Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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I # 6 S A z; —– t U6(x Smarandache ,oK Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO

(New Expanded Edition)

Chinese Branch Xiquan House 2013

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

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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 2013 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: 978-1-59973-214-5

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H " 5 R y: – f + U_ /, 1-3 $V C U A x "&o V _Rb_: , V _g 9 R& * 100190$ ℄: bnw*℄ f*~=x5Dn(sF_5 ) * K2 0 C F_ Q O V b _>_℄&Q | ( Æ _F R &>_℄&% " |&oV_Rb_: ,V_g9R&> (Ip B&> g9 V _ S0 + / * yD Æ _~ / &_ o 7 P - Q Æ _&6D P - Q ~ { Æ _ + *_ t j&*n7 Q qZ' NJ*x5Dn&F_&S0&&> 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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H " 5 R y: – f + b_ /, 4-19 $V t U A x "&o V _Rb_: , V _g 9 R& * 100190 $ ℄: D 1h + & b*℄ f*~=x5DnSKS~=b_ ℄ + asV&{ o &_1( ^~℄x5KoDZ ^ ( Æ _F R&>_℄&^&oV_R B&> g9 C^oAw - pK oDZ0 &b& *℄ 1985 ~ -2006 ~f BV_g 9 + k= V&x*℄ ~7g 9 Sq +)% sV&{ o b_M .# + ls VC: Smarandache * ` hn# +_ 0 NJ*&_", Dn, DV', b_℄, Smarandache K , Smarandache * ` h, b_M .# 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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11 T&a`m ?3a E OP%[-C:&? A dynamic talk on maps and graphs on surfaces–my group action idea Æ"`3 5 4:4^Y [ + B| + ~= 5 - f + eZ'a ;$%o-(N}ZL&yM# z%% eg 2002 1&= ,?O ?7_|,?N}OI ?% o?F & D /4p ^ k o?N}℄ "/`m 2003 ^ = Æ |&?) 2003 6 T f,= ,?O ?7_|,?N}O kN}℄ - [k N} G&u -N}℄ %<s &l GT Ob{rSh -%N}&?aT oj%4fy/ hamiltonian %N}℄ G% | 4lz0[&k X[ `?"*

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5ry l?a o?L[%l |7 "7GL%{v kl* rj [u q(%℄ f&?uqr jS ?vg /W 4fz%& m#" C I Perez oC " Your book is very good. High research you have done. Æ, KF[| ?l4E C " } P g& ÆV Æ Zf%u0b z% T?%'{* T % /ua 2005 &?a`m% F ? = ,?O ?7_|,?N}Oy %l ? d :&`?% A ?w-O|7 l N}℄ f% ZL&m# l3" C Scientia Magna \1lz 8?a 2005 ~#}zZL }8-9f% 9 r| S-?%w-|7 Smarandache +'{&?rm.y`l| ?f% | '% 7w-&{l r 4f % ?; &#Gb_M ^a7I

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H " 5 R y: – f + M C /, 20-32 $V6( I A x "&o V _Rb_: , V _g 9 R& * 100190$ ℄: Vg+RwDZD\ &2 O ℄ )X&2 O ℄ * y ℄XJJ n jAg 9 nL b *℄ f*~=4 < _}& Q4 < g 9 % %4< &\% %(Ig 9 &"i* %_g9%N K < t + g 9 sV&j _ ^ V gDZ&z l ℄ &z x ~ _ V &&* \ E Ff * \ oXg 9 >mDZ lFL + & - P% t + g 9℄ &%C1:oB_} ( D`u& 4X xD V gDZ}S t mj +5 S & b _ d .- ; 5 3 Z' NJ: ?w-O { w-? I? I - Smarandache ' { w-$Æ + -M ,Ne5 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.

AMS(2000): 01A25,01A70 1 2

< + ,8`s -> SE

2010 e-print: www.paper.edu.cn

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k=1

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k=1

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n X

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p&&: D *\ %?\u?\ ?%1TV .Æ%|le +#a &z?LV %u&?L?" ? ulT l% J .y G%&o2(f zz. fT f{ fy&{l ℄aG ?3%?\9~

H " 5 R y: – 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

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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.

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t$

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

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XB U v4 + $_ VY*Y &_ S0+L ~ .-~E+ qZ' NJ*x5Dn&V "&*DM S&*DNâ„„&:xb &w &w / Kz ' 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.

46

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℄ % 4 Æ Ei)o % >&? k ?%7T℄ SI Hh 7℄ [/lX& X5O/:61M℄2:C ulTpV % D~r&7 E℄M_Æb &*V:6r:rz& *'R℄A% i℄&q30b C8B "&*?[%q℄℄℄aa℄ [ 4lTT℄ S &℄h ::^? ,_fC& eG | & * lTT%Sj d u_P:Ci ? DHT{℄h= HCi j8 ?℄% " % 4tt&lg 200-300 e% &.0I Ci 8dQ 1981 rN ?aX5 K ;8%K/[% b < # &lg 700 fe% P&d.I 1A fi$&Wd%&ublX??Wmu &?H}Dlz"O/ X5{ &_{ X5{ \=j - 2m &HT lzX5{ K ;8 &i*K/ [% 20b_ < 4l" j2p{Sf+T% b_ y A G.Birkhoff * S.MacLane % A Survey of Modern Algebra (4th edition) Weixuan Li % 4 < ) ua T M8% y?vg - &l) 1984 Y=`m℄g ?Z W0E ?\ -&? =HTH ?:y/f/u'Y %(lzd%Cd [-S & LGs0b ?\9O$g&yv q 7E%0b = 7Tp lb)*y℄MX 9O \C&" 4 b9Æj* 82 [% ?S<3&?\^) ^) +)9O3 J z \?&? E(ob_ t<& * u?a=? M? %y"%3J /uj : "?\{?8E % k 0 b_ Cg 9 ? \G)? %$uÆEJ& fT {Sf+T b_ y A :%l y \ 4fy℄P$!?% ?sd"&dC?dM ?[-l ^) ?=% ; l l M&=? ?%rfsd&di#?T >3 &`?Q?.y }alTiF %CV ` ?; d lDsd&i* T -=% D J* ? wv%fl ) TF IX[% &_b_ _ NH & #: [( 2 ;n"a(\O&( 2 U(Z(t% ; vn"a v&/u W l F IX[4B W?a l |m &z QGBB `I% &_b_SxNH ?aa 0&?Q, ? C&`?.S | ng&GdG ? l M k!Pk?%oK#Q ?%g(℄ u i℄&?0bg<"g(℄ &fl ℄j?k i℄( ~sg`bEy&lu i&a``ZZ%B :b }:&aB >:b %r,pu S4%lXB >&ldb lXQ& :U,1ua2[ : ;2) " blz 4&t/:[% IU Zy TT?&?sg

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:.y<s ~b &/uYfSG[/ 7 :6U℄ i% E℄ *V":6`*tsd?04_& ℄C(" TT "af%`*tsd #C % E℄ &8#C "l4X℄J[E &iFV &l>= " ( 50cm & 60cm E J=&> d[%;2 ~2 Ei) Evt`Q& * XgEvxf &NV &℄ [/`D u B dxO*s℄Æ% i >h :C i C& & :k?&N\ &lz℄"+W&? k 2a`*tNB : f &+7>dQ&"a iiFV&B HJ%iFC E` y ?&yu E _SG%B &℄ [/lFPd4 ?a2[6/ 1982 Æ%lh&? 7:6 _,B E℄ &℄"+W?{B :dQ&3 l4BBÆ- :6℄ % ?S G4h1&;?:6%℄J E ℄Æ &*uu`*t uu uuu:F)&?llf% Y?&=?G 6 r fGL %a &? " l :*t ' FL > t^ Qb Vs~"E- 7k a=? M%0`*X[%M!dN u$ G= M` ?M!yG%uE ?%D=SX[& f Q Æ_ L n ^& w K x y Q > {Æ– uG 74f " %Q lS l3~ ` %X!&d l3~ % 2005 10 T& ?_za =? MSj%2 & 4X[ uq p&? z?Æf? G%n2j| `G%0' 1982 1&q & *&(&? j\ l M&4fz ℄ % j Æy_&nuAWG "l7%℄A&i*W℄℄G&& &Of t)& nu y_ Zg? Jh&y aQ? )I{r ~ ℄ l SH b)%i5u Aa S2mFab) ? ℄ uy= 7Tp b) i5 Q%&y = 7Tp b)uq> &E(.yj %y_&. ' k 9udZ ? D_ lzG&&y9u i℄ A {*&0`lS%(g 1983 7 TG&j&? }TT%r a .\&d G& G&& Æ U. y&a X5O/:6℄2?&?l[ 9<f i℄,l[ fG&Gzg%b Æ S &G&Æ bÆ?&?9 Æ t 10 fÆ&S= ?& 110 Æ&hA<Æ &.y=klF {% ? ?\ ?n ? j kM&? 9{rf i℄&)fd ? l :TT b &?UE(I ?% x$& ,?E! 7 a ? E%F IX[x &G"?%Æ a 4DE; &4D ddlF ? % :T1&= 7Tp lb)7`mE{7E? p_"& GLOf, )℄ 3 G&%C 8 ℄g7` 7T 7TW Zg

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f%*M, %?\ *?\2a`m&?4{j a`m%td?\ ?&/ uZ z, )℄ b)eX! ?\& yy9udZ% 4f& /ub)w' 99 ?hQ& ?pÆ 4l & ^ jm# `m E{7E? ℄g7` 7TZg?\%td 1983 9 T 9 %&?*l4℄ [ /lX&': "X5 `m%l=& " 4℄ [/ÆfQ uq"X5 D `m%= 7Tp lb)p 4ph&?tf"b)w' K %l Y ? ? Z & k ,)SSI 4f ;?& 1zG&&*dDHzP: Æ t E(l F ? ? ?L8GL&1980 vzG&Gz&?.y:[|℄O & ? M &=Z%1M APut ℄J t 5h℄% KZg&E(d u 3 %,?}z&l [ &+& 10-20%& &? u [lgS& z s, l [&?duh a>& u: t lD r&u f, ? K& b ? u M ?{& d4 u G S&a8 {1d%℄ 39 m +Z 1983 9 T 10 %&? lQ `mE{7E? 7 83-1 C& *l ,) S&7 `m{=)Zg? |l& % _ z?lX k *M, % Zg?\ F? f :u"`m7T℄J?OÆ`Q%lF? & n / `m{7E,Gd&? 4 4*Ob 3E[ PQJ/`m7T℄J?O ( '%u&p_f ?& 2010 ^&?('G* `m7T℄J?O%℄?E & $u u &=%yp `mE{7E? ulFyG=*X7%=)Zg? &Fy4l? ?%& sg:PuG=3J&(%iG=3JUb( l < *y&?4l? Uu d ? t3&ly? _{r i℄ &tdQ& | lTT% "&? X[*z? $!?% ? M Gz℄~"&Xg t3& ?Uu!P t z &L8 ?%F IX[&? `m F ? ?\ & GW?Y? 4Ga`m%z?&yt?a?\ ? J=(; " E G + }4&l4u`m℄g ?%ZW0X[&U(l4u`m℄g ?O%BC&X[&? ?E(0 [& *a?U%2 NL&WG d E( G%a lZW0X[l{ 1984 rÆ k&l), f?% ?? \& du??Q k j G`m{G)E>k?& F&,r& )I& ` ?x ? oN}S%ObE

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l℄℄w'7~ l3=% 2 N&l ",u?Y=ZW0X[?\% ? 4 < aS.f/=%lA e &℄7%0}y_l7℄℄ "%g? I(y& u?%p~ z?*X[%b &?a l M3?&ZK *z?L< lz 6 z l *Gd : r 4Æ&y *$g%fl MX K 3?QG ?a = 7Tp lb){r ~ %sd& s ℄J℄℄w' E ?ug ℄℄m1℄ } &( (!g?_Q Zg? ?\%& / "f/:T* -%?\ l laZgE>e_=& u"=?)If,Zg ? ?\&`F?Zg3 AE( (!g lF? da?\l[ "?w'? Sf%Zg3%f\7S.f\& /" - f/% l?v& V6*lÆ& j `/I F?$g&fl, f/Z(j/ )℄ `mE{7E? ,)M"& 4H *? ? ? 400 fCi%, )℄

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~ u?Rk3j ~ %4lT℄J w' ℄J℄℄ M)?&? ks ℄J℄℄w'E AE(s ℄℄w'E %℄ &.(l SHf℄ %℄℄w'E .a &&8℄JgI .\&SSh*,$F #ooSh&5 ? ? %℄℄w'7~ & SH℄J℄℄w'E &? G 8℄J℄℄w'E %s ℄ &q ℄JhO ℄J[M(z ?&?0M b| Æ&dr℄℄^*% 8℄J ~ J=&al X℄J[%, d&?K5?d n^T ℄ ℄ J=`m%UA ; &i* 0:(Bd%2 & brm&Uba`T Mv E 4!.,_i*℄℄=q3< %C\ W; ),leH# B %E lus5℄℄w'E o2g% ;%, _,L[lUb--7 E 4f%_, =^ h Sh℄℄=%l q) < ℄JgI .\& u T℄J&i? [ T lTF - )oo℄ J%℄℄w'7~ v&`?Q{r ~ y T %B5G4 d sd "m$ ℄ z HN3 1988 1&`m:omOp_U℄J ℄ 1989 ^&?,a `m,j e ℄2&y 8℄J af%uV7 4yd℄J℄℄&?Æ℄3j2: ℄J

d &ZK3j 0dM*s S℄℄w'E ℄ ua T~r:&?a s5`m:omOp_U℄℄w'E sd:&q Z LQ%℄℄w'E s Ey*(y%Sh ;&*?Qs ℄℄w'E ℄℄ pS -9 d sd zLQ%Ey&dik N = 7T℄Jpb)s ℄℄w'E % (* v& l %℄℄w'E (2>SL* s5n , ℄J: G, ℄℄ &SL℄℄ 7=k ℄℄[NÆ ℄℄J7 fV N q)Z

