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Joint invariant signatures
 Found. Comput. Math
, 1999
"... Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, ..."
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Cited by 47 (25 self)
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Dedicated to the memory of Gian–Carlo Rota Abstract. A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated.
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 24 (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
Moving frames and singularities of prolonged group actions
 Selecta Math. (N.S
"... Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have ..."
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Cited by 23 (13 self)
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Abstract. The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie’s theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of “totally singular points, ” and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vectorvalued functions, and methods for the solution to Lie determinant equations.
Differential invariants for parametrized projective surfaces
 Commun. Anal. Geom
, 1999
"... Abstract. We classify the differential invariants and moving frames for surfaces in projective space under the action of the projective group. The role of these results in the analysis of Adler–Gel’fand–Dikii flows that arise in inverse scattering and solitons is explained. 1. Introduction. The diff ..."
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Cited by 11 (9 self)
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Abstract. We classify the differential invariants and moving frames for surfaces in projective space under the action of the projective group. The role of these results in the analysis of Adler–Gel’fand–Dikii flows that arise in inverse scattering and solitons is explained. 1. Introduction. The differential invariants associated with a transformation group acting on a manifold are the fundamental building blocks for understanding the geometry, equivalence, symmetry and other properties of submanifolds. Moreover, the construction of general invariant differential equations and invariant variational problems requires knowledge of the
Moving Frames and Joint Differential Invariants
 REGULAR AND CHAOTIC MECHANICS
, 1999
"... This paper surveys the new, algorithmic theory of moving frames developed by the author and M. Fels. The method is used to classify joint invariants and joint differential invariants of transformation groups, and equivalence and symmetry properties of submanifolds. Applications in classical invarian ..."
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Cited by 3 (0 self)
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This paper surveys the new, algorithmic theory of moving frames developed by the author and M. Fels. The method is used to classify joint invariants and joint differential invariants of transformation groups, and equivalence and symmetry properties of submanifolds. Applications in classical invariant theory, geometry, and computer vision are indicated.
Conformal analogue of the AdlerGel'fandDikii bracket in two dimensions
, 1999
"... In this paper we classify all differential invariants of parametrized curves in R² under the action of O(3; 1). We find a formula for the most general evolution of such curves which are invariant under the action. We show that our formula induces a natural evolution on a generating set of differenti ..."
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Cited by 2 (1 self)
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In this paper we classify all differential invariants of parametrized curves in R² under the action of O(3; 1). We find a formula for the most general evolution of such curves which are invariant under the action. We show that our formula induces a natural evolution on a generating set of differential invariants and we prove that such an evolution is Hamiltonian, giving an explicit expression of its Poisson tensor.
Moving frames: a brief survey
 Symmetry and Perturbation Theory: Proceedings of the International Conference on SPT 2001, Cala Gonone, Sardinia, Italy, May 613, 2001, 143150 (World Scientific
, 2001
"... ..."
Moving Frames
"... 1. Introduction. First introduced by Gaston Darboux, the $theory,of $ moving frames (“$rep\grave{e}res $ mobiles”) is most closely associated with the name of Elie Cartan, [5], who molded it into a powerful and algorithmic tool for studying the geometric properties of submanifolds and their invari ..."
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1. Introduction. First introduced by Gaston Darboux, the $theory,of $ moving frames (“$rep\grave{e}res $ mobiles”) is most closely associated with the name of Elie Cartan, [5], who molded it into a powerful and algorithmic tool for studying the geometric properties of submanifolds and their invariants under the action of a transformation group.In the $1970 ’ s $,