A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

<正>We consider a finite difference scheme for a nonlinear wave equation,whose solutions may lose their smoothness in finite time,i.e.,blow up in finite time.In order to numerically reproduce blow-up solutions,we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations.A numerical blow-up time is defined and is proved to converge,under a certain hypothesis,to the real blow-up time as the grid size tends to zero.     

KAM Theory for Partial Differential Equations

In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.     

Implicit Shape Reconstruction of Unorganized Points Using PDE-Based Deformable 3D Manifolds

<正>In this work we consider the problem of shape reconstruction from an unorganized data set which has many important applications in medical imaging,scientific computing,reverse engineering and geometric modelling.The reconstructed surface is obtained by continuously deforming an initial surface following the Partial Differential Equation(PDE)-based diffusion model derived by a minimal volume-like variational formulation.The evolution is driven both by the distance from the data set and by the curvature analytically computed by it.The distance function is computed by implicit local interpolants defined in terms of radial basis functions.Space discretization of the PDE model is obtained by finite co-volume schemes and semi-implicit approach is used in time/scale.The use of a level set method for the numerical computation of the surface reconstruction allows us to handle complex geometry and even changing topology, without the need of user-interaction.Numerical examples demonstrate the ability of the proposed method to produce high quality reconstructions.Moreover,we show the effectiveness of the new approach to solve hole filling problems and Boolean operations between different data sets.     

傅种孙先生在北京师大附中

1 傅种孙先生简介 1.1 傅先生基本情况介绍 傅种孙先生(1898～1962)1898年2月27日生于江西省高安县.1920年在北京高等师范学校数理部毕业,留母校附中(即现北京师大附中)任教;1921年任母校数理部的讲师;1928年任教授:抗日战争以前,曾先后兼任北京女子师范大学,北平大学女子文理学院,北京大学,辅仁大学教授;1933年当选北平(北京)市数学会理事长秘书;1935年当选中国数学会评议委员兼<数学杂志>编辑;1945年11月至1946年8月在牛津大学、1946年9月至1947年11月在剑桥大学考察;1947～1962年任北京师范大学数学系教授,1956年前曾兼系主任;在1949～1957年间,曾任北京师范大学教务长(3年)及副校长(5年);1952～1957年任北京市人民代表大会代表,中国数学会及其北京市分会常务理事,<中国数学杂志>及其后身<数学通报>总编辑;1962年1月18日病逝于北京.     

Local Unstable Entropy and Local Unstable Pressure for Partially Hyperbolic Endomorphisms*

In this paper, local unstable metric entropy, local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. Specially, two variational principles concerning relationships among the above mentioned numbers are formulated.     

Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.     

A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group

We give the direct method of moving planes for solutions to the conformally invariant fractional power sub Laplace equation on the Heisenberg group.The method is based on four maximum principles derived here.Then symmetry and nonexistence of positive cylindrical solutions are proved.     

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.     

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces.We prove that the operators are bounded on L^p provided that the kernels satisfy weak conditions.     

智慧窗

1 趣解等式七个文字与三个拼音字头组成了一道加法等式,你能否把它们分别换成0～9的数字. (相同文字、字母必须要换相同的数字)使其等式成立吗? (上海市长宁路476弄8号1602室(200042) 张刘福) 2 质数趣题请将5～20各数分别填入圆圈内,使每相邻的两个数之和都是质数.试试看,你能填出     

解析函数与Lipschitz条件

研究了解析函数与Lipschitz条件,得到了如下两个结果:(i)设D是一平面区域,f(z)在D中解析,00,对任意z∈D有|f′(z)|≤md(z,D)k-1,则f∈Lipk(D)且‖f‖k≤cmk,其中c=c(D)是仅与D有关的常数.     

Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .     

