Results 1  10
of
24
Differential Invariant Algebras of Lie Pseudo–Groups
, 2012
"... The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepse ..."
Abstract

Cited by 26 (13 self)
 Add to MetaCart
(Show Context)
The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudogroup acting on the submanifolds of an analytic manifold. Under the assumption of local freeness ofasuitablyhighorder prolongationofthepseudogroup action, wedevelop computational algorithms for locating a finite generating set of differential invariants, a complete system of recurrence relations for the differentiated invariants, and a finite system of generating differential syzygies among the generating differential invariants. In particular, if the pseudogroup acts transitively on the base manifold, then the algebra of differential invariants is shown to form a rational differential algebra with noncommuting derivations. The essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the usual symbol module associated with the infinitesimal determining system of the pseudogroup, and a new “prolonged symbol module” constructed from the symbols of the annihilators of the prolonged pseudogroup generators. Modulo low order complications, thegenerating differential invariants and differential syzygies are in onetoone correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely oncombining the movingframe approach developed inearlier papers with Gröbner basis algorithms from commutative algebra.
A survey of moving frames
 Computer Algebra and Geometric Algebra with Applications. Volume 3519 of Lecture Notes in Computer Science, 105–138
, 2005
"... Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. Acc ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
(Show Context)
Abstract. This article presents the equivariant method of moving frames for finitedimensional Lie group actions, surveying a variety of applications, including geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant flows. 1. Introduction. According to Akivis, [1], the method of moving frames originates in work of the Estonian mathematician Martin Bartels (1769–1836), a teacher of both Gauss and Lobachevsky. The field is most closely associated with Élie Cartan, [21], who forged earlier contributions by Darboux, Frenet, Serret, and Cotton into a powerful tool for analyzing the geometric
Differential Invariants of Conformal and Projective Surfaces
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames. ..."
Abstract

Cited by 16 (13 self)
 Add to MetaCart
We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in threedimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
Invariant Submanifold Flows
, 2011
"... Given a Lie group acting on a manifold, our aim is to analyze th eevolution of differential invariants under invariant submanifold flows. The constructions arebased on the equivariant method of moving frames and the induced invariant variational bicomplex. Applications to integrable soliton dynamic ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
(Show Context)
Given a Lie group acting on a manifold, our aim is to analyze th eevolution of differential invariants under invariant submanifold flows. The constructions arebased on the equivariant method of moving frames and the induced invariant variational bicomplex. Applications to integrable soliton dynamics, and to the evolution of differential invariant signatures, used in equivalence problems and object recognition and symmetry detection in images, are discussed.
Poisson Structures for Geometric Curve Flows in Semisimple Homogeneous Spaces
"... Abstract. We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semisimple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated d ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semisimple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples. 1. Introduction. In 1972, Hasimoto, [6], showed how the evolution of curvature and torsion of space curves under the vortex filament flow is governed by the completely integrable nonlinear Schrödinger equation. Since then, a large variety of integrable soliton equations, including all of the most familiar examples (Korteweg–deVries, modified Korteweg–deVries, Sawada–
Algebraic and Differential Invariants
 FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, BUDAPEST 2011, LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES (403)
, 2011
"... This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants. ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants.
Solving local equivalence problems with the equivariant moving frame method
"... Abstract. Given a Lie pseudogroup action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate mod ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
Abstract. Given a Lie pseudogroup action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the equivariant moving frame constructions extend to submanifold jets where the pseudogroup does not act freely at any order. Once this is done, we review the solution to the local equivalence problem of submanifolds within the equivariant moving frame framework. This offers an alternative approach to Cartan’s equivalence method based on the theory of Gstructures. Key words: differential invariant; equivalence problem; Maurer–Cartan form; moving frame
Differential Invariant Algebras
"... Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including a few surprises and several open questions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their explicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications