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4
( .. . - . - ( . . - . ., ( .. ! 1 - 0 , , . - . (($ . $ 12 ! : . 9 =. - = ! !0 - >!0 . , $ . , . - 0 1 . -$ ( , ( .- 0 - +. $ , - - , , - $ . ( .(0 - ( 0 .- = 0! , , , ? : ! ( , ( - ( ! ( . ! ( . - 9 . - ! - 0 1 , . , -@ .- . . A . B - ( A ( B , - +. $ : = , ! . ! 0 $ -- - ! . ! ( ( - , ( ( - =. ( ,$ , = - ! .- - +. , .$ 6
= , , , ( $ , $ -- - .- ( C $ . 1 - .(1 - , ! , - : ( - - +. , , - - D D-- - $ E . ( -$ ( ,.- = , . , ( -$ . . , . ( -- ( H, , F ( , G . 1 , ( - -2 -2- ( ( -$ . -$ ($ = - , ( * .- 0 - . : . - ( H, .- ( . - -. ( $ - 012 ( H , -$ .-12 ( ( H .- ( !$ = , ! - , = , - ( - , - , : , = $ = (0 . H ! , : ( --1 - - - ! H− . $ = ( $ .- , , 12 ! , ( % . ( ( * - ( H, . - .- - ( - A B .! ( -$ = , . , -- - , ( 9, -$ <,$ , ( , A , . ! , ( B ! .-12 ( - ( !$ ($ ( H -- - . , . : ( $ . . . ! ! $ , ( .- ( '
H, ! ! ( - I : ( . - = 2 - = I, -$ - . -- - N P − I- 1 2 . ! = , ( $ 212 $ .1 - (0 ( 4 . -$ . ( * %$ .( - ( SH, , 1 - . , - . , - ! . : , = . - 0 .- ( -$ = . 2 - . ( = ( SH 1 - . - (( ! : = . - H+ E <( ( 9. . - , , ! " * )*$ )%$ )4$ )6 )* .- - , - -$ . - -$ ! , . 9 ! E = {0, 1}, D = {+1, −1}, D¯ = [−1, 1], Z− =
, R− = . ! , J E n , Dn D¯ n n− , , Rn , (α1 , α2 , . . . , αn ) ∈ E n . , ! . , A = 0 E n B ! a¯ a¯n $ (a1 , a2 , . . . , an ) ∈ R . ! a ¯ a ¯n . K a ¯n n n n ¯n ¯ b R $ ρ(¯a , b ) $ =. 1 n |¯ an | = ai A E n : -- - F i=1 (B$ . a¯n . ! ¯a$ - . Rn (¯a, ¯b). 5 ( ( .. . : 9 ! 1 - . C - K1 K2 , . A: ,B : $ , ! - . 9 ! -. = ! = {1, 2, . . . , n}. E(. ! - C a K1 A K2 B -- - ( . , 8
a ˜ = (α1 , α2 , . . . , αn ), (. αi . * & ($ ,= i C a¯, 1 i n. E $ - K1 K2 , , . A: B$ = a1 , a2 , . . . , an ∈ K1 b1 , b2 , . . . , bn ∈ K2 . = {˜a1 , a˜2 , . . . , a˜n } {˜b1 , ˜b2 , . . . , ˜bn }. G .- : $ - ! 012 A( B 1 ! ( C x ∈ K1 ∪ K2 , : C K1 K2 , , ( .= .- ( - G . ( ! C K : L1 = {a1 , a2 , . . . , an } L2 = {b1 , b2 , . . . , bn }. 9 ! A 2 ( G B C K : L1 L2 . ( ! G . K1 K2 . : L1 L2 . L1 = K1 , L2 = K2 , K1 K2 . , ! G . , $ .- 1, A ,B : , = L1 ⊆ K1 L2 ⊆ K2 , K1 K2 --1 - G . , . L1 L2 , . ( ! ! G . K1 K2 . . , - - . 1 - , : , = - ( G. > . ! . 12 1 1 k : K → E n , -12 1 : K ( $ $ . ( = ! , K1 K2 - $ -- . 1 (1 ( ! A ! B G<. ! . = n ( 9. ( - 9 ! α˜ ∈ E n M = {˜α1 , α˜2 , . . . , α˜m } ⊆ E n ! , . ,@ P(˜ α, M ) =
m
ρ(˜ α, α ˜ i ), ρ(˜ α, M )
i=1
+ , (
α ˜ ∈ E n J M1 , M2 ⊆ E n .
H
1 P(˜ α, M ). m
. - . 12 I ,
* , -1 - , ρ1 = ρ(˜α, M1 ) ρ2 = ρ(˜α, M2 ). 3
% 012 A ∈ D - DB ⎧ ⎪ ρ1 < ρ2 ⎨M1 , α ˜ ∈ M2 , ρ1 > ρ2 ⎪ ⎩ . ρ1 = ρ2 F ( = ,. ! . 12 . , - .- H <. = ˜ 9 ! M1 , M2 ⊆ E n , |M1 | = m1 , |M2 | = m2 di = max ρ(˜α, β), ˜ (. max - α˜ , β ∈ Mi , . = Mi , i = ˜ ". min - α α, β), ˜ ∈ M1 $ β˜ ∈ M2 1, 2, R = min ρ(˜
- =. = M1 M2 . 5. 456758 ... 9 , mi di < , R mi − 1
i = 1, 2
= M1 M2 H <. ,J , di < 1, i = 1, 2 R M2 ! H <. ,
= M1 L $ $ . , *** !$ 2 (-$ !-@ , ,$ A.= ( ! 12 , (B = n n
k,n 9 ! C = α˜ |˜α | n − k Ck,n = α˜ n |˜αn | k 12 -- = -- ( E n . 9 - , *** = = ! . 1 2 ", ) ..0 = 0 C k ,n Ck ,n ! H <. , (. ! (.$ (. . , -1 - @ 1
3k1 + k2 < n,
2
k1 + 3k2 < n.
+ $ H <. : = ! , *%% 0 - @ = C k ,n Ck ,n H <. , (. ! (.$ (. k1 < n2 k2 < n2 . ; 1
2
)% , -$ ( H .-12 ( ( ! n I- = M ⊆ E , . 1 , ! p¯ = |M | α ˜ A . $ p¯ . α∈M ˜ - . , -= , ! , = M . , . B 9 ! p¯1 p¯2 , , = M1 M2 I. '%," q¯ = 12 (¯p1 − p¯2) , - , .- = M1 M2 . + . (1, 1, . . . , 1) ˜1. 5.
% ..0 012 ( H J .-12 ( . (¯ q , (2x − ˜ 1)) = 0
(. q¯ , .- .- , ( = M1 M2 . ( E n . . ! ( n D , , . ( , A Rn B$ . , l(˜x) = 2˜x − ˜1. l(˜ x) ∈ Dn . ! x ¯. H ! $ . E n n D 2 - - , . −1 A . C - −1 B 5 .- Dn . 1 (1 - + $ , p¯ .- = M ⊆ Dn , - - !0 p¯ = |M1 | a ¯. a∈M E $ .-12 ( ( Dn . (¯q, x¯) = 0. + . ( . $ . , 12 *** *%% A ! 1 - - , ***B ", ) .02 9 , mi di < , i = 1, 2 R mi − 1
= 0 Dn = . ! . . (( 2 ( $ .-2 . /
)4 .= - )* . ! H <. : ( ( . , . A.- H <. B . . A.- ! H <. B - 9 ! Ri = max ¯a − P¯i , (. max - a¯ ∈ Mi ,
. = Mi ⊆ Dn $ P¯i , = Mi , i = 1, 2, M q¯ , . = M1 M2 .
% .20 9 , R1 < ¯ q +
P1 2 − P2 2 , 4q
R2 < ¯ q +
P2 2 − P1 2 4q
= M1 M2 H <. , ! H <. .
% .2- = M1 M2 Dn ! H < . , (. ! (.$ (. .- 1, . : , . = {¯a} ⊆ M1 {¯b} ⊆ M2 H <. , . {¯a} {¯b}. . -@ M = M1 ∪ (−M2 ), (. − M2 = {−¯b|¯b ∈ M2 }; ¯ = {x|x = a − b|¯ M a ∈ M1 , ¯b ∈ M2 }.
, , $ =. , = M (M¯ )$ K¯ ∗ ¯ G
$ . , K. )%9, .21 = M1 M2 ∗ Dn ! H < . , (. ! (.$ (. K ⊆ K . G =. - *4' , ", ) .2: K .- . A . B √ d(M ) = M , - - d(M ) < 2n, = M1 M2 Dn ! H <. , )6 . 1 - ,$ - , (0 , ( H = M1 M2 $ (. . .(1 - 2 1 S. 9 ! M . = M. E(. 2 S = {sij } ! - (−1, 1)< =. sij = −1 . 1 0 A * * *& ¯ K(K)
B$ - . i< j < NO , M. 9 ! q¯ = (q1 , . . . , qn )
, .- = M1 M2 . . -@ q∗ = min |qi |$ q ∗ = max |qi |$ b∗ = min(¯ q, a ¯)$ b∗ = max(¯ q, a ¯), (. i i min max - a ¯ ∈ M = M1 ∪(−M2 ) A = M1 M2 .(1 - . , B 9 , . 12 ! , A 1 2 . - .- 2 ,B@
% .-0 I- (0 Δ - = M1 M2 , . , (0 !1 δ, ., @ 2q∗ · n 2q∗ · n 1 <Δ − · δ. b∗ m b∗
", ) .-2 K =. , S 0 !0$ 2qb , = M1 M2 A ! δB " % )* )% .= , ! , , ( -$ . -12 ! ,$ , ( * )* . - ( - . , ( -= : ( ( - 2- . - A B$ - )* (, * +2 . 12 @ .C x ∈ K .=2( 1$ . . ! -$ ( x¯ = k(x) -- - n< ( Dn . E $ . 12- - k -- - = K = D¯ n K → D¯ n , M xi
, 5.= ! $ xi i .- C x ∈ K ,= C x i. = - = -$ : - k - = ( = Dn , = , ! $ ! : 0! C , ,= , , I- ( . .( - - . 12- .! ( $ - . ! , ** ∗
∗
9 ! Φα¯ (¯x) -$ . ( 0 α¯ x ( D¯ n . > . .( !$ Φα¯ (¯x) -- *B ! , A ! α˜ B$ ¯, x ¯) , Φα¯ (¯ x) = fΦ (ρσ α
(. fΦ - . ! - -$ ! σ *$ ρ , - $ =. n |¯a| = |ai | σ = 2, , - . J i=1 %B .. , $ ΦM (¯x), . , x¯ = 0 M = {¯α1 , . . . , α¯m } , - - @ ΦM (¯ x) =
m
Φαi (¯ x);
i=1
4B ,12 $ - fΦ *B -- - A (B ,12 =., Φ .- ( - AΦ . 12 012 R0 : B C x¯ - K1 A K2 B$ 1 1 ΦM (¯ ΦM (¯ R0 (x) = x) − x) > 0 A R0 (¯ x) < 0); |M | |M | 1
1
2
2
B R0 (¯x) = 0) 0 -$ (. M1 M2 - : , = K1 K2 # R0 (¯x) = 0) .- 2 ( ¯ n 1 .-12 1 ( ! D + $ ( ( , -$ , A $ ( ( - , . , K B$ - A $ ( ,$ , , Γ< .B ( , ! , ..-2 , , ( - 9. . , .- . - > . ( !$ ( , A1 A2 : ,$ ¯n ∀x ∈ D
*%
(1)
(2)
sign(R0 (x)) = sign(R0 (x)),
.- 1( ( , A1 A2 1 . A . , : , = B '%," ( AΦ, Φ ( .- - . fΦ (x) = Kx + b, . , ! , ( - G
% 0.- 71 , ( - : ( H. ? - ! ! , (, * , ( , ,. '%," ( $ .-12 .- 1 , . = M1 M2 Dn .-12 ( ( !$ . , ! ( ( ? . - ! . %*6 *%% ", ) 0.1 7 , ( , - --1 - ( ( +, -$ , . ! , ( -$ ! , ( , ( .1 ,
% 0.: 9 ! n 3. E(. 1 ( ( .=., . fΦ . , ( + $ n = 2 = ! 1 1 A .=., . 1B 1 fΦ $ .- ( AΦ . ( - ( ; % Φ = ! - . 12@ m 2 ) ΦM (¯ x) = fΦ ( ρ(¯ αi , x ¯)), i=1
(. fΦ (x) ( , ! -$ $ C .
e−αx $ C x+C *4 1
2
3
($ 012( R0 = ! ! - 012 R1 , .-12 .-12 1 ( ! R1 (¯ x) = ΦM1 (¯ x) − ΦM2 (¯ x) = 0.
