mathematical combinatorics & smarandache multi-spaces (new edition)
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, i.e., beginning as a construction worker, obtained a certification of undergraduate learn by himself and a doctor’s degree in university, after then continuously overlooking these obtained achievements, raising new scientific topics in mathematics and physics by Smarandache’s notion and combinatorial principle for his research, tell us the truth that all roads lead to Rome, which maybe inspires younger researchers and students.TRANSCRIPT
I#�6SA�z;—– tU6(x Smarandache ,oK��t
Let’s Flying by Wing—–Mathematical Combinatorics
& Smarandache Multi-Spaces
Linfan MAO
(New Expanded Edition)
Chinese Branch Xiquan House
2013
��t&ow�/�9.�XXK&�* 100045&oV_Rb_:�,V_g9Rg9nL&�* 100190�*x5DV_Rj��T&g9"#'&�* 100044
Email: [email protected]
I#�6SA�z;—– tU6(x Smarandache ,oK
Let’s Flying by Wing
—–Mathematical Combinatorics
& Smarandache Multi-Spaces
Linfan MAO
(New Expanded Edition)
Chinese Branch Xiquan House
2013
This book can be ordered in a paper bound reprint from:
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 2013 by Chinese Branch Xiquan House and Linfan Mao
Many books can be downloaded from the following Digital Library of Science:
http://www.gallup.unm.edu/∼smarandache/eBooks-otherformats.htm
ISBN: 978-1-59973-214-5
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on Smarandache Geometries & Combinatorial Maps, I� Chinese Branch Xiquan
ii L�y: ihUAbx� – `N�; Smarandache +aiHouse `IC`I&2006 $&" *GLk&u*GL%m�,"?{lT7T℄� �o"?$&�y�G*�?*~ ,?O℄J?�Fqi%�& /ngGL%m�l1G� EX.��vg 2006 `I%9�uqFWE��&�&S=4f?�(9�& l�N}�G TXQPK}&Z b"y?S��*y&a `m?&?�d���%l�℄�&8?y �%l�a�(9�y�?S M%\�&esGlg� &�5=?S%'+�l&8 D*��?L%:VPx�"Let’s Flying by Wing$�&$ I�v�`IC%hR��&S-?� 10 f%?�'{&�:�lT+z *b_M�: Smarandache *`h&S=b_M�u?��a Smarandache B6"sd:&8�%lTa��:7Smarandache �-'{��%,?'{�{_|,?CV<&�}A'{�u�?x2,?N}%VP"Wing of Scientific Research$�g 2010 I� ��� 1985 -2010 ��% 8 89�&m`(yat(bY?*d,�"UBT�Æu 2003 *Zgl4 m< ℄�Ko%yG�l��9���?&�" `mk?&��:af%�?7k?&�Æ�9jJ&��:i#2rW�GL��89�# �4f�}%\^&(%��f+U_ /Æ&(%��f+FQU_/*�x5Dn%&>Æ)z ��"UsT�Æu*Zgy?S&a,jV�J{=?&�_���[SZL%l89��9=yaau���"lT7T℄�&q�0=k?&iq,)S�7T��~ �G��oS*�o?N}�G%��Jy��J=7����?�%? �9=Uau���a`�?w-O�{%X���l�N}�w`��N}℄�%yV)r~��"U5'�H�Æu_ 3 T�E>��=�,�-9at��.?��T&Pl��fQ&?l)ha�.?��?\�7r&yG44&dq9��,�.?�T7r&j 50 �k?i?l)��(Rl= �Rlf%,?N}%�o��&��k7��?�%?�*�?'{%�4& l�,N3 %!g37l�G�nN}th%?� z&lu�{�7k��,?7Cdy��T,?ry%T� ; &�du��?yM�7y?�Æ}%lTQ �9�o�a�n^= &dD3 +#�N})&�au�k{�-f,w-N}&"w-f,�I�-&)2�I?N}&y?f,$Æ�+* -M N}%�T�J&��=&w-'{X �k� t%��&l ℄2�Dk?&�S-��,?jSa`�%N}3 &{l/w�7k��AryluT�?�ry%�Jo& y?�%G4 EX.�
Xj iii�T/sTU\W�2Æu 1985 ?a`mE{7E?�?\ ��=Z�?N}�s,d&P&}a?SCV�%l8ed&`y?S���?:�D %?\*T� T>7E)&T�l�ed�f�:&�duk{^= G=�??\qY%pS&u�`y?S?\�?d(l�OR��A Forward of �SCIENTIFIC ELEMENTS�Æu?*�SCIENTIFIC
ELEMENTS�| �%l87Q�9�/�� Smarandache '{u�℄!g1|pV%lA,?N}'{&l�?w-'{luaysd:&����Æ?=Kxp_|77l%lA.ya�?:fm%'{&{l<D� Smarandache '{*?%�?w-'{a,?N}:��%sd���l=s�pl 0% RMI }�Æuka?�3j;E���?�v-=<�*l�w-�?P�?&`�?:��i�%y_�AzVEl"RMI El$%Y?&�y?S\�r|a�=Z�?N}�:%l89���"UÆ<�Æau���"lT7T℄�&q�1_%�o&iq,)S�℄℄Yg℄J[�7E�4s7p℄J[*�zb)~rq �Z��+p℄J[&) ��?�M *℄J~ Z�?�*lI�E�*%g�d+T 4*lI%&�iJ�9�o�z�`��%�SÆ?&y��l�J=Mf~��gÆ~r*��Nf%;?OV& l1G�%y>2*(lD%X.����v�(vmoUm__hCsT -Smarandache+nJv�Æu���_a ,jV�J{=?[�yE[*?SY?m��?7M *a�ao7n�M�b_:4<_℄�(.�Æ:ZL%l8,K(9�&Sr%uY?p_lp��?%.Sat�Ob�v*l�S-�S=yaY?�8H�℄se"�SmarandacheB6"�Smarandache �+�2�72��+�.V~6"�+)�(&y?` -M �T; f%�l�(�%[-�g 2013 ,EIo���� 2011 -2012 ��%yd 4 89�*�5'T:7j5OsT(vUo<Æu������4H[w-?7�-�d?&�_a�Np[|�?�?,??O*`m7T℄J?O ?O*N}S*yE[�%ZL�9��M�_ÆlxiQ�( �*��}A T� �%w-?,�-*w-�?&SLw-��7w-E����%w-?f�:uÆ?=�Æ�Æ|7ak�,?=%��&�8rM"Kxp_}*w-?=%�`�Æ&fl��6"�I�0MrM1|(9��A'{y^em/���G%=�,?O�?7_|,?N}O�o?ZL&?a4p[��w-?7�-�d:%ZL�Combinatorial speculation and combinatorial conjecture for mathematicsÆ"�8
iv L�y: ihUAbx� – `N�; Smarandache +aiZLr|?G����l?,� �*Hj9Tw�M�_Ælx% �9n$�9�x��TP %'{e"&*y"yA*1?g��=f�℄E^+J�Hn ,)'{e"&ÆQ��℄!g�J& ,� Smarandache B6"*Bd_|&y�8�I��*_|w-Sh%'{&[-�Z��a6"%m,�6"7 - rz$��SlF6"%w-Sh&℄~u Poincare �{%w- F&� +lT 3-+�qx�#z(/l* 3- +�)(�%�I?S y�S` Einstein o1%dn)&9=y?Y?�*+ ℄3 %Bb(&flf%N}%,?�v��W-�$�&1��Q�$ $n8�L7$Æu��s5�G�%gry�N?&}a%gryCV&S-�z�z{1f��%l89��9�a��u -��)�}�M��^?&� g�%KxyV&�y�#G��=��z�z�d"2011-2015$℄��N��9�o�h7!)9|A9�%g�| ��YgkO� ��vq�%(7%�z�z{1Z%t5&[-�%g,?ryrz�|l%%g5V�GrR% Qt5�%gw'7E�{gth*�GÆ��%g -N}�%g�WO7Ey�9O7E)%gryB�r~��L^ATDo&�Q�$0�v��1Æu���G*`m7T℄J?O�z�j��g,S)℄E���z�j -� �?&{$|q�?CV&V[�z�jq� -hG%l89�&g?a=��z�z�d�!)�j�WZ)�}��!)�j�WZ�)Z��^�9�a`&�*lA{1q�d%z�?| -&�z�j -�g�5{1q�%℄7*�%&pS�z�jG`qY7�v,Sz&8v��(Kx��%�(&S-��=k -�Z℄?�� -�fC|�ÆQ�.'(ÆQ)�( -#G�z�j -&_��l -,��z�jf/&�P99f&���{f/=Q& f/=�&�4s�4s�\4sÆ%�z�j -7E�v����$ $q ℄�{��'J5zf"+Æu���G��z�zvn9vQ�7kÆQ�l ?&{v ?�9�?�9Q0})�[vQ��z�zvf℄nk�l�n99�%l89�&a=��z�z�d�}�^?&# �T�%KxyV*���f/=&*dDz^l~�D& z%� TS�D�� z%E�a/`n9=l�℄~xw% T z�9�k����z�zvf℄nk�= 8 n�D& S=l�7x39� T l3%xwf%�vQ&yiT�a���nD9�&fl�| �S�z�jf/�g 9�<��/ z M&�HL�f�^���%��& y>G4 EX.�f�:& +lT�%�S�JPu\A7G`%|l��l�J=&�L b*\A "&*F(%\APE( ".(X:,d b*G`℄U&
Xj v*v �u�S"�=%lT��%�z�_h%\Ad�1fah%G`&.b�Sf��%�wjf,z^&_h%G`� lD�1fahUdG`&b�Z�w�2?\�℄���^V_P-�+Æ'&�-*ZxdmAKI�/.+n&0-�J�%e(+C;Æ&Myy7y?S�?�*2*LeZ Axp O l1pT/`m
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A Forward of�SCIENTIFIC ELEMENTS�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37/y���NyuW* RMI El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39?%℄J&� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 -M r%p_lp��? -Smarandache B6" - . . . . . . . . . . . . . . . . . . . . 75w-?�S`m��?M %�| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97JOe59P&_|h7�z�z{1Z%t5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122{q�?`r&h7�z�j -e_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129��z�zvf℄nk�℄~xw9�vQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139Ax 1985-2012 Æ-Pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
�H"�5R��y:�– f+U_ /, 1-3$VCU��Ax"&oV_Rb_:�,V_g9R&�* 100190$�℄: bnw*℄�f*~=x5Dn(sF_5)�*K20�CF_QOVb��_>_℄&Q|���(Æ_FR&>_℄&%"|&oV_Rb_:�,V_g9R&>�(Ip�B&>�g9�V��_�S0+/*�yDÆ_~�/&_o7P-QÆ_&6DP-Q~�{Æ_+*_tj&*n7Q��qZ'��NJ*x5Dn&F_&S0&&>�Abstract: This paper historically recalls that I passed from a construction
worker to a researcher in the post-doctoral stations of Chinese Academy of
Mathematics and System science, including how to receive the undergraduate
diploma in applied mathematics of Peking University learn by myself, how to
get to the Northern Jiaotong University for a doctorate in operations research
and cybernetics studies with applications, and how to get the post-doctoral
position, which shows that there are not only one way to growth and success
for students in elementary school. Particularly, it is valuable for those students
not getting to a university, or not getting to a favorite university.
Key Words: Construction worker, learn by oneself, growth and success,
doctorate.
AMS(2000): 01A25,01A70℄�p_, ?{l Kx%7T℄�, yG=*X7, lr℄�&lrk?&/ 1999 4 Tf,G�m#�o?4&2003 6 Tf,,Nth{rZgN}�<��J�(, �a_h Q&Zf%u0`7�m�?ualF2�B7=?B7CT%G=�v P{k�G*lT�?��12003 �_lzF(D=j>%~�8�2e-print: http://zikao.eol.cn
2 L�y: ihUAbx� – `N�; Smarandache +ai�h �? , ��?y\, �#Sj4Xf�92�?hG�?\9,, y�0��,�?K,lQ l�7Tb)℄����℄�, ��l 7T℄�, l �i℄��haB��:�Q��, j�lToGZJG��B��s�ulAeoVJ, 7ka=? �0`%�o 6-�(��^%{v, a��℄�%4p1, ?Q `mlF7T?�?\�<�uq℄�, �m %{v�0_�/ul[?f7T��, l[a`m℄g�?l4+E�%,�d?\�?��k{rw-�?%N}�{7T?��G?g?, ?a �E℄��4, G�l ��~ �G�s�℄℄��92, h ℄℄=`m%��; G�?H%%Ob℄���VgX, >m�X{Æ&�u?%El�)��h�, �l���E *e"T���} ��/℄℄f/&lVG�T�%�o��, �?N}dylV=℄��[ "g?r|� >7T��-9, dT�� >℄℄} , �a{7T?�a E℄��4%4H, dFQaT*℄J[��=?��rm, 0`7�� Gik�� %{v4-8, !*u9AN+%EhG%, /u�Dj�g1 ", ��`m{G)E>k?&�, �Gg,?g�f�:, �Am��u�Y%x�, a�Y���%Te, �<� �|��., d �|��t "%(47���x�,L%�o, a7Tg℄�_, , ?t �`m�?*`m{G)E>k?&�,Gdp-Gr%���?Zgg,lg9A*?o?4�Z�J�?`�(Yg%!g���lztd, ?a`m�?0 �oS�&r *n2, r�S&�,rka��P?�, (%Ur|�-9, /u.S�lA�r%T{:�G�)I_T�o?4, o`�Y�Ax �a=?%1e, >h�/℄℄~ , g�l�/�9*Zg3%.\�lz,lz%\A, ?EÆC�pSqY, u,C`I, �9 �y�Hhg�% "�tQ?\�9�1?, ?�/�,�`�?x�?_T�o?4�T�o?4`?Q"u b�%��*EsgSjQJ"EQ%℄��4 #P, WSj�$&�^?l[℄, l[_T?4, �G-9�v , <�?\ b�, �Hh%Sjuu�b�%�g:H7|+Xo, Zg *9nh�, '&3 �>h :3 , ��℄, y+P�vSj�#��:8g:E'&�%; {�n8# %S � G-9, Æ�I, t`�r|��T�o-9E-u= &UuS�(&�u5�kRp%�l�G%�-9�G?&?W���_4E�CM&S-�*�(QÆ�af%�o-9%N}7���J=&J�o2bY��O�N}��~℄���^, at �o?4?, T��#o2fl�ryk%℄3*���/u,I�4�o^�N}uy��%
h-V`Æ0 3ZP*9n, ��,=�,?O{r�o?N}℄��?u`m{G)E>k?&�%lgS�*uk?G�&(F��%?\�v*ÆQ; �T�; %�o���&)I*?_T�o?4d�sd�?�#�%Sjo2(rz, ( ��larz%fm�J=& b�f'&~�n^C��*n^ ���P ^����(lÆ^�#"&�Vgl�D5��, %>?�+�yD� Æ�*�^{, �0&%J, ?"l Kx7T℄�y�G*�l4�?℄����J<�� , u�w7��, �Zf%, lu�I�+%em*�m%℄U�
�H"�5R��y:�– f+b_ /, 4-19
$VtU��Ax"&oV_Rb_:�,V_g9R&�* 100190$�℄: �D1h+��&b*℄�f*~=x5DnSKS~=b_℄+asV&{o&_1(�^~℄x5KoDZ�^���(Æ_FR&>_℄& &oV_R�B&>�g9C^oAw�-� pKoDZ0&b&��*℄�� 1985 ~ -2006 ~f�BV_g9+k=�V&x*℄�~7g9Sq+)%sV&{ob_M��.#+�lsVC: Smarandache *`hn#+_�0��NJ*&_", Dn,DV', b_℄, SmarandacheK�, Smarandache*`h, b_M��.#�Abstract: This paper historically recalls each step that I passed from a
scaffold erector to a mathematician, including the period in a middle school,
in a construction company, in Northern Jiaotong University, also in Chinese
Academy of Sciences and in Guoxin Tendering Co.LTD. Achievements of mine
on mathematics and engineering management gotten in the period from 1985
to 2006 can be also found. There are many rough and bumpy, also delightful
matters on this road. The process for raising the combinatorial conjecture for
mathematics and its comparing with Smarandache multi-spaces are also called
to mind.
Key Words: Student, worker, engineer, mathematician, Smarandache ge-
ometry, Smarandache multi-space, combinatorial conjecture on mathematics.
AMS(2000): 01A25,01A70
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50 W��+$^+�QF�g&So℄�S�Æ�+ÆG4|�Q�g/C� �*K�f �Q*D&&�8�4'�� b*DN℄&Ap�*>�"FAV&?ID�&�xl�jt�ZÆeEGt��1�7&ÆÆ^h�*DvV�K�x-�t~(r7*4zwqm&�D�T-/ne+��KD5)s�dvD&1993 ~&�#;##SDV'�KD_FL+wUDXB�UX{&�SH��K�x-~FL+�"�ÆBT�&�{3�~�ES 1962-2042&dC�*&�r�5j*�~sQl &g9b_Yg9% 80 z�\O~&fx x-% 80z�Æ �Q0��"1996 ' 7 � 30 M$
�H"�5R��y:�– f+M��C /, 20-32$V6(�I��Ax"&oV_Rb_:�,V_g9R&�* 100190$�℄: Vg+RwDZD\�&2O℄��)X&2O℄�*y℄XJJnjAg9nL�b��*℄�f*~=4<_}&��Q4<g9%�%4<&\%�%(Ig9&"i*�%_g9%N�K���<t�+g9sV&j_^VgDZ&z��l℄�&z�x~�_V�����&&*\�EFf�*\��oXg9>mDZ�lFL+&-P%t�+g9℄�&%C1:oB_}�(D`u&�4XxDVgDZ}Stmj+5��S�&b_�d�.-;53Z'��NJ: �?w-O�{�w-?��I?��I�-�Smarandache '{�w-$Æ�+� -M �,Ne5�Abstract: How to select a research subject or a scientific work is the central
objective in researching. Questions such as those of what is valuable? what
is unvalued? often bewilder researchers. By recalling my researching process
from graphical structure to topological graphs, then to topological manifolds,
and then to differential geometry with theoretical physics by Smarandache’s
notion and combinatorial speculation of mine, this paper explains how to
present an objective and how to establish a system in one’s scientific research.
In this process, continuously overlooking these obtained achievements, raising
new scientific objectives keeping in step with the frontier in international re-
search world and exchanging ideas with researchers are the key in research of
myself, which maybe inspires younger researchers and students.
Key Words: CCM conjecture, combinatorics, topology, topological graphs,
Smarandache’s notion, combinatorial theory on multi-spaces, combinatorial
differential geometry, theoretical physics, scientific structure.
AMS(2000): 01A25,01A70
12010 ���<�+�,8`s��->��SE��2e-print: www.paper.edu.cn
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`� `-<2�/ 35`Gf%)| T,fl�b�| D %�� – HT��n^%; ,y?�u?\H%"a����D %"a&U�8ZK="a'�ax`Q&fl�a�n2*S-%y_�k*&?k?$Æ=+D =%/oD *�zq i) xb f(x) ^�Xh [a, b] �Y,ii)f(a) = f(b),iii)f(x) ^MXh(a, b) wY#&f^ (a, b) w���^~; c&:) f ′(c) = 0.Æx�axH%"a'�&$!�D =n2%���n2 i) V"� f(x) a [a, b] :.�|�*|�+,n2 ii) lV"� f(x) a (a, b) :%l7 c �|+,n2 iii)lV"� f ′(c) lD�a��^&X�o4D &�( f ′(c) = 0��d_zu?ed �?%LV(�\ f�:duD &. �H �3g=D v^Bb�T (}Tr%*"l$.\ vF?%$g,"p$Ey?LÆj2ÆQ; �T�; %�o� =ZSQ"&�}Tr%P .B}�T a �p{�p{%`�Ob,?�%b �D��D *\ �`/b �D�*D &HLuT %sd&lFz?PdWyV�&y|B}%uyjT�%\ &H33u�*lAy\l=T="&�`T u| j%�?a?\=QQQJ02!g &(�g27lk% T�8fÆ&`/T u2|(�%r�7lk% T�8fÆ&l�[uq3`k%T �v�ypS�ax,�l�[&bf{ �=r|%~�%l�"%T �v=a℄Æ&�v�q8%�"Rp℄7%T �v�fl�Xfk%�8fÆ�?k%T f/|a&�^~%S ���pk%T '���Ob|ma*"l$ÆYl�Bb%\ %S-?&3.yCY7��S-(y%\ %T�% "�k*&�
∫ √ln(x +
√1 + x2)
1 + x2,.$!
[ln(x +√
1 + x2)]′ =1√
1 + x2%S-&� DuÆ%; �.�"lT*∫ √
ln(x +√
1 + x2)
1 + x2=
∫ √ln(x +
√1 + x2)d ln(x +
√1 + x2)
=2
3ln
3
2 (x +√
1 + x2) + C.
36 L�y: ihUAbx� – `N�; Smarandache +ai"p$�Ul�iFR℄%\ %Tv&aT℄-; =3.yX.�*b"aw-7)l*n∑
k=1
kCkn = n2n−1,.�a{S3g=`
n∑
k=1
Ckn = 2n%"a�v&�d{ �{
1 +n∑
k=1
Cknxk = (1 + x)n`r&}rg` x ��&_� x = 1 �$
n∑
k=1
kCkn = n2n−1.p&&:�D *\ %?\u?\�?%1TV .Æ%|le�+#a&z?LV�%u&?L?"�?� ulT�l%�J�.y�G%&o2(fzz.�fT�f{�fy&{l�� ℄aG�?3%?\9~�
�H"�5R��y:�– A Foreword of�SCIENTIFIC ELEMENTS, 37-38
A Foreword of�SCIENTIFIC ELEMENTS�Linfan Mao
(Chinese Academy of Mathematics and System Science, Beijing, 100190, P.R.China)
Science’s function is realizing the natural world and developing our society in
coordination with its laws. For thousands years, mankind has never stopped his
steps for exploring its behaviors of all kinds. Today, the advanced science and
technology have enabled mankind to handle a few events in the society of mankind
by the knowledge on natural world. But many questions in different fields on the
natural world have no an answer, even some looks clear questions for the Universe,
for example,
what is true colors of the Universe, for instance its dimension?
what is the real face of an event appeared in the natural world, for instance the
electromagnetism? how many dimensions has it?
Different people standing on different views will differently answers these ques-
tions. For being short of knowledge, we can not even distinguish which is the right
answer. Today, we have known two heartening mathematical theories for sciences.
One of them is the Smarandache multi-space theory came into being by purely
logic. Another is the mathematical combinatorics motivated by a combinatorial
speculation, i.e., every mathematical science can be reconstructed from or made by
combinatorialization.
