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The Newton filtration and d-determination of bifurcation problems related to C^0 contact equivalence

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Science China Mathematics 2006年第5期49卷 660-668页

作者： SU Dan ZHANG Dunmu School of Mathematics and Statistics Wuhan University Wuhan 430072 China School of Mathematics and Statistics Wuhan University Wuhan 430072 China

In this paper, from the Newton filtration's point of view, we construct the singular Riemannian metric and use the method in singular theory to study the bifurcation problems, and give the sufficient condition of d-determination of bifurcation problems with respect to C0 contact equivalence. The special cases of the main result in this paper are the results of Sun Weizhi and Zou Jiancheng.

关键词： map germ, Newton filtration, singular Riemannian metric, C0 contact equivalence.

The Einstein-Kahler metrics on Cartan-Hartogs domain of the first type

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Science China Mathematics 2004年第2期47卷 220-235页

作者： WANG An Department of Mathematics Capital Normal University Beijing 100037 China

Let YI be the Cartan-Hartogs domain of the first type. We give the generating function of the Einstein-Kahler metrics on YI, the holomorphic sectional curvature of the invariant Einstein-Kahler metrics on YI. The comparison theorem of complete Einstein-Kahler metric and Kobayashi metric on YI is provided for some cases. For the non-homogeneous domain YI, when K = mn+1/m+n, m ＞ 1, the explicit forms of the complete Einstein-Kahler metrics are obtained.

关键词： Cartan-Hartogs domain, Einstein-Kahler metric, holomorphic sectional curvature, generating function.

Classification of hypersurfaces with parallel Mobius second fundamental form in S^(n+1)

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Science China Mathematics 2004年第3期47卷 417-430页

作者： HU Zejun LI Haizhong Department of Mathematics, Zhengzhou University Zhengzhou 450052, China Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Department of Mathematics Zhengzhou University Zhengzhou China Department of Mathematical Sciences Tsinghua University Beijing China

Let Mn(n≥2) be an immersed umbilic-free hypersurface in the(n+1)-dimensional unit sphere Sn+1. Then Mn is associated witha so-called M(o)bius metric g, and a M(o)bius second fundamental form Bwhich are invariants of Mn under the M(o)bius transformation groupof Sn+1.In this paper, we classify all umbilic-free hypersurfaces withparallel M(o)bius second fundamental form.

关键词： parallel Mobius second fundamental form,hypersurface, Mobius metric, Mobius equivalence.

Some constructions of projectively flat Finsler metrics

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Science China Mathematics 2006年第5期49卷 703-714页

作者： MO Xiaohuan, SHEN Zhongmin & YANG Chunhong LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN 46202-3216, USA Department of Mathematics, Inner Mongolia University, Hohhot 010021, China LMAM School of Mathematical Sciences Peking University Beijing China Department of Mathematical Sciences Indiana University-Purdue University Indianapolis USA Department of Mathematics Inner Mongolia University Hohhot China

In this paper, we find some solutions to a system of partial differential equations that characterize the projectively flat Finsler metrics. Further, we discover that some of these metrics actually have the zero flag ... 详细信息

关键词： Randers metric, (α,β)-metric, Finsler metric, projectively flat metric, Scurvature.

Geometric aspects of the moduli space of Riemann surfaces

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Science China Mathematics 2005年第Z1期48卷 97-122页

作者： Shing-Tung Yau Department of Mathematics Harvard University CambridgeMA 02138USA

We describe some recent progress in the study of moduli space of Riemann surfaces in this survey paper. New complete Kahler metrics were introduced on the moduli space and Teichmuller space. Their curvature properties and asymptotic behavior were studied in details. These natural metrics served as bridges to connect all the known canonical metrics, especially the Kahler-Einstein metric. We showed that all the known complete metrics on the moduli space are equivalent and have Poincare type growth. Furthermore,the Kahler-Einstein metric has strongly bounded geometry. This also implied that the logarithm cotangent bundle of the moduli space is stable in the sense of Mumford.

关键词： moduli space, Teichmuller space, metric, curvature.

Orthogonal separable Hamiltonian systems on T^2

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Science China Mathematics 2007年第12期50卷 1735-1747页

作者： Cheng CHEN Fei LIU Xiang ZHANG Department of Mathematics Shanghai Jiaotong University Shanghai 200240 China Department of Mathematics Shanghai Jiaotong University Shanghai 200240 China Department of Mathematics Shanghai Jiaotong University Shanghai 200240 China

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T2 for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T2 can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way.

关键词： Hamiltonian system, Liouvillian integrability, orthogonal separable, Riemannian metric, dynamics, entropy

Convergence of Fuzzy Set Sequences

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Journal of Donghua University(English Edition) 2002年第4期19卷 29-32页

作者：冯玉瑚 Department of Applied Mathematics Dong Hua UniversityShanghai200051

There are more than one mode of convergence with respect to the fuzzy set sequences. In this paper,common six modes of convergence and their relationships are discussed. These six modes are convergence in uniform metric D, convergence in separable metric Dp or D*p, 1 ≤ p ＜∞, convergence in level set, strong convergence in level set and weak convergence. Suitable counterexamples are given. The necessary and sufficient conditions of convergence in uniform metric D are described. Some comme nts on the convergence of LRfuzzy number sequences are represented.

关键词： Fuzzy set, Hausdorff metric, convergence relationship, LR-fuzzy number