66,897 research outputs found
A Sard theorem for Tame Set-Valued mappings
If is a set-valued mapping from into with closed graph,
then is a critical value of if for some with ,
is not metrically regular at . We prove that the set of critical
values of a set-valued mapping whose graph is a definable (tame) set in an
-minimal structure containing additions and multiplications is a set of
dimension not greater than (resp. a porous set). As a corollary of this
result we get that the collection of asymptotically critical values of a
semialgebraic set-valued mapping has dimension not greater than , thus
extending to such mappings a corresponding result by Kurdyka-Orro-Simon for
semialgebraic mappings. We also give an independent proof of the fact
that a definable continuous real-valued function is constant on components of
the set of its subdifferentiably critical points, thus extending to all
definable functions a recent result of Bolte-Daniilidis-Lewis for globally
subanalytic functions.Comment: 23
Extremal degenerations of polarized Hodge structures
We describe a Hodge theoretic approach to the question: In what ways can a
smooth projective variety degenerate?Comment: v.2 (typos corrected), comments welcom
Monoids in the mapping class group
In this article we survey, and make a few new observations about, the
surprising connection between sub-monoids of mapping class groups and
interesting geometry and topology in low-dimensions.Comment: 36 pages, 18 figure
On the total curvatures of a tame function
Given a definable function f, enough differentiable, we study the continuity
of the total curvature function t --> K(t), total curvature of the level {f=t},
and the total absolute curvature function t-->|K| (t), total absolute curvature
of the level {f=t}. We show they admits at most finitely many discontinuities
- …