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9 E E I M (E, I) E I 2 E M I I X I Y X Y I X, Y I X > Y x X \ Y Y {x} I B E B M

10 E C E C C M r E X E r (X) X X r (X) = X E B M X E Y E X Y X B E F E F F E E E M M M M M M E B M E \ B M M 0 M M M 0 0 M x M B M x B

11 M E r E r

12 E C 2 E E M E C C / C C 1, C 2 C C 1 C 2 C 1 = C 2 C 1 C 2 C e C 1 C 2 C 3 C C 3 (C 1 C 2 ) \ {e} E B 2 E E M E C B B = B 1, B 2 B x B 1 \ B 2 y B 2 \ B 1 (B 1 \ {x}) {y} B

13 E E E E I 2 E E I I X I X I X, Y I I I E C 2 E E C C C 1 C C 2 C C 1 C 2 C 1 C 2 C 1, C 2 C e C 1 C 2 C 3 C C 3 (C 1 C 2 ) \ {e} C E B 2 E E B

14 B B 1, B 2 B x B 1 \ B 2 y B 2 \ B 1 (B 1 \ {x}) {y} B B, {0, 1, 2}, {0, 3, 4}, {0, 1, 2, 3, 4, 5} M Z (M) M X Y M X Y X Y M X Y Z 0 Z Z E (Z) r : Z Z 0

15 Z E r : Z Z 0 M E Z r (Z0) Z (Z1) r (0 Z ) = 0 (Z2) 0 < r (Y ) r (X) < Y X X, Y Z X Y (Z3) X, Y Z r (X) + r (Y ) r (X Y ) + r (X Y ) + (X Y ) (X Y ). r X Y X Y = X Y = X X Y = Y r (X)+r (Y ) r (X)+r (Y ) Z = {, {1}} r r ( ) = 0 (Z2) 0 < r ({1}) < 1 r

16 Z = {, {1, 2}, {1, 2, 3, 4}, {5, 6} {1, 2, 3, 4, 5, 6}}. r r ({1, 2}) = 1 r ({1, 2, 3, 4}) = 2 r ({5, 6}) = 1 r ({1, 2, 3, 4, 5, 6}) = 3 r ({1, 2}) + r ({5, 6}) = 2 < 3 = r ( ) + r ({1, 2, 3, 4, 5, 6}). (E, Z, r) P = (P, ) P P X, Y P P X,Y = {A P : A X, A Y } Z P X,Y W P X,Y W Z Z X Y P X, Y P P E Z 2 E r : Z Z 0 Z M E Z r M Z Z Z 1, Z 2 Z Z 1 Z 2 0 < r (Z 2 ) r (Z 1 ) < Z 2 Z 1

17 Z 1, Z 2 Z Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 r (Z 1 ) + r (Z 2 ) r (Z 1 Z 2 ) + r (Z 1 Z 2 ) + Z 1 Z 2 Z 1 Z 2. Z E Z 2 E E Z Z

18 r : Z Z 0 Z (Z1) (Z2) (Z3) Ax b R Z r Z r r (0 Z ) 0 r (0 Z ) 0 r (Y ) r (X) Y X 1 r (X) r (Y ) 1 r (W Z) + r (W Z) r (W ) r (Z) W Z W Z X, Y Z X Y W, Z Z Ax b x 1 1 V A r (V ) b P (Z) A Z C (Z) P (Z) P (Z) Ax b

19 Z 1 P (Z) Z A Ax b ±1 m n m m ±1 C (Z) P (Z) Z C (Z) Z 1

20 Z 2 C (Z 1 ) = Z C (Z) T Z 2

21 C (Z 2 ) = C (Z 2 ) T Z P (Z) P (Z) Z 3 Z P (Z) Z 4 P (Z 4 ) 11 8 Z P (Z)

22 {1, 2, 3, 4, 5, 6, 7, 8} {1, 2} {3, 4} {5, 6} {7, 8} Z 3 {0,1,2,3,4,5,6,7,8,9,10} {0,7,8,9,10} {2,5,6,8,9} {3,4,5,7} {1,2,6,7,10} Z 4

