阶化Cartan型代数S(m;n)的不可约表示及其约化

设L=S(m;n)是定义在特征P＞3的代数闭合域F上的阶化特殊型李代数,利用已研究L的不可约表示的方法,通过定义L的如下阶化:限制情形定义L=(田)q≥-1 L[q],I,非限制情形定义(L)=(田)q≥-1 (L)[q],I,这里是L的本原P-包络,有表达式(L)=(田)mΣi=1 ni-1Σdi=1 FDpidi,而I是{1,2,…,m)的子集,得到当P-特征标x是正则半单时,在限制李代数情形所有不可约Ux(L)-模都是从不可约Ux(L[0],I)-模诱导的;在非限制的情形,所有不可约U(x)(L),(Upx(L,x))-模都是从不可约L(x)_(L[0],I)-模诱导的,这里(x)是x到(L)*上的平凡扩张.     

单调映射的拓扑度

<正> 本文讨论非线性泛函分析中一类重要映射——单调映射的拓扑度(关于非线性泛函分析,映射度理论,单调映射理论的一般情况,见田方增[1],Nirenberg[7],Barbn[10]).这种广义拓扑度是在逼近正则映射(A-proper)拓扑度理论的基础上建立起来的,我们证明了这种度的一些基本性质,并利用度的方法证明了满足coercive条件的连续单调映射的满射性,这个满射陸结果本身并不是新的,最早可追溯至1956年见关肇直[2](就可微分情形),后来稍一般情形的讨论见E.H.Zarantonello[3],[2]和[3]均采用迭代法求解,更一般的情形的讨论见Minty[4].     

随机泛函分析中可交换映象的随机不动点定理

随机不动点定理在随机泛函分析中是一重要问题.在可分完备的度量空间中的随机不动点定理Bharucha-Reid,王梓坤,?pa?ek,Han?,Itoh及作者等都曾进行过讨论(见[1-5,15-20,21]).在本文中我们对概率分析中可交换映象的随机不动点定理得出了几个新的结果,它推广了前述诸人工作中某些重要结果.在确定性情形也推广了Jungck[6,7,8],Das,Naik[9],Rhoades[10],及Ciric[11]的结果.     

低秩矩阵恢复的混合型增广拉格朗日乘子算法CSCD

1引言低秩矩阵恢复问题,又称为鲁棒主成分分析问题或稀疏低秩矩阵分解问题,是指在较少的观测值的基础上恢复出原始矩阵.该问题来源于许多领域,如协同过滤[1,2,3],机器学习[4],图片对齐[5],信号处理[6]和量子态层析成像[7]等等.在文献[8,9,10]中,低秩矩阵恢复问题可以看作是将向量的稀疏表示推广到低秩矩阵的情形,也就是说当矩阵中某些元素严重缺失时,自动识别出损坏的元素并恢复原始矩阵[11].     

再论三割线定理

贵刊于2008年第九期登载了侯明辉老师的一篇文章《论三割线定理》,见文[1],文中对三割线定理及其逆定理的论述相当精彩,但笔者认为该定理只是一种特殊情形,本文将给出更一般的情形及变化,与各位老师共同探讨交流。     

不连续系统的实用稳定性

实用稳定性的概念,首先在[2]中就连续系统的情形提出.在[3,4]中,继续对连续系统的情形作过讨论。而不连续系统实用稳定性的研究仅见于[5],其中只给出了三个判别准则(定理1～3)。本文系统地研究了不连续系统的实用稳定性,给出比文[5]广泛的各种实用稳定性的概念,得到了判别各种实用稳定性或实用不稳定性的一系列准则。     

具有结点的约束值域的最佳逼近

我们在[1]中讨论了带约束值域的最佳逼近问题,约束函数l(x)和u(x)可以是任意不连续函数,但必须满足条件d(l,u)>0(参阅[1],引理3),当发生d(l,u)=0时,要进行一般的讨论比较困难.1969年G.D.Taytor在[3]的基础上讨论了一类发生l(x)=u(x)的情形.本文将在[1]的基础上讨论另一类发生d(l,u)=0的情形,证明了对于这类情形所建立的理论仍然与[1]中建立的理论,以及与经典的切比晓夫理论存在着完全的类似。     

一个经典不等式的再探究

文[1]得到以下结论: 命题A 若a1,a2,…,an∈R+,且a1+a2+…+an=s,P∈R,且|p|≥2,则 ∑ni=11/api(s-ai)≥np+2/sp+1(n-1).其实,关于上述不等式的研究很多文献早有出现.如:徐丹,杨露老师在2001年就已证明了p≤-2的情形[2],2003年,裘敬华老师将结果推广到p≤-1的情形[3].2010年,翁利帅老师[4]将p的范围推广到p∈(-∞,-1]U[0,+∞),但对于p∈(-1,0)的情形,翁老师自称还“留下一点遗憾”.笔者经一番探究,得到如下结论:     