+- ! (. . ! , %*6$ . . !$ ! ,12 fΦ ( -$ .- , Φ *$%$4 R0 , -- - ( ( $ 2 $ -- - , ( -J = R1 .-12- ( ! 2 , - ( A. , ,0 ρB )% . - = 2 - . ( = 9 ! M1 M2 : , = Dn M = M1 ∪ (−M2 ). ? = = . ! . (−1, 1)< $ , . ! = $ 12 = α ¯ , M. + 9 ! d¯ = α∈M ¯ $ |M1 | = |M2 | d¯ q¯ .- = M1 M2 d¯ = |M | · q¯ A M . - B P B = min(d,¯ α¯ ), (. α¯ ∈ M, . , ! ( - A.- M .- M1 M2 B 9 ! Π = {1, 2, . . . , n} . = M
n K Π ⊆ Π ! . = $ M - , M = Π ! $ -2- , Π, , ! : , Π . 5. 12- - . -- - ! , .- ( H . !0 ( @ .- . ( = M ( - B > 0 ! A 2 B = Π , $ ( - .- M , !0$ C $ (. C $ 0 < C B. 9 - . . ! N P < , 5 . 2 - - A.*6
( H B . - @
%< '"* '%() ") L. , . : , (−1, 1)< , M
1 M2 m × n A ( - B > 0 B , = ! , K n C B. '% 5 2 . = Π , |Π | K ( - .- M1 M2 M1 M2 Π !0$ C Q G
% 00. L. 2 - - 9 . = N P < N P < . 5+ RKGK 9+7S 9 GLT + , ! , = : , , ( .- 0 - , . - : )% .( - ( $ 012( - , . = A : , - ( 2 - = $ ! , .( ( ( 4 ( B I- ( . ! (0 9 ! A = , ( $ , . ( : ( P K(m, n, z) . ! (−1, 1)< m × n, =. , - r −1. + $ ! d¯ = (d, d, . . . , d), (. d = m − 2r. I- (−1, 1)< , M ∈ K(m, n, r), M = (αij ( - B > 0 ( , = S . = S ⊆ {1, 2, . . . , m}, ∀i ∈ S n , d · αij = B. j=1 9 ! ! , A 2 - . ( = B ( A M M , A ( - B B A 0 -B KA (M ), s 2 ! ( ( = s = |S|. 9 !$ ($ K ∗ (M ) ! ! A $ $ *'
, - , M B .- . , M , ( - 2 !0$ B. (M ) '%," RA(M ) = KK (M ) (0 !1 = ( ( A . M. . . 12 1 a, 12 1 ! - ( - B . ,@ A ∗
a=
Bm nd2
+ $ 0 < a 1. 5. . 12 )%9, 001 9 s > 2r .- (0 RA(M ) = ( ( A M ∈ K(m, n, r) @
RA (M ) = min
1− 1−
2r m 2r s
,
1 a
.
" 4 )* )% )* ,1 - : ( SH : $ -2- . 12 : @ *B .( ! , : $ %B A B H <. $ 4B A B ! H <. $ 6B 2 = U$ 'B 2 = UU : %B 4B . . , : , C $ : 6B 'B 2 !0 - ( - . ( = A : ! - = , ( 2 - - $ , )4 (, %B H+ E <( ( SH, 12- : . = )% ( SH 1 . - $ -2 - (($ 2 - - - 0 - =. ! - ! , =. - , ( AL!$ 5< 555 $ $ L. - KB ! <: ! - (. , , *8
! - : $ 212- . - ( - ( SH, = - ( . ,0 1 -$ ! , - ;'V ! ,$ .=2 - ($ , */;&< */;6(( . . < ( ! ( ( . ( 7 $ , W*'$ *8$ *3X .., ! YU 1 A5 $ */;4B$ 1 . A $ */;6B < . ! < ( ! "#@ . . > .- A*/;6B$ . . . . 5 0 . 5 9. 9! - ! $ ,= ( 1 (. ! . > .- > ( 11 2! - $ = " I !0 1 2! .( (
*3
-2- ( -2 1 -. -2 - . : ( ( - H. )* .- - , - -$ . - - I- 0 - . : - ( H, ( $ ( = .- D D ( C $ ( H ( ( - = - - ( - - , ! H . = 9. - $ .12- . , - , . > . : - )4 -2 (, )%$ -2 012 ( H, - = - .- .! 0( . - $ 012 ( H @ .-0 ( ( H ( ! )6 : (, -2 1 - - (0 : , C - G . 1 - = ( A ! 0 B - *;
H > . ! ! . 12 - , C - @
E = {0, 1, }; Z = , J R = . ! , J E n . , n< , J (α1 , α2 , . . . , αn ) E n . , !
. , , A = 0 E n B ! α ¯ $ =- . ! ( !$ α¯ n J . , E n , , . -1 - , , , −, A.C 1 $ C1 $ -$ = - mod 2$ = , -. <J . Z2 - = - . , mod 2 = - * & BJ @ |˜ α |= n
n
αi
i=1
A|˜α|− . α˜
Sρ (˜ αn ) = β˜ ρ(β˜n , α ˜n) = ρ , 0 ρ n
. ρ 0 α˜n
n n
α | = k , 0 k n− Snk = α ˜ |˜
k
E n , n
C k,n =
Sni , 0 k n
i=n−k
-- k 0 E n , C
k,n
=
k i=0
Sni , 0 k n
*/
= -- k 0 E n , A - B F (@ n
ρ(˜ αn , β˜n ) = |˜ αn ⊕ β˜n | =
|αi − βi |
i=1
ρ(˜ αn , β˜n ) . $ , 1 β˜n , -. = @ α ˜ n β˜n ⇔ ∀i(αi βi ) (˜ α0n , α ˜ 1n , . . . , α ˜ kn ) n n ˜ α ˜ β , @
, α˜ n
! A. , kB$ . -12- 0 ,
n α˜kn = β˜n ∀i(ρ(˜αi−1 ,α ˜ in ) = 1),
! 12-$ . !
α ˜ 0n = α ˜n
n ∀i(˜ αi−1 <α ˜ in )
! ! -$ . ! A . $ .- 1, α˜ n β˜n - ! 2 . ρ(˜αn , β˜n );B Zn n - - 0 $ (a1 , . . . , an ) Zn . , ! A B 0 A B ! a an , @ |a| =
n
|ai |
i=1
n
1/2
A. a2i . ! aJ i=1 A=. B @ ρ(an , bn ) = |an − bn | =
n i=1
|ai − bi |;
. $ (p1 , p2 , . . . , pn ) Rn . , ! = ! p¯ p¯n J %& Rn
- , . @ (¯ pn , q¯n ) =
n
pi · qi ;
i=1
¯ p = n
n
1/2 p2i
;
i=1
A=. . B ρ(¯ p , q¯ ) = ¯ p − q¯ = n
n
n
n
n
1/2
2
(pi − qi )
.
i=1
9. ! A ( ( .. W*X 0 1 . -B . : 1 ( H. 9 ! 1 - . C - K1 K2 $ 12 -B , . A: ,B K1 K2 , , ! -. 9 ! -. = ! = 1, 2, . . . , n . E(. ! - C a K1 A K2 B -- - ( . , a˜ = (α1 , . . . , αn ), (. αi . * & ($ ,= i C a, 1 i n. E $ - K1 K2 , , . : $ = a1 , a2 , . . . , am ∈ K1 b1 , b2 , . . . , b m ∈ K2 , . =
a˜1 , a˜2 , . . . , a˜m ˜b1 , ˜b2 , . . . , ˜bm . m1 × n m2 × n, , --1 - A -. B C a1 , a2 , . . . , am b1 , b2 , . . . , bm G .- : $ - ! 012 A ( B 1 ! ( C x . ( ! , : C $ , ( .= > $ ! K1 K2 . = $ K1 ∩ K2 = ∅, K1 ∪ K2 = K. 9 ! χ1 χ2 : = 1
2
1
2
1
2
χ(x) ˜ = χ1 (x), χ2 (x) , x ∈ K.
%*
G - . 12 = $ -12 : = K ( @ k : K → E n , k(x) = x ˜, x ∈ K,
. - 1 χ(x) ˜ = χ(y) ˜ ⇒ (˜ x = y˜).
I- = M1 = {a1 , a2 , . . . , am1 } ⊆ K1 $ ˜1 M ˜ 2 M2 = {b1 , b2 , . . . , bm2 } ⊆ K2 M ˜ ˜ ˜ = {˜a1 , a˜2 , . . . , a˜m1 } {b1 , b2 , . . . , bm2 }, :
, --1 - 12 - 9 ! ΦM˜ ,M˜ : K → E 2 \ {1, 1}, ΦM˜ ,M˜ (˜ x) = ϕ(x), x ∈ K A***B
= $ : A . ( ( - A B .- M˜ 1 M˜ 2 I. ! . '%," - ( A = K1 ⊆ K1 K2 ⊆ K2 1
2
1
2
KM˜ 1 ,M˜ 2 (A, K1 , K2 )
(.
=
ϕ(x), ˜ χ(x)) ˜
x∈K1 ∪K2
K ∪ K 1 2
,
ϕ(x), ˜ χ(x)) ˜ = ϕ1 (x) · χ1 (x) + ϕ2 (x) · χ2 (x).
!0 . - $ (. ( A .- - = K1 = K1 K2 = K2 ; : .- , . ! ! KM˜ ,M˜ (A). '%," K .- , = M1 ⊆ K1 M2 ⊆ K2 KM˜ ,M˜ (A) *&&V$ . , ! A = ( B 0 , U . A.- . , K1 K2 B$ = K1 K2 A<. , A . M1 M2 B J $ (. KM˜ ,M˜ (A) = 100%, .- 1, = M1 ⊆ K1 M2 ⊆ K2 . ( ! 0 A 0 ( AB UU . A.- . , K1 K2 BJ = K1 K2 . ! ! A<. , 1
2
1
1
½
%%
2
2
M1 = K1 M2 = K2 ,
+ . $ = K1 K2 --1 - ! A< . , $ . 1, = K1 K2 , (. K1 ⊆ K1 K2 ⊆ K2 . L $ ! , . $ $ - K1 K2 . - A - χ(x) ˜ 0! . - $ 2 $ BJ , 0! : K1 ⊂ K1 K2 ⊂ K2 . E ( - = , ! , . , A B . 0! . = K1 K2 , , = . ! . , ,0 . - K1 K2 , .- , !$ K1 = K1 K2 = K2 . ($ = ! A ! ( B . 12 1 1 k $ (. = ! = K1 K2 - K˜ 1 K˜ 2 , -- . 1 (1$ ( ! A<. 0 ! . = n< ( 9. 1 ( H A. , $ ( H - ( ( - W*XB 9 ! α˜ n , β˜n ∈ E n , M = {˜α1 , α˜2 , . . . , α˜m } ⊆ E n ! , . . 12 , * γ(˜αn , β˜n ) = n − ρ(˜αn , β˜n ) . $ , , α˜ n β˜n .1 J m
( α˜ ! = M ; ˜ ) 4 Γ(˜α, M ) = γ(α,M
. ( α˜ m ! = M J %
γ(˜ α, M ) =
6
ρ(˜ α, M ) =
i=1
m i=1
γ(˜ α, α ˜i )
- α˜ . α˜ = α ∈ E β˜ ∈ E n , .
ρ(˜ α, α ˜i )
= ( !
M;
˜ = ρ(α, β)
n i=1
ρ(α, βi ) =
n i=1
(α ⊕ βi )
%4
'
. - = M. + . $ 0 @ P(˜ α, M ) =
ρ(α,M ˜ ) m
m
γ(˜ α, αi )
α ˜
.
m n − ρ(˜ α, αi )
γ(˜ α, M ) = i=1 = i=1 = m m m m m n·m− ρ(˜ α, αi ) ρ(˜ α, αi ) i=1 i=1 = =n− = n − P(˜ α, M ). m m
Γ(˜ α, M ) =
+ , ( H . - . 12 A . . : , B I , α˜ ∈ E n , M1 , M2 ⊆ E n ; * , -1 - ,@ B Γ1 = Γ(˜α, M1 ) Γ2 = Γ(˜α, M2 ) B P = P(˜α, M1 ) P2 = P(˜α, M2 ) % 012 ⎧ A D - DB@ ⎪ Γ1 > Γ2 ⎨M1 , B α˜ ⎪M2 , Γ1 < Γ2 ⎩ $ Γ1 = Γ2 ⎧ ⎪ M P1 > P ⎨ 1, B α˜ ⎪M2 , P1 < P2 ⎩ $ P1 = P2 5 .= $ .12 1 . H <. . = 0 E n A0 , ( H B 9 ! M1 , M2 ⊆ E n , |M1 | = m1 , |M2 | = m2 di = α,˜ max ρ(˜ α, α ˜) α ˜ ∈M
. = Mi , i = 1, 2, R = α∈M min ρ(˜ α, α ˜) ˜ ,α ˜ ∈M - =. = M1 M2 A ρ F (B 5.
% .
1
%6
mi di < , R mi − 1
i = 1, 2,
2
i
M1 M2 H M1
di < 1, i = 1, 2, R M2 H
( "# ) I- . ! , . ! $ ( H ! 0 , = M1 M1 , 0 , = M2 M2 . I ! $ ! M1 = {˜α1 , α˜2 , . . . , α˜m } M2 = {β˜1 , β˜2 , . . . , β˜m }. G ( 1 , ( H @ 1
2
(˜ α ∈ M1 ) ⇔ (P(˜ α, M1 ) < P(˜ α, M2 )).
.- -( i = 1, 2, . . . , m1 , ! - . - ,$ @ P(˜ αi , M2 ) =
m2 ρ(˜ αi , M2 ) 1 1 = ρ(˜ αi , β˜i ) R · m2 = R > m2 m2 j=1 m2
>
m1 d1 (m1 − 1) 1 ρ(˜ αi , α ˜ j ) = P(˜ αi , M1 ), m1 m1 j=1
. . ! ! , ( H = M1 : α M1 ). (˜ α ∈ M1 ) ⇒ (˜
E = - - ! ! , ( = M2 . I - , ! N1 ⊆ M1 $ N2 ⊆ M2 n1 $ dn $ n2 $ dn 2 . , = M1 M2 E(. ∀˜α ∈ M1 : 1
2
P(˜ α, N2 ) =
1 1 ρ(˜ α, βi ) R · n2 = R > dn1 > n2 n2 βi ∈N2
>
dn1 1 (n1 − 1) ρ(˜ α, βj ) = P(˜ α, N1 ) n1 n1 βi ∈N1
( $ (∀˜α ∈ M2 ) ⇔ P(˜α, n2 ) < P(˜α, N1 ). E . %'
; + $ . , ***$ 2 !$ 2 (-$ !-@ , , A.= ( (B . 12 @ M1 = {(1, 1, 1, 0), (1, 0, 0, 1)} d1 = 3, M2 = {(1, 1, 0, 0), (0, 1, 1, 0)} d2 = 2, R = 1, d2 P(α1 , M1 ) > P(α1 , M2 ), R = 2, (. α1 = (1, 1, 1, 0).