23 Z Z 2 C (Z 2 ) C (Z 2 ) Z 2 Z C (Z 3 ) Z 3 P (Z) Z 4 8 3

24 P (Z) Z P (Z) P (Z) { z 2 : z Z} Z Z Z

25 M E D E M \ D E \ D D M J M \ D J E \ D M M E D E M/D E \ D M E \ D (M \ D) M M M \ D 1 /D 2 D 1 D 2 E (M) R x x x R x y y z x z x, y, z R

26 x y y x x = y x, y R R x < y x y x y x, y R x y y x R (R, ) y x x < y z R x < z < y Z x, y Z x y x y x y x y x y x y Z z Z z x z y w x w y w Z w z p Z x p y p x q y q q Z p q Z 0 Z 1 Z x Z 0 Z x 1 Z (Z, ) K Z K K R S R S R S x R S y x R y x S y x R y S R S R S {(x, y) : x R, y S} (x, y) R S (z, w) x R z y S w

27 E P R n P = {x : Ax b} m n A b P M N z P z M P F P F = { } x : Ax b, A x = b A l n A b b A n n R P c P = {x : Ax b} cx x P x x Z n Ax = b A x x i = (A i) (A) A i i A b Ax b

28 A m n z Az mn b Az b 3 X n x 1,..., x n m (m + 2n) n A 1 i m i A j 1 x j i X 1 x j i X 0 1 k n (m + 2k 1) 1 k (m + 2k) 1 k b i b 1 l l i X 1 i m (m + 2k 1) 1 (m + 2k) 0 1 k n X = (x 1 x 2 x 3 ) (x 2 x 3 x 4 ) ( x 1 x 4 x 5 )

29 x 1 x 2 x 3 x 4 x x 1 x 2 x 3 x 1 + x 2 + x 3 1 x 1 x 2 x 3 x 1 + x 2 + (1 x 3 ) 1 x 1 x 2 x 3 x 1 + (1 x 2 ) + x 3 1 x 1 x 2 x 3 x 1 + (1 x 2 ) + (1 x 3 ) 1 x 1 x 2 x 3 (1 x 1 ) + x 2 + x 3 1 x 1 x 2 x 3 (1 x 1 ) + x 2 + (1 x 3 ) 1 x 1 x 2 x 3 (1 x 1 ) + (1 x 2 ) + x 3 1 x 1 x 2 x 3 (1 x 1 ) + (1 x 2 ) + (1 x 3 ) 1. z Az b Az b 0 1 X m + 1 m + 2n 0 1 0

30 3 A A ±1 A {Y 1, Y 2,..., Y t } A (ϵ j ) j t ϵ j = ±1 1 j t t i=1 ϵ iy i { 1, 0, 1} m n A m < n m m A ±1 A m A m A A B A B C BC = A Z (Z1), (Z2), (Z3) Ax b A b n

31 1 0 Z 0 b Z 0 b 0 X, Y Z X Y 1 Y 1 X 0 b Y X 1 1 Y 1 X 0 b 1 X, Y Z X Y 1 X Y 1 X Y X Y 0 b (X Y ) (X Y ) P (Z) R Z 0 Z 0 1 Z 1 Z 1 P (Z) P (Z) R Z Ax b P (Z) (Z1) (Z2) (Z3) P (Z) Z Z A X Y Z

32 {1, 2, 3, 4, 5, 6, 7, 8, 9} = 1 Z1 a Z1 = {1, 2, 3, 4, 5} {4, 5, 6, 7} = b Z1 {4} = 0 Z1 Z 1 r (Y ) r (X) 1 r (Z) r (Y ) 1 r (Z) r (X) 1 r (Y ) r (X) < Y X r (Z) r (Y ) < Z Y r (Z) r (X) < Z X = Z Y + Y X p R Z Y X C (Z) C (Z) x b Ax b Z 1

