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. . , . . -
512.5
: , , -
.
! ! Rn, n > 1, " #. $
, %
&! %
! %"
' " %
'
%.
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 LM 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] 7819 '., % '. 4 1: 3 $9* %1 /% &%5; 2.33 1 99011%5;
*2 #. | (.: ./, 1969.
8] - 9 7. *2 #: 7*+0, =$.*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 , < (C1 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+
: : : + snm )(x1 + x2 + : : : + xn)
(2.3)
. . 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 .
,
A ( (8$ /:
x0 + y0 )(yn + xn ) = (x0 + y0 )yn + (x0 + y0 )xn = x0 yn + y + x + y0 xn
(2.5)
0 0
0
0
0
0
(x + y )(yn + xn ) = x (yn + xn ) + y (yn + xn ) = x yn + x + y + y xn :
< ( C x0 yn = x1 : : :xn;1 yn y 0 xn =
= y1 : : :yn;1 xn . > (2.5) ,
3 x0 yn , , 3 y 0 xn , ( x + y = y + x.
(
B .
3. ! !
3.1.
)
S
>(
i
| ( .
i-
,
i /8. G, S |
j -( 8 ( j .
>(% ($
0
(
1
5 2.6 1.1 ,
{
655
#
3.2. " n!-! S xn = 0, S n(2n ;1) = 0.
$%. &, (% ($ ) (-
n!-( , i-( 1 i
n.
>( X | 3 ', / X = m. & X (8 ((( ( ), /0(8 X , X | 0 / %
(. . ( )? X (n) | ((( X , (8 , 1 C n (
, X = X (1) ). A Z-( ( %
) R = Z X R(n) = Z X (n) . I | R. : , ( Z | % (, '-% R=R(n) / (
% 1 j
h
i
h
i
j
h
h
i
i
h
i
h
i
h
h
( ( % ).
A (% $ S n
i
i
S
/
Z-
, S n
/ (((
xn
= 0. > | %. &
S ((8 ((( (% S .
E'( a1 : : : am
S . A / ' : X
S , oe '(3
'(xi1 xi2 : : :xik ) = ai1 ai2 : : :aik :
(3.1)
&, ' | ' ((, 0
'( X (n) )
S n . ,
(
n
)
((
X
8 3
3 R(n) Z,
'
hX i(n) / (
) ' %
(
: R n) S n '(3
2
h
i !
h
h
!
R
X
i
iyi
X
=
i
i
i
i '(yi )
i Z yi
2
X (n) :
2 h
(3.2)
i
3.3. %" R prn q, p q
X ,
N X R, $ p0 sn q0, p0 s q0 S . (&2
h
i
r
p0 q0 , p = J q = J, J |
X .)
. mA (3 p = J = q. >( p = xi(1p) : : :xi(kp) ,
P
q = xi(1q) : : :xi(lq) , r = j r(j ) , j N, r(j ) X , r(j ) = xi(1j) : : :xi(mj)j .
2
h
i 2
h
i
6
j =1
= a (p) : : :a (p) , q 0
i1
ik
2
6
2 h
i
Pm
p0 = '(p)
= '(q ) = a (q ) : : : a (q ) , s =
j s(j ) , i1
il
j =1
s(j ) = '(r(j ) ) = ai(1j) : : :ai(mj)j , ( 0 . >/
(3.1), (3.2)
R (prnq) = R
X
J =fj1 :::jn g(Nm )n
j1 : : :jn pr
(j1 )
: : :r
(jn )
q
. . 656
n = 1 2 : : : n . :/ (
1, , /
R(n) , ( / N
f
X
J (Nm )n
g
j1 : : :jn R (pr(j1 ) : : :r(jn ) q) =
=
X
J(Nm )n
j1 : : :jn p0s(j1 ) : : :s(jn ) q0 = p0 sq0
.
2 (
p = J ( q = J) 8 ( ( (8$1 ).
3.4.
dR(n) (dS (n)) C d, n!-( R ((% $ n!-( S ) / xn = 0 Rd = 0
(S d = 0). D % % ($( 1.1, (& % % | 8 3.2.
K
d(Rl) (n) (d(Sl) (n))
( % l--
/0
1 % ($ ) 1 (3 (. I %
($(8 ( !4].
R In (R) In+ (R), /0
r % Z X = R (% N X
R + (R)
. >( / R(n) , In
In (R) R(n) ,
+ (R) /8 ( Z-( R(n) | . >, In (R) In
n
) pr q , p q
X , r Z X In(R) (
r N X In+ (R)).
3.5. In+ (R) = In(R).
. >/, r Z X rn
+ (R), ( ( (.
/ In
A %
rn,
h
i
h
i 2 h
2
h
i
2
h
i
i
2
A %(
01
BB1
A=B
BB1.
@ ..
0
1
2
.
.
.
0
12
22
.
.
.
:::
:::
:::
..
.
h
i
1
1n C
C
2n C
C
. C
. A
.
0
n n2 : : : nn
/ h = det A = n!(n
1)! : : : 1!. I a
r = a b, a b N X . >/ r+ = a + (h
r = r+ hr; , r+ r; N X . 1
;
;
2
;
h
2
rn = (r+ hr; )n =
;
n
X
i=0
i
;
h
(;1)
b r;
1) ,
=
b,
(
i
i hi fi (r+ r;) =
=
f0 (r+ r; ) +
n
X
i=1
i hi;1
!(;1)
hfi (r+ r;)]
(3.3)
{
657
fi (x y) | (x + y)n i y, . .
( 1 x, y n
x i y. >/
0f (r i 1
0 + r;)
BBf1(r+ r;)CC
F =B
BBf2(r+. r;)CCC :
;
@
A
.
.
fn (r+ r;)
r ir2 )n In+ (R) 0 6 i 6 n, . . AF
In R). 5, 83 %3 %
A ($( % % B , BA = (det A)E ,
E | % (., , !5,
XIII.4]).
+ (R). E /, % BAF = (det A)F = hF / In
f0 (r+ r;) = r+n In+ (R). (3.3) 0 rn + , ( rn
33 % In
In+ (R), E, ( 1 +
+(
%
/ 2
x
2
2
.
3.6. dR (n) = dS (n).
. A
(% $ n!-(S , / xn = 0, '( a1 a2 : : : adR (n)
S . D/, a1 a2 : : :adR (n) (8, ( ( ($ dS (n) (/ .
>/ X = dR (n), R = Z X . A % R /
2
j
j
h
i
Jn (R) = r R k = k(r) N : (n!)k r In (R) =
= r
R k = k(r) N : (n!)k r In+ (R)
f
2
j 9
2
2
f
2
g
j 9
2
2
g
3.5. / ,
Jn (R) | , 0 Jn (R) R(n) , R=R(n) | % (, In (R)
R(n). >( R0 = R=Jn (R). % R0 n
/ x = 0, ( 8 n- R / In (R) ,
, Jn (R). : , R0 | (% n!-(. 2 , n!r
Jn (R), (n!)k(r)+1r In (R) , , r Jn (R).
dR (n) (, (R=Jn (R))d
R (n) = 0, (
x1x2 : : :xdR (n) Jn(R). Jn (R) R(n), R (Jn(R)) = 0
( (/ . >( S , S n ,
+ (R)) = 0. n!-(, , R (In
n
3.3 R (pr q ) = 0 p q
X , r N X , ( ( (.
2
2
2
2
2 h
i
2
h
i
.
#
3.7. " n!-! S xn = 0, S 2n ;1 = 0.
.
1.1 3.6.
M/ . . 658
N
( /0
1 (% $ .
3.8. d(Rl) (n) = d(Sl) (n).
. A (% $ S /
xn = 0, /0 a1 : : : al . >/ X = l, R = Z X . O,
j
j
h
i
R0 = R=In (R) / xn = 0, R0 (l)
/ X .
8 3.4 (R0 )dR (n) = 0,
(l)
8 dR (n) X / In (R). < 3.3 3.5 ' % R : R(n)
S n , +
(3.1), (3.2), ( R (In (R)) = R (In (R)) = 0, ( 8 (l)
(l)
dR (n) /8$1 S ( dR (n) X ) (8, ( ( (/ .
#
3.9. l-' S xn = 0, ( 2ln+1n3 .
'
!
. 2 !4] , 0 % l
-/0
1 (% (
/.
(8 C (.
N / 2.
. @( ( .
"
1] Nagata M. On the nilpotency of nilalgebras // J. Math. Soc. Japan. | 1952. |
Vol. 4. | P. 296{301.
2] Higman G. On a conjecture of Nagata // Proc. Cam. Phil. Soc. | 1956. | Vol. 52. |
P. 1{4.
3] . ., . ., !. "., #$ . !. %, &'
% (). | .: +,, 1978.
4] / . 0. 1#) +2 ({32) 45 6,% // 7,4). 6#. ) . |
1995. | 1. 1, (6. 2. | . 523{527.
5] 82 . 2&#. | .: #, 1968.
' ( ) 2000 .
. . . . . 511.36
: , ,
! , " .
# " ! $ %!
$ $ $ ! " $ " d=b, d b | " , d
' ( ").
