mathematical combinatorics & smarandache multi-spaces

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—– Smarandache Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO Chinese Branch Xiquan House 2010

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

TRANSCRIPT

Page 1: Mathematical Combinatorics & Smarandache Multi-Spaces

�5hm�r^�B—– �O})_ Smarandache uK;d\�

Let’s Flying by Wing

—–Mathematical Combinatorics

& Smarandache Multi-Spaces

Linfan MAO

Chinese Branch Xiquan House

2010

Page 2: Mathematical Combinatorics & Smarandache Multi-Spaces

d\�H℄'�E�#�!VÆ�#r[ 100045H℄{>Æ�>{�D{>CeÆCeF�#r[ 100190r[9SC%>Æ/:C#CeiTk#r[ 100044

Email: [email protected]

�5hm�r^�B—– �O})_ Smarandache uK;

Let’s Flying by Wing

—–Mathematical Combinatorics

& Smarandache Multi-Spaces

Linfan MAO

Chinese Branch Xiquan House

2010

Page 3: Mathematical Combinatorics & Smarandache Multi-Spaces

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

Books on Demand

ProQuest Information & Learning

(University of Microfilm International)

300 N.Zeeb Road

P.O.Box 1346, Ann Arbor

MI 48106-1346, USA

Tel:1-800-521-0600(Customer Service)

http://wwwlib.umi.com/bod

Peer Reviewers:

Y.P.Liu, Department of Applied Mathematics, Beijing Jiaotong University, Beijing

100044, P.R.China.

F.Tian, Academy of Mathematics and Systems, Chinese Academy of Sciences, Bei-

jing 100190, P.R.China.

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

P.R.China.

W.L.He, Department of Applied Mathematics, Beijing Jiaotong University, Beijing

100044, P.R.China.

Copyright 2010 by Chinese Branch Xiquan House and Linfan Mao

Many books can be downloaded from the following Digital Library of Science:

http://www.gallup.unm.edu/∼smarandache/eBooks-otherformats.htm

ISBN: 978-1-59973-032-5

Page 4: Mathematical Combinatorics & Smarandache Multi-Spaces

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Page 5: Mathematical Combinatorics & Smarandache Multi-Spaces
Page 6: Mathematical Combinatorics & Smarandache Multi-Spaces

M SbxN\y�&$\�Æ#xN)?�$�> oC V$�%3=m$V NR�$`qT7R!�#8&$\"���&i`x,\Bkg�i`#x8${=_2G���e#��0��i`��5"�g#�v�uo<Q �4��C#N{N{F ;Æ#h �eqTljXZg0 eoYl&41�C#y℄I�{F7Hg| �#h �p=6v#�Ay)6>�,\�Cy8��3T�6NWQ�k�p *(%IT 17 &#6pNr)%IT 3 &#6NWk�Xjn 4/%IT 8 &#�8Iv%FW! 3 &u.)#,\TBX�96Y#H&�&i�$� ��yiqY68�$$i`3x9E�LL`#Wnx�$#s��D0 N�t,so��$W��Q�d�n3B h j8k$�mg#y�y|p *(v%,#t�i�pesh�1YP~#%Iu�#0M2�$&IÆyW�r)6U�"�C�K��|uS#3{|)U�#6U�%Ik$N)�$~��{{Q#Io�$&I3G℄ZAzN�$K�2-�$K��$|p *(�#%Iu�℄7$K�$#�X7$Q�Æ0)nx�$&I\V�3�Nu$K&:�Zs�$�RaQ8UQxqk$',vs$)#-�vs$)�#��8QV$%vs�9"O[u�$2-%I#......�6�#3WxR`IC v%tsÆ��~�o%3��$2-\{ �wNR�I�&W$`#?bv#k2�i�pe℄�A0)4<3{��"��jgxd7'd71xn�7Ik0(;�)N{�jgx0('0(1x6;Æ)d7�`Q��NgC�\+���sy�60(#"=�#6/�0(ByD�a�/��T3�3sh�0(#h 6lYN��j�si#3�% 7I#����j�Cf|)�`#3�%yiL_#Ai�n7I,\�M~#

Page 7: Mathematical Combinatorics & Smarandache Multi-Spaces

ii K): DnS�X,h�� – �?hn| Smarandache K�.Zfk"���j℄���& 10 *#3$|)[p$|$#s6u\+) 10 &��!~#k|℄n=g=&u.�Bs$#F(x,\TB)�b�A#M�$#�5� �s$nk3�D[�3��$uC ��\.? �A Collection of Selected Papers

on Smarandache Geometries & Combinatorial Maps, kQ Chinese Branch Xiquan

House D?MD?#2006 &!!#"y Cn7I#x Cn��0&"3[N�p %,|vs�#sB 0 Q��s�$`B!��C#84�Cn��0��0�y7i�t�\ 2006&D?�.?W!B`7G)#Xue�=$� .Q#8Nu2-,� �j{(u�#"yD"&$\)� Y#$|U��#3WQ)�w�Nu%I#q3Z�y�Nu$b .QZA$\i`�KI#&e0N\pÆ#s-$\�&!Qp#qÆ�� �(3n�E;EZb��Let’s

Flying by Wing!�QH|kQ�WD?M�4�Qk#�r3b 10 &��D$�&e#XE)N�tj\$�>gm{ Smarandache J�-#e�>gmx3bu&6 Smarandache �`^3GE#!{�N�6QSE� Smarandache ON&es,�V$&e�[NuV$|.u#b$Æ&eox�$��V$2-�;E�Wing of Scientific Research!�ZQ8�8\Æ�.QW$IN� D�I&�4{}Nn^ x 2003 & y�N)|��|%Iu/�&0�?I�.QyÆ�#ogM|U�7$Q��E6��3�7$Q� l.+"#1x�5�mD�)Cn�bN.Q�|)�=�O�(3#��x�3��$uC #��x�3�7Q�$C$[p %,|vs �j\��4{�Nn^ x y�&$\#6+K_�!~$#nBk�r \O��NN.Q�. P$:)I`~N�p %,#!TZg7$#�!"C\�p K�C ,��vs\ovs�2-,���T3ZAT3�Q���$W�x��.�$:)I`ZD�$�r�}e�jhAQSNu2-p�8I`2-%I�>"�uh��4{|(�Gn^ x�& 3 *n~�|�QVKN.6`�z�$`VOu&?I�=&{#3NxI6z�$`�$�P�2JB�)�#F��G&,8z�$`*�2J{ 50��7$n3Nx1:�*�"\�*��V$2-� ��Xubu&7I�Q�$`�x9oQ$&e� }#8NuV2℄\�.m��Nu�r.2-4/�$`ys#?x��[,�7&�V$�M%ZAn�V$DK�8HM0\#bFxbu&3B$`�&$`℄g�N��\�.Q�z$hCA"\#jg℄\z��2-�#s$:)7I[zN�8�r2

Page 8: Mathematical Combinatorics & Smarandache Multi-Spaces

9F iii-#~�r�8℄zN#x�℄$2-#B��8�℄Gpo N> 2-�n�T3#bv#�r&ej|)Ldj`�Ix#?y5�k�73#s�rQSV${7ZD��2-℄\#[?�z,�7&v�DKAx�,$�DK��"�#8&$`�0�y7i��N:�N{�}!. x 1985 &36U�.~pN$r$Kin�&�$2-�e"%u&#O6$\|.y�NN℄%#8&$\T4�$�'�� �$Ko�,\�hpN�#O)Nu℄%�lSE#bFx7I[B|��$$K!<�:�#a�8&$\$K�$%�NuD���A Forward of �SCIENTIFIC ELEMENTS� x3 �SCIENTIFIC

ELEMENTS�\Æy�NN�5�.Q��) Smarandache &ex,�.m[At��NÆ&e#?�$�r&eAx6Y3GE#�x�r&eXZ6�$ElY�&e#[?��) Smarandache &eo3��$�r&e6V$2-Esb�3G����3��L�e,& RMI ag x7I6$)���~���$QHN"u�oNu�r�$�I�#8�$EJOnx�>NsKL9�A�RMI �A!��I#sZ$\KIDl6�&�$2-�E�NN.Q��Qa8QZ�{�V�7�N -SmarandachetJ:Qa xI`nB$|+K_�!~$zk&~ o$\�IYn�$�> o6�>℄ �Ngm�>{HB>�?2�a EO��NNV_ .Q#e��x�IBgNtU�$��\W$��DQHoNu�N�e P�I)��kRH���Smarandache�`^�Smarandache Gp��z��zGp�#.%`^Gp��5#B�8 N> G�0\�)Nu�d�UN�,\T3#v#yD d7Y ~#h B��d7(h� ~}�d&FyD 0(L+#h �yTx,\F|�N�pp�Cj��`�d7%`%2�`�0(#}D#72o#�M7{�&sA#�`�0(syN�`%2�`�%0(#EY#"X�_�$K�%I��6V$Eh�VP�k~#}�y&�g72'�AC8��,#��G�g|H!��� �$Z\Æ�&I�-y� d/JB O N O &gN*U�

Page 9: Mathematical Combinatorics & Smarandache Multi-Spaces

o `�5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i3��$uC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13��$uC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43��rw�uC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20$K�$���℄% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A Forward of�SCIENTIFIC ELEMENTS�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37x|E��x|g�o RMI �A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .393�%3uC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 N> jD�BgNtU�$ -Smarandache �`^ N . . . . . . . . . . . . . . . . . . . . 75d/J 1985-2010 ℄&N��D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Page 10: Mathematical Combinatorics & Smarandache Multi-Spaces
Page 11: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– lY5>77, 1-36|~Oo_d/J�H℄{>Æ�>{�D{>CeÆ#r[ 100190!jT: vh2N�Q$lnR=9SCFA_`>hUr[}4\C�`>y�v{hÆi>y> #yNr�?AG>Et�y> #�TNH℄{>Æ�>{�D{>CeÆ�ys2o#�|�ysCe�%��\$#�Y7�M;}WG>R;7#zj"NlyWG># }NpyWR(eG>YB>Æ�#�s�yiK3ql�!<v$9SCF# >##�#�y�Abstract: This paper historically recalls that I passed from a construction

worker to a researcher in the post-doctoral stations of Chinese Academy of

Mathematics and System science, including how to receive the undergraduate

diploma in applied mathematics of Peking University learn by myself, how to

get to the Northern Jiaotong University for a doctorate in operations research

and cybernetics studies with applications, and how to get the post-doctoral

position, which shows that there are not only one way to growth and success

for students in elementary school. Particularly, it is valuable for those students

not getting to a university, or not getting to a favorite university.

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

doctorate.

AMS(2000): 01A25,01A70%IBg&, 3[N�_q�p %,, Z� j�, Nd%I#Nd7$# 1999 & 4 *�8�r-�vs$)#2003 & 6 *�8V24/[u&I2-�9&T3LS, t6�`M{#"=�xyD��-�3x6NB����$��9,����ipe7I 0 N��$W�12003 %T�}�}�6#P�k-�2e-print: http://zikao.eol.cn

Page 12: Mathematical Combinatorics & Smarandache Multi-Spaces

2 K): DnS�X,h�� – �?hn| Smarandache K�.n`M�$Æ, I�$ K, ��\+y=l�O+�$=0)$KOV, B $ j8k$mg?{|NWp *(%I��X%I, v)N�p %,, N�[5%�n`6{�[E4{4�, +jN�h�0"��f{�[S`xNÆ℄��", �7I6$inyD�/�yME��)bA�eH, 6�X%I� B&�, 3{|U�NBp $r$K�9&W!%I, t�i�eHs$P~�xNz$2p K�, Nz6U�%Ik$N)t~��{{Q$K�$sIo[u�r�$�2-�[p $rÆ0$I�, 3$|)�%Ir), 0)N�K�C ,��eye%K�.m, I e%DY�K�0\0)3j2��D%I���Bp, i^*pe #bx3��A��K�4*, xNuK�� o���AK��\sxe%lj#N.0)�,�/�Qk, �$2-FhYN.5�b3i^T�DlTyHp K�N., F�ATyHe%�\, s6[p $r$|�%Ir)� :&, ZZ�Z℄ %3 �t:�G&1DY, yD��dy0℄��vi�eHy��, . x.WJ���h=0�, xA��xI�i^, �XU�~��~�7$Q�, Æ0\V$I�lSE, bÆYloxQk�qq, 6Qk4{,�8�, �synl#z�, Fynl# �|g�w)�}��qT+&B�/�, 6p I%Ig+&i, 3�|)U�k$oU�~��~�7$Q�"�%�r=D�nx�$&I\V`I.Wo$s$)�"XS)38Q�kI�.m�3&NZ4%, 36U�k$k|vs\XQ�Dogm, D�eQ�V�7I6T�($T, ���DlTN., x�\)NÆyD�fe:��rx�',vs$), hDQkbÆ=��6:��=&�, 4` e%C , <�A l�o&I℄�wK�NZ�NZ�d7, 3h�?�:�!<, yxV9DR, ��yT>�j`<��i^C�{$K���=&�, 3 j8)UQxqk$',vs$)�,vs$)83{"xy�S��th 73\\+{!��{�%Ir)yr7, y�\+\!#bA3Nzi%, Nz',$), Æ0N.��i, 9&$Ky�S, tj`�\+xav�S��<E:�E�jO, M&IÆo.Z4*, &Q℄\�4`yE℄i, 1�i%, Z!7Wo\+Ir��Eq<Eh&QÆ�0\eÆ.q�|��Rn 0N., ik6?, �D�Dl�n�vsN.7Nx"\#�xe�5#Cx-[7I*P�QpÆ0��N.Æ0�#x�)Q�g)~�XV#�NC �m4 �6�vsN.�2-�yIT3#S�b��!QS�92-QkG%I�bA, 6�|vs$)�, �,��b��N{DK7I�W?oQk�x� )yk�o�2-a>Qk�

Page 13: Mathematical Combinatorics & Smarandache Multi-Spaces

n[6?88 3&�o.Z, sj8QV$%[uvs�2-%I�3xU�~��~�7$Q��`I\�h x7$0�#��Æn�$KQHo℄B0\��A0\� ��bu#x� 3',vs$)iQ)3G�3��,�\+b���j, �/��?6�j�lYT3#yDT=&Qn,CAXZ#oCAMp#���f,�Q �N<fy�g#��Bf�}a �, |?Q�y/% �h bAe, syu", 3~N�_qp %,B 0 )N)�$%I`�T39&�℄, yG���a�, t"=�, Ax,RZz�℄Yo�-�L+�

Page 14: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– lY�>77, 4-19

6|�Oo_d/J�H℄{>Æ�>{�D{>CeÆ#r[ 100190!jT: Unp-Ys#vh�Q$lnR=9SCF#�℄R=�>%Y>_%#kH>p���R%9SH�Cq��r�?AG>Et�y> #�H℄{>Æ�|�ysCeZ��℄*'�p��GH�Cq\#hH�B�Q$� 1985 u -2006 ul�|{>CeY0'�%#O�Q$R"Ce#^YXU_%#k�>gmz�Y61_%�{ Smarandache J�-��Yzw\�!<v$H>i, CF,C%k, �>%, Smarandache�k, SmarandacheJ�-, �>gmz��Abstract: This paper historically recalls each step that I passed from a

scaffold erector to a mathematician, including the period in a middle school,

in a construction company, in Northern Jiaotong University, also in Chinese

Academy of Sciences and in Guoxin Tendering Co.LTD. Achievements of mine

on mathematics and engineering management gotten in the period from 1985

to 2006 can be also found. There are many rough and bumpy, also delightful

matters on this road. The process for raising the combinatorial conjecture for

mathematics and its comparing with Smarandache multi-spaces are also called

to mind.

