Mathematical Combinatorics & Smarandache Multi-Spaces

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Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO

Chinese Branch Xiquan House 2010

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Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO

Chinese Branch Xiquan House 2010

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 100190, P.R.China. J.Y.Yan, Graduate Student College, Chinese Academy of Sciences, Beijing 100083, P.R.China. W.L.He, Department of Applied Mathematics, Beijing Jiaotong University, Beijing 100044, P.R.China.

Copyright 2010 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: 978-1-59973-032-5

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Abstract: This paper historically recalls that I passed from a construction worker to a researcher in the post-doctoral stations of Chinese Academy of Mathematics and System science, including how to receive the undergraduate diploma in applied mathematics of Peking University learn by myself, how to get to the Northern Jiaotong University for a doctorate in operations research and cybernetics studies with applications, and how to get the post-doctoral position, which shows that there are not only one way to growth and success for students in elementary school. Particularly, it is valuable for those students not getting to a university, or not getting to a favorite university.

Key Words: Construction worker, learn by oneself, growth and success, doctorate.

AMS(2000): 01A25,01A70

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Abstract: This paper historically recalls each step that I passed from a scaffold erector to a mathematician, including the period in a middle school, in a construction company, in Northern Jiaotong University, also in Chinese Academy of Sciences and in Guoxin Tendering Co.LTD. Achievements of mine on mathematics and engineering management gotten in the period from 1985 to 2006 can be also found. There are many rough and bumpy, also delightful matters on this road. The process for raising the combinatorial conjecture for mathematics and its comparing with Smarandache multi-spaces are also called to mind.

Key Words: Student, worker, engineer, mathematician, Smarandache geometry, Smarandache multi-space, combinatorial conjecture on mathematics.

AMS(2000): 01A25,01A70

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Abstract: How to select a research subject or a scientific work is the central objective in researching. Questions such as those of what is valuable? what is unvalued? often bewilder researchers. By recalling my researching process from graphical structure to topological graphs, then to topological manifolds, and then to differential geometry with theoretical physics by Smarandache’s notion and combinatorial speculation of mine, this paper explains how to present an objective and how to establish a system in one’s scientific research. In this process, continuously overlooking these obtained achievements, raising new scientific objectives keeping in step with the frontier in international research world and exchanging ideas with researchers are the key in research of myself, which maybe inspires younger researchers and students.

Key Words: CCM conjecture, combinatorics, topology, topological graphs, Smarandache’s notion, combinatorial theory on multi-spaces, combinatorial differential geometry, theoretical physics, scientific structure. 1 2

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:u# ' oK\ $Kx$K $ = uyX℄ uN℄ z Z s$n"` x#3n$g $ByxN rN T31XZÆ0 #b = ZLw =, =e = #[? y5Z $℄ $K %

4 A

gl q℄ – A Foreword of SCIENTIFIC ELEMENTS, 37-38 A Foreword of SCIENTIFIC ELEMENTS Linfan Mao (Chinese Academy of Mathematics and System Science, Beijing, 100190, P.R.China)

Science’s function is realizing the natural world and developing our society in coordination with its laws. For thousands years, mankind has never stopped his steps for exploring its behaviors of all kinds. Today, the advanced science and technology have enabled mankind to handle a few events in the society of mankind by the knowledge on natural world. But many questions in different fields on the natural world have no an answer, even some looks clear questions for the Universe, for example, what is true colors of the Universe, for instance its dimension? what is the real face of an event appeared in the natural world, for instance the electromagnetism? how many dimensions has it? Different people standing on different views will differently answers these questions. For being short of knowledge, we can not even distinguish which is the right answer. Today, we have known two heartening mathematical theories for sciences. One of them is the Smarandache multi-space theory came into being by purely logic. Another is the mathematical combinatorics motivated by a combinatorial speculation, i.e., every mathematical science can be reconstructed from or made by combinatorialization. Why are they important? We all know a famous proverb, i.e., the six blind men and an elephant. These blind men were asked to determine what an elephant looked like by feeling different parts of the elephant’s body. The man touched the elephant’s leg, tail, trunk, ear, belly or tusk claims it’s like a pillar, a rope, a tree branch, a hand fan, a wall or a solid pipe, respectively. They entered into an endless argument. Each of them insisted his view right. All of you are right! A wise man explains to them: why are you telling it differently is because each one of you touched the

K)

38

:

DnS X,h ?hn| –

Smarandache

K .

different part of the elephant. So, actually the elephant has all those features what you all said. That is to say an elephant is nothing but a union of those claims of six blind men, i.e., a Smarandache multi-space with some combinatorial structures. The situation for one realizing the behaviors of natural world is analogous to the blind men determining what an elephant looks like. L.F.Mao said Smarandache multi-spaces being a right theory for the natural world once in an open lecture. For a quick glance at the applications of Smarandache’s notion to mathematics, physics and other sciences, this book selects 12 papers for showing applied fields of Smarandache’s notion, such as those of Smarandache multi-spaces, Smarandache geometries, Neutrosophy,

to mathematics, physics, logic, cosmology.

Although

each application is a quite elementary application, we already experience its great potential. Author(s) is assumed for responsibility on his (their) papers selected in this books and not meaning that the editors completely agree the view point in each paper. The Scientific Elements is a serial collections in publication, maybe with different title. All papers on applications of Smarandache’s notion to scientific fields are welcome and can directly sent to the editors by an email.