Why are they important? We all know a famous proverb, i.e., the six blind men
and an elephant. These blind men were asked to determine what an elephant looked
like by feeling different parts of the elephant’s body. The man touched the elephant’s
leg, tail, trunk, ear, belly or tusk claims it’s like a pillar, a rope, a tree branch, a
hand fan, a wall or a solid pipe, respectively. They entered into an endless argument.
Each of them insisted his view right. All of you are right! A wise man explains to
them: why are you telling it differently is because each one of you touched the
38 L�y: ihUAbx� – `N�; Smarandache +aidifferent 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.
�H"�5R��y:�– HHb�rH�%C RMI Hf, 39-44
�m>t�qm!1; RMI ~ Ax(�)PJw�^2%w 83-1 p^!)/eh��*NKN$uW%N}=&XÆkm�l�Bb%�?'{�8lT��yGl��~E���&x�N}��%(9# ��%(9&�uE���-=N}��%sg'{�l/eh��Z���l'{�j�1C��_*1, cos x, sin x, cos 2x, sin 2x, · · · , cos nx, sin nx, · · · %�?(8DM�� f(x)"EDM* 2π& u 2π &.y f�WOG�A~G$5bl*
a0 =1
π
∫ π
−π
f(x)dx, ak =1
π
∫ π
−π
f(x) cos kxdx,
bk =1
π
∫ π
−π
f(x) sin kxdx, (k = 1, 2, 3, · · ·)yG�E���*a0
2+
∞∑
k=1
(ak cos kx + bk sin kx).`/DM��.y��lA����D�%�v&uGDM��&{l~.yyG/y���F5����%D�&(Q&�u* D�7 [a, b] :%DM�� F (x)&B�D�lT�� F ∗ (x)&i#` � x ∈ [a, b]&P( F ∗ (x) = F (x)&l F ∗ (x) uD�;��G:%&y (b − a) *DM%���{ 1[3] ba [−π, π] :8 f(x) = x2 y�*/y��&?L.yg�� f(x) %D�*f ∗ (x) =
{x2, x ∈ [−π, π],
(x − 2kπ)2, x 6∈ [−π, π],�e&k u��&i# |x − 2kπ| ≤ π&l f ∗ (x) uy 2π *DM%DM��&S��0de� 1*1D8q{ �<Y�>M|�%1!1985#�
40 L�y: ihUAbx� – `N�; Smarandache +ai-
6O−π π
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−3π 3π−5π 5πx
y
� 1�^&?L.y8 f ∗ (x) a�fG:yG/eh��*a0 =
1
π
∫ π
−π
f ∗ (x)dx =1
π
∫ π
−π
x2dx =2
3π2,
an =1
π
∫ π
−π
f ∗ (x) cos nxdx =1
π
∫ π
−π
x2 cos nxdx = 4 × (−1)n
n2,
bn =1
π
∫ π
−π
f ∗ (x) sin nxdx =1
π
∫ π
−π
x2 sin nxdx = 0.t$f ∗ (x) =
2
3π2 + 4
∞∑
n=1
(−1)n
ncos nx.y���t/ f ∗ (x) ℄~2&a [−π, π] :&�(
f(x) = f ∗ (x) =2
3π2 + 4
∞∑
n=1
(−1)n
ncos nx.�^&?L�# � f(x) %/ehy�l�-\//eh��=%:�,~&�F�|m�u*�N}%�� f(x) DM�� f ∗(x)
# f ∗(x) %(9# f ∗(x) %(9-
?�?��D� N} f ∗(x)℄k
!II �sI�&D RMI Ig 41�?==.\%Z;\O*�l�;�(�Z;%~v&u$-,?N}�%%�lNyuW�uj��lsg'{��e&?L(�Y?ydNyuWaT3_�$�J=%���NyuW*L[f(x)] = F (p) =
∫ +∞
0
f(x)e−ptdt.SBb(�g|maG(4fT^%(9&��℄(i#$Æ�J\O*���J��g&x��;&?L.#L[f (n)(t)] = pnF (p) − {pn−1f(0) + pn−2f ′(0) + · · · + f (n−1)(0)}."y& l3_�GLt($Æ�J
dny
dxn+ a1
dn−1
dxn−1+ · · ·+ dy
dx+ any = f(x),.b ,NyuW*y(x) → F (p) =
∫ +∞
0f(x)e−ptdt&�.yOGlTy L[y(x)] *0$�%lCt(�J&�^�.yT# L[y(x)]&{l�( y(x) = L−1{L[y(x)]}�{ 2 �NyuW��J y”(x)+2y′(x)+2y(x) = e−x %<sn2 y(0) = y′(0) =
0 %T�� i) ,NyuW L[y(x)] = Y & �J}XNyuW&V� y(0) = y′(0) =
0&#y/ Y %���Jp2 + 2pY + 2Y =
1
p + 1.
ii) T:[%���J�(*Y =
1
(p2 + 2p + 2)(p + 1)=
1
p + 1− p + 1
(p + 1)2 + 1.
iii) Ny�uW�(y(x) = e−x − e−x cos x = e−x(1 − cos x).�ATv.yMF�|m*d [4]*3$Æ�J ���%����
���$�J%T-
?�?NyuW T���JNyuW
42 L�y: ihUAbx� – `N�; Smarandache +aiE-u/eh��&UuNyuW&Ssg'{PuZ��%lA'{%�e�����u�?�v-=%y_�AzVGl&� RMI El [1]:>E~=w-l�H, X +`��Q�, S&zqyy%~=YE#� A {S #�|6#G S∗, fY� S∗ (s~E+b_�vml�#, X∗ = A(X) Elt&�i(s{oG|#��Ym X = A−1(X∗) Elt��l�J.�F�|m**
S S∗
X∗X
- ?�?A
A−1
RMI Elu�?=KxZ�%�vEl�H%�eZ�a�?A%UT�;P.y �{ 3 =Z��?�4p"4pÆ"A<�%TQ�+&ST�; %sg'{�u RMI El%Z��.�F�|m&i?L.y| 2 �l7*�+y_; ��y_; # lA��y_# lA�+y_
-?�?
��|m ���;�+TwaT3_�t($Æ�J &du RMI El%�eZ�&SF�|m*d*3_�Vzt($Æ�J ℄�f~l℄�f~l%X$Æ�J%T
-?�?
,�uW T�J`�y_
!II �sI�&D RMI Ig 43{ 4[2] h�w--=& >T�; %�vu RMI El%�eZ���e(�Y?ldS=%m��"SG��$%�v�F5m���v&�u=lTEr��*{un} = {u1, u2, · · · , un, · · ·} `�/#lW��u(t) =
∑
i≥0
uiti*
eut =∑
i≥0
ui
ti
i!,�PD ui := ui"j�D*Km��&?�D*,m��$&�?x�N}Km��*,m��&_x�:[%`�y_&zVaQ&�.y# �� {un} %lA(9�F�|m** ��; W��; W��%lA(9��%(9
-?�?
`�vl Z;`�vlk*&T+�J un+2 = aun+1 + bun, u0 = A, u1 = B �.y���Am��%�v�Eu(t) =
∑
i≥0
uitil(
atu(t) =∑
i≥0
auiti+1,
bt2u(t) =∑
i≥0
buiti+2.Fy
u(t) − atu(t) − bt2u(t) = u0 + u1t − au0t +∑
i≥0
(ui+2 − aui+1 − bui).X�� {un} %6�y_&�((1 − at − bt2)u(t) = u0 + (u1 − au0)t.
44 L�y: ihUAbx� – `N�; Smarandache +aiFy&u(t) =
u0 + (u1 − au0)t
1 − at − bt2.E 1 − at − bt2 = (1 − r1t)(1 − r2t)&l(
u(t) =1
r1 − r2
(u1 − au0 + r1u0
1 − r1t− u1 − au0 + r2u0
1 − r2t.
)8 u(t) y�GW��&�(un = (B − aA) × rn
1 − rn2
r1 − r2+ A × rn+1
1 − rn+12
r1 − r2.℄~2&`/"i�r%��vT; h%�� {Fn}&( Fn = Fn−1 +
Fn−2&F2 = F1 = 1&t( A = B = 1, a = b = 1 t r1 =1 +
√5
2, r2 =
1 −√
5
2, Fy
Fn =rn+11 − rn+1
2
r1 − r2=
1√5
(
1 +√
5
2
)n+1
−(
1 −√
5
2
)n+1 .�ulTy�D1%li&�� {Fn} = l~Pu��&�l Fn �ux�E �Q|m%��*.℄2h2�
[1] 3j;��?�v=<��[2] )��2�-�w--�:"�[3] {�?�?_��?ÆQ�=�d"�[4] =)Zg?�_/x�E���?�4,"�
�H"�5R��y:�– f+DV /, 45-73
$V�=��Ax(&ow�/�9.&�*&100045)�℄: b��*℄�f�~=x5Dn&-s�.~+��&�-V�"�*D4zDV'�x��℄:xIDV'�w�Ko(l-��8℄��IDV'&�%Ab_�t��DVb�8℄_}S~���T�Kz9.�XXKS~�+k=sV&:=nL�~1(�wSS~=b_DZ}-`&C:=nZ4Xn��XB�U�v4+$_�VY*Y�&_�S0+L�~�.-~E+�qZ'��NJ*x5Dn&V�"&*DM��S&*DN℄&:xb�&w���&w�/�Kz�'�
Abstract: This paper historically recalls each rough step that I passed from
a construction worker to a professor in mathematician and engineer 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.
