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Фундаментальная и прикладная математика (1995 №1) (1995).pdf

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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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yI = hy1 : : : yn i 9. K( () ( M )
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I 4( 4 % )
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* M . ;; - '(Ix) % ; A j xIB
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12 Z4 j= : '(1) , Z4 j= '(2) , *,' ,,-&'(* '(Ix) '( ) % .' mI '( M , ,'*/ ' xI .
4 3( !4 -.
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V , '- UAV = D = diag(d1 : : : dk ) 2. 6)7' '* (+ yIA = xIB %+ %+ 2(,, ( yI , , , ' xI .' .+ 2(,,*) . )7' '*
4
. . , . . , . . (+ yIAV = xIBV yIU ;1D = xIBV . 8 zI = yIU ;1 * )*,
, ' ,% zID = xIBV , % (9-) (+ zi di = xIvIi t () \t" ) ,), ,'* ). ; 2, (( + (*+
/ %*/ 2(,,) ,% , 7+ /+ + '
+*/ (+, )'*+ &'(' zI = yIU ;1 < yI = zIU .
, 1.4. % ,,-&'( '(Ix) %*/ 2(,, .
9- ,,-&'( a j xIIbt < a , bi 2 Z.
= 14 !4 6110] , %9 1 "; ;"4 - xIrI = 0 pk j xIsIt , p 2 Z.
! * ( -9 ( 9). < , ( - '(Ix) (Ix) * ' ! (' ) ,
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- % 9 % * L
! (, * (. - 4 4 (1--) !* ) L ((( - \x = 0" . 4 9 - ' = \A j xIB" = \C j xID" * L ( - '+ = \9I(uIv xI = uI + vI ^ uIA = xIB ^ vIC = xID)" ,
L | ) %( ;"( -.
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9 4) *4 49 4.
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' ' 2 ,' -, ',-+: A j xIB ! AV j xIBV =
= \GC j xIBV " ! C j xIBV = \C j xI(D + HC)" ! C j xID , ( + , 7+ ' (+ ' %)':
yIA = xIB ! yIAV = xIBV ! yIGC = xI(D+HC) ! \(Iy G ; xIH)C = xID" = \IzC = xID" ,
2 zI = yIG ; xIH .
. M ! ! ! 1) ) 2) . C
. . ! M = hxIyI j yIA = xIB i . 0 M j= '(Ix) , M j= (Ix) , !
4( zI = (IxyI)(;HG)t , M zIC = xID . C( ) % , 1 % hxIyIi , %. 2
1.6. ( R 2( (, @+'() - '(Ix) = \A j xIB " ,,&'(. 1 479] - '- Rn 2(*' -' 2(, 5
*+ RnA , ',+ '-+ E , AE = E WA = E + '-* W 2 Rn . 12 ,(' -,( ',-+:
A j xIB ! \E j xIB " ! xIB(I ; E) = I0 , 2 I n n '-. A %,
) xIB(I ; E) = 0I ( xIBE = xIB , E j xIB = \WA j xIB " ! A j xIB .
? 2 '' '(Ix) xIB(I ; E) = I0 . 6' , ,*+ B(I ; E)Rn , ,('
1.7. 6419] % ,,-&'( '(Ix) 2(*' -' R . ,,-&'( xIE = I0 , 2 E ', '-.
4 4 , !;34 1.5, 1.8. ( %3 ;) 6396]. A',-
A1 j xIB1 ^ : : : ^ An j xIBn ! A j xIB ,,-&'( ,)*' -' 2 ( (' %)': Ai j xIBi ! Ai Vi j xIBi Vi =
= \Gi A j xIBi Vi " ! A j xIBi Vi / i A j xIB1 V1 ^ : : : ^ A j xIBn Vn !
! A j xI(B1 V1 + : : : + Bn Vn ) = \A j xI(B + GA)" ! A j xIB .
M ) 1 1 )! ( "
- A1 j xIB1 ^ : : : ^ An j xIBn ! A j xIB * ( R , *4 ;39 49 4 " Ai 4 4 (( %3 ; | " A .
1 4 " -, %9 ( ((, ( .
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, ) * ,'* ,) (2/ . ',
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4236].
4) ( S = fR=I , 2 I ,*+ ,*+ - Rg . @)' .( ( (,+) W -+ (J&) ,(' W -,* W -9* '(. ? '(+ M N W - , ,,-&'( Ia j xI , ) )7' '( N
( yIa = mI , mI 2 M ( 2 )7' M .
S-4 - A j xIB , A 2 S . ; 4
M N S-! , ( ;%4 S- '(Ix) ;%
% mI 2 M N j= '(m)
I M j= '(m)
I . ;3 1 1( (( S-!4 S- ;( 4.
1.10. 670] , . 1 6476] 6396] . =* ( ( '(+ : M N : 1) S -&'( < 2) S -<
3) (- % M K ! N K , 2 K 2 S ''&)'.
1.11. % S - , , '' */ '(+ S = fM j M 2 S g '* 477] .
; 4 M N ( S-3, S-, ! K N 4 4 !, K \ M = f0g , "
7
; M N=K ((( S-. 1, ; M N !
S-( % ( M , ) ; S-3 ! N
((( S- . :!( ; 9, M 6228] ,
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;, ;%4 S-4 ! . . ;3 1
( (, 9 . . ; S- . -,
1! 4 9" % % 6355].
1.12. 670]. =* ( ( '( M ,)*' -' R: 1) T
M S -9 < 2) M . . %+ ,,-&'(*
'(Ix) *, '(M) =
(M) , 2 S(') = f(Ix) j S -&'( ' ! g
2S (')
(M) = fmI j M j= (m)
I g ,,-&'(* (Ix) .
E, . . ( M 1 %! 9 : ;%( ( 49 4 ( M) *
M , ( * M . 3 1 , S-! ( M ) . . "
;%4 49 4 " S . 1,
) 1( ( M , '(M) ( - '(Ix) ! %3 * 4 4 xIrIt = 0 , rI 2 R (. 6397] ( ( )4 ).
@, ! 3! ( " 9 ( S- ( ( 1. 0,
M 6247] %: 33 . . RD-4 !
4 %!;? = , 3 ( ;4 4 %!; 1 ! O 6476].
0 ( ! 4 1. % 6354] , ! Z2 !" Z6x] . . (! ), RD- . 4 665], !( 1 1.12, , !" k6x y] 9 ;39 9 1 14 (
! %4).
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!" RD-9 4 3 !*, . ., 1 9 " *, ( ( ",
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1 RD- !" Z6x].
( ! " -. 1, R ! S-!", ;%4 . . 4 ! R S- , ! ((( ( (4 4 S . P, ;% !" !
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8
. . , . . , . . 1, ( (( S-!" "!; -.
1.13. 670], 6396] . =* ( ( ,)-
- R: 1) R , S -- < 2) % , ,,-&'( R
. + 9- S -&'(, ,,-&'( A j xIB ,
2 A 2 S < 3) %+ . ,. ,*+ '( R S -, < 3) %+ . . ,*+ '( R S -9 < 5) , S - , C
,.
E, , (( 1 ( S = fRg , ; 9"; (9 ". 4
( . 1) ( ! ;3; ; ((
S-!".
1.14. % , S -- S --, 2 S . ,. */ '(+, )'*/ ' '-'.
H ! RD-!" ( !" O), ( 1) , (
RD-!" ( !" O).
1.15. 670] , 6396]. % , RD-- *' RD--'.
: 1 (, % ( !" O . ;3 1;, ! ; 1
% %3 , 4 ) ;3: %
;% !" O RD-!"?
1.16. 667], 6396]. % , - J& (*' ,*') RD--'.
. W -4 !" 14 1.14, 1.13
;%( ( - R ) 4 ;" -
aIt j xIIbt , aI , Ib ) R. 0!, ( W-4 !" R 1 1.13, )!
aIt j xIIbt aI1 j xIB1 ^ : : : ^ aIn j xIBn . %3 ; (1 1.8) "( ( 4 ) 3(( %: aIi j xIBi ! IaiVi j xIBi Vi = \GiaIt j xIBi Vi " ! Iat j xIBi Vi B1 V1 +: : :+BnVn = Ibt +GIat . = IaiVi = GiaIt , Gi = gIi Vi = vIit %" ) R . 0 !, aIt j xIIbt aIi vIit j xIBi vIit
aIi vIit ) !" R . 0! ( 1 1.16 1.15. 2
( RD-!" (!" O) % 6476] (,
% %( "! ( ( 9 "), % !; 4 4.
9
1.17. 6476], 6477] . C''( - R RD--' (-' J&) 2 2, 2 R ''(
,& -, )- R , ''*' ' ''(* - '.
1.18. % , ,,-&'( ''(*' -' . + 9- ,,-&'( a j xIIbt ( RD-&'() 2
2, 2 R ''( ,& (= ''( %( ) -.
@, 1 1.16 %4 % RD-4
!" 9*4 4 ( ) 4
. . 4 . 0, 667] % RD-4 9 9 " ( ) ( % 4( A1 (k) k 4 9), !( ; 6211] , 6212] 19
4 !". E, RD 4 n-4 %
4( An (k) k 4 9 1!
* ; 6461] %!4 " ;%
1 1 ( ( 4. $ 6396],
!" " RD-!" ! RD-!".
0 ;3 RD-" 1( (, ! ) ! *.
1.19. 667] , 6396]. ( * - ' RD-+: ,(-,*, 2(*, * ;)( % 2*/ ,*/ ('. 4178] ,' / -), * * ,* (hnp;) - *' ' %'*/ (( ,, ,* - 2*/ ), - 2*/ . ? % 4344] ), -
End (Qp Zp1) , 2 Qp 2(,, p-/ , RD--.
% 6478] % ( 9 W -". ;3 1 4 4 ) 4.
1.20. 667]. =* ( ( ,) - R: 1) R , - J& < 2) %+ '-* A R )7' ' '*/ (+: G1 + : : : + Gn = I + GA < AGi = gIit hI iA .
Z
E, 3; "9 9 RD-", )
;3 1(.
1.21. 667], 6396]. ( ,) - R RD--' 2 2, 2 R - C, %+
*+ ,*+ '( R )2 ,'( (''( -/ '(+.
? , - (/ ' *+ , ,+.
10
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K! O 6476], 6477] , 9;3 RD-!", ( *: 1) "( ! ;, !(
" A 2) , ! RD-!" (((
!" (( ! 4 ( ;;) . ( * 2) , 4, :
% ;% ! RD-!" " 9 ? ) 1.22. 668] , 6396]. ( R RD - *, /
%* ) (/ (+: 1) J 1 = f0g < 2) J - < 3) R ,(-(
- ( aR Ra Ra aR / a 2 R) < 4) R %. 12 R
-,T-. (F J G%
- R , J 1 ,
T
+1
J , 2 J = J J J = J ,*/ 72/).
2Ord
<
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( ;%4 ) !".
x
2 -
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! !. K, -, , ( 4 ; "; \%;" ; 4 (
! ! %. F( ! ( 4 9, 9 1( .
= -!9 44. ( 9 | ) 4! 1 -. < , '(Ix) = \A j xIB" (() -, 4( 4 ( -
(( : D'(Ix) = \9zI(Ixt = BIz t ^ AIz t = I0t )". < (( ( -, 4( 4 4 -. =, , 4( 4 - a j x, ) 4 -
ax = 0 %. 6390] 1) D2 ' ' A 2) ' ! , D ! D' , \4!".
F" 6277] 4! 9 (9. -9,
, 1 4! 1 ( 9 9 4, , -9, 1 , ( -, , 4; 4 ( %9 ). E, 11
) ;%4 ) \(". $ , ) 4! 1 4! 4, (;( 4 !" 6187] , 6188]. )4 4 %; ( ! ( 9 4 !"),
9( !( ( 6277] % !4 4 4.
- 4 4 h' i ( !4, ' ! , :( ! ') 6' ] * 9 1-- !".
: %34 R 6497] , 3 4 !; . . 14 4 ! M !" 4,
( ) m 2 M M j= (m) ^: '(m) . : 4 , 6D D'] * 9 1--,
3 4 14 . . ! DM )
n 2 DM 4, DM j= D'(n) ^ : D(n) . M ! DM (
4! F" 1 1 . . (,
;3 ! ( 6277]) | %, , !-%! . :4 *4 4
6277] ((( %4 D DM .
1( 4 F" 4 -!4 ( ; ) ( 9 4.
2.1. ( mI M , nI 2 N * %* .' (+ *) ,2 '( M 2 N . 12 mI nIt = 0 %+
2(,, M N , + , ,,-&'( '(Ix) , M j= '(m)
I N j= D'(In) .
= 6395] %%3 ) !, 4 !4 - % ) (.
% 6398] 4! ( 1 !9
%9 ( 4 9 4
. K( 14 - '(Ix) 1 ! F' . . 9 4 mod ; R ; %9 , 1
F'(M) = '(M) ( M 2 mod ; R 4 9 ". 0 F' . . % T 9 9 mod ; R Ab ;%4 . . (= 14) (n-4) %;3 4 . G , ;%4 . .
% T F' =F ( 9 - ' , 9, ! ' . 4 F" ( ! ( 1
. . F 2 T !4 DT , 4;34 . .
9 4 R ; mod Ab . =14 ! ( 6398] (!
)4 4 4 4!; < 6267] *( (-(, *4 %1 , (
4 ( 4 ( % 1 ( 9
(9).
H3 9 -! 9 4 "
F" 6275] (4, (9 3 4 3(;39( !4 ! !" 9 12
. . , . . , . . . . 4. ' % -! ! ! < R 6502] , 6504], , , , ! !4 3(;34( ! * !". $ , 1 %
( 3(;34( !, " ( \4".
$( 3 ;% , (;3 ; 4 ; 4 | !9 4. - 1 ( S-! ( ( S . . 9 4), %3 % 670], ( ( RD-. ) % (( RD- !") M 6228].
!"
2.2. *+ '( M -' R )* &*' ,
%2 .' r 2 R M * ,2(,, r(M), *,* ( (: 1) 0(M) = f0g < 1(M) = M < 2) ab 2 Rc .' a , b , c 2 R , a(M)b c(M) .
;3 1, 6228].
2.3. 1) M'( r(M) = Mr ) &- '( M , )*'( +. D'', Mr = '(M) , 2 '(x) = \r j x" RD-&'(, ,.'( . &- ) ' RD-+ < 2) M N '( N &, &- M , )' &'(+
r(M) = M \ r(N) )* (-+.
@, !" M %; ;; ;
* L , ( !"( ! !*4 ) L , a
!"(, "( 4 !"4 4 %
( M, | %!*4. ;3 1 , !" ; %" 9.
2.4. 665], 6228]. 8 M &*+ '(, + M N , &- M (- + &-+ N , r(M) = M \ Nr / r 2 R .
$ 9 !9 4 % ; ; %
; F , % 4 ((;( ! , | 4, 9(;3 !"; . @, )4 | ) 1( 4, ( | ;( M N , !"( M " N . : % F ((;( 4
!"4 ( ) ).
1, !4 ! M ! ( F ,
( ;% ( K N F ;%4 !4 f : ! M
1( g : N ! M . ;3 1 ( )
( RD-!; .
13
2.5. 665] , 6228]. M*+ '( M 9 2 F 2 2, 2 M RD-9 ' ( &- .
3 4 4 % ! % ; F ; S U 9 9 4 9 4 fR=Rr j r 2 Rg ; %9
. @* ) 1. M! ; M T (M) 2 U , 1) T(M)(R=Rr) = M=r(M) %9 A 2) f : R=Rr ! R=Rs , 4 1 ) t 2 R ,
rt 2 Rs , ) r(M)t s(M) 1 t " %9 T(M)(f) : M=r(M) ! M=s(M) A 3) g : M ! N F , " %1( M=r(M) ! N=r(N) ; % T(M) ! T(N) . :, 1 : F ! U , % ( . @,
M R=Rr = M=Mr ! ;
M ) 1 1( M ; . 0! 1 2.4 ; "; ;% Im()
M ; ( ( M .
% 667] , " 1 4 F 4 9(;39 ) 9 9 fR=Rr j r 2 Rg Ab . )4 ( 1( 1 !
\; 4" ( 4 ; 9 9 9(;39 ) R ; mod Ab .
; ) % 4 *4 ": % ;% - !" - ?
=, !" R ( - , ;%4 4 ! R ( (; 19 4. :
3 ! )9 4 4 () !" R :
1) ;%4 4 ! R . . A 2) ;%4 4 ! R . . A 3) ;%4
4 ! R !-% 6389] . 2;% - !" R
3 6 jRj + ! 19 9 9 4 R , 1. 0 1,
%( - !" %! . ! ! 6389], ;% - !" 4
4, 94 ) ;3: % ;%
- !" ! 4 4 ( ! % 9 19 4 )?
- !" | ) !" 4 4 %!4 0, ! ((;( 9 ". G ! ! (; .
0 - 9 9( ! 6389]: 3 ! 9 19 9 4 4 - !". ' ! % R-F 6506], 4
14
. . , . . , . . ! % . H 1, 3!; ( !", %) 6490].
= F" 6278] " %1; % ; \! !"", %; ( , , *( 1! ( PI-". 6458] , 6459] (! %
4- *4 4 < 3 4 4 4 !; " , (
3 1.
H3 9"; -9 " <4( M 694]: !" R - ! , ;%4 4 ! R ! --2 !, ! ;%4 Q
Q
M Ai ! M Ai ((( ( ;%9 9
i2I
i 2I
4 Ai , i 2 I R . --2 4 ((;( . .
. . 0-!4 9 9 ; C 6423].
=, , M ! --2 ! , M ! 1 - 9 |
! ) \!" \%!*9" . . 4. $
, )4 % , - !" %!
9 !", --2 4 9
4 9(( ! !9 4. H3 ' , . . ! 9( ); )! 4, ! ( ) )9 4 M , N
( %(9 M N) . . ( K M K N K . 3 ( , 4 9 --2.
x
3
-! ) ( ! % -! () 9 " {9 4, 4 (;3 " . . 4 " !" % !" (.
! 4. H M ! mI ) M , { ppM (m)
I % mI M ( % 4
p = p+ : p; , p+ = f'(Ix) j ' { M j= '(m)
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9 xI ( % { ( xI) 9 "4 p = p+ : p;
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{ ( !; )) 1 p+ p; .
15
;%4 %4 { p(Ix) ( M , ! p = ppM (m)
I ( % ) mI 2 M . P, {
p (( 4 4 !;, ) 1( p+ ( 1 1 q p+ , p+ = hqi). -4
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I = p A 2) nI N ( . . ( N ppN (In) = p , 3 ( 3(;3()
1 f : N(p) ! N , f(m)
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1 .
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I , ! M j= (m)
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16
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R, !, I 4, J 4 !" R, I 6= ei R ,
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R 6497] 1 { 1( 3.6.
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7 1-,,{&'( .
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6 v 2 e1 Re2 , I1 = vI2 , J2 = J1 v .
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)'2 (-,2) '(.
. K1 ! ; !. ! M . . 4 !
R ) 0 6= m 2 Me ( e 2 R % ,
I = fr 2 eR : xr = 0g (( 9 ) Me) .
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N(p) 4. 2
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%4 4( A1(k) 4 9?
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-,*' ,(-( -' , '(a1 b1)^: : :^'(an bn) ! '(a b)
2 2, 2 '(ai bi) ! '(a b) 2 i .
21
E, ) 1 ! 673], !;3 !! , % " 4 $(-W.
;3 1 \" 1-{ " - !" 9 9 \".
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R ' */ R ,*' '' /, 1) b 2 aR , f(b) f(a) , &(- f ) < T
2) f(0) = f(1) 6= R < 3) I(f) = fr 2 R : f(r) = g J(f) = f
f(s)g , s2RnI (f )
I(f) ,*+, J(f) *+ %* * - R , ,' s r 6= 0 /
r 2 R n I(f) , s 2 R n J(f) . = ) , (': '(a b) 2 p , a 2= f(b) .
M "( f , ( 3 1, ! 4 " M 6247]. E, ", ;34 1 {, ( !, (4
19 4 .
%( % 1 ", 1 ! . . %
4. 0, ( ( 19 4 666] !!
" .
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1 % *( ). K1 ( $@= (, ! 1( 19 4 N(p) (; 9.
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22
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x
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)+ %2 2 ,' *' .
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!" % .
: )" ( { 9 4 !" ( 1.1) , %4 ! M !" R !
, ;%( 1-{ '(x) ( M ; ;3; M '(M) . , ;%4 !4 !
!-%, ! ( ; ! ( 9 .
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, % ,2(,, Zp1 , 1-,,{&'( \xp = 0" , (!)
Zpn , %(, %( , n > 1 , ,2(,,(.
;3 " % (
! !9 4 ; 9 4 %( ".
23
#$
4.3. 664] , 6277] K( M -' R '' 2 2, 2 '' AnnM -'(, ,' R=AnnM .' (
%, ' . C / -, '/ ''*
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!( - % !! . ( - "
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 ( !.
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? r * / 0 6= r 2 R .
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( ;%4 % % ", 14 ;34
9 9 640] . E, 4 ! !4
! ! . =, ) ( ;%4 %4 % 663]. O4 , ! 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! .
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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]
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%4 4( A1 (k) ( 9 0) % 9 4: (
;%4 -4 N M % N , % M=N k .
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Z
x
5 %# &##
25
) " 4 -! 4 | %!!, %!!, !.
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!-%!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 .
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! , i 6 j) .
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%!( ( ! 1, ! 2 9 4 ;%4 \ %!*4" 3 .
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% ; . 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 !
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26
. . , . . , . . @"!4 ) W9 6443]. H -!" !- !" R "4 K1%
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698] , -!" !" %!4 4 K1% , !. = !" !-%!4 4 C4 6174]. %4 % ! % %!9 4 , , %!9 4 ",
% G 696] , . 1 6122].
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4 " 4 ! ( ; !-9 4
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27
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1 " ! 4 j'i (M)='i+1(M)j % ( 9 i
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2] ' (. (. () * // +. |
1992. | . 4, . 1. | ". 75{97.
3] ') /. 0., 12 (. 3., 4
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7
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6] ' . (. "
476 +2
7 // ". . =. |
1992. | . 33, N 4. | ". 24{29.
7] ' (. >. ?
76 9
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8] '
@ 8. 2
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9] '
7 1. . 9 =: 6, // 0+. +
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10] '
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6 // 0
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1987. | . 25. | ". 3{66.
11] (
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9
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12] 8
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) // ". . =. |
1983. | . 24, N 4. | ". 201{205.
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9 7
76 // . (. 9. | : 1990. | ". 7{30.
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9 // . | 1991. |
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19] 8
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20] 8
9 B. 1., /
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21] =
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22] + C.. * 76* // 4. +. | 1975. | . 18,
N 5. | ". 705{710.
23] B2
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1980.
24] B2
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9 9
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25] B2
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D
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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
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30] F . ?. H+2
7 G* ) * +) 6
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32] F7 '. 0. /
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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
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9 // ". . =. |
1983. | . 24, N 6. | ". 56{65.
41
36] 0
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37] /9
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38] /) 4. . ? 76, K
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40] /
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41] / B. 4. 4
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43] /=
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44] /+9 (. . H G
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47] 4) . 4
7 // "
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48] 4
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49] 4)
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50] 4* . (. 7
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3 """H. | 1986. | . 289, N 6. | ". 1304{1308.
51] 42 .?., "
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53] 4
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55] 4
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56] 4
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1990. | . 31, N 3. | ". 94{108.
57] 4
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42
. . , . . , . . 58] 4
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7
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59] 3
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76
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60] ?K B. . " * ) // "
9 . .
L 1. | 4. : 3. | 1982. | ". 320{387.
61] ? H. 6 . | 4. : 4. | 1986.
62] ? (. . 0Q 7 // . | 1994. |
. 33, N 2.
63] ? (. . 47 76 //
H
7. | 1994.
64] ? (. ., ?) 8. B. 47 K@
76
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65] ?) 8. B. L
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66] ?) 8. B. "+
= 9
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67] ?) 8. B. /
76, 9
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68] ?) 8. B. RD-@
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70] ?) 8. B. 76* :
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72] ?) 8. B. +9 9
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73] ?) 8. B. -
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74] H
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75] H
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76] ")
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77] "
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78] "
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79] "
1. . O ) . | 4. : 3. | 1983.
43
80] "
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) 6)) , . 2. { 4. : 3. | 1991.
82] "
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83] . . 76 G
// :. . =. | 1986. | . 38, N 1. | ". 63{67.
84] K 1. (. 7
) * ) ) // . | 1982. | . 21, N 1. | " 73{83.
85] T
9 . 8. :7 * * 6 // 4.
+. :. . | 1984. | . 13, N 1. | " 156{164.
86] L*9 ". . @1 -
9 76 // ". . =. | 1977. |
. 18, N 4. | " 908{914.
87] O
@ ?. -
= 9
) * // 4. : 4. | 1980.
88] Abian A. Categoricity of denumerable atomless Boolean rings // Stud. Log. | 1972. |
V. 30. | P. 63{68.
89] Adler A. | On the multiplicative semigroup of rings // Comm. Algebra. | 1978. | V.6,
N 17. | P. 1751{1753.
90] Ahlbrandt G., Ziegler M. What is special about Z=4Z(!) // Arch. Math. Logic. | 1992. |
V. 31, N 2. | P. 115{132.
91] Auslander M., Bridger M. Stable module theory // Mem. Amer. Math. Soc. |1976. |
N 94.
92] Azumaya G. Countable generatedness version of rings of pure global dimension zero //
Lond. Math. Soc. Lect. Note Ser. |1992. | N 168. | P. 43{79.
93] Azumaya G. Locally pure-projective modules // Contemp. Math. | 1992. | V. 124. |
P. 17{22.
94] Azumaya G., Facchini A. Rings of pure global dimension zero and Mittag-Le`er modules
// J. Pure. Appl. Algebra. |1989. | V. 62. | P. 209{222.
95] Baldwin J. T. Some EC classes of rings // Zeitschr. math. Log. Grundl. Math. |1978. |
V. 24, N 6. | P. 489{492.
96] Baldwin J.T. Stability theory and algebra // J. Symb. Logic. |1979. | V. 44, N 4. |
P. 599{608.
97] Baldwin J. T., Lachlan A. H. On universal Horn classes categorical in some in|nite power
// Algebra Univ. | 1973. | V. 3. | P. 98{111.
98] Baldwin J. T., Rose B. @0 -categority and stability of rings // J. Algebra. | 1977. | V. 45,
N 1. | P. 1{16.
99] Barwise J., Eklof P. In|nitary properties of abelian torsion groups // Ann. Math. Logic. |
1970. | V. 2. | P. 25{68.
100] Bauditsch A. The elementary theory of abelian groups with m-chains of pure subgroups
// Fund. Math. | 1981. | V. 112, N 2. | P. 147{157.
101] Bauditsch A. On elementary properties of free Lie algebras // ICM{82, Warszawa. |
1983. | Sect. 1. | P. 4.
44
. . , . . , . . 102] Bauditsch A. Corrections and supplement to \Decidability of the theory of abelian groups
with Ramsey quanti|ers" // Bull. Acad. Sci. Polon. | 1983. | V. 31, N 3{4. | P. 99{105.
103] Bauditsch A. Magidor-Malitz quati|ers in modules // J. Symb. Logic. | 1984. | V. 49,
N 1. | P. 1{8.
104] Bauditsch A. Tensor products of modules and elementary equivalence // Algebra. Univ. |
1984. | V. 19, N 1. | P. 120{127.
105] Bauditsch A., Rothmaler Ph. The strati|ed order in modules // Math. Res. | 1984. |
V.20. | Berlin: Akademie-Verlag.
106] Bauditsch A., Seese D., Tuschik P., Weese M. Decidability and quanti|ers elimination //
Chapter VII in: Model-theoretic logic. | New-York: Springer, 1985.
107] Baur W. Decidability and undecidability of the theory of abelian groups with predicates
for subgroups // Compos. Math. | 1975. | V. 31. | P. 23{30.
108] Baur W. @0 -categorical modules // J. Symb. Logic. | 1975. | V. 40, N 2. | P. 213{220.
109] Baur W. Undecidability of the theory of abelian group with a subgroup // Proc. Amer.
Math. Soc. | 1976. | V. 55, N 1. | P. 125{128.
110] Baur W. Elimination of quanti|ers for modules // Isr. J. Math. | 1976. | V. 25, N 1{2. |
P. 64{70.
111] Baur W. On the elementary theory of quadruples of vector spaces // Ann. Math. Logic. |
1980. | V. 19. | P. 243{262.
112] Baur W., Cherlin G., Macintyre A. Totally categorical groups and rings // J. Algebra. |
1979. | V. 57, N 2. | P. 407{440.
113] Bauval A. Une condition n„ecessaire d equivalence „el„ementaire entre ann„eaux de polin^omes
sur des corps // C. R. Acad. Sci. Paris., Ser. A. | 1982. | V. 295, N 2. | P. 31{33.
114] Bazzoni S., Salce L. On nonstandart uniserial modules over valuation domains and their
quotients // J. Algebra. | 1990. | V. 128, N 2. | P. 291{305.
115] Becker Th. Real closed rings and ordered valuation rings // Zeitschr. math. Log. Grundl.
Math. | 1983. | V. 29, N 5. | P. 417{425.
116] Becker J., Denef J., Lipshitz L. Further remarks on the elementary theory of formal power
series rings // Lect. Notes Math. | 1980. | V. 834. | P. 1{9.
117] Becker J., Lipshitz L. Remarks on the elementary theories of formal and convergent power
series // Fund. Math. | 1980. | V. 105, N 3. | P. 229{239.
118] Becker T., Fuchs L., Shelah S. Whitehead modules over domains // Forum. Math. |
1989. | V. 1, N 1. | P. 53{68.
119] Beidar K. I., Martindale W. S. 3rd, Mikhalev A. V. Rings with generalized polynomial
identities. | Marcel Dekker, 1995.
120] Berline Ch. Cat„egoricite en @0 du groupe lin„eare d un anneau de Boole // C. R. Acad.
Sci. Paris., Ser. A | 1975. | V. 280, N 12. | P. 753{754.
121] Berline Ch. Elimination of quanti|eurs for nonsemisimple rings of characteristic p // Lect.
Notes Math. | 1980. | V. 834. | P. 10{19.
122] Berline Ch. Stabilit„e et alg„ebre. 3. Anneaux. Groupe „etude th„eor. stables // Univ. P. et
M. Curie. | 1978-79. (1980). | 3/01 { 3/08.
123] Berline Ch. Rings which admit elimination of quanti|ers // J. Symb. Logic. | 1981. |
V. 46, N 1. | P. 56{58.
0
0
45
124] Berline Ch. Ideaux des anneaux de Peano (d apr„es Cherlin) // Lect. Notes Math. |
1981. | V. 890. | P. 32{43.
125] Berline Ch., Cherlin G. QE-rings in characteristic p // Lect. Notes Math. | 1981. |
V. 859. | P. 16{31.
126] Berline Ch., Cherlin G. QE-rings in prime characteristic // Bull. Soc. Belg. Ser. B |
1981. | V. 33, N 1. | P. 3{17.
127] Berline Ch., Cherlin G. QE-rings in characteristic pn // J. Symb. Logic. | 1983. | V. 48,
N 1. | P. 140{162.
128] Bican L. Notes on purities // Czechosl. Math. J. | 1972. | V. 22, N 97. | P. 525{534.
129] Bo‰a M. Mod„eles universels homog„enes et mod„eles g„eneriques // C. R. Acad. Sci. Paris.,
Ser. A. | 1972. | V. 274, N 9. | P. 693{694.
130] Bo‰a M. Corps -clos // C. R. Acad. Sci. Paris., Ser. A | 1972. | V. 275, N 19. |
P. 881{882.
131] Bo‰a M. Sur l existence des corps universels-homog„enes // C. R. Acad. Sci. Paris. |
1972. | V. 275, N 25. | P. 1267{1268.
132] Bo‰a M. A note on existentially complete division rings // Lect. Notes Math. | 1975. |
V. 498. | P. 56{59.
133] Bo‰a M., Cherlin G. Elimination des quanti|cateurs dans les faisceaux // C. R. Acad.
Sci. Paris., Ser. A. | 1980. | V. 290, N 8. | P. 355{357.
134] Bo‰a M., Macintyre A., Point F. The quanti|er elimination problem for rings without
nilpotent elements and for semisimple rings // Lect. Notes Math. | 1980. | V. 834. |
P. 20{30.
135] Bo‰a M., van Praag P. Sur les corps g„en„eriques // C. R. Acad. Sci. Paris., Ser. A. |
1972. | V. 274, N 18. | P. 1325{1327.
136] Bo‰a M., van Praag P. Sur les sous champs maximaux des corps g„en„eriques d„enomerables
// C. R. Acad. Sci. Paris., Ser. A. | 1972. | V. 275, N 20. | P. 945{947.
137] Bouscaren E. Existentially closed modules: types and prime models // Lect. Notes
Math. | 1980. | V. 834. | P. 31 {43.
138] Brenner Sh. Endomorphism algebras of vector spaces with distinquished sets of subspaces
// J. Algebra. | 1967. | V. 6. | P. 100{114.
139] Brenner Sh. Decomposition properties of some small diagrams of modules // Symp.
Math. | 1974. | V. 13. | P. 127{141.
140] Brenner Sh. On four subspaces of vector space // J. Algebra. | 1974. | V. 29. |
P. 587{599.
141] Brenner Sh., Butler M. On diagrams of vector spaces // J. Austral. Math. Soc. | 1969. |
V. 9. | P. 445{448.
142] Brune H. Some left pure semisimple ringoids which are not right pure semisimple //
Comm. Algebra. | 1979. | V. 7, N 17. | P. 1795{1803.
143] Brune H. On |nite representation type and a theorem of Kulikov // Lect. Notes Math. |
1980. | V. 832. | P. 170{176.
144] Buechler S. The classi|cation of small weakly minimal sets III : modules // J. Symb.
Logic. | 1988. | V. 53, N 3. | P. 975{979.
0
0
46
. . , . . , . . 145] Bunge M. Sheaves and prime model extensions // J. Algebra. | 1981. | V. 68, N 1. |
P. 79{96.
146] Bunge M., Reyes G. E. Boolean spectra and model completions // Fund. Math. | 1981. |
V. 113, N 3. | P. 165{173.
147] Burris S., Lawrence J. De|nable principal congruences in varities of groups and rings //
Alg. Univ. | 1979. | V. 9, N 2. | P. 152{164. Corrigendum: ibid. | 1981. | V. 13,
N 2. | P. 264{267.
148] Burris S., McKenzie R. Decidability and Boolean representations // Mem. Amer. Math.
Soc. | 1981. | N 246.
149] Camps R., Dicks W. On semilocal rings // Israel J. Math. | 1993. | V. 81. | P. 203{211.
150] Camps R., Fachini A. The PrŠufer rings that are endomorphism rings of artinian modules
// Comm. Algebra. | 1994. | V. 22, N 8. | P. 3133{3157.
151] Camps R., Facchini A. Chain rings and serial rings that are endomorphism rings of artinian
modules. | Preprint. | 1993.
152] Camps R., Menal P. Power cancellation for artinian modules // Comm. Algebra. |
1991. | V. 19. | P. 2081{2095.
153] Carson A. B. The model completion of the theory of commutative regular rings // J.
Algebra. | 1973. | V. 27, N 1. | P. 136{146.
154] Carson A. B. Algebraically closed regular rings // Canad. J. Math. | 1974. | V. 26,
N 5. | P. 1036{1049.
