167
—– Smarandache Let’s Flying by Wing —–Mathematical Combinatorics & Smarandache Multi-Spaces Linfan MAO (New Expanded Edition) Chinese Branch Xiquan House 2013

(New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

  • Upload
    others

  • View
    5

  • Download
    0

Embed Size (px)

Citation preview

Page 1: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

��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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

y"�q

�Y�?h�,M��/�g,�k!

Page 5: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:
Page 6: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

% �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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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

GeometriesÆ"<54�Y[C Smarandache K�+F*Qe$%�o?ZL*M�%�7yz &I� SNJ \1%Os Perez �odayV�zZL&yj&ZL%�(uq�G,9��tZL%&b��w-�?VH4|(#!%E��&U(l�N}S��?ZL Map Geometryl[ &=�,?ON}SO�, �&�?�Rs�E��X��=l��&;?*�Iseri + Smarandache K�eG^�4`hDY4�+&H�r4{O%sE-Æ?"*�O%s&*�O%s&iKg9s Iseri `3 Smarandache (I+o81&��E�+Map GeometryD�+`heG+:g&*�Y�DM�<�v&��DM�54�v_-=b_Cv,t�1`a"K*x::gÆ�?%�A|7&l0# ���w-2�?��?%�o�[�X.E�%�4�y:u?��o?ZL�J=%lT%��q��p_1%�o&?{l47T℄�y�G*��?℄��&�l�J=# �4f����?�%y�7#P&SL=? Ml��?X[*��`?a�,�?N}�[%�|�(_) )Ta5?uaE 103 ℄J,\�Ea,jV�J{"o$i5�?�G%�??g&/ 1976 lg�� "�?T^=��%uÆ-5&2u 103 ℄J,\�Ea�Jo%A5�℄664��#�?lg 6elep_�1T?�&�J=?��J�=?*:WbC?�&2u&g�`|&&OJ8k%�iÆ �:WbC?�^=��^&1976 9 T -1978 7 T&?Q :WbC?�T^= (}5)���=�F?�E℄uu�+%&g?W��zE|&0b�b?\&,b--?�f%?�E℄7E&S=(l[ "�Æ?SUN��?��1%,7℄�&�%���Z�����12))��#� E��?�M %u��℄X[��l M�?:Bt-` Goldbach�{~ �1+2Æ%dn&��0-V`Sf%�o;l M&℄~u3R%ZL9?�6(k.#Æ, i?`�?.S�|� ��af,^=?\?q39j?\nA\&31 "q3��?X[�El��?; ��e� �"(�gH_�`I%^)��� )�+ &:[%\ i� ?L?\%E, :%\ b}& "df�?a31��M�f "P�QT

Page 17: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

6 L�y: ihUAbx� – `N�; Smarandache +ai�:[%\ �?a�? 4℄S�2&�f,=??&sg:E( "_��2&l8��Æ "�Q?\&i*a^=4l?M8^l%�?3?�&4p?Ml8�T^=�?3?�)�{^=4p�k?\G=3J&�du?a?Qf,�J=?G=4l?M�k���o2�G=�?z0�t z%E�1977 ���o^=p&&?�?#�<Æ 120 Æ�� (lT(�%%�&�u3j �%X[!*�?��%zd�7(O&l?%T�u��%&l3z�W?<Æ���?�#^=�??\b T:�&lDbf~\ &�x�\ fl� T:%��7�P&�du�T�?%?\�%�� lDb�~9 7x &?���f�ul��?�a�?N}=%�#&<���^)�?%�v.yT�&�`=?SQ"S�r7�v%J0��u}/ T%��#� %^=z?(dap�d9E���:), GL�u?�5h��%i�&_�u� 103 ℄J,\�A5�℄6% 5 4i5&GL?Q�a=�7T4p℄J�_|�{r℄J7E�^=lg?���oT?&�&?m#�iF"%Gz��"�Jo8�oT?&�=%j 100 &Sep �J=??\&j 50 �=a 1 C&51-100 a

2 C&{li#?(td �Jo=?G 80 � 1 C?\��#G= M%z?���~�:�F ��Z����K�[�)��l M�?Gz�`&4#�?X[D=S%�9&G W?<l73� �℄%T �v)���+?��� 1979 1T�o2�G=�?z0&m#�2�4 18 %Gz�=? M?℄STl��?3�TM&*4s�%�k0K�r(��{?8%�k0b_��Cg9��K/[%��q(��3��L��%���q�I3��G�x�E��20b_�"4l"$)�1980 4 T?2m��"�JoD /`X5, ��X5�?_.7E&?.� ?�U�&f%G&j%d ��� 1980 ��G&&<� {&#9��g, t&�E�*G)?� �1980 7 T1?",jV�JoRkHl=�X5�aR4==|% ;�8 Bt-�%�k0b<�(II)& X5?,j#K/[%ZP�b<#���/u�k?\S=%�Æ�N, z �9?\G)�?&��kT�{Sf+T�b_�y��A�=l�\ �=? M%�?E>bEW?Ssg$g�&Zf2&��p?S%�?49E>&l�` E%�?X[la`�FGb�&�b<|:�D %��7�P&,b$!Sa`r%�'�y�`ZGl$�?%��&�uiF}%�f�:&hG4f?S ℄S�?%lT)IEuE[a<3 �g;,& �~

Page 18: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 7��|�lÆ&?S*�0&��`_<aÆl2?\&y?�G&� 60 ÆS��/?SQ"&l0bf;�T*dD*S2O�l�=+�-S2O�t�l--P-,�C{+�v-�=E�32OZ'--P-C{+�q))�z &=? Mf�:du?Sf%�Sl1% M&�uE7?}T�[P� XB}%; �?T�f�:ua=? Ml1G*Zg�?℄��%&�dG*�?*&�N%�z�(m) Q0ÆJa51980 12 T1?���℄�& =�7Tp�lb)��l �i℄&��� ��y�%lor:6O/:67E�*:C4\&*?℄M,q3�C&�:?\�?q34�&D��:laf a X5�-�2m��9?\�(lzaB��:+7�f�&l4℄�X[/?b=?G&&*v^faN%l���y?&�Gl7%B�� &℄�[/lF �?a:[��i& �?a2[6/B����WGL�� T���?F�IX["m*ha8g�?E�$sP%�k0b__NH���k0b_SxNH�l �`S=�Æ�(a`k0T&# G%d4�GdG�?a℄� M�k!P?\�?%oK#Q�?[vg `IL?&���``IC�G��l� C&�G%b�, ?5�k% T`8 %℄7��v)`�g f%�o-C��� <�Hh%℄�4�=&�#�ag1!P?\&7\=? M%����+)�(&��9?\�?e%l��?3g�� ?�M�aB��??\&G/W�?l\j2p�0^Mot%�b_�y�V�,�4:(�zr `m`+&d�GLW?8a� >�?Zg%E, &*/O\o%�b_�y��9m!%�4�xb<�*?:�%���xb�<�)&*??\l��?sd3aa��t�℄��ly?&T�daz'u���^℄�lS&*aYgZf%u'�;y_&l�??\lu?%T� �&� d# Yg%B}&/u�D_��lzG&�&�Gz9��g, t 10 fÆ&�#E�*G� �L8F�IX[?&GQ�ng&"?4DddlFG� �<�#0�f,G�?\&T��!D�r`+r& `m�?%���(P) �+Za5"/�b9.~~ÆÆ%�|&hGYg���GT�&=�7Tp�d�b

Page 19: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

8 L�y: ihUAbx� – `N�; Smarandache +ai)��7G��=)Zg?�p_,�)℄���G&?=k�lF`m7T?�&/ 1983 6 T���=�7Tp�lb),�)℄�GhQ&�&m#�4 1 %&�Gz�Sr%#uj�`m�l=�9O=�%�s&�G�?%?\�"/&��4%,)S&1983 9 T2 1987 7 T?a`mE7?�℄g7`�7TZg7 83-1C?\��[ Mu?f,�?N}%^�L[�a?��?Gz�`&1985 �_a�&8b_g9�:r|"1$/y���NyuW� RMI El&&8b_g9&29-32,1"1985$"2$?\�?%7,ed&&8b_g9&22-23,2"1985$}89��{ 1983 10 TX&q�F�IX[Y?&?a`m℄g�?ZW0X[,�d_|?\�?�?Zg%3J��T Mg??\��b_�y���20�b���&?�b���M�b_���4<�)3J&℄~u�-&"G �&?LlX?\"y�[|�?℄(KX["m{m ℄�?E�$w�%�?�4<�"*Hj�?� Bollobas P$&lX?��j}�, `??Q{r�-N}, a�r7�v:X �<s���a?\hp���k ZW0X[*?g<T��-�Gaei:���1T z%�^&�??WS=}T^i44&;*��=sV^b_ht\^v-Æ&/uGa2H:�d"123$7"132$}T4W�"*��>F"�e++��Æ"yi?�� T��?%g9&�QJ/pV�%N/pV&da>�a�?�v:�A"�e Z�%�J& ??Qa�?N}=7/a`; �D�vT�; X �)I���N}�℄%Sh(9&fl# �%0MuSh�-=%lTBb; �(lh&ZW0X[tQl8r|a�>}Æ__��:%-9�`3F&?4&+K=E�Æ�?!�Tld&�`?"�XR%~EVV>E*d�&�(r~7<b&�B~�b_+g9�Æ�aZW0X[%,�d&q�z.M��"&?�#.y8S= R(G) = 2 �� R(G) = r %~#&�8 c2(G) = 4, 5 �� lF~# cr(G) = 2r, 2r + 1��q�z.�"&�E(# AMS ��^l)P9�,TfT�) (lh&?���g �# cr(G) = 2r, 2r + 1 `lFk=��.�� ��&��faLx*3 %�℄=Dzk&S �h�M`�lT R(G) = 3&ltS=�a7 x i#c3(x) = 8 %k=��&{l�D�EQ%�{�a�lL[U"a� c3(G) = 6, 7%S-&=?y?*ZW0X[% �&!9�`�18-9� l�?Z&��dÆ�aQ���=(l8-9%MJ�u℄(KX[&Gd$!���(?�^

Page 20: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 9lTI�-N}%y&�Q�Zg�GL8?*HjP �?� Erdos (TE1C#�{&X!?�N}ld�q��TT%N}&?# �lTlF(S &<�E�?1T��T�{&�&$%; uqiE�{b����" 1987 �ao7q�4<_℄�(.Æa=9SE��&?{����zd�&�=-9�W�℄(KX[�G�+?����zd�&�ad:`8S f%�ZL�1987 11 T&"=�,?O�;=����N}GY?&?!g���P �I�-��?QG*?%�o�[%=�,?O���?F%�X.E��G7�?8�8-9t`�r|��"� �F�b_�(l8y/w-7)l%9�����N}GMV&G8-9\W�?f%M)�?M�?g:� &s}z1dl$!�?�^lT���8-9a�F�b_�:/ 1990 �lr|�� `mE7?�%X[P$!?℄S�?&℄~u=Zj}%3f�:aB.G=3&X[lFd ~?&i#?(XÆ% "�`m� �)Vl�9nh�&�`m℄g�?7ZW0X[lX[-�?; �����?��M%l��?[-C?\&i* 1986-1987 &�g?������N}G"maI�$OP%�Kac-Moody �b�<qÆ�`m℄g�?XWE�"m|K�?E�$OP%��B�<C,�'�<qÆ)���`/?_h�k}alTiF��%CV �w-; X � ����(z) Q0�r�v1987 8 T&?a �=�7Tp�lb)&Æa�8b)41℄JhS.��s ��G�%3gNObus�℄℄w'E��℄℄�7*T�℄J℄℄�J=`m%��} �?a 1989 -1991 8 T��`m�C?OlM℄J7E~ ,1991 10 T

-1993 12 TT*��^4&��`m��e>�)℄J7E,1994 1 T -12 T =�7Tp�lb)1Æb)S.��,,4&1994 3 TdFQgT*℄J[,1995 1 T -1998 9 T `m:oS.AV,\=�~rp℄J[&8℄J�℄?dC*���-q~Æ℄J,1998 10 T -12 T =K`tF~rp℄J[��lT M�?N}qlV=℄��qlVy_/℄℄��} * &i*`���b-�J�BT℄℄��%N}�`�"5�QSh#P��%N})�g?a℄J℄℄~ =T�� >B�%��} &��ka��℄℄�;r|7T��-9�g?a℄℄���℄℄9~*4�~ �[r|�_f8-9&��_����x5DV*DM��S4��'Q=�*�x5DV*D4�Q=�

Page 21: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

10 L�y: ihUAbx� – `N�; Smarandache +ai4 2�7 "%s��<�*y&?\�?�t �?℄g,9A%{v�0�_�1990 1��#$`m{(���?Zgg,%k?&�&B��X�?m#�?Zgg,9A%{v�/u&?{ 1991 4 T�k��`m{G)E>k?&�& 1991 l�lz(&��Æ?�99��9��?ÆQ�TQ�+�:&-7� |��.u��-�3$Æ�J) 13 K3J&S=4l?M&#℄~"��#Z&%HK3="}K3&� 99 Æ&lK3&� 98 Æ&�q`m�?%l�` X[P� nT&!*&`�^%Gz`k?%�Q"NKT���^& 1993 :L&�S�&?� jt ����?ZgZ,9A&ly �GF(3J Wd1K3��l Mda����%l�?�d��1988 aha{��?���R�&oM�S(�oX�<.Æ,1989 a5Fy����ao7)�4<_℄�(.Æ)�

1989 ~7 �ao7)�4<_℄�(.Æ"�FL!$:Jb�T"��,Æ_$� 1993=M&ZW0X[Q�&�?7GlX/ 1994 8T�NE���ao7j�4<_℄�(.Æ&�^?,8N4�h,%�?N}B�aXQ�?� % �uq\ � hamiltonian �%N}:�q�` 1991r|a���-\1:

Gould E�l8o�9��uy-9%NT&?!9�G�l4y/ hamiltonian�%-9&Æ~a��H<;_R_����b_g9: <�)\1:r|��� 1994 �ao7j�4<_℄�(.Æ%z &?!g�ZW0X[%�?z?&`m�?%3a E�, Gu�����-%Æ���G%lT|72_#��|f?&�u��TdRX&dR81&�t0y3lÆSqÆ. ?/ 1995:L�G�`m{G)E>k?&�,Gd�D%F(3J&�&/ 6 T1�

Page 22: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 11G�lg�w&t �`m�?Gr%���?Zgg,9A* ??o?4&lS=lg�www4�u`m�?%3a E��

1994 ~7 �ao7j�4<_℄�(.Æ"�{�H$& l|�\?% �T Mg?a�F�b_���b_g9: <���y�b_:�'b_�)?�M�:r|� 5 8�?-9�1994-1998 ?a`m:oS.AV,\=�℄J� pNSp℄J[��oL[r|%y/ hamiltonian �%l�-9f�:ua�l M�G%��D~�$A��y+$ &~�:ZES+Æ�� q( >yp�4 ?&X!?�GLve℄�&&$ �vE�E��lz��%td&1995 1a`m�?0 �oS�&r &rm��w--��%&�,r?�?�&(%Ur|�-9�/uT�.SlAT{*�*0_�+�Æ℄FR&>_℄��^{ 1996 X&?%?\yx��oS,?&�*rz�1996 &=�7Tp��x5��%℄P����?g�&��d��V�b_+DV'Æl9f%Z!"052$�y�/ 1998 &�`�?x�? ?O�X.E�%�oS�(&) �e)gT�1999 4 T&?f,�`�?x�??\&�k�?%�oSSI�b4lbe3f&0b���3?\�&�T�o?\� ��b��� ?%℄�y_Ua=�7Tp�lb)&�GL #P?%?\&b�?gd�?\M"E +Sjap%�� �z�?�_T�o?4��^b?\�&?U0b�℄�iy<s�v�#�1999 1 T -2000 6 T&?� =�v?ds7Mb|p℄J[,2000 7 T -2002 � ���z(rj b)~rq �Sj�i >Z�&d7l�T��kz �y}A'+�l&{r}A℄�%Sj\�� jf%�o-9�w(}Tn2&lTu�G)℄�7�D%3J?\�m

Page 23: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

12 L�y: ihUAbx� – `N�; Smarandache +ai#F"Gz&U(lTn2�uba?�M�:r|lD�~%-9�"a?_T�o?4&jU(4fN}℄�E(r|&/u8a=�7Tp�lb)℄�y?��G%l��-N}℄�&��� `Q� ��l�?�M�:r|&g<s�~b��z &da`�?x�?�X.E�,�d&�k��I�-�w-2�%?\7N}�Q?�&j?�`m�?3a E�℄�%�|iF9B��&af,�oL[?\%4p����-�v~`�lTy/w-2���%"S &# ��[%d4��TS ?Qa�b_t�_��:r|�a�G�89�%z &rm.y`�[�%kzh�.SlT(�%5ÆS &��u?Qr|a�b_"m�:%v89��*+���?℄��T�f�℄�; &u{r���?%��b2f%Ey%�2001 &?L�Tz?z =k�?x?OGk*X[%�Yi_^�(YWj�&+�'Æ%3J&q3u�:�:3&,?aGÆd�&HzP(�3%z?&�"hu �&�&)I�l83J-9�t�3?&?���-%�v&S-G%3J��l8y/be?x.'(%-9?�:�&� dE�#n^&�u�GlK3Jlu� Mi &?%�Tz?nT%L8?#� 90 Æ&"lFz?# 80 Æ�4 ��&G WGk}TZK??x�N%�oS 90 ÆGz&l?lu?�?%�a�4z?%ngd&?�#�8-9.yt ��l�?x?Z:r|&�^��W��&oK/_��&S a4p�r|`Q��b$!&�uZK??x%?Sa:[r|9�duiFK}%�?T�� "�89�&*H; :l8�?9���yh'Drm�u?r|%9�= �&yG%l89�&4f?�?QSf℄-%'�,�G� >fl�%N}℄��?�N}�u�^& �&G lD�|H%?��+G&Zf%&u|a �%� H%�f&_�u.y�9�Gl�?�-9aM�:r|�l?la�X.E�%��d&_dE(Bs�Tf�; �"/aT�ojuq(�_f%$gu\*N}Ey&�T�o-9�A cen-

sus of maps on surfaces with given underlying graphsÆ(<Y[>E:q4+54)u5�?k%'+�l�`Q%, Ob����� -`�[:w-2�f%Æ℄���N}&�a��:duhajS%�-9�G??W�� 10 4E�CM&S-�*�.���=(lT(�%%�&� �o-9�w,GdO[%u��P �?��=�,?O%Q`�E��a�wj 20 h&GL8�X.E�"M( �?%-9� H&��X.X[B��M)&�^ED%�w�Ev*Mf%��? �Q`�E�&8?a�o-9=��%�v��r7p��# %ObS

Page 24: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 13-���:a��[%fy))�Gf%�ya%Y?�XgSt?&A'�ld&!*?%'�*�vF&�X.E�yj,�%�T?S(4"%p�&S-(lD -�+&/u���d�`-9%C9&��w-�O�;`?a`l�N}7��

&><b��% W3�z!�K�x��_��zv�\n () )g,Yb?k�#�o-9=U(4f; �{v0bfl�fm&d0blD%Tv� "�fm&�^a�olg?�kp_�4~�o?�

2002 ~7 �?�b_℄Æ.M�_B.�Æ"0℄:$W3Gg\�\n �pv��_��F��K�x

Page 25: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

14 L�y: ihUAbx� – `N�; Smarandache +ai2002 11 T&a`m�?3a E�OP%[-C:&?���A dynamic

talk on maps and graphs on surfaces–my group action ideaÆ"`354:4^Y[+B|+~=�5 - f+eZ'a;$%o-(N}ZL&yM# ��z%%��eg�2002 1&=�,?O�?7_|,?N}OI��?%�o?F�&��D/4p^�k�o?N}℄��"/`m 2003 ^��=Æ�|&?) 2003 6 T�f,=�,?O�?7_|,?N}O�kN}℄��-��[k�N}G&u���-N}℄�%<s�&l�GT�Ob{rSh�-%N}&?aT�oj%4fy/ hamiltonian �%N}℄���G%�|�4lz0[&k�X[�`?"*�{T&y'Z+o7g9DZ&f*G&{FL-hÆ��i#?(Xs% "8�oL[E(N}�%℄�N}�&z nT�{v�y�%N}�;��di#?.yo��[%'�&{l~`l��%N}℄��rf"a�n�u`%��G%b�&?a=�,?O�?7_|,?N}O���zZL*�Active problems in maps and graphs on surfacesÆ(54:4^Y[+B|&~73U+d�)�

R�x�*2004 ~ 8 V7 �ao7~�4<:M�b__℄�(.�Æ">}k-j0$:�0F� "7^ao7q�4<_℄�(.1fT6^~hmw&�1f$D~=Dn$&��t[�jf+��SÆ&�-^V_R�Bg9DZ��

Page 26: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 15�olgy?&?l)a'&�^}T; &�u&p�y'^ow/℄�K.~+�%4<-2O'-�_b_-n7M�-�}T; du��4fz%q3;?%; &*�X.X[%4f,N℄�����I%�vh �-; &4f� H�/u&=�X.X[%�v� SH�?�;&�Zf%��T�A�v&{l�� SG�;~`l��%,NG �G�?a�o?L[℄�%B7&�du?k[%lz�%m|�*y&?a=,OM"fB?\w-��-�;y�%l�ZP&*�|'$Æ�+� =�[� =�+����t�/O�[))&��y�uyN}�� Ob{v&u��w-�vN}q8�?; &yM�k.S�%�|�4l8-9�Riemann Y[ Hurwitz E�+M�:gÆ�u�l'{%�eem��8-9�G?&�"���7n�&oV_R&>�>m:�B_V_℄<�Æ&�?W�w,d`I�5�Wk�o?�D%N}��&?a 2005 4 T�G%�o?ZL�On

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,

Surfaces and Smarandache GeometriesÆ?QG*���Hp�?,� :TwSmarandache �+�8 �9n=%}8�&9n��o?ZL�G?&?�#�8t`�`I&=?`w-�?%�A|7�p�bÆ��"a 2005 ^&I�(l�`IC�?P J, ?�=�o?ZLrW�GL�GL4�9?%|7&7�?aZL=o�(y Smarandache �+%�(�q�o��,9&8 / 2005 6 TaI� American Research Press `IC�l`I�

(6) DsTU���o?N}S�?&"/?uq 42 ?�&`mE(lF�?n,NthM�Iw?{rN}nE?�/u�?%b�&���r�8?)IÆ-a�yj?℄

Page 27: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

16 L�y: ihUAbx� – `N�; Smarandache +ai�%���z(rj b)&��#=�,?O�?7_|,?N}O%z�&?#��*GLve%N}�G{rN}℄��&Fy=k�n�&lu? {8?a℄J7E�;fu\%qY7$g�_"H?a��U2����,nV{O~�K�(l�y/℄J~ %<�&tC�f*!)zG$,purj��%,Ne5; &℄~u,�.gO'{%�|&i#sd,?N}�`hj& �~d .�~`l��G ���rj%,Ne5)IhG�,N℄��y �-9�~�-9%'D�~& M�~d .��f%l��p(%�%N}℄��5�,?N}%�%&�p(%N}lF0b 5-10 % "��r|`-9&a��&�^%,N�Gg�d<�2��a�A�`hj'{�|d&aYg7a,Nth{r�?N}f�:ul^%�?T�%Twu�'^4z�+<℄qmf+b_g9&J~�b_g9+>/Æ&�A|7&# �=�,?O�?7_|,?N}O%#P�0b℄~,`%u&I��42*�?a�o?ZL=o�%�(&u���?N}:lT�%�F7&"y.yi#�?$gp�$*8H�℄s �l^5ry�l?a�o?L[%l�|7�"7GL%{v kl*�rj��[uq(%℄�� f&?uqr �jS�?vg /W���4fz%&�m#"C�I� Perez�oC�"�Your book is very good. High research you have done.Æ,KF[|�?l4E�C�" �}P�g&�ÆVÆ��Zf%u0b����z%�T?%'{*�T��%���/ua 2005&?a`m%�F�?�=�,?O�?7_|,?N}Oy���%l�?�d�:& ?%�A�?w-O|7�l�N}℄�f%�ZL&m#�l3"C�Scientia Magna \1lz�8?a 2005 ~#}zZL�}8-9f%�9� r|�S-?%w-|7� Smarandache�+'{&?rm.y`l|�?f%�|'%��7w-&{l r�4f�%�?; &#Gb_M�����^a7I�2*x�?&�_�k�%N}℄����&z u�Smarandache Multi-spaces

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

Page 28: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 17T+y�%��:&��u6M9�y/6M9&?%|7u2�:%�℄ .�# &*HL ha?L|## %��+:�haG+6"=%6�SM�8i2�:�%68bG&*HLh%6"+�i2��%G�a�7:&? z�rj�%%vA!*2��℄.yx�2�fY�v 6M9%|7�{ -:<&Smarandache �+S� Riemannian �+&{lS�2$U��u`-&�*+fmll)E(�x�l?rm��?%w--|7&l.y`S�SL~io?4f�(f%B77����g u/ 2006 3 TaI�`I�I�2*a�}:bÆ��g &.yX��9d^�2006 8 T, ?���4p[��w-�?7�-�d, a�zd:, ?`?%�?w-O�{�# %����+y�w-�[%l�S f%�ZL, # �7d�%l3"C&W7d�lTBb%Zm, v�ua=�0brln�?w-O%ry!��l�d4�ui=�G*�?p�%lnoq&��(�) E;� +lTG`%,N℄��Pa ���%#P7y�&?d k��?/

1990 a`mSh&Oi7?zalT7Tb)&�m 1993 ^`S�ap_f{r�??\7N}%�J=&# �QkUT�[%y�*�|�2u :�}�?��v=K4���k?�&23 ?�&X���q℄&:p��_�"�4�+f,B�7T℄J?O��C?\&1?�.~~ÆÆ?\dG=℄&) 1979 ?���GL�[��S�r�Z,9A�2u�Aa?�rX0=k?%%*`?.S�JJ%�|&l�vGG`?�A*f% T*C�&Zu`?r:�?N}!�X � .g_%���

