Фундаментальная и прикладная математика (1995, №1)

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70- (14.02.1924{26.05.1989) 515.55 + 515.55.0

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Abstract K. I. Beidar, A. V. Mikhalev, G. E. Puninski, Logical aspects of the theory of rings and modules, Fundamentalnaya i prikladnaya matematika 1(1995), 1{62.

The rather complete review of logical (or model-theoretical) constructions and methods studied and used in ring and module theory is given. The historically /rst results (about algebraic reformulation of model theoretical notions | categoricity, stability) as well as modernones, for example, concerningthe problem of pure-simplicity is under consideration.

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1995, 1, N 1, 1{62. c 1995 , !" \$ "

. 2 3 10 14 22 25 29 33 37 39

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23

#$ 4.3. 664] , 6277] K( M -' R '' 2

2, 2 '' AnnM -'( , ,' R=AnnM . ' ( % , ' . C / -, '/ '' * '( , , ' -, % / *' 2''&)'' . M ! 1 !" R=AnnM ! "; { ( ) ). 0 { (; ( ", ) "( 1 ( ( 4 "( $ 640]. G % ( ! ) ( ! !( - % ! ! . ( - " 4 R 1 { ? H M 4 !4 ! !" (= % !;) R , , ! Mr = M ( 9 0 6= r 2 R ( s 2 R ? s = fm 2 M j ms = 0g . K! (, 9 4 ( ! . 4.4. ( 4 !4 ! ) 662] ( M %*+ *+ ,*+ '( % R % M 9 , % R % D. 12 M '' , M ' ,2(,,* ? r * / 0 6= r 2 R . = 24 6321] ( %4 %4 khx yi 9 ;39 9 ( k %), 4 , (. 0( 1 "( 4 ( ;%4 % % " , 14 ;34 9 9 640] . E , 4 ! !4 ! ! . =, ) ( ;%4 % 4 % 663]. O 4 , ! 9 % khx yi (( ( ! ! ) 4 %4, ) ;3 1(: 4.5. 663] % - R , % 9 *' '' *' *' ,*' '( ', ,+ % D. . : aR \ bR = f0g ( 9 0 6= a b 2 R . K1, -( ? b M %. -1 !, M ! ( % ( mR , 0 6= m 2 M . C { p = fxi ak(ij )b ; xj ak(ij )b = x , i 6= j , k(i j) g fxib = 0g fxr = 0 j mr = 0g fxs 6= 0 j ms 6= 0g . :! ( % %3 (1 1.8) , an bR % ; (; , !, p , ! ( N % ) n , n1 , n2 : : :. 0 nR = mR , E(N) = M K, ! m , mi " ) n , ni M . H mi = mj , p m = 0, 1. 2 H3 C4 6403] ! ! % :

24

. . , . . , . .

4.6. L % 2(,, M '' 2 2, 2 % 1) M = Z(p1p) Q() , 2 p < ! < 2) M = Z(pp) , 2 p > ! , p , .

p

24 6320] 4 ! ( 4, !( % 1) 1( 4.5. $ 663] , ) 4 ( 3 % * " % " .

4.7. ( M %*+ *+ ,*+ '( -' R

%*' - ' C . 12 M '' , ( ( ( ,) % D T , R T EndC M Mt = M , ? t / 0 6= t 2 T .

! ;%4 ! !4 ! !- %, 6251] 1 (; 19 !9 4 ! !" ) . M ( ! !9 4 " ( 1 ;. 3 ( "( 9 4 !" % % 664] . 4.8. ( M *+ ) '*+ '' *+ '( ''( *' -' (= % ) R , ' */ Q . 12 % M ' R-( M = Q , % + - K - R ( R 6= K 6= Q), K End M M '' *+ ( *+ ) '*+) '( K . ? , ' ( jRj 6 2! . :! ( ) 1, ( 662] !; ! ! !" R . ) ! ! ( - 6336] ( 9 4 !". F" 6277] , !4 DM ! ; M ( 9 , % ( - 9) 4 ( ( ! - ( DM . -! M ( % ! ( 2 1), ;%( 1{ '(x) ( N = '(M) ;, N M=N . % ! 24 6320] , " ; ( . %4 ( 1 \ 9" ! (%) !9 4. 6396] , 14 14 ! ( 4 %4 4( A1 (k) ( 9 0) % 9 4 : ( ;%4 -4 N M % N , % M=N k . 1, 4 ( %%34 % 4( 62] % % !.

Z

x

5 % # &##

25

) " 4

-! 4 | %! !, %! !, !.

%9 (. H T ( ( L A 6= M 1 M ) 4 , 1- A ( !4 % '(x Ia) , aI 2 A 4 4 ( L . -1 9 ( !; ) ) 1- A % S(A) . 0( T ( !- %!4, ;% 1 ;%4 ( ( ( 2! ). 0( T ( %!4, jS(A)j = jAj ( 9 1 A 3 > 2jT j . :, ", ( T ( %!4, jS(A)j = jAj ( 9 1 A 9, jAjjT j = jAj . : , 9 ; W9: !- %! ! ) %! ! ) %! !. \!- %!( (" %( ( \ ! " ( (". 3 %( ( %! : ( T %! ! , ( '(Ix yI) T %4 44 ( ( ! 3 ! M T % 14 mI i M , M j= '(mI i mI j )

! , i 6 j) . : , , 9 ( %4 44 (, %!. %! ! ( 4 ! !4 : ;%( %!( ( ! 1, ! 2 9 4 ;%4 \ %!*4" 3 . 0( T ( - 4, ( !; ) ! 3 . G( !1 (!), !1 (!)- ; ;. : ( % ( 9 ( 4 ! ; ( @1 (@0 ) . E , ) 9 ( 4 ( % ; . 2;%( !1- ( ( !- %!. ( ( ! !-) ! T C!-= ) 1 n- ( 1 ) T . ) ( 4 . . A !- T = T(A) 1) ( !( ! A , ! 3 4 " f : ! ! ! , ;%( n-1( A % f(n) ) 2) ( ;% n 2 ! 3

! n-% 4 AutA (% 1 ). , ;%( . . !- 4 4 ! . C( %39 4 %! 1 4 % 9 660] , 6443], 6448]. G!* ! , (39( . ., 1 ( % M 6238]. 9 ) ! ;39 4 D : 1) D !1 - A 2) D !- %! A 3) D %!A 4) ( D )"; A 5) D % 633], 6121], 6175], 6323], 6328], 6413]. E , 6174] ! % | % ;% %! !- %!.

26

. . , . . , . .

@ " !4 ) W9 6443]. H -!" !- !" R "4 K1% %, R = F S , F !", S % ! !" ! % 686], 6174]. K( !" R % ! 9 ) ) !- %! !, %! ! ! (4 9 % 9 4 6167], 6175]. W9 6175] , ;% %! !" (( ( (4 4 !" 1 9 " % (. 0 1 ( !1 - 9 " 6237]. ! ;% %! !" %%9 6431]. $!" " - %! !" - %! 6430]. $ ! !" !- %! ( 1 %!) ! , % 6468]. = %! ! ( 9 ( K; 6210]. G C 698] , -!" !" %!4 4 K1% , ! . = !" !- %!4 4 C4 6174]. %4 % ! % %!9 4 , , %!9 4 ", % G 696] , . 1 6122]. E!% 632] !1 - 9 0 !". , !1 - !" 9 0 | ) 1 ( 9 ) % % 9 0 . : 1 , ( ! !1 - !" 4 9 ( , 1;39 1 1 4 " 9 9. %%9 6431] , !" " !1- !" !1- . C 6412] O 6486] , ( !" 99 !9 " % P !1-

! , P % . C 6414] , 1( ( ( " !( % % !1 - ( ". !1 - !", ((;3( ! !, 6374]. - 4 C * 4 6330] !- !" "4 % ! 9 ) . ) " ( ; 4 % p ) , ! ( . ? 76 9 * 6* 6 // . | 1978. | . 17, N 6. | ". 627{638. 8] ' @ 8. 2 . | 4. : 3. | 1984. 9] ' 7 1. . 9 =: 6, // 0+. + . 4. | 1982. | N 11. | ". 3{11. 10] ' 7 1. ., / 8. ?. 3+2 9 , 6 // 0 *. (0300. | . . 8 . | 1987. | . 25. | ". 3{66. 11] ( . . 3 + 7 2 9 9 * 6 // 4 . + . | 1977. | . 21, N 4. | ". 449{452. 12] 8 . 0. 0 9 ) // ". . =. | 1983. | . 24, N 4. | ". 201{205. 13] 8 . 0. 0 + 9 hnp- 76 // . | 1986. | . 25, N 4. | ". 384{404. 14] 8 . 0. " * 9* 7 ) 6 76 // / 76 . ?7 . . . (. 1. | 1: 1986. | ". 87{102.

40

. . , . . , . .

15] 8 . 0. 0 9 76 9 ) // . (. 7. | : 1988. | ". 31{41. 16] 8 . 0. 9 9 9 76 // . . ". . =. (0300. | 12.09.89. | N 5805{(89. 17] 8 . 0. 8 8 9 7 76 // . (. 9. | : 1990. | ". 7{30. 18] 8 . 0. 9 7 9 9 // . | 1991. | . 30, N 3. | ". 259{292. 19] 8 . 0. 7 9 * * I. ? + . // . | 1992. | . 4, N 1. | ". 98{119. 20] 8 9 B. 1., / 6) 3. >. + 7 ) * // :. . =. | 1982. | . 34, N 2. | ". 151{157. 21] = 3. 6. | 4. : 0 . . | 1947. 22] + C.. * 76* // 4 . + . | 1975. | . 18, N 5. | ". 705{710. 23] B2 C. 1. ? +2 . | 4. : 3. | 1980. 24] B2 C. 1. 9 ). " 9 9 ) . III. | 4. : 3. | 1982. 25] B2 C. 1. ? D -E* // . | 1989. | . 26, N 6. | ". 640{642. 26] F) '. (., / 6) 3. >. G // :. . =. | 1990. | . 42, N 7. | ". 1000{1004. 27] F . ?. 4

+ 6* 6, G * +2 // . 3 """H. | 1976. | . 229, N 2. | ". 276{279. 28] F . ?. ? + 6* 6, G * +2 // ". . =. | 1978. | . 19, N 6. | ". 1266{1282. 29] F . ?. H+2 7 G ) 9*

+) 6 // . (0300. | 20.07.83. | N 4091{83. 30] F . ?. H+2 7 G * ) *

+) 6 // 4. +. :. . | 1983. | . 13. | ". 52{74. 31] F . ?. / 76 7 G

+* +2 ) G ) ) // 4. +. :. . | 1989. | . 14, N 4. | ". 64{71. 32] F7 '. 0. / 76, * @1 - 9 // . | 1974. | . 13, N 2. | ". 168{187. 33] F7 '. 0. 8 76, * 9 // Fund. Math. | 1977. | V. 95, N 3. | P. 173{188. 34] F7 '. 0. 39 9 7 1 // 0+. 2. 9. +. 4. | 1982. | N 5. | ". 75. 35] 0 . . 3 9 // ". . =. | 1983. | . 24, N 6. | ". 56{65.

41

36] 0 . . +2 ) = 7* 6* 9 // ". . =. | 1984. | . 25, N 4. | ". 78{81. 37] /9 (. (. 76 // ? 04 3 :""H. | 1975. 38] /) 4. . ? 76, K 9 76 // 0+. 2. 9. +. 4. | 1977. | N 8. | ". 41{48. 39] / . 0., ? . 8. ( +2 2* ) // :* . . | 1978. | . 33, N 2. | ". 49{84. 40] / ?. ?. " 76 * +. | 4. : 4. | 1971. 41] / B. 4. 4 9 9 ) ) // ". . =. | 1984. | . 25, N 6. | ". 70{75. 42] / B. 4. * 76* 7 = * ) ) // 4. + . . | 3 : 1988. | ". 31{39. 43] /= * 8. (., / 8. ?. 9 ) * 6 // ". . =. | 1991. | . 32, N 6. | ". 87{99. 44] /+9 (. . H G @+ 9* )* // /. . | 4 : 1986. 45] 1K6) (. . 9 + // . (0300. | 29.11.83. | N 6341{83. 46] 1K6) (. . 6 9. 3 + // :* . . | 1989. | . 44, N 4. | ". 99{153. 47] 4) . 4 7 // " 9 9 ) . L. 1. | 4. : 3. | 1982. | ". 141{182. 48] 4 (. ., 4* . (., " 1. ., . . 4 // 0 *. (0300. | 1981. | . 19. | ". 13{134. 49] 4) /. . *7 7 * ) // . . | /: 1985. | ". 69{78. 50] 4* . (. 7 9 // . 3 """H. | 1986. | . 289, N 6. | ". 1304{1308. 51] 42 .?., " 1. . . | 4. : 3. | 1969. 52] 4+ 1, p n > 0 q 8 p, p | k. 2) xq1 : : : xqn y1l : : : yrl z1l : : :zdl 2 st(H ). 0 B && R, B

R-. 0

Z = k!xq1 : : : xqn y1l : : : yrl z1l : : :zdl ] st(H ): 0 8 8 !A5] A = B]t H && RH Z = RH !xq1 : : : xqn y1l : : : yrl z1l : : :zdl ]: 0 v2 (t) : : : vn(t) 2 k!t]. 4 x4 !A5], A 8 & 81 a z ;! a z z = x1 z = yj z = zj a z ;! a (xi + vi (xq1)) z = xi i > 1. (4) 4 !L] (.!A5], 13) 2. . f 2 Z n 0. 1- . . ( q '/+' (4), 2( f * 2 %( * * x1 : : :xqn ' l l l l xqs (5) 1 h h 2 k!y1 : : : yr z1 : : :zd ] n 0: 0 , B

B = i B (i) i = (i1 : : : in) ij > 0 ij 2 Z:

(6)

j = (j1 : : : jn) jr > 0 xqj = xqj1 1 : : :xqjn n :

0,

(6) 2- t H B , . . t(xj xj ) 2 B (j + j 0 ) xj B (j 0 ) B (j + j 0 ): (7) ( ,

(6) %

A = B]t H , A = i A(i), 0

A(i) =

X B(j)]t(xj Hm Sm): 0

j +j =i 0

67

2

+ % & , B H = C A R Z q l @, B

(6) (7). 4 , , R | && B , 1 B . 1. .

Q 2 P(B (0)) s > 2 + max(2 K: dim R) g 2 Aut((A B(0) Q) As ): 3 %+ g K1 (A) %+ K1 (B (0)), g = g1g2 , - g1 2 Aut(QB (0)s ), g2 * -.** 4 ' &5 '/+' E (A B(0) Q. t A). 6.** 4 ' &5 '/+' ( . + ' * '&5 -''/+' &5 A-'. ( A ;! (A B(0) Q) As : (8)

. 0 S = Z n 0, Z sx1. G K: dim S;1 A = K: dim R. 0 g & (A B(0) Q) A , 181 & (8). 0 !S1], 4.4, 1 8 h 2 S , v 2 E (Ah B(0) Q. s Ah ), vg 2 Aut(Q B (0)s ), vg | A

(8). H A

A = j >0 A(j ) A(j ) = i1 +:::+in =j A(i1 : : : in): 4 4 !A5], 8 2 , @ f (5). 0 4 !A2] & g Cg Z . 0 , xq1 Cg . 0 } | Z , } Cg , xq1 2 Z n }. 0 8 2 !A5] A+} ' A+} (0)!x1. ] . . A+} x1, | & A+} (0). 3 , !A1] A+} (0) . 4 , !B2]. 0, h }. G A+} f x1 , . . h A+} (0) . 0 2 3 !A2], .28, 1 2 Cg }. ( } = Z h 2 }, 8. C , h 2 Cg . G , , f = h. 0 F | A, B]t (Ha Hm ) z1 : : : zd;1 . G A ' F !zd . ], | & F , 18 zd . 0 , Cg , A+} 2 3 !A2], .28 , Cg % Cg \ F . 0 % , % Cg \ k!y1l : : : yrl ]. 0 G | l . G A = G!yl . ], A, B Ha Sm y1l : : : yr;1 r | & G, 18 yr . 0 2, 3, 7, 1, . 16, 5, . 30{31 !A2], @, l ]. 0 % , Cg % k!y1l : : : yr;1

68

. .

, Cg \ k 6= 0. 3 8 1 2 Cg , . 0 , xq1 2 Cg . 0 % & (4), 1 x1 x1 + 1, 8 8 g , (x1 + 1)q 2 Cg . H A

A = j >0Aj ] Aj ] = i2 ++in =j A(i1 : : : in ) q 4 !A2] %

Cg Z . G (x1 + 1)q x1 2 Cg (0) 1 = (x1 + 1)q ; xq1 2 Cg (0): 3 8 . 2 1. . L | % ( * -%& ( * ( -% * Al Bl Cl Dl * ' k * ( 5 p. 3 U | . % & -% L s > 2 + max(2 dimk !L L]), GL(s U ) = GL(s U )E (s U ). 6.** E (s U ) ( . + ' & s. 2. . k | * , Q = (qij) 2 Mat(n k), - qii = qij qji = 1 5 i j 4 ' & qij ' + )& k. . AQ | ) -%, *' k 4 ' ' u1 = y1 : : : ur = yr y1;1 : : : yr;1 ur+1 = z1 : : :un = zn;r +' * ' 2' uiuj = qij uj ui yi yi;1 = yi;1 yi = 1: 3 s > 4, GL(s AQ ) = G1E (t AQ ), - G1 | -.** - &5 ' ) diag(1 : : : 1 ay1l1 : : :yrlr ), a 2 k , li 2 Z.

. D , AQ = k]t (Hm Sm ) Hm Sm . 4 , K1 (AQ ) ' K1 (k) Zd, !Q], .122.2 3 , AQ !D].

A1] . . . // . !!!". !. . | 1984. | 48, N 6. | !. 1123{1137. A2] . . - . . / 0 . // ! 1 . | 2.: 245, 1989. | !. 6{49. A3] Artamonov V. A. Projective modules and groups of invertible matrices over crossed products. // Contemp. Math. | 1992. | 131, N 2. | P. 227{235.

69

A4] . . ! 7 18. // . 9 1 :. 4 :. | 2., 9. | 1991. | 9. 29. | !. 3{63. A5] Artamonov V. A. Projective modules over crossed products. // J. Algebra. | 1995. B1] ; 7. 0: . >. ? / 1/ / 11. // 51/ . . | 1993. | 9. 48, N 6. | !. 39{74. L] Lam T. Y. Serre`s conjecture. | Springer Lecture Notes Math. | 1978, N 635. MS] Montgomery S., Small L., Ed. Non-commutative rings. | Springer-Verlag, 1992. Q] Quillen D. Higher algebraic K-theory, Springer Lecture Notes Math. | 1973. | 341. | P. 85{147. S1] ! . . 0: 0 N("), ! N("),

!

", .

. 1. , ! f . 2. ; , . 2 2.3

2.6. . # N , ! A B f (ni ) (A) ! C C 2 N ni ! 1 ! f (ni ) (B) 2 N.

2.7. 1. # # . 2. # + -

# . 3. 0 # | . 4. 2 M | # , # + - . 5. * #

!" # .

2.8. A ; B ( 0- ) " A, -

# B, # f (ni ) (B) f (ni ) (A) ! A. A ; B k- " A, # B, 5 # f (ni ) (B) f (ni ) (A) ! A A 2 A ; B (k ; 1)- . A ; B 1- " A, # B, 5 # A ; B k- k 2 N.

74

. . , . .

2.9. A ; B k- .

0 A ; B k- l- " F (l) (A) ; F (l) (B) . 2 An ! A, Bn ! B, (An ; Bn A ; B) ! 0.

1 (A ; B)- L0 = Li+1 > S ! Ai ; Bi , Ai Bi 2 Li . (! LAB = Li . 9 , ! ! LAB , | , " , A B.

3

?,

, . ( P(n) | m + 1, - ,"" am+1 | . 1 Pk (n), k = 0 :::m: Pm (n) = P(n) Pm;1 (n) = Pm (n + 1) ; Pm (n) .. . (1) Pi;1(n) = Pi (n + 1) ; Pi (n) .. . A , " , P0(n) = n!am+1 | . (! " = P0(n), xi(n) = fPi(n)g x0i(n) = xi(n + 1), (1) 8 x0 = x + x mod 1 >>< m0 m m;1 xm;1 = xm;1 + xm;2 mod 1 (2) .. . >: 0 x1 = x1 + " mod 1 " | , " = n!am+1. ( , #2fP(n)g] = 0 0 6 xm (n) < 1=2. . , (x01 : : : x0m ) (x1 xm ) ( , ). 1 xm = 0 xm = 1=2 . C ! !

k.

75

3.1

3.1. 6 " ro + " w v +

ro(i) =

0 w = v i i

1 wi 6= vi & (w v) w v +

Pi

j =1 ro(j) (w v) = ilim !1 i 3.2. & w v |

!" x0 2 T. 7 (w v) = 0. . A #6] , x0 2 T . A ! f, mes(@U) = 0 , , wn = un , f (n) (x0 ) 62 @U. ; > , ! . 2

3.3. & x x ! !

!"!. 7 (wx wx )

0. . E! , wx wx . F T T. F ! : . ( O T T | ! , (x x ) O. ? | , ! , . ( | > ( ), 6= 0, . 2

4

-

4.1. 8 , M # -

#, M 9. 6 , xn . F k- :

8> (k) 1 n >> xm = .xm + Ck xm;1 + : : : + Ck " mod 1 .. >< ( k ) 1 i >> xi = .xi + Ck xi;1 + : : : + Ck" mod 1 . >> : x(k) = .x1 + C 1" mod 1: 1

k

(3)

76

. . , . .

G A (x1 xm ) B (x1 +Hx1 xm +Hxm ) ! ! M 0 , f (k) (A) f (k) (B) ! ! ! ! . A (3) :

8> (k) m + Ck1Hxm;1 + : : : + Ckn;1Hx1 mod 1 >> Hxm = Hx ... >< (k) i + Ck1Hxi;1 + : : : + Cki ;1Hx1 mod 1 >> Hxi = Hx .. . >: (k) Hx1 = Hx1 mod 1:

(4)

4.2. . ! T 0 # , .

4.3. & T | m- , U | - " l, n. 7 !

nlm

!". . ; ,

lm

S l. ( P(x) = 0 |

S. A (3) , x(ik) | i- k, P (x(1k) : : : x(mk)) = Q(x1 : : : xm k) - , ml k. G

ml x

S, Q(x1 : : : xm k)

ml , , x ! S, , , x . 2

! . I! U x , ,, . 4.4. & 1, ", Hxi Q. 7 A ; B

# , i ; 1

!

B.

. E! , Hxj | j < i. ( Hxj = pj =qjQ| Hxj k m! ij =1 qj . C x(jkl) = xj j < i (3) (4) !

8> x(kl) >> m >> (kl) >> xj >>< (kl) >> x1 (kl) >> Hxm >> >> Hx(kl) >: j(kl) Hx j

= xm + Ckl1 xm;1 + : : : + Ckln " mod 1 .. . = xj + Ckl1 xj ;1 + : : : + Cklj " mod 1 .. . = x1 + Ckl1 " mod 1 = Hxm + Ck l1 Hxm;1 + : : : + Ckln ;1Hx1 mod 1 .. . = Hxj + Ck l1 Hxj ;1 + : : : + Cklj ;1Hx1 mod1(j > i) = Hxj mod 1(j < i)

77

(5)

(x(1kl) : : : x(mkl) Hx(ikl) : : : Hx(jkl)) #6] 2m ; i + 1, , A ; B , ! , i ; 1 B, . 2

4.5. & Hxi | " . 7

Bn ,

!"

!" A B

: HxBi n = nHxBi :

. B0 B1 A B . , Bk

! , Bk+1 f (ni ) (Bk ) f (ni ) (Bk;1) ! Bk . 1 , Bk 2 A ; B k- . 2 4.6. & Hxi | " . 7 B ,

!"

!" A B

: HxBi = 0 6 6 1:

J " ! 4.5 ! , , . C , , Hxi | , , 1, ", Hxi | Q. . , Hxj |

Q " - , i- Mi = ij =1 mj , mj |

x = x ! 1 6 j 6 m : Mj xj = Mj xj .

78

. . , . .

" 4.7. # #,

1=Mi

i-

!

- + . * # 0 6 xm < 1=2 . 1 ! : 4.8. ; , ! (3), # U : 0 6 x1 6 1=2 L("),

!" L(") ! ,

!

!". . E! U , , 2.5 0 , N( 0) , , - 0 , . 1 , N-, , N- , . 4

, . F , N( 0 ), ! - !

n + 1 f n #x1 = q=2 q 2 N, ! , ! , n + 1 , n + 1 , , N( 0 ) , , , !. E, n + 1 , , . ( | ! , , , ! L(") = max(N( 0 ) N( )), , ! k , k f i #x1 = q=2 q = 0 : : : k ; 1. C (")

m

-

! m , , , k f i #x1 = q=2 q = 0 : : : k ; 1, , , , . 2 L Q(k), , T(k) (k > K) k, " :

1 ;km : : : ;km 1 m deg Q(k) = m(m + 1) : Q(k) = : : : ; ; 2 k k 06k 1 0 = x 2. - . = inf f : = g| ( . 0

= (E );1 E , 1 ( % .). 2 1 . # x 3 . # , , , E 0 1 , . # x 4 . Wk

Wk

Xk k

W

x

Nxn

xn

x

Nxn

k

Wk

n

Nxn

Xk

Xk

xn

Xk
1 W0 = x: (1) &3], &11], &12], 4 % 4 , % , Xk ! 4% , P(Xk = 1) = p P(Xk = ;1) = q P(Xk = 0) = r (2) p > 0 q > 0 r > 0 p + q + r = 1: $ !