Key Words: Student, worker, engineer, mathematician, Smarandache ge-

ometry, Smarandache multi-space, combinatorial conjecture on mathematics.

AMS(2000): 01A25,01A70

12006 % 3 ) 26 1�*J^� }#�qb[N�"2e-print: www.K12.com.cn n www.wyszx.cn

Page 15: Mathematical Combinatorics & Smarandache Multi-Spaces

n[�?88 5Z{B OO:&:*=gN2E; 10$00n#QV$%�$�NuV$2-%O�l�#3��On Automorphism Groups of Maps, surfaces and Smarandache Geome-

tries �BaH�8X� Smarandache �kY`BKA!�vs�O�6`6��Ysi#kQ SNJ 2���e Perez vsF6>"bZO�#Y{#O���5W!TC��T�mO��#EQ��r�$�:)<����~��#��Nu2-\�v3O�|Map GeometryN3i#QV$%2-\%w"ÆQ#�$W63 ~�6j)�NSz#03$�Iseri Y Smarandache �k`.��r�-}}Zr�Y#VW+%HswU_E' 3"$�wU_#�RwU_#~-Ce_Iseri Rt Smarandache 2/YjvVO#+�l_YMap Geometry})Y�-`.YKW#n=}ZUngmB��#4�}gmaH��z℄g�>�8Dw�p�ST0J9{KW �3�bÆA�#�Z�|)Q��r�z$W�3�vs{ 8;F~���^�ZEx3Ivs�O�T3�N����!T)Bg�&�/�#3[N)p %,B 0 )�$%I`#bNT3�|)�=Q���$W�>��r7#Hv$i`Nu�$� oW,836{8�$2-Qz�qf�(V) sNÆw3x6� 103 %3{ |N6+K_�!~�[!5Æp$Æ0�p$$I#1976 &`I�vi~p$℄B�x�x℄E�#{�i- 103 %3{ |N6�![���X%����Q�p$`Ii��N-�N)=�$r#�!$��!h$o�i*M$r#{�JTIDlJ#�"q7I�V5℄|)�i*M$r'B�bA#1976 & 9 * -1978 & 7 *#3{|�i*M$r,B ($&�)�QbbB$rNexav��#T��TGZ~�#�DRD$K#�DEr$r�$rNepN#e�N3i^|℄$\�5q)$r����p%I#W?��0�Q�Vn�����Q�vi~��$�> �x�PL� �bNi`o�E+$98 Goldbach}eGD�1+2 �,Z#Q�N�8e�k� Mi`#Wnx�9�O�.$�8W℄p� , n38�$�\)<k���6�8B$K�!�"{$K.�K#℄�i^!�k�$� Æ~Nu�$0\�W�viog�G\:g&nD?�B�n��B�GpÆ#Ez�K\℄vi3n$K�~VÆE�K\D�#\�F=�36℄�AY`k=i^(x{�

Page 16: Mathematical Combinatorics & Smarandache Multi-Spaces

6 K): DnS�X,h�� – �?hn| Smarandache K�.hEz�K\�36p$iyL���#t�8$�#3\Eh�i^4���#?qk|℄i^x{$K#℄66B N$`qBN��$℄$Æ# B$`Aqn�B�$℄$Æ��[B B&Io$K�℄3#bFx36�{�8�!$� N$` 1�Xg[����$*<s�|�Z��h�1977 &�X�[B�Q#3�$�)_℄ 120 ℄�vi�N������#1x|?;>�� . �$�>�j0h/��#?3��hxo#�#N�s`�3_℄���3��B�$$KE ��'�#N�D=GK\#sqTK\�N{ ��'��\��4#bFxn��$�$KKI�tyN�D�GO\�=\#�`Bk=�xNu�$W6�$2-�3�#9&�xB��$�QHXZ�A#t8$\{"eK��QH�SZ�\x� ���Q�vi�Bs$�Æ6��Æ.*��p��, CnCx$r~�-a�50#41xvi 103 %3{ |��X%�� 5 )5Æ#Cn�{C6Qp B%34Nu�[u%3pN�B`I��X�[℄$Q�#3-�)℄�g�0:�og�![q�[℄$Q��{ 100 �Q\&:|�!$$K#{ 50 �?6 1 9#51-100 �6

2 9#[?n�3�4%|�![$� 80 E 1 9$K�Q��i`�s$Zg��Co��8ÆV�=:�So���t��bNi`h�$0:yD#y��$� �\��D#Cyi�3uN�℄�\�W��\QH��s�f3�X) 1979 &=*g[����$*<#-�)�� 18 ��0:�$i`3L�,Nu�$℄�,>#6�SZ��0\�k I��"I}��0\�>$��Ce���Q#���L�QB0���0hD���2u{/0����rqx~~�4\�>�� N�!��1980 & 4 *3{��W~�![:|tUSA, �XSAf�"wpN#3} |$r!r#��Q{�0Y��X 1980 &�Q�Q#9&y e#0"T)\VD�`#th ��$rD��1980 & 7 *�3~+K_�![*71,&�SA�6C,�&��Æ��[|+$9y��0\�B�(II)#|SA��0��Q#�&���BTd��xIo$Ke�|℄Q�, siT�$K���$#sIo�h;r=!f��>�z�7��NuK\�$i`��$~�E~�$\3\sm�#"=�#nX�$\��$2�~�#?b8-~��$� AZD)��D�#CDu��'�� ��\��4#�Ds~eZD���&CZA8"�NÆ�$�Ix#bx℄����lSE#=0�=$\yL��$�N�x��hx~ 6u℄i[\ V#y G|

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n[�?88 17 Y�|#h DnyI63nA �|�Qk!E�I6�!`^�� \>n�℄��E,��CD�#h DnI�`^!�℄��,���6b�E#3ys`�{9��Æ. ��,�XZqT��l<QHY|.>��A��[ NEu#Smarandache GpH[ Riemannian Gp#[?H[+h#PJ a8N#t6plYANxh�|(�?3DY�x3��rNA�#AXZ8e�Hv%5�$�=�5��p��J�b\ÆW 2006 & 3 *6kQD?�kQI�6�OE*z)b\Æ#XZw\�.Q3�2006 & 8 *, 3�X) B �Q�r�$�zNk%, 6bZ%E, 383��$�r�}eA�|�n��GpZA�rQz�Nu�R�)O�, �|)�%`�N�gX, ��%`N��D�lq, �1x6Q�D;Ng�$�r��DK~C. ?bF�oxnQ0 �$�Q�Ngb!uC.

(\) 9$od-pN�0(�V2%I`(�yIW,�r7�>�#3Fy���3 1990&6U��)#a5�3s6N�p *(#0� 1993 &BD\�6Bg=&[u�$$K�2-�T3#�|){7��Qz�>�oqf�{��ET$&p$�hWoSuyp1Io${#23 <1Qj);E�%#EtU;g&n~r)�f�8��p %3$%℄�9$K#=&�h�mugF $KZ[5#x| 1979 &$r�kCnb ℄�\wD)&V.W�{�bÆ63QujZg7$� 83�\)SS�qf#?Wo0�83bÆy2� �omp#"x83;E�$2-~Cj|)yX�~�Ix�

2004 u 8 �{�_i{��(;p6e�/h

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18 K): DnS�X,h�� – �?hn| Smarandache K�.ZEx3[N)_qp %,!T=&�a�/�B ;E�$2-�yVKuC�T39&�℄�LS#t"=�Ax,\Zz�℄Y�6bN�T3#�=� #Hv$�3�� oI�#83;Ebg,\~Cj|)yX�~�_�Ix�3e#bn�Fxn�~��KI��=$�?y_�Fs{Vt_z$iW�O|�p�RH�r[h�i�iuH)�f C%kJ(v:}9S#0X�D#)OrT��>#15 uÆ#9S{�>#℄)i^HYJN�#�),rd_�

1981 u#4HxPYJ(URH�Q5\yp$R℄*C#"usum"Ur[�"9> C[9VP>�#+℄��i�}�>Y)6G$W�Y�-�� �-#)�"�&$H{Æ�%H)NgY�Kac-Mody J�2Bb �r[CPG>NgY����B��gl2Bb >�#�uv��RY>�?2�6?Bh��"9> xPs#J(O$bm���#�\u-#)s�r[h�TÆ�r[ 6G��r[ L>Æ\C%pbm��T�Cq�I?YCq7v#)Ik)t�>�)�"�&$��NH℄gmlmz℄!2B� ��>℄ 3NHB��gl>�?2� �� QNHP℄!HB>�?2� \>�?2���G�6?HfhBh#���mr�>���;�>{gl�>���-��%>Æ>p�\vwW��_Hfh�>Bh�� 1991 u�#J(uv�&r[�4\C�`>y�#VEgl�>�;uZmbY#��X$r[G>�>>y> �Qu0#)py$H{Æ�DY�yCei#�gs�&$r[G>�yiN>y���lY℄ 34 &$#�Vibat�y#-��p-#(ZBppy�y� J(M}�#~z�FY�1#JCr `a�$�>�>>$lRI��� �iC�>YBp#J(Y��CqO0X�5#!�1991 u#JCufup-S#$�k{C~7�ÆlG{ wa!Pk�|�[kj #��r[� L>Æ 100 ��URY|�+lCHXZ#Fgl�+���s�X��k{�T��\< ��H9 H� QC #^Q\< !��9r[ev�8 ok<�� �p#JC�BCe$?mok*HKxfY�wibmlC��#m4$,C%

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n[�?88 19

50 U�[ok*YHK`��#℄℄%G�$G#YG.RQHK��E^��r[�PYUHKlCH#)>Æl$��*lC��#iGr[h�i�iuH) Cs#):1$ZedDqG`l.ds�yJ#GG&�$lC#u�J(p+MRIHs$�PN�!#:Cz6pNFk\ �JC�X_r|J�#1993 u#u�<Uj℄C%k�JCz`�YN5}p|�5pe#)#F$�J(pRv`�Y�i G|��#):�>u�l℄ 1962-2042#j��n#)H,a�$�R 1��#Ce�>}ZCeU 80 &s�2}#lOF%0U 80&� �?y_��1996 p 7 f 30 �!

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��4gl�q℄�A�– lYgm$,77, 20-326|})�Ho_d/J�H℄{>Æ�>{�D{>CeÆ#r[ 100190!jT: {CY�NCq}<7#qM~7<Xp#qM~7�q;p��E\CeF��vh�B�Q$lnR=HB>*#�HKHBCeUO�HB#�UO�2/Ce#T~nO�>CeUY��ki�Bw�YCe_%#�\�{CCqHMk61~7#Mk9�R3>{8��+OH#�x�l`l��xHm℄!Ce$ICq61`�Y#psU5IYCe~7#��p{℄Q>*?2}R6#+r!WO}O={CCq*ln^oYa��℄=#vhz=|Y�pe�_ql�!<v: �$�r�}e��r$�℄$�℄zN�Smarandache &e��r�℄Gp� N> �V2℄��Abstract: How to select a research subject or a scientific work is the central

objective in researching. Questions such as those of what is valuable? what

is unvalued? often bewilder researchers. By recalling my researching process

from graphical structure to topological graphs, then to topological manifolds,

and then to differential geometry with theoretical physics by Smarandache’s

notion and combinatorial speculation of mine, this paper explains how to

present an objective and how to establish a system in one’s scientific research.

In this process, continuously overlooking these obtained achievements, raising

new scientific objectives keeping in step with the frontier in international re-

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

myself, which maybe inspires younger researchers and students.

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

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

differential geometry, theoretical physics, scientific structure.