4 A

gl q℄ – & } RMI , 39-44

bh

4 M f-2 RMI d/J (qZ! 8^ = "8 83-1 a = h)

x JE ox_x#g 2- #=â„„Wq)Nu D $&e qN â„„ K0Nâ„„ h7 E #qT2-E |â„„ #bx7 E N 2 -â„„ 3\&e ?x JE 0x)bN&e x=|â„„ N$1, cos x, sin x, cos 2x, sin 2x, ¡ ¡ ¡ , cos nx, sin nx, ¡ ¡ ¡ ox q `â„„ f (x) N ` 2Ď€ #yx 2Ď€ i#XZj n 0bÆ q!-*p$ 1 a0 = Ď€

K0)7 E $

Z

Ď€

f (x)dx,

âˆ’Ď€

1 bk = π

Z

1 ak = π

Ď€ âˆ’Ď€

Z

Ď€

f (x) cos kxdx, âˆ’Ď€

f (x) sin kxdx, (k = 1, 2, 3, ¡ ¡ ¡) ∞

a0 X + (ak cos kx + bk sin kx). 2 k=1

8 Y `â„„ XZ xNÆ Jâ„„ QH#g0 `â„„ #[?_X ZK0x|E B* Jâ„„ # 5u1x$8 [a, b] E Y `â„„ F (x)# N â„„ F ∗ (x)#n 8-` x ∈ [a, b]#( F ∗ (x) = F (x)# ? F ∗ (x) x E #Z (b − a) ` â„„ T 1 D6 [âˆ’Ď€, Ď€] Eq f (x) = x KI x|E #3nXZT J f (x) $ [3]

2

f ∗ (x) =

(

x2 , 2

x ∈ [âˆ’Ď€, Ď€],

(x − 2kĎ€) , x 6∈ [âˆ’Ď€, Ď€],

b #k xn #n |x − 2kĎ€| ≤ Ď€ #A f ∗ (x) xZ 2Ď€ ` `â„„ #ez jkQGz 1 $ -Ck5Æ % #1, "1 1985 1

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.... ... ... ... ... ... ... ... .. −5π

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.... ... ... ... ... ... ... ... ..

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6

.... ... ... ... ... ... ... ... .. O π

.... ... ... ... ... ... ... ... ..

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-

Smarandache

x

( 6 l EK0x JE $ 1

bA#3nXZq f ∗ (x)

Z 1 π 2 2 f ∗ (x)dx = x dx = π 2 , π −π 3 −π Z Z 1 π 1 π 2 (−1)n an = f ∗ (x) cos nxdx = x cos nxdx = 4 × , π −π π −π n2 Z Z 1 π 1 π 2 bn = f ∗ (x) sin nxdx = x sin nxdx = 0. π −π π −π 1 a0 = π

9s

Z

Ï€

∞ X 2 (−1)n f ∗ (x) = π 2 + 4 cos nx. 3 n n=1

YE f ∗ (x) Wn #6 [−π, π] E#1

∞ X 2 2 (−1)n cos nx. f (x) = f ∗ (x) = π + 4 3 n n=1

bA#3n1 |) f (x) x JKIp 2 x JE E +h#xpzlq1x$ p2- ℄ f (x) J - `℄ f (x) ∗

2- f (x) ∗

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f (x)e−pt dt.

L[f (x)] = F (p) =

0

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~Y#8-N N ` ℄Q3 dn y dn−1 dy + a + · · · + + an y = f (x), 1 dxn dxn−1 dx

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−pt

−1

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p2 + 2pY + 2Y = ii)

Ez n Q31 $ Y =

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(p2

1 p+1 1 = − . + 2p + 2)(p + 1) p + 1 (p + 1)2 + 1

x|$g 1 y(x) = e−x − e−x cos x = e−x (1 − cos x).

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n

u(t) =

X

1

2

n

uiti

i≥0

o eut =

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ti ui , i!

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X

bui ti+2 .

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7 , {u } L>N#1

X i≼0

(ui+2 − aui+1 − bui ).

n

(1 − at − bt2 )u(t) = u0 + (u1 − au0 )t.

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u(t) =

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2

u0 + (u1 − au0 )t . 1 − at − bt2

#A

= (1 − r1 t)(1 − r2 t) 1 u1 − au0 + r1 u0 u1 − au0 + r2 u0 u(t) = − . r1 − r2 1 − r1 t 1 − r2 t

q u(t) KI0vE #1

un = (B − aA) Ă—

K .

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= aun+1 + bun , u0 = A, u1 = B u(t) =

Smarandache

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Wn #8 ~ 5 Xt f0\|I , {F √}#h F =√F + 1− 5 1+ 5 F #F = F = 1 #9 A = B = 1, a = b = 1 r = , r = , BZ 2 2 n

n−2

2

1

Fn =

1

r1n+1

r2n+1

− r1 − r2

√ !n+1 1  1+ 5 =√ − 2 5

n

n−1

2

 √ !n+1 1− 5 . 2

bxN ,)% p5# , {F } -Nh(xn #&? F "xqT7 {lq n

wT z

$ kG Z [1] $QH"u [2] T\ ' rN E [3] ;/k$ $N $℄B Q [4] &I$r'Yqx~~ $ +

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6| po_

d/J (H â„„ ' E # #r[#100045)

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u z # â„„_ f â„„ V % %k#;U > w % V% > *â„„ R a C

0P# V Æ ℄ R aY0 ' %#{ F.Y u p N #℄ R > q* p # { F'rpF v p| 5,yrYF > . u W } #z # Y.Y u qpR lYK3q l !<v$9S F#_ i#l g9_ #l # 9 #' J #' E 0P Ck :

P

$

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! C iC T ! _ = = C RB = i C C C

i

T

Abstract: This paper historically recalls each rough step that I passed from a construction worker to a professor in mathematician and engineer management, including the period being a worker in First Company of China Construction Second Engineering Bureau, an entrusted student in Beijing Urban Construction School, an engineer in First Company of China Construction Second Engineering Bureau, a general engineer in the Construction Department of Chinese Law Committee, a consultor, a deputy general engineer in the Guoxin Tendering Co., Ltd and a deputy secretary general in China Tendering & Bidding Association, which is encouraged by to become a mathematician beginning when I was a student in an elementary school, also inspired by being a sincerity man with honesty and duty. This paper is more contributed to success of younger researchers and students.