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; :B e_&d B,ee_) ℄z&`V"℄℄ 9~ ℄M&NV℄J℄℄GgX lD% &z k d k?d o? e"(O f%eH#Ce_ B ;)&flT ℄℄=< %l ; ℄℄w'E s ?&b0bh ℄℄ J=% ; M E uZ 9 &℄ U;u`z.&? ad ”ao 7 < 4 < _℄ ( .” %l8 9 a?%b d&℄2:~|%z1O?f% , I* Æ&l % :~ u 286 386 )Xl:~ 1989 ) rÆM"&? a`m,j e℄2&a℄2:q3.y = G ?S f:W{1EKqdr & a ,Æ?%lD & #sNb & 4 # &y 1989 1&uV7 D% 4uqsg℄℄ 4&?ay% ~ ℄ dsgS &?I (yh ?&?,\ dlT℄ 2& 9{r ~ ℄ "P$ ℄/ T `m C?O℄J&u?T 7T℄℄ r GÆ&f% p %l T~r 1987 1& kC*vDh. h e&℄℄ NQJ*>& ` " (YgQ"& N# > a l#sd&1989 1&= 7Tp lb) 1hNI `m C?OlMPT 4℄J%7E N& ulT℄℄℄M_Æb &z *V"?Sa 1991 Æ j ?&, K*DÆ7E%lT~r w':4 ?a T℄23j℄℄ ~ &l7E 4l & "(l4`mE {7E? i? }[lg%z? E? sd℄℄& ?:d 5&4ph℄4w',beH?rmsd{X `m =3[)9~ q&℄2 9~ Q Khs s q 4,.h 7"0 i pz T$&7z?lX #E 4%z Q ℄℄ G, .i qlG ?%B7℄ &di?$! V"℄J9~&bs5℄℄ N &;0a℄℄ J= ( %9~957~ ℄&l au ISO9000 9~! "e_%( '{ l &`m{7E,Gda { y 4℄J [9~~ ℄ &`m C?O℄J , ')~r&S=(l~&(& u~ b`d &(& ` 0,z &d db`~ f%&(& b( d') 4 AE( $ ! 8nD~ * 0')&℄JhÆ~9~ % Tz1q9 hd ') h *y&?gM lT|& Sz `:d %')7d')y_& ? ,

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{T L jd* Q 4 * &{T O ~ O * 1 N _RD 5+ X v &o* > Dd* Q E Æt d %z1aQ"?L yr# ')w|[&? ( MYM7& *v u * 0')l d %l h lu& E(af ~ J=i "/`m C?OlM℄JPT 4℄Jz ℄&eH 0 ~ &a y_k#4b & 0b ;a℄J:%eH# u .y,&dD ".y ,dQf,D\ Sh ;?&? pz#C% v& {%eH,b?f %pz#C&S144l 0b% v& ^ NV eH %lz(z( & q3`℄℄h7f% ;& &{ x5N℄ x5*D )M :T ~℄Æ&i?`m1l (E℄%Sh ;uq% ÆY ?SeW℄℄ J=& $ H Vi u 10 T 30 %&h^AZ %y+ 7V* d 6 C V&8>i AJ}i %i #L .y# d 8 C V& % :%7V+ d 15 C V < a H: ; siV7& 4ph :I')&rm T%i L& taIdQ%}hi v& u lz9~rt&.# i AJ}%z1QÆQE &7E O%z18~h Mv Æ|j ?& l$i a T?LTT ,b&I' 4|b<{? B Tz-i%i 1990 1&l4lX℄ %zrL8?&"`m{(lAG)E>k?& & :?&)I5 Zg,r& .y /u&?LlX L E> C; | ?&j8 & D*& ?=k%Zgu ?&sd3Z a L E> &Zgsd*Zg3la`m ? *y&℄ &1&? kd 4p ,T * T %& ,r l GÆ&* V"`m C?O 1991 ?S j,?&℄Jh k N 4p 0b℄℄% T-\~r&S=% 100m b- J℄℄ r%J

b- J ℄℄ v(}A, lAua d EB , a 4_ 44:#e QrC\ Ti & 95S℄℄U~oVuT}7&z lz (z f&Q Gg ; U(lAua2[#e QrC\ Ti &℄G 2E pV?a{IB 4_"C# & 4jFk e℄B& φ = 25mm % HC N>&{2[j jFE = aT2E 4_44&"hua2[iF (|95 ℄℄9~&z NV E # %l Æ & 7u φ = 25mm %HCa℄℄ J=0bKNsO&sOdQ%HCEv_zj &. lz(Q ,℄JGg o

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l 7T N}th kN}S. φ = 48mm %Lee℄B& φ = 48 × 3.5mm %^<E C~f φ = 25mm %HC *Le#>& ^ .aLe℄℄=a φ = 48 × 3.5mm C~ * ~&{l=+℄J℄℄Gg y l 68uqS.` ^t& aLe℄J℄℄=UE(i z^2& b - J aTdE( A "e℄B = 7Tp hR 1 C,Ns ^&hR A "e℄BaT b- J 100m ℄℄ N} f// `m C?O℄J&N} Nk 1 ?% : T aTB~ 105t&' ^ X_ 1.2&E J'^5 126t 95 *y&?f% ~CSh ; jF ;*fYe &fl G TaT _|& aT ?> jFaT_| ?7* ℄zE ℄ FE %aT 16 $ C5 1B &#C:d}TT{&S=:T{*e℄B# &dT{ /444 Vl &y aT J=e℄B L,?>: I M30 . & *;qI:d?>&yt/aaT J=4_*, ,jF_|= Le℄J℄℄ =%lAjF95L 24 TjFk e℄BS{I XÆ4&z a 23 $?> :qIl ℄&y# ?> ouO 8~E y a`m C?O 100m b J aT=# G` q aTd a T 3aT* 4)a T℄z& y 4/E zG 31.40m 44&lei 7 h "&* 8~ ℄℄ %N}~` lDdn `m C?O 100m b- J aTE y m# 1991 = 7Tp , f p):&l ?=Zlg 1 fl 7&(Dd ulT Glu ?`aT_|E S℄℄f% pS& u 1992 Æ~r|a e DV % * 1 N _R 100m gmC#*DÆ* x 5 V N % * 1 N _Rgm C #*DÆv}89 b- J%℄℄&N `m C?OlM℄JO W ℄J℄℄ 7o&5 * % & lg \8g&S >gÆ%℄℄El&?f% e℄ ℄ N&s ℄℄w'E w'f℄&{lV" 1991 S ? 3

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N * x5N℄ : r 1992 fh&`m{ 7 ℄J ℄&[`l G$F Sh&*+=+S℄ J℄℄Gg G%[j&℄~u0b` l> B f%9|&y* > Q &aj%,SH℄7℄℄&{l V℄℄℄M l M&`m{ uq(l ℄℄ 4a %: ℄℄ &.0b}$ & ? vC f%aTy<s℄℄℄ [b n^{ - ; f E(.yX.% qY ?7℄℄^ 4zr 5hlT℄2 | lz&aQ? {a>H% -s d&!*%: u<m{ ;e"% Hj H$ Ti J =A7%C\?Tk,"Cn)I }X ~ #C& 7i QiNI&f l#GlT:D%o?e_ q ;&?E ` `m{ 7 o:%%: & s5` ℄℄ 7w'℄Jf℄ 1992 GÆ&= 7Tp w' (DCD℄ & *=Zlg <H & ?y %(D#u G&qR ℄J[P u *y&?Z%X!u C:R ℄J[ .y b) ,C;&3jy~℄ %zrL8?&"?%R ℄ J[ 0bjld| .y & d a; ,"b) 4 a&$Wh a

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l ℄J℄℄~G uq ka℄J7E ; , z zz t5& |& z s5%z1sg:X D=z % D & ^& z %(dZ 4y1 v z℄ "b)q , w'&b) ,*Æb)A;,Æ~ s z℄℄w'E D* zZ 92 q3ua zjlhie_p7y?& "b) HDd z ℄M)y1,z? Æ \ z92& z℄ b< 4ph:IH7|+ G&4 = ? (l~Ob℄ & u-- zs ℄℄w'E D& u?%p~ Hz z&ÆY℄℄ ?lF.0b1h " .y5 z92%b & G℄ ℄w'E D%s ℄ l &?g? 7 _fT℄J% z℄ `m{EFlM℄J" h7E& z "4Y&Cz,z( zZ ℄℄w'E ) ( t ℄℄ ?&? }h G ℄℄w'E D% s &?WMb| Æ l℄JA;Z l _fh& "q ,% 4zr C 7s &IL4pp?&"z^% z℄ &?. 1h&lGLl _fh&N\ zvh&b)q Æf?*b)q ,4 Æb)q ) T lX? r8,D%d | zd z%& #U(`m7℄ = 7T l % Tb) r8a D% "w' zd & < Q?&" r8l 4℄ G, &l47Z &l4a2H: 7zS ?&"HT z Z g& DU z 7 z%g?z7 ?LZ 3 $& 41T7 z z92%5 l&(%u)I %&(% %Æ&l?L%℄℄w 'E lu :~ Æa 16 0: j[}T z %℄℄w'E &E-u Æ #l&Uu7z " ?t` Æ"%℄℄w'E D&5 s5n ℄J:G ℄℄ Ob℄℄ v Ob ℄%z7&*eH C5D " eH H B,e#Ce_ i kJi J?q)Z )O b (&aDÆ 2f% ;T&S ?&q ,4" z =&? z7 #y" l M&℄Jz1uC zZ %Obn &lF z Pdx UA [t y?%z1 +&Fy zZ lFP+ f&l r8,{=℄J?W=

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l % zqi&i?^ g a( N%8>q e5=& z zu qa DNS8 J=r\lD% &l CdO {1q %_h&Es H8Z r\ O{1hJ-4% du??Qf, z th{r z ℄ %lTObE " $ ℄_ Z 7af)E 1995 rN?&= 7Tp lb)NI ,`m{ 9511 Æ℄J%`m :oS.AV= sd℄J℄℄ N b)fT7a/NTS?9%Sh℄J*_ ,_x℄J℄℄&Fy` T℄J℄℄w'℄~B}&ZKZA b) 4y℄℄ ~ *p~% GGl ~rq &ZA?*~r+p℄J[&p℄J[l"Æ b)p℄J[ u?℄℄ ~ B %lT~r&du?FO %& ℄℄ a7E E *℄℄ r Æ= XÆr\S℄℄ %l T~r

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) 1995 5 T^&a;G s*%--d& ks5 g $ S ?& D GD℄44&w' ℄ 8℄Zg^℄℄ ~rq .( # ~rq *?1T &sd℄℄ J=℄~ =&H1h b*f+l hiC&4phUb 3:C %b &a℄℄pfV95d&?s GD ℄℄℄fV N 8 Ny?lh +2#yf℄ (lz&? f℄ QZ % Ed _SJ%℄81&')Gp*℄8 9~& GnK& * ℄ 89~sg:u'℄ k = & GlF d 8 ') d *?Q8℄J℄℄&?a7E E *℄℄ r Æ= ℄℄ d s d A 6 TdBS &*y0bd ℄JSh℄℄ N&s ℄℄w 'E & kf%Sh℄℄d Sh℄℄ } J?&l 4/s5 &a J=uq G Ssd℄℄,U(l 4/F{C:&7s5rt L& 0biBE Ssd&*y&?=k rl"℄sd&q o?*℄Sh ;&= k ℄x 4V i pV) &E -\&as5 S jf% ℄ ℄& G 8J?sd%℄℄&*?Q℄JSh℄℄ q)Z d sd y &U0b1fSh℄℄^VNNS 7 ~rq 7l4X℄4T }h& ETdQ&℄ u: j G4f & "?:?+iC&?"& G : d Q&?_YGTlz :&?=G ?%Mb|&}T lr&(&lr h& y?T VNNS; ? G z L8G&Gu?*~rq }T {SG℄2m= Q% HT lS=< D %~r. (u f%& z uymGT o%(jtd * ℄ _u: j&WGSG Q NS^& `Gab) lS. Pd( |& Ga b1'& b* Ai%NS Q l\ ztd 4ph:C& Guqa51w'℄ t d ?9℄℄ ~rq 4 &l_;? dD v G k(u|( s5 `3&y 77E 4a TT%Vs--&= 7Tp lb) a 1995 6 T S %z & jNI 8℄JSh _x_,℄℄ N ? T pNS p℄J[&3j8℄JpNS ~ 8℄Jg?( 23 T ÆS 4 7 ℄℄7E&?Lk lN ℄JSh*^_,℄℄ N F Q )ooeH %uL>H5 %D" eHe_& HuL>HB,ee_&S E g "?T G ℄℄l5 FR}T℄℄[D\&Fy2d|i %℄℄9~u " `m:oS.AV= ℄J_p$yduK`:o b)%Mb $&

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1$ XG 5m %( Q C40 i &*V"i T9~& `m =3 [%m &?d 2d1$&,\℄ Ti lL &lT℄ dQ ?&"l4K`:o b)% a?Mb|)? ?N7℄ TV r~ ?&?ba Mb| l &EQu b)+pq [ ℄7Mb|%O G ;?&*dD i ?Ubukd 51,\& ℄ < % ?L8G&7 TSh=%l y1 4%9~uV"℄JSh4 %y1&* lz( Q T C40 i & y1 4o2Q ℄℄&{lV"S9~ G`?% Awgo KJ|!z&d"y J uH"% T7 ℄℄ $ & eHfzD\& & $GV%uO&℄ l eH aG` 4<I&BV k&S hG i T J=`m em ,e? .#,\℄ = ` Æ_b ( zEE& # w'B ℄D"eHyV "℄℄9~ ^&℄J{2:,$ k % eH 2:Sh℄℄ _,$ &`m{7E,Gdw'℄J9~ ')&Zg `m:oS.AV= ℄J *y&b) Æb)p℄J[*3jS.%q 4f &q9 h ℄2Q m1| 6S h &(%6 #Ua℄2 24 f℄ | ')jlh&?;| Æ^')%w4) &Wa7,℄ % l4z? T:Q& GWÆ^% T G;?℄J9~(; 6&?"E (&Ob{ GL>')ld& *℄2 h* I')wI# YY% 4 ph& 4b) j ? f')wQ ℄2d | B1 &w4 ;* n ℄ DK x-Æ?f d |?&GY?"*

nDAfj &fxd ~ e N L&{ - ) * f T * d 71h&{ \~?&f T O ~ O> K Æ?/uÆfGL _l$ ld S & 4b) nT& $!?77,% U( ^l$y_ "/~rq T uÆb)% ` &Æb) "?L T 1996 7 T1&Æb)w'4 S.')&Q `m:oS.AV= ℄J& )``℄i alT $: d%lh t&/u ``~rq CiGG sD 200 C%s & TTu℄JfV*℄℄9~y"%lTT&le G ,$ Sh℄℄ = " &E(# :g&WQ% usD&~rq TC iGGP{ x ?f :u/ 1995 dL m# `m ?*`m{G)E> k?& ,GdGr% ?Zgg,9A* ??o?4&dd a= 7T p lb)4M℄ d Ey*.y"y# b)%B & lzÆb)` ?L T %hs&i?_lz'&EQE({| %lT; & u^o-4z& D*D wDZl >Y )% { + a&)% #;<.- Q u ^ ?o2

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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.

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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.

PJ*8H â„„s -&M- -&B6"&2 +&Smarandache +&.V~6" +&Finsler + |u# AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50 §1.

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(a)

6

6 ?

-

–

6

?

?

(b)

1.4. f%ZJ n M -&8HpS k %vT #6"+ u 11 +%& â„„s k?& S= 4 T +a %M PG&l 7 T +la C & ^#G?L_h # % 4 +9|8H* 0% 7 +$|8H 4 +9|8H % o$- Einstein o1 J&l 7 +$|8H % o$--N/ J& "y# d[ TS- bv 1.1 M- < +1`D~ = ^ Q = ;H Y ~ = 7 U` h R + 4 U` h R D 1.1 * 6":%9 o%bOn2&?L.y# TownsendWohlfarth"% 4 + 50 8He" 7

4

e

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

φ(t) =

t

1 (ln K(t) − 3Îť0 t), m−1

K(t) =

m

m+2

S 2 = K m−1 e− m−1 Îť0 t

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

e Μ = p3 + 6/m. " ς <s dς = S (t)dt&l 50 8H%n2 * > 0 # <s + ;|a&. m = 7 l0 i* 3.04 3

d2 S dς 2

dS dς

>0

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81

{ ?CV<&D 1.1 =%7f : u7lu6"&"y J% ?;

D ^ t~(b_` h &, & Q = ;{ w # ~ = 1 U +` h) |a&* ^% ?6" a&vHlD u?L%3Sj= # % 6"&d u?Laq8 ?=<0 %6"&k*a 3 +t(6"=&HT7. y|m* (x, y, z)&H . S lT+ /)/ 1 %i6" +nJ g&$lT( %; * §2. Smarandache

1 + 1 =?

ak _=&?L$! 1+1=2 a 2 f5Z;e_=&?LU$! 1+1=10& e% 10 f :Uu 2 *a 2 f5Z;e_=.(}TZ;C4 0 * 1&SZ; l* 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

n D& D)/4DÆ%Æ?'{&?L lA z'+% ! %Æ '{ ([18] − [20]) QB T; &B ÆQ 1 + 1 = 2 n 6= 2 ?L$! 1, 2, 3, 4, 5, · · · ^% hGk _ N a T _=&n % %&HT D*j[b f% %? & 2 %? * 3& * 2 = 3 z ^&3 = 4, 4 = 5, · · · ^?L # ′

′

′

z U#

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

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

^l Z;)l {la Ak %Z;e_d&?L. # 1 + 1 = 2 % S- ma&?LauldZ;%D WDlT - S &` ∀x, y ∈ S, D x∗y = z & 'u S : alT 2 CS- A ∗ : S × S → S &i#

82

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∗(x, y) = z.

T% l&?L.y = Ay_a [:|m`Q g= S =% HTC [:%7|m&* S =( n TC&la [: n T et%7, }T7 x, z &"qIln( t[&* alTC y i# x ∗ y = z, ?La n t[:z: ∗y &D*8t[%ÆB&* 2.1 Fm

2.1. qt -Æ v V A` u 1 − 1 % S ` % * G[S] ma&* ?L{ lT <s 1 + 1 = 3 %Z;_|&?L.ygW` 1 + 1, 2 + 1, ¡ ¡ ¡ )^+ x TQ G

b f 2.1 ~ = b , (A; â—Ś) OS ~+ } ^~ = # Ě&#x; : A → A :)_ ∀a, b ∈ A & w a â—Ś b ∈ A &f ^~ = Q~+G c ∈ A, c â—Ś Ě&#x;(b) ∈ A & 5O Ě&#x; S ~# ?L4(|# ydy/ _| (A; â—Ś) 7 G[A] %y_%lTS

bv 2.1 (A; â—Ś) S~ = b ,&f (i) } (A; â—Ś) ^~ = ~# Ě&#x; &f G[A] D~ = Euler 4 { &} G[A] D~ = Euler 4&f (A; â—Ś) D~ = ~Yx , (ii) } (A; â—Ś) D~ = E a + bYx ,&f G[A] & Q = C;+lVS |A| , B&z q (A; â—Ś) / â„„7 S &f G[A] D~ = E a +* 2- 4 KQ = C;j ~ =" :)**C; h + S _ 2- &{ g `/(rTC%~#&.y lA(r % l D`F(Z;S & 2.2 W` |S| = 3 %}AZ;e_

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2.2. 3 TC% vZ;

" 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.

" 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.

`lT - S, |S| = n&.yaS:D n A z%Z;e_ ^?L .yalT -:z D ` h AZ;&h ≤ n l# lT h- BZ;e_ (S; â—¦ , â—¦ , · · · , â—¦ ) aq8 ?=& u l%Z;e_&T ; e) u 2 B Z;e_ lF2&?LD lT Smarandache n- B6"*d b f 2.2 ~ = n- *` h P&E S n =A A , A , · · · , A +% 3

3

1

2

h

1

X

=

n [

2

n

Ai

i=1

KQ =A Ai LE ~(Yx â—¦i :) (Ai, â—¦i) S~ = b & n S

b&1 ≤ i ≤ n aB6"%F d&?L.yfl q8 ?= T ; ~6" %: l# B BT B; B ~6"%: & # u % Sh bf 2.3 Re = S R S~=E + m- *`h &K_p Æ b i, j, i 6= j, 1 ≤ e & w +Yx q L i, j ≤ m, (R ; + , × ) S~ =" K _ p ÆG ∀x, y, z ∈ R ^&fm

i

i=1

i

i

i

L y: ihUA bx

84

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

C

–

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Smarandache

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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,

fO Re S~ = m- * " } _ p Æ b i, 1 ≤ i ≤ m, (R; + , × ) D~ = &fO Re S~ = m- * b f 2.4 Ve = S V S~ = E + m- *` h &, Yx A S O(Ve ) = S Ë™ , · ) | 1 ≤ i ≤ m} &Fe = F S~ = * &, Yx A S O(Fe) = {(+ , × ) | 1 ≤ {(+ i ≤ k} } _ p Æ b i, j, 1 ≤ i, j ≤ k C p ÆG ∀a, b, c ∈ Ve , k , k ∈ Fe, w_ +Yx q ^&f Ë™ , · ) S F +* ` h &, * ^ vS +Ë™ Æ& TvS · Æ, (i) (V ; + i

i

k

i

i=1

k

i

i

i

i

i

i=1

1

i

i

i

i

2

i

i

Ë™ i b)+ Ë™ j c = a+ Ë™ i (b+ Ë™ j c); (ii) (a+

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

fO S* + ** ` h &T S (Ve ; Fe) "y?L$!&M- -=%6"e"f :ulAB6"e" bv 2.2 P = (x , x , · · · , x ) S n- U H` h R &+~ = ; f_ p Æ b s, 1 ≤ s ≤ n&; P { w ~ = s +D` h

_ V y6" R = azds e = (1, 0, 0, · · · , 0), e = (0, 1, 0, · · · , 0), · · ·, e = (0, · · · , 0, 1, 0, · · · , 0) (4 i TC* 1 &S1* 0), · · ·, e = (0, 0, · · · , 0, 1) i # R =% 7 (x , x , · · · , x ) .y|m* 1

2

n

n

n

1

2

n

i

n

1

2

n

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

; F = {a , b , c , · · · , d ; i ≥ 1}&?LD lT % ~6" i

i

i

i

R− = (V, +new , ◦new ),

e V = {x , x , · · · , x } \Kx(&?L D x , x , · · · , x uRl%& . az~ a , a , · · · , a i# 1

1

2

2

n

s

1

2

s

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=

85

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

1

= a2 = ¡ ¡ ¡ = 0new

t az~ b , c , ¡ ¡ ¡ , d &1 ≤ i ≤ s&i# i

i

i

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 .

{l?L# 7 P :%lT s- +i6" ♎ 2.1 P S H`h R &+~=; f ^~=D`hW n

:) R

− − − R− 0 ⊂ R1 ⊂ ¡ ¡ ¡ ⊂ Rn−1 ⊂ Rn

− n

= {P }

K D` h R−i +UbS n − i & 1 ≤ i ≤ n

Y w Y ?& 3.1. Smarandache ?& SmarandacheK ulAy % +&S ( ;rjÆ$% LobachevshyBolyai + Riemann +7 Finsler +&S`r7u zd Kn y +=%` bE ?L gauld y + + Riemann +%pl J y +%b e_"d[ HnbEwG* (1) Q = ;% Q = , +; EY , (2)Q LY n hK, (3) p Æ;S&?&( s p > E+ # ~;Y Z~M, (4) - L 0, §3.

L y: ihUA bx

86

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`N ;

Smarandache

+ai

* [wz -~ : # &K ^ > ~ +~< + w 13 &f ^ ~< , z 4 3.1 = (5)

3.1. ln)t7}n %)tu? e&∠a + ∠b < 180 y?lnbExD* y4HbE&HU.y d [ A6 v* s> E B+~;&8 ^~ : > E+ * k{ ybEbÆyQ& Ll) #S4HbE 8 *bE`m&H : fa 8ulTd *y&4f ? 3o/ j,nbE"a4HbE& l)E(G` /u( { SG E f y4HbE&'Y# %b e_ u &u aBd _ p &Lobachevshy * Bolyai Riemann Æ~ z% E y4HbEm#G` GL % EÆ~u* Lobachevshy-Bolyai E*s> E B+~;& ^ : > E+ * Riemann E*s> E B+~;&* ^ : > E+ * Riemann E# B}%E a/"y.y7l`= +&? d Einstein *Su`-=% o 6& = o1 lT`=6" z^2&?Lu .yfl 9u ybE# % +l ;E(% y + Lobachevshy-Bolyai + Riemann +* Finsler +-9n [16] =T T; ; %T #o/ Smarandache +'{l7l.V~6" +& e` Smarandache + lT(bY?*d&dlN_Y?.V~6" + Smarandache +S a < K K { K *{ K ),A&Æ~ n z%bE7l S=&a < K %bE* ybE"1$-"4$y d[ +lnbE* (P − 1) ^~ ( B+~;&:)- s( ;+ L: O