155] Cegielski P. Le th„eorie „el„ementaire de la multiplication // C. R. Acad. Sci. Paris., Ser. A. |
1980. | V. 290, N 20. | P. 935{938.
156] Cegielski P. La th„eorie d axiomes de la multiplication est cons„equence d un nombre |ni
d axiomes de I‹0 // C. R. Acad. Sci. Paris., Ser. A. | 1981. | V. 293. | P. 351{352.
157] Cegielski P. Th„eorie „el„ementaire de la multiplication des entiers naturels // Lect. Notes
Math. | 1981. | V. 890. | P. 44{89.
158] Cegielski P. Le th„eorie „el„ementaire de la divisibilit„e est |nitement axiomatizable // C. R.
Acad. Sci. Paris., Ser. A. | 1984. | V. 299, N 9. | P. 367{370.
159] Chadzidakis Z. La repr„esentationssen termes faisceaux des mod„eles de la th„eorie
„el„ementaire de la multiplication des entieres naturels // Lect. Notes Math. | 1981. |
V. 890. | P. 90{110.
160] Chadzidakis Z., Cherlin G., Srour G., Shelah S., Wood C. Ortogonality of types in separably closed |elds // Lect. Notes. Math. | 1987. | V. 1292. | P. 71{78.
161] Chase S. U. Torsion free modules over K (xy) // Paci|c J. Math. | 1962. | V. 12,
N 2. | P. 437{447.
162] Cherlin G. The model companion of a class of structures // J. Symb. Logic. | 1972. |
V. 37, N 3. | P. 546{556.
163] Cherlin G. Algebraically closed commutative rings // J. Symb. Logic. | 1973. | V. 38. |
P. 493{499.
164] Cherlin G. Ideals in some nonstandart Dedekind rings // Log. etanal. | 1975. | V. 18,
N 71{72. | P. 379{406.
165] Cherlin G. Model theoretic algebra. Selected topics // Lect. Notes Math. | 1976. |
V. 521.
0
0
0
47
166] Cherlin G. Amalmagation bases for commutative rings without nilpotent elements // Isr.
J. Math. | 1976. | V. 25, N 1{2. | P. 87{96.
167] Cherlin G. Superstable division rings // Log. Coll 77. (Wroclaw). | Amsterdam: 1978. |
P. 99{111.
168] Cherlin G. Lindenbaum algebras and model companions // Fund. Math. | 1979. |
V. 104, N 3. | P. 213{219.
169] Cherlin G. On @0 -categorical nilrings I // Alg. Univ. | 1980. | V. 10, N 1. | P. 27{30.
170] Cherlin G. On @0 -categorical nilrings II // J. Symb. Logic. | 1980. | V. 45, N 2. |
P. 291{302.
171] Cherlin G. Rings of continuous functions: decision problems // Lect. Notes Math. |
1980. | V. 834. | P. 44{91.
172] Cherlin G., Dickmann M. A. Anneaux r„eels des functions continues // C. R. Acad. Sci.
Paris., Ser. A. | 1980. | V. 290, N 1. | P. 1{4.
173] Cherlin G., Dickman M. A. Real closed rings II. Model theory // Ann. Pure. Appl.
Logic. | 1983. | V. 26, N 3. | P. 213{231.
174] Cherlin G., Reineke J. Categoricity and stability of commutative rings // Ann. Math.
Logic. | 1976. | V. 9, N 4. | P. 367{399.
175] Cherlin G., Shelah S. Superstable |elds and groups // Ann. Math. Logic. | 1980. |
V. 18, N 3. | P. 227{270.
176] Cohn P. M. On the free product of associative rings // Math. Z. | 1959. | V. 71. |
P. 380{398.
177] Cohn P. The class of rings embeddable in skew |elds // Bull. Lond. Math. Soc. | 1974. |
V. 6, N 2. | P. 147{148.
178] Cohn P. Right principal Bezout domains // J. Lond. Math. Soc. | 1987. | V. 35, N 2. |
P. 251{262.
179] Contessa M. Ultraproducts of PM-rings and MP-rings // J. Pure. Appl. Algebra. |
1984. | V. 32, N 1. | P. 11{20.
180] Corner A. Each countable reduced cotorsion free ring is an endomorphism ring // Proc.
Lond. Math. Soc. | 1963. | V. 13. | P. 687{710.
181] Corner A. Endomorphism algebras of large modules with distinquished submodules // J.
Algebra. | 1969. | V. 11. | P. 155{185.
182] Corner A. On the existence of very decomposable abelian group // Lect. Notes Math. |
1983. | V. 1006. | P. 354{357.
183] Corner A. Fully rigid systems of modules // Rend. Sem. Mat. Univ. Padova. | 1989. |
V. 82. | P. 55{66.
184] Corner A., GŠobel R. Prescribing endomorphism algebras | a uni|ed treatment // Proc.
Lond. Math. Soc., Ser. 3. | 1985. | V. 50, N 3. | P. 447{479.
185] Coushot F. Toplogie co|nie et modules pur-injectifs // C. R. Acad. Sci. Paris., Ser. A. |
1976. | V. 283, N 6. | P. 277{280.
186] Coushot F. Sous modules purs et modules de type co|ni // Lect. Notes Math. | 1978. |
V. 641. | P. 198{208.
187] Crawley-Bouvey W. Tame algebras and generic modules // Proc. Lond. Math. Soc. |
1991. | V. 63, N 2. | P. 241{265.
0
48
. . , . . , . . 188] Crawley-Bouvey W. W. Modules of |nite length over their endomorphism rings // Lond.
Math. Soc. Lect. Notes Ser. | 1992. | N 168. | P. 127{184.
189] Crivei I. c-pure exact sequences of R-modules // Mathematica (Cluj). | 1975. | V. 17
(40), N 1. | P. 59{69.
190] Dales H. C., Woodin W. H. An intriduction to independence for analyses // Lond. Math.
Lect. Note Ser. | 1987. | N 115.
191] Decker J. C. E. Countable vector spaces with resursive operations I // J. Symb. Logic. |
1969. | V. 34, N 3. | P. 363{387.
192] Decker J. C.E. Countable vector spaces with recursive operations II // J. Symb. Logic. |
1971. | V. 36, N 3. | P. 477{493.
193] Delon F. Ind„ecidabilit„e de la th„eorie des anneaux de s„eries formelles „a plusiers
ind„eterminan„ees // Fund. Math. | 1981. | V. 112, N 3. | P. 215{229.
194] Delon F. Id„eaux et types sur les corps s„eparablement clos // M„em. Soc. Math. France. |
1988. | N 33.
195] Delon F. Ind„ecidabilit„e de la th„eorie des paires imm„ediates de corps valu„es henseliens //
J. Symb. Logic. | 1991. | V. 56, N 4. | P. 1236{1242.
196] Dowbor P., Ringel C. M., Simson D. Hereditary artinian rings of |nite representation type
// Lect. Notes. Math. | 1980. | V. 832. | P. 232{258.
197] Dowbor P., Simson D. Quasi-Artin species and rings of |nite representation type // J.
Algebra. | 1980. | V. 63. | P. 435{443.
198] Driess L. van den. Artin-Schreier theory for commutative regular rings // Ann. Math.
Logic. | 1977. | V. 12, N 2. | P. 113{150.
199] Driess L. van den. A linearly ordered rings whose theory admits elimination of quanti|ers
in real closed |elds // Proc. Amer. Math. Soc. | 1980. | V. 79, N 1. | P. 97{100.
200] Driess L. van den. Quanti|er elimination for linear formulas over ordered and valued |elds
// Bull. Soc. Math. Belg., Ser. B. | 1981. | V. 33, N 1. | P. 19{31.
201] Driess L. van den, Holly J. Quanti|er elimination for modules with scalar variables //
Ann. Pure. Appl. Logic. | 1992. | V. 57, N 2. | P. 161{179.
202] Dugas M. On the existence of large mixed modules // Lect. Notes Math. | 1983. |
V. 1006. | P. 412{424.
203] Dugas M., GŠobel R. On endomorphism rings of primary abelian groups // Math. Ann. |
1982. | V. 261, N 3. | P. 359{385.
204] Dugas M., GŠobel R. Every cotorsion-free algebra is an endomorphism algebra // Math.
Z. | 1982. | V. 181. | P. 451{470.
205] Dugas M., GŠobel R. Every cotorsion-free ring is an endomorphism ring // Proc. Lond.
Math. Soc. | 1982. | V. 45, N 2. | P. 319{336.
206] Dugas M., GŠobel R. Endomorphism algebras of torsion modules II // Lect. Notes Math. |
1983. | V. 1006. | P. 400{411.
207] Dugas M., GŠobel R. Representation of algebras over a complete discrete valuation ring //
Quart. J. Math., Ser. 1. | 1984. | V. 35, N 138. | P. 131{146.
208] Dugas M., GŠobel R. Endomorphism ring of separable torsion free Abelian groups //
Houston J. Math. | 1985. | V. 11. | P. 471{483.
49
209] Dugas M., Irwin J., Khabbaz S. Countable rings as endomorphism rings // Quart. J.
Math. | 1988. | V. 39, N 154. | P. 201{211.
210] Duret J. L. Instabilit„e des corps formellent r„eels // Canad. Nath. Bull. | 1977. | V. 20,
N 3. | P. 385{387.
211] Eisenbud D., Robson J. C. Modules over Dedekind prime rings // J. Algebra. | 1970. |
V. 16, N 1. | P. 67{85.
212] Eisenbud D., Robson J. C. Hereditary noetherian prime rings // J. Algebra. | 1970. |
V. 16, N 1. | P. 86{104.
213] Eklof P. Homogeneous universal modules // Math. Scand. | 1971. | V. 29. | P. 187{196.
214] Eklof P. Some model theory of abelian groups // Fund. Math. | 1972. | V. 37, N 2. |
P. 335{342.
215] Eklof P. Whitehead s problem is undecidable // Amer. Math. Monthly. | 1976. |
V. 83. | P. 775{788.
216] Eklof P. Set theory and structure theorems // Lect. Notes Math. | 1993. | V. 1006. |
P. 275{284.
217] Eklof P., Fisher E. The elementary theory of abelian groups // Ann. Math. Logic. |
1972. | V. 4, N 1. | P. 115{171.
218] Eklof P., Fuchs L. Baer modules over valuation domains // Ann. Math Pure Appl. |
1988. | V. 150. | P. 363{374.
219] Eklof P., Herzog I. Some model theory over a serial ring. // Preprint. | 1993.
220] Eklof P., Meckler A. On endomorphism rings of !1 -separable abelian groups // Lect. Notes
Math. | 1983. | V. 1006. | P. 320{339.
221] Eklof P., Meckler A. Almost free modules. Set theoretic methods. | North.-Holland Math.
Library. | 1988.
222] Eklof P., Mez H.C. The ideal structure of existentially closed algebras // J. Symb. Logic. |
1985. | V. 50, N 4. | P. 1025{1043.
223] Eklof P., Mez H. C. Modules over existentially closed algebras // J. Symb. Logic. |
1987. | V. 52, N 1. | P. 54{63.
224] Eklof P., Sabbagh G. Model completion and modules // Ann. Math. Logic. | 1971. |
V. 2, N 3. | P. 251{295.
225] Ellentuck E. Boolean valued rings // Fund. Math. | 1977. | V. 96, N 1. | P. 67{86.
226] Facchini A. Decomposition of algebraically compact modules // Pacif. J. Math. | 1985. |
V. 116, N 1. | P. 25{37.
227] Facchini A. Algebraically compact modules // Acta Univ. Carol. | 1985. | V. 26. |
P. 27{37.
228] Facchini A. Relative injectivity and pure{injective modules over PrŠufer rings // J. Algebra | 1987. | V. 110. | P. 380{406.
229] Facchini A. Pure{injective envelope of a commutative ring and localizations // Quart. J.
Math., Ser. 2. | 1988. | V. 39, N 155. | P. 307{322.
230] Facchini A. Rings of |nite representation type, rings of pure global dimension zero and
Mittag-Le`er modules // Rend. Sem. Math. Fis. Milano. | 1989. | V. 59. | P. 65{80.
231] Facchini A. Mittag-Le‰Œer modules, reduced products and direct products // Rend. Sem.
Mat. Univ. Padova. | 1991. | V. 85. | P. 119{132.
0
50
. . , . . , . . 232] Facchini A., Levy S., Vamos P. Krull{Schmidt fails for artinian modules // Preprint. |
1993.
233] Facchini A., Puninski G. ‹-pure-injective modules over serial rings // Preprint. | 1994.
234] Faith K. The basis theorem for modules a brief survey and a look to the future // Lect.
Notes Pure Appl. Math. | 1978. | V. 40. | P. 9{23.
235] Fakhruddin S. M. Finitely generated modules // Comment. Math. Univ. St. Paul. |
1981. | V. 30, N 2. | P. 119{123.
236] Feigelstock S. On modules over Dedekind rings // Acta. Sci. Math. Acad. Hung. | 1977. |
V. 39. | P. 255{263.
237] Felgner U. @1 -kategorische Theorien nichtkommutativer Ringe // Fund. Math. | 1975. |
V. 82, N 4. | P. 331{346.
238] Felgner U. KategiritŠat // Yaresber. Detsch. Math. | 1980. | V. 82, N 1. | P. 12{32.
239] Felgner U. Horn-theories of abelian groups // Lect. Notes. Math. | 1980. | V. 834. |
P. 163{173.
240] Ferrandum M. Ultraproducts et anneaux primitifs // Semin. Dubreil. et Pisot. Fac. Sci.
Paris. | 1967. | V. 19, N 2. | P. 16/01{16/09.
241] Fieldhouse D. Aspects of purity // Lect. Notes Pure Appl. Math. | 1974. | V. 7. |
P. 185{196.
242] Fisher E. Abelian structures // Preprint. | 1974/75. | Yale University.
243] Fisher E. Abelian structures I // Lect. Notes Math. | 1977. | V. 616. | P. 270{322.
244] Franzen B., GŠobel R. Nonstandart uniserial modules over valuation domains // Result.
Math. | 1987. | V. 12. | P. 86{94.
245] Fuchs L. Algebraically compact modules over noetherian rings // Indian J. Math. |
1967. | V. 9. | P. 357{374.
246] Fuchs L., Monari-Martinez E. Butler modules over valuation domains // Canad. J.
Math. | 1991. | V. 43, N 1. | P. 48{60.
247] Fuchs L., Salce L. Modules over valuation domains // Lect. Notes Pure Appl. Math. |
1985. | V. 97.
248] Fuchs L., Shelah S. Kaplansky s problem on valuation rings // Proc. Amer. Math. Soc. |
1989. | V. 105, N 1. | P. 25{30.
249] Fuller K. R., Reiten I. Note on rings of |nite representation type and decomposition of
modules // Proc. Amer. Math. Soc. | 1975. | V. 50. | P. 92{94.
250] Garavaglia S. Direct product decomposition of theories of modules // J. Symb. Logic. |
1979. | V. 44, N 1. | P. 77{86.
251] Garavaglia S. Decomposition of totally transcendental modules | J. Symb. Logic. |
1980. | V. 45, N 1. | P. 155{164.
252] Garavaglia S. Forking in modules // Notre Dame J. Form. Logic. | 1981. | V. 22. |
P. 155{162.
253] Garcia J. L., Simson D. On right pure semisimple PI-rings // Bull. Pol. Acad. Sci. |
1994. | V. 42. | to appear.
254] Goldsmith B. On endomorphism rings of nonseparable Abelian p-groups // J. Algebra. |
1989. | V. 127, N 1. | P. 73{79.
0
51
255] Goodearl K. R., Boyle A.K. Dimension theory for nonsingular injective modules // Mem.
Amer. Math. Soc. | 1976. | V. 7, N 177.
256] Gordon R., Robson J. C. Krull dimension // Mem. Amer. Math. Soc. | 1973. | N 133.
257] GŠobel R. Darstellungen von Ringen als Endomorphismringe // Arch. Math. | 1980. |
V. 35, N 4. | P.338 {350.
258] GŠobel R. Endomorphism rings of abelian groups // Lect. Notes Math. | 1983. |
V. 1006. | P. 340{353.
259] GŠobel R. Vector space with |ve distinquished subspace // Res. Math. | 1987. | V. 11. |
P. 211{218.
260] GŠobel R. An easy topological construction for realizing endomorphism rings // Proc. Irish
Acad. Sect. A. | 1992. | V. 92, N 2. | P. 281{284.
261] GŠobel R., Goldsmith B. Essentially indecomposable modules which are almost free //
Quart. J. Math. | 1988. | V. 39, N 154. | P. 213{222.
262] GŠobel R., Goldsmith B. On separable torsion free modules of countable density character
// J. Algebra. | 1991. | V. 144, N 1. | P. 79{87.
263] GŠobel R., May W. For submodules suŽce for realizing algebras over commutative rings
// J. Pure Appl. Algebra. | 1990. | V. 65, N 1. | P. 29{43.
264] GŠobel R., Shelach S. Modules over arbitrary domains // Math. Z. | 1985. | V. 188,
N 3. | P. 325{337.
265] GŠobel R., Wald R. Separable torsion free modules of small type // Houston. J. Math. |
1990. | V. 16, N 2. | P. 271{287.
266] Gruson L., Jensen C. U. Modules alg„ebriqument compacts et functeurs lim(i) // C. R.
267]
268]
269]
270]
271]
272]
273]
274]
275]
276]
277]
Acad. Sci. Paris, Ser. A. | 1973. | V. 276. | P. 1651{1653.
Gruson L., Jensen C. U. Dimension cohomologiques relie„es aux foncteurs lim(i) // Lect.
Notes Math. | 1981. | V. 867. | P. 234{294.
Guichard D. R. Automorphism of substructure lattices in recursive algebra // Ann. Pure.
Appl. Logic. | 1983. | V. 25, N 1. | P. 47{58.
Gute B., Beuter K. (Poizat B., Ziegler M.) The last word on elimination of quanti|ers in
modules // J. Symb. Logic. | 1990. | V. 55, N 2. | P. 670{673.
Haley D. A note on compactifying Artinian rings // Canad. J. Math. | 1974. | V. 26. |
P. 580{582.
Haley D. Equational compactness in rings // Lect. Notes Math. | 1970. | V. 745.
Hermann C., Jensen C. U., Lenzing H. Applications of model theory to representation of
|nite dimensional algebras // Math. Z. | 1981. | V. 178, N 1. | P. 83{98.
Hermida J., S„anchez-Giralda T. Linear equations for commutative rings and determinantal
ideals // J. Algebra. | 1986. | V. 99, N 1. | P. 72{79.
Hermida J., S„anchez-Giralda T. Some sriteria for solvability of system of linear equation
over modules // Lect. Notes Math. | 1988. | V. 1328. | P. 122{134.
Herzog I. The Auslander-Reiten translate // Preprint. | 1991.
Herzog I. Modules with few type // J. Algebra. | 1992. | V. 149, N 2. | P. 358{370.
Herzog I. Elementary duality for modules // Trans. Amer. Math. Soc. | 1993. | V. 340,
N 1. | P. 37{69.
52
. . , . . , . . 278] Herzog I. A test for |nite representation type // Preprint. | 1993.
279] Herzog I., Rothmaler Ph. Modules with generic types 1{3 // Model theory of groups.
Notre Dame Press. Indiana. | 1989. | P. 138{176.
280] Herzog I., Rothmaler Ph. Modules with generic types. Part 4 // J. Symb. Logic. | 1992. |
V. 57, N 1. | P. 193{199.
281] Hirschfeld J., Wheeler W. H. Forcing, arithmetic, division rings // Lect. Notes Math. |
1975. | V. 454.
282] Hitoshi A. On the Whitehead problem // Math. Sem. Notes Kobe Univ. | 1978. | V. 6,
N 2. | P. 363{367.
283] Hodges W. Constructions pure-injective hulls // J. Symb. Logic. | 1980. | V. 45, N 3. |
P. 544{548.
284] HŠonke H.-J. Reele Polynomgleichungen und Ungleichungen und die Methode der Elimination der Quanti|katoren // Weterbildungszenntr. Math. Kybern. und Rechnentechn.
Sekt. Math. | 1975. | N 11. | P. 94{112.
285] Hrushovski E. The Mordell-Lang conjecture for function |elds // Preprint. | 1993.
286] Jensen C. U. Arithmetical rings // Acta Math. Acad. Sci. Hungar. | 1966. | V. 17,
N 1. | P. 115{123.
287] Jensen C. U. Les functeurs d„eriv„es de lim (i) et leurs applications en th„orie des modules //
288]
289]
290]
291]
292]
293]
294]
295]
296]
297]
298]
299]
Lect. Notes Math. | 1972. | V. 254.
Jensen C.U. Peano rings of arbitrary global dimension // J. Lond. Math. Soc. | 1980. |
V. 21, N 1. | P. 39{44.
Jensen C.U. Properti„es homologiques et logiques des anneaux de functins enti„eres // C.
R. Acad. Sci. Paris, Ser. A. | 1980. | V. 291, N 9. | P. 515{517.
Jensen C. U. Applications logiques on theorie des anneaux et des modules // Conf. Coll.
d Algebre a„ Rennes. | 1980.
Jensen C. U. Sur une classe de corps und„ecidables // C. R. Acad. Sci. Paris, Ser. A. |
1982. | V. 295, N 9. | P. 507{509.
Jensen C. U. L indecidabilite d une classe de corps des functions m„emorphes // Preprint. |
1983.
Jensen C. U., Lenzing H. Model theory and representation of algebras // Lect. Notes
Math. | 1980. | V. 832. | P. 302{310.
Jensen C. U., Lenzing H. Algebraic compactness of reduced products and applications to
pure global dimension // Comm. Algebra. | 1983. | V. 11, N 3. | P. 305{325.
Jensen C. U. , Lenzing H. Model theoretic algebra with particular emphasis on |elds, rings,
modules. | New York. | 1989.
Jensen C. U., Simson D. Purity and generalized chain conditions // J. Pure. Appl. Algebra. | 1979. | V. 14. | P. 297{305.
Jensen C. U., Vamos P. On the axiomatizable of certain classes of modules // Math. Z. |
1979. | V. 167. | P. 227{237.
Jensen C.U., Zimmermann-Huisgen B. Algebraic compactness of ultrapowers and representation type // Pacif. J. Math. | 1989. | V. 139, N 2. | P. 251{265.
Jndrup S., Ringel C. M. Remarks on a paper by Skornjakov concerning rings for which
every module is a direct sum of left ideals // Arch. Math. | V. 31. | 1978. | P. 329{331.
0
0
0
53
300] Jurie P.-F. D„ecidabilit„e de la th„eorie „el„elementaire des anneaux booliens a op„erateurs dans
un groupe |ni // C. R. Acad. Sci. Paris, Ser. A. | 1982. | V. 295, N 3. | P. 215{217.
301] Kaplansky I. Elementary divisors and modules // Trans. Amer. Math. Soc. | 1949. |
V. 66. | P. 464{491.
302] Kaplansky I. In|nite abelian groups. | Ann. Arbor , Michigan. | 1954.
303] Kawada J. On KŠothe's problem concerning algebras for which every indecomposable module is cyclic. I{III // Sci. Rep. Tokyo Kyoiku Daigaku. | 1962. | V. 7. | P. 154{230 ‘
1963. | V. 8. | P. 1{62 ‘ 1964. | V. 9. | P. 165{250.
304] Kielpinski R., Simson D. On pure global dimension // Bull. Acad. Polon. Sci. | 1975. |
V. 23, N 1. | P. 1{16.
305] Kikyo H. Remarks on Zilber s results // Tsukuba J. Math. | 1988. | V. 12, N 1. |
P. 235{240.
306] Klingen N. Zur Idealstruktur in nichtstandart Modellen von Dedekindringen // J. reine
und angew. Math. | 1975. | V. 274{275. | P. 38{60.
307] Klushin A., Kozhukhov I. On algebraically compact semigroup rings // Semigroup. Forum. | 1989. | V. 39, N 2. | P. 215{248.
308] Kucera T. Generalizations of Deissler s minimal rank // J. Symb. Logic. | 1988. | V. 53,
N 1. | P. 269{283.
309] Kucera T. Positive Deissler rank and the complexity of injective modules // J. Symb.
Logic. | 1988. | V. 53, N 1. | P. 284{293.
310] Kucera T. Totally transcendental theories of modules: decomposition of models and types
// Ann. Pure. Appl. Logic. | 1988. | V. 39. | P. 239{272.
311] Kucera T., Prest M. Imaginary modules // J. Symb. Logic. | 1992. | V. 57, N 2. |
P. 698{723.
312] Kucera T., Prest M. For concepts from \geometrical" stability theory in modules // J.
Symb. Logic. | 1992. | V. 57, N 2. | P. 724{740.
313] Lavendhomme R., Lucas T. Charactreization of some theories preserved by localization
// Semin. math. Inst. math. pure et appl. Univ. cathol. Louvain. | 1983. | N 1. |
P. VII/1{VII/16.
314] Lawrence J. Primitive rings do not form an elementary class // Comm. Algebra. |
1981. | V. 9, N 4. | P. 397{400.
315] Lawrence J. The theory of 89-elementary conditions on rings // Math. Repts. Sci.
Canad. | 1983. | V. 5, N 1. | P. 41{45.
316] Leason J. J., Butson A. T. Equationally complete (m n)-rings // Alg. Univ. | 1980. |
V. 11, N 1. | P. 28{41.
317] LindstrŠom P. On model completeness // Theoria. | 1964. | V. 30. | P. 183{196.
318] Lipshitz L. The real closure of a commutative regular f -rings // Fund. Math. | 1977. |
V. 94, N 3. | P. 173{176.
319] Lipshitz L., Saracino D. The model companion of the theory of commutative rings without
nilpotent elements // Proc. Amer. Math. Soc. | 1973. | V. 38, N 2. | P. 381{387.
320] Loveys J. Weakly minimal groups of unbounded exponent // J. Symb. Logic. | 1990. |
V. 55, N 3. | P. 928{938.
0
0
54
. . , . . , . . 321] Loveys J. Locally |nite weakly minimal theories // Ann. Pure Appl. Logic. | 1991. |
V. 56, N 2. | P. 153{203.
322] Lu Chin Pi. Purity of linearly topological rings in overrings // J. Algebra. | 1979. |
V. 59. | P. 290{301.
323] Macintyre A. On !1 -categorical theories of |elds // Fund. Math. | 1971. | V. 71, N 1. |
P. 1{25.
324] Macintyre A. On !1 -categorical theories of abelian groups // Fund. Math. | 1971. |
V. 70. | P. 253{270.
325] Macintyre A. Model completeness for sheaves of structures // Fund. Math. | 1973. |
V. 81. | P. 73{89.
326] Macintyre A. A note on axioms for in|nite generic structures // J. Lond. Math. Soc. |
1975. | V. 9, N 4. | P. 581{584.
327] Macintyre A. Combinatorical problems for skew |elds. I. Analogue of Briton s lemma, and
results of Adjan-Rabin type // Proc. Lond. Math. Soc., Ser. 3. | 1979. | V. 39, N 2. |
P. 211{236.
328] Macintyre A., McKenna K., van den Driess L. Elimination of quanti|ers in algebraic
structures // Adv. Math. | 1983. | V. 47, N 1. | P. 74{87.
329] Macintyre A., Point F. The quanti|er elimination problem for rings without nilpotent
elements and for semisimple rings // Lect. Notes Math. | 1980. | V. 834. | P. 20{30.
330] Macintyre A., Rosenstein J. @0 -categicity of rings without nilpotent elements and for
Boolean structures // J. Algebra. | 1976. | V. 43, N 1. | P. 129{154.
331] Marcja A., Prest M., To‰alori C. Classi|cation theory for abelian groups with an endomorphism // Arch. Math. Logic. | 1991. | V. 31, N 2. | P. 95{104.
332] Marcja A., Prest M., To‰alori C. The stability classi|cation theory for abelian-by-|nite
groups and modules over a group ring // J. Lond. Math. Soc., Ser. 2. | 1993. | V. 47,
N 2. | P. 212{226.
333] Marcja A., To‰alori C. Decidability for modules over a group ring // Comm. Algebra. |
1993. | V. 21. | P. 2251{2264.
334] Marubayashi H. Modules over Dedekind prime rings. II. // Osaka. J. Math. | 1972. |
V. 9, N 3. | P. 427{445.
335] Marubayashi H. Pure-injective modules over hereditary Noetherian prime rings with
enough invertible ideals // Osaka J. Math. | 1981. | V. 18, N 1. | P. 95{107.
336] Matlis E. Injective modules over Noetherian rings // Pacif. J. Math. | 1958. | V. 8. |
P. 514{528.
337] May W. Isomorphism of endomorphism algebras over complete discrete valuation rings //
Math. Z. | 1990. | V. 204, N 4. | P. 485{499.
338] McConnell J.C. , Robson J.C. Homomorphisms and extensions of modules over certain
di‰erential polinomial rings // J. Algebra. | 1973. | V. 26, N 2. | P. 319{342.
339] McConnell J. C., Robson J. C. Noncommutative noetherian rings. | New York. | 1988.
340] Meinel K. Superdecomposable modules over Dedekind domains // Arch. Math. | 1982. |
V. 39, N 1. | P. 11{18.
341] Meckler A. H. Primitive rings are not de|nable in L // Comm. Algebra. | 1982. |
V. 10, N 5. | P.1689 {1690.
0
1 1
55
342] Meckler A. H. Proper forcing and abelian groups // Lect. Notes Math. | 1983. |
V. 1006. | P. 285{303.
343] Meckler A. H., Shelah S. The solution Crawley s problem // Pacif. J. Math. | 1986. |
V. 121. | P. 133{134.
344] Menal P., Vamos P. Pure extensions and self-FP-injective rings // Math. Proc. Cambr.
Phil. Soc. | 1989. | V. 105, N 3. | P. 447{458.
345] Metakides G., Nerode A. Recursive enumerable vector spaces // Ann. Math. Logic. |
1972. | V. 11, N 2. | P. 147{171.
346] Michaux C. Sur l „elimination des quanti|ers dans les anneax diff„erentials // C. R. Acad.
Sci. Paris, Ser. A. | 1986. | V. 302. | P. 287{290.
347] Mohamed S. H., MŠuller B. J. Continuous and discrete modules. | Cambridge. | 1990.
348] Molzan B. Now to eliminate quanti|ers in the theory of p-rings // Z. math. Log. Grundl.
Math. | 1982. | V. 28, N 1. | P. 83{92.
349] Monari-Martinez E. On pure-injective modules // CISM courses and lectures. | 1984. |
V. 287.
350] Monari-Martinez E. On ‹-pure-injective modules over valuation domains // Arch.
Math. | 1986. | V. 46. | P. 26{32.
351] Monari-Martinez E. Some properties of Butler modules over valuation domains // Rend.
Sem. Univ. Padova. | 1991. | V. 85. | P. 185{199.
352] Monk I. Elementary-recursive decision procedures. | Doctoral thesis. | Berkeley. |
1975.
353] MŠuller B. J., Singh S. Uniform modules over a serial rings II // Lect. Notes in Math. |
1989. | V. 1448. | P. 25{32.
354] Naud„e C. G., Naud„e G., Pretorius L. M. Equational characterization of relative injectivities
// Comm. Algebra. | 1986. | V. 14, N 1. | P. 39{48.
355] Naud„e C. G., Naud„e G., Pretorius L. M. Pure-injective modules need not be RD-injective
// Quaest. Math. | 1986. | V. 9. | P. 363{364.
356] Nit„a A. Quelques observation sur les ultraproducts d anneaux // Bull. Math. Soc. RSR. |
1976. | V. 20, N 3{4. | P. 325{328.
357] Nit„a A. Ultraproduse ale unor clase de inele // Bul. Inst. poitechn. Gh. Gheorgiu. |
1980. | V. 42, N 1. | P. 13{17.
358] Okoh F. Indecomposable pure-injective modules over hereditary Artin algebras of tame
type // Comm. Algebra. | 1980. | V. 8, N 20. | P. 1939{1941.
359] Okoh F. On pure-injective modules over Artin alebras // J. Algebra. | 1982. | V. 77,
N 2. | P. 275{282.
360] Okoh F. Pure-injective modules over path algebras // J. Pure Appl. Algebra. | 1991. |
V. 75. | P. 75{83.
361] Olin P. Some |rst order properties of direct sum of modules // Z. Math. Log. Grundl.
Math. | 1970. | V. 16. | P. 405{416.
362] Ono H. Equational theories and universal theory of |elds // J. Math. Soc. Jap. | 1983. |
V. 35, N 2. | P. 289{306.
363] Osofsky B. A construction of nonstandart uniserial modules over valuation domains //
Bull. Amer. Math. Soc. | 1991. | V. 25, N 1. | P. 89{97.
0
0
0
56
. . , . . , . . 364] Osofsky B. Constructing nonstandart uniserial modules over valuation domains // Proc.
Conf. in honor Goro Azumaja s 70-th birthday: Providence. | 1991. | P. 151{164.
365] Perlis D. Group algebras and model theory // Ill. J. Math. | 1976. | V. 20, N 2. |
P. 298{305.
366] Petry A. A. A propos des centralisateurs de certain sous-corps des corps g„en„eriques //
Bull. Soc. Roy. Sci. Li„ege. | 1977. | V. 46, N 3. | P. 113{116.
367] Pillay A., Prest M. Forking and pushouts in modules // Proc. Lond. Math. Soc., Ser. 3. |
1983. | V. 46. | P. 363{384.
368] Pillay A., Prest M. Modules and stability theory // Trans. Amer. Math. Soc. | 1987. |
V. 300, N 2. | P. 641{662.
369] Pillay A., Rothmaler Ph. Unidimensional modules: uniqueness of maximal nonmodular
submoduls // Ann. Pure. Appl. Logic. | 1993. | V. 61. | P. 175{181.
370] Pillay A., Sokolovi’c Z’ . Superstable di‰erential |elds // J. Symb. Logic. | 1992. | V. 57,
N 3. | P. 97{108.
371] Pillay A., Steinhorn C. De|nable sets in ordered srtuctures // Bull. Amer. Math. Soc. |
1984. | V. 11, N 1. | P. 159{162.
372] Pillay A., Scowroft Ph., Steinhorn Ch. Between groups and rings // Rocky Mount. J.
Math. | 1989. | V. 19, N 3. | P. 871{885.
373] Podewski K.-R. Minimale Ringe // Math.-phys. Semesterber. | 1975. | V. 22, N 2. |
P. 193{197.
374] Podewski K.-R., Reineke J. An !1 -categorical rings is not strongly minimal // J. Symb.
Logic. | 1974. | V. 3, N 4. | P. 665{668.
375] Podewski K.-R., Reineke J. Algebraically closed commutative local rings // J. Symb.
Logic. | 1979. | V. 44, N 1. | P. 89{94.
376] Podewski K.-R., Reineke J. On algebraically closed commutative indecomposable rings //
Alg. Univ. | 1981. | V. 12, N 1. | P. 123{131.
377] Point F. Elimination des quanti|cateurs dans les L-anneaux -reguliers et application aux
anneaux bireguliers // Bull. Soc. Math. Belg., Ser. B. | 1981. | V. 33, N 1. | P. 93{107.
378] Point F. Sur l elimination lin„eare // C. R. Acad. Sci. Paris., Ser. A. | 1982. | V. 295. |
P. 211{213.
379] Point F. Probl„emes de d„ecidabilit„e pour les th„eories des modules // Bull. Belg. Math.