46 L�y: ihUAbx� – `N�; Smarandache +aim3℄J~ &l)u?�v�#*T��??\7N}%q�QJ&du?�'^4z�+<℄qmf+b_g9&J~�b_g9+>/Æ%�e�x�2009 lz℄J�z)E&?�_*?G<T�w�/�v�&3W:&?��w�/�v�n9�D%��7�P&x��~f�rS%�z�j7kf%�a3%ÆQ*<T& X�?G%eb�d3?&4f?G#'f?&����GLaf�℄�=< %; *Ko&�C?%�0�?S-k`�w�/�v�% Ty�T�%uIqi&ll��ÆQ*T�&?GL��|m 1y%&? ����BtÆ�aaUh%�:&w')E%l)E=�O `?"*�{�℄1f�S�aD�~ $℄Ql&-+^r�T&-+^r�$&$-+rAP&~I*I5^$&$|V�&j_�~��℄_�_L+[ �Æ℄�1_&?{lT7T℄�&iq,)S�℄℄Yg℄J[�7E�4s7p℄J[*�zb)~rq �Z��+p℄J[&) ��?�M *℄J~ Z�?�*lI�E�*%g�d+T 4*lI&7T�y> Ml1bG*lT�?℄��(y&Z7T�Mf~��gÆ~r*��Nf%;?OVV .Æ�_�Q0ÆJ1980�1981 q9}G&&<��9�� Æ�t&�?k�E��,�?�K?\�1981 9 T&X5{7E�%a�E(*G� %&S==kG℄&?���e'&�E(d &.�Y=mu*l�i5a� 2u℄�%�4=�7Tp��b)< ℄�1981 12 T&=�7Tp�lb)f%�|'�℄&y<s℄J7E%0b�� 12 T 25 %&?���℄�&74fSHb)%i5lXQ �� aX5%=�7Tp�lb)��lz��℄�%y(,_f�&b)aX5�-�:6℄2w'�*M}TT%,6E>&�D&(-Q?�Æ-℄A�1982 rN?&y℄,6E>S�&?dÆ- �=�7Tp�lb)1h:^&℄Au�i℄&7?lXÆ 8b)1h%&U( 4-5 4i5�2u =�7Tp�1b)E>,4&"q7=�7Tp�lb)E>,4�T��&b�a℄�:�uld?�GL4�u?&*�i℄?�Ml�\�B&z HT%�,��R(p_fCi&zeD~,_f℄&iSH℄AG�?� {?q℄&*2u�uq℄`I&S���~G��d�u*E�{rk{{r%℄A&i??Qb�Zg?�Jhl9u�S�
h-EWÆ0 47��℄�%�4�Æ�Ei)o %�>&?�k�?%7T℄�SI�Hh7℄�[/lX& X5O/:61M℄2:C��ulTpV�%�D~r&7E℄M_Æb�&*V:6r:rz&�*'R℄A%�i℄&q30b�C8B���"&*?[%q℄℄℄aa℄�[�4lTT℄�S�&℄h�::^?� �,_fC&�eG |�&*� lTT%Sj�d�u_P:Ci�?�DHT{℄h=�HCi�j8�?℄% "� % 4tt&lg 200-300e% &.0I�Ci��8dQ�1981 rN?aX5�K ;8%K/[%�b<#��&lg 700 fe% P&d.I�1Afi$&Wd%&ublX??Wmu�� &?H}Dlz"O/ X5{�&_{X5{�\=j �-��2m�&HT�lzX5{��K ;8 &i*K/[%�20b_�<�4l"�j2p{Sf+T%�b_�y��A��G.Birkhoff* S.MacLane %�A Survey of Modern Algebra�(4th edition)�Weixuan Li %�4<�)�ua�T M8%�y?vg�- &l) 1984 Y=`m℄g�?ZW0E�?\�-&?�=HTH�� �?:y/f/u'Y� %(lzd%Cd�[-S�&�LGs0b?\9O$g&yv�q�7E%0b�=�7Tp�lb)*y℄MX�9O�\C&"�4�b9Æj* 82 [%�?S<3&?\^)��� )�+)9O3J�����z�\?&?���E(ob_�t<&*��u?a=? M?%y"%3J�/uj��: "?\{?8E�%�k0b_��Cg9��?\G)?�%$uÆEJ&��fT�{Sf+T�b_�y��A�:%l�y\ �4fy℄P$!?%�?sd"&dC?dM�?[-l�^)�?=%; �l�l M&=?�?%rfsd&di#?T� >3� & ?Q?.y}alTiF��%CV `�?; d�lDsd&i*�T�-=% D�J*�?�wv%fl���)�TF�IX[%�&_b__NH�&�#:[(�2��;n"a(\O&(�2�U(Z(t%�;�vn"a�v&/uW����l���F�IX[4B�W?a�l��|m��&z �QGBB`I%�&_b_SxNH��?aa�0&?Q,�?� C&`?.S�|�ng&GdG�?�l M�k!Pk?%oK#Q��?%g(℄�u�i℄&?0bg<"g(℄�&fl��℄j?k��i℄(�~sg`bEy&lu��i&a``ZZ%B��:b�}:&aB�>:b�%r,pu�S4%lXB�>&ldb�lXQ&�:U,1ua2[�:�;2)�"��blz 4&t/:[%�IU�Zy��TT?&?sg
48 L�y: ihUAbx� – `N�; Smarandache +ai:.y<s��~b��&/uYfSG[/�7�:6U℄�i%�E℄��*V":6`*tsd?04_&�℄C("�TT "af%`*tsd#C�%�E℄�&8#C�"l4X℄J[E�&iFV�&l>=�" (50cm&� 60cm��E�J=&> d[%;2�~2�Ei)�Evt`Q&*�XgEvxf�&NV�&℄�[/`D�u�BdxO*s℄Æ%�i�>h:C�i��C&�&�:k?&N\�&lz℄"+W&? k�2a`*tNB�:�f�&+7>dQ&"a�iiFV&B�HJ%iFC�E`����y?&yu�E�_SG%B��&℄�[/lFPd4�?a2[6/���
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w�)wnpVn``iDtU�Smarandche+nJv� ∗Ax(%nU^Qa^9�+U^f8Q%�) 100190)�℄. �=?Zn.~�M;&�<t�_℄~���3xt�_+~(Æ,~�<>heory of everything��(1U^K�t�_℄+�� &3=?ZgktS5&�>D�<t�&��<C M- �<+x�t�_℄*1Æ7%&x�(Æ,~�<+t;^3b_℄P-x:��_�+b_�<&'�T+�j&>Dn.~?Z+t�_�-x3t�&��S,g9:qm+0�^3b_℄$P-x3n.~?Z+b_�<�4X&t�_℄Yy*Oo&^��<: M- �<x+*1&b_℄xx�~(YO Sn.~?Zb_+�<&�>DSmarandache*`h�<&,�'+_,�D�<t�_℄�1U+&�g&ZS~(b_�<&,�,[O��<t�+RUwg�)d�b+/wl+^3�,5� �~�<+F"�/�g9+/wd��n#�/w�vC/wg9Sq0, �&YOlM�b_n#^,xsV&3%+:IZ'�b/wZ/3Z}>&^Rolu+~81 [16] &+,�/��
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
1p``n6m�L�a^93;^\�'-!2006 � 8 S%g`%z�>#)+iU�Iz<>YK!2006 � 3 S#�2e-print: %nUM;a℄�%200607-91
�=u��r-o/��[ ` –Smarandche +ai�= 75
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 yetÆ, 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.�PJ*8H�℄s -&M- -&B6"&2��+&Smarandache�+&.V~6"�+&Finsler�+�|u# AMS(2000): 03C05, 05C15, 51D20,51H20, 51P05, 83C05,83E50
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1.3. y-N�Æv�mDUsT!� -M 7fYM N}|aM9"1AM9mi&�:i e�:?/ u*d?/ d hG�p_p�^`m�}A\�mi&"uH��% -&��u Einstein%u`-* Dirac %~io?�u`-u\� o% -&lF�/N}8HM ?,l~io?uy/$|mi��o% -&SL:uo�p(uo�/(uo��,A��ohG�mi&"uH��%sg��o�{:Tp�p_��k&4fM ?�&SL Einstein g�l)3o/|l�,Asg��o&�|lu`-*~io?&7lM ?%�|l -&�9n=q3`m%Theory of Everything�q� 80 f%N}&; l)E(# H<T��; %}7a/��u`-uy/9|8H% -&*�/_�N\_�2M)&S�EM9uq9ÆÆ%,l~io?uy/$|8H% -&*:i�9i�=i)&S�EM9ua2ÆÆ%�ltK=fJ,N}&ÆQ��℄!g�;%l��; &i*<-DQ~+E-zq*Q~&-d�=<--S2OnzO*%,�<--nz"3,&+<-+Ub03d�-�+znz(JrS+ 3 UE-Einstein nSa`%��u`(E *�-t�j7^pÆ7Q�&E-�*+I<*)E * ~=�1+X�wdN����+p�t�5�D*YX�+7l� o1�J&�
Rµν −1
2Rgµν + λgµν = −8πGTµν .S-8H?E , �V�aVS 104l.y 1&<-&p�~;�~=;+p��*L
�=u��r-o/��[ ` –Smarandche +ai�= 79nE�&Robertson-Walker# � o1�J%lA�`DTds2 = −c2dt2 + a2(t)[
dr2
1 − Kr2+ r2(dθ2 + sin2 θdϕ2)].u�%8HD* Friedmann 8H�q�f%h9|*&Hubble a 1929 rm&�℄�U%8HulT%��50�%8H&tV� o1�J%�50�TG�M ?�%Ob��&�S0<sn2
da
dt> 0,
d2a
dt2> 0.?L$!&.
a(t) = tµ, b(t) = tν,�e&µ =
3 ±√
3m(m + 2)
3(m + 3), ν =
3 ∓√
3m(m + 2)
3(m + 3),l Kasner V�
ds2 = −dt2 + a(t)2d2R
3 + b(t)2ds2(Tm)* Einstein 1�J% 4 + m +�6T�lF~Gd&�TT� �W` 4 +�50�T���� "\r`DuWt → t+∞ − t, a(t) = (t+∞ − t)µ,?L# lT 4 +%�50�T&*
da(t)
dt> 0,
d2a(t)
dt2> 0.p_p�i`m% M- -*T�:�; <D�sd��l -�Emi u97lu+� z% p-f&�Sf p T��(4V%i6"&�e p ulT����1-flFD��&2-fD�[f�� 1.4 =W`� 1- f�2- f�SZJ�
80 L�y: ihUAbx� – `N�; Smarandache +ai? 6 ? 6 ? 6-
(a)
? 6 ?6 6
?-(b)� 1.4. f%ZJn M -&8HpS�k %vT�#6"+�u 11 +%&�℄s�k?&S= 4 T��+a��%M��PG&l�� 7 T��+la�����C�&�^#G?L_h # % 4 +9|8H* 0% 7 +$|8H�4 +9|8H�%��o$- Einstein o1�J&l 7 +$|8H�%��o$--N/�J&"y# d[�TS-�bv 1.1 M- �<+1`D~=^Q=;HY~= 7 U`h R7 + 4 U`h
R4���D 1.1 *��6":%9 o%bOn2&?L.y# Townsend-
Wohlfarth"% 4 +�50�8He"ds2 = e−mφ(t)(−S6dt2 + S2dx2
3) + r2Ce2φ(t)ds2
Hm,�e
φ(t) =1
m − 1(ln K(t) − 3λ0t), S2 = K
mm−1 e−
m+2
m−1λ0tt
K(t) =λ0ζrc
(m − 1) sin[λ0ζ |t + t1|],�e ζ =
√3 + 6/m. " ς <s dς = S3(t)dt&l�50�8H%n2 dS
dς> 0* d2S
dς2> 0 �# <s��+�;|a&. m = 7 l0�i* 3.04�
�=u��r-o/��[ ` –Smarandche +ai�= 81{�?CV<&D 1.1 =%7f�: u7lu6"&"y J%�?; u D��^�t~(b_`h&,&Q=;{w#~= 1 U+`h-)�|a&* �^%�?6"�a&vHlD u?L%3Sj= # %6"&d u?Laq8�?=<0�%6"&k*a 3 +t(6"=&HT7.y|m* (x, y, z)&H .�S�lT+��/)/ 1 %i6"�§2. Smarandache +nJ�g&$lT(�%; *
1 + 1 =?ak��_=&?L$! 1+1=2�a 2 f5Z;e_=&?LU$! 1+1=10&�e% 10 f�:Uu 2�*a 2 f5Z;e_=.(}TZ;C4 0 * 1&SZ;�l*0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.n��D&�D)/4DÆ%Æ?'{&?L��lA�z'+%�!�%Æ'{ ([18] − [20]) QB� ��T; &B�ÆQ 1 + 1 = 2 n 6= 2�?L$! 1, 2, 3, 4, 5, · · · �^%�hGk��_ N�a�T�_=&n��%�%&HT�D*j[b�f%�%?��&� 2 %?��* 3&�* 2′ = 3�z^&3′ = 4, 4′ = 5, · · ·��^?L�# �
1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, · · · ;z U# �1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, · · ·�^l�Z;)l�{la�Ak��%Z;e_d&?L.�# 1 + 1 = 2 %S-�ma&?LauldZ;%D��WDlT�- S& ∀x, y ∈ S,D� x∗y = z&�'u S :�alT 2 CS-�A ∗ : S × S → S&i#