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).
. . . ! "1{3] . . '
"4] 1=b, b | . , "3] d=b, d b | , d
- ..
1. / I | , I |
I0 j = j +ij , j = 1 : : : s, | I, ;1 ;2 : : :0 a 2 I, a 6= 0, | , aj 2 I, j = 1 : : : s0
Z
Z
Z
, 2001, 7, 2 3, . 659{671.
c 2001 ,
!"
#$ %
660
. . b d 2 I n f0g, jdj 6= 1,
Z
(z) =
1
X
=0
z
m
a !"1 + 1 ] : : :"s + 1 ] m = s + 1 " + 1 ] = ( + 1) : : :( + ) " + 1 0] = 1
j = fj g, j = j + ij , j = 1 : : : s, fg | . /
1 : : : s , 1 > 1 > 2 > : : : > s > 0:
9
:j = j ;s 1 + j ; 1 + : s: : + s j = 1 : : : s0
: = 1min
::
6j 6s j
/ K(
j ) | j g(x) = (x ; 1) : : :(x ; s),
rl = max
K(
j ) r = min
r
= l
j =l
l
; j, 1 6 j 6 s, j = l , | l, :l = :.
<, p, = 1 : : : N, , , = (d=b) e > 0, p | , , p, N(a) | a I.
. b d 2 I n f0g, jdj 6= 1, p a, p e 6 m, N(p) = p ,
= 1 : : : N s
X
R = hk (k) db hk 2 I 0max
jh j = H > 3
(1)
6k6s k
k=0
>(x) = x;s(ln x);s(1;)(ln ln x)s(r;) :
(2)
Cj = Cj (a b 1 : : : s d p1 : : : pN p1 : : : pN e1 : : : eN ) j = 1 2
:
1) (1) H > 3
jRj > C1>(H):
(3)
2) (1), jRj < C2>(H):
Z
-
Z
2. ' "3, x 3], h00 ( db ) + : : : + hs s ( db ), 661
0 (z) = (z) 1(z) = az dzd (z)
d
d
(4)
j
;
1
j (z) = a
z dz + 1 : : : z dz + j ;1 1 (z) j = 2 : : : s:
? = , , = 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:
F0l = 1 l = 1 : : : s
i=1
(6)
< A; .
1. x y 2 I, p | I , p | " , p, p (y), N(p) = p, t 2 t > p. t
Q
(x ; yi) p pt ] , "] % " .
Z
-
N
i=1
. ? p (y), x;yi, i = 1 : : : p, =
-
B p. / B N(p) = p, I B p p (. 1 "5, . 242]).
C, , i0 , x ; yi0 0 (mod p),
1 6 i0 6 p. 9B , x ; y(i0 + pk), k = 0 : : : " pt ] ; 1,
Z
662
. . 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):
9B (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]
:
s
Y
LtJ (z) = ft (a) Fjll (z) =
N
N
N
= K(a) =
s jY
l ;1
Y
l=1
j
;1
s
l
YY
t
Y
l=1 i=1
k=1
a( + l ; i) l=1 i=1
s jY
l ;2
Y
=z
=z
a( + l ; i) l=1 i=0
s jY
l ;2
Y
l=1 i=0
t
Y
(a ; ak)(z(z)) =
k=1
tY
;1
a( + l ; i) (a ; ak)(z) =
(a ; ak)(z) =
k=0
a( + l ; i) ft;1(a)(z) = zLt;1J1 (z):
(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:
9B
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
-
N
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
;i (
;
1)
Atprjr = gtp(a(jr ; r + i)) a ! i = 1 : : : p
i=0
X
(12)
(. "9, . 53{57]). C A (11), , Aptprjr = 1.
(6) (11) , 664
. . Fjii (z) = ft;p (a) gtp(a(jr ; r )) +
i=1
Y
p
;
1
s
X
Y
+ Atprjr a
( + r ; jr ; ) Fjii (z) =
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
Atprjr Lt;pJr (z) =
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
9B
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 ,
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t ; jr = n + 1, jr = min(j1 : : : js).
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0
0
0
0
N
666
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max(j1 : : : js) ; min
j min j 6 jr + 6 max
j0
>
>
i2J i i2J i
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j
max
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j:
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j 6 jr +
i2J i
i2J i
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j ; jr ; 6 D(J) ; 6 D0 , D(Jr ) 6 D0 .
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9B 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,
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t+j +: : :+j 1
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0
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E 4 . 9
t
Y
t
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(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
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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:
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/
J = (j1 : : : js ) = 1 : : : s
(30)
(
juv = 1 u 6 v ; 1
(31)
0 u > v ; 1:
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(32)
9B (8), (28) i (z),
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+
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, , 1, , a,
b, d, p1 : : : pN , p1 : : : pN , e1 : : : eN , d e (t + l ; 1) p Btli b ; m
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m
m
m
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668
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tm+l ]
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(34), (35) p , B
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i , l : : : vl;1 , vl A
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6i6s tli
tm+l ] d X
s
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i=0
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s
M Rtl , l;1 6= l ( 0 = s + 1).
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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 t > 1 l = 1 : : : s
(37)
Rtl >l (Htl ) >(Htl )
(38)
" >(H) (2).
D f(t) g(t) , f(t) g(t) f(t) g(t).
Z
669
. / =, , 7 "3], , Btli d t!jajttl (lnt)l ;1 i = 0 : : : s
(39)
b
Btli d t!jajttl (ln t)l ;1 :
max
(40)
06i6s
b
9B (36) , = .
"3] (. A (66)) , t
Ltl d tl;s;1;s(t!);sjaj;st d :
(41)
b
b
'
(36) (41) = , (37). CA (38) = , =
8 "3]. / B = 6, A (37) , l = rl . E
. A , q > q0 .
7. t l " )" htli Rtl Rtl+1 : : : Rts Rt;11 : : : Rt;1l .
= 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 .
9B Rtl , Z
670
. . 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
+
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= t:
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i=1
i=0 m
i=l m
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9
(c) , Rtl cRtl. ? 5 (c) | I (c) 6= 0, 1 6 j(c)j = jcjj j 6 jcj j0j. C, =
= j10 j . . / =, , x 5 "3] =
, =, (1) (3).
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"3] | 7 8, =B, "3] | (36), j , j = 1 : : : s, B, "3], B j ( db ). ? , = .
= = 5,
>(H) = 1min
> (H). ? . 6l6s l
= dm D
;
;
= B . . ! , .
1] . . . // . . | 1970. | %. 8, ( 1. | ). 19{28.
2] . . . ,
// . . |
1976. | %. 20, ( 1. | ). 35{45.
3] . . . 1 2
// . 3
. | 1984. | %. 124 (166), ( 3 (7). | ). 416{430.
4] . 6. 7
3
2. // 8 ,. ). , . | 1981. | ( 6. | ). 36{40.
5] 9. . :
2, . ;. <2. %
= . | .: 6, 1985.
6] . :. <?
2. %? . | .: 6, 1987.
671
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
. F3H= J{F? 3
3K // . 3
. | 1994. | %. 185, ( 10. | ). 39{72.
9] 8. L. 2. %
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& ' 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 | (
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(f )(x1 : : : xn) = f (x2 x3 : : : xn x1)
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(.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.
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, 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
1) 9 = f>f 2(x1 : : : xn)+1] (x ; y) x ; 1 ;xyg, f (x1 : : : xn)
! Z"
2) Mf , f (x1 : : : xn) Z.
. @ #
.
1. f (x1 : : : xn) Z. C(# 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 $" (
4).
# f>f 2 (x1 : : : xn) + 1] (x ; y) x ; 1 x + yg $" (
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#
$
# $ $##.
682
. . 1] . . . | .: , 1976.
2] .
. !"# $""%& $""'"# %(&', )&' -
)"(&' "+,,)%' // .'. $. '%'. | 1996. | /. 2,
#&$. 2. | 0. 365{374.
3] %#5 6. . % $"' 7(%. | .: .8'%%, 1993.
& ' 1996 .
, { . . , 512.66
: , , !" , !
! #, %& , ', ! ({*
.
+, ' ,
,, "! +- !! "! !" . ( , !
% ! . -
, . !, / " 0 !
"! '! % , 1"
{2{". # ,& ! '. - ;
+-, ! (ti tj = qij1 tj ti ) ! !" G, ! % !% % "!% . + ' (AF ) A ! !. "! F
G, ! " A F . +
" ! , ' ! +-!. ! ,
'. !
&% '.
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.
, 2001, 7, @ 3, . 683{698.
c 2001 ! "#,
$! %& '
684
. . 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.
1 {
G. # $ $$
% .
&
, ( ) *
+ 2.
, +- $$
% , -,
$$
% + +. % ) .12], 2
3 .6]. ,
4. 5. .5] -
$, % , ( -
+ + . #. 5$ .11] ( % -
. 7
, 8. 9. #$
% .4] (
+ .
8 -
.2] -
(7), <{8% (7<8), G , , <. 7, $ -
% | :
ti tj = qij;1tj ti:
8 %
(. .3]), %+- $$
% , $
, , $
<{8%, % $
, % (+, .
