. . , . .
-
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# . $ , %
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Abstract O. I. Balashov, A. I. Generalov, Projective resolutions of simple modules for a class of Frobenius algebras, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 637{650.
An in/nite series of nongroup symmetric algebras Rn , n > 1, is constructed as quotient algebras of a path algebra of a quiver. For these algebras, it is shown that minimal projective resolution of a simple module may be obtained as the total complex of a double complex of the same shape.
1.
, , . (., , !1]). % G | , K | p > 0, ) p G. % : : : ! P2 ! P1 ! P0 ! K ! 0 | K, * G. + cKG (K) K | ,
s, dimK Pm 6 ms;1 m 2 N > 0. - !2,3] , 2001, 7, 0 3, . 637{650. c 2001 !", #$ %& '
638
. . , . .
, cKG (K) p- G, . . , r, (Z=pZ)r G. 2 , cKG (K) = s, 4 5 , ,
Q(j ) : : : : ! Q(2j ) ! Q(1j ) ! Q(0j ) ! 0 j = 1 : : : s
Ns
* s- Q(j ) j =1 * * K (!4]). % *,
5 s, 54 . 8 !1] ( cKG (K) = 2). 8 , K * char K = 2 A5 , A6 54 45 (. !1, p. 196]). 8 4 * Rn, n > 1, * * , KA5 , KA6 . : 54 , !5], . % ) 5 5 (. 2), *
, Rn , KA5 KA6 5 .
2. % K | . <
R: 1 0 2
(2.1)
K- R = Rn, n > 1, ) 5 , 8 = 0 >< = 0 (2.2) :> ( )n = ( )n . . Rn = K!R]=In , K!R] | * R, In | h ( )n ; ( )2n i. = , R | L , , , R = Pi , Pi = Rei | , i=0
!...
639
ei , i = 0 1 2, | , 54 , R. , R
* * (., , !6]). > R
Si , i = 0 1 2, * * R: Si = Rsi ,
s0 = ( )n = ( ) n s1 = ( )n s2 = ( )n : (2.3) , si | * R. 2* * Pi Si , i : Pi = Rei ! Si = Rsi i (rei ) = rsi r 2 R. 1. Rn | K - , R (2.1) (2.2).
# # # # # # # P0 P0 P0 P0 # # # # P1 P0 P0 P0 P0 # # # # # P0 P0 P0 P0 P2 # # # P1 P0 P0 P2 # # P0 P2 P1 P0 P0 P0
(2.4)
(Qi di) ! S0 . " , , 8 >< (l + 1)P0 m = 3l Qm ' >P1 l P0 P2 m = 3l + 1 (l > 0): : (l + 1)P0 m = 3l + 2 @ , A K (!7]),
*: 1) " A ' " HomK (A K)C 2) 4 K- * : A ! K, Ker , AC 3) 4 * f : A A ! K, f(a bc) = f(ab c) 5 a b c 2 A. D f (ab) = f(a b). E A 54 f , A *. F5 A QF- *, I * 5 (. !7,8]).
640
. . , . .
, !9] 5
4 , 4 , *. 1. # R = Rn $ n > 1. . % B * R ei , i = 0 1 2. J B | K- R. : K A ! K K , ( (x) = 1 x 2 fsi gi=012 0 x 62 fsi gi=012 si 2 R, i = 0 1 2, 5 , (2.3). E * * R I Ker , Soc I Ker C Soc R I Soc R R = = S0 S1 S2 , Soc R I 6= 0. J 4 i, Si Soc R I Ker , (Si ) = 0, (si ) = 0. -, R . : , R , , (ab) = (ba) 5 a b 2 B. E (ab) = 1, ab = si i. 2 si | * , , ab ba 5 L M , ba | * , ba = sj j, (ba) = 1.
3.
1 (5, 2.1]). % A | K- . A- | D(X f), 4 54 :
1) X | * * , fx1 x2 : : : xng. E X xi ! xj , e(xi xj ). N , xi > xj , X 4 ) xi = y0 ! y1 ! : : : ! yt = xj (t > 0). O X 54 : 1a) X
) , 4 x 2 X, x > x, 5 , X 4
C 1b) x1 x2 x3 2 X, x1 > x2 > x3, 4 e(x1 x3)C 2) f * , x 2 X * A- f(x) e(x y) f(e(x y)) 2 Ext1" (f(x) f(y)). N f(x) * *. R f 54 : 3) , x y1 : : : yt 2 X f(y1 ) ' : : : ' f(yt ) ' N. E X
) e(x yi ), i = 1 : : : t, f(e(x yi )) K- * Ext1" (f(x) N). : , 5 ) e(yi x), i = 1 : : : t,
!...
641
f(e(yi x)) K- * Ext1" (N f(x)). 8 , e(x y) f(e(x y)) 6= 0. % , D(X f) | . % * U ( , ) X ) , x 2 U, y < x y 2 U (8x y 2 X). , I
X | . : U | U c = X ; U, 4
, , 4 U, ) , 54 , , U. J , V 5: x 2 V x < y ) y 2 V . E U | X, 545 5 D(U f jU ) D(U f). % * U ( , ) X , U X . ,
X . 8 , x z 2 U, z < y < x y 2 U (8x y z 2 X)C X 5 . 2 (5, 2.2]). * D(X f) | A- M U 7! MU , 54 X M. N M U 7! MU 54 . % U V W | X. J 1) MX = M, M? = 0 U V ) MU MV , 2) MU \V = MU \ MV , MU V = MU + MV , 3) V = U fxg, x 2= U,
EUV :
i 0 ;! MU ;! MV ;! f(x) ;! 0: E , , W = U fyg, y 2= U, x 6= y, EWV W = (iUW ) (EUV ), (iUW ) : Ext1" (f(x) MU ) ! Ext1" (f(x) MW ) | , * iUW , 4) , V = U fxg, W = V fyg, x 2= U, y 2= V 4 e(y x). J (UV ) (EVW ) = f(e(y x)) 2 Ext1" (f(y) f(x)). : 5 4 5 5 54 . 8 , M | * D(X f) X
n , , M n. T
,
M s , 5 N, X
s , x1 : : : xs, f(xi ) ' N (!5, 2.4]). N M D(X f) X. > , 5 W X 4 54 * - MW UV
UV
642
. . , . .
W : M ! MW , , (1){(4) 2. R * MU := Ker W , U | W, MW :=M=MU , U | , W |
(!5, 2.5]). 3 (5, 2.6]). % D(X f) D(Y g) | . ': D(X f) ! D(Y g) | ': X ! Y , * g ' = f , ) . ': D(X f) ! D(Y g) V X, U Y '0 : D(V f) ! D(U g). ' | D(V c f), V c = X ; V | V . ' | D(U g). E W X , , '(W) '0 (V \ W ). E S Y , , ';1 (S) ';0 1 (U \ S) V c . % , "M " N | D(X f) D(Y g) . O A- * : M ! N ) , D- , 4 ': D(X f) ! D(Y g), * (MU ) = N'(U ) 5 U X. E ' , , Ker = MKer ' Im = NIm ' . 2 , W | Y , ;1 (NW ) = M';1 (W ) (!5, 2.7]). , , * D- ( 4 ), D- *. 4 . , | . 4 (5, 3.4]). % , D(X f) D(Y g) | . % U X, V Y | , , 4 ': D(U f) ! D(V g). D(X f) ' D(Y g) | , D(X f) D(Y g) D(U f) D(V g) = '(D(U f)). J
, D(Z h), Z = (X Y )=(x = '(x))
(
h(z) = f(z) z 2 X , ) . g(z) z 2 Z : 5 U X, V Y . 2 (5, 3.5]). L, M , N | A- %
D(X f), D(Y g), D(Z h) . &, $ D- ' 1 : N ! L, 2 : N ! M $ '
'1 : D(Z h) ! D(X f),
!...
'2 : D(Z h) ! D(Y g). ( & W := Z ; (Ker '1 Ker '2 ) Z U := '1 (Z) ; '1 (Ker '2 ) X V := '2(Z) ; '2 (Ker '1) Y: ' )$ ' ;1
643
'1 '2 D(U f) ;! D(W h) ;! D(V g): ; & 12 : N ! L M
D(X ; '1 (Ker '2 ) f) ' D(Y ; '2(Ker '1 ) g). * , 1 : M ! L, 2 : N ! L | D- ' $ '
'1 , '2 . W := Im'1 \ Im'2 , U := ';1 1 (W) ; Ker '1 , V := ';2 1 (W) ; Ker '2 , ' | ) ;1 '1 2 D(U f) ';! D(V g). & ( 1 2): M N ! L D(';1 1 (W) f) ' D(';2 1 (W) g). 5 (5, 5.1]). T , D(X f)
,
5 . E , , 5
D- 54 * 5 D(X f), , D- . E D(X f)
D- 5 , ,
D- . 6 (5]). @) 5 D(X f) ! , , X ( 1 n) , x1 < x2 <: : :< xn . D(X f) ) , , X ( 1 n) , 5 i = 1 : : : n ; 1 4 e(xi xi+1) 4 e(xi+1 xi), ) . 8 ,
( 5 (3) 1) , . 3 (5, 6.1]). &, A- $ . 1) , & )
+ 2) D- , & )
. 7 (5]). : D(X f) ) "# , 5 e(x y) X dimK Ext1" (f(x) f(y)) = 1. 4 (5, 6.5]). &, D(X f) D(X f 0 ) | & , f(x) = f 0 (x) $ x 2 X . A- D(X f) , D(X f 0 ).
644
. . , . .
8 (5]). % D(X f) | .
, ! $ X 54 :
Rad X := fx 2 X j 9 y 2 X : x < yg Soc X := fx 2 X j @ y 2 X : y < xg top X := X ; Rad X: J , Soc X = f g | ,
X , 4
) , top X = f g | ,
, 4
) . 5 (5, 4.1]). &, A- $ . M |
D(X f). RadM = MRad X , Soc M = MSoc X , top M = M=Rad M = Mtop X .
4. !
8 45 5 | 2, ) * Rn , ) R (2.1) , (2.2) ( 3). - * * 4 * | 1. T 54 : , x 2 X f(x) ( * ). 2. A | QF- , $ ' % % , $ + fei g2i=0 | & % % , & P0, P1 , P2, Pi = Aei , $
S1 u S0 u S2 u S0 u : : : u S2 ^
S0
S0 ^
u
S0
S1
u
S0
u
: : : u S1 (4.2) S1 u S0 u S2 u S0 u S1 u : : : u S0 u S1 S2 u S0 u S1 u S0 u S2 u : : : u S0 u S2 (4.3) , Si ' Soc Pi ' Pi= Rad Pi+
(4.1) 8n ,
(4.2) (4.3) | 4n + 1 (n > 1). (2.4), ! S0 . S2
u
(4.1)
645
!...
: 2 5 4 5, . E A = KG, G * A5 A6 , K | ,
*
: * S0 = K S1 S2 (. !10, Appendix]). % P0, P1, P2 * 5 (4.1){(4.3) , n = 1 G = A5 , n = 2 G = A6. J , 54
2. 6. , KG- K G- K , char K = 2,
G, A5 A6 , & (2.4). 3. - & P0, P1 P2 Rn, n > 1, R (2.1) (2.2), $
(4.1), (4.2) (4.3) . . , K- P0 = Re0 fe0 : : : ( )n = s0 = ( )n ( )n;1 ( )n;1 : : : g, dimK P0 = 8n. K- P1 | fe1 : :: ( )n = s1 g, P2 | fe2 : : : ( )n = s2 g. + , dimK P1 = dimK P2 = 4n + 1, dimK R = = 16n + 2. , 45 * Pi, i=0 1 2. F , dimK Ext1R (Si Sj )= = 1, i 6= j i, j 0. R Z 2 Ext1R (S1 S0), Z 2 Ext1R (S0 S1 ), Z 2 Ext1R (S2 S0), Z 2 Ext1R (S0 S2). J , Di , 54 Pi (i = 0 1 2), 5 54 * ( * * 54 * * ): % % % % S1 u S0 u S2 u S0 u % : : : u S2 ^ %
D0 : S0 ^
u
u
S0
% % % % S2 u S0 u % S1 u S0 u : : : u % S1 % % % % S0 u % S2 u S0 u % S1 u : : : u S0 u % S1 % S u % S u % S u % S u % : : : u % S u % S : 0 1 0 2 0 2
%
D1 : S1 D2 : S2
%
646
. . , . .
, * * 4n ( 54 * ). J , 1, * , 2.
5. R 5 4) 2. N * S0 , 5 5 d Q ;! d Q ;! d Q ;!
S ;! 0 ;! 2 1 0 0
(5.1) * (2.4). F , Ext1" (Si Sj ) , i 6= j i, j 0, 5 . % Di = D(Xi fi ) 5 ) , 5 4 fi (e(x y)) 2 Ext1" (fi (x) fi (y)) . R , *, . $ 1. .$ )
$
Di , i = 0 1 2, . . F , 5* , 54 * 5 , . 8 , P1 P2 | . J A | QF- , Pi 5 I . % D(X f), X k , ,
Sc Sd : : : Sh * * Di , ,
- M. E Sc = S1 , Soc M ' S1 , M P1 M ' Radl P1 | l-* (54 ) F ,
l = 4n + 1 ; k. > , Sc = S2 , M ) 5 . % Sc = S0 , Sd ) S1 D(X f)
S0 S1 S0 S2 : : : Sh . N , M RadP0 P0. [ M 0 Rad P0 , 54 * X 0 D0 , 54
, X, M 0 = (P0)X 0 . : , M ' M 0. % 5 ,
'0 : Rad M ! Rad M 0 . J P0 I , 4 ': P0 ! P0, * 'jRad M = '0 . J RadM | 4 * P0 , '0 | , . : , '(M) = M 0 .
M '(M), , Rad M = RadM 0 . % , M 6= M 0. < L := (M + M 0 )= Rad2 M * . Soc L ' Rad M= Rad2 M ' Si i = 0 1 2, top L ' M= RadM M 0 = Rad M 0 ' Sh Sh , i = 1 2 L Pi, L *, 5. E i = 0, L 2
1
0
!...
647
P0, L= Soc L = topL P0= Soc P0,
, Soc(P0 = Soc P0) ' S1 S2 . % *) 5 * * (5.1) * \m = \m (S0 ) = Ker dm;2 S0 ( , d;1 = ). N Q0 = P0 = P (S0 ) | S0 . @ | , D0 5 S0 . + , \1
( * 5 4n ; 1), 545 D0 , S0 : S1 u S0 u S2 u S0 u : : : u S0 u S2 A D A \1 : S0 (5.2) S2 u S0 u S1 u S0 u : : : u S0 u S1 R ) 3 1
D- , . . \1 . 2 (5.2), top \1 ' S2 S1 , , Q1 = P (\1 ) = P2 P1. < \1 , 54 4n ; 1 S0 S1 : : : S0 S2 S0 S2 : : : S0 S1 , 54 \1. R 5 , S2 S1 5 P2 P1 . % D- P2 P1 5 - ( , P2= Soc P2 P1= Soc P1), . 2 1 ) 1 J , Q1 = P2 P1 ( ;! \ 54 . 4n ; 1 S0 S1 : : : S0 S2 S0 S2 : : : S0 S1 P2 P1 \1 . J ( 2 1) \2
\2
u
P1
w P2 u1
w\
\2 | 2. J , \2 54 : S2 ' *' 2 \: S0 ^ S1 R L M P0, top \2 ' S0 , \3 ( * 5 4n ; 2) L M D0 :
648
. . , . .
\3 :
A S1 u A D S0 S2 u
:::
u
S1
u
S0
: : : u S2 u S0 > Q3 = P0 P0 ! \3 \4, * S1 S0 S2 \4 : 4n;1 2 2 4n;1 S0 S0 * ( 4 ), . % , 5 L M L M . % \3t+1 5 \3t+2 45 54 * , * | \3t+1, * | \3t+2: S1 S0 S0 S0 S2 4n;1
P1
1
2
S0
4n;2
P0
2
4n;2
4n;2
2
S0
P0
:::
2
P0
4n;2
S0
4n;1
P2
(5.3)
1
S1 S0 S0 S0 S2 % ,
* *, 4 * S0 ( , * S1 S2 , 54 54 , ). 8 , , Soc(\3t+1) ' S0t+1 ' top(\3t+2), top(\3t+1) ' ' S1 S0t S26t,+1Soc(\3t+2) ' S0t+1 ,
* : l(\ ) = 8n(t + 1) ; 1, l(\6t+4 ) = 8n(t + 1) + 3, l(\6t+2) = 8nt + 3, l(\6t+5) = 8n(t + 1) ; 1 (t > 0). > , * \3t, t > 1, * S0 S0 S0 S0 \3t : 4n;2 2 4n;2 4n;2 2 4n;2 ::: S0 S0 % Soc(\3t) ' S0t , top(\3t ) ' S0t+1 , l(\6t ) = 8nt + 1, l(\6t+3) = = 8n(t + 1) ; 3. 8 * * Q S0 . @ ,
649
!...
" \3t+2 ;! i Q d3t+1 : Q3t+2 ;! 3t+1 45 54 :
P1
S0
S1 ^ P0
P0
S0
S0 ^ P0
P0
S0
P0 :::
S0
S0 ^ P0
P2
S2
5 ": P3t+2 ' P0t+1 ! \3t+2, i (5.3). 5 , (2.4), , , 2 , . %& . R , 4 , * S1 S2 A 2, , 5 , , 5 : : : w P0 w P1 w P1 w P0 w P1 w P1 w P0 w P1 : : : w P0 w P2 w P2 w P0 w P2 w P2 w P0 w P2 . 8 5 5 >. 8. =
.
#
1] Stammbach U. Types of projective resolutions for nite groups // The Hilton Sympos. 1993, Topics in Topol. and Group Theory, Centre de Rech. Math., CRM Proc. and Lect. Notes. | 1994. | Vol. 6. | P. 187{198. 2] Quillen D. The spectrum of an equivariant cohomology ring, I, II // Annals Math. | 1971. | Vol. 94, no. 3. | P. 549{602. 3] Alperin J. L., Evens L. Representations, resolutions and Quillen's dimension theorem // J. Pure Appl. Algebra. | 1981. | Vol. 22, no. 1. | P. 1{9. 4] Benson D. J., Carlson J. F. Complexity and multiple complexes // Math. Zeitschr. | 1987. | Vol. 195, no. 2. | P. 221{238. 5] Benson D. J., Carlson J. F. Diagrammatic methods for modular representations and cohomology // Comm. Algebra. | 1987. | Vol. 15, no. 1/2. | P. 53{121. 6] . ., . !. "# $% $& '. (. ) *+,%$ // -.%/0. %*1 1 2 31*. | 1973. | 4. 7, 53. 4. | 6. 54{69. 7] 78 19 '., % '. 4 1: 3 $9 * %1 /% &%5; 2.33 1 9901 1%5; *2 #. | (.: ./, 1969. 8] - 9 7. *2 #: 7*+0, =$.*1 1 / 211. 4. 2. | (.: (1, 1979.
650
. . , . .
9] Green E. L. Frobenius algebras and their quivers // Can. J. Math. | 1978. | Vol. 30, no. 5. | P. 1029{1044. 10] Benson D. J. Modular represent theory: New trends and methods. | Lect. Notes in Math. Vol. 1081. | 1984. ( 1998 .
{ . .
. . .
512.558
: , .
(
, ! "), !$ ! "% xn = 0. ( ! ) ! !. . * % ) n!- - ! "% xn = 0, . . / % % % ) n!- - % . . ( % % l- "%0!$ % "% xn = 0 % .
) % . . * % S ! "% xn = 0, S n | .
Abstract I. I. Bogdanov, The Nagata{Higman theorem for hemirings, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 651{658.
In this paper the hemirings (in general, with noncommutative addition) with the identity xn = 0 are studied. The main results are the following ones. Theorem. If a n!-torsionfree general hemiring satis6es the identity xn = 0, then it is nilpotent. The estimates of the nilpotency index are equal for n!-torsionless rings and general hemirings. Theorem. The estimates of the nilpotency index of l-generated rings and general hemirings with identity xn = 0 are equal. The proof is based on the following lemma. Lemma. If a general semiring S satis6es the identity xn = 0, then S n is a ring.
1. { , ( 1 2
: : : n , . !1, 2]), $
%. &$ '( .
1.1 ( . 3, 6.1.1]). R i 1 6 i 6 n. , R xn = 0, R2n ;1 = 0. , 2001, 7, 7 3, . 651{658. c 2001 !, "# $% &
. .
652
- !4] /0 1 -(% ($ ). 2 (3 ( % %
ln+1n3 l-/0 1 (% % nn 2ln+1 n3 + n l-/0 1 (% $ .
2
2 2 3 1 a 1 -(%1 $ . (% $
S
5, ,
xn
/
= 0,
Sn
|
% ( ( / ). 2 3, ( , , % % (% $ ( 8 ( , , % (% $ ($ (
n
1 2 ; 1, . ( 3.6 3.7). : , , % /0 1 % (% $ / 8 ( 3.8). < (-
ln+1 n3 l-/0 -(% $ ( .
% 2 3.9).
2. - 2.1.
>(% /
S 0 -
0 % + , 0
S S
1) (
2) (
+) | ( 3 3 0? ) | ((?
3) (/ ( / ( )? 4) 0
a = a0 = 0 8 a S . 2
@/ (% ( ( (% %3? (% (
N, N |
(% % 1 % 1 % .
& (% $ ( 1 / , 8 ( / . 2.
. @ , (%1 /
4) ( 1 ( / ). 0 =
xn
= 0 -
2 , (
xn = x xn;1 = x(xn;1 + 0) = xn + x0 = 0 + x0 = x0, 0x = 0
( . A (% ($ ) /
xn
= 0, (
n > 2 ( ( n = 1. -( (% ($ ) x , ($ ( 3 y , x + y = y + x = 0.
B , (8$
2.2. S xn = 0,
S n | .
{
653
. A /
(x1 + x2 + : : : + xn )n = 0.
x = x1x2 : : :xn. , y, ( x + y = 0, y S n , x S n . / 3 S n (n 1 , 83 S , < ( C 1 1 (
(( 1 1
2
. >/, (% $ / ( / . & (8$
2.3. x, y, x0
x + y = y + x0 = 0, x = x0 , . . y | . . x = x + 0 = x + (y + x0 ) = (x + y) + x0 = 0 + x0 = x0 . 2.4. S | , x1 x2 : : : xN S , x1 + x2 + : : : + xN = xN + xN ;1 + : : : + x1 = 0. i, 1 6 i 6 N ,
xi . . >( 1 6 i 6 N . & si = (xi+1 + xi+2 + + : : : + xN ) + (x1 + x2 + : : : + xi;1 ) ( i = N ( ( , i = 1 | ), s0i = (xi;1 + xi;2 + : : : + x1 ) + (xN + xN ;1 + : : : + xi+1 ). D/ (% 3 i, xi + si = s0i + xi = 0 (2.1) 2
( 2.3 (
si = s0i
(2.2)
( / . - (%
( 8 . (2.1) .
xi+1 + si+1
=
.
(2.2) ( ,
si + xi
=
xi + s0i
=
A 3 ' 3 3 C
xi
s0i+1 + xi+1,
xi
>( +
s0i
=
si
i
+
k xi , (
=
.
B
X = x1 x2 : : : xn . 20 0 xj i < j . 1 m f
g
,
' 3 , 3 / ( .
2.5. (x1 + x2 + : : : + xn)m !
, " . . ( n- ), ! $ m, $ " . . D/ ( (% 3 m. - (m = 1) . >( /
(x1 + x2 + : : : + xn )m+1 (/ , 3.
/ 8 (%
( C /-
s
s
( 1+ 2+
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654
s1 s2 : : : snm
m, ( si xk sj xl , si sj si = sj xk xl : (2.4)
|
. E ,
,
(2.3) ( /
sx
sx
( 1 1+ 1 2+
: : : + s1 xn ) + : : : + (snm x1 + snm x2 + : : : + snm xn)
/ ( , (2.4), . , (.
2.6. S xn = 0, S n | . . , 8 1
x1 x2 : : : xn y1 y2 : : : yn S x = x1 x2 : : :xn y = y1 y2 : : :yn x + y = y + x. >( s1 s2 : : : snn | n ( x1 x2 : : : xn , 2
( 8$ ' .
x
x
: : : xn n
xn xn;
: : :x n
-
> ( 8
+ ) = ( + ( 1 + 2 + 1+ 1) = 0. A 1 / 1, ( , 2.5,
s1 + s2 + : : : + snn = snn + snn ;1 + : : : + s1 = 0 (
-
xi 0 xj
i > j ). 2.4 si ? , x1x2 : : :xn = x. D/, x y ( (8 . >/ x0 = x1 : : :xn;1, y 0 = y1 : : :yn;1 .
,
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1] Nagata M. On the nilpotency of nilalgebras // J. Math. Soc. Japan. | 1952. | Vol. 4. | P. 296{301. 2] Higman G. On a conjecture of Nagata // Proc. Cam. Phil. Soc. | 1956. | Vol. 52. | P. 1{4. 3] . ., . ., !. "., #$ . !. %, &' % (). | .: +,, 1978. 4] / . 0. 1#) +2 ({32) 45 6,% // 7,4). 6#. ) . | 1995. | 1. 1, (6. 2. | . 523{527. 5] 82 . 2&#. | .: #, 1968. ' ( ) 2000 .
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Abstract Z. V. Bulatov, On precise with respect to the height estimates of certain linear forms, Fundamentalnayai prikladnayamatematika,vol. 7 (2001), no. 3, pp. 659{671.
The paper presents precise estimates with respect to the height of linear forms in the values of certain hypergeometric functions at points d=b, where d and b are algebraic numbers of the imaginary quadratic 1eld, and d is not 'very big).
.
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/ K( j ) | j g(x) = (x ; 1) : : :(x ; s), rl = max K( j ) r = min r = l j =l l
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, p, = 1 : : : N, , , = (d=b) e > 0, p | , ,
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-
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661
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, = q + 1 > 1 > 2 > : : : > s > q (5) q | , A . < ,B = "3]. < "3], = B, , 1 : : : s , , (5), j B B,
. / l = l+1 = : : : = ul;1 (j 6= l , j < l j > ul ) ul ; l j B , = , B . / , rl1 : : : rl | (rl1 + : : : + rl = ul ; l), rl1 > rl2 > : : : > rl : ; t Y ft (x) = (x ; ai) t > 1 = z dzd i=0 iY ;1 Fil = ai ( + l ; ) l = 1 : : : s i > 10 =0
J = (j1 : : : js ) jk > 0 k = 1 : : : s s Y LtJ (z) = Ltj1 :::js (z) = ft (a ) Fji i (z) t > 1:
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i=1
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662
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Qt t p. D , (x ; yi) p p ] . E 1 i=1 . (. "3, . 421]), (4) B B, : 1 (z) = az dzd 0 (z) d a z dz + j ;1 j ;1(z) = j (z) j = 2 : : : s (7) d a z dz + s s (z) = z0 (z): 9 B (6) , s X LtJ (z) = BtJl (z)l (z) t > 1 j1 : : : js 2 f0g N
l=0
(8)
BtJl (z) 2 I"z]. 9 J = (j1 ; : : : js ; ), 2 f0g. 2. j1 : : : js T 2 f0g, t 2 , t > T + 2 ji > T + 2, i = 1 : : : s. BtJl (z) = z T +1 Bt;T ;1JT +1 l (z) l = 0 : : : s:
. (7) , (z) B K(a )(z) = z(z) K(z) = z(z+a1 ) : : :(z+as ). / (6) 13 "6, . 332]
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N
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l=1 j ;1 s l YY
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l=1 i=1
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(9)
663
D (9) LtJ (z) Lt;1J1 (z) = B (8) B l (z), l = 0 : : : s, (z) ( . "7, 2] "8, 2 x 4 . 3]), , BtJl (z) = zBt;1J1 l (z) l = 0 : : : s: 9 B BtJl (z) = zBt;1J1 l (z) = z 2 Bt;2J2 l (z) = : : : = z T +1 Bt;T ;1JT +1 l (z) l = 0 : : : s. E . < p p (x) , p = (x), x 2 I n f0g. ' , = p (0) = 1 p. 3. p p, p a, N(p) = p, j1 : : : js 2 f0g, t 2 , t > p Jl = (j1 : : : jl;1 jl + jl+1 : : : js), l = 1 : : : s, 2 f0g. & r, 1 6 r 6 s. d min B + (10) p BtJl db > =0 p :::p p t;pJr l b
l = 0 : : : s, ( p = 1 0 6 6 p ; 1 0 = p: C
-
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N
N
. C , B, LtJ (z) . pQ ;1 / ft (x) = ft;p (x)gtp(x), gtp(x) = (x ; a(t ; i)). A i=0 B B gtp (x) x = a(jr ; r + ), = 0 : : : p, p Y ;1 X gtp (x) = gtp(a(jr ; r )) + Atprjr (x ; x ) (11)
=1
=0
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(12)
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664
. .
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LtJ (z) = ft;p (a )gtp (a ) =1
s Y
=0
= gtp (a(jr ; r ))Lt;pJ (z) + =
p X =0
p X =1
i=1
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Atprjr Lt;pJr (z)
(13)
A0tprjr = gtp(a(jr ; r )). C A (8) B l (z), l = 0 : : : s, (z), (13) p X BtJl (z) = Atprjr Bt;pJr l (z) l = 0 : : : s j1 : : : js 2 f0g: (14) C
N
=0
9 B d d p BtJl b > =0 min (A ) + p Bt;pJr l b : :::p p tprjr
(15)
1 B p (gtp(x)) > 1, x 2 I, , (12) A0tprjr , p (Atprjr ) > 1 (16) 0 6 6 p ; 1. J
, Aptprjr = 1, (15) (16) (10). E . 9 D(J) = max(j1 : : : js) ; min(j1 : : : js), D0 = max(p1 : : : pN ). 4. j1 : : : js 2 f0g, t 2 , t > max(j1 : : : js ) D(J) 6 D0 . 0, % a, b, d, 1 : : : s , e1 : : : eN , p1 : : : pN , p1 : : : pN , p BtJl db > e (t + j1 m+ : : : + js) ; 0 (17)
l = 0 : : : s, = 1 : : : N .
. 9 :tJ = i=1 max (t ; ji ), D = 3 16max p. :::s 6N K :tJ . K . / = , = :tJ 6 D. / T = i=1min j ; 2 = ji0 ; 2, ji0 = ji ; T ; 1, J 0 = (j10 : : : js0 ). :::s i Z
N
N
665
J T > 0. ? 0 6 t ; ji 6 :tJ 6 D, i = 1 : : : s, 1 6 ji 0 = ji ; T ; 1 = ji ; ji0 + 1 6 t ; ji0 + 1 6 D + 1 i = 1 : : : s (18) T + 1 = ji0 ; 1 > t ; (D + 1): / 2 BtJl (z) = z T +1 Bt;T ;1J l (z) l = 0 : : : s: (19) (18), (19) e , = 1 : : : N, , p BtJl db = (T + 1)p db + p Bt;T ;1J l db > > (t ; (D + 1))e + p Bt;T ;1J l db > t + j1 +m: : : + js e + + p Bt;T ;1J l db ; (D + 1) =1 max e = 1 : : : N l = 0 : : : s: :::N (20) (18) t ; T ; 1 6 D + 1 ji0 6 6 D + 1, i = 1 : : : s, , 1 = = 1(a b d 1 : : : s p1 : : : pN p1 : : : pN e1 : : : eN ), B, : p Bt;T ;1J l db > ;1 = 1 : : : N l = 0 : : : s: (21) ' (20) (21), d t + j + : : : + j 1 se ; p BtJl b > (22) 2 m = 1 : : : N, l = 0 : : : s, 2 = 1 + (D + 1) =1 max e . :::N / T 6 ;1. i=1max j 6 t 6 ji0 + D 6 D + 1 :::N i ; 3, ,
a, b, d, 1 : : : s , p1 : : : pN , p1 : : : pN , e1 : : : eN , d t + j + : : : + j 1 se ; p BtJl b > (23) 3 m = 1 : : : N, l = 0 : : : s. ? (22) (23) , :tJ 6 D B (17) 0 = max(2 3). L . / = :tJ 6 n, n 2 . / = , :tJ = n + 1. ? :tJ 6 D A , = , :tJ = n + 1 > D , , t > D > p, = 1 : : : N. / B :tJ t ; jr = n + 1, jr = min(j1 : : : js). 0
0
0
0
0
N
666
. .
3, ; p = p, d p BtJl db > =0min B + p :::p p t;p Jr l b = 1 : : : N, l = 0 : : : s. / = f1 : : : sg n fjr g. ? = 0 : : : p
8 max(j1 : : : js) ; min j min j 6 jr + 6 max j0 > > i2J i i2J i i2J i < j max j < jr + 0 D(Jr ) = >jr + ; min i2J i i2J i > :max ji ; jr ; jr + < min j: i2J i2J i
(24)
J
? max(j1 : : : js) ; min j 6 D(J) 6 D0 , jr + ; min j 6 jr + i2J i i2J i + p ; jr 6 D0 max j ; jr ; 6 D(J) ; 6 D0 , D(Jr ) 6 D0 . i2J i 9 B t ; p > D + min(j1 : : : js ) ; p > D ; p + + max(j1 : : : js ) ; D0 > max(j1 : : : js ) + p :t;pJr 6 n , Bt;p Jr l (d=b) = . , B B e p 6 m, = 1 : : : N, p Bt;p Jr l db > t ; p + j1 +m : : : + js + e ; 0 > t+j +: : :+j 1 s e ; 1 ; = 0 : : : p ; 1 = 1 : : : N l = 0 : : : s > 0 m (25) d t ; p + j + : : : + j + p 1 s e ; > p Bt;p Jrp l b > 0 m t + j + : : : + j 1 s e ; = 1 : : : N l = 0 : : : s: > (26) 0 m / (24), (25) (26) , d t + j + : : : + j 1 s e ; = 1 : : : N l = 0 : : : s: p BtJl b > 0 m E 4 . 9 t Y t Ltl (z) = a ( ; k)l (z) l = 1 : : : s t > 1: (27) k=1
(7) ,
Ltl (z) = Btli (z) =
dX tli k=0
btlikz k
s X i=0
Btli (z)i (z)
2 I"z] dtli = deg Btli (z) dtl = 0max d : 6i6s tli
667 (28)
5. ' =(a b d 1 : : :s p1 : : : pN p1 : : : pN e1 : : : eN ), b tm+l ] d d d Btli b 2 I i = 0 : : : s l = 1 : : : s t > 1:
. = 5 "3] , btlik 2 I k = 0 : : : dtli t > 1 l = 1 : : : s i = 0 : : : s: (29) / J = (j1 : : : js ) = 1 : : : s (30) ( juv = 1 u 6 v ; 1 (31) 0 u > v ; 1: (6), (27), (30) (31) , Ltl (z) = LtJl (z) l = 1 : : : s t > 1: (32) 9 B (8), (28) i (z), i = 0 : : : s, (z) Btli (z) = BtJl i (z) i = 0 : : : s l = 1 : : : s t > 1: (33) / = B 5 "3] dtl (28) " tm+l ], (29) B p I, p (d=b) 6 0,
tm+l ] tm+l ];k b p db Btli db > k=0min (b ) + > 0 (34) p :::dtli p tlik d i = 0 : : : s, l = 1 : : : s, t > 1. ? (31) j1l +: : :+jsl = l ; 1, l = 1 : : : s, (30), (33) 4 , , 1, ,
a, b, d, p1 : : : pN , p1 : : : pN , e1 : : : eN , d e (t + l ; 1) p Btli b ; m > ;1 = 1 : : : N: 9 B (t + l ; 1)e t + l ; 1 t + l e > e ; 1 > e ; 1 ; m m m m Z
Z
C
668
. .
,
b tm+l ] d p d d (35) Btli b > 0 = 1 : : : N = (a b d 1 : : : s p1 : : : pN p1 : : : pN e1 : : : eN ). / tm+l ] htli = d db Btli db i = 0 : : : s l = 1 : : : s t > 1 (34), (35) p , B p I p (htli) > 0. C , (htli) , htli 2 I. E . / l = l6max K ( ), l = 1 : : : s, Kl (i ) | , i6vl ;1 l i i , l : : : vl;1 , vl A l = l+1 = : : : = vl;1 vl 6= l : <
, Htl = 0max jh j t > 1 l = 1 : : : s 6i6s tli tm+l ] d X s Rtl = d db Ltl b = htli i db l = 1 : : : s t > 1 (36) i=0 1 + : : : + s = >l (H) = H ;s (ln H);s(1;l) (ln ln H)s(rl ;l ) l = 1 : : : s: s M Rtl , l;1 6= l ( 0 = s + 1). 6. ( Rtl, t > 1, l = 1 : : : s, : t 1) Htl t! jajj db j m1 tl (ln t)l ;1 = Fl (t), q > q0(a b d 1 : : : s 1 : : : s p1 : : : pN p1 : : : pN e1 : : : eN ) Htl Fl (t), , % , % q, a, b, d, 1 : : : s , p1 : : : pN , p1 : : : pN , e1 : : : eN 2) q "
m1 ;st Rtl tl;s;1;s(t!);s jaj
db
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= 9 "3]. / (36) B, "3], B, 8. R = cRtl = h0 0 ( db ) + : : : + hs s ( db ), hi 2 I, c 2 I , c 6= 0, jcj > > 0, % a, b, 1 : : : s , d, p1 : : : pN , p1 : : : pN , e1 : : : eN .
. / D(z) | Btli (z) Lt;1l (z) Lt;1l;1(z) : : : Lt;11(z) Lts(z) Lts;1(z) : : : Ltl (z) | htli Rt;1l Rt;1l;1 : : : Rt;11 Rts Rts;1 : : : Rtl: 9 "3] , D(z) = z t . 9 B Rtl , Z
670
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d Y l b t+mi 1 ] Y s b tm+i ] = b i=1 d i=l d d t Y l b t+mi 1 ] Y s b tm+i ] = 0 b (42) i=1 d i=l d 0 = 0 (a b 1 : : : s d p1 : : : pN p1 : : : pN e1 : : : eN ) 6= 0. E , l t + i ; 1 X s t + i X s t + i X + = = t: m i=1 i=0 m i=l m / (42) j j = j0j. 9 (c) , Rtl cRtl. ? 5 (c) | I (c) 6= 0, 1 6 j(c)j = jcjj j 6 jcj j0j. C , = = j 10 j .
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7] . I. Galochkin. On e@ective bounds for certain linear forms // New advances in Transcendence Theory / Edited by A. Baker. | Cambridge University Press, 1988. | P. 207{214. 8] D. 8. 6 . F 3 H = J {F? 3 3K // . 3 . | 1994. | %. 185, ( 10. | ). 39{72. 9] 8. L. 2. % = 2 = 3 H = . | .{L.: ?, 1934. & ' 1998 .
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Abstract V. Sh. Darsalia, On the bases of functional systems of polynomials, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 673{682.
The problem of existence and cardinality of bases of complete systems in functional systems of polynomials with natural, integer and rational coe2cients is being solved. We also consider the algorithmic variant of the basis problem.
N, Z Q | ( 0), X 2 fN Z Qg. PX ! ", # $ , %!! $ $ X . & !
X (# . ., . . . . ). U = fu1 u2 : : : um : : :g, (# um | $ X (m = 1 2 3 : : :), | . , # $ U # x y z t, # . , #- $ , , ., r # ! PX : (f )(x1 : : : xn) = f (x2 x3 : : : xn x1) (f )(x1 : : : xn) = f (x2 x1 x3 : : : xn) (.f )(x1 : : : xn;1) = f (x1 x1 x2 : : : xn;1) n > 1 (f ) = (f ) = (.f ) = f n = 1 (rf )(x1 x2 : : : xn xn+1) = f (x2 x3 : : : xn xn+1): , 2001, 7, 3 3, . 673{682. c 2001 , !" #$ %
674
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3 , f (x1 x2 : : : xn) g(xn+1 : : : xn+m ) PX , $ ( (f g)(x2 : : : xn xn+1 : : : xn+m ) = f (g(xn+1 : : : xn+m ) x2 : : : xn): & $ . 4( FX = (PX 5), (# 5 = f . r g, - ( ..)
X . !.. FX I , " # $ # M (M PX ) $ # I (M ), 6 ! ", $ M $ 6 ( $ $ " 5. 7 - M , I (M ) = M , , I (M ) = PX . 7 - M , PX , , # " f PX n M $ I (M ff g) = PX . 7 - M T , M T I (M ) = PX , $ M 0 M $ I (M 0) 6= PX . , , T = PX , M !.. FX . 8 f 2 PX - FX , ff g | $
FX . 9# ( , f (x1 : : : xn) 2 PX W (W X ), f (c1 : : : cn) 2 W $ # c1 : : : cn 2 W : PX W (W X ), # ! % " W . ; ( , W | $ # X , ! " PX , W , , ( PX . 8.. FX - , 6 $ $ # PX , - , 6 - $ $ # PX , $ # PX $ . 7< " # " # #6 $ . 1. , $ PX ? 2. 7" 6 $ PX . 8.. FN -$ #- ", $ , " - , - .$. ! " >2]. 8.. FZ -$ #- " >2], # , $ " : ( , $ # #6 # . 1. n N nf0g . . FZ ,
n. . @ # . 1. n = 1 $ # % ff g, (# f | $ ! , !.. FZ(6 " ! >2]).
675
2. n > 1 Mn = fp1 : : : pn;1 >(t1 ; p1 )2 + : : : + (tn;1 ; pn;1)2 + 1]2f g (# p1 : : : pn;1 | $ $ $ $ , f | $ ! . C f 2 I (Mn ), # , I (Mn ) = PZ . 3 , I (Mn n f>(t1 ; p1)2 + : : : + (tn;1 ; pn;1)2 + 1]2f g) = I (fp1 : : : pn;1g) 6= PZ pi 2= I (Mn nfpi g), 1 6 i 6 n ; 1 ( Mn nfpig .$. ! ", $ pi: # % ( # , ! >(t1 ; p1)2 + : : : + (tn;1 ; pn;1)2 + 1]2 # " fp1 : : : pi;1 pi+1 : : : pn;1g $ t1 : : : tn;1). D # , I (Mn n fpi g) 6= PZ . E, Mn # ! , $ $ # $ " FZ . F , Mn | !.. FZ . C # . G x ; y xy 21 31 : : : p1 : : : , (# p | $ $ , !.. FQ>2]: # , FQ| - -$ #- ! , $ % $ FQ - " ( , , 6 ). 7 $ FQ ? % $ #- #6 # . 2. . . FQ , . . q1 0, qi pi1;1 (i = 2 3 4 : : :), (# pi;1 (i ; 1)- $ $ # $ 2 3 5 7 11 :: :. M = fx ; y xy f1 (x) : : : fn(x) : : :g, (# fn (x) (n = 1 2 3 : ::) | .$. ! n-" $ (. . ! , # $ # " $ " x, $ ( n, %!! | ), $ 6 q1 q2 : : : qn q2 q3 : : : qn+1 . C .$. ! # >1]. C I (M ) # $ $ # x ; y xy 21 13 : : : 1p : : : , I (M ) = PQ. 3 $ , M 9. C FQ| -$ #- !.., 9 . , #6 . 1. 9 = fx ; y xy fi1 : : : fin : : :g, (# fi1 : : : in : : :g f1 : : : n : : :g. 7 (# ( , $ # # fx ; y xy fj1 : : : fjm : : :g, (# fj1 : : : jm : : :g fi1 : : : in : : :g, # $ $ # x ; y xy 1 1 1 . 2 3 : : : p : : : , # , $ " FQ 2. 9 = fx ; y fi1 : : : fin : : :g, (# fi1 : : : in : : :g f1 : : : n : : :g. E ( ! " 9 $ 6 $ " $ $
676
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$ ! xy: $ % ! g1 : : : gk . C (# , $ # # fx ; y g1 : : : gk fj1 : : : fjm : : :g, (# fj1 : : : jm : : :g fi1 : : : in : : :g, $ "
FQ, # $ $ # x ; y xy 21 31 : : : p1 : : : . 3. 9 = fxy fi1 : : : fin g, (# fi1 : : : in : : :g f1 : : : n : : :g. E ( ! " 9 $ 6 $ " $ $ $ ! x ; y: $ % ! h1 : : : hs. C (# , $ # # fxy h1 : : : hs : : : fj1 : : : fjm : : :g, (# fj1 : : : jm : : :g fi1 : : : in : : :g, $ "
FQ, # $ $ # x ; y xy 21 31 : : : p1 : : : . 4. 9 = ffi1 : : : fin : : :g, (# fi1 : : : in : : :g f1 : : : n : : :g. 7 (# 9 .$. ! ", 6 # " $ ": # , I (9) 6= PQ. , $ # $ $ # . C # . 3 -$ #- !.. $ (. . # !.. FN FZ ) ( " # , . . ( , , " .
3. . . FN , . . # $ $ M . C " !.. FN - , - .$. ! ", # , jM j > 3. , # . 1. jM j = 3: (# , $ M !.. FN. 2. jM j > 4: (# - % - % $ # M ( # , ). 3 # ( , $ !.. FN(% , !.. FN$ $ ( < >2]): # , M # . C # . 1. ..
f (x1 : : : xn) Mf = f>f 2(x1 : : : xn) + 1] (x ; y) x ; 1 x + y ;xyg . . FZ . . , #- : g1(x1 : : : xn x y) >f 2 (x1 : : : xn) + 1] (x ; y): g2(x) x ; 1: g3(x y) x + y: g4(x y) ;xy: H , g1(x1 : : : xn x x) = 0, g2 (0) = ;1, g2(;1) = ;2, g2 (;2) = ;3 : : :. , # .
677
1. f (0 : : : 0) = 0: (# , I (Mf ) # ! g5(x y) g1(0 : : : 0 x y) = x ; y, g6(y) g5(0 y) = ;y, g6 (;1) = 1 g7 (x y) g6(g4 (x y)) = xy, . . # $ # f1 x ; y xyg,
$ " >2]: # , Mf | $ . 2. f (0 : : : 0) 6= 0: (# k (f 2 (0 : : : 0) + 1) > 2: $ % ;k + 1 < 0, # , ;k + 1 2 I (Mf ). E g3(k ;k + 1) = 1: g7(x y z ) g4 (x g4(y z )) = xyz : g8(x y) g7(x y 1) = xy: g9(y) g7(1 y ;1) = ;y: g10(x y) g3 (x g9(y)) = x ; y: E, I (Mf ) # $ # f1 x ; y xyg, $ ": # , Mf | $ . ;
# . 2. .. M = fx ; 1 x + y ;xyg . . FZ . . ; ( , .$. ! x ; 1, x + y, ;xy $ W = f;1 ;2 ;3 : : :g, ( % ! " $ W . F , M W , # , I (M ) 6= PZ . ;
# . 3. ..
f (x1 : : : xn) M = f>f 2 (x1 : : : xn) + 1] (x ; y) x + y ;xyg . . FZ . . H , .$. ! >f 2 (x1 : : : xn) + 1] (x ; y), x + y, ;xy W = f0g, # , I (M ) 6= PZ . ;
# . D ( , ! If ! " # x y + cJ f = x + y + x f = x ; y + c f = ;x + y + c f = ;x ; y + c ! If ! " # x y + cJ f 6= x + y + c f 6= x ; y + c f 6= ;x + y + c f 6= ;x ; y + c: 4. ..
f (x1 : : : xn) M = f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 ;xyg . . FZ ! , f (x1 : : : xn) ! Z.
678
. .
. f (x1 : : : xn) Z, . . 6 " c1 : : : cn Z, f (c1 : : : cn) = 0: , #- : g1(x1 : : : xn x y) >f 2 (x1 : : : xn)+1] (x ; y) g2 (x) x ; 1 g3(x y) ;xy: H , g1 (x1 : : : xn x x) = 0 g2 (0) = ;1 g2 (;1) = ;2 g2 (;2) = ;3 : : :: 3 , # . 1. f (0 : : : 0) = 0. C (# , I (M ) # ! g4(x y) g1(0 : : : 0 x y) = x ; y g4 (0 ;1) = 1 g4 (0 y) = ;y g5 (x y) g3 (x g4(0 y)) = xy: E, I (M ) # $ # f1 x ; y xyg, $ ", # , I (M ) = PZ . 2. f (0 : : : 0) 6= 0. C (# k f 2 (0 : : : 0) + 1 | $ . E g2 (k) = k ; 1 g2(k ; 1) = k ; 2 : : : g2 (2) = 1: g6(x) g3 (x 1) = ;x: g7 (x y) g3 (g6(x) y) = xy: g6(;1) = 1 g6 (;2) = 2 g6 (;3) = 3 : : :: E, I (M ) Z, , c1 : : : cn 2 I (M ), $ % >f 2 (c1 : : : cn) + 1] (x ; y) = x ; y 2 I (M ): C , I (M ) # $ # f1 x ; y xyg, $ ", , M | $ . 3 . f (x1 : : : xn) Z, . . # ( a1 : : : an Z $ f (a1 : : : an) 6= 0, (# f 2 (a1 : : : an) + 1 > 2, $ % .$. ! g1 (x1 : : : xn x y) $ " Z( , $ 1). $ # $ # H1, H2 , H3,... ! " PZ . 9 #. H1 = M . E # " $ #. $ H1 : : : Hl , (# Hl+1 $ # $ $ " # g(h1 : : : hm ), (# g | ! M , h1 : : : hm | $ ! , ! Hl . # , 1 Hl = I (M ): l=1
679
D $ 6 " # $ l $ , Hl (l = 1 2 3 : ::) # .$. ! # x y + c, (# c | $ Z. 9 #. # , H1 # .$. !
# x y + c. E # " $ #. Hl # .$. !
# x y + c, (# # , Hl+1 # .$. ! # x y + c. 3 $ $ . Hl+1 # ! # x y + c, , Hl+1 # $ $ g(h1 : : : hm ), " " ! " # x y + c, (# g | ! M , h1 : : : hm | $ , ! Hl . , #6 . 1. g = >f 2 (x1 : : : xn)+1] (x ; y). C (# , m = n +2 g(h1 : : : hm ) = = >f 2 (h1 : : : hn) + 1] (hn+1 ; hn+2 ) $ " Z, # , " " ! # x y + c. $ . 2. g = x ; 1. C (# , m = 1 g(h1 : : : hm ) = h1 ; 1. D # , g(h1 : : : hm ) ! " # x y + c (#, (# h1 ! " # x y + c, . . (# Hl # ! # x y + c. 7 $ $ #$ # Hl # ! # x y + c. $ . 3. g = ;xy. C (# , m = 2 g(h1 : : : hm) = ;h1 h2. D # , g(h1 : : : hm ) ! " # x y + c (#, (# h1 | ! # x y + c, h2 = 1 h1 = 1, h2 | ! # x y + c, . . (# Hl # ! # x y + c. 7 $ $ #$ # Hl # ! # x y + c. $ . E, H1 H2 H3 : : : # ! # x y + c, # , I (M ) # " ! # x y + c, $ % I (M ) 6= PZ . ;
# . 5. ..
f (x1 : : : xn) M = f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 x + yg . . FZ . . @ ! " M = ff 2 (x1 : : : xn) t x ; y 1g $ , I (M ) I (M ). , #- g1(x1 : : : xn t) f 2 (x1 : : : xn) t g2 (x y) x ; y: E g3 (x1 : : : xn x y) g(x1 : : : xn g2(x y)) = >f 2(x1 : : : xn) + 1] (x ; y): g4 (x) g2(x 1) = x ; 1:
680
. .
E, M I (M ), $ % I (M ) I (M ). D # , # # $ M # $ $ M . $ # $ # H1 H2 : : : Hl : : : , % # #
4 ( # H1 = M = = ff 2 (x1 : : : xn) t x ; y 1g) $ 6 " # $ l # , Hl ! ", # # cxy - %!! (. . c = 2k, (# k | $ , 0). 9 #. ; " ! f (x1 : : : xn) c0 + c1 x1 + : : : + cnxn . C (# # , " ! f 2 (x1 : : : xn) c20 + 2c0c1x1 + : : : + 2c0cn xn , # , f 2 t # # cxy - %!! . 4 ! x ; y 1 6 # # cxy (. . # % ! " c = 0). # # , H1 ! ", # # cxy - %!! . E # " $ #. Hl ! ", # # cxy - %!! . C (# # , Hl+1 ! ", #
# cxy - %!! . @ $ $ $ g(h1 : : : hm ) Hl+1 , (# h1 : : : hm | ! Hl . , #6 . 1. g = f 2 (x1 : : : xn) t. C (# , m = n + 1 g(h1 : : : hm ) = = f 2 (h1 : : : hn) hn+1. # , f 2 (h1 : : : hn) # # cxy cx - %!! , $ $ #$ # ! hn+1 # # cxy - %!! . D # , ! f 2 (h1 : : : hn) hn+1, . . $ $ g(h1 : : : hm ), # # cxy - %!! . 2. g = x ; y. C (# , m = 2 g(h1 : : : hm ) = h1 ; h2 . C h1 h2 2 Hl , $ $ #$ # # # cxy - %!! . D # , ! h1 ; h2 , . . $ $ g(h1 : : : hm), # # cxy - %!! . 3. g = 1. C (# , m = 0 g(h1 : : : hm ) = 1. D # , # # cxy - %!! . E, Hl (l = 1 2 3 : : :) ! ", # 1 S # cxy - %!! . % Hl = I (M ) l=1 ! ", # # cxy - %!! . D # , I (M ) 6= PZ . ;
# . 6. ..
f (x1 : : : xn) Mf = f>f 2(x1 : : : xn) + 1] (x ; y) x ; 1 x + y ;xyg
681
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4 9 $ " FZ . # fx ; 1 xyg $ ", I (fx;1 xyg) I (fx;1 x+y ;xyg), fx;1 x+y ;xyg | $
2. # f>f 2 (x1 : : : xn)+1] (x;y) ;xyg $ ", I (f>f 2 (x1 : : : xn)+1] (x ; y) ;xyg) I (f>f 2 (x1 : : : xn)+1] (x ; y) x + y ;xyg), f>f 2 (x1 : : : xn)+1] (x ; y) x + y ;xyg | $
3. # f>f 2 (x1 : : : xn)+1] (x ; y) x ; 1g $ ", I (f>f 2 (x1 : : : xn)+1] (x ; y) x ; 1g) I (f>f 2 (x1 : : : xn)+1] (x ; y) x ; 1 x + yg), f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 x + yg | $
5. D # , # $ # $ " 9 $ " FZ . & , 9 | !.. FZ . 2. f (x1 : : : xn) Z. C (# $ # fx ; 1 x + y ;xyg Mf $ " (
2). # f>f 2 (x1 : : : xn)+ + 1] (x ; y) x + y ;xyg $ " (
3). # f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 ;xyg $ " (
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5). D # , # $ # $ " Mf $ " FZ . & , Mf | !.. FZ . ;
# . 4. . . FZ , . . 3 $ , 6 ( A, " " " $ " # . C (#, , A # Mf = f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 x + y ;xyg, (# f (x1 : : : xn) | $ .$. ! . , # . 1. 9 Mf f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 ;xyg. C (# f (x1 : : : xn) Z(
6). 2. 9 Mf Mf . C (# f (x1 : : : xn) Z(
6). D # , 6 ( , $ 6", $ .$. ! f (x1 : : : xn) Z, . . 6 ( # < $ ( # ! . & $ >3]. C # . , ( ( # # # 4C7 @8, $ ! ,. 9. M# $ # $ $ ## .
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Abstract E. E. Demidov, Schur pairs, non-commutative deformation of the Kadomtsev{Petviashvili hierarchy and skew dierential operators, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 683{698.
The concept of Schur pairs emerges naturally when the KP-hierarchy is treated geometrically as a dynamical system on an in;nite-dimensional Grassmann manifold. On the other hand, these pairs classify the commutative subalgebras of di<erential operators. Analyzing these interrelations one can obtain a solution of the classical Schottky problem or a version of the Burchnall{Chaundy{Krichever correspondence. The article is devoted to a non-commutative analogue of the Schur pairs. The author has introduced the KP-hierarchy with non-commutative time space (ti tj = q;1 tj ti ) and a non-commutative Grassmann manifold, which form ij
# / #==> 93-01-01542.
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a non-commutative formal dynamical system. The Schur pair (A F ) consists of a subalgebra A of pseudodi<erential operators with non-commutative coeDcients and a point F of G such that A stabilizes F . We obtain a transformation law for Schur pairs under non-commutative KP Eows. A way of constructing di<erential operators from a given Schur pair is presented. The commutative subalgebras of di<erential operators of a special type are classi;ed in terms of Schur pairs.
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K, (G + ti . , , K=(t1 t2 : : :) = k..x]] =: K. & k- $ S K, S(x) = x S(ti ) = qi ti: # ) X f = f0 x0 t 2 K 0
= (1 2 : : :) % % ,
( -
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06=0
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H
%
( % qi . 2
687
1.2.
8 G k- $$
@ K, @x = 1 @ti = 0
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( : 1 X n S n;k (@ k f)Dn;k n 2 Z: n D f= k=0 k & , ( VK = fW 2 EK j W = 1 + w1D;1 + : : :g ( +. # #& X
X+ X; $$
% B % C
.
+- ( ) $$
%
: ^EK = pi 2 K P = X piDi 8P 9N M C > 0: v(pi ) > Ci + M i > N i2Z D^K = fP 2 E^K j P; = 0g v % K, G v(x) = 0, v(ti ) = i.
1.3.
# ) W 2 VK D 7! WDW ;1 D -%+ W. . L = D+u1 D;1 +u2D;2 +: : : 2 EK W 2 VK , L = W DW ;1 . C 2 Vk . . , W D = LW -
W = 1 + w1D;1 + : : ::
688 4
. .
wi = Swi + @wi;1 +
8 ,
X ;k ;k;j j uk S (@ wi;1;k;j ): k>1 j >0 j
( )i
w1 = Sw1 ( )1 w2 = Sw2 + @w1 + u1 ( )2 ;1 w3 = Sw3 + @w2 + u1S w1 + u2: ( )3 4 ( )1 , w1 t: w1 = x w1. ) ( )2 tw ; S(t w ) = tu t( )2 2 2 1 x ( )2 0 = @(x w1) + x u1: 4 , )$$ w1 %+ ( )1 x( )2 . 8 ( t( )2 t- % w2 . L + +- . E ( )i;1 ( )i + % wi;1 %+ t- % wi . , ( )i+1 t w ; S(t w ) = @(t w ) + ( ) t( )i+1 i+1 i+1 i x x 0 = @( wi) + ( ) ( )i+1 B C
( )$$ uj $$
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( +- - . . X = adDd + ad;1Dd;1 + : : : Dl ,
! : ai 2 k. . <
,
( , ,
. 8 d + l d + l ; 1 .X Dl ] = 0, ad = S l (ad ) ad;1 = lS l;1 (@ad ) + S l (ad;1 ):
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, X
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689
1.4. " # $ . X = Dd + ad;1Dd;1 + : : : 2 EK ! Y = D + b0 + b1D;1 + : : : 2 EK , Y d = X .
. Y . ( Y1 = D, Y1d = X + O(d ; 1), O(k) % k. ( , (
Yk , Ykd = X + O(d ; k). 8%G Yk+1 = Yk + bD;k+1 , d = (S d;1 (b) + S d;2 (b) + : : : + b)Dd;k + O(d ; k ; 1) = Yk+1 = X + aDd;k + O(d ; k ; 1): < (d ; k)- : X X b0 (1 + q + q2 + : : : + q(d;1))x0 t = a0 x0 t:
0 % q
0
+ 1, )$$ b0 + d = X + O(d ; k ; 1). L
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. % X | ) . 8%G
W, L = WDd W ;1. &+
Y = W ;1XW ( Dd . N Z (Dd ) : Z (Dd ) = k((D;1 )). % D = L1=d , %
. 2 1.6. ()) *
& DK $$
%
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(
L = Dd + aDd;2 + : : :: (L) , ( , B
1, . . + % d
(
d. 4 % . 1.4 , L1=d = D + u1D;1 + u2D;2 + : : :: 8
W 2 VK , L1=d = W DW ;1 . H Y 2 B .Y L] = 0 G .W ;1Y W D] = W ;1 .Y W DW ;1]W = W ;1.Y L1=d ]W = 0
690
. .
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W ;1Y W ( Ek A = W ;1 BW % Ek . & , A
1. (
%, +
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WAW ;1 = B: 2
2. ! " # 2.1. + ,
7 G ( < < .13]. % R % % -
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( GL(R) N = = (nij )ij 6;1 ) R, + (1) (2). L ( ( +, +-+ Frame(R) ( : M 7! MN.
691
+ R-
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8 G R- % V = R((z)) $ % R , z % % +- ) R
. # ( = M(mod GL(R)) R- % U V , (G ) X ui = mji z j : j 2Z
2
. & , GL(R) ( R- U , %,
G 2 G(R). 4 , ( % G(R) B C . 4 , ( , R- % U V % G(R), . 7 %, R = k ) G(k) ) B$ %C (.,
, .10]). 2.3. ,
L % ) VK 8-
, , ( : X 7! (X~ij ) ER (
) R X Di X = X~ij Dj
G?(T), T | K, (G ti . j 2Z
%
G ER (. .3, . 2.7]). 8- , ( W 2 VK (W) 2 G(T) $
; (W) = W~ ;1x=0 i2Z j 6;1 (mod GL(T )): . & % : VK ! G?(T ) % . . .3, . 5.3]. 2 5 ( % (
B C. 4 , % V ; = z ;1R.z ;1] V % % z ;1 . H ; (W ) = W~ ;1x=0 V ; :
692
. .
3. # % 3.1. *
% F % G(R). & #& G(R) aF = a~F F | , +- F . . (A F ), - A #& )$$ R (. . A ER A E^R ) F G?(R), R- , A F, . . AF F . R- , A % F : A = AF =: fa 2 E^R j aF F g: #& )$$ .9]. 3.2. )) 0 ) *
, B ,! DK
A Ek
W 2 VK , B = WAW ;1 . T- +- . ( F = (W), A ( . E
( , AF F . . A
a 2 A. + WaW ;1 = b 2 DK aW ;1 = W ;1b:
) $ % (W), a~ (W) = (W)Yb Yb | B
%- C , i j-) j ; i > ord b. 7
,
~a(j- (W))
% K- (W). < %, aF F. 2 3.3. * )
#0 . , ))
% (A F ) | T - A ET . &
W 2 VK ;1 (F). H a 2 A
WaW ;1 ( DK .
693
. + aF F . 8 $ ) , - B
%- C Ca, +- (
) , ~a (W ) = (W )Ca : < %, (W~ jx=0)~a(W~ ;1 jx=0) = C0a : A
b = WaW ;1 ( b = b; + b+ . 7 %, b; = 0. % + $$
%
Bi > 0, j < 0C , + , (~b; jx=0)i>0 j<0 = 0. 8 ) , 1 ~bij = X i S i;k (b(k) j ;i+k ) k=0 k (~b; )0j (0) = bj (0) = 0 (1) (~b; )1j (0) = S(bj (0)) + b(1) j (0) = 0 bj (0) = 0 . . H , j < 0 k > 0 b(k) j (0) = 0, . . b; = 0. 2 H ( % ( % A E^T . , , $$
%
$ A. ), T- (A F) A, + $$
%
. 3.4. , )
4
( +- : B ,! DK 1, +- + (L), T - (A F ) k 6= A Ek %+ Vk F . 1.6, 3.2, 3.3. 2
4. ' ( { , % % .3].
694
. .
4.1. 1 ))
# ,,
U T % k- $ % +- t1 t2 : : : 1 2 : : :, +- +- : ti tj = qij;1 tj ti i j = ;cqij;1 j i i2 = 0 ti i = c i ti ti j = cqij;1 j ti i < j ti j = (c ; 1) i tj + qij;1 j ti i > j: , % c 2 k | $ , 1. & ( d: U ! U,
-
ti i i 0,
G ( !) $$
U. A K- EK U(EK ) = U T EK +- % : x i = qi;1 ix D i = qi i D ( , , S( i ) = qi i ). R
%, $$
d U ( U(EK ) dx = 0, dD = 0 R . 4.2. 2 {4#,
A EK - + 1-$ !D = E
X
i D i :
dW = ;(W!D W ;1 ); W
(7<8) ! {# . 8 .3] , (7<8) , !D2 = 0, . . qij = cqi;j qji i < j: #
, ) . H , $ + c q1 q2 : : :.
. ' ( {* $ $ + , . e. % W0 2 VK + W(t) 2 VK , W(0) = W0 . + $ $ W(t);1 Y (t) = E(t)W0;1
(SOL)
E(t) = expc
X i>1
ti D i
695
Y (t) 2 D^ (. e. %! ! % Y (0) = 1). P ' , expc (u) = un=.n]c! .n]c = (cn ; 1)=(c ; 1). K
n>0
4.3. 5 ) # ), ,,)
< <
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- , ( % (<8)- G? (k). 7 ( % % $ % (G? (k) (7<8)), , (7<8)
G? (k) G?(T).
5. * " '( -
% (A0 F0) | k- . 4 % < , G
W0 2 VK , (W0 ) = F0. , (7<8),
%
W0 . H ) + + F0 , Ft = (W(t)). A0 ? 4 - ( , At ( % E^T , +- Ft. . (A0 F0) | -, % E(t)A0E(t);1 E^T %$ Ft. . % a0 | ) A0, . . a~0F0 F0: ; 1 & E(t)a0E(t)
at. H
% % (SOL): W(t);1 Y (t) = E(t)W0;1: < %, at W(t);1 Y (t) = atE(t)W0;1 = E(t)a0W0;1 : V ; , ~ a~0 (W0 ) E(t) (W ~ a~t (W(t)) = E(t) 0 ) = (W(t)) a~tFt Ft: R . 2
696
. .
H
% At E(t)A0 E(t);1 . L % . 8 , A0 = AF0 , (AF0 )t AF AF0 E(t);1 AF E(t): & AF0 = E(t);1 AF E(t),
( . 8 At A0. 8 $ , A0
( Ek , At E^T . < %, (At Ft) G + D^K . 8 +
Dt;1 = E(t)D;1 E(t);1 : % X X E(t) = expc ti Di = pn(t)Dn t
t
t
i>1
n>0
( v) + pn(t) = tn + ( t
% X Dt;1 = D;1 + dnDn n>0
dn;1 = (pn ; S ;1 pn) + ( p
6. - A % - ( ( H. <. < . 4 .9] , { ( % < {8 % dW = ;(W!D W ;1 ); W ti i 7! 0, i > 4, +- ( )$$
W . Y ( % $$
% , +- $ + - .
697
4 , (
! = 1 D + 2 D2 + 3 D3 W = 1 + w1 D;1 + w2 D;2 + : : : W = W (x t1 t2 t3 )
dW = ;(W !W ;1 ); W: # +- % : dwi = 3.@ 3 wi + 3S(@ 2 wi+1) + 3S 2 (@wi+2 ) + S 3 (wi+3 )] + + 2 .@ 2wi + 2S(@wi+1 ) + S 2 (wi+2 )] + 1 .@wi + S(wi+1 )] + + 0 wi ; q1;i;1wi+1 1 ; q2;i;2wi+2 2 ; q3;i;3wi+3 3
2 = 2 + q3;1 w1 3 ; 3S 3 (w1) 1 = 3 (S 3 (u2 ) ; 3S 2 (@w1 )) ; q3;1w1 3S 2 (w1) + q3;2 w2 3 ; ; 2 S 2 (w1) + q2;1 w1 2 + 1 0 = 3 (S 3 (u3 ) ; 3S(@w1 ) + 3S 2 (u2 )) ; ; q3;2 w2 3 S(w1 ) + q3;1 w1 3 S 2 (u2) ; 2q3;1w1 3S(@w1 ) + + 2 (S 2 (u2) ; 2S(@w1 )) ; q2;1w1 2S(w1 ) + q1;1w1 1 + + q2;2 w2 2 + q3;3w3 3 ; 1 S(w1 ) u2 = w1S ;1 (w1) ; w2 u3 = w1S ;1 (w2) + w2S ;2 (w1) ; w1 S ;2 (@w1 ) ; w1S ;1 (w1 )S ;2 (w1 ) ; w3:
$$
% $ ( % - -%+
Sij : K ! K, G U X f j = i Sij (f) f 2 K (. .3]).
.
i>j
1] E. E. { . | .: - ! " # " " # $% ", 1995. 2] *. *. + { " % , %- " %"
% // /,%. % # . | 1995. | 0. 29, 1 2. | 2. 73{76.
698
. .
3] *. *. ! , % 6 { // 7 ! 0 . 2
. . . 08. -. 0. 28. 4] %6; 7. <. + , %- ; % - % , %- // /,%. % # . | 1978. | 0. 11, 1 1. | 2. 11{14. 5] 8 . . , %- ; -% %%- 66 %;%- // /,%. % # . | 1978. | 0. 12, 1 3. | 2. 20{31. 6] Burchnall J. L., Chaundy T. W. Commutative ordinary di>erential operators // Proc. Lond. Math. Soc. | 1923. | Vol. 21. | P. 420{440. 7] 6 . ? @-6,%. | .: , 1990. 8] Mulase M. Cohomological structure in soliton equations and Jacobian varieties // J. Di>. Geom. | 1984. | Vol. 19. | P. 403{430. 9] Mulase M. Solvability of the super KP equations and a generalizaton of the Birkho> decomposition // Invent. Math. | 1988. | Vol. 92, no. 1. | P. 1{46. 10] Mulase M. Normalization of the Krichever data // Contemp. Math. | 1992. | Vol. 136. | P. 297{304. 11] Mumford D. An algebraic-geometric construction of commuting operators and solutions to the Toda lattice equation, Korteweg{de Vries equation and related non-linear equations // Proc. Int. Symp. Alg. Geom. Kyoto, 1977. | Tokyo: Kinokuniya, 1978. | P. 115{153. 12] Schur I. UF ber vertauschbare lineare Di>erentialausdrFucke // Sitzungsber. Berliner Math. Gesel. | 1905. | B. 4. | S. 2{8. 13] Sato M., Sato Ya. Soliton equations as dynamical systems on inHnitedimensional Grassmann manifold // Lect. Notes Numer. Appl. Anal. | 1982. | Vol. 5. | P. 259{271. 14] Takasaki K. Geometry of universal Grassmann manifold from algebraic point of view // Reviews Math. Phys. | 1989. | Vol. 1, no. 1. | P. 1{46. ( ) 1997 .
. .
519.21
: , , .
! "# ! ! $ % " ! ! & ! &"# & $ $ ! .
Abstract A. Ya. Dorogovtsev, Periodic in the law solutions of the boundary value problem for heat equation, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 699{712.
We give the existence criterions of periodic in the law solutions of the boundary value problem for an abstract heat equation with a random periodic in the law disturbance and of the stochastic boundary value problem for such equation in a stripe.
1.
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" g C01, A C (R (B )) !
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:
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E sup kvk (t)k a6t6b
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k
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f
2
2
k=1
g
f
g
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Z 2 gk = g(x) sin kx dx k > 1:
0
< "
u(t x) :=
1 X
k=1
vk (t)gk sin kx (t x) Q: 2
B (10) , u Q 1. ( u 1. H !, " u
t, ! '
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1 1 X X u0t (t x) = vk0 (t)gk sin kx = (A(t) k2 I )vk (t) + 2 (t) gk sin kx = k=1 k=1 ;
= A(t)u(t x)
;
1 X
k=1
k2 vk (t)gk sin kx + (t)g(x)
1 X 00 uxx(t x) = vk (t)gk ( k2 ) sin kx k=1
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(6). D ! 2
(7). D 4 . . < ! : (1 0 00 i ut(t x) = uxx(t x) + A(t)u(t x) + g(x) (t) (t x) Q u(t 0) = u(t ) = 0 t R: ( W |
: ( 0 W (t) = iA(t)W (t) t R W (0) = I: & !
! ! , ! (W ( )) z C : z = 1 = ?. K
! A
(A) R = ?. 2 P
2
2
2
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2
j j
g
\
4. "
C
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F) ! G. ( B = H | )
w(t): t R | H - , E w(t) = 90 E w(t) w(s) 2 = t s tr W s t R f
2
k
g
;
k
j ;
j
f
g
706
. .
W . (' t := a(w(s) s 6 t), t R. 1 , (H )- h , '! t R
h(t) t- . : ' ! h ! w ) ) . E
' ! H - ! ! ! . 3. ( " g C01. H - " u = u(t x): (t x) Q
! (0 ut(t x) u00xx(t x) = A(t)u(t x) + g(x)w0 (t) (t x) Q (11) u(t 0) = u(t ) = 90 t R
" u ' t, " u, u0t , u00xx
Q 1 ' s < t
F
j
2
L
2
F
2
f
2
g
;
2
2
u(t x) u(s x) ;
Zt
;
Zt 00 uxx(r x) dr = A(r)u(r x) dr + g(x)(w(t) w(s)) (12) s ;
s
u(t 0) = u(t ) = 90: E ! . 5. " A C (R (H )) | $. &
. (i) ( w $ g C03 (12) t , sup E u(t x) 2 < + : 2
L
2
Q( )
k
k
1
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f
k
j >1 m=1
k
1
. B (ii) (i). ( w " g C03 . D! Z 1 X 2 g(x) = gk sin kx x #0 ]= gk = g(x) dx k > 1 k=1 2
2
0
#0 ].
707
( k N " . ('
Z
2
"k (n) := gk Uk ( )Uk;1 (r) dw(r + n ) n Z:
(13)
2
0
H
"(0)
Z
S" = jgkj2
0
Uk ( )Uk;1(r)WUk;1 (r)Uk ( ) dr:
( (ii)
, 3
vk ((n + 1) ) = Uk ( )vk (n ) + "k (n) n Z (14)
'
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vk (n ): n Z . E
2
f
2
g
f
2
g
Z
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2
0
vk (t): t R '
, '
1 #10,11,13,14]. / " Uk , ' s < t
f
2
g
Zt ; 1 vk (t) = Uk (t)Uk (s)vk (s) + Uk (t)Uk;1 (r)gk dw(r): s
(15)
( , vk (t): t R
f
vk (t) ; vk (s) + k2
Zt
2
g
Zt
vk (r) dr = A(r)vk (r) dr + gk (w(t) w(s)) ;
s
s
, ' ,
Zt
vk (t) vk (s) = (A(r) k2 I )vk (t) dr + gk (w(t) w(s)): ;
;
;
s
: , )! )
:= s = t0 < t1 < : : : < tn = t f
g
(t t)
:= 06max k6n;1 k+1 k
j j
;
(16)
708
. . nX ;1
vk (t) vk (s) = ;
nX ;1
=
j =0 j =0
!
;
(Uk (tj +1)Uk;1 (tj ) I )vk (tj ) + gk
tZj+1
;
nX ;1
=
(vk (tj +1 ) vk (tj )) =
j =0
(A(tj ) k2 )vk (tj ) + gk ;
J1 ( ) :=
nX ;1;
nX ;1 tZj+1 j =0 tj
tj
; 1 Uk (tj +1)Uk (r) dw(r) =
dw(r) + J1( ) + J2( )
Uk (tj +1 ) Uk (tj ) (A(tj ) k2)Uk (tj ) Uk (tj )vk (tj ) ;
;
;
j =0 nX ;1 tZj+1 J2 ( ) := gk (Uk (tj +1 )Uk;1(r) ; I ) dw(r): j =0 tj
( 0 J1( ) 0 1, J2( ) . D ) , 0 j
j !
k
vk (t) vk (s) ;
k !
Zt
!
k
0
(A(r) k2 I )vk (r) dr + gk (w(t) w(s)) ;
s
k !
j j !
;
. D
(16) i . h E
E sup vk (t) . : ! , 06t6
(5) 3 k
X +1
E kvk (0)k2 = tr
j =0
Ukj ( )S" Ukj ( )
= jgk j2 tr / , E vk (0) 2 6 gk 2 k
k
X +1
tr
j =0
6 gk 2 tr j
j
j
j
+X 1 j =0
k
=
Ukj +1( )
Z 0
Uk;1 (r)WUk;1(r) drUk(j +1)( ) :
Z 2 2 j j ; 2 k ; 2 k ( ; r ) ; 1 ; 1 j e U ( ) e U ( )U (r)WU (r)U ( ) drU ( ) 6
X +1 j =0
0
Z
e;2j U j ( ) U ( )U ;1(r)WU ;1 (r)U ( ) drU ( ) 6 L gk 2 j
0
j
709
! L < + (ii). D (15)
1
h
E sup kvk (t)k 06t6
i
Z 2 ; k ( t ; r ) ; 1 sup e U (t)U (r) dw(r) 6 L2 gk 06t6
6 L1 gk + gk E j
j
j
j
j
j
0
) !
! ! #10,15] k L2 . E "
u(t x) :=
1 X
k=1
vk (t) sin kx (t x) Q: 2
B
, , " u 1 Q
t. I ! Q " u0t (t x): (t x) Q , u00xx(t x): (t x) Q
f
f
2
u0x (t x) =
1 X
k=1
2
g
g
1 X vk (t)k cos kx u00xx(t x) = vk (t)k2 sin kx (t x) Q: ;
2
k=1
/
" (16)
)! x #0 ]
Zt s
A(r)u(r x) dr = =
1 X k=1
2
1 Zt X
A(r)vk (r) dr sin kx =
k=1 s
vk (t) ; vk (s) + k2
= u(t x) u(s x) ;
Zt
Zt
vk (r) dr gk (w(t) w(s)) sin kx = ;
s
;
u00xx(r x) dr g(x)(w(t) w(s)): ;
;
s
;
/ ,
(12). B (i) (ii). ( w, " g C03 , u |
t '
! (11),
w, g. : k N ' Z 2 vk (t) := u(t x) sin kx dx t R: 2
2
2
0
/ vk
' 2 H - , E sup vk (t) < + . /! (12) 06t6
(16), Z gk := 2 g(x) sin kx dx k > 1: k
0
k
1
710
. .
/ " Uk , k > 1,
(16) '
Zt
vk (t) vk (s) = (A(r) k2 I )Uk (r)Uk (r);1 vk (r) dr + gk (w(t) w(s)) ;
;
;
s
Zt
vk (t) vk (s) = Uk0 (r)Uk (r);1 vk (r) dr + gk (w(t) w(s)) ;
;
s
(17)
! s < t. B (17) , s < t
(15). : , '! )
:= s = t0 < t1 < : : : < tn = t := 06max (t t) k6n;1 k+1 k
f
g
j j
;
Uk;1 (t)vk (t) Uk;1(s)vk (s) = ;
= !
nX ;1 j =0
(Uk;1(tj +1 )vk (tj +1) Uk;1 (tj )vk (tj )) = J1 ( ) + J2 ( ) (18) ;
J1 ( ) := J2 ( ) :=
nX ;1
#Uk;1(tj +1 ) Uk;1(tj )]vk (tj +1 )
j =0 nX ;1 j =0
;
Uk;1 (tj )#vk (tj +1) vk (tj )]: ;
/ (17) ! J2( ) J3( ) + J4( ), !
J3 ( ) :=
nX ;1
Uk;1 (tj )
tZj+1
Uk0 (r)Uk;1(r)vk (r) dr
j =0 tj nX ;1 J4 ( ) := gk Uk;1 (tj )#w(tj +1 ) ; w(tj )]: j =0
( ,
J1( )
Zt
!
s
(Uk;1(r))0 vk (r) dr J3 ( )
1 ' ,
!
Zt !
s
Uk;1 (r)Uk0 (r)Uk;1(r)vk (r) dr
(19)
0. B
! !
711
J4 ( )
Zt
!
gk Uk;1(r) dw(r)
(20)
s
0. / (18){(20), (15). /! (15) vk ((n + 1) ) = Uk ( )vk (n ) + "(n) n Z !
!
2
Z
"(n) := gk Uk ( )Uk;1 (r) dw(r + n ) n Z: 2
0
C , ' k N
vk (n ): n Z
. : , ) ) !
k
(21)
, !
)
! , ' . ( 3 '!
! ) ej : j > 1 H
2
f
2
g
f
sup
1 X
j >1 m=1
g
Uk ( )m ej 2 < +
k
k
1
'! k N. / ,
(ii). D 5 . 2
#
1] O. Vejvoda et al. Partial Dierential Equations: Time-Periodic Solutions. | Noordho, 1981. 2] . . . !"#$ % & "%#$ '%( )$ $ & . | *.: +%, 1969. 3] .. /. 0 1 ! . 2 ! # 3# 4 #$ #$ $ $ $ . | 5 : 6( 7", 1992. 4] A. Ya. Dorogovtsev. Periodic processes: a survey of results // Theory of Stochastic Processes. | 1998. | Vol. 2 (18), no. 2{4. | P. 36{53. 5] *. >. 67, .. 6. ?% . * ' 1 $. | *.: +%, 1980. 6] A. V. Fursikov. Time-periodic statistical solution of the Navier{Stokes equations // Turbulence Modeling and Vortex Dynamics (Proceedings of a Workshop held at Istanbul, Turkey, 2{6 September, 1996). Lecture Notes in Physics. Vol. 491. | Springer, 1997. | P. 123{147. 7] A. Ya. Dorogovtsev. Stationary and periodic solutions of stochastic dierence and dierential equations in Banach space // New Trends in Probability and Statistics. Vol. 1. Proceedings of the Bakuriani Colloquium in Honor of Yu. V. Prohorov / Eds. V. V. Sasonov and T. Shervashidze. | Vilnius: Mokslas, 1991. | P. 375{390.
712
. .
8] A. Ya. Dorogovtsev. Necesary and suJcient conditions for existence of stationary and periodic solutions of a stochastic dierence equation in Hilbert space // Computes Math. Appl. | 1990. | Vol. 19, no. 1. | P. 31{37. 9] .. /. 0 1 ! . 2 7 ) K "Q!#$ !"#$ % , '%( #$ "%# & ! // . . 3% . | 1990. | R. 41, U 12. | X. 1642{1648. 10] R. F. Curtain and P. L. Falb. Stochastic dierential equation in Hilbert space // J. Dierential Equations. | 1971. | P. 412{430. 11] B. Goldys. On some regularity properties of solutions to stochastic evolution equations in Hilbert space // Colloquium Mathematicum. | 1990. | Vol. LVIII, no. 2. | P. 327{338. 12] .-X. ]. ]% # 4$ #$ & $. | *.: * , 1979. 13] P. Kotelenez. The H_older continuity of Hilbert space valued stochastic integrals with an applications to SPDE // Stochastic dierential systems. Lect. Notes Contr. Inf. Sci. | 1981. | Vol. 36. | P. 110{116. Rt 14] L. Tubaro. Regularity results of the process X (t) = U (t s)g(s) dW (s) // Rendiconti 0 del Sem. Matematico. | 1982. | Vol. 39. | P. 241{248. 15] P. Kotelenez. A submartingale type inequality with applications to stochastic evolution equations // Stochastics. | 1982. | Vol. 8. | P. 139{151. & ' 1998 .
. .
. . .
519.713
:
,
, .
! "!
#$!
! , % "! "! m- !' . ( %!
, $ % ! .
Abstract
A. S. Doumov, On the complexity of gure growing in homogeneous structures, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 713{720.
The paper deals with growing of some classes of .gures in a class of /at homogeneous structures with a cross-like neighbourhood pattern. We estimate the number of cell states that are necessary and su0cient for such growing.
1.
1], . ( ) S ! S = (Zk En V f), # Zk | % k- & , En = f0 1 : : : n ; 1g, V = fa0 : : : ah;1g | ( ( Zk, f | )& h , f : (En)h ! En . + S ! , k = 2. - % Zk + ( S. / % En + (. V , 0 , % ( a S V (a) = (a + a0 : : : a + ah;1 ), ! + ( a, / # ! ( a. )& f )& S. S + )&+ g, !+ % Zk +2+ ( En. 3 a 2 Zk g | S, ( g(a) ( a, g S. / g , 2001, 7, 1 3, . 713{720. c 2001 !, "# $% &
714
. .
S ! g0 , ( g ( 1. ( / g0 g ( A(g). 6& g ! % ( a, g(a) ( # + V (a) ( . 7 g ! 2 # #&. 8 % ; S + )&+ F S, # F(g1) = g2, g1 g2 2 ; % g2 (a) = f(g1 (a + a0 ) : : : g1(a + ah;1)). :& S ! g0 g1 : : :, gi+1 = F(gi), i = 0 1 : : :. g0 ( . gi # () i. 3 g1 g2 | S g2 = F t(g1 ), #, ( g2 2 g1 t, & 2. ;)#& S ! ! g, g(a) = 0 % ( a, , %, (# (, % % ( , ( #. 3 g(a) 6= 0 ( a, )#& & . 7 2 ( % ) T (S g) )#& g S i, g0 = g, g1 = F(g0), g2 = F(g1),.. . S % gi , # % )#& (, # i 2. 8 ! ( (a1 : : : am) ( Zk, ( kai+1 ; ai k = 1, 1 6 i < m. <+ (a1 : : : am ) ! , ! ka1 ; am k = 1. = ( k = 2 (a1 : : : am ) % Zk % Z1 Z2, 0 0+ . >% X = Z1 a1 : : : am )#, #( , + | #& )# X. :# X ! , 2+ a, b, c, a 2 X, b 2 X, c 2= X, +2 ( A, B, C % , # , (! C % % A B. = ( ! )# X . % )#&+ G )# X ( / G = X, +# a (a 2 X) ^ (G(a) = 1), (a 2= X) ^ (G(a) = 0). 8 ! + )# m-# , m ( )#&, +2 / )#, + . ? # , ( )# X S, 2 g0 g1 : : : gm : : : / , # g0 | & )#&, ( A(gm ) = A(gm+1 ) = : : : = X. =( m )# X. ? # , ( KX )# KS
715
) +, ( ( 0 , + )# X 2 KX 2 S 2 KS , / )# X . 8 ! 0 V , # a, ( kak 6 1, . 3 0 0 ) . 7 C(n) | n ( 0 ) . ( ( Vp )# p, ( Vpm | m-# p. = ( # p > p0. 7 L(x) | (p;P m)=2 )&, )& xx5 , F (p m) = C2mi Cpm;2i. A i = m= 2 +2 1. W = Vp W = Vpm ] C(n) , n > L(2p ) ; c
n > L(F (p m)) ; c] n 6 L(2p ) + c
n 6 L(F (p m)) + c]. X W c p. 7 ( % % , m-# (+ # , #( ( W = Vp . 3 2 )#, % ) 2+ . 7 KX = fX1 X2 : : :g | ( (! % )#, 2 # @, +2 ( )# Xi ! i. ( ( KX (m) m / KX . 2. KX (m), m > m0 , C(n) , n > L(m) n 6 L(m) + c.
2. 1
1. ! F(x) | # $ , F(x) > 1, F (1) = 1, ! L(F (x)) = f(x). % $
F (x) = F(x) logk F(x), k | ! , $ ! , , x > x0 # : f(x) ; c 6 L(F (x)) 6 f(x) + c, c | ! , k. . B% ( f(x)f (x)5 = F(x) #) / (, ! k, 5 f(x)f (x) (f(x)5 log f(x))k = F (x) logk F(x):
716
. .
( + ( /# ( A. 7 2 ( +2 &: (f(x) + c)(f (x)+c)5 > f(x)f (x)5 f(x)c > A (f(x) ; c)(f (x);c)5 6 f(x)f (x)5 f(x)c 6 A x > x0 f(x) ; c 6 L(A) = L(F (x)) 6 f(x) + c. < . A / #, ( logk F(x) + ,
2. ! F(x) | # $ , F(1) = 1, L(F(x)) = f(x). % $
F (x) = kF (x), k | ! , x > x0, # : f(x) ; c 6 6 L(F (x)) 6 f(x) + c, c | ! , k.
3. ' jVp j p, p > p0 , ! $ 2p =cp2. . 8 #( 2 ( , ( p & 8. jVp j #( ( jVp0 j )#, #& ( ( ( 1 p) ( p 1). 7 A = (1 p) B = (p 1). ;( jLAB j EF, +2 A B, ( ( & 2q, q = p=8 ; 1, ( ( & ( ( 0# # , & | ). 7 ) # p 2q e;2q p2q 2q (2q)! 2(2q) q jLAB j = C2q = (q!)2 p q ;q p 2 = p2 q : ( 2q e q) = (!+ ( ( EF, LAB ), ( ( + &: 22q 4 28q 2p pq = 2q2 > cp2 c = const : I L(D) ( D ( )# 2 % % 3]. 7 D &, (+ 3 1 2, ( n = L(2p =c0 p2) = = L(2p ) ; c . I% ( L(2p ) + c Vp )#. I /# (. m- 2. 7 ( M m-( ( k , , % dk;1 : : :d0 , # di 2 f0 : : : m ; 1g | &) m-( (. K# gM , ( gM (a) = 0 ( a , ( (0 1) : : : (0 k ; 1), 0
00
717
( a0 : : : ak;1 . ;% / k ( % f : : : m ; 1g, / # + &) m-( (, (! gM (a0 ) 0 ( M, gM (a1 ) | +2 . . ? # , ( ( )#& ( M. 7%, S % # & E 2F /# ( M, . . & 2 )#& gM , +2 ( M, )#& gM=2 , +2 ( M=2. 7 # ( /# & % ( (;1 k ; 1) D0 . :& f, ( # ! )#&+ gM )#&+ gM=2 % + +2 &. L +2 & l, r, d, u (+ ( a , , , . . l = g(a ; (1 0)), r = g(a + (1 0)), d = g(a ; (0 1)), u = g(a + (0 1)). 2# +2 & (+ , ( ( a # . 2 ( @ D0 D1 0 d = D0
@=2] d = D1
(m + @)=2] u | (! &) W0 u | (! &) W1 m + u (! W0 m + u (! W1 r = W0 D0 r = W1 D1
. 7 % ( S +( (N I), # N 2 f ! # "g, I 2 f ! # " g. ? # , ( ( a (N I) # # , N = , # | N = !, . . 8 ! % B ( ( b1 : : : bk (Ni Ii ), ( bi bi;1, 1 < i 6 k. ( /# % ( Zk # .) 7 B, B = b1 : : : bk, Ii ( bi b1 ( b0, b0 2= B. +2 & + + )&+ f S, ( bk ( bk+1 , +2 bk , (! ) /# , % % bk+1 0+ bk + ( 0) ( b0 % f ! # "g: ( b0
718
. .
