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.     

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.     

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.     

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.     

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.     

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.     

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.     

Poisson Stable Solutions for Stochastic Differential Equations with Lévy Noise

In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for semilinear stochastic differential equations driven by infinite dimensional L′evy noise with large jumps.Under suitable conditions on drift,diffusion and jump coefficients,we prove that there exist solutions which inherit the Poisson stability of coefficients.Further we show that these solutions are globally asymptotically stable in square-mean sense.Finally,we illustrate our theoretical results by several examples.     

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.     

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.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

半严格不变凸函数

文章在Banach空间中定义了一种新的广义凸函数—半严格不变凸函数.对于满足局部Lipschitz条件的半严格不变凸函数,得到了它的广义Clarke次微分性质.文中还讨论了半严格不变凸函数与不变凸函数及半严格预不变凸函数之间的关系,得到了半严格不变凸函数的一些性质.     

强预拟不变凸函数与强拟不变凸函数

本文提出了两类新的广义凸函数—强预拟不变凸函数与强拟不变凸函数.讨论了强预拟不变凸函数与强拟不变凸函数间的关系,强拟不变凸函数与强伪不变凸函数间的关系.研究了强预拟不变凸函数在多目标优化中的应用.     

Subvexormal Functions and Subvex Functions

Subvexormal functions and subinvexormal functions are proposed, whose properties are shared commonly by most generalized convex functions and most generalized invex functions, respectively. A necessary and sufficient condition for a subvexormal function to be subinvexormal is given in the locally Lipschitz and regular case. Furthermore, subvex functions and subinvex functions are introduced. It is proved that the class of strictly subvex functions is equivalent to that of functions whose local minima are global and that, in the locally Lipschitz and regular case, both strongly subvex functions and strongly subinvex functions can be characterized as functions whose relatively stationary points (slight extension of stationary points) are global minima.     

集合上的递归函数

本文在Ｊｅｎｓｅｎ和Ｋａｒｐ工作的基础上引进了集合上的递归函数的概念．研究了递归集函数的初步性质，讨论了递归集函数与Ｊｅｎｓｅｎ和Ｋａｒｐ定义的原始递归集函数及递归数论函数之间的关系，并给出了ＺＦＣ的可定义集模型上递归集函数的范式定理．     

A new type of the generalized Bézier curves

In this paper, we improve the generalized Bernstein basis functions introduced by Han, et al. The new basis functions not only inherit the most properties of the classical Bernstein basis functions, but also reserve the shape parameters that are similar to the shape parameters of the generalized Bernstein basis functions. The degree elevation algorithm and the conversion formulae between the new basis functions and the classical Bernstein basis functions are obtained. Also the new Q-Bézier curve and surface...     

广义部分Bent函数和广义Bent函数的关系

Bent函数是一类特殊的布尔函数,因其非线性性和稳定性在密码学和通信等领域有很重要的应用,但它们数量少,不平衡且无相关免疫性,为了弥补Bent函数的不足,Claud Carlet提出了部分Bent函数的概念,部分Bent函数是包含Bent函数的更大的函数类,后来,人们又将这两种函数概念先后都拓广到了环zm^n(m为正整数）上,分别被称为zm^n上的广义Bent函数和广义部分Bent函数,本文利用zp^n(p为素数）上广义部分Bent函数的Chrestenson循环谱特征讨论了zp^n上的广义部分Bent函数和广义Bent函数之间的关系,给出了这两种函数之间的函数关系式和谱值关系式。     

Prox-regular functions in variational analysis

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower- functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

     

Cross-product of Bessel functions: Monotonicity patterns and functional inequalities

In this paper, we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.     

Locally tensor product functions

We present in this paper a family of functions which are tensor product functions in subdomains, while not having the usual drawback of functions which are tensor product functions in the whole domain. With these functions we can add more points in some region without adding points on lines parallel to the axes. These functions are linear combinations of tensor product polynomial B-splines, and the knots of different B-splines are less connected together than with usual polynomial B-splines. Approximation of functions, or data, with such functions gives satisfactory results, as shown by numerical experimentation. AMS subject classification 41A15, 41A63, 65Dxx