Combinatorial Geometry with Applications to Field Theory : Second Edition

By Mao, Linfan

Book Id: WPLBN0002828178
Format Type: PDF eBook:
File Size: 2.70 MB
Reproduction Date: 7/12/2013

Title:	Combinatorial Geometry with Applications to Field Theory : Second Edition
Author:	Mao, Linfan
Volume:	Second Edition
Language:	English
Subject:	Non Fiction, Education, Geometry
Collections:	Authors Community, Mathematics
Historic Publication Date:	2013
Publisher:	World Public Library

Member Page:	Florentin Smarandache
Citation APA MLA Chicago Mao, B. L. (2013). Combinatorial Geometry with Applications to Field Theory : Second Edition. Retrieved from http://self.gutenberg.org/

Description
In The 2nd Conference on Combinatorics and Graph Theory of China (Aug. 16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathematical sciences (abbreviated to CC Conjecture), i.e., a mathematical science can be reconstructed from or made by combinatorialization, implicated in the foreword of Chapter 5 of my book Automorphism groups of Maps, Surfaces and Smarandache Geometries (USA, 2005). This conjecture is essentially a philosophic notion for developing mathematical sciences of 21st century, which means that we can combine different fields into a union one and then determines its behavior quantitatively. It is this notion that urges me to research mathematics and physics by combinatorics, i.e., mathematical combinatorics beginning in 2004 when I was a post-doctor of Chinese Academy of Mathematics and System Science. It finally brought about me one self-contained book, the first edition of this book, published by InfoQuest Publisher in 2009. This edition is a revisited edition, also includes the development of a few topics discussed in the first edition.

Excerpt
1.5 ENUMERATION TECHNIQUES 1.5.1 Enumeration Principle. The enumeration problem on a finite set is to count and find closed formula for elements in this set. A fundamental principle for solving this problem in general is on account of the enumeration principle: For finite sets X and Y , the equality |X| = |Y | holds if and only if there is a bijection f : X → Y . Certainly, if the set Y can be easily countable, then we can find a closed formula for elements in X.

Table of Contents
Contents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . i Chapter 1. Combinatorial Principle with Graphs . . . . . . . . . . 1 1.1 Multi-sets with operations. . . . . . . . . . . . . . . . . . . . .2 1.1.1 Set . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.4 Multi-Set . . . . . . . . . . . . . . . . . . . . . . . . . .8 1.2 Multi-posets . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Partially ordered set . . . . . . . . . . . . . . . . . . . . .11 1.2.2 Multi-Poset . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Countable set . . . . . . . . . . . . . . . . . . . . 16 1.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.1 Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . .18 1.4.2 Subgraph . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4.3 Labeled graph. . . . . . . . . . . . . . . . . . . . 22 1.4.4 Graph family. . . . . . . . . . . . . . . . . . . . .22 1.4.5 Operation on graphs . . . . . . . . . . . . . . . . . . . . 25 1.5 Enumeration techniques. . . . . . . . . . . . . . . . . . . . . .26 1.5.1 Enumeration principle . . . . . . . . . . . . . . . . . . . 26 1.5.2 Inclusion-Exclusion principle . . . . . . . . . . . . . . . . . . . 26 1.5.3 Enumerating mappings . . . . . . . . . . . . . . . . . . 28 1.5.4 Enumerating vertex-edge labeled graphs . . . . . . . . . . . . . . . 30 1.5.5 Enumerating rooted maps . . . . . . . . . . . . . . . . . . . . . . 34 1.5.6 Automorphism groups identity of trees . . . . . . . . . . . . . . . . 36 1.6 Combinatorial principle . . . . . . . . . . . . . . . . . . . . . . 