Smarandache Geometries

SMARANDACHE GEOMETRIES & MAP THEORY WITH APPLICATIONS(I) Linfan MAO

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This book can be ordered in a paper bound reprint from:

Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N.Zeeb Road P.O.Box 1346, Ann Arbor MI 48106-1346, USA Tel:1-800-521-0600(Customer Service) http://wwwlib.umi.com/bod

Peer Reviewers: Y.P.Liu,Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China.

F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100080, P.R.China.

J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, Beijing 100083, P.R.China.

E.L.Wei, Information school, China Renmin University, Beijing 100872, P.R.China.

Copyright 2006 by Chinese Branch Xiquan House 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:1-59973-019-7 Printed in China

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Preface

As a powerful tool, also a useful language, mathematics has become a scientific foundation for realizing or finding new laws in the natural world. This characteristic implies that a more important task for mathematics is to bring benefit to mankind by applying mathematics to or establishing new mathematical systems for solving various mathematical problems in sciences or the natural world.

Solving problems by applying the mathematical method is the center of mathematics. As we known, these 23 problems asked by Hilbert in the beginning of the 20th century have produced more power for the development of mathematics in last century, and the unified filed theory initiated by Einstein in his later years advanced the theoretical physics and helped to bring about the string/M-theory in 80s of the 20th century, which increases the ability of human beings to comprehend the universe.

Sciences are developing, also advancing. A true conclusion in one time maybe a falsehood in another time. Whence, that thinking sciences is an absolutely truth is not right, which includes the mathematics. Modern sciences are so advanced getting into the 21st century that to find a universal genus in the society of sciences is nearly impossible. Thereby a mathematician can only give his or her contribution in one or several mathematical fields. The frequent crossing and combination of different subjects of sciences have become a main trend in realizing our natural world because our natural world itself is overlapping and combinatorial. In this situation, to make the combination of different branches of the classical mathematics so that it can bring benefit to mankind and scientific research is a burning issue of the moment.

What is the mathematics of the 21st century? The mathematics of the 21st century is the combinatorization with its generalization for classical mathematics,

vi

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

also the result for mathematics consistency with the scientific research in the 21st century. In the mathematics of 21st century, we can encounter some incorrect conclusions in classical mathematics maybe true in this time, and even a claim with its non-claim are true simultaneously in a new mathematical system.

For introducing the combinatorization for classical mathematics, this collection contains 10 papers finished by the author or the author with other mathematicians.

All these papers have been published in an e-print form on the internet unless two

The Symposia of 2004 Postdoctoral Forum on Frontal & Interdisciplinary Sciences of Chinese Academy of Sciences, Dec. 2004, Beijing , another reported at The 2005 International Conference on Graph Theory and Combinatorics of China, June, 2005, Zejiang . papers, one reported at

Now we outline contents in each paper.

Combinatorial speculations and the combinatorial conjecture for mathematics is a survey paper submitted to The 2th Conference on Graph Theory and Combinatorics of China . This paper introduces the idea of combinatoThe

rial conjecture for mathematics and the contributions of combinatorial speculations to mathematics such as those of algebra and geometries and to combinatorial cosmoses, particularly for 5 or 6-dimensional cosmos based on the materials in my two monographs published recently, i.e., Automorphism groups of maps, surfaces and Smarandache geometries (American Research Press, 2005) and Smarandache multi-space theory (Hexis, America, 2006).

The mathematics of 21st century aroused by theoretical physicsSmarandache multi-space theory is a paper for introducing the background, approaches and results appeared in mathematics of the 21st century, such as those of Big Bang in cosmological physics, Smarandache multi-spaces, Smarandache geometries, maps, map geometries and pseudo-metric space geometries, also includes discussion for some open problems in theoretical physics. This paper is reported to teachers and students of Wanyuan School in Mar. 2006, also a paper submitted to

The 2th Conference on Graph Theory and Combinatorics of China .

A new view of combinatorial maps by Smarandacheâ&#x20AC;&#x2122;s notion is a speculation paper for combinatorial maps by applying that of Smarandacheâ&#x20AC;&#x2122;s notion, reported at the Chinese Academy of Mathematics and System Science and the

Department of Applied Mathematics of Beijing Jiaotong University in May, 2005.

Preface

vii

This paper firstly introduces the conception of map geometries, which are generalization of these 2-dimensional Smarandache manifolds defined by Dr.Iseri, also includes some elementary properties and classification for map geometries. Open problems for the combinatorization of some results in classical mathematics are also given in the final section of this paper, which are benefit for mathematicians researching or wish researching the combinatorization problem for classical mathematics.

An introduction to Smarandache geometries on maps is a survey paper reported at The 2005 International Conference on Graph Theory and Combinatorics of China . This paper introduces maps, map geometries, particularly the necessary and sufficient conditions for parallel bundles in planar map geometries, which are generalized works for the 5th postulate in Euclid plane geometry.

A multi-space model for Chinese bid evaluation with analyzing and

A mathematical model for Chinese bid evaluation with its solution analyzing are two papers coped with the suggestion of Dr.Perze, the editor of American Research Press after I published a Chinese book Chinese Construction Project Bidding Technique & Cases Analyzing-Smarandache Multi-Space Model of Bidding in Xiquan Publishing House (2006). These papers firstly constructed a mathematical model for bids evaluation system, pointed out that it is a decision problem for finite multiple objectives under the law and regulations system for tendering, also gave a graphical method for analyzing the order of bids. Some open problems for weighted Smarandache multi-spaces and suggestions for solving problems existed in current bids evaluation system in China are presented in the final section.

The number of complete maps on surfaces and On automorphisms and enumeration of maps of Cayley graphs of a finite group are two papers finished in May, 2004 and Nov. 2001. Applying the group action idea, these papers enumerate the unrooted complete maps and non-equivalent maps underlying a Cayley graph of a finite group on orientable and non-orientable surfaces, which generalize a scheme for enumerating non-equivalent embeddings of a graph presented

by White et.al. They are applications of classical mathematics to combinatorics.

A combinatorial refinement of Hurwitz theorem on Riemann Surfaces is a paper by applying combinatorial maps to determine automorphisms of

Riemann surfaces and getting combinatorial refinement for some classical results.

viii

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

The Symposia of 2004 Postdoctoral Forum on Frontal & Interdisciplinary Sciences of Chinese Academy of Sciences . It is the applications This paper is submitted to

of combinatorics to classical mathematics. It is due to this paper that I presented the combinatorial conjecture for mathematics and open combinatorial problems for Riemann surfaces, Riemann geometry, differential geometry and Riemann manifolds in my monograph Automorphism groups of maps, surfaces and Smarandache geometries.

The mathematical steps of mine is a paper for encouraging young teach-

ers and students, reported at my old school Sichuan Wanyuan school in Mar. 2006.

This paper historically recalls each step that I passed from a scaffold erector to a mathematician, including the period in Wanyuan school, in a construction company, in Northern Jiaotong University, in Chinese Academy of Sciences and in Guoxin Tendering Co.LTD. The social contact of mine with some mathematicians and the process for raising the combinatorial conjecture for mathematics is also called to mind.

Certainly, there are rights and obligations for a scientist such as those of the choice of research theme and research methods is freedom without limitation; all scientists are equal before research themes regardless their position in our society; to participate and publish scientific results is freedom, can not be rejected if the results disagrees with or contradicts preferred theory or not favor with the editors, the referees, or other expert censors; every scientist bears a moral responsibility in her or his research, can not allow her/his research work injurious to mankind.

Recently, Prof. Dimtri Rabounski, the chief editor of Progress in Physics , issued an open letter Declaration of academic freedom: scientific human

rights in Vol. 1,2006, to clarify those opinions for scientific research. This is a precondition for sciences in the 21st century and the harmonizing development of the human society with the natural world, also an indispensable path for developing mathematics of 21st century and bringing benefit to mankind by mathematics.

The Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers is a serial collections in publication. Papers on the following 6 fields are welcome.

(1) Metrization of graphs and combinatorics, or the combinatorization for classical mathematics;

ix

Preface

(2) Multi-spaces, including algebraic multi-spaces, multi-metric spaces, nonEuclid geometry, modern differential geometry; (3) Topological graphs and combinatorial maps with applications in mathematics and physics; (4) General relativity theory with its applications to cosmological physics; (5) Mathematics theory in string/M-theory; (6) Other new models for the universe and its mathematical theory.

All these submitted papers can be directly sent to me by email or by post. My

maolinfan@163.com and my post address is Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100080, P.R.China .

email address is:

L.F.Mao Aug. 2006 in Beijing

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

1. Combinatorial speculations and the combinatorial conjecture for mathematics 1

_ _QXVI _ _vZX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. /\</7 o=m !%-'I -Smarandache = 1/\ . . . . . . . . . . . . . . . . . 24 The mathematics of 21st century aroused by theoretical physics . . . . . . . . . . . . . . . .24 3. A new view of combinatorial maps by Smarandache’s notion . . . . . . . . . . . . . . . .47 Smarandache

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4. An introduction to Smarandache geometries on maps . . . . . . . . . . . . . . . . . . . . . . . 73

D = Smarandache ℄7\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5. A multi-space model for Chinese bid evaluation with analyzing . . . . . . . . . . . . . . 84

Æ L7,sI == 1 q ^P A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6. Æ L7,sI = I q ^P A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A mathematical model for Chinese bid evaluation with its solution analyzing . . 104 7. The number of complete maps on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

av h D = !I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8. Automorphisms and enumeration of maps of Cayley graphs of a finite group147

%LSk = Cayley =D =W 3 >!I . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents

9. Riemann

xi

av Hurwitz R / ;_ _Ă&#x2020;E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A combinatorial refinement of Hurwitz theorem on Riemann Surfaces. . . . . . . . . .167 10.

2 = I $ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

The mathematical steps of mine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183 About the author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

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Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

Combinatorial Speculations and the Combinatorial Conjecture for Mathematicsâ&#x2C6;&#x2014; Linfan Mao (Chinese Academy of Mathematics and System Sciences, Beijing 100080) maolinfan@163.com

Abstract: Combinatorics is a powerful tool for dealing with relations among objectives mushroomed in the past century. However, an even more important work for mathematician is to apply combinatorics to other mathematics and other sciences beside just to find combinatorial behavior for objectives.

In the past few years, works of this kind frequently appeared on journals for mathematics and theoretical physics for cosmos. The main purpose of this paper is to survey these thinking and ideas for mathematics and cosmological physics, such as those of multi-spaces, map geometries and combinatorial cosmoses, also the combinatorial conjecture for mathematics proposed by myself in 2005. Some open problems are included for the advance of 21st mathematics by a combinatorial speculation.

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Key words: combinatorial speculation, combinatorial conjecture for mathematics, Smarandache multi-space, M-theory, combinatorial cosmos.

Classification: AMS(2000) 03C05,05C15,51D20,51H20,51P05,83C05,83E50.

1. The role of classical combinatorics in mathematics Modern science has so advanced that to find a universal genus in the society of sciences is nearly impossible. Thereby a scientist can only give his or her contribution in one or several fields. The same thing also happens for researchers in combinatorics.

Generally, combinatorics deals with twofold: Question 1.1. to determine or find structures or properties of configurations, such as those structure results appeared in graph theory, combinatorial maps and design theory,..., etc..

Question 1.2. to enumerate configurations, such as those appeared in the enumeration of graphs, labelled graphs, rooted maps, unrooted maps and combinatorial designs,...,etc..

Consider the contribution of a question to science. We can separate mathematical questions into three ranks: Rank 1 they contribute to all sciences.

Rank 2 they contribute to all or several branches of mathematics.

Rank 3 they contribute only to one branch of mathematics, for instance, just to the graph theory or combinatorial theory.

Classical combinatorics is just a rank 3 mathematics by this view. This conclusion is despair for researchers in combinatorics, also for me 4 years ago. Whether can combinatorics be applied to other mathematics or other sciences? Whether can it contribute to humanâ&#x20AC;&#x2122;s lives, not just in games?

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

3

Although become a universal genus in science is nearly impossible, our world is a combinatorial world. A combinatorician should stand on all mathematics and all sciences, not just on classical combinatorics and with a real combinatorial notion, i.e., combining different fields into a unifying field ([25]-[28]), such as combine different or even anti branches in mathematics or science into a unifying science for its freedom of research ([24]). This notion requires us answering three questions for solving a combinatorial question before. What is this question working for? What is its objective? What is its contribution to science or human’s society? After these works be well done, modern combinatorics can applied to all sciences and all sciences are combinatorization.

2. The combinatorics metrization and mathematics combinatorization There is a prerequisite for the application of combinatorics to other mathematics and other sciences, i.e, to introduce various metrics into combinatorics, ignored by the classical combinatorics since they are the fundamental of scientific realization for our world. This speculation is firstly appeared in the beginning of Chapter 5 of my book [16]: · · · our world is full of measures. For applying combinatorics to other branch of mathematics, a good idea is pullback measures on combinatorial objects again, ignored by the classical combinatorics and reconstructed or make combinatorial generalization for the classical mathematics, such as those of algebra, differential geometry, Riemann geometry, Smarandache geometries, · · · and the mechanics, theoretical physics, · · ·.

The combinatorial conjecture for mathematics, abbreviated to CCM is stated in the following.

Conjecture 2.1(CCM Conjecture) Mathematics can be reconstructed from or made by combinatorization.

Remark 2.1 We need some further clarifications for this conjecture. (i) This conjecture assumes that one can select finite combinatorial rulers and axioms to reconstruct or make generalization for classical mathematics.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

(ii) Classical mathematics is a particular case in the combinatorization of mathematics, i.e., the later is a combinatorial generalization of the former. (iii) We can make one combinatorization of different branches in mathematics and find new theorems after then.

Therefore, a branch in mathematics can not be ended if it has not been combinatorization and all mathematics can not be ended if its combinatorization has not completed. There is an assumption in one’s realization of our world, i.e., every science can be made mathematization. Whence, we similarly get the combinatorial conjecture for science.

Conjecture 2.2(CCS Conjecture) Science can be reconstructed from or made by combinatorization.

A typical example for the combinatorization of classical mathematics is the combinatorial map theory, i.e., a combinatorial theory for surfaces([14]-[15]). Combinatorially, a surface is topological equivalent to a polygon with even number of edges by identifying each pairs of edges along a given direction on it. If label each pair of edges by a letter e, e ∈ E, a surface S is also identifying to a cyclic permutation such that each edge e, e ∈ E just appears two times in S, one is e and another is e−1 . Let a, b, c, · · · denote the letters in E and A, B, C, · · · the sections of successive letters in a linear order on a surface S (or a string of letters on S). Then, a surface can be represented as follows: S = (· · · , A, a, B, a−1 , C, · · ·), where, a ∈ E,A, B, C denote a string of letters. Define three elementary transformations as follows: (O1 )

(A, a, a−1 , B) ⇔ (A, B);

(O2 )

(i) (A, a, b, B, b−1 , a−1 ) ⇔ (A, c, B, c−1 ); (ii) (A, a, b, B, a, b) ⇔ (A, c, B, c);

(O3 )

(i) (A, a, B, C, a−1 , D) ⇔ (B, a, A, D, a−1, C); (ii) (A, a, B, C, a, D) ⇔ (B, a, A, C −1 , a, D −1 ).

If a surface S can be obtained from S0 by these elementary transformations

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

5

O1 -O3 , we say that S is elementary equivalent with S0 , denoted by S ∼El S0 . Then we can get the classification theorem of compact surface as follows([29]): Any compact surface is homeomorphic to one of the following standard surfaces: (P0 ) the sphere: aa−1 ; (Pn ) the connected sum of n, n ≥ 1 tori: −1 −1 −1 −1 −1 a1 b1 a−1 1 b1 a2 b2 a2 b2 · · · an bn an bn ;

(Qn ) the connected sum of n, n ≥ 1 projective planes: a1 a1 a2 a2 · · · an an .

A map M is a connected topological graph cellularly embedded in a surface S. In 1973, Tutte suggested an algebraic representation for an embedding graph on a locally orientable surface ([16]): A combinatorial map M = (Xα,β , P) is defined to be a basic permutation P, i.e, for any x ∈ Xα,β , no integer k exists such that P k x = αx, acting on Xα,β , the disjoint union of quadricells Kx of x ∈ X (the base set), where K = {1, α, β, αβ} is the Klein group satisfying the following two conditions: (i) αP = P −1 α; (ii) the group ΨJ =< α, β, P > is transitive on Xα,β .

For a given map M = (Xα,β , P), it can be shown that M ∗ = (Xβ,α , Pαβ) is also a map, call it the dual of the map M. The vertices of M are defined as the pairs of conjugatcy orbits of P action on Xα,β by the condition (i) and edges the orbits of K on Xα,β , for example, for ∀x ∈ Xα,β , {x, αx, βx, αβx} is an edge of the map M. Define the faces of M to be the vertices in the dual map M ∗ . Then the Euler characteristic χ(M) of the map M is χ(M) = ν(M) − ε(M) + φ(M) where,ν(M), ε(M), φ(M) are the number of vertices, edges and faces of the map M, respectively. For each vertex of a map M, its valency is defined to be the length of the orbits of P action on a quadricell incident with u.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

For example, the graph K4 on the tours with one face length 4 and another 8 shown in Fig.2.1

Fig.1. the graph K4 on the tours can be algebraically represented by (Xα,β , P) with Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} and

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw) × (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv) with 4 vertices, 6 edges and 2 faces on an orientable surface of genus 1.

By the view of combinatorial maps, these standard surfaces P0 , Pn , Qn for n ≥ 1 is nothing but the bouquet Bn on a locally orientable surface with just one face.

Therefore, combinatorial maps are the combinatorization of surfaces.

Many open problems are motivated by the CCM Conjecture. For example, a Gauss mapping among surfaces is defined as follows.

Let S ⊂ R3 be a surface with an orientation N. The mapping N :S → R3 takes its value in the unit sphere S 2 = {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1} along the orientation N. The map N :S → S 2 , thus defined, is called the Gauss mapping.

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

7

we know that for a point P ∈ S such that the Gaussian curvature K(P ) 6= 0 and V a connected neighborhood of P with K does not change sign, N(A) , A→0 A

K(P ) = lim

where A is the area of a region B ⊂ V and N(A) is the area of the image of B by the Gauss mapping N : S → S 2 ([2],[4]). Now the questions are (i) what is its combinatorial meaning of the Gauss mapping? How to realizes it by combinatorial maps? (ii) how can we define various curvatures for maps and rebuilt the results in the classical differential geometry?

Let S be a compact orientable surface. Then the Gauss-Bonnet theorem asserts that Z Z

S

Kdσ = 2πχ(S),

where K is the Gaussian curvature of S. By the CCM Conjecture, the following questions should be considered. (i) How can we define various metrics for combinatorial maps, such as those of length, distance, angle, area, curvature,· · ·? (ii) Can we rebuilt the Gauss-Bonnet theorem by maps for dimensional 2 or higher dimensional compact manifolds without boundary?

One can see references [15] and [16] for more open problems for the classical mathematics motivated by this CCM Conjecture, also raise new open problems for his or her research works.

3. The contribution of combinatorial speculation to mathematics 3.1. The combinatorization of algebra By the view of combinatorics, algebra can be seen as a combinatorial mathematics itself. The combinatorial speculation can generalize it by the means of combinatorization. For this objective, a Smarandache multi-algebraic system is combinatorially defined in the following definition.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Definition 3.1([17],[18]) For any integers n, n ≥ 1 and i, 1 ≤ i ≤ n, let Ai be a set with an operation set O(Ai) such that (Ai , O(Ai)) is a complete algebraic system.

Then the union n [

(Ai , O(Ai))

i=1

is called an n multi-algebra system.

An example of multi-algebra system is constructed by a finite additive group.

Now let n be an integer, Z1 = ({0, 1, 2, · · · , n − 1}, +) an additive group (modn) and P = (0, 1, 2, · · · , n − 1) a permutation. For any integer i, 0 ≤ i ≤ n − 1, define Zi+1 = P i (Z1 ) satisfying that if k + l = m in Z1 , then P i(k) +i P i (l) = P i (m) in Zi+1 , where +i denotes the binary operation +i : (P i (k), P i(l)) → P i (m). Then we know that n [

Zi

i=1

is an n multi-algebra system .

The conception of multi-algebra systems can be extensively used for generalizing conceptions and results in the algebraic structure, such as those of groups, rings, bodies, fields and vector spaces, · · ·, etc.. Some of them are explained in the following. n e = S G be a complete multi-algebra system with a binary Definition 3.2 Let G i i=1

e = {× , 1 ≤ i ≤ n}. If for any integer i, 1 ≤ i ≤ n, (G ; × ) is a operation set O(G) i i i e and any two binary operations group and for ∀x, y, z ∈ G

× and ◦ , × =6 ◦,

there is one operation, for example the operation × satisfying the distribution law to the operation

◦ provided their operation results exist , i.e., x × (y ◦ z) = (x × y) ◦ (x × z),

(y ◦ z) × x = (y × x) ◦ (z × x), e is called a multi-group. then G

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

9

e O(G)), G f ⊂ G e and O(G f ) ⊂ O(G), e call (G f , O(G f )) For a multi-group (G, 1 1 1 1

e O(G)) if G f is also a multi-group under the operations in a sub-multi-group of (G, 1 f ), denoted by G f G. e For two sets A and B, if A T B = ∅, we denote the O(G 1 1

union A

S

B by A

finite group.

L

B. Then we get a generalization of the Lagrange theorem of

f of a finite multi-group G, e there is Theorem 3.1([18]) For any sub-multi-group H e such that a representation set T , T ⊂ G, e = G

M

x∈T

f xH.

f of G, e × ∈ O(H) f and ∀g ∈ G(×), e f For a sub-multi-group H if for ∀h ∈ H, f g × h × g −1 ∈ H,

f a normal sub-multi-group of G. e An order of operations in O(G) e is said then call H − → e an oriented operation sequence, denoted by O (G).

We get a generalization of the

Jordan-H¨older theorem for finite multi-groups.

n S

e = Theorem 3.2([18]) For a finite multi-group G Gi and an oriented operai=1 − → e tion sequence O (G), the length of maximal series of normal sub-multi-groups is a e itself. constant, only dependent on G

e Then G e is a body. If (G ; × ) In Definition 2.2, choose n = 2, G1 = G2 = G. 1 1

e is a field. For multi-algebra and (G2 ; ×2 ) both are commutative groups, then G

system with two or more operations on one set, we introduce the conception of multi-rings and multi-vector spaces in the following. e = Definition 3.3 Let R

m S

i=1

Ri be a complete multi-algebra system with double binary

e = {(+ , × ), 1 ≤ i ≤ m}. If for any integers i, j, i 6= j, 1 ≤ i, j ≤ operation set O(R) i i e m, (Ri ; +i , ×i ) is a ring and for ∀x, y, z ∈ R,

(x +i y) +j z = x +i (y +j z), and

(x ×i y) ×j z = x ×i (y ×j z)

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

x ×i (y +j z) = x ×i y +j x ×i z,

(y +j z) ×i x = y ×i x +j z ×i x

e is called a multi-ring. If for any provided all their operation results exist, then R e is called a multi-filed. integer 1 ≤ i ≤ m, (R; +i , ×i ) is a filed, then R

k S Definition 3.4 Let Ve = Vi be a complete multi-algebra system with binary i=1

operation set

O(Ve )

k S ˙ i , ·i ) | 1 ≤ i ≤ m} and Fe = = {(+ Fi a multi-filed with i=1

double binary operation set O(Fe ) = {(+i , ×i ) | 1 ≤ i ≤ k}. If for any integers i, j, 1 ≤ i, j ≤ k and ∀a, b, c ∈ Ve , k1 , k2 ∈ Fe , ˙ i , ·i) is a vector space on Fi with vector additive + ˙ i and scalar multi(i) (Vi ; + plication ·i ; ˙ i b)+ ˙ j c = a+ ˙ i (b+ ˙ j c); (ii) (a+ (iii) (k1 +i k2 ) ·j a = k1 +i (k2 ·j a); provided all those operation results exist, then Ve is called a multi-vector space on the multi-filed Fe with a binary operation set O(Ve ), denoted by (Ve ; Fe ).

Similar to multi-groups, we can also obtain results for multi-rings and multi-

vector spaces to generalize classical results in rings or linear spaces. Certainly, results can be also found in the references [17] and [18].

3.2. The combinatorization of geometries First, we generalize classical metric spaces by the combinatorial speculation. m f = S M such that each M is a Definition 3.5 A multi-metric space is a union M i i

space with metric ρi for ∀i, 1 ≤ i ≤ m.

i=1

We generalized two well-known results in metric spaces. m f = S M be a completed multi-metric space. For an ǫTheorem 3.3([19]) Let M i i=1

disk sequence {B(ǫn , xn )}, where ǫn > 0 for n = 1, 2, 3, · · ·, the following conditions

hold: (i) B(ǫ1 , x1 ) ⊃ B(ǫ2 , x2 ) ⊃ B(ǫ3 , x3 ) ⊃ · · · ⊃ B(ǫn , xn ) ⊃ · · ·; (ii)

lim ǫn = 0.

n→+∞

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

Then

+∞ T

n=1

11

B(ǫn , xn ) only has one point.

f = Theorem 3.4([19]) Let M f . Then contraction on M

m S

i=1

Mi be a completed multi-metric space and T a

1 ≤# Φ(T ) ≤ m.

Particularly, let m = 1. We get the Banach fixed-point theorem again.

Corollary 3.1(Banach) Let M be a metric space and T a contraction on M. Then T has just one fixed point.

Smarandache geometries were proposed by Smarandache in [25] which are generalization of classical geometries, i.e., these Euclid, Lobachevshy-Bolyai-Gauss and Riemann geometries may be united altogether in a same space, by some Smarandache geometries under the combinatorial speculation. These geometries can be either partially Euclidean and partially Non-Euclidean, or Non-Euclidean. In general, Smarandache geometries are defined in the next.

Definition 3.6 An axiom is said to be Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969).

For example, let us consider an euclidean plane R2 and three non-collinear points A, B and C. Define s-points as all usual euclidean points on R2 and slines as any euclidean line that passes through one and only one of points A, B and C. Then this geometry is a Smarandache geometry because two axioms are Smarandachely denied comparing with an Euclid geometry: (i) The axiom (A5) that through a point exterior to a given line there is only one parallel passing through it is now replaced by two statements: one parallel and no parallel. Let L be an s-line passing through C and is parallel in the euclidean sense to AB. Notice that through any s-point not lying on AB there is one s-line parallel to L and through any other s-point lying on AB there is no s-lines parallel

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

to L such as those shown in Fig.3.1(a).

Fig.2. an example of Smarandache geometry (ii) The axiom that through any two distinct points there exists one line passing through them is now replaced by; one s-line and no s-line. Notice that through any two distinct s-points D, E collinear with one of A, B and C, there is one s-line passing through them and through any two distinct s-points F, G lying on AB or non-collinear with one of A, B and C, there is no s-line passing through them such as those shown in Fig.3.1(b).

A Smarandache n-manifold is an n-dimensional manifold that support a Smarandache geometry. Now there are many approaches to construct Smarandache manifolds for n = 2. A general way is by the so called map geometries without or with boundary underlying orientable or non-orientable maps proposed in references [14] and [15] firstly.

Definition 3.7 For a combinatorial map M with each vertex valency≥ 3, associates a real number µ(u), 0 < µ(u) <

4π , ρM (u)

to each vertex u, u ∈ V (M). Call (M, µ) a

map geometry without boundary, µ(u) an angle factor of the vertex u and orientablle or non-orientable if M is orientable or not.

Definition 3.8 For a map geometry (M, µ) without boundary and faces f1 , f2 , · · · , fl ∈ F (M), 1 ≤ l ≤ φ(M) − 1, if S(M) \ {f1 , f2 , · · · , fl } is connected, then call (M, µ)−l = (S(M) \ {f1 , f2 , · · · , fl }, µ) a map geometry with boundary f1 , f2 , · · · , fl , where S(M) denotes the locally orientable surface underlying map M. The realization for vertices u, v, w ∈ V (M) in a space R3 is shown in Fig.3.2, where ρM (u)µ(u) < 2π for the vertex u, ρM (v)µ(v) = 2π for the vertex v and ρM (w)µ(w) > 2π for the vertex w, are called to be elliptic, euclidean or hyperbolic,

13

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

respectively.

ρM (u)µ(u) < 2π

ρM (u)µ(u) = 2π

ρM (u)µ(u) > 2π

Fig.3. the realization of an elliptic, euclidean or hyperbolic point in R3 On an Euclid plane R2 , a straight line passing through an elliptic or a hyperbolic point is shown in Fig.3.3.

Fig.4. a straight line passes through an elliptic or hyperbolic point Theorem 3.5([17]) There are Smarandache geometries, including paradoxist geometries, non-geometries and anti-geometries in map geometries without or with boundary.

Generally, we can ever generalize the ideas in Definitions 3.7 and 3.8 to a metric space and find new geometries.

Definition 3.9 Let U and W be two metric spaces with metric ρ, W ⊆ U. For ∀u ∈ U, if there is a continuous mapping ω : u → ω(u), where ω(u) ∈ Rn for an integer n, n ≥ 1 such that for any number ǫ > 0, there exists a number δ > 0 and a point v ∈ W , ρ(u − v) < δ such that ρ(ω(u) − ω(v)) < ǫ, then U is called a metric pseudo-space if U = W or a bounded metric pseudo-space if there is a number N > 0 such that ∀w ∈ W , ρ(w) ≤ N, denoted by (U, ω) or (U − , ω), respectively.

For the case n = 1, we can also explain ω(u) being an angle function with

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

0 < ω(u) ≤ 4π as in the case of map geometries without or with boundary, i.e., ω(u) =

  

ω(u)(mod4π), if u ∈ W, if u ∈ U \ W

2π,

(∗)

and get some interesting metric pseudo-space geometries. For example, let U = W = Euclid plane = P

P

, then we obtained some interesting results for pseudo-plane

geometries ( , ω) as shown in the following([17]).

P

Theorem 3.6 In a pseudo-plane ( , ω), if there are no euclidean points, then all P

points of ( , ω) is either elliptic or hyperbolic.

Theorem 3.7 There are no saddle points and stable knots in a pseudo-plane plane P

( , ω).

Theorem 3.8 For two constants ρ0 , θ0 , ρ0 > 0 and θ0 6= 0, there is a pseudo-plane P

( , ω) with ω(ρ, θ) = 2(π −

ρ0 ρ0 ) or ω(ρ, θ) = 2(π + ) θ0 ρ θ0 ρ

such that ρ = ρ0 P

is a limiting ring in ( , ω).

Now for an m-manifold M m and ∀u ∈ M m , choose U = W = M m in Definition 3.9 for n = 1 and ω(u) a smooth function. We get a pseudo-manifold geometry (M m , ω) on M m . By definitions in the reference [2], a Minkowski norm on M m is a function F : M m → [0, +∞) such that (i)

F is smooth on M m \ {0};

(ii) F is 1-homogeneous, i.e., F (λu) = λF (u) for u ∈ M m and λ > 0; (iii) for ∀y ∈ M m \ {0}, the symmetric bilinear form gy : M m × M m → R with gy (u, v) =

1 ∂ 2 F 2 (y + su + tv) |t=s=0 2 ∂s∂t

is positive definite and a Finsler manifold is a manifold M m endowed with a function F : T M m → [0, +∞) such that

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

15

S

(i) F is smooth on T M m \ {0} = {Tx M m \ {0} : x ∈ M m }; (ii) F |Tx M m → [0, +∞) is a Minkowski norm for ∀x ∈ M m .

As a special case, we choose ω(x) = F (x) for x ∈ M m , then (M m , ω) is a Finsler manifold. Particularly, if ω(x) = gx (y, y) = F 2 (x, y), then (M m , ω) is a Riemann manifold. Therefore, we get a relation for Smarandache geometries with Finsler or Riemann geometry.

Theorem 3.9 There is an inclusion for Smarandache, pseudo-manifold, Finsler and Riemann geometries as shown in the following:

{Smarandache geometries} ⊃ {pseudo − manif old geometries} ⊃ {F insler geometry} ⊃ {Riemann geometry}.

4. The contribution of combinatorial speculation to theoretical physics The progress of theoretical physics in last twenty years of the 20th century enables human beings to probe the mystic cosmos: where are we came from? where are we going to?. Today, these problems still confuse eyes of human beings. Accompanying with research in cosmos, new puzzling problems also arose: Whether are there finite or infinite cosmoses? Is just one? What is the dimension of our cosmos? We do not even know what the right degree of freedom in the universe is, as Witten said([3]).

We are used to the idea that our living space has three dimensions: length, breadth and height, with time providing the fourth dimension of spacetime by Einstein. Applying his principle of general relativity, i.e. all the laws of physics take the same form in any reference system and equivalence principle, i.e., there are no difference for physical effects of the inertial force and the gravitation in a field small enough., Einstein got the equation of gravitational field

where Rµν = Rνµ

1 Rµν − Rgµν + λgµν = −8πGTµν .

2 α = Rµiν , α Rµiν

∂Γiµi ∂Γiµν = − + Γαµi Γiαν − Γαµν Γiαi , ν i ∂x ∂x

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

1 ∂gmp ∂gnp ∂gmn Γgmn = g pq ( n + m − ) 2 ∂u ∂u ∂up and R = g νµ Rνµ .

Combining the Einstein’s equation of gravitational field with the cosmological principle, i.e., there are no difference at different points and different orientations at a point of a cosmos on the metric 104 l.y. , Friedmann got a standard model of cosmos. The metrics of the standard cosmos are dr 2 ds = −c dt + a (t)[ + r 2 (dθ2 + sin2 θdϕ2 )] 2 1 − Kr 2

2

2

2

and R2 (t) , gφφ = −r 2 R2 (t) sin2 θ.

1 − Kr 2 The standard model of cosmos enables the birth of big bang model of our cosmos gtt = 1, grr = −

in thirties of the 20th century. The following diagram describes the developing process of our cosmos in different periods after the big bang.

Fig.5. the evolution of our cosmos

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

17

4.1. The M-theory The M-theory was established by Witten in 1995 for the unity of those five already known string theories and superstring theories, which postulates that all matter and energy can be reduced to branes of energy vibrating in an 11 dimensional space, then in a higher dimensional space solve the Einstein’s equation of gravitational field under some physical conditions ([1],[3],[22]-[23]). Here, a brane is an object or subspace which can have various spatial dimensions. For any integer p ≥ 0, a pbrane has length in p dimensions. For example, a 0-brane is just a point or particle; a 1-brane is a string and a 2-brane is a surface or membrane, · · ·.

We mainly discuss line elements in differential forms in Riemann geometry. By a geometrical view, these p-branes in M-theory can be seen as volume elements in spaces. Whence, we can construct a graph model for p-branes in a space and combinatorially research graphs in spaces.

Definition 4.1 For each m-brane B of a space Rm , let (n1 (B), n2 (B), · · · , np (B)) be its unit vibrating normal vector along these p directions and q : Rm → R4 a continuous mapping. Now construct a graph phase (G, ω, Λ) by V (G) = {p − branes q(B)},

E(G) = {(q(B1 ), q(B2))|there is an action between B1 and B2 },

ω(q(B)) = (n1 (B), n2 (B), · · · , np (B)), and Λ(q(B1), q(B2 )) = f orces between B1 and B2 .

Then we get a graph phase (G, ω, Λ) in R4 . Similarly, if m = 11, it is a graph phase for the M-theory.

As an example for applying M-theory to find an accelerating expansion cosmos of 4-dimensional cosmos from supergravity compactification on hyperbolic spaces is the Townsend-Wohlfarth type metric in which the line element is

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

ds2 = e−mφ(t) (−S 6 dt2 + S 2 dx23 ) + rC2 e2φ(t) ds2Hm , where φ(t) =

1 (ln K(t) − 3λ0 t), m−1

m

m+2

S 2 = K m−1 e− m−1 λ0 t

and K(t) = with ζ =

λ0 ζrc (m − 1) sin[λ0 ζ|t + t1 |]

q

3 + 6/m. This solution is obtainable from space-like brane solution and

if the proper time ς is defined by dς = S 3 (t)dt, then the conditions for expansion and acceleration are

dS dς

> 0 and

d2 S dς 2

> 0. For example, the expansion factor is 3.04

if m = 7, i.e., a really expanding cosmos.

According to M-theory, the evolution picture of our cosmos started as a perfect 11 dimensional space. However, this 11 dimensional space was unstable. The original 11 dimensional space finally cracked into two pieces, a 4 and a 7 dimensional cosmos.

The cosmos made the 7 of the 11 dimensions curled into a tiny ball, allowing the remaining 4 dimensional cosmos to inflate at enormous rates.

4.2. The combinatorial cosmos The combinatorial speculation made the following combinatorial cosmos([17]).

Definition 4.2 A combinatorial cosmos is constructed by a triple (Ω, ∆, T ), where Ω=

[

i≥0

Ωi ,

∆=

[

Oi

i≥0

and T = {ti ; i ≥ 0} are respectively called the cosmos, the operation or the time set with the following conditions hold.

(1) (Ω, ∆) is a Smarandache multi-space dependent on T , i.e., the cosmos (Ωi , Oi) is dependent on time parameter ti for any integer i, i ≥ 0.

(2) For any integer i, i ≥ 0, there is a sub-cosmos sequence (S) : Ωi ⊃ · · · ⊃ Ωi1 ⊃ Ωi0

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

19

in the cosmos (Ωi , Oi ) and for two sub-cosmoses (Ωij , Oi ) and (Ωil , Oi), if Ωij ⊃ Ωil , then there is a homomorphism ρΩij ,Ωil : (Ωij , Oi ) → (Ωil , Oi ) such that (i) for ∀(Ωi1 , Oi), (Ωi2 , Oi)(Ωi3 , Oi) ∈ (S), if Ωi1 ⊃ Ωi2 ⊃ Ωi3 , then ρΩi1 ,Ωi3 = ρΩi1 ,Ωi2 ◦ ρΩi2 ,Ωi3 , where

◦ denotes the composition operation on homomorphisms.

(ii) for ∀g, h ∈ Ωi , if for any integer i, ρΩ,Ωi (g) = ρΩ,Ωi (h), then g = h. (iii) for ∀i, if there is an fi ∈ Ωi with

for integers i, j, Ωi any integer i.

T

ρΩi ,Ωi T Ωj (fi ) = ρΩj ,Ωi T Ωj (fj )

Ωj 6= ∅, then there exists an f ∈ Ω such that ρΩ,Ωi (f ) = fi for

By this definition, there is just one cosmos Ω and the sub-cosmos sequence is

− − R4 ⊃ R3 ⊃ R2 ⊃ R1 ⊃ R0 = {P } ⊃ R− 7 ⊃ · · · ⊃ R1 ⊃ R0 = {Q}.

in the string/M-theory. In Fig.4.1, we have shown the idea of the combinatorial cosmos.

