304 research outputs found

    Demazure roots and spherical varieties: the example of horizontal SL(2)-actions

    Full text link
    Let GG be a connected reductive group, and let XX be an affine GG-spherical variety. We show that the classification of Ga\mathbb{G}_{a}-actions on XX normalized by GG can be reduced to the description of quasi-affine homogeneous spaces under the action of a semi-direct product Ga⋊G\mathbb{G}_{a}\rtimes G with the following property. The induced GG-action is spherical and the complement of the open orbit is either empty or a GG-orbit of codimension one. These homogeneous spaces are parametrized by a subset Rt(X){\rm Rt}(X) of the character lattice X(G)\mathbb{X}(G) of GG, which we call the set of Demazure roots of XX. We give a complete description of the set Rt(X){\rm Rt}(X) when GG is a semi-direct product of SL2{\rm SL}_{2} and an algebraic torus; we show particularly that Rt(X){\rm Rt}(X) can be obtained explicitly as the intersection of a finite union of polyhedra in Q⊗ZX(G)\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G) and a sublattice of X(G)\mathbb{X}(G). We conjecture that Rt(X){\rm Rt}(X) can be described in a similar combinatorial way for an arbitrary affine spherical variety XX.Comment: Added Section 4; modified main result, Theorem 5.18 now; other change

    On automorphism groups of affine surfaces

    Full text link
    This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.Comment: Proposition 2.10 from the previous version (published in Algebraic Varieties and Automorphism Groups, ASPM 75) deleted. There is a mistake in the proof kindly indicated by J.-P. Furter; the validity of the result remains open. This does not affect the rest of the pape

    Hyperbolic Models of Homogeneous Two-Fluid Mixtures

    Full text link
    One derives the governing equations and the Rankine - Hugoniot conditions for a mixture of two miscible fluids using an extended form of Hamilton's principle of least action. The Lagrangian is constructed as the difference between the kinetic energy and a potential depending on the relative velocity of components. To obtain the governing equations and the jump conditions one uses two reference frames related with the Lagrangian coordinates of each component. Under some hypotheses on flow properties one proves the hyperbolicity of the governing system for small relative velocity of phases.Comment: 14 page

    Modeling the multiphase flows in deformable porous media

    Full text link
    This work proposes the nonlinear model for the flow of mixture of compressible liquids in a porous medium with consideration of finite deformations and thermal effects. Development of this model is based on the method of thermodynamically consistent systems of conservation laws. Numerical analysis of the model is based on the WENO-Runge-Kutta method of the high accuracy. The model is developed to solve the problems arising when studying the different-scale fluid dynamic processes. Evolution of the wave fields in inhomogeneous saturated porous media is considered

    A variational principle for two-fluid models

    Full text link
    A variational principle for two-fluid mixtures is proposed. The Lagrangian is constructed as the difference between the kinetic energy of the mixture and a thermodynamic potential conjugated to the internal energy with respect to the relative velocity of phases. The equations of motion and a set of Rankine-Hugoniot conditions are obtained. It is proved also that the convexity of the internal energy guarantees the hyperbolicity of the one-dimensional equations of motion linearized at rest.Comment: 7 page
    • …
    corecore