Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.intsys.msu.ru/magazine/archive/v10(1-4)/buyevich-613-638.pdf
Дата изменения: Thu Jun 21 03:39:46 2007
Дата индексирования: Tue Oct 2 00:12:58 2012
Кодировка:

, A- -
. .

, , . (. .) . . . , , . . . . . . ., , , [114]. . . . . . . . Pk , (. , . . , . . , --, . ) . Pk , [15, 711]. , Pk (). [7, 8]. . . [912, 15, 1727]; . . P.. , - - (.-.

614

. .

), . , . . P.. , . ( ) . M P.. , [M], P.. . [912, 15] . . P .. , . . P.. [912] , , , P.. . , P .. . , . . [911], P.. , , .-. . , . . [15] , .-. . [2, 10, 11] .-. . , .-. . , ( , ) , , . -. 1. M P .. -, .-. f .-. M .-. f , f , . [12, 21, 22] , - ,

615

. , - = 1. , 2 .., , , , .-. , . , , , . - , - , .-. - , - ; - , - [1], 1- . , 1 - .-. - . - A- ( ). 1 M P.. A-, -. A- , .-. f .-. , A- M, [M] .-. f1 , f 2 , . . . , f , . . . , 1 .-. f .-. f , .

616

. .

, .-. A- - 1. , , . - .-. , A- , A- .-. , . A- [1923, 2529]. , [19]. , A- A- , A- , A- , A- . , - A- : A- 1 , A- -. , A- - 1. .-. , A- [21, 22, 25]. , - , , A- . , M A- , .-. .-. , .-. M. , .-. M, .-. , , . .-. [2, 12] , .-. .-. g (x) ( ) .-. f (x1 , x2 ) ( .-. ). ~ P.. = [{g (x), f (x1 , x2 )}], ~ . .-. P..

617

- [12]. , .-. .-. g (x) f (x1 , x2 ) . , ~ P.. P.. A- , , .-. ~ P.. .-. . ~ P.. , .-. , . : A- ~ .. P ? , . 1 ([27]). ~ A- P.. ~ P.. ^ P.. , .-. , , . 2 ([27]). ^ A- P.. A- .-. -. A- -; - - ; - 1 ; , -, Z ( ), J ( ), D ( ), M( ), S ( ), L( ) V ( ). -

618

. .

[2023]. , - .-. - . - .. k P , (. ); , k 2 . E k , Ek = {0, 1, . . . , k - 1}. t t 1, Ek Ek t. a = (a(1) . . . a(t)).

E

k

=
t=1

t Ek . , ,

k . , P , . P k Ek k , P . k 1. . f (x1 , . . . , xn ), g (x1 , . . . , xn ) P -, (a 1 , . . . , an ) k Ek f (a1 , . . . , an ) g (a1 , . . . , an ). M P . k M -, . f P M . , - k f . N P - , N k -, . f P \ N N {f } -. h 1, T = (t1 , . . . , th ) t t T Ek = Ek1 в. . .вEkh . T , E T , h R Ek k . . f (x 1 , . . . , xn ) R, {(a 1 , . . . , a1 ), . . . , (an , . . . , an )} 1 1 h h R (f (a1 , . . . , a1 ), . . . , f (an , . . . , an )) 1 1 h h R. . , R, U (R). k 2. , t t Ek в Ek {0, 1, 2, . . . }. a1 Ek1 , a2 Ek2 , t = min{t1 , t2 }.

619

(a1 , a2 ) = 0, a1 (1) = a2 (1), . . . , a1 (t) = a2 (t); (a2 , a2 ) = i (1 i t - 1), a1 (1) = a2 (1), . . . , a1 (t - i) = a2 (t - i), a1 (t - i + 1) = a2 (t - i + 1); (a1 , a2 ) = t, a1 (1) = a2 (1). , (a1 , a2 ) = 0 t1 = t2 , a1 = a2 , (a1 , a2 ) = 0 t1 < t2 , a1 a2 .

