Забыли?

?

Фундаментальная и прикладная математика (2002 №1) (2002).pdf

код для вставкиСкачать
``` . . 517.983.28+517.928
: , , !" , ", # \$\$&' .
() # ! \$\$& L = dtd ; A0 ; BA0 : D(L) C (R Y ) ! C (R Y )
!.# ' C (R Y ) ' )'
\$"&!, /' #! R 0) ' Y . 1!! A0 : D(A0 ) Y ! Y 0#
! , " " ! . iR, A0 , 2 20 1), | A0 ,
B : C (R Y ) ! C (R Y ) | !! )! .
Abstract
A. G. Baskakov, Splitting of perturbated dierential operators with unbounded operator coecients, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002),
no. 1, pp. 1{16.
We obtain some theorems on splitting of di7erential operators of the form
L = dtd ; A0 ; BA0 : D(L) C (R Y ) ! C (R Y )
acting in the Banach space C (R Y ) of continuous and bounded functions de8ned on real axis R with values in the Banach space Y . The linear operator
A0 : D(A0 ) Y ! Y is the generating operator of a strongly continuous semigroup of operators and its spectrum does not intersect the imaginary axis iR. Here
A0 , 2 20 1), is a fractional power of A0 and B : C (R Y ) ! C (R Y ) is a bounded
linear operator.
9 \$! :" 9!" \$ \$ '
!.
, 2002, 8, ; 1, . 1{16.
c 2002 !"#,
\$%
&' (
2
. . Y | , End Y | , Y (kX k1 | X 2 End Y ). !
F (R Y ) (, F ) "
" : Lp = Lp (R Y ), p 2 %1 1], | p ( p = 1), "
( R = (;1 1) )*, "+ Y ,
C = C(R Y ) | )* " L1 , AP(R Y ) |
)* " C.
,
+ ))* A0 = dtd ; A0 : D(A0 ) F ! F A0 : D(A0 ) Y ! Y | "+ %1] fU(t) t > 0g " End Y . .
+ D(A0 ) ))* A0 ++ "
. /*+ x 2 F + D(A0 ), )*+
f 2 F , + + s 6 t " R Zt
x(t) = U(t ; s)x(s) + U(t ; )f() d:
s
1
A0 x = f.
2 "+ ("
3) L = dtd ; A0 ; B0 A0 : D(A) F ! F (1)
A0 , 2 %0 1), | + A0 B0 2 End F . 2 ,
F = C B0 4 (B0 x)(s) =
X
k>1
Z
Bk (s)x(s + hk ) + F (s s ; )x() d
P
R
(2)
Bk 2 C(REnd Y ), hk 2 R, k > 1, kBk kc < 1 )*+ 5: s 7! F (s ),
k>1
5: R ! L1 (R End Y ) ( )*+ ) 4 C(RL1(R End Y )).
. " +" (
"*)
L +
))* , " 4 "+ , 3
" 4 +++
))*
. , ++ , 3 6. , 7. 8, 6. 9. :+, ;. 9. <, ;. ;. ( %2] <. .. / %3]. + (
2) + " ", 2 3 3
"
3 L, ++ L = A0 ; B, A0 "
3 , B = B0 A0 +++ "
.
>
4 5 4 " . (" 4+ ) %2, . IV] + ))*
L = d=dt ; A0 ; B(t), A0 2 End Y B 2 C(R End Y ), +
))* ""
| %4].
>
L (2), A0 | ( 8)) , = 0 B0 | 4+ ))*
)* Q " C(R End Y ), 2. ;. /
%5]
(
. "
8), 1 "
, "+
)* Q +++ ,
4 , "
B0 +++ ))*
.
" 1
%6, 7] 6. 6. 63
%8]. .
3 !. D. < %9] ) %10],
" ))* .
24 , (, . .) "
3 L " * "
%1], " . . +" , "
3
A0 (
. 1).
," , 1. . A0 : D(A0 ) F ! F .
.
, " " %11] (
. 4 %12]).
1. (A0) A0 : D(A0) F ! F (A0 ) = f 2 C : 9
2 (U(1)) j exp j = j
jg:
, (A0 ) , iR C (A0 ). (;A0 ) | %12], (A0 ) .
1. ; " 1 , A0 1 +
(U(1)) \ T = ?
(3)
T = f 2 C : jj = 1g | + 4 " C ( , "
A0 , (A0 ) \ (iR) = ?, ). 2 , " )
+ (3)
, Y = Y1 Y2 , Yk = ImPk , k = 1 2, P1 = P (1 U(1)) | 4
. . ,, 4 1 = f 2 (U(1)): jj < 1g,
P2 = I ; P1 | (I | 4 ),
. . P2 = P(2 U(1)), 2 = (U(1)) n 1 . G ;A0 | ,
(A0 ) = H1 H2, H1 C ; = f 2 C : Re < 0g H2 C + =
= f 2 C : Re > 0g, H2 | 4 Pk = P(Hk A0), k = 1 2.
2. G A0 : D(A0) F ! F , A;0 1 2 End F ++ )
(A;1 f)(t) =
0
Z1
;1
G(t ; s)f(s) ds = (G f)(t) f 2 F t 2 R
G: R ! End Y | )*+ D, + (
G(t) = U(t)P1 t > 0
;U(t)P2 t < 0
+ * kG(t)k 6 M exp(; jtj), M > 0, > 0, t 2 R.
x
(4)
(5)
1. 2 "+ 1. 8 Ai : D(Ai ) X ! X , i = 1 2
(X | ) "+ , U 2 End X , U D(A2 ) = D(A1 ) A1 U x = UA2 x
8x 2 D(A2 ). . U "3
"+ A1
A2 .
!
A " "
3 (A : D(A) X ! X ), J " "+ . . B : D(B) X ! X "+ 3
A, D(B) D(A) + ++ C > 0, kBxk 6 C(kxk + kAxk) 8x 2 D(A). 94 , 3
A, "
LA(X ).
+ "
3 A ; B,
B 2 LA (X ), +++ + D(A) A, , D(B) = D(A) 8B 2 LA(X ). >+ 3 "+ LA (X ) . ( , LA(X )
4 , 4 kBkA = inf C, )
3+ +
C > 0, +
"
J .
; , LA(X ) | . !
(A) (A) "+ " 4 A.
5
2. A | " " LA(X )
I : A ! A, ;: A ! End X | )
(. . ). > (A I ;) "3
+
A, A | "
, 1) A | ( k k), 4 LA(X ) (. . kX k > const kX kA 8X 2 A)L
2) I ; | L
3) (;X)D(A) D(A) A;X ; ;X A = X ; I X 8X 2 AL
4) (;X)Y X;Y 2 A 8X Y 2 A + ++ > 0, k;k 6 maxfkX;Y k k(;X)Y kg 6 kX kkY k 8X Y 2 AL
5) I | I ((;X)I Y ) = I ((I X);Y ) = 0 8X Y 2 AL
6) 8X 2 A 8" > 0 90 2 (A), kX(A ; 0 I);1 k1 < ".
(A I ;) | + + A : D(A) X ! X B 2 A | "
A. (
X0 2 A,
+ (A ; B)(I + ;X0 ) = (I + ;X0 )(A ; I X0 )
(6)
k;X0 k1 < 1 ( U = I +
+ ;X0 ) " A ; B A ; I X0 . ; ,
(6) , X0 | J +
(7)
X = B;X ; (;X)IB ; (;X)I (B;X) + B = 5(X)
A "
. M"
4
4, 3 5: A ! A ( + " + ), (
. %6,13]), 2. kBk kIk < 14 (8)
(7) " X0 , # (6),
\$ I + ;X0 .
3. )
; ++ )
adA : D(adA) End X ! End X + D(adA ), + " X0 2 End X , + D(A) D(A), AX0 ; X0 A : D(A) ! X J D(A) Y0 2 End X ( +
Y0 = adA X0 ).
>
" ))* * , ++ 4 + "4+ X +
X = X1 X2 6
. . "
3 A : D(A) X ! X X1 X2 ,
3
4 i = (Ai ), i = 1 2, "
+ (Ai = AjXi ,
i = 1 2, | 4 A Xi , A = A1 A2 ).
Pi , i = 1 2, | , * "
"4
X , . . Xi = Im Pi , i = 1 2. .
, "
4 i, i = 1 2, , Pi = P (i A), i = 1 2, | ,, 4
i, i = 1 2.
3. 8
+ + A (A I ;) "+ + , :
1) Pi X Pj 2 A, i j = 1 2, + X 2 A, )
I I X = P1X P1 + P2X P2 , X 2 AL
2) Pi (;X)Pj = ;(Pi X Pj ), i j = 1 2, + X 2 A, 3
Pi (;X)Pi = 0, i = 1 2.
,
+ A + " + 3. N "+ A +
A = A11 A12 A21 A22 Aij = fPiX Pj : X 2 Ag, i j = 1 2. !
Xij " ( ) Pi X Pj " Aij , i j = 1 2, X = (P1 +P2 )X(P1 +P2 ) =
= X11 + X12 + X21 + X22, X 2 A.
++ +
+ (7) P1 P2 ( ) "+ 2 " + 3, + Xij , i j = 1 2, X 2 A:
X11 = B12 ;X21 + B11 (9)
X21 = B22 ;X21 ; (;X21 )B11 ; (;X21 )B12;X21 + B21 = 51 (X21 ) (10)
X12 = B11 ;X12 ; (;X12 )B22 ; (;X12 )B21;X12 + B12 = 52 (X12 ) (11)
X22 = B21 ;X12 + B22 :
(12)
24 , + (10) (11) "
+ A21 A12.
O+ "J
, , , 4 (9), (12),
)
, "+ : bij = kBij k, i j = 1 2,
~b12, ~b21 | X 7! B12 ;X : A12 ! A12, X 7! B21;X : A21 ! A21
~b22 | J+ " X 7! (;X)B22 :
A12 ! A12, X 7! B22;X : A21 ! A21. .
, ~b12 6 b12 , ~b21 6 b21.
3. %
d = b11 + ~b22 + 2(b12b21)1=2 < 1:
(13)
&# A ; B A ; P1X P1 ; P2X P2 = A ; X11 ; X22 7
# X | " (7), Xij , i j = 1 2, | " '
(9){(12), U = I + ;X = I + ;X12 + ;X21 , \$
U ;1 = I + (I ; ;X21)(I ; (;X12);X21 );1 ;X12 +
+ (I ; ;X12)(I ; (;X21);X12 );1 ;X21: (14)
( #, ' ' ):
~
21b12 L
kX11 ; B11 k 6 ~ 2b21b12
6 2b
(15)
~
1 ; b22 ; b11 + q 1 ; b22 ; b11
~
21b12 L
kX22 ; B22 k 6 ~ 2b12b21
6 2b
(16)
1 ; b22 ; b11 + q 1 ; ~b22 ; b11
kX21 ; B21 k 6 ~ 2qb21
6 ~2b21 L
(17)
1 ; b22 ; b11 + q 1 ; b22 ; b11
kX12 ; B12 k 6 ~ 2qb12
6 ~2b12 L
(18)
1 ; b22 ; b11 + q 1 ; b22 ; b11
~
kX11 ; B11 ; B12 ;B21k 6 2~b12b21q L
(19)
1 ; b22 ; b11
~
kX22 ; B22 ; B21 ;B12k 6 2b~12b21q (20)
1 ; b22 ; b11
# q = %(1 ; ~b22 ; b11 )2 ; 4b12 b21]1=2.
. ,
(10) +
51 : A21 ! A21. ;3
J B(r1 ) =
= fY 2 A21 : kY k 6 r1g " A21, 51 +. .+
J r1 r1 = rb21. M" + k51 (Y )k 6 rb21 + Y 2 A21 , 51(B(r1 )) B(r1 ), r > 0 + r~b22b21 + rb21 b11 + r2~b12 b221 + b21 6 rb21:
. , r1 4 "+ ~
r1 = rb21 = (1 ; b22 ;~b11 ; q) = 2b21(1 ; ~b22 ; b11 + q);1:
2 b12
8+ Y1 , Y2 " J B(r1 ) *
k51(Y1 ) ; 51 (Y2)k 6 (~b22 + b11 + 4 2 b12b21(1 ; ~b22 ; b11 + q);1 )kY1 ; Y2 k 6
1=2
~
6 ~b22 + b11 + 2(b12 b21)~ (1 ; b22 ; b11 ) kY1 ; Y2 k 6 dkY1 ; Y2 k:
1 ; b22 ; b11 + q
8
. . 2 + (13) 51 +++ 4+ J B(r1 ), 1
(10) J B(r1 ) J X21 , 4 *. !,
(9) J X11 . .* (15), (17), (19) " + 4 X21 J B(r1 ).
6 4+ (11) ( ,
(12)), +" 52 : A12 ! A12. . +++
4+ J B(r2 ), r2 = 2b12(1 ; ~b22 ; b11 + q);1, "+ * (16), (18) (20).
k;X21k1 k;X12k1 6 2 r1r2 = 4 2 b12b21(1 ; ~b22 ; b11 + q);1 < 1,
I ; (;X21 );X12, I ; (;X12);X21 . ;
+ , U = I+;X12 +;X21 = I+;X
(14). >
".
4. + A~ = A ; P1X P1 ;
; P2X P2 A ; B , Xi = Im Pi , i = 1 2, A~ 1
A~ = A~1 A~2 , A~i = Ai ; Pi X jXi, i = 1 2, | 4+ A~ Xi. >
"
, " A ; B + "
A~1 A~2 . ;
, (A ; B) = (A~) = (A~1 ) (A~2 ).
5. ,
A({) = A ; B({), )*+ B : fz 2 C : jz j 6 g ! A +++ ) ( + 3
), 3
B(0) = 0. > " 3 , 0 > 0 )*
U : fz 2 C : jz j < 0 g = S0 ! End X , X : S0 ! A, (A ; B({ ))U ({ ) = U ({ )(A ; P1 X({ )P1 ; P2 X({ )P2 ) j{ j < 0 (21)
U ({ ) = I + ;X({ ), j{ j < 0 , | X({ ) | J
+ (7) B = B({ ) ( (9){(12) + X({ ), { 2 S0 ). M
*, "
+ 4+
X({ ), "+ , )* U ({ ) X({ ) ( ,
Xii ({ ), i = 1 2) )
( )* B({ )).
2 , X1 | , " (21)
, A ; B({ ),
j{ j < 0 , + A~i ({ ), i = 1 2, j{ j < 0 , " X1.
G 1 = (A1 ) = f1 g | " " A, P1 = P(1 A) | x1 | , " 3 3 * , (A ; B({ )) = (A1 ({ )) (A2 ({ )), A~1 (0) = A1 1
A ; B({ ), j{ j < 0 , " 1 ({ ) x1 ({ ), {lim
!0 1({ ) = 1 , {lim
!0 x1({ ) = x1 .
9
x
2. "
" L (1), L = A ; B : D(A) F ! F = F (R Y )
A = A0 = d=dt ; A0 + "
3
B = B0 A0 | "
. M A (4 1). . 3 P1 P2 2 End X , + "4 Y = Y1 Y2, Yk = ImPk ,
k = 1 2, Y (
. "
1 " +).
8 A0 + ;A0 (
. 4).
4 )
, + "
3 ))* A. "+
6. ;A0 | , 1
) fU(t) t > 0g. M" " * %12, . I]
+ A 2 (A) , "+ 4 )* (A ; I);1 f, f 2 F (
. "
2 )
(4))
4 D(A0 ), 0 6 < 1, *
Z
kA0 (A ; I);1 k 6 kA0 G(u)k du 6 C( ")(0 + ") ;1 (22)
R
G : R ! End Y | )*+ D + A ; I, 0 =
= dist(iR (A0)) | + iR (A0 ), 0 < " < 0 | 3 "
, C( ") > 0.
"
+ A
"+
4. : X 2 End F "3
c-
,
" )* t 7! Tk (t)XTk (;t): R ! End F , k = 1 2,
. S T1(t)' = 't ,
't (s) = '(s + t), s t 2 R | "
+ )* " F
(T2 (t)x)(s) = (exp its)x(s), s t 2 R, x 2 X .
1. . B0 (2) c- F = C + )* 5 Bk , k > 1, " )
(T1 (t)BT1 (;t)x)(s) =
(T2 (t)BT2 (;t)x)(s) =
X
j >1
X
j >1
Bj (s + t)x(s + hj ) +
Z1
;1
eih t Bj (s)x(s + hj ) +
Z1
j
;1
F (s + t s ; )x() d
eit(s; ) F(s s ; )x() d:
(23)
10
. . 5. A(0) " Endc F c- "3
, +:
1) A(0) | kX k0, X 2 A(0), + kX k0 > kX k1 8X 2 A(0)L
2) + X 2 A(0) t 2 R X(t) = T(t)XT (;t) 4
A(0), kX(t)k0 = kX k0 )*+ t 7! X(t): R ! A(0) L
3) + C1 C2 2 End Y C1XC2 4 A(0) kC1XC2 k0 6 kC1k1 kC2k1 kX k0 .
2. 6 Endc F +++ .
3. 6 B 2 End F , F = C(R Y), (2), +++
, 1
P
4 kB k0 = kBi k1 + sup k5(s)kL1 . N "
s2R
i>1
A .
4. " Endc F , . . X 2 Endc F , + )*+ t 7! T (t)XT (;t):
R ! End F , +++ .
6. A(0) | + + + " Endc F . !
A(), 2 (0 1), "
" X = X0 A0 , X0 2 A(0), " LA(F ).
2 A() LA(F ) " "
+ 6 (
. * (22)). : A() + , 4
kX0 A0 k = kX0 k0 8X0 2 A(0). A() J
"
+ ))* A
(A0 | ).
5. G | +
" Rn P
* @G A0 = p(x D) =
a(x)D | 1 ))*jj62m
+ 2m (a : G ! C | )*)
+ D(A0 ) " ! W22m (G) L2 (G),
+
" 8 ( " ).
> A0 | (
. %12]) 4 B = P B D : D(d=dt+A0) C = C(RL2(G)) ! C, B 2 End C(R L2(G)),
jj62m
4 B = B0 (A0 ; 0 I) 0 2 (A0 ) = C n (A0 ),
B0 2 End C = 2m2m;1 . 2 , B , jj 6 2m ; 1, (B ')(t x) = b(t x)'(t x), t 2 R, x 2 G, ' 2 C(R L2(G)),
b 2 C(RC(G)).
7. 8+ X = X0A0 , X0 2 A(0), 4
I X = (P1X0 P1 + P2X2 P2)A0 I : A() ! A():
11
< + I " 3) . T, I | .
8+ + )
; = ; : A() ! A(0) End F "
T~(t), t 2 R, " End A(0)
~ 0 = T1 (t)X0 T2 (;t), t 2 R, X0 2 A(0). ; " T(t)X
+ adD , D = d=dt, , +++ "+
1 L "
D0.
6. 8+ B0 2 A, 3 )
(2), " )
(23) , 4 + D(D0 ) D0 : D(D0 ) A ! A = A(0), )* Bi , i > 1, 5: s 7! F(s )
))*
" B_ i , i > 1, 5_ .
1
D0(B0 ) ++ )
Z1 @F
X_
(D0 (B)x)(s) = Bi (s)x(s + hi) + @s (s s ; )x() d:
i>1
;1
(24)
>)
;: A() ! A(0) End F 4
X = X0 A0 " A() J Y 2 A(0) +
adA Y = DY + A0 Y ; Y A0 = X ; I X = X12 + X21
(25)
4 A12(0) A21(0) " A(0), Aij (0) = fPi X0 Pj : X0 2 A(0)g, i j = 1 2, D : D(D) A() ! A()
D(X0 A0 ) = D0(X0 )A0 , D(D) = fX0 A0 2 A(): X0 2 D(D0 )g. ; "4 ", J Y = ;X 4 ;X = Y12 + Y21 Y12 2 A12(0), Y21 2 A21(0),
3 )
Z1
Y12 = ; U(s)P1 T1(s)X0 T1 (;s)P2 A0 U(;s) ds = ;X12 Y21 =
0
Z0
;1
U(s)P2 T1 (s)X0 T1(;s)P1 A0 U(;s) ds = ;X21
(26)
(27)
2 %0 1), (;A0 ) | , = 0 .
< + Y12 Y21 " . !
)
(26), (27) + *
kU()P1A0 k1 6 C1() exp(;1 ), > 0,
kU()P2A0 k1 6 C2()j j exp(2 ), 6 0, C1(), C2 (), 1 , 2 | 4 +, 3
1 2 > 1=2 dist(iR (A0)). 2 Y12 2 A12(0), Y21 2 A21(0) * k;X k1 6 k;X k0 6 kX k +
X 2 A(), + 6 C() dist(iR(A0)); (28)
12
. . (;A0 ) | , 6 C dist(iR (A0));1 (29)
= 0, + C, C() > 0, "+ C1 (), C2 ().
1. % (A() I ;) A = d=dt ; A0.
. G X =;1X0A0 2 A(), X 0 2 A(0),;
+ 2 (A) \ R+
* kX0 A0 (A; I) k1 6 kX k0 kA0 (A; I) 1 k1 . M" * (22)
, kA0 (A ; I);1 k1 , > 0. >
"
, 1) 6) " + 2.
; " + )
I ; + 3 , , 2) 5) " + 2.
8+ 3) " X = X0 A0 , Y = Y0A0 , X0 Y0 2 A(0), " A(). . X;Y (;X)Y
Z1 A0 Z2 A0 , Z1 = ;(X0 A0 )Y0 , Z2 = X0 (A0 ;Y0 )
4 A(0), 3
*
kZ1 k0 6 kX0 k0kY0k0 = kX kkY k
kX;Y k = kZ2 k 6 kX0 kkY0k = kX kkY k
++ * , + ;(X0 A0 ) A0 ;Y0 (
. )
(26), (27) * (28), (29)L 1
"+ "
A0 ).
! 3) " 2, "+ +.
8. (xn) )* " F "+ c-++ )* x0 2 F , nlim
!1 f(xn ; x) = 0 + )* f 2 C(R C ) . 2 1
"+
" c-lim
x = X0 .
n!1 n
9. Xn " End F
"+ c-++ X0 2 End F , X0 x = c-lim
Xx
n!1 n
8x 2 F .
10. : C : D(C) F ! F "+
c-"
, " c-limxn = x, xn 2 D(C), n > 1, c-lim
Cx = y0
n!1 n
, x0 2 D(C) Cx0 = y0.
7. <4 c- X " Endc F +++
c-"
. 1
c-"
+++ C : D(C) F ! F , + C ;1 2 Endc F . ;
, c-"
+++ A = d=dt ; A0.
2. + ;X, X 2 A() # A(0) ' (;X)D(A) D(A), A;X ; (;X)A = adA ;X = X ;I X.
13
. 8
, X0 2 A(0) X0 =
= (A0 ; 0 I);1 Z0 (A0 ; 0 I);1 , Z0 2 D(D0 ), 0 2 (A0 ). > "
D0 4+ " End Y D0(T1 (s)X0 T1 (;s)) = dsd (T1 (s)X0 T1 (;s)) =
= T(s)D0 (X0 )T (;s) , Y12 Y21, 3 )
(26) (27), 4 + D(D) D (
+
)
Z1
D(Y12 ) = ; U(s)P1 dsd (T1 (s)X0 A0 T1 (;s))P2 U(;s) ds =
0
= P1X0 A0 P2 ; A0 Y12 + Y12A0 :
>
"
, Y12 2 D(adA ) \ A(0) adA Y12 = D(Y12 ) + A0Y12 ; Y12A0 =
= P1XP2 , X = X0 A0 . 6 + Y21, 1
;X = Y12 + Y21 2 D(adA ) adA ;X = X ; I X.
X = X0 A0 | " " A(). An = n(nI ; A0 );1, n > 1, +
4
4 " D(D0 )
A(0) ( ), Xn0 0 " A(0) Xn0 0 = AnX0 An , X0 2 D(D0 ),
n > 1, c-+++ X0 . > Xn = Xn0 0 A0 4 D(adA ), (;Xn )' = A;1Xn A' + A;1 (Xn ; I Xn)',
n > 1, ' 2 D(A). (Xn ; I Xn )' = (Xn0 0 ; I Xn0 0)A0 ' c-+ (X ; I X)', (;Xn )A' c-+ (;X)A' (
. )
(26), (27)), " c-"
A , )*+ (;X)'
4 D(A) A(;X)' ; (;X)A' = (X ;I X)'.
:
".
4. %
(;A0 ): D(A0 ) Y ! Y | ,
(A0 ) \ iR = ?, 2 %0 1), B # A(0).
&# 4kB k < 1 ( (A() I ;)) (13) 3 L = d=dt ; A0 ; BA0
L0 = d=dt ; A0 ; (P1B0 P1 +P2 B0 P2)A0 , # B0 2 A(0),
I + ;B0 .
>
"+ 2 3 + A = d=dt ; A0 (A() I ;).
2 + A0 : D(A0 ) Y ! Y | + %14] , +
+
dist((A0 ) iR) > 0 (A0 ) fz 2 C : Rez > g
(30)
+ 2 R. P = P( A0): W ! End Y | +
"+ , 3+ - W 4 " C . O+ (30) A0 "
4 14
. . "+
U(t), t > 0 (U(t) = ft (A0 ), ft () = exp t, 2 C L . %14]).
P1 = P (C ; A0), P2 = P (C + A0), Y = Y1 Y2, Yk = ImPk ,
k = 1 2. U(t)P1, t > 0, U(t)P2, t < 0, +++ )*+
A0 , , "+ " * " %14], , kU(t)k1 6 4M exp 1 t, t > 0, 1 = sup Re , 1 = (A0 ) \ C ; , kU(t)P2k1 6
21
6 4M exp 2 t, t < 0, 2 = inf
Re
,
2 = (A0 ) \ C ; , M = sup kP (H A0)k1 .
22
2
!, + 4+ 1, 1
A;1 .
5. A0 | , ' " , "0 > 0, '# B 2 End F , F = C(R Y ), kBk < "0 ,,) A;B = d=dt;A0 ;B d=dt;A0 ;P1 B0 P1 ;P2B0 P2 ,
# B0 | End F .
. ! ))* A = d=dt ; A0
4 X +
, iR + " A0 , 1
4 A1 = AjF1, F1 = C(R Y1),
Y1 = ImP1, H1 C ; , 4 A2 = AjF2 ,
F2 = C(R Y2), Y2 = Im P2, H2 C + . ( A = End F "4
+
A = A11 A12 A21 A22
, "4
X = X1 X2 . M" "
%6] (
20.3 28.2) , 4+ F0 adA A12 A21 3 4
H1 ; H2 H2 ; H1 (H1 ; H2 = f ; : 2 H1 2 H2 ), 1
+ dist(H1 H2) > 0 F0 . 23
)
I ;: A ! A, 4 I X = P1XP1 + P2XP2 , ;X = F0 (X),
X 2 A12 A21, ;X = 0 + X 2 A11 A22. > (A I ;) + A, 3
= k;k. .3+ 2 3. >
".
\$
. Y | # A0 | \$ # , '#
B 2 Endc F , '# '
4kBk1 < dist(1 2)
1 = (A0 ) \ C ; 2 = (A0 ) \ C + d=dt ; A0 ; B d=dt ; A0 ; P1B0 P1 ; P2B0 P2 ,
# B0 | End F .
. . A0 +++ , 1
)
;: A ! A = Endc F ;X0 = Y12 + Y21 + X0 2 A, Y12 Y21 )
(26), (27) = 0.
15
; " 1 )
k;X0k1 6 maxfkY12k kY21kg 6
6 (1 + 2 );1kX0 k = dist(1 2);1kX0 k, . . 6 dist(1 2);1 .
x
3. \$%,
% %
8. >
5 2. ;. /
%5] + A0 , (, ,
Y ), B 2 End F +++ 4+ ))*
)* B "
C(REnd Y ).
9. G A1 : D(A1) Y ! Y 4
+
+ iR, 2 R, A0 = A1 ; I + +
4 ( 5), d=dt ; A0 ; BA0
(+ kB k) d=dt ; A0 ;
; (P1B0 P1 +P2 B0 P2)A0 , B0 2 A(0), , d=dt ; A1 ; BA0 , d=dt ; A1 ; (P1B0 P1 + P2 B0 P2)A0 .
10. 2 !. D. < %9] (
. 4 ) %10, . I,
x 8]) " ))*
+ +
1))*
(31)
" dx
dt = Ax + "Bx " > 0
A | "+ " End Y B 2 End Y . .+ + , (A) = 1 2 , 1 | , 1 \ 2 = ? 2 |
"
4. 2 1 J
, B 3 A kB kA . G3 4 " 2 "
.
,
A + "B, " > 0, A = LA (Y ),
I X = P1XP1 + P2XP2 , Pi = P(i A), i = 1 2, )
;: A ! End Y
3 " + adA ;X = X ; I X = P1XP2 + P2XP1 , X 2 A (
)
, , %13]). > (A I ;) +++ + A, 1
" 3 "0 > 0, A +"B, " < "0 , A+"P1B0 (")P1 +
+ "P2 B0 (")P2 , B0 : (0 "0) ! A = LA (Y ) | )+ )*+, "+ U(") = I+";B0 ("), 4 +++ )
)* (U : (0 "0 ) ! End Y ). x(t) = U(")y(t) (31) "y(t)
_ = A + "P1B0 (")P1 + "P2 B0 (")P2 , 1 J+ ))* Yi = ImPi, i = 1 2.
16
. . '
1] . . | .: , 1972.
2]
!" #. \$., . %. &'(') *
+,,
!) . ''
. | .: /", 1970.
3] 2+' . 3. 4
'
" 5).
'-
'
. | .: , 1969.
4] 8
. 9., 2+(" 4. :. 3 +' '
*
''
+,,
!) ' "
+(
'" ";,,!
//
,,
!. . | 1979. | . 15, > 5. | 9. 771{783.
5] 2 4. /. (
'" (
'"5 ' '
+
''
. | \$
5+: :+- \$%&, 1972.
6] 8'""
A.
%.
%(
'"
.
|
4
B:
:+- 4%&, 1987.
7] 8'"" A. %. 3 '
+,,
!)5 ' 5(
";,,!
//
A/. | 1992. | . 323,
> 3. |
9. 380{384.
8] A"'
A. A.
''.. . . ".,.-.. | 4
B, 1989.
9] 9. %. A'(
'"
'
// 2"!
\$ . | /'.'": /", 1986. | 9. 206{214.
10] (
'" /.
., 9. %., /5 C . 3
+ 5++"
. | .: /", 1989.
11] 8'"" A. %. D5 ' '
") +,,
!) // 2"!. . 5 B
. |
1996. | . 30, > 3. | 9. 1{11.
12] E
. %
(
'" .(
'" . |
.: , 1985.
13] 8'"" A. %. '
"
'
B
' (
'" // :. A/ 999F. 9
. . |
1986. | . 50, > 3. | 9. 435{457.
14]
,+ /., G!
B. . \$
. . III. | .: , 1974.
) * 2000 .
. . , . . 512.541
: , , , .
!" 2 #: 1) " &
!
" ' 2) "
"#
& .
Abstract
I. H. Bekker, V. N. Nedov, About determinableness of an Abelian group by
its holomorph in the class of all Abelian groups, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 1, pp. 17{25.
For an Abelian group without elements of order 2 the following results were
obtained: 1) a criterion for its determinableness by its holomorph in the class of all
Abelian groups' 2) a criterion for its characteristicness in its holomorph.
:
1) !
2) #
.
%
&
2. ' 1) ( )1] +
,
+#
. -
)2]
,
, . /
, 2),
%
)3], g ! 2g (g 2 G), ,
+
.
: Aut G | G, CG(H) | 3
H G,
t(A) | A (A | 1), H / G
, H | G, Hol G | G,
: A B , | +
A B, Z |
3
.
, 2002, 8, - 1, . 17{25.
c 2002 ,
!
"# \$
18
x
. . , . . 1. ,
1.1. 5 G Hol G def
= hG Aut G +i, 3 +
:
(a ') + (b ) = (a + 'b ') a b 2 G ' 2 Aut G:
1.2. -
, G , G , HolG = Hol G , , G =G.
8
1.1 :
1.3. G Hol G :
1) G def
= h(g ") j g 2 Gi, G | HolG, Aut G def
= h(0 ') j ' 2 Aut Gi = Aut G (" | G).
2) CHol G (G) = G, . . ! G HolG .
1.3 , & ,
+
G G, Aut G Aut G.
1.4 (4]). " G | # 2,
H | Hol G, H | # 2 H = H1 :1, H1 :1 | \$, \$ H , H1 \$% G, '1 g ; g 2 H1
'1 2 :1 g 2 G.
. % (a ') 2 H 2(a ') = (0 "). ; '2 = ", H Hol G, (;g ") + (a ') + (g ") 2 H g 2 H. %&
('g ; g ") 2 H, H | , '('g ; g) = 'g ; g, ,
'2 g = g + 2('g ; g). ; G | &
2, ' = ", , a = 0 H | &
2. <+
+
H = H1 :1 . % (h1 '1) 2 H,
(0 ;") + (h1 '1 ) + (0 ") = (;h1 '1) = (;2h1 ") + (h1'1 ) 2 H, &
(2h1 ") 2 H. < (a ') 2 H (a ')+(2h1 ") = (2h1 ")+(a '),
& '(2h1 ) = 2h1 , , 'h1 = h1, & (h1 ") 2 CHol H (H) = H. ;
(h1 '1) = (h1 ")+(0 '1) (h1 '1) (h1 ") 2 H, (0 '1) 2 H +
H = H1 :1 .
;, H1 G, , (h1 '1 ) 2 H, ' 2 Aut G (0 ') + (h1 '1 ) + (0 ' 1) = ('h1 ''1' 1 ) 2 H, &
'h1 2 H1, , G. = , H Hol G, +
, 'g ; g 2 H ' 2 :1 g 2 G.
0
0
0
;
;
19
1.5. " G = A B Hom(A B) = 0, Aut G ! 0 , 2 Aut A, 2 Aut B , 2 Hom(B A).
. % 2 Aut G. ;, A G, def
= jA 2 Aut A. ?
+
,
= ( j ), A B | 3 G A B . 8
+ , 2 Aut
B, 2 Hom(B A). <
, 2 Aut G 3 0 , .
1.5 +
Aut G 3.
1.6 (2]). "
G | # 2, G |
HolG HolG (. . | Hol G
Hol G ), G 6= G A def:( A : | \$ \$, \$ def
1
1
G G, B = : , B = :. )
:
1) G = A B , G = A B , A A B B (
2) Hom(A B) = 0, Hom(A
B ) = 0( 3) : = "0 Hom("BA) , : = 0" Hom(B" A ) (
4) B = Hom(B A), B = Hom(B A ), A A .
. % 1.4 G | &
2, 1 G = A :, G = A : , , G = A :, G = 1A 1 : ,
A = 1 G \ G A = G \ G , , A = A , G = A B,
G = A B . ; A G, A G,
Hom(A B) = 0, Hom(A B ) = 0. C 1.4 : 0" Hom("BA) ,
: "0 Hom(B" A ) . <+
4). =
B = : = := : =
= 0" Hom("BA) = Hom(B A), B : = Hom(B A ). D
1.6 .
0
0
0
0
;
0
0
0
;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;
;
0
0
0
0
0
0
0
0
0
;
0
0
0
0
0
0
0
0
x
;
0
0
0
0
0
0
0
2. % Hol G Hol G ( | +
), G | &
2, H def
= 1 G . % 1.6 G = A B, G = A B | +
G, G H = A :, : = "0 Hom("BA) .
0
;
0
0
0
2.1 ( !). HolG
0
0
~ HolH , ~ = ( | Hol G Hol H , ! G H).
0
0
20
. . , . . HolG Hol H, 8
< G H
: (0 ') (0 1') ' 2 Aut G :
. Hol G
:
:
0
;1
0
;
0
', ~ = | +
Hol G Hol H. ?
,
+
&
H .
% A B | , 2 Aut A, 2 Aut B. ?
2 Aut Hom(B A), ' = ' (' 2 Hom(B A)),
' = '
(' 2 Hom(B A)). G
, 3
, A B . #
: (Aut A) def
= h j 2 Aut Ai,
def
(Aut B) = h
j 2 Aut B i. I, (Aut A) , (Aut B) | Aut Hom(B A). : + (Aut A) (Aut B) Aut Hom(B A)?
2.2. 5 B Hom(Hom(B A) A) b ! b ,
b ' = 'b (b 2 B, ' 2 Hom(B A)), #
.
2.3. " B = Hom(Hom(B A) A), b ! b , (Aut B) = Aut Hom(B A).
. % 2 Aut Hom(B A), +
: Hom(B A) ! A, #
(') = (')b, b | &
B. ; +
b ! b | ,
(') = (')b = b (') = 'b b 2 B. K +
: b ! b . =
(')b = '
b, +
+
, 1 (')b = '
b, & ' = '
= '
' 2 Hom(B A). ', = = , . . 2 Aut B, , (') = '(
) 2 Aut B. =, , (Aut B) = Aut Hom(B A). D
2.3 .
%
.
" 2.4 (# \$). " G | # 2. * \$
\$ , :
1) G = A B , Hom(A B) = Hom(A Hom(B A)) = 0(
2) B = Hom(Hom(B A) A), %+ (
3) Hom(B A) 6
= B.
. 8
. % 1.6 +
G=AB, 3
Hol G HolG ,
0
0
0
0
;
0
0
0
0
0
0
21
H = A :, : = "0 Hom("BA) . = ~ : Hol G HolH, 2.1, (H +
A Hom(B A)):
A 3 a ! a
"
Hom(Hom(B
A)
A)
B 3 b ! (0 b ) 2 0
"
: 3 ' ! (' ) ' = 0" " ' def
= 2 Hom(B A):
G 1) 3) 1.6 , G 6
= G.
; , , B
= Hom(Hom(B A) A). %+
, & + ~
. <
, (0 ') + (b ") + (0 ' 1) = ('b ; b + b ") ~ (' ") + (0 b) + (;' ") = (' ; b' b), , b ' = ; ' b ' 2 Hom(B A).
K +
b ! ;b , , +
| B Hom(Hom(B A) A).
<. = , +
~ Hol G Hol H 2.1, G = A B, H = A Hom(B A), + (
H +
A Hom(B A)):
A 3 a ! a
"
"
Hom(Hom(B
A)
A)
b
B 3 b ! (0 b )
b 2 0 " 2 0
"
' = "0 '"" ' 2 Hom(B A)
: 3 ' ! (' )
Aut A 3 (0 ) ! (f() )
Aut B 3 (0 ) ! (f(
) ):
'
b ' = ;' b ' 2 Hom(B A). %
, + f(), f(
), . % +
+
~.
; (0 ) + (b ") = (b ")+(0 ), (0 b)+(f() ) = (f() ) + (0 b),
& b f() = f() b 2 B, , f() 2 A.
~
C &
(0 ') &
(0 ), : ' 1 (' ")! ~
, ' 1 (f() )+ (' ")+ (; 1f() 1 ) = ( ' ").
%& :
' = ' ' 2 Hom(B A)!
jA = !
f() 2 A:
0
;
;
;
;
;
22
. . , . . C &
(0 ') &
(0 ), :
' = ' 1 ' 2 Hom(B A), #
+ jA = "
( , (0 ) (a ") ), f(
) = 0 ( , ). <
, 2.3 , Aut Hom(B A) = (Aut B) Aut H = hb j b 2 B 2 Aut A 2 Aut B i. ;
+
, ~ : Hol G ! HolH A 3 a ! a
A) A) B 3 b ! (0 b )
b 2 "0 "b 2 "0 Hom(Hom(B
"
: 3 ' ! (' )
' = 0" '"" ' 2 Hom(B A)
Aut A 3 (0 ) ! (0 ) jA = ' = ' ' 2 Hom(B A)
Aut B 3 (0 ) ! (0 ) jA = " ' = ' 1 ' 2 Hom(B A)
+
. G 6= H :
G = A B, G = A B (Hom(A B)) = Hom(A B ) = 0) G = G , B
6 B,
= B . % Hom(A B) = Hom(A Hom(B A)) = 0 Hom(B A) =
& G 6= H. ;
2.4 .
%
+ , +
.
;
0
0
;
0
0
0
" 2.5 (# \$ \$ %#
&' (& &) , %
G
.
\$ \$
hL
m i
n
L
, G = A B , A =
Ai A , B = Bk , Ai , Bk | i=1
k=1
% 1, Hom(A B) = 0, Hom(A Hom(B A)) = 0 :
1) k 2 1 n ik 2 1 m, %
Hom(Bk Aik ) 6= 0, t(Bk ) t(Aik ) 1 (. . \$\$
1 \$ \$ ), Hom(B Ai ) 6= 0 i 2 1 m,
Hom(B A ) = 0(
2) k 2 1 n Hom(Hom(Bk Aik ) Aj ) = 0, j 6= ik (
n
n
L
L
3) Hom(Bk Aik ) 6
= Bk .
k=1
k=1
. 1) 2) &
2) 2.4, #
+ , A, B | 1, t(A) > t(B) t(A) t(B) 1, B = Hom(Hom(B A) A), #
. % 3) 2.5 &
3) 2.4. ;
2.5
.
%
2.4, + , .
0
0
23
) 2.6. % G = A B , A, B | 1, t(A) > t(B ) (
t(A) | A), B 6 Z, B =6 Hom(B A),
=
t(A) t(B ) 1 (. . 1 +
), G . 5 G , Hol G 6= G , + = HolG , G def
: G = A Hom(B A).
C
, + 3 +
, 2.5. L
) 2.7. % G = A B, B = B , A, B | 1, , 2.6, @ | , G , #
G , Hol G 6= G , +
= HolG , G : G = A Hom(B A).
<
, A | , , )5, . 193],
B
= Hom(Hom(B A) A), #
. %
2.4, .
%
#
, #
.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
@
0
0
0
0
x
3. !
% G | &
2, + , + 1.6.
3.1 (2]). " 2 Aut HolG, H def
= 1G 6= G A, :, A , : |
\$ \$, \$ H G, :
1) G = A B , B def
= 1 : , B 6= 0(
2) Hom(A B) = 0(
3) : = : = "0 Hom("BA) (
4) B : = Hom(B A), A = A.
<
. <
& 1.6, , G = G, 1.6.
% 2 Aut HolG, G | &
2, H def
= 1 G.
3.2. Hol G ~ HolH , ~ def
= ( | Hol G Hol H ,
! G H), #
H .
;
;
0
0
0
0
;
0
0
24
. . , . . <
. <
+
, 2.1, ,
G = G.
" 3.3 (# +(). " G | # 2. * \$% + , :
1) G = A B , B 6= 0, Hom(A B) = 0(
2) B = Hom(Hom(B A) A), %+ (
3) Hom(B A) = B.
<
. 8
. % 1) 3) 3.1. % 2 Aut HolG, H = 1 G 6= G, H = A :, : = 0" Hom("BA) .
C +
+ , H = A Hom(B A) +
~ : Hol G Hol H 3.2 :
A 3 a ! a
"
Hom(Hom(B
A)
A)
B 3 b ! (0 b ) b 2 0
"
: 3 ' ! (' ) ' = 0" " ' def
= 2 Hom(B A):
;
, 2.4, + , B = Hom(Hom(B A) A), #
.
<. ?+
, #
A 3 a ! a
"
Hom(Hom(B
A)
A)
B 3 b ! (0 b) b 2 0
"
: 3 ' ! (' ) b = 0" "b ' = 0" "b b ! b | +
B Hom(Hom(B A) A), #
b ' = ;' b ( ' 2 Hom(B A)), + +
~ : Hol G Hol H | +
Hol G Hol H, H = AHom(B A) (& +
, 2.4 2)). % | Hol G Hol H, 3 G H. ; +
def
= 1 ~ "
Hom(
BA
+
G A :, : = 0 " ) Aut G. ;
, 2 Aut Hol G, G 6= G. ;
3.3 .
" 3.4 (Peremans). " G | g ! 2g (g 2 G), - , , % , \$% + , , :
1) G = A B , B 6= 0(
2) Hom(A B) = 0(
0
;
;
25
3) B = Hom(B A), %+ : B Hom(B A), b 'b , 'b b = 'b(
b ) 2 Aut B \$
b b 2 B .
;
3.3 3.4 &
, G | g ! 2g (g 2 G), .
3.5. " B = Hom(B A), B = Hom(Hom(B A) A), %+
, 2 Aut B ,
% b ! 'b , B
Hom(B A), , % 'b b = 'b(
b ).
. 8
. K +
b ('b ) def
=
def
= 'b (b) b 2 B. ; b 2 Hom(Hom(B A) A), &
b ('b ) = 'b b = 'b (b ). ?+
b b B.
%& 'b b = 'b (
b ). < .
;
2.4 3.3 , + (A B) , :
Hom(A B) = 0, Hom(A Hom(B A)) = 0, B = Hom(Hom(B A) A), #
. ; , , , +
.
0
0
0
0
0
0
0
0
0
0
0
00
0
0
00
0
"
1] Mills W. H. Multiple holomorphs of nitely generated abelian groups // Trans. Amer.
Math. Soc. | 1953. | Vol. 74, no. 3. | P. 428{443.
2] . ., . . !"#\$!% &'(% )** '\$ +,#-. |
.!/: \$1- .!/) ,- &, 1988.
3] Peremans W. Completeness of holomorphs // Yndagations math. | 1957. |
Vol. 19. | P. 608{619.
4] . . 5 )(!"& ,16#&,,% &'(% )** // \$. %/7.
+',. \$&1. 8& !. | 1968. | 9 8. | . 3{10.
5] / :. /,+,% &'(% )**%. .. 2. | 8.: 8#, 1977.
% & 1997 .
. . . . . 519.65.651
: , , , .
!
" #. \$. % & % ' &
( %.
)*
( . #. + " & , & & % ,& , & ' * & % .
Abstract
A. Yu. Golubkov, The tracing of external and internal representation functions
of continuous functions of several variables by superposition of continuous functions
of one variable, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1,
pp. 27{38.
In this article we present e5ective procedures of Kolmogorov's representation of
continuous functions of several variables by superposition of continuous functions of
one variable.
On the one hand, we show internal functions from Sprecher's process in explicit
form, on the other hand, we construct external functions using only the information
about modules of continuity of the initial function of several variables.
. . !3] \$ 2N
X
F(X) = (q (Gq (X))
q=0
, 2002, 8, 7 1, . 27{38.
c 2002 ,
!"
#\$ %
28
. . fGq g, !2], , - :
N
X
Gq (X) = p /(xp + "q)
(0)
p=1
X = (x1 : : : xN ), = (N) = const, " = const, F | - -, F 2 C(!0 1]N ) N > 2, f(q q = 0 : : : 2N g / | , \$ f(q q = 0 : : : 2N g
-4- F , / - F N.
f(q g - !1] 45 fGq g, - (0) !2].
6 4- 7 f(q g fGq g , \$ 8--- - /
f(q g - F \$ 9
- .
6 7 , ,, \$- !2].
4 / ", -45
fGq q = 0 : : : 2N g (0).
| , > 2N + 2, " = ( ; 1);1. :- 4 k i ,
ek (i) = i ;k k = ( ; 2)( ; 1);1 ;k
fEk (i) = !ek (i) ek (i) + k ]; i k > 0g.
< fHk (i);
i k > 0g \$ fJk k > 0g,
- \$ fHk (Jk (i)); i k > 0g -4 4: Hk (Jk (r)) Hk (Jk (l)) 4- r l, r 6= l, k. - - - / 5 fHk (Jk (i)) = /(Ek (i))gki>0 .
< Hk (Jk (i)) = /(Ek (i)) \$, - ,
\$ Hk (Jk (i)) /(Ek (i)) 4 ,.
=- !2], \$ fk g,
4 - - fHk (i)g.
> 0 | \$ \$, (deg()) N. , 0 = 0 1 = 1.
:- , k , \$
D(k) = f; k : : : 0 : : : k g B(k) = f(h1 : : : hN ) 2 D(k)N g n f(0 : : : 0)g.
k , k > 1, k+1 45 -:
...
X
N
p=1
p;1
29
X
N
;
;
;
1
p
;
1
k
+1
k
<
min hp :
B(k)
p=1
? 2 (0 N ;1), , 7 45:
X
N
hp p :
;k+1 6 ;k ;1 min
B(k)
p=1
6- 59 @- !5]: - 4 \$ , deg() = k, \$ Z,
Z(x) = z0 + z1 x + : : : + zk xk , \$ P ,
P(x) = a0 + a1 x + : : : + ad xd , H(P) = max
ja j - fj g j
,: P() = 0, jP()j > C()d H(P )1;k . Qk
Z(x) = zk (x ; ) (x ; j ), \$ C() , - C() =
j =2
Qk
= jzk j;k (1 + jj j);1 .
j =2
N
P
6 \$ deg() = M > N. B\$,
hp p;1 6= 0 p=1
(h1 : : : hN ) 2 B(k) -- X
N
;
;
1
p
;
1
k
min hp > ;Mk ;1 C()N ;1 :
B(k)
p=1
=, , \$, \$
;k+1 < ;Mk ;1C()N ;1 ( ; 1)(N ; 1);1 ;k+1 < ;Mk ;C1 C1 = C1 (N ) > 1 k+1 > Mk + C1 :
: \$ fk g 9 .
,
1
X
Ek = ( ; 2) ;k+j hk (j) = j ;k Hk (j) = !hk (j) hk (j) + Ek ]:
j =1
C , \$ fJk g, ,45 - , \$.
, J0 (i) = i, J1 (i) = i i > 0, fJn g , k, k > 1. :- i0 i0 = i + t, t , \$- 0 : : : ; 1 .
, Jk+1(i0 ) = Jk (i) k+1 ;k + t, t = 0 : : : ; 2, Jk+1(i0 ) = 12 (Jk+1(i + ; 2) + Jk+1 ((i + 1)))
t = ; 1. B ! ] \$ 4 \$ \$.
30
. . - -4 - fJk k > 0g. 45 4 \$ / -
\$ x > 0. :- 7 - 1
X
x = ik ;k :
k=0
| , ik
(1)
B i0
k > 1 \$0 : : : ; 1.
= (i) (ii) !2], - /, - f/(Ek (i)) = Hk (Jk (i)); i k > 0g, --- 45. B\$, f!x k ] ;k g,
-54- k ! 1 x, \$ - /(x) = klim
(J (!x k ]) ;k ):
!1 k
D - (1) - x , \$ !x k ] = i0 k + : : : + ik . C fJk g Jk (i0 k + : : : + ik ) = k ;k;1 Jk;1(i0 k;1 + : : : + ik;1) +
+ ik ;1 12 ( ; 2 + Mk;1 (i0 k;1 + : : : + ik;1)) + ik (1 ; ik ;1 ) (2)
mn | , \$- - Mk;1 -
Mk;1 (i0 k;1 + : : : + ik;1) =
= k ;k;1 (Jk;1(i0 k;1 + : : : + ik;1 + 1) ; Jk;1(i0 k;1 + : : : + ik;1 )): (3)
69 \$-:
J^k = Jk (i0 k + : : : + ik )
M^ k = Mk (i0 k + : : : + ik )
7 M^ 0 = J^0 = i0 .
: - - R, -- j
Y
R(n j) = (im ;1 + im ;2 )
m=n
j > n R(n j) = 1 \$.
6 7 \$- (3) , M^ k;1 = k ;k;1 ik;1 ;1 M^ k;2 + (1 ; ik;1 ;1 )(1 ; ik;1 ;2( ; 1)) +
+ (ik;1 ;2 ; ik;1 ;1 ) 21 ( ; 2 + M^ k;2) : (4)
...
31
: \$ \$9, 7 \$ r+1 ;r M^ r , \$
- (3), (r+1 ; r ) > (N ; 1)r + C1. =, , (4) 1
;
k
k
;
1
^
^
Mk;1 = 2 R(k ; 1 k ; 1)(Mk;2 ; ) + 1 :
C (2) - J^k = k ;k;1 J^k;1 + ik ;1 12 ( ; 2 + M^ k;1) + (1 ; ik ;1 )ik :
6 , \$
kX
;1 1
M^ k;1 = k R(1 k ; 1)21;k +
1 ; 2 R(j j) R(j + 1 k ; 1) k ;j 21;k+j j =1
k
X
J^k ;k = i0 + Pj ;j j =1
Pj = ij ;1 21 ( ; 2 + M^ j ;1 ) + ij (1 ; ij ;1 ):
\$9, jPj ;j j < ( ; 2)N ;1 2;j ;1 + 2;j :
G
45 ,- - - Pj ;j :
Pj ;j = ;j ij (1 ; ij ;1) + ij ;1 21 ( ; 2) +
j ;1 X
+ ij ;1 R(1 j ; 1)2;j +
1 ; 12 R(l l) R(l + 1 j ; 1) ;l 2l;j :
l=1
6 \$ ij 6 ; 2 \$
jPj ;j j 6 ij ;j 6 ( ; 2) ;j \$
(5)
32
. . jPj ;j j 6 1 ( ; 2) ;j + R(1 j ; 1)2;j +
2
j ;1 X
1
+ 1 ; 2 R(l l) R(l + 1 j ; 1) ;l 2l;j : (6)
l=1
( (6) , - R .)
;k (2N + 2);k , k = 1 : : : j , R(m n) 1 = maxR(m n) - (6), 2N + 2 6 , j1 ; 12 R(l l)j 6 12 ( ; 2) l > l l > 1.
C , (6) 45:
j ;1
X
jPj ;j j 6 ( ; 2)(N + 1);j 2;j ;1 + ( ; 2)2;j ;1(N + 1);l + 2;j l=1
jPj ;j j 6 ( ; 2)2;j ;1
j
X
(N + 1);l + 2;j :
l=1
D \$
jPj ;j j < ( ; 2)N ;1 2;j ;1 + 2;j :
(7)
=- (5) (7), \$ 4 .
--- 45- .
/ /(x) = i0 (x) +
1
X
j =1
Pj (x) ;j :
: 7 - 4 , \$ -, - 7 /(x) x - - (1 ; x);1 21 , , .
B, \$ 7 - /
, Pj j - x
\$- .
G fGq q = 0 : : : 2N g \$ \$ \$ - \$ \$ e. , 2 !5].
...
33
< 7 \$ !1].
, k > 0 N- fSkq (Iq )g, q = 0 : : : 2N | , k | , Iq = (iq1 : : : iqN ) - N- \$ .
:- 7 45 fEkq (i)g 45
:
Ekq (i) = !ek (i) ; "q ek (i) + k ; "q]
,
N
Y
Skq (Iq ) = Ekq (iqp ):
p=1
fSkq g
6 !2] , \$ \$ - / fGq ; q = 0 : : : 2N g 4
45 .
1. , k > 0 - , \$ !0 1]N 9- N + 1 ,5 9 k- ffSkq (Iq )g;
q = 0 : : : 2N g.
J fSkqq(Iq ); q = 0 : : : 2N g 45 fEk (i); q = 0 : : : 2N g: , k > 0 , \$ x 2 !0 1] 9
- 2N ,5 9 fEkq (i); q = 0 : : : 2N g. 9 7
.
:, - \$ x 2 !0 1]
x 2 Ekq (i) i q \$, ffx k g + q( ; 1);1 g 6 ( ; 2)( ; 1);1 :
(8)
B f g \$ 4 \$ \$.
<9 !0 1] 45- Tl = !l( ; 1);1 (l + 1)( ; 1);1 )
l , \$- 0 : : : ; 2.
? fx k g 2 T ;2 , q = 1 : : : 2N (8) --, > 2N +2. l 6= ; 2, - fx k g 2 Tl , (8)
-- q = 0 : : : ; 3 ; l q = ; 1 ; l : : : 2N. ?,
q = ; 1 ; l : : : 2N - \$ ; 2 ; l < 2N. \$9
5 .
2. , k > 1 q = 0 : : : 2N - Gq , N
X
Gq (X) = p /(xq + "q)
p=1
34
. . --- -45 - q- !0 1]N , \$ fGq (Skq (Iq ))g -4 4: Gq (Skq (Iq )) Gq (Skq (Iq0 )) 4- Iq 6= Iq0 , Iq Iq0 \$-,
\$ Skq (Iq ) Skq (Iq0 ) !0 1]N .
log(2)
0
0 log(2)= log( ) , C = const.
3. / 2 Lip! log(
) ], . . j/(x) ; /(x )j 6 C jx ; x j
J / , \$, - = 2M ;1 + 2,
M = deg(), -454 Jk (2) Jk (i0 k + : : : + ik ) = k ;k;1 Jk;1(i0 k;1 + : : : + ik;1) +
+ ik ;1 !(1 + ( ; 2);1 Mk;1(i0 k;1 + : : : + ik;1))] + ik (1 ; ik ;1 )
Mk -- (3).
\$- - /0 , /. 7 2 - , 3 9 ;2)
/0 2 Lip! 12 log(
log( ) ].
= 3 /0 \$ / 5 .
: \$ k , - Iq Iq0 ( k) \$ 2 q.
> 2N + 3, \$
Aqk (I) = (i1 ;k ; "q + ;k ( ; 1);1 : : : iN ;k ; "q + ;k ( ; 1);1)
I = (i1 : : : iN ).
69 Tk 45 :
2N
X
Tk : C(!0 1]N ) ! C(!0 1]N ) Tk (F) = (qk (F):
q=0
B Fq | - C(!0
1]N ).
q
q
, (k (F) Gq (Sk (Iq )) F(Ak (Iq ))(N + 1);1 ,
!0 1]N , Gq . = 2 .
: 5- - , 9 !1], 4\$- \$ fkr (F)g1
r=1 , kr (F ) > 1, 5- (I ; (I ; Tkr (F ) ) : : :(I ; Tk1 (F ) ))F ! F r ! 1.
B I | \$ , IF = I.
4 7 \$ - , - fkr (F )g1
r=1 F. :- 7 - - .
G
k(I ; Tk )F kC ( 01]N ) N- \$ \$- \$ !1].
:, \$ j(I ; Tk )F(X)j \$
...
35
X 2 !0 1]N - 1, . .
X
NX
+1
N 0
j(I ; Tk )F (X)j 6 F(X) ;
(qk(i) (F)(X) + (qk (j )(F)(X)
i=1
j =1
fq(i)g | , \$ 5 Iq(i) , - X 2 Skq(i) (Iq(i) ), fq0(j)g | - .
B\$, - Tk , - \$ , \$:
k(I ; Tk )F kC ( 01]N ) 6 N(N + 1);1 kF kC ( 01]N ) + !(F a(k))
(9)
a(k) = ( ; 3)( ; 1);1 ;k (10)
!(F a) = kX ;max
jF(X) ; F(X 0 )j |
X 0 k6a
F. 6 \$ !0 1]N - kX ; X 0 k = max
jx ; x0 j.
fig i i
=45- - !((I ; Tk )F a(r)) \$- r.
D 3 !(Gq a) 6 C(N ; 1)( ; 1);1 alog(2)= log( ) :
a(k) a !(Gq a(k)) 6 C2(N ; 1)( ; 1);1 2;k C2 = const q , \$- 0 : : : 2N.
6 /(Ek (i)) = Hk (Jk (i)) - /, \$, \$ - q = 0 : : : 2N Iq Gq (Skq (Iq )) (jGq (Skq (Iq ))j) (11)
jGq (Skq (Iq ))j = (N ; 1)( ; 1);1Ek :
;
r
r , \$ C22 6 Ek , !(Gq a(r)) 6 jGq (Skq (Iq ))j
(12)
Iq q.
D - Tk , \$ - X X 0 !0 1]N , \$ kX ; X 0 k 6 a(r), (qk (F)(X) = F (Aqk (Iq ))(N + 1);1 + (1 ; )F(Aqk (Iq0 ))(N + 1);1 (13)
(qk (F)(X 0 ) = 0F (Aqk (Iq ))(N + 1);1 + (1 ; 0 )F(Aqk (Iq0 ))(N + 1);1
(14)
0
0
- Iq , Iq 2 !0 1].
36
. . G
j;0j 5 - 59 @- (11). C 0 | 7
,-,
\$,
j ; 0j 6 !(Gq a(r))(C()N ;1 ;k M ; (N ; 1)( ; 1);1Ek );1 :
B \$ C()N ;1 ;k M - , Gq (Iq ) Gq (Iq0 ).
\$ - fk g1
k=0 \$ \$
j ; 0 j 6 C22;r ( (;k M ;C1+1) ; Ek );1 :
(15)
6\$ (14) (13) \$- :
j(qk (F)(X) ; (qk (F )(X 0 )j = j( ; 0)(F(Aqk (Iq )) ; F (Aqk (Iq0 )))j(N + 1);1: (16)
G5 : @- !5] 45- - , \$ \$ - jF (Aqk (Iq )) ; F (Aqk (Iq0 ))j \$ !(F a(r)) - a(r), r > 1. J , 4 , \$ - j ; 0 j r, -45
4 (12).
=, - (15) (16) X X 0 , - kX ; X 0 k 6 a(r) r - C22;r 6 Ek , \$:
(17)
!((qk (F) a(r)) 6 C221;r kF kC ( 01]N ) k+1 (N + 1);1 :
6 Tk (17), :
!((I ; Tk )F a(r)) 6 !(F a(r)) + (2N + 1)(N + 1);1kF kC ( 01])C2 k+1 21;r : (18)
=- !1], , fkr (F)g1
r=1 5 \$.
!1], k1(F ) -- !(F a(k1(F ))) 6 kF kC ( 01N ) =(2N + 2):
? fkn(F )grn=1 - r > 1, kr+1 (F) - !(Fr a(kr+1 (F))) 6 kFr kC ( 01]N ) (2N + 2);1 (19)
Fr = (I ; Tkr (F ) ) : : :(I ; Tk1 (F ) )F. C Fr , (19)
.
G - fkr (F )g, 9 (1), --- - F, (18) -- fFr g.
, \$ ,.
K \$- 5 , \$ fkr (F)g 45 -45 4
C22;kr+1 (F ) 6 Ekr (F ) :
...
69 \$-:
37
!(Fr a(kn(F))) = brn
arn = C2(2N + 1)(N + 1);1 kr (F )+1 21;kn(F )
(20)
0 6 r < n, F0 = F. , br0 = kFr kC ( 01]N ) , arr = N(N +1);1
a0r = 1 0 6 r.
