Giáo Trình Lý Thuyết Đồ Thị

7/21/2019 Gio Trnh L Thuyt Th

1/166

G =;J V =>p= W =y

B =i :

M} OSH H eH


2/166

G =;J V =>p= W =y

B =i :

M} OSH H eH

C = b:

M} OSH H eH

]H H NL NOSH:NNJ://PIL.O.PH//902114
http://voer.edu.vn/c/9c021e14http://voer.edu.vn/c/9c021e14


3/166

MC LC

1. CX EX HG H Ft NOSN u N

2. CX NON H H

3. wH u,O NL_H, u F]H NbH4. N M H u N u N

5. BO H u N NL]H GXS P N^H

6. DH MX H(OH)

7. DH MX E

8. CX NON NIXH N_G EG NL]H u N PY H H

9. _G EG NI O LH NL]H u N

10. _G uH u PY EG NL N^H F]H NbH

11. w N OFL PY u N GNIH12. w N GFNIH

13. CZS PY ZS EOH u N

14. CZS EOH u N

15. ZS H NJ X O NL_H H u N

16. BY NIXH ZS EOH H HN

17. BY NIXH uH u HH HN

18. wH u HH HN RON JXN N GN uH

19. LH J G NLZH NLH M EbH ZG-ON NIXH DEMNL

20. w N NLIH u N EbH ` O NL_H

21. wH u HH HN NN X J uH

22. BY NIXH FOH u NLIH GH

23. 'XN N.wH NvH FObH.wH Ft !IL!OFELMIH

24. ON NIXH N_G FOH u

25. N M Y NIXH FOH NH KOXN

26. N M H H NLIH N J

27. BY NJ H 3

28. BY NJ H 429. BY NJ H 5

30. BY NJ H 6

31. BY NJ H 7

32. CX Y NJ EX

G u`H `J

1/164


4/166

C v ] b + V =>p= W =y

't NOSN u N FY GN F}H P u[ ` N FZO PY ` HO H H H u. H NNH H Ft NOSN u N u u RON PYI HH HvG uO N E 18

HY NIXH F F H S 'HL OLFL. C^H bH FY H u[ M H uN u Y NIXH H NH P X X O NYH J &IHML.

w N u M H u X Y NIXH NLIH HO F}H P EX HO. CH H, uN ` N M H u RX uH X G PaH NLIH PH u N^ G uH. CdHN ` N JZH N X J N ` O EX HO P eH bH N JZH NHH EX HO P O NLd JZH N H u N. CdH N ` N RX uH GXS N^HNLIH GH ` N NLI u NbH NH u P HO S EbH H Gb _H u N GH GXS N^H. w N ` NLH M NL]H X H ` N M H u X Y NIXH H:_G uH u HH HN NYH J NLIH GH I NbH. CdH N ~H aHM H u N u X Y NIXH P FJ F, N E` O, PY JZH NH M I XNLG JXN NH PY NLOSH _Hl

YxNH NGHZA Y %Hx

w N FY GN O NLd L L I G X uH PY X H H X uH HYS. CdH NJZH N X FI u N EX HO 97x PY eH H uH HYI u` uN. w ` N _H OH u N MI F H uH X FI u N EX HO, dH N MH]O P^ M H dH u Gb N GN GH GXS N^H. M N ` GN GH G X

GXS N^H PY X E]H uH NI ( NN FY E]H NI) H X GXS N^H HYS. CdH N` N O H X P NL^ uN HXS N^H X uG PY X E]H NI H dH XuIH H, RG _H 1.

HJ 1. u GH GXS N^H.

2/164


5/166

H NS LH NLIH GH _H 1, GXS N E ` HO HN FY GN E]HNI H dH, E]H NI H I J\J F]H F O PY EbH ` GXS N^H HYIF u H P ^H H`. u GH GXS I NLIH _H 1 u FY ^c ^ } Re. u uH uH H} MO

Yy [+ 1.

wH u N Pb H = (,) I G FY NJ X uH, PY FY NJ X J EbH `N N G JH N EX HO FY X H.

LIH NLH J GXS N^H HYI u` NH ROS]H J NLOSH N HO NbHNH H N J H GXS HYO HO E]H NI. H P u E]H NI XGXS u I NLIH _H 2.

HJ 2. u GH GXS N^H P u E]H NI.

Yy [+ 2.

w u N Pb H = (, ) I G FY NJ X uH, PY FY NJ X J EbH `N N G JH N EX HO FY X H. H 1 PY 2 u FYH FJ HO dH eH NH H P GN J uH.

HJ 3. u GH GXS N^H P E]H NI NbH XI.

3/164


6/166

c LYH G uH u N uO FY u u N, HH EbH J u u N HYI ~H FY uH uN, P_ NLIH u u N ` N ` (I HO H) H H GN J uH HYI u`.LIHGH GXS N^H ` N ` HH E]H NI H GN HXS HYI u` P ^H H` (H HP G u^H NbH XI). H H PS u I NLIH _H 3.& u` u u N EbHN Gb N u GH H PS, P_ ` HH EOS]H(H H GN uH P ^H H ).LIH NLH J HYSdH N H M H uH EX HG u N Pb H, uuH H} H MO:

Yy [+ 3.

u N Pb H = (, ) I G FY NJ X uH PY FY NJ X J EbH` N N G JH N (EbH HN NN J EX HO) FY H. CH u FY EOS]H HO H` ` H = (O, O).

HJ 4. H GXS N^H P E]H NI GN O

CX E]H NI NLIH GH GXS N^H ` N I J\J NLOSH NH NI GN O.CH H, NLIH _H 4 GXS Y ` N HH NH N X GXS u JH,` GN M GXS ` N NH u, aH X E]H NI I J\J NLOSH NH NI O u NS N H ` H H O HO.

u uH uH H} MO.

Yy [+ 4.

wH u N ` H = (, ) I G FY NJ X uH PY FY NJ X J ` N NG JH N EX HO FY X OH.

O NLIH GH ` N ` u E]H NI GN O, N M J M H uH EX HG uu N ` H

4/164


7/166

Yy [+ 5.

w u N ` H = (, ) I G FY NJ X uH PY FY NJ X J ` N NG JH N EX HO FY X OH. OH 1, 2 NH H P eHGN J uH u FY OH FJ.

LIH X JH NJ NI SO dH N M FYG P P uH u N Pb H PY uHu N ` H. _ PS, u I HH H, N M KO N^H N W] E H uH dH.

5/164


8/166

C =>i= ] b

LIH G HYS dH N M NL_H YS GN M NON H H Ft NOSN u N. LN]H, N R\N X NON H Gb N X uH PY H u N Pb H.

Yy [+ 1.

uH O PY P u N Pb H u FY E HO HO (O,P) FY H u N. O = (O, P) FY H u N N H` H HYS FY F]H NO P uH O PY P,I ~H H` FY H uH O PY uH P, uH N X uH O PY P M u FY X uHuO H (O, P).

w ` N N ` PI H]O H F]H NO P GN uH, N u PYI uH H} MO

Yy [+ 2.

uH P NLIH u N Pb H FY M H F]H NO P H` PY M Et OFY (P).

HJ 1. w N Pb H

% 1. \N u N I NLIH _H 1, N `

(/) = 1, () = 4, () = 4, () = 3,

() = 1, () = 3, () = 0

wH 0 FY ^| R l. wH 1 u FY uH NLI. LIH P^ NL]H uH FY uH b FJ, PY FY X uH NLI. B uH ` N^H N MO:

Yy V 1. G7g G = (, E) ^ } R e 7 f. 7 ^P R l / i g ^| o /7 j .

6/164


9/166

C . c LYH G H = (O, P) u N^H GN FH NLIH () PY GN FHNLIH (). u` MOS L NH NN X X uH H FH M H.

% 2. w N P H uH ` FY 6 ` I H]O H?

Gb: I uH Ft 1 N ` 2 = 6. u` MOS L NH X H u N FY 3H.

Hv >b. '@ ^ } R e, ^| l s (a/ P l s) q.

C . PS, PY NH H FY NJ uH F PY NJ uH H u N. `

2G = ? (P) + ? (P) P P

DI () FY H P P FY uH NLIH H]H NH N HN NL]H FY M H. u` MOSL NH N (^H FY NH X uH F) ~H J FY M H, I NN X

M H H` FY M F, H]H NH HYS J G GN M H X M H. _ PS, M uH F J FY M H.

R\N X NON H NH N I u N Pb H.

Yy [+ 3.

O = (O, P) FY OH u N ` H N_ N H` uH O PY P FY E HO, PYH` OH (O, P) H uH O P uH P I ~H H` OH HYS FY u L E uH O PY PYIuH P. wH O(P) M u FY uH uO (O) OH (O,P).

H N H EX HG , u P u N ` H N ` EX HG XH L PY XH PYI GN uH.

Yy [+ 4.

'/ ~7 l @/ ( l ) / ^| @ ^ } P e / ^ }^7 @/ 97 P (^7 P) 9] 7z + () ( - ())

7/164


10/166

HJ 2. w N ` H

% 3. \N u N I NLIH _H 2. `

- (/)=1, - ()=2, - ()=2, - ()=2, - () = 2.

+ (/)=3, + ()=1, + ()=1, + ()=2, + ()=2.

DI G OH (O, P) M u N^H GN FH NLIH XH PYI uH P PY GN FH NLIHXH L uH O H]H N `:

Yy V 2. G7g G = (, E) ^ } P e. 7 ^P

2G = ? +(P) + ? -(P)

P P

N HO N^H N u N ` H EbH J NO PYI H NL]H X OH H`. _ PS, NLIH HO NLH J M NOH NH H HO N KO H NL]H X OH u N. w N Pb H NO u H X KO H NL]H X OH u FY^ } R e ec P u N ` H u[ I.

8/164


11/166

Y^ W,> =;J, W =

Yy [+ 1.

è ^7 ^ 7 ^| ^v ^| , @ ^P M ec, @M ^ } Re G = (, E) L

0 , 1 ,W, -1 ,

@ P = 0 , = , ( 7 , 7+1 ) E, 7 = 0, 1, 2,W, -1.

wH u H` NL]H aH ` N O H H [S X H:

( 0 , 1 ), ( 1 , 2 ), W, ( -1 , )

`| ~7 ^| ^j, Q ^| ~7 ^| 7 / ê ^7. è ^7 P ^| ^j@T 7 ^| 7 ( = ) ê ~7 @O . è ^7 / @O ê~7 ^c v e 9R P f } r f7.

% 1. L]H u N Pb H I NLIH _H 1: , , , @, FY uH u uH u Y 4.CaH , , , EbH FY uH u, I (,) EbH J FY H u N. D[S , , @, , FY O NL_H u Y 4. wH u , , , , , ` u Y FY 5 EbH J FY uH u uH,I H (, ) ` GN NLIH H` 2 FH.

HJ 1. wH u NL]H u N

&X HG uH u PY O NL_H NL]H u N ` H u uH H} IYH NIYH NHN H NLIH NLH J u N Pb H, EX FY N ` d t uH H NL]H XOH.

Yy [+ 2.

è ^7 ^ 7 ^| ^v ^| , @ ^P, M ec, @M ^ } Pe G = (, A) L

9/164


12/166

0 , 1 ,W, -1 ,

@ P = 0 , = , (7, 7+1 ) E, 7 = 0, 1, 2,W, -1.

wH u H` NL]H aH ` N O H H [S X OH:

( 0 , 1 ), ( 1 , 2 ), W, ( -1 , )

`| ~7 ^| ^j, Q ^| ~7 ^| 7 / ê ^7. è ^7 P ^| ^j@T 7 ^| 7 ( = ) ê ~7 @O . è ^7 / @O ê~7 ^c v e 9R P f } r f7.

% 2. L]H u N ` H I NLIH _H 1: , , , @, FY uH u uH u Y 4.CaH , , , EbH FY uH u, I (,) EbH J FY H u N. D[S , , @, , FY O NL_H u Y 4. wH u , , , , , ` u Y FY 5 EbH J FY uH u uH,I H (, ) ` GN NLIH H` 2 FH.

\N GN GH GXS N^H. N ZO uN L FY GXS N^H N E NLIH GH HYS ` NNLI u NbH NH u P HO I FY NL NJ KO E]H H dH I NbH KOGN I PY GXS N^H NLOH H NLIH GH? O M H u N u O H GHGXS N^H HYS (NLIH u` X uH u N NH H P X GXS N^H, aH X HNH H P X E]H H) ZO u` u JXN O NLIH HbH H u N H MO:H N S EbH uH u G J uH u N.

Yy [+ 3.

w N Pb H = (, ) u FY F]H NbH HO FObH N_G u uH u uH N E H`.

PS GXS N^H N E NLIH GH ` N NLI u NbH NH u P HO E PY E u N NH H P GH HYS FY u N F]H NbH.

% 3. LIH _H 2 : w N FY F]H NbH, aH u N FY EbH F]H NbH.

10/164


13/166

HJ 2. w N PY

Yy [+ 4.

'/ ~7 ^ } / ^ } G = (, E) ^ } H = (, F), @ ^P FE.

LIH NLH J u N FY EbH F]H NbH, H` M L[ L NYH GN M u N IH F]HNbH ub GN EbH ` uH OH. H u N IH F]H NbH H PS N M FYX j 7M R u N.

% 4. w N NLIH _H 2 G 3 NYH JH F]H NbH 1, 2, 3.

LIH GH GXS N^H ` N ` HH GXS (H E]H H) GY M H ` H`M H H uH P NLI u NbH NH NLIH GH. CX EX HG NH H P N_HOH HYS u u L NLIH uH H} MO.

Yy [+ 5.

`| ^e ~7 ^| @t v 7z f7 T 7 f 7M 7 P97 ^ } _ j 7M R / ^ }. Cf ^e ~7 j v7z f7 P 97 ^ } _ j 7M R / ^ }.

% 5. LIH u N _H2, uH PY FY uH L HXH, aH X H (,) PY(,@) FY O.

w P u N ` H ` EX HG F]H NbH J NO PYI P N ` R\N uHH NL]H X OH S EbH.

Yy [+ 6.

w N ` H = (, A) u FY F]H NbH GH HO FObH N_G u uH u uH N E H`.

Yy [+ 7.

w N ` H = (, A) u FY F]H NbH SO HO u N Pb H NH HP H` FY Pb H F]H NbH.

c LYH HO u N FY F]H NbH GH N_ H` ~H FY F]H NbH SO, HH uO HF FY EbH FObH udH, H L NLIH P^ H uZS.

% 6. LIH _H 3 u N FY F]H NbH GH, aH FY F]H NbH SO HHEbH FY F]H NbH GH.

11/164


14/166

_H 3. w N F]H NbH GH PY u N F]H NbH SO

N ZO uN L FY E HYI ` N uH H X H GN u N Pb H F]HNbH u ` N NO u u N ` H F]H NbH GH? M u N H PS FYu N uH H u. wH Ft uZS I N N]O OH HH N GN u N ` FYuH H u S EbH.

Yy V 1.

w N Pb H F]H NbH FY uH H u E PY E G H H` HG NL]H^N HN GN O NL_H.

C .

`7w 97z j. M (O,P) FY GN H GN u N. M NH N uH u ` HN O uH P PY H F MOS L (O, P) J HG NL]H ^N HN GN O NL_H.