2004 ~ 8 V:)D��j^>}k-j0%

Page 29: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

18 L�y: ihUAbx� – `N�; Smarandache +aiy:u?{l4Kx7T℄�q�f%&��oy�r:�?N}% �y&���J<�� 7�(&�Zf2lu�S�+%em�a�lT�J=&4fX[&SL=?L[%X[*2*& ?r:�n�S!�X � .g_%�f���?{&��8du�TE>%�%��O*�Q0��gDU_-�S*��tUV�=\~Ko�*>�"FAV&?(lIDV'K�x�Dx5&-)*&�x�V�b_&15 ~t&x5:b_SS�"b&+*w,�&���4Js�1981 ~&2&Æz+K�x%~Ko�sj℄X�~abD&�~��~t%�*KQx_3D^x8z_�&�S�1>�!b_+��I�gq+`h�3(h&��Y7^�&VRSx&?/w+�Kac-Mody �b�<qÆC�*DzÆ_/w+��/�<C,�'�<qÆ_�&%M;*�`+_℄�(.��<b��Qx_3Æz�&K�x��H�N℄L&*j~h&���^�*>��R��*rtÆÆ��*1N_R0DVXH�N℄b�DZ�I9+DZ 5&�tQ?3b_���Y7^��R�&oM�S(�oX�<.Æ��ao7)�4<C,�'_℄�(.Æ��7��&RoX4<_℄�(.Æ0_℄�(.%*Æ.��&�b<b&$ �F�b_���yb_:�'b_����H<;_R_��0Ntq�s&�bb_<b�� 1991 ~3&K�xM;7^�*K20�CF_QO&8F�'b_�℄~(�+S>5)��*Æ_�__>_℄��~k&��Q�&VR�,+k>g9"&*;17^��*Æ_&>"|_QO��f�- 34 z�&\ ,>q5Rk>&�w1h&�*1�Q&>�ÆK�xzDj&i_Æn+`�&KD��15j*�_b_>�f~(&U�Æ^Ogb_+*1&K�x+N℄DZx-)!tS5�1991 ~&KD�v~1hES�n�Vg℄��4'Æa℄�f:�C�BgmC#Æ&%^�*K1N_R 100 �W =Wg�*D&)SH�'"�(N℄��5)�n�VN"+n0~Æ��&xIKo QC Sq�0~Æ$�7x�*jbJ�3+$!ga��1&KD��g9� ~+$^�QP�+QZ�H�*DN℄&~�(DV

Page 30: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h- `Æ0 19

50 W��+$^+�QF�g&So℄�S�Æ�+ÆG4|�Q�g/C� �*K�f �Q*D&&�8�4'�� b*DN℄&Ap�*>�"FAV&?ID�&�xl�jt�ZÆeEGt��1�7&ÆÆ^h�*DvV�K�x-�t~(r7*4zwqm&�D�T-/ne+��KD5)s�dvD&1993 ~&�#;##SDV'�KD_FL+wUDXB�UX{&�SH��K�x-~FL+�"�ÆBT�&�{3�~�ES 1962-2042&dC�*&�r�5j*�~sQl &g9b_Yg9% 80 z�\O~&fx x-% 80z�Æ �Q0��"1996 ' 7 � 30 M$

Page 31: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 21m3,?N}%�b℄�u= �= .yÆG}T$z*lu?S �%= &y�G�[b�%rz&t ?4*d,puRlf%,?N} %= &�ybY��jS&�G�p(℄�&y?,7Ex2aG�℄!gk��o*zd���=&�DN}dD� Bb&Bb%&a/�kRl ℄dD; +#N}�ltN} � u�akÆY%; �?,n�;&l�f{SGuy�;T℄Æ&bY��N}�s&fl#GkR℄%N}�l�G`,?*�℄Cd�(F�f%�p(℄���u,?N}%��|��e&?{�T�{rN}℄�Fr�%&�"w-?=%�-N} �I�-*�I?N}&_"�I?N} $Æ�+* -M N}& ℄�Dk?� ℄a`�%N}3 7��?�?�%N}iJTl7ed&7��e}�ao7n�4<:M�b__℄�(.Z�5f�:&�nN}&�em/?a�ao7n�4<:M�b__℄�(.Æ"2006 8 T&{��?$:F�%�M�n#:b_M��.#Æ"Combinatorial

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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*

Page 34: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 234X4<6M�b_+g9*�^*�d�& uDht3&32O�ZSb_V_Æ℄&&+SL ~&\��*w�fTUd_M�+n&II\~7yM�zd�℄g9&3t3℄�-��K=A"+M�8D.C\�fTII*/dFL�t+d�*fg9+�=d�_b_V_-2O's-_V_g91-n7M�-S2O*/dDRw�n+��7+�tJbSfTxj*ltg9+℄�-2O' Udg9"5wf�#'I)�=�<D_+&hf~ &g��:~�bqÆ— �JJDowM�g9+�I��w3{ow+M�g9&#'�T-~(w|�=b_Æ℄&+?�&*wW�F�&*w�A~7-=+4<6M�d�*�&_�4K��Æ&(�7+d�~Ew�7&*w�S$Yq�<biM�_"~i\5"Kg9�*1#'M�g9+℄��℄�b_6V_+>m"K&*w�Sn��_"f~Ewg�5_�,�b_6V_�7&�DM�_"S|~+5�&g9M�_~7k0d�J*Rw�2�+b_:q&G�3bSn+vf?�i*7*�=b_V_+qm��dO~O**℄��**b_6**V_g9+d�:�q&�&q�M�+M�&"i_b_g9XlM��JW*wS>UY(5��>U<b�b:l~H�i*�"K~7n�+xI&�_=nqmDP-{s+�w-?f�:ulK�?;&u�T,?&m��A|7a��:du �gp��# !z&la��&:p�:_�iv1�4ZJ%1##a�=�f�:uw-?rJ2&pX%��-�f�:�ulKw-%?;���-�eW`�� zw-e"�| z%k�~#&fl!gk�%�v��q�%u&�w-e"�| z~#%~va=��E(# 4"%�N&luYa��?�?[&&�GL%?��?u�K=f��?�&8w-? ��T,?&m%|7dd# ��?�!z&?T�D&*�M��C /Æ&du��A. `�J���?N}&x2,?N}&flG*�?���,?p��("h�m�"U�3T?%�IN}k/ 1999 [{�X.E�?\�I�-��I�-%(�uN}�a�[:%(9&℄~u�aS:% 2- Rnm,(9&7lF�-N} z&�I�-0bN}�� ��?����I?&℄~u�[Æ℄�S(9�[%$g&�?14��*0bFf%�?sd&��N}�I�-%��fQJ/�X.)E�%5i�S?S�� lX[{�X.E�%("�45i&ObN}}℄; &l℄N}�a

Page 35: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

24 L�y: ihUAbx� – `N�; Smarandache +ai�[:m,%(9�HQ�;,�l℄N} zhm,%��; &Obu��WDl℄�[:zX2�%��; �?z N}�}℄; &� �7l{�| ��54Sb+#D2O-:4B|+`�D2O-����Æ℄%{v&?�f`lT�a�[:%F(m,&� Burnside f%zhÆ℄&��rmf�:.y�lT(�bl�;lT�a�[:SG%zX2��&/ua>2���f�:ua�;�%kzh�L%��*�x3���-�vlFEv# lTb}bl&lw-�vl.yrml�℄2�℄b}bl���u 2003-2004 ?r|a�gÆ�dM�[������b_"m���b_t�_��*����'b_:Sx�)M�:9�S %QJ��%�[m,lObN}l�℄(9%m,&*HT[�* % 2- Rnm,)�a(*HQ�;; ��� ��"(lg Gross * Tucker P%�9%��%4<�&X`Q� .Æ?f%!�NT�`r7gu{�| �%�[HQ�;�v&S rm��=%:F�(f4p%��*w-at�:F�f�:u�I?=,;6"%lT℄~#&�8.yaZ�|'�����A{vqh�o# �"a&�i�?�}al���M�:r|%��:F��vhhO$Æ| -(y9���G�o-9�<>E:q4+54��t �o?4?&�#.�l:^&� d( "Qz'_T�o?41���dDN}-F#N}G (u��+-�rmF(℄�G $ s! ℄~uf,=�,?O�?7_|,?N}OY=k�N}G{r�o?N}%}&?l3��a��^lT; &�u�%4<%4-2O'-�^�%_&r��=b_&nD��(��-�T; ) ?%�o?N}S�&℄~uNT�l����I�-��%P���a>�(�"Q&�I�-f�:u{�w-%�vN}�[(a&f%�[C%w-0M���hG���%�I?� d�N}�I�-; �a�I?� Q&�[ �ub3% 2- +�#&SÆ℄; uq# �4"%T��Fy 2- +�#:%; a�I?= ddB}&(y℄�d d`�I?%ry(Z�%�f����I?�ZyV%u n- +�#�l{��CV<&℄~u�℄!gk�V0b&Gs0b�| lF+�%�#Sh�{�o �o?&?l)a��zhÆ℄%�v�� zX2�&��a�[:m,���4f℄�a� u���l����&U.y�` >�.-9���{| �I�-a�I?=%24?&?u-��I�-%N}℄�&<�UyV��[%fy&d#������-7w-?�?�d��du?{ 2005 y?_E���-��I�-�[9�%ObE�

Page 36: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 25�| ��I�-%y�r%&d)I�G�?a`�?w-O�{&*�I�-f�:�u�[%w-O&G�?%�?w-O�{lT(o}"��A{v&n�"u�?w-O�{E")Iema�?%�o?ZL�<54: KleinY[+F*Q�&d�u 2005 ?aI�`I%�54�Y[: Smarandache K�+F*Qe�l 4H�jQ=&�d[�[Q*For applying combinatorics to other branch of mathematics, a good idea is pull-

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

and reconstructed or make combinatorial generalization for the classical mathemat-

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

chanics, theoretical physics, .... "#w{M�_�'3,�b_��&~(S{+wvD^M�_g91&)�^-=M�_&��N+V�&_-=b__V&z�b�N�K��Riemann K�C�_��<t�... 0"K*x6M�:g�$�A'{d)I�G�?N} n- +�#&flh�ka��M�:r|l�y/w-�#�Ssg��zA��[9��P�"U�|?&2005 7l4I�E�%�:tK&i?d��{w-\� Riemannian �+&�flf%Sw-��%N}℄��"_$Smarandache ?&?a=�,?O�?7_|,?N}O%�o?N}/ 2005 5 T1S�&p_am,Nth��?℄�&E(l��4M�I��EMv.�a {r�o?j℄��%l��zb)℄�� �8�?�*(gQ~&T��#4[�&d(lA�?�V$_�%����^I�l�`ICW?Q�l�:i#2&D.b =( Smarandache �+%�(&GL.yhR?aI�`I�?4�9T�a�o?ZL%vA�?w-%|7&!*<� �(g{r�?N}&��8=k`�?ry%0T�p�$�&a�?�;�d73w�/u�1T?�kB�� �o?ZL&o���Æ�(rW�I�v�`Ib)&�L&?a =4H���w-|7[-� Riemannian �[���b)%s} �Ji?&a` Smarandache �+Li Riemannian �+��&7�?N}?o�(y�(&�aaW?uy9nh��� % Smarandache �+N}u�^)&.b�hh`�A�+�.y�r�"/fN}w-��I%E&?\�/��

Page 37: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

26 L�y: ihUAbx� – `N�; Smarandache +aiw-'+�lN}GL%; &rmGLa`%lT0T�; ���[2�4(a��/ua =o����ÆN}S ?rWI�v�`IC�y�&�g /2005 5 TaI��l`I��^&a�o?N}S�z &?�aI�`I�lg?�ZP&dpd��o?N}S�aI�`IZP%g/�If Smarandache �+&rm��[�a�~%N}℄�&uy-9�P�aI�`IF(|&7��?�9N}�? �&��D\� Smarandache �+N}��?� %℄�y_uqa �b)&*� ��V!*?u,C`I%?�&?7=�,?O�?7_|,?N}O(y��8~&�8?aI�`I%vgZP/WGL��4��q�N}&l3z�?.y�9y=�,?O�?7_|,?N}O �r|-9� IZP*��?�d�& ?%N}℄�aa�|�#P�

2- U Smarandache K�Smarandache �+ulAYFl||%�+&�b�a�A�+6"=&lTd .yz Gl7 Gl&nuy}Ay:�l Gl&�aq8�+=u M�I�%&p�#�ulAS�Bd%e_�q8�+iF℄"&$v��(�X(9%�+6"&�7�LFM�%�Æ��u��&p�#.(�X%�u�I%&d�u {%�� $�A�X%�+6"E-ak�V&Uua�℄Cd� .��a�k�VryFn'%Jo&�u�A�X7 �7&"%Bd,l�℄Cd=9���7=���e�y�=4+ %Bd&l�i�℄Cd ℄Pd&�f��℄Cdf�7ry�?d�Ax�%�+T �&*HZ$-k�V*�℄Cd%f�~G&du�?℄��0b��D~�l�| %�/uaIdQ% "e&?N}� Smaran-

dache �+u(S &℄~u Iseri ���[�+hh`%v� Smarandache �+&?�#�alF�[:hh Smarandache �+&/ua`2��+%:��y�uyN}���N}��� Iseri %hh�v&d*??QN}w-$Æ�+d�sd�

Page 38: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 272��+f�:lF(2hh`� 2- + Smarandache �#&l`/lF(%hh` n- + Smarandache �#&��:bD(\���E( +�?S �� &l4I�E�;?��a� Smarandache �+7 Finsler �+�Weyl �+%y_�GL{)�:!* Smarandache �+�8SL?[}A�+&�l)W `�?"a�XR/?a��e"6"7l Smarandache �+%℄�&? ,�. n +�#&�aS:7l��~1*i+|&{llF(2�G� n- + Smarandache �#hh℄��a��:�S�y_&# ��4I�E�%!z&�aS=?s�%lg = ���TS �rj��Hp�?,� :Tw Smarandache �+le ���8�&9n&S=}8u"?�G%�"m$+nJ5'v�N}�Ax�% Smarandache �+&?�g f�:.ylF(2f%�A�?e_%7E&i*=}Ty:%����T�;nuV~6"�alX&flV[S�(%Sh�(9�a�G��8℄--9?&?=?%{vL$�I�2*&GLL8?�A{vf�:ulA℄% Smarandache B6"�lT SmarandacheB6"D�* n T}} z6"%�&S=�� n �/)/ 2��^&Smarandache�+ �ulA℄% Smarandache B6"�{ -:<&�℄�Hz*kIn2%r5&.��Æ2�n{ zCV!gk�&�# l�:[S &lta�℄!gCV Pu��%&��*P %�Hn ,ÆF[�%Æ?! �vD`k�%��!g&* u:[%&v��8uF(!gS %p*&�F(k�%*℄�%��-&�du��:h!*.y"y�S -M %|l -"Theory of Everything$%E��lF(&$ z6"%�� �ulT�-; &Ev��/k�%D~0M� z6"%�.y(4fA�l&l�A�Æ�0bXR/w-nu�-�v�Fy?l)!* SmarandacheB6"ulA�w- -&lt7?yja`%�?w-O�{%{v kl-� �?!*&�?w-O�{ulA.yf/�pl% -&"y.y �pl��%�?w- -l u�a�-%��T$[:�q�N}&�=?%|7f%_|pS&?/ 2006 4 TaI�`I�4pgZP�Smarandache *`h�<�&SL�?y/��B6"�Smarandache �+�2��+�.�[�+)�[%N}S &S=���yjqaI�l�?�M�:r|�al� 9D C�=&��ZPdC-�D(*���+�[%ZP�

Page 39: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

28 L�y: ihUAbx� – `N�; Smarandache +ai"P$5'�|?& -M &℄~u o1 -=��%Ob℄�u Riemannian �+&Fy{b8 Smarandache �+n:[�?w- ��/ -M N}&1|!gk�& �Gl�D(N}℄�uLL k%&�du?l)`I��4E�w'%N}�wa`%>4�0�{ 2006 8 T?a�ao7n�M�b_:4<_℄�(.Æ:a`�?w-O�{&?&?l)a2�7lw- Riemannian �+℄�&a $! 6"sd:a`�w-�#:��*N}`��)|:<&H�u�#aWDw-Shd%6"w-&/u�(��Iw-�#*$Æw-�#&Æ�j�N}w-�#%�I(9&* d- qx(&sg��zA���I℄�)�(&?�N}$Æw-�#%$Æ(9&*�~1��~1��$Æ�p2�Riemannian V~�Sh�J)$Æ(9�N}S |a&�AN}`��(�#*�Sh�B(9&4fS 0bz ���#*�%l�,z��\�| &�du?yM� %&*��uw-%gQ[r�4l8�T��%-9�M�(I+K��<Æ"Geometrical theory on com-

binatorial manifold$4� 50 1e&/ 2007 ^�G�?T�!*�8-9<D�w-$Æ�+%sd&/u� �I� Trans. Amer. Math. Soc. :&s}41^&L8?rj8M�uF%$Æ�+J21&��r �&7�?9�SG�M�?W�4I�2*��&;u�?��MI�y/ Smarandache �#%-9�GLL8?ÆV(lg�+7�I%?�M�I��℄-9&?/u8�89������MJ�u�M%Os&%gl44( %�+?��GW`�4D%C�&�DI��8-9�=?s}�W?rQ�l%I��&�z 0�l�I[�x$��?��$!�gM�ub��%�&$ �89�%�p(&q�7s}[�U�&y?5 50% %��#0�I[���89� 2007 a�gM�:�`�(��zEE&�&�A�p(%9�al�pl "Fg%M�:� M��'%E*luI�%9�℄~u`vD,pupl "Qg&SMJ�*����u`vD,1u`JF�F)�?�-9U},...... )&a?l4z?%7�d&?7I�2*8~&C;��OR?aI�pllgM�&ZK�'�?w-7 -M �[%-9*o�(9���"yj?a��`I�lg-9��b_M�<b\( I )�"Selected Papers on Mathematical Combinatorics$, /u�D���oXb_M�[�Æ"International Journal of Mathematical Combinatorics$%�9� �*|a�gM�[�f=�9O&M��[��/�7N|�f%w-&[

Page 40: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 29�f��w-'{N}�?�M �8H�M%�AF�Æ?'{�I�2*4M�O�&j|�aI��d� �f%V"�F�|l%M�$)r~�"GL�G�?�rluw's,d&�7=�,?O�?7_|,?Op_&X!�y=,OlT�4% �s}&z M��W�k`maS�}��e:�=�,?O�?7_|,?O%(y��4#P&z�y=�,?O~ �� 7�W_|B7fY| �s}&t8(yM�Y?�s, �*�JZr)�(wa�8fY|�}:��gM�/ 2007 _TaI��lp�&*?Qr|��:��w-'{N}q8�?* -M %9���-9r|%�x�&$ w- o1�J7l*N}0b&?,!9f%�w-�#:%p2��&�~�uÆ�Sh�J�iw-�#�Saw-�y6"n�y6"=%m,&y� Lie ��Oi+| -%w-��)N}&# �l�4(�+%S &�^�^��G�w-$Æ�+ -e_%7E℄�&�/ 2009 9 TaI�`I�w-$Æ�+�Sa1-=%��vgZP�+#la%u&��=7lw-�#:Oi+| -F��%�u�I�-=:F��v&�lA����,;6"7$Æ�+(tS-%w-�v�z�"Uv�(v�+N}%r%u*M N}aa 6e"�Smarandache �+ugf*��u`-*�%8Haa�?e"a`%�?` -M %N}k/ 2006 &aZP�Smarandache *`h�<�y?l�&?^�[-� Smarandache �+7��u`-%y_�M -&y�78H?%y_)�(�*� UE(_|7lw-$Æ�+&Ev{�?:`�%8Hf%�;7[-&Fy = u(�%D(N}&(yS dE(t M�:r|� � 2009 &q�1f%N}&w-$Æ�+ -F�uq�7XQ�&(y%�&�~�;uq.y0Jf℄�*y&?,B��k'& Einstein %��u`-*�% 6; �f�:&w-"$Æ$�#*�%1aa�lA�?e"&� +lT�Lw-�#.y �lT�%1&. �0b=S=%HT�# �lTM 1�CFD$&Einstein ��u`-/�%&f�:ulAk��% y�%�1*\r%Æ?'{&�lÆQ"&�u\�M �%%�?�JaF(�&_=%|m#l�l3�vD&w- 6=% Einstein o�J�8n^|��-7q8% Einstein o�J℄-&w- o1�J�*�&�~�J&. �� %�&*w-$Æ�#:%�&�

Page 41: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

30 L�y: ihUAbx� – `N�; Smarandache +aiw- 6&f�:[�flA�℄`k�!g�a�r(%Æ?'{&�dua=�p�l�Æ?P�&*Xi%�'(-�=Fem%�a�A!gd&�℄`k�%#~S f�: u�fS %l�Æ&�lT:[S &l�fS LF�℄!g %b.\%f&*y&?a`aw- 6=&b0b<s Einstein %��u`(E �&U0b<sA�E &x3%"&�u\�w- 6M �%%�?�Ja"�T6"�� S=HTwG6"%uWd#l u&�7�℄`k�%!g*�℄�Æ��ul3%��^&��?%w- o1�J�[��q8% Einstein o1�J�z�Q&j�uq# %l�q8 o1 Einstein �J%T&* Schwarzschild �`DT�Reissner-Nordstrom T)&��w-�v.�&�hhw- Einstein o1�J%l�℄T& w- o1%%*f%e���A'{&�g`ma?%l8y/w-�#:%�&�J%-9&=?&I�l�M ?�M�_y�8��5Æ"Special Report$ZK�`l8?%�M���H+�_NH�Æ"Relativity in Combinatorial Gravitational Fields$%9�&ya/���Aw- o1�7q8 o1%y_�f�:&�Aw-1%'{U.y��/`�|1f%w-&7lw- Yang-Mills �J&fl�/miM N}�?2009 aI�`I%�M�N�K�C,^H<&+�'�vgZP=f%�l�^�VD&��[U(l�sd; 0bT�&d(F�%N}r\6"�&�j5LjY�$y}auT�p_fFr�%N}!�&J�,?N}&℄~uEp(,?N}&bo2(lA,?�NoK�&Ua �lA"%N}'�*?�'{�j �&l�GeqZ!?�uqG*r|-9%�LHÆoÆ� y*&& $��=99% P �/Ep&`�J,?x2�℄CdE(*!���HLnu�9�P&N}S )z/y\ ,nuYf��M�:ll89�&<�l79f��fr|,nu*C(D�*�0Hr|-9b��~l<}�:Z&�,......&yy))&*dD.S�Am�u+#J'%���=(,Ne5&SL,Ns^#R�,N℄�C��v)�[%; &d(,N�GkI49*(g��; ��guq�; &��f�,N�G%℄hb℄���AFWE��,N�G&℄~uN}sd,?�G�<+l)uT �%rf�?%4fz?*t ,Ns^& # �_N}��_�����%?�?�&*GL�gu�&�gb<sSj0��2008 &?aF{[|�?WyE[*N}Slz?�ZL=& ��l4j�`�℄�e%$zNÆ�G"&�℄f�:Æ*1T$z&4l$u�h�/S�; &��i&℄�E:,4p$5lD'{ ��℄f/&i*

Page 42: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-N��DÆ0 31E[�s[),41$luph'{&i*��avzZL=&?L8tC&N}�?,?f�:�uN}Æ?&4/�4j�Æ℄=%41$&�/uqT��q�sd%:$7T�;�* U0b�f℄�i&vD&�N} ��?,?�lypVry �p_lp�&kfUiN}?;u mf%&U *�{rl�+$z%℄�&gT�"S�; _T?�N}�Fy&T�"�f�,N�Gq�; &iGLE?u&�u7l,�p�%�b℄��l,Nq�CM&hR� hR�lG�s^~ �%���<�Hzs^CM�4f(��%Z�f%CM&� $?g(Z_&�GL�lD$!3 F��%3 d*?,ryÆQ�F�l uAX3 F��aF�ZL=%/�nu`F��,N℄�%�T ℄S3 �+��A~v&sohG?�V%ÆOÆ�&y�Ny_r?Kr2rS�q30 l�X>H%3 U (s^hR,ll��p(%3 &℄~ul�y?�a`%l��p(3 &S�;y_ 4&CM�, HnEvA#S`,?ry%��l# hR�?l)E(F����,Ns^&ObEu?%N}℄�&℄~uZP�-9%`I&l)# I�l�thhR��^.y(Zf% "N}l�`�?n,?ry�(�f%3 l �*H&(0b%-9�~r\�?T�k��#&{r,?N}%yObjauN}�`N}℄�%�2�M�*&�N�I�&"%N}'{&l �s�D/a�?n,Nth{r(gN}&�du ℄lT�{r,?N}ha:[v4j�Æ℄=uT$z%lTObzd�Sz&��`,N�G��?E[%C�t5&l)5Pf��?�ry&vA�`hj%'{& ygN}�{rv�Y�B%3 N}& M�d .��~l�(�/,?ry%�3 ��duhG��p�(G >&7R�r���+ QQQ�%lTObE�2007 ?qalT-R=r�lT�`3 SCI <bÆ%ri& ��,NC�e5f%�'D&S=l[Q*d*=nrS� ^owCRw:�+&D{~7*�(q|V_ a��+wx:�*t&�z SCI <bb� a~=_36~=_}&�SV_DZ+ a��^�~�0yOlt,a�&-+��D^b.~�&z Yang-Mills H>D��[+~=�D�hnTd℄g9~7:nzr7Fg�:nzE�Fg�`+*Æ℄�&�0DV_g9+Rwpv&xDb_g9\�+�*�?%�A|7a��h�d( >�Q�2008 1&I�℄g�?d)14d4qp r9&4C�� SCI ,zC�,N℄�&z`S7,Ns^hRwf�?%�4I�2*d 7?[-℄-; ��AC�e_f�:���lA �

Page 43: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

32 L�y: ihUAbx� – `N�; Smarandache +ai&G%-9?���lDG&S�G�?���G%�E&�u| ,?%&du!l%&* SCI ,z�;Obn%u-9 �&&*! B�-9%?���& �&G �|a?��.yfl�~l�℄�&|aT��8-9%�iFflu�Fy&* T�,N�G�℄~u{rsd,?N}%�G�<; & 9u�� SCI -9,zC�,N℄�%�v&?L4} l��%�(���|%�p(℄�[p�I��t�"m�?ZOs Rabounski E�/ 2006 qa8?Z:Æ~���9*v9r|�l��V_g9F*Zi*V_n+n_Æ"Declaration

of academic freedom: scientific human rights$%3,?Cd%b��&� ���,?V%KxyV�G,`*V_g9+n#DM�+&n$d+,V_℄^V_d�g9D�0+�F*+&n_L$D'(nL0HX�,V_<b+q�DF*+&*ynS��u`&6���4n�{iCK<b+q�&*y�<b+ SCI�EI 0l~_,"K0H ,*1V_g9wVSnz�.+'(a, *y�Bo7{nz+�-a3nz'(+g9&�DQ~℄V_DZ}+_�:�v0��℄Cdf, p_lp�&Tv� ._ShJ); %�LB&7�℄��.<`k�% $�&{r�f(�/k��%%jJ(y�y &,?N}Gs0b7k�*�℄Cd�Ary�ezf�&�up_lp�,?kIry*S%N/�℄Cd`���%oq&�&duHT,N�G0b7&�N�dnS�6%�; �dyy97��?�LeZ

Page 44: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�H"�5R��y:�– _�b_+;1�., 33-36

U0tUV℄X�3Ax(�)PJw�^2%w 83-1 p^!)�?=%D�u�QTwl�Bb%�?:%&D lu�Qz�:&"%uHyp"(9$%��}�Æ$ghG��?%sg�(�lT�\ %r%(a/)℄?LZ�F?%D��D ÆQ; �T�; %�o&1�(t2|l/?L?\�?%�J=�Fy&?Lb?"�?&y1u��?\:�D �\ %��?\�v�g9?�y; &S-k?\�?%ed&T�~l!g&7z?LHu8[�_��)UT/�?=%sg:x3"D�Q"a�?\lTD�&�gKlKÆ2ÆQld8D�W`�n^lT:".u.yM�%&U.yXD�M`�T�Q&yt/)| T$� SbB}D�=%rDG� – n2&� ?L T:%y1Fa&*D�=%n2& u�Q�~SH:%sd�k*�ySbÆu�4,S 0 iS,*S 0 +�bÆ&���TD�&�gbXU�4,S 0 iS,*S 0 +�bÆ�TBbn2&* +lT.�P.y�G a + bi %#l&a:�D�=&��b� a 6= 0 %.�&* 2 + 5i, 5 − 4

3i, · · ·&,�b� a = b = 0 %|X~#�x�y:ÆQ&?L$!�ySbÆf�:�u�4,S 0 +SbÆ&*

5i,−34i, · · · )�(��?D�& zIf ��� T&n�<� T&�`G%��� >�&�a?\�J=u�3m��lT:&.(a�% =����ed G%�+�l?L`lT:%�? T&gI�ulTA75f%�J�rf:&.(�?L lT:%�e�� & H% T�d"9l|�:T*�e%�?#��k*& ?$uÆ%z? �#�K3}H&ObE�u`|r%:

1D8q{ �<Y�>M|�%2!1985#�

Page 45: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

34 L�y: ihUAbx� – `N�; Smarandache +aiy�"Hl|m`Q%l\'{�v T ��?k%edu&?\|r�Sz�`%'{�=f$uÆ$g%u\lK5J,&℄~u?��DuÆl�?&�!�edldDuÆ=%��8 → U� → Z?�Æ%'{&H)Iem�$uÆ%sg'{�|r%��� dXÆkm`Q��� _QB7ld|r%D�&vD&`G% T� u�s%l[ T�&lbJ0%f&�(#l&,(�(�`(y:�yiF��w&{lJ, Tuy:&"%p_&�u?\�?%|Bb%�v�k*&?�.�l�& UA %:f%iF&�(.� (a + bi)

f� (b = 0)

1� (b 6= 0)

( � (mn, n 6= 0, m, n ∈ Z)

E �1� (a 6= 0)

s1� (a = 0)�AiF7�w(R/?L=:%?\�JV*�Vry� ��_|%�?$gu4("h%�?\lT:&?℄S'&�?=*dDba`�T:&k*&7��(y%l�:*Tt(�JwVsuy��*�Cvf9: 9u0$�%_�73�~&�i#Tt(�Jw%�J.a"0$�%_�*3�~wG%�|=f%&��(���%:�y�&bi#a��=T0$�G*.�&�o2D�H%�AuW&i#Tt(�Jw=Y4%�AuWa��=P�z�`Q&��.S���%1AuW"Q<u�^(�&e[�[�flA{�e Z�%�?�v$�x��^%'&&i?�Z"2?\(��� �<iÆ%�?:�m�bv%/�UT/D uz�:&"%uHp_%�?\lTD &�gb�HD %�'&

Page 46: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

`� `-<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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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

6O−π π

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

−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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

!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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

!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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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/���

1982 �Æ%lh&?�7:6�_,B���E℄�&℄"+W?{B��:dQ&3 l4BBÆ- :6℄�%�?S�G4h1&;?:6%℄J�E℄Æ�&*uu`*t�uu���uuu:F)&?llf%�Y?&=?G�6�r�� fGL�%a�&? "� l��:*t'FL>�t^�QbVs~"E-�7ka=? M%0`*X[%M!dN u$��G= M`?M!yG%uE�?%D=SX[&�fQÆ_Ln�^&�wK�xyQ>{Æ– �uG� 74f�"�%Q���lS �l3~�`�%X!&d �l3~�%� �2005 10 T&�?_za =? MSj%2� &�4X[uq�p&?�z?Æf? G%n2j|��`G%0'�1982 1&q�&�*&(&? j\����l M&4fz ��℄�%�j��Æy_&nuAWG�"l7%℄A&i*W℄℄G&& &Of�t)&nu�y_�Zg?�Jh&y�aQ?�)I{r��~ ℄��� l�SHb)%i5u��Aa�S2mFab)�?��℄� uy=�7Tp��b)i5 �Q%&y =�7Tp��b)uq>�&E(.yj�%y_&.�'k9udZ��?�D_��lzG&&y9u�i℄�A {*&0`lS%(g�1983

7 TG&j&?��}TT%r�a�.\&d G&����G&&��Æ�U.y&a X5O/:6℄2?&?l[�9<f�i℄,l[�fG&Gzg%bÆ� S�&G&Æ�bÆ?&?9�� Æ�t 10 fÆ&S=�?&� 110Æ&hA<Æ�&.y=klF {%�?�?\�?n���?��j�kM&?�9{rf�i℄&)fd�? ��l:TT �b���&?UE(I �?% x$&��,?E!��7� a���? E%F�IX[x�&G"?%Æ�a��4DE; &4DddlF�? %�:T1&=�7Tp�lb)7`mE{7E?�p_"&�GLOf,�)℄ 3 ���G&%C 8�℄g7`�7T�7TW��Zg

Page 60: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 49f%*M,%?\�*?\2a`m&?4{j�a`m%td?\�?&/uZ ���z,�)℄�� b)eX!��?\&�yy9udZ%�4f&/ub)w'�99��?hQ&��?pÆ� 4l &�^ jm#� `mE{7E?�℄g7`�7TZg?\%td�1983 9 T 9 %&?*l4℄�[/lX&':�"X5� `m%l=&�"�4℄�[/ÆfQ �uq"X5D `m%=�7Tp�lb)p��4ph&?tf"b)w'�K�%l�Y?� ?�Z &�k�,)SSI���4f�;?&���1zG&&*dDHzP:� Æ�t�E(lF�? ?�?L8GL&1980 vzG&Gz&?.y:[|℄O�&�? M�&=Z%1M�APut�℄J�t�5h℄%�KZg&E(d u�3%,?}z&l�[ &+& 10-20%&�&? u�[lgS& z�s,�l�[&?duh�a>&�u:� t� lD�r&u��f,�?�K&�b ?�u�M�����?{&�d4�u��G��S&a8�{1d%℄39�m��+Z1983 9 T 10 %&?�lQ `mE{7E?�7 83-1 C&�*l ,)S&7��`m{=)Zg?�|l&� %�_ z?lX�k�*M,%Zg?\��F?�f�:u"`m7T℄J?OÆ`Q%lF?�&� n�/`m{7E,Gd&?��4�4*Ob�3E[�PQJ/`m7T℄J?O�(�'%u&p_f?&� 2010 ^&?('G*�`m7T℄J?O%℄?E�& $u���u��&=%yp�`mE{7E?�ulFyG=*X7%=)Zg?�&Fy4l??%&sg:PuG=3J&(%iG=3JUb(�l��<�*y&?4l?Uud ?�t3&ly?�� �_{r�i℄�&tdQ& |� lTT% "&?�X[*z?�$!?%�?�M Gz℄~"&Xg� �t3&�?Uu!P�t�z &L8���?%F�IX[&? `m�F?�?\�&�GW?Y?�4Ga`m%z?&yt?a?\�?�J=(; "�E�G�+�}4&l4u`m℄g�?%ZW0X[&U(l4u`m℄g?O%BC&X[&?�?E(0�[&*a?U%2�NL&WG����dE(� G%a��lZW0X[l{ 1984 rÆ�k&l),�f?%�??\&�du??Q�k j�G`m{G)E>k?&�F&,r&�)I&`�?x�?�oN}S%ObE�

Page 61: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

50 L�y: ihUAbx� – `N�; Smarandache +aia`mE{7E?�?\M"&?iSGz?� 3-4 ?&7X[h#4'b& SuE�?��9� -o?���o?�Sho?�7T?*℄℄��%�4X[�l[ "?&l�X[;�?X[&"?�^"%Gz*dDdQ`mE7?��?&�?X[L8GL?%at&"?EQulT℄�&(td`Q��_|%Zg?\&ia℄2:�℄�bp4f&*HT�%�%rvP l^�?\��vK�:x5�4�&S=%Mv�+ �uaG=�?=%le�+sd:&fl��� �[:S &?XQEdDK},7T5�luaMv�+%sd:&a7TlD�4Hs�?f%q)��%.M&i*7T�[�&�uan:44:Hs?�d%����S ,l{=%-\�&lu{C\Fa44f%%q)Hs?f%%![��S �?a=? M%l�#uu�fH%&� %M X[`D?%l��_ZbÆ&�7T�'+B& M��/#l(y&�?\7T5�&b��djM`~atn�&Uby\�zd%�0e&�`?uTZx��bgy(i&,b9ukEQ%�l\���TT?&aX[%ORd&�0lsg:.y�y&{l.yM`l�<sb�%7T���? aG=?\�}�9&sd;iF+%&ly Y=ZW0X[?\N}�4<�l3&,Gi??"�9&*� ��%�-N}BBX�&Gs0bVT�~%��:b�r|%�99n��#� ?���%u�q�Z?O%�:q�>�&Sxe~b� u4f&��99v<#℄~"&�&�9X[`9v,~(R?&HT9v7Pd*?La3ÆQ&( Uj�31�3&fl�<Tl�}Æ�4Æ%9vy�[1ÆQ&i#?�9��o44B&ap�sd�9?\S�&?uq.yTHr|a��M�&*�4<[����}�b_�):[%�9Zg-9��?\�C7T?&i?`�C%hh*|m(��?%�T,lC\i��Sh�CSh*qSh1K3&l,�?fl� T��?%`�&z dfl��T�a7TShE��J=��%o?e"��;�E*�;�v&�du?a =�7Tp�lb){r��~ &aSh�;�[z�s%lTObE�?\7T���7T℄℄��)3J&,km`?aX5O/:6℄J�7�}℄J7E%�s&E-u7Tt��7T���7TG>nLG>&Uu℄℄℄z7℄℄�v&?2>uq(�4f�(!g&F&)I{^=Q%z?b"4f& ���C\�eH�i���\T�Kn�h[�%Q)℄J℄℄�v&aO/:6(%'R~%℄A<�&(%l)I<�,_i*MeH�&~%z?*E0�& $!�8n^�M&?l$!Hf�:Xf_i��%(℄%��&a1dnua![&MXQ#����

Page 62: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 51l℄℄w'7~ l3=%�2�N&l�",u?Y=ZW0X[?\%�?�4<�aS.f/=%lA�e��&℄7%0}y_l7℄℄ "%g?����I(y&�u?%p~��z?*X[%b�&?a�l M3?&ZK*z?L<�lz�6zl�*Gd�:�r4Æ&y�*$g%fl�MX��K3?QG�?a =�7Tp�lb){r��~ %sd&�s�℄J℄℄w'E��?ug ℄℄m1℄��}&(��(!g?_Q Zg?�?\%&�/"f/:T* -%?\�l�laZgE>e_=& u"=?)If,Zg?�?\& F?Zg3�AE(�(!g�lF?�da?\l[ "?w'?Sf%Zg3%f\7S.f\&�/" - f/%�l?v&�V6*lÆ&�j�`/I�F?$g&fl,�f/Z(j/��)℄�`mE{7E?�,)M"&�4H*?�?�? 400 fCi%,�)℄�&z ?HTU( 36 C%Sj��?��� &?Uda X5O/:6℄2&EQUalT7�%�4y℄H8*~&*?:?M"%Sj��u"GLO?� `mUh%�P��rJ�`B{Zg?�lg%?SQ"&*+8a?�F?Zg$g��/℄℄~ &uHT℄`7Zglg%?SPGs0b&$%; , YgQ"&n^*>?SaYg%)℄ "&yiGL�kB52v�℄℄~ &uHTYgGs%b��1987 7 T&?%,)SIS�&a �=�7Tp�lb){r℄J��~ & ��L "&�v��℄J℄℄~ &RkN�X�7T℄J��~ �"_$�℄_�a�1987 7 T?"`mE{7E?�?\S�&Æ- �41℄JhS.��s o-��G&--l4X℄J[3j`m:omO℄J�� �℄JUE(�℄&�yjuq�℄%l�℄J& m{E7�7�l) �S�℄h�& "u�℄� <sb��F5�℄�&�uaEE��0:&8℄℄�J=rS%E�uZ&��9ln�z�aE��0:&yt/?�ah lT; &0b�T�℄fG .y)`�? ℄2%4l~ N�u�Gl�℄J%�℄�& ��℄�5�zd*E�uZ�(�?Kl(`yj~��G%�℄�zV*"au�z��E�uZ&z ÆY�0:l�3�%zV�v�q�a32)`&rm�f(l�Bb%E

Page 63: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

52 L�y: ihUAbx� – `N�; Smarandache +ai�uZ�(E(*fz�a�℄�:&/uf%�i�n,9��℄h�y?r?W��7~ �K�1987 1&w':ZA? `m,j�e℄J{rjM℄Jd ℄�&f%℄℄m1�[Æ4y�-\%7℄JE�&�S"UZK `ml�.i7TS.68�&*&--���KEp�1987 ^&��C*vDh.�h�e& m,j�e℄J�,�`m{4l4uV7~r&EE�%}�OJ� u℄℄��"� `m:omO�℄7ES���7K��&"%q_U℄J&?,a �`m:omO℄2&3j8~r%��~ ��u?Rk3j��~ %4lT℄J�w'�℄J℄℄�M)?&?�ks�℄J℄℄w'E��� �AE(s�℄℄w'E�%℄� &.(l�SHf℄�%℄℄w'E�.a�&&8℄JgI� .\&SSh*,$F� -#ooSh&5�?�?�%℄℄w'7~ &��SH℄J℄℄w'E�&?�G�8℄J℄℄w'E�%s�℄�&q�℄JhO ℄J[M(z�?&?0Mb|Æ&dr℄℄^*%�8℄J��~ �J=&al�X℄J[%,�d&?K5?d�n^T�℄℄�J=`m%UA��; &i*�0:(Bd%2�&�brm&Uba`T�Mv�E��4!.,_i*℄℄=q3< %C\�W; ),leH#��B��%E�lus5℄℄w'E�o2g%�;%, �_,L[lUb--7E�4f%_,��=^�h Sh℄℄=%l��q)�<�℄JgI .\&��u�T℄J&i?�[�T�lTF� - )oo℄J%℄℄w'7~ �v& ?Q{r��~ y�T�%B5G4d�sd�"m$�℄zHN31988 1& m:omOp_U℄J�℄�1989 ^&?,a `m,j�e℄2&y 8℄J�af%uV7�4yd℄J℄℄&?Æ℄3j2:��℄J��d &ZK3j�0dM*s�S℄℄w'E�℄���ua�T~r:&?as5`m:omOp_U℄℄w'E�sd:&q��Z�LQ%℄℄w'E�s�Ey*(y%Sh�;&*?Qs�℄℄w'E���℄℄��pS�-9d�sd��zLQ%Ey&dik�N�=�7T℄Jpb)s�℄℄w'E�%�(*�v&�l���%℄℄w'E��(2>SL*�s5n,�℄J:G,�℄℄��&SL℄℄�7=k�℄℄[NÆ�℄℄J7�fV�N�q)Zt�),�℄℄d �℄��N,�Ob~r℄℄�v,�℄℄p�[�Æ4,�U~0��N,�Ob���℄&SL9~V"�℄�4����℄�G�5��Æ℄℄�℄*=+℄JGg���℄),�5��5|�a�T~r=&q�Sh

Page 64: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 53�;����:B��e_&d���B,ee_)�����℄z& V"℄℄9~�℄M&NV℄J℄℄GgX �lD%��&z kd�k?d��o?e"(O�f%eH#Ce_�B���;)&flT�℄℄=< %l���; �℄℄w'E�s�?&b0bh ℄℄�J=%��; �M E�uZ�9�&℄�U;u`z.&?� �ad ��”ao7<�4<_℄�(.” %l89��a?%b�d&℄2:~|%z1O?f%�,��I*Æ&l� %:~ u 286�386 )Xl:~�1989 �)rÆM"&?�a`m,j�e℄2&a℄2:q3.y �=�G�?S�f:W{1EKqdr�&�a��,Æ?%lD�&#sNb�&�4 # ��&y���� � 1989 1&uV7�D%�4uqsg℄℄ 4&?ay%��~ ℄�dsgS�&?I�(yh�?&?,\ �dlT℄2&�9{r��~ ℄��"P$�℄/�T�`m�C?O℄J&u?T�7T℄℄����r�GÆ&f%��p�%lT~r�1987 1&���kC*vDh.�h�e&℄℄ NQJ*>&�`�"�(YgQ"& N#� >�a�l#sd&1989 1&=�7Tp�lb)1hNI�`m�C?OlMPT�4℄J%7E N&�ulT℄℄℄M_Æb�&z *V"?Sa 1991 �Æ j�?&,����K*DÆ7E%lT~r�w':4�?a�T℄23j℄℄��~ &l7E�4l�&�"(l4`mE{7E?�i?�}[lg%z?�E?�sd℄℄&�?:d�5&4ph℄4w',beH?rmsd{X�`m��=3[)9~�q&℄2� �9~ Q�Khs�s��q�4,.h �7"0�i��pz�T$&7z?lX�#E��4%z��Q�℄℄�G,.i���qlG�?%B7℄�&di?$!�V"℄J9~&bs5℄℄�N�&;0a℄℄�J=�(%9~957~ �℄&l�au ISO9000 9~!"e_%(�'{��l& m{7E,Gda�{�y�4℄J�[9~~ ��℄�& m�C?O℄J�,�')~r&S=(l~&(&�u~ �b`d�&(&�`�0,z &d�db`~ �f%&(&�b(� �d')�4�AE(�$!�8nD~�*�0')&℄JhÆ~9~���%�Tz1q9�hd ')h��*y&?gM�lT|&�Sz�`:d�%')7d')y_&�?�,

Page 65: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

54 L�y: ihUAbx� – `N�; Smarandache +ai�T'),r&�g:l�leE��&�p;|�`���&(%�'��{')pS�d:&���h�# �4D�l4��ad:"!*�{TLjd*Q�4�*�&{TO~O�*1N_RD5+Xv&o*>Dd*Q�E Æt�d%z1aQ"?Lyr# �')w|[&?��(�MYM7&*v �u*�0')l d %l�h�lu&�E(af�~ �J=i��"/`m�C?OlM℄JPT�4℄Jz �℄&eH���0�~�&a�y_k#4b�&��0b�;a℄J:%eH#�u�.y,&dD ".y,dQf,D\�Sh�;?&?���pz#C%�v&��{%eH,b?f%pz#C&S144l 0b%�v&�^��NV�eH���%lz(z(&�q3`℄℄h7f%�;&�&{�x5N℄���x5*D�)M�:T��~℄Æ&i?`m1l� (E℄%Sh�;uq%�ÆY�?SeW℄℄�J=&�$�H�Vi��u 10 T 30 %&h^AZ�%y+7V*�d 6oC V&8>i��AJ}i�%i��#L�.y#�d 8oC V&��%�:%7V+ ��d 15oC V�<�a�H:�;��siV7&�4ph:I')&rm�T%i�����L&�taIdQ%}hi�� �v&�ulz9~rt&.#i��AJ}%z1QÆQE&7E�O%z18~h Mv�Æ|j ?&�l$i��a�T?LTT��,b&I'�4|b<{?B��Tz-i%i���1990 1&l4lX℄�%zrL8?&"`m{(lAG)E>k?&�& �:?&)I5�Zg,r&��.y��/u&?LlX �L�E>�C;| ?&j8�&��D*&�� �?=k%Zgu���?&sd3Z a�L�E>�&Zgsd*Zg3la`m�?�*y&℄�&1&?�kd 4p,T�*�T�%&�,r��lGÆ&*�V"`m�C?O 1991 ?S j,?&℄Jh�k�N4p0b℄℄%�T-\~r&S=% 100m3 �b-�J℄℄� ��r%J����b-�J�℄℄�v(}A, lAua�d��EB��, a�4_44:#e�QrC\��Ti��&�95S℄℄U~oVuT}7&z lz(z���f&Q�Gg�; U(lAua2[#e�QrC\��Ti��&℄G2E�pV?a{IB�4_"C#�&�4jFk�e℄B&�� φ = 25mm %HC�N>&{2[j�jFE =�aT2E�4_44&"hua2[iF(|95�℄℄9~&z NV�E�#�%l�Æ��&�7u φ = 25mm%HCa℄℄�J=0bKNsO&sOdQ%HCEv_zj�&.�lz(Q�,℄JGg�

Page 66: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 55� ��l�7T��N}th�kN}S. φ = 48mm %Lee℄B&��� φ = 48 × 3.5mm %^<E�C~f� φ = 25mm %HC�*Le#>&�^.aLe℄℄=a� φ = 48 × 3.5mm C~�*�~&{l=+℄J℄℄Gg�y l�68uqS.`�^t&�aLe℄J℄℄=UE(i���z^2&�b-�J�aTdE(����A�"e℄B�=�7Tp�hR� 1 �C,Ns^&hR���A�"e℄BaT�b-�J 100m3 �℄℄��N}�f//`m�C?O℄J&N} Nk�1 �?%�:��T�aTB~� 105t&'^ �X_� 1.2&E�J'^5 126t 95�*y&?f%��~CSh�;�jF�;*fYe�&fl�G��TaT_|&�aT��?>�jFaT_|�?7*��℄zE�℄��FE�%aT��� 16 $�C5�1B�&#C:d}TT{&S=:T{*e℄B#�&dT{�/444�Vl�&y�aT�J=e℄BL,?>:�I M30 .�&��*;qI:d?>&yt/aaT�J=4_*,�,jF_|=�Le℄J℄℄=%lAjF95L�24 TjFk�e℄BS{I��XÆ4&z a 23 $?>:qIl�℄&y#�?>�ouO�8~E�y�a`m�C?O 100m3 �b-�J�aT=# G`���q�aTd �a�T��3aT*��4)aT℄z&�y��4/E�zG 31.40m 44&lei 7 h "&*��8~℄℄��%N}~`�lDdn�`m�C?O 100m3 �b-�J�aTE�y�m#� 1991 =�7Tp�,�f�p):&l� ?=Zlg�1fl7&(Dd �ulT��Glu�?`aT_|E��S℄℄f%�pS&��u 1992 Æ~r|a��eDV�%��*1N_R 100m3 gmC#*DÆ*�x5VN�%��*1N_RgmC#*DÆv}89���b-�J%℄℄&N��`m�C?OlM℄JO�W�����℄J℄℄7o&5��*�%�&�lg�\8g&S�>gÆ%℄℄El&?f%��e℄℄�N&s��℄℄w'E��w'f℄&{lV"� 1991 �S�?�RL-NÆ_3J

Page 67: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

56 L�y: ihUAbx� – `N�; Smarandache +ai`m�C?OlM℄J℄℄&i?_|����e℄J℄℄w'�v�?Q&� 1996 ��x5N℄�\1C/Y%C4&P&?``m�C?OlM℄J℄℄w'f%�pS&��u� aGOs%�x5DV*DM��S4��'Q=�=%�_3~(DV*DM�I�SÆ�p_?% 2010 &�?_za �F?�&y ?�u/ 1995 7SH?�-�&Z *�PqC�? �?%z?L8?&"?� =�E�SaT℄z%?��b-�J&uqa 2009 ,b�&*vA20b9�H�&z a�℄z%P�&uq 0b_�RE4l��_SG%�J�&?4��&d_z�� �,�f�%o~�"z$�℄#;��Atq�wV�x`m�C?O 100m3 �b-�J℄℄%G`&i?lRG*�b)%Ob��q<�1991 10 T&?�dQ �=�7Tp�lb)1h4:℄℄^ ��^4&"�llT~r%��~ ryGz �3�T~r%��~ ℄�&SL`mq�Fe�&�o�Ey�� m{�7�*=�=mN}O,NgN�)℄J�`mq�Fe�&�o�Ey�℄℄(}T}7&lTu 50mzd&�Q�10moLo�Q%�P�y��Q℄℄oV,_lT�u4/�Q:�4 62m&G 1.5m"��G 2.3m$%EvSA�oi���{�+$!ga��*V"�Q#P&E����1!�t&lu�Q�!w[d�l$}�����,puShk��&�� C28 ��� UEA .-0��%��i��&S#P)� ≥ B8,1u�Q1��� 3 : 7 [�aj&��=ysd%u�QSh#P�*y&?)V��~h�&�{ ;ZK8a�qPP%�x5t+��b��l NT& �eui��%��95�;��℄(�FJ0%�T&/u�D{Sh���;*℄℄�℄V"}T�[V"&�Q�o�Q%Sh�P��q�Sh���;&?�!�T�Q%Sh*i��kI�~ d�3Sh��&/ub�

Page 68: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 57℄45��D%℄℄�N&�7�℄w'℄℄�� &℄JhO ℄J[���eui����95�;�v&�;#`%S-u�s'i��ShSkI�~d�3 50m zd&�QSh��&4f��/u=?� G%Mb|_�;lz�?*G5��;�I&ez8~�)(y_����&I�lTdI ",B��;�lx&|a 'i��ShkI�sy/#SkI�~�3%��&7yj?%�;S l3&/u�Da�℄V"%jad&x$℄4d i���T�a℄℄�℄�[&?&$�l_�%���℄&SL�QU��eHD4�Q1zG95�i��#P℄℄�#�&!78ÆE� #)�℄�4lz�T −1.68m yd�4%i��&SL�Q1H*l�ÆQp&�SpV<sE�pV?w'y:Qp%�T&*y&aI �44_��>-�Æ,z &*V"SSh#P&Hz�T�0blz(�T�G&5�����g��?��Æ%El&"`�{`D�T&� ��i��U�&f%i��℄G&�V�&�Q�o�Q%Shk��℄℄9~��Q:�4 62m %EvSA�oi���{℄℄j&?g%E��SeH#C�&�� φ = 48 × 3.5mm �~�El>" 600mm × 800mm&� 1200mm%#C��f%�Sh�;&�!#C�<s℄℄b��EvSA�oi��{(lT℄℄y1&h&�ub�VEvSA�o\%��44d�& �i���TrJS44�*y&?b�℄45�E��tf%�^&a�{:HR 1 S�C\`Sf%�44vD�ai���Tj&℄Jhw'%9~')�J=&O ℄J[`8�{%#C�a`�s�&!*�il>NU�&"� O/:6`*tsd%;V7yh%{GV+ f&#C�l>=�" 500mm&� 600mm&b�?.;ld�?.;?Sh#u4�%&��G��V"�i���T�J=&�4X℄J[uk ℄2:f%�) & ℄℄4�?����EvSA�o�N℄�luai��℄G� �NpV?w'&��`D�N%El& {�EvSA�o\s$&Klf%�N&yV"A�o�N dhGi�����o��l�A�q�`m�C?O 100m3 �BT_|E�y�yh 62m 4%EvSA�oi���{℄℄&?uÆ$�n^8o?e"��/℄℄~ &z d�a>�*dD(%℄J[%E�u�V�&E�a/��%o?e" l^�i*eH#C�%E���(#{�;e"&�;F�t&�_ÆV�,l��q9{�;e"&�;<$.\l7&�7f�~GiFIh&z Z(j/=+℄J℄℄Gg�`mq�Fe� 50m zd&�Q%Sh�P�℄℄��&y�S 62m EvSA�oi���{℄℄��?Qy?*5B% ���}8-9&Æ~a�x5V

Page 69: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

58 L�y: ihUAbx� – `N�; Smarandache +aiN�*�x5N℄�:�r�1992 fh& m{�7�℄J�℄&[`l�G$F�Sh&*+=+S℄J℄℄Gg� ����G%[j&℄~u0b`�l>�B��f%9|&y*>��Q���&aj%,SH℄7℄℄&{l�V℄℄℄M��l M&`m{uq(l�℄℄�4a��%:�℄℄��&.0b}$��&�?���vC�f%aTy<s℄℄℄�[b���n^{ -�; f����E(.yX.%qY�?7℄℄^�4zr 5hlT℄2�|�lz&aQ?�{a>H% -sd&!*%:��� �u<m{�;e"%���Hj��H$�Ti���J=A7%C\?Tk,"Cn)I��}X�~�#C&�7i��QiNI&fl#GlT:D%o?e_�q��;&?E�`�`m{�7��o:%%:�&�s5`�℄℄�7w'℄Jf℄�1992 GÆ&=�7Tp�w'��(DCD℄�&*=Zlg�<H&?y %(D#u��G&qR ℄J[P u�*y&?Z%X!u�C:R ℄J[�.y�� b)��,C;&3jy~℄�%zrL8?&"?%R ℄J[ 0bjld|�.y�& d�a; ,"b)�4���a&$Wha���;~`�`dn%14��GFQgT*℄J[&?�uS=lT� �G,"&1T��FQ%.�( �&m�℄J� dz�&�?(T� d &*?�lg 5 &v}4zrulg 8 ���1?aX52m�& t �+p℄J[��x5VN�Osd�`%Mb|��lY��G`m�C?O 100m3 �BT_|E��G`���w'`mq�Fe�&��o�Ey�*`m{�7�)℄J��~ &z T��S=l���}7&?7G4ÆY&Gd`?%l���p�℄�(lD%�T�TQ"T ��z��(DCD&?L8Gb)uO?FZ�(D&�G��ld&?� � $!Gu℄J�(DCM,Gd%O ��TT?? ℄J��d&�GMb|B�G &GL8?&"y*?gTG�℄J[�& FZ|�$!FZ%u℄J[&�G%��p�℄���<sFQn2��?&) b)l4zrL8?&?�$!vz(DCM%l��er~�I"vz(DCM&lb)D%Elu����nÆ&��v}4zrFQ*℄J[&*GLlg "i?41&z &W?b)�?℄J[%�<�.u�lb)q E{ %u&?a℄J����;%Q!LL9��v}4zr�aCMd:&d�`w'M)�Ub)FZ0bFQ%�%��?&tX?%FZ��&Y?�?��a℄℄��p��[�G%l�Ob℄�&�"!*lb)FZ%1

Page 70: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 594�G=&?%n2<s℄J�FQgT%n2&|mz�?FQgT*℄J[�t G%�0&��CM,Gd%lb)q _Æf�&*�7��CMdj&b)��Ci�D%El l3�'j{?&G}XQ"*��I& �T�=n&&K�x+N℄g�DSE+�zq{rS�GK#;�v&o�℄*�x~EGK#;�v�Æd�`"*�K�x+N℄DZf��&o�℄*�DZxX�*�&�-7EF�Æ0lb)q *v}4zro�FQ&d�`y?|m� z`%�0�y?q�eC,�;&z�?L1T�FQgT*℄J[��2r?QG�lb)l���%��&�4q a `m?�7G�"�*�'K�x:DV�"#-2O`�&�O �.�I5�S�j�Æ�lf�:&?7d�`%y_ ulT���G7℄J���%℄�y_&. ����G%��p�℄�f&aGOs%�x5VN�:r|%��-9f&fl# �G%yV&?L&"f�:�E(Zfl�%? �*_&�4j=�7Tp�%XZ�uqvp_f��?T�k���&?a=�7Tp�lb)℄�%_f"&&Fy�ka℄℄���;(�7�&℄~u?Qa`m:oS.AV=�℄J=%`3|m&7Ga�l M`?%yR*#PuÆ �%��^&1993 3 T&a?=ZlgHfl7 &?�7SGl�g,lg<H&ta��~ :(lDGz%zrlXm#�℄J[(D�� ��(Du7℄hwf%�/u&?%℄h{EQ%HT 84 ClR �℄J[%y+℄ht124 C� ��l�T&x�k?&�&?m#�`m{G)E>k?&�,Gd*`m�?p-Gr%���?ZgZ,9A�z a(D�℄h*G�9A1T�[m#��&℄~u℄hAG�HT 124C&i?G�b)�l��%�W`��?du?Q��$!&� lb)�A�Y�G"%�#�NOS�m�uq �u�LB%JV&< V��d� QkSH��[%r�l?a�l M'{Uu��s&y*.b""xN℄℄����*YgT�℄℄��} &�d� ��B}*B�& $� �"�(℄℄Yg�ha\"M&S~ t5���t5*Æ-5V)�ha9P%^�L[&?%�A{v $-8b)%O�&�duhG?y��Da�=�7Tp�lb)%)IE�"&$Z7�rj

1993 dL&b)f%th9P&8℄Jh9*Æb)�*l�q�(;�4�1994 1 T&?a �Æb)� S.��,,4�E(�℄2:%:J&�h�u�d�a0:T�&?zl�# \��"a��HT%℄℄S.')&U

Page 71: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

60 L�y: ihUAbx� – `N�; Smarandache +ai� l�℄J℄℄~G�� ��uq�ka℄J7E�; ,�z�zz�t5& �� �|&� �z�s5%z1sg:X�D=z�%�D��&�^&�z�%(dZ��4y1��v �z℄�"b)q�, �w'&b)��,*Æb)A;,Æ~s��z℄℄w'E��D*�zZ�92�q3ua�zjlhie_p7y?&�"b)���HDd�z��℄M)y1,z?��Æ�\�z92&�z℄�b< 4ph:IH7|+���G&4�=�?� (l~Ob℄�&�u--�zs�℄℄w'E��D&�u?%p~�Hz�z&ÆY℄℄�?lF.0b1h "�.y5��z92%b�&�G℄℄w'E��D%s�℄���l&?g?�7�_fT℄J%�z℄��`m{EFlM℄J"����h7E&�z "4Y&Cz,z(�zZ��℄℄w'E�)�(�t ℄℄�?&? ��}h��G�℄℄w'E��D%s�&?WMb|Æ��l℄JA;Z�l��_fh&�"q�,%�4zr�C�7s��&IL4pp?&"z^%�z℄�&?.�1h&lGLl��_fh&N\���zvh&b)q Æf?*b)q�,4�Æb)q )�T�lX? �r8,D%d�|���zd�����z%&�#U(`m7℄��=�7Tl�%�Tb)��r8a�D% "w'�zd�&��<�Q?&"�r8l4℄��G,�&l47Z�&l4a2H:� �7zS�?&"HT�z�Zg&�DU�z�7��z%g?z7�?LZ � 3 $&�41T7��z�� �z92%5��� l&(%u)I��%&(%��%Æ&l?L%℄℄w'E�lu��:~Æa 16 �0:�j[}T�z�%℄℄w'E�&E-uÆ�#l&Uu7z � "�?t`Æ"%℄℄w'E��D&5�s5n�℄J:G�℄℄���Ob℄℄�v�Ob���℄%z7&*eH��C5D"�eH��H��B,e#Ce_�i����kJi���J?q)Z)Ob�(&aDÆ�2f%�;T&S�?&q�,4"���z�=&?��z7#y"��l M&℄Jz1uC��zZ�%Obn&lF�z�Pdx�UA�[t y?%z1�+&Fy�zZ�lFP+ f&l�r8,{=℄J?W=�7Tp�lb)Q℄℄&!*�^`�i%��d("h&*���lFP$!=�7T℄Jpb)��^&ay?�D=z��J=&?s��uk7`%℄℄w'E��D�z��s&�dG��r8==z��D*=�7Tp�lb)%Obn*`�"v�

Page 72: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 61�l%�zqi&i?^��g a(�N%8>q�e5=&�z�zuqa�DNS8�J=r\lD%���&l �CdO�{1q�%_h&EsH8Z�r\�O{1hJ-4%����du??Qf,�z� th{r�z℄�%lTObE�"$�℄_�Z7af)E1995 rN?&=�7Tp�lb)NI ��,`m{�9511Æ℄J%`m:oS.AV=�sd℄J℄℄ N�b)fT7a/NTS?9%Sh℄J*_,_x℄J℄℄&Fy`�T℄J℄℄w'℄~B}&ZKZA�b)�4y℄℄~ *p~%�GGl�~rq �&ZA?*~r+p℄J[&p℄J[l"Æb)p℄J[� ��u?℄℄��~ � B�%lT~r&du?FO�%&℄℄�a7E�E�� *℄℄�r�Æ=�XÆr\S℄℄������%lT~r�

�*>�"FAV&? ℄2:�$!&?L�T�uaÆb)� ��&�a℄℄~ :,P(lD4h&u�' *P&7 1*�Æ%���"� =�7T℄Jpb)�%�(lvÆ℄℄&b)Ob��{a�T~r~ el:(��F&�=��T�ZA`Qw7~rq �&S |a&�T�&"%--uu�'b%&�du=�7Tp�lb)a`m�G%4lT�-q~Æ℄J��T�rN?�Q �℄2�y ℄JF[%,f℄�UE(�G&℄J℄℄�9daM �J=&.�f%�℄j%l�d ℄�&i*&--7E�4f% 10Kv -:%4_℄�&?E��8-:%sdy�℄℄7��&�G??W℄�℄℄�1995 3 T&℄℄�9��M�&.yf%sd℄℄���� ,f℄�E(?1�G&.b℄2t�r`Q|&5h�`�d`Qu-℄℄&�^�z&l

Page 73: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

62 L�y: ihUAbx� – `N�; Smarandache +ai) 1995 5 T^&a;G�s*%--d&��ks5�����g��$���S�?&�D�GD℄44&w'�℄�8℄Zg^℄℄�� ~rq �.(#� ��~rq *?1T�&sd℄℄�J=℄~�=&H1h�b*f+lhiC&4phUb�3:C�� %b�&a℄℄pfV95d&?s�GD℄℄℄fV�N�8�Ny?lh +2#yf℄�(lz&?�f℄��QZ�%�Ed ��_SJ%℄81&')Gp*℄8��9~&� �GnK&*�℄�89~sg:u'℄�k=�&���GlF �d 8�')��d*?Q8℄J℄℄&?a7E�E�� *℄℄�r�Æ=��℄℄��d�sd�����A� 6 TdBS�&*y0bd ℄JSh℄℄�N&s�℄℄w'E�&�kf%Sh℄℄d �Sh℄℄���}�J?&l�4/s5�&a�����J=uq�G�Ssd℄℄,U(l�4/F{C:&7s5rt L&0biBE�Ssd&*y&?=k�rl"℄sd&q��o?*℄Sh�;&=k�℄x�4V�i��pV)�&E��-\&as5��S�jf%�℄�℄&�G�8J?sd%℄℄&*?Q℄JSh℄℄��q)Zd�sd�y�&U0b1fSh℄℄^VNNS�7�~rq 7l4X℄4T�}h&ETdQ&℄�u: j�G4f�&�"?:?+iC&?"&��G�:�dQ&?_YGTlz��:&?=G� ?%Mb|&}T�lr&(&lr�h&y?T �VNNS; �?�G ���z L8G&Gu?*~rq }T�{SG℄2m=�Q%�HT�lS=< �D�%~r.�(u f%&�zuymGT��o%(jtd�* ℄�_u: j&WGSG�Q��NS^&`Gab)�lS.�Pd(�|&�Ga�b1'& b*��Ai%NS�Ql\���ztd�4ph:C& Guqa51w'℄��t�d ?9℄℄��~rq 4��&l_;?��dD�v�G�k(u|(�s5��`3&y�77E�4a�TT%Vs--&=�7Tp�lb)a 1995 6 T����S�%z & jNI�8℄JSh�_x_,℄℄ N�?� T pNS�p℄J[&3j8℄JpNS��~ �8℄Jg?( 23 TÆS�4�7�℄℄7E&?LklN�℄JSh*^_,℄℄ N�F�Q�)ooeH��%uL>H5�%D"�eHe_&�HuL>HB,ee_&SE��g��"?T��G�℄℄l5�FR}T℄℄[D\&Fy2d|i��%℄℄9~u�"�`m:oS.AV=�℄J_p$yduK`:o�b)%Mb�$&�b)��℄~y�℄Jfy~G&�:2�q3d ℄2e l �(lh&2d

Page 74: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 631$�XG 5m %(�Q� C40 i��&*V"i���T9~& `m�=3[%m�&?d �2d1$&,\℄��Ti���� lL &lT℄�dQ�?&"l4K`:o�b)%��a?Mb|)?�?N7℄��TV�r~?&?ba Mb|�l &EQu�b)+pq�[�℄7Mb|%O �G;?&*dD�i��?Ubukd 51,\&�℄�<�%��?L8G&7TSh=%l�y1�4%9~uV"℄JSh4�%y1&*�lz(�Q�TC40 i��&��y1�4o2Q�℄℄&{lV"S9~�G`?%�AwgoKJ|!z&d"y�J�uH"%�T7� �℄℄ �$ &eHfzD\&�&�$GV%uO&℄���l��eHaG`�4<I&BV k&S hG�i���T�J=`m�em��,e?.#,\℄�=�`�Æ_b�(��zEE& # w'B��℄D"eHyV"℄℄9~��^&℄J{2:,$�k����%�eH��2:Sh℄℄ _,$ & m{7E,Gdw'℄J9~�')&Zg �`m:oS.AV=�℄J�*y&b)�Æb)p℄J[*3jS.%q 4f�&q9�h ℄2Q�m1| 6S�� ��h�&(%6 #Ua℄2 24 � f℄�| �')jlh&?;| �Æ^')%w4) &Wa7,℄�%l4z?�T:Q&�GWÆ^%T���G;?℄J9~(; 6&?"E(&Ob{�GL>')ld&*℄2��h*�I')wI#��YY%�4ph&�4b)��j�?�f')wQ ℄2d�|�B1�&w4�;*�n℄DK�x-Æ?f d�|?&GY?"*�nDAfj�&fxd�~ eNL&{�-)*�fT*d�71h&{\~?&fTO~O>K��Æ?/uÆfGL _l$ �ld�S��&��4b)��� nT& $!?77,%�U(�^l$y_��"/~rq ��T��uÆb)%��`�&Æb)��� "?L�T��1996 7 T1&Æb)w'4�S.')&Q `m:oS.AV=�℄J&)``℄i�alT�$:�d%lh�t&/u�``~rq ���CiGGsD 200 C%s�&�TTu℄JfV*℄℄9~y"%lTT&le�G�,$Sh℄℄��=� �"�&E(# :g&WQ%�usD&~rq ��TCiGGP{ x�?f�:u/ 1995 dLm#�`m�?*`m{G)E>k?&�,GdGr%���?Zgg,9A* ??o?4&dd a=�7Tp�lb)4M℄�d���Ey*.y"y# b)%B�&��lzÆb)`?L�T�%hs&i?_lz'&EQE({| %lT; &�u^o-4z&D*D�wDZl�>Y)%{+ a&)%#;<.- Q u�^�?o2

Page 75: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

64 L�y: ihUAbx� – `N�; Smarandache +ai*k%lS_�`�k�d�u"/�TE&i?d��{ 1996 9 T��kd N}S%,?&��<*y&?l)gf�~vBVg-&>~Ew( 100%+��-{Æ%El&Fya℄2:#i�~"�,\"H~℄�&y+G"b)%j��. ��:a �&j�g1 ".\`3&d N}S,?&��1996 9 T1℄JOeSh℄℄�&?L�kw''GSh\T℄���l

11 TdB&7E�4�k=k_xb)&℄7hO �?YG�{m&*_xb)�℄7hO `&*�tVV&?L�T={�GP �T&*�t&Py*{m&*���a `m&S ��Q�� 11 TdB%`muq4^�&r% =?k�}2Ap&��l2�\&S �{m�����&/u��l2Ap&) �:�&��faN�&."j��:�:�?pb)8�l24/e8��$dl}u�{& �:U�`0&.) �1J&>hyG7V 31oC&:�8%p%d %�&.",8�l2[�Ul7%e8&�^�;ZpY=℄7hO �G�&*�z�&*%�T�?Ql)=?�2r�GT&"?kfGh%p%& �1J&S l�rl�8p%�d�u�lzqi&i? �QDÆ��AgÆ)�9(��% T&d$!��Ki3+Æ��Ki/Æ 1un^lar�

s*&�!% tsC+V`�% ?� a7E�E�� *℄℄�r�Æ=(fFG%#!&HDlz%�r�ÆS.AVd��u"?OP��%���d?&"?X� � &�??0Æ&_ÆrW�d�4f℄� b)� ℄2:%�4 ℄J[Mb�a?RpMb|&GLf�:dG�?a℄J9~~ %(oR��')`℄℄; & gt?%�0&_5?%�0�h �1996 Gh&?L�kw'8℄JR`C:2du:}℄℄�lh&l4 ℄J[L8?&(l[o%C\�erS��_S%r4&RsueHBV k&l℄� u�&h �ld��k#eH&d �T:�i���G�?� ld& Th �0�℄�h "�.y��t�G%Q&??bd s5* & �*GF"�?=�C%℄4� Mb|55

Page 76: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 654C�l &�G?bw'℄�=eH,dQ�=?&q�CSh�;?&?M�l�m��&�GLar4%44�� 12mm ;%CH&z �D���4V�h �?&?ukw'�')&�!E(; ?&L8 ℄J[� l �G"*�*'�&f�A{�Æ ℄J[=(l4 60 f?%X℄J[G2�F&l[ "7p y_ N"&*G`l�; N!��&�GY?%y_4"�(lh&GL8?ba�℄2&d a�℄X��?;|�E&�?;GM� M��=g7E b)lT℄2� &yj8b)l4~rp Q℄2�?&X!?YGlX< &�?� %�'a/&N}S�a�℄2&FyE��G��4X℄J[t?"?&<:����l���h?&?ÆfGQ �=g7E b)&WG�� +&���lT℄J +p ��T℄J�u=�v?d,NgN�&� Uha℄JjMd &m1,f*℄℄�UE(�G�G?QdG�?a`�?x�?_T�o?4M"&a=�v?ds7Mb| p℄J[%gMJ9��*>�"FAV&?BiE�% _x℄℄M"&7E�4)I,D�6A�℄�4_�4��f77E�4%y_&��b) t{pNS�4%|l~ &l)?%?4�S.AVY?eZ?&?L8G �f��lh&?a℄J�')℄℄~G&r �$ &℄4L8?��b)a�$�H:���lT 1m× 1.8m %M&?;℄4&?Lu uE5�0WGA�~!M:&G"��&44*U�5�0A�%&E; �GÆf? �ld��b)��%~!M:&C\s℄� >��"�lh"?w'�r�ÆS.AVd�&ShE�3j�Q ℄2?&�?,I:℄�[ l ℄℄~G�r �$ &?℄2ÆI 6A4_b)��%M:h �ld&�L8I?L5E�A�%8MadD2�&4_�4�C�f>&O℄�f>X�HC\)�I4f�&;?*dD 5-GL℄℄�?L8I&��b)u7E�4,D%& ^&Xg� %{pS�4%~ �S.AVd:&�4Sh℄J[4 1^24C��6A4_b)&"��b)�Afx* w�&^,tM�oOÆ~=J&J\�oOd0�&��RwA�S =w�&��I{+,~b���*:��E fDP-wv~og,t+�Qba�&h�TFL#wvh Æt ShE�3

Page 77: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

66 L�y: ihUAbx� – `N�; Smarandache +aij��^%Q&p ℄J[ad:)Id��6A℄Ju℄ 15 h&z b���b)�`J0')% ��� hs?&��b)f���t&�$!u* %pS�4~ F3��y?&��b)_d A 5pNS,�%r��1997 �h&7E�4)I,D�℄J:%_xK��lK"�El�lK6�℄&z b�U_xb)7865�,D%�S�QgEap-z�g�-z?&���lK67U_xb)%--l) u4"&"��_xb)&Q�?La&?f%AT?&(l79|� # {��Gh&*V"GÆ℄℄&7E�4w'�ob)aja�&��7Vh 30oC�4p�r&Gh4_%l�K�k(�u#�����lK6 ℄2h &���lK6"uGha�&|�7VNGF3& u6e�℄%; & M�h �?."w'U_xb)f%, �(lÆKu#faN�&Ev, &℄4���=K:%[H�>&S rmK��℄��%uÆf�5%H�&/u��℄�K ?%Mb|�? ?_ÆS^&t`l�0l0�dl!,�&x$F(_xb)&a; h �&j&E(?%,� d����lK6#0����d���lK6f��&Q�?�xy_�?"&�gt`�7&=yja�%F(KWdQ&z ��℄%K<s��zd?&?��d�0�,�����lK6.#z�8F(KWdQ&z a6��p��9~~ �LTT?&���lK6x�7E�4&_� �pS�4lX �E')K%�℄9~&�!�℄℄z-Q?&?�!9g`z��S#0��%,��8℄J/ 1998 7 T�℄&g?w'�7E�E�� *℄℄�r�ÆY�*`m{9~ Qp}%(Y&�lzY�-Q&9~)�CD*�|&*?Q8℄JC*�-q~Æ~rd�sd�� 9 T�&?&�`�?x�??xZ?O%�oN}S&/u�k�lr_T?4, lr℄%d �z�)�qT26�QV�v=�v?d,NgN�℄J/ 1997 l~&8?d��g? `m:oS.AV=�℄J&*�}z&7?dK5ÆY&�}z_�?���8℄J%E��7-"d&?%vA�8�U�Æ%OV&W=�v?d���dFJ�|&GL H℄J7E&X!l4H℄J7E%℄J[��"yj?�+ =g7E b)%2�FX℄J[�* �|&*�T℄JjMd �� >℄�&a=�v?ds7Mb|(F"%�|�GL8~?l3z�?�?*=�v?ds7Mb|%p℄J[�� ?uq$!&:�`�?x�?%�o&. ��l x$ UE(rdQ�Fy&?f�:ua 1998 �h��k*=�v?d,NgN

Page 78: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 67�%7E~℄�&. �� ℄JUha`℄℄�4&*��z�zL[& �?hh�℄2�1998 12 T&?t �`�?x�?�oN}S%,?x$ ���?tf,?x$ �b)�r,;� &GLW;d�&"ub�4��N}ld���D; ;��?4f�&�{fa %�v(��h?&b)�r,x$?�M�9&�?gglT��DF=F_�9�X�?��;����:[Va�b) N�?_T�o?4% +��&z E( +Sj�nSj�p�"/�Q?l�{�_T�o?4&?! $B2a�� :gd�k% l��?&�?t �o?4 &4f�"X�2r&P"=�7Tp�lb)E(To& M��U��&*a� &�tudE℄�d(sgSj��_T�o?4%4lsd3f&"a? �hh�=�v?d℄2&lFlD�T}z&=; T���%&�i#?.y j�G�oN}S4l%3J?\�?Q&� 2004 &=�v?d%l���{aE℄Jsd: ,h^&_o7,$&�?Off%� �A`F(��h�&?rm� E�uZ�Y�� )�u?�|7E�4g%l�? ℄2: &lFg:℄�[')ldfy*9~&�?� b)*℄℄�4Æ~?�l�V�r~&_7s7Mb|%z18~ld?9r~�.�&℄� ;b��8℄Jsd5�$E�&�lM 7E2:,$&?}$u*pz97~d %�B$�H�Ti�� &E��℄℄�E(&$B$QiO\#��n; &9~ Q}%z1rm?&b�℄2u℄�9�*y&?5�lF℄JuV7`Qi%VG�v& ��l��9�7&"E��℄℄gl?*%�8℄J_x℄℄j&s7Mb|O 7?lX�D�_x℄J=%�2r&�?Wv?dd4�lTZL&��d4iBDdQ&yt℄Jf,_x℄℄� �h&d4�4mdQ��s7Mb|O G 2Y?"&y Wd4ZLr~&Z12r�(l2# !.� ��&�lzle:Z� 8 2r&. �a����lÆ�:K�xDV'�E� B(Æ�^lÆQ&S d4���4d��G"�.K�U*{+ÆtÆ�*pAv%^B(*L9(&?dd4a`�$�s2[*o[x[����23IEO�s7Mb|O *?dSG�4z1�d4fQ&# %�.u�{T*GÆ&/u_E�AfQ��s7Mb|O 7�?yZ�I�&Yd47�ld��h?%lTdI&d4 ℄2')℄�&?lr,d4 ℄2&lr7G?T�?;Gu� ��:��4+'iÆ&G" ��?"��[-~=41&>D�=�D"|~=>!℄CmS+�h&5[:ARoo�,K=ÆG&ohmD>D4'�k�;G+5[�E[�Æ?IfL8G&7T|

Page 79: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

68 L�y: ihUAbx� – `N�; Smarandache +ai�_x����2R3 4dO&lF�/(V(G�℄~L9%2�&*dW�ÆQF{%��&lv?d�*lT?�(w'&a_x�Q:&Zf2&b��� us�d4ta>�?%�'&r �s7Mb|O ?�"*�R?Æ#+;P�;&{�8℄�<�Eh&f*f℄��Æ�*?�[ "%℄�a�NOtyrN~ ��(lD Q&�i#??Q℄%v��z� b) jNI�l�Mb�7E~r~ &*vT~r%7E�4��~��T?&a`��hK�xt% �=(lb�Æ%n2�&��$OvZW=�v?d,NgN�℄J/ 1999 1�℄&=?,9h�{rlKE(%p$���2000 5 T%lh&��ts7Ml4+O "X����,�Gl�l��zb)&/u�IY??����zb)℄��I"?.yl)�av?d℄�&"?%℄�y_uq\ =�v?dMb|�?L8G&? ℄S�alT2�Er.~&*,NgN�uq�℄�x�I% +&a78b)pq Y?�?%qi?& jQ ���b) ~rq &� ��b)Ua`m^'KxM�eMb&p�%G℄d �1_f��?� _T�o?4f,�4p&uqf,Zg3?\&HD DHdI���[w'%lT[-C�.y�&Wd%�uk?\�[���l�?�%�g�9ZP�)V�&9n&*�Glg-9~d &�di#?(sk "a���zb){r� gN���z� ulA! (4p%`r� %*&S���2�(v�*�|(92℄~f&"(��y"�%l\v%e_�*y& ~rq v%v�*! ��(fFG%b��?B ���zb) &b)pq /�?l\�&�n^O�ow�/�vaX�&??\�H%Ob�(&y� 2000 ���,GÆl����p-�K���"/yj{r�_f%℄℄~ &z ,�7��z&Fy���z�z%v�`?Q" ul2}r�?T�lZuS-w'�z� ℄�&x�f/& (ynD%��*�Pfzf%z'� i*ÆQ&y�J����du?h�.yJ,l`2<T�z�zv%v�%lTObE�lf%℄J~ f/&di#?Qb)lTT�a℄J�z�[w���C&��|�z��7l�℄J�z~r%CzjJ�2001 &b)NI�l�,Mb�℄J&4�?3j8℄J�z��T�,%℄24&�4b)��l_N7?b q1o2=�T~r~"�*y&lTfT% "e&?30b�8�,Mb�7E~ Mb|(y��eZ&7GL%

Page 80: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 69℄��G[-�z92%ObnD�lz&8�Mb�7E~ Mb|O ;?&℄J℄℄�z &� �`mf%s5℄℄%�4�uld&i*aCÆMv=WG� 2 Æ&*���Q℄2')|��s5℄℄�4�?"&.bGa`QW�T�4�Æ&m�~� dz`&�&z�v%=%�K�ÆEl&* (�z��8&dhG�z\A�t?�^"&G!*(! &/u9u�G{a�z92=*�T�4�Æ%b��lh:I&?tf8�_� 8 T℄℄Yg�z%�2*�z92a%! QthM � 7&Ba Mb|&8�Mb�7E~ Mb|%z1Q:Q&"GLBB� ��4��%l�4M92& GL_��z% 8 T℄℄�4f%�4�M)&S=( 3 T�44�M) -Q&0bB�M 7�9��hdI&?�Mb�7E~ Mb|l4+O tf4����92%.Æ2&Y?lX�%! QthM 7�Br �K: &?��{X4����92:(l[Q-A �&*Hb�"1T�44M -Q�& �lT�4��dI&x&� u4M92�8~%&* �l`�dhG |�|�?;�4+O &tye(E(tV�&y"�a��.Æ2:�;tV��Ga Mb|�;tV�??LlX�%! Q�KM 7�9&8���;�<YÆpl%92�3j 7%�G`m�ld��aQ��8�Mb�℄℄�zS�}TT?%lh&Mb�7E~ Mb|O �?O~&�"4��X���92%h4da1&?�Ga�yr&Gl_`?|m��&!*?%dOR��GL℄�%�B�8℄JCz &Mb�7E~ Mb|O ��4%,�&aCzj�Cz,GdGGY?� 5 T�z�4%sg~G&�X!az)n2d&℄J�"l℄℄Yg℄℄�G<�Q?&??bf%�Z�&�L8Cz,Gd&G%Q ��*Cz%n&05�z92=%Czzd*�v`�z92f%CM*iF��zCz&℄℄w'E���%u6z&�S= `m�z� D&�[:�z�% Dduq�0�:�&�� zs$zg�z�&�Cz,Gdf%CM*iF�(l4Z��G�CM?&OJ8G%CMS �? l ��;?&G�e� 4l% 3 $�zu u7E�4{b%&?7�7���4Z�=?t8CM|?W�℄��G��l)u?Fa8%&�u�z�X!S=z%�z�&o2uqCz,GdlzCMy�%�z���z� ��z�,�&f}a,��%CV'&; &f,��f{uhq2�%r&��� �1f.yE(El%~ +r&l��=&nv#%,�� (jf�:uVG,��j�%y��x�4f =&,��a`l�&v&�%b��� �f℄&33u*GL $!? %LB(�� &.b<| &�

Page 81: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

70 L�y: ihUAbx� – `N�; Smarandache +aif�,��P d!P0� ml5p6�℄="℄J℄℄�z&�z��D=z�?7=z�f%8�&b�=z�=�������℄J)~r\"�z��%rS&=z� z��y?&a�z�a`a=z�sd:o� 100 �C%n2d&=z�z���z�%b�&�d��X��"��gl;��? 86M �z?9r~ &64=��gE%�X��W? &?l0�g �; %LB(�?L8G�A%*&v&"�G�^4K�*sV&*.l�d�&�F~n~�{#�6D}p&~E.-}p�S�o1�SnL.q�{^^DNh(ld�"�&�n 100 F&.h�SnL�l�d&.� �{^�=(lU,�Æ64t?d� �; %LB(&;?nDM�?"*�m�-ql℄+)d9�a,P*t-+&oK=(lh&�n℄�{&{�E+�℄Y7:6��Æ64y?�w�?%7�&?Ld"y�J�ky� ��z� �J=&nv���nvM �z92 7um#%! Q�K� &flCY 7 "�V"�zfJ%(l�x�lz&a�G�lF?�M7℄J �z?&%! Q�K%O Y?"*�f$S{.yfw��>M�QYv&�qDq�w�bv 20 ��0M��ÆG,L8?&�2℄J �z&lF 10 h|+�x�y_F��z&GLU # 4d�x��z℄J �z&G`?(fQ�"�&!*?ulTnvMr%��?Q8℄J℄℄�z&bÆCzS ?&l�E=z%�z�'f?& �?r&b�?Tw|GLE�*=z==�%E�?Tw"Cz,GdE(�+GL*=z==�&��� <&Rs(�a���zCzjJ�/u&a%! Q�K%Mb|�'f& �?a Uh&�u�4O O?T�'�G� W�2�`F:Q&�s*5-����%�r%*�(lz&lT!)�h~r�z&2�%! Q�Kl4��I :���&{L8l�z�CzZ� �&W?:Q�?� uqub)+p℄J[�Z�Mb|O �?�{&CzZ�ua%! Q�Kw7%Z�>=Z%&bL8duGL)IL8�z��.y�&*dDUb�!?&�)I;G�GL8?&GLveZ`%CzZ�&aCz�k 30 Æ?yjuEv$!%&aj�?L&�lzZ%Z�����&*Z�Z_|�- +�aj$!CzZ� ��G7�"*�{ =�R�5&wU8℄�{TKo �8℄f&gZ&f��~ &fT>�'8℄a���Æ?"*�Y+ *s*3D{��+&{S�vBw[�/w p�B�zq-n/w&��D^{}p1~E.-�S&o1�n.j{^Ov9Z�=(l&�So℄q)V 29 }$_bjE&��/C(l+ �8℄�ytN3��-`,S+ �8℄f��tX.>{���t~E+�

Page 82: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 71��Æ�4��t?`?"�<< yK&f_n�Æ�8~rCz?&�T�z��E(=z&|a�4���E(��T�z���CzZ� ���4��a %jlh&? G%Mb|� G&z W1G4�a &*q("�T�aG�T44:`r���&?l)℄SLZl�{r�z� gN%2*&� "$RF�"Æ&Æ$�z�zv%v��aGkIgN49���(g!"u~"�z� gN%�bn2�l�{g�GkI49 �&p!*.bu�z�%b��lDbema�z92=&p!*.bu%! Q�Ga`%n2�lDb{���A?{(�/�z� rg%ry&dR4��VvA�w���D�o`wÆ%-A� $&��=%lTBbTN�u`��b��n2u�-vg% ℄&fl���Du��wGL%b�nn2&�uHT{g�0bW�%��� $L^+VU[2002 1&?m#�`�?x�?℄?�o?4&�# �a=�,?O{r�o?N}%td�"a�o?N}a ":?.yk~O&FyN}&1#a���,���zb)℄��?� uq{gN�KA �Z�Mb| +O &�k{r�z�z! N}&,�~rq �|��gN)℄��2003 1 T&?*b)~rq ���DV*Dw��℄Æ&�u?4lz*{g�Gf%%(g)E&<�qY s&�y?��b)H%gN)E&Pu"?*Z�Mb|SG�4z1N�%&?O<�DV*Dw�NH:f��yÆlK3���<�?Qqfl�� aI�l�`IC`I&G�8b)gN)EE�&��u�&oDVx�*Dw�NH:f��y�vg �2004 fh&7E�{1)%l4h4Wb)p℄J[:Q&"7E�l�)Etham�MlT�z�zgNN[d&X!�T�ZK<�w�/�v��b)�4��P4�&/u4�?�< 2 T� 3��u?4lz*�z�z%g<�v%3�<�d �"�h&�`v%=(y0bV�%r~f%�zV�* }a_h%CV �vz<3&?T���vz�3f�:4!l& �=nD�(<�/�lu&��# �7d�%��n���d"a�� �w�/�v�;�℄�U uu�al�v%�D%?\��z�z�zN[dy?&4f{r��[)E%z1d$!�?&�_�?*l�gN)EC�3&?� ��%&u?*�zb)~rq gN)Es�%E�&��&oDVx�*Dw�NH:f��y�&lFS-�z�zv%v�*℄JE�� �℄℄*pM�j<� 1-1.5 h "�

Page 83: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

72 L�y: ihUAbx� – `N�; Smarandache +ai2005 &��ry9P,*�|℄J�z�z%*&w's�=�% FIDICe_92&� 2007 GÆ%�&�n^O�o��*Dw�C;E�bv�*�&�n^O�o��*Dw�bv��GL��T�g%s��l�^J&�?w'l�g�Z�[-�?avzd:a�Ff%,97�&i??Q)I�7��}�zd92%s�℄�&G��}�92�Si�,{%Obs�Z��2007 }�zd92q�� 9 T�,p-r9GÆ�?� ���g &=?`�z�zv%v�% T*f|qYEW{g�G& /�T�z�z%g%ryEsdX �|*�f���/u&?a 2007 1v���zb)%℄�&7=��z�z�dp_&�/ 2008 �ok�lG*8�dl ℄��G�2008 ^&��ry9P,℄N�}�zd92%;�d�&��4s5Z�Æ℄�<&?7��l4z13jhQAM92%;<��<�J=&rm7?lX3jhQAM92%v4z1<#4"&lGL4�<T�/�nT�Æ�� �wvÆ}�%z1<+l��?7�GL�}�WT�<&yX� ;�r%���ry9P,%z1d!*�/�nT�Æ<# "�GL7��l�"?Q<&l��u?%4h&*�/�nT�Æf�:uv%S-℄J℄℄℄7%l\�z�zJ7(�D�?S-v%�D& �/�nT�Æ%n9�(&S-l�f�7kf%�J,l`%<T&� �7d�%KxS���u�lz<�&i?% Qa���z�z%g# �|�aG���{behJ?|℄�~,dd{�Mlzzd92)EC&���;�d�%��{ry9P,%�4z1�C�0&S C:lx27��RK&>�Æ&GL!*?%<�iF$-tC:1��GL%b�&?*)EC<T�hQAM92��z92�d3?&l�z1Ul_ ;��{?~M%z1&;GL{ueQ%X[&��_w�/�DZ�Z��Æ&�l_|m<3�(`GL%℄�4(Zr��l?a��le��h 60 1<�&tC� �����2009 ^&��lT�4dX!*S{r℄J~ %�w'lz*M}h%�z)E&B7u�x�DVDV�O�Saj}�"2008 I$*�&�n^O�o��*Dw�bv�"2007 I$�4lh"L\7T�?%l4E�<T��x�DVDV�O�Saj}�"2008 I$&S=B���z���*-z~ �(��4�4��<3�(B.&/u�3jw')E%}447?8~�?�GL=?\j�W? �ld&L8GL dB.&}44L�Ls�4ph<�3&S=l44`?"*�O{qst+��&f$��?{T�=y'�℄wx*���qr{~M�&>q�{+�℄wx*Yy:�*�&�S{Cd�+D4s9Z�f��y�Æ(��lzqY&?a?Q<3 &PdrgL8?G&?%<

Page 84: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

h-EWÆ0 73� �uTaD&�/a&k<3�(*0b�%7k&*��%<3�(u_a?~ie%�?T�k�=�?N}�*T�%1�&l={r℄J~ �*'R�[& �u{r�?N}%q�#Q& {�a}T�;z m#������=�ZAXn��XBÆuXg&l �DGz&� A�T�%�S�w�ZGrz:f&lu?�k{lT7T℄�ry *_&flfm�S�+%�x*�v�8iaP_f? pSklS "*�o.-qi-�3_&�.i&r.i*7&q.i��b&).ili&*.i�?�B&*6A�Æ?{&�f�:u?LHT�&℄~u(1�%�%�S���z &�du?LHT�%G4�%&0b Xy>�SE>�%B}&*yu��%0Q7X!�

Page 85: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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

Page 86: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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|�

Page 87: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

76 L�y: ihUAbx� – `N�; Smarandache +ai1.2. y-IZUN�Æv�n�℄�e%|*&℄~u Einstein % o1�J&M ?�7l�8H%�℄s -&�A -!*&8HXJ/lTh-/�6aO%�X�a6"&�T6"�(�6���Vn2%uO&i�T�X%�(�~%�6"rS�℄s�-Gsgmi�w��~&G7G�~aOd&sgmi-G�y^�A(�M9&aq�h 137 }%VO#G�_h%%�8H�n Hawking %|7&℄s.S�J℄-/�=^'ZJ%S ([6]− [7])&*� 1.2 Fm�

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

� 1.3. 8H�℄s%�J

Page 88: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=u��r-o/��[ ` –Smarandche +ai�= 77n�℄s -%�;S &8H�Sy?%VO�J�3*d*9\ /8H�℄s�k/lT�6�6"&lTP 137 }j℄s%�Ekl�Æ&H%Xk "* 0�H(ErG%7V*Er�%VV�rjU ��u$%�?*M %�%"a� %~G� "{y℄s�k&6"d{y��0�M��4slaO / " 10−43 ℄&VVu 1093kg/m3&7V= 1032K�� %8H�Xlt`D&.( "�6"*�61�N�_aO / " 10−35 ℄&7V= 1028K&8HrS��z[�&S)xa 10−32 ℄% "�o�� 1050 b�[� X��rn�%mi.S&� <� ou{|l%oÆa`Q&�"/�~�G&po�/o*:uoPU0Æ�&.S%midE(�Æ��l MBi� �7%�J�~f%&hGBi)f/zBi&S?7V=+&)�r%Bi*zBiu<_&��d�.(=i*9il�A zBi% `D%m %8H�=3aO / " 10−6 ℄&7V* 1014K,>3aO / " 10−2 ℄&7V* 1012K,�M"&p���/��*:u��K5�Æ��8H=`m�UAmi&"/7V4G"1010K y:$&mi%S� "Pu|Y%&HLx�uH3`luH\O&Ei� UE`m��TaO / " 1− 10 ℄&7V=2P 1010 − 5× 109K&9i*z9i�:i*�:iu< _&.S��~%�i�=$iy�z=$i&sgmi�kS-GEi(&�~y�i"A%#l`m (�LVD$|pV*8HSh%�oa�ed-)� K;aO / " 3 Æ?&7V=2P 109K&)x0� P 1 ���&(h1GM9-G*�&(z��\,L� ?&(9~%mi�*�i�%iP� � 10−9&"AVV#��/M9VV�~3s~f baO /L� ?&7V=+ 108K&UAmiauH3`=�~ su �uH\O">~%_b�$&{� X, 8H=UAmi�%�V��/:D�"/� 7V#�4G&�i(sk%�~r> +Y #G%Ei&=?�%:iT�&Fy� E(.�`mEi�\O('aO / " 1000− 2000 &7V=2P 105K&M9VV�k�/"AVV�"/8H%0�&�i � +l7"k*lTBB#G%Ei$ P8Æ% X%fKY�liS�4o�l�~*�&"/Æ%5V=8H%0�lK5o�&���i%�4d� ℄o�l�~ ℄*��('L�`�T)��Dr / " 105 &7V=2P 5000K&M97V�k

Page 89: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

78 L�y: ihUAbx� – `N�; Smarandache +ai+/"A7V&yB*yz%sgmi�%i+VP7D��Pq�l?�&"℄s^M.S%�i%�~�= � syr>EiN2yrEi%JV&8H� �f,��i*EiuHÆa%Æ� �&�8HuG��a%&7V�P=*3000K�{� �k&Ei�k#G&�d.�.SFz%C4�2/FB%C4&vua�_�7�#G?&a7���#G%&7�#G?&U7���.S�Uk z%7V�9�q%ZBC4lua9��℄rn�_%3`�℄r=#G%�G1K; / " 108 &"A7V=2P 100K&M97V* 1K�uG��*G�LG:Z 4,D5// " 109 &7V=2P 12K&N6K5#G?L?Q|# %~t�$:Wh / " 1010 &"A7V=2P 3K&�_M97VP 105K�

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

Page 90: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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�

Page 91: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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�

Page 92: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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

Page 103: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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"�+�

Page 104: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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�

Page 105: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�=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.

Page 107: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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.

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

10/2000.

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

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

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

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

384.

[20] F.Smarandache, A Unifying Field in Logic: Neutrosophic Field, Multi-Valued

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

385-438.

Page 108: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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

Page 109: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

98 L�y: ihUAbx� – `N�; Smarandache +ai_�m3p_u8H�r�M%lAsgy_&w-u8H�r�MuHp_%lA|m#l&u�x��$8H�Ml�l%�a�℄Cd=&lT|�*lT��wGlT�v&:(2m&d(i�&fl#G�℄Cd=ysg%u�y_&u�℄#yxU4�%lAV����&=f6�24% ℄aG&lTy�|��}%; �ud[�Tep} *t2t�!�*ZS~=O�+t�&y"�yA*1?%�&{�=f-

� 1-1 t2t�a`wx4 v!�a���=yAÆ&XEl0z`*�{$x~=O�+t�E-O{�&fA{s~�D&{y" Æ[f�vTe&K�a���=y"Æ&l5�(�z`*�"{s{+yAL*=&{$DnE-�K�&{�-~=A Æ�T; dG�|�Ly M�a�%lT; �2012 +b8Zd&�i0&�Z"|$*h^9"�$m#{�?&4f� �T��l; %X!*h{q�{,b�0&�=:Yt�*>�si��E *}�Pu&�{���D�tE? �7u�D% �j&?alg4ZM�: ll7�T} (y%tr&"%ul4�?lgS A %1zw2S �\AE*4lz&aG�?lgH&rg�(FG&q�l4��%�UY?&7l4=��z|�%z �S X 7l�w2y_�}TT?%lh&�S X ;G*�`zf�{A*1?"g�&{�=f-Æ�u|S A �Sy�d;�%lT; &� X ��G a��T; ��P`�&uAD+&{>jj&P2O`�+�Æ�S X "�/u A �f�ia�*�zq{�fA*1?"g�&f�g�=fA �S{zqep�&f$Y\y~=&�f�-~=A&�zqep�&f>\xP-A��Æ/u�S X 7 A _E� Q&w2L��

Page 110: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 994pz&�PL?&,(l4�S Y Q �G%Ir&7G7l�w2y_��l R �&�S Y ,b A a�z^%; �`zf�{A*1?"g�&{�=f$D�={A-Æ(�:zEE&|S A {*�{�fAx$*t&*�>m�GÆ+jhÆ�/ua�!*�zq{�fA*1?"�g�&f�g�={�Æ�S2O-Æ�S�Oln&*�Tj fG��fA.+$&�~N1$ XsK|v Æ�S Y ld�u�u*�{�FL+AL*�^?&&$xnE-��_{yA�t{enQ&_�>Y�t_f�f$/ >{!Æ41z&�2r~��l|+&q�Y?&A ,!g�l4�S Z�j}z%qi&�Gr\&l�I;X?���fA*1?"g�&f�=fÆ&?8nDa��? ?' #ST�/ua}��k�lIf�Pv1�& d;?�{�fA*1?"g�&f�=fÆ�T; &�d�xnd&f px*.*�+Æ��S Z"*�{�?h&f%O*.d{�=d�&f�^*d&{tx*.d& px*.d�=d�+!Æ|S A �℄&y*�/< �l4E(Kq�%��S 4ph&Y?�lg��Q;G*�4�5wf&{T℄P-�-#8h-ÆA lt����{℄0-�-#v hO*B-ÆY?�L8G&�S Z !*G(oK�&}�w2y_�yS��*dD�^lT|(�%; nDa�PE(��S X�Y�Z <��-}SE&a/ A %�7(1T&���=yAÆ���=y"Æ*�fx*=Æ&��lF�`/�!& d=k�fx*=Æ&� ~n^=k& ℄S ��7�%�DÆ a A&la�S X�Y�Z �=&�.(�a�i#���S<�% =��m#y�S �lF2&,?; .yÆG}℄*l℄u.((l�7%; &y .b`(l�7�.y# ��S ,U(l℄u�afT�7&y �`�7 ��&U0 Ddl&�z �afT d & �x�lz=k# ��S �a=�9O=&��u�E1Æ%:�?L3t �^lÆQ�j{K{>K&*KxK,j{*K{>*K&Kx*KÆ&au�A; %z�&*=�9O�`�=�E�Æ&�z^l2r&alT1-u��%&a�lT1-.��u�%&* � �a,?ry�J=&:�4l℄; D*�D(; &�E7S &"�a�D(y_&.y���� Y = f(X) |�`Q,4p℄; %lFOD*�D(; &ua~io?N}�J rm%&�~i%*% �D(&d(�D&*� ENV_�(Æ&��=y(�|(%&u�-�℄E^+JÆ; �Sb9U� {~=Ya+1J�|~=4,b��O6M`+*2_��D

Page 111: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

100 L�y: ihUAbx� – `N�; Smarandache +ai�&g��,D�JY^�D�F*JI&zq*1?2I�O6M`&O6.�hdG�=�D&S�J�Op&�^n M,D>&n��0J+"p<l�℄E^d*o�Ya+1J%4Dp�$Dtg3A-

� 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�

� 2-1 t3-D!��

Page 112: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 101�!"q�4,_p�=&�'"~&~"n&n"�&�"FtÆ%'{&/w�Xi}a9�℄GV �8H�M.S%Æ?'{���=&�℄`ma��"FtÆ&?&��℄`k�%�$�ra��Æ&?&vD&��=%�nÆQJ/=�9O=`�M% T&��Ft%�i�rÆ,�~Æl,�M�e6S�Æ%��Æ&l�'Æ&l,�%ÆvfÆn�K_�4Æ�Fy&,?QJ/�%�n�l� ���Æ��Æ&��%)�`k��%%�$*�V�*:�&�kx�f/)I'Y%S-& �r/�℄�$%�FtÆ&l{}S8HgJ&*B��<-vfÆ%N}&.�x� ('+f%0}�V&�gI�u!g�[lTy��1�l�[&�`k�%!g4}u8HXgvl&�!"q�4l�/w�'Y'&�J',aYa&�JaÆ&*F(�aÆu�℄ Tk�F-&tF(�aÆJ�r,�l�[& �<-vfÆ%!gEvx�)If/'Y�Fy&4}(lA�℄%,?.y�� rt��Æ���℄!gpV%Xga/����r7*�Æ&�d�u�!"q��-n1S&����Æ%J0���V�&�FtÆ.S/�n�&��n&"-Æ&�e%�-Æ,�℄�k!g %rM&�nÆ,�℄; �!g%rM�,?%g9a/"�-Æ!g�nÆ&flWS &DSO&A#Ssl%N/�℄Cd���A!g&f�:du�l}tl& }t Æ&��$*�Hn ,Æ&�℄!gk�&f�:)z/�Hn ,Æ�Q[�%Æ?! *

� 2-2 �J!?�J!?: �;>&-)℄Hn$�nj3Æ,z��.&#�'Æ,K)2OtD�~�& �)=Hn^U &~℄�n9t~1Æ,h)℄Hn.�*7~℄Hn %�Æ, `&�f�'�&Æ,>)~=1Æ�1��1e�+Æ>( Æ7n℄Hn %�Æ,+�D&�*_&*_&Æ,D�bDO Æ7�℄Hn %�Æ,+le&�*_&*_ Æ,D~mÆ&�D Æ

Page 113: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

102 L�y: ihUAbx� – `N�; Smarandache +ai7r℄Hn %�Æ,+TD&�hO +-Æ,__D~SEE Æ7q℄Hn %�Æ,+Æ;&�7�j&Æ,D�Æ3D Æ7)℄Hn %�Æ,+Xk&�_&{TL*_&Æ,*sD~�:$D Æ�)=Hn�M*O&LjFL %+0D��Æ,+tD��1&�~℄ }j*�{T*w��&L_ �{TQ=nj+Æ,*~t&D�S{TQ=n %+DÆ,��+**,℄ Æ�Hn ,Æ�T�Q&l�[/w��℄!gk�%�r(&��℄!gk�#G%,?$gÆ(:[(,�l�[d/w�!grM%_|��en��(%Bb(�vD&b~�T~�℄vA:[%!g& ℄�Æ,ÆgQ[r%��!g�v&f�:�ulA�{xÆ%�v&�*v46�F"%&�4?�`��%!gP`&d�u"&Æ, = { ` }⋃{ �D }⋃{ le }⋃{ TD }⋃{ ; }⋃{ Xk } · · ·= { Æ>( }⋃{ bD }⋃{ &� }⋃{ E }⋃{ 3D }⋃{ :$ } · · ·&��^%!g&p����N!l&��℄%!g�Jf�:�f*y&�!g%urMz� �%�&u(9n℄�%�-&lt�A�-&!g�v lU lD���bf 2.1([10],[18]) _3pÆ>E+�b n ≥ 1&� S1, S2, · · · , Sn D n =��**+b_�,&~= Smarandache *`hE�S S =

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•�

(2) u/: � (R1; {+1, ◦1}), (R2; {+2, ◦2}) D�=**+"&f,_�+*`

Page 114: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 103hE�Sd" (R1

⋃R2; {+1, ◦1} → {+2, ◦2})�lF2& ��� n ≥ 2&?LU.D� n B��n BT)&�"y[-S��Sh�h�&��:l�?�Oba[-����TSh&y�"y F%l��Z;��Sh�`/ n = 2 % Smarandache B6"&SB7a/b�ShG�?_|% z&�A_|z dulABd_|&lFD�*d*bf 2.2([11]) ~=E�3b_�, ∑+jfOS Smarandache �E+&},^*~=�, ∑ &*1��lS6*S&6�(�<��*S�~=w- Smarandache �Ejf+�,OS Smarandache �,&��5&�

∑ DK�`h1&_�+K�OS Smarandache K��Smarandache �+ulAS�Bd�l%�+ ([7], [9]-[11]&[17])&�lT�l.y�Gl, Gl&n�yfA�l Gl&� a��6":Gl&ulAYFl||%�+&�aq8�?=u M�I�%&p�#�ulAS�Bd%e_�q8�?℄"&$v��(�X(9%_|n6"&��&�7�L%�Æ��u��&!*�r�Mu�X%��I%&� $�A�X%�?_|n�+E-ak�V&Uua�℄Cd� .��a&.�ulA�?:% {S n�fS %h-�{�l�[<&�?ry�u"T�Bd�OTBd�J%. ��[%8"�|u��+%pl�?L$!&�y�+"d[HnbE�VSG*(1)�Q=;%Q=,�+;�EY���,(2)Q���LYn�hK,(3)pÆ;S&?&(sp�>E+#~;YZ~M,(4)�-��L�0,(5)s>E��B+~;&8�^~���:��>E+��*����e&y?lnbExD*��H7qK�Æ&�Ll)�#S �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�RiemannÆ~�� z%�E��y4HbE&�"ypl Lobachevshy-Bolyai �+*

Riemannian �+�GL��%�EÆ~u*

Page 115: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

104 L�y: ihUAbx� – `N�; Smarandache +aiL GV: s>E��B+~;&���^����:>E+��*���R GV: s>E��B+~;&*�^��:>E+��*���Riemann �E# B}%Ea/"y.y7l Riemannian �+&?�d

Einstein �*Su`-=% o 6&�= o � Riemannian 6"%�&&fl0M o1�"D� 2.2 } `, Smarandache �+u Lobachevshy-Bolyai �+* Riemannian �+%fl�����D��^v-�7u4D% �lT(�ki*d*{ 2.1 � A�B�C D�[*O�+�=;&z4 2-2 �=�Q5~=K�`h&,;:(J+�<K�&+;�*&��� ∑ S�[(sK�(s A�B�C&~;+��QS+A��f�=K�>D Smarandache K�&�*z *"1$�HK�&+7 1 �K���Q=;%Q=,�+;�EY���Æ*\S&Zi� +D�(spÆ�=;�^~���6P-���Æ�z^42-2(a) &&D�C�E �=;O�&\s D�E �;>�^Q~~��� l1&�-sF�G �=;&∑ &>*�^-s��=;+���

L3

E

LL1

BA

F C

D

(b)(a)

D C E

A BF G

l1

L2

� 2-2 _ Smarandache ?&{3"2$�HK�&+7 5 �K��s>E��B+~;&8�^~���:>E+��*��Æ*\S&Zi� +D�s>E��B+~;&�^6}*�^~���:>E+��*��Æ��z4 2-2(b) &&L D-s; C � AB ��+~; D +��K AE �K3 CD&f-s E ; ∑ &�^~��K3 L +�� L1&�-s*^ AE ��+; F&f ∑ &>*�^�K3 L +��&Gs; A�F +�� L2 ~E: L ���

Page 116: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 105yghh Smarandache�+%℄�" Iseriagp�^�G&Gj��[�y�+_|&hh`�l_��[ Smarandache�+&k*�-�+��+�zA��+*z�+)&0G/ 2002`I%ZP [6]�lF(2a�[:hh Smarandache

2- �#&�2��+%℄�"��a 2004-2006 &XR/�[2�%℄��G&g?a�[:hh`��-�+��+�zA��+*z�+)&�/ 2007-2009lF(2hh Smarandache n- �#&0 [10]-[13]&� �%z1.y)V:�9n&yfl�N}�P�sT5'p_&,rM&"uH�|�uH5P%y_&ulAyKx%Æ?|7���_|,?'{&� Smarandache B6"N}rM &rM"p_|m%Z��?Shu�I��Sa6"=%%*�"_$C?�lTg,4f�:u�-:pCy_%lA�m&Æ*(r�*Er�}A�g9.[-(r�& Er�� �%�.y)V(y9n�E V ulT(r�-&E ⊂ V × V&lpC` (V, E) �D�*lT4G&�DV *B7�-&E *r�-�( &*pAu� G %B7�nr�&:�$$d�G V (G) * E(G) ([1],[3])� /lTWD%� G&=SB7`�*�[:%�I7&}T7&"%r`�*qI}T�I7&"%�t&l�.y# lT�a�[:%�T�k*&� 3-1 =W`���p�� K(4, 4) *��� K6 %�T�

K(4, 4) K6� 3-1 � �{WD}T� G1 = (V1, E1), G2 = (V2, E2)&DHL*Q&,%u�alT 1-1 �A φ : V1 → V2 i# φ(u, v) = (φ(u), φ(v))&�e&(u, v) *� G1 %lnr�f�:&}Tzh%�& uz� z&Sw-Sh��l3�k*&� 3-2 =W`%}T�&�u}Tzh��

Page 117: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

106 L�y: ihUAbx� – `N�; Smarandache +aiu1

v2v3

e1

e2

e3

e4

G1

v1

u2u3

f1 f2

f3

f4

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

21, t

31), (t2, t

22, t

32), · · · , (tn, t2n, t3n)&�e&t1, t2, · · · , tn u n T zf��l.yY"& / z%�� i, j, k, l, 1 ≤ i, j, k, l ≤ n&*qI7 (ti, t

2i , t

3i ) *

(tj , t2j , t

3j) %)t7qI7 (tk, t

2k, t

3k) * (tl, t

2l , t

3l ) %)tu?&lo(

∣∣∣∣∣∣∣∣

tk − ti tj − ti tl − tk

t2k − t2i t2j − t2i t2l − t2k

t3k − t3i t3j − t3i t3l − t3k

∣∣∣∣∣∣∣∣= 0,�^�(�� s, f ∈ {k, l, i, j} i# ts = tf&7 t1, t2, · · · , tn u n T zf�%vBd� �

Page 118: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 107"/G+6"�a)tm,&u`(�&/u�I�-Zf%yV�a�[:% 2- Rnm,&�{�[ S :�>� G ?&Wd%�[A��Iz(/lT 2- +�H�&�B7� {(x1, x2)|x21 + x2

2 < 1}�2ODY[-lTb3EV% 2- +�#D*Y[([9])�� 3-3 =&W`��[*T[�sphere torus� 3-3 A�%/� �lT�[.y|m*lTr}}G`fr#%v-6"&��[.y|m*��rfr#%l$m S&i#HTl$a S =a`m}z�� 3-4 =&W`�T[* Klein B%fr#|m�--

6 6 6 6-�

torus T 2 Klein bottle� 3-4 /�% Klein 0l�K&f�I?=P %�[Æ℄D |a& +lT�[�Iz(/�[&nu p TT[%qx*&nu q TA��[%qx*&�d[�TD �bv 3.2([8],[9]) p�~=Y[*�3 [�(��Y[ ~:

(P0) T[: aa−1;

(Pn) n, n ≥ 1 ="[+�(�:

a1b1a−11 b−1

1 a2b2a−12 b−1

2 · · ·anbna−1n b−1

n ;

Page 119: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

108 L�y: ihUAbx� – `N�; Smarandache +ai(Qn) n, n ≥ 1 =� �[+�(�:

a1a1a2a2 · · ·anan.lT�[D*YE*%&.SfS: +lnn-�trl ?Sv�~7SX� ��uz,z&&.7X� ��uz&lD**YE*�lT�[% EulerHQ χ(S)&�χ(S) =

2, if S ∼El S2,

2 − 2p, if S ∼El T 2#T 2# · · ·#T 2

︸ ︷︷ ︸p

,

2 − q, if S ∼El P 2#P 2# · · ·#P 2

︸ ︷︷ ︸q

.��=%�� p D*.D��[HQ&�* γ(S)&�[%HQD�* γ(S) = 2&qD* .D�HQ&�* γ(S)�� 3-5 Fm�*��� K4 a Klein B:%lT 2-Rnm,��%lTm,dD*54�V�&2�: �lnr.yD�!&"y&qTutte )�"a&2�.y�����vW`. 6 66-

� u3

2

2

(b)(a)

u1

u2

u4u3

u1 u2

1u4

1

� 3-5 K4 � Klein 0RU_*<Obf 3.1([8],[9]) ~=54 M = (Xα,β,P)&E�S^:qA� X +rGzDKx, 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 Yv9�V�&�e%b 1 V"� P .yÆT*e�AT%Hu&b 2 V"�2�%qx(��^&�Im,; alDJV:�O*�l℄��4W; &S=

Page 120: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 109%e�ATD�*B7&l Pαβ =%e�ATD�*[�k*&��� K4 aT[:%m,1

1

2 2?*Y� +sx

yz

u

v

w

--

6 6� 3-6 K4 <Ov/��.y|m*��#l (Xα,β,P), �e&Xα,β = {x, y, z, u, v, w, αx, αy, αz, αu, αv,

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

× (αx, αz, αy)(βx, αw, αu)(βz, αv, βu)(βy, βw, βv).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

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

Page 121: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

110 L�y: ihUAbx� – `N�; Smarandache +ai{D 3.3 .y `&�% .D�y�HQ"�%��)I�D&Fy&N}�%.m,; &�=a�DS.m,�[%y�HQ*.D�y�HQ&�γ(G), γ(G) * γM(G)&S=�D γ(G), γ(G) uu�K}%&rj �`l�℄�℄&*��� Kn&��p�� K(m, n) �D�SHQ�!� 2: >E~=Y[ S&4 G ^Y[ S -d=**Q+B|-6}~s5&_3>E+7b&zE;b��b0&S -d�=**Q+B|6���*��54-�℄; D*m,n zX2���; �D�� G %.D�� .D�m,f~l*

g[G](x) =∑

i≥0

gi(G)xi, g[G](x) =∑

i≥0

gi(G)xi�e&gi(G), gi(G) *� G aHQ* γ(G) + i − 1 * γ(G) + i − 1 :%m,��rj �`l�℄%�℄&*T�� �)�D�m,f~l�j� Bunside &.y4(|"aWDlTqx%(�� G&Sa�[:%F( zhzX.D�2�� rO(G) **rO(G) =

2ε(G)∏

v∈V (G)

(ρ(v) − 1)!

|AutG| ,�e&ρ(v) *B7 v %z�&AutG *� G %kzh��k*&��� Kn���p�� K(m, n) a�[:SG%.D�zX2��Æ~*(n − 2)!n−1, 2(m − 1)!n−1(n − 1)!m−1 (m 6= n), (n − 1)!2n−2.��� Kn, n ≥ 4 a�[:SG% zX.D�2��** nO(K4) = 3&t.

n ≥ 5 l(nO(Kn) =

1

2(∑

k|n

+∑

k|n,k≡0(mod2)

)(n − 2)!

nk

knk (n

k)!

+∑

k|(n−1),k 6=1

φ(k)(n − 2)!n−1

k

n − 1."P$\ - �$#�3�E G ulT�I�&L ulT�-�G :lTz$ θL : V (G)

⋃E(G) → L&��� L =%C` G %B7*rf%z�&k*&� 3-7 =&���� {1, 2, 3, 4} `

K4 f%�zD�

Page 122: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 111

1

2

3

1

1

1

4 2

3

4

2

3 4

4

1 1 2

2 1

2 � 3-7 $#�}T7 - rz$�I� GL1

1 , GL2

2 D*0a+&*�alTzh τ : G1 → G2 i#` � x ∈ V (G1)⋃

E(G1)&( τθL1(x) = θL2

τ(x)�7 - rz$�I�.y�* Smarandache B6" S =n⋃

i=1

Si %lAw-�ISh&�A7 - rz$�I� G[S] D�*d ([11],[12])*V (G[S]) = {S1, S2, · · · , Sn},E(G[S]) = {(Si, Sj), Si

⋂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.

a

b

d

c1

c2

e f

g1 g2

h

g3 g4

a ∩ d c1 ∩ d

b ∩ d c2 ∩ d

d ∩ e e ∩ f

g1 ∩ f g2 ∩ f

g3 ∩ f g4 ∩ f

f ∩ h

� 3-8 N?U�3Y����A�ISh7��%f�#a4 u$. vD&n^"� 3-8 %w-Sh&{�+:8SK5VuGlT℄-/��%#a�-lAk�2{vu&�g&8� 3-8 =HTB7�� 2- +H��&�^# �[��%a#0� 3-9.

Page 123: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

112 L�y: ihUAbx� – `N�; Smarandache +aid

a

b

c1

f

c2

e

g1 g2

g3 g4

h

� 3-9 /�N?IdQ&.yfl�8� 3-9 =wG��%Hlw2"�[0�GlT 3- +�e&�f%��,x&�^�# k�VlT�f%��&0� 3-10.

� 3-10 3- �N?"z$��3�:[vA"lT 1- +�I�u* 2- +�I�&�?u* 3- +�I�&# ?L0 %�f��#a%~v&f�:.yf%lF�?Z�&# n- +�I�([13]-[15])&D�*d:bf 3.2([11],[14]) ~=E- 1- U�%4 G �Q+�%`hOS d- U�%4&TS Md[G]&zqGK

(1)Md[G]\V (Md[G])D-�=MA e1, e2, · · · , em +%&,&Q= ei, 1 ≤ i ≤ m*�3 d- UMT Bd;

(2)_pÆ�b 1 ≤ i ≤ m&�� ei − ei *~=6}�= d- UT Bd MS&KnGM (ei, ei) *�3nGM (Bd, Sd−1)�℄-2&* lT d- +�I�% 1- +�I�*�&�S:E(n�x&lD8 d- +�I�*d- U`�l(d[�Ty/ d- +�I�sg�%S-�

Page 124: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 113bv 3.4([11],[14]) _pÆ~= d-U�%4 Md(G)&- π(Md(G)) ≃ π1(G, x0), x0 ∈G, ��5&_3pÆ~T d- U`&πk(Md(G)) ≃ 1G, 1 ≤ k ≤ d.sg�*�y�%�I6"D*��(`h&"D 3.4 $& �l* d- +�u�qx6"�h/$% Perelman G`T� Poincare �{&/ 2010 m#�o$�?�:��e&G"a�Q=��(+ 3- (I*�3 S3, j��TS &?L.y# d[�Ty/�qx 3- �#%Sh(D *bv 3.5([14]) Q=��(+ 3- (ID~T 3- U`�℄-2&.y&$�L% d-+�I�&�a d-+�I�: ,$ÆSh&*�~1�S:%p_��&)$Æ�+:&℄~2&���z�;w- Riemannian�# (lAG+�I�) :%�&�~ R&# *dS *bv 3.6([13]) � M S~=-�M�(I&R : X (M) × X (M) × X (M) ×X (M) → C∞(M)S~=M�(I+Y8r��f_ ∀p ∈ M&,\�Z (Up; [ϕp])&- R = R(σς)(ηθ)(µν)(κλ)dxσς ⊗ dxηθ ⊗ dxµν ⊗ dxκλ&��&

R(σς)(ηθ)(µν)(κλ) =1

2(∂2g(µν)(σς)

∂xκλ∂xηθ+

∂2g(κλ)(ηθ)

∂xµνν∂xσς− ∂2g(µν)(ηθ)

∂xκλ∂xσς− ∂2g(κλ)(σς)

∂xµν∂xηθ)

+ Γϑι(µν)(σς)Γ

ξo

(κλ)(ηθ)g(ξo)(ϑι) − Γξo

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

∂xκλ )�z�v�(vM N}&�r/�℄%.}6"&*.(a�℄.}6"=&uy%S-�.yXR/f�f%"fn".�"_$(van1h:`hD2O`�-a Einstein &j&N}�GlF�� Newton % 6|&!* "76"u}A z%��M�6"V~mi44& "V~miuOfJ&����` 6|!gM9&SV~mi% 6e"* ((x1, x2, x3)|t)&y `/}Tr2 A1 = (x1, x2, x3|t1) * A2 = (y1, y2, y3|t2)&S 6"R*△(A1, A2) =

√(x1 − y1)2 + (x2 − y2)2 + (x3 − y3)2.�l&Einstein %u`-|a "76"V .Æ&/u(u` 6|&� R4�

Page 125: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

114 L�y: ihUAbx� – `N�; Smarandache +ai/u` 6=%}Tr2 B1 = (x1, x2, x3, t1) * B2 = (y1, y2, y3, t2)&S 6"R*△2s = −c2△t2 + △2(B1, B2),�e&c *�5&{l`m���ma*8Q%*d��*

-6

x

y

z

future

past

� 4-1 �D�5�w�ru` 6%|raO*�) 6&� Minkowskian 6"&S`�%tC*d2s = ηµνdxµdxν = −c2dt2 + dx2 + dy2 + dz2,�e&ηµν * Minkowskian V~&D�*

ηµν =

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

.?L$!& "u�℄�y\�M9ZJ�Jnr2rS�J%lT��&SM ?��ur2rS S�% 0"R��1h#D2O-?�V(�-*�\ 1: 1hDnzSr7Bvi��+7b&:1h�P-`���A|7&f�:�u�` 6|7��\ 2: 1hyDnzSr7i��+~=7b&��:1h�*Y�}��A|7�uu` 6|7&du Einstein u`-|7&�|��M ?V%O�|7�a�A|7d&!* "u(XJ%&�" Einstein o1�J`r&7l%8H�℄se"=&�℄svl0u "%�X�

Page 126: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 115��=U(4fÆ?; 0bf%J0'&&�u1h95DBt^-[N$DnzgB�^-�+#%4D2O-�:Æ�iF"+k�eC,�5�+`�D2O-.(�℄; N}| &�℄��y�!g "%g9�"m$�f9j�w Einstein m�8,?N}=&*+o ��B}VÆlG*#|�ul2_ÆK}%r~&Einstein��u`-%-�&a/So`��℄!g&�*Xio`�℄!gla`�'Æz`ll���&Einstein ��u`-.� �ulT��(!gS &=f "%�rlfl�9f�Einstein ��u`-%f9&uSy/M D%a`%�u(E ([2])*C N}v* ~=t�E7+�V^�-\�7{�&L*�u3\��+\e&E-�*+��I<�V�&�z_u�nO|℄"7l%&Fy&Einstein %�u(E /�%&f�:�u�[aj7*n+Æ�S9�Æ�lÆ?'{%em�*y&Einsteina�u(E sd:&j�$Æ�+ [4] =%�&�~7l��H�V&� ([2])

Rµν −1

2gµνR = κTµν ,�e&Rµν = Rα

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 =

(1 − 2mG

r

)dt2 − dr2

1 − 2mGr

− r2dθ2 − r2 sin2 θdφ2.aysd:&��8H?E &�*�V�S 104l.y. 1&<-&pÆ;L�*&P-�*+E�&Friedmann a:p�1_�7l�zd8He"&�ds2 = −c2dt2 + a2(t)[

dr2

1 − Kr2+ r2(dθ2 + sin2 θdϕ2)],�88HÆG℄1℄* ~y-* da/dt = 0;p|y-* da/dt < 0;,�y-* da/dt > 0.

Page 127: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

116 L�y: ihUAbx� – `N�; Smarandache +aiHubble a 1929 %|#|a&8H�h/0�L[&"y.S�8H�℄s�"&�*<-F"3 137 �~>+~�<-Æ�i�"P$5'y-�℄rmk�V=�a 4 Asg��o&����>z��F��:~��)&Einstein %u`-Ob\� o%*&l~io?luy/$|mi"uH��&�:uo�p(uo�/(uo)�4fM ?�&SL Einstein g�l)3o/|l�,Asg��o&�|lu`-*~io?&7lM ?%�|l -&d�u9n=q3`m%Theory of

Everything&�; l)E(# T�&S%}7a/��u`-uy/9|8H -&*�/_�N\_�2M)&S�EM9uq9ÆÆ%,l~io?luy/$|8H -&*:i�9i�=i)&S�EM9ua2ÆÆ%�=f; N}%J,&�℄!g�;d`m�l��; &k**<-DQ~+E-zq*Q~&-d�=<--S2OnzO*%,�<--nz"3,&+<-+Ub03d�-�D 3 UE-zq<-F"3Æ�i&oOÆ�i >D2O-�D`h~n�-E-�g&��+�~n�-Æ�+D^nzOt�~n�-Æ�......p_p�i`m%j -*T�:�; <D�sd��l -�Emi u97lu+� z% p-f&�Sf p T��(4V%i6"&�e p ulT���&fgI(�(&℄-lXvj��(ZJ�z &j -*8HpS�`�l#(�%1t*<-x"1D~=`hUbS 11 U`h�Æ�i�&,& 4 =�*U^EG+pr�h�&i#B 7 =�*Uf^EGHY�|1&�tISfT��O)%+ 4 U�a<-�O*t+ 7 UNa<-�4 U�a<-w+Z'��� Einstein��H�V&i 7 UNa<-w+Z'���3rZ�V&*�Y�&��<+1`D~=^Q=;HY~= 7 U`h R7 + 4 U`h R4�j -f�:ua��u`-sd:ry%lA�?w- -&�℄m(��Ev`S"fn".&Fy&j -<�uh1_M ?V%N}O�&� (z`Q&!*H ulA��%M -& �j -_z����℄%'{�;&*!g8HgQ[r,`�lT�����=&�0bT�%!g; &un^|#6"+� ≥ 4 %rM%*&*�℄rj%|#�� r/ 3- +6"&a 3- +6"=fm�℄-/?�`��%

Page 128: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

N�`D-`�� `u�-!% 117!g�J&lAy)I%'{&u= z%|#S w-XQ&fl#G`rM%�e!g&�w- 3- +6"&��Ka>&fle�S%*�� 4-2 =W`� K4:�%|#e"%lTki ([9]-[11])�R3 R3

R3 R3

P1 P2

P3 P4

- ? ? �- 6 �6� 4-2 /L�4V�&�e R3

⋂R3 ux3 T% 3- +6"&S+�.y* 1 +�2 +n

3 +&�^&�.y��S/G+6"|#�k*&&$ m T 3- +6"w-&EV (G) = {u, v, · · · , w}&l.y8 Einstein o1�J�� �Aw-6":&*

Rµuνu− 1

2gµuνu

R = −8πGEµuνu,

Rµvνv− 1

2gµvνv

R = −8πGEµvνv,

· · · · · · · · · · · · · · · ,

Rµwνw− 1

2gµwνw

R = −8πGEµwνw.`�%&d.y��w-�vD�StCw-&��=&D�tCw-7F(6"%?%+� m (y�k*& HT6"���`D�z&l.y# Sa�6aO%�`DT&la m = 1&�HT6"% "V~ tµ = t &(

ds2 =

m∑

µ=1

(1 − 2Gmµ

c2rµ

)dt2 −

m∑

µ=1

(1 − 2Gmµ

c2rµ

)−1dr2µ −

m∑

µ=1

r2µ(dθ2

µ + sin2 θµdφ2µ).��[U(4f0fl�N}%℄�&� �%�.yTlT9n [10]-[16]�

Page 129: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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�,

Page 130: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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��%�

Page 131: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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.

[11] Linfan Mao, Combinatorial Geometry with Applications to Field Theory (Sec-

ond edition), Graduate Textbook in Mathematics, The Education Publisher

Inc. 2011.

[12] Linfan Mao, Combinatorial speculation and combinatorial conjecture for math-

Page 132: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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.

[15] Linfan Mao, A generalization of Seifert-Van Kampen theorem for fundamental

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

[16] Linfan Mao, Relativity in combinatorial gravitational fields, Progress in Physics,

Vol.3(2010), 39-50.

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

10/2000.

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

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

Page 133: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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.

