74,479 research outputs found

    A Characterization of Reduced Forms of Linear Differential Systems

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    A differential system [A]:  Y′=AY[A] : \; Y'=AY, with A∈Mat(n,kˉ)A\in \mathrm{Mat}(n, \bar{k}) is said to be in reduced form if A∈g(kˉ)A\in \mathfrak{g}(\bar{k}) where g\mathfrak{g} is the Lie algebra of the differential Galois group GG of [A][A]. In this article, we give a constructive criterion for a system to be in reduced form. When GG is reductive and unimodular, the system [A][A] is in reduced form if and only if all of its invariants (rational solutions of appropriate symmetric powers) have constant coefficients (instead of rational functions). When GG is non-reductive, we give a similar characterization via the semi-invariants of GG. In the reductive case, we propose a decision procedure for putting the system into reduced form which, in turn, gives a constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr

    Point Symmetries of Generalized Toda Field Theories

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    A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.Comment: 25 pages, no figures, Latex fil

    New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory

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    DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors and other tensorial invariants, algebraic classification of curvature, and symmetry reduction of field equations.Comment: 42 page

    [SADE] A Maple package for the Symmetry Analysis of Differential Equations

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    We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund and potential symmetries, invariant solutions, first-integrals, N\"other theorem for both discrete and continuous systems, solution of ordinary differential equations, reduction of order or dimension using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented in the package QPSI by the authors) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given

    Cubic Differentials in the Differential Geometry of Surfaces

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    We discuss the local differential geometry of convex affine spheres in \re^3 and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for a2(2)\mathfrak a^{(2)}_2. We also discuss the global theory and applications to representations of surface groups and to mirror symmetry.Comment: corrected published editio
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