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Pfe(h—– 6[Lk Smarandache C<^r

Let’s Flying by Wing— Mathematical Combinatorics

& Smarandache Multi-Spaces

Linfan MAO

(3rd Edition)

Chinese Branch Xiquan House

2017

^r~ <QtxQYY<Qhthal xQD*~0Aht (AMCA,x ) t xQD*m(ISSN 1937-1055*)~ W~m Ihtt ~v<Q (x *i` lEQ2`Wdht'Email: [email protected]

Pfe(h—– 6[Lk Smarandache C<

Let’s Flying by Wing

—–Mathematical Combinatorics

& Smarandache Multi-Spaces

Linfan MAO

(3rd Edition)

Chinese Branch Xiquan House

2017

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, Bei-

jing 100190, P.R.China.

J.L. Cai, Department of Mathematics, Beijing Normal University, Beijing 100088,

P.R.China.

H.Ren, East China Normal University, Shanghai 200062, P.R.China.

Copyright 2017 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: 978-1-59973-519-1 Price: 65$

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4. guBE9, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

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6. d|3*)NÆÆSÆy["uo . . . . . . . . . . . . . . . . . . . . . . 174

7. d|3ICAMTPBCS-2016ÆÆSÆy[["uo . . . . . . . . . 176

8.Æ?4C' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9. g-_bu~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183N 3 ! +"p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

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1. VÆÆ'CÆ-;~INEÆ;[ (BBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

2. ^JD+Y+_ – Æ-<p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

3. mDaY+_ – Æ-<p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296N 5 ! 4u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

1. F.Smarandache: Æ_Æ:x 'Carq – JÆ'CilY`+_ . . . . 310

2. 1Æs4y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

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4. '"1GkK~MJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

5. Y`+_G^k~r7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .330

6. nyÆu\~E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

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9. Y`i& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Æl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

1. 4z0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

2. Y` 1985-2016 7;s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

M 1 6=_E

aiQ~&LQV L*,V L(?$Ay,+VNAQÆQ!Qs,"Q =a!ei, =a.NJ\&

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gGB6/i 1,2F: *7\r 1HAEMaY7QlW2W_7Q6\<*x$QDQ$,6poQYQO4DDQ$,yp~ <QtxQYY<QhtDD:6*(IJDD:h7 VCKh>NQPC,C-tL6Q,nNtL6$Q[Qh,K;_6$j:QA032EMa,7Q,V,DDAbstract: 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 study in university. 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): 01A25,01A705WL8, hBOvzy|, `WL , O5,O,& 1999 8 4 V6Ht,`X,2003 8 6 V6 $%oBdJ$Q>r , ae.y=,ZGvg&.vhgeOK\` W` V0vW$Pl7?1LORV/y, 5|Æ, ot<LUqAnÆq , ,\S1#TgnR=oOVzye'512003 7u>o?y8Æs,_2http://zikao.eol.cn

1 o. 3S\D 3BW5, eOzy|, O\/yew\#M=Md, >OR%FOw\4HgOûpL%, .?eWPW&vEp2n !v7^, eBW5vW8, h=ov?OKzyHÆ>r\C5, aSPv7^'SI&gOzyA, Oev?JTOX vAmÆ'zZBd(Dv$QBzyH7J[, hoB5X, OA`|FTIA`t, $ÌW'vAdshdvrE5 XO, s'O,!ghvBuTA, OLAB`hpA+s'&IUq,O4R|vEpm?, $QGs<O4W;!9P [Y 2'zyA<`, Gp 2'I+s, 'eBzyHoB5Xvt8, Nq=LFa?[=8XY', &.vm2lePv7^L$P, LgÙ|vBsnv, &gpjJ(P , BWv?pz 9t, 7~ JUO#, !^'A0g!v%, e!m|v- , 14'2W5z, G2W51"o6v<X.Z +8_vEp, ezyJ5L+8P, h"ov?Tv?pz 9tOFrDZYvJ~ J``XZW3h!JvVKrO%, hev?Tro,`< /t, Yot ?e d+ , vuY <`, &g[<O^Yv7:dH=_0,`X, !!êe?[v8 , OyW&I`, k9uW&~1Jv#ÆO#OvGS, h`&HC6, 2g V1, 212 dyk9vP r"=Æ218[, h\&#vmT_0,`X0,`Xh= g22vasLo#~<=I(B=v5X27, 2U<z*,!hOR, O_0X, 7<`$P, >rÆ22, adyv<g/e2vk#t 1! -, J`(, %sOy2#P, XdR, `MV<z79#|k#`%7vds77|tov/`<`, RP\, "dY/R,`<ò<gs,ug0,rgC?.0vm\7v<`7[,U0LX 5,<rL D=e6l,`<`v$Q.T5 W,382XOr$Qm?35!, e"o,`X[, R|ot6O4Y?vlm?&g#gLTp&d$Q/m?vt`(, '#W OBd,`[$Q5hgv?pz 9tvJ<sLg:,Æ7/vÆm

K 1 4<D – 0~wPZB, 5-18gG6/i 1,2F: 10f:,*\r 1HAEMaAxQ"O,b~Q1"EVMQoQYQO4DDQ$,~ <QtIJDD:h~ &15Ls VMQ,*~p\r I 1985 C -2006C1IJ<Qh3,\r hw,bxQD*IS~D-~Y Smarandache I0~C032~Q, Ma, M', xQ", Smarandache(, SmarandacheI0, xQD*IS~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, construction worker, engineer, mathematician,

Smarandache geometry, Smarandache multi-space, combinatorial conjecture

on mathematics.

AMS(2000): 01A25,01A70WRW OO t8tVLO#u 10200 /,W O. $QOoE0,hv On Automorphism Groups of Maps, surfaces and Smarandache

12006 7 3 U 26 K*%?:HoVÆhGE;nD+2www.K12.com.cn ? www.wyszx.cn

6 \3|lzz – ySE+Z Smarandache J1Geometries(V hD~ Smarandache (7[_V*v,`[oE_l.< P,e SNJ `Gvr Perez ,`Ge!oE,<3,oEv0\C Mx1 oEv,0(D#tX.CAv 2,uOL$Q<ehoEoMap GeometryO9P,W O$Q<OfOK,V)#k _ wWO,dh2 Iseri Smarandache (.m6I0N6m,,h3wb7h 2 3w,Kv3w,Q/h Iseri wQ Smarandache 61-,%Map GeometryN&I0.p,HC D*Vo`,>9ND* ho`CxQ~8?~1Izy3EYphv!^ ,mto0(DVhv,`mF4d vx`#gh5,`[oE WvORPC WL(8vEp,hBOXzy|,\L5,!O WtoG02Vv`.7,h2WPOLPFV|he4$Qmv9

(J) =5w-hgeB 103 A5-e+&;Ip()*F7vFJ,& 1976 8JePF=W=vg| M,(EP~ 103 A5-e;I)v=MWffdKtFJPf OirRH,;IW;I)WWe,H,(E555,r%|?v"|oWe,H:W!,1976 8 9 V -1978 8 7 V,h=oWe,H0W (8M)KiW!KH-Ig/eVv,[x = q,|ELEÆ,#E DH6lH-Iz-,WO9P 5|<u\HG`vxz5,sZ. m~/`zzKteP y`vg=kPF!OP07#uFÆ Goldbach673 1+2vg(,0'<#6lTp)P,lgvoE` 7|q.S~, Xh[<.Te e6WÆ[C k3Æ=Æ,(P C ?PFR OLdsV eP06=~tL8W\vzWz=B,#vÆsePhhÆv #vÆsE+,shGGhe(4YTGP +=Q#vÆsheFPLs),a6W[,#~#`P d),R|T5|P =Æ,eWO|Ov7,Wu|/RW7zBWW8zZÆW,!Gghe[=6;IWWO1rBWP)\WM'"ovBs

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1977 8BWi)Wr,htQ| 120 |ePOR vP,Xg)qUmvPFLtmvQE~,RhvQg0nv,OI fUhQ|hotWÆ`2:2,OEG3Æs,' Æs6O4`2:v0*.2&,!Gg/RvÆ4a2OEd3qs.s,[qTGgOLVe$QWvKt,>r=zvm^`p,aW<= A.m^v30*g+&`vKtePvW ae;a`FWz, MhrgH+:CIvG, XgeP 103 A5=MWfv 5 X,Mh[=reWzyW\0Bdz-WJ[BWi)=t,htÆ6v,06;I)|i)=tWv3 100 <o;IWÆ,3 50 2We 1 V,51-100 e2 V,BRXth%o;I)W 80 : 1 VÆKtWPv P;EWq~9EiU*z!OPs,,LtPF`W<v\!,M2PUhO 2slvsm^z'$mhBW 1979 8VP)\WM,t\ 18 v,WPhs0OL20y,4_v*(Æ~(Sv*xQU~hi>[vIxYvU3")svIP='13H 82xQ(OM*z

1980 8 4 Vh(iV;I)WoGvf, BWf'[#z-,hD1oHzH,6l3vzBW 1980 8i,>r2`7,k ~ /b.,a`1LzH/b1980 8 7 Vh+&;I).pdfeBYpqv0KouFÆTv*xV(II),of[#qti>[vtxVt'&gzZÆWv5| , PPÆz,'zZQ-qGMxQ~HUF|WOLÆsWPv 9 U<#~8V2,ZG,W;<v5R 9,R!~ vPFuqÆEY,7EN2:`v0*.2&,#E8pq v%-`4ZOOv5,!gÆ+vUO#,nG<2svOR=Bsg FeP~ ,213o p!-,<L&t \s\Æ,,[7t 60 |&<= ,u|EGd=RLR_22qD-A+G72q|D-7LtL'KD`7A%~2qQA7LtLKdzz P,WPUO#Gg<6l|<mGvP,!g .Rm+z `rvd

8 \3|lzz – ySE+Z Smarandache J1shR|UO#geWPmGLJ5v,!GLhL:~)v3(i) JRw-1980 8 12 VhBW5,oWzyW\Oe'eO\,BWePi,TvpYf*Gfz-sL#VLV,L7#C WV,9#ÆC L9,d!9#uwofÆ$VPÆOew\#V ,OX|PF%72Lhm,sL$UeUF%!`[,OO vw\P,|F%OY2vhe#O\,3vhe.w\8UMhePs0>Tq(-PF('Ly0 JT *tv*xQU**xQ*O'W5|0q?r,toMvhMGhe|P1r`ÆvA47u[$~\_8[,>|\,vM|TO,MvEY, hC?v`/vl m^z!~6lD !P>rdyv5L ,a1eJ(`Æ,ÆWPvW=Bz0,'PÆT vOL~ePhTF0e`TTÆ,M.UhOj32rtH-FU7vxQ~HW4X#L dov?V,G(MhUhK2'Jv ,;qUvxQ~Hv`v6.xVY2gv.x'Vz,LhÆOL# q m5O8`[,R|Ged%gX!5O<,sLe!JZGvg|O,RÆughvR|e ,'2to!Jv`r,&gp BWOt,k ~ /b. 10 G|,a`1LH/bE9q(-PF[,M=bxg, hOKH/b>rS16HÆ,R|m` ||,ov?Yvb`($ Zpw-& *= 10 C `v9,n!JA|Fl,WzyW\e'.HWzJHrO(_?A|F,hsOKv?zyH,& 1983 8 6 VBWWzyW\Oe'O(_?|FNt,t 1 vt, vgjv?!OW`nW`ve,7vÆ&bXvO_<,1983 8 9 VH 1987 8 7 Vhev?yzHJ

1 o. 3yS\D 9.zyJz 83-1 VÆ!9Pgh6$Qv:9seH,,1985 8Ae~xQh#Y(1*%n:7nx4 RMI Bu,~xQh,29-32,1(1985*(2*Æv ~u,~xQh,22-23,2(1985*p` B 1983 8 10 V ,C q(-PF'(,hev?JT92^PFAmÆTJv!RP[ÆxQ~H2 x|G xD*xQhVz,lg<,Mzm,hhO ÆJ#FgTIiPF(')?`T *_pvehV(tjV Bollobas t*,O 73 , h[=Bd<$Q, eA.m^# o#5eÆ:bWzZP92^PFLhp<MeÆ#RsR2 vs,r[xWRsXK,d2 AxQ|r87,&gMeJ[#T(123*.(132*RKx' 2 s V+G<Xhr`v~R,7=I&b#'|&b#,GOem^#!êuoAv ,h[=e$QW&qds'm^pds o=5$QYvopR,6Rtovkgo<WvORÈdsOy,92^PF"=OpYe!KQQÆ#v<` wQ7~$h~A%~vh$0O,'h i4w%9qKJ ,0'[3V*,IJuxQh e92^PFvAm,C d#kt5,hot`|W R(G) = 2 $o R(G) = r vQi,7| c2(G) = 4, 5 $oOYQi cr(G) = 2r, 2r + 1aC d#t5,r`to=!O=&+RGV=oOy,hrfVo6N cr(G) = 2r, 2r + 1 OYW`1'20n,Xte_KL3 vYWdl,2=yXkOR R(G) = 3,RBWJe x Xtc3(x) = 8 vW`,BRB=v67e!O9u5 c3(G) = 6, 7v<,?[`h92^PFvk,0Tp<òOLo,ar&=!eWOp<`v5C|gIiPF,MG8p0h!ORB<$QvJ8,'=bzgME9htjtV Erdos Roi67,Ah1$QOC =RVv$Q,htoOROYp,>r`1rp!R67,a3vds\CB67E06 1987 8 O =ohVQpQ6 e5:Aez,h7dBW!o,XL<`GU

10 \3|lzz – ySE+Z Smarandache J1IiPFM$mhBW!o,'e#/6loE1987 8 11 V,W OI=W`Æ$QF'(,hV0t-<V[=Lhv,`mFvW OKv4d Mzoh|!p<`"dY06!P'xQOp&(DQz\v` vÆ$QF5X,M|<`Uh6l5Th57[/#T,/ GGsR8ph!OR|!p<`e'xQ#& 1990 80\YePv?yzHvPF+8phs,lgW38vUO#e`#W,PFOYG2h,Xth|vP dv? TXOL`(,dv?JT.92^PFO i<dsBW0_`vOLi<VÆ, 1986-1987 8,X[BWÆ$QF('ee*rv Kac-Moody x;Vzv?JTfY ('NiT *rv ~m~V~0A;Vzz!L&h.y1reORÆjv4[(Dds o2F5() Jl3K1987 8 8 V,hoWzyW\Oe',|e/e'$<[A~AF J|rEgTI(;-IImEpI W'vA+she 1989 8 -19918 8 VBWv?;OOz-`41991 8 10 V

-1993 8 12 V=LA>d,BWv?>u9 zz-41994 8 1 V -12 V~WzyW\Oe'|e'<[A d,1994 8 3 VN7=LF41995 8 1 V -1998 8 9 V~v?p<[4AW`< F,/w[LV zL41998 8 10 V -12 V~Wi$D< F!ORP$QzO4W; zO4`_IA+sL?~,0jÆP=IAv$QTho9Av$Qz[eI`Wp 2'`TvA+s,'zZe0I3YzyA<`[eIAIRBi`mYLGp<`,'ABWEM*MD^60A`hEM*M6`h 27 MvT>r<,Æ"oY~ `v7^'Ss'1990 8f2t8v?pJ~ vt,`_s htJ~ `v7^&g,hB 1991 8 4 VzZBWv?pz 9t,3 1991 8O8XOp 1`~1|=B

1 o. 3yS\D 1125<.`I#,< E|mz 13 g,WOtl6KtovtgWg 99 |,Og 98 |,Xsv?TvOLsPF+8o,L!v,v|= U4!,o 1993 8#_8t,hXj"oJ `,R<Pf7K3Bg!OPGeBW0vOLo1988 8ey0)zTBW bo~ D*IDI ;V 41989 8e JkBW O 9ohVQpQ6 z

E 1.2.1 1989 C_# O 9ohVQpQ6 (1q*YzgWd(mQ**719938W,92^PF=b,vh.MO & 19948 8 VdUBBW O pohVQpQ6 ,!h#|HK:+8v$Q`_N =h!Pve \Co hamiltonianv$Q#C 19918YeO<`G#

Gould Op ` 4/<`v$0,h07Ok& hamiltonianv<`,|e*hvQtQÆxQhYVz`G#YBW 1994 8 O pohVQpQ6 v P,hV92^PFvT ,v?TvF , Mg0W<vU|MvOR H.r9h,Xg $BJ4i,J4,|V<3-. h& 19958#_87v?pz 9tOFvKt,& 6 V7JQ,"ov?TZYvJ~ ```X,RWJQ((dXgv?TvF

12 \3|lzz – ySE+Z Smarandache J1

E 1.2.2 1994 C_# O pohVQpQ6 oMp37!RP[e'xQxQhYV>PxQY0AxQz#Y 5 p<`1994-1998 8hev?p<[4AW`\~hF,`9Yv& hamiltonian vOL<ÙO#ge!OP7v N'V,Dp,'QePz2'rXoh,AhdMh$ 5,3oVBsr`dOKrv%,1995 8ev?Tro,`< /,Y'W(D<m?vt hr ,vuY <`&gR|[<O^72b[[QIO4DDQ$!B 1996 8 ,hvÆ` ,`<tL 1996 8,WzyW\EÆvlQK=rh~|,'T xQM'O`6lop(r+t*,\& 1998 8bvmTÒ4d v,`<(j) 5^1999 8 4 V,h6vmTÆ,zZhv,`<<O8eiG,|EBWTÆ2,/R,`Æ'28o2!Phv5ueWzyW\Oe',aMh27hvÆ,EYh/Æ o~B<0vMo: fhd_0,`X!Æ2,hu|EdR+2`Q#Vz71999 8 1 V -2000 8 6 V,h\~W^#zèqF42000 8 7 V -2002 8\~b,q~e'< C`<~z2'T ,GzmR|1r Pz^%Mm\,Bd^5v<ÆÆj6l,`<`QRt,ORg7_?mEvÆ'tÆ6,,uORtXgEe#YOv<`6eh_0,`X:3uG$Q5`Y,&g|eWzyW\Oe'5,[

1 o. 3yS\D 13=87vOL<$Q5,/`=Go0OL#Y,Q#EY P,GevmT4d Am,zZ-<4(DvÆ.$Qs=T:3hv?TF 5v9Æq`W,e6,`9ÆvW8X=p<m^3OR&(DIv6,tomFvh!R[=exQ?~QÆ#Ye7!p` v P,Y'`~Æv op[<OR v+U,!Xg[=YexQy##v$p` B=edpUO5ds,gBdv|xE6l|v2001 8,hh=R PsO PFv 'QQYgG~0Av,C g9#d#,#7e!B,d+lv ,a6$g2t,=TOp<`7[,h=<vm^,DMvTOp&eipv<`#d,ePGòtx,Xg7OgR\oP,hv=R vE9ht 90 |, OY to 80 |XL2N,M3UM?Rglv,`< 90 |,,Rhugve=X vxg,hot!p<``"o0O:o#Y,!XGU~ VCQÆ,eW8XY=E8p,Xggv<e#Y` GgÆ1+vhR|'26!p` ,sLN=2#Op` a,:hIY'!ghYv` Wz5,vOp` ,G[=*Y,v%-#72'6O4v$Q5$QXg!,z52OWNv[?,ZGv,g v#v|G, Xg`P7OL<è#YRhue4d vkm, G`+#!RUOds&e0,`3\CLG8v8V&V$Q|,/R,`< A cen-

sus of maps on surfaces with given underlying graphs(VDE%t2h h)gCh?v%Mm\T=v, rE=p5`<#(D6l|YI$Q,!eO#Gg$e3*v<`7[U0 10 X 5,<rLz!eWOR vP,\~,`<`QOFrvg0tVW OvRk eQ3 20 y,ME94d 5>2hv<`2#,v4dPF`_|5T,!BvQXo^6lhoRk ,|he,`<`W=vmÂ.0_tovrE<4O#e!mv6zz?M6l5v'(P<[,t%O,Lhv%-m^Æ:4d `3Amv=R<L6v0_,<O

14 \3|lzz – ySE+Z Smarandache J1`<[?,&g\rT<`v1,'X(Dn3hqOL$Qzo

E 1.2.3 DDV*Z237N4k(d(&i_#W+( ) P?'h?ot,`<`WuGds47^|E6O4U',G|EOvtI4P dU',!e,`J[zZrX3,`[

E 1.2.4 2002 C_# GixQ" D*&( (/"$*NM#W+ (&_k(d2002 8 11 V,ev?TF rvi<V#,h5 A dynamic

talk on maps and graphs on surfaces–my group action idea(wQ hYhD#pAÆ5 - 1VQAz*vDp$QoE, to0 lvjiV

1 o. 3yS\D 15

2002 8,W O. $QOhv,`[.R,'n&W8zZ,`[$Q5&v? 2003 8 x9,h=o 20038 6 V:6W O. $QOzZ$Q5D5mF|$QF,g0<$Q5v#|:OMR|rEBdo<v$Q,he0,`3vG& hamiltonian v$Q5rMv9Or,|PFXh 2 4r'Q-hMQ,1K(,7-o!Xth#vP |,`9`$Q7v5$Q7, PQbR|7^z_v$Q3!GXth`zmFv%-,BR3OL_v$Q5dU5!-gvMvEY,heW O. $QO5xoE2 Active problems in maps and graphs on surfaces( hYhD#p~ix.F)

E 1.2.5 2004 C 8z_# O ohVYD*xQQpQ6 (!K6&*YÆ'*7(_#O =ohVQpQ6 1109:,n11RNAMa*,F=&1.II,<QtIJhMQ ,`J`[,hO=e%!Rds,Xg4tir' : -CyhVL2qA7'CxQL+Yp7!RdsGg0G lC dhvds,sL4dPFvG $5W-vm^$`<ds,G|2#&g,L4dPFvm^oN3,vZGv|!^m^,BR$oM33OLTv $Xhe,`[95v

16 \3|lzz – ySE+Z Smarandache J1` ,!Ggh?vO_v~L<,heW O `Æ(D<3`2vOLt,TgIE|=B\R\R=BW.;szz,'z/$Q!PrE7^,g=(Dm^$QCds, 1r[<Tv9Op< Riemann D Hurwitz %~D*pXg!O%7veuu'!p<`7[,06z Wo~ <QtDD:pYQmQ<QpV0,XU(O\CU?,`[nv$Qm?,he 2005 8 4 V7v,`[oE On

Automorphisms of Maps & Surfaces\C2g4(D<m?v<`,NUO#\CB`hvG_< ,R!L< ))7hvO^ ,Xge(DV=, (zxQQ<*y3D*Iny3D*E,y3y!^ ehv,`[oEW3zR,Tv5|Ed3,`[oEv,[O Xgehv8VgI0,l_E|=B\R=BWG=(Dm^|Ed6O4$Qvds2005 8 6 VhBW 2005 hVYD*xQ vobj QpQ6 ,'5 An introduction on Smarandache geometries on mapsvoE!poE.heW O5v,`[oE On Automorphism Groups of Maps,

Surfaces and Smarandache Geometries[=LOer?Q i#kSmarandache =B!pz`(WvpB`(,`[oE7[,hot/"d\,Lh(Dv!^ ?b|e306e 2005 8,eOV\,?hQC, hXL,`[oEYUMhMhL\!hv ,zoheoEWyW Smarandache =Bv0C yW4x1,/& 2005 8 6 Vee American Research Press \,0\\

(.) 5ZK,`[$Q[,&h\C 42 B,v?`OKT $%oLf%hBd$Q &ghvEY,V|d5|h=| <3h5vb,q~e','-tW O. $QOv f,hr5LMh$ v$Q|FBd$Q5:K`s!-,Ogh27|hez-3G8&VvC6.8Vs'(d8heiSV5Opr5grOL&`v8,bTGL2F*4Wg 30v $uMds,lg A[Jn%7v9,Xt# $Q5`:j,2d

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Theory(Smarandache "I0~V*,)W=B`<4y`P,=hv(D6l_v$Q.`zhR|v/igT-b#GR/i,v/e|^,vu/eQ&W|Yv|uonBs, W|YE7VN/R/igOtL1+vd, sLW|Y2ovUGUGhh!R/ivMg 3,.N/i 5|Q,QvN M1g 3,G1T& 3!,e!L/iWv5|yRbhh!R/ivR,ahh2oNh,sLNh$ehhtov 3RM`2vm?#,!XgDyR&DyR,hv gW#v|Y21to,sLNh2$ehhNtovm?M#$eM WvN1<y/W#|vN E,sLNh$v MW|ve! #,h2 f 3lv$^LW|Y` WU6m^oDyRv B`<#,Smarandache =Bh) Riemannian =B,BRh)As"_

18 \3|lzz – ySE+Z Smarandache J1k/<,aBU'uO=`KRhY'=hv(D< ,u`h2pG06l`z.$!~\& 2006 8 3 Vee\ei e?#e3!~,`xzi`b2006 8 8 V, hBWW*i(D.<T, e!#, hhv(Dn674tovW=B`4(DmvOL6loE, to.vOI6,U.OR`Ev"^, $XgeW|E O(DnvYp-R!G0gXWL;vOC:-(f) x<!w~BOR`v $5+^2zV|v7.`,hG2l2h&

1990 8ev?,.h eORzye',GS 1993 8<eWLG8BdÆ.$Qv W,to=SRmv`9(E3# 8FsVy1LFXzZ,23 BX !:,#bR!L8WX$m6`TzyO,VÆ,8[s -C `ÆW;,=o 1979 8H:?Mh!*,<0Y `(E!^ehKd vlLh[<33v9,RVFh!^>v`&[,Zgh #$Qp- o2 v5

E 1.2.6 2004 C 8 zY5$OR!K6&37`#ghBOXzy|C G8vgEp,\ #$Qv2~b:- >r.,aZGug|<[?vu'e!OR W,GPF,h2W9vPFi ,h #!|<p- o2 vG65h7,!/Gg/R 9v4

K 1 4<D – 0~C))ZB, 19-31

gG[L!/i 1,2F: <hbMQNMF,2qEFeO,2qEFK<IOzz`,hal*p\r 1HAhVQ<,Id_hVhwyhV,wy61h,yQHyQhw ~($~V?~h,<hMQ~n(D-EF,n(E<Q<IYn~,KA%71KAd* hpMQD-7,L2w=fhEF,~1Y |Q<Q6Nw>,6N<hMQ<I4s5 oC,*CCJ LQA03: (Dn67(D--<Smarandache %7(DE|=B <y` $uMAbstract: How to select a research subject or a scientific work is the central

objective in researching. Questions such as those of what is valuable and 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 re-

search world and exchanging ideas with researchers are the key in research of

myself, which maybe inspires younger researchers and students.

Key Words: CC conjecture, combinatorics, topology, topological graphs,

Smarandache’s notion, combinatorial theory on multi-spaces, combinatorial

differential geometry, theoretical physics, scientific structure.

AMS(2000): 01A25,01A70

12010 7V;_d-yÆqJ42www.paper.edu.cn

20 \3|lzz – ySE+Z Smarandache J1WR $QvxE5gss`|RO2Og<PWvs,`7mFEYv ,"oXL4Wg.m6l $QPvs,`2XO3*,7z0p5, z-&Hq|YVr1pL!eW,p$QR_'2`E, Ev,e&1r.mU;R_ds2?t$QRB$Q213e?vds 3,RGBM/3b?|,2XO$Q[e,6Ri?.lv$Qm\7 |Y,rKG6vz0p5!g $Qv0n:! ,h7XR|Bd$Q5K v,7(DWv<$Qo-<-$Q, -$QoE|=B`<y`$Q,2;h2;q_v$Qs.2v$Qi^O u,.TVi:E 1.3.1 O WohVYD*xQQpQ6 QÆ5UO#,!$Q:-u'&he O WohVYD*xQQpQ6 (2006 8 8 V,)zT*#K5v D*~YxQD*IS~(Combinatorial

speculations and the combinatorial conjecture for mathematics*oEWqv(Dn67,7~BOg r`6l(Dn6l(D`zhR|Z6L!R675O^ $Qv D*~72(1*C (zxQI<Q,MLsPV~$D*ZAD*oy3ExQV~I~,kV~I(2*ZAD*o`CxQ<QK[~W%D*d_t2y3D*p,IQE!xQ<QIY1Aa[ xQD*(Mathematical

Combinatorics*oE57[,=XWJ8z.h6l|i<, Pe?#hG.2=X ,h2'~Orkr Lovasz 6l,Z`hvb`,'eR|\C\v hDY Smarandache(7[_V(Automorphism

Groups of Maps, Surfaces and Smarandache Geometries, e American Research

1 o. 3E++\D 21

Press, 2005 8*Smarandache I0~V(Smarandache multi-Space Theory,e Hexis,Phoenix, AZ,2006 8*# #,& 2009 8ee\D* ~(~xV~0A(Combinatorial Geometry with Applications to Field Theory,e InfoQuest, 2009) t,=(Dm^7(D - - - E|=B - `<y`v`<)\z-5JeFZK5hgÆkBd(DW<$Qv1983 8,h=?eOKzyHY,aR|vG?e&,&gev?JT92^ AmÆ(D,Kt0vgJ#FgOv Bollobas t5ehV(Ii_p*'zOL#Yvt6l161984 8 -1998 8! 15 8W,hOIeBdvopR$Q5>r!OPGÆ -E|iE|i\z,a`3 3$Q,Re<m,u[$Q v^`5WòiHamiltonian Cayley z,'e0#Y <`! vO5|$Qe0#vY,O=&oh_0,` ,GXh1rj7,`9|EezY<`v,6,`<`Q!PK$QgLv,GgJ(v! v$Q5G`nvm?,R|UO#G2N,\!L$Q57[Y YR_,3gLe#Y<`5O^,,O i`,R!^7ô#bRIL8WU' =utR=R1+eP0#L<$QU ,&g#OLrE<<`6lTgIT<<ìo&C,6H`~B`,3g K2Z7OhpXRY5,:OLP<L0<~|X2#O<$Q6Y8|hR|Z\L,#bRIL8W0#<<`vg-.x,N=mIT2Lf $Q<,sL<`YU+>r[=OL(;d8e0zO O hVYD*QpQ6 ,L"<3Og(,a#~#+gOLPUO BW, ve.GVOLi ,L<`Y#-vO-6WLObR[,!^"G65GmEE$_<E2E$Q,E2EX#Ov$Q4[7QEgv20078,hzeOR(D<\ Y wQE*hVh6OR,Wh!Rdsv^,! 2

22 \3|lzz – ySE+Z Smarandache J16hVnD*xQhKUE.F,w>N|3,32qQxQ<Q"W~l[,MF01EJQD*a,M>D*x.FIh,3h3IiL0AIyD*5.*H1K/.7|.F21hA.FCxQ<QL2qA57C<QhOL+Yp72qK/.N?Y9s^zÆ1K-hhEFL2qA&EJh51 t'%AdVNC,[1Ud,X:*n**— zzN :D*hb[*"3 :D*h,t'$BLkpMAxQ"W$,,Khm7,K ,hVnD*.FKv,Q SV,'_.F%_,K"RZ7V*QD^QQ y3h[1t'D^hEF0*xQn<Qpy3,KVaBQ%l QU&xQn<QY9,ND*QIs o,hD*Q*.FzK?*2xQt2,QÆajL$~QKMAxQ<QZ#0J33K[EFK[xQnK[<Qh.FYd,I~ZmD*Yp,yQCxQhO-Yp.K(?=(?V*nxY4"9Qy37ho*,CAaZ#NtL5(DUO#gOgTd,g/R :!^ eO#Ggo~bR:to ,Re0,#bRIL8W$`%v(CeWUO#g(DYI,zPvUO#XgOg(Dvd U2 (DhW2 vrQi,6RVrvm^aT.vg,(DhW2 Qiv3^eW'`toL6vP,RgXe2[[,4Mhv[sh/b^?O,|(D5/R :v Gto0 ,hR|w:L D**[C,G/b!^#e$%0$Q,&H $Q,6RLT ;T6$ieFI*5hv-$QZ& 1999 8FB4d Æ-<-<v>`g$Qe#vpR,lge#v 2- g96pR,.OY<$Q2 ,-<|E$QezWW-,lg|Y4pRmv8V,TLBsL|EÆGv# ,0$Q-<v|TG=I&4dz v4<ePO FB4d v6=X,rE$QYds,OY$Qe

1 o. 3E++\D 23#6vpR4,NI=4OY$Q2 o6vIds,rEgIUOY#WvIdsh P$Q!Yds,ae u7DN5H hxyN2q7Yh#pwYN2q7=W|Yv7^,htORe#vK6, Burnside z`6l o|Y,f2Y'UO#ÒRje\I=ORe#<vW,&gOIUO#geI=v opvj =p<mÔYo^toOR2Ee\,R(Dmû`Y'OLlY2Ee\!Xg 2003-2004 8hYendD*m0xQy#xQ?~QÆ(0AxQYz#` v=Iv6urE$QOLlpRv6,dRrLev 2- g96zJep,NI=dsH 06O~ Gross Tucker tv~`vyhV,%=/#[6l$$0Y ~g7DNv,NI=m^,Y'!eWvL;vW(DwFUO#g-W 3 vORlQi,/èZTgI0!^7^C:8Epto5,GXh!8eOLO#Yv=mônrE|D`<` 7,`< VE%t2h h'"o,`X[,ot-O&,!PGP =d%_0,`X85LR_$Q7Kt$Q#L[?7:Y'K5E2#p&lg6W O. $QOX?|$QFBd,`[$Qv8,hOItQ!ORds,XgyhVwL2qA7'yQ,0nMAxQ~% (U7!Rds=ohv,`[$Q,lg$0OL2-<TVvt5:DOj =,-<UO#g7(Dvm^$Qp,6lv(Dk!Xn$0v-V2d$Q-<dse-V=,2 g2Iv 2- Mi,|Yds\CtoL6vpK` 2- Mi#vdse-W2`r,5G2-vYZTvG65-VZvg n- MiRB4,lg|YVr#|E,A|EDNOYMvioB,ò,`[,hO=e= o|YvmÎ2W,7e#6IG5eeP\POO~,u`T2'z<àe7N-<e-WvX[,hC-<v$Q5,>ru!mv6,GrBW0<.(D!GghB 2005 8`[ `T <4-<m` vrEBs

24 \3|lzz – ySE+Z Smarandache J1DN-<v,\ v,G=Ghq(Dn67,sL-<UO#Xgv(Dn,hv(Dn67ORp25!^7^, g(Dn67Bh=u'ehv,`[oEV hY KleinD7[_,GXg 2005 8hee\v hDY Smarandache (7[_VOt 3(W,7!9p2For applying combinatorics to other branch of mathematics, a good idea is pull-

back measures on combinatorial objects again, ignored by the classical combinatorics

and reconstructed or make combinatorial generalization for the classical mathemat-

ics, such as, the algebra, differential geometry, Riemann geometry, and the me-

chanics, theoretical physics, .... (~ID*Q0AQ'xQ~W,I`ND*Qh1,ZD*Q~$=[9,CxQQ<,n x ~(Riemann (~Q~V?~... y3EnD*p*!^%7G=Gh$Q n- Mi,6R:81reO#YOL&(Di4#~p pm` \eFSjI2005 8.OXe v?#%J,Xhp`B(D? Riemannian =B,'6R6l(D$v$Q5(J*Smarandache jIheW O. $QOv,`[$Q& 2005 8 5 V,re? $%oT5, OVXLfv `^D1oBd,`[35 vOVe'521|5L<J=3,R|otL`,GO^v#Y'v8o!8eOV\,Uh= Ot,wDEW Smarandache =Bv0,Mh`vhee\hL\!R|e,`[oEv$^(Dv ,L>r21<JBd$Q,a/L?Yvrvb|8E,e3 K(&g!8VhzZ`_/`,`[oE,yW5|0YUe$V\e','E:heWt =(D i< Riemannian !Ve'v/ C[,q Smarandache =BK Riemannian =Bj,zoh$Q[yW0,'q Uh/`(ePv Smarandache =B$Q/ez,DE1on!^=BX`Y&G8$Q(D-vBs,hÆÆ&=

1 o. 3E++\D 25(D%Mm\$QMhvds,Y'MhqvORSpds=~LQ&geWyW!5|$Q[YUe$V\,,\,!~&2005 8 5 Vee0\\!,e,`[$Q P,hXee\O~t,G0,`[$Qee\tvG" Smarandache =B,Y'!mJeTv$Q5,/<`t5ee\Æ , shP$Qe ,'p? Smarandache =B$Qah!Pv5\Coe',L2v2#Lhgu V1v,h.W O. $QOm ,'|hee\v$~t.UMh=XmC $Q,OI fh`P`W O. $QOkY<`\tBWo,hv$Q5q .T7

E 1.3.2 2- Smarandache (Smarandache =BgO^):v=B,7EYe!^=B W,ORs` Pm.2m,g`^`#m\2m,!eC=BWg2Lfv,ot!gO^h)\EvuC=BÆ63$LerZpRv=B ,!.|hK[v`8/\,otDrZv:g7ev,G:g`7va28!^rZv=B o<er#,uge|Y,r21Jer#YKQv%p,0g!^rZ.2r.: v\E4R|Y,Wk:T.yUFiJ`4y*Vfv\E,uGX|Y,2;L,G6|Y,64.Yh!^qv=Bz,sLNZDr#|Y,vUOQ+,Gg5|E=m\DNv&ge=vP ,h$Q Smaran-

dache =B\,lg Iseri =~=Bonv$L Smarandache =B,hoteOY#on Smarandache =B,&gq=Bv2:'z/$Q!L$Q$ Iseri vonm^,GLh[=$Q(DE|=BR#

26 \3|lzz – ySE+Z Smarandache J1=BUO#OYpon 2- M Smarandache i,R&OYpvon n- M Smarandache i,O#p| 2'`~B!P,OXe dh1Q Smarandache =B. Finsler =BWeyl =BvMhB=o#L Smarandache =B/h2[^=B,aO=U25%v&heJ= r zm Smarandache =Bv5,hzQ n Mi,'e#zm`MD,BROYp7 n- M Smarandache ion5'Q# h),to=Xe v ,'e?[TvO~Wz !R 3Oer?Q i#k Smarandache =BOiz!pB`(,Wpgh7v(i*B;~ZKKq$Q!^qv Smarandache =B,hfVoUO#ÒYp6l!ûvz-,LR`#vWpt3g4 seO ,6Rbivo4pRe7=pY,<`[,hLhv7Ê8ei ,MhE9h!^7ÛO#gO^lv Smarandache` OR Smarandache` kL n R2 v',W/ n T&z& 2!,Smarandache=B2 gO^lv Smarandache ` B`<#,|Yt1tv,M,D15|B2 4Vr,7toOLr,RBe|YV4+g0nv,!0tv gaK^)vp`$_rv0nV,2grv,$X/gKVv,7KrlLl-v'2D,!GgO#:8L`< <`<y`vO`<(Theory of Everything*vBsaOYp32 v'22 gOR2Dds,o^&rvk2 v'`G^m\,R!^\|X|E%v&(Dg<m^K`hO=L Smarandache` gO^j(D`<,RB.h<3qv(Dn67v7^2RD2 hL,(Dn67gO^Ùq'0mv`<,<`zm0mi_v(D`<R233e2Dv'!RO#C $Q,'Lhv 6l,h& 2006 8 4 Vee\W~tSmarandache I0~V,h2h&W` Smarandache =B=BQ~=Bzmv$Q,WT5<3zeeO#YeOLzÌz[W,!5t<pLW=Bmvt

1 o. 3E++\D 27(\*ZKSjI`<y`,lgzp``<WvrEeg Riemannian =B,K`7E| Smarandache =B#(D`:&`<y`$Q,Vr,37OLp$Q5gKK2rv,!GghO=e=X (;v$QF(qvXfrB 2006 8 8 Vhe O WoD*xQYhVQpQ6 #q(Dn67:[,hO=eatzm(D Riemannian =B5,e4"p # #q(Di2:'5L$QA=#,NXgieU(Dov (D,&gX-(DiE|(Di:|3$Q(Div-pR, d- sp,#~p p-l-z0,[$QE|(DivE|pR,?``2E|rBRiemannian 4omzE|pR$Q,!^$QAeio`pR,G|E P=ivOLA:1| N,!Ggh,Lfov,sL!:g(Dv~= Op!Rm?v< D*61(~V(Geometrical theory on com-

binatorial manifold*dP 50 (H,& 2007 87hR|L!p<`(DE|=Bv# ,&goe Trans. Amer. Math. Soc. #,/L&,E9h 3/&vE|=BCt8:0Y27,zoh1MyhU=Xei db,d#Ly& Smarandache iv<`MhE9h4O~=B.-v!Y<`,h&g|!p` d5C|gyvr,~OXLv=BVMUv[,p!p<`?[/5UhY=0\vv,,a P&O\z8h!:8p!~gEvzv3o!p` vz0p,C ./i[u[,,[C 50% vz7&\z!p` 2007 8e!~#! ,W:!^z0pv` eOL0mP Ækv#r2LfxvBs2Ogv` Y/4Wg0mP Rk,5C|5pr/4g fufz<`r4...... z,ehOX vzo,h.ei ,d1 vhee0mO~,gx(D.`<y`mv<` p` 06<3he~\O~<`2xQD*V*M( I )(Selected Papers on Mathematical Combinatorics*, &gp= xQD*m(International Journal of Mathematical Combinatorics*v~`L!~^)W`n,=G.U.6l(D,

28 \3|lzz – ySE+Z Smarandache J1)=(D%7$Qy`4/i;yv!^ m%7ei LLf W,ee 6lM.ROv9zd<rMh7h!ug(;O,'.W O. Or,A1`W ORXvk/, Pb1r'e??H#W O. OvmL7, f`W O`pJ.b` U6qk/,|'(O-C"dz0e/U6q?#!~& 2007 8 10 Vee0\0,L[=YO#=(D%7$QC`<y`v` Rz<`YvK3o(Dzp`mzm$Q|E,h#06l(Di#vrB5&|om(Di4e(DJn Jn Wv6,`4 Lie prMD`<v(D$z$Q,toOTk[?v,!X47(DE|=B`<uvz-5,'& 2009 8 9 Vee\(DE|=B4e`<Wv$~t?tOqvg,!eWzm(Di#rMD`<K=v0g-<Wm^,7O^p 3 .E|=B%Dv(Dm^eFKqnK=B$Qv vgLy`$Qq PhSmarandache =Bg~Lk/<~l/iq hqvh`<y`v$QZ& 2006 8,etSmarandache I0~V,[O ,h4i< Smarandache =B.k/<vM `<, 4./ivz0sLePu`zm(DE|=B,o^B#~l/i6lI=.i<,K`W3gjvp$Q,G`"o#Yo 2009 8,C 8Gv$Q,(DE|=B`<)\\COz =,v5I=\C`&lUIL<,h#`_zZ% Einstein vk/<~lPdsUO#,(D(E|*iL~l`q O^h,7~BORj(Di`5OR~l`,D2 |ELWvdRi5ORy``bKd8,Einstein k/<\ v,UO#gO^r42`|vfGLUv%7,Ohp ,Xg| y`4vmeKBWv'i\OI$_,(DPWv Einstein zpm/x .7.Cv Einstein zpmY,,(Dzp`mL5m,D2 !Pv5L(DE|i#v5

30 \3|lzz – ySE+Z Smarandache J1 F!Fz4Oug0n%7, e$oEW,hE9b,$Q UO#Xg$Q,X&!X |YWvO,&\CpCF# v#Ozy3u|EWR+2,$_,5$Q2 Lb#YoWLObR,P2$Qdg2'Uv,u2dBdOL|Ov5,p6<Ids ^$QK`,p6TG $|FCzds,XMho[:gzm A;vxE5R $Cz5,v2vu#-`v`%>rd#-5G~vV6l5,a28J_,'uMhXO8ps.R|vsL YU=R3g%s.R|e.RoEWv\ g.R| $5vU;s[?!^3^,en#v| |R, 47 [gdtY<C roOLPvsu2P#-v4ROLz0pvs,lgOLJ8qvOLz0ps,s|O2oX,5#2#o^=N Yv5Rt2ovhO=`.R 0 $#-,rEBsghv$Q5,lgt<`v\,O=toeOL%ov!`ZGvP $QOL YrG6vsR2Ld8>|Ev<`YhR|Z\ot,Bd $Qv,rE3qg$Q$Q5vyA'LfL:~)\1W:6v$Q%7,R24p&eT $%oBd<J$Q,!GgU;OR|Bd $Q$e#$X |YW#ROvORrE,0 $|FT Fv[%M,O=MQ0Y,$^5`:jv%7,2P*g$QBd$L7~%vs$Q,2LfG21d3OLm& YvTs!Ggn00_p',.mYPVVfR=RTvORrEBs2007 8hzeOR<\WYOR wQ SCI V*v,0 $[uM6l%I,WO9p2Aaend :~?P,NIK0'/p<Q'IY:lP\h,n SCI V*x'AQnAQ<,"<QMQ'nvC:V<3-h'e,LnNx-C:,n Yang-Mills xNoA5[aJIhYaxe97XYaxO07XwwEF,VN<Qhb B,NxQhMFohv!^ e2:=8G2'\:2008 8,eJzXdzrY`,k= SCI A[ $5,d. $#-vjhv=Xei G2P.hi<Y,ds!^[uUO#=O^z

K 1 4<D – 0~ÆZB, 32-62

gG!o/i 1F: *p\r 1IAEMa, 30 JCM,d*MM'Eh$tEr.=M'$15V(~" =M'3 i,bw|xQ?~ zY6z|~"Q<3,YAa1C1mAxQMQ<Lw,KYAa6Oa~OJ$?BB6Q,9K~,CmV1CjML%j:QA032EMa,d,*MD^,*Mp,tE|~,15 ~,Rdh3"Abstract: This paper historically recalls each rough step that I passed from

a construction worker to a professor in mathematics and engineer manage-

ment, including the period being a worker in First Company of China Con-

struction 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 Depart-

ment of Chinese Law Society, a consultor, a deputy general engineer in the

Guoxin Tendering Co., Ltd,a deputy secretary general in China Tendering &

Bidding Association and a vice president of China Academy of Urban Gover-

nance, 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 manage-

ment plan, construction technology, construction management, project bid-

ding agency, teacher in bidding, governance.

