第一类弱奇异核Fredholm积分方程的数值求解

第一类弱奇异核Fredholm积分方程由于奇异及本质的不适定性,给求解带来很大难度．本文首先利用克雷斯变换将方程转化,并对转化后的方程进行高斯一勒让德离散,得到一离散不适定的线性方程组,结合正则化方法对该类问题进行数值求解．最后给出了数值模拟,验证了本文方法的可行性及有效性．     

广义Pell方程Ax^2-By^2=4的通解公式

主要运用 Lucas 数的奇偶性,讨论了当 A, B 是适合 A〉1,2-AB 且 AB 非平方数的正整数时,广义 Pell 方程的整数解（x, y）,即给出了方程 Ax^2 -By^2=4适合gcd（x, y）=1的整数解（x, y）的通解公式。     

Solvability of the Dirichlet Problem in W~（2,p ）for a Class of Elliptic Equations with Singular Data

We prove an existence and uniqueness result for the Dirichlet problem for a class of elliptic equations with singular data in weighted Sobolev spaces.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

Weighted pseudo almost periodic solutions of N-th order neutral differential equations with piecewise constant arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.     

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called "Acquistapace–Terreni" conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.     

On solution sequences of the singular Liouville equations with an integral constraint

In this paper,we analyze the blow-up behavior of sequences{uk}satisfying the following conditionsΔuk=|x|2αk V k eukinΩ,(0.1)whereΩR2,V k→V in C1,|Vk|≤A,0a≤Vk≤b,0≤αk→α∈(0,∞),andΩ|x|2αkeuk dx≤Λ1.(0.2)Furthermore,we assume that there exists someq∈(1,2)such that rq 2Br(p)|uk|qdx≤Λ2(0.3)for anyB r(p)Ω.As a result,we give a new proof of the concentration-compactness theorem for the mean feld equation.     

用矩阵方程法求解循环卷积的改进算法

从循环卷积的定义出发,描述了用以m为变量的方法求解循环卷积的步骤.特别详述了其中的矩阵方程法,指出了该方法的不足——含有冗余项,提出了矩阵方程法的改进算法,消除了冗余项,简化了矩阵方程,给出了矩阵方程内部各列的构成规律.实例表明该方法简便、有效.     

Approximation of the Cubic Functional Equations in Lipschitz Spaces

Let G be an Abelian group and let ρ : G×G→[0,∞) be a metric on GLet E be a normed spaceWe prove that under some conditions if f : G→E is an odd function and Cx : G→E defined by Cx(y) := 2 f(x + y) +2 f(x-y) + 12 f(x)-f(2x + y)-f(2x-y)is a cubic function for all x∈G, then there exists a cubic function C : G→E such that f-C is LipschitzMoreover, we investigate the stability of cubic functional equation2 f(x + y) + 2 f(x-y) + 12 f(x)-f(2x + y)-f(2x-y) = 0 on Lipschitz spaces.     

由一道全国高考题引起的思考与探究

(2006全国理2)已知抛物线x2=4y的焦点为F,A、B是抛物线上的两动点,且(→AF)=λ(→FB)(λ＞0).过A、B两点允别作抛物线的切线,设其交点为M.证明(→FM)·(→AB)为定值. 一、初步探究 本题的M点坐标为(x1+x2/2,-1),说明M点都在直线y=-1上,而抛物线的准线恰好为直线y=-1,这是巧合还是必然?