相对函数芽的有限决定性与多参数等变分歧问题开折的通用性及稳定性
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摘要
奇点理论源于上世纪30年代H.M.Morse的临界点理论,Whitney于1955年的关于把平面到平面的映射的奇点的工作使得其成为一个独立的分支,同时R.Thom、J.N.Mather、V I.Arnold和M.Golubitsky等人在此方面都作了很重要的贡献,国内的研究者在李培信研究员的带领下主要有李养成、张敦穆、张国滨、孙伟志、余建明、姜广峰、邹建成、裴东河等以及他们的学生,其成果进一步充实了奇点理论的内涵。
本文研究奇点理论中的几个问题,其中主要讨论相对函数芽及其形变的有限决定性与多参数等变分歧问题开折的通用性及稳定性。
关于相对函数芽与映射芽的研究从1978年P.F.S.Porto的函数芽的相对决定性开始,后来有了很多的讨论,如R.Bulaiich、J.A.Gomez和L.Kushner于1987年在《Bolleaino U M I》上发表的相对映射芽的相对通用性,给出了相对通用性的代数特征和相对Malgrange预备定理,1992年L.Kushner的关于一般代数集芽上相对有限决定性的讨论以及他与B.T.Leme于2000年的有限相对决定性与相对稳定性,2003年唐铁桥、李养成的光滑函数芽的Rr(S;n)-决定性,V.Grandjean于2004年的发表在伦敦数学学会期刊的关于满足相对Lojasiewicz条件有横截孤立奇点的光滑函数芽的无限相对决定性的文章,2004年孙伟志、陈亮、裴东河的相对映射芽的强A_(S,T)有限决定性与通用开折,2004年申海燕、李养成的相对映射芽的相对A_决定性,2006年李养成、梁琼初的接触等价下的相对映射芽的通用形变,2007年孙伟志、高峰、裴东河的K等价下相对映射芽的通用形变,2006年张中峰的硕士论文中也有关于R(S;n)与Rr(S;n)群下的有限决定性的进一步的讨论。
关于等变分歧问题方面,国内外的相关工作特别多,主要讨论分歧问题的开折、分歧问题及其开折的稳定性和分歧问题的识别与分类。当状态空间与靶空间相同时,M.Golubitsky,I.Stewart和D.GSchaeffer引入奇点理论来研究等变分歧问题并给出了通用开折定理。其后,许多学者对此继续研究。利用奇点理论的技巧,A.L.Lavassani和Y C.Lu研究了等变分歧问题及其开折的稳定性。特别在国内,李养成教授带领他的研究生建立了各种形式的通用开折定理,张敦穆教授与其博士刘恒兴讨论了г-等变(r,s)-等价关系和г-等变分歧问题开折的(r,s)-无穷小稳定等等。
本文基于奇点理论,研究了:(1)关于两等价群作用下更一般情形的等变分歧问题的通用开折;(2)含两组状态变量且分歧参数也带有对称性的等变分歧问题的无穷小稳定开折;(3)相对函数芽的有限决定性与决定性范围;(4)相对函数芽形变的有限决定性与决定性范围;(5)对A.L.Lavassani与Y C.Lu用DA代数系统讨论的接触等价下光滑映射芽的有限决定性,本文用避开DA代数系统的方式给出了分析。这里的前四部分内容分别发表在《数学物理学报》(英文版,SCI源期刊)、《湘潭大学自然科学学报》、《应用数学》和《湖南大学自然科学学报》上,最后一个内容也与崔登兰博士以及我的导师进行了充分的讨论。
本文内容大致如下:
第一章在关于两等价群作用下更一般情形的等变分歧问题的通用开折。本章是在前人对开折理论研究的基础上的更一般情形下的讨论,其状态变量分成了两组,分歧参数的对称性也考虑了进来,主要讨论等变左右等价群作用下的通用开折定理以及等变左右等价群下的通用开折与等变接触等价下的通用形变之间的关系。
第二章含两组状态变量且参数带有对称性的等变分歧问题在等变左右等价下的无穷小稳定开折。分歧问题开折的通用性与稳定性是分歧理论研究中的一个重要内容。本章和第一章一样,也是在更一般情形下的讨论,对这类等变分歧问题的开折讨论在等变左右等价下的无穷小稳定性,所得的主要结果将类似文献的结果作为了其特殊情形。
第三章讨论相对函数芽,本章将A.du Plessis的决定性范围与P.F.S.Porto的相对函数芽的有限决定性结合在一起讨论得到了几个新的结果,如推论3.1.8给出了k-右决定的充要条件,由此可以判断右等价下的决定性阶数。
第四章讨论相对函数芽形变。本章是在第三章基础上的进一步分析,对于一类函数芽形变给出了其决定性范围,并分析了其相对稳定性的充分条件。
第五章研究在接触等价群作用下的分歧问题,通过对A.L.Lavassani与Y.C.Lu的一篇文章仔细分析,考虑到DA代数系统需要的代数基础比较多,本章力求避开DA代数系统,来分析文献[47]中与有限决定性有关的引理6.2.2的证明。首先证明出几个引理,然后通过细致分析,运用这些引理给出了引理6.2.2的另一证明。考虑到其证明过程有其独特之处,因而作为一章写在最后。
Singularity theory came from H. M. Morse's critical point theory in the 1930s; in 1955 Whitney wrote the paper "On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane", this work makes it an independent branch, R. Thom, J. N. Mather, V. I. Arnold and M. Golubitsky and others made very important contributions in this regard. In our country under the leadership of Li Peixin researcher there are many scholars such as Li Yangcheng, Zhang Dunmu, Zhang Guobin, SunWeizhi, Zou Jiancheng, YuJianming, Jiang Guangfeng, Pei Donghe, as well as their students, their results enriched the singularity theory.