. 5 Xk \ " k- , . . % 4 0 Xk0 Xk00 . $ (2) , , fXk0 g fXk00 g | % . 8 , 4% ) ( ) . - ! ( !" 0 ). . % fWk g " , % % % x R (9 "). .% R = n ; 1, Nxn " | 0 n (1), . . Nxn = inf fk: Wk = ng: (3) ; % p, x = 0) ! %% xn = Nxn(ENxn);1 , % % x 2 &0 n) ! p q ( 2.1 2.3). 8 % % &11]. ; , % 4 ! r, . . % xn r. ; %, 0 )

83

% , %4 % ( 2.2) (. &5]). . p 6= q % % x 4 , , x % % ) , % , n ; x ! 1 n ! 1. - , x = n ; k (k = const ) % xn k. - , p < q % % , " k ( 2.3). - , , ENxn !" % ) 2.1 ! % &16], r = 0 . . 0 4 " ! | !" 0 . ? ! 2:10

2:30, % 0 . ? (. . &8]), ! 4 p q Xk %! 1 0 n, . . 4 ". * ,

2.2, n (1) n, "!" (EXk = p ; q > 0), 0 %

(p < q). 0 \" " , . 8 ! 3- ) . @ , , P(Xk = 1 = Wk;1 = x) = p(x) P(Xk = ;1 = Wk;1 = x) = q(x) (4) P(Xk = 0 = Wk;1 = x) = r(x): A , ! n1 n2 (n1 6 n2) ,

n;1 ni ! ai i = 1 2 n ! 1

! !" 4 p(x) = p0 q(x) = q0 r(x) = 0 p0 > q0 x 2 &0 n1) p(x) = p2 q(x) = q2 r(x) = 0 p2 > q2 x 2 &n2 1):

(5) (6)

B &n1 n2), % p(x) = p1, q(x) = q1, r(x) = r1 . * , % r1, 0 % . C , ( 3.3),

(4){(6) (p1 > q1, p1 = q1 p1 < q1 ) xn % 4 x, , ) . F (5) !,

n1 n2 \ ! ", . . , , . ;

84

. .

, . G % !" : % Wk % &0 n1), , &n2 n) %4 . A % | % %4% . ! , % % 3- )

% ), ! !" . . % Xk ! , %% q, x n , n ; x ! 1, n ! 1, xn !p 1: (15) ; 1 p = q, n x ! c, 0 6 c < 1, n ! 1, xn !d c

(16)

c | gc(y) gc (y) = (2y3 (1 ; c2)); 21

1

X

m=;1

+ c ; 1) (;1)m+1 (2m + c ; 1) expf; (2m 2y(1 ; c2)

2

g (17)

y > 0. p < q, x , n ; x ! 1, n ! 1, xn !d (18) 1. (! " , !p (15) !d (16), (18) # .)

. A (15){(18) % (.

. &9] . 496), % n % d 0 !" ! n ! 0. . % xn n

'n (s) = Fxn(e;m;1 s ) xn

(19)

1 mxn ! 1 n ! 1,

% (7), (8), xn ! Fxn(e;s ) s.

1. .% p > q. . 4 (1) !" hwx un;x Fxn(z) = 11 + + hwn

h = ba;1 w = (d ; f)(d + f);1 u = 2pz(d + f);1

a b d f ! ) (2). . z = e;s % e;s = 1 ; s + o(s), d = 1 ; r + rs + o(s): . % p 6= q f = j ;1 j(1 + ( 2 (p + q) ; 1)s) + o(s) d + f = 2p(1 + ( ; 1)s) + o(s) d ; f = 2q(1 ; ( + 1)s) + o(s) a = ;2 ;1 ; 2(p ; ;1 )s + o(s) b = ;2q s + o(s): K h(e;s ) = O(s) u(e;s ) = 1 ; s + o(s) 0 < w(e;s ) < 1 lim ' (s) = n;lim (1 ; m;xn1 s + o((n ; x);1))n;x : n!1 n x!1 A , (7), n ; x ! 1 mxn = (n ; x)(1 + o(1)): 5 %, '(s) = e;s ,

87

(20) (21) (22) (23) (24) (25) (26)

xn !d 1: K (15) , ! % . 2. p = q % (1) ) x + e2 v x 2 z n;x Fxn(z) = ee1vvn1 + e v 1 1 2 2n % v1 = (d + f)(1 ; r);1 e1 = z ; v1 v2 = (d ; f)(1 ; r);1 e2 = v2 ; z: ; % (22) , z = e;s f = (2(1 ; r)s + o(s)) 12

88

. .

v1 = 1 + (2(1 ; r);1 s) 12 + O(s) v2 = 1 ; (2(1 ; r);1 s) 12 + O(s) ei = ;(2(1 ; r);1 s) 21 + O(s) i = 1 2: ? (8), n ! 1, n;1x ! c n;2mxn ! (1 ; c2 )(1 ; r);1 : .0 z = expf;m;xn1 sg

K ,

v1n2(z) ! expf(2(1 ; c2 );1s) 12 g v1x2(z) ! expfc(2(1 ; c2 );1s) 12 g:

'n (s) ! '((1 ~ ; c2);1 s)

(27)

'~(s) = cosh fc(2s) 12 g(cosh f(2s) 12 g);1: (28) ? (. . &4]), ) '(s) ~ n0 P(Nn 6 M) < ": (37) ? (35){(37), 4 (34). 4. Nn > 0, ENn ! 1 Nn (ENn );1 ! (38)

n ! 1, , Nn !p 1: . . % P(Nn 6 M) = P(Nn (ENn );1 6 M(ENn );1 ) (39)

! " > 0 y > 0 , P( 6 y) < 2;1", nO , n > nO M(ENn );1 < y, (38), (39) (37), 4. ;,

, 3 | 0 , A() &5] . C 4 9 , % &5] A(+0) = 0. 2. fYk gk>1 | "# "# " m, & ' " " " Nn " 4,

SN (ESN );1 !d n ! 1: (40) . 54 (40) % (14) SN SN Nn ESN = Nn ESN (34), (38) (. &6], . 281). 1. ; !" 3

,

% (40). . % ! . C , , % x n , % , n ; x = k, k > 1. 0 (7), (8) %

8 2 n;k( k ; 1) + k p > q < q

mxn = : k(2n ; k + 1)(1 ; r);1 p = q (41) 2 n ; k q (1 ; ) + k p < q: n

n

n

n

n

n

; (41) , n ! 1

91

k p > q 1 p 6 q: *! , p > q %% mxn

Nxn . 5 ,

%, (n ; x = k, p > q)

(20) " )

0 6 w(z) < 1 0 6 z 6 1, %, Fxn(z) ! uk (z): (42) ; 1 L , u(z) = 1 (z), 1(z) | %4 (6). .

&8] %, u(z) | 0 " ) 1 , !" . K ,

%, % Nxn | 0 k- \ ". .

p = q, % mxn n. G , 0 6 w(z) < 1 4% 0 6 z < 1, w(1) = 1. K 4 (42) . ;, p > q Nxn ! % , " k, xn , . - , ! p < q. 54 (42) %. * !" % , ;1 1 (1) = ;1 < 1. 8 , 1 % %! 1 ; ;1. A % xn % 4 (19), (20), (32), (41). . (32) (41) z = expf;m;xn1 sg, x = n ; k, n ! 1 u(z) ! 1 h(z)wn (z) ! ;k 1 s h(z)wx (z) ! (1 ; k );k 1 s k = 1 ; ;k , %, (43) 'n (s) ! 1 ; k + k (1 + ;k 1 s);1 : ;, !" . 3. x = n ; k, k > 1, ;kn ! 1 & Nxn & 1 (z), 1 (z) = (d + f)(2pz);1 | ) (6). * p > q p < q. + p < q & xn, , (43). mxn !

92

. .

; %, % Nxn % k, r, ) (43) % xn 4% k ;1 = pq;1, r. ? , %

0 4

", . . ! fW^ k g !" 0 . , 0 , % \^" . K, N^xn n fW^ k g,

" x. 10. & N^xn ) x (z2 ; 1) + x (1 ; z1 ) 2 F^xn(z) = n1 (z ; 1) + n(1 ; z ) 1

2

2

1

(10 )

i (z), i = 1 2 | (6). . *

, F^x(z) ! (3), (4) !" : F^0 (z) = z F^1(z) F^n (z) = 1: (40 ) ; % (5) (40), % (10).

20. p 6= q

m^ xn = 2pq 2 ( n ; x ) + (n ; x)

" " 2. p = q m^ xn = (n2 ; x2 )(1 ; r);1 :

(70 ) (80 )

" , 4 (9) V^0 = 1, V^n;1 = m^ n;1n (11). 5 1 3.

10. .$ (15){(18) " # " 1 xn ^xn.

%!, % 1. . 4%, p 6= q 4 (19){(26), (31){(33). G , (20) p > q w^ = w, h^ = (1 ; z1 )(z2 ; 1);1, u^ = u = ;1 1 , p < q % h^ = (z2 ; 1)(1 ; z1 );1 , w^ = 1 ;2 1 , u^ = ;2 1 .

p = q % % % (10 ) " )

4 (27){(30).

93

30. x = n ; k, k > 1. / & ;k -

( n ! 1) N^xn, & & 1 (z). + p < q ^xn ( k 1;k ) k " $ , . A % , % 3.

3

N % ) | % Nxn , n !" 0 fXk gk>1 , !" (1.4), (1.6). @ ,

(1.5) xn = Nxn(ENxn);1 % 1. 1. - p1 > q1. . x < n2 !" Nxn =d Nxn2 + Nn2 n: (1) A % , uk (x) = E(Nxn2 )k k = 1 2: 1. n1 6 x 6 n2 u1(x) = 1 (n2 ; x) + g0 ;1 n1+1 ( x1 ; n1 2 ) (2) 0 6 x 6 n1 ; 1 u1(x) = 0 (n1 ; x) + 0 ( n0 1 ; x0 ) + u1(n1 ) (3)

i = qi p;i 1 i = p;i 1 i = i (1 ; i);1 i = 0 1 (4) 0 = 2 0(1 ; 0);2 g0 = ( 0 ; 1 + n0 1;1 (1 ; 0 ))(1 ; 1 );1: . 5

N0n2 =d 1 + N1n2 Nn2 n2 = 0 , 1 6 x 6 n2 ; 1 Nxn2 =d

1 X

(1 + Nx+in2 )ix

i=;1

(5) (6)

ix = 1 , x i, ix = 0, ix Nx+in2 , i = ;1 0 1.

94

. .

.

, (5), (6) 4 u1(0) = 1 + u1(1) u1 (n2 ) = 0

(7)

v1(x) = 0v1 (x ; 1) + 0 1 6 x 6 n1 ; 1

(10)

u1 (x) = p0 u1(x + 1) + q0 u1(x ; 1) + 1 1 6 x 6 n1 ; 1 (8) u1(x) = p1u1(x + 1) + r1u1(x) + q1u1 (x ; 1) + 1 n1 6 x 6 n2 ; 1: (9)

(2.10) v1 (x) = u1 (x) ; u1(x + 1), 4 (8) (9)

v1 (x) = 1v1 (x ; 1) + 1 n1 6 x 6 n2 ; 1 (7) v1 (0) = 1 v1 (n2 ; 1) = u1(n2 ; 1):

(11) (12)

; (10), (11) 4 (12),

v1 (x) = 0 + x0 (1 ; 0 ) 0 6 x 6 n1 ; 1

v1 (x) = 1 + x1 ;n1+1 (v1 (n1 ; 1) ; 1 ) n1 6 x 6 n2 ; 1: ; % 4 (12), u1(x) =

nX 2 ;1 k=x

v1(k) 1 6 x 6 n2 ; 1

(13) (14) (15)

"%! (13), (14) 4 (2), (3) ENxn2 .

1. " (1.5) 0 6 a1 < a2 6 1, 0 6 x 6 n2 # )# n u1 (x) 6 c1n (16) c1 | $ . . . % N0n =d N0x + Nxn , u1(x) 6 u1(0)

0 6 x 6 n2 ; 1. A , i < 1, i > 0, i = 1 2, (2), (3)

, n ! 1 (17) n;1 u1(0) ! a1 0 + (a2 ; a1 ) 1 > 0 % (16). 2

2

2. # n > n2 + 1 " & ENn2 n > 2 ( n2 ;n2 ; 1) + 2 (n ; n2)

2, 2 2 " (4).

(18)

95

. $ !-

" , Nxn2 . R n2 !

(. &1], &7], &14]). 5 %, ! Nn2 n %

Nn2 n =

X

i=1

i +

X ;1 i=1

i +

(19)

i =d Nn2 n2 +1 , i , n2 , n2 +1 % n, n n2 + 1, % n2 . 5 P( = k) = p (n)(1 ; p (n))k;1 k > 1 p (n) | % n2 + 1 % n %4 , % n2 . *"% , %, , \ " % n n2 . K i > 1, (19) Nn2 n > +

X ;1 i=1

i + =d N~0n;n2 :

(20)

B N~xn;n2 n ; n2, x, p2 q2 . - (20) , ,

, n2 , 4 , !" 0 n2. . n2 , n ; n2, . K , ENn2 n > u~1(0) (21) ~ ) u~1(x) = ENxn;n2 (22) u~1 (x) = p2 u~1(x + 1) + q2u~1 (x ; 1) + 1 1 6 x 6 n ; n2 + 1

u~1(0) = 1 + u~1(1) u~(n ; n2 ) = 0: (23) . % (22), (23) (8) (7), u~1(x) = 2 (n ; n2 ; x) + 2 ( n2 ;n2 ; x2 ) 0 6 x 6 n ; n2 (24) 2, 2 , 2 , 0, 0 , 0 (4), 0 2 > 1, 2 < 0, p2 > q2. K (18) (21) (24)

x = 0.

96

. .

2. n;1n2 ! a2, 0 6 a2 < 1 n ! 1, # )# n

ENn2 n > c2gn c2 > 0 g > 1 | " ".

(25)

. . ! n ; n2 > 2;1(1 ; a2)n n ;n

%4 n. C 2 2 u~1 (0) ! 2 n ; n2 ! 1, (25) ; 1 g = 22 (1 ; a2) c2 = 2;1 2 .

3. " (1.5) 0 6 a1 < a2 < 1, n ! 1 x 6 n2 ; 1

ENxn2 ! 0 ENn2 n ! 1: ENxn ENxn . ; (16) (25) , ENxn2 ! 0 n ! 1 ENn2 n 0 (1) 4 (26).

(26)

(27)

3. + # (1.5) 0 6 a1 < a2 < 1 x 6 n2 ; 1

Nxn2 !p 0 n ! 1: (28) ENxn . . % Nxn2 > 1 x 6 n2 ; 1, (28) B 4

ENxn2 N xn 2 P EN > " 6 "EN xn xn

4 (26).

1. + # " 3 " ( n ! 1) xn n2 n .

Nxn2 + ENn2 n xn =d EN n n xn ENxn 2 "%! (28) 4 (26) (. &6] . 281). 1. ; , 1 % 4 ! % ! x 6 n2. ? 4 , &0 1) ! n1

n2 !" . * n = (n1 n2)

.

97

4. 1

En = 1 ; p2 2 + q2( 1 + n1 2;n1 ( 0 ; 1 + n0 1;1 (1 ; 0 ))): . .

(6) % %,

(29)

n =d 1 + Nn2 +1n2 1n2 + Nn2 ;1n2 ;n21

(30)

En = 1 + p2ENn2 +1n2 + q2ENn2 ;1n2 :

(31)

?

(2) (4), ENn2 ;1n2 = 1 + n1 2 ;n1 ( 0 ; 1 + n0 1 ;1(1 ; 0 )): (32) 5 , ENn2 +1n2 | 0 , % 1, % p2 4 %

%4 4 q2 = 1 ; p2 . ?

(. &8] . 345), " ) Nn2 +1n2

(z) = (1 ; (1 ; 4p2 q2z 2 ) 21 )(2p2z);1 : . % ENn2 +1n2 = 0 (1), 0 (z) = (z)(1 ; 4p2q2z 2 ); 12 z ;1

(33)

ENn2 +1n2 = ; 2 (34) . (32) (34) (31), 4 (29). 5. + # " 3 n ! 1 En ! 1 ; p2 2 + q2 1 (35) En2 ; En ! 2q2(p1 13 ; p2 23 ): (36) . 54 (35) | 0

(29),

i < 1, i = 0 1 n1 ! 1, n2 ; n1 ! 1. A , (30) En2 = p2E(Nn2 +1n2 )2 + q2E(Nn2 ;1n2 )2 + 2En ; 1: (37) . % E(Nn2 +1n2 )2 = 00(1) + 0 (1), "%! (33), (34) % E(Nn2 +1n2 )2 = 2 + 2 22 2( 2 ; 1);1: (38) A (37) , E(Nn2 ;1n2 )2 = u2 (n2 ; 1). ? (6), u2(x) = p0u2 (x + 1) + q0u2(x ; 1) + f2(x) 1 6 x 6 n1 ; 1 (39)

98

. .

u2(x) = p1u2(x + 1) + r1u1(x) + q1u2 (x ; 1) + f2 (x) n1 6 x 6 n2 ; 1

f2 (x) = 2u1(x) ; 1 1 6 x 6 n2 ; 1: S ! (5) !

u2 (0) = 1 + 2u1(1) + u2(1) u2(n2 ) = 0: v2 (x) = u2 (x) ; u2 (x + 1) 4 (39), (40) !" v2 (x) = x0 v2 (0) + 0

x

X

k=1

f2 (k) x0 ;k 1 6 x 6 n1 ; 1

v2 (x) = x1 ;n1+1 v2 (n1 ; 1) + 1

x

X

k=n1

f2 (k) x1 ;k n1 6 x 6 n2 ; 1:

(40) (41) (42) (43) (44) (45)

?

(42), (43) (7), !" E(Nn2 ;1n2 )2 = v2 (n2 ; 1), v2 (0) = 2u1(0) ; 1. 5 (41) (44) % v2 (n1 ; 1) = n0 1;1 (2u1(0) ; 1) + 0

nX 1 ;1 k=1

(2u1(k) ; 1) n0 1;k;1:

(46)

%4 %

(2) (3) )

u1(x) 0 6 x 6 n1, % (46), (47) n;1 v2(n1 ; 1) ! 2 0 1 (a2 ; a1) n1 2 ;n1 v2 (n1 ; 1) ! 0 n ! 1. .0 (45) , lim v (n ; 1) = nlim n!1 2 2 !1 1

nX 2 ;1 k=n1

(2u1(k) ; 1) n1 2;k;1:

(48)

; % (2), ! % (48), (49) v2 (n2 ; 1) ! 2 12 (1 ; 1);1 ; 1 n ! 1: * % (36) (37) 4 (35), (38) (49). 2. + $ (1.5), 0 6 a1 < a2 < 1, n ! 1 n2n !d 1.

(50)

99

. % %-

\ " !" (. . &7] . 50). 4 0 !" . 54 (50) , q~nEn2 (En );2 ! 0 n ! 1 (51) q~n | % n

. *

, q~n = p2 p (n), p (n) 2. - p (n) | 0 %, , % n ; n2 ; 1, % 4 %

p2 . ; % % &8] . 339, (52) q~n = 2;1 (1 ; n2 ;n2 );1 : 5 % 4 (51)

% (35), (36) 5 (52), 2 > 1. K % ) % %. 3. n;1ni ! ai, i = 1 2, n ! 1 (0 6 a1 < a2 < 1), x 6 n2 xn !d (53) 2. . 54 (53)

1 2. 2. . , p1 = q1, , " . 54 (1), , . - 1 !" u1(x) = ENxn2 . 10. n1 6 x 6 n2 u1 (x) = (n2 ; x)( 0 + n0 1 ;1(1 ; 0 ) + 2;11 (n2 ; 2n1 + x + 1)) (20) 0 , 0 1 " (4), 0 6 x 6 n1 ; 1 u1(x) (3). . <

%, 4 (5){(15) ! . F p1 = q1, . . 1 = 1, 4% (11) (14). ;,

v1 (x) = v1 (x ; 1) + 1 n1 6 x 6 n2 ; 1 (110) v1 (x) = v1 (n1 ; 1) + 1 (x ; n1 + 1): (140) K ,

u1 (x) 0 6 x 6 n1 ;1 ) (3), ) (20 ) (140) "%! (15). .

!" 1.

100

. .

10. " (1.5) 0 6 a1 < a2 < 1, # )# n 0 6 x 6 n2

u1(x) 6 c01n2 c01 | $ .

(160 )

. .!0 , 4 (16), ), 0 < 1, 1 > 0. ; (2 ) (3) , n ! 1 n;2 u1(0) ! 2;11 (a2 ; a1) > 0

(17)

(16). < 2, 4% 4 n2, 4 ! 1, . L , 2. . % (160 ) (25) (27), ! 3, 3 1. *

.

40. 1

En = 1 ; p2 2 + q2( 0 + n0 1 ;1(1 ; 0 ) + 1 (n2 ; n1 )):

(290 )

ENn2 ;1n2 = 0 + n0 1 ;1(1 ; 0 ) + 1 (n2 ; n1 ):

(320 )

. 54 (30) (31) !, (20) . % ENn2 +1n2 , , ) (34), (290) .

5. + # " 3 n ! 1

n;1En ! q21 (a2 ; a1)

n;3 En ! (2=3)q212 (a2 ; a1 )3:

(350 ) (360 )

. 54 (350)

(290).

A ! (37){(44), 1 = 1 n1 6 x 6 n2 ;1 v2 (x) = v2 (n1 ; 1) + 1

x

X

k=n1

f2 (k):

(450 )

(46) , (20)

n;2 v2 (n1 ; 1) ! 0 1 (a2 ; a1 )2 n ! 1:

(470 )

n;3v2 (n2 ; 1) ! (2=3)12 (a2 ; a1 )3 :

(490 )

. (450) x = n2 ; 1, , n ! 1 C 0

(350) (37) (360).

101

20. .$ (50) " 2 p1 = q1. . F (51) (350), (360) (52), ,

, (50). C , 3. 30. + p1 = q1 ) (53) " $# " 3.

3. p1 < q1 (1) % x 6 n1 ; 1 Nxn =d Nxn1 + Nn1 n (100)

%

n1

. * uO1(x) = ENxn1 . 100. 0 6 x 6 n1 ; 1 uO1 (x) = 0 (n1 ; x) + 0 ( n0 1 ; x0 ) (300) 0, 0 0 " (4). % (16) 1 uO1 (x)

0 6 x 6 n1 . . % ENn1 n > ENn2 n n1 6 n2, ! (18) 2 (25) 2 ENn1 n. C 0 , 3,

3

1 % n2 n1. A % 4 5,

% % "!" 0 l. .% l1 ! , , 1 6 x 6 l1 ; 1 , p1, q1

r1, l1 6 x 6 l ; 1 4% p2 q2 . . , pi < qi, i = 1 2. * pO(x) = pO(x l1 l) % " l %

x. 6. 1 ) pO(1) = (B + l11 ;1 Al );1 (54) B = ( l11 ; 1)( 1 ; 1);1 (55) l ; l ; 1 1 +1 Al = ( 2 ; 2)( 2 ; 1) : (56) . . ) pO(x) = p1pO(x + 1) + r1 pO(x) + q1pO(x ; 1) 1 6 x 6 l1 ; 1 (57)

102

. .

pO(x) = p2 pO(x + 1) + q2pO(x ; 1) l1 6 x 6 l ; 1

pO(0) = 0 pO(l) = 1: w(x) = pO(x) ; pO(x + 1), (57), (58) w(x) = x1 w(0) 1 6 x 6 l1 ; 1

. (59)

w(x) = x2 ;l1 +1 l11 ;1 w(0) l1 6 x 6 l ; 1: l;

1 X

k=0

(58) (59) (60) (61)

w(k) = ;1

(60) (61) % % (54){(56). 2. *

, l1 = n2 ; n1, l = n ; n1,

! &n1 n), %, pO(1 n2 ; n1 n ; n1 ) | 0 p (n) 2, , %, n1 + 1, % n, % n1 . .% Tx (l1 l) | " % % x.

4. n ! 1 1 6 x 6 n2 ; n1 ; 1

Tx (n2 ; n1 n ; n1) ! Nn1 +xn1 .

(62)

6 ! 2.

. ! Tx . A " uk (x) = E(Tx )k , k > 0. .

(7), (8) (39), (40) % k > 1 uk (x) = p1 uk (x + 1) + r1 u1(x) + q1u1(x ; 1) + fk (x) 1 6 x 6 l1 ; 1

uk (x) = p2 uk (x + 1) + q2uk (x ; 1) + f2 (x) l1 6 x 6 l ; 1 fk (x) =

kX ;1 i=0

Cki (ui (x) ; fi (x)) f0 (x) = 0:

(63) (64) (65)

, (65) , f1 (x) = 1, f2 (x) = 2u1(x) ; 1

(41). . % T0 = 0, Tl = 0, !

