析几何的综合题”.原因是解析几何综合题思路虽然清晰,但是字母多、式子繁,不容易找到一个合适的关系式把这些字母、式子统帅起来进行处理,并且运算量太大,通常学生只能得一些步骤分.但最近这些年的现实是全国各省、市的高考试题中,解析几何综合题(分值14分左右,约占数学总分的9.3％)始终都是高考命题的热点内容之一,向来作为压轴题出现,成为考生能否取得高分的关键.因此,解析几何综合题成为高三年级老师和学生不得不面对、也不得不解决的一个大问题.     

椭圆、双曲线过焦点的弦长公式及其应用

椭圆、双曲线过焦点的弦长问题是解析几何在高考、竞赛中的热点之一,解决这类问题,传统的做法一般是将弦所在的直线方程与椭圆或双曲线方程进行联立,利用焦半径公式或弦长公式｜AB｜＝｜x1-x2｜·√1+k2=｜y1-y2 ｜·√1+1/k2求解,这样求解运算量往往较大,若我们利用椭圆、双曲线过焦点的弦长公式处理此类问题会省时省力,同时也能大大提高解题的准确率.现阐述如下,供同仁参考.