33 0 Z1 a Z1 b Z1 1 Z r 1 r 2 r 3 r 4 b C (Z) Z 4 Z C (Z) x b P = {x R n : C (Z) x b} C P x C x = b n n C C x x i = ( ) C i (C ). C ±1 C x

34 P n n ±1 C T M \ e M e X M \ e X M C M \ e C M e / C F M \ e F F {e} M F M \ e F M F M (F ) x x M (F ) F C x C F {x} x e x M (F ) r (F ) = r (F {x}) F M \ e x = e M (F ) = F {e} F M F {e} F M \ e F M F M F {e} M C e C F {e} F F {e} M Z C (Z) C (Z) 1 = {X 1, X 2,..., X n } C (Z) [ ] a 1 X 1 + a 2 X a n X n =

35 1 X i 0 C (Z)

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37 Z E (Z) x E (Z) Z = {X {x} : X Z} M E (Z) Z Z M U 0,1 M r M U0,1 (X {x}) = r M (X) X Z Z E (Z) x 0 Z Z = {X \ {x} : X Z} M E (Z) Z 0 Z x r M ({x}) = 0 x M Z M \ x r M\x (X \ {x}) = r M (X) X Z

38 Z E (Z) Z E (Z) {x} x / E (Z) Z Z Z M E (Z) Z M U 1,1 M r M U1,1 (X) = r M (X) X Z Z E (Z) x E (Z) \ 1 Z Z E (Z) \ {x} (Z, E (Z) \ {x}) (Z, E (Z)) M E (Z) Z x M Z M \ x r M\x (X) = r M (X) X Z Z E (Z) x x 0 Z x E (Z) \ 1 Z 0 Z 1 Z E (Z) Z Z

39 Z 3 P (Z) Z 0 Z 1 Z X Y Z Y E (Z) r r (X) + r (Y ) r (X Y ) + r (X Y ) + (X Y ) \ (X Y ) = r (E (Z)) + r ( ) + X Y = r (E (Z)) + X Y, r (X) X Y + r (E) r (Y ) X Y + 1. X 0 Z 1 Z X 1 Z X Z \ 1 Z r (1 Z ) { X Y : X, Y Z, X Y, 0 < X, Y < 1 Z } + 2. r Z r (0 Z ) = 0 r (1 Z ) = { X Y : X, Y Z, X Y, 0 < X, Y < 1 Z } + 2. p 1 Z U Z \ {0 Z, 1 Z } p 1, U p U Y = p 2 Y Z r (U) = U 1, Z r (0 Z ) = 0

40 p 2 r (U) 1 U Z \ {0 Z } U = 1 r (U) 0 U > 0 Z r (U) r (0 Z ) + 1 = 1 P (Z) r (1 Z ) > r (U) U Z X, Y Z Y X r (Y ) r (X) Y X. Z Y = 1 Z X = 0 Z Y = 1 Z r (X) = p 1 Y X = 1 r (Y ) r (X) 0 r (Y ) r (X) 1 P (Z) Y = 1 Z r (X) = X 1 1 Z X P (Z) X = 0 Z r (Y ) < Y r P (Z) X, Y Z r (X)+r (Y ) < r (X Y )+r (X Y )+ (X Y ) \ (X Y ) = r (E)+ X Y. X Y r (X) = p 1 p 1 + r (Y ) < r (E) + X Y = p + X Y, r (Y ) < X Y + 1, r (Y )

41 r (X) = p 1 P (Z) r (X), r (Y ) < p 1 r (X) = X 1 r (Y ) = Y 1 X Y P (Z) E P (Z) r P (Z) p U U Z r r C (Z) x b P (Z) Z Z Z C (Z) C (Z) Z C (Z) Z X, Y Z X Y Y = X + 1 P (Z) r P (Z) r (Y ) r (X) 1 r (Y ) r (X) Y X 1 = 0

42 Z 2 X Y X Y X Y = 1 Z Z r : Z Z 0 0, X = 0 Z X 1, X 0 Z r (X) = r (Y ) + 1, X Y 0 Z X 1 Z {r (Y ) : Y X} + 1, X = 1 Z r P (Z) P (Z) X Y r (X) r (Y ) = 1 r (X) r (Y ) = X Y 1 r r n 1 0 Z r P (Z) r r / P (Z) (Z1) r (0 Z ) = 0 (Z2) r (X) < r (Y ) X Y X Y r (Y ) r (X) Y X Y X r (Y ) r (X) = 1 Y X = 1 P (Z) X = 0 Z r (Y ) = Y 1 < Y 0 Z = Y 0 = Y.

