Фундаментальная и прикладная математика (2002, №1)

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517.983.28+517.928

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() # ! $$& 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. 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. 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

; " 1 ) k;X0k1 6 maxfkY12k kY21kg 6 6 ( 1 + 2 );1kX0 k = dist(1 2);1kX0 k, . . 6 dist(1 2);1 .

x

3. $ %, % %

8. > 5 2. ;. / %5] + A0 , (, , Y ), B 2 End F + + + 4 + )) * )* B " C(REnd Y ). 9. G A1 : D(A1) Y ! Y 4 + + iR, 2 R, A0 = A1 ; I + + 4 ( 5), d=dt ; A0 ; BA0 ( + kB k) d=dt ; A0 ; ; (P1B0 P1 +P2 B0 P2)A0 , B0 2 A(0), , d=dt ; A1 ; BA0 , d=dt ; A1 ; (P1B0 P1 + P2 B0 P2)A0 . 10. 2 !. D. 0 A | " + " End Y B 2 End Y . . + + , (A) = 1 2 , 1 | , 1 \ 2 = ? 2 | " 4 . 2 1

J , B 3 A kB kA

. G3 4 " 2 " . ,

A + "B, " > 0, A = LA (Y ), I X = P1XP1 + P2XP2 , Pi = P(i A), i = 1 2, ) ;: A ! End Y 3 " + adA ;X = X ; I X = P1XP2 + P2XP1 , X 2 A ( ) ,

, %13]). > (A I ;) + + + + A, 1 " 3 "0 > 0, A +"B, " < "0 , A+"P1B0 (")P1 + + "P2 B0 (")P2 , B0 : (0 "0) ! A = LA (Y ) | )+ )*+, " +

U(") = I+";B0 ("), 4 + ++ ) )* (U : (0 "0 ) ! End Y ). x(t) = U(")y(t) (31) "y(t) _ = A + "P1B0 (")P1 + "P2 B0 (")P2 , 1 J +

)) * Yi = ImPi, i = 1 2.

16

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" ' 2) " " # &

.

Abstract I. H. Bekker, V. N. Nedov, About determinableness of an Abelian group by its holomorph in the class of all Abelian groups, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 17{25.

For an Abelian group without elements of order 2 the following results were obtained: 1) a criterion for its determinableness by its holomorph in the class of all Abelian groups' 2) a criterion for its characteristicness in its holomorph.

: 1) ! 2) # . %

& 2. ' 1) ( )1] + , +# . - )2] , , . / , 2), % )3], g ! 2g (g 2 G), , + . : Aut G | G, CG(H) | 3 H G, t(A) | A (A | 1), H / G , H | G, Hol G | G, : A B , | + A B, Z | 3 . , 2002, 8, - 1, . 17{25. c 2002 , ! "# $

18

x

. . , . .

1. ,

1.1. 5 G Hol G def = hG Aut G +i,

3 + : (a ') + (b ) = (a + 'b ') a b 2 G ' 2 Aut G: 1.2. - , G , G , HolG = Hol G , , G =G. 8 1.1 :

1.3.

G Hol G : 1) G def = h(g ") j g 2 Gi, G |

HolG, Aut G def = h(0 ') j ' 2 Aut Gi = Aut G (" |

G). 2) CHol G (G) = G, . . !

G HolG

. 1.3 , & , + G G, Aut G Aut G.

1.4 (4]). " G |

# 2, H |

Hol G, H |

# 2 H = H1 :1, H1 :1 |

$, $

H , H1 $% G, '1 g ; g 2 H1 '1 2 :1 g 2 G. . % (a ') 2 H 2(a ') = (0 "). ; '2 = ", H Hol G, (;g ") + (a ') + (g ") 2 H g 2 H. %& ('g ; g ") 2 H, H | , '('g ; g) = 'g ; g, , '2 g = g + 2('g ; g). ; G | & 2, ' = ", , a = 0 H | & 2. 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

. 95, 5 C {9 : (O ; S) + O ! (S) + O2 v1s = O xO ; K1 0 xO2 1 1 s (N ; S) + N ! (S) + N2 v2 = N xN ; K 0 xN2 2 s (O ; S) + CO ! (S) + CO2 v3 = O xCO ; K1 0 xCO2 3 1 (CO ; S) + O ! (S) + CO2 v4s = CO xO ; K 0 xCO2 4 1 s (O ; S) + C ! (S) + CO v5 = O xC ; K 0 xCO 5 1 s (C ; S) + O ! (S) + CO v6 = C xO ; K 0 xCO : 6 95 5- 5 CO 5 : O + (S) ! (O ; S) v7s = 0 xO ; pK1 O 7 1 N + (S) ! (N ; S) v8s = 0 xN ; pK N 8 1 s 0 C + (S) ! (C ; S) v9 = xC ; pK C 9

!

63

s = 0 xCO ; 1 CO : CO + (S) ! (CO ; S) v10 pK 10

95 F {G+ &: s = 2 ; 1 pxO (0 )2 v11 2 O K 11 s = 2 ; 1 pxN (0 )2 (N ; S) + (N ; S) ! N2 + 2(S) v12 2 N K 12 s = O C ; 1 pxCO (0 )2 (O ; S) + (C ; S) ! CO + 2(S) v13 K13 s = O CO ; 1 pxCO (0 )2 : (O ; S) + (CO ; S) ! CO2 + 2(S) v14 2 K14

(O ; S) + (O ; S) ! O2 + 2(S)

95 5- 5 O2 , N2 , NO, CO2 5 : s = 0 xO ; 1 O O2 + (S) ! (O2 ; S) v15 2 pK15 2 s = 0 xN ; 1 N N2 + (S) ! (N2 ; S) v16 2 pK16 2 s = 0 xCO ; 1 CO CO2 + (S) ! (CO2 ; S) v17 2 pK17 2 s = 0 xNO ; 1 NO : NO + (S) ! (NO ; S) v18 pK 18

4 & Aj , (Aj ; S) | , (S) | . 9 5 vis @ p | , xi | 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

55- 5 : O2 + M ! 2O + M v1 = Kpp1 xO2 ; x2O N2 + M ! 2N + M v2 = Kpp2 xN2 ; x2N CO2 + M ! CO + O + M v3 = Kpp3 xCO2 ; xCO xO CO + M ! C + O + M v4 = Kpp4 xCO ; xCxO NO + M ! N + O + M v5 = Kpp5 xNO ; xNxO : 4 & M | & 5. 3 5 & 5 vis ' & vi , 5, vis , 5 5- 5 , (1.3) & 5 (1.4). H , + Kp1 = K 1K 2 Kp2 = K 1K 2 Kp3 = K K1 K 2 Kp4 = K K1 K : (2.3) 11 7 12 8 7 12 8 7 9 13 / , s ; k13 vs ; k14 vs ) RO = ;mO (p(2k1v1s +k3v3s +k4 v4s +k5v5s +k6 v6s ) ; 2k11v11 13 14 s s RN = ;2mN (pk2 v2 + k12v12) (2.4) s ; k14vs ) RCO = ;mCO (p(k3v3s + k4v4s ; k5v5s ; k6v6s ) + k13v13 14 s ): RCO2 = mCO2 (p(k3 v3s + k4v4s ) + k14v14 K vis c (2.1) (2.3) K K p 1 p 2 s 0 2 s 0 2 v1 = pK7 xO ; p xO2 v2 = pK8 xN ; p xN2 K K p 3 p 3 s 0 s 0 v3 = pK7 xO xCO ; p xCO2 v4 = pK10 xO xCO ; p xCO2 K K p 4 p 4 s 0 s 0 v5 = pK7 xO xC ; p xCO v6 = pK9 xO xC ; p xCO K K p 2 p 1 s 2 2 0 2 2 s 2 2 0 2 2 v11 = p K7 ( ) xO ; p xO2 v12 = p K8 ( ) xN ; p xN2 K K p 4 p 3 s 2 0 2 s 2 0 2 v13 =p K7 K9 ( ) xO xC ; p xCO v14 =p K7 K10( ) xO xC ; p xCO2 :

66

. .

" &, ' 5 Ri : K p 1 2 0 2 0 2 RO = ;mO p 2(k1K7 + k11K7 ) xO ; p xO2 + K p 4 0 + (k5K7 + k6K9 + k13K7 K9 ) xO xC ; p xCO + K p 3 0 + (k3K7 + k4K10 + k14K7 K10 ) xO xCO ; p xCO2 K p 2 2 0 2 0 2 RN = ;2mN p (k2K8 + k12K8 ) xN ; p xN2 K p 3 2 0 0 RCO = ;mCO p (k3K7 + k4K10 ; k14K7 K10 ) xO xCO ; p xCO2 + K p 4 0 + (k13K7 K9 ; k5K7 ; k6K9 ) xO xC ; p xCO K p 3 2 0 0 RCO2 = mCO2 p (k3 K7 + k4 K10 + k14K7 K10 ) xO xCO ; p xCO2 : (2.5) 3. 9

5. 3 5 5 C {9 , & 5 RO = ;mO p2 0 2k1K7 x2O ; Kpp1 xO2 + (k5K7 + k6K9 ) xO xC ; Kpp4 xCO + + (k3K7 + k4K10) xO xCO ; Kpp3 xCO2 RN = ;2mN p20 k2K8 x2N ; Kpp2 xN2 RCO = ;mCO p20 (k3K7 + k4K10 ) xO xCO ; Kpp3 xCO2 ; ; (k5K7 ; k6K9 ) xO xC ; Kpp4 xCO RCO2 = mCO2 p20 (k3 K7 + k4 K10) xO xCO ; Kpp3 xCO2 : (3.1) 3

5 5 F {G+ &

!

67

K p 1 RO = ;mO xO2 + p K K p 3 p 4 + k13K7K9 xO xC ; p xCO + k14K7 K10 xO xCO ; p xCO2 K p 2 2 0 2 2 2 RN = ;2mN p ( ) k12K8 xN ; p xN2 (3.2) K p 4 2 0 2 RCO = ;mCO p ( ) k13K7 K9 xO xC ; p xCO ; K p 3 ; k14K7K10 xO xCO ; p xCO2 K p 3 2 0 2 RCO2 = mCO2 p ( ) k14K7 K10 xO xCO ; p xCO2 : 4 , (T < 3000 K) 5 Kpi 1. a) &+ 5 & pi = pxi 5

5 (pi xi Ki 1), 5 5 & . , 5

5 pxO K7 pxNK8 pxiKi , i = 9 10, RO = ;mO p(2k1xO + k5xC + k3xCO ) RN = ;2mN pk2xN (3.3) RCO = ;mCO p(k3xCO ; k5xC ) RCO2 = mCO2 pk3xCO : A 5

5 C (pxCK9 pxiKi , i = 7 8 10) CO (pxCO K10 pxiKi , i = 7 8 9), & (3.4) RO = ;mO pk6xO RN = 0 RCO = mCO pk6xCO RCO2 = 0 RO = ;mO pk4xO RN = 0 RCO = ;mCO pk4xO RCO2 = mCO2 k4 xO : (3.5) 1 F {G+ & 0 5. A pxO K7 pxNK8 pxiKi , i = 9 10, RO = ;2mO k11 RN = ;2mN k12 RCO = 0 RCO2 = 0: (3.6) 3 , 5 C (pxCK9 pxiKi , i = 7 8 10) 5 CO (pxCO K10 pxi Ki , i = 7 8 9) & RO = RN = RCO = RCO2 = 0: (3.7) p2 (0 )2

2k11K72

x2O ;

68

. .

b) 5

5

(pi Ki 1). 3 0 5, C {9 ,

5 F {G+ &, 5. 3 C {9 5 : RO = ;mO p2(2k1K7 x2O + (k5 K7 + k6K9 )xO xC + (k3K7 + k4K10 )xO xCO ) RN = ;2mN p2 k2K8 x2N (3.8) RCO = ;mCO p2((k3 K7 + k4K10)xO xCO ; (k5 K7 ; k6 K9 )xO xC) RCO2 = mCO2 p2 (k3K7 + k4K10)xO xCO : 3 F {G+ & RO = ;mO p2(2k11K72 x2O + k13K7 K9 xO xC + k14K7 K10 xO xCO ) RN = ;2mN p2 k12K82x2N (3.9) RCO = ;mCO p2(k13K7 K9 xO xC ; k14K7 K10 xO xCO ) RCO2 = mCO2 p2 k14K7 K10 xO xCO : . & F ,

5 & . K &

5

5,

5 5. , 5 & 5 5. 5 & C {9 5, F {G+ & . ,

5 . M ', ' 5 1 , & & 0 , 0 &

5 .

1] . ., . ., . . . !"# $% & $ %& %& $$ // ()*+. ,-+. .%. | 1987. | 3 676.

!

69

2] Chen Y.-K., Henline W. D., Stewart D. A., Candler G. V. Navier{Stokes solution with surface catalysis for Martian atmospheric entry // Journal of Spacecraft and Rockets. | 1993. | Vol. 30, no. 1. | P. 32{42. 3] Mitcheltree R. A., Ggno;o P. A. Wake 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+ % . 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) . 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 q1 : (9) D k (* ! (!( 8 () ! 6 7( ((* (7) (! ! !( ) !*! )!*, *! + ) ! * *). B (* * ( () ! 6 7(

89

!8 )! 6, ! ( !!(. (!!( q X

a Sk q + =

!

m=1 m;2 ((kq))

T ( ) =

X

2 e amq T( ) + O(qL)

(10)

(log p)e( p2 ):

p p6 N p;2 (k) pm (q)

) =!(! 6 ! 6 ! 8, ( ! , ! #!*! k, m, q !6 ( (k 2) = (m q) = 1 m ;2 ((k q)),
p6 N pf (!kq])

B ( H(x h) =

X max max y6x (lh)=1

y : log p ; '(h) p6y

pl (h)

) 7( I!, )! !( T( ) p

T( ) = ; =;

Z N

0

pN

pf (!kq])

Z

0

p

d e( y2 ) dy + X logpe( N) = log p dy p p6y p6 N pf (!kq]) X

y + O(H(pN k q])) d e( y2 ) dy + '(k q]) dy

p + '(kNq]) + O(H( N k q])) e( N) =

= '(k1 q]) ; +O

pN

Z

0

p

p

ZN

0

d e( y2 ) dy + pNe( N) + y dy

p

H( N k q])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 ;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 > > >
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 > > > > 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 QQ kq d6Q k6H q>Q kq N1 =

X X

(kq)=d

(kq)=d

1 1 1 1 + 1 X 1X 3=2 3=2 3=2 3=2 d6Q d k6H=d k q>Q=d q Q(! { ((. 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 1. 7 !" O ! t+1 O, s +( ; 1)t 6 t +1, 9 # = 2, s = 1 t = 0, . B'" = 1, ; | " % " 0 # GQ(s s2 ; s), A

(s2 t + s (s ; 1)(t + 1) s ; 2 (s ; 1)2), A (s ; 1)(t +1), s ; 1, ;(s2 ; 2s +2) ' 1, (s ; 1)(st + 1), st . 8 " . ; | " % " % pG(s t),

k, r = s ; , ;m = ;(1 + t) % 1, t(t+1) s(s;+1)(st+) f = s((ss+1) +t+1;) , g = (s+t+1;) . & " !, ; | " % " % pG(s t), " " # F % (s + 1 ; 2)t 6 (s ; 1)(s + 1 ; )2: 7 " " ' % a 2 ; 5 (1] (a] (a]0

# . H

, I = (a] ! " " % pG;1(s ; 1 x), " ( ; 1)t = x(s ; 1). 7 g < f , 6.2 (7] ; | 5 # (i) ' 9 % , , r2 = r = r1 , m1 + m2 = m ; r, g2 = g1 = g ; 1, "

120

. .

r1, ;m1 | (a] f1 , g1 , , r2, ;m2 | (a]0 f2 , g2.

2.3. ( ; I = ;(a) ) , s = 2, t = (2 ; 1)( + 1)2 . . &# (s +2 ; 2)x = (s ; 2)(s +1 ; )2 . 7 " " ' % b 2 I I(b) " " pG;2(s ; 2 y),

" ( ; 2)x = y(s ; 2). 2 ! (s + 1 ; 2)t = (s ; 1)(s + 1 ; )2 ( ; 1) ( ; 1)t x(s ; 1), # (s + 1 ; 2)x = ( ; 1)(s + 1 ; )2. K , (s + 2 ; 2)y = ( ; 2)(s + 1 ; )2. 2 ! # ; 2, #" (s ; 2)(s +1 ; 2)y = ( ; 1)( ; 2)(s +1 ; )2 . B'", ( ; 2)(s +1 ; )2 (s + 2 ; 2)y, # (s ; 2)(s + 1 ; 2) = ( ; 1)(s + 2 ; 2), 9 # s ; 2 = u(s + 2 ; 2) " # u. & ! s + 1 ; 2 = w. 7 " = uw + 1, s = 2uw + w + 1. 5 "# % , s ; 2 = u(w + 1), " , uw = u + 1 ; w. 7 , w = 1 s = 2.

2.4. ( ; | pG2 (s t),

) , s

4, 5 7 % $ a 2 ; (a] GQ(s ; 1 x), x 9, 8 9

. . & # ' F % t = (ss;;1)3 3 , 9 # s ; 3 1, 2 4. & 9 t 27, 32 54, " , s 9, 8 9

.

