viii Contents 4.5.6. Circular minimal nets and nets with constant mean curvature 178 4.6. Exercises 179 4.7. Bibliographical notes 183 Chapter 5. Approximation 187 5.1. Discrete hyperbolic systems 187 5.2. Approximation in discrete hyperbolic systems 190 5.3. Convergence of Q-nets 196 5.4. Convergence of discrete Moutard nets 197 5.5. Convergence of discrete asymptotic nets 199 5.6. Convergence of circular nets 200 5.7. Convergence of discrete K-surfaces 205 5.8. Exercises 206 5.9. Bibliographical notes 207 Chapter 6. Consistency as Integrability 209 6.1. Continuous integrable systems 210 6.2. Discrete integrable systems 213 6.3. Discrete 2D integrable systems on graphs 215 6.4. Discrete Laplace type equations 217 6.5. Quad-graphs 218 6.6. Three-dimensional consistency 220 6.7. From 3D consistency to zero curvature representations and acklund transformations 222 6.8. Geometry of boundary value problems for integrable 2D equations 227 6.8.1. Initial value problem 228 6.8.2. Extension to a multidimensional lattice 231 6.9. 3D consistent equations with noncommutative fields 235 6.10. Classification of discrete integrable 2D systems with fields on vertices. I 239 6.11. Proof of the classification theorem 242 6.11.1. 3D consistent systems, biquadratics and tetrahedron property 242 6.11.2. Analysis: descending from multiaffine Q to quartic r 245 6.11.3. Synthesis: ascending from quartic r to biquadratic h 247
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