1 ( ! H < . . ! , 0 n ( A H . . 0 E n ! , *%% 0 - @ = C k ,n Ck ,n H <. , (. ! (.$ (. k1 < n2 k2 < n2 . 9 ! C k ,n Ck ,n -- k1 0 = -- k2 0 En . G , *** , ", ) ..0 = 0 C k ,n Ck ,n ! H <. , (. ! (.$ (. . , -1 - 1
1
2
2
1
3k1 + k2 < n,
2
k1 + 3k2 < n.
( "# ) + . $ .- . d1 d2 = 0 C k ,n Ck ,n $ = - - R = . A ,0 . , B . 12 - A = !$ k1 < n2 k2 < n2 B@ 1
2
d1 = 2k1
d2 = 2k2
R = n − (k1 + k2 ).
E!$ , - -$ 2ki 2ki di = < = 1, R n − (k1 + k2 ) 2ki
i = 1, 2.
. ! . , *** 9 ! ! . - , - -$ $ 3k1 + k2 n. (. 1 1 0 α n − k1 <( - E n , α ˜ ∈ Snn−k ⊆ C k ,n , .- S2k (˜ α) . 2 · k1 0 α ˜:
1
1
˜ = 2 · k1 . α, β) α) = β˜n ρ(˜ S2k1 (˜
. . $
S2k1 (˜ α) ∩ C k1 ,n = M1 = ∅
%8
1
S2k
1
(˜ α) ∩ Ck2 ,n = M2 = ∅
( H α˜ A : M1 M2 B -- - 0 $ = C k ,n Ck ,n --1 - H <. , . M1 M2 , --1 - ! H <. , 5. . 1
2
H - - - .- , - . ( Z $ . , )* , ( H, $ P. 9 ! M = {˜α1 , . . . , α˜m } ⊆ E n = . , n. L0 ( . , A ,B M m×n$ , α ˜1, . . . , α ˜ n A -.BJj , , M A . - . , . , m$ , j , . α˜1 , . . . , α˜n B S˜j , 1 j n . -. . @ 1
* %
Pj = |S˜j | j, 1 j n$ j , M; n
P =
j=1 m
P =
i=1
4 6
pj =
Pj m
j;
Pj
.
. M A . $
|˜ αi |BJ
. , A ,B
p¯ = (p1 , . . . , pn ) , A B = M A . $ p¯ . - . , -= , ! , = M . , .
, ( ( . , W4XB
G %3
En
.0. 9 ! α˜ = (α , . . . , α ), M = {˜α , . . . , α˜ 1
1
n
p¯ , = M. E(. P(˜ α, M ) =
n
m}
⊆
[αi + (1 − 2αi )pi ].
i=1
n
( "# ) . γ˜ β˜ ˜ ˜
E ,
ρ(β, γ˜) = 1 β < γ˜. 9 ! .- . βi = γi .- i = j A*%%B βj < γj
βj = @
0,
˜ M) = P(β,
γj = 1.
E(. . 12-
m 1 ˜ 1 ˜ ˜ α ρ(β, α ˜ 1 ) + . . . + ρ(β, ρ(β, α ˜i) = ˜m) = m i=1 m
1 (β1 ⊕α11 )+. . .+(βn ⊕α1n )+. . .+(β1 ⊕αm1 )+. . .+(βn ⊕αmn ) = m 1 (β1 ⊕α11 )+. . .+(β1 ⊕αm1 )+. . .+(βn ⊕α1n )+. . .+(βn ⊕αmn ) = = m n 1 1 = ρ(β1 , S˜1 ) + . . . + ρ(βn , S˜n ) = ρ(βi , S˜i ). m m i=1 =
( $
1 1 ρ(˜ γ, α ˜i ) = ρ(γi , S˜i ). m i=1 m i=1 m
P(˜ γ, M ) =
n
# ,- A*%*B$ @
˜ M ) = 1 [ρ(1, S˜i ) − ρ(0, S˜i )] = P(˜ γ , M ) − P(β, m 1 1 2Pj = [(m − Pj ) − Pj ] = [m − 2Pj ] = 1 − = 1 − 2pj . m m m
A*%4B , ! P(˜α, M ). I- : ( ! 1
! A . $ . $ |˜α|B$ . -12 1 α˜ , ˜0 = (0, . . . , 0) : ˜ ˜. 0 = δ˜0 < δ˜1 < . . . < δ˜k = α
%;
5 @
P(˜ α, M ) = P(δ˜k , M )−P(δ˜k−1 , M ) + P(δ˜k−1 , M )−P(δ˜k−2 , M ) +. . . . . . + P(δ˜1 , M ) − P(δ˜0 , M ) + P(δ˜0 , M ).
# ,-$ . 1 ρ(δ˜i , δ˜i−1 ) = 1, 1 i k, - ! , δ˜i−k < δ˜i , = ! - A*%%B$ . $ . -- = , ! @ n
αi (1 − 2pi ) + P(˜ 0, M ).
i=1
1 ˜ 1 0, M ) = P(˜ 0, M ) = ρ(˜ ρ(0, α ˜i) = m m i=1 m
1 ˜ ˜ 1 P = ρ(0, Sj ) = Pj = pi . m j=1 m j=1 m i=1 n
n
n
+ ! @ P(˜ α, M ) =
n
αi (1 − 2pi ) +
i=1
n
pi =
i=1
=
n
[αi (1 − 2pi ) + pi ] =
i=1 n
[αi + (1 − 2αi )pi ].
A*%6B
i=1
7 . 9 ! ! 1 - . = M1 M2 E n ! P¯1 P¯2 , : = I. '%," q¯ = 12 (P¯1 − P¯2) . , ! , .- = M1 M2 . 5.
% .00 012 ( H J .-12 ( .@
q¯, (2˜ x−˜ 1) = 0,
(. q¯ , .- .- , ( H = M1 M2 . %/
( "# ) 9 ! x˜ = (x1, . . . , xn) ! , E n . 5( 1 , ( H , *%*$ @
x ˜ ∈ M1 ⇔
n
⇔
x, M1 ) > 0 P(˜ x, M2 ) − P(˜ n
[xi + p2,i (1 − 2xi )] −
i=1
⇔ n
n
⇔
[xi + p1,i (1 − 2xi )] > 0
i=1
[xi + p2,i (1 − 2xi ) − xi − p1,i (1 − 2xi )] > 0
i=1
(p1,i − p2,i )(2xi − 1) > 0
i=1
⇔
⇔
n i=1
⇔
⇔
2qi (2xi − 1) > 0
⇔
(¯ q , (2˜ x − 1)) > 0 ,
(. ˜1 . , (1, . . . ,1). (
x ˜ ∈ M2
⇔
(¯ q , (2˜ x−˜ 1)) < 0 .
E . -. n , E n = {0, 1}n . . ! . !0 . , A ! E n B n , Dn = {−1, 1}n , , . ( , A Rn B$ . , L(˜x) = 2˜x − 1. L(˜x) ∈ Dn . ! x ¯. H ! $ . E n Dn 2 - 12 - , . −1. , . . ! ( ! . $ , .- E n , Dn . + 0!$ , p¯ A -= B = M = {¯ a1 , . . . , a ¯m } Dn , - - @ m p¯ =
1 a ¯i . m i=1
¾
4&
5 . ($ .-12 ( .@ (¯ q, x ¯) = 0
-- - ( $ .-2 . . - q¯. '%," " ! ΓH , (¯q, x¯) = 0 . , ! H <( !1 A.- = M1 M2 B ; . , *%%$ ! , ( H . 0 Dn , , = ( ΓH . 9= $ q¯ , $ : 2 !0 0 9 ! M = {¯x ∈ Dn |(¯q, x¯ = 0}. -. q¯ = (q1 , . . . , qn ) q¯+ = (|q1 |, . . . , |qn |) = M = {¯ x ∈ Dn |(¯ q+ , x ¯ = 0}. + . $ |M | = |M |. 9 ! ! x ¯x ¯ . Dn . 9= $ ,$ . .= ! = M ( I ! $ !$ $ x¯ < x¯ . , j1 , j2 , . . . , jk .1 $ xj < xj , 1 i k $ . ( $ xj = −1, xj = 1. K ! .= !$ x¯ ∈ M x¯ ∈ M , (¯ q+ , x ¯ ) = 0 (¯ q+ , x ¯) = 0, , - . ! . . @ i
i
0 = (¯ q+ , x ¯ ) − (¯ q+ , x ¯) =
i
i
n
|qi |(xi − xi ) =
i=1
k
2 · |qji | > 0,
i=1
+ 1. . $ = M , G $ W6X$ , E n A Dn . Cn[ ] . I- . ( = .@ n 2
2n [n] Cn2 √ = o¯(2n ), n
n → ∞.
G . , *%%$ = , *** , . ( . 9 ! M1 M2 . = 0 Dn , |M1 | = m1 , |M2 | = m2 , . d1 d2 - R =. A ! - Zn B 4*
", ) .02 9 , di mi < , R mi − 1
i = 1, 2,
= 0 = . ! . . (( 2 ( $ .-2 .
( "# ) 5. ($ H <( ! . = M1 M2 -- - + . : ! H <( - , ! , A. , B ! $ , - , . 12 @ $ , ! . $ 1 - D[D D D$ ( , - - ( - - , A $ .= 2( ,B . S $ , - , , ! 1 A 1B $ .- 2 -= . , 9 ! ! 1 - . , ! , M1 = {¯a1 , . . . , a¯m } M2 = {¯b1 , . . . , ¯bm }, , - - M : . ! ( $ .- 2 . )%9, .02 5 M . ! , M1 M2 , |M1 | = |m2 | ! ( $ .-2 . $ . ( ! ( 1 - H <( .- = M1 M2 . I ! I ! $ ! n¯ ! , . . ,$ .-12 1 ( !$ .-2 1 . E(. .- , M . M1 M2 ! 1 4%
m m |M| = (¯ n, a ¯i ) + (¯ n, ¯bi ) = i=1 i=1 m m ¯, = n ¯, ai − n (−bi ) = i=1
i=1
= |m(¯ n, PM1 − P−M2 )| = 2m|(¯ n, q¯)| 2m¯ q ,
(. q¯ , .- = M1 M2 , A B |M| . ( - $ (. n¯ q¯.
H )* : (, = . ! A ***B . . ! H <. . . = n ( : ($ - ! *%%$ , .= : ( $ 9 ! M = {¯ a1 , a ¯2 , . . . , a ¯n } a ¯ ∈ Dn . I. '%," . = M ! a¯ . , ! R = maxi ¯ai − a¯. $ (. a¯ ! -= , M, , . ( ! . = M, , a¯. 9 ! ! M1 = {¯a1 , . . . , a¯m }$ M2 = {¯b1 , . . . , ¯bm } . = $ p¯1 , A -= B = M1 $ p¯2 , A -= B = M2 q¯ , .- = M1 M2 . + R . = 1
2
M1 ∪ (−M2 ) = {¯ a1 , . . . , a ¯m1 } ∪ {−¯b1 , . . . , −¯bm2 }
! q¯. 5. )%9, .2. K R < q, = M1 M2 H <. , 44
( "# ) I- -( i = 1, . . . , m
1
@
(¯ q, a ¯i = (¯ ai − q¯ + q¯, q¯) = (¯ ai − q¯, q¯) + ¯ q 2 ai − q¯ · ¯ q ¯ q 2 − R¯ q > 0, ¯ q 2 − ¯
, *%% 0 ! , ( H = M1 . ( $ .- 1( j = 1, . . . , m2
(¯ q , ¯bj = −(−¯bj − q¯ + q¯, q¯) q −¯ q 2 + R¯ q < 0, −¯ q 2 + − ¯bj − q¯ · ¯
, *%% . 0 ! , ( H = = M2 . I =. *4* . . 12 9 ! R1 R2 . , = M1 M2 G
% .20 9 , R1 < ¯ q +
p2 2 ¯ p1 2 − ¯ 4¯ q
q + R2 < ¯
p1 2 ¯ p2 2 − ¯ 4¯ q
= M1 M2 H <. ,
( "# ) E =$ . ! =. - *4*$ 0 ! , ( H = M1 M2 . I- -( i = 1, . . . , m1 @ p¯1 + p¯2 , q¯) = 2
2 ¯ p1 − ¯ p2 2 = (¯ ai − p¯1 , q¯) + ¯ q 2 + 4¯ q
2 ¯ p1 − ¯ p2 2 q ) + ¯ q (−¯ ai − p¯1 + ¯ 4¯ q
p2 2 ¯ p2 − ¯ − R1 > 0 A 1B ¯ q · ¯ q+ 1 4¯ q
(¯ ai , q¯) = (¯ ai − p¯1 , +¯ q+
46
( $ .- 1( j = 1, . . . , m2 :
¯ p1 2 − ¯ p2 2 4¯ q
¯ p1 2 − ¯ p2 2 ¯ q ) − ¯ q (bj − p¯2 − ¯ 4¯ q
p2 2 ¯ p2 − ¯ q− 1 ¯ q · R2 − ¯ < 0 A 1B$ 4¯ q
q 2 + (¯bj , q¯) = (¯bj − p¯2 , q¯) − ¯
. ! 0 ( H, = M1 , = M1 . E . + $ =. *4* , . , 9. 1 - ! H <. . = M1 M2 Dn . 9. ! .. ! . $ -2 - , , = W**X 9 ! M ! = Rn A , . < ,B = M , - , , $ ($ x¯ ∈ M y¯ ∈ M . α¯x + (1 − α)¯y, (. 0 α 1. , = M , - , , $ .- =.( x¯ ∈ M 1( α ∈ R, α 0, α¯x ∈ M. I- M ⊆ Rn !0 .=2 M , = , - , M ; . ! C(M ). !0 , , $ .=2 = M ⊆ Rn , . , ! , , $ =. , = M ! Con(M ). = !$ .M = {¯ x1 , x ¯2 , . . . , x ¯k } ⊆ Rn @ C(M ) = {¯ x|x ¯=
k
αi x ¯i , αi 0,
i=1
Con(M ) =
x ¯|x ¯=
k
αi = 1},
i=1 k
αi x ¯i , αi 0 .
i=1
I$ $ ¯0 . 1 ! - 4'
G $ ! 1 - . , = + 2 . = @
Dn .