3. GQ(4 ) t

+ 9 " #" " , ; | " % " GQ(4 t). & # ' $ t + 4 " 20 12, 9 # t 1, 2, 4, 6, 8, 11, 12 16.

3.1. ; 4- 3, % 5-, Ki = = Ki (3), xi = jKij. (1) x0 + x3 = 4t + 3* (2) e 2 K3 , d 2 3 ; (e], 30 = 3 feg, Li = Ki (3), yi = jLi j,

3 \ (e] + (d] \ (e], y0 = t + 2, y2 = t + 4 $ + L0 t + 1 $ + L2. . & " !"# 3 ;;3 2-# % $ 3, # # # % X X X i ; 1 xi = 20t + 1 ixi = 16t + 4 x0 + 2 xi = 6: i i i

121

+ # # , #

# !" . &# x3 6= 0, L = ; ; (d? e? ). 7 " jLj = 13t ; 4, # 3 \ (e] ' L. & ! , # 3 \ (e] = fa b cg (d] \ (e]. &# a ! % % w 2 ((d] \ (e]) ; 3. 7 " (w] " !

4t ; 3 L, j(a] \ Lj = 4t ; 1, (b] (c] " !

4t ; 2 L. C, jL;((a](b](c])j 6 t+1, (w]\L " !

" % (a]

2t ; 4 (b] (c], " , 2t 6 t + 1, t = 1, = 2, , - " d? \ e? " ! # . 7 !" fa b cg ! 4t ; 2 L, 9 # y0 = t + 2. 2

, L2 " ! t ; 2 (d] \ (e] " %

(d] \ (z ], (z ] \ (e] " z 2 fa b cg, 9 # y2 = t + 4. f 2 L0 , j(f ] \ (z ]j = t + 1 " z 2 30 , " , (f ] " !

t + 1 L2 .

3.2. , 3.1, t > 11, x3 = 0. . 2 # . 5 3.1 ! L = (d] \ (e] ; 3. 7 " jLj = t ; 2 3.1 !" L0 ! % % t ; 5 L. ; " ! (4 5)- " M, zi = xi (M), 2.1 z0 + + P ;i;2 1zi = 6. i ; N #% t = 12. 7 " L " ! 102 = 252 9 ; " ! , L0 " !

72 14 = 294 " ! . &# u, w | " L0 , (u] \ (w] " ! 9 " ! L0 L. & ! M = fd e u wO L0g. F ; , z0 + P i;2 1 zi = 6. B'"

L0 ; fu wg ! % % L0 . R, L0 " ! % 9 % " L00 , !" L00 !

" # L0. D " # (6 5)- " fL00O L ; L0g, . : (7], # t = 16 ; . &# t = 11. & " !, %"# L0 , !

' L2 ; L. 0 " L0 ! % % ' L2 ; L, # (5 5)- " , " d, e ! 9 (5 5)- " , . R, ' " L0 !

L2 ; L % % ' " L. D " fd eg L L0 " ! (6 5)- " , . C, L0 %" % % 10 , !" ! ' L. 5 # (5 5)- " ,

122

. .

9 " !

, . . C 3.1, 3.2 "# , - " " " GQ(4 t), t > 8, " ! 3-. F , ! " , " t = 8 3.2 " L0 " ! , ! ' 0 L.

3.3. -% - + ; n- 3 < n < 8. . 2 # . &# a, b | ! (a] \ (b] " ! n-# c1 c2 : : :cn. & ! 3 = fa b c1 : : : cng, xi = xi(3). 7 " N = n + 2, M = 3n, d1 = d2 = n, di = 4 " i > 2. & 2.1

X i ; 1 n2 ; 11n + 26 t + ;n2 + 7n ; 8 : x0 + x = i 2 2 2

D ! % $ 3 < n < 8, ' # n = 4, t 6 2. B" 9 # 6 3. 8 " . &# "

- " ; " ! # .

3.4. 3 | + ; xi = xi(3). (1) ( 3 | , 4-, x0 = 8t + 2. (2) ( 3 $ ( + ), x0 + x3 = 5t + 3, x4 = 0. (3) ( 3 + $, $ ( + ),

x0 + x3 = 2t + 4, x4 = 0. . & 2.1 " # 3 # , - " ; " ! # # .

3.5. ( $ c, d ; - (c] \ (d] 3- eabf , 3 = fa b c d e f g xi = xi(3) xi = 0 i > 3, x0 = 4. ; . & 2.1 x0 + P i;2 1 xi = 4. 2 #, w 2 K3 (3) K4 (3), ! 30 = fwg 3, yi = xi (30P ). ; w 2 K4(3), , # 2.1 " # 30, #, y0 + i;2 1 yi < 0. R, w 2 K3 (3). R , ' " 3 ! # : (w] " ! (1) e, b, f O (2) c, eP, dO (3) c, e, f O (4)Pc, a, dO (5) c, a, e (6) c, a, ;b. + !" 9 # yi = 20t ; 2, iyi = 28t, P 2i yi . P ; + # (1) 2i yi " 21 % , 4t ; 10 , w, 3t ; 5 , ! w. C , P ; i y ! = 7 t + 6. T ! # (3) (5). 2 i

123

(2)

18 7t ; 13P ; i , 9 # P ;+i y#

= 7 t + 5. D $, # (4) (6) # i 2 ; 2 yi = 7t + 4. 7 , y0 + P i;2 1 yi ;t + 2, -

. 8 " . & %" " # 1. &# ; | " %

" GQ(4 t), " !0% 49- # O. 7 " " A = ;;O

# (196 39 2 9).

3.6. A . . &# 3 | " P ; A, Ki = Ki (3), xi = jKi j, Ki0 = Ki \ A, x0i = jKi0j. & 3.4 x0 +P ; i;2 1 xi = 28. & # 2.1 " # 3 A, #, x00 + i;2 1 x0i = 28. R, K0 K3 A. &# w 2= 3, 30 = 3 fwg, yi = xi(30 ), yi0 = jKi(30 ) \ Aj. w2K " # 30 ; A, #, 3 , , P # ;i;1 2.1 P; i ; 1 0 0 y0 + 2 yi = y0 + 2 yi ; 2, . C, x0 = 28, P x1 ;= 180, x2 = 32, x00 = 28, x01 = 147, x02 = 16. &# i 0 w 2 K2 \ A, = 2 di, " di | P ; 3 . & i ; 1 0 # 2.1 " # 3 P;; A, #, y0 + 2 yi = 42 ; , y1 = 102 + 2 , y2 = 95 ; , y00 + i;2 1 yi0 = 41 ; , y10 = 84 + 2 , y20 = 65 ; . D , O " ! 16 K2 33 K1 . B'" O " ! " #' # L3 , 27 L2 21 L1 . & , y2 = y20 = 30.

3.7. A PQ(3 12 9). 0 . &# 3 | P ;i;1A, Ki = Ki (3), xi = jKi j, Ki = 0 0 = Ki \ A, xi = jKi j. & 3.5 x0 + 2; xi = 63. & # 2.1 " # 3 A, #, x00 + P i;2 1 x0i = 57. & 3.6 x3 = 0, , x0 = 63, x0 ; x00 = 6. &# w 2 K2 \ A, 30 = 3 fwg, yi = xi(30 ), yi0 = jKi (30) \ Aj, = P;2i di ,

" di | P; i3;10 . & # 2.1 " # 30 ; A, y0 + 2 yi = 54 ; , y1 = 126 + 2 , y2 = 60 ; , P; #, y00 + i;2 1 yi0 = 51 ; , y10 = 97 + 2 , y20 = 43 ; . D , O " ! 6 K0 , 9 K2 34 K1 . &# (w] " ! K2 \ O. 7 " (w] " ! 3 + K0 \O 10 ; 2 K1 \O. & , jL2 \Oj = 17 = = 10 ; 2 +(9 ; ). C, A " ! , 9 # ' %

13 3-, A # .

3.8. . ; . . D " " , ' % - " (a] \ (b] %. U !" , a b 2 O ( ' A ! " % % O). + "# 3.7 ! ,

124

. .

a 2 A, b 2 O (a] \ (b] " ! cd. & 3.7 A " ! 4-# 3 = fa c d eg, xi(3) = 0 " i > 2. &# Ki = Ki (3). 7 " O " ! 6 K2 , 40 K1 3 K0 . &# w 2 K0 \ A, 30 = 3 fwg, yi = xi (30), yi0 = jKi(30 ) P \ A;j. & i;1y = 50, # 2.1 " # P; 30 ; A, #, y 0 + i 2 y1 = 132, y2 = 58, y00 + i;2 1 yi0 = 44, y10 = 111, y20 = 36. &# (w] " ! K0 \ O. 7 " (w] " ! 3 + K2 \ O 10 ; 2 K1 \ O. & , jL2 \ Oj = 22 = = 10 ; 2 + (3 ; ). 8 , % 1 " .

4. # pG2(5 32)

+ 9 " " , ; | " % " % pG2 (5 32). 7 " ; |

# % (486 165 36 66), "

" # F %. & 9 # " ' % a 2 ; (a] (a]0

# (165 36 3 9) (320 99 18 36) .

4.1. ab | ;. ; 36 $ + (a] \ (b], 128 $ + (b] ; a? , (a] ; b? 192 $ + (a]0 \ (b]0. ( x 2 (a] \ (b], (x] 3 $ + (a] \ (b], 32 $ + (a] ; b?, (b] ; a? 96 $ + (a]0 \ (b]0. . & .

4.2. $ c, d ; . ; 66 $ + (c] \ (d], 99 $ + (c] ; d?, (d] ; c? 220 $ + (c]0 \ (d]0. ( y 2 (c] \ (d], (y] 9 $ + (c] \ (d], 27 $ + (c] ; d?, (d] ; c? 100 $ + (c]0 \ (d]0. . & .

4.3. -% 5- + ;

6-. . &# 5- 3 = fa b e f gg ! 6- ;. 7 " " ' x y 2 fe f gg " (x] \ (y] " ! a, b, " #' # 3 ; fa bg, " % (b] ; a? , (a] ; b? 31 # A = (a]0 \ (b]0. (e] \ (f ] \ (g] " ! # ((b] ; a? ) ((a] ; b? ), " A " ! 3 31 K2 (3) 3 34 K1 (3), , A 192 . R, A " ! # K3 (3), 3 30 K2 (3) 3 35 K1 (3), .

4.4. ( ab | + ;, (a] \ (b] (2 3)- . /, % $ - + I = (a] $ 2. . 2 #, ab | ; (a] \ (b] " ! (2 3)- " fc1 c2O d1 d2 d3g. 7 " (c1]\(c2] " ! a, b, 3

125

d1 : : : d3 (a]\(b], 5 (b];a? , (a];b? 31 # A = (a]0 \(b]0. R, A " ! 51 # (c1] \ (c2] 45 (c1] ; c?2 , (c2] ; c?1 . 2

, (ci ] \ (dj ] " ! a, b, 2 (b] ; a? , (a] ; b? 30 A. & 9 (dj ] \ A " !

7 (c1] \ (c2], 9 # (dj ] \ A " !

23 (c1] ; (c2] (c2] ; (c1 ]. D " ((c1] \ A) ; (c2 ] " !

23 3 ; 3 7 = 48 , .

& # !" " . + # !"

"# . + 4.5{4.7 I |

# % (165 36 3 9), !" 4- " ! % 5- - " " " #.

4.5. ( ab | + I, I 3 $ + (a] \ (b], 32 $ + (b] ; a? , (a] ; b? 96 $ + (a]0 \ (b]0. ( x 2 (a] \ (b] (a] \ (b] , (x] a, b, 2 $ + (a] ; b? , (b] ; a? 30 $ + (a]0 \ (b]0. . & .

4.6. $ c, d I . I 9 $ + (c] \ (d], 27 $ + (c] ; d? , (d] ; c? 100 $ + (c]0 \ (d]0. ( y 2 (c] \ (d] $ y (c] \ (d] ,

(y] 3 ; $ + (c] ; d? , (d] ; c? 28 + $ + (c]0 \ (d]0. . & .

4.7. ( $ I #% 5- Li , - + I 3-. . &# c, d I ! (c] \ (d] " ! 3-# eabf , 3 = fa b c d ef g, Ki = Ki (3), xi = jKi j. & 3.5 x0 = 4, xi = 0 " i > 3, 9 # e, f (c] \ (d] 1 # x0 + x1 + x2 = 159, x1 +2x2 = 194. 7 , x0 = 4, x1 = 116, x2 = 39. &# a, b, c, d e, f ! L1 : : : L6 . 7 " L1 , L6 " ! " K2 , L2 , L5 | " %, L3 , L4 ' K2 . 2

, Li , '0 K0 , " ! " K0 K2 , "# # K0 " K2 .

, x2 = 6 +3 + 2(4 ; 2) +(27 ; 4 + ) = 37, . 8 " . 2 $ #" , ;

- " " ! 3-# %. + 4.8{4.10 " , ab | ;, (a] \ (b] " ! # xyzw, 3 = fa b x y z wg, Ki = Ki (3), xi = jKi j.

4.8. , x0 = xi = 0 i > 4, x1 = 168, x2 = 240, x3 = 72.

126

. .

. & 2.1 x0 + x3 + 3x4 = 72. &# x4 = . 7 " # x1 + x2 = 408+2 , x1 +2x2 = 648+8 . 7 , x2 = 240 + 6 , x1 = 168 ; 4 . 2

, K3 " ! 4 (a] \ (b], 12 (a] ; b? , (b] ; a? , ! ! fx y z wg, 8 (a] ; b? , (b] ; a? , ! fx y z wg, 28 ; 4 a? b? , ! 3-# fx y z wg. B'" x0 = . 2 #, > 0, # r K0 . 7 "

396 2-# % " ufr " u 2 3. 5 "# % , (r] " ! (a] \ (b]

4 K3

26 K2 , (a] ; b?

20 K3

36 K1 K2 , a? b?

28 ; 4 K3

K4 . 7 , 2-# % " ufr 4 + 3(72 ; 4 ) + 2(93 + 3 ) = 402 ; 2 . &# Ki0 = Ki \ (r]0, x0i = jKi0j. & 2.1, % " # 3

(r]0, # x00 + x03 +3x04 = 8. > 2, = 3 (r] " ! 60 K3 " ' % r 2 K0 ( , ' ; !

36 ), = 2, K0 = fq rg (q] \ (r] " ! " K4 64 K3 . D " # x03 = x04 = 0, x00 = 1, . R, = 1, x00 = 0 x03 = 8, x04 = 0, x03 = 5, x04 = 1, 2-# % " ufr

392 393 , .

4.9. 30 = 3 ; fbg, Li = Ki (30), yi = jLi j. L5 = fbg, y0 = 20, y1 = 17, y2 = 395, y3 = 48. . & 2.1 y0 + y3 + 3y4 + 6y5 = 74. + "# 4.8 L5 = fbg yi = 0 " i = 4. 2

, 3.4 (b] " ! 20 K1 , '0 L0 . B'" y0 = 20, y3 = 48. 7 y1 + y2 = 412, y1 + 2y2 = 807, 9 # y2 = 395, y1 = 17.

4.10. u 2 K1 \ (b], L0i = Li \ (u], yi0 = jL0i j. y10 = 85, 0 y2 = 196, y00 + y30 = 34. . + "# 4.9 yi0 = 0 " i > 4. & 2.1, % " # 30 (u]0, # y00 + y30 = 34. 7 y10 + y20 = 285, y10 + 2y20 = 477, 9 # y20 = 192, y10 = 93.

4.11. , ; - , . . + # 4.9, 4.10 "# , 93 = y10 6 y1 = 17, .

4.12. , ; - , . . &# a, b | ! ;, (a] \ (b] " ! # xyz , 3 = fa b x y z g, Ki = Ki (3), xi = jKij. + "# 4.11 xi = 0 " i > 4. & 2.1 x0 + x3 = 40. 7 x1 + x2 = 441, x1 + 2x2 = 807, 9 # x2 = 366, x1 = 75. 2

, K3 " ! 21 # (a] \ (b], 6 (a] ; (b], (b] ; (a], ! fx y z g " #' # a? b?, !#'

127

fx y z g. B'" x0 = 6. &# u 2 K0 , Ki0 = Ki \ (u], x0i = jKi0j. & 2.1, % " # 3 (u]0, # x00 + x03 = 12. 7 x01 + x02 = 303, x01 + 2x02 = 477, 9 # x02 = 174, x01 = 127. 7 , 127 = x01 6 x1 = 75, . 8 " . C 4.12 "# , ; ' , 9 # ; GQ(4 8)- . 7 2 " .