5 % -
) .
H I J. J
L9].
N0 / , , J. J
L8{10]. /
!
"! +, , 0 '! , L1].
1
2
685
(A F ) A k((z)) F k((z)), ( % G?(k). , A ( % F: AF F . 8 -
A C, C ; 1 = Spec A, F F . - +
G , %, (
, A , F (
C. + (
(-
F) A $$
%
.
8 , (
, q-
) $$
% ( )$$
) ( $% . A 7<8-, , % () (A0 F0) B(
C (At Ft) At = A0 ,
At (
D^K $$
% )$$
. 4, $$
% . # | $$
% DK . & -
% (
.
< -
% +, (C F ) + )+: ( -%+
-
) ( -%+ 7). & + + + C. ) F (
% $ , .
, . 5 $$
%
(#&). E+ $: -
#&, #& . . |
., , .7]1. 7( %, (
+ %, | + . 5 %
B
-C
DK . ,
.3], +-
7-
. <
+-
% -
$ . & % DK , ( -G
) .
1 (
, " 0 ! " "%, !
"!, / & %, . 1.1 . 1.3,
%& ! ! ",. * /
Q"R ! , "
/ ,
, , ! . * / , ! 3.4 " !
"! , !
! , 0 ! ".
686
. . DK . % *
, 7-
+ . 8 (
-
% H. <. <, +-
B
%C .
1. 1.1. (, k %. A k-
K $% x t1 t2 : : : +- :
ti tj = qij;1 tj ti xti = qi;1ti x
qij qi 2 k % $. 5 %, qi % k . & T K, (G
+ ti . ,
, K=(t1 t2 : : :) = k..x]] =: K. &
k-
$ S K, S(x) = x S(ti ) = qi ti:
# )
X
f = f0 x0 t 2 K
0 = (1 2 : : :) % %
, (- )
, t = t1 1 t2 2 : : :, f = xf + tf
X
X
xf =
f0 0x0 tf =
f0 x0 t :
0
06=0
2
% ) Bx-%+ fC Bt-%+ fC .
. Sf = f f 2 K, tf = 0.
. ,
, S(xf) = xf X
S(tf) =
f0 q x0 t :
0 6=0
H
% (
% qi . 2
687
1.2. 8
G
k-
$$
@ K, @x = 1 @ti = 0
@(a:b) = @a:b + S(a):@b
a b 2 K. &
, @ S = S @. H, .3], EK . KG
)
+ $%
D;1 )$$
K:
X
X = ai D i i
(
:
1 X
n S n;k (@ k f)Dn;k n 2 Z:
n
D f=
k=0 k
&
, (
VK = fW 2 EK j W = 1 + w1D;1 + : : :g
(
+.
# #& X X+ X; $$
% 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 ( )$$
uj $$
%
w1 : : : wi;1. 4 ) G
wi t-% wi+1.
8
(
+-
-
.
. 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 ):
G
% ad t. 8
t- x-, , , @ad = 0, . . ad | ,
tad;1 = 0. #
, X ad Dd (
, +-
Dl ), + . 2
# % , W1 DW1;1 = W2DW2;1
G
.W2;1W1 D] = 0. <
%, W1 = W2 C C 2 Vk . 2
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, )$$
b0 +
d = X + O(d ; k ; 1). L %. 2
. 4, Yk+1
1.5. &$ . #$
Z (L) L = Dd + ad;1 Dd;1 + : : : EK
;
1=d
k((L )).
. % X | )
. 8%G
W, L = WDd W ;1. &+ Y = W ;1XW ( Dd . N
Z (Dd ) : Z (Dd ) = k((D;1 )).
% D = L1=d , % . 2
1.6. ()) *
& DK $$
% EK .
8 ) DK % . 4
, %, B ,! DK ( 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
. . % Y ( Z (L) , . 1.5, % L;1=d )$$
. <
%, ( Y 2 B
W ;1Y W ( Ek A = W ;1 BW % Ek .
&
, A 1.
( %, + W1 W2 2 VK A1 A2 EK , B = W1 A1W1;1 = W2 A2 W2;1:
% A1 A2 + 1, $
k((D;1 )) = Ek . ) (
is : As ! DK is (X) = Ws XWs;1 (+ (
is : Ek ! EK . 8 ,
W1DW1;1 = W2 DW2;1 %, W1 = W2 C C 2 Vk , A1 = A2.
8
) +- (. .9]).
. %! B ,! DK, 1
(L), % A Ek
W 2 VK , ! Vk , WAW ;1 = B: 2
2. !
" #
2.1. + ,
7G
(
< < .13]. % R % % -
. I = (: : : i;3 i;2 i;1) +- %% , i;n = ;n % n. 7(
? (: : : ;3 ;2 ;1). M | ) (mij ) )
R i 2 Z j 6 ;1. # M I MI (mij )i2I j 6;1. %
Frame(R) % (
)
R,
+- +- :
(1) m;i;i = 1 % i,
(2) mij = 0, i < j i ,
(3) MI | I.
#
(
GL(R) N =
= (nij )ij 6;1 )
R, + (1) (2).
L (
(
+, +-+ Frame(R)
(
: M 7! MN.
691
+ R- % )
G(R) = Frame(R)=GL(R):
# R = k ) (. .14]). & GI (R) (
)
M(modGL(R)) G(R),
MI .
2.2. -./ ,
8
G
R-% V = R((z)) $% R, z %
% +- )
R . # ( = M(mod GL(R)) R-% U V , (G
)
X
ui = mji z j :
j 2Z
2 . &
, GL(R) (
R- U , %, G
2 G(R).
4, ( % G(R) BC . 4
, (
,
R-% U V % G(R), .
7
%, R = k ) G(k) )
B$
%C (., , .10]).
2.3. , L % )
VK 8-
, , (
: X 7! (X~ij ) ER (
)
R X
Di X = X~ij Dj
G?(T), T | K, (G
ti .
j 2Z
% G
ER (. .3, . 2.7]).
8-, (
W 2 VK (W) 2 G(T) $
; (W) = W~ ;1x=0 i2Z
j 6;1 (mod GL(T )):
. &% : VK ! G?(T ) % .
. .3, . 5.3]. 2
5 (
% (
BC. 4
, %
V ; = z ;1R.z ;1] V % % z ;1 . H
; (W ) = W~ ;1x=0 V ; :
692
. . 3. #%
3.1. *
% F % G(R). &
#& G(R) aF = a~F
F | , +- F .
. (A F ), - A #& )$$
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 = ci ti ti j = cqij;1j ti i < j
tij = (c ; 1)i tj + qij;1 j ti i > j:
,
% c 2 k | $, 1. & (
d: U ! U, -
ti i i 0, G
(
!) $$
U.
A K- EK U(EK ) = U T EK +- % :
xi = qi;1 ix Di = qii D (, , S(i ) = qii ).
R
%, $$
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)# ),
,,)
<
< % (7<8)-
W0 7! W(t) (
G?(k) ! G? (T). ( ,
- , (
% (<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<n ).
%
X
Dt;1 = D;1 + dnDn n>0
dn;1 = (pn ; S ;1 pn) + (
p<n ) = (1 ; qn;1)tn + (
t<n).
8 ,
;1
q
;
q
1
;1
;1
;1
;1
2
;1
1
Dt = D + (1 ; q1 )t1 + (1 ; q2 )t2 +
.2]c + q1 ; 1 t1 D + : : ::
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+33 2 = 2 + q3;1 w13 ; 3S 3 (w1)
1 = 3 (S 3 (u2 ) ; 3S 2 (@w1 )) ; q3;1w1 3S 2 (w1) + q3;2 w23 ;
; 2 S 2 (w1) + q2;1 w12 + 1
0 = 3 (S 3 (u3 ) ; 3S(@w1 ) + 3S 2 (u2 )) ;
; q3;2 w23 S(w1 ) + q3;1 w13 S 2 (u2) ; 2q3;1w1 3S(@w1 ) +
+ 2 (S 2 (u2) ; 2S(@w1 )) ; q2;1w1 2S(w1 ) + q1;1w1 1 +
+ q2;2 w22 + 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
fj = 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. , ., , !" #1]. & , '
! . (
", ! . & , ! ) !
, " . ()
""
! , ., , #2], #3] ) #4],
, 2001, & 7, . 3, . 699{712.
c 2001 !,
" #$ %
700
. . !
' . /
) ) '
, ., , #5]. (
1
{/ )
#6]. & ) #7], #8] #9].
( (B ) | )
) , 90 |
B (B ) | ) !
, B , , '
)
. 1'
B - " ""
""
B .
: (B )- " . : A '
(A) (A) '
A.
&
'
! (; P ). < ", 1= 1. >
| .
kk
L
kk
L
F
2. (
'
) ! ! ! ) #7]. ( D (B ) | .
1 (7]). x(n + 1) = Dx(n) + "(n) n Z
(1)
B- x(n): n
Z E x(0) <
"(n): n Z E "(0) <
< + , (D ) z C : z = 1 = ? :
B 1 #7] '
, 2 L
2
f
k
k
j1
f
2
g
2
g
k
k
1
\f
2
E kx(0)k 6
+1
X
j =0
j j
g
Dj E "(0) k
k
k
k
(2)
', (D) ' !
z C: z 61 .