`7w 97z ^. N MO uZS I J\J uH H X H u N u NO u uN ` H F]H NbH GH. M C FY GN O NL_H HYI u` NLIH u N. wH HX H NL]H O NL_H HYS NI GN H u PaH NI H`. O NN X H uN FY u[ u uH H N_ EN Nd N N. F, H FY GN H uHH ` OH uH P ^N HN GN NLIH M X H u[ uH H. I NN N_Gu O NL_H Co H . wH H X H u uH H Co NIGN H NI O NL_H HYS (EbH uH H F X H u[ ` uH H). N NL]H M u FJ F I uH E NN X H u N u uH H. &u` N NO u u N ` H F]H NbH GH.

12/164


15/166

M=


16/166

HJ 2. w N PaH C3, C4, C5, C6

Y =y /.

w N H NO u N CHH X MOH PYI GN uH G H P NN X uH

CH (RG _H 3).

HJ 3. w N XH R 3, 4, 5, 6

Y =y i ^].

w N FJ JH H uH H FY u N P X uH O H 2H RZO H JZH u Y H. uH H` FY E HO HO H RZO H JZH NH H EX HO 1 N._H 4 I NS H P H=1,2,3.

HJ 4. w N FJ JH 1, 2, 3

Y =y + +.

wH u N =(,) u FY J^ HO H NJ uH H` ` N JZH INYH NJ PY MI I G H u N H GN uH HYI u` NLIH PGN uH HYI u` NLIH . & u` N M M H Et O =(? , ) u u N J^ P NJ uH ? .

14/164


17/166

wH Ft MO uZS I J\J HH N GN uH u N ` J FY J^ S EbH.

Yy V 1.`c ^ } ^ } /7 N/ 97 | 97 P 9R / @O ^ 7s.

w EG NL RG GN u N F]H NbH ` J FY J^ S EbH ` N XJ H N

N MO. CI P FY GN uH N E u N. wN =UPW, aH FY NJ X uH E P.& u` X uH E X uH NLIH J NO PYI . &t O NJ X uH H PSFY . _ N HO JXN H ? # N_ u N EbH J FY J^, EN Nd. HF, uN = ? . J N R\N H PS u P o FY NJ X uH E ,. . .

w N J^ =( ? , ) P ? ?= G,?? = H u FY u N J^ uS u PYEt O FY &2,3, &3,3, &3,4 u I NLIH _H 5. & l

HJ 5. w N J^

Y =y k.w N u FY u N JH HO N ` N P H` NL]H GN JH MI I X H H` EbH N HO HIY uH. CX P H PS M u FY O H JH uN.

^ u N &4 FY JH, P_ ` N P H` NL]H GN JH MI I X H H` EbHN HO HIY uH (RG _H 6).

HJ 6. w N &4 FY u N JH

15/164


18/166

N uO uXH FO t HO u N FY JH N_ FObH ` N P H` NL]H GN JH P XH H FY X uIH NH EbH N HO HIY uH (P^ RG X P &4 NLIH_H 6).

w HH N RG GN u N ` J FY u N JH ` N M H uH Ft &OLNIPME,GY u JXN O H` N H GN M EX HG MO: GN J\J H (O,P) uN FY P FI H HYS E u N PY N]G PYI u N GN uH G Q eH P H (O,Q), (Q, O) . u N (,) PY =(,!) u FY uH O HO dH `N NO u N eH GN u N HYI u` H J\J H.

Yy V 2 (K>;+=/;). G7g G ^ } p 7M R 7 ^|, f.G~7 @ 7w / r p } 7/ 7 7x 7y p / G. 7 ^P

L = G-H + 2

C` N H GH uH Ft H KO HJ. \N N^ GH I I XJ H bH NOFL.

16/164


19/166

% . CI FY u N JH F]H NbH P 20 uH, G uH uO ` FY 3. GNJH FYG I H]O JH O H JH u N ?

Gb. DI G uH u N uO ` FY 3, H]H NH X uH FY 3R20=60. u` MOS L M H u N G=60/20=30. _ PS, NI bH N OFL, M GH H N_GFY L=30-20+2=12.

17/164


20/166

Bs> t W =y =; =

w FO NL u N PY N H X NON NIXH EX HO P u N NL]H GXS N^H H JN_G HH O NLd FO N^ J u Gb N u N. H O NLd FO HYI

u O H u N ` NX uH LN FH uH O KO NON NIXH. _ PS, P HF O NLd FO u O H u N J NO PYI NH N_H OH N (Y NIXHPY NON NIXH N). LIH G HYS dH N M R\N GN M JH JXJ H uM H u O H u N NL]H GXS N^H, uH N ~H JZH N^ GN X HH HHH O uG ~H H HH H uG dH.

MA %#hN Kq. MA %#hN %#zNG $}

\N uH u N Pb H =(,), P NJ uH =U 1, 2,. . . ,H? , NJ H =U 1, 2,. .

.,G W. G NLH E u N FY G NLH.A=U , : ,=1, 2,. . . ,H W

X JH N u RX uH NI KO N MO uZS:

, = 0, HO (,) PY

, = 1 , HO (,) , , =1, 2,. . .,H.

% 1. NLH NLH E u N Pb H I NLIH _H 1 FY:

18/164


21/166

HO 1. ` } R e G ` } P e G 1

C = d= + + =;i r:

1) c LYH G NLH E u N Pb H FY G NLH u RH, N FY

,=,, ,=1,2,. . .,H.

H F, G (0,1)-G NLH u RH J H M NH H, ^H RX uH X uXH MuH (aH H` FY: ^H RX uH uH O), P GN uH u N Pb H H uH.

2) H X JH N NL]H aH (N ) G NLH E ^H H uH (uH ).

3) HO Et O

J , ,=1, 2,. . . ,H

FY JH N G NLH

AJ =A.A. . .A

19/164


22/166

J N M

& u`

J , ,=1, 2,. . . ,H

I N M uH u EX HO N uH uH uH KO J-1 uH NLOH H.

NLH E u N ` H u uH H} GN X IYH NIYH NH N.

% 2. w N ` H 1 I NLIH _H 1 ` G NLH E FY G NLH MO:

'O t LH G NLH E u N ` H EbH J FY G NLH u RH.CS ]: L]H uZS dH N R\N uH u N. NLH E u u N ` N RZS HIYH NIYH NH N, EX FY NS P_ 1 PYI P NL^ , HO (,) FY H u N,dH N M E FY M H H uH , .

LIH LN HO PH u H H Ft NOSN u N, G H =(O,P) u N uXH P GN IH M () (aH PN FY (O,P) FY NLH M H . w N NLIH NLHJ H PS u FY u N ` NLH M. LIH NLH J u N ` NLH M, NS P_GY NLH E, u O H u N N M H G NLH NLH M.

C= U,, ,=1, 2,. . .,HW

P

,=(,) HO (,)

PY ,= HO (,)

20/164


23/166

NLIH u` M , NO NH NLH J N, ` N u uN H GN NLIH X X NLMO: 0, +h , -h .

O uG FH HN JH JXJ O H u N H G NLH E (I G NLH NLHM) FY u NL F ZO : uH O,P ` E HO NL]H u N S EbH, dH N JN H GN J\J MI MXH. H uG FH HN JH JXJ HYS FY: EbH J

NO PYI M H u N, N FObH J M H H2 uH P H u FO NL G NLHE H`.

DANH $SCH C_NH (C&NG)

LIH NLH J u N N (u N ` M H G NI G[H N H N: G


24/166

DANH $SCH Kq

LIH LN HO PH u H H Ft NOSN u N, X O H u N HH MX E FY X O H N^ J HN u M H.

LIH X O H HYS, P G uH P u N dH N FO NL H MX X uHE P H`, GY N M Et O FY

&(P)= ? O

: (P,O)

?& u` PaH FJ N H P G GN JH N NLIH H MX HYS NI N N X JHN u MJ RJ NLIH H` M u PN H MO:

@IL O&(P) I. . .

CH H, NL]H ACA' ` N Gb N H MX HYS H MO ( FY O NLd F;@+;$=+;):

22/164


25/166

CIHMN

=1000; ? - / ?

= 100; ? - 7 ?

PL

&:LLS1..G I@ HNL;

LI:LLS1..H+1 I@ HNL;

LIH u` LI HH P NL^ N uO H MX E uH , =1, 2,. . .,H,LIH+1=2G+1.

& u` aH FH KO

@ ()

H

. . . .

H.

C` N NS N O NLd FH N NL]H ACA' H MO

!IL :=LIP NI LIP+1-1 I

BH

:=&;

. . . .

H;

LIH LN HO NON NIXH FYG P P u N dH N NH ROS]H J N H XNI NX: ]G I N GN M H. LIH NLH J HYS O NLd FO eH NL]H FY EbH NOH NH. & u` H]H OSH MH M H H MX E F]H EN ('HEAHHS 'MN) H Gb N NLIH H NL_H HJ H MX E u N N YHJ^G PY u H MX u` L GYH _H MO uZS:

LILG A'MN;

23/164


26/166

CIHMN

GR=100;

SJ

FHE=HI;

HI=LIL

P:HNL;

HRN:FHE;

H;

L

,R,S,G,H,O,P:HNL;

N:FHE;

&:LLS1. .GR I@ FHE;

BH

LN(nCI MI H P H O I N:o); LFH(G,H);

(*&I NI*)

@IL :=1 NI H I &:=HF;

@IL :=1 NI G I

H

QLN(nCI H O P OI O H n,,o:o);

LFH(R,S);

HQ(N); N.P:=R, N.HRN:=&S; &S:=N;

HQ(N); N.P:=S, N.HRN:=&R; &R:=N;

24/164


27/166

H;

QLNFH(nDH M E O H O I N:o);

@IL %:=1 NI G I

H

QLNFH(nDH M H E O H n,,o:o);

N:=&;

QF N.HRNHF I

H

QLN(N.P:4);

N:=N.HRN;

H;

H;

LFH;

H.

% 4. DH MX E X u N NLIH _H 1 u Gb N NLIH _H MO:

wH uO

25/164


28/166

wH uO

_H 2. DH MX E u N Pb H PY ` H 1 I NLIH _H 1

w t LH NLIH X O H HYS dH N H J M H G+H uH P H.LIH X NON NIXH Gb N X JH NJ NI O NLd H MX E PY GNLH NLH M u M H NH ROS]H.

26/164


29/166

D+ )

LIH NLH J u N N (u N ` M H G NI G[H N H N: G


30/166

D+


31/166

C` N NS N O NLd FH N NL]H ACA' H MO

!IL :=LIP NI LIP+1-1 I

BH

:=&;

. . . .

H;

LIH LN HO NON NIXH FYG P P u N dH N NH ROS]H J N H XNI NX: ]G I N GN M H. LIH NLH J HYS O NLd FO eH NL]H FY EbH NOH NH. & u` H]H OSH MH M H H MX E F]H EN ('HEAHHS 'MN) H Gb N NLIH H NL_H HJ H MX E u N N YH

J^G PY u H MX u` L GYH _H MO uZS:

LILG A'MN;

CIHMN

GR=100;

SJ

FHE=HI;

HI=LIL

P:HNL;

HRN:FHE;

H;

L

,R,S,G,H,O,P:HNL;

N:FHE;

&:LLS1. .GR I@ FHE;

29/164


32/166

BH

LN(nCI MI H P H O I N:o); LFH(G,H);

(*&I NI*)

@IL :=1 NI H I &:=HF;

@IL :=1 NI G I

H

QLN(nCI H O P OI O H n,,o:o);

LFH(R,S);

HQ(N); N.P:=R, N.HRN:=&S; &S:=N;

HQ(N); N.P:=S, N.HRN:=&R; &R:=N;

H;

QLNFH(nDH M E O H O I N:o);

@IL %:=1 NI G I

H

QLNFH(nDH M H E O H n,,o:o);

N:=&;

QF N.HRNHF I

H

QLN(N.P:4);

N:=N.HRN;

H;

H;

30/164


33/166

LFH;

H.

% 4. DH MX E X u N NLIH _H 1 u Gb N NLIH _H MO:

wH uO

wH uO

_H 2. DH MX E u N Pb H PY ` H 1 I NLIH _H 1

w t LH NLIH X O H HYS dH N H J M H G+H uH P H.LIH X NON NIXH Gb N X JH NJ NI O NLd H MX E PY GNLH NLH M u M H NH ROS]H.

31/164


34/166

C =>i= = =J p =; W =y

N HO NOH NIXH NL]H u N u RZS H NL]H M OSN NN X uH u N MI I G uH H` u PH NvG udH GN FH. _ PS, P RZS HHH NON NIXH I J\J OSN GN X NH NN X uH u N FY GNPH u KOH NLH NO dN M KOH NZG H]H O HO NX . H NON NIXHH PS dH N M FY NON NIXH N_G EG NL]H u N. LIH G HYS dH N M NO NON NIXH N_G EG H NL]H u N: 'l O 97v 7w K(D F7@ &/@) PY 'l O 97v 7w @ (B@/ F7@ &/@) PYH H dH PYI P GN M Y NIXH NL]H u N.

LIH G HYS dH N M R\N u N Pb H =(,), P uH H PY G H.

CdH N M KOH NZG uH P uXH X O KO X NON NIXH NL]H u N, GYGNNLIH HH u NLH KOH NLH HN FY u J NJ N^H NIXH, N FY M J\J NIXH GYNON NIXH H J N H NLIH N_H OH RO HN u O H H YG E^N uO PYI Y NIXH. LIH X NON NIXH NL]H u N, uO PYI FY u N =(,),P_ PS, E^ N Y NIXH FY M uH H PY M H G u N. & u` u J NJN^H NIXH NON NIXH M u O H H FY YG H M @(H,G) FY M J\JNIXH HO HN H J N H NI NON NIXH u P G u N H uH PY G H.

& MI MXH N u NvH YG HH X NL EbH ZG @(H) PY (H) dH N M MH Et O MO:

@(H)=((H))

O e o R C, " V 0 /

() C () 7 ~7 U ".

H N H PS HO @(H1, H2,. . . ,HE), (H1, H2,. . . ,HE) FY X YG HO H N PN

( 1 , 2 ,. . . ,9) = #(( 1 , 2 ,. . . ,9))

O ^e o C," >0 /

( 1 , 2 ,. . . ,9) U C ( 1 , 2 ,. . . ,9) 7 ~7 1 , 2 ,. . . ,9 V ".

32/164


35/166

O u J NJ N^H NIXH NON NIXH FY ((H)) N_ N M aH H` FY H` ua NH N^H ((H)).

%UM KIoM %HE CHIq& $Q& %#RN Y %Hx

s NH ^H NON NIXH ` N NL_H YS H MO. M N uO N_G EG N GNuH P0 HYI u` u N. O u` H O FY GN uH NO t E P P0 PY FJ F KOX NL_Hu P O. NH KOXN, M N uH R\N uH P. O H NLIH M X uH E PP N_G u uH Q FY u R\N N_ N M R\N uH HYS (H` M NL NYH u[ R\N) PY NuO N H` N M N uO KOX NL_H N_G EG aH HO H EbH aH uH HYI E P P FY R\N N_ N H` LH uH HYS u[ OSN RIH PY KOS NL F NJ N N_G EG N uHGY NL u` N uH u uH P (HO P=P0, N_ EN Nd N_G EG). C` N H` HbG H FYN_G EG NI O MZO N uO N uH P u N H NL]H M N_G EG NI O

MZO N NN X uH R\N E P P. OX NL_H HYS ` N Gb N N N u KOMO uZS:

LIOL D!(P);

(*NG EG NI O MO N O NO H P; H CORN, & F H NIH O*)

BH

GH(P);

CORNP:=@FM;

F@ ()

$@ CORNO NH D!(O);

H; (*H P OSN RIH*)

& u`, N_G EG NI O MZO NL]H u N u N H H NON NIXH MO:

BH

(*$HNFTNIH*)

@ C/:=@;

@

33/164


36/166

@ CORNP NH D!(P);

H.

c LYH FH !(P) M I J\J uH NvG NN X uH NO eH NYH JHF]H NbH P uH P, P_ MO E NvG uH FY FH uH N N D! u P NN

X uH E P H`. N EX, I G E NvG uH P RIH, ?H CORNP uuN F X NL @FM H]H G uH M u NvG udH GN FH. ON NIXH FH FN M NHYH N_G EG N X uH u NvG , P_ PS, H` M R\N KO NN X uH uN (EbH HN NN J FY F]H NbH).

w uXH X u J NJ N^H NIXH N N, NL N HH NS LH M J\J NIXH HN H NLIH O NL_H NON NIXH ( PaH @IL H NL_H ^H) FY H. N D! J N H EbH KOX H FH. H M J\J NIXH H J N HNLIH X N N HYS FY (H+G), I NLIH X N N HYS N J R\N KO NN X HPY X uH u N. S u J NJ N^H NIXH NON NIXH FY (H+G).

% 1. \N u N I NLIH _H 1 G 13 uH, X uH u uXH M N 1 uH 13H MO:

_H 1

& u` X uH u N u uXH M F NI N N dH u NvG NI N NN_G EG NI O MZO Gb N NL]H H _H 2. NN LH X uH NLIH H MXE uH P (&(P)) u MJ RJ NI N N NvH H M.

34/164


37/166

_H 2. C M G (NLIH HI) X uH u uXH F NI N N dH uNvG NLIH NON NIXH N_G EG NI O MZO.

ON NIXH N_G EG NI O MZO NL]H u N Pb H NL_H YS NL]H YH ` NGb N F I u N ` H. LIH NLH J u N ` H, N N D!(P) M IJ\J NvG NN X uH O HYI GY N P ` uH u uH O. w J NJ N^H NIXH NON NIXH FY (H+G).

35/164


38/166

%J p =/ r> ; =; W =y

w t LH NLIH NON NIXH N_G EG NI O MZO uH u NvG YH GOH M YHMG NL NYH u[ OSN RIH. wO u` FY KO NN SO P X uH u NvG M

u EN HJ PYI NLIH HvH RJ (AC&). _G EG NI O LH NL]H u N, HOH` GN X HH H, u RZS H NL]H M NS N HvH RJ (AC&) YHu (). M ]H H PS, uH u NvG YH MG M YH MG NL NYHu[ OSN RIH (N FY YH MG E YH u). N uH M NL NYH u[ OSN RIHHS MO E N R\N RIH NN X uH E ( u NvG) P H`. N ` N GbN H MO:

LIOL B!(P);

(*G EG NI O LIH N O NO H P, H CORN, & F H O I*)

H

$EE:= ;

$EE; (*9 / / / $EE*)

C/:=/;

7 $EE

BH

$EE:; (*/ $EE:*)

GH(J);

F@ ()

$@ CORNO NG

BH

$EE;

CORNO:=@FM;

H;

36/164


39/166

H;

H;

& u`, N_G EG NI O LH NL]H u N u N H H NON NIXH MO:

BH

(*I77/7/7*)

@ C/:=@;

@

@ CORNP NH B!(P);

H.

'J FOH NH N H NLIH N N N_G EG NI O MZO, ` N L u LHFH B!(P) M I J\J NvG uH NN X uH NO eH NYH JH F]H NbHP uH P, PY G uH u N M u NvG udH GN FH. w J NJ N^H NIXH NON NIXH FY (G+H).

% 2. \N u N R\N NLIH _H 1. N NvG uH u N NI NON NIXH N_GEG NI O LH u NLIH HI.

37/164


40/166

_H3. C M G (NLIH HI) X uH u uXH F NI N N dH uNvG NLIH NON NIXH N_G EG NI O MZO.

38/164


41/166

%J W^ W s =;+ = =

LIH G HYS N R\N H H X NON NIXH N_G EG Gb N NLIH X G NL PYIP Y NIXH H NL]H u N: Y NIXH P N_G uH u PY Y NIXH P RX uH

N^H F]H NbH ub N.7

B = =J W^ W + + Ww:

M M PY N FY uH HYI u` u N. [S N_G uH u N M uH N. NL]H u[ JZHN^, N N D!(M) (B(M)) M I NvG NN X uH NO eH GN NYH JH F]HNbH P M. P_ PS, MO E N H RIH N N, HO CORNN=NLO, N_ uO u` `H} FY EbH ` uH u N M uH N, aH HO CORNN=@FM N_ N NO eH NYHJH F]H NbH P M, S H` GN X EX: NH N uH u N M uH N. LIH NLH

J NH N uH u, u HH uH u, N eH N]G O N LOIP u HHuH u NL uH P NLIH uH u N_G EG N M uH P. & u`, u P N N D!(P)H M u ZO FH _ NLIH H` H MO:

$@ CORNO NH

BH

LOIO:=P;

D!(O);

H;

CaH u P N N B!(P) H M u ZO FH @ NLIH H` H MO:

I C/

BH

$EE;

CORNO:=@FM;

LOIO:=J;

H;

39/164


42/166

CS ]: wH u N_G u NI NON NIXH N_G EG NI O LH FY uH u HHHN (NI M H) N M uH N. wO HYS MOS NL NJ N N N NvG uH NI NON NIXHN_G EG NI O LH.

%J = f = + W =y:

[S I N u N G I H]O NYH JH F]H NbH PY NH NYH JH F]H NbH H` FY G HH uH HYI.

DI N N D!(P) (B!(M)) I J\J NvG NN X uH NO eH GN NYH JHF]H NbH P M, H]H M NYH JH F]H NbH u N H M FH uH N N HYS.H u aH F FY X HH X uH NLIH NH NYH JH F]H NbH. eH N]GH $HRP u] HH M NYH JH F]H NbH uH P, PY eH N]GH $HIHHN u uG M NYH JH F]H NbH (H HYS H E NI X NL 0). NGH(P) NLIH X N N D!(P) PY B!(P) ` HG P XH: $HRP:=IHHN,

aH ZO FH @ NLIH X H NL_H ^H uH X N N HYS H u M F HMO:

$HIHHN:=0;

$@ CORNP NH

BH

$HIHHN:=$HIHHN+1;

D!(P); (*B!(P)*)

H;

&N Nd PaH FJ N NLIH H NL_H ^H, $HIHHN I M NYH JH F]HNbH u N, aH H GH $HRP, P I J\J FN E] X uH NO eHGN NYH JH F]H NbH.

CH NL_H ACA' Y NIXH NL]H ` N PN H MO:

? C $ $ D D$ A &$ A $ '$ CAC A A $ &$ D $?

OMM LN;

PL

:LLS1..20,1..20 @I SN;

40/164


43/166

$EE, C/, '@: /@@/1..20 ;

,,H,MIFN,E,M,N: HNL;

NIJ: IIFH;

C: L;

LIOL JMIFO;

BH

LN(nCI MI H O I N:o); LFH(H);

LNFH(nJ MI FO G NLH E:o);

!IL := 1 NI H I

BH

!IL := +1 NI H I

BH

LN(nn,,o,o,,o=o); LFH(,);

H;

,W:=0; QLNFH;

H;

H;

U===========================W

LIOL L@F;

L @:NRN; @H:MNLH;

BH

LN(n CI NH @F O FO:o); LFH (@H);

41/164


44/166

AMMH(@H@H); LMN(@); LFH(@,H);

LNFH(nJ MI FO G NLH E:o);

!IL := 1 NI H I

!IL :=1 NI H I L(@, ,W);

CFIM(@);

H;

U===========================W

LIOL $HMIFO;

BH

LNFH(n NLH E:o);

!IL := 1 NI H I

BH

!IL :=1 NI H I QLN(,:3);

LNFH;

H;

H;

U===============================W

LIOL &NKOFHNIH;

BH

$HMIFO;

$@ MIFN=1 NH QLNFH(nDI N F FH NIHo)

FM

42/164


45/166

BH

LSFH(nH JH FH NIH NO n,,o IG H:o);

!IL :=1 NI H I @ CORN= NH QLN(:3); QLNFH;

H;

LN(nI HNL NJ NOlo#7); LFH;

H;

U========================================W

LIOL B!(:HNL);

(*NG EG NI O LIH N O NO H *);

PL O, O, COI,: HNL;

H

O=1; OI:=1;

OI:=; CORN:=IFN;

F O


46/166

H;

E; ? @@ BF& ?

U==================================W

LIOL D!(P:HNL);

(*G EG NI O MO N O NO H P*);

PL : HNL;

H

CORNP:=MIFN;

!IL O:=1 NI H I

$@ (P,O=1) H (CORNO=0) NH

BH

LOIO:=P;

D!9(O);

H;

H;

U=================================W

LIOL 'HNIH;

BH

? 7 / 7 ?

@IL :=1 NI H I CORN:=0;

MIFN:=0;

@IL :=1 NI H I

44/164


47/166

@ CORN=0 NH

H

MIFN:=MIFN+1;

? BF&(7); ? DF&(7);

H;

&NKOFHNIH;

H;

U===============================W

LIOL &NKOOIH;

BH

$@ LOIN=0 NH QLNFH(n&IH I OIH NO o, M,o H n,N)

FM

BH

LNFH(nDOIH NO n,M,o H n,N,o F:o);

%:=N;

LN(N,o


48/166

H;

LN(nI HNL NJ NOlo#7); LFH;

H;

U============================W

LIOL OIH;

BH

$HMIFO;

LN(nG OIH NO H:o); LFH(M);

LN(n H H:o); LFH(N);

F@ 8:=1 ? 7 / 7 ?

BH

LOI:=0;

CORN:=0;

H;

&7:=1; BF&(); ? DF&(); ?

&NKOOIH;

H;

U============================W

LIOL GHO;

BH

CFLML;

LNFH(n$ D D$ A &$ A $ '$ o);

46/164


49/166

LNFH(nCA D $% A A $ &$ D $o);

LNFH(n===============================================o);

LNFH(n 1. J MI FO NO H JGo);

LNFH(n 2. J MI FO NO @Fo);

LNFH(n 3. &G NL NH FH NIHo);

LNFH(n 4. G OIH O Ho);

LNFH(n 5. INo);

LNFH(n--------------------------------------------------------------o);

LN(nS I JG MI IH O HHl#7);

C:=LES;

LNFH();

H;

U===================================W

? !/7 @@/ ?

BH

LJN

GHO;

M I@

n1o:JMIFO;

n2o:@F;

n3o:'HNIH;

n4o:DOIH;

47/164


50/166

OHNF (=o5o) IL (OJM ()=o);

H.

48/164


51/166

Y =y E>/; W =y H+=

LIH H HYS dH L M H]H O H u N u N FY u N OFL PY uN GFNIH. D uZS, HO EbH ` N^ MOH, NON H u N u eH u OH u u N Pb H PY ` H, PY NON H H M eH u OH H u N Pb H ~H H OH u N ` H.

Y %Hx E&LE#

Yy [+ 1. CO NL_H uH NLIH u N u KO G H H` GN FH u FY O NL_H OFL. wH u uH NLIH u KO G H H` GN FH u FYuH u OFL. w N u FY u N OFL HO H` ` O NL_H OFL, PY FY uN H OFL HO H` ` uH u OFL.

c LYH G u N OFL FObH FY H OFL, HH uO H F EbH FObH udH.

% 1. w N 1 NLIH _H 1 FY u N OFL P_ H` ` O NL_H OFL , , , , , ,. w N 3 EbH ` O NL_H OFL HH H` ` uH u OFL , , , , , , , ,P_ N 3 FY u N OFL. w N 2 EbH ` O NL_H ~H H uH u OFL.

HO 1. ` } G 1 , G 2 , G 3

% 2. w N 2 NLIH _H 2 FY u N OFL P_ H` ` O NL_H OFL , , , , , .w N 3 EbH ` O NL_H OFL HH H` ` uH u OFL , , , , , P_ N 3FY u N H OFL. w N 1 EbH ` O NL_H ~H H uH u OFL.

49/164


52/166

HO 2. ` } H 1 , H 2 , H 3

wO EH H PY u u GN u N FY GN u N OFL u OFL N_G L PYI HvG 1736E bH KOSN Y NIXH ` d H NH N N u` P S X O NYH J&IHML PY uZS FY uH Ft uO N]H Ft NOSN u N.

Yy V 1 (E>/;).` } R e 7M R G ^ } E@ 97 | 97 ~7 ^|/ G ^w P l q.

w H GH uH Ft NL N N H GH u:

B Wr.O G uH u N EbH H H 2 N_ O NL_H.

C .

O ` H FJ N_ EH uH u FY H H]H. _ PS M FY uH u N.

P FY GN uH HYI u` . M RZS H NI KO HJ uH u

P | P1 |P2 |. . .

NLIH u` P1 FY uH E P P, aH P j1 H P+1 # P-F (` N H P+1 H PS FY P_(P) j2). DI NJ uH FY O H , H]H MO GN M O H N J KOSF GN uH u[ RON H NL u`. uH uO N]H H N FY PE. & u`, uIH uH u RZS H HG uH PE FY 1 O NL_H H N_G.

C Wy V:

Cf. M FY u N OFL N FY NH N O NL_H OFL NLIH . & u` G FHO NL_H u KO GN uH HYI u` uH u` NvH F]H 2. GN EX G H u N RON H NLIH udH GN FH, MOS L G uH u N uO ` H.

Y."> ` =/


53/166

KO NN X H . & u` FI E NN X H NO C N NO uGN u N G PH ` FY H. I NN KO HJ, NLIH G NYH JH F]HNbH uO N_G u O NL_H OFL. DI FY F]H NbH H]H NLIH G NYH JH ` ^N HN GN uH OH P O NL_H C. _ PS, N ` N RZS H O NL_HOFL NLIH H MO: N uO N GN uH HYI u` O NL_H C, u NI X H C H HYI J J uH EbH b FJ . O J J uH H PS N Mu NI O NL_H OFL NYH JH F]H NbH uH u`. O u` F NJ Nu NI H C I uH E J J uH EbH b FJ N_ F NI O NL_HOFL NYH JH F]H NbH NH H NLIH P.Pl (RG _H 3). OX NL_H M ENNd E N NL P uH RON JXN , N FY NO u O NL_H u KO G H u NudH GN FH.

_H 3. H I I H GH uH Ft 1.

Hv >b 2. w N Pb H F]H NbH FY H OFL E PY E H` ` EbH KOX 2uH F.

C . PS , HO ` EbH KOX 2 uH F N_ M uH F H` ` N FY 0 I 2. O EbH ` uH F N_ NI uH Ft 1, H` FY u N OFL. M ` 2 uH F FY O PY P. FY u N NO u N H X N]G PYI GNuH G Q PY H (Q,O) PY(Q,P). & u` NN X uH uO ` H,

P_ N NI uH Ft 1, H` ` O NL_H OFL C. IX E O NL_H HYS uH Q PY H E H` N NO u uH u OFL NLIH u N .

M FY u N OFL, N H GH uH Ft N ` N N MO u N_G O NL_H OFLNLIH .

LIOL OFLCSF;

BH

51/164


54/166

&'AC:=; CE:= ;

CIH O F GIN H HI I O I N;

&'AC;

7 &'AC

BH

:=NIJ(AC&); (* R F JH NO O AC&)

I ()

BH

:=H O NH NLIH H M &(R);

&'AC;

(* FI I H (R,S) EI I N *)

():=() ? ? ;

():=() ? ? ;

H

FM

BH

&'AC; CE;

H;

H;

H;

M FY u N OFL, NON NIXH uH H MO uZS I J\J RX uH O NL_H OFLE FYG H NS.

52/164


55/166

%>i= = F;

ON JXN N GN uH O HYI u` N u NI X H H` GN X NO t HNOZH N 2 KO N MO:

(1) IX H u[ u KO uH N RIX HH uH b FJ NI NYH.

(2) G N u KO O E EbH aH X F IH HYI EX.

C = W Wj + =>i= =.

L N]H N L LH N N NL]H ` N N H G . M N u uH GNuH P HYI u`, E u` HO P#O N_ u N IH aH F FY F]H NbH PY udH uH F FY P PY O. I KO NLIH ` uH u OFL N P N O. DI P RIX H uO N]H uH u EbH FYG GN N^H F]H NbH , N u` MOS L NN ` N N H G . O P=O N_ FJ FOH NL]H M PH udH H HYI PH

aH H E P O.

PS aH J L N N NL]H H uH uH u OFL. PS NLIH EbH N aH H u KO E GY N M H H O eH E P O (NLIH NLHJ H F, P FI GN H HYI u` E P GN NLIH M HH H aH F u KO M H uH GN u N EbH F]H NbH, PY uO u` FY GZO NOH P NN ).

CH GH NH N H NLIH uH Ft 1 N NO u EN KO MO uZS I u N `H.

Yy V 2. w N ` H F]H NbH GH FY u N OFL E PY E

D+(P)=- (P), P .

53/164


56/166

Y =y H+=

LIH G HYS dH N R\N Y NIXH NH N H NLIH G NL EX FY N KOHNZG uH uH u KO NN X uH u N, G uH udH GN FH. NS u

NH H H FY EbH uXH E HYS NL]H N N u[ H uH M J NJ IX PH uH KOSN.

Yy [+ 2. wH u KO NN X uH u N G uH udH GN FH u FY uH u GFNIH. CO NL_H N uO N GN uH P HYI u` KO NN X uH aH FG uH udH GN FH L KOS NL P P u FY O NL_H GFNIH. w N u FY u N GFNIH HO H` O NL_H GFNIH PY FY u N H GFNIH HOH` ` uH u GFNIH.

c LYH u N GFNIH FY H GFNIH, HH uO H F EbH aH udH.

% 3. LIH _H 4: 3 FY GFNIH, 2 FY H GFNIH aH 1 EbH FY HGFNIH.

_H 4. w N GFNIH 3, H GFNIH 2 , PY 1.

CI uH HS P N_G GN N]O OH HH N u N GFNIH PH aH FY G, G euZS FY GN PH u NLOH NZG Ft NOSN u N. H N H, I uH HS ~H ` NON NIXH O KO u EG NL GN u N ` FY GFNIH S EbH. CX EN KO NOu JH FH FY uO EH u u GN u N FY u N GFNIH. H FH dH uO `H "HO ` M H u FH N_ FY GFNIH". N EN KO H PS u JXN ONLIH uH Ft MO uZS.

Yy V 3 (D;+ 1952). wH u N Pb H P H>2 uH, G uH ` EbHH H H/2 FY u N GFNIH.

CH GH:

]G PYI u N E uH G PY H dH P NN X uH . M E FY MH HN X uH H N]G PYI u I u N NO u o FY u N GFNIH. M

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L LH E=0. PS, M H F FY E >0. &t O P, J, Q, . . ., P FY O NL_HGFNIH NLIH o, NLIH u` P, Q FY uH aH J FY GN NLIH M X uH G. &u` Q EbH E P P P_ HO H F, N EbH H M H J PY uO u` FY GZO NOHP NN E H HN. H N H uH (Qo H H) E P Q EbH N u FH MOuH Po (E P P) P_ LH E u` ` N NS

P| J| Q | . . . | Po| Qo | . . . | P

P| Po| . . . | Q| Qo | . . . | P

H X uI H uIH O NL_H HG Q PY Po. u` MOS L FY M uH u N o EbH E P Q FY EbH H H M uH E P P (N FY ^N HN ~H FY HH/2+E), uH N M uH o E P Q ^N L FY J H H/2+E. DI EbH ` uHHYI o P EbH E, F P E P Q, I H]H NH M uH u N o (o `H+E uH) EbH ^N H H+2E. ZO NOH NO u u[ H GH uH Ft.

wH Ft MO FY NH KOXN IX uH Ft DLE I u N ` H:

Yy V 4.

M FY u ` H F]H NbH P H uH. O

+ (P)jH/2, m(P) j H/2, P

N_ FY GFNIH.

C` GN M H u N GY N ` N N E HYI FY u N GFNIH. N P^ H PSFY u N uO FI. w N uO FI FY u N ` H GY NLIH u` uH N E H`u H P HO udH GN OH. ]H uO FI RON H H PS P_ u N H PS` N eH u O H EN KO N uO `H OSH, `H YH S N GN NLa HYI GY EbH I J\J IY. ` uH Ft MO:

wH Ft 5.

) u N uO FI FY H GFNIH.

) u N uO FI F]H NbH GH FY GFNIH.

% 4. w N uO FI D5, D6 u I NLIH _H 5.

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_H 5. w N uO FI D5, uO FI F]H NbH GH D6

ON NIXH FN E] NN X O NL_H GFNIH u N

ON NIXH MO uZS u RZS H NL]H M NON NIXH KOS FO I J\J FN E] NN X O NL_H GFNIH u N.

LIOL GFNIH(E);

(* 7 9 / @7 H/7 / 7 / @7 / 7 (1,. . . ,9-1) / 7 G=(,E) 7 / / 9: (), *)

H

@ (9-1)

@ (E =+1) H (S=P0) NH HH(1,. . . , H, P0)

FM

@ CORNS NH

H

E:=S;

CORNS:=@FM;

GFNIH(E+1);

CORNS:=NLO;

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H;

H;

(* H JLILG*)

H

@ C/:=@;

1:=0; (* P0 F GIN H HI I O I N *)

CORNP0:=@FM;

GFNIH(2);

H.

% 5. _H 6 uZS Gb N ZS N_G EG NI NON NIXH P Gb N.

_H 6. w N PY ZS FN E] O NL_H GFNIH H` NI NON NIXH KOS FO

LIH NLH J u N ` EbH KOX HO H NON NIXH NL]H ` N M H u EGNL u N ` J FY GFNIH S EbH.

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C > + W =y

w N Pb H F]H NbH EbH ` O NL_H FY ZS. &X HG ZS FH uO N]Hu CSFS u L PYI HvG 1857, E bH M H dH u uG GN H O NLd

JZH N X J N IX NLIH IX O . CZS aH u M H LH L[NLIH LN HO F}H P EX HO, u N NLIH NH , ZS u M H u RZS HX NON NIXH N X N G, X NON NIXH N , NLOSH FO PY N_G EGl

CQ 'P CSC %TNH CHc% C\ BaN CA CQ

Yy [+1.

ZS FY u N Pb H F]H NbH EbH ` O NL_H. w N EbH ` O NL_Hu FY LH.

PS, LH FY u N GY G NYH JH F]H NbH H` FY GN ZS.

% 1. LIH _H 1 FY GN LH G 3 ZS 1, 2, 3.

HJ 1. H G 3 ZS 1, 2, 3.

C` N H` ZS FY u N Pb H uH H HN. wH Ft MO uZS I N GN M N^H N ZS.

Yy V 1. M =(,) FY u N Pb H H uH. & u` X GH u MO uZS FY

NH uH:

(1) FY ZS;

(2) EbH O NL_H PY ` H-1 H;

(3) F]H NbH PY ` H-1 H;

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(4) F]H NbH PY G H H` uO FY O;

(5) uH N E u H P HO udH GN uH u uH;

(6) EbH O NL_H HH N]G PYI GN H N NO u udH GN ONL_H.

C . M H GH uH Ft NI M u MO:

(1)(2) (3) (4) (5) (6) (1)

(1) (2) I uH H} EbH O NL_H. M H GH H KO HJ NIM uH H I EH uH: H ZS P H uH FY H-1. c LYH EH uH udHP H=1. M H>1. L N HH LH NLIH G ZS ` H uH uO N_G u ^N HNGN uH FY uH NLI (N FY uH ` FY 1). PS, P1, P2 , . . .,PE FY uHu Y HN (NI MH) NLIH . & u` Lc LYH P1 PY PE FY X uH NLI, P_ N P1 (PE)EbH ` H H P N uH HYI NLIH M X uH P2, P3 , . . .,PE (I u N EbH O NL_H), ~H H P N uH HYI EX u N (I uH u uH R\N YHN). 'I P1 PY H (P1 , P2) E N NO u ZS 1 P H-1 uH, GY NI NN KO HJ ` H-2 H. S ZS ` H-2+1 = H-1 H.

(2) (3) H GH H JH H. M EbH F]H NbH. & u` JZH L[NYH Ej2 JH F]H NbH 1, 2, . . . E. DI EbH O NL_H H]H G (=1,2,.. .,E) ~H EbH O NL_H, P_ N G FY ZS. DI u` HO H() PY () NIN N FY M uH PY H , N `:

() = H() m 1, = 1, 2, . . ., E,

MOS L

H-1 = () = (1) + . . . + (E)

= H(1) + . . . H(E) m E

= H() mE < H-1

ZO NOH NO u H N FY F]H NbH.

(3) (4) FI GN H N E E H uH u N P H uH PY H-2 H LcLYH FY u N EbH F]H NbH. S G H NLIH uO FY O.

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(4) (5) DI FY F]H NbH H]H uH N E H` u H P HO GN uHu uH. O ` J uH HYI ` uH u uH EX HO H dH, N_ N u`MOS L u N O NL_H, PY P_ N X H NL]H O NL_H HYS EbH J FY O.

(5) (6) EbH O NL_H, P_ N HO ` O NL_H N_ IX L N_G u JuH u H P HO uH u uH. BZS , HO N]G PYI GN H H uH O PY P HYI u` . & u` H HYS eH P uH u uH H O P P MNI NYH O NL_H NLIH . CO NL_H NO u HYS FY OS HN, P_ HO NO u HOH GN O NL_H N_ MOS L NLIH NL u` J ` MH O NL_H.

(6) (1) M EbH F]H NbH. & u` G ^N L FY 2 NYH JH F]H NbH. _PS, HO N]G PYI GN H H uH NO NYH JH F]H NbH EX HON EbH NO u N]G GN O NL_H HYI . wO u` GZO NOH P NN (6).

wH Ft u H GH.

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C > + W =y

Yy [+ 2. w N PY ZS EOH H` u I NLIH _H 2

_H 2. w N PY X ZS EOH H`

wH Ft MO uZS I N M FH ZS EOH u N uS u &H:

Yy V 2 (C+/). & e K 9 / ^ } -2 .

wH Ft 2 I NS M FH ZS EOH u N FY GN M LN FH. BZS N R\NXJ H NON NIXH N_G EG NI O MZO PY NI O LH NL]H u N u RZSH ZS EOH u N Pb H F]H NbH. LIH NLH J G E N uHu uH G O (N CORNO=NLO) N uH P N_ H (P, O) M u EN HJ PYI ZSEOH. NON NIXH NH H u NL_H YS NLIH N N MO uZS.

LIOL MNLD!(P);

(* 7 97 7 / / / 7 / / / / 9 ' / 7 7 G 7 / / 9. C/ 7 C/, , ' / / *)

H

CORNP:=@FM;

F@ ()

$@ CORNO NH

BH

':=' ? (,);

D!(O);

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H;

H;

(* H LILG *)

H

(* $HNFNIH *)

@ C/:=@;

' :=; (* ' / / / / / 9 *)

D!(LIIN); ( LIIN F H HI I O I N *)

H.

LIOL NLB!(P);

(* NG EG NI O LIH J OH NG NJ H O O EOH O I N PI OIHFH NIH I I H M & *)

H

$:=;

$@;

CORNL:=@FM;

7

BH

;

F@ @()

$@ CORNO NH

BH

$;

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CORNO:=@FM;

':= ' ? (,);

H;

H;

H;

(* H LILG *);

H

@ C/:=@;

' :=; (* ' / / / / / 9 *)

NLB!(LIIN); (* LIIN F GIN H NOS S O I N *)

H.

Cd t:

1. 'J FOH NH N H NLIH JH NL ` N L u LH X NON NIXH Gb N NL]H ` u J NJ N^H NIXH (G+H).

2. CZS EOH N_G u NI N N NLB!() FY ZS uH u HH HN N L uHNN X uH aH F u N.

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=i > =;J ] b + W=y

BY NIXH RZS H ZS EOH u N F]H KOH N uH GN M Y NIXH H HEX Ft NOSN u N: Y NIXH RZS H NJ X O NL_H H u N GY N MR\N NLIH G HYS.

M =(,) FY uH u N Pb H F]H NbH, =(,) FY ZS EOH H`. CXH u N NO ZS EOH N M FY X H NLIH, aH X H aH F M FY H HIY.

Yy [+ 3.

O N]G GN H HIY PYI ZS EOH dH N M NO u udH GN ONL_H NLIH , Et O O NL_H HYS FY C. J X O NL_H

? = ? C : ?

u FY NJ X O NL_H H u N .

M A PY B FY NJ J, N u PYI J\J NIXH MO

A ? B = ( A ? B) ( A ? B).

J A ? B u FY O u RH NJ A PY B.

]H O NL_H H H FH P M EH FY G O NL_H u N uO ` N NOu N X O NL_H H H L NLIH uH Ft MO uZS:

Yy V 3.

M =(,) FY u N Pb H F]H NbH, =(,) FY ZS EOH H`. & u`G O NL_H u N uO ` N O H H FY O u RH GN M X

O NL_H H.

N_G NJ J O NL_H H GN P NLa KOH NLH NLIH PH u N^ GHuH. C N H, NI G O NL_H H u N NH H P GH uH HJZH N^ N M NN FJ u GN JH NL_H NOSH N^H NI uH FON &LI@@: NHO uH N NI GN G PaH FY H EbH. NH JH NL_H NOSH N^HNO u I J\J N^H NIXH O uH N NL]H G uH ZS F uH.

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M RZS H NON NIXH RZS H X O NL_H H NL]H N N N_G EG NIO MZO NL]H u N. ON NIXH ` O NLd NH N H NON NIXH RZS H ZS EOHNI N N N_G EG NI O MZO Gb N NLIH G NL.

ON NIXH RZS H NJ X O NL_H H.

NN LH u N =(,) u Gb N H H MX &(P),P.

LIOL CSF(P);

(* NG EG O NLH I H O NH JH FH NIH O H P; H , HOG, MNE, HR F H NIH O *)

H

:=+1; MNE:=P; HOG:=HOG+1;HRP:=HOG;

@ ()

@ HRO=0 NH SF(O)

FM

@ (O MNE-1) H (HRP>HRO) NH

:=-1;

H;

(* H LILG *)

H

@ I:=0;

HOG:=0; :=0; MNE0:=0;

@

@ $HRP=0 NH SF(P);

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H.

CS ]: w J NJ N^H NIXH NON NIXH P Gb N FY (? ? ? ? ).

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B = > | d=

BY NIXH ZS EOH H HN u N FY GN NLIH M HH Y NIXH N O NL]H u NN_G u H H NLIH HO F}H P EX HO u MH. LIH G HYS dH

N NL_H YS HH NON NIXH H u Y NIXH HYI. L N dH N JXN OH OH Y NIXH.

CI =(,) FY u N Pb H F]H NbH P NJ uH =? 1, 2, . . . ,H? PY NJ H G G H. H u N u XH P GN M EbH ZG (), FY u Y H`. M =(,) FY ZS EOH u N . u Y () ZS EOH FY NH u Y X H H`:

C() = ? ().

BY NIXH uN L FY NLIH NN ZS EOH u N [S N_G ZS EOH P u Y HHN. CZS EOH H PS H PS u FY ZS EOH H HN u N PY Y NIXHuN L u FY Y NIXH ZS EOH H HN.

w GH I I HH H H Y NIXH ZS EOH H HN, uZS, N JXN O Gb _H N N N]O O H`.

B7 K z ^e n. M N GOH RZS H GN NH uHMN H H NYH J MI I YH EX ` N u N N E GN NYH J HYI uH NE GN NLIH X NYH J aH F. N EX NL]H KOH uG EH N ua FY J^RZS H NH uH J H HN. c LYH u N GY uH FY X NYH J aHX H FY X NOSH uH MN H X NYH J NH H P JH XH RZS HN O J FY ZS. _ PZS, Y NIXH uN L H P Y NIXH N_G ZS EOH H HN NL]H uN uS u H uH, G uH NH H P GN NYH J, P u Y NL]H X X H^H FY J^ RZS H uH LS H NYH J NH H (d t FY NLIH YNIXH HYS N NN FY EbH RZS H NOSH uH MN ` X HY JZH NOSH HGHIY X NYH J).

B7 7 f N. CH H GH GN NH G H GXS N^H uXH M N 1uH H. BN J^ H GXS P GXS FY ,, , = 1, 2, . . . ,H ( NbH NH J^HYS J NO PYI u Y XJ H H M H). [S N_G X H GH MI I NH J^ H GH FY H HN.

w Y NIXH ZS EOH H HN, NN H]H ` N FN E] NN X ZS EOH u N PY H NLIH M ZS EOH S ZS EOH H HN. H JXJ H PS, NLIH

NLH J u N uS u, M ua N H HH-2 , PY Lc LYH EbH N N Hu HS P HH u N P M uH YH . N GS FY u P Y NIXH

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ZS EOH H HN dH N u[ ` HH NON NIXH LN O KO u dH. CdHN R\N NLIH M HH NON NIXH H PS: ON NIXH &LOMEF PY ON NIXH LG.

%>i= = K;>


71/166

_H 3. w N PY ZS EOH H HN

FH J uO N]H N FH FN MOH PYI NJ X H (3,5) , (4,6) , (4,5). c LYHHO N]G H (5,6) PYI N_ M NI NYH 2 H (4,5), (4,6) u[ ` NLIH O NL_H._H OH NH N ~H RS L u P H (3,4) FY H NJ NI [S. J NI

N MOH H (1,3), (2,3) PYI PY NO u NJ G 5 H:

= ? (3,5) , (4,6) , (4,5) , (1,3) , (2,3) ?

C^H FY NJ H ZS EOH H HN H N_G.

CH GH N^H udH uH NON NIXH.

c LYH u N NO u NI NON NIXH ` H-1 H PY EbH ` O NL_H, P_ PS NIuH Ft 1 H` FY ZS EOH u N . PS, aH J L LH ` u Y

H HN. M NH N ZS u N GY () < (). &t O E FY H uO N]HNLIH [S X H RZS H NI NON NIXH P Gb N EbH NO . E u` uN IH MH ZS u MOH H E M GN O NL_H C OS HN uKO E. DI O NL_H C J H NO HH EbH NO H]H u N IHNO u N H X NS H H` H E (Et O u N FY o) M FY ZSEOH. I X RZS H (E) i () I u` (o) i (), uH N M H OH o PY u[ NvH N]G 1 MI P M H OH PY . 'J F KOX NL_H NL]H NH GN N ` N H u NYH PY NLIH G NH u Y EbH NvH, N FY ()i (). ZO NOH NO u dH N FY ZS EOH H HN.

P FJ NL_H N H NON NIXH.

& FH N^H NIXH HO HN NON NIXH ^H FY MJ RJ X H uN NI N N EbH G u Y u F H H MOH. w P u N G HH J N H G FI G J\J NIXH u MJ RJ X H u N NYH [S EbHG NI u Y. OS H]H, u RZS H ZS EOH H HN P H-1 H, H` OHN EbH H J MJ N N NIYH X H GY H R\N JH NL]H [S u` L < G H. w FYG P u` N ` N M H X N N MJ RJ H OH uH (J

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ILN). LIH N N HYS, u NI uH uO N]H N GN (G) J\J NIXH, G JH NNJ NI NLIH uH ` N N_G MO N H (FI G). _ PS, P NH HYS NONNIXH M GN N H (G+J) FI G) I P MJ RJ X H. LIH N N N^HNIXH M J H H LN HO MI P G.

H u N NLIH P N H NON NIXH &LOMEF FY P F H H u MOH

ua J ` GN N N O KO EG NL NJ H ? ? W ` O NL_H SEbH. w t LH, X H NLIH X FJ NLOH H M NI NYH GN LH.CH H EI MXN M NI NYH O NL_H P X H NLIH E PY E uH uO H` NO PYI eH GN ZS IH LH H` NL]H. DI u`, HO H EbHNI NYH O NL_H P X H NLIH , N_ H` J H ZS EX HO NLIH . P_N, u EG NL RG ` N MOH H PYI N H EG NL RG H` ` H ZS EX HO NLIH S EbH. N NLIH X JH JXJ O KO u N HP EG NL HYS FY N M JZH I NJ X uH u N L NYH X NJ IH EbHI HO, G NJ RX uH GN ZS IH NLIH (u _H NYH X IP MOH H PYI ). H H, u P u N NLIH P^ 3, uO N]H N ` MXO NJIH 1 JH N: ? 1W , ? 2W , ? 3W , ? 4W , ? 5W , ? 6W . O E MOH H (3,5), N `X NJ IH ?1W , ?2W , ? 3,5W , ? 4W , ? 6W . N 3, N H H (4,5), E u` NJ IH u H F PY H MX X NJ IH FY ? 1W , ?2W , ? 3,4,5,6W . CH ` u YNJ NI FY (4,6), I uO H` NO PYI eH GN NJ IH?3,4,5,6W , H]H H` M NINYH O NL_H NLIH NJ HYS. _ PS H HYS EbH u H. Y NON NIXH M NJN H H NJ NI u EI MXN l

PS, u KOSN PH u N HYS N J RZS H N N: &G NL RG uO O, P H =(O,P) ` NO PYI NJ IH EX HO S EbH, PY NLIH

NLH J ZO NL F FY EH uH, H NJ IH NH H NYH GN NJ. Cd tLH G NJ IH NLIH JZH I ` N FO NL H FY GN ZS ` , PY E u` G M u M H FYG H[H HH N NJ IH NH H.CH NL_H NL]H MFN H NON NIXH &LOMEF P HH HH R\N P H]O ` N PN H MO:

(* r Cp & q &&A' CA w C B$ DA qC C *)

OMM LN;

NSJLLH=LLS1. .50 I@ HNL;

LLG= LLS1...50 I@ HNL;

PL

H,G, GH':HNL;

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73/166

DO, OI, :LLG;

DO, COI, !NL:LLH;

CIHHN:IIFH;

LIOL JF;

L

:HNL;

!HG:MNLH;

!:NRN;

BH

LN(nCI NH @F O FO:o) LFH(@HG);

AMMH(@,@HG); LMN(@);

FH(@,H,G);

!IL :=1 NI G I LFH(@, DO, COI, );

CFIM(@);

H;

LIOL $HOFO;

L :HNL;

BH

LNFH(nI H n,H,o. I H o,G);

LNFH(nDH O DH OI DI o);

!IL :=1 NI G I

LNFH(DO:4, COIW:10, :12);

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H;

LIOL J(!LMN, 'MN:HNL);

L ,E,N1,N2,N3 : HNL;

BH

%:=@LMN;

F (


75/166

LI:=;

F !NLLI>0 I LI:=!NLLI;

!H:=LI;

H;

LIOL HIH(,:HNL);

L R:HNL;

BH

R:=@NL+@NL;

@ @NL>@NL NH

H

@NL:=@;

@NL:=R;

H

FM

H

@NL:=;

@NL:=R;

H;

H;

LIOL &LOMEF;

L

$, 'MN, O,P,L1,L2, H, H:HNL;

73/164


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BH

(* &I NI GH !NL H O S IH P EI NI J *)

@IL := 1 NI H I @NL:=-1;

@IL :=NLOH(G/2) IQHNI 1 I J(,G);

FMN:=G; H:=0; H:=0;

H':=0;

CIHHN:=NLO;

F (H


77/166

H;

(* I O F J *)

DO1:=DO'MN;

COI1:=COI'MN;

1:='MN;

'MN:='MN-1;

H;

$@ HH-1 NH CIHHN:=@FM;

H;

LIOL $HENKO;

L :HNL;

BH

LNFH(n******************************o);

LNFH(n***** &N KO NH NIH ********o);

LNFH(n*******************************o);

LNFH(nDI O S EOH HI HN: n,H');

LNFH(nC H O S EOH HI HNo);

!IL :=1 NI H-1 I

LNFH(n(n,DO:2,o,o,COI:2,o)o);

LNFH(n*******************************o);

H;

BH

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CFLML;

JF;o

$HOFO;

&LOMEF;

$@ IHHN NH $HENKO

FM

LNFH(n DI N EIH FH NIHo);

FH;

H.

!F FO Y NIXH NLIH P^ 3 ` H MO:

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%>i= = !;

ON NIXH &LOMEF FYG P E\G O KO P HH u N YS (u N P M H GkH(H-1)/2). LIH NLH J u` NON NIXH LG N L O KO H. ON NIXH LG aHu FY JH JXJ FZH H H HN. LIH JH JXJ HYS N uO N GN uHNO t u N, uO N]H N H M P uH FZH H H H` HN, H H FY uH S.} FY NLIH M X H E uH M, H (M,S) ` u Y H HN. J NI NLIHM X H E P uH M I S N N_G H ` u Y H HN, H HYS H uHuH N T, PY N NO u ZS JH G 3 uH PY 2 H. OX NL_H HYS M NJN I uH E N NO u ZS G H uH PY H-1 H M ^H FY ZS EOH H HNH N_G.

M u N I G NLH NLH M C = ? ,, , = 1, 2, . . . , H? . NLIH KOX NL_HN H NON NIXH, G u ` N HH `H H uH PY H H MOHPYI ZS EOH, X uH u N M u XH I X H[H. [H GN uH P M

G JH PY ` H P, HLP, NLIH u` P eH u HH u Y H` u Y H HN NLIH M X H H P uH P P X uH ZS EOH uHRZS H (N M FY EIH X N uH P uH NJ uH ZS EOH), H` GN X^H RX

P:= GH ? P,Q : Q ? ( = P,T),

aH HLP HH uH ZS EOH H P HN (HLP:=T).

ON NIXH LG u Gb N uS u NLIH N N MO:

LIOL LG;

BH

(* OI EI NI *)

IH M F GIN H HI I O I N;

H:= ? ? ; ':= ; :=0; /@:=.

F@ H

BH

DP:=M,P;

HLP:=M;

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H;

(* OI FJ *)

MNIJ:=@FM;

QF HIN MNIJ I

H

7 H / /:

=7 ? : H ? ;

H:= H ? ? ? ; ' := ' ? ? (, /@) ? ;

I ? H ? =

BH

=( ,) F S EOH HI HN O I N;

NIJ:=NLO;

H

FM

F@ H

$@ P>O,P NH

BH

P:=O,P;

HLP:=O;

E;

H;

H;

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% 4. _G ZS EOH H HN I u N R\N NLIH P^ 3 NI NON NIXH LG. NLH NLH M u N ` H

BH uZS H[H X uH NLIH X FJ NON NIXH, uH uXH O *FY uH u H u MOH PYI ZS EOH (E u` H[H H` EbH aH H uNLIH X FJ NJ NI, P_ PS N uXH O m u HH uO u`):

M= | d=

LIH G HYS N H]O Y NIXH ` N H P Y NIXH ZS EOH H HN, PY P_ N` N u H X NON NIXH Gb N NLIH G NL

) BY NIXH ZS EOH FH HN. D YH HH NS LH NLIH X NON NIXH NL]H NEbH H M H uH ua P O u Y. _ N ` N XJ H dH u P

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u N ` X H P u Y ` O NO t. _ PS, M N J N_G ZS FH HN (NFY ` u Y () FH HN) N_ H u O NN X u uI PY XJ H GN NLIH NON NIXH P Gb N.

) BY NIXH N_G GH uH P u NH S FH HN. CI F uH ` H HdN. wH ZSH HdN P HdN ` u NH S 1>J ,>0, , =1, 2, . . . , H. = (, ) FY u N

NH H P F uH HYS. [S N_G ZS EOH u N P u NH S

J()

FH HN.

BY NIXH HYS H P Y NIXH N_G ZS EOH P NH u Y H HN NL]H u N Pu Y G H FY mFI J(). PS, M FY ZS EOH H HN NL]Hu N P u Y m FI J(), N `

- ? FI J() i - ? FI J() o

P G ZS EOH o u N . u` MOS L

? FI J() j ? FI J() o

I u`

FI J() j FI J() o

S FY

J() j J() o

P G ZS EOH o. S FY ZS EOH ` u NH S FH HN.

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B = W^ W j d=

LIH X H H N N, PY NIXH N_G uH u HH HN uH GN uN F]H NbH ` GN t H} NI FH. C` N H P Y NIXH H PS HO Y NIXH NN KOH NLH. ^ , Y NIXH H GN YH NL_H NN EG HN (NI N]O OH IEIH X I N H I J^) NL]H GN GH I NbH uH , uHNS I uH EbH; Y NIXH H GN JH JXJ NN EG HN u u L GN NH uH F N NLH NX RON JXN uH NLH GN NLH NX u^, Y NIXH FJ FN bH X bH X bH uIH NLIH GN bH NL_H N bH FH, Y NIXH F HuH NLOSH NH P J^ H HN NLIH GH NbH NH, P.Pl H HS ` LN HOJH JXJ u X Y NIXH H PS. HH, NbH NH, X NON NIXHu RZS H NL]H M Ft NOSN u N N L FY X NON NIXH ` O KO IHN. LIH H HYS dH N M R\N GN M NON NIXH H PS.

CSC KHSI NIuM M Ye&

LIH H HYS dH N R\N u N ` H =(,), VV=H, VV=G P X OHu XH NLH M, H} FY, G OH (O,P) H` u uN NH H P GN MN (O,P) FY NLH M H`. CdH N M uN (O,P) = h , HO (O,P) . O [S

P0, P1, . . ., PJ

FY GN uH u NL]H , N_ u Y H` u uH H} FY NH MO

J ?(P-1, P). =1

N FY, u Y uH u ^H FY NH X NLH M NL]H X OH H`. (Cd tLH HO dH N XH NLH M I NN OH uO H 1, N_ N NO u uH H}u Y uH u H FY M OH uH u H H NLIH X H NL u[R\N).