In this paper several issues related to singularity theory are discussed, and the relative finite determinacy of function germs and the versality, stability of multiparameter equivariant bifurcation problems are two main issues of them.
In 1978 P. F. S. Porto studied the relative function germs, which is the first paper to discuss relative function germs and relative map germs. From then on, there was a lot of discussion related to this issue, such as, in 1987 R. Bulajich, J. A. Gomez and L. Kushner's work "relative versality for map germs" published by the Bollettino UMI, in 1992 L. Kushner's paper about the relative finite determinacy on the algebraic set germs and his work in 2000 of finite relative determination and relative stability, in 2003 Tang Tieqiao and Li Yangcheng's work "R_r(S;n)-determinancy of function germs", in 2004 V. Grandjean's article "Infinite relative determinacy of smooth function germs with transverse isolated singularity and relative Lojasiewicz conditions" published by the journal of London Mathematical Society, then in 2004 Sun Weizhi, Chen Liang and Pei Donghe's paper "Strong Relative A_(S, T) Finite Determinacy of Map Germs and Relation with Relative Versal Unfolding" , in 2004 Shen Haiyan and Li Yangcheng's work "Relative A- determinacy of Relative Smooth map germs", in 2004 Li Yangcheng and Liang Qiongchu's paper "Versal Deformations of Relative Smooth Map-germs with Respect to Contact Equivalence", in 2007 Sun Weizhi, Gao Feng and Pei Donghe's work "versal deformations of relative map germs with K equivalence", and in 2006 Zhang Zhongfeng discussed the finite determicacy of group R(S;n) and R_r(S;n) in his Master thesis.
There are a lot of related works with equivariant bifurcation problems at home and abroad, mainly to discuss the stability, unfoldings of equivariant bifurcation problems, the stability of equivariant bifurcation problems and their unfoldings, classification and recognition of them. When its state space is the same as the target space, the theoretical machinery from singularity theory are introduced by M. Golubitsky, I. Stewart and D. G. Schaeffer to study the equivariant bifurcation problems, and they got the equivariant universal unfolding theorem. Since then, many scholars continue to study this. Applying related methods and techniques in the theory of singularities of smooth map germs, A. L. Lavassani and Y. C. Lu studied the unfolding and stability of equivariant bifurcation problem. Particularly in our country, led by Professor Li Yangcheng, his students gave various versions of versality theorem. Professor Zhang Dunmu and his doctor Liu Hengxing discussed theΓ-equivariant (s , t)-equivalence relation andΓ-equivariant infinitesimally (r, s)-stability ofΓ-equivariant bifurcation problem.