M = M1 ∪ (−M2 ),
M1
M2
¯ = {¯ M a − ¯b | a ¯ ∈ M1 , ¯b ∈ M2 }.
9 ! p¯1 , A -= B = M1 $ p¯2 , A -= B = M2 q¯ = 12 (¯p1 − p¯2 , . = M1 M2 . , . .( !$ . , = M1 M2 --1 - H <. , $ ∀¯ a ∈ M1 , (¯ q, a ¯) > 0,
∀¯b ∈ M2 , (¯ q , ¯b) < 0.
I- . ( ,0 = M : : @ ∀¯ a ∈ M, (¯ q, a ¯) > 0.
. 12 = KR : KR = {¯ x | ∀¯ a ∈ M, (¯ x, a ¯) > 0}.
+ $ O KR 1 A q¯ ∈ KR B . - , , , ¯0. 9 ! ! M1 M2 , . = M1 M2 $ q¯ , .- M1 M2 . E - ! $ . , = M1 M2 H <. , 2 M1 M2 , . ! M1 RM2 . + . $ M1 RM2 ⇔ q¯ ∈ KR . A*4'B > . ! = $ . - - : K A = {a1 , . . . , an } B = {b1 , . . . , bM }
. = < $ . . . ! . 12 = < @ A
B = {a1 , . . . , an , b1 , . . . , bM }.
+ - . - . , - . ,0 0 R. > ! ∅ = M1 ⊆ M1 $ ∅ = M1 ⊆ M1 $ ∅ = M2 ⊆ M2 $ ∅ = M2 ⊆ M2 . E(. .22 K M1 RM2 M1RM2, (M1
48
M1 )RM2 ,
M1 RM2 M1 RM2 , M1 R(M2
M2 ).
( "# ) 9 ! . =. ,J - - ( $ ! = $ 9 !$ p¯1 $ p¯1 $ p¯2 $ p¯2 , A , -= B = M1 $ M2 $ M2 M1 ∪ M1 $ q¯1 = 12 (¯p1 − p¯2 )$ q¯2 = 12 (¯ p1 − p¯2 ) , .- = M1 M2 $ M1 , M2 , M2 , M1 , M1 M2 + . $ , . ! , 0 α 1 p¯ = α¯ p1 + (1 − α)¯ p1 .
E(. @ q¯ =
1 1 (¯ p − p¯ ) = (α¯ p1 + (1 − α)¯ p1 − p¯2 ) = 2 2 1 p1 − p¯2 ) + (1 − α)(¯ = (α(¯ p1 − p¯2 )) = α¯ q1 + (1 − α)¯ q2 2
1 A*4*B , - q1 ∈ KR ,
q2 ∈ K R
5. ! $ q¯ ∈ KR (M1 ∪ M1 )RM2 . G . , , =$ . ! - , , - - M1 RM2 , (. (M1 ∪ M1 )R(M2 ∪ M2 ).
5 .= ! . 12 1
% .2- = M1 M2 Dn ! H < . ,$ (. ! (.$ (. .- 1, . : , . = {¯a} ⊆ M1 {¯b} ⊆ M2 . H <. , 2 {¯a} {¯b}.
( "# ) . ! , , . - ! H <. = $ : ! . ! 9 ! M1 = {¯a1 , . . . , a¯r } M2 = {¯b1 , . . . , ¯bs }
. , . = = M1 M2 9 1 ,$ $ ∀i, ∀j
{¯ ai }R{¯bj }, 1 i r, 1 j s.
+ $ M1 = {¯a1 } ∪ {¯a2 } ∪ . . . ∪ {¯ar } M2 = {¯b1 } ∪ {¯b2 } ∪ . . . ∪ {¯bs }. 9 -- . ! $ Z ∀j
M1 R{¯bj },
1 j s.
43
K2 $ -- . ! $ ! @ M1 RM2 ,
M1 M2 H <. , 2 M1 M2 . E M1 M2 , ! , . = $ . 1 .- = M1 M2 ! 1 H <. ! E . G ! - . 1 $ ! 1 H <. ! . = M1 M2 = ! $ - , : . = 9 ! K = Con(M )$ K¯ = Con(M¯ ), (. = M M¯ .- ! M1 M2 ,0$ K¯ ∗ ! , , $ . , K¯ :
¯∗ = x ¯ (¯ K ¯ | ∀¯ y ∈ K, x, y¯) > 0 . ¯ 0 , , 1 B
A $ + $ K¯ ⊆ K. G )%9, .21 = M1 M2¯ ∗ Dn ! H < . , (. ! (.$ (. K ⊆ K .
( "# ) 1 K ⊆ K¯ ∗ . ! ! .- 12 K. 9: $ ! x¯ ∈ M. G - ! H <. . $ ∀¯a ∈ M1 ∀¯b ∈ ¯ $ ∀¯a ∈ M1 ∀¯b ∈ M2 {¯ a}R{b}. 9 . $ ¯ ¯ ∗. x ¯, (¯ a − b) > 0, , x ¯∈K M2 9.= !$ , - - 1 K ⊆ K¯ ∗ . + $ $ $ G E .-
¯ ∀¯ x ∈ M ∀¯ y∈K
(¯ x, y¯) > 0.
∀¯ x ∈ M ∀¯ a ∈ M1 ∀¯b ∈ M2 (¯ x, a ¯ − ¯b) > 0. ! , a¯ ∈ M1 ¯b ∈ M2 ∀¯ x∈M
(¯ x, a ¯ − ¯b) > 0.
: $ {¯a}R{¯b}. 9 -- $ 1. 1 ! 1 H <. ! M1 M2 . # =. . G =. - = ! . . ! H <. ( , , K . , ! , $ ∀¯x, y¯ ∈ K, (¯x, y¯) > 0; 4;
( . , ! 11 ( ! ( =. : ( #( K . ! φ(K) φ(K) = sup ∠(¯ x, y¯). x ¯,¯ y ∈K
E $ ( K $ φ(K) < π2 . - =. -$ . 12 1 @ ¯ ∗, K∗ ⊆ K
, 12 ( 1 ¯ ⊆ K. K
9: $ , - - 1 K ⊆ K ∗ , 1 K ⊆ K¯ ∗ - =. - , - - $ ! 1 H <. ! M1 M2 ¯ D D D,1 - , K K, , D, - . . K ⊆ K¯ ∗ . 1 = K ⊆ K ∗ : $ K ,$ φ(K) < π2 . E 12 K Dn , φ(K) < π2 ! $ . √ . = M !0$ 2n. E , ", ) .2: K .- . = M = M1 ∪M2 √ , - - d(M ) < 2n, = M1 M2 Dn ! H <. ,
H : ( (1 - ,$ - , (0 , ( H : , = M1 M2 $ (. . .(1 - 2 1 E 2 - ! 1 $ $ ($ $ , 1 - C ,$ .=2 1$ --1 - , =1 0! ,= ( ( C I ( A ( ! ! B 4/
. 12- - k A )*B 0! = (0 !1 ? ,$ - , (0 !1 D . , . , . 12 .! @ * 1 - . = : M1 M2 , , ( H 012 R, : = M1 M2 -- 0 , J % = M1 M2 A $ ! . = : : = K1 K2 - * . - -B .(1 2 1 . ,$ ! ( 1 = : Hˆ 1 Hˆ 2 J 4 : , = Hˆ 1 ∪ Hˆ 2 1 - ( H A ! 012( RB$ ( - ! , , > $ ! M1 , M2 ⊆ Dn $ |M1 | + |M2 | = m$ q¯ , .- M1 , M2 ! M = {˜α − β˜ | α˜ ∈ M1 , β˜ ∈ M2 }. = M1 M2 . (1 - H <. , $ = M A = M1 M2 B = ! 2
, M AM1 M2 B$ . -1 . 1 ! : M A M1 M2 B$ $ $ M = (αij )$ i = 1, 2, . . . , m$ j = 1, 2, . . . , n = {−1, 1} m × nJ . , ! 0 =. sij = −1 . 1 0 A 1 −1 B$ - . i j ( E $ . 0 = , ! ! , 9 ! A −1B 0 S N, m n N=
1 |sij − 1|. 2 i=1 j=1
N E(. δ = m·n . , ! A ! B (0 !1 .- = M1 M2 A$ $ .- M B$ S 6&
0 (0 !1 δ Aδ . = - ! B ! . - , 0 S (0 !1 δ M - ˆ δ = (αij · Sij ), i = 1, . . . , m, i = 1, . . . , n. M
5 12 - M1 M2 Mˆ1 δ Mˆ2 δ , , , - = M, M1 M2 Mˆ δ , Mˆ1 δ Mˆ2 δ , ! - ( H A : , = M1 M2 B = Mˆ1 δ ˆ2 δ A )*B@ M
ˆ δ, M ˆ δ) = KM1 ,M2 (M 1 2
(φ(¯ x), χ(¯ ˜ x))
ˆ δ ∪M ˆδ x ¯ ∈M 1 2
ˆ δ | + |M ˆ δ| |M 1 2
.
? A $ . $ M1 $ M2 δB . ! , : = Mˆ 1δ Mˆ 2δ . 9= ˆ δ, M ˆ δ ), Δ = 1 − min KM1 ,M2 (M 1 2 S
(. - = , 0 (0 !1 δ. Δ A M1 , M2 δB ! = 1 .1 ! , : Mˆ 1δ Mˆ 2δ . ? Δ . , ! A ! B (0 !1 - = M1 M2 A$ $ .- M B$ . , (0 !1 δ. # ! (0 Δ (0 δ. 9. ! .= . ( ! 1 ( 1 9 ! p¯ ∈ Rn ! , $ .12 Rn ( ! Πp¯ (¯p, x¯) = 0. ? ( .- . ( ! . ! @ Π+ p¯ =
p, x ¯) > 0 x ¯ ∈ Rn (¯
Π− p¯ =
p, x ¯) 0 . x ¯ ∈ Rn (¯
6*
9 ! ! x¯ 0 Dn , .=2- = - Π+p¯ . I. '%," K 1- ! . , k Dn z += ¯1 , . . . , x ¯k ), ,.-2- 0 , x ¯, .= Πp¯ , (¯ x=x ¯0 , x 0 x¯ , - ! k 0J x¯ , - ! k 0 I- p¯ = (p1 , . . . , pk ) . . 12 -@ p∗ = min |pi |, i
G
.-. 9 k <
p∗ = max |pi |. i
0 x¯ ∈ Dn ∩Π+p¯ -- ! k 0J 1 p, x ¯) 2p∗ (¯
k min
n
1 , (¯ p, x ¯) 2 2p∗
0 x¯ ! k 0
( "# ) 9 ! 0 x¯i ! z . , i, ,.-2 0 , x ¯. . . !$ . @ 2p∗ |(¯ p, x ¯) − (¯ p, x ¯1 )| 2p∗ .
K ! x¯ ∈ Dn ∩ Π+p¯ , (¯p, x¯) > 0, k < 2p1 (¯p, x¯), .1 z . , k ,.-2 x¯ = x¯0 .- -( i k ∗
i−1 |(¯ p, x ¯) − (¯ p, x ¯i )| = [(¯ p, x ¯i ) − (¯ p, x ¯j+1 )] j=0
i−1
|(¯ p, x ¯j ) − (¯ p, x ¯j+1 )| 2p∗ · i 2p∗ · k < (¯ p, x ¯),
j=0
, (¯p, x¯i ) > 0, ! z .= Π+p¯ , 0 x¯ ! k 0 K k 2p1 (¯p, x¯), $ . $ 2 - ! z 6% ∗
. , k, (¯ p, x ¯) − (¯ p, x ¯k ) =
k−1
[(¯ p, x ¯j ) − (¯ p, x ¯j+1 )]
j=0
k · 2p∗ (¯ p, x ¯),
x¯k ∈ Π−x¯ , , ! 0 , x¯ ! k 0 K = k ] n2 [, 0 $ =2 = , - . , k, ,.-2 0 , x¯, !0$ 2n−1 , (. Π+p¯ !0 2n−1 0 + 1. . ! 0 , x¯ ! k 0 k ] n2 [. 7 . 9 ! ! Mˆ δ , M ! . - , 0 S (0 !1 δ, 0 δ · mn, i< 0 ni , n1 + . . . + nm = N. . -@ b = min (¯ q, α ¯ ), α∈M ¯
B = max(¯ q, α ¯ ). α∈M ¯
? , . , ! ( - + $ , .( b > 0. K = Mˆ δ 0 t 0$ Mˆ δ . - t i1 , . . . , it , .- ,$ ! - *6*$ = ! Nij
1 1 (¯ q, α ¯ ij ) ∗ b, 2q ∗ 2q
9: N=
m
ni
i=1
G t
t
i = 1, 2, . . . , t.