Key Words: Construction worker, entrusted student, construction management plan, construction technology, construction management, project bidding agency, teacher in bidding.

n[D&88

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V ?y

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Qp B4/*( 0i% 1981 & 12 *# Qp B4N*( kM X%#Z_<%3pN D v& 12 * 25 2#3 X)%I# =eD*( 5ÆNj{|)vi6SA

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48

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℄E| %I k$\ Cyy #03 %3 N ez4#6 xqL4 xO?R xg B #3NN ) I#: Ck ; M2C# Wq#3y~ |Ni 7$n V` i+ M A *W u _ R i E'b 7I6 $i` yDo ` FJyao) i`8 3` B x~ $ \ # yWG> qs(d#? NJ( q yWi e – bxCvi =,"T ,bN\y 2|n,8# G #Fy 2|n, - 2005 & 10 *#v34Z$| $i`\+ Qi#b) W! t#3(s$m23|C

{lg)8C *& 1982 & #!TQ oQm#3 (o) bNi`# =si X%I , x$m>N#.x 0)gN %Æ#℄6 e% #D2W` # .x >N &I$rS=#Z ${ x [uK C %I biNueD *( 5ÆW^^ $)e{ B6*( 3 X%IixZ Qp B4/*( 5Æ { #Yi Qp B4/*(W!'m#h XZ x >N#} R 7I g 0) 3A 4 XNZ Q#Z g[5%bÆye uyDN\ wI 1983 & 7 * Q{#3Æ)$ * uY6WwK#0Y Q XÆ Q# ℄ X Z#$|SA't % #3NzT 2[5%&Nz:2 Q0:<2* z Rye&# Q℄ *z #3"T)D ℄ ` 10 =℄#e $Q) 110 ℄# _℄)#XZ" NB e k$ $K $.nx $) bÆ7$#3 T [u2[5%# 2Zk$D bN&1*8M1DT )#3 h |k$ D qs# 8$7 ) vi6')k$-~ 8gs q #C"3 ℄ 6')^ h0\#^ %ZNBk$D 1* # Qp B4N*( U .~pN$r Ng# (CnD2" C 3 K , # 9| r%I xp p 6 &I

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K .

`+& $K h $K 6U #3ye x6U 4%$K $# xO XbZ" C bi*( G X$K#sZY g 0 ,y=# x*( v) . $4 Q 3:℄6 N #bA - )|U .~pN$r%I xp &I$K 4% 1983 & 9 * 9 2#3oN)%, xNj# E)~SAI U ,&#s~b)%, xm2{|)W!~SA :|U Qp B4N*(:| B`#3 2~*( v|mI Nd I |$rO|#Io)"C\\- bu&y=,03# X)=Z Q# jgjZ(E)D ℄ `th N Bk$D 3 3 5Cn#1980 & Z Q0:#3XZE M %r#t3y $`#"O $G (x4z%3 4z = *m&I#h ZD xo & $Z#NQzD J # 10-20%#Xu3yxn `I\#yLzv&5 NQz#3Fx & 4#1xE)D `syN ;#xk 8k$rm# 1DM$rxk$`X#) 3e#bF oxQ rX\#6[Q~ Q W ?0

.

s 1983 & 9 * 10 2#3op{|U .~pN$rp 83-1 9#I N "C\# XU ~ &I$ruNQ D Gg s$NjIo) `+& &I$ K bB$rlSEx~U p %3$%℄D{ NB$r#vi U ~ pN" %#$rG)r o D ℄~ k({! U p %3$% `& x#Bg=& #C 2010 &B#3 0 )U p %3$% WU~ #y sxkb1xMMu > U .~pN$rxNBZ j &I$r#BZ N$&$ # 3\E(x ℄3# ℄ ℄3 D rNu 9&6Y#3 N$& x 0i|$rm℄#`)$Æ)yx4[u[5%)#4%{uy℄ N * i^#$r os$1s~3 $ > 0:Wng# \1 yxm℄#t3 x℄7 m si# 5')k$ 8gs #3|U bB $r$K)#(C 3 IG)C6U s$#Zf36$K $T3 0\ gÆ~ C f)$)#N)xU %Ik$ =: # N)xU %I $% + # `3h kTz#h 3! QJ## Cy)d F h |C $ ?=: A[ 1984 &RLIo#Nx{{23 $$ K#bFx3 { 1 Æ0U ~ ~ 7$Q BQV #sx Q U Qxqk$vs2-\ D h

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Qp B4N*([uK C #6 /P8QzLzv N D h $Kp ~* p e%K ℄3# WYD36SA't %3 T $&%3pN zv#7Nxp 4z p ~* p 0T.B0T# xe% %[ e%QH#3 HW! ) = .m# ux [B { s$Dg =#8 Q Æ > *( m mN %3e%QH#6' t r n %Æ T# Ax T&4℄6 >z#n s$h hkT#ys~n CA #3As~DlSEj2+*( 5n Ix#6 Q.x6 z# j{ n