=u

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87

* , (P − 2$ ^~ ( B+~;&:)- s( ; 8 ^~ : * , (P − 3) ^~ ( B+~;&:)- s( ; 8 ^- + k : * &k ≼ 2 , (P − 4) ^~ ( B+~;&:)- s( ; 8 nb : * , (P − 5) ^~ ( B+~;&:)- s( ;+ p L: K %b e_u D y + 5 nbE=% 1 Tn T& yd lnn nbE ybE=%` bE* (−1) s> E+ p Æ ;*~E ^~ , (−2) ^~ * y n hK, (−3) > E~; ~ = 4b&%*~EY l~ = M, (−4) %*~E 0, (−5) s> E B+~;&*~E ^~ : > E+ * { K %b e_u DA +=%lnn nbE&u d bE * (C − 1) ^ 6 P - { w => E+;, (C − 2) A, B, C S = * O +;&D, E S = **; } A, D, C B, E, C ; O &f( s A, B + :( s D, E + * , (C − 3) Q d w - = **+; { K %b e_u D Hilbert b e_=%lnn nbE b f 3.1 ~ =K OSSmarandache E+&}, ^*~ = ` h &*1 l S 6 *S &6 ( < *S ~ =w - Smarandache E K + K OSSmarandacheK d[ Tkiy d[} N|a Smarandache +uKx a% { 3.1 E A, B, C * y [:1T et%7&D )t* y [:x A, B, C =a"lT7%)t l?L# lT Smarandache + *7 y +b e_uiF&S=}nbEu Smarandache D%

L y: ihUA bx

88

–

`N ;

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+ai

y4HbEma*- s ~ B+~;& ^~ 6 * ^ K 3 ( F E)t L q 7 C t %/)t AB V q +lT a AB :%7a"(ln)t %/ L&lq )t AB :% +l7 a %/ L %)t&* 3.2(a) Fm (ii) bE- s p Æ = **; ^~ ma*- s p Æ = **; ^~ 6* ^ V q }T7 D, E & e D, E 7 A, B, C =%l7& *7 C et&* 3.2(b) Fm&a"(ln)tq D, E l` }Ta)t AB 7 F, G n 7 A, B, C =lT7et%}T7 G, H aq HL%) t&* 3.2(b) Fm (i)

3.2. s- )t%~G

b kY I?=lT4P %D " Q = Y[ 6 }S T[ &6 }* 3^ T[ A ℄ 2p = J&Q = J h 4'~ = 3 [ "b D$ ,6 }* 3^ T[ A ℄ q = J&Q = J4' F kl + : , j > }SYE* Y[ &m ; E S p , }S*YE* Y[ &m ; E S q e.D % 'ulTq)/ [% ~Sf [ZJl ?a `r7u 9u ~% )|:$! [u.D %&l9iA$Ælu .D %&* 3.3 Fm&S= (a) *) %0[&(b) * v-?%9iA$Æ 3.2

3.3. 9iA$Æ%#G

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89

2 u [%lANÆ& Sf ANÆ8 [) ?&# %HT[A z (/H D = {(x, y)|x + y ≤ 1} Tutte / 1973 W` 2 % D & [12] =% 9&2 D /d bf 3.2 ~=54 M = (X , P)&E S^:qA X +rGzD Kx, x ∈ X +nKOG+% A X +~=: % P &KGK [ + K 1 K 2& K = {1.Îą, β, ιβ} S Klein 4- G e & ^ P S : % &G * ^ b k, :) P x = Îąx v 1 *ÎąP = P Îą; v 2 *e Ψ = hÎą, β, Pi ^ X Y ; n D 3.2&2 %B7*[Æ~D *4W P * Pιβ / X :# % e !,r* Klein 4- C K / X :# % ! j Euler-Poincar´eb l&?L# 2

2

ι,β

ι,β

k

−1

J

ι,β

ι,β

ι,β

|V (M)| − |E(M)| + |F (M)| = χ(M),

e V (M), E(M), F (M) Æ~|m2 M %B7 r *[ &χ(M ) |m2 M % Euler HQ&S +)/2 M Fm,%vT [% Euler HQ DlT 2 M = (X , P) u*YE*+&.4W Ψ = hιβ, Pi a X :u.f%& lD*YE*+ Îą,β

I

ι,β

3.4. D a Kelin [:%m, *lTki& 3.4 =W` D a Kelin [:%lTm,&.y 2 M = (X , P) |m*d& e 0.4.0

0.4.0

ι,β

Xι,β =

[

{e, ιe, βe, ιβe},

e∈{x,y,z,w}

P = (x, y, z, w)(ιβx, ιβy, βz, βw)

L y: ihUA bx

90

–

`N ;

Smarandache

+ai

Ă— (Îąx, Îąw, Îąz, Îąy)(βx, ιβw, ιβz, βy).

3.4 =%2 ( 2 B7 v

1

βw), (βx, ιβw, ιβz, βy)}, 4

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

nr e

1

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

y 2 T[ f = {(x, ιβy, z, βy, ιx, ιβw), = {(βw, ιz), (w, ιβz)} &S Euler HQ*

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

2

χ(M) = 2 − 4 + 2 = 0

t4W Ψ = hιβ, Pi a X :.f ^ { CV# D a Klein [:%m, K=f -M N}%0b&?LU.ylF(2&$ a6"y fB [ :%m, aB [:%m,D *d b f 3.3 4 G +C; A E- V (G) = S V , _ p Æ b 1 ≤ T i, j ≤ k &V V = ∅ &1 S , S , ¡ ¡ ¡ , S SV ` h E &+ k = Y[ &k ≼ 1 } ^~ = 1-1 Y# Ď€ : G → E :)_ p Æ b i, 1 ≤ i ≤ k &Ď€| D~ = ' |K S \ Ď€(hV i) &+ Q = ( * 3M D = {(x, y)|x + y ≤ 1} &fO Ď€(G) D G ^ Y[ S , S , ¡ ¡ ¡ , S +* B| D 3.3 = [ S , S , ¡ ¡ ¡ , S %6"44`Bm,( | alA S , S , ¡ ¡ ¡ , S &i#` j, 1 ≤ j ≤ k &S u S %i6" &D* G a S , S , ¡ ¡ ¡ , S :% w w * B| y/ [&(d[%S- bv 3.1 ~ = 4 G ^ T[ P ⊃ P ⊃ ¡ ¡ ¡ ⊃ P ^ z+ w w * B| J U 4 G ^g G = G &:)_ p Æ b i, 1 < i < s, (i) G D [ +, S (ii) _ p Æ ∀v ∈ V (G ), N (x) ⊆ ( V (G )). I

0.4.0

ι,β

k

i

j=1

i

1

j

2

k

hVi i

i

2

i

1

2

k

1

i1 1

i2

2

k

ik

2

2

ij

ij+1

k

1

s

2

s

i

i=1

i

i+1

i

G

j

j=i−1

Y ?& 54 K ua2 sd:h7% Smarandache +&z dup_w- ? 7q8 ?%ÆÆ 2 +%: ga9n [13] =a`&=?a9n [14] − [16] =&â„„~u [16] f% a3%N}&SD *d bf 3.4 ^54 M Q=C; u, u ∈ V (M) 7~=4b Âľ(u), Âľ(u)Ď (u)(mod2Ď€)& 3.3

M

=u

r-o/ [ `

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=

91

O (M, µ) S~ = 54 K &µ(u) S; u + D x b NWU 6 *WU Y[ + Y u s h ~ =6 h K= [ iO ( 54 K n 6 - 3.5 =W` )tk 2 :%B7%~# e% CVÆ* /. ) /. /.&u 2&7 u D* H7 y7* 7

3.5. )tk H7n 7 H7 y7* 7a 3 +6"= u.yfm%& e7%fm(~/ y6"%~#& lDu )%&b 87 u y7 3.6 =W` 1 A7a 3 +6"%fm v, =7 u * H7&v * y7l w * 7

bv 3.1

3.6. H7 y7* 7a 3- +6"%fm - n 54 K &L ^a < K K { K { K

D %"a09n [16] *t/ T&?Ld[Y? [2 +%~# a A~#& .yaB7:-3C i &U.yb qIB7&"%rul Tq9 & ^`fl T [: t_Æ( i*a [2 + =( ^%S-* [ 54 K n S *u s 54 6 u s +;S H; *lTki& 3.7 =M` s/ ,[e%lA [2 +&S=B7

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92

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r:% +|a8B7 2 b%C i +

3.7. lT [2 +%ki 3.8 =M` 3.7 D % [2 +=)t%~#&℄-2& 3.9 = M` 8 [2 +=% Afr#

3.8. [2 +%)t

3.9. [2 +%fr# §4.

f nJ?&

% u`-℄Q 6"a o du %&N2 td k & l7af |#=uq# "f 2 +%'{f :.ylF2D /lT V~6":& a8V~6"%HT7:-3lT ~l7l.V~6" + Einstein

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93

b f 4.1 P U SVCALS Ď ?ALIF&W ⊆ U BMW ∀u ∈ U &N =ZVCKUXO ω : u → ω(u)&\J&BMWâ„„R n, n ≼ 1&ω(u) ∈ R Q> BMW?^R ÇŤ > 0&H=ZVCR δ > 0 DVC v ∈ W , Ď (u − v) < δ Q> Ď (ω(u) − ω(v)) < ÇŤ [N U = W &< U SVCTALIF&ES (U, ω) ,N=Z^ R N > 0 Q> ∀w ∈ W , Ď (w) ≤ N &[< U SVCYGTALIF&ES (U , ω) V ω uC i &{.V~6"?L# Einstein % 6" *t / T&?L[-. [ +t ω *C i %~#& g(d[}T( %S - bv 4.1 s W [ (P, ω) + ; u v *~E ^ HÆ + bv 4.2 ^~=W [ (P, ω) &}* ^ H;&f (P, ω) ,Q=;LS >M; 6 Q = ;LSd Y ; `/ [ t&l(*dS bv 4.3 ^W [ (P, ω) ^ b Y F (x, y) = 0 - s X D &+; (x , y ) K F (x , y ) = 0 K _ p Æ ∀(x, y) ∈ D & n

−

0

0

0

0

(Ď€ −

ω(x, y) dy )(1 + ( )2 ) = sign(x, y). 2 dx

ma&?L_a .V~6": n D 4.1&`lT m- # M * 7 ∀u ∈ M & U = W = M &n = 1 t ω(u) *lT L l?L# # M :%. # + (M , ω) ?L$!& # M :%Minkowski}bD *<s*dn2%lT F : M → [0, +∞) (i) F a M \ {0} :hh L, (ii) F u 1- Vz%& ` % u ∈ M * Îť > 0 &( F (Îťu) = ÎťF (u) , (iii) ` % ∀y ∈ M \ {0} &<sn2 m

m

m

m

m

m

m

m

m

m

gy (u, v) =

1 ∂ 2 F 2 (y + su + tv) |t=s=0 2 ∂s∂t

.

%`D t(" g : M Ă— M → R u D% Finsler( If : u-3 Minkowski | % #& eQ< u # M Ss6":%lT F : T M → [0, +∞) <s*dn2 y

m

m

m

m

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94

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a T M \ {0} = S{T M \ {0} : x ∈ M } :hh L, (ii) ` ∀x ∈ M &F | → [0, +∞) ulT Minkowski | *.V~6" +%lT℄k&` x ∈ M &?L=k ω(x) = F (x) l .V~6" + (M , ω) ulT Finsler #&℄~2&* ω(x) = g (y, y) = F (x, y) &l (M , ω) u Riemann # ^&?L # d S- bv 4.4 WV `hK (M , ω)&~s5&Smarandache K &{w Finsler K & i{ w Riemann K m

(i) F

m

x

m

m

Tx M m

m

m

2

x

m

m

§5.

Oâ„„\_-YbU!

p_lp % -M * ?N}a` ~0bN}%; & e?L T v (v! 5.1 -d =<--S2 On zq * , <-` h- D : C Y ` h - ` .y(E T & Y4(fT8H& u9n [10] = %8H %|7&duM ?VKxI %|7 Einstein ℄Q 6"a o du %&{lA :< y6"a fpV=u a% {fY|#%CV& ℄ |#n# k V=lAul uSgI&E-uG+6"Uu++6" A 4 +6"&9n [16] =`yu( ^ 0M q8$Æ +=j s ~|0 M % vnR/l ℄D%p2 l lF(%N} 6" ?1`.V~ 6" (M , ω) f%N} s/ )6"%N}.yrm&2>a ?:Y4 % 8H% a& ℄rj%|# vEv|# v ( v ! 5.2 n z" 3 +<-Ub%4Dd -D - p_p i -M %ry a ℄9u e Q#G%6"| &{l |f ?%uP l P % -M ? Z)Q f%" f T * ' n z `h+F*V%4Dd Æ a _ -7fY%n2d&bI| T; (lD% K}& * ℄ |# %FRNf j -=!*6"+ u 10&M -=%6"+ u 11 tHAu$%jn9j - uS|r~# l> M ? aN}% F- -%6"+ u 12 K=f A'{&.y7llF%6" + -N} Einstein 1 J&a l7:& ? ra M ? %?[ m

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v (v! 5.3 nzyV &6 " | J E-5 T D ^~("tY 4 U" | 3 U 6 3 U" | 2 U` h2Mf :u Einstein 1 Ja zV n2d%T7T l [&M ? !*2M a % o& q td k & +Me '1,2M= 8d % G*>:"[6]-[7]$,z M ? , #2MuqI z8H z 6%q {& * ℄ T%lsM %a2M \ 72Mu`%& -M =U (lA>M&S℄ u +Me Evf,>M * $ 2M >MTe&? L drm} uk 7%&T %z u W&Fy2M7>M 8ula r ^ ℄&℄~u8 GE0 >f 2M fOM9a2 :uKx a%&* % ) 0MfOM9% -& -uM nu ? 0 X L%sk!g f yQ&o?e"l)!PMeZ J=O u lTsgEl { -:!g 6kB&o2I| ZJ=OuOÆ Q%; & :De%ZJ; "yÆQ% ?; u ([16]) * (1) D Q _& E n 74D E+&n 7D* E+%"K z (2){ 4 B| %dU` h w &g9 , ` h j 7 (3)x 4+ ` h YI _ `M ?Q"&0ba2 :D k9uSO%SM&flrmS 6kB % v ( v ! 5.4 n z^5 T Y y%dt# E6M976 ~l)uM ?V%lT KQ K=f ℄`6"+ !g %9u& T; du#% .\ * 6"+ ≼ 11&vD6M9 lDha ℄ # % +:&d . a2 : H F( PnR/ ℄!g 7fY %aG 2h 7 [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]

BVI*B+7, N K , `m ?`IC, 2001.