. 1. . 1 . a1 a2 , (a1 , a2 ) = 0, D v v1 , v2 , D Ek . c1 , D v 1 , t1 , c2 , D v 2 , t2 . 1 , . . . , 11 c1 2 , . . . , 22 c2 t t 1 1 a1 (1), . . . , a1 (t1 ) a2 (1), . . . , a2 (t2 ). (a1 , a2 ) = t, c1 c2 . (a1 , a2 ) = i (1 i t - 1), 1 2 . ., 1 1 1-i 2-i ; 1-i+1 2-i+1 . t t t t T Ek . T; A A = (a1 , . . . , ah ), A = (a1 , . . . , ah ) Ek A, i, j {1, . . . , h} (ai , aj ) (ai , aj ); A A,

620

. .

i, j {1, . . . , h} (ai , aj ) < (ai , aj ). , A A , A A A A. T A = (a1 , . . . , ah ) Ek . T , A A Ek A, -, A, h -. - . - : (M ) , (m) . , h = 1 (m) = . , i, j {1, . . . , h} (ai , aj ) (a1 , . . . , ah ) (M ) . (ai , aj ) (i, j ); - , i, j (1, . . . , h) , i = j (i, j ) = 0. , , - : i, j, l {1, . . . , h}, i min{ti , tl } - (i, l) j l, min{ti , tj } - (i, j ). (I)

, - T . -, E k . - D , Ek . D v h v1 , . . . , vh . i {1, . . . , h} c i , v v i , ti . i, j {1, . . . , h}, ti,j = min{ti , tj }. ti,j - (i, j ) ci cj , , , (ti,j - (i, j ) + 1)- ci (ti,j - (i, j ) + 1)- cj . (I) , i, j, l {1, . . . , h} , i < j < l, ci , cj , cl , vi vj , vl vj . ci ai (1), . . . , ai (ti ) .

621

ci ai = (ai (1) . . . ai (ti )). , A = (a1 , . . . , ah ) - ; , i, j {1, . . . , h} (ti,j - (i, j ) + 1)- ci cj , A (M ) , A (m) . , - .

. 2. . 2 - D . E4 , (a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 , a12 ) (M ) , a1 = (01), a2 = (033), a3 = (032), a4 = (1000), a5 = (1001), a6 = (1003), a7 = (102), a8 = (21), a9 = (223), a10 = (2220), a11 = (221), T a12 = (3). , - E 4 , T = (2, 3, 3, 4, 4, 4, 3, 2, 3, 4, 3, 1). T Ek . 1, . . . , h -, (a1 , . . . , ah ) (a (1) , . . . , a (h) ) . R . R -, (m) R, -, (a1 , . . . , ah ) R - (a (1) , . . . , a (h) ) R.

622

. .

R , -. A = (a1 , . . . , ah ) \ R, i {1, . . . , h} i i CR (A) Ek : CR (A) , ti a Ek , (ai , a) ~ ~ 1, a(ti ) = ~ (a1 , . . . , ah ) R, ai = a. , ~ i (A), , A \ R, i {1, . . . , h} C R . Zk (). k 2, - h, h 1. R Zk (), -, - . m 1, A 1 = (a1 , . . . , a1 ), . . . , Am = 1 h (am , . . . , am ) \ R. 1 h i1 {1, . . . , h}, . . . , im {1, . . . , h}. , ti1 = · · · = tim , (a11 , a22 ) 1, . . . , (a11 , am ) 1, i i i im
i1 CR (A1 ) . . . C im R

(Am ) = .

Zk ( ) T Zk (), - E k , 1, max{t1 , . . . , th } , , Zk () = . T = (t1 , . . . , th ), h Zk ( ) = k 2, 1. T 1. {(a1 , . . . , a1 ), . . . , (as , . . . , as )} Ek , s 1 1 h h (M ) -, - 1 , . . . , s , q , r {1, . . . , s}, i, j {1, . . . , h} (i, j ) (a
q q (i)

,a

r r (j )

);

R , -. R -, -, -, \ R -. R - . ,
t E1 Ek1 , . . . , Eh E th k

, :

623

) (a1 , . . . , ah ) R, - a
(1)