82 L�y: ihUAbx� – `N�; Smarandache +ai∗(x, y) = z.���T%�l&?L.y��=�Ay_a�[:|m`Q��g= S =%HTC��[:%7|m&* S =( n TC&la�[:� n T et%7,}T7 x, z &"qIln(�t[&* �alTC y i# x ∗ y = z, ?La�nt[:z: ∗y&D*8t[%ÆB&*� 2.1 Fm�
� 2.1. qt�-Æ�vV��A`�u 1− 1 %�� S `�%�* G[S]�ma&* ?L{ lT<s 1 + 1 = 3 %Z;_|&?L.ygW` 1 + 1, 2 + 1, · · · )^+�x��TQ�G�bf 2.1 ~=�b�, (A; ◦) OS�~+}�^~=#� : A → A :)_∀a, b ∈ A&�w a ◦ b ∈ A&f�^~=Q~+G c ∈ A, c ◦ (b) ∈ A&��5O S�~#��?L4(|# ydy/��_| (A; ◦) 7� G[A] %y_%lTS �bv 2.1 � (A; ◦) S~=�b�,&f
(i) } (A; ◦) �^~=�~#� &f G[A] D~= Euler 4�{ &} G[A]D~= Euler 4&f (A; ◦) D~=�~Yx�,�(ii) } (A; ◦) D~=Ea+�bYx�,&f G[A] &Q=C;+lVS |A|,�B&zq (A; ◦) /℄7S&f G[A] D~=Ea+* 2- 4KQ=C;j�~=":)**C; h+�S�_ 2- �&{ g�`/(rTC%~#&.y��lA(r�%�l�D`F(Z;S &� 2.2W`� |S| = 3 %}AZ;e_�
�=u��r-o/��[ ` –Smarandche +ai�= 83
� 2.2. 3 TC%�vZ;�"� 2.2(a) (1+1 = 2, 1+2 = 3, 1+3 = 1; 2+1 = 3, 2+2 = 1, 2+3 = 2; 3+1 = 1, 3+2 = 2, 3+3 = 3."� 2.2(b) (1+1 = 3, 1+2 = 1, 1+3 = 2; 2+1 = 1, 2+2 = 2, 2+3 = 3; 3+1 = 2, 3+2 = 3, 3+3 = 1.`lT�- S, |S| = n&.yaS:D� n3 A z%Z;e_��^?L�.yalT�-:z D�` h AZ;&h ≤ n3 l# lT h- BZ;e_(S; ◦1, ◦2, · · · , ◦h)�aq8��?=&�u�l%Z;e_&T�;�e)�u 2 BZ;e_�lF2&?LD�lT Smarandache n- B6"*d�bf 2.2 ~= n- *`h ∑&E�S n =A� A1, A2, · · · , An +%
∑=
n⋃
i=1
AiKQ=A� Ai LE��~(Yx ◦i :) (Ai, ◦i) S~=�b��&�� n S��b&1 ≤ i ≤ n�aB6"%F�d&?L.yfl���q8��?=��T�;��~6"%:l# B��BT�B;�B�~6"%:&�# u�%��Sh�bf 2.3 � R =m⋃
i=1
Ri S~=E�+ m- *`h&K_pÆ�b i, j, i 6= j, 1 ≤
i, j ≤ m, (Ri; +i,×i) S~="K_pÆG ∀x, y, z ∈ R&�w��+Yx�qL�^&f-
84 L�y: ihUAbx� – `N�; Smarandache +ai(x +i y) +j z = x +i (y +j z), (x ×i y) ×j z = x ×i (y ×j z)C
x ×i (y +j z) = x ×i y +j x ×i z, (y +j z) ×i x = y ×i x +j z ×i x,fO R S~= m- *"�}_pÆ�b i, 1 ≤ i ≤ m, (R; +i,×i) D~=�&fO RS~= m- *��bf 2.4 � V =k⋃
i=1
Vi S~=E�+ m- *`h&,YxA�S O(V ) =
{(+i, ·i) | 1 ≤ i ≤ m}&F =k⋃
i=1
Fi S~=*�&,YxA�S O(F ) = {(+i,×i) | 1 ≤
i ≤ k}�}_pÆ�b i, j, 1 ≤ i, j ≤ k CpÆG ∀a,b, c ∈ V , k1, k2 ∈ F , �w_�+Yx�q�^&f(i) (Vi; +i, ·i) S� Fi +*�`h&,*�^vS�+iÆ&��TvS�·iÆ,(ii) (a+ib)+jc = a+i(b+jc);
(iii) (k1 +i k2) ·j a = k1 +i (k2 ·j a);fO V S*� F + k **�`h&TS (V ; F )�"y?L$!&M- -=%6"e"f�:ulAB6"e"�bv 2.2 � P = (x1, x2, · · · , xn) S n- U�H`h Rn &+~=;�f_pÆ�b s, 1 ≤ s ≤ n&; P {w~= s +D`h�_ V��y6" Rn =�azds e1 = (1, 0, 0, · · · , 0), e2 = (0, 1, 0, · · · , 0),
· · ·, ei = (0, · · · , 0, 1, 0, · · · , 0) (4 i TC* 1&S1* 0), · · ·, en = (0, 0, · · · , 0, 1) i# Rn =% �7 (x1, x2, · · · , xn) .y|m*(x1, x2, · · · , xn) = x1e1 + x2e2 + · · ·+ xnen; F = {ai, bi, ci, · · · , di; i ≥ 1}&?LD�lT�%�~6"
R− = (V, +new, ◦new),�e V = {x1, x2, · · · , xn}� \Kx(&?L�D x1, x2, · · · , xs uRl%&�.�az~ a1, a2, · · · , as i#
�=u��r-o/��[ ` –Smarandche +ai�= 85
a1 ◦new x1 +new a2 ◦new x2 +new · · ·+new as ◦new xs = 0,lD( a1 = a2 = · · · = 0new t�az~ bi, ci, · · · , di&1 ≤ i ≤ s&i#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.{l?L# 7 P :%lT s- +i6"� ♮�� 2.1 � P S�H`h Rn &+~=;�f�^~=D`hW�R−
0 ⊂ R−1 ⊂ · · · ⊂ R−
n−1 ⊂ R−n:) R−
n = {P} KD`h R−i +UbS n − i&�� 1 ≤ i ≤ n�
§3. Y�wY�?&3.1. Smarandache ?&SmarandacheK�ulAy��%��+&S�(�;rjÆ$% Lobachevshy-
Bolyai �+�Riemann �+7 Finsler �+&S`r7u��zd Kn��y�+=%`�bE�?L�gauld�y�+����+�Riemann �+%pl�J��y�+%b e_"d[�HnbEwG*(1)�Q=;%Q=,�+;�EY���,(2)Q���LYn�hK,(3)pÆ;S&?&(sp�>E+#~;YZ~M,(4)�-��L�0,
86 L�y: ihUAbx� – `N�; Smarandache +ai(5)*�[wz-~���:#������&K^>~���+~<��+�w� �13���&f������^�~<��, z4 3.1 �=�
� 3.1. ln)t7}n �%)tu?�e&∠a + ∠b < 180O�y?lnbExD*�y4HbE&HU.y��d[�A6��v*s>E��B+~;&8�^~���:>E+��*���k{�ybEbÆyQ&�Ll)�#S4HbE �8�*bE`m&H :�fa�8ulTd �*y&4f�?�3o/��j,nbE"a4HbE&�l)E(G`�/u(�{�SG�E�f�y4HbE&'Y# %b e_u�� &u��aBd�_�p�&Lobachevshy * Bolyai�Riemann Æ~�� z%�E��y4HbEm#G`�GL��%�EÆ~u*Lobachevshy-Bolyai �E*s>E��B+~;&���^����:>E+��*���Riemann �E*s>E��B+~;&*�^��:>E+��*���Riemann�E# B}%Ea/"y.y7l`=�+&?�d Einstein �*Su`-=% o 6&�= o1 �lT`=6"�z^2&?Lu�.yfl��9u�ybE# �%�+l�;E(%�y�+�Lobachevshy-Bolyai �+�Riemann �+* Finsler �+-9n [16] =T���T; �; %T�#o/�� Smarandache �+'{l7l.V~6"�+&�e` Smarandache �+�lT(bY?*d&dlN_Y?.V~6"�+�Smarandache �+S�a<K���K��{� K�*{K�),A&Æ~n z%bE7l�S=&a<K���%bE*�ybE"1$-"4$y�d[ +lnbE*(P − 1) ���^~����(��B+~;&:)-s(;+��L:��
�=u��r-o/��[ ` –Smarandche +ai�= 87��*��,(P − 2$���^~����(��B+~;&:)-s(;8�^~���:����*��,(P − 3) ���^~����(��B+~;&:)-s(;8�^-�+ k���:����*��&k ≥ 2,(P − 4) ���^~����(��B+~;&:)-s(;8�nb���:����*��,(P − 5) ���^~����(��B+~;&:)-s(;+p���L:��������K���%b e_u�D�y�+ 5 nbE=% 1 Tn�T&���ydlnn�nbE��ybE=%`�bE*(−1) s>E+pÆ�;*~E�^~���,(−2) �^~���*yn�hK,(−3) >E~;�~=4b&%*~EY�l~=M,(−4) ��%*~E�0,(−5) s>E��B+~;&*~E�^~���:>E+��*���{� K���%b e_u�DA��+=%lnn�nbE&u���d�bE�*(C − 1) ���^����6P-��{w�=>E+;,(C − 2) � A, B, C S�=*O�+;&D, E S�=**;�} A, D, C �
B, E, C �;O�&f(s A, B +��:(s D, E +��*��,(C − 3) Q����dw-�=**+;�{K���%b e_u�D Hilbert b e_=%lnn�nbE�bf 3.1 ~=K�OSSmarandache�E+&},^*~=`h&*1��lS6*S&6���(�<��*S�~=w- Smarandache �EK�+K�OSSmarandacheK��d[�Tkiy�d[}�N|a Smarandache �+uKx�a%�{ 3.1 E A, B, C *�y�[:1T et%7&D�)t*�y�[:x�
A, B, C =a"lT7%)t�l?L# lT Smarandache �+�*7�y�+b e_uiF&S=}nbEu Smarandache �D%�
88 L�y: ihUAbx� – `N�; Smarandache +ai(i) �y4HbEma*-s~���B+~;&�^~�6*�^���K3(���F���E)t L q�7 C t�%/)t AB�V�q� +lT a AB :%7a"(ln)t�%/ L&lq�)t AB :% +l7� �a�%/ L %)t&*� 3.2(a) Fm�(ii) bE-spÆ�=**;�^~���ma*-spÆ�=**;�^~���6*�^����V�q�}T7 D, E&�e D, E 7 A, B, C =%l7&*7 C et&*� 3.2(b) Fm&a"(ln)tq� D, E�l` �}Ta)t
AB 7 F, G n 7 A, B, C =lT7et%}T7 G, H � �aq�HL%)t&*� 3.2(b) Fm�
� 3.2. s- )t%~G3.2 b�kY�-�I?=lT4P %D "Q=Y[6}ST[&6}*�3^T[A℄
2p =J&Q�=J h4'~=3["bD$��,6}*�3^T[A℄ q=J&Q=J4'F�kl�+��:,�j��>}SYE*Y[&m;E�Sp,�}S*YE*Y[&m;E�S q��e.D�%�'ulTq)/�[%�~Sf�[ZJl ?a `r7u�9u�~%���)|:$!�[u.D�%&l9iA$Ælu .D�%&*� 3.3 Fm&S= (a) *)�%0[&(b) *v-?%9iA$Æ�
� 3.3. 9iA$Æ%#G
�=u��r-o/��[ ` –Smarandche +ai�= 892�u�[%lANÆ&�Sf�ANÆ8�[)�?&# %HT[A�z(/H� D = {(x, y)|x2 + y2 ≤ 1}�Tutte / 1973 W`�2�%��D�&��[12] =%�9&2�D�/d�bf 3.2~=54 M = (Xα,β,P)&E�S^:qA� X +rGzD Kx, x ∈ X+nKOG+%A Xα,β +~=:�% P&KGK [+K� 1 �K� 2&��K = {1.α, β, αβ} S Klein 4- Ge&�^ P S:�%&G*�^��b k, :)Pkx = αx��v 1*αP = P−1α;�v 2*e ΨJ = 〈α, β,P〉 ^ Xα,β Y;�nD� 3.2&2�%B7*[Æ~D�*4W P * Pαβ ��/ Xα,β :# %e��!,r* Klein 4- C� K ��/ Xα,β :# %�!�j�Euler-Poincarebl&?L#