C , (2), fkr (F )g 5-- 5 \$ fb0r g, . . (2) -
F.
:- 7 (9) (18) \$- \$9
, fkr (F )g:
brr+k 6 br;1r+k + arr+k br;10 (21)
br0 6 br;1r + arr br;10:
(22)
G k1(F) ,, (1), . . b01 6 (2N + 2);1b00:
K \$, \$ r > 1 0 6 n 6 r fcnjgnj=0 , \$9 , \$
cnn = 1
0 6 j < n
n
X
cnj =
aknckj ;1:
k=j +1
K , \$, \$ 7 fkl (F)grl=2 , - \$ kl (F ) - 4 l = 2 : : : r - 4
j ;1
l; 1
l;1
X
X
X
(23)
b0l + ajl cjk;1b0k 6 (2N + 2);1 clj;1 b0j j =1
j =0
k=0
l = 1 : : : r ; 1 5 4
C22;kl+1 (F ) 6 Ekl(F ) :
(24)
6 (21), (22) ,- ffcnjgnj=0; 0 6 n 6 rg \$ 0 6 n 6 r
n
X
bn0 6 cnjb0j 0 6 n 6 r ; 1 bnn+1 6 b0n+1 +
j =0
n
X
j =1
ajn+1
j ;1
X
cjk;1b0k :
k=0
38
. . I. 6 kr+1 (F ) (23), (24), l r + 1
r .
n n
II. fcrj +1 grj +1
=0 ffcj gj =0 ; 0 6 n 6 rg fajr+1g. 7 - \$- , fcnj; n 6 rg, n r + 1.
6, - , \$ 0 6 j < l ajl ! 0
l ! 1 (. (20)) b0l ! 0 l ! 1 (10) F.
G9 (2) br0 6 ((2N + 1)(2N + 2);1)r b00
- 9 , \$ br0 6 (2N + 1)(2N + 2);1br;10 45 - (1).
kr (F) 9 7
. 6 7 \$ .
, 6. 6. 6. N. C \$, 5 , .
1] . . . . "- !!
!"#\$ % // '!" . . | 1963. | . 18, . 5. |
0. 55{92.
2] Sprecher D. A. On the structure of continuous functions of several variables // Trans.
Amer. Math. Soc. | 1965. | Vol. 115. | P. 340{355.
3] . . 5 "6!
" ""
% "! """ 6" !"# ""
% 6 """ !7"\$ // 8 000. | 1957. | . 114, . 5. | 0. 953{956.
4] 6 . :. 5 "6!
" % "! """ 6" !"# % " ;" <! """ // . !
"="". | 1958. |
. 3. | 0. 41{61.
5] >< . :., "!"" ?. ., @6
!% . B. "6"" "C <!". |
.: :#6-
!
! "!", 1995.
& ' 1997 .
. . . . . . . 658.512
: , , , ,
.
, !" # . \$ #% &' ' , & ! ! ), "! ), %)
. . !"*
&
&" " #" & ! # ", )+ !. ,%+* )" "" %". , +*
.
Abstract
A. A. Gorsky, B. Ya. Lokshin, A mathematical model of goods production
and sale for production supervision and planning, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 1, pp. 39{45.
The paper presents a nonlinear dynamicalmodel of goods productionand sale. It
can be used for production planning and as a submodel of more complicated models
of economics, for instance in models of cooperation and competition. The appropriate strategy of production guarantees the desired features of the processes and
maximizes the return. Optimisation can be performed by standard control design methods. Several practical examples with interesting dynamic behavior are
presented.
, 2002, 8, 4 1, . 39{45.
c 2002 !",
#\$
%& '
40
1. . . , . . , !
" " #. % # (., ,
'1, 2]). , " -. / #, /
. . 0
. . . 0
" , . #.
2. 1. " " / / ", , .
%
z_ = u ; n(c)(Y ; v)z ; k1z
v_ = n(c)(Y ; v)z ; k2 v
w_ = cn(c)(Y ; v)z ; u ; k3z
/
(1)
u | " ( # , # )5
z | 5
v | ( /)5
w | (
". "
# )5
Y | #. ( , / . / )5
c | # (c > 1, . 1)5
k1 | !!# 5
k2 | !!# / # 5
k3 | " # / # 5
n(c) | !!# .
0
Y , k1 , k2, k3 . 6 u, c ( " #) " " / . , . 0
z , v, w !#
/ ". " .
0
, |
. 7
. " 41
. 8 , , (1) . !!
#. , /
. 9/ . '1].
:" . , /
" , .". " !
", . ( .) . " . . ", , " " #.
0
" . .. , . /
. , .". .
z_ = u ; n(c)(Y ; v) a +z z ; k1z
v_ = n(c)(Y ; v) a +z z ; k2v
w_ = cn(c)(Y ; v) a +z z ; u ; k3z
(2)
a | .
0
. ", . . # "
. 0
", ./ .
# " !# , / .
/
3. 0 (1), , " " u " # c. 6 " " / # /
. . , . 7, u c / #
, z = z0 = const > 0 v = v0 = const > 0
/ z0 , v0 " / ,
v0 < Y .
7 ", "# . .
< ". ", #. ( "
), . . u = m(Y ; v)z=(a + z ), / m > n, " 42
#
:
. . , . . z1 = 0 v1 = 0
z2 = kk1 Yn +; kak1 v2 = n Yn +; kak1 :
2
2
2
(>. " = m ; n(c).) 0
/.
0
< ak1=4 , #, / ". 6
" . > ak1=4,
/ . 6 "# . !
/ .
7 ", " !
# , .
#.
4. !
"
" #
6 . # " ",
!
# , . " "#
!
./ /
/ . 0
/
"# / " : ".
" | " , . | . 7 /
uk+1 = uk + @w
@u u=uk / | , k | # " ".
6 " " . .:
@w = w(uk ) ; w(uk;1) :
@u
uk ; uk;1
? " , " " ". 0
. ./ " . 5 .
0
/ / ., # (1): "" (1) "
. ".
< " " ,
. . " "# / /
" .
5. # % 43
A" ! ./ " " " (1). A" -. " "
"-" !! / " . , .
".
0
, #, /
/ !# "
". 7 . /
!# / #. , , /". 0
# " / , / B"
C " !
: !
# /
!# .
6" ./ . #. h. D Y + h ; v, !# Y + h ; v ; z . 1. / " (1) h:
z
a + z ; k1 z
v_ = n(c)(Y + h ; v) a +z z ; k2v
h_ = (c)h(Y + h ; v ; z )
w_ = cn(c)(Y + h ; v) a +z z ; u ; k3 z
z_ = u ; n(c)(Y
/
+ h ; v)
(3)
| !!#.
E (3) #
h = h0 = 0 z = z0 v = v0 / z0 v0 | #
(1) / . (3) 5 ". /
" , " z ..
D (3) . / #
h = hr z = zr v = vr zr + vr > Y . (> zr , vr
" / / ). , , . ." / ".
44
. . , . . 6. 6 " . !
. 0
" !!
, / , /
/ / . " / , .,
. , " / (1) v_ = n(c)(Y ; v)z ; k2v(Y ; v):
6
!
" "
/ . D ". "
. , " ud = w_ = 'cn(c)(Y ; v)z ; ud ; k3z ]
/ ud | " . ", | !!#,
ud = d'cn(c)(Y ; v)z ; k3z ]
/ m = 1=(1 + ). >
" . " / T u_ + u = ud :
F . " " !!
#. z_ = u ; n(c)(Y ; v)z ; k1z
v_ = n(c)(Y ; v)z ; k2v(Y ; v)
T u_ + u = d'cn(c)(Y ; v)z ; k3z ]:
(4)
%
, . ." . .
6 .
/ (4)
zn+1 = zn + h(un ; n(c)(Y ; vn )zn ; k1zn )
vn+1 = vn + h(n(c)(Y ; vn )zn ; k2vn (Y ; vn))
un+1 = un + h d'cn(c)(Y ; vn)Tzn ; k3zn ] ; un :
(5)
H ", (5) . . #. 6 , . # " : Y = 2, n(c) = 02,
k1 = 0, k2 = 005, k3 = 002, d = 03, T = 5, h = 1.
> "
, , . " " (
" . ", "
) " : 45
, B"C , # . 0
!# /
.". " / #.
'
1] . ., . ., . ., . . !
! "! #\$, %! &\$ // #. ().
*!% ! !!\$. | 1992. | *. 3. | . 190{193.
2] . . 1\$!\$ !! ! 2 !
. | 1.: . ,
1976.
( ) 1997 .
. . , . . , . . . . . 517.947.42
: , , .
! " # # \$"% #"&% % . ' (" )\$ %, \$ | . + ",-. &% # %
# / . - /#.
0 & ( #"&% % ( 1, ) #" "\$"
.
Abstract
A. S. Gosteva, N. Ch. Krutitskaya, P. A. Krutitskii, A mixed problem in a magnetized semiconductor lm with two periodic systems of cuts, Fundamentalnaya i
prikladnaya matematika, vol. 8 (2002), no. 1, pp. 47{60.
The boundary-value problem for harmonic functions is considered in the exterior
of straight line periodiccuts in a plane. The Dirichlet boundarycondition is speci8ed
on one side of each cut and the skew derivative boundary condition is speci8ed on
another side of each cut. An explicit solution of the problem is obtained with the
help of the theory of analytic complex functions. The uniqueness of the solution
is proved. The problem describes the electric current from straight line periodic
electrodes in a semiconductor 8lm placed in a constant magnetic 8eld.
, 8] ! ! . # ! \$ %&! . '
\$ (, | ! !. * ! \$ !
.
9! ) \$" 9::; 02-01-01067, ;<=>+ YSF 00-17.
, 2002, " 8, ? 1, . 47{60.
c 2002 !",
#\$
%& '
48
. . , . . , . . 1. ,
' , - w = (w1 w2). 1 -,
! \$-. ! &! B, !
(w1 w2). 2 B \$ & B - Ow3. 3, '! 4{7]
div J = 0 J = 7E E = grad U:
(1.1)
9
- J = (J1 J2) | - , U | & '
,
E = (E1 E2) | - '
, 7 | , ! ! ! (') 7 = 1 + 2 ;1 1 | - ,
= B, | - !. ; \$ -, |
, 6= 0.
; (w1 w2) ,
- Ow1 c 2 Ow2 \$ , &. ; - L1 :
+
1
N1
L1 =
L1(2k) L1 (2k) = L1n (2k)
L1n (2k) = fw: w1
k=;1
2 (a1n b1n)
n=1
w2 = 2kg, n = 1 : : : N1, k = 0 1 2 : : :.
* - L2 :
L2 =
+
1
k=;1
L2 (2k + 1) L2(2k + 1) =
N2
L2 (2k + 1)
n=1
n
L2n (2k + 1) = fw: w1 2 (a2n b2n) w2 = (2k + 1)g, n = 1 : : : N2 ,
k = 0 1 2 : : :. #
- \$ - L,
L = L1 L2 . - \$ L1 \$ - Ow1, L2 | \$.
NSm
#
- (amn bmn ) Ow1 \$ \$- Lm ,
n=1
m = 1 2. ( , N1 + N2 > 1.
, - L. *! \$ L1 \$ >+?, ! | >;?. ( L2
\$ ! \$ >;?, ! | >+?. , ! L1 &, ! |
49
-! :
U(w1 2k + 0) = q1+ (w1 ) w 2 (L1 )+ J2 (w1 2k ; 0) = ;q1; (w1) 1 + 2 w 2 (L1 ); k = 0 1 2 : : :.
;\$, ! L2 -! , ! | &:
J2 (w1 (2k + 1) + 0) = ;q2; (w1) 1 + 2 w 2 (L2 ); U(w1 (2k + 1) ; 0) = q2+ (w1 ) w 2 (L2 )+ k = 0 1 2 : : :.
* - & U(w) = U(w1 w2) %&!, 2-
w2, . . , U(w1 w2) = U(w1 w2 + 2k) k = 0 1 2 : : ::
(1.2)
A H0L . 1 -, %& U(w) H0L , :
1) U(w) L, L+ L; , & LC
2) Uw1 (w), Uw2 (w) L, L+ L;
, & L, - \$
, . . "
" > ;1 A > 0, jUw1 j jUw2 j 6 Ajw ; dj jw ; dj ! 0, d | -! &
LC
3) U(w) (1.2) \$
:
jU j < constC jUw1 j = o(1) w1 ! 1:
(1.3)
9
- o(1) \$ - %&, . ; ! (1.1) ! ! .
K. ;! %& U(w) H0L , L U(w1 2k+0) = q1+ (w1) w 2 (L1)+ (1.4a)
Uw2 (w1 2k ; 0) ; Uw1 (w1 2k ; 0) = q1; (w1 ) w 2 (L1); (1.4b)
Uw2 (w1 (2k+1)+0) ; Uw1 (w1 (2k+1)+0) = q2; (w1 ) w 2 (L2); (1.4c)
U(w1 (2k+1) ; 0) = q2+ (w1) w 2 (L2)+ (1.4d)
k = 0 1 2 : : :.
50
. . , . . , . . 3
\$
& L H0L
\$
& L \$
.
\$ \$-, \$ q1+ (w1) 2 C 1(LF1),
+
q2 (w1 ) 2 C 1(LF2), q1; (w1) 2 C 0(LF1), q2; (w1) 2 C 0(LF2), LFm | - Lm Ow1, m = 1 2, 2 (0 1]. 2
C 1(LFm ) \$ %&!, %%& LFm , - C C 0(LFm ) | %&!,
- LFm .
9, , -
- - %& q1+ (w1) q2+ (w2 ), (1.4a) (1.4d) - ' :
(1.5a)
Uw1 (w1 2k + 0) = (q1+ )0 (w1 ) w 2 (L1 )+ +
0
2
+
Uw1 (w1 (2k + 1) ; 0) = (q2 ) (w1 ) w 2 (L ) (1.5b)
+
U(a1n 2k) = q1 (a1n ) n = 1 : : : N1
(1.5c)
+
2
2
U(an (2k + 1)) = q2 (an ) n = 1 : : : N2
(1.5d)
k = 0 1 2 : : :.
1 ( ). K .
. * !
K -, K - - . - D | \$
-, -! ! DF
! %& Z
2
krU kL2(D) = U @U
@ n dl
@D
n | - @ D.
Dd0 = fw: ; d < w1 < d w2 2 ; 2 ; ]g, 2 (0 ), d > 0
.
- U0 | ! K. -
- -
%& U0 , - \$
Dd0 n L , L = L1(0) L2 (1). G
>;? >+? \$ \$- - %&! \$ . Z
@U0
2
krU0 kL2(D10 ) = dlim
!1 0 U0 @ n dl =
@ (Dd nL )
= dlim
!1
Z
@Dd0
b1n
N1 Z @U ; @U + X
@U
0
U0 @ n dl +
U0 @w0 ; U0 @w0 dw1 +
2
2
w2 =0
n=1 1
an
51
+
2
N2 Zbn
X
@U0 + ; U @U0 ; U0 @w
0 @w
2
n=1 a2n
2
dw1:
w2 =
,
:
Z
0
lim
U0 @U
d!1
@ n dl =
@Dd0
= dlim
!1
Zd d
Z @U0 @U0 ;U0 @w
dw
+
U
dw
1
0 @w 1 +
2 w2 =;
2 w2 =2;
;d
;d
2Z;
@U
0
;U0 @w 1
2Z;
@U0 dw = 0
+ dlim
dw
+
U
2
0
!1
@w1 w1 =d 2
w
1 =;d
;
;
2-
%& U0 w2, d ! 1 ! \$
(1.3). , %& U0 (w) (1.4),
:
1
N1 Zbn
X
n=1 a1n
+
=
=
@U0 ; ; U @U0 + U0 @w
0 @w
2
2
w2 =0
dw1 +
N2 Zbn
X
+ ; @U
@U
0
0
U0 @w
; U0 @w
dw1 =
2
2
w2 =
2
n=1 a2n
N1 Zbn
X
@U0 ; U0 @w
2 1
n=1 a1n
w2 =0
N1 Zbn
X
n=1 a1
n
N
1
X
1
N2 Zbn
X
2
dw1 ;
n=1 a2n
@U0 ; U0 @w
2
w2 =
dw1 =
; N2 Zbn @U ; X
@U
0
U0 @w dw1 ; U0 @w0 dw1 =
1
1
w2 =0
w2 =
n=1 a2
2
n
1 fU ; (b1 0)g2 ; fU ; (a1 0)g2] ;
0 n
0 n
n=1 2
N2
X
; 21 fU0;(b2n )g2 ; fU0;(a2n )g2] = 0
n=1
U0 (w) & , H0L , ! (1.4)
=
52
. . , . . , . . U0; (b1n 0) = U0+ (b1n 0) = 0 U0; (a1n 0) = U0+ (a1n 0) = 0 n = 1 : : : N1
U0; (b2n ) = U0+ (b2n ) = 0 U0; (a2n ) = U0+ (a2n ) = 0 n = 1 : : : N2:
0 . G #-, krU0kL2 (D10 ) = 0, . . U0 const D1
0
(1.4) , U0 0 D1 . G , U0 0 ! . - K !!, .
2. , K \$ - , ! ! %&! ,{H-\$
%&! 1{3].
; !
- (w1 w2) W = w1 + iw2 . * h0L %&!. 1 -, %& F(W) h0L , - % ! L, \$
F (W) = o(1), jw1j ! 1, F(w1 + iw2 ) = F (w1 + i(w2 + 2k)), k = 0 1 2 : : :. I& - %! ! L, L
, & L, - \$
1].
- U(w) | K. ,
! ! J{,, \$ , %& V (w) %& K(W) = U(w) +
+ iV (w). L F(W) = KW (W) = Uw1 ; iUw2 \$ ! ! %&!. 9, F(W ) = ;E1 + iE2 & '
E = (E1 E2). 9 K ,{H-\$ - %& F(W).
R. ;! %& F(W ) h0L , :
Re F (w1 + i(2k + 0)) = (q1+ )0 (w1) w1 2 (L1 )+ Re(; + i)F (w1 + i(2k ; 0))] = q1; (w1 ) w1 2 (L1 ); Re(; + i)F (w1 + i((2k + 1) + 0))] = q2; (w1 ) w1 2 (L2 ); Re F (w1 + i((2k + 1) ; 0)) = (q2+ )0 (w1) w1 2 (L2 )+ k = 0 1 2 : : :.
,
% \$ W = w1 + iw2 - Z = x1 + ix2 , Z = eW , W = Ln Z. J L ; = ;1 ;2 , 53
L1 ) ;1 =
N1
;1n ;1n = fx: x2 = 0 x1 2 ea1n eb1n ]g
n=1
N2
L2 ) ;2 = ;2n ;2n = fx: x2 = 0 x1 2 ;eb2n ;ea2n ]g:
n=1
2 ;1, ;2 ; \$ \$- Ox1. % \$ %& F (W) O(Z) = O(eW ). (
+0
1
w1
1
+
Q (x1 ) = (q1+ )0(ln x1) x1 2 ;2 x1 = e w1 w1 2 L 2
(q2 ) (ln jx1j) x1 2 ; x1 = ;e w1 2 L (
;
1
w1
1
Q;(x1 ) = q1; (ln x1) x1 2 ;2 x1 = e w 1 w1 2 L 2
q2 (ln jx1j) x1 2 ; x1 = ;e w1 2 L :
, - Z - ;. 2 ;+ \$-
! \$ , ;; | !. 9, % \$ L+ ;+ , L; | ;; . 9 - \$ ; Ox1.
; Z h0;. 1 -, %& O(Z) h0;, - % ! ;, O(1) = 0, ; , &, \$
.
- F(W) | R, %& O(Z) = F(W) = F(Ln Z)
h0; Z ! ,{H-\$.
R;. ;! - % %& O(Z) ! ;, h0;, ;
Re O+ (t) = Q+ (t) t 2 ;
Re(; + i)O; (t)] = Q;(t) t 2 ;
- : O(0) = 0. G
>+? >;? \$
- - %
%&! ;+ ;; .
. * O(0) = 0, \$ O(Z).
P
O(Z) - % ! ; O(Z) 2 h0;, %F - % ! ;, & O (Z) = O(Z)
0
h; !
-! 1,3]
O (t) = O (t):
54
. . , . . , . . R; ;+ : O+ (t) + O+ (t) = 2Q+ (t)
;; : (; + i)O; (t) + (; ; i)O; (t) = 2Q;(t)
\$ , R; 3,8]. ( ' %& '1 (Z) = O(Z)
F '2 (t) = '1 (t),
'2 (Z) = O (Z) = '1 (Z),
t 2 ;.
C = ; ;++i i g(t) = 1
; (t)
f1 (t) = 2Q+ (t) f2 (t) = 2Q
+ i t 2 ;
% , R;.
S. ;! '(Z) = ('1(Z) '2 (Z)), ! '(Z) h0; ;
'+1 (t) = ;g(t)';2 (t) + f1 (t) '+2 (t) = ;Cg(t)';1 (t) ; f2 (t):
* %& 1(Z) = '1 (Z), 2(Z) = ;'2 (Z), % S 1, 2. ' \$
!, !, 1+ (t) = g(t)2; (t) + f1 (t) 2+ (t) = Cg(t)1; (t) + f2 (t)
t 2 ;:
Q 8], 1 Z j (t) dt t 2 ; j = 1 2:
j (z) = 2i
t;z
+ A2(t0 ) ; A1 (t0)]e i2 t0 2 ;
1 + 2
; (t0 )e;i Q
+
; A2(t0 ) + A1 (t0)]e; i2 t0 2 ;
2 (t0 ) = Q (t0 ) + p
2
1+
1 (t0 ) = Q+ (t0 ) +
L
Q; (t0 )ei p
A1 (t0) = A1 (t0) + i sin 2 H 1(t ) PN1 1 +N2 ;1 (t0)
1 0
;
Z
sin
1
A1 (t0) = H (t2 ) H1(t) pQ (t) 2 + Q+ (t) t ;dtt 1+
1 0
0
;
A2 (t0) = A2 (t0) + i cos 2 H 1(t ) PN2 1 +N2 ;1(t0 )
2 0
;
cos 2 Z
1
A2 (t0) = H (t ) i H2 (t) pQ (t) 2 ; Q+ (t) t ;dtt 2 0
0
1+
;
(2.1)
55
-
\$
H1(t) =
N2
Y
n=1
H2(t) =
N2
Y
n=1
N1
Y
n=1
jt ; ea1n j1; 2 jt ; eb1n j 2 sign(t ; ea1n ) jt + ea2n j 2 sign(t + eb2n )
jt + eb2n j1; 2 2
N1
Y
n=1
jt ; ea1n j 21 ; 2 jt ; eb1n j 21 + 2 sign(t ; ea1n ) jt + eb2n j 12 ; 2 jt + ea2n j 12 + 2 sign(t + eb2n ):
2 PN1 1 +N2 ;1(t0 ) PN2 1 +N2 ;1(t0 ) \$ - N1 + N2 ; 1. * 2 (0 ) \$:
cos = p 2 sin = p 1 2 :
1+
1+
L \$, -%& '(Z) = (1(Z) ; 2(Z)) S. * 3] R; %!
1 Z (t0 ) dt (2.2)
O(Z) = 21 (1(Z) ; 2(Z)) = 2i
t0 ; Z 0
;
; i
(t0 ) = 12 (1 (t0) + 2 (t0 )) = Q+ (t0) + Qp (t0 )e 2 + (A2 (t0 ) ; A1 (t0))e i2 +
1+
cos
sin + ie i2 H (t2 ) PN2 1 +N2 ;1(t0 ) + e i2 H (t2 ) PN1 1 +N2 ;1(t0 ) t0 2 ;:
2 0
1 0
2 PN1 1 +N2 ;1 (t0 ), PN2 1 +N2 ;1 (t0) \$ N1 +N2 ;1
- !
- '%%&, . .
PNm1 +N2 ;1 (t0) =
jm
N1 +X
N2 ;1
j =0
jm tj0 m = 1 2
| '%%&.
I& O(Z) R;, O(0) = 0, ' Z
(2.3)
(t) dtt = 0:
;
56
. . , . . , . . Q \$! '%%& , (t). , (2.3) , #(ZZ ) - % ! ; Z = 0. ( #(ZZ ) O(Z) = 1 Z (t0) dt (2.4)
Z
2i t ; Z t
;
0
0
(2.2), % #& - %! %& #(ZZ ) !
;, ! \$
1].
\$ % \$. *
- - W = Ln Z, F(W ) = O(Z) = KW (W):
C, W | - , W0 | - %
, -
- (2.4), K(W) =
ZW
W0
F(W) dW =
Z
ZZ(W )
Z (W0 )
O(Z) dZ = 1 Z (t )
0
Z
2i
Z
;
ZZ(W )
Z (W0 )
1 1
t ; Z t dZ dt =
(t) ln Z(W ) ; t dt = ; 1 (t) ln(Z(W) ; t) dt + const =
t Z(W0 ) ; t
2i
t
;Z
;
1 (t) ln(eW ; t) dt + const :
= ; 2i
t
1
= ; 2i
;
* - \$. ln(eW ; t) = ln R(w t) + i(w t)
R(w t) = jeW ; tj = jew1 +iw2 ; tj = jew1 cos w2 + iew1 sin w2 ; tj =
= (ew1 cos w2 ; t)2 + e2w1 sin2 w2] 12 :
I& (w t) c - 2k (k = 1 2 : : :) %
w1 cos w2 ; t
ew1 sin w2 :
cos (w t) = e R(w
sin
(w
t)
=
t)
R(w t)
(w t) \$ - - %
- '! %&, t ;. 9, (w t) | %&.
! K. * 3] '! \$ - 1 Z Im(t) ln R(w t) + Re (t)(w t)] dt + D (2.5)
U(w) = Re K(W) = ; 2
t
;
57
D | ,
;
Re (t) = Q+ (t) + Qp(t) cos2 ; sin 2 Im A2(t) ;
1+
1
P
(t) PN2 1 +N2 ;1(t) N
1 +N2 ;1
; cos 2 A1 (t) + cos 2 sin 2
H1(t) ; H2(t)
;
Im(t) = Qp(t) sin2 + cos 2 Im A2 (t) ;
1+
1
(t)
P 2 2 ;1(t) :
2 ;1
; sin 2 A1(t) + sin2 2 PN1H+N(t)
+ cos2 2 N1H+N(t)
1
2
9
- Im A1 (t) = 0, Im A2 (t) = ;iA2 (t), Re A2 (t) = 0.
*
(w t) %& U(w) !. ( U(w) \$, , ! 3]:
Z
Re (t) dtt = 0 n = 1 2 : : : Nm m = 1 2:
(2.6)
m
;n
G \$
(1.3) :
Z
Im (t) dtt = 0:
;
(2.7)
;, - \$ (2.3). 9, ! (2.6), (2.7), . . (2.6), (2.7), (2.3) .
I& U(w) 2(N1 + N2) + 1 - , \$ '%%& jm , j = 0 : : : N1 +N2 ; 1, m = 1 2,
D. Q \$- \$ \$, \$
%& U(w) (1.5c), (1.5d) (2.6), (2.7). 3 ' , 2(N1 +N2 )+1
! \$
! - 2(N1 +N2 )+1 . (!
-, U(w) (2.6),
N1 + N2 :
N1 +X
N2 ;1
j =0
Z
1m 1 ;
Knj
j
N1 +X
N2 ;1
j =0
2m 2 = cm n = 1 : : : Nm m = 1 2
Knj
j
n
(2.8)
j
t dt
pm =
Knj
Hp(t) t m=1 2 n=1 : : : Nm j =0 : : : N1 + N2 ; 1 p=1 2C
;
Z ; (t0 ) cos Q
1
+
m
Q (t0)+ p 2 ; sin 2 Im A2 (t0 ) ; cos 2 A1 (t0) dtt 0 :
cn = ; cos sin 2
2 ;m
n
1+
0
58
. . , . . , . . U(w) (1.5c), (1.5d), N1 + N2 :
N1 +X
N2 ;1
j =0
V1njm j1 +
N1 +X
N2 ;1
j =0
V2njm j2 + D = nm m = 1 2 n = 1 : : : Nm (2.9)
(
Z
2
j
p
= ; 2 Ht (t) lnR(amn (m ; 1) t)] dtt p = sin 2 2 p p = 1
cos 2 p p = 2
p
;
1 Z ln R(am (m ; 1) t ) nm = 2
0
n
;
sin
;
p
Q (t0) + Im A2(t0 ) cos 2 ; A1 (t0 ) sin 2 dtt 0 +
2
1+
0
(
+
1
+ q1+ (an2 ) p m = 1
q2 (an ) p m = 2:
! (2.9) \$ - (2.6),
-, - c - 2k (k = 0 1 2 : : :)
(a1n 0 t) = 0 t < ea1n (a1n 0 t) = t > ea1n (a2n t) = 0 t < ;ea2n 2
(an2 t) = t > ;ean :
;&, (2.7), :
Vpm
nj
N1 +X
N2 ;1
j =0
j1 j1 +
N1 +X
N2 ;1
j =0
j2 j2 = W
(2.10)
Z
tj0 dt0 Hm (t0) t0
;
Z ;
Q
(t
)
0
p
W=;
sin + Im A2(t0 ) cos 2 ; A1 (t0 ) sin 2 dtt 0
2
0
1+
jm = m
;
m (m = 1 2) .
#
2(N1 + N2 ) + 1 ! (2.8), (2.9), (2.10) -
2(N1 +N2 )+ 1 nm , D (n = 0 : : : N1 +N2 ; 1, m = 1 2) - I - 9, c. 60]. , 59
- - . (-
. - mn D | - ! (2.8), (2.9),
(2.10), N1 +X
N2 ;1 1
P N1 +N2 ;1 (t) =
1ntn n=0
N1 +X
N2 ;1 P 2N1 +N2 ;1 (t) =
2ntn n=0
sin 2 1
i
(t0 ) = i cos 2 P 2
2 :
(t
)
+
P
(t
)
e
0
0
N
+
N
;
1
N
+
N
;
1
1
2
1
2
H2(t0 )
H1(t0 )
I& U(w), % (2.5) - (t0 ) ! K. * 1 U(w) 0.
#-, (2.4),
Z O(Z)
Z
O(Z) = U w1 ; iU w2 = Z Z = 2i t(t;0Z) dtt 0 0
0
0
;
Z = eW , -
Z Z
2i
;
0 ) dt0
(t
t0 ; Z t0 0:
0) 0. 3
' % #& ;, (t
!
- - -, ,
cos 2 6= 0 sin 2 6= 0 H1;21(t0 ) 6= 0 t0 2 ;
N1 +X
N2 ;1 N1 +X
N2 ;1 1n tn0 0
2n tn0 0:
n=0
n=0
A
! \$
mn = 0 (n = 0 : : : N1 + N2 ; 1 m = 1 2):
*
-
- - (2.9), D = 0.
L
m \$, , n D (n = 0 : : : N1 +N2 ; 1, m = 1 2) | - !
! (2.8), (2.9), (2.10). #-, (2.8), (2.9), (2.10) - - , -
I - (2.8), (2.9), (2.10) \$! ! .
60
. . , . . , . . . (2.8){(2.10), 2(N1 + N2 ) + 1 2(N1 + N2 ) + 1 .
*- D '%%& PN1 1 +N2 ;1 (t0 ), PN2 1 +N2 ;1 (t0 ) (2.5) (2.8){(2.10). L- %& U(w) (2.5) . ; -, U(w) 2 H0L .
J , U(w) - K. L %&, (2.5), \$ K,
'%%& PN1 1 +N2 ;1 (t0), PN2 1 +N2 ;1(t0 ) D
! ! ! (2.8){(2.10),
. #
2. q1+ (w1 ) 2 C 1(LF1 ), q2+ (w1 ) 2 C 1(LF2 ), q1; (w1) 2 C 0(LF1),
;
q2 (w1) 2 C 0(LF2), 2 (0 1], K , ! " (2.5), D #""\$ PN1 1 +N2 ;1 (t0 ), PN2 1 +N2 ;1 (t0 ) (2.8){(2.10), .
1] . . . | .: , 1968.
2] !" #. \$. % &'(
. | .: #
&)
&, 1963.
3] %
+
, -. .. / &'( "", 0"
&"'", ' 0""
&&)
'"
0)", & , &'( // . )"'
"
. | 1990. | 3. 2, 4 4. |
. 143{154.
4] !8" . .., %
+
, -. .. /8 "'", ", &'(, &", 0"
)
9
("" " (& &)
(, 0"0""'
// . )"'
"
. | 1989. | 3. 1, 4 5. | . 71{79.
5] %
+
, -. .. / 9
("" " 0)"
, 9"'"
&)
(", 0"0""'
"", 0<
// => #. | 1990. | 3. 30,
4 11. | . 1689{1701.
6] ?"(-?
( >. @., %
" . =. #
&
0"0""'
". | .: ,
1990.
7] >'
)
" >. >., >"" .. #., ,
" A. B. -&) 0"0""'
". |
.: .")
&', 1979.
8] !" .. ., %
+ . C., %
+
, -. .. ) &'( &)
(", 0"0""'
"", 0<
&&)
'" 0)", // #'). 0
.
). | 2000. | 3. 6, 0. 4. | . 1061{1073.
9] %
+ . C., D
.. .. @
, 8. | .: > ", 1985.
( ) 1998 .
. . . . . 533.6.011.5+532.526+541.2
: , " " #\$ %, " &"
".
' \$ ( \$ \$ )* #"
#" \$ #\$+ ,\$, + \$\$%+ \$\$(* \$ . ( \$\$ %\$\$ ", #\$, %+, *#*&+ "+ \$\$%"+ ," %.
Abstract
V. L. Kovalev, Catalytical surface boundary conditions for Martian atmospheric entry, Fundamentalnayai prikladnayamatematika, vol. 8 (2002), no. 1, pp. 61{69.
Boundary conditions for a catalytic surface in a dissociated Martian atmosphere
are obtained on the basis of ideally-adsorbed Langmuir theory. The reaction based
on Eley{Rideal shock mechanism and the Langmuir{Hinshelwood reaction based on
the recombination of two adsorbed species are taken into account.
. , !" #
\$ !%\$
& ' & 2{3 &+, & ,1]. / ' 0 1 ,2{4].
0 + .
3 4
& & 0 & & 5 5 ,5{7]. 3 1.
, 2002, 8, 2 1, \$. 61{69.
c 2002 !,
"#
\$% &
62
. . 9 & , 1, & & &-
(
& 5 ) ,2{4].
1. & + ,8]
( Vn )w (ci ; c1i ) + Ji = Ri :
(1.1)
4& ci , Ji | 55 @ Ri | & 5@ , Vn | & & @ w + , 1 + .
A
& +, ( Vn )w = 0 (1.1) Ji = Ri:
(1.2)
5, 0
.
95, 5 C
{9
:
(O ; S) + O ! (S) + O2 v1s = O xO ; K1 0 xO2 1
1
s
(N ; S) + N ! (S) + N2 v2 = N xN ; K 0 xN2 2
s
(O ; S) + CO ! (S) + CO2 v3 = O xCO ; K1 0 xCO2 3
1
(CO ; S) + O ! (S) + CO2 v4s = CO xO ; K 0 xCO2 4
1
s
(O ; S) + C ! (S) + CO v5 = O xC ; K 0 xCO 5
1
s
(C ; S) + O ! (S) + CO v6 = C xO ; K 0 xCO :
6
95 5-5 CO 5:
O + (S) ! (O ; S) v7s = 0 xO ; pK1 O 7
1
N + (S) ! (N ; S) v8s = 0 xN ; pK N 8
1
s
0
C + (S) ! (C ; S) v9 = xC ; pK C 9
!
63
s = 0 xCO ; 1 CO :
CO + (S) ! (CO ; S) v10
pK
10
95 F {G+
&:
s = 2 ; 1 pxO (0 )2 v11
2
O K
11
s = 2 ; 1 pxN (0 )2 (N ; S) + (N ; S) ! N2 + 2(S) v12
2
N K
12
s = O C ; 1 pxCO (0 )2
(O ; S) + (C ; S) ! CO + 2(S) v13
K13
s = O CO ; 1 pxCO (0 )2 :
(O ; S) + (CO ; S) ! CO2 + 2(S) v14
2
K14
(O ; S) + (O ; S) ! O2 + 2(S)
95 5-5 O2 , N2 , NO, CO2 5:
s = 0 xO ; 1 O O2 + (S) ! (O2 ; S) v15
2
pK15 2
s = 0 xN ; 1 N N2 + (S) ! (N2 ; S) v16
2
pK16 2
s = 0 xCO ; 1 CO CO2 + (S) ! (CO2 ; S) v17
2
pK17 2
s = 0 xNO ; 1 NO :
NO + (S) ! (NO ; S) v18
pK
18
4& Aj , (Aj ; S) | , (S) | . 9 5 vis @ p | , xi | 55,
i , 0 | .
H 0
Ki , ki ki; 5 + ;
Q
D
D
k
i
i
+
;
;
i
i
Ki = ; = Ai exp( R T ) ki = Bi exp R T ki = Bi exp R T ki
A
A
A
Qi , Di+ , Di; , RA | 5, 0 5
5, @ Ai , Bi+ , Bi; |
05
& '
.
3 & 5 ,9,10]:
64
. . RO = ;mO p(k1v1s + k4 v4s + k6v6s + k7v7s )
RN = ;mN p(k2v2s + k8 v8s )
s ) + mCO k13vs RCO = ;mCO p(k3 v3s ; k5v5s ; k6v6s + k10v10
13
s
s
s
RCO2 = mCO2 p(k3 v3 + k4v4 ) + mCO2 k14v14
(1.3)
s = 0
R(O2 ;S ) = mO2 pk15v15
s = 0
R(N2;S ) = mN2 pk16v16
s = 0
R(CO2 ;S ) = mCO2 pk17v17
s = 0:
R(NO;S ) = mNO pk18v18
3 5 & ' . "
&, + ,12]
s + k13vs + k14vs = 0
R(O;S ) = k1v1s + k3v3s + k5v5s ; k7v7s + 2k11v11
13
14
s
s
s
R(N;S ) = k2v2 ; k8v8 + 2k12v12 = 0
s = 0
R(C;S ) = k6 v6s ; k9v9s + k13v13
s + k14vs = 0
R(CO;S ) = k4v4s ; k10v10
14
(1.4)
s
R(O2 ;S ) = mO2 pk15v15 = 0
s = 0
R(N2;S ) = mN2 pk16v16
s = 0
R(CO2 ;S ) = mCO2 pk17v17
s = 0:
R(NO;S ) = mNO pk18v18
J
5 (1.4) & (1.3) . 0
&& +
Na
X
0 + i = 1:
(1.5)
i=1
2. ', 5 5-5 CO , O2 , N2 , NO, CO2 ,11,12]. / O = pK7 0 xO N = pK80 xN C = pK9 0 xC CO = pK100 xCO (2.1)
O = N2 = CO2 = NO = 0:
C (1.5) 0 = 1 + pK x + pK x 1+ pK x + pK x :
(2.2)
7 O
8 N
9 C
10 CO
!
65
3+ &+ 5 55-5 :
O2 + M ! 2O + M
v1 = Kpp1 xO2 ; x2O N2 + M ! 2N + M
v2 = Kpp2 xN2 ; x2N CO2 + M ! CO + O + M v3 = Kpp3 xCO2 ; xCO xO CO + M ! C + O + M v4 = Kpp4 xCO ; xCxO NO + M ! N + O + M v5 = Kpp5 xNO ; xNxO :
4& M | & 5.
3 5 & 5 vis ' & vi , 5, vis , 5
5-5 , (1.3) &
5 (1.4). H , +
Kp1 = K 1K 2 Kp2 = K 1K 2 Kp3 = K K1 K 2 Kp4 = K K1 K : (2.3)
11 7
12 8
7 12 8
7 9 13
/ , s ; k13 vs ; k14 vs )
RO = ;mO (p(2k1v1s +k3v3s +k4 v4s +k5v5s +k6 v6s ) ; 2k11v11
13
14
s
s
RN = ;2mN (pk2 v2 + k12v12)
(2.4)
s ; k14vs )
RCO = ;mCO (p(k3v3s + k4v4s ; k5v5s ; k6v6s ) + k13v13
14
s ):
RCO2 = mCO2 (p(k3 v3s + k4v4s ) + k14v14
K
vis c (2.1) (2.3) K
K
p
1
p
2
s
0
2
s
0
2
v1 = pK7 xO ; p xO2 v2 = pK8 xN ; p xN2 K
K
p
3
p
3
s
0
s
0
v3 = pK7 xO xCO ; p xCO2 v4 = pK10 xO xCO ; p xCO2 K
K
p
4
p
4
s
0
s
0
v5 = pK7 xO xC ; p xCO v6 = pK9 xO xC ; p xCO K
K
p
2
p
1
s
2
2
0
2
2
s
2
2
0
2
2
v11 = p K7 ( ) xO ; p xO2 v12 = p K8 ( ) xN ; p xN2 K
K
p
4
p
3
s
2
0
2
s
2
0
2
v13 =p K7 K9 ( ) xO xC ; p xCO v14 =p K7 K10( ) xO xC ; p xCO2 :
66
. . "
&, ' 5 Ri :
K
p
1
2
0
2
0
2
RO = ;mO p 2(k1K7 + k11K7 ) xO ; p xO2 +
K
p
4
0
+ (k5K7 + k6K9 + k13K7 K9 ) xO xC ; p xCO +
K
p
3
0
+ (k3K7 + k4K10 + k14K7 K10 ) xO xCO ; p xCO2 K
p
2
2
0
2
0
2
RN = ;2mN p (k2K8 + k12K8 ) xN ; p xN2 K
p
3
2
0
0
RCO = ;mCO p (k3K7 + k4K10 ; k14K7 K10 ) xO xCO ; p xCO2 +
K
p
4
0
+ (k13K7 K9 ; k5K7 ; k6K9 ) xO xC ; p xCO K
p
3
2
0
0
RCO2 = mCO2 p (k3 K7 + k4 K10 + k14K7 K10 ) xO xCO ; p xCO2 :
(2.5)
3. 9 5.
3 5 5 C
{9
, & 5 RO = ;mO p2 0 2k1K7 x2O ; Kpp1 xO2 + (k5K7 + k6K9 ) xO xC ; Kpp4 xCO +
+ (k3K7 + k4K10) xO xCO ; Kpp3 xCO2 RN = ;2mN p20 k2K8 x2N ; Kpp2 xN2 RCO = ;mCO p20 (k3K7 + k4K10 ) xO xCO ; Kpp3 xCO2 ;
; (k5K7 ; k6K9 ) xO xC ; Kpp4 xCO RCO2 = mCO2 p20 (k3 K7 + k4 K10) xO xCO ; Kpp3 xCO2 :
(3.1)
3 5 5 F {G+
& !
67
K
p
1
RO = ;mO
xO2 +
p
K
K
p
3
p
4
+ k13K7K9 xO xC ; p xCO + k14K7 K10 xO xCO ; p xCO2 K
p
2
2
0
2
2
2
RN = ;2mN p ( ) k12K8 xN ; p xN2 (3.2)
K
p
4
2
0
2
RCO = ;mCO p ( ) k13K7 K9 xO xC ; p xCO ;
K
p
3
; k14K7K10 xO xCO ; p xCO2 K
p
3
2
0
2
RCO2 = mCO2 p ( ) k14K7 K10 xO xCO ; p xCO2 :
4, (T < 3000 K)
5 Kpi 1.
a) &+ 5
& pi = pxi 5 5 (pi xi Ki 1), 5 5
& .
, 5 5 pxO K7 pxNK8 pxiKi , i = 9 10, RO = ;mO p(2k1xO + k5xC + k3xCO ) RN = ;2mN pk2xN
(3.3)
RCO = ;mCO p(k3xCO ; k5xC ) RCO2 = mCO2 pk3xCO :
A
5 5 C
(pxCK9 pxiKi , i = 7 8 10) CO (pxCO K10 pxiKi ,
i = 7 8 9), &
(3.4)
RO = ;mO pk6xO RN = 0 RCO = mCO pk6xCO RCO2 = 0
RO = ;mO pk4xO RN = 0 RCO = ;mCO pk4xO RCO2 = mCO2 k4 xO :
(3.5)
1 F {G+
& 0 5.
A
pxO K7 pxNK8 pxiKi , i = 9 10, RO = ;2mO k11 RN = ;2mN k12 RCO = 0 RCO2 = 0:
(3.6)
3 , 5 C (pxCK9 pxiKi ,
i = 7 8 10) 5 CO (pxCO K10 pxi Ki , i = 7 8 9)
& RO = RN = RCO = RCO2 = 0:
(3.7)
p2 (0 )2
2k11K72
x2O ;
68
. . b) 5 5 (pi Ki 1). 3 0 5, C
{9
, 5 F {G+
&, 5.
3 C
{9
5 :
RO = ;mO p2(2k1K7 x2O + (k5 K7 + k6K9 )xO xC + (k3K7 + k4K10 )xO xCO )
RN = ;2mN p2 k2K8 x2N
(3.8)
RCO = ;mCO p2((k3 K7 + k4K10)xO xCO ; (k5 K7 ; k6 K9 )xO xC)
RCO2 = mCO2 p2 (k3K7 + k4K10)xO xCO :
3 F {G+
& RO = ;mO p2(2k11K72 x2O + k13K7 K9 xO xC + k14K7 K10 xO xCO )
RN = ;2mN p2 k12K82x2N (3.9)
RCO = ;mCO p2(k13K7 K9 xO xC ; k14K7 K10 xO xCO )
RCO2 = mCO2 p2 k14K7 K10 xO xCO :
. & F , 5 & . K
& 5 5, 5 5.
, 5
& 5 5. 5
& C
{9
5, F {G+
& .
, 5 .
M ', ' 5 1 , & & 0
, 0
& 5 .
1] . ., . ., . . . !"#
\$% &
\$
%&
%& \$\$ // ()*+. ,-+. .%. | 1987. | 3 676.
!
69
2] Chen Y.-K., Henline W. D., Stewart D. A., Candler G. V. Navier{Stokes solution
with surface catalysis for Martian atmospheric entry // Journal of Spacecraft and
Rockets. | 1993. | Vol. 30, no. 1. | P. 32{42.
3] Mitcheltree R. A., Ggno;o P. A. Wake <ow about the Mars Path=nder entry vehicle // Journal of Spacecraft and Rockets. | 1995. | Vol. 32, no. 5. | P. 771{776.
4] Gupta R. N., Lee K. P., Scott C. D. Aerotermal study of Mars Path=nder aeroshell //
Journal of Spacecraft and Rockets. | 1996. | Vol. 33, no. 1. | P. 61{69.
5] . ., ? . ,. " ## & #
\$&@ // + \$ \$
!"
\$." #" %& &"#
& . | .:
+- *E, 1987. | ?. 58{69.
6] . ., ? . ,. " ## & #
\$&@ %
"\$ \$ // +. F),. I*. | 1996. | 3 5. | ?. 179{190.
7] . ., ). O., \$ ). )., Q
. +. ) T" #
& ", \$%@U& #
\$& %
"\$ " \$ // +. F),. I*. |
1996. | 3 6. | ?. 133{144.
8] -
*. ). E \$& % " "\$%&
"& // ""
"&
. | 1961. | -. XXV, 3 2. |
C. 196{208.
9] Langmuir I. Monolayers on solids // J. Chemical Society. | 1940. | Vol. 4. |
P. 511.
10] -"
. +. %& [%& // &" [%& #
& . | .: ,
, 1970. | C. 57{72.
11] -"\$
O. * % . F
\$%& "
\$&
" // , \$& . %\$. 1. | .:
, 1977. | ?. 235{284.
12] Wise H., Wood B. J. Reactive colisions between gas and surface atoms // Adv. in
Atomic and Mol. Phys. N. Y. | 1967. | Vol. 3. | P. 291{353.
' ( 1997 .
DQDB-
. . . . . 519.248.2+519.248:62
: DQDB-, - , , , ! , " , # "\$
% &'( ((, )\$ !\$ " %,
!&* .
+ ( (( (&\$ DQDB- (Distributed Queue Dual Bus), &!. & / ( 0(" 1 !. 2& (& & &/*, 1& /! ! "( .
3 4( / !/( "\$ % ,
(' 0/"\$. & &( DQDB-, (&\$5 & /) &'1 & . 6'( !/\$( '
(, &' / )\$ !
" % / &/1 /!.
Abstract
V. G. Konovalov, A mathematical model of the DQDB protocol, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 71{83.
In this paper we consider a mathematical model of the Distributed Queue Bus
(DQDB) protocol intended for data ;ow control in communication networks. We
assume that station's input tra=c is a renewal process. The proposed Markov chain
is the most adequate model for the DQDB protocol among those already studied in
several articles. We derive a positive recurrent condition in the case of two stations.
1. Distributed Queue Dual Bus (DQDB) . 1990 802.6 \$ % & & '( (IEEE). )* +,DQDB-
% .
/ %
, 1 .
DQDB-
% . 2 * 1 3, +
, 2002, ( 8, > 1, . 71{83.
c 2002 ,
!"
#\$ %
72
. . + , * . 5 , *, +, DQDB-
, %
62{4].
:+
, * *
+, , 1 *. 2 ,3 % , ;,- , . < , ,* , % +.
2. = - , DQDB-
( . 61,2,4]). ) DQDB- - -
3 (A B), ; . >% 3 (
1 , ?1 -; *
- *, ,- DQDB- ).
A
A
-
?. 1. @// ', \$!/( & &'1 DQDB-
>%* %, , +* 3* A, 3* % B. : ; ; + * (*) , ;* . : &, * 3 , % , , &
, % ; * (, - 3 A, | *
3 B). C % , ,
* , ( ) * . 3 A , A(i), i |
+ 3 A. 3 B , B(i), i | + 3 B ( 1 3 B ; % 3 A). )
3 DQDB- . 2.
\$, DQDB-
% ,
-; . F %* * *, +, & . G ,*3* 73
DQDB-
A
A
?6
1
2
?6
-
?6
B
B
?6
-
d
?. 2. DQDB-\$
DQDB-
, ,, 1 C 1(1) C 2(1), , ,3,
+.
>%* % , *: , 1 , . 1/. > * , 1 C 1(1) ; 1 , . : & 1 C 1(1) %* +,
3 B , ;* , ( ,3 0) ,3 +
%* , 3 A * ( ;* , ,3
1 C 1 (1) %* , 0).
A(i)
-
C 2
0
- C2
C1
C 1
1,
A(i) !,
1,
B (i) B(i)
?. 3. ? # ", 1&* &( > , * *, . 1/. 2 & 1 C 1 (1) 1 C 2 (1), C 1 (1) 0. 2 74
. . , , 1 C 2(1) ,3
+ (
) * , 3 A * . > , 1 C 2(1) 0, % , 1 3 A -;* * . 2 % , 1 ,3 1 C 2 (1) , 1 C 1 (1) + *
, 3 B , ;* .
A(i)
-
C 1
1,
B (i) -
C2
C1
C 2
1,
A(i) !
B(i)
?. 4. ? # ", 1&* A. #C
: , + ;-
, . ( , 1 C 1 (1) 1 C 2(1), 0, . .
2* * , 1 ;- , , ; 3 B
, * 3 A, ; . K % ;* , %1
, * , * ; . L .
)
3 DQDB-
% % . 5.
= , % *
, * - . L + % , ,
. . , ; , , -; (, ; 3 *). \$ DQDB- ;,- 1*
+
=, - * * .
21 -; .
Xk (j) | , 33 j-* k,
j = 1 2, k > 0.
Sm (j) = Sm(k) (j) | * m- , 33 j-* (1 < m < Xk (j)) k, j = 1 2, k > 0. 5 Xk (j)
75
DQDB-
C 1 =) C 2 * C 1 := 0*
Q=0
-
Q>0
'
#
(
3 I
Y
C 1 := C 1 + B (i) ; (1 ; A(i))
C 2 := C 2 ; (1 ; A(i))
C 1 := C 1 + B (i)
Q>0
Q=0
)
z
C2 > 0
'
#)
(
C2 = 0
"
#
! ! ?. 5. ? DQDB-
- 1 * , k , Sm(k) (j) % ,.
XP
k (1)
XP
k (2)
k =
Sm (1) +
Sm (2) | *, m=1
m=1
33 * * k.
Qk (j) | j- k, j = 1 2, k > 0.
Ck1(1) | k 1, % %-; + % * % , k > 0.
Ck2(1) | k 1, % %-; , ,
, % , 1 ;* , k > 0.
Ak (i), 0 1 , , ,* *, i-* + 3 A k, i = 1 : : : d, k > 0.
Bk (i), 0 1 , , ,* , i-* + 3 B k,
i = 1 : : : d, k > 0.
d | % , % .
P, -; , -
-;* :
76
. . Qk+1(1) = (Qk (1) ; I fCk2(1) = 0g)+ +
Qk+1(2) = (Qk (2) ; (1 ; Ak (d)))+ +
XX
k (1)
m=1
XX
k (2)
Sm (1)
Sm (2)
m=1
Ck1+1(1) = (1 ; Hk (1))((Ck1 (1) ; I fQk (1) = 0g)+ + Bk (1))
Ck2+1(1) = (Ck2 (1) ; 1)+ + Hk (1)((Ck1(1) ; I fQk (1) = 0g)+ + Bk (1))
Hk (1) = I fCk2 (1) = 0gI fQk (1) 6= 0gI f(Qk (1) ; 1)+ + Xk (1) 6= 0g +
+ I fQk (1) = 0gI fXk (1) 6= 0g
Ak+1 (1) = I fCk2(1) = 0gI fQk (1) 6= 0g
Ak+1 (i) = Ak (i ; 1) i = 2 3 : : : d
Bk+1 (d) = (1 ; Ak (d))I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) 6= 0g +
+ I fQk (2) = 0gI fXk (2) 6= 0g
Bk+1 (i) = Bk (i + 1) i = 1 2 : : : d ; 1:
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
: & ,,
M = fQk (j) Ck1 (1) Ck2(1) Ak (i) Bk (i) j = 1 2 i = 1 : : : dg
(2.10)
, fXk (j) k > 0 j = 1 2g fSm (j) m > 0 j = 1 2g | , 1 * ,
+
, =.
), , %,- , +
= M~ = M J, -;* * +
M . J
k + 1 -; :
Jk+1 = Jk + I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) = 0g ;
; I fQk (2) = 0gI fXk (2) 6= 0g: (2.11)
:% % J0 = 0. : Jk , : ;1, 0, 1. : &, ,* k = 0 , Jk : 0 ;1, ,* , *, Jk
0 1.
5
, 1 +
,- = M~ -;- ,,:
M~ = M J = fQk (j) Ck1(1) Ck2(1) Ak (i) Bk (i) Jk j = 1 2 i = 1 : : : dg:
(2.12)
3. * +* S % +* '(x y): R+ R+ n K ! 60 1], K | , -; -;
*:
DQDB-
77
1) 0 6 '(x y) 6 1,
2) '(x + x1 y + y1 ) ; '(x y) 6 C1(x y) max(x1 y1 0) x1 y1 > ;1,
3) '(0 y) = 0, '(x1 y + y1 ) 6 C2(y y1 )x1 x1 > 0, y1 > ;1,
4) '(x 0) = 1, 1 ; '(x + x1 y1 ) 6 C3 (x x1)y1 y1 > 0, x1 > ;1,
5) + C1(x y), C2(y y1 ), C3(x x1) - x y ! 1.
1. S .
. T+ '(x y) = px2x+y2 % R+ R+ n
n 60 2] 60 2] % * S .
), * 1) , 0 6 '(x y) 6 1.
> , (x y) 2= K = 60 2] 60 2] x1 y1 > ;1 x + x1
'(x + x1 y + y1 ) ; '(x y) = p
; p 2x 2 6
2
2
+ y1 )
x +y
8< p x1 (x+ x1) + (y
x1 y1 > 0
2
2
6 : px j+xy1 j+jy1 j
(3.1)
x
=
;
1
y
=
;
1:
1
1
2
2
(x;1) +(y;1)
L, + '(x y) % U
3+ %
. < %* * (x0 y0) U
3+ % ,, C1(x0 y0) = px21+y2 . 2 ,
y2
0
0
j'0x(x y)j = (x2 + y2 )3=2 6 p 21 2 x +y
j
xy
j
6p 1 :
j'0 (x y)j =
y
(x2 + y2 )3=2
x2 + y2
(3.2)
(3.3)
5 ,
j'(x + x1 y + y1 ) ; '(x y)j 6
6 j'(x + x1 y + y1 ) ; '(x y + y1 )j + j'(x y + y1 ) ; '(x y)j 6
6 j'0x( y + y1 )j jx1j + j'0y (x !)j jy1j
| , %; 6x x + x1 ], ! | 6y y + y1 ]. P % (4.2) (4.3), -;+:
j'(x + x1 y + y1 ) ; '(x y)j 6 p jx1j2+ jy1j 2 :
(3.4)
(x ; 1) + (y ; 1)
5 * 2) % .
5 % , -; * 3) 4).
x1
'(x1 y + y1 ) = p
(3.5)
6 x1 (x + x1 )2 + (y + y1 )2 y ; 1
78
. . p
2
2
1 ; '(x + x1 y1) = 1 ; p x + x21 2 6 (x + x1 )x ++ yx1 ; (x + x1) =
1
(x + x1 ) + y1
s
2
= 1 + (x +y1x )2 ; 1 6 x +y1x 6 x y;1 1 (3.6)
1
1
p 2
1 + x 6 1 + x x > 0. ), % (4.1), (4.5)
(4.6) - x y ! 1. 5 , 5) %
.
2. A :
d
X
i=1
A1 (i) ;
d
X
i=1
A0 (i) = I fC02(1) = 0gI fQ0(1) 6= 0g ; A0(d):
(3.7)
. <, & 3 (3.6)
(3.7):
d
X
i=1
A1 (i) ;
d
X
i=1
A0 (i) =
= A1 (1) ; A0 (d) +
d
X
(A1 (i) ; A0 (i ; 1)) = A1 (1) ; A0 (d) =
i=2
= I fC02(1) = 0gI fQ0(1) 6= 0g ; A0(d)
, ,.
3. k > 0 :
Bk+1 (d) + Jk+1 = Jk + (1 ; Ak (d))I fQk (2) 6= 0g:
(3.8)
. < , (4.8) , % (3.8) (3.11), - - , :
Bk+1(d) + Jk+1 = (1 ; Ak (d))I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) 6= 0g +
+ I fQk (2) = 0gI fXk (2) 6= 0g + I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) = 0g +
+ Jk ; I fQk (2) = 0gI fXk (2) 6= 0g:
5 , I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) = 0g = 1, - Ak (d), , , 3
,3, 1, , , ;* 3 A ,
*, I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) = 0g =
= (1 ; Ak (d))I fQk (2) 6= 0gI f(Qk (2) ; 1)+ + Xk (2) = 0g:
,, % Bk+1 (d) + Jk+1 = Jk + (1 ; Ak (d))I fQk (2) 6= 0g:
79
DQDB-
4. 1. fXk(j) k > 0 j = 1 2g, fSm (j) m > 0 j = 1 2g |
"# \$# %# %, "# . ' :
(i) 0 < EX0 (j) < 1 0 < ES0 (j) < 1 j = 1 2V
(ii) E0 = EX0 (1)ES1 (1) + EX0 (2)ES1 (2) < 1:
( M~ = fQk (j) Ck1 (1) Ck2(1) Ak (i) Bk (i) Jk j = 1 2
i = 1 : : : dg ' ".
. L, Ck1(1), Ck2(1), Ak(i), Bk (i), Jk ,
i = 1 : : : d, - k - :
0 6 Ak (i) 6 1 0 6 Bk (i) 6 1 i = 1 : : : d
(4.1)
0 6 Ck1(1) 6 d + 1 0 6 Ck2(1) 6 d + 1 ;1 6 Jk 6 1 - k. (4.2)
W -;- - +-:
V = V (Q(1) Q(2) C 1(1) C 2(1) J A(1) : : : A(d) B(1) : : : B(d)) =
d
X
1
2
= Q(1) + Q(2) + '(Q(1) Q(2)) C (1) + C (1) + J + B(i) +
X
i=1
d
+ (1 ; '(Q(1) Q(2)))
i=1
A(i) (4.3)
+ '(x y) % * S (. 1).
:;* * * + A V = E 0 + (Q0 (1) ; I fC02(1) = 0g)+ ; Q0(1) + (Q0(2) ; (1 ; A0 (d)))+ ;
; Q0 (2) + '(Q1 (1) Q1(2)) (C01(1) ; I fQ0 (1) = 0g)+ + (C02 (1) ; 1)+ +
+ B0 (1) +
d
X
i=1
B1 (i) +J1 ; '(Q0(1) Q0(2))
X
d
+ (1 ; '(Q1 (1) Q1(2)))
i=1
C01(1) + C02 (1) +
d
X
i=1
B0 (i) +J0 +
X
d
A1 (i) ; (1 ; '(W0 (1) W0(2)))
i=1
A0(i) :
(4.4)
21 -; : '(Q1 (1) Q1(2)) = '1, '(Q0 (1) Q0(2)) = '0 .
5, 2, 80
. . A V = E 0 + (Q0 (1) ; I fC02(1) = 0g)+ ; Q0 (1) + (Q0(2) ; 1 + A0(d))+ ;
; Q0(2) ; A0 (d) +I fC02(1) = 0gI fQ0 (1) 6= 0g + '1
+ (C02 (1) ; 1)+ + B0 (1) + J1 +
; '0 C01 (1) + C02 (1) + J0 +
5 S 0 , d
X
i=1
d
X
B1 (i) ;
B0 (i) ;
i=1
E0
d
X
i=1
d
X
i=1
(C01 (1) ; I fQ0(1) = 0g)+ +
A1 (i) ;
A0 (i) :
(4.5)
< 1, - " ; -
Ef0 V 0 > S 0 g < ":
(4.6)
1 < ":
S 00
(4.7)
> , & % " ; S 00 , :%
S = maxfS 0 S 00 g:
(4.8)
: S (4.6), (4.7).
W ;* 1 .
1: Q0(1) 6= 0, Q0(2) 6= 0.
d
X
i=1
A V = E 0 ; 1 + '1 C01 (1) + (C02(1) ; 1)+ + B0 (1) + J1 +
;
d
X
i=1
A1 (i)
; '0 C01 (1) + C02 (1) + J0 +
d
X
i=1
B0 (i) ;
d
X
i=1
B1 (i) ;
A0 (i)
=
( A1(i + 1) = A0 (i), i = 1 2 : : : d ; 1, B1 (i ; 1) = B0 (i), i = 2 3 : : : d)
= Ef0 ; 1g + E ('1 ; '0 ) C01(1) +
d
X
B0 (i) ;
dX
;1
A0(i)
i=2
i=1
2
+
+ Ef'1((C0 (1) ; 1) + J1 + B1 (d) + B0 (1) ; A1 (1)) ;
; '0(C02 (1) + J0 + B0 (1) ; A0 (d))g =
+
( 3)
d
d;1
X
X
1
= Ef0 ; 1g + E ('1 ; '0 ) C0 (1) + B0 (i) ; A0(i) +
i=2
i=1
+ Ef'1((C02(1) ; 1)+ + J0 + 1 ; A0 (d) + B0 (1) ; I fCk2(1) = 0g) ;
; '0(C02 (1) + J0 + B0 (1) ; A0 (d))g =
81
DQDB-
C01(1) + C02(1) + J0 +
d
X
d
X
= Ef0 ; 1g + E ('1 ; '0 )
B0 (i) ; A0 (i) 6
i=1
i=1
(
(4.1) (4.2))
6 E0 + (3d + 3)Ef'1 ; '0g ; 1 =
= Ef0V 0 > S g + Ef0 V 0 6 S g + 3(d + 1)Ef('1 ; '0)V 0 > S g +
+ 3(d + 1)Ef('1 ; '0 )V 0 6 S g ; 1 6
(S , (4.8)V , + '(x y) * 1){5) 1)
6 E0 + " + 3(d + 1)" + 3(d + 1)SC1 (Q0 (1) Q0(2)) ; 1 6
6 E0 ; 1 + (3d + 4)" + 3(d + 1)SC1 (Q0 (1) Q0(2)):
(4.9)
2: Q0(1) = 0, Q0(2) 6= 0.
d
X
i=1
A V = E 0 ; 1 + '1 (C01(1) ; 1)+ + (C02 (1) ; 1)+ + B0 (1) + J1 +
;
d
X
i=1
A1(i) ; '0 C01(1) + C02(1) + J0 +
d
X
i=1
B0 (i) ;
d
X
i=1
A0 (i)
B1 (i) ;
=
( , Q0(1) = 0 1 C02(1) % 0)
= Ef0 ; 1g + E ('1 ; '0 )
X
d
dX
;1
B0 (i) ; A0 (i) +
i=2
i=1
1
+
+ Ef'1((C0 (1) ; 1) + J1 + B1 (d) + B0 (1) ; A1(1)) ;
; '0 (C01(1) + J0 + B0 (1) ; A0 (d))g =
(
3)
X
d
dX
;1
B0 (i) ; A0 (i) +
= Ef0 ; 1g + E ('1 ; '0 )
i=2
i=1
1
+
+ Ef'1((C0 (1) ; 1) + J0 + 1 ; A0(d) + B0 (1) ;
; '0 (C01(1) + J0 + B0 (1) ; A0 (d))g =
= Ef0 ; 1g + E ('1 ; '0 ) C02(1) + J0 +
+ Ef'1((C01 (1) ; 1)+ + 1) ; '0 C01(1)g =
= Ef0 ; 1g + E ('1 ; '0 )
+ Ef'1I fC01 (1) = 0gg 6
d
X
i=1
B0 (i) ;
C01(1) + C02(1) + J0 +
d
X
d
X
i=1
i=1
A0 (i)
B0 (i) ;
d
X
i=1
+
A0 (i)
+
82
. . (, S , (4.8)V +, , 1 (. (4.9), + ,))
6 E0 ; 1 + " + 3(d + 1)" + 3(d + 1)SC1 (0 Q0(2)) + " + Ef'1V 0 6 S g 6
6 E0 ; 1 + (3d + 5)" + 3(d + 1)SC1 (0 Q0(2)) + C2 (Q0(2) S)S:
(4.10)
3: Q0(1) 6= 0, Q0(2) = 0.
A V = E 0 ; A0 (d) + '1 C01(1) + (C02(1) ; 1)+ + B0 (1) + J1 +
;
d
X
i=1
A1(i)
; '0 C01(1) + C02(1) + J0 +
d
X
i=1
B0 (i) ;
= Ef0 ; 1g + Ef1 ; A0(d)g + E ('1 ; '0 ) C01(1) +
d
X
i=1
d
X
i=2
d
X
i=1
B1 (i) ;
A0 (i)
B0 (i) ;
d;1
X
i=1
=
A0 (i) +
+ Ef'1((C02 (1) ; 1)+ + J1 + B1 (d) + B0 (1) ; A1(1)) ;
; '0 (C02(1) + J0 + B0 (1) ; A0 (d))g =
(
3)
d
dX
;1
X
= Ef0 ; 1g + Ef1 ; A0(d)g + E ('1 ; '0) C01(1) + B0 (i) ; A0 (i) +
i=2
i=1
2
+
2
+ Ef'1((C0 (1) ; 1) + J0 + B0 (1) ; I fCk (1) = 0g) ;
; '0 (C02(1) + J0 + B0 (1) ; A0 (d))g = Ef0 ; 1g + Ef1 ; A0 (d)g +
+ E ('1 ; '0 ) C01(1) + C02(1) + J0 +
d
X
i=1
B0 (i) ;
d
X
i=1
A0 (i)
+
+ Ef;'1 + '1 A0 (d)g 6 Ef0 ; 1g + Ef(1 ; '1 )(1 ; A0(d))g +
+ E ('1 ; '0 )
C01(1) + C02(1) + J0 +
d
X
i=1
B0 (i) ;
d
X
i=1
A0 (i)
6
(-; + 1 1 - * + ' +,* S)
6 E0 ; 1 + " + 3(d + 1)" + 3(d + 1)SC1 (Q0 (1) 0) + " + Ef(1 ; '1)V 0 6 S g 6
6 E0 ; 1 + (3d + 5)" + 3(d + 1)SC1 (Q0(1) 0) + C3 (Q0(1) S)S:
(4.11)
5 , ; +
A V 6 E0 ; 1 + 3(d + 2)" + 3(d + 1)C1(Q0(1) Q0(2))S +
(
+ C2(Q0(2) S)S Q0 (2) 6= 0 (4.12)
C3(Q0(1) S)S Q0 (1) 6= 0:
DQDB-
83
5 + C1(Q0 (1) Q0(2)), C2(Q0 (2) S), C3(Q0 (1) S) - Q0 (1) Q0(2) ! 1, ; K, -; :
C1(Q0 (1) Q0(2)) 6 3S(d"+ 1) maxfQ0(1) Q0(2)g > K
(4.13)
C2(Q0 (2) S) 6 S"
Q0(2) > K
(4.14)
"
Q0(1) > K:
(4.15)
C3(Q0 (1) S) 6 S
\$, maxfQ0(1) Q0(2)g > K + AV 6 E0 ; 1 +
+ 3(d + 3)". 5 - E0 < 1, ; > 0, E0 ; 1 + < 0. 2 " = 3(d+1) . : " A V 6 E0 ; 1 + < 0, %,- ,
+
= M~ * .
. \$
, +- '(x y) = px2x+y2 , -; 1, % , % +*
C1(Q0 (1) Q0(2)), C2 (Q0(2) S), C3 (Q0(1) S), , - ,, , K, - (4.13){(4.15).
+1] Mukherjee B., Bisdikian C. A journey through the DQDB network literature // Performance Evaluation. | 1992. | Vol. 165. | P. 129{158.
+2] Tran-Gia P., Stock T. Approximate performance analysis of the DQDB access protocol // Proc. International Teletra6c Congress (ITC), Adelaida, Australia, September
1989* Comput. Networks ISDN Systems. | 1990. | Vol. 20. | P. 231{240.
+3] Sharma V. Some asymptotic results on the DQDB protocol. | Presented in Seminar
on Teletra6c Analysis Methods for Current and Future Telecom Networks, International Teletra6c Congress (ITC), Bangalor, September 1993.
+4] Mukherjee B., Bisdikian C. Alternative strategies for improving the fairness in and
an analytical model of the DQDB network // IEEE Transactions on Computers. |
1993. | Vol. 42, no. 2.
+5] Kalashnikov V. Mathematical Methods in Queueing Theory. | Kluwer Acad. Publ.,
1994.
+6] Kalashnikov V. Topics on Regenerative Processes. | CRC Press, 1994.
+7] Kalashnikov V. Crossing and comparison of regenerative processes // Acta Appl.
Math. | 1994. | Vol. 34. | P. 151{386.
& ' ' 1997 .
,
. . . . . Plovdiv University P. Hilendarski, Complesso Universitario di Monte S. Angelo, 511.2
: .
= 21 + 22 +
1 , 2 , 3 , 4 , 5 | ,
(mod 24).
N
p
p
p
p
p
p
p
+ 24 + 25
0 (mod ), ( 2) = 1 1+2
2
p3
p
p p
k
k
N
5
Abstract
M. B. S. Laporta, D. I. Tolev, On the sum of squares of ve prime numbers
one of which belongs to an arithmetic progression, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 1, pp. 85{96.
We study the equation
= 21 + 22 + 23 + 24 +
are prime numbers, 1 + 2
N
where 1 , 2 , 3 ,
5 (mod 24).
p
p
p
p4
,
p5
p
p
p
p
p
2
p5 0 (mod ), ( 2) = 1, and
k
k
N
1. 1937 . . 9,10] , !"! #! N !!
N = p1 + p2 + p3
(1)
(!! ! !! )*+ + p1 , p2, p3 . 1938 -. .. / 3] , ! ! #!! N 5 (mod 24) (! *
)!! (( ) )*+ !.
, 2002, ( 8, ) 1, . 85{96.
c 2002 !"#,
\$% & %%
86
. . . , . . 5 (! ! )! * ()!6 7( R(N) ! !8 !
N = p21 + p22 + p23 + p24 + p25
(2)
! p1, p2, p3, p4 , p5 | )*! ((. 4]):
5
3=2 ;5
3=2 ;6
R(N) = S(N) ;(1=2)
;(5=2) N L + O(N L log L)
! L = log N, ; ! ((-7# 58! S(N) ! !*8 +<8 !( S(N) 1 N 5 (mod 24).
=8 ! (* !( !! (2) p1 + 2 0 (mod k), (k 2) = 1
N 5 (mod 24). >!! , )
X
Ik (N) =
log p1 log p2 log p3 log p4 log p5
p21 +p22 +p23 +p24 +p25 =N
p1 +20 (mod k)
Y
Sk (N) = 4 (1 + kN (p))
p>2
; !, ! ( -! ap ,
8 N 3
( )p +5(2( ;pN )+1)p2 +5(( Np )+2( ;p1 ))p+1
>
> p
>
(p;1)5
>
>
> 5p2 +10( ;1 )p+1
>
p
<
4
;
kN (p) = > p2 +6( ;(p1 ;)p1)+1
p
>
>
(p;1)3 >
>
>
2
N ;4
;1
4
;
N
>
: (4( p )+1)p +(4( p )+6( p ))p+1 ;
1
(N) = 32
(p;1)4
X
m1 ::: m5 6N
m1 +m2 +m3 +m4 +m5 =N
! p6 jN
! p6 jk pjN
! pjk pj(4 ; N)
! pjk p6 j(4 ; N)
1
pm m m
mm:
1 2 3 4 5
* !( !6<8 !.
. A > 0 B = B(A) > 0, X Ik (N) ; (N) Sk (N) N 32 L;A :
'(k)
N 1=4
k6 logB N
(k2)=1
( ! =8 ) 7],
8 * ! *8 ! ! (1) p1 l
(mod k), (k l) = 1. 5 !! * )! ?!)( !!(, , Visiting Professor Program Programma di scambi internazionali
con Universita ed Instituti stranieri per la mobilita di breve durata di docenti, studiosi e ricercatori, D. R. n. 3251.
2. 87
@ N | ! #!! , ! N 5 (mod 24),
A > 100 | ! ) B = 10000A. @!)(, H |
!8!! )!! H 6 N 1=4L;B .
A!! x, y, !( !8!*!, !! p, p1, p2 , p3 , p4, p5 |
)*! !! m, n, q, k, l, r, h, f | #!*! !!. . *, (n), '(n), (n) 6 7#6 "!, 7#6 58! )!*+ !!!8 n !!. @ (m n) ! 8 <8 !!, m n] | (! !! <!! ! !
m n. B !8!*+ ! x, y (x y) ! !
!8!8 )(8 #!*( ( x y. A (! (!! , ! !. C!! m n (mod k) (* !(
)! )* )<!, m n (k) , ! !( ) !! e(x) = exp (2ix).
@ c ! !8!6 )!6 )6, ( ! ! . ?)(!, = )! ( )
plog x
(log x)e;c
plog x
e;c
:
.* (+ O 6*! A. B(
q X
q
X
a=1
a=1
(aq)=1
=
X0
k6H
=
X
k6H
(k2)=1
:
@
Q = L100A = NQ;1
X0 (N)
E = Ik (N) ; '(k) S(N) k6H
E1 =
q
;1
q6Q a=0
(aq)=1
a; 1 a+ 1
q q q q
| !+ #!,
E2 = ; 1 1 ; 1 n E1 | #!,
Sk () =
X
p
(3)
p6 N
p+20 (k)
(logp)e(p2 ) S() = S1 () M() =
N
X
m=1
1 e(m):
p
2 m
88
. . . , . . (!!(
Ik (N) =
!
1;
Z 1=
;1=
I (i) (N) =
k
C!!,
!
Sk ()S 4 ()e(;N) d = Ik(1) (N) + Ik(2) (N)
E1 =
Z
Ei
Sk ()S 4 ()e(;N) d i = 1 2:
E 6 E1 + E2 X0 (1)
I (N)
k
k6H
(4)
; (N)
(k) Sk (N) E2 =
X0 (2)
k6H
jIk (N)j:
(5)
D! !!(* !! (4), (5) !!
E1 N 3=2L;A E2 N 3=2L;A :
(6)
D )! #! ) E1 (* )(! 8 (! /{- !!( >(!{ . @ !8 #! *
)! *( =8 ! 7] )!( !*+ (!, )!<+ /. . F! (* )!( 8
(! #! E2.
3. "
E1
!, Ik(1) (N) =
!
Ik (a q) =
1=Z(q )
;1=(q )
q XX
q6Q a=1
Ik (a q)
(7)
4
Sk aq + S aq + e ;N aq + d:
(8)
G(( Sk ( aq + ) a, q, , !6<+ (
q 6 Q (a q) = 1 jj 6 q1 :
(9)
D k (* ! (!( 8 ()!6 7(
((* (7) (! ! !( ) !*! )!*, *! +
) ! * *). B (* *( ()!6 7(
89
!8 )! 6, ! ( !!(. (!!(
q X
a
Sk q + =
!
m=1
m;2 ((kq))
T () =
X
2
e amq T() + O(qL)
(10)
(log p)e(p2 ):
p
p6 N
p;2 (k)
pm (q)
) =!(!6 !6 !8, ( ! , ! #!*! k, m, q !6 ( (k 2) = (m q) = 1 m ;2
((k q)), <!! #!! f = f(k m q), ! (f k q]) = 1 ! 6 #! n !! n f (k q]) =! !(! n ;2 (k),
n m (q). B6
X
T() =
(log p)e(p2 ):
p
p6 N
pf (!kq])
B(
H(x h) =
X
max
max
y6x (lh)=1
y :
log p ; '(h)
p6y
pl (h)
) 7( I!, )!!( T() p
T() = ;
=;
Z N
0
pN
pf (!kq])
Z 0
p
d e(y2 ) dy + X logpe(N) =
log p dy
p
p6y
p6 N
pf (!kq])
X
y + O(H(pN k q])) d e(y2 ) dy +
'(k q])
dy
p
+ '(kNq]) + O(H( N k q])) e(N) =
= '(k1 q]) ;
+O
pN
Z
0
p
p
ZN 0
d e(y2 ) dy + pNe(N) +
y dy
p
H( N k q])jjy dy + O(H( N k q])):
(11)
90
. . . , . . ! ) (, !( )(! ! (9) + !6
7( ((. 8, Ch. 2])
p
ZN
(* )(
0
e(y2 ) dy = M() + O(1 + N jj)
M() + O 1 + N H(pN k q]) :
T () = '(k
q])
q
@ (10) )!!! *!! T() * ! (9), )(
ck (a q) M() + O(QH(pN k q]))
Sk aq + = '(k
(12)
q])
!
q 2
X
ck (a q) =
e am
q :
m=1
m;2 ((kq))
B)!!( c(a q) = c1(a q). K* ! (9), (* (!!(
q) M() + O(pNe;cpL)
S aq + = c(a
'(q)
((. 4, Lemma 7.15]).
7( (12), (13) 8 #!
p
NL
a
Sk
q + k * (9), )(
4 Sk aq + S aq + e ;N aq + =
q)4 ck (a q) e ;N a M()5e(;N) +
= c(a
'(q)4 '(k q])
q
(13)
5=2
p
p
+ O Nk e;c L + O(N 2 QH( N k q])):
@=( (8) +(
q)4 ck (a q) e ;N a
Ik (a q) = c(a
'(q)4 '(k q])
q
+O
1=Z(q )
M()5e(;N) d +
;1=(q )
N 3=2 e;cpL + O(q;1 NQ2 H(pN k q])):
k
91
,
q X
(q)
Ik (a q) = '(kbkq])'(q)
4
a=1
!
1=Z(q )
;1=(q )
M()5 e(;N) d +
3=2 p
p
+ O Nk e;c L + O(NQ2 H( N k q])) (14)
q X
ck (a q)c(a q)4e ;N aq :
a=1
* !(, ((. 8, Ch. 2])
bk (q) =
1=Z(q )
;1=(q )
M()5e(;N) d = (N) + O((q)3=2 )
(7) (14) )(
3=2
pL b
(q)
N
k
;
c
Ik (N) = (N) '(k q])'(q)4 + O k e
+
q6Q
X jbk (q)jq3=2 2 X H(pN k q]) : (15)
+ O 3=2 '(k
+
O
NQ
q])'(q)4
q6Q
q 6Q
X
?! (!, bk (q) | () 7# q , !, ! q = pl , p )!, p6 ja, (* (!!( ((. 5, Ch. 7])
8
>
! l > 2
<0
c(a 2l ) = >;1 ! l = 1
:
2e( a4 ) ! l = 2
(
p > 2 l > 1
c(a pl ) = 0 a p
p ( p ) p ; 1 p > 2 l = 1
!
(
p = 1 p 1 (4)
i p 3 (4):
>!! , p > 2 p6 ja )!(, (
a p
ck (a p) = c(a4ap) = ( p )p p ; 1 p6 jk
e( p )
pjk
, 26 jk, (, ck (a 2l ) = c(a 2l ) l a 6 0 (2).
92
. . . , . . , ! (!, (k 2) = 1 )!
8
0
>
>
>
<
p > 2 l > 1
bk (pl ) = >0 p = 2 l > 2
1 p = 2 l = 1
>
>
:
64 p = 2 l = 2
8
>
( Np )p3 + 5(2( ;pN ) + 1)p2 + 5(( Np ) + 2( ;p1 ))p + 1
>
>
>
>
<(1 ; p)(5p2 + 10( ;1 )p + 1)
p
bk (p) = >
2 + 6( ;1 )p + 1)
(p
;
1)(p
>
p
>
>
>
:
4
;
N
;(4( p ) + 1)p2 ; (4( Np;4 ) + 6( ;p1 ))p ; 1
2 < p6 jkN
2 < p6 jk pjN
pjk pj(4 ; N)
pjk p6 j(4 ; N):
)!*<+ 7( , !8 68 )8
c0 > 0
bk (q) = (q)c0 q3 :
(16)
B( kN (q) = bk (q)'((k q))'(q);5 . * )!(, +1
X
q=1
k (q)
6 +. M k (q) () ! q, )
! 58! ((. 2, Th. 286]) *( 7(( bk (pl )
(* 8"!(
+1
X
q=1
k (q) = (1 + k (2) + k (4))
Y
p>2
(1 + k (p)) = Sk (N):
, (15), (16) "!( N 3=2 (N) N 3=2 '((k q))'(k q]) = '(k)'(q)
(* (!( *!
k (N) + O N 3=2 X (q)c0 q3(k q) + O N 3=2 e;cpL +
Ik(1) = (N) S'(k)
'(k) q>Q '(q)5
k
X (q)c0 q9=2
X
p
3
=
2
2
+O H( N k q]) :
4 + O NQ
q6Q '(k q])'(q)
q6Q
F!(, ) + !*! #! '(n) n(log log 10n);1 (n) n" , (* )!(
p
E1 N 3=2LN1 + 3=2LN2 + NQ2 N3 + N 3=2e;c L (17)
93
!
(k q) N = X X (k q) N = X X H(pN k q]):
2
3
3=2
k6H q>Q kq
k6H q6Q k
k6H q6Q
D #! N1 (!!(
X X X
1 + X dX X 1 N1 = d
3=2
3=2
Q<d6H k6H q>Q kq
d6Q k6H q>Q kq
N1 =
X X
(kq)=d
(kq)=d
1 1
1
1 +
1 X 1X
3=2
3=2
3=2
3=2 d6Q d k6H=d k q>Q=d q
Q<d6H d k6H=d k q=1 q
X
X
1 + L X 1 L2 :
L d31=2
(18)
3
=2
3=2
Q1=2
d6Q
Q<d d
q>Q=d q
X
1
X
X
X
D ((* N2 )!(
X X 1 X
X X 1 X
X 1
N2 d
1
1
QL
QL2 (19)
k
k
d
d6Q k6H q6Q
d6Q k6H=d q6Q=d
d6Q
#! N3
N3 =
(kq)=d
X
h6QH
p
!(h)H( N h) ! !(h) =
X X
k6H q6Q
!kq]=h
1:
(20)
-! (!, !(h) OL. ) !!( >(!{ ((. 6, Theorem 15.1]), (11), (20), ! )!!!8 H Q (*
)(
p
N3 NQ2L6;B :
(21)
!! (17), (18), (19), (21) )!!!8 Q, , B +(
E1 N 3=2L;A :
4. "
E2
C<!6 ()!*! ak , *! ) (6 !#! ! E2 =
X0
k6H
Z
Z
X0
E2
k6H
ak Sk ()S()4 e(;N) d = S()4
E2
ak Sk ()e(;N) d: (22)
) !! O"!!, )!(
Z1
E2 sup jS()j
2E2
0
jS()j4 d
4
1=4
3=4 Z1 X
0
a
S
()
d
:
k k 0 k6H
(23)
94
. . . , . . D #! ! (!!(
J=
4
Z1 X
0
ak Sk () d =
=
0 k6H
X0
ki6H
i=1234
L4
L4
ak1 ak2 ak3 ak4
X
X
X
p
p1 :::p4 6 N
p21 +p22 =p23 +p24
pi +20 (ki )
ki6H n :::n 6pN
i=1234 n211+n22 =4n23 +n24
ni +20 (ki )
X
n1 :::n4 6pN
n21 +n22 =n23 +n24
1 = L4
log p1 log p2 log p3 log p4 X
X
p
n1 :::n4 6 N
ki6H
n21 +n22 =n23 +n24 ki jni +2
i=1234
1
(n1 + 2)(n2 + 2)(n3 + 2)(n4 + 2):
) !! abcd a4 + b4 + c4 + d4, )(
J L4
X
p
n1 :::n4 6 N
(n1 + 2)4 = L4 (J1 + J2 )
(24)
n21 +n22 =n23 +n24
!
J1 =
J2 =
X
X
p
n1 n26 N
p
16jh1 jjh2 jh3 h4 62 N
h1 h3 =h4 h2
(n1 + 2)4
X
p
n1 :::n4 6 N
n1;n3 =h1 n4;n2 =h2
(n1 + 2)4 :
n1 +n3 =h3 n4+n2 =h4
B!,
J1 NL15:
(25)
B( !! J20 , *8 6 )!*! h1 , h2 J2 . *
)(
J20 X
X
p
16h1 :::h4 62 N
n1 :::n4 6pN
h1 h3 =h4 h2 n1 ;n3 =h1 n4 ;n2 =h2
h1 h3 (2) n1+n3 =h3 n4 +n2 =h4
X
p
16h1 :::h4 62 N
h1 h3 =h4 h2
(h1 + h3 + 4)4 =
(n1 + 2)4 95
=
=
X
X
rs62pN 16h1 :::h4 62pN
(h1 + h3 + 4)4 =
h1 h3 =h4 h2
(h1 h2 )=r (h3 h4 )=s
X
X
p
p
p
rs62 N l1 l2 6 2 rN l3 l4 6 2 sN
(rl1 + sl3 + 4)4 :
l1 l3 =l4 l2
(l1 l2 )=1 (l3 l4 )=1
F(!, ! l1 l3 = l4 l2 , (l1 l2 ) = 1, (l3 l4) = 1, l1 = l4 l2 = l3 .
@=(
X
X
(rl1 + sl2 + 4)4 =
J20 =
p
p
X
(m1 + m2 + 4)4
p
rs62 N l1 l2 6min( 2 rN 2 sN )
p
m1 m2 62 N
p
rs62 pN p
l1 l2 6min( 2 rN 2 sN )
1
l1 r=m1 l2 s=m2
X
p
m1 m2 62 N
X
(m1 )(m2 )(m1 + m2 + 4)4 X
p
m1 m2 62 N
(m1
)6 +
X
X
p
m1 m2 62 N
(m1 + m2
+ 4)6
X
NL63 + p (l)6
1
p
l64 N +4
m1 m2 62 N
p X m1 +6 m2 +4=l 63
63
NL + N p (l) NL :
l64 N +4
M( ! )( #!( , (*8 J2 #!*( h1, h2 .
, (* 8"!(
J2 NL63:
(26)
/ ! #! ! ) )! ! (23). *
(!!(, Z1
0
jS()j4 d NL7
(27)
((. 3] ) ) !8 (8 !)! L).
!8 #! ((* =)! )*+ !
)!!! Q, , E2, (* +(
p
sup jS()j NL;5A
(28)
2E2
96
. . . , . . ((., )(!, 4,10], ! 1] ) ) !8 !)! L). ,
(23){(28) 6!(, E2 N 3=2L;A :
M!( (*( !!( )6 .
\$
1] Fujii A. Some additive problems of numbers // Banach Center Publ. | 1985. |
Vol. 17. | P. 121{141.
2] Hardy G. H., Wright E. M. An Introduction to the Theory of Numbers, 5th. edition. | Oxford University Press, 1979.
3] Hua L.-K. Some results in the additive prime number theory // Quart. J. Math.
Oxford. | 1938. | Vol. 9. | P. 68{80.
4] Hua L.-K. Additive Theory of Prime Numbers. | Providence, Rhode Island: Amer.
Math. Soc., 1965.
5] Hua L.-K. An Introduction to Number Theory. | Springer-Verlag, 1982.
6] Montgomery H. L. Topics in Multiplicative Number Theory. | Springer-Verlag,
1971. | Lecture Notes in Mathematics, vol. 227.
7] . . !"#\$% \$&"\$' (' # ! )**+ " ,
"+,, !\$ - ." +, \$#!/" # 0*".% ' // .#!+ 1..' 1#"*#".' \$"")"# *. 2. 3. 4".#. | 1997. |
. 218.
8] Vaughan R. C. The Hardy{Littlewood Method, 2nd ed. | Cambridge University
Press, 1997.
9] 2\$' #! . 1. 5 !"#\$ \$"\$' # ! )**+ " , "+, // 36 4447. | 1937. | . 15. | 4. 169{172.
10] Vinogradov I. M. Selected Works. | Springer-Verlag, 1985.
' % ( 1998 .
. . , . . 517.958+523.030
: , , .
! ! " !!#\$ #\$ %&, #'(\$ ! ! *+,- ! .%
!! #/ 0! -*!* ! . !* / " +!#* !* ! # !. !.! ! "' !% %
1*{3.+! *!#* -00,!*, '(
*!, ! *! *. ! ! *%!. 4# !% ! '(% *!,# ! !
!' 1*{3.+! * #* * *
50 1] * !#* !.* .
1" 1*{3.+! 7! **0#\$ !. "*! '! !. !#
-00,!# / +# *!# " % .
Abstract
A. V. Latyshev, A. V. Moiseev, The boundary-value problem for the equations
of radiation transfer of polarized light, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 1, pp. 97{115.
The theory of the solution of half-space boundary-value problems for Chandrasekhar's equations describing the scattering of polarized light in the case of
a combination of Rayleigh and isotropic scattering with arbitrary photon survival
probability in an elementary scattering is constructed. A theorem on the expansion
of the solution in terms of eigenvectors of discrete and continuous spectra is proved.
The proof reduces to solving the Riemann{Hilbert vector boundary-value problem
with a matrix coe;cient. The matrix that reduces the coe;cient to diagonal form
has eight branch points in the complex plain. The de<nition of an analytical branch
of a diagonalizing matrix gives us the opportunity to reduce the Riemann{Hilbert
vector boundary-value problem to two scalar boundary-value problems on the major
cut 50 1] and two vector boundary-value problems on the supplementary cut.
The solution of the Riemann{Hilbert boundary-value problem is given in
the class of meromorphic vectors. The solvability conditions enable unique determination of the unknown coe;cients of the expansion and the free parameters of
the solution.
, 2002, !* 8, = 1, . 97{115.
c 2002 ,
!"
#\$ %
98
x
. . , . . 1. 1
@ I ( ) + I ( ) = 1 !Q( ) Z QT( )I ( ) d :
@
2
0
1
0
0
(1.1)
#\$ %& Q( ) 1=2 2 3
1=2 ; 2)
Q( ) = 3(2(c c++2)2) c 1+(c2+(12); c) (2c) (1
0
3
| , 0 < < +1, | %+ +\$% % - % ,
2 (;1 0) (0 1), I ( ) | %% Il ( ) Ir ( ), ! | \$ , . . \$
+
- .%% , 0 < ! < 1. /% c
%& Q( ) % .
, c 2 00 1], c = 1 % .
. 2 T
.
3
(1.1) %& .
%- 4 01] (. x 18,
(1.1) +% %& P0 ( ) E ( ), %
5 5 ! 1.
6 \$5 5 02,3], 7
5 (1.1) .
. 3 \$ %
5 5 5 (1.1), %
! c: ! 2 (0 1) c 2 00 1].
3
(1.1) \$ 5 5 , 7 \$
7\$
-&. #
%, + \$ (03, 4]), 5 %
5. 3 .% % 5 (%. 02, 3]), 7 % % 4
5 .
, 7 (1.1) c = 1, ! = 1, 02]: c = 1, ! 2 (0 1] | 03]. ;
% , %
5 5 (1.1). 05]. B 06], \$
, \$ \$ >. ?
+ 07]. 08]
\$
H -%& 4, %
5 %5 5
0
0
99
%{C\$. H -%& + . /% FN % (1.1) %
09]. ? \$ %
H -%& 010].
;%%, \$, + %%5 , 011].
D
(1.1) %
\$ \$ Q 1I = Y ;
1
@ Y ( ) + Y ( ) = 1 ! Z K ( )Y ( ) d @
2
0
;
0
(1.2)
0
1
1
2
2
K ( ) = QT ( )Q( ) = 4(c 9+ 2) ( ) (+)9((c +) 2) ( 2)( () ) p
( ) = c 2 + 32 (1 ; c): ( ) = 2c(1 ; 2 ), 7% det K (z ) = 169 2 ( ) =
= 89 c(1 ; 2)2 .
3 7 \$ --&,
C7\$, \$ .
x
2. E \$ (1.2) Y ( ) = exp ; F(
) 2 C (C | % \$), (1.2) %
(
; )F(
) = 21 !
n(
)
(2.1)
1
n1 (
) Z
n(
) = n (
) = K ( )F(
) d
(2.2)
2
;
1
\$ 5 %
5 .
2 5 (2.1) (2.2) 2 (;1 1) % F(
) = F~ (
)n(
)
(2.3)
100
. . , . . F~ (
) = 21 !
P ;1 I + B (
) (
; )
(2.4)
\$ %& (
). #\$ %
Px 1 | H x 1, (x) |
\$--& I, I | %&, B (z ) = K 1 (z )J(z ): %&
Z1
J(z ) = I + 21 !z K ( ) d
(2.5)
;z
;
;
;
1
;
5 %&5, 7 \$ (z ) 5 -&5.
/
% (z ) (z ) = a(z )2C (z ) + 2b(z )C (z ) + c(z )
a(z ) = !2 det K (z ) = 98 c(1 ; z 2 )2 !2 b(z ) = 89 ! (1 ; !)c 1 ; 43 z 2 + z 4 + 49 (1 ; c) c(z ) = 1 ; 89 !c(! ; 2) ; !(1 ; c) ; 43 !c(1 ; !)z 2 -&
Z1
C (z ) = 1 + 21 z d; z
;
1
\$ -& H5 (%. 012]).
/
% -& (z ) = 8(1 ;9cz 2)2 !1 (z )!2 (z )
(2.6)
1
4
4
2
2
2
4
! (z ) = !(1 ; z ) C (z ) + c c(1 ; !) 1 ; 3 z + z + 9 (1 ; c) +
+ 4(c +1 2)c (;1) 1pq = 1 2
;
q(z ) = 16(c + 2)2 z 8 c2 (! ; 1)2 + 23 z 6 c2(! ; 1)(4 ; 3!) +
101
2 z 2 (17!c2 ; 8!c ; 20c2 + 8c +
+ 19 z 4 c2(13!2 ; 38! + 26) + 27
1
2
+ 81 (25c ; 32c + 16) :
3% . I . \$% &% % % (2.6). #%%, (z ) 0, 1 5
\$5 % C . 3%
5
\$ %% -&5 ! (x):
! (x) = ! i 2 !x(1 ; x2) x 2 K = 1 2
! = !(1 ; x2 )2C (x) + 1c c(1 ; !) 1 ; 43 x2 + x4 + 94 (1 ; c) +
+ 4(c +1 2)c (;1) 1pq = 1 2:
;
>% ! (x) \$ \$ 7 0, 1, 9c
8(1 ; x2 )2 !1 (x)!2 (x)
. L
\$, 7 5, % ,
C n 0;1 1] -% (%. 013])
= 21 0arg (z )]
, -% (2.6), = 1 + 2, = 21 0 ] = arg !+ | %, = @ K | %5 0;1 1], 5 5 , +
0: : :] -&, 5 .
? !(z ) = ! (;z ) !M + = ! , x 2 K ( %
+), = 2 0(x)](01):
;
>% \$ -& !+ (x), = 1 2, +\$,
p
;
!1(0) = 91c 5c + 4 + 25c2 ; 32c + 16 > 0
1
2
!1(1) = 3 3 (c + 2) ; !c > 0 c 2 (0 1] ! 2 (0 1]:
102
. . , . . N , 1 = 0. I, c 2 (0 1], ! 2 (0 1]
p
;
!2 (0) = 91c 5c + 4 ; 25c2 ; 32c + 16 > 0
!2 (1 ; 0) = ;0 < 0
(1) = ;1 < 0 (
% (z ) 0, \$ ,
;1 , c (1) = ;1), . . 2 = 2. ?%
%, = 2, , (z ) % x0, % 7 (z ). /+%, . 5
\$. 3 %% ,
\$, (1 + 0) = ;1, + O
; ! (10 + 7!)c + ! (33 ; 26!)c + : : : (jz j ! 1)
(x) = 1 10
70z 2
, (1) > 0. N , x0 2 (1 +1), ;x0 2 (;1 ;1).
2, 5 5
\$ x0 , % (%. (2.3) (2.4))
Y x0 ( ) = exp ; x F~ (x0 )n(x0)
0
F~ (x0 ) = 12 ! xx;0 0
7% %+ , %
J(x0 )n(x0 ) = 0
(2.7)
% det J(x0 ) = (x0 ) = 0.
/
% (1.2). / = 0 %
, . . +%, lim Y ( ) = Y0 ( ) 2 K+ (2.8)
+0
!
Y0 ( ) | --&, % 5 C7\$ 00 1].
3 & % :
lim Y ( ) = 0 2 K :
(2.9)
;
!1
5 (1.2), (2.8), (2.9) % \$ --&5 Y ( ), %+
0 6 6 +1 2 K, C7\$ 00 1] 0 < < +1 --&% %+
0 < < +1
2 K. H --&5 % <.
x
103
3. 3 .% - %
%\$ %{C\$ %% .--&%, +
(%. + x 4) \$
% F(
), 2 (0 1), 5 % %
F~ (x0 )n(x0), x0 > 1.
%% : 5 %& X ( ), B + ( )X + ( ) = B ( )X ( ) 2 K+ (3.1)
1
B ( ) = K ( )J( ). I .5 7% %& B ( ) \$% . 3% 7 % B (z ) = !C (z )I + 2(c + 2)c1(1 ; 2)2 P (z )
;
;
;
.% %& P (z ) p11 = 2c(1 ; 2)(4 2 (! ; 1) ; !(c + 2) + 4)
p
p12 = 2c(1 ; 2 ) 4 2c(! ; 1) ; 38 (1 ; c) p p21 = ; 2c 4 4c(! ; 1) + 23 2(;!c2 ; 6!c + 4! + 10c ; 4) + 38 (1 ; c) p22 = 4 4 c2(1 ; !) + 23 2 c(5!c ; 2! ; 8c + 8) +
2
2
2
+ 9 (;9!c ; 18!c + 10c ; 8c + 16) :
I & B (z ) \$ %& P (z ).
>&, P (z ), \$%7% 1
1
pq p11 ; p22 + pq
p
;
p
;
11
22
2
2
S (z ) = p21
p21
q(z ) = (p11 + p22)2 ; 4 det P (z ) = 16(c + 2)2 z 8 c2(! ; 1)2 +
+ 32 z 6 c2 (! ; 1)(4 ; 3!) + 19 z 4 c2(13!2 ; 38! + 26) +
2 z 2 (17!c2 ; 8!c ; 20c2 + 8c) + 1 (25c2 ; 32c + 16):
+ 27
81
>&--& S (z ) 5 %&5 C , % z , pzM , = 1 2, % % q(z ). #%%, -& q(z ) 5 -& C , .% S (z ) % \$ .
104
. . , . . p
P& q(z ) 5 C n 00 1] p
,
.% % , q(z ).
I . % % q(z ).
3 04], % q(z ) % !, \$ -& %
! c: ! 2 (0 1] c 2 00 1], \$ + 5 q(z ).
/
5 %
. 1. 2 ., 7% % %: \$% % ;1 , ;2, ;3 , ;4 % z1 , ;z1 , zM1 , ;zM1
75 5, % ;5 % %\$ %,
z2 zM2 1, ;6 ;z2 ;zM2 5 ;1.
;R . % ; (%. . 2).
1. 1. / % ( ) = 1 = 0 5? i = i ( 1), @i = @i ( 1), = 1 2,
j = j ( 0 5), @j = @j ( 0 5), = 3 4, | * ( )?
i (0 1), j (0 0 5) | . / %,
i (1 1) j (1 0 5) | / %.
E! #'! *( % . / .
q z
z
z
z
z
!
c
z
z
c
!
z
j
z
!
z
z
!
i
q z
z
z
p
/
75 \$ \$ q(z )
7% 5- C n 00 1] % ;. I+% .. /+%,p | % %
z0 , + C n ;, q(z0 ) = q0, p
q0 = j q(z0 )j exp i '20 105
1. 2. G% !.# #? p
1 , @1 , 2 , @2 | ! ! ( ),
I+ | % , ; , = 1
6, | !.# #,
| *! ;6 .
z
z
z
z
q z
:::
p
'0 \$ % q(z0 ). / %
p
q(z0 ) = q0ei'=2 , ' = 0arg(z ; z1 )(z + z1 ) : : : (z + zM2 )] , z , zM |
% q(z ). #%%, ' = '1 +p'2 + : : : + '8 . E ;5 ;6
+ , 'i = 0, q(z0 ) q0. E \$ ;6 +
(%. . 2), '1 = '2 = '3 = '4 = '5 = '7 = 0 '6 = '8 = 2
p
( +\$),
'
=
4
q(z0 ) = q0, p
, q(z0 ) = q0p
, + p % ;5 . ?% %,
\$ q(z ), z 2 C n ;, q(z0 ) = q0, .
S% %
\$ %& S (z ) %&--& C n 00 1] % ;.
/
7 & % .--& \$
-& (3.1) X (z ) = S (z )U (z )S 1 (z )
(3.2)
1 ; 1 p11 p22 q 1
:
1
2
p
21
S (z ) = pq ;1 12 p11 pp2122 + q 2 %& S (z ) , S 1 (z )B (z )S (z ) = U(z ) = diagfU1(z ) U2(z )g
;
;
;
;
;
;
p
p
106
. . , . . U (z ) = !C (z ) + 4(c + 2)21c(1 ; z 2 )2 (p11 + p22 + (;1)+1 pq) = 1 2:
3 (3.2) % U (z ) = diagfU1(z ) U2 (z )g 5 5 \$5
%&5.
>& X (z ) + \$ 5, 5 C n00 1].
V (3.2), %, ; % \$ \$ , S (z ) % ;:
X + ( ) = X ( ) 2 ;:
(3.3)
/
+ (3.2) (3.1) (3.3), % % %
% \$% :
U+ ( ) U + ( ) = U ( ) U ( ) 2 K +
(3.4)
U + ( )T = TU ( ) 2 ;:
(3.5)
#\$
0 1
+
1
T = 0S ( )] S ( ) = 1 0 2 ;:
;
;
;
;
;
;
# (3.4) .
% % % %
U+ ( ) = UU+ (( )) U ( ) 2 K+ = 1 2
(3.6)
(3.5) | % %
U1+ ( ) = U2 ( ) U1 ( ) = U2+ ( ) 2 ;:
(3.7)
/% 7%, 5 03], % + -&5
U1 (z ) U2 (z ):
(3.8)
U (z ) = expf;A(z ) + (;1) r(z )B (z )g = 1 2:
#\$
;
;
;
;
0
1
1
Z
Z
1
(
)
+
(
)
+
k
1
1
2
A(z ) = ; 2
d B (z ) = 2 1 (r)(+ )(2 (; )z+) k d
;z
0
p
0
-& r(z ) = q(z ) , k 2 Z.
4 k % % %, &
%+ z = 0 z = 1. 657% (0) (1), = 1 2. 3\$% \$ %+ -&% w (z ) (%.
(2.6)) -&% U (z ):
w(z ) = (1 ; z 2 )2 U(z ):
107
/%, arg w+ ( ) = arg U+ ( ), = 1 2. 2\$ \$ x 2,
% (0) = 0, = 1 2, 1 (1) = 0 1 (1) = . ?% %,
1
1
Z
Z
A(z ) = ; 21 1 ( ) + ;2 (z ) ; d B (z ) = 21 1 (r()+)(2;( z) )+ d:
0
0
U (z ), = 1 2, % , . I 5 5
% 7 .
/
% U1 (z ) U2 (z ) U1 (z ) = U1 (z )'(z ) U2 (z ) = U'2((zz)) (3.9)
'(x) | -&, C \$ ;. /
(3.9) (3.7), % '+ ( ) = ' 1( ) 2 ;:
P& '(z ) \$%7% '(z ) = exp(;r(z )R(z ))
(3.10)
Zx1 Zx3 d :
R(z ) =
+
r( )( ; z )
0
0
0
;
0
x2
P& R(z ) 5 -& 03]. / % %, \$5%, % %\$ %
\$
% -&5 (0 1).
2, (3.4) (3.5):
U (z ) = expf;A(z ) + (;1) r(z )(B (z ) ; R(z ))g = 1 2:
(3.11)
? x , = 1 2 3, % %, -& U (z )
% 5 75 . I . +%
-& B (z ) R(z ) O 75 % 5
1
1 Z b( ) 1 d =
2 r( )
Zx1 Zx3 ;
0
0
+
x2
1 d = 1 2 3:
r( )
;
(3.12)
(3.12) \$ x, = 1 2 3.
#%%, % xalpha 2 (0 1) -& '(z ) % (3.10), fU1 U2 g, 7 % (3.11), % % % (3.4) (3.5).
108
. . , . . ?% %, %& X (z ) :
X11 = p11 ;r p22 U1 ;2 U2 + U1 +2 U2 X12 = 2 pr12 U1 ;2 U2 X21 = 2 pr21 U1 ;2 U2 X22 = ; p11 ;r p22 U1 ;2 U2 + U1 +2 U2 :
/ %& X (z ) C % 00 1]. P& pij ,
r(z ) .
x
4. ! " #\$
" 3 .% - , %5
-&5
F(
), 2 (0 1), 5 -&5 F~ (x0 )n(x0), 5
% . I .
+%, -& Y0 ( ), C7\$ KM + , % Z1
Y0 ( ) = a0F~ (x0 )n(x0) + F~ (
)a(
) d
:
(4.1)
0
;% .--& a0 a(
). /
% (4.1) (2.4), % \$ % H
1
Z
1
x
1
(
) d
+ B ( )a( )
0
Y0 ( ) = a0 2 ! x ; n(x0) + 2 ! a
;
0
0
2 K+ :
(4.2)
D%+ . %& 2i !K ( ), %
2i !K ( )Y0 ( ) = i K ( )a0 ! x x;0 n(x0) +
0
Z1
(
) d
+ J( )2i K ( )a( ) 2 K+ : (4.3)
+ i K ( )! a
;
0
3
7% %\$ --&
A(z ) =
Z1
0
a(
) d
;z
(4.4)
109
%5 % 00 1]. L %\$ 5 -&5 A(z ) J(z ) % K+ ,
% -% L&, 7% (4.3) 5 5
%{C\$
J+ ( )0A+ ( ) + a0 x x;0 z n(x0)] ; J ( )0A ( ) + a0 x x;0 z n(x0 )] =
0
0
= 2i K ( )Y0 ( ) 2 K+ (4.5)
%% .--&% G( ) = 0J+ ( )] 1J ( ).
/\$\$ \$% x 3, % (4.5) 5 . I . \$% % (3.1), \$ %+
K ( ) :
0X + ( )] 1 0A+ ( ) + a0 x x;0 n(x0 )] ; 0X ( )] 1 0A ( ) + a0 x x;0 n(x0 )] =
0
0
= 2i 0B + ( )X + ( )] 1 Y0 ( ) 2 K+ : (4.6)
3% . . #%%, -&
U1 (z ) % 0 x2 x1 x3. P& U2 (z ) % x1, x3 5
\$ x2 . D
, X 1 (z ) = S (z )U 1 (z )S 1 (z ), %,
\$ (4.6) % x , = 0 : : : 3.
3% 5 (4.6). ;%
;( ) = (B + ( )X + ( )) 1 :
>& ;( ) % x, = 1 2 3, C7\$ (0 1). #, .% % ;( )Y0 ( ).
D
, U1+ = U1 exp0;i1 + i(1 + 2)]
U2+ = U2 exp0;i2 + i(1 ; 1 + 2 )]
(
(
1
2
(0
x
)
2 (x2 x3) |
1
1 = 0 2= (0 x )
2 = 10 2= (x2 x3)
1
-& , + 1 = (1 ; 2 )2 U+ U+ !+ ( )U+ ( )
+%, .% ;( )Y0 ( ) ! +0 ! 1 ; 0. D
., %, (4.6) %+ \$
\$ % H \$ ;( )Y0 ( ) 5
;
;
;
;
;
;
;
;
;
;
;
;
;
110
. . , . . .5 %%- 3 X
A(z ) = a0 z ;x0x n(x0) + X (z ) W(z ) + i z ;1 x i
0
i
i=0
i i (i = 0 : : : 3) | \$ ,
1
1(z ) Z
W(z ) = 2(z ) =
0
(4.7)
;( )Y0 ( ) d; z :
(4.7) % 0, x0 x1 , x2, x3. D% 7 %
i i (i = 0 : : : 3).
D x0 % %
a0 x0n(x0) + X (x0 ) 00 = 0
% %
x0n1 (x0)a0 + X11(x0 )0 + X12 (x0)0 = 0
(4.8)
x0n2 (x0)a0 + X21(x0 )0 + X22 (x0)0 = 0:
(4.9)
/
% (4.7) U1 (z )
1 F1 (z )
x
0
0
(4.10)
A(z ) = a0 z ; x n(x0) + S (z ) 0 U2 (z ) S (z ) F2 (z ) 0
3
X
F1 (z ) = 1(z ) + z ;ix ;
i=0
3
X
i
i :
z
;
xi
i=0
;
, x1 + \$ S211 (z )F1 (z ) + S221 (z )F2 (z ) = O(z ; x1 ) (z ! x1 )
(4.11)
+ .% (4.10). #%%, % S211 (z )F1 (z ) + S221 (z )F2 (z ) = O(z ; x3) (z ! x3)
x3.
I x1 x3 % (4.11) g1(z ) + zg2;(zx) = O(z ; x) (z ! x) = 1 3
(4.12)
F2 (z ) = 2(z ) +
;
;
;
;
111
i + S 1 (z ) (z ) + X i 2
22
z ; xi
z ; xi
P
= 1 3, %
%%
i 0 3 %,
i = , = 0 : : : 3,
g2 (z ) = 2S2 1 1(z ) + 2 S221 (z ):
2 (4.12) , .
7% %
g2(x1 ) = 0 g2(x3 ) = 0
(4.13)
(4.14)
g1 (x1) + g2(x1 ) = 0 g1 (x3) + g2(x3 ) = 0:
2 5 (4.11) (4.14) %
1 = 1 1 3 = 3 3
(4.15)
X
X
1(x )+ x ;i x ; 2(x )+ x ;i x ; = 0 = 1 3: (4.16)
g1(z ) = S211 (z ) 1 (z ) +
;
X
;
;
;
0
0
0
i
i
#\$ 1 = 1 (x1), 1 = 1 (x1 ), 3 = 1 (x3), 3 = 1 (x3 ), 1
1 (z ) = ; S221 (z ) :
S21 (z )
+ , %, x2 %
2 = 2 2
(4.17)
X i
X i 1 (x2) + x ; x ; 2 2 (x1) + x ; x ; 2 2 = 0:
(4.18)
0
0
0
0
;
;
2
2
i
#\$ 2 = 2 (x2), 2 = 2 (x2 ), 0
2
2
0
i
0
1
2 (z ) = ; S121 (z ) :
S11 (z )
;\$ \$ . 3 03], %&
S 1 (z ) c 2 00 1) 0. /.% %+ \$
\$ + %, x1 ,
x2, x3, . . % X
X 1 (0) + x i ; 0 2 (0) + xi = 0
(4.19)
i
i
0 = 2 (0).
;
;
;
112
. . , . . I a0, i , i (i = 0 : : : 3) % 5 5 (4.8), (4.9), (4.15){(4.19).
V 03] %+ \$, % %
. ?% %, % (4.7), .--& a0 + (4.1), % \$ 5 .
657% .--& a(
) + (4.1). 6 (4.4)
-% L& %%
A+ ( ) ; A ( ) = 2i a( ):
(4.20)
L 5 , \$ A+ ( ) A ( ) %+ 5 (4.7)
\$ 5 K+ -& A(z ), %5 5 C % 00 1], . . a(
) .
2, % . fF(
) 2 (0 1): F~ (x0 )n(x0)g
.
I% %, + 5 -& Y0 ( ), 5 C7\$ 00 1], % % (2.1).
;
;
x
5. & 3 .% - % +%, 5 (1.2), (2.8),
(2.9) % % .
L-%% %.
. (1.2), (2.8), (2.9) , (2.1)
Y ( ) = a0 e
=x
;
Z1
F(x0 )n(x0) + e
;
0
=
F~ (
)a(
) d
:
(5.1)
!
" (5.1) # " a0 - a(
), n(x0 ) .
. 3\$
\$ % % (2.8), (2.9),
% (5.1) +
Z1
Y0 ( ) = a0F(x0 )n(x0) + F~ (
)a(
) d
:
0
(5.2)
3 + (5.2) Y0( ) \$ --&, C7\$ 00 1].
113
L %% % %+% +\$ Y0 ( ) % % , 7% .--& a0 a(
) .
/
% + (5.1) 5 5 Y ( ) = a0 x x;0 e
0
;
=x0
1
Z
1
n(x0) + 2 ! e
=
;
0
+e
a(
) d
+
;
=
;
B ( )a( )( )
2 K+ (5.3)
(
2 K+
( ) = 10 2= K+ :
/+%, , 5 + (5.3) 5 . H.--& a(
) (4.20), \$ 5 A+ A (4.7). ? \$
-& A(z ) , %, ! +0 ! 1 ; 0 + (A+ ( ) ; A ( )) % 5 . I, -& A(z ) C7\$ K+ . 2 + %, + (5.3) H 7\$
5
\$.
I\$
%+ \$ \$
%+ \$ + - % % : \$
03].
;\$ \$, + (5.3) % (1.2).
/ + (5.3) % % (2.8)
(2.9).
3 \$ (1.2), %
;
;
@
@ Y ( ) + Y ( ) = a0e
=x0
;
1
Z
n(x0 ) + 21 ! e
=
;
0
a(
) d
:
(5.4)
657% \$ (1.2):
1 ! Z K ( )Y ( ) d = a e =x0 n(x )I ; a e =x0 n(x )J(x ) +
0
0
0
0
0
2
Z1
Z1
Z1
1
1
1
=
=
+ 2 ! e a(
) d
I ; 2 ! e a(
)J(
) d
+ 2 ! e = a(
)J(
) d
:
0
0
0
(5.5)
/ (5.5) \$
(2.5). 2\$ (5.5) (2.8), %
;
;
;
;
;
114
. . , . . 1 ! Z K ( )Y ( ) d = a e
0
2
=x0
;
1
Z
n(x0 ) + 21 ! e
=
;
0
a(
) d
:
(5.6)
L
5 (5.4) (5.6) \$
%.
x
6. (
)
.
1. / \$
% (5.5) % \$
-% /{S (%. 013]).
2. H \$ (5.3), 5 +
<, 7% & x 2.
3 \$ % %\$ R .
. % .5 , 5 5 % -&5, 5 5 % .
% %{C\$ , %&, %5 .--&, %
%5 . L % (%. 013]) .% %%.
/ 5 % -& % .--& (
%
) %{C\$, 5 % 5 .
3 %%, 5 % 5 5 5 %+ \$ %7 + & 5 +% %
K( ) =
0
n
X
i=1
Li ( )Mi ( )
0
%& Li ( ) Mi ( ) % %\$ .% % \$ % c.
0
*
1] . . | .: . , 1953.
2] \$%
& '. (. ) %
* + &
,
+ // .( /. | 1994. | 1. 34, 2 2. | C. 234{245.
115
3] \$%
& '. (. (
+ &+ 45{678
& * * ++ ,+& &
// .( /. | 1995. | 1. 35, 2 7. |
C. 1108{1127.
4] \$%
& '. (. '
%
&
* 5
7* *
&
,+5 +5 * ,<
+ // 1
. 5
5. =. |
1993. | 1. 97, 2 2 (+87). | . 283{303.
5] Burniston E. E., Siewert C. E. Half-range expansion theorems in studies of polarized
light // J. Math. Phys. | 1970. | Vol. 11, no. 12. | P. 3416{3420.
6] Bond G. R., Siewert C. E. On the nonconservative eqation of transfer for a combination of Rayleigh and isotropic scattering // The Astrophysical Journal. | 1971. |
Vol. 164, no. 1. | P. 96{110.
7] Siewert C. E., Burniston E. E. An explicit closed-form result for the discrete eigenvalue in studies of polarized light // The Astrophysical Journal. | 1972. | Vol. 173,
no. 2. | P. 405{406.
8] Siewert C. E., Burniston E. E. On existence and uniqueness theorems concerning the
H -matrix of radiative transfer // The Astrophysical Journal. | 1972. | Vol. 174,
no. 3. | P. 629{641.
9] Siewert C. E., Maiorino J. R. The complete solution for the scattering of polarized
light in a Rayleigh and isotopically scattering atmosphere // Astrophysics and Space
Science. | 1980. | Vol. 72. | P. 189{201.
10] Siewert C. E. On the half-range orthogonality appropriate to the scattering of polarized light // J.Q.S.R.T. | 1972. | Vol. 12, no. 4. | P. 683{694.
11] 6
5
& 1. '., D&& F. (., D75 . 6. )& 5
5
,
,+& + (
7) // 1
5,5 IK, & ,<
+L. NU, 26{30 +8+ 1981 . | W
&: -& 'F '54, 1989.
12] D
D., X&=
7 . \$
+ + ,
. | .: , 1972.
13] 6*& /. Y. D
&
. | .: F, 1977.
& ' ' 1997 .
. . 519.14
: , ,
, F -
".
#, \$#% " ; GQ(4 12), *
%
O, + , 49 , # " . - . \$
" ; ; O \$
(196 39 2 9). ,
, % " % pG2 (5 32)
# GQ(4 8)-
".
Abstract
A. A. Makhnev, On pseudogeometrical graphs for some partial geometries, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 117{127.
It is shown that a pseudogeometrical graph ; for GQ(4 12), containing a 49-coclique O, is a point graph of generalized quadrangle. Furthermore, the subgraph
; ; O is strongly regular with parameters (196 39 2 9). It is proved that a pseudogeometrical graph for partial geometry pG2 (5 32) is locally a GQ(4 8)-graph.
1. .
a, b | ;, d(a b) !"#
a b, ;i (a) | ", "#\$% ; !
;, " i a. &" ;1 (a)
#"
' a (a]. *
a?
", "#\$% ; fag (a], (a]0 | ; ; a? .
+
' .
, ; # k, '% a ; k. , ; # (v k ),
"
! v , # k !"
! #. , ; | # % 5
\$*
5% " "
#
%, 96-01-00488.
, 2002, 8, 6 1, . 117{127.
c 2002 ! "#,
\$
! %
&
118
. . (v k ), # #'0 (a] \ (b] "
! " ' "# a, b, " 0 2 ;. +
# % # , "
2.
2 " 3 ; Ki (3) !
; ; 3, ! i 3, xi(3) = jKi(3)j.
*
(m1 : : : mn ) % "% " " m1 : : : mn . &", "#\$% (a] \ (b], (-) -", a, b ! (" 2). 5
\$"
, 0 , -% % " (s t), !" "
! s + 1 #, !"
! t + 1 % ( ' "%
) " '% a, !0
% % L, %"
, " 0 a '0 L (
pG(s t)). = 1, 0
# GQ(s t). * # PQ(s t ) \$"
, 0 , % !" "
! s +1 #, !" ! t +1 % ( '
"% ), " '% a, !0
% % L,
%"
"% %, " 0
% a '0
% L, % ' "# ! .
7
% , ' "
!, ! 0
% %. 8
, % %
pG(s t) # v = (s +1)(1+ st ), k = s(t +1),
= (s ; 1)+( ; 1)t, = (t +1). 2 9 "
\$ : " (. (1]) !" (s +1)- L
! L.
+ "% "!
" #
# 0' # ".
1. GQ(4 12) 49-
O. ; . ; ; O
(196 39 2 9).
2. pG2(5 32) GQ(4 8)-
.
2. + 9 "
.
2.1. ; | (v k ),
3 | #
N \$, M 119
\$ d1 : : : dN . (v ; N ) ; (kN ; 2M )+ M + xi = xi(3).
N
2
X
N
N
X
d
i
;1
i
;M ;
=x +
x
i=1
2
0
i=3
2
i
. 5. "
1 (4].
&# ; | "
% " % pG(s t),
"
!0% # O, "'0#' \$ : ( 9 #
O
") \$ @
" , A = ; ; O. 7"
s(s;+1)(st+)
jOj = 1 + st
= (s+t+1;) , 9# t = (s ; 1)(s + 1 ; ).
2.2. ; | pG (s t), O, %% # &
# '
. = 1, t = s2 ; s A = ; ; O (s2 t + s (s ; 1)(t +1) s ; 2 (s ; 1)2 ).
. & 4 (1] A # . K
(s +1)-% ;, K " O "
% x.
7" ' O ; x ! K , 9# ' K ; O ! 1 + ( ; 1)(s + 1 ; ) O. 7 A # , '
A ! 1 + ( ; 1)(s + 1 ; )
O. B'" #' !
" v, k, . C
# k(k ; ; 1) = (v ; k ; 1) # !
" . D
\$, !" ! A ! (t + 1) ; O.
2#, > 1. 7 !" O ! t+1 O, s +( ; 1)t 6 t +1, 9# = 2, s = 1 t = 0, . B'"
= 1, ; | "
% " 0
#
GQ(s s2 ; s), A (s2 t + s (s ; 1)(t + 1) s ; 2 (s ; 1)2),
A (s ; 1)(t +1), s ; 1, ;(s2 ; 2s +2) '
1, (s ; 1)(st + 1), st . 8
".
; | "
% " % pG(s t),
k, r = s ; , ;m = ;(1 + t) % 1,
t(t+1)
s(s;+1)(st+)
f = s((ss+1)
+t+1;) , g = (s+t+1;) .
&
"!, ; | "
% " % pG(s t), " "
# F
%
(s + 1 ; 2)t 6 (s ; 1)(s + 1 ; )2:
7" " '% a 2 ; 5 (1] (a] (a]0 # . H
, I = (a] !
"
" % pG;1(s ; 1 x), "
( ; 1)t = x(s ; 1). 7 g < f ,
6.2 (7] ; | 5 # (i) '
9% ,
, r2 = r = r1 , m1 + m2 = m ; r, g2 = g1 = g ; 1, "
120
. . r1, ;m1 | (a] f1 , g1 , ,
r2, ;m2 | (a]0 f2 , g2.
2.3. ( ; I = ;(a) ), s = 2, t = (2 ; 1)( + 1)2 .