BY NIXH N_G uH u HH HN NL]H u N H NH KOXN ` N JXN O HMO: N_G uH u ` u Y H HN N GN uH RON JXN M uH uH O (u^) N . wH u H PS N M FY uH u HH HN N M uH N aH u Y H` N MEt O FY (M,N) PY aH FY EIH X N M uH N (EIH X uH H} H PS` N FY M ZG). O H EbH NH N uH u N M uH N N_ N M uN (M,N)=h . cLYH, HO H G O NL_H NLIH u N uO ` u Y H, NLIH uH u HH HNEbH ` uH HYI FJ F (uH u EbH ` uH FJ F M FY uH u H).N EX HO NLIH u N ` O NL_H P u Y ZG (O NL_H H PS u HH H

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N FY O NL_H ZG) N_ EIH X GN M J uH HYI u` u N ` N FYEbH RX uH, P_, H X u PaH NI O NL_H HYS GN M u FH FH, N ` N L uH u X uH HYS ` u Y H H N M N I NL HYI. LIHHH NLH J H PS, ` N uN PH u N_G uH u H HH HN, NOS H]HY NIXH uN L M NL H]H J NJ H LN HO, P_ H` Y NIXH R\N M NH NuH u GFNIH NLIH u N H FY GN NLH J L]H.

L N H d t LH HO N EIH X N M uH N, N_ uH u HH HN N MuH N, NLIH NLH J NLH M EbH ZG, ` N N_G u GN X YH. w N_GuH u, H u t FY u P J uH M, N NO t (M N) FObH N_G u uH PMI I

(M,N) = (M,P) + (P,N).

PS, uH P H PS ^H FY uH u NL uH N NLIH uH u HH HN N MuH N. J NI N F ` N N_G u uH O MI I (M,P) = (M,O) + (O,P), . . . NN P N^H EbH ZG X NLH M YH MOS L LH [S N, P, O, . . . EbH uH FJ F PY EN Nd uH M. c LYH [S NO u RX uH (HO FN H N NX uH NLIH H`) uH u HH HN N M uH N. u` N ` NON NIXH MO uZS u N_GuH u HH HN N M uH N E N u Y H`.

LIOL !HN;

(*

wO PYI:

D - 9g ^| ^v i g ^| Q f7 ;

- uH u^;

/,, , X/ @l @~ @M .

wO L:

H NE [S uH RX uH uH u HH HN N M uH N

*)

H

/9:=; /9 ; :=;

QF P M I

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H

O:=uH NI G[H P=O+O,P;

/9;

P:=O;

H;

H;

Cd t LH u J NJ N^H NIXH NON NIXH FY (H2), I u N_G uH O N J R\N KONN X uH u N. N H]H, N ~H ` N M H E NON HH uH uu[ NL_H YS NLIH H 3: eH H GH LOIP, P , u H uH u NLP NLIH uH u N_G EG.

C~H H FO t N]G FY NLIH NLH J NLH M NL]H X H FY EbH ZG, Y NIXHN_G uH u HH HN NL]H u N Pb H ` N H P Y NIXH NL]H u N ` H,H X NS u G H H` H` OH ` H H O HO PeH NLH M FY NLH M X H NH H. OS H]H, NLIH NLH J ` NLHM ZG, P NS H PS ` N H uH O NL_H ZG.

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Y^ W j d= >d= = = = Ww

H FH X NON NIXH N_G EIH X uH M PY N u RZS H H E NONN^H NIXH GY N ` N Gb N u N H MO: N G NLH NLH M O,P, O,P , N N^H

H NL]H P EIH X N M uH NN X uH P . E JXN H

O + O,P < P (1)

H NL]H P M u FYG NN F]H: P + O,P.

OX NL_H u` M EN Nd E HYI dH N EbH FYG NN N]G u N E H NL]H HYI.& u`, Lc LYH X NL G P M I EIH X N uH M uH uH P. & NH E NON N^H NIXH HYS NL]H GXS N^H, H NL]H P M u FY H[H uH P,aH P N^H F X H HYS M u FY N N XH. H NS LH u N^H EIH

X N M uH N, uZS, N J N^H EIH X N M uH NN X uH aH F u N.H HS PH N NON NIXH HYI I J\J N_G uH u HH HN uHFYG P N M O KO H HH NON NIXH N_G uH u HH HN N GN uH uHNN X uH aH F.

u N^H NIXH GY N P Gb N aH RX uH, P_ aH J L N N X uHO PY P u EG NL uO EH (1). N H HYS ` H H LN FH uH O KO NON NIXH.

BZS N M Gb N NOZN NIXH !IL-BFFGH N_G uH u HH HN N uH M uH NN X uH aH F u N. ON NIXH FYG P NLIH NLH J NLH M X OHFY NO t, HH NN LH NLIH u N EbH ` O NL_H ZG.

LIOL !ILBFFGH

(*

wO PYI:

w N ` H =(,) P H uH,

& ^| i ,A,, , , / @l @~ ;

NN: w N EbH ` O NL_H ZG.

wO L:

g ^| ^v i g ^| Q f7 , .

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'@e, , 7 l ^| ^7 @e @ ^e ^7 n i ^v .

*)

H

(* & NI *)

@

H

P:=M,P;

'@:=;

H;

M:=0;

@IL E:=1 NI H-2 I

@ ? ?

@

@ P > O +O,P NH

H

P:=O+O,P;

LOIP:=O;

H;

H;

^H udH uH NON NIXH ` N H GH NL]H M NL]H HOS]H Ft N O KOS

I uH. c LYH FY u J NJ N^H NIXH NON NIXH FY (H3). 'O t LH dHN ` N G N PaH FJ NI E E JXN H NLIH KOX NL_H N H PaH FJNLIH EbH ` H P HYI u X NL. HYS ` N RS L u P E


88/166

P u N N NN H FY M H H MX E &(P), P , u O H u N, E u`PaH FJ NI O H PN F H

!IL O &(P) I

$@ P > O + O,P NHBH

DP:=O+O,P;LOIP:=O;

H;

LIH NLH J HYS N NO u NON NIXH P u J NJ (H,G).

% 1. R\N u N NLIH _H 1. CX EN KO N^H NIXH NI NON NIXH u Gb N NLIHH uZS

HJ 1. H NON NIXH !ILBFFGH

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BH EN KO N^H NIXH NI NON NIXH !ILBFFGH

LIH X G NJ NI dH N M R\N GN M NLH J L]H Y NIXH N_G uHu HH HN GY u dH ` N RZS H HH NON NIXH O KO H NON NIXH!ILBFFGH. w` FY E NLH M NN X OH FY X M EbH ZG I FY E uN EbH ` O NL_H.

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%;^ + =; =;{ i= = D


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LOIP:=M;

H;

:=0; ':= ? ? ; (* ' l | L f 7 *)

(* B FJ *)

7 '

H

O ^| ' g L =7 ? :' ? ;

':=' ? ? ; (* C ^} L / | *)

F@ '

$@ P>O+O,P NH

BH

P:=O+O,P;

LOIP:=O;

H;

H;

H;

Yy V 1. ON NIXH DEMNL N_G u uH u HH HN NL]H u N MO N H

(H2).

C .

L N N H GH FY NON NIXH N_G u uH u HH HN N uH M uH XuH aH F u N. M GN FJ HYI u` X H[H uH I N u Y XuH u HH HN N M uH X uH ` H[H uH, N M HGH LH FH JNJ NI HO uH O* NO u H[H uH (O*) ^H FY u Y uH u HH HN NM uH O*.

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92/166

&t O 1 FY NJ J X uH ` H[H uH aH 2 FY NJ X uH ` H[H NG N

FJ uH R\N. &N Nd G FJ H[H NG N (O*) I N u Y uHu HH HN N M uH O* EbH HG NLH NLIH NJ 1, N FY H` u KO ^N HN GN uH NJ 2. T 2 FY uH uO N]H H PS NL]H uH u HYS. DI NLH M NL]H XOH FY EbH ZG, H]H uIH uH N T uH O* ` u Y '>0 PY

(T) < (O*) m ' < (O*).

BN uH N HYS FY GZO NOH P X RX uH uH O* FY uH ` H[H NG N HHN. S uH u HH HN N M uH O* J HG NLH NLIH 1, PY P_ N, O* FY uY H`. DI FH FJ uO N]H 1 = ? M? PY MO G FH FJ N N]G PYI GN uHO* H]H NN FY (P) I u Y uH u HH HN N M uH P P G P 1 FY udHP FJ uO N]H. I KO HJ MOS L NON NIXH I N uH u HH HN N M uHG uH u N.

BZS N M uXH X M J\J NIXH H N H NI NON NIXH. f G FJ uN_G L uH O H J N H (H) J\J NIXH, PY u XH H[H F ~H H N HGN M FH J\J NIXH ~H FY (H). NON NIXH J N H H-1 FJ, P_ PS N

H N^H NIXH NOH NIXH (H2).

wH Ft u H GH.

& N_G u u Y uH u HH HN P N_ uH u HYS ` N N_G PYIH[H LOIP, P, NI KO N H H dH N u[ R\N NLIH H 3.

% 2. _G uH u HH HN N 1 uH X uH aH F u N _H 2.

HJ 2. H NON NIXH DEMNL

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&N KO N^H NIXH NI NON NIXH u NL_H YS NI H uZS. O PN NYH JH H[H NI N N: P. wH u uXH O * FY uH u H u uH H[H FJ uH R\N, H[H H` EbH H u X NJ NI, P_ NN uXH O -.

Cd t:

O H N_G uH u HH HN N M uH GN uH N HYI u` N_ ` N EN Nd NONNIXH E uH N NL NYH ` H[H uH.

H N H NLIH G 2, YH Gb N NON NIXH NLIH NLH J u N I H MX E. w ` N G N E FH N^H NIXH NLIH P RX uH uH O G FJ, ` N M H NON NIXH MILN (NH N H NLIH H 5 E N HNON NIXH &LOMEF). & u` ` N NO u NON NIXH P u J NJ N^H NIXH FY (GFI H).

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Y =y =; W =y > =;J

BZS N R\N NLH J L]H N Y NIXH uH u HH HN, GY u H`

` N RZS H NON NIXH P u J NJ N^H NIXH (H2), u` FY E u N EbH `O NL_H (aH NLH M NL]H X OH ` N FY X M N NO t). L N N HGH uH Ft MO.

Yy V 2. M FY u N EbH ` O NL_H. & u` X uH H` ` N uXH MMI I G OH u N H N uH ` M H H uH uH ` M FHH, H} FY G OH H` ` M O H H (P, P), NLIH u`


95/166

(* 'N /= - () *)

@

@ () /:=/+1;

$:=;

F@

7 /=0 $ ;

HOG:=0;

7

H

;

HOG:=HOG+1; O:=HOG;

@ ()

H

IP:=IP-1;

I /=0 ;

H;

H;

H;

ON NIXH u RZS H NL]H t NH LN uH H MO: Lc LYH NLIH u N EbH` O NL_H I ~H N_G u uH ` XH PYI H 0 (EbH ` OH u PYI). PS, N uO N uH P1 HO ` OH u PYI H` N P2 N_ N F OSH MH R\N uHP2 . O ` OH N P3 u PYI P2, N_ N F OSH MH R\N uH P3. . .DI u N EbH `O NL_H H]H MO GN M O H FH OSH H PS N J u uH uH EbH ` OHu PYI. IN N]H, N_G X uH H PS u N. c LYH N ` N uXH M dHNI N N NO t N uO N 1. J NI, FI E u N HH uH u[ u uXHM eH X OH u L E dH, N NO u u N G ~H EbH ` O NL_H, PY

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N N u FJ P u N G HYS. OX NL_H u` M u NJ N I uH E NN XuH u N u uXH M.

Cd t:

c LYH NLIH E NI L J OSN KO NN X OH u N E N^H XH

PYI X uH, P_ PS u` N NH (G) J\J NIXH, NLIH u` G FY M OH uN. J NI, G FH uXH M GN uH, u N H P FI uH u[ uXH M eHP X OH u L E H`, dH N F OSN KO NN X OH HYS. OS L u uXH MNN X uH u N dH N M J OSN KO NN X OH u N GN FHH. S u J NJ NON NIXH FY (G).

ON NIXH ` N XJ H u EG NL RG u N ` O NL_H S EbH? PS, HO EN Nd NON NIXH PH aH ` uH u uXH M (HOG


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@IL :=2 NI H I

@ 8 :=7(, 8+/8, );

H;

w J NJ N^H NIXH NON NIXH FY (G), I G OH u N J R\N KO udHGN FH.

CX NON NIXH u Gb N NL]H NH u H H PYI P RZS H HHJH JXJ Y NIXH uO EH P N H HH XH FH, NN FY (LIN PFONIH H PQ HKO) S CD (CLNF JN NI). N N^ uH H I H H HYS u Gb N NLIH N^ uZS.

% 4. N bH GN bH NL_H FH u NYH H bH uIH, uXH M N1 uH H. C` GN M bH uIH GY P N H H` u NH YH MO E GN MbbH uIH HYI u` u[ IYH NYH. w P G IH uIH N N FY N H H NNu IYH NYH H` (=1, 2,. . .,H). D FO P H=8 u I NLIH H uZS

M N uG N uO NH YH N bH bH NL_H FY 0. [S N_G NH u N bHbH NL_H ( Lc G bH uIH J u N uO N H PYI N uG HYI) IbH NL_H u IYH NYH RIH NLIH N uG MG HN ` N u.

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` N RZS H u N ` H H uH O H H P NL_H N N H XbH P H MO: uH u N NH H P GN bH P, HO bH P J u N H NL bH uIH N_ NL]H u N ` OH (,), NLH M NL]H OH HYSu XH H N, RG _H 4 uZS.

_H 4. w N GH I

]G PYI u N uH 0 PY H+1 NH H P M EH u N: uH 0 NH HP bH uIH F E bH, H` J u N H NL NN X bH uIH EX, PYuH H+1 NH H P bH uIH N vH EXH NYH bH NL_H, H` J u NH MO X bH uIH, P N0=NH+1=0 (NL]H N N H H uH 0 P NN XuH ` XH H 0 PY H NN X uH ` XH L H 0 P uH H+1). uN NO u FY . c LYH Y NIXH uN L H P Y NIXH N_G uH u HH HN N uH0 uH NN X uH aH F NL]H u N . DI u N Lc LYH FY EbH O NL_H,H]H u Y NIXH uN L ` N XJ H X NON NIXH Gb N NL]H, H u O NN

X NLH M NL]H X OH NYH O H F, I uH H H H u NIXH NH NLIH NON NIXH CLNFN NYH NIXH N R. &N Nd NON NIXH, dH N NOu P FY u Y uH u Y HN N uH 0 uH uH P. & u` P I N N uGMG HN ` N N uO N H bH uIH P, H` L]H H+1 FY N uG MG HN` N N vH EXH NYH, N FY N uG MG HN ` N IYH NYH NIYH bHNL_H.

CZS uH u Y HN Y NIXH NLIH N^ 4 N_G u NI NON NIXH u LNLIH _H 4.