, ,

uk (0) = 0 uk (l) = 0 k > 1:

(66)

vk (x) = uk (x) ; uk (x + 1) k > 1

(67)

103

4 (63), (64) ) vk (x) = x1 vk (0) + 1

x

X

m=1

vk (x) = x2 ;l1 +1 vk (l1 ; 1) + 2

fk (m) x1 ;m 1 6 x 6 l1 ; 1 x

X

m=l1

fk (m) x2 ;m l1 6 x 6 l ; 1

(68) (69)

% i = qi p;i 1, i = p;i 1 , i = 1 2. . 0 (66), (67) , l;

1 X

m=0

vk (m) = 0:

(70)

F (70) vk (0) (68), %, vk (l1 ; 1), ) !" (69). L % uk (x), % ) uk (x) = ; uk (x) =

xX ;1

vk (m) 1 6 x 6 l1 ; 1

(71)

vk (m) l1 6 x 6 l ; 1:

(72)

m=0 l;

1 X

m=x

7. 1 " )

u1(x) = ; 1 x + 1 (1 ; x1 ) 1 6 x 6 l1 ; 1

u1(x) = 2 (l ; x) + 2 ( x2 ; l2) l1 6 x 6 l ; 1

1 = (v1 (0) ; 1 )(1 ; 1);1 2 = ;2 l1 +1 (v1 (l1 ; 1) ; 2 )(1 ; 2);1 v1(0) = 1 ; pO(1)( 1 l1 + 2 (l ; l1 ) + ( 1 ; 2 )Al ) v1 (l1 ; 1) = 1 ; l11 ;1pO(1)(B( 1 ; 2 ) ; l11 ;1( 1 l1 + 2 (l ; l1 ))) " pO(1), B Al (54){(56).

(73) (74) (75) (76) (77) (78)

, (68){(72) k = 1

(73){(78), .

5. l ! 1

u1 (1) ! ( 1 ; 2 ) ;1 l1 +1 ; 1 :

(79)

104

. .

. . % u1(1) = ;v1(0), (77)

(79). . (65) uk (x) ) x % ui (x)

i 6 k ; 1 1 6 x 6 l ; 1. . k > 1 % ! , 7 k = 1, 0 % !. ? %!, ! 4% Nn1 +1n1 . A !" . 8. k = 1 2 E(Nn1 +1n1 )k = llim E(T (n ; n l))k : (80) !1 1 2 1

. . % u3(1) 6 C < 1 l, (T1(n2 ; n1 l))k k = 1 2 (80) (62). A , % (68){(70) k = 2, u2 (1) = pO(1)(1 Al

lX 1 ;1

m=1

f2 (m) l11 ;m;1 + + 1

9. l ! 1

lX 1 ;1 m=1

f2 (m)(1 ;

l1 ;m ) + 2 1

l;

1 X

m=l1

f2 (m)(1 ;

l;m )):

(81)

2

u2 (1) ! ;1 l1 +1 ( 1 l11 ;1 + 2 ; 1 + 2 12 ( l11 ; 1)( 1 ; 1);1 + (82) + 2 1 ( 1 ; 2 )(1 ; ;1 l1 +1 )( 1 ; 1);1 ; 2 1 (2 1 ; 2 )(l1 ; 1)): . . (81), % (54){(56) (73){(78), , ), l ! 1 1 ! ( 1 ; 2 ) ;1 l1 +1 (1 ; 1 );1 2 l2(l ; l1 );1 ! 2 : K % % . .% On |

. 400. 1 EOn = (1 ; r1 );1 (1 + q1( 0 + n0 1 ;1(1 ; 0 )) + p1 (( 1 ; 2 ) n1 1;n2 +1 ; 1 )): (2900) . ) (30) (31) % 2 1, , r1 > 0. % (1 ; r1 )EOn = 1 + p1ENn1 +1n1 + q1ENn1 ;1n1 : (3100) ? (300), ENn1 ;1n1 = 0 + n0 1;1 (1 ; 0 ): (3200) ; % (79), (80), ENn1 +1n1 = ( 1 ; 2 ) n1 1;n2 +1 ; 1 : (3400) ? 1, (3200) (3400) (3100), (2900).

105

500. + $ (1.5) n ! 1

(1 ; r1 )EOn ! 1 + q1 0 ; p1 1 (1 ; r1 )EOn2 ; (1 + r1)EOn ! 2q1(p0 03 ; p1 13 ):

(3500) (3600)

. ?

%4 , (3500) (2900). 00

.

(31 ) (1 ; r1)EOn2 = p1 E(Nn1 +1n1 )2 + q1E(Nn1 ;1n1 )2 + 2EOn ; 1:

. (80) (82) n ! 1

E(Nn1 +1n1 )2 ! 1 + 2 12 1( 1 ; 1);1

(3700) (3800)

. . 2 1 (38). * % % E(Nn1 ;1n1 )2 . - %, 0 % (46), u1 (k), 0 6 k 6 n1 ; 1 ! ) (300) (3), 0 , % u1 (n1) = 0. % ! % (49), 1 0. ;, 2+q

nlim !1(p1E(Nn1 +1n1 )

2

1E(Nn1 ;1n1 )

; 1) = 2q1(p0 03 ; p1 13) ; (1 ; p1 1 + q1 0)

(3700) (3600). 5 ! 2 (52) q~n = p1pO(1 n2 ; n1 n ; n1 ):

(5200)

; % (54){(56), %, % (51) (3500), (3600) (5200). 5 %, 2 3. 5) 4% !! .

300. + $ " 3 x 6 n1 " (53).

4 !

?

3.3 3:30, 3:300 ! , ) 2 % 4% % x, " . ; % xn % x " , 4 4 ! r0

r2, !" \ " &0 n1)

&n2 n). - , ) 3 , r0 = r2 = 0. K % % "!" 0 , !, % , 4 , 0 !.

106

. .

; xn % ,

(. . &3], &12]). @ % , , , )

. ? %

, ! %

. ? % ! W (t) t M = M = 1. - , 0 % , %

% , ! ! %! , % . T4 (. . &1]), W(t) | 0 % R . 8 , W (t) !" :

i > 0 % , i = 0 , i > 0 + . %! p = ( + );1

%! q = ( + );1 ! i > 1, 0 1 %! 1. K , % ! % R | , fW^ k g !" 0 r = 0. C N^xn #xn, n, % x. *

, #xn = inf ft : W(t) = ng % , xn. . %% , 2xn = N^xn + (n ; x), %, , !" %.

!" 1. ( n ! 1) xn(Exn);1

" 2.1 " ^xn .

;1 ;1 xn = ^ 1 + n ; x EN^xn + 1 + Exn xn n;x EN^xn "%! 2.2 2.1. .

9 %, " ) ! . R %,

ni, i = 1 2 ! ! % , % 4 %4 ( ,

!" ), % . !" "

. - , , W (t) % %

,

, 4 4 ,

!" , %

!

!

107

0 ! !

. . 8 % %,

% , .

"

1] . . , . . . ! " #% & ! | (.: *&+- (,, 1980. 2] . . , . . . #%+ ! " 1 !

/ 3

" 4. 5 #1 1 & | (.: (, 1984. 3] . . , . . . 4 ! & + 1 #%+ // 8 .

! . | 8. XXIX. | N 4. | . 654{668. 4] . ! , . .d2 u ~0 du ~ , & & @ & = u~0 : 2 = Fu(~u 0)~u0. , d

d

I: u ~0 (0) < 0 u~(1) = 0A II : u~0(0) > 0 u~(1) = 0. B . 0 z ! 1 ! 1: z (0) = z 0 , z (1) = 1, !! . !' $ w - 0 . + & w(0) = 0, w(1) = 0. - !! & I , ! & II , 1u & & (1.8), . ! (1.4) & & ' !

! (1.4) 0 u(x ") . !. &, & ! & (1.4) ! &, . . +& u(x ") . . & (1.4) &! O("n+1 ) 0 1] (2], 3]). ) * . 4 '+ 0 II x = 0 I x = 1. ') u0 = u0 > (0). 9 & 0 u~, ! ' ! '(0) ! 1. 9! * . 3, & ! (u0 0) & '! ! 1 , & , 0 & (1:70). 9 & (1.1), (1.2) 0 !. & (1.4). ) u0 = (0). =$ ! ' , ! !! 0 & & . &, @. (1.8), ! 0 '& u~0 . C ., !. ', '& &' &@ #.

2

!

x = 0 & ! u0

= (0):

(2:1)

= 0 (1.6) , (1.7) , (1:70) , (1.8) @ . , u2i , 0 u & , !! 0. D ! 1 u, , ! !! u~0 0 & & . & (1.8) ( #! u~0 '+ * . 5), & 0 & (1.8) '& ' Z1

(F~u'0 (0) + F~x) u~0 d = 0:

(2:2)

0

"2

d2u

dx2

= ;u (u + a(x))

& F = ;u(u + a), Fu = ;2u ; a, Fx = ;ua0 , Fuu = ;2, Fxu = ;a0 , Fxx = ;ua00, F~u = ; a(0) ; 2a(0) + 0 u] = a(0) ; 20 u, F~x = ; a0(0);a(0) + 0u], '(x) = ;a(x), '0 (0) = ;a0 (0). = 0 (2.2) & Z1

f;a0(0) a(0) ; 20u] ; a0 (0) ;a(0) + 0 u]g u~0 d = 0

0

a

0

Z1

(0) 0uu~0 d = a0(0)I = 0 0

113

& I 6= 0, ! ! 0 u > 0, u~0 < 0, > 0. 9! ' (2.2) a0(0) = 0. (2.4). G = a=2, F ( x) = (;3=4)a2 , u22 = ;a00 =a. 0 a (0) = 0 (2.4) ! ! &! & ! & 1

00

a

1

Z Z 2 3 (0) a(0) + (0u) d ] = ; (1 u)2u~0 d : 2 2

0

0

H-##$ a00(0), ! ' 0 !& &, , (> 0). I@& & a00(0), . . ! & ' a(x). 9! ', # & (1.4) !+& 0 ! ' , + 0 & (1.1), (1.2), & u0 = (0), & (1.4). - ! x = 0 &@ ': u0 = (0) + (") (2:10)

& (") # !$ ", !@ '& ! & & (")

= 1" + 2"2 + :

= 0 '& & (1.4). -+ & (1.6) (1.7), (1:70). ) (1.8) '& d21 u d

= F~u1u + (F~u'0 (0) + F~x) = F~u1 u + H1 1 u(0) = 1 1 u(1) = 0:

(2:5)

F + u~0 &' , !! + & 0, Z1 00 u ~ (0)1 u(0) = (F~u'0 (0) + F~x)~u0 2 d

0

1F ( 0) =

C & 1:

Z1

(F~u '0 (0) + Fx)~u0 d :

0 1

Z 1 1 = F~u'0 (0) + F~x] u~0 d : F ( (0):0)

(2:6)

0

F & 2 2u. > 2u (2.3), ! '& 2u(0) = ;2u2 (0) + 2:

114

. .

, (2.4) Z1

F ( (0) 0) 2 ; u 22(0)] =

H2u ~0 d

0

!&

1 R

2

=

0

H2u ~0 d

F ( (0) 0)

+ u22 (0):

(2:7)

E ' & + & + & i (i > 2).

2. ).& 1 & &, . 1. G 1

=

;a Z

1

; 34 a2 (0) a

4 (0) = 3a2 (0) a0

a

f;a0(0);2u ; a(0)] ; ua0(0)g du =

2

a

Z2

Z2

2a0(0) (p u + a) du (u + a) du = p 2 3a (0) ;a a ; 2u ;a p

. . ! 1 = a0(0)= a(0). ).& i, + i u. ' &' 1 u. ,& @ 1 @ # !$@ z : 1 u = z + 0u. 9 & & z & (0) ; '(0) d2z d 2

= F~u z + H1

H1 z (0)

= H1 + 1F~u0 u ; F~ ] (0) ;1 '(0) = 0

z (1)

= 0:

& @ 1 (2.6) Z1

H1 u ~0 d

= 0:

0

9 & z + & & z

Z

= ~ ( ) (~ ) u0

u0 ;2 d

Z

u ~0 H1 d:

1

0

I! Z

1 u = ~ ( ) (~ ) u0

0

u0 ;2 d

Z

1

u ~0H1 d +

1 0 u: (0) ; '(0)

(2:8)

115

. , + (3.2) u~0 , !! + & 0, 1u(0)F ( (0) 0) =

Z1

0

H1 u ~0 d

116

. .

. . 1 u(0) = 1, & 1 & # . (2.6). # !$ 1 u & & (1.1), (3.1) (' (1 u)II ) '& 1 u & & (1.1), (1.2), (2:10) (' (1u)I ). , & , (3.2) (1u)0 II j =0 = ;'0 (0). , + (1u)0 I j =0 = 0. , - + '& , &## $ (1 u)I , & # . (2.8) ( &! I !) d1 u d

I@&

= u~

00

Z

(~u )

0 ;2

0

Z

d 1

(1 u)0I j =0 =

9! '

u ~ H1 d ; u~10 0

Z

u ~0H1 d +

1

1 Fu( (0) 0)~ u0(0) (0) ; '(0)

1 F~ : (0) ; '(0)

= 0:

(1u)II = (1 u)I + C1 u~0

& C1 & ;'0 (0) = C1 u~00(0) = C1F ( (0) 0), . . C1

0 (0) : = ; F (' (0) 0)

(3:3) (3:4) (3:5)

= + & + . Ci (2.9), @ 0 @ & (1.1), (3.1). 9 , 0 & (1.1), (3.1) &! 0 & (1.1), (1.2), (2:10), !. i = i u(0)II . (0")= u00("), . 8+ (3.1) ! du dx u0 0(") ! + "1 +"2 2 + . = 0 !. &, ! ! + '., & # !$., & , 2. E! 0 '& & + . (2.9), ! Ci & @ i .

4

!

. 0 & (1.1), (1.2) & 0 u(x t ") ' !. & ; @u + "2 @

u(0 t ") =

2u

= F (u x) @t @x2 u0 u(1 t ") = u1 u(x 0 ") = u0 (x):

> -. & 0 & (1.1), (1.2) $ 0 (' u ). 0 ., & @' > 0 .& ! (), k u0(x) ; u k < & k u(x t ") ; u k < & t > 0. 8 ' ,

117

, + & - . = > >

F~x (u 0) du:

(4:5)

(0)

9! ', # & 1 + @ 6] & !. < ! & ! (4.5).

3. & (4.5) & &, . 1. 'Z(0)

F~x du =

(0)

;

;a Z

a0(0)u du =

a 2

2 ;a

; a0(0) u2

a 2

= ; 23 a0 (0)a2(0)

119

I@& & , 1 < 0 a0(0) > 0, . . 0 '

. &! O(") $ .

4. > 1, & a = const, ' & 7] . ' . ' # !$., & @ @ 00N =

h

p

p

;2 a ; 12a e a (1 + e a ) + 0

i

0 N:

(4:6)

)& & , @ &@ 0 (4.6) 1) 0 N = 2) 0 N = 3) 0 N =

p

e(3 a )=2

(1 +

, 0 = 5a , 00N(0) = 0. 4

3 e a )

p

e2 a

(1 + p

p

p

;e p

a

3 e a )

, 0 = 0, 0 N(0) = 0. p

e(5 a )=2 ; 3 e(3 a )=2 p

e a )

3

p

+ e(

a )=2

, = ; 34a , 0 0 N(0) = 0.

(1 + * !$ 0 N 2) 0. ' . # !$. . !. &. H! ! 8], a x & ! ! ' @ &! O("), & # . (4.5). , (4.5) &! \1" !-##$ . " + 0 ' ". * !$ 0 N 1) 3) & @ '. 0@ &@@ ' @ # !$@, ! @ ! @. ! ! 1) 0 = O(1) > 0, 0 . !. & .. F a x & ! ! &! O(") ! 0 +. 9! ', & + 0 , !! 0 . !. &, . a0 (0) > 0 . a0 (0) < 0, !! 0 . !. &, .. I& , & . . !. & & ! , 0 ., &+ 1 < 0.

120

. .

121

122

. .

%

1] . ., . . // !# ! | 1987. | (. 42. | N 6. | ,. 831{841. 2] . ., . . !# / 0 !12 | .: 4, 1973. | 272 . 3] . ., . . !# ! 5 0 !1 | .: , 1990. 4] Fife P.C. Singular Perturbation by a quasilinear operator // Lecture Notes in Mathematics. | V. 322. | Springer, 1973. | P. 87{100. 5] . . 8 5! 5! ! # // #. ! !. ! !. 9 | 1992. | (. 32. | N 10. | ,. 1582{1593. 6] . . : # // !# !5 | 1991. | (. 3. | N 4. | ,. 114{123. 7] ; . ;. : # \ " 0! // !# !5 ! 5 > ((5 ! !# # @,A, 26 { 2 9 1994 0.). | ., 1994. | ,. 18{19. 8] . . 8 : # // !# !5 | 1990. | (. 2. | N 1. | ,. 119{125. & ': 1995.

, . .

. . .

, , , , , .

,

.

Abstract E. E. Gasanov, Some instantly solvable in average search problems, Fundamentalnaya i prikladnaya matematika 1(1995), 123{146.

The concept of instantly solvable in average search problem is introduced as that of a problem, which can be solved in the average time equal to the time of answer enumeration plus some constant which is independent of the problem dimension. Examples of instantly solvable in average search problems are given.

1

,

( ) ! " , , ! ( ., , $1, 3]), " ( ( . ) ! * ( * , ( !+ " ( , , !, +! . , * !+! , " . , , ! " , * , + . " " ,

, . - $2] , +

. / ( !(" . 0 ( , * , . / : 1995, 1, N 1, 123{146. c 1995 , !" \$ "

124

. .

* , ,

, , , !- !3 , (, , , , ( , !- ! 3 n- " , ( n- * * , n- - , n > 1. 4 , $2], , + , ! " , , ! * , ! " " "* , . 4 * !*, " , ! " " . 5 !+* , *

( 6) +* + $1]

( 6). 6 "

. 7 * ! ! *, * , *, . . , +* " ! !* !, ! !( . 6 , * ! ! !. 4! | ) " ( * * *. 9 ( ! ! *. 7 ! !* *. ! * ) . 7 ( " 0. :. ;!! 7. 6. !.

2

0 6 ! * . !" X | ( , X hX Pi, | ( ( X, P | . V = fy1 y2 : : : yk g | , V Y , Y | ( ( , ). | X Y , . I = hX V i |

( ), " x 2 X * * " * V , * x. O(y ) = fx 2 X : xyg | " y 2 Y . Nf = fx 2 X : f(x) = 1g, f | , X, . . f : X ! f0 1g.

125

y : X ! f0 1g , N = O(y ) | * ! y. F | ( * , * ( X, ( . G | ( * , * ( X. ! " ! , " * ( !" . ! F = hF Gi ( . ? n | !" , g(x) | ", gn(x)

, X, y

N = fx 2 X : g(x) = ng: n g

@

Gb = fgn : g 2 G n 2 Ng: @ 6 ( " ). 4 ) ! ! (* ) " ) , | ! ". I ) . @ 6 ! !. !" ". 0 , " | " . 0 * ( ! " ). ? , !" * . ;( G, " ,

!+ ) ! !, . /

! . 5 ( , * + ,

( f1 g. / " , )

| ! "* . -, + " , . ;( ! ! ! ( F: /

! * . 6 ( ! ! ! " ( Y: /

! ". !! !(! "

" ( F = hF Gi. II ) . @ ! 6. !" 6 U. " " * ( 1 2), ( 2 3) : : :, ( m;1 m) " 1 m . ? c , $c] ! !. " ( ) , $( )], | 3

126

. .

g()] , | " , g | ",

!+ . " , . ? " x, ", " x 1, + " x. 0 6 * (

! f !+ : = , f (x) 1 (x 2 X) 3 6= !+! 6 * , f (x) 03 6= ( * ! , f (x) , * * . B! 6 6

! " ' (x). C R(U) P (U) L(U) ( R P L) ( , " U

. !" N | " (. . " ( ) 6 U. C hN i ( ,

!+* " ) ( , | U h i ! " ",

!+! ! ). 6 ( , 6 U ! ! J : X ! 2Y , ! !

U !

: J (x) = hf 2 L(U) : ' (x) = 1gi:

6 " . 6 ( , 6 U I = hX V i 8x 2 X J (x) = fy 2 V

: xyg:

6 ( " 6 U x T (U x) = b

X

2RnP

' (x) + a

X 2P

' (x)

a * ! ( " , b | . 1 $1] !+ .

1. Nf , f 2 F Gb,

U x, .

F

= hF Gi T (U x),

127

5 ! ! ", * ! * 1. 6 ( " 6 U ( T (U x), . . T (U) = M T (U x): ? ( ) | 6, ( " ) b P(N' ) | ( ) | 3 a P(N' )= | ) | " . F ", ( " 6 ! ( 6, . .

T(U) = b

X

2RnP

P(N' ) + a

X

2P

P(N' ):

5 ! ! ", a = b = 1. !" 6 U. @, Q(U) 6 U U. !" I. 6 ( " I ( F , q T(I F q) = inf fT(U) : U 2 U (I F ) Q(U) 6 qg U (I F ) | ( * 6 ( F , +* I. C T (I F ) = minfT (U F q) : q 2 Ng ( " I ( F . 6

!! , + !" 6 6. 4 ! " " )* , " !

*

!+* ", * $1], " !+ . !" U | 6, y | " Y . C LU (y) ( " U,

! " y.

1. U " I = hX V i # ,

y 2 V , , O(y ) 6= ?, LU (y) 6= ? W ' = y , y 2 V , , O(y ) = ?, 2L (y) W ' = 0. L (y) = ?, U

U

2LU (y)

2. # I = hX V ri | ", F | , & 1, , U (I F ) 6= ?, T(I F ) >

X

y2V

P(O(y )):

128

. .

/

! 1 3 $1]. - !+! , ! *

, . !" Y | ( , Y . !" X = Y | ( . ( , V = fy1 : : : yk g Y . !" " , . . xy () x = y: )* ! I = hX V i ! " * , . !" X hX Pi. !" gm1 (x) | ", gm1 (x) = i x 2 Xi (i = 1 m)

(1)

X1 X2 : : : Xm | ( X (. . X = X1 X2 Xm Xi \ Xj = ? i 6= j) , P(Xi ) 6 c=m (i = 1 m) c = const, + m. !" x a ga2 (x) = 1 (2) 2 ! a 2 X fa (x) = !"

0 x 6= a 1 x = a a 2 X:

G1 = fgm1 (x) : m 2 Ng G2 = fga2 (x) : a 2 X g F = ffa (x) : a 2 X g F = hF G1 G2i: !" N0 = N f0g | ( * "* . !" 8 0 l = 0 < L1 (l) = : ] log l$+1 l = 1 2 3 | logl + 2 l > 4

! , ( N0 . H !+ .

(3) (4) (5) (6) (7) (8)

3. # I = hX V i | ( ), jV j

= k,

F

| , (1){(7). #

129

s(k m) = 2 k. # L(l) = L1 (l) L1 (l) | , (8) /

1 < T (I F s(k m)) 6 mc k ; mk m L mk + 1 + k + 1: + m ; k + mk m L m

0 , c0 = max(c 1)

1 < T(I F s(k $c0 k])) < 2 T (I F ) 1 k ! 1:

. 0 * ( :) X V. - !+! ! . 5 ! , ( Y , ( X = Y . !" V = fy1 : : : yk g Y: @ X V

xy () (y 2 V )&(x y)&(:(9y0 )((y0 2 V )&(x y0 )&(y0 y))) . . xy, y 2 V , ( x. )* ! I = hX V i . !" ( F

= h? G1 G2i

G1 G2

(1), (2), (4), (5). ( 06l63 : L2 (l) = llog(l + 1) + 1 l > 3

(9) (10)

4. # I = hX V i | , jV j = k. #

| , (9), s(k m) = 2k + m, L(l) = L2 (l), L2 (l) | , (10). / -

F

, 3.

7 , (! ! " , . .

xy () (y 2 V )&(y x)&(:(9y0 )((y0 2 V )&(y0 x)&(y y0 ))): I = hX V i .

130

. .

- !+! , n- ! ! " . !" Y = $0 1]n V = fy~1 : : : y~k g Y (11) X = fx~ = (u1 v1 : : : un vn) : 0 6 ui 6 vi 6 1 i = 1 ng | ( . !" ( X hX Pi P ! p(x). @ X Y !+

(u1 v1 : : : un vn)(y1 : : : yn ) () ui 6 yi 6 vi i = 1 n:

(12)

1 (u1 v1 : : : un vn) = max(1 ]ui m$) : i 2 f1 ng m 2 Ng G1 = fgim

(13)

2 (u1 v1 : : : un vn) = max(1 ]vi m$) : i 2 f1 n ; 1g m 2 Ng G2 = fgim

(14)

1 u 6 a i 2 ui > a : i 2 f1 ng a 2 $0 1]g

(15)

!"

3 (u1 v1 : : : un vn) = G3 = fgia 4 (u v : : : u v ) = G4 = fgia 1 1 n n

1 v 6 a i 2 vi > a : i 2 f1 n ; 1g a 2 $0 1]g: (16)

@ Mab = fx~ = (u1 v1 : : : un vn) 2 X : un 6 b vn > ag: !"