. &# (s +2 ; 2)x = (s ; 2)(s +1 ; )2 . 7" " '%
b 2 I I(b) "
" pG;2(s ; 2 y),
"
( ; 2)x = y(s ; 2).
2! (s + 1 ; 2)t = (s ; 1)(s + 1 ; )2 ( ; 1) ( ; 1)t x(s ; 1), # (s + 1 ; 2)x = ( ; 1)(s + 1 ; )2. K,
(s + 2 ; 2)y = ( ; 2)(s + 1 ; )2.
2! #
; 2, #"
(s ; 2)(s +1 ; 2)y = ( ; 1)( ; 2)(s +1 ; )2 . B'", ( ; 2)(s +1 ; )2
(s + 2 ; 2)y, # (s ; 2)(s + 1 ; 2) = ( ; 1)(s + 2 ; 2), 9#
s ; 2 = u(s + 2 ; 2) " # u.
&! s + 1 ; 2 = w. 7" = uw + 1, s = 2uw + w + 1. 5 "#%
, s ; 2 = u(w + 1), "
, uw = u + 1 ; w. 7 ,
w = 1 s = 2.
2.4. ( ; | pG2 (s t), ), s
4, 5 7 %
\$ a 2 ; (a] GQ(s ; 1 x), x 9, 8 9 .
. & #' F
% t = (ss;;1)3 3 , 9# s ; 3 1,
2 4. & 9 t 27, 32 54, "
, s 9, 8 9
.
3. GQ(4 )
t
+ 9 "
#"
", ; | "
% " GQ(4 t). & #' \$
t + 4 "
20 12, 9# t 1, 2, 4, 6, 8, 11, 12 16.
3.1. ; 4- 3, % 5-, Ki =
= Ki (3), xi = jKij. (1) x0 + x3 = 4t + 3*
(2) e 2 K3 , d 2 3 ; (e], 30 = 3 feg, Li = Ki (3), yi = jLi j, 3 \ (e] +
(d] \ (e], y0 = t + 2, y2 = t + 4 \$ + L0 t + 1 \$
+ L2.
. &" !"# 3 ;;3 2-#
%
\$ 3, # # #
%
X
X
X i ; 1
xi = 20t + 1
ixi = 16t + 4 x0 +
2 xi = 6:
i
i
i
121
+ #
# , # #
!"
.
&# x3 6= 0, L = ; ; (d? e? ). 7" jLj = 13t ; 4, # 3 \ (e] ' L. &!
, # 3 \ (e] = fa b cg (d] \ (e]. &# a !
% % w 2 ((d] \ (e]) ; 3. 7" (w] "
! 4t ; 3
L, j(a] \ Lj = 4t ; 1, (b] (c] "
! 4t ; 2 L. C, jL;((a](b](c])j 6 t+1, (w]\L "
! "% (a] 2t ; 4 (b] (c], "
, 2t 6 t + 1, t = 1, = 2, , -" d? \ e? "
! #.
7
!" fa b cg ! 4t ; 2 L, 9#
y0 = t + 2. 2
, L2 "
! t ; 2 (d] \ (e] "% (d] \ (z ], (z ] \ (e] " z 2 fa b cg, 9# y2 = t + 4. f 2 L0 , j(f ] \ (z ]j = t + 1 " z 2 30 , "
, (f ] "
! t + 1 L2 .
3.2. , 3.1, t > 11, x3 = 0.
. 2# . 5 3.1 ! L = (d] \ (e] ; 3. 7" jLj = t ; 2 3.1 !"
L0 ! %
% t ; 5 L.
; "
! (4 5)-" M, zi = xi (M), 2.1 z0 +
+ P ;i;2 1zi = 6.
i
; N #% t = 12. 7" L "
! 102 = 252 9
; "!
, L0 "
! 72 14 = 294 "!
.
&# u, w | "
L0 , (u] \ (w] "
! 9
"!
L0 L. &! M = fd e u wO L0g. F ; , z0 + P i;2 1 zi = 6. B'" L0 ; fu wg ! %
% L0 . R, L0 "
! %
9
% " L00 , !" L00 ! "# L0. D " # (6 5)-" fL00O L ; L0g,
.
: (7], #
t = 16 ; .
&# t = 11. &
"!, %"# L0 , !
' L2 ; L. 0
" L0 !
%
% ' L2 ; L, # (5 5)-",
" d, e ! 9 (5 5)-",
. R, ' "
L0 ! L2 ; L %
% '
"
L. D " fd eg L L0 "
! (6 5)-",
.
C, L0 %"
%
% 10 , !" ! ' L. 5 # (5 5)-", 122
. . 9 " ! , .
. C 3.1, 3.2 "#
, -" "
" GQ(4 t), t > 8, "
! 3-. F
, !
", " t = 8 3.2 " L0 "
!
, ! ' 0 L.
3.3. -%
-
+ ; n-
3 < n < 8.
. 2# . &# a, b | !
(a] \ (b] "
! n-# c1 c2 : : :cn. &! 3 = fa b c1 : : : cng,
xi = xi(3). 7" N = n + 2, M = 3n, d1 = d2 = n, di = 4 " i > 2. &
2.1 X i ; 1
n2 ; 11n + 26 t + ;n2 + 7n ; 8 :
x0 +
x
=
i
2
2
2
D !
% \$
3 < n < 8, '
# n = 4, t 6 2. B" 9 #
6 3. 8
".
&# "
-" ; "
! #.
3.4. 3 | + ; xi = xi(3).
(1) ( 3 | , 4-, x0 = 8t + 2.
(2) ( 3 \$ (
+
), x0 + x3 = 5t + 3,
x4 = 0.
(3) ( 3 + \$, \$ (
+
), x0 + x3 = 2t + 4, x4 = 0.
. &
2.1 "# 3 #
,
-" ; "
! # #.
3.5. ( \$ c, d ; -
(c] \ (d]
3- eabf , 3 = fa b c d e f g xi = xi(3) xi = 0 i > 3, x0 = 4.
; . & 2.1 x0 + P i;2 1 xi = 4. 2#, w 2 K3 (3) K4 (3), ! 30 = fwg 3, yi = xi (30P
). ; w 2 K4(3), ,
# 2.1 "# 30, #, y0 + i;2 1 yi < 0. R,
w 2 K3 (3). R
, ' " 3 ! #
: (w] "
! (1) e, b, f O (2) c, eP, dO (3) c, e, f O (4)Pc, a, dO (5) c, a, e (6) c, a, ;b. + !" 9 #
yi = 20t ; 2, iyi = 28t, P 2i yi . P ; + #
(1) 2i yi "
21 % , 4t ; 10 ,
w, 3t ; 5 , ! w. C,
P ; i y !
=
7
t
+
6.
T
!
# (3) (5).
2 i
123
(2) 18 7t ; 13P
; i , 9#
P ;+i y#
=
7
t
+
5.
D
\$,
#
(4)
(6)
#
i
2
; 2 yi = 7t + 4.
7 , y0 + P i;2 1 yi ;t + 2, -
. 8
".
&
%"
"
# 1. &# ; | "
%
" GQ(4 t), "
!0% 49-# O. 7" " A = ;;O # (196 39 2 9).
3.6. A .
. &# 3 | "
P ; A, Ki = Ki (3), xi = jKi j,
Ki0 = Ki \ A, x0i = jKi0j. & 3.4 x0 +P ; i;2 1 xi = 28. &
# 2.1
"# 3 A, #, x00 + i;2 1 x0i = 28. R, K0 K3 A.
&# w 2= 3, 30 = 3 fwg, yi = xi(30 ), yi0 = jKi(30 ) \ Aj. w2K
"# 30 ; A, #, 3 , , P
#
;i;1 2.1
P;
i
;
1
0
0
y0 +
2 yi = y0 +
2 yi ; 2, .
C, x0 = 28, P
x1 ;= 180, x2 = 32, x00 = 28, x01 = 147, x02 = 16. &#
i
0
w 2 K2 \ A, =
2 di, "
di | P; 3 . &
i
;
1
0
# 2.1 "# 3 P;; A, #, y0 + 2 yi = 42 ; ,
y1 = 102 + 2
, y2 = 95 ; , y00 + i;2 1 yi0 = 41 ; , y10 = 84 + 2
, y20 = 65 ; .
D, O "
! 16 K2 33 K1 . B'" O "
! "
#' # L3 , 27 L2 21 L1 .
&
, y2 = y20 = 30.
3.7. A PQ(3 12 9).
0
. &# 3 | P ;i;1A, Ki = Ki (3), xi = jKi j, Ki =
0
0
= Ki \ A, xi = jKi j. & 3.5 x0 +
2; xi = 63. &
# 2.1 "# 3 A, #, x00 + P i;2 1 x0i = 57. & 3.6 x3 = 0,
, x0 = 63, x0 ; x00 = 6.
&# w 2 K2 \ A, 30 = 3 fwg, yi = xi(30 ), yi0 = jKi (30) \ Aj, = P;2i di ,
"
di | P; i3;10 . &
# 2.1 "# 30 ; A,
y0 +
2 yi = 54 ; , y1 = 126 + 2
, y2 = 60 ; ,
P;#,
y00 + i;2 1 yi0 = 51 ; , y10 = 97 + 2
, y20 = 43 ; .
D, O "
! 6 K0 , 9 K2 34
K1 . &# (w] "
! K2 \ O. 7" (w] "
! 3 + K0 \O 10 ; 2 K1 \O. &
, jL2 \Oj = 17 =
= 10 ; 2 +(9 ; ). C, A "
! , 9# '%
13 3-, A #.
3.8. . ; .
. D " ", '% -" (a] \ (b]
%. U
!"
, a b 2 O ( '
A ! "
% % O). +"# 3.7 ! ,
124
. . a 2 A, b 2 O (a] \ (b] "
! cd. & 3.7 A "
!
4-# 3 = fa c d eg, xi(3) = 0 " i > 2. &# Ki = Ki (3). 7"
O "
! 6 K2 , 40 K1 3 K0 .
&# w 2 K0 \ A, 30 = 3 fwg, yi = xi (30), yi0 = jKi(30 ) P
\ A;j. &
i;1y = 50,
# 2.1 "# P;
30 ;
A,
#,
y
0 +
i
2
y1 = 132, y2 = 58, y00 + i;2 1 yi0 = 44, y10 = 111, y20 = 36.
&# (w] "
! K0 \ O. 7" (w] "
! 3 + K2 \ O 10 ; 2 K1 \ O. &
, jL2 \ Oj = 22 =
= 10 ; 2 + (3 ; ). 8
, % 1 ".
4. # pG2(5 32)
+ 9 "
"
, ; | "
% "
% pG2 (5 32). 7" ; | # % (486 165 36 66), " "
# F
%.
&9# " '% a 2 ; (a] (a]0 # (165 36 3 9) (320 99 18 36) .
4.1. ab | ;. ; 36 \$
+ (a] \ (b], 128 \$ + (b] ; a? , (a] ; b? 192 \$ + (a]0 \ (b]0. (
x 2 (a] \ (b], (x] 3 \$ + (a] \ (b], 32 \$ +
(a] ; b?, (b] ; a? 96 \$ + (a]0 \ (b]0.
. &
.
4.2. \$ c, d ; . ; 66 \$ + (c] \ (d], 99 \$ + (c] ; d?, (d] ; c? 220 \$ + (c]0 \ (d]0.
( y 2 (c] \ (d], (y] 9 \$ + (c] \ (d], 27 \$ +
(c] ; d?, (d] ; c? 100 \$ + (c]0 \ (d]0.
. &
.
4.3. -% 5- + ; 6-.
. &# 5- 3 = fa b e f gg ! 6-
;.
7" " ' x y 2 fe f gg " (x] \ (y] "
!
a, b, "
#' # 3 ; fa bg, "% (b] ; a? , (a] ; b?
31 # A = (a]0 \ (b]0.
(e] \ (f ] \ (g] "
! # ((b] ; a? ) ((a] ; b? ), " A
"
! 3 31 K2 (3) 3 34 K1 (3), , A 192 . R, A "
! # K3 (3),
3 30 K2 (3) 3 35 K1 (3), .
4.4. ( ab | + ;, (a] \ (b] (2 3)-
.
/, %
\$ -
+ I = (a] \$ 2.
. 2#, ab | ; (a] \ (b] "
!
(2 3)-" fc1 c2O d1 d2 d3g. 7" (c1]\(c2] "
! a, b, 3 125
d1 : : : d3 (a]\(b], 5 (b];a? , (a];b? 31 # A = (a]0 \(b]0.
R, A "
! 51 # (c1] \ (c2] 45 (c1] ; c?2 , (c2] ; c?1 .
2
, (ci ] \ (dj ] "
! a, b, 2 (b] ; a? , (a] ; b? 30 A. & 9 (dj ] \ A "
! 7 (c1] \ (c2], 9#
(dj ] \ A "
! 23 (c1] ; (c2] (c2] ; (c1 ]. D
" ((c1] \ A) ; (c2 ] "
! 23 3 ; 3 7 = 48 , .
&
#
!"
".
+
#
!"
"#
.
+ 4.5{4.7 I | # % (165 36 3 9),
!" 4- "
! % 5-
-" " "#.
4.5. ( ab | + I, I 3 \$ +
(a] \ (b], 32 \$ + (b] ; a? , (a] ; b? 96 \$ + (a]0 \ (b]0. (
x 2 (a] \ (b] (a] \ (b] , (x] a, b, 2 \$ + (a] ; b? ,
(b] ; a? 30 \$ + (a]0 \ (b]0.
. &
.
4.6. \$ c, d I . I 9 \$ + (c] \ (d], 27 \$ + (c] ; d? , (d] ; c? 100 \$ +
(c]0 \ (d]0. ( y 2 (c] \ (d] \$ y (c] \ (d] , (y] 3 ; \$ + (c] ; d? , (d] ; c? 28 + \$ + (c]0 \ (d]0.
. &
.
4.7. ( \$ I #%
5- Li , -
+ I 3-.
. &# c, d I ! (c] \ (d] "
! 3-# eabf , 3 = fa b c d ef g, Ki = Ki (3), xi = jKi j. & 3.5
x0 = 4, xi = 0 " i > 3, 9# e, f (c] \ (d] 1 # x0 + x1 + x2 = 159, x1 +2x2 = 194. 7 ,
x0 = 4, x1 = 116, x2 = 39. &# a, b, c, d e, f ! L1 : : : L6
. 7" L1 , L6 "
! "
K2 , L2 , L5 | "%,
L3 , L4 ' K2 . 2
, Li , '0 K0 , "
!
"
K0 K2 , "# # K0 "
K2 . , x2 = 6 +3 + 2(4 ; 2) +(27 ; 4 + ) = 37,
. 8
".
2 \$ #"
, ; -" "
! 3-#
%. + 4.8{4.10 "
, ab |
;, (a] \ (b] "
! # xyzw, 3 = fa b x y z wg,
Ki = Ki (3), xi = jKi j.
4.8. , x0 = xi = 0 i > 4, x1 = 168, x2 = 240,
x3 = 72.
126
. . . & 2.1 x0 + x3 + 3x4 = 72. &# x4 = . 7"
# x1 + x2 = 408+2 , x1 +2x2 = 648+8 . 7 ,
x2 = 240 + 6 , x1 = 168 ; 4 .
2
, K3 "
! 4 (a] \ (b], 12 (a] ; b? , (b] ; a? ,
! ! fx y z wg, 8 (a] ; b? ,
(b] ; a? , ! fx y z wg, 28 ; 4 a? b? ,
! 3-# fx y z wg. B'" x0 = . 2#, > 0, # r K0 . 7" 396 2-#
% " ufr " u 2 3.
5 "#% , (r] "
! (a] \ (b] 4 K3 26
K2 , (a] ; b? 20 K3 36 K1 K2 , a? b?
28 ; 4 K3 K4 . 7 , 2-#
% " ufr 4 + 3(72 ; 4 ) + 2(93 + 3 ) = 402 ; 2 .
&# Ki0 = Ki \ (r]0, x0i = jKi0j. & 2.1, % "# 3
(r]0, # x00 + x03 +3x04 = 8. > 2, = 3 (r] "
!
60 K3 " '% r 2 K0 (
,
'
; ! 36 ), = 2, K0 = fq rg (q] \ (r] "
! "
K4 64 K3 .
D "
#
x03 = x04 = 0, x00 = 1, . R, = 1,
x00 = 0 x03 = 8, x04 = 0, x03 = 5, x04 = 1, 2-#
% " ufr 392 393 , .
4.9. 30 = 3 ; fbg, Li = Ki (30), yi = jLi j. L5 = fbg,
y0 = 20, y1 = 17, y2 = 395, y3 = 48.
. & 2.1 y0 + y3 + 3y4 + 6y5 = 74. +"# 4.8
L5 = fbg yi = 0 " i = 4. 2
, 3.4 (b] "
! 20 K1 , '0 L0 . B'" y0 = 20, y3 = 48. 7
y1 + y2 = 412,
y1 + 2y2 = 807, 9# y2 = 395, y1 = 17.
4.10. u 2 K1 \ (b], L0i = Li \ (u], yi0 = jL0i j. y10 = 85,
0
y2 = 196, y00 + y30 = 34.
. +"# 4.9 yi0 = 0 " i > 4. & 2.1, % "# 30 (u]0, # y00 + y30 = 34. 7
y10 + y20 = 285,
y10 + 2y20 = 477, 9# y20 = 192, y10 = 93.
4.11. , ; -
, .
. + #
4.9, 4.10 "#
, 93 = y10 6 y1 = 17, .
4.12. , ; -
, .
. &# a, b | !
;, (a] \ (b]
"
! # xyz , 3 = fa b x y z g, Ki = Ki (3), xi = jKij. +"#
4.11 xi = 0 " i > 4. & 2.1 x0 + x3 = 40. 7
x1 + x2 = 441,
x1 + 2x2 = 807, 9# x2 = 366, x1 = 75.
2
, K3 "
! 21 # (a] \ (b], 6 (a] ; (b], (b] ; (a],
! fx y z g "
#' # a? b?, !#'
127
fx y z g. B'" x0 = 6. &# u 2 K0 , Ki0 = Ki \ (u], x0i = jKi0j. &
2.1, % "# 3 (u]0, # x00 + x03 = 12. 7
x01 + x02 = 303, x01 + 2x02 = 477, 9# x02 = 174, x01 = 127. 7 ,
127 = x01 6 x1 = 75, . 8
".
C 4.12 "#
, ; ' , 9# ; GQ(4 8)-. 7
2
".
%
1] . ., . . . // !". "#. | 1987. | (. 24. | . 186{229.
2] Makhnev A. A. Pseudogeometric graphs connected with partial geometries pG(4 R 1)
// Mathem. Forschunginst. Oberwolfach. Tagungsbericht. | 32/91. | P. 11.
3] 01 . . 2 3 (64 18 2 6) // 4#.
. | 1995. | (. 7, 6 3. | . 121{128.
4] Wilbrink H. A., Brouwer A. E. (57,14,1) strongly regular graph does not exist // Proc.
Kon. Nederl. Akad. Ser. A. | 1983. | Vol. 45, no. 1. | P. 117{121.
5] Paine S., Thas J. Finite generalized quadrangles. | Boston: Pitman, 1985.
6] Goethals J.-M., Seidel J. J. The regular two graph on 276 points // Discr. Math. |
1975. | Vol. 12, no. 1. | P. 143{158.
7] Cameron P., Goethals J.-M., Seidel J. J. Strongly regular graphs having strongly
regular subconstituents // J. Algebra. | 1978. | Vol. 55, no. 2. | P. 257{280.
' ( ( 1997 .
{
( ) ]
GF q
x y
. . ,
. -
e-mail: mechvel@botik.ru
519.6+512.62
: , !"# \$!!!,
!! %!, !&! !\$.
'"( LLL \$!) %%!! * . +. ,! -. .. /! (1982) -. . . (1985) ) %!3 !"#
\$!! " 4 ] ) !\$ %!\$ . -. . . !) !# 3
!&! !!!( 6 ((degx )6(degy )2 ) \$ * )((
.
7! . +. ,! -. .. /! \$ # ) %!! !!!(
%!( !&!, %!)! !# !). 8) \$ %! "\$, ! 9! %!! )!%3 %! !!* %!%! 3:36 !# 3: !!!( 6 ((degx )4(degy )3 ).
f
F x y
F
O
f
f
F
O
f
f
Abstract
S. D. Mechveliani, Cost bound for LLL{Grigoryev method for factoring in
( )4x y], Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1,
pp. 129{139.
GF q
The well-known LLL method was accommodated in papers by D. Yu. Grigoryev
and A. L. Chistov (1982) and A. K. Lenstra (1985) for factoring a polynomial in
4 ] over a >nite >eld . A. K. Lenstra derives a cost bound for his method with
the main summand ((degx )6 (degy )2) arithmeticoperationsin . D. Yu. Grigoryev and A. L. Chistov aimed to provide a method of a degree cost bound and did not
consider any detailedestimation. Here we show that this methodallows, after certain
correction, to prove a better bound with the main summand ((degx )4(degy )3 ).
f
F x y
F
O
f
f
F
O
f
f
1. CG] 1982 ( G] ) !" # f Ft x] \$ # !# \$\$ F, #
\$ & # (degx f)(degt f) q s.
, 2002, !\$ 8, ? 1, . 129{139.
c 2002 ! "# \$%&,
' () *
130
. . * &+ ,#.
q | #, q = qs , F = GF(q ) | q ,
A = F t], p | \$ # A,
deg = degx , r = deg f, f = degt f | / A Ax],
v = max vi : 1 6 i 6 n | A,
c(q) | " !# \$ GF (q) =
= Z=(q), " (O((log q)2 )),
c(F) | " !# \$ F
(O(s2 c(q))),
P a xi | \$ !" # Ax],
f=
i
0
0
0
j
j j
fj
j
j
g
06i6r
logDf = logq (r f ), lc f | 1\$ /!!" #,
cont f = # = 234 /!!" f,
a], a b c],... | ( ) / \$ , D, & D,
a : : : : 5(a)] | , ! \$ & 5,
n::m] | " # n m &# , n 6 m #
&, n > m | ,&.
6 , # # !" Ft]x].
3, G,Le] & ,\$ , LLL !" Zx]7 ,+ # , # " Z F t] . 8 , | G] |
( ) 2. 4 \$1
\$ # , 2, \$
, 9.
: \$ , ! Mi] (1989), +: !" Zx] Ft x].
;, Le, 2] # G] : b
1: L (. 9). 9 G] + b, \$
1 : /!!" uij F | 1 h0-sv
9. < Le, 2] b : Le, 1]. G] ,\$ " , Le] ( 2.18) "
O(r6 f 2 + (r3 + f 3) q s)
(CB-Le)
\$\$ F (! 1 ,#). 21\$ " &
, , \$ , # , , ,, ,,& . 4 ,#\$ , !", 11\$ j
j
2
j j
j
j
131
{
&\$ , DoCon Me]. -, , , #: # & Le, 1, 2]. ;+ + , GK] (1985),
+:, , , !". 4 : #1\$ ", # (CB-Le). ? ,
# G], (1) G],
(2) / G, Le] 1:, #1& " # :
/!!". 8 : \$ "
O(r4 f 3 + r4(logq r)3 f 2 + r3 f (logD f)3 s q (log q))
(CB)
\$\$ F. 6# /, # \$ L , 1
\$\$ 1, # "\$ ,? 2. * ,
# , G] # #1 ". &
& # , L. 8 , " #1& , " ,. 9 1 #1.
G] ( 2).
(1) 2 1 h0-lt \$ , 1: L, \$ \$ , B. * / # , L.
(2) * &+ p A 1 p #
.
*: (1) #1& " # : /!!". *: (2) , , , # Ft] " .
j
j
j
j
j
j
2
2. !"
# #
\$
#% ] ] \$ % \$
f
F t
x
F
* ,# 1, 1 /\$ #.
sq. ; , , f Ax].
# 234 \$, :, # #& , f, # degx f > 1,
cont f = 1, lc(lc f) = 1.
p. 2\$ A &+\$ \$ # p. 4#
\$ p GF (q)t], \$ F \$, # p lc f f mod p , . 4 / p GF (q)t], # \$ p F. 2# f mod p 234 \$.
2
2
2
132
. . f. A f mod p (A=(p))x]. B , f . 3: #\$, \$ f mod p = (h1 mod p)(g mod p)
h1 mod p A=(p), deg(g mod p) > 0.
* h1 g
Ax] h1 mod p,
g mod p, &+ /!!", " p, l1 = deg h1 <
< deg f = r.
2
h0. 2\$ \$ h0 f, &+\$
h1 mod p. ;+ \$ h0 , # h1 mod p
h0 mod p. h0 " m 6 r:
m = l1 l1 + 1 : : : h0-k k = min k : p k l1 > m f + r f 7
(A-h0-1)
h0-h h = E:*F& (p f k h1). k<# h k
f
0
j j
0
j
j
j
jg
: h = h1 (mod p), h mod p f mod p , lc h = 1,
deg h = l1 7
h0-l , M 1: L = L(m k) A:
L = v Ax] deg v 6 m h mod pk v mod pk L h xi, pk xj (m + 1) (m + 1) "
M = h xm l1 : : :h pk xl1 1 : : :pk ]
v(0) : : : v(m)] A , #
X
v=
v(i)xi Ax]7
f
2
j
g
;
;
2
06i6m
h0-lt M \$ \$ \$ " B7
h0-sv L # \$ \$
b. 4 / :\$ u Am+1 0 = b = u B,
&+\$
p k l1 > m f + r b :
(A-h0-2)
3.3 , # # u u 6 U. (A-h0-2) : \$& /!!" uij F # ui A. 3: 1 / 7
h0-e , h0 = b=(cont b). B
, " &+ m.
2
j j
j
j
j
2
2
6
j
j j
{
3. #
133
3 1 sq, p G]. ; & 1 sq ,# | 234 Ft]x] , ,. 2, # 234 \$\$ ,
.
; 1 p " #\$
O(r3 f (logDf)3 q (log q ) c(F )):
< , G] q = qs , / ! G] F # , ,,+ #\$ # , # & /\$ .
j
0
j
0
3.1. h0
* h # E: h1 1 h0-h.
#, h = h1 (mod p), h mod pk f mod pk .
h0.1. h0 f Ax], h mod p h0 mod p.
4 , # f , , p
, , # " p .
h0.2 ( ! "LLL]: Proposition (2.5)). f, k, h h0-h g f Ax]. :
(i) h mod p g mod p (A=(p))x]!
(ii) h mod pk g mod pk (A=(pk ))x]!
(iii) h0 g Ax].
2 / 2
1 G].
2
x
3.2. L(m k)
G] , B " u
1 h0-sv: ui 6 i , = max bij : bij B .
2 , ". */ 9 \$ \$ , B, " \$ " M. *
m l1 + 1 M & "
M1 = h xi i (m l1 )::0]]
xi i- ,". 31 # " M2 , \$ l1 ," & & " c pk
j
j
j
fj
j
;
j
2
;
g
j
134
. . , ," . *, " M1 M1 , M2 , + / A
( 4.2), : M1 B1 = xi + T (i) i m::l1]]
T (i) Ax], deg T (i) < l1 , # x ,& , T (i) |
. 6 m l1 + 1 ," B1 ,& #&
" E7 # " B1 T . *# , #
\$ \$ "\$:
! !
E
T
B=
= B1 :
(h0lt)
k
00 p E
B2
B 1:, E, E # " m l1 +1
l1 , 00 | ".
0
0
j
2
0
2
0
;
0
0
0
0
;
3.3. !
L(m k)
I&,\$ L u B u Am+1 . ;
3.4, 9 + \$ u, #
0 = u B def
= b p k l1 > m f + r b :
(A-h0-2)
8 u = 0 u B 6 U U = max i: p k l1 m f > r i :
(uO)
B U > pk 1, B # b. */ 1 U < pk 1. J (uO) , #, \$
/!!" uij F &+ ui u, i 1::(m+1)],
j 0::U ]. 6 U # u . 2# # # # . 2 " B , #
# # U = U. N, . " b , # u B u 6 U.
4 # #
01 0 b b 1
13 14
B
0
1
b
b C
B=B
@0 0 p23k 034CA bij < pk 0 0 0 pk
# ,+\$ #\$.
u = u1 u2 u3 u4]. * 0 = 1. N (uO) , #
u1 u2 6 U. 4 u \$ ," (uO) # u1b13 + u2 b23 + u3pk 6 U. O U < pk 1, , u3 = 0, ,
0 < u3 + pk = u1b13 + u2b23 < U + pk :
6
6
j
j
2
j j
j
j j
f
j j
j j
;
j
j
j;
j
j ;
2
0
2
g
2
0
j j
0
j
j
j
j
j j
j
j j
j
j
;
j
j
j
j
j
j
j
j
j
j;
j
j
j
135
{
*/ u3 < U. J u #:\$ ," c, #, u4 .
j
j
j
j
3.4. # L(m k)
\$ %
f
3, & # : 1 h0, " m = l1 l1 + 1 : : :.
*# b L(m k) (A-h0-2) : \$ \$ h0 f? *# \$ b + m "?
& ('!( 1.2 "G] ')* ! "LLL]).
f, r, h1 , l1 , m | %, k | &, h | () h1 f k, L = L(m h) = L(m k) |
) %.
, 0 = b L p k l1 > m f + r b , h0 b Ax].
4 G], Le, Proposition 2.7].
6
2
j j
j j
j j
+ .
(1) * m l1 ::] % m = deg h0.
(2) + (
) b, %, b = a h0 a A.
4 (1): \$ b L = L(m k)7 & L deg b 6 m. * m < deg h0 , (A-h0-2) . N, #, , b h0 , deg h0 6 deg b 6 m |
#.
4 m = deg h0 b = h0 (A-h0-2). 4\$ , h0, L , # h0 L. * , k p k l1 > m f + r f . ?
h0 f, f > h0 p k l1 > m f + r h0 .
4 (2): h0 b. ?, (1), deg h0 =
= deg b. */ b=h0 A. N, 9 ,.
2
2
2
j j
j
j
j j
j
j j
j
j
j
j
j
j
j
2
4. 4.1. &
* 9 f F t]x] , 1 # O(C-h0-sv + C-p-Berlekamp)
(CB )
\$\$ F, C-h0-sv = r4 f 3 + r4 (logq r)3 f 2 C-p-Berlekamp = r3 f (logDf)3 s q (log q)
0
j
j
j
j
j
j
136
. . ,# 1. /\$ ! (CB) C-h0-sv " 1 h0-sv 9: # 1: L(h1 m k(m)), \$ ,. 8 \$ & m deg h1:: deg h0 ] \$ h0 # f. # C-h0-sv / \$.
; C-p-Berlekamp " p 1 p !" f mod p. 3" (CB ) , , , &+ & 9:
2
0
T = T (1 sq, p, f) +
+ T (1 h0, \$ \$ h1 ):
T (1 sq, p, f) = O(C-p-Berlekamp) = O(C-sq-sepp + C-ftbyp), C-sq-sepp = r3 f (logD f)3 s q (log q) " , f &+ p,
C-ftbyp = r3(logDf) s q | " !" f mod p #
h1. ..
j
j
T (1 h0) ( h0 h1 mod p) =
= O(((deg h0 ) l1 + 1) C-LP(m)) |
" LP(m) m l1 ::], ;
2
C-LP(m) = C-Hensel + C-triang + C-smallV,
C-Hensel = r3 f 2 (logDf)2 | E: F& h1 h = h(k),
C-triang = r3 f 2(logD f)2 | ,,
C-smallV = r3 f 3 + r3 (logq r)3 f 2 | 1 uij .
j
j
j j
j
j
j
j
; C-h0-sv (CB ) T (1 h0) h0 # f. 4 h0 " LP(m) , (deg h0) . T LP(m) O(r3 f 3 + r3(logq r)3 f 2).
: / " (CB), " (C-triang),
(C-smallV) #\$ 9. 3 # " G,Le].
0
j j
!(.
j
j
1. 2, (#) # 9 1 L(m).
2. *E: F& 1 h0-sv, ", 1& " . */ E: &# & ".
137
{
, -* . 3" (CB-Le) Le] # ,#\$ (,# Le] ,# G]):
(p q) (m s) (X x) (Y t) (X degx = deg)
(Y degt = ) (h h1) (F p) (u f ):
. /. ,
q, f, p, r, l1, m, k % p 6 1 + logq (2r f ) = O(logDf)
(PB)
l1 p k < O(r f + r (logq r))
(LPK)
l1 p k < O(r (logq r) f ):
(LPK )
4 (PB) # ( CG]) , #
# tqa t R = resultantx (f df=dx) qa > R .
4 (LPK): , & (A-h0-1) k l1 6 m 6 r (r + m) f < l1 p k 6 (r + m) f + l1 p = O(r f + r p ).
* (PB), #
l1 p k = O(r f + r (logq r) + r (logq f )) = O(r f + r (logq r)):
2", (LPK ) \$1 (LPK).
- 0
sq, p: . # 3.
- 0
f. 4 !" (f mod p) (A=(p))x]
Le] Be, 5] : " O(r3 p s q c(F )). * &
(PB) p , # O(r3(logDf) s q c(F)).
- 0
h0-h. ; E:
h1 mod p (h mod pk ) A=(pk ) " #\$
O(r3 f 2 (logD f)2 c(F )).
!. Le] \$ Yu] "
O( p r f + p 2r3 + k2 p 2l1 (r l1 ))
\$\$ F . < : < O(r3 f p 2 + (l1 p k)2r), : (LPK) l1 p k (PB) p .
!
!
!
!
!
j j
!
!
!
!
j j
j
j j
! j j
j
j j
j j
j
0
j
;
j
j j
j j
j j
j j
j
j j
j
j
j j
j
j
j
j j
j
0
2
j j
j j
2
j
j
j j j
j
j j
j j
j jj j
j j
;
j j
j j
4.2. &' %
(
h0-lt
; , M , B ( 3.2) # #\$
O((deg h0)r2 (logq r)2 f 2 c(F)):
!. 8 1 ml (m + 1) " M1 ( 3.2) B1 / , , #
# T1 " M1 #\$ " E ml ml ,
j
j
0
0
0
138
. . ml = m l1 + 1. 9 # T , ,& l1 ," " B1 . * / A
,: pk . 8 1: L + /. * , \$ " A=(pk ). ? " M1 1 O((m l1 )l1 ) \$\$ +,
, A=(pk ). 3 , #: :
M1 # , \$ # h. <"
M1 = trg(h), # M1 , # \$ "\$ trg:
0
0
;
;
;
0
trg(h') = deg(h') == m ]
_p^k(trg(h'*x - c(h')*h)) ++ h']
6 c(h') /!!" # h' ( x) l1 1, ++ h']
# " ".
2, # h \$ 0 0 0 1 a(1)b(1) c(1)]
A, h # # 0 0 1 0a(2)b(2) c(2)], 0 1 0 0 a(3)
b(3) c(3)],... a(i), b(i), c(i) A, " pk . 8
& & " B1 . I
, # / # h & 1:, # M1 . 6 , # j- 1 h j \$ 1\$ &+\$ 1
l1 &+ & #:. */ 1 h0-lt # #
O((m l1 )l1 ) = O((deg h0)l1 ( p k)2) = O((deg h0 )(l1 p k)2)
\$\$ +, , A=(pk ). ? (LPK ) : ,& ".
;
0
0
0
;
;
j j
j j
0
4.3. &' (
h0-sv
; b L, &+ (A-h0-2) ( 3.3), " O(r3 f 3 + r3 (logq r)3 f 2 )
\$\$ F .
!. ; 3.3, U , , # r U <
< l1 p k m f 6 r(U + 1). 6 L + b, &+\$ 0 = b 6 U. ? # , , # r U < l1 p k. B U > pk 1,
B (A-h0-2), " 1 .
#, 3.3 b u B, u Am+1 ,
u 6 U < pk = p k. */ 1 \$\$ #
uij F , X
ui + 1 = O(r U) = O(l1 p k):
j
j j
6
;
j
j
j j
j
j j
j j
j
j
j
j
j
j ;
2
j j
2
16i6m+1
j
j
j j
; \$ /\$ ? * m l1 + 1 ," " B
\$ uij &. W\$ ," Bc(i) : ,
\$, &+\$ u Bc(i) < U |
()
;
j
j
139
{
l1 , \$ \$ uij . ? U, Bij < p k, \$
, ( ) , 1 p k \$, \$ , 1, #
l1 p k, 1 \$ \$\$ O((l1 p k)3 c(F )).
* (LPK) : O(r( f + logq r)3 ) = O(r3 ( f 3 + f 2 (logq r)3)) = O(r3 f 3 + r3(logq r)3 f 2):
j
j
j j
j j
j j
j j
j
j
j j
j
j
j j
j
j
(
Be] Berlekamp E. R. Factoring polynomials over large nite elds // Mathematics of
Computation. | 1970. | Vol. 24. | P. 713{735.
CG] Chistov A. L., Grigoryev D. Yu. Polynomial-time factoring of the multivariable
polynomials over a global eld. | Preprint E-5-82 of the Leningrad department
of Steklov Mathematical Institute LOMI, USSR, 1982.
G] . . !" #"\$ " "% &"\$"'# ( # )"
**+# ,\$*&- .""/ // 0(*& ".\$"'- *#" 1234 56
777. | 1984. | 8. 137. | 7. 20{79.
GK] Von zur Gathen J., Kaltofen E. Polynomial-time factorization of multivariate polynomials over nite elds // Mathematics of Computation. | Vol. 45. | P. 251{261.
Le] Lenstra A. K. Factoring multivariate polynomials over nite elds // Journal of
Computer and System Sciences. | 1985. | Vol. 30. | P. 235{248.
LLL] Lenstra A. K., Lenstra H. W. Jr., Lovasz L. Factoring polynomials with rational
coe:cients // Math. Ann. | 1982. | Vol. 261. | P. 515{534.
Me] Mechveliani S. D. DoCon, the Algebraic Domain Constructor. Manual and program. | ;* -0 **&/, 1998, 2000. | 7#. ftp.botik.ru:/pub/local/
Mechveliani/docon/.
Mi] Mignotte M. Math<ematiques pour le caclul formel. | Presses Universitaires de
France, 1989. (;% " " /*&/: Mathematics for computer algebra. |
Springer-Verlag, 1992.)
Yu] Yun D. Y. Y. The Hensel Lemma in Algebraic Manipulation. | MIT: Cambridge,
Mass., 1974@ Garland: New York, 1980.
+ #, " 2001 .
. . , . . 517.5
: , .
. , " 1 < p 6 q < +1, K (x) > 0 8x 2 Rn (Af )(x) =
Z
Rn
K (x ; y)f (y) dy = K f
(
Lp Lq , )
C (p qn), Z
C sup jej1=p1;1=q K (x) dx 6 kAkLp !Lq :
e2Q(C )
e
*(
+ Q(C ) | - - .
,
("
/)- "/ je + ej 6 C jej, jej | .
e.
0" 1 < p < q < +1, A (
Lp Lq Q |
- - , )
C (p qn),
Z
C sup 1=p1;1=q K (x) dx 6 kAkLp !Lq :
e2Q jej
e
Abstract
E. D. Nursultanov, K. S. Saidahmetov, On lower bound of the norm of integral
convolution operator, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002),
no. 1, pp. 141{150.
We study the lower bound problem for the norm of integral convolutionoperator.
We prove that if 1 < p 6 q < +1, K (x) > 0 8x 2 Rn and the operator
(Af )(x) =
Z
Rn
K (x ; y)f (y) dy = K f
is a bounded operator from Lp to Lq , then there exists a constant C (pq n) such
that
Z
C sup jej1=p1;1=q K (x) dx 6 kAkLp !Lq :
e2Q(C )
e
Here Q(C ) is the set of all Lebesgue measurable sets of 5nite measure that satisfy
the condition je + ej 6 C jej, jej being the Lebesgue measure of the set e.
, 2002, 8, 6 1, . 141{150.
c 2002 ! " #\$%,
&' (
142
. . , . . If 1 < p < q < +1, the operator A is a bounded operator from Lp to Lq , and
Q is the set of all harmonic segments, then there exists a constant C (pq n) such
that
Z
C sup jej1=p1;1=q K (x) dx 6 kAkLp !Lq :
e2Q
e
Rn | n- . (Af)(x) =
Z
R
K(x ; y)f(y) dy = K f
(1)
n
! " Lp Lq , Lp = Lp (Rn) | \$%.
1 6 p 6 q 6 +1 ' , ( kAkLp !Lq 6
6 kK kLr , 1r = 1 ; p1 + q1 . ) ( "*
+ + K(x) = jx1j . , R
-{\$ Af = n jxf;(yy)j +++ (
, n = 1 ; p1 + q1 .
/') 12] % " kAkLp !Lq 6 C kK kr1 (1 6 p 6
6 q 6 +1, r1 = 1 ; p1 + 1q , Lr1 | 34(), % , ( ', ( ( 5 5 -{\$.
/ " /') +++ !+ 4:
Z
1
(2)
kAkLp!Lq 6 C(p q) sup jej1=p;1=q K(x) dx
e2E
R
e
E | * "*5 (5 "5 \$% * Rn, jej | \$% * e.
7 % "(+ * 4 . ", ( E "+ % " , 8 (2) %!+.
7 ! ".
1. 1 < p 6 q < +1, K(x) > 0 8x 2 Rn. (1)
Lp Lq , C(p q n), Z
C sup jej1=p1;1=q K(x) dx 6 kAkLp !Lq :
e2Q(C )
e
Q(C) | ! ,
"
" je + ej 6 C jej.
. :, ( p = q ( 5 (2) 8 (+ %5 (
( (1), ! " Lp Lp , ( %
" 7. ;. , (. 13]).
143
x = (x1 : : : xn) 2 Rn, y = (y1 : : : yn ) 2 Rn. = " x 6 y % "( xi 6 yi 8i = 1 : : : n. ;+ x 2 Rn y 2 Rn 4
x y = (x1 y1 x2 y2 : : : xn yn ), + ! , (
* . =* %"(
Rn+ = fx 2 Rn j xi > 0 8i = 1 : : : ng
Zn+ = fm 2 Zn j mi > 0 8i = 1 : : : ng:
> 2 R d 2 Rn, d = d = (d1 : : : dn).
d 2 Rn+. ?" Id (z) %"( + 4 ( z 2 Rn , +,
! d1 : : : dn, n
Id (z) = x 2 R jxi ; zi j < d2i :
. h 2 Rn+, m 2 Zn+, Id | .
3*
Qd (x m h) =
Id (x + i h)
06i6m
" ( ", * Id (x).
3* 5 (5 " %"( (" Q.
2. 1 < p < q < +1. (1) Lp Lq , C(p q n), Z
1
C sup jej1=p;1=q K(x) dx 6 kAkLp!Lq :
e2Q
e
1. (1) ( " Lp (Rn) Lq (Rn).
f(x) = ;(e+e) (x) | 5( @4 * ;(e + e). A, ( kf kp = je + ej1=p 6 C 1=p jej1=p. =
kAf kq =
Z Z
RR
n
>
Z q
K(x ; y)f(y) dy dx
n
Z
q
K(x ; y) dy dx
1
1
=q
=q
=
>
Z Z
q
=q
1
K(y) dy dx
:
;e ;(e+e)
e (e+e);x
> x 2 e, e + x e + e. /+ %5 ( x, (, ( x 2 e, e e + e ; x. = 4 K(x) " *+ (
144
kAf kq >
. . , . . Z e
Z
(e+e);x
q
=q
1
K(y) dy dx
>
Z Z
e
e
>
q
K(y) dy dx
1
=q
Z
= jej1=q K(y) dy:
) , ( kAf kq 6 kAk kf kp 6 C 1=p kAk jej1=p:
,,
Z
jej1=q K(x) dy 6 C 1=p kAk jej1=p :
e
e
7 " e B "(
Z
C ;1=p sup jej1=p1;1=q K(x) dx 6 kAk:
e2Q(C )
e
7 (, p = q ( C1kK kL1 6 kAk. = ".
1. x 2 Id" (0), "! ! 2 Rn
Id (!) Id(1+") (! ; x):
. +
x 2 Id" (0) () jxij 6 di2 " 8i = 1 : : : n
z 2 Id (!) () jzi ; !ij 6 d2i 8i = 1 : : : n:
= + " "+ ( z " + Id (!) z 2 Id (!) =) jzi ; (!i ; xi )j 6 jzi ; !i j ; jxij 6
6 d2i + d2i " = di (12+ ") 8i = 1 : : : n:
,, z 2 Id(1+") (! ; x). " % ( z "
.
2. Id (x) 2 Id(1+")(y) , " I i ,
i = 1 2 : : : 2n, jI i \ I j j = 0 i 6= j , jI i j 6 (1 + ")2n;1"jId (x)j
2n
Id(1+") (y) n Id (x) = S I i .
i=1
. Id (x) = 1a1 b1] : : : 1an bn]
Id(1+") (y) = 1a01 b01] : : : 1a0n b0n]:
145
jb0i ; a0ij = (1 + ")di = (1 + ") jbi ; ai j:
= ( 5 * "+, , ! *:
I 1 = 1a01 a1] 1a02 b02] : : : 1a0n b0n]
I 2 = 1b1 b01] 1a02 b02] : : : 1a0n b0n]
I 3 = 1a1 b1] 1a02 a2] : : : 1a0n b0n]
I 4 = 1a1 b1] 1b2 b02] : : : 1a0n b0n]
: : :: : :: : :
I 2i;1 = 1a1 b1] : : : 1ai;1 bi;1] : : :1a0i ai ] : : :1a0i+1 b0i+1] : : : 1a0n b0n]
I 2i = 1a1 b1] : : : 1ai;1 bi;1] : : :1bi b0i] : : :1a0i+1 b0i+1] : : : 1a0n b0n]
: : :: : :: : :
I 2n = 1a1 b1] : : : 1bn b0n]:
2Sn
+, ( Id(1+") (y) n Id (x) = I i jI i \ I j j = 0. =* i=1
+ I i * 4 ! %":
jI i j 6
Y
jb0j ; a0j j(jb0i ; a0ij ; jbi ; aij) 6 (1 + ")dj ((1 + ")di ; di) =
i6=j
i=
6j
2n
Y
=
dj (1 + ")2n;1 " = (1 + ")2n;1" jId (x)j:
Y
j =1
\$ ".
3. 1 < p < q < 1,
Z
1
sup 1=p;1=q K(x) dx < +1
e2Q jej
e
(1) Lp (Rn) Lq (Rn). # C , K , Z
C sup jej1=p1;1=q K(x) dx 6 kAkLp !Lq :
e2Q
e
. ( 1
J = sup jej1=p;1=q
e2Q
Z
e
K(x) dx < 1:
= ( 5 ! ( " Qd (x m h), (
146
. . , . . 2
1
=p
;
1
=q
jQd (x m h)j
Z
Qd (xmh)
;
K(x) dx > J:
(3)
1 1
16n 22n .
" =
/%"( 1m"] = (1m1"] : : : 1mn"]).
k | ( 1m"], + 5 1mi "] > 0, . .
k=
m
X
i=1
sign1mi "]:
) (+ %!, * (, ( i = 1 2 : : : k 1mi "] > 1.
/%"( (" Q (5 " Qd (! m h)
Qd" (0 1m"] h), . . Q = Qd (! m h) + Qd" (0 1m"] h).
*, ( Q * +++ ( ". Q=
(Id (! + i h) + Qd" (0 1m"] h)) =
=
=
=
06i6m
06i6m 06j 6\$m"]
06i6m 06j 6\$m"]
06i6m+\$m"]
(Id (! + i h) + Id" (j h)) =
Id(1+") (! + (i + j) h) =
Id(1+") (! + i h) = Qd(1+")(! m + 1m"] h):
(4)
> x 2 Id" (j h), Qd (! m h) + x Q, ,
Qd (! m h) Q ; x = Qd(1+") (! ; x m + 1m"] h):
= * ! * ( 8+ ( ". x 2 Id" (0) Qd" (0 1m"] h).
=
Qd(1+") (! ; x m + 1m"] h) =
= Qd(1+")(! ; x (m1 m2 + 1m2 "] : : : 1mn"]) h) Qd(1+") (! ; x+((m1 + 1)h1 0 : : : 0) (1m1"] ; 1 m2 +1m2 "] : : : 1mn"]) h) =
= Qd(1+")(! ; x m n) Q1 Q2 : : : Qk Qi = Qd(1+") (! ; x + yi li h) i = 1 : : : k:
:
yi = (0 : : : 0 (mi + 1)hi 0 : : : 0)
li = (m1 : : : mi;1 1mi"] ; 1 mi+1 + 1mi+1 "] : : : mn + 1mn"]):
A, ( jQi \ Qj j = 0 jQi j 6
n
Y
i=1
di (1 + ") 6
n
Y
i=1
jY
;1
i=1
mi 1mj "] di (1 + ") Y
i6=j
n
Y
147
(mi + 1mi "]) 6
i=j +1
mi (1 + ") mj " 6 2n;1jQd(! m h)j ": (5)
= x 2 Id" (0), 1 Id (!) Id(1+") (! ; x), ,
Id (! + i h) Id(1+") (! ; x + i h). B 2 ! +, (
Id(1+") (! ; x) = Id (!) I 1 : : : I 2n jI i \ I j j = 0 i 6= j, i = 1 : : : 2n, j = 1 : : : n, jI i j 6 "(1 + ")n;1 jId (!)j
8i = 1 : : : n. = (" Q^ i, i = 1 : : : 2n, %"( ( "
Y
Q^ i =
I i + j h i = 1 : : : 2n:
06j 6m
jQ^ i j 6 "(1 + ")n;1jQd (! m h)j 6 " 2n;1jQd(! m h)j:
Qd(1+") (! ; x m n) = Qd (! m h) Q^ 1 : : : Q^ 2n:
(6)
/ , (
Qd(1+") (! ; x m + 1m"] h) = Qd (! m h) Q^ 1 : : : Q^ 2n Q1 Q2 : : : Qk (, x 2 Id" (i h), 0 6 i 6 1m"], % x = y + i h, y 2 Id" (0), !+ Qd(1+") ((! ; i h) ; y m + 1m"] h), * Qd(1+") (! ; y m h), 8" 5.
f(x) = ;Q (x) | 5( @4 * ;Q. =, (+ %*+ ! "
5, kAf kq =
Z Z
RR
n
>
>
>
;
n
Z
Z
;Qd" (0\$m"]h) ;Q
Z
Z
Qd" (0\$m"]h) Q;x
Z
Qd" (0\$m"]h)
2n Z
X
i=1 Q^ i
q
K(x ; y)f(y) dy dx
K(y)dy ;
q
q
K(y) dy dx
Z
i=1 Qi
>
=q
1
K(x ; y) dy dx
Qd (!mh)
2n Z
X
=q
1
=q
1
>
>
K(y) dy ;
K(y) dy
q
1
dx
=q
:
(7)
148
. . , . . H" (3) , ( + % ( " QI Z
1 Z K(x) dx 6
2
K(x) dx
1
=p
;
1
=q
1
=p
;
1
=q
I
jQj
jQd (x m h)j
Z
Q'
Q'
I j1=p;1=q 2
j
Q
K(x) dx 6 jQ (x m h)j1=p;1=q d
Z
Qd (xmh)
Qd (xmh)
K(x) dx :
7 % " (5), (6) , (
2jQ^ ij1=p;1=q 6 2(2n;1 ")1=p;1=q 6 1 8n
jQd (x m h)j1=p;1=q
i
1
=p
;
1
=q
^
2jQ j
1
jQd (x m h)j1=p;1=q 6 8n :
=, *+ (7), (
kAf k > 12
Z
q
=q
1
Z
Qd" (0\$m"h]) Qd (!mh)
K(y) dy
=
= 21 jQd" (0 1m"] h)j1=q
Z
Qd (!mh)
, , ( kAf kq 6 kAk kf k = kAk jQj1=q 6 22njQd(x m h)j1=q :
n
K(y) dy : (8)
(9)
n
n
k Y
Y
Y
1
n
jQd" (0 1m"] h)j = "di 1mi"] > " di mi " =
mi " 2 i=1
i=1
i=1
i=1
i;k+1
2n
= "2n 21k jQd (x m h)j 6 "2n jQd(x m h)j: (10)
Y
k
Y
H" (8), (9), (10) (, ( ! C, "+! p, q, n,
(
Z
1
C jQ (x m h)j1=p;1=q K(x) dx 6 kAkLp !Lq :
d
Qd (xmh)
,,
1
C1 sup jej1=p;1=q
e2Q
\$ ".
Z
e
K(x) dx 6 kAkLp !Lq :
149
2. *, ( (1), -
! " Lp Lq , (, Z
J = sup jej1=p1;1=q K(x) dx = +1:
e2Q
e
@4 fKi(x)g+i=11 , 8
>
<i signK(x) jK(x)j > i x 2 I(i:::i) (0)
Ki (x) = >K(x)
jK(x)j 6 i x 2 I(i:::i)(0)
:
0
x 2= I(i:::i) (0):
= i!lim
K (x) = K(x), \$% 5
+1 i
Z
Z
R
R
lim Ki (x ; y)f(y) dy = K(x ; y)f(y) dy:
i!+1
n
n
= Ai f = Ki (x ; y)f(y) dy +++ ( n
R
R
sup
1
e2Q jej1=p;1=q
Z
e
Ki (x) dx < +1:
, 3 ! C, "+!+ Ki (x), + (
Z
C sup jej1=p1;1=q Ki (x) dx 6 kAikLp !Lq :
(11)
e2Q
e
K5{L"
lim kA k
= kAkLp !Lq :
(12)
i!+1 i Lp !Lq
= J = +1, + % M ! (
" Q, (
1 Z K(x) dx > 2M:
jQj1=p;1=q Q
= + 5 ( %85 i ++
Z
1 jQj1=p;1=q Ki (x) dx > M:
Q
7 " M B "(,
(
Z
1
= +1:
K
(x)
dx
lim
sup
i
1
=p
;
1
=q
i!+1
jej
e2Q
e
, (11), (12) kAkLp !Lq = +1. = %", 8 ( , kAkLp!Lq < +1. ,,
J < +1. = 3 ( *+ .
150
. . , . . 1] . ., . ., . . !-
# \$ %. | .: !, 1975.
2] O'Neil R. Convolution operators and L(p q) spaces // Duke Math. J. | 1963. |
Vol. 30. | P. 129{142.
3] 5 . . . | 6: !, 1983.
) !* 1997 ".
. I
. . 517.977
: , , , .
. ! "# \$ x_ = (t)A(t)x, (t) | -*+ .
Abstract
D. M. Olenchikov, Impulse control of Liapunov exponents. I, Fundamentalnaya
i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 151{169.
De1nition of solution of the system x_ = (t)A(t)x, where (t) is Dirac's
delta-function, is introduced by means of non-standard analysis methods.
(., , 1{4]). &
( )
**+
. &+ ,
-
( -
). , - ( (. . ( (
, ) ,
-*+ ,
*+, **+
. . / ) (
( ) ( **+
(., , 5]). , 6, . 145]
34 5 3 *
*
6 (
97-01-00413) +
(
97-04).
, 2002, 8, 9 1, . 151{169.
c 2002 ,
!" #
152
. . .. 3. 4( ( **+
, (
(
(
. 5 - .
/ (, (
( (, *
.
4 (
6
+, -*+ ( ( . (
60- ( 7]). , - ( 8{13].
& ( . ; (
(
6 , ( (
. , (,
( ) ( (2)
( : , ( ) , *
;{6
.
= (
14,15].
1. . ( , (
. >
( ( (
(
, ) , . ? - (
. > ( (
,
) R R, @ ) . 3
, -
R . / , , *+, *+
.
A ) . ., *+ ( R R. & *+ () *+, )
(, *+, (. B - , , (. C
( (
16].
. I
153
1.1. B@ , D E, (. 4 X1 = R, Xi+1 = Xi P(Xi ), P(Xi ) | - Xi .
1.1. X1 ( , -
( .
, .
1.2. / M = S Xi ( i2N
. A
(M ; X1 ) ( M .
4, . ,-, @ . H
a 2 M b 2 M , ( (a b) fa fa bgg,
(a b) 2 M . &
, A B 2 M , A B 2 M . ;
, M ) A B . I
) *+ ( A B . 5
, , M *+
,
*+
*+
. . / . ? (, M , (. ; , M , . . (
, , ,
, -
. . -
, M .
B@ *
( L, *
-
M . ; *
( , ( . A
* ( . &
( L :
1) D=E, D2E, D&E, D:E, D(E, D)E, D9E, DE1K
2) @ , ( x1, x2, x3 ,... K
3) , ( . 5
-
M ( (
@).
I ( .
.
1. B .
1 :4; "<
+ 4; 6 4= 4; *+ . > 6 4 ;
; .
154
. . 2. I .
3. H
T1 T2 : : : Tk | , (T1 T2 : : : Tk ) .
4. H
T1 T2 | , T1(T2 ) .
5. H
T1 | , (T1 ) .
, - 1 2 ( +. B
3 , ( , ), ( ,
. B
4 *+ . B
5 () ( D
)E . ;, . , (
. , 3 4 T1 T2 , (, , ( ( L. >
( - ( (, (
- .
B : (x1 2 x4), x1(x2), +((2 7)), sin(+((x1 ))). , )
( ( *
+ *+, ) .
.
1. H
T1 T2 | , (T1 = T2 ) (T1 2 T2 ) | *
.
2. H
1 2 | *
, (1&2) :1 *
.
3. H
T | , xi , | *
, (9xi 2 (T )()) *
.
B
1 (@ ( , . B
2 (
*
(. B
3 () *
. A . 4
*
, 3. M, xi | xi . N
( xi . = (
4
, , xi T . ? , , *
) ( .
B@ , *
. & , , ( ( M . T , ( ( jT jM , .
1. H
T | , jT jM -
M . A , ( ( -
M .
2. H
T = (T1 T2 : : : Tk ), jT jM = (jT1 jM jT2jM : : : jTk jM ).
3. H
T = T1 (T2 ), jT jM = jT1jM (jT2jM ).
4. H
T = (T1 ), jT jM = jT1jM .
H
3 ( jT1jM @ *+,
( T . C - , T (. ., 1=0. ? (, (
. I
155
| - @ *+.
N
( , @ ( .
B *
@ *
, D
E *
. ., 1=0 = 2=0. ?
*
( , . ;
*
.
1. H
( T1 T2 , *
(T1 = T2 ) (T1 2 T2) .
2. H
( T , | *
, *
(9xi 2 (T )()) .
3. H
*
1 2 , *
(1 &2) :1 .
, ) *
. 5 ( *
M ,
( ( M j= .
1. M j= (T1 = T2 ) , *
(T1 = T2 ) jT1jM = jT2jM .
2. M j= (T1 2 T2) , *
(T1 2 T2 ) jT1jM 2 jT2 jM .
3. M j= (1 &2) , *
(1 &2) , M j= 1 M j= 2.
4. M j= : , *
,
M j= .
5. M j= (9(xi 2 T )()) , - *
M j= (c) c 2 jT jM , (c) (
*
, ( *
( xi c.
N
( M , M j= :. / (, ( *
, .
1.2. L
B 1 2 | *
, T | ( L. ? ( L ( *
:
1 _ 2
-
:(:1&:2 )K
1 ) 2
-
:(1&:2)K
1 , 2
-
(:(1&:2) & :(2&:1))K
8xi 2 T (1 ) -
:(9xi 2 T (:1)):
&
, *
, ( ( , ( ( L, (
( L.
156
. . B, ( ( L .
X Y -
*
8x 2 X (x 2 Y ).
Z = X Y ( *
(8z 2 Z 9x 2 X 9y 2 Y (z = (x y))) & (8x 2 X 8y 2 Y 9z 2 Z (z = (x y))):
1.3. B A | M . A ( (), *
(xi ) ( L, A = fx 2 M : M j= (x)g. B - *
( A.
B *
( ( M , ( M . ?
(, ( L. O
+ ( , - . ? M (
) , , (
( ( L.
., ( L ( , (,
, **+
) . . I -,
L ( - . ., (, , , ( ) , + . .
.
, ( ( ( L. B
-
M , ( L. I -, ( L @ :
( , ( , M . A (
*
( L , , (
, M D E. ; -
) *
@ , M R N (
( . .( ( L + ( *) +,
(
. ; (
( *
+.
1.3. , (
M . 5(
M , , 16].
157
. I
. M , :
X1 X2 X1 P( X1 ) : : : Xi+1 Xi P( Xi ) : : : M = Xi :
N
i2
, , ( . B - . & ' : M ! M , @ 8i ('(Xi ) Xi ).
1.4. B x 2 M . H
x 2 M , '(x) = x, -
x ( . , *
( Da .E
(, -
a .
I , -
M ( . ., 2 '(2). H
, -
, - -
(@(
.
? , M , M (
L ( ( ( M , M ). 5
, , *
, *
. M j= , (, *
M .
4 *
( L. *
( L, ( -
. B - *
( L, ( ( .
., *
8x1 2 R (sin( + x1) < 5) ( *
8x1 2 R ( sin( + x1 ) < 5).
1.5. H
*
( L (
) ( *
( L, *
(
.
. B | ( *
( L. ?
M j= , M j= :
B+ , *
, . A
(
( *
, .
& , ( *
. B+ - *
. A (, ( (, ( (.
B (x) | *
. ? + (
*
(x) x ( @ x. 4 *
158
9x 2
. . X (x). H
(x) x, (x)
x.
B+ (
, (
. 5
- (
@ +.
1.6. ;) r 2 M , r A B, ( , -
a1 : : : an 2 A @ -
b 2 B , 8i (ai b) 2 r ( -
b ) r ai ).
( ). B ) r 2 M , r A B , . ? -
b 2 B , a 2 A ( a b) 2 r.
B+ *(+ , -
@ -
, ) r, @ -
b, ) r -
( A. >, -
b .
;, , , -
b ) r
-
A, A -
.
4 ) r = f(a b) j (a b 2 N) & (a < b)g. . , - ) . , + *(+
!, ) .
1.7. =
! 2 R ( , j!j ) . 5
( -
*
( ( D! .E.
1.8. =
" ( , ) .
x y (, x ; y . H . A ( (
, (, .
B, X2 X1 P( X1 ) . 5
- P = f! 2 N j ! .g. , -
. A ( ( L.