43 (Z2) Y = 1 Z r (X) < r (Y ) 1 Z X Y Z Y X Z Y r (Y ) r (X) Y X Z r (Y ) Z +r (X) r (Y ) > Z Y r (Y ) = r (Z) + 1 r (Z) Z 1 r (Y ) Z (Z2) (Z3) X, Y Z r (X) + r (Y ) < r (X Y ) + r (X Y ) + X Y X Y. Z X r (Z) + r (Y ) r (Z Y ) + r (Z Y ) + Z Y Z Y Z Y X Y (Z2) X Y X Y = 0 Z r (0 Z ) = 0 X Y = r (X) + r (Y ) < r (1 Z ) Z Z 0 Z Z Z X Z Y Z Y = 1 Z Z Y = 0 Z r (Z) < r (X) r (Z) + r (Y ) < 1 Z X, Y Z X Y 0 Z (Z3) X Y 1 Z X Y 1 Z X Y P (Z) P (Z) Z

44

45 M 1 M 2 M 1 M 2 E (M 1 ) E (M 2 ) X M 1 M 2 X = I 1 I 2 I 1 I (M 1 ) I 2 I (M 2 ) M 1, M 2 Z (M 1 ), Z (M 2 ) {X 1 X 2 : X 1 Z (M 1 ), X 2 Z (M 2 )} M 1 M 2 M 1 M 2 F E (M 1 M 2 ) M 1 M 2 F E (M 1 ) M 1 F E (M 2 ) M 2 F (M 1 M 2 ) = {F 1 F 2 : F 1 F (M 1 ), F 2 F (M 2 )} C E (M 1 M 2 ) M 1 M 2 M 1 M 2 F M 1 M 2 F E (M 1 ) F E (M 2 ) F Z (M 1 M 2 ) F E (M 1 ) Z (M 1 ) F E (M 2 ) Z (M 2 )

46 p 2 = {0, 1, 2, 3, 4, 5} r (p 2 ) = 3 p 1 = {0, 1, 2} r (p 1 ) = 2 p 0 = r (p 0 ) = 0 P 6 Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 = {Z 1 Z 2 : Z 1 Z 1, Z 2 Z 2 }. Z 1 Z 2 Z 1 Z 2 Y Z Y Z M 1 M 1 Z (M 1 M 2 ) = Z (M 1 ) Z (M 2 )

47 s 3 = {6, 7, 8, 9, 10, 11} r (s 3 ) = 3 s 1 = {6, 7, 8} s 2 = {9, 10, 11} r (s 1 ) = 2 r (s 2 ) = 2 s 0 = r (s 0 ) = 0 R 6 P 6 6 P 6 R 6 6 R 6 Z P6 Z R6 = (0, 2, 72, 2, 72, 5, 2, 72 ), 5, 4, 5, 6

48 r (x 1 ) r (x 0 ) 2 = 2 0 r (x 3 ) r (x 0 ) 2 = 2 0 r (x 6 ) r (x 0 ) 2 = 6 0 r (x 10 ) r (x 9 ) 1 = 5 4 r (x 11 ) r (x 5 ) 1 = 6 5 r (x 11 ) r (x 8 ) 1 = 6 5 r (x 11 ) r (x 10 ) 1 = 6 5 r (x 2 ) + r (x 4 ) r (x 1 ) r (x 5 ) 0 = r (x 2 ) + r (x 7 ) r (x 1 ) r (x 8 ) 0 = r (x 3 ) + r (x 6 ) r (x 0 ) r (x 9 ) 0 = r (x 4 ) + r (x 7 ) r (x 1 ) r (x 10 ) 0 = r (x 0 ) 0 = 0 Z P6 Z R6 M 1 M 2 M 1 M 2 M 1 M 2 E (M 1 ) E (M 2 )