12010 %mÆPUJM-5_�y�#_UNAH�2e-print: www.paper.edu.cn

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AMS(2000): 01A25,01A70Z{V$2-��D%Ix"\�"\XZ℄0$�ÆZ$Nx$\in�"\#ZÆ0{ D���j#�|$) 0&Bx*��V$2-i�"\#nZ�!QS{7#Æ0IP %I#Z$VpN��Z�,�.m7& � j0�bv#A�2-jgsy�D#�D�#6 1*�;5jg0\yz�2-�?�2-y �n767I�H�0\�$V.4�#?n=[eCa>4�D��℄#�!QS2-�v#�?�07I*W�2-QpÆ08V$o,�M%C�B_��IP %I�bxV$2-�o#A'�b�#3e1�,[u2-%IB;T�#C~�r$�zN2-|℄zNo℄$2-#4~℄$2-|�℄Gpo N> 2-#y5k�73�y5ZD��2-℄\�Q�$`x9�2-�3ON�℄%#�kW-g�>℄ �NHB{gm�>>�?2�Wqp7lSE#bg2-uC℄Y36�>℄ �NHB{gm�>>�?2� �2006 & 8 *#�Ik$!EBI��gm��{�>gmz� �Combinatorial

speculations and the combinatorial conjecture for mathematics!O�ZD��$�r�}e#C-pNm�$V$CXZ��r�.��r�p�3�,"Lg3b�}eMINÆV$2-��gm�� C$�1!zGkRQ�>zY{>#}Z<�p�;H�igmY�Ælgm�wT0J9�+�!tH�z��#NwH�zNW��2!Ælgm��}Zz℄g�>{>�B�5�RlgmHK�3WT0gmKW#�~9�(Y�>{>8��l=F 7℄��>gm �Combinatorial

Mathematics!�O�IÆ�#G)&$`F�3�)=℄UN#si6�E3F�Q�

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

Press, 2005 &!o�Smarandache J�-�B��Smarandache multi-Space Theory,kQ Hexis#Phoenix, AZ#2006 &!3GE# 2009 &6kQD?�gmY��k����BHYgl�(Combinatorial Geometry with Applications to Field Theory,kQ InfoQuest, 2009) &�#�x�rQHÆ0)�r - ℄ - �℄Gp - N> � Np[pN%I�V�4{|(N3x�<[u�r$zN2-�$`�1983 &#3{�6NBp $r�$#t�,��k6�$#x6U�%Ik$=:�~�{{Q$K�r�$#Q�,�xW M$%�k� Bollobas �I��<HB��e5�If!sIK8Nu`KEDl��R�2����1984 & -1998 &b 15 &#3N�6[u2z��/ �2-%I�9&bNi`F$KT℄$��℄9���℄�p�℄3#th�GTS82-#?6zNQz#AT�2-Tz���J�7�z�7=|�z�Hamiltonian zo Cayley z�#s6Q�`KEDlTN.�b`^�N|℄2-0R6Q�`KE�Dl#Nx4�|3',vs`^#Fn3 1 �Æ0vs�3�D*IDlN.��%#�8vsN.hh�biB�2-x7$�#FxI���b`^�2-%IFh��#�Qk#�,lSEFy�FB bu2-%IÆ0�8�$DK�V$DK�jgx#�x36$�`KEDlN.MINÆM%5.#NÆ�J� #?bÆeH|)EtU1g&n�lYj{Ag�'{'u��viQ�$��_i. zN2-J5℄D0R#x�$�Nu�D$�`K8zNN.�)kM�d��k%zNN.:|��#W�h�-p ~#�x��X�vv�� N<�1iD)I`�B�Nu�T\^^ Q�zN�V�Q�!yEQSzN2-�3D�J�3�,o . #EtU1g&n�Q�$��8zNN.�d�xCt1�#Dx�{�)k%$`y$`42-zN#h N.DlJ�)�9&�{Nu$��vj$&6Q�\INZ�>℄HB{gm>�?2� # ��zN$`GD)N�,Z#t3\E(xNu�$`m2NDÆ5�X#e��6�=.mNuI�# N.Dlo)Y3�4�YNgDC��8BgNtU�#bÆ��_�IxF2d��e�)��gzNDyD2-#DyD!EQS$`�2-{F'h/x^���2007

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n[hn%-88 25"bQz��K#F0&�XQ�zN��r$�x9%�bFx3[ 2005 &Z�4hyTzNA℄zNQz.Q��D�h�.�F)℄zN�B ��#Fx�_0)3ZD�$�r�}e#h ℄zNlSE1x�z��r�#0)3��$�r�}eN���Fq�bÆeH#.`"x�$�r�}e��x�℄Y6)3�vs�O��BaH{ Klein8XY`BK�#F1x 2005 &36kQD?��aH�8X{ Smarandache �kY`BKA�NÆ :Q{5#CQzb3�$For 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, .... �;��R�>DFH)53>�N#APS���1IR�>?�/#��I��R�>O4�2��'#��3>>!#+�3�7����Riemann ���B�%>�#(:#���<Q��R�6Æ�!bÆ&eFx�_0)32- n- !9�#�?�& 16QS`KEDlNu>�r9�Ae3\%�s�%Qz.Q���4{-�5'2005 &�N)kQ~���E4"#n3QA�[�r(k Riemannian Gp#s�?�e�r�J�2-%I��V!Smarandache 5'36QV$%�$�NuV$2-%�vs�2- 2005 & 5 *���#�N6�V24/�k$Y%I#h�NWr)$`���hCH} $|[uvs�{%IT�NWXj*(%I�y q�$I wI{G#�,��y��#F�NÆ($��>p)����b&BkQNWD?M�3{TNa�5�m#-}DÆ� Smarandache Gp��5#CnXZ4�36kQD?�3y�D�,6vs�O���Æ�$�r�A�#. 9&y wI[u�$2-#tn�37I8�$DK�k�(t,so#6�$4�7Q�x7�xb&=*3Io��n vs�O�#DX)|℄�5D�)kQ�WD?*(#s�u36Æ :Q�x�rA�UN) Riemannian �z�bW*(�e=MT�5�#ZD Smarandache Gp#℄ Riemannian GpJO#pe32-�DX

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26 K): DnS�X,h�� – �?hn| Smarandache K�.�>�5#sZ)�3a>.Z4*�vi� Smarandache Gp2-avB�#}D /=DbÆGp1XZKD�~=&2-�r�℄��h#3KE�x�r&!Qp2-Cn�0\#DYCnZD�N�$�A0\�xVz�zy5$h�x6ÆDX)b|℄2-�R�D�kQ�WD?M�B #b\Æ2005 & 5 *6kQopD?�bA#6vs�2-��si#316kQD?)N\$�&�#FPQ)vs�2-��6kQD?&��Tt��H Smarandache Gp#DYbQzb6k%�2-%I#a>N.��I6kQD?�5℄#�')3T�2-�$��#sA�(k Smarandache Gp2-�t3bi�%I>NW!$|)*(# )y(��. 3xYV9DR�$`#3�QV$%�$�NuV$2-%�>4{C%#sq36kQD?��\&�.�Cn�G)4{!T2-#N�s`3XZT�ZQV$%�$�NuV$2-%� DlN.�D?&�o�X$�%e#83�2-%IZ))<kr7�

2- ^ Smarandache �kSmarandache GpxNÆ>ZMuA'�Gp#CD�6bÆGp`^#N��\XZsi0��y0�#.xZ$ÆZEQpy0�#b6!�Gpxy$`��#:��bxNÆH[e;�℄N�!�Gp℄�LgQH�u:�C. ��Gp`^#b�,nB`p��3�na�n#:��}�C.��xÆk�#F�x e��tysbÆC.�Gp`^7N67&�#�x6,�M%CyX b6�7&�DKBPR�"�#oxbÆC.�yC�u^�e;&?,�M%"EkQ�S�pQ-bZAS}�;�e;#A_n,�M%y5��#_�),�M%�{�DK�3ZbÆ=g�GpDj)#h D"or7&�o,�M%�lS q#Fx�$%I`�D�x�%Qp.�F��x6�Q{�i^�#32-) Smaran-

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30 K): DnS�X,h�� – �?hn| Smarandache K�. �r�℄9�E��J��ri`#lSE1[2NÆ,�87&.mb64^ �^$&e#bFx6Q5nNu^$�I#6�5��VW℄�B℄Y��6bÆ.mQ#,�87&� %�RlSE�x l�R�N|℄#CN�Pz�R#? l�R#�,�.m|�Dw2�=# Y#3ZD6�ri`#E�D_< Einstein �J a8 � �#��D_<Kq� #q1�"#1x{��ri`> KI��$Q36~n�`^vq|ej��0`^�g�Q�pyg#b�,�87&�.mo,��3�nxN���bA#�J���rj��Q311[)!�� Einstein j��Q3�LT{#�xW!�|�Nu!�j�� Einstein Q3��#6 Schwarzschild �8-��Reissner-Nordstrom ��#�x�rQHXxu�/=�r Einstein j��Q3�NuW�#8�rj��� ��"�bÆ&e#�TDY63�NN>�r9�E��JQ3�N.#:�#kQNa> $�`K�&Z�V7p7 �Special Report!&mKDNN3��gmd��Y�z1�� �Relativity in Combinatorial Gravitational Fields!�.Q# P��)bÆ�rj��A�!�j���>N�lSE#bÆ�r��&e�XZnx8KM���r#p��r Yang-Mills Q3#�?x�5> 2-�32009 &6kQD?��gmY��k����BHYgl��\&��)NuB{Q�#bQz��Nu3G0\�D�A#F��k�2-D `^�7��?KHR!q��$:�,Bg=&B;T�2-~C#S�V$2-#Wnx�P V$2-#Eb��NÆV$_&�T�#��yINÆg�2-&Co$�&e�{y.#Nui℄FO~3QW!0 DlN.����G℄ sjZ 3#ysbv 99%(y��P#8�"V$��,�M%h�)fIx�Dn.xp�p�#2-�R�s K\&.x!2QS`KENN.Q#GIN���1 2Dl&.x Xw-� nyj&DlN.D��%?R℄�#IuI&......#KY��# jg�\bÆYlxz�S&��bv�V2℄�#HvV23�r��V2%IXZQH�Qz�0\#F�V2,�7R2�owI��0\��Tx!\0\#Bk=�V2,��%4E�W�G�B`7G�V2,�#Wnx2-3GV$,�p��Nxx�ym�ul�3��=s$h �y|V23�#y�yWp2-�Wp�XQ���$�x9#h Cn�Tx,#�TD_<\+���2008 &#36� Mk$�&~ o2-\NZ$�O�#jx)N)+i8,�%℄�ÆZ�℄�C"#,�lSE℄ =�ÆZ# NÆ

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��4gl�q℄�A�– >��>Ye℄8�, 33-36

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?��?[f^9 35�xA �&�N{D.�F� �Ix – D�A)CA�0\&B��x$KD�q��T4)� �q�#�n�&m3q�&CZ�D{#�N{�#gmo�N�>N��6#37$�℄z� �QA� $�M^ i) � f(x) �y7- [a, b] W�:&ii)f(a) = f(b)&iii)f(x) �u7-(a, b) p}T#�� (a, b) pAYC�Re c#uX f ′(c) = 0. qTZ�D�q�&C#s~)� gm�Ix�gm i) Kq) f(x) 6 [a, b] EXg<ko<pz&gm ii) AKq) f(x) 6 (a, b) E�N� c g<z&gm iii)AKq) f ′(c) N�b6�bA# 7\AV� #1� f ′(c) = 0�bF4Zx3℄%|�$�0u �K\lSEFx� #}yTDyl℄\� �A�D��\�$���$�N!wK+;B$�sm&�B!* 3n3+�℄B0\��A0\� ��8&\{"#b$���(yX����\�yI�e��e�8l�D{$T�* �� �� oK\�8* �� o� #Dnx�\�3G#N<s$(%�Z"`#B℄���xZ{�T�K\#D��xI NÆ K?:�:�#b8�\x<y���36$K'{'SZ�.m|#�`m�p�7I�\���� #8�\xm<�d�u�p�7I�\���� #NQzx!�87I��\QHXZ:��Z�&5NQz#D=[ÆKDl�n,�Nug��\QHZ��℄#x�u!���~*PW���\QH��N{=l7I���� �37I��\ljl�#bAG��R# IP7I��\&C�b�DlY6$�N!�HNu�D�K\��N�#�XZ?1�bu�N�>�K\��h�i^��6#�

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3

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Page 46: Mathematical Combinatorics & Smarandache Multi-Spaces

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Page 47: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– A Foreword of�SCIENTIFIC ELEMENTS, 37-38

A Foreword of�SCIENTIFIC ELEMENTS�Linfan Mao

(Chinese Academy of Mathematics and System Science, Beijing, 100190, P.R.China)

Science’s function is realizing the natural world and developing our society in coor-

dination with its laws. For thousands years, mankind has never stopped his steps

for exploring its behaviors of all kinds. Today, the advanced science and technol-

ogy 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 natu-

ral world have no an answer, even some looks clear questions for the Universe, for

example,

what is true colors of the Universe, for instance its dimension?

what is the real face of an event appeared in the natural world, for instance the

electromagnetism? how many dimensions has it?

Different people standing on different views will differently answers these questions.

For being short of knowledge, we can not even distinguish which is the right answer.

Today, we have known two heartening mathematical theories for sciences. One of

them is the Smarandache multi-space theory came into being by purely logic. An-

other is the mathematical combinatorics motivated by a combinatorial speculation,

i.e., every mathematical science can be reconstructed from or made by combinato-

rialization.

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

Page 48: Mathematical Combinatorics & Smarandache Multi-Spaces

38 K): DnS�X,h�� – �?hn| Smarandache K�.different part of the elephant. So, actually the elephant has all those features what

you all said. That is to say an elephant is nothing but a union of those claims of

six blind men, i.e., a Smarandache multi-space with some combinatorial structures.

The situation for one realizing the behaviors of natural world is analogous to the

blind men determining what an elephant looks like. L.F.Mao said Smarandache

multi-spaces being a right theory for the natural world once in an open lecture.

For a quick glance at the applications of Smarandache’s notion to mathematics,

physics and other sciences, this book selects 12 papers for showing applied fields of

Smarandache’s notion, such as those of Smarandache multi-spaces, Smarandache

geometries, Neutrosophy, � to mathematics, physics, logic, cosmology. Although

each application is a quite elementary application, we already experience its great

potential. Author(s) is assumed for responsibility on his (their) papers selected in

this books and not meaning that the editors completely agree the view point in each

paper. The Scientific Elements is a serial collections in publication, maybe with

different title. All papers on applications of Smarandache’s notion to scientific fields

are welcome and can directly sent to the editors by an email.