5, P . ?, N, + , P -
. > , ) !. ; ! ; 1 ) . &
, P -
M . ? P 2 P( X1 ), P 2= X2 . O
. I
159
(, Xi Xi;1 P( Xi;1 ) . >(( ( P :
X1 X1
= !X1
X2 = X1 P(X1 ) X2
!X2 = !X1 P(!X1 )
'
Xi+1 = Xi P(Xi )
!
M=
S Xi
N
i2
!Xi+1
Xi+1
S
M=
i2
Xi
N
!M
=
= !Xi P(!Xi )
S !Xi :
N
i2
1.9. A
M ( .
H
X 2 M | , , X | .
= ( , ,
) .
1.10. A
(!M ; M ) ( .
, , X1 = !X1 .
, . P |
) , ) . N , , .
! 1.1. A | , B | ( M L). A \ B .
A , , *
( L, .
>, ( ( .
1. &. R
( -
. >
, (.
2. ,. R
-
M . 5
-
+ .
3. ,). . -
M . ; - ) ( ( ( !M ), ( (
. ; ( , - .
1.4. C
(, ) ). ; , ),
) ,
. / ).
160
. . 1.11. =
( , ) .
B ).
! 1.2 ("#). a 2 R , a = x + ", x , " .
1.12. B a | . , 4 x, x a. =
x ( a. 5
( - x = St(a).
N+ St() ) *+. H
( , ) (
.
1.13. / (x) = fy : x yg , ( x, 6+, ( x.
I ( . 6 , .
) *+ f x0 :
8" 2 R " > 0 9 2 R > 0 8x 2 R (jx ; x0 j < ) jf (x) ; f (x0 )j < "):
B (*) ) ! *+ y = !x
. A *+ , ( @ (
*+ y = x !). I , - *+ , - @ ( + , *+ . B
( ), - ( "- .
! 1.3. !" f # x0 , f ((x0 )) (f (x0 )).
\$%&. B *+ f x0.
* (
x x0 . ? (, f (x) f (x0 ).
5 (, jf (x) ; f (x0 )j < r r. * (
r.
? *+ f , @ > 0,
M j= 8y (jy ; x0j < ) jf (y) ; f (x0 )j < r). B + y = x : M j= jx ; x0 j < ) jf (x) ; f (x0 )j < r. ? x x0 ,
jx ; x0 j < , | .
&
, jf (x) ; f (x0 )j < r. , (
r , f (x) f (x0 ).
B (x x0 ) f (x) f (x0 )). 5, M j= 8" > 0 9 > 0 8x (jx ; x0 j < ) jf (x) ; f (x0 )j < "):
. I
161
* (
". ? (, M j= 9 > 0 8x (jx ; x0 j < ) jf (x) ; f (x0 )j < "). , + ( - *
. B
. 5, M j= 8x (jx ; x0 j < ) jf (x) ; f (x0 )j < ").
,(@ (
() x, jx ; x0j < . ? , x x0. ? f (x) f (x0 ), ,
jf (x) ; f (x0 )j < ", " | . , (
x " (
.
5 , *+ , *+. , ( *
. B
-
( *+ A. I :
8x0 x1 2 A (x0 x1 ) f (x0 ) f (x1 )):
(1)
; (, -
@ (,
| . A ( , *
8" > 0 9 > 0 8y (jy ; x0j < ) jf (y) ; f (x0 )j < ")
, x0 . &
, - *
M r , (
. , (
x0 , - + .
! 1.4. f | !", A | R. f A \$
(1).
! 1.5. %
# !" f A \$
8x0 x1 2 A (x0 . & x0 x1 ) f (x0 ) f (x1 )):
! 1.6. f | !", # # . & & x0 , x1 (x0 x1 ) f (x0 ) f (x1 )).
\$%&. ,(@ (
x0 x1. , @ ( a b], x0 x1 2 a b]. B I f a b], , f (x0 ) f (x1 ).
1.14. , x = (x1 : : : xn) 2 Rn ( 8i 2 f1 : : : ng (xi
.
,
0). x
y (, x ; y |
162
. . ! 1.7. x | # , # # k k kxk 0.
! 1.8. kxk0 0, k k0 | .
x 0.
? (, , .
! 1.9. '
(
x(t t0 x0) # )( x_ = A(t)x,
x(t0) = x0 & # # (t0 x0) , 8t1 x1 (t1 t0 & x1 x0 ) t (x(t t0 x0) x(t t1 x1))):
5(
+ (
1.3.
1.15. /+ A ( , @
-
.
/+ , @ .
1.16. , ( , .
1.17. /+ ( , @ -
.
I + -
.
! 1.10. A | , B | ".
det(A + B ) det(A).
A ( -
+.
B@ -
*
. , *
(, .
=
b fang, a! b ) !.
=
b fan g, ) !, a! b.
B
fang , a! ) !.
B
fang , a! ) ) !.
4 M ( (
DenlargementE. & @ .
. I
2. !""#
163
, ( ( -
enlargement. I
( ( + .
2.1. - !"
2.1. N+ (t) ( - , :
1) (t) > 0 t 2 RK
2) supp( ) (0), *+ K
+R1
3)
(t) dt = 1K
;1
4) (t) (;1 +1).
, @ *+, 2.1 +
-*+, @ ( *+.
2.2. N+ " (t) ( " - , " > 0 |
, 1), 3) 4)
-*+, 2) supp( " ) (;" ").
;, " -*+ -*+.
' 2.1. * -!" (t) " -!"
# ( ").
\$%&. / supp( (t)) , - . ,
" = max(; inf(supp (t)) sup(supp (t))).
() 2.1. (t) | -!". # -
# # !" f
R
+1
;1
f (t) (t) dt f (0).
2.2. #
4(
) , -*+, 17{21].
4 *
(
x_ = A0 (t) +
Xk (t ; ti)Ai(t)x
i=1
x(t0 ) = x0 (2)
164
. . ti | (
, Ai | + -
, x0 | .
2.3. N+ f g ( -
(f g), t ((8i 2 f1 : : : kg t 2= (ti )) ) f (t) g(t)),
ti | -*+ ( (2).
B (t ; ti ) -*+ i (t ; ti). ? *
( (2) ( ( I)
x_ = A0 (t) +
Xk i(t ; ti)Ai(t)x
i=1
x(t0 ) = x0 :
(3)
B () ) ( (3). ;( ( K ( I), ( *
( (2) -*+ i . ;( ) - ( ( X ,
X = fx(): k 2 K x | ) ( kg.
2.4. I
-
xU ( ) ( (2), xU \ X 6= ?.
2.5. N+ x() ( ) ( (2), -
xU, x() 2 xU xU | ) ( (2).
2.1. , ) ( x_ = (t)f (t)x(t), x(;1) = x0 -
*+ x0 exp(f (0))(t), (t) | *+ 3.
2.3. !\$ #
4 ( I):
x_ = (A0 (t) + " (t ; t0)A1 (t))x x(t0 ; ") = x0 (4)
A0 A1 | + -**+,
" | " -*+, t0 , x0 | .
;( @ ) ( x().
' 2.2. +" x() t0 ; " t0 + "].
\$%&. , M
{C
Zt
kx(t)k 6 kx0k exp
t0;"
kA0 ( )k d
+
Zt
t0 ;"
" ( ; t0 )kA1( )k d
:
? kA0( )k kA1( )k | *+, . &
, kx(t)k .
. I
165
;( ( z () ) (
z_ = " (t ; t0)A1 (t0 )z z (t0 ; ") = x0:
(5)
' 2.3. '
(
x(t) z(t) t0 ; " t0 + "].
\$%&. ? -
+ A1(t) | *+, A1 (t) = A1 (t0) + B (t), B (t) | +. ?
d
dt (x ; z)(t) 6 " (t ; t0)kA1(t0)k k(x ; z)(t)k +
+ (kA0(t)k + " (t ; t0 )kB (t)k)kx(t)k:
> , x(t) , k(x ; z )(t)k 6 "1 +
Zt
t0 ;"
kA1 (t0)k " ( ; t0 ) k(x ; z )( )k d
"1 | . B M
{C
(x ; z )(t) 0.
! 2.1. ,
x(t0 + ") y(1), y(t) | (
#
# )(
y_ = A1 (t0 )y y(0) = St(x0):
(6)
\$%&. ;( (
y1 (t) ) (
t
R
y_ = A1 (t0 )y, y(0) = x0 . ? z (t) = y1
" ( ; t0) d ) (t0 ;"
(5). (6) ( ) . ? x0 St(x0), 2.3 , x(t0 + ") z (t0 + ") = y1 (1) y(1).
! 2.2. t0 < t1 < : : : < tk, ( \$
) (
x~(t) (2). \$ x~(t)
.
1. t 2 (;1 t1] !" x~(t) (
x_ = A0 (t)x,
x(t0) = x0.
2. t 2 (ti ti+1], i < k, t 2 (tk +1) !" x~(t) (
x_ = A0 (t)x, x(ti ) = yi (1), yi (t) | (
y_ = Ai (ti )y,
y(0) = x~(ti ).
\$%&. 5 (, (
-*+ i (t) ) x(t) ( (3) -
x~(t). * (
*+ i (t). 5
*+ i (t)
@ "i , -*+ i (t) " -*+. ;( ( " ( "i . 5 + i, t 2 ti + " ti+1 ; "] x(t) x~(t). . i
166
. . (;1 t1 ; "] ) x(t) x~(t) . B x(tm ; ") x~(tm ; "). 5, 8t 2 tm + " tm+1 ; "] (x(t) x~(t)). . tm ; " tm + "]
) x(t) ) ( z_ = (A0 (t) + " (t ; tm )Am (t))z ,
z (tm ; ") = x(tm ; "). B 2.1 ( x(tm + ") y(1), y(t) |
) ( y_ = Am (tm )y, y(0) = St(x(tm ; ")) = x~(tm ). ? + x(tm ; ") x~(tm ; "), *+ x~(t) tm , St(x(tm ;")) = x~(tm ). . tm +" tm+1 ;"]
) x(t) ) ( z_ = A0 (t)z , z (tm + ") = x(tm + ").
? x(tm + ") y(1), (, ) x~(t) t 2 (tm tm+1 ], , ( ) 8t 2 tm + " tm+1 ; "] (x(t) x~(t)). O
, ( ) (, t > tk + " x(t) x~(t). ? (, *+ x(t) x~(t)
-
.
5( (
) ( -*+. B - ) ( ( -*+. .
*
) ( -*+.
2.6. ! ( (2) ( *+ x(t), ) ( (2).
! 2.3 (*+ ). (
x(t) . * (
x(t) # 2.2. * # # t0 (
, -!"# ) (
x(t) (
x_ = A0 (t)x, x(t0) = x0.
! 2.4. (
(2) & #.
\$%&. & ) ( (2) ( ) ( I). B- (
( ( ( ) .
2.4. %
! &&{(!
4 x_ =
k
X
A0 (t) + Ai (t) (t ; ti ) x
i=1
(7)
ti | (
, Ai | + -
.
. I
167
2.7. & *+ x() ( ) (7), t0 x0, x() ) ( (2).
! 2.5. x1() : : : xn() | (
(7),
n | (7). / W (t) " " x1(t) : : : xn(t). t0 6= ti & i > 0 ! /{1
det W (t) = det W (t0 ) exp
Zt
8> 0
>< 0
i(t) = >
>:;trtrAAi(it(it)i)
t0
tr A0( ) d +
Xk i(t)
i=1
ti > t0 t 6 ti ti < t0 t > ti ti > t0 t > ti ti < t0 t 6 ti :
\$%&. *
(
-*+
i (
k
P
) ( I) x_ = A0(t) + Ai (t) i (t ; ti ) x, x(t0 ) = xj (t0) i=1
( x^j (). , x^j () xj () x^j (t0 ) = xj (t0 ). ;( ( W^ (t) + + x^1(t) : : : x^n(t).
B + det W^ (t) = det W^ (t0 ) exp
Zt
t0
k Zt
X
tr A0 ( ) d +
tr Ai ( ) i ( ) d :
i=1 t0
, (
t 2= ft1 : : : tkg. ?
det W^ (t) det W^ (t0 ) exp
Zt
t0
tr A0( ) d +
Xk i(t):
i=1
? xj () x^j () xj (t0 ) = x^j (t0 ), det W^ (t) det W (t). &
Rt
Pk , det W (t) det W (t0 ) exp tr A0( ) d + i (t) . ; i=1
t0
, - @ .
, (
t (
t, -*+. 5
-*+ *
;{6
@ .
168
. . 2.5. %!
"
2.8. C( ) (7) ( ( ).
() 2.2. 2 (
# (7) .
() 2.3. 3 t0 2= ft1 : : : tkg, # (
# (7) # # t0 (
# x_ = A0 (t)x.
, 2.1. '
(
# ! (7).
2.9. /+ W () ( *
+, @ + ( ( ) (7).
() 2.4. +
" .
B ) 14]K , 14]
*
I) n- .
1] . . , !!" // \$ %. | 1978. | * 1. | +. 75{86.
2] 0 +. 1., + \$. 2., 34 +. 5. 3 . | +: +.-7. . 4-, 1983.
3] + \$. 2. 9 4 % % : !!"% , ;0% 4 4% !" 0 // 3!!". . | 1986. |
1. 22, * 11. | +. 2009{2011.
4] . . 94" % 0 //
\$ %. | 1989. | * 1. | +. 23{34.
5] Callot Jean-Louis. Travaux de recherche: Colloq. trajectorian m>em. Georges Reeb
et Jean-Louis Callot, Strasbourg-Obernai, 12{16 juin, 1995 // Prepubl. Inst. rech.
math. avan. | 1995. | No. 13. | P. 183{189.
6] ?4 2. @. A 0. +. 4 . B. D. E 9 // 7% . . | 1984. | 1. 39, . 4.
7] Brunovsky P. Controllability and linear closed-loop controls in linear periodic systems // J. of DiKer. Equat. | 1969. | Vol. 103, no. 1. | P. 296{313.
8] 1 5. Q. 9 ; // 3!!".
. | 1983. | 1. 19, * 2. | +. 269{278.
9] E +. 2., 1 5. Q. 7 4 Q %
. I // 3!!". . | 1994. | 1. 30, * 10. | +. 1687{1696.
10] E +. 2., 1 5. Q. 7 4 Q %
. II // 3!!". . | 1994. | 1. 30, * 11. | +. 1949{1957.
. I
169
11] E +. 2., 1 5. Q. 7 4 Q %
. III // 3!!". . | 1995. | 1. 31, * 2. | +. 228{238.
12] E +. 2., 1 5. Q. U %
// 3!!". . | 1995. | 1. 31, * 4. | +. 723{724.
13] E +. 2., 1 5. Q. + 4
Q // 3!!". . | 1997. | 1. 33, * 2. | +. 246{235.
14] 9 3. . 2 4 !!"% 0 W!!" // B4 !.
A. 1. | B;, 1995. | +. 3{50.
15] 9 3. . E4 Q % // B4 !. A. 2. | B;, 1996. | +. 69{84.
16] 3 . E 4. | .: , 1980.
17] Q \$. X. A !!" // A Y . | 1974. | A. 8. |
+. 122{144.
18] E X. A. 9 4 A E // 3!!".
. | 1979. | 1. 15, * 4. | +. 761.
19] Z \$. Z. 3!!" 4 [. |
.: 2, 1985.
20] \$% \$. A. 9 % % % !" !!"% // 3\$2 +++?. | 1986. | 1. 286, * 5. | +. 1037{1040.
21] 3 A. Y. U [ : !!" 0 !" W!!"% // 3\$2 +++?. | 1988. | 1. 289,
* 2. | +. 269{272.
\$ % % 1997 .
. II
. . 517.977
: , , , .
P1
!" x_ = A0 (t)+ (t ; ti)Ai (t), () |-)* i=1
. +, " ,- , . ./ 0 - .
Abstract
D. M. Olenchikov, Impulse control of Liapunov exponents. II, Fundamentalnaya
i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 171{185.
The systems x_ = A0 (t) + P (t ; ti )Ai (t), where () is Dirac's delta-function,
i=1
are investigated. It is proved that the basic results of Liapunov exponents theory
remain valid for such systems. The theory of impulse control of Liapunov exponents
is developed.
1
, 1]. .
1. 1.1. 1
P
U1 (t ; ti )Ui , i=1
t1 < t2 < : : : < tn < : : :, Ui # \$ , & ti, \$ ' , .
./ 6 . )
)
, (
97-01-00413) *
(
97-04).
, 2002, 8, 9 1, . 171{185.
c 2002 ,
!" #
172
. . u 2 U1 N (u t) * i, ti < t. +
0 6 N (u t) 6 (t ; t1 ) + .
+ U1 & .
. U1 &\$ :
X
ku()k = sup
kAik Ia = fi: ti 2 a a + 1)g:
a>0 i2Ia
1.1. U1 | .
1# *, U1 .
2* u() 2 U1 . 3 4 -
x_ = (A0 (t) + u(t))x t 6 t0 :
(1)
6
&
* 7 #*, A0 (t) #.
1.2. 1.1. 2* f : R ! Rn. 8 ( ;1 +1),
f] = t!lim
+1(ln kf(t)k=t) * 1 (, 1).
1.1. f .
1. kf(t)k 6 kg(t)k, f] 6 g].
2. kf(t)k] = f(t)]" cf(t)] = f(t)] c 6= 0.
3. f] = 6= 1, # " > 0
(a) f(t) = oe(+")t],
;(;")t = +1.
(b) t!lim
+1 kf(t)ke
4. \$% &# & '
%'# () ( & ) &)
, %' '%
#).
5. \$% &# & ' () .
6. \$% #
' #
% .
7. *, & , .
8. f] < 0, t!lim
+1 f(t) = 0.
* 2].
1.2. + % ' x() (1) & )
& %.
. II
173
. > M > 0, kA0(t)k 6 M ku()k 6 M.
? x() | * 7, x0 = x(t0) 6= 0. kx(t0)k = a. i, t 2 t0 A ti] a exp(;(t ; t0 + N (u t))M) 6 kx(t)k 6 a exp((t ; t0 + N (u t))M):
+# 1, 2.2] t0A t1] 7 x() 7 x_ = A0 (t)x, x(t0) = x0, kx(t)k 6 kx(t0)k +
Zt
t0
kA0 ()kkx()k d:
2 B{6, a exp(;(t ; t0 )M) 6 kx(t)k 6 a exp((t ; t0 )M):
2
, i < k . * i = k. 2 1, 2.2] (tk;1A tk ] 7
x() 7 x_ = A0 (t)x, x(tk;1) = y(1), #
y() | 7 y_ = Ak;1(tk;1)y, y(0) = x(tk;1). &
ky(1)k exp(;(t ; tk;1)M) 6 kx(t)k 6 ky(1)k exp((t ; tk;1)M):
D#, \$*& B{6 ky(1)k:
kx(tk;1)k exp(;M) 6 ky(1)k 6 kx(tk;1)k exp(M):
E* , kx(tk;1)k. 7 * # 7# # #. > kx(t)k ae;(t;t0 +N (ut))M ] 6 kx(t)k] 6 ae(t;t0 +N (ut))M ]:
&
;(1 + )M 6 x(t)] 6 (1 + )M.
1.2. +& &) ' (1) ) ( .
2. 2.1. 8 Mnm * n m
( n = m, 7 Mn ), MFnm | (7) n m, RnF |2 (7) .
Vect: Mn ! Rn &, FG . H* A * &
.
174
. . 3 & & &
:
x_ = A(t)x + B(t)u y = C (t)x
(x u y) 2 Rn Rm Rr:
(2)
>&
* 7 #, A, B C ,
# t. + (2) * (A B C).
8 Uab] * * , &Pk
\$ U(t) = Ui (t ; ti ), #
a 6 t1 < : : : < tk < b, Ui 2 Mmr . 8 k
i=1
* * * # N (U). I U(t) = U1 (t ; t1) * ' .
+ * *# & .
2.1. Uab] | .
Pk
>
4 U kU()k = kU k . .
*,
ab]
i=1
i
Uab] .
(7) * * a b] UFab] .
2.2. > & * #& # #, 4 # 3].
X(t s) J7 x_ = A(t)x.
>
4 , &\$ Uab] Mn , k
X
Hab] (U()) =
X(a ti )B(ti )Ui C (ti )X(ti a)
Pk
i=1
#
U() 2 Uab] U(t) = Ui (t ; ti).
i=1
2.1. + (A B C) # a b],
Hab] (Uab] ) = Mn .
2.1. (A B C) # a b] # % #, # & a < t1 < : :: < tn2 < b n
Ui , &
fVect(X(a ti)B(ti)Ui C (ti)X(ti a))gi=12 .
. * .
*. 2* Hab] (Uab] ) = Mn . >*4 *2
Mn . . 4 * Uj (),
fGj gnj =1
Hab] (Uj ()) = Gj . 2 & Uj () 7*
. II
175
#. +
*, \$ ' * fVi ()gki=1 , Uj * ' Vi (). ?#
Hab] (fVi ()gki=1) =
= Mn . >
fH(Vi ())g *& & . K * n2 '. 8 *
, * ' 2
fH(Vi ())gni=1
. I Vi () & Vi (t) = Vi (t ; ti2).
+
*, fQi = Vect(X(a ti )B(ti )Vi C (ti )X(t a))gni=1
. > .
1. > ti . ?#
.
2. . ti &. H Vi . ? Qi ti , F* *G &\$ ti , & * Qi .
Z(t U) 7 J7
Z_ = A(t)Z + B(t)U(t)C (t)X(t b) Z(a) = 0:
(3)
Z(U) = Z(b U).
, # ' &\$
, 2.2 (
* 2.2).
2.2. + (A B C) # a b], \$ l > 0, G 2 Mn 4
* U 2 Uab] , kU()k 6 lkGk Z(U) = G.
2.2. ,
2.1 2.2 (.
. 2 J7 Z(U) = X(b a)Hab] (U()). 2*
(A B C) # 2.1. 2 2.1
2
n
\$ * fVi(t) = Ui (t ; ti )gi=1, fHab] (Vi ())g | Mn . > *& G 2 Mn X(a b)G fHab] (Vi ())g. vi '
# . 3 * n2
P
U(t) = vi Vi (t). 2, Z(U) = G. ? Vi () i=1
G, \$ l, \$ G, kU()k 6 lkGk.
+
*, (A B C) # 2.2.
2* (A B C) # 2.2. ?#
Hab] (Uab] ) = Mn .
2.2 # 3] * . E , ' #.
? Z(U) * , , , * , \$ l * * &\$ ' .
176
. . 2.3. + (A B C) # a b], Z(U ) = Mn . > U | ' a b] , * | Uab] .
> * Z(U) = X(a b)H(U). > Z(U) =
Rb
= X(b a) X(a t)B(t)U(t)C (t)X(t a) dt .
a
2.2. \$% fAig, i 2 I , | a b] Mn.
\$%
Zb
V1 =
i2I a
Ai (t) dt V2 =
i2I t2ab]
fAi (t)g
# # & & . -# V1 V2.
> V2 | ' #*# , V1 | #. +
* &# # #, #* V2.
2.3. #% (A B C) ( . #-
& .
. 2* (A B C) # . >*4 * Gi Mn 4 &\$ Ui (), &\$ & #. ?#
i
Zb
a
X(a t)B(t)Ui (t)C (t)X(t a) dt = X(a b)Gi :
? X(a b)Gi , 2.2 , fX(a t)B(t)Ui (t)C (t)X(t a)g = Mn :
i2I t2ab]
2
&
, 4 f(Uij tj )gnj =1
, X(b tj )B(tj )Uij (t)C (tj )X(tj b) . +
*,
(A B C) # * .
2* (A B C) # * . ,
(A B C) & 2.3 . 2.3 , ' *, (A B C) ' & . >*4 Gi Mn . E
* # , Ui () 2 Uab] , . II
177
Z(Ui ) = Gi. 2
- , \$ Ui , * *-. 2 ' Ui \$& () a b] U^i . J #, &# i Z(U^i ) = G^ i Gi . ? Gi , G^ i . +
*,
Z(U ) = Mn , #
U | ' a b] . +
*, ' , * (A B C) # .
2.3. H ] (U
ab
ab
])
2.1 ( !"). / # &
f1(t) f2(t) : : : fn(t) Y , ) &, & & t1 t2 : : : tn 2 Y ,
&
f1(t1) f1(t2) f1(tn)
f2(t1) f2(t2) f2(tn)
..
. .
. 6= 0:
f (t. ) f (t.. ) . . f (t.. )
n 1
n 2
n n
. * . n
*. 2 n = 1 . 2* n < k
. 2* ffi (t)gki=1 Y . +
*, & \$
;1, det D 6= 0, #
fti gki=1
0 f (t ) f (t ) 1
1 1
1 k ;1
CA :
..
...
D=B
@ ...
.
fk;1(t1 ) fk;1(tk;1)
?#
D \$ . '
' c1 : : : ck;1. ? f1 : : : fk
Y , \$ tk 2 Y , fk (tk ) 6= c1 f1 (tk ) + : : : + ck;1fk;1(tk ). +
*, ' ci 0f (t ) f (t )1
B@ 1 ... 1 . . . 1 ... k CA
fk (t1 ) fk (tk )
, 4 * &.
178
. . Eij , ' ij , * ' P &. M * Uab] , &\$ U(t) = k Eij (t ; tk ), #
k 2 R, Uij .
k
2.3. 0 Uij % -
Uab].
2.4. \$
Uab] # P
E '# , H (U ) = H (U ). ?-
Uij .
ab] ab]
ij
ab] ij
, Hab] (Uab] ) Hab] (Uij ).
H i j. > (f1 (t) : : : fn2 (t)) = Vect(X(b t)B(t)Eij C (t)X(t b)):
>
fl *& & . 6
4 *, ' ff1() : : : fk ()g. 3
* ' :
fk+1(t) = ck+11f1 (t) + : : : + ck+1k fk (t)
:::
fn2 (t) = cn2 1f1 (t) + : : : + cn2 k fk (t):
3 80 1 1 0 0 1 0 0 19
>
>
>
BB 0 CC BB 1 CC BB 0 CC>
>
>
>
BB ... CC BB ... CC BB ... CC>
>
>
=
<B C B C B
C
C
C
B
C
B
0
1
0
fv1 : : : vk g = B
:
:
:
BBc CC BBc CC BBc CC> :
>
>
BB k+1. 1CC BB k+1. 2CC BB k+1. kCC>
>
>
>
.. A @ .. A
.. A>
@
@
>
>
: cn21
cn2 k cn22
V = hv1 : : : vk i & ' .
2.4. ,
Vect(Hab](Uij )) V .
. , Vect(Hab] (Uij )) V . > * U() 2 Uij , Vect(Hab] (U())) 2 V . *
' * ' U(). 2* U(t) = Eij (t ; t1 ). 2
'
0 f (t ) 1
1 1
B
Vect(Hab] (U())) = @ ... C
A = (f1(t1)v1 + : : : + fk (t1)vk ):
2
fn (t1 )
. II
179
?#
Vect(Hab] (U())) 2 V . , Dim Hab] (Uij ) = k. ? f1 () : : : fk () , \$& t1 : : : tk , f (t ) f (t )
1 . 1 . 1 . k ..
.. 6= 0:
..
fk (t1) fk (tk )
3 ' Ul (t) = Eij (t ; tl ), #
l = 1 : : : k.
> Vect(Hab] (Ul )) , ' Dim Hab] (Uij ) = k.
2.5. \$% Q1 Q2 2 Mn | , # Hab](Uab] ) = Mn # % #, # Q1Hab] (Uab])Q2 = Mn.
,
Q1Hab] (Uab])Q2 &% , & Hab](Uab]).
> , Q1 = X(b a), Q2 = X(a b).
& 2.1. 3 0 #] (A B C), #
00 1 0 01
001
B
BB0CC
0 0 1 0C
B
CC
B
.
.
.
.
.
.. .. .. . . .. C
A=B
B=B
BB ... CCC C (t) = (r1(t) : : : rn(t)): (4)
B
C
@0 0 0 1A
@0A
0 0 0 0
1
Hab] (Uab] ) * & E11 = (1). > Q(t) = X(0 t)B(t)E11 C (t)X(t 0) = (qij (t)), #
j
j ;k
n;i X
qij (t) = ((n;t); i)! rk (t) (jt; k)! :
k=1
+
*, (4) # #
* #
, #
qij ( 0 #]).
& 2.2. 3 0 #] x_ i = ai (t)xi + bi (t)u i = 1 : : : n y = c1 (t)x1 + : : : + cn (t)xn:
(5)
> Q(t) = X(0 t)B(t)(1)C (t)X(t 0) = (qij (t)), #
qij (t) =
Rt
= bi(t)cj (t) exp (aj () ; ai ()) d . + (5) # #
*
0
#
, #
qij ( 0 #]).
2.4. # #
2.4. 1 y = C1x1 + : : : + Ckxk x1 : : : xk , Ci A
, \$ ( ) CiA
180
. . * , .
2.5. + fxig -
, \$ * ' ,
* &. Q , & .
2.6. &) fxig & # % #, # ) ) & fSt(xi)g
.
E # &\$ .
2.7. &) fxigni=1 & # % #, # det(x1 : : : xn) 0.
2.6. J fxig y, \$ xi, * y. J hfxi giF .
2.8. +& & % ) &) . 0 &, & & &) & & ) )
&.
2.9. \$% fxignni=1 | , # hfxigiF = RF .
2.10. fyig | % fxj g, hfxj giF = hfyigiF .
2.5. \$% fxig | &) hfxigiF = RnF .
-# fxig,
n (.
. >
fxig & &
fyj g, \$& *# . ?#
hfxigiF = hfSt(yj )giF = RnF . +
*, fyj g Rn.
2.5. % 2.7. + (A B C) l- # , \$ , &# \$&
< t1 < : : : < tn2 < + l Ui , kUi k = 1 k(J1 : : : Jn2 );1 k 6 , #
Ji = Vect(X( ti )B(ti )Ui C (ti )X(ti )):
. II
181
.
*, ' &\$
& 2.8.
2.8. + (A B C) l- # , \$ " > 0, &# \$& < t1 < : : : < tn2 < + l Ui , kUi k = 1 det(J1 : : : Jn2 ) > ", #
Ji = Vect(X( ti )B(ti )Ui C (ti )X(ti )).
2.6. (A B C) l-
# # % #, # # ( & & %'#) (
2 Ui() 2 UF +l] , & fH +l] (Ui ())gni=1 .
. * * ' Ui () l- #.
*. 2* &# \$& '
2
Ui () 2 UF +l] , fH +l] (Ui ())gni=1
& . 2 * l-&
#* . >*4 " * . H * 2. > Ui () 2 UF +l] , fH +l] (Ui ())gni=1 & . + Vi () = Ui ()=kUi ()k. ? Ui
2
H +l] , fH +l] (Vi ())gni=1
& . +
*, *
det(Vect(H +l] (V1 ())) : : : Vect(H +l] (Vn2 ())))
, # * *7 " ( " ). . # \$, *, ti Vi ()
, F* *G
4 , ' & . +
*, (A B C) l-
# .
2.7. (A B C) l-
# # %
#, # # (
#) H +l] (UF +l] ) = MFn.
. 2* (A B C) l- #.
>*4 * . 2 & 2.6 \$& Ui () 2 UF +l] ,
2
fH +l] (Ui ())gni=1
& . ?#
2.10 hfH +l] (Ui ())giF = MFn . ? UF +l] * ,
H +l] (UF +l] ) = MFn .
*. H * . 2
, H +l] (UF +l] ) = MFn. ?#
\$& Uij (), H +l] (Uij ()) = Eij , #
Eij | .
182
. . J
Uij () * # '
. M ' , \$ Uij , V . ?#
hH +l] (V )iF = MFn . +# 2.5 2
V ' fVk ()gnk=1
, & fH +l] (Vk ())g & . &
, * , 2.6 l- #*.
2.11. \$% Q1 Q2 | & & , # H +l] (UF +l] ) = MFn # %
#, # Q1H +l] (UF +l] )Q2 = MFn.
2.8. \$
l-
# l-
# .
. 2* (A B C) l- #.
2 1 x = L(t)z 4 (AL BL CL) J7 XL (t s), #
BL = L;1 (t)B CL = L (t)C(t) XL (t s) = L;1 (t)X(t s)L(s):
(AL BL CL) HLab] . .
, HLab] (U) = L;1 (b)Hab] (U)L(b). M L(t) L;1(t) & t & *. >*4 * .
> 2.7 2.11 , HL +l] (UF +l] ) =
= MFn. +
*, * (AL BL CL)
l- # .
2.12. (A B C) l-
#,
# l1 > l (A B C) l1-
#.
2.9. + (A B C) # , l- # l.
2.10. + (A B C) # , \$ l, (A B C) # & l.
2.13. (A B C) #, #.
.
\$4 ' l- #-
2.11. + (A B C) l- # , 9 9N 8 8G 2 Mn 9U 2 U +l) N (U) 6 N, kU()k 6 kGk Z_ = A(t)Z + B(t)U(t)C (t)X(t )
Z() = 0 Z( + l) = G
7 * Z().
. II
183
2.9. ,
2.7 2.11 (.
. > * #
2
* 2.2 ( N * n ).
> #& . 2* (A B C) l- # 2.11. H N, &\$
& 2.11. 3 * + l] Z_ = A(t)Z + B(t)U(t)C (t)X(t ), Z() = 0. >*4 G *& & . ? 7, 4
U(), H +l] (U()) = X( + l)G, 4 N (U()) 6 N
, kU()k 6 kGk, kGk , U() 2 UF +l] . ? A #, X( + l) , ' * G , H +l] (UF +l] ) = MFn. +
*, 2.7
(A B C) l- #.
2.6. ' J7 x_ = (A(t) + B(t)U(t)C (t))x XU (t s).
2.10. (A B C) # a b], I F : Mn ! Uab] , I , &
F(I) = 0, N (F(H)) 6 n2 XF (H ) (b a) = X(b a)H .
. 6
* U() = F (H) U(t) =
n
X
2
(ui Ui (t ; ti ))
i=1
#
ui 2 R, ti Ui 2.1 , X(a ti )Di X(ti a) Di = B(ti )Ui C (ti ).
& , #
>7 J7:
XU (b a) = X(b tn2 )eun2 Dn2 X(tn2 tn2;1)eun2 ;1 Dn2 ;1 : : :X(t2 t1)eu1 D1 X(t1 a):
3 2H 2 -& f(u) = Vect(X(a b)XU (b a)), &\$& Rn Rn . .
*, f(0) = I f
. * *, I \$ . + R J(u) = (J1 (u) J2(u) : : : Jn2 (u)), & Ji (u) = Vect(X(a tn2 )eun2 Dn2 : : :X(ti+1 ti)Di eui Di : : :X(t2 t1)eu1 D1 X(t1 a)):
M J u . 3
Ji (0) = Vect(X(a ti )B(ti )Ui C (ti )X(ti a)). K ,
*, jJ(0)j 6= 0. ?#
I \$ f ;1 (H), f(u).
184
. . 2.11 (, , !). \$% f : Rn ! Rn
f(x) = f(x0) + J(xk;(xx;0x) )+k (x ; x0),
&. r > 0, & 8x kx ; x0k 6 r ) kx;x k 6 2kJ1; k .
,& ry = 2kJr; k . -# ry f(x0 ) f ;1(y), &. kf ;1(y) ; x0k 6 2kJ ;1k ky ; f(x0 )k.
0
0
1
1
E
& 5].
2.12. (A B C) l-
#, # " > 0 . > 0, & % H 2 Mn,
& kH ; I k 6 , # 2 R . % U(),
kU()k < ", & XU ( + l ) = X( + l )H .
. H * " > 0 * +l]. ? , * 2.10, f(u), &\$ f(u) = Vect(I) + J u + (u). > 2.7
kJ ;1k 6 . ? B(t), C(t) #, X( + l t)
# t 2 + l] ' &, \$ R > 0, \$ , R
kk(uuk)k 6 21 . ?#
2.11 2R \$ U(H), 4 kU k 6 2 kH ; I k.
> * min( 2R 2" ).
D# &\$ .
2.13. (A B C) l-
#, # " > 0 . > 0, & % H 2 Mn,
& kH ; I k 6 , # 2 R . % U(),
kU()k < ", & XU ( + l ) = HX( + l ).
2.7. ) ? 2.13 2.8 & *, 4] *# .
2* 1 (A) > : : : > n (A) | 1 x_ = A(t)x. > j (A) * j- .
2.12. 2 (A B C) & *
, U ! (A + BUC ) U 0 T = f 2 Rn j 1 6 : : : 6 n g, # , Rn.
2.14. x_ = A(t)x #
, (A B C)
#, (A B C) %
.
* * ##
4]. E
* & &\$.
. II
185
2* (A B C) l- #. 2 1 4 #* . 8 * i (A)
i , \$*& * , &# # k > 0 XU ((k +1)l kl) = HX((k + 1)l kl), #
H = diag(exp(1 l) : : : exp(n l)).
. 6], * x_ = A(t)x
k
1 X
U(A) = T>
inf0 klim
!1 kT j =1 ln jX(jT (j ; 1)T j
* k
1 X
!(A) = sup klim
ln jX(jT (j ; 1)T j;1:
T>0 !1 kT j =1
2.13. + (A B C) *
* , &# " > 0
4 > 0, &# 2 R, jj < , \$ U 2 U1 , kU k < ", &\$ U(A + BUC ) = U(A) + .
D# * * .
2.15. (A B C) #, % ) % .
* # *, 4 4].
1] . . . I // .
. . | 2002. | ". 8, %. 1. | &. 151{169.
2] +. ,. - . .. | .: 0
,
1967.
3] , &. 0., "
3. . 4 5%6
. I // 77-. . | 1994. | ". 30, 9 10. | &. 1687{1696.
4] , &. 0., "
3. . 4 5%6
. II // 77-. . | 1994. | ". 30, 9 11. | &. 1949{1957.
5] : ;. ,., <
. ,
. .%. . | .: , 1988.
6] +% +. ., =5 >. <., ?: . ., 0%-
. =. =. " .
@ .. | ., 1966.
\$ % % 1997 .
. . - 519.21
.
: , , -
!
" #
\$ # \$ \$\$ %
. &!
"# ' (. (%
!
.
Abstract
H. Yu. Piliguzova, On rate of approximation of critical excursion probability for
random process by moments method, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 1, pp. 187{194.
The rate of approximation of the number of high level crossings distribution for
a Gaussian process with various spectra have been investigated by Monte Carlo
method. Methods to simulate the paths of these processes have been used. Approximations have been obtained with the aid of Rice method. The result is presented
as a table.
1. ,
. ! , , " #, \$ \$ \$ \$ \$. % \$ . % " \$ . & \$ \$ ', " . & \$
, 2002, 8, 0 1, . 187{194.
c 2002 !,
" #\$ %
188
. . , \$ ', " ,
\$ *-+. , , \$ \$ \$ \$ %. -. ., \$
.
2. !
. X = fXt : t 2 10 T]g | \$ ##\$
M = maxfXt : t 2 10 T]g. %5 " ,
. 11]:
Nuf = ft: t 2 10 T] f(t) = ug | u8
Uuf = ft: t 2 10 T] f(t) = u ft(1) > 0g | \$ u8
Vm (k)(= k(k ; 1) : : :(k ; m +1), k m | \$ \$ 8
A = 1 A 8 | # \$ A.
0 A :
- \$ \$ , "
, " \$ , . 13].
; fM > ug = fX0 > ug + fX0 6 u M > ug fX0 6 u M > ug =
1
S1
P
= fUuX 10 T ] > 1g = fUuX 10 T] = kg, PfX0 6 u M > ug = Pk , k=1
k=1
Pk = PfUuX 10 T ] = kg.
< :
PfX0 6 u M
1
X
1
X
1
X
> ug = kPk ; (k ; 1)Pk = EUuX 10 T ] ; (k ; 1)Pk :
k=1
k=2
k=2
% \$ " :
1
1
X
X
(k ; 1)Pk 6 21 k(k ; 1)Pk = 12 EUuX (UuX ; 1):
k=2
k=2
. , 5 10 T] 1
PfX0 > ug + EUuX ; EUuX (EUuX ; 1) 6 PfM > ug 6 PfX0 > ug + EUuX :
2
; \$ \$ " , " \$\$ ##\$ , ' , \$ 189
" \$ (. 12]):
1
X
PfM > ug = PfX0 > ug +
(;1)m vm!m m=1
vm = EfVm (UuX )g =
Z
Z
dt1 : : :dtm
01]m
Z
R
( +)
Z jY
=m j =1
yj pt1 :::tm t1 :::tm (u : : : u8 y1 : : : ym ) dy1 : : :dym :
@ , 1942 ' \$ # \$ :
T12=2 e;u2 =20 EfUu (0 T)g =
210=2
0 2 | \$ \$ X, 0 = EXt2 , 2 = EXt02
(. 11]). % " \$ \$.
3. < \$ \$, \$ , \$ \$ ' \$
. \$ \$ ( #) B. . r(t) | # . . 5 B:
1
P
r(t) = k2 cos kt. ; \$ #1
k=1
P
\$ " : ~(t) = Ak cos kt + Bk sin kt,
k=1
Ak Bk | \$ \$1\$ P
(0 k2). & D = k2 , k=1
\$ \$ 1, ~(t)D;1=2 . ,, 5 \$ \$ 5\$
\$ " , 1:
(t) = D;1=2
X
N
k=1
Ak cos kt + Bk sin kt t 2 10 T ]:
190
. . %\$ k2 " : k2 = k;d , d 5 : d = f008
108 208 258 308 35g. - , 5 , (t) " \$, # \$\$
B, " \$ #.
@ r(
), \$
" d, Pfmax(t) > ug.
& (t) :
0 = E 2(t) = 1 2 = D;1
4. N
X
k=1
k2 k2 :
. " \$ , 5\$ 10 1], \$ Ak Bk (
'. ;. -, " # ).
N
N
P
. (t) (t) = D;1=2 k k cos kt+k k sin kt, k k |
k=1
\$ \$ \$ \$. & \$ \$ \$ N. &, \$, N ! 1, ,
N \$ \$ \$ . ; \$ \$ 5\$ \$ \$ \$ 1;3 3], \$ \$ / N(0 1) 5
pm 00026, , N = 10, " :
Pfmax(t) > ug =
= Pfmax(t) > u j max(1j1 j : : : N jN j8 1j1j : : : N jN j) > 3g Pfmax(1 j1j : : : N jN j8 1j1j : : : N jN j) > 3g +
+ Pfmax(t) > u j max(1 j1j : : : N jN j8 1j1j : : : N jN j) 6 3g Pfmax(1 j1j : : : N jN j8 1j1j : : : N jN j) 6 3g:
F
191
Pfmax(1 j1 j : : : N jN j8 1j1j : : : N jN j) > 3g 6
N
N
X
X
6 Pfmax(k jk j) > 3g = Pfmax jk j > 3k;1 g 6
k=1
k=1
N
X
p
6 Pfmax jk j > 3 kg 6 00026 + 0000021 0003:
k=1
- , \$
\$ N = 10.
-, \$ \$ (T = 05 | \$ \$ # \$, , # t1 ; t2 = 2k (. ) " ). - " , \$ m = 5000
M1 : : : Mm \$ (t) t 2 10 T] # FD1(u):
(
m
X
1
FD1(u) = 1 ; m I fMk 6 ug I = 1 Mk < u
0 Mk > u:
k=1
5. !
& ' # (t) # FD1. , 1
PfM > ug Pf(0) > ug + EUu ; v2
2
v2 = EfUu (Uu ; 1)g. v2 . & \$ (t1 ), (t2 ), 0(t1 ), 0 (t2 ). . G |
\$ (t1 ), (t2 ), 0 (t1), 0 (t2 ), Gij = Ei j . ;, A(v ) = D;1
N
X
k=1
kv k2 cos k
B(v ) = D;1
" G t1 6= t2 (k = 1 : : : N):
N
X
k=1
kv k2 sin k
192
. . 0 1
1
A(0 )
0
;B(1 )
B A(0 )
1
B(1 )
0 C
C
G=B
@ 0
B(1 ) A(2 0) A(2 ) A :
;B(1 )
0
A(2 ) A(2 0)
- p(yk1 : : : ykn 8 yj1 : : : yjm ) = (2);2 jGj;1=2 exp ;(2jGj);1
X
jk
jk yj yk jk | j k \$ G (k = 1 : : : n,
j = 1 : : : m). Mjk j k \$ G. ;, \$, M11 = M22, M43 = M34 , M41 = M14, M23 = M32 , M33 = M44
M23 = ;M14 " \$
#
\$ #\$ u:
11
ZZ
ZZ
1
1
2
dx1 dx2 x1x2 v2 = (42 jGj1=2)
exp ; jGj u (M11 + M12) dt1 dt2
0 0
00
1
2
2
exp ; 1(x1 + x2)M33 + 2u(x1 ; x2 )M41 + 2x1x2M43 ] :
2jGj
TT
H\$ \$ , I, \$ t1 t2 : : : tn ## A1 A2 : : : An, #
Zb
a
n
X
f(x) dx = 21 (b ; a) Ai f(xi )
i=1
f(t) \$ (. 14]).
J , \$ # f(t1 t2 x1 x2) #\$ t1 t2 x1(t1 t2 x2) \$ # x2(x1).
. \$ \$ x1, x2 I | # F(t1 t2).
< " X1 max , X2 max , \$ , 5 \$ \$ #, \$ (" = 001 5 \$).
& 5 \$ , \$ u = 0. , 11, . 220]
pt(0 0 x1 x2) = 42j1Gj1=2 exp ; 2j1Gj 1M33(x21 + x22) + 2M23x1x2 ] :
;
F(t1 t2) =
=
=
Z1Z1
193
x1x2 pt(0 0 x1 x2) dx1 dx2 =
0 0
11
3
=
2
jGj
(x21 + x22 + 2x1x2)
x
x
exp
;
1
2
42M332
2
00
jGj3=2 cosec2 (')(1 ; ' ctg ')
42M332
ZZ
dx1 dx2 =
= M43=M33 = cos '.
; #, " \$ F(t1 t2) \$ \$\$. , \$ , , \$. . (M33),
" #\$.
; . J , Y=
ZTZT
0 0
ZT
dt1 dt2 F(t1 t2) = 2 F (
)(T ; ) d
e
e | , F (
) = 0, \$ Y .
B F (u) I
(\$ \$ \$ ).
6. *\$ #
, " \$ p1 = 01
p2 = 002. \$ \$ , " \$ ' d " # \$. - FD2 = Pf(0) > ug + EUu 8 FD3 = Pf(0) > ug + EUu ; 21 v2:
, , :
# FD2 # ,
\$ #, . . d = 25 3 35.
K5 p = 018
194
. . 1!
d
p1 ;FD2
p1
p1 ;FD3
p1
p2 ;FD2
p2
p2 ;FD3
p2
0
1
2
25
3
35
0,36 0,23 0,09 0,12 0,03 0,01
0,26 0,21 0,09 0,12 0,03 0,01
0,67 0,27 0,13 0,08 0,15 0,15
0,65 0,27 0,13 0,08 0,15 0,15
# FD3 # " \$ p = 018
p = 002 ,
p = 01, # "5 , FD2 FD3 p = 002 d.
&
1] ., . . . | .: ,
1969.
2] Azais Jean-Marc, Wschebor Mario. A formula to compute the distribution of the maximum of a random process // Publication du Laboratoire de Statistique et Probabilites
07-95.
3] )* +. ,. - *./0 0 // 1. /. . | 1996. | 2. 2, .. 1. | . 187{204.
4] 6. 7. )., ,. 8. 9. .: / / ) .
7. ). 6.. | .: ,;-. <;.-. , 1996.
& ' ( 1997 .
. . . . . 681.3
: , , B-,
F-, , RAP- !! , , , "#!
.
\$ %
F-
& f ! j
g &
( ! ) I ( ! ) = ( n ) ! ( ), ,
* F-, %
, + *
%
& ,1 I ,2 I
I ,k I F2 B3,
,i 2 , ,k I = ,k I ( ! ), a F2 ( ) B3 ( !) | ./0 (
), #
#
.
X
X
Y
Z
V
X
Y
Z
X Y
R
Y
Y
R
V
F
:::
F
W
R
W
W
W
W
Abstract
L. A. Pomortsev, Algebraic interpretation of derivation axioms completeness,
Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 195{219.
The operation ( ! ) I ( ! ) = ( n ) ! ( ) is determined in
the full set f ! j
g of F-dependences over a certain scheme . Let ,
be an F-dependence, which follows from a set of F-dependences. We prove that
, = ,1 I ,2 I I ,k I F2 B3 for some ,1 ,2
,k 2 and ,
where ,k I = ,k I ( ! ). The unary operations F2 and B3 correspond
to axioms of derivation F2 (completion) and B3 (projectivity) pro tanto.
X
X
Y
Y
X Y
Z
V
X
Z
Y
Y
V
R
R
F
:::
W
W
W
:::
F
W
R
W
, () , , , F-
. ! " " # = hX Y i, %
! & R " , & . '
! ( , X Y R | ! R. *&
%, + +, % % %,!
! %
ha b : : : z i -, ! , F-
"
% . # = X ! Y .
, 2002, 8, 6 1, . 195{219.
c 2002 !",
#\$
%& '
196
. . F-
( "
( ) %%,
( , %, ! ( +. % (
+
% 23
( . 4 ,
+( & F-
", %
! (, ! !
23
! ", (,! % %5
3. - F-
,
, , +( & & %
( " %". 6" ! ( %
! F-
+". 6 F-
, +! ,( -"
2
, 3 ! F-
" &!. - ! ,
&
( .
R | !
DR = fdom(A) j A 2 Rg | % ! R, " ! %
2
. 7
" %( 3! %
! ! ! " .
- (R) ! R "
%.
1. r R (R)-
+ r DR . 8
r %
r A2R dom(A)1) .
' + ( %" (R), - (,! . * &-% t 2 r ! X R % %
t(X). 9 t ! X. %! + r ! R % ( r(R), "
r = r(R).
2. r | + !" R X Y R | !. 9+ r F- X ! Y , t1 (X) = t2 (X) ) t1 (Y ) = t2 (Y ) (%" " t1 t2 2 r.
: , &
Y = R, X +
r(R).
3. ; F-
" F %" F-
X ! Y , +, (, F, X ! Y .
: - % & , 23
X ! Y F, ( F j= X ! Y .
1)
| .
197
F-
&! .
4
, F-
&-% ! F , %
" "" " ), , -
" F-
" F (,! , !
F-
.
F1. :
X !X
F2. :
X !Y )X Z !Y
F3. :
X !Y X !Z ) X !Y Z
F4. :
X !Y Z )X !Y
F5. :
X !Y Y !Z )X !Z
F6. : X ! Y Y Z ! W ) X Z ! W
F1, F2 F6, ! ", ( . ( ! F3, F4 F5, " +! !.
4. 6 F1{F6 % A,
A fF1 F2 F3 F4 F5 F6g. &, F-
" F , , , A, (%" F 2 A F-
F . F-
F.
5. F | ! F-
" ! R. #
F , %
F + , | - + , F 23
! ", >&
.
?(, , +, .
6. F-
X ! Y F, (X ! Y ) 2 F + .
4 " 3 6 (F ) X ! Y ) ) (F j= X ! Y ),
, : F- X ! Y F ,
F X ! Y .
+ F1, F2
F6, " F-
. B. 2
(
% , %
.
1 ( ). F-
X ! Y F , F X ! Y . (F ) X ! Y ) , (F j= X ! Y ):
1 C1, . 58, 4.1], &
2
( ).
198
. . 7. F | F-
" !" R.
F-
" P F , 23
P %
(1) F , %
(2) ,! F-
" P " F1{F6.
8. P X ! Y ,
X ! Y 2 P .
: F ) X ! Y , +
& F-
F . ' ,
&
, & 2 1 &
. &
3 " &
. !
B-
.
!.
B1. :
B2. \$:
B3. :
% 1. X !X
X !Y Z Z !U )X !Y Z U
X !Y Z )X !Y
fF1 F2 F6g B- fB1 B2
B3g . ! , fB1 B2 B3g .
B-
, F j= X ! Y &
" F X ! Y , (,(
B-
.
4
B-
2
RAP-
1)
, 9.
9. X ! Y F-
" F , '-
, (, :
(1) F-
| - X ! X,
(2) F-
| - X ! Y ,
(3) F-
, (,
" ", %
F, % X ! Z ,( B2.
RAP- .
K -" 3 (
(, .
2. "
F | F-
. \$ F-
X ! Y F , %
RAP-
X ! Y F .
1) 7 # B-: Re9exivity, Accumulation, Projectivity.
199
2 2, 4.2 C1, . 62], .
, . 1, " 3 , ." . &. B& " %
3, (."
, 3 4.2 C1].
+ %
, %(, %+" 3 .
:-!, ," % &. ?(," & 3 ,& .
:-!, ! %
L A BM , , 3" A ) B.
:-!, 3
3" A1 ) A2 , A2 ) A3 ,... , Ak;1 ) Ak A1 ) A2N ) A3N : : :N ) Ak :
9%
, % . 4
- ) ! ,
(, %
! F-
" &!. 4
& ,, "" . 6 , 3 ".. 7, , 3 ) %, ! " .
: ( ! %
&
2 :. :. % (;6';) ( ,& .
1. RAP-
4
-& ( &
( , (, &
3( " ! F-
" &%
! ".
1.1. P | F . F-
U ! V 2 P , & P
P .
% 1.1. "
P | X ! Y F-
F , P &% F-
% F F-
X ! Y .
' &%( F-
.
( & " !, " %
(," 2
& P & -
F-
U ! V .
200
. . !
P
P %
:: ::: :: :: :: ::: :: :: :
:: :: :: : :: :: :: :: : : : :
1
U !V
1
( (a) : : :
U !V
2
2
:: ::: :: :: :: ::: :: :: :
3
U !V
( (b) : : :
:: :: :: : :: :: :: :: : : : :
X
4
4
W ! Z ( (c) : : :
5
( (a) : : :
W ! Z ( (c) : : :
5
:: ::: :: :: :: ::: :: :: :
:: :: :: : :: :: :: :: : : : :
: " 32
, &
| %
( %
!). 6
, ( "
U ! V , !," . 4 (" 3) ( U ! V % " . & 3
, !,& W ! Z. 9 " U ! V ( (b),
!,( .. :. % ". , (,! F-
" +
F-
" P. % " P (, F-
P .
9%
, F-
, (,
3, . 4
- ( %
)
%
, 3 P | X ! Y F . - . %
% "
& P . Q
, | 3 3", % ,
F-
%
" . 9
P
.
1.2. P | -% .
F-
P . , -% F-
, +(,
" P .
: RAP-
! & % B2 (
) B3 (). : ! 201
X ! , X ! Y .
( ) 2. L P | X ! Y F , (,
B-
, ,
B-
. *
P F-
, , & X ! Y . 9
+
. ,. X ! Y . X ! X -"
, . M.
, F-
, , & P ,( B3 (), . 4
. %
P ! ! F-
", & % P F-
Z ! W , ( Z ! V W ,( B3.
Z ! W P "+& , " B3 B2 (
). B Z ! W B3, F-
Z ! W 0 , & W 0 W . 4
Z ! W 0 % ,( B3 Z ! V W ,
Z ! W, % &
- . ;
. .
Z ! W " & "
B2 0 " . B2 (X 0 ! Y 0 Z 0
Z ! U 0 ) X 0 ! Y 0 Z 0 U 0) Z ! W , " B2, %
:
9>
2: Z ! V W
1: Z ! V W
>
+
B3
Z !W
+
B2
0
0
X ! Y Z W ( X 0 ! Y 0 Z
>=
+
B3
Z ! W
W =W1 W2 >
>>
+
B2
Z ! W U (
W1 ! U (1.1)
4 Z ! W, X 0 ! Y 0 Z W1 ! U P, "
,, ! +( & F-
X 0 ! Y 0 Z W Z ! W U, , &.
: "+ + %
" ! 2
& % &
. B B2 Z ! V W, 9
Z ! V W
2: Z ! V W
W =W1 W2=
1:
+
B2
X 0 ! Y 0 Z V W ( X 0 ! Y 0 Z
+
B2
Z ! V W U (
W1 ! U
(1.2)
(, F-
(1.2) ( (1.1) !" V . B "+& 2
3( (1.2) P , (, B2 % V , % - 202
. . " B3. % %, V % ,( B3 " F-
P, " X !Y.
, : B3 P ( . 9 ! %
, P F-
, %" B2 , , B1. 6
, P "
RAP-"
(1) (2) . B3 . 4
( -
S (, (3)
. P P,
9 &, , F-
# 2 PS X ! , # 2 F. 9 % , , B3, B1 PS .
? -" 3( % & P (-% F-
"
( X , % %
!
,", "
%
% X. 4 %& %
%, F-
" (, " X, P , - . 3
%
5
%
, % ! F-
X ! P. ! T R, U 6= X V 0 V - %
"
F-
" U ! V ) X ! T U V ( X ! T U, %," ! &
P B2 (
).
", &
U ! V 2= F. ' , U ! V %
% B1 - U = V , % P !
,! F-
", - ! U ! V 1 ( " B2). : B2
U ! U X ! T U X ! T UN ) U ! U % P % X ! Y . : , U ! V 1 2= F. &
P U ! V 2 . '
3 % . 4
- +
& P 0, P 0 P , ( + %:
9
U !Vk
2F
>>
>>
U ! V k;1
( #k
: :: :: :: : : ::: :: : :: :: ::: : : : : :: : = P 0:
>>
U !V1
( #2
>>
U !V
( #1
X ! T U V ( X ! T U
: . ( %
#j F-
" P, %!! U ! V j " U ! V j ;1. : " ( B2 ( V k V k;1 : : : V 2 V 1 V:
(1.3)
203
: , &
k 6= 01), % P RAP- &
P 0 X ! T U V P 00:
9>
U !Vk
2F
>>=
X ! T U V k ( X ! T U>
k
;
1
X !T U V
( #k
P 00:
:: :: ::: :: :: :: : : : :: : : :: : : :: ::: :: :: >
>>>
X ! T U V 1 ( #2
X !T U V
( #1
-& . : F-
U ! V J P 0 ,
, , % P n P 0 &! F-
"
X ! ?
: ! X ! T U V P 0 P 00 - % F-
, P , , " X ! (
& ! V , P
" B2. ? . (1.3) - ,
U ! V J % 2" !
. ( P ! !, (,! & & F-
". :( ! +(, U ! V J %
P . : ! "" %.
P1 X ! Y . 4
(, -
% P RAP- P1 P2 .
9%
, k 6= 0 ! Pm Pm+1 +
m F-
", ! XN
) '(P) = 0 < 1 < 2 < : : :. & .
4
n- +
& %
-" 3. B Pn %
F-
, (, X ! Y , %( RAP-
.
*+. P | # 2 P. 4
F-
P % %
U(# P ).
% 1.2. "
P | RAP-
X ! Y F-
F , Q ! S 2 SP | F-
,
k = U(Q ! S P ) V(Q ! S) = X V | ) (
U !V 2F
(U !V P )<k
! (=(( ) F-
, *
&%(
P F-
Q ! S , S V(Q ! S).
(. . 3( k = U(Q ! S P). k = 1
k = 2, &
F !! F-
" ,. Q ! S, . , !
! F-
" U ! V 2 P, ! P + k, 1)
0#
*!, * =
V
V
0
.
204
. . . B Q ! S ,( B3, , U ! V , X = Q = U, U ! V S V . + -&
, , U(U ! V P ) < kN ) V V(U ! V ). V(U ! V ) = V(Q ! S)
N ) 1.2 Q ! S ,( B3.
". (, &
Q ! S B2.
? -" 3( + 2
& P , Q ! S:
X ! S0
+
B2
X ! S ( W1 ! W2
Q Q X, - . (3) 9. : B2 (, !
%: W1 S 0 S = S 0 W2 . F-
X ! S 0
+ P F-
X ! S, - X ! S 0 &
+ k , & S 0 V(Q ! S 0 ). F-
W1 ! W2 ,
. 6 , W2 V(W1 ! W2 )N ) S = S 0 W2 V(Q ! S 0 ) V(W1 ! W2 ) V(Q ! S).
B W1 ! W2 , . U(W1 ! W2 P) < k S = S 0 W2 V(Q ! S 0 ) W2 V(Q ! S).
, 1.1. "
P | RAP-
X ! Y F-
F V(P) = X S V | ) (
U !V 2F
, (( (, , ( ) F-
. \$ X ! Z P , &% %& B2
V(P ) = V(X ! Z), Z V(P) % , Y X , X ! Y
P B2 P .
% 1.3. "
P | RAP-
X ! Y F-
F . \$ F-
X ! Z P , &% %& B2, P F-
U ! V , ( V 6 Z .
(. : %, B2 F-
W1 ! W10 W2 ! W20 (( W2 W10 1) . : F-
W1 ! W10 W20 , " %5 ! " !! F-
". -, U ! V 2 P V " %, (, ( X ! Z, - , 3
, (,
U ! V , %
, ! X ! Z, , P ( 1)
:
; %
! +*
W1
W
0
2
, ; #<
.
205
(. 1.1) F-
. 6! , %
&
1.1, . %
" 3 ( U ! V . %" P
F-
, ! Z.
1.3. 3 (
%
RAP-
) ! F-
" . .
. , " ! F-
", . .
, 1.2. \$ P | RAP-
X ! Y F-
F X ! Z | P , &% %&
B2, Z = V(P ), -
S V
V(P ) = V(X ! Z) = X .
U !V 2F
(. :( S V Z (&
U !V 2P
1.3N ) V(P ) Z. ? - ( ( 1.2, % .
2. 2.1. RAP-
P F-
X ! Y & F-
" F . :
(0) 3 F-
" P N
(1) F-
P | - X ! XN
(2) F-
P X ! Y . 9
." ," F-
,(
B3N
(3) F-
P . ( X ! ( ") ,( B2 ! ,! F-
"N
(4) F-
, ", (, P .
, ( ! F-
" F .
2.1.
1. RAP-
P &% .
