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—– Smarandache Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO (New Expanded Edition) Chinese Branch Xiquan House 2013

Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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This book is for young students, words of one mathematician, also being a physicist and an engineer to young students. By recalling each of his growth and success steps, 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.

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Page 1: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 2: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

��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

Page 3: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 4: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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Page 5: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)
Page 6: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

Q Z�ulg�Wy2*% &ul4��?�M *~ ,?7℄JfK?,*lI%?�x�kIqi& y2*"%Q�y Mu�SyI"% M&u`0QX<fX!7X{&l1G�% M�3Q"#"&�:� K;3K Æ�_4%�&l�l�d�r�&*X{x�f/.yuG {*mf,_Y%�& :��BdEv� �7&*M�Xv�LT� �vk��S%� ;4�?g?al��(�"7Tb)℄�� 17 &a7E�4℄�� 3 &al��"�z� th℄�� 8 &f,%g�dduq 5 &��&�S�L&ha$hd&w�<*y&>�y�? %~tU &maTs*=? M?uC:%���&℄~u�?&�l1bG*lTp�$�%�?��G&%\ji?y�E��,�?%K,& # 7Tb)�℄�&�m %P{�E(�y_&℄�&1&#Xf�?Zg ��X�4a`m,�)℄��<�&�&?Q �`m&a`m℄g�?l4�?E�%,�d&�k�?Zgsd3y��-%?\7N}�?\S�a 7Tb)?&℄�&1!P?\�?&��k?&��G����?Zgg,L[%_|?\,=?yz)?iI�&,`�?x�?_T�o?4&m#�o?4?&,f,=�,?O�o?�J}{r�?N}℄�&......�*_&?uuI$%g~ �d+T 4�E�*℄J��?N}S�[�(/lI%g�Z�?�&l��=&�fm %P{:flG�nJ?jf%Jo�dDu\A-\A�ui#k�G`rf�l��dDuG`-G`�uar��\A?Wd%vln��Sj%Z�� a/G`&Zf2&a/ �G`F0`%&��o7�J�?�E(G`&*afmlTrz%z &?,d*k�D�%rz��t �_h&?Ud zV&� A�k�S%�!&yt�ZG%rz:f�2010 10 T&?a �{�? =?&�a&Sj� 10 %�J{&0 Æ~ 30 f&�%^=z?&dPu�S�L��� P& _;&H8*~�z?L�?Db��f+b_ /Æ%vg9�"A Collection of Selected Papers

on Smarandache Geometries & Combinatorial Maps, I� Chinese Branch Xiquan

Page 7: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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.�

Page 8: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 9: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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&

Page 10: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 11: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

% �7Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i?%�?&� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1?%�?&� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4?%w-. &� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20?\�?%7,ed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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

Page 12: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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

Page 13: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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��%

Page 14: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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�

Page 15: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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

12006 � 3 S 26 $)+iU�Iz<>��ZRYK%2e-print: www.K12.com.cn ) www.wyszx.cn

Page 16: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

h- `Æ0 5m3p OO HHT1_l%:I 10*00 �&=�,?O�?7_|,?N}OZLs�&?%�On Automorphism Groups of Maps, surfaces and Smarandache

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Automorphisms of Maps & SurfacesÆuq usw-n�-��%-9�&Hf�:uq�\�?%4f�-7&l��-7aa#Pf?%lA|7&�uaw-�?� Q&p�~Sb__VLY"KM��6"KM�*x&%"K:g��A|7a?%�o?ZL= ��T�&(�~%℄�0b�~��o?ZL%y?l��ua?%$g|'�&k�$Æ�+� =�+=4f��w-�v0b�fl�N}%�?; �2005 6 T?����2005 4<:M�b_e7��t��eoX_℄�(.Æ&����An introduction on Smarandache geometries on mapsÆ%ZL��8ZL7?a=�,?O�%�o?ZL�On Automorphism Groups of Maps,

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TheoryÆ"Smarandache *B`h�<$& l|����+ -�M 6|&��?%w-|f%�%N}7B7�?T�%8H|u�epV(4fT8H&(%uHÆa&(%luH?&�"/2��℄%�ehhE, 2��℄b{!g|�T8Hul24K}%r, *2��℄ %NfNf��?L�T8H%+�u 3&7SH8H6"(�Æ?&&?&%SH6"+�.�u 3&d.��/ 3��^&a��8H=%�ÆM9z?L�T8H%9~&�?L HL&*HLha?L # % 3