37 1.6.1 Proposition in lgic. . . . . . . . . . . . . . . . . . . . . . .37 1.6.2 Mathematical system. . . . . . . . . . . . . . . . . . . .39 1.6.3 Combinatorial system . . . . . . . . . . . . . . . . . . . 41 1.7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter 2. Algebraic Combinatorics . . . . . . . . . . . . . . . . 47 2.1 Algebraic systems . . . . . . . . . . . . . . . . . . . . .48 2.1.1 Algebraic system. . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.2 Associative and commutative law. . . . . . . . . . . . . . .48 2.1.3 Group. . . . . . . . . . . . . . . . . . . . . . . . . . . .50 2.1.4 Isomorphism of systems . . . . . . . . . . . . . . . . . 50 2.1.5 Homomorphism theorem . . . . . . . . . . . . . . . . 51 2.2 Multi-operation systems . . . . . . . . . . . . . . . . . . . . . 55 2.2.1 Multi-operation system. . . . . . . . . . . . . . . . . . 55 2.2.2 Isomorphism of multi-systems . . . . . . . . . . . . . . . . . . 55 2.2.3 Distribute law. . . . . . . . . . . . . . . . . . . . 58 2.2.4 Multi-group and multi-ring . . . . . . . . . . . . . . . . . . . . . 59 2.2.5 Multi-ideal . . . . . . . . . . . . . . . . . . . . . . . 61 2.3 Multi-modules . . . . . . . . . . . . . . . . . . . . . . . . 62 2.3.1 Multi-module . . . . . . . . . . . . . . . . . . . . 62 2.3.2 Finite dimensional multi-module. . . . . . . . . . . . . . . .66 2.4 Action of multi-groups . . . . . . . . . . . . . . . . . . . . . . . 68 2.4.1 Construction of permutation multi-group . . . . . . . . . . . . . . 68 2.4.2 Action of multi-group . . . . . . . . . . . . . . . . . . . 71 2.5 Combinatorial algebraic systems . . . . . . . . . . . . . . . . . . . . 79 2.5.1 Algebraic multi-system. . . . . . . . . . . . . . . . . . 79 2.5.2 Diagram of multi-system . . . . . . . . . . . . . . . . 81 2.5.3 Cayley diagram . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Chapter 3. Topology with Smarandache Geometry . . . . . . . . . . . . . . 91 3.1 Algebraic topology . . . . . . . . . . . . . . . . . . . . 92 3.1.1 Topological space. . . . . . . . . . . . . . . . . . . . . . . .92 3.1.2 Metric space . . . . . . . . . . . . . . . . . . . . . 95 3.1.3 Fundamental group. . . . . . . . . . . . . . . . . . . . . .96 3.1.4 Seifert and Van-Kampen theorem . . . . . . . . . . . . . 101 3.1.5 Space attached with graphs . . . . . . . . . . . . . . . . . . . 103 3.1.6 Generalized Seifert-Van Kampen theorem . . . . . . . . . . . . 106 3.1.7 Covering space . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1.8 Simplicial homology group . . . . . . . . . . . . . . . . . . . . 115 3.1.9 Surface. . . . . . . . . . . . . . . . . . . . . . . . . .119 3.2 Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2.1 Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . 122 3.2.2 Linear mapping . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2.3 Differential calculus on Rn . . . . . . . . . . . . . . . . . . . . 128 3.2.4 Differential form . . . . . . . . . . . . . . . . . . . . . . . 131 3.2.5 Stokes’ theorem on simplicial complex . . . . . . . . . . . . . . . . 133 3.3 Smarandache manifolds . . . . . . . . . . . . . . . . . . . . . 135 3.3.1 Smarandache geometry. . . . . . . . . . . . . . . . .135 3.3.2 Map geometry . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.3.3 Pseudo-Euclidean space . . . . . . . . . . . . . . . . 143 3.3.4 Smarandache manifold . . . . . . . . . . . . . . . . . 147 3.4 Differentially Smarandache manifolds . . . . . . . . . . . . . . 150 3.4.1 Differential manifold . . . . . . . . . . . . . . . . . . . 150 3.4.2 Differentially Smarandache manifold. . . . . . . . . . . . . . . . . .150 3.4.3 Tangent space on Smarandache manifold . . . . . . . . . . . . . 151 3.5 Pseudo-manifold geometry . . . . . . . . . . . . . . . . . . 154 3.5.1 Pseudo-manifold geometry . . . . . . . . . . . . . . . . . . . . 