1www.cgxh.org, www.mathcombin.com

34 \3|lzz – ySE+Z Smarandache J1v8WgOR2NvshePO`7,s(EgVe1,A)aG0gsL`1Bd?74v^,:Xh[=EdJH3nR1|<75BiUXzz,hzZzy|<dy.|F%O ,of*Gf#V!gORb#wlX'< ,z-L|2LkfY ,5Lv^v\C |WV|w\O6,L[v<+zIq 5ORVY,W#WVzho 40 G,e.,sLePORVv<zGXgLI2hpdVBW 5 2dqKY0(ePLZ,O~ 200-300 Hv,D|g=2X1K=1981 83hef_iKvi>[vxVt',O~ 700 GHvt,GDg 3 $G2*,Bv,&fO [UE!P,hddO*Gofp\, Bfp\p3Æ\(V,dVdOfp\_iK,i>[v2xQ'VOM32r-qGMvxQ~HUF|G.Birkhoff S.MacLane vA Survey of Modern Algebra(4th edition)a[EvhVzXge!RPKv,[$~<,=o 1984 8X?v?JT92^ Æ<,h:LNi50#eP07#,#&Uqgh6$`vJOTi<,|hA|EÆ`n8V, hVCFz-|EWzyW\Oe'LJ` `n0ÆV,=X *=3 1982 8Jv=XT<zWz=Bz`nBW=0Æ[,h8oÈ d,sL!0gheWPt,6v&g,hj9#P Æ~(S v*xQU~hzHvE&| z,'tQ-qGMxQ~HUF|#vOL|ÆsGJ+8phv# 6,V[GLf.hi<OLzdsW# vU,GXthe!OP10#2'2,h[=èORÆj4[dsRO# ,Y<Wv2m%^6O4z0P~ &>Tvq(-PFv~QxQU*,ot#LmI=5~,LmuZjI=5m^,&gTÛ5TObq(-PFL%XUhOb^8Y, PG=M;;\v~QxQ*vhqqfr,[=#vhTOGU\,,Pugzy|vh[<.TxgMGhe!OP1r`vA47u

1 o. 3\D 35ahv~<5g\h|E46~<5?h?\=<#~`E|,OgM\,eBBvw\#E1b,ew6#E1l 4Wg=qdvOWw6,OE1m =,'bz4ge?#tzFh8E 1 oX,&#v|z8|=RV[,h#~#`Q#!=<EY,&gXF%hB.fSYw\O-5Lk5f(9%# &B,\V6=RVP e6l(9%# 7v\O-/7v\WzyW\Oe'OXPF-I,/ky,m6W` f3 50cm,4f 60cmO- W,ovttXzro^"=,sL|W~o^)6d,|F%w!g ^UFKIv\Oy#V9#WV,W:9#uE,+xllO w,h2oe(9%`\#,V d,6e\Æu,w[ vÆ&:`f2!`[,bgO-=Lqvw\,|F%OY+BPhe.\1982 8VBvOy,hB.f2xw\O-5 whBw\#=,joOX;;| of5vT<ML0,dhfv.-I3\,#g(9%#g)nKz,hOO6l'(,?[M?f\ dMKdvw,h28oO*`;24y7|S`*9b7sLeORzy|.?eWPv&PFvAUe2WPhA,vgPF`W<, 6Q.;$%,ik(d<6– !gMeP.G| vp|!O<21s)|vb~,Z21s)Mh5vA2005 8 10 V,ehe a oWP<P,!XPF\Cdb, UhoMv3PhMv>%1982 8,C t>,h0!OP,G PBW5v|j(voU,gx6O v^,UIF[[%, s.z,g(dJH3n`Y=[1=Bd`5!POLNe'vG\(Kee'hBW5Pg`WzyW\Te'k=v,<PWzyW\Te'\CqB'WzyW\e'`jvoU,D1?vEp1hp BWO, 1\!^27L&:O<v<J1983 8

7 V3,hR 2 RVdYe(V#ÆBW[,R|8otuÆ`7of*Gf[,OP4\,OT,ke32r,|e3[,hk ~ /b. 10 G|,W 110 |,:Q|(Q| 120 |*,`sOK`7Td

36 \3|lzz – ySE+Z Smarandache J17L,hO4\,OTT/b8vko=!O8 8V-X d,huòT/b8,8ooA.>Tq(-PFb,M hv|e>`ds,OKT/bv8V,WzyW\Oe'.v?ypz-Hr,O(Mh_? 3 A|F,PVo/HJ.zyzyUPJ6lL 4 8vÆsÆev?,hL7jev?v%Æ,&goBW!O(_?!P,e' ABWÆ'`<1v|LG,L<,e'(;1`Nth|PO,jtov?ypz-HJ.zyJÆv%&g,1983 8 9 V 9 ,hOX|F%O ,x#fzv?vp,'!X|F%U=o\CfW.ov?vWzyW\Oe'5Wy,h"e'(;5gzv'(boHoo,zZL 4 8vO_<<!L8LG|dh,BW 3 ,LR_d+k /b|.a`LOKT/bhhE9Mh,1980 8$,hv,`#FgYOH,ah2Lf,ovGL=+g%U%UMnYvygJ, /bg0 v4[,Om/b5|,3 10-20%,W:h#2g*J<,2e4Om,hGg:L8:O,Xg#/b.'2O1/bg16THg,XHg5h7,!G0g0T<e/b5/ |PvlH&iZp1983 8 9 V 10 ,h0\=ov?ypz-Hz 83-1 V,5LOO_<,.BWv?pWzJHOt/bv=L O zZL 4 8vJÆ!KHUO#gePv?zyO|=vOKWzJH,o&v?pz-OFH=XHdrE FT+=I&v?zyOf%vg,WLG8[,7 2010 8,hnLv?zyO('v?zyT*ve< ,'./HCF.Òi 0ìM~ < eidJ`(=qm?*,28g!Xg==:Wvrv?ypz-HgOK`WL vWzJH,KÒ8v#~#+gW,vWuEjOL><,hO8ugPoH,L72 4\,%=:2 3ORVvP ,HPF X8phvy`,l6,W~X2,ahug`d P,hr%E9>Tvq(-PF,h\o

1 o. 3\D 37v?!KHÆ,AM1Uh'(=XMev?v , heÆ WY+ds6Y M$mX,OXgv?JTv92^PF,uOXgv?JO('v?`T*vuwPF,[h`r ,sL^hzvmUK,W:UMTb`R92^PFuB 1984 83BzZ,O=AmhvÆ,!Ggh[=1rj v?pz 9tJK ,'=bvmT,`$Q<vrEBsev?ypz-HÆ ,hM T 3-4 B,.PF$tL*,g ~1 <p8popzyIAv=XPFO9P [,OLPFdPF, h!6v,LR_=v?yzHYPFE9MhhvwF, hB=gOR|,%=BWvJÆ,e#ezy|E;G,sLdR|-v ^2OÆF`(YEth,Wvk^=B2 geWWvmu=B# #,6O4o~#v, =`R_1+4zyMugek^=Bv# #,ezyO5XAr[6l2=v[y,zy~,Xge,XK#A[2=?v4R~Wv +,ugB<+KeXK6lv2=A[6lvLheWPvTtg/ejAvePvy`PFwhvA $l*,!.R|%M%,2LY2&i\aÆzyM,=,kF.2,uE|ÆTvp/u,!hgR8LE |p,#E1?B=vTÆÆ=RV[,ePFv v,p/#~#` ,BR`kOQ#EYvzyWP,h3eWÆ 8~1,# =ÆV,R<PX?92^PFÆ$QhV,#Xh6~1,sLeP0v<$Q; 4,A|EX0TvO#ezYv~``(KtePH=vgTs#Ovt2.\,;EY2gLGa~`1^tl6,W:~1PF1^#Q.Y,dR1^ +LhhI|,Puj(0,6O4OL+hdhv1^`49A|,Xth~1~ydL%!,eW8# ~1ÆP,h\C`0#Ye~`O,hVmyxQz#vJ<`Ænmzy,Xhnmvon^rv4RÆ<+= o<oo 3 g,u#vh6O4`v`, PG6O4ezyo-I W=vphI=Y-I=m^,!GghoWzyW\Oe'BdA`,eoI=m8evORBsÆzy8zyIAz,##'hef*GfB. 8

38 \3|lzz – ySE+Z Smarandache J1z-ve,sLo<gzy%Uzy8zy_,ugIaIm^,hH'\C8pV,Æ:=BW=v 6G m<+[= $yg,ÆzIm^,he*GfP,vvv^4 ,vu=4 4 k[,v sL`r ,28p/xdk,hu8pNUO# = v%v5,egeL,k =t`wRI(;.`OWv?BIl,u06#ghX?92^PFÆvehVe<[UqWvO^eu,W,v?/u.IP v[4A ,!ghv;< PFvEY,he!OP[,zgL h O 7>).FYTeh,`5L8Vv6O43!g[=hoWzyW\Oe'BdA`,7TI(;-Iv# hgoI'`58,8pV[ oJHÆv,&Uq#=L`<vÆm\ReJ 9uW,gW=6JH,KJ=`8pVOYHeÆO9P [(;<6lJvUÆ.<[UÆ,&`<oUqv\^, fl[,a3&K8V,6RAmUqZj&|:_?v?ypz-HO_ ,Xd8Lh?H 400 G2vO(_?z, PhdVu 36 <zHsY,huof*Gf,B=zeOR8)v=XJe9 Q,sLh# v<zigMh hGov?z$v\lRw;BJHJv<= ,B|eHKJ8V&5W,gdRJJv<+A|E3vds!J= ,Bl'<e!Jv_?P , XMh1r%6h`|E,gdR!JAvEY1987 87 V,hvO_<,oWzyW\Oe'BdA`,3_8P ,XhI`,\/e'OLzyvJA`(J*nTUL

1987 8 7 Vhv?ypz-HÆ,| o$<[A~DAF, DOXPF)qv?pPOeP_u`z,a<3\CwvOL,v?pyzhE O=2vw, g

1 o. 3\D 39w2Q#EYKZw,XgeB-IF#,|I WY<v-IZ,=`de-IF#,`&[|e$`Rds,|EwU+PThovO<~|Xg7OLvwwM5-IZ0,hmO><3|7vw gd-IZ, PF#OL vm^C IT,Y'nUOLÈv-IZ0Ùdew#,&g6lzWx1w,[UUhE`5g1987 8,(;#hov?+&TBd< 3z,6lI'`~3K`4 jvz-I,! ugov?OVEzy<[f dU, D85g1988 8,VHl[,v?+&Tv?pOkvz< ,B-Iv8rPrgI06!Pv?pPOzz-%n%.g&%: vsG,h#ov?pPO,)q/< vA`!gh.)qA`vOR(;7I5T[,hzZTI(;-IeP=`TI(;-Ive,DOLNUI vI(;-I B/~1'2#`,oL 4 O)\ -dp:o,CH vI(;.`,BNI(;-I,h.7/I(;-IvT,C $r~F5> f[,&èqR,YI>>l/A` W,eOLPFvAm,hmxxpI W'vSÂds,F#\Evm,LEY',uEqp`^v-IX4 IWC 5ov<+Wxdsz4R[7\w\v-IugMI(;-I# RloI=v4ox9uuE Dz-X6lx8$òIWvOLl+z>r~12#`,a0g!RXhiOR)\ - kp:vI(;.`m^,[=BdA``4R|v%6dR# (i*n091988 8,v?pPOrGw!,e 1989 8h#ov?+&T<P/0e6lvz5X`< vI,h|)q#pnAz,g)qF5TI(;-I50ge!R< #,heMv?pPOrGI(;-Iv# #,C ZW"NvI(;-IT|voI=,L[=TI(;-ITIA

40 \3|lzz – ySE+Z Smarandache J1<`R# !"Nv|,GX?PWzye'TI(;-IvT)m^,7O7/vI(;-IH'h22MQb42+4I5,h2ImEsI9l|I64Il2=%Uz4 Iz45Il4!rE< Im^4"I~3K4#S<|Il4$rEAMI,h2Rk5MIBiCoMI!,BIMI|~AMIz4%++e!R< W,C oI==8w\u,G=%Wuz_A_a,k5IR,I~ oOv5, P?GzZ=phjn'6l[7vuw\I=z,6RpIW5ovOLAdsI(;-I7[,|E$`I WvAds `-IZw2,5/K-,h!P0ezBW O ohVQpQ6 vOp` ePv,OYg 286386 zP\,,ehvEY,,qv G hP\Rh"RCBW$1989 8 6.4 ,h0ev?+&T,e#C `o#\H<_WB'yg> ,ae 6.4[vOd0,sieU2,X2t2sY, of2o 1989 8,vvz5X\#~IoX,he<vA`5,7[,#oORPBdA`5(\*n5v?;O,ghR|zyIA`~ ?,6lA0_vOR< 1987 8,VzZHl[,I~|=Il',aTh!J= ,~|r2'e!Oie,1989 8,WzyW\Oe'$v?;OOpuvz-~|,!gORIL|2, PLk5<e 19918VBjz,#= w3*Mz-vOR< (;#BPhe!R)qIA`,Rz-XOm,06OXv?ypz-Hh9*Jv %# I,07#T,Wy[W[,Y'# ~W5'fEzRl+,XoR_,5gv$ZTl+5Xx#$`mE(g= Wy*,. O -t-IXv f,GI|Fx#= l+uhv5` ,GXh8pk5R,MIIl2,$|eI W=bKvRM.`MI,R!)g ISO9000 R5u

1 o. 3\D 41v>`%7!O8,v?pz-OFeipzXiR`$5,v?;OhT< WO<?>,EY`:>,Tfr4 P,:GE`6l>,'EK/hTX=`|8p/x_3L&hT,$|RAv=R Gs=yzhTL<,hkOR,vd#:vhT.hT,r[=RhT ,|/#Oi`=,a=P?>vf%iphTT#,!=toOXme# p2 .6"6mK ,33WoQtM O`,-KN6"b&zv G= hhs<dtohT(:,h8oL,sL$2 gL&hTRPzvOR\,'`eUO` WXE 1.4.1 b.oQj&v?;OO 7 RX Pz,[\|YT, Y#tL2!X|EI=e#v[7\#L`W,$QR_P `q3W=6doI=[,h=W7vvm^,7T~v[W[6lW7v,(XKu2|Evm^,!TT[\vOp5C Iot6lI=,W:BEpE*Mz#b|vC6,Xh'`OLPp-IvoI=\C[<SeI W,xO%[= g 10 V 30 ,ey&=ov,|4L 8oC 4 = zWv= 'D` 10oC4,ae9#UO4|o 16oC 4>re= y[W3Iuk,aWy#uhT,Y'yv= i5','Be=vy= O=2=!gORd,Dt= v GO |Bs,.-IOv G $``^|Nq~[,!OO= ey[_RVi5W,R5XN4N[`_y v=

1990 8,OXO 5v dE9h, v?pO^z 9t,

42 \3|lzz – ySE+Z Smarandache J12#,=CJ tX`o!OChe.&g,hhO oS\ 9\dtEY[,qKtT=thsvJg,# oeS\ 9\,J# Juev?TL<,5:(,hzZW8 4 V 9 Vvtz!O8!B,Lk5v?;O 1991 8<j,$zZIlW8|EIv=R j< ,Wv 100m3 jÆPIRood0jÆPIm^^,O^ge5O-w\,eBXK#7f<+y= ,aMIS<A4gR+ , POp\G,ZB~T4uO^m^ge7f<+y= ,?dH-I;4[e15Bh<7\,sKM'`-,,= φ = 25mmvG<576,BjMB`Lq=H-IBXK!^m^v6$geÆ MIR, PO-7\vO5|z,l g φ = 25mm vG<eI W|EmAJ,AJ=vG<`^ j,D1OpZB~!P0OLzyA$Q%ozZ$Q<[ φ = 48mm vj-,,4= φ = 48× 3.5mm vZKo<vW φ = 25mm vG<5Lj76!ejIW`v φ = 48× 3.5mm <5L\,BR|I~<POLf \C<[%,aejIWu`X ,jÆPq=G`= !^_h-,L<,WzyW\v 1 ; $Cz,vhh=!^_h-,q=jÆP 100m3 IA$Q'Uq&v?;O,$Q~|rAohv#/Rq=`P 105t,=b2rZb 1.2,-I%=bC 126t ML<,h6lT<oI=MI=U64,6R7/Rq=,7q=\6Mq= G5a-I5K-Ivq=\= 16 9H<M5\,7v#Rt~,W#t~L-,78,t~&BKq=WvBik, oq= W-,Rj46#/ M30 =,=(s#6, &eq= WBWV4MjIWvO^MMS24 RM'`-,*12rZ3K, Pe 23 96#s)8%, Nt6pn/<-I,\ev?;O 100m3 jÆPq=Wt`C q=ztq=0 q=XXzq=a,,\XX&-I 31.40m XK,3[OiiP 7 yP ,L0/<IA$Q3Og(v?;O 100m3 jÆPq=-I,\t 1991 8WzyW\ A64Wz,RePhWJ:8G

1 o. 3\D 43O ,<wG2 gORAFR\hq=-I4I6l,!Xg 1992 8|YeEMv WoQt 100m3 #!*ME<v WoQt#!*M$p` jÆPvI,7zv?;OOO0UP?I,C Hw!,f8W,I:EvIBu,h6l/uIIl,TI(;-I'(;UI,BRk5 1991 8_<zv?;OOI,XhipuI(;m^1996 8,Ep G,dfD::Q,hv?;OOI(;6l,!Xg[=v/eMrvEM*MD^60A`hWv QM*MD^=20108,eh o!KHP,/H\& 19958.NHD',ZLx+CThv E9h, heP `-Iq=avHjÆP,\Ce 2009 8W,sL$$|E15N P,s avL_,\C2|E .-KO8=LqvP ,vh 8o A64vp(*nev2d;;Ywv?;O 100m3 jÆPIv`,XhOSLe'A41991 8 10 V,h=oWzyW\Oe'$I>~A>d,OOR< vA`Y P\)=R< vA`,h2v?TDuH |nv?phE WWP$QO $J|%zv?TDuH |nIR+ ,ORg 50m 10mSvo9'`4IA44 ORXgX&#md 62m, 1.5m(\5 2.3m*vo=p= T~

E 1.4.2 J=Q\rLk59,-I= 3 po.,Og0L:OOO=o4Wgoo,= C28 :BW UEA #DgDvo= ,9

44 \3|lzz – ySE+Z Smarandache J1z: ≥ B84g5= 3 : 7 ,!eW, ,# vgo9L<,hTXT,'BgK=ltvE?$MtO$0,Tu&= vMI=4MIÆ3v,&gpBoI=IMIk5Rmk5 voo9'C oI=,hn=Rvo= 1y2mIoz,&gEYdCLvIIl,7MI(;IeP,$r~F=Tu&= MI=tv<g4= o1ymI 50m oz,L5&gLh-oMv`eq I=OhMCI=4j,i TB,gORuP #`_I=O,3= o1X#`ymIvz,.h<3vI=OI,&gpeMIk53q,8dz= yeIMIm,h3OvAMI,h2 K[XM= 9I4,L3-4_NzMIOy −1.68m `5Xv= ,h2[O5| ,[;4Po-I;4[(;`# vyL<,eR5XB=CU4 P,Lk5o9,dyr|EOp7,C ~P~U~kEpvBu,v?)wy,4PPs= t,6l= ?d,nk vooIR#md 62m vo=p= T~I3,hl-I[7v\,= φ = 48 × 3.5mm \O-m6 f 600mm × 800mm,4f 1200mmv7v\'6loI=,n7v\Q#IEYo=p= ~ORIs:$,XgEnko=p+vRsXKn,21s= yU%XKL<,hEYdC-I.6ls,eT~#dP 1 q<+6lXKe= y3,$(;vRhT W,r~F/T~v7v\qYk,L\m6U, eP*Gf(9%# vX4.<$v~4V2G,7v\m6W` f3 500mm,4f 600mm,EYh#=Oh#=[L7v\gBiv,'?M5k5= y W,!XPFEo#6lhT,oIBi[:s`o=p75uge= ?dPo7;4[(;,=w7vBu,~0o=p+9,mO6l7, k5=p72n= \5p TRFC v?;O 100m3 =-I`4<$ 62m dvo=p= T~I,h\8x|ph&IA`, PGDO

1 o. 3\D 45LR_vFv-Iky,BsXe&=vph2O[7v\-I=j7~I=h,I=maL|ky4R=s~I=h,I=%E#Ò a.UOQ+Æ:, PZj&|I~v?TDuH 50m voo9'IA, 4 62m o=p= T~IA[=`hH;vkTp<`,|eE<Ep#Y1992 8y,v?phE z,O8O)\o,B|I~RoA|F3,lg|Em62w\6l1, l'\ZBz,q3PNI,BRnkI!OP,v?p\COLIXe=P\IA,D|EO\,r[=jx_(6lq=X`Q#I5EYaxB`<I=oUOG5'``%pvC6h.I>=X do+:ORBO,=[X7ONv`<# ,LP\A2 g~~phvNjdOy= W=Jv<+t'h<=XW\57v,'.= u7,6RiORbvpuC I=,h-Iv?phE 2:#vP\,'MImE(;UI1992 8!B,WzyW\(;A<w5<P,sWJ:Qt8,hv<wgAF,sv`F+2gL<,hA1#v`Foe'A d,)q<<5v dE9h, hvv`F3|EOX`,2Jeds# e'=Xm0e3U:8eA33g(vXAFN7=LF,hXgWOR2 M# ,R|rNv1p2T,uI\2 f,vhR``z,sLh:Jt8,$X d\WJI8!8fÆ\(V,ho\FE<ra1vèqdOgs7v?;O 100m3 =-I'`(;v?TDuH |nv?phE zA`,pWOLIA+ ,h.ML,MGhvOLA5O p ô!A<w,hE9Me'\ h.o<w,vMfO,heP'28pMg\<w5OFr~=RV[ho\z,dMèqTrMP,ME9h, `Lh7=:F.,.o:8p.ovgF,7vA0_5erQ#Nt=8[,=oe'OX dE9h,h:8p$<w5vOLeudQ

46 \3|lzz – ySE+Z Smarandache J1O $<w5,e'vBug ÆvKW,7v$X dN7=LFh5aI,sLMhJP hd8, P,Uhe'0|Fv[5gvOe'C``7ovg,he\A3v:k $X d5#,a1(;5TSe'|EN|Fv.o8[," hv.o8,r%'(h!L8eIA0_m7vOLrE5,'LOe'.ovX|FW,hvtQ#\N7=Ft,D fhN7=LFoMvfr,BW5ve'C`L|5,sL.BW53e'mVLBu2OI%37[,M = 2 =,&vAa~k(dpwNIqne&eA?PA,-$[m%eA?PAa1 2 k(dpMQ1 f,-$[mMQO K,kLq"&re'C`L$X dp.N,a1,[^2 d,[,CiuOv,hhR| PN7=LF!td[=e'OLmv`%!XC`ov?[.M| KYyk(dYM/tL2qwY, =b[&LRUO#,h.a1v3gORA|F.\mv5R\,D2 e d=87vA5G,eMrvE<#YvA<`G,toMvR\,hh: UO#'`Z6O4v.,!XWzyW\e= AWnnvV\C9bWLG8hR|Z\8o,heWzyW\Oe'5vLG8 ,:K`1reIA3LzÆ,lg[=ev?p<[4W`Wv',.MP~Oe'FA dv=X Ghvq7g|2zv!,1993 8 3 V,eWJt8GO P,hX.MOL~ JQt8BeA`#O,v dO tF<wePA<wg.jv&g,hvBB=vdV 84 OSoFv,|. 124o!O8 6 V, t,htv?pz 9tOFv?TrDZYvJ ` Pe<wH`Rmtvm,lgdV 124,Xhe'0OL|v85AhGg[=:8p,<P!|F v077 R'A\Co/e"`v4,%2fXo=MjmpuvR Rhe!OP%7u/e4, LDE66)$IA1L!JpA+sXom`r`,28!PThI!J0$eh,`%M|%M| M4z0$e1Lv:9,hv!^7^'2Dh|h%nÆ%,6S^ J5Y<v3^,!Ggnh,\^

50 \3|lzz – ySE+Z Smarandache J1kp:[=vgn[M5vhT[u,%[gn[%Wu,-IrhR|7IuC RI9d,K`q= vIR/e6v?p<[4W`LWO`givp2"e'vè%O/2"e'ml`6Q+,9#4C o OOy,O=W 5m v>ù C40 = ,Lk5= yR,2'fEzRl+,hoO,A|y= oO_P,OR|=-h, OXivp2"e'vmehèqzhhq-|=Rfd<[,72oèqO,B=g2"e'CFFzèqvr~Mdh,LR_= huEEoA,v|4XlhE9M,zyoWvOLÈ5XvRgk5oBivs,!O>ùyC40 = !LÈ5X,GI,BRk5RMhv!^JJA43 ,G<W3/e v`.b~IoxOP,s[Gd,W:xO4vn,|=OLW[e5Xw,;42r,n= y W''AW[DtA|L5|o! ,2t2(;`_Wh[`k5IR!,B#+OzZX=_vT[#oIoL+OP,v?pz-OF(;RThT,/ov?p<[4W`L<,e'|e'F)q<[vC`L5,s=yo,G'`N`8<G/Àz,vN2tze 24 FP|N`hT3Oy,hdNU>hTv(dq,UezO5vOX RRp,vMUU>vRR\MdhRdsI,h `,rEg7vMh'hTO,sL!=yLhTBt|`vWy,=Xe'm3\[hT(=ooq;A8,(dXd2 +$Nk(d7h6ooq[,MXh 2 u5I1 ,1. d-0l,-KW1KJe910,Mk,1333 h&gUMhoLOOOX=Xe'mL,28ph.zOv|gR_a&< C`5=R|g|e'P;A,|e'm'26hh=R|1996 8 7 V,|e'(;Bi<[hT,=ov?p<[4W`,TL9eR%O#vO$,&gz< C`5mVF`|Z' 200 vZ,!RVg64IR,6vORV,Oi7+OoI tog,x=vmgZ',< C`5=RVF+72

1 o. 3\D 51hUO#\& 1995 8_8tv?Tv?pz 9tOFZYvJ~ ```X,GzeWzyW\Oe'd5dB`Lp~ `Xèe'TU,a!Z'Xh 7 B=`7NvORds,Xg L,NKNiMQ-w',w!v 7=2g!,hL?vO< 5nsG0g&!RBs,Xhp`B 1996 8 9 VzZz$Q<t><,hO=~ AJX-,%~ 100% M-vBu,e#;`36A6d<5, Md6e'vjmD2 9#oV,jJ(P #Æ`,z$Q<t1996 8 9 VruoI7,hhzZ(;Ido$y5!O8 11 V,z-XzZsle',z$r~vhaMd)?Ule'sU-.ku,hh=R?B|F+2U-., L)?U7X1ov?,-Ip11 Vvv?\CLZ,h vPY'tZS,2WOt2j,o)?X8oy,&g)OtZS,zo##,8oUeUy,D6j9#d##Qe'KOtdu8Og#),o#u1&,zo ,Oy,4 31oC,##KvSG2l,D6#KOtjO vu,!:=z;X?z$r~7U dUv=R|[=O=Lh!tdeI^, h'!yvS[o ,O- O-KSG0g!OCi,Xh 6p !hz1_v`,G8p VWJJ VW |ogxOd

E 1.4.4 3~-37 E 1.4.5 3*=9%S37!P,hez--I_Ì ÆoWÆCA,ddOv Æo<[4oh+rzz7[,hWbK//`,r[&R, |YUBXUI_è'Ro#v=X_`FXehP è,MhUO#GheR`mvpvwhTIds,

52 \3|lzz – ySE+Z Smarandache J1bhvfr, Chvfrd$`1996 8!y,hhzZ(;/v#IOy,OX_`FE9h,O:v<+/uY<=[qvUX,qYg[;42r,R|3gI5$ÒXzZ7[,zy#5= MvhdO,R$`frv|$`6X`7Mvp,h72o#U,$MK hLeVvd-oèqMMkOC,vM72(;|L[W=?[,C <oI=[,hkO^f,vMheUXvXKW/ 12mm Xv<[, P/d4$`7[,hE(;hT,n`ds[,E9_`FdOM 2 KA ,1w&_`FWOX 60 GBvPFV( ,O9P .< _2U6,sLMOLdsU$,aMXhvL6Oy,ME9hE^z,zV?PhdNBs,r[dMLf2LfdWz-_è'ORe_`,sL<3/e'OX< _= h,AhXMO 4_`,ah!Pv`%e$Q<^z,K``QM!XPFh [,QQdtOt=y[,hUM=oWz-_è'UM5zm,'dOR~_!RXgW^ $J|%,ePu$e3z,sL'`W.Iu`7MGh[=evmT_0,`X eW^#zèq~AdFv E 1.4.6 s!9~$|$&'37lI ,z-X=AW4BX.z-Xv,!Ve'2BhXvO`,O=hlh5<[4Xho[,hE9M25,`^$Òy,he0hTIQ+, oxOP,dE9h!Ve'exO%[#_zOR 1.2m × 1.8m v(,hdd,hhg2g`CFUM=p(,M ,XK KCF=v,dsMUhO!Ve'_zvp(,<+A;2'06!Oyh(; Æo<[4o,o-I)q|o[,vhaO#5OIQ+ oxOP,hlUOoBe'_zv($O,'E9OhhC-I=v(eR_m,BXW&G',J;G'W%

1 o. 3\D 53[<+zOL5,dhLR_2MCMhIhE9O,!Ve'gz-XAv,?&,W~X2BhXv`<[4#,!XoFL2&k!VBe', !Ve' IK[1;,>0 -qA,,.A -qJ0q,n?I[A1;,I=|~KR< b&1NtLÙ-U>d_gO ,[&7~ò&oo-I)q|!vp,_`Fe#=P15 y, PEY!Ve'T3hTv_ò$Z[,!Ve'Gm(|R,:8pgsL2hX`KI!`[,!Ve' G292ChAld

1997 8Vy,z-X=A#vlgog2eogfW, PEYSle'./fCAv[N/ D /7D [,!Vogf.Sle'v DO=2gL6,6=Vle'[=?hE,h6l[,O 1aKO=2`7!8!y,Lk5!BI,z-X(;ype'q3 H,%04: 30oCW8z3,!yBvOLg'iv!Vogfo$`,!Vogf g!y H,q04UKI,2gf Wvds,2Lf$`hD6(;Sle'6lxÒSgiUeUT,o^x`,dXv|Lg#v[d,Y'g[W=vgUÆmv[8,&gXv|Rohvèqh[L|<&,"Oe`F?wTOpA,8Kle',eogds$`7:3,hvA2?!Vogf7&~Bz!!Vogf5,=|hh ,5"mE,L<3 vKgx=, P_WvgQ#V,C'`6vDN[:1P&zA!VogfDt f|Kgx=, Pef0W;8R`_RV[,!Vogf z-XAR_`hXo2ehTgvWR,nWaDN[,h:0/ f?7&zvA/& 1998 8 7 Vw,[(;z--I_Ì Æo6vv?pR_,v>6,rO6vDN,Rz:L,L[=/L zL< RÙv# !8 9 V,h8bvmTOv,`$Q<,&gzZ 'O4Q$'[Mvzj!,hG$%I!LG8e!Ve'vtG,LO436 =7p5[`3KW^ $J|%& 1997 8m<,/#zm[ 2 ov?

54 \3|lzz – ySE+Z Smarandache J1p<[4WÙ,.hGmx,' 2 ARhBW/v-ImE<5hv$ Qb?BB6W4,UW^#zmÆ39Mh2#z-`,AOX#z-vF06<3h$moWz-_è'vV( PF5L_`W,L!R3z52'5,eW^#zèqÆ6v9Mh [OI f|RhLW^#zèqvAdeFh!P\C8p#vmTv,`$Q<,D2 /b8u`Y=K`,hUO#ge 1998 8VyXzZLW^ $J|%vz-35,D2 ePu$eIXU9,2hyyd1998 8 12 V,hvovmT,`$Q<8,#ge<_?,7|EKeX7&8iI 25500 v_?z, Pq hÆ <zaeh"8de'45 3 P,Mh8, EC=Xe'm$QO:1p323 hL5,`7Ue2lX9<=y[vOy,45 8hd`w,aEYh/OR`M7|vsQUi[:13 ,u21dÆ/Mo#e'2\h_0,`X v~Bz,2q <z0,KÆ zR|\zd<=8=,sO`7d_0,`XRR)e!Sêv1I,h52>eMo#/?,sL1|<v3qgq1p,R6H_0,`XoYg,KK,:OL8hG2Pd%,nn1V1DS $L?Q|A,DD1R<0'Vb?h7,!0g Wr$,rW$ vp`_0,`XvO8# G,6eh2yydW^,OddR 2-3LdspXl!Xth`j7,`$Q<O8v# Æ[=,7 2004 8,W^vOLm7eB# #z- yz+O,vh WpJhBhEq=83/è%KA,Y'eP-IZ6vK/zrghWz-X/vho#P,OY#5hTO64R,r[?_è'IX|[OLfd<, .#zèqv G O[dQX,52=2/# C!O-I,aO3z-#+O,[OgLWzzvO%[y= P,-IIr`3Our+'ds,R_, GY'[,EY/1L<,hCOYvzuvkdm^,eO/1mE,-II/[>l/lI3,#z`r~.hO nlWv=td,r[U

1 o. 3\D 55^dRORoE,RY;%=, j6lI2=y,dXk^=#z`r~eXh , UdoEd<,ot1OttoX2N,!OOi#oItd,D2 ezTOh Yk(dM'% dJ!Ohp,dXikM 2 0K=L;^4`":p,dzqxOT:li5=Jg>%#z`r~M=X G?d6(,tovQ#g K(&&g `|96(#z`r~zoh`Vd1, XdzoO=y[vORu,dohT5,hOad,O.M^hdMg r B6Wl,M h LA ,NA>MypAD;$s r0, ,x tQA9,-095NZA/n%R $&hE9M,zyq0l=J)P|E48,OY&&upl":vm,sLlKU|U=b,R^5LORp|"u,elN#,ZGv|8oEA.dOhvf%, P?#z`r~[ 2 bkU%R?$";V:%o,1K. &0sLe^ $J|%v#z`5,Xhe|O%d|`\0O:,G:Xth[=5v$Ve'jO8|O5Oè%z-< `,sL$R< vz-X?\[,q B[k(dh#A(|~vtj y<K.hW^ $J|%& 1999 8w,?[#1nvO&Bv 2 OF%2000 8 5 VvOy,Kr#zÒXr~ VIOEmOVe'h&g(O'(d!Ve'5!Xr~E9hÒ=e^5, >r#z`zqB'9&=X G,ahv5\oèqhE9M,h2s!,sL $J|%4v 2 OF%W\w Ovzm,e?/e'C`'(hv5Ci[,hj=o!Ve'~< C`,eP!Ve'ue-TTè,5vFWeO G2 30 G|h!P_0,`X6W8,\C6JÆdd3dtuBWmF(;vORi<VX`,Bvg?ÆmF42OLv=~~`t,d TX`(LJ<`3z,P #Æ,!GXth#rP e!Ve'BdJ|5

56 \3|lzz – ySE+Z Smarandache J1W`gO^2JpL;vdW`lL, mp^gp`tlG,< C`^4^2J~GÆEY;o!Ve'P,e'C`.hOjMB.Tv~Da[$ 15a5Òi,hÆNvrE0,`4 2000 8VIOZ3vOL5rD5g &<3Bd LG8vI`, P#B. ,Kìv^h= 2gOt+dhR|uZgDW`5, Uq'v0*2&Gd%|, W3i!Ggh!L8`35^4^vORrEBsRG8vÙq,GXth=e'ORVXem+,'W|B.OL< v%2001 8e'5Oè%,BPh)q/Is!R5Oe0vlX,e'=XmO q-hE2T(pL!R< 36L<,ORGVvP ,h |E?/5Oè%z-`èqmo,.Mhv5|FO i<`tvrE'O,/5è%z-`èqr~dh,IP,1210e6l#IvXO,e|WUMW 2 |,u 5m==hT+Æ/#IXh ,DEMq=U!RXW|,uI|2d,aHd^4Wv VwVPBu,<P|9XnGS,s!^lL&=2D`tM?/IXh! ,MLp`,&g17e`tWL!RX.W|EYOy#u,h"/5ARvIRIXv,t`tel2_,%o``7zE,;oèq,/5è%z-`èqvOX G=p, Mh;;voVBi5OB5`t,MhARvIRIX6lBi5T,WRXBi5T2DN,|`_``zEw!yu,hvè%z-`èqOXr~"Bi5!`tv#tXhO dl2_,%o``zE%/5TgP,hr7 Bi5!`t#O9p,2+,sLN RXB52DN2,uORXTWhe:;,!2gB5`t/3v,)o,#n29hd!Xr~,% `%u ,,61e!#t#W3 vWMoèqW3%u [?hO dl2_,%o`zEw,|!W3 vWv`t?)qzE|FÔXv=/5è%IRV[vOy,e%z-`èqr~Rhi,06VBi5 I!`tv$dGe`,h?Mq4<d,MO h^8Y,Lhv48p0Mh5v^/P, e%z-`èqr~5dO(,3?OF

1 o. 3\D 59(*M15*Yl~H$~2004 8y,z-5p`'OX$dUe'FRp, z-5OV_%oe?_ÒRJ|$i,AR|g15a5è'=Xm+LW,BPhd 2 RFP!ghOLBJ|F4_%o^4hvkz6=y,'d^4W|Efv'd<6le.yv4[$,$UO#L5,2 gL'0%W\ R\,amto.vy x!G eP15a5`5u3eOL^4vÆ.Ò!$i`[,GBd!m_v GG8ph,'ARhLOLJ|_Vh!P=vghLe'< C`J|_Tv 8,7~ ME*M15*Yl~HO,D^4^-I_Ìy=q 1-1.5 yP 2005 8,VY1LOLg3lL,(;TWv

FIDIC u`t,7 2007 8Z3v~Da[$ 5**M152? *A~Da[$ 5**M15*AMh=R|lTOC,r[(;OLJ0V5ohe$5o#qdP 11 Hvx1zoVY1LO[==EYhB.!`tT5,'!`t4 jXA)vrETV`t& 2007 8CVT5O\rDY`Z3h!PrfVo,LR|^4^vÙqC6)UBJ|F, P,(;gT,&/RlJYoY ogG65&g,he 2007 89de'5,.OVlJMr,'& 2008 8_8RZ0\L/MO5|F,'&RV[\~sd2008 8,VY1LOeY1Lz-jpz 9 R5O\Wl!`tvo,=XMV|t,h.2OX G)qN=5`tvt W,Y'.hO )qN=5`tv$X GtL6,RMhBP a5aBY 5` v GUqC6%VOL,217/'`tEhzo! MhxR|, xP2o vVY1LOv GGL a5aBYt26,Mhzo!O h=,R!0ghvd$,sL a5aBYUO#g^4DIl vOjphQb^4 , a5aBYv`DTUOG5El6l35v,o.s0g!O8,XhelJ0v::to.Tq>*pei

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K 1 4<D – SwdS-S RMI g, 63-68

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1

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bk =1

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−π

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x2, x ∈ [−π, π],

(x− 2kπ)2, x 6∈ [−π, π],! ,k g/,Xt |x− 2kπ| ≤ π,u f ∗ (x) g` 2π Ldvd,,>r 1.5.11V~Æ#P+1'1985)

64 \3|lzz – ySE+Z Smarandache J1-

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−3π 3π−5π 5πx

y

E 1.5.1!,hh`| f ∗ (x) eiUg#% K:2a0 =

1

π

∫ π

−π

f ∗ (x)dx =1

π

∫ π

−π

x2dx =2

3π2,

an =1

π

∫ π

−π

f ∗ (x) cosnxdx =1

π

∫ π

−π

x2 cosnxdx = 4× (−1)n

n2,

bn =1

π

∫ π

−π

f ∗ (x) sinnxdx =1

π

∫ π

−π

x2 sinnxdx = 0.8f ∗ (x) =

2

3π2 + 4

∞∑

n=1

(−1)n

ncos nx.<:vv& f ∗ (x) l,e [−π, π] #,X

f(x) = f ∗ (x) =2

3π2 + 4

∞∑

n=1

(−1)n

ncosnx.!,hhXto f(x) v% Kz\X`&% K:Wv# 4 <,)^Xg2Z#Pu+ f(x) + f∗(x)

sn f∗(x) uoQsn f∗(x) uoQ-

??,j #P f∗(x)akk

1 o. UyfU/U RMI i 65WL#`v=nLO30j=v3^,gD $Q4vR7nx0gj!O#~%7! ,hhj'(`7nxe E|mWv7nx2L[f(x)] = F (p) =

∫ +∞

0

f(x)e−ptdt.`Epx'eMGvpR,!LlpXtE|mnLWmx, I=,hhtL[f (n)(t)] = pnF (p)− pn−1f(0) + pn−2f ′(0) + · · ·+ f (n−1)(0).<,~O .pE|m

dny

dxn+ a1

dn−1

dxn−1+ · · ·+ dy

dx+ any = f(x),DEz7nx2y(x) → F (p) =

∫ +∞

0f(x)e−ptdt,X`nOR` L[y(x)] LS8vO.pm,!X`t L[y(x)],BRX y(x) = L−1L[y(x)]R 2 7nxYm y”(x)+2y′(x)+2y(x) = e−x vQ#t y(0) = y′(0) =

0 vf i) z7nx L[y(x)] = Y,m6b7nx,f y(0) = y′(0) =

0,t& Y vWmp2 + 2pY + 2Y =

1

p+ 1.

ii) #vWmX2Y =

1

(p2 + 2p+ 2)(p+ 1)=

1

p+ 1− p+ 1

(p+ 1)2 + 1.

iii) b7n6xXy(x) = e−x − e−x cosx = e−x(1− cosx).!^^`k)^ [4]2bDl~ +uV+

+Dl~u-

??6mw Vl~6mw

66 \3|lzz – ySE+Z Smarandache J1o<g% K:,ug7nx,#~%7+gZjvO^%7veu!Xgm^<Wv*d0Fu,4 RMI Bu [1]:E%AL(5h X wYd_Y S,n<3wA%: A IS : pn:e S∗, I S∗ Y%xQo`s(5: X∗ = A(X) T%-h,IQYhrA: ,s X = A−1(X∗) T%-h!O )^L2

S S∗

X∗X

- ??A

A−1

RMI BugWvm^BuNveue^vSR3+`oR 3 WWMW|M^ v=B,pdsv#~%7Xg RMI Buv)^,Xhh`No!O 2<A r V rsnVsn<A

-??

V VH<<Aje .pE|mP,Gg RMI Buveu,)^2b?-oDl~ k,F;[k,F;[uVDl~u

-??

w l~?

