English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let S be a closed set in R"', and let a C1 Whitney stratification of 5 be given. (Actually, only Whitney (a)-regularity is needed.) Let /: R" -» R"' be a C1 map transversal to all the strata. Assume that the image of / intersects S. Then for each compact set K in R", the Euclidean distances p(x,/-1(S)) and p{f(x\ S), for x in K, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps g which are close enough to/in a C1 sense on a neighborhood of K. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS]. Introduction. In the course of studying quotients of smooth functions, one is led to consider properties of the zero sets of the denominators. In his proof that division of distributions by an analytic function is possible, Lojasiewicz [Loj] studied certain stratifications of the zero set of an analytic function. He proved that the strata satisfy a certain metric condition which he called regular separation. Malgrange [Mai, p. 14] defined an equivalent condition on the closures of the strata. In his terminology, one says that two closed sets X,Y c U, for U an open set in Rm, are regularly situated if either (1) X n y is empty, or (2) given any pair of compact sets K c X, L c Y, there exists a pair of positive constants C and jti such that for every x in K, one has p(x,L)> C{p(x,Xn Y)f (here p denotes Euclidean distance in Rm). This condition is symmetric in X and Y, using the same exponent jti locally ([Loj, p. 91]; see also Lemma 3 below). One gets an equivalent definition if p(x, L) is replaced by p(x, Y).The exponent p can always be increased. It is of necessity > 1 if K intersects L (respectively Y), and can be chosen to equal any positive number otherwise. Lojasiewicz showed [Loj; Mai, p. 62, Corollary 4.4] that any two analytic subsets of R" are regularly situated and that the zero set of an analytic function admits a stratification for which the closures of the strata are regularly situated. From this fact, together with growth conditions for an analytic function near its zero set, Lojasiewicz was able to deduce his division Received by the editors December 31, 1984. 1980 Mathematics Subject Classification. Primary 46E25; Secondary 26E10, 46E10, 46H10.

Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold