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Abstract A. G. Chkhartishvili, E. V. Shikin, Dynamic search of objects. A geometric approach to the problem, Fundamentalnaya i prikladnaya matematika 1(1995), 827{862.
The review of the geometrical methods and constructions used for solving the problems of the dynamic search on the plane and in threedimensional Euclidean space is given. The known results as well as modern ones are under consideration.
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846
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847
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848
. . , . .
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% ( ' '/ - $/ - ) $ $% $ - - % l. 4 " - '$ " '( B , "# n %, p1 = (N +1 1)n : 8. ; '( B '$ , #$# '( A # " R-$% $# " # , , '$ ", # . . 4 " - pk '$ " '( B k " k pk = 1 ; 1 ; (N +1 1)n : , $ -, pk ! 1 k ! 1. ? ', " - '$ " '( B "
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.% A " | ). % 7-'-;-, 1990, = 3751{790. 5] 4+9.+ ?. ., 5 6. 7. ?."8" !9 . ! // 7 ! !% !! + % . ? " 15. 7& " * * 9 . | 1992. | = 2. | ?. 46{50. 6] 4+9.+ ?. . ?."8" !9 % .&, )! . | ). % 7-'-;-, 1992, = 2020-792. 7] 5+ % . 7. C+!%" ! ,% % . * & ! .& )! . | ). % 7-'-;-, 1993, = 3170-793. 8] 2, 3% . 4. (9 !.!* !* & !* %! % ."8 !9 % .&, )! // 7 ! !% !! + % . ? " 1. * . , . | 1992. | = 3. | ?. 7{10. 9] 4+9.+ ?. ., 2, 3% . 4., 5 6. 7. 4!* & %! % ."8 !9 % .&, )! // ; .!.!% *A.+!.! !
\#!9&% !%*" !* "", C, 18{22 %+ 1992 . 2 1. | 1992. | C. 27{28. 10] 2, 3% . 4., 5 6. 7. * & .& )! !9+A " ! !, *+ , )!%,! ", // +% ". | 1993. | ;. 29, = 11. | ?. 1948{1957. 11] 2, 3% . 4., 5 6. 7. $.& )! !9+A " ! !, )!%,! ", // ! " & "{IV. 7 "" %!!A " * * & " 3!, 3 { 8 *" 1993 . | ; .!.!%. | 7!!A, 1993. | C. 205.
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12] 2, 3% . 4., 5 6. 7. ?."8" !9 % )! % #!9&% !! // 7 ! !% !! + % . ? " 1. * . , . | 1994.{ = 2. | ?. 36{41. 13] 2, 3% . 4., 5 6. 7. (9 !.!* !* & !* )!.,!. 3 G . * & , )! // +% ". | 1994. | ;. 30, = 6. | ?. 1097{1098. 14] 2, 3% . 4., 5 6. 7. 4!* & * !. )! %! +!"G8 , " !9H !% // ; .!.!% 7 ! ! 3!-!!% +* )! !, & * * !.* !*
, 9+- G !, 25 "9" { 2 !"9" 1994 . | .: '+&! .-%! ;7 , 1994. | C. 125{126. 15] 2, 3% . 4. ( !* !! ."8 !9 % .& . * & !! )! . | ). % 7-'-;-, 1994, = 1445-794. 16] ?%!!% . . J !% " 3 *! )! !%! .& 8+8 -+!"G8 " 9 !&!* .. | ). % 7-'-;-, 1994, = 1565-794. 17] 2, 3% . 4., 5 6. 7. ( )! , , )! 9 !&!* +!* . // *.* . | 1995. | ;. 58, = 4. 18] C * . . !. )! ) .!% " )!.% A, !9H !%. | .: '+, 1989. 19] K!% 7. ?., C"!% L. ?., !% 6. . ! 3" . "
%9!!* *!* %* % *3, ", // % !* *, . | 1987. | = 8. | ?. 46{55. 20] 6*!!% . '., C"!% L. ?., !% 6. . (9 !.! . ! % *3, ", // % !* *, . | 1986. | = 10. | ?. 32{45. 21] ! " #. ., 4% . M. - )! . | ? 9.: -.-%! ?.- 9+ !! +- , 1992. 22] -%!% . '., !% 6. . J! ! %8G8 " ! !9+A " // % !* *, . | 1992. | = 6. | ?. 30{37. 23] C!) 7. N. () *! +)% )!.% A* !9H !* ) +!
! % & "* // ;, & " 9 . | 1992. | = 1. | ?. 42{48. 24] K!% 7. ?., -%!% . '., !% 6. ., C+ O. . (9 !.! .& +! " % )! % // % !* *, . | 1992. | = 6. | ?. 11{22. 25] P* (. 7%. % ! G !) *!! )! . | .: '+, 1985. 26] # . . J)A.G8" !9 , )!!A.*" )! 8+8 ,, % + ". | ). % 7-'-;-, 1995, = 0000-795. 27] P*!3 . C ) * * & // J ), * *. +. | 1971. | ;. 26, = 5. | C. 243{269. & ' ( 1995 .
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517.95
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Abstract A. O. Abduvaliev, Asymptotic expansions of the Darboux problem solutions for the singularly perturbed hyperbolic equations, Fundamentalnaya i prikladnaya matematika 1(1995), 863{869.
The asymptotic expansions of the Darboux problem solutions for the singularly perturbed hyperbolic equation are constructed. The error estimate for this representations is obtained.
1{3]. $ : 2u 2u @ @ 2 (1) " @x2 ; @y2 + "a(x y) @u @x + k(x y)u = f(x y) (x y) 2 D ujy=0 = (x) 0 6 x 6 1 ujy=;x = '(x) 0 6 x 6 12 (2) " > 0 | , D | , . y = 0 y = ;x y = x ; 1. / 4],
(0) = '(0) 122 3 , 23 (1), (2) . 4 . . I. a(x y) k(x y) f(x y) 2 C 1(D), (x) 2 C 1 0 1], '(x) 2 C 1 0 12 , D | D. II. a(x y) > a0 > 0, k(x y) > k0 > 0 (x y) 2 D, (0) = '(0). 1995, 1, - 4, 863{869. c 1995 , !" \ "
864
. .
7 u = u(x y ") (1), (2) 1 X u = "k u8k (x y) + 9k x y" + Qk y +" x y + Rk x" y" (3) k=0 u8k | , 9k , Qk Rk | 23 3]. 4 (3) (1){(2) 3], . 122 3 1 . 7 122 3 . u0 k(x y)8u0 = f(x y) k(x y)8u1 = ;a(x y) @8 @x k;1 ; @ 2 u8k;2 + @ 2 u8k;2 k > 2 k(x y)8uk = ;a(x y) @8u@x @x2 @y2 23 9k Qk | . : 2 ; @@y92k + k(x 0)9k = pk (x ) = y" p0 0 (4) u8k (x 0) + 9k (x 0) = 0k (x) 0 6 x 6 1 9k (x ) ! 0 ! ;1 y+x k a(;y y) @Q @ + k(;y y)Qk = qk ( y) = " q0 0 (5) u8k (;y y)+Qk (0 y)=0k "(;y) ; 12 6 y 6 0 Qk ( y) ! 0 ! +1 pk qk . 90 : : : 9k;1 Q0 : : : Qk;1 , 0k | ; . < , 23 9k Qk . : k(;y y) hp i 9k (x ) = k (x ) exp k(x 0) Qk ( y) = k ( y) exp ; a( ;y y) (6) k (x ) k ( y) | . / (6) . : N N y i h p X X k " 9k x " y=;x = "k k () exp ; k(0 0) + O("N +1 ) (7) k=0 k=0 N @9 y 1 @9 y X "k @xk x " ; " @k x " = y=;x k=0 (8) N i h p i h 0 X p N k ; 1 = " k () ; k(0 0) k () exp ; k(0 0) + O(" ) k=0
865
N > 0 | . 3 , = x" , k (), k = 0 1 : : : N | . > 23 Rk ( ), k = 0 1 : : :, ? = f( ): < 0 + > 0g @ 2 Rk ; @ 2 Rk + a(0 0) @Rk + k(0 0)R = k @ 2 @2 @ k (9) X @R k ; m 8 = (1 ; 0k )
8m ( ) @ + m ( )Rk;m rk ( ) m=1 8 8 m ( ) m ( ) | m, 0k | ; . A 23 , y = ;x y = 0 D 23 9k Qk . 41 (6) (7) k(0 0) (10) Rk ( 0) = ; k ( 0) exp ; a(0 0) k () 0 < < +1 Rk ( ;) = ; k () exp ;k(0 0)] 'k () 0 < < +1: (11) / 5], (9){(11) ( ?) . : Rk( ) = exp a2 k (+) + i h i h + exp ; a4 (+) 'k ;2 ; exp ; a4 ( ; ) 'k + 2 + a Z+ a hp i + exp ; 2 exp 2 0 k (0 )I81 (0 ; ; )(0 ; +) d0 ; 0 i Z; a 0 hp (+) a ; 2 exp ; 2 exp 4 0 'k 2 I81 ( ; ; 0 )(+) d0 + 0 + h Z i p exp a4 0 'k 20 I81 (+ ; 0 )( ; ) d0 + + (2; ) exp ; a2 0 Z+ + 14 exp ; a2 d0 0 Z; h i ha i 0 +0 0 ; 0 p I0 (0 ; ; )(0 ; +) exp 4 (0 +0 ) rk 2 2 d0 ; 0
866
. .
a Z+ 1 ; 4 exp ; 2 d0 0 Z+ h i i h p 0 0 ; 0 I0 (0 ; ; )(0 ; +) exp a4 (0 +0) rk 0 + d0 2 2 0 (12) 2 I a 1
a = a(0 0), k = k(0 0), = k ; 4 , I81 (z) = z(z) , p I1 (z) | 23 B, > 0 > 0. $ ( 8 p ) > k a a2 > 0 k > min = k ; < a 4 2 4 ( (13) => r 2 ) p a a > k a k > : min a 4 2 2 ; 4 ; k < 0: > , r0 0 k > 1 23 rk
R1 : : : Rk;1 (. (9)), 3 @R k jRk ( )j + @ ( ) 6 k ( )e; ( ) 2 ? ( ) | . < Un . (3): N y x y X y + x k UN (x y ") = " u8k (x y) + 9k x " + Qk x " + Rk " " : (14) k=0
.
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6
ju(x y ") ; UN (x y ")j 6 C"N +1 (x y) 2 D: (15) . 4 w = u ; UN +2 . 4 (1), (2) u = UN +2 +w
122 3 (14), w . . : 2 2 "2 @@xw2 ; @@yw2 = ;"a(x y) @w @x ; k(x y)w + H(x y ") (x y) 2 D (16) (17) w(x 0) = 0 0 6 x 6 1 w(x ;x) = h(x ") 0 6 x 6 21
; H(x y ") = O "N +3 (x y) 2 D ; ; N +2 h(x ") = O "N +3 0 6 x 6 21 @h 0 6 x 6 12 : @x (x ") = O " $ 23 . ( 1 ): # ZM " @w @w 2 2 2 C(t ") = " @x + @y + k(x y)w ds 0 < t 6 12 P
867
(18)
(19)
1- PM, . P (t ;t) M(2t 0) (x y), 1 t 2 0 2 . < , (19) . : 2 @w 2 p Z0 2 @w C(t ") = 2 " @x (y + 2t y) + @y (y + 2t y) + ;t p Z2t 2 @w 2 + k(y + 2t y)(w(y + 2t y)) dy + 2 2 " @x (x x ; 2t) @w @y (x x ; 2t) dx: t
22 3 t, (17) : M d C = Z 4 @w "2 @ 2 w ; @ 2 w + 4 @w k(x y)w + 2k0 (x y)w2 ds + F (t ") 1 x dt @x @x2 @y2 @x P (20) ( ) 2 p 2 @w @w 2 F1(t ") = 2 " @x (t ;t) ; @y (t ;t) + k(t ;t) w(t ;t)] + ( ) (21) p 2 @w p 2 d h 2 @w 2 + 2 2" @x (2t 0) @y (2t 0) 2 " d t (t ") + k(t ;t) h(t ")] : 4 (20) . (16) 2 2 "2 @@xw2 ; @@yw2 , M d C = Z 4 @w ;"a(x y) @w ; k(x y)w + H(x y ") + dt @x @x P @w 0 2 + 4k(x y) @x w + 2k (x y)w ds + F1(t ") =
868
. .
ZM p p H(x y ") 2 + H(x y ")]2 + 2k0 (x y)w2 ds + p ; 2 " a(x y) @w + x @x "a(x y) "a(x y) P + F1(t ): <. d C 6 C + F(t ") (22) dt ZM H(x y ")]2 0 (x y)j 2 j k x = max k(x y) F(t ") = (23) "a(x y) ds + F1(t "): (xy)2D =
P
7 (22) , C(0 ") = 0, Zt C(t ") 6 exp (t ; s)]F(s ") ds: 0
<., (18), (21), (23), 3 C(t "): ; C(t ") = O "2N +5 : (24) 4 (x y) | D. / p2 (Zxy) x ; y @w x ; y @w w(x y ") = w 2 ; 2 " + 2 @x + @y ds x;y ; x;y ( 2 2 ) (17), 3 (24) ; -B : p2 (Zxy) @w @w x ; y w(x y ") ; h 2 " = 2 x;y x;y @x + @y ds 6 ( 2 ; 2 ) " (Zxy) (Zxy) p 2 #1=2 2 @w @w 6 2 ds @x + @y ds 6 x;y ; x;y x;y ; x;y ( 2 2 ) ( 2 2 ) p4 " (xZ;y0) 2 #1=2 p4 2 1 x ; y 1=2 3 @w @w 2 + @y ds 6 2 "2 C 2 " = O "N + 2 : 6 2 @x ( x;2y ; x;2y )
869
<., (18), w(x y ") = O "N + 32 (x y) 2 D: / (15), ; w = u ; UN +2 = u ; UN + O "N +1 : D .
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& ! % ! ! !
@ 2 u ; @ 2 u + "a(x y) @u + "b(x y) @u + k(x y)u = f(x y) "2 @x 2 @y2 @x @y a(x y) > b(x y) > b0 > 0, k(x y) > k0 > 0, (x y) 2 D.
1] . . ! " . | $. !. | 1977. | '. 104 (146), . 3 (11). | 0. 460{485. 2] ' . 5. 6 7 " 89"- ". | . ". | 1970. | '. 25, . 4. | 0. 123{156. 3] : 5. ., . . 5 " . | $.: ", 1990. | 208 . 4] < 5. . =" "
. | $.: =-
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% & 1995 .
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515.12
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Abstract A. V. Arhangelskii, V. V. Fedorchuk, On one-to-one mappings of countably compact spaces onto Hausdor compact spaces, Fundamentalnaya i prikladnaya matematika 1(1995), 871{880.
We deal with the following problem: is it correct that every separable, countably compact, 1rst countable space admits a one-to-one mapping onto some Hausdor3 compact space? Theorems 2 and 3 show that this problem has no positive solution in ZFC.
.
1. 1- ?
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-
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# 0 % &13] \# ". , 0. , 1996 . % 1986 . 0 500 234. ", . 5 , " 0" , , . . 0 0" . 1995, 1, 4 4, 871{880. c 1995 !", #$ \& "
872
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873
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"" D. 0 D Y , \ ", 0 5 0 &1]. L " , 0 0 5 ( " X ) ". 0 D (. &1], 0 5). M - X Y D 0" "0 Z , > - | "0 p. E, p Z . #" , " p " 0 ". ? 0 D 0 , pX : X Y ! X , q: Z ! X , " pX = qp. ? B (X ) = q;1 (X ). N " " Y 0" "0 0 1. ? Bi = p(X fig), i = 0 1. % B (X ) 0 D1, "" | B0 B1 . M - B (X ) D1 0" "0 E (X ), > - B (X ) ! E (X ) | "0 r. #" , 0 D1 . ,", r 0 . ? E (X ) 0 B (X ).
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874
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1- . % !, 4) X , E (X ) 5) X ' , E (X ) ' . . % , " E = E(X ) 0 -
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0 G0 G1, " bi 2 Gi S = G0 G1. 0 OG0 OG1 E . J E " 0 E0 = E n S . 2 , E -0 H S , OG0 OG1 . ; H \ OG0 -0 b0 E , b1. % " E b0 b1. K 2 0.
875
0 (f )(K ) L, f (b0 ), f (b1 ). ? M = f ;1 (L). ? M 0 E , 0", ": . ? b0 b1 2= M , ind M = 0. % , f j M M 0 L. % ": 1- ": 0 , , >0 . ? 0 L " M . ; 1 0. 2 1.
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877
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879
, 0 1, ", " " f (y0 ) f (y1 ), y1 = p(x0 1), " . 2 , " f (y0 ) , Z nff (y0 )g. % " , " Z nff (y0 )g ": 0 ": E1 n fy0 g. 0 2. 5 0, E . H , 0 2. %: 3. 2 ZFC, " nO (nC) , ? 4. " - 0, " nO (nC) , ( ) ? 5. 5 (ind, dim) ": 0 ?
1] . . // ". . | 1976. | '. 99, ) 1. | *. 3{33. 2] . /. 0 . | ".: ", 1986. 3] . . * // 4 . 56 ***/. | 1975. | '. 222, ) 2. | *. 302{305. 4] Ostaszewski A. On countably compact, perfectly normal spaces // J. London Math. Soc. | 1976. | V. 14. | P. 505{516. 5] Franklin S. P., Rajagopalan M. Some examples in topology // Trans. Amer. Math. Soc. | 1971. | V. 155. | P. 305{314. 6] Hajnal A., Juhasz I. A separable normal topological group need not be Lindel:of // General Topology and Appl. | 1976. | V. 6, ) 2. | P. 199{205. 7] Comfort W. W., Ross K. A. Pseudocompactness and uniform continuity in topological groups // Paci;c J. Math. | 1966. | V. 16. | P. 483{496. 8] Comfort W. W. Topological groups // Handbook of Set-Theoretic Topology. | Amsterdam: North-Holland. | P. 1143{1264. 9] Arhangel'skii A. V. On countably compact topologies on compact groups and on dyadic compacta // Topology and Appl. | 1994. | V. 57. | P. 163{181. 10] Hagler J. On the structure of S and C (S ) for S dyadic // Trans. Amer. Math. Soc. | 1975. | V. 214. | P. 415{428. 11] Gerlits J. On subspaces of dyadic compacta // Studia Sci. Math. Hungar. | 1976. | V. 11. | P. 115{120. 12] ?@ B. 5. 0 // ". . | 1977. | '. 103, ) 1. | *. 52{68.
880
. . , . .
13] Nyikos P. On ;rst countable, countably compact spaces III: The problem of obtaining separable noncompact examples // Open Problems in Topology. | Amsterdam: North Holland, 1990. | P. 127{161. ( ) 1995 .
. .
. .
. . .
.
Abstract O. N. Bulycheva,V. G. Sushko, The approximative solution constructing for some problem with nonsmooth degeneration, Fundamentalnaya i prikladnaya matematika 1(1995), 881{905.
The asymptotic solution for some singularly perturbed parabolic equation is constructing when the degenerate equation has the angle characteristic.
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884
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885
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, $ ' vm (! ) # vm (! +0) ; vm (! ;0) = um (! y0 (!) ; 0) ; um (! y0(!) + 0) = m (!) (9) @vm (! +0) ; @vm (! ;0) = @ @ (10) @u @u m ;1 (! y0 (!) ; 0) m ;1 (! y0 (!) + 0) = ; = m (!) @y @y vm (0 ) = 0: (11) 1# $ # , "'' (8) $ #
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886
. . , . .
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890
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(19)
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+ I1(! ")
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2
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.
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#
$ : @w1(! y ") = @y Z ! = ; 2"1p v1 ( ;y0 ()=") p(!) p() (
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: 2Z ! jI1(! y ")j 6 M";1 exp ; 2y ;4"y20!(!)] 0 (; 12 + ";1) (! ;1 ) 12 + y0 ()!]2 d = I + I : exp ; 2(y ; 4"y02(!)) 11 12 !(! ; )
+ ,
jI11(! ")j 6 M";1 exp ; 2y ;4"y20!(!)]
2
:
y > m0 ! ' I12 # #
# , # 1){3), # , ' I11 : y 6 m0 ! 2 1 (y ; y (!)) 0 ; 2 2 jI12(! ")j 6 M" ! exp ; 4"2! :
894
. . , . .
8 , I2 (! y "). y > m0 ! , ' w1(! y "). # # 0 6 y 6 m0 !. $
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, " ! 0 # 0 6 ! 6 (T), 0 6 y < 1
g1(! y ") = O("). 1
: Z !r 2 () (y ; y0 (!) + y0 ())2 y 0 ; 3 jg1(! y ")j 6 M" 0 ! ; exp ; 4"2 ; 4"2(! ; ) d = 2 2Z ! r (y ; y (!) + y ()!) (y ; y (!)) 0 0 0 ;3 d: = M" exp ; 4"2 ! ! ; exp ; 4"2 !(! ; ) 0 y > m0 !, m0 | , $
$ # #
895
z = (!)2"2 (! ; )];1. y 6 m0 !
$ #
2 (y ; y (!)) 0 ; 1 jg1(! y ")j 6 M" ! exp ; 4"2! = O("): 1 ,
,
' g1 (! "), $ # ' w2(! y "): 5. $ # $$ $ .
1. ) ! > 0 q1(! ")
2
jq1(! ")j 6 M! exp ; y4"0 (!) 2! 2 (!) y 0 0 ; 1 jq1(! ")j 6 M! exp ; 4"2!
> 2;1 | . * !
Z 1 Q1(! y ") = 2"p q1( ")y32 p(!) p() exp (! ; ) 0
2 ; (y ; 4"y02(!)(! ;; y)0()) d
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1 2
; (y ;4"y20!(!))
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m0! 6 y < 1. + ,
; (y ;4"y20!(!))
2 (y ; y (!)) 0 ; 4"2!
L0 Q1 = r1 (! y ")
2
896
. . , . .
r1(! y ")
jr1(! y)j 6 M"
!
2 exp ; (y ;4"y20!(!))
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2 (y ; y (!)) 0 exp ; 4"2 !
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0 6 y 6 m0!,
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m0! 6 y < 1,
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m0! 6 y < 1.
; (y ;4"y20!(!))
; (y ;4"y02!(!))
2
2
. + ' Q1(! y ")
, ' r1 (! y ") ' w1 (! y "). + , Q1(! y ") = = w1(! y "), = 1=2. C ' r1(! y ")
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897
8 $ #, ' y0 () ' z 0()
# $
2 20 (T )]. I $ 20 !] $ , $ ' z 0 () , $ Z ! 1 + p2 jz 0()j exp (;z 2 ()) d 6 M! ! 0 Z ! jy0() ; 2y00 ()1 j;1+ exp (;z2()) d 6 M! 21 + : (! ; ) 2 0 # # y > m0 !. ! "
2 @r1 exp (y ; y0 (!)) 6 @y 4"2 ! Z ! j(y ; y0(!)) + y0()! j;1+ ;3 6 M" 3 (! ; ) 2 0 2 (y ; y (!) + y ()!) 0 0 exp ; 4"2!(! ; ) d + Z ! ;1+ 2 y () (y ; y (!) + y ()!) 0 0 0 ; 3 p! ; exp ; 4"2!(! ; ) + M" d = 0 = I1 (! y ") + I2 (! y "): + ,
1 exp ; ! "2 (! ; ) d 0 !; > 0 | .
2 2 = !"+!z "2 z 2 I1(! y ") I2(! y ") :
I1 (! y ") 6 M";2!
Z !
I1 (! y ") 6 M";1 !; 12 + I2 (! y ") 6 M";1 !; 12 + :
2. ) # m1 ! > 0, y > 0 r2(! y ") 2 (y ; y (!)) 0 jr2(! y ")j 6 M" ! exp ; 4"2! 2 @ r2 (! y ") 6 M";2! 12 + exp ; (y ; y0 (!)) @y 4"2 ! ;2 21 +
898
. . , . .
0 6 y 6 m1!,
2 jr2(! y ")j 6 M! exp ; (y ;4"y02!(!)) 2 @ r2(! y ") 6 M";1 !;1+ exp ; (y ; y0 (!)) @y 4"2 ! m1! 6 y < 1, > 2;1 | . * Q2(! y ") = Z1 Z! 1 d (y ; y0 (!) + y0() ; )2 d = 2"p p r2 ( ") p(!) exp ; p() 4"2 (! ; ) !;
; 21 +
0
0
L0 Q2 = ;r2 (! y ")
jQ2(! y ")j 6 M("
;2 32 +
!
+ ! + ) exp 1 2
0 6 y 6 m1!,
jQ2(! y ")j
m1! 6 y 6 1,
6 M("! + ! 12 + ) exp
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0 6 y 6 m1!,
@ Q2 (! y ") 6 M";2 !
@y
m1! 6 y 6 1,
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1 2
2 ; (y ;4"y20!(!))
2 (y ; y (!)) 0 ; 4"2!
2 ; (y ;4"y20!(!))
2 (y ; y (!)) j y ; y (!) j 0 0 p exp ; 4"2 ! +"+ " !
2 @ ;2 ! ; 21 (1 + ";2!) 1 + y0 (!) exp Q (! y ") 6 M" @y2 2 " !
p
2
; (y4"0 (!)) 2!
:
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Z!
Z p!d; r2( ") 0 0 2 p(!) 2 ( ; y ()) (y ; y (!) + y ()! ; !) 0 0 0 exp 4"2 p() exp ; d: 4"2!(! ; )
1 exp Q2(! y ") = 2"p
1
899
# y > m1 ! = 3m0 !, m0 | , , m0 > 0max y0 (!): (T )] 0 " , ,
(T) |
#, $ 6 m0 2 20 (T)] # y ; y0 (!) + y0 ()! ; ! > m0 !: (20) M
' Q2(! y ") 2 jQ2(! y ")j exp (y ;4"y20!(!)) 6 Z ! 1 + Z m0 ! 2 2 (y ; y (!) + y ()! ; !) 0 0 ; 3 p 6 M" d + exp ; 4"2!(! ; ) 0 !; 0 Z ! ; 1 + Z 1 2 (!) + y0 ()! ; !)2 d = exp ; (y ; y04" + M";1 p 2!(! ; ) 0 ! ; m0 !
= I1 (! y ") + I2 (! y "): ? (20), I1 (! y ") $
#: Z ! 3 +
2 I1(! y ") 6 M";3 p exp ; "2 (!!; ) d 0 !; > 0 | . . # , I1 (! y ") 6 M"! : C
p ! ; y + y0 (!) ; y0 ()! = 2" !(! ; )z (21) $ $ I2 (! y ") 6 M! 21 + : # # y 6 3m0 !: ! " #
I1 (! y ") 6 M";2 ! 32 + I2 (! y ") 6 M! 21 + :
. + , @ Q (! y ") = ; 1p Z ! d Z 1 r ( ") p(!) @y 2 4"3 0 (! ; ) 32 0 2 p() 2 (y ; y (!) + y () ; ) 0 0 2y ; y0(!) + y0()! ; ] exp ; d: 4"2 (! ; )
900
. . , . .
. # , ' Q2(! y "),
(y ; y0 (!))2 6 4"2 (! ; ) Z ! 1 +
Z m0 ! 2 d ;5 6 M" jy ; y0(!) + y0() ; j 3 0 (! ; ) 2 0 2 (y ; y (!) + y ()! ; !) 0 0 exp ; d + 4"2 !(! ; ) Z ! ; 1 +
2 d Z 1 jy ; y (!) + y () ; j + M";3 0 0 3 m0 0 (! ; ) 2 + y0 ()! ; !)2 d: exp ; (y ; y04"(!) 2 !(! ; )
@ Q2(! y ") exp @y
(21)
jy ;y0(!)+y0 ();j 6 4"2jzj!;1(! ;)jz0()j+2"!; jy0(!);yj(! ;) jz0()j 3 2
$
#:
;
3 2
1 2
2 @ Q2 (! y ") exp (y y0 (!)) 6 @y 4"2(! ) Z r Z ! 1 +
;
2 d 2"2(! ; )!;1 jz j + 3 0 (! ; ) 2 q 2 + "jy ; y0 (!)j 21 (! ; ) 32 !; 23 ]e;z dz +
6 M";5
Z !
12 + d Z 1 2"2(! ; )!;1 jz j + 3 0 (! ; ) 2 r 2 + "jy ; y0 (!)j 12 (! ; ) 32 !; 23 ]e;z dz = + M";3
= I1 (! y ") + I2 (! y ")
h p
q = 2;y + y0 (!) ; y0 ()!] 2" !(! ; ) h p
i;1
i;1
r = 2m0 ! ; y + y0 (!) ; y0 ()!] 2" !(! ; )
:
8
I1 (! y ")
,
# ' . # y > 3m0 !.
901
+ , " q < 0, p 6 0: 1
: ;3 ;1+
I11(! y ") 6 M" !
Z !
0
p! ; exp 2;r2(! y ")] d 6 3 2
Z !
3 2 p 6 M" ! exp ; "2 (!!; ) d 0 !; > 0 | . ( , #
;3 ;1+
I11 (! y ") 6 M"2 !
Z
! 23 z 4 e;z2 dz 6 M! min f";1 !; 12 g: 0 (! + "2 z 2 )3 1
1 I12 # : ;4 ; 32
I12(! y ") 6 M" !
jy ; y0(!)j
Z !
0
! ; "2(! ; ) d 6 6 M! jy ;"py0!(!)j min f";1 !; 12 g:
1+ exp
1 I2 (! y ") $ : I21 (! y ") 6 M";1!;1 ;2 ; 23
I22(! y ") 6 M" !
Z !
jy ; y0(!)j
0
p! +; d 6 M";1!
Z !
1 2
0
d 6 M";2 !; 21 + jy ; y0 (!)j:
> $ , ;1
I21(! y ") + I22 (! y ") 6 M"
!
j y ; y (!) j 0 p 1+ :
" !
C 0 6 y 6 m0 ! , # I2 . ! 0 6 y 6 3m0 ! I2 (! y ") $ , I1 (! y ") $ # $ : j y ; y (!) j 0 1+
: I1 (! y ") 6 M! 1 + "p! 8 , ' Q2 (! y "): C "
902
. . , . .
: @ 2 Q2 (! 0 ") = @y2 Z ! = ; 4"31p
0 (!
1
; )
3 2
r2( 0 ")2y0(!) ; y0 ()]
2 exp ; (y04"(!)2(!; ;y0)()) d + Z ! Z 1 1 d @r2 2 + y (!) ; y ()] + 2"3p 0 0 3 0 (! ; ) 2 0 @ ; y0())2 d = exp ; ( + y4"0(!) 2 (! ; ) = I1 (! ") + I2 (! "):
( , #
jI1(! ")j exp
y02 (!) 6 M";5 !1+ : 4"2!
I2 (! ") #
' r2(! "):
y02 (!) 6 4"2! Z !
d Z p ;6 6 M" 2"2 !;1 (! ; )jz j +"y0 (!)!; 23 12 (! ; ) 32 ]e;z2 dz + 3 2 0 (! ; ) q Z ! ;1+ Z 1 d 2"2!;1 (! ; )jz j +"y (!)!; 32 12 (! ; ) 23 ]e;z2 dz= + M";4 0 3 0 (! ; ) 2 p = I21(! ") + I22 (! ")
jI2(! ")j exp
q = q(! 0 "), p = p(! 0 ")
. + $ # , " # :
I21 (! ") 6 M";4 ! 12 + 1 + y"0p(!) !
I22 (! ") 6 M" ! 1 + y"0p(!) ! : L #
2. ;2 ; 21 +
903
3. ) y ! > 0, 0 6 y 6 1 q3(! y ")
0 6 y 6 m1!, m1! 6 y < 1,
jq3(! y ")j 6 M"k!l exp ; (y ;4"y20!(!)) jq3(! ")j
6 M"r !s exp
2
2 (y ; y (!)) 0 ; 4"2!
@ q3(! y ") 6 M" ! exp @y
2 ; (y ;4"y02!(!))
@ q3(! y ") 6 M" ! exp @y
2 (y ; y (!)) 0 ; 4"2!
0 6 y 6 m1!,
m1! 6 y 6 1. * 1 Q3(! y ") = 2"p
Z !
0
p!d;
Z
1
0
n > 1
( ; y0 ())n q3( ") p(!) p()
exp ; (y ; y04"(!)2(!+ ;y0)() ; )
2
d
i n 2 jQ3(! y ")j 6 M"n!(n+1)=2 X jy ;"py0!(!)j ("k !l +"r !s ) exp ; (y ;4"y20!(!))
0 6 y 6 m1!,
jQ3(! y ")j
i=0
j ;p j j ;p j
n (y y (!) i X 0 n +1 = 2 1 = 2 k +1 = 2 l + 6 M" ! " ! "i " ! i=0 n X (y y0 (!) i exp + "r !s+n=2 "l ! i=1
m1! 6 y < 1,
; (y ;4"y20!(!))
2
Z ! d Z 1 @q3( ") ( ; y ())n p(!) p Q4(! y ") = 2"1p 0 @ p() 0 !; 0 2 (y ; y (!) + y () ; ) 0 0 d exp ; 4"2 (! ; )
904
. . , . .
n > 2 :
i n 2 jQ4(! y ")j 6 M"n!n+1=2(" ! + " ! ) X j(y ;"py!0(!)j exp ; (y ;4"y20!(!))
i=1
0 6 y 6 m1!,
i n jQ4(! y ")j 6 "n+1=2!1=2 " +1=2! X "i j(y ;"py!0(!)j + i=1 n j(y ; y (!)j i 2 X (y ; y (!)) 0 0
+ n= 2 +" ! exp ; 4"2 ! "p! i=1 m1! 6 y < 1.
C #
3
, # # #
. 6. 1 , ' #
$ $ "'' " ! ! 0 "'' . ( "
. 4. ! (1) { (2)
-
! #
1 X
1
X u(t x ") = "2k u2k (t x) + "k 2vk (t (x k=0 k=1
; x0(t))=") + wk(t x ")]
u2k(t x) | ! # , vk (t ) wk(t x ") | , " " ! 0 x = x0(t). C #
4 $ #
, " #, , 25].
1] . . . . | .: #$, 1989, 336 . 2] . . , +. ,. - . . $ /0 // . / . | 1982. | 2. 119, 3 3. | . 307{324.
905
3] 8. ,. 9$$ , . 8. # . 0 / $ 0 // :$ . . . ; . | 1982. | 2. 22, 3 4. | . 865{870. 4] 8. ,. 9$$ , . 8. # . = $0 $>
;$ ? 0 // @ . # A. | 1982. | 2. 263, 3 4. |
. 786{789. 5] 9. . 9 , 8. B. $ . $ $ 0
// :$ . . .
; . | 1991. | 2. 31, 3 9. | . 1338{1343.
( ) ) 1995 .
- - . . , . .
. . . e-mail:
[email protected]
517.956.226
: , ! \!#$-%&-
&- ".
(! ! ! %) ) !* ! !#$-%&&- +! !* %&& , ) !#$*. -! %! !+) &#$* $% !.! ! ! #! /. 0$#! ! +) + ! # +! 12 ,!1) &#$*.
Abstract
V. F. Butuzov, N. T. Levashova, On a singularly perturbed reaction-diusiontransfer system in the case of slow diusion and fast reactions, Fundamentalnaya i prikladnaya matematika 1(1995), 907{922.
A system of two singularly perturbed reaction-di7usion-transferequations is considered in the case of slow di7usion and fast reactions. By the boundary layer function method, the asymptotics of the solution is constructed with the help of a smoothing procedure. The estimate of the remainder terms of the asymptotics is obtained by using barrier functions.
x
1
, ,
x: @u + b(x) @u ; "2 a (x) @ 2 u = 1 f (u v x t ") 1 @t @x @x2 " (1) @v + b(x) @v ; "2 a (x) @ 2 v = 1 g(u v x t "): 2 @t @x @x2 " % u(x t), v(x t) | , b(x) > 0 | ( "2 ai (x) | ) ( " > 0 | (
1995, 1, 8 4, 907{922. c 1995 !", #$ \& "
908
. . , . .
)( f g , 1=" , , . . , / f g | u v: f = ;p(x t)u + q(x t)v + "f1 (x t ") g = kp(x t)u ; kq(x t)v + "f2 (x t ") p(x t) > 0 q(x t) > 0 k = const > 0: 0 ) u v
f g ) p, q, k . 1 , (1) ", : @u ; "3 a (x) @ 2u = ;p(x t)u + q(x t)v + "f (x t ") L1" 2u] " @u + "b ( x ) 1 1 @t @x @x2 (2) 2v @v @ @v 3 L2" 2v] " @t + "b(x) @x ; " a2(x) @x2 = kp(x t)u ; kq(x t)v + "f2 (x t "):
5 (2) , 6 = (0 < x < 1) (0 < t 6 T )
ujt=0 = '(x)( vjt=0 = (x) / @u = 0( @v = 0: @x x=0 @x x=0 x=1
(3) (4)
x=1
9/ / / ( , , / , ' (x) x=0 6= 0, (x) x=0 6= 0. 0
0
x=1
x=1
1 : (2){(4) 6 = (0 6 x 6 1) (0 6 t 6 T ) " "2
: ) / ) (2) '(x) (x) (3)( ,) a1 (0) = a2 (0). < ,) , : "3=2 . 9 (1) , 21, 2]. = , 21]. . 21] / (2), . . , / / / / . > /, ,
909
21], , , , : , , , 21]. . x 2 : , x 3 . x
2
? : , / / 23]. ) @ . 1 / (2) u = u0(x t) + "u1(x t) + : : :, v = v 0(x t) + "v 1 (x t) + : : :. 0 ) "0 , / / u0 (x t) v0 (x t):
;p(x t)u0 + q(x t)v0 = 0 kp(x t)u0 ; kq(x t)v0 = 0: A / , , : : u0 = q(x t) 0(x t) v0 = p(x t) 0(x t) (5) / 0(x t) | . C , , (2)
| : . 1 u1, v1 : @u0 + b(x) @u0 = ;p(x t)u + q(x t)v + f (x t 0) 1 1 1 @t @x (6) @v 0 + b(x) @v 0 = kp(x t)u ; kq(x t)v + f (x t 0): 1 1 2 @t @x E : ) @u0 @v0 @v 0 0 k @u @t + kb(x) @x + @t + b(x) @x = kf1(x t 0) + f2 (x t 0): 0
u0 v 0 (5), 0(x t): @ @ @ ( x t ) @ ( x t ) 0 0 (x t) @t + b(x) (x t) @x + 0(x t) @t + b(x) @x = h0 (x t) (7) / (x t) = kq(x t) + p(x t) > 0 (8) h0 (x t) = kf1 (x t 0) + f2 (x t 0):
910
. . , . .
.
z0 (x t) = (x t) 0(x t): (9) 9 e (7) @z0 + b(x) @z0 = h (x t): (10) @t @x 0 Z x G t = bds (s) B (x) (10), 0 (0 0), , 6 : t 6 B (x) t > B (x). 1 z0 (x t), , 0(x t), ) , , x = 0 t = 0. = t = 0 // / . 1 / H0 u(x ), H0 v(x ) | / / | / ( = t=") @ H0 u = ;p(x 0)H u + q(x 0)H v @ H0 v = kp(x 0)H u ; kq(x 0)H v (11) 0 0 0 0 @ @ H0 u(x 0) + u0 (x 0) = '(x) H0v(x 0) + v 0(x 0) = (x): (12) > /, , , , H- ! 1: H0u(x 1) = 0 H0 v(x 1) = 0: (13) I , ) (11) , ; (x 0) < 0, / (x t) (8). 0) , : (11), (13), H0 u = C0(x) exp(; (x 0) ) H0v = ;C0 (x)k exp(; (x 0) ) (14) C0(x) | . 0
(14) (5) (12), / , C0(x) 0(x 0). A : ; q(x 0) (x) : (15) 0(x 0) = k'( x()x+0) (x) C0(x) = p(x 0)'(x )(x 0) C , , H0 u, H0v ) , ! 1 (15) 0(x t).
911
1 z0(x t) 0(x t) , t > B (x) , / x = 0. I // x = 0. 0/ , ) /, ". 1 Q1u( t), Q1 v( t) ( = x=") : b(0) @Q@1 u = ;p(0 t)Q1u + q(0 t)Q1v b(0) @Q@1 v = kp(0 t)Q1u ; kq(0 t)Q1v (16) / @u0 + @Q1u = 0 @v 0 @Q1 v (17) @x x=0 @ =0 @x x=0 + @ =0 = 0: > /, , , , Q- ! 1:
Q1u(1 t) = 0 Q1 v(1 t) = 0: (18) J (16) , : , t) < 0, / (x t) (8). ; b(0(0) I, : (16), (18), (0 t) Q v = ;kD (t) exp ; (0 t) Q1u = D1 (t) exp ; b(0) (19) 1 1 b(0) D1 (t) | . 0
(19) (5) (17), , @ 0 (20) (0 t) @ @x (0 t) + 0(0 t) @x (0 t) = 0 / 0(x t) x = 0. 0 : (15) (20), (9) z0 (x t): z0 (x 0) = k'(x) + (x) K(x) (21) @z0 (22) @x x=0 = 0: I, z0 (x t) , t > B (x) z0I (x t), , t 6 B (x) | z0II (x t). = z0I (x t). 9 z0I (0 t). 1 )/ (10) x = 0: @z0I @z0I + b (0) @t x=0 @x x=0 = h0(0 t):
912
. . , . .
. : (22), : )
@z0I @t x=0 = h0 (0 t): 0 (21) x = 0, )/ : z0I (0 0) = K(0): L / , Zt
z0I (0 t) = h0(0 ) d + K(0) M(t): 0
: (10) ) / , z0 (x t) , t > B (x):
z (x t) = M(t ; B (x)) + I 0
Zt
h0 B 1 ( + B (x) ; t) d : ;
;
(23)
t;B (x)
C 0(x t) , t > B (x): I0(x t) = z0I (x t) 1(x t). . , t 6 B (x) z0 (x t) (10) (21). : ) (9), ;
Zt
z (x t) = K B (B (x) ; t) + h0 B 1 ( + B (x) ; t) d II 0
;
;
1
;
0
;
(24)
II0 (x t) = z0II (x t) 1(x t): L , 0(x t), , u0(x t) v0 (x t) (. (5)), , 6. I , z0 (x t), 0(x t), u0(x t), v 0(x t) 6, / t = B (x). = ) z0 , 0 , u0, v0 ( ), , /
) . 0 / . ). 1 D1 (t) t) D1 (t) = b (0)M( 3 (0 t) 2p(0 t)qx(0 t) ; q(0 t)px(0 t)]: C Q1u Q1v . ,) ? / . 0 / H- / , / . ;
913
L (6)
u1 (x t) = u^1(x t) + q(x t) 1(x t) v 1(x t) = v^(x t) + p(x t) 1(x t) / u^1, v^1 | ( : (6))( 1(x t) | . E : u2, v 2 1 (x t) / , (7). . z1 (x t) = (x t) 1(x t), ) @z1 + b(x) @z1 = h (x t) @t @x 1 / h1(x t) | , t = B (x). 0 / H1 u, H1v,
: (11), , z1 (x 0) z1 (x t) = z1II (x t) , t 6 B (x), / Q2u, Q2 v, / , ) , z0 (0 t), / z1 (0 t)
z1 (x t) = z1I (x t) , t > B (x). 0 )/ z1 (x t) , 6, , 1(x t) u1(x t), v 1(x t) ) ,. A
t = B (x). O H1 u, H1v ) ! 1: jH1u(x )j 6 c exp(;{ ), jH1 v(x )j 6 c exp(;{ ), Q2 u( t), Q2v( t) ) ! 1. O "2 Q2u, "2 Q2v, "2 u2 , "2 v2 "2 ) : O("2 ) , , / . 0 ) u2 , v 2 , : , ) . ) 9/ / / . 0 / i(x t) (i = 0 1). 9 ) / ziI , t < B (x) ziII , t > B (x). 0 / , h0(x t) , t < 0, (23) z0I (x t) t < B (x). ?/, / , K(x) = k'(x) + (x) h0(x t) , x < 0, (24) z0II t > B (x). C z0I z0II , , I0(x t) II0 (x t) , , 6. C / , 6 I1(x t) II1 (x t). . / = t ; B" (x)
914
. . , . .
ei(x t ) = Ii (x t)!( ) + IIi (x t)!(; ) (i = 0 1) Z / !( ) = p1 exp(;s2 ) ds. O ei(x t ) / 6 " ! 0 / i(x t) / ,
t = B (x). = , t = B (x) i (x t) ", , ue = ue0 + "ue1 ( ev = ve0 + "ev1 ( i(x t) ei (x t )) (2) (3), (4) O("2 ). = , (2) (3), (4) / / , L1"2ue] + p(x t)ue ; q(x t)ev ; "f1 (x t ") = 2A (t)F (t)( 2 ; 1) exp(; 2 ) + O("3 ) = "2 p 1 q (25) L2"2ev] ; kp(x t)ue + kq(x t)ev ; "f2 (x t ") = 2A (t)F (t)( 2 ; 1) exp(; 2 ) + O("3 ) = "2 p 2 p ; ; 1 1 (t) t a q B i B (t) / A = b(0)K (0), i (t) = b2 (B 1 (t)) (i = 1 2)( Fq (t) = (B 1 (t) t) , Fp (t) = ; 1 (t) t p B = (B 1 (t) t) ( t = 0, x > 0 (t = 0, 6 0): ue(x 0) + H0 u(x 0) + "H1 u(x 0) ; '(x) = "AFq (0)!( ) + O("2 ) (26) v(x 0) + H0 v(x 0) + "H1 v(x 0) ; (x) = "AFp (0)!( ) + O("2 )( e x = 0, t > 0 (t = " , > 0): @ ue + @ ue @ @ 2 @x x=0 @ @x x=0 + @ ("Q1 u + " Q2u) =0 = = K (0)Fq (0)(!(; )) + O(")h( ) (27) @ ve @ @ @ ve 2 @x + @ @x + @ ("Q1 v + " Q2 v) = ;1
;
;
;
;
0
;
;
0
0
x=0
x=0
= K (0)Fp (0)(!(; )) + O(")h( ) / h( ) = O((!(; )) + 2! ( )) = O(exp(;{ t=")). 0
0
0
0
=0
915
1 (25){(27) "S1 u( t) + "2 S2 u( t), "S1 v( t) + "2 S2 v( t) (S0 u( t) 0, S0 v( t) 0). O S1 u, S1 v / , ; ; ;p B 1 (t) t S1 u( t) + q B 1 (t) t S1 v( t) = 0 ; ; kp B 1 (t) t S1 u( t) ; kq B 1 (t) t S1 v( t) = 0 : : ; ; S1 u( t) = q B 1 (t) t 1 ( t) S1 v( t) = p B 1 (t) t 1 ( t) (28) / 1 ( t) | . L : S2 u, S2 v , 1 ( t). < ; Y ( t) = B 1 (t) t 1 ( t) (29) ) @Y ; (t) @ 2 Y = ; p2A (t)( 2 ; 1) exp(; 2 ) (30) @t @ 2 / (t) = kFq (t)1 (t) + Fp (t)2 (t). E S1 u, S1 v, (26), (27) Y ( t): Y ( 0)j 60 = ;A!( ) (31) @Y = A ( ! ( ; )) : @ t=" >0 0 ,, t = " , > 0. % / t = 0, > 0: @Y ( 0) = A(!(; )) : @ >0 0 / , , Y (0 0) = 0 (31). C/ : Y ( 0)j >0 = A!(; ): (32) I Y ( t) : (30) (31), (32). : ) : ;
;
;
;
;
;
;
0
0
Y ( t) = ; pA
Z
Z 2 p exp(;s ) ds + pA t exp ; 4t / t = (x) ds:
t
2
0
0
0
p
2 t0
0
916
. . , . .
= jY ( t)j 6 c exp(;{ 2 ), (28) (29), S1 u( t), S1 v( t). /) E/ / . 1 , "Q1 u( t), "Q1 v( t) t = 0 H0 u(x ) H0 v(x ) / x = 0, / / "P1u( ), "P1 v( ) (P0u = P0 v 0). 1 : @P1 u + b(0) @P1 u = ;p(0 0)P u + q(0 0)P v 1 1 @ @ @P1 v + b(0) @P1 v = kp(0 0)P u ; kq(0 0)P v > 0 > 0 1 1 @ @
P1u( 0) = ;Q1 u( 0) P1v( 0) = ;Q1 v( 0) @P1 u (0 ) = ; @ H0 u (0 ) @P1v (0 ) = ; @ H0 v (0 ): @ @x @ @x . P1( ) = P1 v( ) + kP1u( ). E (14) (19), P1 ( ) @P1 + b(0) @P1 = 0 P ( 0) = 0 @P1 (0 ) = 0 1 @ @ @ P1( ) 0 P1v( ) = ;kP1u( ): (33) 1 P1v @P1 v + b(0) @P1 v = ; (0 0)P v > 0 > 0 1 @ @ H0 v (0 ): P1 v( 0) = ;Q1 v( 0) @P@1 v (0 ) = ; @ @x A : / z0 (x t), : 8 I >
> b(0) > < P1 v ( ) P1v( ) = > > :P II v ( ) 6 1 b(0) / P1I v( ) = = k D1 (0)+ C0 ( ; b(0) )+ 12 b 1 (0)C0(0) x (0 0)( ; b(0) )2 exp(; (0 0) ) (0 0) II P1 v( ) = kD1 (0) exp ; b(0) : 0
;
917
I , P1v P1 u = ;kP1v ) , ! 1, ! 1. % , P1v P1u / , ( ) = =b(0). 0 / P1 v P1 u / , ) , / . . Pe1v( ) = P1Iv( )!() + P1IIv( )!(;) Pe1u( ) = ;kPe1v( ) /
x
; b(0) t ; b(0) = p" = "3=2 | Z exp(;s2 ) ds. / , !() = p1 O Pe1u, Pe1v / , , ue, ev, (2) (3){(4). .: Pe1 v: L2" 2"Pe1v] ; "kp(0 0)Pe1u + "kq(0 0)Pe1v = (34) = "3=2 p2 M (2 ; 1) exp(;2 ; (0 0) ) + O("2 )g( ) (0) , M = kb(0) 2q(0 0) (0) ; p(0 0)' (0)], / = ab22(0) (0 0) { x { t 3 2 g( ) = O( exp(; ; (0 0) )) = O exp ; " ; " ( (35) t = 0, x > 0 ( = 0, 6 0): "2Pe1 v( 0) + Q1 v( 0)] = "3=2 M!() + O("2 )( (36) p x = 0, t > 0 ( = ", > 0): @ Pe1 v + @ Pe1 v @ + @ H0 v M (!(;)) + O "1=2 h() (37) = @ =0 @ @ =0 @x x=0 b(0) ;1
0
0
0
/ h() = O(!(;)) = O exp ; "{3=t2 . 1 / (34){(37) "3=2T3=2 v( ). 1 a1 (0) = a2(0) , / (30){(32): @T3=2v ; @ 2 T3=2 v = ; (0 0)T v ; p2 M (2 ; 1) exp(;2 ; (0 0) ) 3=2 @ @2 T3=2v( 0) 60 = ;M!() T3=2 v( 0) >0 = M!(;):
0
918
. . , . .
< : : Z 2 p T3=2 v( ) = pM exp ; 4 ;
0
0
exp(;s2 ) ds exp(; (0 0) )
p
2 0
/ = . E (33), : T3=2u( ) = ; k1 T3=2 v( ). J , T3=2v( ) T3=2v( ) 6 c exp(;{( + jj)) T3=2 u. ) 0/ x = 1. / H- / x = 1. 1 ) "2 Q2 u( t) "2 Q2 v( t), / / = 1 ";2 x . I @ 2 (i = 1 2), / , / ;b(1) @@ ;ai (1) @ 2 ) , ! 1, , "2 . I ,
: / . 0
x
3
. , :
U (x t ") =
1 X i=0
2"i uei (x t) + "i Hi u(x )] +
+ "Q1 u( t) + "Pe1 u( ) + "S1 u( t) + "3=2 T3=2u( )
V (x t ") =
1 X i=0
2"i evi (x t) + "i Hi v(x )] +
+ "Q1 v( t) + "Pe1 v( ) + "S1 v( t) + "3=2T3=2v( ): . U , V 6 = (0 6 x 6 1) (0 6 t 6 T ) u(x t "), v(x t ") (2){(4) O("2 ), max ju(x t ") ; U (x t ")j = O("2 ) max jv(x t ") ; V (x t ")j = O("2 ):
919
. . Ue (x t ") = U (x t ") + "2 ue2(x t) + "2 Q2 u( t) + "2 Q2 u( t) Ve (x t ") = V (x t ") + "2 ev2 (x t) + "2 Q2v( t) + "2 Q2v( t) / ue2 ev2 / u2 v2 ( )/
u2 v2 i ei). 0 u ; Ue = r1 , v ; Ve = r2. C Ue ; U = O("2 ), Ve ; V = O("2 ), , r1 = O("2 ) r2 = O("2 ) (38) 6. 1 r1(x t), r2(x t) L1"2r1]+p(x t)r1;q(x t)r2 = h1(x t ") L2"2r2];kp(x t)r1;q(x t)r2 = h2(x t ") @ + "b(x) @ ; "3 a (x) @ 2 , i = 1 2. / Li" = " @t i @x @x2 E ) / (35), hi : { t 2 hi(x t ") = O " exp ; " + O("3 ): = / ri (x t) (i = 1 2) : @ri i (0) (1) ri(x 0) = i (x ") @r @x x=0 = i (t ") @x x=1 = i (t "):
I , i = O("2 ), i(1) = O("2 ), i(0) (27) (37)
i(0) (t ") = O "1=2 exp ; "{3=t2 + " exp ; {"t : 1 (38) , : { t { x { t 2 W1 = " exp(Mt)ch(Nx ; d)q(x t) ;A1 exp ; " + A2 exp ; " ; " ; ; A3 exp ; "{3=t2 + A4 exp ; "{3=x2 ; "{3=t2 + A5 W2 = "2 exp(Mt)ch(Nx ; d)p(x t) ;A1 exp ; {"t + A2 exp ; {"x ; {"t ; { t { x { t ; A3 exp ; "3=2 + A4 exp ; "3=2 ; "3=2 + A5
920
. . , . .
/ M , N , d, A1 , A2 , A3, A4 , A5 | , ", A5 > A1 + A3 A1 > A2 + A4 A3 > A4 M > N: (39) 0
(39) W1 , W2 , : I. Wi > 0, i = 1 2.
II.
W1jt=0 = "2 ch(Nx ; d)q(x 0)
;A1 + A2 exp ; {"x ; A3 + A4 exp ; "{3=x2 + A5 > j1(x ")j W2jt=0 = "2 ch(Nx ; d)p(x 0)
;A1 + A2 exp ; {"x ; A3 + A4 exp ; "{3=x2 + A5 > j2(x ")j ,: A5 .
III. 1 ; @W @x
x=0
= ;"2 exp(Mt)(Nsh(;d)q(0 t) + ch(;d)qx (0 t))
;A1 exp ; {"t + A2 exp ; {"t ; A3 exp ; "{3=t2 + + A4 exp ; "{3=t2 + A5 + "1=2 exp(Mt)ch(;d)q(0 t){ A4 exp ; "{3=t2 + { t + " exp(Mt)ch(;d)q(0 t){ A2 exp ; " > j1(0)(t ")j( 2 (0) /, ; @W @x x=0 > j2 (t ")j ,: N , A2, A4 A5 . IV. @W1 2 @x x=1 = " exp(Mt)(Nsh(N ; d)q(1 t) + ch(N ; d)qx(1 t)) { t { { t { t ;A1 exp ; " + A2 exp ; " ; " ; A3 exp ; "3=2 + + A4 exp ; "3{=2 ; "{3=t2 + A5 ; ; "1=2 exp(Mt)ch(N ; d)q(1 t){ A4 exp ; "3{=2 ; "{3=t2 ; { { t ; " exp(Mt)ch(N ; d)q(1 t){ A2 exp ; " ; " > j1(1) (t ")j(
2 /, @W @x
V.
x=1
921
> j2(1) (t ")j ,: N A5.
L1"2W1] + pW1 ; qW2 = L1" 2W1] = = "3=2 exp(Mt)ch(Nx ; d)q(x t){
A3 exp ; {3=t2 ; A4 exp ; {3=x2 ; {3=t2 ;
"
"
"
; b(x)A4 exp ; "{3=x2 ; "{3=t2 + "2 exp(Mt)ch(Nx ; d)q(x t){ A1 exp ; {"t ; A2 exp ; {"x ; {"t ; b(x)A2 exp ; {"x ; {"t ; { t { x ; a1(x)A4 { exp ; "3=2 ; "3=2 + + "3 exp(Mt)2(Mq(x t) + qt(x t)) ch(Nx ; d) +
+ b(x)(Nsh(Nx ; d)q(x t) + ch(Nx ; d)qx(x t))]
;A1 exp ; {"t + A2 exp ; {"x ; {"t ;
; A3 exp ; {3=x2 + A4 exp ; {3=x2 ; {3=t2 + A5 ;
"
"
" { t { x 2 ; a1(x)ch(Nx ; d)q(x t){ A2 exp ; " ; " + O("7=2) = = H1(x t ") > jh1(x t ")j(
/, L2"2W2 ] ; kpW1 + kqW2 = H2(x t ") > jh2(x t ")j ,: M , A1 , A3 A5 . 1 , I{V jri j < Wi 6. L II , t = 0, t=0 jrij < Wi : (40) 0 t0 | : t, : (40), : x0 2 (0 1). 0 , , r1(x0 t0) = W1 (x0 t0 )( jr2(x0 t0)j 6 W2 (x0 t0). C/ (W1 ; r1) (x0 t0)
@ (W ; r ) = 0( @ 2 (W ; r ) > 0( @ (W ; r ) 6 0: @x 1 1 @x2 1 1 @t 1 1
(41)
922
. . , . .
9 / , V (x0 t0)
L1"2W1 ; r1 ] + p(W1 ; r1) ; q(W2 ; r2) = H1(x t ") ; h1(x t ") > 0: = )/ (41) (x0 t0). 0 , , r1 < W1 : , (0 < x < 1( 0 6 t 6 T ). ?/ , : (40) (. . r1 > ;W1 , ;W2 < r2 < W2 ) ) ,. 0 , r1 < W1 : / x = 0, t = t0 . 0 r1 (0 t0) = W1 (0 t0). C/ , III, (0 t0) @ (;W ) > @r1 , @ (r ; W ) > 0. %, (r ; W ) @x 1 1 1 @x @x 1 1 x (0 t0) ) . 0) t = t0 x > 0 r1 > W1 , : . ?/ r1 < W1 / = 1. C , , r1 < W1 , 6. ?/ (40) , 6. C Wi = O("2 ), ri (38). C . . < (3) / (4) / / (0 0), . . ' (0) = 0, (0) = 0, i(x t), , ui (x t), vi (x t) (i = 0 1), P1 u( ) P1v( ) , / 6, S1 u( t), S1 v( t), T3=2u( ), T3=2v( ) . . ) : (2){(4) 0
u(x t ") = v(x t ") =
1 X i=0
0
2"i ui (x t) + "i Hi u(x )] + "Q1 u( t) + "P1 u( ) + O("2 )
1 X i=0
2"iv i (x t) + "i Hi v(x )] + "Q1 v( t) + "P1 v( ) + O("2 ):
"
1] . . // . . | 1993. | . 29, 5. | . 833{845. 2] . ., #$% . . // &' ()$'. %%. %%. . | 1994. | . 34, 10. | . 1380{1400. 3] $'+ ,. ., . . ,$%-)$. %/( $0'(1 %23. | 4.: ($6 6.', 1990.
( ) 1995 .
. .
: , -
.
, ( ) . # $$ - . % & {( . X
S
C X S
X
S
Abstract V. I. Varankina, Maximal ideals in semirings of continuous functions, Fundamentalnaya i prikladnaya matematika 1(1995), 923{937.
The author investigates the conditions on a topological space and a topological semiring of the existence of a canonical homeomorphism between the maximal ideal space of the semiring ( ) and the Tikhonov product of and the maximal ideal space of the semiring . The generalized zero sets and the properties of continuous function semirings are considered. An analog of Gelfand{Kolmogorov theorem is obtained for maximal ideals of semirings of continuous functions with values in topological semiskew/elds. X
S
C X S
X
S
| , | .
! . ". # $. %. & '1] . & '2]. + ,
Y ! { Y Y . + 24 '2] Y Q- T ,/ (y m), y 2 Y m | T . 2 # {& 1995, 1, 0 4, 923{937. c 1995 !, "# \% "
924
. .
, '3]. 5. ". + '4] ! !, # { & . 6
! ! '5]. 6 8. . & '6]. :
! , # : '7]. + / X S , , C (X S ) X S . < / ! !,
- . : '4] . 2
! .
hT + 0 1i, hT + 0i | , hT 1i | , / ! , 0 6= 1
0 x = x 0 = 0. 2 , /, , @ !, . 2 T , /! a 2 T @ a + 1 !. 5 T , , T
. A T , T | ! @ T n f0g . : T U (T ) ! ! ! @ T . : Y T ! C (Y T ) T - ,
Y ,
. " Max T T , , T . , S | , /, : S | ,
S
, U (S ) ! @ . A X S -
, /! A X x 2 X n A f 2 C (X S ), f (A) = f0g f (x) = 1, g 2 C (X S ), g(A) = f1g g(x) = 0. 2 C (X S ) ! ! C . % , T1 - Y , , !, , - .
925
C , /,
S . 1. + , 0 . + : (a) hR+ + i | ! . (b) hR_ _ i | , @ a b a _ b = max(a b), ! . 2. : ,
. 3. A ! D , 1 . 4. A C (Y R+), Y | , - . + U (C (Y R+)) /! ! Y . E
C (Y R+) - , Y .
1
2 T T ! Rad T . 2 T , Rad T = 0: I T , /! a b 2 T ab 2 I a 2 I b 2 I .
1.1. m -
T
a + b 2 m () a 2 m b 2 m:
. 2 , /!
m a + b 2 m a 62 m. A m + T aT = T , , P c + ni=1 si ati = 1 si ti T (i = 1 : : : nP) n c 2 m. 2! ! P
@ i=1 si bti, : @ c + ni=1 si (a + b)ti = 1 + bt ! m, .
1.2. m T , , .
926
. .
. 2 , m . A ab 2 m, a 62 m b 62 m @ a b T . A m , T a + m = T bT + m = T , , t1 t2 T , m1 m2 m, t1 a + m1 = 1 bt2 + m2 = 1. 2 @ . : t1 abt2 + m1 bt2 + t1am2 + m1 m2 = 1: 2 m, 1 2 m. 2 . 1.3. " T # a
,
t $ T # a + t
.
. 6! s = a + t. A s = (1 + ta;1)a 2 U (S). 1.4. U (C ) = C (X U (S)) , , C (X S) .
. . :
@ x 2 X m S ! Mxm = ff 2 C (X S ) j f (x) 2 mg: 6
, Mxm | . 1.1. x 2 X m S Mxm .
. 2 x0 2 X m | S. 2 , Mx0 m . A M C (X S ), Mx0 m M Mx0 m 6= M: F , f (x0 ) = c 62 m f 2 M . A m , cS + m = S , , , / @ s 2 S a 2 m, cs + a = 1. + / g = fs + a 2 C (X S ). 2 f 2 M a 2 Mx0 m M , g 2 M . " U = U (S ) ! S @ g(x0) = 1 2 U , @ ! V = g;1 (U ) ! g X x0 . A X S - , h 2 C (X S ), h(X n V ) = f1g h(x0 ) = 0:
927
6
, h 2 Mx0 m M . A ' = g + h M . 5 x 2 X n V , '(x) = 1 + h(x) | ! @ . 5 x 2 V , g(x) 2 U (S ), @ '(x) = g(x) + h(x) ! 1.3. A ! , ' 2 M !. 2
D
. 1.5. Rad C = C (X Rad S). %
, S ,
C .
. 1.1 , Rad C C (X Rad S). : , . 2 , @ . A
f 2 C (X Rad S ) n Rad C , , , C , M , f . : @ f M M + CfC = C @ g 2 M ti gi (i = 1 : : : n) C , g+
Xn tifhi = 1: i=1
2 x | X . A
Xn ti(x)f (x)hi(x) = 1: i=1 2 f (x) 2 Rad S , Pni=1 ti (x)f (x)hi (x) 2 Rad S . g(x) +
6
, /! x 2 X @ g(x) m S ( m ! 1). F , g(x) 2 U (S ) x 2 X , g 2 U (C ) 1.4. , g ! M | . : T Max T , / D(I ) = fN 2 Max T j I 6 N g I | T . F D : \ D(I ) D(J ) = D(IJ )
D(I ) = DX I
928
. .
2 a 2 T . "
D(I ) = ? () I Rad T D(I ) = Max T () I = T:
D(a) = fN 2 Max T j a 62 N g
!/ / ! Max T , D(I ) = D(a): "
a2I
Z(a) = fN 2 Max T j a 2 N g
!/ / ! @ . F / / Max T ,/ , A Max T
\
A = M 2 Max T
N 2A
N M :
1.6. Max T T1 - .
. 2 T , Max T , Max T .
2 -
. & - - '
f 2 C (X S ) Z(f ) = f(x m) 2 X Max S j f (x) 2 mg:
-
2.1. Z(f ) = ? () f 2 U (C ):
2.2. Z(f ) = X Max S () f 2 Rad C:
2.3. Z(f + g) = Z(f ) \ Z(g):
2.4. Z(fg) = Z(f ) Z(g):
: @ 1.4, 1.5, 1.1 1.2. . 6
I C z- , f 2 I , g 2 C Z(f ) Z(g) g 2 I .
929
2.1. z- C .
. 2 2.4 Z(gf ) = Z(fg) = Z(f ) Z(g) Z(g)
/! g 2 I f 2 C , IC I CI I . ! 2.1. ( C z- , , .
. 2 M | C . 2 , M z - . A f g, f 2 M , g 2 C n M Z(f ) Z(g). A M , gC + M = C , , t 2 C h 2 M , gt + h = 1. C 2.3 2.4, : ;
? = Z(1) = Z(gt + h) = Z(gt) \ Z(h) = Z(g) Z(t) ;\ Z(h) = ;
= Z(g) \ Z(h) Z(t) \ Z(h) : < / f + h 2 M . : Z(f + h) = Z(f ) \ Z(h) Z(g) \ Z(h) = ?: A 2.1 f +h | ! @ C . A ! , D / f + h 2 M . 2.1. ( C . 2 1.2, 2.2. ( C .
3
H ! : X Max S ;! Max C : ;
(x m) = Mxm /! x 2 X m 2 Max S 1.1. 6! Max0 C = (X Max S ) = fMxm 2 Max C j x 2 X m 2 Max S g: 3.1. & )
.
930
. .
. 5 x1 6= x2, S- X Mx1 m1 6= Mx2 m2 /! m1 m2 Max S . 2 m1 6= m2 . A S @ a 2 m1 n m2 . " Mx1 m1 Mx2 m2 /! x1 x2 X , a 2 Mx1 m1 a 62 Mx2 m2 . 3.2. Max0 C Max C .
. 2 f 2 C D(f ) 6= ?. A f 62 Rad C
1.5 f 62 C (X Rad S ). F , f (x) 62 m x 2 X m 2 Max S , . . D(f ) \ Max0C 6= ?. 8 . : A X Max S ! MA = ff 2 C j A Z(f )g:
3.3. M C '
f 2 C f 2 M () MZ(f ) M:
. 2 MZ(f ) M . 2 f 2 MZ(f ) , f 2 M . 6! , f 2 M . + g 2 MZ(f ) . A Z(f ) Z(g). A M z - , g 2 M . +/ MZ(f ) M . 3.4. ;Z(f ) Max C = Z (f ) f 2 C .
. F 3.3 :
;
Z(f )
Max C
=
\ M 2 Max C N M = ;
N 2 Z(f )
= fM 2 Max C j MZ(f ) M g = fM 2 Max C j f 2 M g = Z (f ): 3.1. & )
, X .
. 2 ! /I , X = S X | X . : x 2 X / fx , fx (x) = 1 fx (X n X ) = f0g, X |
@ , , x. 6
, fx 62 Mxm
/! m 2 Max S . 2 M | C ,
ffx gx2X . A M 6 Mxm /! (x m) 2 X Max S | /I, M = C / 2.1. A
fi = fx (i = 1 : : : nP ) @ hi 2 C (i = 1 : : : n) ni=1 hi fi = 1, /, - fi (i = 1 n)T . A /! i = 1 : : : n fSi (X n Xi ) = f0g Xi = X , ni=1 X n Xi = ?. F , X = ni=1 Xi |
X X | . x
i
xi
x
931
6! , X . 2 , C M 6= Mxm /! x 2 X m 2 Max S . H x 2 X . : /! m 2 Max S fm , fm 62 m. 6
, S ,
@ fm (x) (m 2 Max S ), S . A @ @ fi (i = 1 : : : n) s1 : : : sn 2 S , Pni=1 fi (x)si = 1. + / gx = fi si 2 M , ; gx (x) = 1. 2 x 2 X , , Vx = gx;1 U (S ) X X !/ . A X , ; gi (i = 1 : : : k), Vi = gi;1 U (S ) (i = 1 : : : k) !/ X . J g = g1 + + gk M , , /! x 2 X @ g(x) = g1 (x)+ + gk (x) ! S 1.3, g | ! @ C . 2
, /! X Mxm . 2 . 2 3.1 X Max S Max C , @ ; (x m) 2 X Max S ! (x m) = Mxm 2 Max0 C . A X Max S Max C . + ! ;1 , ! , / @ . H, Z(f ) = Max0 C \ Z (f ) /! f 2 C . 2 @ - C Max0 C X Max S . 3.2. " S , .
. 2 (x0 m0) 2 X Max S. A! ,
f 2 C , (x0 m0 ) Mx0 m0 2 D(f ),
U x0 @ a 2 S , (x0 m0) 2 U D(a) D(f ): 6! a = f (x0 ). H, /! S . : m: V =Sn m2D(a)
A V S , a V \ m = ? m 2 D(a). 2 U = f ;1 (V ). K , (x0 m0 ) 2 U D(a). : / U D(a) D(f ). + (y n) 2 U D(a), . . y 2 Y n 2 D(a). A f (y) 2 V , , f (y) \ m = ? /! m 2 D(a). 6 f (y) 62 n. H , (y n) Myn 2 D(f ). 2 .
932
. .
3.3. " X S , ;1jMax C
.
0
. : ;1 X .
2 (x0 m0 ) 2 X Max S . + ! / (x0 m0 ) . E U D(a), U | - x0 X a | @ S , m0 2 D(a). 2 / f 2 C , a x 2 U L f (x) = 0 x 2 X n U: 2 f (x0 ) = a 2= m0 , (x0 m0 ) 2 D(f ). : , D(f ) U D(a). : (x m) 2 D(f ) f (x) 62 m x 2 U , , m 2 D(a). % ;1 . 2 S . + @ ! Max S /
@ , ! (x0 m0 ) 2 X S U D(a), U | x0 , D(a) = fm0 g. A X | S - , , g 2 C , g(x0 ) = 1 g(X n U ) = f0g. : f = ag 2 C
, (x0 m0 ) 2 D(f ). : / D(f ) 2 U D(a). A D(a) = fm0g, a 2 m m 6= m0 , , f (x) = ag(x) 2 m
m 6= m0 . A /! (x m) 2 D(f ) , m = m0 x 2 U . 2 . ! 3.1. S . * '
$ X Max S Max C . " X , | ' $.
. 2 / 3.1 3.2 3.1{3.3. "#. : , /,
S ,
. 2, , X = T = '0 1] ! . % T / _ . 2 X T , /,
m = '0 1). %
, Mxm C (X T ) /! x 2 X .
4
# $ %$
2 S | , 0 . A S , D | . 2 @ -
933
! !,
- f 2 C (X S ) - ! : Z(f ) = fx 2 X j f (x) = 0g, Mx = fg 2 C j g(x) = 0g, x 2 X , . H, 0 . & ,
, /! @
.
4.1. ( M D(f )
, ' h 2 M , Z(f ) \ Z(h) = ?.
. 2 M 2 D(f ). A f 62 M , , 2.1 M + fC = C , , h 2 M t 2 C , h + ft = 1. 2 2.1, 2.3 2.4, : Z(h) \ Z(f ) = ?. 6! , Z(f ) \ Z(h) = ? h 2 M f 2 C . A f 2= M , 2.1 2.3 f + h 2 M ! ! !. F , M 2 D(f ). 4.2. " S $, - Z(f ) Z(g) ' f g $ C (X S ) X -$ .
. 2 Z(f ) Z(g) /. 2 2.1 f + g !. A h = f (f + g);1 t = g(t + g);1 h + t = (f + g)(f + g);1 = 1. A S
, S : S = S1 S2 , S1 S2 | /, - , 0 2 S1 1 2 S2 . A X1 = h;1(S1 ) X2 = h;1(S2 ) - /. K , Z(h) X1 . : , Z(t) X2 . 2 x 2 Z(t). A t(x) = 0 , h + t = 1, h(x) = 1 2 S2 . F , x 2 h;1(S2 ) = X2 . A 2.4 Z(f ) Z(h) X1 Z(g) Z(t) X2 , D
. 4.3. " S $, Max C (X S) .
. 2 M -
D(f ) f 2 C (X S ). 2 4.1 h 2 M , Z(f ) \ Z(h) = ?. A 4.2 X X = X1 X2 , X1 X2 | /, - , Z(f ) X1 Z(h) X2 . 2 0 x 2 X1L f0 (x) = 1 x 2 X2
934
. .
1 x 2 X1L
g(x) = 0 x 2 X : 2 : , D(f0 ) = Z(g). A ! - D(f0 ). " D(f0 ) Z(g). : , N 2 D(f0 ), f0 62 N gf0 = 0 g 2 N / 2.2. + / Z(g) D(f0 ). + N 2 Z(g) n D(f0 ) ! g 2 N f0 2 N , . . 1 = g + f0 2 N . 6
, M 2 D(f0 ) D(f0 ) D(f ). 2 X1 \ X2 = ?, Z(f0 ) = X1 Z(h) X2 , Z(f0 ) \ Z(h) = ? h 2 M . A 4.1 M 2 D(f0 ). A Z(f ) Z(f0 ), , 3.4, Z(f ) = Z(f )MaxC Z(fo )MaxC = Z(f0) D(f0 ) D(f ). 8 . : /! X ,
, / X , 0 X , , - X X (
) - 0 X . E M. M D '2]. ! 4.1. " S | $ X | , Max C (X S ) '
'
,- 0 X X .
. 2 . 2 4.3
Max C | . 2 3.2 ! X | Max C (X S ). 6 , /! f0 1g- f 2 C (X S ) Max C . H, f0 1g S . 2 f : X ;! f0 1g S , X1 = f ;1 (0) X2 = = f ;1 (1). < / g 2 C (X S ), x 2 X1 L g(x) = 10 x 2 X2 : : , Z(f ) Z(g) = Max C . 2 V 2 Max C . 2 fg = 0 2 M , / 2.2 f 2 M g 2 M , !, M 2 Z(f ) M 2 Z(g), M 2 Z(f ) Z(g). A f + g = 1, Z(f ) Z(g) /. 6/ /, Z(f ) Z(g) - Max C . H / fN: Max C ;! S, Nf (M ) = 0 M 2 Z(f )L 1 M 2 Z(g):
935
6
, fN | . 2 Z(f ) \ X = Z(f ) Z(g) \ X = Z(g) x (x) = Mx , fN f Max C . 4.4. S | S-
. * Max C (X S ) '.
. 2 M M 0 Max C . A M + M 0 = C , , h 2 M h0 2 M 0 , h + h0 = 1 , , Z(h) \ Z(h0 ) = ?. 6, h(x) = 1 /! x 2 Z(h0 ). 2 U (0) U (1) S , U (0) \ U (1) = ?. + S - S , / ' C (S S ), ;
;
'(0) = 1 ' S n U (0) = f0g (1) = 1 S n U (1) = f0g: < f = ' h g = h C (X S ). : /! x 2 Z(h) f (x) = (' h)(x) = '(0) = 1. F , Z(h) \ Z(f ) = ?. $ Z(h0 ) \ Z(g) = ?. 2 h 2 M h0 2 M 0 , 4.1 M 2 D(f ) M 0 2 D(g). 6 , D(f ) \ D(g) = ?. A U (0) \ U (1) = ?, /! x 2 X h(x) 2 S n U (0) h(x) 2 S n U (1). + f (x) = 0, . . x 2 Z(f ), x 2 Z(g). F , Z(f ) Z(g) = X , fg = 0. 2 @ D(f ) \ D(g) = D(fCg) = D(0) = ?. F A0 ! !
A. ! 4.2. " S | $ S-
X |
, C (X S ) ' {
'
X X .
. 2 1.6, 3.2 4.4 Max C (X S) | X . 2 @ , ! : X ;! Max C , /, X , . . (x) Mx /! x 2 X . : I ( ! ). 2 p q 2 X p 6= q. A p q / X /, U V . 2 A B | p q, U V
: A U B V . % f g 2 C (X '0 1]), f (A) = f0g f (X n U ) = f1g g(B ) = f0g g(X n V ) = f1g: K , Z(f ) \ Z(g) = ?. 2 Z(f ) A, p 2 Z0 (f ). $ q 2 Z0 (g).
936
. .
S , h: '0 1] ;! S , h(0) = 0 h(1) = 1. J h f h g C (X S ). % !, , ! f g. F @ X ! f0 = f jX 2 C (X S ) g0 = gjX 2 C (X S ).
:
0
Z(f0 )X = Z0 (f ):
0
0
2 Z(f0 )X = Z(f0 ) \ X X Z(f ), Z(f0 )X Z0 (f ). " Z0 (f ) n Z(f0 )X X n X , , Z0 (f ) n Z(f0 )X = ?. A Z0 (f ) Z(f0 )X . 6 /, 0 Z0 (f ) Z(f0 )X . 0 $ Z0 (g) = Z(g0 )X . A
0
p 2 Z(f0 )X Z(f0 )X q 2 Z(g0 )X Z(g0 )X : (p) 2 Z(f0 )Max C Z(f ) (q) = Z(g0 )Max C Z(g): 2 Z(f ) Z(g) /, (p) 6= (q): A .
4.1 4.2 3.4 4.1. S | $ $ S -
X |
. * C (X S ) | #
M p = ff 2 C (X S ) j p 2 Z(f ) K g p 2 K K = 0 X , S $, K = X , S $. 4.2.
X Max C (X R+) ' X . E # { & '3] . 4.3. X |
, Y |
C (Y R+) - (. ; 4, ).
* C X C (Y R+) ' (X Y ),
, X | , ;
Max C X C (Y R+) X Y:
937
;
. + 3.4.3 '9] C X C (Y R+) +
C (X Y R ). 6 4.2. 6, X Y (X Y ) ! ! / X Y '9, . 3.6.D]. + / ! 5. ". + ,
! .
&
1] . ., . . ! #$ $! $! // & '''(. | 1939. | ,. 22, . 1. | '. 11{15. 2] Kaplansky I. Topological rings // Amer. J. Math. | 1947. | V. 69. | P. 153{183. 3] Slowikowski W., Zawadowski A. A generalization of maximal ideals method of Stone and Gelfand // Fund. Math. | 1955. | V. 42, . 2. | P. 215{231. 4] 5#! 6. . ! { $ // 7$ !. . | 1992. | ,. 47, . 5. | '. 171{172. 5] Hery W. J. Maximal ideals in algebras of topological algebra valued functions // Pacif. J. Math. | 1976. | V. 65, .. 2. | P. 365{373. 6] 8. . 9 ! $! T1 - $! $! // & :''(. | 1986. | ,. 30, . 11. | C. 972{974. 7] Gillman L., Jerisan M. Rings of continuous functions. | N.Y.: Springer{Verlag, 1976. 8] Banaschewski B. U= ber nulldimensionale R= ume // Math Nachr. | 1955. | B. 13, . 3{4. | S. 129{140. 9] > (. ?@ D ! D. | .: , 1986. ' ( ( 1995 .
{ . 511.336.6
: , -
{ .
!" " " # # $ " $ $ ! !n + n + % " 1 & nr + ] r ! # % $ & 1 ] & 2 ] ( 1 2 r | ! % " , 0 6 6 1). , $ # Z N :::N
n
< n
x
< ::: < n
x
::: x
0
=
R
(
)k
jI 1 : : : r j
d1 : : : dr
r
# | $ %. , k
)=
(
I 1 : : : r
, "
Z1
i
n 0 = max 1+
nj +
n
dx:
6 0 , # o + r + 1 + ( 2+ 1)
k > k0
k
j x
j=1
0
0
X r
exp 2
k
n
k
r r
r
:
Abstract
A. Zrein, Convergence exponent of singular integral in generalized Hilbert{Kamke problem, Fundamentalnaya i prikladnaya matematika 1(1995), 939{951.
In this article we 2nd exact value of the convergence exponent of singular integral in the problem of simultaneous representation of increasing set of natural numbers n + n + & nr + ] ( 1 2 r by sum of terms & 1 ] & 2 ] r | 1 natural numbers, 0 6 6 1). We consider integral: Z
N :::N
x
x
::: x
n
< n
< ::: < n
0
=
R
(
)k
jI 1 : : : r j
d1 : : : dr
r
where is an unrestricted index and k
(
I 1 : : : r
It is proved that
)=
Z1
X r
exp 2
i
converges when n + 0 = max 1+
k
dx:
and diverges when ( + 1) + 1o r+ 2
k > k0
n
j=1
0
0
j
n + x j
n
6
k
r
r r
:
1995, 1, 4 4, 939{951. c 1995 , !"
# \% ## "
k0
where
940
.
N1 : : : Nr xn1+ ] xn2+ ] : : : xnr+ ] (n1 < n2 < : : : < nr ). # nj | , 0 < < 1. ' ( {* (. 1]) +
= 0. -. .. / 2] 1 n1 n2 : : : nr 1 2 : : : r, , k0 2 3 r(r + 1) + r 6 k 6 r(r + ): 0 2 , , ni = i i = 1 : : : r, k0 = r(r + 1)=2 + max 1 r , , n1 + + nr > r(r + 1)=2, k0 = n1 + + nr + r. 5 2 , 2 3]. 7 f(x) = 1xn1 + + 2xn2+ + + r xnr + (1) 1 : : : r | 1 8 0 6 6 18 n1 < : : : < nr |
. 9 1 Z 0 = I(1 : : : r ) k d1 : : : dr f
g
R
j
j
r
k | 1 , I(1 : : : r )
Z1 I(1 : : : r ) = e2if (x) dx:
(2)
0
. k0 = max n1 + + nr + r r(r + 1)=2 + 1 . : 1) 0 k > k0 2) 0 k 6 k0 . , 0 k0 = max n1 + + nr + r r(r + 1)=2 + 1 : 1) : 1 2 1. (s) 1. f(x) = 1xn1+ +2xn2 + + +r xnr + s (x) = f s!(x) Xn H = H(1 : : : r ) = 0min (x) 1=s, n > nr | " 6x61 s=1 s #. f
f
g
g
j
j
941
I(1 : : : r ) =
Z1 0
e2if (x) dx
"$ % : I(1 : : : r ) 6 6en3 H ;1. : 1 3] (. 2
2 2]), = 2, > + f(x) 1 s (x)
n ( > + f(x) nr 6 n). (s) 2. s (x) = f s!(x) p > p;1 "$: s ( ) 6 ps s = 1 : : : nr : s > 1 s ( ) 6 (2p)s . . 7 s = 1 : : : nr
s ( ) 6 ps < (2p)s. :
+3 s. 7 2 , s( ) 6 (2p)s s = 1 : : : n, n > nr . ? x; f 0 (x) | (nr 1)-1 , (x; f 0 (x))(n) 0 2 . @ 3 2 2 f (n+1) (x) f 0 f 00 : : : f (n) : n X ( 1)s Cns ( + 1) : : :( + s 1)x;sf n;s+1 (x): f (n+1) (x) = j
j
j
j
j
j
j
j
j
j
;
;
s=1
;
;
9 = , ;s < ps 3 f (n;s+1)( ) 6 (n s + 1)!(2p)n;s+1 2 3 +. 7= n X f (n+1) ( ) 6 (n n!s)!s! ( + 1) : : :( + s 1)ps(n s + 1)!(2p)n;s+1 6 s=1 n X 6 n!(n s + 1)2n;s+1pn+1 : j
j
j
j
;
;
;
;
;
s=1
? 0 < < 1, n X n+1( ) < (n +1 1)! n!(n s + 1)2n;s+1pn+1 = s=1 n n X X = (2p)n+1 n n +s +1 1 2;s < (2p)n+1 2;s < (2p)n+1 s=1 s=1 j
j
;
;
, , s ( ) 6 (2p)s s 6 n, n+1( ) 6 (2p)n+1. ? + 2 . j
j
j
j
942
.
. ' # > 0 " s ( ) 6 (1= )s s = 1 : : : nr , s > 1 s ( ) 6 (2= )s .
j
j
j
j
: 2 2 2 p = 1= . 3. f(x) = Pns=1 ixCi , Ci | # -
# . ( x0 ) Bx0 Rr, Bx0 = (1 : : : r ) f (s) (x0 ) 6 As s = 1 : : : r A1 : : : Ar | * # . ;1 Y r Y (Bx0 ) = 2As C1 : : :Cr (Ci Cj ) x0
f
= r(r 2+ 1) C1 ;
j j
j
g
;
s=1
; ;
i<j
Cr .
. 7 > x0 f (s) (x0 ) 11 + 1 1 : : : r : n X f (s) (x0 ) = mis i mis = Ci(Ci 1) : : :(Ci s + 1)xC0 i ;s ;
s=1
;
= (f (1) (x0 ) f (2) (x0 ) : : : f (r) (x0 )) (1 : : : r ) 2 + M = (mis ). C , M 2 , (1 : : : r ) = M ;1 (f (1) (x0) f (2) (x0 ) : : : f (r) (x0 )) ,
(Bx0 ) = det M ;1 j
j
r Y s=1
2As :
(3)
det M, 2 .
. f(x) = 1 xn1+ +
+ r xnr + ,
Bx0 = (1 : : : r ) s(x0 ) 6 C s = 1 : : : n C | , n > nr + 1. (Bx0 ) 6 C1x0 , > 1, C1 = (2Cn!)r ((1 ));r . f
j j
j
g
;
3 > + g(x) = = f (n;r) (x), n > nr + 1.
943
(s) + r xnr + s (x) = f s!(x) . n > nr + 1 ) B(1) Rr: X 2n 1 =s B(1) = (1 : : : r ) s ( ) 6 1 (0 1] :
4. f(x) = 1xn1 + +
j
s=1
j
2
(B(1)) < , . . B(1) # + . 1
. : 2 k = 1 2 : : : Dk = = 2;k 2 2;k ] 4 2k 1, uk = 2;k + 2;2;2k ( = 0 1 : : : 4 2k 1) | . 9 2
;
Bk = (1 : : : r ) s (uk ) 6 C0 s = 1 : : : n P s ;s < . :2 , B(1) S Bk . C0 = 2n 1 s=1 Cn+s 2 k 7 (1 : : :r ) > + f(x) B(1)8 P 2 n 1 =s 3 B(1) 0 < 6 1 s=1 s( ) 6 1, s ( ) 6 1 s 6 2n: () 9 Dk , 21 , uk | 21E 1 E . ?
: ;k
> 2;k 8 uk < 42 2k 6 4 2k : 9 2 i (x) ? 1 : 1 1 1 X X ( s) s i (x) = s! i ( )(x ) = Cis+s i+s ( )(x )s : s=0 s=0 f
j j
j
g
1
2
j
j
j
j
j
;
j
;
;
7 i 6 n, x = uk . ? = 2 s 6 n i+s ( ) 6 1, = , 6 1, ( ) , s 6 2n s ( ) 6 ;s . 7 2,
: s ( ) 6 (2= )s s. F = , , i 6 n n X X i (uk ) 6 Cii+s uk s + Cii+s (2= )i+s uk s: j
j
j
j
j
j
j
s=0
;
j
j
j
j
;
j
s>n
1 uk 1=2, 1 | 4 2k , n X X i (uk ) 6 Cii+s 2;s + Cii+s ;i 2i+s (4 2k );s: j
j
j
s=0
;
j
s>n
944
.
# , i 6 n ;i 6 ;n 6 2kn, = i 6 n n X X i (uk ) 6 Cii+s 2;s + Cii+s 2i;s 2k(n;s) < s=0 s>n n 1 X X X < Cii+s 2;s + 2i Cii+s 2;s < 2n Cnn+s2;s = C0 : j
j
s=0
s=0
s>n
S C , , B(1) k Bk . G 3 2 (Bk ) 6 C1 (uk ) 6 C1(2 2;k ) > 1, C1 = (2Cn!)r ((1 ));r . C = , ,
;
(B(1)) 6
1 4X 2k;1 X k =1 =0
1
X C1 (2 2;k ) = 4 2k C1 2;k =
k =1
= 4 2 C1
1 X 2k(1;) < k =1
1
> 1, . C , 4 .
@ k0 E
r + n1 + + nr 1 + r(r 2+ 1) . 9 p 2 2n X 1 =s B(p) = (1 : : : r ) 0 < 6 1 s ( ) 6 p
j 9
s=1
n > nr + 1. 5. ( p
j
j
8 > < Cpk0 k0 = 1 + r(r + 1) 2 (B(p)) 6 > :Cpk0 ln p k0 = 1 + r(r + 1) 2 6
C > 0 p. . 9 > + f B(p), 3 2 p;1 1]. @ 2n X ; 1 1 =s B0 (p) = (1 : : : r ) p 1] s( ) 6 p j 9
2
s=1
j
j
945
B0(p) = (1 : : : r ) j 9 2 (0 p;1)
2n X
1=s 6 p : j s ( )j
s=1 0 @ , B(p) B0 (p) B (p). @+ B0 (p), = 2 B , = 2 3 : : : 2p.
B (p) = (1 : : : r ) i (u ) 6 C0 pi i = 1 : : : r P
. n ;s C0 = 2r 1 s=0 Cn+s 2 , u = 2p S B . 7 f B (p) u | 21E :2 , B0 (p) 0 p P 1 2 n 3 . ? u 6 4p s=1 s( ) 1=s 6 p, , s( ) 6 ps s 6 2n. 7 2, , s ( ) 6 (2p)s s > 1. * 4,
? 1 i (u ): 1 X i (u ) = Cis+s i+s ( )(u )s f
j j
j
g
2
j
j
j
;
j
j
j
j
j
, j
i(u ) 6 j
;
s=0
1 X s=0
Cis+s (2p)i+s(4p);s = (2p)i
1 X s=0
Cis+s 2;s 6 C0 pi
i 6 r, . . f B . S2p B (B (p)) 6 P2p (B ). ? , , B0 (p) 0 =2 =2 7 3, : Y ;1 r r Y Y r (r +1) (B ) = (2C0 pi) (ni + ) (ni nj ) u < (2C0)r p 2 u 2
i=1
;
i=1
= r(r 2+ 1) r (C1 + ;
;
(B0 (p)) 6 (2C0)r p
i<j
r (r +1) 2
+ Cr ). 7=
2p X =2
( =2p) = 2r; C0r pr+C1 ++Cr
? , + (B0 (p)): 8 r (r +1) > > Cp 2 r + C1 + > < r(r+1) (B0 (p)) 6 > Cp 2 ln p r + C1 + > > :Cpr+C1 ++Cr r + C1 +
2p X =2
:
+ Cr < 1 + r(r 2+ 1) + Cr = 1 + r(r 2+ 1) + Cr > 1 + r(r 2+ 1)
946
.
C p. : 5 + (B0(p)). C , f(x) = 1 xn1+ + 2xn2 + + + r xnr + f(x) B0 (p): 9 > +3 g(x) = f(x=p) x 0 1]. g(s) (x) = p;sf (s) (x=p), = x = p 6 1 2n g(s) (x) 1=s 2n X = p;1 X s ( ) 1=s 6 1: s! s=1 s=1
2
2
j
? , g(x) =
r X
i(x=p)ni + =
j
r X
~ ixni i +
i=1 i=1 ; n ; ~i = ip i 2 B(1),
3
4. G , (B0 (p)) 6 pn1 ++nr +r (B(1)):
C + 2 B0 (p) B0 (p) , B(p) B0 (p) B0 (p), 2 5. ? 3]. 9 s = 0 1 : : : 2 Bs = B(2s+1)=B(2s ). C 1,
Z 1Z 1 X X k 0 = I d1 : : :dr + I k d1 : : :dr < (B(1)) + 6en3 2;ks (Bs ):
j j
s=0
(1)
j j
s=0
s
C 5 , Bs B(2s+1 ), 2
. 2) : 1 , 1 k 6 r + n1 + + nr , r + n1 + + nr < r(r + 1)=2 + 1 ( . . ni = i < 1=r) , 1 k < r(r + 1)=2 + 1. ?, 0 k 6 r + n1 + + nr , 2 1 2], 1 = > ni = i i = 1 : : : r. 7= E , 0 k 6 r(r + 1)=2 + 1, < 1=r ni = i i. : = + p > 450r ;2r = 1 : : : p 3
2 Bp Rr: Bp | = 2 , 2 (1 : : : r ), 3 1 > + f(x) = 1 x1+ + 2x2+ + + r xr+
947
f 0 (x)
f 0 (x) = x ( r (x x )r;1 + r;1 (x x )r;2 + + 1 )
, pr < 6 (2p)r 8 < (Cp)i x = 41 + 2p i r i ; 25 r 2 i < r, C = 4 . . , 2 Bp 3 . C 3, ;
;
j
j
(Bp ) = (r + ) : : :(1 + ) ;1 (2r 1)2r;1C 2 p1++r : 6. ( " * % f(x) = Pri=1 ix+i (0 < < 21 ) (1 : : :r ) Bp "$ % : Z1 e2if (x) dx > 1 : 16p j
j
r (r +1)
;
0
. 7 I =
Z1 0
e2if (x) dx 3
: I = I1 + I2 + I3 + I4 = xZ xZ ; + Z" Z1 = e2if (x) dx + e2if (x) dx + e2if (x) dx + e2if (x) dx
" 0 x ; x + + " = 2 4;12r , D = Ap;1 , A = 420;1 . G P I2 + I4 . ? f 0 (x)
f 0 (x) = x ri=1 i (x ; x )i;1 ,
1 i
: f 0 (x) > x (pr yr;1 (Cp)r;1yr;2 Cp) y = x x , y > D
C C C = f 0 (x) > x pr Dr;1 1 pD (pD)2 (pD)r;1 = x Ar;1 p(1 C(A;1 + A;2 + + A;r+1 )) > 21 x Ar;1 p > 21 " Ar;1 p j
j
j
j
;
j
;
; ;
j
;
;
;;
;
x > ". 1 : 2 ] > + f 0 (x) f 0 (x) > C, Z e2if (x) dx 6 1 2C
948
.
(., , 3], . 15). F , f 0 (x)
(r + 1) , , r I2 + I4 6 "rA+r;11 p < "A2r;1 p < 12 . C , r 20 I2 + I4 < 4;6r 4202r;20 ;r+1 p 6 2427r p 6 2141 p : ? + I1 , 3 > . L 2 Z 2 if (x) 00 ] f (x) > C, e dx 6 12C ;1=2 (. 3], . 14). : f 00 (x)
2
r 1 00 r ;1 f (x) = r x (x x ) r + x x + r 2 r ;2 + r;1 x (x x ) + + + x 1 r x x x: L x 6 " < 81 , x x > 14 " > 81 , = x i = 1 : : : r 1 6 . # , < + i < , , 0 < x i x < 8r < 2" 2x 2x x x x x x 6 " x;1 x x r;2 f 00 (x) > r x x x r;1 2x 1 x;1 > r ;1 > 21 pr x;1 x x r;1 Cx;1 (pr;1 x x r;2 + + p2 x x + p) > ;1 1 r r ;1 r ;1 >x Crp > 2 p x x > x;1 21 pr (1=4)r;1 Crpr;1 ";1 > 1 r ;1 > 12 pr (1=4)r;1 1 2rC 4 p ": ? < 12 , = 2r4;25r 4r;1 > 1 : r ;1 1 2rC 4 > 1 p 450r ;2r 2 j
j
j
p
j
j
j
;
;
;
;
;
;
;
;
;
j
j
;
;
j
j
;
j
;
j
;j
j
;
j
j
;
j
j
;
;
;
;
p
;
;
;
j
j
; ; j
j
j
;
j
949
# , x 6 " j
7=
f 00 (x) > 41r pr ";1=2 > 41r p2 (2 4;12r );1=2 = 210r p2: j
Z" 2 if (x) I1 = e dx 6 12(210r p2 );1=2 = 3212r p < 238 p :
j
j
0
Z
x +
. +, + I3 , I3 = j
j
x
g(x) = f(x ) + x
@ , g(x ) = f(x ),
;
r X s (x x )s : s s=1 ;
f 0 (x) g0 (x) = (x x ) ;
Z
;
x +
7 I30 =
x
;
e2if (x) dx. : =
r X s=1
s (x x )s;1: ;
e2ig(x) dx. ?
Z
x +
I30 ; I3 =
e2ig(x) 1 e2iF (x) dx ;
x
;
F(x) = f(x) g(x). 7 M2 F(x) = (x x )F 0( ) 2 x x . G , x x 6 D r X F (x) 6 D x s ( x )s;1 ;
;
j
;
j
j
;
j
s=1
j
;
j
x < D. ? '(t) = t 0 < < 21 , x
1 , 8 1 ;1
x < 8 x < 4 x :
x 6 8 ? , x x 6 D r X F(x) 6 4D2 s Ds;1 6 4D(2r pr Dr + (CpD)r;1 + + CpD) < 8(2A)r D:
E
j
;
j
j
;
j
s=1
j
j
j
j
;
;
j
j
;
j
j
j
;
j
950
.
7= 3 3 + I30 I3 : xZ xZ + + 0 I3 I3 = 2 sinF (x) dx 6 2 F(x) dx < ;
j
;
j
j
j
j
;
j
; r 2 r +7 r +2 ;2 < 16(2A) 2D < 2 A p = 2r+7 240(r+2);r;2 p;2 < x
x
< 250r ;2r p;2 < (250r ;2r 4;50r 2r )p;1 6 2501 p :
? , + I30 ,
Z
I30 = e2if (x )
;
r X exp 2ix ss ys dy s=1
Z
I300 = e2if (x )
;
Z
r e2i y dy, = x rr . C
:
X I300 I30 = e2i yr 1 exp 2ix ss ys dy 6 s
j
;
j
;
j
;
s
j
s
r ;1 2 32 4;25r r 440;2 X = 6 2151 p : (CA)s < 8DCA < 32CA p p s=1 2 Z Z 2 i y r 00 dy > cos(2yr ) dy = I3 = e ; ; Z Z1 r r r = 2 cos(2y ) dy = 2D cos(2D t ) dt:
< 4D
j
j
0
# ,
0
r r Dr = x rr Ar p;r > 14 Ar > A2r > 1: 2 1 + 1 > 1 r > 2: Z1 cos(2tr ) dt > 1 ;1=r 8 0
951
(. 3], . 18). ? , I300 > 2D 81 (Dr );1=r :
j
j
1. F , = x rr < r < (2p)r , , I300 > 8p @N + : 1 1 1 > 1 1 p;1: I3 > I300 I300 I30 I30 I3 > 8p 215p 250p 8 214 , 1 1 3 1 I > I3 I1 I2 + I4 > 8 214 28 214 p;1 > (16p);1 M 6 , = , j
j
j
j
j j
j;j
j
j; j
;
j;j
j;j
;
j
;
j
0 >
1 X 2n X n>n0 =1
;
j
;
;
;
;
(B2n )(16 2n);k
3 . - (. C. - . .. O .
1] . . { // . . . . | 1984. | #. 48, % 1. | . 3{52. 2] )
. . *+
, + -.
+,
// /. . |
1986. | #. 40, % 3. | . 310{320. 3] . ., *2 . ., 32 4. . # 5 +6 , + -. .2. | /.: 2, 1987.
'
# ( 1995 .
. .
. . .
517.73
: , , , ! .
, " ($ " ! " ") & $ ' " ! " , ( $ , ! ). * & 1963 &. ! & " T-! " .
Abstract
G. A. Kovaleva, Non-existence of closed asymptotic curves on the tubes of negative Gaussian curvature with one-to-one spherical mapping, Fundamentalnaya i prikladnaya matematika 1(1995), 953{977.
It is proved that on the tubes having negative Gaussian curvature and biunique spherical image there is no closed asymptotic curves and by that a Nirenberg's problem is solved which he has posed in relation with the study of rigidity of T-surfaces.
1. 1963 . . 1] ! \ T" C 4 $
, & $' , ' '()'' ! * ! ! & ' , $ ! ! & !
. , & + & $ , () ! & ', $ & . $ )
&
! ) , & , ' . 1967 . -! 2] $ ) ' * ! ! ! & !
! ! & ! , ! *
.
1995, 1, / 4, 953{977. c 1995 !, "# \% "
954
. .
1991 . 1. 2. 3 3] - , & ! &
! | , $ ,
&
!. 5 + & $
! ' ' ' & . 6 ') ! | () . . C 5- ( , , C 1 ). 7 & !. 8 & 4 ' + , ()' -++ * * ' (
:), ' ' * . 5 ' , )( 6 ', & -++ * * ' & & ! & ! & ', . . & ' $ *
( ). < *, 7 ' !
= ( -! $ & ! , '
) ' , ' & ' ' & !, . 2. - s, 0 6 s 6 l |
, r?(s) | -
! &
! ! L. @ $! & r?(s) ) ()'' , . . , $)' ? = r?0 r?00.
( ?(s) ')
!
& ! , ' ! () !' . @ $ , & ?0 6= 0, ?0 ) ' = & & & . B & $ ', & r?00 , , ) ' . 5 ' ( ? ? = ? ?, { & +
?0 = { ?, ?0 = ;?, $ , - ' ?, ?. < ' ? , & ? ?0 = ;{ ? + : 8 , + D ( ) ! + , $ )' * !. 3. @ ' & '
', . .
', ! ' () !' ( !. E & , &
( -!
. 8 , (
( ( & (
955
> 0) ! -! !, &
' & !, + & $ & ,
! $ & & '. <,
! ! !
! ! > 0 & & '. B +
& ' ( . 1). @ ! + : { 1
ZZ I K ds + { g ds = 2: (S )
G K 1 | + , { g | & '
!, s |
. 1 . 1. : & ' ! | - * ' ( , 0 00 { g = (?rjr?r?0j3n? ) n? |
& ! .
& r? = ?, r?0 = ;?, r?00 = ;0 ? ; (;{ ? + ?), r?0 r?00 = 2 ({ ? + ?), n? = ? ( * ,
!), { { g = H ds = ds: I
Zl
@- { g ds = { ds , , 0
Zl
S + { ds = 2 0
(S | ) , &
!
!
!). 5& , & 0 < S < 4, - Zl
;2 < { ds < 2: 0
< ( D ' 4]), & { |
!
956 !,
. .
Zl
j{ j ds > 2, l |
! !, s |
-
0
!. 5( , & { ' , & & . <, {
! ! $ ' , $ ) & , { = 0. 4. @ ('
() ' L1 L2 - r?1(t) r?2 (s), t s |
. B -++ * * ' - $ '
: 5] ZZ 1 I (L1 L2) = 4 (?r2 ;jr?r?1; dr?r?1j 3dr?2) 2 1 L1 L2
' & ( L1
L2 . 5 & & a?(s t) ?a(s t) = r?2 ; r?1 jr?2 ; r?1j
ZZ (1) I = 41 (?aa?s ?at) ds dt: L1 L2
@$ , & ( + * ( + (1) $ @ (?a?b?b ) ; @ (?a?b?b ) (?aa?s a?t ) = @t (2) s @s t ?b |
& ! ' ! ?a . I ! , ?b2 = 1, ?bs ? ?b, ?bt ? ?b , , ?bs ?bt = ?b, , & @ (?a?b?b ) ; @ (?a?b?b ) = (?a ?b?b ) ; (?a ?b?b ). ?a ? ?b, (?a?bs?bt) = 0. B @t s t s s t @s t ? @ ?b = a? M? , M? (s t) | ! ' ! , ja? M j ?bs = a?s M? + a? M? s ; a? M? 2 ja? M? js ja? M? j ja? M? j ?bt = a?t M? + a? M? t ; a? M? 2 ja? M? jt ja? M? j ja? M? j
957
, ,
? ? ? (?at?b?bs ) ; (?as?b?bt) = (?at ?a M ?as ?M2 + a? Ms ) ; ja? M j ? ? ) ; (?a?atM? )(?as M? ) ? ? ? = ; (?as a? M ja??at M?Mj2 + ?a Mt ) = (?a?asM )(?at M ja? M? j2 ? ?a M? ]) (?asa?t a?)(?a M ? a? M? ) = (?as a?t ] M = = (?aa?s ?at): ja? M? j2 ja? M? j2
-! * & ! + & ': A? B? C? ] = B? (A? C? ) ; C? (A? B? ) (A? B? ] C? D? ]) = (A? C? )(B? D? ) ; (B? C? )(A? D? ) $ , & a?s ? a?, a?t ? a? , & , ?as a?t = (?as a?t ?a)?a: <, + (2) . 5 & r?2 ; r?1 = A?, A? M? = B? , ? M? = A? M? = B? a? = jAA?j ?b = jaa?? M? j jA? M? j jB? j
(1) (2) &
"
#
(3)
ZZ @ (?aB? B?s ) ; @ (?aB? B?t ) ds dt 1 (4) I = 4 @t @s B? 2 B? 2
' ' 0 6 s 6 S0 0 6 t 6 T0 $
$ . @ ' + * ' & , B? = 0. 7 &!, B? = 0
!
;, () ! & & '. @ '' + :
, !
' ; ! ! =
2 , I ? ? + B?t dt) ? dB? ) 1 I (?a B I = ; 41 (?a B BsBds = ; ?2 4 B? 2
' ' * -! ,
! , ' .
958
. .
1 . 2.
2 $ , ' & ' ' ; ! !. 8 * + ( $ ! ' $ ' * -! . @ -
$ '
' , $ ' , &$(', & Z Z I = ; 41 + ;
(5)
;;
; ;; | &
, ! ! ;, ' . @ s = s(u), t = t(u) | & ' -! !, u |
, . .
s0u 2 + t0u2 = 1:
B
& ! ! ! ;
ft0 ;s0 g. @ & ' * ;
s~ = s + t0 ~t = t ; s0 :
959
7 $
; : B? (~s ~t) = B? (s + t0 t ; s0 ) = = B? (s t) + (B?s t0 ; B?t s0 ) + 21 2 (B?ss t0 2 ; 2B?stt0 s0 + B?tt s0 2) = ? 2 2? = ddnB + 2 ddnB2 + : G & ': dB? = B? t0 ; B? s0 (6:1) s t dn d2 B? = B? t02 ; 2B? t0 s0 + B? s0 2 : (6:2) ss st tt dn2 I , dB? (~s ~t) = B?u0 du(~s ~t) = ( 0 ) ? 0 2 ? 0 2B ? ? d B 1 d d B 1 d B 2 2 = dn + 2 dn2 + du = dn + 2 dn2 + du: u G !=
++ *
& ++ * u. B ? dB? ) = (?a B ! ?0 2 ?0 ? 1 2 d2B? d a ? d B d B 1 d B 2 = a? + dn + dn + 2 dn2 + dn + 2 dn2 + du = ! ? 0 2 ? 0 ? 1 d2B? d a ? d B d B 1 d B 2 = a? + dn + dn + 2 dn2 + dn + 2 dn2 + du
Z
;
Z ;
? dB? ) (?a B B? 2 =
Zl 0
a? + ddnB! + 2 ddn2 B!2 +
! dB dn
! dB dn
0
+ 2
2
+ 2 ddn2 B!2 +
! d2 B dn2
0
+ du
(7)
l |
! ! ;. ? 2 ddnB 6= 0 &
;, + (7) $ !
! 0
,
$ + (5) &$'.
960
. .
? 2 ddnB = 0
;, + (7) $
& ' . ? @ ddnB ) '
& u0 u1 : : : uk . B ( . 3) Zl 0
=
Z;
u0 !
0
+
Z
u0 +!
;
+
Z;
++
u0 +!
u0 !
Zl
u1 !
uk +!
! > 0 | +
& ($ ' $! & ui ' !i ). ? @ ddnB ) ' =
& , !- ! - & - &
'. @ ' 1 . 3.
! 0
Z;
Z;
u1 ! u2 !
,
u0 +! u1 +!
. ., ' &
$ . 5 '
!- ',
k uZi +! X i=0u
i
;!
.
7 & !- & u0, . .
Z
u0 +!
;
u0 !
. G -
! + *
? dB + d2 B? + 2 d3 B? + 2 = dn 2! dn2 3! dn3 ? 2 ? 2? 2 3B 2 d2B ? ? d3B? d B d B d B
d
= dn + dn dn2 + 3 dn3 + + 4 dn2 + 3 dn3 + = ( 0 2 !) ? ? dB? 00 d B d B = dn + dn dn (u ; u0 )2 + (1 ) ? 2? 0 2 3 2 d2B ? ? d3B? d B d B
d B
+ dn dn2 + 3 dn3 + (u ; u0) + 4 dn2 + 3 dn3 + (2 ) 1 2 $ $ u u0. G $ + B ! * & u0 & & + $.
961
O &
!
da? + dB? + d2B? + dB? 0 + d2B? 0 + = ?a + dn dn 2 dn2 dn 2 dn2 ! ( ! ! 2B ? dB? 0 ? dB? 0 ? d2B? 0 d B
d d B = ?a dn dn + 2 a? dn2 dn + a? dn dn2 + !) ? dB? 0 d a ? d B + 2 dn dn dn + 2 (: : :) + =
? ? 0 = ?a ddnB ddnB
!0
(3 )
(u ; u0 ) + 2 f: : :g + 2 (: : :) + :
@ & 2 ,
I0 (
)
! 00
; + D + (: : :)+ d ;u;u0 ?a uZ0 +! I0 = 0 2 00 ; (3 ) 0 2! 2 ! dB! ! d2 B! d3 B ! ! dB u;u0 2 dB u;u0 1 d B dB + dn dn2 + 3 dn3 + ( ) + 4 dn2 + u0 ;! dn + dn dn 2 ! dB dn
u u0
dB dn
(1)
!
!
!
2B ? dB? 0 1 ? d2B? 0 ? dB? 0 1 d d B d a ? d B D = 2 a? dn2 dn + 2 a? dn dn2 + dn dn dn :
(
( v = u ; u0 . B ;! 6 u ; u0 6 !, ; ! 6 v 6 ! , , Z
u0 +!
I0 =
;
(: : :) du =
u0 !
?a
!
=
Z
;!
! dB dn
02
+
! dB dn
! dB dn
! dB dn
! 00 dB dn
0 0 (3 )
v + 2
(1 )
v + D + (: : :) + ! dB dn
! d2 B dn2
0
+
(2 )
v+
1 4
! d2 B dn2
2 dv:
+
@ ' & ! 0, ! ! 0, & ', &
962
. .
? 0 ? - u 1 2 3 ! u0 , ddnB ! 0, ', & ddnB (u0 ) 6= 0, & : Z1
lim =
!!00 ;1
!
! dB dn
0 2
1 2
a? ddn2 B!2
! dB dn
0
(u0 )
dv = 2 ddn2 B2! (u0)v + 14 ddn2 B2! (u0 ) ! 0 ! 1 d2 B a dn2 ddnB (u0) Z1 2 ? dv = ; dB 0 d2 B ; d2 B 2 H ! 0 2 ( ) dn dB 1 dn2 2 ;1 v + ( dB )0 2 v + 4 ( dndB2)0 2 dn
(u0 )v2 +
0
! dB dn
dn
dn
G ! + *
() :
v+
1 2
0
! 0 2 dB ! dB dn
! d2 B dn2
!
dn
; 14
! dB dn
0 ! dB
! d2 B dn2
= v+
1 2
! d2 B dn2
2
+ 14 0 2 = ! dB
04
dn
2
! 0
dn
! 0 2 dB dB dn
! d2 B dn2
dn
!
+ 41
2 0 dB! ! d2 B dn dn2 ! 0 4 dB dn
lim I0 =
!!00
!
1 2
?a
! d2 B dn2
! dB dn
! 0
dB dn
0 2
1 1 ( ) v + 2 1 dB 10 d2 B arctg 1 dB (0 d2)B = 2 ( dn ) dn2 2 ( dn ) dn2 02 02 d B d B ( dn ) ( dn ) ;1 0 ;
0 d2 B dB dn dn2 02 dB dn
?a ddn2 B2!
= 0 ! dB
dn
! dB dn
! d2 B dn2
(u0):
E & ' & $
$ ( ' ; , & $
).
Z;1
1
+
,
'
<, & k 1 ZZ 1X (? a a ? a ? ) ds dt = ; s t 4 2 i=0
?a
! d2 B dn2
! dB dn
0 dB! dn
963
0
! d2 B dn2
(ui ):
(8)
@$ , & # 1 ;1. 2? ? 0 I' - & , & & u0 ddnB2 ddnB ka?. $ . B
; ( B? = 0,
B?s s0 + B?t t0 = 0: ? P & u0, ! ddnB = 0, . . dB? = B? t0 ; B? s0 = 0: s t dn 8 , -! & B?s = B?t = 0. + (3) B? = A? M? . < ' B? = 0 , & M? = A?, - B?s = A?s M? + A? M? s = A? (M? s ; A?s ) B?t = A?t M? + A? M? t = A? (M? t ; A?t ): < & , & u0
? M? s ; A?s = 1 A ? M? t ; A?t = 2 A:
B B?ss = A?ss M? + 2A?s M? s + A? M? ss = = A? (M? ss ; A?ss ) + 2A?s ( A?s + 1 A?) = A? (M? ss ; A?ss ; 21A?s ) . . B?ss ? A?. Q & , B?tt = A? (M? tt ; A?tt ; 22 A?t), . . B?tt ? A?, B?st = A?st M? + A?s M? t + A?t M? s + A? M? st = = A? (M? st ; A?st) + A?s ( A?t + 2 A?) + A?t ( A?s + 1 A?) = = A? (M? st ; A?st ; 2 A?s ; 1 A?t) . . B?st ? A?.
964
. .
2? < + (6.2) , & ddnB2 ?A?. *, ? 0 dB (u ) = (B? t0 ; B? s0 )0 = 0 s t dn = B?s t00 ; B?t s00 + (B?ss s0 + B?stt0 )t0 ; s0 (B?tss0 + B?tt t0 ) = = (B?ss ; B?tt)s0 t0 + B?st (t0 2 ; s0 2 )
? 0 ? 0 2? . . ddnB ?A?. 8 , ddnB2 ddnB (u0)kA?. R $ . . D (8) ' &', B? = 0 ?
;, ddnB = 0 =
& . @$ , & 2? 3? 4? ? & , B? = 0, ddnB = 0, ddnB2 = 0, ddnB3 = 0
;, ddnB4 = 0 =
& , + & ' & !
a? k X I = 41 (?a?asa?t ) ds dt = ; 12 5 ZZ
;
= =
Z
l
a? +
! 4 d4 B 4! dn4
+
! 5 d5 B 5! dn5
+
d B!5 dn
i=0
I ! , - & Z ? dB? ) (?a B B? 2 =
! d5 B dn5
! 4 d4 B 4! dn4
+
2
! d4 B dn4 ! d4 B dn4
0
0
! 5 d5 B 5! dn5
0
! d4 B dn4
2
+ 5 ddn5 B!5 +
(9)
0
+
+ 5!5 ddn5 B5! + 0 4 0 5 5 4 Z l a? + ddnB4! + 5 ddnB5! + ddnB4! + 5 ddnB5! + ! 4 d4 B 4! dn4
( ui ):
du =
du
? , & + (9) & ' (8) !
ddnB 2? 4? 5?
21 ddnB2 ddnB4 15 ddnB5 . 5. 7 (
( ( L & & ' & > 0. 1 & , & - ' & ' '. @ ! -! !. @ - ', &
965
& ('. = & | , & & ' * , *
& !. 2 ( ) !
90 , -++ * * ' & '. P ( ( -++ * * ' & , ' ( r?1 = r?(t) ; "?(t) r?2 = r?(s) + "?(s) 0 6 s 6 l0 0 6 t 6 l0 : B
A? = r?2(s) ; r?1(t) = r?(s) ; r?(t) + "(?(s) + ?(t)) = ? + "(?(s) + ?(t)): G & ? = r?(s) ; r?(t). < (4)
, B? = A? M? = 0. @ M? = ? N? (s t). N?
, & $
( s = t. B ' (, & * (', A? 6= 0. @- B? = 0, ? I. M? = 0 II. A? k M: (10) @ $ & : 1) ? = 0 (- $ =
s = t), 2) N? = 0 ? 3) ? k N:
(11)
& II A? k M? & , & ? ? + "(?(s) + ?(t)) = ? N: R $' & ' ?, & ?2 + "(? ?(s) + ?(t)) = 0 , & j?j 6 2", & ' " $ = s, t. @ - ' ' js ; tj 6 0. & N? = ?(s) ?(t) ; ?(s) ?(t): (12) ? ?(s) + ?(t)) = 0, ?(s) ; ?(t), ?(s) + ?(t), ?(s) ?(t) @ (N ' , N? $ (13) N? = C1(?(s) ; ?(t)) + C2 ?(s) ?(t):
966
. .
R $' ' (13) ?(s) ; ?(t) ?(s) ?(t), & $ ' ' -++ * C1 C2: C1 = (? (s)? (t))?+ (??(s)? (t)) H C2 = (? (s)? (t))?+ (??(s)? (t)) H (14) 1 ; ( (s) (t)) 1 ; ( (s) (t))
C12 + C22 = 1, . . $ &
C1 = sin C2 = cos
N?
N? = sin (?(s) ; ?(t)) + cos ?(s) ?(t): (15) N? $ )' &' : ) ?(s) = ?(t) | - $ =
s = t, & (
! & & 'H
?(s) = ;?(t) ) sin = 0: . - & ? sin = (? (s)? (t)) +2 (? (s)? (t)) = (? (s) (s)? (t))2 + (? (s)? (t)) = (? (s)? (t)) = 0
(? (s)?(t)) = 0 $ , , & ? k ?(t) ?(s) k ?(t). @ N? $ $ , ! : ? N?1 = N? + pF p = (1 ; (?(s)?(t)))3 (16) F? = h1(?(s) ; ?(t)) + h2 ?(s) ?(t) h1 h2 | . B &
1 ; (?(s)?(t)) > 0 ) ' =
s = t, ' ! -!
. @ p0 | = & p js ; tj > 0 ( & U). B, ', , h1 = p2 , U 0 sin + h1 p = sin + 2pp > 2 + sin 6= 0 0
, , N?1 !. I ! , N?1 = 0, . . N? + pF? = 0. B (15) (16) (sin + ph1)(?(s) ; ?(t)) + (cos + ph2 )?(s) ?(t) = 0:
967
8 $ ' $
(, U !. Q & h2 < 0 , & ' cos + ph2 6= 0, & $ = . B , $ U ? k N?1 . 7 (? ?(s) + ?(t)) = 0: @ ? = 1 (s t)(?(s) ; ?(t)) + 2 (s t)?(s) ?(t) | (17) = . ( $ = ?(s) = ;?(t)
.) 2 & , ? = N?1 , $
$ $ (17), & N?1 ? (?(s) + ?(t)), . . $ '' 1 (?(s);?(t))+2 ?(s)?(t) = (sin +h1 p)(?(s);?(t))+ (cos +h2 p)?(s)?(t)
(1 ; (sin + h1p))(?(s) ; ?(t)) + (2 ; (cos + h2 p))?(s) ?(t) = 0: E ( (1 ; (sin + h1 p))(?(s) ; ?(t)) = 0 (2 ; (cos + h2p))?(s) ?(t) = 0: @ U ?(s) ; ?(t) 6= 0, ' : )
(
1 ; (sin + h1 p) = 0 2 ; (cos + h2p) = 0H )
(
1 ; (sin + h1 p) = 0 ?(s) ?(t) = 0:
7 &! ). 3 h1 h2 , & sin + + h1 p 6= 0 cos + h2p 6= 0, . . = , 1 2 '. <(&' -! , & 1(cos + h2 p) ; 2(sin + h1p) = 0
1 cos ; 2 sin ; 2h1 p = ;h : 2 1p 1 , 2 , cos , sin , p
&
+ *
'
! ! L, h1 = , 1 p 6= 0 U. 8 , ' & ' & . @ | '' . @ h2 < 0, & - $ cos + h2p 6= 0 $ ' ' jh2j > . B )
= !.
968
. .
7 &! ). @ ?(s) = ;?(t) !
. B
(17) , & -!
? = 21 (s t)?(s): I ++ * ' '
, & (
;(s)? (s) ds = (t)? dt ?(s) ds ; ?(t) dt = 2?(s) d1 ; 21 (s)? (s) ds
, & ?(s) k ?(t), . . ?(s) k ?(t), $' ' ?(s), & ;21(s) = 0, . . 1 = 0. 8 , - &! $ $ . & , ?(s) = ;?(t)
? k ?(s), . & ! ! & $ = & & . - & \ ". @ (s0 t0) | & . 7$ ?(s t) N?1 (s t) + B ! * & (s0 t0), ) ' ( r0 , ! R? (! 2 ' ( ) jR? (! 2 j 6 "0 : 1 : ) ? ? (!) ?(s t) = ?(s0 t0)+ ?(s0 )Vs ; ?(t0 )Vt + R? (! 2 = k1 (s0 )+ ?(s0 )Vs ; ?(t0)Vt + R2 : (18) G k1 = 21(s0 t0) 6= 0, Vs = s ; s0 , Vt = t ; t0H
N?1 (s t) = N?1 (s0 t0) + (N?1 )0s (s0 t0 )Vs + (N?1 )0t(s0 t0)Vt + R? (2N!1 ) = = k2 ?(s0 ) + (N?1 )0sVs + (N?1 )0s Vt + R? 2(N!1 )
(19)
k2 = 2(sin + ph1)(s0 t0) 6= 0. G ! N?2 ? N?2 = N?1 + Q = 0 (, ! \ " & . 5 Q? & (s0 t0) () : Q? = e0 (r02 ; Vs2 ; Vt2) e0 | ' . B * , . . $ Vs2 + Vt2 = r02, N?2 = N?1 , ? k N?1 * , ? , N?2 $ r0 , , * r12 6 Vs2 + Vt2 6 r02.
969
@ e0 = r2 e;1 r2 , . . 0
1
N?2 = N?1 + e1 (r0 ;r2V;s r2; Vt ) ?(s0 ): 2
2
0
2
1
@$ , & $ e1 , & r1 ? , N?2 . @ N?2 = c?, (18) (19) & ) C fk1?(s0 ) + ?(s0 )Vs ; ?(t0)Vt + R? (! 2 g= = k2?(s0 ) + (N?1 )0s Vs + (N?1 )0t Vt + R? (2N!1 ) + r2 e;1 r2 (r02 ; Vs2 ; Vt2)? (s0 ): 0 1 (20) R $ ' (20) ?(s0 ) ?(s0 ), & ) ? C fk1 + (R? (! 2 (s0 ))g = (21) = k2 + ((N?1 )0s Vs + (N?1 )0t Vt + R? (2N!1 ) ?(s0 )) ) C fVs ; (? (t0)? (s0 ))Vt + (R? (! 2 ?(s0 ))g = = ((N?1 )0s Vs + (N?1 )0t Vt + R? (2N!1 ) ?(s0 )) + r2 e;1 r2 (r02 ; Vs2 ; Vt2):
0
1
(22)
5 C ' (21). E $ , k1 6= 0, ) ? jR? (! 2 (s0 )j 6 "0 , "0 $ . @ & ' C (22) = () : C f: : :g ; ((N?1 )0s Vs + (N?1 )0t Vt + R? (2N!1 ) ?(s0 )) = r2 e;1 r2 (r02 ; Vs2 ; Vt2): (23) 0 1 ' & ' (23) & (s0 t0) & . @ ! 0. 8 $ Vs2 + Vt2 6 r12
e1 > 0 e1 (r2 ; Vs2 ; Vt2) > e1 (r2 ; r2) = e : 2 r0 ; r12 0 r02 ; r12 0 1 1 ' e1 = 20 , & & , , & ? , N?2 ! r0 * & (s0 t0). Q & \ " & . <, 3) ' (11) ' ' N?2 U. @ $ 2) $ , & U N?2 6= 0. I ! , N?2 = N?1 & ! U (& ! \ " & N?1 6= 0, ( = - . Q? (s0 t0) k ?(s0 ), N?1 (s0 t0) k ?(s0 ). 8 , & ! N?1 (s t) , ?(s0 ), . . N?2 6= 0.
970
. .
7
s = t, . . js ; tj 6 0. B -! & , ?(s) = ;?(t), = 0, . . N?2 = N?1 . 7$ N?2 (s t) ' t = s + . N?2 = N?1 = N? + pF? = N? + ph1(?(s) ; ?(t)) + ph2 ?(s) ?(t) = = ;(s)? (s) + + ph1 (s)? (s) + + ph2 (s)? (s) + = (24) = ;(s)(1 ; ph2 )? (s) + ph1 (s)? (s) + p 8 (s) , h2 < 0, N?2 6= 0. @$ , & -! $ ! N?2 , ! + (15) $ ? N?2 = (sin + ph2)? (?(s) ; ?(t)) + (cos + ph2)? ?(s) ?(t)]: 6 6
< + (14) s = t + & , & cos ;1, , , cos + ph2 6= 0. 7$ ' ?, ?(s) ; ?(t), ?(s) ?(t) ( 3 2 ? = ;? ; 2 { ? ; 6 ({ 0? ; { 2? + { ?) ; (25) 4 ; 24 f({ 00 ; { 3 ; { 2 )? ; 3{ { 0? + (2{ 0 + { 0)?g + 2 ?(s) ; ?(t) = ? + 2 (0 ? ; { ? + 2 ?) + 2 ?(s) ?(t) = ? + 2 (0 ? + { ?) + 3 + 6 f(00 ; { 2 ; 3 )? + (20 { + { 0)? g + 4 + 24 f(300{ + 30 { 0 + { 00 ; { 3 ; { 3)? + + (000 ; 30{ 2 ; 3{ { 0 ; 602 )? g + & ' 3 ? (?(s) ; ?(t)) = ;2 ? ; 2 (0 ? ; 2 ?) + 6= 0 (26) 4 ? ?(s) ?(t)] = ; 12 { (0 ? + 22 ?) ; (27) 5 ; 24 f{ 0(0 ? + 22?) + 30 { ? + ({ 00 + 2{ 3 )? ; 2{ 22 ?g + :
971
7 & &!, { ', . . 4 j{ j > { 0 > 0. B + (27) = ! & $ ' ; 12 { (0 ? + 22 ?) 6= 0 , (26),
? (?(s) ; ?(t)). @- ? N?2
, & & ', , ? N?2 6= 0. & , 5 { = 0, = ! & $ ' (27)
; { 0 (0 ? + 22 ?)
24 { { 0
)(', ' ? N?2 6= 0. @ { 0. @ = (27) 4 5 ? ?(s) ?(t)] = ; 12 { + 2 { 0 (0 ? + 22?) ; 24 { (: : :) + :
B { + 2 { 0 6= 0, * ' ' '. 5 ' ' &!, { + 2 { 0 0. 2 ? ?(s) ?(t)] 6= 0
? (?(s) ; ?(t)), ? N?2 6= 0. I , & ? ?(s) ?(t)] $ )' &
, { + 2 { 0 0. B & , { = 0, - , & ' ! -! & , . . j{ j 6 j{ 1 j. 8 , (. . p) -!
$ $ '. @- $ h~ 1 > h1 , & sin + p~h1 6= 0 !
. B & , & ? N?2 6= 0. < *, $ &!, { + 2 { 0 0 ?(?(s);?(t))
? ?(s) ?(t)]
. @ ? ?(s) ?(t)] = 0 ? (?(s) ; ?(t)), 0 = 0 (s t) 6= 0. B ? N?2 = 0, (sin + p~h1 ) + 0 (cos + ph2) = 0, . . sin + 0 cos + ph1 = ;h : 2 p 0
< ' ;~h2 > ;h2 , & ' . <, js ; tj 6 0 ? N?2 6= 0. P$ & , & h1 h2 & : &
h1 = p2 h2 < 0 , & N?1 6= 0 UH 0 ~h1 ~h2 , & js ; tj 6 0 ' ? N?1 6= 0. < *, jh~2 j > j~h2j , & ? N?1 6= 0 U j~h2 j ~h1. E & h1 h2 . 5 ' , & js ; tj 6 0 $ A? , M? . @ - , . . ? + "(?(s) + ?(t)) k ? N?2 . R $ &
' N?2 & , & (? N?2) = 0. B
972
. .
+ (24) (25) 2 ; ? + 2 { ? + ;(1 ; ph2 )? + ph1 ? + = 0 h 1 1 ; ph2 ; 2 { p h1 = 0, . . 1 = ph2 1 + 2 { h , - $ 2 , h2 < 0, h1 jh2j. <, ' (10) (11) (&
s = t. () ! '
-!
. 6. 7 & &!, B? = A? ? N? ], A? = ? + "(?(s) + ?(t)) N? = ?(s) ?(t) ; ?(s) ?(t): @ B?s B?t ( B?s = A?s ? N? ] + A? ? N? ] + A? ? N?s ] B?t = A?t ? N? ] + A? ;?(t) N? ] + A? ? N?t ]: ?
s = t B?s = B?t = 0, . . ddnB = 0. Q & ' , &
s = t B?ss = B?st = B?tt = 0 2? . . ddnB2 = 0, B?sss = B?sst = B?stt = B?ttt = 0 3 ? , ddnB3 = 0. @ & ' -!
( B?s4 = ;8"{ 2 ? B?st3 = 8"{ 2 ? B?s3 t = 8"{ 2 ? B?t4 = ;8"{ 2 ? B?s2 t2 = ;8"{ 2 ? d4B? = B? 4 t0 4 ; 4B? 3 t0 3 s0 + 6B? 2 2 t0 2 s0 2 ; 4B? 3 t0 s0 3 + B? 4 s0 4 = ;32"{ 2 ?: s s t s t st t dn4 G s0 = t0 = p1 , & ! s = t s0 2 + t0 2 = 1. 2 4B ? d <, dn4 = 0 & , { = 0.
973
@ ' ', &
s = t & ,
{ = 0, (
B?s5 = ;20{ 0 (? + "?) B?s4 t = { 0(20? + 12"?) B?s3 t2 = { 0 (;20? ; 4"?)
B?s2 t3 = { 0 (20? ; 4"?) B?st4 = { 0(;20? + 12"?) B?t5 = 20{ 0(? ; "?)
d5B? = B? 5 t0 5 ; 5B? 4 t0 4s0 + 10B? 3 2 t0 3s0 2 ; 10B? 2 3 t0 2 s0 3 + s s t s t s t dn5 + 5B?st4 t0 s0 4 ; B?t5 s0 5 = p1 5 (;640{ 0 )? : ( 2) B ! 5B 0 ? d4 B? 0 d ? ; 640 p{ 5 ? ; p32 "{ 0 2 ? = ;2560"{ 0 23 : = a? dn5 dn4 ( 2) 2 P B? = A? M? , M? = ? N? . 2 ' M? = ? N?2 , ', N?2 = N? + pF? , p = (1 ; (?(s) ?(t)))3 , . .
s = t & ! pF? ! ' ' ' (&
s = t (, . . ' ' N?2 N?
s = t . @ + (9), ( $ ! k 5B X ? d4B? 0 d 1 I = 2 sgn a? dn5 dn4 (ui ): i=0 (G $ !, & = & . 2 $ ; ) ' ' s = t , !
'
.)
, & -++ * * ' & ',
& & , { = 0, . . & ' . 7. @ '
& (
* ( -++ * * ' & & ! . < , &
! ', $ !
D ) ' , ='
) ' ! ? (
? ( { .
= ! ! ?. : ! ! ? ? )
) ! ! , & ) - ,
2, -++ * * '?
974
. .
! & E1 E2 ( . 4). @ - & ( & ' t
s. @
D & E1 & E2
! , ') ! & & E2 () ! ? ; N = ?(t) ?(s); ? ?(s) . E
( $
) '
1 . 4. !?1 = ?(t) ?(s) ( = ) , & ?(t) ?(s), ( ( ) ) !? 2 = ?(t) ?(s) , & ? 6]. E ; N = !? 1 ; !? 2 = !? = ? +
!. 7 !? & ? N E1, . . E1 E2 !? = ? ; . 7$ ? N? ' 4 ? N? = ? (? (s) ?(t) ; ?(s) ?(t)) = 12 (2{ 2 ? + (2{ 0 ; { 0)?) + : @ ' & - $ ' ! ,
&?
! ? N? ! 0 j? N j
V? = 2{ 2 ? + (2{ 0 ; { 0)?: @ , ! = - , $) ! ! , ? ?. @ | , ! V ' ! ( ? ( . 5). 0 0 ? ? B tg = (V? ) = 2{ 2{;2{ , ) (V ?) V ! ! V =
Zl 0
0
arctg 2{ 2{;2{
0 0
ds:
E
& , { = 0. @ s0 | & . 7$ { (s) ' (s ; s0 ) -! & { (s) = { 0 (s0 )(s ; s0 ) + 1 . 5.
975
, 0 0 0 { 0(s0 ) + = arctg { ; 2 = arctg { 2 ({ 0 (s0 )(s ; s0 ) + ) ; 22 = 0 ; = arctg (s ;1s+)(1 + ) 22 0 , & ! ; 2 s ! s0 ; 0 ! 2 s ! s0 + 0 ( . 6).
1 . 6.
@- ) V ( (;) & & , { = 0, - ) . 7
! ! L ( . 7) 0 00
! & ( kg = (?rjr?r?0j3n? ) . G r? = ?, r?0 = { ?, r?00 = { 0 ? + { (;{ ? + ?), n? = ?, , kg = j{ j > 0, . . ' -!
& ' ' $ ' $ & , { = 0. < , & ! (
&
) ' '() : knn? + kg ?n t?], kn, kg | ' & ' , n? | 1 . 7. , ?t |
& ! ! !.
= & & ' '()' : ? kg ?n ?t] = j{ j ? ? sgn { ] = { : @ ' -! '() ! 0 0 { ; { 0 2 ? = ? ; ? 2 {
{
{
976
. .
- , $) ! ! !. @$ , & ! ? ? V. I ! , 0~ | , 0 ! 0 0 - ' { ; { { ~ ?. B tg = ;{ 2 = { ; 2 , & , ) V~ , = 0 , $ , V . @- , - { ? !, ! ' & ' ' . 5 , & ! '
'. B $ ' - '. @ & s0 n- ', . . -! & $ { = { 0 (s ; s0 )n + { 0 6= 0: B 0 0 n;1 + 0 { n { ( s ; s ) 0 0 ~ = arctg { ; 2 = arctg ({ 0 (s ; s0 )n + ); 2 = 0 n + = arctg (s ; s )(1 + ) ; 2
, & , ' '.
0
(& <. X. 8 '
*
& '. 3 , ! $ ! = ! & ! E. :. @ ', & = , &( &
. 8 ! ' E : & ' .
1] Nirenberg L. Rigidity of a class of closed surfaces // Nonlinear problems. | Madison: The University of Wisconsin Press, 1963. 2] . . ) !" !# ! !!$ % & %!' () * ) %!! // +! 2-* - .* !"! " * %!! '. | / %0 , 1967. &) /%!% * ") % & "% ! ) !" !# ) !!) // 2 . !. | 1968. | +. 3, ". 4. | 5. 403{413. 3] % 8. 9. ! !" !# ! % & %!' () %!! // :0#! * %!! ' 0$ "*% ; *&%!). 2; !) &%! # % 0. | 5.-/ %& %*, < !) !) * 0. "0**!# !) !% ! !. . =. %', 1991. | 5. 15{19.
977
4] Fenchel W. On the di?erential geometry of closed space curves // Bull. Amer. Math. Soc. | 1951. | V. 57, @ 1. | P. 44{54. 5] ! C. . D!EE%'!($ * %!$ ! "*!$ %! . | 2.: G , 1987. 6] 8 *(' G. G. ) % % !# ) !!. | 2., 1967. ' ( 1995 .
. .
517.982+512.625.5+512.546
: , , !
"#.
$ % & , ( # ! )# X . ) ) *) "# *# +, #- X . $ ( ! # , ( X .
Abstract S. V. Ludkovsky, Non-Archimedean free Banach spaces, Fundamentalnaya i prikladnaya matematika 1(1995), 979{987.
The article is devoted to the investigation of free Banach spaces over nonArchimedean 1elds, generated by ultrametrizable spaces X . Their isomor1sms with spaces of integrable functions are proved for compact X . Non-Archimedean free locally convex spaces generated by uniform spaces X are also considered.
1] ()
R C , K .
B(X x K) K ! ! x X (") #$! # % # . & X | $ d(x y) 6 maxfd(x z) d(y z)g ) x, y z X, K * ! , K . $ , ! $ ) *$) # ! ) #$!
1]{ 6]. - . / ) , % $ 1] 0# : , ! B(X K) # % L1 (X F m K) $ (2 )? & X | ! , F | - * X, m | , K | R C . 4, 5] 6 / $! $ ! X 2 (2 ), * X ( 2 4, 5]
! 3], ! $! ! X ). 6 / X. 1995, 1, 2 4, 979{987. c 1995 !" #!" "$ %&',
\) !! !"
980
. .
$! K # $. , #$ X / # % # $ %$0 L(X F m K) m #! K ( . 7 7]). < . $! 2 . 6 # 0 !) $! R C . $ #! d (R(d)) * R(jjK ) =: ;(K) K, #! d(y A) := inf fd(y z): z Ag A X, sp (X K) :=fz = a(1)x(1)+ + + a(n)x(n): a(j) K x(j) X@ j = 1 : : : n@ n Ng. 1. . B(X y K) =: Bm # ", $ 1{ 3: f1g $.$ # ! * v: X ! Bm@ f2g sp (X K) )$ Bm@ f3g ) f: X ! H, . !. kf(x) ; f(z)k 6 d(x z) x, z X f(y) = 0, H | K, $.$ F : Bm ! H, . !. kF k 6 1 ! v(X): F()jv(X ) = f(v;1 ()). B* f f3g # ) ). . 2. . B(X K) =: B # , $ f1g, f2g f4g: ) H ). f: X ! H jf(y)j < 1, y | ! ! X, $.$ F : B ! H, . !. F (x)jv(X ) = f(v;1 (x)) kF k 6 1. C , h(x) = d(x A) A X h(x) = jd(x z) ; d(z y)j ) ). %$0 #! ;(K). 3. . h: X ! ;(K) | , . . jh(x) ; h(y)j 6 d(x y) x y X , ;(K) n f0g .
f: X ! K , . . jf(x)jK = h(x) x X. . < -# $ * (B&) X(n) :=fz X: d(z y) 6 r(n)g n < r(n) < n + 1. D inf fd(x y): x=/=y X g =: q > 0, X . $ q = 0. - ) bS= 1=t > 0 $.$ #F) # X 0 (n) B& X(n t j), . !. fX(n t j): j g = = X 0 (n t), X 0 (n t) X(n), supfjh(x) ; h(z)j: x z X(n j)g < b, . . $ $S;(K) n f0g , X 6 B(m), fX 0 (n t): n Ng X, t # N, a m = w(X) | ! X ( . 8]). , $.$) f(n t ): X(n) ! K jf(n t x) ; f(n t z)j < 2b ) x z X(n t j) jjf(n t x)j ; h(x)j < b ) x X(n), * # f(n x) = tlim !1 f(n t x) . ff(n t x): t Ng, * ). %$0 g, X(n), * ). %$0
X(n) $ g. - , h ). * X X, 6 $ ! X. " * X(n) . G* X 0 (n t) X(n). 4.3.26 -
T
981
) 3.9.4 8] ! fX 0 (n t): t Ng X(n). 6 $ ) n > m x X(m) f(n x) = f(m x), , ff(n x): ng ).$) f(x) X, SfX(n): ng = X. 4. . (X d) | , ;(K) nf0g , Y | X , f: Y ! K | , ! f: X ! K . . $ S | # $ * R h: Y ! S | ). * , h * ). * h: Y ! S. H 3, * , ! X Y . < g(x y) :=(h(x) ; h(y))=d(x y) Y 2 n D =: Q, X 2 n D := P, D :=f(x y) 2 X 2 : x = yg. P #$ , g Q, ! Q # $ P , Y # $ X. " 0 { I (2.1.8 8]) g * X Y n D, Y X n D P, . !. jgj 6 1. J# 6 * x y, * g ($ % 0
# 2.1.8), ! h ). * X. ! h * # h(x) := jf(x)j, # 3 $ $. ). * f: X ! K. 5. . " (X d), y K # # # B(X K), B(X y K) # (Y ; v(y)), Y | # $ # # 1 B(X K) K . & B(X K) # # B(X K),
B(X y K) | B(X y K), X | X . ' X , N(X) B(X K), . . X # B(X K)=N(X) () . . $ ! J | K ;(J) nf0g, X(J) :=fx X: d(x y) (J)g, X(J n) :=fx X(J): d(x y) 6 ng. D ;(K) nf0g , * # J = J(t) ;(J), #$). 1=t- ;(K) \ 0 n], n t N. C X(J) * # $) $ $ d(J@ x z) := inf fs: d(x z) 6 s s 2 ;(J)g. " d(J@ x z) X(J) #! ;(J). B#! !# E(J) ). f: X(J) ! J f(y) = 0, F $ #! * sp X(J) := sp (v(X(J)) J), . !. v(y) = 0 X(J) nfyg | # L sp X(J), # J = J(t). B f5g: kz k := supfjF (z)j: f E(J)g, z(i) = v(x(i)), z = a(1)z(1) + + a(n)z(n), x(i) X(J), a(i) J, i = 1 : : : n, n N, , jF(z)j 6 maxfja(i)j jf(z(i))j: i = 1 : : : ng < 1, kz k | , $.$ ). f(x) jf(x)j = d(x fx(2) : : : x(n)g), f E(J) 3. - % ! J M{ 7] $ R(d) ;(J) sp X(J) B(J) = B(X(J) y J). 4 B(J n) := B(X(J n) y J) # ! ) B(J). 5.13 5.16 7] $.$) ( ) # B(J) 0
982
. .
T(J@ n m): B(J n) ! B(J m) ) n > m, * T (J@ n): B(J) ! B(J n). $ P 0 :=fJ: J | % ! K ;(J) nf0gg, P 0 : J > L, J L ( ;(J) ;(L)). - , ) # ; R (. !. ; n f0g | $ $) ! k $.$ K ;(K) = ; B(K 0 1)=B(K 0 10) = k, B(K 0 r) :=fx K: jxj 6 rg, B(K 0 r0) :=fx K: jxj < rg ( . A.9 9]). ! ) K $.$) % * 6 H(K) :=fh(s): s S g F P (K) :=ff(i): i I g, . !. H(K) % #0
k, jF (K)j = ;(K), a = fh(s)f(i)s(il) : s = s(i l) l = l(i) Z i I 0 g, I 0 | ! * I Q :=fjf(i)s(il) j: i I 0 g
sup Q =:q 2 Q. 6 $ L K $.$ ). * P(K L): K ! L, * # H(L) H(K) F (L) F(K). " P(K L) T (J@ n m) $0 $) ). T (J L@ n m): B(J n) ! B(L m) n > m J L, #! (J n) > (L m), . . P 00 := P 0 N . < B := lim S 0 S 0 := ; := fB(J n)@ T (JL@nm)@ P 00g . !
, . . x :=fx(J n)g
# S 0 ( . . 2.5 8]) kxk := supfkx(J n)k: (J n) # P 00g. $ S 0 % S 00 , !). $ t = t(n) = n J(t) X(J n) /, ). # % . $ Y | K f: X ! Y | ). * , f: X(J n) ! Y (J n) | ). * ) (J n) # P 00. & Y (J n) * * Y , *. cl sp f(X(J n)) | # ! f(X(J n)) K, . !. Y (J n) | J kyk := inf fq: kykY 6 q q ;(J)g. < # J 6 # Y $ H$-O, * , ! (J n) > (L m) ) ). 0 Q(J L@ n m): Y (J n) ! Y (L m). 6 Y 0 := lim ; fY (J)@ Q(J L)@ P g . !
# % Y 2 . " # , f * S 00 $). fY (J n)g ). * f(J n@ x): B(J n) ! Y (J n). $ * # % % $ / n, a f(X(J n)) B 0 (Y f(y) n) ) (J n) # P 00 c(n) kz kY (Jn) 6 kz kB(Jn) 6 kz kY (Jn) z # B 0 (Y 0 n) \ Y (J), nlim !1 c(n) = 1, ; 1 (kT (J L@ n m)k kT (J L@ n m)k) (c(n) 1=c(n)), $.$ 00 ). F := lim n m)@ P 00g. B) ; ff(J n@ x)@ P g: B ! lim ; fY (J n)@ Q(J L@ S $,S ! B # % " B(X y K), fJ: J # P 0g K, f;(J): J # P 0g = ;(K), B 0 (Y y r) :=fx 2 Y j kx ; ykY 6 rg. D B(X y K) $ # * * * id: X ! X. 6. . f: Y ! X | ! -
983
(Y d0) (X d) (. . d0 (x z) > d(f(x) f(z)) x z # Y), $$ F: B(Y K) !B(X K) kF k 6 1. ' f | # ! , F | # ! . ' Y X , B(X K) B(Y K) # # . 7. - L(X m K) #! $ f: X ! K m | !- X #! K, # m S supp m := fA(j): A(j) | m j 2 V g, V | * N 7]. $ (X d) , R(d) ;(J), Ind X = 0
X | ( ) * D@0 , D :=f0 1Pg 8], , ! !- m m(A) = fm(a): a 2 A \ H g, H ! X, H :=fa(n) J: n N
nlim !1 jm(a(n))j = 0g. " B(X J) # % L(X m J) (4.19, 5.16 Ch. 7 2]), ;(J) nf0g , X sp (v(H) J) B(X J). ! # % # T: B(X J) ! L(X m J) : (kT k kT ;1k) c(J) 1=c(J)], c(J) := supfjxj: x J jxj < 1g. R# c0(fB(g): g S g) #! 6 f = ff(g): g S g, S | *, f(g) 2 B(g), kf k := supfkf(g)kB(g) : g S g ) b > 0 * fg: kf(g)k > bg !, B(g) | . 8. . (C d) (X n C d) | # (G d0) & $ d0 (. . 0 < a < b < 1, . . a d(x z) 6 d0(x z) 6 6 b d(x z) x z # X n C ), ,- W G d0(g h) = 1 gW \ hW = ?. m # K $ - Bf(X), . . B(X K)
# L(C m K) c0(fL(gW \X nC m K): g S g), ! $ x m
# X ,- W(x) - S Bf(W(x)), gW \ hW = ? g=/=h # S , fgW : g # S g = G. 0 m - ( ), X , . . B(X K) # L(C m K). . $ X(J n) *, # 5, X . " * m, . !. L(X(J n) m J) B(J n) # % . J# X $, ! $.$ ! S S 00 = fJ ng fX(J n): (J n) # S 00 g X, 6 $
$0
* # m )$ * X, m X(J n) n X(L j) (J n) > (L j) #! J. J #$ / J B(L j), 6 . !
L(X(J n) m J) B(J n), $! # % # L(X m K) B(X K). - , f6g: kf k = supfjf(x)j N(m x): x X g, N(m x) := inf fkV k: V x V g, supp m = H, H ! X, X $.$ / , ).
$) M$%$ ).
984
. .
S . $! * W (x) ! # (X n C), # ) #$ $ G, ) ( ! ) $ d G # ! # % # B(gV K) B(V K) V G, g 2 G. - , G #F) F B& * gW g # S. H* B(gW \ X n C K) # % L(gW \ X n C m K) . - m gW \ X n C, B(gW \ X n C K) # ! ) B(gW K). sp (X K) B(X K) . ) 6 H/ $ $ # % # $. 9. !. $ N # : d(n m) = 1 n=/=m d(n n) = 0, B((N d) K) # % c0 (N K). D N # d0, $ # * N Zp ! *, ;(K) ;(Zp), B(X K) # % L(Zp m K). $! % ! K ! * l1 (N K) Y h, ) m, . !. N(h x) 6 E N(m x) x, E | h m ( . 7]). $ supp m !, Y | K, l1 (N K) K. , B((N d) K) B((N d0) K) # % K- $ . 10. $ X | ) # IP $ d R(d) ;(K). " ! 5] * $ $!. D X , % Xd, $! % #0 / ) 6 fx y: d(x y) = 0g, ) ! $ , #! * d. X fXd: d # IP g
fm(d ) Bf(Xd): d # IP g, $! # $ m A(X) 0 ! * X ( . 10]). " * L(X A m K)
$ # m 0 ! %$0 f(x) = h(z), T (d@ x) = z | % * X Xd, h | $ m(d ) %$0
Xd kf km = khkm(d) . % * T(d d0@ ): Xd ! Xd0 # a(d0 j) # ) #F) F fa(d i): i J g (Xd m(d )), J = J(j) N, m(d ) | # m T (d@ ). . $! m - , A | - . J #$ , L(X A m K) lim ; fL(Xd@ m(d )@ K)g. T ! 5] . ) 5 $ $. 2 M(X K). B) $! 1 # 5] $).$) $. 11. . ' X | -
985
# $ $ IP , K | ;(K) d(X X) d # IP , # m # K A ! X , . . M(X K) # L(X A m K). 12. - $ ! $! . $ X $ $ ) (C): X | ! d, $0 # # * S, !). 6 $ # * ) S = fX(a) (a b ) Ag, X(a) | 6 , X = lim ; S, A | ! *, n(a b ): X(a) ! X(b) | $!%% * . H* ! # * 11, 12], $ d , , X (locally Une), . !. * ! (uniform reUnement, . 3.2 11]). 13. . X (C), B(X K) # L1 (X F m K), . . m | - , m(X) = 1, supp m = X , F | - , F Bf(X). . B(I nK) # % S:=L1(I nF(n) lnK), I = 0 1] P= fR 3 x: 0 6 x 6 1g, I n jx ; yj = fjx(i) ; y(i)j: i = 1 : : : ng, I n 3 x = (x(1) : : : x(n)), ln | 2 I n , F (n) | - , *. $) - $ Bf(I n ). - , %$0 ) g(x t) = 1 0 6 t 6 x g(x t) = 0 x < t 6 1, I 3 x t, , g: I ! L1 (I F(1) l1 K) # ! * . 2 ) %$0 * ). f: I ! K $ Z 1 Z x %$0 F: L1 (I F (1) l1 K) ! K, . !. F(t)g(x t) dt = f(x) = F (t) dt. "-
Q
0
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B(IPn K) # % Y = f(B(I K)): i = 1 : : : ng Y : kz k = fkz(i)k: i = 1 : : : ng, z = (z(1) : : : z(n)), B(I K) 3 z(i), Y 3 z. n n Y # % LP 1 (I F(n) ln K ) =: Z. D f = (f1 : : : fn), jf(x)j = fjfj (x)j: j = 1 : : : ng, 6 $ Z # % Lfj :fI(S)!i :K,i =Z13: :f,: ng =:X. B! , ! X # % S Z. B) $, ! B(I n K) # % S, B(Rn K) # % L1 (Rn F(n) ln K), ln F (n) # Rn. - * # J H# 11, 12], . !. * X(a) | ! , , ! F * X(a n), % I h(a:n)
! , N 3 h(a n). " # % # B(X(a) K) = = L1 (X(a) F (a) ma K)=str-ind-limfL1(X(a n) F (a n) mX (an) K)g (| $ , . II.6.6 13]) fma : (a b ma ) = = mb 8a > bg !$) # $ m, , . .
986
. .
X | <, mX (an) = ma j X(a n), F(a) Bf(X(a)), F(a) \ X(a n) = F (a n) 10]. < ) #/ # . 14. #$ . " $.
# % # L(X m K) $! $ #$ X K * * # . ) V -J -H# 6 # * . 6 * # , ! ). %$0 f: B 0 (L 0 1) ! L
! / B 0 (L 0 1) )$ ). , L | . J# $. 2 , * 14] 15. . K | R, C . $ (X d) d(X X) ;(K) ( ) 12 E(K) 3 B((X d) K) # f: (X d) ! (Y d0) | ! , ! F: B((X d) K) ! B((Y d0 ) K) ( G(K) | 1() ), & E(K) $ $ G(K). 4 12 1() f(x) = f(y) x # X y # Y .
1] . . // . . . | 1986. | %. 20. | . 81{82. 2] ) *. +. . I{III. | ,.: ,, 1976. 3] 12 3. 4. 5 6 // ,. . | 1964. | %. 63. | . 582{590. 4] +9 2 . . 5 , : 5 6 6 6 // ;. 6. . | 1993. | %. 48, < 2. |
. 189{190. 5] +9 2 . . 5 =6>=6 // ;. 6. . | 1994. | %. 49, < 6. | . 207{208. 6] Flood J. Free topological vector spaces // Dissert. Mathem. (Rozprawy Mat.) | 1984. | V. 221. | P. 5{99. 7] Van Rooij, A.C.M. Non-Archimedean functional analysis. | N.Y.: Marcel Dekker Inc., 1978. 8] B5 1. EG . | ,.: ,, 1986. 9] Schikhof W. H. Ultrametric calculus. | Cambridge: Cambr. Univ. Press, 1984. 10] 3 2 K. +., 6 . . , >> 5 6 . | ,.: * , 1983.
987
11] Corson H. H., Isbell J. R. Some properties of strong uniformities // Quart. J. Math. | 1960. | V. 11. | P. 17{40. 12] L= 2 O. ,. 49 T5 =: 6 // % ,,E. | 1979. | %. 40. | . 83{119. 13] X> Z. Z. % . | ,.: ,, 1971. 14] ) O., 3 4. 9 2 > . | ,.: ,, 1972. + #, 1995 $.
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Abstract V. M. Petrogradsky, On types of overexponential growth in Lie PI-algebras, Fundamentalnaya i prikladnaya matematika 1(1995), 989{1007.
The growth function of identities n (V ) for varieties of Lie algebras is studied0 where n (V ) is the dimension of a linear span of multilinear words in distinct letters in a free algebra (V ) of the variety V . The main results are as follows: the description of types of overexponential growth is suggested0 the growth of identities for polynilpotent varieties is found. A complexity function C (V ) is used0 it corresponds to any nontrivial variety of Lie algebras V and is an entire function of a complex variable. c
c
n
F
X
z
1
H | K. F = F(X) | X = fxi j i 2 Ng. ! 0 6= f 2 F # H, ai 2 H i 2 N % : F ! H (xi) = ai i 2 N (f) = 0. I F, ' V , # I, . (# # V (X), V , F(X), # F(V X) = F (X)=V (X) | V . ) *, 8H 2 V 8 !!$ , $ 9 $ $) $ !$) #$ ), $ M22000. 1995, 1, : 4, 989{1007.
c 1995 , ! \# "
990
. .
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,
# . ) * ' % -1]. Pn(V ) F(V X) | # * 0 x1 : : : xn. 1 , Pn (V ) * ' x1 : : : xn (
, F (V X)), ' , '# ' xi i = 1 : : : n . 2 V ' %'3 #
cn(V ) = cn(F(V X) X) = dimK Pn(V ) n 2 N: 4 , 0 xi1 : : : xin # . cn (V ) # ' ' V . 1 3 # . ) # 3
.
1.1 ( , 2], . 1]). V | . C , cn(V ) 6 C n n 2 N:
V ! d, ! C = (d ; 1)2. 2 , 0' 3 * \ " 3 . 7 * | \8' " #
. 4 Nsq Ns2 Ns1 , (sq : : : s2 s1). 9 q = 1, Ns1 , # T1 (x1 : : : xs1 ) = -x1 : : : xs1 ] 0: ( ' ' : -a1 : : : an]=-: : :-a1 a2] : : : ak ].) 9 # Tq;1 (x1 : : : xt),
Nsq;1 Ns1 , Nsq Ns1 # Tq (x11 : : : xtsq ) = = -Tq;1 (x11 : : : xt1) Tq;1 (x12 : : : xt2) : : : Tq;1(x1sq : : : xtsq )] 0 xij 1 6 i 6 t 1 6 j 6 sq | 0 . (
Nsq Ns1 ' H, 3 ' 0=Hq+1 Hq : : : H1 = H, Hi=Hi+1 2 Nsi . : , MV * | 0 ' ' L,
991
H L, '*, H 2 M L=H 2 V . : sq = : : :=s2 =s1 8 Aq . 4 . 2 A2 ' , cn(A2 ) = n ; 1. ; , %'3 cn ' n. < * : 0 , # Ns A, # N2 A -3], -4]. 9 9a b > 1 : 9N an 6 cn(V ) 6 bn n > N " . 4 * , # # 0' 3 . p ?
, @. . ( ', limn!1 n cn (V ) < 2 (. . cn < 2n ' n), . < B ' | 0 0' 3 , ' , '
* . 4* ' 8 -5]. ; * Wk = var(Wk ), # D ' p Wk , 0' 3 . ?
limn!1 n cn(Wk ) = 2k+1k, -6], -7], . '# -8]. !' , * 8 0' 3 8 ,
. -5]. : 3 (. 1.1) # AN2 " -9] (. . ''* 0' *). < '# ' 3 ' %'3 . 4, * F = F(X)
cn(F) = (n ; 1)! <
1.2 (10]). V | #. 8a > 1 9N 8n > N cn (V ) 6 an!n : H. . ; $% ! V : 1 X C (V z) = cnn!(V ) z n z 2 C : n=1 I 0' 3* * %'3 -11]. ! * # % -12], -8]. 1.3 (8]). V #, C (V z) | $ . H. . ; # %'3: q1 (z) = 1J qs+1(z) = exp
Z z 0
qs(z) dz s 2 N:
P P1 n n 2 %'3* f(z) = 1 n=0 an z g(z) = n=0 bn z , f(z) g(z), ' an 6 bn n 2 N.
992
. .
1.4 (8]). V | #, s 2 N, C (V z) qs (z). 4 8 ' 8 * * - -8]. 9 1.4 %'3 # , * , '' 8* %'3 * #. 1.5 (8]). & % s 2 N - V , C (V z) qs(z).
2
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D' 8 , 8 0' PI- J ' # AN2 0' 3 . : ' 8 0' 3 , ' 8 . 48 3 : ' 0' 3 J , 8' ' . ; %'3* Lq (n) q = 2 3 : : : : 8 ; ;1 > > < n! q L (n) = > n! > : ( q ;2) n= (ln n)
> 1J q = 2J > 0J q = 3 4 : : :
ln(1) n = ln n ln(s+1) n = ln ln(s) n s 2 N. : q = 2 '#
%'3 ; ;1
Lq (n) = n! n= > 1J > 1: f(n) | %'3 ,
8 0 %'3 . (n) | %'3 , . c %' . : : f(n) c c (n) () inf f > 0 j f(n) (n)g = c f(n) c c (n) () inf f > 0 j f(n) (n)g =supf > 0 j (n) f(n)g =c f(n) 6c c (n) () inf f > 0 j f(n) (n)g 6 c f(n) g(n) %'3* , 9N 8n > N f(n) 6 g(n). 4 f(n) c c (n) () 8" > 0 9N : c;" (n) 6 f(n) 6 c+" (n) n > N f(n) 6c c (n) () 8" > 0 9N : f(n) 6 c+" (n) n > N: )* * .
993
2.1. V = Nsq Ns1 , q > 2 | -
. 1.
8 s1;1 > < (n!) s1 cn(V ) s1 Lqs1 (n) = > n! : ( q ;2) n=s1 (ln n)
q = 2J q = 3 4 : : :
; s1 ;1
2. cn(V ) s2 L2s1 s2 (n) = n! s1 (s2 )n=s1 q = 2: : * * . 2.2. V | #. V ! m 2 N. cn (V ) 61 Lm1 (n) = (m;n!2) n=1 : (ln n) D' # , 0 % ' 1.4 ' '0%%3 . 2.1. q s 2 N, 2 R+ . 1. Lq (n) + Ls (n) Lq (n), q > s) 2. Lq (n) Ls (n) Lq (n), q > s) 3. Lq (n ; 1) Lq (n): . 4 , Lq (n ; 1) = Lq+"(n)(n)J lim "(n) = 0: n!1 I ' 8 . 2
V | . 2 q = 2 3 : : : ! q : Compq V = inf f > 0jcn(V ) Lq (n)g: 9 0 Compq V = 0 < < 1, ' , Compq;1 V = 1 Compq+1 V = 0. 0
q ! V , (q ) | ! % V . 2 q = 2 = Comp2 V # '# % ! Comp2q V = inf f > 1 j cn(V ) L2 (n)g: : 0 Nsq Ns1
# (q s1) ( (q s1 s2) q = 2). 2 q = 1 '# , Ns
# (1 s). : ' .
994
. .
!" 1. V M % ! (q1 ) (q2 ) . + , VM ! (q1 + q2 )? 9 V = Ns , 4.1. < , # #, . . Compq V = 1 Compq+1 V = 0? 9 2.1, 2.2 3 , 0' 3 3 * . : * # 1.1 ; # , 0 . ) , B ' '# ! . 1 , A(X) | , # ' # X = fx1 : : : xng. A(X n) | * '
, n # X. : ' %'3 A (n) = A (X n) = dimK A(X n) n 2 N: 4 , 3 0 %'3 0' *. 9 A | 3 , Q 8 -13] %'3 A (A) . )' , PI- ' 3 , # # 0' 3 -14]. !'#
-15] V * L = Fk (V ) k * ' %'3 L (n).
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C (A z) =
1 X cn z n n=1 n!
( n = 0 c0 = 1 3
3 *). 3.1 (11]). ; | ( ), !/ ! X = fxi j i 2 Ng. & , ! ( )
995
A ; B ; ! $ ! . ! " a b a 2 A b 2 B ! A B ; (/ )) ( ). A B $/ ! , C (A B z) = C (A z)C (B z).
8 cn (A B) =
n n X i=0
i ci (A)cn;i(B)
'0%%3 # # fx1 : : : xng # . ) ' 8 . 2 :# , H. . ; , . 3.2 (8]). L = F (V X) | # ( #, / ! $% ! ), ! ! X = fxi ji 2 Ng. / %/ U(L) ! ! $% ! , C (U(L) z) = exp(C (L z)). f(z) ' ' . ) Mf (r) = maxjzj=r jf(z)j. ! ' ( # ') -16], -17]: ord f = inf f j 9C1 C2 8r > 0 Mf (r) 6 C1 exp(C2r )gJ typ f = inf f j 9C 8r > 0 Mf (r) 6 C exp(r )g: 2 ' * # %'3 #. ; ' %'3: 0(z) = z i+1 = 1 + (i (z) ; 1) exp(i (z)) i = 0 1 2 : : :
3.3. 0 1 Aq %% $%
! :
C (Aq z) = 0(z) + : : : + q;1(z):
. 4 * -18]. F (xiji 2 N) | . ?' xi R0-. 9 u v | , uv | , u > v
u = u1u2
'# u2 6 v. 0 u v Ri Rj . ! uv Rk -, k = maxfi j g i 6= j k = i + 1 i = j. 0 ' ': 1) u | Ri- v | Rj - i > j, u > v. 2) u v | Ri -. 9
996
. .
u 8 v, u > v. 9 # u = u1u2 , v = v1 v2 , u > v, u1 > v1 u1 = v1 u2 > v2 . 4, R0 = fxi ji 2 Ng. 4 -18], Ri+1- -vi1 vi2 : : : vik ] vi1 > vi2 6 : : : 6 vik k > 2 vi 2 Ri : (1) F (0) = F F (i+1) = -F (i) F (i)]. ! Ri- F (i)=F (i+1) -18]. 2'# '3, C (Ri z) = i (z). ) , C (R0 z) = z. T , Ri+1 #
f(vi1 : : : vik ) j vi2 6 : : : 6 vik k > 2 vi 2 Rig n n f(vi1 : : : vik ) j vi1 6 : : : 6 vik k > 2 vi 2 Rig ' , C (Ri z) = i(z). ! , # '-
3 3.2, 3.1,
C (Ri+1 z) = i (z)(exp(i (z)) ; 1) ; (exp(i (z)) ; i (z) ; 1) = i+1(z) ;1 C (Aq z) = C (F=F (q) z) = Pqi=0 C (Ri z) = 0(z) + : : : + q;1(z): ) ' # . 2 2 %'3 # ' . U'3 # *
' '0%%3 cn .
3.4. & Nc A c 1 X
i
1 + exp(z)(z ; 1) i=1 i . F (X) | . ! ' F 2 * * , # * # -1]. Y = f-xi1 xi2 : : : xik ]jxi1 > xi2 6 : : : 6 xik k > 2g: < * , # %'3 #:
C (Nc A z) = z +
C (Y z) = C (R1 z) = 1 (z) = 1 + exp(z)(z ; 1):
(2)
F1 | , # # A, %'3 * # h(z) = C (A z). ! * *
997
c 1 X
H = F1 =F1c+1
C (H z) = (h(z))n . 2 * , 3n n=1 A- n
%'3 (h(z))n . 3' ' n ' 0 , ' * 0 Q 8 , . -19]. ) % C (H z). 0 %' ' F (Nc A) = F=(F 2)c+1 , c X C (Nc A z) = z + 1i (C (Y z))i i=1
(2) ' . 2 3.5. & L = F(Nc Nd ) $ ! C (L z) | $ d c=d. . F(X) | . ! F d+1 | . : %'3 # # Y
. ; = ;(X) | * 3 # X = fxi j i 2 Ng | 3 . 2 # 0 ; c ' ' F(X) # xi ! xi, i 2 N. @ deg x 0 x 2 ;
. T , S = ; \ F d+1 | * ;. R | V F(X), . . u < v u 8 vJ ' , xi < xj i < j. ! u 2 RnS v 2 R \ S , u < v. I w 2 S S- ' , ' w = u v u v 2 SJ S- . (# Y S- 0 R \ S
#
# F d+1 -1, 2.4.2]. ) Z = R n S = = fw 2 R j deg w 6 dg. 9 w 2 Y w = u v u v 2 R, V u > v, v 2 Z. : # : u 2 Z, u 2= ZJ u = u1 u2 u1 > u2 u2 6 v, u2 2 Z. : # 3 # ' u1 . !' , w 2 Y #
w = -zi1 zi2 : : : zin ] zij 2 Z n > 2J (3) zi1 > zi2 6 : : : 6 zin -zi1 zi2 ] 2= Z -zi1 zi2 ] 2 R: (4)
' '# , 0 P # Y . )3 C (Y z). ) (z) = C (Z z) = di=1 z i =i. Y1 | # * 0 (3), (4) . ! 3.3 d i X (5) C (Y z) C (Y1 z) = 1 + ( (z) ; 1) exp( (z))J (z) = zi : i=1
998
. .
2 3 ' %'3 C (Y z) # Z=fx2Z j deg x=dg, C (Z z) = z d =d. T , zij 2 Z (4) . < # Y = f-zi1 zi2 : : : zin ] j zij 2 Z n > 2 zi1 > zi2 6 : : : 6 zin g Y 3 ' d d C (Y z) C (Z z) = 1 + zd ; 1 exp zd :
(6)
< 3 ' (5),(6) ' , ord C (Y z) = d typ C (Y z) = 1=d. 1 3.4 H = F d+1 =(F d+1 )c+1
C (H z) =
c X C i (Y z)=i: i=1
2 L = F(Nc Nd )
L = F=(F d+1 )c+1 , C (L z) = (z) + C (H z)
' d c=d. 2
# $ ! 3.1. & L = F(Nc Nd ) cn (L) c (n!) d;d 1 n=d : n . @ -17, 3.2.3] 3 * %'3 f(z)= P1 n=1 anz
'
8
p
limn!1n1= n janj = (e )1= : P
n 4, 8 C (L z) = 1 n=0 cn(L)=n!z . )
cn(L) c n;n=d(ce)n=d n!
p
% @ nn n!en= 2n * . 2
# $ ! 3.2. 3 L = F(NcNd ) cn(L) d (n!)
d;1 d :
# $ ! 3.3. & V = Nc A cn (V ) c cn .
4 %!
: ' 3.2 ' -8]: X n! k1 kt cn(U(L)) = k1!(n1!)k1 : : :kt!(nt!)kt cn1 (L) : : :cnt (L) k1 n1 +:::+kt nt=n n1 >n2>:::>nt
999
(7)
%$ &! 4.1. & V = Nc Nd ; d;1
cn (V ) d n! d = L2d (n): . : 3 ' 3.2. Le = F(Nc Nd ), L = F(ANd )J L Le =(Led+1 )2 , # * e > cn(L). T, ' # 3 ' cn (L) * L = F(ANd X), X = fxi j i 2 Ng. H = F(Nd Y ), Y = fyi j i 2 Ng hZ i = hzi j zi 2 Z i | Z = fzi j i 2 Ng. ; L1 = Z wr H = Z H, Z = Z U(H) -1]. @ # : L ! L1 , #
# : xi 7! zi + yi i 2 N. i : Z ! zi U(H) | '3, ' , n (Pn(L fx1 : : : xng)) = zn Pn;1(U(H) fy1 : : : yn;1g)J (8) cn (L) > cn;1(U(H)): 2 3 ' cn (U(H)) (7) ,
t = 2 n1 = d k1 = -n=d]J n2 = 1 k2 | ' , ' (7). 4, cm (H) = = (m ; 1)! 1 6 m 6 dJ cm (H) = 0 m > d. 2 3 ', ' ' *
% @ : ; n! = n! 1;1=d+O(1=n) = (n!)1;1=(d+O(1=n)): (9) cn(U(H)) > dn=d(n=d)!d! (8) n;1 (9) ( 0 8 ' 2.1), 3 '. 2 4 * %':
1. $ f : -a b] ! R+ 0 6 a < b, g00 (x) > 0, g(x) = lnf(x). x+y=max f(x)f(y) = f(b). b xy>0
4.1.
2. f(x) = x!(ln(s) x);x= % , a , f(x) .
1000
. .
3. & $ g(x) = x ln(s) x s 2 N min
y1 +:::+yt =x yi >0
g(y1 ) + : : : + g(yt ) = tg(y)jy=x=t = x ln(s) (x=t):
. 1) ; h(x) = g(x) + g(b ; x)J h0 (x) = = ;g00 ()(b;2x), 2 (x b;x) , h(x) -a b=2]
-b=2 b]. 3) * 0' . 2 4.1. V | #, cn(V ) Lq (n), q > 2, > 1 q = 2 > 0 q > 3. M = Ns V cn (M) Lq+1 (n). ; * . %$ &! 4.2. V | #, cn(V ) L2 (n) > 1. M=NsV cn(M) L3 (n). . L = F(M), H = F (V ) | . 0 <
cn(H) > L20 (n), ' n > N. 9 '# , cn (L) > L30 +o(1) (n), ' , ' # 3 '. 2 ' 8 ,
cm (H) > L2 (m) = (m!)1;1= m > N: ; (7) ' : t = 2 n1 = m > N k1 = -n=m]J n2 = 1, k2 | ' , ' . ! , 3 ' a! < (a=e)a a, n! n! > J cn(U(H)) > n (m!) m (n=m)!m! f(n m)h(n m) (10) n h(n m) = m m n=m m! n= n n=m f(n m)= me
em 1 : ln f(n m) = n lnm ; m1 + ln(n=m) ; m
(11)
: * ' , (11) %' n '
m = (m ) (x) = ln(n=x): (12) T , # (x) # 8 (12) 3*, ' 3 ' 8
1001
' m0 = ln n mi+1 = (mi ) i 2 N lim m = m J i!1 i m0 = ln n > m > m1 =(ln n ; ln ln n ; ln )
ln ln n m = ln n + O ln n :
(13)
' m (11) ' ' (12). < (13), : ln f(n m) > lnf(n m ) = n lnm ; m1 = n lnln n (1 + o(1)) (14) lnn J ln(f(n m )h(n m )) = n lnm ; m1 + ln h(n m ) = n ln + o(1) n! n > N: (15) cn(U(H)) > n= (lnn) (+o(1)) D' 8 , Ns = A, '# (8) cn(M) > cn;1(U(H)). : ' , ' n;1 (15) ' ' . !' , ' # 3 '. 2 * 3 ' #, 0 >
cn(H) 6 L20 (n), n > N. 9 '# , cn (L) 6 L30 +o(1) (n), 0 > # 3 '. Q ' . @ 3 ' cn (U(H)). P P p T , (7)
t < n, ' '' ti=1 ni > ti=1 i > n, ' . (7) ni 6 N
ki > n=(N ln lnn). D' ' , '* # 3 ': n! n! n! ; 3 (16) ki ! > n=(N ln ln n) ! = ; ln n n=o(1) = Lo(1) (n): T, 'P ' (7) #
: ni>N niki > n(1 ; 1= ln ln n). !' , 8 (7) 3 ': X n! Cn k = 1 k1!(n1!) : : :kt!(nt!)kt= J 1 n(1; ln ln n )6k1 n1 +:::+kt nt6n (17) n1>n2 >:::>nt >N N C = max c (H): i=1 ni
1002
. .
2 '# i=1 : : : t # ki !(ni!)ki = > > f(ni ki ni) (
(11) %'3 f(n m) 3 ' a! > (a=e)n ). <
3 ' (14): lnf(ni ki ni) > mi lnln mi (1 + o(1)) mi = niki : ( # N 8, o(1) *8 # . )3 ' (17) ' 3 ' min g(m1 )+ : : :+g(mt )J g(m)=m ln ln m mi >N m1 +:::+mt =en
t X n(1; ln 1lnn ) 6 ne = mi 6n: i=1
@ 4.1 % (17) 3 ': ne=t) = 1 n ln ln n(1 + o(1)): tg(ne =t) = ne ln ln( W (7) ' | * (n), ' 8 ' -20] ; p
(n) 1p exp 2n=3 : 4n 3 I' (17) 2.1 '# , (18) cn(U(H)) 6 (ln n)n=n!(+o(1)) = L3+o(1) (n): @ Ns - W = Z wrNs H, Z = F(Ns Z), Z = fzi j i 2 Ng, H = F (V Y ), Y = fyi j i 2 Ng. <
% ' W = Z H, Z | W , * * * Ns # # -1]: zj hi ,
fhig | * U(H) w u, w 2 Z, u 2 U(H) | # * H Z. ) # : xi 7! yi + zi i 2 N # # : L ! W . ; # Pn(L fx1 : : : xng) Y Z. ( , # zi , # Pn (H fy1 : : : yn g),
, cn (H), 2.1 . < # X' ', , * # : w = -w1 : : : wc] wk = -zik ya(k1) : : : ya(klk) ] 1 6 c 6 s lk > 0 c + l1 + : : : + lc = n fi1 : : : icg fa(k j) j 1 6 k 6 c 1 6 j 6 lk g = f1 : : : ng:
1003
! # # ' -? ' %-: ' , 0 # : w = -w1 : : : wc] wk = zik uk (19) uk 2 Plk (U(H) fya(k1) : : : ya(klk ) g) 1 6 c 6 s lk > 0 c+l1 +: : :+lc =n fi1 : : : icgfa(k j) j 1 6 k 6 c 1 6 j 6lk g = f1 : : : ng: (20) 2 '# c = 1 : : : s
nc zik
cn \ " yj w1 : : : wc, 2.1 # * . 2 (20) w1 : : : wc (19) ': l1 (U(H)) : : : lc (U(H))J l1 + : : : + lc 6 n: Y 3 ' (18) 4.1, ' cn (L) 6 (ln n)n=n!(+o(1)) = L3+o(1) (n) ' . 2 %$ &! 4.3. V | #, cn(V ) Lq (n), q > 3, > 0. M=Ns V cn (M) Lq+1 (n): . L = F(M), H = F (V ) | . 0 <
cn(H) > Lq 0 (n), ' n > N. 9 '# , cn(L) > Lq+1 0 +o(1) (n), ' # 3 '. 2 ' 8 , n! cm (H) > Lq (m) = (q;2) m > N: (ln n)n= ; (7) ' : t = 2 n1 = m > N k1 = -n=m]J n2 = 1, k2 | ' , ' . ! , 3 ' a! < (a=e)a a, n! n! > f(n m)h(n cn (U(H)) > (q;2) n= m) J (21) (ln m) (n=m)!m! h(n m) = n=m m! n n=m f(n m) = (ln(q;2) m)n= em (22) ln f(n m) = n ln(q;1) m + mn (ln(n=m) ; 1):
1004
. .
: * ' , (22) %' n '
m = (m ) (x) = (ln(q;2) x : : :ln(1) x) ln(n=x):
(23)
) # (x) # , 8 (23) 3*, ' 3 ' ' m0 = ln n mi+1 = (mi ) i 2 N limi!1 mi = m J m0 = ln n < m < m1 =(m0 ):
(24)
: # (23) ln(n=m ) (22) , (24), : ln f(n m) > ln f(n m ) = n ln(q;1) m (1 + o(1)) = n ln(q) n(1 + o(1))J (25) ln(f(n m )h(n m )) = n ln(q) n(1 + o(1)) + ln h(n m ) = +no(1) ln(q) nJ cn (U(H)) >
n!
(ln(q;1) n)n=(+o(1))
n > N:
(26)
D' 8 (8), Ns = A, cn(M) > cn;1(U(H)). ' n;1 (26) ' ' ' # 3 '. 2 * 3 ' #, 0 >
cn(H) 6 Lq0 (n), n > N. 9 '# , cn (L) 6 Lq+1 0 +o(1) (n), 0 > # 3 '. Q ' . @ 3 ' cn (U(H)). D' 8 (7)
t < pn. D' ' P 3 ' (16) # # , # (7) ni>N niki > n(1 ; 1= ln ln n). @ 8 (7) 3 ': X n! Cn J n1 k1 ( q ;2) n1) : : :kt !(ln(q;2) nt ) ntkt n(1; 1 )6k1 n1+:::+kt nt 6n k1!(ln ln ln n
n1 >n2>:::>nt>N
N C = max c (H): i=1 ni
(27) 2 '# i = 1 : : : t #
ki!(ln(q;2) ni )ni ki = > f(ni ki ni) (
(22) %'3 f(n m) 3 ' a! > (a=e)n ). <
3 ' (25): (q) ln f(ni ki ni) > mi ln mi (1 + o(1)) mi = niki :
1005
( # N 8, o(1) *8 # . )3 ' (27) ' 3 ' min
mi >N m1 +:::+mt =~n
g(m1 ) + : : :+g(mt )J g(m) = m ln(q) m
t X
n 1 ; ln 1ln n 6 ne = mi 6n: i=1
@ 4.1 % (27) 3 ': (q) tg(ne =t) = ne ln (ne=t) = 1 n ln(q) n(1 + o(1)): D' 8 , (7) 0' (27) , cn (U(H)) 6 (q;1) n!n=(+o(1)) = Lq+1 (28) +o(1) (n): (ln n) ;# # 3 ' cn (L) ' 3 ' *: l1 (U(H)) : : : lc (U(H))J l1 + : : : + lc 6 n: @ 3 ' (28) 4.1 ' cn (L) 6 (q;1) n!n=(+o(1)) = Lq+1 +o(1) (n): 2 (ln n) ' 2.1 ' 3.1, # 4.1 4.1 '3. 2 # $ ! 4.1. V = Nsq Ns1 | , q > 3. C (V z) / . 1 X
. 2 3 * %'3 f(z) = anz n n=1 p
-16]: ordf 6 () 9C 8n: n1= n janj 6 C. : 8
an = Lq (n)=n! = (ln(q;2) n);n=, 0 ' ' . 2 # $ ! 4.2. $
1 1 X X a n n f(z) = z g(z) = exp(f(z)) = bn!n z n : n! n=0 n=0 !, an Lq (n), bn Lq+1 (n).
1006
. .
# # 4.2, 4.3. Z * 8 ' # *: , an = cn(H) Lq (n), ' , bn = cn(U(H)) Lq+1 (n). ' 0 8 ' 8 (7),
1 X cn(H) z n g(z) = C (U(H) z) = '0%%3 f(z) = C (H z) = n! n=0 1 X c (U(H)) n n = n! z , ' 3.2
g(z) = exp(f(z)). n=0
2
' 2.2. ; %'3,
# H. . ; : q1(z) = 1J qs+1(z) = exp
Z z 0
qs (z) dz s 2 NJ qs(z) =
1 X an (s) z n : n=0 n!
2'# '3 an(s) 1 Ls1(n) %' s. 2 s=2
q2 (z)= exp(z), an (2)=1 n 2 N, ' , Zan (2) 1 L21 (n). 1 z X ' s
an (s) 1 Ls1 (n). ! qs(z) dz = bn!n z n 0 n=0
bn = an;1(s) 2.1 bn 1 Ls1 (n). ! 4.2 an(s+1) 1 Ls1+1 (n). V # m, ' # 1.4 , cn (V ) 6 an (m) 1 Lm1 (n) n 2 N ' . 2
1] . . . | ., 1985. 2] Regev A. Existence of polynomial identities in A B // Bull. Am. Math. Soc. | 1971. | V. 77, # 6. | P. 1067{1069. 3] ( ) *. *., +( , . -. T- / , / 0 1 2 /1 1 02 ( 31 , // 45i 6 778. | 1980. | . 3. | 7. 5{10. 4] : ( 7. ;. < 1 3 = 0 1 2 01 , ( ( // . 31( . | 1986. | . 40, # 6. | 7. 713{721. 5] : ( 7. ;. 8 1 3 , // >0 1. (. | 1990. | . 45, # 6. | 7. 25{45. 6] : ( 7. ;. ? 0 1 @ // . (. | 1984. | . 124, # 1. | 7. 57{67.
1007
7] ? . ., . *. < )(, ( / ( / 0,. | * 0 ( , 11 ( 1. ?/C. ; 0 # 168. | 1985. 8] 831/ . ;. 0 ,. | ., 1989. 9] 4 ) ( *. . < 1 3
AN2 01 ( (
2 // D6 778. | 1981. | . XXV, # 12. | 7. 1063{1066. 10] F C( . 6. < 1 3 , // . 31( . | 1988. | . 4, # 1. | 7. 51{54. 11] F2 G., D( D. ; ) 2 = (1 (. | .: 6(, 1990. 12] 831/ . ;. < 1 3 , 0 , // 4 ( F>. | 1988 | # 4. | 7. 75{78. 13] I( ?. ., 7 2( . ., J( *. ;., J C . *. ?25, 3( ( 5 /1. | .: 6(, 1978. 14] ; ( , 4. . < ( / 0 0 1) // >0 1. (. | 1993. | # 5. | 7. 181{182. 15] Petrogradsky V. M. Intermediate growth in Lie algebras and their enveloping algebras // J. Algebra, to appear. 16] J . 4. 4 (10( /, 3. . 1. | .: 6(, 1985. 17] - O . . 10 )( 5 ( 5/ O (5
. | .: 1979. 18] (2 . . 3 / 0 20 / // (. | 1963. | . 2, # 4. | 7. 13{19. 19] Bahturin Yu. A., Mikhalev A. A., Petrogradsky V. M., Zaicev M. V. InQnite Dimensional Lie Superalgebras // Gruyter Expositions in Mathematics 7. | Berlin: Walter de Gruyter, 1992. 20] @ \ F. = 3 ,. | ., 1982. % & 1995 .
. .
-
519.21.219.5 .
:
, ,
Jm+1 = Jm ; "Jm LSm Jm , m = = 0 1 2: ::# " > 0, $ Jm LSm | & RN , LSm = (Sm )Sm , Sm ) *, + , ) ) . , * J0 ) . - )
J~m = kJm k;1Jm . / $ +
0 $
* 1 *. + , + 1 2 *: I: limm!1 J~m = PL # II: limm!1 J~m = ;P # III: Jm = 0 $ m0 , $ PL P | $ ) * L RN , ) ) ** )* . 40+ P+ (") P; (") I II. ) )$ $ J0 + , lim"!+0 P+ (") = 1, lim"!+1 P; (") = 1# & J0 1 0 0 + , P; (") 1.
Abstract A. Yu. Plakhov, On asymptotic behavior of some class of random matrix iterations, Fundamentalnaya i prikladnaya matematika 1(1995), 1009{1018.
In the paper iterations Jm+1 = Jm ; "Jm LSm Jm , m = 0 1 2 :: :# " > 0 are considered. Jm and LSm are selfadjoint operators on RN , LSm = (Sm )Sm , with Sm being independent identically distributed random vectors which satisfy some additional conditions. Initial opetator J0 is nonrandom. Asymptotic behavior of the rescaled operator J~m = kJm k;1 Jm is examined. Problems of this type appear in neural network theory when studying REM sleep phenomenon. It is proven that one of the following three relations holds almost surely: I: limm!1 J~m = PL # II: limm!1 J~m = ;P # III: Jm = 0 starting from some m0 # here PL and P are orthogonal projectors on random subspace L RN and one-dimensional subspace spanned by random nonzero vector , respectively. Denote P+ (") and P; (") the probabilities of asymptotic behaviors I and II. For J0 being nonzero positive semide7nite it is shown that lim"!+0 P+ (") = 1, lim"!+1 P; (") = 1, but if J0 has at least one negative eigenvalue, then P; (") 1. 1995, 1, 8 4, 1009{1018. c 1995 ! "# $%&, # \( "
1010
1
. .
Jm+1 = Jm ; "Jm LSm Jm m = 0 1 2 : :: " > 0
(1)
Jm LSm | RN . " LSm # $ x 2 RN % LSm x = (x Sm )Sm ( ) & ' & . ( Sm $ ' # , & , ' # J0 ' . * & $
# + # ,1]{,3]. Jm # + &# m- , " & ' #. 0 # # # # kJm k;1Jm , k k & ' $ . ( 1 # ' 1 # m ! 1. ( #2 , ' $3 Sm : (i) M(LSm ) = cI # # c > 0, I | ' # , M & ' ' (ii) 8x 2 RN Pf(Sm x) = 0g < 1=2:
(5 , $ RN , ' Sm , 2 1=2.) "& ' J~m = kJm k;1Jm . " & $3+ +.
1. -
:
~ I: mlim !1 Jm = PL ~ II: mlim !1 Jm = ;P III: Jm = 0 m0 : PL P L RN , .
5 & & + 2 3. ( & & , ' $ 1 & # + Jm , m = 0 1 2 : : :, & 2 # I, III. 7
1011
+ & Jm & ' ,
, & & , II. "& ' P+ ("), P;(") P0 (") = 1 ; P+ (") ; P; (") I, II III . 8 $3 & ' 1 + # " ! 0 " ! 1. 2. ! J0 " , lim P (") = 1 "!+0 + lim P (") = 1: "!+1 ; # P; (") 1, J0 " , P0 (") 1, J0 = 0. 5 & 1 # '
& .
2 Jm
& , ' $ 1 & # + Jm , m = 0 1 2 : : :, & 2 # I, III. *& 2 (1) , ' Ker Jm Ker Jm+1 . 8 , ' 2 m0 , Ker Jm & , . . m > m0 Ker Jm = Ker Jm0 . "& ' L
Ker Jm0 . 9 ' , ' L 6= f0g ( ' ' # III). "' , m > m0 Jm L . "& ' J m ' Jm L +
+ 2 (1) J m+1 = J m ; "J m Lm J m m > m0 (2) m = PL Sm . : Lm & , # $3# & L L. " 1%% ;m ;m = 1 ; "(J m m m ) m > m0 , ' 0 < ;m 6 1: (3) ; 1 5# , 2 (2) (J m+1 m m ) = " ;m (1;;m ). 7 m 6= 0, # J m+1 0 < ;m < 1 m = 0, ;m = 1. 0 + J ;m1, ' $ % J ;m1+1 = J ;m1 + ";;m1 Lm
1012
. .
$ . *& + J ;m1 = J ;m10 + "
mX ;1 n=m0
;;n 1Ln :
(4)
" J m m ! 1 & 2 . 1P . *& # (i) & 2+ ' , ' ;1 L = mcI + o(m), m ! 1. 0 , ' L . . mn=0 Sn n ' L PL LSn PL , ' mX ;1 n=m0
Ln = mcIL + o(m) m ! 1
(5)
(& IL | ' # L). ( (3) (4) J ;m1 > J ;m10 + "
mX ;1 n=m0
Ln
(6)
' % (5) ' J m = O(m;1 ) m ! 1: (7) 2 2. A ' P1 jSm j . 20 2 M(jSm j ) = M(Tr LSm ) < 1, $ > 0 m=1 PfjSm j > mg 6 6 ;1 M(jSm j2) < 1, , , . . ' 2 m jSm j2 6 m. ( & jSm j2 = o(m): (8) *& ' + 1 ; "kJ m kjSm j2 6 ;m 6 1, & (7) (8), + , ' . . ;m = 1 + o(1) m ! 1: (9) 5 , m mX ;1 ;1 m;1 X ; 1 ;1 ; 1)j nj2 6 X (;;1 ; 1)jSnj2: 6 (; ; 1)L (; n n n n
n=m0
n=m0
n=m0
" $ , & (9) ' , ' jSnj2 | & ' # ' ' ' , ' , ' . . mX ;1 (10) (;;n 1 ; 1)Ln = o(m): n=m0
1013
A , $ ' (4) 3$ % (5) (10), ' J ;m1 = "mcIL + o(m) J m = ";1 m;1 c;1IL + o(m;1 ):
0 + + Jm , ' % I.
3
9 & " = 1. D , ' , ' 3 . "& ' m 2 & ' Jm . ( 1 & & , ' m0 m0 < 0, . . II. 5 & . "& ' m ' # # , $3# & ' $ m ,
m = (Sm m):
(11)
0 & & m , & , ' ( m m;1) > 0. 7 & ' m ( , & ' m ), ' # & ' + + , $3+ ( m;1) > 0. A & m \ -+ 2 ", \+ 2 ", j m j > , & ' hn ( ) ' + 2+ & ' # m 2 f0 1 : : : n ; 1g.
1. $ 0 > 0 lim (hn ( 0 )=n) > 1=2:
n!1
. *& # (ii) , ' $ 2 RN # ' > 0 p > 1=2, ' Pfj(Sm )j > g > p . "' , ' Pfj(Sm `)j > g > p + ` & ' # . " $ ' ' # % RN 3 +
0 > 0 p > 1=2, ' Pfj(Sm )j > 0 g > p $ ' . ( & Sm m Pfj(Sm m )j > 0 g > p $ & & 2+ ' + lim (hn( 0 )=n) > p, n!1 ' & 1. ( #2 0 -+ 2 & ' m & ' hn ( 0 ) = hn.
1014
2.
. . h
+O (1)
m > 22 m : . * & % (1), + m+1 6 (Jm+1 m m) = m ; m2 2m : (12) ( m , m+1 6 m ; 02 2m0 + 2+ & ' # m > m0 . 0 ' + 2+ m . . ' , limm!1 m = ;1. 8 , m1 > m0 j 02 m1 j > 1, % log2 log2 j 02 m1 j. 5 , ' , ' + 2+ m j 02 m+1 j > 2j 02 m j2 , % log2 j m j > 2hm ;hm1 +log2 log2 j0 m1 j ; 2 log2 0 , ' 2. (N ;1) > (N ) = & ' J , "& ' (1) m > : : : > m m m m ' & ' $
&, + P ;1 (k) , qm = Nk=1 j m j.
3.
qm 6 22m;hm +O(1) : $3$ % : 2 mq m qm+1 6 c 1 + 2 j j + qm + 1 (13) m m $ ' ' 2+ & ' # m, c | . *& % (13) ' qm 6 (N ; 1)j m j, ' 2+ m, , ' + 2+ & ' # m qm+1 6 c1mqm + c (14) c1 = c(1 + (N ; 1) 0;2 ), + & ' # qm+1 6 c,mqm2 + qm + 1]: (15) D ' & ' m`. " Qm , m > m`, % Qm+1 = 2c1 mQm + 2+ m (16) Qm+1 = 2cmQ2m + m (17) ' Qm` , ' c1 m`Qm` > c m`Q2m` > Qm` + 1. * & 2 (14){(17), , ' qm 6 Qm . 8 # ,
1015
log2 Qm log2 Qm = km 2m;hm , + + , ' km + . " $ 3. (13). 5 3 & ' # & # m m+1 & 2 + , , J, S, , , , q Jm , Sm , m , m , m , qm , J 0 Jm+1 . . * & 2 (1), ' Tr J 0 = Tr J ; (J 2 S S) " $
Tr(J 0 )2 = Tr J 2 ; 2(J 3 S S) + (J 2S S)2 :
T0 = T + 2,(J 3S S) ; (J 2S S) Tr J] (18) 2 2 & & ' T = (Tr J) ; TrJ . " '
(18). "& ' S 0 = S ; , (11), ' S 0
. H' , ' (J 2 S S) = 2 2 + (J 2 S 0 S 0 ) (J 2 S 0 S 0 ) 6 6 jS j2 16max ( (k) )2 6 jS j2q2 , + k6N ;1 (J 2 S S) 6 2 2 + jS j2q2 :
(19)
J ' ' (J 3 S S) 6 2 3 + jS j2 w3
(20)
& & ' w = supf0 (nk) j k = 1 : : : N n = 0 1 2 : ::g. * w < 1, , & , f (nk) g ' + & ' (1) 0 . * & (19), (20) ' Tr J > ; q, ' $3$ + + # ' (18): ,: : :] 6 2 2 q + jS j2 q2(; + q) + jS j2w3 :
(21)
" + T, +
T = 2 *&
NX ;1 NX ;1 (k) + (k) (l) k=1 kl=1 (k6=l)
6 ;2 q + q2:
NX ;1 0 (k) 6 ;q0 + 2(N ; 1)w k=1
(22)
1016
. . X (k) 0 (l) 0 X >2
k6=l
f(k) >0g ' & T0 :
(k) 0
X
f(l) <0g
(l) 0 > ;2(N ; 1)q0 w
T0 > 2 0,;q0 + 2(N ; 1)w] ; 2(N ; 1)q0 w:
(23)
H' (21), (22) (23) (18), '
2 0,;q0 +2(N ; 1)w] ; 2(N ; 1)q0 w 6 (; + 2 2)2q+q2 +2jS j2 q2 (; +q)+2jS j2 w3 :
(24) : , & ' (24) ;2 0 ' (12), + & ' + ; 0 > ; + 2 2, + 2 2 3 (N ; 1)w 0 q 1 + 0 ;2(N ;1)w 6 q+ 2(; q+ 2 2 ) +jS j2 q2 ;; ++ 2q 2 + ; jS+j w 2 2 : : & , ' < 0 ' 2+ m. H' q 6 (N ; 1)j j, + 2 2 2 3 q0 1 + (N ; 0 1)w 6 N 2+ 1 q + 1N+jS j2 jq j + jSjj wj + 2(N ; 1)w , + + & ' J ! Jm , J 0 ! Jm+1 . . ' (8) jS j2, ' % (13), ' c & c = (N + 1)=2 + 2(N ; 1)w + 1.
4. m pmq m j m ; m+1 j 6 p : j m j ; qm ; mqm2 . 0 m+1 m+1 = cos m + sin 0 6 6 =2 (25) | ' # ,
# m . *& ' (Jm+1 m+1 m+1) 6 (Jm+1 m m ) c ' % (1), (25) (11) cos2 m + sin2 (Jm ) ; ,cos m m + sin (Jm Sm )]2 6 m ; 2m m2 : " $ , , ' m < 0, j(Jm )j 6 16max j (k)j 6 qm k6N ;1 m j(Jm Sm )j 6 qm jSm j, ' tg 6 j j +2 j 2m
2m;j qqm j;Smq2j jS j2 : (26) m m m m m m
1017
*& (26) ' j m ; m+1 j 6 tg p j m j + 2m m2 ; qm ; qm2 jSm j2 > 2j m m j j m j ; qm ; qm2 jSm j2 + qm jSm j j m ; m+1 j 6 p j m j ; qm ; qm2 jSm j2 m 6= 0, j m ; m+1 j = 0, m = 0, , ' (8), ' 4. *& 1{3 , ' qm = o(j m j ) m ! 1 $ > 0: (27) P *& 1, 4 & 2 (27) + , ' m j m ; m+1 j + , , 3 lim = : (28) m!1 m A , & c 2 # (27) (28) Jm = m P + o( m ) ' II. L 1 & .
4 " # 2
7 J0 & ' , , & & 3, ' & ' # II, . . P; (") 1. 8 ' # J0 = 0
. M ' #, J0 # #. , ' .. & # S0 S1 S2 : : : 3 $ ' "1 "2 , ' 1. " > "1 2 II 2. 0 < " < "2 I. *& 1 + # 2. 1. N & # (ii), , ' Sm 2= Ker J0 , ' ( 2 1/2). 8 , & Sm , . . # & ' m, ' Sm 2= Ker J0 . "& ' m0 2 & + & ' #. "' , Jm0 = J0 , (J0 Sm0 Sm0 ) > 0, % (1) (Jm0 +1 Sm0 Sm0 ) = "(J0 Sm0 Sm0 ) ,1 ; "(J0 Sm0 Sm0 )] : (29) 0 " > (J0 Sm0 Sm0 );1 Jm0 +1 , (29), #, , & & 3, 2 II.
1018
. .
2. "& ' L0
Ker J0 & ' , ' & ' & & : J m | ' Jm L = PL0 Sm . "& ' , , lm 2 & ' Pm;01 m J ;0 1 P > 0. 8 n=0 Ln P | 2 & ' 2 % (5) (8), . . lm = cm + o(m) jSm j = o(m), , "2 > 0, ' $ m jSm j2 < lm + ";2 1 P: (30) 5 , ' " < "2 J m , m = 0 1 2 : ::, $ . 5 & . 0 , ' J 0 J 1 : : : J m . 0 & & 2, ' % (6) m0 = 0, , ' 2 & ' J ;m1 2, ' P + "lm , kJ m k 6 (P + "lm );1 : (31) 5 , ' , ' ;m > 1 ; "kJ m kjSm j2, & (30) (31), + , ' ;m > 0. A , & x 2 L (2) (J m+1 x x) = (J m x x) ; "(J m x m)2 > ;m (J m x x) > 0
J m+1 & . L & , " < "2 Jm . " $ , & & 2, . . ' 2 I. . 0 & ' # + Sm 2 , ' ' & 2+ ' . 01 & 1 # $ , & & 1 ' # S0 S1 S2 : : : A 1 9. . Q '.
$
1] J. J. Hopeld, D. I. Feinstein, R. G. Palmer. \Unlearning" has a stabilizing e ect in collective memories // Nature. | 1983. | V. 304. | P. 158{159. 2] U. Wimbauer, N. Klemmer, J. L. van Hemmen. Universality of unlearning // Neural Networks. | 1994. | V. 7. | P. 261. 3] A. Yu. Plakhov, S. A. Semenov. Neural networks: iterative unlearning algorithm converging to the projector rule matrix // J. Phys. I France. | 1994. | V. 4. | P. 253{260. * !+ 1995 ".
-
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Abstract S. A. Piartli, Locally convex group algebras over compact groups, Fundamentalnaya i prikladnaya matematika 1(1995), 1019{1031.
It is proved that Tannaka-Krein duality theorem can be formulated through mutually conjugated locally convex Hopf algebras constructed by the group algebra C +G] and the Krein algebra of representation functions.
- C G], C | , G !
, " !. $ ,
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6. - (8]), ! 3 | L1 (G d ), l
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1020
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+ ! ! C . : 2 !
', ! 2 ! | =. B G | , . P P & C G] : : a g ! a g;1
P C G],
x y 2 C G], g 2 G: i
i
8p 2 P g 2 G p(g) = 1E 8p 2 P p(x ) = p(x)E 8p 2 P 9q 2 P C > 0 : p(xy) 6 C q(x) q(y)E 8p q 2 P max(p q) 2 P p
p
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(1) (2) (3) (4)
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. B U | C G]. ? U " fx : x 2 U g. - , U | . B U 0 = U \ U . G p | 2 6 ( . 10]) U 0, p(x) = p(x). B " ! , , C G] " ! . : " , ! ! p C G] q, > 0 q(x) 6 , q(y) 6 p(xy) 6 1 x y 2 C G]. -, x y 2 C G] q(x) 6= 0, q(y) 6= 0, x y ) 6 1 q(x) q(y). p(xy) = 12 q(x) q(y) p( q(x) q(y) 2 B p(xy) 6 12 q(x) q(y) x y 2 C G]. - " q1 (x) = max(q(x) q(x )). F , p(xy) 6 12 q1(x) q1(y). B , " C G] 3) ! . B p | C G],
. H 2 sup p(gxh) p1(x) = sup 2 p(g) : 2 gh
G
g
G
: G . ? , p1 | C G], g 2 Gp1(g) = 1. & , sup p(gx h) sup 2 p((h;1xg;1 ) ) p1(x ) = sup2 p(g) = = p1(x) sup p(g) gh
G
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G
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:
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p1 (x) > sup p(x)p(g) E g
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(sup 2 q(g)) (sup 2 r(g)) r(x)E sup 2 p(g) sup q(gx)q(yh) sup p(gxyh) 1 (x) q1(y) 6 C qsup p1(xy) = sup2 p(g) 6 C sup2 p(g) 2 2 2 p(g) p1(x) 6 C C p
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1022
. .
p q r 2 P , p(xy) 6 C q(x) q(y), q(xy) 6 C r(x) r(y) sup q(gxh) q1(x) = sup 2 q(g) : 2 : , " C G],
G 1, ! !. - . + !@ , C G] " ! ! P . - " ! 2
Cd G] C G] G. - , 2 ! (Cd G])0 2 F C (G) ( , 3(Cd G])0 ' (C G])0 ). % f 2 F
! ! ! ! 2 f.^ B f 2 F f^ p 2 P ( ^ < . f 2 F ). - > 0, p(x) < f(x) % x ^ = p(x) f( ^ x ) < p(x) f(x) p(x) ^ j p0(f) = sup jf(x) p(x) < 1: F , 2P F = F . B k k | 2 ! C G] G. - f 2 F , kf k 6 p0(f). % ^ (f x) = f(x) p
q
gh
g
G
G
p
p
p
c
p
c
f 2 F , x 2 Cd G]. " x 2 Cd G] " ! ! L : F ! F x
(L f g) = (f x g): x
^ = (L f 1). F , f(x) 1.2. f 2 F x y 2 C G] p(xy) 6 C q(x)q(y): L f 2 F q0 (f) 6 C q(x) p0(f). x
p
p
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1023
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p
J " .
1.3. F L2 (G dg) f 2 F kf k2 < p0 (f) d(G), dg | ", k k2 | L2 (G dg). % , F C (G) p
. < " .
1.4. T | , G, V | # , kk | V , v 2 V , kvk = 1. $ B > 0, v1 2 V x 2 C G], $ $ : 8p 2 P p(x) 6 B kv1 kE
(i)
T(x)v = v1 :
(ii)
. : G , n = dimV < 1. : T g1 : : : g 2 G, fT(g )v = e g | P V. P % u = c e " kuk0 = jc j. k k0 | V , 4 0 6 B k k. k k. & , B > 0, k kP P P P B v1 = c e , v1 = T c g v. p c g 6 jc j = = kv1k0 6 B kv1 k. J " . % , . < ( . 9, 11]), , 2
f 2 L2 (G dg) Z ~ = f(g)T (g)dg 2 2 f() ^ G^ | " 4 " L2 (G dg) L2 (G), ^ | ! G, L2 (G) ^ 2 ! '() G,
: n
k
k
k
k
k
k
k
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n() tr('()' ()) < 1
n() | ! .
(i) (ii)
1024
. .
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: L2 (G) (' ) =
X
n() tr('() ()):
0 < , " 4 , P ~ k k2 . , f
n() tr(f()T (g)). < k k Mat ( )(C ) (n() tr(AA )) 21 . < , kAB k 6 kAk kB k . ~ 6= 0, f~ | 1.5. f 2 F , 2 G^ f() & '
f . T T Cd G] .
n
B . B " T 2 C G] End H . - C > 0 fx g C G], ! A, x ! 0 kT (x )k > C > 0, kk | H . + fv g H , kv k = 1 kT (x )v k > C. ~ 6= 0, v 2 H , kf()vk = 1. B - f() ~ =v " ! 1 > B > 0 fy g C G], T (y )f()v
p(y ) 6 B 8p 2 P . - x y ! 0. B f 2 F , p 2 P p(xy) 6 C q(x) q(y). H f = L f. % > 0 ! 2 A, > q(x y ) < . - kf k2 6 6 d (G) C p0 (f) , kf k2 ! 0. ! , kf k2 = kf~ k02 > ~ k > C1 C > 0, C1 | > kf~ ()k > C1kf~ ()k = C1kT (x y )f()v , H . B @ .
p
p
x
y
l
p
1.6. x 2 Cd G]. ( T : Cd G] ! ! End H , , T (x) = 0, x = 0. . H f 2 F . : ! ,
~ = 0 , T " f() P ~ k , T Cd G]. - kL f k22 = kT (x)f() " Cd G]. : ! kL f k2 = 0, L F = f0g. < Cd G] , x = 0. - .
x
x
x
1.7. ) Cd G] * .
1025
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2 ! !, ! G ( . 9]), ! ! . H C G], !
! : ! G " C G] T E " p (x) = kT k, k k | . - H ! T , T C G] H . N , T ! ! C G] ( . C G] = C G]). < " " , !. : !
G ! , T G . - " (C G])0 , " *! ! ( . C G]0 ). B F | " " , " ! 2 f^ f G. - F ! ! 3 ! 2 ! G.
w
w
2.1. F | ' (C G])0 R. " , ImF R. B f^ 2 (C G])0 . : ^ f1 : : : g 2 G, # ^ ^ \ KerP T k Ker f, : xf(x) = f (T (x)), T = T k . T | , =f^ ! ! ^ 2 R. ! 4 T . , F (f) P B f 2 R, f(g) = a f (g), f | P4 ^ = f # (T(g)), T = T k . ^ ? f(g) T k , 2 G. 0 ^ , f 2 (C G]) . ^ j = jF(f)(g) ^ j = jf(g)j. %" < F , jf(g) ; 1 F . P + x 2 C G] > 0. x = =1 a g , max ja j = M. G , jf(x) ^ j < , @ @ . jf(g )j < nM n
k
k
k
k
k
n k
k
k
k
k
1026
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2.2. R0 , *- , ' C G] . . . " , C G] ! !. . , $ ( . 10], . 161) , ((C G])0 )0 2 ! C G]. +
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. :, :
C G] = (C G]) = R0 R =R = (C G])0 : w
w
w
w
? C G] R | - , . R
, -" >2. $ , - A " ! >2 (m u T ), m | " , u | , T | " , | , | ( . 12]). G " ( " " ), A(m u T ) - ! ! >2. G " A(m u T ) " , A(m u T ) ! ! >2. 2.3. N - ! A B, " >2 (m u T ) (m0 u0 T0 0 0) , " , x y 2 A, f h 2 B
: A0 = BE w
(i)
B0 (ii) = AE (T0 (f) x y) = (f m(x y))E (iii) 0 (f h T(x)) = (m (f h) x)E (iv) (0 (f) x) = (f (x))E (v)
0(f) = (f u(1))E (vi) 0
(x) = (u (1) x): (vii) 2.4. - A, B , "' (m u T ) (m0 u0 T0 0 0) , . A(m u T )
B(m0 u0 T0 0 0) | - "'. w
1027
-
. + , (i), (ii) : A = A, B = B. - -" >2 , ! (m u T ). < T. B x 2 A w
w
p(x) = max j(f x)j =1 k
k
n
ff g B f ! . - p A A : k
k
w
p (z) = sup j(f h)(z)j w
f h 2 B, , jf(x)j 6 p(x)
jh(x)j 6 p(x).
Pn =1 ak1fk h = k=1 ak2fk , jakj j 6 1. B X pw (T(x)) 6 sup ak1 ak2 (fk1 fk2 T(x)) = jakj j61 k1 k2 X = sup ak1 ak2 (m0 (fk1 fk2 ) x): jakj j61 k1 k2
< f =
P
n k
H q(x) = max j(m0 (f f ) x)j, p (T(x)) 6 6 n2q(x), T | " . ! , j(f m(z))j @ P , , j(T0(f) z)j < . B T0(f) = 1 f h , j(T0(f) z)j < n p (z), p(x) = max(jf (x)j jh (x)j): kt
k
t
w
n
k
k
w
k
k
k
? m. < 2,
A B , | ! (vi), (vii) (v) " >2, . J " . - C G] R >2. B" g g 2 G x y 2 C G], f h 2 R, a a 2 C . k
k
X
X
a g = a g g E m(x y) = xyE u(a) = a eE X X
ag = aE
T
k
X
k
k
k
k
ag = k
k
k
k
k
X
a g ;1 E k
k
1028
. .
m0 (f h)(x) = f(x)h(x)E u0 (a) = a 1 E
0(f) = f(1)E 0 (f)(g) = f(g;1 ): 0. B" T0(f) = P f h , < T P^ ^ ^ f (x)h (y) = f(xy). . 22.23. 9] (. 1, . 446) " ! 4 . B4 @ R
2 ! 12]. : " . 214 . 221 12] , ! (m u T ) (m0 u0 T0 0 0) C G] R >2. B , " ! @ !, ! ! >2 ! G. 2.5. ) "' C G] (m u T ) R (m0 u0 T0 0 0) - . % ! ! " (i{vii). B ! ! - . ! ( . . 30.1 9], . 2, . 189). :
2 !, - ( . " ), " . < R. B A | - F " ! ! : p (x) = j'(x)j, ' | ! ! 2 A C . B - A !. 2.6. A | - F , , . A ' $ ' G, . % - -. ! . + @ : 2.7. - $ : (i) , G "' "' $ '
. (ii) - "', "' G. G
k
k
k
k
1029
-
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2 !. 2.8. G1 G2 | "' '. G1 ' G2.
III
-
3.1. G | , H |
, T G H . $ '
f G $ h 2 H , f(h) 6= 0. G = H . . % . H x 2 GnH U G, Ux \ HU ;1 = ? ( ! G H). B I1 | " Ux, I2 | U, '1 () '2 () | , . 1 2, '1(1 ) 6= 0 '2 (2 ) 6= 0. H f = I1 I2 . f | 2 G, h 2 H
f(h) = "
Z
I1(hg;1 )I2 (g)dg =
Z
G
U
Z
Z
G
U
I1 (hg;1)dg = 0
f(x) = I1 (xg;1)I2 (g)dg = I1(xg;1 )dg 6= 0 ~ = '1 () '2 (), , f , f() ~ kf()k 6 k'1 ()k k'2()k X ~ k 6 X k'1()k k'2 ()k 6 k'1 k02 k'2 k02 kf()
, f f ! .
1030
. .
G f0 2
fP H, f0 0 ^ ! n() tr(f()T ~
! 4 f~ 2 L2 (H), (h)) ! . B @ .
3.2. C G](m u T ) | "' - . : (i) G C G]. (ii) C G](m u T ) "' G. . (i). B p | ! C G], I = fx 2 C G] : p(x) = 0g | C G]. : ! , A = C G]=I | . A G 4 , ! G, " " C G]. ? , C G] " ! p (x) = kT (x)k, " V | " 4 ! G, " C G], " C G] = (C G]) . & , T V , , V " 4 . . , C G] V " . B , F 2 !,
2 (C G])0 - " 4 ! V. -" , F ! ! 2 (C G])0 . - F | - , - -. ! " 2 C G1. - C G] = (C G]) | " 4 g 2 G ! F , 2 " G G1 . ! ,
F , ! f 2 F ! g 2 G, f(g) 6= 0. B " G = G1. ^ ^ < V = G1 = G, " (i). : , C G](m u T ) " ^ ! ! ! p , 2 V = G, >2 G. - . p
p
p
p
w
w
w
-
1031
3.3. G | , A(m u T ) | - "', $ $ : (i) $ | ' G
A. (ii) (G) | - . (iii) (G) A. (iv) 8g 2 GT((g)) = (g) (g). (v) 8g 2 G((g)) = (g;1). (vi) 8g 2 G ((g)) = 1. A(m u T ) ' "' G.
1] . ., . . // . . | 1989. | ".46, % 6. | &. 3{9. 2] *+ &. . ,- . . / 0 . . // . . | 1993. | ". 53, % 2. | &. 110{113. 3] Tannaka T. U4 ber den Dualit4atsatz der nichtkommutativen topologischen Gruppen // Tohoku Math. J. | 1938. | V. 45, % 1. | P. 1{12. 4] 5 . 6. * . 50 0 .+ - 5 . 5 - -- // 78 | 1949. | ". 69, % 6. | &. 725{728. 5] 6. . . 50 0 I // ". ,. | 1963. | ". 12. | &. 259{301. 6] 6. . . 50 0 II // ". ,. | 1965. | ". 13. | &. 84{113. 7] Tatsuuma N. A duality theorem for locally compact groups // J. Math. Kyoto Univ. | 1967. | V. 6, % 2. | P. 187{293. 8] Takesaki M. Duality and Hopf-von Neumann algebras // Bull. Amer. Math. Soc. | 1971. | V. 77, % 4. | P. 553{557. 9] :; , < 00. -0 5 =05 >. 10] 0.. ? 5 >. | .: , 1967. 11] . . C .0 5. | .: 8, 1978. 12] E F. . ,0 0 0 5 -. | .: 8, 1990. ' ( 1995 .
. .
. . .
514.772
: , , , .
1963 . ". # $ , % &&'
T - ' ( ) * ' + % ( -
& , ' &' ' (. % , ) ), , &' (, & &' , ( ( (- , . / ) / ) % , , )*' ) 0- / .
Abstract I. Kh. Sabitov, Some remarks on the tubes of negative curvature, Fundamentalnaya i prikladnaya matematika 1(1995), 1033{1043.
In 1963 L. Nirenberg has showed that the rigidity of a so-called T -surface depends on the nonexistenceof two closed asymptoticlines on the tubes of negativecurvature. In the article we give some conditionssu5cient for nonexistenceof closed asymptotic curves and besides we formulate and comment a number of problems concerning the exterior geometric structure of the tubes of negative curvature. 1.
60- -
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S ; : , . ;i , 2 6 i 6 2g , g > 1 | S , . Sk; S ; , 1 6 k 6 g ,
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8 . . - 1 8 1 , 1*
T - g > 1 T - 1
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x = r(u) cos ' y = r(u) sin ' z = u (a 6 u 6 b 0 6 ' 6 2): @ , ,
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= 0,
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=
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. (+ , $ f (x) x0 (a b) ( ) , f (x) 6 (>) f (x0 ) x0 ( x0 x, f (x) < (>) f (x0 ).)
. C M0 2 ; | - ; v . - dv = 0, - - A = 0.
D 1* . .
M0; p; ;B + B 2 ; AC du ( ). @ B (M0 ) 6= 0 dv(M0 ) = 0, B (M0 ) > 0< , du(M0 ) 6=; 0, ; - p M0 ; dv = ; A B + B 2 ; AC du, . . dv . , , v . M0 - > . 2) C (M0 ) = 0. E, 1 , . : 2) | C 0 M0 ; | 2) | ; * Mn ! M0 , n ! 1, C (Mn ) 6= 0. 2) ; dv = ;(A=2B )du, . . dv . . 2) ; dv = ;(A=2B )du , C = 0, p ; Cdv = ;B + B 2 ; AC du , C 6= 0. dv(M0 ) = 0 du(M0) 6= 0 B (Mp , B (M0 ) > 0, - 0 ) < 0 . < ; ; dv = ; A B + B 2 ; AC du , C 6= 0. , dv M0 . , 1)
C (M0 ) 6= 0.
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1037
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(
L d'2 + 2M d' dz + N dz 2 = 0 2 2 L = (
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'z z ' )=I, N =
zz =I, I = q =
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du dv), -
1 (
, . . 1 . D>
A cos2 ' + 2B cos ' sin ' + C sin2 ' = 0, p
C 2 ; A2 4B B 2 ; AC cos 2' = (A ; C )2 + 4B 2
sin 2
'=
p ;2B (A + C ) 2(A ; C ) B 2 ; AC (A ; C )2 + 4B 2
1039
p
)(A + C 2i B 2 ; AC ) u v) cos 2' + i sin 2' = (C ; A ; 2iB : (A ; C )2 + 4B 2
P(
7 , P = (cos 2
' sin 2') .
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;< Ind; . J Ind; (cos 2 + sin 2 ) = 2 (;), |
, (cos sin ) ;, 1*
. . , (du : dv), (2), , , ( , a = (A ; C 2B ) . 4. / (du : dv), (2), , $$ sin ' du ; cos ' dv = 0 (3) cos ' + i sin ' = exp
1 2
ln P
p
' = 21 arg (C ; A ; 2iB ) A + C 2i B 2 ; AC :
5. / Ind; (L ; N 2M ) 6= 2, ; | , ( ( , ), ( ). 2 , ! , L 6= N M 6= 0. O . > * , " *
1, 1 -
1 . C - " * , * , . , -, 88 \ " . , - , 88 ,
C . C e1 = '1 sin '1), e2 = (cos '2 sin '2 ) | . C e0 = a1 e1 + a2 e2 , e00 = b1 e1 + b2 e2 ,
. = (cos
. .
1040
a1 a2b1 b2 6= 0 a1 b2 ; a2 b1 6= 0. @ 8 -
, 1 1
h
i
p
ab
a2b2 )(A2 + 2B 2 ) 2(a2 b2 ; a1b1 )B B 2 ; AC + (a1b2 + a2b1 )AD du2 + h p + 2 (a1 b1 + a2 b2 )B (A + C ) 2(a2 b2 ; a1 b1 )(C ; A) B 2 ; AC + i + (a1 b2 + a2 b1 )BD du dv + h p + (a1 b1 + a2 b2 )(C 2 + 2B 2 ) 2(a2 b2 ; a1 b1 )B B 2 ; AC + i + (a1 b2 + a2 b1 )CD dv 2 : =a du2 + 2b du dv + c dv 2 = 0 (D 2 = (A ; C )2 + 4B 2 )
( 1 1+
( , ).
a1 : : : b2
-
J . , -88
a1 : : : b2, 1* (4) , du : dv , (4),
, 1* 1 1 , . ;8 - 1* ( , , -88 ).
a1 a2 b1 b2 (4) 1 (4)
2. & C 2- -
(u v), (, R (u v). & (4) A = L, B = M , C = N , R , $ F (u v) : 1) grad F 6= 0 R# 2) F ! R ( !, )# 3) aFv2 ;2bFu Fv +cFu2 = 0, a b c (4) , a1 : : : b2. * . ( , ,
F = const
1 (4) 1 1 R, , * , .
F = const, . .
7 , - 8 ,
| . " , 8
F (u v) * -88 a1 : : : b2, *
(3) 2 , - . . . : 1
T0
* -
Tt , Tt 8 F , a1 : : : b2 *1, *1 t0 , * 8 *1,
1041
(4) . @ . , 1* 8 . . ; 1* . 1* .
4.
C * -
, 1* 8 . . C - , . (
) 8 -
8 . . @
x y z ) 1 #5]: 2x = (1 ; 2 + 2 )L ; 2 M , 2y = (1 + 2 ; 2 )M ; 2 L, z = L + M , 2 2 2x = ;2 N + (1 ; + )M , 2 2 2y = ;2 M + (1 + ; )N , z = N + M , L, M N 1 M ; N = 12 +(L 2++N )2 M ; L = 12 +(L 2++N )2 : (
(5)
(6)
3. & C 2- -
, z = c = const $ ( ) ( , ( ) . * .
. E z = c , . -
f g
1
fz z g
- -
, , > . 1
z + z
< , ., .
@ , (5)
(6),
L 2 + 2M + N 2 > 0: C ( ').
(7) -
L 2 + 2M + N 2 )d 2 + 2 (M ( 2 ; 2 ) ; (L ; N ) )d d' + 2 2 2 2 + (L ; 2M + N )d' = 0:
(
. .
1042
; (7) ,
'
. -
-, , . -
d' < , .,
d' > 0:
(8)
B , - , 8 , 1*1
d' dz .
z
.
C
L 6= 0. @ L d M d d + N d 2 = 0 p p ; ; L d = ;M M 2 ; LN d ;M M 2 ; LN d = N d . C dz dz = ( L + M )d + ( N + M )d p 1 p > L d N d , dz = ( d ; d ) M 2 ; LN = = 2 M 2 ; LNd', (8) z . J L = 0, N 6= 0, . C L = 0, N = 0, . . d = 0 d = 0. @ dz = 2 M d', , -88
2+2
, -
dz > d'. C ; |
| ; (. . , 1 1* ). J 1 L = 0, M 6= 0 N d < d = 0, d = ; 2M dz = ;M d', dz = M d', . . dz d' . C L = 0, L 6= 0. @ , , , . L 6= 0, M 6= 0. J 1 , L 6= 0 M 6= 0. C 1 L 6= 0, 1 2 , , L = 0, M 6= 0. " 0 L 6= 0 M 6= 0. C 0
,
p
p
L d = ;M + M 2 ; LN d =) ;M ; M 2 ; LN d = N d : (9) C . 0 2 , 2 1 M 6= 0. C 0 a0 2 2 , L = 0. - : d = 0 2M d = ;N d . J M > 0, .1 , 1 (9), -
a0 J . M < 0, a0 d = 0, ;
p
M + M 2 ; LN d
=
;N d .
L d
=
;
p ;M + M 2 ; LN d .
. ,
1043
-
dz d'
a0
. ; , , (8),
z -
. . @ .
6. / - $ ! , $ . . C , dz d'
, > -
(5){(6), . . - * 1 , 8 . ( *
dz d' > 1 , dz = 0 () d' = 0 dz = ( L + M )d + + ( N + M )d = 0, (L d + M d )d + (M d + N d )d = 0). . ,
- . * +. '. 7 ?
.
1] . . .
// . . | 1938. | ". 4, % 1. | &. 69{77. 2] L. Nirenberg. Rigidity of a class of closed surfaces // Non linear problems. | University of Visconsin Press, 1963. | P. 177{193. 3] M. Shi,man. On surfaces of stationary area bounded by two circles, or convex curves, in parallel planes // Ann. of Math. | 1956. | V. 63, % 2. | P. 77{90. 4] . / 0, &. 1 -2
. 3 45 5 4 5. | 2 4 , 1981. | 344 . 5] 6. 7. & . 0 5 0 4 8 9 // . . | 1967. | ". 2, % 6. | &. 645{656. ' ( ( 1995 .
. .
( )
521.13
: , ,
, .
! " " #$ " $ % $ & $" j : (j + 3) $ %
$+,&- &- % $ $ # + "+ ! $ . +, -, .
Abstract V. N. Shinkin, The integrable cases of the three-body problem at third-order resonance under the oblateness of the central planet, Fundamentalnaya i prikladnaya matematika 1(1995), 1045{1058.
In the present paper the general spatial and restricted elliptic spatial three-body problem at third-order resonance j : (j + 3) are analytically integrated by quadratures with the help of the Weierstrass functions, using an expansion of the disturbing function up to and including the third-degree terms in the eccentricities and in the inclinations.
j : (j + 3).
! (Liu & Inannen, 1985( Morbidelli & Giorgilli, 1990( S*idlichovsk+y, 1992( Scholl & Froeschle, 1975).
. ! /.0 1 ! 0 . 3 . . (Shinkin, 1993, 1994). 5 ! . 2 : 5 6. 7 5.. 2 : 5 7 6. . 8 . 9 : 6. 0 A B 5. . . 1 : 4 6 ;/
. 1995, 1, 4 4, 1045{1058. c 1995 !", #$ \& "
1046
. .
=. t | , f | 0 , m0 , b, J2 | , . 6 0 ( mk , ak , ek , ik , k , ?k , k , nk , Mk | , 1 ., 0 , 0 , 0, . , , 6
k- . , k = m0 + + mk k = mkk;1 k = fm0 mk k k = aa1 < 1 0 = (0) " = (e21 + e22 + i21 + i22 )1=2 2 (0) 2 2 (0) () 1 0 0 02 = 03 = @b1@() + 2 @ b@ 2 11 = 2 ;
2 (1) @b(1) 1 () 2 @ b1 () ;211 = 2b(1) ( ) 2 1 @ @2 ;
;
(j ;1) (0) j290 = (2 j )b(3j ;1)() 2 @b3 @ () 01 = b1 2() (j ;1) j300 = 12 + j b(3j ;1)() + 2 @b3 @ () (j ) 3 j240 = 133j + 5j 2 43j b(1j )() + 32 + 92j 2j 2 @b1@() + 2 (j ) () 3 @ 3 b(j )() 1 1 + (1 j )2 @ b@ 2 6 @3 (j ) j250 = ( 52 3j 2 + 4j 3 )b(1j )() + 2 2j + 6j 2 @b1@() + 2 (j )() 3 @ 3 b(j )() 1 1 + 12 + 3j 2 @ b@ 2 + 2 @3 (j ) 25 j ( j) j 2 3 2 260 = (4j + 9j + 4j )b1 () 5 + 2 + 6j @b1@() 2 (j )() 3 @ 3 b(j ) () 1 1 (4 + 3j )2 @ b@ 2 2 @3 65j + 7j 2 + 4j 3 b(j )() + 25 + 17j + 2j 2 @b(1j ) () + j 270 = 15 + 4 6 3 1 4 2 @ ( j) ( j) 2 3 3 2 @ b1 () @ b1 () + 11 8 + j @2 + 24 @3 ( 16 m0 j 1 j270 = 270 j = 1( 1270 = 270 3 ;
;
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i
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6
;
;
;
b(j )()
;
;
| // 0 C.
1
1047
D. 6 ! =. Lk , k , 1k , !1k, 2k , !2k: Lk = (k ak )1=2 k = nk t + Mk (0) + k k = 1 2( 2 2 1k = Lk (1 (1 e2k )1=2 ) nk = (kL3 ) 1 + 3J2 ab k k k 2k = Lk (1 e2k )1=2(1 cos ik ) !1k = k !2k = ?k : ;
;
;
;
;
;
=. ! . E . ! .! /.0 ! ! Hori (1966) 0 ! , .
x 1
= . ! 1 = (j + 3)2 j1 2 = 2 L1 = jI1 L2 = (j + 3)I1 + I2 ( (1) ;
;
I1 I1 = L2 J1 1 = 1 ij = L2 ij !ij = !ij m1 1 = n t i j = 1 2( = m 2
(2)
xij = (2ij )1=2 cos !ij yij = (2ij )1=2 sin !ij i j = 1 2(
(3)
z11 = ex11 + fx12 q11 = ey11 + fy12 z12 = fx11 + ex12 q12 = fy11 + ey12 z21 = gx21 + hx22 q21 = gy21 + hy22 z22 = hx21 + gx22 q22 = hy21 + gy22 (
(4)
zij = (2rij )1=2 cos ij qij = (2rij )1=2 sin ij i j = 1 2
(5)
;
0
;
;
;
;
1 | J, I1 | . I1 /.0
2 2 b m b 1 0 n1 L1 1 + J2 a + n2L2 1 + m 21 + J2 a 1 0 2
1048
. .
ni Li | ni Li I1 = I1 , e = (1 f 2 )1=2 h = (1 g2 )1=2 = 1 < 1 2 1 f = (((((a(1) c(1))=b(1) )2 + 1)1=2 + (a(1) c(1) )=b(1))2 + 1)1=2 1 g= (2) (2) (2) 2 1 = 2 (((((a c )=b ) + 1) + (a(2) c(2))=b(2) )2 + 1)1=2 2 m 2 L 1 b (1) 0 a = L 2 + 3J2 a m0 a(2) = a(1) 1 2 1 1 =2 1=2 b(2) = 2011 LL2 b(1) = ;211 LL2 1 1 2 0 c(2) = c(1) : c(1) = 03 + 3J2 ab m 2 m1 ;
;
;
;
;
;
;
;
;
;
= J1 1 . (j + 3)n2 jn1 = 0. J J1 , 1 , rij ij ;
H = H0 + O(1) 2 H0 = !0(3) J21 + 11r11 + 12r12 + 21 r21 + 22 r22 + 3=2 1=2 +3 L2 + 8 j240 (e3 (2r11)3=2 cos(1 + 3 11 ) L 1 1 =2 ) cos(
3e2f 2r11 (2r12 1 + 2 11 + 12 ) + + 3ef 2 (2r11)1=22r12 cos(1 + 11 + 2 12 ) f 3 (2r12)3=2 cos(1 + 3 12 )) + +2 L2 (e2 f (2r )3=2 cos( + 3 ) + + j250 11 1 11 L1 + (e3 2ef 2 )2r11(2r12)1=2 cos(1 + 2 11 + 12 ) + + (f 3 2e2 f )(2r11)1=2 2r12 cos(1 + 11 + 2 12) + + ef 2 (2r12)3=2 cos(1 + 3 12)) + 1=2 +1 L2 (ef 2 (2r11)3=2 cos(1 + 3 11 ) + + j260 L1 + (2e2f f 3 )2r11(2r12)1=2 cos(1 + 2 11 + 12) + ;
;
;
;
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;
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1049
+ (e3 2ef 2 )(2r11)1=22r12 cos(1 + 11 + 2 12 ) ;
;
;
e2 f (2r12 )3=2 cos(1 + 3 12 )) +
+ j270(f 3 (2r11)3=2 cos(1 + 3 11 ) + + 3ef 2 2r11(2r12)1=2 cos(1 + 2 11 + 12 ) + + 3e2 f (2r11 )1=22r12 cos(1 + 11 + 2 12 ) + + e3 (2r12)3=2 cos(1 + 3 12)) + 1=2 L2 1=2 g h 2 +3 L2 + j290 e L1 L1 (2r11)1=2 2r21 cos(1 + 11 + 2 21) L 1=2 1=2 2 2e LL2 g h h + g L1 1 (2r11)1=2 (2r21)1=2 (2r22)1=2 cos(1 + 11 + 21 + 22) + 2 1=2 h + g (2r11)1=22r22 cos(1 + 11 + 2 22 ) + e LL2 1 2 1=2 g h (2r12)1=2 2r21 cos(1 + 12 + 2 21) + f LL2 1 L 1=2 1=2 2 + 2f LL2 g h h + g L1 1 (2r12)1=2 (2r21)1=2 (2r22)1=2 cos(1 + 12 + 21 + 22) 2 1=2 h + g (2r12)1=2 2r22 cos(1 + 12 + 2 22) + f LL2 1 L2 1=2 g h2(2r )1=22r cos( + + 2 ) +2 f + j300 11 21 1 11 21 L1 L 1=2 1=2 2 2f LL2 g h h + g L1 1 (2r11)1=2 (2r21)1=2 (2r22)1=2 cos(1 + 11 + 21 + 22) + 2 1=2 h + g (2r11)1=2 2r22 cos(1 + 11 + 2 22) + + f LL2 1 1=2 2 + e LL2 g h (2r12)1=22r21 cos(1 + 12 + 2 21 ) 1 ;
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1050
. .
L2 1=2 g h L2 1=2 h + g L1 L1 (2r12)1=2(2r21)1=2(2r22)1=2 cos(1 + 12 + 21 + 22 ) + 2 2 1=2 L 1 = 2 h + g (2r12) 2r22 cos(1 + 12 + 2 22 ) = + e L 1 2 = !0(3) J21 + 11r11 + 12r12 + 21r21 + 22r22 + X + 1=2 Ai11 i12 i21 i22 ;
2e
;
(6)
i11 +i12 +i21 +i22 =3" i21 +i22 =02" i11 >0 i12 >0 i21 >0 i22 >0 i11 =2 i12 =2 i21 =2
(2r11) (2r12) (2r21) (2r22)i22 =2 cos(1 + i11 11 + i12 12 + i21 21 + i22 22 )
11 + 12 + 21 + 22 = 0 2 j + 3 2 2 L b b (3) !0 = 3j (j + 3) L 1 + 4J2 a + j 1 + 4J2 a 1 2 1 (1) 2 (1) (1) 2 (1) 2 (1) (1) 2 11 = a e + b 2ef + c f 12 = a f b 2 ef + c e (2) 2 (2) (2) 2 (2) 2 (2) (2) 2 21 = a g + b 2gh + c h 22 = a h b 2gh + c g : =. H0 (6) ! . !0(3) J1 + j1111 + j1212 + j21 21 + j2222 = 1=4O(1) (7) j11 + j12 + j21 + j22 = 3 j21 + j22 = 02 jij > 0 | 0 . E H0 (6) ! Hori (1966) O(1), , .! . . . (7). ; ! !
/.0
1=2W / . @W + O(1) = + 1=2 @W + O(1) J 1 = J1 1=2 @ 1 1 @J1 1 @W @W + O(1) rij = rij ij 1=2 @ + O(1) ij = ij + ij 1=2 @r ij ij ij = 1 i j = 1 2( ;
;
;
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1051
X
W=
Ai11 i12 i21i22
(3) i11 +i12 +i21 +i22 =3" i21 +i22 =02" !0 J1 + i11 11 + i1212 + i21 21 + i2222 i11 >0 i12 >0 i21 >0i22 >0" (i11 i12 i21 i22 )6=(j11 j12 j21 j22 ) i11 =2 (2r11) (2r12)i12 =2 (2r21)i21 =2(2r22)i22 =2
sin(1 + i11 11 + i12 12 + i21 21 + i22 22 ):
(8) = H = H0 + O(1) 2 H0 = !0(3) J21 + 11 r11 + 12r12 + 21r21 + 22r22 + (9) + 1=2Aj11 j12 j21 j22 (2r11)j11 =2 (2r12)j12 =2 (2r21)j21 =2 (2r22 )j22 =2 cos(1 + j11 11 + j12 12 + j21 21 + j22 22): =. J10 = j1111 + j1212 + j2121 + j22 22. 5 (9) .! . 10 ' = + j + j + j + j J = J 1 + J(3) 11 11 12 12 21 21 22 22 1 !0 kij = rij ij jij J 1 'ij = ij i j = 1 2: J J ' H = H0 + O(1) 2 H0 = !0 J2 + 1=2A(P3(J ))1=2 cos ' ! J 10 (0) P3 (J ) = P3 J (3) A = 23=2Aj11 j12 j21 j22 !0 = !0(3) !0 (0) P3 = (11j11 J + k11)j11 (12j12 J + k12)j12 (21 j21J + k21)j21 (22 j22J + k22)j22 (10) kij | H0 (6). = P3(J ) P3(J ) = b0J 3 + b1J 2 b2J + b3. 1. H0 (10) . . = J ( ) . . ! d2 J A2 @P3 (J ) ! J H ! J 2 = 0 0 0 0 2 d 2 2 @J
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1052
. .
)2 A2 P3(J ) ! J 2 H0 !0 J 2=4 = 0: E = (dJ=d 0 2 2 2 5 , dJ 2 = A J 4 + 4A J 3 + 6A J 2 + 4A J + A = P (J ) 0 1 2 3 4 4 d 2 2 2 (11) A0 = !40 A1 = A4 b0 A2 = !06H0 + A6 b1 2 A3 = A4 b2 A4 = 2E + A2 b3 : K1 . (11) (Whittaker & Watson, 1927) J = J0 + ( ( GB3G ) B =2 0 2 3 2 J0 | P4 (J ), | /.0 1, G2 = 3B22 4B1 B3 G3 = 2B1 B2 B3 B23 B0 B32 B0 = A0 B1 = A0 J0 + A1 B2 = A0 J02 + 2A1 J0 + A2 B3 = A0 J03 + 3A1 J02 + 3A2 J0 + A3 : N J /.0 ! , . 6 cos ' sin ' 2 cos ' = 1H=20A(P!0(JJ ))=12=2 3 dJ=d sin ' = 1=2A(P (J ))1=2 : ;
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3
O . . P H0 (6) . , H (9) . (3) J 2 ! 0 H = H0 + O(1) H0 = 2 1 + 11r11 + 12r12 + 21r21 + 22r22: K1 H0
J 1 = const rij = const 1 = 1 (0) !0(3) J 1 ( 0) ij = ij (0 ) ij ij ( 0)( i j = 1 2: ;
;
;
;
1053
. 6 ! /. 2 H0 = !02J + A(J + I11)j11 =2 (J + I12)j12 =2(J + I21 )j21 =2(J + I22 )j22 =2 cos ' (12) j11 + j12 + j21 + j22 6 s (s = 3) Iij = const jij > 0 jij | 0 , ' | / , J | 1 . (j + s)n2 jn1 = 0. Q 6 . , 0 ', ' <
, 0 ', ' . 3. 0 ! / ' H0 (12) j11 = 3, I12 (I12 < 0 < I11 ) | I j12 = j21 = j22 = 0. =. I11 . 2 3A 2 I = 0: I 2 23!A I 4!0 11 0 ;
j
j j ! 1
j
;
j ! 1
j
1
j j ! 1
;
2. " j11 = 3, j12 = j21 = j22 = 0. ) $ I11 > 0, (12) % & (I11 (I12 0). ' ( (J ') & (I11 ) & ( , 0) & ( ). ) * ' H0), & (I12 0), ' H0(J ') 6 H0 (I12 0). $ H0(J ') > H0(I12 . ), $ I11 = 0, (12) & (I11 I11 > 0. ' ( (J ') & (I11 ) & ( , H0). ) * ' H0(J ') 6 0. $ H0(J ') > 0, ' 2 . 3 A $ 4! < I11 < 0, (12) % &0 ) (I12 0), I11 > I12 > I11 . ' ( (J ') & (I11 0) & ((I11 ) & ( , H0), & (I12 0) . ). ) * ' J > I12 H0(J ') 6 H0(I12 $ J < I12 H0(J ') > H0(I12 0) , ' . 3A 2 = I , (12) & $ 11 4!0 ) & (I11 ), I11 > I11 . ' ( (J ') & (I11 ). $ ( & (). ) * ' H0(J ') = H0(I11 H0(J ') = H0(I11 ), ' . 3A 2 > I , (12) J > I $ 11 11 4!0 % & * '. ;
;
ff
ff
g f
g \ f
gg
gg
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6
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1054
. .
x 2 (m1 m2)
= . ! 1 = (j + 3)(n2t + M20 + 2 ) j1 L1 = jI1 ( ;
(13)
;
2 I1 I1 = L1 J1 1 = 1 i1 = L1 i1 !ij = !ij 12 = 2e2 (14) 2 j + 3 n t i j = 1 2( 2 1 = 22 = 2i2 = m 2 m0 j ;
xi1 = (2i1)1=2 cos(!i1 !i2) yi1 = (2i1)1=2 sin(!i1 !i2) xi2 = (2i2)1=2 yi2 = 0 i j = 1 2(
(15)
z11 = x11 + fx12 q11 = y11 + fy12 z21 = x21 + hx22 q21 = y21 + hy22 z12 = x12 q12 = y12 z22 = x22 q22 = y22 (
(16)
;
;
zi1 = (2ri1)1=2 cos( i1 !i2) qi1 = (2ri1)1=2 sin( i1 !i2 ) ri2 = zi22 i2 = !i2 i j = 1 2 1 | J, I1 | . I1 /.0
;
;
2 0 2 n1L1 1 + m2 =m0 221 + J2 (b=a1) (j + 3)n2 1 + 32 J2 ab I1 2 ;
L1 | L1 I1 = I1 , 2 m b 3 12 = 2 J2 a m0 j +j 3 2 2 ;1 f = 12 0 + 1= 3J (b=a21)2m =m 2 = 2 1 0 2 12 2 0 h = 0 1= 3J (b=a11)2 m =m + 2 = = aa1 < 1: 2 1 0 2 12 2 11 ;
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;
J J1 , 1 , rij ij H = H0 + O(1)
(17)
1055
2
H0 = !0(3) J21 + 11r11 + 12r12 + 21r21 + 22r22 + 1=2
+3((2r )3=2 cos( + 3 ) + 8 (j240 11 1 11 3f 2r11(2r12)1=2 cos(1 + 2 11 + 12) +
;
;
+ 3f 2 (2r11)1=22r12 cos(1 + 11 + 2 12) ;
;
f 3 (2r12)3=2 cos(1 + 3 12)) +
+2(2r (2r )1=2 cos( + 2 + ) + j250 11 12 1 11 12 2f (2r11 )1=22r12 cos(1 + 11 + 2 12) + + f 2 (2r12)3=2 cos(1 + 3 12)) +
;
;
+1((2r )1=2 2r cos( + + 2 ) + j260 11 12 1 11 12 f (2r12 )3=2 cos(1 + 3 12 )) + j270(2r12)3=2 cos(1 + 3 12) + ;
(18)
;
+3((2r )1=2 2r cos( + + 2 ) + j290 11 21 1 11 21
;
2(h + 1)(2r11)1=2 (2r21)1=2(2r22)1=2 cos(1 + 11 + 21 + 22) + + (h + 1)2 (2r11)1=2 2r22 cos(1 + 11 + 2 22) f (2r12 )1=2 2r21 cos(1 + 12 + 2 21) + ;
;
;
+ 2f (h + 1)(2r12)1=2 (2r21)1=2 (2r22)1=2 cos(1 + 12 + 21 + 22) f (h + 1)2(2r12)1=22r22 cos(1 + 12 + 2 22)) + +2(f (2r )1=22r cos( + + 2 ) + j300 1 12 21 12 21 2(h + 1)(2r12)1=2 (2r21)1=2(2r22)1=2 cos(1 + 12 + 21 + 22) + + (h + 1)2 (2r12)1=2 2r22 cos(1 + 12 + 2 22))): R (18) (6). J 1 . ! , . 6 ;
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;
12 = 22 = 0 11 = 21 = 1 !0(3) = 3j 2 (1 + 4J2 (b=a1)2 ) 21 = 11 0 2 22 = 12 11 = 2 + 1= 3J22(b=a1) m0 =m2 : ;
;
1056
. .
x 3 # (m1 m2)
= . ! 1 = (j + 3)2 j (n1 t + M10 + 1 ) L2 = (j + 3)I1 ( ;
(19)
2 I1 I1 = L2 J1 1 = 1 i2 = L2 i2 !ij = !ij 11 = 2e1 (20) 2 j n t i j = 1 2( 1 1 = 21 = 2i1 = m 1 m0 j+3 ;
xi2 = (2i2)1=2 cos(!i2 !1i) yi2 = (2i2)1=2 sin(!i2 !1i) xi1 = (2i1)1=2 yi1 = 0 i j = 1 2( ;
;
(21)
z12 = fx11 + x12 q12 = fy11 + y12 z22 = hx21 + x22 (22) q22 = hy21 + y22 z11 = x11 q11 = y11 z21 = x21 q21 = y21 ( ;
;
;
;
zi2 = (2ri2)1=2 cos( i2 !i1) qi2 = (2ri2)1=2 sin( i2 !i1 ) (23) qi2 = (2ri2)1=2 sin( i2 !i1) ri1 = zi21 =2 i1 = !i1 i j = 1 2 1 | J, I1 | . I1 /.0
2 0 2 3 b jn1 1 + 2 J2 a I1 + n2L2 1 + m1 =m0 221 + J2(b=a2) 1 ;
;
;
L2 | L2 I1 = I1 ,
2 m j + 3 b 3 11 = 2 J2 a m0 j 1 1 ;1 f = 21 0 + 3J (b=a )21 2 2 2 m0 =m1 211 3 0 h = 0 3J (b=a)112 m =m + 2 : 2 2 0 1 11 11 ;
;
;
J J1 , 1 , rij ij H = H0 + O(1)
1057
2
H0 = !0(3) J21 + 11r11 + 12r12 + 21r21 + 22r22 + 1=2 +3 (2r )3=2 cos( + 3 ) + + 8 (j240 11 1 11 +2(f (2r )3=2 cos( + 3 ) + + j250 11 1 11 1 = 2 + 2r11(2r12) cos(1 + 2 11 + 12 )) + +1(f 2 (2r )3=2 cos( + 3 ) + + j260 11 1 11 + 2f 2r11(2r12)1=2 cos(1 + 2 11 + 12) + + (2r11)1=22r12 cos(1 + 11 + 2 12 )) + + j270(f 3 (2r11)3=2 cos(1 + 3 11 ) + + 3f 2 2r11(2r12)1=2 cos(1 + 2 11 + 12 ) + + 3f (2r11 )1=22r12 cos(1 + 11 + 2 12) + (24) 3 = 2 + (2r12) cos(1 + 3 12 )) + +3((h 1)2 (2r )1=22r cos( + + 2 ) + + j290 11 21 1 11 21 + 2(h 1)(2r11)1=2 (2r21)1=2(2r22)1=2 cos(1 + 11 + 21 + 22) + + (2r11)1=22r22 cos(1 + 11 + 2 22 )) + +2(f (h 1)2 (2r )1=2 2r cos( + + 2 ) + + j300 11 21 1 11 21 + 2f (h 1)(2r11)1=2 (2r21)1=2 (2r22)1=2 cos(1 + 11 + 21 + 22) + + f (2r11 )1=22r22 cos(1 + 11 + 2 22 ) + + (h 1)2 (2r12)1=2 2r21 cos(1 + 12 + 2 21) + + 2(h 1)(2r12)1=2 (2r21)1=2(2r22)1=2 cos(1 + 12 + 21 + 22) + + f (2r12 )1=22r22 cos(1 + 12 + 2 22 ))): R (24) (6). J 1 1 !
, . 6 ;
;
;
;
;
;
11 = 21 = 0 12 = 22 = 1 !0(3) = 3(j + 3)2 (1 + 4J2 (b=a2)2 ) 0 2 )2m0 =m1 : 22 = 12 21 = 11 12 = 3 + 3J2(b=a 2 ;
;
1058
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Hori G.-I. // Publ. Astron. Soc. Japan. | 1966. | V. 18. | P. 287. Liu L, Inannen K. A. // Astron. J. | 1985. | V. 90. | P. 1906. Morbidelli A., Giogilli A. // Celest. Mech. | 1990. | V. 47. | P. 173. Sidlichovsky M. // Astron. Astrophys. | 1992. | V. 259. | P. 341. Shinkin V. N. // Celest. Mech. | 1993. | V. 55. | P. 249. Shinkin V. N. // Celest. Mech. | 1994. | V. 60. | P. 307. Scholl H., Froeschle C. // Astron. Astrophys. | 1975. | V. 42. | P. 457. Whittaker E. T., Watson G. N. A Course of Modern Analysis. | Cambridge: University Press, 1927. ( ) 1995 .
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Abstract A. A. Yakovleva, A. V. Yakovlev, Homological characterisation of torsion modules, Fundamentalnaya i prikladnaya matematika 1(1995), 1059{1067.
An approach to homological description of torsion modules over group rings is proposed. As example of this approach the authors 5nd a set of parameters completely de5ning the structure of some Galois modules of a local 5eld (completed multiplicative group, group of principal units).
, p- . 5]. # $ 7], & , p- ' ( . ) * , & ' ' p- . 1 , & ' . - e 2 ' 1995, 1, 6 4, 1059{1067. c 1995 !", #$ \& "
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> A, B | Zp F ]-, A0 | A. >, , 2 H 0 (F Hom(A B )). B& X A Zp F ] P : A ! X (: (a) = f 2F a f a 2 A. :, & B 0 B X . C & & Zp F ]- ( : A ! B, 0 , , & ( D : A ! B : D (a) = (a) (a) a 2 A. B& U (A A0 B ) B 0 =D(A0 ). > 1 = ( +Tr ): A ! B | Zp F ]- (, ( | - Zp- ( A B ). B& " ( B 0 = B X , (
"(b a f ) = (b ; (f)(a)) a f
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1061
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f = D 1(a) 8
"D = D 1 " ' ( U (A A0 B )= B 0 =D(A0 ) U (A A0 B 1) = B 0 =D 1(A0 ). , U (A A0 B ) ( : A ! B , ( 8 & & U (A A0 B ).
1. A, B B1 , B, B1 , U (A A0 B ), U (A A0 B1 ) . . > $, B = B1 X1, X1 | & Zp F ]-8 : A ! B, 1: A ! B1 | ( , 0 . B& 1 ( A B = B1 X1 , & $ , & a 2 A 1 (a) 0. > 01 , ( B 0 = B X = B1 X1 X D 10 (A0 ) = f1(a) 0 (a) j a 2 A0 g U (A A0 B ). , $ , , X1 U (A A0 B1 ).
1. A, B | Zp F ]- , A0 | A, | " # # A B . $%# # 2 2 H 0 (F Hom(A B )) & " # ' . Coker | ( , B1 | Zp F ]- , B , B=(A0 ), U (A A0 B1 ) . . > ( : B ! X = A Zp F ],
= . B ( 0 : (A) ! X , 0 ((a)) = (a). > B=(A) = Coker | * , Ext1 F ] (B=(A) X ) ( H 1 (F Hom(B=(A) X )) ( 5], XVI.7). ) X | & Zp F ]- (. 1], 4), .
Z p
Z
Z
Z
Hom F ] (B X ) ! Hom F ] ((A) X ) ! Ext1 F ] (B=(A) X ) , , ( : B ! X , (A) 0 . B ( B X ( (b x) = = b (x ; (b)) (x 2 X b 2 B ). K , D = 0: p
p
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D(a) = ((a) (a)) = (a) ((a) ; (a)) = (a) 0 = ( 0)(a)
1062
. . , . .
a 2 A. > ( ' ( U (A A0 B ) = B 0 =D (A0 ) B 0 =( 0)(A0 ), , , ( B=(A0 ) X . : * 1. L ' . > B , C | Zp F ]-, 2 H 0(F Hom(B C )). B& Y C Zp F ], : Y ! C | ' : (c f ) = c c 2 C , f 2 F . :, & B 0 B Y . C & & Zp F ]- ( : B ! C , , ( D: B 0 ! C ( D(b y) = (b) + (y) b 2 B , y 2 Y . B& V (B C ) Ker D. > 1 = ( + Tr ): B ! C | Zp F ]- (, ( | Zp- ( B C ). B& " ( B 0 = B Y , (
"(b y) = b (y ;
X
f 2F
(f )(b) f )
(b 2 B y 2 Y ) :
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= (b) +
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(f )(b) + (y) ;
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D1 " = D, " ' ( V (B C 1) = Ker D1 V (B C ) = Ker D. B , V (B C ) ( : B ! C , ( 8 & & V (B C ). : , , 1, . 2. B B1 | Zp F ]- , V (B C ), V (B1 C ) . 2. C , B | Zp F ]- , | " # B C . $%# # 2 H 0 (F Hom(B C )) & " # ' . C | ( , B1 | Zp F ]- , B, B=(A0), U (A A0 B1 ) . . > C | * , , Zp- ( 0: C ! B , 0 | $ ( C 8 Zp- ( : Y = C Zp F ] ! B ( : (c f ) = (f 0 )(c)
1063
c 2 C , f 2 F . L (g(c f )) = (gc gf ) = (gf 0 )(gc) = g(f 0 )(g;1 gc) = g (c f ) (c f ) = (f 0 (f ;1 c)) = f 0 (f ;1 c) = ff ;1 c = c = (c f ) (f g 2 F , c 2 C ) , Zp F ]- ( = . 3 ( B 0 = B Y , & : (b y) = (b + (y)) y (y 2 Y b 2 B ). : b 2 B , y 2 Y ( 0) (b y) = ( 0)((b + (y)) y) = (b) + (y) = (b) + (y) = D(b y) ( 0) = D, ' ( V (B C ) = Ker D Ker( 0), , , ( Ker Y . : * 2.
2
H
0 (F Hom(V W ))
> V , W | Zp F ]-. B& $, , & ( : V ! W ' ( i : H i(FV ) ! H i(FW ), $ ( NiVW : H 0(F Hom(V W )) ! Hom(H i (F V ) H i(F W )). $ , ( (. > I | @,P. . & $ Zp F ] !P Zp, , & f 2F uf f ((' f 2F uf . B& I i i I 8 , Zp = I 0 . 3. V = I i W = I i, V | ( , & i " # NVW # " # .
. > V = I i8 H i(F V ) = H 0(F Zp) = Zp=nZp,
H 0 (F Hom(V W )) = H 0 (F Hom(I i W ) ( H i (F W ) = Hom(H i(F V ) H i (F W )). O, ( NiVW . > V | * , W = I i . 3, $ Hom(Hom(V W ) Zp) Hom(W V ) ( . L P: H 0(F Hom(V W )) H 0 (F Hom(W V ) ! Zp=nZp P1 : Hom(H i (F V ) H i(F W )) Hom(H i (F W ) H i(F V )) ! Zp=nZp : > | * H ;i(F X ) H i (F Hom(X Zp)) ! Zp=nZp 8
1064
. . , . .