49 x 11 =p 2 s 3 x 10 =p 1 s 3 x 8 =p 2 s 1 x 5 =p 2 s 2 x 9 =p 0 s 3 x 4 =p 1 s 2 x 7 =p 1 s 1 x 2 =p 2 s 0 x 6 =p 0 s 1 x 3 =p 0 s 2 x 1 =p 1 s 0 x 0 =p 0 s 0 R 6 P 6 Z (M 1 M 2 ) = (Z (M 1 ) \ {E (M 1 )}) Z (M 2 ) Q Z (M 2 ) = {Z E (M 1 ) : Z Z (M 2 )} {E (M 1 )}, M 1 M 2 Q =, r M1 M 2 (X) = r M1 (X) X E (M 1 ) r M1 M 2 (E (M 1 ) Y ) = r M1 (E (M 1 )) + r M2 (Y ) Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 = {X : X Z 1, X E (Z 1 )} {E (Z 1 ) Y : Y Z 2, Y } Q

50 {E (Z 1 )}, 1 Z(M1 ) = E (M 1 ) 0 Z(M2 ) = Q =, Z 1 Z 2 Z 1 Z 2 Y Z Y Z Z (M 1 M 2 ) = Z (M 1 ) Z (M 2 ) Y, Z Y Z Y Z C (Y) = [P 1 Z ] 1 0 C (Z) Z Q 1 Y C (Y) 0 Z C (Z) 0 Z Y Z Y Z

51 C (Y Z) = P 1 Y Z Q X C (Y Z) C (Y Z) C (Y) C (Z) C (Y Z) C (Y) C (Z) X = Y Z Y C (Y Z) Y C (Y) Z C (Y Z) Z C (Z) C (Y Z) Y Z Y Z (ϵ j ) j Y, (ζ k ) k Z ϵ j = ±1 ζ k = ±1 ϵ i Y i ζ i Z i ζ i Z i ζ i Z i ( ζ k ) C (Y Z) X m n Y Z (ζ k ) ϵ m = ζ n C (Y Z) ϵ i Y i + ζ i Z i (η l ) l X η l = ±1 η i X i C (Y Z) Y Z C (Y Z) = ( P 1 Y Z Q 1 Y 0 Z C (Y Z) X C (Y Z) Y Z X = Y Z )

52 Y Z C (Y Z) C (Y) C (Z) 1 Y 0 Z C (Y Z) Y Z 1 Y = 0 Z X Y Z = {1 Y } (ϵ j ) j Y, (ζ k ) k Z 1 Y = 0 Z X m Y Z ϵ m = ζ m (η l ) l X η j = ϵ j 1 j Y η k+ Y = ζ k 1 k Z k m X X j = Y j j X Y +k = Z k k m C (Y Z) η i X i Y Z Y, Z Y Z C (Y Z) T B C (Y Z) T D BD = C (Y Z) T A P 1 Y 0 0 C (Y Z) = Z Q C (Y) C (Z) B Y, B Z D Y, D Z B Y D Y = C (Y) T B Z D Z = C (Z) T Z C (Y Z) C (Z) T B Z B Z

53 B Y B Z C (Y Z) C (Y Z) T 0 Z B Z a B = B Y B Z {a} C (Y Z)T B B B = ±1 B B Y I Y B Y Y + 1 ( I Y 0 0 B Z B Z B ) D T = ( [D Y ] T 0 0 [D Z ] T ) C (Y Z) BD = C (Y Z) B C (Y Z) T D BD = C (Y Z) C (Y Z) C (Y Z)

54 C (Y Z) = ( P 1 Y Z Q ) B = B Y B Z C (Y Z) T ( I Y 0 0 B Z Y ±1 ( I Y B Z B D Z D Z D T = ( ) ) ) [D Y ] T 0 0 [ ] D T Z BD = C (Z) D Z D Z C (Y Z) Y Z Y Z Y Z v u P (Y ) P (Z) v u + x x

