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T1 theorem for Besov and Triebel-Lizorkin spaces

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Science China mathematics 2005年第5期48卷 657-665页

作者： DENG Donggao & HAN Yongsheng department of mathematics, Zhongshan university, Guangzhou 510275, China department of mathematics, auburn university, alabama, 36849, usa Dept. of Math. Zhongshan Univ. Guangzhou 510275 China Dept. of Math. Auburn Univ. Alabama 36849 United States

Using the discrete Calderon type reproducing formula and the PlancherelPolya characterization for the Besov and Triebel-Lizorkin spaces, the T1 theorem for the Besov and Triebel-Lizorkin spaces was proved.

关键词： T1 theorem Besov and Triebel-Lizorkin spaces Calderon reproducing formula Plancherel-Polya inequalities.

Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation

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Acta Mathematica Sinica,English Series 2000年第2期16卷 207-218页

作者： Weinan E department of mathematics and Program in Applied and Computational mathematics,Princeton university,usa department of mathematics and Program in Applied and Computational mathematics Princeton University Princeton USA

A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to *** is particularly important when boundaries are present since vorticitv is typically generated at the boundary as a result of boundary layer *** boundary laver theory,developed by Prandtl about a hundred years ago,has become a standard tool in addressing these *** at the mathematical level,there is still a lack of fundamental understanding of these questions and the validity of the boundary layer *** this article,we review recent progresses on the analysis of Prandtl’s equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes *** also discuss some directions where progress is expected in the near future.

关键词： Boundary layer Blow-up Zero-viscosity limit Prandtl’s equation

A PARALLEL ITERATIVE DOMAIN DECOMPOSITION ALGORITHM FOR ELLIPTIC PROBLEMS

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Journal of Computational mathematics 1998年第2期16卷 141-151页

作者： Dao-qi Yang (department of mathematics, Wayne State university, Detroit, MI 48202, usa) Wayne State Univ Dept Math Detroit MI 48202 USA

An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments arc conducted to show the effectiveness of the method.

关键词： Domain decomposition methods finite element methods parallel computing

AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

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Journal of Computational mathematics 2002年第3期20卷 225-244页

作者： Jian-guo Liu (department of mathematics, university of North Texas, Denton, TX 76203, usa) department of mathematics University of North Texas 美国 Denton

Presents information on a study which analyzed an interior trust-region-based algorithm for linearly constrained minimization problems. Optimality conditions for the linearly constrained minimization problem presented; Vectors for each updating step in the algorithm proposed; Establishment of the convergence properties of the proposed algorithm.

关键词： Nonlinear programming Linear constraints Trust region algorithms Newton methods Interior algorithms Quadratic convergence

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

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Acta Mathematica Sinica,English Series 2000年第3期16卷 405-444页

作者： Guozhen Lu department of mathematics,Wayne State university,Detroit,MI 48202,usa department of mathematics Wayne State University Detroit USA

This paper consists of three main *** of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie *** the extensive research after Jerison’s work[3]on Poincaré-type inequalities for Hrmander’s vector fields over the years,our results given here even in the nonweighted case appear to be *** interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector *** main tools to prove such inequalities are approximating the Sobolev func- tions by polynomials associated with the left invariant vector fields on *** very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own *** the existence of such polynomials is the second main part of this *** results of these two parts have been announced in the author’s paper in Mathematical Research Letters[38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on(εδ)domains. Some results of weighted Sobolev spaces are also given *** construct a linear extension operator which is bounded on different Sobolev spaces *** particular,we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev *** are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

关键词： Poincaré inequalities Extension theorems Interpolation inequalities Anisotropic Sobolev spaces A_p weights (ε,δ)domains Vector fields Polynomials on stratified groups

SYMPLECTIC SCHEMES FOR QUASILINEAR WAVE EQUATIONS OF KLEIN-GORDON AND SINE-GORDON TYPE

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Journal of Computational mathematics 2001年第5期19卷 475-488页

作者： Xiao-wu Lu (department of mathematics and Computer Science, Emory university, Atlanta, GA 30322, usa) Emory university 美国 30322

Presents a class of finite difference methods of first and second order accuracy for the computation of solutions to the quasilinear wave equations. Overview of general symplectic schemes and generating functional; Nu... 详细信息

关键词： Symplectic integration Sine-Gordon equations Finite difference method.

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

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Chinese Annals of mathematics,Series B 2000年第2期21卷 187-200页

作者： WAN DAQING(department of mathematics, university of California, Irvine, CA 92697-3875, usa.) department of mathematics University of California Irvine CA 92697-3875 USA

A vanishing theorem is proved for e-adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibiliy of the poles or equivalently an improvement of the polar part of the AxKatz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

关键词： Pole Zeta function Complete intersection

Rational holomorphic maps from B^2 into B^N with degree 2

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Science China mathematics 2006年第11期49卷 1504-1522页

作者： CHEN Zhihua, Jl Shanyu & XU Dekang department of mathematics, Tongji university, Shanghai 200092, China department of mathematics, university of Houston, Houston, TX 77204, usa department of mathematics Tongji University Shanghai 200092 China department of mathematics University of Houston Houston TX 77204 United States

Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.

关键词： rational holomorphic maps between balls.

ON THE KINEMATIC GEOMETRY OF MANY BODY SYSTEMS

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Chinese Annals of mathematics,Series B 1999年第1期20卷 11-28页

作者： WU-YI HSIANG(department of mathematics,university of CALIFORNIA,berkeley,CA 94720-3840,usa) Univ Calif Berkeley Dept Math Berkeley CA 94720 USA

In mechanics, both classical and quantum, one studies the profound interaction betweentwo types of energy, namely, the kinetic energy and the potential energy. The former can beorganized as the kinematic metric on the configuration space while the latter can be representedby a suitable potential function, such as the Newtonian potential in celestial mechanics andthe Coulomb potential in quantum mechanics of atomic and molecular physics. In this paper,the author studies the kinematic geometry of n-body systems. The main results are (i) theintroduction of a canonical coordinate svstem which reveals the total amount of kinematicsymmetry by an SO(3) x O(n-1) action in such a canonical coordinate representationt (ii) anin depth analysis of the above kinematic system both in the setting of classical invariant theoryand by the technique of equivariant Riemannian geometry; (iii) a remarkably simple formulafor the potential function in such a canonical coordinate system which reveals the well-fittingbetween the kinematic symmetry and the Potential energy.

关键词： Kinematic geometry Many body system Configuration space