154 3.5.2 Inclusion in pseudo-manifold geometry . . . . . . . . . . . . . . . 157 3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Chapter 4. Combinatorial Manifolds . . . . . . . . . . . . . . . 162 4.1 Combinatorial space . . . . . . . . . . . . . . . . . . . . . . . . 163 4.1.1 Combinatorial Euclidean space . . . . . . . . . . . . . . . . 163 4.1.2 Combinatorial fan-space . . . . . . . . . . . . . . . . . . . . . . .173 4.1.3 Decomposition space into combinatorial one . . . . . . . . . . 176 4.2 Combinatorial manifolds . . . . . . . . . . . . . . . . . . . . 179 4.2.1 Combinatorial manifold . . . . . . . . . . . . . . . . 179 4.2.2 Combinatorial submanifold . . . . . . . . . . . . . . . . . . . . 185 4.2.3 Combinatorial equivalence. . . . . . . . . . . . . . . . . . . . .188 4.2.4 Homotopy class . . . . . . . . . . . . . . . . . . . . . . . . 190 4.2.5 Euler-Poincar´e characteristic . . . . . . . . . . . . . . . . . . 192 4.3 Fundamental groups of combinatorial manifolds . . . . . . . . . . . 194 4.3.1 Retraction . . . . . . . . . . . . . . . . . . . . . . 194 4.3.2 Fundamental d-group . . . . . . . . . . . . . . . . . . 195 4.3.3 Fundamental group of combinatorial manifold . . . . . . . . 202 4.3.4 Fundamental Group of Manifold. . . . . . . . . . . . . . .205 4.3.5 Homotopy equivalence. . . . . . . . . . . . . . . . . .206 4.4 Homology groups of combinatorial manifolds . . . . . . . . . . . . . . 207 4.4.1 Singular homology group . . . . . . . . . . . . . . . . . . . . . . 207 4.4.2 Relative homology group . . . . . . . . . . . . . . . . . . . . . . 211 4.4.3 Exact chain . . . . . . . . . . . . . . . . . . . . . 212 4.4.4 Homology group of d-dimensional graph . . . . . . . . . . . . . . 213 4.4.5 Homology group of combinatorial manifodl . . . . . . . . . . . 217 4.5 Regular covering of combinatorial manifolds by voltage assignment . . . . . . 218 4.5.1 Action of fundamental group on covering space . . . . . . . . . . . . . . 218 4.5.2 Regular covering of labeled graph . . . . . . . . . . . . . 219 4.5.3 Lifting automorphism of voltage labeled graph . . . . . . . . . . . . . . 222 4.5.4 Regular covering of combinatorial manifold . . . . . . . . . . . 226 4.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Chapter 5. Combinatorial Differential Geometry . . . . . . . . 232 5.1 Differentiable combinatorial manifolds . . . . . . . . . . . . . 233 5.1.1 Smoothly combinatorial manifold. . . . . . . . . . . . . .233 5.1.2 Tangent vector space . . . . . . . . . . . . . . . . . . . 235 5.1.3 Cotangent vector space. . . . . . . . . . . . . . . . .240 5.2 Tensor fields on combinatorial manifolds . . . . . . . . . . . . . . . . . . 240 5.2.1 Tensor on combinatorial manifold . . . . . . . . . . . . . 240 5.2.2 Tensor field on combinatorial manifold . . . . . . . . . . . . . . . 242 5.2.3 Exterior differentation. . . . . . . . . . . . . . . . . .244 5.3 Connections on tensors . . . . . . . . . . . . . . . . . . . . . 247 5.3.1 Connection on tensor . . . . . . . . . . . . . . . . . . .247 5.3.2 Torsion-free tensor . . . . . . . . . . . . . . . . . . . . . 250 5.3.3 Combinatorial Riemannian manifold. . . . . . . . . . . . . . . . . .250 5.4 Curvatures on connection spaces . . . . . . . . . . . . . . . . . . . 252 5.4.1 Combinatorial curvature operator . . . . . . . . . . . . . 252 5.4.2 Curvature tensor on combinatorial manifold . . . . . . . . . . 255 5.4.3 Structural equation . . . . . . . . . . . . . . . . . . . . 257 5.4.4 Local form of curvature tensor . . . . . . . . . . . . . . . . 258 5.5 Curvatures on Riemannian manifolds . . . . . . . . . . . . . . 260 5.5.1 Combinatorial Riemannian curvature tensor . . . . . . . . . . 260 5.5.2 Structural equation in Riemannian manifold. . . . . . . . . . 263 5.5.3 Local form of Riemannian curvature tensor . . . . . . . . . . . 