Fig.6. an example of combinatorial cosmoses For 5 or 6 dimensional spaces, it has been established a dynamical theory by this combinatorial speculation([20][21]). In this dynamics, we look for a solution in

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

the Einstein equation of gravitational field in 6-dimensional spacetime with a metric of the form

2

2

2

2

ds = −n (t, y, z)dt + a (t, y, z)d

2 X

+b2 (t, y, z)dy 2 + d2 (t, y, z)dz 2

k

where d

P2

k

represents the 3-dimensional spatial sections metric with k = −1, 0, 1 re-

spective corresponding to the hyperbolic, flat and elliptic spaces. For 5-dimensional spacetime, deletes the undefinite z in this metric form. Now consider a 4-brane moving in a 6-dimensional Schwarzschild-ADS spacetime, the metric can be written as ds2 = −h(z)dt2 +

2 z2 X d +h−1 (z)dz 2 , l2 k

where d

2 X

=

k

dr 2 + r 2 dΩ2(2) + (1 − kr 2 )dy 2 1 − kr 2

and z2 M − 3. l2 z Then the equation of a 4-dimensional cosmos moving in a 6-spacetime is h(z) = k +

¨ κ4(6) 2 κ4(6) R R˙ κ 5 + 3( )2 = −3 ρ − ρp − 3 2 − 2 R R 64 8 R l by applying the Darmois-Israel conditions for a moving brane. Similarly, for the 2

case of a(z) 6= b(z), the equations of motion of the brane are

¨ d d˙R˙ − dR 2

q

1 + d2 R˙ 2

−

q

κ4(6) 1 + d2 R˙ 2 ∂z n 2 ˙ ˙ (dn˙ R + − (d∂z n − n∂z d)R ) = − (3(p + ρ) + pˆ), n d 8 q

κ4(6) ∂z a 1 + d2 R˙ 2 = − (ρ + p − pˆ), ad 8 q

κ4(6) ∂z b 2 2 ˙ 1+d R =− (ρ − 3(p − pˆ)), bd 8

21

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

where the energy-momentum tensor on the brane is 1 Tˆµν = hνα Tµα − T hµν 4 with Tµα = diag(−ρ, p, p, p, pˆ) and the Darmois-Israel conditions [Kµν ] = −κ2(6) Tˆµν , where Kµν is the extrinsic curvature tensor.

The combinatorial cosmos also presents new questions to combinatorics, such as: (i) to embed a graph into spaces with dimensional≥ 4; (ii) to research the phase space of a graph embedded in a space; (iii) to establish graph dynamics in a space with dimensional≥ 4, · · ·, etc..

For example, we have gotten the following result for graphs in spaces in [17].

Theorem 4.1 A graph G has a nontrivial including multi-embedding on spheres P1 ⊃ P2 ⊃ · · · ⊃ Ps if and only if there is a block decomposition G = such that for any integer i, 1 < i < s,

s U

i=1

Gi of G

(i) Gi is planar; (ii) for ∀v ∈ V (Gi ), NG (x) ⊆ (

i+1 S

j=i−1

V (Gj )).

Further research of the combinatorial cosmos will richen the knowledge of combinatorics and cosmology, also get the combinatorization for cosmology.

References [1] I.Antoniadis, Physics with large extra dimensions: String theory under experimental test, Current Science, Vol.81, No.12, 25(2001),1609-1613 [2] S.S.Chern and W.H.Chern, Lectures in Differential Geometry(in Chinese), Peking University Press, 2001.

[3] M.J.Duff, A layman’s guide to M-theory, arXiv: hep-th/9805177, v3, 2 July(1998).

[4] B.J.Fei, Relativity Theory and Non-Euclid Geometries, Science Publisher Press, Beijing, 2005.

22

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

[5] U.G¨ unther and A.Zhuk, Phenomenology of brane-world cosmological models, arXiv: gr-qc/0410130.

[6] S.Hawking, A Brief History of Times, A Bantam Books/November, 1996.

[7] S.Hawking, The Universe in Nutshell, A Bantam Books/November, 2001.

[8] D.Ida, Brane-world cosmology, arXiv: gr-qc/9912002.

[9] H.Iseri, Smarandache Manifolds, American Research Press, Rehoboth, NM,2002.

[10] H.Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm. edu/˜smarandache/Howard-Iseri-paper.htm.

[11] M.Kaku, Hyperspace: A Scientific Odyssey through Parallel Universe, Time Warps and 10th Dimension, Oxford Univ. Press.

[12] P.Kanti, R.Madden and K.A.Olive, A 6-D brane world model, arXiv: hepth/0104177.

[13] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Canada, Vol.12(2003) [14] L.F.Mao, On Automorphisms groups of Maps, Surfaces and Smarandache geometries, Sientia Magna, Vol.1(2005), No.2, 55-73.

[15] L.F.Mao, A new view of combinatorial maps by Smarandache’s notion, arXiv: math.GM/0506232.

[16] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research Press, 2005.

[17] L.F.Mao, Smarandache multi-space theory, Hexis, Phoenix, AZ 2006.

[18] L.F.Mao, On algebraic multi-groups, Sientia Magna, Vol.2(2006), No.1, 64-70.

[19] L.F.Mao, On multi-metric spaces, Sientia Magna, Vol.2(2006), No.1, 87-94.

[20] E.Papantonopoulos, Braneworld cosmological models, arXiv: gr-qc/0410032.

[21] E.Papantonopoulos, Cosmology in six dimensions, arXiv: gr-qc/0601011.

[22] J.A.Peacock, Cosmological Physics, Cambridge University Press, 2003.

[23] J.Polchinski, String Theory, Vol.1-Vol.2, Cambridge University Press, 2003.

[24] D.Rabounski, Declaration of academic freedom: Scientific Human Rights, Progress in Physics, January (2006), Vol.1, 57-60.

[25] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

[26] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth, 1999.

Combinatorial Speculations and the Combinatorial Conjecture for Mathematics

23

[27] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297384.

[28] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438.

[29] J.Stillwell, Classical topology and combinatorial group theory, Springer-Verlag New York Inc., (1980).

24

H'^5<GI1 /

Linfan Mao:

kvYk )|.O2t Smarandche eSju gG s (. {G~EGt d{GNj~ 100080)

â&#x2C6;&#x2014;

U

'â&#x201E;&#x201E;z m4;G&R w/ 0Ă&#x2020; H=â&#x201E;&#x201E; rI Ă&#x2020; H eâ&#x201E;&#x201E; 0 2 k~ 4 Ă&#x2020; H= 'T r m4;G G } 1 m7 0Ă&#x2020; 00 0'Ă&#x2020; Ă&#x2020; H= ) ,# I eâ&#x201E;&#x201E; 0'N0 rFH .

2

Theory of everything

2

2

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/

M-

=:pI u &Bj'FH 0 mT>' J m7G&â&#x201E;&#x201E;4;' Ă&#x2020; H`fI l , hOkuI Ă&#x2020; 'ha rFH= :p I lG&â&#x201E;&#x201E;4;'FH 0 +: Ă&#x2020; H= P y 0u M- 0I ' ) FH=[I â&#x201E;&#x201E; 2 a " {G&â&#x201E;&#x201E;4; FH' 0 m7Smarandache3 A 0 hjm'B- 7 0Ă&#x2020; H=Sk~' S {â&#x201E;&#x201E; 2 FH 0 h`f }Q 0Ă&#x2020; ' ~Z S&C ' 9ZJ' r e,X â&#x201E;&#x201E; 0' " g Ok'9Z â&#x201E;&#x201E; L) 9ZTLa0 9 ZOk ), / ay N FHL) hĂ&#x2020; / l#'l; S m 9 Z r S 7a < 'â&#x201E;&#x201E; ?: [16] / ' Y

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The Mathematics of 21st Century Aroused by Theoretical Physics Abstract. Begin with 20s in last century, physicists devote their works to establish a unified field theory for physics, i.e., the Theory of Everything. The

t -FYM EGti/GAM(# 2006 8 j uW }( Ă&#x2020;ZS 66 4 2006 3 jĂ&#x2020; e-print: . {6/ $ 200607-91 1

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25

aim is near in 1980s while the String/M-theory has been established. They also realize the bottleneck for developing the String/M-theory is there are no applicable mathematical theory for their research works.

the Problem is

that 21st-century mathematics has not been invented yet , They said. In fact, mathematician has established a new theory, i.e., the Smarandache multi-space theory applicable for their needing while the the String/M-theory was established. The purpose of this paper is to survey its historical background, main thoughts, research problems, approaches and some results based on the monograph [16] of mine. We can find the central role of combinatorial speculation in this process.

.V WB) 7 /\ M- /\ = 1 D â&#x201E;&#x201E; Smarandache â&#x201E;&#x201E; ( ^ 1 â&#x201E;&#x201E; Finsler â&#x201E;&#x201E; #i8 AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50 §1.

vĂ&#x2020;~k8 1sS9

Xj*e eh 1^Ct wf " t = x, = y = z }%&eh 1 ,C }&aI[swf d L (x, y, z) W h 1^Ct #1wf wK" #1H % t }%&W h 1 ,^C d L (x, y, z, t) wf eh 1 >Hv: 1.1 #1 %&HvI A }p+u %&# 8= W B rh%$ 1 &W B= %&Mv (section) 1.1.

1.1. d LI = #1Hv p+ = o %9M{ p+ = W B 1V v W h 11% 0= p+ = 11 3 & = wK) #1H }1 4 & =, l1 Einstein = # >

26 1.2.

Linfan Mao:

& w ju

H'^5<GI1 /

&pp+IJ = >k oN1 Einstein =79rz </ (?7BW B=) 7 /\ ; /\r% W B CdP%&â&#x201E;&#x201E;UP O h = ynâ&#x201E;&#x201E; O 1 & 1rL Qy< = Hv ' &yn = L = â&#x201E;&#x201E; 1 o B7 _ 8 V 3 | - O hK 8 V _ B ` ; 5 . < 4 u Lâ&#x201E;&#x201E; 137 . = vr BVv = X j W B &p Hawking =>H 7 o L +UPK 7 G%p W= OK ([6] â&#x2C6;&#x2019; [7]) w 1.2 b*

1.2. K 7 G% = p W 1.3 7 WJwEBW B F ) 7 ~( = vV+ &/ o Vvp+> e ;8= W B L

1.3. W B) 7 = L

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27

&p ) 7 /\ = ![OK W B1 ,k = vL )0 wK W B) 7 ~(P%& â&#x201E;&#x201E; 1 %&g 137 . N7 = _( â&#x201E;&#x201E; e= C(#1% 0 eL5S = - ^ [5S )= q ^ NzV C* # = I [</ = FWM{ 2 # = [ #1 7 ~( 1" t+ ! ) h * #1 10 x q ^ 1 10 kg/m - ^ C 8 10 K # = W B yn i V d ~ + L#1 1[ r {* #1 10 x - ^ C 8 10 K W Bo BI 6 > & g u 10 x=#1 ) B 10 < 6 7CBI `p = 8 V o #\l7 9* % = 9 - $ 0 FP L R9 {9[ K 9 Z z) ~ o = 8 V "kL` %#< =V IV>e = L ) [s z =V Y h P u =V >k- ^ C ? > I ==V [ u=V U[ i y lKKB + L 7V [ 4V i l V 8u=V= V d ~ = P# = W B * #1 10 x - ^ % 10 K * #1 10 x -^% 10 K <1 R C { C[ K C D =` ~ W B7 PB' ; 8 V FP- ^ 10 K , 8 V= " # 1 Z 1 _= eo LUp- Ni Up L v _ V #zk P &"* #1 1 â&#x2C6;&#x2019; 10 x -^C/ g 10 â&#x2C6;&#x2019; 5 Ã&#x2014; 10 K 4V[u4V KV [ KV U[# i y o B ) = C V 7 V , u7 V 8 V ~(O _ _ V Y ,C V = r) P (pola >-Q[W B O3 = 9u 1 _) 5m * #1 3 8 - ^ C / g 10 K & g+ 8 g 1 C ) _ L â&#x201E;&#x201E;} < 4 _ %P Y u :â&#x201E;&#x201E; ,_#k L 4 = 8 V I[C V I = Ag % 8 B 10 q ^ sl ) P< 4 q ^ m 1% R * ,_#k - ^ C ?8 10 K ' ; 8 V uUp- N7 5 V \ *V Up L v =i y Ã&#x2020; #C, W B7 ' ; 8 V I = ^ l_P0 R FP #- ^ sl C V L \ 5 = ^qâ&#x201E;&#x201E; _ vr = _ V !k =KV 1d b, 2 #kL P_ V [ X * #1 1000 â&#x2C6;&#x2019; 2000 -^C/g 10 K <4q^~( ) P q ^ FPW B= + C V8% qâ&#x201E;&#x201E;% H 5w%& r = _ V # Z 5 s7C=h ;(b: i '>Rt )i 7_ FP sX ^ â&#x201E;&#x201E;W B= +

iD = ) dC V= Rt"lV a )i V a 7_ X xâ&#x201E;&#x201E;&" B| g #1 10 -^C/ g 5000K <4-^~( â&#x2C6;&#x2019;43

93

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5

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28

Linfan Mao:

H'^5<GI1 /

? P - ^ `= [ ` Y = 8 V I = A ' 2 e R ) g L%# F 7 <o = C V= lC 8 BV \ , ^_ V / o _ V= ^ W B #l[xBC V [_ V Up - = o # + W B H B { = - ^) gC% 3000K #~( _ V ~(r 0 " + o IY = ^W / PI == ^W 1unI enr k uen Yr = enr k 'en Yo B' W V = - ^ wL| = * = ^W}1uwkn7 o nI = - N 7 o7 r = j\m #1 108 - ^ C / g 100K < 4 - ^ % 1K ij; ?j njE&}^ _ #1 109 - ^ C / g 12K g D =r 2ok$>e 8= [d Ă&#x2020;X #1 1010 - ^ C / g 3K nI< 4 - ^ g 105K 1.3. w ju~ 1sS9 /\</V% </ jM{< 4 F} ; < 4 8 V KV e Ă&#x2020; u [K Ă&#x2020; d 3 m !-' PB? ; wE8 V$ 1Up C = /\ l1 Einstein = Ud\[ Dirac = V9 U d \1wE79 = /\ %'CP jW B </ i V 9 1<P >8 V C9 = /\ 0 K 9 RY 9 {Y 9 T ; C93 B8 V$ 1Up C = C9 &-' m ! + ~( } h </ ( 0 Einstein p% &0 9P % T ; C9 %U d \[ V 9 ?7</ =) %/\ /Q 7 s P =Theory of Everything L 80 h = j 1q% & kL ;8 be Pw 1q = Ă&#x2020; H uPE0U d \1<Pg>W B= /\ w6`I g I b X> >* < 4 1; W = i V 9 1<P >W B= /\ w KV 4 V 7V> >* < 4 1-~ W = i V+â&#x201E;&#x201E; T x j * $Bp+r&I Z = %dk1q Aw

i

i z p i

w 7 7zâ&#x201E;&#x201E;'2 zâ&#x201E;&#x201E; pC mw 7 {*9 y #hTw 7

"%h / 'w 7 '|F)rC ' b Ă&#x2020;{' 3 |2 Einstein &p>p = E0U d v_/ SpĂ&#x2020; . z / rp& '=0[> b_/ â&#x201E;&#x201E;mQ0' xO ? Ă&#x2020;h ' Ă&#x2020; 1j7 Y'?7B79r z 1 R¾ν â&#x2C6;&#x2019; Rg¾ν + Îťg¾ν = â&#x2C6;&#x2019;8Ď&#x20AC;GT¾ν .

2

O_W B _/, ? ?{ 10 l.y ) w 7/ â&#x201E;&#x201E;0Ă&#x2020;â&#x201E;&#x201E;m0' T,v 4

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;8 B79r z = % ; â&#x201E;&#x201E; d ~P

Robertson-Walker

ds2 = â&#x2C6;&#x2019;c2 dt2 + a2 (t)[

dr 2 + r 2 (dθ2 + sin2 θdĎ&#x2022;2 )].

1 â&#x2C6;&#x2019; Kr 2

U: = W B ~% Friedmann W B L h = v/>k Hubble u 1929 o P p +mI=WB 1%&u2)X+ = W B 9l^79r z = )X+ P B< / ( =E z [ >ze \ y< da > 0, dt

d2 a > 0. dt2

a(t) = tÂľ ,

b(t) = tν,

3m(m + 2)

3â&#x2C6;&#x201C;

2o #9 z 1 Âľ=

3Âą

q

3(m + 3)

,ν =

q

3m(m + 2)

3(m + 3)

,

} Kasner ^ F ds2 = â&#x2C6;&#x2019;dt2 + a(t)2 d2R3 + b(t)2 ds2 (T m )

% Einstein rz = 4 + m & P %'[ K &PPV ( 4 &)X+

P 0 ^C#1 L ' d ~H{ t â&#x2020;&#x2019; t+â&#x2C6;&#x17E; â&#x2C6;&#x2019; t, a(t) = (t+â&#x2C6;&#x17E; â&#x2C6;&#x2019; t)Âľ ,

2o ;8 %& 4 & = )X+ P 5% da(t) > 0, dt

d2 a(t) > 0. dt2

m !-' P = M- /\%Pw E1q MR B %/\* 8 V V 14Hi1&IV = p-F T p &z[Lt ^=V 1 1 p 1%& I 1-F%'~ 2-F~ F 1.4 7 ( B 1- 2- >p W

30

Linfan Mao:

H'^5<GI1 /

1.4. = p W &p M /\ WB ~(# = &℄r 1&I1 11 &= )7 ~(k > 7 4 & z [&u t = ! Æ i J 7 & z [&}u tua `_ r 2oVv ;8= 4 &g>WB [ V: = 7 & >WB 4 &g>WB = C9 _ Einstein 79r z i 7 & >W B = C9 _ A "Æ z F ;8 Kv &O\ j 1.1 M- 0') 7℄m ;m0s ℄m 7 | A R ' 4 | A R :C R / 1.1 [Ja 1 = w79 = Yvy< 2o , ;8TownsendWohlfarthq = 4 &)X+ W B q 7

4

ds2 = e−mφ(t) (−S 6 dt2 + S 2 dx23 ) + rC2 e2φ(t) ds2Hm ,

1 φ(t) =

V

1 (ln K(t) − 3λ0 t), m−1

K(t) =

1 ζ =

[

d2 S dς 2

m

m+2

S 2 = K m−1 e− m−1 λ0 t

λ0 ζrc , (m − 1) sin[λ0 ζ|t + t1 |]

b#1 ς e \ dς = S (t)dt })X+ W B= y< > 0 y ;8 e \ I ' ![M{ z m = 7 }+ 5 V % 3.04 q

3 + 6/m.

3

dS dς

>0

Ã&#x2020;1 Ã&#x2020;iJ(H'^5<GI 1

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I G ^ B R / 1.1 7=H %$ V1 Hi 1 1 F 7 = I 1q

R

7\ Xâ&#x201E;&#x201E; 2 FH A h / ;m0 %â&#x201E;&#x201E;m 1 |a ' A & vM{ wK = I 1 " u e% R V12ous 7 ;8= 1 "V12ou J I 7 [:L = 1 5wu 3 &Tv 1 7 l& H ,M*% (x, y, z) eV 0R%&&I ) P > P 1 =V 1 §2. Smarandache

eS

=L V%&5 .= 1q 1 + 1 =?

u W lII 7 2o #9 1+1=2 u 2 [ 2 p[sI 7 2oz #9 1+1=10 1= 10 %$ z1 2 5%u 2 [2p[sI 7+ L?&p[^W 0 [ 1 >p[ F}% 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

&p R$ R> P R = QX 2o^C% ; u Q& = # = QX ([18] â&#x2C6;&#x2019; [20]) $ = k , &1q = k A 1 + 1 = 2 6= 2 2o #9 1, 2, 3, 4, 5, · · · = I3 W lII N u &II 7 &pII = FW l&I~%NvYH T= I = k%I 2 = k%I% 3 "% 2 = 3 3 = 4, 4 = 5, · · · 2ol ;8 B â&#x20AC;²

â&#x20AC;²

â&#x20AC;²

1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, · · · ;

#z ;8 B 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, · · ·

%dp[ > ) i u ;W lI = p[sIK 2o + ;8 1 + 1 = 2 =

O\ Pu 2o~:%Kp[ =R0 ( R%& _ S d â&#x2C6;&#x20AC;x, y â&#x2C6;&#x2C6; S, R0 xâ&#x2C6;&#x2014;y = z /Q1 S " u%& 2 ^O_= â&#x2C6;&#x2014; : S Ã&#x2014; S â&#x2020;&#x2019; S ' ;

32

Linfan Mao:

H'^5<GI1 /

â&#x2C6;&#x2014;(x, y) = z.

^C P=z ) 2o ,C ! ; <Iu4v M* $ =L! S 7= l&^C4v =H M* wK S 7 L n &^ }u4v lb n &V2T =H ?& H x, z $ 1;K%yL[T ` wK " u%&^ y ' ; x â&#x2C6;&#x2014; y = z, 2ou y T ` L â&#x2C6;&#x2014;y ~% T `= f = w 2.1 b*

2.1. ;T f zq J / ;d :1 1 â&#x2C6;&#x2019; 1 = " S d : = % G[S] Pu wK2oX 8 %& e \ 1 + 1 = 3 = p[I 2o ,L( 1 + 1, 2 + 1, ¡ ¡ ¡ > ' P L P$

|

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(i) (A; â&#x2014;Ś) â&#x201E;&#x201E;m â&#x201E;&#x201E;l Ě&#x; G[A] 7â&#x201E;&#x201E;m Euler j Q " G[A] 7â&#x201E;&#x201E;m Euler j (A; â&#x2014;Ś) 7â&#x201E;&#x201E;m â&#x201E;&#x201E; P e (ii) (A; â&#x2014;Ś) 7â&#x201E;&#x201E;ms ' F P e G[A] / ;m60' ?{ |A| q (A; â&#x2014;Ś) . .

G[A] 7â&#x201E;&#x201E;ms ' 3 2- jw;m60 â&#x201E;&#x201E; m .& 60 " A' {&B 2- Q " b d PLS&^ = [r ,^C% ; LS =z )F R bLp[OK 2.2 ( B |S| = 3 = ? ; p[sI

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2.2. 3 &^ = ) q p[

F 2.2(a) L

1+1 = 2, 1+2 = 3, 1+3 = 1; 2+1 = 3, 2+2 = 1, 2+3 = 2; 3+1 = 1, 3+2 = 2, 3+3 = 3.

F 2.2(b) L 1+1 = 3, 1+2 = 1, 1+3 = 2; 2+1 = 1, 2+2 = 2, 2+3 = 3; 3+1 = 2, 3+2 = 3, 3+3 = 1.

d %& _

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34

H'^5<GI1 /

Linfan Mao:

(x +i y) +j z = x +i (y +j z),

(x ×i y) ×j z = x ×i (y ×j z)

a0 x ×i (y +j z) = x ×i y +j x ×i z, (y +j z) ×i x = y ×i x +j z ×i x,

| {℄m {℄m x

e R

m-

m-

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3 B F i, 1 ≤ i ≤ m, (R; +i , ×i ) 7℄mx

Z

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˙ i b)+ ˙ j c = a+ ˙ i (b+ ˙ j c); (ii) (a+

i

(iii) (k1 +i k2 ) ·j a = k1 +i (k2 ·j a);

{ 3 x Fe ' k 3 , A 9{ (Ve ; Fe ) F 2o #9 M- /\ 7= 1 q%$ 1% ;= 1 q j 2.2 P = (x , x , · · · , x ) { n- |W8 A R / '℄m0 B F s, 1 ≤ s ≤ n 0 P ℄m s ' G A C J / 5 1 R 7" uL R e = (1, 0, 0, · · · , 0), e = (0, 1, 0, · · · , 0), · · ·, e = (0, · · · , 0, 1, 0, · · · , 0) (E i &^% 1 >R% 0), · · ·, e = (0, 0, · · · , 0, 1) ' ; R 7= q/ H (x , x , · · · , x ) ,M*%

Ve

R 1

2

n

n

n

1

2

i

n

n

1

2

n

(x1 , x2 , · · · , xn ) = x1 e1 + x2 e2 + · · · + xn en

bZ F = {a , b , c , · · · , d ; i ≥ 1} 2o R 0%&k = [ 1 i

i

i

i

R− = (V, +new , ◦new ),

1 V

= {x1 , x2 , · · · , xn }

uL a , a , · · · , a ' ; 1

2

s

V ;Jv 2o* R x , x , · · · , x 1 \ 7 = z " 1

2

s

Ă&#x2020;1 Ă&#x2020;iJ(H'^5<GI

â&#x20AC;&#x201C;Smarandche

4 BĂ&#x2020;1

35

a1 â&#x2014;Śnew x1 +new a2 â&#x2014;Śnew x2 +new ¡ ¡ ¡ +new as â&#x2014;Śnew xs = 0,

}RL a

1

= a2 = ¡ ¡ ¡ = 0new

V " uL b , c , ¡ ¡ ¡ , d 1 â&#x2030;¤ i â&#x2030;¤ s ' ; i

i

i

xs+1 = b1 â&#x2014;Śnew x1 +new b2 â&#x2014;Śnew x2 +new ¡ ¡ ¡ +new bs â&#x2014;Śnew xs ;

xs+2 = c1 â&#x2014;Śnew x1 +new c2 â&#x2014;Śnew x2 +new ¡ ¡ ¡ +new cs â&#x2014;Śnew xs ;

¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡;

xn = d1 â&#x2014;Śnew x1 +new d2 â&#x2014;Śnew x2 +new ¡ ¡ ¡ +new ds â&#x2014;Śnew xs .

i 2o ;8H P = %& s- & V 1

â&#x2122;Ž

Fu 2.1 P {W8 A R / 'â&#x201E;&#x201E;m0 â&#x201E;&#x201E;m G AC n

â&#x2C6;&#x2019; â&#x2C6;&#x2019; â&#x2C6;&#x2019; Râ&#x2C6;&#x2019; 0 â&#x160;&#x201A; R1 â&#x160;&#x201A; ¡ ¡ ¡ â&#x160;&#x201A; Rnâ&#x2C6;&#x2019;1 â&#x160;&#x201A; Rn

.& R

â&#x2C6;&#x2019; n

= {P }

w G A R '|F{ n â&#x2C6;&#x2019; i 1 â&#x2030;¤ i â&#x2030;¤ n â&#x2C6;&#x2019; i

C CH: 3.1. Smarandache H: Smarandache4 1% ;` E w=} â&#x201E;&#x201E; > vS NC #= LobachevshyBolyai â&#x201E;&#x201E; Riemann â&#x201E;&#x201E;V Finsler â&#x201E;&#x201E; > oH 1^C u }q D yb + 5 â&#x201E;&#x201E;7=d :0 2o=L~:%K 5 â&#x201E;&#x201E; Ja â&#x201E;&#x201E; Riemann â&#x201E;&#x201E;= 7 L 5 â&#x201E;&#x201E; = 0/sIFKv 8y0 _ (1) ;m0#;mhT'0 7 ah % % (2);` % %= a #Q (3)a 0{ / 8 b o7'%â&#x201E;&#x201E;0 a S â&#x201E;&#x201E;| (4)Sp % O=&) §3.

^ {

36

Linfan Mao:

H'^5<GI1 /

r

` O pâ&#x201E;&#x201E;` % %u% ` % %&N w râ&#x201E;&#x201E;` % %'â&#x201E;&#x201E; SN' OO " Ã&#x2020;0r % O ` % % â&#x201E;&#x201E; &N, j 3.1 S1

^ r

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1 6 a + 6 ; E zq

3.1. %y & TV?yV4s & TUE b < 180 ` k%y0 ~% 5 E 80 ez ,^CKv O

i

o7 % %q'â&#x201E;&#x201E;0 n â&#x201E;&#x201E;` % %uo7' % % &N W 50 0W,$ po% & v ; > E 80 V: %0 P e d%u: 1%&}q % } h I ( 0 9P^CNTy0 ! { E 80

0 % & kL / P1LpXC>d* + t 5 E 80 4 ;8= 0/sI 1 = 1 " uh g !k-' Lobachevshy [ Bolyai Riemann N^CV = * b + 5 E 80 ; / do^C = * N1 Lobachevshy-Bolyai * o7 % %q'â&#x201E;&#x201E;0 ) ` % %uo7' % % &N Riemann * o7 % %q'â&#x201E;&#x201E;0 % %uo7' % % &N Riemann * ;8= 9 = _5uPF ,?7,f â&#x201E;&#x201E; k > Einstein C %>U d \ 7= 79# !79r %&,f 1 D 2o1 ,[%Xd H 50 ;8 k = â&#x201E;&#x201E; i S _L = 5 â&#x201E;&#x201E; Lobachevshy-Bolyai â&#x201E;&#x201E; Riemann â&#x201E;&#x201E;[ Finsler â&#x201E;&#x201E; /Q [16] 7PwB &1q 1q = Pw ; 9P:C Smarandache â&#x201E;&#x201E;QX i ?7( ^ 1 â&#x201E;&#x201E; 1 d Smarandache â&#x201E;&#x201E; %&5 S wK K%NsS ( ^ 1 â&#x201E;&#x201E; Smarandache â&#x201E;&#x201E;0R U 04 V4 Q k4 [Q4 > T ; N &pV = 0 ?7 > 7 U 04 ^C = 0 % 50

1 - 4 , Kvq â&#x201E;&#x201E;%y0

(P â&#x2C6;&#x2019; 1) ) â&#x201E;&#x201E;` % %Ã&#x2020;d % %q'â&#x201E;&#x201E;0 .&f d0' % %vu `

i

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37

)

%

% %q'â&#x201E;&#x201E;0 .&f d0n â&#x201E;&#x201E;` % %

) `% % (P â&#x2C6;&#x2019; 4) ) u % (P â&#x2C6;&#x2019; 5) ) ` % %&N

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% %q'â&#x201E;&#x201E;0 .&f d0n p#' k

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k â&#x2030;¥2

%

% %q'â&#x201E;&#x201E;0 .&f d0n F` % %

%

% %q'â&#x201E;&#x201E;0 .&f d0' % %vu

V4 ^C=0/sI1 R 5 â&#x201E;&#x201E; 5 y0 7= 1 & I& ^C,K %y Iy0 b + 50 7=d :0

(â&#x2C6;&#x2019;1) o7' 0 â&#x201E;&#x201E;7 â&#x201E;&#x201E;` % % (â&#x2C6;&#x2019;2) â&#x201E;&#x201E;` % % P #Q (â&#x2C6;&#x2019;3) o7â&#x201E;&#x201E;0Ã&#x2020;â&#x201E;&#x201E;m+F â&#x201E;&#x201E;7 a â&#x201E;&#x201E;m| (â&#x2C6;&#x2019;4) % O â&#x201E;&#x201E;7&) (â&#x2C6;&#x2019;5) o7 % %q'â&#x201E;&#x201E;0 â&#x201E;&#x201E;7 â&#x201E;&#x201E;` % %uo7' % % &N Q k4 ^C = 0/sI1 R < â&#x201E;&#x201E; 7= %y Iy0 U:^CK E0 b + (C â&#x2C6;&#x2019; 1) ) ` % %':p % % mo7'0 (C â&#x2C6;&#x2019; 2) A, B, C { m v%'0 D, E { m 0 A, D, C Ã&#x2020; B, E, C 0v% b A, B ' % %ub D, E ' % % &N (C â&#x2C6;&#x2019; 3) ;` % % ) C p m '0 Q4 ^C = 0/sI1 R Hilbert 0/sI 7= %y Iy0

} 3.1 â&#x201E;&#x201E;mt {Smarandache\7' h â&#x201E;&#x201E;m A / ) ! ' ' ) a 2 a T0 ! â&#x201E;&#x201E;m p Smarandache \7t '4 {Smarandache4 Kv &5 V , Kv?_NM{ Smarandache â&#x201E;&#x201E;1;J " u = l 3.1 A, B, C % 54v }&V2T=H R0&T% 54v L A, B, C 7 IV%& H=& T }2o ;8 %& Smarandache â&#x201E;&#x201E; 5%V 5 â&#x201E;&#x201E;0/s IUAI > 7 ?y0 1 Smarandache R=

i i fi{ fi i i

i}

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38

Linfan Mao:

H'^5<GI1 /

i

5 E 80 Pu%f â&#x201E;&#x201E;` % %q'â&#x201E;&#x201E;0 â&#x201E;&#x201E;`' % %`> rd` % %bb + * & T L L H C V4sP & T AB J / Lqâ&#x201E;&#x201E;%& Vu AB =HIVL%y & T4sP L i L&T AB =qâ&#x201E;&#x201E;%H yV " u 4sP L =& T w 3.2(a) b* (ii) 0 f m 0 â&#x201E;&#x201E;` % %Pu%f m 0 â&#x201E;&#x201E; `%%' % %b + J / L?& H D, E 1 D, E V A, B, C 7=%H w H C 2T w 3.2(b) b* IVL%y & T L D, E id q/?&u & T AB H F, G VV A, B, C 7 %& H 2T = ?& H G, H yV " u Leo =& T w 3.2(b) b* (i)

i

i

i

3.2. s- & T = [ +y/ C : 7 %& F | =R /M;m ' {} ' \r } p 2p m< ; m< " A mâ&#x201E;&#x201E;m ; ~ G & ' \r } p q m< ;m< m3 K ' Wuh& r { 7, l7d{ p { 7, l7d{ q 1 R [ = /Q1%& & Pav = [ T avp W %ek~ 8 oH 1 H[ =z [ & > #9 â&#x201E;&#x201E;v1 R [ = i A3N * }1V R [ = w 3.3 b* > 7 (a) %6~ =, v (b) % _ k = A3N * 3.2

o o \ _ i

3.3. A3N *= r

Ă&#x2020;1 Ă&#x2020;iJ(H'^5<GI

â&#x20AC;&#x201C;Smarandche

4 BĂ&#x2020;1

39

D 1av = % ; t 2 T ; t av6~k ;8= l&v y &Pb! D = {(x, y)|x2 + y2 â&#x2030;¤ 1} Tutte P 1973 ( B D =+ I R 0 ^C [12] 7= DX D R 0PK

Q

} 3.2 â&#x201E;&#x201E;m,j M = (X , P) 7d{ ( . X 'Ny t Kx, x â&#x2C6;&#x2C6; X '

tvy' . X 'â&#x201E;&#x201E;m( + P w4 L 't 1 Ă&#x2020;t 2 K = {1.Îą, β, ιβ} { Klein 4- y S P {( + 2 F k, .& P x = Îąx +j 1 ÎąP = P Îą; +j 2 Ψ = hÎą, β, Pi X o &pR0 3.2 D =QH[v NR 0%1{ P [ Pιβ CP X ;8= 2mG9 E% Klein 4- ^k K CP X ;8=G9 4CEuler-Poincar´e0 ) 2o ;8

f

ι,β

ι,β

[

[

k

i

â&#x2C6;&#x2019;1

J

ι,β

ι,β

ι,β

|V (M)| â&#x2C6;&#x2019; |E(M)| + |F (M)| = Ď&#x2021;(M),

1 V (M), E(M), F (M) NM* D M =QH E [v Ď&#x2021;(M ) M* D M = Euler % >I '> P D M bPx = &av = Euler % ~%& D M = (XÎą,β , P) 1 7,' z 1 {k ΨI = hιβ, Pi u XÎą,β 1 K =

}~% 7,'

i

3.4. D u Kelin av = Px %%&5 V 3.4 7 ( B D u Kelin av = %&Px ,^ C D M = (X , P) M*wK 1 0.4.0

0.4.0

ι,β

Xι,β =

[

{e, ιe, βe, ιβe},

eâ&#x2C6;&#x2C6;{x,y,z,w}

P = (x, y, z, w)(ιβx, ιβy, βz, βw)

40

Linfan Mao:

H'^5<GI1 /

Ă&#x2014; (Îąx, Îąw, Îąz, Îąy)(βx, ιβw, ιβz, βy).

3.4 7=D L 2 QH v

1

βw), (βx, ιβw, ιβz, βy)}, 4

= {(x, y, z, w), (ιx, ιw, ιz, ιy)}, v2 = {(ιβx, ιβy, βz,

yE e

1

= {x, ιx, βx, ιβx}, e2 = {y, ιy, βy, ιβy}, e3 =

, 2 &v f = {(x, ιβy, z, βy, ιx, ιβw), = {(βw, ιz), (w, ιβz)} > Euler %%

{z, ιz, βz, ιβz}, e4 = {w, ιw, βw, ιβw} (βx, ιw, ιβx, y, βz, ιy)}, f2

2

Ď&#x2021;(M) = 2 â&#x2C6;&#x2019; 4 + 2 = 0

V 1 {k Ψ = hιβ, Pi u X K l + IG ^;8 D u Klein v = Px +â&#x201E;&#x201E;T/\</ j = z 2oz ,%'v D V u 1, h= av = Px u = av = Px R 0wK } 3.3 j G '60. rp Y V (G) = S V , B F 1 â&#x2030;¤ i, j â&#x2030;¤ T k V V = â&#x2C6;&#x2026; q S , S , ¡ ¡ ¡ , S {? A E / ' k m k â&#x2030;Ľ 1 â&#x201E;&#x201E; m 1-1 El Ď&#x20AC; : G â&#x2020;&#x2019; E .&B F i, 1 â&#x2030;¤ i â&#x2030;¤ k Ď&#x20AC;| 7â&#x201E;&#x201E;mb w S \ Ď&#x20AC;(hV i) / ';m b_ \r|Y D = {(x, y)|x + y â&#x2030;¤ 1} Ď&#x20AC;(G) 7 G S , S , ¡ ¡ ¡ , S ' 3 s R 0 3.3 7 av S , S , ¡ ¡ ¡ , S = 1+ 1d= PxL<Y 2" u% ; D S , S , ¡ ¡ ¡ , S ' ;d q/ I j, 1 â&#x2030;¤ j â&#x2030;¤ k S 1 S =V 1# ~% G u S , S , ¡ ¡ ¡ , S = O 3 s <Pâ&#x201E;&#x201E;v LKv = O\ j 3.1 â&#x201E;&#x201E;mj G } P â&#x160;&#x192; P â&#x160;&#x192; ¡ ¡ ¡ â&#x160;&#x192; P V`P'O 3 s v^ U j G Y G = G .&B F i, 1 < i < s, (i) G 7` ' S (ii) B â&#x2C6;&#x20AC;v â&#x2C6;&#x2C6; V (G ), N (x) â&#x160;&#x2020; ( V (G )).

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M

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41

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3.5. & T L b H Ja H b H 5 H [Ja H u 3 & 1 7 y1 ,%P = 1 H= %PLNP 5 1 = [r V% R 14 &= Ã&#x2020;} H l1 5 H 3.6 7 ( B } ;H u 3 & 1 = %P zq, 7H u % b H v % 5 Hi w %Ja H

3.6. b H 5 H [Ja H u 3- & 1 = %P j 3.1 pW W,j4 / v U 04 V4 Q k4 Ã&#x2020;Q4 R / =! {:/Q [16] %GP/P 2oKvS 4v D â&#x201E;&#x201E; = [r u ; [r VZ ,u QH TG5 V TI z , ^;K QH$ 1 = E1% &; TI d [%X/P4v + IaT! L/0 Awu4v D â&#x201E;&#x201E; 7 L = O\ ` ,j4 | % % ,j' '0{W80 %%&5 V 3.7 7 s B P Tvs = % ; 4v D â&#x201E;&#x201E; > 7QH E = I ' M{ QH 2 < = G5 V TI '

i

42

Linfan Mao:

H'^5<GI1 /

3.7. %&4v D ℄ = 5 V 3.8 7 s B 3.7 R 0 = 4v D ℄ 7& T = [r +U D 3.9 7 s B 4v D ℄ 7= ;h Er

3.8. 4v D ℄ =& T

3.9. 4v D ℄ =h Er N neSH: Einstein = E0U d \ a B 1u79 CK1 a = / CT"V5 %Hu%$>e 7 * ;8! % D ℄ = QX%$ ,%' DR 0P%& ^ 1 u ^ 1 = l& H T%&[ i ?7( ^ 1 ℄ } 4.1 U ρ W ⊆ U ∀u ∈ U §4.

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6#'2<$C7 6#

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Ď&#x2030; : u â&#x2020;&#x2019; Ď&#x2030;(u)

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N > 0

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\ j 4.1 ~` (P, Ď&#x2030;) ' 0 u Ă&#x2020; v â&#x201E;&#x201E;7 W8 d ' % % j 4.2 â&#x201E;&#x201E;m~` (P, Ď&#x2030;) W80 (P, Ď&#x2030;) h;m0v{n| 0';m0v{I 0 d P4v + IaT }LwKOK j 4.3 ~` (P, Ď&#x2030;) F % F (x, y) = 0 f x D /'0 (x , y ) w^ F (x , y ) = 0 wB â&#x2C6;&#x20AC;(x, y) â&#x2C6;&#x2C6; D

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m

m

m

m

m

m

m

m

m

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1 â&#x2C6;&#x201A; 2 F 2 (y + su + tv) |t=s=0 2 â&#x2C6;&#x201A;sâ&#x2C6;&#x201A;t

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44

Linfan Mao:

H'^5<GI1 /

u T M \ {0} = S{T M \ {0} : x â&#x2C6;&#x2C6; M } Cr (ii) d q/ â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; M F | â&#x2020;&#x2019; [0, +â&#x2C6;&#x17E;) 1%& Minkowski v I %( ^ 1 â&#x201E;&#x201E; = %&o5 d q/ x â&#x2C6;&#x2C6; M 2o | Ï&#x2030;(x) = F (x) } ( ^ 1 â&#x201E;&#x201E; (M , Ï&#x2030;) 1%& Finsler Nr oN D wKb Ï&#x2030;(x) = g (y, y) = F (x, y) } (M , Ï&#x2030;) l1 Riemann Nr 2ol ;8 KEO\ j 4.4 ~? A4 (M , Ï&#x2030;) â&#x201E;&#x201E; , Smarandache 4 / Finsler 4 D Riemann 4 m

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juXjS9 5.3 PyRa'_ <2 ,} 7\ â&#x201E;&#x201E; 2 "Ă&#x2020; a 4 |_ 3 |' 3 |_ 2 | A bX%$ 1 Einstein rz uV ^ Fy<K = ? H P % z v </ ( r%b X" uq )= 79 l;CT"V 5 qâ&#x201E;&#x201E;<sVu^xb X7 y > OE %^2 [6]-[7] #</ (NZeb X 1;KV W B V # = S = 5%p+BP = %U</FWub X y b Vb X U d= /\</ 7 z L% ; " X >o 1qâ&#x201E;&#x201E;<sy5 q [x" X wK$~b X " X &s 2 ol o P? y1 W >e = C7 = #l1 b,b X V" X : 1%~ . p+ oN1WUa5z - lV_l N [Bb X h h< 4 u D â&#x201E;&#x201E; 1;J " u = wK I /> 0 s h h< 4= /\ V \1</ 1I y)7Cpo =\ 5r& h ,$ 9 q% & / <sp W7 hVH %& _} /\ r&# _ D{ Z p W7 hHv * $ = 1q V0 R s = p W 1q F * $ = I 1q1 ([16]) (1)_qTw H 7K2j7 7' K27 7' _>Y (2)Jjs #C| AO Okh& A .

(3)I j'& A ; H d </ $M z u D â&#x201E;&#x201E; 5 H>h = < [ io P># _ FW juXjS9 5.4 ,} a # Ă&#x2020; - 2 <4V % & 1</ Q = %&nnwq +â&#x201E;&#x201E; T p+ d 1&Ir& = H &1q"H ; u2Ă&#x2020;q wK 1&I â&#x2030;Ľ 11 j < 4 lV% R u p+ ;8=z [& "V u D â&#x201E;&#x201E; 8 e bL d Z &%Pp+r&K 4V% D = p

i f `

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aP` [1] A.D.Aleksandrov and V.A.Zalgaller, Intrinsic Geometry of Surfaces, American Mathematical Society, Providence, Rhode Island, 1976.

[2] I.Antoniadis, Physics with large extra dimensions: String theory under experimental test, Current Science, Vol.81, No.12, 25(2001),1609-1613 [3]

} [}&e, xY4 Ld, :_ ) * , 2001.

46

Linfan Mao:

H'^5<GI1 /

[4] M.J.Duff, A laymanâ&#x20AC;&#x2122;s guide to M-theory, arXiv: hep-th/9805177, v3, 2 July(1998).

3{, &B0uVW4 , * , :_, 2005.

[6] S.Hawking, )AE-, n * ,2005.

[7] S.Hawking, ~ 'w 7, n * ,2005.

[5]

[8] H.Iseri, Smarandache Manifolds, American Research Press, Rehoboth, NM,2002.

[9] H.Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm. edu/Ë&#x153;smarandache/Howard-Iseri-paper.htm.

[10] M.Kaku, Hyperspace: A Scientific Odyssey through Parallel Universe, Time Warps and 10th Dimension, Oxford Univ. Press.

[11] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Canada, Vol.12(2003) [12] Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 1999.

[13] L.F.Mao, On Automorphisms groups of Maps, Surfaces and Smarandache geometries, Sientia Magna, Vol.1(2005), No.2, 55-73.

[14] L.F.Mao, A new view of combinatorial maps by Smarandacheâ&#x20AC;&#x2122;s notion, arXiv: math.GM/0506232.

[15] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research Press, 2005.

[16] L.F.Mao, Smarandache multi-space theory, Hexis, Phoenix, AZ 2006.

[17] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

[18] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth, 1999.

[19] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297384.

[20] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438.

47

A New View of Combinatorial Maps by Smarandacheâ&#x20AC;&#x2122;s Notionâ&#x2C6;&#x2014;

A New View of Combinatorial Maps

by Smarandacheâ&#x20AC;&#x2122;s Notionâ&#x2C6;&#x2014; Linfan MAO (Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China) e-mail: maolinfan@163.com

Abstract: On a geometrical view, the conception of map geometries is introduced, which are a nice models of the Smarandache geometries, also new kind of and more general intrinsic geometry on surfaces. Results convincing one that map geometries are Smarandache geometries are obtained. Some open problems related combinatorial maps with the Riemann geometry and Smarandache geometries are also presented.

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Key Words: map, Smarandache geometry, model, classification.

AMS(2000): 05C15, 20H15, 51D99, 51M05

1

Reported at the Chinese Academy of Mathematics and System Science and the Department of

Applied Mathematics of Beijing Jiaotong University.

2 e-print: arXiv: math.GM/0506232.

48

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

1. What is a combinatorial map A graph Γ is a 2-tuple (V, E) consists of a finite non-empty set V of vertices together with a set E of unordered pairs of vertices, i.e., E ⊆ V × V . Often denoted by V (Γ), E(Γ) the vertex set and edge set of a graph Γ([9]).

For example, the graph in the Fig.1 is a complete graph K4 with vertex set V = {1, 2, 3, 4} and edge set E = {12, 13, 14, 23, 24, 34}.

Fig.1 A map is a connected topological graph cellularly embedded in a surface. In 1973, Tutte gave an algebraic representation for embedding a graph on locally orientable surface ([18]), which transfer a geometrical partition of a surface to a kind of permutation in algebra as follows([7][8]).

A combinatorial map M = (Xα,β , P) is defined to be a basic permutation P, i.e, for any x ∈ Xα,β , no integer k exists such that P k x = αx, acting on Xα,β , the disjoint union of quadricells Kx of x ∈ X (the base set), where K = {1, α, β, αβ} is the Klein group, with the following two conditions holding: (i) αP = P −1 α; (ii) the group ΨJ =< α, β, P > is transitive on Xα,β .

For a given map M = (Xα,β , P), it can be shown that M ∗ = (Xβ,α , Pαβ) is also a map, call it the dual of the map M. The vertices of M are defined as the pairs of conjugatcy orbits of P action on Xα,β by the condition (Ci) and edges the orbits of K on Xα,β , for example, for ∀x ∈ Xα,β , {x, αx, βx, αβx} is an edge of the map M. Define the faces of M to be the vertices in the dual map M ∗ . Then the Euler characteristic χ(M) of the map M is

A New View of Combinatorial Maps by Smarandache’s Notion∗

49

χ(M) = ν(M) − ε(M) + φ(M) where, ν(M), ε(M), φ(M) are the number of vertices, edges and faces of the map M, respectively. For each vertex of a map M, its valency is defined to be the length of the orbits of P action on a quadricell incident with u.

Fig.2 For example, the graph K4 on the tours with one face length 4 and another 8 , shown in the Fig.2, can be algebraically represented as follows.

A map (Xα,β , P) with Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} and

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw) × (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv) The four vertices of this map are {(x, y, z), (αx, αz, αy)}, {(αβx, u, w), (βx, αw, αu)}, {(αβz, αβu, v), (βz, αv, βu)} and {(αβy, αβv, αβw), (βy, βw, βv)} and six edges are {e, αe, βe, αβe}, where, e ∈ {x, y, z, u, v, w}. The Euler characteristic χ(M) is χ(M) = 4 − 6 + 2 = 0.

Geometrically, an embedding M of a graph Γ on a surface is a map and has an algebraic representation. The graph Γ is said the underlying graph of the map M and denoted by Γ = Γ(M). For determining a given map (Xα,β , P) is orientable or not, the following condition is needed.

50

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

(iii) If the group ΨI = hαβ, Pi is transitive on Xα,β , then M is non-orientable.

Otherwise, orientable.

It can be shown that the number of orbits of the group ΨI = hαβ, Pi in the Fig.2 action on Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} is 2. Whence, it is an orientable map and the genus of the surface is 1. Therefore, the algebraic representation is correspondent with its geometrical meaning.

2. What are lost in combinatorial maps As we known, mathematics is a powerful tool of sciences for its unity and neatness, without any shade of mankind. On the other hand, it is also a kind of aesthetics deep down in one’s mind. There is a famous proverb says that only the beautiful things can be handed down to today, which is also true for the mathematics.

Here, the term unity and neatness is relative and local, also have various conditions. For acquiring the target, many unimportant matters are abandoned in the process. Whether are those matters in this time still unimportant in another time?

It is not true. That is why we need to think the question: what are lost in the classical mathematics?

For example, a compact surface is topological equivalent to a polygon with even number of edges by identifying each pairs of edges along a given direction on it([17]).

If label each pair of edges by a letter e, e ∈ E, a surface S is also identifying to a cyclic permutation such that each edge e, e ∈ E just appears two times in S, one is e and another is e−1 . Let a, b, c, · · · denote the letters in E and A, B, C, · · · the sections of successive letters in linear order on a surface S (or a string of letters on S). Then, a surface can be represented as follows: S = (· · · , A, a, B, a−1 , C, · · ·),

where a ∈ E andA, B, C denote a string of letters. Define three elementary transformations by (O1 )

(A, a, a−1 , B) ⇔ (A, B);

(O2 )

(i) (A, a, b, B, b−1 , a−1 ) ⇔ (A, c, B, c−1 );

A New View of Combinatorial Maps by Smarandache’s Notion∗

51

(ii) (A, a, b, B, a, b) ⇔ (A, c, B, c); (O3 )

(i) (A, a, B, C, a−1 , D) ⇔ (B, a, A, D, a−1, C); (ii) (A, a, B, C, a, D) ⇔ (B, a, A, C −1 , a, D −1 ).

If a surface S0 can be obtained by the elementary transformations O1 -O3 from a surface S, it is said that S is elementary equivalent with S0 , denoted by S ∼El S0 .

We have known the following formula in [8]: (i) (A, a, B, b, C, a−1 , D, b−1 , E) ∼El (A, D, C, B, E, a, b, a−1, b−1 ); (ii) (A, c, B, c) ∼El (A, B −1 , C, c, c); (iii) (A, c, c, a, b, a−1 , b−1 ) ∼El (A, c, c, a, a, b, b).

Then we can get the classification theorem of compact surfaces as follows([14]): Any compact surface is homeomorphic to one of the following standard surfaces: (P0 ) the sphere: aa−1 ; (Pn ) the connected sum of n, n ≥ 1, tori: −1 −1 −1 −1 −1 a1 b1 a−1 1 b1 a2 b2 a2 b2 · · · an bn an bn ;

(Qn ) the connected sum of n, n ≥ 1, projective planes: a1 a1 a2 a2 · · · an an .

Generally, a combinatorial map is a kind of decomposition of a surface. Notice that all the standard surfaces are just one face map underlying an one vertex graph.

By a geometrical view, a combinatorial map is also a surface. But this assertion need more clarifying. For example, see the left graph Π4 in the Fig. 3, which is just the tetrahedron.

Fig.3

52

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Whether can we say it is the sphere? Certainly NOT. Since any point u on a sphere has a neighborhood N(u) homeomorphic to the open disc, therefore, all angles incident with the point 1 must all be 120◦ degree on a sphere. But in Π4 , they are all 60◦ degree. For making them topologically same, i.e., homeomorphism, we must blow up the Π4 to a sphere, as shown in the Fig.3. Whence, for getting the classification theorem of compact surfaces, we lose the angle,area, volume, distance, curvature,· · ·, etc, which are also lost in the combinatorial maps.

Klein Erlanger Program says that any geometry is finding invariant properties under a transformation group of this geometry. This is essentially the group action idea and widely used in mathematics today. In combinatorial maps, we know the following problems are applications of the Klein Erlanger Program: (i)to determine isomorphism maps or rooted maps; (ii)to determine equivalent embeddings of a graph; (iii)to determine an embedding whether exists; (iv)to enumerate maps or rooted maps on a surface; (v)to enumerate embeddings of a graph on a surface; (vi) · · ·, etc.

All the problems are extensively investigated by researches in the last century and papers related those problems are still appearing frequently on journals today. Then, what are their importance to classical mathematics? and what are their contributions to sciences? These are the central topics of this paper.

3. The Smarandache geometries The Smarandache geometries is proposed by Smarandache in 1969 ([16]), which is a generalization of the classical geometries, i.e., the Euclid, Lobachevshy-BolyaiGauss and Riemannian geometries may be united altogether in a same space, by some Smarandache geometries. These last geometries can be either partially Euclidean and partially Non-Euclidean, or Non-Euclidean. It seems that the Smarandache geometries are connected with the Relativity Theory (because they include the Riemann geometry in a subspace) and with the Parallel Universes (because they combine separate spaces into one space) too([5]). For a detail illustration, we need to consider the classical geometries.

A New View of Combinatorial Maps by Smarandache’s Notion∗

53

The axioms system of Euclid geometry are the following: (A1)there is a straight line between any two points.

(A2)a finite straight line can produce a infinite straight line continuously.

(A3)any point and a distance can describe a circle.

(A4)all right angles are equal to one another.

(A5)if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The axiom (A5) can be also replaced by: (A5’)given a line l and a point u exterior this line, there is one line passing through u parallel to the line l.

The Lobachevshy-Bolyai-Gauss geometry, also called hyperbolic geometry, is a geometry with axioms (A1) − (A4) and the following axiom (L5): (L5) there are infinitely many line parallels to a given line passing through an exterior point.

The Riemann geometry, also called elliptic geometry, is a geometry with axioms (A1) − (A4) and the following axiom (R5): there are no parallel to a given line passing through an exterior point.

By the thought of Anti-Mathematics: not in a nihilistic way, but in a positive one, i.e., banish the old concepts by some new ones: their opposites, Smarandache introduced the paradoxist geometry, non-geometry, counter-projective geometry and anti-geometry in [16] by contradicts the axioms (A1) − (A5) in Euclid geometry, generalized the classical geometries.

Paradoxist geometries In these geometries, their axioms are (A1) − (A4) and with one of the following as the axiom (P 5): (i)there are at least a straight line and a point exterior to it in this space for which any line that passes through the point and intersect the initial line. (ii)there are at least a straight line and a point exterior to it in this space for

54

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

which only one line passes through the point and does not intersect the initial line. (iii)there are at least a straight line and a point exterior to it in this space for which only a finite number of lines l1 , l2 , · · · , lk , k ≥ 2 pass through the point and do not intersect the initial line. (iv)there are at least a straight line and a point exterior to it in this space for which an infinite number of lines pass through the point (but not all of them) and do not intersect the initial line. (v)there are at least a straight line and a point exterior to it in this space for which any line that passes through the point and does not intersect the initial line.

Non-Geometries These non-geometries are geometries by denial some axioms of (A1) − (A5), such as: (A1− )it is not always possible to draw a line from an arbitrary point to another arbitrary point.

(A2− )it is not always possible to extend by continuity a finite line to an infinite line.

(A3− )it is not always possible to draw a circle from an arbitrary point and of an arbitrary interval.

(A4− )not all the right angles are congruent.

(A5− )if a line, cutting two other lines, forms the interior angles of the same side of it strictly less than two right angle, then not always the two lines extended towards infinite cut each other in the side where the angles are strictly less than two right angle.

Counter-Projective geometries Denoted by P the point set, L the line set and R a relation included in P × L. A counter-projective geometry is a geometry with the following counter-axioms: (C1)there exist: either at least two lines, or no line, that contains two given distinct points.

(C2)let p1 , p2 , p3 be three non-collinear points, and q1 , q2 two distinct points.

Suppose that {p1 .q1 , p3 } and {p2 , q2 , p3 } are collinear triples. Then the line contain-

A New View of Combinatorial Maps by Smarandache’s Notion∗

55

ing p1 , p2 and the line containing q1 , q2 do not intersect.

(C3)every line contains at most two distinct points.

Anti-Geometries These geometries are constructed by denial some axioms of the Hilbert’s 21 axioms of Euclidean geometry. As shown in [5], there are at least 221 − 1 antigeometries.

The Smarandache geometries are defined as follows.

Definition 3.1 An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom(1969).

A nice model for Smarandache geometries, called s-manifolds, is found by Iseri in [3] and [4], which is defined as follows: An s-manifold is any collection C(T, n) of these equilateral triangular disks Ti , 1 ≤ i ≤ n satisfying the following conditions: (i) each edge e is the identification of at most two edges ei , ej in two distinct triangular disks Ti , Tj , 1 ≤ i, j ≤ n and i 6= j; (ii) each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks.

These vertices are classified by the number of the disks around them. A vertex around five, six or seven triangular disks is called an elliptic vertex, an Euclid vertex or a hyperbolic vertex, respectively.

An s-manifold is called closed if each edge is shared by exactly two triangular disks. An elementary classification for closed s-manifolds by triangulation are made in the reference [11]. These closed s-manifolds are classified into 7 classes in [11], as follows: Classical Type: (1) ∆1 = {5 − regular triangular maps} (elliptic); (2) ∆2 = {6 − regular triangular maps}(euclidean); (3) ∆3 = {7 − regular triangular maps}(hyperbolic).

56

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Smarandache Type: (4) ∆4 = {triangular maps with vertex valency 5 and 6} (euclid-elliptic); (5) ∆5 = {triangular maps with vertex valency 5 and 7} (elliptic-hyperbolic); (6) ∆6 = {triangular maps with vertex valency 6 and 7} (euclid-hyperbolic); (7) ∆7 = {triangular maps with vertex valency 5, 6 and 7} (mixed).

It is proved in [11] that |∆1 | = 2, |∆5 | ≥ 2 and |∆i |, i = 2, 3, 4, 6, 7 are infinite.

Isier proposed a question in [3]: do the other closed 2-manifolds correspond to smanifolds with only hyperbolic vertices?. Since there are infinite Hurwitz maps, i.e., |∆3 | is infinite, the answer is affirmative.

4. Map geometries Combinatorial maps can be used to construct new geometries, which are nice models for the Smarandache geometries, also a generalization of Isier’s model and Poincar´e’s model for the hyperbolic geometry.

4.1 Map geometries without boundary For a given map on a surface, the map geometries without boundary are defined as follows.

Definition 4.1 For a combinatorial map M with each vertex valency≥ 3, endows a real number µ(u), 0 < µ(u) < π, with each vertex u, u ∈ V (M). Call (M, µ) a map geometry with out boundary, µ(u) the angle factor of the vertex u and to be orientablle or non-orientable if M is orientable or not.

The realization of each vertex u, u ∈ V (M) in R3 space is shown in the Fig.4 for each case of ρ(u)µ(u) > 2π, = 2π or < 2π.

ρ(u)µ(u) < 2π

ρ(u)µ(u) = 2π

ρ(u)µ(u) > 2π

Fig.4 As pointed out in Section 2, this kind of realization is not a surface, but it is

57

A New View of Combinatorial Maps by Smarandache’s Notion∗

homeomorphic to a surface. We classify points in a map geometry (M, µ) without boundary as follows.

Definition 4.2 A point u in a map geometry (M, µ) is called elliptic, euclidean or hyperbolic if ρ(u)µ(u) < 2π, ρ(u)µ(u) = 2π or ρ(u)µ(u) > 2π.

Then we have the following results.

Proposition 4.1 Let M be a map with ∀u ∈ V (M), ρ(u) ≥ 3. Then for ∀u ∈ V (M), there is a map geometries (M, µ) without boundary such that u is elliptic, euclidean or hyperbolic in this geometry.

Proof Since ρ(u) ≥ 3, we can choose the angle factor µ(u) such that µ(u)ρ(u) < 2π, µ(u)ρ(u) = 2π or µ(u)ρ(u) > 2π. Notice that

0 <

2π < π. ρ(u)

Whence, we can also choose µ(u) satisfying that 0 < µ(u) < π

♮

Proposition 4.2 Let M be a map of order≥ 3 and ∀u ∈ V (M), ρ(u) ≥ 3. Then there exists a map geometry (M, µ) without boundary, in which all points are one of the elliptic vertices, euclidean vertices and hyperbolic vertices or their mixed.

Proof According to the Proposition 4.1, we can choose an angle factor µ such that a vertex u, u ∈ V (M) to be elliptic, or euclidean, or hyperbolic. Since |V (M)| ≥ 3, we can also choose the angle factor µ such that any two vertices v, w ∈ V (M)\{u} to be elliptic, or euclidean, or hyperbolic as we wish. Then the map geometry (M, µ) makes the assertion holding.

♮

A geodesic in a manifold is a curve as straight as possible. Similarly, in a map geometry, its m-lines and m-points are defined as follows.

Definition 4.3 Let (M, µ) be a map geometry without boundary. An m-line in (M, µ) is a curve with a constant curvature and points in it are called m-points.

Examples for an m-line on the torus and Klein bottle are shown in Fig.5.

58

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Fig.5 If an m-line pass through an elliptic point or a hyperbolic point u, it must has the angle

µ(u)ρ(u) 2

a=

with the entering line, not 180◦ , which are explained in Fig.6.

µ(u)ρ(u) 2

< π

a=

µ(u)ρ(u) 2

> π

Fig.6 The following proposition asserts that map geometries without boundary are Smarandache geometries.

Proposition 4.3 For a map M on a locally orientable surface with order≥ 3 and vertex valency≥ 3, there is an angle factor µ such that (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

A New View of Combinatorial Maps by Smarandache’s Notion∗

59

Proof According to Proposition 4.1, we know that there exist an angle factor µ such that there are elliptic vertices, euclidean vertices and hyperbolic vertices in (M, µ) simultaneously. The proof is divided into three cases.

Case 1. M is a planar map Notice that for a given line L not pass through the vertices in the map M and a point u in (M, µ), if u is an euclidean point, then there is one and only one line passing through u not intersecting with L, and if u is an elliptic point, then there are infinite lines passing through u not intersecting with L, but if u is a hyperbolic point, then each line passing through u will intersect with L, see also the Fig.7, in where, the planar graph is the complete graph K4 and the points 1, 2 is elliptic vertices, the point 3 is euclidean and the point 4 hyperbolic. Then all m-lines in the filed A do not intersect with L and each m-line passing through the point 4 will intersect with the line L. Therefore, (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5), (L5) and (R5).

Fig.7 Case 2. M is an orientable map According to the classification theorem of compact surfaces, We only need to

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

prove this result for the torus. Notice that on the torus, an m-line has the following properties ([15]): If the slope ς of m-line L is a rational number, then L is a closed line on the torus. Otherwise, L is infinite, and moreover L passes arbitrarily close to every point of the torus.

Whence, if L1 is an m-line on the torus, not passes through an elliptic or hyperbolic point, then for any point u exterior L1 , we know that if u is an euclidean point, then there is only one m-line passing through u not intersecting with L1 , and if u is elliptic or hyperbolic, then any m-line passing through u will intersect with L1 .

Now let L2 be an m-line passes through an elliptic or hyperbolic point, such as the m-line in Fig.8 and v an euclidean point.

Fig.8 Then any m-line L in the shade filed passing through the point v will not intersect with L2 . Therefore, (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

Case 3. M is a non-orientable map Similar to the Case 2, by the classification theorem of the compact surfaces, we only need to prove this result for the projective plane. An m-line in a projective plane is shown in the Fig.9, in where, case (a) is an m-line passes through euclidean points, (b) passes through an elliptic point and (c) passes through a hyperbolic

61

A New View of Combinatorial Maps by Smarandache’s Notion∗

point.

Fig.9 Now let the m-line passes through the center in the circle. Then if u is an euclidean point, there is only one m-line passing through u, see (a) in the Fig.10. If v is an elliptic point and there is an m-line passes through it and intersect with L, see (b) in Fig.10, assume the point 1 is a point such that the m-line 1v passes through 0, then any m-line in the shade of (b) passing through the point v will intersect with L.

Fig.10 If w is a hyperbolic point and there is an m-line passing through it and does not intersect with L, see Fig.10(c), then any m-line in the shade of (c) passing through the point w will not intersect with L. Since the position of vertices of the map M on the projective plane can be choose as we wish, the proof is complete.

♮.

4.2 Map geometries with boundary The Poincar´e’s model for the hyperbolic geometry hints us to introduce the map geometries with boundary, which are defined as follows.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Definition 4.4 For a map geometry (M, µ) without boundary and faces f1 , f2 , · · · , fl ∈ F (M), 1 ≤ l ≤ φ(M)−1, if (M, µ)\{f1 , f2 , · · · , fl } is connected, then call (M, µ)−l = (M, µ) \ {f1 , f2 , · · · , fl } a map geometry with boundary f1 , f2 , · · · , fl and orientable or not if (M, µ) is orientable or not.

A connected curve with constant curvature in (M, µ)−l is called an m− -line and points m− -points.

Two m− -lines on the torus and projective plane are shown in Fig.11 and Fig.12.

Fig.11

Fig.12 The map geometries with boundary also are Smarandache geometries, which is convince by the following result.

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A New View of Combinatorial Maps by Smarandache’s Notion∗

Proposition 4.4 For a map M on a locally orientable surface with order≥ 3, vertex valency≥ 3 and a face f ∈ F (M), there is an angle factor µ such that (M, µ)−1 is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

Proof Similar to the proof of Proposition 4.3, consider the map M being a planar map, an orientable map on a torus or a non-orientable map on a projective plane, respectively. We get the assertion.

♮

Notice that for an one face map geometry (M, µ)−1 with boundary, if we choose all points being euclidean, then (M, µ)−1 is just the Poincar´e’s model for the hyperbolic geometry.

4.3 Classification of map geometries For the classification of map geometries, we introduce the following definition.

Definition 4.5 Two map geometries (M1 , µ1 ), (M2 , µ2 ) or (M1 , µ1 )−l , (M2 , µ2 )−l are called to be equivalent if there is a bijection θ : M1 → M2 such that for ∀u ∈ V (M), θ(u) is euclidean, elliptic or hyperbolic iff u is euclidean, elliptic or hyperbolic.

The relation of the numbers of unrooted maps with the map geometries is in the following.

Proposition 4.5 If M is a set of non-isomorphisc maps with order n and m faces, then the number of map geometries without boundary is 3n |M| and the number of map geometries with one face being its boundary is 3n m|M|.

Proof By the definition, for a map M ∈ M, there are 3n map geometries without boundary and 3n m map geometries with one face being its boundary by Proposition 4.3. Whence, we get 3n |M| map geometries without boundary and 3n m|M| map geometries with one face being its boundary from M.

♮.

We have the following enumeration result for non-equivalent map geometries without boundary.

Proposition 4.6 The numbers nO (Γ, g), nN (Γ, g) of non-equivalent orientable, nonorientable map geometries without boundary underlying a simple graph Γ by denial the axiom (A5) by (A5), (L5) or (R5) are

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

3|Γ| O

n (Γ, g) =

Q

(ρ(v) − 1)!

v∈V (Γ)

,

2|AutΓ|

and (2β(Γ) − 1)3|Γ|

Q

(ρ(v) − 1)!

v∈V (Γ)

N

n (Γ, g) =

, 2|AutΓ| where β(Γ) = ε(Γ) − ν(Γ) + 1 is the Betti number of the graph Γ.

Proof Denote by M(Γ) the set of all non-isomorphic maps underlying the graph Γ on locally orientable surfaces and by E(Γ) the set of all embeddings of the graph Γ on locally orientable surfaces. For a map M, M ∈ M(Γ), there are

3|M | |AutM |

different

map geometries without boundary by choosing the angle factor µ on a vertex u such that u is euclidean, elliptic or hyperbolic. From permutation groups, we know that |AutΓ × hαi | = |(AutΓ)M ||M AutΓ×hαi | = |AutM||M AutΓ×hαi |.

Therefore, we get that

nO (Γ, g) =

3|M | |AutM| M ∈M(Γ) X

X |AutΓ × hαi | 3|Γ| |AutΓ × hαi | M ∈M(Γ) |AutM|

=

X 3|Γ| = |M AutΓ×hαi | |AutΓ × hαi | M ∈M(Γ)

3|Γ| |E O (Γ)| |AutΓ × hαi | Q 3|Γ| (ρ(v) − 1)! v∈V (Γ) = .

2|AutΓ| =

Similarly, we get that

nN (Γ, g) =

=

3|Γ| |E N (Γ)| |AutΓ × hαi | Q (2β(Γ) − 1)3|Γ| (ρ(v) − 1)! v∈V (Γ)

2|AutΓ|

.

A New View of Combinatorial Maps by Smarandache’s Notion∗

This completes the proof.

65

♮

For the classification of map geometries with boundary, we have the following result.

Proposition 4.7 The numbers nO (Γ, −g), nN (Γ, −g) of non-equivalent orientable, non-orientable map geometries with one face being its boundary and underlying a simple graph Γ by denial the axiom (A5) by (A5), (L5) or (R5) are respective

nO (Γ, −g) =

Y 3|Γ| 2d(g[Γ](x)) [(β(Γ) + 1) (ρ(v) − 1)! − |x=1 ] 2|AutΓ| dx v∈V (Γ)

and

nN (Γ, −g) =

Y 2d(g[Γ](x)) (2β(Γ) − 1)3|Γ| [(β(Γ) + 1) (ρ(v) − 1)! − |x=1 ], 2|AutΓ| dx v∈V (Γ)

where g[Γ](x) is the genus polynomial of the graph Γ ( see [12]), i.e., g[Γ](x) = γm (Γ) P

gk [Γ]xk with gk [Γ] being the number of embeddings of Γ on the orientable sur-

k=γ(Γ)

face of genus k.

Proof Notice that ν(M) − ε(M) + φ(M) = 2 − 2g(M) for an orientable map M by the Euler characteristic. Similar to the proof of Proposition 4.6 with the notation M(Γ), by Proposition 4.5 we know that

nO (Γ, −g) =

φ(M)3|M | |AutM| M ∈M(Γ) X

=

(2 + ε(Γ) − ν(Γ) − 2g(M))3|M | |AutM| M ∈M(Γ)

=

X 2g(M)3|M | (2 + ε(Γ) − ν(Γ))3|M | − |AutM| |AutM| M ∈M(Γ) M ∈M(Γ)

=

(2 + ε(Γ) − ν(Γ))3|M | |AutΓ × hαi |

−

X g(M)|AutΓ × hαi | 2 × 3|Γ| |AutΓ × hαi | M ∈M(Γ) |AutM|

=

(β(Γ) + 1)3|M | X (Γ)|M AutΓ×hαi | |AutΓ × hαi | M ∈M

X

X

X

|AutΓ × hαi | |AutM| M ∈M(Γ)

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

−

X 3|Γ| g(M)|M AutΓ×hαi | |AutΓ| M ∈M(Γ)

=

m X (β(Γ) + 1)3|Γ| Y 3|Γ| (ρ(v) − 1)! − kgk [Γ] 2|AutΓ| |AutΓ| k=γ(Γ) v∈V (Γ)

γ (Γ)

Y 2d(g[Γ](x)) 3|Γ| = [(β(Γ) + 1) (ρ(v) − 1)! − |x=1 ].

2|AutΓ| dx v∈V (Γ)

Notice that nL (Γ, −g) = nO (Γ, −g)+nN (Γ, −g) and the number of re-embeddings of an orientable map M on surfaces is 2β(M ) (see also [13]). We have that

nL (Γ, −g) =

2β(M ) × 3|M | φ(M) |AutM| M ∈M(Γ) X

= 2β(M ) nO (Γ, −g).

Whence, we get that

nN (Γ, −g) = (2β(M ) − 1)nO (Γ, −g) Y (2β(M ) − 1)3|Γ| 2d(g[Γ](x)) = [(β(Γ) + 1) (ρ(v) − 1)! − |x=1 ].

2|AutΓ| dx v∈V (Γ) This completes the proof.

♮

4.4 Polygons in a map geometry A k-polygon in a map geometry is a k-polygon with each line segment being m-lines or m− -lines. For the sum of the internal angles in a k-polygon, we have the following result.

Proposition 4.8 Let P be a k-polygon in a map geometry with each line segment passing through at most one elliptic or hyperbolic point. If H is the set of elliptic points and hyperbolic points on the line segment of P , then the sum of the internal angles in P is (k + |H| − 2)π −

1 X ρ(u)µ(u).

2 u∈H

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A New View of Combinatorial Maps by Smarandache’s Notion∗

Proof Denote by U, V the sets of elliptic points and hyperbolic points in H and |U| = p, |V | = q. If an m-line segment passes through an elliptic point u, add an auxiliary line segment in the plane as shown in Fig.13(1). Then we get that ρ(u)µ(u) .

2 If an m-line passes through an hyperbolic point v, also add an auxiliary line angle a = angle1 + angle2 = π −

segment in the plane as shown in Fig.13(2). Then we get that angle b = angle3 + angle4 =

ρ(v)µ(v) − π.

2

Fig.13 Since the sum of the internal angles of a k-polygon in the plane is (k − 2)π, we know that the sum of the internal angles in P is X ρ(u)µ(u) ρ(u)µ(u) )− ( − π) 2 2 u∈U v∈V 1 X = (k + p + q − 2)π − ρ(u)µ(u) 2 u∈H 1 X = (k + |H| − 2)π − ρ(u)µ(u).

2 u∈H

(k −2)π +

This completes the proof.

X

(π −

♮

As a corollary, we get the sum of the internal angles of a triangle in a map geometry as follows, which is consistent with the classical results.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Corollary 4.1 Let △ be a triangle in a map geometry. Then (i) if △ is euclidean, then then the sum of its internal angles is equal to π; (ii) if △ is elliptic, then the sum of its internal angles is less than π; (iii) if △ is hyperbolic, then the sum of its internal angles is more than π.

5. Open problems for applying maps to classical geometries Here is a collection of open problems concerned combinatorial maps with these Riemann geometry and Smarandache geometries. Although they are called open problems, in fact, any solution for one of these problems needs to establish a new mathematical system first.

5.1 The uniformization theorem for simple connected Riemann surfaces The uniformization theorem for simple connected Riemann surfaces is one of those beautiful results in the Riemann surface theory, which is stated as follows([2]).

If S is a simple connected Riemann surface, then S is conformally equivalent to one and only one of the following three: (a) C (b)

S

∞;

C;

(c) △ = {z ∈ C||z| < 1}.

We have proved in [11] that any automorphism of a map is conformal. Therefore, we can also introduced the conformal mapping between maps. Then, how can we define the conformal equivalence for maps enabling us to get the uniformization theorem of maps? What is the correspondent map classes with the three type (a) − (c) Riemann surfaces?

5.2 Combinatorial construction of an algebraic curve of genus A complex plane algebraic curve Cl is a homogeneous equation f (x, y, z) = 0 in P2 C = (C 2 \ (0, 0, 0))/ ∼, where f (x, y, z) is a polynomial in x, y and z with coefficients in C. The degree of f (x, y, z) is said the degree of the curve Cl . For a Riemann surface S, a well-known result is ([2])there is a holomorphic mapping ϕ : S → P2 C such that ϕ(S) is a complex plane algebraic curve and

A New View of Combinatorial Maps by Smarandache’s Notion∗

g(S) =

69

(d(ϕ(S)) − 1)(d(ϕ(S)) − 2) .

2

By map theory, we know a combinatorial map also is on a surface with genus.

Then whether can we get an algebraic curve by all edges in a map or by make operations on the vertices or edges of the map to get plane algebraic curve with given k-multiple points? and how do we find the equation f (x, y, z) = 0?

5.3 Classification of s-manifolds by maps We present an elementary classification for the closed s-manifolds in the Section 3.

For the general s-manifolds, their correspondent combinatorial model is the maps on surfaces with boundary, founded by Bryant and Singerman in 1985 ([1]). The later are also related to the modular groups of spaces and need to investigate further themselves. The questions are (i) how can we combinatorially classify the general s-manifolds by maps with boundary? (ii) how can we find the automorphism group of an s-manifold? (iii) how can we know the numbers of non-isomorphic s-manifolds, with or without root? (iv) find rulers for drawing an s-manifold on a surface, such as, the torus, the projective plane or Klein bottle, not only the plane.