F1 = ffab : Nf = Mab 0 6 a 6 b 6 1g

(17)

F2 = f:f0a : a 2 $0 1] f0a 2 F1 g

(18)

ab

G5 = fga5 (u1 v1 : : : un vn ) = !"

F

1 u 6 v < u + a n n n 2 ! : a 2 $0 1]g

= hF1 F2 G1 G2 G3 G4 G5i:

(19) (20)

131

5. # " I = hX V i | n- # ,

(11){(12), F | , (13){(20), n > 1. #

R(I) =

X

y 2V

P(O(y )):

/ p(x) 6 c,

R(I) < T(I F (4 k + 2 + (1 + 6 $logk]) c0 ) (k (k + 1)=2)n;1) 6 R(I) + 4 n + 1 c0 = max(1 c=2n;1):

/ $3].

3 ! " #$

0 ) ! * , " 3. 6 ( " !(.

2. # L1(l) L2(l) | , , k m 2 N m m X X rj (k m) = maxf Lj (li ) : l1 2 N0 : : : lm 2 N0 li = kg: i=1

/

k

rj (k m) = k ; m m

i=1

Lj

k

k k m + 1 + m m Lj m :

5 " : ?

" ! L1 (l) ("! ! " , !+ : 0 6 x 6 4 L1 (x) = x logx + 2 x > 4 ! L2 (l) | !+ :

x

0 6 x 6 3 log(x + 1) + 1 x > 3 ! ! !! ! . ,

+ , ! " )* ! , !" . : " ! . L2 (x) =

132

. .

( , , . .

k ; mk m Lj mk + 1 + i=1 (21) + mk m Lj mk (j = 1 2) li0 (i = 1 m) !+! 2 , " * " !*. : + ( ", l10 ; l20 > 2. !" 0 0 0 0 l100 = l1 +2 l2 l200 = l1 +2 l2 : rj (k m) =

m X

Lj (li0 ) >

l100 + l200 = l10 + l20 l100 ; l200 6 1. H ! Lj (x) !, ! !* ! Lj (l100 ) + Lj (l200 ) 6 Lj (l10 ) + Lj (l20 ) j = 1 2: (22) ? (22) , ! , m X Lj (li0 ) | " 3 ( , li00 = li0 (i = 3 m) i=1 ! m X i=1

Lj (li00 ) =

m X i=1

Lj (li0 ) = rj (k m) j = 1 2:

? li00 (i = 1 m) , " * 1, ! (21), ( ",

! , ! " * , ! (22), m X ! l1(n) : : : lm(n) Lj (li(n) ) = rj (k m) j = 1 2 i=1 , 1. H ! (21), " ". "! 3. " Um0 , +! , !+ . 0 " ! 0 , Um0 . 0! 0 m , 1 m, , 0 " gm1 (x). !" Vi = Xi \ V li = jVi j i = 1 m: ; i i . 5 * * i, Vi 6= ?, !+! !!. 0! i li , ] logli $, . ? ) " ! , * , !+

! !, ! ! ( ",

133

" ), , ! )* , ! . ! Di . @, ) " ) Vi . (9 , ! ! ! !, \ ", \ " !( ) ! .) !" " ! Di . @ V ( ,

!+* " , !+ . !" 0 , ! ( , ) . !" y = ymax y: 2V 0

@, ! Di , * * , !+ ! , ( ! !, * + !, 1, ! | 2, " gy2 (x): 0 Di , * + ", , ( ! ! !, !+ ! " y, fy (x). !! " jV j = k " Um0 . / " ( F . ( , Um0 ! I = hX V i: 0 " " . !" ! " y 2 V . 5 ", ' (x) = fy (x). H fy (x), N' (x) Nf (x) = fyg: ( , ' (y) = 1. !" y 2 Vi (i 2 f1 mg) . . ( ! Di . 0 ", + ] logli $+1 , ) , , ". H gm2 (y) = i ) " i- , * + , " 1. fy (x), " fy (y) = 1: ( , " "* 1. 0 " " )* ( 0 ). ? ( 0 ) , * + , y 2 V , , gy2 (y)=1,

y

0

y 6 y = ymax y0 : 2V 0

0

? ( 0 ) , * + , y > y gy2 (y) = 2, " ( 0 ) * !* 1. H , , ' (x) = fy (x). O " " ! 1, ! , " Um0 ! I = hX V i: @, Um0 m X Q(Um0 ) 6 m + (2 li ; 1) = 2 k:

i=1

( " ! Um0 .

134

. .

- " x 2 Xi . li = 0, Di | ! T(U0 x) = 1. ? li > 0, , , ) U0m ( , ' (x) = 1) ! ! ", !+! ", " " , ! " ) , . ) , ] logli $ "* , * !* . H , T(U0m ) 6 2+] logli $6 1 + L1 (li ) L1 (l) | ! ,

(8). , ! ! 2, T (U0m )

= = =

Z

M T(U0m x) = T (U0m x) P(dx) = X m Z m X X T(U0m x) P(dx) 6 (1 + L1 (li )) P(Xi ) = i=1 Xi i=1 m m X X 1 + L1 (li ) P(Xi ) 6 mc L1 (li ) 6 1 + r(k m) mc i=1 i=1

=

= mc k ; mk m L mk + 1 + + m ; k + mk m L mk + 1:

!" c0 = max(c 1). 0 " m = $c0 k]. H m > k. ? m = k, k=m = 1

r(m k) = 0 L1 (2) + k L1(1) = k L1 (1): ? m > k, k=m = 0 r(m k) = k L1 (1) + (m ; k) L1 (0) = k L1(1): 6 " ,

T(I F ) 6 T(U0m ) 6 1 + $c c k] k L1 (1) < 2: 0

? " m = k (k), (k) ! 1 k) ! 1 (k) > 1 k, k T(I F ) 6 1 + k c (k) 1: 6 ! , T(I F ) > 1, ( * " * , " ! k > 0. C " ". * , .

135

1. - !+! . !" Y = $0 1] X = $0 1] V = fy1 : : : yk g Y: @ " , . . xy () x = y: !" X hX Pi P " p(x). !" X1 = fx 2 X : 0 6 x 6 1=mg:Xi = fx 2 X : i;m1 < x 6 x mi g i = 2 m (23) H " gm1 (x) = max(1 ]x m$) !

(1). ? p(x) 6 c = const, P(Xi ) 6 c=m, * ! * 3.

" , " 3, " ) . - " $0 1] m *

(23). ;( ( ( V , + , (+* ) . H" " - x V , !, ! !+ . @ ! " , ( x. ? min($x m] + 1 m). H" (, , !+ * . 6 2, " ! !, ( V m . ? m " m = k, !, (! "

V , ! " "! ( ". 7 " ! , ( + 1 ( 1 , 2 ( !( ) ( " ! * , . 2. !" X = Y = f1 : : : N g. @ " . !" hX Pi | X, | ( * (, P ! " ( X, . . x 2 X P(x) = 1=N: !" m 2 N3 r = N ; m $N=m]3 X1 : : : Xm | (, Xi = fx 2 X : 1 + (i ; 1) ($N=m] + 1) 6 x 6 i ($N=m] + 1)g i = 1 r : r ($N=m] + 1) + 1 + (i ; 1 ; r) $N=m] 6 x 6 6 r ($N=m] + 1) + (i ; r) $N=m]g i = r + 1 m gm1 (x) = i, x 2 Xi i 2 f1 mg: H P(Xi ) 6 ($N=m] + 1)=N < 2=m

" * ! * 3 c = 2. Xi =

fx 2 X

136

. .

4 & #

0 " 4. " Um1 " Um0 . 0 " ! 0 , . 0! 0 m , 1 m, , 0 " gm1 (x). !" Vi = Xi \ V li = jVi j i = 1 m: ; i i . 5 * * i, Vi 6= ?, !+! !!. 0! i Di li + 1 ] log(li + 1)$. @, ) Di , ( ), " ) Vi . 5 " ! Di V y ,

(, Um0 . @, ! Di ( ! ! !, * + !, 1, ! | 2, " gy2 (x): !" i 2 f1 mg. @ j(i) , j(i) > i, jVj (i)j > 0 !+! j 0 : jVj j > 0 j 0 > i j 0 < j(i), . . j(i) ( *! ! ( Vj (i). ? ( , j(i) = 0. H" ( Di ! ! ! ! ( Dj (i), j(i) 6= 0. 5 ( i, li = 0, ! i ( Dj (i), j(i) 6= 0. ! " ! " Um1 . 5 " , " Um1 ! ! I = hX V i, "!, " Um0 ! * , . @, Um1

0

Q(Um1 ) = m +

m X i=1

(2 (li + 1) ; 2) = m = +2

m X i=1

li = m + 2 k:

( " Um1 . - " x 2 X. li = 0 T (Um1 x) = 1. ? li > 0, x Um1 ! ! ", !+! " ( Di . ) 1+] log(li +1)$ "* , T(Um1 x) 6 1+] log(li + 1)$: @ !, T(Um1 x) 6 1 + L2 (li ):

137

H 2 3 !( 4. C " ".

5

5 " 5 ). 6 !, . . n = 1, ! ! (n > 1). 0 ! ( ( " ( F = hF1 F2 G1 G3 G5 i . . ( G2 G4 !( " . 0 ! Y = $0 1], V " ( $0 1]. !" V = fy1 y2 : : : yk g, y1 6 y2 6 6 yk , . . V | ( , ! ! * ) . 4( ! 2. 6, * " . 0 " !, , 0 . 0! , * ! " , ! . ; 1 , | 2 . !" m , (. " g15=m (u v) ( G5. 4 , n = 1, ) ! ! *. F ! ! 1, ! | 2. 0! 1 D, " Di " 3. 7 , D ! " !+ . 6 " k ( , k = jV j), ] logk$, . ? ) " ! , * , !+ ! !, ! | ! ( ", " ), , ! )* , ! . ! D. @, ) " ! ! * . 4 , , , ! ! ! !, \ ", \ " !( ) ! . @ i- i ! " yi . !" | " ! D. ; 3,

V ( ,

!+* " , !+ . !" 0 | , ! ( , ) . !" y = ymax y: 2V 0

@, ! D, * * , !+ ! , (

138

. .

! !, * + !, 1, ! | 2, " gy3 (x) ( G3. 0 D, * + ", , ( ! ! !, !+ ! " y, fyy (x) ( F1 . 5 ! " ( ( "* D " " D, ( * | " D. H" ( i (i = 1 k ; 1) ! , !+ i+1, ! fy +1 y +1 2 F1. / ( (k ; 1)- ". H" 2 ( , ) , ! , * + ) ! m ; 1 , 1 m ; 1,

, 2 " g11m 2 G1 ( , m | ). @ , 2 ! m ; 1 , * " g11m ( " m . ; , * + 2 + i, i0 . 0 ( S = fs1 : : : sm;1 g, , si | V , ys | ( " i=m (i = 1 m ; 1), !+!, si = 0. 0 " k * , , * " 01 : : : 0k . ;( ! ! i (i = 1 k) " yi ( ( " yi ! ! " i 0i ). ( 0i (i = 2 k) ! , !+ 0i;1, ! fy 1 y 1 2 F1. / ( (k ; 1)- ". H" ( i0 !+ . ? si 6= 0, i0 ! , !+ 0s , ! fy y 2 F1. ? si < k, i0 ! , !+ s +1 , ! fy +1 y +1 2 F1. / ( , * +* i0 (i = 1 m ; 1), ". !! 6 U0 . ( , " U0 ! ! " I = hX V i. ;( " yi 2 V

! ! " U0 , . . LU0 (yi ) = f i 0ig. H O(y ) = Nf ( i 0i ! " , fy y , N' _' O(yi ):

i

i

i

i;

i;

si

i

si

i

si

si

yy

i

i

i

0 i

H , 1, ", 8yi 2 V N' _' O(yi ) i

0 i

, ( , ", 8x 2 O(yi ) ' (x) = 1, ' (x) = 1, . . i, 0i !+! + ". i

0

i

139

@ Aa = fx = (u v) : u 6 v 6 u + ag: 0 " "! " yi 2 V . - !, x = (u v) 2 A1=m \ O(yi ). / , v ; u < 1=m u 6 yi 6 v. ( , i !+! + ". 0 ! g15=m (x) = 1 " (0 1 ) ! 1. @ , " (0 2 ) ! 0. 4! ", 3 4 D (!+ 1 ) !+! + ", !+ 1 j , , " yj , ( u V , (+ ! $u v] ( !+!, yi 2 $u v]). H , : u 6 yj 6 yi 6 v. @ !, " , !+ yj yi , +. H , + x , !+ i . @ (, ' (x) = 0, 0i ( " " (0 2 ), " x, !( , 0. - " !, x = (u v) 2 (X nA1=m ) \ O(yi ), . . v ; u > 1=m u 6 yi 6 v. 0 ) ! g15=m (x) = 2, " (0 1 ) ! 0, (0 2 ) 1. !" j 2 f1 m ; 1g , j=m | ( u . F ", g11m (x) = j. H v ; u > 1=m, j=m ( ! $u v]. - !. 1) yi 6 j=m. H u 6 yi 6 ys 6 v, ys | ( j=m " V . @ !, " , !+ j0 0s 1. @ " ", " , !+ 0s 0i, ( ! 1, 0 6 yi 6 ys 6 v. @ (, ) ! ' (x) = 0, ! ! s +1 , ) * i. 2) yi > j=m. H u 6 ys +1 6 yi 6 v. @ !, " , !+ j0 s +1 , 1, " , !+ s +1 i, ( 1 x. 7 !+ ! ! ' (x) = 0. H , 8yi 2 V 8x 2 X : xyi U0 !+! + x ", !+ - ( ) " i 0i. / , " U0 ! I. " ( " U0 . 0

i

j

j

j

j

j

i

j

j

j

j

0

i

140

. .

- " x 2 A1=m . 0 ) ! T(U0 x) 6 1 + (] logk$;1) + 2 + jJ (x)j: "

! g15=m 0 . 0 , * + ! +! ", +! "! " D. H"

! !* ,

!+* D, !+ , ! + ". C

! ,

!+* , * + ", * * ( ! (! "). - " !, x 2 X nA1=m . H T (U0 x) 6 1 + 1 + 2 + jJ (x)j: "

! g15=m , | g11m 2 , " | !* , * , * + j0 , ! + 2 . , , , ,

! ,

!+* , * + ", * * . ; , yi , , ! " i 0i ! " ! 1, " , (

!

. H" ( " ( " U0 . T (U0 ) = M T (U0 x) = P(A1=m ) (2+] logk$) + + P(X nA1=m ) 4 + M jJ (x)j 6 X 6 P(A1=m ) ($logk] ; 1) + 4 + P(O(y )) 6

y2V X 6 c ($logk] ; 1) m ; m12 + 4 + P(O(y )) 6 y 2V X 6 2c ($logmk] ; 1) + 4 + P(O(y )):

2

y2V

X

"! " M jJ (x)j = P(O(y )) ! . y2V , U0 . Q(U0 ) 6 2 + (2k ; 1) + (k ; 1) + (k ; 1) + m + 2m: "

! , * + 0 . 0 " D. H"

! * .

141

| ) , * + 2 . , , ", , * +* i0 (i = 1 m). 0 " m = 2 c $log k] ! T (U0 ) 6 5 +

X

y 2V

P(O(y ))

Q(U0 ) 6 4 k ; 1 + 6 c$logk] !( 5 ! n = 1. 5

" , . !" ( V = fy1 : : : yk g, ( " . 6 ! . ? *! c ! , m " m = 2 c $logk], ( c , ( " , , c = 2. V ( S = fs1 : : : sm;1 g, . @ , " (. H" " ! !- ! x = (u v) !+ . 6 x. ? ", 1=m, ( V * * ( u ". 4 , V | v * , " " v. H ) !, , log k . ? v ; u > 1=m, +" ! g11m ! j j=m, + $u v]. H", sj , V | u. ; " " ( " u, , sj + 1, V | v * , " " v. H

, ) ! , 4 * ( v ; u 1=m, ! g11m , 1 , , 1 , ). @ " ", m , ( " ! 1, *! ! , , ) . , , , ! "! " log k, *" ( S.

6 ) " 5 n > 1. @ x~ = (u1 v1 : : : un vn) z~i = (ui vi) X1 = f(u v) : 0 6 u 6 v 6 1g

142

. .

pi (u v) =

Z | X1

Z X1}

{z

n;1

p(~x) d~z1 d~zi;1 d~zi+1 d~zn

p1i (u) = p2i (v) = F ", pi (u v) 6 c

Z

Z1 Z uv 0

pi (u v) dv pi (u v) du :

Z

d~z1 d~zi;1 d~zi+1 d~zn = 2nc;1 X1

| X1 {z } n;1

p1i (u) 6 2nc;1 (1 ; u) 6 2nc;1 p2i (v) 6 2nc;1 v 6 2nc;1 : -" ! ! ( . ( , ( n- " (n ; 1)- . !" S V , 1 6 i1 < < il 6 n: @ Pi1 :::i (S) = f(yi1 : : : yi ) : (y1 : : : yn) 2 S g l

l

( S i1 : : : il . @ W i = f(y0 y00 ) : y0 y00 2 Pi (V ) y0 6 y00 g i = 1 k: @ jW ij 6 k (k + 1)=2. @ Z i = f(y11 y21 : : : y1i y2i ) : y1j y2j 2 Pi (V ) y1j 6 y2j j = 1 ig i = 1 k:

5 ( (y0 y00) 2 W i (

Syi y = fy~ = (y1 : : : yn) 2 V : y0 6 yi 6 y00 g: 0

00

@ V 1 = V M 1 = P1 (V 1 ) My1 = fy0 2 V 1 : y0 > yg V i (y11 y21 : : : y1i;1 y2i;1 ) =

i\ ;1

Syj y i = 2 n

j j j =1 1 2

143

M i (y11 y21 : : : y1i;1 y2i;1) = Pi (V i(y11 y21 : : : y1i;1 y2i;1 )) i = 2 n Myi (y11 y21 : : : y1i;1 y2i;1 ) = fy0 2 M i (y11 y21 : : : y1i;1 y2i;1) : y0 > yg i = 2 n: H n- ! ! " ( " !+ . !" x~ = (u1 v1 : : : un vn) 2 X | " . 6 ! ! (y0 y00) W 1 , y0 | ( u1 ( M 1 , y00 | ( v1 My1 . ? , x~ !, ( ", ( x~0 = (u2 v2 : : : un vn ) (n ; 1)- " ( Py2 :::y (V 2(y0 y00 )), ( V 2(y0 y00) ( y~ = (y1 : : : yn) 2 V , u1 6 y1 6 v1. H n ; 1 " , . @ 6, ! I ( . 6 6 Um1 , +! ! ! ( M 1 , " 4, m " m = $c0 k]. H Q(Um1 ) = 2k + $c0 k] T (Um1 ) < 2. Um1 " gm1 g11m , gy2 g13y . 0 " " Um1 . !" !

! y 2 M 1 . @, ! ! ! ! ! y. " Um1y , +! ! ! ( My1 ( , My1 6= ?), m " m = $c0 jMy1j]. H Q(Um1y ) = 2 jMy1j + $c0 jMy1j] 6 jMy1j (2 + c0 ) T(Um1y ) < 2. Um1y " gm1 g12m, gy2 g14y . H" ( " Um1y , . . " Um1y " ! , ! ! . , , ( 0 Um1y ! ! 0 ! y0 ! (y y0 ). /! ! ! " ( V 2 (y y0 ). ! ( Um1 . !! " U1 . @ k(k + 1)=2 ",

Z 1 . / " x~ = (u1 v1 : : : un vn) 2 X * " ! ! (y0 y00 ) 2 Z 1 , y0 | ( u1 M 1 , y00 | ( v1 My1 , , , !+!, . . ! * " ( V 2(y0 y00) V , ! (12). H Um1 " 1 !", 0

n

0

T(U 1) = T(Um1 ) + max T(Um1y ) 6 2 + 2 = 4: y

144

. .

F ", Q(U 1 ) 6 (2 + c0 )k + (2 + c0 )

X y 2V 1

jMy1j = k(k + 3)(2 + c0)=2:

- " U 1. !" !

! (y z) 2 W 1 . @ , V 2(y z) ! . H" ( , !+ ( V 1 " U 1, " ( V 2(y z). ) g11m g13y g12m g14y g21m g23y g22m g24y . ( ! ! " ) , ! (y0 y00), ! (y z y0 y00 ). 2 . @ !! " Uyz H" , ! ! ! ! ! (y z) 2 , . . " U 2 ! . ( Uyz yz ! ( U 1 . !! " U 2 . 2 6" U 2 (k(k + 1)=2) ",

! ( Z 2 , ( (y11 y21 y12 y22 ) ( V 3(y11 y21 y12 y22 ). F ", T(U 2 ) 6 8

1 Q(U 2 ) 6 Q(U 0 ) + (2 +0c0)

11 X X 2 1 1 2 1 1 My (y1 y2 )AA 6 @M (y1 y2 ) + @ 1 1 1 1 1 2 (y1 y2 )2Z y2M (y1 y2 ) ! k(k + 1) k(k + 1) k(k + 1) 2

6 (2 + c0 ) k +

+ (2 + c0) k 2 + 2 k(k + 1) 2! k(k + 1) 6 (2 + c0 ) : 2 (k + 2) + 2

2

6

5 U 2 ! " U 3 . . 4 (n ; 1)- ! " U n;1, ! " (k(k + 1)=2)n;1 ",

! ( Z n;1. / " x~ = (u1 v1 : : : un vn) * " (y11 y21 : : : y1n;1 y2n;1) 2 Z n;1, y1i | ( ui M i (y11 y21 : : : y1i;1 y2i;1), y2i | ( vi My11 (y11 y21 : : : y1i;1 y2i;1) (i = 1 n ; 1), (, , !+!. ? ( , ! !, . . , * ". ? ", ( (y11 y21 : : : y1n;1 y2n;1 ) ( V n (y11 y21 : : : y1n;1 y2n;1), " U n;1 * " x~ ( ( V , ! (n ; 1)- (12). i

145

F ", T (U n;1) 6 4(n ; 1) Q(U n;1)

6

n;2 k(k + 1) n;1! k(k + 1) k + 6 2 2 n;2 n;1!

Q(U n;2) + (2 + c0)

6 (2 + c0 ) (k + 2) k(k 2+ 1)

+ k(k 2+ 1)

:

H" " " . - " Un;1 . !" ! (y11 y21 : : : y1n;1 y2n;1), ( V n (y11 y21 : : : y1n;1 y2n;1 ). ) ( " U , +! ! ! "

yn . /! " !, ! !+ . ?

" l = jV n(y11 y21 : : : y1n;1 y2n;1 )j ) 4 l ; 1 + 3 2 $log l] c=2n;1 6 4 k ; 1 + 6 c $logk]=2n;1: @ (, T(U ) 6 R(I) + 5: @, " ! , ! ! ! " U . )! ( Un;1 !! " Un. H Un;1 (k (k + 1)=2)n;1 ", Q(Un) 6 (4 k + 2 + (1 + 6 $logk]) c0) (k (k + 1)=2)n;1: F ", ! " Un ! I, x~ = (u1 v1 : : : un vn ) +" U n;1 * ( ( V , ! (n ; 1)- (12), +" , !+ U n;1,

!+ ) ! (!, ) ( , ! + (12). (, x~ !+! " !", !+ U n;1, ) , x~ ! " U , T(U n) 6 4 (n ; 1) + 5 + R(I) = 4 n + 1 + R(I): 7 " ! 2 T (I F ) > R(I) " .

146

. .

* 1] . . // . | 1991. | ". 3, %. 2. | '. 69{76. 2] . . +% % , , -./ , 0 // 12 . | " +: 40- " 2 , 1990. | '. 11{17. 3] . .,

6/ 7. 8. % ,

n-

0 -

+ // 9% % / ( "0% X ;2 ). | ' : 40- ' 2 , 1993. | '. 48{49.

& ': 1995.

CSL- . . ,

, , , H . \" " A ( H A- ) CSL-, , H , & " L , ". ' CSL-, .

Abstract

Ju. O. Golovin, Property of the spatial projectivity in the class of CSL-algebras with atomic commutant, Fundamentalnaya i prikladnaya matematika 1(1995), 147{159.

This work continues to study spatial homological properties of, generally speaking, nonselfadjoint, re.exive operator algebras in a Hilbert space H . A \lattice" criterion of spatial projectivity of an algebra A (i.e. the projectivity of H as left Banach A-module) is obtained in the class of indecomposable CSL-algebras: the existence of immediate predesessor of H as element of the lattice of invariant subspaces. Also, the direct product of indecomposable CSL-algebras A , 2 0, is a spatial projective algebra i1 the algebra A is spatial projective for every .