263 5.6 Integration on combinatorial manifolds . . . . . . . . . . . . .265 5.6.1 Determining H (n,m) . . . . . . . . . . . . . . . . . .265 5.6.2 Partition of unity . . . . . . . . . . . . . . . . . . . . . . 266 5.6.3 Integration on combinatorial manifold . . . . . . . . . . . . . . . . 268 5.7 Combinatorial Stokes’ and Gauss’ theorem. . . . . . . . . . . . . . . . 274 5.7.1 Combinatorial Stokes’ theorem. . . . . . . . . . . . . . . . 274 5.7.2 Combinatorial Gauss’ theorem . . . . . . . . . . . . . . . . 278 5.8 Combinatorial Finsler geometry . . . . . . . . . . . . . . . . . . . . 282 5.8.1 Combinatorial Minkowskian norm. . . . . . . . . . . . . 282 5.8.2 Combinatorial Finsler geometry . . . . . . . . . . . . . . . 283 5.8.3 Inclusion in combinatorial Finsler geometry . . . . . . . . . . 284 5.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Chapter 6. Combinatorial Riemannian Submanifolds with Principal Fiber Bundles . . . . . . . . . . . . . . . . . 289 6.1 Combinatorial Riemannian submanifolds . . . . . . . . . . . . . . . . . . 290 6.1.1 Fundamental formulae of submanifold . . . . . . . . . . . . . . . . 290 6.1.2 Local form of fundamental formulae . . . . . . . . . . . . . . . . . . 294 6.2 Fundamental equations on combinatorial submanifolds . . . . . . . . . . . 296 6.2.1 Gauss equation. . . . . . . . . . . . . . . . . . . . . . . . .296 6.2.2 Codazzi equaton . . . . . . . . . . . . . . . . . . . . . . . 297 6.2.3 Ricci equation. . . . . . . . . . . . . . . . . . . . . . . . . .298 6.2.4 Local form of fundamental equation . . . . . . . . . . . . . . . . . . 298 6.3 Embedded combinatorial submanifolds . . . . . . . . . . . . . 300 6.3.1 Embedded combinatorial submanifold . . . . . . . . . . . . . . . . 300 6.3.2 Embedded in combinatorial Euclidean space . . . . . . . . . . 303 6.4 Topological multi-groups . . . . . . . . . . . . . . . . . . . . 309 6.4.1 Topological multi-group . . . . . . . . . . . . . . . . 309 6.4.2 Lie multi-group . . . . . . . . . . . . . . . . . . . . . . . . 315 6.4.3 Homomorphism on lie multi-group . . . . . . . . . . . . 321 6.4.4 Adjoint representation . . . . . . . . . . . . . . . . . 323 6.4.5 Lie multi-subgroup . . . . . . . . . . . . . . . . . . . . . 323 6.4.6 Exponential mapping. . . . . . . . . . . . . . . . . . .324 6.4.7 Action of Lie multi-group . . . . . . . . . . . . . . . . . . . . . 328 6.5 Principal fiber bundles . . . . . . . . . . . . . . . . . . . . . . 332 6.5.1 Principal fiber bundle . . . . . . . . . . . . . . . . . . 332 6.5.2 Combinatorial principal fiber bundle . . . . . . . . . . . . . . . . . 334 6.5.3 Automorphism of principal fiber bundle . . . . . . . . . . . . . . 336 6.5.4 Gauge transformation . . . . . . . . . . . . . . . . . . 338 6.5.5 Connection on principal fiber bundle . . . . . . . . . . . . . . . . . 341 6.5.6 Curvature form on principal fiber bundle . . . . . . . . . . . . . 346 6.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Chapter 7. Fields with Dynamics. . . . . . . . . . . . . . . . . . .351 7.1 Mechanical fields. . . . . . . . . . . . . . . . . . . . .352 7.1.1 Particle dynamic . . . . . . . . . . . . . . . . . . . . . . . 352 7.1.2 Variational principle . . . . . . . . . . . . . . . . . . . 355 7.1.3 Hamiltonian principle . . . . . . . . . . . . . . . . . . 357 7.1.4 Lagrange field . . . . . . . . . . . . . . . . . . . . . . . . . . 355 7.1.5 Hamiltonian field. . . . . . . . . . . . . . . . . . . . . . .360 7.1.6 Conservation law. . . . . . . . . . . . . . . . . . . . . . . 362 7.1.7 Euler-Lagrange equation . . . . . . . . . . . . . . . . . . . . . . 364 7.2 Gravitational field . . . . . . . . . . . . . . . . . . . . . . . . . . 365 7.2.1 Newtonian gravitational field . . . . . . . . . . . . . . . . . . 