0 , bk+1 bk , !, , . . 7#, ( 0 / ( bk+1 . 2 ( (Ni Ii ) 0 i=1 (N1 x) i>1 (Ni Ii;1) u = (N #) (" ) r = (N ) (! ) d = (N ") (# ) l = (N !) ( ) (x = g(b0 ), g(b0 ) | ( b0 , g(b0 ) 2 f ! # "g, ( x = 0.) . = ! + )#&+ V , + ( / A(V ) | )# p. = % ( )#& V 0 ( 0 & ( ! ( A. M#(, % ( & ( , % ( ( & % ( & ( , ( ( B, C, D. L, ( 2 ( ( A, B, C, D # . N ( A, B, C, D + #& )#& V ( % VAB , VBC , VCD , VDA . O, !, % VAB . ( # ( ( ai , 1 6 i 6 jVAB j: ( A | ( a1 , ++ A ( | ( a2 ,. .. , ( B | ( ajVAB j . >% VAB % ( jVAB j ; 1, i- # 0, ( ai+1 ai , 1, ai+1 ! % . M#( % % VBC , VCD , VDA . K , % )# p % (! hX a b ci, # X | ( p ; 4, a, b, c | & (, a 6 b 6 c 6 p ; 4, jVAB j, jVBC j, jVCD j . 7# +2 2 )#& V . 7 )#& gM ( M, ( # X (! hX a b ci, +2 )#&+ V . = & )& /# ( 2, %# ( R0 , , % (! D0 D1 #, ( ( M (. 7 ( R1, +2 ( R0, % f ! # "g +2
719
)& f(r u), # r u (+ # # . r D0 D1 D0 D1 D0 D1 D0 D1 u L1 L1 L2 L2 L3 L3 L4 L4 f(r u) ! # # " " ! I, ( R1 E(F , ## & ( R1 , / 0. ( ( R2 # ( R1. 7 EF R2 % G, ( (!( . B (!( 3]. 7& )& , ( (+ & )#& % . 7 ( R2 t L1 , ( R3 2 G, +2 ( R2, L1, ! t1 = 4jLAB j. 7 t + t1 + 1 ( R2 ! L2 , ( R3 | L2, ! ( t2 = 4jLBC j. A . . ;/))& 4 & | ( ( R0 D0 D1 . = # ( R1 ( (, ( & ) #& (. 7 /# % E# F ( (( (!( . .) E F . 7 E F & ( #& , ( #. I 2 / & ( # )# ( . !"" # ". 8 & ( , ( ) )#& gM , +2 ( M, ( # #& )#, ( , ( #, ( % ! & )#& % ( G 4jLAB j, 4jLBC j 4jLCD j . O0 ( % 3], # , ( 2 )#& gM , M 6 D, & )#&, ( n = L(D)+c1 (. = 0 ( D = 2p . I 2! ) ( c2 , % # 2 S & )#& % , 2 p. (= / c2 2 , #( 3], % ( + % & )#&, 2 mm4 , # m = L(2p ), mm4 > p.) + ( n = L(2p ) + c (. K .
720
. .
3. 2
8 L(m) 2 % 3]. I% ( L(m) + c . 2 #, +2# i )# Xi , ( 2 0 K +#, , ( i, ( ( ( 0#, (, +2 )# Xi , . K 0 2] % 0 , # ( # . Q )& , E F ( )# Xi / )#, # R!, ! 2 (. K , ! & ( , ( #, ( & )#& g0 )#&+, +2+ i. - 3] L(i) + c . K .
1] . ., . ., . . . | .: ! , 1985. 2] . ., . ., . . &' ( () '. | .: ! , 1990. 3] , . . & '- ' ./ 0 () ' ) // 3. 4. . | 2000. | 5. 6, (4. 1. | . 133{142.
' ( ( 1996 .
. .
519.21
: , ,
, !.
"
! #! $ %
$ & $ $$ . ' ( ! ) ! $ ! $ , * % $ $ $ ! % & ) & & %! $ .
Abstract
S. V. Ekisheva, Limit theorems for sample quantiles of associated random sequences, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 721{734.
The Bahadur representation of the empirical distribution function for associated strictly stationary random sequence is considered. It is used for proving asymptotic normality of sample quantile, the functional central theorem and the functional law of the iterated logarithm for sample quantile.
.
!
, , . # $ %4] , (
, )
. $ m- ,
(%12]), (%8]), - / (%13]) c - / (%14]). , 2001, ! 7, 2 3, . 721{734. c 2001 , ! "# $
722
. .
1.
fXj gj 2N| , (3 F P). 4 , = (1 : : : m ) , $ ) f g: Rm ! R, E f()g(), E f(), E g() ,
cov(f() g()) > 0: $ , $ . F(x) | ) Xj ,
, f(x) | . $ 0 6 Xj 6 1, j 2 N. 8 0 < p < 1 9p p- F (x), 9p = inf fx 2 %0 1]: F (x) = pg: 8 n Xn1 6 Xn2 6 : : : 6 6 Xn1, (X1 X2 : : : Xn). : p- Znp = Zn : Zn = Xnr , r = %np]+1. #( : Jn = t 2 %0 1]: 9p ; n; 125 6 t 6 9p + n; 125 n 2 N n X Fn(x) = n1 I fXi 6 xg | ! ) i=1 p2 = cov(I fX1 6 9p g I fX1 6 9p g) + 2
1 X
k=2
cov(I fX1 6 9p g I fXk 6 9p g)
Yn (t) = n 21 (Fn (t) ; F(t)) t 2 %0 1]: < , , p2 = nlim (n VarFn (9p )). !1
1 ( ). fXj gj 2N|
, F(x) |
f(x), 0 6 Xj 6 1, j 2 N. p 2 (0 1) p2 > 0 (1) 1 X n7 cov(X1 Xn) < + n=1
1:
n supfj(Fn(t) ; F (t)) ; (Fn (9p ) ; p)j : t 2 Jn g 6 Cn; 85 lnn:
(2) (3)
723
= C, Ci $ . 2. ! 1 , f(x) !
9p , 0 < f 0 (x) < M < +1 " . n 1 n 2 (Zn ; 9p )f(9p ) + n 12 (Fn(9p ) ; p) 6 Cn; 18 ln n (4)
n ! 1 n 12 f(9p ) (Z ; 9 ) !d N(0 1): (5) n p p
? C%0 1] ) %0 1]
(X Y ) = sup jX(t) ; Y (t)j t2 01]
X Y
2 C%0 1]. : fWn(t) t 2 %0 1]g: Wn (0) = 0 Wn nk = kf(9p)(Zpkn; 9p ) k = 1 : : : n p k + 1 k Wn (t) = Wn n + (nt ; k) Wn n ; Wn nk k + 1 k t 2 n n k = 0 : : : n ; 1:
3 ( !). # ! 2 , n ! +1
fWn(t) t 2 %0 1]g ! ! $ C%0 1]. # D%0 1]
(%nt] + 1)f(9p )(Z nt]+1 ; 9p ) p Hn(t) = t 2 %0 1] n > 3 p 2n lnlnn / @
K = x(t) t 2 %0 1]: x(t) =
Zt 0
h(z) dz
Z1 0
(h(z))2 dz = 1 :
4 ( ! # $ $
!). %$ fHn n > 3g D%0 1] K .
724
. .
2.
% 1 (&15], * (22.15) &1]). fXj gj 2N|
, & ! ! (3 F P), F (x) | ' , F (x) | . $ 0 6 Xj 6 1, j 2 N, p2 > 0 ' $ , 1 X n 132 + cov(X1 Xn ) < +1: n=1
j
E Yn(t)
; Yn(s)j4 6 C ;n; ; + jt ; sj 1 2
3
6 5
s, t & & %0 1]. % 2 (&1]). fXj gj2N !! 2. s 2 (0 1) sup jYn (t) ; Yn(s)j 6 3 1max jY (s + iq) ; Yn(s)j + qn 12 6i6m n s6t6s+mq
m 2 N, 0 < s + mq < 1. # %1]
- / . A
. % 3 (&9, 1]). d 2 N & fn n 2 Zd+g, ( ! ! !' !! > 1. $ ' > 1 fun > 0 n 2 Zd+g, ! R
& Zd+ R = R(b1 : : : bdB m1 : : : md ) = = fn = (n1 : : : nd ) 2 Zd+ : bj < nj 6 bj + mj 8j j = 1 : : : dg b1X X bdX +md +m1 E ::: n1 :::nd 6 un : n1 =b1 +1 nd =bd +1 n2R d R & Z+ bX bdX +pd 1+p1 6 E max : : : max : : : n 1 :::nd 16p1 6m1 16pd 6md n =b +1 n =b +1 1 1 d d X d 5 (1 ; ) = ; d ) un : 6 2 (1 2 n2R
;
725
% 4 ( ! ! 1 &7], *-*! 3.1 &6]).
X , Y |
,
h: R ! R, g: R ! R ! ! & , (! " & &, !, !$ , . j cov(h(X) g(Y ))j 6 M1M2 cov(X Y ) + ; @h M1 = max sup @x sup @h x2R x 2R @x ; @g+ M2 = max sup @x sup @g @x : x2R
x2R
% 5 (# 3 &3], * &5]). fj gj 2N|
, E j = 0, j 2 N, ' r > 2, > 0, > 0, ' : sup E jk jr+ < +1 u(n) = sup
k2
N
k2 m : jm;kj>n
NX
mX +n
sup E m2N0
k=m+1
cov(k m ) = O(n; ): r
k = O(n (r ) )
(
+ ; 2);1 0 6 < 0 (r ) = r ; (1 + )(r r > 0 2
0 = (r+2)(r;2) .
3. " 3.1. 1
8 $ n
An = !: sup j(Fn(t) ; F(t)) ; (Fn(9p ) ; p)j > 4n; 58 ln n = t2Jn
= !: sup jYn(t) ; Yn (9p )j > 4n; 18 ln n : t2Jn
726
. .
#( fnkgk2N, nk = exp k. =) k 2 N 5 5 8 , m = %n 24 ] + 1. < 2 q , m $ qk = nk;+1 k k k k+1 : P
An 6 P n 6max sup jYn (t) ; Yn(9p )j > 4nk;+18 ln nk 6 k n
1
1
jY (t) ; Yn(9p )j > 4n;k+18 lnnk + 6 P nk 6max sup n
;8 + P n 6max sup j Y (t) ; Y (9 ) j > 4n n n p k+1 lnnk 6 n
6 P nk 6max max jY (9 + lqk ) ; Yn (9p )j > nk;+18 lnnk + n
n ln n : n p k n p k k+1 k n
(6)
k
: 0 6 s < t 6 1, i 2 N, j = 1 : : : mk , i (s t) = (I fXi 6 tg ; F(t)) ; (I fXi 6 sg ; F(s))B ij = i (9p + (j ; 1)qk 9p + jqk ) ij = i (9p ; jqk 9p ; (j ; 1)qk ) j > mk ij = 0, ij = 0. < n 2 N, l = 1 : : : mk
/ : X X l n n X 1 1 jYn(9p + lqk) ; Yn(9p)j = pn i(9p 9p + lqk) = pn ij i=1 i=1 j =1
n n l jYn(9p ; lqk) ; Yn(9p)j = p1n X i(9p ; jqk 9p) = p1n X X ij : i=1 i=1 j =1
$ (6), P
X l n X
3
An 6 P n 6max max ij > nk8+1 ln nk + k n
3
+ P n 6max max ij > nk8+1 lnnk : k n
: 3. ? R = = R(i1 j1B i2 j2), nk 6 i2 ; i1 < nk+1 . D j1 > mk , E
X i2
j2 X
i=i1 +1 j =j1 +1
ij
4
= 0:
727
D j2 6 mk ,
fXj g 1 ( = 21 )
X 4 iX 4 j2 j2 i2 2 ;i1 X X E ij = E ij = i=i1 +1 j =j1 +1 i=1 j =j1+1 iX 4 j2 j1 iX 2 ;i1 X 2 ;i1 X =E ij ij = i=1 j =1 i=1 j =1 2 = (i2 i1 ) E Yi2;i1 (9p + j2qk ) Yi2 ;i1 (9p + j1 qk ) 4 6 3 ; 6 C(i2 i1 )2 (i2 i1); 32 + (j2 j1 ) 65 nk;+14 6 2C(i2 i1 ) 43 (j2 X 65 ; 65 5 10 uij = (2C) 6 (i2 i1 ) 9 (j2 j1 ) 6 ij2R
;
;
j
;
;
;
; ;
;
j
;
; j1 )
6 5
=
uij = (2C) 65 i 19 . 4 , R c j1 6 mk , j2 > mk ( $ $ X i2
E
j2 X
i=i1 +1 j =j1+1
ij
4
X i2
=E
mk X
i=i1 +1 j =j1 +1
6 P
4
ij
4
6
X i2 X mk
i=i1 j =j1
uij
65
6
X
56
ij2R
uij :
E E ij . < , 3 ij2R P
An 6
E
max n 6n
k+1
P 4 n P l ij k j =1 i=1
max 16l6m
+
3 (ln nk )4 nk2+1 nk 6n
6
! C1, C2 k. : ,
1 1 X X
An 6 C2 k;4 < + P k=1 k=1 nk 6n
1:
< , {G , / n 1 (3). < 1 .
728
. .
3.2. 2
H $ (4). 8 ! Bn , n 2 N: Bn = fZn 6 9p ; n; 21 ln ng = = f r Xi i = 1 : : : n 9p ; n; 12 lnng = =
X n
I fXj 6 9p ; n; 21 ln ng > r =
j =1 X 1 n
= n (I fXj 6 9p ; n; 12 ln ng; F (9p ; n; 12 ln n)) > nr ; F(9p ; n; 21 ln n) : j =1 =) n ; j = I Xj 6 9p ; n; 12 ln n ; F 9p ; n; 12 ln n : = , f;j g | . I , f;j g 5. 8 , E(;j ) = 0 8j 2 N, sup E j ; j jr+ < +1 8r > 0
N
j2
j; j j 6 2. 1 P , cov(1 k) $ 1 k=1 P C3 cov 31 (X1 Xm ). : $ k 2 N ) hk (x) m=1 : 8 > 1 x 6 9p ; n; 12 ln n > < 0 x > 9p ; n; 12 ln n + ak hk (x) = > 1 ; > :1 ; x;$p +n 2 ln n 9p ; n; 21 ln n < x 6 9p ; n; 12 ln n + ak ak $ fak g . < 4 k / n ( f(x)
; cov I
X1 6 9p ; n; 21 ln n I Xk 6 9p ; n; 12 ln n 6 6 j cov(hk (X1 ) hk (Xk ))j + 3Mak 6 a;k 2 cov(X1 Xn) + 3Mak : (7) D cov(X1 Xk ) = 0, cov(1 k ) = 0, (7) 1
ak > 0. D cov(X1 Xk ) 6= 0, ak = cov 3 (X1 Xk ) (7) k j cov(1 k)j 6 C3 cov 13 (X1 Xk ):
729
< , K( (2)
m2N m X
k=1
cov(1 k) 6 C3
m X
k=1
k 73 k; 37 cov 13 (X1 Xk ) 6
6 C3
X m
k=1
k7 cov(X1 Xk )
13 X m
k=1
k; 27
23
6 C4 :
1 P
H , cov(1 k ) . E , k=1 1 X
k=m
cov(1 k ) 6 C3
6 C3
X 1
1 X
k=m
cov 13 (X1 Xk ) 6
k7 cov(X1 Xk )
31 X 1
k=m 5, = 1. 2 2
k=m
k; 72
23
6 (C5 m; 25 ) 23 6 C6m; 23 : (8)
# r = < (8) ( 5 c 0 = 32 . L , $ $ ,
mX 5 +n 2 sup E k m>0 k=m+1
6 7n 54 :
(9)
= , ,
r ; 21 n ; F (9p ; n lnn) = = F (9p ) ; F(9p ; n; 21 ln n) = f(9p )n; 12 lnn(1 + o(1)): (10) $ , / n Zn > 9p ; n; 12 ln n: (11) = , (10) Bn $ : Bn = Zn 6 9p ; n; 21 ln n =
X n
i=1
p
i > n f(9p ) n lnn
n ! 1 n ! 1. : = n> inf1fn g, ! (10) f(x) > 0. = , (9)
3 fi i 2 Ng d = 1, = 25 , = 45 , ui = (C7) 45 , i 2 N. fnk g $, 1. < 3 (10)
730 P
. .
p
n X
Bn 6 P n 6max i > f(9p ) nk ln nk 6 k n
6 C8nk4+1 (f(9p ) ln nk ); 52 n;k 4 6 C9k; 25 5
5
! C9 k. :
1 1 X
X P Bn 6 C9 k; 52 k=1 nk 6n
< +1
{G (11). E , / n
Zn 6 9p + n; 12 ln n: (12) < , (11) (12) / n 1 jZn ; 9p j < n; 21 lnn: (13) = ,
Fn(Zn ) = nr = p + O(n;1)
9p (13), / n jFn(Zn) ; F (Zn) + f(9p )(Zn ; 9p)j = = p + O(n;1) ; p ; f(9p )(Zn ; 9p ) ; 12 f 0 (Mn )(Zn ; 9p )2 + + f(9p )(Zn ; 9p ) 6 C10n;1 ln2 n (14) Mn | , 9p < Mn < Zn 9p > Mn > Zn . L (14) 1 ,
1 n 2 f(9p )(Zn ; 9p ) + n 21 (Fn (9p ) ; p) 6 n 12 f(9p )(Zn ; 9p ) + + n 21 (Fn(Zn ) ; F(Zn )) + n 12 (Fn (9p ) ; p) ; n 12 (Fn (Zn ) ; F(Zn)) 6 6 Cn; 18 ln n (15) / n, Zn 2 Jn / n . A (4). 8 (5) , (15) 1 n 2 f(9p )(Zn ; 9p ) ; n 12 (p ; Fn (9p )) 6 Cn; 18 ln n (16) / n, p ; Fn(9p ) n
731
p ; I fXi 6 9p g,
(%10]). H 1 X
n=1
cov(I fX1 6 9p g I fXn 6 9p g) 1 P
$, cov(1 k) k=1 1. = , n ! 1 n 21 (p ; Fn(9p )) !d N(0 1): p L1 (16) , n 2 f(9p )(Zn ; 9p ) $ ,
n 21 f(9p ) (Z ; 9 ) ! d N(0 1) n p p n ! 1. A 2. 3.3. 3
? ( fWn (t) t 2 %0 1] n 2 Ng, ( : Wn (0) = 0 Wn nk = k(p ; Fpkn(9p )) k = 1 : : : n p k k + 1 Wn (t) = Wn n + (nt ; k) Wn n ; Wn nk k k + 1 t 2 n n k = 0 : : : n ; 1: 8 fWn (t)g )
%11], p ; I fXi 6 9p g , , p2 < +1. = , n ! 1 C%0 1]: Wn (t) !d W (17) W | $. 8 $ , n ! 1 P
(Wn Wn) ! 0: (18) 8 ! fkng, kn 2 N, , 1 ; 2 kn ! +1, knn ! 0 n ! 1. L
(Wn Wn) 6 supk jWn(t)j + supk jWn(t)j + k sup jWn(t) ; Wn (t)j: 06t6 nn
06t6 nn
n n
6t61
732
. .
!
kf(9p )(Zk ; 9p ) 6 p sup jWn (t)j = 16max k6kn p n 06t6 knn p ) max jZ ; 9 j 6 2knf(9 p) ! 0 p p 6 knf(9 k p 1 6 k 6 k n n n p p n ! 1. 8, (17) , fWn (t)g C%0 1] , , n ! 1 P supk jWn (t)j ! 0: n
06t6 n
4 , fWn (t)g fWn(t)g (4)
sup jWn (t) ; Wn (t)j = 6t61 kf(9p )(Zk ; 9p ) k(Fk (9p ) ; p) ; 81 ln n p p + 6 C n = k max 12 6k61 n n
kn n
n
p
p
/ n . < (18) , fWn(t)g fWn (t)g . A . 3.4. 4
? : (%nt] + 1)(p ; F nt]+1(9p)) t 2 %0 1] n > 3: p Hn (t) = p 2n lnlnn 8
p ; I fXi 6 9g ) ) %2], , , E jp ; I fXi 6 9gj3 < 1 !
!)) G {K (8)
/ u(n) 6 C13n; 23 : = , $ fHn n > 3g D%0 1]
$ K. $ , n ! 1
(Hn Hn ) ! 0: (19) 8 , (16) , C14 < +1, 1 k 2 N jpkf(9p )(Zk ; 9p) ; pk(p ; Fk(9p ))j 6 C14:
733
<
(Hn Hn ) = sup jHn(t) ; Hn (t)j 6 06t61 p 1 6q 2 sup %nt] + 1f(9p )(Z nt]+1 ; 9p ) ; 2p ln ln n 06t61 ; p%nt] + 1(p ; F nt]+1(9p )) 6 p p 6q 1 sup kf(9p )(Zk ; 9p ) ; k(p ; Fk (9p )) 6 C15(lnln n); 21 2p2 ln ln n k2N (19). < , fHn n > 3g fHn n > 3g
D%0 1] $ . < . H $ ) NKI. E $ $ ) E. #. .
#
1] . . | M.: , 1977. 2] " #. $. % &" ' ( ) & +" (" // % . ( . . | 1995. | T. 1, . 3. | . 623{639. 3] " #. $. & + // 2 " (. | 1993. | 2. 38, (. 2. | . 417{425. 4] Bahadur R. R. A note on quantiles in the large samples // Ann. Math. Statist. | 1966. | Vol. 37. | P. 577{580. 5] Birkel T. Moment bounds for associated sequences // Ann. Probab. | 1988. | Vol. 16, no. 3. | P. 1184{1193. 6] Birkel T. On the convergence rate in the central limit theorem for associated processes // Ann. Probab. | 1988. | Vol. 16, no. 4. | P. 1685{1698. 7] Bulinski A. V. On the convergence rates in the CLT for positively and negatively dependent random 6elds // Probability Theory and Mathematical statistics / Eds. I. A. Ibragimov and A. Yu. Zaitsev. | Gordon and Breach Publishers, 1996. | P. 3{15. 8] Dutta K., Sen P. K. On the Bahadur prepresentation of sample quantiles in some stationary multivariate autoregressive processes // J. Multivariate Analysis. | 1971. | Vol. 1. | P. 186{198. 9] Moricz F. A general moment inequality for the maximum of the retrangular partial sums of multiple series // Acta Math. Hung. | 1983. | Vol. 41. | P. 337{346. 10] Newman C. M. Normal 7uctuations and the FKG-inequalities // Commun. Math. Phys. | 1980. | Vol. 74. | P. 119{128.
734
. .
11] Newman C. M., Wright A. L. An invariance principle for certain dependent sequences // Ann. Probab. | 1981. | Vol. 9. | P. 671{675. 12] Sen P. K. Asymptotic normality of sample quantiles for m-dependent processes // Ann. Math. Statist. | 1968. | Vol. 39. | P. 1724{1730. 13] Sen R. K. On the Bahadur representation of sample quantiles for sequences of -mixing random variables // J. Multivariate Anal. | 1972. | Vol. 2. | P. 77{95. 14] Yoshihara K.-I. The Bahadur representation of sample quantiles for sequences of strongly mixing random variables // Statistics and Probability Letters. | 1995. | Vol. 24. | P. 299{304. 15] Yu H. A Glivenko{Cantelli lemma and weak convergence for empirical processes of associated seduences // Probab. Theory Relat. Fields. | 1993. | Vol. 95. | P. 357{370. % 1997 .
. .
e-mail: [email protected]
517.9+62.50
: , !"#$% & '!"(#, ) *" &, $ (.
+ ", -' . / ) &( ! ' ! , -' .) ( ), , ,, , ! -& ( !". # " &") ! /&' # *"/ !" ". 0&(( # * " ( ) ( , &( ) # ), & # '!"(#.
Abstract Yu. V. Zaika, Integral observation operators of nonlinear dynamical systems, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 735{760.
In terms of functionaldependence we obtain a descriptionof observable functions in nonlinear dynamical systems, which are analytical by phase variables. For measurements processing the integral operators are used. An analog of duality theory known for linear problems of observation and control is developed.
x
1.
, . ! " . # $ . # %. & '. '. & (1]. + . # $ " " . , - , 2001, 7, 5 3, . 735{760. c 2001 , ! "# $
736
. .
$ - . - . % U Rn x_ = f(x) y = g(x) f : U ! Rn g: U ! Rm (1.1) $ "
. #- " f, g , | $ U. 7 (0 T] UT = fx(T)g U, 9 " (1.1) x(: x T ) (x(T : x T) = x 2 UT ) (0 T ]. 7 y(: x T ) = g(x(: x T )): (0 T] ! Rm x = x(T ). % ; - < . 7 y(: x T ) , (0 T] - " y() x T . , 9 . " " y() $ x(T) UT . = y(t), t 2 (0 T ], T < T, . > x0 = x(0) 2 U0 , " . ? . # x(T) ( x(0)) 9 . , " $ $ 9 $ m = 1. ' 9 $ . # Y. Inoye (2] , (1.1) ( (f g)) x(T ) y(i) (t ), t 2 (0 T]. ? 9 . ;# 9< n- fy(: x T ) j x 2 UT g . ? - &. =. , (. (4,5] ). &. #. & (5] , " $ (f g) " x(T ) y() 2n+1 y(tj ). = y(t) $ , " y(t). ? $ '. '. & (1]. f = Fx, g = Gx, F, G |
737
" n n, m n. = - V_ (t) = ;F 0V (t) + G0k(t) V (0) = 0 (1.2) k() V (T ) = h, y() " x(T) h: h0x(T) = hk yiL2 8x(T ) 2 Rn. , h 2 Rn, y() h0x(T), DT = fV (T )g. # ( " ), (f g) = (F G) , - (1.2) (DT = Rn). D '. =. & (6, 7] $- . y() " ': UT ! R1
ZT
'(x(T )) = k( y()) d 8x(T ) 2 UT
0
(1.3)
: - @v (t x) + @v (t x) f(x) = k(t g(x)) v(0 x) = 0 (1.4) @t @x k( ) v(T x) = '(x), x 2 UT . ' 9 . # (f = Fx, g = Gx, k = k0 (t)y) v(t x) = V 0 (t)x, V (t) (1.2). # , " $ - . # , (1.4) k, v 9 . % $ .
x
2.
? (1.3): '(x(T)) = hk yiL2 , L2 = L2 ((0 T ] R1), m = 1. ! , " k() y() (0 T]. E " y() 7! hk yi ( ) hk yi " . 7 . , y(). > k1() : : : kp(), " y() " Ji (y()) = hki yiL2 " x(T) y() $ (J1(y()) : : : Jp (y())) x(T) 2 UT :
738
. .
# ; < y() 9 y(t), $ . ! " : " , (f g) ( G x(T ) 7! y()), " hki yi x(T) ? 7 ki(), i = 1 p, y() (x(T ) 2 UT ). = ki() $ , p? k() . J x(T) 7! y() 7! hk yi " '(x(T)) = hk yi. & '()? # - . ? , 9 9 , $, " '(x(T)). & '() k( ), (1.3), ? ' $ $ | . - . 1. E " ': UT ! R1 - M UT , $ " K '(x) = K(y(: x T )), x 2 M. ' ' M , - '(x) x = x(T) " y(: x T), x 2 M. ' (f g) " '(x) = xi , i = 1 n, UT . & " ' M ' 9 M~ (M M~ UT ), - - UT n M~ . ? M(M) M " '. ?~ M(M) M M. ~ , M(M) 2. N M(M) - 'i 2 M(M), i = 1 p, " '(x) = F' ('1 (x) : : : 'p(x)) 8' 2 M(M) 8x 2 M: ? , M(M) ( " ) " . y(: x T ) (x 2 M) '1 (x) : : : 'p (x), " x = x(T) y() . ' (f g) M UT ('1 (x) : : : 'p(x)) $ x 2 M. = , ( $ ). O , Ki | " , $ " 'i 2 M(M) 1, fki i > 1g | L2 = L2 ((0 T ] R1) " . P i 2 M(UT ) M(M),
739
i (x) = hki y(: x T )i, x 2 UT . i(x) = = Fi ('1 (x) : : : 'p (x)) 8i > 1 8x 2 M. 7 , 'i (x) = Ki (y(: x T )), i = 1 p, i (x), i > 1, . # fki i > 1g ('1 (x) : : : 'p (x)) $ y(: x T), x 2 M, " y(: x T) p- ('1 (x) : : : 'p(x)), x 2 M. ~ M~ M. D " 'i (x) M(M) E " Ki 1 , Ki(y()) = y(ti ), Ki(y()) = y(i) (t ). ? ( ) " K(y()) = hk yi. ? , " (x) = hk y( x: T)i UT , . . 2 M(M) 8M UT 8k(). k() 9 L2 . 1. M(M) | M UT (f g) : f 2 C ! (U Rn), g 2 C ! (U R1). " # # M $ UT $ $ L2 fki i > 1g ki (), ' : UT ! R1 (' (x) = hki y( x T)i, = 1 p) $ $ M(M). . ? UT UT $ " Ri(x1 x2) = i (x1 ) ; i (x2 ) = hki y(: x1 T) ; y(: x2 T)i xj 2 UT : # + , Ri W = UTc UTc C 2n , UTc UT C n . D Ri(z 1 z 2 ) = hki y(: z 1 T ) ; y(: z 2 T )i z j 2 UTc : , y(: z T ), z 2 UTc , , 9 x_ = f(x) &9 x(T ) = z 2 UTc C n . 9 (0 T] , f, g U c U C n , f, g " " . ? Zi " R1i W . P $ T Ri Z = Zj W j =1 $ i1 : : : ip ,
\p Z \ (M M) = Zi \ (M M): =1
7 (8, . 53]. ! Ri (x1 x2) = 0 = 1 p, xj 2 M Ri(x1 x2) = 0, i > 1, fki i > 1g
. .
740
y(: x1 T ) = y(: x2 T). ? ('1 (x) : : : 'p (x)) = (hki1 yi : : : hkip yi) $ y(: x T ) x 2 M: ? , " ' 2 M(M) '(x) = K(y(: x T)) = = F'('1 (x) : : : 'p (x)), x 2 M. P . P fki i > 1g L2 | fhki y(: x T)i i > 1g $ y(: x T ), x 2 UT ( Y = fy() j x(T ) 2 UT g). > " ki() jki(t)j 6 kS = const, ki() . . T " p f, g, M, fki i > 1g. . % " C ! (W) W " , - fRi i > 1g. D | " " Ri " C ! (W ). & , $ , " hki yi M = UT ( , M UT ). # , - " " (9, . 50], xS 2 UT $ P" = fz 2 C n j kz ; xSk = max jz ; xSi j < "g UTc (P" \ Rn UT ) i i " Ri1 : : : Riq Rj (z 1 z 2) =
q X j (z 1 z 2 )Ri (z 1 z 2 ) =1
j > 1 (z 1 z 2) 2 P" P" j 2 C ! (P" P"):
P (Ri (x1 x2) = 0 = 1 q xj 2 M = P" \ UT ) ) ) (Ri (x1 x2) = 0 i > 1) ) y(: x1 T ) = y(: x2 T): N M(M) " ' (x) = i (x) = hki y(: x T)i, = 1 q, ('1 (x) : : : 'q (x)) $ y(: x T) x 2 M = P" \ UT : = " ( , k() 2 fki i > 1g), $ (M = UT , p = 2n + 1). 2. fki i > 1g | $ L2 . % $ 2n + 1 fri() i = 0 2ng, 'i (x) = hri y(: x T )i, i = 0 2n, $ $ M(UT ) ( M(M) # M UT ). & rj () (0 T] ' fki i > 1g.
741
. # $ (8, . 54]. ffg2I | " , n- U. P $ Z U, - $ " gi 2 C ! (U), i = 0 n, $ Z. V " . # - gn. Ui | U, $ Z, ai 2 Ui n Z | . O i - " fi , S fi (ai) 6= 0. U - G Kj (Kj Kj +1, K - s, K Ks ) " j cj , X 1 j ;j jcj fj (z)j < 2 8z 2 Kj ck fk (ai ) > 2 jcifi (ai)j 8i 6 j: (2.1) k=1 P % ci fi U " , gn . gn (ai ) 6= 0 8i, , dim(Zgn \ Ui ) < n, Zgn | gn U. ? gn;1 : : : g0 (8] " : gs jZ 0 > s Zgn \ : : : \ Zgs U Z. > $ g0 : : : gn Z. ? (. 1) " Ri : W = UTc UTc ! C Ri(z 1 z 2) = i(z 1 ) ; i (z 2 ) i > 1 $ i (x) UT
UTc UT C n : i(z) = hki y(: z T )i, z = x(T ) 2 UTc . 9 fRi i > 1g U = W C 2n 9 ", " cj jcj Rj (z 1 z 2)j = jhcj kj y(: z 1 T) ; y(: z 2 T )ij 6 6 kcj kj kC ky(: z 1 T) ; y(: z 2 T )kL1 6 2;j 8(z 1 z 2) 2 Kj
$ cj (2.1). D P P ciRi W " , ci ki C(0 T ]. ! " C(0 T] r2n : : : r0, $ . > $ " qi(z 1 z 2) = hri y(: z 1 T ) ; y(: z 2 T)i i = 0 2n T1
W Z = Zj (Zj | Rj W ). # fki i > 1g j =1 y(: x1 T) 6= y(: x2 T ), xj 2 UT , " : (hr0 y(: x1 T )i : : : hr2n y(: x1 T)i) 6= (hr0 y(: x2 T)i : : : hr2n y(: x2 T )i):
742
. .
! y(: x T ) $ ('0 (x) : : : '2n(x)), x 2 UT , 'i (x) = hri y(: x T )i, 'i M(UT ). V fri i = 0 2ng : - fki i > 1g, as, Kj , " c ,
$ . P . = " ki (t) = ti , rj () $ ( ) (0 T]. O ki (). jh ij 6 k kC k kL1 &9 {N , - rj () L2 . ' (f g) M UT (y(: x T ) $ x 2 M) , L2 ( Y = fy()g) fki i > 1g $ " Ri(x1 x2) M M f(x x) j x 2 M g. O rj () x(T ) 2 M " i = hri yi, i = 0 2n. ' (f g) , M(M) M UT . # , u(x(t)) (t ; T t] x(t), " u(x). = - k() " jk(t)j 6 ` = const, - k() . P " , ;" <. # $ . # g : U ! Rm, m > 1, 1, 2 $ . ! : y() (t 2 (0 T ], T < T ) x = x(T ) 2 UT ('(x(T ))). # " " yj 0T ] $ yj 0T ] ( ). # " f = f(t x), g = g(t x), t , - , . O " (1.3) k(t y) = 0, t > T . # $ $ " hk y(: x T)i &9 x = x(T) 2 UT . O T 6 T , (T = T ). % " x_ = f(t x) y = g(t x) (2.2) U = (t1 t2) U, (0 T ] (t1 t2). #- " f, g U $
743
x U t 2 (t1 t2). , , f = f c j g = gc j f c (t ) 2 C ! (U c ) gc (t ) 2 C ! (U c ) f c 2 C((t1 t2) U c C n ) gc 2 C((t1 t2) U c C m ) U c | U C n . D $ , 9 &9 (10]. UT U 9 x(: x T ), x = x(T) 2 UT , (0 T ]. P k() 2 Lm2 = L2 ((0 T] Rm) " (x) = hk y(: x T )iLm2 $ UT ( UTc | UT C n ). , 9 | . ? M (M) M UT " : ' 2 M (M) , '(x) = K(y(: x T )) x 2 M y: (0 T ] ! Rm: ? 2 - . # $ y() 2 Y = fy: (0 T ] ! Rm j x(T) 2 UT g. & k() - | (0 T ]. - - (T T]. 1 {2 . ( (2.2) # M $ UT $ $ Lm2 (0 T ) ( Y ) - fki i > 1g ki (), ' (x) = hki y(: x T)iLm2 (x 2 UT , = 1 p) $ $ M (M). ) $ # # k() 2 fki i > 1g M = UT , p = 2n + 1. 1. # 1, 10{20 M = UT , , UT 9 x(T) 2 U^ (0 T]. U^ U cl UT . ! Y (Y ) Rp. 7 , ; < " (" y() Lm2 ) (1.1), (2.2) ( UT ). ? , ; <. # (2.2) y() " - " t. 0
x
0
3. " #$
% x 2 (1.3), ; <. # '() k( ) , 9 .
. .
744
% U = (t1 t2) U " (2.2) : f g fx0 gx0 2 C(U). (0 T] UT x(T ) , $ W = f(t x) j t 2 (0 T ] x 2 x(t: UT T)g Wg = f(t y(t)) j t 2 (0 T] x(T ) 2 UT g: # Q Wg " k( ): Q = Q(k) Rm+1, k ky0 2 C(Q). P " v(t x) =
Zt
0
k( y(: x t)) d
(3.1)
C 1(W ) 9 " (t0 x0) = (t x) 2 W . 2. 7 v 2 C 1(W) , v( ) ~ # W~ W~ W v 2 C 1 (W). G (t x(t)), $ 9 x(: x(T ) T ) (x(T) 2 UT ), (t y(t)) Q k( ). , t (0 T], vt0 (0 x), (T x) . ! . , v( ) 9 @v (t x) + @v (t x) f(t x) = k(t g(t x)) (t x) 2 W (3.2) @t @x v(0 x) = 0, x 2 U0 = x(0: UT T). O , - (t x) 2 W (t, x ) 9 x() x(t) = x ( 2 (;" t + "), " = "(t x) > 0). P , , d @v (t x) f(t x) Lf v(t x) = d v( x()) = @v (t x) + @t @x =t ( - (3.1)),
Z
Z
Z
0
0
v( x())= k(s y(s: x() )) ds= k(s y(s: x(0) 0)) ds= k(s y(s: x(t) t)) ds
0
d v( x()) = k(t y(t: x t)) = k(t g(t x)): d =t = . O vS 9 (3.2) ( ). 7 , vS(t x(t)) const. > W (0 T], vS(0 ) = 0. vS(t x) = 0, (t x) 2 W .
745
, " (3.1) (3.2) (6,7] $. = (3.1) t = T, (1.3). = '(x(T)), (3.2) v(T x) = '(x), x 2 UT . (3.2) v(t ) ' T. # $ +. >. Z %. N . = k( ) (k ky0 2 C(Q)) 9 (1.3), v( ) (3.2) v(0 x) = 0, x 2 U0 = x(0 UT T ), v(T x) = '(x), x 2 UT . ? , k( ) 9 , ,
(3.2) x 9 x(t: x(T) T), x(T) 2 UT , t (0 T ] ( v(t _ x(t))), (1.3). !, (1.3) v(0 ) = 0, v(T ) = '. V (3.2) (k v). 3. V (3.2) 9 W, $ G (x(T) 2 UT ). ' : 9 x(: x t) (t x) 2 (0 T ] U~ (UT U~ U) (0 t] ~ P x(T ) 2 UT U. 1 ~ (3.1) v 2 C ((0 T] U) (3.2) (0 T] U~ , . ? Q k( ) f(t g(t x)) j t 2 (0 T ] x 2 U~ g. 4. O k( ) (k ky0 2 C(Q)) k(t ) = 0, t > T 2 (0 T ). & , k(t y) = k0(t)y - " k(t). ' v( ) (3.1) , (3.2) t = T , t = tj . 9 k( ) . ? " (3.2) . O : V_ (t) = ;A(t)V (t) + B(t)K(t) V (0) = 0 @v (t )f(t ) V (t) = v(t ): x(t: UT T ) ! R1 A(t)V (t) = @x (3.3) V_ (t) = @v @t (t ) K(t) = k(t ) B(t)K(t) = k(t g(t )): = ( 3), (3.3) | ~ # ; < C 1 (U). v(t ) ( " x) t 2 (0 T ]. O (1.3) K()
. .
746
~ P , 9 V (T) = ' (x 2 UT , x 2 U). DT = fV (T) = v(T )g C 1(UT ). DT M(UT ). ? jk(t y)j 6 kS . # `k(t y) `, (1.3) `. ! 9 $ 9 (11]. # $ " . ! 9 ( "- ) (3.3): (f g) DT wi : UT ! R1, i = 1 p, (w1 (x) : : : wp (x)) $ x 2 UT , . . x = H(w1(x) : : : wp(x)). = " ': UT ! R1 ('(x) = K(y(: x T))), ' 2 DT , " ' = H' (w1 : : : wp) UT '(x(T)) = H'
ZT
0
ZT
k1( y()) d : : : kp ( y()) d :
0
# f = F (t)x, y = G(t)x, k = k0 (t)y " 0 h x(T) (UT = Rn) fh = V (T )g (1.2), . . L Vi (T), i = 1 p, p 6 n. V (1.2) L = Rn, (Vi0 (T)x i = 1 n) $ x 2 UT (Rn), p = n. (3.3) - (f g) ; " < '(x(T )) = v(T x(T)) ; < , " . & DT , ( , f , g ). > , k(t y) = k0 (t)y, DT 9 Lfy(: x T) j x 2 UT g ( dimDT = dim L). - ( (3.3)) " . 3. N DT = fV (T ) = = v(T ): UT ! R1g M UT - wi 2 DT , i = 1 p, w(x) = Hw (w1(x) : : : wp(x)) 8x 2 M 8w 2 DT : , - (3.2) ((3.3)) M UT , wi , i = 1 p, M $ (w1 (x) : : : wp(x)) $ x 2 M (. . " wi - " x 2 M).
747
, M UT DT M. O Lm2 (0 T ) fki i > 1g - " vi (T ) 2 DT , vi (T x) = hki y(: x T)i. ;& " E < hki yi w (x) M (vi (T x) = Hi (w1(x) : : : wp(x)), x 2 M) , , (w1 (x) : : : wp(x)) $ y(: x T ), x 2 M. N M
M~ M. % x 2 $ $ " . % " (2.2) ( T = T). # - 9 $ (2.2). P M UT Lm2 (0 T ) - " fki i > 1g ki (), " w (x) = hki y(: x T )i = vi (T x) (x 2 UT ki (t) = 0 t > T = 1 p) M DT (T T ). # 1 M = UT . N k() 2 fki i > 1g (0 T ] fkj ()g (j = 1 p, p = 2n + 1) k(), $ wj = vj (T ), j = 1 : : : 2n + 1, DT M = UT . # $ . 3. * M (M) # $ M H(M) = fH(w1 : : : wp)g # - $ wi, i = 1 p, M DT = fv(T ): UT ! R1 j T < T ) k(t ) = 0 t > T g: + , # k(t y) = k0(t)y M = UT , p = 2n + 1. % ' H(UT ) M $ M (M). (f g) # $ ( T = T) M UT # #, # , (3.2) M . - DT . # " DT = fV (T)g (1.2): DT = L(K), L(K) | " " K = (G0 F 0G0 : : : F 0n;1G0). - (1.4) ((3.3)). !, " $ (f g) ( (1.1)). O $ m = 1, T = T , 9 " . & , k(t y) = k(t)y. , " x 2 . DT = fv(T ): UT ! R1 j (f g) (1:1) k(t y) = k(t)yg
. .
748
" " Lif g.
@ (Li g(x)) f(x) x 2 U (x 2 U): L0f g(x) = g(x) Lif+1 g(x) = @x T f # f, g " " , Lif g . # (3.3) Lif g = Ai B (A = @()=@x f, B = g, BK(t) = k(t)g) " " : (f g) = (F G) Lif g(x) = GF ix, F 0j ;1G0 | j- " K. y(i) (t) Lif g(x(t)). P Lif g(x) = y(i) (T ) , i = 0 n ; 1 ( ) 9 x UT . G M Rn Rn M. ' " . # (1.3) : " . - , $ . 4 ( 9, . 44]). ( # J Hn z0 2 C n n $ (I) O C n z0 h1 : : : hr z0 $ J (II) ($ C n ) # # fPi i > 1g z0 , ($ cl P1 O)(III) %i , i > 1, : Pi h ^h 2 J 1 : : : r , Pi , Pi h(z) =
r X j (z)hj (z) j =1
k j kPi = sup j j j 6 %i khkPi j = 1 r: z2Pi
(3.4) (3.5)
$ . E xS 2 UT 9 [ = fx 2 Rn j kx ; xSk = max jxi ; xSij < g UT cl [ UT : i
# " yj 0T ] $ yj t1 t2] (0 6 t1 < t2 6 T ):
749
$ $ , y(: x T) (;" T + ") [, " > 0, ( ; T) x ; xS. , x ; xS [ Li (x) = Lif g(x) = y(i) (T: x T ) w(x) = v(T x) = hk y(: x T)i " Lci : P ! C , wc : P ! C : Lci j = Li j wc j = wj P = fz 2 C n j kz ; xSk = max jz ; xSij < g: i i
Lci , wc P ( 9 ). % J " Hn " xS, - fLci : P ! C i > 0g. D J | " L^ cj " Hn. Oi z0 = xS ( Pi $ C n ), " h1 : : : hr , %i , i > 1, 4. E s > 1, p > 1 cl Os P Os j > p
pX ;1 c Lj (z) = j (z)Lc (z) k j kOs 6 %kLcj kOs (3.6) =0 !
j 2 C (Os C ) = 0 p ; 1 j > p % > 0: D , (3.4) " Lcj Oi , i > 1, " h1 : : : hr . , , xS " Lc -
J ( " | " ). , $ %, $ j, " (3.5). 7 y(t: x T ) = L0 (x) + (t ; T )L1 (x) + (t ; T)2 L22(x) + : : : t 2 (;" T + ") x 2 Os \ Rn Lj , j > p, " (3.6) ;- " < L0 : : : Lp;1 . y(t: x T ) = v(T x) =
pX ;1 i=0
pX ;1 i=0
i (t x)Li (x) Li = Lif g
i(x)Li (x) i(x) = hk i ( x)i
x 2 Os \ Rn t 2 (;"0 T + "0 ) 0 < "0 < ":
(3.7)
. .
750
+ " , $ $
" i (t x), i (x) $ &9 jLcj (z)j N ~ j! 6 T~j z 2 P T 2 (T + "0 T + ") N = const " Os " j (3.6) % 6= %( j). 5. # (f g) : T , x | xS 2 UT , p = p(Sx). ' y(t: x T ) t ; t , t 2 (0 T]. P , i (t x), Li (x) x = x(T) x = x(t : x T) xS = x(t : xS T ), t 2 (t ; " t + " ) = I . I (0 T] T . # " T $ . #, t y(t: x T ) (y(t: x t )) 6 T. T - x(T) x(T) = xS. - $ . 5. (f g) U Rn, $ (0 T] UT $ x(T) (UT | xS 2 U). " # UT ' DT = fv(T ): UT ! R1 j k(t y) = k(t)yg (3.7), # i (t x) (t0 t00) UT (0 T] UT , i (x) | UT (i 6= i (k())). = UT - $ Lr (x) =
r;1 X (x)L (x) 2 C ! (UT ) =0
(3.7) p = r. p > r Lr+1 : : : Lp;1 L0 : : : Lr;1 " f. 9 " . # , ;" " < Ai B = Lif g " i " . = L0 : : : Lp;1 $ UT , (f g) UT . m > 1 hk yiLm2 = hk1 y1 i + : : : + hkm ym i (3.7) , i | (i1 : : : im).
751
% $ $ . f = F x, g = Gx (" ) y(t: x T) = G expfF (t ; T)gx =
1 j X (t ; T )j GFj! x : j =0
O j > p, p > rank(G0 F 0G0 : : : F 0n;1G0), GF j " G GF : : : GF p;1. >
( | , | ), pX ;1 pX ;1 1 n j 8t 2 R 8x 2 R y(t: x T ) = j (t)GF x = j (t)Lj (x) j =0 j =0 p;1 X v(T x) = hk yi = hk j iLj (x): j =0
P " j (t) (T ) = (0 (T ) : : : p;1 (T ))0 = e1 = (1 0 : : : 0)0 _ (T ) = e2 : : : (p;1) (T ) = ep j (t x), (3.7): (T x) = e1 , @=@t(T x) = e2 : : :. #, . O expfFtg = 0(t)E + : : : + + p;1 (t)F p;1. 7 ( m > 1) p $ " F. j (t) 9 p- . 6. = " , $ Li = Lif g . # 9 v(T x) = c0 +c1L1 (x)+c2 L2 (x)+: : :, ci = hk ( ; T)i i=i!. P , . $ AiB = Lif g DT , y(tj : T) ( " : " ). > DT T, k(t) = 0, t > T , DT ( ). # , DT , 9 (f g) = (x2 x3). - " k(), $ . O " i (t x) (3.7). , (3.7) | p (
. .
752
$ i 0), i " Lj . . E (3.7) " j (t x), $ 4 (T, UT ): 1 j X j (t x) = (t ;j!T) + j (x) (t ; !T ) j = 0 p ; 1 =p j (x) = Re j (x) t 2 (;"0 T + "0 ) x 2 UT : E L2 (0 T) fki i > 1g. P wi(x) = vi (T x) = hki y(: x T )i, x 2 UT , 0w (x)1 0hk i : : : hk i1 0 L0(x) 1 @w12(x)A = @hk21 00i : : : hk12 pp;;11iA B@ ... CA : (3.8) ::: ::: ::: ::: Lp;1 (x) $. % ; (3.8) ' hki j ( x)i x 2 UT p $ . O , : ; c = 0, x = x^ 2 UT , c = (c0 : : : cp;1)0 6= 6= 0 ( ). P hki c0 i = 0 i > 1, = (0 : : : p;1)0 . # fki i > 1g t c0 (t x^) =
pX ;1 j =0
cj (t ;j!T) + (t ; T )p j
pX ;1 j =0
cj pjp!(^x) + : : : = 0:
? cj = 0. . % , 9 p, 5. 6. (f g) U . . L2 (0 T ) fki i > 1g . / , UT , ki1 : : : kiq , : 1) ' wi (x) = vi (T x) = hki yi (x 2 UT , = 1 q) $ $ DT (M(UT ))2) $ DT $ w~i (x) = hki + i yi $ ki kL1 < "~. . , UT xS 2 U ( UT ). O $ T . % J~ " H2n " (Sx xS) 2 UT UT , - fRLci : P P ! C i > 0g RLci (x1 x2) = Lci (x1) ; Lci (x2 ): 7 5.
q;1 X RLcj (z 1 z 2 ) = ~j (z 1 z 2)RLc (z 1 z 2) =0 j > q k ~j kO~s 6 %~kRcjkO~s %~ 6= %~( j)
753 (3.9)
O~s (Sx xS) P P , (3.6). , UT UT O~s \ R2n. P ( ) (3.8) wi (x) Rwi(x1 x2) = wi(x1 ) ; wi(x2 ) = hki y(: x1 T) ; y(: x2 T)i, L (x) | RL (x1 x2) = L (x1 ) ; L (x2 ), p | q. O 2 Ry = y(t: x1 T) ; y(t: x2 T) = RL0 + (t ; T)RL1 + (t ; T )2 RL 2 + ::: RLj , j > q, " RL0 : : : RLq;1 (3.9) ( ). E i1 : : : iq (Sx xS) $ " ;: (Rwi1 : : : Rwiq ) = (RL0 : : : RLq;1) R (x1 x2) 2 UT UT det R(Sx xS) 6= 0: D " R hki j ( x1 x2)i (j = 0 q ; 1, = 1 q). $ ki kL1 < "~ " R~ hki +i j i (Sx xS) - UT UT . 7 , , 9 UT . ? ~ det R~ 6= 0 (Rw~i1 : : : Rw~iq ) = (RL0 : : : RLq;1 ) R (3.10) (x1 x2) 2 UT UT Rw~i (x1 x2) = hki + i Ryi ki kL1 < "~: ! Rw~i (x1 x2) = 0, = 1 q, RLj (x1 x2) = 0, j = 0 q ; 1. ' (3.9) RLj (x1 x2) = 0, j > 0, . . y(: x1 T) = y(: x2 T ). (w~i1 (x) : : : w~iq (x)) $ y(: x T ) x 2 UT : E " w~i DT M(UT ) UT " '. D T. 9 UT xS 2 U - y(t) (t ; T) (t0 t00) (0 T ] x(T ) 2 UT . $ 9 (0 T ] . # 0 6 t1 < t2 < : : : < tr 6 T , UT i )) y(t: x(T ) T ) = L0 (x(ti )) + (t ; ti )L1 (x(ti)) + (t ; ti )2 L2 (x(t 2 + :::
754
. .
Ii = (ti ; "i ti + "i ), S I $ i (0 T]. % (sj sj +1 ] (s0 = 0, s1 2 I1 \ I2,.. ., sr = T) 9 UT , (3.8) (i > 1, x(T ) 2 UT ):
r X (hki j 0( x(tj ))ij : : : hki jpj ;1 ( x(tj ))ij ) j =1 (L0 (x(tj )) : : : Lpj ;1(x(tj )))0 : ! j h ij (sj ;1 sj ]. Z " ; G - (L0 (x(t1)) : : : Lp ;1 (x(t1 )) : : : L0(x(T )) : : : Lpr ;1(x(T)))0 : $
wi (x(T)) =
1
" x(T) 2 UT . O UT UT (Sx xS), 9. ?" 9 UT (UT = UT (Sx f g fkig), q 6= q(fkig), "~ 1). D . P . %. / (f g) UT , x(T) 2 UT $ q = hki yi $ vi (T x) = , = 1 q. 0 $ $ ki (). 7. ? UT $ fkig. ', , - " hk yi q^ = q^(fki g). , . O UT , T UT UT ` , (3.10). O G - " q^ = q` ^ rank R^ = q UT UT , R~ " R, "~ 1. ' - - RLj (x1 x2) = 0, j = 0 q ; 1, y(: x1 T ) = y(: x2 T). " y(t) (t ; t ) (t 2 (0 T], t 2 I = (t ; " t + " )) k(t), (t; t+ ] I (. 5). O fki i > 1g L2 (t; t+ ) ( Y ). m > 1 6 , L2 Lm2 . O 6 fki i > 1g Lm2 (0 T ) ( Y ),
ki(t) = 0, t > T . _ k() D T DT , D T DT . " (f g) f, g . # ~ $ . % - (3.2) (0 T] U. %9 x(: x t) (t x) 2 (0 T] U~ ~ (0 t], (x(T) 2 UT U) U~ ( 3). (2.2) $ :
r X x_ = f(t x) + i (t)hi (t x) hi hix0 2 C(U): i=1
755 (3.11)
E " i (t) , , ji(t)j 6 S = const. ~ k( ) (3.2) 9 (1.3) (v(T x) = '(x), x 2 UT (U)) @v i ~ (3.12) @x (t x) h (t x) = 0 (t x) 2 (0 T] U i = 1 r: P - (3.2) f (3.11). % (3:2) . % $- 9 x(: x(T) T ) ~ x(T ) 2 UT , - (0 T ] U. x (3:2) t (0 T ]: '(x(T)) =
ZT
0
~ k( y(: x(T) T )) d x(T ) 2 UT x() 2 U:
(12] - : " k( ) $ = (1 : : : r )0 , - y(t) = g(t x(t)). '(x(T)) $ , - x(T) 2 UT . # t . # (3.3) (3.12) P (t)V (t) = 0 (@v=@x(t ) H(t ) = 0, H = (h1 : : : hr )). !, $ hi , v(T ) = ', vx0 (t ). , . ? P V $ , $ . ? . > 9 $- , (3.12) hi . D (' k).
x
4. " #
? . % U = = (t1 t2 ) U (2.2) . N , ( ): f(t 0) = 0, g(t 0) = 0, 0 2 UT U. ?
. .
756
k(t y) = k0 (t)y. # Q t 2 (0 T] " v(t x), f(t x), g(t x) x. & " (0 T ], " v(t x) 9 k(t). , t 2 (0 T ]. - (3.2) x ( ): p @v(p) (t x) + X @v(i) (t x) f (p;i+1) (t x) = k0(t)g(p) (t x) @t (4.1) i=1 @x v(p) (0 x) = 0 x 2 Q t 2 (0 T ] p > 1: & w(p) () p- w(p) ( : : : ) w(p) (x) w(p) (x : : : x), x 2 Rn. # (t | ) (4.1) 9 p X i @v(p) (t x : : : x) + X v(i) (t x : : : f (p;i+1) (t x : : : x) : : : x) = @t i=1 j =1 = k0 (t)g(p) (t x : : : x): # " xi1 : : : xip :
X i ( p ;i+1)0 E :::F (t) : : : E V (i) (t) = G(p)0(t)k(t): (4.2) i=1 j =1 7 t 2 (0 T ], p > 1, | ( , ) (13] ", j F (p;i+1)0(t) X V_ (p) (t) + p
i , E | " n n, V (s)0(t)X (s) = v(s) (t x : : : x) = v(s) (t x) G(s)(t)X (s) = g(s) (t x : : : x) = g(s) (t x) F (s)(t)X (s) = f (s) (t x : : : x) = f (s) (t x) X (s) = x : : : x X (1) = x: O G - V = (V (1)0 : : : V (p)0 : : :)0 V_ (t) = ;F 0(t)V (t) + G 0 (t)k(t) V (0) = 0 (4.3) n - " F (t) (F (1) F (2) : : :), $ n2 | (0 F (1) E + E F (1) : : : F (p) E + E F (p) : : :) : : : G = (G(1) G(2) : : :): 7 , (f g) 9 X_ = F (t)X y = G (t)X X = (x0 X (2)0 : : :)0 : (4.4)
757
(4.3) X - , v(t x) = V 0 (t)X. 7 (1.1), (1.2) , ; < (4.3), (4.4) " F , G . # " ( ) F 0iG 0 " " Lif g(x) = GF i X. ' " k( ) $ , k(t 0) = 0. O y ( - (t y) (t1 t2) P (t1 t2) (0 T] P C m ). P " (4.2)
(1) 0 E : : : F (t) : : : E V (p) (t) + j =1 pX ;1 X q ( p ;q+1)0 + E :::F : : : E V (q) (t) ; q=1 j =1 ; G(p)0 k(1) (t) ; (G(2)0 G(1)0 + G(1)0 G(2)0) K (2)(t) ; : : : ; X ; G(i )0 : : : G(ip; )0 K (p;1) (t) = G(1)0 : : : G(1)0 K (p) (t) i +:::+ip; =p ( s ) 0 K (t)Y (s) = k(s) (t y) Y (s) = y : : : y:
V_ (p) (t) +
X p
1
1
1
1
(4.3) " F (t) , k(t) K(t) = = (k(1)0 K (2)0 : : :)0 , m G (G(1) G(2) : : :), $ m2 | (0 G(1) G(1) G(2) G(1) + G(1) G(2) : : :) : : : V 0 (t)X = v(t x): O - '(x) ('(0) = 0) '(x) '(1) (x)+: : :+'(r) (x) = Wr0 Xr , Xr = (x0 : : : X (r)0 )0 , 9 Vr (T) = (V (1)0 : : : V (r)0 )0 Wr Kr (t) = (k(1)0 : : : K (r)0 )0. N - " - $ Vr (t), V (i) , K (i) , i > r. D 9 V (j ) (T ) W (j ), j = 1 r, . # '(x(T ))
ZT
0
kr (t y(t)) dt kr (t y) = Kr0 (t)Yr kr | r y.
% , - 9 r . # , , r = 2. , ( " ) (14].
. .
758
# Hj0 V (t) = 0, Hj hj , " F f. # " , " f, g. + . # 1 (x) : : : N (x) (x 2 U~ | 3). " hj , j = 1 r, hj (t g(t x)) : N X ~ hj (t g(t x)) bj (t) (x) x 2 U: =1
+ ,
N N X X @ '(x) d (x) A = @x f(t x) a (t) (x): =1 =1
!$ k, v
k(t y)
r N X X kj (t)hj (y) v(t x) v (t) (x): =1 j =1
- " (x) V_ (t) = ;A0 V (t) + B 0 k(t) V (0) = 0 V (T ) d d = fdj g V = fvj g k = fkj g A = faij g B = fbij g: # T
ZX r '(x(T )) kj (t)hj (t y(t)) dt: 0 j=1
O kj (t) = 0, t > T . # " $ . ' " $ - 9 . #-
v(t x): v(0 x) = 0 v(T x) = '(x) x 2 U~ ( , t'(x)=T). O 1v1 (t x) + : : : + N vN (t x) vi (0 x) = vi (T x) = 0 (vi = t(t ; T)i (t)i (x)): + , k(t y) = 1 k1(t y) + : : : + M kM (t y).
- (3.2), R(t x: 1 : : : M ). =- ~ # $ (0 T] U. . Z (k v) - " (15], .
759
? : yi (t) = xi(t), i = 1 m. E . # - | " t x1 : : : xm , L v = 0 v(0 ) = 0 v(T ) = '
@v @v @ Lv = @x @t + @x f i = m + 1 n : i # " k(t y1 : : : ym ) (1.3) Lf v. # " . % " - " " L . -, | , 9 9 , - ( " ). P .
&
1] . . . | .: , 1968. 2] Inoye Y. On the observability of autonomous nonlinear systems // J. Math. Anal. Appl. | 1977. | Vol. 60, no. 1. | P. 236{247. 3] '( . ). *+, - ( ( ,.( // * (( (/ . | 1994. | 0 12. | C. 59{69. 4] '( . ). 2 ,.( + 45/ ( // * (( (/ . | 1996. | 0 4. | C. 38{45. 5] 6 . 7. 8 - 6 // 9* ''':. | 1987. | T. 296, 0 5. | C. 1069{1071. 6] . ). ( ( ; - / (/ // 75 / ; . 75. 8. | <6- = +. -(, 1986. | C. 118{125. 7] . )., < <. 8;-5 (5 6-/ ( . | > (: ?, 1990. 8] @ ). . 5 ( - ( . | .: , 1985. 9] B . ? ; + / 5/ 5/. | .: , 1965. 10] +( B. *., = . ,5 5/ DD; 45/ . | .: <=, 1958. 11] Curtain R. F., Zwart H. An introduction to inEnite dimensional linear systems theory. | Springer, 1995. 12] 4 . '. <4 ,.5 (5 // 9* ''':. | 1970. | T. 191, 0 6. | C. 1224{1227. 13] = ( G. ( ;. | .: , 1982.
760
. .
14] *6 . 7., K L. 7., : *. G. 75- (45 (5 > 6- 6 (6 ( ( 4+ . | .: <6- *<, 1989. 15] N . . = ; ( 5- . | .: , 1971. % & ' 1998 .
W . .
-
517.53
: , ,
.
W ! " # " % & .
Abstract M. D. Kovalenko, About Borel transformation in the class W of quasi-integral functions, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 761{774.
The class W of quasi-integral functions is introduced and the properties of the Borel trasformation in this class are investigated.
1. W
( ), . . 1] !" # !$ ! " (%&') fK (): = + i = tei 0 < jtj < 1 jj < 1g G(), ! , - . . () 2], 0"
W, 0 ! 1 !.
Z1
;1
jG()j2d < 1:
(1.1)
1 ! W . 2$ # "3 !"! --, W . 2$ # 3]. 5! g(!) - " , ! 6! 1] G(). ' ( . # 2 . 1,2]), G() 2 W, . g(!) !" # . / ISTC, 415-96. , 2001, & 7, 1 3, . 761{774. c 2001 , !" #$ %
762
. .
!$ ! " fK (!): ! = x + iy ! = uei' 0 < juj < 1 j'j < 1g, 2 !# 2 ! " "# f;: x = 0 jyj 6 1g. G() g(!) - !"! %&' K () K (!),
" 2$ .$ C () C (!), !2$ ! 2 , !!2, - , $ #. %! { C () "3 -2 ! 0 -" , 2$ .," ! . 6 " , ! # , # , . . f{ : = 0 6 0g. : !! C (!) 2-!" T--! 2# !! T , -! 2# ! " " "# ; " f` : y = 0 x 6 0g. & S | ! 2# !, $2 ,# !! T !$ "2# 3 " ! (! # ! ). 1 # (1.1) !" 2 !-! . 6!. 1] " -2 ! 2 ," -!": 1 Z g(!)e! d! (Re > 0) G() = (1.2) 2 i
8> R1 ;!S < 0 G()e d (Re ! > 0) g(!) = >:; R1 G(ei )e! d (Re ! < 0 6 arg ! 6 ) 2
(1.3)
0
%3" - 2 ! S !$ " 3 " -!" !! ` - " . S !! ;, . 2" 2 " ! W 2]. 5! 1 gy (y) = lim g(iy + ") ; g(iy ; ")] (" > 0) supp gy (y) 2 ;< (1.4) "!0 2 1 g(ue;i ) ; g(uei )] supp gx (;u) 2 ` (1.5) gx (;u) = 2 i
- " g(!) !!$ ; ` . 1 " !" 2 (1.2) " , ! . G(): G() = Gy () + Gx () (Re > 0) (1.6) Gy () = Gx () =
Z1
;1 1
Z 0
gy (y)eiy dy
;
( 2 C ())
gx ( u)e;u du
(Re > 0)
(1.7) (1.8)
W
763
( . Gx(), , ! 3. - C () n `). @ ", ! ! (1.3) ! " (1.5) gx (;u) !" -!, . ! (1.8)
;
gx ( u) =
!
Z1 0
;
Gx ( t)e;tu dt
(u > 0)
(1.9)
1 G (te;i ) ; G (tei ) (1.10) x 2 i x - Gx () !! { . %"!" !2 (1.7){(1.9). : # &{: ! 2,3] . Gy () !. (1.1) , gy (y) 2 L2 (;). & - 2" #" - ! # Gx (), gx (;u). C " , Gx( ) 2 L2 (0 1) , gx (;u) 2 L2(`). : "" , Gx( ) 2 L2(0 1), Gx (;t) 2 L2 (0 1). D !" 2 (1.9) ! 2$ E # . !-! . ' 4] ", gx (;u) 2 L2 (`). &! 0", . Gx() " - " ! " - ! , , . Gx(;t) ! , . gx (;u) -2 - !# # "! juj;1. F-! !3 . 3, !" 2 (1.8). " 2E, !!2 { `, - , # G() g(!), -, " -2 ! 2 -2" ", 2$ .," ! " 2$ .$ C () C (!) . : , !! ` ! x > 0, !! { | > 0, . Gx () " (1.8) "3 , ! ( ! -!0. uei ):
;
Gx ( t) =
Gx () =
G
Z
1ei
i
gx(u)e;ue d(uei )
(Re < 0):
(1.11)
0
1 2i (1.12) 2 i g(u) ; g(ue )] supp gx (u) 2 0 1) | g(!) !! x > 0. !" -!, . ! (1.11) ! . ",
! ! (1.3): gx (u) =
gx(u) =
Z1 0
Gx (t)e;tudt
(u > 0)
(1.13)
764
. .
1 2i (1.14) 2 i Gx(te ) ; Gx(t)] | Gx() !! > 0. H , ! # Gx() (1.11) gx (u) - ," #" ( 2" " 2E . !2 # Gx() (1.8) gx (;u)): Gx ( ) 2 L2 (;1 0) , gx(u) 2 L2 (0 1). 1 " -!", " , !3 . . Gx() (1.1) Gx (t) =
, , , gx (;u) 2 L2(`), gx (u) 2 L2 (0 1).
1 W "3 ! ""2 W " " ( !. ,# (1.1)), !"2 &{: ! ""2 1 " , !" . . ! G() 2 W !" , ! g(!) (1.4), (1.5), (1.12). 1. : !" # (1.2) Gy (), ! S .. !! ;, "3 -2 ! Gy () =
Z1+r
lim r !0
;1+r
gy (y)eiy dy +
1 I g(!)e! d! + 1 I g(!)e! d! : (1.15) 2 i 2 i c+
c;
! 2 !2 -! . ! 3 ." c+ c; " ! r > 0, $2 ," , 2 y = 1 ! ;. I G() 2 W, . g(!) " - ! " $ ! ; , , " ! 2 !2 ! 2 . I 3 G() 2= W, - " ! , # , ! . # 6! . g(!) " 2 - $ fx = 0 y = 1g. : " . gy (y) " $ y = 1 !!22 - # 0 ! 2$ (!. !2$ ! ! - G( )). J !3 . 2 2" ! 5]. 2. : ! 2$ # ""2 1 . 1 , , " " #" (1.1). . g(!), ! . 6! G() 2 W,
! .. " " (1.4), (1.5) ( (1.12)) !! T ," -!": (1.16) g(!) = gy (!) + gx (!) (! 2 C (!) n T )
W
Z1 gy (y) g (!) = ;
dy (! 2 C (!) n ;) ; ! ;1 Z1 g (;u) g (!) = du (! 2 C (!) n `): y
iy
x
u+!
x
0
765
(1.17) (1.18)
&! (1.16) . ! ! (1.6) !" (1.3). " " Gx () 3 ! .. " " Gx(;t) Gx(t). : , .. (1.9) (1.8), " Gx () =
Z1 G (;t) x
+t
0
(1.19)
dt:
2.
& h (y) 2 L2 (;1 1) ( h (y), -, !., "3 -2 -2"), H ( ) 2 L2 (;1 1) | 0 !-! !. : 0" - . h(u) =
Z1
H ( )e
;u
d
;
(u > 0)
h( u) =
;
Z0
H ( )eu d
(u > 0): (2.1)
;1
0
1 ". , ! &!., ! . !" 2 (1.8) !# !" 2 (2.1):
Z1
;
gx( u)h(u) du =
0
Z1
Gx( )H ( ) d:
(2.2)
0
&!"" (1.11) = " " !! uei !! " !" # u > 0. 1 "
;
Gx ( ) =
;
Z1
gx (u)eu du
( < 0):
(2.3)
0
& . !" # (2.3) !# !" # (2.1), " ,0 ! &!.:
Z1 0
;
gx(u)h( u) du =
Z0
;1
;
Gx ( )H ( ) d:
(2.4)
766
. .
@ 2. ! (2.2) (2.4) !$ . !" # !! . u > 0 !" # x, ! 0 # # , # , " ! &!.
Z1
;
gx ( x)h(x) dx =
;1
Z1
jj
Gx( )H ( ) d:
;1
(2.5)
F -2 ! &!. . # !! "2$
!" !
Z1
;
gx ( x)h(x) dx = 2
;1
Z1
gx(y)h (y) dy:
(2.6)
;1
G"", ! gx (y) - !-! ! Gx(j j). @ 2. ! (2.6) 2" ! " &!.
Z1
Z1
;1
;1
2 gy (y)h (y) dy = !$ " !
Z1
;1
;
gx ( x)h(x) dx + 2
Z1
;1
Gy ( )H ( ) d
Z1
gy (y)h (y) dy =
;1
(2.7)
Gy ( ) + Gx(j j)] H ( ) d (2.8)
! (2.6) " , ! &!.: Z1 Z1 2 gy (y) + gx (y)]h (y) dy = Gy ( ) + Gx(j j)]H ( ) d: (2.9) ;1
;1
D " E. -, , "3 , G( ) = Re G( ) ! > 0 . Re G( ) 0 . 1 Gy ( )+ Gx (j j) = Re G( ), ! (2.9) !-! ,#, - " 2# : 2
Z1
;1
gy (y) + gx (y)]h (y) dy =
Z1
;1
ReG( )]H ( ) d:
(2.10)
%"!" 3 2 2 2$ ! &!.. 1. & h (y) = 21 e;iy ( | , 2# !"!). 1 H ( ) = = ( ; ) | -! ! # , h(u) h(;u), ! ." (2.1), ! 2 ( ;u ( u e ( > 0) e ( < 0) h(u) = h(;u) = (2.11) 0 ( < 0) 0 ( > 0):
W
767
: " ! &!. (2.2) (2.4) !!, . !" 2 (1.8) (2.3) , ! (2.10) " ,
3 !" : Re G() =
Z1
gy (y) + gx (y)]eiy dy:
(2.12)
;1
&!# 0" E .$ (2.11) !" # u > 0 !" # x, ! 0 # # , # . & h(x) - " ! E (x). F , , ! ( ;x e ( x > 0) E (x) = (2.13) ; x ;e ( x < 0): 1 , . ! " (2.8), " ! ! . Re G():
Z1
Re G() =
;
gx( x)E (x) dx +
;1
Z1
gy (y)eiy dy:
(2.14)
;1
2. & H ( ) = e;! . & !" (2.1) $ " h(u) = (u + !);1 . D ! &!. (2.2) " gx(!) =
Z1 g (;u) x
0
u+!
du =
Z1
Gx( )e;! d
(Re ! > 0):
(2.15)
0
F , !, .,# ! (2.15), ... " ! 3 " !, ., !, - gx (!).
3. &! 0" !2$ 3 2$ !"! !-! . 6!. . 1. Gx() = ln | " " . & !" (1.3) $ " , ! # 6! : gx (!) =
Z1 0
ln t e;t! dt = ; ln !!+ (Re ! > 0):
(3.1)
G = (C | . . J#!). . gx (!), , ! 3. - C (! ) n ` ( . ! 0 ", !! ` ! x 6 0). eC
768
. .
: " ln . @ # !"!" ,# !, .2# ! , $2 ," !! ` 2# ! 3 cr " ! r !" ! $ #, ! 0 2$ !$ " 3 " -!" !! `: 1 2 i
Z;r
;i
gx(ue;i )eue
;1
d(ue;i )
; 21 i
Z;r
i
gx (uei )eue d(uei ) +
;1
Z Z1 Z 1 1 ! ; u + 2 i g(!)e d! = gx (;u)e du ; 2 (ln r + i' + ) d' = cr r ; Z1 e;u =;
r
u
du
; (ln r + ) = Ei(;r) ; (ln r + ):
(3.2)
G ! gx (;u) = ; u1 - (1.5) gx (!) !! uei 6 r, ! Ei(;r) | ! . . .. &!$ . (3.2) ! ! r ! 0 ( " ! . 6] . Ei(;r)), " " ln . @ gx (;u) gx (!) " !! ` "3 ! "2 --,0 2$ # ," -!": 1 + (u) : (3.3) g (;u) = ; x
u
& (3.3) !" (1.8), " ln , (3.3) !" (1.18) 0 gx (!) (3.1), ! 0 # - C (!) n `. 2. & Gx ( + a) (a > 0 , ) | " " # . = ;a. F- " gax(!) , ! . 6! Gx( + a). & !" (1.3) $ " gax (!) =
Z1
Gx( + a)e;! d = ea! gx (!)
(Re ! > 0)
(3.4)
;a
gx (!) | ., ! . 6! Gx (), " ,# . ! . 1 . gx (!) - C (! ) n `, . gax (!) # 3 -. L 2. ! (3.4), !" 2 (1.18) " , ! gx (!) - C (! ) n `: gax(!) = ea!
Z1 g (;u) x
0
u+!
d!
(3.5)
W
769
gx (;u) | gx (!) !! `. !" , ! (3.4) ! !" . gax(!) !! ` gax (;u) = e;au gx (;u): (3.6) & , !"!, . Gx( + a) = ln( + a). 1 ln ! + (! 2 C (!) n `) | g (!) = ;e;a! (3.7) ax
!
., ! . # 6! , (3.7) !! ` ! 1 gax(;u) = ;e;au + (u) : (3.8) u
3. & . Gx (t) = ln ja + tj | , . ln(a + t) (a > 0 , ). M !! . .", # 0" gax (!) =
Z1 0
a! ln ja + tje;t! dt = ln a ; e !Ei(;a!) (Re ! > 0):
(3.9)
. gax (!), , ! 3. - C (!) n ` " ! ! j!j ! 0 (3.7). @ , $ # !! ` , " (3.8) ! 2 (3.7) (3.9). D "" (3.7) (3.9) ! . . ! 2" !" ": . (3.7) | !" (3.5), . (3.9) . # (3.8) !" (1.8). G"", (3.7) (3.9) . . 4. . Gx (t) = ln ja ; tj | , . ln(a ; t). & !" (1.3) $ " ln a ; e;a! Ei (a!) (! 2 C (!) n `) (3.10) g (! ) = ax
!
Ei (a!) | ! . . . 6]. &! j!j " ! ln ! + : g (!) ;e;a! ax
!
@ , (3.10) !! ` ! au 1 g (;u) = ;e + (u) : ax
u
! 0 ".
(3.11) (3.12)
5. %"!" Gx (t) = ln ja2 ; t2 j | , ln(a2 ; t2 ). H , ., ! . # 6! , ! "" # (3.9) (3.10), 0 !! ` | "" (3.8) (3.12).
770
. .
L 2. ! . ! (2.6) h (y) = , !"2 ! . ln ja2 ; t2 j: ln j
a2
; j=
Z1
t2
1 ;ity 2 e
;
gax ( x)E (tx) dx
;1
(;2 ; 1 x + (x) ch ax (x > 0) gax (;x) = ; 2 1 + (x) ch ax (x < 0)
, " (3.13) (3.14)
x
ln j
a2
; j=; t2
Z1 cos ay
;1
jyj
+ 2 (y) eity dy:
(3.15)
6. & ! P () = ei ln | , ! 1. F - C () n { ( " ", !! { 6 0). . P () ... !"!" ! 3 , ! "2$ ! . " " . . P () ! 3 W, !. (1.1). &!"!" , ! 3,# W, "3 3, !"!, . sin ln( 2 ; 2 ). & !" " (1.3) # 0" , ! 6! P (): p(!) =
Z1 0
p(!) =
;
ln t e;(!;i)tdt = ; ln(!!;;i)i + (Re ! > 0)
Z1 0
ln(te )e;(;!+i)t dt = ln(;! ;+!i)+i i +
(3.16)
(Re ! < 0 2 6 arg ! 6 ): (3.17) &. !" $ (3.16), (3.17) ! = "+iy (" > 0) !$ . ! ! " ! 0, ! " py (y) (1.4) p(!) " "# . : !" $ --,0 2$ # "3 ! 1 ; i + (y ; 1) (0 6 y 6 1): py (y) = y;1 2 & !" (1.5) # 0" p(!) `: 1 : px (;u) = ; i+u 1 " -!", . p(!) " # C (! ), !! # ! " "# 0 6 y 6 1 `.
W
771
1! !" " (1.7), (1.8) "3 Py () Px () P ():
Z1 1 Py () = ; i + (y ; 1) eiy dy = ;Ei(;i) ; ln ]ei ( 2 C ()) y ;1 2 0 Z1 1
Px() =
;
0
u+i
;
e;udu = Ei( i)ei
( 2 C () n { ):
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Abstract I. B. Kozhukhov, On potential properties of a semigroup connected with generation of one-sided congruences, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 775{782.
A property is ful2lled potentially in a semigroup if is ful2lled in some supersemigroup . We 2nd necessary and su3cient conditions for a given pair ( ) of elements of to belong to the right congruence generated by a given set of pairs ( i i ), 2 , potentially. In connection to this we investigate potential solvability of the equation systems of the form = , = , = . P
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8 1 a b S ' / / 1& / S, . . 1& /, %4 / (a b). (c d) j= (a b), , (c d) j= (a b), T S. $ (c d) j= (a b) , ( (c d) j= (a b) . 9 1 f(a b ) j i 2 I g j= (a b), 1& , %4 (a b ), i 2 I, % (a b), f(a b ) j i 2 I g j= (a b), 1 :
T S. ; ) ) ( (. 0 ). $ ' 1& ax = b, ax = bx, ax = by ) & ' ( ) . <' , & ' ( ) ) ' ( ' ( % 2, 3, 4). $ ' / /
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a | 4 1 , ' : S 1 ! S 1 | % a, . . s' = sa s 2 S 1 . < % S ! T(S 1 ), a 7! ' , ' : % . $ ( % 1 a ' , S T(S 1 ). , %4 / 1 a, , , ' ' hai. X = fx j 2 Ag | % ). @ /0/ S: a x = b x (i 2 I) c x = d (j 2 J) (1) a b c d 2 S | . A A I J | ' % , . $ 3] ' , & 1 1 ) 1 ' (. 3, 8.5]). 8 (
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% . 1. S : (i) (1) (. . - T S) (ii) (1) T(S 1 ). . C & (ii) ) (i) . 8 % (i) ) (ii). T S | , (1)
( x 2 T. < 1 ' 2 T(S 1 ): s 2 S 1 % s' = sx , sx 2 S,
s' = 1, sx 2 T n S. , 1 ' / S ab
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(1). 8 , a x 2 S, b x 2 S, 1 a ' = = a x = b x , a x 2 T n S, b x 2 T n S, ' , a ' = 1 = = b ' . 3 &, c x = d 2 S, c ' = c x = d . . ? S | , T(S1) , 1 & ' ( (1) % S / , 1 ' ( (% , % ).
) 1& 4
& , (1), . 9 , S | a b u1 : : : u v1 : : : v | 4 : 1 . @ a = u1x1 v1 x1 = u2 x2 ::: (2) v ;1x ;1 = u x v x = b: 5 , 4 /0 : 8p1 : : : p p10 : : : p0 2 S 1 9 p u = p0 u = p v ;1 = p ;1u ;1 : : : p2v1 = p1u1 ) p1 a = p01 aD (3 ) p0 v ;1 = p0 ;1 u ;1 : : : p02v1 = p01u1 8q : : : q q0 : : : q0 2 S 1 9 q v = q0 v0 = q u +1 = q +1v +1 : : : q ;1u = q v ) q b = q0 bD (4 ) q0 u +1 = q0+1 v +1 : : : q0 ;1u = q0 v 8r1 : : : r +1 2 S 1 (r1 u1 = r2v1 r2u2 = r3v2 : : : r u = r +1 b ) r1a = r +1 b): (5) 2. S ! a b u1 : : : u v1 : : : v : (i) (2) (ii) (2) T(S 1 ) (iii) S (5), # # i 2 f1 2 : : : ng (3 ), (4 ). . * (i) , (ii) ' % 1. 8 % & / (i) ) (iii). T S | , (2) ' ( , x1 : : : x 2 T | 1 , /0 ( 1 . (5). r1 u1 = r2v1 , r2 u2 = r3v2 : : :, r u = r +1v . E r1 a = r1u1 x1 = r2v1 x1 = r2u2x2 = r3 v2x2 = : : : = = r u x = r +1v x = r +1b. E (3 ). / i
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778
. .
' &
. E p1a = p1 u1x1 = p2 v1 x1 = = p2 u2x2 = : : : = p v ;1 x ;1 = p u x = p0 u x = p0 v ;1 x ;1 = p0 ;1 u ;1x ;1 = = p02v1 x1 = p01u1 x1 = p01 a. F (4 ) ' . 8 % (iii) ) (ii). (31 ){(3 ), (41 ){(4 )
(5). < 1 x1 : : : x 2 T(S 1 ). i 2 f1 2 : : : ng. ? s 2 S 1 1 1 0 1 p1 p2 : : : p 2 S 1 , s = p u p v ;1 = p ;1 u ;1 p ;1v ;2 = p ;2u ;2 : : : p2 v1 = p1u1 (6) sx = p1aD (7)
s 2 S 1 0 q q +1 : : : q , s = q v q u +1 = q +1v +1 : : : q ;1u = q v (8) sx = q bD (9)
% ) , sx = 1: (10) % x . $ % , 1 s 2 S 1 0 / p1 : : : p p01 : : : p0 , / (6), % s = p0 u p0 v ;1 = p0 ;1u ;1 : : : p02 v1 = p01u1 : E ' (3 ) , p1a = p01a, . . (7) ' p1 : : : p . 9 ' (9) q : : : q . % , 1 s 0 / p1 : : : p q q +1 : : : q , /0 (6) (8). E
: p1 u1 = p2 v1 , p2u2 = p3 v2,.. ., p ;1u ;1 = p v ;1, p u = s = q v , q u +1 = q +1 v +1 ,... , q ;1u = q v , (5) p1a = q b, . . (7) (9) /. C, 1 x . < , 1 x1 : : : x / (2). p 2 S 1 . E 1 s = pu1 0 p1 = p 1 (6), , (7)
p(u1 x1) = = (pu1)x1 = sx1 = pa, a = u1x1. 9 ' v x = b. < ' v ;1x ;1 = u x . p 2 S 1 . % s = pu . 8
' . 1- : 0 p = p p ;1 p ;2 : : : p1, /0 ( (6). E (7) sx = p1 a. 5 , p ;1 : : : p1 1 t = p ;1u ;1 / % , p : : : p1 1 s, 1 , ' (7) i
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779
1 t, tx ;1 = p1 a. </ p(u x ) = sx = p1a = tx ;1 = = p ;1u ;1x ;1 = p v ;1 x ;1 = p(v ;1 x ;1), ' , ' 1 p, u x = v ;1x ;1. 2- : 1 s = pu % s = qv 0 q = q q +1 : : : q , /0 (8). E (9) sx = q b. % q ;1 = p t = pv ;1. E
: t = pv ;1 , pu = q v , q u +1 = q +1v +1 ,.. ., q ;1u = q v , . . / (8). </ (9) tx ;1 = q b. 2 , p(u x ) = = sx = q b = tx ;1 = p(v ;1x ;1), ' 1 p u x = v ;1x ;1. 3- : 1 s = pu 0 p = p p ;1 : : : p1, q q +1 : : : q . E (10) sx = 1. % , pv ;1x ;1 6= 1. ? 1 t = pv ;1 0 J& 'K pv ;1 pu = q v q u +1 = q +1v +1 : : : q ;1u = q v , 1 & J& / 'K s = pu , / 3- . 8
, pv ;1 x ;1 6= 1 t = pv ;1 J& 'K, J& )K: pv ;1 = p ;1u ;1, p ;1v ;2 = p ;2u ;2,.. ., p2v1 = p1u1 . 8 1 & s = pu , J& )K 1 s. < & ' 0 3- , ' . E ' , pv ;1 x ;1 6= 1 ' % , ' , pv ;1 x ;1 = 1. 2 , pu x = sx = 1 = pv ;1 x ;1. $ ' 1 p
u x = v ;1x ;1. % ' . @ . <
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s 2 S 1 , % (sa )' = sb , 1 ' % ' ' . 5 ' ' (iii). , ' | ( (11). s 2 S 1 . E s(a ') = sb . $ ' 1 s
a ' = b . i
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780
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M | ' % , T(M) | ' a b 2 T(M) i 2 I. 4. $ : (i) (11) T(M) (ii) 8i j 2 I 8p q 2 M (pa = qa ) pb = qb ). 8 ' , ' % 3.
4 / ( j= j= . 5. S ! a b a b ( 2 A) : (i) f(a b ) j 2 Ag j= (a b) (ii) f(a b ) j 2 Ag j= (a b) T(S 1 ). . 8 ' & / (i) ) (ii). % , (i), T S | , f(a b ) j 2 Ag j= (a b). E 0 / 1 x1 : : : x 2 T,
a = u1x1 v1 x1 = u2 x2 ::: (12) v ;1x ;1 = u x v x = b fu v g = fa b g i = 1 2 : : : n. ? x1 : : : x 2 T, (12) % ( %
, 1 (2)). / & ' ( . </ % / 1 0 ) x1 : : : x 2 T(S 1 ), / 1 . * ' , f(a b ) j 2 Ag j= (a b) T(S 1 ). . ( j= . . (a b) j= (c d) (c d) j= (e f). % 5 ' , (a b) j= (c d) (c d) j= (e f) T(S 1 ). </ (a b) j= (e f) T(S 1 ), ' , (a b) j= (e f). C' % 5 2 . ) S | a b s t ( 2 A) | ! . + f(s t ) j 2 Ag j= (a b) , n u1 : : : u v1 : : : v , , - fu1 v1g = fs 1 t 1 g,.. ., fu v g = fs t g, (3 ), (4 ) ( i = 1 2 : : : n) (5). i
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781
$ ' / . A , ) ( (a b) j= (c d) /0
: 8s 2 S 1 (sa = sb ) sc = sd) ( ' % ' (12) % s). 6. ) S | a b s 2 S. + (s s2 ) j= (a b) , a b 2 sS 1 hsia \ hsib 6= ?.
.
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/ a = u1x1 v1 x1 = u2 x2 : : : v ;1x ;1 = u x v x = b fu v g = fs s2g i = 1 2 : : : n. C' , a b 2 sS 1 . 8
, fu v g = fs s2 g, su = v , sv = u . 1 s u = s 1 v k > 1. A ( 1 /0 : s u = s + v , " 2 f1 ;1g. k > n. E s a = s u1x1 = = s + 1 v1 x1 = s + 1 u2x2 = : : : = s + 1 + + b, . . shai \ shbi 6= ?. . a = sx, b = sy s a = s b ) x y 2 S 1
i j > 1. < , (s s2 ) j= (sx s2 x), . . (s s2 ) j= (a sa). 8
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Abstract
B. O. Kuliev, Fuzzy coalition structures and some particular cases: Stackelberg and Cournot structures, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 783{796.
The paper provides a formalism for the notion of fuzzy coalition and coalition structuresof 9rms as the general case of coalitionsand coalitionstructures. We study the properties of fuzzy coalition structures and introduce operations on coalitions. We investigate an important particular case, the coalitions and coalition structures, for which two axiom systems are proposed, depending on the type of interaction between coalitions: Stackelberg system and Cournot system. We de9ne the notion of stability for coalition structures, study the properties of the stable coalition structures and propose a criterion to determine them. We give a connection between Stackelberg and Cournot structures and their comparison.
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. .
$2, 5, 6]). * + , - , $3, 4], . $1] , + - . 0 , . * - 0 , , - 0 1 +, - , - .
2. , ! * 1 N . 2.1. 20 | , -4 s N , . . - f(n j s(n))g 8n 2 N s(n) | - 0 , 1 - $0 1], - n-
0 s. 2.2. 20 | , S 0 s, 1 : X 8n 2 N : s(n) = 1: s2S
81 0 s P s(n) ( - jsj)9 n2N supp s = fn 2 N : s(n) > 0g 0 s. * x | :4 , 0 s :4 jNsj x, . . :4 0 . ; n - 0 s s(n), 01 - 0, | , 0. 20 s 4 - , = const | -, 0. * 0 0 -
, 0.
785
2.1. s, S, N, . . X jsj = N: s2S
. <
1 jsj = P = s(n), jsj = s(n) = s(n) = 1 = N. n2N s2S s2S n2N n2N s2S n2N 2 $1], -, - 0 ( -, 0) 0 1 - x, . . p(x) = c ; bx, c b > 0. >, S 0 s P
P
P
P
jsj. = s2S P P P
P (s x) = p(x) jNsj x ; jNsj x ; = = jNsj x(p(x) ; ) ; = jNsj x(c ; bx ; ) ; = b jNsj x(d ; x) ; d = c;b . * n- X X X P (n x) = ( sj(snj) P (s x) ; s(n)) = sj(snj) P (s x) ; s(n) = s2S s2S s2S X s(n) X s(n) jsj bx ( d ; x) X = ( b x ( d ; x) ; ) ; = s ( n ) ; ; = N s2S s2S jsj N s2S jsj X X = bx(dN; x) ; sj(snj) ; = bx(dN; x) ; ; sj(snj) : s2S s2S 2.1.
0 = bd;2bN . . * 0 S x P (S x) = P P (s x) = P (b jNsj x(d ; x) ; ) = bx(d ; x) P jNsj ; N . s2S P s2S s2S j s j > 2.1 N = N . ?-, P (S x) = bx(d ; x) ; N . 8s2S S 0 = bd;2bN , ,
- 0 . @ ,
- x0 Px0 (S x0) = b(d ; x0) ; bx0 ; N 0 Px00(S x0) = ;2 + 0. @ P 0 - f (n) = sj(snj) , F (S ) = ff (n): n 2 N g. s2S
786
. .
< -. 1. 8 : P (n x) = bx(Nd;x) ; ; P N1 . s2S 2. = 0: P (n x) = bx(Nd;x) ; ; (- + , - ). 2.2. P sj(snj) > N1 , s2S
n- !
, " fs 2 S : s(n) > 0 jsj < N g = ?: . > P sj(snj) > N1 P s(n) = N1 . = s2S s2S 1- n-1 , - 0 s0 2 S , - s0 (n) > 0 js0 j < N . = sj0s(0nj) > s0N(n) , , P s(n) 1 P s(n) = 1 . jsj > N N
s2S
s2S
2.3.
s(n) s2S jsj P
6 1,
n- !
" , fs 2 S : s(n) > 0 jsj > 1g = ?: . = s 2 S sj(snj) 6 1, P P f (n) = sj(snj) 6 s(n) = 1. = 1- n-1 s2S s2S , - 0 s0 2 S , P -Ps0 (n) > 0 js0 j > 1. s 0 (n) = s0 < s0 (n), , sj(snj) < s(n) = 1. s2S
s2S
0 0 0 1 1 . * S | 0 , si , i = 1 2 : :n: l, | 0. =o l l S P 0 : 0 si 0 s = n j si (n) , , i=1 1 : 0 S 0 = fS n fsi : i = 1 2 : : : lgg s0 . < 0 | 0, : .
2.2. # " s0 $-
% si , i = 1 2 : : : l.
. @ , l
X
i=1
si =
l
X X
n2N i=1
si (n) =
l
X X
i=1 n2N
si (n) =
l
X
i=1
jsi j:
2.4. & ' ( $ ".
787
. @- 0, , - , - P (s1 x) < P (s1 s2 x) P (s2 x) < P (s1 s2 x). @ , 0 P (s1 x) = b jsN1 j x(d ; x) ; , P (s2 x) = b jsN2 j x(d ; x) ; , : , 0 P (s1 s2 x) = b js1jN+js2 j x(d ; x) ; . > - js1 j < js1 j + js2 j js2 j < js1 j + js2 j, , . + - - -4 0. 2.3. B 0 0 S - 0 , - si (n): N ! f0 1g $0 1], 0 -4 . - 0 | , . C 0 4 1. <- 0 - :1 . 20 | , 1 :1 0. * 0 (l1 m1 9 l2 m2 9 : : : 9 lk mk ), m1 > m2 > : : : > mk , l1 | - 0 1- m1 , l2 | - 0 2- m2 . . *- 4 4 , - 0 :1. C k . 20 i- 1 i- . 2.4. 0 -1 1 : ) ( ) : 1 - 0 9
) 0 :1 0 1 , 0, , , 9
) S S 0 , 0 S 0 - +1 , - S ( ). >0 , 1: - ) : 1 09 - ) 1 0, 9 - ) .
, 0 1 - 0 . 1 - , - 0 ( 0 1 , 1 . .). | , + 0 . 8 - , - , -, - 0 . D , , , 1 0 .
788
. .
2.5. B , 0 , - , - | - . < , - . E 0: E 2 . - 1 (- ). F , , . 1. ( , ). 2. * - 1- - 0 ( 0, 1 4 ). 3. < 0 :4 . 4. * 0. = 4 (1- ) E 2 . E 0 + 1 0 + 1 E . G- 2 E , 11 :4 0 E . 5S. E (l1 m19 l2 m29 : : : 9 lk mk ) 0 1- 1 :4 2d , 0 2- | d4 . ., - 0 k- 1 :4 2d . E 4 2d . 2 0 1 + 1 2 , 0 + 9 + - 2 . *- 1 . 5K. 20 1- 1 l1 d+1 , - 4 s1 = l1 d+1 , 0 2- 1 d 1 =
l2s+1 (l1 +1)(l2 +1) . ., - 0 k- , 1 Q d . E 4 d (l +1) =1 Q (l +1) . k
k
k
k
i=1
i
i
i
; 0 - . * 0 1.
3.
789
0 E , . . 0 , 0 5S. = 0 , 0, - 11 . 3.1. ) (l1 m19 l2 m29 : : : 9 lk mk ) * !, r- ,
"
bd2
2k+r lr mr .
. G r- b xr y, xr | , y | . D4
xr . E 5S E r- -
, 2d . > 3 , - 0, - , 2 dl . D 0, 4
xr 2 ldm . 1 - y, 5S, 2d . = , r- bd2 2+lm . - 1 0 . E 1 0 0 1 0. = , - - E , . 3.2. ) (l1 m19 l2 m29 : : : 9 lk mk ) i = 1 2 : : : k ; 1 li+1 mi+1 < mi , li+1 mi+1 = mi li > 3: . * li+1mi+1 > mi. = li+1mi+1 > mi - i = 1 2 : : : k ; 1, 0 (i + 1)- : 1, , 0 0 i, 0 i 0 (i + 1). = , : 4 0 - 1 , . ?-, - li+1 mi+1 + , - mi . B li+1 mi+1 = mi , 0 (i +1) : 01 1 mi 4 0 i- ,
2 2 ;1bd(l 2+1)m > 2 +1 2 bdl +12 m +1 . - - - , - 0 (k ; 1)- . = li+1 mi+1 = mi , - , - li < 3. ?-, - , li+1 mi+1 = mi , li > 3. > , , -, - - 0 - 0 0. r
r
r
r
r
r
k
k r
r
r
i k
i
i
i
k
i
i
790
. .
3.1. 20 (1 m19 1 m29 : : : 9 1 mk), m1 > m2 > : : : > mk , 0 - (m1 m2 : : : mk ). E 3.1. ,
r- , 2
0 2 +bd m . 3.3. + " (m1 m2 : : : mk) | . - mi > 2k;i+2, i-
!- . . k r
r
.
. G i- 0 2 bd m . = 2
i+k
- , 0, +
--. : 1) mk > 1, -- 0
0 k + 19 1) mk = 1, -- 0 1 0 k. D 4 1 . B mi ; 1 > mi+1 , 0 i- (i +1)- - , mi ; 1 = mi+1 , i- (i + 1)- , - -4 - . > , , - -4 -4 - . >, 4 : ) mk > 1, mi ; 1 > mi+1 , (k +1)9
) mk > 1, mi ; 1 = mi+1 , k9 ) mk = 1, mi ; 1 > mi+1 , k9 ) mk = 1, mi ; 1 = mi+1 , (k ;2 1). * --
, : ) 2 +1bd2 +1 9 ) 2bd22 9 ) 2 bd22 2 9 ) 2 ;1bd22 ;1 2 . * , c , , + , - , i- 0. -4 4 i- 0. k
k
k k
k k
i
k
k
3.1. / ,
: (2k+1 : : : 16 8 4). 3.1. ) i- $ "% .
. < 1 E (m1 m2 : : : mk ). G 0 i- 0 - 2i+k mi . * S = fmj : j 2 J 9 J f(i + 1) (i + 2) : : : kgg r 0 + . * 1 r 1 1 6 r 6 (k ; i). * : 0 i- , r 0 S 0 i- , 0 r - . = ,
P i + k ; r 0 0 - 2 mi + mj . = j 2J
791
mj < mi j 2 J , 2i+k;imi + P mj < 2i+k;r (r + 1)mi . j 2J i+k;r (r + 1)mi 6 2i+k mi , r > 1. I1 - 1 - 2 2i+k;r mi + P mj < 2i+k mi , . . , 01 i- j 2J , - : 1 1 . D 4 E (m1 m2 m3 ) 1 : 0 0 . D 0 (m1 + m2 m3), (m1 + m3 m2) (m1 m2 m3 ). G , 01 , 0 , 0 - 24(m1 +m2 ), 24(m1 +m3 ), 2 8 m1 . @ , m1 > m2 , m1 > m3 , - 0 2 4 (m1 + m2 ) < 2 4 2m1 2 4 (m1 + m3 ) < 2 4 2m1. * -
2 8 m1 . E , 2 4 (m1 + m2 ) < 2 8 m1 2 4 (m1 + m3 ) < 2 8 m1 . = : , , , - 0 + , ( ). D , 4 1 E (8 2 1). 20 : 0 - + . @ , 0
: , 01 - 482 = 64 2410 = 80. = 0 : 0 .
3.4. . . . * , . . - 4 -
(m1 m2 m3). I , -
0 0 - 2 8 m1 , 4 8 m2 , 8 8 m3 . @ - 1- 2- 0 : 1 (1 m1 + m2 9 1 m3), , - . > 3.1
, - 1- 0 : , , 2- 0 , . ?-, 2 4 (m1 + m2 ) > 4 8 m2 m1 + m2 > 4m2 - 4 m1 > 3m2 . F -, - 1- 3- 0 : (1 m1 + m3 9 1 m3), 1 : 2 4 (m1 + m3 ) > 8 8 m3
792
. .
m1 + m3 > 8m3 :
?-, m1 > 7m3 . >, - 4 : m1 > 3m2 , m > 7m3 . E , - 4 m1 > m2 + m3 . = , - 2- 3- 0 : (1 m19 1 m2 + m3 ), 4 4 (m2 + m3 ) > 8 8 m2 m2 + m3 > 4m2 - m2 > 3m3 . >, - 4 m1 > 3m2 , m1 > 7m3 4 m2 > 3m3. I 1- 0 , J L 01 m2 m3 , - 4 -. = + 0 , , 0 m2 , - (1 m1 ; m2 9 2 m29 1 m3), 0 m3 , - (1 m1 ; m3 9 1 m29 2 m3). = 3.4 .
3.5. +
-
" .
. @ , 1- 1 - 4 . + . B , 3.4, + . = , , - . 3.2. - !
12, .
. @ , - - 0 . I 1 3.1 - 12 , | + 4. 3.6. (l1 m19 l2 m2) : ) l2 m2 < m1 , l2 m2 = m1 , l1 > 31 ) l1 6 41 ) l2 6 41 ) m2 6 41 ) m1 6 10. . G ) 3.2.
793
@ ). B l1 > 4, - 0 1- : 1 (1 2m1 9 (l1 ; 2) m19 l2 m2), , , 01 1- , 1 28bd22m1 ,
+1, - 24bdl12m1 . ?-, - l1 6 4. @ ). B l2 > 4, - 0 2- : 1 (l1 m19 1 2m29 (l2 ; 2) m2), , , 01 2- , 1 48bd22m2 ,
+1, - 44bdl22m2 . ?-, l2 6 4. @ ). * m2 > 1 - 0 2-
--, 4 + , . . 4 4 l2 m2 > 64, l2 m2 > 4. ?-, - l2 m2 6 4. I1 ) m2 6 4. @ ). = m1 > m2 , m1 > 1. * m1 > 1 m2 > 1 - 0 1- --, 4 + , . . 2 4 l1 m1 > 64, l1 m1 > 4. ?-, - l1 m1 6 4. I1 ) m1 6 4. * m1 > 1 m2 = 1 - 0 1- --, 4 + , . . 2 4 l1 m1 > 4 4 (l2 +1), l1 m1 > 2(l2 +1). ?-, - l1 m1 6 2(l2 +1). I1 ) ) m1 6 10. , , 0 + , 3.3. 2 2 .
. E (2 29 2 1) - . , 1 : ) (2 29 1 2)9
) (1 49 1 2)9 ) (1 49 2 1)9 ) (1 29 4 1)9 ) (1 39 1 29 1 1). * , - - 0 1- 0 2- . I -4 1, - 0 , , .
4.
0 2 , . . 0 , 0 52, , 0 E . = 0 , 0, - 11 . 4.1. ) (l1 m19 l2 m29 : : : 9 lk mk ) * !, r- ,
" Q bd2 Q . m (l +1) (l +1) r
r
i=1
k
i
i=1
i
794
. .
. G r- b xr y, xr | , y | . D4
xr . E 5K 2 0 r- , Q r(dl +1) . E 4 xr =1 m Q dr(l +1) . 1 - y, =1 5K, Q k(dl +1) . = , r- =1 bd2 Q Q m r(l +1) k(l +1) . i
r
i
i
i
i
r
i=1
i
i
i=1
i
- 1 0 . 4.1. 20 (l m) 0 . E 2 4.1 0 bd 2 . (l+1) m 4.2. + " N = l m = l0 m0, l > l0 >0 1, 0 (l m) * , (l m ),
" " ! "% , .
. G bd22 p0 = 0 bd22 0 , - - , p = (l+1) m (l +1) m - (l +1)2 m > (l0 +1)2 m0 . < - , - l m = l0 m0 = N , (l ; l0 )N > m0 ; m, - , - + N , + N . 4.1. ) l 6 m. >, , , 0, 0 - -+ , - - , -+ | 0 0 , . . , , -+ 2 | 0 + . E 1 , -, , - - | 4.2 0 : 1
1. 4.3. + (l m) * : ) l > 1 m > 161 ) l > 3. . G bd22 . > 4.2 , - l 6 m. (l+1) m - ) , , 0 --. @ ,
795
(l ; 1 m9 1 m ; 219 1 1), -- d bd 4l , 16l22 . * 4 + , - , 2 (l + 1) m > 16l l > 1, m > 16. C . - ) : 0. @ , (1 2m9 l ; 2 m)9 , 0 1- 2d , 2(ld;1) , 2 0 1- 22(lbd;1)2 m . * 0 + , - : , 8m(l ; 1) < (l + 1)2 m, (l ; 3)2 > 0, l > 3. C . * - l 6 3, - 11 0 m l = 3 2 1. @ l = 3 16m 6 16 9,
m 6 99 l = 2 9m 6 164, m 6 79 0, l = 1 4m > 16, m 6 4. >, - 1 2 : (1 m), m 6 49 (2 m), m 6 79 (3 m), m 6 9. I - - . E (3 m) -4 m - , (2 32m ) -+ 4.2 , 0 .
5.
< 01 (l1 m19 l2 m29 : : : 9 lk mk ).
5.1. ) 3 " ! "%
4 * ! *
" . . * (l1 m19 l2 m29 : : : 9 lk mk) | 0 . = 2 r- 2 bd , E , Q (l +1) Q (l +1)m r
k
i
j
r
r k Q bd . ?-, , - 2r 2k lr mr 6 Q ( l + 1) (lj + 1)mr , i 2 2 lr mr i=1 j =1 Q r k r Q Q 2r 2k lr 6 (li + 1) (lj + 1). = li > 1 i, 2r;1 6 (li + 1) i=1 j =1 i=1 k Q 2k;1 6 (lj + 1), - , - 4lr 6 (lr + 1)2 . < j =1j 6=r 4 4lr 1 -. *- 0 6 lr2 + 2lr + 1 ; 4lr , . . 0 6 (lr ; 1)2 : ( ) 2
r k
i=1
j =1
796
. .
. M , -r li > 1, ( ) . @ , lr > 1, 2r;1 < Q (li + 1), i=1 k li > 1 - i 6= r, 2k;1 < Q (lj + 1). B li = 1, j =1 j 6=r ( ) - 0 . C . ? , - 1 , - 0 2 l1 = l2 = : : : = lk = 1, , 0 E .
%
1] . . // . . . | 2001. | #. 7, & . 2. | . 433{440. 2] *& +. ,. * -. | *., 1998. 3] Cournot A. Recherches sur les principles mathematique de la theorie des richesses. | Paris, 1938. 4] Von Stackelberg H. Marktform und Gleichgewicht. | Wien, Berlin, 1934. 5] 1 ., 2 3. & & . | *.: ,-*, 1997. 6] 7 8 . ., 29 :. *. #; 9< & & . | *.: *, 1998. ' ( ) 2001 .
L-
. .
513.83
: L- , E1 - , ,
, .
! " ! ! L- !# # $ ! " . %
& " L- ' !# $ !# . (
$ )-
, !# & &* ! " !# # . * " !# )-
L- # .
Abstract S. V. Lapin, L-theory for local systems of homotopy coalgebras and the exact sequence for surgery, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 797{828.
In this work various versions of L-theory for local systems of homotopycoalgebras over simplicial complexesare developed. The relation to L-theory for local systems of chain complexes is examined. We give a construction of )-spectra whose homotopy groups are equal to the corresponding bordism groups of local systems of homotopy Poincare coalgebras. By means of these )-spectra the exact sequence for surgery in L-theory for homotopy coalgebras is obtained.
1] . . . . ! " !#$! %!. !& # '#& #( )!( K-, ! ! ( $ !+ ! $ $ , " %$ ! %(. " ( $, 2] . .#( % L-& #$ $ ! ! " !#$! %!. .#$( L- $ !, , 2 & 3 & 4 %556 ( 7 96-01-00158). , 2001, 7, 7 3, . 797{828. c 2001 , !" #$ %
798
. .
! ( ( L- { . 3 ( + L-, %$( $! !!, % "$! " ! ( " 4{5{ : : : ;! S Top (M) ;! M8 G=Top] ;! Ln () " n-! ! % M, = 1(M), n > 5. !! ", . .#! 3] " A L (Z]) ;! @ S (M) ;! : : : : : : ;! Sn+1 (M) ;! Hn(M L ) ;! n n Ln (Z]), %!' ( " 4{5{ . $ ( " .# & ! & "%! #$ $ ! "( & . , !+ " S Top (M) ( (. 5 ! L- $ ! ! & )( L- ( ( " . ! ' !& "!$ # L- $ ! #$ ! " !#$! %!. ! ' "& % $ $ L- $ ! ! & % L-( $ ! #$ ! . "!, !, ' "& # <- , ! ! ! $ + & $ "%! $ ! ! . ! ( " %$ , L- ! " . 1. L-
)! ' !& $ # L- $ ! #$ ! " !#$! %!, %( . .#! 2] (+ !. 4]). K | !! # "#(. Z- "$( #( ! !$ K-!"( fCi d: Ci ! Ci;1g "! %$ $!, Ci , % & ! !+ $ " i 2 Z, & $!. @ & $ "$ K-!"$ #$ ! % ! % Chain(K). X | !# %. . !! X &, A! ( & ! $ 2 X, !'%!! +
L-
799
+ ! . 3 "! , !+ , % X %" $( ($( ". B " " +" n-! ! 2 X "$! %! " " !+ (n ; 1)-!$ ( f@0 @1 : : : @n g: )! X ! $" !( !'%! f@i j 2 X 0 6 i 6 dim()g , " ", +" !'%! )( . E( !( K-!"$ #$ ! " !#$! %! X %$ $( ' C : X ! Chain(K): '%!! $ ! " X & $ % & '. @ & $ ! K-!"$ #$ ! " !#$! %! X % ! % ChainK X]. E& ! C : X ! Chain(K) " " $ "$ K-!"$ #$ ! C = fC] d]: C] ! C];1 j 2 X g #$ +( @i : C] ! C@i] 0 6 i 6 dim() "& ,! @i @j = @j ;1@i , i < j. , !'%! f : C ! B $ ! " X " ( !( #$ +( f = ff]: C] ! B] j 2 X g @i f] = f@i ] @i , 0 6 i 6 dim(). X 0 | # "%" !# % X B( X) = f^0^1 : : : ^p 2 X 0 j 0 1 : : : p g | # %%" ! 2 X. @ ! !! ( !$ #$ ! " X + $( ' C : X ! Chain(K), " C] = G (B( X)) | ) K-!"$( #( ! # ( %%"$ B( X) ! 2 X. G n | K-!"$( #( ! !# % " ! Gn . + i : Gn;1 ! Gn, 0 6 i 6 n, (n ; 1)-! ! i-& n-! ! "& #$ + i : G n ;1 ! G n 0 6 i 6 n
800
. .
$ %$ ,! j i = i j ;1, i < j. . !! !( #$ ! X CnX] = C] n > 0: dim()=n
H$ + @i : C] ! C@i] , 0 6 i 6 dim(), "& #$ + @i : CnX] ! Cn;1X] 0 6 i 6 n " $ $& , @i @j = @j ;1@i , i < j. B! %!, fCnX] @i : CnX] ! Cn;1X] 0 6 i 6 ngn>0 | ) " !#$( A ( !. 5,6]) Chain(K).
( ( !$ C = fC] j 2 X g 2 ChainK X] %$ $( "$( K-!"$( #( ! X CX] = Cn X] K G n = n>0
" , ) +" ,! @i x y x i y x y 2 CnX] K G n ;1 n > 0: B! %!, CX] | ) # %# ( !. 5, 6]) " !# A fCnX] @i : CnX] ! Cn;1X] 0 6 i 6 ngn>0 Chain(K). E ", !'%! f : C ! B 2 ChainK X] "# # + fX]: CX] ! BX] 2 Chain(K) "! %$ ( !'%! f. = 1(X) | '"! !# % X, K] | # $ , X~ | !# % $ " X #( p: X~ ! X. . !! !( K]-!"$ #$ ! ~ = X Cp(~)] ~ 2 X ~ n > 0: Cn X] dim(~)=n
H$ + @i : Cp(~)] ! C@ip(~ )], 0 6 i 6 dim(p(~)) "#& #$ + K]-!"( ~ ! Cn;1X] ~ 0 6 i 6 n @i : CnX] $ %$ ,! @i @j = @j ;1@i , i < j.
L-
801
4 ( ( ( !$ C = fC] j 2 X g 2 ChainK X] %$ $( "$( K]-!"$( #( ! ~ = X Cn X] ~ K G n = CX] n>0
" , ) +" ,! ~ K G n n > 0: @i x~ y x~ i y x~ y 2 CnX] '%! $ ! f : C ! B 2 ChainK X] " # + K]-!"( ~ CX] ~ ! BX] ~ fX]: "! %$ ( ( !'%! f. '%!$ f g : C ! B ChainK X] %$& !$!, ! # ! s = fs]: C] ! B]+1 j 2 X g !+" !'%!! f g, . . d] s] + s] d] = f] ; g] @i s] = s@i ] @i " & ! 2 X, 0 6 i 6 dim(). 5" , , ! !+" !'%!! ChainK X] ,! ) . '%! $ ! f : C ! B " X %$ #( ) &, ( !'%! $ ! g: B ! C, !%# f g g f !$ &! +" $! !'%!! ChainK X]. E ! C = fC] j 2 X g 2 ChainK X] %$ (K X)- "(, " +" ! 2 X & i 2 Z %" ( "$( "!" C()i C]i, + X C()i ! C]i
"! $! '! C : X ! Chain(K), %!'%!! K-!"( " & 2 X. N% ! % Chain(K X) & " & ChainK X], A! ( + (K X)- "$ $ !$ " X.
802
. .
. ! $, ! C : X ! Chain(K), " C] = = G (B( X)), (K X)- "( ( !( " X. !! ", @B( X) = f^0^1 : : : ^p 2 X 0 j 0 1 : : : p g | # # ( %%"$ ! 2 X Gi(B( X)8 @B( X)) = Gi (B( X))=Gi (@B( X)) | "$( K-!" $ i-!$ !#$ #(. O + C()i = Gi (B( X) @B( X)) ! %+ !& !! X C]i = C()i :
. !! " (K X)- "$ $ ! #$ ! " !#$! %! X. C = fC] d]: C] ! C];1 j 2 X g 2 Chain(K X): B " " % i 2 Z !! X X C]i = C()i d](C()i) C()i;1 :
P% ) ", " & ! 2 X " "$( $( K-!"$( #( ! (C() d()), " d(): C() ! C();1 | ) ! d], & +" ! +& . , f = ff]: C] ! B] j 2 X g 2 Chain(K X): B " % %+ !& !! X C]i = C()i ",
f](C()i )
X
B()i :
, " & ! 2 X " # + f(): C() ! B() $ ! (C() d()), (B() d()), !( # + f], &( +" ! +& .
L-
803
C : X ! Chain(K) | (K X)- " ! #$ ! " %! X. . !! $( ' C $( ' C : X op ! Chain(K) " X op | , "( X. N"! $( "$( K-!"$( #( ! (C(X) d(X)), C(X) = lim ;! C] " !( " " ( X op . 5" ", X X C(X)i = C()i d(X)(C()i ) C()i;1: 2X
@! , !'%! f : C ! B Chain(K X) "# " # + f(X) = ; lim ! B(X) ! f] : C(X) X f(X)(C()i ) B()i :
3 (K X)- "( ( !$ C : X ! Chain(K), " C] = = G(B( X)) C()i = Gi (B( X)8 @B( X)), ! C(X) #$! ! ! G (X 0 ) # "%" X 0 !# % X. X | !# %. K-!" M %! X-" $!, %" %+ !& !! X M= M() 2X
" fM()g | !( "$ K-!"(, " ! ! 2 X. '%!! X-" $ K-!"( f : M ! N %$ ( K-!"$( !!'%!, X f(M()) N():
@ & X-" $ K-!"( "! % % ModX (K). X | !# %. H( ! C = fCi di : Ci ! Ci;1g 2 Chain(K) %$ X-" $!, Ci 2 ModX (K) di 2 ModX (K) " & i 2 Z. '%!! f = ffi g : C ! B X-" $ ! %$ # + f, fi 2 ModX (K) i 2 Z:
804
. .
@ & X-" $ #$ ! "! % % ChainX (K). R!( !+" !'%!! f g: C ! B 2 ChainX (K) %$ # ! s = fsi g, i 2 Z, !+" f g 2 Chain(K), si 2 ModX (K). E ", , ! !+" !'%!! ChainX (K) ,! ) . #( ) ChainX (K) " "$! %!. . !$( $, ' lim ;! : Chain(K X) ! ChainX (K) %!'%!! (. !! ", X-" $( #( ! K-!"( X X C = C = C() d(C() ) C();1 2X
"% " (K X)- "& & ! #$ ! C] = fC]] d]: C]] ! C]];1 j 2 X g " !#$! %! X, " X C]]i = C()i
$ #$ + @i : C]] ! C]@i] 0 6 i 6 dim() %"& +! ! !. 5" ", C(X)] = C B](X) = B, " C 2 Chain(K X), B 2 ChainX (K). , # + X-" $ ! K-!"( "% " !'%! & (K X)- "$ $ !. f : C ! B 2 Chain(K X), = 1(X). + %, + f(X): C(X) ! B(X) #( ) & ChainX (K) " ", " " +" ! 2 X # + f(): C() ! B() #( ) &. @! , f(X) | # ) ChainX (K), ~ CX] ~ ! BX] ~ fX]: !'%! f #( ) & K]-!"(.
L-
805
. !! C]X] ( !$ C] 2 Chain(K X), " C 2 ChainX (K). N"! # +
: C]X] ! C](X) (x y) = x "(y), " ": G n ! K, n > 0, | !# # ! G n . 5" , #( )~ | ( !$ &. , C]X] C] 2 Chain(K X), " C 2 ChainX (K). @! , ~ = lim Cp(~)] C](X) ;! ~ " p: X ! X | # $, !( " " ( X~ op . B " # + ~ ! C](X) ~
~ : C]X] "! +, $,, #( ) & K]-!"(. ("! & #( "( ChainX (K). ) !! "! , X $! " $! !#$! %!. ", " +" ! 2 X !+ f 2 X j dim() = dim() + 1g " $!. . ! $ ) !+ " % , %,! " f 0 1 : : : s : : :g: X M= M() 2 ModX (K): 2X
N"! $( ' D : ModX (K) ! ChainX (K) X (DM)i = DM()i 2X 8P < M() i = ; dim() DM()i = : 0 i 6= ; dim() " M() = homK (M() K) | ) +$( K-!" " M(). R $( d: (DM)i ! (DM)i;1
806
. .
%" '!( d = s : + +
P (;1)s , " s
s>0
X
M()
X
s
!
X
M()
s
X
M()
|
M()
! !. P % #& ! # ! , "$! %! "+! $( ' D : ModX (K) ! ChainX (K) " ' D : ChainX (K) ! ChainX (K) $( %$ #( "( & ChainX (K). H( ! (DC) 2 ChainX (K) %$ "$! #! ! C 2 ChainX (K). 3 "! % % (DC);i = C i (Df);i = f i i 2 Z: M N 2 ModX (K). . !! #( ! homX (DM N) , " homX (DM N)n | ) K-!" !'%! ModX (K) n. E ", ! ! %!'%! X X X homX (DM N)n = M() K N() : dim()=n
+ %! %" " %!'%! K-!"$ #$ ! T : homX (DM N) ! homX (DN M) : C B 2 ChainX (K). P%!'%! T " "+ " # %!'%! T : homX (DC B) ! homX (DB C) : O + C = M, B = DM, " M 2 ModX (K) ChainX (K) ! #( %!'%! T : homX (DM DM) ! homX (D2 M M) : S ) %!'%! 0-# id: DM ! DM $! 0-#! e(M): D2 M ! M:
L-
807
+ , # + e(M) " % $ ' e: D2 ! 1, " $& "& : 1) e(DM) D(e(M)) = 1: DM ! D3 M ! DM, 2) e(M): D2 M ! M | # ) . . !! %$ %" $ ! " X. B%$! %"! $ ! C B 2 ChainK X] %$ ! C K B = f(C K B)] j 2 X g 2 ChainK X] " (C K B)] = C] K B] : O C B 2 Chain(K X), C K B 2 Chain(K X). !! ", + X X (C K B)()i = C( )s K B( )t i 2 Z =\ s+t=i
! %+ !& !! X (C K B)]i = (C K B)()i i 2 Z:
. !! (C K B)X] ( !$ C K B, " C B 2 2 Chain(K X). 5" %, (C K B)X] = homX (DC(X) B(X)) :
("! & % $ L- Chain(K X). C 2 Chain(K X) W | % KZ2]-!" K: 1;T 0 ; KZ2] ; KZ2] 1+ ;T KZ2 ] ; : : : " T | %& Z2 . . !! Q- $ Qn(C) = Hn(homK Z2 ] (W 8 (C K C)X])) n 2 Z " KZ2 ]-!" (C K C)X] "# ( T !+( ( ! C K C. '%! f : C ! B Chain(K X) "# !!'%! Q- f n : Qn (C) ! Qn(B), n 2 Z. O f | # ) Chain(K X), !!'%!$ f n & %!'%!!. @ ! ( ' 2 Qn(C) " ( ) #( f's 2 (C K C)X]n+s s > 0g " $ $& , d('s ) = (;1)n ('s;1 + (;1)s 's;1 T ) s > 0 ';1 = 0:
808
. .
. !$( $, #( %!'%! (C K C)X] = homX (DC(X) C(X)) %!'%!! KZ2 ]-!"(, " KZ2]-!" homX (DC(X) C(X)) " ! (f)T = D(f) f 2 homX (DC(X) C(X)) T 2 Z2 : B! %!, Hn(homK Z2 ] (W8 (C K C)X]) = Hn(homK Z2 ] (W 8 homX (DC(X) C(X)))): )! ! ( ' 2 Qn (C) " ! !'%! X-" $ K-!"( f's : C(X)n;i+s ! C(X)i s > 0 i 2 Z g " $ $& , d('s ) = (;1)n ('s;1 + (;1)s 's;1 T ) s > 0 ';1 = 0 " d('s) = dC 's + (;1)n+s;1 's dDC (X ) : H( ! C 2 Chain(K X), !!$( ! )!! ' = f's g 2 Qn(C), %$ n-!$! $! ! ! ! " X, !'%! '0 : C(X) ! C(X)n; 2 ChainX (K) #( ) & ChainX (K). '%!! ( ! ( ) &) f : (C f'Cs g) ! (B f'Bs g) n-!$ $ ! " X %$ ( !'%! ( # ) ) Chain(K X), f n (f'Cs g) = f'Bs g, n 2 Z. 3 n-!$ $ ! (C f'Cs g) (B f'Bs g) " X " ! !! (C f'Cs g) (B f'Bs g) = (C B f'Cs 'Bs g): " %!! # ! (C f'Cs g) ! ! ! (;(C f'Cs g)) = (C f;'Cs g): %" %$( !'%! f : C ! B Chain(K X) . !! $ Q- $ Qn+1(f) = Hn+1 (homK Z2 ] (W8 C((f K f)X]))
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809
" C((f K f)X]) | # + (f f)X], KZ2]-!" C((f K f)X]) "# ( T !+( %! %". @ ! ( 'f 2 Qn+1 (f) " ( ) #( f( 's 's) 2 (B K B)X]n+1+s (C K C)X]n+s s > 0g " $ $& , d( 's) = (;1)n+1 ( 's;1 + (;1)s 's;1T) + (;1)n+s (f K f)X]('s ) d('s) = (;1)n+1 ('s;1 + (;1)s 's;1 T ): B !& KZ2 ]-!"$ #$ %!'%!$ (B B)X] = homX (DB(X) B(X)) (C B)X];1 = homX (DC(X) B(X));1 !+ , ! ( 'f 2 Qn+1 (f) " "! !( ! !'%! X-" $ K-!"( f 's : B(X)n+1;i+s ! B(X)i s > 0 i 2 Z g f's : C(X)n;i+s ! C(X)i s > 0 i 2 Z g " $ $& , d( 's) = (;1)n+1 ( 's;1 + (;1)s 's;1T) + (;1)n+s f(X) 's Df(X) d('s) = (;1)n+1 ('s;1 + (;1)s 's;1 T ): '%! f : C ! B 2 Chain(K X), !!$( ! )!! 'f = f( 's 's)g 2 Qn+1(f), %$ (n + 1)-!( ( ( ( " X, !'%! ( '0 '0): C(f(X) ) ! B(X)n+1; 2 ChainX (K) ( '0 '0)(g h) = '0(g) + f(X)('0 (h)) (g h) 2 B(X)i C(X)i;1 #( ) & ChainX (K). %" (n + 1)-! (f : C ! B f( 's 's)g) " X. B " (C f(;1)s's g) n-!$! $! ! ! ! " X, $( %$ #( $ (f : C ! B f( 's 's)g). n-!$ $ ! $ (C f'Cs g) (B f'Bs g) " X %$& "$!, (n+1)-! " X, #( (C f'Cs g) (;(B f'Bs g)) = (B C f'Cs ;'Bs g): N, " n-!$ $ ! " X ,! ) . + " n-!$ $ !
810
. .
% ! % Lnl (K X). N# !( !!$ %! # %"& Lnl (K X) ( $. H( ! C 2 Chain(K X), !!$( ! )!! ' = f's g 2 Qn(C), %$ n-!$! $! ! ! ! " X, ~ C(X) ]X] ~ ! CX] ~ n; '0 ]X]: !'%! '0] 2 Chain(K X) K]-!"( #( ) &. '%! f : C ! B 2 Chain(K X), !!$( ! )!! 'f = f( 's 's )g 2 Qn+1(f), %$ (n + 1)-!( ( ( ( " X, ~ C(f(X) )]X] ~ ! BX] ~ n+1; ( '0 '0 )]X]: !'%! ( '0 '0)] 2 Chain(K X) K]-!"( #( ) &. & n-!$ $ ! " , " n-!$ $ ! . N, " n-!$ $ ! " X ,! ) . + " n-!$ $ ! % ! % Lng (K X). N# !( !!$ %! # "& Lng (K X) ( $. (C f'sg) | n-!$( $( ( ! " X. B !'%! '0 ) & ChainX (K), ~ | # ) K]-!"(. '0]X] )! (C f'sg) n-!$! $! ! ! ! , ", " ( !!'%! A: Lnl (K X) ! Lng (K X) n 2 Z: E$( ( ! (C ') " X %$ ~ ( !$ !$!, CX] C 2 Chain(K X) !$! #$! ! ! K]-!"(. " ! " X #. n-!$ !$ $ ! $ (C 'C ) (B 'B ) " X %$& "$!, (" (n + 1)-! ! " X, #( (C 'C ) (;(B 'B )) = (C B 'C ;'B ): N, " n-!$ !$ $ ! " X ,! ) . + " (n ; 1)-!$ !$ $ ! " X % !
L-
811
% S n (K X), n 2 Z. N# !( !!$ %! # "& S n (K X) !!( $. R$ Lnl (K X), Lng (K X) S n (K X), n 2 Z, %$ !+" ( "( ( " & A Ln (K X) ;! S n (K X) ;! : : :: : : : ;! S n+1 (K X) ;! Lnl(K X) ;! g Chain(K]) | $ "$ K]-!"$ #$ ! , " K] | K- "( $ . # Lnl (K X) "!, %! & Chain(K X) & Chain(K]), $ Ln (K]), n 2 Z. U ( " ( !!'%! : Lng (K X) ! Ln (K]) n 2 Z " = 1(X). O K = Q, #$ !!'%!$ Q & %!'%!! " & n 2 Z. " ( $, $ Lnl (K X) & ( ( ! ( Hn(X L ), n 2 Z, "!( $! <- ! L . P% ) ", K = Q % $, " # ! " A : : : ;! S n+1 (K X) Q ;! Hn(X L ) Q ;! A Ln (K]) Q ;! S n (K X) Q ;! : : :: ;! V " " %$ #( ( " & " . 2. L-
)! ' "& % $ $ L- $ ! ! & % L-( $ ! #$ ! . 5!! $ #, %$ ! "$ ! ( $ ( !. 7, 8]). !! ! !( ! E = fE (j)gj >1 %$ !( K-!"$ #$ ! E (j), $ "( & !! $ Wj . '%!! !! !( f : E 0 ! E 00 + !( #$ +( ff(j): E 0(j) ! E 00(j)gj >1 $ "( ! !! . 3 !! !( E 0 E 00 " !! !( E 0 E 00 = f(E 0 E 00)(j)gj >1
812
. .
" (E 0 E 00)(j) | '-! " Wj -! , +" #$! ! ! X E 0(k) E 00(j1 ) : : : E 00(jk ) j1 +:::+jk =j
! ,& ) ( !. 8]). 4% -%" #$!, . . " %$ !! !( E , E 0, E 00 ! ! %!'%! E (E 0 E 00) (E E 0 ) E 00:
!! !( E %$ "(, %" ( !'%! !! !( : E E ! E !!$( "$! !+!, ( 1) = (1 ). @! , ! ( )! 1 2 E (1)0, (1 ej ) = ej ej 2 E (j) j > 1 (ek 1 : : : 1) = ek ek 2 E (k) k > 1: '%!! " + !'%!$ !! !( , $ "$! !+!. @ ! !! "$ " E C = fE C (j)gj >1 , # #$! ! ! C 2 Chain(K). 3"! . C j | j- % " K # ! C. N"! K-!"$( #( ! E C (j) , E C (j) = homK (C8 C j ) " homK (C8 C j ) | #( ! K- !!'%! C ! C j , ! E C (j)0 = homK (C8 C j )0 | ) K-!" #$ +(. 3( !! ( $ Wj E C (j) " "( ! Wj C j ( !+(, $ $! ,! % . N" !+ C : E C E C ! E C %" '!( C (g g1 : : : gk ) = (g1 : : : gk ) g " gi 2 E C (ji ), 1 6 i 6 k, g 2 E C (k). )! 1 2 E C (1)0 = homK (C8 C)0 +" +. 3 ! +$! !! "$ + " E = fE(j)gj >1 . 3"! . G n | #( ! K-!"( " !# % ! n-! ! Gn. H$
L-
813
+ i : G n ;1 ! G n , 0 6 i 6 n, "#$ +! (, "& #$ ! G = fG n gn>0 " !# A ( !. 5,6]) Chain(K). (G )j = f(G n )j gn>0 | j- % " K ! G . N #$ +( ( i )j , 0 6 i 6 n, #$ ! (G )j " !#$! A! Chain(K). N% ! % E(j) #& %#& " !# ! (G )j , . . E(j) = hom(G 8 (G )j ) " hom(G (G )j ) | #( ! " !#$ K- !!'%! E(j)0 = hom(G 8 (G )j )0 | K-!" " !#$ #$ +( G ! (G )j . N" !+ : E E ! E " +, " "$ E C . E ", " +" j > 1 #( ! E(j) Wj - "$! # $!. . !! 0-!$( # 0 2 E(2)0 = hom(G 8 G K G )0 $( %" %& (0 1 : : : n) 2 G nn, n > 0, '!( n X 0(0 1 : : : n) = (0 1 : : : i) (i i + 1 : : : n): i=0
H( ! C 2 Chain(K), !!$( ! %"$! !'%!! " : E ! E C %$ ! ( (, E- (. O C ! ( (, $( !'%! "% " Wj -)$ #$ + j : C K E(j) ! C j j > 1 1(c 1) = c, j (1 ) = g (k 1 : : : 1), " k X g = (j1 : : : jk ) U js = j s=1
U | " & +. '%!! E- f : C ! B %$ # +, Bj (f 1) = f j Cj j > 1: @ & K-!"$ E- % ! % ECoalg(K). X | !# % G (X) | ! !#$ #( % X )''#! K. 5 #! !
814
. .
G (X) $! %! " E- $. !! ", x 2 Gn(X) | %&, 2 E(j) x: G n ! G (X)] | # + K-!"(, " %&& (0 1 : : : n) 2 G nn )! x. N"! Wj -)$ #$ + j : G(X) K E(j) ! G (X)j j > 1 %& x 2 Gn(X) j (x ) = (x : : : x)(0 1 : : : n): N+ j , j > 1, "& !'%! " : E ! E " (X ) " (2)(0) | " !# " +. B! %!, #( ! G(X) ! ( (. X , !# + "# !'%! & E- . ! & L- $ ! ! " !#$! %!. X | " !# %, "! ! & ( !. %" 1). 1. E( !( K-!"$ ! " !#$! %! X %$ $( ' C : X ! ECoalg(K): '%!! $ ! E- " X + $ % & '. @ & $ ! K-!"$ ! " !#$! %! X % ! % ECoalgK X]. E& ! C 2 ECoalgK X] " " ! C = fC] 2 ECoalg(K) j 2 X g !'%! E- @i : C] ! C@i] 0 6 i 6 dim() "& ,! @i @j = @j ;1@i , i < j. , !'%! f : C ! B 2 ECoalgK X] " ( !( +( ! f = ff]: C] ! B] j 2 X g @i f] = f@i ] @i , 0 6 i 6 dim().
L-
815
X 0 | # "%" !# % X B( X) X 0 | # %%" ! 2 X. "$! !! ( !$ K-!"$ ! " X + $( ' C : X ! ECoalg(K), " C] = G (B( X)) | #( ! # ( %%"$, !&( !& $, E- $. 2. ( ( !$ C 2 ECoalgK X] %$ CX] 2 Chain(K) ( !$ #$ ! , ( % C ! E- $. = 1(X) | '"! !# % X, K] | K- $ , X~ | $ " X. 3. 4 ( ( ( !$ C 2 ECoalgK X] ~ 2 Chain(K]) ( !$ %$ CX] #$ ! , ( % C ! E- $. . E ", CX] ( !$ C 2 2 ECoalgK X] ! & ! ( $, # %# " !#( E- $ ~ . E- ( ( !. 8]). " ( CX] 4. E ! C 2 ECoalgK X] %$ (K X)- "(, , !! ! #$ ! , + Chain(K X). & " & ECoalgK X], A! ( & (K X)- "$ $ !$ ! " X, % ! % ECoalg(K X). !! (K X)- "( ( !$ E- + ! $, ! C = fC] = G (B( X)) j 2 X g: C = fC] j 2 X g | ! ! " %! X Cj ] : C] K E(j) ! C] j 2 Chain(K) | $ + E- $ C]. . !! & ! C E(j) 2 ChainK X], " (C E(j))] = C] K E(j) : H$ + Cj ] , j > 1, "& !( !'%! Cj : C E(j) ! C j 2 ChainK X] j > 1
816
. .
" Cj ] = jC ] , C j | % ( !$ C, !!( A ChainK X]. B (C E(j))X] = CX] K E(j) j > 1 !'%!$ "#& #$ + Cj X]: CX] K E(j) ! (C j )X] j > 1: . !$( $, )! 0 2 E(2)0 " # + C2 X](0): CX] ! (C C)X] 2 Chain(K): O C 2 ECoalg(K X), ( !. %" 1) ! ! %!'%! (C C)X] = homX (DC(X)8 C(X)) : )! " % n-! # n 2 CX]n " # # \n : C(X) ! C(X)n; 2 ChainX (K) \n = (C2 X](0))(n ): N!!, " ! $ n-!$ # n n0 2 CX]n & # \n \n0 & # !$! ChainX (K). 5. E ! C 2 ECoalg(K X), !! ! ! ! ( fng 2 Hn(CX]), %$ n-!( ( ! ( ( " !#$! %! X, # \n : C(X) ! C(X)n; #( ) & ChainX (K). R! ( fng %! '"!$! ! n-!( ( E- $ (C fng) " X. 6. E ! C 2 ECoalg(K X), !! ! ! ! ( fng 2 Hn(CX]), %$ n-!( ( ! ( ( " !#$! %! X, ~ C(X) ]X] ~ ! CX] ~ n; \n]X]: !'%! \n] 2 Chain(K X) K]-!"( #( ) &. R! ( fn g %! '"!$! ! n-!( ( E- $ (C fng) " X. '%!! n-!$ $ ( $ ) ! f : (C fnC g) ! (B fnB g) " X + !'%!$ f : C ! B $ ! E- " X, f fnC g = fnB g:
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817
f : C ! B 2 ECoalgK X]. . !! & ! C(f) = fC(f)] j 2 X g 2 ChainK X] " C(f)] = C(f]) | # + f]. $ + Cj ] Bj ] , j > 1, K-!"$ E- C] B] "& #$ +( jC (f ]) : C(f]) K E(j) ! C(f]j ) j > 1: . !! & ! C(f) E(j) 2 ChainK X] (C(f) E(j))] = C(f)] K E(j) : H$ + Cj (f ]) , j > 1, "& !( !'%! Cj (f ) : C(f) E(j) ! C(f j ) 2 ChainK X] j > 1 " Cj (f ) ] = Cj (f ]) , C(f j ) | %( !'%! f. B (C(f) E(j))X] = C(f)X] K E(j) !'%!$ jC (f ) , j > 1, "#& #$ + Cj (f ) X]: (C(f))X] K E(j) ! (C(f j ))X] : . !$( $, # 0 2 E(2)0 " # + C2 (f ) X](0): (C(f))X] ! (C(f f))X] : @! C2 (f ) X](0) "$! #$! +! (C(f f))X] ! C((f f)X]) ! + \ : (C(f))X] ! C((f f)X]) 2 Chain(K): O f : C ! B 2 ECoalg(K X) n+1 | $( (n + 1)-!$( # ! (C(f))X] , )! \(n+1 ) 2 C((f f)X])n+1 +, %$ $,, " #& \n+1 : C(f(X) ) ! B(X)n+1; 2 ChainX (K): 3 ! $ # n+1 n0 +1 2 (C(f))X]n+1 & # & !$! ChainX (K). 7. '%! f : C ! B 2 ECoalg(K X), !!$( ! ! ! ( fnf +1 g 2 Hn+1((C(f))X]), %$ (n+1)-!( ( ! ( ( " X, # \nf +1 : C(f(X) ) ! B(X)n+1;
818
. .