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x) d{ [℄N#F x)l ℄N K %[#8Kqe% % %`# _%3e%0\j|)N Ix#si7IFIo$% x $ s >r,℄N { [P8 # ? Ae% | NuK 0\ e% vNPeÆ #E DI e%T3 K 0\ C NPg" #%I 8a8 -#3bio60Y X”> ℄ hNH B> ? 2 ” NN .Q 63 D Q#% s D3 ) 8 6?oik#?vi x 286 386 p 1989 & 3 `^#3o6U +KkR% #6% E! XZM|W r$\62 i[z.m Q;T#t6 ;+ N #h vJ S# r)y yWY#ZS` |) 1989 & #n pK |)W!3\e%| )#36Y K C %IF3\ #x Æ >4* #3 (|)QN %

#T [uK C %I !bCje U f$%%3#x3 ,p e%K V;k0 # K P N h 1987 & #QWIo?e; 4 v4K #e%-?{!eH#t8k Q kI{"#-?0&yH 6bN vQ#1989 & # Qp B4N*(=I 5 )U f$%N`b r)%3 pN-?#bxN e%%`g℄ S# si Kq$\6 1991 & L I$# x 0 l C pN N h vE,636b % |?e%K C #?pNr)NQ#og N)U .~ pN$r℄3 $ `I s$ ~$<3Ge%#o EQkÆ# B`% v E > DY3G" | DY)e1Uz %!℄#% |) %\)|mIE ey!℄|) wI Q/ HW*( BZy !# s$Njl NPr) s` )_e%, w*( !℄A0)3 %I#Fn3s~)Kq%3 %#Ee e%P #F 6e%T3 t % C de#?brx ISO9000 %. q℄N m &e bN&#U ~pN" %6 ~IKr)%3 z %C J%I#U f$%%3,8)b h #e NhQm#1xC `D8QEQm#yD `k&si#QEFD8C ` Qm#sD QD Zb r)G h ,s ~n CgG nyb #%3I℄C % K G s G`0Yb 4* Y#3T )N l#(eLsDEQE b Zb >N#& ,8 G b V #Y,xE8 N-hGa#t:8lgD) kQm `& ~

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BZr,#e ) Ay D QH#bAkk _) > [* NZ L J ! 8e%/p P8#Xu[ 9S 9Sl C `KED ) k% ℄#n38Y Nu0i Ne /P8W!2 H $\kRe%T3 # Æ<>y"*( x 10 * 30 2#`o Ov2B ,. 2Q 6 C .#CT*( z OaX *( N$MXZN2Q 8 C .#t v2 E ,. |)2Q 11 C . 9&6<>EX ) 5K,#t B` E;b #DYy *( | $#s 6 Q{ $`*( y(;#bx NZ %u9#} Y*( z O s {℄B h# NP% s C%I CH ℄ ?- #bNÆ*( 6y B * | E# |) E + y sE℄ *( 1990 & #N)Nj%I su 53#"U ~ NÆ ~ 7$Q # yxE$#x -[&IV Q 1XZ) x#3nNj| H ~ 4(0 F #0[)Q k oQ xÆ 3" &Ixnx $#3G℄O 6 H ~ 4#&I3Go&I℄A6U k$ Y#%Iu #3Io0Y B &+*ao/*a Q V bN& L# )KqU f$% 1991 &$\ 8$#%3IIoP B& De% G EVh #e 100m y.W Fe%6|)eu23 Q y.W F Ne%QH $Æ, NÆx6 NQ|fN{ [, 6 N,+ ) Er FG Æ y *( #t ee% h .x #siNZ Lx[*=#Lm0\k; NÆx6 zr FG Æ y *( # NP . 6tR |,+ r[#W L+L v # x φ = 25mm Ix #[ z xL+ 3 NZ℄ NP,+) #gIx6 z℄ 5℄ Ne% %#si _fN Nr[ N|℄\x#! x φ = 25mm 6e%T3 D # Q{ hH4Z x#} NZ L m8%30\ biQ Nup K 2-4/Io2-\ φ = 48mm v #" o

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Z Æ L8 .x 1x$ [CIr,#s *( 5x # ? 0N / $℄N !TP8#3NPD)U ~x/D E d[# se D)e%Q/ v%3le 1992 & L# Qp B4 vK w-X %I#h &`I _:&# 3Yi w-0xK # %3 (yx Y#3O G x XE %3 1XZ) |*(K V(0#|?Yh%I su 53#"3 % 3 DbNQl1XZ)#y%b60\&"*(G)4{o6QH &6 K 4 GDyD,Z =)K Z ℄ %3 #31xe N yTC "#= ,CZ X yk#3P%34y%s`#(3 0Y#h 3 `I 5 &# $)suW`I 8 &) b& 3$SA{ W# f|4t:%3 9S{ eÆb C* )NT hÆ0U f$% 100m N ℄NuNPs0(nx vU V ℄r u h * RoU ~x/D %3K C #si A)e N uK #3 Cy H#CF83 NuK P %I N ) O ^ O|)bZK w-X #3 5C*(WD3OO)w-#(C7`NQ#3v isys~Cx%34w-XV" % - G * 3|%34I%# CC * 8VCi#C 53#"Z 3 ℄ E%3 #MOOl s~OO x %3 #Æ0 K P %Iv&_<Z gm G& #x|*(N)su 53#3 s~ Zw-XV Nu:℄u E" Zw-XV#N*( Ax pQ0 #C( $)suZ %3 #h Cn`Ii^℄3 =&#si# 3*( U%3 p Xx(N* (! he| x#36%34K 4 Z ##"T) $)su 6XV% E#Æb vV ) *(OO DZ , ~* # j3 OO~*# I)3bu&6e%K P QzÆ0 Nu D%I#s". N*(OO = ), #3 gm_<%34Z ℄ gm#lqs`3Z ℄ %3 m|C `k# XXV" % N*(! g℄2B#h b XXV%{# 3