%

1] . ., . . . // !" . "#. | 1987. | (. 24. | . 186{229. 2] Makhnev A. A. Pseudogeometric graphs connected with partial geometries pG(4 R 1) // Mathem. Forschunginst. Oberwolfach. Tagungsbericht. | 32/91. | P. 11. 3] 01 . . 2 3 (64 18 2 6) // 4# . . | 1995. | (. 7, 6 3. | . 121{128. 4] Wilbrink H. A., Brouwer A. E. (57,14,1) strongly regular graph does not exist // Proc. Kon. Nederl. Akad. Ser. A. | 1983. | Vol. 45, no. 1. | P. 117{121. 5] Paine S., Thas J. Finite generalized quadrangles. | Boston: Pitman, 1985. 6] Goethals J.-M., Seidel J. J. The regular two graph on 276 points // Discr. Math. | 1975. | Vol. 12, no. 1. | P. 143{158. 7] Cameron P., Goethals J.-M., Seidel J. J. Strongly regular graphs having strongly regular subconstituents // J. Algebra. | 1978. | Vol. 55, no. 2. | P. 257{280. ' ( ( 1997 .

{ ( ) ] GF q

x y

. .

,

. - e-mail: [email protected]

519.6+512.62

: , !"# $!!!, !! %!, !&! !$.

'"( LLL $!) %%!! * . +. ,! -. .. /! (1982) -. . . (1985) ) %!3 !"# $!! " 4 ] ) !$ %!$ . -. . . !) !# 3 !&! !!!( 6 ((degx )6(degy )2 ) $ * )(( . 7! . +. ,! -. .. /! $ # ) %!! !!!( %!( !&!, %!)! !# !). 8) $ %! "$, ! 9! %!! )!%3 %! !!* %!%! 3:36 !# 3: !!!( 6 ((degx )4(degy )3 ). f

F x y

F

O

f

f

F

O

f

f

Abstract S. D. Mechveliani, Cost bound for LLL{Grigoryev method for factoring in ( )4x y], Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 129{139. GF q

The well-known LLL method was accommodated in papers by D. Yu. Grigoryev and A. L. Chistov (1982) and A. K. Lenstra (1985) for factoring a polynomial in 4 ] over a >nite >eld . A. K. Lenstra derives a cost bound for his method with the main summand ((degx )6 (degy )2) arithmeticoperationsin . D. Yu. Grigoryev and A. L. Chistov aimed to provide a method of a degree cost bound and did not consider any detailedestimation. Here we show that this methodallows, after certain correction, to prove a better bound with the main summand ((degx )4(degy )3 ). f

F x y

F

O

f

f

F

O

f

f

1. CG] 1982 ( G] ) !" # f Ft x] $ # !# $$ F, # $ & # (degx f)(degt f) q s. , 2002, !$ 8, ? 1, . 129{139. c 2002 ! " # $%&, ' () *

130

. .

* &+ ,#. q | #, q = qs , F = GF(q ) | q , A = F t], p | $ # A, deg = degx , r = deg f, f = degt f | / A Ax], v = max vi : 1 6 i 6 n | A, c(q) | " !# $ GF (q) = = Z=(q), " (O((log q)2 )), c(F) | " !# $ F (O(s2 c(q))), P a xi | $ !" # Ax], f= i 0

0

0

j

j j

fj

j

j

g

06i6r

logDf = logq (r f ), lc f | 1$ /!!" #, cont f = # = 234 /!!" f, a], a b c],... | ( ) / $ , D, & D, a : : : : 5(a)] | , ! $ & 5, n::m] | " # n m &# , n 6 m # &, n > m | ,&. 6 , # # !" Ft]x]. 3, G,Le] & ,$ , LLL !" Zx]7 ,+ # , # " Z F t] . 8 , | G] | ( ) 2. 4 $1 $ # , 2, $ , 9. : $ , ! Mi] (1989), +: !" Zx] Ft x]. ;, Le, 2] # G] : b 1: L (. 9). 9 G] + b, $ 1 : /!!" uij F | 1 h0-sv 9. m f + r f 7 (A-h0-1) h0-h h = E:*F& (p f k h1). k m f + r b : (A-h0-2) 3.3 , # # u u 6 U. (A-h0-2) : $& /!!" uij F # ui A. 3: 1 / 7 h0-e , h0 = b=(cont b). B , " &+ m. 2

j j

j

j

j

2

2

6

j

j j

{

3. #

133

3 1 sq, p G]. ; & 1 sq ,# | 234 Ft]x] , ,. 2, # 234 $$ , . ; 1 p " #$ O(r3 f (logDf)3 q (log q ) c(F )): 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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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

x_ =

k X A0 (t) + Ai (t) (t ; ti ) x i=1

(7)

ti | ( , Ai | + - .

. I

167

2.7. & * + x() ( ) (7), t0 x0, x() ) ( (2). ! 2.5. x1() : : : xn() | ( (7), n | (7). / W (t) " " x1(t) : : : xn(t). t0 6= ti & i > 0 ! / {1 det W (t) = det W (t0 ) exp

Zt

8> 0 >< 0

i(t) = > >:;trtrAAi(it(it)i)

t0

tr A0( ) d +

Xk i(t) i=1

ti > t0 t 6 ti

ti < t0 t > ti

ti > t0 t > ti

ti < t0 t 6 ti : $%&. * ( -* + i ( k P ) ( I) x_ = A0(t) + Ai (t) i (t ; ti ) x, x(t0 ) = xj (t0) i=1 ( x^j (). , x^j () xj () x^j (t0 ) = xj (t0 ). ;( ( W^ (t) + + x^1(t) : : : x^n(t). B + det W^ (t) = det W^ (t0 ) exp

Zt t0

k Zt X tr A0 ( ) d + tr Ai ( ) i ( ) d : i=1 t0

, ( t 2= ft1 : : : tkg. ? det W^ (t) det W^ (t0 ) exp

Zt t0

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Xk i(t): i=1

? xj () x^j () xj (t0 ) = x^j (t0 ), det W^ (t) det W (t). & Rt Pk , det W (t) det W (t0 ) exp tr A0( ) d + i (t) . ; i=1 t0 , - @

. , ( t (

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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] . . 94 " % 0 // $ % . | 1989. | * 1. | +. 23{34. 5] Callot Jean-Louis. Travaux de recherche: Colloq. trajectorian m>em. Georges Reeb et Jean-Louis Callot, Strasbourg-Obernai, 12{16 juin, 1995 // Prepubl. Inst. rech. math. avan. | 1995. | No. 13. | P. 183{189. 6] ?4 2. @. A 0. +. 4 . B. D. E 9 // 7% . . | 1984. | 1. 39, . 4. 7] Brunovsky P. Controllability and linear closed-loop controls in linear periodic systems // J. of DiKer. Equat. | 1969. | Vol. 103, no. 1. | P. 296{313. 8] 1 5. Q. 9 ; // 3!!". . | 1983. | 1. 19, * 2. | +. 269{278. 9] E +. 2., 1 5. Q. 7 4 Q % . I // 3!!". . | 1994. | 1. 30, * 10. | +. 1687{1696. 10] E +. 2., 1 5. Q. 7 4 Q % . II // 3!!". . | 1994. | 1. 30, * 11. | +. 1949{1957.

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11] E +. 2., 1 5. Q. 7 4 Q % . III // 3!!". . | 1995. | 1. 31, * 2. | +. 228{238. 12] E +. 2., 1 5. Q. U % // 3!!". . | 1995. | 1. 31, * 4. | +. 723{724. 13] E +. 2., 1 5. Q. + 4 Q // 3!!". . | 1997. | 1. 33, * 2. | +. 246{235. 14] 9 3. . 2 4 !!" % 0 W!!" // B4 ! . A. 1. | B;, 1995. | +. 3{50. 15] 9 3. . E 4 Q % // B4 ! . A. 2. | B;, 1996. | +. 69{84. 16] 3 . E 4. | .: , 1980. 17] Q $. X. A !!" // A Y . | 1974. | A. 8. | +. 122{144. 18] E X. A. 9 4 A E // 3!!". . | 1979. | 1. 15, * 4. | +. 761. 19] Z $. Z. 3!!" 4 [. | .: 2 , 1985. 20] $% $. A. 9 % % % !" !!" % // 3$2 +++?. | 1986. | 1. 286, * 5. | +. 1037{1040. 21] 3 A. Y. U [ : !!" 0 !" W!!" % // 3$2 +++?. | 1988. | 1. 289, * 2. | +. 269{272. $ % % 1997 .

. II . .

517.977

: , , , .

P1

! " x_ = A0 (t)+ (t ; ti)Ai (t), () | -) * i=1 . +, " ,- , . ./ 0 - .

Abstract D. M. Olenchikov, Impulse control of Liapunov exponents. II, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 171{185.

The systems x_ = A0 (t) + P (t ; ti )Ai (t), where () is Dirac's delta-function, i=1 are investigated. It is proved that the basic results of Liapunov exponents theory remain valid for such systems. The theory of impulse control of Liapunov exponents is developed.

1

, 1].

.

1. 1.1.

1 P

U1 (t ; ti )Ui , i=1 t1 < t2 < : : : < tn < : : :, Ui # $ , & ti, $ ' , . ./ 6 . ) ) , (

97-01-00413) * ( 97-04). , 2002, 8, 9 1, . 171{185. c 2002 , !" #

172

. .

u 2 U1 N (u t) * i, ti < t. + 0 6 N (u t) 6 (t ; t1 ) + . + U1

& . . U1

&$ : X ku()k = sup kAik Ia = fi: ti 2 a a + 1)g: a>0 i2Ia

1.1. U1 |

.

1 # *, U1 . 2* u() 2 U1 . 3 4 - x_ = (A0 (t) + u(t))x t 6 t0 : (1) 6

& * 7 #*, A0 (t) # . 1.2.

1.1. 2* f : R ! Rn. 8 ( ;1 +1),

f] = t!lim +1(ln kf(t)k=t)

* 1 ( , 1 ).

1.1. f . 1. kf(t)k 6 kg(t)k, f] 6 g]. 2. kf(t)k] = f(t)]" cf(t)] = f(t)] c 6= 0. 3. f] = 6= 1, # " > 0 (a) f(t) = oe(+")t], ;(;")t = +1. (b) t!lim +1 kf(t)ke 4. $ % &# & ' %'# ( ) ( & ) & ) , %' '% # ). 5. $ % &# & ' ( ) . 6. $ % #

' #

% . 7. * ,

& , . 8. f] < 0, t!lim +1 f(t) = 0. *

2].

1.2. + % ' x() (1) & )

& %.

. II

173

. > M > 0, kA0(t)k 6 M ku()k 6 M. ? x() | * 7 , x0 = x(t0) 6= 0. kx(t0)k = a. i, t 2 t0 A ti] a exp(;(t ; t0 + N (u t))M) 6 kx(t)k 6 a exp((t ; t0 + N (u t))M): +# 1, 2.2] t0A t1] 7 x() 7 x_ = A0 (t)x, x(t0) = x0, kx(t)k 6 kx(t0)k +

Zt

t0

kA0 ()kkx()k d:

2 B{6 , a exp(;(t ; t0 )M) 6 kx(t)k 6 a exp((t ; t0 )M): 2 , i < k . * i = k. 2 1, 2.2] (tk;1A tk ] 7 x() 7 x_ = A0 (t)x, x(tk;1) = y(1), #

y() | 7 y_ = Ak;1(tk;1)y, y(0) = x(tk;1). & ky(1)k exp(;(t ; tk;1)M) 6 kx(t)k 6 ky(1)k exp((t ; tk;1)M): D#, $*& B{6 ky(1)k: kx(tk;1)k exp(;M) 6 ky(1)k 6 kx(tk;1)k exp(M): E * , kx(tk;1)k. 7 * # 7# # #. > kx(t)k ae;(t;t0 +N (ut))M ] 6 kx(t)k] 6 ae(t;t0 +N (ut))M ]: & ;(1 + )M 6 x(t)] 6 (1 + )M.

1.2. + &

&) ' (1) )

( .

2. 2.1.

8 Mnm

* n m ( n = m, 7 Mn ), MFnm | ( 7

) n m, RnF |2 ( 7

) . Vect: Mn ! Rn &, 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 Uab]

* * , &Pk $ U(t) = Ui (t ; ti ), #

a 6 t1 < : : : < tk < b, Ui 2 Mmr . 8 k i=1

* * * # N (U). I U(t) = U1 (t ; t1)

* ' . + *

*#

& .

2.1. Uab] |

. Pk > 4 U kU()k = kU k . . *, ab]

i=1

i

Uab] . ( 7

) * * a b] UFab] . 2.2.

> & * #& # #, 4 # 3]. X(t s) J7 x_ = A(t)x. > 4 ,

&$ Uab] Mn ,

k X Hab] (U()) = X(a ti )B(ti )Ui C (ti )X(ti a)

Pk

i=1

#

U() 2 Uab] U(t) = Ui (t ; ti). i=1 2.1. + (A B C) # a b],

Hab] (Uab] ) = Mn . 2.1. (A B C) # a b] # % # , # & a < t1 < : :: < tn2 < b n Ui , & fVect(X(a ti)B(ti)Ui C (ti)X(ti a))gi=12 . . * . *. 2* Hab] (Uab] ) = Mn . >*4 *2 Mn . . 4 * Uj (), fGj gnj =1 Hab] (Uj ()) = Gj . 2

& Uj () 7*

. II

175

# . + *, $ ' * fVi ()gki=1 , Uj * ' Vi (). ?# Hab] (fVi ()gki=1) = = Mn . >

fH(Vi ())g *& & . K

* n2 ' . 8 * ,

* ' 2 fH(Vi ())gni=1 . I Vi () & Vi (t) = Vi (t ; ti2). + *, fQi = Vect(X(a ti )B(ti )Vi C (ti )X(t a))gni=1 . > . 1. > ti . ?# . 2. . ti &. H Vi . ? Qi ti , F * *G &$ ti , & * Qi . Z(t U) 7 J7 Z_ = A(t)Z + B(t)U(t)C (t)X(t b) Z(a) = 0: (3) Z(U) = Z(b U). , # ' &$ ,

2.2 ( * 2.2). 2.2. + (A B C) # a b], $ l > 0, G 2 Mn 4 * U 2 Uab] , kU()k 6 lkGk Z(U) = G. 2.2. , 2.1 2.2 ( . . 2 J7 Z(U) = X(b a)Hab] (U()). 2* (A B C) #

2.1. 2 2.1 2 n $ * fVi(t) = Ui (t ; ti )gi=1, fHab] (Vi ())g | Mn . > *& G 2 Mn X(a b)G fHab] (Vi ())g. vi '

# . 3 * n2 P U(t) = vi Vi (t). 2 , Z(U) = G. ? Vi () i=1 G, $ l, $ G, kU()k 6 lkGk. + *, (A B C) #

2.2. 2* (A B C) #

2.2. ?# Hab] (Uab] ) = Mn .

2.2 #

3] * . E , ' #. ? Z(U) * , , , * , $ l * * &$

'

.

176

. .

2.3. + (A B C) # a b], Z(U ) = Mn . > U | ' a b] , * | Uab] . > * Z(U) = X(a b)H(U). > Z(U) = Rb = X(b a) X(a t)B(t)U(t)C (t)X(t a) dt . a

2.2. $% fAig, i 2 I , | a b] Mn. $% Zb V1 =

i2I a

Ai (t) dt V2 =

i2I t2ab]

fAi (t)g

# # & & . -# V1 V2.

>

V2 | ' #*# , V1 | #. + *

&# # #, #* V2. 2.3. # % (A B C) ( . # -

& .

. 2* (A B C) # . >*4 * Gi Mn 4 &$ Ui (), &$

& #. ?# i Zb a

X(a t)B(t)Ui (t)C (t)X(t a) dt = X(a b)Gi :

? X(a b)Gi ,

2.2 , fX(a t)B(t)Ui (t)C (t)X(t a)g = Mn : i2I t2ab]

2 & , 4 f(Uij tj )gnj =1 , X(b tj )B(tj )Uij (t)C (tj )X(tj b) . + *, (A B C) # * . 2* (A B C) # * . , (A B C)

& 2.3 .

2.3 , ' *, (A B C) '

&

. >*4 Gi Mn . E * # , Ui () 2 Uab] ,

. II

177

Z(Ui ) = Gi. 2 - , $ Ui , *

*-. 2 ' Ui $& ( ) a b] Uî . J #, &# i Z(Uî ) = G^ i Gi . ? Gi , G^ i . + *, Z(U ) = Mn , #

U | ' a b] . + *, '

, * (A B C) # . 2.3.

H ] (U ab

ab

])

2.1 ( ! "). / # & f1(t) f2(t) : : : fn(t) Y , ) &, & & t1 t2 : : : tn 2 Y , & f1(t1) f1(t2) f1(tn) f2(t1) f2(t2) f2(tn) .. . . . 6= 0: f (t. ) f (t.. ) . . f (t.. ) n 1 n 2 n n

. * . n *. 2 n = 1

. 2* n < k

. 2* ffi (t)gki=1 Y . + *, & $ ;1, det D 6= 0, #

fti gki=1 0 f (t ) f (t ) 1 1 1 1 k ;1 CA : .. ... D=B @ ... . fk;1(t1 ) fk;1(tk;1) ?# D

$ . ' ' c1 : : : ck;1. ? f1 : : : fk Y , $ tk 2 Y , fk (tk ) 6= c1 f1 (tk ) + : : : + ck;1fk;1(tk ). + *, ' ci 0f (t ) f (t )1 B@ 1 ... 1 . . . 1 ... k CA fk (t1 ) fk (tk ) , 4

* &.

178

. .

Eij , ' ij , * ' P &. M * Uab] , &$ U(t) = k Eij (t ; tk ), #

k 2 R, Uij . k

2.3. 0 Uij %

-

Uab].

