74,479 research outputs found
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system 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 is non-reductive, we give a similar characterization via
the semi-invariants of . 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
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
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
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
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 . We also discuss the global theory and applications to
representations of surface groups and to mirror symmetry.Comment: corrected published editio
- …