E1 , . . . , a

(h)

Eh ;

) i, j {1, . . . , h}, a Ei , a Ej (i, j ) (a, a ). R - . A = (a1 , . . . , ah ) \ R, i {1, . . . , h} Qi (A) Ek : Qi (A) , R R \ R (a1 , . . . , ai-1 , a , ai+1 , . . . , ah ) , (ai , a ) 1, a (ti ) = . , A \ R, i {1, . . . , h} Qi (A) . R m 1, P = {R1 , . . . , Rm } - . P T -, l {1, . . . , m}, l {1, . . . , m}, A \ Rl , i i A \ Rl , i {1, . . . , h} CRl (A), CR (A)
i , , , C Rl (A) = , Qi l (A) = Ek , R Ek j {1, . . . , h} , t i = tj , (i, j ) 1, Qj l (A). R R - , A \ R, i {1, . . . , h}. i i ER (A) Ek , CR (A), i CR (A) , Qi (A) . R Jk (). k 2, - h, h 3. R Jk (), m 1 R - R 1 , . . . , Rm , {R1 , . . . , Rm } T -, A \ R1 1 2 CR1 (A), CR1 (A) , i {3, . . . , h} i , CR1 (A) . l 1, s1 , . . . , sl {1, . . . , m}, A1 = (a1 , . . . , a1 ), . . . , Al = (al , . . . , al ) 1 1 h h \ Rs1 , . . . , \ Rsl . i1 {1, . . . , h}, . . . , il {1, . . . , h}
l

ti1 = · · · = til , (a11 , a12 ) i i

1, . . . , (a1 , al l ) i i

1.

624

. .

, s1 , . . . , sl , E
i1 Rs
1

(A1 ) . . . E

il R

sl

= .

Jk ( ) T Jk (), - E k , T = (t1 , . . . , th ), h 3, max{t1 , . . . , th } , , Jk () = . Jk ( ) = 1, k > 2, 2, k = 2. Dk (). k 2, - h, h 2. R Dk (), m 1 R - R 1 , . . . , Rm , {R1 , . . . , Rm } T -, A \ R1 1 2 CR1 (A), CR1 (A) , h 3 i {3, . . . , h} i (A) . C R1 ) h = 2. R1 . . . Rm
(M )

= .

) h 3, l 1, s1 , . . . , sl {1, . . . , m}, A1 = (a1 , . . . , a1 ), . . . , Al = (al , . . . , al ) 1 1 h h \ Rs1 , . . . , \ Rsl , i, j {1, . . . , l} (a i , aj ) = 1. 11 i1 {1, . . . , h}, . . . , il {1, . . . , h}. , ti1 = · · · = til , (a11 , a22 ) i i E
i1 Rs
1

1, . . . , (a11 , al l ) i i
il R
sl

1,

(A1 ) . . . E

(Al ) = ,

j1 , . . . , jq {1, . . . , l} , ij1 {1, 2}, . . . , ijq {1, 2}, {Aj1 , . . . , Ajq } - E Dk (), T = (t1 , . . . , th Dk ( ) =
R
sj 1

(Aj1 ) . . . E

R

sj q

(Ajq ) = .

Dk ( ) ), h 2, max{t1 , . . . , t 1, k > 2,

T - E k , , Dk () = . h} 2, k = 2.

625

T k 2, t 1, T = (t, t), t Ek , t - , t (1, 2) = 1. Mk ( ) (Mk ( ) = k 2, 1). R Mk ( ) , R t , t , R , T Ek k t-1 t-1 . k 2, Sk ( ) (Sk ( ) = k 1). R Sk ( ) , R t , t t , R , Ek , p 2, t R, a E k (a, R (a)) R , (a1 , a2 ) R, a2 = R (a1 ). t t 1, t E k (), E k . , t t a Ek , a . t t , t , a, a Ek a a , (a, a ) 1. T4 = (t, t, t, t), N T4 Ek , (a1 , a2 , a3 , a4 ) , i, j {1, 2, 3, 4} (ai , aj ) 1. k = pm , p , m 1, G = Ek , p. 1, Lk ( ) (Lk ( ) = m , p k = p , m 1). R L k ( ) , t , t . k = pm . R N ; (a1 , a2 , a3 , a4 ) N R,

a1 (a1 (t)) a1 (a2 (t)) = a1 (a3 (t)) a1 (a4 (t)), R . T - , t 2, T = (t, t), t Ek , t t (1, 2) = 2. ~ Vk ( ) (Vk ( ) = k 2, 2). R Vk ( ) , R t ,