1 o. UyfU/U RMI i 67R 4[2] :W(D<W,2'pdsvm^g RMI Buveu! j'(OWv,(<,*vm^KZ,m^,XgLORo,2un = u1, u2, · · · , un, · · · &i\w:u(t) =

∑

i≥0

uiti

eut =∑

i≥0

ui

ti

i!,'Q ui := ui(3wL,,[wLA,*,r[ $Q,A,, #v,d0=,X`to un v^pR)^L2 r v9 rv9uoQuoQ

-??

?t \<?tl,V|m un+2 = aun+1 + bun, u0 = A, u1 = B X`=!^,vm^-u(t) =

∑

i≥0

uitiu

atu(t) =∑

i≥0

auiti+1,

bt2u(t) =∑

i≥0

buiti+2.K`

u(t)− atu(t)− bt2u(t) = u0 + u1t− au0t+∑

i≥0

(ui+2 − aui+1 − bui).Wb un v,X(1− at− bt2)u(t) = u0 + (u1 − au0)t.

68 \3|lzz – ySE+Z Smarandache J1K`,u(t) =

u0 + (u1 − au0)t

1− at− bt2 .- 1− at− bt2 = (1− r1t)(1− r2t),uu(t) =

1

r1 − r2

(u1 − au0 + r1u0

1− r1t− u1 − au0 + r2u0

1− r2t.

)| u(t) zw:,Xun = (B − aA)× rn

1 − rn2

r1 − r2+ A× rn+1

1 − rn+12

r1 − r2.l,&! vX*$dso$v Fn,s Fn = Fn−1 +

Fn−2,F2 = F1 = 1, A = B = 1, a = b = 1 B r1 =1 +√

5

2, r2 =

1−√

5

2, K`

Fn =rn+11 − rn+1

2

r1 − r2=

1√5

(

1 +√

5

2

)n+1

−(

1−√

5

2

)n+1 .!gOR'|Tv\, Fn W~O<+g/,rR Fn mg o`=^v 2HF2P\

[1] jUm^[2] V;(D<#M[3] -T|WM[4] WzJH 8+M

K 1 4<D – PTwP~Hb, 69-72

66GQI3\Wvkg=kOLÈv2:v,ùg=d2:: v/er(pR*v!5|8Vov#~0RQÆsv ve&_?hhKvk`|dspdsv1p,%O&hhÆv WK`,hhE6,sgiÆ2:`Æsv0nÆm^~`[X<ds,D?Ævu, LF5V,. he/ iJF5zWv#~2: k= ÆORk,mmh|O/kUxOR2:(g`kv,u`Wbkk=R=,`&=`*E`rkWv, – t,!Phh`2:vsKe,sLkWvt,g=\N2:v# l >xg 6Q 0 QQK 0 x,i!Rk,xEz 6Q 0 QQK 0 x!RÈt,sL~BOR#+`T a + bi vi\,e# kW,LP a 6= 0 v#, 2 + 5i, 5 − 43i, · · ·,#P a = b = 0 v.6Qi `#|,hh8p >xUO#Xg 6Q 0 x,

5i,−34i, · · · zLk,"217i`,>1`,aMv58oUr,!eÆ Wg0 'AOR2:,DevPY:1$0uoMv[?RhhOR2:vr`,~1XgORx6v dU#,DehhoOR2:veu5P,Nv`:` #=LeuviAl,E&|v ot!g+#,rEBsXg.,v2:`4NR'=vOj%7m^`2h?vug,Æ.,4dv%7?E&|8Vv&VRmx3,lg7&|O [,$uO&|Wv ~; → ?$ → Fsv%7,N=u'E&|v#~%7.,v5!PG|#^=!P =Ò.,vk,

1V~Æ#P+2'1985)

70 \3|lzz – ySE+Z Smarandache J1$_,Mv`X23g4v`,RE3vG,Li\,#02:W`Æ%,BR3`/2:: vr,!gÆv.`Evm^l,7#O ,S^v2:6lÆ,X# (a+ bi)

U (b = 0)

(b 6= 0)

` (mn, n 6= 0, m, n ∈ Z)

o` (a 6= 0)

4 (a = 0)!^Æ.%v&hhL2:vÆ?344Yiv8VgL6$vÆOR2:,hs%WLR_Eq!R2:,l,.*vOL2:.pm(uA/WlC^UR#31S8v. <,!Xt.pm(v eS8v <(vW6l,!X*v2:<2,EXte*WS8L1,XkNv=^x,Xt.pm(W[v=^xe*W+1d=,!X[<*v^x(p>g!j, m^)O^BeuoAvm^* !v%,Xh1Z6ÆL|8o pv2:iXKGz/F5z`gd2:: v/ervÆOR`,xED#`vf%,M6l=`46O4EDN`v5 – Npxvds4,[:gÆNv5i`v5,u/gL5%-q=,6O4nt<vl,hE|W?`Wv>U`2

1 o. SVySJd 71 n i) x f(x) !C0 [a, b] I4ii)f(a) = f(b)4iii)f(x) 0C0(a, b) :t, (a, b) :nU c,= f ′(c) = 0. qNv5%-,8p`Wtv5t i) k5 f(x) e [a, b] #P.T.F?4t ii) uk5 f(x) e (a, b) #vO c P.?4t iii)uk5 f ′(c) OJe!,WbzUF`,X f ′(c) = 0!G ghuov"upÆsUO#Gg`,D2 N2A~W`$`EsR v2(O*#ÆfKv8V4(W*|hh|dspdsv1pW<= ,!R v+2^rs^2zr7r7vArEA ve`k`Æs&e`k`,Nhgsv# ,OY +U`f,, ^rvg`3 vÆs,N g5LO^|ÆR??B,!sg.2jvheÆWR=R3Vo,fVzm?vs `S,&sgt.mvdzm?vs `S,OmgC ?vsm^W`q4Om,EGBWYv|vOL6vsm^Wqb?|,$LCv.0l vsm^d6O4U?v `Sh?vsUq,!3vn1z0?vs%-!rE'e2(O*OL`EvÆsv<[, `H7.!L<vÆsvQvP l,Y

∫ √ln(x+

√1 + x2)

1 + x2,8p

[ln(x+√

1 + x2)]′ =1√

1 + x2v<,Y2&|vdsXR2∫ √

ln(x+√

1 + x2)

1 + x2=

∫ √ln(x+

√1 + x2)d ln(x+

√1 + x2)

=2

3ln

32 (x+

√1 + x2) + C.(W*KzOLÆ.lvÆsv^,eY,dsW `%pE5(DQz\2

n∑

k=1

kCkn = n2n−1,

72 \3|lzz – ySE+Z Smarandache J117~Wn∑

k=1

Ckn = 2nv5m^,X7oB

1 +

n∑

k=1

Cknx

k = (1 + x)nY, x Ym, x = 1 X8n∑

k=1

kCkn = n2n−1.:,2:`ÆsvÆgÆvRu2|vOu?tqm hfvg,hh6q2gOROv X`7v,Gd#G0G7G|,BR:12;qvÆR

M 2 OoVL

CC S~2 (zxQQ<*y3D*Iny3D*E,y3y

K 2 MmTK – )H7 wPb>, 74-97

`R1%!S 6L\Oo— 4 1+1 x~??Hg 2 '2F: EJ<Q" l,21 G<QND*Q/+<Q,IJ?0U3Y:Q~0AQ<Qh,YpQaxe97X$ ,<Q~[xQD*Æ5C<QZ#,I 1+1H xQ.F3,lxQD*C~f[ ?~ax 7nHYY7XB6VP,b z 5","I0fQga:QO%Yi~O%YoKfoD7XMsY$aps-.y3 lÆ5k~ *IY<QI,aWQu,H!p,O0QK[kFV03: ?[~[ 7 7e9s7nHY7XB6VPYWYQ

Abstract: Many scientists show clearly that the leader of science is combi-

natorics in the 21th century, which implies that the notion of the existence

of common connection in things will be extensively applied to the scientific

research for understanding the nature and our society. This notion is called

mathematical combinatorics. Beginning from 1 + 1 =? and other interesting

questions, this popular report introduces this trend of science development to

all audience such as those of the complexity of things in the nature and our so-

ciety, contradictory system with natural reality, circular economy,· · ·, including

the multiverse interpretation on superposition of particles, the philosophical

meaning of the fable of the six blind men with an elephant, Schrodinger’s cat,

the importance of non-solvable equations with the reality, industrial systems

and the input-output model. This report adapts to different levels of audi-

ence for its combination of the science with that of culture, edutainment and

explain the profound notion in simple words.

12015 7 6 U 15 dNÆ'C3#PN (AMCA) :HT5"zV["unD

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 75

Key Words: Theory of everything, complexity of the universe and soci-

ety, understanding limitation, contradictory system, reality, circular system,

teaching and learning.

AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35§1. WA~gPz 3 |0vFMpWO& 1+1 z&=v`F, 2.1.1 K^

E 2.1.1~d2 1+1 2q7\dQ 37gPQ2 1+1 2q7\dKQ 3&~z 2 Z&zQ2 1+1 W7\dQ 3&gP!2 W,W-RQ 6 8&er,# `FgFL+TRyWvI hh8p,S4FgTzi; g0)( CvOXteO0W,Mqoe~OX,`<Qvd<`Q#,OXPFd!X,`<2 1+1 wKQ 27M`721NAV*wxQDDa, .q7h.F,~%L,F%N1U 1+1 KQ 2&gMT//OJ[ve\,t5 1+12z& 2`7o$RPF[ =2 1+1 NQ 2,q7h.FRA/,\,!R|`1 <`Q"o,`Xzi; 2 (a.11+1 NKNQ 2,1 1+1 Pz7\[dNQ 2,kN>f7\dNKQ 2 n[1U 1+1 Q 2,1UE3,1U1+1 KQ 2,1UE3ezi; =,1+1 `z& 2,G`2z& 2&$_,xQ,1+1 Q87OL\8v 1+1 =\mt2

76 \3|lzz – ySE+Z Smarandache J11 + 1 = 2(eU R # C Wm*,1 + 1 = 10(e 2 6M 2- 3Wm*,1 + 1 = 1(e`?/W*!L\+g0nv!!L8,,#ulOR 1+1 ds,I&tVuFÆ$Qv (AQ 4 Ux7BA x[$, 1 + 1 .F,7FrJH67,l26 = 3 + 3,8 = 3 + 5,10 = 3 + 7 = 5 + 5,12 = 5 + 7,· · ·,zzf,uFÆ! $Qv2g 1 + 1 z&=dsRgOR<W5ds

E 2.1.2L 1 `L,$_,O 1 + 1 = 2tVi>[e2xQ'V(Wqo2 x(shu,4 *pQx (shu,3 ),1 A 1 A x,QNs 1, 2, 3, 4, 5,· · ·<,1 + 1 = 2 I& (3 :),7 1 [ 1 R,wL 23OR 1 wL#,[OR 1 wL[Phh 2.1.2 Wv!_l,! ,1LIA n5v#~X,!.!L K[vIA,9 +Le## # Ix(shu,3 *vf%,!tore<# #6O4-,Xto/ Z` NU R # C,#vW^=r/# Iu, 1 + 1 = 2ag,L 1 `L2D X Y,uOY 1 + 1 6= 2, 2.1.3 K^.

E 2.1.3

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 77OY,hh |X⋃ Y | = |X|+ |Y | − |X⋂Y |K`,EIu$Xtz\ 1 + 1 = 2,7 |X⋃Y | = |X|+ |Y | meB3e |X⋂Y | = ∅j!O2D,hhpOLjvIdsR 1 47 Q_#xQ$\*6\,~\*~ 95 ~ 14 a,xQ~ 95 ~ 21 a,z.K 95 ~L 22 a0.2zE 95 ~LJa7K X = \*~ 95 ~Q,Y = xQ~ 95 ~Qu|X⋃ Y | = 47− 22 = 25 |,ge 95 |`#v|L

|X⋂

Y | = |X|+ |Y | − |X⋃

Y |= 14 + 21− 25 = 10,7 10 |v1`t,e 95 |`#$R:,(DWÒROYB`,l h~,7& n R,2D

X1, X2, · · · , Xn,∣∣∣∣∣

n⋃

i=1

Xi

∣∣∣∣∣ =

n∑

i=1

(−1)s+1∑

i1,i2,···,is⊂1,2,···,n

∣∣∣Xi1

⋂Xi2

⋂· · ·⋂

Xis

∣∣∣ .(DWuORÈvIB`,7I#B`2gsK ns n+ 1 i:5 (n?u) vp n A:< (n$K) ~,-qnLA:< (n$K) ~LinKJ:5 (n?u) !OB`,j=o,mLGÈ$.l,!R$<eWXG vGq s n(r − 1) + 1 A?uvp n A,5~,nLA,5~?uxKQ r AR 2 0l/ 1500 AU$[tSe:IL 5 AzN[z[hU ,G 4 Rb V <,1 8,G 366 y(Æ8*,$_,G 4 × 366 = 1464 R|Xtdy) 4 R|,a'e 1500 > 1464,<,e/O6QP,,' 5 R| V <R 3 ~ InL 2 Aa ZxN|J

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 79

E 2.1.5|YUb#,7/ivV/i;yv#`p, 2.1.6 K^,E 2.1.66hhhh1vV,7 1I+h,w+I?!OR#~ds+ 2N

2.1 lG". bKd8,?C&Æ#AvOR_o#/i;zpvJe,t;zp4( 2.1.7*

E 2.1.7b<|hI= uxWvO/i64 V1 = 7.9km/s, v^WvW/i64 V2 = 11.2km/s v^U;v/i64 V3 = 16.7km/s,

80 \3|lzz – ySE+Z Smarandache J1.y|n\ /iv*v# azp[<vB%pO=1w?Czy`VQ%2t[,98v?Ctzp=I& v k'3b2YE 2.1.8As"_vk/<L,yRvJeGX P Y<,RzpUO#gP3( 2.1.8*nvP3`\R=BWvrB Γµ

αβ 6lk<,As"_!^=Bnv%7KkzpvB%p,'e Γµαβ kM# #,zmkMvzp`<,7zp`m

Rµν − 1

2Rgµν + λgµν = −8πGT µν ,! ,Rµν = Rµαν

α = gαβRαµβν , R = gµνR

µν |L Ricci ,Ricci ,G = 6.673× 10−8cm3/gs2,κ = 8πG/c4 = 2.08× 10−48cm−1 · g−1 · s2,R?Cvzp`<2 gO::,`<Y-As"_zp`mv (t, r, θ, φ) Q#Wwt,W/& 1915 8toWwds2 = f(t)

(1− 2mG

r

)dt2 − 1

1− 2mGr

dr2 − r2(dθ2 + sin2 θdφ2).Y,,>- - jUe#bR 1935-1936 8tods2 = −c2dt2 + a2(t)

[dr2

1−Kr2+ r2

(dθ2 + sin2 θdφ2

)],!eWv a(t)wL/i 4se<# #,UrRq/ih,'|/i|LY,7bW/i da/dt = 0,vH/i da/dt < 0 g/i da/dt > 0N,hh 3v/i$eTg,a'2NKg!,6H|=N4d8[/i'TvHTYL

2.2 j['. |Y`6O ,|Y,g#`Gv!L8,?,GXv2;q,OR,v*|vdsXg!R-z+s2

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 81H)Hsb/. QA`.,r$r\[1w-f7

E 2.1.9Q fr\,Pmd2 RA`.b73 ,I5,r&Vu:1Q fr,mHX|d2 zr\.K,RNab79 ,iLA\&!RdsG*|h,2LfQvORds=83,O~+Tx !Oud, OXTJ< A vzA2OeMTJt8,dJFKvPYC OXy`T '(,M.OX&?wa.TvG< XzmzARV[vOy,G< XdM2 %n1$\[1y-,f7!g*< A ~<,Nd4vORds 6,,,t2qwY&G< X &g A T^Q2 n$1\[1y-,1nXf1\&ne ,1R3A,k1iLA\n(e ,1tL\ G< X [TF,. A vzAE1Wge!td d_8[,#OXG< Y .MzmzArRPP2S,G< Y GE A Q2 %n1$\[1-,f1RNf\7# ,*< A 72 hP1\VK=,Ku5(eo&gQp2 n$1\[1y ,1nXf 2q7G<TO,T-<M 1\ J=,(C31R1;J8&G< Y w2 7\.Kv$~,Rab7uMCr\|yR;7,M|C1,/+E!geWzAO81!P ,A C|'(#VOXG< Z3vCiO=vMY7o,O_Od h ($1\[1y-,f,

82 \3|lzz – ySE+Z Smarandache J11'q\Z8?Q%2t&g A e|0\"XQ^ ,2dh $1\[1yf!Rds, . u.,1[K \ZG< Z 2 v$,1>rK .A.FmK.,IhK .,[K .A.F!*< A T, L\&5oOX`4C%v|WyOk,'(|X =dM2 6L51,"tL>:o7A OX5, "VL>8&q\J7'(|E9M,G< Z LMA4%,|zAX<LR_!OR.jvdsx_Q+`vG< XY Z Qf.7QBs,e&!OdsvQER,7 fr\ fr K,er&|p,OY2s K,a2xs,U;0n.vpf2e A,ReG< XYZ wW,7DeMvQXtmQfPm:1zA4vve rr\[1-fds,|Y,WUrfVvds,QEgO^2npds,.FvEn2npds/,2.3 S2TR%WIX'. ORBORB>4R($_,'I0wYN|87ky`W,LB>r7AWuOv y,UyuO,IxB>, 2.1.10 K^rR,|YNEnY',EneGWlL,7` P'eRZGR XK#,!.Uyu%T/K

E 2.1.10LiA\kEnv2np,ty`V q[=`Mv%7h,7O%Yi, 2.1.11 K^

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 83

E 2.1.113X Fy. IAfivpAQgr 10wKd/y5,X:*,5iy57H>*,nK$X* 10w,1 0V!eMA,5,i 1ka[0,5,;IYÆi&\O%Y.2-ifiwN RNgXi,7LkEnvGW'A,gQ vXg)gds,p=*,kEnlL,7ne |'Ov25|3N3D8pe O 'v25aO_Nt/n'e (x, y, z) XK,Ne!O v25L 1,M L 0,7/nv*,eN LHo/ !^kgY-Bh 2.1.10 $kv,O=`=21y`VQf!,=o1956 8,ty`VS vOX,`< H. ? l?"CTq,`<` Wave Mechanics Without Probability,qEnGWlLv` k,7 vXv . iRWrJe, iv$RW$eOR W, $RW$eOR , 2.1.12 K^,

E 2.1.12D2 |YD1'WOR R\H. ? l!^kOY%Mm\

84 \3|lzz – ySE+Z Smarandache J1v` k'`to"CTOF,S1 ,`<`QQOF'EYx1<`&W8`_qeS Am,H. ? lehex1<`WOL [& 1957 8`_?"CTq,`<`,,\<`Q t,`Xa!td H. ? lvR ÆT,IX,`J[2 2L+#`<y`R?A$Q8[,H. ? lv` kto RL`<y`O$#M,`<y`Y4/iV[<m T92007 8,<Qx az-OR\,v|gT p` `R: H. ? lv` `< < 50 d8

E 2.1.13|Ydyv88 0;( gaWvV|88TAiO(2.1.13*e!R;(W,!RV|EY"nTAioTA%RJTI2v||rTALORu>Æ6wO1:gOW, 2.1.13 K^MhdR|+`?rR.<2w.NC! OXNMh 2 vAa[$e1&K|,N"wK[Q$6,!LvAa$1&>I$_,$v<5gaz1&N2q87gMhdR|TA88v'2D21&= 4 H5 ⋃1 A"5 ⋃1 AtV

⋃2 A`- ⋃1 6& ⋃1 H|5 OY,YeOP t\88ORdy T vl- µ1, µ2, · · · , µn,R νi, i ≥ 1g$S88vl-ue?/#,dy T XgT =

(n⋃

i=1

µi)⋃(⋃

k≥1

νk)

(2.1)

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 85ReP t v88 T =n⋃

i=1

µi 3gORx6?|Y8dyv# g,sLD`L|Yv8q |veavYG'27iQ#!^|EAs"_zmMo2 AxQ%OiC0w7XB6,&.N-qKT%4h[,i&T% ,OKC07XB6 ~1|Eh|Y8RY,21e$ hm7RKE2§3. 5tp -2B)2014 8 5 V 5 ,heE,WTp:1CxQF0DZ 1983C,n1EQ?Q"~QmxQ,QN'Q,Ep,'MQxsvr'gtdQ BollobasehVL?E?AwrTV,QN&:W&+5xvAn,TTd:.12xQn(r87:,&d(123*$(132*ArT,=1FDxQNH!Iw$s?&K%xQP; 92^PFhv,UO#0gpv[< ,GXg ,p[<&xhh`RSs|#9G 12 3$_,e#Ss v9x 6lk,UXg

(123)→ (132)→ (312).

E 2.1.14Y,lu|wv1!x0mi#vwx,


E 2.1.15Nhvwx|L (12) (13), (24), (1234) z$R:,hh`ton wp,7UOR n 2D Ωn = 1, 2, · · · , n,#vORKx σ kL

σ =

(1 2 · · · n

1σ 2σ · · · nσ

).l,n = 4 P,

σ =

(1 2 3 4

2 4 1 3

)XgORKx Ωn #vRKx σ τ,^kLiστ = (iσ)τ ,! ,i = 1, 2, · · · , nl,e

σ =

(1 2 3 4

2 4 1 3

), τ =

(1 2 3 4

2 1 4 3

)P,στ =

(1 2 3 4

2 4 1 3

)(1 2 3 4

2 1 4 3

)=

(1 2 3 4

1 3 2 4

).K Ωn #vKKxov2DL Sn`65,Sn Wv~f a, b, c e pR2

(1) Q#D4 (ab)c=a(bc);

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2 o. +J9 ySd – J 1+1 3r8eLR 2 4 87

(3) ~f σ ∈ Sn Je6 σ−1 ∈ Sn Xt σσ−1 = σ−1σ = IUO#,! vσ−1 =

(1σ 2σ · · · nσ

1 2 · · · n

).Q##RtvW=wLVwp Sn v~fOR2D,Q##pR,uwLKxpl,k|wwvKxp = 〈(12)〉,k0mi2vKxp = 〈(13), (24), (1234)〉,WKxR,/2 Z eW^=oW^p (Z; +),ae^=2oprR,`U#eW^^=rop,Lo,phh8p,wgO^Re! K`,pI&;ywvA

E 2.1.16 ,hhuY'. Pwvi( 2.1.17*,!eW,c1 = 11, c2 = 1221, c3 = 123321, c4 = 12344321,

c5 = 12344321, c5 = 1234554321, c6 = 123456654321,

c7 = 12345677654321, c8 = 1234567887654321,

c9 = 123456789987654321.

c1 c111 11111 111

1111 111111111 11111

111111 1111111111111 1111111

11111111 11111111111111111 111111111

1121

123211234321

12345432112345654321

1234567654321123456787654321

12345678987654321

c2

11

c2c3 c3

c4 c4c5 c5

c6 c6c7 c7

c8 c8

c9 c9

E 2.1.17

88 \3|lzz – ySE+Z Smarandache J1uì<=W,l,(1) t (R; +,×) o,7 R WW^^Q#x4D4,^W^Q#| 4,Wôxp,7 a + b = b + al,/2 Z eW^ +^ ×=Xot (Z; +,×);

(2) 3 (F ; +,×),7 F eW^^=rLxpvt (F ; +,×)l,`3 (N ; +,×)U3 (R; +,×) #3 (C; +,×) zrL3OR2D A e= LWv3qgWvKe= ,7~f a, b ∈ A,O a b ∈ Ae=2P,o^=WklL,a`= 2- 6lkORhgOR 2- (V ;E), 2 V 2 E ⊂ V × V o,2.1.18 K^,g-??zUOdyvA

K(4, 4) K6E 2.1.18OR`ke~#,Xt3e6 $/uwLw h, 2.1.19 K^,NUO#ghh<WovS^vA

E 2.1.19kP,|E2 vV\=b2 v), \|!LV\GM+OÂ1u,Xg=+^)X`7M

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E 2.1.20~z &=Ko,TvWOY,hh`e#i<ds-W,ORgOR2Io#v 2- Mi,l, <W rvWtz+&( 2.1.20*

E 2.1.21

90 \3|lzz – ySE+Z Smarandache J1$_,|Hl *wAP8?hh`bO A4 F,|L 1 9. 2 9r[,LR 1 9DeO ,to 1 Ru, LR 2 9,f,<Pv 2 9&u# 2 RGdDeO ,hhXtoORt, 2.1.21 K^-W!OR<,7 AD *: QAd.UJ'17BLOR G ke#,Xt: 3e6 /wLe#v6l,4 7i K4 eW;s#v6r 2.1.226 66- u3

2

2

(b)(a)

u1

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1u4

1

E 2.1.22hh8p,V|A$R;(W,C$XNvAz,TAv-o/gOÆ, 2.1.23 1L$RK^,! ,t1 = ,e1, e2 = TI,h =,b = 2,l1, l2, l3, l4 = %, 4 t2 = RJ,BR /se33eNh=r$_,1|H:t,Y(.Tw 3- 1&8? x,hh`*~ R2,|ÆWvd*~1x,7GS B2 bWRto~TA,r[, LdRGS*. R2 2=$Rm?1x,7dRGSnGuu,'e R3 WI` x,LdRGuu*TAv5X,!,hhXtoOR 3- MTA, 2.1.23 K^E 2.1.23

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 91

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n⋃i=1

µi g T v~$ (2.1) vx6?

E 2.1.24$_,nHYNK 7J?B8? QEgv&Ydy T1, T2, T3, T4 T ′1, T

′2, T

′3, T

′4 vlL| Rm(

(LESN4 )

x+ y = 1

x+ y = −1

x− y = −1

x− y = 1

(LESS4 )

x = y

x+ y = 2

x = 1

y = 16lk#r,m( (LESN4 ) o,sLm x + y = −1 . x + y = 1 \E,m x − y = −1 . x − y = 1 \E4 P,m( (LESS

4 ) x = 1 y = 1$_,1<AlJ? T ′1, T

′2, T

′3, T

′4 N x = 1, y = 1 QJ? T1, T2, T3, T4 2qKNb? er21,sL (x, y) = (1, 1) 2 gdy T ′

1, T′2, T

′3, T

′4 e~ R2 #v=. ,R T1, T2, T3, T4 e~ R2 #` R\rR,Nh$UJe& R2#, 2.1.25 K^

92 \3|lzz – ySE+Z Smarandache J1-6

Ox

y

x+ y = 1

x+ y = −1x− y = 1

x− y = −1

A

B

C

D -6

x

y

x = yx = 1

y = 1P

x+ y = 2

O

(LESN4 ) (LESS

4 )E 2.1.25hh8p,yR|L0yRdyRuyRW,uyRg'&0dyR vO^W W,K 'o

E 2.1.26oyRv#~ – "h& 1964 8y`V . 3UR?U. 7CN.mq,= SU(3) p\8;6l|YRto'\8!^Tv"q2#(u*(d*(s* (c*(b*(t*"vdnd"l,`gR"(Rd"(*(v#Dn,'OR0"ORd"o,+g#Dn

E 2.1.27

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 93$_,|BFTA*Q387hh8p,| n*%lLvmL mi~∂ψ

∂t= − ~2

2m∇2ψ + Uψ,W ψ Lnv*,,~ = 6.582× 10−22MeV s LI ,

∇2 =∂2

∂x2+

∂2

∂y2+

∂2

∂z2.K`,y`W"v%plL= m|Tb/ nTAoNO%Yo,ABF5nl5w ~oDNf7# m(OY2,GXg , 'l;m(g\Em(rR,"eU6WO=`Y',hh6H29;(,TAN_?yIheLr`'l;m(g\Em(,CeX``^$`7"gO^yRovY-m,Nh$UvJe,G|E|Yoz`h)\Ev`<6lk#bR 908W'vSmarandache (,wL_V(Ze\El-,7ORe`e!Y=BW` Pm2m,`G^m\2m,7!^=BRC=BW *|p7EY,LyRov=B`<Rzp,sL/i;y~1Xe2rPWW,)|EO^urZv6lk

§5. 9U n 7RKJe5[Æ2014 8 5 V 5 ,heE,WuTp21CxQvF0DZC*Aa~=dCMQ,: Mathematics

on non- mathematics<rV*,g-xQN;nHY,LnHY,nKf ~oDN7Xh:,NtQD*xQ~ Ma* ,7XYNVP,LnHxQN7XB6,NxQ(5J<[vi<d,7B8<[yoCz,,[oyvi 2.1.28 K^

94 \3|lzz – ySE+Z Smarandache J1lqo r8u? qb ` k4/- z6 - - - ??-- V%-E 2.1.28CFW,[|g|JCFvmvrEm^, 2.1.29 K^

Fi(x)

M1

M2

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x2i

xni

P1

P2

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---xi1

xi2

xin

W1 W2 Ws

? ? ?E 2.1.29l,nWORkndv%ph,73*";lh

A→ X,

2X + Y → 3X,

B +X → Y +D,

X → EW,A,B,X, Y |L+^yRvB4nd%p8,Z? A B sv|T,u X Y `qE|m(

∂X

∂t= k1∆X + A +X2Y − (B + 1)X,

∂Y

∂t= k2∆Y +BX −X2Y6lk

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 95B Ma* :Yozv[J<WgO^tCF,NEY#([y'y*L<[(I*,BU'y'y,\Ps&r,BRU'|r4=,Fn, 2.1.30 K^zu r1 r2 rn $ k5q/M-- - - - -?? ? - ? 6XN 8_8\S E 2.1.30B#,# [J<WvE|m(OYg2,7&Wv\E!G ,7g|Yv<[<,lLke#GTGg\E<,|YE7$0Lidy~$,$QCWv\E,n\EL/,!,GD!:1Lidy~$

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+((k(^((kg(

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E 2.1.31

96 \3|lzz – ySE+Z Smarandache J198L?kv:AL2 <-=mQQ(m*⇒ v-Q(Q*⇒Æ-QKo(J~*⇒ =-YM(y*⇒ 9-T(AlT*⇒ -I$$^,KU!Gghh.y|:_?vByg6.1 55zSWÆiÈ 2(1) iW=B25IE&|4z# 8VA1%7m^,EGd=RLR_,2qD-A+G,N2q72q|%7%~+,f' 2q.F7LtLK )yz;

(2) EG3Æs,LiI=5A1.A, hv;

(3) EL5LOR/uÆ,2qyW~BOR|7,sLNh6,6Rm`r zÈl,(DQz\ n∑

k=1

kCkn = n2n−1 `=%^5,aK28p3|gxY'!RQz\vUO#,B

1 +n∑

k=1

Cknx

k = (1 + x)nY, x Ym, x = 1,rXn∑

k=1

kCkn = n2n−1.

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∂U

∂x,!X?CW4vOYi\

∂U

∂x= −md

2x

dt2.

2 o. +J9 ySd – J 1+1 3r8eLR 2 4 97 ,j1e\ E =1

2m−→p 2 + U

i~∂ψ

∂t= Eψ, −i~∇ψ = −→p 2ψ,L to m

i~∂ψ

∂t= − ~2

2m∇2ψ + Uψ.

6.3 j5Se!R8Vrer?4VVffv8W,e 2,jer?Li,_ 8V:g<::-,zo36` 2(1) #.Æ/D,LE?PF, uE/ez4(2) E. 2vÆ/D4(3) F8E.er?Æ/Der?#vQ i9z,+gH0ÆvK065, PGg38VvKm^,z``rW 9g/R 9v# h3be`dXPF +1AUuo!O 9,PF Wg< Q+2l,|E PY .Rmvr1%p,sLD .v%D,:g 9|v`:-

K 2 MmTK – C)PBlbwP>6, 98-122[L6g0f=6oL\—— I r$r\[1-f.F3F: )~~'~+P:U;PgÆ~X\+xgPI>/X~K`+Bo)mbwP>gÆ#v~P)k~-+?GB&oSmarandacheH/#mGX\+HxgPX\C) ^~+:U#gH/~"oH/ - &4gH/~C) ^+=8MPoincareR~C)&+b' 3- gX50ZaP8 3- sKG~xP +Qkxg+C)50DUBEinstein&Æw~Xo+)H9km''`X6W Y"ZT+[wPDE~~Æ8+xPx2gÆ~;Pn_03: C)xgSmarandache H/&Æw+;Pn_

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, 01A27JWRrg/i;d;yvO^#~,(Dg/i;d;y/ervO^'i\,g| 88/i;yRnmve|Y,W,OR*|ORG|(ORV,#(,G,6Ri|Y,W,#~vE,g|12012 7 8 U 30 10 U 9 d/iyEfSÆÆÆN?u>yxSÆ_ÆNnD2www.paper.edu.cn, www.nstl.gov.cn

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100 \3|lzz – ySE+Z Smarandache J1 2q7G<TO,T-<M 1\ J=,(C31R1;J8&G< Y OXw2 7\.Kv$~,Rab7uMCr\|yR;7,M|C1R/+E!,!tdQ dO81!,C|'(,A #VOXG< Z3vCi,vMY,O_Od h ($1\[1y-,1f,h/x_Q.? Q%2t&ge|zZ0\"XQ^ ,2dh $1\[1y-,1f!Rds, . u.,1[K \ZG< Z 2 v$o,1>rK .A.F,1mK.,IhK .,[K .A.F!*< A T, L\&5oOX`4C%v|Wy,'(|OkX =dM2 6L51,"tL>:o7A OX5 "VL>8&q\J7'(|E9M,G< Z LMA4%,|zAX<LR_!OR.jvdsx_Q+`vG< XYZ Qf.7QBs,e& A vQER,7 fr\ fr K,erOY|&|p,2s K,a2xs,U;0n.vpf2e A,ReG< XYZ wW,7DeQXt!LG<QfvPY:1t,TOY, ds`|Y2OYgDJOQEvds,<PDEJOQEX`to0n4uOYgJeGRQE,<P35QE27i,u|Uu,e PJeGRUP,21 Osto0neW`nW,!Xg O1v2:hh o!Ohp 33,K334K3K3,3K3,)g!^dsvd,sLW`n AIq ,7 Otd,eOR`Dg0nv,eOR`D1XgNv,sL2Pe Y W,# OYdswLnpds,7Bs.: Jenp,`=, Y = f(X) =4WYdsvOYnwLunpds,gep$Q PY'v,7lLv2np,G|w:L T%7<Q),!eW,Wpv,gH< O%Yids3X Fy IAfivpAQgr 10wKd/y5,X:*,5iy57H>*,nK$X* 10w,1 0V!eMA,5,I:i1,ka[0,5,;IYÆi&\O%Y.2-ifiwN RNgXi,7

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G,G• ⊂ G L G W= • v2,u (G; ), (G•; •) gRp,BG = G

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(2) U: (R1; +1, 1), (R2; +2, 2) NAK[P,C0I0%P (R1

⋃R2; +1, 1 → +2, 2)

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Riemannian =BMh=vY-|g2L yk: E%bu|,nUPbuYE%buKwQ

R yk: E%bu|,KUbuYE%buKwQRiemann Y-to`rvBse&<`zm Riemannian =B,[

Einstein L/<WvzpP,7Lzp5 Riemannian v5,6Rkzp`k 2.2 2+, Smarandache =Bg Lobachevshy-Bolyai =B Riemannian =Bv6O4$'NU87QEgv&_ORjl2R 2.1 ABC NwK[uvA,nh 2-2 $B6MA(I0,YYzT(~w[,kbu ∑wY/vY ABC~bu_|*A(N Smarandache (,~Hnd2(1*TT(~ 1 PV IvAwvA&$%'buK,FQ [N Y AUPbuntLbunh2.2.5(a) ~,DCE vA[u,p DE UPbu l1,kFG A,∑ ~KUAbu

L3

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E 2.2.5 J Smarandache jIRR(2*TT(~ 5 PV E%bu|,UPbuYE%buKwQK,FQ [N E%bu|,Un<KUPbuYE%buKwQnh 2.2.5(b) ~,L N C $ AB u D bu/ AE w3Q CD, E ∑ ~UPw3Q L bu L1,kK AE u F,∑ ~KUw3Q L bu, AF bu L2 %Y L wQ,on Smarandache=Bv5 Iserie~bR7,Mj~Jn=

106 \3|lzz – ySE+Z Smarandache J1B,onO~ Smarandache=B,lH<=Bu=Bd*=Bd=Bz,rM& 20028\vt [6]OYpe#on Smarandache

2- i,7=Bv55e 2004-2006 8,%v&v57,[e#onH<=Bu=Bd*=Bd=Bz,'& 2007-20098OYpon Smarandache n- i,r [10]-[13],8e v G`TX# `(, 6O4$Q\5ZKr,Ady: /e9/eMQv,gO^,v = %7,7 Smarandache ` $QdyP,dy r'vAog-4e WvlL(J*)EOR$hUO#g2D#WvO^^,|L,o,^~`Di<,,o,8e v|`TX`(- V gOR,2D,E ⊂ V × V,uW (V,E) XkLORhG,'wV L 2D,E L2DP,L;g G v 22,# 9GK V (G) E(G) ([1],[3])&ORUv G,L L~#v- ,R : vLsR- : v.,uX`toORe~#vl, 2.2.6 WU7iW5 K(4, 4) 7i K6 v

K(4, 4) K6E 2.2.6 ZERUR G1 = (V1, E1), G2 = (V2, E2),wNh[_,AvgJeOR 1-1 * φ : V1 → V2 Xt φ(u, v) = (φ(u), φ(v)),! ,(u, v) L G1 vOUO#,R ov,3gK2 ,(Do7iOIl, 2.2.7 WUvR,XgR o

u1

v2v3

e1

e2

e3

e4

G1

v1

u2u3

f1 f2

f3

f4

G2E 2.2.7 Z>&EORh7y P,AvgOR2DP = G1, G2, · · · , Gn, · · ·BQ#e o φ x2,7~f G ∈ P Gφ ∈ P<`i<vsp6pip,!uCJ7z, 4nvOL#~B,s4.mM,lgv op45z!mv# 8V,`B02\vOL~,lbDtv [1],`4 Gary Chartrand Linda Lesniak Dtv [3] z(i*I*E-gORAe- v6- G gOR,S gOR- ,uOR-, AvgORsv 1-1 * τ : G→ S,Xt.: ,Ge Q([5]*e MT&z& 3 P,# v6`U'L=.96,7EYKv6oM [rL=.9,!Xg!R`2XK 3.1([10]) A n _7hhbu#pwTTI0 Rn, n ≥ 3U UO#3|5 n = 3 vQiUv G,e . (t, t2, t3) #s n R (t1, t

21, t

31), (t2, t

22, t

32), · · · , (tn, t2n, t3n),! ,t1, t2, · · · , tn g n R2 Uu`65,&2 v/ i, j, k, l, 1 ≤ i, j, k, l ≤ n,s (ti, t

2i , t

3i )

(tj , t2j , t

3j) v=..s (tk, t

2k, t

3k) (tl, t

2l , t

3l ) v=./,u

∣∣∣∣∣∣∣∣

tk − ti tj − ti tl − tkt2k − t2i t2j − t2i t2l − t2kt3k − t3i t3j − t3i t3l − t3k

∣∣∣∣∣∣∣∣= 0,!X/ s, f ∈ k, l, i, j Xt ts = tf,. t1, t2, · · · , tn g n R2 Uvb^\E

108 \3|lzz – ySE+Z Smarandache J1&M Je=.6,/j,&g-<ZGve#v 2- g96,7B S #d G [,Bv$r- &OR 2- MzGS,4 2 (x1, x2)|x21 + x2

2 < 12qND7OR2Io#v 2- MiwLD([9]) 2.2.8 W,UWtsphere torusE 2.2.8 AGU~fOR`^LORGivD ,7`^LKGiv+ S,XtdRe S W)' 2.2.9 W,Ut Klein vGi^--

6 6 6 6-

torus T 2 Klein bottleE 2.2.9 UG Klein &hs$-Wtv|Y`,~BOR- &W,g p Rtvs,g q R*~vs,7!R`XK 3.2([8],[9]) (AD[bQdv5*D[:

(P0) =: aa−1;

(Pn) n, n ≥ 1 APY$:

a1b1a−11 b−1

1 a2b2a−12 b−1

2 · · ·anbna−1n b−1

n ;

(Qn) n, n ≥ 1 A 7wY$:

a1a1a2a2 · · ·anan.ORwL%v,*#~BOD. Oe[^?. 4Pm?/ 4d:,. 4Pm?/d,uwLK%ORv Euler,N χ(S),7χ(S) =

2, if S ∼El S2,

2− 2p, if S ∼El T2#T 2# · · ·#T 2

︸︷︷︸p

,

2− q, if S ∼El P2#P 2# · · ·#P 2

︸︷︷︸q

.!eWv/ p wL?,N,KL γ(S),Wv,NkL γ(S) = 2,qwL2?,N,KL γ(S) 2.2.10 K^7L7i K4 e Klein #vOR2- g96vOR6GwL hf,#~fO`kL,<,CTutte z|5,`=Wm^U. 6 66-

u3

2

2

(b)(a)

u1

u2

u4u3

u1 u2

1u4

1

E 2.2.10 K4 Klein &dFJA<YXQ 3.1([8],[9]) A h M = (Xα,β,P),%t2|* X Æe%Kx, x ∈ X, ;V[e| Xα,β AtrT P,/eAdV~ 1 $V~ 2, K = 1.α, β, αβ Klein 4- eV,$% P trT,KUPMx k, = Pkx = αx"K 12αP = P−1α;"K 22V ΨJ = 〈α, β,P〉 Xα,β 8f,! ve` 1 k5 P `|LiUtv&,e` 2 k5vsp!,-6dseO4#XnLOYWKxds,W

110 \3|lzz – ySE+Z Smarandache J1viUtkL ,R Pαβ WviUtkLl,7i K4 et#v61

1

2 2?*Y +sx

yz

u

v

w

--

6 6E 2.2.11 K4 <YgUX`^LWi\ (Xα,β,P), ! ,Xα,β = x, y, z, u, v, w, αx, αy, αz, αu, αv,

αw, βx, βy, βz, βu, βv, βw, αβx, αβy, αβz, αβu, αβv, αβw,P = (x, y, z)(αβx, u, w)(αβz, αβu, v)(αβy, αβv, αβw)

× (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv). 2Lu1 = (x, y, z), (αx, αz, αy), u2 = (αβx, u, w), (βx, αw, αu),u3 = (αβz, αβu, v), (βz, αv, βu), u4 = (αβy, αβv, αβw), (βy, βw, βv),2L e, αe, βe, αβe, where, e ∈ x, y, z, u, v, wKZ5H h,AvXgdOR+g9 r ∈ Kx vje` 1 2,`L 55H h7[_VNwfV 1M&e#v6,rEYds2b/ 1: E%Ah G $%n<K%D S, G N#p S?!Yds\!R`Q2XK 3.3([5]) C AYh G,#p%DYK%D^?*MxC0,r GRO(G), GRN (G) ~9 G #p%D$K%D^?,

GRO = [γ(G), γM(G)], GRN (G) = [γ(G), β(G)],, β(G) = |E(G)| − |V (G)|+ 1

2 o. E+SDndyS8 – Js~sg~/G44 111B` 3.3 `,v2?,T,NvB=n,K`,$Qv6ds,2Wen6v,F,N?,T,N,7γ(G), γ(G) γM(G),Wn γ(G), γ(G) g/e1+v, 33%OLlY,7i Kn,7iW5 K(m,n) n,Nb/ 2: E%AD S,h G D S LJAK[_#p7n<| ,CQE%_x,n%x'x,S LJAK[_#pn5HK5H h7!YdswL62WIdsk G v?2?6G<\L

g[G](x) =∑

i≥0

gi(G)xi, g[G](x) =∑

i≥0

gi(G)xi! ,gi(G), gi(G) L G e,NL γ(G) + i − 1 γ(G) + i − 1 #v6 33%OLlvY,tnzn6G<\j Bunside z`,`L 5UORsvj G,e#vK2 oW? rO(G) L2rO(G) =

2ε(G)∏

v∈V (G)

(ρ(v)− 1)!

|AutG| ,! ,ρ(v) L v v,AutG L G v opl,7i Kn7iW5 K(m,n) e#<v?W|L(n− 2)!n−1, 2(m− 1)!n−1(n− 1)!m−1 (m 6= n), (n− 1)!2n−2.7i Kn, n ≥ 4 e#<v2W?L2 nO(K4) = 3,B

n ≥ 5 unO(Kn) =

1

2(∑

k|n

+∑

k|n,k≡0(mod2)

)(n− 2)!

nk

knk (n

k)!