These s-manifolds only using triangulations of surfaces with vertex valency in {5, 6, 7}. Then what are the geometrical meaning of the other maps, such as, the 4-regular maps on surfaces. It is already known that the later is related to the Gauss cross problem of curves([9]).

5.4 Map geometries As we have seen in the previous section, map geometries are nice models of the Smarandache geometries. More works should be dong for them. (i) For a given graph G, determine properties of map geometries underlying G. (ii) For a given locally orientable surface S, determine the properties of map

70

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

geometries on S. (iii) Classify map geometries on a locally orientable surface. (iv) Enumerate non-equivalent map geometries underlying a graph or on a locally orientable surface. (v) Establish the surface geometry by map geometries.

5.5 Gauss mapping among surfaces In the classical differential geometry, a Gauss mapping among surfaces is defined as follows([10]): Let S ⊂ R3 be a surface with an orientation N. The mapping N : S → R3 takes its value in the unit sphere S 2 = {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1} along the orientation N. The map N : S → S 2 , thus defined, is called the Gauss mapping.

We know that for a point P ∈ S such that the Gaussian curvature K(P ) 6= 0 and V a connected neighborhood of P with K does not change sign, N(A) , A→0 A where A is the area of a region B ⊂ V and N(A) is the area of the image of B by K(P ) = lim

the Gauss mapping N : S → S 2 . The questions are (i) what is its combinatorial meaning of the Gauss mapping? How to realizes it by maps? (ii) how can we define various curvatures for maps and rebuilt the results in the classical differential geometry?

5.6 The Gauss-Bonnet theorem Let S be a compact orientable surface. Then Z Z

S

Kdσ = 2πχ(S),

A New View of Combinatorial Maps by Smarandache’s Notion∗

71

where K is the Gaussian curvature on S. This is the famous Gauss-Bonnet theorem for compact surface ([2], [6]). The questions are (i) what is its combinatorial meaning of the Gauss curvature? (ii) how can we define the angle, area, volume, curvature, · · ·, of a map? (iii) can we rebuilt the Gauss-Bonnet theorem by maps? or can we get a generalization of the classical Gauss-Bonnet theorem by maps?

5.7 Riemann manifolds A Riemann surface is just a Riemann 2-manifold, which has become a source of the mathematical creative power. A Riemann n-manifold (M, g) is a n-manifold M with a Riemann metric g. Many important results in Riemann surfaces are generalized to Riemann manifolds with a higher dimension ([6]). For example, let M be a complete, simple-connected Riemann n-manifold with constant sectional curvature c, then we know that M is isometric to one of the model spaces Rn , SRn or HRn . Whether can we systematically rebuilt the Riemann manifold theory by combinatorial maps? or can we make a combinatorial generalization of results in the Riemann geometry, for example, the Chern-Gauss-Bonnet theorem ([6])?

References [1] R.P.Bryant and D.Singerman, Foundations of the theory of maps on surfaces with boundary,Quart.J.Math.Oxford(2),36(1985), 17-41.

[2] H. M.Farkas and I. Kra, Riemann Surfaces, Springer-Verlag New York inc(1980).

[3] H.Iseri, Smarandache manifolds, American Research Press, Rehoboth, NM,2002.

[4] H.Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm. edu/˜smarandache/Howard-Iseri-paper.htm.

[5] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Canada, Vol.12(2003) [6] J.M.Lee, Riemann Manifolds, Springer-Verlag New York,Inc(1997).

[7] Y.P.Liu, Advances in Combinatorial Maps, Northern Jiaotong University Publisher, Beijing (2003).

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Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

[8] Yanpei Liu, Enumerative Theory of Maps, Kluwer Academic Publisher, Dordrecht / Boston / London (1999).

[9] Y.P.Liu, Embeddability in Graphs, Kluwer Academic Publisher, Dordrecht / Boston / London (1995).

[10] Mantredo P.de Carmao, Differential Geometry of Curves and Surfaces, Pearson Education asia Ltd (2004).

[11] L.F.Mao, Automorphism groups of maps, surfaces and Smarandache geometries, American Research Press, Rehoboth, NM,2005.

[12] L.F.Mao and Y.P.Liu, A new approach for enumerating maps on orientable surfaces, Australasian J. Combinatorics, vol.30(2004), 247-259.

[13] L.F.Mao and Y.P.Liu, Group action approach for enumerating maps on surfaces,J.Applied Math. & Computing, vol.13, No.1-2,201-215.

[14] W.S.Massey, Algebraic topology: an introduction, Springer-Verlag,New York, etc.(1977).

[15] V.V.Nikulin and I.R.Shafarevlch, Geometries and Groups, Springer-Verlag Berlin Heidelberg (1987) [16] F. Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

[17] J.Stillwell, Classical topology and combinatorial group theory, Springer-Verlag New York Inc., (1980).

[18] W.T.Tutte, What is a maps? in New Directions in the Theory of Graphs (ed.by F.Harary), Academic Press (1973), 309 325.

73

An Introduction to Smarandache Geometries on Maps

An Introduction to Smarandache Geometries on Mapsâ&#x2C6;&#x2014; Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China) maolinfan@163.com

Abstract: A map is a connected topological graph cellularly embedded in a surface. On the past century, works on maps are concentrated on its combinatorial counterpart without metrics, such as, the embedding of graphs and the enumeration of maps. For returning to its original face, the conception of map geometries is introduced, which are nice models of the Smarandache geometries, also a new kind of intrinsic geometry of surfaces. Some properties of parallel bundles in planar map geometries are obtained in this paper. Open problems related combinatorial maps with the differential geometry, Riemann geometry and Smarandache geometries are also presented for further applications of combinatorial maps to classical mathematics.

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Reported at The 2005 International Conference on Graphs and Combinatorics, June 26,

2005 Zhejiang, P.R.China

74

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Key Words: map, Smarandache geometry, map geometry, planar map geometry, parallel bundle.

AMS(2000): 05C15, 20H15, 51D99, 51M05 1. Questions for a combinatorial problem When we research a mathematical problem, the following four questions should be asked firstly by ourself, which is the same for a combinatorial problem.

• What is its contribution to combinatorics?

• What is its contribution to mathematics?

• What is its contribution to sciences?

• Is its contribution local or global?

The topic introduced in this report has stood a trial by the four questions.

2. What are Smarandache geometries?

Definition 2.1 An axiom is said Smarandachely denied if the axiom behaves in at least two different ways within the same space, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache geometry is a geometry which has at least one Smarandachely denied axiom (1969).

F. Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

L.F.Mao, Automorphism groups of maps, surfaces and Smarandache geometries, American Research Press, Rehoboth, NM,2005. Also see the web page: www. gallup. unm. edu/ smarandache/Linfan.pdf

• Applications to classical geometries The axioms system of Euclid geometry is in the following: (A1)there is a straight line between any two points.

(A2)a finite straight line can produce a infinite straight line continuously.

(A3)any point and a distance can describe a circle.

(A4)all right angles are equal to one another.

(A5)if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, then the two straight lines, if produced

An Introduction to Smarandache Geometries on Maps

75

indefinitely, meet on that side on which are the angles less than the two right angles.

The axiom (A5) can be also replaced by: (A5’)given a line and a point exterior this line, there is one line parallel to this line.

The Lobachevshy-Bolyai-Gauss geometry, also called hyperbolic geometry, is a geometry with axioms (A1) − (A4) and the following axiom (L5): (L5) there are infinitely many line parallels to a given line passing through an exterior point.

The Riemann geometry is a geometry with axioms (A1)−(A4) and the following axiom (R5): there is no parallel to a given line passing through an exterior point.

• Further applications (1)Relativity Theory (Because they include the Riemann geometry in a subspace) (2)Parallel Universes (Because they combine separate spaces into one space) L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Ca- nada, Vol.12(2003)

• Iseri’s model for Smarandache geometries An s-manifold is any collection C(T, n) of these equilateral triangular disks Ti , 1 ≤ i ≤ n satisfying the following conditions: (i) Each edge e is the identification of at most two edges ei , ej in two distinct triangular disks Ti , Tj , 1 ≤ i, j ≤ n and i 6= j; (ii) Each vertex v is the identification of one vertex in each of five, six or seven distinct triangular disks.

H.Iseri, Smarandache manifolds, American Research Press, Rehoboth, NM,2002.

3. What is a map?

A combinatorial map is a connected topological graph cellularly embedded in a surface.

Definition 3.1: A combinatorial map M = (Xα,β , P) is defined to be a basic per-

76

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

mutation P, i.e, for any x ∈ Xα,β , no integer k exists such that P k x = αx, acting on Xα,β , the disjoint union of quadricells Kx of x ∈ X (the base set), where K = {1, α, β, αβ} is the Klein group, with the following two conditions holding: (i) αP = P −1 α; (ii) the group ΨJ =< α, β, P > is transitive on Xα,β .

W.T.Tutte, What is a maps? in New Directions in the Theory of Graphs (ed.by F.Harary), Academic Press (1973), 309 325.

Y.P.Liu, Advances in Combinatorial Maps(in Chinese), Northern Jiaotong University Publisher, Beijing (2003).

Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publisher, Dordrecht / Boston / London (1999).

• Orientation: If the group ΨI = hαβ, Pi is transitive on Xα,β , then M is non-orientable.

Otherwise, orientable.

• An Example of Maps: K4 on the torus.

Fig.1 M = (Xα,β , P):

Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw}

77

An Introduction to Smarandache Geometries on Maps

P = (x, y, z)(αβx, u, w)(αβz, αβu, v) × (αβy, αβv, αβw)(αx, αz, αy)(βx, αw, αu) × (βz, αv, βu)(βy, βw, βv) Vertices: v1 = {(x, y, z), (αx, αz, αy)} v2 = {(αβx, u, w), (βx, αw, αu)} v3 = {(αβz, αβu, v), (βz, αv, βu)} v4 = {(αβy, αβv, αβw), (βy, βw, βv)} Edges: {e, αe, βe, αβe}, e ∈ {x, y, z, u, v, w} Faces: f1 = {(x, u, v, αβw, αβx, y, αβv, αβz), (βx, αz, αv, βy, αx, αw, βv, βu)} f2 = {(z, αβu, w, αβy), (βz, αy, βw, αu)} 4. Map geometries Definition 4.1 For a combinatorial map M, endows a real number µ(u), 0

<

µ(u) < π, with each vertex u, u ∈ V (M). Call (M, µ) a map geometry without boundary, µ(u) the angle factor of the vertex u and to be orientablle or non-orientable if M is orientable or not.

L.F.Mao, A new view of combinatorial maps by Smarandache’s notion, arXiv: Math.

GM/0506232.

• A realization of a vertex u, u ∈ V (M) in R3 space.

ρ(u)µ(u) < 2π

ρ(u)µ(u) = 2π

Fig.2

ρ(u)µ(u) > 2π

78

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Theorem 4.1 For a map M on a locally orientable surface with order≥ 3, there is an angle factor µ such that (M, µ) is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

Definition 4.2 For a map geometry (M, µ) without boundary and faces f1 , f2 , · · · , fl ∈ F (M), 1 ≤ l ≤ φ(M)−1, if (M, µ)\{f1 , f2 , · · · , fl } is connected, then call (M, µ)−l = (M, µ) \ {f1 , f2 , · · · , fl } a map geometry with boundary f1 , f2 , · · · , fl and orientable or not if (M, µ) is orientable or not.

• An one face map geometry (M, µ)−1 with boundary is just the Poincar´e’s model for the hyperbolic geometry if we choose all points being euclidean.

Theorem 4.2 For a map M on a locally orientable surface with order≥ 3 and a face f ∈ F (M), there is an angle factor µ such that (M, µ)−1 is a Smarandache geometry by denial the axiom (A5) with the axioms (A5),(L5) and (R5).

• Map geometries are a generalization of s-manifolds.

• Enumeration results for map geometries: Theorem 4.3 The numbers nO (Γ, g), nN (Γ, g) of non-equivalent orientable, nonorientable map geometries without boundary underlying a simple graph Γ by denial the axiom (A5) by (A5), (L5) or (R5) are 3|Γ| O

n (Γ, g) =

Q

(ρ(v) − 1)!

v∈V (Γ)

2|AutΓ|

,

and (2β(Γ) − 1)3|Γ| N

n (Γ, g) =

Q

v∈V (Γ)

2|AutΓ|

(ρ(v) − 1)! ,

where β(Γ) = ε(Γ) − ν(Γ) + 1 is the Betti number of the graph Γ.

Similarly, we can also get enumeration results for map geometries with boundary.

5. Parallel bundles in planar map geometries Definition 5.1 A family L of infinite lines not intersecting each other in a planar geometry is called a parallel bundle.

79

An Introduction to Smarandache Geometries on Maps

Fig.3 Theorem 5.1 Let (M, µ) be a planar map geometry, C = {u1v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1 and the angle functions on them are f1 , f2 , · · · , fl , respectively, also see the Fig.4.

Fig.4 Then a family L of parallel lines passing through C is a parallel bundle iff for any x, x ≥ 0,

f1′ (x) ≥ 0 ′ ′ f1+ (x) + f2+ (x) ≥ 0 ′ ′ ′ f1+ (x) + f2+ (x) + f3+ (x) ≥ 0

············ ′ ′ ′ f1+ (x) + f2+ (x) + · · · + fl+ (x) ≥ 0.

Theorem 5.2 Let (M, µ) be a planar map geometry, C = {u1v1 , u2 v2 , · · · , ul vl } a

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

cut of the map M with order u1 v1 , u2 v2 , · · · , ul vl from the left to the right, l ≥ 1 and the angle functions on them are f1 , f2 , · · · , fl . Then the parallel lines parallel the initial parallel lines after them passing through C iff for ∀x ≥ 0,

f1′ (x) ≥ 0 ′ ′ f1+ (x) + f2+ (x) ≥ 0 ′ ′ ′ f1+ (x) + f2+ (x) + f3+ (x) ≥ 0

············ ′ ′ ′ f1+ (x) + f2+ (x) + · · · + fl−1+ (x) ≥ 0

and f1 (x) + f2 (x) + · · · + fl (x) = lπ.

• Linear criterion Theorem 5.3 Let (M, µ) be a planar map geometry, C = {u1v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2v2 , · · · , ul vl from the left to the right, l ≥ 1.

Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor µ satisfies the following linear inequality system ρ(v1 )µ(v1 ) ≥ ρ(u1 )µ(u1 ) ρ(v1 )µ(v1 ) ρ(v2 )µ(v2 ) ρ(u1 )µ(u1 ) ρ(u2 )µ(u2) + ≥ + d(u1 v1 ) d(u2 v2 ) d(u1 v1 ) d(u2v2 ) ············

ρ(v1 )µ(v1 ) ρ(v2 )µ(v2 ) ρ(vl )µ(vl ) ρ(u1 )µ(u1) ρ(ul )µ(ul ) + +···+ ≥ +···+ . d(u1 v1 ) d(u2 v2 ) d(ul vl ) d(u1 , v1 ) d(ul , vl ) Corollary 5.1 Let (M, µ) be a planar map geometry with M underlying a regular graph, C = {u1 v1 , u2v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2v2 , · · · , ul vl

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An Introduction to Smarandache Geometries on Maps

from the left to the right, l ≥ 1. Then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle iff the angle factor µ satisfies the following linear inequality system µ(v1 ) ≥ µ(u1 )

µ(v1 ) µ(v2 ) µ(u1) µ(u2 ) + ≥ + d(u1 v1 ) d(u2v2 ) d(u1 v1 ) d(u2v2 )

············

µ(v2 ) µ(vl ) µ(u1 ) µ(u2 ) µ(ul ) µ(v1 ) + +···+ ≥ + +···+ d(u1 v1 ) d(u2 v2 ) d(ul vl ) d(u1v1 ) d(u2 v2 ) d(ul vl ) and particularly, if assume that all the lengths of edges in C are the same, then

µ(v1 )

≥

µ(u1 )

µ(v1 ) + µ(v2 )

≥

µ(u1 ) + µ(u2)

······ ··· ······ µ(v1 ) + µ(v2 ) + · · · + µ(vl )

≥

µ(u1 ) + µ(u2) + · · · + µ(ul ).

Theorem 5.4 Let (M, µ) be a planar map geometry, C = {u1v1 , u2 v2 , · · · , ul vl } a cut of the map M with order u1 v1 , u2v2 , · · · , ul vl from the left to the right, l ≥ 1. If for any integer i, i ≥ 1, ρ(ui ) µ(vi ) ≤ , ρ(vi ) µ(ui) then under the linear assumption, a family L of parallel lines passing through C is a parallel bundle.

• A example of parallel bundle:

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Fig.5 More results for parallel bundles can be seen in: Linfan Mao, Parallel bundles in planar map geometries, e-print: arXiv: math.GM/0506386, also appearing in Scientia Magna, Vol.1(2005), No.2,120-133.

6. Open Problems • The uniformization theorem for simple connected Riemann surfaces: If S is a simple connected Riemann surface, then S is conformally equivalent to one and only one of the following three: (a) C (b)

S

∞;

C;

(c) △ = {z ∈ C||z| < 1}.

Problem 6.1: How can we define the conformal equivalence for maps enabling us to get the uniformization theorem of maps?

Problem 6.2 What is the correspondence class maps with the three type (a) − (c) Riemann surfaces?

• The Gauss-Bonnet Theorem Let S be a compact orientable surface. Then

An Introduction to Smarandache Geometries on Maps

Z Z

S

83

Kdσ = 2πχ(S),

where K is Gaussian curvature on S. Problem 6.3 What is its combinatorial meaning of the Gauss curvature?

Problem 6.4 How can we define the angle, area, volume, curvature, · · ·, of a map?

Problem 6.5 Can we rebuilt the Gauss-Bonnet theorem by maps? Or can we get a generalization of the classical Gauss-Bonnet theorem by maps?

• Map Geometries Problem 6.6 For a given graph, determine the properties of map geometries underlying this graph.

Problem 6.7 For a given locally orientable surface, determine the properties of map geometries on this surface.

Problem 6.8 Classify map geometries on a given locally orientable surface.

Problem 6.9 Enumerate non-equivalent map geometries underlying a graph or on a locally orientable surface.

Problem 6.10 Establish the surface geometry by map geometries.

84

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

A Multi-Space Model for Chinese Bids Evaluation with Analyzingâ&#x2C6;&#x2014; Linfan Mao Chinese Academy of Mathematics and System Sciences, Beijing 100080, P.R.China maolinfan@163.com

Abstract. A tendering is a negotiating process for a contract through by a tenderer issuing an invitation, bidders submitting bidding documents and the tenderer accepting a bidding by sending out a notification of award. As a useful way of purchasing, there are many norms and rulers for it in the purchasing guides of the World Bank, the Asian Development Bank, ¡ ¡ ¡, also in contract conditions of various consultant associations. In China, there is a law and regulation system for tendering and bidding. However, few works on the mathematical model of a tendering and its evaluation can be found in publication. The main purpose of this paper is to construct a Smarandache multi-space model for a tendering, establish an evaluation system for bidding based on those ideas in the references [7] and [8] and analyze its solution by applying the decision approach for multiple objectives and value engineering.

Open problems for pseudo-multi-spaces are also presented in the final section.

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A Multi-Space Model for Chinese Bids Evaluation with Analyzing

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3S Keâ&#x201E;&#x201E;' 0^ ' 9 ZJ' r_q S 7a < ' ?: [7][8] I hb?^ ' Smarandache 3 AD; mCJ t TLĂ&#x2020;? & r 0Bh_>~UY I g 'FH 0 P \ |r~ 3 A'4mFH â&#x201E;&#x201E; {_â&#x201E;&#x201E; jm / 'L)rCJ t ' T,

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Key Words:

b

tendering, bidding, evaluation, Smarandache multi-space,

condition of successful bidding, decision of multiple objectives, decision of simply objective, pseudo-multiple evaluation, pseudo-multi-space.

AMS(2000): 90B50,90C35,90C90 §1. Introduction The tendering is an efficient way for purchasing in the market economy. According to the Contract Law of the Peopleâ&#x20AC;&#x2122;s Republic of China (Adopted at the second meeting of the Standing Committee of the 9th National Peopleâ&#x20AC;&#x2122;s Congress on March 15,1999), it is just a civil business through by a tenderer issuing a tendering announcement or an invitation, bidders submitting bidding documents compiled on the tendering document and the tenderer accepting a bidding after evaluation by sending out a notification of award. The process of this business forms a negotiating process of a contract. In China, there is an interval time for the acceptation of a bidding and becoming effective of the contract, i.e., the bidding is accepted as the tenderer send out the notification of award, but the contract become effective only as the tenderer and the successful bidder both sign the contract.

In the Tendering and Bidding Law of the Peopleâ&#x20AC;&#x2122;s Republic of China (Adopted at the 11th meeting of the Standing Committee of the 9th National Peopleâ&#x20AC;&#x2122;s Congress on August 30,1999), the programming and liability or obligation of the tenderer, the bidders, the bid evaluation committee and the government administration are stipulated in detail step by step. According to this law, the tenderer is on the side of raising and formulating rulers for a tender project and the bidders are on the side of response each ruler of the tender. Although the bid evaluation committee is organized by the tenderer, its action is independent on the tenderer. In tendering and bidding law and regulations of China, it is said that any unit or person can not disturbs works of the bid evaluation committee illegally. The action of them should consistent with the tendering and bidding law of China and they should place

86

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

themselves under the supervision of the government administration.

The role of each partner can be represented by a tetrahedron such as those shown in Fig.1.

Fig.1 The 41th item in the Tendering and Bidding Law of the Peopleâ&#x20AC;&#x2122;s Republic of China provides conditions for a successful bidder: (1) optimally responsive all of the comprehensive criterions in the tendering document; (2) substantially responsive criterions in the tender document with the lowest evaluated bidding price unless it is lower than this bidderâ&#x20AC;&#x2122;s cost.

The conditions (1) and (2) are often called the comprehensive evaluation method and the lowest evaluated price method. In the same time, these conditions also imply that the tendering system in China is a multiple objective system, not only evaluating in the price, but also in the equipments, experiences, achievements, staff and the programme, etc.. However, nearly all the encountered evaluation methods in China do not apply the scientific decision of multiple objectives. In where, the comprehensive evaluation method is simply replaced by the 100 marks and the lowest evaluated price method by the lowest bidding price method. Regardless of whether different objectives being comparable, there also exist problems for the ability of bidders and specialists in the bid evaluation committee creating a false impression for the successful bidding price or the successful bidder. The tendering and bidding is badly in need of establishing a scientific evaluation system in accordance with these laws and regulations in China. Based on the reference [7] for Smarandache multi-spaces and the mathematical model for the tendering in [8], the main purpose

87

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

of this paper is to establish a multi-space model for the tendering and a scientific evaluation system for bids by applying the approach in the multiple objectives and value engineering, which enables us to find a scientific approach for tendering and its management in practice. Some cases are also presented in this paper.

The terminology and notations are standard in this paper. For terminology and notation not defined in this paper can be seen in [7] for multi-spaces, in [1] − [3] and [6] for programming, decision and graphs and in [8] for the tendering and bidding laws and regulations in China. §2. A multi-space model for tendering Under an idea of anti-thought or paradox for mathematics :combining different fields into a unifying field, Smarandache introduced the conception of multi-spaces in 1969([9]-[12]), including algebraic multi-spaces and multi-metric spaces. The contains the well-known Smarandache geometries([5] − [6]), which can be used to General Relativity and Cosmological Physics([7]). As an application to Social Sciences, multi-spaces can be also used to establish a mathematical model for tendering.

These algebraic multi-spaces are defined in the following definition.

Definition 2.1 An algebraic multi-space A1 , A2 , · · · , Am X

=

P m [

with multiple m is a union of m sets

Ai ,

i=1

where 1 ≤ m < +∞ and there is an operation or ruler ◦i on each set Ai such that (Ai , ◦i) is an algebraic system for any integer i, 1 ≤ i ≤ m.

Notice that if i 6= j, 1 ≤ i, j ≤ m, there must not be Ai

T

Aj = ∅, which are just

correspondent with the characteristics of a tendering. Thereby, we can construct a Smarandache multi-space model for a tendering as follows.

Assume there are m evaluation items A1 , A2 , · · · , Am for a tendering Ae and there

are ni evaluation indexes ai1 , ai2 , · · · , aini for each evaluation item Ai , 1 ≤ i ≤ m.

By applying mathematics, this tendering can be represented by Ae =

m [

i=1

Ai ,

88

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

where, for any integer i, 1 ≤ i ≤ m, (Ai , ◦i) = {ai1 , ai2 , · · · , aini |◦i } is an algebraic system. Notice that we do not define other relations of the tendering Ae and evaluation indexes aij with Ai , 1 ≤ i ≤ m unless Ai ⊆ Ae and aij ∈ Ai in this

multi-space model.

Now assume there are k, k ≥ 3 bidders R1 , R2 , · · · , Rk in the tendering Ae and

the bidding of bidder Rj , 1 ≤ j ≤ k is 

e = Rj (A)

   Rj    

A1

 

A2   

···   Am





=

      

Rj (A1 )

  

Rj (A2 )   ···

Rj (Am )

  

.

According to the successful bidding criterion in the Tendering and Bidding Law of the People’s Republic of China and regulations, the bid evaluation committee needs to determine indexes i1 , i2 , · · · , ik , where {i1 , i2 , · · · , ik } = {1, 2, · · · , k} such that there is an ordered sequence e ≻ R (A) e ≻ · · · ≻ R (A) e Ri1 (A) i2 ik

e R (A), e · · · , R (A) e of bidders R , R , · · · , R . Here, these for these bidding R1 (A), 2 k 1 2 k

bidders Ri1 , Ri2 and Ri3 are pre-successful bidders in succession determined by the bid evaluation committee in the laws and regulations in China.

Definition 2.2 An ordered sequence for elements in the symmetry group Sn on {1, 2, · · · , m} is said an alphabetical sequence if it is arranged by the following criterions: (i) (1, 0 · · · , 0) P for any permutation P ∈ Sn . (ii) if integers s1 , s2 , · · · , sh ∈ {1, 2, · · · , m}, 1 ≤ h < m and permutations

(s1 , s2 , · · · , sh , t, · · ·), (s1 , s2 , · · · , sh , l, · · ·) ∈ Sn , then (s1 , s2 , · · · , sh , t, · · ·) ≻ (s1 , s2 , · · · , sh , l, · · ·)

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

89

if and only if t < l. Let {xσi }n1 be a sequence, where σ1 ≻ σ2 ≻ · · · ≻ σn and σi ∈ Sn for 1 ≤ i ≤ n, then the sequence {xσi }n1 is said an alphabetical sequence.

Now if xσ ≻ xτ , xσ is preferable than xτ in order. If xσ xτ , then xσ is preferable or equal with xτ in order. If xσ xτ and xτ xσ , then xσ is equal xτ in order, denoted by xσ ≈ xτ .

We get the following result for an evaluation of a tendering. e ∈ O × O × O × · · · for Theorem 2.1 Let O1 , O2 , O3 · · · be ordered sets. If Rj (A) 1 2 3

any integer j, 1 ≤ j ≤ k, then there exists an arrangement i1 , i2 , · · · , ik for indexes 1, 2, · · · , k such that e R (A) e · · · R (A). e Ri1 (A) i2 ik

Proof By the assumption, for any integer j, 1 ≤ j ≤ k, e ∈ O × O × O × ···.

Rj (A) 1 2 3

e can be represented by Whence, Rj (A)

e = (x , x , x , · · ·), Rj (A) j1 j2 j3

where xjt ∈ Ot , t ≥ 1. Define a set St = {xjt ; 1 ≤ j ≤ m}.

Then the set St ⊆ Ot is finite. Because the set Ot is an ordered set, so there exists an order for elements in St . Not loss of generality, assume the order is x1t x2t · · · xmt , e R (A), e for elements in St . Then we can apply the alphabetical approach to Ri1 (A), i2 e and get indexes i , i , · · · , i such that · · · , Rik (A) 1 2 k

e R (A) e · · · R (A). e Ri1 (A) ♮ i2 ik

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

If we choose Oi , i ≥ 1 to be an ordered function set in Theorem 2.1, particularly, let O1 = {f }, f : Ai → R, 1 ≤ i ≤ m be a monotone function set and Ot = ∅ for t ≥ 2, then we get the next result.

Theorem 2.2 Let Rj : Ai → R, 1 ≤ i ≤ m, 1 ≤ j ≤ k be monotone functions. Then there exists an arrangement i1 , i2 , · · · , ik for indexes 1, 2, · · · , k such that e R (A) e · · · R (A). e Ri1 (A) i2 ik

We also get the following consequence for evaluation numbers by Theorem 2.2.

Corollary 2.1 If Rj (Ai ) ∈ [−∞, +∞] × [−∞, +∞] × [−∞, +∞] × · · · for any integers i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ k, then there exists an arrangement i1 , i2 , · · · , ik for indexes 1, 2, · · · , k such that e R (A) e · · · R (A). e Ri1 (A) i2 ik

Notice that in the above ordered sequence, if we arrange Ris ≻ Ril or Ril ≻ Ris further in the case of Ris ≈ Ril , s 6= l, then we can get an ordered sequence e ≻ R (A) e ≻ · · · ≻ R (A), e Ri1 (A) i2 ik

and the pre-successful bidders accordance with the laws and regulations in China. §3. A mathematical analog for bids evaluation For constructing an evaluation system of bids by the multi-space of tendering, the following two problems should be solved in the first.

Problem 1 For any integers i, j, 1 ≤ i, j ≤ m, how to determine Rj (Ai ) on account of the responsiveness of a bidder Rj on indexes ai1 , ai2 , · · · , aini ? e on account of Problem 2 For any integer j, 1 ≤ j ≤ m, how to determine Rj (A)

the vector (Rj (A1 ), Rj (A2 ), · · · , Rj (Am ))t ?

Different approaches for solving Problems 1 and 2 enable us to get different mathematical analogs for bids evaluation.

3.1. An approach of multiple objectives decision

91

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

This approach is originated at the assumption that Rj (A1 ), Rj (A2 ), · · · , Rj (Am ), 1 ≤ j ≤ m are independent and can not compare under a unified value unit. The objectives of tendering is multiple, not only in the price, but also in the equipments, experiences, achievements, staff and the programme, etc., which are also required by the 41th item in the Tendering and Bidding Law of the People’s Republic of China.

According to Theorems 2.1 − 2.2 and their inference, we can establish a programming for arranging the order of each evaluation item Ai , 1 ≤ i ≤ m and getting

m e R (A), e · · · , R (A) e of a tendering A e = S A , as an ordered sequence of bids R1 (A), 2 k i i=1

follows:

STEP 1 determine the order of the evaluation items A1 , A2 , · · · , Am . For example, for m = 5, A1 ≻ A2 ≈ A3 ≻ A4 ≈ A5 is an order of the evaluation items A1 , A2 , A3 , A4 , A5 .

STEP 2 for two bids Rj1 (Ai ), Rj2 (Ai ), j1 6= j2 , 1 ≤ i ≤ m, determine the condition for Rj1 (Ai ) ≈ Aj2 (A2 ). For example, let A1 be the bidding price. Then Rj1 (A1 ) ≈ Rj2 (A1 ) providing |Rj1 (A) − Rj2 (A1 )| ≤ 100(10 thousand yuan).

STEP 3 for any integer i, 1 ≤ i ≤ m, determine the order of R1 (Ai ), R2 (Ai ), · · · , Rk (Ai ). For example, arrange the order of bidding price from lower to higher and the bidding programming dependent on the evaluation committee. e R (A), e · · · , R (A), e which need an apSTEP 4 alphabetically arrange R1 (A), 2 k

e ≈ R (A) e in order. For example, arrange proach for arranging equal bids Rj1 (A) j2

them by the ruler of lower price preferable and get an ordered sequence e ≻ R (A) e ≻ · · · ≻ R (A) e Ri1 (A) i2 ik

e R (A), e · · · , R (A). e of these bids R1 (A), 2 k

Notice that we can also get an ordered sequence through by defining the weight

functions

and

e = H(ω(A ), ω(A ), · · · , ω(A )) ω(A) 1 2 m

ω(Ai) = F (ω(ai1 ), ω(ai2 ), · · · , ω(aini )).

For the weight function in detail, see the next section.

92

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Theorem 3.1 The ordered sequence of bids of a tendering Ae can be gotten by the

above programming.

Proof Assume there are k bidders in this tendering. Then we can alphabetically e R (A), e · · · , R (A) e and get arrange these bids Ri1 (A), i2 ik e R (A) e · · · R (A). e Ri1 (A) i2 ik

e ≈ R (A), e we finally Now applying the arranging approach in the case of Rj1 (A) j2

obtain an ordered sequence

e ≻ R (A) e ≻ · · · ≻ R (A). e Ri1 (A) ♮ i2 ik

Example 3.1 There are 3 evaluation items in a building construction tendering Ae with A1 =price, A2 =programming and A3 =similar achievements in nearly 3 years.

The order of the evaluation items is A1 ≻ A3 ≻ A2 and Rj1 (Ai ) ≈ Rj2 (Ai ), 1 ≤ i ≤ 3 providing |Rj1 (A1 ) − Rj2 (A1 )| ≤ 150, Rj1 (A2 ) and Rj2 (A2 ) are in the same rank or the difference of architectural area between Rj1 (A3 ) and Rj2 (A3 ) is not more than 40000m2 . For determining the order of bids for each evaluation item, it applies the rulers that from the lower to the higher for the price, from higher rank to a lower rank for the programming by the bid evaluation committee and from great to small e R (A), e amount for the similar achievements in nearly 3 years and arrange Rj1 (A), j2

1 ≤ j1 , j2 ≤ k =bidders by the ruler of lower price first for two equal bids in order

e ≈ R (A). e Rj1 (A) j2

There were 4 bidders R1 , R2 , R3 , R4 in this tendering. Their bidding prices are

in table 1. bidder A1

R1

R2

R3

R4

3526 3166 3280 3486 table 1

Applying the arrangement ruler for A1 , the order for R2 (A1 ), R3 (A1 ), R4 (A1 ), R1 (A1 ) is R2 (A1 ) ≈ R3 (A1 ) ≻ R4 (A1 ) ≈ R1 (A1 ).

93

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

The evaluation order for A2 by the bid evaluation committee is R3 (A2 ) ≈ R2 (A2 ) ≻ R1 (A2 ) ≻ R4 (A2 ). They also found the bidding results for A3 are in table 2. bidder A3 (m2 )

R1

R2

R3

R4

250806 210208 290108 300105 table 2

Whence the order of R4 (A3 ), R3 (A3 ), R1 (A3 ), R2 (A3 ) is R4 (A3 ) ≈ R3 (A3 ) ≻ R1 (A3 ) ≈ R2 (A3 ).

is

e R (A), e R (A) e and R (A) e Therefore, the ordered sequence for these bids R1 (A), 2 3 4

e ≻ R (A) e ≻ R (A) e ≻ R (A). e R3 (A) 2 4 1

Let the order of evaluation items be A1 ≻ A2 ≻ · · · ≻ Am . Then we can also get the ordered sequence of a tendering by applying a graphic method. By the terminology in graph theory, to arrange these bids of a tendering is equivalent e to find a directed path passing through all bidders R1 , R2 , · · · , Rk in a graph G[A]

defined in the next definition. Generally, the graphic method is more convenience in the case of less bidders, for instance 7 bidders for a building construction tendering in China.

m S Definition 3.1 Let R1 , R2 , · · · , Rk be all these k bidders in a tendering Ae = Ai . i=1

e = (V (G[A]), e E(G[A])) e Define a directed graph G[A] as follows.

e = {R , R , · · · , R } × {A , A , · · · , A }, E(G[A]) e =E V (G[A]) 1 2 k 1 2 m 1

S

E2

S

E3 ,

where E1 consists of all these directed edges (Rj1 (Ai ), Rj2 (Ai )), 1 ≤ i ≤ m, 1 ≤ j1 , j2 ≤ k and Rj1 (Ai ) ≻ Rj2 (Ai ) is an adjacent order. Notice that if Rs (Ai ) ≈ Rl (Ai ) ≻ Rj (Ai ), then there are Rs (Ai ) ≻ Rj (Ai ) and Rl (Ai ) ≻ Rj (Ai ) simultaneously. E2 consists of edges Rj1 (Ai )Rj2 (Ai ), 1 ≤ i ≤ m, 1 ≤ j1 , j2 ≤ k, where Rj1 (Ai ) ≈ Rj2 (Ai ) and E3 = {Rj (Ai )Rj (Ai+1 )|1 ≤ i ≤ m − 1, 1 ≤ j ≤ k}. e for Example 3.1 is shown in Fig.2.

For example, the graph G[A]

94

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Fig.2 Now we need to find a directed path passing through R1 , R2 , R3 , R4 with start vertex R2 (A1 ) or R3 (A1 ). By the ruler in an alphabetical order, we should travel starting from the vertex R3 (A1 ) passing through A2 , A3 and then arriving at A1 .

Whence, we find a direct path correspondent with the ordered sequence e ≻ R (A) e ≻ R (A) e ≻ R (A). e R3 (A) 2 4 1

3.2. An approach of simply objective decision

This approach is established under the following considerations for Problems 1 and 2. e seek Consideration 1 In these evaluation items A1 , A2 , · · · , Am of a tendering A,

the optimum of one evaluation item. For example, seek the lowest bidding price in a construction tendering for a simply building or seek the optimum of design scheme in a design project tendering, etc..

Consideration 2 The value of these evaluation items A1 , A2 , · · · , Am is comparable which enables us to measure each of them by a unify unit and to construct various weighted functions on them. For example, the 100 marks and the lowest evaluated price method widely used in China are used under this consideration.

3.2.1. The optimum of one objective Assume the optimal objective being A1 in a tendering Ae = mine the acceptable basic criterions for all other items

m S

Ai . We need to deteri=1 A2 , A3 , · · · , Ak , then arrange

R1 (A1 ), R2 (A1 ), · · · , Rl (A1 ) among these acceptable bids R1 , R2 , · · · , Rl for items A2 , A3 , · · · , Ak in Ri , 1 ≤ i ≤ k. For example, evaluating these items A2 , A3 , · · · , Ak

95

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

by qualification or by weighted function on A2 , A3 , · · · , Ak up to these criterions, then arrange these acceptable bids R1 , R2 , · · · , Rl under their response to A1 and e R (A) e if R (A ) ≈ R (A ). According to Theorem 3.1, we get the order of Ri (A), i i 1 j 1

the following result.

Theorem 3.2 The approach of one optimal objective can get an ordered sequence e of bids for a tendering A.