星形变换的周期

关于线段映射的周期点之周期,有著名的Sarkovskii定理[3]、[2]。P.Stefen[4],L.Block及合作者[1]对这一定理给出了简单的证明。其后,关于圆周的变换,L.Block[2]在有不动点的情形下得到了类似的结果。对于较复杂一些的图形,这方面的问题似乎还没有人讨论过。本文在星形的情形下,做了初步讨论,证明了有关的两个定理。 设有有限个紧致线段,这些线段有一个公共端点O,且除O外任何两个线段无另外的公     

二次插值样条及二次样条有限元解的超收敛性

一 一维二次样条(等距节点情形)的渐近性态 [0,1]上函数f的二次插值样条s(x)∈C~1[0,1],且s(0)=f(0),s(1)=f(1),s(x_i+1/2)f(x_i+h/2),其中h=1/N,x_i=ih,在(x_i,x_(i+1))上为二次多项式,(i=0,1,     

高阶广义KDV方程和KP方程的数值解法

采用一种线性隐格式来解3-阶和5-阶的广义Korteweg-De Vries(KDV)方程,对这种方法做一推广,就能应用到它的二维形式广义Kadomtsew-Petviashvili(KP)方程．这种方法是无条件稳定的,且是无损耗的．数值实验描述了一个线性孤波运动的情形以及两个孤波交互的情形,从结果来看,它们满足孤立子解的两个守恒-动量守恒和能量守恒．     

Numerical solution for the KdV equation based on similarity reductions

The present paper is devoted to the development of a new scheme to solve the KdV equation locally on sub-domains using similarity reductions for partial differential equations. Each sub-domain is divided into three-grid points. The ordinary differential equation deduced from the similarity reduction can be linearized, integrated analytically and then used to approximate the flux vector in the KdV equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two test problems to show the behavior of the solution of the problems. The computed results are compared with other numerical results.     

A Schwarz-based domain decomposition method for the dispersion equation

We propose a Schwarz-based domain decomposition method for solving a dispersion equation consisting on the linearized KdV equation without the advective term, using simple interface operators based on the exact transparent boundary conditions for this equation. An optimization process is performed for obtaining the approximation that provides the method with the fastest convergence to the solution of the monodomain problem.     

An Initial Boundary-Value Problem in a Half-Strip for the Korteweg–De Vries Equation in Fractional-Order Sobolev Spaces

Abstract

An initial boundary-value problem in a half-strip with one boundary condition for the Korteweg–de Vries equation is considered and results on global well-posedness of this problem are established in Sobolev spaces of various orders, including fractional. Initial and boundary data satisfy natural (or close to natural) conditions, originating from properties of solutions of a corresponding initial-value problem for a linearized KdV equation. An essential part of the study is the investigation of special solutions of a “boundary potential” type for this linearized KdV equation.     

Exact solutions to the combined KdV and mKdV equation

This paper presents two different methods for the construction of exact solutions to the combined KdV and mKdV equation. The first method is a direct one based on a general form of solution to both the KdV and the modified KdV (mKdV) equations. The second method is a leading order analysis method. The method was devised by Jeffrey and Xu. Each of these methods is capable of solving the combined KdV and mKdV equation exactly.     

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.     

Fractional green function for linear time-fractional inhomogeneous partial differential equations in fluid mechanics

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.     

Explicit Complex-Valued Solutions of the Korteweg–de Vries Equation on the Half-Line and on the Whole-Line

The paper deals with the initial-value problems for the Korteweg–de Vries (KdV) equations on the half-line and on the whole-line for complex-valued measurable and exponentially decreasing potentials. The time evolution equation for the reflection coefficient is derived and then a one-to-one correspondence between the scattering data and the solution of the KdV equation is shown. Families of exact solutions of the KdV equation are represented for the class of reflection-free potentials, in which the inverse scattering problem associated with the KdV equation can be solved exactly. Some helpful examples of soliton solutions of the KdV equation are provided.     

Soliton interaction for the extended Korteweg-de Vries equation

Soliton interactions for the extended Korteweg-de Vries (KdV)equation are examined. It is shown that the extended KdV equationcan be transformed (to its order of approximation) to a higher-ordermember of the KdV hierarchy of integrable equations. This transformationis used to derive the higher-order, two-soliton solution forthe extended KdV equation. Hence it follows that the higher-ordersolitary-wave collisions are elastic, to the order of approximationof the extended KdV equation. In addition, the higher-ordercorrections to the phase shifts are found. To examine the exactnature of higher-order, solitary-wave collisions, numericalresults for various special cases (including surface waves onshallow water) of the extended KdV equation are presented. Thenumerical results show evidence of inelastic behaviour wellbeyond the order of approximation of the extended KdV equation;after collision, a dispersive wavetrain of extremely small amplitudeis found behind the smaller, higher-order solitary wave.     

单物种人口模型指数隐式Euler方法的振动性

本文研究了用以描述单物种人口模型的延迟Logistic方程的数值振动性.对方程应用隐式Euler方法进行求解,针对离散格式定义了指数隐式Euler方法,证明了该方法的收敛阶为1.根据线性振动性理论获得了数值解振动的充分条件.进而还对非振动数值解的性质作了讨论.最后用数值算例对理论结果进行了验证.