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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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Cited by 31 (17 self)
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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.
MaurerCartan Forms and the Structure of Lie Pseudo–Groups
, 2005
"... This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations fo ..."
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Cited by 25 (10 self)
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This paper begins a series devoted to developing a general and practical theory of moving frames for infinitedimensional Lie pseudogroups. In this first, preparatory part, we present a new, direct approach to the construction of invariant Maurer–Cartan forms and the Cartan structure equations for a pseudogroup. Our approach is completely explicit and avoids reliance on the theory of exterior differential systems and prolongation. The second paper [60] will apply these constructions in order to develop the moving frame algorithm for the action of the pseudogroup on submanifolds. The third paper [61] will apply Gröbner basis methods to prove a fundamental theorem on the freeness of pseudogroup actions on jet bundles, and a constructive version of the finiteness theorem of Tresse and Kumpera for generating systems of differential invariants and also their syzygies. Applications of the moving frame method include practical algorithms for constructing complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, analyzing invariant variational principles, and solving equivalence and symmetry problems arising in geometry and physics.
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 ..."
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Cited by 23 (3 self)
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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
Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations
 J. Math. Phys
, 2005
"... ABSTRACT. A new method of constructing structure equations of Lie symmetry pseudogroups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudogroups, is presented, and illustrated by the examples of the Kadomtsev–Petviashvili and Kort ..."
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Cited by 16 (14 self)
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ABSTRACT. A new method of constructing structure equations of Lie symmetry pseudogroups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudogroups, is presented, and illustrated by the examples of the Kadomtsev–Petviashvili and Korteweg–deVries equations. 1.
Moving frames for pseudo–groups. II. Differential invariants for submanifolds
, 2004
"... Abstract. This paper is the second in a series that develops a theory of moving frames for pseudogroup actions. In this paper, we define a moving frame for free pseudogroup action on the submanifolds, illustrated by explicit examples. Our methods, based on the consequential moving frame connection, ..."
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Cited by 6 (4 self)
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Abstract. This paper is the second in a series that develops a theory of moving frames for pseudogroup actions. In this paper, we define a moving frame for free pseudogroup action on the submanifolds, illustrated by explicit examples. Our methods, based on the consequential moving frame connection, provides an effective means for explicitly 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. 8. Introduction. In this, the second paper in a series on Lie pseudogroups, we will develop the foundations of a general theory of moving frames for submanifolds under the action of a prescribed pseudogroup. Our setup will be the same as described in the first paper in this series, [41], and we will continue to use the previous notation and referencing without further
Moving frames for pseudo–groups. I. The Maurer–Cartan forms
, 2002
"... Sur la théorie, si importante sans doute, mais pour nous si obscure, des ≪groupes ..."
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Cited by 5 (3 self)
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Sur la théorie, si importante sans doute, mais pour nous si obscure, des ≪groupes
On the Structure of Lie PseudoGroups
"... structure equations, essential invariant. ..."
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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
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.
Symmetry, Integrability and Geometry: Methods and Applications On the Structure of Lie PseudoGroups ⋆
"... doi:10.3842/SIGMA.2009.077 Abstract. We compare and contrast two approaches to the structure theory for Lie pseudogroups, the first due to Cartan, and the second due to the first two authors. We argue that the latter approach offers certain advantages from both a theoretical and practical standpoint ..."
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doi:10.3842/SIGMA.2009.077 Abstract. We compare and contrast two approaches to the structure theory for Lie pseudogroups, the first due to Cartan, and the second due to the first two authors. We argue that the latter approach offers certain advantages from both a theoretical and practical standpoint. Key words: Lie pseudogroup; infinitesimal generator; jet; contact form; Maurer–Cartan form; structure equations; essential invariant