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Hausdorff dimension of Julia set of a polynomial with a Siegel disk

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Science china Mathematics 2006年第9期49卷 1284-1296页

作者： SHEN Liang LMAM, school of mathematical sciences, peking University,{4., china LMAM School of Mathematical Sciences Peking University Beijing 100871 China

Let f(z) = e2πiθz(1+z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension o... 详细信息

关键词： Hausdorff dimension Siegel disk porous set

On Riemann-Finsler geometry

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Chinese Science Bulletin 1998年第6期43卷 447-450页

作者： MO Xiaohuan school of mathematical sciences, peking University,{4., china school of mathematical sciences Peking University Beijing China

The history of Finsler geometry is reviewed and briefly recent development in Finsler geometry and its application is completed systematically. Furthermore, an interesting open problem has been proposed in this field.

关键词： Finsler space Chern connection Hilbert form Minkowski potential projective sphere bundle.

A quantitative program for Hadwiger's covering conjecture

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Science china Mathematics 2010年第9期53卷 2551-2560页

作者： ZONG Chuanming school of mathematical sciences,peking University,{4. 100871,china 1. school of mathematical sciences Peking University Beijing 100871 China

In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its *** to now,this conjecture is still open for all n *** 1933,Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller *** to now,this conjecture is open for 4.n *** this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies,propose a four-step program to attack them,and obtain some partial results.

关键词： convex body Hadwiger’s conjecture Banach-Mazur metric β-net Borsuk’s conjecture

BRILL-NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

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Chinese Annals of Mathematics,Series B 2002年第4期23卷 531-538页

作者： TAN XIAOJIANGschool of mathematical sciences, peking University,{4., china school of mathematical sciences Peking University Beijing 100871 China

Let X be an arbitrary smooth irreducible complex projective curve, E (?)X a rank two vector bundle generated by its sections. The author first represents ?as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H?X, [D1] |D2) is a. collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using / and the Brill-Noether matrix of D1+D2, the author constructs a 2g × d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?) Im{H?X, E) (?) H0(X,[D2]) (?) H0(X, [D2]|D2)} From this and H0(X,E) = H0(X,[D1]) (?)Im{H0(X,E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.

关键词： Brill-Noether matrix Vector bundle Effective divisor

On multiply of sections of line bundles on compact Riemann surfaces

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Science china Mathematics 2008年第4期51卷 721-727页

作者： TAN XiaoJiang LMAM and school of mathematical sciences peking University {4. 100871 china 1. LMAM and school of mathematical sciences Peking University Beijing 100871 China

In this paper,we give some conditions on the surjective of multiply maps H^0(R,L)×H^0(R,K)→H^0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.

关键词： compact Riemann surfaces line bundles 14.20 14.24

Wavelets associated with Hankel transform and their Weyl transforms

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Science china Mathematics 2004年第3期47卷 393-400页

作者： PENG Lizhong MA Ruiqin LMAM, school of mathematical sciences, peking University,{4., china 1. LMAM School of Mathematical Sciences Peking University 100871 Beijing China

The Hankel transform is an important transform. In this paper, westudy the wavelets associated with the Hankel transform, thendefine the Weyl transform of the wavelets. We give criteria of itsboundedness and compactne... 详细信息

关键词： Bessel function, Hankel transform, wavelets, Weyltransform, boundedness, compactness.

Oblique derivative problem for general Chaplygin-Rassias equations

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Science china Mathematics 2008年第1期51卷 5-36页

作者： WEN GuoChun LMAM, school of mathematical sciences, peking University,{4., china LMAM School of Mathematical Sciences Peking University Beijing 100871 China

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4. where Γ(? {y > 0}) ∈ C μ 2 (0 4.are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y 4.4.\frac{{\partial u}}{{\partial \nu }} = \frac4.{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4., \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4.respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4.as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H

关键词： oblique derivative problem equations of mixed type nonsmooth degenerate line 35J70 35L80 35N99

Global well-posedness for the 2D critical dissipative quasi-geostrophic equation

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Science china Mathematics 2007年第4期50卷 485-494页

作者： Zhi-fei ZHANG school of mathematical sciences, peking University,{4., china school of mathematical sciences Peking University Beijing 100871 China

We study the initial value problem for the 2D critical dissipative quasi-geostrophic equation. We prove the global existence for small data in the scale invariant Besov spaces ? p,1 2/p , 1 ? p ? ∞. In particular, for p = ∞, our result does not impose any regularity on the initial data. Our proofs are based on an exponential decay estimate of the semigroup $e^{{ - t\kappa ( - \Delta )}^{\alpha}} $ and the use of space-time Besov spaces.

关键词： quasi-geostrophic equation Besov space Littlewood-Paley decomposition well-posedness 35Q35

Maslov-type index and brake orbits in nonlinear Hamiltonian systems

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Science china Mathematics 2007年第6期50卷 761-772页

作者： Duan-zhi ZHANG school of mathematical sciences, peking University,{4., china 1. school of mathematical sciences Peking University Beijing 100871 China

In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian s... 详细信息

关键词： brake orbit Maslov-type index relative Morse index dual variational method 58E05 70H05 34.25