+∑

k|(n−1),k 6=1

φ(k)(n− 2)!n−1

k

n− 1.(\*P - syEI*E- G gOR-,L gOR2DG #OR9 θL : V (G)

⋃E(G)→ L,7= L Wv G v 6lK,l, 2.2.12 W,=/ 1, 2, 3, 4 K4 6l


2

3

1

1

1

4 2

3

4

2

3 4

4

1 1 2

2 1

2 E 2.2.12 yEER - 9- GL1

1 , GL2

2 wL',JeOR o τ : G1 → G2 Xt~f x ∈ V (G1)⋃E(G1), τθL1

(x) = θL2τ(x) - 9-`5L Smarandache ` S =

n⋃i=1

Si vO^(D-o,!^ - 9- G[S] k ([11],[12])2V (G[S]) = S1, S2, · · · , Sn,E(G[S]) = (Si, Sj), Si

⋂Sj 6= ∅, 1 ≤ i, j ≤ n,'2D Si, 1 ≤ i ≤ n K ,Si

⋂Sj K (Si, Sj) ∈ E(G[S])l,&TA,K a = ,b = ,c1, c2 = TI ,d = A ,e = .

,f = 2 ,g1, g2, g3, g4 = % ,h = RJ ,u-or 2.2.13.

a

b

d

c1

c2

e f

g1 g2

h

g3 g4

a ∩ d c1 ∩ d

b ∩ d c2 ∩ d

d ∩ e e ∩ fg1 ∩ f g2 ∩ f

g3 ∩ f g4 ∩ f

f ∩ h

E 2.2.13 9FI*&TA!^-o.TAvUOiL2/. $_,x 3-8 v(Do,B=B#|mx0ORY,&TAvi.7O^r7^g,x,| 2.2.13 WdR = 2- MGSbW,!to~TAvir 2.2.14.

a

b

d

c1

c2

e f

g1 g2

g3 g4

h

E 2.2.14 %9=,`6O4| 2.2.13 W(TAvdO(t~gOR 3-MWu,'6l\5xl,!Xtor#OR$UvTA,r 2.2.14.

E 2.2.14 3- Y9(*YI*E#$^OR 1- M-L 2- M-,r[L 3- M-,tohhrov$UTAiv3^,UO#`6lOYA,to n- M-([13]-[15]),k:XQ 3.2([11],[14]) A!L 1- yh G d_yI0 d- yh, Md[G],neA

(1)Md[G]\V (Md[G])NLsA0| e1, e2, · · · , em ,~vA ei, 1 ≤ i ≤ m[bQ d- 0= Bd;

(2)C Mx 1 ≤ i ≤ m,'i ei− ei HAn<A d- = Bd D,/WeD (ei, ei) [bQWeD (Bd, Sd−1)

114 \3|lzz – ySE+Z Smarandache J1Y,,OR d- M-v 1- M-LÆ,7#`-K,uw/ d- M-Ld- tu!R& d- M-#~pv<XK 3.4([11],[14]) C A d-yh Md(G),L π(Md(G)) ≃ π1(G, x0), x0 ∈G, >9 ,CQ 9 d- t,πk(Md(G)) ≃ 1G, 1 ≤ k ≤ d.#~pL~bpv- wLhYI0,` 3.4 8,~fO d- MÆgs K>"v Perelman s`p Poincare 67,& 2010 8ttU"T! ,M5vAhY 3- 61[bQ S3, j!R,hh`to!R&s 3- ivop`2XK 3.5([14]) vAhY 3- 61N9 3- tY,,`3jv d-M-,7e d-M-#zE|o,`4#vr5zE|=B2:,l,=6I=(D Riemanniani (O^M-) #v5 R,to2XK 3.6([13]) M ALsD*61,R : X (M) ×X (M) ×X (M) ×X (M)→ C∞(M)AD*61DQ,C ∀p ∈ M,S5F (Up; [ϕp]),L R = R(σς)(ηθ)(µν)(κλ)dx

σς ⊗ dxηθ ⊗ dxµν ⊗ dxκλ,,R(σς)(ηθ)(µν)(κλ) =

1

2(∂2g(µν)(σς)

∂xκλ∂xηθ+∂2g(κλ)(ηθ)

∂xµνν∂xσς− ∂2g(µν)(ηθ)

∂xκλ∂xσς− ∂2g(κλ)(σς)

∂xµν∂xηθ)

+ Γϑι(µν)(σς)Γ

ξo

(κλ)(ηθ)g(ξo)(ϑι) − Γξo

(µν)(ηθ)Γ(κλ)(σς)ϑιg(ξo)(ϑι),/ g(µν)(κλ) = g( ∂∂xµν ,

∂∂xκλ )KqnKy`$Q,\,&|Yvr ,sLDe|Yr W,/v<:`%v&UO6l5U5Q(J*nKw;10YI0N2qwY7e Einstein :3,$Q|FOY= Newton vP,LP . g^2 vBy 4nXK,P 4nn6,7=qPVyR,4nvPhL ((x1, x2, x3)|t),<P&Rdt A1 = (x1, x2, x3|t1) A2 = (y1, y2, y3|t2),P PL

(A1, A2) =√

(x1 − y1)2 + (x2 − y2)2 + (x3 − y3)2.

2 o. E+SDndyS8 – Js~sg~/G44 115rR,Einstein v/2

-6

x

y

z

future

past

E 2.2.15 ;DjC/Pv.,WL~=P,7 Minkowskian ,v.Ld2s = ηµνdx

µdxν = −c2dt2 + dx2 + dy2 + dz2,! ,ηµν L Minkowskian 4,kLηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

.hh8p,P g|Y`| yR% dtY< vORB,y`)kgdtY<ovP Pa10yN2q7#.<22P 1: 10Naxe9JAQr_x,Y10tLwY!^ ,UO#XgqP 2P 2: 10Naxe9QrA_x,k'Y10K~!^ Xg/P ,Gg Einstein /< ,WeWy`#vr e!^ ,LP g Iv,7 Einstein zp`mY,zmv/iTrhW,Tr$OgP vz6

116 \3|lzz – ySE+Z Smarandache J1!eWuGds|E6l3%,Xg10NJ?sLn7RNaxN|N#7'ywN2q7'Ys^~~&0wYN2q7D!Yds$QN,|Y:1,\VoP v~R(i*4Qeqj Einstein WU $QW,B nJ<RLXgOtL|1+vdQ,Einsteink/<vPT,e&|YV,0P|YVRq y Ouer,Einstein k/<1~rgOR\5pV,?P v$UR6O416Einstein k/<vUR,g&y`4qvMpB` ([2])2v'sK2 A?~%Oo$LS5_6Y~.KiQS5YM,!Lw[7m1f,6g|Qbr6zmv,K`,Einstein vMpB`\ v,UO#Xg CzOKam"!O%7vu'L<,EinsteineMpB`# #,jE|=B [4] Wv5zm'xo,7 ([2])

Rµν −1

2gµνR = κTµν ,! ,Rµν = Rα

µαν = gαβRαµβν , R = gµνRµν |LRicci ,, Ricci DQ5,

κ = 8πG/c4 = 2.08× 10−48cm−1 · g−1 · s2e$W,Einstein zp`mvWw Schwarzschild 4,7ds2 =

(1− 2mG

r

)dt2 − dr2

1− 2mGr

− r2dθ2 − r2 sin2 θdφ2.e<# #,/iB`,72n9 104l.y. 1,[ ~ *w[,tLoq ,Friedmann e#bRL8Wzm/ih,7ds2 = −c2dt2 + a2(t)[

dr2

1−Kr2+ r2(dθ2 + sin2 θdϕ2)],'|/i|YY2a$lG2 da/dt = 0; "lG2 da/dt < 0;lG2 da/dt > 0.

Hubble e 1929 vN,/i0$&g9,<[</iTrY ,72[ sQ 137 CF[ (\*ZKlG|YY'r#WJe 4 ^#~5p,7'?'"Ys"z,Einstein v/<rE|TzplL,Rpug&En /e5,78p;>8p>8pzGy`V,h2 Einstein ~|O=Ip&O!+^#~5p,7O/<p,zmy`vTO`<,GXg`(WC 'vTheory of

Everything,adsO=`top,v+ e&k/<g&U/i`<,wGU;J(z,Y-yRgs|3v4Rpug&E/i`<,RWz,Y-yRg^|3v?ds$Qv3,|YV3G'OL_ds,l2[ Nb7nK,LJA[ 72qax3Kw&[ 7axi~[ xQJ7BN 3 b7n[ sQ,-q[N2q7BNI0;$Lb7nX, ;$LgNax3h ;$L......WLbR'v"`<Lp# ds# !O`<Y-n2gR RgM2 v p-,7* p Rm?d4v ,! p gOR0/,~1dp,Y,OWF"5dp% P,"`<L/i0<5O v`F2[ <1NAI0x 11 I0:,~ 4 Ao# ,j,Q1| 7 Ao#$D ,|11uM3w 4 6z[ $3K= 7 z[ 4 6z[ :QA* Einstein'xo,Q 7 z[ :QA*eAo,HCY,h~V1INAvA$DA 7 I0 R7 4 I0 R4"`<UO#gek/<# #YvO^(D`<,|Y'Ao^5U5Q,K`,"`<>rg:L8y`#v$Qr,a2Pd:t,LN2gO^$0vy``<,2 "`< z-|Yv%73,LV/i~= AORm?!eW,5|EpvVds,gxN M ≥ 4 vdylL,sL|Y 3vNA3,& 3- M ,e 3- M WU'Y,&V|TAv

118 \3|lzz – ySE+Z Smarandache J1V ,O^,=v%7,gL2 vN(D =,6Ridyv/uV,7(D 3-M ,7w3zi,6R4lL 2.2.16WU K4#~lNhvORl ([9]-[11])R3 R3

R3 R3

P1 P2

P3 P4

- ? ? - 6 6E 2.2.16 %%2f,! R3

⋂R3 2g `v 3- M ,M`L 1 M2 M

3 M,!,X`&M Nl,3 m R 3- M (D,-V (G) = u, v, · · · , w,u`| Einstein zp`m$o!^(D #,L

Rµuνu− 1

2gµuνu

R = −8πGEµuνu,

Rµvνv− 1

2gµvνv

R = −8πGEµvνv,

· · · · · · · · · · · · · · · ,

Rµwνw− 1

2gµwνw

R = −8πGEµwνw.v,G`=(Dm^Y.(D,!eW,Y.(D.K vvM m l,dR =Ww6,u`toe$WvWw,ue m = 1,7dR vP 4 tµ = t P,

ds2 =

m∑

µ=1

(1− 2Gmµ

c2rµ

)dt2 −

m∑

µ=1

(1− 2Gmµ

c2rµ

)−1dr2µ −

m∑

µ=1

r2µ(dθ

2µ + sin2 θµdφ

2µ).!muG|6O4$Qv5,8e v|`0O0`( [10]-[16]

2 o. E+SDndyS8 – Js~sg~/G44 119j5?R(% g|Y`pVrv[y,Y'Vr4,6RG6|YYhr,.rMYXWLObR YvrEm?, PGT $5q_v~

E 2.2.17 lGVM$_,0'n(MF,n(\AMxQEFCaY7X!Z#QA70' 2q|$,IJxQQU$h87(1*?P,'d$K 2qQxQ? CxQBL*Hb? UO#g,&CF# #v#Ozy$^GA&6LQ;*ev7^g2'Uv|YiW,$LtVv<=+LNy,pVvPVvzprCW =2a(g,=#2aT;,= 2aO|,a&2a$ZX,4[q2a3uN%a,"K(,pIFC7gP|<T,,[,| ?4 <-L=QLmQQ,v-Q,Æ-QKo,=-QYM,9-QT,-QI$$^,KU,d,jmVUm8,O1L |*2tGK`,7EL$0v5,O lG,LY $`2WEpvgp.p`(2*Dz`.,ZKK 2qhA.F,'L2qA7 !RdsgdOR5vds |tR,VdyXR3,a P,dyvVGXRZ&r$_,BBV#1!^+,6R1 ga$^\5vB. ~= q[)[vV.7(Dg# E8p, | 2g v,Ridyv$U`:g vEB!O4Y, $s`|LtY2L 1 I: CxQ~W~A.FL'e4L 2 I: CxQ~WZ#L'e4

2 o. E+SDndyS8 – Js~sg~/G44 121q,ra%v1pGT4y;,a.< P,tIlt|g2 <Izds["`,!.|Y d48r4v28E,Bd GH&r4v% 4C7L<,dR 5+OR|3%vds,!Xg,<Q(N2q7<QNaxy|,7XO7QEg#v Lr|&|YVr, $Q*|.rMYm?6l,!G0g5|EBNvds,R!O W,(DoYLzm;d;yr,nm-q #~%7m^2_[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 Sci-

entific, 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, Math-

ematics Magazine, Aurora, Canada, Vol.12(2003)

[8] Y.P.Liu, Introductory Map Theory, Kapa & Omega, Glendale, AZ, USA, 2010.

[9] Za, Automorphism Groups of Maps, Surfaces and Smarandache Geometries

(Second edition), Graduate Textbook in Mathematics, The Education Publisher

Inc. 2011.

[10] Za, Smarandache Multi-Space Theory (Second edition), Graduate Textbook

in Mathematics, The Education Publisher Inc. 2011.

[11] Za, Combinatorial Geometry with Applications to Field Theory (Second

edition), Graduate Textbook in Mathematics, The Education Publisher Inc.

2011.

[12] Za, Combinatorial speculation and combinatorial conjecture for mathemat-

122 \3|lzz – ySE+Z Smarandache J1ics, International J.Math. Combin. Vol.1(2007), No.1, 1-19.

[13] Za, Geometrical theory on combinatorial manifolds, JP J.Geometry and

Topology, Vol.7, No.1(2007),65-114.

[14] Za, Graph structure of manifolds with listing, International J.Contemp.

Math. Sciences, Vol.5, 2011, No.2,71-85.

[15] Za, A generalization of Seifert-Van Kampen theorem for fundamental groups,

Far East Journal of Mathematical Sciences Vol.61 No.2 (2012), 141-160.

[16] Za, 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 Proba-

bility, Set, and Logic, American research Press, Rehoboth, 1999.

K 2 MmTK – wPC)XI>A, 123-1476[Lkoq% 1,2F: axe9UwBnHkO.YGiz,InyGiz6MwJ?0nH7$Y7,n~ (AGiz*i<BFJ?QB67,Pnga-Aal~$:Qy~|XQ,7XiJ?U1,3, (A7XJ?*N!L%I0d_[1OeA?y 3O"I0pC,s5J?BO=aIIxQYyxQD*,_ExQ[~V,xQD*,I0yhCxQYy3y,=[1eAb 3O,xQe[QA6*(,QlQA6,nNyhxQ~V~xQ?~?~&<Q0A,bKf xn ~oD G- f,Banach n Hilbert−→G - 6I0~"I0oD"I0f,~&<Q,nt 5,"I0fQ'1I$?xQo0A,uQxQBFJ?3s7pQxQN!;nHY,g-xQD*NxQD*y,k[1BFnHYxnHY3,K!L<Q3,"J?0UnHNaxe9$pQxJ?703: J?B,zi,nH,yh, Banach I0,Smarandache "I0,−→G - 6,o"I0f,21I,xQD*Abstract: There are 2 but contradictory views on our world, i.e., contin-

uous or discrete, which result in that only partially reality of a thing T can

be understood by one of continuous or discrete mathematics because of the

universality of contradiction and the connection of things in the nature, just

as the philosophical meaning in the story of the blind men with an elephant.

Holding on the reality of natural things motivates the combination of contin-

uous mathematics with that of discrete, i.e., an envelope theory called math-

1Mathematical combinatorics with natural reality, International J.Math.Combin., Vol.2,2017220177 4 U 18 d2017 Spring International Conference on Applied and Engineering Math-

ematics"4uSÆnD

124 \3|lzz – ySE+Z Smarandache J1ematical combinatorics which extends classical mathematics over topological

graphs because a thing is nothing else but a multiverse over a spacial structure

of graphs with conservation laws hold on its vertices. Such a mathematical

object is said to be an action flow. The main purpose of this report is to

introduce the powerful role of action flows, or mathematics over graphs with

applications to physics, biology and other sciences, such as those of G-solution

of non-solvable algebraic or differential equations, Banach or Hilbert−→G -flow

spaces with multiverse, multiverse on equations, · · · and with applications to,

for examples, the understanding of particles, spacetime and biology. All of

these make it clear that holding on the reality of things by classical mathe-

matics is only on the coherent behaviors of things for its homogenous without

contradictions, but the mathematics over graphs G is applicable for contradic-

tory systems because contradiction is universal only in eyes of human beings

but not the nature of a thing itself.

Key Words: Graph, Banach space, Smarandache multispace,−→G -flow, ob-

servation, natural reality, non-solvable equation, mathematical combinatorics.

AMS(2010): 03A10,05C15,20A05, 34A26,35A01,51A05,51D20,53A35.

§1. WAOY,ORdy T v~$ANeb##v d'e|=JeW,R2<gL|YKNogL|YV`rR,|YORdy~$vVQ>&Nm^, 4Nb#v` – s^b#,7LORdy TvlLgORQ>P t vs, f(t),g,o, x1, x2, · · · , xn,! n ≥ 1,N.dyv~$1/VL;I-

(a) (b)E 2.3.1

2 o. ySE+ZKC 125$_,1GiNI,RNy? er+2g,sLe|Y-W,LJe^pdy,#Jespdyl, 2.3.1(a) WOÆ#vg^v,a 2.3.1(b) (pep-#vlYugsvBiW#,LidylL PG6s.^vY,7=svm^$Q^ds,g=^vm^$Qsdsl,- x y |LORXP(v 2- U,lxr “i$rm( 2.3.2*Wvd(X* (P*,E 2.3.2u Lotka-Volterra =E|mkNhvlL ([4])

x = x( λ− by),y = y(−µ− cx).

(1.1)Y,,ORspdsI=%U'vK?I=+g%v&^m^,sLI=%v=^1(UR#g^v!mORWpl 2.3.3K^g,7O^>KU'vs<,Lidy~$|Es.^vOE 2.3.3Rey`W,ORnlL dye?CB R3 gAs"_B R4 WvW, ψ(t, x) N µ1, µ2, · · · , µn onvE|m ([2])

F (t, x1, x2, x3, ψt, ψx1, ψx2

, · · · , ψx1x2, · · ·) = 0 (1.2)Kk

126 \3|lzz – ySE+Z Smarandache J1 = ,ORy`Ag#`v,6H.MdtAQ`$_,OY,o (1.2) $LfNA z 5 P GiBb? er2g,sLB#,m (1.2) 3gQ>& P eP t vlLN µ1, µ2, · · · , µn 6lkv,'2g/ul,pW Schrodinger m ([24])

i~∂ψ

∂t= − ~2

2m∇2ψ + Uψ (1.3)kvEnlLXgORU5,sLN,En` P'^ZG1W,GXgWW,0 2.3.4 W Schrodinger Kdv,-Ar A10w,5~iN RNgXi,Ohh6H|2Nm (1.3)#ORgEn,sLdR31'O^nW

E 2.3.4Z6O4,1<dAlo (1.2) n BF z 5 P *Q3b? er21,sLm (1.2) g#&ORL2Y-,7eCpW+&n P v gOR=B ,epWgOR`Rzmv, P,NuQ>&Ngen2ugn06lN_l= ,OR| H2O RLBOR<B(, 2.3.5 K^

E 2.3.5

2 o. ySE+ZKC 127Y- H2O gOR =B ,Ne|2NLB<BvlL,uM31tLB H<B O .| H2O MOIvb,aM6o|056lN,M|tLB H <B O 2 vN,3wL|zi,[wL:zi<P,Qb2 vNzm| H2O%pm,3g (1.3) R[uLB H <B O #vRE|m

−i~∂ψO

∂t=

~2

2mO

∇2ψO − V (x)ψO

−i~∂ψH1

∂t=

~2

2mH1

∇2ψH1− V (x)ψH1

−i~∂ψH2

∂t=

~2

2mH2

∇2ψH2− V (x)ψH2

(1.4)

($_,+ANBF~5 H2O *Q3no,(1.3) RN (1.4)?!OdsvQg1+v,sL (1.3) 31kLB H<B O .|H2O MOIv%plL,Rm (1.4) eCW`,72 ([17])~`vrE v,e&NORdy T v~$e|Y-WgOR\Exhp ,Ng2m(e=B#ov` 6R,k`dyv~$|E|YozO^_v`< –xQD*,7#& -#v,'NYv`,`4e#~nzp`4M Wvl,-v Banach Hilbert

−→G - 4#vj,|, E|mov2m(#v=B, 4e#~n p<yM #vz~`Wv19gv,p19 &`( [1],<yv &`( [4],(D=Bv &`( [8],#~nv &`( [23] [24],Smarandache` v &`( [25], P,~`i<vK'AlLrY-$UJe

§2. |gygRIyL*U| sNdy~$.C$Uv#~,a'|/ 2.1 % R, MR ~9J?B|$xQe9B6|MR ⊂ R, MR 6= R. (2.1)U f,Cvoz# go\Ep,7CgORMOIv/rR,dy \Eo$2e<,C$U231g

128 \3|lzz – ySE+Z Smarandache J1dy~$2v2,7 (2.1) \MR ⊂ R, MR 6= R. # sUO#gORs>rrj,a<z,|YLir~$,7 MR = R,|EeC`<# #ozO^hB`<,7h\EJev`<

2.1 "x,hhi<R%7hT1. xR. !g CWORtvd,)/u5|o,rR|Y Liv+g5|Rudy/uv;fe!RdW,!RV|EYdTA1uv2 5XRnTAvioTA%RJTI2v||rTA2iLORu>Æ6wO1:gOW, 2.3.6 K^Mh`?rR.<2w.NC! OXNMh 2[$e1&K[N"wIQ$K[6,!LvAa$>I<,!XNE9V|gvTAiL1&= 4 H5 ⋃1 A"5 ⋃1 AtV

⋃2 A`- ⋃1 6& ⋃1 H|5

E 2.3.6ApJ:QN2q? |YVdy~$v ,0dWV|88TAviO |YORdy T v88gORx6 _l= ,YeP t,\C88 T vl- µ1, µ2, · · · , µn,R νi, i ≥ 1 g$S88vl-,ue

2 o. ySE+ZKC 129?/#,dy T XgT =

(n⋃

i=1

µi)⋃(⋃

k≥1

νk)

(2.3)ReP tv88 T =n⋃

i=1

µi3gORx6?! vm (2.3)OYwLSmarandache"I0 ([8], [25]),7=dy^l-v(DVdy TT2. EverettBW;~. H.Everett & 1957 8Xm (1.3) v*,U` k [3],%UQEnvGWlL Y-Nv*,.Nn/eRY,MU2 W,/$e2 vp,'/Bm

(1.3) v<!,Env`W =XAgOR 2- |Q/i,XAzv 2 7wÆO ([16], [17]), 2.3.7 K^

E 2.3.7

Everett !^nGWv` kz0pv1<3RYvO^)V,!^VLNO_6lN,Xz *,?%L+HO^WRNWCGvooUO#,Lu>Æ6w:2 vy` ,V|AW$XNE9V|vTAi (2.2) Xg Everett` ,K` Everett &nGWlLv` kUO#^)&V|AdWT3. >9". yRv|pO=GX|hyRov#~,7#~n,l"K45,h24NW', 4dnz ([23],

[24])v,uyRu'&yR.dyR: ,NhO5|5|yRo,RM5|udyRo ([26], [27])l,e Sakata, Gell-Mann Ne’eman "hW, R"oR'uOR"ORd"o, 2.3.8K^,$ PUORtR"(vnh

E 2.3.8rR,U6WO=ò"hh6H21;( Schrodingerm (1.3)g|T"lLv)emg,2qa?~~&Kt 5,nTA35~wBQA5*QoNo (1.3) 8? BsXe&|YvNBUrO31NonvM lLRuRulLe#,m (1.3) |TnlLP,eU#LnALOR=B gOR.mv`,XnvOLllLt2oKk,L<,y`V6O4Y-nv05o,02.3.8 K^rR,!O3ê?/#gL|)v,sLhh6Ho^Ng!R=B `,gn~1ugWvRo,7"2.2 |gygyL*U||h7/vÒRdy T,7e\ (2.3) eP t P T = T o,uq2e T vV#Je\ErR,!|Y= =g21v,0eWO~u tvt5,7prCW,Pe 1 K;(v ,xz4;?[>,L?[!O,!G P^)\EvpB`m (2.1)vki\`,$XNVvTAi (2.2) gOR7/VadXV|v8u2g$_,'+ANPTeY? !OdsvQEQ>&NKvr$!XV|TAv8UO#gEvg0N8,R$XNv8ugUvg2N8|h|EU8TAi,$XNv8g0nv,a|EBEO8TA,$!XV|BTA1u2 5X8TAig0nvK`,LLiORdy T v~$,|h|EOR7/vBRu,7\58odyv7/8!^Nm^wLw3zi([17]),BRdyNW,ORND1NdyO^lLAvl+, 2.3.9 K^,$ ,RN|WRLBOR<BvlL P6lN

2 o. ySE+ZKC 131

1 ? Y?I>

O1

P1

O2

P2

O3

P3

E 2.3.9!=,$XNTAiv8 (2.2) UO#g!XV|TA1u2 5X~lN8vD8 ,Everett GWv` 8, 4 Sakata, Gell-Mann Ne’eman v"hz,+&!^8m^!G ,=0NORdy T tovOg` ,7` $$JeRu Max Tegmark e`( [28] W|Yv I − IV $+^Qiag,#&` ~lNbozvm (1.2),RLBOR<Bmozvm (1.4) OYg2v ([17])RB,OY, p<yW=E|m (1.2) ozR`#^pvE|mGg2v$_,|IxQs5J? T B6Qw MR = R 8? !Ods,6vQgs.^vO,7=(Dm^|CWvunL/([13]),!XgxQD*,O^Lir$Uvlm^,sLN`Kndy Jev\EL/§3. 5ZK3.1 yXEOR$hG gOR 2- ( (V,E),Q#t V 6= ∅ E ⊂ V × V,! ,V E +g,2D,|wL G v 22,KL V (G)E(G)- T LOR- ,JeORsv 1 − 1 * φ : G → T ,XtDE p, q 6∈ V (G),u φ(p) 6= φ(q),uw G `6o- T 5LORll,b T = R3,uWv-wLI0h 2.3.10 WvlXgOR mmu C4 × C4,

132 \3|lzz – ySE+Z Smarandache J1v1 v2

v3

u1 u2

u3

v4

u4

e1

e2e3e4

e5 e6

e7e8

e9

e10e11

e12 E 2.3.10 G #vOR9 L,kLOR* L : V (G)⋃E(G) → L,! v LwL92l, 2.3.10 v92L L = vi, ui, ej, 1 ≤ i ≤ 4, 1 ≤ j ≤ 12f,V|AWV|hnvTAoUO#gOÆ, 2.3.11 K^,

t1

e1

e2

h b

l1

l2

l3

l4

t2

E 2.3.11! ,t1 = ,e1, e2 = TI,h = ,b = 2,l1, l2, l3, l4 = %, 4t2 = RJ$_,I0 R3 ~|Hh 2.3.11 ~5%twI01&8? x,|h`L 2.3.11 Wvd1 =,7 ∀e ∈ E(GL),3x e→Guu,r[,5 x,XtdRGuuxLTA/ i,!Xe R3 WtoOR 3- MTAv =Bi, 2.3.12 K^

E 2.3.12

2 o. ySE+ZKC 133LLidy~$,# i<UO#^)|heL5LOR([20]) R2g35LO^96l$Q,sL

Rn ~5%h ⇔ J?:QD*d_!vp,5%h5wN2q,'vvNK[b? P,5%hNe9J?7XB6Lo`,RNvvN5J[? UO#,eCWO=5LO^9,Ru[|hY'!^:,7=Wv6l, -m2qd? 5%dLQ~f7XJ?b? Qgv ([6], [7]),sL!3vZj&OLAvdyl-,ly`Wv4Ankdy~$,6RLi~$3.2 t#ZF G - - F : R

n × Rm → R

m LOR Ck, 1 ≤ k ≤ ∞ *,B F (x0, y0) = 0,!

x0 ∈ Rn,y0 ∈ Rm B (∂F j/∂yi(x0, y0)) gOR m×m u&n*,u,`,OJe x0 vz3 V ⊂ Rn, y0 vz3 W ⊂ R

m OR Ck *

φ : V → W Xt T (x, φ(x)) = 0,7e!^Qi,m (1.2) gvb F1,F2, · · · ,Fm L m RQ#,`tv*,B~f/ 1 ≤i ≤ m,SFi

⊂ Rn LORXt Fi : SFi

→ 0 v=Bihhe Euclidean Rn W3m(

F1(x1, x2, · · · , xn) = 0

F2(x1, x2, · · · , xn) = 0

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

Fm(x1, x2, · · · , xn) = 0

(3.1)

vpR,! n ≥ 1B=B#,m( (3.1)obp&2 m⋂

i=1

SFi= ∅ 6= ∅$_,(3.1) ;f71NC~fJ?BtL8? QEgv,sLm( (3.1) o332 m⋂

i=1

SFi= ∅ RudylLUO#,o<2,dylLe'2 m⋃

i=1

SFi6lk,!G0 (2.2) \|TTAiO

134 \3|lzz – ySE+Z Smarandache J1l,Ydy T1, T2, T3, T4 T ′1, T

′2, T

′3, T

′4 vlL| Rm(

(LESN4 )

x+ y = 1

x+ y = −1

x− y = −1

x− y = 1

(LESS4 )

x = y

x+ y = 2

x = 1

y = 16lk#r,m( (LESN4 ) 2,sLm x + y = −1 . x + y = 1 \E,m x − y = −1 . x − y = 1 \E,7` x0, y0 Xt (LESN

4 ) m4rR,m( (LESS4 ) x = 1 y = 1$_,1<AlJ? T ′

1, T′2, T

′3, T

′4 N

x = 1, y = 1 QJ? T1, T2, T3, T4 2qKNb? er21,sL (x, y) = (1, 1) 2 gdy T ′1, T

′2, T

′3, T

′4 e~ R2 #v=. ,R T1, T2, T3, T4 e~ R2 #` R\rR,Nh$UJe& R2 #, 2.3.13 K^

-6O

x

y

x+ y = 1

x+ y = −1x− y = 1

x− y = −1

A

B

C

D -6

x

y

x = yx = 1

y = 1P

x+ y = 2

O

(LESN4 ) (LESS

4 )E 2.3.13Y La,b,c = (x, y)|ax+by = c, ab 6= 0L R2#v 2,hhL 8p,dyT1, T2, T3, T4 T ′

1, T′2, T

′3, T

′4v=.lL|L'2 L1,−1,0

⋃L1,1,2

⋃L1,0,1

⋃L0,1,1 L1,1,1

⋃L1,1,−1

⋃L1,−1,−1

⋃L1,−1,1,L<,hhzkXQ 3.1 oD (3.1) G- f%A5%h GL,#|$'|~9%

V (G) = SFi, 1 ≤ i ≤ n;

E(G) =(SFi

, SFj) nC Mx 1 ≤ i, j ≤ n SFi

⋂SFj6= ∅,5: %

L : SFi→ SFi

, (SFi, SFj

)→ SFi

⋂SFj

.

2 o. ySE+ZKC 135l,m( (LESN4 ) (LESS

4 )v G-|L CL4 KL

4 , 2.3.14K^L1,−1,−1 L1,1,1

L1,−1,1L1,1,−1

L1,0,1 L1,−1,0

L0,1,1L1,1,2

A

B

C

D P

P

P

P

P P

CL4

KL4E 2.3.14XK 3.2 nC Mx i, 1 ≤ i ≤ n L Fi ∈ C1 $ Fi|(x01,x0

2,···,x0n) = 0,kN

∂Fi

∂xi

∣∣∣∣(x0

1,x02,···,x0

n)

6= 0,oD (3.1) N G- f〈et, e2t〉〈e2t, e3t〉

〈e3t, e4t〉

〈e4t, e5t〉〈e5t, e6t〉

〈e6t, et〉

〈e2t〉

〈e3t〉

〈e4t〉〈e5t〉

〈e6t〉

〈et〉

E 2.3.15Q#` 3.2 2m(v(Do8e v0,`BX`( [9]-

[14] RoZG2Wm E|qE|mv(Dl,-(LDES1

m) L.pE|m(,

x− 3x+ 2x = 0 (1)

x− 5x+ 6x = 0 (2)

x− 7x+ 12x = 0 (3)

x− 9x+ 20x = 0 (4)

x− 11x+ 30x = 0 (5)

x− 7x+ 6x = 0 (6)

136 \3|lzz – ySE+Z Smarandache J1! ,x =d2x

dt2,x =

dx

dtL#,!Rm(2,a&E|m (1)− (6),|# et, e2t, e2t, e3t, e3t, e4t, e4t, e5t, e5t, e6t e6t, et, G- 2.3.15 K^,W,9 〈∆〉 ^ ∆ Wvv.p

3.3 I*EdF5- (A ; 1, 2, · · · , k) LORW,7 ∀a, b ∈ A , 1 ≤ i ≤ k a i b ∈ A , P,- −→G L6e T Wv?9 −→GL

A ^K9 L : E(−→G)→ A[<vBQ#tv −→GL

:

R1 : C ∀−→GL1

,−→G

L2 ∈ −→GL

A,% −→GL1i−→GL2

=−→G

L1iL2,,C ∀e ∈ E (−→G)$Mx 1 ≤ i ≤ k,L L1 i L2 : e→ L1(e) i L2(e)l, −→C 4 #vuR1 2.3.16 K^,! ,a3=a1ia2, b3 =b1ib2,

c3=c1ic2, d3 =d1id2- ?6 ?6 ?6i

v1 v2

v3v4

v1 v2

v3v4

v1 v2

v3v4

a1

b1

c1

d1

a2

b2

c2

d2

a3

b3

c3

d3

- -E 2.3.16uuR1, −→GL1 i −→G

L2

=−→G

L1iL2 ∈ −→GL

AOY,~f/ 1 ≤i1, i2, · · · , is ≤ k,

−→G

L1 i1−→G

L2 i2 · · · is−→G

Ls+1 ∈ −→GL

A ,7 −→GL

A GgORW,B (A ; 1, 2, · · · , k) &= i gxv,! ,/ i, 1 ≤ i ≤ k,uN&= i Ggxvl, k = 1 B(A ; 1) gORp,u −→GL

A GgORp!,e T W,hhXLW(A ; 1, 2, · · · , k) -o −→G #fo# - −→GL

A 33g −→G #vW-,'`Ddyv~p,7yRyQ4,GXge O ORu&#vyQ,D1gv1<,Lidy$UGX|heQ#yRyQ4te −→G #- (A ; 1, 2, · · · , k),72R2 :

∑

l

F(v)−l =∑

s

F(v)+s , X,F(v)−l , l ≥ 1 F(v)+

s , s ≥ 1 |hg0

2 o. ySE+ZKC 137o(ts v ∈ E(−→G ) rF^!^%7,\mI|s.^(DeO ,[<O^_v,wLQA6XQ 3.3([19]) AQA6 (−→

G ;L,A) NyI0 S ~ALh −→G,vP' (v, u) Banach I0 B d*5: L : (v, u) → L(v, u), A<5

A+vu : L(v, u)→ LA+

vu(v, u) $ A+uv : L(u, v)→ LA+

uv(u, v),nh 2.3.17 $B,-u vL(u, v)A+

uv A+vuE 2.3.17[1,C ∀(v, u) ∈ E (−→G) L L(v, u) = −L(u, v) $ A+

vu(−L(v, u)) = −LA+vu(v, u),=C ∀v ∈ V (−→G),L

∑

u∈NG(v)

LA+vu (v, u) = cv, 3O,n,h 2.3.18 ~,

- ---

--v

u1

u2

u3

u4

u5

u6

L(u1, v)

L(u2, v)

L(u3, v)

L(v, u4)

L(v, u5)

L(v, u6)

A1

A2

A3

A4

A5

A6E 2.3.18# v 3O−LA1(v, u1)− LA2(v, u2)− LA4(v, u3) + LA4(v, u4) + LA5(v, u5) + LA6(v, u6) = cv,,cv # v 3| ,F cv = 0nU,5gO^ Pes^l-vl, 2.3.19 WvgOR.,

138 \3|lzz – ySE+Z Smarandache J1--v u

(x, y)t

(x, y)t

(x, y)t

(x, y)t

A1

A2

A3

A4

B1

B2

B3

B4E 2.3.19#v yQ4LqE|m

a1∂2x

∂t2+ b1

∂2y

∂t2− a3

∂x

∂t+ (a2 − a4)x+ (b2 − b3 − b4)y = 0

c2∂2x

∂t2+ d2

∂2y

∂t2− d4

∂y

∂t+ (c1 − c3 − c4)x+ (d1 − d3)y = 0

,! ,A1 = (a1∂2/∂t2, b1∂

2/∂t2), A2 = (a2, b2), A3 = (a3∂/∂t, b3), A4 = (a4, b4),

B1 = (c1, d1), B2 = (c2∂2/∂t2, d2∂

2/∂t2), B3 = (c3, d3), B4 = (c4, d4∂/∂t)er,'uKv+eQ# yQ4vteOR −→G #-,u −→G eOLlv(Do2 &.p A,!v-ue~fOR-#+`6l,7hhXK 3.4([20]) , (A ; +, ·) 1OIK,−→G !( T rTJ<,#x∀(v, u) ∈ E(

−→G),A+

vu = A+uv = 1AWVWR1R2G,(−→GL

A ; +, ·) &1OIK;-,' V rA3TH,W (−→

GL

A ; +, ·) rA3 dimA β(

−→G )4' V rA3EH,W (−→

GL

A ; +, ·) rA3NEH,X,β(

−→G ) = |E(

−→G)| − |V (

−→G )|+ 1 <

−→G r Betti 3O^lv5 (−→

G ;L, 1A

),7 ∀(v, u) ∈ E(−→G ) A+

vu = A+uv = 1A,OYwL−→G -6,KL −→GL, P,e −→G #-v.p (−→

GL

A ; +, ·)jKL −→GA

§4. Banach−→G - `;~jBW;~

4.1 Banach−→G - `;~OR Banach Hilbert |LU3 R #3 C !7ig ‖ · ‖ 0& 〈·, ·〉 v.p A,7 A W~fOR Cauchy xn,A WOJe

x Xt

2 o. ySE+ZKC 139

limn→∞

‖xn − x‖A = 0 n< limn→∞

〈xn − x, xn − x〉A = 0 z −→G - #vg∥∥∥−→GL

∥∥∥ =∑

(v,u)∈E(−→G )

‖L(v, u)‖ ,W,‖L(v, u)‖ ^ L(v, u) eOR Banach Hilbert A Wvg,u A `e- −→G #6l-,toXK 4.1([15]) ' A 1O Banach ,Wx%QO=< −→G,−→GA v1O Banach ,Oi,' A 1O Hilbert ,W −→GA N1O Hilberthh3 Banach Hilbert−→G - #v5=-Tg −→GA #vOR= ∀−→GL1

,−→G

L2 ∈ −→GA λ, µ ∈ F,T(λ−→G

L1

+ µ−→G

L2)

= λT(−→G

L1)

+ µT(−→G

L2),uw T :

−→G

A → −→GA Lu75!,hh`L Hilbert # Frechet Riesz &.psj,^`$o Hilbert

−→G - −→GA #,toXK 4.2([15]) , T :

−→G

A → C 1OIKLyWpV?OO −→G L ∈ −→GA/qx ∀−→GL ∈ −→GA,T T(−→G

L)

=

⟨−→G

L,−→G

L⟩f,WJe.psj,l,E|=&|=+g.psj,l,- A LOR2D

∆ = (x1, x2, · · · , xn) ∈ Rn|ai ≤ xi ≤ bi, 1 ≤ i ≤ n#vN, f(x1, x2, · · · , xn) (v Hilbert ,7e0&

〈f (x) , g (x)〉 =

∫

∆

f(x)g(x)dx, f(x), g(x) ∈ L2[∆]v, L2[∆],! x = (x1, x2, · · · , xn)` 4.1, A -L Hilbert−→G - −→GA, P,A #v ~5x~5

D =n∑

i=1

ai

∂

∂xi

∫

∆

140 \3|lzz – ySE+Z Smarandache J1 ∀(v, u) ∈ E(−→G),| D

−→G

L=−→G

DL(v,u) ∫

∆

−→G

L=

∫

∆

K(x,y)−→G

L[y]dy =

−→G

∫∆

K(x,y)L(v,u)[y]dy-o −→GA #,! ,~f/ 1 ≤ i, j ≤ n,ai,∂ai

∂xj

∈ C0(∆),B K(x,y) :

∆×∆→ C ∈ L2(∆×∆,C) Q#∫

∆×∆

K(x,y)dxdy <∞.XK 4.3([15]) >|5a D :−→G

A → −→GA |5a ∫

∆

:−→G

A → −→GA v1 −→GA*rIK5al, ∆ = [0, 1],b f(t) = t, g(t) = et, K(t, τ) = t2 +τ 2,R −→GL L 2.3.201v −→G - ,ueE|&|=5[,hhto 2.3.20 !v −→G - =6--

--=- ?6?6 -= =-

---- ?6 -6

D

∫[0,1]

t

t

t

t

et

et et

et

t

t

tt

et

et

et et

et

et

et et

1

1

1

1

a(t)

a(t) ?a(t)

a(t)

b(t)

b(t)

b(t) b(t)

E 2.3.20! ,a(t) =t2

2+

1

4,b(t) = (e− 1)t2 + e− 2.

