14,835 research outputs found
Multigraded regularity, a*-invariant and the minimal free resolution
In recent years, two different multigraded variants of Castelnuovo-Mumford
regularity have been developed, namely multigraded regularity, defined by the
vanishing of multigraded pieces of local cohomology modules, and the resolution
regularity vector, defined by the multidegrees in a minimal free resolution. In
this paper, we study the relationship between multigraded regularity and the
resolution regularity vector. Our method is to investigate multigraded variants
of the usual a*-invariant. This, in particular, provides an effective approach
to examining the vanishing of multigraded pieces of local cohomology modules
with respect to different graded irrelevant ideals.Comment: Final version to appear in J. Algebra; 24 page
Sharp Strichartz estimates for water waves systems
Water waves are well-known to be dispersive at the linearization level.
Considering the fully nonlinear systems, we prove for reasonably smooth
solutions the optimal Strichartz estimates for pure gravity waves and the
semi-classical Strichartz estimates for gravity-capillary waves; for both 2D
and 3D waves. Here, by optimal we mean the gains of regularity (over the
Sobolev embedding from Sobolev spaces to H\"older spaces) obtained for the
linearized systems. Our proofs combine the paradifferential reductions of
Alazard-Burq-Zuily with a dispersive estimate using a localized wave package
type parametrix of Koch-Tataru.Comment: Some typos fixe
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