626

. .
(m)

t , : (a1 , a2 ) t R, a1 (t) = a2 (t), R ; (M ) t , (a1 , a2 ) t R, Ek a1 (t) = a1 (), a2 (t) = a2 () R . k 2, 1, Wk ( ) Zk ( ), Jk ( ), Dk ( ), Mk ( ), Sk ( ), Lk ( ), Vk ( ).
k 3. k 2, 1. M P - , M R Wk ( ), M U (R).

4. k 2, 1. N - k . R W ( ) , P k N = U (R). 5. k 2, 1. R, R Wk ( ). . ) R, R {Zk ( ), Jk ( ), Dk ( ), Mk ( ), Sk ( ), Lk ( ), Vk ( )}. U (R) = U (R ); ) R, R L k ( ), Vk ( ), R = R . U (R) = U (R ); ) R, R Z k ( ), Jk ( ), Dk ( ) . U (R) = U (R ); ) R, R Mk ( ), Zk ( ), Jk ( ), Dk ( ) h. U (R) = U (R ) 1, . . . , h , (a 1 , . . . , ah ) R (a (1) , . . . , a (h) ) R ; (a1 , . . . , ah ) R (a -1 (1) , . . . , a -1 (h) ) R; ) R, R Sk ( ). U (R) = U (R ) , R , R , R, R , .

627

, Zk (1), Jk (1), Dk (1), Mk (1), Sk (1), Lk (1) Z , J , D , M, S , L, k - [7, 8, 11]. , , [7, 8] 3, 4, 5 . k - [24]. . . , (, . . P .. , 1, 2), , , (, , k - k ). [1, 2, 5, 11] , , . , k - k 3 , , k - . . . P.. , -, A- k .-. P .. (1) .-. , , . M k . k 2, 1. , - Mk , , - . - , k P.. (1). - Mk Wk ( ) , .-. [25, 26]. T k 2, t 1, h k , T = (t, . . . , t) , Ek , -. - k -, S {1, . . . , t} : ) t S ; )
h

628

. .

q S q q {l1 , . . . , lk } , q q {1, . . . , k } (lm , ln ) = t - q + p {1, . . . , t} \ S , (i, j ) = t

l {1, . l 1, -p+1

. . , h} {1, . . . , h} q q {l1 , . . . , lk } m, n m = n; ) S = {1, . . . , t}, i, j {1, . . . , h}.

. 3. k - - k = 3, h = 27, t = 5, S = {1, 3, 5} . 3. k = 2, t1 > 1, t2 > 1, t2 < t1 , h1 2, h2 2, T T = (t1 , . . . , t1 , t2 , . . . , t2 ), E2 , -.
h
1

h

2

- , T 2- - 1 2 , 1 E2 1 , 2 T E2 2 , T1 = (t1 , . . . , t1 ), T2 = (t2 , . . . , t2 ) i, j {1, . . . , h1 },
h
1

h

2

l, m {h1 + 1, . . . , h1 + h2 } : (i, j ) =

1

(i, j ), (l, m) =

2

(l - h1 , m - h1 ), (i, l) = (j, m),

(i, l) > max{ (i, j ), (l, m) + t1 - t2 } + t2 - t1 .

629

. 4.

. 5.

- . 4, 5. . 5 (M ) h = 12, t1 = 7, t2 = 6. k 2, t 1. Ck (t). R

630

. .