|V (M)| − |E(M)| + |F (M)| = χ(M),�e V (M), E(M), F (M) Æ~|m2� M %B7��r�*[�&χ(M) |m2� M % Euler HQ&S�+)/2� M Fm,%vT�[% Euler HQ�DlT2� M = (Xα,β,P) u*YE*+&.4W� ΨI = 〈αβ,P〉 a Xα,β :u.f%&�lD*YE*+�� 3.4. � D0.4.0 a Kelin �[:%m,�*lTki&� 3.4 =W`�� D0.4.0 a Kelin �[:%lTm,&.y��2� M = (Xα,β,P) |m*d&�e
Xα,β =⋃
e∈{x,y,z,w}
{e, αe, βe, αβe},
P = (x, y, z, w)(αβx, αβy, βz, βw)
90 L�y: ihUAbx� – `N�; Smarandache +ai× (αx, αw, αz, αy)(βx, αβw, αβz, βy).� 3.4 =%2�( 2 B7 v1 = {(x, y, z, w), (αx, αw, αz, αy)}, v2 = {(αβx, αβy, βz,
βw), (βx, αβw, αβz, βy)}, 4 nr e1 = {x, αx, βx, αβx}, e2 = {y, αy, βy, αβy}, e3 =
{z, αz, βz, αβz}, e4 = {w, αw, βw, αβw}y� 2T[ f2 = {(x, αβy, z, βy, αx, αβw),
(βx, αw, αβx, y, βz, αy)}, f2 = {(βw, αz), (w, αβz)}&S Euler HQ*χ(M) = 2 − 4 + 2 = 0t4W� ΨI = 〈αβ,P〉 a Xα,β :.f��^�{��CV# � D0.4.0 a Klein[:%m,�K=f -M N}%0b&?LU.ylF(2&$�a6"y�fB�[:%m,��aB�[:%m,D�*d�bf 3.3 �4 G +C;A�E-�� V (G) =
k⋃j=1
Vi, ��_pÆ�b 1 ≤
i, j ≤ k&Vi
⋂Vj = ∅&1 S1, S2, · · · , Sk SV�`h E &+ k =Y[&k ≥ 1�}�^~= 1-1 �Y#� π : G → E :)_pÆ�b i, 1 ≤ i ≤ k&π|〈Vi〉 D~='|K
Si \ π(〈Vi〉) &+Q=�(�*�3M� D = {(x, y)|x2 + y2 ≤ 1}&fO π(G) D G^Y[ S1, S2, · · · , Sk +*B|�D� 3.3 =�[ S1, S2, · · · , Sk %6"44`Bm,(�|���alA��Si1 , Si2 , · · · , Sik&i#` ��� j, 1 ≤ j ≤ k&Sij u Sij+1
%i6" &D* G aS1, S2, · · · , Sk :%ww*B|�y/�[&(d[%S-�bv 3.1 ~=4 G ^T[ P1 ⊃ P2 ⊃ · · · ⊃ Ps �^��z+ww*B|�J��4 G �^g�� G =
s⊎i=1
Gi&:)_p�b i, 1 < i < s,
(i) Gi D�[+,(ii) _pÆ ∀v ∈ V (Gi), NG(x) ⊆ (
i+1⋃j=i−1
V (Gj)).
3.3 Y�?&54K�ua2�sd:h7% Smarandache �+&z dup_w-�?7q8�?%ÆÆ�2��+%:�ga9n [13]=a`&=?a9n [14]− [16]=&℄~u [16] f%�a3%N}&SD�*d�bf 3.4 ^54M Q=C; u, u ∈ V (M)�7~=4b µ(u), µ(u)ρM(u)(mod2π)&
�=u��r-o/��[ ` –Smarandche +ai�= 91O (M, µ) S~=54K�&µ(u) S; u +��Dxb�NWU6*WUY[+Y�ush~=6hK=[iO(54K�n��6-���� 3.5 =W`�)tk�2�:%B7%~#�e%� CVÆ*�/.�)/.��/.&u�2&7 u D*�H7��y7*��7�� 3.5. )tk��H7n��7�H7��y7*��7a 3 +6"=�u.yfm%&�e7%fm(~/�y6"%~#&� lDu�)%&b87�u�y7�� 3.6 =W`��1A7a 3 +6"%fm�v, �=7 u *�H7&v *�y7l w *��7�
� 3.6. �H7��y7*��7a 3- +6"%fmbv 3.1 -��n�54K�&L�^a<K���K��{� K��{K��D %"a09n [16]�*t/ T&?Ld[Y?�[2��+%~#�a�A~#& .yaB7:-3Ci��&U.yb�qIB7&"%rulTq9��&�^`fl� T�[:���t_Æ(���i*a�[2��+=(�^%S-*�[54K�nS��*us546us+;S�H;��*lTki&� 3.7 =M`�s/�,[e%lA�[2��+&S=B7
92 L�y: ihUAbx� – `N�; Smarandache +air:%�+|a8B7 2 b%Ci��+�� 3.7. lT�[2��+%ki� 3.8 =M`�� 3.7 D�%�[2��+=)t%~#&℄-2&� 3.9 =M`�8�[2��+=%�Afr#�� 3.8. �[2��+%)t
� 3.9. �[2��+%fr#§4. �f�nJ?&Einstein%��u`-℄Q�6"a o��du��%&N2�td k�&�l7af�|#=uq# "f�2��+%'{f�:.ylF2D�/lTV~6":&�a8V~6"%HT7:-3lT�~l7l.V~6"�+�
�=u��r-o/��[ ` –Smarandche +ai�= 93bf 4.1 P U SVCALS ρ ?ALIF&W ⊆ U�BMW ∀u ∈ U&N=ZVCKUXO ω : u → ω(u)&\J&BMW℄R n, n ≥ 1&ω(u) ∈ Rn Q>BMW?^R ǫ > 0&H=ZVCR δ > 0 DVC� v ∈ W , ρ(u − v) < δ Q>ρ(ω(u)− ω(v)) < ǫ�[N U = W&< U SVCTALIF&ES (U, ω),N=Z^R N > 0 Q> ∀w ∈ W , ρ(w) ≤ N&[< U SVCYGTALIF&ES (U−, ω)�V� ω uCi�� &{.V~6"?L# Einstein %��6"�*t/ T&?L[-.�[�+t ω *Ci��%~#&�g(d[}T(�%S-�bv 4.1 sW�[ (P, ω) +�; u � v *~E�^�HÆ�+���bv 4.2 ^~=W�[ (
∑, ω) &}*�^�H;&f (
∑, ω) ,Q=;LS>M;6Q=;LSdY;�`/�[���t&l(*dS �bv 4.3 ^W�[ (
∑, ω) �^�bY� F (x, y) = 0 -sX� D &+;
(x0, y0) �K�� F (x0, y0) = 0 K_pÆ ∀(x, y) ∈ D&(π − ω(x, y)
2)(1 + (
dy
dx)2) = sign(x, y).ma&?L_a .V~6":�nD� 4.1&`lT m- �# Mm * �7 ∀u ∈ Mm& U = W = Mm&n = 1 t ω(u) *lT�L���l?L# �#
Mm :%.�#�+ (Mm, ω)�?L$!&�# Mm :%Minkowski}bD�*<s*dn2%lT�� F :
Mm → [0, +∞)�(i) F a Mm \ {0} :hh�L,(ii) F u 1- Vz%&�` �% u ∈ Mm * λ > 0&( F (λu) = λF (u),(iii) ` �% ∀y ∈ Mm \ {0}&<sn2
gy(u, v) =1
2
∂2F 2(y + su + tv)
∂s∂t|t=s=0
.%`D�t(" gy : Mm × Mm → R u�D%�Finsler(If�:�u-3� Minkowski |�%�#&�eQ<�u�# Mm�Ss6":%lT�� F : TMm → [0, +∞) �<s*dn2�
94 L�y: ihUAbx� – `N�; Smarandache +ai(i) F a TMm \ {0} =
⋃{TxMm \ {0} : x ∈ Mm} :hh�L,
(ii) ` � ∀x ∈ Mm&F |TxMm → [0, +∞) ulT Minkowski |���*.V~6"�+%lT℄k& � x ∈ Mm&?L=k ω(x) = F (x)�l.V~6"�+ (Mm, ω) ulT Finsler �#&℄~2&* ω(x) = gx(y, y) =
F 2(x, y)&l (Mm, ω) �u Riemann �#��^&?L�# d�S-�bv 4.4 WV�`hK� (Mm, ω)&~s5&Smarandache K�&{w FinslerK�&�i{w Riemann K��§5. O℄\_-YbU!�p_lp�% -M *�?N}a`��~0bN}%; &�e?L ���T�v�(v!� 5.1 -d�=<--S2Onzq�*�,�<-`h-�D�:��CY`h-`-��.y(E�T��&���Y4(fT8H&��u9n [10] =�%8H%|7&duM ?VKxI�%|7�Einstein ℄Q�6"a o��du��%&{lA��:<�y6"a�fpV=u �a%�{fY|#%CV&�℄ �|#n#� k�V=lAul uSgI&E-uG+6"Uu++6"�A 4 +6"&9n [16] =`yu(�^�0M�q8$Æ�+=j�s�~|0M��%�vnR/l�℄D%p2�l�lF(%N}��6"�?1`.V~6" (Mm, ω) f%N}�s/�)6"%N}.yrm&2>a�?:Y4�%8H%�a&��℄rj%|#�vEv|# �v�(v!� 5.2 nz"3+<-Ub%4Dd�-D�-�-p_p�i -M %ry�a��℄9u�eQ#G%6"|&{l�|f�?%uP�l�P % -M ?�Z)Q f%"�fT��*�'nz`h+F*V%4Dd�Æ�a�_ -7fY%n2d&bI|�T; (lD%K}&*�℄ �|# %FRNf��j -=!*6"+�u 10&M- -=%6"+�u 11 tHAu$%jn9j -�uS|r~#�l>�M ?��aN}% F- -%6"+�u 12�K=f�A'{&.y7llF%6"+� -N} Einstein 1�J&a�l7:&�?�ra�M ?�%?[�
�=u��r-o/��[ ` –Smarandche +ai�= 95v�(v!� 5.3 nzyV&6"|�JE-5TD��^~("tY� 4 U"| 3 U6� 3 U"| 2 U`h-2Mf�:u Einstein 1�Ja zV�n2d%T7T�l�[&M ?�!*2M�a�% o&�q�td �k�& +Me '1,2M=�8d%�G*>:"[6]-[7]$,z M ?�,�#2MuqI z8H� z 6%q{&*�℄�T%lsM �%a2M��\�72Mu`%& -M =U(lA>M&S℄�u +Me�Evf,>M��* $�2M�>MTe&?L�drm}��uk�7%&T %z �u�W&Fy2M7>M�8ular��^�℄&℄~u8�GE0�� ��>f�2M�fOM9a2�:uKx�a%&*��%��)��0MfOM9% -& -uM nu�?�0 X�L%sk!g�fyQ&o?e"l)!PMeZJ=O u�lTsgEl�{ -:!g 6kB&o2I| ZJ=OuOÆQ%; &� :De%ZJ; �"yÆQ%�?; u ([16])*(1)�D�Q�_& En74D E+&n7D* E+%"K�z�(2){4B|%dU`hw&g9,�`h��j7�(3)x4+�`hYI�_�`M ?Q"&0ba2�:D�k9uSO%SM&flrmS 6kB�%�v�(v!� 5.4 nz^5TYy%dt#E-6M976�~l)uM ?V%lT�KQ �K=f�℄`6"+�!g%9u&�T; du#%�.\�* 6"+� ≥ 11&vD6M9� lDha�℄ # %��+:&d .�a2�: H�F(��PnR/�℄!g��7fY��%aG�2h�7
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ematics Magazine, Aurora, Canada, Vol.12(2003)
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�H"�5R��y:�– M�_C,_��b_t�+ $, 97-121
6(U;8k6PtU)wVp=—— ��y"�yA*1?��=fÆd�3��u
(%nU^Qa^9�+U^f8Q%�)%100190)�℄: a*.vk*%℄2Æ;U^f8*�+m"%F?$3YRAsg �*��% 9H��a^s~f8�2$*Y&�a%���J<0`*m"dF%$)~�{�Smarandache)_g�Mb�+%�Bz?$3YR+�+L��P*m"%Æ;�f+3℄_g*A{�_g: - ��|3B+}r_g*L��P%��CPoincare-"*L���%Fo�}< 3- T��''H)�2}S 3- T_/,[*?$^�p%#���℄?$3%FL�'H�wM�~;�B+^Einstein��G*L�/%a%R����y�o℄m9Ef8�plM�%�[a^\�*)vUU%!h!Jf8*U^�u��NJ: L�m"�?$3�Smarandache )_g���G%U^�u�Abstract: By showing several thought models, this paper introduces the
Smarandache multi-space and Smarandache system underlying a topological
graph, i.e., d-dimensional graph step by step., and surveys applications of
topological graphs to modern mathematics and physics, particularly, the idea
of M-theory for the gravitational field. The last part of this paper discusses
how to select a topic for research on the coordinated development of man with
nature.