nij
j=1
b t. 2q ∗
2q ∗ 2q ∗ N = m · n · δ. b b
+ 1. .- Δ = m1 max t S δ
2q ∗ δ. b
64
5 . ( ,$ 66* nij
9:
1 1 (¯ q, α ¯ ij ) ∗ B. 2q ∗ 2q t j=1
nij
B t, 2q ∗
G .- Δ = m1 max t, . $ @ S Δ
1 1 N 2nq∗ · 2q∗ > − . m B B m
E
% .-0 I- (0 Δ - = M1 M2 , . , (0 !1 δ, ., @ 1 2q ∗ n 2q∗ n − <Δ δ. B m b
+ . . , *6% > . ( !$ = M1 M2 ! 0 δ, (0 ! = Mˆ 1δ Mˆ 1δ 1 G . ! , *6% . , ", ) .-2 K =. , S 0 !0$ 2qb , = M1 M2 A ! S B ∗
66
! " # $
-= : ( ( . ! - 2- . -$ )* (, 9 ! D¯ n = [−1, 1]n . $ D¯ n ! n , Rn , 0 , ( 1 = ¯ n , . Dn . +2 . 12 @ .- C x ∈ D =2( 1$ . . ! -J ( ¯ n . E x ¯$ x ¯ = k(x), -- - n ( D $ . 12- - k - = ¯n : K = D ¯ n, k: K→D
, 5.= ! $ x¯i i .- C x ∈ K , = C x i. = - = . ! -$ : - k - = ( = 0 Dn , = , ! ! : 0! C , ,= , A B , . - . . - . ,$ 0!$ 6'
( H ρ .( - n n Z , ρ(¯x, y¯) = |xi − yi |. i=1 9= $ $ 012 ( - H = @ ( H = .-12 ( ( ! H. I ! $ ! M1 M2 ! , = 0 Dn $ M1 , M2 ⊆ Dn , M1 = {¯a1 , . . . , a¯m }, M2 = {¯b1 , . . . , ¯bm } x ¯ ! - Dn . G @ 1
2
ρ(¯ x, M1 ) =
m1 1 ρ(¯ x, a ¯i ) = m1 i=1
m1 n n m1 1 1 |xj − aij | = |xj − aij | = m1 i=1 j=1 m1 j=1 i=1 ⎛ ⎞ m1 m1 n 1 ⎝ = (1 − xj ) + (1 + xj )⎠ = m1 j=1 i=1,a =1 i=1,a =−1
=
ij
ij
n pj + 1 1 − pj 1 (1 − xj ) + m1 · (1 + xj ) = m1 · = m1 j=1 2 2
n n 1 + pj 1 − pj − (1−xj , pj ) = n−(¯ x, p¯1 ), 1 − xj · = = 2 2 j=1 j=1 m 1
(. p¯1 = (p1 , p2 , . . . , pn ) = m1 a¯i , i=1 A -= B = M1 . ( $ 1
ρ(¯ x, M2 ) = n(¯ x, p¯2 ),
m 2
(. p¯2 = m1 ¯bi , A -= B i=1 = M2 . E! . $ 2
ρ(¯ x, M2 )−ρ(¯ x, M1 ) = n−(¯ x, p¯2 )−n+(¯ x, p¯1 ) = (¯ x, p¯1 −p¯2 ) = 2(¯ x, q¯),
(. q¯ = 12 (¯p1 − p¯2 ) , .- = M1 M2 . + 1. . $ ( H H <( ! .- = M1 M2 : 68
9. ! 1 ( - +0 $ $ .1$ =2 1 ( -$ , . ! , W*%X I- =. .( - ! . 1 ! $ 1 D D = , ! , , φ1 φ2 -$ . ( : , ( ( 012 ( .- - ,= - φ1 − φ2 . $ ( , --1 - D , D@ =.- . - - A ! ( ! (B 12 D D . +. . ! . ! - 0! , ( , 9 !$ $ Φα¯ (¯x) -$ . ( α¯ (¯x) D¯ n . , . .( !$ Φα¯ (¯x) -- B ! , A ! α¯ B$ (. Φα¯ (¯ x) = fφ (ρ(¯ α, x ¯)), (. fφ - . ! -
-$ ρ D¯ n . = . ! ! - = ! Φα¯ (¯x) = fΦ (ρi (¯α, x¯)), (. ! i *$ ρ Zn i = 2, ρ . Rn . B .. , $ ΦM (¯x), . , x ¯ = M = {¯ α1 , . . . , α ¯ m } , - - @ Φm (¯ x) =
M
Φα¯ i (¯ x);
i=1
B ,12 $ - fΦ B -- - , 12 '%," 9 Φ, . -12 - B< B$ . , ! + $ ( ( , -$ , A $ ( ( - , . , 1 W*XB$ -$ $ ( ,$ , , Γ< . W%XB ( , ! 63
, ..-2 , ,
12 012 =., Φ .- , ( - AΦ , . 12 ( , I , = M1 , M2 ⊆ Dn , |M1 | = m1 , |M2 | = m2 x¯ ∈ D¯ n . * , -1 - . , , φ1 φ2 $ . , x¯ = M1 M2 @ φ1 =
1 ΦM1 (¯ x), m1
φ2 =
1 Φm2 (¯ x). m2
% 012 R0 : ⎧
φ1 > φ2 φ1 < φ2 . . $ φ1 = φ2 =., ( - AΦ . , ! ( A -B$
A = AΦ Φ . ! ( I. . '%," I ( - A1 A2 . , ! : , $ .- , 012 =. , = - , , . ( ( !1 . . ( 0 - : =. .- - ! . ! , . - 9 ! f (x) g(x) . . ! ( ( x. 5= $ . ,$ 2 1 a, b R, a > 0, .- x R , - - @ ⎪ ⎨M1 , x ¯ ∈ M2 , ⎪ ⎩
g(x) = a · f (x) + b.
'%," I Φ1 Φ2 . , $ 12 ! , fΦ fΦ . , 6; 1
2
9 . . 12 0.. K , Φ1 Φ2 . ,$ ( , - AΦ AΦ : , I ! $ .- Φ1 Φ2 .- 1( x¯ ∈ Dn @ R0 (¯ x = φ1 (¯ x) − φ2 (¯ x) = A.- ( AΦ B = 1
2
1
1 1 1 1 Φ1,M1 (¯ x)− Φ1,M2 (¯ x) = Φ1,α¯ (¯ x)− Φ1,β¯ (¯ x) = m1 m2 m1 m2 ¯ 1 = m1
α∈M ¯ 1
α∈M ¯ 1
β∈M2
1 ¯ x fΦ1 (ρ(¯ α, x ¯)) − fΦ1 (ρ(β, ¯)) = m2 ¯ β∈M2
A , a > 0 b R)
1 1 ¯ x [afΦ2 (ρ(¯ α, x ¯) + b] − [afΦ2 (ρ(β, ¯) + b] = m1 m2 ¯ α∈M ¯ 1 β∈M2 ⎛ ⎞ 1 1 ¯ x =a [fΦ2 (ρ(¯ α, x ¯) + b − a ⎝ [fΦ2 (ρ(β, ¯)⎠ − b = m1 m2 ¯ =
α∈M ¯ 1
β∈M2
1 1 = a( Φ2,M1 (¯ x) − Φ2,M2 (¯ x) = m1 m2 x) − φ2 (¯ x)) = a · R0 (¯ x) .- ( AΦ2 . = a(φ1 (¯
G ( a > 0 . - . $ ( , : , E ( , - . ! =. ( 0! !1 . . ( 0 - : $ . -1 ! , , : , ( -$ . . ! 0! ( , . ,. . ( ( 9 ! ρ Zn , Φ .- - . !
fΦ , fΦ (x) = kx + b, K < 0. 5 ( ( . . ! AL(Z ) , Φ . , ! 6/ n
, AU B 9 ! ρ Rn A. B Φα¯ (¯ x) = fΦ ρ2 (¯ α, x ¯)
.- - . ! fΦ , fΦ (x) = kx + b, k < 0.
5 ( ( . . ! AL(R ) , Φ . , ! , AUU B $ AL ( AL(Z ) ∪ AL(R ) ; ( , : ( . , ! , ( E 1, . A ,12B , $ . $ . ,$ - ! %**$ $ 1, . ( AL(Z ) : , $ .- fΦ A 12( (
U<( ΦB . . $ , ( AΦ . B , ( H. E $ ( H -- - . : , ( AL(Z ) . ( $ 1, . ( AL(R ) = : , $ .- A 12( ( UU< ΦB ( AΦ A , ( E B -- - . : , ( AL(R ) . E $ 0.0 ( H -- - . : , ( AL(Z ) , E . : , ( AL(R ) . + $ , ( E = , ! . $ ( B - , H, (. ρ , 12
. - . = ! - . ρ, - - . Aρ2 B # ! . ! . 12 ,J 0.2 ( , - H E : , '& n
n
n
n
n
n
n
n
n
( "# ) 9 ! M1 = {¯α1, . . . , α¯m }$ M2 = {β¯1, . . . , β¯m
. ! , = Dn x¯ = (x1 , . . . , xn ) ! - Dn . , . - x ¯ . = M1 A . ! ! : -$ . , )* (, * , ( H B 1
P(¯ x, M1 ) = = =
2
}
m1 m1 1 1 ρ2 (¯ αi , x ¯) = ¯ αi − x ¯2 = m1 i=1 m1 i=1
m1 m1 1 1 ¯ αi −¯ p1 +¯ p1 −¯ x2 = (¯ αi −¯ p1 +¯ p1 −¯ x, α ¯ i −¯ p1 +¯ p1 −¯ x) = m1 i=1 m1 i=1 m1 1 [(¯ αi − p¯1 , α ¯ i − p¯1 ) + 2(¯ p1 − x ¯, α ¯ i − p¯1 ) + (¯ p1 − x ¯, p¯1 − x ¯)] = m1 i=1 m1 1 = (¯ αi − p¯1 , α ¯ i − p¯1 )+ m1 i=1
+2
m1
(¯ p1 − x ¯, α ¯ i − p¯1 ) + m1 (¯ p1 − x ¯, p¯1 − x ¯) =
i=1
=
=
m1 m1 m1 1 2 1 (¯ αi , α ¯ i )− (¯ αi , p¯1 )+(¯ p1 , p¯1 )+2(¯ p1 −¯ x, (¯ αi −¯ p1 ))+ m1 i=1 m1 i=1 m1 i=1 m1 1 + (¯ p1 , p¯1 ) − 2(¯ p1 , x ¯) + (¯ x, x ¯) = α ¯ i = p¯1 = m1 i=1 m1 1 ¯ αi 2 − 2(¯ p1 , p¯1 ) + (¯ p1 , p¯1 ) + (¯ p1 , p¯1 ) − 2(¯ p1 , x ¯) + ¯ x2 = m1 i=1 p1 , x = ¯ αi 2 = n, i = 1, 2, . . . , m1 = n − 2(¯ ¯) + ¯ x2 .
( $ P(¯ x, M2 ) = n − 2(¯ p2 , x ¯ + ¯ x2 .
012 ( E .- - '*
,= 1 1 ΦM2 (¯ x) − ΦM1 (¯ x) = P(¯ x, M2 ) − P(¯ x, M1 ) = m2 m1 = n−2(¯ p2 , x ¯)+¯ x2 −n+2(¯ p1 , x ¯)−¯ x2 = 2(¯ p1 − p¯2 , x ¯) = 4(¯ q, x ¯), R0 (¯ x=
: .- .-12 1 ( ! (¯ q, x ¯) = 0. 9 . -- = -- - H <( !1$ 1 ( H A $ ( .1 B 5. ! $ ( , H E : , 7 . G %*% %*4 . ,
% 0.- 71 , ( - : ( H. ( H, = ! ,0$ .- 1, . = M1 M2 0 Dn .-12 ( ( ! H A $ (. , q¯ .- = M1 M2 . , $ ( ! H ,=. - B > 2 ! , ( - A = ( ,=. - ( B .. '%," ( $ .-12 .- 1 , . = M1 M2 0 Dn . -12 ( ( !$ . , ! ( ( ? . - ! . , %*6 ", ) 0.1 7 , ( , - --1 - ( ( 9= $ =. . - . , : ( - - ( ( A . = 1 ( ,B 2 . - -!@ ( .=., . , ! , - . %*' . 2
% 0.: 9 ! n 3. E(. 1 ( ( .=., . ! . , ( '%
( "# ) 9 ! AΦ ( ( $ Φ ( .- . .=., . !
f. E ( AΦ .-12 ( .- 1, . = M1 M2 ( !$ .- = M1 = {¯a, ¯b} M2 = {¯c}, (. a¯ = (−1, 1, %−1, .&'. . , −1()$ ¯b = (1, −1, −1, . . . , −1)$ % &' ( n−2
a ¯ = (1, 1, −1, . . . , −1), % &' (
(
AΦ
n−2
( !
n−2
.- - 012 ( AΦ :
Π1 ,
Φa¯ (¯ x) + Φ¯b (¯ x) − 2Φc¯(¯ x) = 0.
Dn . !1 Π(2) , .- 0 - x3 = t − 1$ x4 = t − 1$ $xn = t − 1, (. t ∈ [0, 2] , . . . OXY $ x¯ = (x, y, t − 1, . . . , t − 1) Π(2) : . , (x, y). 7( - -$ (0, 0) Π(2) .= ( Π1 , $ $ (1, 1) .= + 1. . $ . - ! Π(2) ( ! Π1 1 - - l A.-2 , . $ . Π(2) B ! . - 5 U ρ = ρ2E , (. ρE . Rn .