4.2 t#FBW;~f,eUvZt Schrodinger m (1.3) D1toEn P vO^W,rR,En P $eGWW,D`kGX H.Everett %Uz` , 4 Sakata, Gell-Mann Ne’eman zEnv"hUO#,

2 o. ySE+ZKC 141` 4.1− 4.3 `X|he −→GA WY.pm (3.1) v` ,BRLMhv5UORkl,hh`3y)mm#v Cauchy ds,7m∂X

∂t= c2

n∑

i=1

∂2X

∂x2ie?t X|t=t0 e −→GRn×R W,7 −→G #v Hilbert R

n×R #v,6Rtoy)mmv` XK 4.4([15]) x ∀−→GL′

∈ −→GRn×R R _rO 0 j3 c,Vn9 X|t=t0 =−→G

L′

∈ −→GRn×R G,'x ∀(v, u) ∈ E (−→G),L′ (v, u) V Rn _1L#1Tr,W>|zl∂X

∂t= c2

n∑

i=1

∂2X

∂x2i*r Cauchy D7V −→GRn×R _Ou P,H.Everett & Schrodinger m (1.3)v` k2gv,RUR#g 2.3.21 K^vOQ# ψ1 = ψ11 +ψ12, ψ11 = ψ111 +ψ112, ψ12 = ψ121 +ψ122,

· · · v 2- QÆ (5r`( [16], [17])63Yo o 7ψ1

ψ11 ψ12

ψ111 ψ112 ψ121 ψ122

E 2.3.21m( (3.1) 2g.pv,hh21=` 4.1 − 4.3 to −→G#v` a −→G e^lvo,le|,DE (3.1)e A W,hh`e Hilbert−→G - −→GA WY#v Cauchy ds,'toOYp,/ =kEnv`W ([17])<yGp, 4oz Einstein zpPv` h

142 \3|lzz – ySE+Z Smarandache J1XK 4.5([15]) '< −→G 1":r,#T$| −→G =l⋃

i=1

−→C i /qx ∀(v, u) ∈

E(−→C i

), 1 ≤ i ≤ l,T L(v, u) = Li (x),Px%Q[3 1 ≤ i ≤ l,Cauchy D7

Fi (x, u, ux1

, · · · , uxn, ux1x2

, · · ·) = 0

u|x0= Li(x)Va ∆ ⊂ Rn *r Hilbert A _,Wn −→GL ∈ −→GA,#x ∀(v, u) ∈

E(−→C i

),L (v, u) = Li(x) r Cauchy D7

Fi (x, X,Xx1, · · · , Xxn

, Xx1x2, · · ·) = 0

X|x0=−→G

LOu` 4.5 q $Q` ,lg Einstein zpmRµν − 1

2Rgµν + λgµν = −8πGT µν` vO^m^,! Rµν = Rµαν

α = gαβRαµβν , R = gµνR

µν |L Ricci,Ricci 5,G = 6.673× 10−8cm3/gs2, κ = 8πG/c4 = 2.08× 10−48cm−1 ·g−1 · s2

E 2.3.22hh8p,Einstein k/<ge R4 #zmvrR,n[ x n > 4,10|hBF[ 1I8? UO#,/iM > 4,sL|v NMg 3,<PKN+gdy~$e|!R8#v,7|Rn =

m⋃i=1

R4i ∣∣∣∣

m⋂i=1

R4i

∣∣∣∣ = 1,! vNiORP M,0 2.3.22 W#^v3- Myue Euclidean ~ R2 #vO

2 o. ySE+ZKC 143e!^Qi,hh`= R4 v7i KLm kP ([7]-[8])l,m = 4,ue+R v ∈ V

(KL

4

) #r Einstein zpme R4 #,hh`=WwORORv!Lm,= 2.3.23 $on7i KL4 ,

SF1SF2

SF3SF4E 2.3.23! ,~f/ 1 ≤ i ≤ 4,SFi

^ Schwarzschild Pds2 = f(t)

(1− rs

r

)dt2 − 1

1− rs

r

dr2 − r2(dθ2 + sin2 θdφ2)v=B f#v m = 4 3gORY-hhH.28p m vnA?gG'Y,,` 4.5,hhG`to,6He R4 Wto Einstein zpmv` er,hhG28pW#OR:g/iP~$- ?y6 Sf1

Sf2

Sf3

Sf4

v1 v2

v3v4 E 2.3.24XK 4.6([15], [19]) Einstein 'oRµν − 1

2Rgµν = −8πGT µνI0 −→GC L;xJA −→G - 6f,n,!LJ~f −→G =

m⋃i=1

−→C i /'*ZA Schwarzschild 1I5%h −→G

144 \3|lzz – ySE+Z Smarandache J1l,- −→G =−→C 4hhL to Einsteinzpmv −→C 4-, 2.3.24K^<P,/iPUO#gOR 2.3.25 #^vt

E 2.3.25OY, −→G g m R?e −→C i, 1 ≤ i ≤ m v',u` 4.6 fTEinstein zpPL mRte Wov=B , (DoL −→Gf, −→G e|eB3e −→G g Eulerian !,` 4.1− 4.5 u`&<yW n R^p(vSyx/ubpv$Q,toXK 4.7([21]) A!L*e −→GL0 4? −→GL NMI,%nMICy,%n/vnUA Eulerian- ~f

(−→G⋃←−

G)L

=

s⊕

i=1

−→H

L

i=C Mx 1 ≤ i ≤ s,Eulerian 5h −→HL

i 3ofN,%nCy,%,,(−→G ⋃←−G)L Nh −→G %Lh −→G ⋃←−G,5: L : V (

−→G⋃←−G) → L

(V (−→G )), L : E

(−→G⋃←−G)→ L

(E(−→G⋃←−G)) % L :

(u, v) → 0, (x, y), yf ′, (v, u) → xf, (x, y), 0,C5,C ∀(u, v) ∈ E(−→G),L :

(u, v)→ xf, (x, y), yf ′,nh 2.3.26 $B,x y y- x

-u v

xf (x, y) yf ′0

(x, y)yf ′

xf

(x, y)0

u vE 2.3.26

2 o. ySE+ZKC 145;-,< (−→G⋃←−G)L r`| s⊕

i=1

−→H

L

i uR(−→G⋃←−

G)L

=s⋃

i=1

−→H i,X,x%Q[3 1 ≤6= j ≤ s,−→H i 6= −→H j,

−→H i

⋂−→H j = ∅ 6= ∅XK 4.8([21]) '4TJ< (−→

G⋃←−G)L pVO Eulerian `|

(−→G⋃←−

G)L

=

s⊕

k=1

−→H

L

k#x%Q v ∈ V (−→H

L

k ), *r2zlVs −→HL0

k rIKzl AvXv = 0hv×hv_Y Av r6Z~ λi b Reλi < 0,X,1 ≤ i ≤ hv,Wn −→GL0 r.F −→GL 1[8ÆCur,X,V (−→H

L

k ) = v1, v2, · · · , vhv,

Av =

av11 av

12 · · · av1hv

av21 av

22 · · · av2hv

· · · · · · · · · · · ·av

h1 avh2 · · · av

hhv

j3Y,Px%Q[3 1 ≤ k ≤ s,T Xk = (xv1, xv2

, · · · , xvhv)T

§5. qEQs.^#RZj&`dy~$!OdsU2 ,sLhhvb# Pes.^l-2 \EI&|YV\,pRudy~$Lidy~$GX|hn\EL/,Y'$^L\EeVdy~$#`[?vN~ ,6ReC# #zmhB`<,R!X|Es.^vO,7(D,sLCWvuUO#g- −→G #v(5r`( [13]*2_[1] R.Abraham and J.E.Marsden, Foundation of Mechanics (2nd edition), Addison-

Wesley, Reading, Mass, 1978

146 \3|lzz – ySE+Z Smarandache J1[2] A.Eistein, Relativity: the Special and General Theory, Methuen & Co Ltd.,

1916(Wp~*[3] H.Everett, Relative state formulation of quantum mechanics, Rev.Mod.Phys.,

29(1957), 454-462[4] Fred Brauer and Carlos Castillo-Chaver, Mathematical Models in Population

Biology and Epidemiology(2nd Edition), Springer, 2012(Wp~*[5] &, <yWvd3m,~ <Q. xQ$, Vol.45(2015), 1619-

1634[6] Za,Combinatorial speculation and combinatorial conjecture for mathe-

matics, International J.Math. Combin. Vol.1(2007), No.1, 1-19[7] Za,Geometrical theory on combinatorial manifolds, JP J.Geometry and

Topology, Vol.7, No.1(2007),65-114[8] Za,Combinatorial Geometry with Applications to Field Theory, The Edu-

cation Publisher Inc., USA, 2011[9] Za,Non-solvable spaces of linear equation systems, International J.Math.

Combin., Vol.2(2012), 9-23[10] Za,Global stability of non-solvable ordinary differential equations with

applications, International J.Math. Combin., Vol.1 (2013), 1-37[11] Za,Non-solvable equation systems with graphs embedded in Rn, Pro-

ceedings of the First International Conference on Smarandache Multispace and

Multistructure, The Education Publisher Inc. July, 2013[12] Za,Geometry on GL-systems of homogenous polynomials, International

J.Contemp. Math. Sciences, Vol.9 (2014), No.6, 287-308[13] Za,Mathematics on non-mathematics - A combinatorial contribution, In-

ternational J.Math. Combin., Vol.3(2014), 1-34[14] Za,Cauchy problem on non-solvable system of first order partial differen-

tial equations with applications, Methods and Applications of Analysis, Vol.22,

2(2015), 171-200[15] Za,Extended Banach

−→G -flow spaces on differential equations with applica-

tions, Electronic J.Mathematical Analysis and Applications, Vol.3, No.2 (2015),

59-91[16] Za,A new understanding of particles by

−→G -flow interpretation of differen-

tial equation, Progress in Physics, Vol.11(2015), 193-201

2 o. ySE+ZKC 147

[17] Za,A review on natural reality with physical equation, Progress in Physics,

Vol.11(2015), 276-282[18] Za,Mathematics after CC conjecture – combinatorial notions and achieve-

ments, International J.Math. Combin., Vol.2(2015), 1-31[19] Za,Mathematics with natural reality–action flows, Bull.Cal.Math.Soc.,

Vol.107, 6(2015), 443-474[20] Za,Labeled graph – A mathematical element, International J.Math. Com-

bin., Vol.3(2016), 27-56[21] Za,Biological n-system with global stability, Bull.Cal.Math.Soc., Vol.108,

6(2016), 403-430[22] J.D.Murray, Mathematical Biology I: An Introduction (3rd Edition), Springer-

Verlag Berlin Heidelberg, 2002[23] Y.Nambu, Quarks: Frontiers in Elementary Particle Physics, World Scientific

Publishing Co.Pte.Ltd, 1985(Wp~*[24] Quang Ho-Kim and Pham Xuan Yem, Elementary Particles and Their Inter-

actions, Springer-Verlag Berlin Heidelberg, 1998[25] F.Smarandache, Paradoxist Geometry, State Archives from Valcea, Rm. Valcea,

Romania, 1969, and in Paradoxist Mathematics, Collected Papers (Vol. II),

Kishinev University Press, Kishinev, 5-28, 1997[26] F.Smarandache, A new form of matter – unmatter, composed of particles and

anti-particles, Progess in Physics, Vol.1(2005), 9-11[27] F.Smarandache and D.Rabounski, Unmatter entities inside nuclei, predicted by

the Brightsen nucleon cluster model, Progress in Physics, Vol.1(2006), 14-18[28] M.Tegmark, Parallel universes, in Science and Ultimate Reality: From Quantum

to Cosmos, ed. by J.D.Barrow, P.C.W.Davies and C.L.Harper, Cambridge

University Press, 2003(Wp~*

K 2 MmTK– U>&Y~V,FwP, 148-169LroLXnGjvKw6SmarandcheB;~Kq 1,2F. IAGW-C 0>,~V?~Q"bpQE?~Q~V,Theory of everything? ?~Q"Md,QAG*=,N~V?~~h~V~ M- ~V<?~Q"[19w,E~V4QxQ"tLEYCwC0xQ~V,A&L,NW-G?~QEh ,ktvhYZ#yQxQ"RtLEW-GxQ~V6,?~Q"<K40,h~VY M- ~VE[1,xQ"E [W-GxQ~V,NSmarandacheI0~V,0ACPN~V?~Q"$?,nX,QxQ~V,,rX~V?~??lJ*(QY l~Vsh.F~o`~h, I~3-D*xQ~<~wp*QA*FUQQ<!|x - [16] ~Q~U!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

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

12006 7 3 U8 Ud*%?:HVÆ?NÆVnC)wPXgUPoP5b'x/+(ySÆ)nD2;U)t+200607-91

2 o. W([X.H yS –Smarandche J1W 149

monograph [16] of mine. We can find the central role of combinatorial specu-

lation in this process.032/iTr`<,M- `<, ,=B,Smarandache=B,Q4 =B,Finsler=BIE AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50

§1. lGl".,F5b/1.1. nKw;HW =d&|T,K2d = x, & = y, = z,uORHW `RB6l|T,76 (x, y, z)4%W =d&P |T,KP L t,uOR%W `=6 (x, y, z, t) |THW 4nr 2.4.1|P 5ORng,u|YeORPov/iiWUO#g/R/ivOR (section)

E 2.4.1 S5Y10.I|Yv<[<Uq,|Y<v/i .#%W gOIv,7|Y<v g 3 Mv,W#P ,ug 4 Mv, !Xg Einstein vP1.2. lG2pF9mKqQb|Y-8vU,lg Einstein vzp`m,y`Vzm/ivTr`<,!^`<L,/i I&OR:,&$WvrZW ,!R e$12#tvn,X!RrZv1vW Y<rD#~nks1,1W,#~nD,=^jyR,eC : 137 d8v0ni.yv:M/iQb Hawking v ,r[< Y,&W&\%v ([6]− [7]), 2.4.2 K^

150 l)e: \3|lzz – ySE+Z Smarandache J1

E 2.4.2~`* 2.4.3 W,5| /iTrzZv0n.g,=H[<.y|YNtov/i

E 2.4.3 [ QbTr`<vI=,/i <`[v0n TI22P 9/iTrzZ&OR$W ,ORQ 137 d83rv BZW,Nv ZP L 0No,v4o,Tvu4 3u21\8vy`v4 ePvQ+P B<rzZ, GB<5jg3T,D9w< 9P 10−43 ~,u4g 1093kg/m3,4o 1032K!Pv/

2 o. W([X.H yS –Smarandche J1W 151irZRBw,DP $`9?Jw< 9P 10−35 ~,4o 1028K,/iY<p,=Ke 10−32 ~vP 0yT 1050 ypz |vn[<,!P>rzp\BOvp|^=,a&1 ,;pp8p+uS|z,[<vnG`\|!OP`2yQv T6l,n`8G&d`,[4|,z v`d`/5K,XDWRR=2od`v2wv'Pv/i=Rw< 9P 10−6 ~,4L 1014K4>Rw< 9P 10−2 ~,4L 1012K4! ,;5585mx\|z/iW'S^n,&4L(1010K `#*,nv<JP +g.7v,Nh /ejR/en,B!Pu`'iw< 9P 1− 10 ~,4HQ 1010− 5× 109K,RdR0/5PK,[<TvWE`4dWE,#~nzZDB>,1`*vi\' (|hbIEb#/iovEpe! D)?$!w< 9P 3 |Y,4HQ 109K,=KgoQ 1 8TF,:yRDL$,>dCG4_FP[,RvnvQPo 10−9,*u4rT&yRu4TR^aXw< 9_FP[,4|o 108K,S^ne/ejWs12#\21/en('vK2*,B!P , /iWS^nv4X[&b&!P4rL,#rv1 A~B7givB,L[vid,K`eP`1'BYn:w< 9P 1000− 2000 8,4HQ 105K,yRu4zZT&*u4&/ivg,oP~BO (lOR;;ivB*P+|s&lz vGSKRX*dyTR1lF,&&l64?/ivgRmxyT,!Lv*dGX2;yTR12;lFn:4o$i=D,C 9P 105 8,4HQ 5000K,yR4zZ|&*4,,`,Kv#~nv?kQTQC OQ;8,r[<vv1Xo2#` AB6H*YBv4,/i!PX6B/e|^v&[PW,7/iv,4TQL3000KB!PzZ,BzZi,aGD1[<ÆKv5H&Æ`v5,$ge Q i[,eQ 05iv,Q i[,SQ 05[<S2 v4k vZ`5ugek_ rY vjrYWiv

152 l)e: \3|lzz – ySE+Z Smarandache J1$! 9P 108 8,*4HQ 100K,yR4L 1KI2N%fpP,99P 109 8,4HQ 12K,Umxihh[=NovQF: 9P 1010 8,*4HQ 3K, yR4Q 105K1.3. lG9mKqW,F5b/`<y`.U6y`$QyR^yRn,7 e#" u" d oWLbR'^| n: /e5v`<,!Xg Einsteinv/< Dirac vp/<g| zpv`<,OY&$Q/iy`4Rpg&En5pv`<,h28p;>8p>8p!+^5pon: /e5v#~5pB#RbRWL8WzZ,Gy`V,h2 Einstein ~|O=Ip&O!+^#~5p,7O/<p,zmy`vTO`<,7`(WC 'vTheory of EverythingC 80 G8v$Q,dsO=`toGQpdsv+ e&k/<g&U/iv`<,wGU;J(z,Y-yRgs|3v4Rpg&E/iv`<,RWz,Y-yRg^|3vRB^?3$Q,U=|YV3vOL_ds,[ Nb7nK,LJA[ 72qax3Kw&[ 7axi~[ xQJ7BnaxYze 3 b7Einstein Qbqvk/pB`2$L?~O _6Y~!Lw[1zKB`2AXC:7$' (?~0NKC~zmzp`m,7

Rµν −1

2Rgµν + λgµν = −8πGTµν .D/iB`, 799 104l.y 1,[ ~ ($A (o*;q9,Robertson-Walkertozp`mvO^Ww

ds2 = −c2dt2 + a2(t)[dr2

1−Kr2+ r2(dθ2 + sin2 θdϕ2)]./v/iwL Friedmann /iC G8vy`U,Hubble e 1929 Y',|YZzv/igORmW6gv/i,bYzp`mvW6gy`VvrEm?,7|Q#t

da

dt> 0,

d2a

dt2> 0.

2 o. W([X.H yS –Smarandche J1W 153hh8p,a(t) = tµ, b(t) = tν,! ,

µ =3±

√3m(m+ 2)

3(m+ 3), ν =

3∓√

3m(m+ 2)

3(m+ 3),u Kasner 4

ds2 = −dt2 + a(t)2d2R3 + b(t)2ds2(Tm)L Einstein `mv 4 +m M$OYQ+,!R'21U 4 MW6ga=P Uwx

t→ t+∞ − t, a(t) = (t+∞ − t)µ,hhtoOR 4 MvW6g,sLda(t)

dt> 0,

d2a(t)

dt2> 0.WLbR'v M- `<Lp# ds# !O`<Y-n2gR RgM2 v p-,7* p Rm?d4v ,! p gOR0/1-OYw5h,2-w5 2.4.4 WU 1- 2- 4%

? 6 ? 6 ? 6-(a)

? 6 ?6 6

?-(b)E 2.4.4 *Qb M `<,/i0<zZPv$RWi Mg 11 Mv,TrzZ[,W 4 Rm?Me5jv3&/,R2 7 Rm?Mue5jmHF,!ihh.ytov 4 MU/i2rv 7 ME/i4 MU/i0

154 l)e: \3|lzz – ySE+Z Smarandache J1v5pD Einstein zp`m,R 7 ME/i0v5pD7m,<to!R<XK 1.1 M- ~V1INAvA$DA 7 I0 R7 4 I0R4` 1.1 #vkzpv2nt,hh`toTownsend-

Wohlfarthhv 4 MW6g/ihds2 = e−mφ(t)(−S6dt2 + S2dx2

3) + r2Ce

2φ(t)ds2Hm,!

φ(t) =1

m− 1(lnK(t)− 3λ0t), S2 = K

mm−1 e−

m+2

m−1λ0tB

K(t) =λ0ζrc

(m− 1) sin[λ0ζ |t+ t1|],! ζ =

√3 + 6/m. bP ς Q# dς = S3(t)dt,uW6g/ivt dS

dς> 0 d2S

dς2> 0 rtoQ#?I=, m = 7 ugsL 3.04B4,` 1.1 Wv UO#2g Rg ,<z3vdsg NU|xQI0,~vA1A 1 I07=o,!v Je,$NO2ghh <Wtov ,G2ghheCW5r v ,le 3 M.p W,dR `^L (x, y, z),N21h)ORMT&z& 1 v §2. Smarandache B;~x3ORjvds2

1 + 1 =?erW,hh8p 1+1=2e 2 6M=uW,hhu8p 1+1=10,! v 10 UO#ug 2sLe 2 6M=uWDR=5 0 1,=uL0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

2 o. W([X.H yS –Smarandche J1W 155Qb :z&v%7,hh=O d%MvV6v%7 ([18]− [20]) =`_[!Rds, _| 1 + 1 = 2 6= 2hh8p 1, 2, 3, 4, 5, · · · !vor Ne!RW,Qbv4,dRwL32vv[P,7 2 v[PL 3,KL 2′ = 3 ,3′ = 4, 4′ = 5, · · ·!hhXto1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, · · · ; Puto1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, · · ·!OL=z\BRe!^rv=u,hhD1to 1 + 1 = 2 v<'e,hhO=vkUOR2D S, ∀x, y ∈ S,k x∗y = z,f%g S #JeOR 2 D* ∗ : S × S → S,Xt

∗(x, y) = z.=vm\,hh`L!ê~#^=xL S WvdR~#v ^, S W n R,ue~#Xb n R2i.v 4R x, z : sO?.9,JeOR y Xt x ∗ y = z, hhe!.9## ∗y,wL/.9vf`, 2.4.5 KÊ 2.4.5 u~Lo`f!^g 1− 1 vK S vL G[S]'e,hh7oORQ# 1 + 1 = 3 v=,hhÙ 1 + 1, 2 + 1, · · · z?' =7XQ 2.1 A xY (A; ) hrUA: : A→ A =C

∀a, b ∈ A,i a b ∈ A,UAe c ∈ A, c (b) ∈ A,w0 h: hhL to`&W (A; ) . G[A] vvOR

156 l)e: \3|lzz – ySE+Z Smarandache J1XK 2.1 (A; ) A xY,(i) r (A; ) UAh: , G[A] NA Euler hh[,r G[A]NA Euler h, (A; ) NAhY(ii) r (A; ) NAO xY, G[A] ~vA#-9 |A|4C|,n (A; ) IO, G[A] NAO 2- h/vA# *AP=K[#[0'wC 2- ',h[X&,RvQi,`=O^,vm\K=,

2.4.6 U |S| = 3 v^=u^7/ - ^ 6 - 1

23

1

23

+2 +2

+2

+1 +1

+1

+3

+3 +3

+2

+2+2

+3 +3

+3

+1 +1

+1

(a) (b)E 2.4.6 3 Ae#`h 2.4.6(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. 2.4.6(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.OR2D S, |S| = n,`e#k n3 ^2 v=u!hhX`eOR2D# Pk h ^=,h ≤ n3 RtoOR h- `=u(S; 1, 2, · · · , h)eCWW,pgOv=u,t3uzrg 2 `=uOY,hhkOR Smarandache n- ` XQ 2.2 A n- I0 ∑,% n A|* A1, A2, · · · , An

∑=

n⋃

i=1

Ai

2 o. W([X.H yS –Smarandche J1W 157/vA|* Ai *% i = (Ai, i) A xIY, n PMx,1 ≤ i ≤ ne` v)\,hh`6O4$CWWpt34? v2:Rto`p t 34`? v2:,'to/vWoXQ 2.3 R =m⋃

i=1

Ri A m- I0,/C Mx i, j, i 6= j, 1 ≤

i, j ≤ m, (Ri; +i,×i) AP/C e ∀x, y, z ∈ R,iw0d*U,L(x+i y) +j z = x+i (y +j z), (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, R A m- PrC Mx i, 1 ≤ i ≤ m, (R; +i,×i) NA, RA m- XQ 2.4 V =k⋃

i=1

Vi A m- I0,|* O(V ) =

(+i, ·i) | 1 ≤ i ≤ m,F =k⋃

i=1

Fi A,|* O(F ) = (+i,×i) | 1 ≤

i ≤ krC Mx i, j, 1 ≤ i, j ≤ k ~ e ∀a,b, c ∈ V , k1, k2 ∈ F , iC0dU,(i) (Vi; +i, ·i) Fi I0,#` +i,5 ` ·i4(ii) (a+ib)+jc = a+i(b+jc);

(iii) (k1 +i k2) ·j a = k1 +i (k2 ·j a); V F k I0, (V ; F )<hh8p,M- `<Wv hUO#gO^` hXK 2.2 P = (x1, x2, · · · , xn) n- TTI0 Rn ~AC Mx s, 1 ≤ s ≤ n, P A s 5I0U fJn Rn WJe# e1 = (1, 0, 0, · · · , 0), e2 = (0, 1, 0, · · · , 0),

· · ·, ei = (0, · · · , 0, 1, 0, · · · , 0) ( i RL 1,(L 0), · · ·, en = (0, 0, · · · , 0, 1) Xt Rn Wv~f (x1, x2, · · · , xn) `^L(x1, x2, · · · , xn) = x1e1 + x2e2 + · · ·+ xnen

158 l)e: \3|lzz – ySE+Z Smarandache J1b3 F = ai, bi, ci, · · · , di; i ≥ 1,hhkOR_v? R− = (V,+new, new),! V = x1, x2, · · · , xn2Gp,hhY x1, x2, · · · , xs g.mv,7Je a1, a2, · · · , as Xt

a1 new x1 +new a2 new x2 +new · · ·+new as new xs = 0,u a1 = a2 = · · · = 0new BJe bi, ci, · · · , di,1 ≤ i ≤ s,Xtxs+1 = b1 new x1 +new b2 new x2 +new · · ·+new bs new xs;

xs+2 = c1 new x1 +new c2 new x2 +new · · ·+new cs new xs;

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ;

xn = d1 new x1 +new d2 new x2 +new · · ·+new ds new xs.BRhhto P #vOR s- M Gq 2.1 P TTI0 Rn ~AUA5I0G#R−

0 ⊂ R−1 ⊂ · · · ⊂ R−

n−1 ⊂ R−n= R−

n = P /5I0 R−i x n− i, 1 ≤ i ≤ n

§3. JEjJEjI3.1. Smarandache jISmarandache(gO^,jvuJ=B,0*3 38v Lobachevshy-

Bolyai =BRiemann =B. Finsler =B,Y g=dsmbWJn=BWve-hhxOJn=B4=BRiemann =Bv0m Jn=Bve`u!te-(2(1)IvAwvA&$%'bu4(2)vPbu.;sj4(3) ~$,Y (E%1Qn4(4)$LbU.w4

(5)[w:nLPbuY1PbuwQ,/Pbuf$Q:U[$QbU,:Pbu$fwQ, nh 2.4.7 $B

E 2.4.7 PbuYPKw3buwQ! ,∠a + ∠b < 180O,[Oe-wLJnte-,Nu`=!^ m^2E%bu|,UPbuYE%buKwQBJne-e3`=,|hO=otte-2/5Le-',N#dUe/gORsL<,GVIp&=3+e-5te-,aO=``&g|7MY-WvJnte-,h6tove`ug7z,gJe\ELTbR,Lobachevshy BolyaiRiemann |=2 vY-bWJnte-t`Mh=vY-|g2Lobachevshy-Bolyai Y-2E%bu|,nUPbuYE%buKwQRiemann Y-2E%bu|,KUbuYE%buKwQRiemann Y-to`rvBse&<`zm\R=B,[ Einstein L/<WvzpP,7Lzp`5OR\R ,hhg`6O4d1Jne-to_v=BR*3BvJn=BLobachevshy-Bolyai =BRiemann =B Finsler =B7`( [16] Wp!Rdsdsvptp& Smarandache =B%7RzmQ4 =B,! Smarandache =B5ORjE'(,O '(Q4 =BSmarandache =Bh)_V(x(h 7(h(z+^,|Qb2 ve-zmW,_V(=ve-LJne-(1*-(4*`4~BOe-2(P − 1) nUPbu$'bu|,='bu*YP

160 l)e: \3|lzz – ySE+Z Smarandache J1buKwQ4(P − 2*nUPbu$'bu|,='UPbuYPbuKwQ4(P − 3) nUPbu$'bu|,='ULs kPbuYPbuKwQ,k ≥ 24(P − 4) nUPbu$'bu|,='U;xPbuYPbuKwQ4(P − 5) nUPbu$'bu|,=' (bu*YPbuwQx(=veùgJn=B 5 e-Wv 1 RR,7=Òe-bWJne-Wve-2(−1) E% K%UPbu4(−2) UPbuK<;sj4(−3) E%$A6x,K%F-An4(−4) bUK%w4(−5) E%bu|,K%UPbuYE%buKwQh 7(=veùg*=BWvOe-,/= e-bW2(C − 1) nUPbuntLbuAE%4(C − 2) A,B,C vAK[u,D,E AK[r A,D,C $

B,E,C v[u,Y A,B buYY D,E buKwQ4(C − 3) vPbunJLAK[h(=veùg Hilbert eùWvOe-XQ 3.1 AVSmarandache%,r[AI0~[17m-nK,nno7mKAL Smarandache %V(Smarandache(!Rl`4F Smarandache =BgJevR 3.1 - A,B,C LJn~#R2i.v ,k=.LJn~#

A,B,C W)6OR v=.uhhtoOR Smarandache =BsL.Jn=Beù/Æ,We-g Smarandache v

(i) Jnte-'eLPbu|,UPnKUbuw3Q'PbuKbWY-=. L C C B~l&=. ABfC ~BOR2e AB #v )6O=.~l& L,RC =. AB #v~BO r2Je~l& L v=., 2.4.8(a) K^(ii) e- AK[UPbu'eL AK[UPbunKUbubWfC R D,E,! D,E . A,B,C WvO , C i., 2.4.8(b) K^,)6O=.C D,ER~fRe=. AB F,G 2. A,B,C WOR i.vR G,H r2JeC Nhv=., 3.2(b) K^

E 2.4.8 s- bu7\3.2 x~JE7-WORLtv` vADn<=,n<[bQ=zI

2p A,,vA,[0ZAA(|5*w4n<[bQ=zI qA,,vA,ZA 6a'iYw *<%D, ?%p4:<K%D, ?% q! ?vf%gOR2=&v?*%Oe[oY g1?vm?=#8pWg?v,RLk"Uug2?v, 2.4.9 K^,W (a) LkzvF,(b)LD[vLk"U

E 2.4.9 6a1

162 l)e: \3|lzz – ySE+Z Smarandache J1gvO^l|,e*!^l||kz[,tovdR$r &GS D = (x, y)|x2 + y2 ≤ 1Tutte & 1973 8UvWk,=[12] Wv1,k&XQ 3.2A h M = (Xα,β,P),%t2|* X Æe% Kx, x ∈ X;V[e| Xα,β AtrT P,/eAdV~ 1 $V~ 2,K = 1.α, β, αβ Klein 4- eV,$% P trT,KUPMx k, =Pkx = αx"K 12αP = P−1α;"K 22V ΨJ = 〈α, β,P〉 Xα,β Qbk 3.2,v |kLKx P Pαβ 5& Xα,β #toviUp4L Klein 4- p K 5& Xα,β #tovpjEuler-Poincaree\,hhto

|V (M)| − |E(M)|+ |F (M)| = χ(M),! V (M), E(M), F (M) |^ M v 222,χ(M) ^ M v Euler ,N,?z& M K6v$Rv Euler ,NwOR M = (Xα,β,P) gK%,Kxp ΨI = 〈αβ,P〉 e Xα,β #g.v,uwL%E 2.4.10 h D0.4.0 Kelin D#p5LORl, 2.4.10 WU D0.4.0 e Kelin #vOR6,`= M = (Xα,β,P) ^,! Xα,β =

⋃

e∈x,y,z,w

e, αe, βe, αβe,

P = (x, y, z, w)(αβx, αβy, βz, βw)

× (αx, αw, αz, αy)(βx, αβw, αβz, βy).

2 o. W([X.H yS –Smarandche J1W 163 3.4 Wv 2 v1 = (x, y, z, w), (αx, αw, αz, αy), v2 = (αβx, αβy, βz,βw), (βx, αβw, αβz, βy), 4 e1 = x, αx, βx, αβx, e2 = y, αy, βy, αβy, e3 =

z, αz, βz, αβz, e4 = w, αw, βw, αβw`4 2R f2 = (x, αβy, z, βy, αx, αβw),

(βx, αw, αβx, y, βz, αy), f2 = (βw, αz), (w, αβz), Euler ,NLχ(M) = 2− 4 + 2 = 0BKxp ΨI = 〈αβ,P〉 e Xα,β #.!XBW4to D0.4.0 e Klein#v6^?`<y`$Qv|E,hhu`OYp3e `4G`#v6e`#v6kXQ 3.3 h G #|*!LG~ V (G) =

k⋃j=1

Vi, C Mx 1 ≤

i, j ≤ k,Vi

⋂Vj = ∅,O S1, S2, · · · , Sk 9I0 E ~ k AD,k ≥ 1rUA 1-1 I: π : G → E =C Mx i, 1 ≤ i ≤ k,π|〈Vi〉 NAp/

Si \ π(〈Vi〉) ~vAYq[bQnZ D = (x, y)|x2 + y2 ≤ 1, π(G) N GD S1, S2, · · · , Sk #pk 3.3 W S1, S2, · · · , Sk v XK`69eJeO^PSi1 , Si2, · · · , Sik,Xt~f/ j, 1 ≤ j ≤ k,Sij g Sij+1

v P,wL G eS1, S2, · · · , Sk #v:#p&W,v<XK 3.1 Ah G = P1 ⊃ P2 ⊃ · · · ⊃ Ps Uxwf:#pn.vnh G UUG~ G =

s⊎i=1

Gi,=C Mx i, 1 < i < s,

(i) Gi Nw4(ii) C ∀v ∈ V (Gi), NG(x) ⊆ (

i+1⋃j=i−1

V (Gj)).

3.3 JEjI h(ge# #ozv Smarandache =B, PGgr(D.CvAU=Bv2:xe`( [13]Wq,?[e`( [14]− [16]W,lg [16] 6lIv$Q,kXQ 3.4 hM vA# u, u ∈ V (M)VA6x µ(u), µ(u)ρM(u)(mod2π), (M,µ) A h(,µ(u) u U"5x[EnKEDDu6AnAQ' h(;'inL'i

164 l)e: \3|lzz – ySE+Z Smarandache J1 2.4.11 WU=.' #v vQi! v34|LT&8z&84F&8,/, u wL*G Jn

E 2.4.11 bu6wnnD*G Jn e 3 M WrgÙ'v,! vU'&Jn vQi,72Og~=v,u/ XgJn 2.4.12 WU!^ e 3 M vU'm^, W u L*G ,v LJn R w L E 2.4.12 wnTT$D 3- I06mXK 3.1 Li;i h(~*U_V(x(h 7($h(`v5r`( [16]L&`,hh'(~=BvQie!^Qi,23è #!*s,,uÈYs : vgORs,,!6O4`~#W.L|fke~=BW!v<2w h(;<buK6 hn6TT5LORl, 2.4.13 Wk#&0+uvO^~=B,W #v?/ 2 yvs,?

E 2.4.13 Aw h(5 2.4.14 Wk 2.4.13 kv~=BW=.vQi,Y,,2.4.15 Wk/~=BWv=^Gi

E 2.4.14 w h(bu

E 2.4.15 w h(J'1§4. \^[;~jIEinstein vk/<;( ezp5g3v,6H.G2l2,!O eUONW\Cto5U=Bv%7UO#`OYk&OR4 #,7e/4 vdR #!*OR?RzmQ4 =BXQ 4.1 , U Ow ρ rw,W ⊆ Ux%Q ∀u ∈ U,)

166 l)e: \3|lzz – ySE+Z Smarandache J1pVOLS+ ω : u → ω(u),X,x%Q[3 n, n ≥ 1,ω(u) ∈ Rn /qx%Qr\3 ǫ > 0,pVO3 δ > 0 Os v ∈ W , ρ(u − v) < δ /qρ(ω(u)−ω(v)) < ǫW) U = W,k U OBw, (U, ω)4)pV\3 N > 0 /q ∀w ∈W , ρ(w) ≤ N,Wk U OTBw, (U−, ω)f ω gs,P,BQ4 hhto Einstein v3 L&`,hhi<Q~=BB ω Ls,vQi,xRjv<XK 4.1 w (P, ω) u $ v K%UTTbuXK 4.2 Aw (

∑, ω) ,rKUTT, (

∑, ω) vA*wnnvA*D&~W.,uXK 4.3 w (

∑, ω) U xDu F (x, y) = 0 C D ~

(x0, y0) n/vn F (x0, y0) = 0 /C ∀(x, y) ∈ D,(π − ω(x, y)

2)(1 + (

dy

dx)2) = sign(x, y).'e,hh oQ4 #Qbk 4.1,OR m- i Mm ~f ∀u ∈ Mm,b U = W = Mm,n = 1 B ω(u) LORj,uhhtoi

Mm #vQi=B (Mm, ω)hh8p,i Mm #vMinkowskijxkLQ#tvOR, F :

Mm → [0,+∞)(i) F e Mm \ 0 #$$j4(ii) F g 1- v,7~fv u ∈Mm λ > 0, F (λu) = λF (u)4(iii) ~fv ∀y ∈Mm \ 0,Q#t

gy(u, v) =1

2

∂2F 2(y + su+ tv)

∂s∂t|t=s=0

.vw.ph gy : Mm ×Mm → R g0vFinsler61UO#Xg!* Minkowski gvi,eu=Xgi Mm4A #vOR, F : TMm → [0,+∞) 'Q#t(i) F e TMm \ 0 =

⋃TxMm \ 0 : x ∈Mm #$$j4

(ii) ~f ∀x ∈Mm,F |TxMm → [0,+∞) gOR Minkowski g5LQ4 =BvORll,~f x ∈ Mm,hhs ω(x) = F (x)uQ4 =B (Mm, ω) gOR Finsler i,l,b ω(x) = gx(y, y) =

F 2(x, y),u (Mm, ω) Xg Riemann i!,hhXto <XK 4.4 9I0( (Mm, ω),| ,Smarandache (~ Finsler(,IQ Riemann (§5. )FJ?'Fb/WLObRv`<y`L$QqT|E$Qvds,! hh3_=Rb/ 5.1 LJA[ 72qaxZmK &[ I07NY'~DI0Lw7Lr`oR W,erX[GR/i,!Xg`( [10] W~l/iv ,Ggy`#v Einstein ;( ezp5g3v,B^fk#Jn e$Ub#Wg2JevBU6Nv4,|Y31NNtor#W^/R2g~1,o<gM ug|M *o 4 M , ( [16] W<\L4kCE|=BWjA?Dk3vm^Q>&OLlvrBuOYpv$Q3 rQ4 (Mm, ω) 6l$Q#&u~= v$Q`Y',H'e#[~l/ivJe,a|Y 3vNm^o^Nob/ 5.2 axi[ xwNJ7NLs7WLbR`<y`vY0ev|Y1-8=iv :,BR9vLOLtv`<y`VZ=(2v 1nKYyaxI07H9wNJee.`<.U6vt,EBN!RdsOv1+,sL|Y2oN2ov UG"`<WL Mg 10,M-`<Wv Mg 11 Bt^\8v"k"`<rg.,QiR'y`V0e$Qv F- `<v Mg 12 ?!^%7,`zmOYv M`<$Q Einstein `m,e!O #,V ey`Vv[b/ 5.3 ax<d[|nyp/,b7 =NU?I 4

168 l)e: \3|lzz – ySE+Z Smarandache J1yp 3 nI 3 yp 2 I07J(UO#g Einstein `me2 4tv Om,y`VLJ(Je Tvzp,Xs.G21l2,~Byu2nAJ(Wr|#LAr([6]-[7]*4 Py`V#6NJ(gs2 /i2 Pv>~,sL|YvOAy`4eJ(0rGK.J(/v, <y`WuO^O(,l-g~Byuro^6O(0YzJ(O(Ru,hhXY'rgyQv,zv PXgP,K`J(.O(/gOd!|Y,lg/3Fo|\`2F`6J(GWyReW#gJev, zakGWyRv`<,2<gy`grSz |hv#rVG8`=,phO=`yu%WW2!OR#~BuB`<#VP'G,BN%WWnU=vds,72buv%ds<U=vdsg ([16])2(1) d_Q,T%+hN,%,+NK,%y3~x(2)Ih#pwJI0:,hwI0.IO(3)EhwI0*Qy`= ,|EeW#1r1Wv<y,6RY'P'G4b/ 5.4 ax =3wi?yb7DyR.D1O=gy`#vORygps ?|Y MVv1,!RdsGtm#` M ≥ 11,$_DyRX2O$e|Ytovm?M#,G21eW#oNK!L+Q>&|YV~.U6Avq2_

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[18] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Proba-


[19] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic,

Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297-

384.

[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.

K 2MmTK– *8NCETMMS-2015wPPoP5\b/ #?~MK, 170-173

Y_ÆG86g666nC6 a:\0ÆfGTG`h<h,TVk#6&5Lr ,R[h?o(;,WUSQ^8Y& P,SXWs vi e R~o^8Y&hh8p, v`e&Vr,6RX|Y,vYDr4,v,v5e&Lir$Uq |vepm^,<, e|&|Yv,UqUqY^2z`<Am, ,`<v$6G^2zUqh6.y, YvOR#tl-g2 QYi 64,6RmI4M YvOL_[e,l,/e5eCF<W, 4(DeC4 YWvzhAdOX.!%|,^, P?M.Æ8V.m^h/bdOX.2s<RgLv,[,=|~oGQ`YYTV& j2e4 i4 Y[eTz\#vpB`Talk in Inaugural Session of NCETMMS-2015

Ladies and gentlemen, good morning!

It is my pleasant duty to extend my sincere thank to the organizer - Calcutta

Mathematical Society on behalf of a chief guest, thanks also go to all attendants to

be here for a common interest: mathematics and mathematical sciences.

As we known, the function of science is realizing the natural world and develop-

ing our society in coordination with natural laws, and the function of mathematics

is provide quantitative analysis tools or ways for holding on the reality of things by

observation. Thus, sciences should be served to the social practice of human beings.

2 o. ,:NCETMMS-2015ySSqR7_d1"%AOM 171

The practice can not do well without theory, and also, the theory can not

develop without practice test. Today, a main character for developing of science

is its cross promotion and common development on different branches of science,

also called to be interdisciplinary, which results in many new developing trends

in mathematics and other sciences, for examples, the applications of interaction

systems to economy or ecology, and combinatorics to classical mathematics and

mathematical sciences.

I wish each attendant will use this opportunity to fully exchange well, display

your results and learn more from others. I believe you will not come in vain on this

conference.

Finally, I wish the conference a complete success! Thanks to everyone!

Y_ÆG86g666nC6 a:\rÆfGTG`h<h,TVu6&x,R[h|IWUSQ`(;~o,|I<`WvC& P,<`CGE0n[,|EKzvg,,Wvg dR2gS=N2 g5|<-#vOR ,'2fT $5v\ Rg_v 5eB<WyEpvÆ 'L:~),D!,5:g5|<,\vCje<,hGA8K =hhL,dXV+egVR23gI=,sL%7gl%vA)R~1X&ger,V|\1Yds,!eW,LR_5s!RdsR2g$Rds6l$Q'24bp&5v1p, Pbp&5v[?,gsL5L!RdsÈ,[?!L8,OLJ8?hIY$QdsUO#,'.,4 WJeGG`pvdsÈv,2gp$QR_,$gjv,Rp2$QR_u.<2 ,ZWÈB#,tRBu` v5p$QR_, 42$QR_2(1) ds~1p|7WORds[?;

172 \3|lzz – ySE+Z Smarandache J1(2) dsvp$%O|7[?4(3) dsvp$%/RY[?;

(4) dsvp$%OP Y[?;

(5) dsvpVr|YY[?eh=,VR(,`Ev,q233gdpORAds,e$ vk!,Rg(&/R |YL<,e?6l$Qv P,?$Q3RGOM Vv$Q4,6R$%2 |7 i YhL~ i4 Y[eT. Y!^[e 4,K`5R(g`EB[?vYYTV& j2e4 i4 Y[eT\#vpB`Talk in Valedictory Session of NCETMMS-2015

Ladies and gentlemen, good afternoon!

Please allow me congratulating to Calcutta Mathematical Society for organized

this successful conference, and congratulating to all winners in the contest.

On the first, please allow me to say some words to the winners. It should be

remembered that achievements only represent the past, not the future. It is only a

shine point on your life road, not implies the terminal but a new starting point. You

should be worked ever hard on mathematics or mathematical sciences after then,

and then you will be the winner in your whole life.

I would like also to say some words to young scholars in here. I believe a

mathematician should be a philosopher, not just only calculating because the notion

is the guide to action, and the mathematics himself is a philosophy. Certainly, a

mathematician is solving mathematical problems in all his times. Why you research

this problem, not that problem is not only dependent on your ability but also on

your philosophy and values on the world. Your think it is important and you think

it is valuable to do in your life.

In recent years, many young scholars asked me providing open problems for

them. In fact, there are many and many unsolved problems in mathematics and

mathematical sciences today. However, to decide what to do is simple, to decide not

2 o. ,:NCETMMS-2015ySSqR7_d1"%AOM 173

to do what is a little different but more important. By a philosophical view, there

are 5 guide lines for deciding whether a problem is valuable for you:

(1) valuable for solving a problem in a mathematical branch;

(2) valuable for developing a mathematical branch;

(3) valuable for developing the whole mathematics;

(4) valuable for developing science at a times;

(5) valuable for understanding the nature, beneficial to human progress.

In my opinion, the most important for a mathematician is not just solving an

abstract problem with only self-interest but should contribute to all sciences and

human beings. Along with your own research, you should look at more researches of

other scientists in your or not in your subjects or fields, and then push forward the

promotion and development for different sciences together. So, I think this National

Conference on Emerging Trends in Mathematics and Mathematical Sciences coincide

with the development pace of sciences. It is a valuable and important conference

for all mathematicians.

Thanks for you attention! Thanks to everyone!

K 2 MmTK– *8Æ nzxwPPoP5\b~MK, 174-175

Y_ÆM8vJ_66 a:\0ÆfGTG`h<g,TVk#6&5LT ,hLee! ,L^!Ri ve sTVvo=&.y,hh^b#vV,7s^b#WO~tvz,y|,gOXtVPKT!~v 42 ody=v^%7,7p<O,O<W,W<,<;y. ,'WyRo`<,7r#;y#~no,UO#GgO^b#^%7,',\mI!oTVi<vrs,7^'., YORrEl-,Xg2 vQY,i G6,Gw:LQ ,'<[< YvOL_[e,l, %7e`4 WvzhAdOX.j!%|,'R|$Q, P?M.ÆZG.m^,'3bBW~oq2o`Re,[,=|~oGQ &YYTV& j2e4 +*O^Tz\#vpB`Talk in Inaugural Session of ICDM-2016

Ladies and Gentlemen, Good Morning!