Key Words: Combinatorial principle, topological graph, Smarandache
multi-space, gravitational field, research notion.
AMS(2010): 01A25, 01A27
1p. 2012 � 8 S 30 $) 10 S 9 $ }`�MoZ{>�>+>>N)_l6S\I>Nb>NYK2e-print: http://www.nstl.gov.cn
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� 1-2 Sb9U��T; f�:7j[%�y"yAd�*1?g��=fÆ; z`ll&*�77H�vi;i%'Y�Vsuy�f�:&E-uk�VUu�℄Cd&�k���D(,?a�%; |>&�~%&nu*�℄!gk�*�℄Cd0b&0�Cdry0bT�v� �D(; &�ulA,?_|ry%'{&up_lp�,?ry%Ob��&lw-'{*�v&l*�A,?%_|ryaa�(%�?sd�m�jTLe�<:7�rdDu,?-,?uy�*(�&x��`k�*�℄Cd%�$u\&!gk�*�℄Cd�%�0}�V&S`�a/ (!gk�&*�℄v�k�&7k��AryaaÆQ ℄*� % -X�
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n⋃
i=1
Si�(1) uF* � (G1; ◦), (G2; •) D�=**+e&f,MS+*`hOSde&TS (G1
⋃G2; {◦, •}).WQ&&lT��ualT�- G :D��}A��Z; {◦, •}&E G◦ ⊂
G, G• ⊂ G * G =`Z; ◦ n • �n%i�&l (G◦; ◦), (G•; •) u}T�&tG = G◦
⋃G•�℄~2&* (G; ◦) u?W�&S�4C�* 0&(G \ {0}; •) *�&t<sÆ-%*_pÆ x, y, z ∈ G&- x• (y ◦z) = (x•y)◦ (x•z)� (y ◦z)•x = (y •x)◦ (z•x)&f~=de>D~=�b��C"~+&zq (G \ {0}; •) xD~=�%e&KGK^��7&~=de>D~=�b��"y.y `, ��uq8����e�;%��&l��=%�~, a/u�D��Æ-%&y�`Z;�n%�- G◦, G•�
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e1
e2
e3
e4
G1
v1
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f1 f2
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G2� 3-2 � ��lT4N# P&,%ulT��-P = {G1, G2, · · · , Gn, · · ·}t<sa�zh φ uWd u&�` � G ∈ P ( Gφ ∈ P�"y.y[-�%qx(�m,(�xi(&*�V ���N�)&y��D�%l�sg��&*qxV�Rl��95�&℄~u�%kzh��S��)���[%sd$g&.y�&���`I%l�3g&k*N-*j--P% [1]&y� Gary Chartrand* Linda Lesniak -P% [3] )�"m$�3��I�ulTZ��a�I6"%m,�E G ulT�&S ulT�I6"&llT�I�, ,%ulTq9% 1-1 �A τ : G → S&i#r7r&"yfaB7?&"[5]$��6"+��/)/ 3 &:��%m,.yfm*)t[m,&�b�F(%rm, G+6"?�*)t[&��ud[�TD *bv 3.1([10]) pÆ~= n �n�4Y��B|%�H`h Rn, n ≥ 3�_ f�: 0"a n = 3 %~#� WD%� G&a6"�t (t, t2, t3) :=k n T7 (t1, t
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× (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv).SB7�*u1 = {(x, y, z), (αx, αz, αy)}, u2 = {(αβx, u, w), (βx, αw, αu)},u3 = {(αβz, αβu, v), (βz, αv, βu)}, u4 = {(αβy, αβv, αβw), (βy, βw, βv)},r�* {e, αe, βe, αβe}, where, e ∈ {x, y, z, u, v, w}�F5��54&,%�urgzDlT,CRnC r ∈ Kx %2��j�b 1 * 2&.y4(|"a��54+F*QeD�ze 1M�y/�a�[:%m,n2�&Ob(}℄; *!� 1: >E~=4 G �YE*6}*YE*Y[ S, G D�YB| S?�℄; u"d[�TD a�*bv 3.3([5]) _pÆ~=�(4 G&,YB|+YE*Y[:*YE*Y[m;LS�bXh&G} GRO(G), GRN(G) ��S G YB|+E*Y[�*YE*Y[m;&f
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⋂Sj 6= ∅, 1 ≤ i, j ≤ n},���- Si, 1 ≤ i ≤ n z�D7&Si
⋂Sj z�r (Si, Sj) ∈ E(G[S])�k*&`/��&� a = { HT }&b = { hi }&c1, c2 = { ng }&d = { �� }&e = { i
}&f = { Ui }&g1, g2, g3, g4 = { }&h = { /; }&lS�ISh0� 3-8.
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√(x1 − y1)2 + (x2 − y2)2 + (x3 − y3)2.�l&Einstein %u`-|a "76"V .Æ&/u(u` 6|&� R4�
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µαν = gαβRαµβν , R = gµνRµν Æ~*Ricci r�, Ricci Y8��,
κ = 8πG/c4 = 2.08× 10−48cm−1 · g−1 · s2�a�6aOd&Einstein o1�J%�`DT Schwarzschild V�&�ds2 =
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118 L�y: ihUAbx� – `N�; Smarandache +ai&�Jw4GCao�,?u�℄ (!gk�%.M&rm�!gk��%&fl�f�℄ryv�k�&7k��Ary�G�p_lp�,?ry%Ob��&z d`��,N℄��a`��%m|�
� 5-1 y-l)1vD&�(z�\�&z� \\e+b_℄�_n:Fg9Aqm+Z'-�(�2Ot+?��Bb__��g9v-"1$Q*�z&b9�v {S2O_b_? {_b_�-C\E? �?f�:uÆ?&�/q�sd:%:$7T�vA�X!/?"�?G*?�4<%{vu mf%�q|�℄ih&v�P Æ?�%Sj�AP4|=&*!�%Xi�)�%8i)��!"q�=�q�$nlH&q�$nl*&qY$n e&_\ �$n?qj&t) 8$nK��D&n&S$*Sl&\℄ Z�Æ�uXi�S��&y?R`�ry�� h&Æ,8i�o.-qi-�3_&�.i&r.i*7&q.i��b&).ili&*.i�?�B&*6AÆ&D&��&8�)�;��)&lIV=e� #1�Fy&{bG*��%�?℄��&o2(lA��'a�Æ&* �,?� ��o%�o7���"2$" 6F&5'jv {S2Owg9�=d�&�-2O'- �T; uHlT�?℄��o2[`%; ��??,Æ#Qa&!grM�QJ0&�z &`rM%!gd�Q�/:[�vD&*+{!g:9u�AaG&fl9u�Hn ,ÆvA��%t7�Æ,ÆgQ[r�E ~y�( :�Æ%!gS �-w-usd�b$!&,?Æ, ur%&l#G`rM%�f T�u,?%n/�{�lCV`r&,N3 .yÆ*H℄*Z 1 u: _h~b_��&h~=d�-a�,Z 2 u: _h~b_��qm-a�,
N�`D-`�� `u�-!% 119Z 3 u: _b_��qm-a�,Z 4 u: _V_qm-a�,Z 5 u: _nzr7Fg-a��Fy&�� ℄N}3 �/:�H℄=%ul℄&K5$_�+ �%3 &=k`aG�℄!gk���(�+%3 u0b%���b�HT�`S?�SI�N &nS z℄7f%�N�?S Mba�?ll�;d!f%?,sd&�*�*�Æ&z `7�?(y%uy?,&* -o?� -M � -O?)sd3f%���T&y`�?sd�Suy$g(lT�[% T*��&�^alg?{�?,?N} ��ayVSN}�;fy%z &yV z?,`SN}�;%�|&�yw-'{*sd&fm?,"%?&�|*�J&�rzÆ(Æ&fl*�℄!gk��v�k�l~`dn�"3$' ^^&X[(T -=�<>D��E&_L�<>~E� rt��E-�℄!g%�r()I�3,NG %�r(&4}rmlTN}G d�&,�lJd&*�℄%N}G PualD%n2dm#%&�n2�Cn��SGn2� &q8S .�� Glnu(n2%Gl�Fy& b!*q8S ��LGl&�Cn2&nu&$q8S a zn2d%℄i&fl# S��ulAKx%N}���z &d bQ�Æ#&��u�&Æ#%!gd�a�r(&bA/!P� &A/m|Æ#S-l Q��uLd%;?OV�"4$:�4"&?�d* f)%+Sq%O*yLTE&%OSE-,?N}E(�|rz&_h%G &" Dah�dG*�L%Kx!g&vA*l m#%G l�kkF��*n"ZÆu,?N}%�.�*y&b(9Qk?&b �Dk?&"�m#Gz&*v �u�"�~�Æ&z &1|C�G `,?ry��℄!gk�%Bb(& SP99f&*.( ℄':�LL&�k�%N}�J&�(.�*�℄!gk��`dn�"5$+T�(&�(4Z f.S�n+S>XsE&fyV:,�n�(=ng9S>E-m�,?N}&u"n'T�6o\O*�e6 %Sk&l�B�3 f�:uqG*��N}3 &4f?�<ha z2;��yfzl; nuh; %N}�Fy&'lT�%�oT���,?=%B�; �A _(.�(�*y&b�HTN}�Gb(lAS(%�O&OJ',���,N�e&f�Tz%N}G &z &�T z�;%y�G &℄~uN}'{*�v&�{=m#℄Ælryk%N}&fl#Gp�G l u�G�?���℄Cdf, �p_lp�&,?��# �5ry&�℄Sj��%�
120 L�y: ihUAbx� – `N�; Smarandache +aiaG&�`k�℄�%�od�#Vop&�7yz &TvB���:0�� ._ShJ<Q); %�LB&�7�℄��.<`k��%% $�&{r�f(�/k��%%jJ*�V�#(y�*y&HT,?℄��P[lT0J'%; &��u&V_+'?l+D2O-V_D�YO�nz'(&O�Fgj7-�7uak%�,?��%N/�℄!gk�&,?N}o2Sf�7k��Ary��f%&�d�u�?℄��0bI| %; &l�l�J=&w-?Esd*7l�r�Mp_&�l�Iy_aasg'{*�v�2h�7[1] J.A.Bondy and U.S.R.Murty, Graph Theory with Applications, The Macmillan
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�H"�5R��y:�– ����)9&�,Qxw�/�KHYK<�, 122-128X.�%��'2��R�%Æ%o9�M8%Ax(&ow�/�9.&�* 100045)�℄: v�.�JGXJ;�%�_�v�.�2HC�}�n\����-JN����Am�u^�%C��1�,O4�%}�'k"��a��%}^v�.�JG�fPk%'r^Jyplk��Jy�Uw��JydM;�9J��JyL��Jy�y;P�mK9JyE6�v�.�~;���Jy�}��Jya�w�/ 8 <�Z% i|v�.�JG!V!J,��}�
Abstract: The operation system of bidding market is important for protect-
ing the State’s interests, the social and public interests, the legitimate rights
and interests of parties to tender and bid activities, improving the economic
effects and ensuring the quality of projects. Combining its normalizing devel-
opment with those of market researches in recent years, this paper explores
how to make such a system effectively operating through by aspects such as
those of the goal of bidding business, construction of system rules, supervision
by the government, tenderer and bidder organization, institutions, personnel,
self-regulation system, bidding theory and culture. All of these discussions are
beneficial for the development of tendering and bidding business in China.{1u{1q�%^e&E({1�E({1q���%_P[H=�dXÆ4D�{1ahJ-4=%sd(��&lx�7lv%*��5V%{1Z%t5&luV"{1b�?|&+P{1q�(Z%%g�n2��z�z5V&�*?�CdO�q�7E=�OhJ-4�aGq���V"~r9~%lA?|5V&a�fq�e59P�)>*�|�z�z{1e_&A�*K;*A?|%*)�[#���Mr%Gz& ?�q�7Er\�Bb��&gu#G�"�z���z� th��z�*Oe&%!�K Q%{1e_�=f�z�z5V��|'%J,&d[�`l�e5=%; &*�z�z1D8q{.� <� – 3^�y~y�%2011 �0 2 L.2e-print: http://www.qstheory.com.