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

W , _ < : K , ,?`IC, `m, 2005. [6] S.Hawking, 1 hn 8, F{, `IC,2005. [7] S.Hawking, q W +<-, F{, `IC,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.

H " 5R y : – M _ C , _ b_t + $, 97-121 6( U ;8k 6 Pt U) wV p= —— y " y A *1? =fÆd 3 u (% n U^Qa^9 +U^f8Q% )%100190)

℄: a*.vk*%℄2Æ;U^f8* +m"%F?$3YRAsg * %^9H a^s~f8 2$*Y& a% J<0`*m"d F%$)~ { Smarandache)_g Mb +% Bz?$3YR+ +L P *m"%Æ ; f+3℄_ g * A{ _ g : - | 3 B + }r_ g *L P % CPoincare-"*L %F o } < 3- T ' ' H) 2} S 3- T_/,[*?$^ p %# ℄?$3%FL 'H w M ~; B+^Einstein G*L /%a%R y o℄m9Ef8 plM % [a^\ *)vUU%!h!Jf8*U^ u NJ: L m" ?$3 Smarandache )_g G%U^ u Abstract: By showing several thought models, this paper introduces the Smarandache multi-space and Smarandache system underlying a topological graph, i.e., d-dimensional graph step by step., and surveys applications of topological graphs to modern mathematics and physics, particularly, the idea of M-theory for the gravitational field. The last part of this paper discusses how to select a topic for research on the coordinated development of man with nature.

Key Words:

Combinatorial principle, topological graph, Smarandache

multi-space, gravitational field, research notion.

AMS(2010): 01A25, 01A27

p. 2012 8 S 30 $) 10 S 9 $ }` MoZ{ > >+>>N)_l6S\I >Nb>NYK 1

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6

6

torus T 2

-

6

6

Klein bottle

3-4 / % Klein 0l K&f

I?=P % [Æ℄D |a& +lT [ Iz(/ [&nu p T T[%qx*&nu q TA [%qx*& d[ TD bv 3.2([8],[9]) p ~ = Y[ * 3 [ ( Y[ ~: (P0 )

T[:

aa−1 ;

(Pn ) n, n ≥ 1

=" [ + ( : −1 −1 −1 −1 −1 a1 b1 a−1 1 b1 a2 b2 a2 b2 · · · an bn an bn ;

L y: ihUA bx

108 (Qn ) n, n ≼ 1

`N ;

–

Smarandache

+ai

= [ + ( : a1 a1 a2 a2 ¡ ¡ ¡ an an .

lT [D*YE*%&.SfS: +lnn- trl ?Sv ~7S X uz,z&&.7X uz&lD**YE* lT [% Euler HQ χ(S)& χ(S) =

  2,       2 − 2p,

if S âˆźEl S 2 ,

if S âˆźEl T 2 #T 2 # ¡ ¡ ¡ #T 2 , | {z } p

    2 − q,   

2

2

if S âˆźEl P #P # ¡ ¡ ¡ #P 2 . {z } | q

=% p D*.D [HQ& * Îł(S)& [%HQD * Îł(S) = 2&q D* .D HQ& * Îł(S) 3-5 Fm * K a Klein B:%lT 2Rnm, %lTm,dD*54 V &2 : lnr.yD !&"y&q Tutte ) "a&2 .y vW`. 4

-

u1

6 1

u2 u3

u4

(a)

u4

2 u1 u3

6

1

2 (b)

3-5 K Klein 0RU_*<O 4

u2

b f 3.1([8],[9]) ~ = 54 M = (X , P)&E S^ : q A X +rGz D Kx, x ∈ X, +n KO G+% A X +~ =: % P &KG K [ + K 1 K 2 & K = {1.Îą, β, ιβ} S Klein 4- G e & ^ P S : % &G * ^ b k, :) P x = Îąx v 1 *ÎąP = P Îą; v 2 *e Ψ = hÎą, β, Pi ^ X Yv9 V & e%b 1 V" P .yÆT*e AT%Hu&b 2 V" 2 %qx( ^& Im,; alDJV: O* lâ„„ 4W; &S= Îą,β

ι,β

k

−1

J

ι,β

N `D-` `u -!%

109

%e ATD *B7&l Pαβ =%e ATD *[ k*& K aT[ :%m, 4

-

1 w u

6

+s v Yzy*x ?

2

1

6

2

-

3-6 K <Ov/ 4

.y|m* #l (X

α,β , P),

e&X

α,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}

&

P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw)

SB7 *

× (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv).

u1 = {(x, y, z), (αx, αz, αy)},

u3 = {(αβz, αβu, v), (βz, αv, βu)},

u2 = {(αβx, u, w), (βx, αw, αu)},

u4 = {(αβy, αβv, αβw), (βy, βw, βv)},

r * {e, αe, βe, αβe}, where, e ∈ {x, y, z, u, v, w} F5 54&,% urg zDlT,CRnC r ∈ Kx %2 j b 1 * 2&.y4(|"a 54 +F* Q e D z e 1 y/ a [:%m,n2 &Ob(}℄; * ! 1: >E~=4 G YE*6}*YE*Y[ S, G D Y B| S? ℄; u"d[ TD a * bv 3.3([5]) _pÆ~= (4 G&,YB|+YE* Y[ :*YE* Y[ m ; LS b X h &G } GR (G), GR (G) S G Y B| +E* Y[ *YE* Y[ m ; &f M

O

N

GRO = [γ(G), γM (G)],

, β(G) = |E(G)| − |V (G)| + 1

GRN (G) = [γ(G), β(G)],

L y: ihUA bx

110

–

`N ;

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+ai

{D 3.3 .y `& % .D y HQ" % )I D&Fy&N } %.m,; & =a DS.m, [%y HQ*.D y HQ& Îł(G), Îł(G) * Îł (G) &S= D Îł(G), Îł(G) uu K}%&rj `l â„„ â„„&* K & p K(m, n) D SHQ ! 2: >E~=Y[ S &4 G ^Y[ S -d =**Q+B|-6}~s 5&_3>E+7b&z E;b b0&S -d =** Q + B| 6 * 54 â„„; D*m,n zX2 ; D G %.D .D m, f~l* X X M n

gi (G)xi ,

g[G](x) =

i≼0

ge[G](x) =

i≼0

gei (G)xi

e&g (G), g (G) * G aHQ* Îł(G) + i − 1 * Îł(G) + i − 1 :%m, r j `l â„„ % â„„&*T ^ ) D m,f~l j Bunside & .y4(|"aWDlTqx%( G&Sa [:%F( zhzX.D 2 r (G) ** Q i

i

O

2Îľ(G)

O

r (G) =

v∈V (G)

(Ď (v) − 1)!

|AutG|

,

e&Ď (v) *B7 v %z &AutG * G %kzh k*& K p K(m, n) a [:SG%.D zX2 Æ~* n

(n − 2)!n−1 ,

2(m − 1)!n−1(n − 1)!m−1 (m 6= n),

(n − 1)!2n−2 .

K , n ≼ 4 a [:SG% zX.D 2 ** n ≼ 5 l( n

1 X n (Kn ) = ( + 2 O

k|n

nO (K4 ) = 3

&t.

X

n−1 n X (n − 2)! k φ(k)(n − 2)! k + ) n n . n−1 k k ( k )! k|(n−1),k6=1 k|n,k≥0(mod2)

"P$\ - $# 3 E G ulT I &L ulT - G :lTz$ θ : V (G) S E(G) → L& L =%C` G %B7*rf%z &k*& 3-7 =& {1, 2, 3, 4} ` K f% zD L

4

N `D-` `u -!%

111

2 1

1 2

1 2 4 1

3

4

2

3

1

2 1

4 2

3

4

1

3-7 $# }T7 - rz$ I G , G D*0 a +&* alTzh Ï„ : G → G i S #` x ∈ V (G ) E(G )&( Ï„ θ (x) = θ Ï„ (x) 7 - rz$ I .y * Smarandache B6" Se = S S %lAw- I Sh& A7 - rz$ I G[S]e D *d ([11],[12]) * L1 1

1

1

L2 2

1

L1

2

L2

n

i

i=1

e = {S1 , S2 , · · · , Sn }, V (G[S]) \ e = {(Si , Sj ), Si Sj 6= ∅, 1 ≤ i, j ≤ n}, E(G[S])

e k*&`/ - S , 1 ≤ i ≤ n z D7&S T S z r (S , S ) ∈ E(G[S]) & a = { HT }&b = { hi }&c , c = { ng }&d = { }&e = { i } &f = { Ui } &g , g , g , g = { } &h = { /; } &lS ISh0 3-8. i

i

j

1

1

2

a a∩d b∩d b

3

i

j

2

4

g1

c1 c1 ∩ d

d∩e e d c2 ∩ d c2

g2

g1 ∩ f e∩f

g3 ∩ f g3

g2 ∩ f

f

3-8 N?U 3Y

f ∩h

h g4 ∩ f g4

A ISh7 %f #a4 u$. vD&n^" 3-8 %w-S h&{ +:8SK5VuGlTâ„„-/ %#a -lAk 2{vu& g& 8 3-8 =HTB7 2- +H & ^# [ %a#0 3-9.

L y: ihUA bx

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g1

–

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Smarandache

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g2

c1 a

d b

e

f

h

c2 g3

g4

3-9 / N?

IdQ&.yfl 8 3-9 =wG %Hlw2" [0 GlT 3- + e& f% ,x& ^ # k VlT f% &0 3-10.

3-10 3- N?

"z$ 3 :[vA"lT 1- + I u* 2- + I & ?u* 3- + I &# ? L0 % f #a%~v&f :.yf%lF ?Z &# n- + I ([13]-[15]) &D *d: bf 3.2([11],[14]) ~=E- 1- U %4 G Q+ %` h OS d- U %4& f [G] &z q G K TS M f [G]\V (M f [G]) D- = M A e , e , · · · , e +%&, & Q = e , 1 ≤ i ≤ m (1)M * 3 d- UM T B ; (2)_ p Æ b 1 ≤ i ≤ m & e − e *~ =6 } = d- U T B MS&K n GM (e , e ) * 3nGM (B , S ) â„„-2&* lT d- + I % 1- + I * & S:E(n x&lD 8 d- + I *d- U` l(d[ Ty/ d- + I sg %S- d

d

d

1

2

m

i

d

i

d

i

i

d−1

i

d

N `D-` `u -!%

113

bv 3.4([11],[14]) _pÆ~= d- U %4 Mf (G)&- Ď€(Mf (G)) ≃ Ď€ (G, x ), x ∈ f (G)) ≃ 1 , 1 ≤ k ≤ d. G, 5&_3 p Æ~T d- U`&Ď€ (M sg * y % I6"D* (` h &"D 3.4 $& l* d- + u qx6" h/$% Perelman G`T Poincare {&/ 2010 m# o$ ? : e&G"a Q = (+ 3- (I* 3 S , j TS &? L.y# d[ Ty/ qx 3- #%Sh(D * bv 3.5([14]) Q = (+ 3- ( ID~T 3- U` â„„-2&.y&$ L% d- + I & a d- + I : ,$ÆSh&* ~1 S:%p_ &)$Æ +: &â„„~2& z ;w- Riemannian # (lAG+ I ) :% & ~ Re &# *dS * f Ă— bv 3.6([13]) Mf S~ = - M ( I&Re : X (Mf) Ă— X (Mf) Ă— X (M) f) → C (M) f S~ = M ( I + Y8 r f_ ∀p ∈ M f&, \ Z (U ; [Ď• ]) & X (M - Re = Re dx ⊗ dx ⊗ dx ⊗ dx & & d

k

d

d

1

0

0

G

3

∞

p

ĎƒĎ‚

(ĎƒĎ‚)(Ρθ)(¾ν)(κΝ)

Ρθ

¾ν

p

κΝ

2 2 2 2 e(ĎƒĎ‚)(Ρθ)(¾ν)(κΝ) = 1 ( ∂ g(¾ν)(ĎƒĎ‚) + ∂ g(κΝ)(Ρθ) − ∂ g(¾ν)(Ρθ) − ∂ g(κΝ)(ĎƒĎ‚) ) R 2 ∂xκΝ âˆ‚xΡθ ∂x¾νν ∂xĎƒĎ‚ ∂xκΝ âˆ‚xĎƒĎ‚ ∂x¾ν ∂xΡθ Ξo Ξo + Î“Ď‘Κ (¾ν)(ĎƒĎ‚) Γ(κΝ)(Ρθ) g(Ξo)(Ď‘Κ) − Γ(¾ν)(Ρθ) Γ(κΝ)(ĎƒĎ‚)Ď‘Κ g(Ξo)(Ď‘Κ) ,

K g(¾ν)(κΝ) = g( ∂x∂

¾ν

, ∂x∂κΝ )

z v ( v M N}& r/ â„„%.}6"& *.(a â„„.}6"=&uy%S .yXR/f f%"fn". "_$( van 1h:`hD2 O ` -a Einstein &j&N} GlF Newton % 6 |&!* "76"u}A z% M 6"V~mi44& "V~miuO fJ& ` 6|!gM9&SV~mi% 6e"* ((x , x , x )|t)&y `/}Tr2 A = (x , x , x |t ) * A = (y , y , y |t )&S 6"R* 1

1

1

2

â–ł(A1 , A2 ) =

3

1

2

1

2

3

p

(x1 − y1 )2 + (x2 − y2 )2 + (x3 − y3 )2 .

