Combinatorial Geometry with Applications to Field Theory

2011·2.6 MB·English

by  Mao, Linfan.

Graduate Textbook in Mathematics
LINFAN MAO
COMBINATORIAL GEOMETRY
WITH APPLICATIONS TO FIELD THEORY
Second Edition
X A X
B
+
E F G H
-
C
X D X
The Education Publisher Inc.
2011
Linfan MAO
Academy of Mathematics and Systems
Chinese Academy of Sciences
Beijing 100190, P.R.China
and
Beijing Institute of Civil Engineering and Architecture
Beijing 100044, P.R.China
Email: [e-mail]
Combinatorial Geometry
with Applications to Field Theory
Second Edition
The Education Publisher Inc.
2011
This book can be ordered from:
The Educational Publisher, Inc.
1313 Chesapeake Ave.
Columbus, Ohio 43212, USA
Toll Free: 1-866-880-5373
E-mail: [e-mail]
Website: www.EduPublisher.com
Peer Reviewers:
F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Bei-
jing 100190, P.R.China
J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, Beijing 100083,
P.R.China
R.X.Hao and W.L.He, Department of Applied Mathematics, Beijing Jiaotong Uni-
versity, Beijing 100044, P.R.China
Tudor Sireteanu, Dinu Bratosin and Luige Vladareanu, Institute of Solid Mechanics
of Romanian Academy, Bucharest, Romania
Copyright 2011 by The Education Publisher Inc. and Linfan Mao
Many books can be downloaded from the following Digital Library of Science:
http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htm
∼
ISBN: 978-1-59973-155-1
Printed in America
Preface to the Second Edition
Accompanied with humanity into the 21st century, a highlight trend for developing
a science is its overlap and hybrid, and harmoniously with other sciences, which
enables one to handle complex systems in the WORLD. This is also for develop-
ing mathematics. As a powerful tool for dealing with relations among objectives,
combinatorics, including combinatorial theory and graphtheory mushroomed in last
century. Its related with algebra, probability theory and geometry has made it to an
important subject in mathematics and interesting results emerged in large number
without metrics. Today, the time is come for applying combinatorial technique to
other mathematics and other sciences besides just to find combinatorial behavior
for objectives. That is the motivation of this book, i.e., to survey mathematics and
fields by combinatorial principle.
In The 2nd Conference on Combinatorics and Graph Theory of China (Aug.
16-19, 2006, Tianjing), I formally presented a combinatorial conjecture on mathe-
matical 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 de-
veloping 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 Chi-
nese 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
ii CombinatorialGeometrywithApplicationstoField
topics discussed in the first edition.
Contents in this edition are outlined following.
Chapters 1 and 2 are the fundamental of this book. In Chapter 1, we briefly
introduce combinatorial principle with graphs, such as those of multi-sets, Boolean
algebra, multi-posets, countable sets, graphs and enumeration techniques, including
inclusion-exclusion principle with applications, enumerating mappings, vertex-edge
labeled graphs and rooted maps underlying a graph. The final section discusses the
combinatorial principle in philosophy and the CC conjecture, also with its implica-
tions for mathematics. All of these are useful in following chapters.
Chapter 2 is essentially an algebraic combinatorics, i.e., an application of com-
binatorial principle to algebraic systems, including algebraic systems, multi-systems
with diagrams. The algebraic structures, such as those of groups, rings, fields and
modules were generalized to a combinatorial one. We also consider actions of multi-
groups on finite multi-sets, which extends a few well-known results in classical per-
mutation groups. Some interesting properties of Cayley graphs of finite groups can
be also found in this chapter.
Chapter3isasurveyoftopologywithSmarandachegeometry. Terminologiesin
algebraic topology, such as those of fundamental groups, covering space, simplicial
homology group and some important results, for example, the Seifert and Van-
Kampen theorem are introduced. For extending application spaces of Seifert and
Van-Kampen theorem, a generalized Seifert and Van-Kampen theorem can be also
found in here. As a preparing for Smarandache n-manifolds, a popular introduction
to Euclidean spaces, differential forms in Rn and the Stokes theorem on simplicial
complexes are presented in Section 3.2. In Section 3.3-3.5, these pseudo-Euclidean
spaces, Smarandache geometry, mapgeometry, Smarandachemanifoldwithdifferen-
tial, principal fiber bundles and geometrical inclusions in pseudo-manifold geometry
are seriously discussed.
Chapters 4 6 are mainly on combinatorial manifolds motivated by the com-