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. Y P 0 ZP 0 f : T ! R, , f jU 2 C(U) U 2 U 0 . 7 c=d 2 Qcl , fc dg C d | C. . f 2 Y P 0, , f(t) c(t)=d(t) 1 t 2 V coz d f(t) 0 t. 6 1 7! f8 Qcl Y 0 Y P 0=R0 Z 0 . 7 , ! 1 . 7 k k 6 m=n, m1 + n = "21 m1 ; n = "22 ! "1 "2 Qcl . 7 "1 = a1 =b1 "2 = a2 =b2 C. 6 (md + nc)b21 = = a21d (md ; nc)b22 = a22 d. = 1 t 2 V \ coz b1 \ coz b2 m=n + c(t)=d(t) = a21(t) m=n ; c(t)=d(t) = a22 (t). > , jf(t)j 6 m=n. N , kf8k 6 m=n. 41, kf8k < m=n. 6 5 W 2 U 0 , , jf(t)j < m=n 1 t 2 W \ V . 4 m=n ; c(t)=d(t) > 0 m=n + c(t)=d(t) > 0 (md(t) + nc(t))d(t) > 0 (md(t) ; nc(t))d(t) > 0. W \ V ! (md+nc)d = = a21 (md ; nc)d = a22 a1 a2 C. = 1 ! m1 + n = (a1=d)2 m1 + n = (a2 =d)2. G, k k 6 m=n. 4 , 1 , Y 0 Z 0 Z 0 Z 0 . 7 f8 2 Z 0 . 6 f 5 fGnk j kg T , , Un fGnk j kg 2 U 0 !(f Gnk ) < 1=n 1 k. G n. 7 Gnk = coz gk gk 2 C , 0 < fk 6 1. . Cki gk 1(]1=(i + 1) 1=(i ; 1)#) Dki gk 1 (]1=(i + 2) 1=(i ; 2)#). . gki ((gk ; 1=(i + 2)) _ 0) _ ((1=(i ; 2) ; gk ) _ 0). 6 gki(t) > 1=(i + 1)(i + 2) 1 t 2 Cki. . fki ((i + 1)(i + 2)gki) ^ 1. F, coz fki = Dki fki(Cki) = f1g. 7 xki inf ff(t) j t 2 Dki g. 4 fn T , fn (t) supfxkifki(t) j i kg 1 t 2 Un fn (t) 0 t 62 Un . O Dki \ Dkj 6= ?, ji ; j j 6 5. > , fDki j k ig Un . 7! fn Un . O t 2 Un , t 2 Cki , , f(t) > fn(t) > xki > f(t) ; 1=n. N , kf8 ; f8n k 6 1=n. 6 1 , Y 0 Z 0 . = , ff8n g Z 0, 8 kfn ; f8m k < 1=m n > m. 6 5 fUm g U 0, , supfjfn(t) ; fm (t)j j t 2 Um g < 1=m n > m. 7 5 1 5 gn 2 Y P 0 Vn 2 U 0, , gnjVn 2 C(Vn ) supfjfn(t) ; gn(t)j j t 2 Vn g 6 1=n. 9 , Wm Um \ Vm 1 . 7 t 2 Wm . 6 jgn(t) ; gm (t)j < 3=m n > m. 4 f T, f(t) 0 1 t 62 U1 , f(t) gm (t) 1 t 2 Wm n Wm+1 f(t) limgm (t) 1 t 2 \Wm . 7 t 2 Wm . O t 2 Wm+i n Wm+i+1 , jf(t) ; gm (t)j < 3=m. O t 2 \Wm , jf(t) ; gm (t)j < 4=m. 4 , f 2 ZP 0. , kf8 ; f8m k 6 5=m, 1 . 7 . > 5 ! - C Q8 cl .

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, , f jU 2 C(U). 6 f 2 ZP . . ! p f8 2 A. 7 U = coz c 0 < c 2 C. . E hc8i A. 7 q g8 2 A qE AD . 7 , q 2= AD , TD \ cozm g 6 inI m. B , TF n coz2m g 2 I F D. F, TF \ cozl c 6= ? l. 7, I . 7 TG n coz2l c 2 I G F. 6 1 , TG n (coz2m g \ coz2l c) 2 I . 7 TD \ cozk (gc) 2 I 1 k. > , TG \ coz2m g \ coz2l c 2 I . G, TG 2 I , . > , E r- . . ' 2 homA (E A), , 'e ep. 6 cf 2 C, 'c 2 uC. F, p '. 7 q g8 2 A q'E AD . 7 , qp 6 inAD , TD \ cozm (gf) 62 I m. 7 TF n coz2m (gf) 2 I F D. 1 , TG n coz2l c 2 I l G F . 6 1 , TG n (coz2m (gf) \ coz2l c) 2 I . 7 TD \ cozk (gfc) 2 I 1 k. 7! TG \ coz2m (gf) \ coz2l c 2 I . G, TG 2 I , . > , qA AD . 6 1 , r- '. 7! p 2 Z oc (uC). 7 p f8 2 A. . f Un fn 5 . ! pn f8n . 6

1.2.2 , p c- A, Z oc (uC). 6 1 , Z 0 - crb - Z oc . = , A aZ oc - . 7 E hfei gi | - r- A, ei f8i , ' 2 homA (E A) j'ej 6 z jej. . ! 'ei g8i . 1 , fi > 0. = fi gi 5 {in fXink j k 2 King in fYinl j l 2 Lin g, , !(f Xink ) < 1=n !(g Yinl ) < 1=n. . xink inf ffi(t) j t 2 Xink g, xink supffi (t) j t 2 Xink g, yinl inf fgi(t) j t 2 Yinl g yinl supfgi(t) j t 2 Yinl g. . 1 f~in fxink xink Xink j kg g~in fyinl yinl Yinl j lg. 7 Xink \ Xink1 6= ? Xink \ Xink2 6= ?. O t 2 Xink \ Xink1 , 0 6 fi (t) ; xink < 1=n, 0 6 xink ; fi (t) < 1=n, 0 6 fi (t) ; xink1 < 1=n 0 6 xink1 ; fi (t) < 1=n jxink ; xink1 j < 2=n jxink ; xink1 j < 2=n. 3, jxink2 ; xink j < 2=n jxink2 ; xinkj < 2=n. 7! jxink1 ; xink2 j < 5=n 0

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x y. . Zim cozm fi . 6 1 m, 1 n > m 1 k, , Xink \ Zim=2 6= ?, xink > fi (s) ; 1=n > > 2=m ; 1=n > 1=m, s 2 Xink \ Zim=2 . O t 2 Xink , fi (t) > fi (s) ; 1=n > 1=m. 7! m n > m 1 h~ imn fzinkl zinkl Zinkl j (k l) 2 Mimn g, zinkl yinl =xink, zinkl yinl =xink, Zinkl Xink \ Yinl Mimn f(k l) 2 Kin Lin j Zinkl \ Zim=2 6= ?g, Z~imn fZinkl j (k l) 2 Mimn g Zim him : Z~imn ! R, , him (t) gi (t)=fi (t). = (k l) 2 Mimn jzinkl ; zinkl j 6 (jyinl xink ; yinl xinkj + jyinl xink ; yinl xinkj)=xinkxink 6 6 m22(jyinljjxink ; xinkj + jxinkjjyinl ; yinl j) 6 6 m2(zkfik=n + kfik=n) = = m kfi k(z + 1)=n imn 0

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kfi k supffi (t) j t 2 T g. F, imn ! 0 n ! 1 1 i m. 7 n > m t 2 Zinkl (k l) 2 Mimn . 6

6

jhim (t) ; zinkl j m2 (jgi(t)xink ; gi (t)fi (t)j + jgi(t)fi (t) ; yinl fi (t)j) < < m2 (z kfi k=n + kfi k=n) = imn 0

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3 jhim(t) ; zinkl j < imn . H !, s 2 Zinkl , jhim (t) ; ; him (s)j 6 jhim(t) ; zinkl j + jzinkl ; him (s)j < 2imn . 7 Xink = Oink Rink, Yinl = Pinl Sinl . 7! Zinkl = = Qinkl Tinkl , Qinkl Oink \ Pinl . = 1, 5 r 2 N, i m, n = n(r i m) > m, , imn < 1=r. = ! n Zinkl Mimn zinkl zinkl 1 Zirkl , Mimr , zirkl zirkl . 6 ei1 'ei2 = ei2 'ei1 , 5 fUi1 i2 j 2 Ng U 0 , jfi1 (t)gi2 (t) ; fi2 (t)gi1 (t)j < 1= 1 t 2 Ui1 i2 . . 1 ~hr fzirkl zirkl Qirkl j (k l) 2 Mimr m ig. 7 (k1 l1 ) 2 2 Mi1 m1 r , (k2 l2 ) 2 Mi2 m2 r Q Qi1 rk1 l1 \ Qi2rk2 l2 6= ?. (m1 m2 ), , m1 m2 = < 1=r. 6 5 t 2 Q \ Ui1 i2 . 7! jhi1 m1 (t) ; hi2 m2 (t)j = jgi1 (t)fi2 (t) ; gi2 (t)fi1 (t)j=fi1 (t)fi2 (t) < m1 m2 = < 1=r jzi1 rk1 l1 ; zi2 rk2 l2 j 6 jzi1 rk1 l1 ; hi1 m1 (t)j + jhi1m1 (t) ; hi2m2 (t)j + jhi2 m2 (t) ; ; zi2 rk2 l2 j < 3=r. 3 . . Ur fQirkl j (k l) 2 Mimr m ig 2 T . 7 , Ur . 7 G coz c. . D 2 Jb , , cl D G. 6 E | r- , 5 i, , ei 62 AD . 7! 5 m, , TD \ Zim=2 62 I . n = n(r i m). 6 G \ Zirkl 62 I (k l) 2 Mimr G \ Qirkl 6= ? , , G \ Ur 6= ?. 4 h h U fUr j r 2 Ng, h (t) supfzirkl j t 2 Qirkl (k l) 2 Mimr g h (t) inf fzirkl j t 2 Qirkl (k l) 2 Mimr g. =

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7 t 2 Ur . 6 h (t) ; h (t) > inf fzi1 rk1 l1 ; zi2 rk2 l2 j t 2 Qi1 rk1 l1 \ \ Qi2 rk2 l2 (k1 l1) 2 Mi1 m1 r (k2 l2 ) 2 Mi2 m2 r g > ; 3=r. H , h (t) ; ; h (t) 6 fzi1 rk1 l1 ; zi2 rk2 l2 g < 3=r. > , h h mod I . 7 fs tg Qirkl (k l) 2 Mimr . 6 00

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h (t) ; h (s) 6 h (t) + 3=r ; h (s) 6 zirkl ; zirkl + 3=r < 4=r 0

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h (t) ; h (s) > h (t) ; h (s) ; 3=r > zirkl ; zirkl ; 3=r > ; 4=r . > , !(h Qirkl) < 4=r. 3 !(h Qirkl) < 4=r. B , h h ZP 0 . . ! p h8 p . = , ei1 = 'ei1 . 7 t 2 Qi2 rk2 l2 \ Ui1 i2 r (k2 l2) 2 Mi2 m2 r . 6 0

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jgi1 (t) ; fi1 (t)h (t)j = = jgi1 (t)fi2 (t) ; fi1 (t)fi2 (t)h (t)j=fi2 (t) (jgi1 (t)fi2 (t) ; gi2 (t)fi1 (t)j + jgi2 (t)fi1 (t) ; fi1 (t)fi2 (t)h (t)j)=fi2 (t) < < m2 =r + kfi1 kjgi2 (t) ; fi2 (t)h (t)j=fi2 (t) = = m2 =r + kfi1 kjhi2m2 (t) ; h (t)j = = m2 =r + kfi1 k(jhi2m2 (t) ; zi2 rk2 l2 j + + j inf fzi2 rk2 l2 ; zirkl j t 2 Qirkl (k l) 2 Mimr gj) m2 =r + kfi1 k(1=r + 3=r) = (m2 + 4kfi1 k)=r (i1 i2 m2 r): 0

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= 1, 5 u 2 N, i1 , i2 m2 , r r(u i1 i2 m2) , (i1 i2 m2 r) < 1=u. = ! r Qi2 rk2 l2 , Ui1 i2 r Mi2 m2 r 1 Qi2uk2 l2 , Ui1 i2u Mi2 m2 u . 6 t 2 Qi2uk2 l2 \ Ui1 i2 u jgi1 (t) ; fi1 (t)h (t)j < 1=u 1 (k2 l2) 2 Mi2 m2 u . . Vi1 u fQi2 uk2l2 \ Ui1 i2 u j (k2 l2) 2 Mi2 m2 u m2 i2 g. B , jgi1 (t) ; fi1 (t)h (t)j < 1=u 1 t 2 Vi1 u . 7 , Vi1 u . 7 G = coz c. . D 2 Jb, , cl D G. 6 E r- , 5 i2 , ei2 62 AD . > , 5 m2 , , TD \ Zi2 m2 =2 62 I . r = r(u i1 i2 m2 ) n = n(r i2 m2 ). 6 Zi2 nk2l2 \ G 62 I ! n (k2 l2) 2 Mi2 m2 n. N , G \ Qi2uk2 l2 62 I (k2 l2) 2 Mi2 m2 u , , G \ Qi2 uk2l2 \ Ui1 i2 u 6= ?. 6 1 , 5 1 , 'ei1 = ei1 p = ei1 1 i1 . G, '. 7 b = g8 2 A b'E AD . 7 , 5 , , TD \ coz (gh ) 62 I . 6 , 5 ? 6= F D, TF n coz2 (gh ) 2 I . 7! 5 U 2 U 0 , 5 . G, TF \ U coz2 (gh ) \ U coz2 (gh ). 7 TF \ U \ Qirkl 6= ? i, m (k l) 2 Mimr . > , TF \ Zim \ coz2 (gh ) 62 I , ! TF \ coz2m (gh fi ) 62 I . u , kgk=u < 1=4m. 6 TD \ coz2m (gfi h ) \ Viu 62 I . 7 t . 6 0

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gi (t) > fi (t)h (t) ; 1=u g(t)gi (t) > g(t)fi (t)h (t) ; g(t)=u > 1=2m ; kgk=u > > 1=4m, t 2 coz4m(ggi ). G, TD \ coz4m (ggi ) 62 I , b'ei 2 AD . > , . 6 1 , bp 2 AD bA AD , r- '. . . 6 fi = fi+ ; fi

fi+ fi ZP 0, ! e+i f8i+ ei f8i . 6 E = hfe+i ei gi, . 7 . 41 E^ crb- u^ : C A^ Z oc E~ ~ Z oc - crb - u~ : C A. 0

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. #21]. 41 Z oc (ûC) B. 7 a | ! B, 5 5 E^ huÊC i F^ hu^FC i A^ ^ A), ^ , '^ '^ 2 homA^(E ûEC u^C ^ â r- '. ^ 7 EC = ffi g FC = fgj g. . E huEC i F huFC i A Z 0 . 7 q g8 2 A q(E F ) AD . 7 , q 62 AD , TD \ cozm g 2 R0 m. 7 TL n coz2m g 2 R0 L D. . V fcoz fi j fi 2 EC g, W fcoz gj j gj 2 FC g U V W . 7 G coz c 2 I , R 2 Jb G \ U \ TR = ?. 6 1G \ 1 U \ HR = ? ^ A^R . 6 E^ F^ r- , u^c 2 A^R , , u^c(E^ F) coz(c ) \ HR = ?. 4 G \ TR = ?. > , U s- . 7! TL \ U 6= ? TL \ cozl (fi + gj ) 6= ? l. 6 R0 s-5 , R0 . 7 TM n coz2l (fi + gj ) 2 R0 M L. 6 1 , TM n (coz2m g \ coz2l (fi + gj )) 2 R0 . 7 TD \ cozk (g(fi + gj )) 2 R0 1 k. > , TM 2 R0 , . 7! E F r- . P 7 e pi1 :::ik f8i1 : : : f8ik | ! E. u^ 1 'û^fi 1 P hi. 6 '^ ûfi = aûfi , coz hi coz fi . . ! " pi1 :::ik u(fi2 : : :fik hi1 ) 2 E . 6 f8i " = euhi . 7 P pfji1h:::jj l=f8j1f:j:h:if,8jl. . ! e e = ! P pj1:::jl u(fj2 : : :fjl hj1 ). 6 f8i = euhi f8i (" ; ) = 0. 7! " ; ; 2 E \ E = f0g. 6 1 , ! ", ! e, e. B ! , ' 2 homA (E A), 'e ". 6 'f8i = uhi 2 uC. ;

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4 f : T ! R, f(t) hi (t)=fi (t) 1 t 2 Vi coz fi f(t) 0 1 t 62 V . . Wj coz gj . 7 , U . 6 5 D 2 Jb , cl D T n V . 6 TD \ Vi = ?, pi 2 AD 1 i. > , E AD . 3 F AD . B r- E F 18 2 AD , . G, . 7 f jU 2 C(U). > , f 2 ZP 0 . . ! p f.8 7 , a f mod R0s (H). 7 s 2 1V . 6 (ûfi )(s) 6= 0 i. 7! (ûfi )(s)(f )(s) = (ûfi a)(s). > , (f )(s) = a(s). 7 s 2 1W. 6 (ûgj )(s) 6= 0 j. 7! (ûgj )(s)(f )(s) = 0. 6 ^ '^, u^gj 'Ê^ = f0g u^gj a = 0. G, (f )(s) = a(s). 6 1 , ! s 2 1U. 6 ! s- , f a. G, a p. B , , , 1 ! a 2 B 5 ! p 2 A , a p. ^ 6 5 fam B g , 7 a 2 A. ja(s)am (s)j < 1=m 1 s 2 H. . ! pm am fm 2 pm . 6 , jam ; anj < < (2=m)1 jpm ;pn j < (2=m)18 1 n > m. c- A 5 ! p f8 2 A , jp ; pm j 6 (2=m)18. G, 5 fImn j ng R0 , jjf(t) ; fm (t)j _ (2=m) ; ; 2=mj < 1=n 1 t 62 Imn . 4 jf(t) ; fm (t)j < 1=n + 2=m. 7

4 5 fJmn j ng R0s (H), , jam (s) ; fm ( s)j < 1=n 1 t 62 Jmn . 7! 1 s 62 1Inn Jnn ja(s) ; f( s)j 6 ja(s) ; an(s)j + jan(s) ; fn ( s)j + jfn( s) ; f( s)j < 5=n. G, a p. 7 . ! 1. 2 crb-+ u^ : C A^ 3 E^ 3 asb-'(3 ^ & ^ : T H^ . 5) 2 2 '2, 31, ' &2 R0s (H) 0 ^ 63& v^ 3 u^ : C A u: C Z , ) ' crb -+ &+ ). ! 2. crb-4+ u: C Z 0 ( +& E^. ;

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A 3 E~ 3 asb -'(3 : T H . 5) () . & p 2 Z 0 2 2 62% a 2 A, p a mod R0s (H).

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. 41 Z 0(uC) B. 7 U, f, p c 2.

172

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4 V 1U s- . . E huci A. B s- V r- E. 4 ' 2 homA (E A), 'uc u(cf). 6 u : C A Z oc - , 5 2 homA (A A), r- 5 '. . ! a 1 2 B. 6 (c )(f ) = u(cf) = uca = (c )a, f( s) = a(s) 1 s 2 V . G, p a mod R0s (H). 7 p f8 2 Z 0 . . Un fn 1. . ! pn f8n . 7 5 1 5 an 2 A , fn( s) = an (s) 1 s 2 Vn 1Un . 7 n > m. 6 jan (s) ; am (s)j < 2=m s 2 Vn \ Vm , jan ; am j < (2=m)1. 7! 5 a 2 A , ja ; anj 6 (4=n)1. > , jf( s) ; a(s)j 6 5=n s 2 Vn. 7! p a. 7 . ! 1. 2 crb-+ u~ : C A~ 3 E~ 3 asb-'(3 ~ & ~ : T H~ . 5) 2 2 '&, 31, ' &2 R0s (H) 0 ~ 63& v~ 3 u: C Z u~ : C A, ) ' crb-+ &+ ). ! 2. crb-4+ u: C Z 0 &+& E~. ;

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7 1 c- C Q8 cl c- C Z0 0 v. Q A fZD j D 2 Jbg c- Z 0 A f( Q8 cl )D j D 2 Jb g c- Q8 cl , ( Q8 cl )D v 1 ZD0 . 6 v crb- (C Lb ) (Z 0 A) (C Lb) ( Q8 cl A). 4 5 1. = + /,-0

&-&( C Q8 cl 3&6 &2 Z 0 -+ C Z 0 &, crb-( , ' Z oc jaZ oc % C .

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= crb- 5 c- . . E C, , 5 C E f0g. . QH

1 - ' 2 HomC (E C) E. C !

QH '1 2 HomC (E1 C) '2 2 HomC (E2 C) '1 + '2 2 HomC (E1 \ E2 C) '1 '2 2 HomC (E1E2 C), ('1 + '2 )c '1 e + '2 e ('1 '2 )e '1 ('2 e). C 1 QH ! , '1 '2 , '1 jE1 \ E2 = '2 jE1 \ E2.

173

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&-&( % C. #5] ! 1 3 62% -6 & ' + C Q8 1 ( 1! T ), 1 15 . 7 ! 1 #7], #12], #22] #23]. . U fU 2 G j cl U = T g 1 R fR T j 9U 2 U (R T n U)g ( ) . 9 P T S -& &, P = G R G 2 G R 2 R. 4 1 > ( . #1] . 1, x 8, . Y' #2] .16.1.7). . S- 1 SP . L f : T ! R Z- , f 2 O(T SP ). c-H O(T SP ) 1 ZP . . c- Z ZP=R. L- c- u : C Z Z -+& % C. ". < () ' )2 ) ' T c-+ C Q8 3&6 c-+ C Z . > 5 ! - , ! 1 , 5 2. 4+ /,-0

&-&( C Q8 3&6 &2 Z -+ C Z &, crb-( , ' Z c jaZ c % C . 6 1 , \ " C Q8 cl \ 8 "( @0 - ) C Q.

3 $ !

, , C Q8 cl , 5 .

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.

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174

. .

, . . G 6= 0 1 G. 7 LN 1 1 - 1 T (#3] . IX, x 1, . 9). L f : T ! R - )2&, ' 4&2,

=1 f

1 T, 5 - 1

. . T RI , - . , R RI =LN . L- u: C R +& 4& % C. > fU 2 G 0 j T n U 2 LN g 1 U 0. fR 2 P j 9U 2 U 0 (R T n U)g 1 R0 . 9 P T S 0 -& &, P = G R G 2 G 0 R 2 R0 . S 0 - T 1 SP 0 .

" 5. 2 f : T ! R ) 62%. 5) . 1

2 2 : ) f 2 RI 9 () f 2 O(T SP 0 )9 ) 2 2 ' & fUn 2 U 0 j ng * 1 '1 {n fGnk 2 G 0 j kg, , !(f Gnk ) < 1=n. . ) ) ). 7 f 2 RI . 7 , 5 n , 1 1 { fQk g T, 5

- 1

, !(f P PQk) > 1=n. 6

=1 S(f {) supff(t) j t 2 Qk gQk s(f {) inf ff(t) j t 2 Qk gQk S(f {) ; s(f {) > T=n, . > , 1 n 5 1 {n fQnk j kg, , !(f Qnk ) < 1=n. 6

- , Qnk 5 Gnk Qnk , Qnk n Gnk 2 LN . 7! Un fGnk j kg

U 0 . ) ) 1). .1 , 5 f xni , xni+1 ; xni < 1=4n. . Qni f 1 (]xni 1 xni+1#), Hni fGnk j Gnk \ Qni 6= ?g Rni (T n Un ) \ Qni. 6 S 0 - Pni Hni Rni 1 T !(f Pni) < < 1=n. 1) ) ). . Q f 1 (]x y#) Qn f 1 (]x + 1=n y ; 1=n#). 6 Q = Pn, Pn = fPnk j Pnk \ Qn 6= ?g fPnk 2 SP 0 j kg | T, , !(f Pnk ) < 1=n. > , Q = G R

5 G 2 G R 2 LN . G, Q 0 1 . 7! f 1 (x) 0 1 1 x 2 R. 7 f #;z z]. 6 X fx 2 #;z z] j f 1(x) 62 LN g 1 . 7! n xn0 ;z < : : : < xni < : : : < xnp z , xni ; xni 1 < 1=n xni 62 X. . 1 T , 5

Qni f 1 (]xni 1 xni#) Sni f 1 (xni) 2 LN . H , Qni 1 Gni 2 G Rni 2 LN . .1 fGni Rni Sni j ig ;

;

;

;

;

;

;

;

;

;

: : :

175

1 {n . B S(f {n ) ; s(f {n) < T=n. G, f 2 RI . 7 . ! 01. 2 ff gg RI . 5) f g mod LN , f g mod R . . . h f ; g 2 RI . 7 jh(t)j < 1=n 1 t 62 Rn 2 LN . 7 5 5 Un fGnk j kg , !(h Gnk ) < 1=n. 7 t 2 Un . 6 t 2 Gnk k. s 2 Gnk n Rn. 6 jh(s)j < 1=n, jh(t)j < 2=n. ! 2. 4+ 4& c-+& u: C O(T SP 0 )=R0 . B , . C O(T SP 0 )=R0 , , Z 0 - C O(T SP 0 )=R0 ! , , . H cr-1 ! . 3.2 % " .

H K T -&' 1&, G \ K 62 LN 1 G, 5 K. >

- T 1 J . C . . T T fTK j K 2 J g , TK K. C &&' 1& '1 & ' T . C c- C &&' 3& L fCK j K 2 J g , CK fc 2 C j TK \ coz c = ?g. . u: (C L ) (A A) 1 cr -+& % C. . c- R = O(T SP 0 )=R0 5 A: J ! ! C (R ) , A(K) fF 2 R j 8n (TK \ cozn f 2 R0 )g. 6 (C L ) (R A) - cr - . 3 1 3. 4+ 4& C R &, cr -( , ' Z oc jaZ oc % C .