: ! . - !
f ! [0, 4n]. 9= $ [2, 4n − 6]$(n 3), - f -- - 9 ! : E(. T ∈ [2, 4n−6] f = 0. L $ (. f (T ) = 0 A f ,12 -B 9 ! x¯ = (x, y) ! - Π(2) A ! . x ¯ = (x, y, t − 1, . . . , t − 1)B$ (. t , . -12 2 + (n − 2)t2 = T. G @ A(x, y) = ρ2E (¯ a, x ¯ = (x + 1)2 + (y − 1)2 + (n − 2)t2 ; ¯ = (x − 1)2 + (y + 1)2 + (n − 2)t2 ; B(x, y) = ρ2 (¯b, x E
'4
C(x, y) = ρ2E (¯ c, x ¯ = (x − 1)2 + (y − 1)2 + (n − 2)t2 ;
$ A(0, 0) = B(0, 0) = C(0, 0) = T. # - l A 12 Π(2) ( Π1 B . . 12 @ Φa¯ (¯ x) + Φ¯b (¯ x) − 2Φc¯(¯ x) = 0
f (A(x, y)) + f (B(x, y)) − 2f (C(x, y)) = 0
$ ! 1 ! . ( F (x, y),
A%**B 9. ( ! , ,
F (x, y) = 0
L $ F (0, 0) = 0. - , . ,@
Ax = 2(x + 1), Ax (0, 0) = 2, Ay = 2(y − 1), Ay (0, 0) = −2; Bx = 2(x − 1), Bx (0, 0) = −2, By = 2(y + 1), By (0, 0) = 2; Cx = 2(x − 1), Cx (0, 0) = −2, Cy = 2(y − 1), Cy (0, 0) = −2; Fx = f(B) Ax + f(B) Bx − 2f(C) Cx ; (0, 0)Ax (0, 0)+f(B) (0, 0)Bx (0, 0)−2f(C) (0, 0)Cx (0, 0) = Fx (0, 0) = f(B)
= 2f (T ) + (−2)f (T ) − 2 · (−2)f (T ) = 4f (T ) Fy = f(B) Ay + f(B) By − 2f(C) Cy ;
Fy (0, 0) = (−2) · f (T ) + 2f (T ) − 2 · (−2)f (T ) = 4f (T ) = 0.
+ 1. . $ - W'X 0 A%**B .- (0, 0) 1 . - x = 0 , - - . 12 @ y (0) = −
Fx (0, 0) = −1. Fy (0, 0)
, , , . ,@
Axx = 2, Bxx = 2, Cxx = 2, = 2, Cyy = 2, Ayy = 2, Byy
'6
Fxx = f (A)(Ax )2 + f (A)Axx + f (B)(Bx )2 + f (B)Bxx − , − 2f (C)(Cx )2 − 2f (C)Cxx (0, 0) = f (T ) · 4 + f (T ) · 2 + f (T ) · 4 + f (T ) · 2− Fxx
− 2f (T ) · 4 − 2f (T ) · 2 = 0,
Fyy = f (A)(Ay )2 + f (A)Ayy + f (B)(By )2 + f (B)Byy − , − 2f (C)(Cy )2 − 2f (C)Cyy Fyy (0, 0) = f (T ) · 4 + f (T ) · 2 + f (T ) · 4 + f (T ) · 2−
− 2f (T ) · 4 − 2f (T ) · 2 = 0, = f (A)Ax Ay + f (A)Axy + f (B)Bx By + f (B)Bxy − Fxy , − (−2)f (C)Cx Cy − 2f (C)Cxy Fxy = f (T )(−4) + f (T )(−4) − 2f (T ) · 4 = 16f (T ) = 0.
G ! .- - . y(x) x = 0: y (0) = −
(0, 0) + 2y (0) − Fxy (0, 0) + y 2 (0)Fyy (0, 0) Fxx = F y(0, 0) 32f (T ) f (T ) =− = 8 = 0, 4f (T ) f (T )
A , T B$ $ y = y(x) ! . Π(2) A $ . $B - l. G $ [2, 4n − 6] . - ! - - f - + .= 1 ! . - [0, 4n] f˜ = $ f f˜ .1 . - . 12 . = 0 Dn @ ¯ M1 = {¯ a, ¯b} M2 = {¯ c, d}, (. a ¯ = (1, 1, 1, . . . , 1), ¯b = (1, . . . , 1, −1, −1), % &' ( n−2
''
c¯ = (−1, 1, 1, . . . , 1), d¯ = (1, −1, 1, . . . , 1), n 3. % &' ( n−2
E 1 ( AΦ ( $ . = M1 M2 A ,=. 1B ( ! ˜ Φ , ( AA ˜Φ ∈ Π . + A ˜ AL(R ) B$ Φ ( .- - .
! f˜, . ,0 5( . 1 %*'$ = .- = M1 M2 1 ,=. 1 ( ! Π . 7( . !$ ( Π Π .1 I ! $ . $ ! , f f˜ .1 [2, 4n − 6], . ! $ $ . .1 ,$ . , = M1 M2 , ˜ M ΦM = Φ ˜ M $ : : = ΦM = Φ . .= , . ! ( Π Π . + 1. . . ( A = 01 2 ( AΦ A˜ Φ B E $ ( , ˜ Φ 1 . $ @ AΦ A n
1
1
2
∀¯ x ∈ Π (Π )
2
1 1 1˜ 1˜ ΦM1 (¯ x) − ΦM2 (¯ x) = Φ x) − Φ x). M1 (¯ M2 (¯ 2 2 2 2
A%*%B 9 ! x¯t . (−1 + t, −1 + t, . . . , −1 + t), (. t ∈ [0, 2]
, + $ 1 t ∈ [0, 2] xt .= ( Π A P i B . -@ a(t) = ρ2E (¯ a, x ¯t ) = (2 − t)2 n; b(t) = ρ2 (¯b, x ¯t ) = (2 − t)2 (n − 2) + 2t2 ; E
c, x ¯t ) = (2 − t)2 (n − 1) + t2 ; c(t) = ρ2E (¯ ¯x ¯t ) = (2 − t)2 (n − 1) + t2 . d(t) = ρ2 (d, E
. - UB n 4. : @
2 < 4 c(t) 4(n − 1) A%*4B 9.= !$ f f˜ .1 x ∈ [0, 2) ∪ (4n − 6, 4n], f (x) − f˜(x). 2 < 4 b(t) 4(n − 2) < 4n − 6,
K
'8
Bx ∈ [0, 2), - A%*%B I ! $ ! t $ a(t) = x. + . $ , t > 1 .- ( 2 < c(t), d(t) n < 4n − 6. A%*6B , 1 1 A%*%B .- x ¯t @ ΦM1 (¯ xt ) − ΦM2 (¯ xt ) = f (ρ2E (¯ a, x ¯t )) + f (ρ2E (¯b, x ¯t )) − f (ρ2E (¯ c, x ¯t ))− 2 ¯ ¯t )) = f (a(t)) + f (b(t)) − f (c(t)) − f (d(t)). − f (ρ (d, x E
( @
˜ m (¯ ˜ m (¯ Φ xt ) − Φ xt ) = f˜(a(t)) + f˜(b(t)) − f˜(c(t)) − f˜(d(t)). 1 2
9 - , ,= -$ A%*4B A%*6B , x : f (x)+f (b(t))−f (c(t))−f (d(t)) = f˜(x)+ f˜(b(t))− f˜(c(t))− f˜(d(t)).
+ 1.$ f˜(x) = f (x). 9 K $ Bx ∈ (4n − 6, 4n], $ ( - 2 $ . !$ = [0, x − 2] f f˜ .1 + $ 1 t, $ a(t) ∈ (4n − 6, 4n] : A%*'B a(t) − c(t) = 4(1 − t) > 2 9 ! t $ a(t) = x. E(. : t c(t) = d(t) x − 2. 5 A%*4B , x @ f (a(t)) = f (x) = f˜(x) = f˜(a(t)), f (b(t)) = f˜(b(t)), f (x(t)) = f (d(t)) = f˜(c(t)) = f˜(d(t)).
9-- B %*% .- xt , - ! UUB n = 3. : . ! . f f˜ = [0, 2) ∪ (6, 12]. B K f (x) = f˜(x) x ∈ (6, 12], $ ( - 2 $ = !$ = [2, x − 6] f √ f˜ .1 9 ! t $ a(t) = x. + . $ t < 2− 2. 9 : @ 2 < b(t) 4, 2 < c(t) = d(t) 8, a(t) − c(t) > 6. A%*8B '3
5 A%*'B$ A%*8B , x, @ f (a(t)) = f (x) = f˜(x) = f˜(a(x)) f (b(t)) = f˜(b(t)), f (c(t)) = f (d(t)) = f˜(c(t)) = f˜(d(t)).
9-- UB A%*%B .- xt , B K f (x) = f˜(x) x ∈ [0, 2), t, $ a(t) = x (t > 1), @ 2 < b(t) 8; 2 < c(t) = d(t) 4.
$ ,- . UUB$ @ f (b(t)) = f˜(b(t)), f (c(t)) = f (d(t)) = f˜(c(t)) = f˜(d(t)), f (a(t)) = f (x) = f˜(x) = f˜(a(t)).
+- ! -- .- xt A%*%B E $ f f˜ .1 . - [0, 4n], ! - - f . - 5 % ρ = ρH Zn . I ! ( . ! U$ : $ - . , - A =. -B$ . ( $ - 0! . , - + ! . - ! . ! . [0, 2n]. 9= $ [2, 2n − 2] (n 3) f . 9 ! : T ∈ [2, 2n − 2] $ f (T ) = 0. , t . -12 @ 2 + (n − 2) · t = T.
0 @ A(x, y) = ρ(¯ a, x ¯) = (x + 1) + (1 − y) + (n − 2)t = x − y + T ; B(x, y) = ρ(¯b, x ¯) = (1 − x) + (y + 1) + (n − 2)t = y − x + T ; C(x, y) = ρ(¯ c, x ¯) = (1 − x) + (1 − y) + (n − 2)t = −x − y + T.
';
# - l . %** I- , . , . 12 -@ Ax = 1 = Ax (0, 0), Ay = −1 = Ay (0, 0), Bx = −1 = Bx (0, 0), By = −1 = By (0, 0), Cx = −1 = Cx (0, 0), Cy = −1 = Cy (0, 0), Fx = f (A) · Ax + f (B) · Bx − 2f (C) · Cx , Fx (0, 0) = f (T ) − f (T ) + 2f (T ) = 2f (T ) = 0, Fy = f (A) · Ay + f (B) · By − 2f (C) · Cy , Fy (0, 0) = −f (T ) + f (T ) + 2f (T ) = 2f (T ) = 0,
E Fy (0, 0) = 0, 0 A%**B .- A - B (0, 0) 1 y = y(x), .- - . @ y (0) = −
Fx (0, 0) = −1. Fy (0, 0)
, , . , A$ B $ C , 1$ .- , , . , F @ Fxx (0, 0) = 0, Fyy (0, 0) = 0, Fxy (0, 0) = f (T )(−1) + f (T )(−1) − 2f (T ) = −4f (T ) = 0.
I- - y (x) x = 0 @ y (0) = −
f (T ) −2(−4) · f (T )) = −4 = 0, 2f (T ) f (T )
$ y = y(x) . -- - - l. G $ [0, 2n − 2] - f - 9 ! f˜
.= 1 ! . [0, 2n]. 9= $ f = f˜. 9 -- . A . , - B =. - - *$ . : A%*%B I$ 0 @ a(t) = ρ(¯ a, x ¯t ) = (2 − t)n; b(t) = ρ(¯b, x ¯t ) = (2 − t)(n − 2) + 2t;
'/
c(t) = ρ(¯ c, x ¯t ) = (2 − t)(n − 1) + t; ¯x d(t) = ρ(d, ¯t ) = (2 − t)(n − 1) + t.
7( . !$ t ∈ [0, 2] - b(t)$ c(t) d(t) .= [2, 2n − 2] An 3B 9: $ . = !$ x ∈ [0, 2) ∪ (2n − 2, 2n] f (x) = f˜(x) , ! t $ , , - ! a(t) = x, . !@ f (a(t)) = f (x) = f˜(x) = f˜(a(x)) f (b(t)) = f˜(b(t)), f (c(t)) = f (d(t)) = f˜(c(t)) = f˜(d(t)).
9-- !0 A%*%B .- xt , E $ ! - - f . - % G $ $ ! - - ( ( AΦ - 5. ! $ ( AΦ - - - , ( E . + $ n = 2 = ! 1 1 A .=., . 1B 1 fΦ , .- ( AΦ . ( - ( $ ( AΦ Φ, .- , ! f (x) = (2 − x)3 A ρ ! - Z2 B$ . .- 1, = 0 M1 M2 . D2 . - ! A ,=. B .-12 1 - 1 E$ $ .- = M1 = {(−1, 1)} M2 = {(1, 1), (1, −1)} .-12 (x, y) ∈ ¯2 : D
ΦM1 (x, y) =
1 ΦM2 (x, y) 2
2f (ρ(x, y), (−1, 1)) = f (ρ((x, y), (1, 1))) + f (ρ((x, y), (1, −1))) 2(−x + y)3 = (x + y)3 + (x − y)3 √ 3 3(y − x)3 = (x + y)3 , 3(y − x) = x + y
. - 8&
√ 3 3+1 x, y= √ 3 3−1
A : ( ( AΦ ! -- , ( $ : ( B 1 ( . ; % ( Φ = ! - . 12@ 2 )ΦM (¯ x) = fΦ (σα∈M ρ(¯ α, x ¯)), ¯
(. ! , fΦ ,1 - C . ($ R0 =
e−αx $ C x+C ! ! - 012 R1 , .-12 .-1 2 1 ( ! 1
2
3
R1 (x) = ΦM1 (x) − ΦM2 (x) = 0.
