Hodge-to-de Rham degeneration for stacks

We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find a good integral model of a stack (a so-called spreading), which, unlike in the case of schemes, need not to exist in general. To address this problem we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary we deduce a (non-canonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action

Long-term modulation of galactic cosmic rays in high-energy region

The results of continuous registration of the cosmic ray muon intensity on the ground and underground at 7, 20 and 60 m w.e. depths in Yakutsk for 1957 to 1983 are presented. In years of the solar activity maximum in cosmic ray density were found two minima or two-stepped decrease. The second intensity minimum appears to be not the result of the inversion of the general magnetic field of the Sun but caused by the increase of the solar wind speed

Bright vector solitons in cross-defocusing nonlinear media

We study two-dimensional soliton-soliton vector pairs in media with self-focusing nonlinearities and defocing cross-interactions. The general properties of the stationary states and their stability are investigated. The different scenarios of instability are observed using numerical simulations. The quasi-stable propagation regime of the high-power vector solitons is revealed.Comment: 6 pages, 7 figure