2.4. $

Uab]

# P E '#

, H (U ) = H (U ). ?-

Uij .

ab] ab]

ij

ab] ij

, Hab] (Uab] )

Hab] (Uij ).

H i j. > (f1 (t) : : : fn2 (t)) = Vect(X(b t)B(t)Eij C (t)X(t b)): >

fl *& & . 6

4 *, ' ff1() : : : fk ()g. 3 * ' : fk+1(t) = ck+11f1 (t) + : : : + ck+1k fk (t) ::: fn2 (t) = cn2 1f1 (t) + : : : + cn2 k fk (t): 3 80 1 1 0 0 1 0 0 19 > > > BB 0 CC BB 1 CC BB 0 CC> > > > BB ... CC BB ... CC BB ... CC> > > = : > > BB k+1. 1CC BB k+1. 2CC BB k+1. kCC> > > > .. A @ .. A .. A> @ @ > > : cn21 cn2 k cn22 V = hv1 : : : vk i & ' . 2.4. ,

Vect(Hab](Uij )) V . . , Vect(Hab] (Uij )) V . > * U() 2 Uij , Vect(Hab] (U())) 2 V . * ' * ' U(). 2* U(t) = Eij (t ; t1 ). 2 ' 0 f (t ) 1 1 1 B Vect(Hab] (U())) = @ ... C A = (f1(t1)v1 + : : : + fk (t1)vk ): 2 fn (t1 )

. II

179

?# Vect(Hab] (U())) 2 V . , Dim Hab] (Uij ) = k. ? f1 () : : : fk () , $ & t1 : : : tk , f (t ) f (t ) 1 . 1 . 1 . k .. .. 6= 0: .. fk (t1) fk (tk ) 3 ' Ul (t) = Eij (t ; tl ), #

l = 1 : : : k. > Vect(Hab] (Ul )) , ' Dim Hab] (Uij ) = k.

2.5. $% Q1 Q2 2 Mn | , # Hab](Uab] ) = Mn # % # , # Q1Hab] (Uab])Q2 = Mn. ,

Q1Hab] (Uab])Q2 & % , & Hab](Uab]).

> ,

Q1 = X(b a), Q2 = X(a b). & 2.1. 3 0 #] (A B C), #

00 1 0 01 001 B BB0CC 0 0 1 0C B CC B . . . . . .. .. .. . . .. C A=B B=B BB ... CCC C (t) = (r1(t) : : : rn(t)): (4) B C @0 0 0 1A @0A 0 0 0 0 1 Hab] (Uab] ) * & E11 = (1). > Q(t) = X(0 t)B(t)E11 C (t)X(t 0) = (qij (t)), #

j j ;k n;i X qij (t) = ((n;t); i)! rk (t) (jt; k)! : k=1 + *, (4) # # * # , # qij ( 0 #]). & 2.2. 3 0 #] x_ i = ai (t)xi + bi (t)u i = 1 : : : n y = c1 (t)x1 + : : : + cn (t)xn: (5) > Q(t) = X(0 t)B(t)(1)C (t)X(t 0) = (qij (t)), #

qij (t) = Rt = bi(t)cj (t) exp (aj () ; ai ()) d . + (5) # # * 0 # , # qij ( 0 #]). 2.4. # #

2.4. 1 y = C1x1 + : : : + Ckxk x1 : : : xk , Ci A , $ ( ) CiA

180

. .

* , .

2.5. + fxig -

, $ * ' , * &. Q , & . 2.6. &) fxig & # % # , # ) ) & fSt(xi)g

.

E #

&$

. 2.7. &) fxigni=1 & # % # , # det(x1 : : : xn) 0. 2.6. J fxig y, $ xi, * y. J hfxi giF .

2.8. +& & %

) &) . 0 & , & & &) & & ) ) & . 2.9. $% fxignni=1 | , # hfxigiF = RF . 2.10. fyig | % fxj g, hfxj giF = hfyigiF . 2.5. $% fxig | &) hfxigiF = RnF . -# fxig, n (. . >

fxig & & fyj g, $& *# . ?# hfxigiF = hfSt(yj )giF = RnF . + *, fyj g Rn. 2.5. %

2.7. + (A B C) l- # , $ , &# $ & < t1 < : : : < tn2 < + l Ui , kUi k = 1 k(J1 : : : Jn2 );1 k 6 , #

Ji = Vect(X( ti )B(ti )Ui C (ti )X(ti )):

. II

181

. *,

' &$

& 2.8. 2.8. + (A B C) l- # , $ " > 0, &# $ & < t1 < : : : < tn2 < + l Ui , kUi k = 1 det(J1 : : : Jn2 ) > ", #

Ji = Vect(X( ti )B(ti )Ui C (ti )X(ti )). 2.6. (A B C) l-

# # % # , # # ( & & %'#) ( 2

Ui() 2 UF +l] , & fH +l] (Ui ())gni=1 . . * * ' Ui ()

l- #. *. 2* &# $ & ' 2 Ui () 2 UF +l] , fH +l] (Ui ())gni=1 & . 2 * l- & #*

. >*4 " * . H * 2. > Ui () 2 UF +l] , fH +l] (Ui ())gni=1 & . + Vi () = Ui ()=kUi ()k. ? Ui 2 H +l] , fH +l] (Vi ())gni=1 & . + *,

* det(Vect(H +l] (V1 ())) : : : Vect(H +l] (Vn2 ()))) , # * *7 " ( " ). . # $, *, ti Vi () , F * *G 4

, ' & . + *, (A B C) l- #

.

2.7. (A B C) l-

# # % # , # # ( #) H +l] (UF +l] ) = MFn.

. 2* (A B C) l- #. >*4 * . 2

& 2.6 $ & Ui () 2 UF +l] , 2 fH +l] (Ui ())gni=1 & . ?#

2.10 hfH +l] (Ui ())giF = MFn . ? UF +l] * , H +l] (UF +l] ) = MFn . *. H * . 2 , H +l] (UF +l] ) = MFn. ?# $ & Uij (), H +l] (Uij ()) = Eij , #

Eij | .

182

. .

J Uij () * # ' . M ' , $ Uij , V . ?# hH +l] (V )iF = MFn . +# 2.5

2 V ' fVk ()gnk=1 , & fH +l] (Vk ())g & . & , * , 2.6 l- #*.

2.11. $% Q1 Q2 | & & , # H +l] (UF +l] ) = MFn # % # , # Q1H +l] (UF +l] )Q2 = MFn. 2.8. $

l-

# l-

# .

. 2* (A B C) l- #. 2 1 x = L(t)z 4 (AL BL CL) J7 XL (t s), #

BL = L;1 (t)B CL = L (t)C(t) XL (t s) = L;1 (t)X(t s)L(s): (AL BL CL)

HLab] . . , HLab] (U) = L;1 (b)Hab] (U)L(b). M L(t) L;1(t) & t &

*. >*4 * . > 2.7

2.11 , HL +l] (UF +l] ) = = MFn. + *, * (AL BL CL) l- # . 2.12. (A B C) l-

# , # l1 > l (A B C) l1-

# .

2.9. + (A B C) # , l- # l. 2.10. + (A B C) # , $ l, (A B C) # & l. 2.13. (A B C)

# ,

# .

.

$4 '

l- #-

2.11. + (A B C) l- # , 9 9N 8 8G 2 Mn 9U 2 U +l) N (U) 6 N, kU()k 6 kGk Z_ = A(t)Z + B(t)U(t)C (t)X(t ) Z() = 0 Z( + l) = G 7 * Z().

. II

183

2.9. , 2.7 2.11 ( . . > * # 2

* 2.2 ( N * n ). > #& . 2* (A B C) l- #

2.11. H N, &$

& 2.11. 3 * + l] Z_ = A(t)Z + B(t)U(t)C (t)X(t ), Z() = 0. >*4 G *& & . ? 7, 4 U(), H +l] (U()) = X( + l)G, 4 N (U()) 6 N , kU()k 6 kGk, kGk , U() 2 UF +l] . ? A # , X( + l) , ' * G , H +l] (UF +l] ) = MFn. + *, 2.7 (A B C) l- #. 2.6. '

J7 x_ = (A(t) + B(t)U(t)C (t))x XU (t s). 2.10. (A B C) # a b], I

F : Mn ! Uab] , I , & F(I) = 0, N (F(H)) 6 n2 XF (H ) (b a) = X(b a)H . . 6

* U() = F (H)

U(t) =

n X 2

(ui Ui (t ; ti )) i=1 #

ui 2 R, ti Ui 2.1 , X(a ti )Di X(ti a) Di = B(ti )Ui C (ti ).

& , #

> 7 J7: XU (b a) = X(b tn2 )eun2 Dn2 X(tn2 tn2;1)eun2 ;1 Dn2 ;1 : : :X(t2 t1)eu1 D1 X(t1 a): 3 2H 2 -& f(u) = Vect(X(a b)XU (b a)),

&$& Rn Rn . . *, f(0) = I f . * *, I $ . + R J(u) = (J1 (u) J2(u) : : : Jn2 (u)), &

Ji (u) = Vect(X(a tn2 )eun2 Dn2 : : :X(ti+1 ti)Di eui Di : : :X(t2 t1)eu1 D1 X(t1 a)): M J u . 3 Ji (0) = Vect(X(a ti )B(ti )Ui C (ti )X(ti a)). K , *, jJ(0)j 6= 0. ?# I $ f ;1 (H), f(u).

184

. .

2.11 (, , ! ). $% f : Rn ! Rn

f(x) = f(x0) + J(xk;(xx;0x) )+k (x ; x0), &. r > 0, & 8x kx ; x0k 6 r ) kx;x k 6 2kJ1; k . , & ry = 2kJr; k . -# ry f(x0 )

f ;1(y), &. kf ;1(y) ; x0k 6 2kJ ;1k ky ; f(x0 )k. 0 0

1

1

E

& 5]. 2.12. (A B C) l-

# , # " > 0 . > 0, & % H 2 Mn, & kH ; I k 6 , # 2 R . %

U(), kU()k < ", &

XU ( + l ) = X( + l )H . . H * " > 0 * +l]. ? , * 2.10, f(u), &$ f(u) = Vect(I) + J u + (u). >

2.7 kJ ;1k 6 . ? B(t), C(t) # , X( + l t) # t 2 + l] '

&, $ R > 0, $

, R kk(uuk)k 6 21 . ?# 2.11 2R $ U(H), 4 kU k 6 2 kH ; I k. > * min( 2R 2" ). D# &$ . 2.13. (A B C) l-

# , # " > 0 . > 0, & % H 2 Mn, & kH ; I k 6 , # 2 R . %

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? 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) %

.

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. 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)

# ,

%

)

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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 >. 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

( +)

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yj

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3.

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F

191

Pfmax(1 j1 j : : : N jN j8 1j 1j : : : N j N j) > 3g 6 N N X X 6 Pfmax(k jk j) > 3g = Pfmax jk j > 3k;1 g 6 k=1 k=1 N X p 6 Pfmax jk j > 3 kg 6 00026 + 0000021 0003: k=1

- , $ $ N = 10.

- , $ $ (T = 05 | $ $ # $ , , # t1 ; t2 = 2k (. ) " ). - " , $ m = 5000 M1 : : : Mm $ (t) t 2 10 T] # FD1(u): ( m X 1 FD1(u) = 1 ; m I fMk 6 ug I = 1 Mk < u 0 Mk > u: k=1

5. ! & ' # (t) # FD1. , 1 PfM > ug Pf (0) > ug + EUu ; v2 2 v2 = EfUu (Uu ; 1)g. v2 . & $ (t1 ), (t2 ), 0(t1 ), 0 (t2 ). . G | $ (t1 ), (t2 ), 0 (t1), 0 (t2 ), Gij = E i j . ; , A(v ) = D;1

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k=1

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192

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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 )

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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, #

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n X f(x) dx = 21 (b ; a) Ai f(xi ) i=1

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198

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1

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202

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203

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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- ,

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! 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,

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F(X ! Y ) | F- ", . 4 , F(X ! Y ) hF 0 F i. Q, %

"+

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# ## % * (= ) # ! &.

207

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% (, % (,( 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

%

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, , m, - . ' ! Y i = Y j F (Y i ) = F (Y j ) (%! i j > m. , Y m = Z. ? - " 3( 2 : 8V 2 Z(P 0) 9i > 0 V Y i : (2.7) 0 (2.7) % 3" U(X ! V P ), , (2.1)

3( F- " P 0. * (2.7) , U(X ! V P 0 ) = 0, - V = X. Q ". ! . & k > 0 ! # 2 P 0, ! U(# P 0) < k, (2.7). 6! - & , , (2.7) # 2 P 0, " U(# P 0) = k. 4 . , & P # = X ! V B3. B X ! W | F- , " #,

208

. .

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: Q ! S 2 P 0 Q ! S 2 F (X ! Y ). " Q ! S 2 P 0, & , , Q = X U(X ! S P 0 ) < U(X ! V P 0) = k: : & " i j > 0, W Y i S Y j . 9 . (2.5) q = maxfi j g W S Y q N ) V = W S Y q N ) (2.7) V . , 3, Q ! S 2 F(X ! Y ). ; - % ( W Y i , & % & i > 0 , % 3N ) W Y i+1 , Y i Y i+1 (2.5). 6 Q W W Y i Q Y i N ) Q ! S 2 F(Y i)N ) S Y i+1 N ) V = W S Y i+1 N ) (2.7) V . 4 - (2.7) + . Z 2 Z(P 0), Z , m = 0 1 : : :, Z Y m . ? & " (. (2.2)), ( Y i V(P) = Z (% & i 2 C0 m]N ) Z = Y m . % , m %

3" (2.5) (2.6) % 3

& 3 Y i F(Y i) i = m. : " (2.6) - F(Y m ) = F(Z). ; &

( F F (Z). Q ! S 2 F n F(Z). X Z, Q 6 X. (% Q ! S F- %

(3) 9 RAP- X ! N , - " 3 B3 Q ! S . , Q ! S " " F- , P, " P . B,. ! " (2.8), %

% %," " Q ! S " F- X ! V ,( & " F- X ! W . %

, ! W , i 2 C0 m], W Y i N ) Q Y i ( Q W)N ) Q ! S 2 F(Y i ), F(Y i ) F(Z).

( F F(Z). ? . & (

209

(2.6) F(X ! Y ) = F 0 F F(Z)N ) F (X ! Y ) = F (Z), F(Z) F (X ! Y ), . : - F- F(X ! Y ). 4 +

, % ;, %

F- " F 0 = fX ! X g F 1 : : : F m (% & i 2 C1 m] F i = F(Y i )nF(Y i;1). :

(2.6) % F 0 : : : F m ,( % F- ",

: F (X ! Y ) =

i

F i F i \ F j = ? 8i j 2 C1 m]:

F(X ! Y ) 3

( 2 3( (#), & (#) = i # 2 F i: : % , .

" + ;, % , (% " F- " # #0 2 F(X ! Y ) % # ; #0 , (#) < (#0): 9 - " + X ! Z

& F F- ", " . F- X ! Y . :+ F- " RAP- P , + ;: F (X ! Y ) = fU 0 ! V 0 = X ! X U 1 ! V 1 :: ::: :: :: : (2.9) U i ! V i :: ::: :: :: : U k ! V kg (, (2.9) F (X ! Y ) (2.6) . X %

- , 23 2 3( (#), # 2 F(X ! Y ) (i) = (U i ! V i ) (% & i 2 C0 k]. - , & 2 3 , % (,

F- (2.9) . % , .

3, & , 2 . 6

& (i) 2 C0 m] (% & i 2 C0 k], & m | "

(2.5) (2.6), & , , Z = Y m . 4 % ,. 3

2 3,

210

. .

% , ( % % " (% & j 2 C0 m]

(j) = maxfi j U i ! V i 2 F j g: 8 3 j 2 C0 m] % F j . " (2.9) F- , ," % F j , - (j) 2 C0 k] (% & j 2 C0 m]. : " &

. ",

" & , 2 (% " & i j 2 C1 m] 3" i < j ) (i) < (j): ( - % % % " ,

i 6 j ) (i) 6 (j). : ,. , - " , (, "- (% & i 2 C0 k] ! ( % :

( (i) ; 1) < i 6 ( (i)): (2.10) 9%,

( (,( X ! Z % (2.9), F(X ! Y ) " i 2 C1 k] F- Si X ! S i , & S i = V j . " (

j =0 ,.

