Contents ix 6.11.4. Synthesis: ascending from biquadratics hij to multiaffine Q 249 6.11.5. Putting equations Q = 0 on the cube 251 6.12. Classification of discrete integrable 2D systems with fields on vertices. II 252 6.13. Integrable discrete Laplace type equations 256 6.14. Fields on edges: Yang-Baxter maps 261 6.15. Classification of Yang-Baxter maps 266 6.16. Discrete integrable 3D systems 272 6.16.1. Fields on 2-faces. 272 6.16.2. Fields on vertices. 276 6.17. Exercises 279 6.18. Bibliographical notes 286 Chapter 7. Discrete Complex Analysis. Linear Theory 291 7.1. Basic notions of discrete linear complex analysis 291 7.2. Moutard transformation for discrete Cauchy-Riemann equations 294 7.3. Integrable discrete Cauchy-Riemann equations 297 7.4. Discrete exponential functions 300 7.5. Discrete logarithmic function 302 7.6. Exercises 307 7.7. Bibliographical notes 308 Chapter 8. Discrete Complex Analysis. Integrable Circle Patterns 311 8.1. Circle patterns 311 8.2. Integrable cross-ratio and Hirota systems 313 8.3. Integrable circle patterns 316 8.4. za and log z circle patterns 319 8.5. Linearization 324 8.6. Exercises 326 8.7. Bibliographical notes 327 Chapter 9. Foundations 331 9.1. Projective geometry 331 9.2. Lie geometry 335 9.2.1. Objects of Lie geometry 335 9.2.2. Projective model of Lie geometry 336 9.2.3. Lie sphere transformations 339
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