#( ) & ChainX (K). R! ( fnf +1g %! '"!$! ! (n + 1)-!( ( E-$ (f fnf +1g) " X. 8. '%! f : C ! B 2 ECoalg(K X), !!$( ! ! ! ( fnf +1 g 2 Hn+1((C(f))X]), %$ (n+1)-!( ( ! ( ( " X, ~ C(f(X) )]X] ~ ! BX] ~ n+1; \nf +1]X]: !'%! \nf +1] 2 Chain(K X) K]-!"( #( ) &. R! ( fnf +1g %! '"!$! ! (n + 1)-!( ( E-$ (f fnf +1 g) " X. f : C ! B 2 ChainK X]. . !! & & " 0 ;! B ;! C(f) ;! C;1 ;! 0 $ ! ChainK X]. & ( " 0 ;! BX] ;! (C(f))X] ;! CX];1 ;! 0 #$ ! " ! & "& & " ! %$&! !!'%!! : H ((C(f))X]) ! H;1(CX]): !, ) " 9], ! "& +". 1. (n + 1)- ( ) (f : C ! B fnf +1 g) X . C 2 ECoalg(K X), fnC g = fnf +1g, n- ( ) E - X . R#( (n+1)-!( ( ( () ! ( $ (f : C ! B fnf +1g) " X %! n-!& & ( &) E- (C fnC+1g = fnf +1 g): 3 n-!$ $ ( $ ) ! (C fnC g) (B fnB g) " X " ! !! (C fnC g) (B fnB g) = (C B fnC nC g): " %!! # ( ( () ! ( $ (C fnC g) " X ! ! ( ( () E- $ (;(C fnC g)) = (C f;nC g) " X.
L-
819
9. n-!$ $ ( $) ! $ (C fnC g) (B fnB g) " X "! %$ "$!, (n + 1)-! ( ) E- " X, #( (C fnC g) (;(B fnB g)): !, ) " 9], ! "& +". 1. n- ( ) X . ! n- ( ) E - X $ . + " n-!$ $ ( $ ) ! " X % ! % (LE)nl (K X) ( % (LE)ng (K X)). N# !( !!$ %! # %"& (LE)nl (K X) ( (LE)ng (K X)) !!( $. B & n-! E- (E-) " X n-!( ( E- ( ( E-() " X, " ( !!'%! A: (LE)nl (K X) ! (LE)ng (K X) n 2 Z: X , X = pt | " , !!'%! A +" $! %!'%!!. +! , n-! ( ) ! (C fnC g) " X "% " n-!$( $( ( $() ( ! (C f'Cs g) " X. B #( ! E(2) KZ2 ]- "$! # $!, !$( $, 0-!$( # 0 2 E(2)0 " " !( )! s 2 E(2)s , s > 0, $ %$ ,! d(s) = s;1 + (;1)s s;1 T T 2 Z2 : S" !( %"& Z2 -) + in: W ! E(2) , " W | % KZ2 ]-!" K. O %" " !( )! 0s 2 E(2)s , s > 0, " $ d(0s) = 0s;1 + (;1)s 0s;1 T 0 = 00 ", & #$ + in in0 & !$! " KZ2 ]. V!$ s 2 E(2)s , s > 0, "& !!'%!$ K-!"( s \s : CX] ! (C C)X]+s
820
. .
" \s = C2 X](s ). $ % !!'%! \s, s > 0, '"!! # nC , ! $ , \s(nc ): C(X) ! C(X)n;+s s > 0: B! %!, "% " $( ( $() ( ! (C f'Cn g) " X, " 'Cs = \s(sC ). , (n+1)-! ( ) E- " X "% " (n + 1)-!& & ( &) & " X. 10. E& ! & " X %! !(, &( ( $( ( ! " X !$!. " !( ( E-$ " X #$. R "%! (n ; 1)-!$ !$ $ ! " X % ! % (SE)n (K X). X , ! ! "$( !!'%! (SE)n+1 (K X) ! (LE)nl (K X) n 2 Z: B! %!, " & n 2 Z "$ $ !!'%!$ Fl : (LE)nl (K X) ! Lnl(K X) Fg : (LE)ng (K X) ! Lng(K X) F : (SE)n (K X) ! S n (K X) " $ " !!$ A ! (LE)n (K X) (SE)n+1 (K X) ;;;;! (LE)nl (K X) ;;;; g ?? ?? ? Fg ? Fl y Fy y A S n+1 (K X) ;;;;! Lnl (K X) ;;;;! Lng (K X) !!$. !, ) " 9], ! "& +". 2. X = pt K = Q. $ n 2 Z % Fl , Fg , F $ % . %& ) ' +!, +" n-! ! (C C 2 Hn(CX]) = Hn(C(X))) " X " !( K-!"$ (n ; jj)-!$ E- f(in : @C] ! C] C ]) j 2 X g
L-
821
" jj = dim() C ] 2 Hn;jj(C(in )). 3( , ! 2 X ! ,$ v0 : : : vjj. B " ! # +( 0 1 : : : jj = " i 2 X | ) i-!$( ! , "!$( ,! v0 : : : vi . . !! + pr d C( ) d d @ : C(X) ;! C(0) ;! 1 ;1 ;! : : : ;! C();jj " pr | " #, d: C(i) ! C(i+1);1 | ! d ! C(X) , & +& i i+1 , 0 6 i 6 jj ; 1. E ", + @ (;jj) "''#! ! C(X) C() . )! )! @ (nC ) (n ; jj)-!$! #! ! C() . B ! C() = C]=@C] , " X @C]i = C()i i 2 Z
# ) C(in ) + E- in : @C] ! C], "% " ! ( C ] = f@ (nC )g 2 Hn;jj(C(in )): 5" ", !( fin C ] j 2 X g " ( !( (n ;jj)-!$ E- " ", " (C C ) n-!( ( ! ( ( " X. 3. L-
)! ' "$ <- $, ! ! ! $ & & $ L- $ ! E- . ! ( " %$ " L- ! . 5!! "!$ #, & ! ( ! ( !#$ %( )''#! <- ( !. 2]). X | " !# % F = fFn Fn+1 !
822
. .
$( <- $ $ " !#$ !+ . !# % X " " " !# !+ X = fXi g, i > 0, " Xi | ) i-!$ ! % X. N% ! % X+ " !# !+ X t pt, " pt | " !# . <- F % X "& F - ! ( <- (F )X+ = f(Fn)X+ j n 2 Z g " (Fn )X+ | '# " !# !+ , p-!$! ! ! + $ " !#$ + X+ Gp ! Fn. S" Gp | " !# !+ , & "! %& p-! ! , X+ Gp | ! %" " !#$ !+ X+ Gp . , %" !#$ %( (X Y ), " Y X, " F - ! ( <- (F )(X Y ) = f(Fn)(X Y ) j n 2 Z g:
!& <- (F )X+ "& $ ! ( % X )''#! <- F : H n (X8 F ) = ;n((F )X+ ) = X+ 8 F;n] n 2 Z " ;n((F )X+ ) | ) (;n)- ! <- (F )X+ , X+ 8 F;n] | ! $ " !#$ +( % X+ F;n
" ( $, " !# % X <- F " F - ! ( <- j n;j X+ ^ F = flim ;! < (X+ ^ F ) j n 2 Z g " X+ ^ Fn;j
j
| " ! %" $ " !#$ !+ X+ Fn;j , <j (X+ ^ Fn;j ) | " !# !+ j-$ X+ ^ Fn;j . R! ! ( % X )''#! <- F %$& $ ;j Hn(X8 F ) = n(X+ ^ F ) = ; lim ! n+j (X+ ^ F ) n 2 Z: j
O % X $!, $ ! ( H(X8 F ) !+ ! ! ( )''#! F . !! ", fv0 v1 : : : vm g | " !+ , % X. N"! !# + in: X @Gm+1 in(vi ) = (i)
L-
823
" @Gm+1 | # " ! Gm+1 , (i) | , ! Gm+1 !! i. Wm | !# %, !& "! i-!! ! 2 Wm " +" (m ; i)-! ! 2 @Gm+1 , ! " ", " . 5" ", % Wm %!' @Gm+1 . . !! !#$ %( (Wm X), " X = f 2 Wm j 2 @Gm+1 n X g Wm (F;m ) | ! ( <- m (F;m )(#m X ) = f(Fn;m )(# X ) j n 2 Z g: N"! + <- W : (F;m )(#m X ) ! X+ ^ F !%#& m w (X ^ F;m )#m ' <m (X ^ F;m ) ! lim <j (X ^ F;j )=X ^ F (F;m )(# X ) ! + + + + ;! j
" + w %" ! " + 4(" Wm ! X+ ^ (Wm =X) ( !. 10]). E ", + <- W ! ( ) &, "#$( !!'%! ! H m;n (Wm X8 F ) = n((F;m )(#m X ) ) ! n(X+ ^ F ) = Hn(X8 F ) " ( %!'%! S-"( ". ! "& <- (LE) (K). %" 2 $ L- $ ! ! . 3( $! %! !+ ! L-& %$!$ $ ! ! " !#$! %!. E$! !! ! " %! X "& & $ '$ C : X ! ECoalg(K) !'%!! ! + $ % '. @ & $ ! K-!"$ ! " X % ! % ECoalgX K]. ECoalgX K] $" " ECoalg(X K), A! ( + (X K)- "$ $ !$ % ECoalgX K]. 4 (X K)- " " ! % ECoalgX K] "%#( (K X)- " $ ! % ECoalgX K]. N" n-!( ( ! ( $ (( E-$) ECoalg(X K) " "( $! %! &! ! ECoalg(K X). 5" ", ECoalg(X K) % % X $! %!, . .
824
. .
" !# + %( f : X ! Y " $( ' f : ECoalg(Y 8 K) ! ECoalg(X8 K): N% ! % (LE)n (K)m , m > 0, n 2 Z, !+ , )!! & n-!$ $ ! $ ECoalg(Gm 8 K), ! ( ( + E- . @$ '$ ( i ) : ECoalg(Gm 8 K) ! ECoalg(Gm;18 K) "#$ +! ( i : Gm;1 ! Gm , "& $ + @i : (LE)n (K)m ! (LE)n(K)m;1 0 6 i 6 m " $ $& , @i @j = @j ;1@i i < j. B! %!, (LE)n (K) = f(LE)n(K)m @ig 0 6 i 6 m m > 0 $! " !#$! !+ ! " +" ' n 2 Z. %" !#$ %( (X Y ), " Y X. N% ! % ECoalg(X Y 8 K) & " & ECoalg(X8 K), A! ( & ! $ C, C() = 0 " ! 2 Y . 1. & ' ( (LE) (K) = f(LE)n (K) j n 2 Z g <- ' ( n((LE) (K)) = (LE)n (K) n 2 Z: m!"#!$. +! , <((LE)n(K)) = (LE)n+1(K). G = f(a0 a1 : : : ai) j 0 6 a0 < a1 : : : < ai 6 mg | " % ! Gm . + m+1 : Gm ! Gm+1 , m+1 (a0 a1 : : : ai ) = = (a0 a1 : : : ai m + 1), +" ! Gm m-!( & ! Gm+1 , +( ,$ (m + 1) 2 Gm+1 . V!! !+ <((LE)n (K))m & , "&, n-!$ $ ! $ (C fnC g), " fnC g 2 Hn(CX]), ECoalg(Gm+1 Gm f(m + 1)g8 K), . . n-!$ $ E- $ (C fnC g) ECoalg(Gm+1 8 K), C((m + 1)) = 0 C((a0 a1 : : : ai)) = 0 0 6 a0 < a1 < : : : < ai 6 m. N"! (Gm 8 K)- "& & ! B = fB(a0 a1 : : : ai)] j 0 6 a0 < a1 < : : : < ai 6 mg
L-
825
B((a0 a1 : : : ai)) = C((a0 a1 : : : ai m + 1)) . E- $ B(a0 a1 : : : ai )] "# ( E- $ C(a0 a1 : : : ai m + 1)] . $ $ %$&, Hn(CGm+1]) = Hn+1 (BGm ]): P% ) ", (B fnB+1 g), " fnB+1g) = fnC g), (n+1)-!( ( ! ( ( ECoalg(Gm 8 K). 4 C((m + 1)) = 0 C((a0 : : : ai)) = 0 0 6 a0 < a1 < : : : < ai 6 m %$&, n-! ! (C fnC g) ECoalg(Gm+1 8 K) (n + 1)-! ! (B fnB+1 g) ECoalg(Gm 8 K) "& " " "% , . . <((LE)n (K))m = (LE)n+1 (K)m . B! %!, (LE) (K) <- !. " !# !+ (LE)n (K) "%$ !, ) " 2]. R! $ m ((LE)n (K)), m > 0, n 2 Z, & ! ! ( ) m-!$ ! (C fnC g) % (LE)n (K), @i (C fnC g) = 0, 0 6 i 6 m. 5" ", %$ m-!$ ! $ % (LE)n (K) & (m + n)-!$! ! ! ! ECoalg(K), , ! !+" ! ! ! ,! " !+" &! E- ! . P% ) ", m ((LE)n (K)) = (LE)n+m (K). B! %!, n((LE) (K)) = (LE)n (K), n 2 Z. X | !# %. . !! !+ (LE)nl (X8 K)m , )!! & n-!$ $ ! $ ECoalg(X Gm 8 K), ! ( ( + E- . + !#$ %( 1 i : X Gm;1 ! X Gm , 0 6 i 6 m, "& $ + @i : (LE)nl (X8 K)m ! (LE)nl(X8 K)m;1 0 6 i 6 m @i @j = @j ;1@i , i < j. ! 1 !, $ " !#$ !+ (LE)l (X8 K) = f(LE)nl (X8 K) j n 2 Z g <- ! $ " !#$ !+ n((LE)l (X8 K)) = (LE)nl (X8 K) n 2 Z " $ (LE)nl (X8 K) "& ! (LE)nl (K8 X) "( $! %!. S!! , (LE)l (X8 K) (LE) (K)- ! ! <- ! ((LE) (K))X+ !# % X. !! ", n-!& & ! &
826
. .
ECoalg(X Gm 8 K) !+ " !( n-!$ $ E- ECoalg(Gjj 8 K), "( " +" ! 2 X Gm , jj = dim(), ! ! ! ! % X Gm & E- % ) !( . )! !+ " !#$ +( " (X Gm )+ ! (LE)n (K) " ! n-!$! $! E- ! ECoalg(X Gm 8 K), . . (LE)nl (X8 K) = ((LE)n (K))X+ : B! %!, (LE)nl (X8 K) = H ;n(X8 (LE) (K)) n 2 Z: X | !# %, !& m ,, (LE)nl (K8 X)p , n 2 Z, p > 0, | !+ , )!! + (n ; m)-!$ $ ! $ ECoalg(Wm Gp X Gp 8 K), " Wm X | !#$ %, !$ $,. + i : Gp;1 ! Gp, 0 6 i 6 p, "& $ ( @i : (LE)nl (K X)p ! (LE)nl (K X)p;1 0 6 i 6 p: ! 1 !, $ " !#$ !+ (LE)l (K X) = f(LE)nl (K X) j n 2 Z g <- ! $ " !#$ !+ n((LE)l (K X)) = (LE)nl (K X) n 2 Z: S!! , (LE)l (K X) ! ) (LE) (K)- ! ! <- X+ ^ (LE) (K) !# % X. 3( ,m +" !$! $, ! (LE)l (K X) ((LE) (K))(# X ) ! ! & ) 4( " W : ((LE) (K))(#m X ) ! X+ ^ ((LE) (K)) ! !& ) <- . B! %!, (LE)nl (K8 X) = Hn(X8 (LE) (K)): 3 % X, !& m ,, !! (LE)g (K8 X) = f(LE)ng(K8 X)) j n 2 Z g " (LE)ng (K8 X) | " !# !+ , p-!$! ! ! & $ ! $
L-
827
ECoalg(Wm Gp X Gp 8 K). ! 1 !, (LE)g (K8 X) " ( <- n ((LE)g (K8 X)) = (LE)ng (K8 X)) n 2 Z: @! , !! <- (SE) (K8 X) = f(SE)n(K X) j n 2 Z g " (SE)n (K X) | " !# !+ , p-!$! ! ! + (n ; 1)-!$ $ !$ E- $ ECoalg(Wm Gp X Gp 8 K). 5" ", n((SE) (K8 X)) = (SE)n (K8 X) n 2 Z: <- $ (LE)l (K8 X), (LE)g (K8 X)) (SE) (K8 X) %$ !+" ( ( " & A (LE) (K8 X) ;! @ (SE) (K8 X) (LE)l (K8 X) ;! g " A | " + ( !. %" 2), + @ +"( n-!( ( ! ( ECoalg(Wm Gp X Gp 8 K) # )( E- $, "!& +, 11]. 4% " <- "# "& & " ! ) <- . N+" ! $ (LE)l (K8 X), (LE)g (K8 X)), (SE) (K8 X) &! (LE)- !, ! "& +". 2. ) A : : : ;! (SE)n+1 (K X) ;! Hn(X (LE) (K)) ;! A (LE)n (K X) ;! @ (SE)n (K X) ;! H (X (LE) (K)) ;! : : :: ;! n;1 g V "& & " $ %! ( " & L- $ ! ! . X , ' %$ E- $ "# + )( ( " & " .# ( !. %" 1). 9] $ %, (LE)n (Q) = = Ln (Q). P% ) ", " & % X !!'%! Fl : (LE)nl (Q X) ! Lnl (Q X) n 2 Z !$( %" 2, %!'%!!. B! %!, ! (LE)nl (Q M) = M8 (G=Top)Q ] " M | ! n-! ! %, n > 5, (G=Top)Q | # %# ! " + BTop ! BG , '#& $ $ ' .
828
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!
1] . ., . .
K- // #$% . . &. &. | 1978. | #. 18. | . 140{168. 2] Ranicki A. A. Algebraic L-theory assembly. Preprint. 1990. 3] Ranicki A. A. Exact sequences in the algebraic theory of surgery. | Princeton Univer. Press, 1981. | Math. Notes. Vol. 26. 4] Ranicki A. A., Weiss M. Chain complexes and assembly // Math. Z. | 1990. | No. 204. | P. 157{185. 5] May J. P. Simplicial objects in algebraic topology. | Princeton: Van Nostrand, 1967. 6] Rourke C. D., Sanderson B. J. 3-sets I: Homotopy theory // Qart. J. Math. Oxford. | 1971. | Vol. 2, no. 22. | P. 321{338. 7] 4. . 5 67 8 // . 9. | 1981. | #. 115 (157), < 1 (5). | . 146{158. 8] 4. . = 7 > > 69 // ?&. @
F. . . | 1985. | #. 49, < 6. | . 1103{1121. 9] G . 4. = % 9 $& E1 - 69 7 L-6 % // . 9. | 1995. 10] Whitehead G. W. Generalized homology theories // Trans. Amer. Math. Soc. | 1962. | Vol. 102. | P. 227{283. 11] Ranicki A. A. The algebraic theory of surgery. I. Foundations // Proc. London Math. Soc. | 1980. | Vol. 40, no. 1. | P. 87{192.
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Abstract Yu. G. Nikonorov, On the asymptotic of the mean value points for some nite di erence operators, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 829{838.
The paper provides a proof of some asymptotic estimations of the mean value points for /nite di0erence operators of n-times di0erentiation.
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A. G. Pinus, On faithful conditional identities and conditionally complete conditional varieties, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 839{847.
It is proved that for any faithful conditional variety M with unique constant algebra there exist a conditionally complete conditional variety Mc and a polyinjective functor F which isomorphically embedds the embedding category M into the embedding category Mc .
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,
.
1] . . . // . | 1958. | $. 120, ( 1. | . 29{32.
847
2] D. Pigozzi. On the structure of equationally complete varieties, I // Collog. Math. | 1981. | Vol. XLV, no. 2. | P. 191{201. 3] D. Pigozzi. On the structure of equationally complete varieties, II // Trans. Amer. Math. Soc. | 1981. | Vol. 264, no. 2. | P. 301{319. 4] . -. .. / 0 12 // 34 0. | 1996. | $. 156. | . 59{78. 5] . -. .. 6 7 0 8 9 // 9 0 0. 1 . | 1997. | $. 38, ( 1. | . 161{165. 6] . -. .. 4 12 : // . | 1998. | $. 37, ( 4. 7] J. Plonca. On a method of construction of abstract algebras // Fund. Math. | 1967. | Vol. 61, no. 2. | P. 183{189. ( ) 1998 .
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Abstract S. V. Pchelintsev, The structure of weak identities on the Grassman envelopes of central-metabelian alternative superalgebras of superrank 1 over a eld of characteristic 3, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 849{871.
The work is devoted to clarify the structure of weak identities of central-metabelian alternative Grassmann algebras over a :eld of characteristic 3. Canonical systems of weak identities n and n are constructed: ( n;2 ) , n;1 n ] = 4 + 2 4 + 31 n := ,, 1 2 ] 3 ] ( 4 ) := , ] ( ) ( ) , ] = 4 4 +3 n 1 2 3 n;2 n;1 n ( ! 3 % 3, 97-01-00785, 00-01-00399. ff
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n > n
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. .
(1) A q St(A) Aq = (0), n, f n # gn := x1 x2]R(x3) : : :R(xn;2) xn;1 xn] n = 4k 4k + 3> (2) A $ N St(A) AN = (0), f n > 7 # fn := x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn] n = 4k + 2 4k + 3: % %& # % 3 ( , % #( %& " % / % a b] c] = 0). B # % / , % "( % / % % % ( ! % &&.%& ( 1& . K " % % % # %(3 ) . 1 &%& ( ( "() % / %. 2 &73 # % % ! % (. B& 9% #( %& %(! " # T- " ! 1 % - % " ! % % ! "(, / 3 a b] c]. 9% " / %& % / &# "() % / % ( 1& . 3 4 %& % % () " D , % () 1 gn (n = 4k 4k + 3) fn (n = 4k + 2 4k + 3) .% ( # &. 6
x
x
x
x
1.
. " % L " !M, % , %& 1 % - % " % % & " ) % % 3. 5 % .7 " # &, # ( " % , / !%, , 13]. ; / % ! ! % % %, % / ) # 1 % # .%& " % " # & !. 1 . . 5 , % & ! % % ! " ( .7 % / % ) # 1: ab c] = ab c] + a c]b + 3(a b c) (1) (xy a b) + (x y a b]) = x(y a b) + (x a b)y (2) (ab)(ca) = a(bc)a (1 % % / % ) (3) (a b) c (a c) b = a b c]]+ 2(a b c) (4) 2 y x] + y x y] = 0 (5) 3 3 y z zy A(2) : (6)
;
2
1. v w A2, a b A. " % 2
853
2
) (va)w = v(w a) ) v a] w] = 0 ") v(wa) = w(va) ) (vRa Rb)w + v(wRa Rb) = 0 ) v a] b] A(2) /) (va)(wa) = 0 ) (v a) b (va)b A(2) #) v a]w b] = 0: . K % ) ") .% # % / % !, ! % % % 1 % ! % " %. ) H % , % % / % % % % 1 % % / % (3) vT (a)T (a) A(2), %. . . A(2) 1& vT (a)T (b) % & ( a b. ; . A(2) v a] b] = (va av)b b(va av) (va)b + (bv)a b(va) + b(av) (v a b) + (b v a) (b a v) = 3(v a b) = 0: ) @ & (4) c = v &&
% ' ) % % 1 % , (a b) v (a v) b = a b v]] + 2(a b v) 2(a v) b 2(v a b) A(2) 9% % % "
% ' . ) (% % # % / % (1), % ) 1 % ! % " %: v a] xy] = xv a] y] + v a] x]y = 0: ) N& %& % % / % (2) 1 % ! % " %. /) 6 %( & . % % %, 1 % % / % % / % 1 % ! % " %, (va)(wa) = v((wa) a) = v(wa2 + awa) A A(2) = (0): #) ? % / % (1) % ): v a]w b] = vw b] a] = 0: 2
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2
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2
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2
2
;
2 . - ! " !. 2. &
fn := x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn] (n > 6)
. % % % ' . 1 . x y] y]R(z) A(2) . B !%% , . A(2) : y y x]]z = (y y) x (y x) y z ( (4)) = = 2 y2 x + (y x) y z = y2 x + (y x)y z ( (5)) = = 0 ( (6)): 2 f
f
;
g
f
g
g
854
. .
2 . id lA (A2 A]) A2 A]+A(2). @ % w A2 > % % / % (1) : w x]y = wy x] wy x] A2 A] + A(2) yw x] = yw x] y x]w A2 A] + A(2) : 3 . C w A2> % x y] z] w t] = 0. O% % / % # ( 1, #). 4 . fn (x1 x2 : : : xn;1 w) = 0, w A2 . , ( 1, ) fn (x1 x2 : : : xn) x1 x2]x3] id lA (xn;1 xn]) % , % 2 3
2
;
2
;
2
2
2
2
fn (x1 x2 : : : xn;1 w) x1 x2] x3] id lA (A2 A]) x1 x2]x3] A2 A] + A(2) = (0): 5 . B / % % 1 fn (x1 x2 : : : xn;1 xn). ?# % 1 (% % % % / & & () %3 ) (). 8 % % fn x3, x4 % # % / % (1) ( 1, ). 8 % % ( x4 : : : xn;2 % # ! % % % 1 % ! % " %. 5 1, % % xn;2 xn;1 (% % # ( 1, ), % / % (1) % 4 . 8 % % ( , 9% !% " % % % . @
% ' & fn = 0 " % 4. 2 2
.
f
6
g
x
) ' id lA (A2 A]) $ ") w A2 , ( x1 x2] x3]R(x4) : : :R(xn ) w w x1]R(x2) : : :R(xn;2) xn;1 xn] xi ) a x1] x2]R(x3) : : :R(xn) a2 = 0, n > 4 ) ( a x1] x2]R(x3) : : :R(xn;1) (a xn) n > 4 xi . . 6% / ) (% % # % 2 ( 2 ( 1, #). 6% / ") (% % # ) % / % (1). @ %( ) ) .%& # "), 1, 2 1 % % 2- / 3 ! "(. 3. ) fn = 0, n = 4k + 5 4k + 4 (k > 1). # "3 %. 1 . @ % n = 4k + 9 (k > 1). ; ( 1 9 % r := x1 x2] x3]R(t1) : : :R(t4k ) x4 x5] 2
855
/ % "(% # r = x1 x2] x3]R(t1 ) : : :R(t2k ) x4 x5]R(t2k+1) : : : R(t4k ) = = x1 x2]R(t1) : : :R(t2k ) x3] x4 x5]R(t2k+1) : : :R(t4k ) = p x] q p = x1 x2]R(t1) : : :R(t2k ), x = x3, q = x4 x5]R(t2k+1) : : :R(t4k ). B , ( 2 1& p x] q % & ( , % p x] q = q x] p. ; % / % (1) p x] q = q x] p = q p x] = p x] q p x] q] A2 A] A2] = (0) ( 1, ). @ 9% 2p x] q = = 0, # %, r = 0. 2 . @ % % n = 4k + 8 (k > 0). ; 9 % r := x1 x2] x3]R(t1) : : :R(t4k+3) x4 x5] / % "(% # r = p x]y q p=x1 x2]R(t1) : : :R(t2k+1), x=x3, y=t2k+2 , q=x4 x5]R(t2k+3) : : :R(t4k+3). H % , % 9 %( p q # &% % 2k+3 (), # %, 1& r % . % , / & ( 7 , p x]y q = q x]y p = 2q x]y p = 2q x] (p y) = = q x] (py) ( q x] p y] = 0 %&) = = q py x] = py x] q = p x]y q % , 2p x]y q = 0, r = 0. 2 3 . " ! " . 5 3 (, % &7 !& % " ( " . 4. A | , R (A) | . " ) A0 A] = 0, ") A0 = A A]R (A). . K % # (! !(! 9 % kn = a b]T (x1) : : :T(xn) / 1 ! n, % kn t] = 0, kn = a b]R(x1) : : :R(xn ). G 1 n = 0 % . % / , % / % (1) kn t] = kn;1xn t] = kn;1 t]xn = 0 kn+1 = knL(xn+1 ) = knR(xn+1): 2
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2
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;
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;
856
. .
5. * A # # gn := x1 x2]R(x3) : : :R(xn;2) xn;1 xn] gn (n = 4k 4k + 3, k > 1) A. .
1 . H % , % ( 4 # (! 9 % # % % % % ! ! " 1 9 % % h: h := (x1 x2]R(x3) : : :) (y1 y2 ]R(y3) : : :): ? # & . % % %, % / % (1) 4 & v w A0 , x A 2w(vx) = w(v x + v x]) = (wx)v w x]v = (wx)v: (7) H %, % ( / & % % % 1 h / % (! % % , 9% # % # / % % & 1 h # 1 % g. 2 . B / , % 1& gn % ( . H % , % % " ! " 1& x1 x2]R(x3) : : :R(xn;2) % ( x3 : : : xn;2. % , 1& gn % ( x3 : : : xn;2. B& # % % % % gn ( x2 , x3 % / % (1) % % /% x2 = x3 %, %
xz t]R(a1) : : :R(an ) y1 y2 ] = 0: @ A2 A]A A2 A]+A(2), % xz t]R(a1) : : :R(an) A2 A]+A(2) . @ && % / % (1) % / % % %, A2 A]A A] A2 A A] A] A2 A A] A] = 0: @
% ' & gn = 0 (n = 4k 4k + 3, k > 1) " % 3. 2 6. * gn (n = 4k + 1 4k + 2, k > 1)
. . @ % z ^= a b], t = c d]> Rn R^n | % ( ( n> v = zR2k , w = tR2k . ; ( 1, ) zR4k+1 t = zR2kR(x) tR^2k = = (vx)w = (wx)v ( % %) = = (xw)v = (xv)w ( ! % % %) = = (vx)w ( ( 4, ))
% 2(vx)w = 0, % , zR4k+1 t = 0 & k > 0. 2
2
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2
6
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x
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;
857
@ ) & % . % &, # % , % A0 A0] = (0). B !%% , p q A0 , % ( 4, ) % / % (5) 2
px q] = 2p x q] = 2p q x] + 2x q p] = 2p q x]
% , A0 A0] A A] A0] = (0). K / & ,
zR4k+2 t = zR2k R(x)R(y) tR^2k = ((vx)y)w = = (vx)(wy) = (wy)(vx) ( % %) = = (vx)(wy) ( vx wy] = 0 # ): ;
H %, (vx)(wy) = 0 zR4k+2 t = 0. 2
x
2.
1 . ' ( St(A). P # St(A) ( " # T- , / 3 (! a b] c], &.7!
" # % ) ". . " ! "( A # 3 !( ( % () # / % Xn = x1 x2 : : : xn : 1) t1 t2] t3]R(t4) : : :R(tn;2) (x1 xi ), i = n, tj Xn x1 xi ( tj & ( # % . > 2) x2 x3] x4]R(x5) : : :R(xn;2) (x1 xn;1 xn)> 3) x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn]> 4) x1 x2] x3]R(x4) : : :R(xn). 7. Pn(A) St(A) n > 7 . . G" # # F ! . " 9 % fn, % ( , ( 7 , " # .% &&.%& "( % / % . Q % % .7)
% ' !: f
6
2
nf
g
g
\
St(A) A2 ] = (0) ( ( 2, % 2 ( 1, ))> St(A) A0 F St(A) = A A] A]R (A) + F:
@ % # % % ( % % % . 1 . @ % w A3 , a b c A. ; . F 2
2
wa b] c] a b] c]w:
858
. .
, % / % (1) wa b] c] = w b]a c] = w b]a c]+ w b] c]a = = w b]a c] ( ( 1, )) = = wa c] b] = a c] b]w ( ( 1, )) = = a c] b](vx) ( w = vx, v A2 ) = = 2a c] b](vx) a c] b](v x) . F = = (a c] b]x)v ( ( 1, )) = = (a b] c]x)v ( ( 2, % 1 ) = = 2a b] c](v x) ( ( 1, )) a b] c](vx) . F = = a b] c]w: 2 . K % # (! !(! s % , ' ! 7, # St(A). ; / % %, % ( . F ) 9 % s % s = a b] c]R(z) : : :R(t). C # s % %
% R(z), % 9 %( a, b, c /( / % / % " () / .7) X, # ) / "( "( / % A2 , % ( 1, ) 9 % a b] c] /% A(2) , s = 0. C / % R(z) : : :R(t) / & # s %, %
% / % (1) / % %, % c X, % 1 % ! % " % / % %, % b X. ! % a A5 . @ % % 1 , % ( . F) 9 % s && %& ! ! " 1 ! 9 % ) y1 y2] y3]R(y4 ) : : :R(yn )> ") y1 y2] y3]R(y4 ) : : :R(yn;2 ) (yn;1yn ). H % , % % % ") / % % ) y1 y2] y3]R(y4 ) : : :R(yn;2 ) (yn;1 yn). ?% , / (! 9 % a b] c]R(z) : ::R(t) % ! ! " 1 % ) ), % () yi Xn . 3 . ?# % 1 ( 2 % / % ! % % % (% %, % ) % ) / % % ( ( & % () 9 % / % X. % %, / % ) ! ( / %& # ) ( % 3) 4). K % % # % ). C & x1 % %& 9 % y1 y2 : : : yn;2, % " # & "7 % /
% %, % yn;2 = x1, , && 1, ), % ) / !
( #% # y1 y2] y3]R(y4) : : : R(yn;3) ((yn;1 yn ) x1), % ( , ( 1, ) # 1 % / % (6), ! ( / .%& # % / % ), % () yn;1 = x1. @ , %
x2 x3] x4]R(x5) : : :R(xn;2)R(xn;1) (x1 xn) ! ( / %& # ( % 1) 2). ;
;
;
2
;
2
2
2
2
859
% & w = x2 x3] x4]R(x5) : : : R(xn;2), ( 1, ), %& ) # ( 2 % / % (4) (wxn;1) (x1 xn) = w ((x1 xn) xn;1) = w ((x1 xn;1) xn +2(x1 xn xn;1)) 3 ) () ! ( / .%& # ( % 1) 2). ; ( %. # . 2 2 . ! " ! " . 9% % ( / . % ! % ( 3 ' )%
" #& " 1 !. 8. ) f n A gn (n = 4k 4k + 3). . . " % / % f % ! ! " 1 9 % xi xj ]R(y1) : : :R(y ) xk xl ]R(z1) : : :R(z ): (8) 6 %( &, % ( " # 9 %( % ! "(, ( 5 , % / (! # (8) % %. # % gn, % , f % %& % gn ( /% . G & n (% .% # ( 6. 2 % / % #( %& .7 % ( 8. 9. A q St(A) Aq = (0). " n0 , f n > n0 # gn n = 4k 4k + 3. . @ % m > q +3. 7 8 " % / % f % m % f = gm + v | ! ! " 1 gm v % 4): gm = x1 x2]R(x3) : : :R(xm;2 ) xm;1 xm ] v := x1 x2] x3]R(x4) : : :R(xm ) ( % 1){3) ( . ( 1, ). @ " % / % /% A(2), % " A ( % / % vt = 0, # %, % & " % / % f % n > m, , % & ) &7 & % f = gn. 2 % & # %( , % 1 gn % ( % & ( . 3), 3 ' )%
" #& " 1 !. . . *
3 gn# := x1 x2]R(x3 x4) : : :R(xn;3 xn;2) xn;1 xn] n = 4k
.
x
860
. .
B& % % # %%, % % / % gn# &&.%& "% ( , %. . " 7 .%& % % ! # () 1( 1, " ! % ! 1 % - % " ! % % ! " A % / % gn# gn .%. H %, # & % % / % " A# , ! # A ' 1(. 2 3 . ( . B % , % " ! " A & ." N St(A) AN = (0): @ % f Pn(A), n > 7, | " % / % "( A. B !' / & # "3 # /() &: f St(A) f = St(A): I. ?% , % f St(A). ; ( 7 f % ! ! " 1 .7 : vi := t1 t2] t3]R(t4) : : :R(tn;2) (x1 xi) tj Xn x1 xi , 2 6 i 6 n 1> qn := x2 x3] x4]R(x5) : : :R(xn;2) (x1 xn;1 xn)> fn := x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn]: H % , % % 4) / % ) % # f, " "( "( ( % St(A) AN = (0) & ) &7 N. ?% , % nX ;2 f = 2v2 + ivi + n;1vn;1 + qn + fn : (9) 6
2
2
2
2
2
nf
g
;
i=3
# 3 (! i, 3 6 i 6 n 2, % xi = w A2 % (9). ; 1 f, vj , qn, fn, vi , " 7 .%& , % , i = 0. H %, % / % (9) % f = 2v2 + n;1vn;1 + qn + fn : (10) H % , %
v2 = x3 x4] x5]R(x6) : : :R(xn) (x1 x2) Vn;1 = x2 x3] x4]R(x5) : : :R(xn;2)R(xn) (x1 xn;1) % , & (10) xn;1 = xn = a, n;1x2 x3] x4]R(x5) : : :R(xn;2)R(a) (x1 a) = 0
% w((x1 a) a) = 0, w = n;1x2 x3] x4]R(x5) : : :R(xn;2). ; w(x1 a2 ) = 0, # %, n;1St(A) An = (0), %. . n;1 = 0. % , (10) % f = 2v2 + qn + fn : (11) ;
2
861
@ & % (11) xn;2 = xn = a, x2 x3] x4]R(x5) : : :R(xn;3)R(a) (x1 xn;1 a) = 0
% w((x1 xn;1 a) a) = 0, w = x2 x3] x4]R(x5) : : :R(xn;3). H %, w A6 = 0, %. . St(A) An = (0) = 0. % , %
f = 2v2 + fn : @ & % x1 = w A2, 0 = 2x3 x4] x5]R(x6) : : : R(xn) (w x2 ) % , 2 St(A) An = (0), % 2 = 0. ; " # , " % / % f 9 % " A % / % fn . II. @ % % f = St(A). ; 7 8 f % ! ! " 1 gn = x1 x2]R(x3) : : :R(xn;2) xn;1 xn] n = 4k 4k + 3 ( . 6)> vi := t1 t2] t3]R(t4) : : :R(tn;2 ) (x1 xi ) tj Xn x1 xi , 2 6 i 6 n 1> qn := x2 x3] x4]R(x5) : : :R(xn;2) (x1 xn;1 xn)> fn = x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn] n = 4k+2 4k+3 ( . 3)> % 4) / % ) % # f, " "( "( ( % St(A) AN = (0) & ) &7 N. H % , % ( gn fn % ) % f % n = 4k +3. ?% , % nX ;2 f = gn + 2v2 + ivi + n;1vn;1 + qn + fn (12)
2
2
2
nf
g
;
i=3
3 = 0 % & I, # %, n = 4k 4k + 3. @ % && ( / & # & I, , % % (12) # / ' i = 0 & ." i: 3 6 i 6 n 2, % xi = w A2 1 f, vj , qn, fn , vi ,
" 7 .%& , % , % / % (12) % f = gn + 2v2 + n;1vn;1 + qn + fn : (13) @ & (13) xn;1 = xn = a, n;1 = 0. H && (13) 2 v2 pn := x3 x4] x5]R(x6) : : :R(xn) (x1 x2), f = gn + pn + qn + fn : (14) @ & % (14) x1 = x2 = a, 6
;
2
2 x3 x4] x5]R(x6) : : :R(xn) a2 + + a x3] x4]R(x5) : : :R(xn;3)R(xn;2) (a xn;1 xn) = 0: (15)
862
. .
H % , % %& # ( 2 / (15) && %& % ! 1 ! ( % xi . @ & "(n) := ( 1)n , : a x3] x4]R(x5) : : :R(xn;3)R(xn;2) (a xn;1 xn) = = "(n 4)x3 x4] x5]R(x6) : : : R(xn;2)R(a) (a xn;1 xn) = = "(n)x3 x4] x5]R(x6) : : :R(xn;2) (a2 xn;1 xn) ( ( 1, ) % / % (2)) = = "(n)x3 x4] x5]R(x6) : : :R(xn;2)R(xn;1)R(xn) a2 ( ( 1, )) % , % (15) % 2 x3 x4] x5]R(x6) : : :R(xn) a2 "(n)x3 x4] x5]R(x6) : : :R(xn;2)R(xn;1)R(xn) a2 = 0 %. ., % & w = x3 x4] x5]R(x6) : : : R(xn), ( + "(n))w a2 = 0,
% (% % ( + "(n))St(A) An = (0), # %, (16)
+ "(n) = 0: @ & % (14) x2 = x3 = a, x1 a]R(a) : : :R(xn;2) xn;1 xn] + a x4] x5]R(x6) : : :R(xn) (x1 a) = 0 % / % (1) 2 x1 a2]R(x4) : : :R(xn;2) xn;1 xn] + + a x4] x5]R(x6) : : : R(xn) (x1 a) = 0: (17) B , %& # ( 2 % / %
a x4] x5]R(x6) : : :R(xn) a2 = 0 1& a x4] x5]R(x6) : : : R(xn) (x1 a) % ( xi . 6 %( & 9% # &, a x4] x5]R(x6) : : :R(xn ) (x1 a) = x1 x4] x5]R(x6) : : :R(xn) a2 = = x4 x5] x1]R(x6) : : :R(xn) a2 = = x4 x5]R(x6) : : :R(xn) x1] a2 ( % / % (1)) = = x4 x5]R(x6) : : :R(xn) a2 x1] ( % / % (1)) = = a2 x1] x4 x5]R(x6) : : :R(xn) (% A2 A] A2] = (0) 1, )) = = a2 x1] xn;1 xn]R(x4) : : :R(xn;2) ( % /% ( xn;1, xn % &.%& ( % ) = ;
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;
863
= x1 a2] xn;1 xn]R(x4) : : :R(xn;2) = = 2n;5 x1 a2]R(xn;2) : : :R(x4) xn;1 xn] ( ( 2, 2 ( 1, )) = = "(n)x1 a2]R(xn;2) : : :R(x4) xn;1 xn] = ( % ( ) / ) = "(n) (n 5) x1 a2]R(x4) : : :R(xn;2) xn;1 xn] ( ( (n) := 1 n = 4k n = 4k + 1 1 n = 4k + 2 n = 4k + 3 (n) % # % (n n 1 : : : 3 2 1)) = = "(n)x1 a2]R(x4) : : : R(xn;2) xn;1 xn] ( & n # /( ' # & n = 4k 4k+3 (n 5)= 1). ?% , a x4] x5]R(x6) : : :R(xn ) (x1 a) = "(n) x1 a2]R(x4) : : :R(xn;2) xn;1 xn] % , % (17) % ;
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;
2(x1 a2]R(x4) : : :R(xn;2) xn;1 xn] + + 2"(n) x1 a2]R(x4) : : :R(xn;2) xn;1 xn] = 0: G%. + "(n) = 0: (18) 5 1, / (14) xn;2 = xn;1 = a. ; x1 x2]R(x3) : : :R(xn;3)R(a) a xn] + + x2 x3] x4]R(x5) : : :R(xn;3)R(a) (x1 a xn) = 0: (19) @ " # / # (). B& x1 x2]R(x3) : : :R(xn;3)R(a) a xn] = x1 x2]R(x3) : : :R(xn;3) a2 xn] ( 1, ) % / % (1). B& % : x2 x3] x4]R(x5) : : :R(xn;3)R(a) (x1 a xn) = = x2 x3] x4]R(x5) : : :R(xn;3) (x1 a2 xn) ( ( 1, ) % / % (2)) = = x1 x2] x4]R(x5) : : :R(xn;3) (x3 a2 xn) ( ( x1, x2, x3 % & 1 ) = = x1 x2]R(x5) : : :R(xn;3) x4] (x3 a2 xn) ( % / % (1) % ( / % x1 x2]) =
864
. .
= x1 x2]R(x5) : : :R(xn;3) xn] (a2 x3 x4) ( # % % % 9 % x3 a2 x4 xn # & %& # ) = = x1 x2]R(x5) : : :R(xn;3) xn] a2R(x3 x4) = = x1 x2]R(x5) : : : R(xn;3) xn]R(x3 x4) a2 ( % / % (2)) = = x1 x2]R(x5) : : : R(xn;3) xn]R(x3)R(x4 ) a2 = = x1 x2]R(x3)R(x4 )R(x5) : : : R(xn;3) xn] a2 ( % / % (1) % % ( xi ) = = x1 x2]R(x3) : : : R(xn;3) a2 xn] ( % / % (1)): ; # % (19) ( + )x1 x2]R(x3) : : :R(xn;3) a2 xn] = 0: H &, % % / % x1 x2]R(x3)R(x4 )R(x5) : : :R(xn;3) xn] a2 = 0 3 % % St(A) An = (0), (20) + = 0: ?# % (16) (18) (% % = , %. # (20) = 0. @ % # ' % # % % ! % (. 2 4 . /" / . @ &% " % / % % &# &% ! 1, .7 ! / . % % %() " 10,13, 14]. 5 , % !(! f #( %& ! 1 !, % & % .7 & : ) f | % & 1& ( > ") f(xT (y) y x3 : : : xn) = f(x y x3 : : : xn)T (y), R = L, L = R. H % , % / " % / % "( A && %& ! 1 !. 6 / " % & "() % / % # () 1!. 5 3 .7 % % / &. . &( ( f A(2) - A f(A2 A2 A : : : A) = (0) f(A3 A : : : A) = (0): . % f(x1 x2 : : : xn) " % f(x1 x2 x3). ?# & (% .% % f(xy x t) = f(yx x t) = 0, % , 1 f(x y t) f(xy z t) % ( ( . ; 1& f(xy ab t) % ( , % , f(xy ab t) = f(ay xb t) = f(ab xy t): ? # & % % f ( ( , f(xy ab t) = f(ab xy t) ;
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% (% % 2f(xy ab t) = 0, # %, f(xy ab t) = 0. 8 % , f((xy)z a b) = f(az xy b) = 0 % % " . 2 ; " # , 1& f(x1 x2 : : : xn) # % % % && %& "( % / % "( A, % 1& f(x1 x2 x3 : : : xn+1) | " % / % "( A.
