Combinatorial geometry with application to field theory

✍ Scribed by Mao L.

• Publisher: IQuest
• Year: 2009
• Tongue: English
• Leaves: 499
• Category: Library

✦ Synopsis

This monograph is motivated with surveying mathematics and physics by CC conjecture, i.e., a mathematical science can be reconstructed from or made by combinatorialization. Topics covered in this book include fundamental of mathematical combinatorics, differential Smarandache n-manifolds, combinatorial or differentiable manifolds and submanifolds, Lie multi-groups, combinatorial principal fiber bundles, gravitational field, quantum fields with their combinatorial generalization, also with discussions on fundamental questions in epistemology. All of these are valuable for researchers in combinatorics, topology, differential geometry, gravitational or quantum fields.

✦ Table of Contents

Preface......Page 4
Contents......Page 8
1. Combinatorics with Graphs......Page 18
§1.1 SETS WITH OPERATIONS......Page 19
§1.2 PARTIALLY ORDERED SETS......Page 29
§1.3 COUNTABLE SETS......Page 33
§1.4 GRAPHS......Page 36
§1.5 ENUMERATION......Page 44
§1.6 REMARKS......Page 52
2. Fundamental of Mathematical Combinatorics......Page 55
§2.1 COMBINATORIAL SYSTEMS......Page 56
§2.2 ALGEBRAIC SYSTEMS......Page 63
§2.3 MULTI-OPERATION SYSTEMS......Page 70
§2.4 MULTI-MODULES......Page 78
§2.5 ACTIONS OF MULTI-GROUPS......Page 84
§2.6 COMBINATORIAL ALGEBRAIC SYSTEMS......Page 95
§2.7 REMARKS......Page 103
3. Smarandache manifolds......Page 105
§3.1 TOPOLOGICAL SPACES......Page 106
§3.2 EUCLIDEAN GEOMETRY......Page 128
§3.3 SMARANDACHE N-MANIFOLDS......Page 143
§3.4 DIFFERENTIAL SMARANDACHE MANIFOLDS......Page 157
§3.5 PSEUDO-MANIFOLD GEOMETRY......Page 162
§3.6 REMARKS......Page 167
4. Combinatorial Manifolds......Page 171
§4.1 COMBINATORIAL SPACES......Page 172
§4.2 COMBINATORIAL MANIFOLDS......Page 189
§4.3 FUNDAMENTAL GROUPS OF COMBINATORIAL MANIFOLDS......Page 204
§4.4 HOMOLOGY GROUPS OF COMBINATORIAL MANIFOLDS......Page 213
§4.5 REGULAR COVERING OF COMBINATORIAL MANIFOLDS BY VOLTAGE ASSIGNMENT......Page 225
§4.6 REMARKS......Page 236
5. Combinatorial Diff
erential Geometry......Page 239
§5.1 DIFFERENTIABLE COMBINATORIAL MANIFOLDS......Page 240
§5.2 TENSOR FIELDS ON COMBINATORIAL MANIFOLDS......Page 248
§5.3 CONNECTIONS ON TENSORS......Page 255
§5.4 CURVATURES ON CONNECTION SPACES......Page 260
§5.5 CURVATURES ON RIEMANNIAN MANIFOLDS......Page 269
§5.6 INTEGRATION ON COMBINATORIAL MANIFOLDS......Page 274
§5.7 COMBINATORIAL STOKES’ AND GAUSS’ THEOREMS......Page 283
§5.8 COMBINATORIAL FINSLER GEOMETRY......Page 293
§5.9 REMARKS......Page 295
6. Combinatorial Riemannian Submanifolds with Principal Fibre Bundles......Page 299
§6.1 COMBINATORIAL RIEMANNIAN SUBMANIFOLDS......Page 300
§6.2 FUNDAMENTAL EQUATIONS ON COMBINATORIAL SUBMANIFOLDS......Page 307
§6.3 EMBEDDED COMBINATORIAL SUBMANIFOLDS......Page 311
§6.4 TOPOLOGICAL MULTI-GROUPS......Page 320
§6.5 PRINCIPAL FIBRE BUNDLES......Page 345
§6.6 REMARKS......Page 361
7. Fields with Dynamics......Page 364
§7.1 MECHANICAL FIELDS......Page 365
§7.2 GRAVITATIONAL FIELD......Page 379
§7.3 ELECTROMAGNETIC FIELD......Page 395
§7.4 GAUGE FIELD......Page 409
§7.5 REMARKS......Page 427
8. Combinatorial Fields with Applications......Page 430
§8.1 COMBINATORIAL FIELDS......Page 431
§8.2 EQUATION OF COMBINATORIAL FIELD 8.2.1 Lagrangian on Combinatorial Field.......Page 439
§8.3 COMBINATORIAL GRAVITATIONAL FIELDS......Page 455
§8.4 COMBINATORIAL GAUGE FIELDS......Page 469
§8.5 APPLICATIONS......Page 479
References......Page 485
C......Page 494
G......Page 495
M......Page 496
R......Page 497
Y......Page 498