On behalf of a honour guest, it is my pleasant to welcome all attendants to be

here for a common interest: discrete mathematics.

Today, we are faced with 2 views on our world, i.e., continuous or discrete, and

there is a well-known ancient book, i.e., TAO TEH KING in China, written by a

philosopher Lao Zi. In the 42 chapter he delivered a generation for things in the

world with a discrete notion, thus, Tao gives birth to one, and one gives birth to

two, and two gives birth to three, and three gives birth to everything. Certainly, the

2 o. ,:pÆ|zySSqR7_d1%AOM 175

modern theory on material constitution, i.e., everything is built up of elementary

particles is in fact a discrete notion on things in the world, which finally results in

the main topics of this conference: discrete mathematics.

Today, there is also a main character in science developing, i.e., cross promotion

and development each other on different branches of science, also called, the inter-

disciplinary, which results in many new developing trends in science, particularly,

the applications of discrete notion to mathematics and mathematical sciences.

I wish each attendant will use this opportunity to fully exchange well, display

his results and learn more from the others. I believe also that each attendant will

not return in vain attending this conference.

Finally, I wish the conference a complete success!

Thanks to everyone!

K 2 MmTK– *8ICAMTPBCS-2016wPP5\b/ #?~MK, 176-179

Y_Æ6 J+[V6oLq%HR6~>GZ_8v6 a:\0ÆfGTG`h<h,TVk#6&5LT: ,R[hsKBW,sTVLORi ve ,XgeM ,lg-%py`<WnWvR~T.y,hh+8p,|YY.r/M,3qg0n`r,ir45LU'!OYEv# ,L q |ve.m^hh+8p,`<^2zUq,B`<31Uq[<C!^%7,hLg0nvIg,sLNg`dsLm?vOg ,lgey`<Wn,`4eJ, WvUO#,LPYhh6HL+\|O<$Qog&ug4.y,hhufo YvO^rEl-,Xg2 QD,i Y,w:LQ ,!Ol-mI YWvOL_[e,lgl,(D%7e. WvjXg!hAKWjWUSQq v!T%|,sLG[<ZG%7,[<ZG6v$QhGABWj!%^R|$Q, P,?MWoZG%7.`:,[,h=|~TGQ &YYTV& j2e4 xQy*Q?~,$IQY~0A QpQ6 z\#vpB`Talk in Inaugural of ICAMTPBCS-2016

Ladies and gentlemen, good morning!

On behalf of a honorary guest, it is my pleasant to welcome all attendants

2 o. ,:ICAMTPBCS-2016ySSqR7_d1"%AOM 177

to be here for a common interest: application of mathematics in other sciences,

particularly, in topological dynamics, physical, biological and chemical systems .

Today, we all known that the development of human beings must be in coordi-

nated with that of nature. Its condition is the correct understanding of the nature,

holds on laws of the nature. As the basis of this objective, mathematics provides

quantitative tools and methods for Sciences.

We all known a theory can not be without the practice, and it can be only from

the practice. By this view, I think applied mathematics is the source of mathematics

because it is problem oriented, particularly, in physics, biology and chemistry with

their applications to industry and social science. And sometimes, we can not even

distinguish a research work is on applied mathematics or pure mathematics.

Today, we should notice also a main character in science developing, i.e., cross

promotion and development each other on different branches of science, also called

interdisciplinary, which results in new developing trends in sciences, particularly, the

mathematics. For example, the applications of combinatorial notion to mathematics

and mathematical sciences.

I wish all attendants will use this opportunity by CMS to exchange well because

more exchanges will bring about more ideas and good results. I also wish you display

results and learn more from the others.

Finally, I wish the conference a complete success!

Thanks to everyone!

Y_Æ6 J+[V6oLq%HR6~>GZ_8v6 a:\rÆfGTG`h<h,TVu6& dy ,he! oGL6v $5, 46v $%7,<,[hxe! WUSQ`(; xQy*Q?~,$IQY~0A QpQ6 ^|IhO=L,ORV~1egORV,UO<%hh!R

178 \3|lzz – ySE+Z Smarandache J1b#~IR2gBdI=,sL~1gOg&hh!Rb#vY !gh+=o4od+OLJ8?hd,Ah1rUMhOL$Qs ` $Q!UO#gO<LjdQahBS3 ,sL.yv`4 WTvSpds MhsRo|h q ,l,eh0yk|UWUSQ Fvh~|~vdO ,[OW+TvSpdse Yv.y,p$QR_gO<Ljv5,ap2$QR_u.<2 ,rR $QZ`EhR|L,V= , Evq2gdp$L3gR|'=R|8e vds,Re=m^M 3g(,2gI=Rg0n,6R:1m&|Y,eBdO<$Qv P,$Q|FeG,GM$Q|Fv$Q5,o<Nh&g2&R|v$Q3,6R$%2 i Y0g3o!LJ,hL~gO`vT,dOXBW+g?tvYYTV&YY& j2e4 xQy*Q?~,$IQY~0A QpQ6 \#vpB`Talk in Valedictory of ICAMTPBCS-2016

Ladies and gentlemen, good afternoon!

I have heard many good works and good ideas in the past 3 days here. So,

please allow me congratulating to CMS for organized this successful conference, the

International Conference on Applications of Mathematics in Topological Dynamics,

Physical, Biological and Chemical Systems.

I always believe an applied mathematician should be a philosopher, thinking

about our world all the times, not only just calculating because mathematics itself

is a philosophical subject for our world.

This is my 4th time come to India for mathematical conference. Each time,

many young scholars asked me to provide a few open problems for them. In fact,

it is easy but I never do so because there are many and many opened problems

in mathematics and mathematical sciences for them today, for example, on the

2 o. ,:ICAMTPBCS-2016ySSqR7_d1"%AOM 179

last section of each chapter of the 3 books of mine donated to the library of CMS

yesterday. However, to decide what to do is simple, and to decide not to do what is

a little different but more important.

In my opinion, the most important for an applied mathematician is not just

solving a mathematical problem with only self-interest, but should contribute to

other sciences by mathematics, not only just calculating but also the creative to

mathematics, and then the beneficial to human beings. Along with a research, one

should look over more researches of other scientists in your or not in your fields, and

then push forward the promotion and development for different sciences altogether.

Considering all of these things, I think this conference is success and it is valuable

for all attendants.

Thanks for you attention! Thanks to everyone!

K 2 MmTK–;PdKhFk, 180-182

66q!7.B v`1e&Vr'GX|Y,.rMY=-8`=,|YB=`C bIr4.y, Av64Xt|Y`Lir#WOLdt,ar#WJeG|Yo^8EQ,6H/iWOL =Lnvds,l,[ BwN2q7>9 ,[ I0xNJ7AJA7XBwN2q7?x,'I0xNJ7e2 m`v|!LdsvQ2 r8Vvl\,hhP6Ho^U;#ORQg0nvrR.y,hh\C8E^xv|`vj%7ORgV?/mv Smarandache ` `<,ORug(D67,7 (zxQ<Q.D*InND*EpzvxQD*2q~N87hh+8ORtv;(,7gae!R;(W,!RV|EY"TA1uv2 5X,nTAvi"oTA%RJTIv||rTAiAORuO9>O9Æ6ORwO1:,gO9pMhG2,+7 M|?v ,[,Mh+owCv.<W.PT& OXN?Mhko2 2qCK|,h"QvAa3NIK[Q$6,<LvAa$$L>I!UO#z&g ,TAvig!RV|88vTAiv'2D,7!LI0D*d_ SmarandacheI0|Yr#dylLvVY,&V|AZa,`zeOez0WA,Smarandache I0Ne97XiJ?3PT~VLX0%6(DSmarandache %74ey``4M #v,~v2 12 p` ,|`?0^3,h2Smarandache ` ,Smarandache =B,WN zey`?//izmv>r!LLz,ahh\C`8ooNhv T4pdX5r<`\/ q~,~'2fT/7i fMv 5gez\vD,dM1=2 v,aEgOIv,7$%(D4 Smarandace %7v$Q.4<,Dbi

2 o.=SfLkHm 181(D4 Smarandace %74 v` ,s<8e v5=tm\CUDW~BOR/ j255p -As_A Foreword ofScientific Elements

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 ques-

tions. 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 different part of the elephant. So, actually the elephant has all those features what

182 \3|lzz – ySE+Z Smarandache J1you 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. Dr.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.

K 2 MmTK – 0XH/~oI, 183-188

gkC<G. 1F2 2qNI072qONxQD*7<.N<Qe9o`,[1ONp*<QZ#~,xQD*N:I0yd_ Smarandache I0,$KLQZAxQo`T%J?3*\r 1$e9I0,>9NLI0yd_I0– xQD*,uQ CCaxe97X,~' boSmarandache I0Yd_ QpQ6 ~HJAbstract: What is a multispace? And what is the mathematical combina-

torics? Both of them are the philosophic notions for scientific research. In

where, the mathematical combinatorics is such maltispaces underlying com-

binatorial structures in topological spaces. So these notions are useful ap-

proaches for determining the behavior of things in the world by mathematics.

I review the multispaces that I known, particularly, the mathematical combi-

natorics in this paper, and explain why it is important for one understanding

things in the world. Some interesting things in preparing the First inter-

national Conference on Smarandache multispace and Multistructure are also

included.hgÆ(D.<,lge#v6(D`<v>re,`,`[9GzCC(D%7,$Q OL(Dds,v hamiltonianpRv^`5Cayley vjs, 4W.2WIdsz,aB2003 8o 2004 8!8P ,hv $5+GRS23,sL2N(Dv\. gR_!8hGO=e%!Rds4D*QCe97XO0Z#QAN2q7'CxQZ#OL+Yp7!Rds7hv,`[oE#tm`E,sL21B%7#pds,

1Proceedings of the First International Conference on Smarandache Multispace & Multistruc-

ture+The Education Publisher Inc.,2013, 132-135.

184 \3|lzz – ySE+Z Smarandache J1,\hv,`[$Q2T6,Nh+4|YVrv 6eh,`9uLG $5`Y,|$H Riemann Klein ,'I=OL_v(D2gOt+dC!O%-,2004 8 11 VH 2005 8 2 V,h7W Ov,`[oEAutomorphism Groups of Maps and Klein Surfacs(. Klein v op*,Y g|e#vOL6, 4I$H Riemann Klein ,'<=(Dm^p#vOL=BdsoEU|v=8og&Wv,&43h?L\!!poE,GL7e#O K(,lgWqv(D Y5o7(Dn67 (CC-Conjecture) 72 (zxQQ<y3D*EnD*p&g,!,`[oEC h=RVv1T3,& 20058 3V|YeOV\,\5C W,/q,oEWv Riemann =B,UO#`eZvgI,7 Smarandache =BW$Q,BRLOR(DdsnLOReUO=Bfkvds,'zoheoEWzW Smarandache =B0,'.oENOI!Xg 2005 8hee American Research Press \vAutomorphism

Groups of Maps, Surfaces and Smarandache Geometries(. Smaran-

dache =B op) $~t5Smarandache =B%7UO#XgO^` %7,lgWOR=Be`` Pm2m,g`G^m\2m%7,gheG8v 9W`'v,sLCWB=2i<OR\Ev,O=LNHw\E4,73$Qo\E,aUO#,\Eer#Wo$2e,!Ggm^21LpK|YVrhrdsvBs,sL|YVS8b#z & V|A^)vp`,7S8dyviVLi,JOm^gDK\5V,7V2Dv'2,!Xg Smarandache ` $_,LnHY0|BF,'OL+3ew7A$Q

Smarandache ` 2 gOR2Dds,2toUG[?vUO#,OR` OReudy,$_2D: OJe^r,70^O^ -oL<,he Smarandache ` # #qxQD*,7U -ov Smarandache ` %7,'& 2006 8ee\Smarandache Multi-Space Theory(Smarandache ` `<*O,C6l(D$!gO#O~ Smarandache ` `<v

2 o. 3ZJ1qK 185t52006 8,e O WoD*QYhVQpQ6 #,hjT(;q v 15 |YoEP ,Lhv(Dn676l\ ,'?[|!oE Combinatorial speculation and combinatorial conjecture for mathematics((D%74(Dn67*YUOLO?Y,o#!poE[=Ler?#M#Q itj1k D*QO;vz`(RhvSmarandache Multi-Space TheoryOC 6O43,u& 20118eeOV 9\,\,L$Q< 8?tfvg,Smarandache ` %7dvg/u,Rhv(Dn67dvgr,7dy: Jer!v%D:g|YVr#dyYv%7,Gg vYm?,sLNu'vg/ur,!eW, -o,oY|L|TdylLv#~e,!Ggh: 10 8O=eBd(D$QvORrEBs|vGpXtM| 21nLiOR|,!GnM|N~`OR|,eb|y,S3.S['v TdV v|!,28,!0g|eV#vrpXro Smarandache ` ,h~|XgOR)elhORJgJ.zy,&geWOVThzyI!JBd 10 G8A`5,aR|O=Ip&$Q,sLR|G?g,e,`.,`[+gBdvLv'Age|G#1hv Linfan Mao,ov#~#+gOLh.vd<,h2he2~`\vOLtYv<`z,aeQ4#1 k(d,ov=+gd<,sLh P\~WMsdO<,!1GgWlH&&ghOR|X'^Ueb2 |aq,1N$xQMQ<4 aq,1N$15Zb~VMQ<,K`h~1XgOR I0L<,2011 8,OLei zoh& 2013 8 28-30 (; bo Smarandache I0~d_ Qph; ,'ee?#xo82First International Conference

On Smarandache Multispace and Multistructure

Month: June 2013

Date: June 28–30

Name: First International Conference on Smarandache Multispace and Mul-

tistructure

Location: Academy of Mathematics and Systems, Chinese Academy of Sci-

186 \3|lzz – ySE+Z Smarandache J1ences, Beijing 100190, People’s Republic of China.

Description: The notion of multispace was introduced by F. Smarandache

in 1969 under his idea of hybrid mathematics: combining different fields into a

unifying field, which is closer to our real life, since we don.t have a homogeneous

space, but many heterogeneous ones. Today, this idea is widely accepted by the

world of sciences. S-Multispace is a qualitative notion and includes both metric

and non-metric spaces. It is believed that the smarandache multispace with its

multistructure is the best candidate for 21st century Theory of Everything in any

domain. It unifies many knowledge fields. In a general definition, a smarandache

multi-space is a finite or infinite (countable or uncountable) union of many spaces

that have various structures. The spaces may overlap. A such multispace can be

used, for example, in physics for the Unified Field Theory that tries to unite the

gravitational, electromagnetic, weak and strong interactions. Other applications:

multi-groups, multi-rings, geometric multispace.

Information: http://fs.gallup.unm.edu/multispace.htm Pee_FT?x6O4ob2Date and Location: 28-30 June 2013, Chinese Academy of Sciences, Beijing,

P. R. China

Organizer: Dr.Linfan Mao, Academy of Mathematics and Systems, Chinese

Academy of Sciences, Beijing 100190, P. R. China, [email: [email protected]]

American Mathematical Society’s Calendar website:

http://www.ams.org/meeting/calendar/2013 jun28-30 beijing100190.html

The notion of multispace was introduced by F. Smarandache in 1969 under his

idea of hybrid mathematics: combining different fields into a unifying field, which is

closer to our real life, since we don’t have a homogeneous space, but many hetero-

geneous ones. Today, this idea is widely accepted by the world of sciences.

S-Multispace is a qualitative notion, since it is too large and includes both

metric and non-metric spaces.

It is believed that the smarandache multispace with its multistructure is the

best candidate for 21st century Theory of Everything in any domain. It unifies many

knowledge fields.

2 o. 3ZJ1qK 187

In a general definition, a smarandache multi-space is a finite or infinite (count-

able or uncountable) union of many spaces that have various structures. The spaces

may overlap.

A such multispace can be used for example in physics for the Unified Field

Theory that tries to unite the gravitational, electromagnetic, weak and strong in-

teractions. Or in the parallel quantum computing and in the mu-bit theory, in

multi-entangled states or particles and up to multi-entangles objects.

As applications we also mention: the algebraic multispaces (multi-groups, multi-

rings, multi-vector spaces, multi-operation systems and multi-manifolds, also multi-

voltage graphs, multi-embedding of a graph in an n-manifold, etc.), geometric mul-

tispaces (combinations of Euclidean and Non-Euclidean geometries into one space

as in Smarandache geometries), theoretical physics, including the relativity theory,

M-theory and cosmology, then multi-space models for p-branes and cosmology, etc.

Papers will be published in the Proceedings of the Conference.LrgOo,~/AROL2o<3,GneF43j RIz 10 GX.hr,7BW!o,a &h(;!o`Cz,&g|oAR,YU!L[,D1UQ/Ez`6e!LÆ`,Lo<`\YUh,=gMhBW!oR2Eo,W,4$QWòv Vasantha Kandasamy UhYO21ovI8b2,uvev Smarandache YUhOOvb,6O4P21ooEv8f4FX gq,=bogLÆ Smarandache ` .ò`<,sLMh;uo!O3 Yv T$%5 bo Smarandache I0~d_ Qph; & 2013 8 6 V 28 ev?zyTz,/ , !LOv` v,7eW_Ò!vo,'to59g21vOtdR!ov(;,h2`3K.BP#o[.Wzd<,Zgtov?zyTCF.ÒOdG+Wd,-Wd, 4eH$Q<zQ+vWz Tp7 !=X GvG5,!oG+&ev?zyT_è<,l?!Li ^8YUO#,Smarandache ` %7.W Ma*%7OI,#.Pv%7/,!Ggh!L8.p$v $Q%7,72<Qh?:Q,>9N~ :Q,sLW2g$ A ,SAS,RgO^

188 \3|lzz – ySE+Z Smarandache J1Vrv%7:K`!vV,gsL 2004 8heW OBd$QP5ovhoo,g $0prC,'uWv V to"YRpvhee\vO~t5Combinatorial Geometry with Applications to

Field Theory((D=B4e`<Wv,2009 8) WO,gi<y|W=9& Vv%7,'|-5Ldy0^o,-E|=BWv hzp`g`6l(D'|lL,R!0ghv(D%7e $QWvUlK`,Bd $Q,2#Wg2lv,sLDW|v%7g%7,$Qdy3mmpg2:QEav_,xQEa*,W%D,Xe` # #[<$QdyGpve – xQD*,!gO^ $Qv=i,&|YVr212 gOtL|mvdQ,!Gg(; bo Smarandache I0~d_ Qph; v=%s

M 3 Æ*!o

Zb6yQWxDM,ZbdDI

K 3 (m – HHs(<+X\^D04`4Vw~2us, 190-196nR37-~'!zCz&b7 1F: 04`4Vw~2us+aZvX04`4h)M?!bUZ#mI`)_K+MC1Æ+T'x) )BB04`4Vw g6[+XB2Y"'42s8D2,/usX22C2Hu^ kX26N04`4UHX2%OH#2)HD 8 n+Bi04`4VwwFx2K6!03 Vw~2us2Y"'4s8D2,/2CHu^X`k04`4UHX2%O2)HAbstract: The operation system of bidding market is important for protect-

ing the State’s interests, the social and public interests, the legitimate rights

and interests of parties in the bid activity 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, government supervi-

sion, tenderer and bidder organization, institutions, personnel, self-regulation

system, bidding theory and culture. All of these discussions are beneficial for

the development of bidding business in China.

Key Words: Operation system, goal of bidding business, system rule,

supervision, bidder organization, bidding theory, bidding data and culture.IE2 F21.p`gp`CFvbu, p`X`p`CFfvL*tWi|pèI KWv# p5,R zm^4 M4vp`l%M,ugk5pè~ ,Mp`CFKlvptM4,5Lh,rkCFz-WnI KqCFKmk5< RvO^ M4,e$6CFuM1L_9gpù,=oUS lLzmbt|p v,,hCFz-YÈ5,J\i1www.qstheory.com.

3 o. JJu*>.Z^`F26b6Xy4wu 191|W`%o|Lru,l25g_,vpù?M4gIv3,Gp+OLuMWvds,M42O2Gg4 u_,l2_,vlG.RX4lJ$;mkd'AeOL3\uÆ,p`M.%M+`Kl4p`rub4eO4#lG,oM.Y'AuÆ"`4J|:>wz-l\Ol,[P|:l\zz,!Lds21toKp,=MQdJvYlJgO^#5g#\vlJ,K`,ozgvp`l%M,OR5gOR\g21pv,= %7,/Rl%MWvuB.`._,z6llJl!eW,2gm,|E6O43nuM1L,2<1,L$L1r p`lJ4pvd<UlJ(;p`dp,BRi2UlJgzm!JrpJQ^Cvvp`l%ML<, MlJYlAmlJvY,zmp`Kl%Mea~`D:8p`$Q+, lJY lJM4z-lJ_,%M.llJ(;lJBJ%o|F.lJ4`<ulJbnlJ`nz-z 8 Rm,gpÒ6lbI, XM4K|&h,rkCFz-,G6p`vt2z-JmQ n26,k555my5s yBy9-F9:.gpÒ,245glL HÈv5V#v2O`45g ep`Wv\5jm,XG8=p`l%MO=Xg 131vN\,RCFjmv[X,2!2|l2|F25L75L'APY<K`,gpÒ~1XEY6O43nCFuM1L,BR:1gpÒRgpÒvEe&6O4YM4ep`nI Kv&.5,K`,lJY L EjLG15a5Wx3vt,1 EB,L<,lJMAm5gl2_,5g3nuM1L, pY'5g\5jm$^.CFuM1L2Mv4fV, |VCFz-LT\,22Ix!OY gzmp`Yi%.eF%H6^,S yBy9-FBFi-lJM4z-|LRO,Og^4M4z-,WglJM4z-

194 \3|lzz – ySE+Z Smarandache J1EG6W`NvO(i*4HdlkBJ>wz-LE3e3,#E3[zp!zL<,|EBRmBJ>wz-6ll21=qm?<_?3o'lHJ|3,eCF 4`YJz-=q~ <_?m?4 P,xgzH4$Q%o6l=qm?!`,`|:v_?4Il7ezHz-=qJvlp$Q,ez 9J_? /W-K=qJ, hCFz-Y,?,q ez=qJ8VvHJ<2=qBJ|F`2007 8,BV|d5VY1LOrDY=qJA|F<J~[glF<J~tUI`^,& 2009 82010 8`(;iF<J~t,H.\ 18000 G|tF<J~5,Lg=qlLRO# aLnkdJvY,3<J~tKK2r,2&qBJ|F<J5R,G2&BJ|F>J`L<,|E6O4zm=q|F|Fv<JM4,' jSOBJ|Fv``^,BJ|FMM4P 9M4>M42DNN&%M,BRk5BJ|Fv<J5RD,Y|E3=qV>wz-zmO7J~<Jpr vV>w,gM4KUIv`EkV|LY,OgV,Q||RQ^754Wg=q`<m^ $QV,=q`<m^ 6l$Q$Y', >x'AeOL\ÆOLV75[`2/oELt,EN|W`%o,LIEL|z4.RV6HjVV!0|FGve,H^LOL|.V+eCR|L<,|E4jIl$%V:`#|2DV!z-5,VtlO`,'6O4YVIv54 P,zmV_>M42DNN&%M,`"NVv`,'

3 o. JJu*>.Z^`F26b6Xy4wu 195W;<JprJ|5R_?,nkM4t`KUIgO<UqpÆ;vM4,`<$QS[&UqO=gOR2.vpsL<,|EBSROoz=q $.oM vV>w,LlJ pJq J`<fr(\*%HUplJ4gnk%ed|Q^B.%b2Qv`Ek5, PGgzm &l)&ev=QbaG8=,p`rub[O=l\v`<.Uq$QL<,|EelJ|# #,(;J0V$Q'zmb[%M4'm4zW`%o||OFVzp`ru|Fvb[54e<# #,zW`%o||OFVvb05,06lÆ,'Z3?UybMPvlJ2Jzmp`rulL4 P,lJ(;EW;p`ruv4_,,6O47lJ4hT,_,%Mz, p0l2_,pv2# yByKq2 yBy55mF`.`<u|LRm?,Og^4# ,7^4M4,WgCF`<# ,|EDECF,F<`z/ ,O=`toJ0v`rL<,|E 2)',kv!J2zG^m\,-m<$Q#-,2lJ(; $%ozY3$Qs,v`<.lJYWOL`Ts$Q4xg $|F`<lJWvy ds6l$Q4 P,Ezm`TEl$Q%M,(;/3V023Y<vOL`TEl6l|&$Q,Y3$QoEAmlJY.%HxQk yBy3KFBFMH|.rMYgfv Y,YvOR`EAm%7L<,E='W A,bAzA0_$Q,jm6IQh`:, Q# aY7X!Z#v,Y bAe=qWvzYKSoX,|E$Qz-vd<ÆG,OgSYb!z-,lJM4!W`%o!|!=

196 \3|lzz – ySE+Z Smarandache J1qBJ|F!z#~b!4Wg=qM4=qei~SH^lLK/eE~Svz-4gBJ%o< 4 b4+g(;$QMlJImEIA,zmlJbIM44lJIAY3`pzd<?tqm|hfvg,lJbnz-,lgM4z-v P,77p5,Rh_U5gT$'ed|H^lLq jtf%H_Qk yBy=j[HQFz32FBF2=qM4eh\CUILG8,h,rkCFz-,=oUS lLzmYÈ5,aPH.,|hDEOqo,^4v=+gYg=qSzps,.T9lJe|h`WviA!r.'< H^G5,aZG.lJ`nl\OL uj|hv & ($``, Gop'=q< uDL<,E`z-,rk>`[?uL# ,$%lJUybv`nuz-,6RqBJ|F%7pr5R `n5?,r1=q|Fe,#viA P,LÆ#M4e,rkA4`yR`z-WYv5,E` PJ85\ÆyvBu,)op\p`rulJBJ%o4|Fv6d(6`4H^dtvl2fd$Zz<5,BRÆmM4e|h`WviA#K ,ozpÒ,g6O4YM4epÌn KW# p5,gp`rulLl2_,_U,GXp`t2z-vO<_ML<,|EekhCFz-v# #,(;V|$Q<5,'%2lJ(;!JRmv&.p,6R:1ozgvp`l%M,Lh,rkCFz-3ZTg(

K 3 (m – HÆPT8,Y+ D04YaUHX, 197-206

5!o6Æ_-n-'!zÆ*Lr3~ 1,2F: 04YaUMsPLU~&P& UId\G?76GKU0~+fs04Ya5;~U<JIè04YaUHX+d8VwÆs8 04Ya~Æ5x+)HÆPT8, B04Yas1UHXx2X\.L+<3;;HkZTe04YaBVwÆ~P032 VwÆ04YaUHXLÆ:jAbstract: 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 in-

struction 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.

Key Words Market system, bidding purchasing, theoretical system, con-

sumption choice, economic substance.IE: F22.=qgp`CFM46lpÌn KvO^M. m\,UR,g U'=qv5Lp`CFvOÊCF%,=q|E/yECF4,/yCFvCzsQ,'e<# #,6lns,6RU'qCFKm,k5< Rv 5LO^M. ,=qu\)G6,A64vq~X=qv!^CFp,p=q`<gÈCFL# vO^,s`<,gO^#&Op,F# #vn`<, B=q`<v#~~|eu Wb|Rm|6l\ 11<pan+2012 7 2 U 17 2www.ctba.org.cn

198 \3|lzz – ySE+Z Smarandache J1J y )KqF`pTs – K n,s yz2qN15Zb~V7=q`<g&=qAvOYpvu ,h2=qOYp2:v5\ ,4=q|Lr=qM4gp`CFWvO^M.p M4,UIX^2zp`CF,|E/p`CF4(J* nF`p,(TV8p,p`CFgApèI KW pp5vO^CFuMp`CFpèI KWv5,e&<[R_B<[[B|zds,Qp`=pp`CFe`+R#~l-2p`ruvrpp`rugECFv#~(E5ep`CFt,~B!JMCF(;R|,gUv|rfvp`ru,1rrC),hYhQp`v~zpp`v~zp,Ap`%vK<[C|Cze1#g~zv, z:lf,7 mGmed|: /e~z,'Lp`x,/z[xBup`%vM.pp`%vM.pgp`CFv[y,'LM.pM.%pvO,GXSYp`ru$$Qp`Q+,|p`b,p`|Y,np`%,hp`|Y4n p`lv^Mpp`lv^Mpgp`CFvW~EY Mpgzm7p`CF%0 Ov^4u,h2p`m^>^Tm,>`gk5p`l ve~ O(i* yByF9F M45LhCFz-WnI KqCFKmk5< RvO^M.p M4,e$6CFuM1L_9gpù,=oUS lLzmbt|p v,,hCFz-YÈ5,J\ip`,!Op`p`rupùp`^,kz#~E5op`ruvp`rug||! ,|g 1 R,|gGRW`%o|vdO(,W`|4OF|Q^(z,O(,C`tWvm^,

3 o. JSV:.[. F26[ WJZ 203,z,wL#4OY21=N,D1 U;$;,wL4,l0_1pb:z&#,`=m^n,a&4,,Km^Xg%v&P`<s56lP,C6ls!^m\Xt|hÒYpzm=qs5 v,6Rn=q s5P6ls=qs5vPÒLKm^2tp 1: E3pC6pJv3qgXbI|,h2A?I16`^4|z=q |Y2 ,pJ=vIA2 ,aOY|QbACFA?,|[X`14|E16d<,6Rn/Y[AvhgI,!OpJ L+4 pJ,u+4=q|vC6.``5R&G`#Xv[=q,`QbiI?6lC6Ptp 2: a2pns5 AB Æ,UqWO^jm^,7^,!^m^g=,'t6s,mEvO^m^ v^s5Gs5 0.618^W|^Fibonacci ^z,!eW,jvg 0.618 ^tp 3: Lx.p$L21 2#A=Ævs5,=VpupJ%Mns5 P!^pJ%ML`YVJe,#`YVpupJe,BRQ#=q|Y VpJOY=` : mVpJ(pJ(F|L,è.oP`=vm^,C xIJxvBupJ4ipJQb,pJb,pJ/d<,EP,(;i<,$QpJ, PiV4VQbR|VC6,6lpJ4 VpJ,eV2upJfr, ,\pJXf,VpJUO#`^L*Y- n Rs5 A1, A2, · · · , An|6lP,uV|E7OR n×nv* [aij ]n×n,! ,aij ,1 ≤ i, j ≤ nL Ai, Aj vpJ$?,7 Ai > Aj P aij = 14d:,Ai < Aj P aij = 0f,! kv*er.#vrL 0,BQ# aij = 1 u aji = 0 vteV(| n ≥ 2 P,2 V7v*2O7iO,<P|E|KVvLOR*,6R7 s5vPm^^,O^g*Y4O^gWfY,7!*dRVOvf

3 o. JSV:.[. F26[ WJZ 205! ,U - 2I44V - 0<44G- 2I40<4v9(p`tI|1pv9*4H- 0<4 v(||F5R.M0<4v*4I - omvCl<,demWS1kv5|QblJ3< l ,=oz4D[x0z , D`nomWv* AXAYU V,g[W|bv# p5! ,bv2/`n g3q,Rs)=*.=qUO/DugW~# ^W|,7fMn h | (i*"74'[evtvP 0,7`1v1pwL[vp,! ,tA[XPvtIt+t4l OQ Oh9v(p,e6e-e-#lY,p'X2O,KÀvtgR_4P A[v7~|P 4?[~|P vyW,['v25|yW,v,pl,_fv(plY 5 8v(p,7g OQ Oh9[,['v25WT4`1uA[+4vezv`14Ak5AKEY[`1vG'AAv|,=9o[pAv|[ph2[v'SpMxp-IpTE5,! ,'SpA[XopXzd,aB4= ,~B[21 100% 2Y<4MxpA[Y<[,1rm% MdMxP,XgMxp4-Ip,A-I[P,|3[v Xp G5p,Cp,TnvEY6l-I[p[,`X25AP A,!LAh24GK5~ro5P ~rGK3P K4z,_l= ,[L N,WXz T > t v[L u(t),u/k[4^LP (T > t) =

u(t)

N.OY,`#5 n t6,e# ? u(t)/N beO? p +:R%,RBt6RG,R%4RF,uw? p L/[vp,KL R(t) = ppgp.`v# :O,ge`<Am .ÙqC6RM,'?$QAY`4C6v*2;x2;7v,OY|RO27p# 4Jp# pEYv[! ,p# gAp.èjAmfk

K 3 (m – 04YaÆCG, 207-218

!zÆ*!o_gbv 1F 04Ya+au+04`4#KuÆ+M2`LH_N)HTxNb2iO +Vw~ aP+uÆ&o`4JusC+xP5lC1Æ+T'x~Æ:j+jm'5l?:jQKDIjE-E+mÆ~b2iÆ25ln_>~gÆ)?04`4_Y) +HL2G04Yat~Æ+/WFUH/wPUD04YaC-+G04Yat~C_>#n_+6:04Ya2~Æx+&s04Ya'>tH\H~ÆnY"03 04Ya2uutwH/-CwFvXC4 Abstract The bidding purchasing, including the process of contracting party

with performance is a kind of competitive mechanism of the actor following

on the code of conduct, such as those of laws, culture, customs, morality and

disciplines in the market, and then increases economic benefits and ensures

the quality of the purchased project. However, there are many basic questions

should be researched beforehand such as those of items in the code of conduct,

economic purpose with effective methods. In this paper, we analyze the com-

modity utility of bidding goods under choice behavior, apply the scheduling

theory, multi-space theory to establish a mathematical model of bidding and

then give the methods on economic substance for guiding the development of

bidding purchasing to selection on the maximization of commodity utility in

China.

Key Words Bidding purchasing, code of conduct, commodity utility, util-

ity function, multi-space model, preferred function, order, optimization, bid

approval.IE: F22.

1-+2013 7 2

210 \3|lzz – ySE+Z Smarandache J1Y|CÆnW|,'.W|C`t`t/D i y )b"F n^ Kbp&NUCzU=vKTF,.CzKtQ#8=l,P'Pv|KT,aOR0elwHSv|KuF!^Q#8.Czr8,rq67&1pvQOR`pCz,CFq65Y-27zpY-,7Cz13KQ#|Ev 4q6se?/#vOIp,7e X > Y, Y > Z P,O X > Zvq64q6o,p,7Gv q6T&'v q6OR`pvCzslL,A`Y K,TnL K,Tnv 5LsA`qK1pL==QvCzslLCFW,&CzXO X vQ#4 u(X) 2 ^k`<,7#K`<K`<,|=2 vKY-2`^yk uAF'Qu |,Nu |x#KY-vURg K`=P'W,7^ X, Y vCz|L x, y,u X, Y vK,|L u(x), u(y),B

u(λX⋃µY ) = λu(x) + νu(y), (2.1)W,λ, µ L X . Y : vvW,7 λ R X vKz& µ R Y vKK`,#K`<vURgY-2 : evW,6R`eK#6lY!^Y-`nnAvs5,d4u&`4zm!,#K`<XLn ,TKdsnL#v,nds-^yk uAK<%2,i<HGn,Gx7BKY-.#KY-/d,7 K21=6lP'WRD16lÆÆ,7^ X, Y vK|L u(X), u(Y ),|hD1nK

u(X) u(Y ) u(X) u(Y ) (2.2)R& X, Y,uKL X Y g X Y,! ,9 , |^K K%Q KDQ

3 o. 26[ BEI 211K`<W, X, Y vK/ ,7 u(X) = u(Y ),uw X, YeCzs#oVqgzKY- s2DL Ω,e 2D Ω #z ǫ- oVq2D2∆ǫ(Ω) = X ∈ Ω|u(X) = ǫ. (2.3)<, 8 u−1(ǫ) = ∆ǫ,7Cze 2D Ω WvsUR#g#2Dvss,!$L21jkvs5,|F5R`1pJUpzÆfk KOY`1[N`4M|Y,5e|YY|Yzp,W `1OYh2XtXEYApz,[N|YrEe&CzqK1p,7==Qt=q |Ly|Y,WyK,7yQ#|hyR`n<^|Yvp,h2yv#~`1.v`1.e`1,7Xpap4K,|EWbCFvYVmWdl[J2J<[p3\02p`Kev025tz,6l%$Q,4z< vp`|Y+z-[mE<[a-zhz-mEz-tu=mE;|CFKmtI,9`41[<v%zm6li3vT$Q<5,' -In,6RQbVg7vz--IFn4R|Kv>`g||v1p,h2A5R(;`.M1pz,Ks` gJ5?|1p=q ,7y|IvK``1EY[NpzA6l|2yKy`1yAABn4[NOYgp`[,7yW<[4oA vKz,h2f[Vzzk%z6z=qkzzz4ph2yp`4y Up,7W<[4k%pz|K|Kv>`e&|1p,OY|F|1p,7|F5RJ,|tI-z-Im^(;Qzs5n4|[N2[2Am[p`[^,Wz-3|[NOY>l2Am[4|puh2||Fp|tI-z-Im^(;|FQvpz!eW,|Fp7||Fv<J5Rpr5?,|tI-z-Im^pugk|1pv# pt

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A =m⋃

i=1

Ai =m⋃

i=1

ai1

⋃ai2

⋃· · ·⋃

ain(i)

=

m⋃

i=1

⋃

(

j = 1)n(i)aij ,

3 o. 26[ BEI 213g^LK*A =

a11 a12 · · · a1n(1)

a21 a22 · · · a2n(2)

· · · · · · · · · · · ·am1 am2 · · · amn(m)

=qlLv^4Q,UR#gEYOFC`tWbvm^, k R R1, R, · · · , Rk CK6lP,'nK,vLW,72b"^2" E% k u R1, R2, · · · , Rk ~vuAG

Ri =

ai11 ai

12 · · · ai1n(1)

ai21 ai

22 · · · ai2n(2)

· · · · · · · · · · · ·ai

m1 aim2 · · · ai

mn(m)

Cu R1, R, · · · , Rk hE%AGy3XG,T%AIDu,Z3Mx s, 1 ≤ s ≤ k,=uAeA u(Rs) ≻ u(Ri), Rs ≻ Ri,,1 ≤ i ≤ k/ i 6= s y )a2Kq

4.1 -0UOR2D S 4#OR ≻,~f x, y ∈ S, x ≻ y,uw x& y,d:,uw x Æ& y,KL x ≺ y,l,/2D#v −2 < −1 <

0 < 1 < 2 < 5 Xg2Dv −2,−1, 0, 1, 2, 5 #vO^,7TF,Ra ⊂ a, b ⊂ a, b, c ug2D a, b, c #2vO^,7h)OR2D S #vQ#`Rt2

(1) x x ~f x ∈ S m4(2) x ≺ y ⇔ y ≻ x ~f x, y ∈ S m4(3) x ≺ y, y ≺ z ⇔ x ≺ z ~f x, y, z ∈ S muw S LORq2ORq2#,#~fRr`6lÆ,uwLi2OY,hh`5!R<2

214 \3|lzz – ySE+Z Smarandache J1XK 1 E%A|*A =

m⋃

i=1

Ai,r|* A1, A2, · · · , Am NpG|,/|* Rk 1 (Ak1, A

k2, · · · , Ak

m),,1 ≤ k ≤ n, R1, R2, · · · , Rn UGwY=pG|U hhe2D R1, R2, · · · , Rn #k ≻ 2x,~f/ k 6= l, 1 ≤ k, l ≤ n, Ak1 ≻ Al

1,k Rk ≻ Rl4, Ak1 = Al

1, · · · , Aks = Al

s a A(s + 1)k ≻ A(s + 1)l uk Rk ≻ Rlhh5!kvXt2D R1, R2, · · · , Rn LORq2,7Q#q2e` ≻ vk,hh3|5, Rk ≻ Rl B Rl ≻ Rs,u Rk ≻ Rs mUO#,Wbk, t′,u 8 Al

1 = As1, · · · , Al

t′ = Ast′ a Al

t′+1 ≻ Ast′+1,

Akt′+1 ≻ As

t′+14 t = t′,u 8 Ak1 = Al

1, · · · , Akt = Al

t a A(t + 1)k ≻ A(t + 1)s4Y,, t < t′,u 8 Al1 = As

1, · · · , Alt = As

t a A(t + 1)l ≻ A(t + 1)s,e A(t + 1)k ≻ A(t + 1)s#78 Rk ≻ Rs m f,=q vl-,, P|G,,K`# `Wvm, n rL,/WW!v<,7e,q2WOJe.Tl,e,i2#OJe,T!,` 1,hh<XK 2 E%A|* A n%~ 4.1r|* A1, A2, · · · , Am NOG|,H A1, A2, · · · , Am GwYY R1, R2, · · · , Rn %UIe

4.2 ^Bn Kvi\? (A1, A2, · · · , Am)WSl-v,l A1 ≻ A2 ≻· · · ≻ Am, 4n Ai, 1 ≤ i ≤ m vi\? (ai1, ai2, · · · , ain(i) WSAv,l ai1 ≻ ai2 ≻ · · · ≻ ain(i) z,gss v# e=, Ai → Rk KP,U2 R #vC` 2 =t,!GgUqWCz6=?,k KvBs|z`fvg,KeOK,7CzeOP 0yWX CzKtvKy ∆(f(Ω))/∆Ω g

3 o. 26[ BEI 215lv KLsB 2 E,uOKl4^) 2 m d2f(Ω)

dΩ2) < 0 vt,7K, f(Ω) gu.p,,B

2 mF& 0CF3JeT2Z=,?kvs5,1pUp5Rz,<POY=C6pJpJVpJzm^ns5l-vÆ! ,C6pJv3qg\s5XbvI|,h2A?I16`^4|z,pJg=,'t6s,mEvO^m^,RVpJugQbVR|VC6,ORRVpJvm^l,Y- nRs5 A1, A2, · · · , An 6lP,-IORlr=s5 A1, A2, · · · , AnKv n×n *([4]*[aij ]n×n,! ,aij , 1 ≤ i, j ≤ n L Ai, Aj vpJ$?,7 Ai ≻ Aj P aij = 14d:,Ai ≺ Aj P aij = 0Y- m RV7v* [M1], [M2], · · · , [Mm]r[6l-dRVv!f?L p1, p2, · · · , pm ≤ 1,I=* [bij ]n×n

= p1[M1] + p2[M2] + · · ·+ pn[Mn] . n(Ai) =n∑

j=1

bij,,[Qbn(Ai), 1 ≤ i ≤ n vTF6ls5P

4.3 hTeKK`<Le OKv oOR 2,7(2.3*\,K`,3Q>K,I=?n P,.UOCzlLOVql,CzW X Y, X v[NL 99 , Y v[NL 100 4`[NpK,u X vK& YaUqWLR_'Cz6 Y,sLCzq6 Y v2iR2IÆG7&v 1 2K`,4Ì=?n KU%UOY,oVq2DoJn vO n Mi(Oô#2Iv- ,dR rJeOR\53 & n MJn *,l 2- MJn v=.,3- M v~z=q| KP,sL Us5,,2|%vE|=Bz|e6l n Mi|,R`jn,7BCz6Y,%Os5,lg$L`=?, f(Ω) kvs5noVq2D,7LQ# ‖f(A) − f(B)‖ ≤ C v A, B oVq,! C gCzsPnvVq ,l, [NVqe 0.8 ;`0v rLoVq,ue A B MtrQ#Czs,g A v[NL 99.8 ;,B v[NL 100 ;P, AB vKX/ ,7 u(A) = u(B)B=B#,!ôVq2DUO#g` f(A) LW`,C L_Kv Wu

216 \3|lzz – ySE+Z Smarandache J1jRR`_=R3 Kv=qlR 1 z-< -I,h2mE-I4-II-I,3o[-ImE1p.pRs5,B-ImE ≻ 1p.pW,o[,o[DVv-Ivz,u*`p4-ImE K,~3\ 1l|tI1-Iz6lDÆ,CQ|MR|41p.p L,4R|F-ItIzt,C L = L1 × L2 × L3 I=,W L1 L|FB,Ji,| 15 `#v 1.5,8-14 |v 1.0,7 |`v 0.54L2 L|F2iB,4R|FJ,v L2 = 1.2,d' 1 RoJ,v,W 0.94L3 L1pB,4R|Fo22iv 1.0,d' 1 R22iv,W 0.8,dORt lJ<v,W 1.2,[,I=|D|? T = K × L,' |T1 − T2| ≤ 20 v/ ,C-ImEt|v|6lPD|?/ ,uCo[|oP~< ABCD I+R|,C5,+R|o[reVvvzgI0,-ImEt|QL 90,92,84,79,1p.pQL1.94,1.8,1.8 1.44,\,ABCD Dt| T QL 174.6,165.61,151.2 113.76f,A . B,B . C v,\|?Ve 20 `0,/ ,a B -ImE& A,A v-ImE& C,,\PL B v ≻ A v ≻C v ≻D v!,Q$m BAC LOWWY|=q OY==p[5LnÆvKs,R 2 %-z=p[^6l5,7= N = K × (B/C)× 100I=|t|,! ,K Lp,-zpW pRt,W p mE|F4|[|z,CQ|MR|[` 1004BL[,C L-zXV |NA − NB| ≤ 5 vR|/ ,<PCo[t||oP| 1234 BWC5vr 1| 1 2 3 4p 0.9 0.85 0.92 0.8V 50 54 48 52o[(;* 36 38 32 34 1

3 o. 26[ BEI 217| 1234 v N ?I=QL 64.8,59.81,61.33,52.31f,| 1 3 v|?| 2 3 v|?VF& 5,,\PL 3 v ≻1 v ≻2 v ≻4 v!,Q$mvWY|L| 3| 1 | 2UqW,uCrs5sY- s5L A1 ≻ A2 ≻ · · · ≻ An,~f/ i, 1 ≤ i ≤ n R Ai Aj vVeOgI0P, Ai s5vÆnPR 3 I< ,=[I(;1p.p 2Rs5,L[ ≻ I(;1p.p[, 5B2|&~v[|oP4I(;1p.p, 5v,C nK = K1K2×BI=,! ,B LI(;-It|4K1 L|F1pp,C K1 = L1×L2×L3I=,W L1 L'``|FB,4R|FJi,| 12 `#v1.5,6-11 |v 1.0,5 |`v 0.54L2 L|F2iB,4R|F#>5,J,v 1.0,d'ORo#>5oJ,v,W 0.94L3 L|F1ppB,4R|Fo22iv 1.0,d' 1 R22iv,W 0.9,d'ORt /Jv,W 1.14K2 LP= .z-zp,~r-z765n,B ∣∣nK

1 − nK2

∣∣ ≤ 10 P,C 'DfBuP/< !R| 123456 r 45,[QL 2580;2450 ;2545 ;2410 ;2510 ;2560 ;,r2|&~4nK I=?QL 158.24154.8151.36146.2141.04134.16f,|123,234, 4 56 v nK ?/ ,,\PL22 v ≻4 v≻3 v ≻1 v ≻5 v ≻6 v!,Q$mvWY|L| 2| 4 | 3 J*q K,Tng=qvsE,Gg=q%/BvCF4U'=q K,TnvxEds,gny| KvUum^a K.|q6,nOK?gOt+d,aBKY,n Y KvÆugOt vd~`Uvl,7geo^n K?t,:,k=q K,'b<nÆvm^

K 3 (m – 04z&HX~wP->eG, 219-235!z(3~G6#gBv 1F. 04M04`Y,u9+ 4`9Pu+04`Y,4XXhRu~)Z2[Æ04PYa~n?+Fh6sjC#LÆ.Wb)Yaf2B04Ya~2)x2|\$+JMH?VwYa2 v_N+0\~04`4_N_HXmB+B04`4-z&HX~gÆa4)^>+P\~UHX)~~'~PP; w,~~ [7], [8]+D04z&HX~ Smarandache H/-+?YI'4& n_#&dLÆUBx2>eG+D??/P04`45;~wPU))H9B9'X2~z&HXx2:U+f,T~-Ee&k03215,a5,5,SmarandacheI0,~5PA,J(5'd,h(5'd,<*'Abstract. A bidding is a negotiating process for a contract through by

a tenderer issuing an invitation, bidders submitting bidding documents and

the tenderer accepting a bidding by sending out a notification of award. As

a useful way of purchasing, there are many norms and rulers for it in the

purchasing guides of the World Bank, the Asian Development Bank, · · ·, also in

contract conditions of various consultant associations. In China, there is a law

and regulation system for bidding. However, few works on the mathematical

model of biding and its evaluation can be found in publication. The main

purpose of this paper is to construct a Smarandache multi-space model for

biding, establish an evaluation system for bidding based on those ideas in the

references [7], [8] and analyze its solution by applying the decision approach for

multiple objectives and value engineering. Open problems are also presented

in the final section of this paper.