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Abstract: The bidding theory is such an economically purchasing theory
established on known theories such as those of consumer behaviors, operation
research, game theory, multi-statistics and reliability analysis ,...,etc. to in-
struction of bidding practice. For letting more peoples understand the essence
of bidding purchasing in the market system and know this theory, this paper
systematically introduces it from micro-economy, which will enables the reader
comprehends the function of bidding in market economy of China.�z�ju{1q�5Vdf%{1hJ�O-4%lAz�?|�l&Sf9&ux��z�z�Jfm�jz%��*{1q�d%lA$|q�jJ&�z�j0b{�$|q�?�%&{�q�?%��=kP�&�aysd:&f%�O=k&flfmaGq��&V"~r9~%rz��*lAz�?|&�z�jU�3f�fCd��f�%j *id��z�j%�Aq��(&�D��z�j -uy$|q�?*sd%lA(r=k -&ulAs/lz(��sd:%�O -&yd{�z�j -%sg N��erz*=z.�ÆQ1T�[Æ~f%/��1D8q{.� (�i�VY�%2012 � 2 S 17 $2e-print: www.ctba.org.cn
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�H"�5R��y:�–�w�/�v4*�����|M>��y, 139-147
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Abstract: The semantic meaning of an article in a law or a regulation is
determined by the meaning of its word groups, order and in which situation
it is. It is well-known that to grasp the semantic meaning of an article in ten-
dering and bidding laws and regulations is foundation of its normally carrying
out and self-regulation of partners. In this paper, we distinguish the seman-
tic meaning of some special words groups in Article 4th, Article 8th, Article
22nd, Article 32nd, Article 34th, Article 52nd, Article 54th and Article 82nd
in the Enforcement Regulation of China Tendering and Bidding Law, which
are applicable for normally purchasing in China.?L$!&9QualD9vn2d��%�lb�� TlT98n9Æ%��&l�[0b�Tx9w-& TS|$��&�l�[l0b T98�9Æ%ka��&�J$��&*9Æ%�� "9Æ|$��*Ska�Wez�D&�9Æ:d9*at�f/=& z^lTv%n9& z%�*dDd( z% T-SXJ�a/ z%�&SCdf/� T�oy�FhTv z&`v%El�v%�l=%l�y1(xw%9�5(j/SkIj���#�gÆ% T&{lhG�( y|&E :�Æ�f�:& +lTv%n9� SFa1��y�i}b�%2013 �3 2 L2e-print: www.ctba.org.cn
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4. On algebraic multi-group spaces, Scientia Magna, Vol.2,No.1(2006), 64-70.
5. On multi-metric spaces, Scientia Magna, Vol.2,No.1(2006), 87-94.
6. On algebraic multi-ring spaces, Scientia Magna, Vol.2,No.2(2006), 48-54.
7. On algebraic multi-vector spaces, Scientia Magna, Vol.2,No.2(2006), 1-6.
8. -M r%p_lp��?� Smarandache B6" -&&oVN<b^�*200607-91�9. �zC�e_%�?e"��TÆQ, &oVN<b^�*200607-112�10. Combinatorial speculation and the combinatorial conjecture for mathematics,
arXiv: math.GM/0606702 and Sciencepaper Online:200607-128.
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math.GM/0605495.
12. Smarandache B6"�uy�?w- -, 0|��$�:s�Smarandache d�g9�, High American Press&2006.
2007 '1. Geometrical theory on combinatorial manifolds, JP J. Geometry and Topology,
Vol.7, 1(2007), 65-113.
2. An introduction to Smarandache multi-spaces and mathematical combinatorics,
Scientia Magna, Vol.3, 1(2007), No.1, 54-80.
3. Combinatorial speculation and combinatorial conjecture for mathematics, 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-
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9. DVx�(lw�4T�<:4s, American Research Press, 2007.
2008 '1. Curvature Equations on Combinatorial Manifolds with Applications to Theoret-
ical Physics&International J. Mathem.Combin., Vol.1, 2008,16-35.
2. Combinatorially Riemannian Submanifolds&International J. Math.Combin., Vol.2,
2008, 23-45.
3. Extending Homomorphism Theorem to Multi-Systems&International J. Math.Combin.,
Vol.3, 2008,1-27.
4. Actions of Multi-groups on Finite Sets&International J. Mathe.Combin., Vol.3,
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'2+L�y 1985-2012 �=2m2 153
2009 '1. Topological Multi-groups and Multi-fields&International J. Math.Combin., Vol.1,
2009, 8-17.
2. Euclidean Pseudo-Geometry on Rn&International J. Math.Combin., Vol.1, 2009,
90-95.
3. �J�z�z{1 ℄r��|&&ox��&2009 1 T 24 %�4. ���z�j�G(g��&�'�E�&, -�w�4Tf��y�"+Os$&=��N`IC&2009�5. Combinatorial Fields - An Introduction, International J. Math.Combin., Vol.3,
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6. Combinatorial Geometry with Application to Field Theory, InfoQuest Press, 2009.
2010 '1. Relativity in Combinatorial Gravitational Fields, Progress in Physics, Vol.3,2010,
39-50.
2.�2010 ~w�'�zg�QO���#�–��z�j7kÆQ�"Os$&=��N`IC&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.
5. Let’s Flying by Wings-Mathematical Combinatorics & Smarandache Geome-
tries(in Chinese), Chinese Branch Xiquan House, 2010.
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edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011.
3. Combinatorial Geometry with Applications to Field Theory(Second edition),
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4. Smarandache Multi-Space Theory(Second edition), Graduate Textbook in Math-
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6. JOe59P&_|h7�z�z{1Z%t5&UD�<H&2011 2 T 28%&&ow�/�&3"2011$�7. mx�z%q�%*ÆQ�{1` &&ow�/�&5"2011$�8. |l{1?|�l&��z�zrg4"ry&&ow�/�&12"2011$�2012 '1. ,?h7�z�j -e_&&ow�/�&1-2"2012$.
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6)Non-solvable spaces of linear equation systems, International J. Math.Combin.,
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[~o� 1558�LZ��s*|&℄?�o��?�o?&E��G�℄J[&I��?dC-G&I�����?w-\1�Os�2006 ,=I��Who’s Who�&E>��=�,�-9at��.?�&m*��r9,O~�d=��z�z�d+T 4�=�,?O~ �� 7�W_|B7fY|N}�G& m7T℄J?O$(E��N}S�[�1962 12 T 31 %`S/,jV"\{&1978-1980 ,jV�J=?G 80� 1 C?\,1981 -1998 a=�7T4p℄J�℄�&g?� ���G���^4�,4�~rp℄J[)(,1998 10 T -2000 6 T =�v?ds7Mb|p℄J[,2000 7 T -2007 12 T���z(rj b)&i ~rq �Z�Mb|O *+p℄J[)(,S" 1999 -2002 `�?x�?_T�o?4&2003 -2005 =�,?O~ �� 7�W_|B7N}|{r�o?N}℄�,2008 1 T2_&=��z�z�d+T 4�f%N}℄�Ob�=a�?� -M *℄J7E)�;&g?a���l�P ?��M:r| Smarandache �+�$Æ�+7 -M �w-2���-*℄J7E~ -9 60f8&aI�`I�1g�??�ZP�}g℄J�z�j -ZP*lg-9�&a��`Ilg�?-9��{ 2007 10T�k&Os���?w-| �MATHEMATICAL COMBINATORICS �(INTERNATIONAL
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Abstract: This book is for young students, words of one mathe-
matician, also being a physicist and an engineer to young students.
By recalling each of his growth and success steps, i.e., beginning
as a construction worker, obtained a certification of undergradu-
ate learn by himself and a doctor’s degree in university, after then
continuously overlooking these obtained achievements, raising new
scientific topics in mathematics and physics by Smarandache’s no-
tion and combinatorial principle for his research, tell us the truth that�all roads
lead to RomeÆ, which maybe inspires younger researchers and students. Some
papers on scientific notions and bidding theory can be also found in this book,
which are beneficial for the development of bidding business in China.)��℄: �tkf�Vx�1)$�%tk3~�>�Lb)}b+>6\IeJ>+)kH$>�wÆjHph%_x�1)!$P���wÆk^�8~, `��R!kS6S\�%wÆj>�Ff+>f%�xy(>h$H�%+F(>�^S�n>3%x�`>P+>M|<�\�Cj>%�\��v,) Smarandache &zVj`_�$l{2be$�>�LbM|$ÆI%;�����&.'?C $F�%�%_Yf�$x�>RF��'xf)W-����<yb�����$�7�y�ib,d^6D8~%_�I<��y~yqf3!qx�D�g�
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ISBN 9781599732145
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