2

3

2

l&Einstein %u`-|a "76"V .Æ&/u(u` 6|& R ` 4

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114

/u` 6=%}Tr2 B

= (x1 , x2 , x3 , t1 )

1

*B

2

–

`N ;

= (y1 , y2, y3 , t2 )

Smarandache

+ai

&S 6"R*

â–ł2 s = −c2 â–łt2 + â–ł2 (B1 , B2 ),

e&c * 5&{l`m ma*8Q%*d * z

6

future

-y x

past

4-1 D 5 w r

u` 6%|raO* ) 6& Minkowskian 6"&S` %tC* d2 s = Ρ¾ν dxÂľ dxν = −c2 dt2 + dx2 + dy 2 + dz 2 ,

e&Ρ * Minkowskian V~&D * ¾ν

Ρ¾ν



−1 0 0 0

  0 =  0  0



 1 0 0  . 0 1 0   0 0 1

?L$!& "u â„„ y\ M9ZJ Jnr2rS J%lT &S M ? ur2rS S % 0"R 1 h #D2 O-? V( -* \ 1: 1 h D n zS r 7B v i +7b&:1 h P - ` A|7&f : u ` 6|7 \ 2: 1 h yD n zS r 7i +~ = 7b& :1 h *Y } A|7 uu` 6|7&du Einstein u`-|7& | M ?V% O |7 a A|7d&!* "u(XJ%& " Einstein o1 J`r& 7l%8H â„„se"=& â„„svl0u "% X

N `D-` `u -!%

115

=U(4fÆ?; 0bf%J0'&& u1 h 95DBt ^ -[N $ D n zgB ^- + #%4D2 O- :Æ iF"+ k e C , 5 + ` D2 O-.( â„„; N}| & â„„ y !g "%g9 "m$ f9 j w Einstein m 8 ,?N}=&*+o` B}V ÆlG*#| ul2_ÆK}%r~&Einstein u`-%- &a/So` â„„!g& *Xio` â„„!gla` 'Æ z`ll &Einstein u`-. ulT (!gS &=f " % rlfl 9f Einstein u`-%f9&uSy/M D%a`% u(E ([2]) * C N}v* ~=t E7+ V^ -\ 7{ &L* u3\ + \e&E- *+ I< V & z_u n O|â„„"7l%&Fy&Einstein % u(E / %& f : u [ aj 7 * n +Æ S9 Æ lÆ?'{%em *y&Einstein a u(E sd:&j $Æ + [4] =% & ~7l H V& ([2]) 1 R¾ν − g¾ν R = ÎşT¾ν , 2

e&R

¾ν

Æ~*Ricci r , Ricci Y8 , ¡ s a 6aOd&Einstein o1 J%

ι = R¾ιν = g ιβ Rι¾βν , R = g ¾ν R¾ν 4

Îş = 8Ď€G/c = 2.08 Ă— 10

−48

−1

cm

`DT Schwarzschild V & 2

ds =

2mG 1− r

¡g

−1

dt2 −

2

dr 2 − r 2 dθ2 − r 2 sin2 θdφ2 . 2mG 1− r

aysd:& 8H?E & * V S 10 l.y. 1&<-&pÆ;L *& P - * +E &Friedmann a:p 1_ 7l zd8He"& 4

ds2 = −c2 dt2 + a2 (t)[

88HÆG℄1℄* ~ y -* da/dt p| y -* da/dt , y -* da/dt

= 0; < 0; > 0.

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

L y: ihUA bx

116

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a 1929 %|#|a&8H h/0 L[&"y.S 8H ℄s "& *<-F"3 137 ~> +~ <-Æ i "P$5' y ℄rmk V= a 4 Asg o& >z F : ~

)&Einstein %u`-Ob\ o%*&l~io?luy/$|mi"uH & :uo p(uo /(uo) 4fM ? &SL Einstein g l)3o/|l ,Asg o& |l u`-*~io?&7lM ?% |l -&d u9n=q3`m%Theory of Everything & ; l)E(# T &S%}7a/ u`-uy/9|8H -&* /_ N\_ 2M)&S EM9uq9ÆÆ%,l~io?luy/ $|8H -&*:i 9i =i)&S EM9ua2ÆÆ% =f; N}% J,& ℄!g ;d`m l ; &k** <-DQ~+ E-z q *Q~&-d = <--S2 On zO*% , <-n z" 3 , &+<-+Ub03d - D 3 U E-z q <-F"3Æ i&oO Æ i >D2 O- D` h ~n - E- g & + ~n -Æ +D^ n z Ot ~n -Æ ...... p_p i`m%j -*T : ; <D sd l - Emi u 97lu+ z% p-f & Sf p T (4V%i6"& e p ulT &fgI( (&℄-lXvj (ZJ z &j -*8HpS ` l#( %1t* <-x"1D~ = ` h UbS 11 U` h Æ i &, & 4 = *U^ E G +pr h &i # B 7 = *Uf^ E GH Y |1& tISf T O)% + 4 U a <- O* t + 7 UN a <- 4 U a <- w +Z' Einstein H V&i 7 UN a <- w +Z' 3rZ V&* Y & < +1 `D~ = ^ Q = ;H Y ~ = 7 U` h R + 4 U` h R j -f :ua u`-sd:ry%lA ?w- -& ℄m( Ev`S"fn".&Fy&j -< uh1_ M ?V%N}O & (z`Q &!*H ulA %M -& j -_z ℄%'{ ;&*!g8HgQ[r,` lT =& 0bT %!g; &un^|#6"+ ≥ 4 %rM%*& * ℄rj%|# r/ 3- +6"&a 3- +6"=fm ℄-/? ` % Hubble

7

4

N `D-` `u -!%

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!g J&lAy)I%'{&u= z%|#S w-XQ&fl#G`rM% e!g& w- 3- +6"& K a >&fle S%* 4-2 =W` K : %|#e"%lTki ([9]-[11])

4

P1

?

?

P2

-

R3

R3

-

R3

R3

6

P3

6

P4

4-2 /L 4

V & e R T R ux3 T% 3- +6"&S+ .y* 1 + 2 +n 3 +& ^& .y S/G+6"|# k*&&$ m T 3- +6"w-&E V (G) = {u, v, · · · , w} &l.y8 Einstein o1 J Aw-6":&* 3

3

1 Rµu νu − gµu νu R = −8πGEµu νu , 2 1 Rµv νv − gµv νv R = −8πGEµv νv , 2 ···············, 1 Rµw νw − gµw νw R = −8πGEµw νw . 2

` %&d.y w- vD StCw-& =&D tCw-7F(6" %?%+ mb (y k*&`HT6" `D z&l.y# Sa 6aO % `DT&la mb = 1& HT6"% "V~ t = t &( µ

2

ds =

m X µ=1

2Gmµ 1− 2 c rµ

m 2Gmµ −1 2 X 2 2 dt − (1 − 2 ) drµ − rµ (dθµ + sin2 θµ dφ2µ ). c r µ µ=1 µ=1 2

m X

[U(4f0fl N}%℄ & % .yTlT9n [10]-[16]

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& Jw4GCao ,?u â„„ (!gk %.M&rm !gk %&fl f â„„ryv k &7k Ary G p_lp ,?ry%Ob &z d` , Nâ„„ a` %m|

5-1 y-l)1

vD& ( z \ &z \\e+b_℄ _ n :F g 9Aqm+Z' ( 2 O t+? Bb__ g9 v"1$Q* z&b9 v {S2O_b_? {_b_ -C \E? ?f : uÆ?& /q sd:%:$7T vA X!/?" ?G*? 4<%{v u mf% q| ℄ih&v P Æ? %Sj AP4|=&*! %Xi ) %8i) !"q = q $nlH &q $n l * &qY $n e&_\ $n ?qj&t ) 8 $n K D & n &S $ *S l &\ ℄ Z Æ u Xi S &y?R` ry h&Æ,8i o.-qi- 3_& .i

&r.i* 7 &q.i b &) .ili&* .i ? B&*6AÆ&D&

&8 ) ; )&lIV=e #1 Fy&{bG* % ?℄ & o2(lA ' a Æ&* ,? o% o7 "2$" 6F&5'jv { S2 O wg9 = d & -2 O '- T; uHlT ?℄ o2[`%; ??,Æ#Qa&!grM QJ0& z &`rM%!gd Q /:[ vD&*+{!g:9u AaG&fl9 u Hn ,ÆvA %t7 Æ,ÆgQ[r E ~ y ( : Æ%!gS -w-usd b$!&,?Æ, ur%&l#G`rM% f T u, ?%n/ { lCV`r&,N3 .yÆ*H℄* Z 1 u: _ h ~b_ & h ~ = d - a , Z 2 u: _ h ~b_ qm- a ,

N `D-` `u -!%

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Z 3 u: _b_ qm- a , Z 4 u: _V_qm- a , Z 5 u: _ n z r 7F g - a Fy& ℄N}3 /: H℄=%ul℄&K5$_ + %3 & =k`aG ℄!gk ( +%3 u0b% b HT `S? S I N &n S z℄7f% N ?S Mba ?ll ; d!f%?, sd& * * Æ&z `7 ?(y%uy?,&* -o? -M -O ?)sd3f% T&y` ?sd Suy$g(lT [% T* & ^alg?{ ?,?N} ayVSN} ;fy%z &yV z?, `SN} ;% |& yw-'{*sd&fm?,"%?& |* J& rz Æ (Æ&fl* ℄!gk v k l~`dn "3$' ^^&X [( T -= < >D E &_ L < >~E r t E- ℄!g% r()I 3,NG % r(&4}rmlTN}G d &, lJd& * ℄%N}G PualD%n2dm#%& n2 Cn SGn2 &q8S . Glnu(n2%Gl Fy& b!*q 8S LGl& Cn2&nu&$q8S a zn2d%℄i&fl# S ulAKx%N} z &d bQ Æ#& u &Æ#%!gd a r(&bA/!P &A/m|Æ#S-l Q uLd%;?OV "4$: 4" &? d * f)%+Sq%O* y LTE &%OS E-,?N }E( |rz&_h%G &" Dah dG* L%Kx!g&vA*l m#%G l kkF *n" Z Æu,?N}% . *y&b(9Qk?&b Dk?&" m#Gz& *v u " ~ Æ&z &1|C G ` ,?ry ℄!gk %Bb(&`SP99f& *.( ℄': LL& k %N} J& (. * ℄!gk `dn "5$+T ( & ( 4Z f . S n +S> X s E &f y V : , n ( = n g9S> E-m ,?N}&u"n'T 6o\O* e6 %Sk&l B 3 f :uqG* N}3 &4f? <ha z2; yfzl; nuh; %N} Fy&'lT % oT ,?=%B ; A _( . ( *y&b HTN} Gb(lAS(% O&OJ', ,N e& f Tz%N}G &z & T z ;%y G &℄~uN}'{* v& {=m#℄Ælryk %N}&fl#Gp G l u G ? ℄Cdf, p_lp &,? # 5ry& ℄Sj %

L y: ihUA bx

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aG& `k ℄ % od #Vop& 7yz &TvB :0 . _ShJ<Q); % LB& 7 ℄ .< `k %% $ &{r f( /k %%jJ* V #(y *y&HT,?℄ P[ lT0J '%; & u&V_+' ? l +D2 O-V_D Y O n z'(&O F g j 7- 7uak% ,? %N/ ℄!gk &,?N}o2Sf 7k Ary f%& d u ?℄ 0bI| %; &l l J=&w -?Esd*7l r Mp_& l Iy_aasg'{* v 2h 7 [1] J.A.Bondy and U.S.R.Murty, Graph Theory with Applications, The Macmillan Press Ltd, 1976. [2] M.Carmei, Classical Fields – General Relativity and Gauge Theory, World Scientific, 2011. [3] G.Chartrand and L.Lesniak, Graphs & Digraphs, Wadsworth, Inc., California, 1986. [4] S.S.Chern and W.H.Chern, Lectures in Differential Geometry (in Chinese), Peking University Press, 2001. [5] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons, 1987. [6] H.Iseri, Smarandache Manifolds, American Research Press, Rehoboth, NM,2002. [7] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Mathematics Magazine, Aurora, Canada, Vol.12(2003) [8] Y.P.Liu, Introductory Map Theory, Kapa & Omega, Glendale, AZ, USA, 2010. [9] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Second edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011. [10] Linfan Mao, Smarandache Multi-Space Theory (Second edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011. [11] Linfan Mao, Combinatorial Geometry with Applications to Field Theory (Second edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011. [12] Linfan Mao, Combinatorial speculation and combinatorial conjecture for math-

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ematics, International J.Math. Combin. Vol.1(2007), No.1, 1-19. [13] Linfan Mao, Geometrical theory on combinatorial manifolds, JP J.Geometry and Topology, Vol.7, No.1(2007),65-114. [14] Linfan Mao, Graph structure of manifolds with listing, International J.Contemp. Math. Sciences, Vol.5, 2011, No.2,71-85. [15] Linfan Mao, A generalization of Seifert-Van Kampen theorem for fundamental groups, Far East Journal of Mathematical Sciences Vol.61 No.2 (2012), 141-160. [16] Linfan Mao, Relativity in combinatorial gravitational fields, Progress in Physics, Vol.3(2010), 39-50. [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.