2. \$ X ! X | F-
&% RAP-
P n = (P) | F-
P , (1) n F-
P &
C0 n ; 1]+
(2) ( P F-
n=2 + 2+
(3) F-
, n=2 ; 2, &
F ,
F-
P .
(. * 2 1, - 1. &
206
. . " 1.3 , % !( RAP-
P , ,( F-
X ! Y F. % , %
, (, -& , &
P, , (
P ! F-
" &!. :%, &, P F-
" . , %
(, P , . (%! #1 #2 2 P (, :
#1 #2 ) U(#1 P ) > U(#2 P ):
(2.1)
?. F-
" P . , % U(X ! X P) = 0.
6 &( 1.2, , ! P ( F-
: X ! Z, ,( B2, P %, | X ! Y . ? %, . ( Z Y , X ! Y X ! Z B3. :, B3 P . : - P ,. - "
F-
X ! Y , ," X ! Y % Y = Z.
, X ! Y X ! Z B3, %
&,
-" ! " !" (," F-
".
+( %
%
"
%" P , & " X ! Z "
X ! Y " F-
!. 6 1.2, & RAP-
P, Z = V(P) = X V:
(2.2)
U !V 2F
K( F-
X ! Z, (
(2.2) X ! V(P), X ! Z | (" F-
X ! Y .
Y 0 = X, F 0 = fX ! X g F (X ! Y ) = P n P 0, & P 0 | P F-
" ((,! B2 B3),
F(X ! Y ) | F-
", . 4 , F(X ! Y ) hF 0 F i. Q, %
"+
3, ,. ., F-
"
F(X ! Y ) ( & F-
F 0 = fX ! X g %+ X ! Y F-
" F 0 .
Q
hF 0 F 0i %5 F 0 F 0, X ! X F-
" F (X ! Y ) = hF 0 F 0i1). F-
" F(X ! Y )
" F-
X ! Y F.
1) =
! #
## %
*
(=
) # !
&.
207
: (3) 9 RAP-
! F-
" ( !" X. , - X ! Y , " P P 0. : - P 0 2
( Z(P 0) = fZ 1 Z 2 : : : Z s g ! ! " F-
" P 0 .
B. % , P 0. ? P 0 Z(P 0) X ! Z(P 0) = P 0, +2
X ! fZ 1 Z 2 : : : Z s g = fX ! Z 1 X ! Z 2 : : : X ! Z s g:
" ! W , W R ! F-
"
F(X ! Y ) % F-
", W:
F(W) = fU ! V j U ! V 2 F(X ! Y ) U W g:
(2.3)
!
Yi =
V i = 1 2 : : :
(2.4)
U !V 2F (Y i;1 )
%
(, %
(,( 3
X = Y 0 Y 1 : : : Y i : : ::
(2.5)
: , X ! X 2 F(Y 0 ) = F(X)N ) X Y 1 . 4 ,
W 1 W 2 ) F (W 1) F (W 2). - F (Y 0) F(Y 1 )N ) Y 1 Y 2 N
) F(Y 1) F (Y 2 )N ) Y 2 Y 3N ) : : :. Q
%
F-
":
fX ! X g = F(Y 0 ) F(Y 1 ) : : : F (Y i ) : : ::
(2.6)
: !" ! % R (2.5) (2.6) %
, . , , m, -
. ' ! Y i = Y j F (Y i ) = F (Y j ) (%! i j > m. , Y m = Z. ? -" 3(
2 :
8V 2 Z(P 0) 9i > 0 V Y i :
(2.7)
0
(2.7) % 3" U(X ! V P ), , (2.1) 3( F-
"
P 0.
* (2.7) , U(X ! V P 0 ) = 0, -
V = X.
Q
". !. & k > 0 !
# 2 P 0, ! U(# P 0) < k, (2.7). 6! -& , , (2.7) # 2 P 0, "
U(# P 0) = k. 4
. , &
P # = X ! V B3. B X ! W | F-
, " #, 208
. . U(X ! W P 0 ) < U(# P 0) = k (2.1), V W
B3N ) (2.7) V .
6%
!" F-
# = X ! V ,( B2
Q!S
(2.8)
B2 + Q W V = W S
X !W ) X !V
%! .
:-!, !! # F-
" % X ! . B" ! (2.8) X ! W ,
X ! W 2 P 0.
:-!, U(X ! W P 0) < U(X ! V P 0) = k &
(2.1).
:-!, Q ! S , : Q ! S 2 P 0 Q ! S 2 F (X ! Y ).
" Q ! S 2 P 0, &, ,
Q = X U(X ! S P 0 ) < U(X ! V P 0) = k:
: & " i j > 0, W Y i
S Y j . 9
. (2.5) q = maxfi j g W S Y q N
) V = W S Y q N ) (2.7) V .
, 3, Q ! S 2 F(X ! Y ). ; - % ( W Y i , & %
& i > 0 , %
3N ) W Y i+1 , Y i Y i+1 (2.5). 6 Q W W Y i Q Y i N ) Q ! S 2 F(Y i)N
) S Y i+1 N ) V = W S Y i+1 N ) (2.7) V . 4
- (2.7)
+
.
Z 2 Z(P 0), Z , m = 0 1 : : :, Z Y m .
? &" (. (2.2)), ( Y i V(P) = Z (%&
i 2 C0 m]N ) Z = Y m . %
, m %
3" (2.5) (2.6) %
3& 3
Y i
F(Y i) i = m. : " (2.6) - F(Y m ) = F(Z).
; & ( F F (Z).
Q ! S 2 F n F(Z). X Z, Q 6 X. (%
Q ! S F-
%
(3) 9 RAP-
X ! N ,
-" 3 B3 Q ! S . , Q ! S "" F-
, P, " P . B,. !" (2.8), %
%
%," " Q ! S " F-
X ! V ,( &" F-
X ! W . % , ! W , i 2 C0 m], W Y i N
) Q Y i (
Q W)N ) Q ! S 2 F(Y i ), F(Y i ) F(Z). ( F F(Z). ? . & (
209
(2.6) F(X ! Y ) = F 0 F F(Z)N ) F (X ! Y ) = F (Z), F(Z) F (X ! Y ), .
: - F-
F(X ! Y ). 4 + , %
;, %
F-
"
F 0 = fX ! X g F 1 : : : F m
(%& i 2 C1 m] F i = F(Y i )nF(Y i;1). :
(2.6) % F 0 : : : F m ,( % F-
", :
F (X ! Y ) =
i
F i F i \ F j = ? 8i j 2 C1 m]:
F(X ! Y ) 3( 23( (#), &
(#) = i # 2 F i:
: % , ." + ;, % , (%" F-
" # #0 2 F(X ! Y ) % # ; #0 , (#) < (#0):
9
-" + X ! Z & F F-
", " . F-
X ! Y . :+ F-
" RAP-
P , + ;:
F (X ! Y ) = fU 0 ! V 0 = X ! X
U 1 ! V 1
:: ::: :: :: :
(2.9)
U i ! V i
:: ::: :: :: :
U k ! V kg
(, (2.9) F (X ! Y ) (2.6) . X%
- , 23 23( (#), # 2 F(X ! Y )
(i) = (U i ! V i ) (%& i 2 C0 k]. - ,
&
23 , %
(,
F-
(2.9) . %
, . 3, &
, 2
. 6 &
(i) 2 C0 m] (%& i 2 C0 k], & m | "
(2.5) (2.6), &, , Z = Y m .
4
% ,. 3
23, 210
. . %
, ( %
%" (%& j 2 C0 m]
(j) = maxfi j U i ! V i 2 F j g:
83 j 2 C0 m] %
F j . " (2.9) F-
, ," % F j , - (j) 2 C0 k] (%& j 2 C0 m]. :
" & . ", " &, 2
(%" & i j 2 C1 m] 3"
i < j ) (i) < (j):
( - %
% % ", i 6 j ) (i) 6 (j).
: ,. , - " , (, "- (%& i 2 C0 k] ! ( %
:
((i) ; 1) < i 6 ((i)):
(2.10)
9%,
( (,( X ! Z %
(2.9), F(X ! Y ) " i 2 C1 k] F-
Si
X ! S i , & S i = V j . " (
j =0
,. "), ( %
alt(P ), &
"
(3) 2.1, %
(
(. ? -" 3( & i 2 C1 k] ( U i S i;1 , %
(, B2 ,"
F-
X ! S i;1 ," U i ! V i . ( U i ! V i 2 F (i)N ) U i ! V i 2 F(Y (i) )N ) U i Y (i) (2.3).
: (2.4) ! Y j . ?&
Y (i) %( %5 ! " F-
", !,! F(Y (i);1). 4
(S
i);1
F (Y (i);1) =
F i , - ; i=0
F (i);1 " F (Y (i);1) %. B. (2.9) .
( (S
i);1)
iS
;1
((i);1)N ) Y (i) =
V i V i = S i;1 , (2.10)
i=0
i=0
((i) ; 1) 6 i ; 1. 9%5 ( , U i Y (i) ,
% U i S i;1 .
6
, %
. , . ! X ! Z. 4
+
(, -
alt(P ) %! " X ! Y , . " 3
" F(X ! Y ), - 2.1 .
3. !" #
\$
211
% 3.1. \$ R | ( X Y Z V R | (, X ! Y Z ! V ) X (Z n Y ) ! Y V:
(3.1)
(. : -& & . + " .
1. ' + 2
23
" . :% t t0 2 r(R) &-% + r(R), ( F-
X ! Y Z ! V , , t t0 t(X (Z n Y )) =
= t0(X (Z n Y ))N ) t(X) = t0 (X)N ) t(Y ) = t0(Y ) (
X ! Y ). &, t(X (Z n Y )) = t0(X (Z n Y ))N ) t(Z n Y ) = t0 (Z n Y )N ) t(Z) = t0(Z) (
t(Y ) = t0 (Y ))N ) t(V ) = t0(V ) (
Z ! V )N ) t(Y V ) = t0(Y V ),
t(Y ) = t0(Y ).
2. :" % F-
&
.
1: Z ! V
(
)
2: X Z ! V
(F2- 1)
3: X ! Y
(
)
4: X Z ! Y
(F2- 3)
5: X Z ! Y V
(F3-
2 4)
6: X ! Z \ Y
(F4- 3)
7: X (Z n Y ) ! Y V (F6-
5, 6
Z = (Z n Y ) (Z \ Y ))
*+. # | -% 23
!" R. ? #Y % %
. , #Z | , . . #Y #Z R # = #Z ! #Y.
-. .. [(%( F-
& "( & F2 (), (,& . (
. : % %( F-
, %
& !
F2 .. & %
!
" 23
! " + . , " 3.1 (, .&
B2 (
)1) F6 (
)2) .
1) >
, n = ? ;# ( ! ! ) ! ), * #
B2.
2) : , (3.1) #
#+
, "##
F6. ?
( n ) ! #*! ( n ) ! , * #
F6, *
!! B3/F4 ( !).
Z
Y
X
Z
Z
Y
X
Y
Y
Y
Y
U
X
Z
Y
Z
Z
U
V
X
Y
V
212
. . , (3.1) F-
"
2
" !" &%
( 3( I:
#1 I #2 = #1Z (#2 Z n #1 Y) ! (#1Y #2Y)
( . . Q
, %
I
.
. , . ". . -&
3.1 & F-
" F 2
3
#1 #2 2 F ) #1 I #2 2 F + :
*+. F-
# F-
,
W ! W, & W R, % # I W =
= # I (W ! W) W I # = (W ! W) I #.
6
-" 3 # B1/F1 (2).
,/ - ..
I. 9
3 I %
. Z ! U (
F-
X ! Y , ! X = Y Z \ U. : , -
(X ! Y ) I (Z ! U) = X (Z n Y ) ! Y U = Z ! U:
? % ,. B X ! Y , (X ! Y ) I (Z ! U) = (X ! Y ) Z X U Y . : -
" 3 &
F-
Z ! U.
?
", ! 3, , %
X = Y Z \ U %
F-
X ! Y Z ! U, - (X ! X) I (Z ! U) = Z ! U = (Z ! U) I (X ! X).
: , ! F-
" 3" ? ! ?.
II. 9
3 I , %, ! #1 = X ! Y #3 = Z ! U (#1 I #2 )Z = X (Z n Y ) =?
=? Z (X n U) = (#2 I #1)Z, , . % W R (X ! Y ) I W, W I (X ! Y ) (W n Y ) I (X ! Y ):
(X ! Y ) I W = X (W n Y ) ! Y W
W I (X ! Y ) = X W ! Y W
(W n Y ) I (X ! Y ) = X (W n Y ) ! Y (W n Y ) = X (W n Y ) ! Y W:
6 , ((X ! Y ) I W)Y = ((W n Y ) I (X ! Y ))Y =
= (W I (X ! Y ))Y, ((X ! Y ) I W)Z = ((W n Y ) I (X ! Y ))Z (W I (X ! Y ))Z (3.2)
(X ! Y ) I W = (W n Y ) I (X ! Y ) =? W I (X ! Y ):
213
: 3 (3.2), +, ,
W \ Y W \ X, , , (W n X) \ Y = ?1) .
III. 9
3 I , (#1 I #2 ) I #3 = #1 I (#2 I #3).
- F-
! #1 = X 1 ! Y 1 , #2 = X 2 ! Y 2 #3 = X 3 ! Y 3 : #1 I #2 = X 1 (X 2 n Y 1) ! Y 1 Y 2, -
(#1 I #2 ) I #3 = (X 1 (X 2 n Y 1 )) (X 3 n (Y 1 Y 2 )) ! (Y 1 Y 2 ) Y 3 =
= X 1 (X 2 n Y 1 ) (X 3 n (Y 1 Y 2 )) ! Y 1 Y 2 Y 3 :
#2 I #3 = X 2 (X 3 n Y 2 ) ! Y 2 Y 3, -
#1 I (#2 I #3 ) = X 1 ((X 2 (X 3 n Y 2)) n Y 1 ) ! Y 1 (Y 2 Y 3 ) =
= X 1 (X 2 n Y 1 ) (X 3 n (Y 2 Y 1 )) ! Y 1 Y 2 Y 3 :
: " 3
( #1 I #2 I : : : I #k F-
" #i = X i ! Y i , i 2 C1 k]. #(k) = #1 I #2 I : : : I #k
Y 0 = ?. 9
+ #(k) = #(k ; 1) I #k ,
3" k
#(k) =
k i
;1
i=1
j =0
Xi n
k
Yj !
i=1
Y i:
(3.3)
3 #(1) = X 1 ! Y 1 , & " ( (3.3) k = 1. , !
(3.3) #(k ; 1), , " ! :
#(k ; 1) I #k =
=
=
k
;1
k
;1
i=1
i
;1
i=1
k j =0
i
;1
Xi n
Xi n
i=1
j =0
Xi n
i
;1
j =0
Y j Xk n
Y
k
j
!
i=1
Yj
k
!
k
;1
i=1
i=1
Xi !
Y i I (X k ! Y k ) =
k;1 i=1
Yi Yk =
Y i = #(k):
1) /
( n ) I ( ! ) =
I ( ! ) %, (( n ) I ( ! ))C =
= ( I ( ! ))C, , * %
, ( n ) = , = ( ) n ( ( n )) = ?. , = ( n ( ( n ))) ( n ( ( n ))) = ( n ( ( n ))),
( n )D ) = ( n ) n ( n ) = ( n ) n ( n ( \ )).
: # E
& # = ? ( n ) ( n ) ;
D ) ( n ( \ )) ( n ( \ ))D \ \ .
= ( n ) \ = ? #
= . %
. > 2 ,
2 2 n (, 2 2 )D ) 2 n D ) 2 ( n ) n ( n ) = D
) . G, #! 2 , 2 n (
!, 2 ) 2 n D
) 2 D ) 2 ( n )\ = D ) .
W
W
W
X
Y
X
Y
W
X
Y
Y
T
X
Y
X
X
W
X
X
Y
W
T
W
W
Y
Y
W
W
X
X
X
W
W
W
S
s
Y
S
t
X
W
W
X
s
W
X
t
W
X
Y
W
Y
W
S
s
W
t
Y
s = X
T
S
t
T
S
s = W
W
X
X
X
W
X
X
X
X
W
W
W
Y
W
Y
W
Y
Y
X
Y
T
Y
W
W
W
Y
T
Y
Y
T
W
W
Y
T
s
s
W
t
X
W
W
Y
t = W
S
T
Y
214
. . IV. 9
3 I , # I # = #. : , # = X ! Y , # I # = X (X n Y ) ! Y Y = X ! Y = #:
% 3 I %%,
", &
LM #1 I #2 I : : : I #k (, LM. ,
, #p = #q ! p q 2 C1 k], ! p < q _k = #1 I : : : I #q I : : : I #k . 4
+" 3( _k = #1 I : : : I #q;1 I #q+1 I : : : I #k (, " - 3 .
F-
_k %
S , % #p = #q V p = V q N ) _k Y = Y i .
i6=q
iS
;1
Y j , i 2 C1 k]. j
Sk
Sk
(3.2) _k = T i ! Y i . 6 i=1
i=1
#p = #q U p = U q . p < q, T p T q N , !
T q " _k Z 2 - . %
_k Z . m 6 q !
! T m Y q .
m > q !
Y p T m , ,. Y q . : Y q T m & m 2 Cq + 1 k]. : : " _k Z (, T i = X i n
%( ( :
_k =
k i=1 i6=q
Xi n
i
;1
j =0 j 6=q
Yj
:
3.1. X ! Y | F-
!" R. X hhX ! Y ii %
-& fX ! Y g F2 () F4 ().
:. :
hhX ! Y ii = fU ! V j U X Y V g:
4
. hhX ! Y ii F-
X ! Y . F-
" F :
hhF ii = S hh#ii.
#2F
*+. 9%
2 F-
"
F = f#i = X i ! Y i j i 2 C1 n]g:
& . %
#h1 2 : : : ki = #1 I #2 I : : : I #k , &, " i 2 C1 k] 215
mi 2 C1 n]. : F-
#hm1 m2 : : : mk j F i
% %5 % k F-
" F. :, , % ,.
%
#hm1 m2 : : : mk i = #hm1 m2 : : : mk j F i:
: , (%& i 2 C1 n] ( #hii = X i ! Y i ,
(, F-
F, , , ! F-
.
?!- 3 I #hm1 m2 : : :mk i
, mi , i 2 C1 k], k 6 n.
3.1. "
F | F-
( R. ,
F + = hhf#hm1 m2 : : : mk j F i I W j m1 m2 : : : mk 2 C1 n] W Rgii:
(. 9%
( & F . -& :
F + = F .
:( F F + , - % F F + . 4
, F + F-
", ! F . :% # 2 F + , hW ! W #1 #2 : : : #k i, & #i 2 F &
i 2 C1 n], | F-
", " &
2.1 (,
P # F . : ! " + & P ( %" F-
, ", (
P . (. P ). !
F-
" B2 (
) , F-
. : , " P " F-
_1 , (," 2, _1 = W I #1. ?(,
F-
_2 _1 &N ,
_2 = _1 I #2N ) : : :N ) _k = _1 I #k N ) # = _k B3. : B3 () _k " 3,
. ?%
,
# = W I #1 I #2 I : : : I #k B3 = W I #hm1 m2 : : : mk j F i B3
& mi 2 C1 n] (%& i 2 C1 k] #i F. _ =
= #hm1 m2 : : : mk i. ?
_ I W W I _ ((
(3.2), (_ I W)Z (W I _)N ) # 2 hh_ I W iiN ) # 2 F N ) F + F :
? 3.1, 2
, 3.1, & + % F1{F6 B1{B3.
216
. . , 3.1. -
&% (
.:
fI () B1=F1 (2
) F2 () B3=F4 (
)g:
3
:
1) F-
, (. 6) F ,
F-
F , &% . fI B1=F1 F2 B3=F4g, *:
F + = f#1 I #2 I : : : I #k I W F2 B3 j #1 #2 : : : #k 2 F W RgN
2) ( ) , . , 1). -( (-
. , #1 I #2 I : : : I #k I W F2 B3
k 6 (F) (. . 196), . I (F ) , B1=F1 ( I W ), F2, B3/ F4 . ! . F2, B3/ F4 & , +
3) 1 1) 2) &
F-
,
(, (. 3) F .
0+ - .. : (%" 23
" X ! Y " " " 2
. ' ! 2. : , + r &
" " t1 t2 2 r %," %" 3
t1(X \ Y ) = t2 (X \ Y ) ) t1 (X \ Y ) = t2(X \ Y )
(,
% ( & , %
.
? ,( B3/F4 () X \ Y %
,
X ! Y X ! Y n X. : % %
-" 3 " F-
" F fX ! Y j X Y Rg
+
.
3.2. 4
F-
" `(R) =
= fX ! Y j X Y Rg ( 3( 0 ( ) #0 = #Z ! #Y n #Z (%" # 2 `(R).
B F fX ! Y j X Y Rg | F-
",
. % F 0 = f#0 j # 2 F g.
3.3. ;
23
! " F G fX ! Y j X Y Rg -
, F + = G+ (. 5).
217
% 3.2. 5
F-
F F 0 .
(. :( F 0 F + %
" B3/F4
().
9%
: F-
" W = #Z ! #Z # 2 F . 1) 3.1 , #0 I W 2 F 0+ , #0 I W =
= #Z ! #Y #ZN ) # = #0 I W B3 2 F 0+ N ) F F 0+ .
+
3" F F 0+N ) F + F 0+N ) F + = F 0+ F 0 F + :
3.4. 4
! F-
" . %
(
3( B #1 B #2 = (#1 I #2)0 (%! #1 #2 2 fX ! Y j
X Y Rg.
% 3.3. #1 B #2 = #01 B #02 &( #1 #2 2 fX ! Y j X Y Rg.
(. : 3! , #1 = X ! Y #2 = Z ! V . &
(#1 B #2 )Y = (Y V ) n (X (Z n Y )) =
= (Y n (X (Z n Y ))) (V n (X (Z n Y ))) =
= (Y n X) ((V n (Z n Y ) n X) = (Y n X) (((V n Z) (V \ Y )) n X) =
= (Y (V n Z) (V \ Y )) n X = (Y (V n Z)) n X:
F-
" _1 _2 2 F 0 ,
:
(_1 B _2 )Y = _1 Y (_2 Y n _1 Z)
-
_1 B _2 = _1 Z (_2 Z n _1 Y) ! _1Y (_2 Y n _1 Z):
(3.4)
4
, #01 = X ! Y n X #02 = Z ! V n Z, - (3.4)
(#01 B #02 )Z = X (Z n (Y n X)) = X (Z n Y ) (Z \ X) = X (Z n Y ) = (#1 B #2 )Z:
6 (3.4) , (#01 B #02 )Y = #01 Y (#02 Y n #01Z) = (Y n X) ((V n Z) n X) = (Y (V n Z)) n X
(
#01 Y = Y n X, #02 Y = V n Z #01Z = X), , (#1 B #2 )Y =
= (#01 B #02)Y. (#1 B #2)Z = (#01 B #02)Z, 3.3 .
3.5. F-
X ! Y . , X \ Y = ?.
,/ - B.
0. 9
3 B ` = fX ! Y j X Y R X \ Y = ?g, F-
". 6 , #1 #2 2 ` )
#1 B #2 2 `.
218
. . (. 6 (3.4) (#1 B #2 )Z \ (#1 B #2)Y = (#1 Z (#2 Z n #1Y)) \ (#1Y (#2Y n #1Z)) =
= (#1 Z \ (#1Y (#2Y n #1Z)) ((#2 Z n #1 Y) \ (#1 Y (#2 Y n #1Z)) =
= (#2 Z n #1Y) \ (#1Y (#2Y n #1Z)) = (#2 Z n #1Y) \ (#2Y n #1Z) #2Z \ #2 Y = ?1):
III. 9
3 B , (#1 B #2 ) B #3 = #1 B (#2 B #3 ).
(. : 3 B
(#1 B #2) B #3 = (#1 B #2)0 B #3 = (#1 I #2 )00 B #03 =
= (#1 I #2 )0 B #03 = ((#1 I #2 ) B #3 = ((#1 I #2) I #3)0 =
= (#1 I (#2 I #3))0 = #1 B (#2 I #3 ) = #01 B (#2 I #3)0 =
= #01 B (#2 B #3 )00 = #01 B (#2 B #3)0 = #1 B (#2 B #3 ):
IV. 9
3 B , # B # = #. 6 % &, , 3 I, - , LM #1 B : : : B #k (, LM, #p = #q
! p q 2 C1 k], ! p < q, #1 B : : : B #q B : : : B #k = #1 B : : : B #q;1 B #q+1 B : : : B #k :
(. , , " 3, #1 B : : : B #k = (#1 I : : : I #k )0 , & . !- 3 I IV.
3 B & 3.1.
, 3.2. -
&% (
.:
fB B1=F1 (2
) F2 () B3=F4 (
)g:
3
:
1) F-
, (. 6) F ,
F-
F , &% . fB B1=F1 F2 B3=F4g, *:
F + = f#1 B #2 B : : : B #k I W F2 B3 j #1 #2 : : : #k 2 F W RgN
2) ( ) , . , 1). -( (-
. B, #1 B #2 B : : : B #k I W F2 B3
1)
\$
#+< # & B & I.
219
k 6 (F), . B (F) , , B1/ F1 ( I W),
F2, B3/ F4 . ! . F2, B3/ F4
& , +
3) 1 1) 2) &
F-
,
(, (. 3) F .
(. *", %+!, % "
3.2, " (%! # #1 : : : #k 2 F +! # = #0 I (#Z ! #Z) B3 #1 B : : : B #k = (#1 I : : : I #k )0.
4
" 3.1 3.2 2 ( .
6
( F-
( ( ( (`(R) | . 3.2)
A\$ = f`(R) I B1=F1 F2 B3=F4g A% = f`(R) B B1=F1 F2 B3=F4g:
A\$ A% 1) , . I fB B1=F1 F2 B3=F4g
. B fI B1=F1 F2 B3=F4g:
%
1] . . | .: , 1987.
2] .
!"# \$. | .: , 1968.
(
) ) 1997 .
1)
A | D A | #
*E
.
T-
. . 512.552
: , , T- ! .
" , # T- p > 0 ! .
Abstract
I. Yu. Sviridova, T-prime varieties and algebraic algebras, Fundamentalnaya i
prikladnaya matematika, vol. 8 (2002), no. 1, pp. 221{243.
We show in the paper that any non-matrix T-prime variety of associative algebras
with unit over a +eld of characteristic p > 0 is generated by an algebraic algebra of
bounded index over some +eld.
.
0
3]. # #
. \$ #%# #
. T-
p > 0 .
# F hX i | F p > 0, ( ( X. #
; | T- F hX i, F* hX i = F hX i=; | ( , #% T-# ;. ,
;n = ;\F hx1 : : : xni, F*n = F hx1 : : : xni=;n # #
n.
T- ; T-
, ;1, ;2 F* hX i , ;1 ;2 = 0, # , 2002, 8, , 1, . 221{243.
c 2002 ,
!
"# \$
222
. . ;1 = 0, ;2 = 0. \$ < T-
, T-
. .
/# # C F , C = P () + J(C), P () \ J(C) = f0g, J(C) | 3
C, P (C) | , % # .
( # m
L
P () = Di , Di = Mni (F) | ni i=1
F . , ei # Di . , ni = 1 1 6 i 6 m, # C # .
5 # . . # P () = F : : : F. 6, ei C, .#
mP
+1
m
P
J(C) =
ei J(C)ej , em+1 = 1 ; ei . , C | ij =1
i=1
m
P
, em+1 = 0 1 = ei .
i=1
7 ei rej , r 2 J(C), i 6= j, 5 . C, . ei rei , r 2 J(C), 5.
85 , .
( C (C) = dimF P () = m, (C) | J(C).
9 (C) ( C.
: , %
# # 2.
; <. <. ;. / 1] #%
# Cn, n Cn n F*n . = < 2, Cn .
( . / Xt = fx1 : : : xtg # #% ", (
u1 x1u2x2 : : :xtut+1 , xl 2 X, ul 2 F hX i:
X
": u1 x1u2x2 : : :xtut+1 7;!
(;1) u1 x(1)u2 x(2) : : :x(t)ut+1:
2S (t)
. # u1"u2 " : : :"ut+1 = "(u1 x1u2 x2 : : :xt ut+1), " 2 E(Xt ).
6, u1"u2 " : : :"ut+1 , " 2 E(x1 : : : xt), tP
;1
/, xi = ij yj ,
j =1
T- 223
ij 2 F, % . ?
, xi = zi +
tP
;1
+ ij yj #
j =1
X
u1 "u2 " : : :"ut+1 = u1(i)"(i) u2(i)"(i) : : : "(i) ut+1(i)
"(i) 2 E zi1 : : : zil t;1
X
j =1
(i)
il+1 j yj : : :
t;1
X
j =1
itj yj l > 0:
85 , #
# x y]k = 0 k. 3, #
## # #.
; gmt 2 F hX i:
gmt (x1 : : : xm+1 y1 : : : ym ) = "1 : : :"t y1 "1 : : : "t y2 : : :ym "1 : : :"t "i 2 E(x1 : : : xm+1). 6, m1 > m t1 > t gm1 t1 = 0 gmt = 0.
1. C | , (C) = m, (C) = t. C gmt = 0.
. ; # #
.
m
P
C gmt . = ci 2 C, ci = ij ej +ri, ij 2 F ,
j =1
ri 2 J(C), 5 , #
gmt (1 : : : m+1 m+2 : : : c2m+1 ) =
X
= "(i1 ) : : :"(it ) cm+2 "(i1 ) : : : "(it ) cm+3 : : :c2m+1 "(i1 ) : : :"(it ) (i)
(i)=((i1 ) : : :(it )), (ij )=(ij1 : : :ijm+1 ), "(ij ) 2 E(rij1 : : :rijl vijl +1 : : :vijm+1 ),
j
j
ri 2 J(C), vi 2 P (C).
/ t . C, , #
#
t .
. = , #
, t |
C, X
gmt(1 : : : m+1 m+2 : : : c2m+1) = ri1 ri2 : : :rit = 0:
(i)
< C #
# gmt = 0. @ . 2
# # # # C # ( m, C #
#
#% #
:
224
. . 1) (C) > m,
2) fi1 : : : im+1 g f1 : : : (C)g .
rj 2 J(C) ei1 r1 ei2 : : :eim rm eim+1 = 0.
2. ! " " C , (C) > m, (C) 6 t. C #
m
, C gmt = 0.
. 3, C ( m #
# gmt = 0. # (C) = m1 > m, ei1 r1ei2 : : :eim rm eim+1 = 0
i1 : : : im+1 .
ri 2 J(C). ; # # .
C
gmt (x1 : : : xm+1 y1 : : : ym ). = . cl 2 C m
m
P1
P1
cl = li ei + ei rijl ej , rijl 2 J(C), i=1
ij =1
#
X
gmt(c1 : : : cm+1 cm+2 : : : c2m+1) = g~mt(bi1 : : : bin )
(i)
g~mt | gmt, bk 2 f
liei ei rijl ej j
l = 1 : : : 2m + 1C i j = 1 : : : m1g.
D
, ei ej = 0, i 6= j, #
#
# . ##
.
g~mt (bi1 : : : bin ), #
bl ( #% .
ei1 ril11 i2 ei2 : : : eim rilmm im+1 eim+1 , i1 i2 : : : im im+1 |
.
( L f1 : : : m1 g. ; # CL C, (# . ei , ei rej , r 2 J(C), i j 2 L.
6, CL , . (CL ) = card L, (CL ) 6 (C). = g~mt(bi1 : : : bin ), bi 2 CL, ( card L 6 m.
,
, # #
. #, 1 CL #
# gmt = 0. = ,
gmt(c1 : : : cm+1 cm+2 : : : c2m+1) = 0
ci 2 C. ?
, gmt = 0 C.
3 #
# xj = eij , yj = eij rj eij+1 fi1 : : : im+1 g f1 : : : m1 g rj 2 J(C) gmt = 0. D
ei ej = 0 i 6= j, gmt(ei1 : : : eim+1 ei1 r1 ei2 : : : eim rm eim+1 ) =
= "1 : : :"t ei1 r1 ei2 "1 : : :"t ei2 r2 ei3 : : :eim rm eim+1 "1 : : :"t =
= ei1 r1ei2 : : :eim rm eim+1 "i 2 E(ei1 : : : eim+1 ):
T- 225
= C #
gmt = 0, ei1 r1ei2 : : :eim rm eim+1 = 0. ?
, C ( m. @ . 2
= ( C # ( C, C (
m1 < (C), # ( C #C
# ( !(C).
:
1 2 #% #
.
1. C " #
m , C gmt = 0 t gm1 t1 = 0 t1 , m1 < m.
. # ( C m (C) = t.
= 1, 2 C #
# gmt = 0. , C gm1 t1 = 0 t1
m1 < m. = ( ( (, (C) > m > m1 . = 2 C ( m1 < m, #, !(C) = m.
3 #
. , t = (C). (C) = m1 < m, 1 C #
# gm1 t = 0,
#
. ?
, (C) > m. / , C ( m1 m1 < m,
# 2 #
# gm1 t = 0. = , (C) = m, (C) > m, 2 C | ( m. . #
!(C) = m.
?
. 2
, g*mt (x y) = gmt (x x2 : : : xm+1 y : : : y), g*mt (x y) = "1 : : :"t y"1 : : :"t y : : :y"1 : : :"t "i 2 E(x x2 : : : xm+1 ):
3. C | , (C) = t. \$
C g*mt1 = 0
t1 , C gmt = 0.
. ; # # # # C, (C) = m1 , (C) = t. # C #
# g*mt1 = 0 t1 m. 3, C #
# gmt = 0. F m1 6 m, 1 C #
# gmt = 0.
mP
+1
,
m
#
x
=
iei ,
1 > m.
m
i=1
P
y = ei ri ei+1 , i | . F, ri | i=1
. C, g*mt1 (x y) = 0. = 226
. . ei ej = 0, i 6= j, #
g*mt1 (x y) = "1 : : :"t1 e1 r1 e2 "1 : : :"t1 e2 r2e3 : : :em rm em+1 "1 : : :"t1 =
X
Q i d(1) d(2) d(m+1) =
(;1) 1 2 : : :
m+1 e1 r1e2 r2 e3 : : :em rm em+1 = 0
i 2S (m+1)
mP
+1
Pt
"i 2 E(v1 v2 : : : vm+1 ), vj =
ji ei , d(j) = i (j). # i=1
i=1
.
i #, e1 r1e2 : : :em rm em+1 = 0.
( # i1 : : : im+1 . .# C ( m 2
#
# gmt = 0.
3
#
, g*mt = 0 gmt = 0. @ . 2
# < | , ; | <, F*n | n, #% T-# ;, Cn | , T Cn] = T F*n]. ; # # # C2 .
.
= < # # ( C2 # !(<).
,
, (. = , # # . m.
1. < | ,
%
"&
:
1) !(<) = m(
2) &
k > 2, !(Ck ) = m(
3) < gmt = 0 t gm1 t1 = 0 t1, m1 < m(
4) !(Cn ) = m " n > 2.
. ( #% :
1) ! 2) ! 3) ! 4) ! 1):
1) ! 2) 4) ! 1) , .# , 2) ! 3),
3) ! 4).
2) ! 3). # Ck | , T Ck] = T F*k ], !(Ck ) = m, (Ck ) = tk . = 1
Ck #
# gmtk = 0, , #
g*mtk (x y) = 0. = , g*mtk (x y) 2 T F*k ], , g*mtk 2 ;.
T- 227
; # C2m+1 , (C2m+1 ) = t. = 3 C2m+1 gmt = 0. ?
,
gmt 2 T F*2m+1]. = gmt 2 F hx1 : : : x2m+1 i, . , gmt 2 ;.
= , < #
# gmt = 0.
, < #
gm1 t1 = 0 m1 , t1 ,
( m1 < m. = < g*m1 t1 = 0. = k > 2, Ck #
# g*m1 t1 = 0, , 3
# gm1 tk = 0. : !(Ck ) = m, 1 Ck
#
# gm1 tk = 0 tk , m1 < m.
?
, 5 . # 3) .
3) ! 4). ; # Cn # n > 2,
# T Cn] = T F*n]. # < #
gmt = 0, g*mt 2 ;.
?
, g*mt = 0 Cn, 3
Cn #
gmtn = 0, tn = (Cn ).
F Cn #
# gm1 t1 = 0 t1 m1 < m, # #% # ,
< #
# gm1 t = 0, t = (C2m1 +1 ).
: . #
3).
= , Cn #
# gmtn = 0 #
gm1 t1 = 0 t1 , m1 < m. ? 1 !(Cn ) = m.
. 2
, 1 I #% #
.
2. gmt = 0 m t.
f = f(x1 : : : xn) xi # T C], C | F-, f xi C f jxi =y1 +y2 = f jxi =y1 + f jxi =y2 :
?
, f = f(x1 : : : xn) xi # T - ; , f jxi =y1 +y2 = f jxi =y1 + f jxi =y2 + g(y1 y2 x1 : : : xn) g 2 ;:
f = f(x1 : : : xn) # ;, f # ; x1 : : : xn.
6, f | xi , degxi f = pk k. 3
, # f # T C], f 2= T C]. 2 F C (
+ )m f = f jxi =(+)xi = f jxi =xi + f jxi =xi = (
m + m )f
228
. . m = degxi f. # f 2= T C], #, (
+ )m =
= (
m + m ). = F | , F .
# m = pk ,
char F = p.
; f(x1 : : : xn) 2 F hX i xji 2 X, 1 6 j 6 degxi f, i = 1 : : : n. ,
degxn f
degX
x1 f
X j
j
f
x1 : : :
xn j =1
j =1
f, f. ,
, #
f # ; .
#, f ;.
; T-
< m > 2. 1 < #
# gmt = 0 t. 6I# s, ps > t. ,
, < #
# gmps = 0.
#
P = F he1 e2 : : : em j ei ej = 0 i 6= jC e2i = ei i:
= P = Fe1 : : : Fem | 1 = e1 + : : : + em ei | P . ; P (( ( B = P F (F hX i)# ( # | (
). \$ , P (F hX i)# B B | ,
( 1 = e1 + : : : + em .
, P (X) B, ( X. . ei uej , i 6= j,
u 2 P (X), # 5, . ei uei , u 2 P (X), | 5.
(B) = m # B ( m. # C | , (C) = m.
K B , ': X ! C
I ': B ! C, '(ei ) = ei ,
1 6 i 6 m. , . C = B. 7 , . B #
xi 2 X bi 2 B. 7 X # . L#
, . b 2 B xi 2 X, degxi b > 0.
# V | B. ; . b 2 B, # b xi 2 X. 7 b
# xi # V , b xi B bjxi =yi +zi = bjyi + bjzi (mod V ):
T- 229
7 b 2 B # # V , b xj 2 X # V . < # # T- , b xi # V b 2= V , degxi b = pk k.
# ;(B) | B, ( . f(b1 : : : bn), bi 2 sB, f(x1 : : : xn) 2 ;C P | , ( fxpi j xi 2 X g. ; #
A = B=(;(B) + P):
3, A < F.
, ;1 = T A], ;1 (B) | B, (
. f1 (b1 : : : bn), bi 2 B, f1 (x1 : : : xn) 2 ;1.
K A , ; ;1. 3
, # f(x1 : : : xn) 2 ;. = b1 : : : bn 2 B
f(b1 : : : bn) 2 ;(B). ?
, ai 2 A I ai = *bi bi 2 B, f(a1 : : : an) = f(*b1 : : : *bn) =
= f(b1 : : : bn) = 0. 6, f 2 ;1 . ,, , #
;(B) ;1 (B).
( T- ;0 F hX i, ;0 (B) | B, ( . f(b1 : : : bn), bi 2 B,
f(x1 : : : xn) 2 ;0.
4. )
xj 2 X gi g 2 ;0 (B) ;0 (B).
. = g 2 ;0 (B), g = g^(b1 : : : bkP), g^ 2 ;0 ,
bi 2 B. g^ g^(x1 : : : xk) = g^i, g^i |
i
g^. P
= F | , g^i 2 ;0. / ., # bi = ij uij , uij |
j
I
X fel gml=1 ,
X
X X 1 1
X
g^(b1 : : : bk ) = g^i
j uj : : : kj ukj = l(j ) g~l (u1j1 : : : ukjk ):
i
j
j
6 g~l | g^i . ?
, g~l 2 ;0 l. = uij | I
X fel gml=1
g~l , . g~l (u1j1 : : : ukjk ) #
xi .
, xj 2 X gi . g 2 ;0 (B) # .
g~l (u1j1 : : : ukjk ), % #
xj 2 X. = g~l 2 ;0, . g~l (u1j1 : : : ukjk )
;(B). ?
, gi ;0(B). @ . 2
230
. . , . #
;(B), ;1 (B).
5. %
f 2 ;1(B) X
s
f = h + g h = li uli xpi vli g 2 ;(B)
xi 2 X uli vli | * X fej gmj=1 li 2 F: (1)
%, f xj 2 X , &
(1), %
h g xj degxj h = degxj g =
= degxj f .
. ; . f 2 ;1 (B). =
^ ^b1 : : : ^bn), f^ 2 ;1 , ^bi 2 B.
f = f(
6, h 2 F hX i h = 0 A , .
bi 2 B h(b1 : : : bn) 2
^ ^b1 : : : ^bn) 2 ;(B) + P.
2 ;(B) + P. ?
, f^ 2 ;1 , f = f(
= , B f = h0 + g0 , h0 2 P,
g0 2 ;(B). 7 X
s
h0 = li uli xpi vli xi 2 X uli vli | I
X fej gmj=1 :
(2)
P
# f , g0 = gi0 , gi0 | i
xj 2 X g0 , ( 4 gi0 2 ;(B). ?
, B X 0
gi = f ; h0 :
(3)
g0
i
? .
i ( , #
xj
f. , . . g. ,
ps v (2)
h # u
x
l
l
i
i
i
s
. h0 , degxj f = degxj uli xpi vli xj 2 X.
B (3) g = f ; h, . g 2 ;(B), h 2 P . ?
, f = h + g, h 2 P ,
g 2 ;(B), . h, g #
xj , ( degxj h =
= degxj g = degxj f. @ . 2
, Nk k
# . : Nk ( 5 . #
I1 I2 | . Nk . i1 #
.# I1 n (I1 \ I2), I1 n (I1 \ I2 ) #, i1 = 0 #C , i2 # .# I2 n (I1 \ I2),
I2 n (I1 \ I2 ) #, i2 = 0 #. = I1 6 I2 , i1 6 i2 . @ , Nk ( #. 5.
T- 231
1. F I1 I2 , I1 6 I2 .
2. # ( . i 2 I2 , j 2 I1 j < i, I1 < I2 .
3. \$ . Nk # .
= Nk | # , .
#
# . :
. I 2 Nk . # # (I). ,
, 1 6 (I) 6 2k , ( (?) = 1, (f1 : : : kg) = 2k .
# I | # s N.
, PI , ( fxpi j i 2 I g.
F I = ?, # , PI | #
.
; . f 2 ;1(B). # f . , (f) #
xi, degxi f > ps .
: % #, , f x1 : : : xn. . degxi f > ps 1 6 i 6 (f) degxi f < ps i > (f).
5
X X
s
f=
li uli xpi vli + g
(4)
i2I li
P P u xps v 2 P , ( h g g 2 ;(B), h =
li li i li
I
i2I li
xj . f. , i 2 I
degxi f = degxi h > ps. ?
, I f1 : : : (f)g.
# I | i, h 2 PI . (4) . f I 2 N(f ) .
= . f # ind(f) = (I). ,
, 1 6 ind(f) 6 2(f ) . # xi . f ( OP, i 2 I,
f # O5P.
6. ! f 2 ;1(B). f 2 PI , (f) > j > i
" i 2 I . J 2 N(f ) , J 3 j ,
(J) > ind(f) xj | +,- f .
. ; ind(f) = (I 0). = f 2 PI , I 0 6 I. # 2 N(f ) , J 2 N(f ) , j 2 J, J > I. ?
, J > I 0 . ,
fj g > I 0 . 6, j 2= I 0 , xj | O5P . @ . 2
, Di B: Di = fej ej xi el j xi 2 XC
1
S
j l = 1 : : : mg. = D = Di .
i=1
. f 2 B, % x1 : : : xn, #
D-
, #
#%# #
: i 2 f1 : : : (f)g ( di 2 Di , f jxi =di = f (mod ;(B)).
:, D-
. ;(B). ;
^ 1 : : : xn) 2 F hX i di 2 Di , f(x
232
. . ^ 1 : : : dn) D-
. 3
, . f = f(d
%, , f ^ i1 : : : eir eir+1 xr+1 ejr+1 : : : ein xnejn ):
f = f(e
= f xr+1 : : : xn xl
^ i1 : : : eir : : : eil xl ejl : : :)jxl=eil xl ejl =
f jxl =eil xl ejl = f(e
^ i1 : : : eir : : : eil eil xl ejl ejl : : :) = f(e
^ i1 : : : eir : : : eil xl ejl : : :) = f:
= f(e
7. f | %
;1(B). .
xi, degxi f > 0, f jxi =ej = f
(mod ;(B)), f 2 ;(B).
. , f 0 = f jxi =ej , f ; f 0 2 ;(B). =
f | xl 2 X ., f 0 xl . degxi f 0 = 0, degxi f > 0. 6, f f 0 | . f ; f 0 , 4 f 2 ;(B) f 0 2 ;(B). @
. 2
8. f | %
;1 (B). . &
di = ei1 xiei2 , i1 6= i2 , f jxi =di = f (mod ;(B)), xi +,
-.
. ; (4) . f:
X X
s
p
f=
lj ulj xj vlj + g:
j 2I
lj
, i 2 I. ? # xi = ei1 xiei2 . f.
D
, ei ej = 0, i 6= j, (ei1 xiei2 )k = 0 k > 1, #
X X 0 ps 0 0
X
s
lj ulj xj vlj + g =
f jxi =ei1 xi ei2 = li u0li (ei1 xiei2 )p vl0i +
li
j 2I nfig lj
X X 0 ps 0 0
=
lj ulj xj vlj + g = f (mod ;(B))
j 2I nfig
lj
u0lj = ulj jxi=di , vl0j = vlj jxi =di , g0 = gjxi =di 2 ;(B). = .
P P 0 ps 0
lj ulj xj vlj 2 PI nfig, I n fig < I, # 5 j 2I nfig lj
. f, 5# . ?
,
i 2= I, xi | O5P . @ . 2
Q# # # 2 S(m) q (y1 : : : ym ) = e(1) y1 e(2) y2 : : :ym;1 e(m) ym 2 B:
T- 233
# ylj 2 X, 1 6 j 6 , 1 6 l 6 m,
() = (1 : : : ), j 2 S(m). (
q() = q1 (y11 : : : ym1 )q2 (y12 : : : ym2 ) : : :q (y1 : : : ym ):
F = 0, # q(0) = 1.
9. yl x z 2 X | . 2 S(m) " i 2 f1 : : : mg
e(1) y1 : : :e(m) ym (ei xei )ps z =
= e(1) y1 : : :yj ;1(ei xei )ps yj : : :e(m) ym ei z (mod ;(B)) j = ;1(i):
. ? # s < #
# gmps = 0. ; # xl = el , 1 6 l 6 m,
xm+1 = ei xei , yl = e(l) yl e(l+1) , 1 6 l 6 m ; 1, ym = e(m) ym ei gmps (x1 : : : xm+1 y1 : : : ym ). # "i 2 E(e1 : : : em eixei ), "0i 2 E(ei ei xei )
(j) = i. D
, ei ej = 0 i 6= j, #:
gmps (e1 : : : em eixei e(1) y1 e(2) : : : e(m) ym ei ) =
= "1 : : :"ps e(1) y1 e(2) : : :yj ;1 ei "1 : : :"ps ei yj : : :e(m) ym ei "1 : : :"ps =
= (;1) (e(1) y1 e(2) : : :yj ;1 ei "01 : : :"0ps ei yj : : :e(m) ym ei "01 : : :"0ps ) =
X
ps s
s ;l
p
l
l
p
= (;1)
(;1) e(1) y1 : : :yj ;1 (ei xei ) yj : : :ym (ei xei )
=
l=0 l
= (;1) (e(1) y1 e(2) : : :yj ;1 ei yj : : :e(m) ym (ei xei )ps ;
s
; e(1) y1 e(2) : : :yj ;1(ei xei )p yj : : :e(m) ym ei ):
, #, e(1) y1 : : :e(m) ym (ei xei )ps z =
= e(1) y1 : : :yj ;1(ei xei )ps yj : : :e(m) ym ei z+(;1) gmps (e1 : : : e(m) ym ei )z =
= e(1) y1 : : :yj ;1(ei xei )ps yj : : :e(m) ym ei z (mod ;(B))
gmps 2 ;. @ . 2
; , > 0 1 6 6 2:
(5)
D I: (
1 1 ) < (
2 2), 1 < 2 ,
1 = 2, 1 < 2 .
, #
H # I# : H ! N0:
(
1) = 0
> 0
;
1
(
) = (
; 1) + (
; 1 2 ) + 1 > 0 1 < 6 2:
234
. . 6, I# #% : (
1 ) 6 (
),
1 6 C (
1 ) 6 (
) , 1 6 6 2.
= f 2 ;1(B) ((f) ind(f)) #
#
(5), . # H I# .
10. ! " D- f 2 ;1 (B), " " ;(B), ((f) ind(f)) = . S(m) () = (1 : : : )
q() f 2 ;(B):
. 3 #
# ((f)ind(f)).
. # ind(f) = 1. 7 , . f
#%
# (4), I = ?, f = g 2 ;(B).
. , #
D-
f 0 2 ;1(B), # ;(B), ((f 0 ) ind(f 0 )) < ((f) ind(f)), q( ) f 0 2 ;(B),
0 = ((f 0 ) ind(f 0 )).
# f #
#
, ind(f) > 1, = ((f) ind(f)).
3, q() f 2 ;(B).
: % #, # , f x1 : : : xn degxi f > ps 1 6 i 6 (f), degxi f < ps i > (f). #
f | .. ;
f (4):
f = h + g h 2 PI (I) = ind(f) g 2 ;(B):
. . h, g xj 2 X f. = ind(f) > 1, h 2= ;(B).
6, f | D-
, h D-
. 3
, # xj , 1 6 j 6 (f), #%
# dj 2 Dj , f jxj =dj = f (mod ;(B)). =
hjxj =dj = (f ; g)jxj =dj = f jxj =dj ; g1 = f + g2 = h + g3
g1 = gjxj =dj 2 ;(B), ;(B) | C g2 g3 2 ;(B).
= f h , #, h | D-
.
# i | . I. , I 0 = I n fig s
p
h , % xi . = i 2 I, degxi h > ps . .#
# h xi #%
. di 2 Di . = h 2= ;(B) xi | OP , 7
8 di = ei xi ei .
X
X X
s
s
p
p
0
li uli xi vli +
h = hjxi =ei xi ei =
lj ulj xj vlj =
x =e x e
0
0
0
0
0
0
li
j 2I
0
lj
i
i 0 i i0
235
T- =
X
X
s
li li (ei xiei )p vl0i +
li
j 2I
u0
0
0
X
lj
0
s
u0 xp v0
lj lj j lj = h (mod ;(B))
u0lj = ulj jxi =di , vl0j = vlj jxi =di , j 2 I. . s
u0lj xpj vl0j 2 PI j 2 I 0 .
( # # 2 S(m). =
0
q (y1 : : : ym ) h0 =
X
X X
s 0
s 0
p
0
0
p
= li q (y1 : : : ym )uli (ei xiei ) vli +
lj q (y1 : : : ym )ulj xj vlj :
0
li
0
j 2I
lj
0
j 0 = ;1(i0 ). , Wlj . # # :
s
Wlj = q (y1 : : : ym )u0lj xpj vl0j = q wj1 (ei xi ei )k1 wj2 : : :(ei xiei )kr 1 wjr wjl | , % #
# xi . 6, j = i, (
kl > ps. F j 2 I 0 , l kl < ps wj1 wjr 2 PI .
5 yx = xy + y x], Wlj #% :
Wlj = e(1) y1 : : :yj ;1 (ei yj : : :e(m) ym wj1 ei )(ei xi ei )k1 (ei wj2 ei ) : : :
(ei wjr 2 ei )(ei xi ei )kr 2 (ei wjr 1 ei )(ei xiei )kr 1 wjr =
degxi h
X
=
e(1) y1 : : :yj ;1 (ei xi ei )k wk0 0
0
0
0
;
0
0
0
;
0
0
0
0
;
0
0
0
k=0
w0
0
0
;
0
0
0
0
0
0
0
;
0
k xi, ei xiei # . ei uei 2 ei Bei . 6, j = i, . w00
Wli # .
, # ps
s
p
ei uei ei xiei : : : ei xi ei ] = ei uei (ei xiei ) ]. = . # . O
P . F j 2 I 0 , Wlj .
w00 2 PI . . # # , w00 | . O
P .
?
,
Rh = q (y1 : : : ym ) h = X li Wli + X X lj Wlj + g =
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
=
+
degxi h
X
k=0
degxi h
X
k=0
li
j 2I
0
lj
e(1) y1 : : :yj ;1 (ei xi ei )k w100k +
0
0
0
e(1) y1 : : :yj ;1 (ei xiei )k w200k + g =
0
0
0
236
. . =
degxi h
X
k=0
e(1) y1 : : :yj ;1 (ei xi ei )k (w100k + w200k ) + g
0
0
0
w100k | . Wli C w200k # Wlj , j 2 I 0 C g 2 ;(B). , wk00 = w100k + w200k .
= h = f ; g, g 2 ;(B) f # ;(B), h
. ?
, . Rh. 3
, q #, q h . # h , degxi h = pr , r | # .
; # xi = xi +1 . hR . ,
, ei uei 2 ei Bei
ei uei ei xei ]jx=x+1 = ei uei ei xei + ei ] = ei uei ei xei ]:
.# .
wk00 #, wk00jxi =1 = 0, wk00 xi C wk00jxi=xi +1 = wk00. # h . wk00 xi
k = pr . D
., pr
X
Rhjxi=xi +1 = e(1) y1 : : :yj ;1 (ei xiei + ei )k wk00 + g1 =
0
0
0
0
0
0
0
0
0
0
0
0
k=0
0
0
0
0
0
0
0
0
0
= e(1) y1 : : :yj ;1 (ei xi ei + ei )pr wp00r + e(1) y1 : : :yj ;1 ei w000 +
r ;1
pX
+
e(1) y1 : : :yj ;1 (ei xiei + ei )k wk00 + g1 =
0
0
0
0
k=1
0
0
0
0
0
0
= e(1) y1 : : :yj ;1 (ei xi ei )pr wp00r + e(1) y1 : : :yj ;1 ei wp00r +
r ;1
pX
00
e(1) y1 : : :yj ;1 (ei xiei )k wk00 +
+ e(1) y1 : : :yj ;1 ei w0 +
0
0
0
+
0
0
l=0
0
0
k=1
r ;1 k;1
pX
X
k=1
pr
X
0
0
0
k e y : : :y (e x e )l w00 + g =
1
k
l (1) 1 j ;1 i i i
0
0
0
e(1) y1 : : :yj ;1 (ei xi ei )k wk00 + e(1) y1 : : :yj ;1 ei wp00r +
k=0
r ;1 k;1 pX
X k
l
+
e(1) y1 : : :yj ;1 (ei xiei ) wk00 + g1 =
l
k=1 l=0
r ;1 k;1 pX
X k
l
R
R
= h + hjxi =1 +
e(1) y1 : : :yj ;1 (ei xi ei ) wk00 + g1 l
k=1 l=0
g1 2 ;(B). = hR # ;(B) xi , # =
0
0
0
0
0
0
0
0
0
0
0
T- r ;1 k;1
pX
X
237
k e y : : :y (e x e )l w00 2 ;(B):
(6)
j ;1 i i i
(1) 1
k
k=1 l=0 l
#
(6) #, e(1) y1 : : :yj ;1 ei wk00 2 ;(B)
1 6 k 6 pr ; 1.
3
, # . . , k0 k, 1 6 k 6 pr ; 1 e(1) y1 : : :yj ;1 ei wk00 2= ;(B).
= ;(B) | , #
e(1) y1 : : :yj ;1 ei wk00 2 ;(B)
#, l
(e(1) y1 : : :yj ;1 ei wk00)jyj 1 =yj 1 (ei xi ei )l = e(1) y1 : : :yj ;1 (ei xiei )l wk00 2 ;(B):
?
,
r ;1 k;1 pX
X k
l
e(1) y1 : : : yj ;1 (ei xiei ) wk00 2 ;(B)C
l
k=k +1 l=0
k kX
;1 X
k e y : : :y (e x e )l w00 2 ;(B):
gR =
j ;1 i i i
(1) 1
k
k=1 l=0 l
# h , degxi wk00 = pr ; k. ; xi
. gR. = degxi e(1) y1 : : :yj ;1(ei xiei )l wk00 = l + degxi wk00 = pr ; k + l > pr ; k0 l > 0 k < k0, gR pr ; k0 xi . gRk = e(1) y1 : : :yj ;1ei wk00 . 4 gR ;(B). , gRk 2 ;(B), # k0 .
= , 1 6 k 6 pr ; 1 . gRk 2 ;(B). ?
, gRk jyj 1 = yj 1 (ei xi ei )k = e(1) y1 : : :yj ;1(ei xiei )k wk00 2 ;(B). .#
. hR Rh = e(1) y1 : : :yj ;1 ei w000 + e(1) y1 : : :yj ;1 (ei xi ei )pr wp00r (mod ;(B)):
; :
00 + e y : : :y
00 :
e(1) y1 : : :yj ;1ei w000 = e(1) y1 : : :yj ;1 ei w10
j ;1 ei w20
(1) 1
00 . O
P . 5 9 6 w10
#
.
wk00, #
X
00 =
e(1) y1 : : :yj ;1ei w10
j1j2 (e(1) y1 : : :yj ;1ei 0
0
0
0
0
0
0
0
0
0
0;
0;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0;
0
0;
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
j1 j2
s
ei yj : : :e(m) ym wj1 ei ei xi ei : : : ei xiei ] (ei xi ei )p ]uj2 ) +
0
0
0
0
0
0
0
0
0
238
. . X
(j )e(1) y1 : : : yj ;1ei yj : : :e(m) ym uj1 ei uj2 ei (ei xiei )ps ]uj3 =
(j )
X l
=
j1 j2 (e(1) y1 : : :yj ;1 (ei xi ei )l yj : : : e(m) ym wj1 (ei xiei )ps u0j2 ;
+
0
0
0
0
0
0
j1 j2 l
s
; e(1) y1 : : :yj ;1 (ei xi ei )p +l yj
0
0
0
0
0
0
0
: : :e(m) ym wj1 ei u0j2 ) +
s
+ (j )e(1) y1 : : : yj ;1ei yj : : :e(m) ym uj1 ei uj2 ei (ei xiei )p ]uj3 =
(j )
X l
=
j1 j2 (e(1) y1 : : :yj ;1 (ei xi ei )ps +l yj : : :e(m) ym wj1 ei u0j2 + g2 ;
X
0
0
0
0
0
0
+
(j )
0
0
0
0
j1 j2 l
s
; e(1) y1 : : :yj ;1 (ei xi ei )p +l yj
X
0
0
0
0
0
0
0
: : :e(m) ym wj1 ei u0j2 ) +
(j )(e(1) y1 : : :yj ;1 (ei xi ei )ps yj : : : e(m) ym 0
0
0
0
0
0
0
0
0
uj1 ei uj2 ei ei ]uj3 + g(j )) = 0 (mod ;(B)):
00 2 ;(B). ?
gl g(j ) 2 ;(B). = e(1) y1 : : :yj ;1ei w10
0
0
0
0
0
,
00 (mod ;(B)) (7)
Rh = e(1) y1 : : :yj ;1 (ei xiei )pr wp00r + e(1) y1 : : :yj ;1 ei w20
00 | . O
P , w00 wp00r #
# xi C w20
20
#
# xi # ei xiei . ei uei , u 2 B.
;
Rh1 = e(1) y1 : : :yj ;1 ei wp00r = hR jxi =1 (mod ;(B)):
(8)
= hR 2 ;1 (B) ;(B) ;1(B), hR 1 2 ;1(B). # (8) # ;(B) . Rh #, Rh1 # ;(B). = Rh | D-
, hR 1 | D-
. degxi hR 1 = 0, , (Rh1 ) = (Rh) ; 1 < (Rh) = (f). = #
q( ) hR 1 2 ;(B)
0 = ((Rh1 ) ind(Rh1 )) S(m)
(0 ) = (10 : : : 0 ). = 0 6 1 , 1 = ((f) ; 1 2(f );1), 1 (1 ) = (11 : : : 11 )
q(1 1 ) hR 1 2 ;(B):
= , #
q(1 1 ) e(1) y1 : : :yj ;1 (ei xiei )pr wp00r = (q(1 1 ) Rh1 )jyj 1 =yj 1 (ei xi ei )pr 2 ;(B):
00 ). 6 w00 | .
; Rh2 = q(1 1 ) (e(1) y1 : : :yj ;1 ei w20
20
00
O
P . 7 , w20 2 PI . ?
, Rh2 2 PI , PI | . / , (7) #
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0;
0
0
0
0
0
0
0;
0
T- 239
r
hR 2 = q(1 1 ) hR ; q(1 1 ) e(1) y1 : : :yj ;1 (ei xi ei )p wp00r (mod ;(B)) =
= q(1 1 ) hR (mod ;(B)):
, #, hR 2 2 ;1(B)C hR 2 | # ;(B)
D-
, Rh | D-
, q(1 1 ) . . degxj Rh2 = degxj f 1 6 j 6 n, , (Rh2 ) = (f).
