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 Tr. Mat. Inst. Steklova, 2004, Volume 247, Pages 186–201 (Mi tm17)

Discrete Connections and Difference Linear Equations

S. P. Novikov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Following earlier works, we develop here a nonstandard discrete analogue of the theory of differential-geometric $GL_{n}$-connections on triangulated manifolds. This theory is based on the interpretation of a connection as a first-order linear difference equation—the “triangle equation”—for scalar functions of vertices in simplicial complexes. This theory appeared as a byproduct of the discretization of famous completely integrable systems such as the 2D Toda lattice. A nonstandard discretization of complex analysis based on these ideas was developed earlier. Here, a complete classification theory based on the mixture of abelian and nonabelian features is given for connections on triangulated manifolds.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 247, 168–183

Bibliographic databases:
UDC: 515.332+515.165.7

Citation: S. P. Novikov, “Discrete Connections and Difference Linear Equations”, Geometric topology and set theory, Collected papers. Dedicated to the 100th birthday of professor Lyudmila Vsevolodovna Keldysh, Tr. Mat. Inst. Steklova, 247, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 186–201; Proc. Steklov Inst. Math., 247 (2004), 168–183

Citation in format AMSBIB
\Bibitem{Nov04} \by S.~P.~Novikov \paper Discrete Connections and Difference Linear Equations \inbook Geometric topology and set theory \bookinfo Collected papers. Dedicated to the 100th birthday of professor Lyudmila Vsevolodovna Keldysh \serial Tr. Mat. Inst. Steklova \yr 2004 \vol 247 \pages 186--201 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm17} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2168170} \zmath{https://zbmath.org/?q=an:1109.39020} \transl \jour Proc. Steklov Inst. Math. \yr 2004 \vol 247 \pages 168--183 

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This publication is cited in the following articles:
1. Vankerschaver J., “Euler–Poincaré reduction for discrete field theories”, Journal of Mathematical Physics, 48:3 (2007), 032902
2. Zivaljevic R.T., “Combinatorial Groupoids, Cubical Complexes, and the Lovasz Conjecture”, Discrete & Computational Geometry, 41:1 (2009), 135–161
3. Proc. Steklov Inst. Math., 273 (2011), 238–251
4. 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
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6. Dabaghian Y., “Maintaining Consistency of Spatial Information in the Hippocampal Network: A Combinatorial Geometry Model”, Neural Comput., 28:6 (2016), 1051–1071
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