H" 5R y : – )9 & , Qx w / KHYK < , 122-128 X. % '2 R % Æ %o9 M 8% A x (& o w / 9 . & * 100045) ℄: v . JGXJ; % _ v . 2HC } n\ JN A m u ^ %C 1 , O 4 %} ' k " a % } ^v . JG f P k%' r ^Jypl k Jy U w Jy d M ; 9J JyL Jy y;P mK9JyE6 v . ~; Jy } Jya w / 8 < Z %^i|v . JG!V!J, } Abstract: The operation system of bidding market is important for protecting the State’s interests, the social and public interests, the legitimate rights and interests of parties to tender and bid activities, improving the economic effects and ensuring the quality of projects. Combining its normalizing development with those of market researches in recent years, this paper explores how to make such a system effectively operating through by aspects such as those of the goal of bidding business, construction of system rules, supervision by the government, tenderer and bidder organization, institutions, personnel, self-regulation system, bidding theory and culture. All of these discussions are beneficial for the development of tendering and bidding business in China.

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H " 5R y : – - P _ Vlq&Qx w 4 T < , 129-138 M_B U TgEp' R %1 w 2 A x (& o w / 9 . & * 100045) ℄: v 3S~;C92. [d~ ; Xh^ J;~; F+ R x XRM x/ U ,~ ; HR*%& C "v 3 S 3 r *~ ; R9 B m v 3S~; %q6JG,O U v 3S*,O3"% a ,O^ U, ^v 3S9p~; !J +kp ~% 9Q|x } <f~M ~ v 3S^JG,O*G& Abstract: The bidding theory is such an economically purchasing theory established on known theories such as those of consumer behaviors, operation research, game theory, multi-statistics and reliability analysis ,...,etc. to instruction of bidding practice. For letting more peoples understand the essence of bidding purchasing in the market system and know this theory, this paper systematically introduces it from micro-economy, which will enables the reader comprehends the function of bidding in market economy of China.

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&, 7 {kIj `r7SG 7 T G n#GpO&F#S ` pO (j lz " - ( $u, 7 a%J=k Ev GP ( &. x z f% -=(lT4P %ki& W 5n3&a Tki=& v}T k} ln z& " s&b } =2>(lT !z& lvOEvW GLDz *y&'*zd } &z aa d[ Tn2 GL=k* z q { ni { +* 4 P n&{ > . S l B- H i 5 )nTF , z q { * ni { +* 4 n&f { { {S 2 E ET& 10 ~ E,z q { T = n L n&f n L . ET& * . {S 2 E ET&{ 8 ~ E, z q n L* n&oO _ L . N N+ hTss& 1 ~ E ^& v}T {k j y Orz`r&Æ~=k `k y(j% & U>& ** ` U>lk U>&l8d 10 !&f 2 W z j z SLz 7- }A& =& z &"% uz &l z 7=z &"l*- 1. [ Whka ): z j J=& z zL[` /z &" z s5% z92= D%-zn2 =zn2*{1n2~G D&1 (t|l/ z jf/& u z s5 z92&y z 7 z & D-z#P l0bB7& $%r~ a Dk n2& z z & V yd Tr~* W %= JV` z jS % | z jz =% W " } Æ (wG&lu{1aW~G,puT z z o2bÆ% W&k* z% e&{1aW D z jz %y JV z jrzfm%. (&ld D zz%&℄~u z92bÆ%z%7f z%%l3(&)I y_ j jq ` z jS % | jq rzui z m#y %rz& 7 zf aGuS-& { z fo`rf%k =k z % A=k Qa & z 5E ) Qa &u 2 z j%q d Q",z&& z jq a &℄~u z n kIf oz & dhG z aq [ A k*{1: ( z ; + *Czsd % l&{#l: z %q a & E(S

134

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- zf aGQngS5kIfoz &d)I 3 z Ev5SkIfo5 Ez )&. x #&numz 'z) [m#y Z +& ^ j/ aGCdq Cd f ` z jS % | z ju{1q dlA OhJ -4%5V&7Cd f V .Æ&i# z jgIU%3f JCd f &ZP7ry A%1?% v* [% N Ob|ma0bk =k i ℄Cd7k ryu A% k =k (gf %z%&℄~u`

J ℄S 7ry( f %.> gf ) z JV` z jS % | z ux z m# % z Qf&=k12 & z m#% Q"&=z m#% Q+ Fy z j J=& z fm jrz% vs/ z %z &=z m# %y )lu*> zz %y JV&y" uz lu = &"%- & mx z a Kv=&'*z"/Æ~a z "yG v} &i#SE (mktd&flfm }T P!z%r% z j=&"/ : M R ℄& z z J=k a- n2 Fy&fm z j y O&5E%=zn2 F- %g n2& i z "- m#% >/)/SRl z% &z p{1 Q&7l{1M C t5 2. Whk'7): z j- & z 7=z gE-z&y #%-z J=% %*&S)I | z jz%%fm&ObB -zgE Æ-* lÆ1Tr~ -zElEl z j-z"=z % z92* z Sr`%=z x$ #G&25 z92 D%f9(b *n2& -zz% ~ 9~ DnZ[ #%Mr 27* l &Pj *T % v)y =z z92 / z92b %N f%El& =&-zz%* ~ 9~ #% Mr 27* l &Pj *T % v)QJ/ z92&n=z % z92 / z92 D%N & Dn Z[lQJ/=z % z92& z a=z ?0z% 0b S#0%pm -z#%=% Ob(1A& - - O7 *41 + D)& 5 r %dniVf% Æ- -z lÆ- `.g~-z l%~ Æ- AÆ- y=+y z#0*rz&5 (j/95n*> l% JV{l*> ` l0b#0%

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135

pm*y rz%ElQÆ--z l "P$ $Y * d * P v 6F\ L sR z jC ,zÆ*}â„„&lâ„„u.y)I#~%,z&k* Q g z)&D*ku~, lâ„„ )I#~&. "I â„„n â„„&D*ku~&k* p o ?) `/ku~&.y ? v O& `/ku~&y( v uXR/ 7 -` = 4f% 7&57y_f%=k A li# L.ylF(27l z j 4"%7y_&fl D z jrz 4 7f%=k j 4% 7(ydl ( v* wq 1: ^ [ !q qY %jau(i | ÆQ&SLS,z +| t 9fM v S ÆQ) jrz0 z&S %| ,z z& lF0n S q ,z +&` ÆQ.>i ` 0b9fr~&fl D8â„„. >,z%v |'& l J B ,? &UB j %qY7 4 9 `/fzB.i %.> j&.yn iz| +f%qY 7 wq 2: sR q D 4 A B " y_&f/=(lA( v& =v& A vu y> Yz =ky 7%lA v 3 % =v( 4nf 4 0.618 v pÆv Fibonacci =v)& =y( %u 0.618 v wq 3: 1Ef3q `v x k,z)IiF7y_% 4&. Z e t5 D 4" 7 A t5 .yr\Z Zg s&,.yr\Z e s&{l<s j0 rz Z lF yd J: GlZ w wGG *T &yta( .y ; v&5 b dbÆ %El , n &ÆY & T uyr~&ob &w'[ -&N} zd&y Geg, Z n T ?g qY*zd&f% , epZ S &` Z e 0&ay i V &Z f :.y|m* = E( n T 4 A , A , ¡ ¡ ¡ , A 0f% 7&lZ 0b GlT nĂ—n % = [a ] & e&a &1 ≤ i, j ≤ n * A , A "y_% +& A > A a = 1 ,z&&A < A a = 0 V 1

ij nĂ—n

i

j

i

j

ij

2

ij

i

j

ij

n

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& eD % aO`Ct:%C * 0&t<s* a = 1 l a = 0 %n2 Z w n ≼ 2 & zZ G% = lD l^&y 0b 8F(Z % = |ep*lT = &fl Grz 4% 7 ep v (}A&lAu *ep, lAu Æ *ep& -3HTZ lD%Æ + p , p , ¡ ¡ ¡ , p ≼ 1& ? ; * ij

1

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[bij ]nĂ—n = p1 [M1 ] + p2 [M2 + ¡ ¡ ¡ + pn [Mn ].

zS 7nR/ zrz 4% 7&? uj %sd f/=&ar z 4 7sd:` z 7(}A v&lAurz v& rz . rz "rz" alAV~y_$*frz v, lAu : gv& a j 4* zS #G%( :& D zS "%l*|m y_%( &fl=k jrz

P $0 v d Qn8. m k F U m R7$ : =zS %. (ÆQ&,=z #%-z&fm=zS %.'(ÆQ&S L=z #P o*z%.'(ÆQ}T [&u z j J=& z k = k=z &fm z jz%%ja&duÆQ z jS & ℄ z jz% .fm(%lA v "_$)$Jk F 1H C =z u . &fl k#Pfm jz%ulT.\%; &B = z w'Sh GSh749 E E℄ aG ~ 7 A o %N9~ # l o&y A41 o)Jfr~&ulA8"%ku~ fC| ? =%Sh Je"&x D u~", ℄~uv Ev)I#~u~% aShy _, flY"e"u &y *+A %'{, *A#*C Cd,?&℄~u ~ ?=%l ,zaa lAs/| ?%h-ÆQ v& A v.( / `=z f%. ÆQ Sh J"#~e"*She"wG&S=&#~e" %t( Jw* X = AX U + C Y = AY V + D,

e&X - J,z"*{1aWTv,z$wG% ~,Y - S,z"*%N 2$wG% ~,U - Jku~,V - Sku~,A - J,z7 Jku~&" X

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137

%y_&u J,za Jku~:% i3' ,A - S,z7 Sku~ &"%y_&u S,za Sku~:% i3' ,C - J,z X %O+ ~,D- S,z Y %O+~,She" %t( Jw** Y

V = GU + HV + I,

e&U - Jku~,V - Sku~,G- Jku~` Sku~% |"*{ 1Tv`%N o% |$,H- Sku~"%y_"*%N G℄497SG Sku~%y_$,I - Sh J% ~&z a J=0 dTw% Æ n %g 2;*~r℄7& h7 - C *, ) J&,?- 2 DSh J=% A A U * V &uC =z . %sd(℄ e& * Drzuja&l i3' 7 z jf uS-luXg ` : 1A=z & ℄65h8 NS8*%N8 "m$7.ki N .>n_|a D%n2* D% " & G D` % oD*.>n_ |%.'(& e& Dn2,.>n_|i %Tvn2*℄Gn2,k*z l> zl"$%`=&aG5b *a5 :%j&S.'(|m l^&F yo2,a D%n2udD, D ",.> D% G N ",=f.> N "%o &.>`mt %:&8o &` %&S.'(d= k*& `6 %`=*%j 5 %`=& tuzl> zl"$?&? `mt %:&d , D` l,.>B D%S2 %` V",z Fb .> ` %f>*S ,z%G+&)I | .>.'(,z%G+ .>n_|.'(SL.>n_|%y ( .+,(*E .'(1 b4& e&y (,.>i Et (ni d4& {_|℄JCVQ"& +. > . 100% rSt ,.+,(,.>rSt ?& k tBO2x + Gn+, bt & u.+,(,E .'(&,E .>n_| &XÆ&$ .>%|i (*| (&5 .'(y O%b f%E .>n_|.'(C &.yi :&,zn ",z& ,zSL.'V \ & Et ℄ " \ j " ( V)& kQ"&* .> ~* N&S=i d T > t %.> ~* u(t)&l84.>.'V.D~|m* X

Y

P (T > t) =

u(t) . N

lF2&* B. n z Y& : i+ u(t)/N :D2all + p 5h J& lt Yz Qf& J&VQ &lD + p *8.>%.'(& * R(t) = p

138

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.'(zdu.'(℄J7~ %sd&l&ua -, dx pS℄J7 ~ f/qYl5D& =fN} ryy qY% 4 ℄,E ℄ 7 %S &lFÆ1T$z* .'(sdzd,Zg.'(sdzd*(.'(b %.>zd e&.'(sdzdu,`.'(℄J7~ ( , %sdzd,Zg.'(sdzdull ℄.>e %.'(zd,(.'(b %.>zdu,UA(.'(,z)b % e.>zd "P$)$Y Un8 =zS %{1V sduCdM e_7E, {1OeyM El*~ hr%Xg Mf Elb `rOea`rjJ=bMf& 1 & N t&f% z , 7 %iÆj& yA G *Cdj % lQm (j, Q& 6kbP&LQ5v% D* r %PD#% N&z &a r PD a n EP?1|~#rSB 9u & nMf %b D r %Æj N*j &{l$uU r j {1M e_&"!)M YgM *T M 1 ÆhG&S=&!)M uCdM %s`&T M uCdM %sd&lyy1 yjR*y | o%uYgM {1M e_7ESL* M 9O7E, v%v 7E, M C zd E4, M W L7E, ~ Q %Ne_7E, h7M {1t5, )℄M g*M o, h7YgM Tv)

H " 5R y : – w / v4* |M> y, 139-147 % Æ %rd^ | (K6 {g # , A x (& o w / 9 . & * 100045) ℄: ui a*= )+{L= {LL =2JD%h?bliv . u a= Ci|v 3SJR% BJyE6*= 9p a^ v . u3) 6q i m-m -m -q p-m p-q i-m %*}6 {L%F^= !J *}6{L!J x% ?bli+ a= %!hi| "v 3S3r Abstract: The semantic meaning of an article in a law or a regulation is determined by the meaning of its word groups, order and in which situation it is. It is well-known that to grasp the semantic meaning of an article in tendering and bidding laws and regulations is foundation of its normally carrying out and self-regulation of partners. In this paper, we distinguish the semantic meaning of some special words groups in Article 4th, Article 8th, Article 22nd, Article 32nd, Article 34th, Article 52nd, Article 54th and Article 82nd in the Enforcement Regulation of China Tendering and Bidding Law, which are applicable for normally purchasing in China.