+- ! . ! , %*6 $ . !$ ! A (B ,12 fΦ , ( -$ .- , 1), 2 ), 3) R0 , -- - ( ( $ 2 $ . - ( L, = R1 .-12- ( ! . 2 ( A.- , ,0 ρB
$ $ ! , . -$ 0 , $ A B -- - , 0 , $ -$ ! , : ! ? . ! $ $ = , . E $ - 1 .- 0 - . - $ ! ., 0 - . - A = 12 - ! ,B $ $ = . , ( - : . ! . ! 8*
: ( .- ( H . - = - . 0 - . 2 - . ( ( - $ - . - . - = , ! .- 0( ( -$ 2 . ( ( - .- 0 - .(1 -$ $ 0! , : . , 9. . 9 ! M1 = {¯α1 , α¯2 , . . . , α¯m } M2 = {β¯1 , β¯2 , . . . , β¯m } m : , = Dn = {−1, 1}n $ p¯1 = m1 α¯i $ 1
2
1
m 2
1
i=1
q¯ =
, , -. = M1 M2 = p¯2 =
1 m2
i=1
β¯i
p¯1 −p¯2 2
M = M1 ∪ (−M2 ) = {¯ α1 , α ¯2 , . . . , α ¯ m1 , −β¯1 , −β¯2 , . . . , −β¯m2 },
= . , ! : , , ! . M = M1 ∪ (−M2 ) = {¯ α1 , α ¯2, . . . , α ¯ m1 , α ¯ m1 +1 , . . . , α ¯ m1 +m2 },
> . .( !$ = M1 M2 H <. ,$ ¯ < 0, ∀¯ α ∈ M1 (¯ q, α ¯ ) > 0 ∀β¯ ∈ M2 (¯ q , β)
(. (¯q, x¯) ! .-12 H <( I ( $ .- : ( = M , -1 - @ ∀˜ α ∈ M (¯ q , α) ˜ >0 A%%3B ( - 2 $ = !$ , qi , i = 1, 2, . . . , n q¯ = ! ,$ ∀i qi > 0, q¯ = (q1 , q2 , . . . , qn ).
I ! $ .- ( i qi < 0, = , i , M, . = , A%%*BJ = , qi = 0, : $ i ( = , ! 8%
+ $ , q¯ . : = M A , - m1 m2 B$ 2 (-$ - ! ( - 2 A $ $ ., - = %%* - . 2 B$ . . ! , . : , = M1 M2 @ m1 = m2 . q¯ α ¯, A%%;B d¯ = α∈M ¯
, . - , A ,B M m × n A(. m = m1 + m2 = 2m1 B$ = M . , A ,B$ (. -. . . , ,1 - . E ,$ -2 1 −1 . , ! (−1, 1)< + . ! =$ , . ( !$ : . $ = A $ M B . -12 , A $ M B$ : = - ( I$ ( q¯ . ! ! d¯ = (d1 , d2 , . . . , dn ). + $ , : ( ! , $ .-2 m. I. ¯α '%," B = α∈M min (d, ¯ ) , - ( ¯ - .- : ( = M. 9 ! Π = {1, 2, . . . , n} . = AB K Π = {i1 , i2 , . . . , ik } ⊆ Π ! A -. 1 B . = $ .- α¯ = ¯ ∈ Dk . ! $ (α1 , α2 , . . . , αn ) ∈ Dn α , i1 , i2 , . . . , ik α¯ $ α¯ = ¯ (αi , αi , . . . , αi ). > . ( !$ α α¯ - = AB Π $ <. ( $ . - A B ( $ .- = M = {¯α1 , α¯2 , . . . , α¯m } A.- , M B M (M ) . ! = {¯α1 , α¯2 , . . . , α¯m A 12 1 B$ , 84 1
2
k
, J . = ( !$ = M A M B = Π $ <. ( $ . - + 1 . 2 - - = ! ! I- . ( : ( = M ( - B > 0 ! A 2 B . = Π , $ ( - .- M
M Π !0$ C, (. C $ 0 < C B.
9 - . -- - ! , .M( H . !0 ( . $ , $ -. , . .- (0, 1)< @ =. 0 , W*&X$ . 0 − 1< ( ( - W8X . G . : . N P < , W3$;$/X 9. A . : .- ! B ! , . - : 9. . - Z = {zi |i ∈ I} - = .. ! , . $ 12 - . ( . ( , , . $ , 1 ! . . 12 @ D.D D D 5 =. . Z = - ! 1 1 Ln : Z → N
1$ 1 D DA , B . .. ! , . Σ k : Z → Σ∗ A%%/B Ln k ! : ,$ ! 2 1 . p p , ∀i Ln (zi ) p(|k(zi )|) |k(zi )| p (Ln (zi )) A. ! |α| . α ∈ Σ∗ B . 12- - k . A%%4B - . 1 . Z . - -, ZZ ⊆ Σ∗ , -2( $ , --1 - . .. ! , . D.D 86
"- $ . , ( 0 . Z A -, ZZ B ! -$ 2 p, .- 1 .. ! . zi ∈ Z - , ( . zi A . k(zi )B ( p(Ln (zi ))A( p(Ln (zi ))B ( .- - - $ , . , ( ! - . ( A : ! B . ( . zi 1 . : @ ! , : D (., -D ( = ( 0 - R ( : $ (. . , A.- . (zi , R) B - -$ D.D D D.- . zi . 0 - R. . , ( 0 . Z .1 .. ! . zi , $ B k(zi ) ∈ ZZ 2 : D (., -D 0 - R, . : D D ( - D.DJ B k(zi ) ∈ ZZ 2 : ( 0 - R, , . , : D.D K . ! B . - p, - , . : A.-2 D.DB ( p(Ln (zi )), (- $ . , ( D0 D. Z ! - ; 9 - . ( A . (B ( = $ , ! ! . $ $ . A . B 0 , E!1 ( , . P N P = . ! ! @ • P . $ , ( , ! 0 , . , ( ! -J • N P . $ , ( , ! D0 ,D . , ! - 8'
9 ! 1 - . . - Z Z . "- $ . Z ! . - . Z, 2 = F : Z → Z,
, . , ! -$ ∀i (k (zi ) ∈ ZZ ) ⇔ (k(F (zi )) ∈ ZZ ) .
L. ! k : Z → Σ∗ k : Z → Σ∗ . 12 . Z Z. K . - Z .= N P .1 . ( . Z N P ! - . ! . Z Z, . Z -- - N P < . 9 N P < . = = ! . 12- . ( W8X@
() 9 ) ") L. , ( G = (V, E) = ! k |V |. '% G .= =
0 2 kQ I ( $ $ 2 . = V ⊆ V, |V | k . 0 , V . , E? > ($ ( : . ( - , ( A . (B = -- - N P < . 5 ! . 2 - - A.( H B . - @
= >? > ; ") L. , . : , (−1, 1)< , M
1 M2 m × n A ( - B > 0B$ , = ! , k n C B. '% 5 2 . = Π , |Π | k ( - .- M1 M2 M1 M2 Π !0$ C? = ! ,0$ . . : , (−1, 1)< M1 M2 m × n ! . 1 : (−1, 1)< , M 2m × n. 88
+ =$ d¯ = (d1 , d2 , . . . , dn ), -2 ! , $ ( - B . , M ( , ! , , ! - Ln (M ) = m · n (max 2m, B 2m · n).
G
i
% 00. L. 5+ RKGK 9+7S 9 GL + -- - N P < I ! $ . : - (−1, 1)< M 2m × n, .-12- A k rB .. ! 1 . $ . , ( D (., -D : . = Π $ |Π | k, . : -$ ( Ln (M ) = m · n = , ! d¯ = (d1 , d2 , . . . , dn ) !$ , : d1 , d2 , . . . , dn : =. , M . $ !0$ C.
5. ! N P < 1 . KL G5G+K + \K5E+ .- ( 0 . 9 ! . ! , ( G = (V, E). 5. . 12 @ " G = (V, E) . , = 2 !0$ k (. ! (.$ (. B ( 2GA ∪ G = 2 !0$ 2k; B ( G = 2 k; B k |V | − Z A.= k |V2 | B 9: $ ( - 2 A -- N P < ,B$ = !$ |E $ |E| = 2m, ( |V | 8 .- ( G = ,- !$ - = k 0 9 ! ( G . . MG ,
, 1 0 v1 , v2 , . . . , vn , e1 , e2 , . . . , e2n . 9 (−1, 1)< M , MG , -- . 1, . , −1. S $ : = . ! ! - A Ln (G) = m · nB 83
: , M1 M2 = - ! ! 11 = 11 A. ! . ! = ,B , M E ( G $ =. , M 1, =. −1. I- , M @ B : d = m − 6 > 0; B ( - B = (n − 4)d > 0. 9= C = (k − 2)d. + $ C B A k n − 3B] G $ ! - .. ! - . (M1 , M2 , k, C) 5+ RKGS 9+7S 9 GL + I. . M1 M2 . !
M.
9 ! V = 0 . ( G |V | = k. + . = 0 V $ |Π | = k. E . 0 , V - , $ =. , M A M Π B . −1. + 1. . $ ( - .- M !0$ (k − 2)d = C. + $ ! Π . = $ |Π | k, . ( ( - .- M !0$ C. 9 ! |Π | < k. E =. , M ! −1, - α¯ M .= −1 .- ¯α (d, ¯ ) (|Π | − 2)d < (k − 2)d = C,
Π . 9: |Π | = k. K - α¯ M .= . −1, - ! @ ¯α (d, ¯ ) = (k − 4)d < c.
+ 1.M . $ =.- M .= . −1, : $ = 0 V Π -- - , , = ( G, |V | = k. E , ! - . ! N P < . KL G5G+K +\K5E+ . 5+ RKGK 9+7S 9 GL + E 0 8;
! . $ 1. . N P 0 . E . N P < . 5+ RKGK 9+7S 9 GL + , ! , = : , ( A .= $ P = N P B .- 0 - . A . - B$ .- 0 - 12
. . , - A $ $ 1 B . - , ! - ! ! 0 . ( . ! - < . ($ - ! ! $ = 9 : , , 1 - , : = -$ .- , ( (0 $ 12 : ( $ - $ ! , ( A.= , - . B . - , = . - .- ( - 0 . > . .( ! (−1, 1)< , M m × n ( - , @ A , B .- ! d¯ = (d, d, . . . , d), (. d ∈ N. P K(m, n, r) . ! (−1, 1)< m × n, =. , - r . + $ M ∈ K(m, n, r), d = m − 2r. 9 .- M ∈ K(m, n, r) . 12 1 1 . @ ! = A BΠ , . ( ( - .- , M !0 A 1 . , B . ! $ - , M ∈ K(m, n, r) : * K α¯ , M, Σ(¯α) . ! : + $ .- 1, . α¯1 α¯2 , M , - Σ(¯ α1 ) ≡ Σ(¯ α2 ) ≡ n(
mod 2);
8/
% B ( - .- M, $ . $ B ≡ 0( mod d). + b = f racBd; 4 . < ! , . , ! . , $ 0 - , ( - !0 . (J 6 $ . 1 ( , -- - . , $ . = 1 ( - + $ ! 0 = , ! . . , J ' l< $ . 1 ( , !0$ l -- . , $ . = 1 ( - + $ l< , . $ 2 (-$ !J 8 . = S ⊆ {1, 2, . . . , m} . , ! ( , = .- , M, ∀i ∈ S d · (¯ αi ) = B, (¯αi ) = b. + $ ∀i ∈ S
(¯ αi ) b + 2
. . ( . M A.- i ∈ S B@ (¯αi b A. ! α¯i , M i$ α¯i - -BJ 3 U . = $ V . = , M, M (U, V ) . ! . $ 1 U V K . = U V $ .( -$ ( = +. ! , = 0 1 . $ , , ! A , ,B ( , . ! . . ( . - , +. .
, .. . ! - ! - 3&
D D@ = - -$ (.$ $ = ! l !0 ($ . , 1, l , l < l, . ( !0 1 9. - : , ! A= ! !0B l ! !- = ! !0$ l !0 - ( -$ D D , ( - l = , ! .! ! , A ( - ,0 ! = ( ( $ . - ( ,B + - l = 2, , - ! . 12 - 2 K M ∈ K(m, n, r) . . $ (. . . . ($ . I ! $ ! .- M . .
i1 , i2 , i3 ! ! ( = .- , M. K S = ∅, . $ . 1( .= 1 , B , M . K S = ∅, =. . , M (S, {i1 , i2 , i3 }) . 1 A . i1 , i2 , i3 . B + 1. . $ $ $ . M (S, {i1 , i2 }) 1 = ! $ : $ .
i1 , i2 , M . ; K M ∈ K(m, n, r) . . l 4 .- ( ( = S |S| 4, (. . . . ($ . .$ ! .- , M . . i1 , i2 , . . . , il . .,. 2 . !$ . . . ( . . , M (S). M (S, {i1 , i2 , . . . , il })
. M. + $ =. : .
, 2l . A . ( , i1 , i2 , . . . , il . .- M B 9.= !$ M -- - 2< $ : $ =. . , M (S, {i1 , i2 , . . . , i} ) 3*
. .- =. , ij , ij , (1 j1 = j2 l) . M (S, {i1 , i2 , . . . , i} ) . - . . G . ( , @ l(l − 1) > 4, |S| · C 2 Cl2 |S| l l 1
l 2
2
[ 2 ]([ 2 ] − 1)
1 |S| 4. G . 12 , . 12 )%9, 000 9 ! M ∈ K(m, n, r) 2< S ( = K |S| 4, M - # =. %%% = ! . |S| 5, $ . $ - |S|, , . 12 $ 2 (-$ =
>%%
1 1 1 1 M = 1 1 −1 −1 −1
1 1 −1 −1 1 1 −1 1 −1 1 1 −1 −1 1 1 −1 1 1 −1 −1 −1 −1 −1 1 −1 −1 1 1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 −1 −1 −1 1 −1 −1 −1 −1 1 1 −1 −1 −1 1 . −1 1 1 1 1 1 1 1 1 1 1 1
L. ! d = 3$ B = 5$ b = 1$ S = {1, 2, 3, 4, 5, 6}$ |S| = 6$ M -- - %< $ . {2, 3, 4, 5} . ! . 12 1 ( .- 2 - - @ F+I@ M ∈ K(m, n, r), B > 0 ( - * , ! ( = S J % K S = ∅, 4$ 6J 4 #. ! 1 , *J 6 9 !$ -- - 2- M (S) 2< K . 8$ 'J 3%
' #. ! . , . . , *J 8 ,. ! - k 0 - . |S|. ; , ( A ! - . 4$' 9 ! A , ( $ , + KA (M ) |SA (M )| ! ( , M ∈ K(m, n, r). 9 !$ ( K ∗ (M ) ! ! A ! B A $ $ , - B .- $ 0 .- , M . 2 - I.
