. .
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 : j j = 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)): j j < 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). 9 4 , 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 9 0 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 2 b (15) ~ 1 ; b22 ; b11 + q 1 ; b22 ; b11 ~ 21b12 L kX22 ; B22 k 6 ~ 2b12b21 6 2 b (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 ; 4 b12 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 + r b21 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 ) = f 1 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 . 2 3 ) 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
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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.
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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 , ! "# $
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1.1. 5 G Hol G def = hG Aut G +i,
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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
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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
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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
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% Hol G Hol G ( | + ), G | & 2, H def = 1 G . % 1.6
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: = "0 Hom("BA) .
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. Hol G
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' , ~ = | + 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
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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:
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' = ' 1 ' 2 Hom(B A), # + jA = " ( , (0 ) (a ") ), f( ) = 0 ( , ). <
, 2.3 , Aut Hom(B A) = (Aut B) Aut H = h b 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 . % + , + .
;
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" 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
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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 . ;
;
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24
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< . < + , 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), #
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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
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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 + j j 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) 6 9 $-: 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 ). 6 9 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) G 5 : @- !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 ) :
...
6 9 $-:
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. G 9 (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 | 5 5 @ Ri | & 5@ , Vn | & & @ w + , 1 + . A & + , ( Vn )w = 0 (1.1) Ji = Ri: (1.2)
5, 0
. 9 5, 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 9 5 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
9 5 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)
9 5 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 | 5 5, 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; | 0 5 & ' . 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
5 5- 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
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] | (! !!
(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 j j 6 q 1 : (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)),
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 logp e( 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])j jy dy + O(H( N k q])):
(11)
90
. . . , . .
! ) (, !( ) (! ! (9) + ! 6 7( ( (. 8, Ch. 2]) p
ZN
(* ) (
0
e( y2 ) dy = M( ) + O(1 + N j j)
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
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 QQ=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
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
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 ;
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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. (., D 75 . 6. ) & 5 5 , ,+ & + ( 7 ) // 1 5, 5 IK, & ,< +L. NU , 26{30 +8+ 1981 . | W &: -& 'F '5 4, 1989. 12] D D., X&= 7 . $ + + , . | .: , 1972. 13] 6*& /. Y. D & . | .: F , 1977. & ' ' 1997 .
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519.14
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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.
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" % %, " 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
. .
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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
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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
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1. CG] 1982 ( G] ) !" # f Ft x] $ # !# $$ F, # $ & # (degx f)(degt f) q s. , 2002, !$ 8, ? 1, . 129{139. c 2002 ! " # $%&, ' () *
130
. .
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logDf = logq (r f ), lc f | 1$ /!!" # , cont f = # = 234 /!!" f, a], a b c],... | ( ) / $ , D, & D, a : : : : 5(a)] | , ! $ & 5, n::m] | " # n m &# , n 6 m # &, n > m | ,&. 6 , # # !" Ft]x]. 3, G,Le] & ,$ , LLL !" Zx]7 ,+ # , # " Z F t] . 8 , | G] | ( ) 2. 4 $1 $ # , 2, $ , 9 . : $ , ! Mi] (1989), +: !" Zx] Ft x]. ;, Le, 2] # G] : b 1: L (. 9 ). 9 G] + b, $ 1 : /!!" uij F | 1 h0-sv 9 . < Le, 2] b : Le, 1]. G] ,$ " , Le] ( 2.18) " O(r6 f 2 + (r3 + f 3) q s) (CB-Le) $$ F (! 1 ,#). 21$ " & , , $ , # , , , , ,,& . 4 ,#$ , !", 11$
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&$ , 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
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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
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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
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fj
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2
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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
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0
;
0
0
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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
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j ;
2
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135
{
*/ u3 < U. J u #:$ ," c, #, u4 . j
j
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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
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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 )) $$ F . < : < O(r3 f p 2 + (l1 p k)2r), : (LPK) l1 p k (PB) p . !
!
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4.2. &' % ( h0-lt
; , M , B ( 3.2) # #$ O((deg h0)r2 (logq r)2 f 2 c(F)): ! . 8 1 ml (m + 1) " M1 ( 3.2) B1 / , , # # T1 " M1 #$ " E ml ml , j
j
0
0
0
138
. .
ml = m l1 + 1. 9 # T , ,& l1 ," " B1 . * / A ,: pk . 8 1: L + / . * , $ " A=(pk ). ? " M1 1 O((m l1 )l1 ) $$ +, , 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 $$ # uij F , X ui + 1 = O(r U) = O(l1 p k): j
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; $ /$ ? * m l1 + 1 ," " B $ uij &. W$ ," Bc(i) : , $, &+$ u Bc(i) < U | () ;
j
j
139
{
l1 , $ $ uij . ? U, Bij < p k, $ , ( ) , 1 p k $, $ , 1, # l1 p k, 1 $ $$ 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
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(
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 | 3 4 (), % , ( ', ( ( 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)
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) !* 1997 ".