= Rh2 2 PI , 6 ind(Rh2 ) < ind(f). = , # 00 = ((Rh2 ) ind(Rh2 ))
S(m) (00 ) = (100 : : : 00 )
q( ) hR 2 2 ;(B):
= ind(Rh2 ) 6 ind(f) ; 1 ;(B) | B, 2 = ((f) ind(f) ; 1) (2 ) = (12 : : : 22 ) q(2 2 ) hR 2 2 ;(B):
#, S(m) () = (12 : : : 22 11 : : : 11 ) , = ((f) ind(f) ; 1) +
+ ((f) ; 1 2(f );1) + 1 = ((f) ind(f)),
0
0
0
0
00
00
00
q() f = q(2 2 ) q(1 1 ) q f = q(2 2 ) q(1 1 ) q h (mod ;(B)) =
= q(2 2 ) q(1 1 ) Rh (mod ;(B)) = q(2 2 ) Rh2 (mod ;(B)) = 0 (mod ;(B)):
?
, q() f 2 ;(B). @ . 2
2. ;1 = ;.
. / , A
#, ; ;1 . .# , ;1 ;.
3, # ; f ,
f 2 ;1 , # f 2 ;. ( # ; f(x1 : : : xn) 2 ;1 . , = (f), = ( 2 ). =
n* = 2 m + n.j ;
# Cn, # T Cn] = T F*n]. #
j
0
(Cn ) = t, xi1 xi2 yi3 zj 2 X | , 1 6 i1 6 n,
1 6 i2 6 m, 1 6 i3 6 m ; 1, 1 6 j 6 .
; h = gm;1t(x11 : : : x1m y11 : : : ym1 ;1)z1 : : :z;1 gm;1t(x1 : : : xm y1 : : : ym ;1 )z f(x01 : : : x0n):
3, Cn #
# h = 0.
= !(<) = m, 1 !(Cn ) = m. ?
,
(Cn) > m, (Cn) = mn > m, i1 : : : im+1 2 f1 : : : mng ri 2 J(Cn) ei1 r1ei2 : : : rm eim+1 = 0.
240
. . m. Cn h = 0. =
m
Pn l
Pn l
cl 2 Cn cl = i ei +
ei rij ej , rijl 2 J(Cn), i=1
ij =1
#
X
h(c1 : : : cn) = ~h(bi1 : : : bik )
(i)
~h | h, bi 2 f
liei ei rijl ej j l = 1 : : : n* C
i j = 1 : : : mng.
7# ## . ## .
~h(bi1 : : : bik ), #
bi #
5 . ei1 ril11i2 ei2 ,.. . , eim rilmm im+1 eim+1 , i1 i2 : : : im im+1 |
. ~h(bi1 : : : bik ), # bil 2 C(i) . C(i) | Cn, ( . ei , ei rej , r 2 J(Cn), i j 2 fi1 : : : im g.
= #, , h = 0
Cn ( m. #
C Cn, (C) = m.
f = f(x1 : : : xn) #% . Di . # # . f^ = f(d1 : : : dn) di 2 Di . = f 2 ;1 ,
f^ 2 ;1(B). ,
, f^ D-
, f # ;, f^ # ;(B). = ^ S(m)
10 f^ = ((f^) ind(f))
(f^) = (1 : : : f^)
q(f^f^) f^ 2 ;(B):
^ 6 , ind(f)
^ 6 2 , xi = di f = (f)
S(m) q() f^ 2 ;(B):
; ': X ! C, '(xl ) = bl bl 2 fei ei rij ej j i j = 1 : : : mg. , ' I ': B ! C, e1 (1) r11 : : :r1m;1e1 (m) bn+1 : : :e (m;1)rm;1 e (m) bn+ f(b1 : : : bn) =
^ = 0: (9)
= '(q() f)
;m
#
# .
C h.
m
P
P
l
= cl = iei + ei rijl ej , rijl 2 J(C), i=1
ij =1
#
X
h(c1 : : : cmn ) = h~ (bi1 : : : bik )
(i)
T- 241
~h | h, bi 2 f
liei ei rijl ej j l = 1 : : : mn C
i j = 1 : : : mg.
## ## . # f # ;, .
g~1 (b11 : : : b1k1 )b1k1 +1 : : : g~ (b1 : : : bk )bk +1 f(b01 : : : b0n)
(10)
g~i | gm;1tC j,
1 6 j 6 , ( , # bjl 2 CL, 1 6 l 6 kj , CL |
C ( m ; 1. 7 # 1.
| # .
(10), 1 6 j 6 #
bj1 : : : bjkj # 5 . ei1 r1ei2 ,.. . ,
eim 1 rm;1 eim , i1 i2 : : : im | . K (9) #, #.
=
h = 0 ( m Cn. ?
, . Cn. h
T F*n] \ F hx1 : : : xni, , h 2 ;.
h % T- (;g ) ;f , ;g | T-,
( gm;1t, ;f | T-, ( f. = , (;g ) ;f ;. = < T-
, #, ;g ;, ;f ;. , 1 < #
# gm;1t = 0 t. ?
, ;f ;,
f 2 ;.
= , f #
f 2 ;1 #, f 2 ;. ?
, . f 2 ;1 . 3
, # f 2= ;. ;1, ( # ;
% ;, # 5. . 2
3. \$
A .
. B | , 1 = e1 + e2 + : : : + em .
; . b 2 B,
b = 1e1 + 2e2 + : : : + m em + u u 2 P (X):
= ~b = (b ; 1)(b ; 2) : : :(b ; m ) 2 P (X).
m
P
3
,
b
=
b
;
,
b
=
(
j ; i)ej + u =
i
i
i
P
j =1
= ij ej + u = vi + uC
;
j 6=i
~b = b1 : : :bm = Y(vi +u) = Y vi +X vi1 : : :u : : :vil = X vi1 : : :u : : :vil 2 P (X)
m
m
i=1
i=1
(i)
(i)
242
. . m
Y
i=1
vi =
m X
Y
( ij ej ) = 0:
i=1 j 6=i
# T | B, ( .
b1 b2], b1 b2 2 B. = (~b)ps 2 P (mod T). = < | , 3 ( ( J = J(F* hX i) T- F hX i, ( #. <. ;. / 2] J | - . , .# N. 6, t 2 T t = f(b1 : : : bn),
f 2 J, bi 2 B. ?
, tN = (f(b1 : : : bn))N 2 ;(B), f N 2 ;. =
((~b)ps )N = (p + t)N = p0 + tN 2 P + ;(B)
p p0 2 P, t 2 T.
; .ma 2 A. # a = *b P
I, b 2 B. = b = i ei + u, A
i=1
((a ; 1) : : :(a ; m ))N ps
= (p0 + tN ) = 0:
?
, A | m ps N. . 2
2, 3 # #
#.
. T-
p > 0 .
. # < | T-
F, char F = p > 0.
@ , !(<) = m.
F m = 1, 1 < #
# g1t = 0 t. ; , #% x1 = 1
g1t(x1 x2 y):
g(x y) = g1t(1 x y) = y x : : : x]:
= < | , 7 y x : : : x] = 0 g1t(x1 x2 y) = 0 , , <. . # 3] < 5
F .
; # m > 1. =, 5, # A, # A | F, T A] = ;. ?
, < A.
= . 2
T- 243
<
# <. ;. /# % .
.
1] . . //
. . . . | 1990. | . 54, ' 4. | . 726{753.
2] Kemer A. R. On the radical of relatively free algebra // Abstracts of Ring Theory
Conference. | Miskolc, Hungary, 1996. | P. 29.
3] Sviridova I. Yu. Varieties and algebraic algebras of bounded degree // Abstracts of
Ring Theory Conference. | Miskolc, Hungary, 1996. | P. 60{61.
% & ' 1997 .
. . 514.763.2
: , , !" -
", \$%.
& ! " !"!' '% () 2- * ! + M @M 6= ?.
*-+! " '" !% **%!% .!!/
!") () 2- '-") p- ! !' *\$- !+" +% M !"* % @M .
(! *\$% '+" '! !\$'% - ) *! ( \$%).
Abstract
S. E. Stepanov, On an application of the Stokes' theorem in global Riemannian geometry, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1,
pp. 245{262.
Applying the Stokes' theorem we have deduced the Weitzenbock's formula for
symmetric 2-forms on a compact Riemannian manifold M with boundary @M 6= ?.
Using the formulawe have proved that Killing symmetric2-forms and Killing p-forms
on a Riemannian manifold M of non-positive sectional curvature and convex boundary @M must be either parallel or zero. Finally, we have applied our results to
the global theory of projective and umbilical maps.
x
1. 1.1.
(. 1, . 77{83]) ()*) + , *,
- )-. .
+ / ) - , + , +- - *,- )-, 0- . 1, 0 2. 3-
4. 5
(. 2] 3]), )-, 7*, / 8
Z
(div X) dv = 0
M
, 2002, 8, 6 1, . 245{262.
c 2002 !",
#\$
%& '
246
. . ) ))) 2
Z
Z
d= = =:
M
@M
(1.1)
>
. ?
m-
M p- !, ) )) (p + 1);1 d! = r! ) 1 6 p 6 m ; 1, r 7 d . A) p = 1 ) (
M ) ). 2*,) *, (. 2, . 59]):
Z
fFp (! !) ; (p + 1)jr!j2g dv = 0
(1.2)
M
+
Fp (! !) + +) , / ?++ ) M.
. (1.2) , + ) ,* - p-
M 0 ) + Fp (! !).
E, + + +
Fp (! !) , ) (. / 4, . 985{987]).
F
) p-
-
7 (. 5, . 339]) 2-, ) ' = 0, * (., , 1, . 54{55]) + ) ' = sym(r'). A) 0 + 2-
' *,) *, (. 1, . 592, 613]):
Z 1 1X
:::m
2
2
2
K
(e
e
)(
;
)
;
jr
'
j
;
j
'
j
dv = 0:
(1.3)
2 i<j i j i j
M
G | )0 ) K(ei ej ) | ) ) M 2-
Tx M + x 2 M, ) ei ej fe1 : : : em g, + 'x (ei ej ) = i ij (. 1, . 54{55,
592]).
. (1.3) , + ) ,* + 2-
' (' 6= ') M
0 ) K ) M. E, + )) ))) + + ) )) ) - p-
.
1.2. .+) + , - ) ) - p-
, I.->. 3
7
247
(. 6]) - - ,
- - p-
- +) - . J ) 7 ) d ) d , 0 D, : > +) p = 1, +
.
E
) 0) I.->. 3
, 0 + /
(. 7] 8]), + )
0 )
/ p- (. 9] 10]). 40) )) r! = p +1 1 d! + m ; 1p + 1 g 2 d !
(1.4)
*
p
X
(g 2 d !)(X0 X1 : : : Xp ) = (;1)i g(X0 Xi)d !(X1 : : : Xi;1 Xi+1 : : : Xp )
i=1
) *- - X0 : : : Xp M. +,
) p = 1 (1.4) ) (. 6] 2, . 47]).
A
) p-
! ) + ,) p-
, * 0 * p-
, p-
(., , 11]), *, * +
, 0 * . + ! 2 ker D \ ker d, +
! 2 ker D \ ker d. ., , , + ) (., , 2, . 43])
p- ! )) ! 2 ker d \ ker d.
K
) p-
! *, (. 7]):
Z
fFp(! !) ; p(p + 1)jD1 !j2 ; (m ; p)jD2 !j2g dv = 0
(1.5)
M
1 d! D2 ! = 1 g 2 d !.
p+1
m;p+1
. (1.5) , + , D1 ! =
)*, ,* - p-
M, 0 ) + Fp (! !). E, + 9] ) 1 6 p 6 m2 ] + 0.
1.3.
x 2 ), ) + 2-
M @M 6= ?, ) ) / 2 (1.1) *, (., , 3]):
Z
Z
(div X) dv = g(X N ) dv0
(1.6)
M
@M
248
. . ) ) X M + @M
) N .
x 3 - ) )) ,* + 2-
0 ) ) M. A, p-
! +* 2-
, + ) ) )) ,*, ) , p-
!. E* +* * ) ) - p-
(. 11] 12]).
2 + ) 0 ))) , + 2m-
M, , 2-
p !, + /
2-
( 2mj!j);1!, * M
) K 6 0 M ))) ). A) p-
*,
0) (. 11]).
A
) * * 0 - (., , 13]). x 4 0) -
+ 0)- ) , ) ,
+ 0 + * +* 2-
.
x
2. 2- " 2.1. > M m-
(m > 2) @M. > ))) (. 14, . 97])
M 0 f : @M ! M.
>
/ ) M (., , 15, . 252]) * * @M ,* ) 0 + x 2 @M
0 + @M Nx 2 Tx M.
> ) ) U 0 = U \ @M ) U M + x 2 @M * xi = xi (u1 : : : um;1 ). > fi () @x U 0. L * @M
0) f fi = @u
0
0
g g = g(f f ).
M) ) 7, + fi = g0 gij fj , Nj N i + fj fi = ji :
(2.1)
249
G gij | + g ) M, g0 |
+ g0 ) @M.
7 ) ) @M 0 M ) )
8
(r0X f )Y 0 = Q0(X 0 Y 0)N
(2.2)
(., , 3, . 92{93])
r0X N = ;f BX 0 (2.3)
Q = Q0 N | ) ) @M B | , ) Q0(X 0 Y 0) = g0 (BX 0 Y 0 )
) - - @M - X 0 Y 0.
3 , + M (
.
16, . 285]), ) ) Q0 ) @M 7* * N - 0 +- - ))) , Q0 (X 0 X 0 ) > 0 ) - X 0 2 Tx @M 0 + x 2 @M. > 0, 0 M @M ,* , - @M (. 16, . 286]).
F ) M , )
) Q0 ) @M 0) , , * @M Q0 ), *+ 7 + x 2 @M, Q0(X 0 X 0 ) > 0 ) - X 0 2 Tx @M.
2.2. ?
+) ) S2 M +- - 2-
M. > +* 2-
' 2 C1 S2 M * @M ) M,
) * Y 0 2 C1 T@M '(N f Y 0) = 0:
>
/ ) @M.
) ) 8 (2.2) (2.3), +
(2.4)
(r')(f X 0 N fY 0) = '(f BX f Y 0):
? (2.4) 7) :
(ri 'i1 i2 )N i1 fi fi2 = 'i1 i2 fi1 Qfi2 :
(2.5)
2
* * + (2.5) fl fj2 'lj2 , ) 0 (2.1), +
(ri'i1 i2 )'ii2 N i1 = Q fi fj 'ii2 'ji2 :
(2.6)
0
0
250
. . X X j = 'ki2 rk 'ji2 ; 'ji2 ri 'ii2 ) (. 17] 11])
1X
:::m
div X = 21
K(ei ej )(i ; j )2 + j 'j2 ; jr'j2 ; j'j2:
i<j
p
)
2 (1.6), dv = det g dx1 ^ : : : ^ dxm p
dv0 = det g0 du1 ^ : : : ^ dum;1.
> +) 2-
', 0*, ) ) ) M, +
2 (1.6) ) 0 g(Z N ) = ('ji2 rj 'ii2 )N i :
(2.7)
>
) 7 (2.6) (2.7), +
g(Z N ) = Q(') = Q (f ') (f ') :
(2.8)
>) , 7 2 (1.6) *, :
Z 1 1X
Z
:::m
2
2
2
2
0
2 i<j K(ei ej )(i ; j ) + j 'j ; jr'j ; j'j dv = Q(')dv : (2.9)
M
@M
A *,)
2.1. M m- @M . 2- ' 2 C1 S2 M , , ! (2.9).
?
+, ) @M ) ' = g ) = C1 @M. L 2-
' ) @M, '(N f X 0 ) = 0 ) * X 0 2 C1 T@M. >
/ g(Z N ) = N i ri = 0.
>/ *,)
2.2. M m- @M . 2- ' 2 C1 S2M , @M ' = g
2 C1 @M g M , Z 1 1X
:::m
2
2
2
2
K
(e
e
)(
;
)
+
j
'
j
;
jr
'
j
;
j
'
j
dv = 0:
(2.10)
2 i<j i j i j
M
x
251
3. \$
" 3.1. ?
+- - 2-
S2 M M @M.
2
+) 2-
' 2 C1 Sp M ) (.
5, . 339]), (rX ')X = 0 ) X 2 C1 TM. A) 0 2-
' + ; M,
t 2 J R, '(X X) = const )
X = ddt; .
3 (. 0 4, . 976]) * K ( ), * M )) cK 6 0 ( K > 0), *+ 7
+ x 2 M, K(X Y ) < 0 ( K(X Y ) > 0) )
*- - +- X Y 2 Tx M.
?
M @M +*
* 2-
', *,*) @M. 2
*,)
3.1. m- M @M ! 2- ',
@M .
(1) " M # , M , ' = g = const.
(2) " M , ' = g = const.
(3) " M # , ' = 0.
. >, + K 6 0 * M. J ) Q0
) @M ) (2.9) * ) + 2-
' M. Q ' 6= g )
= const, >. R. U
(. 18]) M
))) ), U 0 + x M )) ) U = U1 : : : Ur - - M1 M2 : : : Mr , r | + +- - + 2-
'
*,- m1 = dimM1 ,. . ., mr = dimMr .
252
. . Q 0 ) + x 2 M, K(X Y ) < 0 ) *- +- - X Y 2 Tx M, 0 (2.9) 1 = : : : = m = = const.
L
0 0 ) Q0 @M + 0
) M. . 3.1 2.2 )
3.2. m- M @M ! 2- ', ' = g 2 C1 @M g . " # ,
\$ , ' = g = const M .
3.2.
M
(rX0 !)(X1 X2 : : : Xp ) + (rX1 !)(X0 X2 : : : Xp ) =
= 2g(X0 X1 )V(X2 : : : Xp ) ;
p
X
; (;1)i g(X0 Xi)V(X1 X2 : : : Xi;1 Xi+1 : : : Xp ) +
i=2
+ g(X1 Xi)V(X0 X2 : : : Xi;1 Xi+1 : : : Xp )
V = ; m ; 1p + 1 d!
(3.1)
(3.2)
))) )*, (., , 9] 10]) ) p-
M. 19] ), +
(3.1) *
(d!)(X0 X1 : : : Xp ) = (p + 1)(rX0 !)(X1 X2 : : : Xp ) +
p
X
+ (;1)i g(X0 Xi)V(X1 : : : Xi;1 Xi+1 : : : Xp ) (3.3)
i=1
0 (1.3).
G +* 2-
' U 'ij = gl2 k2 : : :glp kp !il2 :::lp !jk2 :::kp = !ii2:::ip !ji2 :::ip (3.4)
!i1 :::ip 7 p-
. L
(3.3) ):
253
rk 'ij = rk !il2 :::lp ] !jl2 :::lp + !il2 :::lp rk !jl2 :::lp ] +
+ m ; 1p + 1 fgkj Vi ; gkiVj ; (p ; 1)(Vikj ; Vjki)g:
G ) *, +):
Vl = gl2 k2 : : :glp kp !ll2 :::lp rk !kk2 :::kp W
i
Vjlk = gl3 k3 : : :glp kp !jll3 :::lp ri!kk
3 :::kp :
>
)), + ) *- X Y Z 2 C1 TM
(rX ')(Y Z) + (rY ')(Z X) + (rZ ')(X Y ) =
= m ;2p + 1 fg(X Z)V(Y ) + g(Y Z)V(X) + g(X Y )V(Z)gW (3.5)
;p+2
2p
(')Y = ; m
(3.6)
m ; p + 1 V(Y )W Y (tr ') = m ; p + 1 V(Y ):
L
, ) ) (3.5) (3.6), 0 ), +
+) 2-
= ' ; 1p (tr ')g ))) . A
3.3. " m- M % p- !
(0 < p < m), M 2-
= ' ; p1 (tr ')g ', (3.4).
. ., + ) +) M ) +- - + ,) - - (. 20, . 158]). 2
* 0 , / , +) 2-
M + - +- (.
20, . 158{159]).
p-K
! * (. 3, . 126]) @M, !(N f X20 : : : f Xp0 ) = 0
(3.7)
0
0
) *- - @M - X1 : : : Xp . A) 0 (2.1) 'i1 j1 N i1 fj11 = !i1 i2 :::ip N i1 !ji21:::ip fj11 =
= !i1 i2 :::ip N i1 fi22 : : :fipp !j1j2 :::jp fj11 fj22 : : :fjpp g2 2 : : :gp p :
R ) p-
! ) ) @M ) M, +) 2-
', )) (3.4), ) @M.
3.3. M r' = 0 ) + 2-
(3.4), rk 'ij = (rk !ii2:::ip )!ji2 :::ip + !ii2:::ip (rk !ji2 :::ip ) = 0
254
. . rj!j2 = 0. *
+
, ' = g ) = const, + 7, +
!ij2:::jp !jj2 :::jp = m1 j!j2gij
(3.8)
) j!j2 = const.
> +*, - (3.4) !i1:::ip 0 p-
! ) 1 6 p 6 m ; 1. L
) + 2-
= ' ; p1 (tr ')g, , = 0.
) (2.9). >
Q0 > 0 Z 1X
:::m
K(ei ej )(i ; j )2 dv 6 0
M i<j
(2.9) 1X
:::m
K(e0i e0j )(0i ; 0j )2 = 0W
(3.9)
r'0 = r' ; p1 r(tr ')g = 0W
(3.10)
i<j
()(X) = (')(X) + 1p X(tr ') = 0
(3.11)
) ij = 0i ij fe1 : : : em g * X 2 C1 TM.
F
0 + , + ) K K 0 * M. >
(3.9) + 01 = : : : = 0m = 0 = 0 g. L ;m tr ', = ' ; 1p (tr ')g *+, + 0 = pmp
; m (tr ')g + 1 (tr ')g = 1 (tr ')g:
' = + p1 (tr ')g = p mp
p
m
4
, 0, + 1 6 p 6 m ; 1, (3.10) tr ' = const. 2, ) p-
! + ), 0 ) (3.8).
* V0 = 0. 4 /,
' = r(tr ') = 0, , , (3.10) / , + r' = 0.
>
+ * >. R. U
(. 18]) ) M ) ).
Q, , M 0 + , ) p-
! (2.9) , + ! = 0. >)
, 0 255
3.4. m- M @M ! @M % p- ! 1 6 p 6 m ; 1.
(1) " M # , j!j = const M , ! (3.8).
(2) " M , ! (3.8).
(3) " M # , ! = 0.
. Q )- ) 3.4 ) 2-
!
) @M ) M (3.8) / ), 0 grad j!j 2 C1 T @M, +) * *
) - (1) (2) 0 +, (3)
.
*+ 2-
- 2m-
M. G, + + 2-
!
* (. 15, . 126]).
3.5. 2m- M
! 2- ! . " & & (1) K M ,
(2) M K 6 0 M ,
p M \$ 2-
( 2mj!j);1!.
. >, + M
) - 7 , ) ) M ) 2-
! 0 +)) 1 j!j2g ) j!j2 = const :
glk !il !jk = 2m
ij
10 , + J (1 1) p
Jji = ( 2mj!j);1gil !lj
*, : J 2 = ;Id g(JX JY ) = g(X Y ) ) *X Y 2 C1 TM. 2, M ))) (. 21, . 139{140]) 2-
p
= = ( 2mj!j);1!. 256
x
. . 4. \$
&
" 4.1. > M N m n-
) + g g0 )) Y-Z
r r0. ?
0 f : M ! N C1 . E+ +
f ;1 TN M, E 0 = Tf (x) N + x 2 M.
A
f : TM ! TN ))) 0, fx
/ ) Tf (x) N T M ) 0 + x
) M f ))) + 1-
M +) f ;1 TN.
U M x1 : : : xm
U 0 N y1 : : : yn , +
f(U) U 0 , 0 f : M ! N ) )
ya = ya (x1 : : : xm )
a b c : : : = 1 2 : : : n. > , + Xi = @x@ i Ya0 = @y@ a , f = (f Xj ) dxj = (fja Y[a0 ) dxj
a
) fia = @y
+ Y[a0 ) f ;1 TN, @xi n 0
)- (Y[a )x = (Ya0 )f (x) ) - x 2 U. A r[ f = (fija Y[a0) dxi dxj (. 22{24])
2 ya
a k
b c 0a
fija = @x@ i @x
(4.1)
j ; fk ;ji + fi fj (;cb f)
) ri Xj = ;kjiXk r0Ya0 = ;0acb Yb0 dyc .
. (4.1) , + r[ f ))) +
+ M +) f ;1 TN, r[ f 2
2 C1 (f ;1 TN S2 M). / r[ f 0) f : M ! N.
+, f : M ! N ))) 0, (r[ f )(X Y ) =
= Q(X Y ) ) f(M) X Y 2 C1 TM.
1
g0 f ;1 TN g[0 , ) )) g[x0 = gf0 (x) ) 0 + x 2 M. >
/
) r[ 0 ) (., , 24]) +, r[ 0g[0 = 0.
3
) g[0 M 0 f
0 f,
+ g = f g[0 gij = fia fjb gab
+ Rang(fx ) = Rang(gx ) 0 + x 2 M. Q g = g ) 2 C1 M, 0 f ) , 257
= const | . Q 0 rg = 0, 0 )
(. 25]).
Q 0 f : M ! N 0* +* ; ) M
+* ;0 = f(;) ) N, 0 ) (., , 23]).
4
+ r[ f 0) *, (. 23{25]):
r[ f = V f + f V
(4.2)
V | ) ) 1-
M.
Q 0, + 0) +) ; ) M 0) +* ;0 / -
)) ) )
-, 0 f ) (. 25]). A) 0) f )) r[ f = 0:
(4.3)
A0, + *,)
4.1. # f : M ! N N g0 . " # f | , = 2(m1+1) lndet(g )] e;4 g ! M
2-.
. > f : M ! N | 0 . Q N | + g0, ,
/ + 7, M ) + g = f g[0 , (4.2) +)) *
(rZ g )(X Y ) = 2V(Z)g (X Y ) + V(X)g (Z Y ) + V(Y )g (X Z)
(4.4)
) - X Y Z 2 C1 TM. F
), +
(rZ g )(X Y ) + (rX g )(Y Z) + (rY g )(Z X) =
= V0 (Z)g (X Y ) + V0 (X)g (Y Z) + V0 (Y )g (Z X) (4.5)
V0 = 4V.
>
0, + f : M ! N ))) , ,
G = det(g ) 6= 0. E+ +
Gij + ) / gij (g ) x1 : : : xm U M. L ) X 2 C1 TM X(G) = Gij rX gij (4.2) X(ln G) = 2(m + 1)V(X), *+, +
V = 2(m1+ 1) grad(ln G):
(4.6)
258
. . / +, / )), + ' = e;4 g ) = 2(m1+1) ln G ) *
(rX ')(X X) = 0:
R ' ))) + 2-
. 4.2. Q + 4.1 3.3, 0 4.2. f : M ! N | M @M N , & f #.
(1) " M # ,
M f |
#, f (
(2) " M , f .
. > ) ) f : M ! N +
) @M ) M g N g0 0, 0
+ x 2 @M ) - X Y 2 Tx M 2 C1 @M g0 (f X f Y ) = (x)g(X Y ).
2 4.1 ' = e;4 g ) = 2(m1+1) lndet(g )] + ))) + 2-
M. >
/ ) @M )
M * m
det(g ) = m , , ' = m2+1
g, 2 C1 @M. >/ 2-
' = e;4 g + ) M @M ) (2.10), ' = 0.
>
, + M * 0* * , (2.10) r' = ' = 0. +m
P
, +, + (tr ')(X) = (Ei )'(Ei X) )
i=1
fE1 : : : Emg - M.
+ x ) M fe1 : : : em g Tx M, + 'x (ei ej ) = i ij
1P
:::m
) i > 0. L ' = 0 +, + (ei )
k = 0.
k6=i
R = grad = 0 , , | . >)
, - *, : rg = 0.
> , >. R. U
(. 18]), +
-. + M ))) ), f | 0. +, M ))) ), g = g )
= const , , f ))) .
259
Q 0 M 0* * , (2.10) , + ' = g ) = const. >
7
g = e4 g 0 ) g (4.4). 3 = 0. > 0 0 ), ) +, f | ). . A 0 ) - 27] f ) , +- +)- f 0 - ) ) 0) f.
E
0 f : M ! N ) (. 13]), r[ f = m1 g f
(4.7)
) J
{Y
0 f = trg r[ f (., , 22, . 11{12]).
.
+, N | g0 f : M ! N | ), f g0 = g,
(4.7) ) f +* *
(. 1, . 58{59]). - (4.7)
))) (. 26, . 126]) , +.
4.3. " M N , M .
. ?
M g = f g[0 .
+ + 0) f : M ! N g )
*
(4.8)
(rZ g )(X Y ) = V(X)g(Y Z) + V(Y )g(X Z)
) V(X) = m1 g[0 (f X f ) - X Y Z 2 C1 TM. . ) (4.8), +, , +
V = 12 grad(tr g ) = 21 grad jf j2:
(4.9)
Q f | / +) ), det(g ) 6= 0. 2 F. 2. 2* (. 26, . 122]) + M 0 ) g 2 C1 S2 M, )*, * (4.8), ))) - + ,) f 0 : M ! M 0 M 0 . 4.3.
260
. . F ) g + = g ; (tr g )g, (4.8) ) * (rX )(X X) = 0 ) * X 2 C1 TM. 2, )))
+ 2-
. L , 4.4. M N g g0 , f : M ! N #.
) = g ; (tr g )g g = f g[0 2- M .
Q + 4.4 3.3, 0 4.5. f : M ! N | # M @M N , & f #.
(1) " M # ,
M f |
#, f (
(2) " M , f .
. 2 4.4 ' = g ; (tr g ) +
+ 0) ))) + 2-
M. >
/ ) @M ) M * ' = (1 ; m)g, 2 C1 @M. >/ 2-
' = g ; (tr g ) +
) M @M )
(2.10), ' = 0.
>
, + M * 0* * , (2.10) r' = ' = 0. +, +, + = 0 )- (4.8), rg = 0.
> , >. R. U
(. 18]), +
-. + M ))) ), f | 0. +, M ))) ), g = g )
= const , , f ))) .
Q 0 M 0* * , (2.10) , + ' = g ) = const. >
7
g = ( + tr g )g 0 ) g (4.8). 3 = 0. > 0 0 ), ) +, f | ). 4.4.
'
261
1] . . . 1, 2. | .: , 1990.
2] ., !. "
#\$ . | .: %&, 1957.
3] Yano K. Integral formulas in Riemannian geometry. | New York: Marcel Dekker,
1970.
4] Wu H. The Bochner technique // Proc. Beijing Symp. Di,er. Geom. and Di,er.
Equat. (Aug. 18{Sept. 21, 1980). Vol. 2. | New York: Science Press & Gordon{Breach, 1982. | P. 929{1071.
5] 3 4. 5. #
6 7"
. | .: 5,
1982.
6] 7
9
:.-;. <37\$6 =>
?@ " 3
4 // C6? 3
3
" 3. | .: , 1985. | !. 260{279.
7] Stepanov S. E. The seven classes of almost symplectic structures // Webs & Quasigroups. | Tver': Tver' State University, 1992. | P. 93{96.
8] Stepanov S. E. A class of closed forms and special Maxwell equations // Proc. Conference on Di,erential Geometry (Budapest, July 27{30, 1996). | Budapest, 1996. |
P. 113.
9] Kashiwada T. On conformal Killing tensor // Natural Science Report, Ochanomizu
University. | 1968. | Vol. 19, no. 2. | P. 67{74.
10] Tashibana Sh. On conformal Killing tensor in a Riemannian space // Tohoku Math.
Journ. | 1969. | Vol. 21. | P. 56{64.
11] !X
" !. Z. [ X3
5
36 ;. . \@" " @ // %". "6. 7#
. "5. 3@. | 1996. | ] 9. | !. 53{59.
12] !X
" !. Z. <36 \$\$
@3X@
3 3
@3 // 6 5@\$5" ^57
5
3#@ 3
3. _. %. &#"@ `!"3
3 {#@ X\$| (
9, 4{6 {"\$
1997 .). | 9, 1997. | !. 114.
13] Stepanov S. E. On the global theory of some classes of mappings // Annals of Global
Analysis and Geometry. | 1995. | Vol. 13, no. 3. | P. 239{249.
14] _3 }. \$ 5"\$9
6 @3X\$@
6 3
. |
.: , 1971.
15] ~7\$
@ }., =
;. 4{{
>\$9
3 \$
. | .: ,
1975.
16] 3\$ 4. 5. }3
" 3 " >\$3. | .: , 1971.
17] !X
" !. Z. !33#@ 6 @3X@
3 3
"3 3
// . 3@. | 1992. | . 52, ] 4. | !. 85{88.
18] \@" ;. . ;
6 X\$ "@" " " X5@ " Riemann'"6 X
" // %". {.-33. -", 9. | 1925. | . 25. |
!. 86{114.
19] Yamaguchi S. On a theorem of Gallot{Meyer{Tachibana in Riemannian manifolds
of positive curvature operator // TRU Math. | 1975. | Vol. 11. | P. 17{22.
20] &. ;. }3
" 3. | .: %&, 1948.
21] \., _357 . [
"6 5{{
>\$9
3. . 2. | .:
_7@, 1981.
262
. . 22] 4"5" ., !" . . "
6 X
" 3
#@ ^
// X 3. 7@. | 1993. | . 48, ] 3. | !. 3{96.
23] Har' El Zvi. Projective mappings and distortion theorems // J. Di,erential Geometry. | 1980. | Vol. 15. | P. 97{106.
24] Nore T. Second fundamental form of a map // Ann. mat. pure et appl. | 1987. |
Vol. 146. | P. 281{310.
25] Yano K., Ishihata Sh. Harmonic and relatively ane mappings // J. Di,erential
Geometry. | 1975. | Vol. 10. | P. 501{509.
26] !
@" _. !. 5#@ ^
3
"6 X
". | .: _7@, 1979.
27] Mikes J. Global geodesic mappings and their generalization for compact Riemannian
space // Proc. Conf. on Di,. Geom. and its Appl. (Opava, August 24{28, 1992). |
Opava, 1992. | P. 143{149.
( ) * 1997 .
. . . . . 511.51
: , .
(
1+ 2+ 3= 1+ 2+ 3
3+ 3+ 3= 3+ 3+ 3
1
2
3
1
2
3
3+ 3+ 3 = 3+ 3+ 3
1
2
3
1
2
3
x
x
x
y
y
y
x
x
x
y
y
y
x
x
x
y
y
y :
Abstract
A. V. Ustinov, On some cubic equations, Fundamentalnaya i prikladnaya
matematika, vol. 8 (2002), no. 1, pp. 263{271.
The paper considers the
( structure of solutions of the system
1+ 2+ 3= 1+ 2+ 3
3+ 3+ 3= 3+ 3+ 3
1
2
3
1
2
3
and the equation
3+ 3+ 3 = 3+ 3+ 3
1
2
3
1
2
3
x
x
x
y
y
y
x
x
x
y
y
y
x
x
x
y
y
y :
2] 51, 52 ! " !. \$
% "& '
& " &" %
! ! . () % '
%
"
&, " '" & )
) ) . *
&,
z
;x 0
2
2
2
2
y
;w
x y + y z +z w + w x = x
w
.
W
%
& &!
3
z
y
+ X3 + Y 3 + Z3 = 0
= x + y + z + w
X =
x ; y ; z + w
Y = ;x + y ; z + w
Z = ;x ; y + z + w
W
, 2002, 8, , 1, . 263{271.
c 2002 ,
!"
#\$ %
264
. . &1 ' )
)
w
;
z
y
3z ;3y
w
3x = 0:
;
x
w
)
' &
&
(
x1 + x2 + x3 = y1 + y2 + y3
(1)
3
3
3
3
3
3
x1 + x2 + x3 = y1 + y2 + y3 :
& 1 ) 5 "& &)& " &" ) !
! . 6&
(1) %"
%! '
!. 1] &%
"
!) % , " '
"& &'
%, %
)
&& & (1), "% ' " )
&1 & & & &. 7&
2 % &1) 8& '
& &%
.
) % " ) &" !
.
9& , ' ) 3
3
3
3
3
3
x1 + x2 + x3 = y1 + y2 + y3 :
(2)
:)& %
& & (1) (2),
! %
" &! x1 , x2, x3 )
%
"& &! y1 , y2 , y3 " ) . (
%5&
&. ; %
& &!
aj = xj ; yj bj = xj + yj
(j = 1 2 3)
(3)
& 1)
3
3
3
2
2
2
a1 + a2 + a3 = 3a1 a2 a3 + (a1 + a2 + a3 )(a1 + a2 + a3 ; a1 a2 ; a1 a3 ; a2 a3 )
&
(1) & )
(
a1 + a2 + a3 = 0
(4)
2
2
2
a1 b1 + a2 b2 + a3 b3 + a1 a2 a3 = 0
(2) )& '
%&:
2
2
2
a1 b1 + a2 b2 + a3 b3 + D a1 a2 a3 = 0
(5)
)
3
3
3
a1 + a2 + a3
D =
3a a a :
1 2 3
\$ & (4) (5) ')& %
&, !) !)& && x1 , x2, x3 , y1 , y2 , y3 '
% & (1) (2) . (
')& %
&.
1. 1 2 3 1 2
(4). 2
2
1 = ( 3 ; 2)
2
2
2 = ( 1 ; 3)
2
2
3 = ( 2 ; 1)
| 2 Z
2 Z, a a a b b b3
d1 d2 d3
= d1 + d2 d3
a
d
d b2 = d2 + d1 d3 a
d
d b3 = d3 + d1 d2 :
. 6& (4) &1 )
8
a1 + a2 + a3 = 0
>
>
>
a
d
<
>
>
>
:
d
b1
a1
b3
;2
b
a2
b1
b
a3
;3
b2
;1
265
b
= 0:
(6)
(7)
; a1 a2 a3 b1 b2 b3 | 5 . ?& &1 " (d1 d2 d3) )) & 8
>
< d1 a1 + d2 b3 ; d3 b2 = 0
(8)
;d1 b3 + d2a2 + d3b1 = 0
>
:
d1 b2 ; d2 b1 + d3 a3 = 0:
:)& "
d1, d2, d3 &
&, a1 , a2, a3, b1 , b2 , b3 | &&. ? )&1 & (8) d1, d2, d3 1, & & (7) "&
(
a1 + a2 + a3 = 0
(9)
2
2
2
a1 d1 + a2 d2 + a3 d3 = 0:
B & &1 &
! %) (a1 a2 a3) (1 1 1) (d21 d22 d23). )! & ' , 8& "
&&
d1 = "1 d d2 = "2 d d3 = "3 d
("1 "2 "3 = 1):
* )
% (8) )
a1 = "1 "3 b2 ; "1 "2 b3 a2 = "1 "2 b3 ; "2 "3 b1 a3 = "2 "3 b1 ; "1 "3 b2 " %&1, ' ' %
" "1 , "2 , "3 ) & . ;8& % & (9) (a1 a2 a3) ) )%
", " ) &1 , %) (1 1 1) (d21 d22 d23):
2
2
a1 = (d3 ; d2 )
2
2
(10)
a2 = (d1 ; d3 )
2
2
a3 = (d2 ; d1 ):
266
. . ; %! a1 , a2, a3 & b1 , b2, b3 ) 1 ) &
8
; d3b2 + d2b3 = ;d1 a1
>
<
d3 b1
; d1b3 = ;d2 a2
>
:
;d2 b1 + d1b2
= ;d3 a3:
C
& 1)
, )
)
)
&& "
)) & ' )). D
&1 % )
b1 = d2 d3 b2 = d1 d3 b3 = d1 d2 :
; 8& ' %
) &:
b1 = d1 + d3 d2 b2 = d2 + d1 d3 b3 = d3 + d1 d2 " (10) ) 1) &. 7&
)
%
.
; & a1 a2 a3 b1 b2 b3 & (4). :)& )
" && . (c1 c2 c3 d1 d2 d3) %5& & )
) 8 , &
& & &, !)
"% " (a1 a2 a3 b1 b2 b3), 1 ! (4).
2. 1 2 3 1 2 3 | (4) 1, 2, 3 . 8
>
< 1 1 + 2 3 ; 3 2 = 0
(11)
; 1 3 2 2+ 3 1 =0
>
:
1 2; 2 1 3 3 =0
, (0 0 0 1 2 3) 1 2 3 1 2 3.
a a a b b b
d
d
d
a a a b b b
d a
d b
d b
d b
d a
d b
d b
d b
d a
d d d
. ? "% " (a1 a2 a3 b1 b2 b3) &1 & & & (0 0 0 d1 d2 d3), '& %
" t
) 1 (
a1 + a2 + a3 = 0
2
2
2
a1 (b1 + td1 ) + a2 (b2 + td2 ) + a3 (b3 + td3 ) + a1 a2 a3 = 0:
;
8 ! t, "& &
8
a1 + a2 + a3 = 0
>
>
>
<
2
2
2
a1 d1 + a2 d2 + a3 d3 = 0
(12)
>
a1 b1 d1 + a2 b2 d2 + a3 b3 d3 = 0
>
>
:
2
2
2
a1 b1 + a2 b2 + a3 b3 + a1 a2 a3 = 0:
267
E% & (12) ), " (a1 d1 a2d2 a3d3) ) & (d1 d2 d3) (b1 b2 b3). )! & ' , % 8 )
'
;a1 a2 a3 = a1b21 + a2 b22 + a3 b23 = 0
" %&1 ) . ;8& ) 1
&
8
>
<b2 d3 ; b3 d2 = a1 d1
(13)
b3 d1 ; b1 d3 = a2 d2
>
:
b1 d2 ; b2 d1 = a3 d3
6= 0. 7
d1 d2d3 6= 0, & &1& % " a1 , a2, a3
% (13). ;)
! "5 & (12), "&
; b1 d3 b2 + b1d2 ; b2 d1 b2 + a a a =
+ b3d1d
1 2 3
2
3
d3
2
= ; (b2 d3 ; b3 d2)(b3 d1 ; b1d3)(b1 d2 ; b2d1 ) + a a a =
b2 d3
;
b3 d2
d1
2
b1
d1 d2 d3
1 2 3
2
= (1 ; )da3dd3da2d2a1 d1 = (1 ; 2 )a1 a2a3 = 0:
1 2 3
()
), " = 1, (11).
6 ) , (11), , )&1
! d1, d2, d3
)
, "&
2
2
2
a1 d1 + a2 d2 + a3 d3 = 0:
? 1 ! )&1 b1 , b2 , b3 1, "&
a1 b1 d1 + a2 b2 d2 + a3 b3 d3 = 0:
6 )
, (0 0 0 d1 d2 d3) & )
) (a1 a2 a3 b1 b2 b3). 7&
)
%
.
1. (1)
! " #.
. & ' ) " d1 , d2, d3 . ;
, (0 0 0 d1 d2 d3) (d23 ; d22 d21 ; d23 d22 ; d21 d2d3 d1d3 d1d2), ') %
& (1) )1
!).
2. (1) !-
" ".
. (0 0 0 d1 d2 d3) & &.
'
% , (0 0 0 d1 d2 d3) (d23 ; d22 d21 ; d23 d22 ; d21 d2d3 d1d3 d1d2), &1 '
& '
%&, "' !)
!)& && x1 = y2 .
268
. . 6 ) & (2), & D "
& " &, %
& &! a1 , a2 , a3 .
p
2 Q 1 2 3 1 2 3 | (2), 1 2 3 =
6 0. 1 2 3 1 2 3 2 Z
2 Q, 1 = ( 2 3 ; 3 2 )( 2 3 ;
2 3)
2 = ( 3 1 ; 1 3 )( 1 3 ;
1 3)
=
(
;
)(
;
3
1 2
2 1
1 2
1 2)
1= ( 1 2;
1 2)( 1 3 ;
1 3)
2= ( 1 2;
1 2)( 2 3 ;
2 3)
3= ( 1 3;
1 3)( 2 3 ;
2 3)
. ; (2). ? &1 )
p
;2
1
p3
;3 2
p1 = 0
;1 3
2
3. D =
a a a b b b
b b b
s s s t t t
k
a
k s t
s t
Dt t
a
k s t
s t
s s
Dt t
a
k s t
s t
s s
Dt t
b
k s s
Dt t
s s
Dt t
b
k s s
Dt t
s s
Dt t
b
k s s
Dt t
s s
Dt t
:
D
a
b
b
a
b
7)
&
s s
8
>
<
b
D
b
:
b
a
D
p
+pb32 ; b23 = 0
;b3 1 + a2 D2 +pb13 = 0
>
:
b2 1 ; b1 2 + a3
D3 = 0
')
& &! 1 2 3 Q(pD). ; j = sj + tj pD, sj tj 2 Z(j = 1 2 3). 1)& p
) & ) 1 8 1 D .
;8& &
a1
D1
8
>
a1 s1
>
>
>
>
a2 s2
>
>
>
<
= b2t3 ; b3t2
= b3t1 ; b1t3
a3 s3 = b1 t2 ; b2 t1
(14)
>
a1 t1 D = b2 s3 ; b3 s2
>
>
>
>
>
a2 t2 D = b3 s1 ; b1 s3
>
>
:
a3 t3 D = b1 s2 ; b2 s1 :
&1
t1 , t2 , t3 )
! a1 , a2, a3 %
)! 5!, "&
8
>
<b2 s1 s3
( ; D t1 t3 ) = b3 (s1 s2 ; D t1 t2)
b3 (s1 s2 ; D t1 t2 ) = b1 (s2 s3 ; D t2 t3 )
>
:
b1 (s2 s3 ; D t2 t3 ) = b2 (s1 s3 ; D t1 t3 ):
(15)
269
? ' 8! ! ! ' )
% ' , ' s1 s2 = D t1 t2 s1 s3 = D t1 t3 s2 s3 = D t2 t3 :
6 )
, t1t2 t3 6= 0 ' '
p
s1
= s2 = s3 = D
t1
t2
t3
" " &. ? 1 t1 t2 t3 = 0, )
% &! 1 , 2, 3 . ; 3 = 0 "
&
1
2
=b =
b1
a2
2
p
D
b3
" %&1. 7
' " , % & (15) b1 , b2, b3
) )%
" " ) & ), %
&. F & % & (14) !) " a1, a2 , a3, 1 ') & 1 ). 7&
)
%
.
p
2 Q 1 2 3 1 2 3 | (2), 1 2 3 =
6 0. p
p
p
4. D =
a a a b b b
b b b
(0 0 0 1 2 3) = (0 0 0 s1 + t1
D s2
+ t2
D s3
+ t3
D
)
\$ , 1 , 2 , 3 8
p
>
1+ 3 2; 2 3 =0
< 1
p
(16)
;
3 1
2
2+ 1 3 =0
p
>
:
2 1; 1 2 3
3 =0
a
D
b a
b b . ; (0 0 0
b b D
b a
D
:
1 2 3 ) | )
) . 7)
8
a1 b1 s1 + a2 b2 s2 + a3 b3 s3 = 0
>
>
>
<
a1 b1 t1 + a2 b2 t2 + a3 b3 t3 = 0
(17)
>
a1 s1 t1 + a2 s2 t2 + a3 s3 t3 = 0
>
>
:
2
2
2
2
2
2
a1 (s1 + Dt1 ) + a2 (s2 + Dt2 ) + a3 (s3 + Dt3 ) = 0:
E% & (17) ), " (a1 s1 a2s2 a3s3 ) ) & (b1 b2 b3) (t1 t2 t3). )! & ' , % 8 ) 2
2
2
a1 b1 + a2 b2 + a3 b3 = 0
(18)
%&1 ) . G
",
8
>
<a1 s1 = (b2 t3 ; b3 t2 )
(19)
a2 s2 = (b3 t1 ; b1 t3 )
>
:
a3 s3 = (b1 t2 ; b2 t1 ):
a1 a2 a3 b1 b2 b3
270
. . F
", &
(a1 t1 a2t2 a3t3 ), % & (17) "&
8
>
<a1 t1 D = (b2 s3 ; b3 s2 )
(20)
a2 t2 D = (b3 s1 ; b1 s3 )
>
:
a3 t3 D = (b1 s2 ; b2 s1 ):
? & (19) )&1 s1 , s2 , s3 1 & & (20), )&1& t1, t2 , t3 , "&
0 = a1 (s21 + Dt21 ) + a2(s22 + Dt22 ) + a3 (s23 + Dt23 ) = ( ; )
b1
b2
b3
s1
s2
s3 :
t1
t2
t3
? ' (b1 b2 b3), (s1 s2 s3 ) (t1 t2 t3) ' & , %
)! ! & (17) )
' (18).
;8& " ) , = . ? && 1 , 2 , 3 , ) ! "& &
8
p
>
< a1 D1 + b3 2 ; b2 3 = 0
p
;
b3 1 + a2
D2 + b1 3 = 0
p
>
:
b2 1 ; b1 2 + a3 D3 = 0:
?5 ) ) 1 , 8&
p
; 3
a1
D1
b 1
b2 1
p
b3 2
a2
;
D2
b1 2
;
b2 3
=
p
b1 3
a3
D3
= 2 (a1 b21 + a2b22 + a3b23 ) + a1a2 a3D = (1 ; 2 )a1a2 a3D = 0:
()
= 1.
6 ) , (16), , &1
! ' (b1 b2 b3) (t1 t2 t3) )
, "& & (17). D5 ) 1) &
. 7&
)
%
.
3. (2) !! #.
4. (2)
!
(
x1
3
x1
= y1
+ x32 + x33 = y13 + y23 + y33 p
% " % Q
D
.
%
" )
%
& ) 1 2.
271
1] Chondhry A. Symmetric Dioph. Systems // Acta Arith. | 1991. | Vol. 59. |
P. 291{307.
2] Dickson L. E. Introduction to the Theory of Numbers. | Chicago, Univ. of Chicago
press, 1931. ( : . . ! "# \$%. | &'%,
1941.)
& ' ( 1997 .
. . 512.556
: , , .
!" (m 2 Rm
8m 2 R M ) % & '
. & . & '(, ,
, R ! ) ( , R % !*& (
:
1) R (), 2) R ( , 3) R ) () ( ) % ! )') , 4) R % %(
! ", 5) R ! ( ')(
( (, R " & &
), 6) R R ') R = 0.
Abstract
A. V. Khokhlov, On existence of unit in semicompact rings and topological rings with niteness conditions, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 1, pp. 273{279.
We study quasi-unitary topological rings and modules (m 2 Rm 8m 2 R M ) and
multiplicative stabilizers of their subsets. We give the de4nition of semicompact
rings. The proved statements imply, in particular, that left quasi-unitariness of a
separable ring R is equvivalent to existence of its left unit, if R has one of the
following properties: 1) R is (semi-)compact, 2) R is left linearly compact, 3) R is
countably semicompact (countably left linearly compact) and has a dense countably
generated right ideal, 4) R is precompact and has a left stable neighborhood of
zero, 5) R has a dense 4nitely generated right ideal (e. g. R satis4es the maximum
conditionfor closed right ideals), 6) the module R R is topologically4nitely generated
and R = 0.
( ) R R = 0, RR = R , r 2 Rr 8r 2 R ( 1] "left D-regular#, 2] | "left s-unital#, , 2002, 8, 5 1, . 273{279.
c 2002 ,
!"
# \$% ##&
274
. . , 3], )
QU-). +, , -
QU- -
. -
. , () QU- . . / QU- ,
0 , , /0 3]. 1 2 0 + : R (r 2 Rr2 8r 2 R) () R | QU- =)
R () R 0 () R | QU- =) R (r 2 (Rr)2 ) () R |
QU-.
4 )
3] , ,-
5
() . /
,
6
) )5 , 2
. 7 8
9 )
2
)
( 1.1, 1.2).
:
3] , , . ; 2 2,
,
, 0.1. R , R :
1) R (
)
(. 2.1),
2) R (. 2.2), , R # ,
3) R (
)
\$ (, ),
4) R \$ ,
5) R #,
6) R \$ (,
R # ),
7) \$ R %,
8) R R \$ R = 0,
9) R = 0 \$ #
\$ R
%.
1. , , 49 0 R, T | , R MT | ). L < R M ,
"L | R M # (L < R R | R), L / R | ,
"# | 9 (),9), a b a + b ; ab.
275
A A B R N R MT N , N | 2
NP, A B fa b j a 2 A b 2 B g, A N fan j A 2 A n 2 N g,
~ fn ; an j a 2 A n 2 N g.
AN ai ni j ai 2 A ni 2 N , AN
1.1. +- () )
2
A R MT 2
SA fr 2 R j rm = m 8m 2 Ag (AS ft 2 T j
mt = m 8m 2 Ag). B SA 6= ?, A - ) .
A
)C
: 9) A R 2
2 , AF fm 2 R j am = m 8a 2 Ag, FA fm 2 R j ma = m
8a 2 Ag, AFM fm 2 R M j am = m 8a 2 Ag.
A 9) A R MT SA SA SA, R SA = SA, AS T = AS . A
A B R MR B S SA B S \ SA, AS SA AS \ SA.
1.2. 2
B (R M ) SfrMF j r 2 Rg (B (MT )) ) () 8
R MT ) ()
)
M .
1.3. E R MT ( R RR ) ) () QU-, B (R M ) = M (B (MT ) = M ),
. . m 2 Rm 9) m 2 M .
4 (Soc M = M ) 2 (Soc(M=L) 6= 0
8L < M ) 3].
1
2 , )
6 "0#
2
R ( , ()
. .)
) 6
.
1.4. F
X R. +- 2
X -
)
8
BX (R M ) fm 2 R M j 9r 2 X : rm = mg = SfrMF j r 2 X g X -)
R M , R M ( R) | ( )
X , BX (M ) = M (BX (R R) = R BX (RR ) = R).
: + ()) 2
(
) . /
R X R ,-
, R (
2.1).
+2 9
95 , , )
2
3].
~ N . '
#
1.1. & R X X X , N BX (R M ), XN
A N SA \ X 6= ?, \$# N hN i BX (R M ).
.
~ N () XN N .
1. B N RSM | ,
PXN
P
~ i Ni , XN
~ N.
2. B N = Ni , Ni , Ni Z, XN
~ .
3. X | () X ; X X X X () X ; X X XX
4. 2
R | -.
276
. . 1. & X | R, R M N , N | ) X -
M , X . '
SA \ X 6= ? # A N .
2. * R M -
X R,
SA \ X 6= ? # A M .
3. + QU-
SA 6= ? # A R.
4. & X R, BX (R R) = R R M = RM ( ,
M = R). '
SA \ X 6= ? # A M .
5. * R X X X , QX (R M ) X # R M | (
,) X
. + R MT QX (R M ) QX (MT ) | R T -
. + R , X QX (R R) ( QX (RR )). - ### , ,
\$ BX (R R) (BX (RR )).
1.2 (3]).
1. * X | QU-
R, V W R M , S V \ X 6= ?, S W \ X 6= ?,
S
(V W ) \ X = S V \ S W \ X 6= ? S (V + W ) \ X 6= ?.
2. * X | # R, X B (R R), V W R M ,
S V \ X 6= ?, S W \ X 6= ?, S (V W ) = S V \ S W 6= ? S (V + W ) 6= ?.
1. -)# \$ # QU-
.
. A) M , , 2
T1-
QU- M . 1 6 () () SM 6= ?. G
| 6 M , SM = ? 2 (
)
, T1- 6).
2. + , \$, # .
3. * B (R R) X X ; X , BX (R R) < RR .
4. & X | (
) QU-
R. '
BX (R R) = B (X R) | (B (RX ) | ) QU-
, \$ X ,
() R.
1.1 (3]).
1. * 9A R M : A = 0 & SA 6= ?, R # . &
R = 0 .
2. * M = 0 SM 6= ?, SM = feg e | R.
2. 277
B R M | , , R (T0 - | ), SA BMF | 2
9) A M , B R S A] = SA.
2.1. & R , X R, X . '
R X () R e 2 X .
!". S r \ X | 2
X S
9) r 2 R. F 9 4 1.1 T S f r \ X j r 2 Rg S
. 4 X 5
e 2 f r \ X j r 2 Rg = R \ X . 2
I 9) A R M SA | ) 2 "# "#, SA 6= ?, ))5 2.1 2.1. I,9 - (,-
) , 9) (,-
) ) 6
,. / - (,-
) , - 9) (,-
) ) 6
2
,
95 ) 2 "# "#, ,.
2.2. * R M | , (
) -
X R, 9e 2 X :
em = m # m 2 M (SA \ X 6= ? # A M ). & M = 0
e | R.
!".
1. F 9 2 1.1 J fSA \ X j A M jAj < 1g | 2
X , ) ,.
F8
J | ) 6
- X 9e 2 T J =
= SM \ X . F M = 0 e | R 1.1.
2. F
A = fan j n 2 Ng, Am fan j n 6 mg. I SAm \ X SAm+1 \ X
S
Am \ X 6= ? 9 2 1.1. F8
J fSAm \TX j m 2 Ng |
,-
) 6
- X SA = J 6= ?. 2
1. .
2. -
QU-
, \$ \$ (, ), .
. 75
9
, (
, ) , ) .
3. * X | # -
R, QX (R M ) < M (. 5 1.1) ( X ).
278
. . 4. * R X , X | QU-
, BX (R R) | (. 4 1.2) ( X ).
2.1. + \$
X R SA \ X 6= ? # A R.
!". G, 2.1. 2
F2
X , ) ) , 9) V fx + V j x 2 X g
2 , X . 4 ,-
5].
2.2. & R \$
X . '
R # () R # V .
!". 9xi 2 X : X Sfxi + V j i = 1 : : : ng. F 9 1
1.2 S (xi + V ) 6= ?, 9e 2 S X . I 9) r 2 R
r = xr = (ex)r = e(xr) = er. 2
2.2. E (,-
) ,
- 9) (,-
) ) 6
2
,.
. 4 , 4{6], 2.2 )
,
) .
L 9
.
2.3. * R M | QU-
(
) R, SM 6= ? (SA 6= ? # A M ).
1. * R R | QU-
,
R .
2. * R R | QU-
R \$ (,
R ), R .
!".
1. F 9 2 1.1 J fSA j A M jAj < 1g | 2
R, ) ,.
I SA = S + A 9) A M , S 2 SA, J | ) 6
2 T
T R. 4 9e 2 J = fS m j m 2 M g.
2. F
A = fan j n 2 Ng, Am fan j n 6 mg. I SAm SAm+1 SAm 6= ? 9 2 1.1. F8
J fSAm j m 2 Ng | ,-
) 6
2 R SA = T J 6= ?. 2
279
E , , 2-, 9) , , 2 .
2.4. * R \$
QU-
R M , R # .
1. * R R | \$ QU-
, R .
!". I M = Tf m j m 2 M g = 0,T m | , 5
, A M , ,
f m j m 2 Ag = 0,
. . A = 0. F 9 2 1.1 SA 6= ?. I 1.1 R )
. ; 1.1 , 9)
QU- R = 0. 2
2.3. & R MT | R-
\$ T -
% N . '
N = M , . . 9e 2 R : em = m 8m 2 M .
!". N = eF 9) e 2 S N , ) N eF | )
T -. F2, ,
eF 6= M m 2= eF . 4)
s 2 R , ,
s(em ; m) = em ; m, , m = (s + e ; se)m, . . m 2 uF ,
u s e. e 2 S N =) u s e 2 S N . F8
eF = N = uF . M
,
m 2 uF n eF , , eF = M . 2
2. R () R | QU-
\$ fP < RR j S P 6= ? P ] = P g %.
!" # 0.1. N
2 1{5, 8, 7 9
2.2, 2.3, 2.4 2.2, 2.3, 6 3
1.1, 9 3]. 2
!
1] Ramamurthi V. S. Weakly regular rings // Canadian Math. Bulletin. | 1973. |
Vol. 16, no. 3. | P. 317{321.
2] Tominaga H. On s-initial rings // Math. J. Okayama Univ. | 1976. | Vol. 18,
no. 2. | P. 117{134.
3] . . !"#\$% #"&%' &(% %") ) *) // +. ,#%*. | 1997. | -. 61, ! 4. | .. 596{611.
4] 0& . 1., "20 +. 1., +% . . %"%% %03 !42%* *%) #"&%'. | 56%, 1988.
5] 70& +. 1. 5#!* % *) (%8. | 56%, 1991.
6] Zelinsky D. Linearly compact modules and rings // Amer. J. Math. | 1953. |
Vol. 75. | P. 73{90.
'
#
( ) 1998 .
CCC-
. . , . . . . . . . 521.13
: , , , { !
, " # \$ .
%
" #\$ ,& #\$
"
" '
" \$(
) ', \$ ! !''! ! . *\$, '
+(
" , )'
,! ", " "
&
- { !
.
Abstract
V. L. Shablov, V. A. Bilyk, Yu. V. Popov, Status of the CCC method within
the frame of the rigorous many-body Coulomb scattering theory, Fundamentalnaya
i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 281{287.
The convergent close-coupling method (CCC), which is now widely used for calculations of chargedparticles scatteringamplitudes, is considered from the viewpoint
of the rigorous many-body Coulomb scattering theory. It is shown that the approximate scattering amplitude calculated within the frame of the method does not
converge to a solution of the Lippmann{Schwinger equation.
, , !, " . \$"
% "
%, & (%&, &
, , 2002, 8, 3 1, . 281{287.
c 2002 !
" #\$,
%&
' (
282
. . , . . , . . . .) & & (, , , ) +" .
\$
%
++ &
, % & & . , % , & , && -& +.
" , & -& & ! + & % & . ., , & , +"& &
, + & %& & &.
/ + . 0+ , +" " + & , % " & , % +"
. / % & +" & +. 0 % +
, , , , .
1+ " & CCC (convergent close-coupling) 21], , & , % +"
&
5{-&. 7 & 22].
8 ( ! | , + 22, 3]) ,
+"+ & H , %- H = K1 + v1 + K2 + v2 + v12 = K1 + K2 + V:
(1)
283
CCC-
/ (1) Ki | %& %, vi | & , v12 | %& . / 21] 5{-&
T2 j'0~k0i = V2j'0~k0i + V2 G2(E + i0)T2 j'0~k0 i:
(2)
/ (2) j'0 i | -, j~k0i | +" , V2 = v1 + v12 G2(Z ) | ?: G2(Z ) =
= (Z ; K1 ; K2 ; v2 );1. 8 (2) Z Z h~k jV j ~k0 ih~k0 jT j' ~k i
X
h~k1f jT2j'0~k0 i = h~k1f jV2j'0~k0 i +
d~k0 1 f 2 i k 2 i 2 0 0 : (3)
E ; "i ; 2 + i0
i
0
/ (3) i (K2 +v2 )jii = "i jii
'i %& "i < 0, & , j~k2 i, %&
" = k2=2.