55 Y Z P (Y Z) P (Y Z)

56

57 P = (P, ) P = (P, ) a b P b a P M E F M F M F M E M e F e F e (E F ) {e} e F F E F Z E M E \ Z M Z E Z = {E \ Z : Z Z} E X Z X Z X X E X Y X, Y Z Y X Z

58 Z Z Z Z Z Y X Z X Y Z ( 0 Z X Y 1 Z ( 0 Z X Y 1 Z X, Y Z X Y ( 0 Z X Y X Y X Y 1 Z ( 0 Z X Y X Y X Y 1 Z C (Z) C (Z ) ( 0 Z 1 Z ) ) ) ) )

59 ( 0 Z 1 Z C (Z ) C (Z) C (Z) ) Z E P (Z) r P (Z) r P (Z ) r r (X) = X r (M) + r ( X ) X Z r (M) = r (1 Z ) + E \ 1 Z. r (0 Z ) = 0 0 Z 0 Z = E \ 1 Z 1 Z r (0 Z ) = E \ 1 Z (r (1 Z ) + E \ 1 Z ) + r (1 Z ) = 0. X Y X, Y Z Y X r ( X ) r ( Y ) X Y 1, r ( X ) ( ) X + 1 r Y Y E r (M) E ( ) ( ) X r (M) + r X + 1 E Y r (M) + r Y,

60 X r (M) + r ( X ) + 1 Y r (M) + r ( Y ) r (X) + 1 r (Y ). r ( Y ) + 1 r ( X ) Y r (M) + r ( Y ) Y + 1 X r (M) + r ( X ) X r (Y ) Y + 1 r (X) X r (Y ) r (X) Y X 1. r (Z2) X, Y Z X, Y Z Z r ( X ) + r ( Y ) r ( X Y ) + r ( X Y ) + X Y X Y, X Y = X Y X Y = X Y r ( X ) + r ( Y ) r ( X Y ) + r ( X Y ) + X Y X Y. X Y = X + Y X Y r ( X ) + r ( Y ) + X + Y r ( X Y ) + r ( X Y ) + X Y + X Y, r ( X ) + r ( Y ) + X + Y r ( X Y ) + r ( X Y ) + X Y + X Y, X + r ( X ) + Y + r ( Y ) X Y + r ( X Y ) + X Y + r ( X Y ) + X Y X Y

61 r (X) + r (Y ) r (X Y ) + r (X Y ) + X Y X Y. r (Z3) r P (Z ) r r P (Z) P (Z ) r P (Z) r X Z X Z r (1 Z ) r (1 Z ) = 1 Z r (M) + r (E \ 1 Z ) r (1 Z ) = E \ 0 Z r (1 Z ) E \ 1 Z + r (0 Z ) r (1 Z ) = 1 Z 0 Z r (1 Z ). r (X) = X r (M) + r ( X ) r (M) = r (1 Z ) + E \ 1 Z r (X) = X r (1 Z ) E \ 1 Z + r ( X ). r ( X ) r ( X ) = X r (M ) + r (X), r (M ) r (1 Z ) + E \ 1 Z r ( X ) = X r (1 Z ) E \ 1 Z + r (X), r (1 Z ) r ( X ) = X 1Z + 0 Z + r (1 Z ) 0 Z + X r (1 Z ) E \ 1 Z + r ( X )

62 r ( X ) = E 1 Z E \ 1 Z + r ( X ) r ( X ) = r ( X ). r r r r P (Z ) P (Z) r P (Z) P (Z) n C (Z) x b (Z) r X 1, X 2 Z X 2 X 1 r (X 2 ) r (X 1 ) = 1 X 1, X 2 Z X 1 X 2 r (X 1 ) + 1 = r (X 2 ) r (M) X1 X1 X2 X2 X1 r (M) + r (X1 ) X1 + 1 = X2 r (M) + r (X2 ) X2 r ( X 1 ) X1 + 1 = r ( X 2 ) X2 r ( X 1 ) r ( X 2 ) = X 1 X 2 1. C (Z ) x b (Z ) Y 1, Y 2 Z Y 2 Y 1 r (Y 2 ) r (Y 1 ) = Y 2 Y 1 1