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yTx Z #x Z 7Iu℄ 5o D6 / #h o u℄ 5x +63 5 3 ,o 3 $2-I , k#?3[u%3C I r 3#M Ix[u $2- !Nr #ye"6$ 4 si- ) bv ' * pF v p| x \#?yik 0:#Ai n , ,\M7k" j ℄ #Ax3 1[N p %,DK|6 # ?lY,\Zz |(oQH b56bg=<i: 7IN\i"$ tm p u~ p t> #Qm~ # m ~ Æ#um~6: ^ #3 m~ # m~ ) ( # w l 3e#blSE x3nj ,#Wnx k , ,\y[ si#bFx3nj , 0 K I# Djj H&Ae~ ` #h &xQW ${ G

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B w > % R ;B t 9 w > R R B Theory of everything +I 5 Jw > % Y z #t W z# b 05# ! #+i} B w H B M- B Y: w > %Bp ^ sU#9 +IGD R B Y n e t > % Np 9 { = z g Y > B #l ) R Y #i} m R z#Y w >Y ℄9 Æ$ #ND C e { !Y_ t > % Np 9 m R z#Y > B r!W#w > %} q 02# B{ M- B 9 YBp# > % O 9 $R I} Z 7℄ m R z# >Y B #+i}SmarandacheJ - B # gl Yz 2} B w > %( 5Y#OB#q℄ R I > B # X D B w Y 5 5 N X| vhYN Nf Y t DaLZ+ R B Y is^ C eYN N j7 N N Z N NC e# \, H} Z w1g > : %H UYKoq l vhN N 9 t q* ( W P 1eY R vVO [16] HY % u

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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

E5=\b Mfl =zGA= >1 2006 % 8 )"_ " Hj# n*J^ } #N 2006 % 3 ) e-print: G \ z A g "200607-91 1

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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$ kRH N#M- N# `^# zGp#SmarandacheG p##.%`^Gp#FinslerGp P% AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50

§1. 1.1.

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%K`^ x m {y#Q$ = x, m = y # = z #AN %K`^XZx = {y#CJj (x, y, z) &"K`^ x m i^{y#6RQ i^g% t#AN "K`^XZ xJj (x, y, z, t) {y %K`^Aeg k z 1.1 qi^MIN g #A, 6 N iZM| KlSEx n N z (section)

z 1.1. JjN i^g , \ \+ljl #, \+ `^ Ez"K`^xN #C , \+ `^x 3 ! #6RXEi^g%#Ax 4 ! , b1x Einstein i `A

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h % y < Wy a ( H% Q~E !#[bij, Æ 5 .1 / ~ bi,.0&y #H5 <1 % 2 ;-p18 0 5 #3 ` 5 I #B Z vih X DY 5 B 8rÆy )i^ 1000 − 2000 & #,.v & 105K#> u.Iok n Ku. ~ HW #H5|g-p N 6 N

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nK,. #B oB 3\ 5 ℄ z K 7| k&!T N5 & #~ R HB` \ H5 %1 v |) y < Z2 ; 5W 9 D 5 3. # b i1 8)H5o 5a ℄ Yin #C g 0)x #,.k& v 3000K [biIo # 5Io 0 #t F } \ 2 2 # x6 N | 0 #6| | 0 #| 0 # | | \) 7 y s ,. "Tj " 2Ax6" R D . N J, R D 0 F<Jo ) i ^ 10 & #nK,. v & 100K#> ,. 1K PF! *F KF1 k>+s? )) i ^ 10 & #,. v & 12K#J` n 03n {A | $ nzA ) i ^ 10 & #nK,. v & 3K# N> ,.& 10 K 1.3. `vxaiQaZs{ N3 N> l < > 2 - l > ~ =Æ> 5 #C 5 e EiY u oQiY d /0 Bgt U BDY)$Æ{ 5u ^ a I x N #b1x Einstein a8No Dirac %5 $ a8Nx{ j N #N<x 2 - > $& ?%5 $ x> A 5I x N #H v U mU ;mU b +ÆI x /0) 5u ^ a I x 3\ I x [E tU Bg&nIo # => $W#H v Einstein \,Nx u N b+Æ 3\ I x #C u N a8No%5 $#p > $ ku N N #C .Z

! DY Theory of Everything !T 80 =& 2 - #0\ N xh | _ A 0\ 6 J a8Nx> ~A N #6 i tN J ? N w % #e Y N> x â„„ z &?%5 $ x> A N #6 5 5 5 #e Y N> x >â„„ z ? A :2S8 2 - #m{), .m4 N u 0\ #â„„ 6 }M}\RY E'M ^ \R #p|Y = }M'â„„q M F w U )}M' F i HY}MY^ \t|Y',YMF A Hâ„„Y 3 ^ E' Einstein P 7eZD J a8 $(pw Y= G^ y Hop BY/wo t $ R = D Y7 pV 1 i d YG k w !g} }7 Y p ) j Q3 #C 8

9

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r $ , C u # +uâ„„ 10 l.y p #}MHG k Re i R = eYG k t 4

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s #Robertson-Walker |) j Q3 N Æ 8- ds2 = −c2 dt2 + a2 (t)[

dr 2 + r 2 (dθ2 + sin2 θdĎ•2 )]. 1 − Kr 2

a n - Friedmann !T=& `.A #Hubble 6 1929 DY #, 3! x N 2 dX 3HW #9Q j Q3 X 3HW 0)> $W D Qk #C e _<g m da > 0, dt

d2 a > 0. dt2

a(t) = tÂľ ,

b(t) = tν,

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p

p 3m(m + 2) 3 ∓ 3m(m + 2) ,ν = , 3(m + 3) 3(m + 3)

ds2 = −dt2 + a(t)2 d2R3 + b(t)2 ds2 (T m )

