Quantum geometry of three-dimensional lattices

VV Bazhanov, VV Mangazeev… - Journal of Statistical …, 2008 - iopscience.iop.org
We study geometric consistency relations between angles on three-dimensional (3D)
circular quadrilateral lattices—lattices whose faces are planar quadrilaterals inscribable into …

Lattice geometry of the discrete Darboux, KP, BKP and CKP equations. Menelaus' and Carnot's theorems

WK Schief - Journal of Nonlinear Mathematical Physics, 2003 - Taylor & Francis
Möbius invariant versions of the discrete Darboux, KP, BKP and CKP equations are derived
by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional …

Desargues maps and the Hirota–Miwa equation

A Doliwa - Proceedings of the Royal Society A …, 2010 - royalsocietypublishing.org
We study the Desargues maps, which generate lattices whose points are collinear with all
their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the …

Hermite–Padé approximation and integrability

A Doliwa, A Siemaszko - Journal of Approximation Theory, 2023 - Elsevier
We show that solution to the Hermite–Padé type I approximation problem leads in a natural
way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of …

Discrete differential geometry. Consistency as integrability

AI Bobenko, YB Suris - arXiv preprint math/0504358, 2005 - arxiv.org
A new field of discrete differential geometry is presently emerging on the border between
differential and discrete geometry. Whereas classical differential geometry investigates …

Non-commutative lattice-modified Gel'fand–Dikii systems

A Doliwa - Journal of Physics A: Mathematical and Theoretical, 2013 - iopscience.iop.org
We introduce integrable multicomponent non-commutative lattice systems, which can be
considered as analogues of the modified Gel'fand–Dikii hierarchy. We present the …

Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

A Doliwa, P Grinevich, M Nieszporski… - Journal of mathematical …, 2007 - pubs.aip.org
We present the sublattice approach, a procedure to generate, from a given integrable lattice,
a sublattice which inherits its integrability features. We consider, as illustrative example of …

Difference systems in bond and face variables and non-potential versions of discrete integrable systems

P Kassotakis, M Nieszporski - Journal of Physics A: Mathematical …, 2018 - iopscience.iop.org
Integrable discrete scalar equations defined on a 2D or 3D lattice can be rewritten as
difference systems in bond variables or in face variables, respectively. Both the difference …

The pentagon relation and incidence geometry

A Doliwa, SM Sergeev - Journal of Mathematical Physics, 2014 - pubs.aip.org
D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we
show that such a map provides solutions to the functional dynamical pentagon equation. We …

[HTML][HTML] The B-quadrilateral lattice, its transformations and the algebro-geometric construction

A Doliwa - Journal of Geometry and Physics, 2007 - Elsevier
The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa's discrete BKP
equation within the quadrialteral lattice (QL) theory. After discussing the projective-geometric …