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:	Mathematics, Geometry, Topology, Cosmology, Statistics, Calculus, Math, Mathematical Analysis, Managerial Economics, Algebra, Electromagnetism, Classical Mechanics, Authors Community, Physics, Engineering, Favorites from the National Library of China, Geography, Technology, Economics, Literature, Sociology, Most Popular Books in China, Law, Government, Favorites in India, Education
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