Example 3.2 The optimum of design scheme is the objective in a design project tendering Ae which is divided into 5 ranks A, B, C, D, E and other evaluation items

such as human resources, design period and bidding price by a qualifiable approach if the bidding price is in the interval of the service fee norm of China. The final order of bids is determined by the order of design schemes with qualifiable human resources, design period and bidding price and applying the ruler of lower price first for two equal design scheme in order.

There were 8 bidders in this tendering. Their bidding prices are in table 3. bidder

R1

bidding price

R2

R3

R4

R5

R6

R7

R8

251 304 268 265 272 283 278 296 table 3

After evaluation for these human resources, design period and bidding price, 4 bidders are qualifiable unless the bidder R5 in human resources. The evaluation result for bidding design schemes is in table 4. rank

A

B

C

D

E

design scheme

R3 R6

R1

R2 R8

R7

R4

table 4 Therefore, the ordered sequence for bids is g

e ≻ R (A) e ≻ R (A) e ≻ R (A) e ≻ R (A) e ≻ R (A). e R3 (A) ≻ R6 (A) 1 8 2 7 4

Example 3.3 The optimum objective in a tendering Ae for a construction of a

dwelling house is the bidding price A1 . All other evaluation items, such as qualifications, management persons and equipments is evaluated by a qualifiable approach.

96

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

There were 7 bidders Ri , 1 ≤ i ≤ 7 in this tendering. The evaluation of price is by a weighted function approach, i.e., determine the standard price S first, then calculate the mark N of each bidder by the following formulae ( S=

7 P

i=1

Ai − max{Ri (A1 )|1 ≤ i ≤ 7} − min{Ri (A1 )|1 ≤ i ≤ 7} ,

5

Ri (A1 ) − S | × 100, 1 ≤ i ≤ 7, S where, if Ri (A1 ) − S > 0 then t = 6 and if Ri (A1 ) − S < 0 then t = 3.

Ni = 100 − t × |

After evaluation, all bidders are qualifiable in qualifications, management persons and equipments. Their bidding prices are in table 5. bidder

R1

A1

R2

R3

R4

R5

R6

R7

3518 3448 3682 3652 3490 3731 3436 table 5

According to these formulae, we get that S = 3558 and the mark of each bidder as those shown in table 6. bidder mark

R1

R2

R3

R4

R5

R6

R7

96.70 91.27 79.12 84.16 94.27 73.84 89.68 table 6

Therefore, the ordered sequence of bids is e ≻ R (A) e ≻ R (A) e ≻ R (A) e ≻ R (A) e ≻ R (A) e ≻ R (A). e R1 (A) 5 2 7 4 3 6

3.2.2. The pseudo-optimum of multiple objectives This approach assumes that there is a unifying unit between these evaluation items A1 , A2 , · · · , Am in an interval [a, b]. Whence it can be transformed into case 3.2.1 and sought the optimum of one objective. Not loss of generality, we assume the unifying unit is ̟ and ̟(Ai ) = fi (̟), 1 ≤ i ≤ m,

97

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

where fi denotes the functional relation of the metric ̟(Ai) with unit ̟. Now the objective of tendering turns to a programming of one objective

max F (f1 (̟), f2 (̟), · · · , fm (̟)) or min F (f1 (̟), f2(̟), · · · , fm (̟)), ̟

̟

where F denotes the functional relation of the tendering Ae with these evaluation items A1 , A2 , · · · , Am , which can be a weighted function, such as a linear function F (f1 (̟), f2(̟), · · · , fm (̟)) =

m X

fi (̟)

i=1

or an ordered sequence. According to Theorem 3.2, we know the following result.

Theorem 3.3 If the function F of a tendering Ae only has one maximum value in

e 1 ≤ i ≤ k after [a, b], then there exists an ordered sequence for these bids Ri (A), e and R (A) e when F (R (A)) e = F (R (A)), e i 6= j. determined how to arrange Ri (A) j i j

The 100 marks and the lowest evaluated price method widely used in China

both are applications of this approach. In the 100 marks, the weight function is a linear function F (f1 (̟), f2 (̟), · · · , fm (̟)) =

m X

fi (̟),

i=1

with 0 ≤ F (f1 (̟), f2 (̟), · · · , fm (̟)) ≤ 100, fi ≥ 0, 1 ≤ i ≤ m. In the lowest evaluated price method, each difference of an evaluation item Ai , 2 ≤ i ≤ m is changed to the bidding price ̟(A1 ), i.e., fi = (R(Ai ) − S(Ai ))̟(A1 ), 1 ≤ i ≤ m, where S(Ai ) is the standard line for Ai , ̟(Ai) is one unit difference of Ai in terms of A1 . The weighted function of the lowest evaluated price method is

F (̟(A1 ), f2 (̟(A1 )), · · · , fm (̟(A1 ))) = (1 +

m X

(R(Ai ) − S(Ai )))̟(A1 ).

j=2

For example, we can fix one unit difference of a technological parameter 15, i.e., ̟(A1 ) = 15 ten thousand dollars in terms of the bidding price.

98

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

§4. Weighted functions and their construction We discuss weighted functions on the evaluation items or indexes in this section.

First, we give a formal definition for weighted functions.

Definition 4.1 For a tendering Ae =

m S

i=1

Ai , where Ai = {ai1 , ai2 , · · · , ain }, 1 ≤ i ≤ m

with k bidders R1 , R2 , · · · , Rk , if there is a continuous function ω : Ae → [a, b] ⊂

(−∞, +∞) or ω : Ai → [a, b] ⊂ (−∞, +∞), 1 ≤ i ≤ m such that for any integers e > R (ω(A)) e or R (ω(A)) e = R (ω(A)) e as R (A) e ≻ R (A) e l, s, 1 ≤ l, s ≤ k, Rl (ω(A)) s l s l s e ≈ R (A) e and R (ω(A ) > R (ω(A )) or R (ω(A )) = R (ω(A )) as R (A ) ≻ or Rl (A) s l i s i l i s i l i

Rs (Ai ) or Rl (Ai ) ≈ Rs (Ai ), 1 ≤ i ≤ m, then ω is called a weighted function for the tendering Ae or the evaluation items Ai , 1 ≤ i ≤ m.

According to the decision theory of multiple objectives([3]), the weighted func-

tion ω(Ai ) must exists for any integer i, 1 ≤ i ≤ m. but generally, the weight e does not exist if the values of these evaluation items A , A , · · · , A function ω(A) 1 2 m

can not compare. There are two choice for the weighted function ω(Ai ).

Choice 1 the monotone functions in the interval [a, b], such as the linear functions.

Choice 2 The continuous functions only with one maximum value in the interval [a, b], such as ω(Ai ) = −2x2 + 6x + 12 or ω(Ai) =

  

x, −x + 4,

if

0 ≤ x ≤ 2, if

x ≥ 4.

As examples of concrete weighted functions ω, we discuss the tendering of civil engineering constructions.

4.1. The weighted function for the bidding price Let A1 be the bidding price. We often encounter the following weighted function ω(A1 ) in practice. ω(Ri (A1 )) = −ς ×

Ri (A1 ) − S +ζ S

where, S=

R1 (A1 ) + R2 (A1 ) + · · · + Rk (A1 ) k

99

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

or S= or

  

R1 (A1 )+R2 (A1 )+···+Rk (A1 )−M −N , k−2 R1 (A1 )+R2 (A1 )+···+Rk (A1 ) , k

k ≥ 5, 3≤k≤4

R1 (A1 ) + R2 (A1 ) + · · · + Rk (A1 ) × (1 − A%). k Where T ,A%,k, M and N are the pre-price of the tender, the percentage of T in S, S = T × A% +

the number of bidders and the maximum and minimum bidding price, respectively, Ri (A1 ), i = 1, 2, · · · , k denote the bidding prices and ς, ζ are both constants.

There is a postulate in these weighted functions, i.e., each bidding price is random and accord with the normal distribution. Then the best excepted value of this civil engineering is the arithmetic mean of these bidding prices. However, each bidding price is not random in fact. It reflects the bidder’s expected value and subjectivity in a tendering. We can not apply any definite mathematics to fix its real value. Therefore, this formula for a weighted function can be only seen as a game, not a scientific decision.

By the view of scientific decision, we can apply weighted functions according to the expected value and its cost in the market, such as (1) the linear function Ri (A1 ) − N +q M −N in the interval [N, M], where M, N are the maximum and minimum bidding prices ω(Ri (A1 )) = −p ×

p is the deduction constant and q is a constant such that Ri (ω(A1 )) ≥ 0, 1 ≤ i ≤ k.

The objective of this approach is seek a lower bidding price.

(2) non-linear functions in the interval [N, M], such as T+

ω(Ri (A1 )) = −p ×

ω(Ri (A1 )) = −p ×

Ri (A1 ) − q

k+1

k P

Ri (A1 )

j=1

Ri (A1 ) −

k+1

+

q,

q

k+1

R1 (A1 )R2 (A1 ) · · · Rk (A1 )T

R1 (A1 )R2 (A1 ) · · · Rk (A1 )T

+q

100

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

or

ω(Ri (A1 )) = −p ×

Ri (A1 ) − r

r

R21 (A1 )+R22 (A1 )+···+R2k (A1 )+T 2 k+1

R21 (A1 )+R22 (A1 )+···+R2k (A1 )+T 2 k+1

+q

etc.. If we wish to analog a curve for these bidding prices and choose a point on this curve as ω(Ri (A1 )), we can apply the value of a polynomial of degree k + 1 f (x) = ak+1 xk+1 + ak xk + · · · + a1 x + a0 by the undetermined coefficient method. Arrange the bidding prices and pre-price of the tender from lower to higher. Not loss of generality, let it be Rj1 (A1 ) ≻ R( j2 )(A1 ) ≻ · · · ≻ T ≻ · · · ≻ Rjk (A1 ). Choose k + 2 constants c1 > c2 > · · · > ck+1 > 0, for instance k + 1 > k > · · · > 1 > 0. Solving the equation system

Rj1 (A1 ) = ak+1 ck+1 + ak ck1 + · · · + a1 c1 + a0 1 Rj2 (A1 ) = ak+1 ck+1 + ak ck2 + · · · + a1 c2 + a0 2 ··························· Rjk−1 (A1 ) = ak+1 ck+1 + ak ckk + · · · + a1 ck + a0 k Rjk (A1 ) = a0 we get a polynomial f (x) of degree k + 1. The bidding price has an acceptable difference in practice. Whence, we also need to provide a bound for the difference which does not affect the ordered sequence of bids.

4.2. The weighted function for the programming Let A2 be the evaluation item of programming with evaluation indexes {a21 , a22 , · · · , a2n2 }. It is difficult to evaluating a programming in quantify, which is not only for the tender, but also for the evaluation specialists. In general, any two indexes of A2 are not comparable. Whence it is not scientific assigning numbers for each index since we can not explain why the mark of a programming is 96 but another is 88.

This means that it should qualitatively evaluate a programming or a quantify after a qualitatively evaluation. Its weight function ω(Ri (A2 )), 1 ≤ i ≤ k can be chosen as a linear function

101

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

ω(Ri (A2 )) = ω(Ri(a21 )) + ω(Ri (a22 )) + · · · + ω(Ri (a2n2 )).

For example, there are 4 evaluation indexes for the programming, and each with A, B, C, D ranks in a tendering. The corespondent mark for each rank is in table 7. index

a21

a22

a23

a24

A

4

2

2

1

B

3

1.5

1.5

0.8

C

2

1

1

0.5

D

1

0.5

0.5

0.3

table 7 If the evaluation results for a bidding programming Ri , 1 ≤ i ≤ 4 are ω(Ri (a21 )) = A, ω(Ri (a22 )) = B, ω(Ri (a23 )) = B and ω(Ri(a24 )) = A, then the mark of this programming is

Ri (ω(A2 )) = Ri (ω(a21 )) + Ri (ω(a22 )) + Ri (ω(a23 )) + Ri (ω(a24 )) = 4 + 3 + 1.5 + 1 = 9.5.

By the approach in Section 3, we can alphabetically or graphicly arrange the order of these programming if we can determine the rank of each programming.

Certainly, we need the order of these indexes for a programming first. The index order for programming is different for different constructions tendering. §5. Further discussions m S e = 5.1 Let Ae = Ai be a Smarandache multi-space with an operation set O(A) i=1

e is also a Smarandache {◦i ; 1 ≤ i ≤ m}. If there is a mapping Θ on Ae such that Θ(A)

e Θ) is called a pseudo-multi-space. Today, nearly all geometries, multi-space, then (A,

such as the Riemann geometry, Finsler geometry and these pseudo-manifold geometries are particular cases of pseudo-multi-geometries. e For applying Smarandache multi-spaces to an evaluation system, choose Θ(A)

e is an being an order set. Then Theorem 3.1 only asserts that any subset of Θ(A)

102

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

order set, which enables us to find the ordered sequence for all bids in a tendering. e is continuous and Θ(A) e ⊆ [−∞, +∞], then Θ is a weighted Particularly, if Θ(A)

function on Ae widely applied in the evaluation of bids in China. By a mathematical

e Θ) is valuable to research. Some open problems are view, many problems on (A,

presented in the following.

e Θ), particularly, for these Problem 5.1 Characterize these pseudo-multi-spaces (A, e = cases of Θ(A)

n S

e = [ai , bi ], Θ(A)

i=1

n S

e = (Gi , ◦i ) and Θ(A)

i=1

n S

(R; +i, ◦i ) with (Gi , ◦i )

i=1

and (R; +i , ◦i) being a finite group or a ring for 1 ≤ i ≤ n.

e be a group, a ring or a filed. Can we find an ordered sequence Problem 5.2 Let Θ(A) e for a finite subset of A?

e be n lines or n planes in an Euclid space Rn . Characterize Problem 5.3 Let Θ(A)

e Θ). Can we find an arrangement for a finite subset of these pseudo-multi-spaces (A, e A?

5.2 The evaluation approach in this paper can be also applied to evaluate any multiple objectives, such as the evaluation of a scientific project, a personal management system, an investment of a project, · · ·, etc..

References [1] G.Chartrand and L.Lesniak,Graphs & Digraphs, Wadsworth, Inc., California, 1986.

[2] T.Chen, Decision Approaches in Multiple Objectives(in Chinese), in The Handbook of Modern Engineering Mathematics, Vol.IV, Part 77,1357-1410. Central China Engineering College Press, 1987.

[3] P.C.Fishburn, Utility Theory for Decision Making, New York, Wiley, 1970.

[4] D.L.Lu, X.S.Zhang and Y.Y.Mi, An offer model for civil engineering construction, Chinese OR Transaction, Vol.5, No.4(2001)41-52.

[5] L.F.Mao, On Automorphisms groups of Maps, Surfaces and Smarandache geometries, Sientia Magna, Vol.1(2005), No.2, 55-73.

[6] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research Press, 2005.

[7] L.F.Mao, Smarandache multi-space theory, Hexis, Phoenix, AZ 2006.

[8] L.F.Mao, Chinese Construction Project Bidding Technique & Cases Analyzing–

A Multi-Space Model for Chinese Bids Evaluation with Analyzing

103

Smarandache Multi-Space Model of Bidding,Xiquan Publishing House (Chinese Branch), America, 2006.

[9] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

[10] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth, 1999.

[11] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297384.

[12] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438.

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A Mathematical Model for Chinese Bids Evaluation with Its Solution Analyzing

Abstract. A tendering is a negotiating process for a contract through by a tenderer issuing an invitation, bidders submitting bidding documents and the tenderer accepting a bidding by sending out a notification of award. As a useful way of purchasing, there are many norms and rulers for it in the purchasing guides of the World Bank, the Asian Development Bank, ¡ ¡ ¡, also in contract conditions of various consultant associations.

1

e-print:

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In China, there is a law and regulation system for tendering and bidding. However, few works on the mathematical model of a tendering and its evaluation can be found in publication. The main purpose of this paper is to construct a Smarandache multi-space model for a tendering, establish an evaluation system for bidding based on those ideas in the references [7] and [8] and analyze its solution by applying the decision approach for multiple objectives and value engineering. The final section discusses some questions for the bids evaluation system already existed in China today. Some suggestions for solving these

b b b q bq

questions are presented in this section.

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2

k

j



e = Rj (A)

.

   Rj    

A1

  

A2   

···   Am



=

      

Rj (A1 )

  

Rj (A2 )   ···

Rj (Am )

  

8!O H'^5<GI1 /

108

:

7 qpz2[J Æ L L q >U< q FF R= j R7 Lp zq % 4 e R (A), e · · · , R (A) e `: j R GL z &p Lp R , R , · · · , R = L R (A), i , i , · · · , i 1 {i , i , · · · , i } = {1, 2, · · · , k} ' ; > L " u 1

1

2

2

1

k

1

k

2

2

k

k

e ≻ R (A) e ≻ · · · ≻ R (A), e Ri1 (A) i2 ik

i Ri , Ri [ Ri l1 7 J q W q Fj R= & Æ8 =7 Li p 1

2

|

3

} 2.2 F. {1, 2, · · · , m} ' ^+ S _qa z _>'XC { I 1XC (i) B + P ∈ S (1, 0 · · · , 0) P (ii) F s , s , · · · , s ∈ {1, 2, · · · , m}, 1 ≤ h ≤ m + (s , s , · · · , s , t, · · ·), (s , s , · · · , s , l, · · ·) ∈ S

n

n

1

1

2

2

1

h

h

2

h

n

|

(s1 , s2 , · · · , sh , t, · · ·) ≻ (s1 , s2 , · · · , sh , l, · · ·)

w^ t < l {x } 7℄mC σ ≻ σ ≻ · · · ≻ σ w σ ∈ S

C {x } {℄m I 1XCC x ≻ x x Cnr x x x x C r x q x x w x x > x u x C 2o ;8 Kv &<P Æ LOK = %'vOK j 2.1 . O , O , O · · · { C. B F j, 1 ≤ j ≤ k, R (A)e ∈ O × O × O × · · ·, O 1, 2, · · · , k '℄ 2 XCTL i , i , · · · , i .& n 1

σi

σ

τ

τ

σ

| |

n σi 1

σ

σ

2

C

3

τ

σ

τ

2

| i σ

2

i

τ

n

σ

b 3

e R (A) e · · · R (A). e Ri1 (A) i2 ik

&p* d q/ I j, 1 ≤ j ≤ k,

9 R (A)e ,M*

n

τ

1

1

1

e ∈ O × O × O × ···, Rj (A) 1 2 3

j

e = (x , x , x , · · ·), Rj (A) j1 j2 j3

1 xjt ∈ Ot, t ≥ 1 R 0 _

j

1

2

k

τ

_ (GIE<1 VZ

109

St = {xjt ; 1 ≤ j ≤ m},

}L S

>%

t

⊆ Ot

%LS _ 5% O %h 9 S 7= ^ " u t

x1t x2t · · · xmt ,

} ,^C XJ =zqd R i , i , · · · , i ' ;

e

e

e

i1 (A), Ri2 (A), · · · , Rik (A)

1

2

k

V ;Jv

t

[s

` k ;8 GL

e R (A) e · · · R (A). e Ri1 (A) ♮ i2 ik

zbR/ 2.1 7 O , i ≥ 1 %rLh = TI _ oN D O = {f }, f : A → R, 1 ≤ i ≤ m % .P TI Vz t ≥ 2 O = ∅ }F R / 2.1 # j 2.2 R : A → R, 1 ≤ i ≤ m, 1 ≤ j ≤ k { 4 F, O 1, 2, · · · , k '℄ 2 XCTL i , i , · · · , i .& 1

i

i

b

t

j

i

1

2

k

e R (A) e · · · R (A). e Ri1 (A) i2 ik

oN D R / 2.2 LwKÆ\ Fu 2.1 B F i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ k p R (A ) ∈ R × R × R × · · ·

O 1, 2, · · · , k '℄ 2 XCTL i , i , · · · , i .&

b

J /u E

j

1

2

i

k

e R (A) e · · · R (A). e Ri1 (A) i2 ik

7 z&p%$[ [%Xg R Ri q }2ol , ;8 e \7 J q W q F ^ =

s

≈ Ril , s 6= l

#=

z

e ≻ R (A) e ≻ · · · ≻ R (A), e Ri1 (A) i2 ik

i & q Æ8 7 Li p 3.

z R<℄ 2t

u %NÆL LI q ?7 Æ L7,sIz Pw,K?&1q

110

8!O H'^5<GI1 / :

b

b

S9 1 B F i, j, 1 â&#x2030;¤ i, j â&#x2030;¤ m X_qg R ' a , a , · · · , a / 'g 7 R (A )?

S9 2 B F j, 1 â&#x2030;¤ j â&#x2030;¤ m X_q, (R (A ), R (A ), · · · , R (A )) e ZA R (A)? d , ?&1q = V r& * $ Ã&#x2020; L7,sIV = I zq 3.1. y ` ;\ ; sI = I oH 1r%1q 2 7= R (A ), R (A ), · · · , R (A ) Up \ 7 V ^C % = , ' - [s d A P Ã&#x2020; LV . 1 %&V = ,% # ^ 7 LpL>V = DW 4 #Ã&#x2020;[?/%9 "1 7 qpz2[J Ã&#x2020; L L q E T!%y 7 Ly<b Q ^ = &p %NR/ 2.1â&#x2C6;&#x2019;2.2 >Ã&#x2020;\ 2o ,%'vD?7Ã&#x2020;LZ Ae = S A e R (A), e = l&7,Z A , 1 â&#x2030;¤ i â&#x2030;¤ m = I zq [ i w R hY L R (A), e = >j R X C wK · · · , R (A) 1 | 7b?+J A , A , · · · , A 'XCT m = 5 ) A â&#x2030;» A â&#x2030;&#x2C6; A â&#x2030;» A â&#x2030;&#x2C6; A m7b?+J A , A , A , A , A 'â&#x201E;&#x201E; 2 XCT 2 | B m 'g R (A ), R (A ), j 6= j , 1 â&#x2030;¤ i â&#x2030;¤ m 7 R (A ) â&#x2030;&#x2C6; A (A ) '`H A g ? 7 |R (A) â&#x2C6;&#x2019; R (A )| â&#x2030;¤ 100 ty B R (A ) â&#x2030;&#x2C6; R (A ) 3 | B F i, 1 â&#x2030;¤ i â&#x2030;¤ m 7 R (A ), R (A ), · · · , R (A ) 'XCT B7B m ? =be Bg ? mo*#h 7XC) e R (A), e · · · , R (A) e 'XC > A 7 4 | I 1XCL 7 R (A), ! Cz R (A)e â&#x2030;&#x2C6; R (A)e )'XCTL _q g ? n 'z e R (A), e · · · , R (A) e 'Ml d 'XCT

_>XC P &# R (A),

b

ini

j

j

i1

i2

i

1

j

2

j

j

m

t

j

1

j

2

j

j

m

m

i=1

1

i

i

2

k

1

2

3

2

i b d bT â&#x201E;&#x201E; bo bT o F

4

5

1

2

j1

j1

i

2

j2

3

i

4

j2

5

1

i

2

1

1

j1

j1

2

1

2

i

i

1

2

k

i

bT

k

j2

1

j2

1

j2

1

j1

1

m

k

e â&#x2030;» R (A) e â&#x2030;» · · · â&#x2030;» R (A). e Ri1 (A) i2 ik

e = H(Ï&#x2030;(A ), Ï&#x2030;(A ), · · · , Ï&#x2030;(A )) J / , 7 y ,^C R 0fTI Ï&#x2030;(A) 1 2 m [ Ï&#x2030;(Ai) = F (Ï&#x2030;(ai1), Ï&#x2030;(ai2), · · · , Ï&#x2030;(ain )) =zq $w R s Z <PfT I = 3zW:K%N i

b

b

j 3.1 Brâ&#x201E;&#x201E;m +J Ae C C a&#g 'MlXC C * L k ' Lpa) L }F XJ q 2o ;8

_ (GIE<1 VZ

111

e R (A) e · · · R (A). e Ri1 (A) i2 ik

s4C P [ R

e

j1 (A)

#=

e â&#x2030;&#x2C6; Rj2 (A)

zq 2o `:;8 %

e â&#x2030;» R (A) e â&#x2030;» · · · â&#x2030;» R (A). e Ri1 (A) â&#x2122;® i2 ik

b bT I b wi bT I bb b p b bT d b d gbT?d X _q 7' B g b T 'XC{

l 3.1 Hmr %r +J 7 3 mb?+J _ { A = g ? A = 7BT A = g a R O+J\, hC| { A â&#x2030;» A â&#x2030;» A 7 |R (A ) â&#x2C6;&#x2019; R (A )| â&#x2030;¤ 150 R (A ) u R (A ) & Ã&#x2020; R (A ) u R (A ) ' *F r 40000m ) R (A ) â&#x2030;&#x2C6; R (A ), 1 â&#x2030;¤ i â&#x2030;¤ 3 ) 7 g ?XCo*#h 7BT XCob ? =_q 7Ã&#x2020;a R O+J \,a *FC _>XC a0 Cg P1 XCa ?*n 'z _>X C d+Jvp 4 mg R , R , R , R > g hg ? 1 g R R R R A (ty) 3526 3166 3280 3486 1

2

3

j1

j2

1

j2

1

1

2

j1

2

3

1

2

3

2

j2

j1

i

3

j2

j1

2

3

i

4

1

2

3

4

1

1.

,

b

A1

R2 (A1 ) â&#x2030;&#x2C6; R3 (A1 ) â&#x2030;» R4 (A1 ) â&#x2030;&#x2C6; R1 (A1 ).

b }{$B A 'b T { R (A ) â&#x2030;&#x2C6; R (A ) â&#x2030;» R (A ) â&#x2030;» R (A ) ) I!hBb +J A 'g T 2 g R R R R

b

2

3

b

1

2

A3 (m )

zhXCT {

d 3

d

2

2

2

2

1

3

2

4

250806 210208 290108 300105 2.

a R O+J' *F

R4 (A3 ) â&#x2030;&#x2C6; R3 (A3 ) â&#x2030;» R1 (A3 ) â&#x2030;&#x2C6; R2 (A3 ).

X _q I 1XC'z P XCT {

4

2

8!O H'^5<GI1 /

112

:

e ≻ R (A) e ≻ R (A) e ≻ R (A). e R3 (A) 2 4 1

* 7,Z = <I% A ≻ A ≻ · · · ≻ A } ;h L7,sIz ,^C # q j R ^C \ 7= DX j R`: = % 4> Puw K R 0 = G[A]e 7 j R %y LbL Lp R , R , · · · , R = L[R i d P LpIV h= [r1! G4 = } 3.1 B℄mp k mg R , R , · · · , R >g ' +J Ae = S A 7d℄mj G[A]e 1

2

m

1

b

1

2

k

2

k

p b b

m

i=1

i

e = {R , R , · · · , R } × {A , A , · · · , A }, V (G[A]) 1 2 k 1 2 m e =E E(G[A]) 1

_

S

E2

S

E3 ,

} _

h / E oSpp, (R (A ), R (A )) w 1 ≤ i ≤ m, 1 ≤ j , j ≤ k w R (A ) ≻ R (A ) {&"C > R (A ) ≈ R (A ) ≻ R (A ) ) R (A ) ≻ R (A ) Æ R (A ) ≻ R (A ) E , E v{ , h / E o R (A )R (A ), 1 ≤ i ≤ m, 1 ≤ j , j ≤ k w R (A ) ≈ R (A ); E = {R (A )R (A )|1 ≤ i ≤ m − 1, 1 ≤ j ≤ k} 5w v5 3.1 d : = L[ % 1

j1 j

i

i

j2

i

l

i

1

2

j

j1

i

j2

i

s

}

2

i

i

3

j1

i

l

i

j

2

j2

i

_

3

j

1

2

i

s

j1

i

i

j

j2

i

i

i+1

2. 5 3.1 d : = L[ Pu 2oz u 2 7jR%yF R (A ) R (A ) %CH LbL L p=L[R &p XJ 2o:F R (A ) o L A , A `k8% A l ;8 <I R (A)e ≻ R (A)e ≻ R (A)e ≻ R (A)e 3.2. y ` ;\ . Lw sI P d 1q 1 [ 2 = ,K? ; r& 2

3

3

2

4

1

1

3

1

2

1

3

1

_ (GIE<1 VZ b b?+J ~H℄mb?+J' n ℄2BmI %r b ) ~ T ? n D ℄2 8 b ~ 8 T I n) bb?+J rp? \ ? D a o me℄' ? BSpb?+J_> _D n FBh_> \ QD&& b? n'J b ! O b b?^ z m' L Y YÆfb ' *b b ?L)

113

1

P

2

P

P

<

/ B P

P / B

A1 , A2 , · · · , Am

B

&

A1 , A2 , · · · , Am

/

+ 2

/

J

,

. L ` Dv ;zq ,Q^=.%7,Z `D 2:>d7,Z % L * ÆLZ m S Ae = Ai b Q ^ =` D7,Z % A1 }z d >d7,Z A2 , A3 , · · · , Ak

i=1 1 `?= K R lku d A2, A3, · · · , Ak K = L, % R1, R2, · · · , Rl 7d R1(A1), R2(A1), · · · , Rl (A1) [s [ i w R Æ8 =7 Li p| . 5w d 7,Z A2, A3, · · · , Ak ^C_% 2 7 ^CfTI =zqd A2, A3, · · · , Ak [s' 1 _% R T lk&p_% = L R1, R2, · · · , Rl d7,Z A1 = L[ [s P 2R Ri(A1) ≈ Rj (A1) # Ri(A)e V Ri(A)e = zq & p R / 3.1 2o ;8 ,KOK 3.2.1.

b b b I bZ I b bT b f b p bI T d F b bT d gbT?d f b gb ℄Z' {i^ b Z~q hTg b ' T? fÆ {℄ Z v^ Z~ Bg b 8T I 'b T d

j 3.2 Br℄m +J Ae J P n a&#g T 'MlXC l 3.2 Hr 8T +J Ae aB~ 8T Pn{J vY{ A, B, C, D, E 8 5 m 7 {℄ 8 5 f4 L ℄7'`H g ? r = 7' X C B'℄?O{ l 7 8T b? & ) ?* n 'z _>XC d+Jvp 8 mg >g h ? 3 g

R R R R R R R R g ? (ty) 251 304 268 265 272 283 278 296 1

2

3

4

5

6

7

8

3.

85

R5

4

g T

bI

d

A

B

C

D

E

R3 R6

R1

R2 R8

R7

R4

4.

I

8T b T

114

8!O H'^5<GI1 / :

X g

b '

P XCT {

g

e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A). e R3 (A) â&#x2030;» R6 (A) 1 8 2 7 4

b o

bo

l 3.3 Heb = r %r +J Ae Bg F l +J~ { Ã&#x2020;7BA b , '( _>?l A b 'TL vp 7 mg R , 1 â&#x2030;¤ i â&#x2030;¤ 7 > g h?lb m F'TL 2< 7b ( C ? S 8Pg &Y N

Z v pb b ( S=

7 P

i=1

b b

1

i

Ai â&#x2C6;&#x2019; max{Ri (A1 )|1 â&#x2030;¤ i â&#x2030;¤ 7} â&#x2C6;&#x2019; min{Ri (A1 )|1 â&#x2030;¤ i â&#x2030;¤ 7} ,

5 Ni = 100 â&#x2C6;&#x2019; t Ã&#x2014; |

Ri (A1 ) â&#x2C6;&#x2019; S | Ã&#x2014; 100, 1 â&#x2030;¤ i â&#x2030;¤ 7, S

Ri (A1 ) â&#x2C6;&#x2019; S > 0 t = 6 Ri (A1 ) â&#x2C6;&#x2019; S < 0 t = 3 ) 7 &Y& ) *?n z XC f b 7 mg ' F l +J~ {Ã&#x2020;7 B A â&#x201E;&#x201E; %r N$ 8v^ Z~ g ' ? 5

F ZZ

b

g

A (ty) 1

b b T d

R1

R2

R3

R4

R5

R6

R7

3518 3448 3682 3652 3490 3731 3436

d gbT?d Ã&#x2020;g b &Y 5.

_qa 8PTL &# S = 3558

b

g

&Y

N1

N2

N3

N4

N5

N6

N7

96.70 91.27 79.12 84.16 94.27 73.84 89.68

b

zg XCT {

d T?&Yd 6.

e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A) e â&#x2030;» R (A). e R1 (A) 5 2 7 4 3 6

3.2.2.

( Y _7, ` D

_ (GIE<1 VZ

115

;zq * 7,Z A1, A2, · · · , Am u%&%I`1 [a, b] " u % =^ R i ,v% 3.2.1 =[rQ^. L ` D V ;Jv 2o ,* > % ^ % ̟ V>Up1 = <I% ̟(Ai ) = fi (̟), 1 ≤ i ≤ m,

1 fi M*TI<I }Pu =Æ Ll L v B . LFt max F (f1 (̟), f2 (̟), · · · , fm (̟)) ̟

min F (f1 (̟), f2 (̟), · · · , fm (̟)), ̟

1 F M* Æ LZ Ae V7,Z A1, A2, · · · , Am 1 = <I ,1fTI w

TvTI

F (f1 (̟), f2(̟), · · · , fm (̟)) =

m X

fi (̟)

i=1

1 ; <I 4C R / 3.2 2o ;8 KEOK j 3.3 F F A [a, b] z℄'P & f _℄ 7 R (A)e = e i 6= j ) R (A) e u R (A) e 'XCTL ~ J b?&#g R , 1 ≤ i ≤ k R (A), e 1 ≤ i ≤ k 'MlC B +J Ae g T R (A), N7JJ ÆL#^C=# 2' [ 7 =`? 7L,q %$ y1 %7,sI = :C u# 2' 7 2o CTvTI j

b

b i

j b

j

i

i

i

F (f1 (̟), f2 (̟), · · · , fm (̟)) =

m X

fi (̟),

i=1

V 0 ≤ F (f (̟), f (̟), · · · , f (̟)) ≤ 100, f ≥ 0, 1 ≤ i ≤ m i u 7 =`? 7 L, q7 >d7,Z A , A , · · · , A = L1ly [ L,% A 1

2

m

2

i

3

m

1

fi = (l(Ai ) − l0 (Ai ))̟(A1), 1 ≤ i ≤ m,

1 l0 17,Z = 7, RH ̟(A1 ) 1 . +1l), ) L i ^C = fTI%

8!O H'^5<GI1 /

116

:

F (̟(A1), f2 (̟(A1)), · · · , fm (̟(A1 ))) = (1 +

m X

(l(Ai ) − l0 (Ai )))̟(A1).

i=2

5wFR2 & } <;v DaIIz^ =) Ll% R' # )>7L,% 10 ̟(A ) = 10 ^ 1

72F %Nn\ R 0P7,Z 7 ) L = fTI 2o=L( >I R 4.

0

b

p b b

} 4.1 Br℄mp k mg R , R , · · · , R ug ' +J Ae = S A 'b?+J A = {a , a , · · · , a }, 1 ≤ i ≤ m ℄m E F ω : Ae → [a, b] ⊂ (−∞, +∞) ' ω : A → [a, b] ⊂ (−∞, +∞), 1 ≤ i ≤ m .& F e ≻ R (A) e ' R (A) e ≈ R (A) e )p R (ω(A)) e > R (ω(A)) e ' l, s, 1 ≤ l, s ≤ k R (A) e = R (ω(A)) e ' R (A ) ≻ R (A ) ' R (A ) ≈ R (A ), 1 ≤ i ≤ m )p R (ω(A)) R (ω(A ) > R (ω(A )) ' R (ω(A )) = R (ω(A )) ω { +J Ae 'b?+ J A , 1 ≤ i ≤ m '℄m F fTI1 R 7 Æ LZ = FP%$- 7 Lp = I 1LS = &p h Lw /\ [3] d q/ I i, 1 ≤ i ≤ m, fTI ω(A ) % R" u 0 %'D z7,Z A , A , · · · , A =,'VrL Av }fTI ω(A)e V"u <P ω(A ) ,L,K? ; | zq (1) A [a, b] 'Ml 4 F %? F) (2) A [a, b] rpz℄ P & ' E F % F G F) Kv2o,- Æ L%5 d fTI ω [s[%X A 4.1. u Q 61

A % L5, %$- 7 s[ 8 ^CwK zq ![fTI ω(A ) = L![7L R , S i

i1

i2

1

2

k

m

i=1

in

i

l

l

s

s

l

i

s

l

l

i

l

i

i

s

s

s

i

l

i

|

l

i

s

s

b i

i

i

1

2

m

i

1

1

S=

R1 (A1 ) + R2 (A1 ) + · · · + Rk (A1 ) k

i

_ (GIE<1 VZ S=

uLL B #^C

  

117

R1 (A1 )+R2 (A1 )+···+Rk (A1 )−M −N , k−2 R1 (A1 )+R2 (A1 )+···+Rk (A1 ) , k

S = T × A% +

k ≥ 5, 3≤k≤4

R1 (A1 ) + R2 (A1 ) + · · · + Rk (A1 ) × (1 − A%). k

1 Ri (A1), i = 1, 2, · · · , k M* L5, k % Lp = &I M N N% ` `? L5, T %L B ,% A% %L B ,%u7L R , 7 b f = }fT I Ri(ω(A1)), 1 ≤ i ≤ k = ![0)% Ri (A1 ) − S +ζ S

Ri (ω(A1 )) = −ς ×

1 ς % sI i ζ %%&' ω(A1) ≥ 0 = sI %![ zq * 5 , q1 h W w ^CI/ ! =zq ,[ ` K℄%$- ,% = % Lp5, = [D4yI 0 %$ Æ L 7= L5,V1% ; ℄ .< >5, L mqL Lp =E >e [< ' ; < ' u} h [rKV ^C R z q $d b, w G ^ B ;zq VL1% ; 7" i} w F} &p Æ Lp = ,%< [7rs[ ,^CwKfTI (1)

A [N, M] '%? F

z F D {℄m.

