mathematical combinatorics & smarandache multi-spaces
DESCRIPTION
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
�5hm�r^�B—– �O})_ Smarandache uK;d\�
Let’s Flying by Wing
—–Mathematical Combinatorics
& Smarandache Multi-Spaces
Linfan MAO
Chinese Branch Xiquan House
2010
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
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
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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
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�
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
��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��$W�12003 %T�}�}�6#P�k-�2e-print: http://zikao.eol.cn
2 K): DnS�X,h�� – �?hn| Smarandache K�.n`M�$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�
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+�
��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
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{�
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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�$�R 1��#Ce�>}ZCeU 80 &s�2}#lOF%0U 80&� �?y_��1996 p 7 f 30 �!
��4gl�q℄�A�– lYgm$,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: �$�r�}e��r$�℄$�℄zN�Smarandache &e��r�℄Gp� N> �V2℄��Abstract: How to select a research subject or a scientific work is the central
objective in researching. Questions such as those of what is valuable? what
is unvalued? often bewilder researchers. By recalling my researching process
from graphical structure to topological graphs, then to topological manifolds,
and then to differential geometry with theoretical physics by Smarandache’s
notion and combinatorial speculation of mine, this paper explains how to
present an objective and how to establish a system in one’s scientific research.
In this process, continuously overlooking these obtained achievements, raising
new scientific objectives keeping in step with the frontier in international re-
search world and exchanging ideas with researchers are the key in research of
myself, which maybe inspires younger researchers and students.
Key Words: CCM conjecture, combinatorics, topology, topological graphs,
Smarandache’s notion, combinatorial theory on multi-spaces, combinatorial
differential geometry, theoretical physics, scientific structure.
12010 %mÆPUJM-5_�y�#_UNAH�2e-print: www.paper.edu.cn
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kCkn = n2n−1.:u#�'�� oK\�$Kx$K�$�=�uyX℄�uN℄�z�Z�s$n"`�x#3n$g�$ByxN�rN�T31XZÆ0�#b��=ZLw�=,�=e�= #[?� y5Z��$℄�$K�%�
��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
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.
��4gl�q℄�A�– &������}�� RMI ��, 39-44
��4��M�f-2 RMI bhd/J(qZ!�8^=�"8 83-1 a=h)x�JE�ox_x#g��2-#=℄Wq)Nu�D��$&e�qN�℄�K0N℄�h7�E�#qT2-E�� ��|℄�� �#bx7�E�N2-℄��3\&e�?x�JE�0x)bN&e��x=|℄�N$1, cos x, sin x,
cos 2x, sin 2x, · · · , cos nx, sin nx, · · · �ox q�`℄� f(x)�N�` 2π#yx2π i#XZj�n��0bÆ q!-*p$
a0 =1
π
∫ π
−π
f(x)dx, ak =1
π
∫ π
−π
f(x) cos kxdx,
bk =1
π
∫ π
−π
f(x) sin kxdx, (k = 1, 2, 3, · · ·)K0)7�E�$a0
2+
∞∑
k=1
(ak cos kx + bk sin kx).8Y�`℄�XZ�xNÆJ℄�� �QH#g0�`℄�#[?_XZK0x|E��B*J℄��� # 5u1x$8� � [a, b] E�Y�`℄� F (x)#��� N�℄� F ∗ (x)#n�8-` x ∈ [a, b]#(� F ∗ (x) = F (x)#? F ∗ (x) x� ����E�#Z (b − a) �`�℄��T 1[3] D6 [−π, π] Eq f(x) = x2 KI x|E�#3nXZTJ f(x) �� $f ∗ (x) =
{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$1�-Ck5Æ%�#1,�"1�1985 �
40 K): DnS�X,h�� – �?hn| Smarandache K�.-
6O−π π
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−3π 3π−5π 5πx
y
( 1bA#3nXZq f ∗ (x) 6�l�EK0x�JE�$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�
'���� �~�� 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|$g�1�y(x) = e−x − e−x cos x = e−x(1 − cos x).bÆ�HXZ�pzlq6Q [4]$
42 K): DnS�X,h�� – �?hn| Smarandache K�.��℄Q3 l℄��n�℄�
l℄��℄Q3��-
?�?x|g� �n�Q3x|g�
7Nxx�JE�#�xx|g�#e3\&e(x"JO�NÆ&e�:℄nx�b1x�$QHN�>NsKL9�A#A RMI �A [1]:>lR=bpf��� X YR�HK�D S#M^q(UR=}lj\ A :S j\N jG S∗, �}Z� S∗ A_RlY�>��^f�j� X∗ = A(X) ?l1Æ#�~A_�J�tj\|}^ X = A−1(X∗) ?l1Æ�bNT3Xxpzlq $
S S∗
X∗X
-?�?
A
A−1
RMI �Ax�$_i0x�QH�A�D�:℄0x6�$Æ���4�(XZM|�T 3 &��$� B� B℄�Æu���BGp#e�A0\�3\&e1xRMI �A�0x�:xpzlq#n3nXZ�F�M|bN�$
'���� �~�� 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 $
44 K): DnS�X,h�� – �?hn| Smarandache K�.�,0\ vE�0\
vE��Æ ��,� �-
?�?8nHA 088nHA�6#��℄Q3 un+2 = aun+1 + bun, u0 = A, u1 = B 1XZ�xbÆ�℄��QH�N
u(t) =∑
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.
'���� �~�� 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~~��$� +��
��4gl�q℄�A�– lYC%77, 46-74
6|�po_d/J(H℄'�E�#�#r[#100045)jT: vh�B�Q$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.
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[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$rS=?�g,\�
48 K): DnS�X,h�� – �?hn| Smarandache K�.4Æ%Ip�,�m�;5�bY�xT#3Io)3�p %,\-�j`�%, xNj#|SA't��=`%�E9�bxN�t�ionh�#pN%`g℄�S# K��D��j#I r�%Æ�[5%#!��DX9q{�[fg# �z��%e%Z)%Iz� N�*%I��#%4XEs�34|)+g=�#����<)#h viN�*�\+\F1xgb1�z�3A�j*[%47:�z�0[�$��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\
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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
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��
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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
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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$��kRH N#M- N#�`^#�zGp#SmarandacheGp##.%`^Gp#FinslerGp��P% AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50
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ds2 = −dt2 + a(t)2d2R
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Wohlfarth�� 4 !X3HW����ds2 = e−mφ(t)(−S6dt2 + S2dx2
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82 K): DnS�X,h�� – �?hn| Smarandache K�.[�$|.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$:NQ08�� ���N�?r S#8 ∀x, y ∈ S,� x∗y = z#`&x S Eb6N� 2 ��rsK ∗ : S × S → S#n�
�Cx�e�[�nS{$�? –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�
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�$%����Ak%`^��'?�|�%������A�k%`^��'#s�|an�n��/��Y 2.3 _ R =m⋃
i=1
Ri ℄R=StY m- J�-#-zG^1� i, j, i 6= j, 1 ≤
i, j ≤ m, (Ri; +i,×i) ℄R=~-zG^� ∀x, y, z ∈ R#?N�gY�%H^tC�#�p
�Cx�e�[�nS{$�? –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 #�#$�℄�·i &(ii) (a+ib)+jc = a+i(b+jc);
(iii) (k1 +i k2) ·j a = k1 +i (k2 ·j a);� V ℄J� F WY k J�#�-#�℄ (V ; F )�~Y3ns~#M- N�`^��lSExNÆ�`^����Q 2.2 _ P = (x1, x2, · · · , xn) ℄ n- ^}��- Rn HYR=e��zG^1�s, 1 ≤ s ≤ n#e P kbR= s Y_�-�4\ "`2|`^ Rn b6j03 e1 = (1, 0, 0, · · · , 0), e2 = (0, 1, 0, · · · , 0),