Page 49: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– &������}�� RMI ��, 39-44

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cos 2x, sin 2x, · · · , cos nx, sin nx, · · · �ox q�`℄� f(x)�N�` 2π#yx2π i#XZj�n��0bÆ q!-*p$

a0 =1

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∫ π

−π

f(x)dx, ak =1

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∫ π

−π

f(x) cos kxdx,

bk =1

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∫ π

−π

f(x) sin kxdx, (k = 1, 2, 3, · · ·)K0)7�E�$a0

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∞∑

k=1

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{x2, x ∈ [−π, π],

(x − 2kπ)2, x 6∈ [−π, π],b�#k xn�#n� |x − 2kπ| ≤ π#A f ∗ (x) xZ 2π �`��`℄�#ezjkQGz 1$1�-Ck5Æ%�#1,�"1�1985 �

Page 50: Mathematical Combinatorics & Smarandache Multi-Spaces

40 K): DnS�X,h�� – �?hn| Smarandache K�.-

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−3π 3π−5π 5πx

y

( 1bA#3nXZq f ∗ (x) 6�l�EK0x�JE�$a0 =

1

π

∫ π

−π

f ∗ (x)dx =1

π

∫ π

−π

x2dx =2

3π2,

an =1

π

∫ π

−π

f ∗ (x) cos nxdx =1

π

∫ π

−π

x2 cos nxdx = 4 × (−1)n

n2,

bn =1

π

∫ π

−π

f ∗ (x) sin nxdx =1

π

∫ π

−π

x2 sin nxdx = 0.9sf ∗ (x) =

2

3π2 + 4

∞∑

n=1

(−1)n

ncos nx.YE��� f ∗ (x) Wn�#6 [−π, π] E#1�

f(x) = f ∗ (x) =2

3π2 + 4

∞∑

n=1

(−1)n

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

�| f ∗(x) � ��| f ∗(x) � �-

?�?J� 2- f ∗(x)�W�

Page 51: Mathematical Combinatorics & Smarandache Multi-Spaces

'���� �~�� RMI �� 41�$3w2�08(� 5N4�� r08�GH#xorV$2-KI��?x|g�ox�xbN3\&e�b�#3n r�IZQx|g�6��N��℄Q3�nx�x|g�$L[f(x)] = F (p) =

∫ +∞

0

f(x)e−ptdt.e�D �TlY6C��=f}� �#buW n��℄Q3(� n�Q3��T#qTP8#3nX�L[f (n)(t)] = pnF (p) − {pn−1f(0) + pn−2f ′(0) + · · · + f (n−1)(0)}.~Y#8-N�N���` �℄Q3

dny

dxn+ a1

dn−1

dxn−1+ · · ·+ dy

dx+ any = f(x),}Dj8x|g�$y(x) → F (p) =

∫ +∞

0f(x)e−ptdt#1XZ�0N�Z L[y(x)] $s��N�` Q3#bA1XZ�� L[y(x)]#[?1� y(x) = L−1{L[y(x)]}�T 2 xx|g��Q3 y”(x) + 2y′(x) + 2y(x) = e−x �_<gm y(0) = y′(0) = 0���I i) j8x|g� L[y(x)] = Y#8Q3$0�x|g�#"` y(0) = y′(0) =

0#�> Y �n�Q3p2 + 2pY + 2Y =

1

p + 1.

ii) �Ez�n�Q31�$Y =

1

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

1

p + 1− p + 1

(p + 1)2 + 1.

iii) �x|$g�1�y(x) = e−x − e−x cos x = e−x(1 − cos x).bÆ�HXZ�pzlq6Q [4]$

Page 52: Mathematical Combinatorics & Smarandache Multi-Spaces

42 K): DnS�X,h�� – �?hn| Smarandache K�.��℄Q3 l℄��n�℄�

l℄��℄Q3��-

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S S∗

X∗X

-?�?

A

A−1

RMI �Ax�$_i0x�QH�A�D�:℄0x6�$Æ���4�(XZM|�T 3 &��$� B� B℄�Æu���BGp#e�A0\�3\&e1xRMI �A�0x�:xpzlq#n3nXZ�F�M|bN�$

Page 53: Mathematical Combinatorics & Smarandache Multi-Spaces

'���� �~�� RMI �� 43

Gp>N0\ n�>N0\�|Æn�>N�|ÆGp>N

-?�?

n�lq n�P8Gp�z6��N�` �℄Q3i#Fx RMI �A�:℄0x#epzlq6Q$�N�hZ` �℄Q3 Wl=hpWl=hp� �℄Q3��

-?�?

{�g� �Q38n>NT 4[2] �n�rN#yH�A0\�QHx RMI �A�:℄0x�b� r�INQe��℄��\0℄�!�QH�B*�℄�QH#1x3N�7^�,${un} = {u1, u2, · · · , un, · · ·} 8n�pvE�

u(t) =∑

i≥0

uitio

eut =∑

i≥0

ui

ti

i!,s&� ui := ui�{`- _�℄�#�`- {�℄�!#&�qT2-_�℄�o{�℄�#4qTEz�8n>N#L9${#1XZ�|�, {un} �Æ ��pzlq $

Page 54: Mathematical Combinatorics & Smarandache Multi-Spaces

44 K): DnS�X,h�� – �?hn| Smarandache K�.�,0\ vE�0\

vE��Æ ��,� �-

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i≥0

uitiA�

atu(t) =∑

i≥0

auiti+1,

bt2u(t) =∑

i≥0

buiti+2.BZ

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

i≥0

(ui+2 − aui+1 − bui). 7�, {un} ��L>N#1�(1 − at − bt2)u(t) = u0 + (u1 − au0)t.BZ#

u(t) =u0 + (u1 − au0)t

1 − at − bt2.N 1 − at − bt2 = (1 − r1t)(1 − r2t)#A�

u(t) =1

r1 − r2

(u1 − au0 + r1u0

1 − r1t− u1 − au0 + r2u0

1 − r2t.

)q u(t) KI0vE�#1�un = (B − aA) × rn

1 − rn2

r1 − r2+ A × rn+1

1 − rn+12

r1 − r2.

Page 55: Mathematical Combinatorics & Smarandache Multi-Spaces

'���� �~�� RMI �� 45Wn�#8~�5���Xt�f0\|I��, {Fn}#h� Fn = Fn−1 +

Fn−2#F2 = F1 = 1#9� A = B = 1, a = b = 1 � r1 =1 +

√5

2, r2 =

1 −√

5

2, BZ

Fn =rn+11 − rn+1

2

r1 − r2

=1√5

(

1 +√

5

2

)n+1

−(

1 −√

5

2

)n+1

.bxN��,)%�p5#�, {Fn} -Nh(xn�#&? Fn "xqT7 �{lq���$wTkGzZ[1] �����$QH"u��[2] T\�'���rN�E��[3] ;/k$�$N��$℄B��Q��[4] �&I$r'Yqx~~��$� +��

Page 56: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– lYC%77, 46-74

6|�po_d/J(H℄'�E�#�#r[#100045)jT: vh�B�Q$l�R=9SCF#℄_QmuYz�#�℄_�i�lC�PC%k�9_M �9 C%ki'�H��f℄��V%�! C%k#;U��>�w�iC%T�V%>*℄Ra�Ci0P#�!VÆ�℄RaY0'_%#{=F.Yup���N#℄R=�>Cq*pR#B{=F'rpF�v�p|i5,yrYF>.uW�}�#z��#�Y.Yu�qpRlYK3ql�!<v$9SCF#_�i#lCg9_�#lC��#�9T�#'�J�#'�E�0PCk�

Abstract: This paper historically recalls each rough step that I passed from

a construction worker to a professor in mathematician 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 Committee, a consultor, a deputy general engineer in the

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

& Bidding Association, which is encouraged by to become a mathematician

beginning when I was a student in an elementary school, also inspired by be-

ing 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.

Page 57: Mathematical Combinatorics & Smarandache Multi-Spaces

n[D&88 47Z{%3C #Nxx3WoIro�,�$$K�2-�!N{!#Fx3�l��P/Y#;�!lY�>Ce#dR;�>CeY(7 �:℄|(�2009 &NZ%3XjC*#3nB $�u��'�E���#℄RE#31�'�E���g.K���\��4#qTk%lSD\�Xj�0/��)P��℄Bou�#jj)$��-��Q℄�#�=$�0�23# ^,6Cn6lS%I�|�0\ou/#l(3�`k�3�r7I8�'�E���� �ZA�,��R!�#NNI)℄Bo�h#$�n^^lqy�Y#$|)�,6℄5 �6$!I�CE#�vC*�C*��-83"$�s=~pl�Q�S�$R�?~1�#pY�H,�#pY�H,?#�pY%\j#Ro�oa�?#?NT$#�\+R?=~z$>�Ya�� %I=g&#3[N�p %,#�!"C\�e%kI%3 �pNr)3p:%3 oXj*(h�! �&W�t:%3 #x|?�$�> o%3C &W$` NR�~�oIv%tsÆ� NR#��,H&i`��D0 N��$%I`�>#"��,4lG,�\℄Guo� ?l��$K.uyX℄�V�?y��1980�1981 &��$&�Q#9&C"T)D�℄�`#t3o h j8k$rm$K�1981 & 9 *#SA~pNi6v&h� �rD��Q\"��%#3�X)℄b#th�ZD�#} !:��oNu5Æ6vi{�%I�r)Qp B4/*(�0i%�

1981 & 12 *#Qp B4N*(�kM�X%#Z_<%3pN��D�v& 12 * 25 2#3�X)%I#��=eD*(�5ÆNj{|)vi6SA�Qp B4N*(�bNZ�X%I�&�+g=,#*(6SA 9*��%��v) `$�*�8�~�#s-Qmr���℄E%Æ�1982 &R��#%8�~���#3Z℄E|)Qp B4N*(=I16#%Æx[5%#�3Nj℄|�*(=I�#�� 4-5 )5Æ�{�i-Qp B4=*(~�V�#7"F�Qp B4N*(~�V�i)�X�#D�6%IE[:NQ3�Cny[:3#h [5%${`N&�(ol#sij*��8Xw��Bg=�z#!k�%+g=�#℄eD%Æ��3vie$�%#h {�1x�%DR#eK�[%��tFoxh h [u7Ie[u�%Æ#n3�{D�&I$rS=?�g,\�

Page 58: Mathematical Combinatorics & Smarandache Multi-Spaces

48 K): DnS�X,h�� – �?hn| Smarandache K�.4Æ%Ip�,�m�;5�bY�xT#3Io)3�p %,\-�j`�%, xNj#|SA't��=`%�E9�bxN�t�ionh�#pN%`g℄�S# K��D��j#I r�%Æ�[5%#!��DX9q{�[fg# �z��%e%Z)%Iz� N�*%I��#%4XEs�34|)+g=�#����<)#h viN�*�\+\F1xgb1�z�3A�j*[%47:�z�0[�$��Æ�vi�ÆyfV#N\ 200-300G�Æ#}� G|z1 [Q{�1981 &R�36SA��Æ�[��Q#���BTd�#N\ 700 =G���#F} )=k=z!# Q�#57Nj�x����bi#3j$�NZ~'t|SA~�#4[SA~�(&{� 9��{�W#j*�NZSA~���Æ�[Æ#℄6�Q#��4\�>dB� N��{0�;r=!f���>�z�7���G.Birkhoffo S.MacLane ��A Survey of Modern Algebra�(4th edition)�Weixuan Li ��HB��1x6b�i`[��B��\zNÆ#Nx| 1984 &!:U�%Ik$=:�~�$KzN#3�3D,!�vio�E>ljxb< ��Nj0�M%kUN��#,n[��D$K.�sm#Zyn!NpN��D�Qp B4N*( %Cj).�wK9#~G)�.� {o 82 �k$\u℄#$KB�n��B�Gp�.�℄3��X)GZwK�#31��h�bD4�mu#h box36$i`$�Bg�℄3�x�x�Ei^$K"I}~���0\�>$��Ce��$K��$r��5℄~3#s�2�h;r=!f��>�z�7��E�Nu K\��=%(s~3��$3Gg#Q9�F$`Y3UNNuB��$�0\�?bNi`#$�$�Gl3G#Fn�3,TyH℄�Æ#8�{3XZO6N�℄�JO�|.M8�$0\iQ)N�3G#℄6���N�y�Q3o�$L�H��N{nx��,8gs� ��H>�>4\�+�#��Ez�u�QP8.q��d�#�u�Q��" f�P8QH.q�QH#x�I`y)Nd��8gs� yl1�3$)Nd�lq�|#siO{CD?��H>�>�%�+�(3ZZ`k#�{�(3yÆX#83�\)<k4�#CF0)3bNi` 1℄77$��Tr��t3�\w%Ix[5%#3�DT�g\w%I#�?� �+37I�[5%�Gh3\(D* #Nx4[5#6CC���{�[ED O/#6{��ED ;&BxGr��N {��#NQD �j{#s/!&=x6�zkE�em�p�~*DNZ|)#fEz�,�!�2 )G�*�#33\