−
binatorial principle on topological or smooth manifolds. In Chapter 4, we discuss
topological behaviors of combinatorial manifolds with characteristics, such as Eu-
clidean spaces and their combinatorial characteristics, topology on combinatorial
manifolds, vertex-edge labeled graphs, Euler-Poincar´e characteristic, fundamental
groups, singular homology groups on combinatorial manifolds or just manifolds and
Preface iii
regular covering of combinatorial manifold by voltage assignment. Some well-known
results in topology, for example, the Mayer-Vietoris theorem on singular homology
groups can be found.
Chapters 5 and 6 form the main parts of combinatorial differential geome-
try, which provides the fundamental for applying it to physics and other sciences.
Chapter 5 discuss tangent and cotangent vector space, tensor fields and exterior dif-
ferentiation on combinatorial manifolds, connections and curvatures on tensors or
combinatorial Riemannian manifolds, integrations and the generalization of Stokes’
and Gauss’ theorem, and so on. Chapter 6 contains three parts. The first concen-
trates on combinatorial submanifold of smooth combinatorial manifolds with fun-
damental equations. The second generalizes topological groups to multiple one, for
example Lie multi-groups. The third is a combinatorial generalization of principal
fiber bundles to combinatorial manifolds by voltage assignment technique, which
provides the mathematical fundamental for discussing combinatorial gauge fields in
Chapter 8.
Chapters7and8introducetheapplicationsofcombinatorialmanifoldstofields.
For this objective, variational principle, Lagrange equations and Euler-Lagrange
equations in mechanical fields, Einstein’s general relativity with gravitational field,
Maxwell field and Abelian or Yang-Mills gauge fields are introduced in Chapter 7.
Applying combinatorial geometry discussed in Chapters 4 6, we then generalize
−
fields to combinatorial fields under the projective principle, i.e., a physics law in
a combinatorial field is invariant under a projection on its a field in Chapter 8.
Then, we show how to determine equations of combinatorial fields by Lagrange
density, to solve equations of combinatorial gravitational fields and how to construct
combinatorialgaugebasisandfields. Elementary applicationsofcombinatorialfields
to many-body mechanics, cosmology, physical structure, economical or engineering
fields can be also found in this chapter.
This edition is preparing beginning from July, 2010. All of these materials are
valuable for researchers or graduate students in topological graph theory with enu-
meration, topology, Smarandache geometry, Riemannian geometry, gravitational or
quantum fields, many-bodysystem andgloballyquantifying economy. Forpreparing
this book, many colleagues and friends of mine have given me enthusiastic support
and endless helps. Without their help, this book will never appears today. Here I
iv CombinatorialGeometrywithApplicationstoField
must mention some of them. On the first, I would like to give my sincerely thanks
to Dr.Perze for his encourage and endless help. Without his suggestion, I would do
some else works, cannot investigate mathematical combinatorics for years and finish
this book. Second, I would like to thank Professors Feng Tian, Yanpei Liu, Mingyao
Xu, Fuji Zhang, Jiyi Yan and Wenpeng Zhang for them interested in my research
works. Their encourage and warmhearted support advance this book. Thanks are
also given to Professors Han Ren, Junliang Cai, Yuanqiu Huang, Rongxia Hao,
Deming Li, Wenguang Zai, Goudong Liu, Weili He and Erling Wei for their kindly
helps and often discussing problems in mathematics altogether. Partially research
results of mine were reported at Chinese Academy of Mathematics & System Sci-
ences, Beijing Jiaotong University, Beijing Normal University, East-China Normal
University and Hunan Normal University in past years. Some of them were also re-
portedatThe 2ndand3rd Conference on Graph Theory and Combinatorics of China
in 2006 and 2008, The 3rd and 4th International Conference on Number Theory and
Smarandache’s Problems of Northwest of China in 2007 and 2008. My sincerely
thanks are also give to these audiences discussing mathematical topics with me in
these periods.
Of course, I am responsible for the correctness all of these materials presented
here. Any suggestions for improving this book and solutions for open problems in
this book are welcome.
L.F.Mao
July, 2011
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
§
vi CombinatorialGeometrywithApplicationstoField
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
§
Contents vii
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