63 Y 1, Y 2 Z Y 1 Y 2 r (Y 2 ) r (Y 1 ) = Y 2 Y 1 1 r (Y 2 ) Y = r (Y 1 ) Y 1, E r (M) E Y 2 r (M) + r (Y 2 ) + 1 = E Y 1 r (M) + r (Y 1 ) Y 2 r (M) + r (Y2 ) + 1 = Y 1 r (M) + r (Y1 ), r ( Y 2 ) r ( Y 1 ) r ( Y 2 ) + 1 = r ( Y 1 ). C (Z ) x b (Z ) Z 1, Z 2 Z r (Z 1 ) + r (Z 2 ) = r (Z 1 Z 2 ) + r (Z 1 Z 2 ) + Z 1 Z 2 Z 1 Z 2 Z 1, Z 2 Z r (Z 1 ) + r (Z 2 ) = r (Z 1 Z 2 ) + r (Z 1 Z 2 ) + Z 1 Z 2 Z 1 Z 2 r (Z 1 ) + r (Z 2 ) = r (Z 1 Z 2 ) + r (Z 1 Z 2 ) + Z1 Z 2 Z1 Z 2, Z1 + Z1 r (Z 1 )+r (Z 2 )+ Z1 + Z2 = r (Z1 Z 2 )+r (Z 1 Z 2 )+ Z1 Z 2 + Z1 Z 2. r (Z 1 )+r (Z 2 )+ Z1 + Z2 = r ( Z 1 Z 2 )+r (Z 1 Z 2 ) + Z1 Z 2 + Z1 Z 2

64 Z1 + r (Z1 ) + Z2 + r (Z2 = ( ) Z1 Z 2 + r Z 1 Z 2 + ( ) Z1 Z 2 + r Z 1 Z 2 + Z1 Z 2 Z1 Z 2 r ( Z 1 ) + r ( Z 2 ) = r ( Z 1 Z 2 ) + r ( Z 1 Z 2 ) + Z1 Z 2 Z1 Z 2. C (Z ) x b (Z ) 0 Z C (Z) x b (Z) 0 Z C (Z ) x b (Z ) C (Z) x b (Z) r C (Z ) x b (Z ) r C (Z ) C (Z) r P (Z ) r P (Z ) Z

65 Z Z e M e E (M) Z M \ e Z Z {e} M Z (M \ e) Z (M)

66 b a c a b c d e e d M 1 M 2 {a, b, c, d, e} {a, b, c, d} {a, b} {a, b} {a, b} M 1 M 2 M 1 \ e M 2 \ e

67 F 6 F 6 \ D F 5 F 2 F 3 F 4 F 2 \ D F 3 \ D F 4 \ D F 1 Z E X Z X W Z D E X = {W \ D : W W} W X Z E X Z W Z X D E X = {W \ D : W W} W W X = W \ D X X W X Z Z X X W X W X S Z S X = {X X : Z X S} S X M \ e M/e M

68 {a, b, c, d, e, f, g, h, i} d c a b e g h i f {a, b, c, d} {a, b, e, f} {a, b} {g, h, i} N N Z E S, T Z Z S Z T X Z X S X X T X Z X S T X S X > X T X s S X t T X s t t T X S X X S X > X T X t T X X S X t t X S X S X X S X S X X Z ϕ : X Z ϕ (X) Z ϕ (Y ) = ϕ (X X Y ) ϕ (X) Z ϕ (Y ) = ϕ (X X Y ) X, Y X

69 1 Y β n α 0 Y β 2 β 1 Y n Z X X Z Z N N \ a X = Z (N \ a) {a, b} {a, b, c, d} {a, b, e, f} {b, c, d} {b, e, f} Z (N \ a) Z(N) X = Z (N) \ {{a, b}} Z (N) {a, b, c, d} {a, b, e, f} Z (N) {a, b} Z(N) X