365 7.2.2 Einstein􀀀s spacetime. . . . . . . . . . . . . . . . . . .366 7.2.3 Einstein gravitational field . . . . . . . . . . . . . . . . . . . . 368 7.2.4 Limitation of Einstein􀀀s equation . . . . . . . . . . . . . 371 7.2.5 Schwarzschild metric . . . . . . . . . . . . . . . . . . . 371 7.2.6 Schwarzschild singularity . . . . . . . . . . . . . . . . . . . . . . 376 7.2.7 Kruskal coordinate . . . . . . . . . . . . . . . . . . . . . 377 7.3 Electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . 378 7.3.1 Electrostatic field . . . . . . . . . . . . . . . . . . . . . . 378 7.3.2 Magnetostatic field . . . . . . . . . . . . . . . . . . . . . 380 7.3.3 Electromagnetic field . . . . . . . . . . . . . . . . . . . 383 7.3.4 Maxwell equation . . . . . . . . . . . . . . . . . . . . . . 385 7.3.5 Electromagnetic field with gravitation . . . . . . . . . . . . . . . . 389 7.4 Gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . 391 7.4.1 Gauge scalar field . . . . . . . . . . . . . . . . . . . . . . 391 7.4.2 Maxwell field. . . . . . . . . . . . . . . . . . . . . . . . . . .393 7.4.3 Weyl field . . . . . . . . . . . . . . . . . . . . . . . 394 7.4.4 Dirac field . . . . . . . . . . . . . . . . . . . . . . 396 7.4.5 Yang-Mills field . . . . . . . . . . . . . . . . . . . . . . . . 399 7.4.6 Higgs mechanism. . . . . . . . . . . . . . . . . . . . . . .401 7.4.7 Geometry of gauge field . . . . . . . . . . . . . . . . 404 7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Chapter 8. Combinatorial Fields with Applications . . . . . . . . . . . . 411 8.1 Combinatorial fields . . . . . . . . . . . . . . . . . . . . . . . . 412 8.1.1 Combinatorial field. . . . . . . . . . . . . . . . . . . . .412 8.1.2 Combinatorial configuration space . . . . . . . . . . . . . 414 8.1.3 Geometry on combinatorial field. . . . . . . . . . . . . . .417 8.1.4 Projective principle in combinatorial field . . . . . . . . . . . . 418 8.2 Equation of combinatorial field. . . . . . . . . . . . . . . . . . . . .420 8.2.1 Lagrangian on combinatorial field . . . . . . . . . . . . . 420 8.2.2 Hamiltonian on combinatorial field . . . . . . . . . . . . 423 8.2.3 Equation of combinatorial field . . . . . . . . . . . . . . . . 427 8.2.4 Tensor equation on combinatorial field . . . . . . . . . . . . . . . 432 8.3 Combinatorial gravitational fields . . . . . . . . . . . . . . . . . . 435 8.3.1 Combinatorial metric. . . . . . . . . . . . . . . . . . .435 8.3.2 Combinatorial Schwarzschild metric . . . . . . . . . . . . . . . . . . 436 8.3.3 Combinatorial Reissner-Nordstr¨om metric . . . . . . . . . . . . 440 8.3.4 Multi-time system. . . . . . . . . . . . . . . . . . . . . .443 8.3.5 Physical condition. . . . . . . . . . . . . . . . . . . . . .445 8.3.6 Parallel probe . . . . . . . . . . . . . . . . . . . . . . . . . . 447 8.3.7 Physical realization . . . . . . . . . . . . . . . . . . . . 448 8.4 Combinatorial gauge fields . . . . . . . . . . . . . . . . . . 450 8.4.1 Gauge multi-basis . . . . . . . . . . . . . . . . . . . . . . 451 8.4.2 Combinatorial gauge basis. . . . . . . . . . . . . . . . . . . . .452 8.4.3 Combinatorial gauge field . . . . . . . . . . . . . . . . . . . . . 454 8.4.4 Geometry on combinatorial gauge field . . . . . . . . . . . . . . . 456 8.4.5 Higgs mechanism on combinatorial gauge field . . . . . . . . 458 8.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 460 8.5.1 Many-body mechanics . . . . . . . . . . . . . . . . . .460 8.5.2 Cosmology . . . . . . . . . . . . . . . . . . . . . . 462 8.5.3 Physical structure . . . . . . . . . . . . . . . . . . . . . . 465 8.5.4 Economical field . . . . . . . . . . . . . . . . . . . . . . . 466 8.5.5 Engineering field . . . . . . . . . . . . . . . . . . . . . . . 467 References . . . . . . . . . . . . . . . . . . . . . . . . 469 Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479