' 1] 2] 3] 4]

. . . 1. | .: , 1966. Semadeni Z. Banach spaces of continuous functions. | Warszawa: Polish. Sci. Publ., 1971. . !"# " #, . III{V, IX. | .: , 1977. Arens R. F. Operations induced in function classes // Monatsh. Math. | 1951. | V. 55, N 1. | P. 1{19. 5] Fine N. J., Gillman L., Lambek J. Rings of quotients of rings of functions. | Montreal: McGill Univ. Press, 1965.

176

. .

6] ()# !. *+ ) , . | .: , 1971. 7] -. /. . 0" + "*" # #, #" # " )#" "#" ) )*" 2#" - " ) ,# 2"3. ) , "##3 "3. 4" + // 5#. )#). " . | 1980. | . 35, 3. 4. | 8. 187{188. 8] Dashiell F., Hager A., Henriksen M. Order-Cauchy completions of rings and vector lattices of continuous functions // Can. J. Math. | 1980. | V. 32, N 3. | P. 657{685. 9] Zaharov V. K. On functions connected with sequential absolute, Cantor completion and classical ring of quotients // Per. Math. Hung. | 1988. | V. 19, N 2. | P. 113{133. 10] 0# . 9#: *+, ) , #

. | .: , 1977. 11] -. /. . cr-: 2 *+ "##3 "3. 4" + // ; . 9 888, "3) *+ ) 2"3. *+ "##3 "3. 4" + , #"3) "#" #) ? #" ) @, 4-8# " // 5#. )#). " . | 1990. | . 45, 3. 6. | 8. 133{134. 13] -. /. . 5" #*" - =)# ) # ? #" # ? #" # 9#" ". #3 "##3 "3. 4" + // 0" +. ". # >. | 1990. | . 24,

3. 2. | 8. 83{84. 14] -. /. . 8 = )#>, ? #" #) (## ? #" #) # # )#>, # EQ ) ) =) // != . 9 888 ' 5 + + 3. 3. 3 62], 63], *. @ $ 67]

A. ;

69] ( + ' 5 p- 5 . 61] :. :. C ' ). A 66] :. :. E ' :. :. C ' / ' 5 . ( $ / ' A(X ) $ 1 : : : n 2 K , + '(xi ) ; i 1 2 A+ (X ) ' i = 1 : : : n. C $ / '0 A(X ), '0 (xi) = '(xi ; i 1) = '(xi ) ; i 1 i = 1 : : : n. 2+ , + '0 $ / A+ (X ), + '0 / A+ (X ) +, ' / A(X ). 3+ , + J (') = J ('0 ), +, + 1 $ %& . 2. ' A+ (X ) , J (') U (X ). A # $ / A+ (X ) $ / A(X ), %& + $ . A / ' 1 2 0 . A %& . 3. U (X )

, U (X ) . $ 4. ' A(X ) , J (') U (X ) .

179

2

1. a b 2 A(X ) i = 1 : : : n U (X )

-

@ (ab) = @b (a 1) + @a (1 b): @xi @xi @xi

. F # a b, + +, a b % ( , ). ( a. . a = 1, + . . a | 1, a = xj , @x@ i . , + ' a, ', + a # k, a k. @ a ' a = a1a2 . %

@ (a1 a2 b) = @ (a2 b) (a 1) + @a1 (1 a b) = 1 2 @xi @xi @x@bi @a 2 1 = @x (a2 1) + @a (1 b ) ( a 1) + (1 a ) (1 b) = 1 2 @xi @xi i @b (a a 1) + @a2 (a b) + @ (a1 a2 ) ; @a2 (a 1) (1 b) = = @x 1 2 @xi 1 @xi @xi 1 i @b (a 1) + @a2 (a b) + @a (1 b) ; @a2 (a b) = = @x @xi 1 @xi @xi 1 i @b (a 1) + @a (1 b) = @x @xi i + . 2. a1 : : : am 2 A(X ) i = 1 : : : n

m @a @ (a1 am ) = X j @x @x (a1 aj ;1 aj +1 am ): i

j =1

i

. ( m. A + m = 1 + , + m = 2 1. m > 2. , + # + . F , a = a1 b = a2 am , 1

%

+ @ (ab) = @b (a 1) + @a (1 b) = @xi @xi @xi @ ( a 2 am ) 1 = (a1 1) + @a @x @x (1 a2 am ) = i 0m i 1 X @a = @ @xj (a2 aj ;1 aj +1 am )A (a1 1) + i j =2

180

. . , . . 1 + @a @xi (1 a2 am ) = m @a X j (a a a a ) + @a1 (1 a a ) = = 1 j ;1 j +1 m 2 m @x @xi j =2 i m @a X j (a a a a ): = 1 j ;1 j +1 m @x j =1 i

3. a 2 A(X ), ' | " A(X ). # i = 1 : : : n

n @'(x ) @'(a) = X @a j @xi j =1 @xi ' @xj :

. F # a, +

+, a | . . a = 1, + # %. a = xj1 xjm . F '(a) = '(xj1 ) '(xjm ), 2 m @'(x ) ; @a = X jk '(x ) '(x ) '(x ) '(x ) = j1 jk;1 jk+1 jm @xi k=1 @xi m @'(x ) X jk = @x '(xj1 xjk;1 xjk+1 xjm ) =

=

i k=1 n @'(x ) @a X r ' @x @xr : i r=1

3 !

4. ' | " A+ (X ). #

; J (' ) = J (') ' J ( ) :

. 3 n @'(x ) @ (x ) @' (xi ) = X j ' i @xk @x @x k j j =1 + .

5. ' | A+ (X ). # J (') -

U (X ).

181

. E | + , 1 | / = ';1 | / , '. F 4 , + ; J (') ' J ( ) = J (' ) = J (1) = E ; ; ; ; ' J ( ) J (') = ' J ( ) (J (')) = ' J ( ') = ' J (1) = E ; J (');1 = ' J (';1) : E , + 5 '

2 . $ #, $ , + . 6. ' | " A+ (X ), | A+ (X ) J (') U (X ). # J ( ')

U (X ).

. 5 J ( ) U (X ). $

4 ; ; ; J (');1 J ( );1 J ( ') = J (');1 J ( );1 J ( ) J (') = ; ; = J (');1 J (') = ; = J (');1 J (') = (E ) = E ; ; ; J ( ') J (');1 J ( );1 = J ( ) J (') J (');1 J ( );1 = ; = J ( ) J (') J (');1 J ( );1 = = J ( ) (E ) J ( );1 = E ; J ( ');1 = J (');1 J ( );1 :

4 # $ %

7. ' | " A+ (X ), J (')

U (X ), j = 1 : : : n

'(xj ) =

n X i=1

cij xi + hj

ij 2 K, hj | " A+ (X ), $ % & . # % ' " bij 2 K, i j = 1 : : : n, & " , n X (xj ) = bij xi i=1

182

. . , . .

, j = 1 : : : n '(xj ) = xj + h0j h0j | " A+ (X ), $ % & .

. ( % $ bij 2 K + '(xj ): '(xj ) =

=

X n

i=1 n X n X

! X n

cij xi + hj =

i=1

cij (xi ) + (hj ) = n X n X

!

cij bkixk + (hj ) = bkicij xk + (hj ): i=1 k=1 k=1 i=1 F $ (hj ) % + , bij 2 K , + $ / , $ bij , / ,

0 B=B @

0 C=B @

b11 b1n .. . . . .. . . bn1 bnn

1 CA

1

c11 c1n .. . . . .. C . . A: cn1 cnn BC = E . @ $ / '0 A+ (X ) , + '0 (xj ) =

' j = 1 : : : n. F '0 (xj ) =

=

X n

n X i=1

cij xi

! X n

cij (xi ) = i=1 ! X n X n n X cij bkixk = bkicij xk = kj xk = xj i=1 k=1 k=1 i=1 k=1

i=1 n X n X

cij xi =

'0 = 1, | / A+ (X ). H , + C K . @ / : U (X ) ! K , + (1 xi) = (xi 1) = 0 ' i = 1 : : : n, (1 1) = 1. F j =c @'@x(xj ) = cij + @h ij @xi i (J (')) = C . C J (') U (X ). $ C (J (');1 ) = (J (')) (J (');1 ) = (J (') J (');1 ) = (E ) = E

C K .

183

5 '

( K 0 , + K K 0 , + + A0 (X ) = K 0 K A(X ) U 0(X ) = K 0 K U (X ) % % K 0 . C A(X ) A0 (X ), U (X ) U 0 (X ). 8. ' | " A+ (X ), & i = 1 : : : n '(xi ) = xi + hi + hi | " A (X ), '% & . ( $

' " A0 (X ) K 0 - . # a 2 U 0(X ) " '(a) $ U (X ), a $ $ U (X ).

. C , + a 2= U (X ), '(a) 2= U (X ). a = ar + ar+1 + + as;1 + as + + am ai | $ a i i ( ), $ ar ar+1 : : : as;1 U (X ), as U (X ). ( + s ; r. 2

| + r = s. " '(ai ), i > r % r. F + '(xi ) xi % i, r '(ar ) ar . $ r '(a) ar , '(a) 2 U (X ) , + ar 2 U (X ), + + % s = r. , + ' s ; r < k, $ a, s ; r = k > 0. F ar 2 U (X ), '(ar ) 2 U (X ),

'(a ; ar ) = '(a) ; '(ar ) 2 U (X ). H a ; ar 2= U (X ), %

% , + '(a ; ar ) 2= U (X ). + + . @h 2 U (X ) i = 1 : : : n. 9. h 2 K 0 K A+ (X ) A0 (X ), @x i # h 2 A+ (X ).

. ( % i = 1 : : : n i , & % b c 2 U (X ) bxi c. 5 , + % a 2 A(X ) + @a = @xi

X

bc i (bc)=a

b c:

@h $// $ $//

b c @x i @h 2 U (X ), $// ' ' i (b c) h. F @x i h, & ' xi , K .

184

. . , . .

10. K K 0 | . ) 2 " K 0 , " "

K.

.

, + J (') U (X ).

7 & / ' A+ (X ), , + ' i = 1 : : : n '(xi ) = xi + hi hi | $ A+ (X ), %& ' +. 6 J ( ') U (X ). $ , # & , + , + + '(xi ) xi ' i = 1 : : : n. . J (') U (X ), U 0 (X ), % ' / A0 (X ). @ / A0 (X ), '. F ; J (') ' J ( ) = E: F J (') ; U (X ), , $ ' J ( ) U (X ). I 8 , + $ J ( ) U (X ), @ (xi ) 2 U (X ) @xj ' i j = 1 : : : n. H 9 (xi ) 2 A+ (X ) ' i = 1 : : : n, (A+ (X )) A+ (X ). F + A+ (X ) $ / A+ (X ), ', + , ' / A+ (X ). 11. ) 2 " ' & $ K,

K.

. ' | $ / A+ (X ), 0 . 2 + + K0 K , $// , +%& 0 $ / ' . A0 (X ), U0 (X ) | K0, . F '(A+0 (X )) A+0 (X ), + ' A+0 (X ). K0 + , 2, + ' A+0 (X ) / , & / A+0 (X ), + % '. 2+ , + , K -

A+ (X ), + / A+ (X ), '.

6 '

185

K | , %& Ki , i I . ( i 2 I + Ai (X ) Ui (X ) + % % Ki %& ' ei X , ei | Ki . F K - A(X ) ' ' Ai (X ), U (X ) | ' Ui (X ). ( i 2 I $ / i: K ! Ki . " $ / K ei Ki . " / i $ / i: A(X ) ! Ai (X ) i: U (X ) ! Ui (X ), i(xj ) = ei xj . E , + $ / ai = i(a), i 2 I , + $ a. E , + + +' . " / i U (X ). 12. K

Ki , i 2 I, 2. # 2 K.

. 2 + + Ei +% Ki , Ei = i(E ). ' | $ / A+ (X ), , + J (') | U (X ) . ( i 2 I $ / 'i A+i (X ), ' j = 1 : : : n 'i (ei xj ) = i'(xj ): 5 , + i(J (')) = J ('i ). F i (E ) = Ei, J (') U (X ), i (J (')) = J ('i ) Ui (X ) % i 2 I . % $ +, + $ / 'i % / . E+ , i 2 I & / i A+i (X ), 'i . / i $ / A+ (X ), a 2 A+ (X ) i (a) = i (i(a)) ( + , $ (a) & ). 2+ , + $ / . < , ' i = 1 : : : n i '(a) = i i'(a) = i 'i i(a) = i(a) +, + '(a) = a,

/ , '. E+ , ' / . 13. ) 3 Ki , i 2 I, K, '% .

186

. . , . .

.

A | U (X ), B | , BA = E . 2 + Bi = i (B ), Ai = i (A), i 2 I . F Ei = i(E ) = i(BA) = i(B )i (A) = Bi Ai

Ai Ui (X ). % Ai , Ci Ui (X ), , + Ai Ci = Ei. @ C U (X ), %, + i (C ) = Ci , + i (AC ) = Ai Ci = Ei AC = E .

7 *

14. K | , J | K, & 2 - K = K=J. # 2 K.

. A(X ) U (X ) |

K . " / ' A+ (X ) $ / ' A+ (X ).

$ / U (X ) U (X ) J (') ' J ('). F U (X ) $ ' U (X ), J (') U (X ). $ & $ / A+ (X ), '. $ / A+ (X ), , + (xi ) = (xi ) ($ , $ / $// K

%& ' ' ' K ). @ $ / = '. C , + ' i = 1 : : : n (xi ) = xi + hi $ hi $// J , (a) ; a 2 2 J A+ (X ) ' a 2 A+ (X ). C , + / A+ (X ) ( % + , + ', , / A+ (X )). 2+ , + %. m |

J . i, + J m;i A+ (X ) (A+ (X )) ' i = 0 1 : : : m. " +, + (A+ (X )) = A+ (X ), | / . 2

: i = 0. A $ + J m A+ (X ) = 0 A+ (X ) = 0 (A+ (X )): , , + J m;i+1 A+ (X ) (A+ (X )). F %' 2 J m;i a 2 A+ (X ) (a) = (a) 2 (a + J A+ (X ))

$

187

(a) 2 (A+ (X )) + J m;i+1 A+ (X ) (A+ (X )):

15. K | , J | K,

& - K = K=J 3. # 3 K.

. C + A(X ), U (X ), , & . - m |

J . A U (X ) , BA = E . F (B )(A) = (E ), (A) U (X ) . % C 0 U (X ), , + (A)C 0 = (E ). C | , , + (C ) = C 0. F A C = E + D, $ D J U (X ). H Dm = 0,

(E + D)(E + (;D) + (;D)2 + + (;D)m;1 ) = E ; (;D)m = E E ; D . $ A .

8 ,

2. ' A+ (X )

, J (') U (X ).

. I , + 2 +, K | (. 65], 68]). K | . F +' Q(K ) K , K Q(K ), 10 2 K . K |

' $ . F K , 10, 12 2 K . K | + . F K | H ,

- R

. J - K=R

' $ . 14 2 K . H , 11 2 % K . 3. U (X )

, U (X ) .

. E , + U (X ) ,

, %. @ +, K . K . $ A(X ), , , .

188

. . , . .

E , + + Mn (A(X ) K A(X )) / Mn (A(X )) K A(X ). A $ Z- Mn (A(X )) K A(X ) =

1 M

i=0

Ai

, + % a 2 Mn (A(X )) % b a b 2 Al l | b. H A0 = Mn (A(X )) K K / Mn (A(X )). $ $ A0 $

. 5% $ c 2 Mn (A(X )) K A(X ) 1 X c = ci i=0

ci 2 Ai , + + ci + . H , +

1 X d = di c, dk =

X

i1 ++it =k

i=0

;1 ;1 ;1 ;1 (;1)t c;1 0 ci1 c0 ci2 c0 c0 cit c0 :

F $ c +, # + + $ dk + . 2 $ c $ $ %. F 3 +, K | . . K , K0 | , U0 (X ), , U (X ), $ . F %, U0 (X ). . K , U (X ) %

. F % K , %& . . K |

' $ , K , 13 + '

' $ . . K + , , + #, / - K=R

' $ , # 15. H , + , $ , ' & .

189

#

1] . . .

, (p-) ! // # . $. | 1992. | (. 47. | N 5. | ,. 187{188. 2] #. #. # . /$ ! // 6-/ 1 23 / $4 / 3/ 5 $ . ( 3 $ . | $, 1990. | ,. 32{33. 3] #. #. # . 8 3 943 $ : ! // ,$. . . | 1993. | (. 34. | N 6. | ,. 179{188. 4] J. S. Birman. An inverse function theorem for free groups // Proc. Amer. Math. Soc. | 1973. | V. 41. | P. 634{638. 5] W. Dicks, J. Lewin. A Jacobian conjecture for free associative algebras // Comm. Algebra | 1982. | V. 10. | P. 1285{1306. 6] A. A. Mikhalev, A. A. Zolotykh. An inverse function theorem for free Lie algebras over commutative rings // Algebra Colloquium, to appear. 7] Ch. Reutenauer. Applications of a noncommutative Jacobian matrix // J. Pure Appl. Algebra | 1992. | V. 77. | P. 169{181. 8] A. H. Scho;eld. Representations of Rings over Skew Fields // London Math. Soc. Lecture Note Ser. | 1985. | V. 92. 9] V. Shpilrain. On generators of L=R2 Lie algebras // Proc. Amer. Math. Soc. | 1993. | V. 119. | P. 1039{1043. ' (: 1995.

. .

. . .

511.361

, " # $ "

" " #% . & #% (

)"

", *$ "

" #.

Abstract P. L. Ivankov, On linear independence of the values of some functions, Fundamentalnaya i prikladnaya matematika 1(1995), 191{206.

Arithmetical properties of the values of hypergeometric functions satisfying a homogeneous di1erential equation are under consideration. Using an e1ective construction of Pade approximation of the second kind it is possible to take into account speci2c character of the homogeneous case.

1

! ! . #1], #2], #3]. ) #4] ! ! !. + ! ! , ! . , ! ! ! ! , ! ! ! ! ! . - I | ! Q a(x) = (x + 1) (x + r ) b(x) = (x + 1 ) (x + m ) 1 = 0 b1(x) = (x + 2 ) (x + m ) m > 2 r < m a(x)b(x) = 0 x = 1 2 3 : : : 6

3 * # "" 4 54"

" ", N MHS000. 1995, 1, N 1, 191{206. c 1995

!" , #$%" \' %%"

192

. .

(z) = 1 +

1. b(x) I#x],

1 X

=1

z

a(x) : x=1 b(x) Y

(1)

2

1 : : : r Q (2)

1 2 : : : m r q (q 6 m r) , {1 : : : {q 2

;

q X = 1 q1 {1 l=1 l ;

I, = 0. , i j Z, i = 1 : : : r j = 1 : : : m, h1 : : : hm | I H = max( h1 : : : hm ) > H0((z) I ") 2

6

;

62

j

X =1

m

j

j

j

hj (j ;1)( ) > H

j

;

(m;1)(m;r)+q m;r;q

;

":

(3)

5 1 , #3]. ) #5] ! ! (1) ! , a(x) b(x) . 7 ! ! 6 #6], ! ! (j ) (z), j = 0 1 : : : m 1, ! ! ! , (3), (m 1) m. ) , ! ! ( ! , (z) ! ! ! ! ) . ! ! 1. 2. 1, (2 ), 1 I, 2 : : : r Q. # # , m r > 21 (m 1) + q:

m 2(m;1)(m;r)+m;1+2q X " (j ;1) 2(m;r);m+1;2q >H h ( ) j ;

;

2

2

;

;

;

j =1

;

$ % # , 1.

193

3. r = 1, 1 I Q, 2 : : : m j = 2 : : : m I, = 0.

2

2

n

I, 1 j

2

;

2

Q Z, n

6

X =1

m

j

hj (j ;1) ( ) > H 1;2m;"

% ( ),: : : , (m;1) ( ) 1.

2 -

j (z) =

jY ;1 l=1

(z + l ) j = 1 : : : m + 1

! ! , ! ! 9 , ! 1 . : !

;n = j (z s) ;

sY ;1 x=0

b(z x) ;

nY ;s x=1

a(z n + x) ;

j =1:::m s=01:::n =01:::m(n+1);1:

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Abstract A. Yu. Kolesov, N. Kh. Rozov, Construction of periodic solutions of a Boussinnesq type equation using the method of quasi-normal forms, Fundamentalnaya i prikladnaya matematika 1(1995), 207{220.

Using the asymptotic method of quasi-normal forms the dynamic characteristics of the following boundary value problem are analyzed: utt ; uxx ; "a2 uxxtt = "uxxt + "ut ; u2 ut ujx=0 = ujx=1 = 0 = const > 0 0 < " 1:

1] ,

! . # $ $ ! !

,

! .

1

% !

utt ; uxx ; "a2 uxxtt = "uxxt + "ut ; u2 ut

(1:1)

ujx=0 = ujx=1 = 0

(1:2)

/0 1 / & " "# ! &. 1995, 1, N 1, 207{220. c 1995 !", #$ \& "

208

. . , . .

0 < " 1, = const > 0. $ - 2] utt ; uxx ; a2 uxxtt = 0 . ! / ! $ . 0 / / , ! (1.1) "uxxt, "ut . $ . ;u2 ut, !

, $ ! (1.1), (1.2). ! ! (1.1), (1.2) . E E, E | . $ W22 (0 1), ! (1.2). 2 . . $ ! $ 3 , !$ (1.1), (1.2) 3

$ !, I + "a2 B, B = ;d2 =dx2, $ utt (1.1). . ! !

$ E $ ., . 0 $! ! t 3 $ ! (1.1), (1.2). 4, ! 1] , .

! . 3 ! $! 3 ! (1.1), (1.2), . . $ (1 + "a2 n2 2 ) 2 + "(n2 2 ; 1) + n2 2 = 0 n = 1 2 : : : (1:3) " = 0 in , !, $ . 6

/$ ! $ ! (1.1), (1.2) . . $ ! 7{-{9. ,

$ 3], 4] ! $ $ !3$ . . 2 ! , ! . ! 3 t x.

2

;. . . 0 (1.1), (1.2), ! p u = "u0(t s x) + "3=2 u1(t s x) + s = "t (2:1) 1 p X (2:2) u0 = 2 n21 2 zn (s) exp(in t) + z % // ? . | 1974. | ). 10. | .. 1778{1787.

( ): 1995.

. .

70- (14.02.1924{26.05.1989)

512.55

K | , A K | .

! " # $ $ A- . % &! $ . , $# ' ( & ! ' $ $ . ) $!, p- ' *{, ( & ! # $ $ . # $ A- ' & $ $## q-- , q | $ , !# . , ' $ ' - . % !, #' q-' /! ( & $(

$ , q ## # ' / ' $ ' $ $! ' $ ' q 2 ; . # # & # q-- ' $ ' - # ' &! #. 1, $ & # $ ' - $ A- # , A ## # $ : $ ' ( & $ ,. $ # # $ /! ($ ! $ # - !, #' $' /!, !' $ , ( & $( $ /! #).

Abstract Z. S. Lipkina, Locally convex modules, Fundamentalnaya i prikladnaya matematika

1(1995), 221{228.

Let K be a non-archimedean valued 7eld, A K be its integer ring. This paper is devoted to the study of the locally convex topological unital A-modules. These modules are very close to the vector spaces over non-archimedean valued 7elds. In particular, the topology of these modules can be determined by some system ; of semipseudonorms. Monna demonstrated that p-adic analogue of Hahn{Banach theorem can be proved for the locally convex vector spaces over non-archimedean valued 7elds. One can give the de7nitions of q-injectivity, where q is the seminorm which is determined on this module, and of the strong topological injectivity. It means that any q-bounded homomorphism can be extended with the same seminorm, where q is a some 7xed seminorm in the 7rst case, and an arbitrary seminorm q 2 ; in the second one. 1995, 1, N 1, 221{228. c 1995 !, "# \% "

222

. .

The necessary and su8cient conditions of q-injectivity and strong topological injectivity for torsion free modules are given. At last, the necessary and su8cient conditions for topological injectivity of a locally convex A-module in the case when A is the integer ring of the main local compact nonarchimedean valued 7eld are the following ones: a topological module is complete and Baire condition holds for any continuoushomomorphism (here topological injectivity means that any continuous homomorphism of a submodule can be extended to a continuous homomorphism of the whole module).

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3 $ ,

. ) % $ ', % p0 %/%, %$ p. ; , - + x a 2 LK - p (x a) = p0(x)jaj. E , p | % $ $ LK , , p (x 1) = p0 (x) 6 p(x). D $ ' , ( s $ , % ' % , Z(L) %, 2 . 4%' ' L N | ' $% A- % , # &, %$ ;(L) ;(N). F ( f 2 HomA (L N) $ q- , q 2 ;(N), %*$%& %$ p 2 ;(L) M > 0, , 8x 2 L q(f(x)) 6 Mp(x). 0 + $ , $ ' %$ q 2 ;(N), ( f $ . 1. :**5,* &'( &9 *( - '& ) ), ) ). 8* ), f | '&&- )**5,* q0 | ', ' (*, (4(* - ' ('*, ;(N), ( ( ' (* p0 , (4(* ' ('*- , ;(L) , q0(f(x)) 6 p0(x). . E , ( $ $ '. 4%' ' f | $ ( q 2 ;(N). 4 %$ q ,, $% '& % , V N. D%*$% $% , ' % , U L, ,, f(U) V . C ' % , U , %$ % p 2 ;(L). 0 x 2 L, %' x 2 U. ; , $ ' " > 0 $ b 2 A, , x 2 Ub jbj 6 p(x) + ". C& %, f(x) 2 f(U)b V b, , q(f(x)) 6 jbj 6 p(x) + ". D $ ' , q(f(x)) 6 p(x). . - %- , $ $, x2@ U. %$- - , 1 $, .