Page 59: Mathematical Combinatorics & Smarandache Multi-Spaces

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1982 &�L�N`#3�����+�{�[fN%I#%^�F3[{�[EQ{#J|N)℄E|��%I�k$\�Cyy�#03���%3�Nez4#6�xqL4��xO?R��xg�B�#3NN�)�I#:�Ck��;��M2C#��Wq#3y~�|Ni�7$nV`�i+M�A�*Wu_RiE'b�7I6$i`�yDo� �`�FJyao)��i`83`�B��x~�$��\� #��yWG>qs(d#?NJ(qyWie – bxCvi��=,"T���,bN\y 2|n,8#�G�#Fy 2|n,��-�2005 & 10 *#v34Z$|$i`\+��Qi#b)� W!�t#3(s$m23|C���{lg)8C�*&�1982 &�#!TQ�oQm#3 �(o)�bNi`#�=si�X%I�,�x$m>N#.x��0)gN��%Æ#℄6�e%����#D2W`�#.x�>N�&I$rS=#Z�${� x�[uK�C %I�biNueD*(�5ÆW^^�$)e{�B6*(�3�X%IixZQp B4/*(5Æ� {�#YiQp B4/*(W!'m#h�XZ�x�>N#} R7I�g�0)�3A�4�XNZ�Q#Z�g[5%bÆye uyDN\�wI�1983 &

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2005 &#QWDK��" KM%3Xjvj #�veyQ� FIDIC℄N.m#C 2007 &=z��HvF[Ji℄�[lC'�^<�dh7�o�HvF[Ji℄�[lC'�h7��CnY)G�,Tey)NaB�#&��vNuI�&WUN�36�Z%EZ)�=���pe#n3�{x���)b$a.m�ey%I#0)b$a.mAenx{���Dey&W�2007 &$a.m!QW 9 �|"�rD.=z�3biy&`m|#338XjvjHIHK� �olN!<~�[I,�#8n�XjvjI�DK7U%j|KMo_�Ix�x#36 2007 &�V�)Xj*(�%I#�QXjvjv%�N#s 2008 &�&Qoop0 �v%N�%I,��2008 &B#QWDK��"A�b$aj0.m� G%e#Y)G)e�&W℄%�u#3�5�N)s�|?4��V.m� u��uT3#DY�3Nj|?4��V.m��)s�u�yg#?Cn,6u��E�F66 ����g� $Q�s�G�Nu�3peCnb$Q��,u#Zwgy| G���QWDK��"�s�F. �E�F66 u�yg�CnpebNQ~3{u#?box3��I#h �E�F66 lSExHI�r%3e%W��NVXjvj3� K��3�rHIK�#8�E�F66 �g.�5#�rNulS/��)S8|D�u�#|)�%`�_i�p�oxbNZuK#n3��Z6Q�XjvjI�|)<kZ��)t~*-4!x℄%IC"%Fe6CNZj0.mC*9#k�X G%e�)t~DK��"�G)s�l(`k#�R�dNX�pe�2J�y= #Cn. 3�uK℄�orm�d%�nCn�D�#3 C*9u�)4��V.m�Xj.m�Q℄�#Nus��N4/0)t~xCC�s�#0Cn[��Y{�� #�)z'�E�Cq-�~$ #sN4lqu℄�58Cn�%Iy�lD�bN&36Q�N-I)� 60 �uK#m�g|)��,�2009 &B#WDN�r)FG� e[u%3C �,�vNZ `$`�XjC*#��x�9_C%C%#0M�)Y ��2008 ?!o�HvF[Ji℄�[lC'�h7��2007 &?!� N`~U?p k$�N)~�u�)�9_C%C%#0M�)Y ��2008 ?!#eLA)Xj�PZorsC �5�G)��q�u℄�5�w#x7|?�vC*�$)���3C%�3(Cn3$K^Q�3M)NQ#�5Cny%�w#$)��B�BU� B`uÆ℄#eN)��83"$�ws�_ÆY=_#l�,L)sR"=�k=~pLJ$�H^WvsRu=#i��sY=~pL�}q{)J$# ℄sB|=Y}r4�qiY��z� �)bNZ!<#36�{u℄i#(%uT�5$�#3�u

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74 K): DnS�X,h�� – �?hn| Smarandache K�. yTx�Z�#xZ�7Iu℄�5o�D6�/�#h o�u℄�5x+63�5���3�,o 3�$2-I �,��k#?3[u%3C I r��3#MIx[u�$2-�!Nr�#ye"6$�4�si-�) ��bv�'*pF�v�p| x \#?yik�0:#Ai�n�,�,\M7k"��j℄�#Ax3 1[N�p %,DK|6�#�?lY,\Zz�|(oQH�b56bg=<i:�7IN\i"$�tmpu~p�t>#Qm~�# m~�Æ#um~6:^#3m~��#�m~�)(�#�wl� 3e#blSEx3nj�,#Wnx��k�,�,\y[�si#bFx3nj�,�0�KI#�DjjH&Ae~�`���#h &xQW�${�G��

Page 85: Mathematical Combinatorics & Smarandache Multi-Spaces

��4gl�q℄�A�– �Bw�d�Y�mRz#�>, 75-97

Rb9R[�| W�8�O�SmarandchetJ:Qa ∗d/J(G\z= �=z�Cz=Bd "qZ 100190)jT. �W=z#�muJuv#�Bw�>%R;B�t9�w�>YRIGDR�B#�Theory of everything�+I�5��Jw�>%Yz��#tW=z#b05#!#+i}�Bw�H�B� M- �BY:��w�>%Bp^sU#9�+IGDR�BYne�t�>%Np9�{=�zgY�>�B#l)RY{�#i}�mRz#Yw�>Y℄9��Æ$#ND��Ce{�!Y_��t�>%�Np9���mRz#Y�>�B�r!W#w�>%}q�02#��B{ M- �B9�YBp#�>%O9�$RI}Z 7℄�mRz#�>Y�B#+i}SmarandacheJ�-�B#�glYz�2}�Bw�>%(�5Y#OB#q℄RI�>�B#�"/X�D�Bw�Y55NWÆX|�vhYNNfY�t�DaLZ+R�BY�is^�CeYNNj7����NN��Z�NNCe#^\, �H}Zw1gm�>����:�_%H�UYKoql�vhNN9�tq*(W�P℄1eYRvVO [16] HY��%�

The Mathematics of 21st Century Aroused by

Theoretical Physics

Abstract. Begin with 20s in last century, physicists devote their works to

establish a unified field theory for physics, i.e., the Theory of Everything. The

1E5=\b�Mfl�=zGA=�>1��2006 % 8 )"_�"�Hj# n*J^� }#N��2006 % 3 ) �2e-print: G\z�Ag��"200607-91

Page 86: Mathematical Combinatorics & Smarandache Multi-Spaces

76 K): DnS�X,h�� – �?hn| Smarandache K�.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 estab-

lished. The purpose of this paper is to survey its historical background, main

thoughts, research problems, approaches and some results based on the mono-

graph [16] of mine. We can find the central role of combinatorial speculation

in this process.!>v$��kRH N#M- N#�`^#�zGp#SmarandacheGp##.%`^Gp#FinslerGp��P% AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50

§1. `v`llI�s{�N3 1.1. 8QÆJ%K`^�x��m��{y#Q$� = x, m = y#� = z#AN�%K`^XZx=����{y#CJj (x, y, z)&"K`^�x��m���i^{y#6RQi^g% t#AN�"K`^XZ�xJj (x, y, z, t) {y�%K`^Aeg�kz 1.1�qi^MIN�g���#A,�6N�iZM|����KlSExn����N��z (section)�

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

Page 87: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 77

1.2. `vu{xaiQaP7,��v&�A�#Wnx Einstein �j��Q3#> $Wp�)���kRH N#bÆ N. #��j!N��, `-K�C.�-`^#b�`^:� ` ���gm�g�#nb�C.��� %��`^D\)RH�r03\�5�zW %#�,� %-KQ#3\�5r0)BBGÆ r>�#6!T� 137 ^&�9��0)�`�jR���P7 Hawking �A�#RH�\T3�,�oA0"��R ([6]− [7])#6z 1.2 Bq�

z 1.2. �oA�0"z 1.3 # P{�)��~kRHIo�9��HW#x��\�`,�A �|���T3�

z 1.3. ��kRH�T3

Page 88: Mathematical Combinatorics & Smarandache Multi-Spaces

78 K): DnS�X,h�� – �?hn| Smarandache K�.P7kRH N�P8�R#��u\Z��9�T3k�6Q$y� )��kRHIoN� `�`^#N�& 137 ^&{RH���o,� #D�joi^ 0�D�7^��,.o7^k�u.��{�y xWs��$o> �KI"�vi� q�i^[YRHIo# ^F[YB=HWwk�vOIÆy )i^ 10−43 |#u.x 1093kg/m3#,.v| 1032K�bi���C.?�8-#}�i^� ^o `��x%VÆy )i^ 10−35 |#,.v| 1028K#��D\)�ZPU#ex(6 10−32 |�i^�Dk) 1050 X�PUjj)���,��5�\#bi9&j�W[uN��℄�D{#t~ %T�#���;�o�U�(�$℄I#�\��5Fh��℄�bNi`�5�y�|�T3k%�#=0�5K=L�5#e�,.v�#�����5oL�5a�Q~#17Q)}�5o�5?G�My|L�5�y8-�Yi����{{Æy )i^ 10−6 |#,. 1014K&|{Æy )i^ 10−2 |#,. 1012K&b`^#�Ix�;Ixo�UIx�n�℄I���DY)�Æ�5#~,.y��1010K ZE!#�5�\bi^(x<1�#DnqTa�J,?a�(�#�5bi�hDY���Æy )i^ 1− 10 |#,.v�& 1010 − 5× 109K#�5oL�5��5oo�5a�iQ~#�\)k%�H5��5ZAL�5#3\�5Io�r0�5m# %ZH5nK��pDY (,nQ��At�o���/�/�6b�%r)�$JoÆy )i^ 3 ℄�#,.v�& 109K#x(HW|& 1 H&kp#��=0>�r0 X#mLnnd&Bpi�#��%��5�oH5��℄&g|) 10−9#nKu.0&k>�u.�U{���2�Æy )Bpi�#,.v�| 108K#�Æ�56a�J,h %y<Wy a�(��H%�Q~E�!#[bij, ���Æ�5�� .1�/��~bi,.0&y�#H5�<1� %2;-p18�0��5#3�`��5I�#BZvih�X DY�5�B�8rÆy )i^ 1000− 2000 &#,.v�& 105K#>�u.IoknKu.�~���HW#H5|g-pN���6N��0��5!i(qh�jj�=_�tn?net�Dk? %ep#~�3.:���HW?�nDk#buH5�t�F1y5Dk? %y5ep�8rw D��s'jsN )i^ 105 &#,.v�& 5000K#>�,.Io

Page 89: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 79�nK,.#B�oB�3\�5��℄zK7|��k&!TN5�&#~RHB`�\�H5� %1v|)y<Z2;�5W�9D�5�3.#��bi1�8)H5o�5a�℄���Yin#C��g0)x��#,.k&v 3000K�[biIo#�5Io�0#tF} �\���2������2#�x6�N�|��0�#6|��|�0�#|��0�#�|��|�\)�7ys�,.�"Tj�"��2Ax6"��RD.�N�J,�RD�0��F<Jo )i^ 108 &#nK,.v�& 100K#>�,. 1K�PF!�*F�KF1k>+s?))i^ 109 &#,.v�& 12K#J`�n�03n�{A |� $�nzA� )i^ 1010 &#nK,.v�& 3K#�N>�,.& 105K�1.3. `vxaiQaZs{�N3 N> �l<> 2-l�>�~=Æ>��5#C�5 e�EiY u oQiYd /0�BgtUBDY)$Æ{��5u^a�Ix� N#b1x Einstein �a8No Dirac �%5�$�a8Nx{�j�� N#N<x2-��> $&?%5�$x>�A�5Ix�� N#Hv�U���mU��;mU��b+ÆIx�/0)�5u^a�Ix�3\Ix��[E�tUBg&nIo#�=> $W#Hv Einstein \,Nx��uNb+Æ3\Ix�#CuNa8No%5�$#p�> $�kuN N#C.Z!�DY�Theory of Everything�!T 80 =&�2-#0\Nxh��|�_�A�0\���6J a8Nx>~A��� N#6itN�J?N�w%�#eYN>�x��℄z�&?%5�$x>�A��� N#6�5��5�5�#eYN>�x�>℄z��?�A:2S82-#m{),�.m4��Nu�0\#℄6}M}\RYE'M^�\R#p|Y=}M'℄qMF�w�U�)}M'F�i��HY}MY^�\t|Y',YMF�A�H℄Y 3 ^E'

Einstein P7eZD�J a8 � $(pw�Y=�G^�y�Hop�BY/wo�t� $�R=D�Y7�pV1�id�YGkw�!g}�}7�Yp�)j��Q3#CRµν −

1

2Rgµν + λgµν = −8πGTµν .�r��$� , Cu#+u℄ 104l.y p#}MHGkReiR=eYGk��t

Page 90: Mathematical Combinatorics & Smarandache Multi-Spaces

80 K): DnS�X,h�� – �?hn| Smarandache K�.s��#Robertson-Walker�|)j��Q3�NÆ�8-�ds2 = −c2dt2 + a2(t)[

dr2

1 − Kr2+ r2(dθ2 + sin2 θdϕ2)].an���- Friedmann ���!T=&�`.A�#Hubble 6 1929 DY#,�3!���xN�2dX3HW���#9Q�j��Q3�X3HW�0)> $W��DQk#Ce�_<gm

da

dt> 0,

d2a

dt2> 0.3ns~#:

a(t) = tµ, b(t) = tν,b�#µ =

3 ±√

3m(m + 2)

3(m + 3), ν =

3 ∓√

3m(m + 2)

3(m + 3),A Kasner .K

ds2 = −dt2 + a(t)2d2R

3 + b(t)2ds2(Tm) Einstein �Q3� 4 + m ! `��N< qQ#b��sy �D 4 !X3HW��t�xi^(T8-g�t → t+∞ − t, a(t) = (t+∞ − t)µ,3n�|N� 4 !�X3HW�#h

da(t)

dt> 0,

d2a(t)

dt2> 0.BgtU�DY� M- N �AE�0\��)3G�bN NYN�5yx��?x!�ys� p-a#C72 p �Qk��.�5`^#b� p xN�on��1-aN<-I#2-a-IXa�z 1.4 �D) 1- ��2- �Ae0"�

Page 91: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 81

? 6 ? 6 ? 6-(a)

? 6 ?6 6

?-(b)z 1.4. ��0"P7 M N#��P\Ioi�����`^!�x 11 !�#kRHIo�#e 4 �Qk!6B=�wS�4P#?5� 7 �Qk!A6B=>��?p#bA�03n�`M�|� 4 !~A��oMyk� 7 !�A���4 !~A����Ix�or Einstein j��Q3#? 7 !�A����Ix�orxYQ3#~Y�|Qzb��N��Q 1.1 M- �BYp�}R=�O=eq8R= 7 ^�- R7 Y 4 ^�- R4�nx� 1.1 o��`^E�"j����gm#3nXZ�|Townsend-

Wohlfarth�� 4 !X3HW����ds2 = e−mφ(t)(−S6dt2 + S2dx2

3) + r2Ce2φ(t)ds2

Hm,b�

φ(t) =1

m − 1(ln K(t) − 3λ0t), S2 = K

mm−1 e−

m+2

m−1λ0t�

K(t) =λ0ζrc

(m − 1) sin[λ0ζ |t + t1|],b� ζ =

√3 + 6/m. �i^ ς _< dς = S3(t)dt#AX3HW���gm dS

dς> 0o d2S

dς2> 0 C�|_<��zP8l�#: m = 7 AHWh5 3.04�

Page 92: Mathematical Combinatorics & Smarandache Multi-Spaces

82 K): DnS�X,h�� – �?hn| Smarandache K�.[�$|.u#� 1.1 ��lSEyx�?x`^#~YjS��$0\x }�C�+MRI�>�-#�HO=ekb-R= 1 ^ZWY�-'x�l�#6RbA��$`^b6#�DN�yx3n2�\+M�|�`^#Fyx3n6!��$�kT�`^#�66 3 !` `^#j��XZlq (x, y, z)#DyX H[N�!�k� 1 �5`^�§2. Smarandache tJ:�TQHN� r�0\$

1 + 1 =?67&�N#3ns~ 1+1=2�6 2 ��08℄N#3n�s~ 1+1=10#b�� 10 lSE�x 2�h 6 2 ��08℄N}�$�08�2 0 o 1#e08KA 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.P7�k�uk��^� �^$&e#3n�xNÆ�L&!��<$� &e ([18] − [20]) {��Mpb�0\#��℄B 1 + 1 = 2 . 6= 2�3ns~ 1, 2, 3, 4, 5, · · · bA��/07&�N N�6b��N#P7���KI#j��- {z�12����T�#C 2 ��T� 3#Q 2′ = 3�sA#3′ = 4, 4′ = 5, · · ·�bA3n1�|)

1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, · · · ;si��|)1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, · · ·bANu08�p�[?6bÆ7&��08℄NQ#3n} �| 1 + 1 = 2 ��N�Y6#3n$:NQ08�� ���N�?r S#8 ∀x, y ∈ S,� x∗y = z#`&x S Eb6N� 2 ��rsK ∗ : S × S → S#n�

Page 93: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 83

∗(x, y) = z.�xz��Qp#3nXZxz3bÆ>N6VzElqD{��T3 S �j��xVzE��lq#6R S � n ��#A6VzE1� n �y-`��&$�� x, z u^��Ng�k`3#6Rb6N�� y n� x ∗ y = z, 3n6bg`3EjE ∗y#- �`3���#6z 2.1 Bq�z 2.1. �`Av�QH"`bÆ8nx 1− 1 ��Q S 8n�z G[S]�Y6#6R3neY|N�_< 1 + 1 = 3 �08Nu#3nXZT�D 1 + 1, 2 + 1, · · · �BzsqTz�{Æ0��Y 2.1 R=J��D (A; ◦) ℄MRYOC�R=j\ : A → A uXz

∀a, b ∈ A#?N a ◦ b ∈ A#�C�R=\RY� c ∈ A, c ◦ (b) ∈ A#�ga ℄MRj\�3ny5℄�|ZQ>n�Nu (A; ◦) �z G[A] �>N�N��R��Q 2.1 _ (A; ◦) ℄R=J��D#�(i) O (A; ◦) WC�R=MRj\ #� G[A] }R= Euler H��7#O G[A]}R= Euler H#� (A; ◦) }R=MR�%�D�(ii) O (A; ◦) }R=S>YJ��%�D#� G[A] HO=keY1u℄ |A|&=Q#M^ (A; ◦) W�;=#�#� G[A] }R=S>YJ 2- H-O=ke mR=~uX�Bke7-Y{℄�z 2- {#�7℄B�8�^��� �#XZ�xNÆ�^z�QpK�DB�08�R#z 2.2�D) |S| = 3 �$Æ08℄N�

Page 94: Mathematical Combinatorics & Smarandache Multi-Spaces

84 K): DnS�X,h�� – �?hn| Smarandache K�.

z 2.2. 3 ���XH08z~z 2.2(a) �1+1 = 2, 1+2 = 3, 1+3 = 1; 2+1 = 3, 2+2 = 1, 2+3 = 2; 3+1 = 1, 3+2 = 2, 3+3 = 3.~z 2.2(b) �1+1 = 3, 1+2 = 1, 1+3 = 2; 2+1 = 1, 2+2 = 2, 2+3 = 3; 3+1 = 2, 3+2 = 3, 3+3 = 1.8N�?r S, |S| = n#XZ6eE� n3 Æys�08℄N�bA3n1XZ6N�?rEsi� D h Æ08#h ≤ n3 ?�|N� h- �08℄N(S; ◦1, ◦2, · · · , ◦h)�6!�n�$#%xrN�08℄N#����℄�Cx 2 �08℄N�N<�#3n� N� Smarandache n- �`^6Q��Y 2.2 R= n- J�- ∑#l_℄ n =�m A1, A2, · · · , An Y�

∑=

n⋃

i=1

Ai-O=�m Ai Wtl_$RI�% ◦i uX (Ai, ◦i) ℄R=J�8�#+� n ℄21�#1 ≤ i ≤ n�6�`^�p[Q#3nXZ�N{�J!�n�$%����Ak%`^��'?�|�%������A�k%`^��'#s�|an�n��/��Y 2.3 _ R =m⋃

i=1

Ri ℄R=StY m- J�-#-zG^1� i, j, i 6= j, 1 ≤

i, j ≤ m, (Ri; +i,×i) ℄R=~-zG^� ∀x, y, z ∈ R#?N�gY�%H^tC�#�p

Page 95: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 85

(x +i y) +j z = x +i (y +j z), (x ×i y) ×j z = x ×i (y ×j z)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 ℄R= m- J~�OzG^1� i, 1 ≤ i ≤ m, (R; +i,×i) }R=�#� R℄R= m- J���Y 2.4 _ V =

k⋃i=1

Vi℄R=StY m-J�-#��%�m℄ O(V ) = {(+i, ·i) | 1 ≤

i ≤ m}#F =k⋃

i=1

Fi ℄R=J�#��%�m℄ O(F ) = {(+i,×i) | 1 ≤ i ≤ k}�OzG^1� i, j, 1 ≤ i, j ≤ k �G^� ∀a,b, c ∈ V , k1, k2 ∈ F , ?NzgY�%H^C�#�(i) (Vi; +i, ·i) ℄� Fi WY�#�-#��#&�℄�+i #�#$�℄�·i &(ii) (a+ib)+jc = a+i(b+jc);

(iii) (k1 +i k2) ·j a = k1 +i (k2 ·j a);� V ℄J� F WY k J�#�-#�℄ (V ; F )�~Y3ns~#M- N�`^��lSExNÆ�`^����Q 2.2 _ P = (x1, x2, · · · , xn) ℄ n- ^}��- Rn HYR=e��zG^1�s, 1 ≤ s ≤ n#e P kbR= s Y_�-�4\ "`2|`^ Rn b6j03 e1 = (1, 0, 0, · · · , 0), e2 = (0, 1, 0, · · · , 0),

· · ·, ei = (0, · · · , 0, 1, 0, · · · , 0) ( i �� 1#e� 0), · · ·, en = (0, 0, · · · , 0, 1) n� Rn �-`� (x1, x2, · · · , xn) XZlq (x1, x2, · · · , xn) = x1e1 + x2e2 + · · ·+ xnen�� F = {ai, bi, ci, · · · , di; i ≥ 1}#3n� N���k%`^

R− = (V, +new, ◦new),b� V = {x1, x2, · · · , xn}�yd_i #3nY� x1, x2, · · · , xs x*��#C:b6j% a1, a2, · · · , as n�

Page 96: Mathematical Combinatorics & Smarandache Multi-Spaces

86 K): DnS�X,h�� – �?hn| Smarandache K�.a1 ◦new x1 +new a2 ◦new x2 +new · · ·+new as ◦new xs = 0,A�� a1 = a2 = · · · = 0new �b6j% bi, ci, · · · , di#1 ≤ i ≤ s#n�

xs+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.[?3n�|� P E�N� s- !5`^� ♮*a 2.1 _ P ℄}��- Rn HYR=e��C�R=_�-8&R−

0 ⊂ R−1 ⊂ · · · ⊂ R−

n−1 ⊂ R−nuX R−

n = {P} -_�- R−i Y^�℄ n − i#+� 1 ≤ i ≤ n�

§3. �(^�(5'3.1. Smarandache 5'Smarandache�k xNÆBJO�Y2Gp#e�5\��{�s� Lobachevshy-

Bolyai Gp�Riemann Gp� Finsler Gp#eDD�x�xL�\�g�n2|Gp�8n*N�3n�T$:NQ2|GpA��Gp�Riemann Gp�P�T3�2|Gp�* ℄N~Qzb:g*N�0$(1)�O=eUO=�)Yezl}Zd;�&(2)O;;�q}Zs�D�&(3)ZG^e℄H)#A_Gk>lY-Re}ZqR&(4)(p;Bq�\&

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(5)B XpMpR;;�{-";;��?#-�$R;;�YR�(?Y"pB7i�t";B#�s";;�z�+R��?, MH 3.1 (x�

z 3.1. Ngx`�$gyVx`axb�#∠a + ∠b < 180O�B�Ng*Nq- 2| :*N#D�XZ�xQzbÆ��QH$_>l;�QYRe# C�R;;�{>lY;���?�7[2|*N*zZ{#,nNx��e :*Nyn�I *NDY#DME�l6n�xN��\� Y#�=�$W���x{+g*Nq� :*N#tNxh�0(�x�,exeCYNn^2| :*N#b<�|�* ℄NxkÆY#xkb6e;�g/tU#Lobachevshy o Bolyai�Riemann ℄n�xys�YN�n2| :*N-�0(�Cn�x�YN℄nx$Lobachevshy-Bolyai YN$_>l;�QYRe#AYC�";;�{>lY;���?�Riemann YN$_>l;�QYRe#�C�;�{>lY;���?�RiemannYN�|����h6~YXZp�`Gp#�`Z Einstein x ea8N�j�i`#C3j��MIN�``^�sA�#3nxkXZ�N{��g2|*N�|��Gp?\����2|Gp�Lobachevshy-Bolyai Gp�Riemann Gpo Finsler Gp'.Z [16] �A)b�0\�0\��A��nx Smarandache Gp&e?p�#.%`^Gp#b�8 Smarandache GpIN� D�I6Q#QN�4�I#.%`^Gp�Smarandache GpH[uB�k���k��\h�ko��k�+Æ#℄nP7ys�*Np��e#uB�k�x�*N 2|*N�1!-�4!ZAQz-pNg*N$(P − 1) AYC�R;;�i,;�QYRe#uX℄_,eY;�t{+;

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88 K): DnS�X,h�� – �?hn| Smarandache K�.;���?&(P − 2!AYC�R;;�i,;�QYRe#uX℄_,e C�R;;�{+;;���?&(P − 3) AYC�R;;�i,;�QYRe#uX℄_,e C�p�Y k;;�{+;;���?#k ≥ 2&(P − 4) AYC�R;;�i,;�QYRe#uX℄_,e Cs�;;�{+;;���?&(P − 5) AYC�R;;�i,;�QYRe#uX℄_,eYGk;�t{+;;��?���k�x�* ℄Nxk�2|Gp 5 g*N� 1 �.��#C�xZQNg.�g*N�n2|*N�8n*N$(−1) _>lYG^"e�RlC�R;;�&(−2) C�R;;��qs�D�&(−3) >lReiR=r�#��Rl}Zx1R=&(−4) ;B��Rl�\&(−5) _>l;�QYRe#�RlC�R;;�{>lY;���?��\h�k�x�* ℄Nxk�KqGp�Ng.�g*N#an�xQ�*N�n$(C − 1) AYC�";;� Np;�kb"=>lYe&(C − 2) _ A, B, C ℄Q=�J�Ye#D, E ℄"=�Be�O A, D, C i

B, E, C QeJ�#�A_ A, B Y;�{A_ D, E Y;���?&(C − 3) O;;�A|bp"=�BYe���k�x�* ℄Nxk� Hilbert * ℄N�Ng.�g*N��Y 3.1 R=H_ ℄Smarandache�lY#O��BR=�-HBp��1#� �#�# AYZ"IZW�w���#��R=bp Smarandache �lH_Y�k ℄Smarandache�k�Qzb��5ZAQz$p�l� Smarandache Gpx_ib6��T 3.1N A, B, C 2|VzE=�y-`��#� x` 2|VzEqT A, B, CrgN���x`�A3n�|N� Smarandache Gp�h �2|Gp* ℄Na℄�#e$g*Nx Smarandache k���

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(i) 2| :*NY6 ℄_R;;�QYRe#C�R; �C�;� 0t,;;�B�n�YNx` L !T� C �Vx` AB�"`!T-pN�y6 AB E��rg�Ngx`V L#?!Tx` AB E�-pN�Cyb6V L �x`#6z 3.2(a) Bq�(ii) *N℄_G^"=�BeC�R;;�Y6 ℄_G^"=�BeC�R;;� �C�;��n�"`!T$�� D, E#b� D, E � A, B, C �N�#6� C -`#6z 3.2(b) Bq#rg�Ngx`!T D, E�?8-`$�6x`

