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The Versal Unfolding of Smooth Map-germs under a Subgroup of Left-Right Equivalence Group

Author: TangZuo
Tutor: GuoRuiZhi
School: Hunan Normal University
Course: Basic mathematics
Keywords: a Subgroup of Left-Right Equivalence Group Product Inte-gration Versal Unfolding
CLC: O189.3
Type: Master's thesis
Year: 2011
Downloads: 6
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Abstract


It is well know that a versal unfolding of a map-germ is one of the most important question for discussion in singularity, it is the core of the catastrophe theory.If F is a versal unfolding of a map-germ f, each unfolding subjected to a perturbation of f can be induced by F. In order to study the versal unfolding of a map-germ, we must, firstly, analysis the characteristics of its tangent space. But the tangent space is different under different equivalence groups.So it is active and meaningful to equivalence groups and study the corresponding versal unfolding.In the thesis we define a subgroup of a left-right equivalence group give some concepts on the equivalence of map-germs and the isomorphism of unfold-ings under this subgroup,we prove the triviality lemma,geometrical lemma and algebraic lcmma,we give a necessary and sufficient condition for an unfolding of a map-germ to be versal.This thesis consists of four chapters. It is arranged as follows.In chapter one, we mainly introduce the background of our research and some know results at home and abroad.In chapter two,we introduce some basic notions and some basic conceptsThen we define a subgroup of left-right equivalence group, an equivalence between two map-germs and an isomorphism between two unfoldings under this subgroup.Lastly we prove the triviality lemma with help of product integration theory,so we obtain the corresponding tangent space and the formly tangent space.In chapter three, we give some important lemmas including the geometri-cal lemma and the algebraic lemma under this subgroup,and study the versal unfolding theory under this subgroup. In chapter four, we give some corresponding corollarys.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Topology ( the situation in geometry ) > Analytic topology
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