Soc., Ser. B. | 1986. | V. , N . | P. {.
380] Point F. Decidability of the theory of modules over the dihedral algebra // Preprint. |
1990.
381] Point F., Prest M. Decidability for theories of modules // J. Lond. Math. Soc. | 1988. |
V. 38, N 2. | P. 193{206.
382] Praag P. van den. Sur les centralisateurs des corps de type |ni dans les corps existellement
clos // C. R. Acad. Sci. Paris., Ser. A. | 1975. | V. 281, N 21. | P. 891{893.
383] Prest M. Model completions of some theories of modules // J. Lond. Math. Soc., Ser. 2. |
1979. | V. 20. | P. 369{372.
384] Prest M. Quanti|er elimination in modules // Bull. Soc. Math. Belg., Ser. B. | 1981. |
V. 33, N 1. | P. 109{120.
0
0
57
385] Prest M. Elementary equivalence of ‹-injective modules // Proc. Lond. Math. Soc.,
Ser. 3. | 1982. | V. 45. | P. 71{88.
386] Prest M. Existentially complete prime rings // J. Lond. Math. Soc., Ser. 2. | 1983. |
V. 28, N 2. | P. 238{246.
387] Prest M. Rings of |nite representation type and modules of |nite Morley rank // J.
Algebra. | 1984. | V. 88, N 2. | P. 502{533.
388] Prest M. The generaized RK -order, orthogonality and regular types // J. Symb. Logic. |
1985. | V. 50, N 1. | P. 202{219.
389] Prest M. Duality and pure-semisimple rings // J. Lond. Math. Soc. | 1988. | V. 38,
N 3. | P. 403{409.
390] Prest M. Model theory and modules. | Cambridge. | 1988.
391] Prest M. Wild representation type and undecidability // Commun. Algebra. | 1991. |
V. 19, N 3. | P. 919{929.
392] Prest M. The category of modules and decidability // Preprint. | 1991.
393] Prest M. Decidability for modules | summary // Preprint. | 1992.
394] Prest M. Remarks on elementary duality // Preprint. | 1993.
395] Prest M. Tensor product and theories of modules // Preprint. | 1994.
396] Prest M., Puninski G., Rothmaler Ph. Rings described by various purities // Preprint. |
1994.
397] Prest M., Rothmaler Ph., Ziegler M. Absolutely pure and Œat modules and indiscrete
rings // Preprint. | 1992.
398] Prest M., Rothmaler Ph., Ziegler M. Extension of elementary duality // Preprint. | 1992.
399] Prestel A., Ziegler M. Non-axiomatizable classes of V -topological |elds // J. fŠur reine und
angew. Math. | 1980. | V. 316. | P. 211{214.
400] Puninski G. Uniserial rings with local endomorphosm rings of |nitely presented indecomposable modules and pure-injective modules over them // Preprint. | 1993.
401] Puninski G., Wisbauer R. ‹-injective modules over left duo and left distributive rings //
Preprint. | 1994.
402] Rangaswamy K. M. An aspects of purity and its dualization in abelian groups and modules
// Symp. Math. | 1979. | V. 23, N 4. | P. 307{320.
403] Reineke J. Minimale Gruppen // Z. math. Log. Grundl. Math. | 1975. | V. 21. |
P. 357{359.
404] Reineke J. On algebraically closed models of theories of commutative rings // Stud. Found.
Math. | 1982. | V. 104. | P. 223{234.
405] Retzla‰ A. Direct summands of recursively enumerable vector spaces // Zeitschr. math.
Log. Grundl. Math. | 1979. | V. 25, N 4. | P. 363{372.
406] Richman F. Finite dimensional algebras over discrete |elds // Stud. Log. Found. Math. |
1982. | V. 110. | P. 397{411.
407] Ringel C.M. In|nite dimensional representation of |nite dimensional algebras // Sympos.
Math. | 1979. | V. 23. |P. 321{412.
408] Ringel C. M. Kawada's theorem // Lect. Notes in Math. | 1981. | V. 874. |P. 431{447.
58
. . , . . , . . 409] Roose J. E. Sur la structure de c„ategories spectrales et coordon„esee von Neumann des treilis
modulaires et compl„ement„es // C. R. Acad. Sci. Paris, Ser. A. | 1967. | V. 265. | P. 42{
45.
410] Roose J. E. Sur la d„ecomposition born„ee des objects injectifs dans les cat„egories de
Grothendieck // C. R. Acad. Sci. Paris, Ser. A. | 1968. | V. 266. | P. 449{452.
411] Rose B. Model theory of alternative rings // Notre Dame J. Form. Logic. | 1978. |
V. 19, N 2. | P. 215{243.
412] Rose B. The @1 -categoricity of strictly upper triangular matrix over algebraically closed
|elds // J. Symb. Logic. | 1978. | V. 43, N 2. | P. 250{259.
413] Rose B. Rings which admits elimination of quanti|ers // J. Symb. Logic. | 1978. | V. 43,
N 1. | P. 92{112. Corrigendum: ibid. | 1979. | 1979. | V. 44, N 1. | P. 109{110.
414] Rose B. On the model theory of |nite-dimensional algebras // Proc. Lond. Math. Soc. |
1980. | V. 40, N 1. | P. 21{39.
415] Rose B. Prime quanti|er eliminable rings // J. Lond. Math. Soc., Ser. 2. | 1980. | V. 21,
N 2. | P. 257{262.
416] Rose B. Preservation of elementary equivalence under scalar extension // J. Symb.
Logic. | 1982. | V. 47, N 4. | P. 734{738.
417] Rosenstein J. G. On GL2 (R) where R is a Boolean ring // Canad. J. Math. | 1972. |
V. 15, N 2. | P. 263{275.
418] Rothmaler Ph. Some model theory of modules I. On total trancendence of modules // J.
Symb. Logic. | 1983. | V. 48, N 3. | P. 570{574.
419] Rothmaler Ph. Some model theory of modules. II. On stability and categoricity of Œat
modules // J. Symb. Logic. | 1983. | V. 48, N 4. | P. 970{985.
420] Rothmaler Ph. Stationary types in modules // Z. math. Log. Grundl. Math. | 1983. |
V. 29, N 5. | P. 445{464.
421] Rothmaler Ph. Some model theory of modules III. On in|teness of sets de|nable in modules // J. Symb. Logic. | 1984. | V. 49, N 1. | P. 32{46.
422] Rothmaler Ph. A trivial remark on purity // Seminarber. Fachber. Math. Humbold Univ.
Berlin | 1991. | N 112. | P. 112{127.
423] Rothmaler Ph. Mittag-Le`er modules and positive atomicity // Preprint. | 1994.
424] Rothmaler Ph., Tuschnik P. A two cardinal theorem for homogenous sets and the elimination of Malitz quanti|ers // Trans. Amer. Math. Soc. | 1982. | V. 269, N 1. |
P. 273{283.
425] Sabbagh G. Sur la puret„e dans les modules // C. R. Acad. Sci. Paris, Ser. A. | 1970. |
V. 271. | P. 865{867.
426] Sabbagh G. Aspects logiques de la puret„e dans les modules // C. R. Acad. Sci. Paris,
Ser. A. | 1970. | V. 271. | P. 900{912.
427] Sabbagh G. Sous-modules purs, existellement clos et „el„ementaires // C. R. Acad. Sci.
Paris, Ser. A. | 1971. | V. 272. | P. 1289{1292.
428] Sabbagh G. Embeddings problems for modules and rings with applications to model companions // J. Algebra. | 1971. | V. 18. | P. 390{403.
429] Sabbagh G. Endomorphisms of |nitely presented modules // Proc. Amer. Math. Soc. |
1971. | V. 30. | P.75{78.
59
430] Sabbagh G. Cat„egoricite en @0 -stabilite: constructions les pr„eservant et conditions de
chaine // C. R. Acad. Sci. Paris, Ser. A. | 1975. | V. 280. | P.531{533.
431] Sabbagh G. Cat„egorocite en stabilit„e quelques exemples parmi les groups et anneaux //
C. R. Acad. Sci. Paris, Ser. A. | 1975. | V. 280. | P.603{606.
432] Sabbagh G., Eklof P. De|nability problems for modules and rings // J. Symb. Logic. |
1971. | V. 36, N 4. | P. 623{649.
433] Salce L., Zanardo P. A result by Kulikov that does not extend to modules over general
valuation domains // Proc. Amer. Math. Soc. | 1991. | V. 111, N 3. | P. 643{649.
434] Salles D. Dimension pures de modules // Lect. Notes Math. | 1982. | V. 924. | P. 423{
441.
435] Sanchez G. Exactness of complexes and systems of equations // Rev. Real. Acad. Cienc.
Exact. Fis. Natur. Madrid. | 1988. | V. 82, N 1. | P. 165{167.
436] Saracino D. On the nonexistence of universal commutative rings // Comm. Algebra. |
1987. | V. 15, N 4. | P. 1977{1979.
437] Saracino D., Wood C. QE commutative nilrings // J. Symb. Logic. | 1984. | V. 49,
N 2. | P. 644{651.
438] Saracino D., Wood C. Finite QE rings in characteristic p2 // Ann. Pure. Appl. Logic. |
1985. | V. 28, N 1. | P. 13{31.
439] Saracino D., Wood C. Finite quati|er-eliminable rings in characteristic 4 // Lect. Not.
Pure. Appl. Math. | 1987. | V. 106. | P. 329{384.
440] Saracino D., Wood C. Homogeneous |nite rings in characteristic 2n // Ann. Pure Appl.
Logic. | 1988. | V. 40, N 1. | P. 11{28.
441] Shamsuddin A. Minimal pure epimorphism // Comm. Algebra. | 1991. | V. 19, N 1. |
P. 325{331.
442] Shelah S. In|nite abelian groups, Whitehead s problem and some constructions // Isr. J.
Math. | 1974. | V. 18. | P. 243{256.
443] Shelah S. The lazy model theoretician s guide to stability // Log. et Analyse. | 1975. |
V. 18, N 71. | P. 241{308.
444] Shelah S. Whitehead groups not be free even assuming CH . I // Isr. J. Math. | 1977. |
V. 28. | P. 193{203.
445] Shelah S. Whitehead groups not be free even assuming CH . II // Isr. J. Math. | 1980. |
V. 35. | P. 257{285.
446] Shelah S. On endo-rigid, strong @1 -free abelian groups in @1 // Isr. J. Math. | 1981. |
V. 40. | P. 291{295.
447] Shelah S. Nonstandart uniserial module over a uniserial domain exists // Lect. Notes
Math. | 1986. | V. 1118. | P. 135{150.
448] Shelah S. Classi|cation theory and the number of nonisomorphic models. | Amsterdam. | 1990.
449] Shelah S. Kaplansky test problem for R-modules // Isr. J. Math. | 1991. | V. 74, N 1. |
P. 91{127.
450] Simmons G. E. Varities of rings with de|nable principal congruences // Proc. Amer. Math.
Soc. | 1983. | V. 87, N 3. | P. 397{402.
0
0
60
. . , . . , . . 451] Simmons G. E. De|nable principal congruences and R-stable indentities // Proc. Amer.
Math. Soc. | 1986. | V. 97. | P. 11{15.
452] Simmons G. E. The structure of rings with de|nable principal congruences // Trans. Amer.
Math. Soc. | 1992. | V. 331. | P. 165{179.
453] Simmons G. E. Finite rings in varities with de|nable principal congruences // Proc. Amer.
Math. Soc. | 1994. | V. 121, N 3. | P. 649{653.
454] Simmons J. Cyclic purity: a generalization of purity for modules // Houston J. Math. |
1987. | V. 13, N 1. | P. 135{150.
455] Simson D. Pure-semisimple categories and rings of |nite representation type // Fund.
Math. | 1978. | V. 100. | P. 211{222. Corrigendum: J. Algebra. | 1980. | V. 67,
N 1. | P. 254{256.
456] Simson D. Right pure semisimple hereditary rings // Lect. Notes Math. | 1980. |
V. 832. | P. 573{578.
457] Simson D. Partial coxeter functors and right pure semisimple rings // J. Algebra. |
1981. | V. 71, N 1. | P. 195{218.
458] Simson D. An Artin problem for division ring extensions and pure semisimplicity conjecture // Preprint. | 1994.
459] Simson D. On right pure semisimple hereditary rings and an Artin problem // Preprint. |
1994.
460] Skornyakov L. A. Finite axiomatizibility of the class of faithful modules // Colloq. Math. |
1979. | V. 42. | P. 365{366.
461] Sta‰ord J. T. Module structure over Weyl algebras // J. Lond. Math. Soc. | 1978. |
V. 18. | P. 429{442.
462] Stegbauer W. A generalized model companion for a theory of partially ordered |elds //
J. Symb. Logic. | 1979. | V. 44, N 4. | P. 643{652.
463] Stephenson W. Modules whose lattice of submodules is distributive // Proc. Lond. Math.
Soc., Ser. 3. | 1974. | V. 28, N 2. | P. 291{310.
464] Szmielew W. Elementary properties of abelian groups // Fund. Math. | 1955. | V. 41. |
P. 203{271.
465] To‰alori C. Aleune osserwayione sugli anelli commutative esistenxialemente chiiusi //
Bull. Unione Math. Ital., Ser. B. | 1979. | V. 16, N 3. | P. 1093{1102.
466] To‰alori C. Sul model-complemento di certe theorie di coppie di anelli // Boll. Unione
Math. Ital., Ser. B. | 1980. | V. 17, N 3. | P. 1439{1456.
467] To‰alori C. Sheaves of pairs of real closed |elds // Rend. Sem. Math. Univ. Polites.
Torino. | 1980. | V. 38, N 3. | P. 77.
468] To‰alori C. Stabilit„a, categoricit„a ed eliminazione dei quanti|catori per una classe de
anelli locali // Ann. Univ. Ferrara. | 1982. | V. 28, N 7. | P. 39{53.
469] Tofalori C. Theoria dei modelli per aleune classi di anelli locali // Bull. Unione Math.
Ital. | 1982. | P. 89{105.
470] To‰alori C. Some decidability questions for lattices over a group ring. | Preprint. |
1994.
471] Trlifai J. Von Neumann regular rings and the Whitehead property of modules // Comment.
Math. Univ. Carolin. | 1990. | V. 31. | P. 621{625.
61
472] Trlifai J. Non perfect rings and the theorem of Eklof and Shelah // Comment. Math.
Univ. Carolin. | 1991. | V. 32, N 1. | P. 27{32.
473] Vilemaire R. Theories of modules closed under direct products // J. Symb. Logic. |
1992. | V. 57, N 2. | P. 515{522.
474] Vogt F. On modules with |nite U-rank types // 8-th Conf. Model. therie. | Berlin. |
1990. | N 110. | P. 148{158.
475] War|eld R. B. Relative injective modules // Manuscript. | 1969.
476] War|eld R. B. Purity and algebraic compactness for modules // Pacif. J. Math. | 1969. |
V. 28. | P. 699{719.
477] War|eld R. B. Decomposability of |nitely presented modules // Proc. Amer. Math. Soc. |
1970. | V. 25. | P. 167{172.
478] War|eld R. B. Serial rings and |nitely presented modules // J. Algebra. | 1975. | V. 37,
N 3. | P. 187{222.
479] Weispfenning V. Model completeness and elimination of quanti|ers for subdirect products
of structures // J. Algebra. | 1975. | V. 36, N 2. | P. 252{277.
480] Weispfenning V. Negative-extentially complete structures and de|nability in free extensions // J. Symb. Logic. | 1976. | V. 41, N 1. | P. 95{108.
481] Weispfenning V. Quanti|er elimination for modules // Arch. Math. Log. Grundl. Math. |
1985. | V. 25. | P. 1{11.
482] Weispfenning V. Existential equivalence of ordered abelian group with parameters // Arch.
Math. Logic. | 1990. | V. 29, N 4. | P. 237{248.
483] Wheeler W. H. Model companions and de|nability in existentially complete structures //
Isr. J. Math. | 1976. | V. 25, N 3. | P. 305{330.
484] Wheeler W. H. A characterization of companionable universal theory // J. Symb. Logic. |
1978. | V. 43, N 3. | P. 402{429.
485] Wheeler W. H. Amalgamation and elimination for theories of |elds // Proc. Amer. Math.
Soc. | 1979. | V. 77, N 2. | P. 243{250.
486] Wheeler W. H. Model theory of strictly upper triangular matrix rings // J. Symb. Logic. |
1980. | V. 45, N 3. | P. 455{463.
487] Wilkie A. J. On discretely ordered rings inwhich every de|nable ideal is principal // Lect.
Notes Math. | 1981. | V. 890. | P.297{303.
488] Winkler P. M. Model completeness and Skolem expansions // Lect. Notes Math. |
1975. | V. 498. | P. 408{463.
489] Wisbauer R. Semisimple and pure semisimple functor rings // Comm. Algebra. | 1990. |
V. 18, N 7. | P. 2343{2354.
490] Wisbauer R. On modules with the Kulikov property and pure semisimple modules and
rings // J. Pure Appl. Algebra. | 1991. | V. 70. | P. 315-320.
491] Wisbauer R. Foundations of module and ring theory. | New York. | 1991.
492] Wood C. The model theory of di‰erential |elds of characteristic p 6= 0 // Proc. Amer.
Math. Soc. | 1973. | V. 40, N 2. | P. 577{584.
493] Wood C. Prime model extension for di‰erential |elds of characteristic p 6= 0 // J. Symb.
Logic. | 1974. | V. 39, N 3. | P. 469{472.
62
. . , . . , . . 494] Wood C. The model theory of di‰erential |elds revisited // Isr. J. Math. | 1976. |
V. 25, N 3{4. | P. 331{352.
495] Wood C. Notes on the stability of separable closed |elds // J. Symb. Logic. | 1979. |
V. 44, N 3. | P. 412{416.
496] Zayed M. An application of a theorem of Ziegler // Monatsch. Math. | 1990. | N 4. |
P. 327{331.
497] Ziegler M. Model theory of modules // Ann. Pure Appl. Logic. | 1984. | V. 26, N 2. |
P. 149{213.
498] Zimmermann W. Einige Characterisierungen der Ringe uŠber denen reine Untermoduln
directe Summande sind // Bayer. Acad. Wiss. Math.{Natur. Kl. S., Ser. B. | 1972,
Abt. II. | P. 77{79.
499] Zimmermann W. -projective Moduln // J. reine angew. Math. | 1977. | V. 292. |
P. 117{124.
500] Zimmerman W. Rein injektive direkte Summen von Moduln // Comm. Algebra | 1977. |
V. 5, N 10. | P. 1083{1117.
501] Zimmerman W. (‹-)algebraic compactness of rings // J. Pure Appl. Algebra | 1982. |
V. 23, N 3. | P. 319{328.
502] Zimmerman W. Existenz von Auslander-Reiten-Folgen // Arch. Math. | 1983. | V. 40 |
P. 40{49.
503] Zimmermann W. Auslander-Reiten sequences over Artinian rings // J. Algebra. |
1990. | V. 119. | P. 366{392.
504] Zimmermann W. On Coxeter functors over tensor rings with duality condtions // Hokk.
Math. J. | 1990. | V. 19, N 1. | P. 89{101.
505] Zimmermann-Huisgen B. Rings whose modules are direct sum of indecomposables modules
// Proc. Amer. Math. Soc. | 1979. | V. 77, N 2. | P. 191{197.
506] Zimmermann-Huisgen B. Strong preinjective partitions and representation type of Artinian ring // Proc. Amer. Math. Soc. | 1990. | V. 109, N 2. | P. 309{322.
507] Zimmerman-Huisgen B., Zimmerman W. Algebraically compact rings and modules //
Math. Z. | 1978. | V. 161. | P. 81{93.
508] Zimmerman-Huisgen B., Zimmerman W. On the sparcity of representations of rings of
pure global dimension zero // Trans. Amer. Math. Soc. | 1990. | V. 320, N 2. |
P. 695{711.
Q
& ': 1994.
. . 70- (14.02.1924{26.05.1989)
. . . K1 -
. , , !
" .
Abstract
V. A. Artamonov, Automorphisms of decomposable projective modules, Fundamentalnaya i prikladnaya matematika 1(1995), 63{69.
A stabilizationtheorem for the functor K1 over some crossed products with a cocommutative bialgebra is proved. In particular this result holds for quantum polynomials whose
multiparameters are the roots of unity.
A P(A) (
) A-. !B1], P0 : : : Pn 2P(A) E (P0 : : : Pn) % &
P = P0 : : : Pn . (, % &
1 + ' P , ' %& P , '(Pi ) Pj
0 6 i 6= j 6 n Pt ker ', t 6= i. ( , (1 + ');1 = 1 ; '. + , P1 = : : : = Pn = A, !A2], !A3]
E (P0. n A) = E (P0 A : : : A).
/ An A- n.
0, A 1 A = B]t H B , H .
0 Q 2P(B ) g | & A-
(A B Q) An:
(1)
3
, & g & Q B n & E (A B Q. n A). 4 , % &
E (A B Q. n A) %&
A- (1) A-. 3
+ , +
! , 93011-1544, INTAS, 93-2618.
1995, 1, N 1, 63{69.
c 1995 !,
"#
\% "
64
. . . 7 & K1 1 81 !S2].
3
!S2].
/ k . + ,
8 9, . + 8 . 3, % !A3]. + , 81 :&, !A3], !A4].
1 + % & 8 , ,
1 . 0 H | k, 81 k-, ; : H ! H H | H , k. /
Xh
h
(1)
h(n) hi 2 H
% h 2 H H ! H n, ;. 0, B , 81 k-. =& k- H B ! B , h b ! hb %&' &'
(' H B , h(ab) =
X(h
h
(1)
a)(h(2)b) 1b = b h1 = "(h)
h 2 H , b 2 B . =& k- t : H H ! B 2-)',
H B , (1) t 8
hom(H H B ) !A4].
(2) t(1 h) = t(h 1) = "(h).
(3) f g h 2 H
X !f
fgh
(1)
t(g(1) h(1))]t(f(2) g(2)h(2) ) =
(4) f g 2 H b 2 B
X!f
fg
(1)
(g(1)b)]t(f(2) g(2)) =
X t(f
fg
X t(f
fg
(1)
(1)
g(1))t(f(2) g(2) h).
g(1))!(f(2)g(2) )b]:
0, H - C & : C ! H C . 3 Xa
a
(1)
: : : a(n) a(0) 2 H n C
65
% a 2 C & C ! H nC .
&' *+' A = B]tH C k- B C (. !A4], !Be])
(a c)(b d) =
X a(c
cd
(1)
b)t(c(2) d(1)) c(0)d(0):
(2)
(1){(4) % , 1 1 | %
A.
4 !A3], !A5] st(H ) @8 H , 18 %
h 2 H , t(f g) = t(g f ) = "(f )"(g) fb = "(f )b:
(3)
g 2 H b 2 B . 0 !A3] !A5], 5, st(H ) H.
1. *', %*-% H ''., -% B *&' '.' ( )( //( *-%( R. . H *&' '.' st(H ). *', K | ) // *-% C, *'
(K ) st(H ) K C | *&( '. K. 1- A = B]tH C
*&' '.' ( )( //( *-%( RH K.
. C (2) (3) , RH K A. 0 8
8 6 !A5] B = RH b1 + : : : + RH bn C = Ku1 + : : :Kur
bi 2 B ui 2 C . 0 8 4 !A5]
A=
X(RH K)(bi uj ):
3 , RH K &&. 0 . 2
0 8 1 , % . 0 k | p > 0 H |
, 81 H = Ha Hm Sm Ha | 81 9 x1 : : : xn. Hm | y1 : : :yr .
Sm | z1 : : : zd .
D, ;(xi ) = xi 1 + 1 xi "(xi ) = 0.
;(yi ) = yi yi "(yi ) = 1. ;(zi ) = zi zi "(zi ) = 1:
66
. . 0 1 B H
8 81 @ 9, @ , + An (B ) .
0, 1
8 q l, 1) q > 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. 38 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):
38 . 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] . . !
718. // . 91:. 4:. | 2.,
9. | 1991. | 9. 29. | !. 3{63.
A5] Artamonov V. A. Projective modules over crossed products. // J. Algebra. | 1995.
B1] ; 7. 0: <-:. | 2.: 2, 1973.
B2] Bass H. Introduction to some methods of algebraic K-theory, CBMS, N 2. | Amer. Math.
Soc. | 1974.
Be] Beattie M. On the Battner-Montgomery duality theorem for Hopf algebras. // Contemp.
Math. | 1992. | 124. | P. 23{28.
D] = >. >. ? / 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: <-:. // . 91:. 4:. | 2.,
9. | 1982. | 9. 20. | !. 71{152
S2] !
. . ? 1.
G G
G 11 . 0
. //
. !!!". !.. | 1977. | 41, N 2. | !. 235{252.
' (: 1994.
. . , . . -
,
, . , , , ! , " #1{6]. (
( ) | , ,"" -
| . . ( = ( n ) 2 N) " f ( )g . . n = #2f ( )g]. 1 ( ) . 1 :
1.1. ( ), , , ( ) = ( ).
P n
w
w
P n
k
w
w
n
P n
T k
w
Q k
P
k
T k
Q k
Abstract
A. Ya. Belov, G. V. Kondakov, Inverse problems of symbolic dimamics, Fundamentalnaya i prikladnaya matematika 1(1995), 71{79.
Let ( ) be a polynomial with irrational greatest coe6cient. Let also a superword
( = ( n ) 2 N) be the sequence of 7rst binary digits of f ( )g, i.e. n = #2f ( )g],
and ( ) be the number of di8erent subwords of whose length is equal to . The main
result of the paper is the following:
Theorem 1.1. For any there exists a polynomial ( ) such that if ( ) = , then
( ) = ( ) for all su"ciently large .
P n
W
w
W
n
P n
T k
W
n
T k
Q k
w
P n
k
Q k
deg P
n
k
1 , . , , , ! , " #1{6].
( P(n) | , ,"" -
. . w (w = (wn) n 2 N) " fP(n)g, . . wn = #2fP(n)g]. 1 T(k) k w. ."
.
1.1. Q(k), P , , k T(k) = Q(k).
#
1995, 1, N 1, 71{79.
c 1995 $
%& '(,
)
\+ "
72
. . , . . 2
2.1
(), . ( M !
f : M ! M !
U M.
4 x x, f(x) : : : f (n) (x) : : :, , x w, : wn = 1, f (n) (x) 2 U wn = 0, f (n) (x) 62 U.
(
w
" , | (M f U x) " w. 6
, U | !
, mes(@U) = 0 M | . 4 wb w x, f(x) : : : f (n) (x) ; n 2 N.
."
.
2.1. vf -
!" x , !
x #
V , !
x 2 V vxb = vf .
$
w !" x , !
| !" x .
( \, " -
\
, ".
2.2. & v | , #
-
!" v (. . # ).
)
# w ( !
# ).
2.2
2.3. *
+
G : (M1 f1 ) ! (M2 f2 ) # g : M1 ! M2 , f1 M
M1 !
2
g#
g#
f2 M
M1 !
2
.
(
,", " " .
9- "- , f ,
!
f.
1
, " .
73
2.4. *
#
V , # !
+ (
+).
2.5. & #
U | , 1. ! ! !"!.
2. !
" > 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 <:::<km 6k 1 1 : : : m X
1
1
1
P T (k) = Q(k) !
- k.
79
!. A
!
!
4.8 ! N. O. 6
.
1] . . . // .
! """#. ". . | 1961. | '. 25. | ". 749{754.
2] -. , .. !. #
/
. |
.: !1, 1985.
3] -. 4. 5/
1. #
// 6
. 1. | 1993. | '. 48. | N 4. | ". 131{166.
4] R. N. Izmailov, A. A. Vladimirov. Dimension of aliasing structures // Int. J. of Systems.
Appl. in Comp. Graphics. | 1993. | V. 17. | N 5.
5] M. Morse, G. A. Hedlund. Symbolic dynamics II: Sturmian trajectories // Amer. J.
Math. | 1940. | V. 62. | P. 1{42.
6] H. Weyl. U8 ber der gleichverteilung von zahlen mod. 1 // Math. Ann. | 1916. | V. 77. |
P. 313{352.
- &.: 1995.
. . . . . | , .
" . # x 1 % .
( )
( = max(0 ;1 + ) > 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 <
n
Abstract
E. V. Bulinskaya, On high-level crossing for a class of discrete-time stochastic processes, Fundamentalnaya i prikladnaya matematika 1(1995), 81{107.
The aim of this paper is to study the asymptotic behaviour of the :rst passage time for
some discrete-time stochastic processes arising in applied probability.
The paper is organized as follows. The systems' description is given in x 1 along with
the main results.
The integer-valued random walks with impenetrable (as well as re<ecting) barrier at
origin
= max(0 ;1 + ) > 1 0 =
are treated in x 2. The main object of investigation is
= inf f :
= g, the :rst
over<ow time in terms of inventory theory. The limit distribution of normalized random
variable
=
(E
);1 is obtained for all the initial states and possible values
of E for the case of the three-valued i. i. d. random variables (demand and supply
in batches of :xed volume). The domain of model's stability with respect to initial state
and system's parameters is established as well.
The in<uence of two-level control policy on system's behaviour is dealt with in x 3. It
is proved, in particular, that
is asymptotically exponential if E
0 in a su=ciently
wide band in the neighbourhood of the absorbing boundary .
The directions of further investigations and various possibilities of application are given
in x 4.
Wk
Wk
Xk k
W
Nxn
xn
Nxn
x
k
Nxn
Wk
x
Xk
Xk
xn
Xk <
n
1995, 1, N 1, 81{107.
c 1995 !,
"#
\% "
n
82
. . 1 n (
n ! 1) , !"
. $% , , , , , . (. . &2]).
$ !"
. 1- ) . * %
! ) 2 3, ) 4 ! "
. -
) ) . .
% ) .
) ! . . | 0 !"
0 Wk = max(0 Wk;1 + Xk ) k > 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)
;%
< % % ! &10] ( q > 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 ! , %% < .
4- ) . , %
, % n1 n2 !
. $ ! "
.
2
;, fWk g !"
0 , 4
(1.1), , !"
(1.2).
8 , !" p q , %! r .
? % , % !
(
n ! 1) Nxn , 4
(1.3), % n % x.
. "! )
! Fxn(z) = Ez N . A "
% )
.
xn
1. 0 6 x 6 n 0 6 z 6 1
a(d + f)x + b(d ; f)x (2pz)n;x Fxn(z) = a(d
+ f)n + b(d ; f)n
(1)
a = 2qz ; d ; f b = d ; f ; 2qz
d = 1 ; rz f = (d2 ; 4pqz 2) 21 :
(2)
Fx(z) = pzFx+1 (z) + rzFx (z) + qzFx;1(z)
(3)
. ;%
) !
F0 (z) = pzF1 (z) + (q + r)zF0(z) Fn(z) = 1:
(4)
85
? (. . &8]), " 4
(3) Fx (z) = A(z)x1 (z) + B(z)x2 (z)
(5)
i (z), i = 1 2 | pz2 ; (1 ; rz) + qz = 0:
(6)
- ! % (4){(6) 4
(1), (2), % 1.
.
mxn = ENxn.
2. p 6= q, r > 0
mxn = q 2 (
n ; x ) + (n ; x)
= qp;1 , = (p ; q);1 .
p = q, mxn = (n ; x)(n + x + 1)(1 ; r);1:
(7)
(8)
% .
. % | %%, 0 &15] x = 0,
0 (1).
, mxn = Fxn
I (7) &10] 4
pVx = qVx;1 + 1
(9)
Vx = mxn ; mx+1n (10)
V0 = p;1 Vn;1 = mn;1n:
(11)
A (8) % (9){(11).
K
) (. . &1]), Nxn
=d
0
NX
xn
k=1
Yk :
(12)
L% =d . -
Yk , k > 1 ! P(Yk = i) = (1 ; r)ri;1 i > 1:
(13)
0 | n \
5 Nxn
", "
%
fYk g, !"
x
86
. . 4!"
p0 = p(1 ; r);1 q0 = q(1 ; r);1.
;%
%
0 EY
ENxn = ENxn
1
(14)
0 ,
% &16], r = 0, . . ENxn
;
1
(7), (8), EY1 = (1 ; r) .
K
, % 2 .
mxn % " )
, % (
n ! 1)
xn . (-
, xn = Nxnm;xn1 .)
1. p > 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 !
< 'n (s) = Ee;s '(s) = Ee;s n ! 1 s > 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 = ;2qs + 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)
~ < )
3
1 @
;
;
;
1
(29)
g(t) = 2 2 @ 1 (2 2 2 t) 1 1 ( ) | 0-)
1 (z t) = (t); 12
=2 2 c
+
1
X
m=;1
(;1)m expf;(z + m ; 2;1)2 t;1 g:
(30)
. (27){(30) (16), (17).
3. $
% !! % p < q.
5 (7) mxn = q 2 n (1 + o(1))
(22), (23) , d + f = 2q(1 ; ( + 1)s) + o(s)
d ; f = 2p(1 + ( ; 1)s) + o(s)
(31)
a = 2qs + o(s)
b = 2 ;1 + 2(p ; ;1 )s + o(s):
" )
(20), % (21) % u = 2pz(d ; f);1 , w = (d + f)(d ; f);1 , h = ab;1.
; (31) , h(e;s ) = q 2 s + o(s) w(e;s ) = (1 + O(s))
(32)
u(e;s) = 1 + O(s)
0
x;n(1 + O(
;n))x s
;n n;x
'n (s) = 1 +1 + (1 +
O(
;n))n s (1 + O(
)) :
89
(33)
K
, ! > 1, p < q, (33)
'(s) = (1+s);1 | < % 1, % 1.
. 1 % 4
! r (12) % )
. ? 0 (12), Nxn | 0 Nxn
Yk , k > 1. K
, % , % 0 !.
Nxn Nxn
: %, -, 0 | 0 n , Nxn Nxn
% , , -, (13) .
? , 0 %, (. . &5] . 285). ?, % &5] % ! % %. * .
.% % fYk gk>1 | %% EYk = m < 1. N
p
Nn > 0 n %
fYk g Nn ! 1 n ! 1.
*
N
X
SN = Yk :
n
n
k=1
3. "# $# fYkg fNng
Nn;1 SN !p m
(34)
n
n ! 1.
, &5]. A (34) %, %4
(. . &6], &9]),
)
Nn;1 SN eitm .
. ) n
E expfitNn;1 SN g =
n
A,
0 k ! 1
1
X
k=1
P(Nn = k)(Eeitk;1Y1 )k :
(35)
EeitY1 = 1 + itm + o(t)
Eeitk;1 Y1
k
! eitm:
(36)
90
. . p
K
Nn !
1 n ! 1 : ! " > 0 0 < M < 1
n0 = n0 (" M), n > 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) ! a10 + (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) B4
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 ; p22 + q21 (35)
En2 ; En ! 2q2(p113 ; 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 + 222 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) ! 20 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) ! 212 (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 ; p22 + 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 + 212 (
l11 ; 1)(
1 ; 1);1 +
(82)
+ 21 (1 ; 2 )(1 ; ;1 l1 +1 )(
1 ; 1);1 ; 21 (21 ; 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 + q10 ; p1 1 (1 ; r1 )EOn2 ; (1 + r1)EOn ! 2q1(p003 ; p113 ):
(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 + 212 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(p003 ; p113) ; (1 ; p11 + q10)
(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) % % , , 44
, !"
, %
!
!
107
0!
! . . 8 % %, % , .