70 {Y 1, Y 2,..., Y t } (ϵ j ) j t ϵ j = ±1 1 j t t i=1 ϵ iy i L C Y = {Y 1, Y 2,..., Y t } C (ϵ j ) j t ϵ j = ±1 1 j t Y p Y q ϵ p = ϵ q Y s = 0 Y Y Y r = 1 Y Y Y Y s Y r ϵ s = ϵ r 0 Y, 1 Y, α Y ϵ j = ϵl 1 j, l t 0 Y 1 Y Y ϵ j = ϵl 1 j, l t Y m = α Y 0 Y, 1 Y / Y ϵ j ϵ m j m t i=1 ϵ iy i C (Y n ) p q Y Y ϵ p = ϵ q 0 4 α β k 1 k n 0 Y 1 Y

71 Y Y Y s = 0 Y Y r = 1 Y α / Y β k / Y 1 k n 0 Y 1 Y ϵ s = ϵ r 1 β k 1 k n 0 Y 0 Y 1 Y Y ϵ j = ϵ l j, l β k 0 Y β k 0 Y Y β k 1 Y α 0 Y α 1 Y Y m = α β k 0 Y, 1 Y / Y ϵ j ϵ m j m α β k Y 0 Y, α, β k α, β k, 1 Y ϵ j ±1 0 Y, α, 1 Y ϵ j ±1 0 Y, β k, 1 Y Y 0 Y 1 Y ϵ j ±1 β k = Y u ϵ u ϵ s ϵ r ±1 Y ϵ j 0 Z Z

72 b 6 c 6 a 5 b 4 b 5 c 4 c 5 a 2 a 3 a 4 b 2 b 3 c 2 c 3 a 1 b 1 c 1 Z Z 3 Z 1 Z 2 Z 3 Z C (Z) 2 Z 1 Z 2 Z Z W Z W = {Z 1, Z 2, Z 3, Z 4, Z 5, Z 6 } Z i b i Z 2 Z 3 Z 1 Z 1 Z 6 Z 2 Z 3 Z 4 Z 5 C (Z) Z 1 Z 2 Z 3 Z

73 2 Z W Z W = {Z 1, Z 2, Z 3, Z 4, Z 5, Z 6 } Z i c i Z 4 Z 2 Z 3 Z 5 Z 3 C (Z) 2 Z 2 Z 3 Z 4 Z Z 2 Z 1 C (Z) Z Z 2 S T Z s 1 s 2 S T s 1 < s 2 s 3 S T s 1 < s 3 < s 2 x S x T x s 1 s 2 x Z W Z Z Z Z W Z Z

74 Z Z 1 d 1 = {1, 3, 5, 6, 11, 12} d 2 = {0, 1, 5, 6, 9, 10, 13} d 3 = {0, 6, 7, 8, 10, 12, 13} d 4 = {1, 4, 6, 7, 8, 10, 13} d 5 = {0, 2, 3, 5, 7, 8, 9, 11, 13} d 6 = {1, 2, 3, 5, 6, 11, 12, 13} E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. P (Z 1 ) Z 1 r 1 = (0, 5, 6, 6, 6, 6, 6, 7) r 2 = (0, 5, 5, 6, 6, 6, 6, 7) M 1 r 2 2 Z 1 Z 1 P ( Z 1) r 1 = (0, 5, 5, 6, 6, 6, 7) r 1 2 = (0, 5, 6, 6, 6, 6, 7) r 3 = (0, 4, 6, 6, 6, 6, 7) r 4 = (0, 5, 6, 6, 6, 5, 7) ( r 5 = 0, 9 2, 11 2, 6, 6, 11 ) 2, 7

75 E d 1 d 2 d 3 d 4 d 5 d 6 Z 1

76 E \ {2} d 1 d 2 d 3 d 4 d 5 \ {2} Z 1 Z 1

77 Z 1

78

79

80

81

82

83

84

85

86

87 M E C E M C E Z M C Z C = r (Z) + 1

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