Page 134: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

����*:(�-Ryx�0�LIZL=� 1235V |l�!`fK,ze Q�%! QC3%�\7Q4,%g�℄*2�VGm�al��;�2�UiF�`&{1z�t5}y(Z%,{1OeM�k%alDJV:�\&E7z��1��zm�UiFLB,Zg��^K7E�u|l�N&?����u))&��; * �# (T�&d)I5P�z�zrg4"ry��z�z%gulA��K��2�%%g&Fy&h7�|(7%�z�z{1Z%t5&�'lT�KnlT2�u .�T�%&o2��_|,?'{&`�TZ%t5=%�l��7��~ �7 Q�)f%%g_|�N���=&!)ug�&0bfl�JOe59P&\u!)(�&=v��kx�{1AN�%gk%T�%r~?W%gw'n{1�T�&{l#G!)9|A9�%g�| ��YgkO� �nvq�%(7%{1Z%t5�*y&x�s5%gry�N,�%g4"ry&7l�z�z{1(Z%t5GaFP�g9S-h`�z�z{1AN~G&x�`%gryrz�%g5V7E�%g Qt57%��%gw'�%g{gth��G7%gk%��z�z -e_�%g�WO*%g9O7E) 8 T�[& �|�z�z{177f%VD&yi�z�z5V(2%N/?�CdO�q�7E&�f{1%r!7E�_�W-^A�$�&V#jTo�$$k�v�$ $n8&PUN:��|�z�z{177&!)�S�K%*Xf2yBb%���!g:% |ly��K"a�z�z{1=%��j�&ifQ�z�z{1Z%t5l)�u�fKfuÆ%Q�&lq�j�%�i&!Y Æ�%!�G �*n(�*m� (rS�Fy&�|�z�z{177gI�b�fl�JOq�e59P&{l���|�z�z{177�l�|�z�z{177%n/a/fl�r\�z�z5Va{1�OhJ-4%u|��&Fy&�z�z%gryrz�*�xj}-W+w�/�KHYK<�&Sfo�./�-Px��vÆ&*y&%g�A,��K*%! Q�K�JOe59P&!�$_�K��j�*vA7q�e59P �A%k�g&y%N��q�7E*��&bb'��lryrz�|� ��z�z{1ry�m�L�uBUL^$f&kQ}�$ $n8&PU+℄��%g5V7EÆ*}T$z&luv%5V7E&pu%gzd5V7E�1�%gv%5V7E��z�z{1=&��ldS&�V*,~Æ%m�al

Page 135: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

124 L�y: ihUAbx� – `N�; Smarandache +ai�2{o#FKx&2�VGO�hal�2�,B�K�&77l��|l%�z�z{1Z%t5a!lS�*y&Gs0bw'��z�zv�*v')℄�&fl��7��zv%e_%7E℄�&�i�z�zv%5V%|l�y{v%5V$[&~�l�2���K% �*nQÆ�*%m��2�%gzd5V&u�DlT%gu�4"ry�u�GÆ%Bbz1�_lHM"&�z�z%gzd5V7E �G�℄J℄℄℄�zd�z92�%5D&`�|� �{1Oe%%gzd��J)�0GÆ�*y&Gs0bw'�G%gl�sg5V&*�z�z%g℄��|��J�%Nzd*��,��0)l_�%g92&y<s%gry0b,z &u|�y����?�7-�&7v�!)��j~ th�%gw')f%?��N[&�x��M�z�zG$-R��z�zG�y�?�?��N[d&_����jZ� ���MZ <�n<?)�l& ,n��l�%&(%�z�z�l�Mv&fl�7%gzd5V%7E℄��P�B�XUId7$&k�v�$ $n8&PUu���%! Qu�z�z5V�KM�K��K�Æ%l~BbV��℄&du��j��Cdbej�*�r�-vÆ�%(oV�&*y&Gs0by�NO7Ev;!)*[t&7llAGrR%%! Qt5&aG%!�Gnv Q%�o&�-S�_bvÆ*�i�vÆ)%*,bVDT��z�z�eb*�Æ��*�eNÆ)�z�z�;%X�}; , l�7%!ty�an�y_nSGj�y_&aabe?|%N%th&n�z� thb�GYg95℄�&s℄S7%!ty?'%j�y_���l�V�&*F`�y�)V�u&g7l��2���%g&t(j|l%%! Qt5& T��*�eNÆf%�gMVD&0b`Sf℄ 7SGV�f%`iÆQ&fl5D`lA$-?�q�7E0b%%! Qe5&��%%! QJ7��(%zdO~ &{laG%! Q�G49�ANrm&2��ydl�%!�K�Gv<A�%��z�zjJ&�X�eN�yÆ`�r��b"hm� (rS,l`�z�zjJ=`m%l�ak&v%*&��UA "���N4 & #%v%-3S%%! Q(j�l�2�%!�GN2=`[�z� [Czd�m�z�n�z� th�5�*lA�3%�,QJ&/u�z�Czd`m�l�Ygz�%�n�S�"C,SPt&,�,S+nLt~ww6�xvÆ%xm��*y&0baG%! Q�Gnv%!�o*rR{!%49&7l%! Q�G*E*|Æt5,z

Page 136: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

����*:(�-Ryx�0�LIZL=� 125 *tyb(B7�(=k2`%! Q�G%*v%**zf% *&A�T�(N%*%&v&�)�z�L^5�QVk�v�$ $n8&P+k5EU&N�_�|�z�z{177&{1U�Oe�Z4SA��v-�Æusd&��=&%gw'{k%�[`{1U�OeXfu|% ���&SBb(E09s&�l�V{2_0�w7�%g�2���K%�z�z%gw'&:?/{1ry�*y&Gs0b7l4��%g|l,��UV�{�Æ�~ �Zg�`�(�a�%�z�z%gw'e_�fl#G���z�z%gw'e_&�x�%gw'N}%gryB�r~&GÆB�r~%gry,��0�%N/!)�%g*dGu�z�z�dEl%n/�*y&�z�z�dbu|I���G!),�r~&OR!)�yuy℄�&z &**>�z�z%!.��%!8172%rS&.a�z�z�dEl�z�z��ATth&ng����af,%!.�n%!81j&g%x��d��ATthf%AT�*Z"2%N/dG&�z�z�d�7%!�K7l|"%gx!&� z�dG8�,z &%!�Kd��o#P7l�Agxt5&*H`#G~ � Q7�7�*le%{1Z%e5�("h�&�L^7�%J�U{'& V�1�v�$ $n8&PUQ}w�L{gth*�G&u�V%g.P9ry%�bn2�"_$L^7�,d=fvDh.�hiVKo�&�z� th�~o45VFB� ��|�&M- 2009 1&��{r�z� gN%th"SL!)�jth$uq� 6000 �|+��z� th"j�z�m�d%�Kx&����F+���D%%N���zd�N2���NT� gN�l�thaNT� gN?&_��SGv�lmi�Z[,�l�[&l�~r%�z� 9~d%�+d&y(�rJ7*� �(���m�|a&0bw'N}�z� .P9ryt5&y�a�~*�;ÆÆ<sq�7E0�n2d&v�95�z� th�~,z &5�{1�X�P%El&ng�z� thryel%p�&�ZgO*�eO��ry&a8�z� th�XÆXFÆ,z &**zYg3�&�`m%�z� {1d,5Vf%N}&?19uvA��g�8�,S8DÆ~ el&b�f�z� hQ%|l�

Page 137: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

126 L�y: ihUAbx� – `N�; Smarandache +ai"m$L^i'QV{g^K7E�b&$�j&,b&$S? o~e �*y&0b{1T�[`{g^K7Ef%_|�N*1��z�j��?S)℄&$ m%G�ZgÆÆ&aq�?�℄?�v?�~ ?℄Zg�E�z�jg,S)℄��,z &ngG)?��N}thf%�z�j��#o��o��%)℄,(�N2�GaG)?��E�z�jZg%.%(N}&aG)E>Zg)℄r =E4�z�jZg&yv�q�7Ery&�Cdaa� �z�jZg$g%G�lgS�2��z�j{g�G~ 2007 &E���r����ry9P,p-Ær���z�jZg���G(g��C� %�D�*��z[(g��&�f℄Mv�&/ 2009 �2010 G`w'�}z���z[(g��&�&2_u( 18000 f�m#��z[(g��" &*�|�z�j%*d�lDsd��*�V�z�zrg.P94"ry& ((g��&�LL k& t/aG{g�G(g49&d t/`{g�G*g~ �*y&0bfl�7l�z�j�G��z�G%(gd,5V&�-\`Ll_�{g�G%~ Mv&*{g�GV"5V��9E>5V�&(5V* -Q|Æt5&{lV"{g�G%(g49$-Cdry0b�3��z�jZ�^K7E7ll#Zg��G�(g!"��%CzZ�^K&u�z�z5V(f℄%BbV��Z�Æ*}℄&luCzZ�&n�z�?�nv�GCz℄�,pu�z�j -��v�E?N}Z�& (y�z�j -��v�E?f%_|N}�ANrm&Cz�JsHÆm�al�2�#FKx�l�Z��GCz℄�?y g�CzZL*n2&b �z�n�z� th&GiDbCzVN�,|T~Z�N2j�!gZ�>��Gf%�s&&v*l��z�7CzZ�mxX�q���*y&0b(�I�(�N2�J���V�y:�`!)o-CzZ�>7E℄�& CzZ��%|l~ &�fl�r\Z�hJ%��,z &7lCzZ�)E�&(5V* -Q|Æt5&yLQCzZ�%~ &��pS(g!"*gN49)℄&�VCz5V#y(f℄�

Page 138: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

����*:(�-Ryx�0�LIZL=� 127�z�zul~f/(Fp%5V&S -N}:?/f/l)ulT �%Q �*y&0b{UT$[h7�z�jE?�,N*���8AT%Z�^K&*%g,?� aaZg -ngC�0�"P$L^4Æ%gk%u�V�z�zjJ�r�nv�7�z�zjJ�M�#P%BbV"&z du7l�SA5��(AUXÆ%)In��fQ& �z�z{1Oe��C�l)�u_|% -7f/N}�*y&0ba%g%Nzdsd:&w'(yg�Z�N}�7l�z�z��C�t5,�K��y`�z� th��z���z��Cz,GdZ�){1Oe��G%��C�℄�,aysd:&�y�z� th��z���z�*Cz,GdZ�%M�p�℄�&`p��f%|�&�GÆ�Mf���{�%%g! �{1Oe%*,z &%gw'b�p`�z�z{1Oe%k% Q&fl��7%gk%')*Cd Qt5)&yR�%! Qo~% s���$ $v��1k�$ $n8jTo�U6F�z�z -e_Æ*}T��&luv%sd&��z�zv%5V&puq� -sd&0bo-$|q�?���-�~ ?*Z℄?)uy,?&l)E(# g�?�%B}�*y&0bx�!)�D�Cd�R�Yg℄�)fA�l&El�z�zZ~N}s^&"!)�%gw'�,Nth)DMrÆN}3 &hR�z�z -7%gry=l�B�3 N},ng,N�G`�z�z -*%g=%�7; f%N},z &b7lB�7kN}t5&w'uy�;Z�`����z�z�;rS%l�B�7kf%ÆQ�d�*N}&rÆN}ZL,�%gry�6�L^F.-QVk�$ $n8�vU+℄b��7k��Aryu�%,?ry|`Cdry%lTBb,�'{�*y&b��m�,?��&*�W��G �y�z�z��p�N}&8�'3�z�hJNP"�z�z &y<s�n:Fg9AqmÆ%Cdryrz��W��a�z�j=%���rL0 4&0bN}�7E%r~Ff&luU℄�>7E&*�z�z%g5V>��z� th>��z�>��z�j{g�G>)sg�>,pu:i�z�j5V�:i�z�jbe�L*�

Page 139: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

128 L�y: ihUAbx� – `N�; Smarandache +aiz�z&v%*� bL�L%7E,1u{gth�z��z~r�a�8�_|,,uw'N}5D%g|��7*|�,z&7l%g�|�5V�%g|�,zrÆ!)r~�+#a&�LV�%u&%g�WO7E&℄~u:i�z5V7E%z &�G��z�z. 6(℄�& �'� *�K)h>��r�&v%*aa�tjn2���L^�-QVk�$ $n8)U2.f$EH!�1U+℄�5�z�j5Va?�uqf℄�_f& ?�CdO�q�7E&A�*K;*A?|%*)�[r\�Bb��&� 2_%&�L.bla �z�z&T�%�APu1��znu�z�jXWf*A)Q &|�2�|��z�z%ga�L�=%#���v�7>�~r&v��(y&�Zf27%g9O�u*l�Gej��L%��.Æ��j&Æ� &�fZ!>��z�j~r�6[(y�*y&by7ECdO�(��+e_*sd&�J%gMf��%9Oe_7E&flaG{g�G'{!"49*,?9O4℄&?19u�z�j�GaCd:%#��z &*�k�z�z5VaCdO�oK9a*M99a7E=r\%��&b!P��[ZvS/�ss�'S�Æ%El&;l�Z!�I{1Oe*%g{gth��G%gfrw�gfG y�&vr2%%!�!rhsS )~℄�&{l�l�z�z5Va�L�=%�℄#��o:F�&_|h7�z�z{177&ufl�r\�z�z5Va{1hJ�O-4=sd(��&�|{1Oe%**%! Q� *&�i{1r!7E%l~���*y&0baV�?�q�7E%sd:&w'Z�XÆN}�-"&�AJ!)�%gw'�Yg1T�[%u|(&fl��_|h7�|(7%�z�z{1Z%t5&*?�CdO�q�7E~`Z�dn�

Page 140: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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

Page 141: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

130 L�y: ihUAbx� – `N�; Smarandache +ai_��$0�v�U6�K* – Y'n8^A&�*�$ M2ODw�4T�<-�z�j -uy/�z�jZ�%�lF(%B�&SL`�z�jlF(:%ya/�&�S�jS ÆQ�����z�j5Vu{1q�=%lAz�(?|5V&Sf℄�a �{1q�&0b{A{1q��%�"_$n8^AUz 6�����$!&{1q�u,{1ANahJ-4=Xsd(��%lAq�e5�{1q��`{1ANahJ-4=%��&a/S.dD�*+S.*.>��); &n'{1ANQT��{1q��(yd,Tsg℄�*�{1Oe%kO(�{1Oeu$|q�%sgwGb4�a{1q�n2d& +Yg�SGq�w'nT�&o2u +Sf%��(XÆkOÆ%{1Oe&�kkOq��k3�H�k?ry�k?P���{1y_%�)(�{1y_%�)(&,{1jJ%F(S.��q���%N�*���aI�:u�)%&E()��℄Æ&�?|��nf��r�&"uH�)&|m*{1?Wy_&{A)�?WEl��{1jJ%z�(�{1jJ%z�(u{1q�%v(.M&|m*z�Fo*z�Jo%|l&�iU℄{1Oe!�N}{1~G&ÆQ{1�W&�T{10�&OT{1�l&v�{10��SuO��{1Z%%v5(�{1Z%%v5(u{1q�%Xgb��v5(u7l*�7{1q�jJ�377%v%e_&SL{1lv**v}��[&(�uV"{1Z%�J%b�?|77�"m$�$ $n8UzN℄{�z�z5V�*?�q�7E=�OhJ-4�aGq���V"~r9~%lAz�(?|5V&a�fq�e59P�)>*�|�z�z{1e_&A�*K;*A?|%*)�[#���Mr%Gz& ?�q�7Er\�Bb��&gu#G��z�z{1&�l{1"{1Oe�{11e�{1v�*CdV�)sgb4hG���z�z{1Oe��z�z%{1Oeu�z�*�z���e&�z�u 1 T&�z�ufT��z� thI��z�%`r,�&� �z��z,Cz,Gd"�z�nvw7&I�S,�&5��z92=%Czzd*�v&

Page 142: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�.Q`�Wmr(Ryx�5U�=�� 131�z92f%_|%CM*iF&�GCzZL*�+=z==�)gC�0���z�z{11e��z�z{11e�?|1e&u℄J�pMn%N���z�z{1v��*�`vxl���-zv����z�zv����z�zvf℄nk�)�|��z{1Oe?|%*%v%�v������D)&u{177%CdV�����%! Q�K�%! Q�K�*�z�z{1%*v�&nv)h�z�zjJ=%&v%*&SjJu+G{177%BbV��"P$�$0�v�ka�Sev��*lA{1q�d%z�?| -&�z�j -�g�5{1q�%℄7*�%&pS�z�jjJ%G`qY7�v,Sz&8v��(Kx��%�(&S-��=k -�Z℄?�� -�fC|�ÆQ�.'(ÆQ)�( -#G�z�j -&_��l -,��z�jf/&�P99f&���{f/=Q& f/=�&�4s�4s�\4sÆ%7E�v���=&{1q�usd&��=k -u,�&lZ℄?�� -�fC|��.'(ÆQ)luk�=k℄�&d.(�^�G%�z�j -&�$-?�{1q�f�&fl.�/,��z�jf/�m��$0�v�Ue�$$ – �^ABh��>$'�"_$�$0�k_*�{R�L��z�ju��z�(�lfm�jz%%lA��=k%*&�A=k%*ux��z��zbLn�z_� r`bP_�&�z�5�z92b�s��a�zM- "j6?�z92f%bP&�z�5Cz,Gd�G%CzZL*�+%=z==��D=z���Sr`=zx$ "N�$&�?�z�*=z�5�-zPD#%�N&�G=z~rfm�jz%��l�j�J.yÆGyd}TL[*Z 1 Wh*[�Wh, <�$ $Wh�lL[ObSLydr~*��z�r`�zbLn�z_� &r��z92,��z�5�z92%b�s5�a�z92�D%�zM- "j6?�z92,��z�w'�z,�Cz,GdCz,��z�nCz,Gd%CzZL*�+%=z==��D=z�,��z�*=z�5��z92*=z�%�z92gE [-z�

Page 143: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

132 L�y: ihUAbx� – `N�; Smarandache +aiZ 2 Wh* �Wh&<0�$U 5Wh�lL[b��z�*=z�n�Mf��El#%-zPD�N&fm�jz%&flfmaGq���V"~r9~%�z�jr%���=&�z�z�Ju�j#l&M�#Pfm�jz%u�(&Fy�z�ju�z�z�J*-z#P�J%|le&p� �Æa&�l&aGq���V"~r9~�1 fh&�du7l�z�z5Vf%�j%n/��jq��(}A&lAu$|q��&�lAu9|q����e&�j$|q��u{�j�Tem#%)Iq��&l�j9|q��lu,�j%*aCdq�e_=X%��y�"y.S%q���Fy&�j�#%CdhJQ>&�j�QG&z &�jS %Cd0�<sJVQG&�j�dQ"��j$|q��*9|q��.y�d[}Tbl�;*�CW = CB − CS − CP

CH = (CB +∑

i

Ci) − (CS + CP +n∑

i=1

CTi)�e&CW - pmO%�j$|q��&CH - pmO%9|q��&CB - �jA;i�CS - �jGg�&CP - -z�Q&Ci- pmO%4 i ~Cd��&CTi

- 4 i T�z�nz��%)IGg���j�%r%ufm�j$|q��*9|q��y�O&��D<s max CW * max CH %�z�na�8*=z�nG?a�8�[-q���a �r��&�e%r��&u,lAM>%��~Ho�lT�4Fo�%<sJV�r�%��u|ml�4%ku~%uO~F X%u~%uO~�ar��=&ku~ulM>%��~&lu~lu<sJVn����=kf�:uA;P�d%lA�O=k, �S���%�O�����=kulA�O=k&�j�lFy �S$|q�� CW y�O*ja&�`/nvo2f%�z%~r&lb��j� �9|q�� CH y�O&�b��jS aG�Cdq��l �uaG�z�n=z����"m$�$)U'7):%a�):��uap�nf��)%`�&j�`�% )uWk%`# )&~`(j/k%� %lA (%*&Æ*-���*-���}℄�F5-��

Page 144: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�.Q`�Wmr(Ryx�5U�=�� 133�&,�7�{kIj�`r7SG�7�T �G��n#GpO&F#S `pO��(j�lz���"-�(��$u,�7�a%J=k Ev�GP�(��&.�x�z�f%�����-=(lT4P %ki&�W5n3&a�Tki=&��v}T�-k}�ln�z&�" s&b}�=2>(lT�!z&�lvOEvWGLDz�*y&'*zd��}�&z aa�d[�Tn2�GL=k*zq{ni{+*4Pn&{>.�SlB-Hi5)nTF�,�zq{*ni{+*4n&f{{� {S2�E��ET& 10 ~E,zq{T�=nLn&f�nL.�ET&�*. {S2�E��ET&{ 8 ~E,zq�nL*n&oO_��L.� NN+ hTss& 1 ~E��^&�v}T�{kj�y�Orz`r&Æ~=k�`ky(j%� &�U>&** `�U>lk U>&l8d 10 !&f� 2 W�����z�j��z SLz���7-���}A&��=&�z�&"%��uz���&l�z�7=z�&"l*-����1. [�Whka�):�z�j�J=&�z�zL[`�/z���&"�z�s5%�z92=�D%-zn2�=zn2*{1n2~G�D&1�(t|l/�z�jf/&u�z�s5�z92&y��z��7�z��&�D-z#P�l0bB7&$%r~�a�Dk�n2& ��z�z� &�V�yd�Tr~*��W�%=�JV`�z�jS %�|��z�jz���=%�W�"}�Æ�(wG&lu{1aW~G,puT �z�z�o2bÆ%�W&k*z%��e&{1aW�D��z�jz�%y�JV��z�jrzfm%.�(&ld��D�zz%&℄~u�z92bÆ%z%7f�z%%l3(&)Iy_ �j�����jq��`�z�jS %�|��jq�rzui�z�m#y���%rz&��7�zf�aGuS-&�{�z�fo`rf%k�=k��z�%�A=k�Qa�&�z�5E�� )�Qa�&u�2�z�j%q���d�Q",z&&�z�jq�� a�&℄~u ���z�nkIfoz�&�dhG�z�aq����[� A��k*{1:��(�z�;���+�*Czsd�%�l&{#l: �z�%q��a�&�E(S

Page 145: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

134 L�y: ihUAbx� – `N�; Smarandache +ai-�zf�aGQngS5kIfoz�&d)I�3�z�Ev5SkIfo5Ez� )&.�x��#&numz�'z)�[m#y�Z�+&�^ j/aGCdq����Cd��f�`�z�jS %�|��z�ju{1q�dlA�OhJ-4%5V&7Cd��f�V .Æ&i#�z�jgIU%3f�JCd��f�&ZP7ry �A%1?%���v*�[% N��Ob|ma0bk�=k�i�℄Cd7k�ryu�A%�k�=k�(gf��%z%&℄~u`�J�℄S�7ry(�f��%.>�gf��)��z�JV`�z�jS %�|��z�ux��z���m#��%��z�Qf&=k12�&�z�m#%���Q"&=z�m#%���Q+�Fy�z�j�J=&�z�fm�jrz%�vs/�z�%z�&=z�m#��%y� )lu*>�zz�%y�JV&y" uz�lu�=�&"%-�&�mx�z�a��Kv=&'*z"/Æ~a z�"yG�v}�&i#SE(mktd&flfm��}T�P!z%r%���z�j=&"/ ���:�M R��℄&�z�z��J=k��a-�n2�Fy&fm�z�j��y�O&5E%=zn2��F-���%g�n2&�i�z�"-�m#%��>/)/SRl�z%��&z �p{1 Q&7l{1M�C�t5�2. �Whk'7):�z�j-���&��z�7=z�gE-z&y�#%-z�J=%��%*&S)I�|�z�jz%%fm&ObB�-zgE���Æ-*�lÆ-1Tr~������-zElEl��z�j-z"=z�%�z92*�z��Sr`%=zx$ #G&25��z92�D%f9(b�*n2&�-zz%��~�9~��DnZ[�#%Mr�27*�l�&Pj *T���%�v)y�=z��z92�/�z92b�%N�f%El&��=&-zz%*�~�9~�#%Mr�27*�l�&Pj *T���%�v)QJ/�z92&n=z�%�z92�/�z92�D%N�&��Dn�Z[lQJ/=z�%�z92&��z�a=z�?0z% 0b�S#0%pm��-z#%=%���Ob(1A&�-����- O7���*41�+D)&�5��r���%dniVf%��Æ-��-z�lÆ-��`.g~-z�l%~ Æ-��AÆ-�y=+y�-z#0*rz&5�(j/95n*>�l%�JV{l*>�`�l0b#0%

Page 146: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�.Q`�Wmr(Ryx�5U�=�� 135pm*y�rz%ElQÆ--z�l�"P$�$Y�*d*Pv��6F\L�sR��z�jC�,zÆ*}℄&l℄u.y)I#~%,z&k*����Q�gz)&D*ku~,�l℄ �)I#~&.�"I ℄n�℄&D*ku~&k*p��o��?)� /ku~&.y���?�v�O&�`/ku~&y(�v�uXR/�7 -`�=4f%�7&57y_f%=k��A�li#�L.ylF(27l�z�j4"%7y_&fl�D�z�jrz4�7f%=k��j4%�7(ydl�(�v*wq 1: ^[!qqY� %jau(i��|�ÆQ&SLS,z�+|��t��9fMv�S ÆQ)��jrz0� z&S� ��%|�,z z&�lF0nS���q�,z�+& �ÆQ.>i�`��0b9fr~&fl�D8℄.>,z%v�|'&�l� �J�B�,?� &UB��j�%qY7� 49� /fzB.i�%.>�j&.yniz|��+f%qY�7�wq 2: sRq�D4 A�B "��y_&f/=(lA(��v&��=v&�A�vu��y>�Yz�=ky��7%lA�v�3�%�=v(�4nf4 0.618v�pÆv�Fibonacci �=v)&��=y(�%u 0.618 v�wq 3: 1Ef3q`v� �x��k,z)IiF7y_%4&.��Z��e� t5�D4"�7��A� t5�.yr\Z�Zg�s&,.yr\Z��e� �s&{l<s�j0�rz�Z�� lF��yd�J: �GlZ�� w�� wGG��*T�&yta(�� .y���;�v&5���b��dbÆ%El� ,���� n&ÆY� �&�T� uyr~&ob &w'[-&N}� zd&y�Geg,�Z�nT�?g�qY*zd&f%� ,�epZ�� S & �Z��e� �0&ay�� �i����V�&Z�� f�:.y|m*�=����E( n T4 A1, A2, · · · , An0f%�7&lZ�0b�GlT n×n%�=�� [aij ]n×n&�e&aij &1 ≤ i, j ≤ n* Ai, Aj "y_%� �+&� Ai > Aj aij = 1,z&&Ai < Aj aij = 0�V

Page 147: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

136 L�y: ihUAbx� – `N�; Smarandache +ai�&�eD�%��aO`Ct:%C�* 0&t<s* aij = 1 l aji = 0 %n2��Z�w�� n ≥ 2 & zZ��G%�=�� lD��l^&y 0b8F(Z�%�=�|ep*lT�=��&fl�Grz4%�7�ep�v(}A&lAu���*ep,�lAu�Æ�*ep&�-3HTZ�lD%Æ+ p1, p2, · · · , pn ≥ 1&�?�;��*[bij ]n×n = p1[M1] + p2[M2 + · · ·+ pn[Mn].�zS �7nR/�zrz4%�7&?�uj�%sd�f/=&arz4�7sd:`�z�7(}A�v&lAurz��v&��rz���.�rz��"rz"�alAV~y_$*frz��v,�lAu�:�gv&�a�j4*�zS #G%(��:&�D�zS "%l*|m��y_%(��&fl=k�jrz�P��$0�v�d�Qn8.��m�kFUmR7$�:�=zS %.�(ÆQ&,=z�#%-z&fm=zS %.'(ÆQ&SL=z�#P�o*z%.'(ÆQ}T�[&u�z�j�J=&�z�k�=k=z�&fm�z�jz%%ja&duÆQ�z�jS & ℄�z�jz%.fm(%lA�v�"_$)$JkF1HC�=z�u�.�&fl�k#Pfm�jz%ulT.\%; &B�=z�w'Sh��GSh749�E �E℄�aG�~ 7�A�o�%N9~�#�l�o&y��A41��o)Jfr~&ulA8"%ku~�fC|�?=%Sh�Je"&x�Du~", ℄~uv�Ev)I#~u~%�aShy_, flY"e"u���&y�*+A�%'{, *A#*C�Cd,?&℄~u~ ?=%l�,zaa�lAs/|�?%h-ÆQ�v&�A�v.(�/`=z�f%.�ÆQ�Sh�J"#~e"*She"wG&S=&#~e"��%t(�Jw*