Q3 4 + m ! ` N< qQ #b sy D 4 !X3H W t x i ^ ( T 8- g Einstein

t → t+∞ − t, a(t) = (t+∞ − t)Âľ ,

3n | N 4 ! X 3HW #h

da(t) > 0, dt

d2 a(t) > 0. dt2

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z 1.4. 0" P 7 M N # P\Ioi ` ^ ! x 11 ! #k R HIo # e 4 Qk!6 B = wS 4 P #?5 7 Qk!A6 B => ?p #b A 03n `M | 4 !~A oM yk 7 ! A 4 !~A I x or Einstein j Q3 #? 7 ! A I x or xYQ3 # ~ Y |Qzb N Q 1.1 M- B Yp }R = O= eq8R = 7 ^ - R Y 4 ^ - R nx 1.1 o ` ^ E " j g m# 3nX Z |TownsendWohlfarth 4 ! X 3HW 7

4

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

b φ(t) =

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K(t) =

m

m+2

S 2 = K m−1 e− m−1 Îť0 t

Îť0 Îśrc , (m − 1) sin[Îť0 Îś|t + t1 |]

b Μ = p3 + 6/m. i ^ ς _< dς = S (t)dt#A X 3HW g m o > 0 C |_< z P 8 l #: m = 7 AHW h 5 3.04 3

d2 S dς 2

dS dς

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[ $| . u# 1.1 l S E y x ?x` ^#~ Y j S $ 0\

} C +MRI > - # H O= ek b- R = 1 ^ZWY -' x l #6Rb A $ ` ^ b6 # D N y x3n2 \ + M | ` ^#Fy x3n6! $ k T ` ^# 66 3 !` ` ^ #j X Zl q (x, y, z) #D y X H [ N ! k 1 5` ^ tJ: TQH N r 0\ $ §2. Smarandache

1 + 1 =?

67& N #3ns~ 1+1=2 6 2 08â„„N #3n s~ 1+1=10#b 10 lSE x 2 h 6 2 08â„„N } $ 08 2 0 o 1#e08 KA 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

P 7 k uk ^ ^ $ &e #3n xN Æ L&! <$ &e ([18] − [20]) { Mpb 0\ # â„„B 1 + 1 = 2 . 6= 2 3ns~ 1, 2, 3, 4, 5, ¡ ¡ ¡ b A /07& N N 6b N #P 7 KI #j - {z 12 T #C 2 T 3#Q 2′ = 3 s A#3′ = 4, 4′ = 5, ¡ ¡ ¡ b A 3n1 |)

si |)

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

1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, ¡ ¡ ¡

b AN u08 p [?6bÆ7& 08â„„NQ #3n} | 1 + 1 = 2 N Y6 #3n$:NQ08 N ? r S #8 ∀x, y ∈ S, x∗y = z # ` &x S Eb6 N 2 r s K ∗ : S Ă— S → S #n

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∗(x, y) = z.

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z 2.1. ` A v QH " ` bÆ8 n x 1 − 1 Q S 8 n z G[S] Y6 #6R3neY| N _< 1 + 1 = 3 08Nu #3nX Z T D 1 + 1, 2 + 1, ¡ ¡ ¡ Bz s qTz { Æ0 Y 2.1 R = J D (A; â—Ś) â„„MRYOC R = j\ Ě&#x; : A → A uXz ∀a, b ∈ A #?N a â—Ś b ∈ A # C R = \RY c ∈ A, c â—Ś Ě&#x;(b) ∈ A # ga Ě&#x; â„„MRj\ 3ny5 â„„ | Z Q> n Nu (A; â—Ś) z G[A] >N N R Q 2.1 _ (A; â—Ś) â„„R = J D # (i) O (A; â—Ś) WC R = MRj\ Ě&#x; # G[A] }R = Euler H 7 #O G[A] }R = Euler H # (A; â—Ś) }R = MR % D (ii) O (A; â—Ś) }R = S>YJ % D # G[A] H O= keY1uâ„„ |A| & =Q #M ^ (A; â—Ś) W ; = # # G[A] }R = S>YJ 2- H- O= ke m R =~ uX Bke7-Y{â„„ z 2- { # 7â„„B 8 ^ #X Z xN Æ ^z QpK DB 08 R #z 2.2 D) |S| = 3 $Æ08â„„N

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z 2.2. 3 X H08z

~ z 2.2(a)

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.

~ z 2.2(b) 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.

8 N ? r S, |S| = n#X Z 6eE n Æ y s 08â„„N b A 3n 1X Z 6 N ? rEsi D h Æ08 #h ≤ n ? | N h- 08â„„N (S; â—¦ , â—¦ , · · · , â—¦ ) 6! n $ #%xr N 08â„„N # â„„ Cx 2 08â„„N N< #3n N Smarandache n- ` ^ 6Q Y 2.2 R = n- J - P#l_â„„ n = m A , A , · · · , A Y 3

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Cx e [ nS{$ ? –Smarandche K

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

Z

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(x ×i y) ×j z = x ×i (y ×j z)

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

Re â„„R = m- J ~ OzG^1 i, 1 ≤ i ≤ m, (R; + , × ) }R = # Re â„„R = m- J Y 2.4 _ Ve = S V â„„R=StY m- J - # % mâ„„ O(Ve ) = {(+Ë™ , · ) | 1 ≤ S F â„„R = J # % m â„„ O(Fe) = {(+ , × ) | 1 ≤ i ≤ k} O i ≤ m} #Fe = zG^1 i, j, 1 ≤ i, j ≤ k G^ ∀a, b, c ∈ Ve , k , k ∈ Fe, ?NzgY %H ^ C # Ë™ , · ) â„„ F WY # - # # & â„„ +Ë™ # # $ â„„ · & (i) (V ; + i