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K % % . " A = A0 + A1 F ) % % 3, .7 . .7 " #( 9 %(: x, ei (i > 2), h4k , h4k+3 (k > 1). O %( 3 %( # 3 3 %( , 9 %( 3 %( 9 % x # 3 3 %( . 3 " # &: U0 = Esp ei i 0 (mod 2) U1 = Esp ei i 1 (mod 2) H0 = Esp hi i 0 (mod 2) H1 = Esp hi i 1 (mod 2) Esp X | ! % % , / 3 X. G # & " #() 9 % : 1) x x = e2 , ei x = ei+1 (i > 2)> 2) x e4k = e4k+1 (k > 1)> 3) x e4k+1 = e4k+2 (k > 1)> 4) x e2 = e3 , x e4k+2 = e4k+3 + h4k+3 (k > 1), 5) x e4k+3 = e4k+4 h4k+4 (k > 0)> 6) ei ej = "(j)hi+j , i + j 0 (mod 4) i + j 3 (mod 4)> "(4k + 1) = "(4k + 2) = 1, "(4k) = "(4k + 3) = 1> 7) # &, % ( # ( ( 7) % ), % .%& ( , %. . H0 + H1 Ann(A)> ei ej = 0 i + j 1 (mod 4) i + j 2 (mod 4): H % , % 9 % e2 % % ."( 9 % "(. B , %, % " A / %& 3 %( 9 % x. G && %& !, %. . ! / % &% % # 9 % % % ! x. 8 / (! " #(! 9 % && %& % % 9% ! !, 3 " # 9 % % % % x. h
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H % , % % & " && %& 1 % - % " !, % A(2) = A2 A2 = H0 + H1 Ann(A). B / % , % ! " G(A) = G0 A0 +G1 A1 % ! "( A ( gn n = 4k 4k +3 % ( % &. B& 9% % % %, % # () # &) n 9 %( x x]sRnx ;4x x]s a b]s | % %
% ( % &. ; x x]s = 2x2 = 2e2, % % % %, %
e2 Rnx ;4 e2 = 0: ? : e2 Rxn;4 = en;2, en;2 e2 = hn = 0. 10. + A2 . . H % , % " A2 ! / %& F 9 % e, h ( ( & % %( 7 (). @ h Ann(A), % % % % % ( # & ei ej ej ei . # /( : 1) i + j 0 (mod 4), 2) i + j 3 (mod 4). 1) C i + j 0 (mod 4), % % % # % ) i 0 (mod 4), j 0 (mod 4)> ") i 2 (mod 4), j 2 (mod 4)> ) i 1 (mod 4), j 3 (mod 4)> ) i 3 (mod 4), j 1 (mod 4). B& / # % ), ") # & "(i), "(j) ( " #() % 3 %(. B& % ), ) % 3 %(, # & "(i), "(j) % /(. 2) C i + j 3 (mod 4), % % % # % ) i 0 (mod 4), j 3 (mod 4)> ") i 3 (mod 4), j 0 (mod 4)> ) i 1 (mod 4), j 2 (mod 4)> ) i 2 (mod 4), j 1 (mod 4). B& / # % # & "(i), "(j) ( # " #() % && %& 3 %( . 2 11. + A , . . a b]s c]s = 0 j a j j b j a b]s = ab ( 1) ba | # a, b, a | # a. . ( 10 % % % % ! a = ei b = c = x:
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867
?# % "1( / & (% %: e2i x]s = e2i x] H1 Ann(A)> e2i+1 x]s = e2i+1 x H0 Ann(A) a b] = ab ba | % % , a b = ab+ba | ! # . 2 12. + A , . . : (a b c) + ( 1)jbj jcj (a c b) = 0 (a b c) + ( 1)jaj jbj(b a c) = 0 a, b, c | # , a | # a, (a b c), , (ab)c a(bc). # "3 % , # %, % % &) / %& ( 9 % % x ei . @ 9% " % 9 %( % ei , &! 1 % (ei ej ek ) !. ( 1 % (, / 7 9 % x 9 % % ei . @ & ( &, " % # &, / 7 9 %( h # &% "(. ) @ / w = e4 h4, (e2 x x) = (e2 x) x e2 (x x) = e4 e2 e2 = w> (x e2 x) = (x e2 ) x x (e2 x) = e3 x x e3 = = e4 ( e4 h4 ) = 2e4 + h4 = e4 + h4 = w> (x x e2) = (x x) e2 x (x e2 ) = e2 e2 x e3 = = h4 ( e4 h4) = e4 + 2h4 = w: & ( 1 % (, &%, % ( ( % / % % % % & % ! % ! 9 % . ") 6 %( & e4k e2 = e2 e4k = 0, (e4k x x) = (e4k x) x e4k (x x) = e4k+2 e4k e2 = e4k+2> (x e4k x) = (x e4k ) x x (e4k x) = e4k+1 x x e4k+1 = = e4k+2 + e4k+2 = e4k+2 > (x x e4k) = (x x) e4k x (x e4k ) = e2 e4k x e4k+1 = e4k+2 = e4k+2 : ) @ % w = e4k+3 h4k+3. 6 %( & 10 % e4k+1 e2 = = h4k+3, (e4k+1 x x) = (e4k+1 x) x e4k+1 (x x) = = e4k+3 e4k+1 e2 = e4k+3 h4k+3 = w> (x e4k+1 x) = (x e4k+1) x x (e4k+1 x) = e4k+2 x x e4k+2 = = e4k+3 (e4k+3 + h4k+3) = 2e4k+3 h4k+3 = w> 2
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(x x e4k+1) = (x x) e4k+1 x (x e4k+1) = e2 e4k+1 + x e4k+2 = = e4k+1 e2 + e4k+3 + h4k+3 = e4k+3 + 2h4k+3 = w: ;
) @ % w = e4k+4 h4k+4. 6 %( & 10 % e4k+2 e2 = = h4k+4, ;
(e4k+2 x x) = (e4k+2 x) x e4k+2 (x x) = e4k+4 e4k+2 e2 = = e4k+4 h4k+4 = w> (x e4k+2 x) = (x e4k+2) x x (e4k+2 x) = (e4k+3 + h4k+3) x x e4k+3 = = e4k+4 ( e4k+4 h4k+4) = 2e4k+4 + h4k+4 = w> (x x e4k+2) = (x x) e4k+2 x (x e4k+2 ) = e2 e4k+2 x (e4k+3 + h4k+3) = = e4k+2 e2 x e4k+3 = h4k+4 ( e4k+4 h4k+4) = e4k+3 + 2h4k+3 = w: ;
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) 6 %( & % e4k+3 e2 = e2 e4k+3 = 0, (e4k+3 x x) = (e4k+3 x) x (x e4k+3 x) = (x e4k+3) x = e4k+4 x x e4k+4 = (x x e4k+3) = (x x) e4k+3 = x e4k+4 = e4k+5:
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e4k+3 (x x) = e4k+5 e4k+3 e2 = e4k+5 > x (e4k+3 x) = ( e4k+4 h4k+4) x x e4k+4 = e4k+5 e4k+5 = 2e4k+5 = e4k+5 > x (x e4k+3) = e2 e4k+3 x ( e4k+4 h4k+4) =
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U1 = Esp ei i 1 (mod 2) > V1 = Esp gi i 1 (mod 2) > H1 = Esp hi i 1 (mod 2) h
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x U0 U1 V0 V1 x e2 V1 V0 U1 + H1 U0 U0 U1 H0 H1 H1 U1 U0 H0 H1 V0 U1 + V1 + H1 H0 H0 H1 V1 U0 + V0 + H0 H1 H0 H0 3 %( % # &%( & . G # & " #() 9 % : 1) x x = e2 > 2) ei x = ei+1 , gi x = gi+1 + ( 1)i ei+1 + ei;1 e2 > 3) x ei = gi+1 , x g2i+1 = e2i+2 , x g2i = e2i+1 + e2 e2i;1> 4) i + n 1 (mod 2), % ) e2i e2n = ( 1)n h2(i+n), ") e2i e2n+1 = = ( 1)n+1 h2(i+n)+1> 5) e2i+1 e2n+1 = e2i+2 e2n > 6) ) g2i g2n = e2i e2n , g2i+1 g2n+1 = e2i+2 e2n, ") i + n 1 (mod 2), % g2i g2n+1 = ( 1)nh2(i+n)+1 > 7) ) g2i e2n = e2i e2n, g2i+1 e2n+1 = e2i+2 e2n, g2i+1 2n = ( 1)n h2(i+n)+1 , i + n 1 (mod 2), ") i + n 1 (mod 2), % e2i+1 g2n = e2i g2n+1 = = ( 1)i+1 h2(i+n)+1> 8) # &, % ( # ( ( 7) % ), % .%& ( . ?# % "1( / & (% .%
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U0 V0 = U1 U0 = U1 V1 = V0 U1 = V1 V0 = (0)> A(2) = H0 + H1: H % , % % & " && %& 1 ! , 3 % 4]. B , " A / %& 3 %( 9 % x. G && %&
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% % 9% ! !, 3 " # 9 % % 9% % % x. B / % , % ! " G(A) = G0 A0 + G1 A1 % ! "( A ( fn % ( % & n = 4k + 2, 4k + 3. B& 9% % % %, % # () # &) n 9 %( x x]s x]sRnx ;5x x]s, a b]s | % % , % ( % &. ; x x]s = 2x2 = 2e2 x x]s x]s = 2x2 x]s = 2(e2 x x e2 ), % % % %, % (e2 x x e2 )Rnx ;5 e2 = 0. @ . A(2) :
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870
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(e2 x x e2 ) x = (e3 g3) x e4 (g4 e4 ) = 2e4 g4 = 2(e4 + g4) ep R2x = (ep x) x = ep+1 x = ep+2 gp R2x = (gp x) x (gp+1 + ( 1)p ep+1 ) x (gp+1 x + ( 1)p ep+1 x) (gp+2 + ( 1)p+1 ep+2 ) + ( 1)p ep+2 = gp+2 % , k;1) 2(e4k + g4k) 2(e4 + g4 )R4( x (e4 + g4) x 2e5 + g5 2(e4 + g4 )R4xi;3 2(e4i+1 + g4i+1 ) (e2 x x e2 )R4xi;3 e2 = 2(e4i + g4i) e2 = 4e4i e2 = e4i e2 = h4i+2 = 0 (e2 x x e2 )R4xi;2 e2 = 2(2e4i+1 + g4i+1) e2 = 2g4i+1 e2 = 2h4i+3 = 0: B / , % % & " && %& 1 % - % " ! % % !. I % & % " % , " H0 Ann(A). B& % % % / ( % ") ( 1 % ( % ( # % %(. 2 ;
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Abstract V. N. Remeslennikov, The dimension of algebraic sets over a free metabelian group, Fundamentalnayai prikladnaya matematika, vol. 7 (2001), no. 3, pp. 873{885.
The aim of the paper is to estimate the dimension of algebraic sets over a nonabelian free metabelian group.
2]. !
. " # $ ( &) F . ( # & x 1 u-$ $ , $ $ $ u-. + u-$ G $ (G) (G) (G) = ((G) (G)). - . , x 2 $ 2.16. ': G1 ! G2 | - u- ker ' 6= 1. (G1 ) < (G2). ( ! # $ & Y F n, F | r > 1. 1 GY | Y , , r 6 (GY ) 6 n+r r ; 1 6 (GY ) 6 n+r ; 1, . . GY $ $ n r $ F . 3 # & , 2001, 7, , 3, . 873{885. c 2001 , ! " #
874
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$ : G $ $ $. , $ ! : qn(x) = 8x (xn = 1 ;! x = 1) n 2 N: (1.2) 3 , $ 8x y (x y x] = x y y] = 1 ;! x y] = 1): (1.3) + ! & : 1)
875
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! $ .$ F: 1) F M0 = 2) Fit(F ) = F 0, F 0 | F= 3) A = F= Fit(F) | rank(A) = rank(F )= 4) Fit(F) & ZA. " # , M0,
U (u-$). E G ; u-, (i) G M0= (ii) Fit(G) ZA = A = G= Fit(G). "; !; | (D- : 8x t z ((x 6= 1 ^ x y] = x z] = 1) ;! y z] = 1): (1.4) 1 # $ G, < $ # $ $. " , , G , ; & , & # . 1.2. &' " u- G " "" ( - . . G G Fit(G). , g | # G, g? | A = G= Fit(G). a 6= 1, a b] = a c] = 1 # a, b, c G. , a 2 Fit(G). D (1 ; ?b)a = (1 ; ?c)a = 0 Fit(G). " $ (ii) u-$ , b c 2 Fit(G) b c] = 1 G. , a 2= Fit(G). D a b] = a c] = 1,
876
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& , gi 2 Xn . 7 supp() = fgi j ni 6= 0g, supp+ () = fgi j ni > 0g, supp; () = fgi j ni < 0g
C() # . 1 supp+ () = ? supp; () = ?, C() = ?. " $ # gi 2 supp+ (), gj 2 supp; () bl = gi gj;1 # b Xn , l > 0. C() # b, $ (gi gj ). K , # , C() = fb1 : : : bkg. Dk Q det = (1 ; bi ) ( # R). 7 i=1 ": ZXn ! Z & & &. "; : # g h1 : : : hm G g"1 h1 +:::+"m hm (g"1 )h1 (g"2 )h2 : : :(g"m )hm , "i 2 f1 ;1g, i = 1 : : : m. 0 6= 2 ZXn. 3< $ . $, $ . 1 "() 6= 0 8y z x1 : : : xn (y z] = 1 ;! y z] = 1): (1.5.a) 1 "() = 0 8y z x1 : : : xn (y z] = 1 ;! y z]det = 1): (1.5.b) 1.4. ) " ' u- (1.1){(1.5). * , " G (1.1){(1.5), G " "" u- . . G u-. D $ $ $ (1.1){(1.4). $ G $. 3 !
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877
G | -; G= Fit(G) n. , G < 0 6= 2 ZXn # x1 = a1 : : :, xn = an , y = b, z = c. D m = b c] 6= 1 m 2 Fit(G). D Fit(G) | , xi ! ai $ b c](a1:::an ) = 1, . ., $ $, (?a1 : : : ?an) = 0, a?i | $ ai An = G= Fit(G). 1 "() 6= 0, ZAn. 1 "() = 0, (?a1 : : : ?an) , . ()
! $ i, j, gi 2 supp+ (), gj 2 supp; () gi(a1 : : : an) gj (a1 : : : an) Fit(G). ( , ! # b C() , b(a1 : : : an) = 1 Fit(G), . (1.5.b) # a1 : : : an b c. , , $ G $ $ $ (1.1){(1.5). +, G u-. D , $<, , G -; , An = G= Fit(G) | n G . + , Fit(G) | ZAn, . . 0 6= m, Ann(m) = 0. # , m = b c]. , 2 Ann(m), 2 ZA. D m 6= 1 G, "() = 0 $ (1.5.a), (1.5.b) y z]det = 1 G. k Q det = (1 ; bi), bi 2 C(), i = 1 : : : k. 7 m0 = m, i=1 Ql (1;bi) ml = y z]i=1 , l 6 k. mj | $ # $ m0 : : : mk . D m1l ;bj+1 = 1, h | bj +1 G. 3, h 2= Fit(G), h | $ # An . D $ G $ (D, h & # $ Fit(G). 7 H = hFit(G) hi. D H | , < Fit(G), , Ann(m) = 0. m | $ # Fit(G). D ! h 2= Fit(G), m h] = m1 6= 1 G ( G $ &, (D). + , 2 Ann(m). D m = 1, m(1;h) = m1 = 1. D m1 | , = 0. 2 1.5. G | u- N | Fit(G). - G=N " "" u- . . 1 N = Fit(G), G=N | , u-. 0 < N < Fit(G). , G=N M0 . + # $ $ (1.3) G=N. a? = a + N, ?b = b + N, (?a ?b ?a) = 1 (?a ?b ?b) = 1 G=N. D a, b # a?, ?b $ (a b a) 2 N, (a b b) 2 N. 1 (a b) 2= N, (a b)a;1 2 N, a 2= Fit(G), N $. #-
878
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a 2 Fit(G), $
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x
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2.1.
" # $ $ $ $ cite2,5, $ . 2.1. G | . E H $ G- , G H.
G- = , . ': H1 ! H2 G- $ G- , '(g) = g g 2 G. 2.2. X | G | . D G F (X), F (X) | X, $ G- GX]. 2.3. - Gn = f(g1 : : : gn) j gi 2 Gg $ n- G= n = 1, G1 = G. S GX], X = fx1 : : : xng. 1 $ $ S = 1 G < # $. 2.4. S GX], X = fx1 : : : xng. D VG (S) = fp 2 Gn j f(p) = 1 f 2 S g $ G, ;$ S. 2.5. S GX] G Y = VG (S). D $ Rad(S) = Rad(Y ) = ff 2 GX] j f(p) = 1 p 2 Y g: 3 ; Rad(S) $ S ( Y ). K , Rad(S) GX]. 2.6. @- GY = GS = GX]= Rad(S) $ Y .
Gn, Z Gm | G. D ': Y ! Z . Y Z, ! f1 : : : fm 2 Gx1 : : : xn], p 2 Y '(p) = = (f1 (p) : : : fm (p)) 2 Z. P Y Z $ .$, ! . $ ': Y ! Z, : Z ! Y , ' = 1Y ' = 1Z . 2.8. ASG G . , ;$ $<. 7 AGG | $ , | . G-. $ ($ $ G-). ASG AGG # $ 2]. 2.9. E G , n > 0 S Gx1 : : : xn] ! S0 S, VG (S) = VG (S0 ). ! (. #2]). ; = , ; . -$ ; Gn $ ! ", $ $ Gn . ( , . (3 Y X $ , Y > $ , $ Y .) 2.10. Y | Gn . # dimY Y $ ! & $ Y = Y0 > Y1 > : : : > Ym (2.1) ! , 1 . $ , $ & G-. '1 '2 'm GY0 ;! GY1 ;! : : : ;! GYm (2:10) 'i | $ G-#. $ ( . 2]). ( , Y $ & $ G-#. $ $ . E $ $ G? 7 # ! . 2.11 (#2]). G | " . " I D I , " GI =D '+:
2.7. Y
879
880
. .
(1) " " G GI =D# (2) " - " G- GI =D Y G. " G- $ G- . $, G- G- . 1 K |
G-, G- $ G-uncl(K) K, G- G-qvar(K), G- G-var(K). V K!
-;$ G- K ( . 5]). " # ! $. 2.12 (#5]). G | " . (1) - G-qvar(G) " "'" G# (2) - G-uncl(G) " "'" G. !. D F ; , ; $ 2.11 2.12. , G | F , G 2 (F -uncl)! 2.12, G 2 uncl(F), , 1.4 G u-. 2.2.
G | -; u-. "; ; $ (G), (G), (G). D G? = G= Fit(G) ? + , , (G) = rank(G). ? Fit(G) Fit(G) & ZG, ? $ $ ZG- . m | ? ! Fit(G), $ ZG, (G) = m. I $ (G) $ ! $ $ ZG? Fit(G). W (G) $ G ((G) (G)). 7 U!
-;$ u- ; f(G) j G 2 U! g. 1 G1 G2 2 U! , (G1) 6 (G2), (G1) < (G2) (G1 ) 6 (G2 ) , (G1) = (G2). ' 2.13. H | - " u- G. (H) 6 (G). . D H \ Fit(G) 6 Fit(H), ! >$ . H=H \ Fit(G) H= Fit(H). D H=H \ Fit(G)
881
. $ G= Fit(G) , $. 2 ' 2.14. ': G1 ! G2 | U! (G1) = (G2 ). (G2 ) 6 (G1 ) , ker ' 6= 1, (G2 ) < (G1 ). . 1 G2 | , G1 | , ker ' 6= 1, rank(G2) = (G2 ) < rank(G1 ) = (G1 ). , G2 | , , 1) ker ' 6 Fit(G1 ) 2) ker ' | $ Fit(G1 ). " , ker ' 66 Fit(G1 ), , '(Fit(G1)) 6 Fit(G2), (G2) < (G1 ), . + , Fit(G2 ) = Fit(G1)= ker '. 1 , , Fit(G2) < Fit(G1 )= ker ', ! # g 2= Fit(G1 ), '(g) 2 Fit(G2), # (G2) < (G1 ). 3&, ker ' $ , Fit(G2 ) = Fit(G1 )= ker ' & ZA, A = G1= Fit(G1 ) = G2 = Fit(G2). I ., G2 | u-. 2 ' 2.15. % ': G1 ! G2 | U! , (G2) 6 (G1 ). . D '(Fit(G1)) 6 Fit(G2), $
. 2 2.16. % ': G1 ! G2 | U! ker ' 6= 1, (G2) < (G1). . 2.15 (G2) 6 (G1). 1 (G2) < (G1), (G2 ) < (G1) . 1 (G2) = (G1 ), 2.14 (G2 ) < (G1 ), # (G2) < (G1). 2 2.17. % G | U! , '+ " n '+ , (G) 6 n. % , , G , (G) 6 n ; 1. $ ; #. 1. + , G Fn n > 1. D Fit(Fn ) = Fn0 , Fn= Fit(Fn) = Fn=F 0 = An , , (Fn) = n. " - Fn < An wr An , x 1, , (Fn ) 6 n. +, (Fn ) = n ; 1. F 0 ZAn # xi xj ], i < j, i j 2 f1 : : : ng. 7 N Fn0 , ;$ (n ; 1) # x1 xj ], j = 2 : : : n. ( ! - , # # $ $ ZAn, , N | $ ZAn- n ; 1. B &, ! # $ F 0
N . $ "{W
# x y z $. 2. (G) = (Fn) = n. D 2.14 (G) 6 (Fn ) = n ; 1, # .
882
. .
3. (G) < n. D # , Fit(G) Fit(Fn)
$ & 2.14. V $ # , ; $ $ (m n), 1 6 m < n, Fmn, ; $ : xi xj ] = 1, i j > m= xi xj xk ] = 1, Fmn = x1 : : : xn i j 2 f1 : : : ng, k > m. (2.2)
E Fmn ! . n ; m = s, xk+1 = t1 ,... , xn = ts Fmn. 7 N = unclhxi xj ] t1 : : : ts j i j 2 f1 : : : mgi Fmn . ' 2.18. - . " N =Fit(Fmn ) Fmn =N | " m. . " Gmn = Fm i T , i | , Fm | m x1 : : : xm, T | $ ZAm- t1 : : : ts , &, Fm T . f ;1 tf = f? t, f 2 Fm , f? | f Am , t 2 T. K , Gmn # x1 : : : xm t1 : : : ts. , , ! Fmn ! ! Gmn . ': Fmn ! Gmn . , ker ' = 1. " , f 2 Fmn '(f) = 1, f 2 N. +, f Fmn f = f0 t1 1 : : :ts s , f0 2 Fm0 , i 2 ZAm, i = 1 : : : s. + # $ xi xk ], 1 6 i 6 m, k > m, . xi xk ] = t1k;xi , Fmn N ZAm- . # '(f) = 1, f0 = 1, 1 = : : : = s = 0 ZAm. + , , Fit(Fmn ). K , N 6 Fit(Fmn), N | . 1 f 2= N, f = x1 1 : : :xmm a, i 2 Z, a 2 N i 6= 0. " # f $ N, f 2= Fit(Fmn ). 3&, .- Fmn=N m. ( , (Fmn) = m. D N S Fm0 $ Fm , (Fmn ) = (Fm ) + s = = m ; 1 + s = n ; 1. " , Fmn u-. ' 2.19. u- G n '+ (G)=m
(aj ) = 1, j > m. # ! y1 : : : yn $ Fn,
883
(yj ) 2 Fit(G), j > m. 7 $ ! (y1 ) : : : (yn ) G $ $ < Fmn (2.2). 2 "; $ 3. (G) = m < n. D 2.19 G .- Fmn (G) = (Fmn). B &, 2.14 (G) 6 (Fmn ) = n ; 1. 2 2.3. !
F
$ # .$ 7]. F | r > 2 x1 : : : xr , Y | F n GY | . Q 2.11 GY n-; F - F = F I =D . D ;$ I. 3, F F F . ' | - Fr Mr = Ar i T , Ar | r a1 : : : ar , T | $ ZAr- t1 : : : tr . 7 Mr , Ar , T, (ZAr), Z Mr , Ar , T, ZAr, Z . D, $<. D ' < ': Fr ! Mr . ( Mr Mr . G , Mr = Ar i T, T | $ (ZAr)- t1 : : : tr . 7, $ # Ar . a = a1 1 : : :ar r , i 2 Z, i = 1 : : : r. !. (ZAr) $ ! .$ & (ZAr): 1) # $ Ar $ # &= 2) & ZAr & (ZAr). 2.20. Y | F n GY | " . GY | u- (GY ) > r, (GY ) > r ; 1. . GY u- 2.1. 7 Ar = F=F 0 Am = GY = Fit(GY ). , m > r. GY | , GY F - F 2].
| $ F -#. GY F. D (Fit(GY )) = F 0 ( <, # $ F, ! F 0, $ F 0 Fit(GY )), & #. : Am ! Ar . 7 m > r. G , | F -. , Ar | Am . + , 2.11 ! : GY ! Fr ,
!; - , $<, Mr . GY # x1 : : : xr ( ! F ) g1 : : : gn,
884
. .
(xi ) = a0i (gj ) = b0j
ti i = 1 : : : r 1 mj b 2 A m 2 T j = 1 : : : n: j r j r 1
7 B Ar , ; # a1 : : : ar b1 : : : bn. K , B ' Am , $<, Ar | Am . D # $ t1 : : : tr Tr & (ZAr) ZB | & (ZAr) # , TB = (ZB)t1 + : : : + (ZB)tr $ ZB- t1 : : : tr . +, # $ m2 = x1 x2],... , mr = x1 xr ] $ ZB- Fit(GY ). -$ ( $ 2.17), m2 : : : mr $ ZAr- TAr . D Ar | L B, B = Ar C. D TB = cTAr . 7 $ , c2C (GY ) > r ; 1. 2 2.21. Y | F n F r, GY | " . dimY 6 ((G) ; r + 1)(n + 1). - , dimF n 6 (n + 1)2. . F = G | r > 1 # $ a1 : : : ar G. GX], X = = fx1 : : : xng, | G- n, Rad(Y ) | Y . D GY = GX]= Rad(Y ) . D GY , ': GX] ! GY
n + r. B $ 2.17
, (Fn+r ) = n + r (Fn+r ) = n + r ; 1. ( , 2.17 (GY ) 6 n + r (GY ) 6 n + r ; 1. D 2.20 &$ # : (GY ) > r, (GY ) > r ; 1. # r 6 (GY ) 6 n + r, r ; 1 6 (GY ) 6 n + r ; 1, . .
& n. t | & G0 = GY ! G1 ! : : : ! Gt $ #. $ $ , ! Y . D 2.16 (Gi) < (Gi;1), i = 1 : : : t. 3, (G) = ((G) (G)) $ . . 7 , t 6 ((G) ; r + 1)(n + 1). 2 !. - $ $, $ $ , $ & $ Y . . $ $ dimY Y F n . B # : dimF n
$ F n.
885
1] . . , . . . G- G- // . | 2000. | #. 39, & 3. | '. 249{272. 2] G. Baumslag, A. Myasnikov, V. Remeslennikov. Algebraic geometry over groups. I // J. Algebra. | 1999. | Vol. 219. | P. 16{79. 3] O. Chapuis. Contributions a la theorie des groupes resolubles. | Universite Paris VII. These de Doctorat Mathematiques. | 1994. 4] N. Gupta. Free group-rings. | Providence: Amer. Math. Soc. 5] A. Myasnikov, V. Remeslennikov. Algebraic geometry over groups. II // J. Algebra. | 2000. | Vol. 234. | P. 225{276. 6] V. Remeslennikov, R. St.ohr. On the quasivariety generated by a non-cyclic free metabelian group. | Preprint. | 2000. 7] # /. '0 1 0 1 / . 2. 1. | .: 3 , 1982. $ % 2001 .
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517.518.126
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Abstract A. P. Solodov, Riemann-type denition for the restricted Denjoy{Bohner integral, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 887{895.
The generalization of the restricted Denjoy integral is studied for the case of Banach-valued functions. The equivalence between this integral and the HL-integral de4ned with the use of generalized Riemann sums is proved.
, . ! " | " %&{( ) , & ",
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%& ) , * ( . ,3]). 0 " 1 {* ( . ,6, . 17 104]). ,11] , & , 1 {* , . , " &
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%&{( ( . ! +( ( ()) !556 ( + 99{01{00354, 00{15{96143). , 2001, # 7, 8 3, ". 887{895. c 2001 ! "#, $% &' (
888
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) , HL- ( 2.2). < , 1. 1. = ( . ,2]), & * , 1 {* . > " " %&{( HL- ,1]. ! " 8 . " 8 , & , & " , " " > 0. % & & ( . 2.1)
( ) , Wu Congxin Yao Xiaobo ,9], , 7 8 ( 3.2).
1.
> " X | , R | " , ,a b] | . , 8 " ( . ,10, VII]). 1.1. @ F : ,a b] ! X VB - & E ,a b], M > 0, Di ni=1 , n P E, !(F Di) < M. (E " !(F P ) = i=1 = sup kF (t) ; F(s)k |
F & P.) ts2P
1.2. @ F : ,a b] ! X AC - & E ,a b], " > 0 > 0, Di ni=1 , n P E jDij < , n i=1 P !(F Di) < ". (E " jP j & P .) i=1 1.3. @ F : ,a b] ! X VBG - & E ,a b], E c K & , & F VB - . 1.4. @ F : ,a b] ! X ACG - & E ,a b], E c K & , & F AC - .
{ 889
1.5. > " F : ,a b] ! X. A 2 X -
F t0 , ; F(t0) = A: lim F(t)t ; t!t0 t0 ( " X X. 1.6 (. 7, . 11]). @ F : ,a b] ! X wAC - & E ,a b], & x 2 X x F AC - E. 1.7 (. 7, . 22] 12, . 102]). F : ,a b] ! X AC - E ,a b] , VB - wAC - E . % M %& ( . ,10, VII]). 1.8 (. 9]). @ f : ,a b] ! X , AC - F : ,a b] ! X, F 0(t) = f(t) . . . < , ,12, . 93]. 1.9 (. 7, . 45]). @ f : ,a b] ! X { , ACG - F : ,a b] ! X, F 0(t) = f(t) . . , ) ( . ,2], ,4] ,11]). > " I | " , & ,a b]. T ,a b] (k Dk ) 2 R I , k = 1 : : : n, : 1)
Di Dj , i 6= j, n S 2) Dk = ,a b]. k=1 1.10. ) T ,a b] - , & ( D) 2 T 2 D ( ; () + ()): 1.11. ) T ,a b] - , & ( D) 2 T D ( ; () + ()): 1.12. @ f : ,a b] ! X
,a b], I 2 X : " > 0 & " (),
890
. .
- * T ,a b] X f(k )jDk j ; I < ": T
I
f ,a b], Rb I = (H) f dx. a 1.13. @ f : ,a b] ! X ,a b], I 2 X : " > 0 & " (), - 7 8 T ,a b] X < ": f( ) j D j ; I k k T
I
f ,a b], Rb I = (M) f dx. a 1.14. @ f : ,a b] ! X HL- ,a b], F : ,a b] ! X : " > 0 & " (), - * T ,a b] X kf(k )jDkj ; F(Dk)k < ": T
(E " F (D) |
F D.) 1.15. @ f : ,a b] ! X ML- ,a b], F : ,a b] ! X : " > 0 & " (), - 7 8 T ,a b] X kf(k )jDkj ; F(Dk)k < ": T
( ,9] ML- " 7 8 .) M ", HL- ML- & * 7 8 . " 8 & . 1.16. f : ,a b] ! X HL- ,a b]. F : ,a b] ! X !. 1.17. f : ,a b] ! X " Rb . HL- f dx = 0. a
{ 891
> & , ( . ,5] ,8]). 1.18. @
f : ,a b] ! R , I : " > 0 & " () ,a b], - * T X n f (k ) Dk i=1
j j;
I < ":
1.19. @
F : ,a b] ! X ACG -! & E ,a b], E c K & , & F AC - . 1.20. f : ,a b] ! R . f ! # , $ ! F : ,a b] ! R ACG - . . B ,2] ,8] " , & , 1.20, " " . ! ,
" ( . ,2] ,8]), 1.20 & .
2. ,9] ". 2.1. X | $ . f : ,a b] ! X &$ , ML- . Q8 | " " %&{( . 2.2. X | $ . f : ,a b] ! X ' {&$ ! , HL- .
! "#$%$& .
(= > " f HL- F. > & , f %&{( . % " . 1. F F 0 = f . . 2. F ACG - . > & & .
892
. .
1. % & " n & Pn ,a b] : t 2 Pn , " " D(it) 1 i=1 , t (t) kF (D(it)) ; f(t)jD(it)jk > jDni j : (1) R , & , F F 0(t) 6= f(t), 1 S & Pn . > & , jPnj > 0. & " n=1 () HL- " = jPnj=(2n). 1 D(it) (t ; (t) t + (t)) t 2 Pn & Pn . > ( . ,10]) & " Dr , 1 6 r 6 k, k X jD(rtr)j > jP2nj : (2) ! K (1) (2),
r=1
kF (D(rt )) ; f(tr )jD(rt )jk > jP2nnj r=1 (). 1 ", jPnj = 0 n. > S 1 k X
r
r
Pn = 0. 1 ", F n=1 F 0 = f . . 2. % " , ,3], , " . & " () HL- " = 1 ( & " () < 1). % & " m " l, a + l=(2m) 6 b. >& m k, k 6 l + 1, k ; 1 k 1 k Em = t 2 a + 2m a + 2m \ ,a b]: kf(t)k 6 m (t) > m : 1 lS +1 S
! , ,a b] = Emk . > ", F m=1 k=1 VB - & & Emk . @ " m k, & Emk . ) " ,ci di] ni=1 c Emk . S & Emk , (ci ,ci di]) ni=1 - * . ; F ( 1.16),
ui vi 2 ,ci di],
kF (vi) ; F(ui)k = !(F ,ci di]): (3)
{ 893
> (3), & Emk - " * , n X i=1
!(F ,ci di]) =
+ + +
n X i=1 n X i=1 n X i=1
n X i=1
n
kF (vi) ; F (ui)k 6 X kF (vi) ; F(ci)k + n X
i=1
kF(ui) ; F(ci)k 6 kF(ui) ; F(ci) ; f(ci )(ui ; ci)k + i=1 n X
kf(ci)k(ui ; ci) + kF(vi) ; F(ci) ; f(ci)(vi ; ci)k + i=1
kf(ci)k(vi ; ci) < 1 + m m1 + 1 + m m1 = 4:
S , F VB - Emk . ) fx f kx k = 1g. ! , jx f(t)j 6 kf(t)k. HL- f
* . > 1.20 x F ACG - . < , ,a b] K & Qn , & F wAC - . ;
, & Emk \ Qn F VB - wAC - . > 1.7, , F AC - Emk \ Qn, " ACG - ,a b]. 1 ", " %&{( . =) > & ", f %&{( , " ACG - F , F 0(t) = f(t) . . > & , f HL- c F. ! D &
t 2 ,a b], F F 0 (t) = f(t). ; E = ,a b] n D
". 1.17 & ", f(t) = 0 t 2 E. ; F ACG - , E K & En, & F AC - . @ " " > 0. >& " () . T 2 D, & " () > 0 , kF(D) ; f()jDjk < 2(b "; a) jDj 2 D ( ; () + ()): (4)
; F AC - En, n > 0, Di mi=1 , m P En jDij < n , i=1
894
. . m X i=1
!(F Di) < 2n"+1 :
(5)
7& En ", & " & " () En , ( ; () + () c 2 En " ( 2En
;
() + ()) < n :
(6)
S , (x) . ) " - * T ,a b]. ! K 8 (5) (6) , f(t) = 0 t 2 En, X (7) kf(k )jDkj; F (Dk)k < 2n"+1 : k 2En
S " (4), X kf(k )jDkj; F(Dk)k < "2 : k 2D
(8)
S , (7) (8), & ", n X
k=1
kf(k )jDkj; F(Dk)k < ":
;
, f HL- .
3. ! " # $
,11] . 3.1. X | $ . ) ! " : 1) X | + 2) f : ,a b] ! X # , HL- + 3) f : ,a b] ! X ,- , ML- . ! K 2.1, 2.2 3.1, ". 3.2. X | $ . ) ! " : 1) X | + 2) f : ,a b] ! X # , ' {&$ ! +
{ 895
3) f : ,a b] ! X ,- , &$ . U " " 1 . U. " " .
%
1] Canoy Jr. S. R., Navarro M. P. A Denjoy-type integral for Banach-valued functions // Rend. Circ. Mat. Palermo. | 1995. | Vol. 44, no. 2. | P. 330{336. 2] Cao S. S. The Henstock integral for Banach-valued functions // SEA Bull. Math. | 1992. | Vol. 16, no. 1. | P. 35{40. 3] Gordon R. Equivalence of the generalized Riemann and restricted Denjoy integral // Real Analysis Exchange. | 1986{1987. | Vol. 12, no. 2. | P. 551{574. 4] Gordon R. The McShane integration of Banach-valued functions // Illinois J. Math. | 1990. | Vol. 34. | P. 557{567. 5] Kurzweil J., Jarnik J. Equiintegrability and controlled convergence of Perron-type integrable functions // Real Analysis Exchange. | 1991{1992. | Vol. 17. | P. 110{139. 6] Pfeer W. The Riemann approach to integration. | Cambridge: Cambridge University Press, 1993. 7] Solomon D. W. Denjoy integration in abstract spaces // Memoirs of the AMS. | 1969. | No. 85. 8] Wang P. Equiintegrability and controlled convergence for the Henstock integral // Real Analysis Exchange. | 1993{1994. | Vol. 19. | P. 236{241. 9] Wu Congxin, Yao Xiaobo. A Riemann-type denition of the Bochner integral // J. Math. Study. | 1994. | Vol. 27, no. 1. | P. 32{36. 10] . !" #. | $.: %&, 1949. 11] # ' ( ). *. % !" #+ , ! $-. '# / 0 ( 1 2 +0 34 5 . // $!. 16! . | 7 82! . 12] , ## 9., : ## 8 ;. . :4 5 #< +. # 1 8 #4" 488+. | $.: %&, 1962. ) !* + 1997 .
(1 2) B
. .
512.554.5
: , -
.
! " # # # #
(1 2) & '
. ( ! " # # #
) & '
. B
Abstract M. N. Trushina, Irreducible alternative superbimodules over the simple alternative superalgebra B (1 2), Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 897{908.
This text is devoted to irreducible alternative superbimodules over the alternative superalgebra (1 2). The classi0cation of unitary irreducible right rightalternative representations of the alternative superalgebra (1 2) over an algebraically closed 0eld is obtained. B
B
.
. . 1]. # 1994 . &. '. 2] )
*+ , , 2 3. & . . 3] , , . / + , , B(1 2). 2 ,
* , B(1 2). & 1 2 1334 00{01{00339. , 2001, 7, 5 3, . 897{908. c 2001 , !" #$ %
898
. .
3, ( , ), / + ,+ 4 B(1 2) 6- , B(1 2), 4 , +/ . k t] , 4) t9 + t6 ; (t3 + 1), | . # - ,) M0 8 1 + t3 , t2 + t5 , t4 + t7 . 9 X 4 + ;t, Y + D + ;1 ( + 1)X 5 , D | * , | . , , B(1 2)
, +
* , B(1 2). 2
, , , , , +
* 2].
1 . .
: .. ( ) A = A0 + A1 k 3 .. B(1 2), A0 = k 1, A1 = k x + k y x y = 1. ' .. B(1 2) x ! ;y, y ! x ) . , .. A = A0 + A1 M = M0 + M1 k, 4 M A ! M 4 (m a b) + (;1)jaj jbj(m b a) = 0 m 2 M, a b 2 A0 A1 , jaj | ) , 8 a. # + , , . . m 1 = m. ', .. A = A0 + A1 M = M0 + M1 k, / = M + A , .. , 4 (a b c) x] = (ab c x) + (bc a x) + (ca b x) +/ , ( . 3]), , , B(1 2).
2 . .
#) M: X = R(x) Y = R(y) ' = X 2 = Y 2 = XY R(x) 4 8 x. , , M B(1 2) , , , X Y ] = ;1: ()
...
899
# ,
* 4 .. B(1 2) (m x y) = (mx)y ; m(xy) = (mx)y ; m? (m y x) = (my)x ; m(yx) = (my)x + m: # 4 - , (m x y) = (m y x) 4 , : (mx)y ; m = (my)x + m. @ , 3, = (). @ 4 +
* . ) = , ,, . 1. : X 2 Y ] = X? (1) ' ] = '? (2) ] = ;? (3) ' ] = 1 ; : (4) . (1) X Y ] = ;1 4 ab c] = a b c] + a c]b +/ ,
* . # , X 2 Y ] = X X Y ] + X Y ]X = ;2X = X: ' ' ] = X 2 XY ] = X X 2 Y ] = X(X X Y ]) = X 2 = '? ] = Y 2 XY ] = Y 2 X]Y = (Y Y X])Y = ;Y 2 = ;? ' ]= X 2 Y 2 ]=X X Y 2 ]=X (Y X Y ])=X Y =2XY + Y X]=1 ; : 2 2. A | , M,
' . " ) XY Y X 2 A# ) '3 | $ A. . 4 () (4). , ) ,, '3 ' .
(2){(4), , '2 ] = ' ' ] = ' ' = ;'2 ? '3 ] = '2 ' ] + '2 ]' = '2 ' ] + f' ' ]g' = = '2 ' ] + ' ' ]' + ' ]'2 = '2 (1 ; ) + '(1 ; )' + (1 ; )'2 = = ;f'2 + '' + '2 g = ;f' '] + '2]g = ' ' ] + '2 ] = = '2 ; '2 = 0: 2
900
. .
3 . . 1. , M -
, . ,, ) , M0 = 0 (
). C M1 A1 + A1 M1 M0 = 0. & () , M1 = M1 X Y ], 8 M1 = 0. , ,. # , = ,, . 2. , N0 M0 , ' , . . N0 A- M0. 4 N1 == N0 x + N0 y. C N0 + N1 | , M. 3. E 4 L = fm 2 M0 j m' = m = 0g , A- M0 .
4 . "
.. (1 2). B
M : ) L = M0 ? ) L = 0. F
L = M0 . , m0 2 M0 . 4 p1 = m0 x q1 = m0 y: C p1x = m0 ' = 0 q1y = m0 = 0: (4) 0 = m0 ' ] = m0 (1 ; ), . . m0 = m0 ,
() p1 y = m0 = m0 q1x = m0 Y X = m0 (1 + ) = ;m0 : C , ,. & 8 ,,
) L = 0. # M : M0 '3 = 0 M0 '3 = M0 : , , M0 ' = M0 .
, 4 : 1) M0 '3 = 0 M0 3 = 0? 2) M0 '3 = M0 M0 3 = 0? 3) M0 '3 = 0 M0 3 = M0 ? 4) M0 '3 = M0 M0 3 = M0 : 3. % , & M 1) 2). " 0 6= m0 2 M0 , ' & m0 = m0 Y X = 0. . , M 1) 2), (90 6= m) m = 0. , , = ,
...
901
(Y X)2 + Y X = Y XY X + Y X = Y X Y ]X + Y 2X 2 + Y X = = ;Y X + Y 2 X 2 + Y X = Y 2 X 2 , (mY X)Y X = ;(mY X). 8 mY X = 0, n = mY X nY X = ;n. , 8 8 n 4 n = 0. , , n = (mY X) = mY XY Y = mY (Y X + X Y ])Y = = mY 2 XY ; mY 2 = m( ; 1) = 0: , 8 n 6= 0 nY X = ;n n = 0: ,, n = n: n = nXY = n(Y X ; 1) = ;2n = n: 4, 8 m0 4 , n'. : n'Y X = n'( + 1) = n' + n' = n' + n(' + ' ]) = 2n' + n' = 0? n' = n ' ] = n(1 ; ) = 0 . . n' = 0: C L = 0, n 6= 0, n = 0, n' 6= 0. 2 % . ( 3 (90 6= m0 2 M0) m0Y = 0. . , m0 + + 3. C m0 Y = 0 , Y = X Y 2 ], (). 2 4. ) mY X = 0, i 8 > i 0 (mod 3)? < 0 mX i Y = >;mX i;1 i 1 (mod 3)? : mX i;1 i 2 (mod 3): . ) *+ i. i = 1 mXY = m(Y X ; 1) = ;m: , 4 i,
4 mX i+1 Y : ) i 0 (mod 3), i + 1 1 (mod 3) mX i+1 Y = mX i (Y X ; 1) = ;mX i ? ) i 1 (mod 3), i + 1 2 (mod 3) mX i+1 Y = mX i (Y X ; 1) = ;2mX i = mX i ? ) i 2 (mod 3), i + 1 0 (mod 3) mX i+1 Y = mX i (Y X ; 1) = mX i ; mX i = 0: 2
902
. .
5 . ' 1).
# 3 ) 8 0 6= m0 2 M0 , m0 Y = 0. C m0 = ;m0 . , m0 '2 = 0. 8
m0 ', m0 '2: m0 ' = m0 ( ' ] + ') = m0 (' + ') = m0 ( + 1)' = 0? m0 '2 = m0 '2 ] = m0 ( ' ] ') = m0 f'(1 ; ) + (1 ; )'g = = 2m0' ; m0 ' = 2m0 ' + m0 ' = 0: :, , m0 '2 2 L = 0, . . m0 '2 = 0. , m0 ' = m0 ' ] = m0 (1 ; ) = ;m0 : C , 1). #) : () 8 ) m1 = m0 ' m2 = m0 ? ( ) 8 ) n1 = m2 x n2 = m1 x: @ 4, n1x = m2 X 2 = m0 ' = m1 ? n1y = m2 XY = m0 = ;m0 = ;m2 ? m1 y = m0 'Y = m0 X 2 Y = m0 X = n1 : C m0 '2 = 0, n2' = m0 'X' = m0 '2 X = 0: , 4, n2 = 0. 8 ,, n2 = 0: n2 = m1 X = m0 'X = m0 X 3 Y 2 = 0: , m2 y = m0 y = 0, M 1): M0 = km1 + km2 , M1 = kn1, 4 8 m1 m2 n1 + * 00 0 01 00 0 11 R(x) = @0 0 1A ? R() = @0 0 0A : 1 0 0 0 ;1 0 , +/ 8 : m1 = x m2 = ;y n1 = 1:
...
