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Families of functions dominated by distributions of C -classes of mappings

Goo Ishikawa (1983)

Annales de l'institut Fourier

A subsheaf of the sheaf Ω of germs C functions over an open subset Ω of R n is called a sheaf of sub C function. Comparing with the investigations of sheaves of ideals of Ω , we study the finite presentability of certain sheaves of sub C -rings. Especially we treat the sheaf defined by the distribution of Mather’s 𝒞 -classes of a C mapping.

Felix Klein's paper on real flexes vindicated

Felice Ronga (1998)

Banach Center Publications

In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.

Final forms for a three-dimensional vector field under blowing-up

Felipe Cano (1987)

Annales de l'institut Fourier

We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). These situations are preserved by permissible blowing-ups and its structure is simple from the view-point of the integral branches. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation.

Fonctions composées différentiables : cas algébrique

Jean-Claude Tougeron (1980)

Annales de l'institut Fourier

Soit f un morphisme propre et de Nash d’un ouvert Ω de R n dans un ouvert Ω ' de R p . Nous démontrons que l’image par f * de l’algèbre C ( Ω ' ) des fonctions réelles C dans Ω ' est fermée dans C ( Ω ) munie de sa topologie habituelle d’espace de Fréchet. Ce résultat généralise, dans le cas algébrique, un résultat de G. Glaeser sur les fonctions composées différentiables.

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