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e-print: www.ctba.org.cn

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142

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6 1$Y f nk 4H_,n D v T"Kw +( l &w n FP% 5 u3 3 uw K =& \ n &K = ( *) 3 3 u Æ&z D& / n 6 }, u` n _ v T"Kw +( l + q - +& ^& \ n K = ( h l w n FP% u3 3 uw Zl ,Zl > & _%w / 3 IÆ =& _%w / 3 IÆ, z #f% z jdlTTN& D =z gE-z bmulArgAL& yI 97℄ % 99e e% K=& \ n Æ& CdL$Cz,Gd +%=z== &yI C d Q ebvQ%&uL$` uCd GUu7 z zjJ(y%j y _ {I Cd Q | z z%*`r& z zjJ I Cd G Q l ℄ z~r&k*B 4 TV%~ry SG(℄ b t Cd Gb %_ z~r&~r(9 t Cd Gb &Fy& e% K =_,Æ Ob `7 z zjJ(y%j y_ &`b zy .y Cd Gb %_ z~r&.y ,DGY 2) Cd Gb & z&&l.y `_ z b &yI z Sj y_ % )$ %B Gv nk 4:_pn D* v T"Kw +( l +w / 3 IO{w / v + j E&_ & q bS4#N $&K * y 4 Z )= *7 : +&w / & n 5& v*>w 6 } =& _& q bS4#N $Æ,=z n=z (rS f9(u O&k*=z * z A&S z D SG z *=z &nu`= zz% D 9~ #%Mr)-zf9( (hG | vD& )= *Æ% udD- )= *ÆlF T*r? %fm% *r%2h Mv&Fy&w'Cz,GdB Czn z B zP.y}* fm z j% ℄ vD& nk 4:_pn=%xw )= *Æ E( + Fy& nk 4:_pn=% )= *Ælx2S- z zv 4 _,n T z zv 4 _,n D v T"Kw +( l O{ v j E&& n5+& { v j E+& v , 5/ n &*> E& n 6 } {

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v*>"Kw Æ& e% *> E& n Æ& a=zE jadnvB D=z %%*&(}A~#* ECz,Gda`% [CzZL* +%=z == ( & nvo2f% z%~r&< S z zjJ&z z zv D& 0`CzZL*=z== hGf9( |&y 09 =zS ., ECz,Gda`CzZLn +=z== E & B Cz. D Cz S &y w'Cz,Gdf%B Cz&B GCzZL* +=z== & ? D=z ob & z .ynvB w7Cz,GdCz V &B C zlDuECz,Gda`%CzZL* +%=z== E jad i & l&&z41_:n4pD D% p ℄ = n *) v - $ + s V q Æ Cz,Gda`% [CzZLn +=z== E & tB Cz# D CzS & z w'B z D=zS & z 2w'B z& z B rÆ zbL z_ )f%bP_ & z 5 % bP_ z& z B w' z Cz* D=z %%* Fy& nk 4 :_pn=%xw )= *Æ&,:[%4 A~#& ECz,Gda`% [ CzZL* +%=z== ( & < z zjJ&z z zv D`= zS hG f9( |& 0`CzZL*=z== hGf9( |&y 0 z 9 =zS .&l S B Cz*B z}A ℄ ^& n k 4:_pn% u* v T"Kw ( l +w / 3 IO{w / v + j E&_& q bS4#N $&K * y 4 Z )= *7 : &G * y DH VL . l+X [ 5 : p +& \ n *> E& q +&O{w / v j E+w KSn5&O{w / v j E+/ KS/ &* 1&O{w / v j E+& q n5& v*> 6 *>M w ` z zv * nk n9Zf%vQ. 09n [3] 2h 7 [1] ry*9P,Gdv ) NOv5Mb| ^) * *v *)& & n^ O o w / v4* F &= N`IC&`m&2012. [2] A x&D V x ( l w 4 T < :4 s &American Research Press, USA, 2007

x 0 w5+ }N? z

147

A x d &w / v b y C f y&= 7Tâ„„g`IC&` m (2013 8`I) [4] *9 * 6 &>i_ - >i >v > &,?`IC&`m&2012. [5] a7nOs&v _&,?`IC&`m&2010 [3]

H " 5R y : – K x 1985-2012 ~< 1 l1, 148-154

t 1985-2012 }( /%

' 1. /y NyuW RMI El&&8b_g9&29-32,1"1985$ 2. ?\ ?%7,ed&&8b_g9&22-23,2"1985$ 1990 ' 1985

1.

The maximum size of r-partite subgraphs of a K3-free graph,

F b_, 4

"1990),417-424. 1992 ' 1. `m C?O 100m3 BTâ„„â„„& e D V&1"1992$ 2. "75B-P$`mq Fe 50m zd& QSh#Pâ„„â„„&x 5V N &4"1992$ 1993 ' 1. "75B-P$`mq Fe 62m EvSA oi {â„„â„„&x 5V N &1"1993$ 1994 ' 1. "7ZW0-P$R(G)=3 %k= % ShN}&y b_: 'b_&Vol. 10"o $(1994),88-98 2. Hamiltonian graphs with constraints on vertices degree in a subgraphs pair, H < ;_R_ &Vol. 15"o $(1994),79-90 1995 ' 1. "75B-P$`mq Fe & Q#Pi Shâ„„â„„&x 5 N â„„&5"1995$

'2 + L y 1985-2012 =2m2

149

& QSh#P℄℄ && o 4'V N S q Æ|="95 I$&1995 3. a4Lee℄B{r BT℄℄ && o 4'V N S q Æ|="95 I$&1995 1996 ' 1. (WDLxk= %y r &{b>DV N Æ__ &Vol. 23"o $(1996), 6-10 2. i eE ÆQ ;&x 5V N &2"1996$ 1997 ' 1. n^s G$7T℄℄4 G 7&x 5b a &11"1997$ 2.

1998

'

1. A localization of Dirac’s theorem for hamiltonian graphs, 2(1998),188-190.

b_g9:

< &Vol.18,

i eE ÆQ ;&x 5 N ℄&9"1998$ 3. n^s G$7T℄℄4 G 7&x 5V N &1"1998$ 1999 ' 1. ? lM℄J℄℄w'pE &/Y%Os*x 5 D V* D M S 4 'Q =&= 7T℄g`IC&1999 2. & Q℄J℄℄w'E &3 *Os*x 5 D V* D M S 4 'Q=& = 7T℄g`IC&1999 3. `mq Fe & Q#Pi Sh℄℄, = 7T℄Jpb)s*x 5 D V * D 4 Q=(2), = 7T℄g`IC&1999 2000 ' 1. O Fan n2%lT &Y # '}Æ__ "F g V_u$&Vol. 26 &3(2000), 25-28 2. `m:oS.AV= ℄℄9~957~ &x 5V N &2"2000$ 3. `m:oS.AV= _x℄J℄℄&x 5 D V* D 4 Q=(7), = 7T℄g `IC&1999 2.

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150

–

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+ai

' 1. "7 X.-P$ Vo %l℄ % OXÆn2&Y # '}Æ__ "F g V_u$&Vol. 27&2(2001), 18-22 2. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, s ' }Æ__ "F g V_u$, 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

'

1. A census of maps on surfaces with given underlying graphs, A Doctorial Dissertation, Northern Jiaotong University, 2002. 2. On the panfactorical property of Cayley graphs,

b_g9:

<, 3(2002), 383-

3. E{b? 2.'(% $ Ne"&& oK / _ , 3"2002$, 88-91 4. Localized neighborhood unions condition for hamiltonian graphs, s '}Æ_ _ "F g V_u), 1(2002), 16-22 2003 ' 1. (with Liu Yanpei) New automorphism groups identity of trees, b_"m, 5(2002), 113-117 390

2. (with Liu Yanpei)Group action approach for enumerating maps on surfaces, J. Applied Math. & Computing, Vol.13(2003), No.1-2, 201-215.

"7 X.-P$ %.D m,%zX. (&b_t _ , 3"2003$,287293 4. "7 -P$B7 a ≼ 2 % On27 Vo & s '}Æ__ "F g V_u$, 1(2003),17-21 2004 ' 3.

1. (with Yanpei Liu)A new approach for enumerating maps on orientable surfaces, Australasian J. Combinatorics, Vol.30(2004), 247-259.

"7k -P$Riemann [: Hurwitz D %w- && o V_R&> > m: B_V_â„„ < < b A &2004 12 T,75-89 2.

'2 + L y 1985-2012 =2m2 2005

151

'

1. (with Feng Tian)On oriented 2-factorable graphs, J.Applied Math. & Computing, Vol.17(2005), No.1-2, 25-38. 2. (with Liu Yanpei and Tian Feng)Automorphisms of maps with a given underlying graph and their application to enumeration,Acta.Math.Sinica, Vol.21, 2(2005), 225236.

&oV_R&> 5 &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

'

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.

& oD V x ( l * D w N H :f y –Smarandache *` h w e G,

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.

-M r%p_lp ? Smarandache B6" -&& o V N < b^ *200607-91 9. zC e_% ?e" TÆQ, & o V N < b^ *200607-112 8.

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

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–

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Smarandache

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

B6" uy ?w- -, 0| $ :s Smarandache d g9 , High American Press&2006. 2007 ' 12. Smarandache

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.

D V x ( l w 4 T < :4 s, American Research Press, 2007.

2008

'

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

&

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

&

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.

'2 + L y 1985-2012 =2m2 2009

153

'

&

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

&

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

J z z{1 â„„r |&& ox &2009 1 T 24 % 4. z j G(g & ' E &, - w 4 T f y "+Os$& = N`IC&2009 3.

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

'

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

~w ' zg QO # – z j7kÆQ "Os$&= N`IC&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. 5. Let’s Flying by Wings-Mathematical Combinatorics & Smarandache Geometries(in Chinese), Chinese Branch Xiquan House, 2010. 2011

'

1. Sequences on graphs with symmetries, International J. Math.Combin., Vol.1, 2011, 20-32. 2. Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Second edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011. 3.

Combinatorial Geometry with Applications to Field Theory(Second edition),

Graduate Textbook in Mathematics, The Education Publisher Inc. 2011. 4. Smarandache Multi-Space Theory(Second edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011.

154

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5. Graph structure of manifolds with listing, International J.Contemp.Math. Science, Vol.5,2011, No.2, 71-85.

JOe59P&_|h7 z z{1Z%t5&U D < H&2011 2 T 28 %&& o w / &3"2011$ 7. mx z%q %*ÆQ {1` && o w / &5"2011$ 8. |l{1?| l& z zrg4"ry&& o w / &12"2011$ 2012 ' 1. ,?h7 z j -e_&& o w / &1-2"2012$. 2. {q ?`r&h7 z j -e_& 4 T A~ &2012 2 T 17 % 6.

3. A generalization of Seifert-Van Kampen theorem for fundamental groups, Far East Journal of Math.Sciences, Vol.61 No.2 (2012), 141-160.

) z j G(g & ' E – w 4Tf y "Os$&=

N`IC&2012 5 )Linear isometries on pseudo-Euclidean space, International J. Math. Combin.,

4

Vol.1, 2012, 1-12.

)

6 Non-solvable spaces of linear equation systems, International J. Math.Combin., Vol.2, 2012, 9-23.

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^4 ,4 ~rp℄J[)(,1998 10 T -2000 6 T = v?ds7M b|p℄J[,2000 7 T -2007 12 T z(rj b)&i ~rq Z Mb|O *+p℄J[)(,S" 1999 -2002 ` ?x ?_T o ?4&2003 -2005 = ,?O~ 7 W_|B7N}|{r o?N }℄ ,2008 1 T2_&= z z d+T 4 f %N}℄ Ob =a ? -M *℄J7E) ;&g?a l P ? M:r| Smarandache + $Æ +7 -M w-2 -*℄J7E~ -9 60 f8&aI `I 1g ?? ZP }g℄J z j -ZP*lg-9 &a `Ilg ?-9 { 2007 10 T k&Os

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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, i.e., 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 topics in mathematics and physics by Smarandache’s no-

tion and combinatorial principle for his research, tell us the truth that

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all roads

lead to Rome , which maybe inspires younger researchers and students. Some papers on scientific notions and bidding theory can be also found in this book, which are beneficial for the development of bidding business in China.

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