Key Words: Tendering, bidding, evaluation, Smarandache multi-space,

condition of successful bidding, decision of multiple objectives, decision of

simply objective, pseudo-multiple evaluation, pseudo-multi-space.IE: F22.

1;U)t+200607-112

220 \3|lzz – ySE+Z Smarandache J11. XBgp`CFWvO^l:Kv=qm\QbWi|iD ^(T*i|WT,1999 8 3 V 15 *Wv,UO#g| eEARYmAQAR,|QbAR,AQ,|YW8IAQvO^d%% ~1oO^D ^Uv !^m\nvD ,D m.D <KOvP V,7|?W|YW8fTD m,aDmQ^/D [:D <KWi|i^(T*i|WT,1999 8 8 V 30*W4B.%v||OFl2_,`5gSmvq~fj.k|,W|gqQ^< ,MuvOm4|ugQbu9vOm,.uvDpuQ^(zvmOF=U>r^4OF|Q^(z,alL2Q>&|, P~BXR|2tu^4=9v ,g/.mvm# me%WvlLD^4,l2_,`5gQ^UIv_,W^4uv`= 0+u, 3.4.1 SmlL6liA|

E 3.4.1Wi|i^+LOW|eD t:O2(1*1r,T,4Q#`tWvS<D[4

3 o. 26|(JZyS/gI 221(2*1rQ#`tvURpEY,'BC5v[N,|4ag[N|&~[v2! vt(1*(2*UO#Xg rov <*f` I5'`!O PG=êW0UlvuMgO^G uM,243[Ns5, Pu3AzUp|F+mEvÆzs5rR, 3 rovmû|jn,2d3G pJv m^/3vds,DuÛO#vQ|MR|bW,RC5v,|[ûLjv,|[^bWB2<!^G uM2 : gJe[?p,X 3|VJ5ReUWXJeGds,=n< W~G$WG$$_,Be^4#vtzmO^ v[uXe|:5~`vrE ve&Qb5_:ee\vt [7] W Smardanche ` `< [8] Wvh,=G pJ[?`<,zmOÔv[uvh,'6l`<|Uq, YO^ v[u,6RQ#UO4`v|E~`W` v1r`( [7],NlpJ<mv1r [1]-[3] [6],W^4^mv1r`( [8]2. !zCzG6#Qbqv h _V%7,Smarandache & 1969 8q` v2:([9]-[12]*,!O2:h2W` `4 ^,[h2 3O#L)(v Smarandache =B([5]-[6]*,`=&k/</iy`([7]*R5L`4 vORUO,N#)6=on[uvhXQ 2.1 A m- I0 ∑,% m A|* A1, A2, · · · , Am ,

1 ≤ m < +∞,∑

=

m⋃

i=1

Ai/vA|* Ai *% n i = (Ai, i) A xIY,n PMx,1 ≤ i ≤ mf, i 6= j, 1 ≤ i, j ≤ m,! '2OEY Ai

⋂Aj = ∅,.< vl 06b<ÒR< on Smarandache ` hYOR< A W-K m R[< A1, A2, · · · , Am,dR[<

222 \3|lzz – ySE+Z Smarandache J1Ai W#-K ni R5A ai1, ai2, · · · , aini

,! ,1 ≤ i ≤ m= ,!R< LA =

m⋃

i=1

Ai,W,~f/ i, 1 ≤ i ≤ m,

(Ai, i) = ai1, ai2, · · · , aini|iLO^Wuf, Ai ⊆ A aij ∈ Ai 2,hh'`< A`4A aij . Ai, 1 ≤ i ≤ m v'eY/< k, k ≥ 3 R| R1, R2, · · · , Rk BW,|

Rj, 1 ≤ j ≤ k vQ+gRj(A) = Rj

A1

A2

· · ·Am

=

Rj(A1)

Rj(A2)

· · ·Rj(Am)

. Wi|i^4/^vnW|m^UR#|EQb| R1, R2, · · · , Rk v R1(A), R2(A), · · · , Rk(A) ,\ni1, i2, · · · , ik,! i1, i2, · · · , ik = 1, 2, · · · , k,XtJeP

Ri1(A) ≻ Ri2(A) ≻ · · · ≻ Rik(A),R Ri1 , Ri2 Ri3 XgW^4^nvQ$mvWY|XQ 2.2 Mx| 1, 2, · · · , m OIrT Sn dhy3XG9XG2(i) C rT P ∈ Sn,(1, 0 · · · , 0) P4(ii) rMx s1, s2, · · · , sh ∈ 1, 2, · · · , m, 1 ≤ h ≤ m,rT (s1, s2, · · · , sh, t, · · ·),

(s1, s2, · · · , sh, l, · · ·) ∈ Sn,(s1, s2, · · · , sh, t, · · ·) ≻ (s1, s2, · · · , sh, l, · · ·)n/vn t < lr xσin1 NAG#, σ1 ≻ σ2 ≻ · · · ≻ σn / σi ∈ Sn,G# xσi

n1 A9XGG#r xσ ≻ xτ, xσ GDQ xτ4r xσ xτ, xσ GK%Q xτOr xσ xτ/ xτ xσ,1 xσ Y xτ [G

3 o. 26|(JZyS/gI 223hhto!R&PvOYpXK 2.1 |* O1, O2, O3 · · · OG|,rC Mx j, 1 ≤ j ≤ k, Rj(A) ∈O1 × O2 × O3 × · · ·, UU5 1, 2, · · · , k XGo` i1, i2, · · · , ik =

Ri1(A) Ri2(A) · · · Rik(A).U QbY-,~f/ j, 1 ≤ j ≤ k,

Rj(A) ∈ O1 × O2 ×O3 × · · · , Rj(A) `^Rj(A) = (xj1, xj2, xj3, · · ·),! xjt ∈ Ot, t ≥ 1k2D

St = xjt; 1 ≤ j ≤ m,u St ⊆ Ot L,2DsL Ot Li2, St WvJeP,2Gp-Lx1t x2t · · · xmt,u`=Pvm^ Ri1(A), Ri2(A), · · · , Rik(A) 6lP,,[to

i1, i2, · · · , ik,XtRi1(A) Ri2(A) · · · Rik(A). b` 2.1 W Oi, i ≥ 1 Leiv,2D,l,O1 = f, f : Ai →

R, 1 ≤ i ≤ m L,,B t ≥ 2,Ot = ∅,u` 2.1 8XK 2.2 Rj : Ai → R, 1 ≤ i ≤ m, 1 ≤ j ≤ k h!x, UU51, 2, · · · , k XGo` i1, i2, · · · , ik =

Ri1(A) Ri2(A) · · · Rik(A).l,` 2.2 $<Gq 2.1 rC Mx i, j, 1 ≤ i ≤ m, 1 ≤ j ≤ k,L Rj(Ai) ∈ R×R×R×· · ·,UU5 1, 2, · · · , k XGo` i1, i2, · · · , ik =Ri1(A) Ri2(A) · · · Rik(A).

224 \3|lzz – ySE+Z Smarandache J1fe# PW,QbUOQ+6O4Q Ris ≈ Ril , s 6= l PvPm^,uhhX`toQ#W^4ÊYvPRi1(A) ≻ Ri2(A) ≻ · · · ≻ Rik(A),BRQ^$mWY|

3. !z(3~G6e#Oh# #zm[u|Ep`Rds2b/ 12C Mx i, j, 1 ≤ i, j ≤ m,| a5a Rj g5 ai1, ai2, · · · ,aini

~a5T% Rj(Ai)?b/ 22C Mx j, 1 ≤ j ≤ m,| (Rj(A1), Rj(A2), · · · , Rj(Am))t\A Rj(A)?`#Rdsv2 V,U=[u2 v4m^3.1. hy.2!ûvY gLds 2 Wv Rj(A1), Rj(A2), · · · , Rj(Am) /e.m,21=Ov[?6l#&24gEOR6v[N, PEYW|6vA5RJ,Ùp,!GgWi|i^+LOWtKYvQb#O` 2.1−2.24$<,hhÒYpzm< A =

m⋃i=1

AivdR[< Ai, 1 ≤ i ≤ m vPm^,6Rpi5 R1(A), R2(A),

· · · , Rk(A) vP,n4j2L 1 2 T%'( A1, A2, · · · , Am XGd,n m = 5 1,A1 ≻A2 ≈ A3 ≻ A4 ≈ A5 N'( A1, A2, A3, A4, A5 XGdL 2 2 CAK[a5 Rj1(Ai), Rj2(Ai), j1 6= j2, 1 ≤ i ≤ m,T%Rj1(Ai) ≈ Aj2(A2) PA A1 7a5Æ',n%n |Rj1(A)−Rj2(A1)| ≤100 ea, Rj1(A1) ≈ Rj2(A1)L 3 2C Mx i, 1 ≤ i ≤ m,T% R1(Ai), R2(Ai), · · · , Rk(Ai) XGdnCp5ZA",Ca5Æ'ZAHw2T%XGL 4 2h9XG`T% R1(A), R2(A), · · · , Rk(A) XGBT%-m[G7\ Rj1(A) ≈ Rj2(A) 1XGo`,n a5Æ'<Dfh

3 o. 26|(JZyS/gI 225y3XG,I:w R1(A), R2(A), · · · , Rk(A) f?XGdRi1(A) ≻ Ri2(A) ≻ · · · ≻ Rik(A).f, #Wr`=kf, ω(A) = H(ω(A1), ω(A2), · · · , ω(Am)) ω(Ai) = F (ω(ai1), ω(ai2), · · · , ω(aini

)) vm^=p/u|<P&f,von5rOXK 3.1 CQA15( A,rGwa5af?XGU Y- k S|BW,uP^hhtoPRi1(A) Ri2(A) · · · Rik(A). j' Q+ Rj1(A) ≈ Rj2(A) PvPm^,hh,\to"NP

Ri1(A) ≻ Ri2(A) ≻ · · · ≻ Rik(A). R 3.1 RI< nR[< ,QL A1 = o[4A2 = AmE4A3 = |:8Y,< J,,L A1 ≻ A3 ≻ A2e |Rj1(A1) − Rj2(A1)| ≤ 150Rj1(A2) . Rj2(A2) h/ Rj1(A3) .Rj2(A3) v&V2T& 40000m2 P Rj1(Ai) ≈ Rj2(Ai), 1 ≤ i ≤ 3, Puo[P|o,AmEPVQbhn:8Y,< J,`&G'6lP, 4 ,\P 'DfvBu6lP/< i 4 R| R1, R2, R3, R4 BW,o[ 1| R1 R2 R3 R4

A1(;) 3526 3166 3280 3486 1!,Qb#Mvu, A1 vPLR2(A1) ≈ R3(A1) ≻ R4(A1) ≈ R1(A1).OF A2 v5L R3(A2) ≈ R2(A2) ≻ R1(A2) ≻ R4(A2), PY'5< A3 v 2

226 \3|lzz – ySE+Z Smarandache J1| R1 R2 R3 R4

A3(m2) 250806 210208 290108 300105 2PL

R4(A3) ≈ R3(A3) ≻ R1(A3) ≈ R2(A3).!,QbPvBu,[PLR3(A) ≻ R2(A) ≻ R4(A) ≻ R1(A).Y-[< vL A1 ≻ A2 ≻ · · · ≻ Am,u!^G [uu`=#5J^nP=<Wv1,n,\PvURz &ekv G[A] WnO K| R1, R2, · · · , Rk v?-,R!&|2GvQigL|jv

R2(A1) R3(A1) R4(A1) R1(A1)

R4(A3) R3(A3) R1(A3) R2(A3)

R3(A2) R2(A2) R1(A2) R4(A2)

- -- -- -

-- --

E 3.4.2XQ 3.1 CAL k Aa5a R1, R2, · · · , Rk _#a515( A =m⋃

i=1

Ai,%Ah G[A] nd2V (G[A]) = R1, R2, · · · , Rk × A1, A2, · · · , Am,E(G[A]) = E1

⋃E2

⋃E3,~ E1 H$LL' (Rj1(Ai), Rj2(Ai)) _, 1 ≤ i ≤ m, 1 ≤ j1, j2 ≤ k /

Rj1(Ai) ≻ Rj2(Ai) w*G,r Rs(Ai) ≈ Rl(Ai) ≻ Rj(Ai),[1 Rs(Ai) ≻

3 o. 26|(JZyS/gI 227

Rj(Ai) $ Rl(Ai) ≻ Rj(Ai)4E2, E3 *;',~ E2 H' Rj1(Ai)Rj2(Ai), 1 ≤i ≤ m, 1 ≤ j1, j2 ≤ k _, Rj1(Ai) ≈ Rj2(Ai); E3 = Rj(Ai)Rj(Ai+1)|1 ≤ i ≤m− 1, 1 ≤ j ≤ k_l= ,l 3.1 v?r 3.4.2'e,hh|Ee 3.4.2 WnO R2(A1) R3(A1) L ,C K|v?-QbP,hh R3(A1) Y,C A2, A3,,[oP A1!Xto R3(A) ≻ R2(A) ≻R4(A) ≻ R1(A)

3.2. >y.2 pJu#&ds 1 2 v`^V2L 1 A215'( A1, A2, · · · , Am ~,(?A'(IDInAE*M151,(?Æ'ID4Q15~,(?olIDL 2 A215'( A1, A2, · · · , Am !L'e7,IQZA9C$L'(y3I,yQrCLxCy3IXQl<*'ID(5nm~ :15'IY~zAw~t[~$I5'`3.2.1. ,n!^m^`YvO[< ,,eM[< L Y-<

A =m⋃

i=1

Ai KYv,[< L A1,u|EM[< A2, A3, · · · , Ak -K,|v#,r[e A2, A3, · · · , Ak v, -L R1, R2, · · · , RlW R1(A1), R2(A1), · · · , Rl(A1) 6lP,6Rp$mvWY|,l[< A2, A3, · · · , Ak =DNM5=f,vm^ A2, A3, · · · , Ak6lR|,-KDN#.,r[QbDNv R1, R2, · · · , Rl [< A1 vQ+6lP,'M Ri(A1) ≈ Rj(A1) P Ri(A) . Ri(A) vPm^Qb` 3.1 hhto`XK 3.2 CQA15( A,h(5IDwa5df?XGR 3.2 -ImE< A`Y-ImE,L ,i|L A, B, C,

D, E ItRh|F z-IdQ#Ovto[X&Vvbz[v40LDN,'e-ImE[h/ PC [|

3 o. 26|(JZyS/gI 229| N1 N2 N3 N4 N5 N6 N7t| 96.70 91.27 79.12 84.16 94.27 73.84 89.68 6PL2R1(A) ≻ R5(A) ≻ R2(A) ≻ R7(A) ≻ R4(A) ≻ R3(A) ≻ R6(A).

3.2.2. QD[,!^m^Y-[< A1, A2, · · · , Am eORU\ [a, b] #JeOv4#,BR`nL 3.2.1 vQiY ,2Gp,hh`Y-O4L ,B/e vL(Ai) = fi(), 1 ≤ i ≤ m,! fi ^,,u'evXn l

max

F (f1(), f2(), · · · , fm())min

F (f1(), f2(), · · · , fm()),! F ^< A .[< A1, A2, · · · , Am v,`gf,,.p,

F (f1(), f2(), · · · , fm()) =m∑

i=1

fi()g^j` 3.2,hhto XK 3.3 rx F C0 [a, b]UIe,yOv% Ri(A) =

Rj(A), i 6= j 1 Ri(A)Y Rj(A)XGo`,<*'wa5a Ri, 1 ≤ i ≤ kC15( A a5d Ri(A), 1 ≤ i ≤ k f?G 3W0P=vQ|MR|C5v,|[^UO#rg!O[uveQ|MR|W,hh.p,F (f1(), f2(), · · · , fm()) =

m∑

i=1

fi(),

230 \3|lzz – ySE+Z Smarandache J1B 0 ≤ F (f1(), f2(), · · · , fm()) ≤ 100, fi ≥ 0, 1 ≤ i ≤ mReC5v,|[^W,|M[< A2, A3, · · · , Am vqVr=[N A1,7fi = (li − l0)(A1), 1 ≤ i ≤ m,! l0 g[< v[# , (A1) gXqVW[A,R=vf,L

F ((A1), f2((A1)), · · · , fm((A1))) = (1 + l2 + · · ·+ lm)(A1).l,eRuspABÆ|YvAVO?P,yW[N 10,7 (A1) = 10 ;4. FCg0'!Oi<k&[< 5A#vf,,hhxUkXQ 4.1 CQAL kAa5a R1, R2, · · · , Rk _Ya515( A =

m⋃i=1

Ain'( Ai = ai1, ai2, · · · , ain, 1 ≤ i ≤ m,rUAIx ω : A →[a, b] ⊂ (−∞, +∞) n ω : Ai → [a, b] ⊂ (−∞, +∞), 1 ≤ i ≤ m,= Mxl, s, 1 ≤ l, s ≤ k,Rl(A) ≻ Rs(A) n Rl(A) ≈ Rs(A) 1L Rl(ω(A)) > Rs(ω(A)) nRl(ω(A)) = Rs(ω(A)),n< Rl(Ai) ≻ Rs(Ai) n Rl(Ai) ≈ Rs(Ai), 1 ≤ i ≤ m 1LRl(ω(Ai) > Rs(ω(Ai)) n Rl(ω(Ai)) = Rs(ω(Ai)), ω 15( A n'( Ai, 1 ≤ i ≤ m ALxf,g5< v# &UO5W|vg,v,QbG pJ`<([3]*,~f/ i, 1 ≤ i ≤ m, f, ω(Ai) OJe4aOY,[< A1, A2, · · · , Am v[?2ep,uf, ω(A) 2Je& ω(Ai) ``^sm^2

(1)C0 [a, b] f?h!x,nu7x4(2)C0 [a, b] !LIeIx,n9ux2 Fxhh`Ll,f, ω 6l6O4|4.1. jzEB- A1 Lo[,UO5WC 5o=m^I=f, ω(A1)x

3 o. 26|(JZyS/gI 231I=#[ S,S =

R1(A1) + R2(A1) + · · ·+ Rk(A1)

kS =

R1(A1)+R2(A1)+···+Rk(A1)−M−N

k−2, k ≥ 5,

R1(A1)+R2(A1)+···+Rk(A1)k

, 3 ≤ k ≤ 4eP= ∗

S = T ×A% +R1(A1) + R2(A1) + · · ·+ Rk(A1)

k× (1−A%).! ,Ri(A1), i = 1, 2, · · · , k ô[,k L|vR,MN |L,,|o[,T L[N,A% L[Ne#[WKfùf, Ri(ω(A1)), 1 ≤ i ≤ k vI=e\L

Ri(ω(A1)) = −ς × Ri(A1)− S

S+ ζ! ,ς L| ,R ζ LORX ω(A1) ≥ 0 v !OI=m^Y-o[hg0W|3,<=Ìvm^`=,:UO[NvL|o[v=~raUO#,Wvo[2gO^?%dt,o[ X`|vrLAA?,!Â?eGQi21=m^=PK`B pJ4,!^m^2 gOÛRu pJuQb|v[NAp`lQ,`=f,2

(1) \ [N, M ] #v.pf,Ri(ω(A1)) = −p× Ri(A1)−N

M −N+ q,! ,p L| ,R q LORX Ri(ω(A1)) ≥ 0, 1 ≤ i ≤ k v !^mêo[5PY[N,|

(2) \ [N, M ] #vu.pf,,Ri(ω(A1)) = −p×

Ri(A1)−T+

k∑j=1

Ri(A1)

k+1

+q,

1K12012 7 2 U 1 ;JuTHksK[8B< l

232 \3|lzz – ySE+Z Smarandache J1Ri(ω(A1)) = −p× Ri(A1)− k+1

√R1(A1)R2(A1) · · ·Rk(A1)T

k+1√

R1(A1)R2(A1) · · ·Rk(A1)T+ q

Ri(ω(A1)) = −p×Ri(A1)−

√R2

1(A1)+R22(A1)+···+R2

k(A1)+T 2

k+1√R2

1(A1)+R22(A1)+···+R2

k(A1)+T 2

k+1

+ qz,! p, q )k #3Po[vEqV,uu`Yo[v k + 1 G<\.,7Yo[L k + 1 G<\f(x) = ak+1x

k+1 + akxk + · · ·+ a1x + a0v?,r[Qbo[4[N|ov,-L Rj1(A1) ≻ R(j2)(A1) ≻

· · · ≻ T ≻ · · · ≻ Rjk(A1),Q k + 2 R c1 > c2 > · · · > ck+1 > 0,l

k + 1 > k > · · · > 1 > 0,m(Rj1(A1) = ak+1c

k+11 + akc

k1 + · · ·+ a1c1 + a0

Rj2(A1) = ak+1ck+12 + akc

k2 + · · ·+ a1c2 + a0

· · · · · · · · · · · · · · · · · · · · · · · · · · ·Rjk−1

(A1) = ak+1ck+1k + akc

kk + · · ·+ a1ck + a0

Rjk(A1) = a0Rto k + 1 G<\ f(x),6RnP4t|m^af,o[eUO5W[qV,K`=!^m^ |EGTvo[qV29P

4.2. teEB-mEv[< L A2,5AL a21, a22, · · · , a2n2U;ORmEvÆgÆ1+v,!2ag|R(,VG<mEv5A g2v,K` =5A!*2 |?v3^UO#2bv,sLhhL+kNLR_ORmEt 96 |RORmED1t 92 |mEv5Z=p5ep# #v5,f, Ri(ω(A2)), 1 ≤ i ≤ k`bL.p,

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

3 o. 26|(JZyS/gI 233RdR5Avf, ω(a2i), 1 ≤ i ≤ n2 Z=p5# #v5,2 v:2 vf,l< vmE5i- 4 R5A,dRA|L A, B, C, D +R:,t| 7 K^5A a21 a22 a23 a24

A 4 2 2 1

B 3 1.5 1.5 0.8

C 2 1 1 0.5

D 1 0.5 0.5 0.3 7OR Rivp5L Ri(ω(a21)) = A,Ri(ω(a22)) = B, Ri(ω(a23))

= B,Ri(ω(a24)) = A,u/|vmEt|LRi(ω(A2)) = Ri(ω(a21)) + Ri(ω(a22)) + Ri(ω(a23)) + Ri(ω(a24))

= 4 + 3 + 1.5 + 1 = 9.5.BP4,DE5mEv:,j#OvP#5J^X`nS|mEvPer,e!:3|n5A v,2 v< ,mE5Avg2Ov5. k*K+rG 0Wi|i^C !8GvUq,\CeW3^Mnz-mNOT4,a^?^4vUI,G0p+OLeUO5Wpvds,24P6lg,en.^4[vwpR! ,hh=Rds6lOL4|.i<b/ 1: yjK>3KE0 oKY,gLD `|vw9R2g v,!: vXe^4O#gL|Nva:8vUq,W0TO^`i\Wv0,` w WvD `v[e,=n `~WTMGKE7Kp6!Rds,Qb^4X,VN.D `vds,W;||OFF_5gv^Mnz-4J`<vÆ,SI

3 o. 26|(JZyS/gI 235j,a!8=rWi|i^vUq,!gO<dRg v53`[1] G.Chartrand and L.Lesniak,Graphs & Digraphs, Wadsworth, Inc., California,

1986.

[2] u<,G pJm^,m MxQ`h, IV m 77 p,1357-1410,iWO\,,1987[3] P.C.Fishburn, Utility Theory for Decision Making, New York, Wiley, 1970.

[4] D.L.Lu, X.S.Zhang and Y.Y.Mi, An offer model for civil engineering construc-

tion, UmMf, Vol.5, No.4(2001)41-52.

[5] Za, On Automorphisms groups of Maps, Surfaces and Smarandache geome-

tries, Sientia Magna, Vol.1(2005), No.2, 55-73.

[6] Za, Automorphism Groups of Maps, Surfaces and Smarandache Geome-

tries, American Research Press, 2005.

[7] Za, Smarandache multi-space theory, Hexis, Phoenix, AZ,2006.

[8] Za,~ ME(*M15*Yl~H –Smarandache I015.,Xiquan Publishing House (Chinese Branch), America, 2006.

[9] F.Smarandache, Mixed noneuclidean geometries, eprint arXiv: math/0010119,

10/2000.

[10] F.Smarandache, A Unifying Field in Logics. Neutrosopy: Neturosophic Proba-


[11] F.Smarandache, Neutrosophy, a new Branch of Philosophy, Multi-Valued Logic,

Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic), 297-

384.

[12] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued

Logic, Vol.8, No.3(2002)(special issue on Neutrosophy and Neutrosophic Logic),

385-438.

K 3 (m – U04Ya8\vX, 236-247

r!zÆ*d:1~F 04YaML2+Vws16 aP+uÆ&o`4Jus+xPC+XmI`OSd8vAvX+OSCm^X04`4h)-vv04YaÆ04`04bu^ 4`z4kb2Q,/PyIH#vX+POSd8vAvX+r404YaÆ5x+:ÆM5l04Ya:j~s1+B+)B04YaÆ~04X`404Xz404Xtb04`4XKu04`4X,/04`4XbxN 6 vX+B04Ya ~6x20G+5;!BvXEePIg~2+i&s04Ya5;+5l04Ya 03 15u ~a55&Lv3R-0 Abstract The bidding purchasing is a process of consumption choice, i.e.,

bidding in the contracting and then preferential selection. Its success is ba-

sically on the understanding of the relation between participants in bidding.

Generally, the participants include the tenderee, tendering agent, bidder, bids

evaluation committee and administrative supervision. Clearly, the correct un-

derstanding the relation between participants is the base of bidding purchasing

for its economic substance in China. In this paper, we differentiate and ana-

lyze 6 basically relations in bidding, i.e., tendering and bidding, tendering and

bids evaluation, tendering and its agent, bidding and contracting performance,

bidding and supervision, bidding and social progress with their function on

the bidding result for normal guidance on bidding purchasing. We also list

some behaviors of violation on civil ruler caused by misunderstanding on these

basic relations in practice in this paper.

Key Words tendering, agent, bidding, bids evaluation, good faith perfor-

mance, administrative supervision, social responsibility.IE: F21.

1-+2014 7 5

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238 \3|lzz – ySE+Z Smarandache J1Om,| N_,7e.O|/$P,13o*R4Om,| N,7xX%,3d217?,E-1$Lm7, a! _v%7,gW|` *Q,U'D5iv8_M^LT,|eWb< l 2Q|EM`t,nKURpEYt`44/D vrE',h2< gIAEY|N5Tvo[EYkzKURpEYt,`4/D vrE',7 W,|grmX,e`tWnKURpEYt,6REY|eQ#URpEYtv# 6lM.!î\,sLg|qtEY|9,|210nV.v5,. n`t/vD W' tPW KwPv,2|OLQ^|2lvk|, ;MÛW|,l,OL`tW 6MY15*AMFq 3% :,d1+KV!M, ,;MEY|W[|OLW0|hUMm|z4;M|W[N,l,|zBz 5000 ;|vI< ,,[nL 3000 ;||W[,g=#[I=Pv5,`tb 5t*' = La5Æ'pw*e 90%REYW[ 10%,g|W[;MEYCO4[,u2/D Moz4|D 2Q%,lgp`y[*%O2| UW|\R23W|1p,l,ORIlz-L 36 RVvz-< ,`t *[L30?'B*,d1+KV!MzlL,. n|W|/vD 21K2l,sL|M`tD t-KP`fVo.W|g _v,g j[f[vD5,4X?jm,Tn'21U'=q^WL,|eC`tvEYM`t, P,`te`tqvURpEYt59,7%W,|gBX,>è&Q#`tURpEYt,6l KD[,a28,!ÊYg`|” yb” L# vUqW,|`|&~vo[M, D5Ym\sbWzQi,.|l &,Z.21fVoW[.|g” z” v,WIe&4|rLbWw9,RuU'=q,6R=bu^w9bW,2t2Q#,g2Qt7lR2q^|4 '~5,2'"f,7jt2Qtvl+|[,D /PEY|L2W[U*Mt,Z6, < EN|L2W[,g&2QX

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3 o. W26[ :_xZ 243OFgefp7efp,7ei`fp,lX|gV%,gV%QÔ(v"uR|,7l2fÔF|(z,7OF2eefppR,u,OFXV%(zRu|(zOL2%oN~L,OF<qe&Md "$ V[,OF(z4`tv5Æ 2%o)q,mIOL\'2[|B.,iH|Q^B.fjvN~5^,q^=qM4E28, " V[$15a5i*nJa*`Lg%ed|Q^/B,u4OFU'vi!^N~VvBs,e&OL2%or`^LV H15aI BtLwQzn<#7Cb`WaRLwQzDS"hn<15 ~v_"S:ww"h~T%ÛIl+L! 5l "l0nI5"S:ww"h~v$FoT%UR,V!~1'2Wl2d<,E3g25gvR(zv” VI!” |Q< l |E|,u25g|v,G|\Ru25g\e!O #,25gW`%o(zV!v<qe&k5VJ5R.1p, 6m"2p[, X||vVC~5,W~Rv\, P,!eWGuG3Ods,l,Bk5?%bvVJ5RQ#< 5|E,BqV<Jpr,6Re06l5Æ,2BLq VI,#l'l2q~zd<r|6O4$Q yByjn~=q,h2Q.2QR , vgeQtzM.%M,6RU'qCFKm,k5< RvEK`,gQ^6l< 2Qv3Kt, ve&i2QQb,26l,/vD H^,oK42QugU' 5v , 2Qv3gi\,o 50|'L2Qq QbMQ,EY%vB.| 6 &,7 Id<2QW|by,2t1x,LEQlLD ^L+,EQgAM|mD vf%^,/f%ê

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248 \3|lzz – ySE+Z Smarandache J1^,~ EÆ,2013-3-1[4] Za,=qlLQ`<|,15Ya5,1(2013)[5] Za,=qCFK4s|,15Ya5,2(2013)[6] Za,=q< %|.M,15Ya5,3(2013)[7] Zav,15Zb~Vt',WzyJ\,,2013[8] Zaa,15a5`P*1H~l~H,WzyJ\,,2013

K 3 (m – 04Ya'iGXLs, 249-261

!zÆ*vk=7F 04Ya'i+04YauXKu5lYatÆso04YauJ;u:mD'xÆP'4`gJ6~?;6?m+auÆB04Yai88J+mI`gb2uu)w_kEb+QaKuÆ6WbJ6+mI`J%Ku!0~'i)?04`4_NY) + 5'iU+H04Yau#KuNÆG04Ya'i!iz&isi_>#n_+B04Ya'ibLXCRK~6!03 04Ya'iuJ;uJmKuiisi4qAbstract. The risk in a bidding purchasing project appears in 3 cases in

China, i.e., can not conclude a treaty, invalid treaty on laws of China or harm-

ful results in the process, efficiency, benefit or the objective, including insuf-

ficient knowing on the risks in bidding and some participants violate code of

conduct for income themselves by contrary to laws, and also these risks in the

performance of the treaty caused by the nature or the society and bad faith

of a few persons. Consulting the control theory of project risks, we analyze

the cause, evaluation and propose means of prevention on risks in bidding

purchasing project from the whole process of contracting and performing in

China. All of these discussions are beneficial for project management of bid-

ding purchasing in China.

Key Words: Bidding purchasing, project risk, can not conclude a treaty,

incorrect treaty, risk in performance, risk control.IE: F21.=qgp`CFW,jp`v# p5, | M.nI K,6ly|=q,6RU'qCFKm,k5=q< REvOR ,eLU'L ReO,|F-zQ1-+2013 7 3

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3 o. 26[ )kIZNu 259%pvgI2Qm\['Q4SY'2DNR6lvD *4SY'WJev2Q+B,g`ted|/vD WS6l%pvQ4 W|2b2Qv,l,IWv4R< C`rE%U-z2o`zlL,`ted|/vD WS%pQ,XW.2Q2OI4!D 2QWv%,`ted|/vD WS=bMI6l%| z|nW|3,nOF$mvWY|gURpQ#`tEY,Y'2Q#v,7OF`Q^2l^4!*v<q,EYOFQ^$mDNvWY||.W|/D P,mO>W|v`t`tWuURpEYtvq^4,u< 2Q1nv%4,'D Ûd<! |fvg,`tuURptvq^>r`Ûd<,aW|`2|vEY,'B|21`W|2uURptvQEYRa/D ,u|e/v^4q~K`,$L|EYW|Q#vEYt,0n3^g%`tvURpEYt,R21[m/D PÛ(\* y )n~u=q< 2Q% 2 ^,O^g=qQ2eiv=q2Q%,O^g2Q Wrs5|Ls5mIv%!eW,rs5mI%Arpv2unmIv< ;[F|F>,Tp,'Wz4|Ls5mI%A&|Y%mIv< 2Q%,sh22lL%,7(;R|lLvf GlLmIv< ;[FG|F>,W|hH^|hn< R|Æz42U%,72\nN.zz v,%gmIv< ;[FG|F>4CF%,7V,CF2Jn, 4C`2p`=N[N*% Yng5nzmIv< 2Q%4 A%,7< AEYJel+,g.XttIA642OIRmIv< ;[FG|F>,l-ImE2D`,l-In,g.IA642MRoÛI, 4IaA[zmIv< 2Q21;[FGz4!(;%,7< =Rjm|%%jm_X2OImIv< 2Q2MM2np(;s5mIv< U'21,g;[FGz=qQ2e%voge&8EQ2env%s541U=

3 o. 26[ )kIZNu 261&ed|L/B!LP:1Y53. =qlL||/B,lLg,7`nÆ3pr2JR4^4vQPeprCLT o (*,z;L,tL # o aJ *, ?44 24+,zJL,R=q06gO^C,ed|A%v&!^CtvmK`,ged|lL,og=q%v>`e&lL| 6 &vi &l,)&ev,Q%M,e |# #7^Mz-,6R:1GXdJvY,!Gge,`WvOR>`ds2_

[1] Hr,(|~mAmo[M],v?: Wz8J\,,2004 8[2] Za,$%p`2; ?g [N],~ EÆ,2009-01-24[3] Za,3nuM1L,ozp`l%M [EB/OL],v?2?N~V,2011-02-28,http://www.qstheory.cn/lg/zl/201102/t20110228 69927.htm

[4] Za,ME(15Zb~VY6<[M],Rehoboth: American Research

Press, USA, 2007[5] Za,BCFY,oz=q`<u [N],RZb&PÆ,2012-

02-17[6] Za,grulL,G6lJvY - ^p`Jevds4p`^ [N],~ EÆ,2013-03-1[7] Za,=qlLQ`<| [J],15Ya5,2013 8 1 [8] Cz, ~Q2!G`Q<!WUY[M],v?2 \,,2010

K 3 (m– V)+-UZÆAzXÆunze, 262-270

[%##%t<LkMu(,F ,BFHN<Æ~ey~x+jXBZ0`"ÆmqPX;5O7Z#=V=,p+5l`X6W Y"76)Y+)ÆPn_+sPL`)^DV~1X~x)ÆP-+D,Vzz&f4HX+f, PPP :jMN<Q_; LX!-03 W*,.W~m I'IYPPP ;kAbstract Chinese urbanization has obtained great progress, promotes the

economic development of China in the past years. However, more urban dis-

eases also appear in the meantime such as those of housing, traffic jam, pollu-

tion, flood, mental disease etc. problems in the rapid growth of urban popu-

lation. Then, how can we realize the developing of human beings with that of

the nature? In this paper, we establish a dynamic model on the operation and

control of urban by the notion that human beings should be harmony with the

nature, and also present the evaluation index system on urban governance of

modernization. It also points out that the PPP is in fact a market mechanism

for the adjustment and transformation of government of China in this paper.

Key Words Dynamic model of urban operation, urban governance for mod-

ernization, evaluation system, PPP’s objective.IE: F22. $Q`dsLm?ehh$QORdsvPY,ozOR D`vh6l$Q,g$QvxE5&0vTWhypeozORR_v h6l$Q.7Bp/e5<W[ORyp,ypgOR%Wv<W,hh`=<W%p=$QyplWvOds,6Rypvz-l6l`,U'|.rMYKZvy)n,Xg|T?yp2!OR !Xenyp|12016 7 11 U 26 dVzX PPP Y"U/"unD

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266 \3|lzz – ySE+Z Smarandache J1m?,`G6,e~0k,y6|ZLY ,sy;,p,`pSmuM%M6,Epz-Wl,rkdJZWjv3F2015 8W7yp5oUO#gehh d 10 G8ypFsYv6N ,ypYgCI|?yp2`CJC/nLypz-v ,|En ,ypE ItIb1p/h,!RtIb1pLG8`=vTVTBo,[, V.`ypY4,60ypYAm%7AU36yp5!v`::gypYv0n`:,>`eR0W d 10 G8ypFshYmIvyp%,sLyp%eR=RWj2015 W7yp5oUhhAypYm?,$.< PhhXORds,ypgORR_vlW,xkl-7Bp`4<W,ypgOR%Wv<W&,2015 8W7yp5on ,yp5gOR,E`|LW`vY%7,Ym?E`2QY,),,,SJ,3y,qR,Em#Q,.`r,r,kdr,1yp<WtI,!Rq^vUO#gEY.rMYyp5LOR%Wv<W,6,l,|pS?,u<|v, P,uOR6,7|Y<[<vyy'y5!,h2y&uzy'yÆZ Æ- E 3.7.62qYOaY7X!Z#7B #,UO#NXgEYyRyQ$_,|Y%x_.ryQ7er,tN|Y%rtIvD%4!L8,LG|e$Q<WtIoORyp =kyp!R<WvPY,yp\3`AORvmv=B ,NvX2O/z,agNvDgOIyQv,z& 0, 3.7.7 K^

3 o. X+-/W\C|Zwp|g 267

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.$_,2qNWQ7TV+8p GDP yd5,NgO^bW[yUO#,TvCFbg21Wv,sLb: gO?/,WD`[XG 2Nv2: .8 6.5% gyd,W23X: vr?/vUO#,yd5/Q#|.rvMY$_,|QN2q,|BF7UO#,yd5X/yp!R h −→G YE ,!XgQ#|.rMYtvCFyd5e!^ypU` ,XO2'Tv, zM2d,2'Tvoy'en[p`#:K`'T!J, RCFYm\gFsv,2Q#|.rMY,2Q#Zz|Y. n Uv!^ryQ,&r,YCFvm, 8|.rMY!O#~`:KIôypei|mu%M. PPP ~D5vypU`,x,|CW7ypovA4 2015 8v,EirQÛm%,Q^lz-U`yp,G6ypUùU`1p'Wn,EviQ^pJvuM%M,LebB.V<5%uznLyp`TpJv^,E7ypUùqypU`1p,ppyp%vds,2;q=yptIR|<vRW`` 2014 8Z3&W;Wl_hN!z-vfr(W`Y[2014]65 9),EY7`TpJfr-2M4,+4eijm|pbA1jmvpJd<,E _l58^<5zGî\,jbN!vfrzo,y;pJ4ebB.4,xg|TW,2MOF,2Bd, W F.N!zD5$Q,bIzmpJ5gN!frvd/%MEzmvi2JuM4,+u4^4^2, T1LmE, T2JMI, T< zpJd<S3,E6llp<5,btICFzmv%u, P, r2 N!uoEvD|Æ,W;2J>lQ+,UIK,9vu,zm5gN!ufrvd/ezzM4,vipJRN10%M,bI205u.N!mu/Dv

3 o. X+-/W\C|Zwp|g 2692Ju\,y;uvp p! vmu,AvgCW``# `tA4mvV_hN!vuypU`'Wnu+4`Rm2ORmgypLvuhhB=OLL`,lg2=`,jvj,z-vz-,L|CypU`'Wnv|Ezu,6l1L4WRypvDDuypv>`gypuQ#y6|ZvEY,u~1gSu: /eD,i $6ypvY,U'ypEl,ypOL#~vud<:

1. ypYORypg2gY,`g2gY,!eWv>`XgIzY.p/n 42. ypgUù.1pùM,'eO ùM+g2ùMypU`~1vg,iU,72,eiU!vORuL<,xgypùg2giU,lg,evB.pJgto|Y,7uQ#2Q#4guWvdR|dR%oB.yp`g2gto|Y,ug2ThTB43. ei| UK5yplW,2g &+,ugypvr,A^QVz,7 UK5Q#2Q#,e|E44. ypvpJ%M,K5445. !Jg,vd/UO#,2 ypgQ#.r/MY!vO^-o2 yp?vl ,Ezm!W-o,7h,6lb|,'QypY6l%W/,zmypY_l= ypYA.ypzY0 Q# yQgyRyQvypYWvh/b`C I$`tr[,Ì=dOR dOR[$#v( R(v) '/Rypv(W2

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270 \3|lzz – ySE+Z Smarandache J1r[,I=yp NHI ANHI

(−→G)

=η(−→

G)

∑

(u,v)∈E(−→G )

f(u, v).