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2

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n

n

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

F = {a , b , c , · · · , d ; i ≥ 1}#3n N k%` ^ i

i

i

i

R− = (V, +new , ◦new ),

b V = {x , x , · · · , x } y d_ i #3n Y x , x , · · · , x x* #C :b 6j% a , a , · · · , a n 1

1

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a1 â—Śnew x1 +new a2 â—Śnew x2 +new ¡ ¡ ¡ +new as â—Śnew xs = 0,

1

= a2 = ¡ ¡ ¡ = 0new

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i

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xs+1 = b1 â—Śnew x1 +new b2 â—Śnew x2 +new ¡ ¡ ¡ +new bs â—Śnew xs ; xs+2 = c1 â—Śnew x1 +new c2 â—Śnew x2 +new ¡ ¡ ¡ +new cs â—Śnew xs ; ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡; xn = d1 â—Śnew x1 +new d2 â—Śnew x2 +new ¡ ¡ ¡ +new ds â—Śnew xs .

[?3n | P E N s- !5` ^ ♎ *a 2.1 _ P ℄ } - R HYR = e C R = _ -8 & n

uX R

− − − R− 0 ⊂ R1 ⊂ ¡ ¡ ¡ ⊂ Rn−1 ⊂ Rn

− n

= {P }

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(^ (5' 3.1. Smarandache 5' Smarandache k x N ÆBJO Y2 G p #e 5\ { s LobachevshyBolyai G p Riemann G p Finsler G p #eDD x x L \ g n2| G p 8 n *N 3n T $ : N Q2| G p A G p Riemann G p P T3 2| G p * â„„N ~ Qzb:g*N 0 $ (1) O= eU O= )Yezl}Zd; & (2)O ;; q}Zs D & (3)ZG^eâ„„H) #A _ G k> lY - Re}ZqR & (4)(p;Bq \ & §3.

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Qa8Q3 5.3 F qMEW TN q p E'a4W} C RIiw}Z 4 ^TN 3 ^ 3 ^TN 2 ^ -' w%lSEx Einstein Q36y s.Kg m Q f N Qz #> $W . w%b68k j #1 H` Fy #-p>℄ y S8w% C qZ $-0 ;P [6]-[7]!&si> $W } w%x y s y si` " #h , ) N > KI6w% Cdt w%a8 # N> N Æ 4 % #eWlx-p>℄C7H 8 4 % 6R>Iw% 4 % ℄ #3 n1 % DY$`Cx7 | #D j si1x6< #B Z w% 4 % n x N$ u b A , #W n x d 7 q y p )w% =K> 6 Ex_ i b6 #6 s tZ =K> N #y Nx> . x $ C$ j j,n <1.m =& Z { # $ N x ℄ 7>℄0 " K yg b N 3\ A [ NE.mi`L> #b F0" K g m { 0\ #Cy / ℄ 0"0\ ~ Ym{ $ 0\x ([16]) $ (1)UnH K > #?l i "H}ilY #i "} ilY T0 (2):H&NU|^ - p #Ce -} zY= (3)9 HY - o > 8> $ {" # D 6 E ) Y 1 g eK \> # ?DYei`L> KI Qa8Q3 5.4 F a4W}Z(UWwE E' . > . % N xx> $ N *m \ A :2, 8` ^ ! .m g#b 0\ Fg 2 d w2 6R` ^ ! ≼ 11# g.> 1 yN I6 , M | Qk!E #Fy X 6 EY|D B bu( P | , .m V l <K Z kG1 [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]

+_Ro+!|, Y k =_, U k $ D ? M, 2001.

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

\ L E, z B { } k, V $ D ? M, U , 2005. [6] S.Hawking, p-2t, V K D ? M,2005. [7] S.Hawking, ^ | Y}M, V K D ? M,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/˜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 Smarandache’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.

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p 1. 8 ; Fr ! UuA:z N 4 | =â„„g m#8 ) k G>>p `B{>e!#Vol. 27 #2(2001), 18-22 2. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, om k G>>p `B{>e!, 4(2001), 13-18 2001

3. (with Liu Yanpei)An approach for constructing 3-connected non-hamiltonian cubic maps on surfaces,OR Transactions, 4(2001), 1-7. 2002

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H℄{>Æ y s p 7 #2005.6.

3. On Automorphisms of Maps and Klein Surfaces,

4. A new view of combinatorial maps by Smarandache’ notion, e-print: arXiv: math.GM/0506232. 5. Automorphism Groups of Maps, Surfaces and Smarandache Geometries, American Research Press, 2005. 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.2,120133. 2006

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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, No.1 (2006), 93-101. 2. Smarandache Multi-Space Theory, Hexis, Phoenix, American 2006. 3.

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Xiquan Publishing House, 2006. 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. 6. On algebraic multi-ring spaces, Scientia Magna, Vol.2,No.2(2006), 48-54. 7. On algebraic multi-vector spaces, Scientia Magna, Vol.2,No.2(2006), 1-6. 8.

N> j D Bg N t U $ Smarandache ` ^ N #H â„„ { B h

$200607-91 9. X j X Z â„„N $ A â„„B, H â„„ { B h $200607-112

10. Combinatorial speculation and the combinatorial conjecture for mathematics, arXiv: math.GM/0606702 and Sciencepaper Online:200607-128.