Based on singularity theory this paper discussed several problems as follows: (1) Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups; (2) Infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group; (3)Relative finite determinacy of smooth function germs; (4)Relative determinacy of deformations of function germs under the action of group; (5) the determinacy which discussed using DA algebra systems by A. L. Lavassani and Y. C. Lu, is analyzed without DA algebra systems in this paper. The first four parts are published in Acta Mathematica Scientia (English version, SCI), Natural Science Journal of Xiangtan University, Mathematica Applicata and Journal of Hunan University (Natural Sciences) respectively, and the last part is discussed carefully with doctor Cui Denglan and my tutor.
The detailed contexts are as follows:
Chapter 1 discusses versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups. For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry, the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs. The corresponding results in the reference can be considered as its special cases. A relationship between the versal unfolding w. r. t. left-right equivalence and the versal deformation w. r. t. contact equivalence is established.
Chapter 2 discusses infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group. Applying the related methods and techniques in the singularity theory of smooth maps, infinitesimal stability of unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry is characterized. And the existence of infinitesimal stable unfolding of such a class of bifurcation problems is discussed.
Chapter 3 studies relative finite determinacy of smooth function germs.Based on the works of P. F. S. Porto and A. du Plessis, it deals with the relative right determinacy of smooth function germs of n variables, which makes the same values on an algebraic set germ in R~n. In this chapter the criteria on range of determinacy of such function germs are obtained. Some results in the chapter generalize or improve the corresponding ones in the reference.
Chapter 4 discusses relative determinacy of deformations of function germs under the action of group. Based on the work of P. F. S. Porto and Andrew du Plessis, it deals with the relative right determinacy of deformations of function germs which are under the group R(S; n). In this chapter criteria on range of determinacy of deformations and a sufficient condition of the stability of deformations are obtained.
Chapter 5 discusses the finite determinacy under contact equivalent group without DA algebra systems. Firstly, we studied carefully an article which is written by A. L. Lavassani and Y. C. Lu, and then considered that we need many basic algebra concepts to understand DA algebra systems, we tried to give another method to prove the finite determinacy without DA-algebra systems, lastly we obtained another method to prove lemma 6.2.2 in their paper, which means that we can prove the finite determinacy without DA-algebra systems. Considered that there has a special method in the proof, we put this in the last charpter.
本文研究奇点理论中的几个问题,其中主要讨论相对函数芽及其形变的有限决定性与多参数等变分歧问题开折的通用性及稳定性。
关于相对函数芽与映射芽的研究从1978年P.F.S.Porto的函数芽的相对决定性开始,后来有了很多的讨论,如R.Bulaiich、J.A.Gomez和L.Kushner于1987年在《Bolleaino U M I》上发表的相对映射芽的相对通用性,给出了相对通用性的代数特征和相对Malgrange预备定理,1992年L.Kushner的关于一般代数集芽上相对有限决定性的讨论以及他与B.T.Leme于2000年的有限相对决定性与相对稳定性,2003年唐铁桥、李养成的光滑函数芽的Rr(S;n)-决定性,V.Grandjean于2004年的发表在伦敦数学学会期刊的关于满足相对Lojasiewicz条件有横截孤立奇点的光滑函数芽的无限相对决定性的文章,2004年孙伟志、陈亮、裴东河的相对映射芽的强A_(S,T)有限决定性与通用开折,2004年申海燕、李养成的相对映射芽的相对A_决定性,2006年李养成、梁琼初的接触等价下的相对映射芽的通用形变,2007年孙伟志、高峰、裴东河的K等价下相对映射芽的通用形变,2006年张中峰的硕士论文中也有关于R(S;n)与Rr(S;n)群下的有限决定性的进一步的讨论。