& ! ! ""
! ) ""
. ( f
2
j j
g
701
> 0 "
A C (R (B ))= A(t + ) = A(t) t R
. / "
A " U : R (B ),
( 0
U (t) = A(t)U (t) t R
U (0) = I:
C
I | . / " U , , U (t) !
) '! t R.
( | B - = (t): t R , sup E (t) < + .
2
L
2
!L
2
2
P
f
2
g
06t6
2. x0(t) = A(t)x(t) + (t) t R
k
k
1
(3)
x(t): t R B sup E x(t) < + 06t6
, (U ( )) z C : z = 1 = ?:
: ! , ! 2 (3) ' ). ( ! 2
f
2
g
k
k
\f
1
2 P
2
j j
g
2 P
Z
"(n) := U ( )U ;1(s)
(n + s) ds n Z
2
0
x(n ): n Z (1) D = U ( ) 1. D! x(t): t R (3)
' f
2
g
f
2
g
Zs
x(n + s) = U (s)x(n ) + U (s)U ;1 (t)
(n + t) dt n Z s #0 ]: (4)
2
0
2
(
!
#8] '
! ! ! !)
. ( H | )
!)
D | (H ).
3 (3, 8]). (1) x(n): n Z "(n): n Z H-
L
f
f
2
g
2
g
702
. . , ! ej : j > 1 H
sup
1
X
j >1 m=0
f
k
g
Dm ej 2 < + :
k
1
B 3 '
, E kx(0)k2 = tr
X
+1
j =0
Dj S" Dj
(5)
!
S" | "(0), A | , '
A.
3. ( Q := R #0 ] Q( ) := #0 ] #0 ].
1. B - " u = u(t x): (t x) Q
f
2
g
t , u " ! .
E ., ,
#3,4,8]. 1'
)
F
G F
G.
('
G10 := g : #0 ] C g(0) = g() = 0 C 1(#0 ])=
G30 := g : #0 ] C g(k)(0) = g(k)() = 0 k = 0 1 2 C 3 (#0 ]):
: " g C01, A C (R (B )) ! ! :
(0
ut (t x) u00xx(t x) = A(t)u(t x) + (t)g(x) (t x) Q
(6)
u(t 0) = u(t ) = 90 t R:
2. B - Q " u ! (6), " u, u0t u00xx 1 Q (6).
('
h
n
i
o
E sup (t) < + :
1 := f
!
j
f
!
j
g\
g\
2
2
L
2 P
;
2
2
P
2 P
06t6
k
k
1
E ! .
4. " A C (R (B )) | $. &
.
2
L
703
(i) ' $ g C01 1 (6) t
h
i
u, E sup u < + , 2
k
k
2 P
1
Q( )
2
k +i j k 2 N
fe
#0 2] (U ( )):
(7)
3
(ii) ' (7) , $ g C0 1 (6) h
i
t , E sup u < + .
2
g 2
2 P
k k
Q( )
1
. B (ii) (i). ( (7) , " g C03 1 . : '! k > 1 ! 2
u0k (t) = (A(t) k2I )vk (t) + (t) t R k N
(8)
vk (t): t R
B , sup E vk (t) < + . C
, vk , k > 1,
06t6
#3], )! n > 2 (v1 v2 : : : vn) . & )
)
vk , k > 1. (
k0 | , '
(U ( )e;k02 ) ' ! z C : z 6 1 .
: '! k N " Uk ( 0
Uk (t) = (A(t) k2I )Uk (t) t R
Uk (0) = I:
2
2 P
;
2
2
f
k
k
k
2
g
1
k
f
2
j j
g
2
;
2
( Uk (t) = U (t)e;k2 t , t R. : '! k > k0 2
Z
"k (n) := Uk ( )Uk (s);1 (n + s) ds n Z
2
0
:
L
E k"k (0)k 6 2 1 2 k ; k0
(9)
!
L1 k. B (4), (9) (7) , '! k > k0
sup E vk (t) 6 k2 L2 k2 06t6
0
!
L2 k. H
!, ) 0 6 a 6 b 6 , n Z
k > k0 '
k
k
;
2
704
E sup kvk (t)k
a6t6b
. . 6
Zt
;
1
6 E sup Uk (t)vk (n ) + E sup Uk (t)Uk (s) (n + s) ds 6
a6t6b
a6t6b
0
Zt
6 L3 E vk (n ) + E sup Uk (t)Uk (s);1 ds sup (s) 6
k
k
k
k
k
a6t6b
k
0
06s6
k
k
6 k2 L4 k2 (10)
0
!
L3 , L4 k.
(
" sin kx x #0 ]: k > 1 C01, ,
1
X
g(x) = gk sin kx x #0 ]= gk : k > 1 C ;
f
2
2
k=1
g
f
g . ( 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, ! '
, , ., , #3].
I! 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
;
1. D ), u ! (6).
705
:'
. ( w | (6) Z
2
wk (t) := w(t x) sin kx dx t R k > 1:
2
0
D! " wk , k > 1, (8). ( ""
J
wk , k > 1, ) w (7) (8) 2 , u = w.
B (i) (ii). ( k N R u ! (6)
" g(x) = sin kx, x #0 ], . : k N '
2
2
2
2 P
Z
2
vk (t) := 2 u(t x) sin kx dx t R:
2
0
( , vk 1, . #3] .
B (6) , vk , k > 1, (8). <
(8) . :
, ) ) (8) , ! ) (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
\f
2
j j
g
\
4. " C
! 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
(ii) ( ! ej : j > 1 H 1
X
sup e;2m U ( )m ej 2 < + :
g
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)
'
"k (n): n Z vk (n ): n Z . E
2
f
2
g
f
2
g
Z
vk (n + s) := Uk (s)vk (n ) + gk Uk (s)Uk;1 (r) dw(r + n ) s #0 ] n Z
2
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 $. | *.: +%, 1980.
6] A. V. Fursikov. Time-periodic statistical solution of the Navier{Stokes equations //
Turbulence Modeling and Vortex Dynamics (Proceedings of a Workshop held at
Istanbul, Turkey, 2{6 September, 1996). Lecture Notes in Physics. Vol. 491. |
Springer, 1997. | P. 123{147.
7] A. Ya. Dorogovtsev. Stationary and periodic solutions of stochastic dierence and
dierential equations in Banach space // New Trends in Probability and Statistics.
Vol. 1. Proceedings of the Bakuriani Colloquium in Honor of Yu. V. Prohorov / Eds.
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712
. . 8] A. Ya. Dorogovtsev. Necesary and suJcient conditions for existence of stationary and
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Math. Appl. | 1990. | Vol. 19, no. 1. | P. 31{37.
9] .. /. 0 1!. 2 7) K"Q!#$ !"#$
% , '%(#$ "%# & ! // . . 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
t201]
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 )(Znt]+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<nk+1 t2Jn
nk6n<nk+1
1
1
1
jY (t) ; Yn(9p )j > 4n;k+18 lnnk +
6 P nk 6max
sup
n<nk+1 $ <t6$ +n; 125 n
p
p
;8
+ P n 6max
sup
j
Y
(t)
;
Y
(9
)
j
>
4n
n
n
p
k+1 lnnk 6
n<n
k
k+1
$p ;n; 125 6t<$p
6 P nk 6max
max jY (9 + lqk ) ; Yn (9p )j > nk;+18 lnnk +
n<nk+1 16l6mk n p
; 18
+ P n 6max
max
j
Y
(9
;
lq
)
;
Y
(9
)
j
>
n
ln
n
:
n
p
k
n
p
k
k+1
k n<nk+1 16l6 m
1
(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<nk+1 16l6mk
nk 6n<nk+1
j =1 i=1
X
l
n X
3
+ P n 6max
max ij > nk8+1 lnnk :
k n<nk+1 16l6mk
j =1 i=1
: 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<n
k
k+1
P
4
n P
l
ij k j =1 i=1
max 16l6m
+
3
(ln nk )4 nk2+1
nk 6n<nk+1
P
4
nP
65
l
k+1 m
Pk
n P
E
max
max
u
ij
nk 6n<nk+1 16l6mk j =1 i=1 ij
i=1 j =1
+
6
C
1
3
3
(ln nk )4nk2+1
(ln nk )4 nk2+1
6 C2 ln;4 nk = C2 k;4
6
!
C1, C2 k. : , 1
1 X
X
An 6 C2 k;4 < +
P
k=1
k=1 nk 6n<nk+1
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<nk+1
i=1
nk 6n<nk+1
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<nk+1
k=1
< +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
; Fnt]+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 )(Znt]+1 ; 9p ) ;
2p ln ln n 06t61
; p%nt] + 1(p ; Fnt]+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: zaika@krc.karelia.ru
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 ))). # " " yj0T ] $ yj0T ] ( ). # " 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 ) k(t )=0 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
kj kPi = sup jj 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
# "
yj0T ] $ yjt1 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) kj 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 , Lv = 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)
2i
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) =
2i
-" 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
2i 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)
2i 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)
2i 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)
2i
2i
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)
%"!" 32 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$ 32$ !"! !-!. 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 !" ! $ #, ! 02$ !$" 3" -!" !! `:
1
2i
Z;r
;i
gx(ue;i )eue
;1
d(ue;i )
; 21i
Z;r
i
gx (uei )eue d(uei ) +
;1
Z
Z1
Z
1
1
!