AB � F, G .y� A, B, C N��-`�$�� G, H Cyb6!TDn�x`#6z 3.2(b) Bq�

z 3.2. s- x`� q3.2 �f��('℄$N�y���� "O=8X *℄4X# *B�t�4XWP; 2p=p#O"=p7-ÆlR=QX�T_!��& *B�t�4XWP; q =p#O=pÆlFwp�IY{J{�� m�$*℄}l�8X#�<l_℄ p&s*℄�}l�8X#�<l_℄ q�b�X�k�`&xN�Qx�z�k%72�z0"N��$|DD�xk�gk%�Qk�xAEs~�zxX�k�#?\℄5#mAxyX�k�#6z 3.3 Bq#e (a) dI�~z#(b) Ir��\℄5#m�

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Page 100: Mathematical Combinatorics & Smarandache Multi-Spaces

90 K): DnS�X,h�� – �?hn| Smarandache K�.�zx�z�NÆ�℄#v72bÆ�℄q�zdI�#�|�j�zkCsB�9 D = {(x, y)|x2 + y2 ≤ 1}�Tutte 1973 &�D)�z�n�� #�x[12] ���#�z� Q��Y 3.2 R=aH M = (Xα,β,P)#l_℄��3�m X Y �j( Kx, x ∈ X YsHJ�Y�� Xα,β WYR=�vC� P#-Ge�XYH� 1 iH� 2#+�K = {1.α, β, αβ} ℄ Klein 4- �A#(d P ℄�vC�#��C�21� k, uXPkx = αx� Q 1$αP = P−1α; Q 2$A ΨJ = 〈α, β,P〉 � Xα,β W}"�P7� 3.2#�z���oz℄n� �� P o Pαβ Ix Xα,β E�|�-WM~&d Klein 4- �% K Ix Xα,β E�|�M~��xEuler-Poincare*p#3n�|

|V (M)| − |E(M)| + |F (M)| = χ(M),b� V (M), E(M), F (M) ℄nlq�z M ���?�d?oz?#χ(M) lq�z M � Euler r�#e�z��z M B}8����z� Euler r��-N��z M = (Xα,β,P) x�}l�Y#:��% ΨI = 〈αβ,P〉 6 Xα,β ExXw�#kA- }l�Y�z 3.4. z D0.4.0 6 Kelin �zE�}8I N��5#z 3.4 �D)z D0.4.0 6 Kelin �zE�N�}8#XZ�x�z M = (Xα,β,P) lq6Q#b�

Xα,β =⋃

e∈{x,y,z,w}

{e, αe, βe, αβe},

P = (x, y, z, w)(αβx, αβy, βz, βw)

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× (αx, αw, αz, αy)(βx, αβw, αβz, βy).z 3.4 ��z� 2 �� v1 = {(x, y, z, w), (αx, αw, αz, αy)}, v2 = {(αβx, αβy, βz,

βw), (βx, αβw, αβz, βy)}, 4 gd e1 = {x, αx, βx, αβx}, e2 = {y, αy, βy, αβy}, e3 =

{z, αz, βz, αβz}, e4 = {w, αw, βw, αβw}ZA 2�z f2 = {(x, αβy, z, βy, αx, αβw),

(βx, αw, αβx, y, βz, αy)}, f2 = {(βw, αz), (w, αβz)}#e Euler r� χ(M) = 2 − 4 + 2 = 0���% ΨI = 〈αβ,P〉 6 Xα,β EXw�bA1[n�|.�|z D0.4.0 6 KleinzE�}8�A:2 N> 2-��D#3n�XZN< �QHz6`^ZA=��zE�}8�z6��zE�}8� 6Q��Y 3.3 _H G Yke�mopy� V (G) =

k⋃j=1

Vi, +�zG^1� 1 ≤ i, j ≤

k#Vi

⋂Vj = ∅#s S1, S2, · · · , Sk ℄u#�- E HY k =8X#k ≥ 1�OC�R= 1-1 �:j\ π : G → E uXzG^1� i, 1 ≤ i ≤ k#π|〈Vi〉 }R=XN-

Si \ π(〈Vi〉) HYO=�A�B�t� D = {(x, y)|x2 + y2 ≤ 1}#� π(G) } G�8X S1, S2, · · · , Sk WYJ&N�� 3.3 �z S1, S2, · · · , Sk �`^)�8�}8�qf�vb6NÆ6,Si1 , Si2 , · · · , Sik#n�8-`n� j, 1 ≤ j ≤ k#Sij x Sij+1

�5`^i#- G 6S1, S2, · · · , Sk E�pbJ&N�>�z#�Qz��N��Q 3.1 R=H G �4X P1 ⊃ P2 ⊃ · · · ⊃ Ps C�� �YpbJ&NO,ROH G C��y� G =

s⊎i=1

Gi#uXzG^1� i, 1 < i < s,

(i) Gi } XY&(ii) zG^ ∀v ∈ V (Gi), NG(x) ⊆ (

i+1⋃j=i−1

V (Gj)).

3.3 �(5'aH�k x6�z3GE/p� Smarandache Gp#siFx�N�r�$�!��$�+m��zGp��'�T6.Z [13] ZD#:�6.Z [14]− [16] #Wnx [16] �)P��2-#e� 6Q��Y 3.4 �aHM O=ke u, u ∈ V (M)W#xR=r� µ(u), µ(u)ρM(u)(mod2π)#

Page 102: Mathematical Combinatorics & Smarandache Multi-Spaces

92 K): DnS�X,h�� – �?hn| Smarandache K�. (M, µ) ℄R=aH�k#µ(u) ℄e u YB _ ����7 ��78XWY8�7_ R= �=X~ ,aH�ks{J p{J�z 3.5 �D)x`LT�zE���� �b���℄|.℄ k(��(Ap(#an�#� u - ����2|�o����z 3.5. x`LT���.�������2|�o���6 3 !`^CxXZlY�#b���lY�n2|`^� �#CyN�xVx�#EY��1x2|��z 3.6 �D)b=Æ�6 3 !`^�lYQH, z� u ���#v 2|�? w ����

z 3.6. ����2|�o���6 3- !`^�lY�Q 3.1 pJ�sJaH�kHtC�uB�k���k��\h�ki��k�� �q�k.Z [16]� f �#3nQz�IVz�zGp� ��6bÆ �#y�XZ6��Ev |h5℄�#�XZD�����u^�dxN���℄�#bA8�N{ �VzEn��`g℄�` �℄66Vz�zGp�bA��N$ XaH�ks3;��7_aH 7_Ye℄}�e�I N��5#z 3.7 �D)3o+z℄�NÆVz�zGp#e��dE��zl���� 2 X�|h5℄�z�

Page 103: Mathematical Combinatorics & Smarandache Multi-Spaces

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z 3.7. N�Vz�zGp��5z 3.8 �D)z 3.7 � �Vz�zGpx`� �#�,�#z 3.9 �D)�Vz�zGp�GÆ=d��z 3.8. Vz�zGp�x`

z 3.9. Vz�zGp�=d�§4. /�YJ:5'Einstein �J a8N55)`^6j�IxQx���#W�H`Fy��#bN�6lSA W!�|ql��zGp�&elSEXZN<�� N�.%`^E#C6�.%`^�j��Ev N�k%?p�#.%`^Gp��Y 4.1 . U 8A �'8 ρ ��'"�#W ⊆ U�*C ∀u ∈ U#,�IA

Page 104: Mathematical Combinatorics & Smarandache Multi-Spaces

94 K): DnS�X,h�� – �?hn| Smarandache K�.&=E- ω : u → ω(u)#K$#*CL3 n, n ≥ 1#ω(u) ∈ Rn 0�*C�M3ǫ > 0# �IA 3 δ > 0 �A � v ∈ W , ρ(u− v) < δ 0� ρ(ω(u)−ω(v)) < ǫ�J, U = W#� U 8A 9�'"�#�8 (U, ω)&,�IM3 N > 0 0�∀w ∈ W , ρ(w) ≤ N#J� U 8A G�9�'"�#�8 (U−, ω)�"` ω x|h5℄�i#[#.%`^3n�| Einstein ���`^� f �#3nUN#VzGp� ω |h5℄�� �#�T�Qz$� r��N��Q 4.1 _` X (P, ω) WY"e u i v �RlC�}�^_WY;���Q 4.2 �R=` X (

∑, ω) W#O�C�}�e#� (

∑, ω) �O=et℄Ne O=et℄�8e�8Vzn��`#A�6Q�R��Q 4.3 �` X (

∑, ω) WC�J�8� F (x, y) = 0 ℄_7� D HYe (x0, y0)O-RO F (x0, y0) = 0 -zG^ ∀(x, y) ∈ D#(π − ω(x, y)

2)(1 + (

dy

dx)2) = sign(x, y).Y6#3n4$|#.%`^E�P7� 4.1#8N� m- 9� Mm o-`� ∀u ∈ Mm#� U = W = Mm#n = 1 � ω(u) N�H�℄��A3n�|9�

Mm E�#9�Gp (Mm, ω)�3ns~#9� Mm E�Minkowski �� _<6Qgm�N�℄� F :

Mm → [0, +∞)�(i) F 6 Mm \ {0} EIIH�&(ii) F x 1- hZ�#C8-`� u ∈ Mm o λ > 0#� F (λu) = λF (u)&(iii) 8-`� ∀y ∈ Mm \ {0}#_<gm

gy(u, v) =1

2

∂2F 2(y + su + tv)

∂s∂t|t=s=0

.�8-�` � gy : Mm × Mm → R xo���Finsler2/lSE1xv ) Minkowski M��9�#:℄{u1x9� MmAe�`^E�N�℄� F : TMm → [0, +∞) s_<6Qgm�

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(i) F 6 TMm \ {0} =⋃{TxM

m \ {0} : x ∈ Mm} EIIH�&(ii) 8-` ∀x ∈ Mm#F |TxMm → [0, +∞) xN� Minkowski M��I #.%`^Gp�N�W�#8-` x ∈ Mm#3n"� ω(x) = F (x)�A#.%`^Gp (Mm, ω) xN� Finsler 9�#Wn�#6R� ω(x) = gx(y, y) =

F 2(x, y)#A (Mm, ω) 1x Riemann 9��bA#3n1�|Q��N��Q 4.4 `u#�-�k (Mm, ω)#R a#Smarandache �kHkb Finsler �k#�~kb Riemann �k�§5. LTBViRE{3 BgNtU� N> �$2-ZD)k%�D2-�0\#b�3n�,6G��Qa8Q3 5.1 p|Y=}M'℄qMF����$�)}M�-'+}�{d�R8�-pR'R&XZ�7����#v&1/��=���#b1x.Z [10] V���A�#Fx> $�_i��A��Einstein 55)`^6j�IxQx���#[Æ` Eu2|`^6 lt�xyb6��[l<A �|.#,�� A . �|7&�Æa?yxe\R#7Nx�!`^�x�!`^sK| 4 !`^#.Z [16] 8YW�uB{Z��!��℄Gp�x�k%\Z����QHP|NuW���TKA�N< �2-��`^n(�8#.%`^ (Mm, ω) �2-�3YVx`^�2-XZDY#�H6�$E/�V���b6#t,��{�A QH7HA |�Qa8Q3 5.2 F�i�Y}M^�U`}|Y'}�p�'BgtU� N> �DKo6(,��g�v&{�0�`^A'#[?qf2�$�g��Nu��� N> $W"x5y'�"�lReA�6VF��-Y`nuU`}|Y �6v� N�l<�gmQ#D��b�0\�N��u�#h ,�My|�A y|��CJ=)�V N. `^!�x 10#M- N�`^!�x 11 �:ÆWs�V."V NCxe<^ ��?H�> $Wo62-� F- N�`^!�x 12�A:2bÆ&e#XZp�N<�`^!� N2- Einstein �Q3#6bN�E#�$W;6)> $W��z�

Page 106: Mathematical Combinatorics & Smarandache Multi-Spaces

96 K): DnS�X,h�� – �?hn| Smarandache K�.Qa8Q3 5.3 F�qMEW TNqpE'a4W}�C�RIiw}Z� 4^TN 3 ^ � 3 ^TN 2 ^�-'w%lSEx Einstein �Q36ys.KgmQ�f���NQz#> $W. w%b68k�j�#1�H`Fy ��#-p>℄y�S8w%CqZ$-0 ;P�[6]-[7]!&si> $W�} w%x��ys���ysi`��"#h ,�)��N�> KI6w%�Cdt��w%a8�# N> ��NÆ4%#eWlx-p>℄C7H�84%��6R>Iw%�4%�℄#3n1%DY$`Cx7�|�#Dj�si1x6<#BZw%�4%n�xN$u�bA,�#Wnx�d�7�q�yp���)w%�=K>�6��Ex_ib6�#6����s��tZ�=K>�� N#yNx> .x�$C$jj,n�<1.m�=&Z{#�$��Nx℄7>℄0"KygbN�3\�A�[ NE.mi`L>#b���F0"Kg�m{�0\#Cy/�℄�0"0\�~Ym{��$0\x ([16])$(1)UnHK�>#?li"H}ilY#i"}�ilY�T0���(2):H&NU|^�-p#Ce���-}zY=�(3)9�HY��-�o�>�8> ${"#�D6��E)Y 1�geK�\>#�?DYei`L>KI�Qa8Q3 5.4 F��a4W}Z(UWwEE'.>��. %Nxx> $��N�*m�\�A:2,�8`^!�.m��g#b�0\Fg�2dw2�6R`^!� ≥ 11#�g.>�1yN�I6,�M�|�Qk!E#FyX 6��EY|D�B�bu(P|,�.m�V�l<K��Z��kG1�

[1] A.D.Aleksandrov and V.A.Zalgaller, Intrinsic Geometry of Surfaces, American

Mathematical Society, Providence, Rhode Island, 1976.

