Results 11  20
of
23
Discrete moving frames and discrete integrable systems
 Foundations of Computational Mathematics, Volume 13, Issue 4 (2013), Page 545582
"... Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differentialdifference systems. In particular, we show that in the centroaffine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices respectively under Miura transformations. We also show that a specified invariant map of polygons in the centroaffine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere. Dedicated to Peter Olver in celebration of his 60th birthday 1
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
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
Invariant Variational Problems and Invariant Flows via Moving Frames
, 2011
"... www.math.umn.edu/~olver ..."
(Show Context)
Differential Invariants of Maximally Symmetric Submanifolds
, 2009
"... Let G be a Lie group acting smoothly on a manifold M. A closed, nonsingular submanifold S ⊂ M is called maximally symmetric if its symmetry subgroup G S ⊂ G has the maximal possible dimension, namely dimG S = dim S, and hence S = G S ·z 0 is an orbit of G S. Maximally symmetric submanifolds are cha ..."
Abstract
 Add to MetaCart
(Show Context)
Let G be a Lie group acting smoothly on a manifold M. A closed, nonsingular submanifold S ⊂ M is called maximally symmetric if its symmetry subgroup G S ⊂ G has the maximal possible dimension, namely dimG S = dim S, and hence S = G S ·z 0 is an orbit of G S. Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries.
THEME Table of contents
"... 3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3 ..."
Abstract
 Add to MetaCart
3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3
THEME Algorithms, Certification, and CryptographyTable of contents
"... 3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3 3.4. Symbolic numeric analysis 3 ..."
Abstract
 Add to MetaCart
3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3 3.4. Symbolic numeric analysis 3
3.2. Algebraic Geometric Modeling 2 3.3. Algebraic Geometric Computing 2
"... c t i v it y e p o r t 2009 Table of contents ..."
(Show Context)
Lagrangian Curves in a 4dimensional affine symplectic space
, 2013
"... Lagrangian curves in R 4 entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian ..."
Abstract
 Add to MetaCart
(Show Context)
Lagrangian curves in R 4 entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in R 4 and determine Lagrangian geodesics.
Modern Developments in the Theory and Applications of Moving Frames
, 2014
"... Abstract. This article discusses recent advances in the general equivariant approach to the method of moving frames, concentrating on finitedimensional Lie group actions. A few of the many applications — to geometry, invariant theory, differential equations, and image processing — are presented. ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. This article discusses recent advances in the general equivariant approach to the method of moving frames, concentrating on finitedimensional Lie group actions. A few of the many applications — to geometry, invariant theory, differential equations, and image processing — are presented.