;
u
+ 2i 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 --,02$ # ," -!":
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 -.
L2. ! (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
. . L2. !. ! (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(!) ""# . : !"$
--,02$ # "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 { ):
C !"!" !
Q() = e;i ln 2= W. C!. # 6!
. q(!) " q(!) = ; ln(!!++ii)+ . H, - "# C (!), !!# ! ""# ;1 6 y 6 0 `.
1 3 " .
, Q():
Qy () = ; Ei(i) ; ln ] e;i ( 2 C ())
Qx () = Ei(i)e;i ( 2 C () n { ):
1! sin ln 2 W "3 ! ""2 (1.6) #
W "" ," -!":
sin ln = 1 P () ; Q()] =
2i
Ei(
i
;i
i
;i
= ; ;i)e 2;iEi(i)e + sin ln + Ei(;i)e 2;iEi(i)e :
G ( !
) 2!3, ., !2$ -$, | W, .. !- ! . -# ""
, !.
,
(1.1) (, ""
" #" !3 ! 0# 2E ""#).
H, ., !. 6!
sin ln , ! .. "" # p(!) q(!). @ , "# C (!), !!# ! ""#
;1 6 y 6 1 `.
4. !
& G() 2 W g' (!) | ., !. # 6!
. 6 " , !! `' , -
,# g'(!), !2" " frei('+) : r > 0 ; 6 ' 6 ' 6= 2 g.
F-" T' = `' ; !!, 2# !! `' ;, S' |
!2# !, $2
,# T' , !$ "2# 3" ! - !32# T' . : " " (1.2)
772
. . " ,
!":
1 Z g (!)e! d! (Re ei' > 0):
G() =
2i '
(4.1)
S'
@.. ! (4.1) ! !!. S' !! T' -. !
1 fg (uei(';) ) ; g (uei('+) )g supp g (;uei' ) 2 ` (4.2)
g'(;uei' ) =
'
'
'
2i '
g' (!) !! `' , 32$ !-!# " ! (1.6) G(), !" Gy () ! .. # 3 !" (1.7), "" Gx () | !"
Gx () =
Z
1ei'
;
i'
g'( uei' )e;(ue ) d(uei' ):
0
(4.3)
J !" "3 ! :
Gx (ei' ) = ei'
Z1
;
i'
g' ( uei' )e;u(e ) du
0
(Re(ei' ) > 0):
(4.4)
: , !". 2!3 (4.4) = te;i' , "
Gx (t) = e
i'
Z1
;
g' ( uei' )e;ut du:
0
(4.5)
M !"2 (4.4) ! ' = 0 ' = . !"2
(1.8) (2.3).
@ " 2 ! !!. S' !" (4.1) !3
- ! -" 2# ! ! C1 . 1
. g' (!) - C (! ) n T' , ! !
Z
Z
1
1
!
!
G() =
(4.6)
2i g' (!)e d! = 2i g' (!)e d!:
& !" E
S'
1
g' (!) = ;
2i
:". ! "
; p ;1 ! = e
i'
Z
C1
Z1
0
1
g'(p)
dp:
p !
;
i'
e(p;!)te dt:
(4.7)
(4.8)
773
W & .. (4.8) (4.7), "
1
g' (!) =
2i
Z
Z1
g' (p) e
i'
C1
i'
e(p;!)te dt
0
=e
i'
dp =
Z1
i'
e;!te
1 Z
0
2i
i'
g' (p)epte dp dt:
C1
G"., -$ " ! ! ! (4.6) G(tei' ), " !" -!,. ! (4.1):
g' (!) = e
i'
Z1
G(tei' )e;te
i' !
(Re(!ei' ) > 0)
dt
(4.9)
0
Z
1ei'
g'(!) =
G(tei' )e;te
i' !
d(tei' ):
(4.10)
0
&" !", 2!3
,
g' (!) ! 0 !! T' , !" (1.16). 1 . g' (!) - C (!) n T' , (4.7) Z g'(p)
1
g ' (! ) = ;
(4.11)
2i p ; ! dp:
S'
@.. ! S' !! T' !$ . " (1.4) (4.2) g' (!) !! T' , " " ! g'
Z1 gy (y)
(!) = ;
;1
iy
;!
dy +
Z
1ei'
0
;
g'( uei' )
d(uei')
uei' + !
(! 2 C (!) n T' ):
(4.12)
&" ! !2 --,. ! &!.. &
2 L2(;1 1), H () 2 L2(;1 1) | !-! ! # . F-"
h (y )
h'(ue
i'
)=
Z
1e;i'
H (te;i' )e;ue
i' (te;i' )
d(te;i' )
(4.13)
0
h' (uei' ) = e;i'
Z1
0
H (te;i' )e;ut dt:
(4.14)
774
. . M ! # (4.13) (4.14) 2 . !"2 (2.1).
6 " , Gx() | "" , !.
,.
(1.1). :E ! " (4.13) h' (uei' ) !"# (4.4) . Gx(te;i' ), ! !# !"2 !. !!. " ! &!. Z
1ei'
;
g'( ue
i'
)h' (ue ) d(ue ) =
i'
i'
0
Z
1e;i'
G(te;i' )H (te;i' ) d(te;i'):
(4.15)
0
I . ! " (4.14) . h(ue;i') !"# (4.5) . Gx(t), ! &!. !-! i'
e
Z1
;
g'( ue
0
i'
)h' (ue ) du = e
i'
;i'
Z1
G(t)H (te;i' ) dt:
(4.16)
0
M 2 ! ' = 0 ' = . ! &!. (2.2) (2.4) ( $, h(u) H ( ) ,2).
C " -2 2 ! --,..
1!. !-!. 6!. W 2$ # !
3
! ! ! 2$ 2$ !E# !2$ $ !2$ ! ! ("., !"!, 7]).
"
1] A. Puger. U ber eine Interpretation gewisser Konvergenz- und Fortsetzungseigenschaften Dirichlet'scher Reichen // Commentarii Mathem. Helv. | 1935/36. | B. 8,
89. | S. 89{129.
2] . . . ! "#$ %!&' ()"%$. | *.: ,-.., 1956.
3] /. -. 0'1. "% # 2# #"3%. | *.: /)", 1965.
4] *. 0. 25, . 6. 782. *2# & 2# ()"%$ "#3!"#9# 3#9#. | *.: ,-:*, 1958.
5] . -. ;)9, 6. <. "#!. :2& ()"% (1" 2'". | *.:
/)", 1971.
6] ,. $23, 0. > $. 6&? 2% 2& ()"%. .. 2. :)"% !@, ()"% 8#!E"#9# %! , #2#9#!5& 3#9#E!&. | *.:
/)", 1974.
7] *. G. J#!"#, L. 6. 78. <#!)#!# # $23 # #2#E#$
!&. .#E# ? // G#"! & 0/. | 1997. | .. 356, 6. | L. 763{765.
& ' (
1998 .
,
. . 512.533
: , !" ! .
#!"! ! ! , ! ! " . $"
%&
'
! , ' ( ) ! * !" , *
+" ( i i ), 2 , . , ! ! !"!
= , = ,
= .
P
S
T
S
a b
a b
S
i
I
ax
ax
b ax
bx
by
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
T
a b
S
P
S
S
a b
i
I
ax
b
ax
bx
ax
by
P | - . , P S , T S (. 1, . 4, . 4.6]). $ , %
& '
( ). *. +. , ' 2], xay = c, xby = d a 6= b & '
(. * ' , / % % , /0/ ) ) 1&. 2
(
& ) 1&. 3
, sa = sb sc 6= sd,
1&, %4
(a b),
% (c d). 5 ' , , /0)
) ) 1&, : 1 ( '
) 1
&
.
, 2001, 7, 4 3, . 775{782.
c 2001 ,
!" #$ %
776
. . 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). $ '/
/ (
j= j= (
%
5, 6, 7).
; ' ' S 1 , / S &. 8 %
X T(X) | ' : X ! X. ' '
%
0
x() = (x). ? S | 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 (
0
%
.
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
S
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S
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777
(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 ). /
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i
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. %
'.
@ . <
/ , '
%
2, 1
(3 ), (4 ) 0
. A
, %
' 0, a x = b (i 2 I)
(11)
. . % ( '
/. 5 (
, %
I '
'
.
3. S :
(i) (11) (ii) (11) T(S 1 )
(iii) 8i j 2 I 8s t 2 S 1 (sa = ta ) sb = tb ).
. *
(i) (ii) ' %
1. 8%
&/ (i) ) (iii). sa = ta . E sa x = ta x,
. . sb = sb . 3 ' &/ (iii) ) (ii). <
%
': S 1 ! S 1 /0 ': 1
) sa , s 2 S 1 , % (sa )' = sb , 1
' %
' '. 5
' ' (iii).