"
1] . . , . . . !" #%
&! | (.: *&+- (,, 1980.
2] . . , . . . #%+
! "
1 !
/ 3
" 4. 5#1 1 & | (.: (, 1984.
3] . . , . . . 4 ! & + 1 #%+
// 8 . !
. | 8. XXIX. | N 4. |
. 654{668.
4] . !
, . <+. 8# "
1 #&
, . 1. | (.: 4,
1969.
5] . . +
. = , 6 &+. | (.: 4, 1988.
6] . =!. ( ! ! + . | (.: (, 1975.
7] . 3. . ! + +%
. | .: ! ! +%
, + +. . . +
| (.: ?+ &, 1983.
8] . @. +
A %
, . I. | (.: (, 1964.
9] . @. +
A %
, . II. | (.: (, 1967.
10] (. B+%. ! !
" C
1
D. | (,,
. 1, ! !. !1. | 1993. | N 1. | . 97{101.
11] E. V. Bulinskaya. Boundary crossing problems for some applied probability models. Dwudziesta trzecia ogHolnopolska konferencja zastosowaHn matematyki, Zakopane-KoHscielisko,
20{27. IX. 1994. | P. 19.
12] E. V. Bulinskaya. On optimal capacities of some inventory systems / Proc. Second Int.
Symp. on Inventories. | Budapest, 1982. | P. 639{648.
13] E. V. Bulinskaya. The asymptotic behaviour of some inventory systems / Proc. Third Int.
Symp. on Inventories. | Budapest, 1984. | P. 459{472.
14] V. Kalashnikov. Topics on regenerative processes. | CRC Press, Boca Raton, Ann Arbor,
London, Tokyo, 1994.
15] R. A. Khan. On cumulative sum procedures and the SPRT with applications // J. R.
Statist. Soc. | 1984. | V. 46. | N 1. | P. 79{85.
16] W. Stadje. Asymptotic behaviour of a stopping-time related to cumulative sum procedures
and single-server queues. // J. Appl. Probab. | 1987. | V. 24. | P. 200{214.
' (: 1995.
,
. . . . . e-mail: VAB@R11523.phys.msu.su
. , ! "#$ $ . % ! $"&.
Abstract
A. B. Vasilieva, On the solution of singular perturbed problems having boundary layer
of spike type, Fundamentalnaya i prikladnaya matematika 1(1995), 109{122.
The singular perturbed second order equation is considered. The boundary condition
causing a boundary layer of spike type is given. The asymptotic approximation for such
solution is obtained and its stability is investigated.
1 "2
d2u
dx2
= F (u x) 0 6 x 6 1
u(0 ") = u0
F (u x) = 0
u = '(x) u = (x) '(x) < (x)
u(1 ") = u1 :
, @F
> 0
@u u='(x)
(1:1)
x
(1:2)
2 0 1] !
@F
< 0
@u u=(x)
! #!$ (x), Z(x)
F (u x) du = 0:
(1:3)
'(x)
, # -$ . $ - -&#/ $ ( 93 { 011 { 1690).
1995, 1, N 1, 109{122.
c 1995 ,
!"
\$ "
110
. . &' $ 1] &
!.
) *
. 1 '+ #! F (u x) #! x. & S1
du
& S2 . , - #. ! (u = u0) d
d2 u
d
2
= F (u x) (x #!) (*
. 2).
/& ! # 0 & (1.1), (1.2) & #!$. 2], 3] &
u
= u2(x ") + u(
") + Qu(
1 ")
(1:4)
& u2 . &, u Qu . & ! x = 0 x = 1
, = x=" 1 = (x ; 1)=":
u
2(x ")
= u20(x) + "u21 (x) + u(
") = 0 u(
) + "1u(
) + Qu(
1 ")
=
(1:5)
Q0u(
1 ) + "Q1 u(
1) + :
. & & :
u
20
u
22
= '(x) u21 0
= F (2u1 x) u2000 u2n = F (2u1 x) u200n;2 + Fn 0
0
(1:6)
& Fn = Fn(2u0 (x) : : : un;1(x) x) ! + #!$ x.
. & u u0. 8 &' u,
!! !$ x = 1 !.
) u0 = u0a < (0). 9
& & #!$.
d20 u
= F ('(0) + 0u 0)
d
2
0u(0) = u0 ; '(0) 0u(1) = 0:
(1:7)
,& u~ = '(0) + 0 u, d2 u
~ = F (~u 0)
d
2
u
~(0) = u0 u~(1) = '(0):
(1:70)
<& ' & +. ) u0 & & ! I II (*
. 3), & ! +. ., &
!
(! . !, &' & '+, '
). = - & & & #. ! (1.4).
> &. (0u)I = (~u)I ; '(0), & &
. (0 u)II = (~u)II ; '(0), 111
&.0 ! (1.4) '& & - & . )0 & 1u (&! & ' , &! I
II )
d2 1u
d
2
= Fu (~u 0)1 u + Fu(~u 0)'0(0) + Fx (~u 0)]
= Fu(~u 0)1u + H1
1u(0) = 0 1 u(1) = 0:
(1:8)
=@ && & 0.. >.d2 u
~0
du
~
, && @ & = u~0 : 2 = Fu(~u 0)~u0. , d
d
I: u
~0 (0) < 0 u~(1) = 0A II : u~0(0) > 0 u~(1) = 0. B. 0 z ! 1 ! 1: z (0) = z 0 , z (1) = 1, !!
. !'$ 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
<& &.0 F~u = Fu (~u 0) F~x = Fx (~u 0). E
' '&
' & & ' !
&!.
112
. . F & 2u &
d22 u
d
2
2
= F~u2 u + F~u 2 '00 (0) + Fuu22(0) +
1
+ F~uu
2('0 (0))2 + (1 u)2] +
2
1
+ F~ux(
'0 (0) + 1u)
+ F~xx
2 =
2
~
= Fu 2 u + H2
(2:3)
2 u(0) = ;2u2(0):
F+ ' u~0 , . Z1
u
~
0
(2 u)
00
d
01
2u 0
1
2u 0
= = u~ ( ) j ; u~ ( )j +
0
00
0
= ;F ((0) 0)2u2 (0) +
Z1
Z1
u
~0002 u d
=
0
~0 2u d
F~uu
0
Z1
F~u2 uu
~0 d
=
0
9! ',
Z1
H2 u
~0 d
:
0
;F ((0) 0)2u2 (0) =
Z1
H2 u
~0 d
:
(2:4)
0
C , ' , & (2.2).
1. > "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
<, 1u & &, ! !! ! (2.8) ' C u~0,
& C , -+ 0 & (2.5).
9! + $ '& @' 0
. 9! ', # 0 (1.4) ('& , !$ x = 1 . '.) & &. )& + , ! (1.4) + '
("C1 + "2 C2 + )~u0, & Ci .
1.0 (1.1),
1
(1.2), u = (0) + ("), u < (1) u
2(x ") + !u(
") + Qu(
1 ") + ("C1 + "2 C2 + )~u0:
(2:9)
"# # (2.9) # % (1.1) % (2:10) % O("n+1 ) &
#, # & x = 1 #%# '&
(
&% .
3 # <&& (2.1) ! & x = 0
du
dx
(0 ") = 0
u(1 ") = u1
(3:1)
'& ! 0 & #!$. !+ & (1.4). - 0 (1.6), (1.7), (1:70) @ +, !! . !. &. ) (1.8) d21 u
= F~ u + H ( u)0 = ;'0 (0):
(3:2)
d
2
u
1
1
1
=0
(0 ' &@ x , x, , ).
)&& & (3.2) & ! (1.8). 9 +
+ ! &, &@ 2 u. 9! ', & (1.1), (3.1)
(! u1 '& ') & ' ! & (2], 3]), + #. & (1.4) &
2. ) # (1.4 ) u(x ") (1.1), (3.1) # (1.4) ### # '
# &
#
, . . - u(x ") # (1.4)
&
#
O("n+1 ) x 2 0 1]. .&
# # (1.1), (3.1) &- -, &
1.
>., + (3.2) u~0 , !! + & 0, 1u(0)F ((0) 0) =
Z1
0
H1 u
~0 d
116
. . . . 1 u(0) = 1, & 1 & #. (2.6). #!$ 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
, + & -. <& '& !. H , '!. & + ! @ u . , &.0 '& , - &!.
G (. . 4]), 0 . !. & (1.1), (1.2), u0 < (0) ( ! + $ x = 1), ..
' . 0 & (1.1), (1.2), (2:10), !
& 0 & (1.1), (3.1). & .
' , , - 0 !!
0 . !. & !! 0 . !. &. /&
, x = 1 & u0 < (0), u~0 (0) < 0 .
!$ + &. . 5].
K & !! & ! & x = 0, & . &
6].
I ' . & & ! ' .
&@. & L{B
"2
d2N
dx2
= f Fu (u(x ") x) + g N
(4:1)
N(0 ") = 0 N(1 "):
/& ! N & (1.4), &
= 0 + "1 + :
(4:2)
H! 6], N , & 0 N d20 N
d
2
= Fu(~u 0) + 0 ] 0N
0N(0) = 0 0 N(1) = 0:
I@& &, 0 = 0, 0 N = u~0 . >
d21N
~1
H
d
2
= F~u 1 N + H~ 1 + 1 u~0
n
= F~uu 1u + '0 (0)
] + F~ux
o
u
~0 1N(0) = 0 1 N(1) = 0:
, + , + u~0 , ~ (0)1N(0) = 0 =
u00
Z1
0
~ 1u~0 d
+ 1
H
Z1
(~u0)2 d
:
(4:3)
0
I@& ! & 1 , ! !
+ ! (4.3).
118
. . ' - ' ' 6]. , d2(1 u)0
= F~u(1 u)0 + H10 2
H1
=
0
n
F~uu
d
o
1u + '0 (0)
] + F~ux
u~0 + F~u '0 (0) + F~x :
F+ u~0 , F ((0) 0)(
)j
0
1 u =0
=
Z1
H10 u
~0 d
:
0
9! !!
~1
H
'& Z1
1
0 2
(~u )
d
Z1
=;
0
= H10 ; (F~u'0 (0) + F~x)
H1 u
~ d
0 0
Z1
+ (F~u '0 (0) + F~x )~u0 d
=
0
0
Z1
= ; F ((0) 0)(1u) j =0 + (F~u'0 (0) + F~x)~u0 d
:
0
0
I@&
2
3;1 2
Z1
1
2
u0 d
5
= 4 (~ )
3
Z1
; F ((0) 0)(1u)0 j =0 + (F~u '0 (0) + F~x )~u0 d
5 :
4
0
(4:4)
0
(3.2) , 'Z(0)
Z1
(F~u '0(0) + F~x )~u0 d
= ;'0 (0)F ((0) 0) +
(0)
0
!
1
=
8
2
>
(0)
>
< Z
6
4
>
>
:
2
'(0)
1
F~x(u 0) du
Zu
31
2
F (u 0) du5 du
'(0)
7
9;1
>
'Z(0)
>
=
>
>
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 .
H | C. !
. " , .
$ , H . H (H). '
\
" *
( + ) (H). , a e a e H +* . - E F (
E F ), E F * H, \E F" (
\E F ").
B
B
2 2 3
3
4
, 93{01{00156, 8
9 3
, M95000.
1995, 1, N 1, 147{159.
c 1995 ,
!" \$ "
. . 148
11], L , * x y * * ** x y * **
x y. L1 3 L , x y L1 , x y x y L1 . 43 ,
* ** ** .
3, : (
) H +* E F
, { . 7 3 3 { 3 , , 0 H .
43 (
) , *
* { , !. 43 , .
, A (H) LatA , , + A: ,
L Alg L , *+ L (Lat Alg * +
\Lattice" \Algebra"). 9
, Alg L { ,
+ 1, LatA | 3. 12], L 3 , L = Lat Alg L 3 * . , A , *+ L , A = Alg Lat A * . ,
3 *+* * ! : <
= = . , +
, + 3, , * .
43 L , L * , E (E F ) F (E F) * E F L
( E F F E). CSL (CSL | \commutative subspace lattice" =
) | <
3 . 7
3 N Alg N . , A | CSL-, a A ,
* E LatA pap = ap, p | E , , (1 p)a(1 p) = (1 p)a. 7 3 p? 1 p, E ? H E. 9
, 3 (
) A. , 3 <
, (. 13] 14]).
? , A , + e , e f = ( e)f, f H, A ( ^
_
2
^
_
E
2 E
E
f g
B
)
)
^
?
^
2
2
;
;
;
;
2
149
g < (e f) g = (g e)f). B A M, M | H, a f, a A, f M. D L | <
3, A M
L. 9
, 3, + M. CSL- A ( 15] |
), Lat A .
<
.
B A | Alg Lat A | * <
, A ,
Lat A ,
A Alg LatA C - * <
. " , <
* <
, , |
\" <! 3, , <
, * (
< 2).
B * ! \; ", *
(
<
) 3. $, ( ) 4
16] G<, =
H
15] + 3, *+ ! ! <
H
<< 17]. D !
! 4
( , 3 , *).
B L | 3, E L. B E; 2
E; =
F L: F
_f
2
6
E:
g
7 3 E; L. , E F,
E; F; . H , H; | 3
H
Lat A, + H; = H. -, * 4
2
E; =
F L: F
_f
2
E
g
E; E, 3
E 3 L.
$ !.
1.
f
E
g
{ !
_(E
;
)=
_ !
E
:
;
L" . . 150
B 18] 3
3 - .
J*+ , H
<< 17], !.
2. (+
16] CSL- 3
15]). L | $ . %
e f Alg L , $
E L , f E , e (E; )? .
2
2
2
.
3.
L | $ " E L.
2
&
(E;)? = e H : e f Alg L $ '
f E :
f
2
2
2
g
, 3 3, 2 , H
<< ( ).
. $
* K.
1) -< E L. B e (E; )? ,
* f E e f Alg L, L. B f E, F L. D F E,
g F, (e f)g = (e g)f E F F { e f. D F E, F E; e (E;)? , (e f)g = 0.
J
, F . K, e f L , e f Alg L. B f { E, , e K, (E;)? K.
2) , . , f H 3, + f, Ef (
+ L:
Ef = F : f F ). B f E e K | e f Alg L * K. B F Ef Ef |
3, + f, F f. D
g F , (e f)g = (e g)f F (e g) = 0. '
, e F . B (Ef ); F L,
+ Ef , , e ((Ef ); )? . ' e | K, , K ((Ef ); )? f E ,
V
W
?
K
((Ef ); ) . B
( (Ef ); )? , *
f 2E
f 2E
W
1 (( Ef ); )? = (E; )? . , 3
f 2E
3
.
1. L | $ " L~ | $ ' $
Lat Alg L. &
$ E
E; $ '
E L ~.
, L L
2
2
2
2
2
2
2
6
2
2
2
2
^f
2
g
2
2
2
2
6
?
2
2
2
7!
2
151
7 , 3, (E; )? Alg L = Alg L~ .
2.
L | $ . ($ Alg L , H; 6= H .
. 1) B H; = H (H;)? = 0 < 6
6
f g
e (H; )? . B 2 e f, f H,
Alg L.
2) B e e f, f H, Alg L. B 3 e (H;)? ,
H; = H.
2
6
B A | * A A-mod. J , *+ + , 19] 110] . , Nc | N
X | A-, : A c X X !
: (a ^ x) = a x
A- , < ! A-mod
(
N
c
, X A X).
B A | . " , H A 3
! a e, a A, e H,
a e. G, A , A- H. 7 3 CSL- (
) , ( +
. 111]), (
<
112], 113] | <
| 114], 115]).
2
2
. . 152
1 CSL-
116], (H) .N, H < : H
(H) c H, ! +
, (H) ! . O - , + !
. B , *+ .
1. A | $ , H; = H . &
A
B
;! B
B
6
.
. ? * 1 , -
3 L Lat Alg L. B * 2 A !
e f, f H. N +
, , e = 1. J < : H
A c H, (f) = (e f) ^ e. 9
, | < A-, = 1H , H .
B <.
1. ) A | $ CSL- H , H | A-
, H; = H .
. D H; = H, 1 H + !.
_ Q 3
B H; = H. $
L H L.
, 1 .
4. L | $ , H; = H . &
k
k
;!
6
6
n f
g
'* :
1. *'
E F L_ , H = E F "
2
_
2. L $
' (
)
', H $$$ ( , H 3-
3 , H $
| . 11]).
. B, 1) , * E F L_ E F = H. B 3 , =< ($ $
2
_
6
153
),
(. 11]), L * ! (= ).
B, H 3
E0 3. B * E0 F = H * F L,_ 3
, F E0. J
, E0 = H;, *
.
, 3
( , , 3 <
, c 1).
_
6
2
5.
A | CSL-,
E . &
:H
!
A | A-
, E L, p |
2
(A E ? ) Ap? :
. B e
E ? p? A, (e) = (p? e) =
Ap? , (E ? ) Ap? , A E ?
2
2
= p?(e) p? A = p? Ap?
Ap? .
7 3 , R. 7. J
+* 2 3. Q (. 117], 1. 26).
2
6. (118], 1.4 19], IV.4.5]).
A | , X | A-
, '* 0
(. 119]). &
$ '
x 2 X $ A-
: X ! A , (x) 6= 0.
B
3 ! 3 <, !.
A | CSL-, *' E1 E2
L 2 = H (A (E1)?) (A (E2)?) = 0. &
A 2.
1
2
, E _ E
.
^
. B : H
5
_
6
A | < A-. B
((A (E1)? ) (A (E2)? )) (Ap?1 ) (Ap?2 )
E2 = H, p?1 p?2 = 0 E1
!
^
Ap?1 Ap?2 = 0:
\
\
J
, ((A (E1)? ) (A (E2 )? )) = 0. B , 6 H .
^
. . 154
3.
A | CSL-, * N L
, _fE : E 2 N_ g = H , ^fA (E )? : E 2 N_ g 6= 0. &
A .
.
B
V
T :H
A { W <V A-. B 5 ( (A (E )? ))
(A(p )? ), E = H, (E )? = 0, T
(p )? *. J
, (A(p )? ) = 0 ( , T(A(p )?), a(p )? = a * * x H (p )? x
a
V
*, a x = 0) (A (E )? ) + .
B , H (
6).
!
2
2
4.
N
E
A | CSL-, * L E : E
N_ = H , , $ F L_ _N F; E = H . &
A .
_f
2
2
g
_
2
6
.V Q 3, , (A (E )? ) = 0. , < f F e (E )? (F;)? . B 2 e f,
V
+ e f, A. J
, f (A (E )? ).
2
6
2
^
2
4 2.
, 0 $ -
H; = H .
6
. ,
CSL-. B F; F * 3, E F 3 , (
, 1, 3
*).
_
! 1. B, H; = H.
1) B + E1 L,_ E2 = A (E1)? = H. ' (E1)? E2,
E1 E2 = H. B A , E2 L (A (E2)? ) E2 = (A (E2 )? ) (A (E1)? ) = 0. B * 2 H .
2) B * E L_ A E ? = H , +
E F L_ , E F = H (
E ? F , A E ? F = H). ' H; = H, 4 3.
2
6
_
^
^
2
2
6
_
6
155
2 CSL-
B H , U | L A (H ) | . 0
$
H P e 2< , (e ) 2 :
P
< (e ) (f ) >= (e f )
L H . $ Q A
(a ) 2 : sup a <
! (a ) 2 = sup a :
(e ) 2 (a e ) 2 .
L
7 H , . B, , A | <
, A = Alg L ,
L = (L U) | 3, 3 L (
, 3,
* H H). , Q A = Alg L,
<
<
. D CSL-, CSL- ( , , 120]).
< <
. B M M 0 |
<
LM LM | 3 . ' LM LM * M M 0 . D M | CSL-, 3 LM , M 0 .
B A | CSL- A | , + ,
T A (
, LA = L?A = E ? : E LA ) D = A A |
A ( D | <
, *+ CSL- 3 LD = (LA LA )) C = D0 ( \core" | , !
) , C | <
. " , <
, * , W ( ).
7+ , , , , * , . $
,
, * CSL- .
W , *+ (+*+ 121]):
f
g
2
B
f
k
k
1g
f
k
k
k
k
k
1g
k
7!
L
2
0
0
L
f
2
g
F
F
7. (122]) ) AT| CSL-, Lr | $ !
Lat A ( Lr = LA L?A ), A0 = W (Lr ).
. . 156
B , .
1) W (Lr ) A0, Lr A0 .
2) B T A0 A0 D0 = C, C { , T | ( <, A0 <
, : , , | ., , 123]). 4 T
Z
T = dE(
)
2
(T )
, E(!), ! { T, * A (. ) , E(!) E(!)?
3 L A Lr , T W (Lr ).
' , CSL- <
( , I1 ). ? 3,
Lr { 3. (11]), 3 , 3 , (). ' *+
3.
1
$ $
.
K
, Lr , A0 (
<
). $* , A0 < ( 3 < ). 7+
A. K , H * H=
MH H | 3 L.
$
A Ap , p | H , A A , , H , CSL- ( 3 L = E LA : E H ).
f
g
f
2
4.
g
A | CSL- . &
! '* $:
1.
2.
3.
4.
157
A "
$ A "
$ '
Lr (H ); = H ( \; " \;", 3 L )
$ '
Lr (H ); = H .
6
6
. 1) = 2) B A <
)
E = H Lr L
p. B H * A- H = E E ? , A-
H A- E. 4 : A Ap : a ap.
B * a A ap = pap, (ab)p = a(bp) = a(pbp) = (ap)(bp)
| (
, *X
) < . B *
e E a e = a (p e) = (ap) e, A-
E !
A-
+* < ( a e = (a) e). $ IV.1.7. 19],N, , , !
Nc Ap
: E E c A A-mod
!*
~
:
E
E
N
c
Ap-mod, ~ = ( 1E ).
2) = 3) 7 1.
3) = 4) B 3), H { Lr .
D E L, E H , E = (E H ) (E H ? ) (H ); H ? = H. B
(H ); E, , (H ); = H.
4) = 3) D H Lr (H ); = H , (H ); ( E L : E H ) H ? = H H ? = H.
3) = 1) " , 3) A +* !
N N
: A cH
H : A c H H.
7 H ? < e P
e f, f H , A. B f H | *, f = f ,
f
H P, 3
P f 2=
f 2 . B (f) = (e f ) ^ e . ,, | . B , f * Xm
f = f i
2
!
7!
2
2
!
!
)
)
2
6
^
_
^
_
6
6
)
_f
2
g
_
_
)
;!
!
2
2
2
k
k
k
k
i=1
* i i. '
(f) =
Xm (ei
i=1
fi ) ^ ei =
Xm ei
i=1
k
fi
fi
k
^ fi ei :
k
k
. . 158
-, * m- !
k
Xm iei
i=1
k
fi
fi
k
k k
Xm (i
i=1
k
fi ei )
k
k
vuXm
u
6t
i=1
k
fi
2= f k
k
k
, (f) f -
, f (. 19], II.2.44). $ , . ', | < A-, , .
!, + , *+ 3
, .
k
k
k
k
5.
E | ! CSL- A " p | E . &
E $$$ A-
, Ap, Ap B(E) .
-, , 4, E 3.
. 4 3 4 < : A Ap a ap 3 IV.1.7. 19]: A- E !
, E A Ap.
B N E Ap + < Ap-
: E A c E, ! . -, < A- ((a e) = (a (p e)) = ((ap)
e) = ap (e) =
N
c
= a(p e) = a(e) * a A e E), ~ : E A E ~ = (i ^ 1E ) ,
i | N Ap A ! ~ : A c E E, E A.
7 *
* 3 W. 9. = 114] 4 + .
!
7!
!
2
2
!
!
5.
1 4
, , $
' .
' , A | <
(= CSL- 3 ), * * A.
W W. 9. = R. 7. J
+
.
159
1] . . | .: , 1984.
2] P. R. Halmos. Reexive lattices of subspaces // J. London Math. Soc. | (2) 4 (1971). |
P. 257{263.
3] W. E. Arveson. Operator algebras and invariant subspaces // Ann. Math. | 100 (1974). |
P. 433{532.
4] K. R. Davidson. Commutative subspace lattices // Indiana Univ. Math. J. | 27 (1978). |
P. 479{490.
5] F. Gilfeather, A. Hopenwasser, D. R. Larson. Reexive algebras with 'nite width lattices:
tensor products, cohomology, compact perturbations // J. Funct. Anal. | 55 (1984). |
P. 176{199.
6] J. R. Ringrose. On some algebras of operators // Pros. London Math. Soc. |
(3) 15 (1965). | P. 61{83.
7] W. E. Longsta(. Strongly reexive lattices // J. London Math. Soc. | (2) 11 (1975). |
P. 491{498.
8] A. Hopenwasser, R. Moore. Finite rank operators in reexive operator algebras // J. London
Math. Soc. | (2) 27 (1983). | P. 331{338.
9] ). *. +,-./. -,
0 123043 5,
6.3 ,
13 | .: 789-0
;, 1986.
10] ). *. +,-./. 2304 5,2-0224 ,
14 | .: , 1989.
11] <. =. ,02. >/ 5.2.022/ 502. 28,?-/ CSL,
14 50 // ; | 49:4 (1994). |@. 161{162.
12] <. =. ,02. -,
6. .0/.0 ,B1043 -9,/ 29 28904- 524- ,
1- // -. N-. | 41 (1987). | @. 769{775.
13] <. =. ,02. Q.2.022 5,..B 2T02.B 28,?-/ CSL,
14 262/ 24. Q52.
14] A. Ya. Helemskii. A description of spatially projective von Neumann algebras | J. Oper.
Theory, 1994.
15] A. Ya. Helemskii. The spatial atness and injectivity of Connes operator algebras | Extracta Mathematicae. | 9 No. 1 (1994). |P. 75{81.
16] Z. 7. >,-2, <. [. @,020. = -,
3 5243 ,
1 // [.2
;. -., -32. | 2 (1981). | @. 55{58.
17] B. L. Osofsky. Homological dimensions of modules // Regional Conference Series in Mathematics. | Providence, 12 (1973).
18] <. [. @,020. 5024 12304 ,
14 // 780. ) @@@]. @. -. |
43 (1979). |@. 1159{1174.
19] A. Grothendieck. Produits tensoriels topologiques et espaces nucleaires // Mem. Amer.
Math. Soc. | 16 (1955).
20] ^. _.-B. C -,
14 3 59.0,2 | .: , 1974.
21] E. Christensen. Derivations of nest algebras // Math. Ann. | 229 (1977). | P. 155{161.
22] F. Gilfeather, D. R. Larson. Commutants modulo the compact operators of certain CSLalgebras // Topics in modern operator theory, Basel, 1981. | P. 105{120.
23] ;. ]92. `2z2,B24/ 2,8 | .: , 1975.
&
': 1995.
, . . -
70- (14.02.1924{26.05.1989)
512.552
--
, , ,
!, -
! #!, $
!% &. . &. '
(
)
( * +%.
Abstract
V. K. Zakharov, Connection between the classical ring of quotients of the ring of continuous functions and Riemann integrable functions, Fundamentalnaya i prikladnaya matematika 1(1995), 161{176.
The small Fine-Gillman-Lambek extension generated by the classical ring of quotients,
and the Riemann extension generated by Riemann -integrable functions are both characterized as divisible envelopes of the same type of the ring of all bounded continuous
functions on the Aleksandrov space. This shows the similarity of these extensions that are
rather di2erent by their origin.
C T,
, BM BM 0 1, ! (#1]
x 31' #2] 18.1.2), B B 0 !
(#2] 15.6) ! (#1] x 32),
R , -
. (#3] IV, x 5, . 16, 17),
L , -
01 (#3] IV, x 6.3), UM (#3] V, x 3.4), 3 C
00
1
! *
3
( .
! 1995, 1, N 1, 161{176.
c 1995 " #$ %&,
' \) "
162
. . (#4]' #2] 27.2) . . 4, 5
, . . c-%&. 6 1, C 1 .
! '(& )() ' * c-+, c-% C.
7 ! 1 1 1 #5] (#6] x 2.3,
x 4.6). 3 #5] ) + -. C B +8 5 & /,-0&-&( C Q,
C Q C Q C
(. #7]). 9 (- ) 1 #7]{#9]
& ) + -. C B 0 &1& +& /,-0&-&( C Q8 cl , 5 C Qcl C Qcl C.
7 5 , c- ! 15 , , , #10], !
-1 1 C -1 C.
= 1 1
#11]{#16], 1 &, ( 2') ' % C. > 5 ! #14] 1 + () C L 3&& + -. C BM10 ') , #15] 1 ' + C C t 2 3&&
+ C UM 1 .
1 : + 4& C R & + /,-0&-&( C
Q8 cl .
@, !, ( 1
), 1 .
= 1- , ' #17], 1 #18], #19]. =
1 1 5 | +
8
/,-0&-&( C Q.
0
1
cr -!
as-"
7 1.1{1.4 #20]. 7! 1
1.4.
163
: : :
7 Us (H) \ H Us (H) 1 Us0 (H). B fR H j 9U 2 Us0(H)(R H n U)g 1 R0s (H). 4 s-5.
C 1, O(T X ) 5 . 7! .
1. 2 f 2 O(T X ) f(t) > 0 () t 2 T . 5) 22
62% g 2 O(T X ) , f = g2 .
p
. 4 g : T ! R, g(t) f(t) 1 t 2 T. 7 f 5 fXnk 2 X j kg , !(f Xnk ) < 1=n. .1 , 5 f xni , xni+1 ; xni = 1=n. . Pni f 1 (#xni xni+1]) Yni fXnk j Xnk \ Pni 6= ?g. 7 s 2 Yni. 6
s 2 Xn Xnk , Xnk \ Pni 6= ?. t
. 6 f(s) = f(s) ; f(t)+f(t) < 1=n+x
ni+1 = xni+2. 3 f(s) > ; 1=n + f(t) > xni 1. p7!p!(g Yni ) < pxni+2 ; pxni 1. pF,
p p p # xni 1 xni+2] #0 xn3].
7! !(g Yni ) < xn2 = 3= n. G, g 2 O(T X ). 0 .
2. 2 pn f8n 2 A O(T X )=I fn 2 O(T X ). 5) .1
2 2:
) 22 ' fnk 2 N j k 2 Ng , '2 ) = 1 () n > n 9
fqkn 2 A j k 2 N n > nk g , k(p2n + qkn
k
() 22 ' fnl 2 N j l 2 Ng , ' fIln 2 I j l 2 N n > nl g , jfn(t)j < 1=l () n > nl ()
t 62 Iln .
. ) ) 1). 7 qkn g8kn gkn 2 O(T X ) 2 O.
6 k n 5 fIkni 2 I j i 2 Ng, , pfn (t)+
p
2 (t) ; 1=k < 1=i 1 t 62 I
+ gkn
kni. i = k. 6 jfn(t)j < 2= k
1
n > nk pt 62pIknk . = l k =
= k(l), , 2= k < 1=l. = ! k nk Iknk
1 nl Iln . 7 n > nl t 62 Iln . 6 jfn(t)j < 1=l.
1) ) ). = k l = l(k), , 1=l2 < 1=k. = ! l nl Iln 1 nk Ikn.
7 n > nk . . hkn 2 O, , hkn(t) (1=k ; fn2 (t)) _ 0
1 t 2 T. 6 5 gkn 2 O, , 1=k ; fn2 (t) =
2 (t) 1 t 62 I . 7! k(f 2 (t) + g2 (t)) = 1 !
t. .
= gkn
kn
n
kn
2 ) = 1 ! qkn g8kn 2 A. 75 , k(p2n + qkn
1 n > nk . 0 .
;
;
;
;
2 $ !
%-&-'"
7 T . .
T a- T fcoz c j c 2 C g, C 1 c- 164
. . T. 6 C = O(T T ) , ,
C c- a- (T T ).
H , G (T ) G T,
F (T ) F G 0(T ) = T . 7 ! G 0 (T ) F 0 (T ) 1 G 0 F 0 .
2.1
2.1.1 asb-
crb-
7 Jb | -1 T , . .
, 1 G 2 G 5 D 2 Jb,
D G.
. a- T 5 Tb : Jb ! F ,
Tb (D) cl D. C (31& '1& ' T. 71
: (T T Tb ) (H H H) 1 asb -'1& '(3& asb -'(3& ' T.
. c- C 5 L : Jb ! C (C) ,
Lb(D) fc 2 C j TD \ coz c = ?g. C (31& 3&&
% C. cr-H (C Lb) cr- as- (T T Tb). . u : (C Lb) (A A) 1 crb-%1& +& crb -+& % C.
3. 1 Tb ( ,& , &.
. 7 F T0 n T ? 6= D 2 Jb. . fD g
! Jb , D D T0D \ F = ?. 7, D = top D .
7 ? 6= E 2 Jb E D. . T0 E0 , E = E0 \ T, U T0 n F. 6 E0 \ U 6= ?, 5 T0 G0 6= ? , clT0 G0 E0 \ U. . ? 6= G G0 \ T. = 5 ? 6= D~ 2 Jb
, D~ G. 6 clT0 D~ U , D~ = D . H
, D~ E. 0 .
B ! , 1 crb- 5 asb -1.
2.1.2 s- asb-
7 : T H | asb -1. > fU 2 G j cl U = T g 1 U ,
U \ G 0 | U 0 . fR T j 9U 2 U (R T n U)g
( ) 1 R R0 . 4 s-5.
4. 0 2 : T H | asb-'(3. 5) 1R Rs(H) 1
0
R Rs (H).
. 7 V = 1U D 2 Jb. 6 5 fD g Jb , cl D D \ U D D. 7 ? 6= E 2 Jb ;
;
;
165
: : :
E D. 6 5 , E \ D 6= ?. 7! 5 !
? 6= F 2 Jb , F E \ D . >, D = top D . 7! HD
HD . 6 TD U, HD V \ HD . G, V \ HD HD .
7! V s-. 0 .
!. ;1 h 1R Rs(H)i = Rs(H) h 1R0 R0s(H)i = R0s (H) ;
s-&.
;
4 1.4.2 , a p mod Rs (H) a 2 O(H H) p 2 F=R
- a 2 OD , p 2 (F=R)D . 3 5
.
2.2 --
7 u : C Qcl 1 C, 5
!
1 c=d c d C, , d (. #6] x 4.6). . Qcl Qcl , 5
! a 2 Qcl , 5 n = n(a) 2 N, ,
n1 ; a n1 + a . . u : C
Qcl cl
), ) % 1* u : C Q . H Qcl ,
15 , c-.
. Qcl kak, 1 m=n, ! m1 + na m1 ; na #7]. . c- Q8 cl , Qcl
! . c-. u : C
Q8 cl 1 L,
M 01 #5] (. #7] #8]), ! 1 &1& +& /,-0&-&(. 7 1 #5] 3 62%-6& ' + C
Q8 cl , 1 #7].
2.3 !"
-#
--
. U 0 fU 2 T j cl U = T g R0 fR T j
9U 2 U 0 (R T n U)g. 9 X T S 0 -&&, X = G R
G 2 T R 2 R0 . . S 0 - 1 SP 0 .
L f : T ! R Z 0 -62%,, f 2 O(T SP 0). c-H O(T SP 0)
1 ZP 0. . c- Z 0 ZP 0=R0 . L--
c- u: C
Z 0 Z 0 -+& % C.
" 1. < () ' )2) ' T c-+
C
Q8 cl 3&6 c-+ C
Z 0.
. 4 Z 0 kf8k, -0
supfjf(t)j j t 2 U g U 2 U .
