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Flatness of CR submanifolds in a sphere

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science China mathematics 2010年第3期53卷 701-718页

作者： JI ShanYu 1, & YUAN Yuan 2 1 department of mathematics, university of Houston, Houston, TX 77204, usa 2 department of mathematics, rutgers university, Piscataway, nj 08854, usa 1. department of mathematics University of Houston Houston TX 77204 USA2. Department of Mathematics Rutgers University Piscataway NJ 08854 USA

In Euclidean geometry, for a real submanifold M in E n+a , M is a piece of E n if and only if its second fundamental form is identically zero. In projective geometry, for a complex submanifold M in CP n+a , M is a piece of CP n if and only if its projective second fundamental form is identically zero. In CR geometry, we prove the CR analogue of this fact in this paper.

关键词： CR submanifolds sphere CR second fundamental form proper holomorphic mappings between balls

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.

INTERIOR ERROR ESTIMATES FOR NONCONFORMINGFINITE ELEMENT METHODS OF THE STOKES EQUATIONS

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Journal of Computational mathematics 1999年第5期17卷 475-494页

作者： Xiao-bo Liu(department of mathematics and computer science Clarkson university Box 5815, Potsdam,NY 13699-5815, usa) Clarkson Univ Dept Math & Comp Sci Potsdam NY 13699 USA

Interior error estimates are derived for nonconforming stable mixed finite element discretizations of the stationary Stokes equations. As an application, interiorconvergences of difference quotients of the finite element solution are obtained forthe derivatives of the exact solution when the mesh satisfies some translation invariant condition. For the linear element, it is proved that the average of thegradients of the finite element solution at the midpoint of two interior adjacenttriangles approximates the gradient of the exact solution quadratically.

关键词： Interior error estimates Nonconforming element Stokes equations.

Equivalence problem for Bishop surfaces

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science China mathematics 2010年第3期53卷 687-700页

作者： HUANG XiaoJun 1, & YIN WanKe 2 1 department of mathematics, rutgers university, New Brunswick, nj 08902, usa 2 School of mathematics and Statistics, Wuhan university, Wuhan 430072, China 1. department of mathematics Rutgers University New Brunswick NJ 08902 USA2. School of Mathematics and Statistics Wuhan University Wuhan 430072 China

The paper has two parts. We first briefly survey recent studies on the equivalence problem for real submanifolds in a complex space under the action of biholomorphic transformations. We will mainly focus on some of the recent studies of Bishop surfaces, which, in particular, includes the work of the authors. In the second part of the paper, we apply the general theory developed by the authors to explicitly classify an algebraic family of Bishop surfaces with a vanishing Bishop invariant. More precisely, we let M be a real submanifold of C 2 defined by an equation of the form w = zz + 2Re(z s + az s+1 ) with s≥ 3 and a a complex parameter. We will prove in the second part of the paper that for s≥ 4 two such surfaces are holomorphically equivalent if and only if the parameter differs by a certain rotation. When s = 3, we show that surfaces of this type with two different real parameters are not holomorphically equivalent.

关键词： equivalence problem Bishop surface Chern-Moser theory elliptic and hyperbolic complex tangents normal form and modular space

Presenting degenerate Ringel-Hall algebras of type B

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science China mathematics 2012年第5期55卷 949-960页

作者： FAN LiJuan1 & ZHAO ZhongHua2,1department of mathematics,Beijing Normal university,Beijing 100875,China 2department of mathematics and computer science,School of science,Beijing university of Chemical Technology,Beijing 100029,China 1. department of mathematics Beijing Normal University Beijing 100875 China2. Department of Mathematics and Computer Science School of Science Beijing University of Chemical Technology Beijing 100029 China

We give a presentation for the degenerate Ringel-Hall algebras of type B by studying the corresponding generic extension monoid *** an application,it is shown that the degenerate Ringel-Hall algebras of type B admit m... 详细信息

关键词： degenerate Ringel-Hall algebras generic extension monoid algebras Auslander-Reiten quivers Frobenius morphism

On characterizations of isometries on function spaces

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science China mathematics 2008年第4期51卷 620-631页

作者： LI SongYing~(1+) RUAN YingBin~2 1 department of mathematics university of California Irvine CA 92697 usa 2 School of mathematics and computer science Fujian Normal university Fuzhou 350007 China 1. department of mathematics University of California Irvine CA 92697 USA 2. School of mathematics and computer science Fujian Normal University Fuzhou 350007 China

The paper gives characterization for an isometric isomorphism on little Bloch space, VMOA and holomorphic Besov space over the unit ball B n in ? n .

关键词： ismometry Besov spaces 32A17 47A15

A priori estimates for classical solutions of fully nonlinear elliptic equations

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science China mathematics 2011年第3期54卷 457-462页

作者： CAO Yi1, LI DongSheng1 & WANG LiHe1,2 1College of science, Xi’an Jiaotong university, Xi’an 710049, China 2department of mathematics, The university of Iowa, Iowa City, IA 52242-1419, usa 1. College of science Xi’an Jiaotong University Xi’an 710049 China2. Department of Mathematics The University of Iowa Iowa City IA 52242-1419 USA

For the fully nonlinear uniformly elliptic equation F(D2u) = 0, it is well known that the viscosity solutions are C2,α if the nonlinear operator F is convex (or concave). In this paper, we study the classical solutions for the fully nonlinear elliptic equation where the nonlinear operator F is locally C1,β a.e. for any 0 2,α. Moreover, the C2,α norm of u depends on n,F and the continuous modulus of D2u.

关键词： fully nonlinear classical solutions C2 α estimates

Eigenvalue approximation from below using non-conforming finite elements

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science China mathematics 2010年第1期53卷 137-150页

作者： YANG YiDu 1 ,ZHANG ZhiMin 2 & LIN FuBiao 11 School of mathematics and computer science,Guizhou Normal university,Guiyang 550001,China 2 department of mathematics,Wayne State university,Detroit,MI 48202,usa 1. School of mathematics and computer science Guizhou Normal University Guiyang 550001 China2. Department of Mathematics Wayne State University Detroit MI 48202 USA

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.

关键词： non-conforming element eigenvalue lower bound

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS, III

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Journal of Computational mathematics 1996年第2期14卷 273-290页

作者： P.A. Farrell(department of mathematics and computer science, Kent State university, usa)P.W. Hemker(CWI Center for mathematics and computer science, Amsterdam, The Netherlands)G.I. Shishkin(IMM, Institute of mathematics and Mechanics, Ural Branch of the Kent State university 美国 Center for mathematics and computer science 荷兰 Institute of mathematics and Mechanics 俄罗斯

In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small *** also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are *** the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions

关键词： III DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS