[1]

Baker M. and Norine S.,
Riemann–Roch and Abel–Jacobi theory on a finite graph,
Adv. Math. 215 (2007), no. 2, 766–788.
Google Scholar

[2]

Bobenko A. and Springborn B.,
A discrete Laplace–Beltrami operator for simplicial surfaces,
Discrete Comput. Geom. 38 (2007), 740–756.
Google Scholar

[3]

Bobenko A. I.,
Introduction to compact Riemann surfaces,
Computational approach to Riemann surfaces,
Lecture Notes in Math. 2013,
Springer, Berlin (2011), 3–64.
Google Scholar

[4]

Bobenko A. I., Mercat C. and Schmies M.,
Period matrices of polyhedral surfaces,
Computational approach to Riemann surfaces,
Lecture Notes in Math. 2013,
Springer, Berlin (2011), 213–226.
Google Scholar

[5]

Bobenko A. I., Mercat C. and Suris Y. B.,
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green’s function,
J. reine angew. Math. 583 (2005), 117–161.
Google Scholar

[6]

Bobenko A. I., Pinkall U. and Springborn B.,
Discrete conformal maps and ideal hyperbolic polyhedra,
preprint 2010, http://arxiv.org/abs/1005.2698v1.

[7]

Bohle C., Pedit F. and Pinkall U.,
Discrete holomorphic geometry I. Darboux transformations and spectral curves,
J. reine angew. Math. 637 (2009), 99–139.
Google Scholar

[8]

Bücking U.,
Approximation of conformal mappings by circle patterns,
Geom. Dedicata 137 (2008), 163–197.
Google Scholar

[9]

Chelkak D. and Smirnov S.,
Universality in the 2D Ising model and conformal invariance of fermionic observables,
Invent. Math. 189 (2012), 515–580.
Google Scholar

[10]

Chelkak D. and Smirnov S.,
Discrete complex analysis on isoradial graphs,
Adv. Math. 228 (2011), no. 3, 1590–1630.
Google Scholar

[11]

Courant R., Friedrichs K. and Lewy H.,
Über die partiellen Differentialgleichungen der mathematischen Physik,
Math. Ann. 100 (1928), 32–74.
Google Scholar

[12]

Duffin R. J.,
Discrete potential theory,
Duke Math. J. 20 (1953), 233–251.
Google Scholar

[13]

Duffin R. J.,
Distributed and lumped networks,
J. Math. Mech. 8 (1959), 793–826.
Google Scholar

[14]

Dynnikov I. A. and Novikov S. P.,
Geometry of the triangle equation on two-manifolds,
Mosc. Math. J. 3 (2003), no. 2, 419–438.
Google Scholar

[15]

Ferrand J.,
Fonctions préharmoniques et fonctions préholomorphes,
Bull. Sci. Math. (2) 68 (1944), 152–180.
Google Scholar

[16]

He Z.-X. and Schramm O.,
On the convergence of circle packings to the Riemann map,
Invent. Math. 125 (1996), no. 2, 285–305.
Google Scholar

[17]

Isaacs R.,
A finite difference function theory,
Univ. Nac. Tucuman. Revista A. 2 (1941), 177–201.
Google Scholar

[18]

Kenyon R.,
Conformal invariance of domino tiling,
Ann. Probab. 28 (2000), no. 2, 759–795.
Google Scholar

[19]

Lelong-Ferrand J.,
Représentation conforme et transformations à intégrale de Dirichlet bornée,
Gauthier-Villars, Paris 1955.
Google Scholar

[20]

Mercat C.,
Discrete Riemann surfaces and the Ising model,
Comm. Math. Phys. 218 (2001), no. 1, 177–216.
Google Scholar

[21]

Mercat C.,
Discrete period matrices and related topics,
preprint 2002, http://arxiv.org/abs/math-ph/0111043.

[22]

Mercat C.,
Discrete polynomials and discrete holomorphic approximation,
preprint 2002, http://arxiv.org/abs/math-ph/0206041.

[23]

Mercat C.,
Discrete Riemann surfaces,
Handbook of Teichmüller theory, vol. I,
IRMA Lect. Math. Theor. Phys. 11,
European Mathematical Society, Zürich (2007), 541–575.
Google Scholar

[24]

Meyer M., Desbrun M., Schröder P. and Barr A. H.,
Discrete differential-geometry operators for triangulated 2-manifolds,
Visualization and mathematics III,
Springer, Berlin (2003), 35–57.
Google Scholar

[25]

Pakharev A., Skopenkov M. and Ustinov A.,
Through the resisting net,
Mat. Prosv. 18 (2014), 33–65.
Google Scholar

[26]

Pinkall U. and Polthier K.,
Computing discrete minimal surfaces and their conjugates,
Experiment. Math. 2 (1993), no. 1, 15–36.
Google Scholar

[27]

Rivin I.,
Euclidean structures on simplicial surfaces and hyperbolic volume,
Ann. of Math. (2) 139 (1994), no. 3, 553–580.
Google Scholar

[28]

Rodin B. and Sullivan D.,
The convergence of circle packings to the Riemann mapping,
J. Differential Geom. 26 (1987), no. 2, 349–360.
Google Scholar

[29]

Schramm O.,
Circle patterns with the combinatorics of the square grid,
Duke Math. J. 86 (1997), 347–389.
Google Scholar

[30]

Skopenkov M.,
The boundary value problem for discrete analytic functions,
Adv. Math. 240 (2013), 61–87.
Google Scholar

[31]

Skopenkov M., Prasolov M. and Dorichenko S.,
Dissections of a metal rectangle,
Kvant 3 (2011), 10–16.
Google Scholar

[32]

Skopenkov M., Smykalov V. and Ustinov A.,
Random walks and electric networks,
Mat. Prosv. 16 (2012), 25–47.
Google Scholar

[33]

Smirnov S.,
Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits,
C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239–244.
Google Scholar

[34]

Smirnov S.,
Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model,
Ann. of Math. (2) 172 (2010), no. 2, 1435–1467.
Google Scholar

[35]

Springborn B., Schröder P. and Pinkall U.,
Conformal equivalence of triangle meshes,
ACM SIGGRAPH 2008,
ACM, New York (2008), article no. 77.
Google Scholar

[36]

Stephenson K.,
Introduction to circle packing. The theory of discrete analytic functions,
Cambridge University Press, Cambridge 2005.
Google Scholar

[37]

Thurston W. P.,
The geometry and topology of three-manifolds,
lecture notes 2002, http://library.msri.org/books/gt3m/.

[38]

Troyanov M.,
Les surface euclidiennes à singularités coniques,
Enseign. Math. 32 (1986), 79–94.
Google Scholar

[39]

Wilson S.,
Conformal cochains,
Trans. Amer. Math. Soc. 360 (2008), no. 10, 5247–5264.
Google Scholar

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.