[2] I.Antoniadis, Physics with large extra dimensions: String theory under experi-

mental test, Current Science, Vol.81, No.12, 25(2001),1609-1613

[3] +_Ro+!|, Y��k=_, U�k$D?M, 2001.

Page 107: Mathematical Combinatorics & Smarandache Multi-Spaces

�Cx�e�[�nS{$�? –Smarandche K�.�C 97

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

[5] \LE, �zB{�}�k, V$D?M, U�, 2005.

[6] S.Hawking, p-2t, �VKD?M,2005.

[7] S.Hawking, ^|�Y}M, �VKD?M,2005.

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

[9] H.Iseri, Partially Paradoxist Smarandache Geometries, http://www.gallup.unm.

edu/smarandache/Howard-Iseri-paper.htm.

[10] M.Kaku, Hyperspace: A Scientific Odyssey through Parallel Universe, Time

Warps and 10th Dimension, Oxford Univ. Press.

[11] L.Kuciuk and M.Antholy, An Introduction to Smarandache Geometries, Math-

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

[12] Y.P.Liu, Enumerative Theory of Maps, Kluwer Academic Publishers, Dordrecht/

Boston/ London, 1999.

[13] L.F.Mao, On Automorphisms groups of Maps, Surfaces and Smarandache ge-

ometries, Sientia Magna, Vol.1(2005), No.2, 55-73.

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

math.GM/0506232.

[15] L.F.Mao, Automorphism Groups of Maps, Surfaces and Smarandache Geome-

tries, American Research Press, 2005.

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

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

10/2000.

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

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

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

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

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��4gl�q℄�A�– J( 1985-2010 �uBOf9, 98-103

d\� 1985-2010 �qbxo`1985 p1. x|E��x|g�A RMI �A#HV�>Ce#29-32,1�1985!2. $K�$���℄%#HV�>Ce#22-23,2�1985!1990 p1. The maximum size of r-partite subgraphs of a K3-free graph, mr�>, 4�1990),417-424.

1992 p1. U��f$% 100m3 �N�℄e%#w`C%#1�1992!2.��Xr�!U��V�℄r 50mj0�u8�/NYe%#9S{�#4�1992!1993 p1. ��Xr�!U��V�℄r 62m 7I��n�*(�k"e%#9S{�#1�1993!1994 p1. ��=:�r�!R(G)=3 �7�z���/2-#;A�>{gl�>#Vol.

10�DK!(1994),88-98�2. Hamiltonian graphs with constraints on vertices degree in a subgraphs pair, -��%>Æ>p#Vol. 15�DK!(1994),79-90�1995 p1. ��Xr�!U��V�℄r�u8NY*(��/e%#9S��#5�1995!

Page 109: Mathematical Combinatorics & Smarandache Multi-Spaces

+:%K) 1985-2010 �wCPg: 99

2. �u8�/NYe%K�#H℄rl{�#^G<g�95 ?!#1995�3. �xk9)��v��[u�N�℄e%K�#H℄rl{�#^G<g�95?!#1995�1996 p1. ���B(7�z�Bkd�#{Uh_{�G>>p#Vol. 23�DK!(1996),

6-10�2. *(�U��h℄BAS�#9S{�#2�1996!�1997 p1. CAey�Æp e%,�S�Q/#9SU>#11�1997!1998 p1. A localization of Dirac’s theorem for hamiltonian graphs,�>Ce{�B#Vol.18,

2(1998),188-190.

2. *(�U��h℄BAS�#9S��#9�1998!3. CAey�Æp e%,�S�Q/#9S{�#1�1998!1999 p1. $rN`%3e%�v:NP#Gaj�e$9SC%lCg9_�r�gl��#Qp %ID?M#1999�2. �u8%3e%�vNP#�Wo�e$9SC%lCg9_�r�gl��#Qp %ID?M#1999�3. U��V�℄r�u8NY*(��/e%, Qp %3:*(e$9SC%lCr���(2), Qp %ID?M#1999�2000 p1. 4|� Fangm�N��J#8)k G>>p�`B{>e!#Vol. 26#3(2000),

25-28�2. U���\��.�e%�% ��C #9S{�#2�2000!�3. U���\��.�+{%3e%#9SC%lCr���(7), Qp %ID?M#1999�

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100 K): DnS�X,h�� – �?hn| Smarandache K�.2001 p1. ��8;Fr�!UuA:z�N���4|�=℄gm#8)k G>>p�`B{>e!#Vol. 27#2(2001), 18-22�2. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, omk G>>p�`B{>e!, 4(2001), 13-18�3. (with Liu Yanpei)An approach for constructing 3-connected non-hamiltonian

cubic maps on surfaces,OR Transactions, 4(2001), 1-7.

2002 p1. A census of maps on surfaces with given underlying graphs, A Doctorial Disser-

tation, Northern Jiaotong University, 2002.

2. On the panfactorical property of Cayley graphs, �>Ce{�B, 3(2002), 383-

390�3. .~*x�TXR ��ÆK���#H℄H7>p, 3�2002!, 88-91�4. Localized neighborhood unions condition for hamiltonian graphs, omk G>>p�`B{>e), 1(2002), 16-22�2003 p1. (with Liu Yanpei) New automorphism groups identity of trees,�>T!, 5(2002),

113-117�2. (with Liu Yanpei)Group action approach for enumerating maps on surfaces, J.

Applied Math. & Computing, Vol.13(2003), No.1-2, 201-215.

3. ��8;Fr�!z�X�k}8�j X� #�>w�>p, 3�2003!,287-

293�4. ��8fr�!��;� ≥ 2 �4|�gm�UuA:z#omk G>>p�`B{>e!, 1(2003),17-21�2004 p1. (with Yanpei Liu)A new approach for enumerating maps on orientable surfaces,

Australasian J. Combinatorics, Vol.30(2004), 247-259.

2. �� r�!Riemann �zE Hurwitz � ��r�J#H℄{>Æ�ys$I{?�>{>�B/Bh�#2004 & 12 *,75-89�

Page 111: Mathematical Combinatorics & Smarandache Multi-Spaces

+:%K) 1985-2010 �wCPg: 101

2005 p1. (with Feng Tian)On oriented 2-factorable graphs, J.Applied Math. & Computing,

Vol.17(2005), No.1-2, 25-38.

2. (with Liu Yanpei and Tian Feng)Automorphisms of maps with a given underlying

graph and their application to enumeration,Acta.Math.Sinica, Vol.21, 2(2005), 225-

236.

3. On Automorphisms of Maps and Klein Surfaces,H℄{>Æ�ysp7#2005.6.

4. A new view of combinatorial maps by Smarandache’ notion, e-print: arXiv:

math.GM/0506232.

5. Automorphism Groups of Maps, Surfaces and Smarandache Geometries, Ameri-

can Research Press, 2005.

6. On automorphism groups of maps, surfaces and Smarandache geometries, Scien-

tia Magna, Vol.1(2005), No.2, 55-73.

7. Parallel bundles in planar map geometries, Scientia Magna, Vol.1(2005), No.2,120-

133.

2006 p1. with Yanpei Liu and Erling Wei) The semi-arc automorphism group of a graph

with application to map enumeration, Graphs and Combinatorics, Vol.22, No.1

(2006), 93-101.

2. Smarandache Multi-Space Theory, Hexis, Phoenix, American 2006.

3. H℄C%9_�flC'��+{Y��z –Smarandache J�-'�`.,

Xiquan Publishing House, 2006.

4. On algebraic multi-group spaces, Scientia Magna, Vol.2,No.1(2006), 64-70.

5. On multi-metric spaces, Scientia Magna, Vol.2,No.1(2006), 87-94.

6. On algebraic multi-ring spaces, Scientia Magna, Vol.2,No.2(2006), 48-54.

7. On algebraic multi-vector spaces, Scientia Magna, Vol.2,No.2(2006), 1-6.

8. N> jD�BgNtU�$� Smarandache �`^ N#H℄{�Bh��$200607-91�9. XjXZ℄N��$��A��℄B, H℄{�Bh��$200607-112�10. Combinatorial speculation and the combinatorial conjecture for mathematics,

arXiv: math.GM/0606702 and Sciencepaper Online:200607-128.

Page 112: Mathematical Combinatorics & Smarandache Multi-Spaces

102 K): DnS�X,h�� – �?hn| Smarandache K�.11. A multi-space model for Chinese bidding evaluation with analyzing, arXiv:

math.GM/0605495.

12. Smarandache �`^Aa>�$�r N, k℄S�Op�e�Smarandache j7Ce�, High American Press#2006.

2007 p1. Geometrical theory on combinatorial manifolds, JP J. Geometry and Topology,

Vol.7, 1(2007), 65-113.

2. An introduction to Smarandache multi-spaces and mathematical combinatorics,

Scientia Magna, Vol.3, 1(2007), No.1, 54-80.

3. Combinatorial speculation and combinatorial conjecture for mathematics, Inter-

national J. Math.Combin., Vol.1,2007, 1-19.

4. Pseudo-manifold geometries with applications, International J. Math.Combin.,

Vol.1,2007, 45-58.

5. A combinatorially generalized Stokes theorem on integration, International J.

Math.Combin., Vol.1,2007, 67-86.

6. Smarandache geometries & map theory with applications(I), Chinese Branch Xi-

quan House, 2007.

7. Differential geometry on Smarandache n-manifolds, in Y.Fu, L.Mao and M.Bencze

ed. Scientific Elements(I), 1-17.

8. Combinatorially differential geometry, in Y.Fu, L.Mao and M.Bencze ed. Scien-

tific Elements(I), 155-195.

9. C%9_�f'�ÆL�B{r4, American Research Press, 2007.

2008 p1. Curvature Equations on Combinatorial Manifolds with Applications to Theoret-

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

2. Combinatorially Riemannian Submanifolds#International J. Math.Combin., Vol.2,

2008, 23-45.

3. Extending Homomorphism Theorem to Multi-Systems#International J. Math.Combin.,

Vol.3, 2008,1-27.

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

2008, 111-121.

Page 113: Mathematical Combinatorics & Smarandache Multi-Spaces

+:%K) 1985-2010 �wCPg: 103

2009 p1. Topological Multi-groups and Multi-fields#International J. Math.Combin., Vol.1,

2009, 8-17.

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

90-95.

3. �"Xjvj~�y5;kKM#H℄9_p#2009 & 1 * 24 2�4. �QXj�0,�wI�VQ�r{~~u+ -�'�ÆLY��z��t�e!#QP�D?M#2009�5. Combinatorial Fields - An Introduction, International J. Math.Combin., Vol.3,

2009, 01-22.

6. Combinatorial Geometry with Application to Field Theory, InfoQuest Press, 2009.

2010 p1. Relativity in Combinatorial Gravitational Fields, Progress in Physics, Vol.3,2010,

39-50.

2.�2010 u'�k:P� y�$�=T�–�Xj�0/�℄B���e!#QP�D?M#2010 &�3. A Combinatorial Decomposition of Euclidean Spaces Rn with Contribution to

Visibility, International J. Math. Combin., Vol.1, 2010, 47-64.

4. Labeling, Covering and Decomposing of Graphs – Smarandaches Notion in

Graph Theory, International J. Math. Combin., Vol.3, 2010, 108-124.

Page 114: Mathematical Combinatorics & Smarandache Multi-Spaces

[Æ$�#%$vs��$vs�#kQ�$%XN�#kQ�QS�$�r2���e�2006 &8"kQ�Who’s Who�#~�|�QVKN.6`�z�$`#Y QWD�"�Cv%QXjvjv%tsÆ��QV$%C �A��FNu��l<�2-,�#U�p %3$%`w~��2-\{ �1962 & 12 * 31 2D\+K_�?~#1978-1980 &+K_�!$� 80E 1 9$K&1981 & -1998 &6Qp B%34%I#T�q-TK���K�6��V��h�:%3 �w&1998 & 10 * -2000 & 6 *-QH$%3pC*�:%3 &2000 & 7 * -2007 & 12 *Q�Xj�^?-*(#�-h�! �&WC*��-ot:%3 �w&e^ 1999 & -2002 &UQxqk$',vs$)#2003 & -2005 &QV$%C �A��FNu��2-�[uvs�2-%I&2008 & 1 *��#QXjvjv%tsÆ��=&�2-%I�D?6�$� N> o%3pN�4�#T�6Q��Nu��$�K>EDl Smarandache Gp��℄Gp� N> ��r�z�zNo%3pNC N. 60=N#6kQD?T=\�$$�&��$\%3Xj�0 N&�oN\N.?#6lQD?N\�$N.?�[ 2007& 10*Io#�eQS�$�r\Æ�MATHEMATICAL COMBINATORICS �(INTERNATIONAL

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Page 115: Mathematical Combinatorics & Smarandache Multi-Spaces

Abstract: This book is for young students, words of one mathematician, also being a

physicist and an engineer to young students. By recalling each of his growth and success

steps, beginning as a construction worker, obtained a certification of undergraduate learn

by himself and a doctor’s degree in university, after then continuously overlooking these

obtained achievements, raising new scientific objectives in mathematics and physics by

Smarandache’s notion and combinatorial principle for his research, tell us the truth that�all roads lead to Rome , which maybe inspires younger researchers and students.

s1jT awM[x��%#[� "wM(>�#�=�nB�U#�$2<l#U�MQ�#_pS6Q �"7�%#[!���H_pSMM+-P, #a(d}M�o�$+"pS6# /[U#H"rYr�#��Q`P7�#q&+ur#("Y�5�zU#1,x4j�62"x4�w?qn Smarandache %d�6HYC��dM\[��#�=�1,�S2"t�(ff}BpPW��/�tB"7x�Ij��%#[/�9�8�nhHw�

7815999 730325

ISBN 1-59973-032-4

53995>