关于等变分歧问题方面,国内外的相关工作特别多,主要讨论分歧问题的开折、分歧问题及其开折的稳定性和分歧问题的识别与分类。当状态空间与靶空间相同时,M.Golubitsky,I.Stewart和D.GSchaeffer引入奇点理论来研究等变分歧问题并给出了通用开折定理。其后,许多学者对此继续研究。利用奇点理论的技巧,A.L.Lavassani和Y C.Lu研究了等变分歧问题及其开折的稳定性。特别在国内,李养成教授带领他的研究生建立了各种形式的通用开折定理,张敦穆教授与其博士刘恒兴讨论了г-等变(r,s)-等价关系和г-等变分歧问题开折的(r,s)-无穷小稳定等等。
本文基于奇点理论,研究了:(1)关于两等价群作用下更一般情形的等变分歧问题的通用开折;(2)含两组状态变量且分歧参数也带有对称性的等变分歧问题的无穷小稳定开折;(3)相对函数芽的有限决定性与决定性范围;(4)相对函数芽形变的有限决定性与决定性范围;(5)对A.L.Lavassani与Y C.Lu用DA代数系统讨论的接触等价下光滑映射芽的有限决定性,本文用避开DA代数系统的方式给出了分析。这里的前四部分内容分别发表在《数学物理学报》(英文版,SCI源期刊)、《湘潭大学自然科学学报》、《应用数学》和《湖南大学自然科学学报》上,最后一个内容也与崔登兰博士以及我的导师进行了充分的讨论。
本文内容大致如下:
第一章在关于两等价群作用下更一般情形的等变分歧问题的通用开折。本章是在前人对开折理论研究的基础上的更一般情形下的讨论,其状态变量分成了两组,分歧参数的对称性也考虑了进来,主要讨论等变左右等价群作用下的通用开折定理以及等变左右等价群下的通用开折与等变接触等价下的通用形变之间的关系。
第二章含两组状态变量且参数带有对称性的等变分歧问题在等变左右等价下的无穷小稳定开折。分歧问题开折的通用性与稳定性是分歧理论研究中的一个重要内容。本章和第一章一样,也是在更一般情形下的讨论,对这类等变分歧问题的开折讨论在等变左右等价下的无穷小稳定性,所得的主要结果将类似文献的结果作为了其特殊情形。
第三章讨论相对函数芽,本章将A.du Plessis的决定性范围与P.F.S.Porto的相对函数芽的有限决定性结合在一起讨论得到了几个新的结果,如推论3.1.8给出了k-右决定的充要条件,由此可以判断右等价下的决定性阶数。
第四章讨论相对函数芽形变。本章是在第三章基础上的进一步分析,对于一类函数芽形变给出了其决定性范围,并分析了其相对稳定性的充分条件。
第五章研究在接触等价群作用下的分歧问题,通过对A.L.Lavassani与Y.C.Lu的一篇文章仔细分析,考虑到DA代数系统需要的代数基础比较多,本章力求避开DA代数系统,来分析文献[47]中与有限决定性有关的引理6.2.2的证明。首先证明出几个引理,然后通过细致分析,运用这些引理给出了引理6.2.2的另一证明。考虑到其证明过程有其独特之处,因而作为一章写在最后。
Singularity theory came from H. M. Morse's critical point theory in the 1930s; in 1955 Whitney wrote the paper "On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane", this work makes it an independent branch, R. Thom, J. N. Mather, V. I. Arnold and M. Golubitsky and others made very important contributions in this regard. In our country under the leadership of Li Peixin researcher there are many scholars such as Li Yangcheng, Zhang Dunmu, Zhang Guobin, SunWeizhi, Zou Jiancheng, YuJianming, Jiang Guangfeng, Pei Donghe, as well as their students, their results enriched the singularity theory.
In this paper several issues related to singularity theory are discussed, and the relative finite determinacy of function germs and the versality, stability of multiparameter equivariant bifurcation problems are two main issues of them.
In 1978 P. F. S. Porto studied the relative function germs, which is the first paper to discuss relative function germs and relative map germs. From then on, there was a lot of discussion related to this issue, such as, in 1987 R. Bulajich, J. A. Gomez and L. Kushner's work "relative versality for map germs" published by the Bollettino UMI, in 1992 L. Kushner's paper about the relative finite determinacy on the algebraic set germs and his work in 2000 of finite relative determination and relative stability, in 2003 Tang Tieqiao and Li Yangcheng's work "R_r(S;n)-determinancy of function germs", in 2004 V. Grandjean's article "Infinite relative determinacy of smooth function germs with transverse isolated singularity and relative Lojasiewicz conditions" published by the journal of London Mathematical Society, then in 2004 Sun Weizhi, Chen Liang and Pei Donghe's paper "Strong Relative A_(S, T) Finite Determinacy of Map Germs and Relation with Relative Versal Unfolding" , in 2004 Shen Haiyan and Li Yangcheng's work "Relative A- determinacy of Relative Smooth map germs", in 2004 Li Yangcheng and Liang Qiongchu's paper "Versal Deformations of Relative Smooth Map-germs with Respect to Contact Equivalence", in 2007 Sun Weizhi, Gao Feng and Pei Donghe's work "versal deformations of relative map germs with K equivalence", and in 2006 Zhang Zhongfeng discussed the finite determicacy of group R(S;n) and R_r(S;n) in his Master thesis.
