Discrete Moving Frames and Discrete Integrable Systems

Discrete Moving Frames and Discrete Integrable Systems

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...

Discrete Integrable Systems and Geometry

Hyperdeterminants as integrable discrete systems

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability (understood as 4d-consistency) of a nonlinear difference equation defined by the 2×2×2 hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this ...

Duality for discrete integrable systems

A new class of discrete dynamical systems is introduced via a duality relation for discrete dynamical systems with a number of explicitly known integrals. The dual equation can be defined via the difference of an arbitrary linear combination of integrals and its upshifted version. We give an example of an integrable mapping with two parameters and four integrals leading to a (four-dimensional) ...

Discrete Integrable Systems and the Moyal Symmetry

In the study of nonlinear systems, completely integrable systems play special roles. They are not independent at all, but are strongly correlated with each other owing to the large symmetries shared among themselves. We like to know how far we can extend such systems without loosing integrability. The question could be answered if we know how much we can deform the symmetries characterizing the...