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Moving frames for Lie pseudo–groups
 Canadian J. Math
"... Abstract. We propose a new, constructive theory of moving frames for Lie pseudogroup actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relation ..."
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Abstract. We propose a new, constructive theory of moving frames for Lie pseudogroup actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications. Mathematics subject classification.
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 ..."
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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.
Algorithms for differential invariants of symmetry groups of differential equations
 Found. Comput. Math
"... Abstract. We present new computational algorithms, based on equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudogroups of differential equations and establishing the structure of the induced differential invariant algebra. The Korteweg–deVries and Kadomtsev– ..."
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Abstract. We present new computational algorithms, based on equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudogroups of differential equations and establishing the structure of the induced differential invariant algebra. The Korteweg–deVries and Kadomtsev–Petviashvili equations are studied to illustrate these methods. 1.
Symmetry group analysis of the shallow water and semigeostrophic
"... The twodimensional shallow water equations and their semigeostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetr ..."
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Cited by 8 (0 self)
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The twodimensional shallow water equations and their semigeostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semigeostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semigeostrophic equations in such a way that a welldefined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an ‘infinite linear cascade’. 1.
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. ..."
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Cited by 4 (4 self)
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This article highlights a coherent series of algorithmic tools to compute and work with algebraic and differential invariants.
Recursive Moving Frames
, 2011
"... A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of ..."
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A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of large scale applications of the equivariant moving frame method.
Recent Advances in the Theory and Application of Lie Pseudo–Groups
"... Abstract. This paper surveys several new developments in the analysis of Lie pseudogroups and their actions on submanifolds. The main themes are direct construction of Maurer–Cartan forms and structure equations, and the use of equivariant moving frames to analyze the algebra of differential invaria ..."
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Abstract. This paper surveys several new developments in the analysis of Lie pseudogroups and their actions on submanifolds. The main themes are direct construction of Maurer–Cartan forms and structure equations, and the use of equivariant moving frames to analyze the algebra of differential invariants and invariant differential forms, including generators, commutation relations, and syzygies. 1. Introduction. Inspired by Galois ’ introduction of group theory to solve polynomial equations, Lie founded his remarkable theory of transformation groups for the purpose of analyzing and solving differential equations. In Lie’s time, abstract groups were as yet unknown, and hence he made no significant distinction between finitedimensional and infinitedimensional
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 ..."
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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
PERSISTENCE OF FREENESS FOR LIE PSEUDOGROUP ACTIONS
"... Abstract. The action of a Lie pseudogroup G on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J n of submanifolds of M. We prove in this paper that both the local and global freeness of the action of G on J n persist under prolongation in the jet order n. Our results un ..."
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Abstract. The action of a Lie pseudogroup G on a smooth manifold M induces a prolonged pseudogroup action on the jet spaces J n of submanifolds of M. We prove in this paper that both the local and global freeness of the action of G on J n persist under prolongation in the jet order n. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the pseudogroup action on J ∞. 1.
Pseudo–Groups, Moving Frames, and Differential Invariants
, 2007
"... Abstract. We survey recent developments in the method of moving frames for infinitedimensional Lie pseudogroups. These include a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for pseudogroups, and new algorithms, based on constructive ..."
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Abstract. We survey recent developments in the method of moving frames for infinitedimensional Lie pseudogroups. These include a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for pseudogroups, and new algorithms, based on constructive commutative algebra, for establishing the structure of their differential invariant algebras.