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~&"

Page 148: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�.Q`�Wmr(Ryx�5U�=�� 137%y_&u�J,za�Jku~:%i3'��,AY - �S,z7�Sku~&"%y_&u�S,za�Sku~:%i3'��,C - �J,z X %O+~,D- �S,z Y %O+~,She"��%t(�Jw**V = GU + HV + I,�e&U - �Jku~,V - �Sku~,G- �Jku~`�Sku~%�|"*{1Tv`%N�o%�|$,H- �Sku~"%y_"*%N�G℄497SG�Sku~%y_$,I - Sh�J%��~&z��a�J=0�dTw%�Æ�n%g�2;*~r℄7&��h7��-�C�*,�)�J&,?- 2�DSh�J=%�� AX�AY�U * V&uC�=z�.�%sd(℄���e&����� *�Drzuja&li3'��7�z�jf�uS-luXg�:�1A=z�&�℄65h8�NS8*%N8�"m$7.kiN.>n_|a�D%n2*�D% "�&�G�D`�%�oD*.>n_|%.'(&�e&�Dn2,.>n_|i� %Tvn2*℄Gn2,k*zl>�zl"$%`=&aG5b�*a5�:%j&S.'(|m� l^&Fyo2,a�D%n2udD,�D ",.>�D%�G N ",=f.> N "%o�&.>`mt�%:&8o�& �%&S.'(d=�k*&�`6%`=*%j� 5 %`=&�tuzl>�zl"$?&?�`mt�%:&d��,�D`�l,.>B��D%S2� %`����V",z�Fb�.>`�%f>*S��,z%G+&)I�| .>.'(,z%G+�.>n_|.'(SL.>n_|%y�(�.+,(*E�.'(1�b4&�e&y�(,.>i�Et�(ni��d4&�{_|℄JCVQ"& +.> .� 100% rSt�,.+,(,.>rSt�?&�k�tBO2x�+Gn+,�bt�&�u.+,(,E�.'(&,E�.>n_| &XÆ&$.>%|i�(*|��(&5�.'(y�O%b�f%E��.>n_|.'(C�&.yi�:&,zn ",z&��,zSL.'V�\&���Et�℄� "���\j "�(V)&�kQ"&* .>�~* N&S=i��d T > t %.>�~* u(t)&l84.>.'V.D~|m*P (T > t) =

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�

Page 149: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

138 L�y: ihUAbx� – `N�; Smarandache +ai.'(zdu.'(℄J7~ %sd&l&ua -,�dx�pS℄J7~ f/qYl5D&�=fN}���ryy�qY%�4 ℄,E� ℄�7%S &lFÆ1T$z*�.'(sdzd,Zg.'(sdzd*(.'(b�%.>zd��e&.'(sdzdu,`.'(℄J7~ �(��,���%sdzd,Zg.'(sdzdull�℄.>e�%.'(zd,(.'(b�%.>zdu,UA(.'(,z)b�%�e.>zd�"P$)$Y�Un8��=zS %{1V�sduCdM�e_7E, �{1OeyM�El*~��hr%Xg�Mf��Elb�`rOea`rjJ=bMf& �1��& Nt&f%��z�,�7�%iÆj& yA�G�*Cdj�%�lQm(j,����Q& 6kbP&LQ5v%�D*�r�%PD#%�N&z &a�r�PD a�n�EP?1|~#rSB�9u &�nMf��%b��D�r�%Æj�N*j &{l$uU��r�j��{1M�e_&"!)M��YgM�*T�M�1�ÆhG&S=&!)M�uCdM�%s`&T�M�uCdM�%sd&lyy1�yjR*y��|o%uYgM��{1M�e_7ESL*�M�9O7E,�v%v�7E,�M�C�zdE4,�M��W�L7E,�~ � Q�%Ne_7E,�h7M�{1t5,�)℄M��g*M��o,�h7YgM�Tv)�

Page 150: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�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

Page 151: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

140 L�y: ihUAbx� – `N�; Smarandache +aiv%e_=&5�9QSh%0}y_&S��u�D*. T%&��=%y1a/8l�v%� �Tv%e_=� T&l ��l�v%� T��v%&*v%"uuHp_*�|%�g9����z�zvf℄nk�"yd(D�nk�$=l�℄~xw&�r5�S9�|'%xwf%vQ&yORT�d�=��nk�u�n99�&fl�| ��z�jf/�_�L�Id.��|Æ�nk�4,n`�z�z%! Q�K%(jf%Æ℄&�D�ovRqm)9,S�#�9Aaow�/�DZ&_o℄*Æx�(l+DVw�/�3I4*eNlC�ovRDz�A~��6�Q�x���(YW�"'�g���v0,S& {jE+� �D_-`w�/�3I4*eNÆ&�e�jE+� Æ&,�NO℄g*�WO�U�Ew7E�?xZ�q!��j�8N)�KaSD(j�D�Eth*D�Gs5)��EÆ=%�E� Æ&���EÆ�D=3(�z�z%! Q(j%&nv`�z�zjJf℄ Q&z&&l �`�z�zjJf℄ Q&*�NO�K#%(�nuS�(v%o%��EÆ�D��nk�4,nz �D��H5�n^��qm)9,S�#�9AK�X�+w�/�DZ��H5�n^��-`,S {jE+� �D&_w�/�3I4*eN&�vCsw�/�3I&+OvKS��H5�n^��_,�[,S-`w�/�3I+eN� �D#-jE+&�,jEÆ&��=&�#-jEÆ=%�#Æuu`/�NO�K(jÆ℄lQ&�o�y:2��`!)`S�z�z%! Q(jÆ℄ z/�NO�KÆ℄%&*%82��`!)%Æ℄&z&&E(Æ℄�D&nu(jÆ℄z/�NOÆ℄%&*%�NO`S�K�z�z%! Q%(jÆ℄�m�\��$�O�nk�4:n4lD*_��zn2&�D�o-C�nb[6}/#5℄+�v�T"Kw�+(l&��KMw�,�- �QI ~+&YuRw�*"~$N℄�[�-�VwU6}UFg"3��&�-��?^/�nYI\e,"n$4'KMw��<+'n(l�*�g+��sÆÆ��g&!*���Æn�sÆÆ&fla`_��z%Oeu�z�l uSG�r�n�%! Q�K,Sz&��z�zvf℄nk��|%lTObr~u�z�1��z&k*&8nv��b��z%~rf%_��z&Fy&�nk�

Page 152: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�x�0�w5+ ��� }N?��z 1414:n4pD�D�->i7n(��QI&[3��7*�jE+(l&*(l�����,S^�����(l1ZlrE,,�(l*w�n�R-`K�eN,SZlrEÆ&�!Dnv.y��_��z%� u�z�lu~rM4�(d�K*%! Q�K�� z%�& T���Æ*�sÆÆzdu l3%&b��|�z�8nv��b��z%~rf%_��z%*�&�eU0bJ'%u&"/�z�7~rM4�(d�Knu%! Q�K T���Æ*�sÆÆUV l&u�d`mnv.yf%_��z&�!D�K 3!D&flhG�z~r&a�~�o℄����.KO���w�/�3I�Bn+�v_�&�2-P5�&~(l#�Æ%lvn/�-{��z�zv�f℄yQ�z��%! Q�Kf�%*Q &4.�`m8�z�lVOG�KMw�Æ�A�l#llB)_��z%`�&k*&�!)�jv��)I8b��z�*�!)�j%Ob�l�f�:&b��z�_��zP�/z�(�j�l&S+� a/�z�u℄D%n� ℄D%�l��:[ T�}A�j�l&?�b��z%`�&!*b��z%�z� $!�z�u�&.y(�-�z�*�z�%mx%*&�A TgI��/�-T:\Æ&&z�B4S�D&v7S�$Æ%v%El�Fy&nvfm�nk�4:n�D&�_��zaCdO�{1q�=r\S�(��&o2`_��z(lA��!g*��z�z%��a/j��z�z{1%sd(AN��&�OhJ-4&flVG��j��Cdbej�*�z�zjJ�r�%-vÆ�&aGq��&V"~r9~&�u?�7l�z�z5V%n/�*y&�z�zjJ�r�&y�~rM4�(d�K*%! Q�K&�}a�lGV �_��z&(j/{1q�ry�(j/v%r%fm%&���5�nk�4:n�D&nvw'_��zl �l12pA�I<Æf%b��z����z�zv��E(�Æb��z*_��z�f*fNÆ& aS4_ln�D�ovRqmS�,S E+o℄*;(l�%�F$X��Kn^�� E+5�*;(l*E�KMw�+&-ovRqmS�,S6}%�F$X��Kn^����&Y"KuRw�Æ&���B7~r*V�k;��)b{�`!)�D%2�B7~r��b��z&�tu��B7~rn2�B7~r& vtb��z%&q�4dd.yw'_��z�aB7~rsd:&�nk�4:n8b��z~r|'M� ��o-C�nb[6}/#5℄+�v�T"Kw�+(lÆ&�a�<sS4lD�D%}Tn2%&q�~rM4�(d

Page 153: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

142 L�y: ihUAbx� – `N�; Smarandache +ai�K*%! Q�K!D&.y��_��z&z^u�AEl%em����z�zv�4_Pn�D�w�n4'uRw��<+&��* 3=E�[�w�(l+y��CA�{+�E+vn6},�M�ql/�uRXÆ&�e%�z��~b�u�3 =Æ&y<s�zz�0b&_�n2u�E�[�w�(l+y��CA�{Æ&flfm�z�jr%�Fy&*fm�nk�4:n�D%_��z&l�[&{�z�Q"&aSq�{1A)&�D<s��z�zv�4_Pn�D%ka�z��~(r&k* 5 T&nuq�#;&�z��9�lD�~&k* 6 T �fm��2-P5�Æ%�jn/ &���f%_��z&�0bV�%u&_��z� lDu��uR 3 =/�nÆ�z&y"_�$-n2%F(ka�z�&k* 5 T�z��z&*|�rV2 ,z�t5&`�z�k��D=zS (j,�l�[&{~rM4�(d�K&y�%! Q�KQ"&.b�z�=k_��z%`r7u*�aGq���V"~r9~&.bS �a�� - n2r5��Wka�z�&���!DSf%_��z&*b��z*_��zPuv%!.%�z#l&�I<Swx�vÆ&�z�z%�wxÆ�r%&uaGq��&V"~r9~�P��$ $4 k2�|Y�O��$�Oji��nk�4p_pn�D�?^/�n6},��u`�n_C;E�bv-��+&��^��C;E��Rbv �1h 2 u>�l,_w�bv-��+&��^/� �1h 10 u>�l�w�n��FP%�� u3 3 uwZl��,Zl��>&��_%w�/�3IÆ���=&xw�_%Æ, u-l[ "&*{hQAM92�kr`2F��F�M-&nu{�z92�kr�2�z��zM-&S "q9�;& u�a���.j&�z� #w'F��hQF�M-nu�z��zM-& #f%�z��e0fl�[-%; u&�w�/�3I_%&D�RwhKC;�R �1h6D/� �1h-u�0bP4&0}��(9&y��z��.u�B�r`hQAM92&n��z92L|�,9lD�hQAM92��z92%L|�,9.��|hQF�92s5t s 3 %%&nu.��|�z92s5 s 15 %%&�z���u� Pa?hQAMF�92nu�z92%M- ",* �!hQAM92��z92�(&zv%�%!v�%p5(�D&�|�hQAMS n�ka�z��z%&nvo2f%�z%~r%�z���

Page 154: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�x�0�w5+ ��� }N?��z 143a,9hQAM92n��z92?B��z.z�0r+'v�O8$*B;� $J5 $J��z�zv�4_:n�D�w�n*)*���v�� �^/�n&*)_?^/�n4K/N�AÆ&�nk�41_pn�D�w�n*)*��+�v��� ?^/�n6}/�nÆ&S="P$z �D #y�,�*���v��� ?^/�n6}/�nÆ��e&- ,-A! nr & - &� -A! n -Ar �q3t �^lÆQ���+*~E�v&�v+*~E��Æ&vD&�z�zv�D%�*���vÆ=%�*��Æ1Tl,dD�0bvQ��V�&�e%�*��Æ u�,% -A! n -Ar &l(℄~n2&*`r%*.b &zv%El*v%�l�u4.%*&Fy&.�av%F�dVD�e�*��Æ%����z�zjJ0b{Ab��b��b�*Mf��%v%El���z�zv�=E(k�u�n2�/ - n2&Fy&��z�zv�4_:n=%�*���vÆ&,&z�z�zv%El%n2&�&z�KM�K��K��Z4A'ÆEl%n2,�nk�41_pn=%�*���vÆ&,8nk�%"l$�"�$A�en2&y�SG&z�KM�K��K��Z4A'ÆEl%n2& �%&"P$=%�,�*���vÆ,"l$�"�$&�%&&z�KM�K��K��Z4A'ÆEl%n2�&� $J2F $8$�nk�41_,n�D�:w�n�^�u`�Yy $w�K�N+vn�,�M�6}=n&*)7^/�Æ��e&�:w�n�^�u`�Æ,7�z��a`rj�y_� v=%j�y_�&,7��%v%y_nllr2�rfn�(Æj�Ny_%�&k*&y/`v%;L\m&j�y_��u,7\m�(�Iy_n�.y_%�&*2m&-�&hu�&uÆ�&uN��7�z��aj�y_%�(yd�A~#*�~ y_�&�f%�N�w'����95�p�)yMfmw'�erz%�&*=�7T℄Jp)7=�7T4l℄J�%y_&=�7T℄Jpb)u?�%~ y_�,�sÆy_�&*�z�%ib)��sn9sb)),�uNy_&�uÆ�7uN�&*7�z��a[Py_nuj8Ss;nu�%�),�SGnv7�z��aj�y_%��+#V�%u&�Yy $w�K�NÆur2 �D℄9&7yu`�&��

Page 155: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

144 L�y: ihUAbx� – `N�; Smarandache +aigÆ, 100%&�*YyÆ, 0%&l�YyÆlY/ 0% 2 100% &"�lF!*&�aj�y_lD�a�|�zb�(%ka.�&��z���&zv%v�p5(�D%�v�G8v��SGw'n�T�=z&yfm�z�j�y�O&��A�Eu�-T:\Æ�Fy,�nk�41_,n4lD�Yy $w�K�NÆ=%�YyÆ& ulF��:%ka.�&lu,&z�zb�(%rf&d�u"&7�z��aj�y_&t�a�|�zb�(rf%%v�n�SGw' #�z&* �z&S�z}*E��kQ"&hA8Kow�&,DKo�7[�b[KoD�Y/�-�7u4D% .y�z&� #�|�zb�(&��z��z�J=& #`m`Sib)��s�9sb)%t�{�%*&*hQAM92��z92&zv%v�%p5(�D`S℄�{�&y�Cz��D=z��J=`S℄�{�%*)�z &* �z�%ib)��s�9sb)����z&�z�% +℄��GP7S�aj�y_&5���z�zv�41_Pn*�nk�4,_�n�D&0baq&�y �z� #��|��Cz�� $�O>?wx���z�zv�41_�n�D� �VL.YwU/�n_/�bv&w�*_ +wxZ�w+YO6}j_Æ&�nk�4H_pn�D�/�bv&-w�*_ +wx�_�bG6}Sxw& �VL.rSRw/�nZl�wYO�j_+&��X[(�(/�nÆ���=&xw�w�*_ ÆulT�,:&,FS�%�� �D� a�� |S&�e(}Tr~0bvQ��g,�nk�4H_pn�D=%�_�bG6}SxwÆ; ;�w�*_ Æ-�7u4D% Fy&<�z^lTxw&�nk�4H_pn=%�w�*_ Æ7��z�zv�41_�n% ��l3&��z�zv�=%�w�*_ Æ*�,&��nk�4H_pn=%�w�*_ Ælu,�_�bG6}SxwÆ&�r~%�� �D� a�� |S�Sz&!*�w�*_ Æ%Oe.�uCz,Gd&�du�nk�4H_pn=�D� �VL.*)d=6}0#/�nZlYO�j_&*)U/�n/I�l+YO�j_Æ%E&�b��/�n+YO�j_��4'X[I<,%*)Ll/�bv+}P6})�/�bv+4#NwxÆ&*Y4I��Ll/�bv+}P6})�/�bv+4#NwxÆ%L|�"a92%%*)I&z��z�zv�4p_:n�^w�bvwU��/�bv+ �1h�t�+/�bv&w�n��CPÆ%�D�

Page 156: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�x�0�w5+ ��� }N?��z 1456�1$Y��f�nk�4H_,n�D��v�T"Kw�+(l&w�n��FP% ��5 u3 3 uwK=&��\n&K=(*)�3 3 uÆ&z �D&�/�n6},��u`�n_�v�T"Kw�+(l+ ��q-��+&��^&��\nK=(h�l�w�n��FP%�� u3 3 uwZl��,Zl��>&��_%w�/�3IÆ���=&�_%w�/�3IÆ,�z� #f%�z�jdlTTN&��D=z��gE-z�bmulArgAL&�yI����97℄�%��99e��e%�K=&��\nÆ&��CdL$Cz,Gd�+%=z==�&yI�Cd Q��ebvQ%&uL$`�uCd�GUu7�z�zjJ(y%j�y_��{I�Cd Q��|�z�z%*`r&�z�zjJ�I�Cd�G Q��l�℄�z~r&k*B���4����TV%~ry�SG(℄b� t�Cd�Gb�%_��z~r&~r(9 t�Cd�Gb�&Fy&�e%�K=_,Æ�Ob�`7�z�zjJ(y%j�y_�& b��zy�.y�Cd�Gb�%_��z~r&.y��,DGY��2)�Cd�Gb�&z&&l.y `_��z�b�&yI��z��Sj�y_�%�����)$%BGv�nk�4:_pn�D*�v�T"Kw�+(l+w�/�3IO{w�/�v���+jE&_&��qbS4#N $&K*y4Z)=*7:�+&w��/��&�n5&���v*>w�6} ����=&�_&��qbS4#N $Æ,=z�n=z�(rS�f9(uO&k*=z��*�z� A&S �z��D�SG�z�*=z�&nu`=zz%��D�9~�#%Mr)-zf9(�(hG��|�vD&�)=*Æ%��udD-�)=*ÆlF T*r?�%fm%*r%2h Mv&Fy&w'Cz,GdB�Czn�z�B��zP.y}*fm�z�j%���℄�vD&�nk�4:_pn=%xw�)=*Æ�E( +���Fy&�nk�4:_pn=%�)=*Ælx2S-��z�zv�4�_,n�� T���z�zv�4�_,n�D��v�T"Kw�+(lO{vjE&&�n5+&���{vjE+&��v�,5/�n&*> E&�n6}�{

Page 157: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

146 L�y: ihUAbx� – `N�; Smarandache +aiv*>"Kw�Æ&�e%�*> E&�nÆ&�a=zEjadnvB��D=z�%%*&(}A~#*�ECz,Gda`% [CzZL*�+%=z==�(&�nvo2f%�z%~r&<�S�z�zjJ&z�z�zv�D&��0`CzZL*=z==�hGf9(�|&y 09�=zS �.,�ECz,Gda`CzZLn�+=z==�E&�B�Cz.�D��CzS &y �w'Cz,Gdf%B�Cz&B��GCzZL*�+=z==�&�?�D=z��ob &�z�.ynvB�w7Cz,GdCz�V�&B�CzlDuECz,Gda`%CzZL*�+%=z==�Ejad��i�&�l&&z41_:n4pD�D%�p��℄�=n*)�v-�� $ �+sV��qÆ��Cz,Gda`% [CzZLn�+=z==�E&�tB�Cz# ��D��CzS &�z�w'B��z���D=zS &�z�2w'B��z&��z�B�rÆ�zbL��z_� )f%bP_�&�z�5�%bP_��z&�z�B�w'�z�Cz*�D=z�%%*�Fy&�nk�4:_pn=%xw�)=*Æ&,:[%4�A~#&�ECz,Gda`% [CzZL*�+%=z==�(&�<��z�zjJ&z�z�zv�D`=zS hG�f9(�|&��0`CzZL*=z==�hGf9(�|&y 0�z�9�=zS �.&l S�B�Cz*B��z}A�℄��^&�nk�4:_pn%���u*�v�T"Kw�(l+w�/�3IO{w�/�v���+jE&_&��qbS4#N $&K*y4Z)=*7:�&G*y�DH �VL.�l+X[ ��5�:p+&��\n*> E&��q+&O{w�/�v���jE+w�KSn5&O{w�/�v���jE+/�KS/�&*1&O{w�/�v���jE+&��qn5&���v*> �6*>M�w����z�zv�*�nk�n9Zf%vQ.�09n [3]�2h�7[1] ��ry*9P,Gdv�)��NOv5Mb|�^)� *�*v *)&&�n^O�ow�/�v4*��F�&=��N`IC& m&2012.

[2] Ax&DVx�(lw�4T�<:4s&American Research Press, USA,

2007�

Page 158: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�x�0�w5+ ��� }N?��z 147

[3] Ax�d��&w�/�v�b�yCf��y&=�7T℄g`IC&`m (2013 �8`I)�[4] *9�* 6�&>i_ - >i�>v�>�&,?`IC& m&2012.

[5] a7nOs&v�_&,?`IC& m&2010�

Page 159: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

�H"�5R��y:�– K�x 1985-2012 �~<1l1, 148-154

��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$`mq�Fe�&�Q#Pi��Sh℄℄&x5N℄&5"1995$

Page 160: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

'2+L�y 1985-2012 �=2m2 149

2. &�QSh#P℄℄��&&o4'VNSqÆ|="95 I$&1995�3. ���a4Lee℄B{r�BT℄℄��&&o4'VNSqÆ|="95I$&1995�1996 '1. (WDLxk=��%y�r�&{b>DVNÆ__�&Vol. 23"o�$(1996),

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,

2(1998),188-190.

2. i���eEÆQ��;&x5N℄&9"1998$3. n^s�G$7T℄℄4��G�7&x5VN&1"1998$1999 '1. ?�lM℄J℄℄w'pE�&/Y%Os*x5DV*DM��S4��'Q=&=�7T℄g`IC&1999�2. &�Q℄J℄℄w'E�&3�*Os*x5DV*DM��S4��'Q=&=�7T℄g`IC&1999�3. `mq�Fe�&�Q#Pi��Sh℄℄, =�7T℄Jpb)s*x5DV*D4�Q=(2), =�7T℄g`IC&1999�2000 '1. ��O Fann2%lT��&Y#'}Æ__�"FgV_u$&Vol. 26&3(2000),

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�

Page 161: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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.

2. On the panfactorical property of Cayley graphs, b_g9: <, 3(2002), 383-

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.

2. "7k�-P$Riemann �[: Hurwitz D %w-��&&oV_R&>�>m:�B_V_℄<�<bA&2004 12 T,75-89�

Page 162: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

'2+L�y 1985-2012 �=2m2 151

2005 '1. (with Feng Tian)On oriented 2-factorable graphs, J.Applied Math. & Computing,

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

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

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

236.

3. On Automorphisms of Maps and Klein Surfaces,&oV_R&>��5&2005.6.

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

math.GM/0506232.

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

can Research Press, 2005.

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

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

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

133.

2006 '1. with Yanpei Liu and Erling Wei) The semi-arc automorphism group of a graph

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

(2006), 93-101.

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

3. &oDVx�(l*Dw�NH:f��y –Smarandache *`hw�eG,

Xiquan Publishing House, 2006.

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

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

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

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

8. -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.

Page 163: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

152 L�y: ihUAbx� – `N�; Smarandache +ai11. A multi-space model for Chinese bidding evaluation with analyzing, arXiv:

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-

tific Elements(I), 155-195.

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,

2008, 111-121.

Page 164: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

'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,

2009, 01-22.

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.

2011 '1. Sequences on graphs with symmetries, International J. Math.Combin., Vol.1,

2011, 20-32.

2. Automorphism Groups of Maps, Surfaces and Smarandache Geometries (Second

edition), Graduate Textbook in Mathematics, The Education Publisher Inc. 2011.

3. Combinatorial Geometry with Applications to Field Theory(Second edition),

Graduate Textbook in Mathematics, The Education Publisher Inc. 2011.

4. Smarandache Multi-Space Theory(Second edition), Graduate Textbook in Math-

ematics, The Education Publisher Inc. 2011.

Page 165: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

154 L�y: ihUAbx� – `N�; Smarandache +ai5. Graph structure of manifolds with listing, International J.Contemp.Math. Sci-

ence, Vol.5,2011, No.2, 71-85.

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$�2012 '1. ,?h7�z�j -e_&&ow�/�&1-2"2012$.

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.,

Vol.2, 2012, 9-23.

Page 166: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

[~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

BOOK SERIES)&8\| uqaI�s}`I� 10 ��a���7� 2 g�"7T℄J~ ℄�"�7T℄J℄℄w'E�fk���"�*�7T℄J℄℄fk�"�"II * VII$%s�,u�=K�`e*�zd℄℄�zhQAM92�"2007 I$*�=K�`e*�zd℄℄�z92�"2007I$�Si�,{%Obs�Z�,2008 X� ���z[(g��&�'�E�,�,Gd,G���z�j7kÆQ"2009 $�'�E�+Os&��z�j7kÆQ"2012 $�'�E�Os�

Page 167: (New Expanded Edition) - fs.unm.edufs.unm.edu/Linfan-Chinese3.pdf · (New Expanded Edition) Chinese Branch Xiquan House 2013. This book can be ordered in a paper bound reprint from:

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�

7321457815999

ISBN 9781599732145

90000 >