Ck (t) , T = (t, . . . , t ), h k k 3 h 4 k = 2, k - T - Ek , R = (m) . k = 2, t1 > 1, t2 > 1, t2 < t1 . C2 (t1 , t2 ). R C2 (t1 , t2 ) , T = (t1 , . . . , t1 , t2 , . . . , t2 ), h1 2, h2 2, h
1

h

h

2

T - E 2 , R = (m) . Ck ( ) k 3 Ck (t), t (Ck ( ) = 1). C2 ( ) C2 (t) C2 (t1 , t2 ), t, t1 , t2 , t , t1 > 1, t2 > 1, t2 < t1 , t1 . (C2 ( ) = , 2). , Ck ( ) k .-. , k P.. (1). L2 ( ) ( 3, 4, 5).

6 ([25]). k 2, 1, M M k . M - , : ) k 3, M , Ck ( ); ) k = 2, M , C2 ( ) L2 ( ). 7 ([25]). k 2, 1, N - , .-. ( . ). : ) k 3, R C k ( ) , N = U (R). ) k = 2, R C 2 ( ) L2 ( ) , N = U (R).

631

( 1), A- .-. . A- A- , A- Wk ( ), 1. , 8. k 2, M Mk . M A- , 1 : ) k 3, M , Ck ( ); ) k = 2, M , C2 ( ) L2 ( ). , C k ( ) , , , , Zk ( ), Jk ( ) Dk ( ). [26] 8 9. k 2 A- M .-. M k , k M \ P.. (1) . .-. , M2 , , k = 2. , 2 2 P.. , P.. (1), 2 . , P .. , . . [2]. , . . [17], . . 2 , M2 , P.. (1), .-. fu (x1 , x2 ) .-. .

632

. .

2 .-. , P .. (1), M2 ; , 2 P.. [28]. 2 .-. f (x1 , . . . , xn ) P.. , Q = {q0 , q1 , . . . , qp } .-. f , q 0 . (x1 , . . . , xn , z ) .-. f , t 1. , q j .-. f t- q i , a1 , . . . , an t E2 , (a1 , . . . , an , qi ) = qj . i {0, 1, . . . , p}, t 1. Qi (t) Q, , t- qi ; Qi (0) = qi . , q 0 , q1 , . . . , qp .-. f (x1 , . . . , xn )

Fq0 (x1 , . . . , xn ), Fq1 (x1 , . . . , xn ), . . . , Fqp (x1 , . . . , xn ).
2 .-. P.. , Fq0 (x1 , . . . , xn ), Fq1 (x1 , . . . , xn ), . . . , Fqp (x1 , . . . , xn ) 2 , P .. . 2 2. , P.. P.. 2 m 1, l {0, 1, . . . , m - 1}, f (x1 , x2 ) P.. . , , .-. f Q. .-. f (x1 , x2 ) B (m, l)-, Q Q

) l = l (mo d m) , Q0 (l ) Q = ; ) Qi (m) Q qi Q qj Qi (m) , Fqj (x1 , x2 ) , 0 i, j p. ~ , .-. f (x1 , x2 ), , B (1, 0)-. , .-. 2 P.. (2) m l B (m, l)- . 2 N P.. , s 1. N(s) .-. N,

633

2 2 x1 , . . . , xs . N P.. (s), f (x1 , . . . , xn ) P.. . , .-. f (x1 , . . . , xn ) N, f1 (x1 , . . . , xs ), . . . , fn (x1 , . . . , xs ) N .-. f (f1 (x1 , . . . , xs ), . . . , fn (x1 , . . . , xs )) N. 2 M P.. N, .-. M N. 0 0 1 1 G, H0 , H1 , D0 , D1 , D0 , D1 2 P.. (1). .-. g (x) G , , . .-. h0 (x) = y1 H0 , h1 (x) = y2 H1 , : t y1 (t) = const t x(1), . . . , x(t - 1); y 2 (t) i {1, . . . , 2 } x(2i); t y 1 (t) i {1, . . . , t-1 } x(2i + 1); 2 y2 (t) = const x(1), . . . , x(t - 1). .-. d0 (x1 ) = y1 , d0 (x1 ) = y2 , d1 (x2 ) = y3 , 0 1 0 0 0 1 1 d1 (x2 ) = y4 , D0 , D1 , D0 , D1 1 , : y1 (1) = const, y2 (1) = const, y3 (1) = x2 (1) y3 (1) = x2 (1), y4 (1) = const. t 2. y1 (t) = x1 (t) y1 (t) = x1 (t), y2 (t) = const, t x1 (t - 1) = 0; y1 (t) = const, y2 (t) = x1 (t) y2 (t) = x1 (t), t x1 (t - 1) = 1; y3 (t) = x2 (t) y3 (t) = x2 (t), y4 (t) = const, t x2 (t - 1) = 0; y3 (t) = const, y4 (t) = x2 (t) y4 (t) = x2 (t), t x2 (t - 1) = 1. . 6 .-. h0 (x), h1 (x), d0 (x), d1 (x), d0 (x), d1 (x), 0 0 1 1 0 1 0 1 H0 , H1 , D0 , D0 , D1 , D1 .

