郭瑞芝,劉 林
(湖南師范大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,湖南長(zhǎng)沙 410081)
相對(duì)映射芽通用開(kāi)折的唯一性和穩(wěn)定性
郭瑞芝,劉 林
(湖南師范大學(xué)數(shù)學(xué)與計(jì)算機(jī)科學(xué)學(xué)院,湖南長(zhǎng)沙 410081)
定義了相對(duì)映射芽及其相對(duì)開(kāi)折在左右等價(jià)下的穩(wěn)定性和無(wú)窮小穩(wěn)定性,證明了這兩種穩(wěn)定性是等價(jià)的.建立了相對(duì)映射芽開(kāi)折的一種與相對(duì)映射芽左右等價(jià)相容的等價(jià)關(guān)系,得到了同一軌道中相對(duì)映射芽的通用開(kāi)折在這種等價(jià)意義下是唯一的結(jié)論,給出了相對(duì)映射芽開(kāi)折穩(wěn)定的一個(gè)充分必要條件.
相對(duì)映射芽;相對(duì)開(kāi)折;穩(wěn)定性;唯一性
文獻(xiàn)[1]研究了映射芽的相對(duì)通用性,給出了映射芽的相對(duì)通用開(kāi)折定理;文獻(xiàn)[2-4]給出了接觸等價(jià)下相對(duì)映射芽的通用形變定理;文獻(xiàn)[5]研究了相對(duì)映射芽的強(qiáng)有限決定性,并給出了通用開(kāi)折定理;文獻(xiàn)[6-7]分別研究了函數(shù)芽的相對(duì)有限決定性和相對(duì)穩(wěn)定性;文獻(xiàn)[8]研究了含兩組狀態(tài)變量的等變分歧問(wèn)題在接觸等價(jià)下開(kāi)折的唯一性與穩(wěn)定性.本文受上述研究的啟發(fā),定義了相對(duì)映射芽及其相對(duì)開(kāi)折在左右等價(jià)下的穩(wěn)定性與無(wú)窮小穩(wěn)定性,證明了這兩種穩(wěn)定性是等價(jià)的;建立相對(duì)映射芽開(kāi)折的一種等價(jià)關(guān)系,這種等價(jià)關(guān)系與相對(duì)映射芽的等價(jià)關(guān)系是相容的,得到了同一軌道中相對(duì)映射芽的通用開(kāi)折在這種等價(jià)意義下是唯一的,并給出了相對(duì)映射芽開(kāi)折穩(wěn)定的一個(gè)充分必要條件.
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Uniqueness and stability on versal unfoldings of relative map germs
GUO Rui-zhi,LIU Lin
(College of Mathematics and Computer Science,Hunan Normal University,Changsha 410081,China)
Stability and infinitesimal stability on relative map germs and their unfoldings are defined under the left right equivalence.The equivalence of these two kinds of stability are proved.An equivalence relation on unfoldings of relative map germs are constructed,which is compatible with the left right equivalence of relative map germs.It is obtained that the versal unfolding of the relative map germs in the same orbit is unique.A sufficiency and necesary condition of an unfolding of a relative map germ to be stable is given.
relative map germs;relative unfoldings;stability;uniqueness
O 192;O 177.91
110·3155
A
1000-1832(2011)03-0032-07
2010-11-05
國(guó)家自然科學(xué)基金資助項(xiàng)目(10971060).
郭瑞芝(1962—),女,博士,教授,主要從事奇點(diǎn)和分歧理論研究.
陶 理)