, ' | (
(11). s 2 S 1 . E s(a ') = sb .
$ ' 1
s a ' = b .
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780
. . 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).
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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= ?.
.
(s s2 ) j= (a b). E ) x1 : : : x 2 S 1
/ 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
,
(s s2 ) j= (s2 s3 ) j= (s2 x s3x) = (sa s2 a). 2
, (s s2 ) j= (a s2a).
% %
', (s s2 ) j= (a s a). 9 '
(s s2 ) j= (b s b). E s a = s b, (s s2 ) = (a b).
7. a b s S (s s2 ) j= (a b) -, :
(i) 8x y 2 S 1 (xs = ys ) (xa = ya&xb = yb))
(ii) hsia \ hsib 6= ?.
.
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:::
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j
(s s2 ) j= (a b). E %
5
j= (a b) %
6 '
, hsia \ hsib 6= ? a b 2 sT(S 1 ), ', xs = ys ) xa = ya xs = ys ) xb = yb.
. (i) (ii). <
1
' 2 T(S 1 ), (xs)' = xa, xs
= xb x 2 S 1 ' 1
u' u
u 2 S 1 n S 1 s '
'
. F
(i) %
' psi. E x 2 S 1 | '
1
, T(S 1 ) / s' = a, s
= b, . .
a b 2 sT(S 1 ). 2
, %
6 (s s2 ) j= (a b) T(S 1 ),
', (s s2 ) j= (a b).
.
(s s2 )
T(S 1 ).
782
. . 1] . . // . . 2. | .: , 1991. |
". 11{191.
2] $ %. &. '( $)* $+,$* //
",- +$+. (. | 1980. | . 21, 0 1. | ". 168{180.
3] Clark C. E., Carruth J. H. Generalized Green's theories // Semigroup Forum. |
1980. | Vol. 20. | P. 95{127.
& ' 1997 .
:
. . . . . 519.865.5+519.8:33
: , ,
, !"# .
$ # %& '() *( % #, ) *( . -./
) ( *( , . ' . . -. *) *) ) | * *, . *( & ( &.) . '.* . * | !"# . 1'. ' ) *( . -./ ) )*(*( '. )
( '.. 2. *( !"# &" . .
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.
1. , . , (.
, 2001, 7, : 3, . 783{796.
c 2001 !,
"#
$% &
784
. . $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 - ) .
, 01 - 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 . D0, 4
xr 2 ldm . 1 - y, 5S, 2d . = , r- bd2
2+lm .
- 1 0
.
E1 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 , JL 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 + .
E1 , -, , - - | 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] 78 . ., 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
"$!. . !$ ) !+ "
%, %,! "
f0 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]))
L- 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:
L- 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 !
<Fn j n 2 Z g |
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 )(XY ) = f(Fn)(XY ) 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 <F;n;1 . "&
$ ! $ !#$ %( (X Y ) )''#! <- F :
H n (X Y 8 F ) = ;n((F )(XY ) ) = (X Y )8 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
. . !
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. . 5 678 // .
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.
& 1997 .
. . 517
: n- , , .
"# $
% & & & n- .
Abstract
Yu. G. Nikonorov, On the asymptotic of the mean value points for some nite dierence 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.
1,2] . . ! " ! "# "# " " ! $" %#" ! ! $" &"'. (
" ! " ! !") "" -" $$"*". ) ! ' +
. | 0 1] , - (0 1), 2 (0 1 )
2 (0 ), n
X
(;1)n;k nk ( ) = n (n) ( )
n
f
n
x
=n
nx
C f kx
k=0
x f
:
. " "*# ! &"' , "+# !) 1 = , 2 = 2 ,.. ., n = ' !,
! $* . %
n
X
( ) k( )
()=
L
x
x
x
x
x
nx
f
L t
k=1
f kx l
t , 2001, 7, 1 3, . 829{838.
c 2001 !"#,
$% &' (
830
. . ( ; 1) ( ; k;1)( ; k+1) ( ; n )
( k ; 1) ( k ; k;1)( k ; k+1) ( k ; n)
2 +# ! 2 0 1], !# "*, " "* $" &"' ! !
(n)
( ) = ( ) + !( ) ( ; 1)( ; 2 ) ( ; n)
' " ( ), ! " ] "' i .
5 = 0, 2 (0 ), "# ! $"
n
(n) ( )
X
(;1)n;1 !
(0) =
(
)
+
(;1)n ! n
(
;
1)
1(
;
1)
(
;
)
!
k=1
k (t) =
t
l
x
x
x
t
::: t
::: x
x
t
x
x
x
::: t
x
::: x
f
L t
t
n
x
t
x
::: t
a b
:
x
a b
x
t
nx
n
f
x
f t
t
x
k k
:::
::: k
f
f kx
n
n x :
n
6 , ! nk = k!(nn;! k)! , ") "'+ "'.
(, ! "'
n
P
(;1)n;k nk ( )
C
C f kx
k=0
n
x
' """ ! !" "" -" $$"*" $* .
7! !" ( ) ")++ " ! $"" "' " (0 ). 6 " # ! ! !# $* " " ! 0
$* .
8 "+ !" " !
(1)
( ) = lim ( )
n
f
x
nx
f
n
T
f
x!0
x
x
:
2 ', ! ! (1) + "+ *
) $*#, +) "+ " ! 0.
7, ! $* , " $* + , |
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519.48
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Abstract
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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5
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A | Mc- CA | A, $B B . # T 1 Mc- A1 A2 CA1 = CA2 . Q
CA (A 2 Mc) C. 5
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% C.
# F(A) ( ) $
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846
. . $
$
t1t2 (x y), F(A) 2 Mc A 2 M. =
, , B $ h: A1 ! A2 M- $
F(h): F (A1) ! F(A2), a 2 A1 F (h)(a) = h(a), a 2 C
F(h)(a) = a F(h)(e) = e, . & .
O
, M $ $
. 5
A | ,
$% !
C, M = IS(A)e | , $B A. Q,
M ( M $
). 4
, $ M.
5
F | , %
$
!
, G | % :
F . & $ IS(A)e GF IS(A)e A = GF(A) ) GF(C). & F (A) F (C) | %
. Q, $
. 5 ,
% .
5 %
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. 5
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F F (A) F (B) $
( ) , F (A) j= f 2 (c) = c & f(c) 6= c,
F (B) j= f 3 (c) = c & f(c) 6= c & f 2 (c) 6= c.
4 " ( ), , .
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] . -. .. 67 0
89 // 9 00.
1. | 1997. | $. 38, ( 1. | . 161{165.
6] . -. .. 4 12 :
// . | 1998. | $. 37, ( 4.
7] J. Plonca. On a method of construction of abstract algebras // Fund. Math. | 1967. |
Vol. 61, no. 2. | P. 183{189.
( ) 1998 .
-
1
3
. . 512.554.5
: , , -
.
!"# "$ % -
% % & !
%3. ( ) % n n :
( n;2 ) , n;1 n ]
= 4 + 2 4 + 31
n := ,, 1 2 ] 3 ] ( 4 )
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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
f
g
x x
x x
x
R x
R x
g
:::R x
:::R x
fg
x
x
x
g
x
n
n
k
k
k
k
:
, 2001, 7, ; 3, . 849{871.
c 2001 !,
"#
$% &
850
. . It is proved that for any in:nitie system of nonzero weak identity there is number 0 ,
since which each of identities of the given system of a degree
0 is equivalent
to one of canonical identities n or n .
As consequence the variety of alternative algebras with unit over a :eld of characteristic 3 which has not :nal bases of identities is speci:ed.
It is proved also, that the class of weak identities of a rather high degree coinside
with the class of mufang functions.
n
n > n
f
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fn := x1 x2] x3]R(x4) : : :R(xn;2) xn;1 xn]:
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%
, & (10) xn;1 = xn = a, n;1x2 x3] x4]R(x5) : : :R(xn;2)R(a) (x1 a) = 0
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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) /% )
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(13)
@
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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, ))
%
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(15) % 2
x3 x4] x5]R(x6) : : :R(xn) a2
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%. ., %& 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)
2x1 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) % ;
;
;
;
;
;
;
;
;
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.
@
%
#'% #%%
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(. 2
4 . /" /. @
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") f(xT (y) y x3 : : : xn) = f(x y x3 : : : xn)T (y), R = L, L = R.
H%, %
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"
%
/%
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1!. 6/ " %
& "() %
/% # () 1!. 5 3 .7
%
%/&.
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. %
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)
;
;
;
;
;
865
% (%% 2f(xy ab t) = 0, # %, f(xy ab t) = 0. 8
%
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%
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. 2
; "#
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%% &&%& "( %
/%
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1& f(x1 x2 x3 : : : xn+1) | "
%
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# #%% ! %
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# .
M
x
3. ,
"# gn
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% %
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%
9%
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9% % %
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h
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. . H%, %
%
& " &&%& 1%
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!,
% 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).