"), ( % alt(P ), &

" (3) 2.1, %

( (. ? - " 3( & i 2 C1 k]

( U i S i;1 , % (, B2 ,"

F- X ! S i;1 ," U i ! V i . ( U i ! V i 2 F (i)N ) U i ! V i 2 F(Y (i) )N ) U i Y (i) (2.3). : (2.4) ! Y j . ? & Y (i) % ( %5 ! " F- ", ! ,! F(Y (i);1). 4 (S i);1 F (Y (i);1) = F i , - ; i=0 F (i);1 " F (Y (i);1) % . B. (2.9) . ( (S i);1) iS ;1 ( (i);1)N ) Y (i) = V i V i = S i;1 , (2.10) i=0 i=0

( (i) ; 1) 6 i ; 1. 9%5

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3. ! " # $

211

% 3.1. $ R | ( X Y Z V R | ( ,

X ! Y Z ! V ) X (Z n Y ) ! Y V: (3.1) (. : - & & . +

" . 1. ' +

2 2 3 " . :% t t0 2 r(R) & -% + r(R), ( F- X ! Y Z ! V , , t t0 t(X (Z n Y )) = = t0(X (Z n Y ))N ) t(X) = t0 (X)N ) t(Y ) = t0(Y ) ( X ! Y ). & , t(X (Z n Y )) = t0(X (Z n Y ))N ) t(Z n Y ) = t0 (Z n Y )N ) t(Z) = t0(Z) ( t(Y ) = t0 (Y ))N ) t(V ) = t0(V ) ( Z ! V )N ) t(Y V ) = t0(Y V ), t(Y ) = t0(Y ). 2. : " % F- & . 1: Z ! V (

) 2: X Z ! V (F2- 1) 3: X ! Y (

) 4: X Z ! Y (F2- 3) 5: X Z ! Y V (F3- 2 4) 6: X ! Z \ Y (F4- 3) 7: X (Z n Y ) ! Y V (F6-

5, 6 Z = (Z n Y ) (Z \ Y )) *+. # | -% 2 3 ! " R. ? #Y % % . , #Z | , . . #Y #Z R # = #Z ! #Y. -. .. [(%( F- & "( & F2 ( ), (,& . ( . : % %( F- , % & ! F2 .. & %

! " 2 3 ! " + . ,

" 3.1 (, .& B2 ( )1) F6 (

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Y

X

Z

Z

Y

X

Y

Y

Y

Y

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Z

U

V

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212

. .

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2

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F- , W ! W, & W R, % # I W = = # I (W ! W) W I # = (W ! W) I #. 6

- " 3 # B1/F1 (2 ).

,/ - ..

I. 9 3 I % . Z ! U ( F- X ! Y , ! X = Y Z \ U. : , - (X ! Y ) I (Z ! U) = X (Z n Y ) ! Y U = Z ! U: ? % ,. B X ! Y

, (X ! Y ) I (Z ! U) = (X ! Y ) Z X U Y . : - " 3 & F- Z ! U. ? ", ! 3, , %

X = Y Z \ U %

F- X ! Y Z ! U, - (X ! X) I (Z ! U) = Z ! U = (Z ! U) I (X ! X). : , ! F- "

3" ? ! ?. II. 9 3 I , %, ! #1 = X ! Y #3 = Z ! U (#1 I #2 )Z = X (Z n Y ) =? =? Z (X n U) = (#2 I #1)Z,

, . % W R (X ! Y ) I W, W I (X ! Y ) (W n Y ) I (X ! Y ): (X ! Y ) I W = X (W n Y ) ! Y W W I (X ! Y ) = X W ! Y W (W n Y ) I (X ! Y ) = X (W n Y ) ! Y (W n Y ) = X (W n Y ) ! Y W: 6 , ((X ! Y ) I W)Y = ((W n Y ) I (X ! Y ))Y = = (W I (X ! Y ))Y, ((X ! Y ) I W)Z = ((W n Y ) I (X ! Y ))Z (W I (X ! Y ))Z (3.2) (X ! Y ) I W = (W n Y ) I (X ! Y ) =? W I (X ! Y ):

213

: 3 (3.2), +, , W \ Y W \ X, , , (W n X) \ Y = ?1) . III. 9 3 I , (#1 I #2 ) I #3 = #1 I (#2 I #3). - F- ! #1 = X 1 ! Y 1 , #2 = X 2 ! Y 2 #3 = X 3 ! Y 3 : #1 I #2 = X 1 (X 2 n Y 1) ! Y 1 Y 2, - (#1 I #2 ) I #3 = (X 1 (X 2 n Y 1 )) (X 3 n (Y 1 Y 2 )) ! (Y 1 Y 2 ) Y 3 = = X 1 (X 2 n Y 1 ) (X 3 n (Y 1 Y 2 )) ! Y 1 Y 2 Y 3 : #2 I #3 = X 2 (X 3 n Y 2 ) ! Y 2 Y 3, - #1 I (#2 I #3 ) = X 1 ((X 2 (X 3 n Y 2)) n Y 1 ) ! Y 1 (Y 2 Y 3 ) = = X 1 (X 2 n Y 1 ) (X 3 n (Y 2 Y 1 )) ! Y 1 Y 2 Y 3 : :

" 3 ( #1 I #2 I : : : I #k F- " #i = X i ! Y i , i 2 C1 k]. #(k) = #1 I #2 I : : : I #k Y 0 = ?. 9 + #(k) = #(k ; 1) I #k , 3" k #(k) =

k

i ;1

i=1

j =0

Xi n

k

Yj !

i=1

Y i:

(3.3)

3 #(1) = X 1 ! Y 1 , & " ( (3.3) k = 1. , ! (3.3) #(k ; 1), , " !

: #(k ; 1) I #k = = =

k ;1

k ;1

i=1 i ;1

i=1 k

j =0 i ;1

Xi n

Xi n

i=1

j =0

Xi n

i ;1

j =0

Y j Xk n

Y

k j !

i=1

Yj

k !

k ;1 i=1

i=1

Xi !

Y i I (X k ! Y k ) =

k;1 i=1

Yi Yk =

Y i = #(k):

1) / ( n ) I ( ! ) = I ( ! ) %, (( n ) I ( ! ))C = = ( I ( ! ))C, , * % , ( n ) = , = ( ) n ( ( n )) = ?.

, = ( n ( ( n ))) ( n ( ( n ))) = ( n ( ( n ))), ( n )D ) = ( n ) n ( n ) = ( n ) n ( n ( \ )). : # E & # = ? ( n ) ( n ) ; D ) ( n ( \ )) ( n ( \ ))D \ \ . = ( n ) \ = ? # = . % . > 2 , 2 2 n (, 2 2 )D ) 2 n D ) 2 ( n ) n ( n ) = D ) . G , #! 2 , 2 n ( !, 2 ) 2 n D ) 2 D ) 2 ( n )\ = D ) . W

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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

, 3.1, & + % F1{F6 B1{B3.

216

. .

, 3.1. - &% ( .: fI ( ) B1=F1 (2 ) F2 ( ) B3=F4 ( )g: 3 : 1) F- , (. 6) F , F- F , &% . fI B1=F1 F2 B3=F4g, *: F + = f#1 I #2 I : : : I #k I W F2 B3 j #1 #2 : : : #k 2 F W RgN 2) ( ) , . , 1). -( (- . , #1 I #2 I : : : I #k I W F2 B3 k 6 (F) (. . 196), . I (F ) , B1=F1 ( I W ), F2, B3/ F4 . ! . F2, B3/ F4 & , + 3) 1 1) 2) & F- , (, (. 3) F . 0+ - .. : (% " 2 3 " X ! Y " " " 2 . '

! 2. : , + r &

" " t1 t2 2 r %," % " 3 t1(X \ Y ) = t2 (X \ Y ) ) t1 (X \ Y ) = t2(X \ Y ) (, % ( & , % . ? ,( B3/F4 ( ) X \ Y % , X ! Y X ! Y n X. : % % - " 3 " F- " F fX ! Y j X Y Rg + . 3.2. 4 F- " `(R) = = fX ! Y j X Y Rg ( 3( 0 ( ) #0 = #Z ! #Y n #Z (% " # 2 `(R). B F fX ! Y j X Y Rg | F- ", .

% F 0 = f#0 j # 2 F g. 3.3. ; 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)

$ #+ 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. ; 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)

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 # < | , ; | 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) = gî, gî | i

g^. P = F | , gî 2 ;0. / ., # bi = ij uij , uij | j I X fel gml=1 , X X X 1 1 X g^(b1 : : : bk ) = gî

j uj : : : kj ukj = l(j ) g~l (u1j1 : : : ukjk ): i

j

j

6 g~l | gî . ? , g~l 2 ;0 l. = uij | I X fel gml=1 g~l

, . g~l (u1j1 : : : ukjk ) # xi . , xj 2 X gi . g 2 ;0 (B) # . g~l (u1j1 : : : ukjk ), % # xj 2 X. = g~l 2 ;0, . g~l (u1j1 : : : ukjk ) ;(B). ? , gi ;0(B). @ . 2

230

. .

, . # ;(B), ;1 (B). 5. % f 2 ;1(B)

X s f = h + g h = li uli xpi vli g 2 ;(B) xi 2 X uli vli | * X fej gmj=1 li 2 F: (1) % , f xj 2 X , &

(1), % h g xj degxj h = degxj g = = degxj f .

. ; . f 2 ;1 (B). = ^ ^b1 : : : ^bn), f^ 2 ;1 , ^bi 2 B. f = f( 6 , h 2 F hX i h = 0 A , . bi 2 B h(b1 : : : bn) 2 ^ ^b1 : : : ^bn) 2 ;(B) + P. 2 ;(B) + P. ? , f^ 2 ;1 , f = f( = , B f = h0 + g0 , h0 2 P, g0 2 ;(B). 7 X s h0 = li uli xpi vli xi 2 X uli vli | I X fej gmj=1 : (2) P # f , g0 = gi0 , gi0 |

i xj 2 X g0 , ( 4 gi0 2 ;(B). ? ,

B X 0 gi = f ; h0 : (3) g0

i

? . i ( , # xj f. , . . g. , ps v (2) h #

u x l l i i i s . h0 , degxj f = degxj uli xpi vli xj 2 X. B (3) g = f ; h, . g 2 ;(B), h 2 P . ? , f = h + g, h 2 P , g 2 ;(B), . h, g # xj , ( degxj h = = degxj g = degxj f. @ . 2 , Nk k

# . : Nk

( 5 . # I1 I2 | . Nk . i1

# . # I1 n (I1 \ I2), I1 n (I1 \ I2 ) #, i1 = 0

#C , i2

# . # I2 n (I1 \ I2), I2 n (I1 \ I2 ) #, i2 = 0

#. = I1 6 I2 , i1 6 i2 . @ , Nk

( #

.

5 .

T-

231

1. F I1 I2 , I1 6 I2 . 2. # ( . i 2 I2 , j 2 I1 j < i, I1 < I2 . 3. $ . Nk # . = Nk | #

, . # # . : . I 2 Nk . # # (I). , , 1 6 (I) 6 2k , ( (?) = 1, (f1 : : : kg) = 2k . # I | # s N. , PI , (

fxpi j i 2 I g. F I = ?, # , PI | # . ; . f 2 ;1(B). # f

. , (f) # xi, degxi f > ps . : % # , , f x1 : : : xn. . degxi f > ps 1 6 i 6 (f) degxi f < ps i > (f). 5 X X s f=

li uli xpi vli + g (4) i2I li P P u xps v 2 P , ( h g g 2 ;(B), h = li li i li I i2I li xj . f. , i 2 I degxi f = degxi h > ps. ? , I f1 : : : (f)g. # I | i, h 2 PI . (4) . f I 2 N(f ) . = . f # ind(f) = (I). , , 1 6 ind(f) 6 2(f ) .

# xi . f ( OP, i 2 I,

f # 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, (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, (

#. 0

.

. # < | T-

F, char F = p > 0. @ , !( ) 7 (2.6) (2.7), + g(Z N ) = Q(') = Q (f ') (f ') : (2.8) > ) , 7 2 (1.6) *, : Z 1 1X Z :::m 2 2 2 2 0 2 i<j K(ei ej )(i ; j ) + j 'j ; jr'j ; j'j dv = Q(')dv : (2.9) M

@M

A *,) 2.1. M m-

@M . 2- ' 2 C1 S2 M , , ! (2.9). ? + , ) @M ) ' = g ) = C1 @M. L 2- ' ) @M, '(N f X 0 ) = 0 ) * X 0 2 C1 T@M. > / g(Z N ) = N i ri = 0. >/ *,) 2.2. M m-

@M . 2- ' 2 C1 S2M , @M ' = g 2 C1 @M g M , Z 1 1X :::m 2 2 2 2 K (e e )( ; ) + j ' j ; jr ' j ; j ' j dv = 0: (2.10) 2 i<j i j i j M

x

251

3. $ "

3.1. ? +- - 2- S2 M M @M. 2 +) 2- ' 2 C1 Sp M ) ( . 5, . 339]), (rX ')X = 0 ) X 2 C1 TM. A) 0 2- ' + ; M, t 2 J R, '(X X) = const ) X = ddt; . 3 ( . 0 4, . 976]) * K

( ), * M )) cK 6 0 ( K > 0), *+ 7 + x 2 M, K(X Y ) < 0 ( K(X Y ) > 0) ) *- - +- X Y 2 Tx M. ? M @M +* * 2- ', *,*) @M. 2 *,) 3.1. m-

M @M ! 2- ', @M . (1) " M #

, M , ' = g = const. (2) " M

, ' = g = const. (3) " M #

, ' = 0.

. > , + K 6 0 * M. J ) Q0 ) @M ) (2.9) * ) + 2- ' M. Q ' 6= g ) = const, >. R. U ( . 18]) M ) )) ), U 0 + x M )) ) U = U1 : : : Ur - - M1 M2 : : : Mr , r | + +- - + 2- ' *,- m1 = dimM1 ,. . ., mr = dimMr .

252

. .

Q 0 ) + x 2 M, K(X Y ) < 0 ) *- +- - X Y 2 Tx M, 0 (2.9) 1 = : : : = m = = const. L 0 0 ) Q0 @M + 0 ) M. . 3.1 2.2 ) 3.2. m-

M @M ! 2- ', ' = g 2 C1 @M g . "

#

,

$ , ' = g = const M . 3.2.

M (rX0 !)(X1 X2 : : : Xp ) + (rX1 !)(X0 X2 : : : Xp ) = = 2g(X0 X1 )V(X2 : : : Xp ) ; p X ; (;1)i g(X0 Xi)V(X1 X2 : : : Xi;1 Xi+1 : : : Xp ) + i=2

+ g(X1 Xi)V(X0 X2 : : : Xi;1 Xi+1 : : : Xp )

V = ; m ; 1p + 1 d!

(3.1) (3.2)

) )) )*, ( ., , 9] 10]) ) p- M. 19] ), + (3.1) * (d!)(X0 X1 : : : Xp ) = (p + 1)(rX0 !)(X1 X2 : : : Xp ) + p X + (;1)i g(X0 Xi)V(X1 : : : Xi;1 Xi+1 : : : Xp ) (3.3) i=1

0 (1.3). G +* 2- ' U 'ij = gl2 k2 : : :glp kp !il2 :::lp !jk2 :::kp = !ii2:::ip !ji2 :::ip (3.4) !i1 :::ip 7 p- . L (3.3) ):

253

rk 'ij = rk !il2 :::lp ] !jl2 :::lp + !il2 :::lp rk !jl2 :::lp ] + + m ; 1p + 1 fgkj Vi ; gkiVj ; (p ; 1)(Vikj ; Vjki)g:

G ) *, +): Vl = gl2 k2 : : :glp kp !ll2 :::lp rk !kk2 :::kp W i Vjlk = gl3 k3 : : :glp kp !jll3 :::lp ri!kk 3 :::kp : > )), + ) *- X Y Z 2 C1 TM (rX ')(Y Z) + (rY ')(Z X) + (rZ ')(X Y ) = = m ;2p + 1 fg(X Z)V(Y ) + g(Y Z)V(X) + g(X Y )V(Z)gW (3.5) ;p+2 2p (')Y = ; m (3.6) m ; p + 1 V(Y )W Y (tr ') = m ; p + 1 V(Y ): L , ) ) (3.5) (3.6), 0 ), + +) 2- = ' ; 1p (tr ')g ) )) . A 3.3. " m- M % p- ! (0 < p < m), M 2- = ' ; p1 (tr ')g ', (3.4). . . , + ) +) M

) +- - + , ) - - ( . 20, . 158]). 2 * 0 , / , +) 2- M

+ - +- ( . 20, . 158{159]). p-K ! * ( . 3, . 126]) @M, !(N f X20 : : : f Xp0 ) = 0 (3.7) 0 0 ) *- - @M - X1 : : : Xp . A) 0 (2.1) 'i1 j1 N i1 fj11 = !i1 i2 :::ip N i1 !ji21:::ip fj11 = = !i1 i2 :::ip N i1 fi22 : : :fipp !j1j2 :::jp fj11 fj22 : : :fjpp g2 2 : : :gp p : R ) p- ! ) ) @M ) M, +) 2- ', ) ) (3.4), ) @M. 3.3. M r' = 0 ) + 2- (3.4),

rk 'ij = (rk !ii2:::ip )!ji2 :::ip + !ii2:::ip (rk !ji2 :::ip ) = 0

254

. .

rj!j2 = 0. * + , ' = g ) = const, + 7, + !ij2:::jp !jj2 :::jp = m1 j!j2gij (3.8)

) j!j2 = const. > + *, - (3.4) !i1:::ip 0 p- ! ) 1 6 p 6 m ; 1. L ) + 2- = ' ; p1 (tr ')g, , = 0. ) (2.9). > Q0 > 0 Z 1X :::m K(ei ej )(i ; j )2 dv 6 0 M i<j

(2.9) 1X :::m

K(e0i e0j )(0i ; 0j )2 = 0W

(3.9)

r'0 = r' ; p1 r(tr ')g = 0W

(3.10)

i<j

( )(X) = (')(X) + 1p X(tr ') = 0

(3.11)

) ij = 0i ij fe1 : : : em g * X 2 C1 TM. F 0 + , + ) K K 0 * M. >

(3.9) + 01 = : : : = 0m = 0 = 0 g. L ;m tr ', = ' ; 1p (tr ')g *+ , + 0 = pmp ; m (tr ')g + 1 (tr ')g = 1 (tr ')g: ' = + p1 (tr ')g = p mp p m 4 , 0, + 1 6 p 6 m ; 1, (3.10) tr ' = const. 2 , ) p- ! + ), 0 ) (3.8). * V0 = 0. 4 /, ' = r(tr ') = 0, , , (3.10) / , + r' = 0. > + * >. R. U ( . 18]) ) M ) ). Q, , M 0 + , ) p- ! (2.9) , + ! = 0. > ) , 0

255

3.4. m-

M @M ! @M % p- ! 1 6 p 6 m ; 1. (1) " M #

, j!j = const M , ! (3.8). (2) " M

, ! (3.8). (3) " M #

, ! = 0. . Q )- ) 3.4 ) 2- ! ) @M ) M (3.8) / ), 0 grad j!j 2 C1 T @M, +) * * ) - (1) (2) 0 +, (3) . *+ 2- - 2m- M. G , + + 2- ! * ( . 15, . 126]). 3.5. 2m- M

! 2- ! . " & & (1)

K M , (2) M

K 6 0 M , p M $ 2- ( 2mj!j);1!.

