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Fast Fourier-Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs

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Science China mathematics 2010年第1期53卷 1-22页

作者： WANG Bo1,WANG Rui2 & XU YueSheng3,4,1Academy of mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China 2School of Information Science and Engineering,Graduate university of the Chinese Academy of Sciences,Beijing 100190,China 3department of mathematics,syracuse university,syracuse,ny 132.4,usa 4department of Scientific Computing and Computer Applications,Sun Yat-sen university,Guangzhou 510275,China 1. Academy of mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China2. School of Information Science and Engineering Graduate University of the Chinese Academy of Sciences Beijing 100190 China3. Department of Mathematics Syracuse University Syracuse NY 13244 USA4. Department of Scientific Computing and Computer Applications Sun Yat-sen University Guangzhou 510275 China

We propose a fully discrete fast Fourier-Galerkin method for solving an integral equation of the first kind with a logarithmic kernel on a smooth open arc,which is a reformulation of the Dirichlet problem of the Laplace equation in the *** optimal convergence order and quasi-linear complexity order of the proposed method are established.A precondition is *** this method with an efficient numerical integration algorithm for computing the single-layer potential defined on an open arc,we obtain the solution of the Dirichlet problem on a smooth open arc in the *** examples are presented to confirm the theoretical estimates and to demonstrate the efficiency and accuracy of the proposed method.

关键词： Dirichlet problem open arc singular boundary integral equations Fourier-Galerkin methods logarithmic potentials

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.

Arithmetic of Shimura curves

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Science China mathematics 2010年第3期53卷 573-592页

作者： ZHANG Shou-Wu department of mathematics, Columbia university, New York, ny 10027, usa 1. department of mathematics Columbia University New York NY 10027 USA

This is the note for a series of lectures that the author gave at the Centre de Recerca Matemtica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

关键词： Shimura curves CM-points L-series

The maximal super-singular integral operators on product spaces

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Science China mathematics 2011年第12期54卷 2615-2626页

作者： WANG Hui1 & CHEN JieCheng2,3, 1department of mathematics, Xidian university, Xi’an 710071, China 2department of mathematics, Zhejiang Normal university, Jinhua 321004, China 3department of mathematics, Zhejiang university, Hangzhou 310027, China 1. department of mathematics Xidian University Xi’an 710071 China2. Department of Mathematics Zhejiang Normal University Jinhua 321004 China3. Department of Mathematics Zhejiang University Hangzhou 310027 China

We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ∈ S（R^n ×R^m）, where 0 ≤ α,β〈∞, b1 b2 ∈ L∞（R＋1 ）,Ω satisfies certain cancellation conditions and Ω∈L1（S^n-1×S^m-1）in the case α,β〉0;Ω∈L（log＋L）（S^n-1×S^m-1）in the case αβ=0 and α＋β 〉0. It is proved that, for 1〈p〈∞.T＊Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p（R^n×R^m）to the Lebesgue space L^p（R^n×R^m）.

关键词： super-singular integral maximal operator product spaces

ω-Gorenstein injective modules and dimensions

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Science China mathematics 2011年第3期54卷 421-436页

作者： ZHU HaiYan1,2 & DING NanQing3 1department of mathematics, Zhejiang university, Zhejiang 310027, China 2department of mathematics, Zhejiang university of Technology, Zhejiang 310023, China 3department of mathematics, Nanjing university, Nanjing 210093, China 1. department of mathematics Zhejiang University Zhejiang 310027 China2. Department of Mathematics Zhejiang University of Technology Zhejiang 310023 China3. Department of Mathematics Nanjing University Nanjing 210093 China

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.

关键词： Wakamatsu tilting module ω-Gorenstein injective module ω-Gorenstein injective dimension cotilting module

On conformal measures for infinitely renormalizable quadratic polynomials

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Science China mathematics 2005年第10期48卷 1411-1420页

作者： HUANG Zhiyong~1 JIANG Yunping~(2,3) & WANG Yuefei~3 1. School of Information,Renmin university of China,Beijing 100872,China 2. department of mathematics,Queens College,City university of New York,ny 11367,usa department of mathematics,Graduate School,City university of New York,New York,ny 10016,usa 3. Academy of mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China 1. School of Information Renmin University of China 100872 Beijing China 2. department of mathematics Queens College City University of New York 11367 NY USA 3. department of mathematics Graduate School City University of New York 10016 New York NY USA 4. Academy of mathematics and Systems Science Chinese Academy of Sciences 100080 Beijing China

We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in t... 详细信息

关键词： Julia set, conformal measure, three-dimensional puzzle, infinitely renormalizable quadratic polynomial.

On the Birch-Swinnerton-Dyer Conjecture of Elliptic Curves E_D:y^2=x^3-D^2x

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Acta Mathematica Sinica,English Series 2000年第2期16卷 229-236页

作者： Delang Li department of mathematics,Sichuan Union university,Chengdu 610064,P.R.China Ye Tian department of mathematics,Columbia university,New York,ny 10027,usa department of mathematics Sichuan Union University Chengdu 610064 P. R. China department of mathematics Columbia University New York NY 10027 USA

We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves.

关键词： Elliptic curve BSD conjecture Graph 2-component

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

The Willmore functional and the containment problem in R^4

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Science China mathematics 2007年第3期50卷 325-333页

作者： Jia-zu ZHOU School of mathematics and Statistics, Southwest university, Chongqing 400715, China department of mathematics, Polytechnic university, Brooklyn, ny 11201, usa School of mathematics and Statistics Southwest University Chongqing 400715China

Let∑be a convex hypersurface in the Euclidean space R4 with mean curvature H. We obtain a geometric lower bound for the Willmore functional∫∑H2dσ. This bound is an invariant involving the area of∑, the volume and Minkowski quermassintegrals of the convex body that∑bounds. We also obtain a sufficient condition for a convex body to contain another in the Euclidean space R4.

关键词： mean curvature scalar curvature kinematic formula Minkowski quermassintegrals convex body convex hypersurface 52A22 53C65 51C16