Computation of Weyl groups of G-varieties

Let G be a connected reductive group. To any irreducible G-variety one associates a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of G-varieties (affine homogeneous vector bundles of maximal rank, affine homogeneous spaces, homogeneous spaces of maximal rank with discrete group of central automorphisms) we compute Weyl groups more or less explicitly.Comment: 82 pages, v2 56 pages, the paper is rewritten: all material related to Hamiltonian actions was removed to AG/0701823v2, AG/0703296v2. Some propositions and their proofs are modifie

Late-Night Thoughts About the Significance of a Small Count of Nuclear or Particle Events

Reconciliation of frequentist and Bayesian approaches to elementary treatment of data in nuclear and particle physics is attempted. Unique procedure to express the significance of a small count in presence of background is henceforth proposed and discussed in some detail.Comment: 18 pages, 9 figures, titles to subsections added, typos correcte

The dynamics of thin gas layer moving between two fluids

The dynamics and stability of a thin gas layer moving between two fluid layers moving in the same or opposite direction is studied. The linear evolutionary equations describing the spatial-temporal dynamics of the interface perturbations between gas and two fluid layers are derived for the flat two-dimensional case. Integral correlations across the layer are obtained, and the various kinds of time dependent base states are found. A linear stability is considered for the system using non-stationary equation array derived. The equation array consists of the two one-dimensional non-stationary equations of a seventh and fourth order. The results of the numerical study of the governing evolution equations support the results of the analysis for more simple limit cases. It is found that the thin sheet gas flow in-between two liquid layers is unstable and the peculiarities are found and discussed together with some applications available.Comment: 15 pages, 1 figure, 10 reference

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group $G$ on an irreducible normal affine Poisson variety $X$. We study the invariant moment map \psi_{G,X}:X\to \g, that is, the composition of the moment map $\mu_{G,X}:X\to g:=Lie(G)$ and the quotient morphism g\to g\quo G. We obtain some results on the dimensions of fibers of $\psi_{G,X}$ and the corresponding morphism of quotients X\quo G\to g\quo G. We also study the "Stein factorisation" of $\psi_{G,X}$. Namely, let $C_{G,X}$ denote the spectrum of the integral closure of $\psi_{G,X}^*(K[g]^G)$ in $K(X)^G$. We investigate the structure of the g\quo G-scheme $C_{G,X}$. Our results partially generalize those obtained by F. Knop in the case of the actions on cotangent bundles and symplectic vector spaces.Comment: v1 46 pages, v2 37 pages, major corrections are made, Theorem 1.5 and its proof are removed, v3 38 pages, final version to appear in Math.

Classification of multiplicity free Hamiltonian actions of complex tori on Stein manifolds

A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus T\cong (\C^\times)^n on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an \'{e}tale map Y\to \t^*, a set of divisors on $Y$ and elements of H^2(Y,\Z)^{\oplus n}, H^2(Y,\C). We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.Comment: 12 pages, v2 minor corrections mad

Lifting central invariants of quantized Hamiltonian actions

Let G be a connected reductive group over an algebraically closed field K of characteristic 0, X an affine symplectic variety equipped with a Hamiltonian action of G. Further, let * be a G-invariant Fedosov star-product on X such that the Hamiltonian action is quantized. We establish an isomorphism between the center of the associative algebra K[X][[h]]^G and the algebra of formal power series with coefficients in the Poisson center of K[X]^G.Comment: v1 9 pages, v2 final version 10 page

The Theory and Applications of Parametric Excitation and Suppression of Oscillations in Continua: State of the Art

The results by development of physical, mathematical and numerical models for parametric excitation and suppression of oscillations on the interfaces separating continuous media, for carrying out computing, physical and natural experiments by revealing the new phenomena and parametric effects, and for their use in improvement the existing and creation the perspective highly efficient technological processes are presented. Scientific novelty of this work consists in development of the theory and applications of parametric excitation and suppression of oscillations on the boundaries of continua on the samples of three tasks classes: flat and radial spreading film flows of viscous incompressible liquids, conductive as well as non-conductive ones; surfaces of phase transition from a liquid state into a solid one; and heterogeneous granular media. The external actions considered are: alternating electromagnetic, vibration, acoustic and thermal fields. Along with linear the non-linear parametric oscillations are investigated (including strongly non-linear) too and the results of theoretical studies are confirmed and supplemented with the corresponding experimental data. The general and specific peculiarities of parametrically excited oscillations and the new parametric effects revealed are discussed for technical and technological applications. First the general statement and substantiation of the problems studied is considered, and then the various parametric oscillations in continua are analyzed from common methodological base. Also the assessment of a current state of the problems, analysis of their features, prospects of further development and the main difficulties of the methodological, mathematical and applied character are presented.Comment: 25 pages, 233 reference

Demazure embeddings are smooth

We prove Brion's conjecture stating that the closure of the orbit of a self-normalizing spherical subalgebra in the corresponding Grassmanian is smoothComment: 7 page

Invariant Ideals and Matsushima's Criterion

Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima's criterion: a homogeneous space G/H is an affine variety if and only if H is reductive.Comment: 6 page

The Kempf-Ness theorem and Invariant Theory

We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.Comment: 3 page