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h- `Æ0 19

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Page 31: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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

Page 32: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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speculations and the combinatorial conjecture for mathematics$ZL=a`%�?w-O�{&� +lK�?,?�.yf%w-Onf%w-B7�?T�Z℄"=�T�{ �lA,?N}%�M�n#Æ�*"1$_p�~Sb_�+V_&Y\e-��K��M�jf4'M��<"K*x��zs3K��n#&��K��wg�"2$4'M��vY_-=b_V_**��^~EM��Q:q"KM�:g&�ix>+b_V_���f=nO S�b_M�Æ"Combinatorial

Mathematics$�ZL��?&�4=y?�q7?f%�XÆ[-&z a�:?d7���4E�&SLm ���?pOO[ LovaszE�f%�?�&Z!D�?%��&�aT�uq`I%�54�Y[: SmarandacheK�+F*Qe�"Automorphism

Groups of Maps, Surfaces and Smarandache Geometries, I� American Research

Page 33: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

22 L�y: ihUAbx� – `N�; Smarandache +aiPress, 2005 $*�Smarandache *`h�<�"Smarandache multi-Space Theory,I� Hexis&Phoenix, AZ&2006 $sd:&/ 2009 aI�`I�M�N�K�C,^H<&+�'�(Combinatorial Geometry with Applications to Field Theory,I� InfoQuest, 2009) ZP&��w-�v�G�w- - �I - $Æ�+ - -M % -F�7E℄��_�"U5'T?uFg{rw-?=�-N}%?��1983 &?QmalF7T?��?&�T�%1�a/�?&/ua`m℄g�?ZW0E�,�d?\w-�?&�#T%uy�[|?O%Æ% Bollobas P��?�4<�"℄(Kw�$��y`l�M�:r|%S �ff%9f�1984 -1998 � 15 =&?l3a{rf�%Sh(9N}℄��<��l Md?\��I?�$Æ�#�$Æ#l)3J&�E(~�J,N}&la�-�[&lg?N}��%a�&�k=���E1C#��Hamiltonian �* Cayley �)&�a��M�:r|�-9��M"%l�ÆN}G a��M�:%r|&l)P9 ?_T�oM"&di?�k j�G�oL[0bb�r|-9%�~&f,�o-9�w�� F(N}ukM%&dug1%��M"%N}℄�dE(a�%��&T�f�:d | y���N}℄��G?`�?ry�,?ry(dD�& u=a?�M�:r|-9 �lACdN!&lA�vDÆ&l�A{v �:p�:_�ifmXQlu#QQQK}�� ��?�VKx!*�-N}N(|`G &/u�?Vl�Ob?�M�`�--9f%��|'�4��~�--9e ÆJ&N2E( + "& u�*�^Nq�ÆlÆQ�r����yhl�XgS��*���-��+d�Y :���-N}fJr���?T�k�!*&:p�:_�i��?�V`�--9%�4uzJq�&H)I�3��~?� M�_N}�-&*-9r|N}��<�?Ql�?�w'H}a����lz�ao4<:M�_℄�(.Æ&*S�-?�~`�lDdn&�sg:Pul�X?�ÆflO5i��&Sr%a7f!gl�2*&*-9r|*Ds^hRln`��f,p_lp�?&�AS�f��d%�$AS$��vD�-b bN}&b bY:��?�%N}�t-�7u4D%�2007&?qalTw-?-Rer��`3�C4<g9+nQÆlTri&�|�?`�T; % v&�e\�*d*

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

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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.

Page 47: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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~�

Page 48: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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

Page 49: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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.

Page 50: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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#�

Page 51: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

40 L�y: ihUAbx� – `N�; Smarandache +ai-

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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

Page 52: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

!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

Page 53: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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_

Page 54: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

!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.

Page 55: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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,"�

Page 56: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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.