There are a lot of related works with equivariant bifurcation problems at home and abroad, mainly to discuss the stability, unfoldings of equivariant bifurcation problems, the stability of equivariant bifurcation problems and their unfoldings, classification and recognition of them. When its state space is the same as the target space, the theoretical machinery from singularity theory are introduced by M. Golubitsky, I. Stewart and D. G. Schaeffer to study the equivariant bifurcation problems, and they got the equivariant universal unfolding theorem. Since then, many scholars continue to study this. Applying related methods and techniques in the theory of singularities of smooth map germs, A. L. Lavassani and Y. C. Lu studied the unfolding and stability of equivariant bifurcation problem. Particularly in our country, led by Professor Li Yangcheng, his students gave various versions of versality theorem. Professor Zhang Dunmu and his doctor Liu Hengxing discussed theΓ-equivariant (s , t)-equivalence relation andΓ-equivariant infinitesimally (r, s)-stability ofΓ-equivariant bifurcation problem.
Based on singularity theory this paper discussed several problems as follows: (1) Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups; (2) Infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group; (3)Relative finite determinacy of smooth function germs; (4)Relative determinacy of deformations of function germs under the action of group; (5) the determinacy which discussed using DA algebra systems by A. L. Lavassani and Y. C. Lu, is analyzed without DA algebra systems in this paper. The first four parts are published in Acta Mathematica Scientia (English version, SCI), Natural Science Journal of Xiangtan University, Mathematica Applicata and Journal of Hunan University (Natural Sciences) respectively, and the last part is discussed carefully with doctor Cui Denglan and my tutor.
The detailed contexts are as follows:
Chapter 1 discusses versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups. For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry, the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs. The corresponding results in the reference can be considered as its special cases. A relationship between the versal unfolding w. r. t. left-right equivalence and the versal deformation w. r. t. contact equivalence is established.
Chapter 2 discusses infinitesimally stable unfolding of a class of equivariant bifurcation problems under equivariant left-right equivalent group. Applying the related methods and techniques in the singularity theory of smooth maps, infinitesimal stability of unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry is characterized. And the existence of infinitesimal stable unfolding of such a class of bifurcation problems is discussed.
Chapter 3 studies relative finite determinacy of smooth function germs.Based on the works of P. F. S. Porto and A. du Plessis, it deals with the relative right determinacy of smooth function germs of n variables, which makes the same values on an algebraic set germ in R~n. In this chapter the criteria on range of determinacy of such function germs are obtained. Some results in the chapter generalize or improve the corresponding ones in the reference.
Chapter 4 discusses relative determinacy of deformations of function germs under the action of group. Based on the work of P. F. S. Porto and Andrew du Plessis, it deals with the relative right determinacy of deformations of function germs which are under the group R(S; n). In this chapter criteria on range of determinacy of deformations and a sufficient condition of the stability of deformations are obtained.
Chapter 5 discusses the finite determinacy under contact equivalent group without DA algebra systems. Firstly, we studied carefully an article which is written by A. L. Lavassani and Y. C. Lu, and then considered that we need many basic algebra concepts to understand DA algebra systems, we tried to give another method to prove the finite determinacy without DA-algebra systems, lastly we obtained another method to prove lemma 6.2.2 in their paper, which means that we can prove the finite determinacy without DA-algebra systems. Considered that there has a special method in the proof, we put this in the last charpter.
引文
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[3]Bruce J W,du Plessis A A,Wall C T C.Determinacy and unipotency.Invent Math[J],1987,88:521-554.
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[6]Cui Denglan,Li Yangcheng.On recognition of(D6,Z2)-equivariant bifurcation problems[J].Mathematica Applicata,2007,20(3):452-457.
[7]He Wei,Li Yangcheng.光滑函数芽的相对有限决定[J],应用数学,2008,21(2):277-282
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