815 research outputs found
Differential forms on free and almost free divisors
We introduce a variant of the usual KƤhler forms on singular free divisors, and show that it enjoys the same depth properties as KƤhler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the GaussāManin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family Formula of discriminants of a versal deformation Formula of a singularity of a mapping
Milnor number equals Tjurina number for functions on space curves
The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor
Tjurina and Milnor numbers of matrix singularities
To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites f ā¦ F with isolated singularities is studied, where f : Y āāC is a function with (possibly non-isolated) singularity and F : X āāY
is a map into the domain of f, and F only is deformed. The corresponding T1(F) is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that
Ļ = Ī¼(f ā¦ F) ā Ī²0 + Ī²1,
where Ļ is the length of T1(F) and Ī²i is the length of ToriOY(OY/Jf, OX). This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When f has CohenāMacaulay singular locus (for example when f is the
determinant function), relations between Ļ and the rank of the vanishing homology of the zero locus of f ā¦ F are obtained
Partial normalizations of coxeter arrangements and discriminants
We study natural partial normalization spaces of Coxeter arrangements and discriminants
and relate their geometry to representation theory. The underlying ring structures arise from Dubrovinās
Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also
describe an independent approach to these structures via duality of maximal CohenāMacaulay fractional
ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter
group. Finally, we show that our partial normalizations give rise to new free divisors
Linear free divisors and Frobenius manifolds
We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the GauĆāManin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure
Vanishing topology of codimension 1 multi-germs over and
We construct all e-codimension 1 multi-germs of analytic (or smooth) maps (kn, T) [rightward arrow] (kp, 0), with n [gt-or-equal, slanted] p ā 1, (n, p) nice dimensions, k = or , by augmentation and concatenation operations, starting from mono-germs (|T| = 1) and one 0-dimensional bi-germ. As an application, we prove general statements for multi-germs of corank [less-than-or-eq, slant] 1: every one has a real form with real perturbation carrying the vanishing homology of the complexification, every one is quasihomogeneous, and when n = p ā 1 every one has image Milnor number equal to 1 (this last is already known when n [gt-or-equal, slanted] p)
Partial normalizations of coxeter arrangements and discriminants
We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovinās Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We alsodescribe an independent approach to these structures via duality of maximal CohenāMacaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors
On the Symmetry of b-Functions of Linear Free Divisors
We introduce the concept of a prehomogeneous determinant as a possibly nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the b-function are symmetric about ā1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under which this symmetry persists.
Combined with Kashiwara\u27s theorem on the roots of b-functions, our symmetry result shows that ā1 is the only integer root of the b-function. This gives a positive answer to a problem posed by Castro-Jimenez and Ucha-Enrquez in the above cases.
We study the condition of strong Euler homogeneity in terms of the action of the stabilizers on the normal spaces.
As an application of our results, we show that the logarithmic comparison theorem holds for reductive linear Koszul free divisors exactly when they are strongly Euler homogeneous
The Hall instability of weakly ionized, radially stratified, rotating disks
Cool weakly ionized gaseous rotating disk, are considered by many models as
the origin of the evolution of protoplanetary clouds. Instabilities against
perturbations in such disks play an important role in the theory of the
formation of stars and planets. Thus, a hierarchy of successive fragmentations
into smaller and smaller pieces as a part of the Kant-Laplace theory of
formation of the planetary system remains valid also for contemporary
cosmogony. Traditionally, axisymmetric magnetohydrodynamic (MHD), and recently
Hall-MHD instabilities have been thoroughly studied as providers of an
efficient mechanism for radial transfer of angular momentum, and of density
radial stratification. In the current work, the Hall instability against
nonaxisymmetric perturbations in compressible rotating fluids in external
magnetic field is proposed as a viable mechanism for the azimuthal
fragmentation of the protoplanetary disk and thus perhaps initiating the road
to planet formation. The Hall instability is excited due to the combined effect
of the radial stratification of the disk and the Hall electric field, and its
growth rate is of the order of the rotation period.Comment: 15 pages, 2 figure
Boundedness properties of fermionic operators
The fermionic second quantization operator is shown to be
bounded by a power of the number operator given that the operator
belongs to the -th von Neumann-Schatten class, . Conversely,
number operator estimates for imply von Neumann-Schatten
conditions on . Quadratic creation and annihilation operators are treated as
well.Comment: 15 page
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