634

. .

. 6. .-. h0 (x), d0 (x), d0 (x) , 0 1 .-. h1 (x), d1 (x), d1 (x) . 0 1 m 1, l {0, . . . , m - 1}, H, D0 , D1 2 2 P.. (1), B (m, l) P .. (2) , : N H N H0 = , N H1 = , N G = ; 0 0 N D0 N D0 = , N D1 = , N H0 = ; 1 1 N D1 N D0 = , N D1 = , N H1 = ; 2 (2) N B (m, l) P.. N, N .-. , B (m, l)-; N H D0 D1 , N B (m, l), f (x) N, f (x1 , x2 ) 2 N , g (x) P.. (1) .-. g (f (x)), g (f (x1 , x2 )) ~ ~ ~ N, N .
2 2 10. M P.. , P.. (1) M. , M , , M A-, , H, D0 , D1 m 1 , B (m, 0), . . . , B (m, m - 1). 2 2 11. M P.. , P.. (1) M. , 2 P.. M , M , H, D0 , D1 .

12. ) 2 2 P.. , P.. , , -

635

.-. , , H, D 0 , D1 ;
2 ) P .. , .-. , D0 , D1 , m > 1 , B (m, 0), . . . , B (m, m - 1); 2 ) P .. , .-. , H, D1 , m > 3 , B (m, 0), . . . , B (m, m - 1); 2 ) P .. , .-. , H, D0 , m > 3 , B (m, 0), . . . , B (m, m - 1).

, , 10, . , . f (x1 , x2 ) .-. , : .-. f (x 1 , x2 ) x1 x2 , x1 &x2 , x1 + x2 , x1 + x2 + 1; m 1 , .-. f (x1 , x2 ) m- , x1 x2 , x1 &x2 ; l {0, . . . , m - 1} , l- , x 1 x2 , x1 &x2 .

636

. .

2 2 , {f (x1 , x2 )} P.. (1) P.. . , .-. f (x 1 , x2 ) , , 1 f (x 1 , x2 ) C2 ( ). , t 0 .-. f (x 1 , x2 ) t- , . 1 f (x1 , x2 ) 2 L2 ( ). , P.. (1) {f (x1 , x2 )} A-. , .-. f (x 1 , x2 ) l {0, . . . , m - 1} B (m, l)-. f (x1 , x2 ) B (m, 0), . . . , B (m, m - 1). x 1 x2 , x1 &x2 , x1 + x2 , x1 + x2 + 1 , x1 0 = x1 , x1 &0 = x1 , x1 + 0 = x1 , x1 + 0 + 1 = x1 , 0 x2 = x2 , 0&x2 = x2 , 0 + x2 = x2 , 0 + x2 + 1 = x2 . , , f (x 1 , x2 ) , 2 H, D0 , D1 . , {f (x1 , x2 )} P.. (1) .

. 10 11 2 , P .. 2 . , , P .. {fu (x1 , x2 )}, fu (x1 , x2 ) .-. , . , f u (x1 , x2 ) , H, D0 , D1 . , 11 . . [17].

[1] . . k - // . . . . . 51. .: , 1958. [2] . . . .: , 1986.

637

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