Abstract:
A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time, we started studying first-order difference “black-and-white triangle operators (equations)” on triangulated surfaces with a black-and-white coloring of triangles. In the present work, we develop a theory of these operators and equations showing their similarity to the complex derivatives $\partial$ and $\bar\partial$.
Key words and phrases:
Discrete connection, discrete analog of complex derivatives, triangle equation, first order difference operator.
This publication is cited in the following 34 articles:
Decio Levi, Miguel A. Rodríguez, “Non-Existence of S-Integrable Three-Point Partial Difference Equations in the Lattice Plane”, SIGMA, 19 (2023), 084, 7 pp.
Dmitry Chelkak, Benoît Laslier, Marianna Russkikh, “Dimer model and holomorphic functions on t‐embeddings of planar graphs”, Proceedings of London Math Soc, 126:5 (2023), 1656
M. O. Nesterenko, “Special Exponential Functions on Lattices of Simple Lie Algebras and the Allotropic Modifications of Carbon”, Ukr Math J, 74:3 (2022), 395
Bobenko A.I., Bucking U., “Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals For Ramified Coverings of the Riemann Sphere”, Math. Phys. Anal. Geom., 24:3 (2021), 23
I. A. Dynnikov, “Bounded discrete holomorphic functions on the hyperbolic plane”, Proc. Steklov Inst. Math., 302 (2018), 186–197
Bobenko A.I., Guenther F., “Discrete Riemann Surfaces Based on Quadrilateral Cellular Decompositions”, Adv. Math., 311 (2017), 885–932
D. V. Egorov, “The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis”, Siberian Math. J., 58:1 (2017), 78–79
Ulrike Bücking, Symmetries and Integrability of Difference Equations, 2017, 153
Bobenko A., Skopenkov M., “Discrete Riemann Surfaces: Linear Discretization and Its Convergence”, J. Reine Angew. Math., 720 (2016), 217–250
Alexander I. Bobenko, Felix Günther, Advances in Discrete Differential Geometry, 2016, 57
I. A. Dynnikov, “On a new discretization of complex analysis”, Russian Math. Surveys, 70:6 (2015), 1031–1050
N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187
Cassatella-Contra G.A., Manas M., Tempesta P., “Singularity Confinement For Matrix Discrete Painlevé Equations”, Nonlinearity, 27:9 (2014), 2321–2335
P. G. Grinevich, S. P. Novikov, “Discrete $SL_n$-connections and self-adjoint difference operators on two-dimensional manifolds”, Russian Math. Surveys, 68:5 (2013), 861–887
Tempesta P., “Integrable Maps From Galois Differential Algebras, Borel Transforms and Number Sequences”, J. Differ. Equ., 255:10 (2013), 2981–2995
Skopenkov M., “The Boundary Value Problem for Discrete Analytic Functions”, Adv. Math., 240 (2013), 61–87
Cimasoni D., “Discrete Dirac Operators on Riemann Surfaces and Kasteleyn Matrices”, J. Eur. Math. Soc., 14:4 (2012), 1209–1244
Bobenko A.I., Mercat Ch., Schmies M., “Period Matrices of Polyhedral Surfaces”, Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics, 2013, 2011, 213–226
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