Riesz函数演算的Lipschitz性质

设A是具有单位的复Banach代数,Ω为复平面C上的一个区域,γ是复平面上的任一可求长的封闭曲线且其内部区域ins(γ)Ω,证明了存在A的子集A_δ~γ,使得对于Ω上的任一解析函数f,Riesz函数演算f:xf(x)是从A_δ~γ到A中的Lipschitz映射即f∈L~1(A_δ~γ,A)且其Lipschtiz常数(L_1(f)■M_f(γ)Γ)/(2πδ~2)．作为这一结果的应用,研究了算子值的根式函数TT~(1/m)及绝对值函数T|T|的Lipschitz性质．最后,证明了:若f为一个复值整函数,则对任一非空有界集EA,有f∈L~1(E,A)．     

亚纯函数结合其导数的值分布

研究了亚纯函数结合其导数的值分布问题,得到了一个有趣的不等式,此不等式概括了方-杨和I.Lahiri和S.Dewan的结果,应用此不等式还得到关于θ(a(z);φ)的一个估计,这里φ(z)=α(z)f~nM[f],M[f]=(f′)~(n_1)(f″)~(n_2)…(f~((k)))~(n_k),n_1,n_2,…,n_k,n为非负整数满足:n_1+n_2+…+n_k≥1,α(z),a(z)(≠00,∞)为f的小函数.     

一类具分布时滞的微分方程的正周期解的存在性与多解性

讨论具分布时滞的微分方程x′(t)=-a(t,x)x(t)+∫-0τf(t,r,x(t+r))dr,x′(t)=a(t,x)x(t)-∫0-τf(t,r,x(t+r))drx′(t)=-g(t,x(t))+∫0-τf(t,r,x(t+r))dr,x′(t)=g(t,x(t))-∫0-τf(t,r,x(t+r))dr正周期解问题,利用锥不动点定理,获得了这类问题正解存在性和多重性的充分条件,推广了已有文献的相关结果.     

图的k符号边控制数

设G=(V,E)是一个图,一个函数f:E→{-1,+1},如果对于G中至少k条边e有sum from e'∈N[e]f(e')≥1成立,则称f为图G的一个k符号边控制函数.一个图的k符号边控制数定义为γ_(ks)^/(G)=min{∑_(e∈E(G))f(e)|f为图G的一个k符号边控制函数}.主要给出了一个图G的k符号边控制数γ_(ks)/(G)=min{∑_(e∈E(G))f(e)|f为图G的一个k符号边控制函数}.主要给出了一个图G的k符号边控制数γ_(ks)^/(G)的若干新下限,并确定了路和圈的k符号边控制数.     

关于等角写像的边界性质

<正> 一个函数f(x)在[a,b]上定义,我们记 f(x)∈H_x~a(0<α≤1),表示f(x)是满足α级 Lipschitz-H(?)lder 条件的函数,下标表示所涉及的自变量,而f(x)∈H_x~(1-0表示f(x)之连续模满足条件ω(δ,f)≤kδ|log δ|,(k为常数).我们记(?)是其连续模满足条件     

计算方程重根及其重数的算法

给出一种计算方程重根及重数的迭代算法,分别具有平方收敛和线性收敛.(i)迭代:x_(n+1)=x_n-f x_n (f'(x_n))/((f'(x_n))~2-(f(x_n)f~n(x_n)),m_n=((f'(x_n)))~2/((f'(x_n))~2-f(xn_)f″(x_n)),n=0,1,2,…,重数m≈mn;(ii)加速迭代:x_(n+1)=x_n-(f~((m-1))(x_n))/(f(~m)(x_n)).     

非线性两点边值问题的存在唯一性

对于非线性两点边值问题x″=f(t,x,x′),　x(0)=A,　x(1)=B,我们在f,fx,fx′,β(t),α′(t)都连续,且fx≥-β(t), -α(t)≤fx′≤M(1+|x′|),max β(t)-α2(t)+2α′(t)4t∈[0,1] ≤λ2<π24,α(1)≤2λcotλ,这些条件之下证明解之存在且唯一.     

一类奇异拟线性椭圆方程正解的多重性

研究奇异拟线性椭圆型方程{-div(|x|~(-ap)|▽u|~(p-2)▽u) + f(x)|u|~(p-2) = g(x)\u|~(q-2)u + λh(x)|u|~(r-2),x R~N,u(x) 0,x∈ R~N,其中λ0是参数,1pN(N3),1rpgp*=0a(N—p)/p,p*=Np/{N~pd),aa+l,d=a+l-60,权函数f(x),g(x),h(x)满足一定的条件.利用山路引理和Ekeland变分原理证明了问题至少有两个非平凡的弱解.