A+
CCC- (3) +
N Z
X
h~k 'N jV j'N ~k0ih~k0 'N jT N j' ~k i
d~k0 1 f 2 i k 2 i 2 0 0 :
h~k1'Nf jT2N j'0~k0i = h~k1'Nf jV2j'0~k0i +
E ; "Bi ; 2 + i0
i=1
(4)
/ (4) j'Ni i + &
X
IN = j'Ni ih'Ni j
0
i
j'0 i IN . 0
, +
h'Ni jK2 + v2 j'Nj i = "Bj ij & %& "Bi &
( " + ). E , (4) + &
% h~k'Ni jT2N j'0~k0i. / N
X
h~k2 j'Ni ih~k1'Ni jT2N j'0~k0 i:
(5)
i=1
/ h~k1 f jT2j'0~k0i h~k1f jT2N j'0~k0i. / 21] , h~kf jT2 j'0~k0i = Nlim
h~k jT N j' ~k i:
(6)
!1 f 2 0 0
284
. . , . . , . . A , (6) %&, " & .
H (4) , %
h~k1'Ni jT2N j'0~k0i + ~k1 i, +" ( % , , 24,5]).
E (5), CCC
+ . / , , N
X
lim
h~k2; j'Ni ih~k1'Ni jT2N j'0~k0i = h~k1~k2; jT2 j'0~k0i:
(7)
N !1
i=1
0 (7) , , " , ,
& & E %&:
E ! (k12 + k22)=2.
0 & % T2 . % + % T2 (Z ), %& &. 0 T2 (Z ) +"
? G(Z ) = (Z ; H );1 :
G2 (Z )T2 (Z ) = G(Z )V2 :
/ T (Z ):
T (Z ) = V2 + V G(Z )V2 :
E&
G0 (Z )T (Z ) = G(Z )V2 = G2(Z )T2 (Z )
& G0(Z ) = (Z ; K1 ; K2 );1 | ? , (Z ; H2 )G0(Z )T (Z ) = T2(Z ):
, h~k1~k2; jT2 (Z )j'0~k0 i Z %& %& %
h~k1~k2; jG0(Z )T2 (Z )j'0~k0 i, h~k1~k2; jT2(Z )j'0~k0i = h~k1~k2; jG0(Z )T (Z )j'0~k0i:
(8)
1
2
2
/ (8) = Z ; 2 (k1 + k2 ) | %& . , % (8) ! 0 "+ & %& % h~k1~k2 jG0(Z )T (Z )j'0~k0i 24,6{8]:
h~k1~k2jG0(Z )T (Z )j'0~k0 i ;1+i M (~k1 ~k2 ~k0 Z ) + i R(~k1 ~k2 ~k0 Z ) (9)
&
= ; k1 ; k1 + ~ 1 ~ |
1
2 jk1 ; k2 j
CCC-
285
, M R &
%& . \$
%
M (~k1 ~k2 ~k0 E + i0) E = "0 +
+ k22=2 = (k12 + k22)=2 + + 24,7]:
; 2 + iA) M (~k ~k ~k E + i0)
t(~k1 ~k2 ~k0 E + i0) = exp(;(1
(10)
1 2 0
; i)
& A = ; k1 ln 2k12 ; k1 ln2k22 + ~ 1 ~ ln j~k1 ; ~k2 j2
1
2
jk1 ; k2j
& 8 L0; , +"& 26, 7,10]. , (9) % T (Z ), +"& + 5{-&
, %& & &+" & , +" &
& +" . ,
(9) 24,6] , , 27,9], &
M (~k1 ~k2 ~k0 Z ) R(~k1 ~k2 ~k0 Z ).
& ;1+i ? & , + CCC- .
, (9) (8) , , ! 0 %
h~k1~k2; jT2 (Z )j'0~k0 i + h~k1~k2; jT2(Z )j0~k0i I ()M (~k1 ~k2 ~k0 Z ) + N (~k1 ~k2 ~k0 Z )
(11)
N (~k1 ~k2 ~k0 Z ) (11) %&, I () Z
2
0 2 ;1+i
k
;
2
0
0
~
~
~
2 = Z ; k21 :
(12)
I () = dk2hk2 jk2i 2 ; 2
Z1
Z
~; ~0
1
I () = B (i 1 ; i) dx x;1+i d~k0 hk2 jk2ik2 2 =
x + 2 ; 2
0
1
Z h~k; j~ri exp;ip2(x + 2 )r
1=2 Z
(2
)
;
1+
i
= ; B (i 1 ; i) dx x
d~r 2
: (13)
r
0
0
H& r (13) "+ M 211], 1
; p
;i2
+ i2) Z dx x;1+i k2 + 2(x + 2) 2i2 I () = e; 22 2 B (i;(1
(14)
1 ; i)
(x + )1+i2
0
286
. . , . . , . . & 2 = ;1=k2. H 212],
& , +" & I () 2 );(1 + i2 ; i)
i;i2 (2k22)i2 exp(; 2;(1
:
; i)
0, %
(8) + h~k1~k2; jT2(Z )j'0~k0i N~ (~k1 ~k2 ~k0 Z ) +
+ i;i2 (2k22)i2 e;( 2 (2 ;)+iA) ;(1 + i2 ; i) t(~k1 ~k2 ~k0 Z ) (15)
& %& N~ (~k1 ~k2 ~k0 Z ) & .
(15) , + N (4) CCC 5{-&, " &.
, CCC ,
5{-& (3) ( N ), . . 213].
7
5{-& +
, % & (15). M, % , +" "+ Z %& E .
, % 214, 15]. , %
, %
(
) "! & (15):
(d)LS = f (x)(d)exp &
x O x = ; = 1 ; 1 :
f (x) = exp(22x
2
);1
k1 j~k1 ; ~k2j
H , , , "
, %& +"& %, & k1 k2 , f (x) 1. P &
Q , (&)
5{-& %& + .
CCC-
287
R & \$ RS (& 97-0-6.1-32),
S1ME, S ! \$
RS ( 108-39(00)-,).
1] Bray I., Stelbovics A. T. Convergent close-coupling calculations of electron-hydrogen
scattering // Phys. Rev. A. | 1992. | Vol. 46, no. 11. | P. 6995{7011.
2] Bencze G., Chandler C. Impossibility of distinguishing between identical particles
in quantum collision processes // Phys. Rev. A. | 1999. | Vol. 59, no. 4. |
P. 3129{3132.
3] Bray I. Reply to Possibility of distinguishing between identical particles in quantum
collision processes // Phys. Rev. A. | 1999. | Vol. 59, no. 4. | P. 3133{3135.
4] . ., . !. "#\$%& \$%'& ((&#'& )& ('(\$* #(%)'+ ,&-##.+ /(\$'0. | .: 2, 1985.
5] . !. \$*\$'/(' %4%(. #\$%%5 \$%'' ((&#'& )& ('(\$*. \$+ ,&-##.+ /(\$'0 // 6. \$. '#-\$ 72 8. | 1963. |
6. 63.
6] Chandler C. The Coulomb problem. A selective review // Nucl. Phys. A. | 1981. |
Vol. 353. | P. 129c{142c.
7] 9:)% ;. ., <'). ;. 7., %4% =. ;. \$% ,%)#\$#.+ '#\$>)#.+
##'5 ,/ % ((&#'' \$+ /(\$'0 ( )%#%('* ,'*%5(\$'* //
#. ' 4'). *\$. | 1998. | 6. 4, .4. 4. | . 1207{1224.
8] Shablov V. L., Bilyk V. A., Popov Yu. V. The momentum representation of
the two-body Coulomb Green's function in n-dimentional space // Journal de
Physique IV (France). | 1999. | Vol. 9, no. Pr6. | P. 59{63.
9] Shablov V. L., Bilyk V. A., Popov Yu. V. The multichannel Coulomb scattering
theory and its applications to (e 2e) reactions // Journal de Physique IV (France). |
1999. | Vol. 9, no. Pr6. | P. 65{69.
10] 8' ., 5*%# <. \$%. (%*##%5 *\$*\$'/(%5 E',''. 6. 3. 6%'&
((&#'&. | .: ', 1983.
11] Nordsieck A. Reduction of an integral in the theory of Bremsstrahlung // Phys.
Rev. | 1954. | Vol. 93, no. 4. | P. 785{787.
12] %H . ;. 7('*4\$%\$': '#\$>). ' &.. | .: 2, 1987.
13] 8' ., 5*%# <. \$%. (%*##%5 *\$*\$'/(%5 E',''. 6. 1. #0'%#)#.5 #)',. | .: ', 1977.
14] Popov Yu. V. Investigation of a three-charged-particle break-up scattering amplitude // J. Phys. B. | 1981. | Vol. 14. | P. 2449{2457.
15] Popov Yu. V., Bang I., Benayoun J. J. A(xx + y)B processes with Coulomb interaction in the Mnal state // J. Phys. B. | 1981. | Vol. 14. | P. 4637{4647.
)
"* 2001 .
. . 512.541
: , , , ! .
"# ! \$ % H1 (' ), )
| +! \$! , ' 6 Aut .
G
G
G
Abstract
E. V. Shaposhnikova, On vanishing of the rst cohomology groups over separable
Abelian groups, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1,
pp. 289{300.
Conditions of vanishing of the 1rst cohomology groups H1 (' ), where
mixed separable Abelian group, ' 6 Aut , are obtained.
G
G
is a
G
1,2] . . !" #! ! , \$ ! ! % . &' " . . !
! %
!" H 1 () G), ) 6 Aut G, \$% %!%" !'
"! "!
!+%!, % !\$ !".
3] \$ " !"
H 1 () G) !#" \$" " " G, -" ! !+%!! 2 ) 6 Aut G, .
-! " !+%! ! / !! !!. & 0! ! \$ H 1 () G) !# ! \$ ! ! "
2-!", Z(2k), k 2 N, - ! !+%!!.
1\$ " | " \$ !" !# ! \$ ! ! ! "
" 2-!". -! 0
%
!' +!
\$ : \$ " !"
H 1 () G), ) 6 Aut G, ! %'!" "
G = G1 G2 (Hom(G1 G2) = 0)
(1)
, 2002, 8, 2 1, . 289{300.
c 2002 ,
!"
#\$ %
290
. . ! ! G1 ! !+%!, '- ), G2 ! " !+%!.
7 %' (1) ! H 1 () G) (! 2.4),
%' (Hom(G2 G1) = Hom(G1 G2) =
= 0) | ! H 1 () G)
( 2.5).
&/! . & :-!
! ()-!
!) %
! !
\$ . ! ! \$.! Z(:) (Z())), : ()) | !
\$ . & !" : :-!
! A +-
H 1 (: A) =
= Z1(: A)=B 1 (: A), Z1(: A) | -/ !!+%!
% : A, \$ '" % : A, f(xy) = f(x)+xf(y),
x y 2 :< B 1 (: A) | -/ !!+%!, \$ -/ !!+%! f(x) = a ; xa, a | + "
0! A.
=!#
% \$", !' / 0! !' '\$ ! !, -
!" !!" 1. & " 1 ! , %!+
Q " Z(p1).
>!+%! ' G % !, 'g 6= g 0! 0 6= g 2 G ( \$ ' ' ! #\$ " 0!
G). >!+%! ' \$ , " ; ' |
!!+%!.
(1]). f 2 Z1() G) , f 2 Im(" ; ) 2 C()).
?% C()) % . ). @!! \$"#! \$%\$ % .
?% )H % !+%! H G,
. !+%!! % ). >!+%! H, . " !+%!! ' 2 ), ! %\$ % 'H . B H |
G, 'H \$ !+%! ' H.
\$"#! \$%\$ ! \$.
7 " " " \$ 0 ! A1 ! A2 ! A3 ! 0
:-!
" \$\$ !" 4, . 87]
0 ! H 0 (: A1) ! H 0 (: A2) ! H 0 (: A3) !
! H 1 (: A1 ) ! H 1 (: A2 ) ! H 1 (: A3 ) ! : : :: (2)
F\$ H 0 (: A) = A = fa 2 A: a = a 2 :g | !' 0! A, ' \$ " :.
291
&
\$ ; | !\$ :. J \$\$ 5, . 449]
0 ! H 1 (:=; A; ) ! H 1 (: A) ! H 1 (; A) ! H 2 (:=; A; ) ! H 2 (: A): (3)
F\$ H 1 (; A) | !' 0! H 1 (; A), ' \$ " :. :-!
\$ H 1 (; A)
-! %!: '!
!!
f +
+ B 1 (; A), f 2 Z1(; A), !" ! f, f] = f;1 ] 0! 2 :, 2 ;.
x
1. H 1 ( )
-
G
p
= !! !" .!
p-
!. &
\$ G = Z(pk), p > 2, k > 1, ) | !+%! G. %, Aut Z(pk) = Z((p ; 1)pk;1) k
k
;
2
p > 3, Aut Z(2 ) = Z(2) Z(2 ) 6, . 35]. &\$ !!, " ' " " !+%! ' Z(pk) ! 1 + pr s, (p s) = 1, 1 6 r < k, \$
" !+%! \$ , 0!+%! " ; ' !!+%!!. 7 0 \$ !
, " ; ' !+%!. 7 p-
Z(pk) 0 %, " ; ' !,
%! ! p.
1.1. Z(pk) , ! p.
. F!! %
,
!+%! Z(2k) l
! 2 , l 6 k ; 2. !! 3 % 7] ', p-
!+%! p.
> !' %\$, p-
!+%! p. , Z(pk) !+%! pl , l 6 k ; 1. 7"\$, \$ ' |
" !+%! Z(pk) o(') = pl , l 6 k ; 1. &\$
Z(pk) pk , ' %/ 0! Z(pk)
pk pl . L % % s . ki, i = 1 s, ., , " , pl , " | !\$#!
-!
!
.,s \$ ki = pj , 1 6 j 6 l, i = 1 s. J!
P
%!, pk = 1+k1 +: : :+ks = 1+ pj , \$ pk 1 (mod p). &.
i=1
7, !+%! Z(pk), p > 3, ! ! %' Aut Z(pk) = h'1 i h'2 i, o('1 ) = p ; 1, o('2 ) = pk;1. M' '\$, !+%! '2 , o('2 ) = pk;1.
292
. . N, !+%! '1 !. -, " !+%! m, mj(p ; 1), . &'! , \$
' | " !+%! m, mj(p ; 1), \$ ' ! 1 + pr s, (p s) = 1, r < k. J !! m
P
(1 + pr s)m ; 1 0 (mod pk ). O ! Cmi (pr s)i 0 (mod pk ),
i=1
\$ pr s(m + Cm2 (pr s)2 + : : : + Cmm (pr s)m ) 0 (mod pk ). L !,
m < p r < k.
&
\$ \$ ' | %\$ " " !+%! Z(pk). = !!
%' Aut Z(pk) !+%! ' 'n1 1 'n2 2 , n1 n2 2 N. & 0!, 'n1 1 6= ", 'n1 1 | " !+%!, p ; 1, 'n2 2 | " !+%!. 7, " " !+%! Z(pk) ' ! " 0! ., '(g) = (1 + pr s)g = g, o(g) = p. O g = '(g) = 'n1 1 'n2 2 (g) = 'n1 1 (g) 0! g % . Z(pk). L !+%! 'n1 1 . F, 'n1 1 = ". J! %!, " "
!+%! ' Z(pk) ! pl , l 6 k ; 1.
1.2. " ' | n Z(pk). \$
1) p 6= 2 " + ' + : : : + 'n;1 = 0, ' , Ker(" + ' + : : : + 'n;1) = Im(" ; '), ' %
2) p = 2 " + ' + : : : + 'n;1 = 0, Ker(" + ' + : : : + 'n;1) =
= Im(" ; ').
. 1) &
\$ p 6= 2. B !+%! ' Z(pk)
, " ; ' | !!+%!, o(') = n, (" ; ')(" +
+ ' + : : : + 'n;1 ) = 0. J " + ' + : : : + 'n;1 = 0.
B ' | " !+%!, ! 1 + pr s,
(p s) = 1, !! 1.1 ! pl , l 6 k ; 1. &'!, 0! Ker(" + ' + : : : + 'n;1 ) = Im(" ; '). Ker(" + ' + : : : + 'n;1) Im(" ; ') .
&' ! %\$ , '! , 0! , ! ;r 6 k ; l. 7 0 ! (1 + pr s)p 1 (mod pk ), %, ;
;
pP
pP
;
p ; C i;1 pir si =
r
p
i
r
i
l 6 k ; r. !! (1 + p s) ; 1 =
Cp ; (p s) =
i p ; ;1
k
k
;
pP
r
r
k
r
i=1
k
k
r
r
i=1
k
r
k
r
=
pk p i Cpi;;1 ;1 si . ! ' ' ! i=1
pk . 7"\$, i = p , 2 N, r(i ; 1) = r(p ; 1) >
> p
; 1 > (1 + 1) ; 1 > 1 + ; 1 = , r > 1, i > 1, p > 2.
&"/! %\$
. &
\$ 0 6= x 2
2 Ker(" + ' + : : : + 'p ;1), \$ (" + ' + : : : + 'p ;1)x = 0. L!+%!
k r
r (i
;1)
k
r
l
l
293
" + ' + : : : + 'p ;1 ! 1 + (1 + pr s) + : : : + (1 + pr s)p ;1 =
;1 p ;1 ;1
;1
pP
pP
pP
P i ir i
p Ci
= pl +
Cj p s = pl + pir si Cpi+1 = pl + pir si i+1
p ;1 =
l
l
l
l
l
i=1 j =i
l;1
pP
1 Ci
pl 1 + pir si i+1
l ;1 ,
p
i=1
l
l
i=1
l
l
i=1
=
%\$ i + 1 = p , 2 N, (p ) = 1,
ir > , ir = r(p
; 1) > p
; 1 > (1 + 1)
; 1 > ,
p > 2. , 0!+%! " + ' + : : : + 'p ;1 !
;1
pP
1
pl 1 + pir si i+1
Cpi ;1 , ' ! , ! 1, l
l
l
i=1
l;1
pP
1 Ci
p. J ! 0 = ("+'+: : : +'p ;1)x = pl 1+ pir si i+1
p ;1 x,
i=1
x 2 Z(pk). =\$, pl x = 0, \$
!! | , %! p. O x = pk;l tg, hgi = Z(pk), t 2 Z, (t p) = 1. F,
r
k
r
k
x 2 Im(" ; ') = p sZ(p ) = p Z(p ), k ; l > r. J! %!,
Ker("+'+: : :+'n;1) = Im(" ; ') !+%! ' Z(pk) p 6= 2.
2) &
\$ \$ ' | !+%! Z(2k) o(') = 2l , l 2 N. J
;1
2P
1
0!+%! "+' +: : : + '2 ;1 ! 2l 1 + 2ir si i+1
C2i ;1
i=1
/ ! # !. & r > 1 !! |
/ : i + 1 = 2
, ir = (2 ; 1)r > 2
; 1 > , r > 1. &0!
, 1), Ker(" + ' + : : : + '2 ;1) = Im(" ; '), !+%! ! 1 + 2r s, r > 1. B ' ' ! 1+2s, Im(" ; ') = 2Z(pk), x 2 Ker("+'+ : : : + '2 ;1 ), " + ' + : : : + '2 ;1 6= 0, / x 2 2Z(2k). F, " + ' + : : :+ '2 ;1 = 0,
Ker(" + ' + : : : + '2 ;1 ) = Im(" ; ') !+%! Z(2k).
1.3. " G | &! p-
, ) | .
1. " p 6= 2 H 1 () G) = 0.
2. " p = 2 H 1 () G) 6= 0 , ) ) = h'i,
' " + ' + : : : + '2 ;1 = 0, o(') = 2l , '
H 1 () G) = Z(2)% ) ) | &!, ' H 1 () G) = Z(2)
1
H () G) = Z(2) Z(2).
. 1. &
\$ G = Z(pk), p > 3. J Aut G = Z((p;1)pk;1),
!+%! ) 6 Aut G .. &0!
f' 2
2 Im(" ; ') -/ !!+%! f 2 Z1() G), f 2 Im(" ; ') !+%! 2 h'i. B ' | "
!+%!, " ; ' | !!+%!, %, !+%!. =\$, H 1 () G) = 0.
l
l
l
l
l
l
l
l
l
l
l
294
. . &
\$ \$ ' | " !+%!, 0! ) = h'i - ' !+%!. &'!,
f' = x 2 G -/ !!+%! f 2 Z1() G).
J f'n = (" + ' + : : : + 'n;1)x = 0, n = o('). O !! 1.2
x 2 Im(" ; '), \$ -/ " !!+%! f !. =\$, H 1 () G) = 0.
2. &
\$ G = Z(2k), k 2 N. J Aut G = Z(2) Z(2k;2). B ) = h'i 2
;
1
"+'+: : :+' 6= 0, !! 1.2 !! Ker("+'+: : :+'2 ;1 ) = Im(" ; ').
O, -! , ! H 1 () G) = 0. B ' ) = h'i
" + ' + : : : + '2 ;1 = 0, ! " -/ " !!+%!
f : ) ! G, ' f' = x, x 2= Im(" ; '). J " ; ' 2= Aut G !+%! ' Z(2k), " 0! x 6= 0 "/.
& 0! f" = f'2 = (" + ' + : : : + '2 ;1 )x = 0, \$, "\$, f |
-/ " !!+%!. , 0! H 1 () G) 6= 0. &'!, 0! H 1 () G) = Z(2). &
\$ 0 6= f1 f2 2= B 1 () G) f1 6= f2 . \$
f1 ' = x1, f2 ' = x2, x1 x2 2= Im(" ; '). % %\$ !! 1.2
, "+'+: : :+'2 ;1 = 0, !+%! ' !
1 + 2s, (s 2) = 1. J (f1 ; f2 )' = (x1 ; x2) 2 Im(" ; '). F, (f1 ; f2 ) |
" -/ " !!+%!, \$ f1 + B 1 () G) = f2 + B 1 () G).
J! %!, H 1 () G) = Z(2).
O\$ !\$ ., ) .".
&
\$ ) = )1 )2 = h"i h'i, o(') = 2l , l 6 k ; 2. \$%
!
!" \$\$ (3). !!
0 ! H 1 ()2 G1 ) ! H 1 () G) ! H 1 () G) ! H 2 ()2 G1 ) ! H 2 () G):
Q!! H 1 ()2 G1 ). &
0! G, ' \$ )1 = h;"i, | 0 0! .\$ G, \$ G1 = Z(2). R
!+%! )2 \$ "
G1 , 0! .
Z(2k) ' \$ !+%! 0" : !+%! 2 Aut Z(2k) ! 1 + 2r s, r > 1,
(2 s) = 1, (1 + 2r s)g = g, g 2 G1 . F, H 1 ()2 G1 ) = Hom()2 G1 ) =
l
= Hom(Z(2 ) Z(2)) = Z(2) 6= 0. J\$ !" \$ %!, H 1 () G) 6= 0, ) | ..
7, !! H 1 ()1 G). &'!, Z1()1 G) G %
!+ )-!
. - , Z1()1 G) = G :
f = f(;") 2 G, f 2 Z1()1 G), (f1 + f2 ) = (f1 + f2 )(;") = f1 (;") + f2 (;"),
f1 f2 2 Z1()1 G). - , %!+%! )-!
\$ ". 7"\$, !, ('f) = '(f) ' 2 ), f 2 Z1()1 G). &
\$
'f = f1 , f = f(;") = g, f1 = f1 (;") = g1 . & 0! f1 (;") = ('f)(;") =
= 'f(;") = 'g, \$ 'g = g1. !! ('f) = f1 = g1 '(f) = 'g.
, Z1()1 G) = G )-!
. S, B 1 () G) | )-!
\$
l
l
l
l
l
l
295
Z1()1
G): f 2 B 1 () G) f(;") = 2a, a 2 G, ('f)(;") = 2'a 2 2G
' 2 ). %!+%! .
%!+%! )-!
"
B 1 () G) 2G )-!
" Z1()1 G) G. J H 1 ()1 G) = G=2G = Z(2) 1
H ()1 G) = (G=2G) = Z(2).
N., !! H 2 ()2 G1 ). J !+%! )2 ., 5, . 162] H 2 ()2 G1 ) = (G1 )2 =(" +
2
;
1
1
1
2
+ ' + : : : + ' )G . !! (G ) = Z(2), (" + ' + : : : + '2 ;1)G1 =
= (" + ' + : : : + '2 ;1 )g = 2l g = 0, G1 = hgi, ohgi = 2. O
H 2 ()2 G1 ) = Z(2)=0 = Z(2).
J\$ ! \$\$ !' \$ %
Z(2) !
! : 0 ! Z(2) ! H 1 () G) !
Z(2) ! H 2 () G). B
1
Im = 0, H () G) = Z(2). B Im = Z(2), Im = 0. J H 1 () G) 2
2 Ext(Z(2) Z(2)). & 0! H 1 () G) | 2-
, ;" 2 ). F,
H 1 () G) = Z(2) Z(2). , H 1 () G) = Z(2) H 1 () G) = Z(2) Z(2),
!+%! ) .. J! %.
8] M! ! " !" H 1 (Aut G G) %\$" " " G. J! 1.3 / H 1 () G), G | . p-
, ) | %\$ / !+%!.
l
l
l
x
2. H 1 ( )
! G
7 \$ "/ " !" H 1 () G) " G,
!-" ! %' G = G1G2 , ) 6 Aut G.
& 0! ! ! G1 !' !\$ !+%!,
'- ), ! ! G2 ! " !+%! 2 2 C()).
U! , , !+%! ) '
!+%! , . ! ! G1 G2. J
) = V h ()1 )2), V = W 6 Hom(G2 G1), )1 )2 | !+%! G1 G2 , . !+%!! % ).
J G1 ", " !+%!
G .
" !+%! G1 +-
G=G1 = G2. &0!
! ! G1 G2 G ' )-!
! !' \$ !" H 1 () G1) H 1 () G2). , !! \$\$ 0 ! G1 ! G ! G2 ! 0.
J ! \$\$
0 ! H 0 () G1) ! H 0 () G) ! H 0 () G2) !
! H 1 () G1) ! H 1 () G) ! H 1 () G2) ! : : ::
296
. . F % " !" H 1 () G) % !" H 1 ()1 G1) H 1 ()2 G2), % !. &0!
% % !'
! H 1 () G1) H 1 ()1 G1), H 1 () G2) H 1 ()2 G2).
&
\$ G | %\$ " :-!
\$. 7 !!
" G % " : G ( \$ !!+%! : : ! Aut G) \$ % :-!
G 9, . 111]. F,
% " !!+%! : : ! Aut G. O%! (:) = ) 6 Aut G.
! % \$ 0! 2 : ! \$ " G ( ! 0!!), g 6= g 0 6= g 2 G.
. B | " 0! % C(:), () | "
!+%! % C()).
@ %\$, !! %\$ :-!
.
> !,
2.1. " G | :- : ( G ' 2 C(:). f 2 Z1() G) , f = a ; a ' a 2 G.
. &
\$ f = a ; a 0! 2 C(:)
0! a 2 G. J %\$ 0! 2 : !! = , f ; f = f ; f, (" ; ())f = (" ; ())(" ; ())a.
= /! %! ! f = (" ; ())a = a ; a. =\$, f 2 B 1 () G).
2.2. " G | :-, : : ! Aut G, (:) = ).
1. ) H 1 (: G) = 0, H 1 () G) = 0.
2. ) : ( ' 2 C(:) ) 6 :, H 1 (: G) = 0 , H 1 () G) = 0.
. 1. &
\$ H 1 (: G) = 0. &'!, -
0
f 2= B 1 () G). Q!! ' f : : ! G, % -!
%!: f = f 0 () 2 :. &!, f | -/ "
!!+%!. 7"\$, 1 2 2 : !!, " , f1 +
+ 1f2 = f 0 (1) + (1)f 0 (2), " , f(1 2) = f 0 ((12 )) =
= f 0 (1)+(1 )f 0 (2). , f 2 Z1(: G). & 0! f " .
J ! 0! a 2 G f = f 0 (()) = a ; a =
= (" ; ())a ! 2 : , \$, ! ' = () 2 ).
F, -/ " !!+%! f 0 " H 1 () G) = 0.
2. &
\$ \$ H 1 () G) = 0. &'!, -
f 2= B 1 (: G).
J ) 6 :, f .
-/ " !!+%! f 0 : ) ! G,
f 0 ' = f', ' 2 ). &\$
f 0 | " -/ " !!+%!, f 0 () = (" ; ())a 0! a 2 G. N f 0 () = f()
() | " 0! % . :. J !! 1.4
f 2 B 1 (: G). &.
297
\$"#! ' % # !'
!
!" H 1 ()G0 G0) H 1 ()G0 G), G0 | ! ! G.
2.3. " G = G1G2 2 2 )\Aut G2.
1. H 1 ()1 G) = 0 , H 1 ()1 G1) = 0.
2. ) ) H 1 ()2 G) = 0, H 1 ()2 G2) = 0. ) ) H 1 ()2 G2) = 0, 1
H ()2 G) = H 1 ()2 G1) = Hom()2 G1).
. = '!, H 1 ()i G) = 0 /
1
H ()i Gi) = 0, i = 1 2. !! Z1()i Gi) 6 Z1()i G), i = 1 2, 0!
f 2 Z1()i Gi), f 2 B 1 ()i G). J f'i = (" ; 'i )a = (" ; 'i )ai 'i 2 )i, a 2 G, ai 2 Gi, i = 1 2.
=\$ ' % , \$ !
! ! ! !+%!!, '-! ). 7 ! ! G1 !. XH 1 ()1 G1) = 0
/ H 1 ()1 G) = 0Y ! !, \$ ! ! G2 ! " !+%! 2 2 ). 7"\$, -/ " !!+%! f : )1 ! G, f'1 2 G1 '1 2 )1 9],
\$ Z1()1 G1) = Z1()1 G). 7 ! ! G2 0 , H 1 ()2 G) !' \$ " " H 1 ()2 G2).
!! , " \$ )2 -!
"
0 ! G1 ! G ! G2 ! 0 \$\$ !"
0 ! H 0 ()2 G1) ! H 0 ()2 G) ! H 0 ()2 G2) !
! H 1 ()2 G1) ! H 1 ()2 G) ! H 1 ()2 G2) ! : : :
H 0 ()2 G2) = G22 = 0. O H 1 ()2 G1) = H 1 ()2 G) = Hom()2 G2),
!+%! )2 \$ "
G1 .
J\$ !' \$ ! !" H 1 () G)
! %'!" " G = G1 G2 , \$ ! ! G2 ! " !+%!, '-"
).
O%! % J 0! G, ' \$ !+%! % ), \$ J = G.
2.4. " G = G1 G2, Hom(G1 G2) = 0, 2 2 ) =
= V1 h ()1 )2) 6 Aut G.
1. ) H 1 ()1 G1) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = 0, H 1 (V G1) = 0, 1
H () G) = 0.
2. ) H 1 () G) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = 0.
. 1.1 !!" 2.2 H 1 ()2 G2) = 0
0 H ()2 G2) = 0. J % " \$
0 ! H 0 () G1) ! H 0 () G) ! H 0 () G2) !
! H 1 () G1) ! H 1 () G) ! H 1 () G2) ! : : :
, H 1 () G) = H 1 () G1), \$
H 0 () G2) = 0.
298
. . % # / " ! !, H 1 () G1) = 0. 7 0 !! \$\$ (3) \$ G1 | )-!
\$,
V ). !!
0 ! H 1 ()1 )2 G1) ! H 1 () G1) ! H 1 (V G1) ! H 2 ()1 )2 G1) ! H 2 () G):
&'!, 0" \$ H 1 ()1 )2 G1) = 0. J
H 1 () G1) = 0.
J )1 )1 )2 , !! " ' \$ (3),
!/" )-!
G1 ,
0 ! H 1 ()2 J) ! H 1 ()1 )2 G1) ! H 1 ()1 G1)1 2 !
! H 2 ()2 J) ! H 2 ()1 )2 G1):
F\$ H 1 ()2 J) = 0 ! , H 1 ()1 G1)1 2 = 0, \$
H 1 ()1 G1) = 0. F, H 1 ()1 )2 G1) = 0.
2. &
\$ H 1 () G) = 0. % \$ (2) %
, H 1 () G1) = 0. O !! 2.2 H 1 ()1 G1) = 0. = /
! 0 !\$ H 1 ()2 J) = 0 % \$ (3), / #, %\$ 1.
. 1
1) Z H ()2 G2) = 0 1 ! 2.4, G1), % !! 1.2, ! ! H 1 () G) ' , ) ' " !+%! , .
-" " !+%! \$
G2, G1. &0!
1 !
\$ ! ! !.
2) &! ! 2.4 H 1 (V G1) = 0. L %\$
, ! H 1 (V G1) = 0. 7"\$, !+%! V
"
G1 \$. &0!
H 1 (V G1) = Hom(V G1). B
0 6= f 2 H 1 (V G1) , ('f) = 'f(';1 ') = f 2 V ' 2 ). O, , , H 1 (V G1) '
!!+%! % V G1, % f 2 G1 " 6= '2 2 ) !
f = f(';2 1 '2 ), 6= ';2 1 '2 . N!, G = Z(p) R,
R | % 1, pR 6= R, !+%!
) 6 Aut G, " V 6= f"g, H 1 (V G1) = 0 , ! H 1 (V G1) 6= 0, \$
! V = Hom(G2 G1) = Z(p).
1
J H (V G1) = Hom(Z(p) Z(p)), " " !!+%! |
!!+%!.
3) N! % 2 ! 2.4 % \$ H 1 () G). &/! 0 /
. &
\$ G = Z(pk) R, R | % 1,
) 6 Aut G, J 6= 0. J
) p = 2 H 1 () G) 6= 0<
H 1 ()1
299
) p 6= 2 )2 6= h;"i H 1 () G) 6= 0 %! , %' G ! ! H 1 ()1 Z(pk))Q H 1 ()2 R). 7"\$, J | .
, )2 6 Aut R = Z(2) Z, : = fp: pR = Rg. F, )2 6= 2)2, p2
)2 6= p)2 p 6= 2, )2 6= h;"i. =\$, Hom()2 J) = H 1 ()2 J) 6= 0,
\$ # ! H 1 () G).
%' G = G1 G2, 2 2 ) 6
6 Aut G, " H 1 () G).
2.5. " G = G1 G2, Hom(G1 G2) = Hom(G2 G1) = 0,
2 2 ) 6 Aut G. * H 1 () G) = 0 , H 1 ()1 G1) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = Hom()2 J) = 0.
. 7\$ / # " % ! 2.4, ! H 1 (V G1) = Hom(h"i G1 ) = 0.
N!\$ " H 1 ()1 G1) = 0, Hom()2 J) = 0 ' % ! 2.4. O/ %\$ !\$ H 1 ()2 G2) = 0. &'! , \$ H 1 () G) = 0 H 1 ()2 G2) 6=
6= 0. J !! 2.2 !! H 1 () G2) 6= 0. &
\$ f2 2= B 1 () G2). & 0!
f2 2 2= (" ; 2)G2 !!" 2.1. U! , f2 2 Z1() G) (" ; '2)G = (" ; '2 )G2 , \$ f2 2 2= (" ; 2)G. L %, f2 | " -/ " !!+%! % ) G. &. =\$,
H 1 () G) = 0, H 1 ()2 G2) = 0. = %.
B !+%! ) ' " !+%! ,
.
-" !+%! G1, G2 , H 1 ()2 J) = 0 ' , J = 0, H 1 () G) ! ! ! !" ! ! ! ! H 1 ()1 G1) H 1 ()2 G2), % ! ' +!.
2.6 (1]). " G = G1 G2, Hom(G1 G2) = Hom(G2 G1) = 0,
2 ) 6 Aut G. * H 1 () G) = 0 , H 1 ()1 G1) = 0, H 1 ()2 G2).
B ' ! ! G1 ! !+%! % ),
H 1 () G) !' \$ , ! H 1 ()1 G1) H 1 ()2 G2) .
2.7. " G = G1 G2, Hom(G1 G2) = Hom(G2 1G1) = 0,
2 2 ) 6 Aut G. ) H 1 ()1 G1) = H 1 ()2 G2) = 0, H 1 () G) = H ()2 J) =
Hom()
J).
=
2
. J )1 ) = )1 )2, !! \$\$
0 ! H 1 ()2 G1 ) ! H 1 () G) ! H 1 ()1 G) ! H 2 ()2 G1 ) ! H 2 () G):
300
. . 0" \$ H 1 ()1 G) = 0, \$
!
()1 G1) = 0 / !!" 2.3 H 1 ()1 G) = 0. F,
() G) = H 1 ()2 G1 ). S, G1 = J G2 . & H 1 ()2 G2) = 0.
& !! 2.3 G = J G2 , ! ! H 1 () G) = H 1 ()2 J) =
= Hom()2 J).
. &
\$ G = Z(pk) R, R | % 1 2R = R, pR = R, \$ %' .
!!" !+%! %\$!/! ) = h'i Aut R,
' | !+%! Z(pk), /! ' p 6= 2 "+'+: : :+'2 ;1 6= 0, o(') = 2l , p = 2. J\$ !" 2.7 H 1 () G) = H 1 (Aut
QR J) = Hom(Aut R J) = Hom(Aut R Z(p)) = Z(p),
Aut R Z
(2)
Z
,
:
=
f
p:
p
R
=
R
g
.
J!
%!,
=
p2
Z(p) " " !" " !#"
\$" ".
%-\$ %
!" !# !
\$ ! !, !' #%' %
\$ % 3] +!
\$ H 1 () G) !# \$ G = G1 G2, ! ! G1 ! !+%!, !+%! ) 6 Aut G:
G1 | , 0! )1 = Aut G1, .
p-
, )1 | %\$ / !+%! ( ,
G1 !' \$ " 2-
")<
G2 | !# \$ ! !+%!!
2 2 )2 .
H1
H1
l
"
1] . . // . !. . . "#. | 1983. | ( 3. | ). 3{11.
2] . . , - - #. // . !. . . "#. | 1986. | ( 2. | ). 3{12.
3] 0! 1. 2. , # 3 - ! . | 4. 256 12.03.97, ( 748-297.
4] ;. ). ; . | ".: 5, 1987.
5] " ). >. | ".: ", 1966.
6] 5. ?# ##. @. ", A, .. | ".:
5, 1966.
7] ;B- ). C. , - #.- - // > . | 6: - 6. -#, 1976. | ). 3{10.
8] Mills W. H. The automorphisms of the holomorph of a Dnite Abelian group // Trans.
Amer. Math. Soc. | 1957. | Vol. 85, no. 1. | P. 1{34.
9] 5. ?# ##. @, X. > . |
".: 5, 1987.
& ' ' 1997 .
{
. . -
,
. . . . . 517.43
: {, -
".
# "\$
{
f(iR);1M 2 ; 'q(x)M ; q00 (x)]gy = ;My
y(1) = y0 (1) = 0
+ M = d2 =dx2 ; 2 , q(x) | " " ", R | "
-.
, | // "
. 01 2
/" " "2"3+ /1"
" 2
/14 ". 5. " " \$
/"",
2"
/12"
" 2
/14 \$6". M 2.
Abstract
M. I. Neiman-zade, A. A. Shkalikov, On the computing of the eigenvalues
of the Orr{Sommerfeld problem, Fundamentalnaya i prikladnaya matematika,
vol. 8 (2002), no. 1, pp. 301{305.
The paper deals with the Orr{Sommerfeld problem
f(iR);1M 2 ; 'q(x)M ; q00 (x)]gy = ;My
y(1) = y0 (1) = 0
where M = d2 =dx2 ; 2 , q(x) is the velocity pro9le, R and are Reynolds and wave
numbers, respectively. We approve the Galerkin method to compute the eigenvalues
of this problem provided that the basis for the method consists of the eigenfunctions
of the operator M 2.
1 (D2 ; 2 )2 ; q(x)(D2 ; 2 ) ; q (x)] y = ;(D2 ; 2 )y
00
iR
y(1) = y (1) = 0
(1)
(2)
{ !. D = dxd , | , R | %&, q(x) | !
(
jxj 6 1. ) ) *& (+ ),
0
, 2002, 8, : 1, . 301{305.
c 2002 !",
#\$
%& '
302
. . -
, . . ,
) ( &
!
.&
%& 1], ) %
, /
0
2], ) 1. 2! 3].
3 *& | (
,
), &
(1), (2) +& ,. ,
)
, , , 4
, , ),
56
7 8, )6
%&.
9 8
, 6
,
), & { !, , 8 :;
. 6 , <
,) )
.
9 , , ; )
), & , ;
)
(
& =
), . >,& ,) )
8,, ;, (+ *.
; , L0 , L1 , M , (
L0 y = (D2 ; 2)2 y
L1 y = ;iRq(x)(D2 ; 2 ) ; q(x)]y
My = ;(D2 ; 2)y
)
D(L0 ) = D(L1 ) = D(M ) = fy j y 2 W24;1 1] y(1) = y (1) = 0g
W2k ;1 1], k > 0, | ?).
)
; 8 ), !8<
& L0 :
L0yk = k yk k = 1 2 : : :
, + ,
L2 (;1 1): kyk k = 1.
y = L0 1 8 (1), (2) 8+=8 8:
(I + S ; K ) = 0 S = L1 L0 1 K = ML0 1:
0
;
;
;
, 8, L0 1=2 , n 1=2 n 2 n ! 1. 2 , L1L0 1=2 ML0 1=2 ,, ,
S K ,, ) , ( 1&{0
S1=2+" +) " > 0. , S K | , ,, 8
; ),& (. 4, . 4])
C() = det k((I + S ; K )yk ym )kkm=1 :
(, 8
C() 8 ), { !. . 8 & ) 8=
& (. 4, . 4]).
;
;
;
;
1
1. Cn() = det k((I + S ; K )yk ym )kNkm=1
;
{
303
(. . Cn | n). jCN () ; C()j ! 0 N ! 1
G C .
6
& :8
<, 8 8+=
& 8.
2. G Cn ()
C().
D ; )
:;
6 8. )
, )=
:;
,
), & (. 5, . 4, x 18] 6 8 8 , , ),& & ),&
, 8+=& & 8 I + S ; K . 9, ( )
I +S &
(I ;(I +S ) 1 K )y = 0
(I ; K (I + S ) 1 )y = 0 ()= K )
, K 1 ). (I + S ) 1 8) & , S (
%& )6
R )=
; 8=, 6. >; ,6 )
:;
8
, )8 )=
&.
>
, ,
8& CN (). G
yk (x) = yk (x) = c1k (cos k x1 ; ch k x1 ) + c2k (sin k x1 ; sh k x1)
x1 = x+1
2 . H
k 8
, ch k cos k ; 1 ( 6 8 8
=8 (
+=
)(
&). 1k c2k + , 8
& 1 8
.
., , 8
,, (Syk ym ) = (L1 L0 1yk ym ) = k 1 (L1 yk ym )
(Kyk ym ) = (ML0 1yk ym ) = k 1(Myk ym )
* , ((=
L1 ) !8<
& yk (x)
,
+ , 8 ,& . >; 8& C() , 8+=
): )= <8
(Kyk ym ), ), kN1 8+= < <
, QR-(
&.
1 8 1, 2 )(, 8, ,
), & (1), (2) 8 !8<
& q(x) = x (
98*) q(x) = x2
(
>8&) . ,
= 1. H
N , 60. %& , ,
R = 7000 R = 15000. 8 & )8
< ), & ,, ;
;
;
;
;
;
;
;
;
304
. . -
, . . ;, ,+, (, <
,. 9 )
| .
1
0
;
a
1a
q
1
0
;
a
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
1a
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
-"
. 1. q(x) = x, R = 2000 (
/), R = 4000 (
/)
0
1b
b
q
q
q
q
q
q
qq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
b
0
q
q
q
q
q
q
qq q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
1b
q
q
q
q
q
-"
. 2. q(x) = x2 , R = 2000 (
/), R = 4000 (
/)
%) , ( %KKL M 01-01-00958 M 00-15-96100.
{
305
1] Draizin R. G. and Reid W. H. Hydrodynamic Stability. | Cambridge, 1981.
2] H
nningson D. S., Reddy S. C. and Schmidt P. J. Pseudospectra of the Orr{Sommerfeld operator // SIAM J. Appl. Math. | 1993. | Vol. 53, no. 1. | P. 15{47.
3] Trefethen A. E., Trefethen L. N. and Schmid P. J. Spectra and pseudospectra for pipe
Poiseuille ow // Comp. Meth. Appl. Mech. Engr. | 1999. | P. 413{420.
4] . ., . . !"#\$ " %\$& '\$( )
%" " \$'*%"+ ),%
,%". | .: .
/0
, 1965.
5] ,,'*,0\$ . 2., !
\$00 . ., 0 3. 3., 4/%\$50\$ 6. 7., 8%50 7. 6. 3\$'\$9 :\$ )
%( /
"\$. | .: .
/0
, 1969.
( ) * 2001 .
A^-
. . . 517.51
: A-
, , -
.
!"# \$.
1. % f | ' R, f (x) ! 0
x ! 1 ' 2 L(R) | ' '. +!
Z
Z
(A^ ) f^(x)'-^(x) dx = (L) f (x)-'(x) dx:
R
R
P
+1
2. % f (x) = n=;1 k eikx , ! k 2 C , fk g | !+1
, k ! 0 (k ! 1), g(x) =
j eijx ,
j=;1
+1
!
jj j < 1. +!
j=;1
2
+1
(A) f (x)-g(x) dx =
m -m
m
=
;1
0
P
P
Z
X
Z2
+1
0
m=;1
(A) f (x)g(x) dx =
X
m ;m :
Abstract
Anter Ali Alsayad, A^ -integration of Fourier transformations, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 307{312.
The following theorems are proved.
Theorem 1. Let f be a function of bounded variation on R, f (x)
(x ! 1), and ' 2 L(R) be a bounded function. Then
Z
Z
(A^ ) f^(x)'-^(x) dx = (L) f (x)-'(x) dx:
R
P
R
!
0
+1
Theorem 2. Let f (x) =
k eikx , where k 2 C , fk g is a sequence
n=;1
+1
with bounded variation, k ! 0 (k ! 1), and let g(x) =
j eijx , where
j=;1
P
, 2002, \$ 8, 3 1, . 307{312.
c 2002 !,
"#
\$% &
308
P
+1
jj j < 1. Then
j=;1
Z2
+1
0
m=;1
(A) f (x)-g(x) dx =
and
X
Z2
+1
0
m=;1
(A) f (x)g(x) dx =
X
m -m
m ;m :
!, , #. %. & '1] , A-
. * + A^ -
+,
- ,
, + .
.
+
A-
, / , 0. 1
/ '2]
3. 4. 5 '3].
1. -
f ! A-
6 E , :
1) %, E fx 2 E : jf (x)j > ng = o(1=n),
2) + %,
R
R
Z
I = nlim
(L) 'f (x)]n dx
!1
(
E
'f (x)]n = f (x) +
jf (x)j 6 n
0
+
jf (x)j > n:
;
R I A-
f + 6 E , I =
= (A) f (x) dx.
E
R
R
2. -
f ! A^-
, :
1) E = + 1) +
1, 6 + 2) %, , L^ ,
1:
Z
ZB
(L^ )
f (x) dx = B!lim+1(L) f (x) dx<
(A^ )
f (x) dx = nlim
(L^ )
!1
ZR
R
A!;1
ZA
'f (x)]n dx:
R
309
A^ - (. . ). fakg1k=1 | -
. 1
X
a
0
f (x) = 2 + ak cos kx
k=1
f (x) =
1
X
k=1
ak sin kx
A- '; ] -A .
1. f | !
, f (x) ! 0
x ! 1 ' 2 L( ) | !
. Z
Z
^
^
@
(A ) f (x)'^(x) dx = (L) f (x)'@(x) dx:
R
R
R
R
! +.
1. a) f '(x) ! 0 (x ! 1), A
'4].
,) .6 f '. . , +
'k k ]nk=1 +
!
n
X
j(f ')(
k ) ; (f ')(k )j 6
k=1
Z1 X
n
6
jf (
k ; t) ; f (k ; t)j j'(t)j dt 6 Var
Rf (t) k'(t)kL:
;1 k=1
2. .6, / + /
:
f '(x) = f^ '^(x):
A
ZM
;M
f
M
R
;M
'(x)e;2iyx dx =
ZM Z1
;M ;1
f (x ; t)'(t)e;2iyx dx dt =
=
+ -,
.
ZM
;M
R
f '(x)e;2iyx dx, ';M M ] | . 1
Z1
;1
'(t)e;2iyt dt
ZM
;M
;2iy(x;t) M
f (x ; t)e;2iy(x;t) d(x ; t) = f (x ; t) e
;2iy ;M +
f (x ; t)e;2iy(x;t) dx
ZM e;2iy(x;t)
;M
2iy
df (x ; t):
310
# ( , t, M ) +
M ! 1.
* + 6
, x, t, M .
# %, + +, +
f '(x) = Mlim
!1
=
R
ZM Z1
f (x ; t)'(t)e;2iyx dt dx =
;M ;1
1
Z
'(t)e;2iyt dt
;1
lim
ZM
M !1
;M
f (x ; t)e;2iy(x;t) dx = '^(y) f^(y):
3. * '5] , f (x) | f (x) ! 0 +
x ! 1, ! +,
- f^() =
R1
= (L^ ) f (t)e;2it dt +
6= 0 f (x) ;1
+ +,
- +
+
A^ -
: f (x) =
R1
= (A^ ) f^()e2ix d , f (x) = (f (x + 0) + f (x ; 0))=2,
;1
. . , , . #
R
Z
(f ')(x) = (A^ ) f '(y)e2iyx dy:
4. # + x = 0, +
(f
R
Z
')(0) = (A^ )
f '(y) dy
Z
= (A^ )
f^ '^(y) dy =
R
Z1
;1
f (t)'(;t) dt:
#6
(t) = '@(;t). 1 ! +,
- ^(y) = '@^(y). E
, Z1
;1
f (t)(;t) dt = (A^ )
Z1
;1
f^ ^ dy:
R
R
F
, (A^ ) f^ '@^ dt = f '@ dt. 1 .
R
R
3 1 .
C
2. f (x) = n=P;1 keikx, k 2 , fkg | +1
, k ! 0 (k ! 1), g(x) = P j eijx ,
+1
j =;1
P j
j j < 1. j =;1
+1
A^ - Z2
(A) f (x)@g (x) dx =
0
Z2
(A) f (x)g(x) dx =
0
+X
1
m=;1
+1
X
m=;1
311
m @m
m ;m :
. 1. .6 +1
P
| +
: k eikx +P
1
k
=
;1
j eijx,
j =;1
* ,
+X
1
n=;1
einx
+1
X
k=;1
k n;k =
+1
X
n=;1
einx
+X
1
k=;1
n;k
k :
X
+1 X
+1
+1
+1
+1
X
X
X
n;k k ;
n+1;k k 6
j
k j
jn;k;n+1;k j < +1:
n=;1 k=;1
k=;1
k=;1
k=;1
P
# fkg | + j
k j
+P
1
,, +
, n;k k ! 0 +
n ! 1.
k=;1
+P
1
+P
1
2. # f (x) =
eikx g(x) =
eijx. 1
f (x)g(x) =
k=;1
+1 X
+1
X
k
n=;1 k=;1
j =;1
+1
X
n;k
k einx =
j
k=;1
k eikx
# #. %. & '1] '6, . 659]
Z
+1
X
0
k=;1
(A) f (x) g(x) dx =
1 g@(x) =
Z2
n=;1
n;k ei(n;k)x:
;k k :
+1
X
@;k eikx
k=;1
(A) f (x) g@(x) dx =
0
+1
X
+1
X
k=;1
k @k :
1 .
* 6 , , +
1. #. %/ + ,.
312
1] . . A-
// . . | 1954. | #. 35 (77), ) 3. | *. 469{490.
2] Titchmarsh E. C. On conjugate functions // Proc. London Math. Soc. | 1929. |
Vol. 29. | P. 49{80.
3] 0
1. 2. 3
4 5 6. | .-.: 32#8,
1936.
4] 8 9. 1., *
6 9. 1., *
:. ;. 6 <. #. 2. |
.: 8<-
, 1987.
5] 1 1. A^-
5<
= > // >.
5. . | 1997. | #. 3, 45. 2. | *. 351{357.
6] : 2. 0. # 4. | .: ?8>, 1961.
' ( ( 1997 .
. DRDO
, 512.48
: , .
! "# (BIBD | balanced incomplete block design) (STS |
Steiner triple system). () *
STS +
)
*!! )
!. , - STS | )
P - *+
. .
Abstract
S. Chakrabarti, New algebraic structure of Steiner triple systems, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 313{318.
Steiner triple system (STS) is a balanced incomplete block design (BIBD). The
well-known algebraic structures of STS are Steiner quasigroup and Steiner loop.
A new algebraic structure of STS called Steiner P -algebra has been developed and
some of its properties have been described here.
1. (
) A. A f1 : : : wg.
1.1. "#, , % -
. &
, A. &
"# , .
1.2. '" ( :
n | "# A+
r | ,
, ,
, r
+
, 2002, 8, 2 1, . 313{318.
c 2002 !",
#
\$%
&
314
. w | ,
A+
k | , ( "
, A.
1.3. &
"# , nr = wk.
0 x y | , "#.
1.4. 1
, ( x y x y xy . 2 "# BIBD | "#, xy = x y 2 A. 1
,
"#.
3# "#, (
"#. BIBD r = 3, % = 1.
1.5. , STS, BIBD c r = 3, = 1.
4 , STS A | ,
SA % , A, % ,
A SA . 5 A SA , w = jAj 1 (mod 6) w = jAj 3 (mod 6). 7
, jSA j = jAj(jA6 j;1) .
8
9 . 0 A STS % , SA . 0 A
, x y 2 A
(
xy = z fx y z g 2 SA +
(1)
x x = y:
1.6 (A, p. 363]).
A (1) x(xy) = y.
, A STS.
8
, ,# , :. ;. <#
P (x y z) = (x=(y n y))(y=z):
1.6 # "
P(x y z) = y(xz):
(2)
( STS #
# P (x y z). , c " >W]. @ % , 315
, ( , . 2
,#-
>BS].
2. 0 A | "
, SA | A STS.
2.1. 2 # P A ( :
a) fx1 x2 x3g 2 SA , P(x1 x2 x3) = minfx1 x2 x3g+
b) fx1 x2 x3g 2= SA x1 6= x3, P (x1 x2 x3) = a, fx1 x3 a1g fx2 a1 ag 2 SA
, a1 2 A+
c) x1 = x3 6= x2, P (x1 x2 x3) = a, fx2 x1 ag 2 SA +
d) x1 = x3 = x2 , P(x1 x2 x3) = x1.
0 (A SA ) # P P - hA P(SA )i.
2.2. xy | A (1) P (x y z) 2.1. " (
P(x y z) = minfx y z g fx yg 2 SA y(xz)
fx y z g 2
= SA :
!" #. C a) b) 2.1 . c) 1.6 P (x1 x2 x1) = x2x1 = x2 x21. d) P(x1 x1 x1) = x1 = x1x21.
@ , % # P(x y z) <#
(2).
2.3. # P - \$ hA P(SA)i % P(x x y) = P (y x x) = y.
!" #. ', fx x yg fx y xg fy x xg SA . 0
,
2.2 1.6 P(x x y) = x(xy) = y P(y x x) = x(yx) = x(xy) = y:
; 2.3 , P - 9 ,#-
,#-
. 2 , P (x y x) = yx2 = yx = xy. @ ,
# P - 9 hA P(SA )i # 9 hA xyi.
316
. \$" # 2.4.
B | A (1). " B P(x y z) 2.1.
2.5.
& P - \$ hA P(SA )i F |
' ' fx1 x2 x3g A, P(x1 x2 x3) =
minfx1 x2 x3g, F = SA .
=
!" #. 2.1, a), , SA F. ', , fx1 x2 x3g 2 F P (x1 x2 x3) ,
x1 x2 x3. 0
,
F ,
, ( ,
x 6= y 2 A. @
P(x x y) = P (x y x) = P(y x x) = minfx yg:
(3)
; 2.3 (3) , y < x. 8
2.1
# P, c), , P (x y x) = a, fy x ag 2 SA , ,
a = y. E
, 9 ,
y x a .
;, F , . 0 F n SA ,
x1 x2 x3. @
P (x1 x2 x3) = minfx1 x2 x3g.
C , b) 2.1 , P (x1 x2 x3) = a, fx1 x3 a1g fx2 a1 ag 2 SA ,
a1 a 2 A. 2 a 6= x2. 0 a = x1 < x2. @
fx1 x3 a1g fx2 a1 x1g2 SA ,
SA , x2 6= x3. C
, a = x3
fx1 x3 a1g fx2 a1 x3g 2 SA , SA , x1 6= x3 . @ , F SA , ,
F = SA .
2.6. P - \$ hA P(SA)i ( , ) ' . " A )' U V , U hA P (SA )i. * , u 2 U , v 2 V , 1) u < v+
2) >u] U , >v] V +
3) j>u]j = j>v]j ; 1+
4) SA '(' ) :
a) fx c dg, c 2 V , d 2 U , ) ( x c (mod )+
b) fx y z g, x y z 2 U , x y z 2 V , ' ' ( ' -', -.
!" #. 0 | ,# A a b 2 A. ' ab : >a] ! >b] ab(x) = P(x a b). @ x a
(mod ), P (x a b) P (a a b) = b (mod ) 2.3. 0
,
ab (x) 2 >b]. 7
, ba (ab (x)) = P (P (x a b) b a) = P (t b a), t = P(x a b) 2 A.
317
0 (a b) 2= . 0
, ( , xab 2 >a],
fxab a bg 2 SA . @
P (xab a b) = minfxab a bg 2 >b]. 0
,
P(xab a b) = b < a. 5 x 2 >a] n xab, 2.1, b), t = P (x a b), % , t1 2 A, fx b t1g fa t1 tg 2 SA :
(4)
0
, P (t b a) = t0 2 >a]. 5 ft b ag 2 SA , b = t1 fx b bg 2 SA , . ;, ft b ag 2= SA , ,
(
, t01 2 A, ft a t01g fb t01 t0g 2 SA :
(5)
0
0
; (4), (5) , t1 = t1 fx b t1g fb t1 t g 2 SA . 0
STS
, x = t01, . . ba (ab (x)) = x x 2 >a] n xab. 7
, ba (ab (xab)) = ba (b) = P(b b a) = a. @ , (
, xab , b < a j>b]j = j>a]j ; 1:
(6)
', (6) , a b ,
-
. '" , a U ,
b 2 A, (6). 0
V = A n U. @
SA ,
, 4a), 4b).
0 x y z 2 U. G
, , , a 2 A, a > x y z. @
j>x]j = j>y]j = j>z]j:
(7)
0
, z < x >z] 6= >x]. 5 y 2 >x] fx y z g 2 SA , j>z]j = j>x]j ; 1, (7). 2 4b) x y z 2 U. :
x y z 2 V 4b).
0
, U . 0 x y z 2 U. 0
2.3 , fx y z g 2= SA x 6= z. 0
2.1, b), % , a1 2 A, fx z a1g fy a1 ag 2 SA , %
P(x y z) = a. 0
, a 2 V . 0
4b) , a1 2 V . 8
fx z a1g 2= SA 4b). @ ,
a 2 U, U .
\$" # 2.7. b < a | ( A, | (
hA P(SA )i U V | ) . " >b] = ab (>a]).
* , a < c, >c] = ac (>a]) , a c 2 U , a c 2 V . & a 2 U , 2 V , >c] = ac (>a]) xac, fxac a cg 2 SA .
\$" # 2.8. m | -
( A 0 | 0
( hA P (SA)i. & >m] >m] , 0 .
!" #. 5 b 2 A, >b] = mb(>m]) mb(>m]0) = >b]0.
2.9. 0 | 0 0 ( hA P(SA)i, ) )
. U V U V | A, )
318
. 0 .
& U = U 0 , = 0 . , , U U 0 (. 2.6,
1)). " U 0 = A, . . 0 | - ( A, )
(' .
!" #. 0 a 2 A >a]
= >a]0. 5 a 2 V \ V 0, = 0
0
0
0
2.7. , U = U . 2
, U = U , V = V = 0 2.7 , a U V .
0 U U 0 a 2 U 0 \ V . @
>c] = >c]0 c < a. @ >a] = >a]0 c 2 U 0 c < a, , , j>c]0j = j>a]0j. 2 , c < a (8)
j>c]j = j>c]0j = j>a]0j = j>a]j:
@ a 2 V , c 2 U % c < a, (8).
0
, a 2 U U 0 . 5 c 2 V \ U 0, >c] = ac (>a]) xac = ac (>a]0) xac = >c]0 xac
2.7, a c 2 U 0. C
, (xac c) 2= 0 . ,
y 6= xac , % (y xac) 2 0 . @
ca (xac ) = P(xac c a) 2 >a] = >a]0 .
7
, P (xac c a) P (y c a) (mod )0 , . . P(y c a) 2 >a]0. 2
, ac (P(y c a)) 2 ac (>a]0) = >c]0, a c 2 U 0 . @
, y 2 >c]0. 7
, (y xac ) 2 0 , ,
(xac c) 2 0 . 8
,
, (xac c) 2= 0 . ;, , , xac
, ,# 0 .
:
P. N. Sundarm, SAG , R. K. Khanna, E.SAG %.
C "
. :. :
, .
A] . . // / .
. . . . 2. | ": \$%, 1991. | ). VI, . 295{367.
BS] Burris S., Sankappanavar H. P. A course in universal algebra. | Springer-Verlag,
1978.
W] Wallis W. D. Combinatorial designs. Pure and Appllied Mathematics. Vol. 118. |
Marcel Dekker Inc., 1988.
' ( (
1998 .
```
Документ
Категория
Без категории
Просмотров
6
Размер файла
5 474 Кб
Теги
2002, фундаментальной, pdf, прикладное, математика
1/--страниц
Пожаловаться на содержимое документа