. I . .
517.977
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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.
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) ( (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
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k X A0 (t) + Ai (t) (t ; ti ) x i=1
(7)
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. I
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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
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? 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 , - @
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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] . . 9 4 " % 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.
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. II . .
517.977
: , , , .
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! " 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 &, F G . 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 U ab]
* * , &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. U ab] |
. Pk > 4 U kU()k = kU k . . *, ab]
i=1
i
U ab] . ( 7
) * * a b] UF ab] . 2.2.
> & * #& # # , 4 # 3]. X(t s) J7 x_ = A(t)x. > 4 ,
&$ U ab] Mn ,
k X H ab] (U()) = X(a ti )B(ti )Ui C (ti )X(ti a)
Pk
i=1
#
U() 2 U ab] U(t) = Ui (t ; ti). i=1 2.1. + (A B C) # a b],
H ab] (U ab] ) = 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 * H ab] (U ab] ) = Mn . >*4 *2 Mn . . 4 * Uj (), fGj gnj =1 H ab] (Uj ()) = Gj . 2
& Uj () 7*
. II
175
# . + *, $ ' * fVi ()gki=1 , Uj * ' Vi (). ?# H ab] (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 U ab] , kU()k 6 lkGk Z(U) = G. 2.2. , 2.1 2.2 ( . . 2 J7 Z(U) = X(b a)H ab] (U()). 2 * (A B C) #
2.1. 2 2.1 2 n $ * fVi(t) = Ui (t ; ti )gi=1, fH ab] (Vi ())g | Mn . > * & G 2 Mn X(a b)G fH ab] (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. ?# H ab] (U ab] ) = Mn .
2.2 #
3] * . E , ' # . ? Z(U) * , , , * , $ l * * &$
'
.
176
. .
2.3. + (A B C) # a b], Z(U ) = Mn . > U | ' a b] , * | U ab] . > * 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. $% Zb V1 =
i2I a
Ai (t) dt V2 =
i2I t2 ab]
fAi (t)g
# # & & . -# V1 V2.
>
V2 | ' # *# , V1 | # . + *
&# # #, # * V2. 2.3. # % (A B C) ( . # -
& .
. 2 * (A B C) # . >*4 * Gi Mn 4 &$ 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 t2 ab]
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 U ab] ,
. 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 * U ab] , &$ U(t) = k Eij (t ; tk ), #
k 2 R, Uij . k
2.3. 0 Uij %
-
U ab].
2.4. $
U ab]
# P E '#
, H (U ) = H (U ). ?-
Uij .
ab] ab]
ij
ab] ij
, H ab] (U ab] )
H ab] (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> > > = : > > 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(H ab](Uij )) V . . , Vect(H ab] (Uij )) V . > * U() 2 Uij , Vect(H ab] (U())) 2 V . * ' * ' U(). 2 * U(t) = Eij (t ; t1 ). 2 ' 0 f (t ) 1 1 1 B Vect(H ab] (U())) = @ ... C A = (f1(t1)v1 + : : : + fk (t1)vk ): 2 fn (t1 )
. II
179
?# Vect(H ab] (U())) 2 V . , Dim H ab] (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(H ab] (Ul )) , ' Dim H ab] (Uij ) = k.
2.5. $% Q1 Q2 2 Mn | , # H ab](U ab] ) = Mn # % # , # Q1H ab] (U ab])Q2 = Mn. ,
Q1H ab] (U ab])Q2 & % , & H ab](U ab]).
> ,
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 H ab] (U ab] ) * & 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
. .
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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)
HL ab] . . , HL ab] (U) = L;1 (b)H ab] (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 ! U ab] , 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 .
: , ,
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$ # $ $ $ % . &! "# ' ( . (% ! .
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(1j 1 j : : : N j N j8 1j 1j : : : N j N j) > 3g Pfmax(1 j 1j : : : N j N j8 1j 1j : : : N j N j) > 3g + + Pfmax (t) > u j max(1 j 1j : : : N j N j8 1j 1j : : : N j N j) 6 3g Pfmax(1 j 1j : : : N j N j8 1j 1j : : : N j N j) 6 3g:
F
191
Pfmax(1 j 1 j : : : N j N j8 1j 1j : : : N j N j) > 3g 6 N N X X 6 Pfmax(k j k j) > 3g = Pfmax j k j > 3k;1 g 6 k=1 k=1 N X p 6 Pfmax j k 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 = E i 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- ( " ( ) % %, ( , % , ! ( + . % ( + % 2 3 ( . 4 ,
+ ( & F- ", %
! ( , !
