487 research outputs found
A Jacobian module for disentanglements and applications to Mond's conjecture
Given a germ of holomorphic map from to ,
we define a module whose dimension over is an upper bound
for the -codimension of , with equality if is weighted
homogeneous. We also define a relative version of the module, for
unfoldings of . The main result is that if are nice
dimensions, then the dimension of over is an upper bound of
the image Milnor number of , with equality if and only if the relative
module is Cohen-Macaulay for some stable unfolding . In particular,
if is Cohen-Macaulay, then we have Mond's conjecture for .
Furthermore, if is quasi-homogeneous, then Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe
that to prove Mond's conjecture, it suffices to prove it in a suitable family
of examples.Comment: 19 page
Isolated singularities of binary differential equations of degree n
We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincar'e-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D1, D2 and D3
Theory of lossless polarization attraction in telecommunication fibers: erratum
An erroneous procedure of averaging the components of the Stokes vector of a polarization scrambled beam over the Poincare sphere introduced in our earlier paper [J. Opt. Soc. Am. B 28, 100-108 (2011)] has been corrected
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