6 . ' 2).
903
G) M0 M M0 ' = M0 M0 3 = 0: , Ker ' = 0. ,
3 / 0 6= m0 2 M0, m0 Y = 0. F
, ', +/ n- M0 : ' (t) = ts + s;1 ts;1 + : : : + 0 . M , M0 4 A- , A | , +/ M0 , 4) ' . , mi = m0 'i . C m0 m1 : : : A- , , , M0 . , , ' , s, M0 m0 m1 : : : ms;1. , dimM0 = deg ' (t): 5. * n ' 3, '(t) = f(t3 ), f(t) |
& . . @ 4 Y X nX ;1 m0 'n = ; i m0 'i i=0 , 4, , n 3: 8 > nX ;1 < 0 i 0 (mod 3)? n i
m0 ' = ; i m0 ' i = >; i 2i 1 (mod 3)? : i 2i 2 (mod 3): i=0 , , Ker ' = 0, n 3 , , 8* , t, 4 3. 2 C , 4 , m0 : : : mn;1 M0 ' : n ; 1 Xl mi ' = mi+1 (i 6 n ; 2) mn;1 ' = ; 3i m3i l = 3 ? (5) i=0 ( mi = ;mi;1 i 1 (mod 3)? (6) 0 i 0 2 (mod 3)? 8 > <;mi i 0 (mod 3)? mi = m0 'i XY = > 0 i 1 (mod 3)? (7) : mi i 2 (mod 3):
904
. .
6. ) k &' + ' ,
M0 = km0 + km1 + km2 M1 = kn0 + kn1 + kn2 ' m0 x = n0 m1 x = n1 m2 x = n2 n0 x = m1 n1x = m2 n2 x = m0 m0 y = 0 m1 y = n0 m2 y = ;n1 n0y = ;m0 n1 y = 0 n2 y = m2 : . 4, , n ) ,= 3, , . F (5){(7) +, + 8 x y 8 mi . F
8 q = m0 + t3m3 + t6 m6 + : : : + t3l m3l n ; 1 l = 3 . : t3 t6 : : : tl ) , 8 q 4
A- , M0 . # (6) (7) q = 0, q = ;q, , , ,, + 8 q'i . F
8 q'3 : q'3 = m3 + t3 m6 + : : : + t3l;3 m3l ; t3l
Xl i=0
3i m3i :
, 4 ' 0
( , , ). , 8 q'3 ; 0 t3l q 4 , q'3 + 0 t3l q = (1 ; 3 t3l + 0 t3t3l )m3 + + (t3 ; 6 t3l + 0 t6 t3l )m6 + : : : + (t3l;3 ; 3l t3l + 0 t23l )m3l : 4 , / , +, , q '3 , , 4) A- , | ) . , , , / =
1 ; 3 t3l + 0 t3 t3l = 0 t3 ; 6 t3l + 0 t6t3l = 0 :: :: :: :: :: :: :: :: :: :: :: :: : t3l;3 ; 3l t3l + 0 t23l = 0: + , t3l;3 t3l;6 : : : t3,
, , , t3l , 1, =
0l , , l+1. k , ,
= k. , , dimM0 = 3, , , 5 , ' m' (t) = t3 ;
...
905
8 M0 + m0 m0' m0 '2 . 4 n0 = m0 x n1 = m1 x n2 = m2 x: , = (5){(7) + , 8 .. B(1 2) M, ) 4 8 n0 n1 n2. 4, n0 n1 n2 . , 0 n0 + 1n1 + 2 n2 = 0, . . 0m0 x + 1 m1 x + 2m2 x = 0, 0m0 X + 1m0 X 3 + 2m0 X 5 = 0: Y , 4: ;0 m0 + 2m0 '2 = 0: m1 m1 m2 + . J ,, 8 x y , . 9 , ,, , M . , N | , M. C N0 A- M0 . , N0 , 5 , 4 , ,= 3, , N0 = M0 . 2 . @ , , , ' ) t3 ; . . & Z3 ) 8 / 12- , .. B(1 2).
7 . ' 3).
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906
. .
+ m0 = m0 , m0 '3 = m0 . ' , / V ( ) ) 8 m0 , m0 = m0 , m0 '3 = m0 , m0 3 = m0 . ,, V ( ) , '. L 4 , + = (2){(4): (m0 ') = m0 ( ' ] + ') = m0 (' + ') = m0 ' + m0 ' = ( + 1)m0 '? (m0 ) = m0 ( ] + ) = m0 (; + ) = ;m0 + m0 = ( ; 1)m0 : , V ( ) , ', , V ( )' V ( + 1 ), V ( ) V ( ; 1 ). ,
/ V ) 8 m0 2 V ( ),
m0 = m0 m0 '3 = m0 m0 3 = m0 m0 ' = m0 (8) 8 . 7. & $ m0 (8). " ' m0 , m0 ', m0 '2 + + & ' M0 M. . @ +
: m0 2 V (), m0 ' 2 V ( + 1), m0 '2 2 2 V ( ; 1), , . 4 ,, W 8 8 A- . , W , ', , , , . : = = ;1 ('2 )('), , W , . 9 + M M0 = W . 2 8. ) M | 4), & ' M0 ' + E0 = (m0 m1 m2), & ' E1 = (m0 x m1x m2 x) + + & ' M1
X Y '+
+ 01 0 01 00 1 01 MatX (E0 E1) = @0 1 0A MatX (E1 E0) = @ 0 0 1A 0 0 1 0 0 0 0 0 +1 1 0 0 1 0 MatY (E0 E1) = @ ; 1 0 0 A MatY (E1 E0) = @ 0 + 1 0 A 0 0 ;1 0 0 a 6= 0 1, b 6= 0. . : / E0 ,) m0 , m1 = m0 ', m2 = m0 '2 . , 4 X E1 .
...
907
, X = X Y X] + 'Y = X + 'Y , m0 , m1 , m2 + , +/ , + 1, ; 1 , ) 4 m0 y, m1 y, m2 y E1: m0 y = ;1 (m0 '3 Y ) = ;1 (m0 '2 )('Y ) = ;1 m2 (X ; X): & m2 = ( ; 1)m2 , , , m0 y = ;1 ( ; 1 ; 1)m2 X = ;1 ( + 1)m2 X: ' , m1 y = m0 'Y = m0 (X ; X) = ( ; 1)m0 x m2 y = m1 'Y = m1 (X ; X) = ( + 1 ; 1)m1 x = m1 x: ) +, X Y + . 2 # 4 . 9. ) X Y + , '+
, M 4). . M ,, A- , N0 , 4) 8 n0 = t0m0 + t1 m1 + t2 m2 ti 2 k
M0 . 4 , ,, + m0 2 N0 . : n0 = t0m0 + ( + 1)t1 m1 + ( ; 1)t2 m2 2 N0 : C n1 := n0 ; n0 = t1 m1 ; t2 m2 2 N0 : , n1 = ( + 1)t1 m1 ; ( ; 1)t2 m2 2 N0 : & *, t2m2 = ( + 1)n1 ; n1 2 N0 : N t2 6= 0, m2 2 N0 , , m0 = ;1 m2 ' 2 N0 . , , t2 = 0. N t1 6= 0, m1 2 N0 , , , m0 2 N0. N t1 = 0, t0 6= 0, , m0 2 N0 . 2 10. , ,
8 (1 1 ) (2 2), + - ' , ' (1 = 2 _ 1 = 2 + 1 _ 1 = 2 ; 1) ^ (1 = 2 ):
908
. .
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, , . = , . 2 . :
, , , , 1 = 2 1 = 2 + 1, / , / m0 e1 . N 4 1 = 2 1 = 2 ; 1, / , / m0 e2 .
9 . - . ' / .
k t] , 4) t9 + t6 ; (t3 + 1), | . , 4 * . # - ,) M0 8 1 + t3 , t2 +t5, t4 +t7 . #) X Y : X 4 ;t, Y = D + ;1 ( + 1)X 5 , D | * , | . J ,, 2) 4).
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Abstract D. V. Juriev, Octonions and binocular mobilevision, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 909{924.
This paper, being addressed to a wide scope of theorists, algebraists and geometers, as well as to applied scientists who specialize in computer graphics, machine vision, mathematical psychology of visual perception and are involved in elaboration of real-time interactive videosystems, is devoted to the interrelation of two objects: the :rst of them is the non-associative algebra of octonions, a classical structure of pure mathematics, the second one is mobilevision, a recently elaborated technique of real-time interactive computer graphics. General aspects of nonclassical computer descriptive geometry and operator (quantum-:eld) methods in the theory of real-time interactive videosystems are also discussed.
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1] Saaty T. L. Speculating on the future of Mathematics // Appl. Math. Lett. | 1988. | Vol. 1. | P. 79{82. 2] Mandelbrot B. Fractals. Form, chance and dimension. | San Francisco: Freeman, 1977. 3] Mandelbrot B. The fractal geometry of nature. | San Francisco: Freeman, 1982. 4] Fractal geometry and computer graphics / Ed. J. L. Encarna~cao. | Springer, 1992. 5] Peiten H. O., Jurgens H., Saupe D. Fractals for the classroom. | Springer, 1992. 6] . . ! " # # ! $ : " -! $& $ '# ( )*$ +,$ // -./. | 1992. | -. 92, &!. 1. | 0. 172{176. 7] Rheingold H. Virtual reality. | New York, Tokyo: Summit, 1991. 8] Virtual reality: an interantional directory of research projects / Ed. J. Thompson. | Westport: Meckler, 1993. 9] Kalawsky R. S. The science of virtual reality and virtual environments. | Addison{Wesley, 1993.
923
10] Virtual reality: applications and explorations / Ed. A. Wexelblat. | Boston: Acad. Publ., 1993. 11] Burdea G., Coi2et Ph. Virtual reality technology. | J. Wiley & Sons, 1994. 12] / ) 4., #$5 6. 75! $8 # # * $ , $ 5 # !5#9 $ '#:5"#9 #55$; # 9 // 65#9 $. <* $. | 1996. | -. 17, &!. 2. | 0. 64{79. 13] Visualization in human-computer interaction / Eds. P. Gorny and J. Tauber. | Springer, 1990. 14] Scienti=c visualization of physical phenomena / Ed. N. M. Patrikalakis. | Springer, 1991. 15] Scienti=c visualization: techniques and applications / Ed. K. W. Brodlie. | Springer, 1992. 16] Focus on scienti=c visualization / Eds. H. Hagen, H. Muller and G. M. Nielson. | Springer, 1993. 17] Visualization in scienti=c computing / Eds. M. Grave, Y. Le Lous and W. T. Hewitt. | Springer, 1994. 18] >&:" 0. ?., . . -# $'@ #:5"# 5 *" *& " , ! " # , ## ! $ // -./. | 1993. | -. 97, &!. 3. | 0. 336{347. 19] . . ! " # # ! $ : " -! $& $ '# * #, +,$ {? $; ! " # &9 G-'#!)*$ #!$ 9 // -./. | 1994. | -. 98, &!. 2. | 0. 220{240. 20] . . )!$"5 ! " # ' ) # # " ! " # # ! $ // -./. | 1994. | -. 101, &!. 3. | 0. 331{348. 21] Juriev D. Algebraic structures of quantum projective =eld theory related to fusion and braiding. Hidden additive weight // J. Math. Phys. | 1994. | Vol. 35. | P. 3368{3379. 22] . # 5 /. 0# 8 #8 @ < #,. 6# I#!&, !! # ! ' )) @5!: #. | ..: Q ;# # 5 8, 1990. 23] Belavin A. A., Polyakov A. M., Zamolodchikov A. B. In=nite conformal symmetry in two-dimensional quantum =eld theory // Nucl. Phys. B. | 1984. | Vol. 241. | P. 333{380. 24] Mack G. Introduction to conformal invariant quantum =eld theory in two and more dimensions // Nonperturbative quantum =eld theory / Eds. G. t'Hooft et al. | New York: Plenum, 1988. 25] Hadjivanov L. K. Existence of primary =elds as a generalization of the Luscher{Mack theorem // J. Math. Phys. | 1993. | Vol. 34. | P. 441{453. 26] . . ?$'@ Vert(C vir\ c) ^# &9 ! ;$ $'@& # 5 // ?$'@ # $#8. | 1991. | -. 3, &!. 3. | 0. 197{205. 27] . . " ( ) # ! $ " " @5" : ) " ))* # ' ) # // _.4. | 1991. | -. 46, &!. 4. | 0. 115{138. 28] Goddard P., Olive D. Kac{Moody and Virasoro algebras in relation to quantum physics // Int. J. Mod. Phys. A. | 1986. | Vol. 1. | P. 303{414. 29] Halpern M. B., Kiritsis E. General Virasoro construction on a`ne g // Mod. Phys. Lett. A. | 1989. | Vol. 4. | P. 1373{1380, 1797.
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30] Morozov A. Yu., Perelomov A. M., Rosly A. A., Shifman M. A., Turbiner A. V. Quasi-exactly-solvable quantal problems: one-dimensional analogue of rational conformal =eld theories // Int. J. Mod. Phys. A. | 1990. | Vol. 5. | P. 803{832. 31] Halpern M. B. Recent developments in the Virasoro master equation. | Berkeley preprint, UCB-PTH-91/43. | 1991. 32] Juriev D. The explicit form of the vertex operator =elds in two-dimensional sl(2 C )-invariant =eld theory // Lett. Math. Phys. | 1991. | Vol. 22. | P. 141{144. 33] . . $ 55#(#" I# ^# &9 ! ;*) , sl(2 C )-# # , ## ! $ // -./. | 1991. | -. 86, &!. 3. | 0. 338{343. 34] } ) $ ;:#" ?. >. - : ;*9: 5 #: S -) #I " &9 5 $# ) ;$# 5# *5-~ ; // 6#5) +-/. | 1977. | -. 25, &!. 10. | 0. 499{502. 35] 0"$ # . ., / ;; . . -)9 #:5"#, ! ;9 ; " # '#*)&) ) ;$ ) " , ## ! $ // ?4 000Q. | 1978. | -. 243, &!. 6. | 0. 1430{1433. 36] - 9 ;< . ?., / ;; . . &, ) ; @ , 8 ; :# # XYZ-) ;$ ~,8 @' // _.4. | 1979. | -. 34, &!. 5. | 0. 13{63. 37] Manin Yu. I. Quantum groups and non-commutative geometry. | Preprint CRM-1561. | Montreal, 1988. 38] Q^ #9# 4. ., - 9 ;< . ?., / ;; . . # '*!! # $'@ # // ?$'@ # $#8. | 1989. | -. 1, &!. 1. | 0. 178{206. 39] . . Watch-dog ((" & >$ "# { $ " $I # " # *! $ )&9 5 9 5 #:5"#9 ;# )#:5"#9 #; 5#5 ) 9 // -./. | 1996. | -. 106, &!. 2. | 0. 333{352. 40] 0$ . ?. &, @
" " " 5#5 ) 5 ! ) // -./. | 1996. | -. 106, &!. 2. | 0. 264{272. 41] Freudenthal H. Oktaven, Ausnahmengruppen und Oktavengeometrie // Geom. Dedicata. | 1985. | Vol. 19. | P. 7{63. 42] Juriev D. Noncommutative geometry, chiral anomaly in the quantum projective sl(2 C )-invariant] =eld theory and jl(2 C )-invariance // J. Math. Phys. | 1992. | Vol. 33. | P. 2819{2822\ (E). | 1993. | Vol. 34. | P. 1615. ' ( ) 1996 !.
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Abstract
G. G. Amosov, An approximation modulo s2 of isometrical operators and cocycle conjugacy of endomorphisms of the CAR algebra, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 925{930.
We investigate the possibility of approximation modulo s2 of isometrical operators in Hilbert space. Further we give the criterion of innerness of quasifree automorphisms of hyper4n4te factors M of type II1 and type III generated by the representations of the algebra of canonical anticommutation relations (CAR). The results are used to describe cocycle conjugacy classes of quasifree shifts on hyper4nite factors of M.
1. V h. ( . 1, . I]) ! h = h0 h1 h0 , #$ V , h1 , #$ V . % V jh1 & | ( , n = dimker V V ( . 1, . I]1], 2, ]). !* +$ ( , sp +$ , -) 1. . * U h +$ V h, U ; V 2 s2 .
, 2001, 7, 5 3, . 925{930. c 2001 !, " #$ %
926
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1. V n > 0 . . 2 , U V , +$ , 3 #, + , ,, , , ( . 3]). % , # +$5 U V *# , W , U = WV , W ; I 2 s2 . 6 , W 0 = UV + + (I ; UU )(I ; V V ) W 0 ; I 2 s2 . 8 , # ker W ker W , + + . ! * , 9, +$ ker W ker W $+$# ker W . , W = W 0 9 , . : , V # + P , # V jh0 , . 6 , 1 ,# +$
. . V n > 0 P. . . (eik )16k6N 6+1 , V 0 h0 , , # . !*
,5 k , jk j 6 1, 1 6 k 6 N , ; k = rk eik , 1 6 k 6 N , , rk , 1 6 k 6 N , , , , , , N X (1 ; rk ) < +1: (1) k=1
(1) h0 ( h = H 2 , ##+$# , , ; < fk () = (1 ; k );1 , 1 6 k 6 N , h, * N Q k; jk j h = Bh ### :#3 B () = 1 1; k=1 k k , h0 : h = h0 h1 . ; < (gk )16k6N , < ; < (fk )16k6N . !* V h ; V jh0 gk = eik gk 1 6 k 6 N V jh1 = S jh1 (2) S ( h. . , S ; (Sf )() = f (), f 2 h, , BS = SB :#3 * B : h ! h1 , S jh1 & S . , S jh1 ### , n = 1.
s2
927
? , V 0 h0 V 0 = V00 V10 , V00 n = 1 V10 | , . % V00 & V , ; (2). 8 , , S V . ? , V ; S = V jh0 ; Ph0 S jh0 ,# #< * ( . 2, . 152]). 1. ? , V , , A& d = i(V ; I )(V + I );1 | # *, . 8 . *# # *, B 2 s2 , # *, d + B , . , U = (d +B+ iI )(d +B ; iI );1 , , * U ; V 2 s2 .
,. 2. 6 +# , # 5
< ,5 3 (AC8) * #5 ( . 4,5,7,8]). A(h) C - AC8 , , h. % +$ A(h) ##+# & , 1, a(f ), a (f ), f 2 h, #+$ AC8: a (f )a(g) + a(g)a (f ) = (g f )1, a(f )a(g) + a(g)a(f ) = 0, f g 2 h. F , a (f ) a(f ), f 2 h, , , , # #
G F (h) , , h , H. 65 # ; <# ! (a (f )a(g)) = (g f ), f g 2 h, | ; 0 < 6 1=2, # # ! A(h). M = (A(h))00. ? J;{. {8 (J.8), +$ #+ ! . % , 0 < < 1=2 M ### ;, W -; III , = 1; , = 1=2 | II1 ( . 5]). 2 ( ,) V # , -& ; (- ; ) B (V ) , M , +$ * +$ ; B (V )( (a(f ))) = (a(V f )), f 2 h. L C - 5
< ,5 3 A(h h). G
0 < 6 1=2 * , 1=2 (1; )1=2 P = 1=2 (1; )1=2 1; h h. A # !P A(h h) ### , !P (j (x)) = = ! (x), x 2 A(h), j , ; C - A(h) A(h 0). %# <# ,# $ ## ! , # !P ,# $, ( . 6]). L , A(h h) H = F (h) F (h). J , , h,
(a(f 0)) = a((1 ; )1=2f ) ; + 1 a( 1=2Jf )
(a(0 f )) = a( 1=2f ) ; ; 1 a((1 ; )1=2Jf )
928
. .
f 2 h. ? ; = ; , ;2 = 1, F (h), + # , # ;a(f ) = ;a(f );, f 2 h, ;H = H. !
# !P 5 : !P (x) = hH H (x)H Hi x 2 A(h h): % , # C - A(h 0) ### J.8, +$ #+ ! . W V , , , h. !* h h , W 0 = W V . :
, , W - ; B (W ) ; ; M # W - ; B (W 0 ) B(H), B (W 0 ) # # ; , +$5 C - , (A(h h)) ; ( (a(f ))) = (a(W 0 f )), f 2 h h, W - B(H) = (A(h h))00 , & , ; . 2. W - ! B (W ) " W - ! B (W 0 ) , W ; V 2 s2 . 2. * , ,5 ; , 6.3 , 7]. , ; , (A(h h)) , N* W 0, W 0 P ] = W 0 P ; PW 0 2 s2 , W 0PW 0 ; P 2 s2 . 3 ### , # J.8
, A(h h) H, +$ , ## !P !W PW , & , ( . 6]). F , *# ; ; . : (A(h h))00 ! (A(h h))00 , (x) = (x), x 2 A(h h). . (A(h h))00 = (A(h h))00 = B(H), + W - ; , , , H *# , W , , #+$ ; . 3. ! 8] , ;; < , A(h), * +$5 ; (a(f )) = a(df ), f 2 dom d, d , # *, , , d 2 s1 . 8 , ; , , , N* ed , d 2 s1 , , *
ed ; I 2 s1 . 8 ,, # , + ; = e , | ;; < . . , , W ,
; , M , , , N* W , ( . 5]). 3. - ! B (W ) ! ! M , 0 < 6 1=2, $ 0
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929
W, , % W ; I 2 s2 . . 2 5] , W ; I 2 s2 5 # , , , ; B (W ), , N* W , , , . , *# 5 * # # + . 3 ( ). C ; B (W ), +, B (W )() = U U , U 2 M . # ,#, +$
M0 ##+# , 1, b(f ) = ; ; (a(0 f )), b (f ) = (a (0 f )); ;, f 2 h. ? , (; ;)2 = I , (; ;) = ; ;, , U ; ;U ; ; , 1 ;1. , U (a(0 f ))U = (a(0 f )) U (a(0 f ))U = = ; (a(0 f )), f 2 h. 8 , ; A(W I ) ;1
A(W (;I )) ;1 , , 2 W ; I 2 s2 W + I 2 s2 . ! ;W ; I 2 s2 . 2 5] , & # , , W - ; B (;W ) , .
, , W - ; B (W ) B (;W ) ##+# . 8 , W - ; B (;I ) = B (W )B (;W ) | , . 5]. U (a(0 f ))U = = (a(0 f )) W ; I 2 s2 . 4. . , -& ; W - , M ,# , +1 T n (M) = C1 ( . 9]). S , n=1 , B (S ) ### ( . 10,11]). 2. . * -& ; , W - , M < # *, , *# , U 2 M , () = U ()U . 8 + 1 1 3 , 4. V n > 0 S, & ! B (V ) ' " B (S ). 2 *, < C0 - 5 < # * ,5 +$ + , 4 ( . 11,12]): 4 . C0- % (Vt )t>0 ! n > 0 C0- % % (St )t>0 , & ! (B (Vt ))t>0 ' " (B (St ))t>0 . 0
930
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1] . ., . ! . | $.: $, 1970. 2] . *. +, ! . | $.: -, 1980. 3] * . . +0 - !1 . | $.: $, 1971. 4] Evans D. Completely positive quasifree maps on the CAR algebra // Commun. Math. Phys. | 1979. | Vol. 70, no. 1. | P. 53{68. 5] Murakami T., Yamagami S. On types of quasifree representations of Cli7ord algebras // Publ. RIMS. | 1995. | Vol. 31, no. 1. | P. 33{44. 6] Powers R. T., Stormer E. Free states of canonical anticommutation relations // Commun. Math. Phys. | 1970. | Vol. 16, no. 1. | P. 1{33. 7] Araki H. Bogoliubov automorphisms and Fock representations of canonical anticommutation relations // Contemp. Math. | 1985. | Vol. 62. | P. 21{141. 8] Araki H. On quasifree states of CAR and Bogoliubov automorphisms // Publ. RIMS. | 1971. | Vol. 6, no. 3. | P. 385{442. 9] Powers R. T. An index theory for semigroups of -endomorphisms of B(H) and II1 factor // Canad. J. Math. | 1988. | Vol. 40, no. 1. | P. 86{114. 10] - :. ;. : ! *- 0 - <-= // >$. | 1996. | ?. 51, 0 . 2. | . 145{146. 11] - :. ;. @0 @ 01 @ 1 // < !. . @ . 1. 1. | $.: $ ?G, 1994. | . 211{220. 12] : . . K! , 0 01 - - // : ! . $ . @., . . $. -. | * , 1997. | . 17{18. 13] : . . K , L ! 01 E0 - -- // ? 0 N!. @. $ ?G. | Q -0, 1997. | . 56. & ' 1998 .
. .
512.546+515.124.32
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Abstract
S. A. Antonian, Extension of the pseudo-compact group actions, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 931{934.
It is proved that for a given pseudo-compact Hausdor1 group G, every continuous action : G X ! X on a metrizable space X has a unique extension to a continuous action ~ : G X ! X , where G is the Stone{Cech extension of G.
, G | . G- . 1] - ! G " ", " G # . . : G X ! X X ~ : G X ! X . & ' . & X Y C (X Y ) * * #" f : X ! Y , + - " ". 1. X - , Y , C (X Y ) . 1
. , ' X = S Xn , # Xn X | n=1 . & # n > 1 * pn : C (X Y ) ! C (Xn Y ) * # #, . # f 2 C (X Y ) # f jXn . , ' # pn , ' , 2001, 7, 2 3, . 931{934. c 2001 , !" #$ %
932
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pn : C (X Y ) ! C (Xn Y ) # . 2 p(f ) = p('), * n=1 S1 # f ' # Xn , ' + X , n=1
. . f = '. 3' K C (X Y ) | # , pjK : K ! p(K ) " p(K ). K 4 2, 5.10]. ' , K # . 2. G X , ! - . " G ( #, ).
. , ' : G X ! X | 4 " . 7 * # ~ : G ! C (X X ), ~ (g)(x) = (g x), g 2 G, x 2 X , 3, . 244]. , ' * ~(G) , ( 1). 4 * # ~ : G ! ~(G) . G 4 ' 2]. 9 , G . 3. | G- X . " d(x y) sup (gx gy) ( g 2G G) X .
. 7 , d | (x y) 6 6 d(x y). 7 + ', - d- '. , # * . 3 " x 2 X ' ' fxng X , (xn x) ! 0, d(xn x) . 3 " > 0 * " ' fyk g fxng * ' d(yk x) > ", k = 1 2 : : :. : "+ * ' ' fgk g G, (1) (gk yk gk x) > 2" k = 1 2 : : ::
S1
; G- Z = G(yk ) G(x), G(a) * k=1 * a 2 X . , ' N | " G Z , . . N = fg 2 G< gz = z 8z 2 Z g. 3 Z 4 " ( # , G) - G=N . , ' Z - , G=N , 2 G=N ' " . , ' g~k = p(gk ), p : G ! G=N | =. , ' G=N | , ' ' fg~k g ' g~ 2 G=N . " G=N Z . H G=N U Z
g~ x , (~hy g~x) < 4" h~ 2 H y 2 U: (2)
933
3 (yk ) x) ! 0, yk 2 U r. , ' g~ | ' fg~k g, g~m 2 H m > r. 3 (2) , (~gm ym g~x) < "=4 (~gm x ~gx) < "=4. ' , (~gm ym g~mx) < "=2, (1), * g~m ym = gm ym , g~m x = gm x. > . & G- X C (G X ) * * #" f : G ! X " . 2 | X , 4 # " (f ') = sup (f (g) '(g)) * " X . g2G , C (G X ) C (G X ) * # # f 2 C (G X ) # F : G ! X . ? C (G X ) " G, " (gf )(x) = f (xg), g x 2 G, f 2 C (G X ). ? ' 4 " C (G X ) = C (G X ). 4. % G- X i(x)(g) = gx, x 2 X , g 2 G i : X ! C (GX ).
. 3 * X . 3 (x y) = (i(x) i(y)) x y 2 X , . . i | . @ ' i ' * i(X ) .
. ; = * #" G X ! G C (G X ) = G C (G X ) ! X | # * # G * # i 4, | 4 * # , . . (g f ) ! f (g) g 2 G, f 2 C (G X ). 7* ~ = 4 " = , # ~ : G X ! X " . " ~ , , G X G X . A 4 ' ~ . 3 . 9 , # G-4 * # f : X ! Y G- * # G-4 . , 4 , G- (G X ) G- G X ~), G- * # f : X ! Y G- * # f : X ! Y , G- MG G- MG . @ " " G- = " " . , ' , * * " G- X " ". @ , * " x 2 X * # f : G ! G(x) * G(x), + f (g) = gx, g 2 G, Gx = f ;1 (x). 9 , Gx | -
934
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1] Comfort W. W., Ross K. Pseudocompactness and uniform continuity in topological groups // Pacic J. Math. | 1966. | Vol. 16, no. 3. | P. 483{496. 2] . . ! "#$ ! % % && '( (% ) // *% ( (. #. | 1984. | +. 39, !)%. 5. | -. 11{50. 3] / 0. 1234 % 4. | 5: 5, 1986. 4] Antonian S. A. Equivariant embedding into G-AR's // Glasnik Mat. | 1987. | Vol. 22. | P. 503{533. & ' 1997 .
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Abstract V. V. Dubrovskii, E. M. Gugina, A new approach to Fourier method in mixed problem for one singular dierential operator, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 935{938.
The article provides an evident example of a new approach to the substantiation of Fourier analysis for a singular di:erential equation in partial derivatives, whose solution is based on the orthogonal polynomial system of Legendre.
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& % . (3) u(x t) 2 C 2 1(Q) ( . 71, c. 10], Q = 7;1 + "8 1 ; "] 708 T ], 0 < " < 1), ./.
. (1) . (2). = % & %
'(x) ( , (3)): '(x) 2 C 27;18 1]: (4) u(x t) (1){(3) / . & %
t 708 T ] / x 2 7;18 1] 1 X u(x t) = T (t)v (x) (5) n
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& % sin nx =1 . B% 71, c. 16]. C , (4) (6), % (3). S ( t) = S1 ( t)+ + S2 ( t)+ v0 ( t) | (5) ( v (x) 72]). " ./ | ., %./. && %: 1 r 2n +1 1 X (; n ) O(1) S2 ( t) = X e(; n )v (cos ) S1 ( t) = e 3 2n n2 n(sin )3 2 =1 =1 n r 1 2n + 1 1 X (; n ) cos((n + 2 ) ; 4 ) t 6= 0 v0 ( t) = e 2n n2 (sin )1 2 =1 n
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. B - & % . (7), x = cos , ./ : p 1r 2 X 2n + 1 ( ) = 2 cos 2 + sin 2 =1 2n cos n + 1r X 2n + 1 + cos 2 ; sin 2 sin n : 2n =1 %% (4) (6), ./ 8 q1 q2 : 1 r 2n + 1 1 r 2n + 1 X X 2n cos n = q1 ( ) =1 2n sin n = q2 ( ): =1 % & % , v0 ( 0): cos( 2 ; 4 )v1 ( ) + cos( 2 + 4 )v2 ( ) v0 ( 0) = (sin )1 2 ( - v0 ( 0) 2 C 2 1(Q)), 1 r 2n + 1 Z Z X v1 ( ) = 2n n2 ; d q1 ( ) d =1
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: (i) ( - N S , . S=N = T 0, T 0 | T (' ) + " +"/ (ii) RT | , R0 N | .
Abstract A. V. Zhuchin, Local contracted semigroup rings, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 939{944.
The local contracted semigroup rings R0 S over non-radical rings R (R = R=J (R) 6= f0g) are under consideration. The following main statement is proved. Let R be a ring, R 6= f0g, S be a semigroup with zero. The ring R0 S is local if and only if: (i) there exists a nil ideal N S such that S=N = T 0 is a semigroup T (without zero) with adjoint zero/ (ii) RT is local, R0 N is radical.
R ( , ), - J (R) ( , ). !2] $ % 0 ( ' ' % ) % p > 0, !11] ) ) . * + $ R0S ) , ' , *$ , !11]. - R | , S | / z . 0 R0S - RS Rz . 1 , ' R0S !RS = x 2 RS j P ri = 0 x = P ri si RS . 3 ' , ' R0 S
R- , *+ % ) S , z / R0S .
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6 ) | $* + . 1.1. R | , R 6= 0, S | . R0S , : 1) - N S , S=N = T 0 , T 0 | T ( !) " 2) RT | , R0N | . 1.1 , $* + . 1.2 ( 10]). R | , S | . # a 2 S \ J (R0S ), a | $ . 1.1. - R0 S | . 7 ' , ' R = R1 S = S 1 , R S $ , R0S | R10 S 1 . ) e 2 R0S , + ' e | R0S=J (R0 S ), N = fa 2 S j ea 2 J (R0S )g. - , ' N | S , R0N | . + , ea 2 J (R0S ), eae ; ae 2 J (R0 S ) , , ae 2 J (R0 S ), ) e(ab) e(ba) 2 J (R0S ) $ ) b 2 S .
, a 2 N ra 2 R0N , r 2 R, (ra)e = r(ae) = (r 1)(ae) 2 J (R0 S 1 ) \ R0S = J (R0 S ), (ra)e ; ra 2 J (R0 S ), ) ra 2 J (R0 S ) R0 N | . - T | N S . 8 s t 2 T , es et 2= 2= J (R0S ) (es)(et) 2= J (R0S ), ) e(st) 2= J (R0S ) st 2 T . 9 R0S=R0N = R0S=N = RT | , R0N J (R0 S ), R | + RT . 0 1.2 N | - , R0 N + R0N . 6 , RT | , R0N | , R0 S | , RT = R0S=R0 N R0 S . 1.3. R | , R 6= 0. # ! R ! S | ! , R0S , : 1) N S , S=N = T 0 , T 0 | T ( !) " 2) RT | . . 8 R0S , R | (. 1.1), ) RS = Rz R0S | , , S ( % ) ' !1,3].
2.
941
) $* , $* % % . 2.1. R | , R 6= 0, S | . RS , : 1) R | " 2) B S , ! !% ! , S=B = N | " 3) RB | , R0N | . 2.1 % +. 2.2 ( 1, 8]). R | , R 6= 0, S | . # RS | , S | ! . 2.3 ( 8]). R | , S | . # RS | , S | . . 0 $ 2.2 S ' , ) $ ) s 2 S +/ ' n, ' sn | S (!9, 1.9]). 9 + % , sn = e, e | RS . 6$ , ' ) sn + S , , S | . 2.4. R | , R 6= 0, S | . # RS | , R | , S | ! eSe | ! % e 2 E (S ). . 0 $ 2.2 S ' , R | + RS . - e 2 E (S ), , ' , RSe =J (RSe ) = RS=J (RS ), Se = eSe, J (RSe ) = e J (RS ) e, RSe e RS e RS . 9 RS | e , RSe |
+, eSe | , 2.3. #$ . ; % % !4] , ' RS | , eSe = feg, char R = 0, eSe | p-, char R = p > 0. - S + 2.4 $ % +. < , ' e f 2 E (S ) $ , ef = fe = e f . 2.5. S | ! . & % !: 1) ! % e 2 E (S ) eSe | " 2) S "
942
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3) S ' % - . . = 2) $ 3). + , 2) 3), !5]. 6 , 3), , ' , $ + S
+ , , ' + , , ' ' S . - , ' 1) $ 2). - 1) ef = fe = e, e f 2 E (S ), fef = e fef 2 fSf . 9 fSf | + f , fef = f , ) e = f . 6 , 2) e 2 E (S ). 8 a 2 eSe, * ' k, ' ak | (!9, 1.9]), eak = ak e = ak , ) ak = e eSe | . 2.1. - RS | , 2.4 2.5 R | * B + 2) 2.1.
, RB , R | * ) RB ! R. 9 RB=J (RB ) = RS=J (RS ), J (RB ) = J (RS ) \ RB RS=J (RS ) | . ? R0N = RS=RB , RS = = RB + J (RS ). 6 , 1){3) 2.1, RS=J (RS ) = RB=J (RB ) Ann(e), Ann(e) | e RB RS . 9 Ann(e) = RS=(RB + J (RS )) | , , Ann(e) = 0 RS | . #$ . ? ' $ 2.4, B + +, char R = 0, + + p-+, char R = p > 0. 2.6. R | , R 6= 0, S = M (G I C P ) | ! . RS , RG | . . 8 e 2 E (S ), eSe = G, ) RG |
(. 2.4). D ) RG ! RG J (R)G RG, '/ J (R)G J (R)S J (RS ), RS | . - , ' RG \ J (RS ) J (RG). + , x 2 RG \ J (RS ), x 2 R1G \ J (R1 S ) = eR1 S )e = = J (R1G), , x 2 RG \ J (R1G) = J (RG). ; , J (R)G J (RG), ) RG RG. 6 , RG | . 8 S , S = f0g RS = RG | , ) ' , ' S . 6' , RS = R0 S 0 = M (RG I C P ) | RG ) '- + P (!9, 5.17]). 9 M (J (RG) I C P ) J (M (RG) I C P )) (., , !12, 2]), M (RG I C P )
943
, M (RG I C P ). ; % PRG !4] , ' RG P=R R = D | , '/ J (RG) = x 2 RG j ri 2 J (R) x = ri gi , ) M (RG I C P ) = M (D I C P ) = DH , H = M (f1g I C P ) | ( ) P D). 6 , ' (!DH )3 = 0, DH (e ; f ) DH = 0 $ % e f 2 H . #$ . % . , 2.1, 2.3, 2.4, 2.5 $ ,
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' , R0S ' S ( 1.3). 3.1 ( 11]). R | , R 6= 0, S = M (G I C P ) | ! . # ! R ! S | ! , RS | , : 1) R | " 2) G = feg, char R = 0, G | ! p- , char R = p > 0. . 0 2.6 RS RG.
, % 3.1 RG 1) 2) !4]. 3.2. R | , R 6= 0, S | . # ! R ! S | ! , R0S , : 1) R | " 2) ! N B S S , N S=B | , B=N = T 0 | ! " 3) T | ! ! ! , char R = 0, T | ! p- , char R = p > 0. 3.2 ' 1.1, 2.1, 3.1, ' , ' - ' + . #$ . !11] ' , ' S
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1] Okninski J. Artinian semigroup rings // Comm. Algebra. | 1982. | Vol. 10. | P. 109{114. 2] Okninski J. Semilocal semigroup rings // Glasgow Math. J. | 1984. | Vol. 25. | P. 37{44. 3] Okninski J. Semigroup algebras. | New York: Marcel Dekker, 1991. 4] Renault G. Sur les anneaux de groupes // C. R. Acad. Sci. | 1971. | Vol. 273, no. 2. | P. 84{87. 5] Schwarz S., Krajnakova D. O totalne nekomutativnych pologrupach // Mat.-Fyz. Casopic Slovensk. Akad. Vied. | 1959. | T. 11, no. 2. | S. 92{100. 6] . . ! "#$%&$""'! #% (& "# 0-"& '! "#$%&$"" // ) . * +,. | 1975. | -. 18, . 2. | /. 203{212. 7] . . / &0 ' "#$%&$""' +#12 // /,(. . . | 1977. | -. 18, . 2. | /. 296{303. 8] 3$4, 5. 6. 7#$#+#1' "#$%&$""' +#12. | ". 68-. | 1981. | . 1874-81. 9] 9#,::& 5., 7& ;. 5#% (&,4 +< &,< "#$%&$"". -. 1. | ).: ),&, 1972. 10] 9 $! . =. > "' "#$%&$""' +#12 , ,! ((? ,<. | ". 68-. | 1979. | . 1862-79. 11] <,+ 5. @. B+#1' "#$%&$""' +#12 // E$ . , "&,+#. . | 1995. | -. 1, . 4. | /. 1115{1118. 12] <,+ 5. @. & ,+#1'! "#$%&$""'! +#12! // ) . * +,. | 1985. | -. 37, . 3. | /. 452{459. ' ( ) 1996 .
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Abstract O. M. Lar'kina, On the identities of the product of normal subgroups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 945{949.
The article presents the proof of the statement: the product of two normal subgroups generates a Cross-variety if and only if the intersection of Cross-varieties generated by these normal subgroups is nilpotent.
, N1 , N2 ( , N1 N2 ), . ! "# # M1 M2 M = M1 M2, X = fG = N1 N2 N1 G N2 G N1 2 M1 N2 2 M2g: % X0 | X, # - .
1. varX = var X0 . % V | varX. ( w 2= V # A 2 X, # w . ) # * a1 : : : an, w(a1 : : : an) 6= 1. A 2 X, , A = N1 N2 j = 1 : : : n aj = n1j n2j , n1j 2 N1 , n2j 2 N2 . ) B, # nkj (k = 1 2, j = 1 : : : n), w. ) # M1 n1j M2 n2j . -: .
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. 4 , * e, 5 6 m 5 6 c e, m, c (71, 5]). :# * e, m, c M1 M2. ! " # . ; , M1 M2 M1M2, 6 # # (72, . 337]). > exp(M1) = e1 , exp(M2 ) = e2 , exp(M1 M2) 6 e1 e2 .
2. M1 M2 c1 c2 M = M1 M2 c = c1 + c2 . 2 1 M X0 . ) G X0 , p- P 6 c. G = N1 N2 , Nk G, Nk 2 Mk , k = 1 2. % P1 = P \ N1, P2 = P \ N2, P3 = = P \ N1 \ N2 = P1 \ P2. ( P1, P2 P3 p- N1 , N2 N1 \ N2 (72, . 345]) P. > jGj = mp , jN1j = m1 p , jN2j = m2 p , jN1 \ N2 j = m3 p , m1 , m2 , m3 p, jGj = jN1=N1 \N2 j jN2=N1 \N2 j jN1 \N2 j =(m1=m3 )p; (m2 =m3)p ; m3 p =mp : 2 , = + ; . ) jP1P2j = jP1=P3j jP2=P3j jP3j = p; p ; p = p: ( , P1P2 # p- # G. ) # # s, t 6 s + t (73, . 151]). % * P1P2 ( , P) c = c1 + c2. : # 5 # X0
6 # , 6 c. ( * c 5 var X0 (71, . 195]). @ var X0 5 H=K 6 d 6 d,
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H=K * h1 : : : hd+1 , 7h1 : : : hd+1] 6= 1. 2 , var X0 5 6 c. 2 ( 5 " #. 3 An * n. 4 Ap Aq , p q | , (74]).
3. M1 = M2 = Ap Aq (p q ) M1 M2 . ) Gi = ha1 b1 : : : ai bi j aqj = bqj = 1 Gi 2 7ak aj ] = 7bk bj ] = 7ak bj ] = 1 k 6= j 7a1 b1] = : : : = 7ai bi]i: A Gi Vi Zp ( Gi), dimVi ! 1, jGij ! 1, . B Hi = Vi h Gi, Vi i. C , fHig . : Hi 2 M1 M2, Gi q: Ai = ha1 : : : ai ci Bi = hb1 : : : bi ci, c = 7a1 b1] = : : : = 7ai bi]. 2
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. ) # # G H, # . D H=Z(H) 6 * # # p- " # q (75]). @ Ap Aq (76]), * var(G) var(H=Z(H)) = Ap Aq . 2 . # M1 M2 * : 1) M1 \ M2 E 2) Ap Aq 6 M1 \ M2. ( , # . F 6 , " vi , : v2 = 7x1 x2 (x;1 1x2)y1 2 ] vn = 7vn;1 xyn (x;1 1xn)y1 : : : (x;n;1 1xn )y ;1 ]: n
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5. M1 M2 M1 \ M2 M1 M2 vr vs M1 M2 vm m = m(r s) . % , M1 M2 vi 6 i. 2 1 Gi 2 X0 , vi 6 Gi. B # Gi Ki , # vi (Gi). @ # 5 # # : Ki = Li1 Li2 Lim(i) Lis = Li : > Li , 4 M1 \ M2 , . 2 , Li = Zp p. C , Gi, 6 # # Ki Si = Gi=Ki , G~ i = Ki h Si . # , Gi = G^ i , Ki = K^ i . ) Gi G^ i DG Gi G^ i , DK = DG \ (Ki K^ i ). % G~ i = (Ki K^ i )DG =DK : (Dk Ki K^ i Ki .) ( G~ i # : G~ i = Ki h Si , Si = DG =DK , G~ i 2 var Gi , G~ i (i) ~ Nk = Ki h Sk(i) 2 Mk , k = 1 2, Sk(i) = Nk(i) =Ki , G~ i 2 X0 . % * , Gi # G~ i . , , Si # Ki , " Ci Ki Gi Ki , Ci \ Si = Di , Di Gi # 5 Gi =Di = Ki h Si =Di 2 X0 , (i) * vi Gi =Di. % S1 S2(i) , , # Ki . > p q S1(i) S2(i) # * q, M1 M2 Ap Aq , Ap Aq # # # # (76]), M1 \ M2 . C , :3 (exp(S1(i) ) exp(S2(i) )) = pe(i), e(i) > 0. ( Ni = N1(i) \ N2(i) | p- . K # Ni Gi Ki . () # # p- " .) C , Ki | " Ni Ki Gi. ( Ni = Ki , Si # Ki . t = maxfr sg vt Gi=Ki , * vt Gi Ki . #
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N1(i) N2(i) Ki . ) vt+1 #: N1(i), | N2(i) . 2 , m = m(r s) vm Gi . % Gi . 2 ; 6 . 2 . 4 M1 Mn ( # ) , Mi Mj i, j.
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. | .: , 1969. . . . . | .: ! , 1967. . $. % &, '. $. %! &. () &* + . | .: ! , 1982. L. G. Kovacs, M. F. Newman. Just-non-Cross varieties // Proc. Internat. Conf. Theory of groups. Austral. Nat. Univ. Canberra, 1965. | Gordon and Breach, 1967. 5] (. '. 1 2+. *, &) 2* ! + *3 )4 %5 *. + +! . 3. $ * +2*. | .: 1959. 6] G. Higman. Some remarks on varieties of groups // Quart. J. Math. Oxford. | 1959. | Vol. 10, no. 2. | P. 165{178. ' ( 1996 .
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