. > , + M ) - 7 , ) ) M ) 2- ! 0 +)) 1 j!j2g ) j!j2 = const : glk !il !jk = 2m ij 10 , + J (1 1) p Jji = ( 2mj!j);1gil !lj *, : J 2 = ;Id g(JX JY ) = g(X Y ) ) *X Y 2 C1 TM. 2 , M ) ))

( . 21, . 139{140]) 2- p = = ( 2mj!j);1!.

256

x

. .

4. $ & "

4.1. > M N m n- ) + g g0 )) Y -Z r r0. ? 0 f : M ! N C1 . E+ + f ;1 TN M, E 0 = Tf (x) N + x 2 M. A f : TM ! TN ) )) 0 , fx / ) Tf (x) N T M ) 0 + x ) M f ) )) + 1- M +) f ;1 TN. U M x1 : : : xm U 0 N y1 : : : yn , + f(U) U 0 , 0 f : M ! N ) ) ya = ya (x1 : : : xm ) a b c : : : = 1 2 : : : n. > , + Xi = @x@ i Ya0 = @y@ a , f = (f Xj ) dxj = (fja Y[a0 ) dxj a ) fia = @y + Y[a0 ) f ;1 TN, @xi n 0 ) - (Y[a )x = (Ya0 )f (x) ) - x 2 U. A r[ f = (fija Y[a0) dxi dxj ( . 22{24]) 2 ya a k b c 0a fija = @x@ i @x (4.1) j ; fk ;ji + fi fj (;cb f) ) ri Xj = ;kjiXk r0Ya0 = ;0acb Yb0 dyc . . (4.1) , + r[ f ) )) + + M +) f ;1 TN, r[ f 2 2 C1 (f ;1 TN S2 M). / r[ f 0) f : M ! N. +, f : M ! N ) )) 0 , (r[ f )(X Y ) = = Q(X Y ) ) f(M) X Y 2 C1 TM. 1 g0 f ;1 TN g[0 , ) )) g[x0 = gf0 (x) ) 0 + x 2 M. > / ) r[ 0 ) ( ., , 24]) + , r[ 0g[0 = 0. 3 ) g[0 M 0 f 0 f, + g = f g[0 gij = fia fjb gab + Rang(fx ) = Rang(gx ) 0 + x 2 M. Q g = g ) 2 C1 M, 0 f ) ,

257

= const | . Q 0 rg = 0, 0 ) ( . 25]). Q 0 f : M ! N 0* +* ; ) M +* ;0 = f(;) ) N, 0 ) ( ., , 23]). 4 + r[ f 0) *, ( . 23{25]): r[ f = V f + f V (4.2) V | ) ) 1- M. Q 0, + 0) +) ; ) M 0) +* ;0 / - )) ) ) -, 0 f ) ( . 25]). A) 0) f )) r[ f = 0: (4.3) A0 , + *,) 4.1. # f : M ! N N g0 . " # f | , = 2(m1+1) lndet(g )] e;4 g ! M 2- .

. > f : M ! N | 0 . Q N | + g0, , / + 7, M ) + g = f g[0 , (4.2) +)) * (rZ g )(X Y ) = 2V(Z)g (X Y ) + V(X)g (Z Y ) + V(Y )g (X Z) (4.4) ) - X Y Z 2 C1 TM. F ), + (rZ g )(X Y ) + (rX g )(Y Z) + (rY g )(Z X) = = V0 (Z)g (X Y ) + V0 (X)g (Y Z) + V0 (Y )g (Z X) (4.5) V0 = 4V. > 0 , + f : M ! N ) )) , , G = det(g ) 6= 0. E+ + Gij + ) / gij (g ) x1 : : : xm U M. L ) X 2 C1 TM X(G) = Gij rX gij (4.2) X(ln G) = 2(m + 1)V(X), *+ , + V = 2(m1+ 1) grad(ln G): (4.6)

258

. .

/ +, / )), + ' = e;4 g ) = 2(m1+1) ln G ) * (rX ')(X X) = 0: R ' ) )) + 2- . 4.2. Q + 4.1 3.3, 0 4.2. f : M ! N |

M @M N , & f # . (1) " M #

, M f |

# , f ( (2) " M

, f .

. > ) ) f : M ! N + ) @M ) M g N g0 0 , 0 + x 2 @M ) - X Y 2 Tx M 2 C1 @M g0 (f X f Y ) = (x)g(X Y ). 2 4.1 ' = e;4 g ) = 2(m1+1) lndet(g )] + ) )) + 2- M. > / ) @M ) M * m det(g ) = m , , ' = m2+1 g, 2 C1 @M. >/ 2- ' = e;4 g + ) M @M ) (2.10), ' = 0. > , + M * 0* * , (2.10) r' = ' = 0. +m P , +, + (tr ') (X) = (Ei )'(Ei X) ) i=1 fE1 : : : Emg - M. + x ) M fe1 : : : em g Tx M, + 'x (ei ej ) = i ij 1P :::m ) i > 0. L ' = 0 +, + (ei ) k = 0. k6=i R = grad = 0 , , | . > ) , - *, : rg = 0. > , >. R. U ( . 18]), + -.

+ M ) )) ), f | 0 . +, M ) )) ), g = g ) = const , , f ) )) .

259

Q 0 M 0* * , (2.10) , + ' = g ) = const. > 7 g = e4 g 0 ) g (4.4). 3 = 0. > 0 0 ), ) + , f | ). . A 0 ) -

27] f ) , +- +)- f 0 - ) ) 0) f. E 0 f : M ! N ) ( . 13]), r[ f = m1 g f (4.7) ) J {Y 0 f = trg r[ f ( ., , 22, . 11{12]). . +, N | g0 f : M ! N | ), f g0 = g,

(4.7) ) f +* * ( . 1, . 58{59]). - (4.7) ) )) ( . 26, . 126]) , + . 4.3. " M N , M .

. ? M g = f g[0 . + + 0) f : M ! N g ) * (4.8) (rZ g )(X Y ) = V(X)g(Y Z) + V(Y )g(X Z) ) V(X) = m1 g[0 (f X f ) - X Y Z 2 C1 TM. . ) (4.8), +, , + V = 12 grad(tr g ) = 21 grad jf j2: (4.9) Q f | / +) ), det(g ) 6= 0. 2 F. 2. 2* ( . 26, . 122]) + M 0 ) g 2 C1 S2 M, )*, * (4.8), ) )) - + , ) f 0 : M ! M 0

M 0 . 4.3.

260

. .

F ) g + = g ; (tr g )g, (4.8) ) * (rX )(X X) = 0 ) * X 2 C1 TM. 2 , ) )) + 2- . L , 4.4. M N g g0

, f : M ! N # . ) = g ; (tr g )g g = f g[0 2- M . Q + 4.4 3.3, 0 4.5. f : M ! N | #

M @M N , & f # . (1) " M #

, M f |

# , f ( (2) " M

, f .

. 2 4.4 ' = g ; (tr g ) + + 0) ) )) + 2- M. > / ) @M ) M * ' = (1 ; m)g, 2 C1 @M. >/ 2- ' = g ; (tr g ) + ) M @M ) (2.10), ' = 0. > , + M * 0* * , (2.10) r' = ' = 0. +, +, + = 0 )- (4.8), rg = 0. > , >. R. U ( . 18]), + -.

+ M ) )) ), f | 0 . +, M ) )) ), g = g ) = const , , f ) )) . Q 0 M 0* * , (2.10) , + ' = g ) = const. > 7 g = ( + tr g )g 0 ) g (4.8). 3 = 0. > 0 0

), ) + , f | ). 4.4.

'

261

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?@ " 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] \@" ;. . ;

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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] Mikes J. Global geodesic mappings and their generalization for compact Riemannian space // Proc. Conf. on Di,. Geom. and its Appl. (Opava, August 24{28, 1992). | Opava, 1992. | P. 143{149. ( ) * 1997 .

. .

. . .

511.51

: , .

( 1+ 2+ 3= 1+ 2+ 3 3+ 3+ 3= 3+ 3+ 3 1 2 3 1 2 3 3+ 3+ 3 = 3+ 3+ 3 1 2 3 1 2 3 x

x

x

y

y

y

x

x

x

y

y

y

x

x

x

y

y

y :

Abstract

A. V. Ustinov, On some cubic equations, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 263{271.

The paper considers the ( structure of solutions of the system 1+ 2+ 3= 1+ 2+ 3 3+ 3+ 3= 3+ 3+ 3 1 2 3 1 2 3 and the equation 3+ 3+ 3 = 3+ 3+ 3 1 2 3 1 2 3 x

x

x

y

y

y

x

x

x

y

y

y

x

x

x

y

y

y :

2] 51, 52 ! " !. $ % "& ' & " &" % ! ! . () % ' % " &, " '" & ) ) ) . * &, z ;x 0 2 2 2 2 y ;w x y + y z +z w + w x = x w

. W

% & &!

3

z

y

+ X3 + Y 3 + Z3 = 0

= x + y + z + w X = x ; y ; z + w Y = ;x + y ; z + w Z = ;x ; y + z + w

W

, 2002, 8, , 1, . 263{271. c 2002 , !" #$ %

264

. .

& 1 ' ) ) w

;

z

y

3z ;3y w 3x = 0:

;

x

w

) ' & & ( x1 + x2 + x3 = y1 + y2 + y3 (1) 3 3 3 3 3 3 x1 + x2 + x3 = y1 + y2 + y3 : & 1 ) 5 "& & ) & " &" ) ! ! . 6& (1) %" %! ' !. 1] &% " ! ) % , " ' " & & ' %, % ) & & & (1), "% ' " ) & 1 & & & &. 7 & 2 % &1) 8& ' & &% . ) % " ) &" ! . 9 & , ' ) 3 3 3 3 3 3 x1 + x2 + x3 = y1 + y2 + y3 : (2) :)& % & & (1) (2), ! % " &! x1 , x2, x3 ) % "& &! y1 , y2 , y3 " ) . ( % 5& &. ; % & &! aj = xj ; yj bj = xj + yj (j = 1 2 3) (3) & 1)

3 3 3 2 2 2 a1 + a2 + a3 = 3a1 a2 a3 + (a1 + a2 + a3 )(a1 + a2 + a3 ; a1 a2 ; a1 a3 ; a2 a3 ) & (1) & ) ( a1 + a2 + a3 = 0 (4) 2 2 2 a1 b1 + a2 b2 + a3 b3 + a1 a2 a3 = 0

(2) )& ' % &: 2 2 2 a1 b1 + a2 b2 + a3 b3 + D a1 a2 a3 = 0 (5) ) 3 3 3 a1 + a2 + a3 D = 3a a a : 1 2 3

$ & (4) (5) ')& % &, ! ) ! )& && x1 , x2, x3 , y1 , y2 , y3 ' % & (1) (2) . ( ')& % &.

1. 1 2 3 1 2 (4). 2 2 1 = ( 3 ; 2) 2 2 2 = ( 1 ; 3) 2 2 3 = ( 2 ; 1)

| 2 Z

2 Z,

a a a b b b3

d1 d2 d3

= d1 + d2 d3 a d d b2 = d2 + d1 d3 a d d b3 = d3 + d1 d2 : . 6& (4) & 1 ) 8 a1 + a2 + a3 = 0 > > > a

d

< > > > :

d

b1

a1

b3

;2

b

a2

b1

b

a3

;3 b2

;1

265

b

= 0:

(6)

(7)

; a1 a2 a3 b1 b2 b3 | 5 . ?& & 1 " (d1 d2 d3) ) ) & 8 > < d1 a1 + d2 b3 ; d3 b2 = 0 (8) ;d1 b3 + d2a2 + d3b1 = 0 > : d1 b2 ; d2 b1 + d3 a3 = 0: :)& " d1, d2, d3 & &, a1 , a2, a3, b1 , b2 , b3 | &&. ? ) & 1 & (8) d1, d2, d3 1, & & (7) "& ( a1 + a2 + a3 = 0 (9) 2 2 2 a1 d1 + a2 d2 + a3 d3 = 0: B & & 1 & ! %) (a1 a2 a3) (1 1 1) (d21 d22 d23). )!

& ' , 8 & " && d1 = "1 d d2 = "2 d d3 = "3 d ("1 "2 "3 = 1): * ) % (8) ) a1 = "1 "3 b2 ; "1 "2 b3 a2 = "1 "2 b3 ; "2 "3 b1 a3 = "2 "3 b1 ; "1 "3 b2 " %& 1 , ' ' % " "1 , "2 , "3 ) & . ; 8 & % & (9) (a1 a2 a3) ) ) % " , " ) & 1 , %) (1 1 1) (d21 d22 d23): 2 2 a1 = (d3 ; d2 ) 2 2 (10) a2 = (d1 ; d3 ) 2 2 a3 = (d2 ; d1 ):

266

. .

; %! a1 , a2, a3 & b1 , b2, b3 ) 1 ) & 8 ; d3b2 + d2b3 = ;d1 a1 > < d3 b1 ; d1b3 = ;d2 a2 > : ;d2 b1 + d1b2 = ;d3 a3: C & 1) , ) ) ) && " ) ) & ' ) ) . D & 1 % ) b1 = d2 d3 b2 = d1 d3 b3 = d1 d2 : ; 8 & ' % ) &: b1 = d1 + d3 d2 b2 = d2 + d1 d3 b3 = d3 + d1 d2 " (10) ) 1) &. 7 & ) % . ; & a1 a2 a3 b1 b2 b3 & (4). :)& ) " & & . (c1 c2 c3 d1 d2 d3) % 5& & ) ) 8 , & & & &, ! ) "% " (a1 a2 a3 b1 b2 b3), 1 ! (4).

2. 1 2 3 1 2 3 | (4) 1, 2, 3 . 8 > : 1 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 >
: 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 > > >
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 > : 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 : 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). D5 ) 1) & . 7 &

) % . 3. (2)

!! #. 4. (2) ! (

x1

3 x1

= y1 + x32 + x33 = y13 + y23 + y33

p

% " % Q

D

.

% " ) % & ) 1 2.

271

1] Chondhry A. Symmetric Dioph. Systems // Acta Arith. | 1991. | Vol. 59. | P. 291{307. 2] Dickson L. E. Introduction to the Theory of Numbers. | Chicago, Univ. of Chicago press, 1931. ( : . . ! "# $%. | &'%, 1941.) & ' ( 1997 .

. . 512.556

: , , .

!" (m 2 Rm 8m 2 R M ) % & ' . & . & ' (, , , R ! ) ( , R % !* & ( : 1) R () , 2) R ( , 3) R ) () ( ) % ! )') , 4) R % % ( ! ", 5) R ! ( ')( ( ( , R

" & &

), 6) R R ') R = 0.

Abstract

A. V. Khokhlov, On existence of unit in semicompact rings and topological rings with niteness conditions, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 273{279.

We study quasi-unitary topological rings and modules (m 2 Rm 8m 2 R M ) and multiplicative stabilizers of their subsets. We give the de4nition of semicompact rings. The proved statements imply, in particular, that left quasi-unitariness of a separable ring R is equvivalent to existence of its left unit, if R has one of the following properties: 1) R is (semi-)compact, 2) R is left linearly compact, 3) R is countably semicompact (countably left linearly compact) and has a dense countably generated right ideal, 4) R is precompact and has a left stable neighborhood of zero, 5) R has a dense 4nitely generated right ideal (e. g. R satis4es the maximum conditionfor closed right ideals), 6) the module R R is topologically4nitely generated and R = 0.

( ) R R = 0, RR = R , r 2 Rr 8r 2 R ( 1] "left D-regular#, 2] | "left s-unital#,

, 2002, 8, 5 1, . 273{279. c 2002 , !"

