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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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1435-5345
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Volume 2005, Issue 583

Issues

Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function

Alexander I. Bobenko
  • Institut für Mathematik, Fachbereich II, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany.
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Christian Mercat
  • Département de Mathématiques, Université Montpellier II, Place Eugéne Bataillon, 34095 Montpellier Cedex 5, France.
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yuri B. Suris
  • Institut für Mathematik, Fachbereich II, TU Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany.
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2005-11-07 | DOI: https://doi.org/10.1515/crll.2005.2005.583.117

Abstract

Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Bäcklund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to ℤ d , where d  is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. We introduce the d-dimensional discrete logarithmic function which is a generalization of Kenyon’s discrete Green’s function, and uncover several new properties of this function. We prove that it is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle patterns along the family of isomonodromic discrete power functions.

About the article

Received: 9. Februar 2004

Published Online: 2005-11-07

Published in Print: 2005-06-27


Citation Information: Journal für die reine und angewandte Mathematik, Volume 2005, Issue 583, Pages 117–161, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crll.2005.2005.583.117.

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