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Discrete moving frames on lattice varieties and lattice based multispace

Beffa, Gloria Marì, Mansfield, Elizabeth L. (2016) Discrete moving frames on lattice varieties and lattice based multispace. Foundations of Computational Mathematics, . ISSN 1615-3375. E-ISSN 1615-3383. (doi:10.1007/s10208-016-9337-5)

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In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we 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, we construct multispace, a generalization of the jet bundle that also generalizes Olver’s one dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We 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. We 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 our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.

Item Type: Article
DOI/Identification number: 10.1007/s10208-016-9337-5
Uncontrolled keywords: discrete moving frame, discrete invariants, local and global syzygies of invariants, multispace, discrete and smooth Maurer–Cartan invariants, finite difference calculus of variations, discrete integrable systems
Subjects: Q Science > QA Mathematics (inc Computing science) > QA440 Geometry
Q Science > QA Mathematics (inc Computing science) > QA440 Geometry > QA611 Topology
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Elizabeth L Mansfield
Date Deposited: 29 Sep 2016 15:09 UTC
Last Modified: 09 Jul 2019 13:30 UTC
Resource URI: (The current URI for this page, for reference purposes)
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