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The numbers of periodic orbits hidden at fixed points of 2-dimensional holomorphic mappings

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Science China Mathematics 2010年第3期53卷 863-886页

作者： ZHANG GuangYuan department of mathematical sciences, tsinghua university, beijing 100084. China 1. department of mathematical sciences Tsinghua University Beijing 100084 China

Let △n be the ball |x| < 1 in the complex vector space C n , let f :△n→ C n be a holomorphic mapping and let M be a positive integer. Assume that the origin 0 = (0, . . . , 0) is an isolated fixed point of both f and the M-th iteration f M of f. Then the (local) Dold index P M (f, 0) at the origin is well defined, which can be interpreted to be the number of virtual periodic points of period M of f hidden at the origin: any holomorphic mapping f 1 :△n→ C n sufficiently close to f has exactly P M (f, 0) distinct periodic points of period M near the origin, provided that all the fixed points of f M 1 near the origin are simple. Therefore, the number O M (f, 0) = P M (f, 0)/M can be understood to be the number of virtual periodic orbits of period M hidden at the fixed point. According to the works of Shub-Sullivan and Chow-Mallet-Paret-Yorke, a necessary condition so that there exists at least one virtual periodic orbit of period M hidden at the fixed point, i.e., O M (f, 0)≥1, is that the linear part of f at the origin has a periodic point of period M. It is proved by the author recently that the converse holds true. In this paper, we will study the condition for the linear part of f at 0 so that O M (f, 0)≥2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that O M (f, 0)≥2 for all holomorphic mappings f :△2 → C 2 such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f M .

关键词： complex dynamical systems periodic points fixed point indices

On mean curvatures in submanifolds geometry

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Science China Mathematics 2008年第6期51卷 1127-1134页

作者： GE JianQuan department of mathematical sciences,tsinghua university,beijing {4.China department of mathematical sciences Tsinghua University Beijing 100084 China

By using moving frame theory,first we introduce 2p-th mean curvatures and(2p+1)-th mean curvature vector fields for a *** then give an integral expression of them that characterizes them as mean values of symmetric functions of principle *** we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.

关键词： moving frame mean curvature tube 53A07 53C4.

On fixed-points and singular values of transcendental meromorphic functions

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Science China Mathematics 2010年第3期53卷 887-894页

作者： ZHENG JianHua department of mathematical sciences, tsinghua university, beijing 100084. China 1. department of mathematical sciences Tsinghua University Beijing 100084 China

In this paper we make a further discussion of a relationship between the number of fixed-points and distribution of singular values along the round annuli centered at the origin of a transcendental meromorphic function. To attain our purpose we first establish a fundamental inequality for the modulus of derivative of a holomorphic covering mapping whose image is an annulus by virtue of the hyperbolic metric. The inequality is of independent significance. We make a simple survey on some domain constants for hyperbolic domains.

关键词： meromorphic functions fixed points singular points hyperbolic metric covering mapping

A Hierarchy of Multidimensional Hénon-Heiles Systems

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Acta Mathematica Sinica,English Series 2000年第3期16卷 527-534页

作者： Yunbo Zeng department of mathematical sciences,tsinghua university,beijing {4.P.R.China department of mathematical sciences Tsinghua University Bejing 100084 P. R. China

A hierarchy of multidimensional Hénon-Heiles(M-H-H)systems are constructed via the x-and t_n-higher-order-constrained flows of KdV *** Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV *** using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.

关键词： Higher-order constrained flow Multidimensional Hénon-Heiles systems Lax representation Jacobi inversion problem

Upper bound and formula for class numbers of abelian function fields

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Progress in Natural Science:Materials International 2006年第3期16卷 321-323页

作者： MA Lianrong and ZHANG Xianke (department of mathematical sciences, tsinghua university, beijing 100084. China) department of mathematical sciences Tsinghua University Beijing 100084 China

For any abelian function field K (i.e. any subfield of a cyclotomic function field L=k (∧P) over the rational function field k ) with conductor being an irreducible polynomial over a finite field of odd characteristic, by studying the Carlitz-module structure and the character group of K, an explicit upper bound and a calculating formula of the relative divisor class number h- (K) of K are given. Our calculated results of K develop Rosen’s recent results of L .

关键词： Carlitz module cyclotomic function field class number character abelian function field.

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

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Acta Mathematica Sinica,English Series 2002年第3期18卷 539-548页

作者： De Rong QIU Center for Advanced Study, tsinghua university, beijing 100084. P. R. China Xian Ke ZHANG department of mathematical sciences, tsinghua university, beijing 100084. P. R. China Center for Advanced Study Tsinghua University Beijing P. R. China department of mathematical sciences Tsinghua University Beijing P. R. China

We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.

关键词： Elliptic curve Rational point Torsion group Mordell group Parameterization

Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds

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Science China Mathematics 2005年第5期48卷 618-648页

作者： LI Yuxiang department of mathematical sciences, tsinghua university, beijing 100084. China 1. department of mathematical sciences Tsinghua University 100084 Beijing China

Let (M,g) be a compact Riemannian manifold without boundary, and (N,g) a compact Riemannian manifold with boundary. We will prove in this paper that the and can be attained. Our proof uses the blow-up analysis.

关键词： Moser-Trudinger inequality extremal function n-Lapalace Green function.

QUASI-STATIONARY DISTRIBUTIONS FOR THE RADIAL ORNSTEIN-UHLENBECK PROCESSES

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Acta Mathematica Scientia 2008年第3期28卷 513-522页

作者：叶俊 department of mathematical sciences Tsinghua University

The purpose of this article is to obtain the quasi-stationary distributions of the δ（δ 〈 2）-dimensional radial Ornstein-Uhlenbeck process with parameter -λ by using the methods of Martinez and San Martin （2001）. It is described that the law of this process conditioned on first hitting 0 is just the probability measure induced by a （4.- δ）- dimensional radial Ornstein-Uhlenbeck process with parameter -λ. Moreover, it is shown that the law of the conditioned process associated with the left eigenfunction of the process conditioned on first hitting 0 is induced by a one-parameter diffusion.

关键词： Radial Ornstein-Uhlenbeck process quasi-stationary distribution quasiinvariant

Heat kernel estimates on effective resistance metric spaces

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Progress in Natural Science:Materials International 2007年第7期17卷 775-783页

作者： Hu Jiaxin Ji Yuan Wen Zhiying (department of mathematical sciences,tsinghua university,beijing {4.China) [a]department of mathematical sciences Tsinghua University Beijing China

We obtain two-sided estimates of heat kernels on effective-resistance metric spaces by using an analytic *** applications,such as on post-critically finite self-similar sets and on products of effective resistance spa... 详细信息

关键词： Heat kernel effective resistance Dirichlet form Green’s function