166
. . . 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 .
;
;
167
: : :
2.4 % &
& Z 0 -
C , R0 s-5 (Tb R0 SP 0)
. 7! 1 A : Jb ! C (Z 0 ), , A(D) ff8 2 Z 0 j 8n(TD \ cozn f 2 R0)g 5 c- Z 0 , u : (C Lb )
.
(Z 0 A) - crb--
" 2. 4+ u: C
1* '
Z 0 Z oc j aZ oc &1& crb -%&
Z oc .
. 41 Z 0 A R0 I . 7 U 2 U 0 f : T ! R |
0
, , 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
00
0
00
0
00
0
00
00
0
0
0
0
00
00
0
00
0
00
0
00
00
168
. . jxink1 ; xink2 j < 5=n. 3 1, Yinl \ Yinl1 6= ? Yinl \ Yinl2 6= ?,
00
0
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
0
00
00
0
0
00
00
00
00
0
00
00
0
0
0
0
00
0
00
00
0
00
0
0
00
0
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
00
0
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. = T n U.
00
0
0
0
00
0
0
0
00
00
00
0
00
0
00
00
0
00
0
169
: : :
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
0
00
0
00
0
00
0
00
0
h (t) ; h (s) 6 h (t) + 3=r ; h (s) 6 zirkl ; zirkl + 3=r < 4=r
0
0
00
0
00
0
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
0
0
00
0
0
00
00
0
00
0
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
6
0
6
0
0
0
00
6
00
6
0
= 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
0
0
0
0
0
0
0
0
170
. . 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
0
;
;
;
;
;
2.5 (
" ) E^ Z 0 -
! , Z 0 - u: C Z 0 1
crb- Z oc .
" 3. 2 crb-+ u^ : C A^ 3 E^ 3 asb-'(3 : T H . 5) (, 62% a 2 A^ 22 .& p 2 Z 0
,, a p mod R0s (H).
. #21]. 41 Z oc (^uC) B. 7 a | ! B, 5 5 E^ hu^EC i F^ hu^FC i A^ ^ A),
^ , '^
'^ 2 homA^(E
^uEC u^C ^ ^a 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 '^u^fi
1P hi. 6 '^
^ufi = a^ufi , 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.
;
;
;
171
: : :
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 (^ufi )(s) 6= 0
i. 7! (^ufi )(s)(f )(s) = (^ufi a)(s). >,
(f )(s) = a(s). 7 s 2 1W. 6 (^ugj )(s) 6= 0 j. 7!
(^ugj )(s)(f )(s) = 0. 6 ^ '^,
u^gj '^E^ = 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) 22 '2, 31, ' &2 R0s (H)
0
^
63& v^ 3 u^ : C A u: C Z , ) ' crb -+
&+ ).
! 2. crb-4+ u: C Z 0 (+& E^.
;
;
;
;
2.6 + E~ Z 0-
! , Z 0 - u : C
crb - Z oc .
Z 0 A 3 E~ 3 asb -'(3 : T H . 5) () .& p 2 Z 0 22 62%
a 2 A, p a mod R0s (H).
" 4. 2 crb-+ u : C
. 41 Z 0(uC) B. 7 U, f, p c 2.
172
. . 7 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) 22 '&, 31, ' &2 R0s (H)
0
~
63& v~ 3 u: C Z u~ : C A, ) ' crb-+
&+ ).
! 2. crb-4+ u: C Z 0 &+& E~.
;
;
2.7
%" -
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 .
;
2.8
. --, )"
= 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
: : :
> x 2.3 #6] (% ) '1& %& 1* % C Q QH= ! '8 1-
' 2 QH.
6 , 2.2, )2 u: C Q ') %
8 Q 1* u: C Q. . c- Q,
kak. c-. u: C Q8 1 #5] (. #7]),
! 1 +& /,-0&-&( % 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 .
3.1 #!"
-#
.
1 1 1 . 7 | T , -
- 5- - B 1
T (#3] . IX, x 3, . 2). = 1 , T 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
) 22 ' & 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 01. 7! f 1 (x) 01 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<. |
1987. | . 294, N 3. | 8. 531{534.
12] -.
/. . 8
=* )#>, "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<, #. )#). |
1990. | . 54, N 5. | 8. 928{956.
15] -.
/. . <?#"# 9#" *+ "##3
"3. 4"+ // 9# "=. | 1992. | . 4, 3. 1. | 8. 135{153.
16] -.
/. . *+ 2"3. ,#)3# 2 *+ "##3
"3. 4"+:
;: : : ,. 4=.-). ". | 8.-T., 1991. | 210 .
17] 9#",
9. ;. Additive functions in abstract spaces I{III // #). . | 1940. |
. 8. | 8. 303{348Y 1941. | . 9. | 8. 563{628Y 1943. | . 13. | 8. 169{238.
18] -.
/. . T= [," "
9#",
>#)# "3# // !=
. <9 , #. )#). | 1992. | . 56, N 2. | 8. 427{448.
19] -.
/. . 2## =3, #
EQ# 2#) ?#") *+ "##3
"3. 4"+ // /#" . "-, #. 1. | 1990. | N 1. |
8. 44{45.
20] -.
/. . 82#"-,#)# ?#"# ?#"# \ *+ ".
#3 "##3
"3. 4"+ ,#) 2 // 9# "=. | 1993. |
. 5, 3. 6. | 8. 121{138.
21] -.
/. . ;#)* " 2#"-"3# ,#3 2#" " ),#
// #). =)#. | 1981. | . 30, N 4. | 8. 481{496.
22] -.
/. . 0"+"*" .#=+ E, #"3# #?# 4"+ ) \ =")*"3. 4"+ ), 2"3. "##3
"3.
4"+ // . . ). -
. | 1982. | . 45. | 8. 68{104.
23] Zaharov V. K. On functions connected with absolute, Dedekind completion and divisible
envelope // Per. Math. Hung. | 1987. | V. 18, N 1. | P. 17{26.
! $+: # 1994.
. . , . . . . . , .
Abstract
A. A. Zolotykh, A. A. Mikhalev, Endomorphisms of free associative algebras over commutative rings and their Jacobian matrices, Fundamentalnaya i prikladnaya matematika
1(1995), 177{189.
A matrix criterion for an endomorphism of the free associative algebra of &nite rank
over a commutative ring with the unity element to be an automorphism is obtained.
1 K | 1,
X = fx1 : : : xng, A(X ) | X K , A+ (X ) | K , U (X ) = A(X ) K A(X ) | K - c (a b) (c d) = ca bd. " # A(X ). $ A(X ) U (X ) % K -
, &
' ' ' .
( i = 1 : : : n K -
@x@ i , %&
, @ (xj a) = (1 a) + (x 1) @a 2 U (X )
ij
j
@xi
@xi
@a ij | *, i = 1 : : : n, a 2 A(X ). " @x
i
+
$ a 2 A(X ).
'( ) * , 93-0111543, .( , M22000.
1995, 1, N 1, 177{189.
c 1995 !,
"#
\% "
178
. . , . . .
' | $/
A(X ), , $ , ' $/
U (X ), + + J (') 0
( U (X )) $/
':
0 @'(x1)
@'(xn ) 1
@x1 C
B @x. 1 .
.. C
..
J (') = B
B@ ..
. C
A:
@'(x1 ) @'(xn )
@xn
@xn
2 1. ' A(X ) , J (') U (X ).
2
, + +, K | , 1 3. (
4. 5
65] n = 2, :. ;//
68] % n.
( ' + , + $/
/
, + ( . <
64]> ' 5
+ + 3. 3. 3
62], 63], *. @$ 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-/ 123
/ $4/ 3/ 5$ . (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:
<
! !
, | !
s !
j s.
1. & r m n
;1
YY Y
;n =
i=1 j =1 s=0
(i j s)n;s
;
;
Y
(z z )
;
>
Q> (z z ) ,
0 6 6 m(n + 1) 1, $ .
. - j = j (r1 x1 x2 ) , !
z ;
;
r
1
Y
i=1
(z + i n + x1) =
;
rX
1 +1
j =1
j j (z x2)
;
! 0 6 r1 6 r 1, 1 6 x1 6 n, 1 6 x2 6 n. < ! ;n, ! s j, ! (s j). = ;n
, s = 0. ) (0 m) (1 1),
;
194
. . (1 2),: : :, (1 r), j (r 1 n 1), j = 1 : : : r. <
(0 m) ! ;
m (z )
n
Y
x=1
a(z n + x) b(z )
;
;
= (r m )m (z )
;
nY
;1
x=1
nY
;1
x=1
a(z n + x)
;
a(z n + x)
;
rY
;1
i=1
rY
;1
i=1
(z + i) =
(z + i ) = 0 1 : : : (n + 1)m 1:
;
) r m ! ! j = m 1 m 2 : : : 1 (0 j) (0 j + 1), !
! r j . > (0 m) (1 1),: : :, (1 r 1), j (r 2 n 1), j = 1 : : : r 1, ! r;1 m .
- ! j = m 1 m 2 : : : 1 (0 j)
(0 j + 1), ! ! r;1 j .
-! ! r , , - ,
s = 0, ! ;
;
;
;
;
;
;
;
;
;
;
j (z )
nY
;1
x=1
a(z n + x) j = 1 : : : m
;
! ! r m
YY
i=1 j =1
(i j ):
;
? ! , s = 1 ( !
) !. - ! ! , , , s = 1. -! s = 0 1 : : : n 1, , 9 ! 9 !
;
;n =
r m
YY
(i j )n
i=1 j =1
;
j (z s)
;
sY
;1
x=0
b(z x)
;
n;Y
s;1
x=1
a(z n + x)
;
s=01:::n j =1:::m
=01:::m(n+1);1:
- s = 0, (0 m) (1 1),: : :, (1 r), j (r 1 n 1 1), j = 1 : : : r . !.
= ! s = 0 1 : : : n 2. - s = 0 . !. ) ;
;
;
;n =
nY
;1 Y
r Y
m
s=0 i=1 j =1
(i ; j ; s)n;s
j (z s)
;
sY
;1
x=0
b(z x)
;
s=01:::n j =1:::m
=01:::m(n+1);1:
195
@! ! ! ! )!! z0 z1 ,: : :zm(n+1);1 (., , ! 334 #7]), ! ! ! ! . B 1 !
.
)! !9 ! ! i j Z.
;
62
2. W(z) | ' (n + 1)m 1. $ wjs, j = 1 : : : m s = 0 1 : : : n , # z ;
W(z) =
n m
XX
s=0 j =1
wjsj (z s)
;
sY
;1
x=0
b(z x)
;
nY
;s
x=1
a(z n + x)
;
) wjs Kjs()W()d
1 I
wjs = 2i
Q s;1
Q ;s
s) x=0 b( x) nx=1
a( n + x)
; j +1 (
Kjs() = 1 s = 0 j = 1 : : : m
j
X
1
Kjs() = a( s + 1) k k ( s)
k=1
)%% k ;
;
;
;
;
a(z + 1) =
mX
+1
k=1
k k (z)
# # z ; | , , -
, # , $ '1 () =
, '2 () =
n
Y
x=0
n
Y
x=1
b( x)
;
a( n + x)
;
#
' $
; i j Z.
. < ! wjs ! 1 i j Z. ?
! ;
62
;
62
196
. . ! ! !
5 #4] 9 , i j
. B 2 !
.
-
1 (x) 1 l (x) = l (x)(xx mn) l = 2 : : : m + 1
;
Rl (z) =
1
X
=n
z l ()
mn
;1
Y
x=0
( x)
;
Y
;n
x=1
a(x)
Y
x=1
(b(x));1 l = 1 : : : m
d (z) j = 1 : : : m:
j (z) = j z dz
5!
Rl (z) =
X
!
m
j =1
Plj (z) =
(4)
Plj (z)j (z)
n
X
s=0
(5)
pljs z s
;1
Kjs()l () mn
x)
x
=0 (
d
Q s;1
Qn;s
(
s)
b(
x)
a(
n + x)
; j +1
x=0
x=1
Kjs () ; ! 2. D !
(5), !, > n, 9 z . E 1
pljs = 2i
I
;
Q
;
;
Y
x=1
b(x)
;
Y
;n
x=1
(a(x));1 , ! , l ()
mn
;1
Y
x=0
( x) =
;
m n
XX
j =1 s=0
pljs j ( s)
;
sY
;1
x=0
b( x)
;
nY
;s
x=1
a( n + x):
;
<! ! ! 2. 5
, > n (5) !. <! !
!, ! . 301 #4]. : (5) !
.
3. 1 6 j k 6 m, 0 6 s 6 n. I
()k ( s)
1
Jks = 2i Kjn
j +1 ( n)
;
;
;
s;1 b( ; x) Qn;s a( ; n + x)
x=0 Q
x=1
d
n;1 b( ; x)
x=0
Q
1 k = j , s = n k s. * Kjn()
; 2.
197
. - s = n. 7 ! Kjn() j
X
l=1
!, l l ( n) = a( n + 1)
;
;
mX
+1
;
l=j +1
l l ( n)
;
+1
1 I k ( n) d mX
1 I l l ( n)k ( n) d:
Jkn = 2i
n)
; j +1(
l=j +1 2i ; j +1( n)a( n + 1)
) , ! , . . . ! ! (
l > j + 1)
, !, ! k = j . 5
, s = n !. - 0 6 s < n.
5! 9 9 !
Pj
Qn;s
I
(
n)
n + x)
1
l
l
l
=1
x=2 a(
Jks = 2i
d:
Qm
Q n;1
n) l=k ( + l s) x=s+1 b( x)
; j +1 (
B
, , !
, ! ! 9 , . . ; ! , . B 3 !
.
-
(z) = Plj (z) lj =1:::m :
4. & Qmn
mn
rmn
=1 a(x n)
(6)
(z) = ( 1) Qn;1xQ
r b(x ) z :
i
x=0 i=1
. @!,
R (z)
P12(z) : : : P1m (z) 1
(z)1 (z) = : :: :: :: ::: : : : : :: :: ::: : : : : :: : R (z)
Pm2 (z) : : : Pmm (z)
m
(z) z = 0 !
9, mn (. . ! Rl (z)). <!,
(z) = Cz mn ! C , !, !, z n
Plj (z), . . !
;
;
;
;
;
;
;
;
;
;
;
j
;
j
;
;
;
;1
x) d 1 I Kjn() l () mn
x
=0 (
:
d = 2i
Qn;1
n)
; j +1(
x=0 b( x)
lj =1:::m
Q
;
;
;
198
. . @ j- j .
5! I
I
m
Y
1
d = (2i)m : : :
;1
;m j =1
Q
mn;1( ; x)
x=0 j
j +1 (j ; n)
!
Kjn(j )
l (j ) lj =1:::m d1 : : :dm Qn;1
x=0 b(j x)
j
j
;
! ;j | j , ! , ! ;. @!
j
l (j ) lj =1:::m
j
! )!! 1 : : : m (. !
1). ? ! d ! ;n 1, z = , = 0 1 : : : mn 1 z = j , = mn 1 + j, j = 1 : : : m. 5! 1
;
;n =
nY
;1 Y
r Y
m
(i j s)n;s
s=0 i=1 j =1
;
;
;
(mn 1)!(mn 2)! : : :1!
;
;
m mn
;1
Y
Y
j =1 x=0
(j x)
;
Y
( ):
m>>>1
;
E m
Kjn(j )
Qn;1
j =1 j +1 (j n) x=0 b(j x)
Y
;
;
;1 : : : ;m . < j
l (j ) lj =1:::m =
j
!
Y
( )
m>>>1
;
m
1 I : : :I ; Y
Kjn(j )
d : : :dm =
;1
(2i)m ;1 ;m n j =1 j +1 (j n) Qnx=0
b(j x) 1
;
=
nY
;1 Y
r Y
m
;
(i j s)n;s (mn 1)!(mn 2)! : : :1!d:
s=0 i=1 j =1
;
;
;
(7)
;
@! ! d. ?, j ! ! ! ;n . - j 3 , !, , !, ! !
m, ! m . - 199
. - , (7), !
d1 = j ( s)
;
sY
;1
x=0
b( x)
;
nY
;s
x=1
a( n + x)
;
:
s=01:::n;1 j =1:::m
=01:::mn;1:
, ! ! ! a( n + 1), ( ! ) !
1 n n 1 z = 0 1 : : : mn 1. -
(7) 9 !
;
;
mn
;1
Y
=0
a( n + 1)(mn 1)! : : :1!
;
;
;
nY
;2 Y
r Y
m
(i j s)n;1;s =
s=0 i=1 j =1
=
;
;
nY
;1 Y
r Y
m
(i j s)n;s (mn 1)! : : :1!d:
;
s=0 i=1 j =1
;
;
7 ! ! ! ! . B 4
!
.
3 @ lj (z) ! Plj (z) !
(z). 5! lj (z) | , ! (m 1)n.
5. +
;
lj (z) =
(mX
;1)n
s=0
ljs z s mn a(x ; n)
x=1
n;1 Qr b(x ; )
k
x=0 k=1
ljs = (;1)rmn Q
Q
1 I Uls ( + mn)
2i ; l+1 ( + mn s)
Q
(8)
j () mn
x
=1 b1 ( + x)d
Q(m;1)n;s
Qs;1
a( + x)
x=0 ( x)b1( + mn x) x=1
;
;
;
1 s = 0 l = 1 : : : m
l
X
Uls () = >
1
:
a( n s + 1) k=1 k k ( s) )%% k 8
>
<
;
;
;
a( n + 1) =
;
mX
+1
k=1
k k ()
200
. . # # . , ; # '1 () =
'2() =
(mY
;1)n
( x)
;
x=0
(mY
;1)n
x=1
a( + x)
#
' ( ; 2).
. E j- ! (z) j (z) , ! (4) !
. - P11(z)
: : : R1(z) : : : P1m (z) : :: :: : :: :: :: :: : :: :: :: :: :: : :: :: :: = (z)(z):
j
Pm1 (z) : : : Rm (z) : : : Pmm (z)
: ! j- ,
m
X
Rl (z)lj (z) = j (z)(z):
(9)
l=1
-! ! z, . . ! ! !
z . - !
! Rl (z). 7
Q ;n
1
X
x=1Qa(x)
Rl (z) =
z l () ( mn)!
b (x) =
x=1 1
=mn
Q
Q(m;1)n
1
mn
X
z
a(x)
=1 a(mn n + x) :
x
=1
= Qmn b (x)
z l ( + mn) !xQ
b (mn + x)
x=1 1
x=1 1
=0
;
;
-! (9) ! ! , , > (m 1)n z +mn ;
Q ;(m;1)n
;1)n X
m
m;1)n
a(mn ; n + x) (mX
x=1Qmna(x) xQ
=1
ljs l ( + mn ; s) x=1 b1(x) x=1 xb1(mn + x)
s=0 l=1
(m;1)
n;s
sY
;1
Y
Q(
x=0
( x)b1 ( + mn x)
;
;
H z +mn (9) , !,
Y
Cj () a(x)
x=1 b(x)
x=1
a( + x):
201
! C | z mn (6). @! ! ! ljs , ! > (m 1)n:
;
(mX
;1)n X
m
s=0 l=1
ljs l ( + mn s)
;
sY
;1
( x)b1 ( + mn x)
x=0
;
;
(m;1)
n;s
Y
x=1
a( + x) = Cj ()
mn
Y
x=1
b1 ( + x):
(10)
-
! ,
! ! (
> (m 1)n). 5 ! 5 ! 2.
B 5 !
.
;
4 ?
1. -
Rl (z) =
1
X
=n
z l;1
mn
;1
Y
x=0
;n a(x)
( x) Qx=1 b(x)
l = 1 : : : m
Q
;
x=1
a(x)
Y
j
;1
j = 1 : : : m:
j (z) = z =0
x=1 b(x)
5!, !, lk kj , !
1
X
l;1 =
m
X
lk k ()
k=1
m
X
j () =
-
jk k;1
k=1
(lk )lk=1:::m (jk )jk=1:::m !, -
! !. 7 (5) !,
m
X
Rl (z) = Plj (z)j (z)
(11)
j =1
!
Plj (z) =
m
m
X X
k1=1 k2 =1
lk1 k2 j Pk1k2 (z):
(12)
?
, Plj (z) I. ? z , = n n + 1 : : : n + m(n + 1) 1 ;
202
. . (11) 9 Y
x=1
b(x)
Y
;n
x=1
(a(x));1 :
) m n
sY
;1
nY
;s
j
;1
pljs ( ; s)
b( ; x) a( ; n + x) =
j =1 s=0
x=0
x=1
mn
;1
Y
l
;1
=
( ; x) = n n + 1 : : : n + m(n + 1) ; 1
x=0
(pljs | z s Plj (z)). , l XX
, ! pljs , 1 !, !
( ). -
I, pljs ! .
: !
(z) = Plj (z)lj =1:::m :
7 (12) !, (z) = lk lk=1:::m Pk1k2 (z) k1 k2 =1:::m kj kj =1:::m = Pk1k2 (z) k1 k2=1:::m :
7 4 , = 0 () = 0. - ! ! h1 1 () + + hm m ()
! h1 : : : hm | I. 5. .
() = 0, ! ! () h1 : : : hm
, ! . - !
! ! :
h1
h2 : : : hm D() = P::21::():: :: P::22:::(): : :: :::: :::: P::2::m ::() = 0:
Pm1 () Pm2 () : : : Pmm ()
- ! D() . ? !
D() n
. : ! D() . ?, (12) ! !
! ! (., , !
. 108{109 #8]), , ! ! D() !
m 1 ! () 3 j
j
j
j
6
j
j
j
j
6
6
6
j
j
! 1
;
203
! lk kj ( 2m). 5. . lk kj
! , n, 9 9 !
m 1 ! (). <! 5. 7 !, Qmn
ljs = ( 1)rmn Qn;1xQ=1r a(x n)
x=0 k=1 b(x k )
(13)
Q
;s;1
b1 ( + x) d
1 I UlsQ( + mn)j () l ( + mn s) mn
x
=1
Q(m;1)n;s
s ( x)
2i ;
a( + x)
x=0
x=1
!
l
Y
l () = b1 () ( + l );1 :
;
;
;
;
;
;
k=2
@ J 1=2i (13).
- 1 ! 2 : : : m r 1 ! ! ! 2 : : : r . >9 J !
, . 7
s U (x + mn) (x) (x + mn s) Qmn;s;1 b (x + x0)
X
J = Qsls (x x0 ) j l Q(m;1)n;s x =1 0 1
=
a(x + x )
x=0 x =0
x =1
;
;
0
;
x 6=x
s Qmn;s;1 Qm ( + x + x0 )
X
x =1
l=r+1 l
=
(n!)m;r
x=0
0
0
0
Q
0
mn;s;1 Qr ( + x + x0)U (x + mn)
l
ls
x =1
l=2
Qs
(x
; x0 )
x =0
x 6=x
(x)l (x + mn ; s)(n!)m;r :
j
Q(m;1)n;s
a(x + x0 )
x =1
0
(14)
0
0
0
:! !
6 #6], , Q1 I , Qmn;s;1 Qm
0
l=r+1 (l + x + x ) Z s = 0 1 : : : (m 1)n x = 0 1 : : : s
Q1 x =1
I
(n!)m;r
" > 0
Q1 = O(n(q +")n ):
? , ! ! (14) 9
Q2 , , 3 #4]
" > 0 ! Q2 = O(n"n):
5
, ! lj () , ! Qn;1 Qr
b(x k )
(m;1)n
x=0
Qmnk =1
(15)
a(x n) Q1Q2 L
0
2
;
;
x=1
;
204
. . ! Q1 = 0, Q2 = 0, 0 = L N, L ZI , " > 0
Q1 Q2 = O(n(q +")n ):
@!, , D() I, ! ! D():
D() > n;(q +")n
(16)
( , , ! , n
! (15) ! nn ). @
(16) ! " > 0 ! ! 9 n.
@
! ! D() ! . 5. .
(z) ! ! , !
j (), j = 1 : : : m ! ! ! 1 (). 5! D() !
L
h2 : : : hm : : : P2m() D() = 1() R:2::():: :::P:22: :()
:
::
:::
:: : : : : :: :: 1
Rm () Pm2() : : : Pmm ()
Pm
! L = j =1 hj j (). 7 ! ! " > 0 ! 9 n Rj () 6 n((1;m)(m;r)+")n
Plj () 6 n(m;r+")n ! D() :
D() 6 Ln(m;1)(m;r+")n + Hn;(m;r;")n H = max( h1 : : : hm ):
H ! (16), ! !
L, ! !, H ! :
(m;1)(m;r)+q
":
m;r;q
L >H
6
6
6
2
2
j
j
! 1
j
j
j
j
j
j
j
j
j
;
j
j
j
;
7 ! ! ! ! 1.
. - !
1 ! j (z)
1 (z) j = 2 : : : m z = ( ! ). ) !,
11 (z)j (z) 1j (z)1 (z) =
P (z)
: : : P2j ;1(z) R2(z) P2j +1(z) : : : P2m(z) 22
= :: : : ::: :: :: :: :: ::: :: :: :: ::: :: :: :: :: ::: :: :: :: ::: :: : : :: ::: = rj (z) j = 2 : : : m
P
m2 (z) : : : Pmj ;1(z) Rm (z) Pmj +1(z) : : : Pmm (z)
;
205
!, ! ( !) ! (z).
?
2 3 !
1.
: 9 9 J (13). ) 2 J=
s
X
x=0
mn;s;1 Qm ( + x + x0)
((m ; 1)n ; s)!
x =1
l=r+1 l
Q(m;1)n;s
(n!)m;r
(1 + x + x0)
x
=1
Qmn;s;1 Qr
x + x0)Uls (x + mn) j (x)l (x + mn ; s)(n!)m;r :
x =1
l=2 (l + Q
Q (m;1)n;s Qr
((m ; 1)n ; s)! sx =0 (x ; x0)
x =1
l=2 (l + x + x0 )
x 6=x
Q
0
0
0
0
0
0
7 ! ! ! ,
! Q1 I, " > 0
Q1 6 n(q + 21 (m;1)+")n
! ! 9 n 9 1. >, 6 #6] !
! Q1 ! ! , !
, 9 !
, ! . ?9 !
2 !
1.
) 3 , 1 I
1 I ;
J = 2i
(17)
2
;1 2i
! ! ! , (13) ;1 !
, ;2 , ! !
j
j
;
(m;1)
n;s
Y
( + 1 + x)
x=1
( 3 r = 1), , ! !
s
Y
x=0
( x)
;
9. 7 ;1 = , ! n"n " > 0 (.
! !
1 #3]). ) 1
206
. . (17) 9 ! ,
;2. 7
n;s
X
1 I = (m;1)
Uls ( Q
1 x + mn)j ( 1 x)
s ( x x0 )
2i ;2
1
x =0
x=1
Q
;s;1 Qm
0
l ( 1 x + mn s) mn
x
=1
k=2 (k 1 x + x ) :
Q(m;1)n;s
(x x0)
x =1
;
;
0
;
;
;
;
;
;
;
;
;
0
0
;
;
x 6=x
0
7 9 9 ! 6 #6] k 1 Q, !
, 9 ;2 ;
2
n 12 (m;1+")n
( " > 0 ! 9 n). ?9 !
!
1.
1] C. Osgood. Some theorems on diophantine approximations // Trans. Amer. Math. Soc. |
1966. | V. 123. | P. 64{87.
2] . . . ! "#$ % %# ! &'#&$%#! " // )%$% $% | 1970. | +. 8. | N 1. | ,. 19{28.
3] -. .. //. ! "#$ % " 0$$# ##1 $ '#$%#$ // )%$% $%. | 1991. | +. 49. | N 2. |
,. 55{63.
4] -. .. //. 3 #"$%! /%/! &'#&$%#! "
// )%$. 3#. | 1991. | +. 182. | N 2. | ,. 283{302.
5] G. V. Chudnovsky. Pad4e approximations to the generalized hypergeometric functions. I //
J. math. pures et appl. Ser. 9. | 1979. | 58. | N 4. | P. 445{476.
6] . . . 3 #"$%! /%/! %# ! ! &'#&$%#! " // ,3. $%$. 5#. | 1976. | +. 17. | N 6. | ,. 1220{1235.
7] . 6. -##7/. ,3# 8 ' &3#. | ).: :, 1984.
8] . . 0#<. 0# / < &3# . | ).: :, 1968.
)% *: 1995.
. . . . . . . 517.926
! " #$ & & utt ; uxx ; "a2 uxxtt = "uxxt + "ut ; u2 ut ujx=0 = ujx=1 = 0 = const > 0 0 < " 1:
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 n22 )
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 n212 zn (s) exp(int) + z<n (s) exp(;int)] sin nx
n=1
(z1 z<1 z2 z<2 : : :) 2 l2 | \ ", / "3=2. u1 ! !
@ 2 u1 ; @ 2 u1 = @ u + @ 2 u0 ; 2 @u0 ; u30 + a2 @ 3 u0 @t2 @x2 @t 0
@x2
@s 3
@x2@t
u1 jx=0 = u1 jx=1 = 0
209
$ 3 2- ! t $ zn n2 X zk zp zm ; n2 X zk zp z<m +
2z_n = (1 ; ia2 n33 ; n22 )zn + 6
4
2 2 2
64 Pn k2p2 m2
Rn k p m
n2 X zk z<p z<m ; 2zn X jzk j2 ; 3 jz j2z n = 1 2 : : : (2:3)
+ 6
4
2 2 2
4 k6=n k4 24n4 n n
Qn k p m
Rn = f(k p m) : k p m 2 N n = k + p + mg
Pn = f(k p ;m) (k ;m p) (;m k p) : k p m 2N n = k + p ; m k 6= m p 6= mg
Qn = f(k ;p ;m) (;p k ;m) (;p ;m k) : k p m 2 N n = k ; p ; mg:
0 1
X
p
v = vn(s) exp(iny) + v<n(s) exp(;iny) vn = ;izn = 22 n2
(2:4)
n=1
, ! ! (2.3) \!" - . !
3
2vs = a2 vyyy + vyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 )
(2:5)
v(s y + 2) v(s y) M(v) = 0
(2:6)
M | ! y. 6 , D (2.4){(2.6), !:
u0 = v(s y1 ) ; v(s y2 ) y1 = t ; x y2 = t + x
(2:7)
v = v(s y) | . 3 ! (2.5), (2.6).
0 3], 4] ! (2.5), (2.6), $ $ 7{ F, .$ $ $
! (1.1), (1.2). G , ! (2.3) |
\!" . . (1.1), (1.2). 0 ! .$ , ! !$ . 2$., ! (2.5), (2.6) !, ! $ .
0 , ! ! (2.5), (2.6) ! 3 v = v0(!0 s + n0 y) n0 2 N v0 (y + 2) v0 (y):
(2:8)
H! (2.1), ! ( !. "5=2 ) ! 3
p
u = "u0( x) + "3=2u1 ( x) = (1 + "0)t 0 = !0=n0 (2:9)
210
. . , . . $ ! (1.1), (1.2). I . (. (2.7))
u0 ( x) = v0 (n0 ( ; x)) ; v0 (n0( + x))
(2:10)
2- ! u1( x) ! u1 (t s x). H, / 3 ! ! ! 3 ! (1.1), (1.2), .
1. (2.5), (2.6) (2.8),
. " (1.1),
(1.2) (2.9), (2.10) ! .
3 ! 4, ! ! n $ (1.3) p $ . "n2 , n ! 1 ! (;=2a2) i( 1="2 ; 2=4a2=a).
J \3" . . , 4], 5] ! . - , / !$ ! (1.1), (1.2) ! $. 0/, , ! .
4$ / | $ $! $ !
(1 + 2"0 )h ; hxx ; "a2 hxx ; "hxx ; "h +
(3:1)
+ "(u20 ( x)h) = 0 hjx=0 = hjx=1 = 0
!$ (1.1), (1.2) ! 3 (2.9) ! "2 3. K 6] /.$ 3$ $ .
/ ! , ! (3.1) h ! h exp(;"),
> 0 . , ! $. !$ !$ V + I, $ .
I . V | , $ V h a2 h000 + h00 ; 20h0 + 1 ; M(v02 (n0 y)) ; v02 (n0 y)]h ;
(3:2)
; 2Mv0(n0y)h]v0 (n0y) + Mv02 (n0y)h] h(y + 2) h(y) M(h) = 0:
4, ! $ !$ $
( !, , !, ! v00 (n0y)).
E P span fsinrx
r = 1 : : : r0g, r0 . 211
, .$3. L C(E) ! ! ;1 < < +1 $ u( x) !
E, : p
1
X
2 u () sin mx = sup ju ()j
u( x) =
m
m
2
2 m
m=0 m
1 !1=2
X 2
kukC (E ) =
m
:
m=0
L C 1(E) ! C(E), $ u( x), ! @u=@ 2 C(E)M $
kukC 1 (E ) = kukC (E ) + k@u=@ kC (E ):
0 C 2 (E) !.
0 (3.1), ! h1 = Ph, h2 = h ; h1, $ (1 + 2"0)h1 + P BP h1 + "a2 PBPh1 + "P BPh1 ; " ;
(3:3)
; "(1 + )h1 + "Pu20( x)(h1 + h2 )] = 0
(1 + 2"0)h2 ; h2xx ; "a2 h2xx ; "h2xx ;
(3:4)
; "(1 + )h2 + "(I ; P)u20( x)(h1 + h2 )] = 0
B = ;d2 =dx2. .$ . 0 6 6 1 h2
(3.3) h1 (3.4). 0!3 C 2(E) .$ N. 0, ! !
" kN;1 kC (E )!C 2(E ) 6 N="M
(3:5)
. .$3 N N1 . . | ", r0 , !
! .
6 (3.3) (3.4) , ! N = diag fN1 N2 g = 0. 7 , $ "0 = "0 (r0 ), ! 0 < " 6 "0
kN;1
(3:6)
1 P kC (E )!C 2(E ) 6 N1 =":
2 . !$ (3.3) h2 = 0 h1 = C0() + "C1 ()] h1 = C00 () + "C10 ()] = (1 <1 : : : r0 <r0 )
C0() = e1 exp(i) e1 exp(;i) : : : er0 exp(ir0 ) er0 exp(;ir0 )]
e1 : : : er0 | ! Rr0, 2- ! C1()
r0 2r0 7] . . !.
! "2 _ = "D
(3:7)
212
. . , . . D | V + I $
exp(iry), r = 1 : : : r0, W22. 4 ., ! (3.7) ! (3.6) , $ 4].
4. F. sin rx, r = 1 2 : : :, !
3 $ (1.3) ! r0 !
.3, , !
kN;1
2 (I ; P )kC (E )!C 2(E ) 6 N2 ="r0 :
(3:8)
% . (3.3), (3.4) C(E). 6. N1 N2 , $ . $. 0 , ,
! ! (3.6), (3.8) ! .3 r0
$. 4 $ |
(3.5).
6 . 6] , ! 3$ ! (3.1) /. , $ $!
(3.7). ;$ $! ! .3 r0 (3.2) (
. 4]).
I3$ / | . ! 3
$ ! (1.1), (1.2) | $ 5] . (1.1) = (1+"0 +"2 )t, = (") | ! ", p
u = "u1( x) + "3=2u2 ( x) + "3=2 h
u1 u2 (2.9). . N(" )h = F( x " h h )
(3:9)
N | !$ (3.1) $$ .$ , 2- ! F( x " u v) ,
!
kF( x " 0 0)kC (E ) 6 N3 "
(3:10)
3
=
2
kF( x " u1 v1 ) ; F ( x " u2 v2 )kC (E ) 6 N4 " fju1 ; u2 j + jv1 ; v2 jg (3:11)
N4 = N4 (R), R = maxfjuj j jvj j j = 1 2g.