'%,"
*
RA (M ) =
KA (M ) K ∗ (M )
(0 !1 = ( ( A . M ; % RA =
inf
M ∈K(m,n,r)
RA (M )
(0 !1 = ( ( K(m, n, r). + RA (M ). I- : K ∗ (M ) $ KA (M ) 9 ! M ∈ K(m, n, r) ( - B > 0.
002
mb . 2 Π = {i1 , i2 , . . . , iK ∗ (M ) }
K ∗ (M )
I ! $ !
! $ .- ( ( - .- M
, M Π !0$ B. E(. .- =. α¯ ∈ M @ d · Σ(¯ α ) B, Σ(¯ α ) b 34
9: @
A%%*&B
Σ(¯ α ) mb
α ¯ ∈M
, -- A%%6B $ @
A%%**B
Σ(¯ α ) = K ∗ (M )d
α ¯ ∈M
G A%%6B A%%'B , @ K ∗ (M ) · d mb
K ∗ (M )
mb . d
+ 1. . %%4 90 , %%4 <. ( I- : ( 0 A%%6B A%%'B .- . , M :
Σ(¯ α) mb
α ¯ ∈M
Σ(¯ α) = nd,
α∈M ¯
nd . b nd m B m . . . 12 1 a, 12 1 ! bm - ( B D . D ,@ a = Bm nd = nd . + $ 0 < a 1. E $ %%4 $ K ∗ (M ) an. 9 ! ! - KA (M ) |SA (M )| , ! , ( A = M. 00- 9 S > 2r 2
2
KA (M )
1− 1−
2r m 2r S
· an.
.$ ! Π $ |Π | = KA (M ) $ . ( A M - , M : Π . 9 . 1 = SA (M ), = ! @ ∀i ∈ SA (M )
36
dΣ(¯ αi ) = B,
Σ(¯ αi ) = b.
A%%*%B
5. 12 = . @
∀i ∈ SA (M ) dΣ(¯ αi ) KA (M ) · d, α ∈M
A%%*4B
(α ) KA (M ).
$ , -- α¯ ∈M Σ(¯α ) = d · KA (M ). 5 %%8 %%3 1. @
D · KA (M ) S · b + (m − S)KA (M ).
G A d = m − 2rB 9 S > 2r
(S − 2r)KA (M ) S · b. S b. S − 2r
an(m − 2r) 2r and = = an 1 − b= . m m m KA (M )
+ 1.
KA (M )
1− 1−
2r m 2r S
a · n,
! . ! + $ = S m, - %%6 = = , %%4 E$ $ m = 10, S = 8, r = 2 KA (M )
4 1 − 10 6 4 a · n = 5 a · n, 1− 8
( 65 - = . , %%4 %%6 , . 12 (0 = ( ( A. )%9, 001 9 S > 2r .- (0 RA(M ) = ( ( M ∈ K(m, n, r) ) A * . RA min a1 , 1− 1− I ! $ S > 2r %%4 %%6 , 2r m 2r S
RA (M )
1− 2r m 1− 2r S
a·n
a·n
=
1− 1−
2r m 2r S
.
3'
n 5 . ( ,$ . $ RA a·n = a1 . + 1. . =. I- (0 ( K(m, n, r) . ! S = |SA (M )|, |SA (M )| 2. 7( . !$ A m 3 r 1B$ M (S) 6 - , ( A -- - 2< $ : ' . . . , A. ( . B + 1. . $ SA (M ) 3. ? - $ ! - =. %%'$ ! (0 RA K(m, n, 1). ", ) I- (0 RA = ( ( A K(m, n, 1)
2 RA 3 1 − m
38
.
: ( . - A , ( $ . ( * %$ A.( - ( SH, , 1 - . , - . - ! . : , = $ . - 0$ $ = . 2 - . ( = )* ,1 - : , ( SH )% ( SH 1 . - (( ! : ! - , $ ( -$ = - ( SH, . ,0 1 -$ , - ,
% & $ ' $&
SH
: ( , - - ( $ - . ( * % . ( H , ( - +0 $ $ . 12 : ,
.( ( ( . H+ E < ( , 33
I- . , : , = M1 M2 A.- , |M1 | = |M2 | = mB 1 - 12 (−1, 1)< , A ,B M1 M2 . m × n, , - A B MM = M1 ∪ (−M2 ) 2m × n , - - , Q(n) = (q1 , . . . , qn ) A , .- : , = M1 M2 B 9 , - qi = 0 , MM .- - A . MASSTB i, - MZAPB K .- ( i qi < 0, i = - MZAP, MM i - - . = , A )% (%B 9. , -1 - Q(n), ! ( 0 - ( , --1 - = ! , ? , Q(n) . -1 - -. , - 5 12- , MM . 1 ( ,$ =. , Q(n) -
! " # H $% !# 9 = - , MM Q, .-1 - , SV(I) = min (Q, ai ),
(.
1im
SN(J) =
min
(Q, aj ),
m1 j2m
ai (aj ) , MM, 12- M1 (M2 ),$ 1 i m (m + 1 j 2m), I J MM, , . ( - 12
K ! min{SV(I), SN(J)} > 0, , M1 M2 A = M1 M2 B H <. ,J .- , MM I J A = ! B . $ . , 1 - H <. L. ! = .- - = , 0 =. $ ! , -- - , A)6 (*B 3;
& ! " # H $% !#
+ A BMRAZ m2 × n, 12 1 = M¯ )4 (, * 9 = - , MM 1 MRAZT, ( % .-1 - , MIMSSV(I) MIMSSV(J) K ! min(MIMSSV(I), MIMSSV(J)) > 0, , M1 M2 A = M1 M2 B ! H <. , A *46BJ .- , MM I J A = ! B . $ . , 1 - ! H <. A $ .- , MM . MASKA.
' ( ) !*# 1 #% $+ !,"
: 6 ' = , - A 2 -B .- , MM A ! B H < . = M1 M2 . 9 : ! - = ( ( . 2 - - A(%$ )%$ *&4B I- , MM, , , ( ,$ ! Qi 2 Q i ( , G ! -@ D ( - .- MM; SIGMA(I) - . , MM I Q; SB(I) ! SIGMA(I) i ( ,J SIGMAM(I) = SIGMA(I) − SB(I).
5 : I .= ( = S (. ! (.$ (. D − SIGMAM(I) SB(I) < D − SIGMAM(I) + Qi .
9 ! IGRAN $ -2 ( ( = S. 5. i ( , , 1 3/
$ , ( ( = S −1. 9 ! L ( ( i ( , A . i ( , $ 2 - - . 1 i + 1 ( ,B #.- , MM L. + : $ ( - .- 2 , !0$ . , ( D 9( A MASST 1 . , B - SB(I) SB(I) ± Qi : MM(I, L) −1, , - +, 1, , - −. + , . - , MM A !0 - ( -B ! ( ( 2 - - . $ 2 = . ( - , . ( , ( . ! 1< 1 . ,
( ) !*# #% %+ !,"
I- , 2 - - % ( . , . , MM, = ! . - , - - ! 12 ( E- 2 - - ( 6 - - $ , = J * .- . - ( ( = D − SIGMAM(I) SB(I) < D − SIGMAM(I) + 2Qi ;
% . i ( , , 1 - $ , ( ( = - = , $ ! , A *BJ 4 SB(I) = SB(I) + σ · Qi , (. σ ! 0, 2 −2, ($ % 9. $ 1 ! - : '$ . , ! 2 − r ;&
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1 . 12 1 = 1 A $ , : B $ . : , = M1 M2 ; 9 .= : ( . . = A $ B . : , = $ : ( - = ! ! - - =. ! ,0 - - = ! - 0 . ( ( - H+ E ( : ( ( SH, . - = $ ! ! 1 ! A∗BJ : $ , , . 12 ($ . =. . , ,0 ( ,0 - ;*
% # #
& ($ # $" . ( .( ( ( 1 - . - (( ! < : ( SH : . , : , = $ - )* : (, I- , $ . , =$ ! !
! - ! A^B . 9. - - 12 = , : - 012( ( SH, : SH1 , SH2 , SH4 , SH5 = SH ( , $ 12 12( : .= A ^ B . , ! ! , - .- + $ 012 : ( : 012 ( H A , *%%B$ 1 A ( SH B .(1 - ! . , : , = $ ! ( =., ( : , ,$ ( . + =$ ! . , : , = : % 4 . . . : = $ : 6 ' . $ : ( - .- 12 - = = ! ! - A )* : (,B >%% A 0 - ! A_B A`aB - ! , =. - : ! ! (( - -$ . - W*4X - . ! .- - 5 $ .=2 1 - <! , . . 12@ * . ! . J % . . ! ;%
I- - =. - <! , . ! ! . - (( - 0$ -2- 3' , $ C. , ( , A $ $ , .$ (12 , ( .$ , . $ =! , = , .$ , = . -J ( - . , .B : , C A . W*4XB , , , . 12 - <! ,b =. - L!-@ ( $ > # . $ 5 $ , ( $ .= $ I.( A _c`aB . $ L # . $ 9 $ + $ >.= $ #! A `a_B E $ - : , = . - - . - M1 M2 6 × 75 =.- . MM 12 × 75 ! , ( SH ,- ! A : %B$ : , = -- - . , $ : 6 ' 0 2 . ( = . 46 A !0 - . ( ( B - .M1 M2 B@ I! 0 2 - = A . - ( SH ( !0 ( ( C B$ . !$ ! 2 - ( $ . ( A : = B W*4X ! *8 A . : , - B$ - - ! , =. - . ! = ! .- : *8 9: . ( SH , ! . , = , : , =. - L!-$ , = M1 M2 6 × 16. * : , ( SH : , , C. , . MM 12 × 16, ( % : A . - , B , , . , . , A , $ , M1 M2 : . , -- !B 12- - MM 8×16. 6 : A . - B , *< - . 8 × 4, ' : % ( - . 8 × 3 A )* : (,B E $ ( SH = SH5 . ;4
! , =. $ , 2 ( 4 ! , =. ! ! , - ! , =. , @ 5< 555 $ $ L. K A $ $ , , W*4XB ! , - ! , =. . , . 12 $ (. = .- - . , ! , - = , ( $ d+d ! $ d−d 0 $ d⊕d ! , =.( - QA.- - ⊕ 0 ( ! 12 B + $ ( ( SH ( SH1 ( SH5 = SH 2 - . , -$ - , 3&V . ;'V
" " "
;6
SH1
⊕
⊕ ! ! !
⊕ ⊕ ! ! !
⊕ ⊕ !
SH2
SH4
SH5 = SH
⊕ ! ⊕ ⊕ ! ! ! ! ! ⊕ ⊕ ! ! ! ! ! ⊕ ⊕ ⊕ ⊕ ! ⊕ ⊕ ⊕
>%% 0 , - (. , , : - , , . 83 , $ ,12 . < (( ! <: - A $ $ $ ! ( -$ (. ! $ , .B 9. : = ! W*6X$ . , - 1 .= @ : , = , , , ' * A , B ' % A , B + - : , = . M1 M2 5 × 67 =.- . ( SH. ! ( , % 4 : ,- !$ , M1 M2 --1 - . , $ ! . , $ $ SH1 = SH2 = SH3 , : : : , , ! 5 12-
MM 6 ' : ! A ! 0 - ( -B . , 10 × 17 A : * - . ! % ( $ SH4 = SH5 = SH B ! , C ( .C-- ! .- - - %& ! , - . , % E = * . $ ( SH1 ( SH5 = SH - Q A 3&V . /'VB , . ! , ! . =.1 ( ,0 -$ . , )* : (,$ ,1 , 1 : ! , ( SH.
SH1,2,3 SH4,5 = SH
! " # $ % & ' ( ! " # $ % & ' ( Q ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) &(* ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) '#*
;'
W*X $ LK$ .- > + <( .. . ( . 5 D9 , D$ , 4*$ $ D D$ */38$ '<44 W%X \ eG ( .. . - D9 , D$ , 44$ @ $ */3;$ '<8; W4X 9- G + . . - D ( ., . D$ "!$ . "E#$ */;%$ ;/<*%* W6X " "9$ 5= 5 . . $ @ $ */33$ 483 W'X ! 5 ( $ U @ $ */;4$ 686 W8X ": $ I= I , ! , 0 , . 0 , . @ $ */;%$ 6*8 W3X 5. ! , . @ $ */3'$ *8<4; W;X f F I=$ #! I= 9 , ! , ( @ $ */3/ W/X 5 5= ! . ,. $ $ , *% @ $ */3'$ '<*' ;8
W*&X I=" - @ $ */88$ *'% W**X . I -- ( - , , " . $ */6;$ 4;8 W*%X $ > ? 7 E , . ! , . . 1 . , , $ */86$ %'$ g8$ /*3< /48 W*4X 5 - 5 . , -. ( . 1 , (( - ! , =. 7(< , 0 - (( . @ $ */3'$ '<;% W*6X .- >$ P= G I - , ., ( @ "#$ */;' W*'X + . ( - ( - I - = - @ "#$ */;' W*8X $ > + , ( - E , YU 1 $ 5 $ */;4 W*3X . 2 - ( ( - I - = - @ "#$ */;'
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7 , ( , - h$ G. ! i . , . ! "#$ ;;
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9. ! %'&6%&&' ( H 8&×/& *]*8 +C '$' L 3 E= %&& : G. ! i9G < ! "#$ ( $ !, (, 7 - . ! 1 .- ! ! GI g &6&'/ %&&%%&&* ( + ( . ( !