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11. A multi-space model for Chinese bidding evaluation with analyzing, arXiv: math.GM/0605495. 12. Smarandache

`^Aa> $ r N, kâ„„S Op e Smarandache j

7Ce , High American Press#2006. 2007 p

1. Geometrical theory on combinatorial manifolds, JP J. Geometry and Topology, Vol.7, 1(2007), 65-113. 2. An introduction to Smarandache multi-spaces and mathematical combinatorics, Scientia Magna, Vol.3, 1(2007), No.1, 54-80. 3. Combinatorial speculation and combinatorial conjecture for mathematics, International J. Math.Combin., Vol.1,2007, 1-19. 4. Pseudo-manifold geometries with applications, International J. Math.Combin., Vol.1,2007, 45-58. 5. A combinatorially generalized Stokes theorem on integration, International J. Math.Combin., Vol.1,2007, 67-86. 6. Smarandache geometries & map theory with applications(I), Chinese Branch Xiquan House, 2007. 7. Differential geometry on Smarandache n-manifolds, in Y.Fu, L.Mao and M.Bencze ed. Scientific Elements(I), 1-17. 8. Combinatorially differential geometry, in Y.Fu, L.Mao and M.Bencze ed. Scientific Elements(I), 155-195. 9.

C %9_ f '

2008

Æ L B {r4, American Research Press, 2007.

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1. Curvature Equations on Combinatorial Manifolds with Applications to Theoret-

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ical Physics International J. Mathem.Combin., Vol.1, 2008,16-35.

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2. Combinatorially Riemannian Submanifolds International J. Math.Combin., Vol.2, 2008, 23-45.

#

3. Extending Homomorphism Theorem to Multi-Systems International J. Math.Combin., Vol.3, 2008,1-27.

#

4. Actions of Multi-groups on Finite Sets International J. Mathe.Combin., Vol.3, 2008, 111-121.

+: % K) 1985-2010 wCPg: 2009

103

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1. Topological Multi-groups and Multi-fields International J. Math.Combin., Vol.1, 2009, 8-17.

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2. Euclidean Pseudo-Geometry on Rn International J. Math.Combin., Vol.1, 2009, 90-95.

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5. Combinatorial Fields - An Introduction, International J. Math.Combin., Vol.3, 2009, 01-22. 6. Combinatorial Geometry with Application to Field Theory, InfoQuest Press, 2009. 2010

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1. Relativity in Combinatorial Gravitational Fields, Progress in Physics, Vol.3,2010, 39-50.

u ' k:P y $ =T – X j 0 / ℄B e!# Q P D ? M #2010 &

2. 2010

3. A Combinatorial Decomposition of Euclidean Spaces Rn with Contribution to Visibility, International J. Math. Combin., Vol.1, 2010, 47-64.

4. Labeling, Covering and Decomposing of Graphs – Smarandache s Notion in Graph Theory, International J. Math. Combin., Vol.3, 2010, 108-124.

[Æ $ #% $v s $v s #kQ $% XN #kQ Q S $ r2 e 2006 &8"kQ Who’s Who #~ | QV K N.6` z $ ` # Y Q W D " Cv % QX j v j v % tsÆ QV $ %C A FNu l < 2 -, #U p %3 $ % ` w ~ 2 -\{ 1962 & 12 * 31 2D\ +K_ ? ~ #1978-1980 &+K_ ! $ 80 E 1 9$ K &1981 & -1998 &6 Q p B%34%I #T q-T K K 6 V h :%3 w &1998 & 10 * -2000 & 6 *- QH$%3pC * :%3 &2000 & 7 * -2007 & 12 *Q Xj ^?-*( # -h ! & WC * -ot:%3 w &e ^ 1999 & -2002 & U Q x qk $ ', v s $ ) #2003 & -2005 & QV $ %C A FNu 2 - [u v s 2 -%I &2008 & 1 * # QX j v j v % tsÆ =& 2 -%I D? 6 $ N> o%3 p N 4 #T 6Q N u $ K>ED l Smarandache G p ℄ G p N> r z z No%3pNC N. 60 =N #6kQD ? T=\ $$ & $ \ %3Xj 0 N& oN\N.?#6l QD ?N\ $N.? [ 2007 & 10 *Io # e QS $ r\Æ MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES) # V\Æ W !6kQ e= D ? ) 10 > 6Q ) 2 \k p %3C p %3e% vN P l nx o p %3e%l II o VII! e y &x , -oQ j 0 e%Xj4 V. m 2007 ?!o , -oQ j 0e%X j . m 2007 ?!A en x { De y& W&2008 &jq- QX j w I VQ r{ ~ ~ {{" % " X j 0 / ℄B N Vr{ ~~ t e# 2010 &X j w I VQ wK{{ u X j 0 / ℄B e

Abstract: This book is for young students, words of one mathematician, also being a physicist and an engineer to young students. By recalling each of his growth and success steps, beginning as a construction worker, obtained a certification of undergraduate learn by himself and a doctor’s degree in university, after then continuously overlooking these obtained achievements, raising new scientific objectives in mathematics and physics by Smarandache’s notion and combinatorial principle for his research, tell us the truth that

all roads lead to Rome , which maybe inspires younger researchers and students.

s1jT aw M[ x % # [ "w M ( > # = nB U # $2<l # U M Q # _pS6Q "7 % # [! H_pS M M+-P, #a (d }M o $ + "pS6# /[U#H"rYr # Q`P7Æ #q&+ ur #( "Y 5 zU #1, x4j 62 "x4 w ?qn Smarandache %d 6HYC dM\[ # = 1 , S2 "t ( ff}BpPW / tB "7x Ij % # [/ 9 8 n h H w

ISBN 1-59973-032-4

53995>

9 781599 730325

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