218 research outputs found
Characterization of distributions having a value at a point in the sense of Robinson
We characterize Schwartz distributions having a value at a single point in
the sense introduced by means of nonstandard analysis by A. Robinson. They
appear to be distributions continuous in a neighborhood of the point.Comment: 5 page
Definable triangulations with regularity conditions
In this paper we prove that every definable set has a definable triangulation
which is locally Lipschitz and weakly bi-Lipschitz on the natural simplicial
stratification of the simplicial complex. We also distinguish a class T of
regularity conditions and give a universal construction of a definable
triangulation with a T condition of a definable set. This class includes the
Whitney (B) condition and the Verdier condition.Comment: 17 page
Tameness of complex dimension in a real analytic set
Given a real analytic set X in a complex manifold and a positive integer d,
denote by A(d) the set of points p in X at which there exists a germ of a
complex analytic set of dimension d contained in X. It is proved that A(d) is a
closed semianalytic subset of X.Comment: Published versio
Nowhere minimal CR submanifolds and Levi-flat hypersurfaces
A local uniqueness property of holomorphic functions on real-analytic nowhere
minimal CR submanifolds of higher codimension is investigated. A sufficient
condition called almost minimality is given and studied. A weaker necessary
condition, being contained a possibly singular real-analytic Levi-flat
hypersurface is studied and characterized. This question is completely resolved
for algebraic submanifolds of codimension 2 and a sufficient condition for
noncontainment is given for non algebraic submanifolds. As a consequence, an
example of a submanifold of codimension 2, not biholomorphically equivalent to
an algebraic one, is given. We also investigate the structure of singularities
of Levi-flat hypersurfaces.Comment: 21 pages; conjecture 2.8 was removed in proof; to appear in J. Geom.
Ana
Tameness of holomorphic closure dimension in a semialgebraic set
Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
The {\L}ojasiewicz exponent of a set of weighted homogeneous ideals
We give an expression for the {\L}ojasiewicz exponent of a set of ideals
which are pieces of a weighted homogeneous filtration. We also study the
application of this formula to the computation of the {\L}ojasiewicz exponent
of the gradient of a semi-weighted homogeneous function (\C^n,0)\to (\C,0)
with an isolated singularity at the origin.Comment: 15 page
From error bounds to the complexity of first-order descent methods for convex functions
This paper shows that error bounds can be used as effective tools for
deriving complexity results for first-order descent methods in convex
minimization. In a first stage, this objective led us to revisit the interplay
between error bounds and the Kurdyka-\L ojasiewicz (KL) inequality. One can
show the equivalence between the two concepts for convex functions having a
moderately flat profile near the set of minimizers (as those of functions with
H\"olderian growth). A counterexample shows that the equivalence is no longer
true for extremely flat functions. This fact reveals the relevance of an
approach based on KL inequality. In a second stage, we show how KL inequalities
can in turn be employed to compute new complexity bounds for a wealth of
descent methods for convex problems. Our approach is completely original and
makes use of a one-dimensional worst-case proximal sequence in the spirit of
the famous majorant method of Kantorovich. Our result applies to a very simple
abstract scheme that covers a wide class of descent methods. As a byproduct of
our study, we also provide new results for the globalization of KL inequalities
in the convex framework.
Our main results inaugurate a simple methodology: derive an error bound,
compute the desingularizing function whenever possible, identify essential
constants in the descent method and finally compute the complexity using the
one-dimensional worst case proximal sequence. Our method is illustrated through
projection methods for feasibility problems, and through the famous iterative
shrinkage thresholding algorithm (ISTA), for which we show that the complexity
bound is of the form where the constituents of the bound only depend
on error bound constants obtained for an arbitrary least squares objective with
regularization
Approximation of holomorphic maps from Runge domains to affine algebraic varieties
We present a geometric proof of the theorem saying that holomorphic maps from
Runge domains to affine algebraic varieties admit approximation by Nash maps.
Next we generalize this theorem.Comment: 24 pages; Proposition 3.3 of v1 is replaced in v2 by a much simpler
Proposition 4.2; Proof of Proposition 3.6 of v1 is simplified (see
Proposition 4.3 in v2); Proof of the main theorem is simplified, its outline
and generalization are adde
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