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Y^ W j d= + =d= b m Ww

c LYH N ` N Y NIXH N_G uH u HH HN NN X J uH u NH X M H H FH NON NIXH Gb N G NL, NLIH u` N M H M FH FN FY

X uH u N. c LYH, E u` N NO u NON NIXH P u J NJ (H4) (HO M

H NON NIXH !ILBFFGH) I (H3) u P NLH J NLH M EbH ZG Iu N EbH ` O NL_H. LIH NLH J NH KOXN, M H NON NIXH !ILBFFGHH FH EbH J FY X FYG NN HN. uZS N M Gb N GN NON NIXH Y NIXH NL]H

P u J NJ N^H NIXH (H3): NON NIXH !FIS. ON NIXH u Gb N NLIH N NMO uZS.

LIOL !FIS;

(* _G uH u HH HN NN X J uH

wO PYI: w N I G NLH NLH M ,, , =1, 2,. . ,H.

`j @/:

NLH uH u HH HN X J uH

,=, , = 1, 2. . .,H,

NLIH u` , I u Y uH u HH HN N uH uH uH .

NLH HH uH u

J,, , = 1, 2.. . , H,

NLIH u` J, HH uH u NL uH NLIH uH u HH HN N uH .

*)

H

(* & NI *)

@IL :=1 NI H I

@IL :=1 NI H I

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H

,:=.;

J.:=;

H;

(* B FJ *)

@IL E:=1 NI H I

@IL :=1 NI H I

@IL :=1 NI H I

@ ,>,E+E, NH

H

,+,E+E,;

7,8>9,8;

H;

H;

c LYH u J NJ N^H NIXH NON NIXH FY (H3).

&N Nd H HYS dH L NL_H YS GN X N H NON NIXH DEMNL NL]H HbHH MF:

(* CHdb"G '%\"H '\! `d"G `I "Gm" "Hh' ' `{"H & `u" `{"H ' 'HE#

'Hk' '#[" DIJ&'%A *)

OMM LN;

IHMN GR=50;

PL

98/164


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H, M, N:HNL;

IH:L;

LOI:LLS1..GR I@ SN;

: LLS1..GR I@ HNL;

: LLS1..GR,1..GR I@ HNL;

@HF: LLS1..GR I@ IIFH;

JLIOL JMIFO;

PL

@:NRN;

@HG:MNLH;

,:HNL;

H

QLN(nI NH @F O FO H I:o);

LFH(@HG);

MMH(@,@HG);

LMN(@);

LFH(@,H);

@IL :=1 NI H I

@IL :=1 NI H I L(@, ,;

FIM(@);

H;

JLIOL $HMIFO;

99/164


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PL

,:HNL;

H

QLNFH(nI H O I N:o,H);

@7(Y!/ @/ 9/ /:Z);

@IL :=1 NI H I

H

@IL :=1 NI H I QLN(,:3,o n);

QLNFH;

H;

H;

LIOL $HENKO;

L

,:HNL;

H

QLNFH(nDOIH HH HN NO n,M,o H n,N);

QLN(N,oo);

QF M I

H

:=LOI;

QLN(,oo);

H;

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H;

LIOL DEMNL;

L

,P,GHJ:HNL;

BH

LN(nG OIH NO M=o);FH(M);LN(n H N=o);FH(N);

!IL P:=1 NI H IBH

P:=M,P;

LOIP:=M;

!FFP:=@FM;

H;

LOIM:=0;DM:=0;!HFM:=NLO;

F HIN @HFN I (* BOI FJ *)BH

G O F H I HH NG NI HI HN

GHJ:=GRHN;

@IL P:=1 NI H I

@ (HIN @HFP) HM GHJ>P) NH

H

O:=P;

GHJ:=P;

H;

@HFO:=NLO;

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@ HIN @HFN NH

@IL P:=1 NI H I

@ (HIN @HFP) H (O+O,P


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CIH:=LES;

LNFH(IH);

CM IH I@

n1o:JMIFO;

n2o: H

$HMIFO;

DEMNL;

$HENKO;

n3o:RN;

H;

HNF @FM;

H.

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B = > W` =; `

BY NIXH FOH u NLIH GH FY GN NLIH M Y NIXH N O NL]H u N N_G uHH H H LH L[ NLIH N N ~H H HH H H Nd P NLIH Ft NOSNN J. BY NIXH u u RON PYI uO HvG 1950, PY H F]H P N]H NO HYNIXH FY !IL PY !OFELMIH. LIH H HYS dH L M NL_H YS NON NIXH!IL PY !OFELMIH u Y NIXH uN L PY H]O GN Mb H H Y NIXH.

M_NG. L&NG %#NG M_NG. BPI %SN L&NG CC Y_I

Yy [+ 1. GH FY u N ` H =(,), NLIH u` OS HN GN uH MEbH ` OH u PYI FY uH JXN, OS HN GN uH N EbH ` OH u L FYuG NO PY G OH =(P,Q) u XH P GN M EbH ZG () =(P,Q) FYE HvH NbH KO OH .

w NOH NH I P NL_H YS N M KO LH HO EbH ` OH (P,Q) N_ E HvHNbH KO (P,Q) u XH H 0.

Yy [+ 2. M I GH =(,). GH @ NLIH GH =(,) ;FY XHR @: |

+XH I G OH =(P,Q) GN M N EbH ZG @()=@(P,Q), FY

FOH NL]H OH , NI G[H X uO EH MO:

'OH NL]H OH EbH PN KOX E HvH NbH KO H`:

0i@()i(),

wO EH ZH H FOH NL]H G uH GH: H FOH NL]H X OH u PYI uHP H NH FOH NL]H X OH u L E uH P, HO P#M, N:

DP @(P) = ? @(Q,P) - ? @(P,Q) = 0 Qx-

(P) Q x+

(P)

NLIH uI x - (P) m NJ X uH GH GY N u` ` OH uH P, x + (P) - NJ X uH GH GY N P ` OH uH H`:

x - (P) = ? Q : (Q,P) ? ,

x + (P) = ? Q : (P,Q) ? .

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X NL FOH @ FY M

F(@) = ? @(M,Q ) = ?? @(Q,N).

Qx + (M) Qx - (N)

B = > W` =; `:

CI GH (,). [S N_G FOH @* NLIH GH P X NL FOH PF(@*) FY FH HN.'OH H PS N M FY FOH u NLIH GH.

BY NIXH H PS ` N RON H NLIH LN HO H H N N. CH H E HRX uH H u FH HN aH PH N HdN GN H u I NbH.LIH P^ HYS F Y NIXH FOH u M I N X uIH uH ubH RHN PY dH NI NYH " J" NH H P aH I NbH R\N NI HdN uH. N P^ EX FY HO R\N u N NH H P GN NH uH H H O.LIH u` X H NH H P X OH, uG JXN ` N I FY NO O, uG NOFY , aH HH uG H X H FY X HdN u N. & HvH NbH KO X OH NH H P NN H X H. CH J N_G FOH O FH HN `N G N NYO O PYI .

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L= j=.Y^ =X >.Yy VF;*F>/;


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- @(P,Q) i 0 P *Q

MOS L PF(@)i(,*). B u u H GH.

Hv >b 1. X NL FOH u NLIH GH EbH PN KOX E HvH NbH KO FXN N J HN NLIH GH.!IL PY !OFELMIH u[ H GH LH X NL FOH uNLIH GH udH H E HvH NbH KO FXN N J HN. w ` N JXN O PYH GH EN KO HYS dH L M H N]G GN M EX HG. M @ FY GN FOHNLIH GH = (,). GH =(,) N RZS H u N ` NLH M NL]H OH @= (, @), P NJ OH @ PY NLH M NL]H X OH u RX uH NI KO N MO:

O =(P,Q) P @(P,Q) =0, N_ (P,Q) @ P NLH M (P,Q);

O =(P,Q) P @(P,Q) =(P,Q), N_ (Q,P) @ P NLH M @(P,Q);

O =(P,Q) P 0


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() (). M H F, N_G u uH NvH FOH . & u` N ` N NvH X NLFOH H X NvH FOH NI uH . wO u` GZO NOH P N^H u FOH @.

() (). M EbH N_G u uH NvH FOH. &t O FY NJ NN X uH

` N uH u N uH M NLIH u N @, PY uN

*

=. & u` (,

*

) FY FXN N, PY@(P,Q)=0 P G P*, Q H]H

PF(@) = @(P,Q) - @(P,Q) = @(P,Q) P P * P Q

* Q Q*

P , Q *, I (P,Q) @, H]H @(P,Q) = (P,Q). S

PF(@) = @(P,Q) = (P,Q) = (,*) P P Q * Q

*

() (). I u 1, PF(@) i (,*) P G FOH @ PY P G FXN N (,*). _

PS, N uH N PF(@) = (,*) MOS L FOH @ FY FOH u NLIH GH.

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%>i= = =J > W`

wH Ft 1 FY M u RZS H NON NIXH FJ MO uZS u N_G FOH u NLIH GH:BN uO N FOH P FOH NL]H NN X OH H 0 (N M FOH H PS FY FOH

EbH), PY FJ F FJ MO uZS I uH E NO u FOH GY u P H` EbH aHuH NvH:

B FJ NvH FOH (!IL-!OFELMIH): _G uH NvH u P FOH H `. vHFOH NI uH .

& u[ ` FOH u, FXN N J HN ` N NI N N Gb N NLIH H GHuH Ft 1. u NON NIXH !IL-!OFELMIH ` N Gb N NLIH N N MO uZS:

LIOL R!FIQ;

(* ON NIXH !IL-!OFELMIH *)

H

(* & NI: BN uO N FOH P X NL 0 *)

@IL O I

@IL P I @(O,P):=0;

MNIJ:=@FM;

QF HIN MNIJ I

@ NH

FM MNIJ:=NLO;

H;

w NvH FOH NLIH @ ` N N_G EG NON NIXH NI O LH (S N_G EG NIO MZO) N uO N uH M, NLIH u` EbH H RZS H u N NH GH @.!IL!OFELMIH u H NON NIXH XH H[H NN MO uZS u Y NIXH FOH u NLIH GH (` N N uO N FOH EbH), MO u` N M NvH FOH H X N_GX uH NvH FOH. w N_G uH NvH FOH N M XJ H JH JXJ XH H[H

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I X uH. uH NLIH KOX NL_H N H NON NIXH M GN NLIH NLH NX: ` H[H, ` H[H R\N, ` H[H u[ R\N. [H GN uH P G 2 JH PY `GN NLIH 2 H MO: +J(P), z (P) I -J(P), z (P). H N HN +J(P) (-J(P)) L FY H NvH (G) FOH NI OH (J(P), P) (OH P, J(P)) aH JH N z (P) L FH FH HN ` N NvH I G FOH NL]H X OH uH NvH FOH NM N P. wO N]H ` uH M u E NI PY H[H H` FY R\N, aH NN XuH aH F FY uO ` H[H. M N XH H[H I NN X uH E P H` PY H[H uH M M NL NYH u[ R\N. J NI, N G uH P ` H[H R\N N F XH H[HI NN X uH ` H[H E H` PY H[H uH P NL NYH u[ R\N. OX NL_H MFJ F I uH E I FY uH N NL NYH ` H[H I FY H[H NN X uH `H[H uO FY u[ R\N HH uH N PH EbH ` H[H. LIH NLH J N HN N N_Gu uH NvH FOH, aH NLIH NLH J N u P FOH uH R\N EbH NHN uH NvH FOH (N FY FOH u[ FY u). E N N_G u uH NvH FOH,N F NvH FOH NI uH N_G u, MO u` RIX NN H[H PY u P FOH G NOu F M H J\J XH H[H X uH u N_G uH NvH FOH. ON NIXH M EN

Nd E HYI u P FOH uH ` NLIH GH EbH N_G u uH NvH FOH. N N N_G uH NvH FOH PY NvH FOH ` N Gb N H MO.

LIOL !HN;

(* N XH H[H N_G uH NvH FOH

JP, z P FY H[H uH P;

m H MX X uH ` H[H HH R\N;

O,P m E HvH NbH KO OH (O,P), O, P ;

@O, P m FOH NL]H OH (O, P), O, P . *)

H

JM:=M;

z M:=+h ;

=? U MW ;

N!IOH:=NLO;

F I

BH

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; (* 'S O N *)

!IL P I

H

$@ (@O,P 0) NH

BH

JP:=O;

z P:=GHUz O, @P,O W ;

= ? U PW ; (* J P PYI H MX uH ` H[H *)

$@ P = N NH RN;

H;

H;

H;

N!IOH:=@FM;

H;

LIOL $H!FIQ;

111/164


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(* vH FOH NI uH NvH *)

H

P:=JN; O:=N; NH:=z N;

QF O M I

H

@ P>0 NH @P,O := @P,O + NH;FMH

P := -P;@O,P := @O,P - NH;

H;O := P; P := JO;

H;

H;

ON NIXH !IL!OFELMIH u N H H N N:

LIOL R!FIQ;

(*ON NIXH !IL!OFELMIH *)

H

(* & NI: BN uO N FOH P X NL 0 *)

@IL O I

@IL P I @O,P := 0;

MNIJ:=@FM;

QF HIN MNIJ I

H

@HJN;

@ JN!IOH NH $H!FIQ

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FM MNIJ:=NLO;

H;

H;

M FY E HvH NbH KO NN X EOH u N FY X M HOS]H. & u`MO G FH NvH FOH, X NL FOH M NvH F]H ^N HN FY 1. u` MOS L NON NIXH!IL!OFELMIH M H MO EbH KOX PF(@*) FH NvH FOH PY I N FOH u

NLIH GH. wH N, Lc LYH @*(O,P) M FY M HOS]H u P G OH (O,P) . u` L ` X EN KO MO:

Yy V 2 (wH Ft P FOH u NLIH GH PY FXN N J HN). 'OH u NLIHGH H E HvH NbH KO FXN N J HN.

Yy V 3 (wH Ft P N^H HOS]H). O NN X E HvH NbH KO FY X M HOS]HN_ FOH N_G u u P FOH NL]H X OH FY X M HOS]H.

OS H]H, HO X E HvH NbH KO FY X M LN FH N_ X NL FOH u~H ` N FY LN FH PY E u` NON NIXH Gb N NL]H M ua N H LN HO NvH FOH. ^ NLIH _H 2 M GH I I uO HYS. _H 2() Gb N GHH R\N P X E HvH NbH KO NL]H X OH. _H 2() Gb N FOH NL]H X OH(M N ]H H OH) MO E N H NvH FOH NI uH NvH FOH (M, ,, N). _H 2() Gb N FOH NL]H X OH MO E N H NvH FOH NI uH

NvH FOH (M, , , N). c LYH, MO 2.106 FH NvH FOH NI uH (M, , , N) PY (M, ,, N) GN X FOZH J]H N NO u FOH u.

_H 2.^ N N u P NON NIXH !IL!OFELMIH.

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H N H, HO X E HvH NbH KO FY X M Pb N, H N aH RZS H uP^ u I NON NIXH EbH H, PY N H HO [S X X NL FOH RZS H NINON NIXH N N_ H` aH EbH N uH X NL FOH u. PS, GOH NONNIXH FYG P O KO, P F H uH NvH FOH H u NH YH N M HNH.

GIHM PY &LJ L LH HO uH NvH FOH u H FY uH HH HNN M uH N NL]H u N @. wO u` ` N N H, HO NLIH N N N_G uH NvH!HN Gb N NL]H, H MX u N H (H} FY N NH N_G uH NvH N N N_G EG NI O LH) N_ NON NIXH M EN Nd MOEbH KOX GH/2 FH M H uH NvH FOH. O u t LH, N_G EG NI O LHNL]H u N ua N H (G+H), N_ NON NIXH NO u M ` u J NJ N^H NIXH

FY (HG2).

