Morse引理的一个推广

设E_n是在0∈Rⁿ的C^∞函数芽环,M是En中唯一的极大理想.如果f∈M²且其二阶Hessain是非退化的,则f同构于它的二阶Hessain,这就是著名的Morse引理.本文将讨论两个变元的C^∞函数芽,得到:(1)若f∈M³?E_xy,且其三阶Hessain是非退化的,则f同构于它的三阶Hessain.(2)若f∈M⁴?E_xy  

函数芽的有限R_r~*(S;n)-决定性

光滑映射芽的有限决定性是奇点理论中一个重要专题 .对函数芽的有限决定性问题 ,主要是在右等价群及其一些子群作用下来讨论的 .本文在 [1]和 [4 ]的基础上讨论函数芽在右等价群的正规子群 R*n (S;n)作用下的有限决定性 ,并组出函数芽有限 R*r (S;n) -决定的一个充分必要条件 .  

Existence of Moduli for Bi-Lipschitz Equivalence of Analytic Functions

We show that the bi-Lipschitz equivalence of analytic function germs (², 0)(, 0) admits continuous moduli. More precisely, we propose an invariant of the bi-Lipschitz equivalence of such germs that varies continuously in many analytic families f _t : (², 0)(, 0). For a single germ f the invariant of f is given in terms of the leading coefficients of the asymptotic expansions of f along the branches of generic polar curve of f.  

Seeking invariants for blow-analytic equivalence

We introduce some blow-analytic invariants of real analytic function-germsand discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.  

"Whitney引理的推广及应用"一文中的错误

本文用一个十分简单的例子说明[1]对整体的Borel定理的证明是错误的.为此， 还须介绍函数芽和函数芽序列一致收敛的概念，并给出一个判定引理.  

V-BILIPSCHITZ DETERMINACY OF WEIGHTED HOMOGENEOUS ANALYTIC FUNCTION-GERMS ON WEIGHTED HOMOGENEOUS REAL ANALYTIC VARIETIES

In this article, we provide estimates for the degree of V bilipschitz determinacy of weighted homogeneous function germs defined on weighted homogeneous analytic variety V satisfying a convenient Lojasiewicz condition.The result gives an explicit order such that the geometrical structure of a weighted homogeneous polynomial function germs is preserved after higher order perturbations.  

两类二元函数芽的一个共同性质及其应用

本文主要研究二元C^∞函数芽环中函数芽的性质问题.利用Mather有限决定性定理和C^∞函数的右等价关系，获得了带有任意4次至k次齐次多项式p_i（x，y），q_i（x，y）（i=4，5，…，k）的两类函数芽f₁=x²y+∑_i=4^kp_i（x，y），f₂=xy²+∑_i=4^kq_i（x，y）（k ≥ 5）的一个共同性质：若M_k²⊂M₂J（f_j）（j=1，2）且f₁，f₂的轨道切空间的余维分布均为c_i=1（i=4，5，…，k-1），则对这里的i，p_i（x，y）中xy^i-1，yⁱ的系数和q_i（x，y）中x^i-1y，xⁱ的系数均为零.最后，利用该性质，给出了f₁，f₂和一类余维数为7的二元函数芽的标准形式.  

On the Multiplicity of a C^∞-differentiable Function-germ

Risler ＆ Trotman in 1997 proved that the multiplicity of an analytic function germ is left-right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman ＆ Trotman, we generalize the work of them to prove that the multiplicity of a C^∞ differentiable function germ is also left-right lipschitz invariant.