6 $ , ! Nh = 0 ! 3 @(u0 ( x) + "u1( x))=@. / "2 6] , . !, ! . K, ! !.
$ ! (3.9), $ $ / .
% N C 2(E), 2- ! $. 7 , ! 3 Nh = G( x) $ . $ ! G ! 3 g
213
$ $ !. %3 hG / , @hG =@ . g. 7 /, khG kC 2(E ) 6 N5 ";1 kGkC (E )
(3:12)
3$ Nh = 0, / (3.5).
% !. (3.9) . G (! G @ 2 (u0 + "u1 )=@ 2
g), $ . , N.
# (3.10){(3.12) . / $ . 2- ! h( x " ).
6 (3.12) , ! $ O3 $ .$ $. 0 . $
! (3.9), ! = p(") + "P(" )
p(") !, P O3. J
.
4 # 6 ! $ $ ! (2.5), (2.6) . = 1=a2. / ! s=a2 ! s ! $ !
3
2vs = vyyy + fvyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 )g
(4:1)
v(s y + 2) v(s y) M(v) = 0
$ $ . . .
; $ . 2 , (4.1)
v = v0 ( ') + v1 ( ') + = s ' = ('1 '2 : : :)
1
2 2 X
v0 = n() exp(i'n ) + <n () exp(;i'n ) 'n = n y ; n 2 s n=1
n , !
1
X
n4jnj2 < 1:
n=1
(4:2)
(4:3)
(4:4)
214
. . , . . 0 / $ , v1 ! 1
1
3
X
X
3 3 @v1
; n @' ;
3 nkm @' @@'v1@' = f( ')
(4:5)
n nkm=1
n k m
n=1
.
@ 2 v0 + 1 ; M(v2 )]v ; v03 + 1 M(v3 )
0
f( ') = ;2 @v
+
(4:6)
0 0
@
@y2
3 3 0
2- ! ' ' . . .
4, ! - 'n = 'k 'm 'p . n, k, m, p 3 (4.5) $ !, ! (4.6) const exp(i'n ), n = 1 2 : : :. 0
, n 1
n = (1 ; n2) ; (3j j2 + 4 X j j2) n = 1 2 : : :
2 d
n
n
n
k
d
k=1
k6=n
.$ $ ! (4.1). .$3, , . $ n = jn j2:
1
dn = (1 ; n2) ; (3 + 4 X
k2 ) n = 1 2 : : : :
(4:7)
n
n n
d
k=1
k6=n
4, ! ! (2.3) (4.7) ! ! , $ ! w22, .$, (4.4).
6 . 4] , ! /. $!
! ! ! n1 : : : nm , n1 < : : : < nm , ! m-$ $ $
! (4.1) $ $!.
H! (4.7), , ! .3 ! (4.1) . 0 > 1 $! , < 1 $!$ , (4.7) $ $ $
1 = (1 ; )=3. 2 ! ! 1=k2, k = 2 : : :,
3 $! vk (!k s + ky ) (2.8),
! (4.7) $ $ $ k = (1 ; k2)=3. 0 .$3 .3 vk .
215
k ; 1 $! ( ! ! 1=(4k2 ; 3m2 ), m = 1 : : : k ; 1), < 1=(4k2 ; 1) $!.
J , ! 0 ! $! $ ! (4.1) ! . 0 Q $,
!.
5 % .
4 (1.1) ! "a2 uxxtt, . . , $ !
utt ; uxx = "uxxt + "ut ; u2ut (5:1)
ujx=0 = ujx=1 = 0
(5:2)
$ $ . ! $ ! (1.1), (1.2). ! Q
. W22 W21 , W22 , W21 | $, ! (5.2).
2 . .$ !$ 3 $
3$ ! (5.1) w1 = Lu, w2 = ut, L | !$ $ . ;d2 =dx2. E = W21 W21 ! w_ = A0 Lw ; "A1 L2w + "A2 w + f(w) w = colon(w1 w2)
(5:3)
0
1
0
0
0
0
0
A0 = ;1 0 A1 = 0 A2 = 0 1 f(w) = ;w2 (L;1 w1)2 :
4. F., , ! A0 L ; "(A1 L2 ; A2 ) E . . 0 Q . (5.3) . $, !
$ .
I, ! = 0 ! (5.1), (5.2) !,
> 0 $! Q 3 ( $
2 + "(2 n2 ; 1)
+ n22 = 0, n = 1 2 : : :) | ! (Re n ! ;1 n ! 1) ! (
n ! ;1=" n ! 1). 4 , !, ! . 3$ ! (5.1), (5.2) t. S . x , . $, t = 0.
2 ! 3$ ! (5.1), (5.2) . . . 4, ! . .
. 2, . . a2 = 0.
!, (2.5), (2.6) ! . 216
. . , . . 3
2vs = vyy + 1 ; M(v2 )]v ; v3 + 13 M(v3 )
(5:4)
v(s y + 2) v(s y) M(v) = 0:
(5:5)
0 , ! ! (5.4), (5.5) v(y) 6 0. J $ fvg = fv(y +c) c 2 Rg. 0 v(y) (2.1), ! ( . 2) ! 3
$ ! (5.1), (5.2).
2. 1=2 (n + 1)2 6 6 1=2n2 n.
(5.4), (5.5) n fvk g,
k = 1 : : : n, fv1 g , -
. "! " (5.1), (5.2) (2.1), (2.7) ! .
4, ! . 3 ! (5.4), (5.5) (5.1), (5.2) ! $ . 3 .
0/ 3. !. , $
! (5.4), (5.5).
(5.4) M(v3 ) = 0, 1 ; M(v2 ) = c > 0, v = pcz(y) ! M . ! $ p
(z 0 )2 = 6c (!1 ; z 2 )(!2 ; z 2 ) !1 2 = 3 9 ; 6 2
(5:6)
p
0 < < 3=2. 4!,p , ! z(y c) 3 (5.6),
z jy=0 = !2, z 0 jy=0 = 0. 4, ., ! ! T = T ( c). T , ! 3 ! (5.4), (5.5) n = 1 2 : : : T( c) = 1=n c(1 + M(z 2)) = 1
(5:7)
, c.
p
p p 4! , ! T(0 c) = = c, T( 3=2 c) = 1, @T=@ > 0. 0/ 2 n2 < 1 (5.7) ! 3 . = n (c),
!
@T ;1
0
n (c) = 2nc @
> 0 2 n2 < c 6 1:
(5:8)
0 = n(c) (5.7), !
p Z1
s2 ds
:
(5:9)
'(c) = 1 '(c) = c + 2n!2 6c p
(1 ; s2 )(!1 ; !2 s2 )
0
7 , '(2 n2 ) = 2 n2 < 1, '(1) > 1, (5.8) , ! '0 (c) > 0.
0/ (5.9) ! c = cn.
217
! $ fvk g . $, vk (y) = pck z(y k (ck ) ck ), k = 1 : : : n, $ $! . !
h00 + qk (y)h ; 2M(vk (y)h)vk (y) + M(vk2 (y)h) = h
(5:10)
h(y + 2) h(y) M(h) = 0
(5:11)
2
qk (y) = ck ; vk (y), ! $ ! (5.4), (5.5).
4 , !, -, ! ! (5.10), (5.11) , !. (5.10) L02
! $ ! $ V M -, = 0 !, !
vk0 (y).
. #
= 0 (5.10), (5.11) | .
. I, ! L(h) = 0, L = d2=dy2 + qk (y),
( !. ) ! 3 vk0 (y), ($ vk0 (y)) 3 @z(y (ck ) ck )=@ @T=@ > 0 ! . I , ! ! M(vk vk0 ) = M(vk0 ) = 0 3. !
L(h1 ) = vk (y) h1(y + 2) h1 (y)M
(5:12)
L(h2 ) = ;1 h2(y + 2) h2 (y):
(5:13)
; , !, ! ! 3 (5.10) = 0 h = 1 h1(y) + 2 h2 (y) + 3 vk0 (y)
(5:14)
(5:15)
1 = 2M(vk h) 2 = M(vk2 h)
3 .. 0 (5.14) (5.15), 1 , 2 ! $ (2M(vk h1 ) ; 1)1 + 2M(vk h2)2 = 0
(5:16)
M(vk2 h1 )1 + (M(vk2 h2) ; 1)2 = 0:
J , . . Q .
2 h1 (y) . !
(;1=2k 6 y 6 1=2k)
L(h1 ) = vk (y) h1(;1=2k) = h1(1=2k) = 0:
(5:17)
4, ! vk (y) > 0 ;1=2k < y < 1=2k L(vk ) = ; 23 vk3 vk (;1=2k) = vk (1=2k) = 0
218
. . , . . . L ! (5.17) .. S.
6, ! (5.17) ! 3, Q 3 h1 (y) ,
! h1(y) = h1(;y), h1(y) < 0 jyj < 1=2k. 0 1=2k 3=2k]
! . ! y = 1=2k, . ! 2=k, !, ! , 3 $ ! (5.12).
0 h2 (y) $ !
L(h2) = ;1 h02 (0) = h02(1=2k) = 0
(5:18)
! 3, !, 3 !$ (5.18) $ $ ! ! ;1=2k 0] ! 1=k ., ! ! 3
L(h) = 0, $ vk0 .
0, ! h2 (y) < 0 0 6 y 6 1=2k. 2$., ! (5.18) vk0 0 1=2k], ! h2 (0)vk00 (0) = vk (0), ! vk (0) > 0, vk00 (0) < 0 h2 (0) < 0. 0 ., ! $ 0 < y0 6 1=2k, ! h2 (y) < 0 0 6 y < y0 ,
h2 (y0 ) = 0. H (5.18) vk 0 6 y 6 y0 , !:
Z y0
Z y0
2
3
0
0 6 vk (y0 )h2 (y0 ) = 3
vk h2 dy ;
vk dy < 0:
0
0
0 3 h2 (y) ! ., ! 3 $ ! (5.13).
2 3 . ! (. (5.16)) ., !
M(vk h2) = 0, M(vk2 h2 ) < 0, M(vk h1 ) < 0, sign h1 = ;sign vk , h2
!, . ! 1=k.
6 $ , ! $ $! $ fvk g 2 (0 1=2k2 ). # , ! / $ . . $ v 0 .3 . 0/
. ! (5.10), (5.11) 2(k ; 1) . !$. J .
0 = 0 3 $ ! (5.4), (5.5) (, , v 0)
s ! 1 ., . . !! .
; $ $ ! (5.1), (5.2) . !. ,
$ .
. = 0 (5.1), (5.2), C -, % .
F!$ /$ : $ .$
( ) $ .
219
6 (
% !
P (@=@t @=@x)u = 0 u(t x + 2) = u(t x)
(6:1)
P(v w) | $ !. 0 , ! P(i! in) = 0
n = 0 1 : : : $ $ . ! = !n. J (6.1) 3
u = expfi(!n t + nx)g n = 0 1 : : : :
(6:2)
; 2], ., ! (6.2) $, n m jcn ; cm j > 0
(6:3)
cn = !n=n, c0 = 0. 4, ! (6.3) . ! . 6! ! utt ; uxx = 0, cn.
; . . 7 , 1] !., ! !
! $! , , !
. ! . 0! / , $ , , ! ! $ (6.2)
! , - ! $! , .
2 3 $ ! (1.1),
(1.2) $ $. 2$., ! .3 a2 $
! (1.1), (1.2) . . .3 ! $! , ! .3 a2, ,
! .3, $ (a2 = 0) $!$ . 7 , !$ $
! 3$ t x, | ! 3$.
! , ! ! . . $ !
utt ; uxx + "a2 uxxxx = "uxxt + "ut ; u2ut
ujx=0 = uxxjx=0 = ujx=1 = uxx jx=1 = 0
! !$ $ . EI = "a2 , . . .$ $ .
! (2.5), (2.6).
220
. . , . . )
1] . . . , ! " // $. "%
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( ): 1995.
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512.55
K | , A K | .
!"
# $ $
A-. %
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$ /! #).
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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B
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,, q-( .
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.
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6 jjpq p(x ; tc)jaj;1 = jjpq p(x ; ya + ya ; tc)jaj;1 6
6 jjpq jaj;1 maxfp(x ; ya) p(ybc ; tc)g 6 maxfra (x) rb(t)g
'% p(ybc ; tc) 6 p(yb ; t)jcj % jjpq jaj;1p(ybc ; tc) 6 rb (t).
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227
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6 jajra(;x) = jajjjpq jaj;1p(x + ya) = jjpq p(x + ya):
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q(x ; bn ) > rn. 4
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, $# n > N(x).
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- ' %% q ,%&
%%, -$:
a = 0
q@((x a)) = q(x)
'(y)jaj a 6= 0 x = ya:
O,, (%/, q@ $ $ +
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$
', q(y ; bn ) = q(t ; bn ).
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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 , , $# n
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$
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. &'(&- A-*( +, ( ') <&* ) ), ) 5 ' (
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, M '/ A, , - bi %*$% 5' $ + $
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$
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4,, % J, - -' ( $ ( $ %
, N. . #' %
$ $, 4 .
1]
2]
3]
4]
. . | .: , 1971.
A. F. Monna. Analyse non-archimedienne. | Berlyn, 1970. | 320 p.
O. Goldman, C. Sah. Locally compact rings of special type // J. Algebra. | 1969.
A. W. Ingleton. Hahn{Banach theorem for nonarchimedean valued #elds | Proc. Cambridge
Phil. Soc. | 1952. | V. 48. | P. 41{45.
' (: 1994.
,
. . . , !
. " , , !
# # # QF - $-"
.
Abstract
A. A. Nechaev, Finite quasi-Frobenius modules, applications to codes and linear recurrences, Fundamentalnaya i prikladnaya matematika 1(1995), 229{254.
A simple exposition of the main properties of the quasi-Frobenius modules over -nite
commutative rings with identity elements. The presented results show the special role of
such modules in the theory of linear recurrences and in the theory of linear codes over rings
and modules. In particular it is proved that the general weight functions of the linear code
over a ring and the dual code over the corresponding QF -module are connected by the
Mac-Williams identity.
, !
" #3], #6], #7], #10], #11], #13],
#15].
, ",
, , ! -
. . |
, , . 0 , ! , , . , , , , !
, .
. .
/ / 0
, 93-012-492
1995, 1, N 1, 229{254.
c 1995 ,
!" \$ ""
"
230
. . 2 , . 2 , , R | e, R M |
R-. 3
! , ! 2 M ! a 2 R , a = a.
. N M K R M N N M . 0
a = (a1 : : : aN ) 2 RN ~ = (1 : : : N ) 2 M N a ~ = a11 + : : : + aN N 2 M . 3 RRN RN ? K = fa 2 RN : aK = 0g
, K R. 9
L < R RN M :
MN ? L = f
~ 2 MN : L
~ = 0g:
,
, -
M N ? (RN ? K) K RN ? (M N ? L) L
(0:1)
RN ? (RN ? L) L:
(0:2)
, , M = R | , !
. : R M ( R) QF -
(QF -), N = 1 (0.1) (
(0.2)) K < R M L / R. ,, (0.1) N 2 N , ,
R M (. #13] 7.1). ; R M . < , QF - , :-=
(. x7).
. 2 k 2 N : Nk0 ! M k-
M . > = (z), z = (z1 : : : zk) | N0 . ? M <k> k- M . ,
k-
A(x) Pk = R#x] P= R#x1 : : : xk ].
3 i = (i1 : : : ik ) 2 Nk0 xi = xi11 : : :xikk . ; A(x) = i2Nk0 aixi . 3P
A(x) = , 2 M <k> (z) = i2Nk0 ai(z + i). ; M <k> Pk -. @, 2 M <k> k- (k- ) M , F1(x1) : : : Fk (xk ) 2 Pk , , , Fs (xs) = 0 s 2 1 k. I / Pk , , . <
I / Pk | S = Pk =I . 2 I / Pk LM (I) = f 2 M <k> : I = 0g
231
k- - M I . 2 B M <k> Pk An(B) = fA(x) 2 Pk : A(x)B = 0g:
,
, An(B) | Pk . 2 #10].
0.1. !" LM <k> k- M Pk -
M <k> . $ % % I / Pk k- - LM (I)
Pk -
LM <k> . & M | Pk M <k> , An(M) | Pk M LM <k> . (
)
An(LM (I)) I
LM (An(M)) M: 2
(0:3)
0
, I | Pk , S = Pk =I ! h = H(x) + I , H(x) 2 Pk , M < Pk M <k> I , M S -, 2 M h h = H(x). @ QF - .
0.2. * + "
.
(a) R M QF -
.
(b) . (0.3) I / Pk M < Pk M <k> .
(c) $ % % I/ Pk LM (I) QF -
S = Pk =I , M < Pk M <k> QF -
S = Pk = An(M).
(d) $ I1 I2 Pk M1 M2 Pk M <k> LM (I1 + I2) = LM (I1 ) \ LM (I2 )
LM (I1 \ I2 ) = LM (I1 ) + LM (I2 )
An(M1 + M2) = An(M1) \ An(M2)
An(M1 \ M2) = An(M1 ) + An(M2):
B
, x6, ! .
,
, #9], #12], #8], #14]. 0 . ;
. @ , - .
1 B
, ! 1 : : : t R M +, M = R1 + : : : + Rt . = ! 232
. . M = R (1 : : : t). ,
(R M) R M . ,
, (R M) = t, jM j 6 jRjt . : R M
R-
% t, t ,
R. B 1.1. R M | % t %
%
, %
(R M) = t jM j = jRjt . 2
= (R M) R M .
< R , N = N(R) ( ! R). #1], R , , :
_ t:
R = R1+_ : : : +R
(1:1)
G
es | Rs , e = e1 + : : : + et ei ej = 0 i 6= j , R M :
_ t Ms = es M s 2 1 t:
M = M1 +_ : : : +M
(1:2)
3
!, Rs = es R = Res , Ms Rs - s 2 1 t.
1.2. (1.1), (1.2) (R M) = maxf
(Rs Ms ) s 2 1 tg:
2 2 , 1 : : : r | R M ,
es 1 : : : es r | Rs Ms H (1s) : : : (rs) | Rs Ms s 2 1 t, R M (t)
(1)
(t)
1 = (1)
1 + : : : + 1 : : : r = r + : : : + r : 2
3 R | . ; R = R=N(R) R = GF (q) q = pr p r. = !
R M 1.3. & R | K | RM ,
M = K + NM , M = K .
2 3 Nn = 0 n 2 N . ; M = K + NM =
= K + N(K + NM) = K + NK + N2M = K + N2 M = : : : = K + Nn M = K: 2
1.4 (
) & RM 6= 0, M 6= NM: 2
f = M=NM , 2 M e = +
= M
f. : RMf + NM 2 M
f aI = a + N 2 R aIe = af. 3
R = R=N , e 2 M
f
f, .
! R M R M
233
1.5. & R | , f
(R M) = dimR M
1 : : : t 2 M M = R (1 : : : t)
f = R (e1 : : : et):
M
(1:3)
(1:4)
2 J (1.4) !
M = R (1 : : : t) + NM . 3, B, !
(1.3). 2
B , , .
1.6. $ R M R + "
:
(a) R M | .
f = 1.
(b) NM = 0 dimR M
(c) R M = R R.
(d) jM j = q.
2 (a) ) (b). ; R M | , M 6= 0, , 1.4, NM 6= M . B NM = 0, R M f = 1.
, dimR M
(b) ) (c) ) (d) .
(d) ) (a). =
1.5, M 6= 0, jM j > 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 | <. ;
ord !i = di M =< !1 > +_ : : : +_ < !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
9 5, "
Z-.
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
. . , R Q = R R 5.
2 : - . 3 1.10, ' : R I ! R R I / R : R R ! R R . R , a 2 I ! '(a)
'(a) : R ! Q=Z. 3
! R- R , a 2 I r 2 R '(a)(r) = '(a)(re) = (r'(a))(e) = '(ra)(e). 0
! : I ! Q=Z
!(a) = '(a)(e) a 2 I:
; 1.11 !b : R ! Q=Z, !. 3
: R ! R (a)(r) = !b (ar) a 2 R r 2 R:
B , | R-, a 2 I (a)(r) = !b (ar) = !(ar) = '(ar)(e) = '(a)(r), . . '. 2
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 ! . < , , Rb = R=I , K | b 3, R ? K = I . G
r 2 R ? K , !
R.
rb = r + I Rb K . B 1.7 br = b0 , . . r 2 I R ? K I . , (2.1).
B, K < R R I = R ? K . ,
R R R , 1.8. ;
I | R , K , , -, , R ? I = K: 2
2 QF - R , .
2.3. (1.1), (1.2) R M QF -
%
%
, %
R1 M1 : : : Rt Mt QF -
.
237
_ t , Ks = es K |
2 P K < R M K = K1 +_ : : : +K
Rs Ms .
< , _ t ? Kt
R ? K = R1 ? K1 +_ : : : +R
_ t ? (Rt ? Kt ):
M ? (R ? K) = M1 ? (R1 ? K1 )+_ : : : +M
3! M ? (R ? K) = K Ms ? (Rs ? Ks ) = Ks s 2 1 t. 9
, I / R _ t , Is = es I | Rs , R ? (M ? I) = I
I = I1 +_ : : : +I
Rs ? (Ms ? Is ) = Is s 2 1 t: 2
3
QF -
. " . '
3 - QF - . B
, R M , R ? M = 0.
,
, QF - . 2 R M R QF - S(M) = M ? N(R), 1.6 R M R M . 0
, NS(M) = 0, S(M) R = R=N .
3.1. $ % R M R + "
:
(a) R M = R R .
(b) R M | QF -
.
(c) S(R M) | R M .
(d) dimR S(R M) = 1.
(e) S(R M) | (
) R M ( R M ).
(f) !
R M " .
2 (a) ) (b) 2.2.
(b) ) (c) 3 2 S(M) n 0. ; N (R ? R) 6= R. @,
R ? R = N (b) R = M ? (R ? R) = M ? N = S(M).
(c) ) (d) ) (e) ) (f) 1.5, 1.6.
(f) ) (a) 3 K | R M . ;
R K | , 1.6 R-
, ' : R K ! R R . ; 1.12 R R |
"
, ' : R M ! R R . 3
! Ker = 0, (f) K Ker . @, | . G
(M) 6= R , R ? (M) 6= 0, R R QF - 2.2. B | , R ? M = R ? (M),
R ? M =
6 0 R M . @, |
. 2
238
. . 3.2. QF -
R -
R R .
2 3 2.3 1.9, , R | , ! !
(a) (b) 3.1. 2
3.3. QF -
R Q 5.
2 @. 3.2 1.12. 2
B
.
3.4. & RQ QF -
, % N 2 N L < R RN K < R QN RN ? (QN ? L) = L QN ? (RN ? K) = K
(3:1)
jLj jQN ? Lj = jRjN jKj jRN ? Kj = jRjN :
(3:2)
2 3 3.2 , Q = R . = , 1.9, QN = (RN ) . 3
! K = QN ? L RN , L , RN ? K | ! RN , K , , 2.2, , RN ? K = L , . . (3.1). ; QN ? L RN =L , 1.7 jQN ? Lj = jRN =Lj , . . (3.2). = (3.1) (3.2) , RN = (QN ) 1.8. 2
3 End(R M) | !
R M ( !
). 2 ! r 2 R rb !
R M , 2 M rb() = r.
b B rb %, Rb = R(M)
End(R M). ,
, R M | (
! ),
b = R. B RM E -
, R(M)
b
End(R M) = R(M). ,
, , R R, E -.
3.5. (a) 8 R Q E -
.
(b) $ % K < R Q % % ' : R K ! R Q
+ r 2 R , ' = rbjK | % % rb K .
(c) & (b) ' | , r " , . . ' " R Q.
2 (a) 3 2 End(R Q). J
K < R Q2 K = f( ()) : 2 Qg . 3 L = R2 ? K | K R2 , I = fb 2 R : 9a 2 R (a b) 2 Lg:
239
,
, I | R. G
I 6= R, K = Q ? I | R Q, (0 ) 2 K , Q2 ? L . ,, R Q QF -, 3.4 Q2 ? L = K . @, (0 ) 2 K
2 K , K . ;
, I = R. B L (;r e), r 2 R, K = Q2 ? L , 2 Q ;r + e() = 0, . .
() = r = rb.
(b) @ 3.3 ' : R Q ! R Q, (a) = rb r 2 R.
(c) 2
, R (1.1) .
; R Q K :
_ t Qs = es Q s 2 1 tH
Q = Q1 +_ : : : +Q
_ t Ks = es K s 2 1 t:
K = K1 +_ : : : +K
3
! Qs Ks Rs = es R, ' '(Ks ) Qs s 2 1 t, '(Ks ) = '(es K) = es '(K) es Q. ; , (c) , R | .
G
K = 0, (c) , r = e. 2
, K 6= 0. 0
, ' | , R ? '(K) = R ? K , R Q | QF -, '(K) = Q ? (R ? '(K)) = Q ? (R ? K) = K .
@, ! r 2 R, -
' = rbjK , rK = K . 3 , r 2 R , r 2 N(R) ( R | ) rK NK , rK = K
1.4. 2
4 ) *# QF -
=
1.2, QF -
, R . = ! . ,
!, S(R)
R, R.
4.1. R Q QF -
R, S(R Q) =
= R! dimR S(R) = t. 7%
(R Q) = t:
(4:1)
& a1 : : : at | R S(R), + 1 : : : t 2 Q aij = ij ! i j 2 1 t
(4:2)
%
ij | 8, Q = R1 + : : : + Rt:
(4:3)
RQ
240
. . 4 , (4.3) , + a1 : : : at R S(R) (4.2).
2 @ 1.5 (R Q) = dimR Q=NQ. 3! (4.1) jQ=NQj = jS(R)j:
; R Q | , S(R) = R ? N = R ? NQ. 3 3.4,
: jQ=NQj = jR ? NQ=R ? Qj = jR ? NQj = jS(R)j . J (4.1)
.
3 a1 : : : at | R S(R), . .
_ t:
S(R) = Ra1+_ : : : +Ra
; R
_ s;1+Ra
_ s+1 +_ : : : +Ra
_ t s 2 1 t
Is = Ra1+_ : : : +Ra
Q ? Is 6 Q ? Ras , Q ? Is Q ? Ras , Is = R ? (Q ? Is ) R(Q ? Ras) = Ras , ! a1 : : : at R. , ! 1 : : : t 2 Q s 2 (Q ? Is ) n (Q ? Ras ) s 2 1 t.
; -
ass 6= 0 ak s = 0 k 2 1 t k 6= s:
3
! as s 2 Q ? N = R!, as 2 R ? N . @, as = us !, us 2 R . ; , ! s = u;s 1 s (4.2). 2 (4.3). 3 R1 + : : : + Rt = K . ; K | R Q,
, R ? K = 0, K = R Q, R Q |
QF -. 3 R ? K 6= 0. ; R ? K / R Ra. 3
S(R) R, a = uI1 a1 + : : : + uItat , uI1 : : : uIt 2 R. ; a 6= 0, uIs 6= I0 s 2 1 t. B (4.2) as = uIs ass = uIs ! 6= 0, , , a 2= R ? K .
3
. @, (4.3).
3, , 1 : : : t 2 Q (4.3). 3
S(R) (4.2). 3
Ks = R1 + : : : + Rs;1 + Rs+1 + : : : + Rt s 2 1 t:
3, S(R) ? Ks 6= 0. ; S(R) = R ? NQ, S(R) ? Ks = S(R) \ (R ? Ks) = (R ? NQ) \ (R ? Ks), 2.1(c)
S(R) ? Ks = R ? (Ks + NQ). G
S(R) ? Ks = 0, . . R ? (Ks + NQ) = 0, Ks + NQ = Q ( R Q | QF -), B Q = Ks , , (R Ks ) 6 t ; 1 < (R Q). = bs 2 (S(R) ? Ks ) n 0 s 2 1 t.
; bs j = 0 j 2 1 t j 6= s. < , bs s 6= 0, (4.3) bs Q = 0 bs = 0. 3
! bss 2 S(R)Q = R!, . . bs s = rs! 241
rs 2 R . 3 as = rs;1 bs s 2 1 t, !
a1 : : : at 2 S(R), (4.2). R R S(R), rI1a1 + : : : + rItat = 0 -
rI1 : : : rIt 2 R,
(Ir1a1 + : : : + rItat )s = rIsas s = 0, (4.2) rIs! = 0, . . rIs = 0 s 2 1 t.
2
4.2. $ QF -
R Q R + "
:
(a) R Q | .
(b) R Q = R R.
(c) R | QF -.
2 (a) ) (b) ) (c) .
(c) ) (a) =
2.3, , R | . = ! (c) , 3.1, S(R) | ,
. . dimR S(R) = 1, 4.1 (R Q) = 1: 2
5 ,
QF -
-
= QF - (
2.2)
, .
B R M , : M ! Q=Z, R M . ;
+ R M . 3
.
5.1. : : M ! Q=Z + %
%
, %
;
8 2 M 6= ) ) 9r 2 R ((r) =
6 (r))
(5:1)
2 3 | 2 M 6= . ; (R( ; )) 6= 0,
. . ! r 2 R , (r( ; )) 6= 0. R ,
(r) 6= (r). B, (5.1) K < R M K 6= 0.
= 2 K n 0. 3 ! r 2 R , (r) 6= (r0_ ) = 0. @, (K) 6= 0: 2
, .
5.2. 7 R M %
%
, %
QF -
.
2 3 R M | QF -. ; 3.2 , M = R , 1.8 R-
' : R R ! R M = R R , ! r 2 R '(r) : R ! Q=Z, ! ! 2 R '(r)(!) = !(r).
242
. . M '(e) R , ! 2 R ! 6= ,
!(r) 6= (r) r 2 R. R , '(e)(r!) 6= '(e)(r), 5.1 '(e) | R M . ; , u 2 R '(u) R M .
3 , R M . 2
, R (1.1) R M (1.2).
5.3. !
RM %
%
, %
"
Rs Ms .
2 3 | R M . ; Ks Rs Ms R M , (Ks ) 6= 0, jMs Ms Rs Ms . B, Rs Ms s | . ; M , 1 2 M1 : : : t 2 Mt (1 + : : : + t) = 1 (1) + : : : + t(t ), R M . 2
, K < R M K 6= 0, es K 6= 0 s 2 1 t, ! s 2 es K < Rs Ms ,
s (s) 6= 0. : s 2 K (s ) = s (s) 6= 0: 2
;
, , R | .
3, -, N | R R = R=N | q = pr
!, S = S(M) = M ? N | R M . 2
, R M . 3, R M | QF -. =
3.1, !
, dimR S = 1. 3 = jS G = Ker . ; (S +) | ! p-, (S) 6= 0, (S) | p
Q=Z, , G | p S . ,
L(G)
R S , G, . . - , G. 3 u1 : : : ur | R Zp .
5.4. $ % G < (S +) L(G) =
\r
s=1
u;s 1G:
(5:2)
. , dimR S > 1, G | % p S , L(G) 6= 0.
2 ,
, 2 L(G) R G. 3! 2 L(G), Tsr 2 1 t us 2 G, . . 2 u;s 1G.
@, L(G) s=1 u;s 1 G. B, 2 u;s 1 G s 2 1 r, u1 : : : ur 2 G, G | (S +), (c1 u1 + : : : + cr ur ) 2 G
c1 : : : cr 2 Zp , . . R G 2 L(G). R
(5.2). =
, G | p S , L(G) | , pr = q, dimR S > 1, jSj > q2 L(G) 6= 0: 2
243
;
, dimR S > 1, G = Ker L(G), L(G) | R M , | R M , . @,
dimR S = 1, 3.1 R M | QF -. 2
6
QF -
= , , QF - R, , , 2.2. B
.
B .
3
F = fi1 : : : iN g Nk0
(6:1)
. ? M F : F ! M .
< % #F ] =
= ((i1 ) : : : (iN )) 2 M N , , , R M F R M N .
0
, k-
2 M <k> ! F : #PF ] = ((i1) : : : (iN )).
% A(x) = i2Nk0 ai xi 2 Pk F (A) = fi 2 Nk0 : ai 6= 0g:
2 ! F (6.1) % % A(x) F AF = (ai1 : : : aiN ) 2 RN I / Pk F IF = fAF : A(x) 2 I F (A) Fg < R RF :
2 R- M M <k> M#F ] R M F M#F ] = f#F ] : 2 Mg:
@ k-PJ3 M F .
6.1. M Pk -
M <k> , F Nk0 | -
. 7%
% Pk A(x) =
X i
aix
i2F
244
. . M , 8 2 M AF #F ] = 0:
(6:2)
(
I = An(M) / Pk F IF = RF ? M#F ]:
(6:3)
2 N
(6.2) 8 2 M A(x) = (0) = 0:
(6:4)
3, 8 2 M A(x) = 0:
(6:5)
3 2 M A(x) = 6= 0. ; (j) 6= 0 j 2 Nk0 . ; M
Pk -, 1 = xj M -
1 2 M A(x)1 = 1 1 = xj 1 (0) = (j) 6= 0
(6.4). @, (6.2) (6.5). 3 . 2
6.2. R Q QF -
M < Pk Q<k> . k7%
An(M) = 0 , "
% F N0 M#F ] = QF :
(6:6)
2 I = An(M) , F Nk0 IF = 0. 3, (6.3), !
RF ? M#F ] = 0. ; R Q QF -, 3.4
M#F ] = QF ? (RF ? M#F ]) = QF ? 0 = QF :2 .
2
, M | Pk - M <k> . =
0.1
! , I = An(M) , . . ! (6:7)
Fs (xs) = xms s ; fm(ss);1xsms ;1 ; : : : ; f1(s) xs ; f0(s) 2 Pk s 2 1 k:
3, I I
!
S = S#m1 : : : mk ] = S#m] = 0 m1 ; 1 : : : 0 mk ; 1
R M R- M#S].
2 H(x) 2 Pk Res(H(x)=;;!
F (x)) = Res(H(x)=F1(x1 ) : : : Fk(xk ))
245
P
A(x) 2 Pk A(x) = i2 aixi , H(x) (F1 (x1) : : : Fk(xk )) / Pk . R . 3 Res(xis =Fs(xs)) | xis Fs(xs ), i 2 Nk0
H i(x) =
; H(x) =
Yk
s=1
Res(xiss =Fs (xs)) =
X ij
hj x :
j2
P h xi i
X i i
Res(H(x)=F) =
hiH (x ):
i
k
2N0
(6:8)
(6:9)
,
, H(x) 2 I Res(H(x)=F) 2 I . 3! I F1 : : : Fk A(x) 2 I : F (A) S:
(6:10)
;
, I / Pk , (6.7), I .
0
, M LM (F) = LM (F1(x1 ) : : : Fk (xk )).
6.3. k- 2 LM (F) % #S] (z) =
X z
hj (j)
j2
(6:11)
%
hzj | (6.8). $ % #S] 2 M +
k- 2 LM (F) #S] = #S]. (
! #S] R-
= R M#S]:
RM 2 3 = xz . ; (z) = (0). ; xz = H z (x), H z (x)
(6.8). @, (0) (6.11), . . (6.11) . 3 #S] 2 M . 3
k- 2 M <k> (z) =
X z
hj (j):
(6:12)
j2
; #S] = #S], H i(x) = xi i 2 S. =
2 LM (F)
(6.12) (6.8). 3 6.3 . 2
246
. . J
LM (I) k-PJ3, I / Pk . > - , I (6.7), LM (I) = LM (I)#S] = f#S] : 2 LM (I)g:
@ 6.1.
6.4. 2 LM (F) " LM (I)
%
%
, %
% A(x) (6.10) A #S] = 0:
(6:13)
(
LM (I) = M ? I :
(6:14)
2 ; I (6.7) (6.10), 2 LM (F) 2 LM (I) , A(x) (6.10). 3 A(x) = . ; (0) = A #S], 2 LM (I), (0) = 0, . . (6.13). B, 2 LM (F) (6.13) A(x) (6.10). 2
, = A(x) 6= 0. ; (j) 6= 0 j 2 Nk0 , k-
1 = xj = xjA(x) 1(0) 6= 0. J
A1 (x) = Res(xj A(x)=F). 3 A1 (x) (6.10), Fs(xs ) = 0 s 2 1 k, A1 (x) = 1 1(0) = A1 #S] 6= 0, (6.13). @, = 0 2 LM (I).
J (6.14) . 2
@ 0.2 . 2 ! : R Q QF -, I / Pk
M = LQ (I) QF - S = Pk =I (! | (a) ) (c) 0.2). 3- , I (6.7). 3 M1 | S M , An(M1) = J .
; S ? M1 = J=I M ? (S ? M1 ) = LQ (J). ;
, M ? (S ? M1) = M1 , M2 = LQ (J)
M1 . ; M1 M2 (6.7), 6.3 , M2#S] = M1#S]. @ 6.4 M2#S] = Q ? J , 6.1 J = R ? M1 #S]. ; R Q QF -, 3.4 M2#S] = M1#S].
2, I1 / S S ? (M ? I1 ) = I1 . I1 I1 = J1 =I , J1 / Pk . ;
M ? I1 = LQ (J1 ) S ? (M ? I1 ) = J2=I , J2 = An(LQ (J1)). J
J2 = J1 !
J2 = J1 . 3 -
J2 = R ? (Q ? J1 ), 6.1 6.4.
; 0.2 QF - S c . = S
R (
, ,
247
). ; S = R#1 : : : k ] | -
R, , S = Pk =I , I = fA(x) 2 Pk : A(~ ) = 0g . 3
! !
s s = xs + I , s 2 1 k. ; S | , I |
. @ Q = LR (I) S -, 0.2 S Q QF -, R QF - (. . R R |
QF -).
2 QF - S = R#x]=I, R = Z4 ,
I = (x2 2x). : S = R#] | 8 !
a0 + a1 , a0 2 0 3, a1 2 0 1. @ S QF - Q = LR (I) = R1 + R2 , 1 = (1 0 0 : : :) 2 = (0 2 0 0 : ::)
. S Q S(Q) = R!, ! = (2 0 0 : ::).
@ S Q S (. . S ) SxQ
(0)
?
% ?
? (1 ) (1 +x 2) (2)
- ?? %
?y &
() ( +? 2) (2)
& ?
y .
"
"
(!)
.
(0 2)
(0)
S
,
, QF - S Q S S
= R#x1 : : : xk ]=I
(6:15)
R | QF -. G
(6.15), R , 0.2 2.2 QF -
S Q k-PJ3 R R :
Q = LR (I):
(6:16)
@
, QF - S Q S k-PJ3
, 2.2, , S (6.15). < , QF - k-PJ3 \-
" , S S , . . S S . 3 ! , Q = LR (I), R = Zd | . = ! d S = Pk =I , ! : (S +) ! (Q=Z +)
S < 1=d > Q=Z. R (Zd +). ;
, S 248
. . ! : (S +) ! (R +). J 2 LR (I) . 3
!
A 2 S A = A(x)+ I . M b : S ! R, , 8A 2 S b(A) = (0) = A(x):
@
! b S L(I) ! S S .
= , QF - S Q (6.16), (i)
2 Q R, b 2 S , , :
8A 2 S b(A) = (0)(e) = A(x)
(6:17)
. . b(A) (0) 2 R e. 3 , b : S ! Q=Z (S +), .
6.5. ( ! b , (6.17), S -
S Q ! S S .
2 ; jQj = jS j = jS j , , 1 2 2 Q , b1 6= b2 . 3 1 (j) 6= 2 (j) j 2 Nk0 .
3 s = xj s s 2 1 2. ; 1(0) 6= 2(0), , 1(0)(r) 6= 2(0)(r)
r 2 R, . . r1(0) r2(0) e . R , A(x) = rxj b1(A) 6= b2 (A), . .
b1 6= b2 : 2
7 / QF -
3 QF - .
7.1. $ R M + "
:
(a) R M QF -
.
(b) % N 2 N L < R RN K < R M N M N ? (RN ? K) = K
(7:1)
N
N
R ? (M ? L) = L:
(7:2)
(c) R M | , % N 2 N % % K < R M N (7.1).
2 (a) ) (b) 3.4. (b) ) (c) .
(c) ) (a). 3 2.3 , R | . =
! R M | QF -, , 3.1(b,c), M ? N = S(R M) K . ;
R ? K = N , M ? (R ? K) = S(M) 6= K , (c). 2
249
< , HmN R
K < R M N , K -
M N Hx# = 0. 3 H | R, H , H < R RN . ;, , H | K , K = M N ? H:
(7:3)
= , ,
7.2. RM | . 7%
RM
R , R M
QF -
.
2 G
R M QF - K < R M N , (7.1) H R, H = RN ? K , K . G
K < R M N R, . .
(7.3) H < R RN , (7.1), 7.1(a,c) R M QF -. 2
2 L < R RN R !
, , QF -.
0 -
R M , a = a a 2 R 2 M . @, KmN R M L < R RN , L -
RN Ky# = 0# . ,
,
K | L , K < R M N , R K , RN ? K = L
(7:4)
L R M ,
(7.4) K = M N ? L . = ! K R K . 7.1 7.3. L < RRN QF -
R Q. 2
0
, ! . B
, R M =
= R Q + R Q QF -, R M .
0 -
R. 3 R R(N ) | N R.
B KmN R M HN n R, b# 2 R(N ) Hx# = b#
(7:5)
250
. . -
R , Kb# = 0# . M- ,
H K !
,
! d -
(7.5) -
d = n ; rang H = rang K + n ; N . R
.
7.4. HN n R KmN QF -
R Q. H | R,
"
H , K | Q, "
K , D(H) ) (7.5)
D(H) = jHj;1jRjn = jKj jRjn;N :
(7:6)
2 @ 7.3, KmN Q , b# 2 R(N )
b# 2 H , Kb# = 0# :
:
K | , b# 2 H (7.5). 3 (7.6) . ; K Q, H , 3.4 jKj jHj = jRjN .
, (7.6). 2
=
-
#5], .
3 jRj = r R = f
1 : : : r g . ; + L < R RN
Z
X s1(u) sr(u)
WL (x1 : : : xr ) =
x1 : : :xr (7:7)
u2L
st (u) | u , t . 9
, jM j = m M = f1 : : : m g , K < R M N :
X 1 ( ~ ) m ( ~ )
(7:8)
WK (y1 : : : ym ) =
y1 : : :ym ~ 2K
t( ~ ) | ~ , t . @, L < R RN K = M N ? L "
!-., : (M +) ! (C )(
(M +) (C ) C ) , (7:9)
W (y : : : y ) = 1 W (
M (y) : : : M (y))
K
1
m
M
(y) =
jLj
m
X
=1
L
1
r
(
)y 2 1 r:
(7:10)
= M = R = GF (q) ! - #5]. = M = R = Z4
#15]. 2 #2].
251
3 , : M ! C ( ) x1 ! : M ! Q=Z ( ): !(x) (x) = exp(2i!(x))
, ! ! M (M +). 3!,
R M + ,
(K) 6= 1 K < R M , 5.2
7.5. 7 R M + %
%
, %
. 2
3
, QF - , .
7.6. & R M | QF -
| % + -
, % % L < RN "
!. (7.9). & R M | , , + L < R RN , (7.9) % % (M +).
2 2 , #5]. @ .
7.7. & | + P RM , % % K < R M 2K () = 0.
2 3 (K) = G | C . 3!
X
j X g = 0: 2
() = jjK
G j g 2G
2K
= K = M N ? L WK (y1 : : : ym ) =
X
~ 2K
m ( ~ ) :
f( ~ ) f( ~ ) = y11 ( ~ ) : : :ym
2 u 2 RN fb(u) =
X
~ 2M N
(u ~ )f( ~ ):
(7:11)
(7:12)
252
. . 7.8. & | + RM , Xb
X
f(u) = jLj f( ~ ):
(7:13)
u2L
~ 2K
2 3 (7.12), Xb
u2L
f(u) =
X X
u2L
~ 2M N
!
(u ~ ) f( ~ ):
(7:14)
G
~ 2 K , (u ~ ) = 1 u 2 L , (7.14) jLj . G
~ 2 M N n K , K = fu ~ : u 2 Lg
R M , 7.7
X
u2L
jLj
(u ~)=
jK j
X
2K
() = 0:
; (7.13) (7.14). 2
2 7.6 - . (7.11), 7.8, Xb
W (y : : : y ) = 1
f(u):
(7:15)
K
1
m
jLj u2L
(7.12), t( ~ ), -
fb(u) =
X
N Y
1:::N 2M l=1
(ul l )y11 (l ) : : :ymm (l) t(l ) =P1, Q l = t , t(l ) = 0, l 6= t . ,, , b u) =
f(
N X
Y
l=1 l 2M
(ul l )y11 (l ) : : :ymm (l )
! Y
N X
m
=
l=1 =1
(ul )y :
3 s (u), -
b u) =
f(
. . (7.10)
m
Yr X
=1 =1
(
l )y
!s(u)
b u) = M1 (y)s1(u) : : :
Mr (y)sr (u):
f(
!
253
;
, fb(u) (7.15) xs11(u) : : :xsrr (u) (7.7) : x = M
(y), 2 1 r. ;
(7.9) (7.15) (7.7).
2 7.6 .
3
N = 1, L = R. ; WL (x1 : : : xr ) = x1 + : : : + xr . > ,
M = f1 : : : m g , 1 = 0. ; K = M ? L | R M , WK (y1 : : : ym ) = y1 . 3
, R M | , . 3 | R M . ; 7.5 K < R M , (K) = 1.
@, M = f 2 M : (R) = 1g d > 2 !.
3 M = f1 : : : d g . ; (7.9) 1 W (
M (y) : : : M (y)) = 1 (
M (y) + : : : + M (y)) =
r
r
jLj L 1
jRj 1
0
1
r X
m
m X
X
X
= jR1 j
(
)y = jR1 j @ (
)A y :
=1 =1
=1 2R
3
! 2 M , (R ) = 1, X
2R
(
) = jRjH
2= M , (R ) = G | C , X
jX
(
) = jjR
Gj g2G g = 0:
2R
@,
M
WL (
M
1 (y) : : : r (y)) = y1 + : : : + yd 6= y1 = WK (y)
. . (7.9) . 2
/
#1] 9 :., : . =
. | M.: :
,
1972. | 160 .
#2] Ericson Th., Zinoviev W. Spherical codes. Manuscript. To appear.
#3] <
9. @. B 9. 9. P
< L 2 // N
. . | 1994. |
;. 49. | N 5. | @. 165{166.
#4] P . < . | :.: :
, 1971. | 279 .
254
. . #5] :-=
_. 2., @! B. 2. 9. ;
, -
. |
:.: @, 1979. | 744 .
#6] B 9. 9. P
// 2
. . | 1991. | ;. 3. | N 4. | @. 107{121.
#7] B 9. 9. P
// N
. . | 1993. | ;. 48. | N 3. | @. 197{198.
#8] _ <. 9: , , . ;. 2. | :.: :
, 1979. | 464 .
#9] Azumaya G. A duality theory for injective modules (Theory of quasi-Frobenius
modules) // Amer. J. Math. | 1959. | V. 81. | N 1. | P. 249{278.
#10] Kuzmin A. S., Kurakin V. L., Mikhalev A. V., Nechaev A. A. Linear recurrences
over rings and modules // J. of Math. Science / Contemporary Math. and it's Appl.
Thematic survays. | 1994. | Vol. 10.
#11] Kuzmin A. S., Nechaev A. A. Error correcting codes on the base of linear recurring
sequences over Galois rings // IV-th Int. Workshop. Algebraic and Combinatorial
Coding Theory. Proceedings. Novgorod, 1994. | 132{135.
#12] Morita K. Duality for modules and its applications to the theory of rings with
minimum condition // Sci. Repts Tokyo Kyoiku Daigaku. | 1958. | A6, N 15
May. | P. 83{142.
#13] Nechaev A. A. Linear codes over „nite rings and QF -modules // IV-th Int. Workshop. Algebraic and Combinatorial Coding Theory. Proceedings. Novgorod, 1994. |
154{157.
#14] Wisbauer R. Grundlagen der Modul- und Ringtheorie. | Munchen: Verl. Reinhard
Fischer, 1988. | VI, 596 s.
#15] Hammous A. R., Kumar P. V., Calderbrank A. R., Sloane N. J. A., Sole P. The
Z4 -linearity of Kerdock, Preparata, Goethals and related codes. Manuscript. 1993.
&" ': 1995.
T- . . . . . T-
R F
2 , , e, eRe = eF .
Abstract
A. E. Pentus, T-ideal of generalized identities for a class of primitive algebras with
involution, Fundamentalnaya i prikladnaya matematika 1(1995), 255{262.
The T-ideal of generalized polynomial identities of any primitive algebra R with involution is found, provided R is an algebra over an algebraically closed *eld F of characteristic
di+erent from 2, the ring R contains no primitive symmetric idempotents, and there exists
an idempotent e such that eRe = eF .
T-
, . 7], #. $. %
1] . '(
8]. * , , + , + , ,
(,, + 4].
., F | , (R ) | F , X = fx1 x2 : : :g,
X = fx1 x2 : : :g, Y = X X | + . . +, RF hY i, (
R F hY i |
F, Y .
3
+, RF hY i | | , ( X
ri0 yj1 ri1 : : : yjm rim i
i 2 F , rik 2 R yjk 2 Y .
4 , +, R +, RF hY i
(: r 7! r r 2 R, xi 7! xi , xi 7! xi, i = 1 2 : : :.
*
f(x1 x2 : : : xk xl1 xl2 : : : xlm ) 2 RF hY i ( ) +, R, f(a) = 0 a
( A = f(r1 r2 : : : rn r1 r2 : : : rn) j ri 2 Rg, n = maxfk mg.
+ ,, +, R ( +, RF hY i, + (, ( G (R).
1995, 1, N 1, 255{262.
c 1995 !,
"#
\% "
256
. . 3:( ; +, RF hY i (, ,
;(r) = r <
r 2 R RF hY i ;(f ) = ;(f)] f 2 RF hY i.
$
I +, RF hY i (
T - , ;(I) I <:( ; +, RF hY i.
., f(Y ) | ( +, RF hY i. . T(R= f(Y )) , T - +, RF hY i, . $ , T (R= f(Y )) | , T - , f(Y ).
>
, . $
e
e0 +, R ( ,, ee0 = e0 e = 0. $
e
(
), , , .
?+
, (s)
+,
Mn (C) (n = 2m) n n { + +, C. '(,
+ n n { m m { (. A
(
:
A B s = u;1 A B t u = Dt ;B t C D
C D
; C t At
0
;I
u = I 0 I | m m { .
. R | F ! 2, | R. R ! ! , %
e = e2 , , eRe = eF . & G (R R) = T (R= S2 (e= Y )) + T(R= Invs (e= Y ))
S2 = ex1 ex2e ; ex2 ex1 e, Invs = e x1e + e x1e.
. $( 2.13.19 9, . 302] , x x = 0 <
x 2 Re <
r (
R e r e + e re = 0, < Invs +, R.
$( +
, S2 | , <
T(R= S2 (e= Y ))+T (R= Invs (e= Y )) G (R R). . + +(
, +
.
%
, , L = Re ++ +
U D = EndR L = eRe = eF . 4 +(
, , + U +
D, +,+
+(
, +
F, (
,. ?
+,
R <
b b0, +
, e = eb0 e be 9,
2.13.18, . 301], ( U ,
-
:, hr1 e r2ei = eb0 e r1 r2e.
H:+
+
+, R h(Yh ). *
, +
+<::
( F fi T- 257
a1yk1 : : :am ykm am+1 , yki 2 Yh i = 1 : : :m, aj 2 R i = 1 : : :m + 1.
., Vh | F- R, <
1, e,
e , eb0 e , e be, +
+<::
ai h <
ai .
I
1.4.2 5, . 46{ 47], <
v + +, R, v = v2 = v vxv = x <
x Vh \ Soc(R).
. < +, vRv +, (t t)- F +
t > 0, <
v | . 3
e , +
, , :( +, vRv, ,
, < <
e11 .
$ +,
R ( +,
. 4 , + :( F
+, , +
+ +
+, e -
, e11. A+ (, vRv |
+, (
t t, t = 2m +
+ .
., E = feij j 1 6 i j 6 tgP| F
+,
, +, e = e11 v = tk=1 ekk . J, +
Pt ++ vxv = x <
x Vh \ Soc(R), Vh \ Soc(R) ij =1 eij F.
A
, <
+, R S = f1=s1 s2 : : : sq g, +,
1) <
F-
( Vh \ Soc(R)=
2) +
, +
=
P ( <
3) Vh = qk=1 sk F +
V
\ Soc(R).
h
P
P
A Vh qk=1 sk F + tij =1 eij F, < h(Yh )
P
, ++ + pi=1 fi i, i 2 F, fi i = 1 : : : p | +<::
( S E.
*( ( Mon fi h.
' ( Mon ++ <
1, S E Yh . . fi 2 Mon
(
E-, <
, S Yh . *( E- ( Mon ( G0. %
, 1 ++ E- ( Mon.
., f | (, ( Mon. *
, ( :.
I. f(Yh ) = g, g | E- ( G0. *( ( GI
+ E- .
II. f(Yh ) = g0 eij g1 , g0, g1 | E-
( G0. *( ( GbII g0 (, ), (
GeII | g1 (+
) II.
III. f(Yh ) = g0ei0 j1 g1ei1 j2 g2 : : :ein;1 jn gn ein jn+1 gn+1. *( ( GbIII g0 (, ), ( GeIII | gn+1 (+
), ( GmIII | E- g1 g2 : : : gn III.
F
+ E- GI GbII GeII b
e
GIII Gm
III GIII +
, +
( Yh . . + + , 258
. . , +
, + (
< . *( , ( +
+( <
( GI , GbII ,
GeII , GbIII , GmIII , GeIII , ( GfI , GbfII , GefII , GbfIII , GmfIII , GefIII .
:, :
L0c = f g] j g 2 GI g f i g] j g 2 GeII GeIII 1 6 i 6 tg
f g j] j g 2 GbII GbIII 1 6 j 6 tg
0
Ls = f i g] j g 2 GfI GefII GefIII 1 6 i 6 tg f i g] j g hN sf s 2 S h 2 GbfII GbfIII GmfIII f 2 GfI GefII GefIII 1 6 i 6 tg
00
Lc = f i g j] j g 2 GmIII 1 6 i j 6 tg
L00s = f i g j] j g 2 GbfII GbfIII GmfIII 1 6 i j 6 tg f i g j] j g Nhsf s 2 S h 2 GbfII GbfIII GmfIII f 2 GbfII GbfIII GmfIII 1 6 i j 6 tg
L0 = L0c L0s
L00 = L00c L00s L = L0 L00:
H
, gN g = sk0 yl1 sk1 : : :yln skn (: gN = skn yln : : :sk1 yl1 sk0 .
H:+
(, + L00. I
+ i g j] ( L00 <
e1i gej m+1 +, RF hY i. (
; m m + 1 6 i 6 2m
i = ii +
m 1 6 i 6 m
1 m + 1 6 i 6 2m
"(i) = ;
1 1 6 i 6 m:
. (i j) = "(i)"(j). + ,, (e1i gej m+1 ) = (j m + 1)(1 i) e1j g ei m+1 . H
+
, g = gN"(g), 8
< 1 g +
+ <
"(g) = : s 2 S
;1 :
. <
e1j gNei m+1 + j gN i ]. > + L00, i g j] j gN i ]. H+ <
(, < , + ( (+. ( L00 L000 (: F
L000 +, +(F
( = ( + , <
(, (,
,F <
( F + L00)= (,
+
+ i g i], g = ;g.
., F L] F L0] | +, :, ( L L0 = L0 L000 .
*
+ ( Mon ( F L] F L0].
I Pf (L) = Pf0(L0) = f] = g], II Pf (L) = Pf0(L0 ) = g0 i] j g1], Pf (L) = g0 i0 ] j1 g1 i1 ] : : : jn gn in ] 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
. . & %
v 2 H ?, , (1) t1 v : : :tn v X- (2) hv ti vi = i i = 1 : : : n.
U0 )
1] . . 14- . . |
1977. | $%. | &. 2. | (.8.
2] * (. +. | ,.: ,, 1965.
3] 234 +. 5. 6 373% 8 (n n)-;3 ; <;
% // ><6 ;3;. 4 | 1992. | . 47. | N 2. | (. 187{188.
4] 234 +. 5. (3 T- 6 373% <;3%6 % <; // ><6 ;3;. 4 | 1992. | . 47. | N 6. | (. 227{228.
5] Beidar K. I., Mikhalev A. V., Salavova C. Generalized Identities and Semiprime Rings with
Involution // Mathematische Zeitschrift | 1981. | B.178. | S. 37{62.
6] Chuang Chen-Lian. *-DiEerential Identities of Prime Rings with Involution // Trans. Amer.
Math. Soc. | 1989. | V.316, N 1. | P. 251{279.
7] Littlewood D. E.| Identical relations satisHed in an algebra // Proc. London Math. Soc.
| 1931. | V.32, N 2. | P. 312{320.
8] Rosen J. D. | -Generalized polynomial identities of Hnite-dimensional central simple
algebras // Israel J. Math. | 1983. | V.46, N 1{2. | P. 97{101.
9] Rowen L. H. Ring Theory Vol. 1. | Academic Press, New York, 1988.
' (: * 1994.
Q- . . Q- | X Q- , ' : X ! S S , ' OX = OS ,
(X=S ) = 1 ;KX '-. & '( ) *. ,
. . ' F (( j;KX + ' hj. ,
, S - ) , X=S
, S | An . & . X S .
Abstract
Yu. G. Prokhorov, On general elephant problem for three-dimensional Q-Fano ber
spaces over a surface, Fundamentalnaya i prikladnaya matematika 1(1995), 263{280.
We consider Q-Fano 7ber spaces X=S over a surface, i. e., a three-dimensional variety
X with terminal Q-factorial singularities and a projective morphism ' : X ! S onto a
normal surface S such that ' OX = OS , (X=S ) = 1 and ;KX '-ample. In this situation
we discuss Reid's conjecture on general elephants, i. e. on general members of the linear
system j;KX + ' hj. We prove that the surface S has only cyclic quotient singularities,
besides if for X=S the elephants conjecture is true, then singularities of S are Du Val
singularities of the type An . In the last case some conditions on singularities of X and S
are obtained.
. 5], 8], 14]. $%&
&
% '
:
1 ( , Reid.) ' : X ! S | X Q- , ' OX = OS ;KX '-
(. .
' : X ! S ). "# # #
h 2 Pic(S) $ # j ; KX + ' hj % #&
(. . #
) .
( . ( ( ( 93{11{1539) *. ( M-90 000)
1995, 1, N 1, 263{280.
c 1995 ,
!"
\$ "
264
. . / %, & 1 0
1 (. 12], 10] ' : X ! S 0
').
& , 0 0 0 % = ;1 '
3
"Q-5
%", . . 0
% "1" 0%& 60
1 . 7
Q-' S 0
X Q-30
, %& 0
3 ' : X ! S S 6 2
0, ' OX = OS , ;KX '-
(X=S) = 1 (. .
' : X ! S | 60
dimS < dimX).
9 Q-5
1
(. 1] dimS = 0):
2. ' : X ! S | Q-' . "# $
Q-'
'0 : X 0 ! S # (
S , # 1
, #
X ; ; ! X0
#'
# '0
S
=
S
#
X ; ; ! X 0 | (
.
& Q-5
. :
,
% 0 '
60
% 0
0 16]. ;
0 0 '
% 0 Q-5
. ,
%&
3 (.) ' : X ! S | Q-' # #
S . "# S % #&
.
7 0
(. 2.4), % Q-5
' : X ! S dimS = 2, S 1 30
. 9, $0 17], 15], 6], x3 '0 0 "6
" '
X=S ; ; ! X1 =S1 ; ; ! Xn =Sn 0, Xn =Sn 1. 0
', (x4) % Q-5
, 0 1. 90
, 1 %0 An . 7 , s 2 S | 0 An , n > 2, ';1 (s) 0 0
> 1 | 6 '00 30
0 n + 1. > Q-5
0, 00 (ii), (iii).
Q- 265
1 .
% C.
. 9 % Q-
1
0 ( 0 ) S 00
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 | '00 , %& Cn . 0
0'% 1
'00 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], 030'% %' (. . 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.
()# $ &. 030' . ;
, , | 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 '00:
1 (Ui nfxig) = ZI (Xx ) (5)
I(X x) - 0 0 x 2 X (. 1.5).
> , (S s) | '00 30
, G - '00 . : 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$. : 0 2.6 , G =
= Zn : '0;1 (s0 ) 0
, . . G : ; top
1
. @ 2.5 & G-
3
;. ;% , G 0%& 0
C10 C20 '0;1 (s0 ), , 0. 9 0 x0 = C10 \ C20
0 x = (x0 ), 0 0 G0 G 0 2 C10 C20 . I G0 = Zn=2 | 0
'00, , & & G0-
0 x01 2 C1,
x01 6= x0. @ x1 = (x01 ) | 0 0 0 r I 2(Ss) . top
1 2.8. X # ';1(s) % #
6 2. "# s 2 S #&
A1 .
' 2.9. ' : X ! S | Q-' #
S h 2 Pic(S) | # 1
. "# # n 0 j ; KX + ' (nh)j . 1
, '(Bs j ; KX + ' (nh)j) | (
S .
0$. 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 6330
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 6330
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 6330
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 |
'00 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) | '00 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, %&
'00 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(2i=n), k 2 N, (n k) = 1. (
X = (P1 C2 )=G, S = C2 =G. "# &$ : X ! S
Q-'. <
X ';1 (0) #
n1 (1 k ;k), S 0 #&
& An.
Q- 279
/ 4.4. X 0 | P2xyz C2uv , # x2 +y2 +z 2 f(u v) = 0, #
f(u v) 2 m4 f(u v) . 4 '0 : X 0 ! C2 & &. <
X 0 #
& & & x0 '0;1 (0) # (x y z u v) = (0 0 1 0 0), cA. 4
#
#
G = Z2 X 0 C2 #&$ : (x y z u v) ! (x ;y z ;u ;v): X = X 0 =G, S = C2 =G. "# ' : X ! S Q'. ,#
G- X 0 - x0 . #
#
, X % # x cAx=2 ';1 (0). -
S 0 #&
& A1 .
" Q
1] Alexeev V. General elephants of -Fano 3-folds // Compositio Math. | 1994. | V. 91. |
P. 91{116.
2] Barth W., Peters C., Van de Ven A. Compact complex surfaces. | Springer-Verlag, Berlin{
Heidelberg{New York{Tokyo, 1984.
3] Catanese F. Automorphosms of rational double points and moduli spaces of surfaces of
general type // Compositio Math. | 1987. | V. 61. | N 1. |P. 81{102.
4] Cutkosky S. Elementary contractions of Gorenstein threefolds // Math. Ann. | 1988. |
V. 280. | P. 521{525.
5] ., ., . !" #$#" !%" | .: ,
1993.
6] Corti A. Factoring birational maps of threefolds after Sarkisov | Preprint, 1992.
7] Kawamata Y. Crepant blowing-ups of 3-dimensional canonical singularities and its application to degenerations of surfaces // Ann. Math. | 1988. | V. 127. | P. 93{163.
8] Kawamata Y., Matsuda K., Matsuki K. Introduction to the minimal model program //
In \Algebraic Geometry, Sendai, 1985", Adv. Stud. in Pure Math. | 1987. | V. 10. |
P. 283{360.
9] Koll)ar J., Miyaoka Y., Mori S. Rationally connected varieties // J. Algebraic Geometry |
1992. | V. 1. | P. 429{448.
10] Koll)ar J., Mori S. Classi*cation of three-dimensional +ips // J. Amer. Math. Soc. |
1992. | V. 5. | N 3. | P. 533{703.
11] ,-. ./0 %1# #$#02 !$$32%4 | .: . 1971.
12] Mori S. Flip theorem and the existence of minimal models for 3-folds // J. Amer. Math.
Soc. | 1988. | V. 1. | N 1. | P. 117{253.
13] Reid M. Minimal models of canonical threefolds // In \Algebraic Varieties and Analitic
Varieties (S. Iitaka, ed.)", Adv. Stud. in Pure Math., vol. 1. | Kinokunya, Tokyo and
North-Holland, Amsterdam, 1983. | P. 131{180.
14] Reid M. Young persons guide to canonical singularities // In \Algebraic Geometry, Bowdoin, 1985", Proc. Symp. Pure Math. | 1987. | V. 46. | P. 345{414.
15] Reid M. Birational geometry of 3-folds according to Sarkisov. | Preprint, 1991.
16] #3 7.8. . %9#%92 4 ## // :;3. <=
>. . %. |
1982. | ?. 46. | . 371{408.
280
17]
18]
19]
20]
. . Q
Sarkisov V.G. Birational maps of standard -Fano *berings. | Preprint, 1989.
Shokurov V.V. 3-fold log models. | Preprint, 1994.
$! ?. ?" 3%3. | .: , 1981.
Koll)ar J. et al. Flips and abundance for threefolds // Ast)erisque 211 | 1992.
& ': 1995.
,
,
. . . . . , S , S , S . , , , !" , , ".
Abstract
I. Kh. Sabitov, Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself, Fundamentalnaya i prikladnaya matematika 1(1995), 281{288.
It is proved that any surface S isometric to a given compact surface S and disposed
su+ciently close to S generates a quasi-conformal mapping of S onto itself. On the base
of this result it is proved that a compact surface admitting sliding bendings onto itself is
topologically a sphere or a torus and its intrinsic metric is of rotation type.
: .
1. | , , !
, . #, , , , , , .
%, M | , ds2 = 2 (x y)(dx2 + dy2 ):
(1)
, ,
- , N 93-01-00154, \2
,
", N 1.4.15.
1995, 1, N 1, 281{288.
c 1995 !,
"#
\% "
282
. . -, S S | M R3 C 2 !, S S , , S S S . . - r S :
r = r ; hn
(2)
r = r + U
(3)
r | - S , n | S , h | ( ) S S , U | 1, S S . . , , S . %, P 2 S (2) P 2 S , S , S P . 2 P (3)
P~ 2 S , M , P . 4, z : M ! M , z (P ) = P~ . 5,
, 6.
5 (x y) | P~ , ( ) | P
M . 4 z . 7
, ! , , S S C n, n > 2, 0 6 6 1, . % (2) (3) r(x( ) y( )) + U (x( ) y( )) = r( ) ; h( )n( ):
(4)
: ! r1 = @r=@ , r2 = @r=@, Fi (x y; ) (r(x y) + U (x y))ri ( ) ; r( )ri ( ) = 0 i = 1 2
det (@ (F1 F2)=@ (x y)) 6= 0 ( 1 U ) x y 1 C n;1. 2 (2) , 1 h( ) 2 C n;1.
2 S ( ). 5
ds 2 = (2 + 2hL + (2HL ; K2 )h2 + h2 )d 2 +
(5)
+ 2(2hM + 2HMh2 + h h )dd + (2 + 2hN + (2HN ; K2 )h2 + h2 )d2 , , H K | , L, M N | 1
S . 4 , ds 2,
(x y), (1). 4, (1) (5), 1
z = x + iy = z (
), = + i, 6
@z
@z
(6)
@ = = q(
) @
283
@ +i @ , @ = 1 @ ;i @ ,
@@
= = 21 @
@ @
2 @ @
(L ; N + 2iM )h + H (L ; N + 2iM )h2p+ 2(@h=@ =)2
q(
) =
?1 + 2Hh + (2H 2 ; K )h2 + 2j@h=@
j2;2 + @( )]2( )
@ = 1 + 4Hh + (4H 2 + 2K )h2 + 4j@h=@
j2;2 +
+ 4KHh3 + K 2 h4 + 2(Lh2 ; 2Mh h + Nh2 )h;4 +
+ ((2HL ; K2 )h2 ; 4HMh h + (2HN ; K2 )h2 )h2 ;4 :
(B, @, h , 1.)
:
, (5) ( ) (x y) (1), , 1 h( ) z (
) 2 (x( ) y( )) =
p
2 ; K )h2 + 2j@h=@
j2;2 +
1
+
2
Hh
+
(2
H
@( ) : (7)
2
= ( )
2
2j@z=@
j
2 , S 3 P ( ) ! P ( ) 2 S S 3 P ( ) !
! P~ (x y) 2 S ! z = z (
) 6 (6).
. P ! P~ S .
1. S S | , , R3
C n, n > 2, 0 6 6 1. S S , (2) @ > 0. S S , !"
! (6) (7).
D . 1. B S ! , , C m , m > 1, 0 6 6 1.
2 1 h , C 1. 2. E S ! S ! (6) 1 q(
),
. ?1].
D . 1. 5 S , 1 U (x y; t), t | 1, S (t)
t S S , . E P P~ (t) 2 S , S (t) S P . 6 (z ), z (
): P (t) 2 S P~ 284
. . 1 S \" P~ 1 S . 2. . C 1 , , 1 C 1, ! S
C 2 !. 3. . , S S (t) t ( S (t) C 2), M M , S (t) .
2. 5 ! : , , (, -, 1 , 1 )?
. , , ?2], , (
?3]). . ?2] , \ 1" , . I
.
2. R3 g . g 6 1, , "
.
. . S 1 h (2) , (6) 1 g = 0.
D, z = z (
) , , g > 2 z (
)
. J g = 0. 2 S , z (
t), t | 1,
-
. 4 , z (
t) t = t0 . .
M , = 0; z = a
=(a + b
). . (7) 1
K = 2 4
a
(8)
K a + b
= K(
)jjaaj+4 b
j :
5 ! 1 1 K(
) 1=j
j4:
2 (8) ! ;a=b K(;a=b) = jb=aj4:
285
5 (8) = a
~=(a + b
~), 4
K a +a
2b
= K(
)jaja+j4 2b
j :
(80)
: ~ ;a=2b, K(;a=2b) = 16jb=aj4:
5 n- (80 ) ! ;a=nb K(;a=nb) = n4 jb=aj4:
D, b(t0) 6= 0, n ! 1 K(0) = 1, , .
5 z (
t) = ca((tt))
++ db((tt)) t .
2 a(0)=d(0) = 1, b(0) = c(0) = 0, z = a(ct()t
)
++b(1t) . 5 , , , t, S M . D t = t0
S 1 , = 0 = 1, z = a
. D t = t0 , , , 1
, . L 1(t) 2 (t) z (
t). 5
c(t) 0. 2 1 = 1 S , 2 (t) = b(t)=(1 ; a(t)), a(t) 6= 1 t 6= 0, , , !, . 4 M , ~ = ; b=(1 ; a). 4 z : z~ = z ; b=(1 ; a). . ~ ! z~ z~ = a(t)
~. 2 1 K K(a
~) = K(
~)=jaj2
(800)
( , K~ , 1). 5 ~ ! 1 K(
~) C=j
~j4, C = Const.