Page 57: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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�

Page 58: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 59: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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�H"�5R��y:�– �<t��q+n.~?Zb_, 74-96

w�)wnpVn``iDtU�Smarandche+nJv� ∗Ax(%nU^Qa^9�+U^f8Q%�) 100190)�℄. �=?Zn.~�M;&�<t�_℄~���3xt�_+~(Æ,~�<&GTheory 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

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�=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

§1. y-��"J�DUsT!�1.1. (vanuO6"��4�C�G\�&�*4 = x, C = y&G = z&llTuO6".y�1T��f%\�&��z (x, y, z),JO6"��4�C�G� "\�&* � "u~* t&llTJO6".y���z (x, y, z, t) \��uO6"�SuO0� 1.1�8 " �lTuO�G&l�℄allT 0 %8H#Of�:u�T8H%lTM[ (section)�

� 1.1. �z_% "uO�℄%S.Sjf/|a&�℄Sj%8H6"7:[JO6"ul3%&��℄Sj%6"u 3 +%&* �: "u~&lu 4 +%, ��u Einstein % 6|�

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� 1.2. �=^'%ZJ� 1.3 =&ya\��8H"�℄s�k%VO70�&)2.S_h�℄|## %8H�J�

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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

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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&*

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? 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�

Page 92: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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#

Page 93: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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_�

Page 94: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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-

Page 95: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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#

Page 96: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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,

Page 97: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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:��

Page 98: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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%�

Page 99: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Page 100: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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)

Page 101: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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)&

Page 102: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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� 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

Page 103: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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� 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"�+�

Page 104: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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�

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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 ?�%?[�

Page 106: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�=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

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

Mathematical Society, Providence, Rhode Island, 1976.

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

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

[3] BVI*B+7, N�K���, `m�?`IC, 2001.

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96 L�y: ihUAbx� – `N�; Smarandache +ai[4] M.J.Duff, A layman’s guide to M-theory, arXiv: hep-th/9805177, v3, 2 July(1998).

[5] �W�, �_<:��K�, ,?`IC, `m, 2005.

[6] S.Hawking, 1hn8, F{,�`IC,2005.

[7] S.Hawking, qW�+<-, F{,�`IC,2005.

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

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

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

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

Warps and 10th Dimension, Oxford Univ. Press.

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

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

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

Boston/ London, 1999.

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

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

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

math.GM/0506232.

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

tries, American Research Press, 2005.

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

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10/2000.

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

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385-438.

Page 108: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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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Page 114: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

N�`D-`�� `u�-!% 103hE�Sd" (R1

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Page 131: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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

Press Ltd, 1976.

[2] M.Carmei, Classical Fields – General Relativity and Gauge Theory, World Sci-

entific, 2011.

[3] G.Chartrand and L.Lesniak, Graphs & Digraphs, Wadsworth, Inc., California,

1986.

[4] S.S.Chern and W.H.Chern, Lectures in Differential Geometry (in Chinese),

Peking University Press, 2001.

[5] J.L.Gross and T.W.Tucker, Topological Graph Theory, John Wiley & Sons,

1987.

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

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

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

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

[9] Linfan Mao, Automorphism Groups of Maps, Surfaces and Smarandache Ge-

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

Publisher Inc. 2011.

[10] Linfan Mao, Smarandache Multi-Space Theory (Second edition), Graduate Text-

book in Mathematics, The Education Publisher Inc. 2011.

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N�`D-`�� `u�-!% 121

ematics, International J.Math. Combin. Vol.1(2007), No.1, 1-19.

[13] Linfan Mao, Geometrical theory on combinatorial manifolds, JP J.Geometry

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

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

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

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

Page 133: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�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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Page 140: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�H"�5R��y:�– �-P_�Vlq&Qxw�4T�<��, 129-138

M_BUTgEp'�R�%1�w��2Ax(&ow�/�9.&�* 100045)�℄: v�3S~;C92.[d~;�Xh^�J;~;� F+R�x�XRM�x/U,~;HR*%&�C�"v�3S3r*~;�R9B m��v�3S~;��%q6JG,O�U�v�3S*,O3"%a�,O^�U, ^v�3S9p~;��!J��+kp��~%�9Q|x�}<f~M�~�v�3S^JG,O*G&�

Abstract: The bidding theory is such an economically purchasing theory

established on known theories such as those of consumer behaviors, operation

research, game theory, multi-statistics and reliability analysis ,...,etc. to 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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X = AXU + C

Y = AY V + D,�e&X - �J,z"*{1aWTv,z$wG%�~,Y - �S,z"*%N�2$wG%�~,U - �Jku~,V - �Sku~,AX - �J,z7�Jku~&"

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u(t)

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

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Page 150: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

�H"�5R��y:�–�w�/�v4*�����|M>��y, 139-147

��%Æ%rd^�|��(K6{g#,Ax(&ow�/�9.&�* 100045)�℄: ui�a*=�)+{L=��{LL����=2JD%h?bliv�.�u�a=�Ci|v�3SJR%�BJyE6*=��9p�a �v�.�u3)���6q�i�m-m��-m��-q�p-m�p-q�i-m�%*}6��{L%F^=�!J���*}6{L!J��x%�?bli+�a=�%!hi|�"v�3S3r�

Abstract: The semantic meaning of an article in a law or a regulation is

determined by the meaning of its word groups, order and in which situation

it is. It is well-known that to grasp the semantic meaning of an article in 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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��t 1985-2012 }(�/%�1985 '1. /y���NyuW� RMI El&&8b_g9&29-32,1"1985$2. ?\�?%7,ed&&8b_g9&22-23,2"1985$1990 '1. The maximum size of r-partite subgraphs of a K3-free graph, F�b_, 4"1990),417-424.

1992 '1. `m�C?O 100m3 �BT℄℄&�eDV&1"1992$2."75B-P$ mq�Fe� 50mzd&�QSh#P℄℄&x5VN&4"1992$1993 '1. "75B-P$`mq�Fe� 62m EvSA�oi���{℄℄&x5VN&1"1993$1994 '1. "7ZW0-P$R(G)=3 %k=��% ShN}&y�b_:�'b_&Vol.

10"o�$(1994),88-98�2. Hamiltonian graphs with constraints on vertices degree in a subgraphs pair, �H<;_R_�&Vol. 15"o�$(1994),79-90�1995 '1. "75B-P$`mq�Fe�&�Q#Pi��Sh℄℄&x5N℄&5"1995$

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6-10�2. i���eEÆQ��;&x5VN&2"1996$�1997 '1. n^s�G$7T℄℄4��G�7&x5ba&11"1997$1998 '1. A localization of Dirac’s theorem for hamiltonian graphs,b_g9: <&Vol.18,

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25-28�2. `m:oS.AV=�℄℄9~957~ &x5VN&2"2000$�3. `m:oS.AV=�_x℄J℄℄&x5DV*D4�Q=(7), =�7T℄g`IC&1999�

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150 L�y: ihUAbx� – `N�; Smarandache +ai2001 '1. "7�X.-P$�Vo �%l℄�%��OXÆn2&Y#'}Æ__�"FgV_u$&Vol. 27&2(2001), 18-22�2. (with Liu Yanpei)On the eccentricity value sequence of a simple graph, s'}Æ__�"FgV_u$, 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 '1. A census of maps on surfaces with given underlying graphs, A Doctorial Disser-

tation, Northern Jiaotong University, 2002.

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

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

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

3. "7�X.-P$�%.D�m,%zX.�(&b_t�_�, 3"2003$,287-

293�4. "7��-P$B7 a ≥ 2 %��On27�Vo �&s'}Æ__�"FgV_u$, 1(2003),17-21�2004 '1. (with Yanpei Liu)A new approach for enumerating maps on orientable surfaces,

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

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

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2. {q�?`r&h7�z�j -e_&��4TA~�&2012 2 T 17 %�3. A generalization of Seifert-Van Kampen theorem for fundamental groups, Far

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

4)���z�j�G(g��&�'�E� –�w�4Tf��y�"Os$&=��N`IC&2012�5)Linear isometries on pseudo-Euclidean space, International J. Math. Combin.,

Vol.1, 2012, 1-12.

6)Non-solvable spaces of linear equation systems, International J. Math.Combin.,

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Page 166: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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Page 167: Mathematical Combinatorics & Smarandache Multi-Spaces (new edition)

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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