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# %
# & "(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:
6
6
2
;
j j
;
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
;
2
;
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;
;
;
;
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;
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868
. . (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:
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
) 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:
;
;
;
;
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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;
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%
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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
. . 2.21. Y | F n
F r > 1, GY | " . dimY 6 ((GY ) ; r + 1)(n + 1)# ,
dimF n 6 (n + 1)2 .
" $ $ $ 7. 8 3] 6]. " , u-$ -$, ;$ 3], x 1 $ 3]. -$ # : 1) $ F < , , 3], , .
# $ $= 2) $
$ 3] >;, $ .
x
1. u-
1.1. g h | #
$ G. g h] = g;1 h;1 gh. G |
(. . < 2), 8x y z t x y]z t]] = 1:
(1.1)
+ , G , G M, G? = G=M | . " # M
?
ZG-
(ZG? | &
& G) , ZG? M : g 2 G, m 2 M,
g?m = g;1 mg.
7 Fit(G) @ G ( G, ; $ $ ). +
$ G Fit(G) . +
< $ : 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)
+
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(1.3) #
a, b.
$ M . 7
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1. $ M0 .
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7 Fn n. @
$ $ $ -
4]. 7 x?i , i = 1 : : : n, $ ! An ( n). -$ . Fn An , (xi) = x?i, i = 1 : : : n. C $
$
; ZA
n-
n ti , i = 1 : : : n. D (xi ) = x0i t1i $ < : Fn ! An wr An (
$ An An). 7 $ $$
! $ .$ 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
. . a b c]] = 1 b c] = m 2 Fit(G). (
, (1 ; a?)m = 0 Fit(G) m = 0
M (
m = 1 G). 2
1.3. )" ' u- G A = G= Fit(G) | ".
. # a 2 G, t 2 N , a 2= Fit(G), at 2 Fit(G). D m = at 6= 1, G | . D a m] = 1, (1 ; a?)m = 0, . 2
u- , $ , . . $ .
I 3] 6]. B
$ 6], $ ; , .
, < $ 3]. ";,
6], $ . +
n > 1 Xn n X = fx1 : : : xng,
R = ZXn. 0 6= | #
R =
k
X
i=1
nigi |
()
& , 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 ! , , 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
. . a 2 Fit(G), $ $ , b 2 Fit(G). 3
a b] = 1 2 N.
, Fit(G=N) . D N | $ , # Fit(F=N) = Fit(G)=N. K, Fit(G)=N Fit(F=N). h + N 2
2 Fit(G=N).
D m 2 Fit(G) m h]+N = N, m h] 2 N,
m(1;h) 2 N. D Fit(G) 6= N, ! m 2= N. " N m(1;h) 2 N , h 2 Fit(G). 2
x
2. 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<n.
G """ Fmn .
. : Fn ! G | $ G $ . D F= Fit(G) ' Am '(Fn0 ) Fit(G),
& : An = Fn=Fn0 ! Am . (
, ! a1 : : : an $ An , (ai ) = ai , i = 1 : : : 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] # /. '01 0 1/ . 2. 1. | .: 3,
1982.
$ % 2001 .
{
. . . . . 517.518.126
: , { , HL-.
!""#"$ % " &',
()*+ ($# (#,% $ # ("". +"$ -". - HL-,
()$# ( (#% %*+ "## !#.
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.
, .
! " | " %&{( ), & ",
*
.
" %&
), *
(. ,3]). 0
" 1
{*
(. ,6, . 17 104]). ,11] , &, 1
{*
, . , " & %&{( *
. 7 "8, " " . % , *
, %&{( (.
! +( ( ()) !556 (+ 99{01{00354, 00{15{96143).
, 2001, # 7, 8 3, ". 887{895.
c 2001 ! "#,
$%
&' (
888
. . 3.2). ; , %&{( -
), 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. E4 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).
C + *, , , /,+ .. B(1 2) 2).
8 . ' 4).
# 8 ' . :,, 4 X Y , , .
4, . C ) m0 , +/ () + . F V () = fm0 2 M0 j m0 = m0 g
/ , +/ )
, .
, '3 + A,
V () , '3 . ,
V () / V ( ), 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
. . . 4, , , +/
(1 1 ) (2 2 ), 1 2 , 2 + 1, 2 ; 1,
1 2 . C , ' ) t3 ; , 1 = 2 . , , | 1 2, 2 + 1, 2 ; 1. , ) , 8, e0 e1 e2 m0 = 0 e0 + 1 e1 + 2 e2 :
C, (m0 y) = (m0 )y:
(m0 y) = f ;1 (1 + 1)m2 xg = ;1 (1 + 1)fm0 '2 X g =
= ;1 (1 + 1)m0 '2 X = ;1 (1 + 1)(
0 e2 x + 1 e0 x + 2 e1 x)
(m0 )y = (
0 e0 + 1 e1 + 2 e2 )y = ;1 (2 + 1)
0 e2 x + (2 ; 1)
1 e0 x + 2
2 e1 x
8 e0 e1 e2 ;1 (1 + 1)
0 = ;1 (2 + 1)
0 ?
(1 + 1)
1 = (2 ; 1)
1 ?
(1 + 1)
2 = 2
2 ,, . = , . 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).
1] . . | :
!" #$, 1993.
2] )
. $. ! *
+ +
,. | .. . . . -..-. . , 1994.
3] 0
. ). )* , .
+-
. | 1 *.
& ' 1998 .
. . 519.689.6+512.554.2
: , , !" , -# $, %%&,
! ''! ( .
), ' # , ' , , '&%*+' (*# ,, %, $'# '" %( '
%*+"' %# !" '' ( , '+ %'% " -: !# % " | ''&
, ''$' ' $'# , # |
01 2, '( % " # (*# ,. 3'1*' 1 + '! ''$'# (*# $(# ! (-!) ! !" '' ( .
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
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;
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917
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2.2. % &%', &%' )* +20]
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1 sl(2 C )
h = (qR;1 + 1)=2. M 1 z 6 sl(2 C )
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= Lk ( u)k;2, Lk qR- 1:
?Li Lj ] = (i j)Li+j (i j > 1= i j 6 1)
+1)(t+3h;1)2 :
?L2 L;2 ] = '(L0 + 1) '(L0 1) '(t) = t((tt+2
h)(t+2h;1)
() 7 J 1 F0E0- qR - , J (u) := lu (J ) =
P
= Jk ( u)k;1, Jk qR- :
Jk = J T ;k fk (t) ?J J ] = c
T f(t) = f(t + 1)T ?T J ] = ?f(u) J ] = 0
fk (t) = t : : : (t k) k > 0
;1
fk (t) = ((t + 2h) : : :(t + 2h k + 1));1 k 6 0 h = qR 2+ 1 :
;
;
;
;
;
;
;
;
918
. . A, qR - qR- 5- sl(2 C )- 1 Vh
(h = (qR;1 + 1)=2) 6 sl(2 C ) 1 2 . M
5 %
, k
Jk = @zk J;k = (
+ 2h) : : :(
z + 2h + k 1) =
L2 = (
+ 3h)@z2 L1 = (
+ 2h)@z L0 = + h L;1 = z
+ 3h
L;2 = z 2 (
+ 2h)(
+ 2h + 1) = z@z =
;
qR - C - 6:
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R
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?37,38].
;
1
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0 H | 1 sl(2 C ) P | C , H. / P sl(2 C )-
_ P = sl(2 C )- A(u u)
?6]. Y A(u u)
_ % % (u)
_ k,
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M % , H Virasoro master
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i 6 N. > P(
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cut
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;
1
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). A Lcut
1 , L0 , L;1 %
sl2 cut
cut
?L0 L;1] = L;1 ?Lcut
1 L0] = L1 ?L1 L;1 ] = h(L0 )
h(x) = P (x1+1) P (1x) . Y sl2 .
0
( N = 1) $
% & A(u u)
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j j
j j
j j
j j
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922
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G2-. ` $ &
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(
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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.
924
. . 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. | Montrƒeal, 1988.
38] Q^#9# 4. ., -9;< . ?., /;; . . # '*!! # $'@
# // ?$'@ # $#8. | 1989. | -. 1, &!. 1. | 0. 178{206.
39] . . Watch-dog „(("& >$"#{$"$I #"#
*!$ )&9 595#: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 !.
s2
. . - 517.98
: , !
, " "#$%.
& ' (#% ') s2 ( ( %( (. (
( (
(
(' (*
( *
*( M
II1 III , #'$ '(
"
!
(,-). /% %)" '" " "#$
(
(' '(
( *
*( M.
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
. . 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
0
s2 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
. . C C. !. :
# , # C. .. - < .
!
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! , 001 - - // : !
. $. @., . . $. -. | *, 1997. | . 17{18.
13] : . . K , L !01 E0 --- // ?0 N!. @. $?G. | Q -0, 1997. | . 56.
& ' 1998 .
. . 512.546+515.124.32
: , .
, ! ! "#! G % ! : G X ! X X ' % !
~ : G X ! X , G | -" * G.
Abstract
S. A. Antonian, Extension of the pseudo-compact group actions, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 3, pp. 931{934.