! 2 3 ! ", (,!
% %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 . : - % & , 2 3 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 2 3 ! ",
> & . ?(, , +, . 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 , 2 3 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 )
! (=( ( ) 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) % , .
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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- %
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% %," " 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 +
, % ;, %
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: F (X ! Y ) =
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210
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211
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212
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213
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" 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
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i=1
j =0
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i=1
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: #(k ; 1) I #k = = =
k ;1
k ;1
i=1 i ;1
i=1 k
j =0 i ;1
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214
. .
IV. 9 3 I , # I # = #. : , # = X ! Y , # I # = X (X n Y ) ! Y Y = X ! Y = #: % 3 I % %, ", & L M #1 I #2 I : : : I #k (, L M. , , #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
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216
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! 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. ; 2 3 ! " 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, - , L M #1 B : : : B #k (, L M, #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 ( O P, i 2 I,
f # O5 P. 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 | O5 P
. @ . 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 | O5 P
. @ . 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 | O P
, 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) ln det(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) ln det(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 $ "@ " " " X 5@ " 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 ane 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
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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
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y
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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 > : 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
+pb3 2 ; b2 3 = 0 ;b3 1 + a2 D 2 +pb1 3 = 0 > : b2 1 ; b1 2 + a3 D 3 = 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
D 1
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 >
( ; 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 > : a3 s3 = (b1 t2 ; b2 t1 ):
a1 a2 a3 b1 b2 b3
270
. .
F " , & (a1 t1 a2t2 a3t3 ), % & (17) "& 8 > : 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 D 1 + b3 2 ; b2 3 = 0 p ;
b3 1 + a2 D 2 + b1 3 = 0 p > :
b2 1 ; b1 2 + a3 D 3 = 0: ?5 ) ) 1 , 8 &
p ; 3
a1
D 1
b 1
b2 1
p
b3 2 a2
;
D 2
b1 2
;
b2 3
=
p
b1 3 a3
D 3
= 2 (a1 b21 + a2b22 + a3b23 ) + a1a2 a3D = (1 ; 2 )a1a2 a3D = 0:
( ) = 1. 6 ) , (16), , & 1 ! ' (b1 b2 b3) (t1 t2 t3) ) , "& & (17). D 5 ) 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 - . F 2, , 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., + % . . %"% % %0 3 !4 2%* *%) #"&%'. | 5 6 %, 1988. 5] 70& +. 1. 5#!* % * ) (% 8. | 5 6 %, 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 %&
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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
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%' (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
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l
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l
l
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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 G 1 ) ! H 1 () G) ! H 1 () G) ! H 2 ()2 G 1 ) ! H 2 () G): Q ! ! H 1 ()2 G 1 ). & 0! G, ' $ )1 = h;"i, | 0 0! .$ G, $ G 1 = Z(2). R ! +%! )2 $ " G 1 , 0! . Z(2k) ' $ ! +%! 0" : ! +%! 2 Aut Z(2k) ! 1 + 2r s, r > 1, (2 s) = 1, (1 + 2r s)g = g, g 2 G 1 . F , H 1 ()2 G 1 ) = Hom()2 G 1 ) = 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
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295
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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 G 1 ). J ! +%! )2 . , 5, . 162] H 2 ()2 G 1 ) = (G 1 ) 2 =(" + 2 ; 1 1 1 2 + ' + : : : + ' )G . !! (G ) = Z(2), (" + ' + : : : + '2 ;1)G 1 = = (" + ' + : : : + '2 ;1 )g = 2l g = 0, G 1 = hgi, ohgi = 2. O H 2 ()2 G 1 ) = 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
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! +%! , . ! ! 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
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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) = G2 2 = 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 G 1 ) ! H 1 () G) ! H 1 ()1 G) ! H 2 ()2 G 1 ) ! 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 G 1 ). S , G 1 = 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
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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
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".
# "$ { 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 !", #$ %& '
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5 6 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 < ) , & , , ;
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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%5 0 7. 6. 3$'$9 : $ ) % ( / " $ . | .: . /0 , 1969. ( ) * 2001 .
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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
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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^ - + , - ,
,
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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
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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 , f kg | +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 j n;k; n+1;k j < +1: n=;1 k=;1 k=;1 k=;1 k=;1 P # f kg | +
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
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! " # (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 .