NHI ART ypl.rRR2M,NgOR)AypU`^2ziU,$Lr^2ziUTV8p,hh'eeV.p$v2,~D5,g~D5 PPP %M,1uX!JD5$~D5gORR_%M,~D5UO#gOR p`%Md?MQ,=|$Qyp/RU`uU`1pQ#2Q#EY<3;25Uv PPP < tRtLTR4joPPP \UO#gO^p`%M,NvEg$%2<12x,PPP ~1gO^p`jvm\,#L|E2sf,#Lg2X,~D5Zj&p`Y,UI!O^%M[`$%2<1,lN2.p`v#,PPP $%2ql2K5ql21p,Xgj2v`%o,OR%o`vd<`.,~D5q ,X` 2qK|=q , P,Aj2e|F,q|Fv<J5Rpr5?,r[KU`)l2` 2vRXlXNXzds,q`dK5,sLp`CFxEvgK5 gvei;2v7p,6O4l'2`Czv7,lKei;2p,BR|i<dYvei;27,6RyWei[|v U1p,!:gV= PPP %Mv[,7U'VU`uU`1p'Wn

K 3 (m – UZrtXÆARDe(<~Vwus – 1)P, 271-280

#%#kt Gb7 – QqL_F HÆHs(<~!#PUZrtXÆA~RDe(<+)B?(<~Vwus+ PPP 1)PX2QzlbH+ )=|#ux2\oG+D,VwD+Z0+f,l2_Ns8 +?s8Vw~P)~Bwi03 UZrtUZÆAPPP usVwiAbstract The essence in the deepening economic reform of China is the

supply-side reform on public goods and services in recent years. In this paper,

we discuss one of its market mechanism, i.e., the PPP on the administrative

governance combining with Chinese characters and experience in other coun-

tries, present some suggestions on its market ruler. We also remaind risks of

PPP’s operating under Chinese laws update.

Key Words Public goods, public service, PPP mechanism, market ruler,

risk.IE: F21.~D5gDKMvO^~'iW,h2ei~.ei~ei~.&~ &~.&~,7!J ~D5W,ei~ ei~.&~,#wLei~.,~D5rE&# -IedJ3,EgYp`%Mqei[ei| U1p.K5,Q#|pbmydvyR`n|E,0n$`2p`v,eOL~rkV`lvC6,a!LV2uCFuM`n.h~RVq,4v *h2O1r!K4v,&~D5gp`CFW!J rED5\,o|GoL<,~`DhCFuM1L,~D5%Mv`:3^,DW|`ÆÆ,|e$%ei[.| UL1L2<1Wv&.5hgI4gd<,bi1)v,%4ogMI,6R7l2oU`,G6Ktv^U2z-1-+2016 7 10

3 o. W\tvZCTGg*>Xywu – 3+R 273WT~emp!Jv,~Cvr$;lJ,EUl R~0R2~0>E2R-|,$6eiI Kp`n,Sî\vl2$;!mORhl,XgBWi|ip5q'[,[i^[(zvW-e',i-6lz-,7L >E24 p2e-Iv1L,h2,lPs-wp->Doz-I,1L3g” ”,1L[z-Mdz%lUORR!J,aefvLI*Wi3,ei;2\i7x;K<QVU`'Wnv>`gozOR/eM,XE5 E5./u M, LAm,|M5,,\U'E5p1,Tn`1,nfvLI*WiA,i3n1Lv g7YW,rkM4,$6VUùU`1p'Wn,h20n$`V.,2.p`p`., v,>`g0n$`2p`v,Z6pèI KWvpp5, 425U<1,Tl'2I= K,$%Cpù[NM.U'Km,TnK5,n0n$`2p`vgozOR7vp`CFuM3q1Lzs`=,2.p`vdseO4#\toKU`,ei[.|v UB2!Jq ,mxL7'W!JM41Mv!J UaPH.,pùu27,p`M.2|,2fpT5kd<`,u^4=Pwp`CF%p`_2oXz w$Q$W$dsÆ,<JprlGGblL"`,5gm"ujmkdz9CFllLTJe,2èO4#A[&,Y,21Q#V'WnY|EL<,2Bp`CFWOLEd|Wi^=, ” KL,K2L”,pqpÙ.`~L<,WiW7&i3n1L4`TdsvpW,2<1U1Lm?,7E #'Z#&SGRd5*t%$6*,#'Wxi*-|,#'CxV[BDS,L<,|E6O4W%2<1,,T,4l'2Ed|`,lgp`%M1rKvp`ru1rQ^rpvlJ(;W'%o1r4`v,+e&, P,n%o.<1,7pJ>l_,%M,ozÛ20n$`2.p`,lgei[.iei| UL1L,URg$Lr$;lJUl R~0R2~0>E2R-|,3n

276 \3|lzz – ySE+Z Smarandache J1 -IedJ< ,Gl\Ov^4^<,Ei<~D5M4-I, 7h~D5p`u1. ~D5m^~D5,lgei~.,~D5,~D5Sm,h22q ei|v!J, 4,~Wv!Ju!JlLg, MdVjm,eijmed|D^fm'l^4gW,'`gv~D5^hUqW,2.,~D5OY| 3 Y,7lC2qK|fD5UqW,x\|OR~D5glC2qK|ugfD57&z-< R(,!Rds6QUO#,ORz-< i<dh2< z-oyzt~D5rLD5,R[z-zt`h),G`2h)eD50l,m3gD5,uUO#Xg< fD5lCvUR,e&2EeOP0lU,~mCf,x(:,)tv~D5XglC,d:u21wLlCRg2qK|l,'l~D5(PPP*\W,O&MTOTBOTBTOBOOTROTDBFO zD5\W) O,7t,+glC,R DBFTBT z\2) O,2glCRg4v2qK||Efvg, >E2e2'2gOR PPP 2:RI01,sL0OLr$;lJUlv R~0R2~0>E2R-|1LUR,XgL2q vei[.|!JlC,'C~vmOl#ei;2`k1<,!e# -IedJ3T+\C71LUqW,.<OR~D5glCug2qK|'`fk,sL~D5v ve&D5Sm B. [42.,~D5Wv e&yWei[.|,Q#|pbmydvyR`n|EaB;k#,2qK|&=q,7Czs,RlCg2lU,~m6l< Clf,W R. 2o$e2~zX,2j&~D5,lgei[.|v U.mfmk<,LkdVjm,eijmed|D^fm,g~D5lL,LEDW`nCFuM1L,2q ei[.|v!J,~Wu!J,`4D5d<fmkz6lm^,Z3

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280 \3|lzz – ySE+Z Smarandache J15. G6t2z-~D5\e%,~&.p,$%CFYv P,G)OL|Ez``rds,l,Bog'm2.,~m6l LZ?g yQ,BogL,~mD%Vjm,eijmv 9g la,og=Dî\,3u^ v7L<,D~D5!OlCF\4W`nl ,WS 6l=o!,ozÛ2Û,~D55LDKMe~OO^'iW,gh#~CFM4vÈU'i\, P,Gg3nCFuM1L,0n$`2.p`,YpèI Kvpp5,$6VUùU`1p'WnvO^Èw9,L<,DhCFuM1L nl ,B~rkVC6$Q.g,Ìl4js UI,`6O43nCFuM1L,G6CFvY

K 3 (m – 2 PPP )i)H, 281-284

( PPP*?g`RF: )k!zus -~ PPP )+~PX)H~vX+m PPP ~PYa)Bwi+=8M_Ni+&_Ns$;Q#`1`v03: PPP )PPP Ya)HiAbstract In this paper, we introduce the PPP’s intention in the national

governance of the United Kingdom of Great Britain in 70s of last century, dis-

cuss its operation with that of Chinese culture and bidding risks, particularly,

those risks on Chinese laws update for recalling the government and investors.

Key Words PPP’s intention, PPP’s purchasing, Chinese culture, risk.IE: F21.he0$Q=L8=q,h2^4=qCF`</t,K`,o<g`<ugUq,h+LÆ3u,sLG.=q/vVM4h+B. Ig<5^o~D5 PPP,h7 Rds2Og PPP v[gR_,Wg PPP =q4%ds,g PPP .W`ndsx,z PPP v[,UO#gLp2p`vds'ep`#oTGg PPP < ,2o PPP %Moge4R_PPP %MvEg p`%M,0n$`2Xp`vdsTV8p,1Lzs 30 G8,O=e$`2Xp`v#dso PPP,,kg~2e#bRL8W5Y ,[gR_.7ePv~x/AU|(;O V,g$Q2.p`#ds,Mh$Q$Lds.7hh8p,e)`\ei[.|2q $Qm<`eP~2q vei[.|,ddg2g2=q 7Mh$Q[l|YOYg2q ,!OYW#|Y,OYg2Eq ,uOYg` 2qK|m\,p2 U1p2#p`K5|ds,p`ruq12017 7 3 U 17 danUZ1ogqvRU/"u/

282 \3|lzz – ySE+Z Smarandache J1 4WYgo|2q ,1p`ruq ,7p`nK`,Wz PPP %M~1gORV8,!R8gR_.7Xg=p`j%M,3nCFuM1L,$%2<1./,BRU'VUù.U`1p'WnPPP %M.VU`'Wnrs06OI,7 ,iUvm\, _`,ei|v Um\er,=2qK|m\q vei[.|,XX PPP 2,2|E2q v,|E2<1,Up`ruq ,2/vH%o|F!gh7 vORds,7z PPP %Mv[gU'VUù.U`1p'Wn,>rhhep`#oTv PPP < ,aUO#,NgO^VU`w9WRds,7 PPP < =q4%dsh$Q=q6=L8,:3zeWM\~sd,g$Q=qe PPP =q ,h2< =q,Bd8zZ,V5gSO`t, PPP < G56lgh~1s$Q< %OR PPP < ,Bz-o,dd,W)Tv%,xg%,u< >l%,uhhH.u 2N^+^%x_p PPP < %7Q^Qg3qhhoLG PPP =q=q2J,;2vG6,Y1vG6,'27iOI,vu/e\Eo=q,W5^4,Og^, Xg2=q^hh PPP =q,OM4-I,J0LGv.<R5OZ3v`th+,J0.<hGaxg,e'l^4M4,121Q^7 PPP < v=q7hLQ^`7,R2g2`,D2 9< >lK5,|E3nCFuM1Le3,!.<g`fkv,|EGB3nCFuM1L,GB< K>l,< %#`, I3F , KLL&o~D5 PPP,hhOR< ,ORz-< ,Bz-oRt,)v`\g$|JPPP \vgi<d`:,EYe!^`:,`_V< =qK<,.'l^4M4'l5kM4'`\E,Dg2U6aho/5OvLG=q`^,hR|L,!L5OZ3v`ê'l^4hgI,M,2gs:+#v!Xnhhe PPP < =q3,TvQ PPP < ,,PPP =qlL.'e^4M4,H^,rROLmre!3,eTpUIhhuQÛ,$5hhv^4l2^! ORds,Xghh5gZ3v `^121x1|Ol,x1^47hL2l,sLWv

3 o. 4 PPP +k+J 283m^^n,X^q21Hd#X^,!gm^v#~BuK`,ehh PPP < =qe'lv^4M4g`7v,Dg2Uj,L5`#,aq21K5R^8^4J0LG|ei<, g5=qM4W!R26,$R26,UO#2g26,g>lK5|ageQÛTwF,Q4^4=q+1U',>rLm26,:J0e?#i<v,E'Ods,u32,[G2,Xg7=!^m\hhv^4M4,!eWTv4evH^dt)W,=nD oK,yTD %,|\H^q~er, 4G|E.Pg6,hhD_ie,4^4Wv/'6l/, VU`'Wn,GXPPP =qZg,|=q%hoR PPP < !,;2OR,< 10000 GR,-L 13 ;d1!,VY1LGOR,< 3700 GR,-L 6 ;dGO !eWORLv'A,Xg< lWTaA5L|,d8Iv,T2|:O1!!Rh`TUg2gn,1gLB=vOLPv BOT < G%o!R PPP IW7g|:O,!RGL|,T PPP < `z,|,BsXe& PPP < nU%PPP < Bz-o,,d`g 30 8hh%=N, OR|!R1p3`y&=oLl,ty`0u=n,k tyvTG2OnK` ,vOR|=N30 8v PPP < %gOt`2ovdQ!eWOR>`ds,Xg< Vds,o#OY< hD3 PPP '`$0top7gf&z< g2gXhD3 PPP G2O,2 v\2 v3Vq,K` PPP < vVdsgs,>è&yK?[,a'eOL5g$vyK?[m^g2gp!Rds7q`p,nTv PPP < yK?[e i\, ÒLm2LPPP < 5LO^_v zw9,hO, PPP , LXL- =m2g!O^%-,Kp2 PPP < UIU=v%&!Rds`p,[rds,Xg%O2U

PPP < ~1R v,Og|i<dv~,Wgqi<dvei|v U1p,!R v:ghhz PPP %Mv v,RyK?[XgEU'!R vXg PPP%M.W`nds3yheVY1O5rOXm, W~# -IOF0e_,AR~OLe,dhLR_`dUFh h2,sL2#RbRz BOT %M ,ugB 2014 8

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Everything with Combinatorics on Nature1

– A Report on the Promoter Dr.Linfan Mao

of Mathematical Combinatorics

Florentin Smarandache

(Mathematics & Science Department of University of New Mexico, Gallup, NM 87301, USA )

http://fs.gallup.unm.edu/FlorentinSmarandache.htm

E-mail: [email protected]

The science’s function is realizing the natural world, developing our society

in coordination with natural laws and the mathematics provides the quantitative

tool and method for solving problems helping with that understanding. Generally,

understanding a natural thing by mathematical ways or means to other sciences

are respectively establishing mathematical model on typical characters of it with

analysis first, and then forecasting its behaviors, and finally, directing human beings

for hold on its essence by that model.

As we known, the contradiction between things is generally kept but a math-

ematical system must be homogenous without contradictions in logic. The great

scientist Albert Einstein complained classical mathematics once that “As far as the

laws of mathematics refer to reality, they are not certain; and as far as they are

certain, they do not refer to reality.” Why did it happens? It is in fact result in the

consistency on mathematical systems because things are full of contradictions in na-

ture in the eyes of human beings, which implies also that the classical mathematics

1International J.Math.Combin. Vol.1,2016, 103-133, www.research gate.net, www.scribd.com,

www.viXra.org and www.academia.com

314 \3|lzz – ySE+Z Smarandache J1for things in the nature is local, can not apply for hold on the behavior of things

in the world completely. Thus, turning a mathematical system with contradictions

to a compatible one and then establish an envelope mathematics matching with the

nature is a proper way for understanding the natural reality of human beings. The

mathematical combinatorics on Smarandache multispaces, proposed by Dr.Linfan

Mao in mathematical circles nearly 10 years is just around this notion for establish-

ing such an envelope theory. As a matter of fact, such a notion is praised highly

by the Eastern culture, i.e., to hold on the global behavior of natural things on the

understanding of individuals, which is nothing else but the essence of combinatorics.

Linfan Mao was born in December 31, 1962, a worker’s family of China. After

graduated from Wanyuan school, he was beginning to work in the first company

of China Construction Second Engineering Bureau at the end of December 1981 as

a scaffold erector first, then appointed to be technician, technical adviser, director

of construction management department, and then finally, the general engineer in

construction project, respectively. But he was special preference for mathematics.

He obtained an undergraduate diploma in applied mathematics and Bachelor of Sci-

ence of Peking University in 1995, also postgraduate courses, such as those of graph

theory, combinatorial mathematics, · · ·, etc. through self-study, and then began

his career of doctoral study under the supervisor of Prof.Yanpei Liu of Northern

Jiaotong University in 1999, finished his doctoral dissertation “A census of maps

on surface with given underlying graph” and got his doctor’s degree in 2002. He

began his postdoctoral research on automorphism groups of surfaces with co-advisor

Prof.Feng Tian in Chinese Academy of Mathematics and System Science from 2003

to 2005. After then, he began to apply combinatorial notion to mathematics and

other sciences cooperating with some professors in USA. Now he has formed his own

unique notion and method on scientific research. For explaining his combinatorial

notion, i.e., any mathematical science can be reconstructed from or made by combi-

natorization, and then extension mathematical fields for developing mathematics, he

addressed a report “Combinatorial speculations and the combinatorial conjecture

for mathematics” in The 2nd Conference on Combinatorics and Graph Theory of

China on his postdoctoral report “On automorphism groups of maps, surfaces and

Smarandache geometries” in 2006. It is in this report he pointed out that the moti-

vation for developing mathematics in 21th century is combinatorics, i.e., establishing

5 o. ySd E+88Y – ySE+u>l)eEE 315

an envelope mathematical theory by combining different branches of classical math-

ematics into a union one such that the classical branch is its special or local case, or

determining the combinatorial structure of classical mathematics and then extend-

ing classical mathematics under a given combinatorial structure, characterizing and

finding its invariants, which is called the CC conjecture today. Although he only re-

ported with 15 minutes limitation in this conference but his report deeply attracted

audiences in combinatorics or graph theory because most of them only research on

a question or a problem in combinatorics or graph theory, never thought the con-

tribution of combinatorial notion to mathematics and the whole science. After the

full text of his report published in journal, Prof.L.Lovasz, the chairman of Inter-

national Mathematical Union (IMU) appraise it “an interesting paper”, and said

“I agree that combinatorics, or rather the interface of combinatorics with classical

mathematics, is a major theme today and in the near future” in one of his letter to

Dr.Linfan Mao. This paper was listed also as a reference for the terminology combi-

natorics in Hungarian on Wikipedia, a free encyclopedia on the internet. After CC

conjecture appeared 10 years, Dr.Linfan Mao was invited to make a plenary report

“Mathematics after CC conjecture – combinatorial notions and achievements” in the

International Conference on Combinatorics, Graph Theory, Topology and Geometry

in January, 2015, surveying its roles in developing mathematics and mathematical

sciences, such as those of its contribution to algebra, topology, Euclidean geometry

or differential geometry, non-solvable differential equations or classical mathemat-

ical systems with contradictions to mathematics, quantum fields and gravitational

field. His report was highly valued by mathematicians coming from USA, France,

Germany and China. They surprisingly found that most results in his report are

finished by himself in the past 10 years.

Generally, the understanding on nature by human beings is originated from

observation, particularly, characterizing behaviors of natural things by solution of

differential equation established on those of observed data. However, the uncer-

tainty of microscopic particles, or different positions of the observer standing on is

resulted in different equations. For example, if the observer is in the interior of a

natural thing, we usually obtain non-solvable differential equations but each of them

is solvable. How can we understand this strange phenomenon? There is an ancient

poetry which answer this thing in China, i.e., “Know not the real face of Lushan

316 \3|lzz – ySE+Z Smarandache J1mountain, Just because you are inside the mountain”. Hence, all contradictions are

artificial, not the nature of things, which only come from the boundedness or unilat-

eral knowing on natural things of human beings. Any thing inherits a combinatorial

structure in the nature. They are coherence work and development. In fact, there

are no contradictions between them in the nature. Thus, extending a contradictory

system in classical mathematics to a compatible one and establishing an envelope

theory for understanding natural things motivate Dr.Linfan Mao to extend classical

mathematical systems such as those of Banach space and Hilbert space on oriented

graphs with operators, i.e., action flows with conservation on each vertex, apply

them to get solutions of action flows with geometry on systems of algebraic equa-

tions, ordinary differential equations or partial differential equations, and construct

combinatorial model for microscopic particles with a mathematical interpretation

on the uncertainty of things. For letting more peoples know his combinatorial no-

tion on contradictory mathematical systems, he addressed a report “Mathematics

with natural reality – action flows” with philosophy on the National Conference on

Emerging Trends in Mathematics and Mathematical Sciences of India as the chief

guest and got highly praised by attendee in December of last year.

After finished his postdoctoral research in 2005, Dr.Linfan Mao always used

combinatorial notion to the nature and completed a number of research works. He

has found a natural road from combinatorics to topology, topology to geometry,

and then from geometry to theoretical physics and other sciences by combinatorics

and published 3 graduate textbooks in mathematics and a number of collection of

research papers on mathematical combinatorics for the guidance of young teachers

and post-graduated students understanding the nature. He is now the president of

the Academy of Mathematical Combinatorics & Applications (USA), also the editor-

in-chief of International Journal of Mathematical Combinatorics (ISSN 1937-1055,

founded in 2007).

Go your own way. “Now the goal is that the horizon, Leaving the world can

be only your back”. Dr.Linfan Mao is also the vice secretary-general of China Ten-

dering & Bidding Association at the same time. He is also busy at the research

on bidding purchasing policy and economic optimization everyday, but obtains his

benefits from the research on mathematics and purchase both. As he wrote in the

postscript “My story with multispaces” for the Proceedings of the First International

5 o. ySd E+88Y – ySE+u>l)eEE 317

Conference on Smarandache Multispace & Multistructure (USA) in 2013, he said:

“For multispaces, a typical example is myself. My first profession is the industrial

and civil buildings, which enables me worked on architecture technology more than

10 years in a large construction enterprise of China. But my ambition is mathemat-

ical research, which impelled me learn mathematics as a doctoral candidate in the

Northern Jiaotong University and then, a postdoctoral research fellow in the Chi-

nese Academy of Sciences. It was a very strange for search my name on the internet.

If you search my name Linfan Mao in Google, all items are related with my works on

mathematics, including my monographs and papers published in English journals.

But if you search my name Linfan Mao in Chinese on Baidu, a Chinese search engine

in China, items are nearly all of my works on bids because I am simultaneously the

vice secretary-general of China Tendering & Bidding Association. Thus, I appear

2 faces in front of the public: In the eyes of foreign peoples I am a mathematician,

but in the eyes of Chinese, I am a scholar on theory of bidding and purchasing. So I

am a multispace myself.” He also mentioned in this postscript: “There is a section

in my monograph Combinatorial Geometry with Applications to Fields published in

USA with a special discussion on scientific notions appeared in TAO TEH KING, a

well-known Chinese book, applying topological graphs as the inherited structure of

things in the nature, and then hold on behavior of things by combinatorics on space

model and gravitational field, gauge field appeared in differential geometry and the-

oretical physics. This is nothing else but examples of applications of mathematical

combinatorics. Hence, it is not good for scientific research if you don’t understand

Chinese philosophy because it is a system notion on things for Chinese, which is

in fact the Smarandache multispace in an early form. There is an old saying, i.e.,

philosophy gives people wisdom and mathematics presents us precision. The organic

combination of them comes into being the scientific notion for multi-facted nature

of natural things on Smarandache multispaces, i.e., mathematical combinatorics.

This is a kind of sublimation of scientific research and good for understanding the

nature.”

This is my report on Dr.Linfan Mao with his combinatorial notion. We there-

fore note that Dr.Linfan Mao is working on a way conforming to the natural law of

human understanding. As he said himself: “mathematics can not be existed inde-

pendent of the nature, and only those of mathematics providing human beings with

318 \3|lzz – ySE+Z Smarandache J1effective methods for understanding the nature should be the search aim of mathe-

maticians!”As a matter of fact, the mathematical combinatorics initiated by him in

recent decade is such a kind of mathematics following with researchers, and there

are journals and institutes on such mathematics. We believe that mathematicians

would provide us more and more effective methods for understanding the nature

following his combinatorial notion and prompt the development of human society in

harmony with the nature.

K 5 1r – PoP&(, 319-323

8K6I_X_uUnfolding the Labyrinth:

Open Problems in Physics, Mathematics, Astrophysics,

And Other Areas of Science(Pages: 28–30)

Authors: F.Smarandache, V.Christianto, Fu Yuhua, R.Khrapko, J. Hutchison

Publisher: Hexis, USA, 2006.

2.10 Smarandache Geometries and Degree of Negation in Geometries

We now present a more general class of geometries extracted from [1].

Definition An axiom is said Smarandachely denied if the axiom behaves in at

least two different ways within the same space (i.e., validated and invalided, or only

invalidated but in multiple distinct ways).

A Smarandache Geometry is a geometry which has at least one Smarandachely

denied axiom (1969).

Notations Lets note any point, line, plane, space, triangle, etc. in a Smaran-

dacheian geometry by s-point, s-line, s-plane, s-space, s-triangle respectively in order

to distinguish them from other geometries.

Applications Why these hybrid geometries? Because in reality there does not

exist isolated homogeneous spaces, but a mixture of them, interconnected, and each

having a different structure.

The Smarandache geometries (SG) are becoming very important now since

they combine many spaces into one, because our world is not formed by perfect

homogeneous spaces as in pure mathematics, but by nonhomogeneous spaces. Also,

320 \3|lzz – ySE+Z Smarandache J1SG introduce the degree of negation in geometry for the first time [for example

an axiom (or theorem, or lemma, or proposition) is denied in 40% of the space

and accepted in 60% of the space], that’s why they can become revolutionary in

science and this thanks to the idea of partially denying and partially accepting of

axioms/theorems/lemmas/propositions in a space (making multi-spaces, i.e. a space

formed by combination of many different other spaces), similarly as in fuzzy logic

(or in neutrosophic logic - the last one is a generalization of the fuzzy logic) the

degree of truth (i.e. for example 40% false and 60% true).

Smarandache geometries are starting to have applications in physics and engi-

neering because of dealing with non-homogeneous spaces.

In the Euclidean geometry, also called parabolic geometry, the fifth Euclidean

postulate that there is only one parallel to a given line passing through an exterior

point, is kept or validated

In the Bolyai-Gauss geometry, called hyperbolic geometry, this fifth Euclidean

postulate is invalidated in the following way: there are infinitely many lines parallels

to a given line passing through an exterior point.

While in the Riemannian geometry, called elliptic geometry, the fifth Euclidean

postulate is also invalidated as follows: there is no parallel to a given line passing

through an exterior point.

Thus, as a particular case, Euclidean, Bolyai-Gauss, and Riemannian geome-

tries may be united altogether, in the same space, by some Smarandache geome-

tries. These last geometries can be partially Euclidean and partially Non-Euclidean.

Howard Iseri [3] constructed a model for this particular Smarandache geometry,

where the Euclidean fifth postulate is replaced by different statements within the

same space, i.e. one parallel, no parallel, infinitely many parallels but all lines pass-

ing through the given point, all lines passing through the given point are parallel.

Mao Linfan [2, 3] showed that SG are generalizations of Pseudo- Manifold

Geometries, which in their turn are generalizations of Finsler Geometry, and which

in its turn is a generalization of Riemann Geometry.

References

[1] Kuciuk L., Antholy M., An Introduction to Smarandache Geometries, Mathe-

matics Magazine, Aurora, Canada, Vol. 12, 2003.

5 o. ÆSqR(B* 321

[2] Mao, Linfan, An introduction to Smarandache geometries on maps, 2005 In-

ternational Conference on Graph Theory and Combinatorics, Zhejiang Normal

University, Jinhua, Zhejiang, P. R. China, June 25-30, 2005.

[3] Mao, Linfan, Automorphism Groups of Maps, Surfaces and Smarandache Ge-

ometries, partially post-doctoral research for the Chinese Academy of Science,

Am. Res. Press, Rehoboth, 2005.Neutrosophy, Paradoxism and Communication(Pages: 10–12)

Authors: F.Smarandache, Dan Valeriu Voinea and Elena Rodica Opran

Publisher: Editura SITECH Craiova, Romania, 2014.

3.1 Mathematics

What is the mathematics of the 21st century? - is wondering professor Linfan Mao. The mathematics of the 21st century is the combinatorization with its generaliza-

tion for classical mathematics, also the result for mathematics consistency with the

scientific research in the 21st century. In the mathematics of 21st century, we can

encounter some incorrect conclusions in classical mathematics maybe true in this

time, and even a claim with its non-claim are true simultaneously in a new mathe-

matical systemasserts Linfan Mao: Mathematics of 21st Century–A Collection of

Selected Papers. The mathematics of 21st century aroused by theoretical physics Smarandache

multi-space theoryasserts Linfan Mao, is a paper for introducing the background,

approaches and results appeared in mathematics of the 21st century, such as those

of Big Bang in cosmological physics, Smarandache multi-spaces, Smarandache ge-

ometries, maps, map geometries and pseudo-metric space geometries, also includes

discussion for some open problems in theoretical physics(Linfan Mao, 2006).

Definition A Smarandache Geometry is a geometry which has at least one smaran-

dachely denied axiom.

322 \3|lzz – ySE+Z Smarandache J1 An axiom is said smarandachely denied if in the same space the axiom be-

haves differently (i.e., validated and invalided; or only invalidated but in at least

two distinct ways). Therefore, we say that an axiom is partially negated, or there

is a degree of negation of an axiomaccording Smarandache’s theory (Linfan Mao,

2011).

Thus, as a particular case, Euclidean, Lobachevsky-Bolyai- Gauss, and Rieman-

nian geometries may be united altogether, in the same space, by some Smarandache

geometries. These last geometries can be partially Euclidean and partially Non-

Euclidean. It seems that Smarandache Geometries are connected with the Theory

of Relativity (because they include the Riemannian geometry in a subspace) and

with the Parallel Universes.

The most important contribution of Smarandache geometries was the introduc-

tion of the degree of negation of an axiom (and more general the degree of negation

of a theorem, lemma, scientific or humanistic proposition) which works somehow

like the negation in fuzzy logic (with a degree of truth, and a degree of falsehood)

or more general like the negation in neutrosophic logic (with a degree of truth, a

degree of falsehood, and a degree of neutrality (neither true nor false, but unknown,

ambiguous, indeterminate) [not only Enclids geometrical axioms, but any scientific

or humanistic proposition in any field] or partial negation of an axiom (and, in gen-

eral, partial negation of a scientific or humanistic proposition in any field) (Linfan

Mao, 2006)

These geometries connect many geometrical spaces with different structures

into a heterogeneous multi-space with multi-structure.

In particular, a Smarandache geometry is such a geometry in which there is at

least one Smarandachely denied rule, and a Smarandache manifold (M;A) is an n-

dimensional manifold M that supports a Smarandache geometry. In a Smarandache

geometry, the points, lines, planes, spaces, triangles, · · · are respectively called s-

points, s-lines, s-planes, s-spaces, s-triangles, · · · in order to distinguish them from

those in classical geometry.

Howard Iseri constructed the Smarandache 2-manifolds by using equilateral tri-

angular disks on Euclidean plane R2. Such manifold came true by paper models in

R3 for elliptic, Euclidean and hyperbolic cases. It should be noted that a more gen-

eral Smarandache n-manifold, i.e. combinatorial manifold and a differential theory

5 o. ÆSqR(B* 323

on such manifold were constructed by Linfan Mao (Iseri, 2006).

Nearly all geometries, such as pseudo-manifold geometries, Finsler geometry,

combinatorial Finsler geometries, Riemann geometry, combinatorial Riemannian ge-

ometries, Weyl geometry, Kahler geometry are particular cases of Smarandache ge-

ometries.[Dr.Linfan Mao,Chinese Academy of Sciences,Beijing,P.R.China,2005-2011]

Multi-space unifies science (and other) fields; actually the whole universe is a

multi-space. Our reality is so obviously formed by a union of many different spaces

(i.e. a multi-space, www.gallup.unm.edu/ smarandache/TRANSDIS.TXT).

Unfortunately there is not much theory behind multi-space (only some research

done about Smarandache Geometries, that are a particular type of multi-space

formed as unions of geometrical spaces). So, we can unite nano-scale space with

our world scale and with cosmic scale, or we can unify the unorganic nanoscale with

organic nanoscale, and so on. The domain is open to develop a multispace theory!

(Smarandache, Christianto, Fu Yuhua, Khrapko, Hutchison, 2006)

Since 2002, together with Dr. Jean Dezert from Office National de Recherches

Aeronautiques in Paris, worked in information fusion and generalized the Dempster-

Shafer Theory to a new theory of plausible and paradoxist fusion (Dezert-Smarandache

Theory): http://fs.gallup.unm.edu/ DSmT.htm. In 2004 he designed an algorithm

for the Unification of Fusion Theories and rules (UFT) used in bioinformatics,

robotics, military.

K 5 1r – 0Xj' , 324-326

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^r 1Za,*,1962 8 12 V 31 <&+&r;p41980 8J&+&;Ip;IW4,`,`[,e<F,OInternational

J. Mathematical Combinatoricsr,v?zyOe< 1981 8 -1998 8eWzyW\5, 1983 8 9 V -1987 8 7 Vv?ypz-HJ.zyJÆ,1991 8 3 V -1995 8 6 VBWv?pz 9Jt,~ `v?TZYv`X419998 4 VvmT,FB4d _0,`X$Q-<.(D42002 8 3 V7,`<`VDE%t2h h(A census of maps

on surfaces with given underlying graphs*, 8 11 Vt,`X42003 8 6 V eW O`b.pJ$QqBd,`[$Q5,2005 8 5 V7,`[oEV hY Klein D7[_(On Automorphisms of Maps and Klein

Surfaces*2007 ee0 International J.Mathematical Combinatorics'~r$Q5+4 Smarandache =B'WE|=B(D<`<y`zGR3,[e02OLty,Graphs and Com-

binatoricsAustralasian J.CombinatoricJ.Appl.Math & ComputingJP

J.Geometry and TopologyActa Mathematica SinicaInternational J. Mathe-

matical CombinatoricsMagna ScientiaxQ?~QÆ'QQÆ~ VCQÆEpEgOz#Y<`tLGp,ee\ ~t~y`<`2,e~\O~<`2Linfan Mao is a researcher fellow of Chinese Academy of Mathematics and

Systems, a professor of Beijing Institute of Civil Engineering and Architecture and

a deputy general secretary of the China Tendering & Bidding Association, in Bei-

jing. His main interesting focus on Mathematical Combinatorics and Smarandache

multi-spaces with applications to sciences, particularly on topology, differential ge-

ometry and theoretical physics. He is the Editor-in-Chief of the journal International

1www.paper.edu.cnC<P;6k, www.baidu.comv8O, www.baike.comA)v;, www.baike.so.com360 v;www.sogou.comv;

340 \3|lzz – ySE+Z Smarandache J1J.Math.Combin. published quarterly in USA.

He was born in December 31,1962 in Deyang of Sichuan in China. Graduated

from Sichuan Wanyuan School in July, 1980. Then worked in the China Construc-

tion Second Engineering Bureau First Company, began as a scaffold erector then

learnt in the Beijing Urban Construction School from 1983 to 1987. After then he

came back to this company again. Began as a technician then an engineer from

1987 to 1998. In this period, he published 5 papers on graph theory. He got his BA

in applied mathematics in Peking University in 1995 learnt by himself. In 1999, he

leaved this engineering company and began his postgraduate study in the Northern

Jiaotong University, also worked as a general engineer in the Construction Depart-

ment of Chinese Law Committee. In 2002, he got a PhD with a doctoral thesis A

Census of Maps on Surface with Given Underlying Graphs under the supervision of

Professor Y.p.Liu. From June, 2003 to May, 2005, he worked in Chinese Academy

of Mathematics and Systems as a post-doctor and finished his report: On Automor-

phisms of Maps, Surfaces and Smarandache Geometries. He published more than

50 papers on Smarandache multi-spaces, graphs, topology, geometry, physics and

construction., also published 7 books in USA and England.^r 2Za'LW O. $QO$Q|F,ahÆ,rvWypU`'Wn$QOOdx$QF,O(D4$QO(AMCA,e*Od,v?zyTe< , xQD*m(ISSN 1937-

1055*rM& 2002 8tvmT,`X,2005 87W O,`[$QoEMv$Qe rE2We(DSmarandache ` 4 v,h2(D<W-=BE|m/e5<y`<y`~l CF'.,Me# 3\CezYIL(p<`,T5t`2l,2011 8e 9\,e'\v hD~ Smarandache (7[_VD*(~xV~0ASmarandache "I0~Vz 3 54t,2007 8e$Q\,\vME(15Zb~VY6<,2013 8WzyJ\,\v15Zb~Vt215a5`P*1H~l~Hz,`4OLtv<2www.engii.org, www.mathcombin.com

5 o. l)e8m 341`,l,Me xQD*mYv D*~~xQD*S~(2007* xxQxQ(2014*, 4 2015 8eOQG8iWv#UCZxQ Y_Yv xQY7XB6 - QA6~Vz,z O#vLinfan Mao is a researcher of Chinese Academy of Mathematics and System

Science, a vice-president, also a chief professor of China Academy of Urban Gover-

nance, the president of Academy of Mathematical Combinatorics with Applications

(AMCA, USA), an adjunct professor of Beijing University of Civil Engineering and

Architecture, and the editor-in-chief of International J.Mathematical Combinatorics

(ISSN 1937-1055). He got his Ph.D in Northern Jiaotong University in 2002 and fin-

ished his postdoctoral report for Chinese Academy of Sciences in 2005. His research

interest is mainly on mathematical combinatorics and Smarandache multi-spaces

with applications to sciences, includes combinatorics, graph theory, algebra, topol-

ogy, geometry, differential equations, interaction system, biological mathematics,

theoretical physics, parallel universe and economy. Now he has published more than

80 papers and 9 books on mathematics and engineering management on these fields,

such as those of Automorphism Groups of Maps, Surfaces and Smarandache Geome-

tries, Combinatorial Geometry with Applications to Field Theory and Smarandache

Multi-Space Theory, 3 books in mathematics by The Education Publisher Inc. of

USA in 2011, Theory and Practice in Construction Project Bidding & Purchase

by American Research Press in 2007, The Foundation of Bidding Theory and The

Provisions of Clauses of the Law on Tendering and Bidding of P.R.China with Cases

Analysis in Chinese by China Architecture & Building Press in 2013, and famous pa-

pers, such as those of Combinatorial Speculation and Combinatorial Conjecture for

Mathematics, Mathematics on Non-mathematics on International J.Mathematical

Combinatorics respectively in 2007 and 2014, and Mathematics with Natural Reality

– Action Flows on Bulletin of Calcutta Mathematical Society in 2015, an established

journal of more than 100 years.^r| 3BWMsd,,`4W O,`[, :3www.zgcsb.org.cn

342 \3|lzz – ySE+Z Smarandache J1:F,e<F,e xQD*mr,v?zyTe< $Q<mF=qJz-OFr~zeWho’sWho,e~ O(D$QO(e*Od, ~v<QQ (ev*:FG8v$Q5rE2We# . <y`%p=qz-< `z3,g24=qCF< `zG LO1vDpGALOo5ToE,$Q%74ZgtoO lhvh[e02OLty#Y(D-=B`<y`tCF4=qz-`z<`IL(p,ee[\~$Q< 8~=q`<tO~<`2,e~\O~<`24e0\zH=qJ~ <t 815Zb~Vt215a5`P*1H~l~Hz 2 5=q`<t,B.~Da[$ 15a5`6*P I,`4ThwMEM*MD^60A`hEM*M6`h(II VII*vT4g~Da[$ 5**M152? *A(2007 \*~Da[$ 5**M15*A(2007 \*4XA),i5"~x5*(\3*(20108\*4XA)vrETV2008 8 \~iF<J~tm 8AmOFOF15Zbl~H(2009 C*m 8r,15Zbl~H(2012 C*m 8r2015 8+(zWMlCJOFM[B.LGR< IJl.`,'5LW`%o< C`7QR=q< ,Vu9` I_`=qz5L08v.=q< ` PPP < JlV,MB 2000 8 e0Bd=q2=q PPP< Jl`.8,+Q(`,bPot;(|

K 5 1r – Mai`Qu\, 343-343

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Od 'gz– j' 1985-2016 BU'F, 344-352

^r 1985-2016 rIk1985 1. %n:7nx4 RMI Bu,~xQh,1(1985*,29-322. Æv ~u,~xQh,2(1985*,22-231990 1. The maximum size of r-partite subgraphs of a K3-free graph, 'xQ, 4(1990),417-4241992 1. v?;O 100m3 =I,EM,1(1992*2.(.H;Dt*v?TDuH 50m o9I,E<,4(1992*1993 1. (.H;Dt*v?TDuH 62m o=p= T~I,E<,1(1993*1994 1. (.92^Dt*R(G)=3 vW`veo$Q,>PxQY0AxQ,Vol.

10(y*(1994),88-982. Hamiltonian graphs with constraints on vertices degree in a subgraphs pair, *hvQtQÆ,Vol. 15(y*(1994),79-901995 1.(.H;Dt*v?TDuH 9= oI,Ep,5(1995*2. o9IA,~ 6A<A(95 \*,19953. =TAXj-,Bd=IA,~ 6A<A(95

&H3l)e 1985-2016 EW)H 345\*,19951996 1. U_KW`v,T,Kg5<QQÆ,Vol. 23(y*(1996),

6-102. = Bs|4oU,E<,2(1996*1997 1. xTOzyIBiodmE,EgO,11(1997*1998 1. A localization of Dirac’s theorem for hamiltonian graphs,xQhYV,Vol.18,

2(1998),188-1902. = Bs|4oU,Ep,9(1998*3. xTOzyIBiodmE,E<,1(1998*1999 1. HOI(;-I,fD:r2EM*MD^60A`h,WzyJ\,,19992. I(;-I,Vr2EM*MD^60A`h,WzyJ\,,19993. v?TDuH 9= oI, Wzye'2EM*M6`h(2), WzyJ\,,19992000 1. \5n FantvOR$,D!'jQQÆ(7X<Q*,Vol. 26,3(2000),

25-282. v?p<[4W`IRM.`,E<,2(2000*3. v?p<[4W`lI,EM*M6`h(7), WzyJ\,,19992001 1. (.4dDt*!uUCvOY_v\5n|t,D!'jQQÆ

346 l)e: \3|lzz – ySE+Z Smarandache J1(7X<Q*,Vol. 27,2(2001), 18-222. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, -3'jQQÆ(7X<Q*, 4(2001), 13-183. (with Liu Yanpei)An approach for constructing 3-connected non-hamiltonian

cubic maps on surfaces,OR Transactions, 4(2001), 1-72002 1. A census of maps on surfaces with given underlying graphs, A Doctorial Disser-

tation, Northern Jiaotong University, 20022. On the panfactorical property of Cayley graphs, xQhYV, 3(2002), 383-

3903. ype?BpvOlh,~ VCQÆ, 3(2002*, 88-914. Localized neighborhood unions condition for hamiltonian graphs, -3'jQQÆ(7X<Q), 1(2002), 16-222003 1. (with Liu Yanpei) New automorphism groups identity of trees,xQy#, 5(2002),

113-1172. (with Liu Yanpei)Group action approach for enumerating maps on surfaces, J.

Applied Math. & Computing, Vol.13(2003), No.1-2, 201-2153. (.4dDt*v?6vWp,xQ?~QÆ, 3(2003*,287-

2934. (.Dt* f^ ≥ 2 v\5nt.!uUC,-3'jQQÆ(7X<Q*, 1(2003),17-212004 1. (with Yanpei Liu)A new approach for enumerating maps on orientable surfaces,

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403-430

Abstract: This book is for students and young scholar, words

of a mathematician, also a physicist and an economic scientist to

them through by the experience himself and his philosophy. 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, promoting mathemat-

ical combinatorics for contradictory system on the reality of things

and economic systems, and after then continuously overlooking these obtained

achievements, raising new scientific topics in mathematics,physics and economy

by Smarandache’s notion and mathematical combinatorics in his research, tell us

the truth that all roads lead to Rome, which maybe inspires students and young

scholars in mathematical sciences, physics and economy, and also beneficial to the

development of bidding business and state governance in China.=_F: fNSTI7h?I7Æu+fNW1Æx_?BEyFKN0uÆ0Bh3;a"?I7huo4N_, h?VIZ+NQyx\+Æ6zÆI+&_yÆhu0?yÆG^/+_ÆW+OOxBXY_uz99,+_3dZ<Æ#PV`X6W Y"+6d#P[DuO+ilÆ'C&aq#T+&1:Æg+1:<Æ'C? Smarandache $6T>pÆx_?BEur5k#P+?yfSuI7Æ;?I7Æz9-/^d)f?$o4Vu.u4?<pBE_;-HbJ#P_+?#$V IuX?HmbJ+6UT_n1nl#

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