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.     

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.     

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     

Hölder,Sobolev,weak-type and BMO estimates in mixed-norm with weights for parabolic equations

We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 10,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation?tu?Δu+u=f.     

ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OFΓANDζ

In this paper,we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function.It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.     

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex.A proper total-k-coloring of G is called neighbor sum distinguishing if∑_c(u)≠∑_c(v)for each edge uv∈E(G),where∑_c(v)denote the sum of the color of a vertex v and the colors of edges incident with v.The least number k needed for such a total coloring of G,denoted byχ∑"is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjecturedχ∑"(G)≤Δ(G)+3 for any simple graph with maximum degreeΔ(G).By using the famous Combinatorial Nullstellensatz,we prove that above conjecture holds for any triangle free IC-planar graph with△(G)≥7.Moreover,it holds for any triangle free planar graph withΔ(G)≥6.     

Functionals for multilinear fractional embedding

A novel representation is developed as a measure for multilinear fractional embedding.Corresponding extensions are given for the Bourgain–Brezis–Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy–Littlewood–Sobolev inequality. Smoothing estimates are used to provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Regularity of viscous compressible real heat conductive gas with density-dependent viscosity in one dimension

We are concerned with the regularity of viscous compressible real heat conductive gas with density-dependent viscosity for Dirichlet boundary problem. Using the interpolation inequality and the embedding theorem, we obtain some delicate inequalities which are crucial to lift the regularity of the solutions.     

Log-Sobolev inequalities and regions with exterior exponential cusps

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log-Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy-type inequality. Our main application is to show that domains which have exterior exponential cusps, and hence have no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusps are not too sharp. Finally, we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp.     

欧氏空间中完备极小子流形

在一个类似于稳定不等式的条件下,得到了欧氏空间中完备极小子流形的Bernstein型定理.我们的结果部分推广了Li H.Z.和Wei G.X.的定理.     

非绝对模糊积分,绝对可积性与积分的绝对值不等式

关于模糊数值函数的绝对可积性及对应的积分不等式,无论在何种绝对值意义下是值得讨论的. 本文借助于模糊数空间到具体的 Banach 空间上的嵌入定理和模糊非绝对积分的刻划定理, 讨论了模糊数值函数的绝对可积性及对应的积分不等式,得到了若干个充分必要条件, 并举出了一些反例.     

B值行独立随机元阵列的完全收敛性

本文研究了B值行独立的随机元阵列完全收敛性，在对Banach空间B没有p型的要求下，根据Banach空间上的A．de Acosta．不等式与Hoffman-JФrgensen不等式，得到了几个有意义的结果．     

Embedding theorem for bounded deformations in domains with cusps

An embedding theorem for the space of L²‐bounded deformations in a planar domain with an external cusp is established. These kinds of embedding inclusions appear naturally in the modeling process of the motion of a rigid body in a fluid, filling a bounded domain. Copyright © 2013 John Wiley & Sons, Ltd.     

Theta functions on the Kodaira-Thurston manifold

We construct an analog of the classical theta function on an abelian variety for the Kodaira-Thurston nilmanifold, which is defined as a (nonholomorphic) section of a special complex line bundle over the Kodaira-Thurston manifold. The theta functions we introduce are used for a canonical symplectic embedding of the Kodaira-Thurston manifold into a complex projective space (an analog of the Lefschetz theorem).