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The first eigenvalue of Laplace operator under powers of mean curvature flow

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Science china Mathematics 2010年第7期53卷 1702-1709页

作者： ZHAO Liang center of mathematical sciences, zhejiang university, hangzhou 310027, china 1. center of mathematical sciences Zhejiang University Hangzhou 310027 China

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along powers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in R n+1 , we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k -flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.

关键词： Laplace operator H k -flow eigenvalue monotonicity

TRIPLE　INTERACTIONS　OF　CONORMAL　WAVES　FOR　HIGHER　ORDER　SEMILINEAR　HYPERBOLIC　EQUMIONS

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Chinese Annals of Mathematics,Series B 1995年第1期16卷 23-32页

作者： FANG DAOYUAN (center for mathematical sciences, zhejiang university, hangzhou 310027, china.) 浙江大学数学科学中心浙江杭州

The interaction of three conormal waves for semi-linear strictly hyperbolic equations of third order is considered. Let Σi, i = 1, 2, 3, be smooth characteristic surfaces for P= Da(D_t￣2 -△) intersecting transversally at the origin. Suppose that the solution u to Pu = f(t, x ,y)D u), ≤2 is conormal to Σi, i = 1, 2, 3, for t < 0. The author uses Bony's second microlocajization techniques and commutator arguments to conclude that the new singularities a short time after the triple interaction lie on the surface of the light cone Γ over the origin plus the surfaces obtained by flow-outs of the lines of intersection Γ ∩ Σi and Σi∩ Σj, i, j = 1, 2, 3.

关键词： Hyperbolic equation Conormal wave Interaction Characteristic surface.

On a Class of Finitely Supported Scaling Functions with Properties of Interpolation and Symmetry

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Acta Mathematica Sinica,English Series 2001年第2期17卷 333-338页

作者： Ze yin ZHANG center for mathematical sciences. zhejiang university. hangzhou 310027. P. R. china center for mathematical sciences Zhejiang University Hangzhou 310027 P. R. China

In this paper, a class of finitely supported orthogonal filters with interpolation and sym metry is constructed.

关键词： Filter Scaling function

Asymptotic Behavior of Gibbs Functions for M-Band Wavelet Expansions

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Acta Mathematica Sinica,English Series 1999年第2期15卷 165-172页

作者： Daren Huang Zeyin Zhang, center for mathematical sciences, zhejiang university, hangzhou 310027, P. R. china center for mathematical sciences Zhejiang University Hangzhou P. R. China

In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.

关键词： Gibbs phenomenon Wavelet expansion Filter Scaling function Vanishing moment

Complete hypersurfaces in a 4-dimensional hyperbolic space

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Applied Mathematics(A Journal of Chinese Universities) 2009年第3期24卷 370-378页

作者： XU Hong-wei1 ZHAO En-tao2 center of mathematical sciences,zhejiang university,hangzhou 310027,china center of mathematical sciences Zhejiang University Hangzhou 310027 China

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4（-1）,whose scalar curvature is bounded from below.

关键词： complete hypersurface mean curvature quasi-Gauss-Kronecker curvature

ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS

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Applied Mathematics(A Journal of Chinese Universities) 2007年第2期22卷 153-162页

作者： Leng Yan Xu Hongwei zhejiang university, center of mathematical sciences Eangzhou 310027, china zhejiang university, center of mathematical sciences Eangzhou 310027, china zhejiang university Center of Mathematical Sciences hangzhou 310027 China

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.

关键词： complete submanifold, rigidity theorem, mean curvature, second fundamental form pinchedRiemannian manifold

Lipschitz continuity of best approximations and Chebyshev centers

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Chinese Science Bulletin 1998年第3期43卷 185-188页

作者： LI Chong 1 and WANG Xinghua 2 1. center for mathematical sciences, zhejiang university, hangzhou 310027, china 2. Department of Mathematics, hangzhou university, hangzhou 310028, china center for mathematical sciences Zhejiang University Hangzhou China Department of Mathematics Hangzhou University Hangzhou China

The Lipschitz continuity results are given for best approximations and Chebyshev centers. Consequently, the open problem presented by Septycki and Vleck is solved completely.

关键词： best approximation Chebyshev center uniform convexity.

On the Cardinal Spline Interpolation Corresponding to Infinite Order Differential Operators

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Acta Mathematica Sinica,English Series 1994年第3期10卷 315-324页

作者： Chen Dirong Department of Mathematics Beijing Normal university Beijing,100875 and center for mathematical sciences zhejiang university hangzhou,310027 china 1. Department of Mathematics Beijing Normal University 100875 Beijing 2. center for mathematical sciences Zhejiang University 310027 Hangzhou China

This paper discusses some problems on the cardinal spline interpolation correspond- ing to infinite order differential *** remainder formulas and a dual theorem are es- tablished for some convolution classes,where the kernels are PF ***,the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of average n-Kolmogorov widths are obtained for the convolution class.

关键词： Math On the Cardinal Spline Interpolation Corresponding to Infinite Order Differential Operators lim Chen 卜成

EMBEDDED 2-SPHERES IN INDEFINITE 4-MANIFOLDS

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Chinese Annals of Mathematics,Series B 1996年第3期17卷 257-262页

作者： GAN DANYAN(Depedmellt of Mathematics and center for mathematical sciences, zhejiang university hangzhou310027, china.)(Project supported by the National Natural Science Foundation of china and by the zhejiang NaturalScience Foundation.) zhejiang UNIV DEPT MATHHANGZHOU 310027PEOPLES R CHINA zhejiang UNIV CTR MATH SCIHANGZHOU 310027PEOPLES R CHINA

The main results are as *** be an element of H2, where γ, δ1,…δn. arestandard generators. Suppose2≤n≤9.(1) If for some i and some i, then ζ can be represented by a smoothly embedded 2-sphere.(2) If ζ is a characteristic homology class, then ζ cannot be represented by a smoothlyembedded 2-sphere except for ζ2 = 16l+1 - n, l = 0 and (3) If ζ= dη for some d∈Z, where η is a primitive ordinary homology class with η2 = 0,then ζ is eqivalent to (d; d, 0,..., 0) and can be represented by a smoothly embedded 2-sphere.

关键词： Representing Characteristic Ordinary Primitive