Sur le problème de la division

Error Bounds: Necessary and Sufficient Conditions

“…Since the fundamental works of Hoffman [21] and Lojasiewicz [36], the notion of (local) error bound plays a key role in variational analysis. Having a closed set A and a function f with the property that A = {x| f (x) ≤ 0}, the principal question reads as follows: For a givenx ∈ A, does there exist a neighborhood U ofx and positive constants c, β such that…”

Idéaux et fonctions différentiables. I

“…Nous démontrons la : soit 3 un idéal propre de k[[y\\. Alors : hta0).fe [[x]])<ht (5). [6] ou Bourbaki [1] a) Pour toute suite exacte 0 --> P' --> P --> P" --> 0 de modules sur A, la suite 0 -> P' ®^ OTc -> P (g^ 31Z -> P" ®A ^ -^ 0 est exacte.…”

“…& (Î2) est fermé (Hôrmander, [2]) ; plus généralement, si ^ est analytique sur ft, l'idéal ^ . 8> (S2) est fermé (feojasiewicz, [5] , qui peuvent être rendues linéaires par changement de coordonnées. D'après le théorème 2, les idéaux /i .â(î2), f^ .…”

Characterizations of Ultradistributions with Compact Support and Decomposition by Support

“…This is possible positively for hyperfunctions and quasianalytic ultradistributions (see Hörmander [11,12]) but, in general, not true for the Schwartz distributions and nonquasianalytic ultradistributions (Lojasiewicz [16], Hörmander [11]). Lojasiewicz [16] gave the geometric condition for Kx and K2 that such a decomposition is possible. Here we give a necessary and sufficient condition that such a decomposition theorem holds for the nonquasianalytic ultradistributions, which will be verified along the context of Malgrange [17].…”