11,187 research outputs found
Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities
In this paper we extend Y.Eliashberg's -principle to arbitrary generic
smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient
condition for a continuous map of smooth manifolds of the same dimension to be
homotopic to a generic map with a prescribed Thom-Boardman singularity
at each point and with no other critical points. In dimension 3 we
rephrase these conditions in terms of the Stiefel-Whitney classes and the
cohomology classes of the given loci of folds, cusps and swallowtail points.Comment: 28 pages. Some new corrections adde
Uniform convergence of Vapnik--Chervonenkis classes under ergodic sampling
We show that if is a complete separable metric space and
is a countable family of Borel subsets of with
finite VC dimension, then, for every stationary ergodic process with values in
, the relative frequencies of sets converge
uniformly to their limiting probabilities. Beyond ergodicity, no assumptions
are imposed on the sampling process, and no regularity conditions are imposed
on the elements of . The result extends existing work of Vapnik
and Chervonenkis, among others, who have studied uniform convergence for i.i.d.
and strongly mixing processes. Our method of proof is new and direct: it does
not rely on symmetrization techniques, probability inequalities or mixing
conditions. The uniform convergence of relative frequencies for VC-major and
VC-graph classes of functions under ergodic sampling is established as a
corollary of the basic result for sets.Comment: Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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