Bg bT ?b ) ~?l *

Ri (A1 ) − N + q, M −N

Ri (ω(A1)) = −p ×

p { Y (2)

q

B

Ri (ω(A1 )) ≥ 0, 1 ≤ i ≤ k

P

A [N, M] 'V%? F T+

Ri (ω(A1)) = −p ×

Ri (ω(A1 )) = −p ×

'

z

' F 2 TL

Ri (A1 ) − q

k+1

k P

Ri (A1 )

j=1

Ri (A1 ) −

k+1

+

q,

q

k+1

R1 (A1 )R2 (A1 ) · · · Rk (A1 )T

R1 (A1 )R2 (A1 ) · · · Rk (A1 )T

+q

8!O H'^5<GI1 /

118

:

Ri (ฯ (A1 )) = โ p ร

Ri (A1 ) โ r

r

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DF%>= 7 } hE(B7L'a &p> E ?" a 7 =zq &Kz B%d7 OK d LpV04[-P;kb > " q "F.< =o 27 L > >_5 0 * E = uP7L'a a = $d[> K #W 4 V% d %& L=" a ss ; V = 7"OK E>1u^C R 7 zq #6Q ; *{N 2 l )R7L K ( = #>' [7L = < 5Y D Pw &1 q 0>)duP %d5 q v = 7 ) LE(B7L'a d R 7 P a8 0 >V l1%<! Æ = .[ ) $K ( = W 4 ? V4 Pz 7LOK u 2 N7L K (#>W 4 ? V4 ip 7L = y<K ))U%℄ = j9 ^ 7LL R [s >v v h Lw % . Lw 5 (1% ; Lb = Pw g #" _ q W q F 9 ,G $y< C#hv[ s h Lw z ) = 7

S9 3: n o % J %dD ` =Æ L L?/- q7 R 2 F R B%d +N = Z D{^C R 7 wNbE R 7 = Np17,Z $ 1 " u % =^ <I , v / i x )h I7,Z $ 1PV " u % ^ <I OKz ÆL/< 7= 7 L R=} v[7L'a =E > v" R d%dV v = 7,Z ^C R v i V1 R 7 =zq ,Lb D Pw %&1q %' i 7L'a a d %& D z DF = 7 OK1 ,^C73 \ >z ) % 2& = Aw" a %& z = _%V 1%&Dy z z1%&_% z d 7L K ( i 1 5 q = u ; y<K d %d5 q v = 7,Z oN1 DY = 7 ^C R v z )7 5(1Pw 2 N7LOK = Lb< ` 27 zL} h J 0= - z [sr jVQ i F v 7rs% F v

_ (GIE<1 VZ

121

aVĂ&#x2020;L W= ' z s% ^C w =z ) 2RĂ&#x2020; L7,sI\1Pw1q = ) i &%P q2 v? "&%PaV Ă&#x2020; L W= ' z & q -. w\l 5 . 0 O $B{ 7 qpz2[J Ă&#x2020; L L q = %9M{ 1%Zt< i 3 = - bQa [1] G.Chartrand and L.Lesniak,Graphs & Digraphs, Wadsworth, Inc., California,

s

1986.

[2]

} h Lw zq ! r FH; , E IV u E 77 0 1357-1410 q 7 - f * 1987

[3] P.C.Fishburn, Utility Theory for Decision Making, New York, Wiley, 1970.

[4] D.L.Lu, X.S.Zhang and Y.Y.Mi, An offer model for civil engineering construction, Chinese OR Transaction, Vol.5, No.4(2001)41-52.

[5] L.F.Mao, On Automorphisms groups of Maps, Surfaces and Smarandache geometries, Sientia Magna, Vol.1(2005), No.2, 55-73.

[6] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research Press, 2005.

b I

[7] L.F.Mao, Smarandache multi-space theory, Hexis, Phoenix, AZ 2006.

[8]

gG s / r I +J%r 7uu Y â&#x20AC;&#x201C;Smarandache 3 A D; Xiquan Publishing House (Chinese Branch), America, 2006.

b

[9] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119, 10/2000.

[10] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Probability, Set, and Logic, American research Press, Rehoboth, 1999.

[11] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297384.

[12] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 385-438.

122

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

The Number of Complete Maps on Surfacesâ&#x2C6;&#x2014; Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China)

Yanpei Liu (Institute of Applied Mathematics, Beijing Jiaotong University, Beijing,100044, P.R.China)

Feng Tian (Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China)

Abstract: A map is a connected topological graph cellularly embedded in a surface and a complete map is a cellularly embedded complete graph in a surface. In this paper, all automorphisms of complete maps of order n are determined by permutations on its vertices. Applying a scheme for enumerating maps on surfaces with a given underlying graph, the numbers of unrooted complete maps on orientable or non-orientable surfaces are obtained.

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Key words: embedding, complete map, isomorphism, automorphism group, Burnside Lemma.

Classification: AMS(2000) 05C10,05C25, 05C30 1

Finished in May, 2004, e-print: arXiv: math.GM/0607790

The Number of Complete Maps on Surfaces

123

1. Introduction All surfaces considered in this paper are 2-dimensional compact closed manifolds without boundary, graphs are connected and simple graphs with the maximum valency ≥ 3 and groups are finite. For terminologies and notations not defined here can be seen in [21] for maps, [20] for graphs and in [2] for permutation groups.

The enumeration of rooted maps on surfaces, especially, the sphere, has been intensively investigated by many researchers after the Tutte’s pioneer work in 1962 (see [21]). Comparing with rooted maps, observation for the enumeration of unrooted maps on surface is not much. By applying the automorphisms of the sphere, Liskovets gave an enumerative scheme for unrooted planar maps(see [12]). Liskovets, Walsh and Liskovets got many enumeration results for general planar maps, regular planar maps, Eulerian planar maps, self-dual planar maps and 2-connected planar maps, etc (see [12] − [14]).

General results for the enumeration of unrooted maps on surface other than sphere are very few. Using the well known Burnside Lemma in permutation group theory, Biggs and White presented a formula for enumerating non-equivalent embeddings of a given graph on orientable surfaces[2] , which are the classification of embeddings by orientation-preserving automorphisms of orientable surfaces. Following their idea, the numbers of non-equivalent embeddings of complete graphs,complete bipartite graphs, wheels and graphs whose automorphism group action on its ordered pair of adjacent vertices is semi-regular are gotten in references [15] − [16], [20] and [11]. Although this formula is not very efficient and need more clarifying for the actual enumeration of non-equivalent embeddings of a graph, the same idea is more practical for enumerating rooted maps on orientable or non-orientable surfaces with given underlying graphs(see [8] − [10]).

For projective maps with a given 3-connected underlying graph, Negami got an enumeration result for non-equivalent embeddings by establishing the double planar covering of projective maps(see [18]). In [7], Jin Ho Kwak and Jaeun Lee obtained the number of non-congruent embeddings of a graph, which is also related to the topic discussed in this paper.

Combining the idea of Biggs and White for non-equivalent embeddings of a graph on orientable surfaces and the Tutte’s algebraic representation for maps on

124

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

surface[19],[21] , a general scheme for enumerating unrooted maps on locally orientable surfaces with a given underlying graph is obtained in this paper. Whence, the enumeration of unrooted maps on surfaces can be carried out by the following programming: STEP 1. Determined all automorphisms of maps with a given underlying graph; STEP 2. Calculation the the fixing set F ix(ς) for each automorphism ς of maps; STEP 3. Enumerating the unrooted maps on surfaces with a given underlying graph by this scheme.

Notice that this programming can be used for orientable or non-orientable surfaces, respectively and get the numbers of orientable or non-orientable unrooted maps underlying a given graph.

The main purpose of this paper is to enumerate the orientable or non-orientable complete maps. In 1971, Biggs proved[1] that the order of automorphism group of an orientable complete map of order n divides n(n − 1), and equal n(n − 1) only if the automorphism group of the complete map is a Frobenius group. In this paper, we get a representation by the permutation on its vertices for the automorphisms of orientable or non-orientable complete maps. Then as soon as we completely calculate the fixing set F ix(ς) for each automorphism ς of complete maps, the enumeration of unrooted orientable or non-orientable complete maps can be well done by our programming.

The problem of determining which automorphism of a graph is an automorphism of a map is also interesting for Riemann surfaces or Klein surfaces - surfaces equipped with an analytic or dianalytic structure, for example, automorphisms of Riemann or Klein surfaces have be given more attention since 1960s, see for example,[3] − [4], [6], [17], but it is difficult to get a concrete representation for an automorphism of Riemann or Klein surfaces. The approach used in this paper can be also used for combinatorial discussion automorphisms of Riemann or Klein surface.

Terminologies and notations used in this paper are standard. Some of them are mentioned in the following.

For a given connected graph Γ, an embedding of Γ is a pair (J , λ), where J is a rotation system of Γ, and λ : E(Γ) → Z2 . The edge with λ(e) = 0 or λ(e) = 1 is

125

The Number of Complete Maps on Surfaces

called the type 0 or type 1 edge, respectively.

A map M = (Xα,β , P) is defined to be a permutation P acting on Xα,β of a disjoint union of quadricells Kx of x ∈ X, where, X is a finite set and K = {1, α, β, αβ} is the Klein group, satisfying the following conditions: i) ∀x ∈ Xα,β , there does not exist an integer k such that P k x = αx; ii) αP = P −1 α; iii) the group ΨJ = hα, β, Pi is transitive on Xα,β .

According to the condition ii), the vertices of a map are defined to be the pairs of conjugate of P action on Xα,β and edges the orbits of K on Xα,β , for example, for ∀x ∈ Xα,β , {x, αx, βx, αβx} is an edge of the map M. Geometrically, any map M is an embedding of a graph Γ on a surface, denoted by M = M(Γ) and Γ = Γ(M) ( see also [19] − [21] for details). The graph Γ is called the underlying graph of the map M. If r ∈ Xα,β is marked beforehand, then M is called a rooted map, denoted

by M r . A map is said non-orientable or orientable if the group ΨI = hαβ, Pi is transitive on Xα,β or not.

For example, the graph K4 on the tours with one face length 4 and another 8 , shown in the following Fig.1,

Fig.1 can be algebraically represented as follows: A map (Xα,β , P) with Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} and

126

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw) × (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv) The four vertices of this map are {(x, y, z), (αx, αz, αy)}, {(αβx, u, w), (βx, αw, αu)}, {(αβz, αβu, v), (βz, αv, βu)} and {(αβy, αβv, αβw), (βy, βw, βv)} and six edges are {e, αe, βe, αβe} for ∀e ∈ {x, y, z, u, v, w}.

1 2 Two maps M1 = (Xα,β , P1 ) and M2 = (Xα,β , P2 ) are said to be isomorphic if 1 2 1 there exists a bijection τ : Xα,β −→ Xα,β such that for ∀x ∈ Xα,β ,τ α(x) = ατ (x),

τ β(x) = βτ (x) and τ P1 (x) = P2 τ (x). τ is called an isomorphism between them. If M1 = M2 = M, then an isomorphism between M1 and M2 is called an automorphism of M. All automorphisms of a map M form a group, called the automorphism group of M and denoted by AutM. Similarly, two rooted maps M1r , M2r are said to be isomorphic if there is an isomorphism θ between them such that θ(r1 ) = r2 , where r1 , r2 are the roots of M1r , M2r , respectively and denote the automorphism group of M r by AutMr . It has been known that AutMr is a trivial group.

According to their action, isomorphisms between maps can divided into two classes: cyclic order-preserving isomorphism and cyclic order-reversing isomorphism, defined as follows, which is useful for determining automorphisms of a map underlying a graph.

For two maps M1 and M2 , a bijection ξ between M1 and M2 is said to be cyclic 1 order-preserving if for ∀x ∈ Xα,β ,τ α(x) = ατ (x), τ β(x) = βτ (x), τ P1 (x) = P2 τ (x)

and cyclic order-reversing if τ α(x) = ατ (x), τ β(x) = βτ (x) τ P1 (x) = P2−1 τ (x).

Now let Γ be a connected graph. The notations E O (Γ), E N (Γ) and E L (Γ) denote the embeddings of Γ on the orientable surfaces, non-orientable surfaces and locally surfaces, M(Γ) and AutΓ denote the set of non-isomorphic maps underlying Γ and its automorphism group, respectively.

2. The enumerative scheme for maps underlying a graph A permutation p on set Ω is called semi-regular if all of its orbits have the same length. For a given connected graph Γ, ∀g ∈ AutΓ, M = (Xα,β , P) ∈ M(Γ), define an extended action of g on M to be

127

The Number of Complete Maps on Surfaces

g ∗ : Xα,β −→ Xα,β , ∗

such that M g = gMg −1 with gα = αg and gβ = βg.

We have already known the following two results.

Lemma 2.1[21] For any rooted map M r , AutM r is trivial.

Lemma 2.2[2],[21] For a given map M, ∀ξ ∈ AutM, ξ transforms vertices to vertices, edges to edges and faces to faces on a map M, i.e, ξ can be naturally extended to an automorphism of surfaces.

Lemma 2.3 If there is an isomorphism ξ between maps M1 and M2 , then Γ(M1 ) = Γ(M2 ) = Γ and ξ ∈ AutΓ if ξ is cyclic order-preserving or ξα ∈ AutΓ if ξ is cyclic order-reversing.

1 Proof By the definition of an isomorphism between maps, if M1 = (Xα,β , P1 ) 2 1 is isomorphic with M2 = (Xα,β , P2 ), then there is an 1 − 1 mapping ξ between Xα,β 2 and Xα,β such that (P1 )ξ = P2 . Since isomorphic graphs are considered to be equal,

we get that Γ(M1 ) = Γ(M2 ) = Γ. Now since (P2 )−1 = (P2 )α .

We get that Γξ = Γ or Γξα = Γ, whence, ξ ∈ AutΓ or ξα ∈ AutΓ.

♮

According to Lemma 2.3, For ∀g ∈ AutΓ, ∀M ∈ E L(Γ), the induced action g ∗ ∗

of g on M is defined by M g = gMg −1 = (Xα,β , gPg −1).

Since P is a permutation on the set Xα,β , by a simple result in permutation group theory, P g is just the permutation replaced each element x in P by g(x). ∗

Whence M and M g are isomorphic. Therefore, we get the following enumerative theorem for unrooted maps underlying a graph.

Theorem 2.1 For a connected graph Γ, let E ⊂ E L(Γ). Then the number n(E, Γ) of unrooted maps in E is n(E, Γ) =

X 1 |Φ(g)|, |AutΓ × hαi | g∈AutΓ×hαi

where, Φ(g) = {P|P ∈ E and P g = P}.

128

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Proof According to Lemma 2.1, two maps M1 , M2 ∈ E are isomorphic if and ∗

only if there exists an isomorphism θ ∈ AutΓ× < α > such that M1θ = M2 .

Whence, we get that all the unrooted maps in E are just the representations of orbits in E under the action of AutΓ × hαi. By the Burnside Lemma, we get the following result for the number of unrooted maps in E n(E, Γ) =

X 1 |Φ(g)|. |AutΓ × hαi | g∈AutΓ×hαi

♮

Corollary 2.1 For a given graph Γ, the numbers of unrooted maps in E O (Γ), E N (Γ) and E L(Γ) are nO (Γ) =

X 1 |ΦO (g)|; |AutΓ × hαi | g∈AutΓ×hαi

(2.1)

nN (Γ) =

X 1 |ΦN (g)|; |AutΓ × hαi | g∈AutΓ×hαi

(2.2)

nL (Γ) =

X 1 |ΦL (g)|, |AutΓ × hαi | g∈AutΓ×hαi

(2.3)

where, ΦO (g) = {P|P ∈ E O (Γ) and P g = P}, ΦN (g) = {P|P ∈ E N (Γ) and P g = P}, ΦL (g) = {P|P ∈ E L(Γ) and P g = P}.

Corollary 2.2 Let E(S, Γ) be the embeddings of Γ in the surface S, then the number n(Γ, S) of unrooted maps on S with underlying gΓ is n(Γ, S) =

X 1 |Φ(g)|, |AutΓ × hαi | g∈AutΓ×hαi

where, Φ(g) = {P|P ∈ E(S, Γ) and P g = P}.

Corollary 2.3 In formulae (2.1)-(2.3), |Φ(g)| = 6 0 i and only if g is an automorphism of an orientable or non-orientable map underlying Γ.

Directly using these formulae (2.1)-(2.3) to count unrooted maps with a given underlying graph is not straightforward. More observation should be considered.

The following two lemmas give necessary conditions for an induced automorphism of a graph Γ to be an cyclic order-preserving automorphism of a surface.

129

The Number of Complete Maps on Surfaces

Lemma 2.4 For a map M underlying a graph Γ, ∀g ∈ AutM, ∀x ∈ Xα,β with X = E(Γ), (i) |xAutM | = |AutM| ; (ii) |x<g> | = o(g), where, o(g) denotes the order of g.

Proof For a subgroup H < AutM, we know that |H| = |xH ||Hx |. Since Hx < AutMx , where M x is a rooted map with root x, we know that |Hx | = 1 by Lemma 2.1. Whence, |xH | = |H|. Now take H = AutM or < g >, we get the assertions (i) and (ii).

♮

Lemma 2.5 Let Γ be a connected graph and g ∈ AutΓ. If there is a map M ∈ E L (Γ) such that the induced action g ∗ ∈ AutM, then for ∀(u, v), (x, y) ∈ E(Γ), [lg (u), lg (v)] = [lg (x), lg (y)] = constant, where, lg (w) denotes the length of the cycle containing the vertex w in the cycle decomposition of g and [a, b] the least common multiple of integers a and b.

Proof According to Lemma 2.4, we know that the length of any quadricell uv+ or uv− under the action of g ∗ is [lg (u), lg (v)]. Since g ∗ is an automorphism of map, therefore, g ∗ is semi-regular. Whence, we get that [lg (u), lg (v)] = [lg (x), lg (y)] = constant.

♮

Now we consider conditions for an induced automorphism of a map by an automorphism of graph to be a cyclic order-reversing automorphism of surfaces.

Lemma 2.6 If ξα is an automorphism of a map, then ξα = αξ.

Proof Since ξα is an automorphism of a map, we know that (ξα)α = α(ξα).

That is, ξα = αξ.

♮

Lemma 2.7 If ξ is an automorphism of M = (Xα,β , P), then ξα is semi-regular on Xα,β with order o(ξ) if o(ξ) ≡ 0(mod2) or 2o(ξ) if o(ξ) ≡ 1(mod2).

130

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Proof Since ξ is an automorphism of map by Lemma 2.6, we know that the cycle decomposition of ξ can be represented by Y

(x1 , x2 , · · · , xk )(αx1 , αx2 , · · · , αxk ),

ξ=

k

where,

Q

k

denotes the product of disjoint cycles with length k = o(ξ).

Therefore, if k ≡ 0(mod2), we get that ξα =

Y

(x1 , αx2 , x3 , · · · , αxk )

k

and if k ≡ 1(mod2), we get that ξα =

Y

(x1 , αx2 , x3 , · · · , xk , αx1 , x2 , αx3 , · · · , αxk ).

2k

Whence, ξ is semi-regular acting on Xα,β .

♮

Now we can prove the following result for cyclic order-reversing automorphisms of maps.

Lemma 2.8 For a connected graph Γ, let K be all automorphisms in AutΓ whose extending action on Xα,β , X = E(Γ), are automorphisms of maps underlying the graph Γ. Then for ∀ξ ∈ K, o(ξ ∗) ≥ 2, ξ ∗ α ∈ K if and only if o(ξ ∗) ≡ 0(mod2).

Proof Notice that by Lemma 2.7, if ξ ∗ is an automorphism of a map underlying Γ, then ξ ∗ α is semi-regular acting on Xα,β .

Assume ξ ∗ is an automorphism of the map M = (Xα,β , P). Without loss of generality, we assume that P = C1 C2 · · · Ck , where,Ci = (xi1 , xi2 , · · · , xiji ) is a cycle in the decomposition of ξ|V (Γ) and xit = {(ei1 , ei2 , · · · , eiti )(αei1 , αeiti , · · · , αei2 )}, ξ|E(Γ) = (e11 , e12 , · · · , es1 )(e21 , e22 , · · · , e2s2 ) · · · (el1 , el2 , · · · , elsl ), and ξ ∗ = C(αC −1 α),

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The Number of Complete Maps on Surfaces

where, C = (e11 , e12 , · · · , es1 )(e21 , e22 , · · · , e2s2 ) · · · (el1 , el2 , · · · , elsl ). Now since ξ ∗ is an automorphism of a map, we get that s1 = s2 = · · · = sl = o(ξ ∗ ) = s.

If o(ξ ∗ ) ≡ 0(mod2), define a map M ∗ = (Xα,β , P ∗ ) with P ∗ = C1∗ C2∗ · · · Ck∗ , where, Ci∗ = (x∗i1 , x∗i2 , · · · , x∗iji ), x∗it = {(e∗i1 , e∗i2 , · · · , e∗iti )(αe∗i1 , αe∗iti , · · · , e∗i2 )} and e∗ij = epq . Take e∗ij = epq if q ≡ 1(mod2) and e∗ij = αepq if q ≡ 0(mod2). Then we get that M ξα = M. Now if o(ξ ∗) ≡ 1(mod2), by Lemma 2.7, o(ξ ∗ α) = 2o(ξ ∗). Therefore, for a chosen quadricell in (ei1 , ei2 , · · · , eiti ) adjacent to the vertex xi1 for i = 1, 2, · · · , n, where, n = the order of the graph Γ, the resultant map M is unstable under the action of ξα. Whence, ξα is not an automorphism of a map underlying Γ.

♮

3. Determine automorphisms of complete maps Now we determine all automorphisms of complete maps in this section by applying the results gotten in Section 2.

Let Kn be a complete graph of order n. Label its vertices by integers 1, 2, ..., n.

Then its edge set is {ij|1 ≤ i, j ≤ n, i 6= j

and ij = ji}. For convenience, we use

ij denoting an edge ij of the complete graph Kn and ij = j i , 1 ≤ i, j ≤ n, i 6= j.

Then its quadricells of this edge can be represented by {ij+ , ij− , j i+ , j i− } and

Xα,β (Kn ) = {ij+ : 1 ≤ i, j ≤ n, i 6= j}

α=

Y

[

{ij− : 1 ≤ i, j ≤ n, i 6= j},

(ij+ , ij− ),

1≤i,j≤n,i6=j

β=

Y

(ij+ , ij+ )(ij− , ij− ).

1≤i,j≤n,i6=j

Recall that the automorphism group of Kn is just the symmetry group of degree n, i.e., AutKn = Sn . The above representation enables us to determine all automorphisms of complete maps of order n on surfaces.

Theorem 3.1 All cyclic order-preserving automorphisms of non-orientable complete maps of order≥ 4 are extended actions of elements in

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E[s ns ] ,

E

[1,s

n−1 s ]

,

and all cyclic order-reversing automorphisms of non-orientable complete maps of order≥ 4 are extended actions of elements in αE[(2s) 2sn ] ,

αE[(2s) 2s4 ] ,

αE[1,1,2] ,

where, Eθ denotes the conjugate class containing element θ in the symmetry group Sn Proof Firstly, we prove that the induced permutation ξ ∗ on complete map of order n by an element ξ ∈ Sn is an cyclic order-preserving automorphism of a non-orientable map, if, and only if, ξ ∈ Es ns

[

E

[1,s

n−1 s ]

Assume the cycle index of ξ is [1k1 , 2k2 , ..., nkn ]. If there exist two integers ki , kj 6= 0, and i, j ≥ 2, i 6= j, then in the cycle decomposition of ξ , there are two cycles (u1 , u2, ..., ui ) and (v1 , v2 , ..., vj ).

Since [lξ (u1 ), lξ (u2)] = i and [lξ (v1 ), lξ (v2 )] = j and i 6= j, we know that ξ ∗ is not an automorphism of embedding by Lemma 2.5. Whence, the cycle index of ξ must be the form of [1k , sl ].

Now if k ≥ 2, let (u), (v) be two cycles of length 1 in the cycle decomposition of ξ. By Lemma 2.5, we know that [lξ (u), lξ (v)] = 1.

If there is a cycle (w, ...) in the cycle decomposition of ξ whose length greater or equal to two, we get that [lξ (u), lξ (w)] = [1, lξ (w)] = lξ (w).

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The Number of Complete Maps on Surfaces

According to Lemma 2.5, we get that lξ (w) = 1, a contradiction. Therefore, the cycle index of ξ must be the forms of [sl ] or [1, sl ]. Whence, sl = n or sl + 1 = n.

Calculation shows that l = n

n s

n−1 . s

or l =

following three types [1 ], [1, s

n−1 s

That is, the cycle index of ξ is one of the n s

] and [s ] for some integer s .

Now we only need to prove that for each element ξ in E

[1,s

n−1 s ]

and E[s ns ] , there

exists an non-orientable complete map M of order n with an induced permutation ξ ∗ being its cyclic order-preserving automorphism of surface. The discussion are divided into two cases. ξ ∈ E[s ns ]

Case 1

Assume the cycle decomposition of ξ being ξ = (a, b, · · · , c) · · · (x, y, · · · , z) · · · (u, v, · · · , w), where, the length of each cycle is k, and 1 ≤ a, b, · · · , c, x, y, · · · , z, u, v, · · · , w ≤ 1 n . In this case, we can construct a non-orientable complete map M1 = (Xα,β , P1 )

as follows.

1 Xα,β = {ij+ : 1 ≤ i, j ≤ n, i(j}

[

Y

P1 =

{ij− : 1 ≤ i, j ≤ n, i 6= j}, (C(x))(αC(x)−1 α),

x∈{a,b,···,c,···,x,y,···,z,u,v,···,w}

where,

C(x) = (xa+ , · · · , xx∗ , · · · , xu+ , xb+ , xy+ , · · · , · · · , xv+ , xc+ , · · · , xz+ , · · · , xw+ ), xx∗ denotes an empty position and

αC(x)−1 α = (xa− , xw− , · · · , xz− , · · · , xc− , xv− , · · · , xb− , xu− , · · · , xy− , · · ·). ∗

It is clear that M1ξ = M1 . Therefore, ξ ∗ is an cyclic order-preserving automorphism of the map M1 .

Case 2

ξ∈E

[1,s

n−1 s ]

We assume the cycle decomposition of ξ being ξ = (a, b, ..., c)...(x, y, ..., z)...(u, v, ..., w)(t),

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where, the length of each cycle is k beside the final cycle, and 1 ≤ a, b...c, x, y..., z, u, v, ..., w, t ≤ n . In this case, we construct a non-orientable complete map M2 = 2 (Xα,β , P2 ) as follows.

2 Xα,β = {ij+ : 1 ≤ i, j ≤ n, i 6= j}

[

{ij− : 1 ≤ i, j ≤ n, i 6= j},

Y

P2 = (A)(αA−1 )

(C(x))(αC(x)−1 α),

x∈{a,b,...,c,...,x,y,...z,u,v,...,w}

where, A = (ta+ , tx+ , ...tu+ , tb+ , ty+ , ..., tv+ , ..., tc+ , tz+ , ..., tw+ ),

αA−1 α = (ta− , tw− , ...tz− , tc− , tv− , ..., ty− , ..., tb− , tu− , ..., tx− ),

C(x) = (xa+ , ..., xx∗ , ..., xu+ , xb+ , ..., xy+ , ..., xv+ , ..., xc+ , ..., xz+ , ..., xw+ ) and

αC(x)−1 α = (xa− , xw− , .., xz− , ..., xc− , ..., xv− , ..., xy− , ..., xb− , xu− , ...). ∗

It is also clear that M2ξ = M2 . Therefore, ξ ∗ is an automorphism of the map M2 .

Now we consider the case of cyclic order-reversing automorphisms of a complete map. According to Lemma 2.8, we know that an element ξα, where, ξ ∈ Sn , is an cyclic order-reversing automorphism of a complete map only if, ξ∈E

[k

n1 k

,(2k)

n−n1 2k

]

.

Our discussion is divided into two parts.

Case 3

n1 = n

Without loss of generality, we can assume the cycle decomposition of ξ has the following form in this case.

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The Number of Complete Maps on Surfaces

ξ = (1, 2, · · · , k)(k + 1, k + 2, · · · , 2k) · · · (n − k + 1, n − k + 2, · · · , n).

Subcase 3.1

k ≡ 1(mod2) and k > 1

According to Lemma 2.8, we know that ξ ∗ α is not an automorphism of maps since o(ξ ∗ ) = k ≡ 1(mod2).

Subcase 3.2

k ≡ 0(mod2)

3 Construct a non-orientable map M3 = (Xα,β , P3 ), where X 3 = E(Kn ) and

P3 =

Y

(C(i))(αC(i)−1 α),

i∈{1,2,···,n}

where, if i ≡ 1(mod2), then

C(i) = (i1+ , ik+1+ , · · · , in−k+1+ , i2+ , · · · , in−k+2+ , · · · , ii∗ , · · · , ik+ , i2k+ , · · · , in+ ),

αC(i)−1 α = (i1− , in− , · · · , i2k− , ik− , · · · , ik+1− ) and if i ≡ 0(mod2), then

C(i) = (i1− , ik+1− , · · · , in−k+1− , i2− , · · · , in−k+2− , · · · , ii∗ , · · · , ik− , i2k− , · · · , in− ),

αC(i)−1 α = (i1+ , in+ , · · · , i2k+ , ik+ , · · · , ik+1+ ).

Where, ii∗ denotes the empty position, for example, (21 , 22∗ , 23 , 24 , 25) = (21 , 23 , 24 , 25 ).

It is clear that P3ξα = P3 , that is, ξα is an automorphism of map M3 .

Case 4

n1 6= n

Without loss of generality, we can assume that

ξ = (1, 2, · · · , k)(k + 1, k + 2, · · · , n1 ) · · · (n1 − k + 1, n1 − k + 2, · · · , n1 ) × (n1 + 1, n1 + 2, · · · , n1 + 2k)(n1 + 2k + 1, · · · , n1 + 4k) · · · (n − 2k + 1, · · · , n)

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k ≡ 0(mod2)

Subcase 4.1

Consider the orbits of 12+ and n1 + 2k + 11+ under the action of < ξα >, we get that |orb((12+ )<ξα> )| = k and |orb(((n1 + 2k + 1)1+ )<ξα> )| = 2k.

Contradicts to Lemma 2.5. k ≡ 1(mod2)

Subcase 4.2

In this case, if k 6= 1, then k ≥ 3. Similar to the discussion of Subcase 3.1, we know that ξα is not an automorphism of complete map. Whence, k = 1 and ξ ∈ E[1n1 ,2n2 ] .

Without loss of generality, assume that

ξ = (1)(2) · · · (n1 )(n1 + 1, n1 + 2)(n1 + 3, n1 + 4) · · · (n1 + n2 − 1, n1 + n2 ).

If n2 ≥ 2, and there exists a map M = (Xα,β , P), assume the vertex v1 in M being v1 = (1l12 + , 1l13 + , · · · , 1l1n + )(1l12 − , 1l1n − , · · · , 1l13 − ) where, l1i ∈ {+2, −2, +3, −3, · · · , +n, −n} and l1i 6= l1j if i 6= j.

Then we get that (v1 )ξα = (1l12 − , 1l13 − , · · · , 1l1n − )(1l12 + , 1l1n + , · · · , 1l13 + ) 6= v1 .

Whence, ξα is not an automorphism of map M, a contradiction.

Therefore, n2 = 1. Similarly, we can also get that n1 = 2. Whence, ξ = (1)(2)(34) and n = 4. We construct a stable non-orientable map M4 under the action of ξα as follows.

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The Number of Complete Maps on Surfaces

4 M4 = (Xα,β , P4 ),

where,

P4 = (12+ , 13+ , 14+ )(21+ , 23+ , 24+ )(31+ , 32+ , 34+ )(41+ , 42+ , 43+ ) × (12− , 14− , 13− )(21− , 24− , 23− )(31− , 34− , 32− )(41− , 43− , 42− ).

Therefore, all cyclic order-preserving automorphisms of non-orientable complete maps are extended actions of elements in E[s ns ] ,

E

[1,s

n−1 s ]

and all cyclic order-reversing automorphisms of non-orientable complete maps are extended actions of elements in αE[(2s) 2sn ] , This completes the proof.

αE[(2s) 2s4 ]

αE[1,1,2] .

♮

According to the Rotation Embedding Scheme for orientable embedding of a graph formalized by Edmonds in [5], each orientable complete map is just the case

of eliminating the signs +, - in our representation for complete maps. Whence,we also get the following result for automorphisms of orientable complete maps, which is similar to Theorem 3.1.

Theorem 3.2 All cyclic order-preserving automorphisms of orientable complete maps of order≥ 4 are extended actions of elements in E[s ns ] ,

E

[1,s

n−1 s ]

and all cyclic order-reversing automorphisms of orientable complete maps of order≥ 4 are extended actions of elements in αE[(2s) 2sn ] ,

αE[(2s) 2s4 ] ,

αE[1,1,2] ,

where,Eθ denotes the conjugate class containing θ in Sn .

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Proof The proof is similar to that of Theorem 3.1. For completion, we only need to construct orientable maps MiO , i = 1, 2, 3, 4 to replace these non-orientable maps M1 , i = 1, 2, 3, 4 in the proof of Theorem 3.1.

In fact, for cyclic order-preserving case, we only need to take M1O , M2O to be the resultant maps eliminating the signs + - in M1 , M2 constructed in the proof of Theorem 3.1.

For the cyclic order-reversing case, we take M3O = (E(Kn )α,β , P3O ) with P3 =

Y

(C(i)),

i∈{1,2,···,n}

where, if i ≡ 1(mod2), then

C(i) = (i1 , ik+1 , · · · , in−k+1 , i2 , · · · , in−k+2, · · · , ii∗ , · · · , ik , i2k , · · · , in ), and if i ≡ 0(mod2), then

C(i) = (i1 , ik+1 , · · · , in−k+1 , i2 , · · · , in−k+2, · · · , ii∗ , · · · , ik , i2k , · · · , in )−1 , where ii∗ denotes the empty position and M4O = (E(K4 )α,β , P4 ) with P4 = (12 , 13 , 14 )(21 , 23 , 24 )(31 , 34 , 32 )(41 , 42 , 43).

It can be shown that (MiO )g∗ = MiO , i = 1, 2 and (MiO )ξα = MiO for i = 3, 4. ♮ All results in this section are useful for the enumeration of complete maps in the next section.

4. The Enumeration of complete maps on surfaces We firstly consider the permutation and its stabilizer . The permutation with the following form (x1 , x2 , ..., xn )(αxn , αx2 , ..., αx1 ) is called a pair permutation. The following result is obvious.

Lemma 4.1 Let g be a permutation on the set Ω = {x1 , x2 , ..., xn } such that gα = αg. If

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The Number of Complete Maps on Surfaces

g(x1 , x2 , ..., xn )(αxn , αxn−1 , ..., αx1 )g −1 = (x1 , x2 , ..., xn )(αxn , αxn−1 , ..., αx1 ), then g = (x1 , x2 , ..., xn )k and if

gα(x1 , x2 , ..., xn )(αxn , αxn−1 , ..., αx1 )(gα)−1 = (x1 , x2 , ..., xn )(αxn , αxn−1 , ..., αx1 ), then gα = (αxn , αxn−1 , ..., αx1 )k for some integer k, 1 ≤ k ≤ n.

Lemma 4.2 For each permutation g, g ∈ E[k nk ] satisfying gα = αg on the set Ω = {x1 , x2 , ..., xn }, the number of stable pair permutations in Ω under the action of g or gα is 2φ(k)(n − 1)! , |E[k nk ] | where φ(k) denotes the Euler function.

Proof Denote the number of stable pair permutations under the action of g or gα by n(g) and C the set of pair permutations. Define the set A = {(g, C)|g ∈ E[k nk ] , C ∈ C

and C g = C or C gα = C}. Clearly, for ∀g1 , g2 ∈ E[k nk ] , we have

n(g1 ) = n(g2 ). Whence, we get that |A| = |E[k nk ] |n(g).

(4.1)

On the other hand, by Lemma 4.1, for any pair permutation C = (x1 , x2 , ..., xn ) (αxn , αxn−1 , ..., αx1 ), since C is stable under the action of g, there must be g = (x1 , x2 , ..., xn )l or gα = (αxn , αxn−1 , ..., αx1 )l , where l = s nk , 1 ≤ s ≤ k and (s, k) = 1. Therefore, there are 2φ(k) permutations in E[k nk ] acting on it stable. Whence, we also have

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|A| = 2φ(k)|C|.

(4.2)

Combining (4.1) with (4.2), we get that 2φ(k)|C| 2φ(k)(n − 1)! = . |E[k nk ] | |E[k nk ] |

n(g) =

♮

Now we can enumerate the unrooted complete maps on surfaces.

Theorem 4.1 The number nL (Kn ) of complete maps of order n ≥ 5 on surfaces is n

X X 1 X φ(k)2β(n,k)(n − 2)!

2α(n,k) (n − 2)! k n (Kn ) = ( + + ) n 2 k|n k|n,k≡0(mod2) n−1 k k ( nk )! k|(n−1),k6=1 L

n−1 k

,

where, α(n, k) = and

  

 

and nL (K4 ) = 11.

n(n−3) , 2k n(n−2) , 2k

(n−1)(n−2) , 2k (n−1)(n−3) , 2k

β(n, k) = 

if

k ≡ 1(mod2);

if

k ≡ 0(mod2),

if

k ≡ 1(mod2);

if

k ≡ 0(mod2).

Proof According to (2.3) in Corollary 2.1 and Theorem 3.1 for n ≥ 5, we know that

nL (Kn ) =

X X 1 ×( |Φ(g1 )| + 2|AutKn | g1 ∈E n g2 ∈E

+ h∈E

X

[k k ]

|Φ(g2 α)|

n [(2s) 2s ]

|Φ(h)|)

n−1 [1,k k ]

= +

X X 1 × ( |E[k nk ] ||Φ(g1 )| + |E[l nl ] ||Φ(g2 α)| 2n! k|n l|n,l≡0(mod2) X

l|(n−1)

|E

[1,l

n−1 l ]

||Φ(h)|),

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The Number of Complete Maps on Surfaces

where, g1 ∈ E[k nk ] , g2 ∈ E[l nl ] and h ∈ E

[1,k

are three chosen elements.

n−1 k ]

Without loss of generality, we assume that an element g, g ∈ E[k nk ] has the following cycle decomposition. n n g = (1, 2, ..., k)(k + 1, k + 2, ..., 2k)...(( − 1)k + 1, ( − 1)k + 2, ..., n) k k and P=

Y

× 1

Y

2

,

where, Y

1

= (1i21 , 1i31 , ..., 1in1 )(2i12 , 2i32 , ..., 2in2 )...(ni1n , ni2n , ..., ni(n−1)n ),

and Y

2

= α(

Y

1

−1

)α−1

being a complete map which is stable under the action of g, where sij ∈ {k+, k−|k = 1, 2, ...n}.

Notice that the quadricells adjacent to the vertex ”1” can make 2n−2 (n − 2)! different pair permutations and for each chosen pair permutation, the pair permutations adjacent to the vertices 2, 3, ..., k are uniquely determined since P is stable under the action of g.

Similarly, for each given pair permutation adjacent to the vertex k + 1, 2k + 1, ..., ( nk − 1)k + 1, the pair permutations adjacent to k + 2, k + 3, ..., 2k and 2k + 2, 2k + 3, ..., 3k and,...,and ( nk −1)k + 2, ( nk −1)k + 3, ...n are also uniquely determined because P is stable under the action of g.

