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Ritt's theorem and the Heins map in hyperbolic complex manifolds

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Science China Mathematics 2005年第Z1期48卷 238-243页

作者： Marco Abate Filippo Bracci Dipartimento di Matematica Università di Pisa Largo Pontecorvo 556127 PisaItalyDipartimento di Matematica Università di Roma "Tor Vergata"Via della Ricerca Scientifica00133 RomaItaly

Let X be a Kobayashi hyperbolic complex manifold, and assume that X does not contain compact complex submanifolds of positive dimension (e.g., X Stein). We shall prove the following generalization of Ritt's theorem: every holomorphic self-map f: X →X such that f(X) is relatively compact in X has a unique fixed point τ(f) ∈ X, which is attracting. Furthermore, we shall prove that τ(f) depends holomorphically on f in a suitable sense, generalizing results by Heins, Joseph-Kwack and the second author.

关键词： holomorphic self-map, fixed point, Wolff point, Ritt's theorem, Heins map, Stein manifold.

Maschke-type theorem and Morita context over weak Hopf algebras

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Science China Mathematics 2006年第5期49卷 587-598页

作者： ZHANG Liangyun College of Science, Nanjing Agricultural University, Nanjing 210095, China Department of Mathematics, Nanjing University, Nanjing 210008, China College of Science Nanjing Agricultural University Nanjing China Department of Mathematics Nanjing University Nanjing China

This paper gives a Maschke-type theorem over semisimple weak Hopf algebras,extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.

关键词： weak Hopf algebra, smash product, Maschke-type theorem, Morita context.

Stone's representation theorem in fuzzy topology

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Science China Mathematics 2003年第6期46卷 775-788页

作者：刘应明张德学 Department of Mathematics Sichuan University Chengdu 610064 China Correspondence should be addressed to Liu Yingming

In this paper, a complete solution to the problem of Stone's repesentation theorem in fuzzy topology is given for a class of completely distributive lattices. Precisely, it is proved that if L is a frame such that 0 ∈ L is a prime or 1 ∈ L is a coprime, then the category of distributive lattices is dually equivalent to the category of coherent L-locales and that if L is moreover completely distributive, then the category of distributive lattices is dually equivalent to the category of coherent stratified L-topological spaces.

关键词： distributive lattice, L-topological space, Stone's representation theorem, frame.

A note on the uniqueness theorem of meromorphic mappings

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Science China Mathematics 2006年第3期49卷 360-365页

作者： YAN Qiming CHEN Zhihua Department of Mathematics Tongji University Shanghai 200092 China Department of Mathematics Tongji University Shanghai 200092 China

In this article, some uniqueness theorems of meromorphic mappings are proved.

关键词： value distribution theory, uniqueness theorem, meromorphic mappings.

On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space

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Science China Mathematics 2002年第12期45卷 1538-1547页

作者：张文俊刘太顺 Department of Mathematics Shenzhen University Normal College Shenzhen 518060 China Department of Mathematics Univ. of Sci. and Technol. of China Hefei 230026 China

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ?n, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.

关键词： Banach space, quasi-convex mapping, growth theorem, covering theorem.

Boundary value problems of discrete generalized Emden-Fowler equation

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Science China Mathematics 2006年第10期49卷 1303-1314页

作者： YU Jianshe GUO Zhiming College of Mathematics and Information Sciences Guangzhou University Guangzhou 510006 China College of Mathematics and Information Sciences Guangzhou University Guangzhou 510006 China

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.

关键词： discrete generalized Emden-Fowler equation, boundary value problem, Saddle Point theorem, discrete variational theory.

Splitting and cone avoidance in the D.C.E. degrees

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Science China Mathematics 2002年第9期45卷 1135-1146页

作者：李昂生 S.B.Cooper Department of Pure Mathematics School of Mathematics University of Leeds Leeds LS2 9JT United Kingdom Institute of Software Chinese Academy of Sciences Beijing 100080 China

It is shown that the Cooper splitting theorem for the n-c.e. degrees is not compatible with cone avoidance: For any n > 1, there exist n-c.e. degree a, c.e. degree b such that 0 < b < a and such that for any n-c.e. degrees x,y, if x ∨ y = a, then either b ≤ x or b ≤ y. This provides a new type of elementary difference between the classes of c.e. and d.c.e. degrees, implementable at lower levels of the high/low hierarchy.

关键词： Cooper splitting theorem, cone avoidance.

The Roper-Suffridge extension operator and classes of biholomorphic mappings Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

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Science China Mathematics 2006年第11期49卷 1539-1552页

作者： GRAHAM Ian KOHR Gabriela Department of Mathematics University of Toronto TorontoOntario M5S 2E4Canada Faculty of Mathematics and Computer Science Babes-Bolyai University 1 M. Kogǎlniceanu Str. 400084 Cluj-NapocaRomania

In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs)and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent.

关键词： convex mapping, starlike mapping, Roper-Suffridge operator, growth theorem, distortion theorem, covering theorem, Loewner chain, parametric representation,extreme points, linear invariant family.

New characterizations of Hajlasz-Sobolev spaces on metric spaces

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Science China Mathematics 2003年第5期46卷 675-689页

作者：杨大春 Department of Mathematics. Beijing Normal University Beijing 100875 China

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, including metric spaces and fractals. These Sobolev spaces include the well-known Hajtasz-Sobolev spaces as special models. The author establishes various characterizations of (sharp) maximal functions for these spaces. As applications, the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces. Moreover, some embedding theorems are also given.

关键词： Sobolev space, Lipschitz-type space, embedding theorem, maximal function, space ofhomogeneous type.