· · ·, ei = (0, · · · , 0, 1, 0, · · · , 0) ( i �� 1#e� 0), · · ·, en = (0, 0, · · · , 0, 1) n� Rn �-`� (x1, x2, · · · , xn) XZlq (x1, x2, · · · , xn) = x1e1 + x2e2 + · · ·+ xnen�� F = {ai, bi, ci, · · · , di; i ≥ 1}#3n� N���k%`^
R− = (V, +new, ◦new),b� V = {x1, x2, · · · , xn}�yd_i #3nY� x1, x2, · · · , xs x*��#C:b6j% a1, a2, · · · , as n�
86 K): DnS�X,h�� – �?hn| Smarandache K�.a1 ◦new x1 +new a2 ◦new x2 +new · · ·+new as ◦new xs = 0,A�� a1 = a2 = · · · = 0new �b6j% bi, ci, · · · , di#1 ≤ i ≤ s#n�
xs+1 = b1 ◦new x1 +new b2 ◦new x2 +new · · ·+new bs ◦new xs;
xs+2 = c1 ◦new x1 +new c2 ◦new x2 +new · · · +new cs ◦new xs;
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ;
xn = d1 ◦new x1 +new d2 ◦new x2 +new · · ·+new ds ◦new xs.[?3n�|� P E�N� s- !5`^� ♮*a 2.1 _ P ℄}��- Rn HYR=e��C�R=_�-8&R−
0 ⊂ R−1 ⊂ · · · ⊂ R−
n−1 ⊂ R−nuX R−
n = {P} -_�- R−i Y^�℄ n − i#+� 1 ≤ i ≤ n�
§3. �(^�(5'3.1. Smarandache 5'Smarandache�k xNÆBJO�Y2Gp#e�5\��{�s� Lobachevshy-
Bolyai Gp�Riemann Gp� Finsler Gp#eDD�x�xL�\�g�n2|Gp�8n*N�3n�T$:NQ2|GpA��Gp�Riemann Gp�P�T3�2|Gp�* ℄N~Qzb:g*N�0$(1)�O=eUO=�)Yezl}Zd;�&(2)O;;�q}Zs�D�&(3)ZG^e℄H)#A_Gk>lY-Re}ZqR&(4)(p;Bq�\&
�Cx�e�[�nS{$�? –Smarandche K�.�C 87
(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{+;
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���
�Cx�e�[�nS{$�? –Smarandche K�.�C 89
(i) 2| :*NY6 ℄_R;;�QYRe#C�R; �C�;� 0t,;;�B�n�YNx` L !T� C �Vx` AB�"`!T-pN�y6 AB E��rg�Ngx`V L#?!Tx` AB E�-pN�Cyb6V L �x`#6z 3.2(a) Bq�(ii) *N℄_G^"=�BeC�R;;�Y6 ℄_G^"=�BeC�R;;� �C�;��n�"`!T$�� 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��$|DD�xk�gk%�Qk�xAEs~�zxX�k�#?\℄5#mAxyX�k�#6z 3.3 Bq#e (a) dI�~z#(b) Ir��\℄5#m�
z 3.3. \℄5#m��0
90 K): DnS�X,h�� – �?hn| Smarandache K�.�zx�z�NÆ�℄#v72bÆ�℄q�zdI�#�|�j�zkCsB�9 D = {(x, y)|x2 + y2 ≤ 1}�Tutte 1973 &�D)�z�n�� #�x[12] ���#�z� Q��Y 3.2 R=aH M = (Xα,β,P)#l_℄��3�m X Y �j( Kx, x ∈ X YsHJ�Y�� Xα,β WYR=�vC� P#-Ge�XYH� 1 iH� 2#+�K = {1.α, β, αβ} ℄ Klein 4- �A#(d P ℄�vC�#��C�21� k, uXPkx = αx� Q 1$αP = P−1α; Q 2$A ΨJ = 〈α, β,P〉 � Xα,β W}"�P7� 3.2#�z���oz℄n� �� P o Pαβ Ix Xα,β E�|�-WM~&d Klein 4- �% K Ix Xα,β E�|�M~��xEuler-Poincare*p#3n�|
|V (M)| − |E(M)| + |F (M)| = χ(M),b� V (M), E(M), F (M) ℄nlq�z M ���?�d?oz?#χ(M) lq�z M � Euler r�#e�z��z M B}8����z� Euler r��-N��z M = (Xα,β,P) x�}l�Y#:��% ΨI = 〈αβ,P〉 6 Xα,β ExXw�#kA- }l�Y�z 3.4. z D0.4.0 6 Kelin �zE�}8I N��5#z 3.4 �D)z D0.4.0 6 Kelin �zE�N�}8#XZ�x�z M = (Xα,β,P) lq6Q#b�
Xα,β =⋃
e∈{x,y,z,w}
{e, αe, βe, αβe},
P = (x, y, z, w)(αβx, αβy, βz, βw)
�Cx�e�[�nS{$�? –Smarandche K�.�C 91
× (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)#
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�
�Cx�e�[�nS{$�? –Smarandche K�.�C 93
z 3.7. N�Vz�zGp��5z 3.8 �D)z 3.7 � �Vz�zGpx`� �#�,�#z 3.9 �D)�Vz�zGp�GÆ=d��z 3.8. Vz�zGp�x`
z 3.9. Vz�zGp�=d�§4. /�YJ:5'Einstein �J a8N55)`^6j�IxQx���#W�H`Fy��#bN�6lSA W!�|ql��zGp�&elSEXZN<�� N�.%`^E#C6�.%`^�j��Ev N�k%?p�#.%`^Gp��Y 4.1 . U 8A �'8 ρ ��'"�#W ⊆ U�*C ∀u ∈ U#,�IA
94 K): DnS�X,h�� – �?hn| Smarandache K�.&=E- ω : u → ω(u)#K$#*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�
�Cx�e�[�nS{$�? –Smarandche K�.�C 95
(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�
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> $��N�*m�\�A:2,�8`^!�.m��g#b�0\Fg�2dw2�6R`^!� ≥ 11#�g.>�1yN�I6,�M�|�Qk!E#FyX 6��EY|D�B�bu(P|,�.m�V�l<K��Z��kG1�
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1992 p1. U��f$% 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.
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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,
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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$9SC%lCr���(2), Qp %ID?M#1999�2000 p1. 4|� Fangm�N��J#8)k G>>p�`B{>e!#Vol. 26#3(2000),
25-28�2. U���\��.�e%�% ��C #9S{�#2�2000!�3. U���\��.�+{%3e%#9SC%lCr���(7), Qp %ID?M#1999�
100 K): DnS�X,h�� – �?hn| Smarandache K�.2001 p1. ��8;Fr�!UuA:z�N���4|�=℄gm#8)k G>>p�`B{>e!#Vol. 27#2(2001), 18-22�2. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, omk G>>p�`B{>e!, 4(2001), 13-18�3. (with Liu Yanpei)An approach for constructing 3-connected non-hamiltonian
cubic maps on surfaces,OR Transactions, 4(2001), 1-7.
2002 p1. A census of maps on surfaces with given underlying graphs, A Doctorial Disser-
tation, Northern Jiaotong University, 2002.
2. On the panfactorical property of Cayley graphs, �>Ce{�B, 3(2002), 383-
390�3. .~*x�TXR ��ÆK���#H℄H7>p, 3�2002!, 88-91�4. Localized neighborhood unions condition for hamiltonian graphs, omk G>>p�`B{>e), 1(2002), 16-22�2003 p1. (with Liu Yanpei) New automorphism groups identity of trees,�>T!, 5(2002),
113-117�2. (with Liu Yanpei)Group action approach for enumerating maps on surfaces, J.
Applied Math. & Computing, Vol.13(2003), No.1-2, 201-215.
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2. �� r�!Riemann �zE Hurwitz � ��r�J#H℄{>Æ�ys$I{?�>{>�B/Bh�#2004 & 12 *,75-89�
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[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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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.
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7815999 730325
ISBN 1-59973-032-4
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