Now for an orientable embedding M1 of Kn , all the induced embeddings by exchanging two sides of some edges and retaining the others unchanged in M1 are the same as M1 by the definition of maps. Whence, the number of different stable embeddings under the action of g gotten by exchanging x and αx in M1 for x ∈ U, U ⊂ Xβ , where Xβ =

S

n

{x, βx} , is 2g(ε)− k , where g(ε) is the number of

x∈E(Kn )

orbits of E(Kn ) under the action of g and we substract 2+

1+

1+

1 , k + 1 , 2k + 1 , · · · , n − k + 1

1+

n k

because we can chosen

first in our enumeration.

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Notice that the length of each orbit under the action of g is k for ∀x ∈ E(Kn ) if k is odd and is

k 2

k

for x = ii+ 2 , i = 1, k + 1, · · · , n − k + 1, or k for all other edges

if k is even. Therefore, we get that g(ε) = Whence, we have that

  

ε(Kn ) , k ε(Kn )− n 2 k

if

k ≡ 1(mod2);

, if

k ≡ 0(mod2).



n  α(n, k) = g(ε) − = k 

and

n(n−3) , 2k n(n−2) , 2k

if

k ≡ 1(mod2);

if

k ≡ 0(mod2),

n

|Φ(g)| = 2α(n,k) (n − 2)! k ,

(4.3)

Similarly, if k ≡ 0(mod2), we get also that n

|Φ(gα)| = 2α(n,k) (n − 2)! k

(4.4)

for an chosen element g, g ∈ E[k nk ] .

Now for ∀h ∈ E

n−1

, without loss of generality, we assume that h = (1, 2, ..., k)(k+

[1,k k ] 1, k + 2, ..., 2k)...(( n−1 − 1)k + 1, ( n−1 − 1)k + 2, ..., (n − 1))(n). k k

Then the above state-

ment is also true for the complete graph Kn−1 with the vertices 1, 2, · · · , n−1. Notice that the quadricells n1+ , n2+ , · · · , nn−1+ can be chosen first in our enumeration and they are not belong to the graph Kn−1 . According to Lemma 4.2, we get that |Φ(h)| = 2β(n,k) (n − 2)!

n−1 k

×

2φ(k)(n − 2)! , |E n−1 | k [1,k

(4.5)

]

Where β(n, k) = h(ε) =

  

ε(Kn−1 ) k ε(Kn−1 ) k

− −

n−1 k n−1 k

= =

(n−1)(n−4) , 2k (n−1)(n−3) , 2k

if

k ≡ 1(mod2);

if

k ≡ 0(mod2).

Combining (4.3) − (4.5), we get that

nL (Kn ) =

X X 1 × ( |E[k nk ] ||Φ(g0 )| + |E[l nl ] ||Φ(g1 α)| 2n! k|n l|n,l≡0(mod2)

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The Number of Complete Maps on Surfaces

+

X

|E

l|(n−1)

[1,l

n−1 l ]

||Φ(h)|) n

n

X n!2α(n,k) (n − 2)! k X 1 n!2α(n,k) (n − 2)! k = + ×( n n n 2n! k k ( k )! k k ( nk )! k|n,k≡0(mod2) k|n

+

X

k|(n−1),k6=1

n! k

n−1 k

)! ( n−1 k

×

2φ(k)(n − 2)!2β(n,k) (n − 2)!

n−1 k

(n−1)!

)

n−1 k k ( n−1 )! k n k

X X φ(k)2β(n,k)(n − 2)!

1 X 2α(n,k) (n − 2)! + = ( + ) n n 2 k|n k|n,k≡0(mod2) n−1 k k ( k )! k|(n−1),k6=1

n−1 k

.

For n = 4, similar calculation shows that nL (K4 ) = 11 by consider the fixing set of permutations in E[s 4s ] ,E[1,s 3s ] , E[(2s) 2s4 ] ,αE[(2s) 2s4 ] and αE[1,1,2] .

♮

For orientable complete maps, we get the number nO (Kn ) of orientable complete maps of order n as follows.

Theorem 4.2 The number nO ((Kn ) of complete maps of order n ≥ 5 on orientable surfaces is n

X X 1 X (n − 2)! k φ(k)(n − 2)! n (Kn ) = ( + ) n n + 2 k|n k|n,k≡0(mod2) k k ( k )! n−1 k|(n−1),k6=1 O

n−1 k

.

and n(K4 ) = 3.

Proof According to the Tutte’s algebraic representation of maps, a map M = (Xα,β , P) is orientable if and only if for ∀x ∈ Xα,β , x and αβx are in a same orbit of Xα,β under the action of the group ΨI = hαβ, Pi. Now applying (2.1) in Corollary 2.1 and Theorem 3.1, similar to the proof of Theorem 4.1, we get the number nO (Kn ) for n ≥ 5 as follows n

X X 1 X (n − 2)! k φ(k)(n − 2)! nO (Kn ) = ( + ) n n + 2 k|n k|n,k≡0(mod2) k k ( k )! n−1 k|(n−1),k6=1

and for the complete graph K4 , calculation shows that n(K4 ) = 3.

n−1 k

.

♮

Notice that nO (Kn )+nN (Kn ) = nL (Kn ). Therefore, we can also get the number nN (Kn ) of unrooted complete maps of order n on non-orientable surfaces by Theorem 4.1 and Theorem 4.2.

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Theorem 4.3 The number nN (Kn ) of unrooted complete maps of order n, n ≥ 5 on non-orientable surfaces is n

X 1 X (2α(n,k) − 1)(n − 2)! k n (Kn ) = ( + ) n 2 k|n k|n,k≡0(mod2) k k ( nk )!

N

φ(k)(2β(n,k) − 1)(n − 2)! + n−1 k|(n−1),k6=1 X

n−1 k

,

and nN (K4 ) = 8. Where, α(n, k) and β(n, k) are same as in Theorem 4.1.

For n = 5, calculation shows that nL (K5 ) = 1080 and nO (K5 ) = 45 based on Theorem 4.1 and 4.2. For n = 4, there are 3 unrooted orientable maps and 8 non-orientable maps shown in the Fig.2.

Fig.2

The Number of Complete Maps on Surfaces

145

All the 11 maps of K4 on surfaces are non-isomorphic.

Noticing that for an orientable map M, its cyclic order-preserving automorphisms are just the orientation-preserving automorphisms of map M by definition.

Now consider the action of cyclic order-preserving automorphisms of complete maps, determined in Theorem 3.2 on all orientable embeddings of a complete graph of order n. Similar to the proof of Theorem 4.2, we can get the number of non-equivalent embeddings of complete graph of order n, which is same as the result of Mull et al. in [15].

References [1] N.L.Biggs, Automorphisms of imbedded graphs, J.Combinatorical Theory 11,132138(1971).

[2] N.L.Biggs and A.T.White, Permutation Groups and Combinatoric Structure, Cambridge University Press (1979).

[3] E.Bujalance, Automorphism groups of compact planar Klein surfaces, Manuscripta Math,56,105-124(1986).

[4] E.Bujalance, Cyclic groups of automorphisms of compact non-orientable Klein surfaces without boundary, Pacific J. Math, vol.109,No.2,279-289(1983).

[5] Edmonds.J, A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc 7 (1960) [6] W.J.Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J.Math.Oxford (2),17,86-97(1966).

[7] Jin Ho Kwak and Jaeun Lee, Enumeration of graph embeddings, Discrete Math, 135(1994), 129 151.

[8] L.F.Mao and Y.P.Liu , On the roots on orientable embeddings of graph, Acta Math. Scienta Ser.A,vol.23, No.3,287-293(2003).

[9] L.F.Mao and Y.P.Liu, Group action for enumerating maps on surfaces, J.Appl.

Math. & Computing,vol.13,No.1-2,201-215(2003).

[10] L.F.Mao and Y.P.Liu, New approach for enumerating maps on orientable surface, Australasian J.Combinatoric (accepted).

[11] L.F.Mao, Y.P.Liu and F. Tian, Automorphisms of maps with given underlying graph and its application to enumeration, Acta Math. Sinica, vol.21,

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Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

No.2(2005), 225-236.

[12] V.A.Liskovets, Enumeration of non-isomorphic planar maps,Sel.Math.Sov., 4;4 (1985), 303 323.

[13] V.A.Liskovets, A reductive technique for Enumerating non-isomorphic planar maps,Discrete Math, 156(1996), 197 217.

[14] V.A.Liskovets and T.R.S.Walsh, The enumeration of non-isomorphic 2-connected planar maps,Canad.J.Math., 35(1983), 417 435.

[15] B.P.Mull,R.G.Rieper and A.T.White, Enumeration 2-cell imbeddings of connected graphs,Proc.Amer.Math.Soc., 103(1988), 321 330.

[16] B.P.Mull, Enumerating the orientable 2-cell imbeddings of complete bipartite graphs, J.Graph Theory, vol 30, 2(1999),77-90.

[17] K. Nakagama, On the order of automorphisms of a closed Riemann surface, Pacific J. Math., vol.115,No.2,435-443(1984) [18] S.Negami, Enumeration of projective -planar embeddings of graphs, Discrete Math, 62(1986), 299 306.

[19] W.T.Tutte, What is a maps? in New Directions in the Theory of Graphs (ed.by F.Harary), Academic Press (1973), 309 325.

[20] A.T.White, Graphs of Group on Surfaces- interactions and models, Elsevier Science B.V. (2001).

[21] Yanpei Liu, Enumerative Theory of Maps, Kluwer Academic Publisher, Dordrecht / Boston / London (1999).

147

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Groupâ&#x2C6;&#x2014; Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing 100080, P.R.China)

Yanpei Liu (Institute of Applied Mathematics, Beijing Jiaotong University, Beijing,100044, P.R.China)

Abstract: A map is a connected topological graph Î&#x201C; cellularly embedded in a surface. In this paper, applying Tutteâ&#x20AC;&#x2122;s algebraic representation of map, new ideas for enumerating non-equivalent orientable or non-orientable maps of graph are presented. By determining automorphisms of maps of Cayley graph Î&#x201C; = Cay(G : S) with AutÎ&#x201C; â&#x2C6;ź = G Ă&#x2014; H on locally orientable, orientable and non-orientable surfaces, formulae for the number of non-equivalent maps of Î&#x201C; on surfaces (orientable, non-orientable or locally orientable) are obtained . Meanwhile, using reseults on GRR graph for finite groups, we enumerate the non-equivalent maps of GRR graph of symmetric groups, groups generated by 3 involutions and abelian groups on orientable or non-orientable surfaces.

d [ i i i

D DL ! Cayley D D ?-F M2 z. â&#x201E;&#x201E;m,j7â&#x201E;&#x201E;m bj ' 2- ts jm Tutte \ ' ,j' F 1TL \ 8Fj 7, u 7, )?s '8PTL b 7( j{ Cayley j Î&#x201C; = Cay(G : S) AutÎ&#x201C; â&#x2C6;ź = G Ă&#x2014; H ',j H w &# 8P Î&#x201C; 7,u 7, )?s '8Pt0 ) mp# ' GRR 1

Q

[

Finished in Nov.,2001, e-print: arXiv: math.GM/0607791

i

d

148

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

|i

}

1'` ! T >r^8F ( j{B 3 ms*y" '

Ă&#x2020;N ' GRR j' 7, 7,' )?s

i

Key words: embedding, map, finite group, Cayley graph, graphical regular representation, automorphism group, Burnside Lemma.

Classification: AMS(1991) 05C10,05C25,05C30 1. Introduction Maps originate from the decomposition of surfaces. A typical example in this field is the Heawood map coloring theorem. Combinatorially, a map is a connected topological graph Î&#x201C; cellularly embedded in a surface. Motivated by the four color problem, the enumeration of maps on surfaces, especially, the planar rooted maps, has been intensively investigated by many researchers after the Tutteâ&#x20AC;&#x2122;s pioneer work in 1962 (see [10]). By using the automorphisms of the sphere, Liskovets gives an enumerative scheme for unrooted planar maps[8] . Liskovets, Walsh and Liskovets got many enumeration results for general planar maps, regular planar maps, Eulerian planar maps, self-dual planar maps and 2-connected planar maps, etc

[7]â&#x2C6;&#x2019;[9]

.

Applying the well-known Burnside Lemma in permutation groups and the Edmonds embedding scheme[2] , Biggs and White presented a formula for enumerating the nonequivalent maps (also a kind of unrooted maps) of a graph on orientable surfaces(see [1],[14],[19]), which has been successfully used for the complete graphs, wheels and complete bipartite graphs by determining the fix set Fv (Îą) for each vertex v and automorphism Îą of a graph[14]â&#x2C6;&#x2019;[15],[19].

Notice that Biggs and Whiteâ&#x20AC;&#x2122;s formula can be only used for orientable surfaces.

For counting non-orientable maps of graphs, new mechanism should be devised.

In 1973,Tutte presented an algebraic representation for maps on locally orientable surface([10],[17]â&#x2C6;&#x2019;[18]) . Applying the Tutteâ&#x20AC;&#x2122;s map representation, a general scheme for enumerating the non-equivalent maps of a graph on surfaces can be established (Lemma 3.1 in section 3), which can be used for orientable or non-orientable surfaces. This enumeration scheme has been used to enumerate complete maps on surfaces (orientable,non-orientable or locally orientable) by determining all orientationpreserving automorphisms of maps of a complete graph[13] . In orientable case, result is the same as in [14]. The approach of counting orbits under the action of a

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

149

permutation group is also used to enumerate the rooted maps and non-congruent embeddings of a graph[6],[11],[16] . Notice that an algebraic approach for construction non-hamiltonian cubic maps on every surface is presented in [12]. The main purpose of this paper is to enumerate the non-equivalent maps of Cayley graph Γ of a finite group G satisfying AutΓ = R(G) × H ∼ = G × H on orientable, non-orientable or locally orientable surfaces, where H is a subgroup of AutΓ. For this objective, we get all orientation-preserving automorphisms of maps of Γ in the Section 2. The scheme for enumerating non-equivalent maps of a graph is re-established in Section 3. Using this scheme, results for non-equivalent maps of Cayley graphs are obtained. For concrete examples, in Section 4, we calculate the numbers of non-equivalent maps of GRR graphs for symmetric groups, groups generated by 3 involutions and abelian groups. Terminologies and notations used in this paper are standard. Some of them are mentioned in the following.

All surfaces are 2-dimensional compact closed manifolds without boundary, graphs are connected and groups are finite in the context.

For a finite group G, choose a subset S ⊂ G such that S −1 = S and 1G 6∈ S, the Cayley graph Γ = Cay(G : S) of G with respect to S is defined as follows: V (Γ) = G; E(Γ) = {(g, sg)|g ∈ G, s ∈ S}.

It has been shown that Γ is transitive, the right regular representation R(G) is a subgroup of AutΓ and it is connected if and only if G = hSi. If there exists a Cayley set S such that Aut(Cay(G : S)) = R(G) ∼ = G, then G is called to have a graphical regular representation, abbreviated to GRR and say Cay(G : S) is the GRR graph of the finite group G. Notice that which groups have GRR are completely determined (see [4] − [5] and [21] for details).

A map M = (Xα,β , P) is defined ] to be a permutation P acting on Xα,β of a disjoint union of quadricells Kx of x ∈ X , where K = {1, α, β, αβ} is the Klein group, satisfying the following conditions: (i) for ∀x ∈ Xα,β , there does not exist an integer k such that P k x = αx; (ii) αP = P −1 α; (iii) the group ΨJ = hα, β, Pi is transitive on Xα,β .

According to the condition (ii), the vertices of a map are defined to be the pairs of conjugate of P action on Xα,β and edges the orbits of K on Xα,β . For

150

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

example, {x, αx, βx, αβx} is an edge for ∀x ∈ Xα,β of M. Geometrically, any map M is an embedding of a graph Γ on a surface ( see also [10], [17] − [18] ), denoted by M = M(Γ) and Γ = Γ(M). The graph Γ is called the underlying graph. If r ∈ Xα,β is marked beforehand, then M is called a rooted map, denoted by M r .

For example, the graph K4 on the tours with one face length 4 and another 8 shown in Fig. 1,

Fig.1 can be algebraically represented as follows: A map (Xα,β , P) with Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw} and

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw) × (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv) The four vertices of this map are {(x, y, z), (αx, αz, αy)}, {(αβx, u, w), (βx, αw, αu)}, {(αβz, αβu, v), (βz, αv, βu)} and {(αβy, αβv, αβw), (βy, βw, βv)} and six edges are {e, αe, βe, αβe} for ∀e ∈ {x, y, z, u, v, w}.

1 2 Two maps M1 = (Xα,β , P1 ) and M2 = (Xα,β , P2 ) are called to be isomorphic if 1 2 1 there exists a bijection τ : Xα,β −→ Xα,β such that for ∀x ∈ Xα,β ,τ α(x) = ατ (x),

τ β(x) = βτ (x) and τ P1 (x) = P2 τ (x) and τ is called an isomorphism between them. Similarly, two maps M1 , M2 are called to be equivalent if there exists an 1 isomorphism ξ between M1 and M2 such that for ∀x ∈ Xα,β , τ P1 (x) 6= P2−1 τ (x).

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

151

Call ξ an equivalence between M1 and M2 . If M1 = M2 = M, then an isomorphism or an equivalence between M1 and M2 is called an automorphism or an orientationpreserving automorphism of M. Certainly, an orientation-preserving automorphism of a map is an automorphism of map preserving the orientation on this map.

All automorphisms or orientation-preserving automorphisms of a map M form groups, called automorphism group or orientation-preserving automorphism group of M and denoted by AutM or AutO M, respectively. Similarly, two rooted maps M1r and M2r are said to be isomorphic if there is an isomorphism θ between them such that θ(r1 ) = r2 , where r1 , r2 are the roots of M1r and M2r , respectively and denote the automorphism group of M r by AutMr . It has been known that AutMr is the trivial group.

Now let Γ be a connected graph. The notations E O (Γ), E N (Γ) and E L(Γ) denote the embeddings of Γ on the orientable surfaces, non-orientable surfaces and locally orientable surfaces, M(Γ) and AutΓ denote the set of non-isomorphic maps underlying a graph Γ and its automorphism group, respectively.

Terminologies and notations not defined here can be seen in [10] for maps and graphs and in [1] and [20] for groups.

Notice that the equivalence and isomorphism for maps are two different concepts, for example, map M = (Xα,β , P) is always isomorphic to its mirror map M −1 = (Xα,β , P −1 ), but M1 must not be equivalent to its mirror M −1 . We establish an approach for calculating non-equivalent maps underlying a graph and concrete results in the sequel sections.

2. Determining orientation-preserving automorphisms of maps of Cayley graphs For C = {(x1 , x2 , · · · , xl ), (αxl , αxl−1 , · · · , αx1 )}, the permutation Θ = (x1 , x2 , · · · , xl )(αxl , αxl−1 , · · · , αx1 ) is called a pair permutation. Denote by {C} the set {x1 , x2 , · · · , xl , αx1 , αx2 , · · · , αxl } and g |Ω1 the constraint of permutation g action on Ω1 for Ω1 ⊂ Ω. Then we get the following result.

Lemma 2.1 Let Γ be a connected graph. Then (i) For any map M ∈ M(Γ), if τ ∈ AutM, then τ |V (Γ) ∈ AutΓ; (ii) For any two maps M1 , M2 underlying the graph Γ, if θ is an isomorphism

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

mapping M1 to M2 , then θ |V (Γ) ∈ AutΓ.

Proof According to the Tutte’s algebraic representation for maps, we can assume that M = (Xα,β , P) with X = E(Γ). For ∀x, y ∈ V (M), we know that x = {(e1 , e2 , · · · , es ), (αes , αes−1, · · · , αe1 )};

y = {(e1 , e2 , · · · , et ), (αet , αet−1 , · · · , αe1 )}.

Now if e = xy ∈ E(G), there must be two integers i, j, such that ei = βej = e or βei = ej = e. Whence, we get that (i) if τ ∈ AutM, then V (Γ) = V (M) = V τ (M) = V τ (Γ) and xτ = {(τ (e1 ), τ (e2 ), · · · , τ (es )), (ατ (es ), ατ (es−1), · · · , ατ (e1 ))};

y τ = {(τ (e1 ), τ (e2 ), · · · , τ (et )), (ατ (et ), ατ (et−1 ), · · · , ατ (e1 ))}.

Therefore, eτ ∈ {xτ } ∩ β{y τ } or eτ ∈ β{xτ } ∩ {y τ }.

Whence, xτ y τ ∈ E(Γ) and τ |V (Γ) ∈ AutΓ. (ii) Similarly, if θ : M1 −→ M2 is an isomorphism, then θ : V (Γ) = V (M1 ) −→ V (M2 ) = V (Γ) and eθ ∈ {xθ } ∩ β{y θ } or eθ ∈ β{xθ } ∩ {y θ } Whence we get that eθ = xθ y θ ∈ E(Γ) and θ |V (Γ) ∈ AutΓ.

Lemma 2.2 For ∀g ∈ AutM, ∀x ∈ Xα,β of a map M, (i) |xAutM | = |AutM| ; (ii) |x≺g≻ | = o(g), where, o(g) denotes the order of g.

♮

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

153

Proof For any subgroup H ≺ AutM, we know that |H| = |xH ||Hx |. Since Hx ≺ AutMx by definition, where M x is the rooted map with root x, and AutMx is trivial, we know that |Hx | = 1. Whence, |xH | = |H|. Now take H = AutM or hgi, we get the assertions (i) and (ii).

♮

For ∀g ∈ AutΓ, M = (Xα,β , P) ∈ M(Γ), define an extending action of g on M by g ∗ = g |Xα,β : Xα,β −→ Xα,β , ∗

such that M g = gMg −1 and gα = αg, gβ = βg. A permutation p on set Ω is called semi-regular if all of its orbits have the same length. Whence, an automorphism of a map is semi-regular. The next result is followed by Lemma 2.1 and the definition of extending action of elements in AutΓ gives a necessary and sufficient condition for an automorphism of a map to be an orientation-preserving automorphism of this map.

Theorem 2.1 For a connected graph Γ, an automorphism ξ ∗ of map M is an orientation-preserving automorphism of map underlying Γ if and only if there exists an element ξ ∈ AutΓ such that ξ ∗ = ξ |Xα,β .

Now for a finite group G, let Γ = Cay(G : S) be a connected Cayley graph respect to S. Then its edge set is {(g, sg)|∀g ∈ G, ∀s ∈ S}. For convenience, we use g sg denoting an edge (g, sg) in the Cayley graph Cay(G : S). Then its quadricell of this edge can be represented by {g sg+ , g sg−, (sg)g+ , (sg)g− } and Xα,β (Γ) = {g sg+|∀g ∈ G, ∀s ∈ S} ∪ {g sg−|∀g ∈ G, ∀s ∈ S};

α=

Y

(g sg+ , g sg−);

g∈G,s∈S

β=

Y

(g sg+ , (sg)g+)(g sg− , (sg)g− ).

g∈G,s∈S

The main result of this section is the following.

Theorem 2.2 Let Γ = Cay(G : S) be a connected Cayley graph with AutΓ = R(G) × H. Then for ∀θ ∈ AutΓ, the extending action θ |Xα,β is an orientationpreserving automorphism of a map in E(Γ) on surfaces.

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

Proof The proof is divided into two parts. First, we prove each automorphism of the graph Γ is semi-regular and second, construct a stable embedding of Γ for ∀θ ∈ AutΓ. (i) For ∀g ∈ AutΓ, since AutΓ = R(G) × H, there must exist γ ∈ R(G), δ ∈ H such that g = γδ = δγ. Now for ∀x ∈ G, the action of elements in hgi on x are as follows. xg = (xδ )γ = xδ γ; 2

2

2

2

xg = (xδ )γ = xδ γ 2 ; ························; n

n

n

n

xg = (xδ )γ = xδ γ n ; ························.

Therefore, the orbit of hgi acting on x is 2

n

xhgi = (x, xδ γ, xδ γ 2 , · · · , xδ γ n , · · ·).

That is, for ∀x ∈ G, |xhgi | = [o(δ), o(γ)]. Whence, g is semi-regular. (ii) Assume that the automorphism θ of Γ is θ = (a, b, · · · , c) · · · (g, h, · · · , k) · · · (x, y, · · · , z), where the length of each cycle is κ = o(g), G = {a, b, · · · , c, · · · , g, h, · · · , k, · · · , x, y, · · · , z} and S = {s1 , s2 , · · · , st } ⊂ G. Denote by T = {a, · · · , g, · · · , x} the representation set of each cycle in θ. We construct a map M = (Xα,β , P) underlying Γ with Xα,β (Γ) = {g sg+|∀g ∈ G, ∀s ∈ S} ∪ {g sg−|∀g ∈ G, ∀s ∈ S};

P=

Y

Y

(Cx )(αCx−1 α−1 ),

g∈T x∈C(g)

where C(g) denotes the cycle containing g and let x = θf (g), then Cx = (θf (g)θ and

f (s

1 g+)

, θf (g)θ

f (s

2 g+)

, · · · , θf (g)θ

f (s

t g+)

)

155

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

αCx−1 α−1 = (αθf (g)θ

f (s

t g−)

, αθf (g)θ

f (s

t−1 g−)

, · · · , αθf (g)θ

f (s

1 g−)

).

It is clear that M = θMθ−1 . According to Theorem 2.1, we know that θ |Xα,β is an orientation-preserving automorphism of map M. Combining (i) with (ii), the proof is complete.

♮

According to the Rotation Embedding Scheme for orientable embeddings of a graph formalized by Edmonds in [2], each orientable complete map is just the case of eliminating the signs

+, - in our representation of maps. Whence,we get the

following result for orientable maps underlying a Cayley graph of a finite group.

Theorem 2.3 Let Γ = Cay(G : S) be a connected Cayley graph with AutΓ = R(G) × H. Then for ∀θ ∈ AutΓ, the extending action θ |Xα,β is an orientationpreserving automorphism of a map in M(Γ) on orientable surfaces.

Notices that a GRR graph Γ of a finite group G satisfies AutΓ = R(G). Since R(G) ∼ = R(G) × {1AutΓ }, by Theorems 2.2 and 2.3, we get all orientation-preserving automorphisms of maps of GRR graphs of a finite group as follows.

Corollary 2.1 Let Γ = Cay(G : S) be a connected GRR graph of a finite group G. Then for ∀θ ∈ AutΓ, the extending action θ |Xα,β is an orientation-preserving automorphism of a map in M(Γ) on locally orientable surfaces.

Corollary 2.2 Let Γ = Cay(G : S) be a connected GRR of a finite group G. Then for ∀θ ∈ AutΓ, the extending action θ |Xα,β is an orientation-preserving automorphism of a map in M(Γ) on orientable surfaces.

3. The enumeration of non-equivalent maps of Cayley graphs According to Theorem 2.1, we can get a general scheme for enumerating the nonequivalent maps of a graph Γ on surfaces.

Lemma 3.1 For any connected graph Γ, let E ⊂ E L(Γ), then the number n(E, M) of non-equivalent maps in E is n(E, M) =

X 1 |Φ(g)|, |AutΓ| g∈AutΓ

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Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

where, Φ(g) = {P|P ∈ E and P g = P}.

Proof According to Theorem 2.1, two maps M1 , M2 ∈ E are equivalent if and ∗

only if there exists an automorphism g ∈ AutΓ such that M1g = M2 , where,g ∗ = g |Xα,β . Whence, all non-equivalent maps in E are just the representations of the orbits in E under the action of AutΓ. By the Burnside Lemma, the number of non-equivalent maps in E is n(E, M) =

X 1 |Φ(g)|. |AutΓ| g∈AutΓ

♮

Corollary 3.1 The numbers of non-equivalent maps in E O (Γ), E N (Γ) and E L (Γ) are n(E O (Γ), M) =

X 1 |ΦO (g)|; |AutΓ| g∈AutΓ

(3.1)

n(E N (Γ), M) =

X 1 |ΦN (g)|; |AutΓ| g∈AutΓ

(3.2)

n(E L (Γ), M) =

X 1 |ΦL (g)|, |AutΓ| g∈AutΓ

(3.3)

where, ΦO (g) = {P|P ∈ E O (Γ) and P g = P}, ΦN (g) = {P|P ∈ E N (Γ) and P g = P}, ΦL (g) = {P|P ∈ E L(Γ) and P g = P}.

Corollary 3.2 In formula (3.1)-(3.3), |Φ(g)| = 6 0 if, and only if g is an orientationpreserving automorphism of map of graph Γ on an orientable, non-orientable or locally orientable surface.

The formula (3.1) is obtained by Biggs and White in [1]. Applying Theorems 2.2 − 2.3 and the formulae (3.1) − (3.3), we can enumerate the non-equivalent maps underlying a Cayley graph Γ of a finite group G satisfying AutΓ = R(G) × H on orientable surfaces, non-orientable surfaces and locally orientable surfaces.

Theorem 3.1 Let Γ = Cay(G : S) be a connected Cayley graph with AutΓ = R(G) × H. Then the number nTM (G : S) of non-equivalent maps underlying Γ on locally orientable surfaces is nLM (G : S) =

X |G| 1 |Eξ |2α(S,ξ)(|S| − 1)! o(ξ) , |G||H| ξ∈OG

157

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

where OG denotes the representation set of conjugate class of AutΓ , Eξ the conjugate class in AutΓ containing ξ and α(S, ξ) =

  

|G||S|−2|G| , 2o(ξ) |G||S|+2l−2|G| , 2o(ξ)

if

ξ∈Θ

if

ξ ∈ ∆.

where, Θ = {ξ|o(ξ) ≡ 1(mod2) ∨ o(ξ) ≡ 0(mod2), 6 ∃s ∈ S, t ∈ G such that s = tξ

o(ξ) 2

}, ∆ = {ξ|o(ξ) ≡ 0(mod2)∧∃si ∈ S, ti ∈ G, 1 ≤ i ≤ l(ξ), l(ξ) ≡ 0(mod o(ξ) ) such 2

that si = tiξ

o(ξ) 2

}.

Proof Notice that ΦL (ξ) is a class function on AutΓ. According to Theorem 2.2 and Corollary 3.1, we know that

nLM (G : S) = =

X 1 × |ΦL (ξ)| |AutΓ| ξ∈AutΓ X 1 |ΦL (g)|. |G||H| ξ∈R(G)×H

(3.4)

Since for ∀ξ = (µ, ν) ∈ AutΓ, ξ is semi-regular, without loss of generality, we can assume that ξ = (a, b, · · · , c) · · · (g, h, · · · , k) · · · (x, y, · · · , z), where the length of each cycle is o(ξ) = [o(µ), o(ν)], P=

Y

Y

(Cx )(αCx−1 ),

g∈T x∈C(g)

being a map underlying the graph Γ and stable under the action of ξ, C(g) denotes the cycle containing g and T is the representation set of cycles in ξ. Let S = {s1 , s2 , · · · , sk } and x = ξ f (g), then Cx = (ξ f (g)ξ

f (s

1 gν1 )

, ξ f (g)ξ

f (s

2 gν2 )

, · · · , ξ f (g)ξ

f (s

t gνk )

),

(3.5)

with νi ∈ {+, −}, 1 ≤ i ≤ k.

Notice that the quadricell adjacent to the vertex a can make 2|S|−1(|S|−1)! pair permutations, and for each chosen pair permutation, the pair permutations adjacent

158

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

to the vertex x, x ∈ C(a) are uniquely determined by (3.5) since P is stable under the action of ξ.

Similarly, for each given pair permutation adjacent to a vertex u ∈ T , the pair permutations adjacent to the vertices v, v ∈ C(u) are also uniquely determined by (3.5) since P is stable under the action of ξ.

Notice that any non-orientable embedding can be obtained by exchanging some x with αx, x ∈ Xα,β (M) in an orientable embedding M underlying Γ. Now for an orientable embedding M1 of Γ, all the induced embeddings by exchanging some edge’s two sides and retaining the others unchanged in M1 are the same as M1 by the definition of embedding. Therefore, the number of different stable maps under the action of ξ gotten by exchanging x and αx in M1 for x ∈ U, U ⊂ Xβ , where Xβ =

S

{x, βx} , is 2

|G|

ξ(ε)− o(ξ)

, where ξ(ε) is the number of orbits of E(Γ) under

x∈E(Γ)

the action of ξ, and we subtract

|G| o(ξ)

because we can choose ab+ , · · · , g a+ , · · · , xa+

first in our enumeration.

Since the length of each orbit under the action of ξ is o(ξ) for ∀e ∈ E(Γ) if o(ξ) ≡ 1(mod2) or o(ξ) ≡ 0(mod2) but there are not s ∈ S, t ∈ G such that s = tξ

o(ξ) 2

and is

o(ξ) 2

for each edge tsi i ti , 1 ≤ i ≤ l(ξ), if o(ξ) ≡ 0(mod2) and there

are si ∈ S, ti ∈ G, 1 ≤ i ≤ l(ξ), such that si = tξi

o(ξ) 2

(Notice that there must be

l ≡ 0(mod o(ξ) ) because ξ is an automorphism of the graph Γ) or o(ξ) for all other 2 edges. Whence, we get that ξ(ε) =

  

ε(Γ) , o(ξ) ε(Γ)−l(ξ) o(ξ)

+

2l(ξ) , o(ξ)

if

ξ∈Θ

if

ξ ∈ ∆.

Now for ∀π ∈ AutΓ, since θ = πξπ −1 ∈ AutΓ, we know that θ(ε) = ξ(ε).

Therefore, we get that α(S, ξ) =

  

|G||S|−2|G| , 2o(ξ) |G||S|+2l(ξ)−2|G| , 2o(ξ)

if

ξ∈Θ

if

ξ ∈ ∆.

and |G|

|ΦT (ξ)| = 2α(S,ξ) (|S| − 1)! o(ξ) .

Combining (3.4) with (3.6), we get that

(3.6)

159

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

nTM (G : S) = and the proof is complete.

X |G| 1 |Eξ |2α(S,ξ)(|S| − 1)! o(ξ) , |G||H| ξ∈OG

♮

According to the formula (3.1) and Theorem 2.3, we also get the number nO M (G : S) of non-equivalent maps of a Cayley graph Cay(G : S) on orientable surfaces.

Theorem 3.2 Let Γ = Cay(G : S) be a Cayley graph with AutΓ = R(G) × H. Then the number nO M (G : S) of non-equivalent maps underlying Γ on orientable surfaces is nO M (G : S) =

X |G| 1 |Eξ |(|S| − 1)! o(ξ) , |G||H| ξ∈OG

where,the means of notations Eξ , OG are the same as in Theorem 3.1.

Proof

By Corollary 3.1, we know that nO M (G : S) =

X 1 × |ΦO (ξ)|. |G||H| ξ∈R(G)×H

Similar to the proof of Theorem 3.1 by applying Theorem 2.3 and Corollary 3.1, we get that for ∀ξ ∈ R(G) × H, |G|

|ΦO (ξ)| = (|S| − 1)! o(ξ) .

Therefore, nO M (G : S) =

X |G| 1 |Eξ |(|S| − 1)! o(ξ) . |G||H| ξ∈OG

♮

Notice that for a given Cayley graph Cay(G : S) of a finite group G, nO M (G : S)+ L nN M (G : S) = nM (G : S). Whence, we get the number of non-equivalent maps

underlying a graph Cay(G : S) on non-orientable surfaces.

Theorem 3.3 Let Γ = Cay(G : S) be a Cayley graph with AutΓ = R(G) × H. Then the number nN M (G : S) of non-equivalent maps underlying Γ on non-orientable surfaces is nN M (G : S) =

X |G| 1 |Eξ |(2α(S,ξ) − 1)(|S| − 1)! o(ξ) , |G||H| ξ∈OG

160

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

where OG denotes the representation set of conjugate class of AutΓ, Eξ the conjugate class in AutΓ containing ξ and α(S, ξ) is the same as in Theorem 3.1.

Since R(G) ∼ = R(G)×{1AutΓ } and the condition s ∈ S, t ∈ G such that s = tξ turns to s = tξ

o(ξ) 2

o(ξ) 2

t−1 when AutΓ = R(G), we get the number of non-equivalent maps

underlying a GRR graph of a finite group by Theorems 3.1 − 3.3 as follows.

Corollary 3.3 Let G be a finite group with a GRR graph Γ = Cay(G : S). Then the numbers of non-equivalent maps underlying Γ on locally orientable, orientable and non-orientable surfaces are respective |G| 1 X nLM (G : S) = |Eg |2α1 (S,g) (|S| − 1)! o(g) , |G| g∈OG nO M (G : S) = and nN M (G : S) =

|G| 1 X |Eg |(|S| − 1)! o(g) |G| g∈OG

|G| 1 X |Eg |(2α1 (S,g) − 1)(|S| − 1)! o(g) , |G| g∈OG

where OG denotes the representation set of conjugate class of G, Eg the conjugate class in G containing g and α1 (S, g) =

  

|G||S|−2|G| , 2o(g) |G||S|+2l(g)−2|G| , 2o(g)

if

g ∈ Θ′

if

g ∈ ∆′ .

where, Θ′ = {g|o(g) ≡ 1(mod2) ∨ o(g) ≡ 0(mod2), ∀s ∈ S, s 6∈ E

g

o(g) 2

} and ∆′ =

{g|o(g) ≡ 0(mod2), ∃ti ∈ G, 1 ≤ i ≤ l(g), l(g) ≡ 0(mod o(g) ) such that ti g 2

o(g) 2

t−1 i

∈ S}.

Corollary 3.4 Let G be a finite group of odd order with a GRR graph Γ = Cay(G : S). Then the number nLM (G : S) of non-equivalent maps of graph Γ on surfaces is nLM (G : S) =

|G||S|−2|G| |G| 1 X |Eg |2 2o(g) (|S| − 1)! o(g) . |G| g∈OG

4. Examples and calculation for GRR graphs Hetze and Godsil investigated GRR for solvable, non-solvable finite groups, respectively. They proved[4],[21] that every group has GRR unless it belongs to one of the following groups:

161

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

(a) abelian groups of exponent greater than 2; (b) generalized dicyclic groups; (c) thirteen ”exceptional” groups: (1) Z22 , Z23 , Z24 ; (2) D6 , D8 , D10 ; (3) A4 ; (4) ha, b, c|a2 = b2 = c2 = 1, abc = bca = cabi; (5) ha, b|a8 = b2 = 1, bab = b5 i; (6) ha, b, c|a3 = c3 = b2 = 1, ac = ca, (ab)2 = (cb)2 = 1i; (7) ha, b, c|a3 = b3 = c3 = 1, ac = ca, bc = cb, c = a−1 b−1 abi; (8) Q8 × Z3 , Q8 × Z4 .

Based on results in previous section, the constructions given in [4] − [5] and Corollary 3.2, we give some calculations for the numbers of non-equivalent maps underlying a GRR graph on surfaces for some special groups.