# $% ## &

274

. .

, 3], ) QU- ). +, , - QU- - . - . , () QU- . . / QU- , 0 , , /0 3]. 1 2 0 + : R (r 2 Rr2 8r 2 R) () R | QU- =) R () R 0 ( ) R | QU- =) R (r 2 (Rr)2 ) () R |

QU- . 4 ) 3] , ,- 5 ( ) . / , 6 ) )5 , 2 . 7 8 9 ) 2 )

( 1.1, 1.2). : 3] , , . ; 2 2 , , ,

0.1.

R , R : 1) R (

) (. 2.1), 2) R (. 2.2), , R # , 3) R (

) $ ( , ), 4) R $ , 5) R #, 6) R $ ( , R # ), 7) $ R % , 8) R R

$ R = 0, 9) R = 0 $ # $ R % .

1. , ,

49 0 R, T | , R MT | ) . L < R M , "L | R M # (L < R R | R), L / R | , "# | 9 (),9), a b a + b ; ab.

275

A A B R N R MT N , N | 2 NP, A B fa b j a 2 A b 2 B g, A N fan j A 2 A n 2 N g, ~ fn ; an j a 2 A n 2 N g. AN ai ni j ai 2 A ni 2 N , AN

1.1. +- () ) 2 A R MT 2 SA fr 2 R j rm = m 8m 2 Ag (AS ft 2 T j mt = m 8m 2 Ag). B SA 6= ?, A - ) . A )C : 9) A R 2 2 , AF fm 2 R j am = m 8a 2 Ag, FA fm 2 R j ma = m 8a 2 Ag, AFM fm 2 R M j am = m 8a 2 Ag. A 9) A R MT SA SA SA, R SA = SA, AS T = AS . A A B R MR B S SA B S \ SA, AS SA AS \ SA.

1.2. 2 B (R M ) SfrMF j r 2 Rg (B (MT )) ) () 8 R MT ) () )

M .

1.3. E R MT ( R RR ) ) () QU- , B (R M ) = M (B (MT ) = M ),

. . m 2 Rm 9) m 2 M . 4 (Soc M = M ) 2 (Soc(M=L) 6= 0 8L < M ) 3]. 1 2 , ) 6 "0# 2 R ( , ( ) . .) ) 6 .

1.4. F X R. +- 2 X - ) 8 BX (R M ) fm 2 R M j 9r 2 X : rm = mg = SfrMF j r 2 X g X -)

R M , R M ( R) | ( ) X , BX (M ) = M (BX (R R) = R BX (RR ) = R). : + () ) 2 ( ) . / R X R ,- , R ( 2.1). +2 9 95 , , ) 2 3]. ~ N . ' # 1.1. & R X X X , N BX (R M ), XN A N SA \ X 6= ?, $# N hN i BX (R M ). . ~ N () XN N . 1. B N RSM | ,

PXN P ~ i Ni , XN ~ N. 2. B N = Ni , Ni , Ni Z, XN ~ . 3. X | () X ; X X X X () X ; X X XX 4. 2 R | - .

276

. .

1. & X |

R, R M N , N | ) X - M , X . ' SA \ X 6= ? # A N . 2. * R M -

X R, SA \ X 6= ? # A M . 3. + QU-

SA 6= ? # A R. 4. & X R, BX (R R) = R R M = RM ( , M = R). ' SA \ X 6= ? # A M . 5. * R X X X , QX (R M ) X # R M | (

,) X . + R MT QX (R M ) QX (MT ) | R T - . +

R

, X QX (R R) ( QX (RR )). - # ##

, ,

$ BX (R R) (BX (RR )). 1.2 (3]). 1. * X | QU-

R, V W R M , S V \ X 6= ?, S W \ X 6= ?, S (V W ) \ X = S V \ S W \ X 6= ? S (V + W ) \ X 6= ?. 2. * X | # R, X B (R R), V W R M , S V \ X 6= ?, S W \ X 6= ?, S (V W ) = S V \ S W 6= ? S (V + W ) 6= ?. 1. -)# $ # QU-

. . A) M , , 2 T1- QU- M . 1 6 () () SM 6= ?. G | 6 M , SM = ? 2 ( )

, T1- 6). 2. +

, $, # . 3. * B (R R) X X ; X , BX (R R) < RR . 4. & X | ( ) QU-

R. ' BX (R R) = B (X R) | (B (RX ) | ) QU-

, $ X , ( ) R. 1.1 (3]). 1. * 9A R M : A = 0 & SA 6= ?, R # . & R = 0 . 2. * M = 0 SM 6= ?, SM = feg e | R.

2.

277

B R M | , , R (T0 - | ), SA BMF | 2 9) A M , B R S A] = SA. 2.1. & R , X R, X . ' R X () R e 2 X . ! ". S r \ X | 2 X S

9) r 2 R. F 9 4 1.1 T S f r \ X j r 2 Rg S . 4 X 5 e 2 f r \ X j r 2 Rg = R \ X . 2 I 9) A R M SA | ) 2 " # "#, SA 6= ?, ))5 2.1

2.1. I , 9 - (,- ) , 9) (,- ) ) 6 ,. / - (,- ) , - 9) (,- ) ) 6 2 , 95 ) 2 " # "#,

,. 2.2. * R M | , ( )

-

X R, 9e 2 X : em = m # m 2 M (SA \ X 6= ? # A M ). & M = 0 e |

R. ! ". 1. F 9 2 1.1 J fSA \ X j A M jAj < 1g | 2 X , ) ,. F8 J | ) 6 - X 9e 2 T J = = SM \ X . F M = 0 e | R 1.1. 2. F A = fan j n 2 Ng, Am fan j n 6 mg. I SAm \ X SAm+1 \ X S Am \ X 6= ? 9 2 1.1. F8 J fSAm \TX j m 2 Ng | ,- ) 6 - X SA = J 6= ?. 2 1.

. 2. -

QU-

, $ $ ( , ), . . 75 9 , ( , ) , ) . 3. * X |

# -

R, QX (R M ) < M (. 5 1.1) ( X ).

278

. .

4. * R X , X |

QU-

,

BX (R R) |

(. 4 1.2) ( X ).

2.1. + $ X

R SA \ X 6= ? # A R. ! ". G , 2.1. 2 F2 X , ) ) , 9) V fx + V j x 2 X g 2 , X . 4 ,- 5]. 2.2. & R $ X . ' R # () R # V . ! ". 9xi 2 X : X Sfxi + V j i = 1 : : : ng. F 9 1

1.2 S (xi + V ) 6= ?, 9e 2 S X . I 9) r 2 R r = xr = (ex)r = e(xr) = er. 2

2.2. E (,- ) , - 9) (,- ) ) 6 2 ,. . 4

, 4{6], 2.2 ) , ) . L 9 . 2.3. * R M | QU- ( )

R, SM 6= ? (SA 6= ? # A M ). 1. * R R | QU- ,

R . 2. * R R | QU- R $ ( , R ), R . ! ". 1. F 9 2 1.1 J fSA j A M jAj < 1g | 2 R, ) ,. I SA = S + A 9) A M , S 2 SA, J | ) 6 2 T T R. 4 9e 2 J = fS m j m 2 M g. 2. F A = fan j n 2 Ng, Am fan j n 6 mg. I SAm SAm+1 SAm 6= ? 9 2 1.1. F8 J fSAm j m 2 Ng | ,- ) 6 2 R SA = T J 6= ?. 2

279

E , , 2-,

9) , , 2 . 2.4. *

R

$ QU- R M , R # . 1. * R R |

$ QU- ,

R . ! ". I M = Tf m j m 2 M g = 0,T m |

, 5 , A M , , f m j m 2 Ag = 0,

. . A = 0. F 9 2 1.1 SA 6= ?. I 1.1 R ) . ; 1.1 , 9)

QU- R = 0. 2 2.3. & R MT | R- $ T - % N . ' N = M , . . 9e 2 R : em = m 8m 2 M . ! ". N = eF 9) e 2 S N , ) N eF | ) T - . 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., +% . . %"%% %03 !42%* *%) #"&%'. | 56%, 1988. 5] 70& +. 1. 5#!* % *) (%8. | 56%, 1991. 6] Zelinsky D. Linearly compact modules and rings // Amer. J. Math. | 1953. | Vol. 75. | P. 73{90. '

# ( ) 1998 .

CCC- . . , . .

. .

. . .

521.13

: , , , { ! , " # $ .

% " #$ ,& #$ "

" ' " $ ( ) ' , $ !

!' '! ! . *$, ' + ( " , )' , ! ",

" " & - { ! .

Abstract V. L. Shablov, V. A. Bilyk, Yu. V. Popov, Status of the CCC method within the frame of the rigorous many-body Coulomb scattering theory, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 1, pp. 281{287.

The convergent close-coupling method (CCC), which is now widely used for calculations of chargedparticles scatteringamplitudes, is considered from the viewpoint of the rigorous many-body Coulomb scattering theory. It is shown that the approximate scattering amplitude calculated within the frame of the method does not converge to a solution of the Lippmann{Schwinger equation.

, , !, " . $ " % " % , & (%&

, & , , 2002, 8, 3 1, . 281{287. c 2002 ! " #$, %& ' (

282

. . , . . , . .

. .) & &

( , , , ) +" . $ % ++ &

, % & & . , % , & , & & - & + . " , & - & &

! + & % & . ., , & , +"& & , + & % & & & . / + . 0 + , +" " + & , % " & , %

+" . / % &

+" & +. 0 % + ,

, , ,

. 1+ " & CCC (convergent close-coupling) 21], , &

, % +" & 5 {- &. 7 & 22].

8 ( ! | , + 22, 3]) , +"+ & H , % - H = K1 + v1 + K2 + v2 + v12 = K1 + K2 + V: (1)

283

CCC-

/ (1) Ki | %& % , vi | & , v12 | % & . / 21] 5 {- & T2 j'0~k0i = V2j'0~k0i + V2 G2(E + i0)T2 j'0~k0 i: (2) / (2) j'0 i | - , j~k0i | +" , V2 = v1 + v12 G2(Z ) | ? : G2(Z ) = = (Z ; K1 ; K2 ; v2 );1. 8 (2)

Z Z h~k jV j ~k0 ih~k0 jT j' ~k i X h~k1f jT2j'0~k0 i = h~k1f jV2j'0~k0 i + d~k0 1 f 2 i k 2 i 2 0 0 : (3) E ; "i ; 2 + i0 i 0

/

(3) i (K2 +v2 )jii = "i jii

'i %& "i < 0,

& , j~k2 i, %& " = k2=2. A+ CCC- (3) + N Z X h~k 'N jV j'N ~k0ih~k0 'N jT N j' ~k i d~k0 1 f 2 i k 2 i 2 0 0 : h~k1'Nf jT2N j'0~k0i = h~k1'Nf jV2j'0~k0i + E ; "Bi ; 2 + i0 i=1 (4) / (4) j'Ni i + & X IN = j'Ni ih'Ni j 0

i

j'0 i IN . 0 , + h'Ni jK2 + v2 j'Nj i = "Bj ij & %&

"Bi & ( " + ). E , (4) +

& % h~k'Ni jT2N j'0~k0i. /

N X h~k2 j'Ni ih~k1'Ni jT2N j'0~k0 i: (5) i=1

/ h~k1 f jT2j'0~k0i h~k1f jT2N j'0~k0i. / 21] , h~kf jT2 j'0~k0i = Nlim h~k jT N j' ~k i: (6) !1 f 2 0 0

284

. . , . . , . .

A , (6) %& , " &

. H (4) , %

h~k1'Ni jT2N j'0~k0i + ~k1 i, +" ( % , , 24,5]). E (5), CCC +

. / , , N X lim h~k2; j'Ni ih~k1'Ni jT2N j'0~k0i = h~k1~k2; jT2 j'0~k0i: (7) N !1 i=1

0 (7)

, , " , , & & E %&

: E ! (k12 + k22)=2. 0 & % T2 . % + % T2 (Z ), %& &. 0 T2 (Z ) +" ? G(Z ) = (Z ; H );1 : G2 (Z )T2 (Z ) = G(Z )V2 : / T (Z ): T (Z ) = V2 + V G(Z )V2 : E & G0 (Z )T (Z ) = G(Z )V2 = G2(Z )T2 (Z ) & G0(Z ) = (Z ; K1 ; K2 );1 | ? , (Z ; H2 )G0(Z )T (Z ) = T2(Z ): ,

h~k1~k2; jT2 (Z )j'0~k0 i Z %&

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CCC-

285

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) ; 1+ i = ; B (i 1 ; i ) dx x d~r 2 : (13) r 0

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R & $ RS (& 97-0-6.1-32), S1ME, S ! $ RS ( 108-39(00)-,).

1] Bray I., Stelbovics A. T. Convergent close-coupling calculations of electron-hydrogen scattering // Phys. Rev. A. | 1992. | Vol. 46, no. 11. | P. 6995{7011. 2] Bencze G., Chandler C. Impossibility of distinguishing between identical particles in quantum collision processes // Phys. Rev. A. | 1999. | Vol. 59, no. 4. | P. 3129{3132. 3] Bray I. Reply to Possibility of distinguishing between identical particles in quantum collision processes // Phys. Rev. A. | 1999. | Vol. 59, no. 4. | P. 3133{3135. 4] . ., . !. "#$%& $%'& ((&#'& )& ('($* #(%)'+ ,&-##.+ /($'0. | .: 2, 1985. 5] . !. $*$'/(' %4%(. #$%%5 $%'' ((&#'& )& ('($*. $+ ,&-##.+ /($'0 // 6. $. '#-$ 72 8. | 1963. | 6. 63. 6] Chandler C. The Coulomb problem. A selective review // Nucl. Phys. A. | 1981. | Vol. 353. | P. 129c{142c. 7] 9:)% ;. ., )#.+ ##'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*%# ). ' &.. | .: 2, 1987. 13] 8' ., 5*%# !' % $, p- ! +%! p. , Z(pk) ! +%! pl , l 6 k ; 1. 7"$, $ ' | " ! +%! 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

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= %$ 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

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i=1

l;1 pP

1 Ci p. J ! 0 = ("+'+: : : +'p ;1)x = pl 1+ pir si i+1 p ;1 x, i=1 x 2 Z(pk). = $, pl x = 0, $ !! | , % ! p. O x = pk;l tg, hgi = Z(pk), t 2 Z, (t p) = 1. F , r k r k x 2 Im(" ; ') = p sZ(p ) = p Z(p ), k ; l > r. J ! %!, Ker("+'+: : :+'n;1) = Im(" ; ') ! +%! ' Z(pk) p 6= 2. 2) & $ $ ' | ! +%! Z(2k) o(') = 2l , l 2 N. J ;1 2P 1 0! +%! "+' +: : : + '2 ;1 ! 2l 1 + 2ir si i+1 C2i ;1 i=1 / ! # !. & r > 1 !! | / : i + 1 = 2 , ir = (2 ; 1)r > 2 ; 1 > , r > 1. &0! , 1), Ker(" + ' + : : : + '2 ;1) = Im(" ; '),

! +%! ! 1 + 2r s, r > 1. B ' ' ! 1+2s, Im(" ; ') = 2Z(pk), x 2 Ker("+'+ : : : + '2 ;1 ), " + ' + : : : + '2 ;1 6= 0, / x 2 2Z(2k). F , " + ' + : : :+ '2 ;1 = 0, Ker(" + ' + : : : + '2 ;1 ) = Im(" ; ') ! +%! Z(2k). 1.3. " G | &! p- , ) | . 1. " p 6= 2 H 1 () G) = 0. 2. " p = 2 H 1 () G) 6= 0 , ) ) = h'i,

' " + ' + : : : + '2 ;1 = 0, o(') = 2l , ' H 1 () G) = Z(2)% ) ) | &!, ' H 1 () G) = Z(2) 1 H () G) = Z(2) Z(2). . 1. & $ G = Z(pk), p > 3. J Aut G = Z((p;1)pk;1), ! +%! ) 6 Aut G . . &0! f' 2 2 Im(" ; ') -/ !! +%! f 2 Z1() G), f 2 Im(" ; ') ! +%! 2 h'i. B ' | "

! +%!, " ; ' | !! +%!, % , ! +%!. = $, H 1 () G) = 0. l

l

l

l

l

l

l

l

l

l

l

294

. .