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-$ +# 5 $ ,$ ,, $ - % . 4%' J = (b 2 A yb 2 S) | $ '/ A. C ( f(b) = (yb), b 2 J. A% ', ( f ,$ ,, q- , , $ % % $, "+, %*$% + z 2 L, , f(b) = (yb) = zb , $# b 2 J. C ' - : S + yA ! L

(x + ya) = (x) + za x 2 S a 2 A: + z ,, J. 0 J = 0, z $, , z 1 - & -$ $ - # 5 $, $5. 0 J 6= 0, + z ,, % $ "+, , $ + % + z 1 - - & $ - # 5 $. , %' s = yb 2 S, rb (s) = 0, '% p(s ; yb) = 0. D $ ' , (s) b;1 -, $ +# 5 $. 4 '% (s) = zb, (s) b;1 = z 1.

227

D $ ' , q( (x + ya)) = q((x) + za) = q (((x) + za) 1) = q ((x) a;1 + z 1)jaj 6 6 jajra(;x) = jajjjpq jaj;1p(x + ya) = jjpq p(x + ya): L , j jpq = jjpq . 2 ' , ( % $ + z. L , S = N, ' % $ . ; , '$ # ', q- 1$ % , L % q-(, . 4 - . 4%' , $ ' ' 5 $ / $ bn % rn % . . , - x 2 L , N(x), , , $# n > N(x) q(x ; bn ) > rn. 4 - '(x) = q(x ; bN (x) ). C$ , q(x ; bn) = q(x ; bN (x) ) , $# n > N(x). 4%' L1 = L K | , , %

. 4 - ' % % q , %& %

%, -$: a = 0 q@((x a)) = q(x) '(y)jaj a 6= 0 x = ya: O ,, (% /, q@ $ $ + y. , x = ya = ta, y ; t 2 Z(L). D $ ' , q(y ; bn ) = q(t ; bn ). 4 - ', q@ | % . 4%' x + y = u(a + b), x = x1a, y = y1 b, a b 2 A, x y x1 y1 u 2 L. . q@(x + y a + b) = '(u)ja + bj, q@(x a) = = '(x1)jaj, q@(y b) = '(y1 )jbj. ' N , '(x1) = q(x1 ; bN ), '(y1 ) = q(y1 ; bN ), '(u) = q(u ; bN ). . q@((x a) + (y b)) = q@(x + y a + b) = '(u)ja + bj = = q(u ; bN )ja + bj = q((x + y) ; bN (a + b)) 6 maxfq(x ; bN a) q(y ; bN b)g = = maxfq(x1 ; bN )jaj q(y1 ; bN )jbjg = maxfq@((x a)) q@((y b))g: . - $,,, q@((x a)b) = q@((x a))jbj, a b 2 K. 4%' ' e : L ! L | -$ - f : L1 ! L | - ( e, &* % - %& % %. 4 - f(0 ;1) = (y 0). . , $# x 2 L f(x 1) = (x ; y 0). D $ ' , q(x ; y) 6 q@(x 1) = '(x). , x = bn , , $# n q(bn ; y) 6 '(bn) 6 rn. D $ ' , 5 $ % , $ $ %. . &'( &- A-*( +, ( ' ) jRj = q. 3! (c) , R M . 2 2 R M M = HomZ(M Q=Z) | (M +) (Q=Z +) 1. L M M . 3 , #4].

1.7. 4 % (M +) = (M +): $ % 2 M n 0 + ! 2 M , !() 6= 0. 2 3 M =< 1 > +_ : : : +_ < t > | (M +)

, ord i = di i 2 1 t. , !i

M , !i(j ) = ij d1i , ij | +_ : : : +_ < !t > = M. G 2 M n 0,

= 1 + : : : + t i 2< i >, i 2 1 t i 6= 0, . . i = ci i 0 < ci < di . B !i() = !i(i ) = ci =di 6= 0: 2

234

. .

, ! r 2 R ! 2 M r! : M ! Q=Z (r!)() = !(r) 2 M: B , r! 2 M , M R-. 0 , 1.7 R M R M . , 1.8. (+ R- = R M . RM 2 0 ' : M ! M , 2 M '() : M ! Q=Z '()(!) = !() ! 2 M : B , ' | (M +) (M +). G

6= 0, 1.7 ! 2 M , !() 6= 0,

'() 2 M n 0, . . ' | , 1.7 jM j = jM j , ' | . , , ' | R-, . . '(r) = r'() r 2 R. 2 ! 2 M , r'() r! , (r'())(!) = '()(r!) = (r!)() = !(r) = '(r)(!): 2 1.9. & M = M1 M2 | , + R- M = M1 M2 : 2 3 ! ~! = (!1 !2) 2 M1 M2 !~ : M ! Q=Z, ! ~ = (1 2) 2 M1 M2 ! ! ~ ( ~ ) = !1(1 ) + !2(2). , , ! ! ~ (M +), ! M1 M2 . ; , M1 M2 M , 1.7 ! ,

. , , ! | R-, r 2 R ! M : (r !~ )( ~ ) = ~! (r ~ ), (r ! ~ )( ~ ) = !1(r1)+!2 (r2) = (r!1 r!2)( ~ ): 2 : R M 5 , R L

K ' : R K ! R M : R L ! R M . M- . 1.10 ( %& ) ! R M 5 % % , % % I / R % % ' : R I ! R M + 2 M , '(a) = a a 2 I (. . + % : R R ! R M , "+ '). 2 2 ., , #4]. 2

235

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1.11. 6 Q=Z5 . 2 G I = Zm | Z ' : I ! Q=Z| , : Z! Q=Z, (1) = '(m)=m. 2

1.12. & RM | , R M | 5

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2

. " # . %

= - I / R R M K < R M R,

M ? I = f 2 M : I = 0g R ? K = fa 2 R : aK = 0g: , , M ? I | R M R ? K | R. B

, I J / R K L < R M - R ? (M ? I) I M ? (R ? K) K R ? (K + L) = (R ? K) \ (R ? L) M ? (I + J) = (M ? I) \ (M ? J) R ? (K \ L) (R ? K) + (R ? L) M ? (I \ J) (M ? I) + (M ? J):

(2:1) (2:2) (2:3)

236

. .

: R M , QF - , I /R

K < R M - (2.1) . 0 , K < R M Rb = R=R ? K , ! ba = a+(R ? K) 2 Rb 2 K ba = a. 9 M ? I R=I .

2.1. R M QF - , I J / R K L < RM . 7 %

+ " . (a) ! K QF - Rb = R=R ? K . (b) ! M ? I QF - R=I . (c) . (2.3) .

2 (a) 3 K 0 < RbK Bb = Rb ? K 0 . ; Bb = B=R ? K , B = R ? K 0 . b = 0g = ,, , K ? Bb = f 2 K : B 0 = f 2 K : B = 0g = K ? B = K \ (M ? B) = K \ K = K 0 , . . b = Bb K ? (Rb ? K 0 ) = K 0 . 9 Rb ? (K ? B) b b B / R. N (b) . (c) ; R M QF -, K = M ? I L = M ? J , I = R ? K J = R ? L, , (2.2), K \ L = M ? (I + J). , R ? (K \ L) = R ? (M ? (I + J)) = I + J = (R ? K) + (R ? L). 9 M ? (I \ J) = (M ? I) + (M ? J): 2 @ QF - R . 2.2. ! RR RR QF - . 2 3 I / R K = R ? I . ; K ! : R ! Q=Z I Ker ! . +

L00, i g j] j gN i ]. H+ < (, < , + ( (+. ( L00 L000 (: F L000

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T-

259

jn+1gn+1 ] 2

F L]. > , Pf (L)

+ j g j], g = g , Pf0(L0 ) = 0. Pf0(L0) (: Pf0(L0 ) = g0 i0 ] j1 g1 i1 ]0 : : : jn gn in]0 jn+1gn+1],

jk gk ik ] 2 L0 0 jk gk ik ] = ;"(g )(i m + 1)(1 j ) ijk gNgk jik ]] . k k k k k k Pp P h = i=1 fi i Ph0(L0) = pi=1 Pf0i (L0)i . ? +( (+( , 4]), Ph0(L0) = 0, h(Yh ) 2 T (R= S2 (e= Y )) + T(R= Invs (e= Y )), < , (+ , +( , , +( ,, 0 (Ph0(L0 )) = 0 :( 0 : F L0] ! F F - 2, . 143]. H:+ ( , + :( 0 . < i g j] 2 L00 n L000 (: 8 0 i g j]

i g j] 2 L000 > > < "(g)(i j)0 j gN i ] i g j] 2 L00 n L000 i g j] = > i g j]

L000 : 0

i = j g = g : . , , Ind | ( + ( Yh n = jIndj. , < : 1) a 2 R= 2) a1 : : :an n = jIndj, ai 2 R= 3) < z10 2 Re < zgi 2 Re, 1 6 i 6 t, g GbfII GbfIII GmfIII zg0 2 Re, + g GfI GefII GefIII = 1){3) . 3 ( 3) ( , z-< , (g i), + z-< | + z-< . ZR1 : sg 2 GbII GbIII 1 6 i 6 t, aszgi = 0 sg i]e11 ZR2 : sg 2 GI , aszg0 = 0 sg]e11 ZR3 : aei1 = 0 1 i]e11 i = 1 : : : t ZR4 : xr sg 2 GfI GefII GefIII xr sg 2 GfI GefII GefIII , ar szg0 =zxr sg0 ar szg0 =zxr sg0

xr sg 2 GbfII GbfIII GmfIII xr sg 2 GbfII GbfIII GmfIII , ar szgi =zxr sgi i = 1 : : : t ar szgi =zxr sgi i = 1 : : : t ZR5 : ar ei1 = zxr i i = 1 : : : t ar ei1 = zxr i i = 1 : : : t ZR6 : g 2 GmIII , e1j zgi = j g i]e11 j = 1 : : : t, i = 1 : : : t ZR7 : g 2 GeII GeIII , e1j zg0 = 0 j g]e11.

260

. .

( ZR1{ZR7 s 2 S, xr xr | ( Yh , g | E- , ei1, i = 1 : : :t | E. $ ,( , , ,, + fi 2 Mon afi (a1 : : : an)z10 = 0 (Pf0i (L0))e11 . .< ah(a1 : : : an)z10 = 0 (Ph0(L0 ))e11 , , + ,+ h(Yh ) ,

0 = 0 (Ph0(L0 ))e11 < 0 (Ph0(L0 )) = 0. A+ (, :( 0 : F L0] ! F , 0 (Ph0 (L0)) = 0, +(. < ai i = 1 : : : n, ZR4 ZR5, , O 6, P 3], < (+ , + z-< . ( ( + z-< : Z1 = f(ysg i) j y | ( Yh s 2 S g | E- g, Z2 = f(y i) j y | ( Yh g. ? , LIz : < szgi , (ysg i) 2 Z1 + y 2 Yh sg 2 GbII GbIII 1 6 i 6 t, < szg0 , sg 2 GI , < + ( Z2 < fej 1 j 1 6 j 6 tg eF- (= CH1 : hzxr i ej 1i = hei1 zxr j i zxr i , (xr i) 2 Z2 , zxr j , (xr i) 2 Z2 = CH2A :hzxr sgi ej 1i = hszgi zxr j i zxr sgi, (xr sg i) 2 Z1, zxr j ,

(xr j) 2 Z2 = CH2B :hzxr i szgj i = hei1 zxr sgj i zxr sgj , (xr sg j) 2 Z1 , zxr i,

(xr i) 2 Z2 = CH3 : hzxr sgi pzhj i = hszgi zxr phj i zxr sgi, (xr sg i) 2 Z1 , zxr phj , (xr ph j) 2 Z1 . , < wgi ( Re, (g i) | + z-< ( ( , w-< ), + , LIw : < swgi , (ysg i) 2 Z1 + y 2 Yh sg 2 GbII GbIII 1 6 i 6 t, < swg0 , sg 2 GI , w-< + P ( PZ2 eF- ( U0 = s2S tj =1 sej 1 F U= WR1 : hwgi sek1i = hswgi ek1i = hsek1 wgii = hek1 swgii = 0 w< wgi , s 2 S ek1, 1 6 k 6 t ( E= WR2A : s 2 S s = s , hwgi swgi i = hswgi wgii = 0 w-< wgi = WR2B : hwgi sw Phj i P= ;hswhj wgii = "(s)hswgi whj i = ;"(s)hwhj swgii = = ; tk=1 tl=1 hek1 sel1 i k g i] l h j] + e11"(g)"(i) i gNsh j] w-< , + i j + , s 2 S= WR3 : hwg0 P swhj iP= ;hswhj wg0i = "(s)hswg0 whj i = ;"(s)hwhj swg0i = = ; tk=1 tl=1 hek1 sel1 i k g] l h j] ; e11"(hs)"(j) j Nhsg] w-< , + j ( + , | , s 2 S=

T-

261

WR4 : hwg0Pswh0 iP= ;hswh0 wg0i = "(s)hswg0 wh0i = ;"(s)hwh0 swg0 i = = ; tk=1 tl=1 hek1 sel1 i k g] l h] w-< , + i j + . > w-< , z-< (: P zg0 = wg0 + tj =1 ej 1 0 j g] P zgi = wgi + tj =1 ej 1 j g i] i = 1 : : : t:

. + z-< ( , ZR6 ZR7 + . ? + ,, ( LIw WR1 {WR4 + LIz , CH1 {CH3 . . + w-< . P ++-, + w-< ( +. w1, LIw , WR1 WR2 (WR4), ,( , : + + + . . , w-< i < P k

P . J, wk , 1) wk 2 ( tj =1 s2S sej 1 F)? = 2) sm 2 S hwk sm wk i , + + ( eF, WR2A , WR2B WR4, + ( mkk = 3) < wj , j < k sm 2 S hwj sm wk i , + + ( eF, WR2B WR3 , + ( mjk = 4) < fswj js 2 S 1 6 i 6 kg , eF - ( P P U0 = s2S tj =1 sej 1 F . 4 < wk ++ < wk0 wk0 , wk0 wk0 2 Re 0 wk ,+ 1) 3). < wk0 ,( + 9, A 2.1.6, . 151], + e R. . ,

P R- P U = Re, X = eRe = eF , U0 = H = tj =1 s2S sej 1 F + T w1 + + T wk;1 + Twk0 , T = s1 F + sq F, < wk0 2 Re, +, wk0 2 H ? , s1 wk0 : : : sq wk0 eF- ( U0 hwk0 sm wk0 i = mkk ; hwk0 sm wk0 i m = 1 : : : q. + ,, < wk = wk0 + wk0 1) { 4). .+ ,+ CH1 {CH3 LIz , , O , < a1 : : : an. < a 2 R, ZR1{ZR3 , . A+ (, < , ( F +( , .

.

X | ! ' h i U . H U0 | U . t1 : : : tn | X- ! ! h i ! U , !, T = t1 X + + tn X ! % ti = ;ti i = 1 : : : n0, ti = ti i = n0 +1 : : : n, 1 6 n0 6 n. 1 : : :n | % X, , n0 +1 = 0 : : : n = 0.

262

. .

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U0 )

1] . . 14- . . | 1977. | $% . | &. 2. | (.8. 2] * (. + . | ,.: , , 1965. 3] 234 +. 5. 6 373% 8 (n n)-;3 ; 2, ';1 (s) 0 0 > 1 | 6 '0 0 30

0 n + 1. >

Q-5 0, 00 (ii), (iii).

Q-

265

1 .

% C. . 9 % Q- 1 0 ( 0

) S 0 0 I(S) := minf m 2 NjmKS { / g

(1)

0 0 S. > (S s) | 0

. : ( ., , 5], 6.11) (S s) 30 0 (S 0 s0 ) (C2 0) 0 G GL(2 C),

%& 0 1. 0 (S 0 s0) ! (S s)

, 0 G GL(2 C2) | # (S s) ( Itop (S s)). 6 0 I(S s) 0 det : G ! C . ;

(S s) % 0 0 , 0 G SL(2 C. * &

s 2 S, 0 3 Cn =G, G = Zn | '0 0 , %& Cn . 0

0'% 1 '0 0 30

( ., , 2]). / 0

0 3 30 C2 =Zn , Zn : C2 ' 1 exp( 2iq 0 n ) (q1 n) = (q2 n) = 1: 2 0 exp( 2iq n ) > q10 q20 q { 0 ' , 0 6 qi < n 0 6 q < n qiqi0 1 modn q q1 q20 modn. @ 0

S = C2 =Zn % (n q), 6

& P Anq . > Se ! S | 1

S E = Ei Se | 0% . 9

3 ; " ":

i i i

:::

i i

c1 c2 c3 cr;1 cr ci ( ci = ;Ei2 ) % nq ' % : n =c ; q 1 c2 ;

1

: 1 c3 ; : : : ; c1 r

! 1.1. (i) + #&

An

An+1n. (ii) , q0 | , 0 6 q0 < n q0 q10 q2 modn, 0 qq 1 modn ( qn & # # cr : : : c2 c1. 0

. .

266

"# " . , % 0

0 %& C3 :

(S s) C3 An An Dn n > 4 E6 E7 E8

uv + yn+1 , z 2 + x2 + yn+1 z 2 + x(y2 + xn;2) z 2 + x3 + y4 z 2 + x(y3 + x2) z 2 + x3 + y5

(2)

$ / 3], 0

30'% %' (. . 3 0 2) : S ! S, %& 0 % 0

(S s). 0 0

(S s) (2) 0 , ,

. 7 , 30

% % 0:

%' (S s) An Dn En E6 E6 Dn Dn A2k+1 A2k+1 An A2k A2k+1

30 (S s)=

(x y z) ! (x y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (x ;y z) (x y z) ! (x ;y ;z) (x y z) ! (;x y ;z) (u v y) ! (;u v ;y) (u v y) ! (;u ;v ;y)

A2 E7 A1 D2n;2 Ak Dk+3 A2n+1 A2k+1k A4k+42k+1

(3)

# $ % &. 7' 0 3 : S 0 ! S Q- 1

, KS = KS . > : S 0 ! S | 0 3 % 0

0 : Se0 ! S 0 | 1

. @ 0'

0 : Se0 ! S 1 S. B0 , 0 0 s 2 S 3 ' 3 ;, : 0

y y i y y

C 3 ; |

3 1 0 : Se0 ! S, 3 ;0,

& 1 | 6

3 0 : Se0 ! S 0 , a 1 % & 3 0 . 1

Q-

267

S

A5 s, 3

% (' %) 0 % S 0 ;1 (s) 0 A2 . ! 1.2. + , % , # # ,

(s0 ) = s, Itop (S s) > Itop (S 0 s0). -

# , # : (S 0 s0) ! (S s) | . D

0 0 '

. $ %&

. ' 1.3. : S 0 ! S | % . # ( , (S 0 =S) = 1. #

C 0 S 0 | ( # ) & # s = (C 0 ) | Anq . . & # e Se Se0 ;! 0 # # 0 S ;! S # : Se ! S 0 : Se0 ! S 0 | % S S 0 , . " # ( % #&$ : (i) e | , . . 0 : Se0 ! S | % S , C 0 KS > 00 (ii) e | # s0 Se, C 0 KS < 0 (. . : S 0 ! S # ). D3 : S 0 ! S (i) % . @0 3 0%, , ' 1

3 , % 0 3 . ( ) # $ &. 0

30'

. ; ,

, | 6 0 ( . (1)). * 1.4 (,13], ,14]) (i) " # 1 | cDV - , . .

$ & #&

. (ii) 1(# (X x) # I = = I(X x) > 1 (X 0 x0) # 1 # I , # &$ X 0 # x0 ( ). (iii) , (X x) | , $ # j ; KX j

x #&

& . 0

0

. .

268

! 1.5. (i) 211] (X x) | cDV - . " # X nfxg # . 4&# # , # & (X x) # 1 (X nfxg) | # I(X x). (ii) (X x) | F 2 j ; KX j | # &$ x #&

& . " # (X 0 x0) ! (X x) # (F 0 x0) ! (F x), x F 0 2 j ; KX j. 4&# , Itop (F x) = I(X x) Itop (F 0 x0). -

, Itop (F x) = I(X x) # #, # (X 0 x0) , . . (X x) | . 0

2 Q-

/ % 2.1 (,4]) ' : X ! S | Q-' # S . # ( , X

% # 1. " # S | ' : X ! S | ( ( ) . ' 2.2. ' : X ! S | Q-' # . " # (# ';1(s), s 2 S #

. 0 $ . > Z | ';1 (s) IZ | 0 -

Z. > ' 0 0 ! IZ ! OX ! OZ ! 0 , R1' (OX ) (0 00 ;KX '- 8], 1-2-5). > H 1 (Z OZ ) = (R1' (OZ ))s = 0, . . pa (Z) = 0. > ' : X ! S |

Q-5 . $ %& ( . 9], Proposition 3.1) 0 ,

S : s 2 S | 0,

& 0 H S, H

0 ';1 (s) ,

, H ! S | 0 3 0 ';1 (s). $ 5], 6.7, s 2 S -

. D 00 6 2.4. / % 2.3. ' : X ! S 3 s | Q-' # # . " #

# X X 0 ;! 0 #' #' (4) # 0 S ;! S # # : S 0 ! S | , '0 : X 0 ! S 0 | ( Q-' . 5 G = Gal(S 0 =S) # X 0 ,

Q-

269

'0 : X 0 ! S 0 | G- , X = X 0 =G : X 0 ! X = X 0 =G | . -

, # G X 0 # ( | # > 1 X . 6 ' X 0 =S 0 X=S. 0 $ . > s 2 S | 0, 0 0 # : (S 0 s0) ! (S s) . > G = Gal(S 0 =S). > X 0 | ' X S S 0 . @ (4) X = X 0 =G. > 00 G = Gal(S 0 =S) S 0 s0 = #;1(s), '

G : X 0 , ' '0;1(s0 ). ; % , X 0 1

G X 0 0 0 x01 x02 : : : x0m 2 '0;1(s0 ) X 0 ( ., , 5], 6.7). @0 ,

1 0 . / % 2.4. X=S | Q-' # # . " # S ( % . 0 $ . 0 S, 6 , ' : X ! S 3 s | 0

Q-5 . F

0 0 0 X 0 =S 0 3 s0 ! X=S 3 s 2.3. > x01 x02 : : : x0m 2 X 0 | 0 0 G xi := (x0i). 9 0 0 xi

% 0% 0 xi 2 Ui X. @ 3 Ui nfxig '0 0: 1 (Ui nfxig) = ZI (Xx ) (5) I(X x) - 0 0 x 2 X ( . 1.5). > , (S s) | '0 0 30

, G - '0 0 . : 2.3 (5) , & G- 0 C 0 := '0;1(s0 ). 9 C 0 | ' 0 ( 2.2). > , 0 0 Cj0 C 0 , Cj0 ' P1 G. @ G PGL(2) , , %& ( ., , 19]): G = Dn | 6 0 2n, n > 3, G = A4, G = S4 G = A5. , , G GL(2). , A4 , S4 A5 % , Dn 0 1 (6 0 2 Dn % ). @0 , G C 0 0 0 . F

3 ; C 0. 9 G : ; 1 . D %& . i

' 2.5. ; | , &$ #

. # ( , Aut(;)

% ;. " # $ Aut(;)- # ;0 ;

. .

270

. .

@@ ;; @ s; @@ @ .

1 2.6 ( $ .) ' : X ! S | Q-'

# # s 2 S | . " # ';1 (s)

# # rItop (S s) # r 2 N. -

,

# G C 0

# ( , r > 2. 0 $ . 2.4 , G = Zn C 0 % 0 % 0 Ci0 ' P1, Zn : P1 0, 6 % Zn : X 0 0 % 0. 1 0 1.5. 1 2.7. # ( , # 2.6 ';1(s) ( # x # I(X x), # $ Itop (S s). " # Itop (S s) . -

, $ Ci ';1 (s), # ($ x, (

$ # x1 # r I 2(Ss) # r 2 N. 0 $ . : 0 2.6 , G = = Zn : '0;1 (s0 ) 0 , . . G : ; top

1 . @ 2.5 & G- 3 ;. ; % , G 0%& 0 C10 C20 '0;1 (s0 ), , 0. 9 0 x0 = C10 \ C20 0 x = (x0 ), 0 0 G0 G 0 2 C10 C20 . I G0 = Zn=2 | 0 '0 0, , & & G0- 0 x01 2 C1, x01 6= x0. @ x1 = (x01 ) | 0 0 0 r I 2(Ss) . top

1 2.8. X

# ';1(s) % # 6 2. " # s 2 S #&

A1 . ' 2.9. ' : X ! S | Q-' # S h 2 Pic(S) | # 1 . " # # n 0 j ; KX + ' (nh)j . 1 , '(Bs j ; KX + ' (nh)j) |

( S . 0 $ . 7 & , h . &% 0 % C 2 jhj F = ';1 (C). @ F | 0 , 2.2 3 ' : F ! C | ' 0 .