B (800) , ja(t)j = 1, . . a(t) = ei(t) . 2 a(t) 6= 1 t 6= 0 a(0) = 1, t = t0 , (t0 ) = 2, |
1. 5, S ! ~
1
a(t), b(t) . 2 (800)
K(e2i ~) = K(
~), 2n , 1 K , ' 2 ?0 2) K(ei' ~) = K(
~), . .
.
286
. . J c(t) 6 0. 2 , c(t) 6=p0. L z (
t) p 1(t) =
= (a ; 1 + (a ; 1)2 + 4bc)=2c 2 (t) = (a ; 1 ; (a ; 1)2 + 4bc)=2c. 4 S
~ = (
; 1 )=(
; 2 ) z~ = (z ; 1 )=(z ; 2 ),
z~ = A
~, p
p
A = (a + 1 ; (a ; 1)2 + 4bc)=(a + 1 + (a ; 1)2 + 4bc):
2 1 (t) 6= 2(t), A(t) 6= 1, A(0) = 1, ! , (800).
J g = 1. 2 M , z (
) z = + B . 4, (7) K:
K(
+ B ) = K(
):
(9)
# ,
K(
) = K(
+ n!1 + m!2 )
(10)
n m | 1
, !1 , !2 | , ! . 5 !1 !2 ( 0) ( ) ,
> 0. 5, B (t) = B1 + iB2 , B (0) = 0, . -, B (t), 0 6 t 6 ", " > 0 ;, . . z = x + B ;, . . (9)
1 K . J, ,
= 0. L K(B (t)) = K(0) = Const. 2 , " 1 = B (t1 ) 2 = B (t2 ), 0 < t1 < t2 < ",
T1 T2 = 0 1 1 2 K(
1 ) = K(
2) = K(0). 2 1 2 , , ;, K(
) K(0), 1 T1 T2 , 1 2 , K K(0). 5
, 1 ! T1 T2 . . ! K = K(0). 6 t1 t2
, T1 T2 , ! "- ". 2 , K = Const, . . , R3 .
%, . 5 B = B1 + iB2 ,
B1 (t) = ac(t), B2 (t) = bc(t), a b | , a2 + b2 = 1. . b ; a = Const 1 K = Const, , K K( ) = f (x),
x = b ; a. 4 u = b ; a, v = a + b, ds2 = 2 (u)(du2 +dv2 ), . . S .
287
: 1. I
, ,
K = Const, , , , .
J .
1. b = 0. 2 K( ) = K(), . . K = Const, , .
O .
2. b ; a = 0 (a b) k ( ). #1 K b ;a = Const , . . K( ) = f (x), x = ; a=b. 2 (10) K( ) = f ( ; a=b) = f ( + m + n ; a( + m )=b):
(11)
. ! f ( ; a=b) = f ( ; a=b + n ):
4, K , , .
B b ; a = Const .
3. 5
b ; a = Const . % (11) m = 0 f (x) = f (x + n ), . . f T1 = . 4 , n = 0 f (x) = f (x + m(b ; a )=b), . .
T2 = (b ; a )=b. O , K Const. D, m0 n0 2 Z, m0 (b ; a ) = n0 , a=b = = ; (n0 )=(m0 ):
(12)
4, a : b ! , ! (12). . | K = Const | = a=b, , ( ), . . 1
, 1. D, R3 3 . 6 , , .
D. % , ! : S C 1, 1 , 1
! .
3. : 2 C 1 ,
1
, . .
. - , 2: , h = th1 + o(t), t ! 0, 1 h1 0, , z = z (
) z = z0 (
), 2.
288
. . 1] . . . | .: , 1959. | 628 .
2] M. Spivak. A comprehensive introduction to Di&erential Geometry, v. 5. | Berkeley: Publish
or Perish. | 1979. | 661 p.
3] E. Rembs. In)nitesimal Verbiegungen von Fl*achen in sich // Math. Nachr. | 1957. |
B. 16. | S. 134{136.
' (: 1995.
. . 70- (14.02.1924{26.05.1989)
A | , , a 2 A n, an A Aa.
A | .
Abstract
A. A. Tuganbaev, On left distributivity of some right distributive rings, Fundamentalnaya i prikladnaya matematika 1(1995), 289{300.
Let A be a right distributive right nonsingular ring. Assume that for every element a 2 A there exists a natural number n such that an A Aa. Then A is a left
distributive ring.
, |
. . MA , M ! ", ! # A. $1] , ' A | , " a ' n, ' an A Aa,
A | . ( , # , $2]. * 1.
1. % A | '% ( )% ( *,
, ') -, a . % n, anA Aa. /) A | '% *.
1 . + ! ! " . + (), () . . , . ' ( ') ' (
') , /#
! '! (
'!) " . (0 ', 1 .) , '.
1995, 1, N 1, 289{300.
c 1995 !,
"#
\%
"
290
. . , ! '. ' (
4), / '! ! ! . 5 r(B) l(B)
' / B A. M ' J(M), End(M), max(M), Z(M) ' /, "1, / ! ! ,
.
(/ T A : () , T | / A, /# /#
A7 () ' , ! " a 2 A, t 2 T
" b 2 A, u 2 T , ' au = tb7 () , ! ! " a 2 A, t 2 T, ' ta = 0, " u 2 T,
' au = 07 () , ! ! " a 2 A,
t 2 T, ' a2 = 0, ta = 0, " u 2 T , ' au = 0.
(/ T A / ! #! ! "! (. $3, . 51]) : () T | ' / A7
() # AT 1 fT f : A ! AT ,
' " f(T ) AT , AT = ff(a)f(t);1 j a 2 A t 2 T g,
Ker(f) = fa 2 A j at = 0 t 2 T g. "! ! AT '! A T, fT | ' 1, a 2 A, B | A, aT , BT fT (a),
fT (B)AT . ; T = A n M, M | A, AM , fM ,
aM , BM AT , fT , aT , BT . <' / ! T , '! TA, ' 1 Tf : A ! TA,
Ta, TB TA, / / Tf(B), B | A, ', T = A n M, M | A, T M. + A (),
() M # AM (MA).
1. % T | ( ( ,%( (,-
* A. /):
() T ' (, AT %%1
(') * A 2. -, %2, , ,,%% '2, -,, 3 T , AT %%1
() ') t 2 T %% % n, r(tn ) =
= r(tn+1 ), AT %%1
()) A | * %, ,, (24 %, AT %%.
. = () () () (), () (). / (). ( a 2 A, t 2 T , b at,
' ta = 0. = 0 = tb = b2 bx = 0 x 2 T.
( u tx 2 T. = au = 0 T . / (). (
ta = 0, a 2 A, t 2 T. > n, ' r(tn ) = r(tn+1). =
291
T ' , u 2 T, b 2 A, ' tnb = au. (" tn+1b = tau = 0, b 2 r(tn+1 ) = r(tn ). = au = 0, T ().
2. % A | %* *. /):
() * A (2. 2. %2 ') (, ( ,, (,, (3 24 -, ,-
( %, (3 -4 -, ', %), (
%1
(') ,,2 (2 2 * A ( (2 (, , * A , 4 ,,24 (24 (
,, 4 ,,24 ( (24 )1
() * A )%1
()) A , (, A | (2. ( , (,,
(3 1
() A (, A | '1
() A , (, A | (2. ( 1
() A , ( (, A | , (
'1
(3) r(a) = r(an ) ') a 2 A 4 %24 n1
() '. ,,2. 2. (2. M 3 A (
(2. , . ,,2, (2, ,1
() T | ( ( ,%( (, A,
AT %% %*2, *,.
. ( (), (), (), () $4, . 286, 288, 316, 321].
( () (). ( () () , '
# . ( (/) () ' $4, . 319]. / (). B, '
r(a) r(an). ( b 2 r(an). = an bn = 0. ( () (ab)n = 0. (" ab = 0, r(an) r(a). / (). = / (
) ', ' M / (
)
, M / ' , # ()
' . / (). ( 1() AT #.
( f : A ! AT | ' 1, a 2 A, t 2 T, q = f(a)f(t);1 ,
' q2 = 0. = T ' , u 2 T , b 2 A,
' au = tb. = 0 = q2 f(tu) = f(a)f(t);1 f(au) = f(ab), ab 2 Ker(f). ("
abs = 0 s 2 T. = () r(a) , ' bs 2 r(a),
0 = atbs = a2 us = a2 x, x us 2 T. ( () r(a) = r(a2 ), ax = 0,
a 2 Ker(f), q = 0.
3. 5 ,% MA 2 %:
() M | '%2. ,%1
(') (62 ,% M '%21
() 2-(2 (,% ,% M '%21
292
. . ()) '24 -, m n 2 M .% -,2 a b * A,
1 = a + b, maA + nbA mA \ nA1
() '24 -, m n 2 M .% -,2 a b c d 2 A,
1 = a + b, ma = nc, nb = md.
. 0
() ) () ) () () ) () '. () ) ().
( f m+n, T mA \ nA. = fA = fA \ mA+fA \ nA, " b d 2 A, ' fb 2 mA, fd 2 nA, f = fb + fd. = nb = fb ; mb 2 T,
md = fd ; nd 2 T . ( a 1 ; b, z a ; d = 1 ; b ; d. = 1 = a + b,
fz = f ; fb ; fd = 0, ma = md + mz = md + fz ; nz = md ; nz, nz = ;mz 2 T,
ma 2 T7 a b | ". () ) (). ( F, G, H | M,
f 2 F \ (G + H), f = m + n, m 2 G, n 2 H. ' '
f 2 F \ G + F \ H. ( a b 2 A, ' 1 = a + b, ma 2 nA,
nb 2 mA. = fb = mb + nb 2 F \ G, fa = ma + na 2 F \ H, f = fb + fa 2 F \ G +
+ F \ H.
4. % T | , (24 3,. * A, Q AT ,
f : A ! Q | . ),,63,. /):
() '24 -, q1 : : : qn 2 Q .% -,2 t 2 T ,
a1 : : : an 2 A, qi = f(ai )f(t);1 ( i = 1 : : : n1
(') B | (2. * A, BT = ff(b)f(t);1 j b 2 B
t 2 T g1
() a 2 A, B | (2. A f(a) 2 BT , at 2 B )
t 2 T1
()) (B + D)T = BT + DT , (B \ D)T = BT \ DT '24 (24 B D
* A1
() N | (2. * Q, E f ;1 (N), E | (2. A
N = ET 1
() G | (2. (,% ) ,% f(A)Q, .
. -, t 2 T , Gf(t) f(A) , , 2. ,% f(A)G
3,6 (,% ,% % Gf(t) * f(A)1
() * A '% , * Q f(A) '%2
, (, f(A)Q | '%2. 2. ,%1
(3) * A '% (, Q | '% (
*1
() * Q '% (, (B \ (D +E))T = (B \ D +B \ E)T
'24 (24 B D E * A1
() * A= Ker(f) , ( (, (), Q |
, ( (, () *1
() T ,, 24 3,., Q = TA,
(, 4 ),,63, A ! Q, A ! TA (1
(,) T = A n M , ) M | (2. A, Q | * J(Q) = MT MM 1
() L N | (2 2 * Q L ) N , f ;1 (L),
;
1
f (N) | (2 2 * A f ;1 (L) ) f ;1 (N)1
293
() A | ( ( () *, Q | ( ( () *1
(() Q | ( *, (DE)T = DT ET '24 (24
D E * A.
. (). 0 / / ', ' n = 2. (
qi = f(di )f(ti );1, di 2 A, ti 2 T , i = 1 2. = / T '
, u1 2 T, u2 2 A, ' t1 u1 = t2u2 t. = u1 t1 2 T,
t 2 T. = f(ti );1 = f(ui )f(t);1 , qi = f(ai )f(t);1 , ai di ui 2 A. ( ()
()7 () | (). ( () (), (), (). ( () # (), (). ( ()
(). ( (/) (), 3 1! ! . ( ()
(), (). ( () (). ( () # (), (). () 1 Q = TA $3,
. 51], '! 1 / T. / (). ' , ' " MT , " Q n MT . ; MT / ", 1 2 MT , ' () # " t 2 T , ' 1 t = t 2 M \ T, ' '. ( a 2 A,
t 2 T , f(a)f(t);1 2 Q n MT . = a 2 A n M = T , f(a) | ", f(a)f(t);1 | ". ( () (). ( ()
(). / (
). ( x 2 D, y 2 E, t 2 T. ;1 f(y) 2 (DE)T . = Q ' ' f(x)f(t)
P
1
, i=0 f(t);i f(y)Q ' /.
(" ;n;1 f(y) = Pn f(t);i f(y)qi . =
"
q
:
:
:
q
2
Q,
'
f(t)
0
n
i=0
P
P
f(t);1 f(y) = ni=0 f(t)n;i f(y)qi , f(x)f(t);1 f(y) = ni=0 f(xtn;iy)qi 2 (DE)T .
5. % A | 3%, ( *. /):
() a 2 A, B | (2. A aM 2 BM 4 M 2 max(AA ), a 2 B1
(') a 2 A aM = 0 4 M 2 max(AA ), a = 01
() B D | (2 2 * A BM = DM 4 M 2 max(AA ),
B = D1
()) %* * A '4, , '2 ') M 2 max(AA ) * AM '2 %*2,1
() B . (2. A, BM | * AM ')
M 2 max(AA ), B | * A1
() ') M 2 max(AA ) * AM (, A |
( *1
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. (). ( H fh 2 A j ah 2 B g. > , ' H = A.
. = H / M 2 max(AA ). (
T A n M. = aM 2 BM , 4() at 2 B t 2 T. = t 2 T \ M ' '. ( (), ()
294
. . ().
() ! 2(), ' | (). ( () () , '
BM (AB)M AM BM = BM . ( () (). /
(/). ( () ' , ' MN | AN N 2 max(AA ). ; M = N, 4() MN = J(AN ) | AN . ( M 6= N. = M + N = A, AN = NN + MN = J(AN ) + MN , MN = AN | AN .
6. 5 ,% MA *, A 2 %:
() M | '%2. ,%1
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mA \ nA = 0, .% -,2 a b 2 A, 1 = a + b, ma = nb = 01
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'3 ,% F 1
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m n 2 M, H mA + nA, f : H ! H=(mA \ nA) | "
1,
B r(f(m)) + r(f(n)). , ' B = A. = "
a 2 r(f(m)), b 2 r(f(n)), ' 1 = a + b. = ma 2 nA, nb 2 mA, 3
M . , ' B 6= A. = B D 2 max(AA ).
(" / f(mA), f(nA) 1, 1
A=B T. = H 1, 1 T T, ' ' .
7. % MA | '%2. ,% *, A. /):
() 0 = Hom((N + T )=T (N + T)=N) = Hom(N=(N \ T) T=(N \ T)), '24
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295
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. (). ; F N=(N \ T ), G T=(N \ T ), / 1 F = (N + T )=T, G = (N + T )=N , ' 6
0 = Hom(F G). ( () 6. +/ 1, ' 6, F G | F G | 1 M, F G 1. = (), ()
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().
8. % M | '%2. ,%, N | (,% M , f | -,63, ,% M . /):
() f ;1 (N) + N = M , f(M) N 1
(') f(M) N + f(N), f(M) N 1
() M = N + f(N), M = N 1
()) ,% M ,,2 (,% ( 2, End(M=J(M)) | %* *1
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7, ,% Ker(f)1
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. (). ( T f ;1 (N). ( h(m + T ) = f(m) + N 1 h : M=T ! M=N. ( 7() h = 0. =
T = M, f(M) N. / (). ( m 2 M. ( n x 2 N, ' f(m) = n + f(x). = m = (m ; x) + x 2 T + N, M = T + N. (
() f(M) N. ( () () ' f(N) f(M).
/ (). 0 () , ' M . = End(M=J(M)) 1 "1 ! 1 M " . / (). ' , ' Ker(f n+1 ) | # Ker(f n ) n.
( H Ker(f n ), L | Ker(f n+1 ), ' L \ H = 0. = f n+1 (L) = 0, f(L) H. ( () L Ker(f) H. = L = L \ H = 0, / . ( () ().
296
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= M N ', a 2 N, b 2 M. = A = M + N,
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# '! ', ()
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= B 2 L. / (). ; M | , 8() M | , A=M | , M | ' . (
T A n N, x 2 A, t 2 T . > # " u 2 T , '
xu 2 tA. ( 3 " a b c d 2 A, ' 1 = a + b, xa = tc,
tb = xd. ; a 2 T, / / u a. ( a 2 A n T = N. =
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7(), ().
10. % N | ( (2. '%) ( * A, T A n N . /):
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*,.
297
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-, h(a) )% h(A).
298
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( 12 AN # . (
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/ (), () , ' / ' / . * ,
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() A | '% ( )% ( *1
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,, (, ') ( () N ( * 24
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15. % A | *( ( , '. /) A |
*( *.
. ( 2(/) A . ( a b | " A. > , ' Aa Ab '. = A , "
f g 2 A, ' fa = gb. = A | , fA gA '. (, , f = gd, d 2 A. = gda = gb, da = b 2 Aa.
16. % A | '% ( %* *. /)
2 %:
() A | '% *1
(') 6* * A ( ',% ,,,% (,% % '%2, *,1
() 6* * A ( ',% ,,,% (,% % 3%,2, ,2, *,1
()) A | 3%, *, , 6* ( ',% ,,,% (,% % ,2, *,1
299
() ') ,,) ) M * 24 MA %% ,2, *,1
() ') ,,) ) M * 24 MA %% *(. ( '.
. 0
() ) () 14. 0
() ) () '. () ) (). 0 2() , ' 1 ' , 9(/) 14. () ) (). ( M | . (
2() M / ' H. ( h : A ! A=H |
"
1. ( 2() h(A) | , # '! h(M)h(A) Q. = h(A) | , '
1 f : h(A) ! Q 1. ( 5(/) h(M) | ' h(A), M | ' A. ( 13() H = fa 2 A j ta = 0 t 2 A n M g.
(" 1 g fh : A ! Q ' 1,
## Q '! A A n M. () ) ().
( M | . ( 5(/)
M | ' . ( 13() # ' 1 f : A ! AM , g : A ! MA. ( H Ker(f). ( 13()
H | ' . ( 4() H = Ker(g),
' A=H ' . ( 4() MA ' . () ) (). ( M | . (
5(/) M | ' . ( 14
AM # . ( = MA. (" MA |
MA # ' . ( 4() AM ' . ( 15 MA | .
17. % * A ,,2 (2 2 , ') -, x . % n, xnA Ax. /):
() * A '. ,,2. 2. M ( (2, ,, (, ') ( () N , A n N
(2, ,%(2, (,,, A=N |
, '1
(') * A -,2 %2, , *, A | 3%, *, , ') ( () N
* 24 NA %% 2, *,.
.
(). , ' M | . = MA = A P
1 = ni=1 mi ai , mi 2 M, ai 2 A. ( P ' d, ' mdi Ami M i = 1 : : : n. ( H ni=1 mdi A. = H 6= A.
(" H / B. = B | , A=B | ' mdi 2 H B , ' mi 2 B P
n
i = 1 : : : n. = 1 = i=1 mi ai 2 B, ' '. (" M |
, A=M | , M | ' . ( T A n N, t 2 T , a 2 A.
300
. . ( " b 2 A, ' tna = bt. = N | ' , / T , tn 2 T . (" T ' . ; a 2 T , b 2 T . (" A=N . ().
F/ () 1().
1. ( 14 A | , . ( 17() A |
. G R A ' 2() , 17() |
. ( 16 A .
1] Grater J. Strong right D-domains // Monatsh. Math. | 1989. | V. 107. | P. 189{205.
2] . . !"# $ %! // &$. '$. |1990. |
. 47. | N 2. | ). 115{123.
3] Stenstrom B. Rings of quotients | Berlin e. a.: Springer, 1975.
4] %- .. ., /01 2. &. /%! ! 0 0. | &.:
3, 1979.
' (: 1994.
. . . . 70- (14.02.1924{26.05.1989)
. : 1) "# $
# # R- # # -%# (##) # %
' 2) R V -.
Abstract
G. M. Brodski, A. G. Grigorjan, Ring properties of endomorphism rings of modules,
Fundamentalnaya i prikladnaya matematika 1(1995), 301{304.
A certain method of studying ring properties of endomorphism rings of modules is
justi0ed. As an example of its applications the equivalence of the following conditions is
proved: 1) the right annihilator of every proper 0nitely generated (principal) left ideal in
any endomorphism ring of an injective right R-module contains a nonzero idempotent' 2)
the ring R is a semiartinian right V -ring.
, . f : X ! Y Dom f = X , Cod f = Y , Coim f = X= Ker f
! coim f "! # Dom f ! CoimPf . $! "!
: 0
: X ! Y | ' Im H = fIm h j h 2 H g
Ker H = \ fKer h j h 2 H g " H End(X )' Ep(R) Mon(R) | * #- * R-' L(X )
L (X ) | * * !-* X ' l(A)
r(A) | A . +
,1] . / X (!), ( ) 0
"!
R .
1 , , | , 2 R * * , (")
* R- * R-, , ! , , * *
, (") . $ #
R
X Y
R
R
R
R
R
R
f
R
R
R
R
R
1995, 1, N 1, 301{304.
c 1995 ,
!
\# "
302
. . , . . | (* * ) X 2 , , ! ( ) X " . 5 D( ), C ( ), S ( ), T ( ), " R D( ) = fDom f j f 2 g, C ( ) = fCod f j f 2 g,
S ( ) = fCod f j f 2 2 # Dom f ! Ker f g. 5
#, 1 : Ep(R) Q "
R' 2 : 0 0 2 "* R X 2 D( ) ' 3 : coim f 2 "* R (f ) 2 , f : X ! Y
* X Y ( 2 6)' 4 : C ( ) = S ( ). "
. 1 | , 1) "* X Y X (Y ) = \PfKer f j f 2 ' Dom f = X ' Ker f Y g (Y ) = fIm f j f 2 ' Cod f = X ' Im f Y g'
2) ( ), | * , | " , (, ) -#, "* R,
X 2 D( ) " (!-, ) I
E = End(X ) : I 2 () X= (Im I ) 2 '
3) ( ), | * , | " , (, ) -, "* R,
X 2 C ( ) " (!-, ) J
E = End(X ) : J 2 () (Ker J ) 2 .
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1. % &'() * R, ,&- . -//0'/ &. (0., -.) -,&&( ( ) ( 1:
1) * ,/0'/ 3- /1 ' D( ) . (-&(., -(.) &(. 3 ./ 4
2) /1 ' C ( ) 3 ./ .
1 . % &'() * R, /- . -//0'/ &. (0., -.) -/&( ( ) ( 1:
1) * ,/0'/ 3- /1 ' C ( ) . (-&(., -(.) (. 3 ./ 4
2) /1 ' D( ) 3 ./ .
8 1 1 ! 1. 1) ) 2). $ S ( ) U 2 C ( ) = S ( ) X 2 D( ) # f : X ! U g : X ! Ker f , ! f 2 , , (Ker f ) = Ker f . $#
X= Ker f = X= Im(ig) = X= Im('E) = X= (Im('E)), i : Ker f ! X |
U=
, E = End(X ) ' = ig 2 E . 9!
, ! 'E 2 ( ) -#, !, ! U 2 .
2) ) 1). $ X 2 D( ) , E = End(X ) I | (!-, ) E . 8 # , R
R
R
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303
# X ! X= (Im I ) , ! X= (Im I ) 2 C ( ) . + , ! ( ) (, ) -#, ,
! I 2 .
: FR , PG , PR , SF , FT , TL , CIC , IN , CC , CT * "*, * * "2*, * *, * * "
, * !* *, * *, * !*
0* "2*, * 0*, * !* *, * * * R- " FR, PG, PR . . $
1 = feR j e2 = e 2 Rg,
2 = 10 = fRg,
3 = fr(J ) j J 2 L( R)g,
4 = fr(J ) j J 2 L ( R)g,
5 = 7 = fI 2 L(R ) j l(I ) = 0g,
6 = fI 2 L(R ) j (9e = e2 2 R n f0g) eI = 0g,
8 = fI 2 L(R ) j (9e = e2 2 R) l(I ) = Reg,
9 = fRe j e2 = e 2 Rg,
11 = fl(I ) j I 2 L(R )g,
12 = 14 = fJ 2 L( R) j r(J ) = 0g,
13 = fJ 2 L( R) j (9e = e2 2 R n f0g) Je = 0g,
15 = fJ 2 L( R) j (9e = e2 2 R) r(J ) = eRg,
1 = 8 = PR,
2 = 7 = 10 = 14 = f0g,
3 = TL,
4 = SF ,
5 = fU j Hom (U R) = 0g,
6 = fU j (9K 2 L(U ) n fU g) U=K 2 PR g,
9 = 15 = IN ,
11 = CT ,
12 = fU j (8Q 2 IN ) Hom (Q U ) = 0g,
13 = fU j (9K 2 L(U )) 0 6= K 2 IN g,
1 = 2 = 6 = ff 2 Ep(R) j Dom f 2 PR g,
3 = ff 2 Ep(R) j Dom f 2 PG g,
4 = ff 2 Ep(R) j Dom f 2 FR g,
5 = ff 2 Ep(R) j Dom f 2 FT g,
7 = ff 2 Ep(R) j Dom f Cod f g,
8 = 1 \ 7 ,
9 = 10 = 13 = ff 2 Mon(R) j Cod f 2 IN g,
11 = ff 2 Mon(R) j Cod f 2 CIC g,
12 = ff 2 Mon(R) j Cod f 2 CC g,
14 = ff 2 Mon(R) j Cod f Dom f g,
15 = 9 \ 14.
; R
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304
. . , . . . . -//0'/ ,& & 1 6 i 6 8
/ & 9 6 i 6 15. 5 1 6 i 6 4, ( ) 0. -,&&.. 5 5 6 i 6 8, ( ) &. -,&&..
5 9 6 i 6 11, ( ) 0. -/&.. 5 12 6 i 6 15,
i
i
i
i
i
i
i
i
i
i
( ) &. -/&..
5 1, 1 | , *, ,
2. 6( 1:
1) * ,/0'/ 3- 7- &- R-/1 &(. 1 - 3- -&- (--) - 1. /&4
2) R &1(/ &(/ V -*/.
. ! 1 10 10 - ( ) = :(10 10) )
) (13 13), = 10 13 = 10 13 S , ! 13 ,2].
? " ?. 8. /* ".
i
i
i
S
S
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S
S
S
1] . : , . . 1. | .: , 1977.
2] Dung N. V., Smith P.E. // J. Pure and Appl. Algebra. | 1992. | V. 82. | P. 27{37.
% &: 1994.
E-
. . . . . 511.36
jP (f1 () : : : fs ())j
!! " E-!" f1 (z) : : : fs(z) $%
"
&%&', P (f1 (): : : fs ()) 6= 0.
Abstract
A. I. Galochkin, Lower bounds of polynomials on values of algebraically dependent
E-functions, Fundamentalnaya i prikladnaya matematika 1(1995), 305{309.
In the paper a lower bound of the modulus of a polynomial jP (f1 () : : : fs())j with
integer coe-cients on the values of E-functions f1 (z) : : :, fs(z) at an algebraic point is
obtained, provided P (f1 (): : : fs ()) 6= 0.
I | , K | I, ZK | K. f1 (z) : : : fs (z) |
KE- (
x 1 3 #1]), %&' ( yi0 = Qi0 (z) +
s
X
j =1
Qij (z)yj i = 1 s Qij (z) 2 K(z):
(1)
+ E- f1 (z) : : : fs (z) , C(z) % &
Qij (z), -
(#1] 3) ,% f1 () : : : fs () , , % , . , (., , #2],
#3], #4]).
1% ,
%, E- , (. #1] 11 12), &% % %, %,&' E-. 2 %' % , E- ,
. ,% '% . .
% %, E-%.
. " & &% " &%' /'% !. 0% N MHS000.
1995, 1, N 1, 305{309.
c 1995 !,
"#
\% "
306
. . . f1 (z) : : : fs (z) | KE-, (1), | , ! Qij (z) (1),
{ = #K() : I] > 2. # P (x1 : : : xs ) 2 ZI #x1 : : : xs]
P (f1 () : : : fs ()) = 0, l+1 l
jP (f1 () : : : fs ())j > CH ; { d (2)
d H | P , l | $ C(z) f1 (z) : : : fs (z)
(1 6 l 6 s), C | ' , C $ d,
f1 (z) : : : fs (z), | f1 (z) : : : fs (z) ( : $ ' ! ).
6 ,, , % '
, 2 K | , K , K().
2 &' ,%. 7% Fi(z) =
1
X
=0
Fi z Fi 2 K i = 1 v
(3)
A(x1 : : : xv ) 2 K#x1 : : : xv ] , , Ak] (F1 () : : : Fv ()),
k = 1 {, ,% , ,
.
A(x1 : : : xv ), Fi (3) &' % K.
8 (2) , x 6 12 #1], , , P k] (f1 () : : : fs ()) 6= 0 k = 1 {:
9
, % % ,
6%. : %% &'%
. (#1], 11, x 2). KE-
f1 (z) : : : fs (z) (1) ( Qi0(z) 0, i = 1 s) $ C(z), 2 K
Qij (z), i j = 1 s. L(x1 : : : xs ) = h1 x1 + + hs xs hj 2 ZK H = maxj jhj j 6= 0 j = 1 s:
# " > 0
max Lk] (f1 () : : : fs ()) > C1 H 1;s;"
k=1{
C1 > 0, $ ", f1 (z) : : : fs (z).
(4)
E-
307
< % &'%
. kj , k = 2 {) j = 1 v | . # X > 1 h1 : : : hv $ ZK , , h1k] k1 + + hvk] kv < C2X 1; {{;v1 k = 2 {=
0 < max jhjk]j 6 X
k=1{ j =1v
(5)
hjk] | , ' 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
Lk ] (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 ()) = L1] (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),
jbjk]()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 .
<( )1 4 * *4.
5. % R | %) / .%, ' Q,
1 2 R, n > 3, a 2 GLn(R). 0. %% t 2 R, ), t;1 2 Q 2(1 r 2 R, 1 6 i 6= j 6 n 3, a;1 (1 + t r t eji ) a En(R).
. 3 (/ Q ., aii Q = ei Q . e2i = ei :
(1)
+( t1 2 R, t;1
1 2Q
t1 ei aik 2 R t1(1 ; ei ) aik 2 R i k:
(1) aii bi = ei , . bi 2 Q. +( t2 2 R, t;1
2 2Q (2)
bi aij (a;1 )jj t2 2 R:
(3)
+4), c = 1 + a;1 (t2 r t2 eji )a 2 En (R) r 2 R 1 6 i 6= j 6 n.
6) *
6. 6]. % n > 2, w 2 Rn ejj , v 2 ejj Rn, v w = 0 2 ekk w = 0,
2 v ekk = 0 (1 j k. 0. 1 + w v 2 En (R).
313
+4)
v1 = t1 ei eji a
v2 = t1 (1 ; ei ) eji a
(4)
w = a;1 t2 r ejj :
3. (2) , , v1 v2 w 2 Rn v1 w = v2 w = 0:
(5)
>(, c = (1 + w v1)(1 + w v2 ). ?) ., (1) v2 eii = 0 *
)) 6, 1 + w v2 2 En (R). +4)
w1 = ;bi aij (a;1 )jj t2 reij + (a;1 )jj t2 r ejj
(6)
w2 = w ; w1 . (3), w1 w2 2 R, * bi , v1 w1 = 0.
(5) , v1 w2 = v1 (w ; w1) = 0 1 + w v1 = (1 + w1 v1)(1 + w2 v1).
< (6) (4) *): ekk w1 = 0, ejj w2 = 0, . k 6= i j , * )) 6,
c = (1 + w1 v1 ) (1 + w2 v1 ) (1 + w v2 ) 2 En(R).
3) )1), t = t2 t1 *), a;1 (1 + t r t eji) a 2 En (R).
+4 5 .
4. + *4 5
;
;
1 a;1 (1 + t r t eji ) a = 1 a;1 (1 + t r t eji) a
1 (1 + t r t eji ) = 1(1 + t r t eji):
+4) 1(b) = 1 (a) c = a b;1 . 3.
c (1 + t r t eji) c;1 = (1 + t r t eji)
4 / (/ Rn. +5 ))), *), 1 + t2 r t2 eji = c (1 + t2 r t2 eji ) c;1:
P
+( T = t8 eii . 3. C T C ;1 = T C T S T C ;1 = T S T T (C S C ;1 ; S ) T = f0g
S 2 Rn. 6 T ;1 2 Qn , , C / Rn 1(C ) = 1. 3.
1(b) = 1(a) 1 (a) = 1(b) = 1(a) a 2 H . 3) 4 .
;)), .. (/ R >. . A 7] ,
En(R) | )(15 ( GLn (R) * n > 2.
14 *(( . . ) .
1] . ., . . ! ""#
-
%
&# // "%. ". -%. (. 1. %., . | 1983. | N 3. | (.
61{72.
314
. . 2] & 1. . ! ""#
%
&# // (
.
%. 2. | 1985. | 4. 26. | N 4. | (. 49{67.
3] . . %
! ""#
%
&#
//
2!. . . 4
" !. %
. | 6"
", 1991. |
(. 24.
4] "
. 6. !
# "! !
7 # // %. ". | 1987. |
4. 134. | N 1. | (. 42{65.
5] . ., . . 8
9% %
# ! PI-&#
//
"". . | 4
"
, 1985. | (. 20{24.
6] ("
. . 8 "%% "#
& ! &# //
. 6 (((;. (. %. | 1977. | 4. 41. | N 2. | (. 235{252.
7] <%
(. . 6% ""% 9% ! // , %
7 ". | .: !- >. 1986. | (. 86{90.
& ': 1994.
. . . . . 512.55
, A((x)) A !" : 1) A((x)) | % & 2) A((x)) | ' ' & 3) A | ' ' .
Abstract
K. I. Sonin, Regular rings of Laurent series, Fundamentalnaya i prikladnaya matematika 1(1995), 315{317.
The following conditions for the ring A((x)) of Laurent series over a ring A are equivalent: 1) A((x)) is a regular ring& 2) A((x)) is a semisimpleArtinian ring& 3) A is a semisimple
Artinian ring.
. A((x)) A ( Pi=;s aix!!
i , s > 0, ai 2 A). $, ! 1
1. :
(1) A((x)) (2) A((x)) (3) A .
& 1 () (. & ) 1{3 A((x)) R.
1. R , A .
. & a A , g R, aga = a, R , A R. .
!! x ), ag0a = a, g0 | g. 0 g0 ( A, A .
2. R , A ! "# $!
% $.
. &
, , , ei | A. 0(
Pi=0 eei0xie1 :::
z= 1
R, ( z . 0
1995, 1, N 1, 315{317.
c 1995 !,
"#
\% "
316
. . P
j
R | , , f = 1
j =;s aj x a;s 6= 0,
zR = fR. . !! ) x ) fz = z , , a;s+i es + : : : + a0 ei + : : : + ai e0 = ei ) i = 0 1 : : :.
&( ei , < 1. 3 f ( zR, P ab0jexij = be;it 06=60,i , g = 1
zg = f . 4 !j =;P
t
t
0
! x :
P a0 = i=0 eibi. - ek, k > 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.
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