& $ $ ' | " ! +%!, 0! ) = h'i - ' ! +%!. & '!, f' = x 2 G -/ !! +%! f 2 Z1() G). J f'n = (" + ' + : : : + 'n;1)x = 0, n = o('). O !! 1.2 x 2 Im(" ; '), $ -/ " !! +%! f !. = $, H 1 () G) = 0. 2. & $ G = Z(2k), k 2 N. J Aut G = Z(2) Z(2k;2). B ) = h'i 2 ; 1 "+'+: : :+' 6= 0, !! 1.2 !! Ker("+'+: : :+'2 ;1 ) = Im(" ; '). O , -! , ! H 1 () G) = 0. B ' ) = h'i " + ' + : : : + '2 ;1 = 0, ! " -/ " !! +%! f : ) ! G, ' f' = x, x 2= Im(" ; '). J " ; ' 2= Aut G ! +%! ' Z(2k), " 0! x 6= 0 "/. & 0! f" = f'2 = (" + ' + : : : + '2 ;1 )x = 0, $, "$, f | -/ " !! +%!. , 0! H 1 () G) 6= 0. & '!, 0! H 1 () G) = Z(2). & $ 0 6= f1 f2 2= B 1 () G) f1 6= f2 . $ f1 ' = x1, f2 ' = x2, x1 x2 2= Im(" ; '). % % $ !! 1.2 , "+'+: : :+'2 ;1 = 0, ! +%! ' ! 1 + 2s, (s 2) = 1. J (f1 ; f2 )' = (x1 ; x2) 2 Im(" ; '). F , (f1 ; f2 ) | " -/ " !! +%!, $ f1 + B 1 () G) = f2 + B 1 () G). J ! %!, H 1 () G) = Z(2). O $ ! $

., ) .". & $ ) = )1 )2 = h"i h'i, o(') = 2l , l 6 k ; 2. $% ! !" $$ (3). !! 0 ! H 1 ()2 G1 ) ! H 1 () G) ! H 1 () G) ! H 2 ()2 G1 ) ! H 2 () G): Q ! ! H 1 ()2 G1 ). & 0! G, ' $ )1 = h;"i, | 0 0! .$ G, $ G1 = Z(2). R ! +%! )2 $ " G1 , 0! . Z(2k) ' $ ! +%! 0" : ! +%! 2 Aut Z(2k) ! 1 + 2r s, r > 1, (2 s) = 1, (1 + 2r s)g = g, g 2 G1 . F , H 1 ()2 G1 ) = Hom()2 G1 ) = l = Hom(Z(2 ) Z(2)) = Z(2) 6= 0. J $ !" $ % !, H 1 () G) 6= 0, ) | . . 7 , ! ! H 1 ()1 G). & '!, Z1()1 G) G % ! + )-! . - , Z1()1 G) = G : f = f(;") 2 G, f 2 Z1()1 G), (f1 + f2 ) = (f1 + f2 )(;") = f1 (;") + f2 (;"), f1 f2 2 Z1()1 G). - , %! +%! )-! $ ". 7"$, !, ('f) = '(f) ' 2 ), f 2 Z1()1 G). & $ 'f = f1 , f = f(;") = g, f1 = f1 (;") = g1 . & 0! f1 (;") = ('f)(;") = = 'f(;") = 'g, $ 'g = g1. !! ('f) = f1 = g1 '(f) = 'g. , Z1()1 G) = G )-! . S, B 1 () G) | )-! $ l

l

l

l

l

l

295

Z1()1

G): f 2 B 1 () G) f(;") = 2a, a 2 G, ('f)(;") = 2'a 2 2G ' 2 ). %! +%! . %! +%! )-! " B 1 () G) 2G )-! " Z1()1 G) G. J H 1 ()1 G) = G=2G = Z(2) 1 H ()1 G) = (G=2G) = Z(2). N ., ! ! H 2 ()2 G1 ). J ! +%! )2 . , 5, . 162] H 2 ()2 G1 ) = (G1 )2 =(" + 2 ; 1 1 1 2 + ' + : : : + ' )G . !! (G ) = Z(2), (" + ' + : : : + '2 ;1)G1 = = (" + ' + : : : + '2 ;1 )g = 2l g = 0, G1 = hgi, ohgi = 2. O H 2 ()2 G1 ) = Z(2)=0 = Z(2). J $ ! $$ !' $ % Z(2) ! ! : 0 ! Z(2) ! H 1 () G) ! Z(2) ! H 2 () G). B 1 Im = 0, H () G) = Z(2). B Im = Z(2), Im = 0. J H 1 () G) 2 2 Ext(Z(2) Z(2)). & 0! H 1 () G) | 2- , ;" 2 ). F , H 1 () G) = Z(2) Z(2). , H 1 () G) = Z(2) H 1 () G) = Z(2) Z(2), ! +%! ) . . J ! % . 8] M! ! " !" H 1 (Aut G G) %$" " " G. J ! 1.3 / H 1 () G), G | . p- , ) | %$ / ! +%!. l

l

l

x

2. H 1 ( ) ! G

7 $ "/ " !" H 1 () G) " G, !-" ! %' G = G1G2 , ) 6 Aut G. & 0! ! ! G1 !' !$ ! +%! , ' - ), ! ! G2 ! " ! +%! 2 2 C()). U ! , , ! +%! ) '

! +%! , . ! ! G1 G2. J ) = V h ()1 )2), V = W 6 Hom(G2 G1), )1 )2 | ! +%! G1 G2 , . ! +%! ! % ). J G1 ", " ! +%! G . " ! +%! G1 + - G=G1 = G2. &0! ! ! G1 G2 G ' )-! ! !' $ !" H 1 () G1) H 1 () G2). , !! $$ 0 ! G1 ! G ! G2 ! 0. J ! $$ 0 ! H 0 () G1) ! H 0 () G) ! H 0 () G2) ! ! H 1 () G1) ! H 1 () G) ! H 1 () G2) ! : : ::

296

. .

F % " !" H 1 () G) % !" H 1 ()1 G1) H 1 ()2 G2), % !. &0! % % !' ! H 1 () G1) H 1 ()1 G1), H 1 () G2) H 1 ()2 G2). & $ G | %$ " :-! $. 7 !! " G % " : G ( $ !! +%! : : ! Aut G) $ % :-! G 9, . 111]. F , % " !! +%! : : ! Aut G. O% ! (:) = ) 6 Aut G. ! % $ 0! 2 : ! $ " G ( ! 0!!), g 6= g 0 6= g 2 G. . B | " 0! % C(:), () | "

! +%! % C()). @ % $, !! %$ :-! . > !, 2.1. " G | :- : ( G ' 2 C(:). f 2 Z1() G) , f = a ; a ' a 2 G. . & $ f = a ; a 0! 2 C(:) 0! a 2 G. J %$ 0! 2 : !! = , f ; f = f ; f, (" ; ())f = (" ; ())(" ; ())a. = /! % ! ! f = (" ; ())a = a ; a. = $, f 2 B 1 () G). 2.2. " G | :- , : : ! Aut G, (:) = ). 1. ) H 1 (: G) = 0, H 1 () G) = 0. 2. ) : ( ' 2 C(:) ) 6 :, H 1 (: G) = 0 , H 1 () G) = 0. . 1. & $ H 1 (: G) = 0. & '!, - 0 f 2= B 1 () G). Q ! ! ' f : : ! G, % -! %!: f = f 0 () 2 :. & !, f | -/ " !! +%!. 7"$, 1 2 2 : !!, " , f1 + + 1f2 = f 0 (1) + (1)f 0 (2), " , f(1 2) = f 0 ((12 )) = = f 0 (1)+(1 )f 0 (2). , f 2 Z1(: G). & 0! f " . J ! 0! a 2 G f = f 0 (()) = a ; a = = (" ; ())a ! 2 : , $, ! ' = () 2 ). F , -/ " !! +%! f 0 " H 1 () G) = 0. 2. & $ $ H 1 () G) = 0. & '!, - f 2= B 1 (: G). J ) 6 :, f . -/ " !! +%! f 0 : ) ! G, f 0 ' = f', ' 2 ). &$ f 0 | " -/ " !! +%!, f 0 () = (" ; ())a 0! a 2 G. N f 0 () = f() () | " 0! % . :. J !! 1.4 f 2 B 1 (: G). & .

297

$"#! ' % # !' ! !" H 1 ()G0 G0) H 1 ()G0 G), G0 | ! ! G. 2.3. " G = G1G2 2 2 )\Aut G2. 1. H 1 ()1 G) = 0 , H 1 ()1 G1) = 0. 2. ) ) H 1 ()2 G) = 0, H 1 ()2 G2) = 0. ) ) H 1 ()2 G2) = 0, 1 H ()2 G) = H 1 ()2 G1) = Hom()2 G1). . = '!, H 1 ()i G) = 0 / 1 H ()i Gi) = 0, i = 1 2. !! Z1()i Gi) 6 Z1()i G), i = 1 2, 0! f 2 Z1()i Gi), f 2 B 1 ()i G). J f'i = (" ; 'i )a = (" ; 'i )ai 'i 2 )i, a 2 G, ai 2 Gi, i = 1 2. = $ ' % , $ ! ! ! ! ! +%!!, ' -! ). 7 ! ! G1 ! . XH 1 ()1 G1) = 0 / H 1 ()1 G) = 0Y ! !, $ ! ! G2 ! " ! +%! 2 2 ). 7"$, -/ " !! +%! f : )1 ! G, f'1 2 G1 '1 2 )1 9], $ Z1()1 G1) = Z1()1 G). 7 ! ! G2 0 , H 1 ()2 G) !' $ " " H 1 ()2 G2). !! , " $ )2 -! " 0 ! G1 ! G ! G2 ! 0 $$ !" 0 ! H 0 ()2 G1) ! H 0 ()2 G) ! H 0 ()2 G2) ! ! H 1 ()2 G1) ! H 1 ()2 G) ! H 1 ()2 G2) ! : : : H 0 ()2 G2) = G22 = 0. O H 1 ()2 G1) = H 1 ()2 G) = Hom()2 G2), ! +%! )2 $ " G1 . J $ !' $ ! !" H 1 () G) ! %'!" " G = G1 G2 , $ ! ! G2 ! " ! +%!, ' -" ). O% ! % J 0! G, ' $ ! +%! % ), $ J = G. 2.4. " G = G1 G2, Hom(G1 G2) = 0, 2 2 ) = = V1 h ()1 )2) 6 Aut G. 1. ) H 1 ()1 G1) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = 0, H 1 (V G1) = 0, 1 H () G) = 0. 2. ) H 1 () G) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = 0. . 1.1 !!" 2.2 H 1 ()2 G2) = 0 0 H ()2 G2) = 0. J % " $ 0 ! H 0 () G1) ! H 0 () G) ! H 0 () G2) ! ! H 1 () G1) ! H 1 () G) ! H 1 () G2) ! : : : , H 1 () G) = H 1 () G1), $ H 0 () G2) = 0.

298

. .

% # / " ! !, H 1 () G1) = 0. 7 0 !! $$ (3) $ G1 | )-! $, V ). !! 0 ! H 1 ()1 )2 G1) ! H 1 () G1) ! H 1 (V G1) ! H 2 ()1 )2 G1) ! H 2 () G): & '!, 0" $ H 1 ()1 )2 G1) = 0. J H 1 () G1) = 0. J )1 )1 )2 , !! " ' $ (3), !/" )-! G1 , 0 ! H 1 ()2 J) ! H 1 ()1 )2 G1) ! H 1 ()1 G1)1 2 ! ! H 2 ()2 J) ! H 2 ()1 )2 G1): F$ H 1 ()2 J) = 0 ! , H 1 ()1 G1)1 2 = 0, $

H 1 ()1 G1) = 0. F , H 1 ()1 )2 G1) = 0. 2. & $ H 1 () G) = 0. % $ (2) % , H 1 () G1) = 0. O !! 2.2 H 1 ()1 G1) = 0. = / ! 0 !$ H 1 ()2 J) = 0 % $ (3), / #, % $ 1.

. 1

1) Z H ()2 G2) = 0 1 ! 2.4, G1), % !! 1.2, ! ! H 1 () G) ' , ) ' " ! +%! , . -" " ! +%! $ G2, G1. &0! 1 ! $ ! ! !. 2) &! ! 2.4 H 1 (V G1) = 0. L % $ , ! H 1 (V G1) = 0. 7"$, ! +%! V " G1 $. &0! H 1 (V G1) = Hom(V G1). B 0 6= f 2 H 1 (V G1) , ('f) = 'f(';1 ') = f 2 V ' 2 ). O , , , H 1 (V G1) ' !! +%! % V G1, % f 2 G1 " 6= '2 2 ) ! f = f(';2 1 '2 ), 6= ';2 1 '2 . N ! , G = Z(p) R, R | % 1, pR 6= R, ! +%! ) 6 Aut G, " V 6= f"g, H 1 (V G1) = 0 , ! H 1 (V G1) 6= 0, $ ! V = Hom(G2 G1) = Z(p). 1 J H (V G1) = Hom(Z(p) Z(p)), " " !! +%! | !! +%!. 3) N ! % 2 ! 2.4 % $ H 1 () G). & /! 0 / . & $ G = Z(pk) R, R | % 1, ) 6 Aut G, J 6= 0. J

) p = 2 H 1 () G) 6= 0< H 1 ()1

299

) p 6= 2 )2 6= h;"i H 1 () G) 6= 0 % ! , %' G ! ! H 1 ()1 Z(pk))Q H 1 ()2 R). 7"$, J | . , )2 6 Aut R = Z(2) Z, : = fp: pR = Rg. F , )2 6= 2)2, p2 )2 6= p)2 p 6= 2, )2 6= h;"i. = $, Hom()2 J) = H 1 ()2 J) 6= 0, $ # ! H 1 () G). %' G = G1 G2, 2 2 ) 6 6 Aut G, " H 1 () G). 2.5. " G = G1 G2, Hom(G1 G2) = Hom(G2 G1) = 0, 2 2 ) 6 Aut G. * H 1 () G) = 0 , H 1 ()1 G1) = 0, H 1 ()2 G2) = 0, H 1 ()2 J) = Hom()2 J) = 0. . 7 $ / # " % ! 2.4, ! H 1 (V G1) = Hom(h"i G1 ) = 0. N !$ " H 1 ()1 G1) = 0, Hom()2 J) = 0 ' % ! 2.4. O / % $ !$ H 1 ()2 G2) = 0. & '! , $ H 1 () G) = 0 H 1 ()2 G2) 6= 6= 0. J !! 2.2 !! H 1 () G2) 6= 0. & $ f2 2= B 1 () G2). & 0! f2 2 2= (" ; 2)G2 !!" 2.1. U ! , f2 2 Z1() G) (" ; '2)G = (" ; '2 )G2 , $ f2 2 2= (" ; 2)G. L % , f2 | " -/ " !! +%! % ) G. & . = $, H 1 () G) = 0, H 1 ()2 G2) = 0. = % . B ! +%! ) ' " ! +%! , . -" ! +%! G1, G2 , H 1 ()2 J) = 0 ' , J = 0, H 1 () G) ! ! ! !" ! ! ! ! H 1 ()1 G1) H 1 ()2 G2), % ! ' + !. 2.6 (1]). " G = G1 G2, Hom(G1 G2) = Hom(G2 G1) = 0, 2 ) 6 Aut G. * H 1 () G) = 0 , H 1 ()1 G1) = 0, H 1 ()2 G2). B ' ! ! G1 ! ! +%! % ), H 1 () G) !' $ , ! H 1 ()1 G1) H 1 ()2 G2) . 2.7. " G = G1 G2, Hom(G1 G2) = Hom(G2 1G1) = 0, 2 2 ) 6 Aut G. ) H 1 ()1 G1) = H 1 ()2 G2) = 0, H 1 () G) = H ()2 J) = Hom() J). = 2 . J )1 ) = )1 )2, !! $$ 0 ! H 1 ()2 G1 ) ! H 1 () G) ! H 1 ()1 G) ! H 2 ()2 G1 ) ! H 2 () G):

300

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0" $ H 1 ()1 G) = 0, $ ! ()1 G1) = 0 / !!" 2.3 H 1 ()1 G) = 0. F , () G) = H 1 ()2 G1 ). S, G1 = J G2 . & H 1 ()2 G2) = 0. & !! 2.3 G = J G2 , ! ! H 1 () G) = H 1 ()2 J) = = Hom()2 J). . & $ G = Z(pk) R, R | % 1 2R = R, pR = R, $ %' . ! !" ! +%! %$!/! ) = h'i Aut R, ' | ! +%! Z(pk), /! ' p 6= 2 "+'+: : :+'2 ;1 6= 0, o(') = 2l , p = 2. J $ !" 2.7 H 1 () G) = H 1 (Aut QR J) = Hom(Aut R J) = Hom(Aut R Z(p)) = Z(p), Aut R Z (2) Z , : = f p: p R = R g . J ! %!, = p2 Z(p) " " !" " !# " $" ". % - $ % !" !# ! $ ! !, !' #%' % $ % 3] + ! $ H 1 () G) !# $ G = G1 G2, ! ! G1 ! ! +%! , ! +%! ) 6 Aut G: G1 | , 0! )1 = Aut G1, . p- , )1 | %$ / ! +%! ( , G1 !' $ " 2- ")< G2 | !# $ ! ! +%!! 2 2 )2 . H1 H1

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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 .

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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) | "

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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

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y(1) = y (1) = 0

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# 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)

Новый Мир ( № 1 2002)

Новый Мир ( № 1 2002)

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Газета День Литературы # 65 (2002 1)

Газета День Литературы # 65 (2002 1)

Association for Jewish studies 2002-26(1)

Association for Jewish studies 2002-26(1)

Рецензия на сборник 'Фантастика-2002-1'

Рецензия на сборник 'Фантастика-2002-1'

Journal of Arabic Literature (33-1, 2002)

Journal of Arabic Literature (33-1, 2002)

JWSR Volume VIII, Number 1 (2002)

JWSR Volume VIII, Number 1 (2002)

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The Elliot Wave Theorist (June 1, 2002)

2002 (132)

2002 (139)

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