It is proved that for a given pseudo-compact Hausdor1 group G, every continuous action : G X ! X on a metrizable space X has a unique extension to a
continuous action ~ : G X ! X , where G is the Stone{Cech extension of G.
, G | . G- .
1] - ! G " ", " G # .
. : G X ! X X ~ : G X ! X .
&
' .
&
X Y C (X Y ) * *#" f : X ! Y , + -" ".
1. X -
, Y , C (X Y ) .
1
. ,' X = S Xn , # Xn X | n=1
. &
# n > 1 * pn : C (X Y ) ! C (Xn Y ) *# #, . # f 2 C (X Y ) # f jXn . ,
' # pn , ' , 2001, 7, 2 3, . 931{934.
c 2001 ,
!"
#$ %
932
. . Q1
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
. . =
' G. A
' " " 1], *', . & ' .
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 .
. . , . . 517.95
.
: , , !
" #
$ %&
'&, (. ). *
+, %
,,
! , , + +
#
. (. ). -
& & .
"
/ #
#$ '%, ' %
,,
!
, # + $ 0
#
. %
' ' #
'% .
&
,,
!& . 1. (. , 2 %
, ""0 ' -
%$ .
,,
! .
( $ "
& + ,,
! .
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.
, 1922 . ". #. $%, && & . ". #. ' 80{90-
% , % , - ,
& % &% && .
, % ./ && . :
, 2001, 7, ; 3, . 935{938.
c 2001 ,
!
"#
$
936
. . , . . 2
(x t) + p(x)u(x t)
; @u(@tx t) = ;(1 ; x2) @ u@x(x2 t) + 2x @u@x
(1)
p(x) 2 C 2 7;18 1] | &%, p(;1) = p(1) = 0.
; < t = 0:
u(x 0) = '(x)
(2)
&%.
(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
n
=1
n
7;18 1] @ %-&&
T (t) = (u(x t) v (x)). = % (
% (5)
), (4) ./ :
n
n
1
X
n
=1
j j < 1
(6)
n
| %-&& &%
(x) = p(x)'(x) ; '00 (x)
(7)
1
708 ] &% 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
n
n
n
n
n
t
=
n
n
n
t
=
n
t
n
937
...
., %
% % ./ &% && :
1 r 2n + 1 cos((n + 1 ) ; )
X
2
4 :
v0 ( 0) =
(8)
2
12
2
n
n
(sin
)
=1
, (8) % &%. 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
n
=
n
n
n
n
n
n
n
n
n
=
n
n
1
X
r
0
Z
0
Z
n
; d q2( ) d:
n
0
0
2n + 1
2n
=1
D. ./ .
. (1){(3)1 ,
P
'(x) 2 C 27;18 1] j j < 1. =1
u(x t) , ! " 7;18 1] #$.
v2 () = n
n
n
1] . . . | ".: "$%, 1991.
938
. . , . . 2] )*
. ., . +. * ,
,, --.
, // )* 01. | 1994. | 3. 30,
6 1. | 7. 35.
% & 2001 .
. . ( )
512.552.7
: , .
" " R (R = R=J (R) 6= f0g). '
( .
) R | , R 6= f0g, S | +". R0 S , : (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 . ,
, 2001, " 7, 4 3, . 939{944.
c 2001 !,
"#
$% &
940
. . ', E (S ) | S , R1 S 1 |
R S /
.
1. 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
. . 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 $ ,
F
G F
G. +.
3. ) 1.1, 2.1, 2.6 '% . <, ' R | R '
, 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 % % % RS . 0
$*+ , ' % % ) .
944
. . . - K | , char K = 0, S | 0- . 8 K0 S , 3.2 S | /
/
(
)
).
0 + , K0 S = M (K I C P ) | K .
8 P '+ K (
, P , ' ', ), K0 S (!7, 10, 12] !6, 1]).
1] Okninski J. Artinian semigroup rings // Comm. Algebra. | 1982. | Vol. 10. |
P. 109{114.
2] Okninski J. Semilocal semigroup rings // Glasgow Math. J. | 1984. | Vol. 25. |
P. 37{44.
3] Okninski J. Semigroup algebras. | New York: Marcel Dekker, 1991.
4] Renault G. Sur les anneaux de groupes // C. R. Acad. Sci. | 1971. | Vol. 273,
no. 2. | P. 84{87.
5] Schwarz S., Krajnakova D. O totalne nekomutativnych pologrupach // Mat.-Fyz.
Casopic Slovensk. Akad. Vied. | 1959. | T. 11, no. 2. | S. 92{100.
6] . . ! "#$%&$""'! #%(& "# 0-"& '! "#$%&$"" // ). *+,. | 1975. | -. 18, . 2. | /. 203{212.
7] . . /&0' "#$%&$""' +#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 .
. . . . . 512.543
: , .
, ! , " $ %$ , & !, ! %$ ', &%$
( % , .
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 . -:
.
, 2001, 7, 0 3, . 945{949.
c 2001 !,
"#
$% &
946
. . B = M1 M2 , Mk Nk , Nk A, Mk
2
, var X = var X0 . 2
3 .
2
Mk . ( B
M1 M2
2
X0 .
-
, M1 \ M2
.
. 4 , *
e, 5
6 m 5
6 c e, m, c (71, 5]). :# *
e, m, c M1 M2.
!" # . ;
, M1 M2 M1M2, 6 # # (72, . 337]).
> exp(M1) = e1 , exp(M2 ) = e2 , exp(M1 M2) 6 e1 e2 .
2.
M1 M2
c1 c2
M = M1 M2
c = c1 + c2
. 2 1 M
X0 . ) G X0 , p- P 6 c.
G = N1 N2 , Nk G, Nk 2 Mk , k = 1 2. %
P1 = P \ N1, P2 = P \ N2, P3 =
= P \ N1 \ N2 = P1 \ P2. ( P1, P2 P3 p-
N1 , N2 N1 \ N2 (72, . 345]) P. > jGj = mp , jN1j = m1 p , jN2j = m2 p , jN1 \ N2 j = m3 p ,
m1 , m2 , m3 p, jGj = jN1=N1 \N2 j jN2=N1 \N2 j jN1 \N2 j =(m1=m3 )p; (m2 =m3)p ; m3 p =mp :
2
, = + ; . ) jP1P2j = jP1=P3j jP2=P3j jP3j = p; p ; p = p:
( , P1P2 # p-# G. ) # # s, t 6 s + t
(73, . 151]). %*
P1P2 ( , P) c = c1 + c2. :
# 5
# X0 6 # , 6 c. ( *
c 5
var X0 (71, . 195]). @ var X0 5
H=K
6 d 6 d,
-
.
,
947
H=K *
h1 : : : hd+1 , 7h1 : : : hd+1] 6= 1. 2
, var X0 5
6 c. 2
( 5
"
#. 3 An *
n. 4 Ap Aq , p q | , (74]).
3.
M1 = M2 = Ap Aq (p q
)
M1 M2
. ) Gi = ha1 b1 : : : ai bi j aqj = bqj = 1 Gi 2
7ak aj ] = 7bk bj ] = 7ak bj ] = 1 k 6= j 7a1 b1] = : : : = 7ai bi]i:
A Gi Vi
Zp ( Gi), dimVi ! 1, jGij ! 1, . B
Hi = Vi h Gi, Vi i. C
, fHig . : Hi 2 M1 M2, Gi q: Ai = ha1 : : : ai ci Bi = hb1 : : : bi ci, c = 7a1 b1] = : : : = 7ai bi]. 2
4.
| , .
", #$ % % ,
# p q
A p Aq
.
. ) # # 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
n
n
n
948
. . (
vi iE vi # 5
(71, . 5]). ) vi i.
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 . #
#
,
,
,
-
#
-
, .
-
949
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.
1]
2]
3]
4]
. . . | .: , 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] (. '. 12+. *, &) 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 .
18 2001 60 ,
, !" " ! "#,
$ " !
% & &.
"% , ( ). * %& #" + "
&( )&( !: , & &( &( , $
(( +&( , & -%,
!! & .+!( | & ).
0 " $ & !& 1 2 3". 4 "
%+"( % % +# (" $, !& ) $ .
5" " ) "% 1 2 3" !! ! "
%&( %$&( &( !( #. 1 2 3" % 6 7 . 0. . !# ( %
", . 8& % " # 0#%& ! 9, %"( % &
9 $ $!& (%"& & ( 9-" 9%. 4!) & ) : ", " , "
9, 1" !, &( 9+ (<- %, =( 5, 7)
%), , . 3 !& % ++" 2&( ?5" @, # $ % , !& 9 + . % " ++, " 3 =&. : !, & $ ++, !& $.
1# A = ( - =& %" %,
"%& " : ". 0 ( 9 " !& % ! $" ( + "
! #, 9+ ) +&
& +&.
952
D" 9%& 1 !& !& % " " 0 E!, % !& % . 0 +
9%& !& %, " & +, (" 6 !. - , #)& 6 !& (" :) !& &, %$ !"& !& 9 %".
& 9 &, " !& %, " " (& & (
% " .
7" ) ?A" " @ ! %"# #! )# ( ! ".
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