Calculation 4.1

Symmetric group Σn

¯ denotes a partition of the integer n: (k) ¯ = k1 , k2 , · · · , kn ) Using the notation (k) ¯ the least common multiple of the intesuch that 1k1 +2k2 +· · ·+nkn = n and lcm(k) ¯ = [1(k1 times), 2(k2 times), gers 1(k1 times), 2(k2 times), · · ·, n(kn times), i.e, lcm(k) · · · , n(kn times)] . Godsil proved that[5] every symmetric group Σn with n ≥ 19 has a cubic GRR with S = {x, y, y −1}, where x2 = y 3 = e. Since |Σn | = n!, we get that the numbers of non-equivalent maps underlying a cubic GRR graph of Σn are nLM (Σn : S) =

|Σn | X α (S,g) X α(S,g)+ n!

1 1 o(g) × 2 1 × 2! o(g) = × 2 n! g∈Σn n! g∈Σn

and

nO M (Σn : S) =

X |Σn | 1 × 2! o(g) n! g∈Σn n!

¯ X X 2 lcm(k) n!

1 n! ¯ lcm(k) = × × 2 = , n n Q k Q k n! (k) ¯ ¯ i i ki ! i i ki ! (k)

i=1

and

i=1

162

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

nN M (Σn : S) =

X n!

1 × 2 o(g) (2α1 (S,g) − 1) n! g∈Σn

For the case n = 6m + 1, we know that[5] x = b1 if m ≡ 1(mod2) and x = b2 if m ≡ 0(mod2), where

b1 = (1, 4)(2, n)(3, n − 1)(n − 6, n − 3)(n − 5, n − 2) ×

m−2 Y

(6r, 6r + 3)(6r + 1, 6r + 4)(6r + 2, 6r + 5)

r=1

and b2 = b1 (n − 12, n − 9).

Notice that b1 ∈ E[13 23m−1 ] and b2 ∈ E[15 23m−2 ] . We define the sets A1 , B1 , A2 and B2 as follows.

A1 = {g|g ∈ Σn , o(g) ≡ 1(mod2) or o(g) ≡ 0(mod2) but g

B1 = {g|g ∈ Σn , o(g) ≡ 0(mod2) but g

o(g) 2

o(g) 2

6∈ E[13 23m−1 ] },

6∈ E[13 23m−1 ] }

and

A2 = {g|g ∈ Σn , o(g) ≡ 1(mod2) or o(g) ≡ 0(mod2) but g

B2 = {g|g ∈ Σn , o(g) ≡ 0(mod2) but g

o(g) 2

o(g) 2

6∈ E[15 23m−2 ] },

6∈ E[15 23m−2 ] }.

For ∀θ ∈ Σn , if ζ ∈ Ai or Bi , i = 1 or 2, it is clear that θζθ−1 ∈ Ai or Bi .

Whence, Eζ ⊂ Ai or Bi . Now calculation shows that

l(g) =

Therefore, we have that

        

3!(n − 3)!!,

if

g ∈ E[13 23m−1 ]

5!(n − 2)!!,

if

g ∈ E[15 23m−2 ]

0,

otherwise.

163

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

nLM (Σn : S)|m≡1(mod2) =

P

n!

2α1 (S,g)+ o(g)

g∈Σn

n!

X n! ¯ 1 n! α1 (S,(k))+ ¯ lcm(k) = × × 2 n Q n! (k) k i ¯ i k!

=

X

2

i i=1 n! ¯ α1 (S,(k))+ ¯ lcm(k) n Q

¯ (k)

,

i=1

iki ki !

where, ¯ = α1 (S, (k)) and

 

n! ¯ , 2·lcm(k) n!+12(n−3)!! , ¯ 2·lcm(k)



nLM (Σn : S)|m≡0(mod2) = =

if

E(k) ¯ ⊂ A1

if

E(k) ¯ ⊂ B1

P

2

n! α′1 (S,g)+ o(g)

g∈Σn

X

n!

2

n! ¯ α′1 (S,(k))+ ¯ lcm(k)

n Q

¯ (k)

i=1

, iki ki !

where ¯ = α1′ (S, (k))

Calculation 4.2

  

n! ¯ , 2·lcm(k) n!+240(n−5)!! , ¯ 2·lcm(k)

if

E(k) ¯ ⊂ A2

if

E(k) ¯ ⊂ B2 .

Group generated by 3 involutions

Let G = ha, b, c|a2 = b2 = c2 = ei be a finite group of order n. In [5], Godsil proved that if (AutG)S = e, where S = {a, b, c}, then G has a GRR Cay(G : S).

Since any element of order 2 must has the form txt−1 , t ∈ G and x = a, b or c. We assume that for ∀t ∈ G, tx 6= xt, for x = a, b, c. Then for ∀g ∈ G, l(g) = Therefore, we get that

  

n,

if

o(g) ≡ 0(mod2)

0,

if

o(g) ≡ 1(mod2).

164

Linfan Mao: Mathematics of 21st Century–A Collection of Selected Papers

α1 (S, g) =

nLM (G : S) =

  

n , 2o(g) 3n , 2o(g)

P

if

o(g) ≡ 1(mod2)

if

o(g) ≡ 0(mod2), 3n

2 2o(g) +

o(g)≡1(mod2)

P

2 2o(g)

P

2 o(g) (2 2o(g) − 1)

5n

o(g)≡0(mod2)

,

n

nO M (G : S) =

P

n

2 o(g)

g∈G

n

and

nN M (G : S) = Calculation 4.3

P

n

n

2 o(g) (2 2o(g) − 1) +

o(g)≡1(mod2)

n

3n

o(g)≡0(mod2)

n Abelian group

Let k = |S|. It has been proved that an abelian group G has GRR if and only if G = (Z2 )n for n = 1 or n ≥ 5. Now for the abelian group G = (Z2 )n = hai × hbi × · · · × hci, every element in G has order 2. Calculation shows that  

l(g) = 

Whence, we get that

α1 (S, g) =

  

2n ,

if

g∈S

0,

if

g 6∈ S.

(k − 2)2n−2, k2

n−2

,

if

g 6∈ S

if

g ∈ S.

Therefore, the numbers of non-equivalent maps underlying a GRR graph of (Z2 )n on locally orienatble or orientable surfaces are nLM ((Z2 )n : S) = =

X |G| 1 × 2α1 (S,g) (k − 1)! o(g) |G| g∈(Z2 )n X X 1 k2n−2 2n−1 (k−2)2n−2 2n−1 × ( 2 (k − 1)! + 2 (k − 1)! ) 2n g∈S g6∈S,g6=e n−2

=

2k2

n−1

k(k − 1)!2 n−2

+

2(k−2)2

n

n−2

+ (2n − k − 1)2(k−2)2 2n

(k − 1)!2 , 2n

n−1

(k − 1)!2

165

On Automorphisms and Enumeration of Maps of Cayley Graphs of a Finite Group

and n nO M ((Z2 ) : S) =

X 2n 1 o(g) × (k − 1)!

2n g∈(Z2 )n n

=

n−1

(k − 1)!2 + (2n − 1)(k − 1)!2 2n

.

References [1] N.L.Biggs and A.T.White, Permutation Groups and Combinatoric Structure, Cambridge University Press (1979).

[2] Edmonds.J, A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc 7 (1960) 646.

[3] X.G.Fang, C.H.Li, J.Wang and M.Y.Xu, On cubic graphical regular representation of finite simple groups (preprint, Peking University,2001).

[4] C.D.Godsil, On the full automorphism group of a graph, Combinatorica, 1(1981), 243-256.

[5] C.D.Godsil, The automorphism groups of some cubic Cayley graphs,Europ.J. Combin. 4(1983),25-32 [6] Jin Ho Kwak and Jaeun Lee, Enumeration of graph embeddings, Discrete Math, 135(1994), 129-151.

[7] V.A.Liskovets, Enumeration of non-isomorphic planar maps,Sel. Math. Sov, 4;4(1985), 303-323.

[8] V.A.Liskovets, A reductive technique for Enumerating non-isomorphic planar maps, Discrete Math, 156(1996), 197-217.

[9] V.A.Liskovets and T.R.S.Walsh, The enumeration of non-isomorphic 2-connected planar maps,Canad.J.Math, 35(1983), 417-435.

[10] Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publisher, Dordrecht / Boston / London (1999).

[11] L.F. Mao and Y.P.Liu, New automorphism groups identity of trees, Chinese advances Math, 1(2003).

[12] L.F. Mao and Y.P.Liu, An approach for constructing 3-connected non-hamiltonian cubic map on surfaces, Chinese OR Transactions, 4(2001),1-7.

[13] L.F. Mao and Y.P.Liu, The number of complete maps on surfaces, arXiv: math.GM/0607.

166

Linfan Mao: Mathematics of 21st Centuryâ&#x20AC;&#x201C;A Collection of Selected Papers

[14] B.P.Mull,R.G.Rieper and A.T.White, Enumeration 2-cell imbeddings of connected graphs, Proc.Amer.Math.Soc, 103(1988), 321-330.

[15] B.P.Mull, Enumerating the orientable 2-cell imbeddings of complete bipartite graphs, J.Graph Theory, vol 30, 2(1999),77-90.

[16] S.Negami, Enumeration of projective -planar embeddings of graphs, Discrete Math, 62(1986), 299-306.

[17] W.T.Tutte, Graph Theory, Combridge university Press (2001).

[18] W.T.Tutte, What is a maps? in New Directions in the Theory of Graphs (ed.by F.Harary), Academic Press (1973), 309-325.

[19] A.T.White, Graphs of Group on Surfaces- interactions and models, Elsevier Science B.V. (2001).

[20] M.Y.Xu et al, Introduction to finite groups (II), Science Press, Beijing (1999).

[21] M.Y.Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math, 182(1998), 309-319.

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α,β

C1 = {x, y, z, u, v, w, αx, αy, αz, αu, αv, αw, βx, βy, βz, βu,

βv, βw, αβx, αβy, αβz, αβu, αβv, αβw}

i

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw) × (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv)

D = 4 & QH % {(x, y, z), (αx, αz, αy)}, {(αβx, u, w), (βx, αw, αu)}, {(αβz, αβu, v), (βz, αv, βu)}

[ {(αβy, αβv, αβw), (βy, βw, βv)} 6 yE% {e, αe, βe, αβe},

1 e ∈ {x, y, z, u, v, w}. ?& D M1 = (Xα,β1 , P1) [ M2 = (Xα,β2 , P2) ~% w' z " u%& 1 − 1 = τ : Xα,β1 −→ Xα,β2 '; ∀x ∈ Xα,β1 ,τ α(x) = ατ (x), τ β(x) = βτ (x) V τ P1(x) = P2 τ (x) ~ τ % ?& D 1 = %& w z M1 = M2 = M, } M1 V M2 1 = 3~% D M = H w D M = bL W 3 uÆ_p[K3 %&k ~% D M =W 3k "% AutM +U D ?&L) D M1r M2r ~% 3 z eo$1"u%& D 3 θ '; θ(r1) = r2 1 r1 , r2 NM*D M1r M2r = ) "L) D M r =W 3k% AutM r 2o* #9 k AutM r %4 t k Riemann av D =W 3V av = }G9 =W 3k $ 1L%&C #= <I &p Jones [ Singerman ( = R [av =D /\[ Tucker d av k C = j 2o , #9 Riemann av =W 3V D W 3 $ 1 = KE<I F 7 B j Riemann av =_ _ zq

Riemann

A

Hurwitz

8Ă&#x2020;(O m

171

j 2.1 G { Riemann S 'â&#x201E;&#x201E;m H w

S â&#x201E;&#x201E;m,j M .& G {,j M ' H w [ , S â&#x201E;&#x201E;m( j{ Cayley j',j M . G &7 M ' H w

Fu 2.1 S â&#x201E;&#x201E;m,j M AutS AutM. Fu 2.2 S ',j M 'S W 2 |AutM| â&#x2030;¤ C C u,j M |'â&#x201E;&#x201E;m F |AutS| â&#x2030;¤ C.

e

[8][14]

â&#x2C6;&#x2014;

3.

â&#x2C6;&#x2014;

z C >, E t 4

, Î&#x201C; = (V, E) %%&; = 5 . > W 3k"% AutÎ&#x201C; b X = E(Î&#x201C;) }> 4 ^.Q X R 0% Îą,β

Xι,β =

[

{x, ιx, βx, βιβx},

xâ&#x2C6;&#x2C6;X

1 K = {1, Îą, β, ιβ} % Klein 4 ^k d q/^ â&#x2C6;&#x20AC;g â&#x2C6;&#x2C6; AutÎ&#x201C;, R 0 g u XÎą,β = C! g|X wK 1 X Îą,β

=

B y â&#x2C6;&#x20AC;x â&#x2C6;&#x2C6; X , x = y, 7d (Îąx) = Îąy, (βx) = βy 0 (ιβx) = ιβy.

D M = (X , P) dW 3 g â&#x2C6;&#x2C6; AutM [q/QH â&#x2C6;&#x20AC;u, v â&#x2C6;&#x2C6; V (M), g| : u â&#x2020;&#x2019; v, z u = v }~ g %>[ W 3 z u = v }~ g % u [ W 3 d q / g â&#x2C6;&#x2C6; AutM -#g '{=,' '{Q,' wQ, H wu=, H w' * {Q, H w D=, H wu=, H w Q, H wuQ, H w' *v{ =,H w G AutM R0 G G % G 7=>[W 3V k }G { G / ''F{ 2 'G

QH v =M*% v = (x , x , ¡ ¡ ¡ , x )(Îąx , ¡ ¡ ¡ , Îąx , Îąx ), "F v = yk% hvi } dD =W 3k LKEv 4 ~j 3.1 G AutM {,j M ' H w G , B â&#x2C6;&#x20AC;v â&#x2C6;&#x2C6; V (M), (i) â&#x2C6;&#x20AC;g â&#x2C6;&#x2C6; G,g {=, H w G hvi, {K

(ii) G < v > Ă&#x2014; hÎąi C (i) D M = (X , P) FPq/ â&#x2C6;&#x20AC;g â&#x2C6;&#x2C6; G %>[ W 3 9 d â&#x2C6;&#x20AC;v â&#x2C6;&#x2C6; V (M), h â&#x2C6;&#x2C6; G , L v = v * RQH E(G)

g

ι,β

g

g

g

V (M )

ι,β

g

~ ~

â&#x2C6;&#x2019;1

g

+

+

1

v

v

ι,β

v

h

2

Ď (v)

Ď (v)

2

1

8!O H'^5<GI1 /

172

:

v = (x1 , x2 , · · · , xρ(v) )(αxρ(v) , αxρ(v)−1 , · · · , αx1 ).

}L [(x1 , x2 , · · · , xρ(v) )(αxρ(v) , · · · , αx2 , αx1 )]h = (x1 , x2 , · · · , xρ(v) )(αxρ(v) , · · · , αx2 , αx1 ).

%$ z h(x1) = xk+1, 1 ≤ k ≤ ρ(v) }L h = [(x1 , x2 , · · · , xρ(v) )(αxρ(v) , αxρ(v)−1 , · · · , αx1 )]k = v k .

z h(x ) = αx

ρ(v)−k+1 , 1

1

≤ k ≤ ρ(v)

}L

h = [(x1 , x2 , · · · , xρ(v) )(αxρ(v) , αxρ(v)−1 , · · · , αx1 )]k α = v k α.

0 z h = vk α - # vh = vα = v−1, # h V1>[ W 3 9L h = v k , 1 ≤ k ≤ ρ(v) Gv 7= ^y% v =z r ξ % Gv 7 ^ =` _r ) I } D E Gv = v ξ hvi %F v ξ = yk (ii) d ∀g ∈ Gv L v g = v [(x1 , x2 , · · · , xρ )(αxρ , αxρ−1 , · · · , αx1 )]g = (x1 , x2 , · · · , xρ )(αxρ , αxρ−1 , · · · , αx1 ).

+UP (i) =! { #" u I s, 1 ≤ s ≤ ρ ' ; g = v g = v α 9L g ∈ hvi g ∈ hvi α s

Gv hvi × hαi .

s

♮

~j 3.2 Γ {℄m bj G AutΓ w ∀v ∈ V (Γ),G hvi × hαi G X ' S m7 ' C qb%& 4 ^.Q x ∈ X 2o ! { G = {1 } %$ g ∈ G , }L x = x oN D ><: =QH u u g CKV W L u = u

O

α,β

v

α,β

x

G

g

u = (x, y1 , · · · , yρ(u)−1 )(αx, αyρ(u)−1 , · · · , αy1),

}FP G

x

g

u

hui × hαi

9

Riemann

A

Hurwitz

8Æ(O m

173

g xg = x, y1g = y1 , · · · , yρ(u)−1 = yρ(u)−1

[

(αx)g = αx, (αy1 )g = αy1 , · · · , (αyρ(u)−1 )g = αyρ(u)−1

dq%&<:P QH u = 4 ^.Q e , e C =R 0 #

g u

u

= eu

&pW 3k AutΓ u X

α,β

(βx)g = βx, (βy1)g = βy1 , · · · , (βyρ(u)−1 )g = βyρ(u)−1

V (αβx)g = αβx, (αβy1)g = αβy1, · · · , (αβyρ(u)−1 )g = αβyρ(u)−1

qb%&^ y ∈ Xα,β y <: =QH % w, }F = ; v # Γ 7 " u%y;K u [ w =9 R P (u, w) = uv1v2 · · · vsw V ;Jv βyk <:P QH v1, }F (βyk)g = βyk Gv hv1i × hαi # q%&<:P QH v1 = 4 ^.

Q

+UD, zq℄%&<:P QH v = 4 ^.Q e u g CKVW (e ) = e , } , ! {q℄%&<:P QH v = 4 ^.Q e u g CKV W. & +Æ ;8 q℄%&<:P QH w = 4 ^.Q e u g CKV W, oN D L 1

ev1 , egv1

= ev1

i

vi

vi

i+1

vi

g

vi+1

w

g

y = y.

9 # g = 1 i L G = {1 } ♮ Pu 2o ! { =W 3k% D W 3k = y< j 3.1 Γ {℄m bj G AutΓ G 7a Γ {( j',j H w ' Z`H7B ∀v ∈ V (Γ), 7 G G hvi × hαi C &p7/ 3.1(ii) # y<1D = P ! {y< = v F7/ 3.2 # G u X = C1, } = d ∀x ∈ X L |G | = 1 9^ x u G CK=G 9 t ^ |x | = |G ||x | = |G| ∀x ∈ X u G CK= G 9 t ^ y% |G|

G u V (Γ) C2L s yG9 O , O , · · · , O 1 O = {u , u , · · · , u }, O = {v , v , · · · , v },· · ·,O = {w , w , · · · , w } 2o3z Γ 7 ' QH= ;8 u G CKV W=1 { P G

x

G

v

α,β

α,β

G

G

x

1

2

1

2

l

s

1

2

2

t

x

α,β

s

1

1

2

k

8!O H'^5<GI1 /

174

:

J / d ∀u ∈ V (Γ) F |G| = |Gu||uG| # [k, l, · · · , t]| |G| 1 [k, l, · · · , t] M * k, l, · · · , t =` _0<I 2o=Lj R G 9 Om−1 1 7 ' QH= qb QH u1 ∈ O1 0 RV k Gu Q = M*% {1G, g1, g2g1, · · · , i=1 gm−i} 1 m = |Gu | Γ 7 <:P QH u1 g ) =L^C,K zqd N(u g ) = ^[s = 4 ^.Q _% N(u b 4 ^ 1 1 g ) C G a a .Q ua1 ∈ N(u N C ;8m−14 ^.Q V A1 = 1 u u u1 [ αu1 m−1 Q Q {ua1 , g1 (ua1 ), · · · , gm−i (ua1 )} [ αA1 = {αua1 , αg1 (ua1 ), · · · , α gm−i (ua1 )} J /F i=1 i=1 T =W 3ku 4 ^.Q = C R 0 # A1 αA1 = ∅ A1 7= ^ % m−1 1

1

1

− → g ) \ (A S αA ) = ∅ } N(u g ) 7= ^ = g )\ A . z N(u OKl1 z N(u S g )\(A S αA ) ^C G CP u ; (A αA ) 6= ∅ b 4 ^.Q u ∈ N(u 8 A = {u , g (u ), · · · , Q g (u )} [ αA = {αu , αg (u ), · · · , α Q g (u )} A S A 7= ^ % Q − → A1 = ua1 , g1 (ua1 ), · · · , gm−i (ua1 ) i=1

1

1

1

1

1

b 1

2

1

1

m−1

b 1

i=1

1

1

b 1

m−i

1

b 1

1

1

u1

b 1

2

1

b 1

1

m−1

b 1

i=1

b 1

m−i

2

m−1 m−1 −−−[ −−→ Y Y A1 A2 = ua1 , g1 (ua1 ), · · · , gm−i (ua1 ); ub1, g1 (ub1 ), · · · , gm−i (ub1 ) i=1

i=1

−−−[ −−→ g )\(A S A S αA S αA ) = ∅ } A S A 7= ^ = z N(u OK% A A S S S S S g ) \ (A g ) \ (A } N(u A αA αA ) 6= ∅ ,b 4 ^.Q u ∈ N(u A S αA αA ) %' D z* ;8 4 ^.Q V A , A , · · · , A , 1 ≤ r ≤ 2k #* −−−[ −−−−− [−−−− [ −−→ g ) \ (A S A S · · · S A S αA S αA S jR B> A A · · · A z N(u S g )\(A S A S · · · S A S αA S αA S · · · · · · αA ) 6= ∅ } ,b 4 ^.Q u ∈ N(u S αA ) R 0 4 ^.Q V 1

1

1

1

2

1

2

1

2

1

2

1

2

c 1

2

1

1

2

r

2

1

r

d 1

1

1

2

2

r

Ar+1 = {ud1 , g1 (ud1 ), · · · ,

m−1 Y

gm−i (ud1 )}

i=1

αAr+1 = {αud1 , αg1 (ud1 ), · · · , α

m−1 Y

gm−i (ud1 )}

i=1

A 7^= r+1

FR

r+1 S j=1

Aj

7^=

m−1 Y −−→ Ar+1 = ud1 , g1 (ud1 ), · · · , gm−i (ud1 ).

OK%

i=1

1

2

r

1

1

1

2

1

r

r

1

2

2

Riemann

A

Hurwitz

8Æ(O m

175

− −−→ r+1 [

j=1

−− → r− [ −−→ Aj = Ai ; Ar+1 . i=1

9L g)=( N(u 1

k [

Aj )

j=1

[

(α

k [

Aj )

j=1

g) 7 V A F u u0 RV−−k−→G CK i; #j R B 4 ^.Q _ N(u S A ^= OK [ A R 0 QH u = % k

e 1

k

j=1

1

u1

k

j

j

j=1

1

̺u1 = (C)(αC −1 α),

1 C = (ua1 , ub1, · · · , ue1; g1 (ua1 ), g1 (ub1 ), · · · , g1(ue1 ), · · · ,

m−1 Y

(ua1 ),

i=1

m−1 Y

(ub1 ), · · · ,

i=1

m−1 Y

(ue1 )).

i=1

d q/ QH ui ∈ O1, 1 ≤ i ≤ k h ∈ G ' ; h(u1) = ui } R 0 QH ui =

̺ % ui

̺ui = ̺hu1 = (C h )(αC −1 α−1 ).

}FP O 1 G u V (Γ) = G 9 9L 1

(

k Y

̺ui )G =

i=1

k Y

̺ui .

i=1

+UD R0G 9 O , · · · , O 7QH= ̺ L 2

v1 , ̺v2 , · · · , ̺vl , · · · , ̺w1 , ̺w2 , · · · , ̺wt

s

(

l Y

i=1

̺vi )

G

=

l Y

̺vi .

i=1

·····················

8!O H'^5<GI1 /

176

:

(

t Y

Ěşwi )G =

i=1

PR01{ P=(

k Y

Ěşwi .

i=1

Ěşui ) Ă&#x2014; (

i=1

t Y

l Y

Ěşvi ) Ă&#x2014; ¡ ¡ ¡ Ă&#x2014; (

i=1

t Y

Ěşwi )

i=1

}FP O , O , ¡ ¡ ¡ , O 1 G u Î&#x201C; = CG 9 L 1

2

s

PG = (

k Y

i=1 k Y

= (

Ěşui )G Ă&#x2014; ( Ěşui ) Ă&#x2014; (

i=1

l Y

Ěşvi )G Ă&#x2014; ¡ ¡ ¡ Ă&#x2014; (

i=1 l Y

t Y

Ěşwi )G

i=1

Ěşvi ) Ă&#x2014; ¡ ¡ ¡ Ă&#x2014; (

i=1

t Y

Ěşwi ) = P.

i=1

R 0 D M = (Xι,β , P) } G % M =W 3k

â&#x2122;Ž

j 3.2 Î&#x201C; {â&#x201E;&#x201E;m bj G AutÎ&#x201C; G 7a Î&#x201C; {( j',j=, H w ' Z`H7B â&#x2C6;&#x20AC;v â&#x2C6;&#x2C6; V (Î&#x201C;), 7 G G hvi 7K

C )p7/ 3.1(i) # y<1D = J /u R / 3.1(i) ! { 7 3 ' Q H = rs zq # G %$ 1 M = >[ W 3k â&#x2122;Ž 5% yk =V ksl1 yk F R / 3.2 , ;8 ,KĂ&#x2020;\ Fu 3.1 Î&#x201C; {â&#x201E;&#x201E;m bj Î&#x201C; ' â&#x201E;&#x201E;mK H w G v{ a Î&#x201C; {( j',j=, H w

Fu 3.2 B F n â&#x201E;&#x201E;ma n SK j{( j'0 /,j M . & Z { M ' H w

R / 3.1 [ 3.2 ( B =W 3 V k%,>% =D W 3k =

D y< /Q [6] 7, Gardiner > p ! {Bu y< z[%X ^ G u = G v }av " u%&, % =d ~ D v

n

C >, (R%& Î&#x201C; >q/%;V P 5w }V e F n Z > ~% Î&#x201C; = P - V X = E(Î&#x201C;) R 0 X 7=V A rL v 4 P z A = 1 Î&#x201C; 4.

ι,β

Riemann

A

Hurwitz

8Æ(O m

177

= P - V d P D =W 3k 2oLKEOK

j 4.1 Γ {℄m bj G AutΓ w ∀v ∈ V (Γ), 7 G G hvi × hαi B 4 y t. X / Sprp? - P ' G . A w '. A(P ) p

Q

}

α,β

v

[|v G ||v ∈ V (Γ)] | |G|

Æ [a, b, · · ·]

d1

Y

|G|||A||A|

' P 0t F C )p 1 {k 7= %&C # OK d ∀v ∈ V (G) L |G| = |G ||v | 9 | | |G| i L a, b, · · ·

v

|v G

G

[|v G ||v ∈ V (Γ)] | |G|

NF7/ 3.2 #k G u 4 ^.Q _ X = C1, }= ∀x ∈ X L |G | = 1 P Vk G u _ A(P ) = C J /z A ∈ A(P ) }FP G ≤ AutΓ 9 ∀g ∈ G,A ) ∈ A(P ) A ⊆ A(P ) { $ G u A(P ) = C1 C = i 2o ,^C G d A(P ) 7= 4 ^.Q[s + d ∀x, y ∈ A(P ) R0 x ∼ y 2 VZ 2" u g ∈ G ' ; x = y )p |G | = 1 |x | = |G| # G u X C = q℄%yG 9= t ^ y% |G| NFP G u A(P ) = C1 C = 9 # G u A(P ) C = q℄%yG 9= t ^ "% |G| J / A(P ) 7 2L |A||A| & 4 ^.Q i L α,β

α,β

x

g

G

g

G

x

α,β

|G| | |A||A|.

♮

bv4 P % Γ 7E=Æ IV) P 2 =V }LKE &LC = Æ\ Fu 4.1 Γ {℄m bj G AutΓ w ∀v ∈ V (Γ), 7 G G hvi × hαi Bj Γ / 3 b F{ 2 '+. T r p

_

DBj Γ /

_

|G| | (l|T r2 |, l = |T | =

3 b'+. T r1 p

v

2

|T | ≥ 1, T ∈ T r 2 , ).

2

8!O H'^5<GI1 /

178

:

e _

|G| | (2l|T r 1 |, l = |T | =

|T | ≥ 1, T ∈ T r 1 , ).

2 φ(i, j) M /

[ , G {,j M ' H w

f F{ j ' F p (a, b, · · ·)

d1

d1,j

{

{ i h / i

|G| | ((2i − j)φ(i, j), i, j ≥ 1)

' P t F Fu 4.2 Γ {℄m bj G AutΓ w ∀v ∈ V (Γ), 7 G G hvi × hαi T r {D. T , p tl

a, b, · · ·

d1j _F{ ' Γ

/

l

v

|G| | (2ltl , l ≥ 1),

G DmF

Fu 4.3 Γ {℄m bj G

p

AutΓ

d 1j { '60mF :C EO\ 2o , ;8D

w ∀v ∈ V (Γ), 7G G

v

hvi × hαi

|G| | (2ivi , i ≥ 1),

vi

Γ

/

i

Riemann

W 3 = Q

j 4.2 Γ {℄m bj (i) G AutΓ 7( j{ Γ ',j M, g(M) ≥ 2 '=, H w

p |G| ≤ 84(g(M) − 1) (ii)

G

AutΓ

7( j{ Γ ',j M, g(M) ≥ 2 ' H w

p |G| ≤ 168(g(M) − 1),

g(M) {,j M

' l C R 0 D M = 4y ν(M) V4yv φ(M) % ν(M) =

1 X iνi ν(M) i≥1

Riemann

A

Hurwitz

8Ă&#x2020;(O m

179

Ď&#x2020;(M) =

1 X jĎ&#x2020;j Ď&#x2020;(M) jâ&#x2030;Ľ1

1 ν(M) Ď&#x2020;(M) Ď&#x2020;(M) [ Ď&#x2020;j NM* M 7=QH I vI % i =QH I [ % j = vI ) }L ν(M)ν(M) = Ď&#x2020;(M)Ď&#x2020;(M) = 2Îľ(M) 9 # ν(M) = 2Îľ(M [ Ď&#x2020;(M) = ν(M ) 2Îľ(M ) Ď&#x2020;(M )

)p Euler 0)

ν(M) â&#x2C6;&#x2019; Îľ(M) + Ď&#x2020;(M) = 2 â&#x2C6;&#x2019; 2g(M)

1 Îľ(M), g(M) NM* D M = EI[ % }L Îľ(M) =

2(g(M) â&#x2C6;&#x2019; 1) .

2 2 (1 â&#x2C6;&#x2019; ν(M â&#x2C6;&#x2019; Ď&#x2020;(M ) ) )

b I k = â&#x152;&#x2C6;ν(M)â&#x152;&#x2030; [ l = â&#x152;&#x2C6;Ď&#x2020;(M)â&#x152;&#x2030; }L Îľ(M) â&#x2030;¤

2(g(M) â&#x2C6;&#x2019; 1) .

1 â&#x2C6;&#x2019; k2 â&#x2C6;&#x2019; 2l

5% 1 â&#x2C6;&#x2019; â&#x2C6;&#x2019; > 0 # k â&#x2030;Ľ 3, l > & K [ # 1 â&#x2C6;&#x2019; 21 VZ 2 (k, l) = (3, 7) (7, 3) # > W 7 i L 2 k

2 l

2k kâ&#x2C6;&#x2019;2

2 k

â&#x2C6;&#x2019;

2 l

=`)' %

Îľ(M â&#x2030;¤ 42(g(M) â&#x2C6;&#x2019; 1)).

#

)p R / 4.1 # |G| â&#x2030;¤ 4Îľ(M) # zk G %>[ = } |G| â&#x2030;¤ 2Îľ(M). 9 |G| â&#x2030;¤ 168(g(M) â&#x2C6;&#x2019; 1))

Vk G %>[ = } |G| â&#x2030;¤ 84(g(M) â&#x2C6;&#x2019; 1)).

> W 7 2 VZ 2 G = AutM, (k, l) = (3, 7) (7, 3) d Riemann av =W 3k L

Fu 4.4 B l g â&#x2030;Ľ 2 ' Riemann S p

â&#x2122;Ž

8!O H'^5<GI1 /

180

:

4g(S) + 2 ≤ |Aut+ S| ≤ 84(g(S) − 1)

Æ 8g(S) + 4 ≤ |AutS| ≤ 168(g(S) − 1),

C

)p R / 4.2 [Æ\ 2.2 # |AutS| [ |Aut S| = Q Pu ! {>K Q uq/%& %% g ≥ 2 = Riemann av 2o3z%& d ~ D M = (X , P ) 1 k = 2g + 1 wK +

k

k

k

Xk = {x1 , x2 , · · · , xk , αx1 , αx2 , · · · , αxk , βx1 , βx2 , · · · , βxk , αβx1 , αβx2 , · · · , αβxk }

Pk = (x1 , x2 , · · · , xk , αβx1 , αβx2 , · · · , αβxk )(βxk , · · · , βx2 , βx1 , αxk , · · · , αx2 , αx1 ).

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# d qâ&#x201E;&#x201E;%&V R [av S, g(S) â&#x2030;Ľ 3 L |Aut+ S| â&#x2030;¤ 42(g(M) â&#x2C6;&#x2019; 2)

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> > W 7 2 VZ 2 > QH % 3, vt% 7 u$ QH % 7 i vt% 3 V%V R [ =d ~ D + D 1 " u z " u1 V R [[r+U L5S h & 4. Bujalance u/Q [3] 7 ( BV R [ = Klein av y W 3 =`) L V Riemann av = [r+U [%X _ _ = G ^ d j>rL )d ~v =D O 3v 4 PC> s Klein av1L/0 = %&1q aP` [1] R.D.M.Accola, On the number of automorphisms of a closed Riemann surface, Tran.Amer.Math.Soc.,vol.131,398-408(1968).

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[2] N.L.Biggs and A.T.White, Permutation Groups and Combinatorial Structures, Cambridge University Press (1979).

[3] Emilio Bujalance, Cyclic group automorphisms of compact non-orientable Klein surfaces without boundary,Pacific J. Mathematics, vol.109, No.2,279-289(1983).

[4] B.P.Chetia and K.Patra, On metabelian groups of automorphisms of compact Riemann surfaces, J.London Math.Soc, Vol.33,No.2, 467-472(1986).

[5] Hershel M.Farkas and Irwin Kra, Riemann Surfaces, Springer-Verlag New York inc(1980). Ë&#x2021; an Ë&#x2021; [6] A.Gardiner,R.Nedela,J.Sir´ Ë&#x2021; and M.Skovera, characteristion of graphs which underlie regular maps on closed surfaces, J.London Math. Soc.(2)59(1999),100108.

[7] W.J.Harwey,Cyclic groups of automorphisms of compact Riemann surfaces, Quart J.Math. Oxford, vol.17, No.2, 86-97(1966).

[8] G.A.Jones and D.Singerman, Theory of maps on orientable surfaces, Proc.London Math.Soc. (3)37(1978),273-307.

L ) I _ _ D \ * :_ 2001 [10] L ) _ _ D [L : z E ) * :_ 2003 [9]

[11] C.Maclachlan, Abelian groups of automorphisms of compact Riemann surfaces, Proc.London Math.Soc., vol.15,No.3,699-712(1965).

[12] A.M.Macbeath, Generators of the linear fractional groups, Proc.Sympos.Pare.

Math.,XII(1969).

[13] Kenji Nakagawa, On the orders of automorphisms of a closed Riemann surface, Pacific J. Mathematics, vol.115, No.2,435-443(1984).

[14] T.W.Tuker, Finite group acting on surfaces and the genus of a group, J.Combin.

Theory, Ser.B34(1983),82-98.

[15] W.T.Tutte, What is a Map ?, in New Directions in the Theory of Graphs( ed.

by F.Harary), Academic Press (1973), 309-325.

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1.A census of maps on surfaces with given underlying graphs, A Doctorial Dissertation, Northern Jiaotong University,2002.

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6.On automorphism groups of maps, surfaces and Smarandache geometries, Scientia Magna, Vol.1(2005), No.2,55-73.

7.Parallel bundles in planar map geometries, Scientia Magna, Vol.1(2005), No.2,120-133.

8.On algebraic multi-ring spaces, arXiv:math.GM/05010478.

9.On algebraic multi-vector spaces, arXiv:math.GM/05010479.

2006

1.with Yanpei Liu and Erling Wei) The semi-arc automorphism group of a graph with application to map enumeration, Graphs and Combinatorics, Vol.22,

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4.On algebraic multi-group spaces, Scientia Magna, Vol.2,No.1(2006), 64-70.

5.On multi-metric spaces, Scientia Magna, Vol.2,No.1(2006), 87-94

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8.Combinatorial speculation and the combinatorial conjecture for mathematics, arXiv: math.GM/0606702 and Sciencepaper Online:200607-128, will also appear in Scientia Magna.

9.A multi-space model for Chinese bidding evaluation with analyzing, arXiv: math.GM/0605495, will also appear in Scientia Magna.

10.Smarandache

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Linfan Mao is a researcher in the Chinese Academy of Mathematics and System Science, also a consultor in the Guoxin Tendering Co., Ltd, in Beijing. His main interesting focus on Smarandache geometries with applications to other sciences, combinatorial map theory and differential geometry. Now he has published more than 35 papers and 2 monographs on Smarandache geometries, combinatorial maps, graphs and operation research.

He was born in December 31,1962 in Deyang of Sichuan in China. Graduated from Sichuan Wanyuan School in July, 1980. Then worked in the China Construction Second Engineering Bureau First Company, began as a scaffold erector then learnt in the Beijing Urban Construction School from 1983 to 1987. After then he came back to this company again. Began as a technician then an engineer from 1987 to 1998.

In this period, he published 8 papers on Architecture Technology, also the co-author in three books on architecture technology and 5 papers on Graph Theory. He got his BA in applied mathematics in Peking University in 1995 learnt by himself.

In 1999, he leaved this engineering company and began his postgraduate study in the Northern Jiaotong University, also worked as a general engineer in the Construction Department of Chinese Law Committee. In 2002, he got a PhD with a doctoral thesis A census of maps on surface with given underlying graphs under the supervision of Professor Yanpei Liu. From June, 2003 to May, 2005, he worked in the Chinese Academy of Mathematics and Systems as a post-doctor and finished his post-doctor report: On Automorphisms of maps, surfaces and Smarandache geometries. Now he is a researcher in the Chinese Academy of Mathematics and Systems.

Abstract: The mathematics of the 21st century is the combinatorization with its generalization for classical mathematics, also a result for mathematics consistency with the scientific research in the 21st century. This collection contains 10 papers finished by the author or the author with other mathematicians for introducing mathematics of the 21st century, including the combinatorial conjecture for mathematics, Smarandache multispaces, map geometries, enumeration of maps and a application of multi-spaces to bids evaluation system in China, which can be seen as a combinatorial speculation for classical mathematics, are also benefit for researchers working in mathematics of the 21st century.

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