Discrete moving frames on lattice varieties and lattice-based multispaces
type: Item
instance of: scholarly article
title: Discrete moving frames on lattice varieties and lattice-based multispaces
zbMATH Open document ID: 1390.14101
author: G. Marí Beffa
author: E. L. Mansfield
published in: Foundations of Computational Mathematics
publication date: 2018-03-13T00:00:00Z
review text: The aim of this paper is to extend the theory of the discrete moving frame in two different ways.The first is to consider a discrete moving frame defined on a lattice variety, which can be thought of as the vertices, or 0-cells, together with their adjacency information, in a discrete approximation of a manifold. The authors describe their associated cross sections and define Maurer-Cartan invariants and local syzygies. They describe the equivalence classes of global syzygies that result from the first fundamental group of the variety. The authors consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, they construct multispace, a generalisation of the jet bundle that also generalises Olver's one-dimensional construction. Using interpolation to provide coordinates, they prove that it is a manifold containing the usual jet bundle as a submanifold and show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. The authors prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In their last chapter the authors discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, they discuss a more general result concerning the discretisation of smooth moving frames, and the continuum limit of equicontinuous families of discrete moving frames, with an example. More precisely, they use the Arzela-Ascoli theorem to give a general convergence result for an equicontinuous family of moving frames. This provides a rigorous foundation to a variety of examples involving the di…
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 14H70
Mathematics Subject Classification ID: 17B80
Mathematics Subject Classification ID: 49M25
Mathematics Subject Classification ID: 53A55
Mathematics Subject Classification ID: 53C99
Mathematics Subject Classification ID: 58A40
Mathematics Subject Classification ID: 58A15
zbMATH DE Number: 6849392
zbMATH Keywords: discrete moving frame
zbMATH Keywords: discrete invariants
zbMATH Keywords: local and global syzygies of invariants
zbMATH Keywords: multispace
zbMATH Keywords: discrete and smooth Maurer-Cartan invariants
zbMATH Keywords: finite difference calculus of variations
zbMATH Keywords: discrete integrable systems
MaRDI profile type: Publication
full work available at URL: s10208-016-9337-5
OpenAlex ID: W2539747288
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Discrete moving frames on lattice varieties and lattice-based multispaces (DOI)
DOI: 10.1007/S10208-016-9337-5