X N N_G uH NvH E\I F\I H, H N u[ RZS H u NON NIXH

P u J NJ N^H NIXH NN H H: (H2G) (DH, 1970). (H

3) (&LTHIP, 1974),

(H2G2), (CLEMES, 1977), (HG FI H) (FNIL, - LLH, 1980).

EN Nd G HYS P^ GH I I NON NIXH !IL!OFELMIH MO uZS. _H3() I GH eH P NbH KO NN X OH PY FOH X NL 10 NLIH H`. M PN ]H H G OH FY E HvH NbH KO OH (M NLIH HI) PY FOHNL]H OH. wH NvH FOH ` H (M, P3, P2, P6, P7, N). N^H z (N) = 1, X NLFOH NvH N 10 F]H 1. _H 3 () I FOH NO u MO E NvH FOH NI uHNvH N_G u.

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_H 3. vH FOH NI uH NvH

'OH NLIH _H 3 () u[ FY u. 'XN N J HN

=U M, P2, P3, P5W, = UP4, P6, P7, NW .

X NL FOH u FY 11.

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M= = >=

LIH JH HYS N H]O L GN M H Y NIXH P FOH NH KOXN GY P dH `N H P Y NIXH FOH u NL_H YS NL]H.

M` r> Ws = Ws =>.

\N GH P J uG JXN M1, M2, . . . , M JPY Ku GN O N1, N2,. . . , NK. M LH FOH` N u N GN uG JXN N E uH NN X uG NO. BY NIXH N_G FOH u NX uG JXN uH X uG NO ` N u P Y NIXH P GN uG JXN PY GN uGNO H X u PYI GN uG JXN M PY GN uG NO N PY H H M P NN X uG JXN PY H H X uG NO P N.

_H 4 GH I I X u GH P HO uG JXN PY HO uG NO P GH ` GN uG JXN PY GN uG NO. & HvH NbH KO OH H M P uG JXNME M H +h HO EbH ` H P FH JXN uG JXN ME, PY HO FH JXN ME H E N_ OH (M, ME) ` E HvH NbH KO FY E. C~H H PS, uP X OH H NE P uG NO N, M E HvH NbH KO (NE, N) M FY HI EbH H NO NI FH NO uG NO HYS ` H S EbH.

LH J GN M uG NO HH "YH" N GN M uG JXN N ` Y NIXH HOFOH FY GN Y NIXH J NJ H LN HO MI P Y NIXH FOH u uGJXN M PY uG NO N.

HJ 4. H P HO uG JXN PY NO.

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B = b X = >+ + > Ww.

M NLIH u N , HIY E HvH NbH KO X OH (O,P), G uH P aH ` E HvH NbH KO X uH FY (P), PY ua NH FOH u PYI uH P EbHu PN KOX (P), N FY.

@(Q,P) i (P)P

CH J N_G FOH u M PY N NLIH GH H PS.ZS H GN GH o MI

I: G uH P NH H P 2 uH P+, P- NLIH o, G OH (O,P) NLIH H

P OH (O-,P+) NLIH o, G OH (P, ) NLIH H P G OH (P-, Q+) NLIH o.

IY L, G OH (P+, P-) NLIH o ` E HvH NbH KO FY (P), N FY H E HvH

NbH KO uH P NLIH .DI FOH u PYI uH P+J u KO OH (P+, P-) P EHvH NbH KO (P), H]H FOH u NLIH o M H FOH u NLIH P EHvH NbH KO X OH PY X uH.

_H 5. _H 5 I P^ GH P E HvH NbH KO OH PY uH.

_H 5 FY GH o NH H ` E HvH NbH KO NL]H X OH.

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M` =; W b X = >+ + > y m + +.

\N GH GY NLIH u` G OH (O, P) ` E HvH NbH KO (H NL]H FOHNL]H OH) (O,P) PY H FOH FY (O,P). BY NIXH uN L FY FO ` NH N FOHNH N^ N M uH N, N FY FOH ? @(O,P): O, P ? NI G[H N]G LYH O

(O,P) i @(O,P) i (O,P) , O, P,

S EbH?

w PYI GH uH JXN M PY uH NO N PY RZS H GH NI KO N:

OH (O,P) GY (O,P) g 0 M NH H P 2 OH (M, P) P (O, N) P E HvHNbH KO FY (O,P). G (O,P) u (O,P) N FY NS E HvH NbH KO OH (O,P) (O,P) m(O,P) aH H H` uN H 0. IY L N]G PYI OH (N,M) P (N,M)= h .

_H 6. H P E HvH NbH KO H J^.

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_H 6() I P^ GH P E HvH NbH KO X OH H J^.w N NH H u I NLIH _H 6().

&t O * = ? (O,P).

(O,P)g0

Yy V 4.

1) O FOH FH HN NLIH GH N M uH N H * N_ NH N FOH NH N^NLIH .

2) O FOH FH HN NLIH GH N M uH N FY EX * N_ EbH NH N FOHNH N^ NLIH .

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M= .

C` G YH NL GN PeH KO] H. w P G YH NL N N X b X GY H NP t. E HYI N_ ` N N X uXG NLIH u` YH NL HYI ~H MXHOS]H P X b X GY G_H P t. ` N RZS H u N P X uH O N XYH NL PY X b X, aH X OH O N M P t X YH NL P X b

X. & u` N NO u GN u N J^.^ . C` 4 YH NL ? 1, 2, 3,4?P Y 5 b X ? 1, 2, 3,4, 5?. P t INLIH H MO

w N NH H u I NLIH _H 7.

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_H 7. H NH H P Y NIXH uXG PeH KO]

w PYI uG JXN M PY uG NO N. M P NN X uH O N X YH NL, PYH N P NN X uH O N X b X. N X OH u N uO ` E HvHNbH KO H 1. BN uO N FOH 0, N N_G FOH u NLIH GH RZS H uNI NON NIXH !IL-!OFELMIH. uH Ft P N^H HOS]H, FOH NL]H X OH FY X M

I 1. c LYH FY HO FOH u NLIH u N ` X NL GR = G, N_ Y NIXH ` F, PY X OH P FOH H 1 M L X N uXG NI G[H uO EHuN L. F, HO Y NIXH ` F N_ GR = G. BY NIXH P uXG PeH KO]FY GN NLH J L]H Y NIXH P J \J NL]H u N J^ GY u H` `N RZS H NON NIXH O KO H.

B = r v =~ W` v >.

CI NJ G JH N =? T1, T2, . . . , TG?. M PY

FY [S X NJ IH . D[S G H JH N EX HO : u FY NH X u H OH [S u[ I HO H N_G u GN IXH P { NJ ? 1, 2,. . .,H? MI I < 1, 2, . . . , H> FY NH X u H JZH N [S PY , N FY uO EH MO u NIG[H: A ? B{ (), = 1, 2, . . ,H. ZS H GH =(,) P NJ uH

= ? M, N? ? ? R1, R2, . . . ,RH?? ? O1, O2, . . . ,OH?? ? P1, P2, . . . ,PH?? ? S1, S2, . . . ,SH?.

NLIH u` uH R NH H P NJ A, uH S NH H P NJ B, X JH N O, SNH H P JH N T. J X OH GH u RX uH H MO

= ? (M,R): 1iiH? ? ? (R,O): P T A, 1iiH, 1iiG? ? ?(O,P):1iiG? ? ?(P, S):P T B, 1iiH, 1iiG? ? ? (S, N): 1iiH?

& HvH NbH KO NN X OH u uN H 1. D YH NS LH NHu H OH [S PY NH N E PY E NLIH GH =(,) N_G u FOHP X NL H. w R\N M NH N FOH H PS ` N M H NON NIXH N_G FOH u N M uH N NLIH GH =(,).

'r = = =~ ^> ; ;`.

LIH G HYS N M NL_H YS NON NIXH u RZS H NL]H NON NIXH N_G FOH u u GN Y NIXH N O L L FY Gb _H NIXH I GN M Y NIXH NO N J.

\N Y NIXH N O L L:

@(R1,R2,...,RH) =

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(1)

P uO EH

NLIH u` ? 0,1? , = 1, 2, . . . , G; =1, 2, . . . H, J mHOS]H H, = 1, 2, . . .,G.

BY NIXH (1)-(3) FY Gb _H NIXH I HO Y NIXH N O N J N N. D uZSN H L GN PY P^ uH _H.

B =


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w PYI H M R = 1, HO NL^ NO H FYG P JaH ,R=0, HO H F,=1,2, . . .,G, =1, 2,. . .,H, E u` NS Gb _H NIXH I Y NIXH uN L ^H FY YNIXH (1)-(3), NLIH u` J=1, =1, 2, . . .,G.

B Wr 2.

BY NIXH (1)-(3) ` JH XH N O E PY E

C . wO EH H u FY H H]H P_ N M NH N JH XH YNIXH MOS L X N uH N NLIH (4) u N H ^N HN H O uH N.

w H GH uO EH u, H L LH HO uO EH (4) u N H N_ YNIXH FObH ` JH XH. PS, M uO EH (4) u N H. & u` HO EtO $+=? 1iiH: =1?,N_ ? $+?jJ , =1 ,2 , . . . ,G.DIu`HO $ $+, ? $? =J, =1,

2,. . . ,G,N_ * = (R*)GRH P X NYH JH u RX uH NI bH N

R* = 1 , $, R* =0, $, = 1, 2, . . .,G,FY JH XH Y NIXH (1)-(3). B u uH GH.

DI (4) FY uO EH H u Y NIXH (1)-(3) ` JH XH, H]H NLIH JH NJ NI N MFObH NN LH uO EH HYS u N H.

BZS N L LH P Y NIXH (1)-(3) ` N H P P GN M O HY NIXH FOH u NLIH GH. L N, P G M HOS]H H E, RZS HGH (E) =(,) P NJ uH

= ? M? ? ? O :=1, 2,. . . ,G? ? ? Q : =1, 2, . . .,H? ? ? N?

NLIH u` M FY uG JXN, N FY uG NO, PY NJ OH

= ? (M,O):=1, 2,. . .,G? ? ? (O,Q):=1, 2,. . . ,G; =1, 2, . . .,H? ? ? (Q,N): =1, 2, . . .,H? .

OH u XH P E HvH NbH KO K() NI KO N MO:

K(M,O)=J, =1, 2,. . .G,

K(O,Q)=, =1, 2,. . .G; =1, 2, . . .,H;

K(Q,N)=E, =1, 2, . . .,H.

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_H 8 L X RZS H GH (E).

_H 8. H (E).

B u MO uZS I NS G F]H FOH u NLIH GH (E) PY JH XH Y NIXH (1)-(3).

B Wr 3.

M u P M HOS]H H E HYI u`, FOH u HOS]H NLIH GH (E) ` X

NL FY { . & u` * = (R*)GRH P X NYH JH u RX uH NI bH N

R* = ? * (O,Q), =1, 2,. . .,G; =1, 2,. . . ,H.

FY JH XH Y NIXH (1)-(3).

C . PS, I FOH u NLIH GH ` X NL { PY FY FOH HOS]HH]H

? * (M,O)=J, =1, 2,. . .,G,

? * (O,Q) ? 0,1? , =1, 2,. . .,G, =1, 2,. . .,H,

N u` MOS L

S * FY JH XH Y NIXH (1)-(3). B u u H GH.

B Wr 4.

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M * =(R*) FY JH XH N O PY E* FY X NL N O Y NIXH (1)-(3). & u`

FOH u NLIH GH (E*) ` X NL { .

C . DI X NL FOH u NLIH GH (E*) EbH PN KOX { , H]H u

H GH u N H L FOH P X NL { NLIH GH (E*).

ZS H FOH ? * NI bH N MO:

? * (M,O) = J, ? * (O, Q) = R*,

D YH EG NL u ? * FY FOH NLIH GH (G) ` X NL { . B u u HGH.

B Wr 5.O E=G N_ FOH u NLIH GH (G) ` X NL { .

C . 'J FOH NH N H NLIH B u 4, N H L FOH P X NL

{ NLIH GH (G). PS, M * = (R*)GRH FY JH XH Y NIXH (1)-(3)

RZS H NI bH N (5). ZS H FOH ? * NI bH N H H NLIH HGH u 4, N ` FOH P X NL { . B u u H GH.

u 3 PY 4 MOS L P Y NIXH (1)-(3) H P P N_G X NL E* HOS]H H

H HN MI I FOH u NLIH GH (E*) ` X NL {. B u 5 I N NS X NL

E* 1,G. _ PS u Y NIXH (1)-(3) N ` N XJ H JH JXJ N_G EG H

JZH NL]H uIH 1,G u N_G X NL E*, NLIH u` G H GN Y NIXH FOH u. w Y NIXH N_G FOH u NLIH GH ` N M H X NON NIXH uN H u[ H` NL]H. u` MOS L EN KO MO

Yy V 5.

BY NIXH (1)-(3) u H NON NIXH u N P u J NJ N^H NIXH FYFI2G.!, NLIH u` ! FY u J NJ N^H NIXH Y NIXH N_G FOH u NLIHGH (E).

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B =i ^] 3

B 1 : C r =; b

CI GN H HN NYH R b PObH ( aH, N). b PObH GNM HOS]H H (NLIH EIH N 1 uH 255). N GH H FY NJ J NN Xb ` eH X NL M MI I dH u u MH HO H X u KO X b ` OHH PY ` eH X NL M uH R\N. w GN GH FY N u aH, N b uON]H NO GH NI N N OSN N NLX MH J PY N NL]H ROH . DH N^ GN GH FY M b NO GH u`. ^ H:

` 4 GH, GH Nb GYO RXG (X NL X b FY 2) ` u 1, 3 PY H N^ FY 7.

CH RX uH:

GH GH.

H ` H N^ FH HN ( Lc X NL H N^ PY u GH).

D FO PYI I NLIH @F PvH H (N]H @F u N YH J^G) ` H:

A1, 1 A1, 2...A1,

A2, 1 A2, 2...A2,

......................................

A, 1 A, 2...A,

NLIH u` A, FY X NL M b , , X M NL]H eH GN aH X HO ^N HNGN O NLH.

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A, = 1/0 PY A, H 0 P G , A, = 1 E PY E GbH J u SRIH NL E GbH (HYS EN Nd GbH J NL HYS N uO GbH ). bH J S NL GbH HO ` GN [S GbH 1, 2, l, E MI I N, N+1 = 1 , 1


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^ 2

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B 4: HJ > L+=

_H PObH F NH J 4 FY G NLH PObH E^ N 4R4 GY G aH PY G N H` uO FY GN IXH P X X A, B, C, D. _H PObH F NH u FYNH uH HO N _H HYS N ` N NO u _H E H M H X J\J H uMO: 1) u aH; 2) u N; 3) u N]H X.

' : _H PObH F NH

A B C D A B C D

1 = B D A C 2 = B C D A

C A D B C D A B

D C B A D A B C

FY NH uH, P_ u aH 3 PY 4 1 N NO u

A B C D

B D A C

D C B A

C A D B

NJ uH u N 3 PY 4 N NO u

A B D C

B D C A

D C A B

C A B D

O eH, u N]H C PY D I HO (H} FY NS C D PY NS D C) NNO u 2.

> f>: CI 1, 2 FY _H PObH F NH J 4. [S RX uH RG _H PObHu[ I ` NH uH S EbH?

D v> : @F PvH H '

Giáo Trình Lý Thuyết Đồ Thị

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