Q-

271

90 , , % . 9 0, C \ '(Bs j ; KX + ' (nh)j) = ? , 60 ,

F \ Bs j ; KX + ' (nh)j = ?: : %

H 0(OX (;KX + ' (nh)) ! H 0(OF (;KX + ' (nh)) ! H 1 (OX (;KX + ' ((n ; 1)h)):

@0 00 n 1 ;2KX + (n ; 1)' h , / {5 H 1 (OX (;KX + ' ((n ; 1)h) = 0. > 3 KF = (KX + ' h)jF . @0 , H 0(OX (;KX + ' (nh)) ! H 0(OF (;KF + ' ((n + 1)h)) ! 0:

; % F \ Bs j ; KX + ' (nh)j = Bs j ; KF + ' ((n + 1)h)jF j. > L F | % 0 'jF : F ! C. 0, 0 , L \ Bs j ; KF + ' ((n + 1)h)jF j = ?. > % 2.1 0 s 2 (S n'(Sing(X))) ';1 (s) 0 0 ( 0 ). ; % 'jF : F ! C |

0 0 0 L 0

L2 = 0 ;1. : 6, , ;2KF ; L + ' ((n + 1)h)jF

633 0 K n 0. : & % H 0(OF (;KF + ' ((n + 1)h)) ! H 0(OL (;KF + ' ((n + 1)h)) ! ! H 1 (OF (;KF ; L + ' ((n + 1)h)):

> F H 1(OF (;KF ; L + ' ((n + 1)h)) = 0. @0 , L \ Bs j ; KF + ' ((n + 1)h)jF j = Bs j(;KF + ' ((n + 1)h))jL j. @0 00 L ' P1 , Bs j(;KF + ' ((n + 1)h))jL j = ?. L 0 .

3 " j;KX + '(h)j

* 3.1. ' : X ! S | Q-' # S h 2 Pic(S) | # # . # ( , $ j;KX + ' (h)j

(

#&

. " # $ # X0 = X ; ; ; ! X1 = X 0 ; ; ; ! X2 ; ; ; ! : : : ; ; ; ! Xn

# '0 ='

; =1

# '1 ='

0

2 ;

# '2

3 ;

n ;

# ' (6) n

S0 = S S1 = S S2 ::: Sn # (# 'i : Xi ! Si | Q-' # (# # # ( # # # #&$ # 0

. .

272

# :

; ; ; ! Z0 #p #q Z

Xi;1 # 'i;1 Si;1

= i

Xi # 'i Si

(7)

; ; ; ! Z0 #p jjq Z

(8) Xi;1 Xi # 'i;1 # 'i Si;1 ; Si # p : Z ! Xi;1 , q : Z 0 ! Xi | # , ( Z ; ; ! Z 0 | & 1, # (8) i : Si ! Si;1 | , (Si =Si;1 ) = 1 (. . i | , 1.3). i

0 30 8],20]. > X | 0 , H | X (. . H 0 ) c 2 Q, c > 0 | 0 0

. 1 1 f : Y ! X

X H. > HY | ,

H Y . > , KX + cH Q-/ . @ X KY + cHY = f (KX + cH) + ai Ei (9) i

ai 2 Q Ei | 0% . I , (X cH) ( , , ) , i ai > 0 (

ai > 0, ai > ;1, ai > ;1). : , 6 1 1 f ( . 20]). M X Q-30

, KX + cH Q-/ % c , 6 (X cH), , (X c0H) % c0 6 c. M (X cH) 1

, ' ' (10) e = e(X cH) = #fi jai 6 0g 0 1 f, 6 # (X cH). ; , e(X cH) = 0 0 , 0 (X cH)

. 9 (X H) 0 ' c(X H) = max fc j (X cH) { 0 0 g

(11)

Q-

273

(X H). O clog (X H). D

(X cH) %& : (*) X 1 Q-30

R (**) H | X 0 R (***) 0 c 2 Q (X cH) 1 0 0

. : 20], 0, 0 , 00 (*{***) 0% ( ., , 1]). , %& / % 3.2. X H | , (*,**) c = c(X H) | (. (11)). " # $ p : Z ! X , (i) (Z cHZ ) # (*{***), # HZ | H Z 0 (ii) p : Z ! X | # ( < ),

(Z=X) = 10 (iii) KZ + cHZ = p (KX + cH)0 (iv) e(Z cHZ ) = e(X cH) ; 1. D3 p : Z ! X 3.2 # (X H). 9 ' : X ! S |

Q-5 . 9 h 2 Pic(S) H = j ; KX + ' (h)j. > 2.9, h , jHj 0 . ' 3.3 (,1], ,7], 8.8) = #&$ : (i) $ # H 2 H

% #&

0 (ii) $ # H 2 H

% 0 (iii) (X H)

0 (iv) (X H)

. $ %& , 0 . ' 3.4. ' : X ! S | Q-' # h(1) h(2) h(0) = h(1) + h(2) 2 Pic(X) | # ,

H(k) = j ; KX + ' (h(k))j, k = 0 1 2 & # ( . " # , (9), X KY + cH(k) Y = f (KX + cH(k) ) + a(ik)Ei k = 0 1 2 i

(1) (0) (2)

a(0) i > ai , ai > ai . ! 3.5. ai X c & , ( , # # h 2 Pic(S) , h nh, n 2 N.

. .

274

( #. > , & 6 H 2 H

, % 0 . @ (X H) | 0 0 , . . c = c(X H) < 1. F

0 p : (Z HZ ! (X H). > E Z | 0% . > 00 KZ + cHZ = p (KX + cH) = p ((1 ; c)KX ) + p ' (h)

(12)

KZ +cHZ

633 0 S. 7 , (KZ +cHZ ) = ;1 > 0 (Z cHZ ) S. > 0 (KZ + cHZ )-3 60 60 1 . D , 0 , :

Z ; ; ! Z0 #p #q (13) X X0 #' # '0 S = S 0 0 C q : Z ! X | 60 0 0 KX + cHX , '0 : X 0 ! S {

Q-5 . 0

0

Z ;; ! #p X #' S ;

Z0 jjq X0 # '0 S0

(14)

C '0 : X 0 ! S 0 |

Q-5 , : S 0 ! S | ' 3, (S 0 =S) = 1 (. . | 0, 00 1.3). % (X 0 cHX ) | 0 0 , . . c(X 0 HX ) > c(X H): (15) > E Z | p- 0% , E 0 E 00 |

E Z 0 X 0

( E 00 = 0, dimq(E 0 ) < 2). @ 2.9 '(p(E)) = s 2 S | 0 ('0 (E 00)) = ('0 (q(E 0 ))) = s. $ 3 '0 : X 0 ! S 0 % 1, 6 (13) dimq(E 0 ) < 2, . . E 0 | q- 0% . (14), , dimE 00 = 2, '0 (E 00) = C 0 | 0 0, (C 0 ) = s. (13) % 3 0

0

KZ + cHZ = q (KX + cHX ) + a0E 00 a0 > 0 0

0

0

(16)

0

( ., , 8], 0 5-1-6). ; % e(X 0 cHX ) = e(Z 0 cHZ ). > % 0 (10) 0 00 ai 3 (9) 0

0

Q-

275

% 3 ( . 20], th. 2.23) % (13), (14): e(X 0 cHX ) = e(Z 0 cHZ ) 6 e(Z cHZ ) 6 e(X cH) ; 1: (17) 7 , 0 00 c | 0 0 (X H) c < 1, KZ + HZ = p (KX + H) ; bE = p ' h ; bE b > 0 b 2 Q h 0 b h. ; % ;KZ + q '0 h = HZ + bE 0 ;KX + '0 h = HX + bE 00 : (14) '0 (E 00) = C 0, E 00 = '0;1 (C 0). >6 0

0

0

0

0

0

HX = ;KX + '0 ( h ; bC 0 ): 0

0

9 h S 0

633 0 K , 6 h , h0 := h ; bC 0 0 . > 1 0

0'% 0 (X 0 c0HX ) . . D

Q-5 'i : Xi ! Si , Hi = j; KX +'ihi j 0 , hi 2 Pic(Si ) , ' c0 = c c1 = c0 c2 : : : ci = c(Xi Hi) ci+1 : : : (18) (13), (14): X0 = X ; ; ! X1 = X 0 ; ; ! X2 ; ; ! : : : # # # (19) S0 = S ; S1 = S 0 ; S2 ; : : : > (15) (18) | %&. 7 , ci = ci+1 , (17) e(Xi ci Hi) > e(Xi+1 ci+1 Hi+1). @0 , 0 1 0 j > i 0, e(Xj ci Hj ) = 0 cj > ci . ; 0, (19) . ., 0 1 c(Xn Hn) > 1. 9 6 0, limci > 1. ' 3.6. # (19) %

'n : Xn ! Sn , $ Hn = j ; KX + 'n hn j

% #&

. $ . > % 0 6]. := limci 6 1 0 , (Xi Hi) | 0 0 i 0. > 6 &

%& / % 3.7 (,20], ch. 18, ,18]) A R | ( (X H), # X | Q- , H | # ( X . " # A # & &$ # . 0

i

n

276

. .

9 0 , (Xi Hi) | i 0. 6 0 %& ai 6 0 3 (9).

! 3.8. ? Q-' # ' : X ! S ;4KS ' (KX2 )+T' , # T' S | # ,

(# ' (. 29], (3.10.1)). A , , ( # i (6). B , # ,

i | (ii) 1.3, S = S0 S1 : : : Sn & #&

(. # 4.2).

4 $

6 3 , ' : X ! S 3 s | 0 Q-5

, 0 1. * 4.1. ' : X ! S 3 s | Q-' # . # ( , # ' : X ! S , . . $ F 2 j;KX j # , #

% #&

. " # & % #&$ ( : (i) s 2 S 0 (ii) s 2 S | #&

A1 0 (iii) X

';1 (s) % # x1 x2 # > 1, & # n > 3 & , (S s), (F x1), (F x2) | #&

An;1.

1 4.2. ' : X ! S | # , #

2. " # S

% #&

An. 0 $ . > n = Itop (S s) | 0 0 . D

, n > 2, (i), (ii). @0 00 ;KX f = 2, f | & ', 'F : F ! S & 0 3. > '1 0 '2 'F : F ;! F ;! S | 30' U . @ '1 : F ! F 0 | ' , '2 : F 0 ! S | 0 3. > 3 KF = 0. ; % , 3 '1 : F ! F 0 0 F 0 0 % 0

0 ';2 1 (s). 9 0 00 deg '2 = 2, #(';2 1 (s)) 6 2. F

#(';2 1(s)) = 2, . . ';2 1 (s) = fx01 x02g | 0. @ '1 0

s 3 0

(S s), (F 0 x01), (F 0 x02) 3 . : 2.4 , 6

| % 0 An;1. : 0 3 ( . 1.2) , F 0 ';1 1(x0i ), i = 1 2 0

Ak k 6 n;1, ,

Q-

277

k = n ; 1 0 0 xi x0i, '1 : (F xi) ! (F 0 x0i) | 3. % 2.6 % 1.5 n = Itop (S s) 6 I(X x) 6 Itop (F x) % 0 x 2 X 0 > 1. ; % ,

1

%

x01, x02 ';1 1 (x0i ) = xi | 0, n = Itop (S s) = I(X xi ) = Itop (F xi) % 1.5 (X xi ), i = 1 2 | '0 0 30

0 n. > 00 F \ ';1(s) = fx1 x2g, 0 0 > 1 X | 6 0 x1, x2. D (iii) . F

#(';2 1(s)) = 1, . . ';2 1 (s) = fx0g |

0. > ';1 1 (x0) = ';1(s) \ F , , 6 | 6 0, 0 . M ';1 1 (x0) = ';1 (s) \ F = fx1g | 0, '1 | 3 Itop (F 0 x0) = Itop (F x1), x1 |

0 0 > 1

X % 2.7 6 0 Itop (S s) = 2. D

(ii) . @0 , , ';1 1(x0 ) = ';1 (s) \ F | 0 , '1 | 3 Itop (F 0 x0) > Itop (F x1). D3 (F 0 x0) ! (S s) 3 30' f1 g 0 2. > 00 (S s) | '0 0 30

0 n = Itop (S s) > 2, F 0 S % 1 %& ( . (3)) : N=o 1 2 3 4 5

(F 0 x0) Itop (F 0 x0)

(S s)

n = Itop (S s)

E6 A2k+1 Am A2k A2k+1

A2 Ak A2m+1 A2k+1k A4k+42k+1

3 k + 1 k > 1 2m + 2 2k + 1 4k + 4

24 2k + 2 m+1 2k + 1 2k + 2

$ % 2.6 C = ';1 (s) & 0 0 0 > n, x1 . 7 , , I(X x1 ) = r1Itop (S s) = r1 n Itop (F x1) = m1 I(X x1 ) Itop (F x1) 6 Itop (F 0 x0) (20) 0 r1 m1 2 N. , Itop (S s) 6 Itop (F 0 x0): (21) F

'. 1) @0 00 Itop (S s) = 3, I(X x1 ) = 3r1, X, 0 x1 , & & 0 0 x2 0 3r2, r2 2 N ( . 2.7). ; , x1 x2 2 F, 6 F 6' F 0 . . 3 '1 : F ! F 0 | 3. F

%& '1 3 ; ( . . 1). >3 ;00 ; & 1 | 6

3 0 0 ';1 (s) \ F 6= ? 6

0 0 . $ , 00 Itop (F x1), Itop (F x2), 3 ;0 ; 1 , %& 0 x1 2 F, x2 2 F 0 ;01, ;02 0, j;01j > 3r1 ; 1, j;02j > 3r2 ; 1. V , ; E6 , ;01 ;02 | A2 ; 1 %& :

. .

278

y y i y y i . . F % 0 x1, x2 A2 , % % 1.5 X % 0 0 > 1 | 0 x1 , x2, %& '0 0 30

0 3. D (iii) n = 3. 2) > 1 % '1 : F ! F 0 | 3, 6 Itop (F x1) < Itop (F 0 x0) = 2k+2 : (21), (20) Itop (F x1) = I(X x1 ) = = Itop (S s) = k + 1. F

0 3 '1 : F ! F 0 %& 3 ; ( . . 1). >3 ;00 ; & 1 | 6

3 0 0 ';1 1 (x0 ) 6 0 0 . ; % 3 ;0 ; 1 , %&

F 0 . M ;0 , F 1 % 0, , X 1 0 0 > 1. @ % 2.7 Itop (S s) = 2 (ii). 9 , ;0 0 ;01, ;02, 0

% 0 x1 2 F , x2 2 F. @0 , ;

y y ;01

:::

i i i ;00

:::

y y ;02

@0 00 Itop (F x1) = k + 1, j;01j = k. ; , 2k + 1 = j;j > j;01j + j;02j. M j;02j = k, Itop (F x2) = k + 1 Itop (F x1) = Itop (F x2) = Itop (S s) = = I(X x1) = I(X x2) = k + 1, % 1) (iii). M j;02j < k, Itop (F x2) < k + 1 2.7 k + 1 , , j;02j = k;2 1 . 7 , 0 x1 , x2 Itop (F x2) = k+1 2 0 C0 ';1(s). L , 3 ;00 1 . j;00j = j;j ; j;01j ; j;02j = 2k + 1 ; k ; k;2 1 = k+3 2 > 1. > . 3) L 0 00 5) (21). 4) > 1 % '1 | 3, (20), 6

2k + 1 = Itop (S s) 6 I(X x1) 6 Itop (F x1) 6 Itop (F 0 x0) = 2k + 1. ; % Itop (F x1) = Itop (F 0 x0), . . '1 | 3, . @ 0 . $ %& 0 %,

Q-5 0 00 (ii) (iii) 4.1 & %. / 4.3. P1 C2 ! C2 { # . D## #

2 #&$ : (u v) ! ("k u ";kv) G = Zn C2uv P1xy Cuv k k ; k (x y u v) ! (x " y " u " v), # " = exp(2 i=n), k 2 N, (n k) = 1. ( X = (P1 C2 )=G, S = C2 =G. " # &$ : X ! S Q-' . 1 h1 : : : hv $ ZK , ,

h 1k] k1 + + h vk] kv < C2X 1; {{;v1 k = 2 {= 0 < max jh jk]j 6 X k=1{ j =1v

(5)

h jk] | , ' hj K, C2 > 0 $ K kj . . ? , ,

#K : Q] = #K : I] #I : Q] = 2{:

. , ZK Z , 2{ .

. ?

, , hj , .

. , Z. @ & , (5) , & & . 2 , % ( % , m = 2({ ; 1) n = 2{v , , % 5.1 , #5] , ,( (5) ' 7 .

. M N | ,

(. 1 , % f11 : : : fss j 2 Z+ 1 + + s 6 M (6) f1 (z) : : : fs (z). 2 K(z), % . , % , , F1(z) : : : Fv (z) M = N (7) E1(z) : : : Em (z) M = N + d (8) d = deg P . ( N m = g(N + d) v = g(N) g(x) 2 R#x] deg g(x) = l (9) ( , % x 11 4 #1]). 2 1 2 4 #1] , , , (7) (8) , % , (6) M = N M = N+d ,

&' , . , % K(z), % z = % %% , . , ,

% ( (1), % %% & , .

.

308

. .

& L (F1 () : : : Fv ()) = h1F1 () + + hv Fv () hj 2 ZK % {v L k ] (F () : : : F ()) < C X 1; {;1 k = 2 {= 1 v 3

k] 0 < max jh j 6 X kj j

(10)

X > 1 (, , C3 C4 : : : , %, , %' X H. ( X

,

{v ; ({ ; 1)m > 0 (11) L (F1 () : : : Fv ()) = L 1] (F1() : : : Fv ()) 6= 0 | (12)

k ]

,

(10), (4) % L (F1() : : : Fv()), k = 1 {, " s = v %%. P(x1 : : : xs ) | , &'

% . C , , ( ,

R(z) = L (F1 (z) : : : Fv (z)) P (f1 (z) : : : fs (z)) = = a1(z)E1 (z) + + am (z)Em (z) aj (z) 2 K(z) (13) % %% & a1 (z) : : : am (z). 2

% , P (f1 () : : : fs ()) 6= 0. C , (12), R() 6= 0. . T (z) 2 ZK #z],

S(z) = T (z)R(z) = b1(z)E1 (z) + + bm (z)Em (z) (14) bj (z) 2 ZK #z] bj () 2 ZK j = 1 m bj () = 0 . ,

(13), jb jk]()j < C4 XH k = 1 {= j = 1 m:

. ( -

& " > 0 max jS k] ()j > C5(XH)1;m;" C5 = C5 ("): k=1{

2 (10), (13) (14)

(15)

jS 1] ()j < C6 X jP j {v

jS k] ()j < C7HX 1; {;1 k = 2 {

P = P (f1 () : : : fs ()), (15) , % X jP j + HX 1; {{;v1 > 2C8(XH)1;m;" :

(16)

E-

309

2 X , %

HX 1; {{;v1 = C8(XH)1;m;" :

(17)

. , H, , X, ,

% (12), | P , aP ( a. C , (16) (17) "

% (11) : {mv

jP j > C9H ; {v;({;1)m :

(18)

D, (9) ,

{v ; ({ ; 1)m = {g(N) ; ({ ; 1)g(N + d) =

= g(N + d) ; {dg0 ( ) 2 (N N + d): (19) 2 N = {ld ( , , %' g(x), , , %, % f1 (z) : : : fs (z). C , (19) : {v ; ({ ; 1)m > 05c({ld)l

c > 0 | ( . g(x). C (11) {mv < 4c{({ld)l {v ; ({ ; 1)m , (18) . F

,

#6].

1] . . . | .: , 1987. 2] Lang S. A transcendence measure for E-functions // Matematika. | 1962. | V. 9. | P. 157{161. 3] . . . )* + , - E-. // ,. -, | 1967. | . 2. | N 1. | /. 33{44. 4] . 1. 2 . ) , -, - E-. // ,. -, | 1968. | . 3. | N 4. | /. 377{386. 5] . 1. 3 4 ,. 5* 67 8* + | .: 1- - 29, 1981. 6] . 1. 2 . On some equations connected with E-function // Diophantische Approximationen 26.09 bis 02.10.1993, Tagungsbericht 43. | Math. Forschungsinstitut Oberwolfach, 1993. | S. 20. ' (: 1995.

. .

70- (14.02.1924{26.05.1989)

R | Q, 1 2 R, n > 3, H | GLn (R), En (R), | ! " P H , P En (R). $ ! " P H .

Abstract

I. Z. Golubchik, Isomorphisms of projective groups over associative rings, Fundamentalnaya i prikladnaya matematika 1(1995), 311{314.

Let R be a two-sided order in a regular ring Q, 1 2 R, n > 3, H a subgroup of the linear group GLn (R) containing the elementary subgroup En (R), an automorphism of the projective group P H which is identical on P En (R). Then is identical on the group P H .

. . . . 1] 1. % R S | ' ( ' '), 1=2 2 R, 1=2 2 S , n > 3, m > 3 ' : GLn (R) ! GLm (S ) | +,-+, .%//. 0. % % ' ( ,/ ( e f ' , ' Rn Sm , ') +,-+, 1 : e Rn ! f Sm ') +,-+, 2 : (1 ; e) Rn ! ! (1 ; f ) Sm , , 1 A 2 En(R) ; '(A) = 1 (e A) + 2 (1 ; e) A;1 : '. . () 2] ) 1 * m > 2. + ( PGLn(R) | -. ** GLn (R) * / , 1 : GLn(R) ! PGLn(R), 2 : GLm (S ) ! P GLm (S ) | .))-)1. . . 3] * )1 1. 2. % R S | ' ( ' 1 1=2, n > 3, m > 2, En(R) G GLn (R), Em (S ) H GLm (S ), H | , () GLm (S ) ' : PG ! PH | +,-+, .%//. 0. % % ' ( ,/ ( e f ' , ' Rn Sm , ') +,-+, 1 : e Rn ! f Sm ') +,-+, 2 : (1 ; e) Rn ! (1 ; f ) Sm , , ; ; '1(A) = 2 1 (e A) + 2 (1 ; e) A;1 *+ + INTAS. 1995, 1, N 1, 311{314. c 1995 , !"

\$ "

312

. .

1 A 2 En(R). ; ; + ( A 2 GLn(R) 1 (A) = ';12 1 (e A) + 2 (1 ; e) A;1 . 3. | )-) . **1 PG, 415 P En(R). 1 . 6. ) 4] , /1 (/ R, 1 )4 *( 415 )-) . **1 PGLn(R), 415 PEn(R). 3) ) 8 / 9 . . . . . 5]

3. % R | PI- ' 1, n > 3, H | /.%// GLn (R), En(R), | ,-+, / ) .%//( PH , () PEn(R). 0. ) .%// PH . 6*)), (/ Q 1 . 1) )1 65), . a 2 Q x 2 Q, . a x a = a. + (/ R Q 1 ) *),

(8a 2 Q)(9c b t s 2 R)t;1 s;1 2 Q a = b t;1 = s;1 c]: ;1) () 5 1

4. % R | % ) / .% , ' Q, 1 2 R, n > 3, H | /.%// GLn(R), En(R), | ,-+, / ) .%//( PH , () P En(R). 0. ) .%// PH . t, ek = a0 ek = ti=0 ei bi ek . &( ek ,

, ek 0. 6 , A ( ) .

3. e | $ A eA | $$ ' ' A, eR | $$ ' ' R.

. 0 f | eR. 6 , m R m, (, ,, fR = fx P 1 ,P f = i=0 ai xi a0 6= 0. 0 (, R , j 1 g= 1 j =0 bj x , fg = e, , fbj 2 Agj =0 , a0 b0 = e a0 bk + a1 bk;1 + : : : + ak b0 = 0 k > 1. & ( , bj . 6 , ef = f , , , eai = ai ) i. 3 eA , b0, a0b0 = e. 0 , b0 : : : bj ;1 , , ;a1 bj ;1 ; : : : ; aj b0 ( eA, eA , , bj , a0 bj = ;a1bj ;1 ; : : :; aj b0. 3 , , f 2 eR fR = eR, eR.

1. (1))(3). 71] , , : (a) ( ( ) 9 (b) . 4 1 A, 2 | (b). (3))(2). A = ni=1 ei A, fei gni=1 | , fei Agni=1 | P ;. 3 f R f = ni=1 fi , fi = ei f , , R = ni=1 ei R, fei Rgni=1 | e ( 3) e R, R | . (2))(1). ) , , 72]. < , $ ) ) . = $, a b, a = a2 b. . , . 1. : (1) A((x)) (2) A((x)) # $ ( (3) A # $ ( .

317

. 4 (1))(3) 1 ! ,

. & (3))(2) , A | , A((x)) | . 3 A | , 1 A((x)) , . 4 (2))(1) . ; ;. . ?) ;. ;. 3 .

1] . : , . .2. | .: , 1979. 2] ! ". . | .: , 1971.

' (: 1995.

1995)

1995)

1995)

Building Internet Firewalls - 1 edition (September 1995)

Building Internet Firewalls - 1 edition (September 1995)

1995 E with Technical Corrigendum 1

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Ajs Review 1995: Fall, No 1

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Историко-математические исследования. Вторая серия, выпуск 1, №1, 1995

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