By James William Bruce (auth.), D. Siersma, C. T. C. Wall, V. Zakalyukin (eds.)

Singularities come up certainly in an immense variety of various components of arithmetic and technology. accordingly, singularity concept lies on the crossroads of paths that attach a few of the most crucial components of purposes of arithmetic with a few of its so much summary regions.

the most objective in so much difficulties of singularity thought is to appreciate the dependence of a few items of research, geometry, physics, or different technology (functions, types, mappings, vector or tensor fields, differential equations, types, etc.) on parameters.

The articles accumulated the following should be grouped lower than 3 headings. (A) Singularities of actual maps; (B) Singular advanced variables; and (C) Singularities of homomorphic maps.

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Proof. Let el, ... , ep be generators for A. By hypothesis each can be written ei = bi + L, mijej, with the bi E B, and mij E M. So writing e for the vector (el, ... ,ep)T, b = (bt, ... ,brf and m for the matrix (mij) we find e = me+b, so (I - m)e = b. Now det(I -m) is invertible, by hypothesis, so multiplying by the adjoint of this matrix we see that each ei lies in B. 14. (i) If M~+l C TRd+M~+2 then M~+l c TRd (and f is r-R-determined). E(n,p) c TKd (and f is r-K-determined). 15. E(n,p). Then f is formally r-Adetermined.

8. (1) If the I-jet is non-zero we can reduce to (x, 0) by the rank theorem. (2) There is one class of non-zero 2-jets, namely (x 2 , 0). A complete transversal for this as a k-jet in Jk+I(I,2) is (x 2 ,ax k+1) if k is even, and em pty if k is odd. If a i= 0 we scale to (x 2, x2m+I), k = 2m, and all subsequent transversals are empty. (3) Now consider homogeneous 3-jets. Again there is only one non-zero type, (x 3, 0). Thinking of this as a 3k-jet (resp. (3k + I)-jet, (3k + 2)jet) a complete transversal is (x 3,ax 3k +l ) (resp.

In particular, the stability of the wave front of an exponential mapping in the simplest case in subriemannian geometry [1, 2] was proved in [3, 4] using flag contact group techniques. 2. 1. BASIC DEFINITIONS By a flag in an affine space An (here A is either R or q we mean a set Fs of nested affine subspaces (or germs of submanifolds) lL~l C lL~2 C ... c lL:s c An of dimensions kl < k2 < ... < ks ::; n. We study the local contact of a distinguished flag Fs with the image of a smooth mapping 1 of a smooth manifold Mm to An, considering the appropriate spaces of germs.