By Allen Hatcher

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The given row converges in the class C ~ 2. r k ( d h ( x k) + N k I Xk 12r From these constructions points x k singularities dgk(Xk)) = n - ~ n ; t. it follows that the mapping of the type ~p+1, e 1' [3, ~ h + has at . In this case, every germ N(h + ~ , x k) is equisingular to the germ at the origim of coordinates of zero set of non-degenerate iable. Thus, N(h + ~ singular point, quadratic form of (p+l) var- , x k) is either a germ of hyperboloid in its or a germ of the single-point set (if the corresponding quadratic form is elliptic).

I. Regular Branching and Regular Deformations of Nonlinear Fredholm Equations, Deposited in VINiTI 53 on August 14, 1980, No. 3617-80. 33. , and Mallet-Paret, J. Applications of Generic Bifurcation, I- Arch. Rational Mech. Anal, 1975, vol. , 1976, vol. 62, N 2. 34. A. Bifurcation Theory for Fredholm Operators, Mem. Amer. Math. , 1976, N 174. 35. S. Bifurcation Problems for Nonlinearly Elastic Structures, Applications of Bifurcation Theory, New York, 1977. 36. H. Some Math. Problems from Plate and Shell Theory, Lecture Notes in Pure and Applied Mathematics, 1977, vol.

40 Im studying stratified mappings, R. Thom [41] formulated some theorems that are directly related to the considered problem of finite definiteness. In proving one of them he actually formulated certain conditions of finite definiteness of equations. R. Zachepa [31~ who obtained some results concerning finite definiteness of Fredholm equations. Necessary and sufficient conditions for r-definiteness are also ascertained in ~ 31] . The theorems of ~31] were used ia [30~ for studying "small" solutions to the nonlinear Karman equation from the theory of thin elastic shells.