$ (. 6]). C | & & (, Hom(X Y ) ( Hom(Y X ) & X , Y ). K &, P1 (NiVW x NiWV y) = = P(x y) x 2 H 0 (F Hom(V W )), y 2 H 0 (F Hom(W V ). - $ , NiWV | (8 & $ , ( NVW $ & . K 3 .
3
> k | * Qp p- K=k | * 1 1 F . - K K F -8 & KD Q p- , .. ( K =(K )p . C K , KD Zp F ]-. ) ' ( : KD ! Zp8 U ' K . 3 F - K Zp F ]- U . & KD , U ( . ), H 1(F0 KD ) = 0, H 2(F0 KD ) = Zp=(F0 : 1)Zp F0 F (., , 5]). C&, , KD U Zp- 8 &, E p- 1, $, K , ' p- . > pm | Q Da | Q $ , . : & f 2 F & lf ' , f aD = lf aD. > A0 | & Zp F ]- 1 & , a0 A $ A0 , a, $ * pm a = Pf 2F lf f ;1 a0 . O, A | * 8 & Zp F ]- ( AE ! E , & $ , a0 0, a | Da. O,
Ker = A0 8 A0 | & Zp F ]-, H i (F A), H i(F A=A0) = H i(F E ) & $ . > 3 H 0(F Hom(A I 2 )) $ ( H 2 (F A) = H 2 (F E ) H 2 (F I 2) = = Zp=nZp = H 2 (F KD )8 $ & ( : H 2 (F E ) ! H 2(F KD ) H 0(F Hom(A I 2 )). 3. (1). KD & K & D " & K=E Zp F ]- 2 W0 = U (A2A0 I 2 ) W = U (A A I 2 ), & | & " # H (F E ) ! H (F KD ), )
E ! KD . j
1065
(2). * & + ) U K * D Zp)) |
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" # H 0(F W ) = H 0 (F K=E % . (3). , KD , U ' E , , %' ' (K : Qp) K .
. (1). L
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D A) ! D E ) ! Ext2 F ] (K=E D A0 ) Ext1 F ] (K=E Ext1 F ] (K=E p
p
p
, * 0 ! A0 ! A ! E ! 0. > D * E | KD , K=E , ( D A0)), , A0 | & H 2(F Hom(K=E Zp F ]-. , | (, , D * 0 ! A ! P ! K=E ! 0, & D ! 0. R , * 0 ! E ! KD ! K=E D ( : B ! K , , P ;! K=E D 0 ;! A? ;! ? ;! 0 ? ? y
y
D 0 ;! E ;! KD ;! K=E ;! 0: > $, P I 2 . 1 ( : P ! KD S , A0 | & Zp F ]-. > P KD : H 1(F0 P ) = 0, H 2(F0 P ) = Zp=(F0 : 1)Zp F0 F 8 KD , P | * . $ $, 5], , , & Zp F ]- X ( I 2 ! P X , Y | * . 9 * , Y Zp F ]- P X . , I 2 Y , P X ( . ) ' & P I 2 ( Qp & Qp F ]- . R * P (. 2]), I 2 , ' & I @ J ( , I J | Zp & ( 1], 9). & , Y | , & Zp F ]-, KD , I 2 .
1066
. . , . .
D ( B , KD ( P=(A0), K=E D P=(A). 1 , K U (A A0 I 2 ) D W = U (A A I 2 ) . , K=E (2). > $ 2. > E & ( W0 ! Zp, Hom(W0 Zp) Hom(W0 =E Zp) = Hom(W Zp) ( . ) W | * , H 0(F Hom(W Zp)) Hom(H 0 (F W ) H 0(F Zp)) & 3 $ . (3). > * , Zp F ]-, , & 8 * , & Zp- KD , U (K : Qp) + 1 (K : Qp). 3 . D ), . =& * , H 0(F K=E 0 H 0 (F K D ) ! H 0 (F K=E D ) ! H 1(F E ) ! H 1(F KD ) : H 0(F E ) ! > @ 8 D ) * Ext1 (H 1(F E ) Coker 0 ). B& H 0(F K=E 1 H (F E ), Coker 0 8 $ , & & Ext1 , , * D ) H 0(F W ) , $ H 0 (F K=E W . 9 , $ ( : H 0(F W ) ! Zp =nZp , , ' ( H 0(F KD ) ! H 0 (F W ) & H 1 (F E )8 * 0 ! T1 ! T ! T2 ! 0 ' T1, T2 , , T $ $ & ' , , Ext1(T2 T1 ), , * , * * Ext1 (H 1(F E ) Coker 0).
1] . . , . . .
. II. !
" # ! // % & . ' . . | 1959. | . 7. | ,. 72{87. 2] . . . # 0 & p- &2 # // . 3 . & . 45 ,,,6. | 1965. | . 80. | ,. 16-29. 3] . . ! 0 # p- &2 # // . 3 . & . 45 ,,,6. | 1965. | . 80. | ,. 30{44. 4] H. Cartan, S. Eilenberg. Homological Algebra. | Princeton, 1956. 5] J. Tate. Higher dimensional cohomology groups of class ;eld theory // Ann. Math. | 1952. | V. 56. | P. 294{297.
1067
6] 4. %. < . = // " . 45 ,,,6. , . . | 1964. | . 28. | ,. 645{660. 7] 4. %. < . > & & # p- & & 0 & & ! ?" & // " . 45 ,,,6. , . . | 1970. | . 34. | ,. 893{907. ( ( 1995 .
. .
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512.554.32
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) ! * # # # #$ $ #$ A, B, C, D. + "# #
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Abstract I. A. Yanson, Some direct limits of the classical Lie algebras and the projective limits connected with them, Fundamentalnaya i prikladnaya matematika 1(1995), 1069{1084.
The present article considers some concrete classes of direct limits of the classical simple Lie algebras of the types A, B, C, D. With each such direct limit one can connect the projective limit of sets of dominant integral weights, the projective limit of the invariants of joint action in the algebras of the polynomial functions and sometimes the projectivelimit of the root lattices. The article is devoted to the study of structural mappings in these projective limits and to the proof of non-triviality of the latters.
1
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0 , S(+ )(p) 2 S(g )g . 2 , ' # S(i ) S(i ) #
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1 , j(S(+ )(j 1 (p)) 2 S(h )W . : ./", # ; , , S(+ )(p) = = S(+ )(j (j 1(p))) = j(S(+ )(j 1(p))). & , S(+ )(p) 2 S(h )W . 0 , S(+ )(S(h )W S(h )W . <, 1.1. +: g ;! g | g g K ,
g = n h n+ , g = n h n + . W W | ! g g " h h ,
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1072
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1073
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1074
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$ ("i ) | h , " (Hi). 0 " R # g h R = f "i "j "k j 1 6 i j k 6 l j < kg.
& H, 2 R, : H"i = 2Hi, H"i "j = Hi ; Hj , H"i+"j = Hi + Hj .
8 " R B = f1 = "1 ; "2 2 = "2 ; "3 : : : l 1 = "l 1 ; "l l = "l g. (2) ) 8 ' # +(1) q +q #1# : +(1) q : so (2l + 1 K) ;! so (2l + 1 K), l ; l = q,
;
;
0
;
0
80A 2sxt B 19 0A 2sxt 0 B 19 < = B y 0 0 x C> = (1) C +q :@ y 0 x A = B 2l+1+2q B @ 0 0 0 0 A> C 2syt D C 2syt 0 D | {z } 2l+1+2q
+(2) 2p : so (2l + 1 K) ;! so (2p(2l + 1) + 1 K),
0A BB BB 0 (2) +2p (A) = B BB 0 BB 0 @0 0 |
... 0 0 0 0
0 0 0 A 0 0 0 0 0 0 0 0 0
{z
0 0 0 0 A 0
2p(2l+1)+1
0 0 0 0 ...
19
0 > > CC> 0C > > = 0C 0C C> 2p(2l+1)+1 B 0C C>
CA> > > A }
+(2) 2p+1 : so (2l + 1 K) ;! so ((2p + 1)(2l + 1) K),
0A 19 0 > = B C . . +(2) (A) = (2p+1)(2l+1) : @ A . 2p+1 > 0 A | {z } (2p+1)(2l+1)
8 ' A 2 so (2l + 1 K).
1075
(C) ) g = sp (2l K) | /, /, Jat J = ;J 1 at J = a, ;
0 19 1 = 0 J = s 0 s = @ A l: 1 | {z 0} 0 s
l
: 1
B a = CA ;sA ts A B C D 2 Mat(l K) B = sB t s, C = sC ts. < #, B C | % /, " .
) 8 h ' % sp (2l K). F % Hi = Eii ; E { {, i = 1 2 : : : l E{ = 2l + 1 ; i.
$ ("i ) | h , " (Hi). 0 R # g h R = f 2"i "j "k j 1 6 i j k 6 l j < kg.
& 13# % H , 2 R, H2"i = Hi, H"i "j = Hi ; Hj , H"i "j = Hi + Hj .
8 " R B = f1 = "1 ; "2 2 = "2 ; "3 : : : l 1 = "l 1 ; "l l = 2"l g. ) 8 ' +(1) q : sp (2l K) ;! sp (2l K), q = l ; l, " # # 13 :
;
;
;
0
+(1) q
A
B C ;sAt s
;
0
0A 0 B 19 = = @ 0 0 0 A 2l+2q : C 0 ;sAt s | {z } 2l+2q
8 ' +(2) q : sp (2l K) ;! sp (2ql K) # # :
1 0A 0 B 0 C B . . A B BBB 0 . . A 0 . . B CCC C BC +(2) q C ;sAt s = B BB . 0 D . 0 CCC .. A @ .. 0
D = ;sAt s.
C 0
D
1076
. .
(D) ) g = so(2l K) | /, " , a 2 so (2l K) , a = ;Sat S, 0 19 1 = 0 s = @ A 2l 1 0 | {z } 2l
$ , so (2l K) ' /
A B C D
A B C D 2 Mat(l K), B = ;sB t s, C = ;sC t s, D = ;sAt s, s | ' /, (C) ).
) 8 h , % 3 , ' % so (2l K). C Hi = Eii ; E { {, i = 1 2 : : : l E{ = 2l + 2 ; i, 1 h.
("i ), ' | % h , " (Hi ). & " R # g h " "i "j , 1 6 i j 6 l, i < j.
8 H 2 R H"i "j = Hi ; Hj , H"i+"j = Hi + Hj .
8 " R B = f1 = "1 ; "2 2 = "2 ; "3 : : : l 1 = "l 1 ; "l l = "l +"l+1 g. ) 8 ' +(1) q : so (2l K) ;! so (2l K), q = l ; l, # # ' , (C). 8 ' +(2) q : so (2l K) ;! so (2lq K) # # ' , (A). $ g | % , +(qk) : g ;! g , q 2 N, k 2 f1 2g | % . . 8 g h B " R ( h ) ' , % # g. 83 , , g ' , g, 13 # ; . 0 # (2) , - , ' # +(1) q +q # , , - , #
#1#
' #. &. ' . !i (!i )
# . g (g ), # h (h ) B (B ).
;
;
0
;
0
0
0
0
0
0
0
0
0
0
0
0
1077
2.1. % +(qk): g ;! g . & 0
m g (( l + 1 2l + 1 2l 2l A, B, C, D
). * # :
! 1 6 i 6 l ! s 6= 0 (2) (A) +(1) q (!i ) = 0i l + 1 6 i 6 l + qB +q (! i ) = 0i s = 0
0
0
i = rm + s, 0 6 s < m.
8 ! i 6 l ; 1 < i (B) +(1) q (!i ) = :2wl l 6 i 6 l ; 1 !l i = l : , i = rm8+ s, 0 6 s < m, s = 0 > < !0s 6 s 6 l ; 1 (2) +2p (!i ) = > 2!l 1s = :!m s l + 2l 6 s 6s =2ll + 1 8 0 s = 0 > > < !s 1 6 s 6 l ; 1 (2) +2p+1 (!i ) = > 2!l s = l s = l + 1 i 6= pm + l > :!m s l + 2 6 s 6 2l
0
0
0
0
;
0
;
!l i = pm + l:
(C)
1 6 i 6 l l + 1 6 i 6 l + q = l : , i = rl +s, 0 6 s < l, ! + r! 0 < s < l +(2) q (! i ) = s r! l s = 0: l
! +(1) q (!i ) = 0i
0
0
0
8 ! > <wl 1 +i wl (1) (D) +q (!i ) = > 2wl : !l
i 6 l ; 2 i = l ; 1 l ; 1 < i < l ; 1 i = l ; 1 i = l : , i = rm8+ s, 0 6 s < m, 0 s = 0 > > ! 1 6 s 6 l ; 2 < s +(2) (! ) = ! + ! s = l ; 1 s = l + 1 i l 1 l 2p > 2! l > : !m s ls +=2l6 s 6 2l ; 1 = m ; 1
0
0
;
;
;
0
0
0
1078
. .
8 0 > > !s > > 2!l !m s > > > : !!l l1
s = 0 1 6 s 6 l ; 2 s = l ; 1 s = l + 1 r 6= p s = l r 6= p l + 2 6 s 6 2l ; 1 i = pm + l ; 1 i = pm + l:
;
0
;
;
. 0 % # 1 ' ,
1 % # ; . ' +(2) q
(). < " +(2) q Hi- Hi - . I # Ei Eii. (2) +(2) q (Hi ) = +q (Ei ; Em+1 i ) = =E i +E l+i + +E (q 1)l+i ; E (q+1)l i+1 ; E (q+2)l i+1 ; ; E 2ql i+1 = 8 > (q + 1)l + 1 ; i = 2ql + 1 ; ((q ; 1)l + i) = (q ; 1)l + i > <(q + 2)l + 1 ; i = 2ql + 1 ; ((q ; 2)l + i) = (q ; 2)l + i => :: :: : :: :: :: :: :: : :: :: :: :: : :: :: :: :: :: : :: :: :: :: :: : :: :: :: : > :2ql + 1 ; i = {: ;
0
0
0
0
;
0
;
0
;
;
2 | def = 2ql + 1 ; j. & # , ' ' 13 : = (E i ; E {) + (E l+i ; E l+{ ) + + (E (q 1)l+i ; E (q 1)l+{ ) = = H i + H l+i + + H (q 1)l+i : 0
0
0
0
0
0
0
0
0
;
;
;
J 1 6 i 6 l ; 1, (2) +(2) q (Hi ) = +q (Hi ; Hi+1) = = H i + H l+i + + H (q 1)l+i ; H i+1 ; H l+i+1 ; ; H (q 1)l+i+1 = = (H i ; H i+1 ) + (H l+i ; H l+i+1 ) + + (H (q 1)l+i ; H (q 1)l+i+1 ) = = H i + H l i + + H q l i: 0
0
0
0
0
0
0
0
0
0
0
;
0
0
+
0
0
0
0
0
;
;
;
( ;1) +
J ' i = l, , # " ' ", ' , +(2) q (Hl ) =
qX1 ;
r=1
r H
0
rl + H rl+1 + + H rl+l 0
0
0
0
0
;1
! + qH : ql 0
0
1079
#' ' +(2) : h ;! h . $ q 1 6 j 6 l ; 1,
(2)
+q
(! i )(Hj ) = ! i (+(2) q (Hj )) = ! i 0
0
J j = l,
0
qX1 ;
r=0
0
! qX1
H rl j = 0
0
+
;
r=0
irl+j :
+(2) (! i )(Hj ) = +(2) (! i )(Hl ) = ! i (+(2) q q q (Hl )) = Pq 1 + qH = =!i r H + H + + H ql rl rl Pq 1 r=1 rl l = r=1 r4 irl + irl+1 + + i(r+1)l 1 ] + q iql :
0
;
0
0
0
0
0
0
0
0
+1
0
+ ;1
0
0
;
;
0 ' # , # . ! i, , ' 1 . . < # 2.1 . , # 13 ' " . , 13" . 0
2.2.
8 i 1 6 i 6 l < 1 (1) ( + 2 + + l ) i = l + 1 ; (A) +q ( i) = : l+1 1 2 l 0 l + 2 6 i 6 l + q + 1:
0
, i = rm + s, 0 6 s < m, s s 6= 0 +(2) q ( i ) = ;( + + 1 2 + l ) s = 0:
0
1 6 i 6 l . , i = rm 8+ s, 0 6 s < m (m = 2l + 1), * l > 2 s 0 < s 6 l > < l + 1 6 s 6 2l m s +(2) 2p ( i ) = >;2(1 + 2 + + l ) s = 0 i 6= pm : ;(1 + 2 + + l) i = pmB 8 s 0 < s 6 l < m s l + 1 6 s 6 2l +2(2)p+1 ( i ) = : ;2(1 + 2 + + l ) s = 0: 8 1 6 i 6 l ; 1 <1 i ( ) = i = l (C) +(1) q i : 20l .
(B)
+(1) q ( i ) = 0i
0
;
0
0
0
;
1080
. .
, i = rl +8s, 0 6 s < l, s 0 < s 6 l ; 1 < +(2) ( ) = ; ( + + + ) q i l s = 0 i 6= pl : 1 2l i = pl: 1 6 i 6 l ; 1 i (D) +(1) 1 ( i ) = 1 (l ; l 1) i = l i = l + 1: 2 8 1 6 i 6 l ; 1 < i (1) , q > 2, +q ( i ) = : 12 (l ; l 1 ) i = l 0 l < i 6 l : i = rm + s, 0 6 s < m (m = 2l), * l > 3
0
0
;
0
;
0
8 X l 2 > > ; 2 k ; l 1 ; l s = 0 r 6= p > > k =1 > s 0 < s 6 l ; 1 < (2) +2p ( i ) = > l ; l 1 s = l l + 1 6 s 6 2l ; 1 > 2l s > l 2 X > > :; k ; 2l 1 ; 2l i = pmB 8k=1X l 2 > > ; 2 k ; l 1 ; l s = 0 > > k=1 < s 0 < s 6 l ; 1 +(2) 2p+1 ( i ) = > l ; l 1 s = l r 6= p > > l + 1 6 s 6 2l ; 1 2 l s > : l i = pm + l: ;
;
0
;
;
;
;
;
;
0
;
;
$ " .
2.1. +(qk): g ;! g | ( A, B, C, D !. : 1. -* # : (A) ) +(qk) (=+ (g )) = =+(g) * q 2 N k 2 f1 2g. ) +(1) q (R(g h )) = R(g h) * q 2 N: (2) +q (R(g h )) = R(g h) * q 2 N: (B) ) +(1) q (=+ (g )) = =+ (g) * q 2 N. +(2) 2p (=+ (g )) = f = 1 !1 + + l 1 !l 1 + 2 l !l j i 2 Z>0 g * p 2 N. +(2) 2p+1 (=+ (g )) = =+ (g) * q 2 N. 0
0
0
0
0
0
6
0
0
;
0
;
1081
) +(qk) (R(g h )) = R(g h) * q 2 N, k 2 f1 2g: . ! *
! (A).
0
0
(C) (D) ) +(1) q (=+ (g )) = f = 1 !1 + + l 2 !l 2 + l 1 (!l 1 + !l ) + l !l j i 2 Z>0 g * q 2 N. (2)
0
;
;
;
;
+2p (=+ (g )) = f = 1 !1 + + l 2 !l 2 + l 1 (!l 1 +!l )+2 l !l j i 2 Z>0 g * p 2 N. +(2) 2p+1 (=+ (g )) = =+ (g) * p 2 N, k 2 f1 2g. ) /
) ( ). 2. A, B, C
0
;
S +(qk)
;
;
;
0
S(h )W = S(h )W * q 2 N k 2 f1 2g: 0
0
D * : S +(1) S(h )W = algS (h )W fg2 g4 : : : g2l 2 g2l g 6= S(h )W q S +(2) S(h )W = algS (h )W fg2 g4 : : : g2l 2 g2l g 6= S(h )W 2p W = alg pl 2p+1 g 6= S(h )W : S +(2) S( h ) 2p+1 S (h )W fg2 g4 : : : g2l (;1) eg
0
0
;
;
0
0
0
0
.
1. , '3 # 1 . , 1 2.1 2.2. 2. I' . 8 #. $ g | % ; . 8 % h 2 h '
Km , m, ' , | % # g. 2 ; " % . (h) T m + g1 (h)T m 1 + + gm (h): < (. 43, VIII.13]), : A (g = sl (l + 1 K)) ./ g2 g3 : : : gm W ' 1 S(h ) B B (g = so (2l + 1 K)) ./ g2 g4 : : : g2l 2 g2l ' 1 S(h )W , % g1 = g2 = = g2l 1 = 0B C (g = sp (2l K)) S(h )W ' # ' 13 g2 g4 : : : g2l, ./ g1 g3 : : : g2l 1 1B , /, D (g = so (2l K)) S(h )W ' # ' 13 g2 g4 : : : g2l 2 eg, eg | % ... % . S (h), S | % # / 2l, " " # /, # B , (;1)l ge 2 = det (h) = g2l B ; , ./ g1 g3 : : : g2l 1 1. ;
;
;
;
;
;
1082
. .
8 ./# ./ fi : h ;! tr ( (h))i, i = 1 2 : : : m, h 2 h. C ./, , 1 " , f1 : : : fr '1# g1 : : : gr , , g1 : : : gr '1# f1 : : : fr ( r 2 f1 2 ::: mg). $%, # S(h )W , ' # ./" fk k = 1 m ./" gk k = 1 m. 9 # ,
S +(qk)
(
(f i ) = 0
fi k = 1 qki k = 2
q. C (k # #W 1 ) W S +q S(h ) = S(h ) # A, B, C. 8 D # , ... " ' # S +(qk) . 9 # 1,:
0
0
S +(1) (ge ) = 0 q (2) e p S +2p (g ) = g2l = (;1)lp eg2p lp 2p+1 = eggp : S +(2) 2p (ge ) = (;1) eg 2l
0
0
0
0
S +(1) S(h )W = S +(1) (alg fg 2 g 4 : : : g 2l 2 ge g) = q q (1) = S +q (alg fg 2 g 4 : : : g 2l 2 f 2l ge g) = = S +(1) (alg ff 2 f 4 : : : f 2l 2 f 2l ge g) = q = alg S (h )W ff2 f4 : : : f2l 2 f2l 0g = = alg S (h )W fg2 g4 : : : g2l 2 g2l = (;1)l ge2 g 6= S(h )W :
0
0
0
0
0
0
0
;
;
0
0
;
0
0
0
;
0
0
;
0
0
2 , ./# f 2l ' #
# # W - ", , '
# ' ' 13 . I ; # # % ' ' . 0
0
S +(2) S(h )W = S +(2) 2p 2p (alg ff 2 f 4 : : : f 2l 2 f 2l ge g) = = alg S (h )W ff2 f4 : : : f2l 2 f2 l (;1)lp ge2p g = = alg S (h )W fg2 g4 : : : g2l 2 g2l = (;1)l eg2 (;1)lp eg2p g = = alg S (h )W fg2 g4 : : : g2l 2 g2lg 6= S(h )W B
0
0
;
;
;
0
0
0
;
0
0
1083
S +(2) S(h )W = S +(2) 2p+1 2p+1 (alg ff 2 f 4 : : : f 2l 2 f 2l ge g) = = alg S (h )W ff2 f4 : : : f2l 2 f2l (;1)lp ge2p+1 g = = alg S (h )W fg2 g4 : : : g2l 2 g2l = (;1)l eg2 (;1)lp ge2p+1 g = = alg S (h )W fg2 g4 : : : g2l 2 g2l egg2pl g 6= S(h )W : 0 , , .
0
0
;
;
;
0
0
0
;
0
0
2.2. g = lim g(i), i 2 I , I | ;!
, g(i) | (
, # +ji, i j 2 I , j > i, *
+ji = +(qk ) +(qk ) +(qkt t ) , k1 k2 : : : kt 2 f1 2g, q1 q2 : : : qt 2 N, t > 1, ( t i j , kr = kr (i j), qr = qr (i j) * r = 1 2 : : : t t = t(i j). 1) =+ (g) = lim =+ (g(i) ) 6= (0), 1
1
2
2
;
W (i)
2) J = lim S(h(i) ) 6= (0), , !
! # ! gi ( i ) ( i ) g , lim S(g ) 6= (0). 3) * i j 2 I , j > i, k1(i j) = = kt (i j) = 2,
3 lim R(g(i) h(i)) ( ! . ,
B, ( * k1 k2 : : : kt:
;
( )
;
;
. 1. I ' # 3 1 2.1 ' # 2.1. 8 # A C =+ (g) 6= (0) - , ' # % #
#1# 1, /#. & B D ' , ' # # 1, /#. : % # 2.1 # ,# % ' =+ (g). 2. # ' #, #
#1# # ' # 2.1. 8 1 # M. ?. F .
1] . . . | .: 1978.
1084
. .
2] Yu. Bahturin, G. Benkart. Highest weight modules over locally "nite Lie algebras. To appear. 3] $. %&. '&(( ). | .: 1978. & ' ' 1995 .
. .
-
, .
512.552.4+512.554.32+512.664.2
: , - , , .
! "# "# ( . . , %&# % " ' %&# (!) # ). + %& : ! A | "# . . & / U , 0 / " A ! ' U . . 0 / U , & ' # , '" , / ") 1 0 ! ' 0
# - "# # ( ! 0 ' 1 # # 0 , | ! ).
Abstract A. Ya. Belov, Classication of weakly Noetherian monomial algebras, Fundamentalnaya i prikladnaya matematika 1(1995), 1085{1089.
We describe weakly Noetherian (i.e. satisfying the ascending chain condition on two-sided ideals) monomial algebras as follows. Let A be a weakly Noetherian monomial algebra. Then there exists a Noetherian set of (super-)words U such that every non-zero word in A is a subword of a word belonging to U . A 6nite set of words or superwords U is said to be Noetherian, if every element of U is either a 6nite word or a product of a 6nite word and one or two uniformly-recurring superwords (in the last case one of these superwords is in6nite to the left and the other one to the right).
. , ( ) (. 10]). $% % (&% ' &' ' ( %& % ) ) &* . + , W | .. % % , AW *% % * % (. & 2) , % , (. .% % % . /& W | . 0% AW | ) , 1 ( ' % v = 0, % v | , ' % W . 3 fWi g | , AfW g | , 1 ( ' i
1995, 1, 7 4, 1085{1089. c 1995 ! "#, $% ! \' "
1086
. .
% v = 0, % v | , ' % % fWi g. 6 U ( . . ), 8k 9n(k) , ( & % k U % * ' & U % n(k). %&' & *% :
1. W : ) k N (k), W k W N (k) ) V
W , W
V . 2 (. 5]). ) W | " , u 6= v | " . # 9r t: rut W rvt 6 W . ) $ U = fui gni=1 | "" % & " & W , 8i 1 6 i 6 n, 9ri ti : riui ti W 8j 6= i ri uj ti 6 W . ) $ I 6= 0 | AW , I , , , ' AW =I " . 2 3 ( ). M | ' A. #
V , "
" % M. 2 uW = Ws. # uW | " u1 . W r | u.
4.
un r,
. 3 W | uW , 8k uk W uk+1 W . 6 % , uW | u1=2 . 2 $ & .. . + ( U , ( ), , *%( ) U , ,
& % .. . , *%( ) U % * % & %&' : 1. 9 2. % uW , % juj < 1, W | .. 9 3. % Wu, % juj < 1, W | .. 9 4. % W1uW2 , % juj < 1, W1 W2 | .. 9 5. & .
5. ( A
U , A = AU .
1087
()
)( % %&
6. A
, . . & % U " . ) *
% ', . . | + , 2 | + & %, 3 | + & %, 4 | " +, " . 2 . % 5 &' & *% .
7. B & cvi c, jvij ) 1 " & i c | " vi . # B
. 2
8. ) A | . # c 2 Wd(A) '(c), u 2 Wd(A) cvc u '(c) &
c. ) $ " & W & c, , , & W '(c) & c. & & . ) $ & W & c , " % "%+, & W '(c) & c. 2 , ( , & % . + % %&'
9. , "" * & . . &
. = %* %&' ( .
10. fWi g1 i=1 | " "" * & . . & . # " k " v Wk "" & fWij g1j =1, & & v.
$ 9. /& fWi g1 i=1 | % ) .. A. / 1 % &' & fWi g1 i=1 , | fWi gj =1 . %., & j
&' % vi , % * %& %&. $ *%' % , ( *% . 2
1088
. .
% 10. / %* . 0% % ' % v fWig1 i=1 * & N (v), i > N (v) v % * Wi . > & 7, i > N (v) *%( & Wi % '(v) % * v. /& ui | % Wi % i. 6 & & U ,
*% % % ui . . ' v U
*%( & % '(v) U % * v. 0 * % ' % c ' Wi . /)& U & % * % Wi . + .. (. & 1) ( ' ' %. ? , U ) Wi , % , Wi ) *%& (. /& & . 2 >1 , % N (A) % . @*%( ( % % .. , % & * % . /& fcigki=1 | &' N (A) (- A ). / & ci & , 7 %&' & *% . 11. ) k, v % Wd(A) % " u k, v = sut, s t | " . . % A ( . . s t N (A)). ) & .& k u 2 N (A) k % Wd(A) & u. 2 (/& ) %& % 1 ( & uw = su . %* 4.) . ) % ( ( A %&
12.
.
juj = k, u 2 N (A).
u, "
# , " . # " &
2
= ( () & * (. . % 5 )( (. 2
1] . ., . ., . ., . ., . ., " # . . $$% &% // " . . 22. "# . )" . *+ . | .: , 1984. | . 3{115.
1089
2] & . ., & . ., 4 5 . . ++ &% // " . . +. ) #. +. 6 +. ) . . 18. | .: , 1988. | . 5{116. 3] $ . . 8 + 5 ) $ 5 ) "# , 9 5 + + :$+ $ // $ . *;. . +., +. | 1985. | < 4. | . 75{77. 4] ;> $ . . +# $+)5$ + "# // " . . +. ) #. +. 6 +. ) . . 57. | .: , 1990. | . 5{177. 5] Belov A., Gateva T. Radicals of monomial algebras // Proceedings of Taivan-Moscow Algebra Workshop. To appear. 6] Gateva-Ivanova T. Noetherian properties of skew polynomial rings with binomial relations // Repts / Dep. Math. Univ. Stockholm. | 1991 | < 8. | P. 1{22. 7] Gateva-Ivanova T. Algorithmic determination of the Jacobson radical of monomial algebras // Lect. Notes Comput. Sci. | 1989. | < 378. | P. 355{364. 8] Gateva-Ivanova T., Latyshev V. On recognizable properties of associative algebras // J. Symb. Comput. | 1988. | V. 6, < 2{3. | P. 371{388. 9] Luca A., Varricchio S. Combinatorial properties of uniformly recurrent words and applications to semigroups // Int. J. of Algebra and Comput. | 1991. | V. 1, < 2. | P. 227{245. 10] Okninski J. On monomial algebras // Arch. Math. | 1987. | V. 45. | P. 417{423. 11] Okninski J. Semigroup algebras. | Marcel Dekker, 1991. | 357 p. 12] Restivo A., Reutenauer C. Some applications of a theorem of Shirshov to language theory // Inf. and Contr. | 1983. | V. 57, < 2{3. | P. 205{213. 13] Restivo A., Reutenauer C. Rational languages and the Burnside problem // Theor. Comput. Sci. | 1985. | V. 40, < 1. | P. 13{30. 14] Rowen L. H. Polynomial Identities in Ring Theory. | New York: Acad. Press, 1980. ) * 1995 .
Lp . .
. . .
517.5
: L , .
p
, A(t)eint B (t)e;i(n+1)t 1 Lp , 0 .
Abstract
B. T. Bilalov, On isomorphism of two bases in Lp , Fundamentalnaya i prikladnaya matematika 1(1995), 1091{1094. If the function system A(t)eint B (t)e;i(n+1)t 1 is a base in Lp then it is 0 isomorphic to the classic exponent system.
(1) A(t)eint B (t)e;i(n+1)t 1 0 Lp (; ), p 2 (1 +1), A(t), B (t) | - ! 1 "; ]. % ! (1) ei(n+ sign n)t +;1
& '( "1]{"3]. + , ' "4] , ' - ! Lp
+1 int - e ;1 . + ' ! ., ' , (1) ' ' Lp , - . / , '( ' ! (1) Lp ' "5]. 1, . . A(t) B (t) | (; ), supvrai t jA(t)j1 jB (t)j1 6 M < +1: (2) , (1) Lp (; ), p 2 (1 +1), ! , ! !
Sf = A(t)
1 X
1 X
0
1
(f einx)eint + B (t)
(f e;inx )e;int
1995, 1, % 4, 1091{1094. c 1995 !, "# \% "
(3)
1092
(f g) =
. .
Z
;
f g6 dx.
. 7 & (1) ' ' Lp (; ). /' S , ! (3). 8, S | ! , Lp Lp . 1 ' ! - 1 . (2) , S | ! L . eint +;1 p %
, (3) , S "eint] = A(t)eint S "e;i(n+1)t] = B (t)e;i(n+1)t n = 0 1: 9 & : , . 8g 2 Lp Sf = g
; . < (1) ' ' Lp , A(t)
1 X
an eint + B (t)
1 X
0
bn e;int = g(t)
(4)
1
an, bn | ' &! - !. /'
F ( ) =
1 X
an n G( ) =
1 X
0
6bn+1 n
j j = 1:
0
P int P1 b e;int ( .. L (; ), , < . ! 1 n p 0 an e 1 Z Z 1 X F ( ) k d = an n+k d = 0
jj=1 Z
jj=1
0
G( ) k d =
1 X 0
jj=1
6bn+1
Z
jj=1
n+k d = 0
8k > 0:
(5)
1 . (5) "6], F (z ) G(z ) > Hp , F + ( ) = F ( ) G+ ( ) = G( ) F + ( ), G+ ( ) | ! . F (z ) G(z ). < ' , .: . A(t) F + (eit) + B (t) e;it G+ (eit ) = g(t)
; . Hp . 8g 2 Lp . ? "6], . 8F (z ) 2 Hp
lim
Z
r!1;0
;
jF (reit) ; F + (eit)jp dt = 0:
Lp
1093
/ , 1 Z F + (eit) e;int dt = an n > 0 0 n < 0 2
;
1 Z G+ (eit) eint dt = 2
,
;
F + (eit ) =
1 X
bn+1 n > 0 0 n < 0
(6)
an eint
0 1 X ; it + it e G (e ) = bn+1e;i(n+1)t: 0
/' 8,
@(z ) = z G(z ): 1 Z @+ (eit ) eint dt = 2
;
bn n > 1 0 n < 1
(7)
@(z ) 2 Hp . +
f (t) = F + (eit ) + @+ (eit ): 1 !: (6), (7) , 1 Z f (t)e;int dt = 2
;
an n > 0 b;n n < 0:
< (4) , Sf = g. + & ! %( , ! '! S ;1 . < . 9 !: ' ' & 9. B. C , ! ' - , : D. 1. B
' : !( &.
1] . . // . . | 1982. | . 37, !". 5 (227). | . 51{95. 2] ' ! (. ). // * + ,. | 1984. | . 275, . 4. | . 794{798.
1094
. .
3] *!/ 0. 0. 12 '3 ''/" . '4!" 5 ''/78" 55/ 3" '/'/'!: * . : : : . 5 2.-. . | ., 1986. 4] . . // * + ,. | 1989. | . 301, . 5. | . 1053{1056. 5] 1 '! 1. . // * 55/. /! 7. | 1990. | . 26, . 1. | . 10{16. 6] ., 1950. | 336 .
' ( 1995 .
. .
512.552+512.553
: , AB 5 , -
.
! " "#, $ " %" " " &, $& . '"" "&# % (# % $& $ % ( ) * ) +" &,).
Abstract G. M. Brodski, A duality theory with applications to endomorphism rings of nitely cogenerated injective cogenerators, Fundamentalnaya i prikladnaya matematika 1(1995), 1095{1099.
It is shown that the Morita equivalence of rings has a dualization di2erent from the Morita duality. We consider applications of the developed duality theory to studying endomorphism rings of 3nitely cogenerated injective cogenerators.
, , . (. . ! ) . ! #11]. & ' Mod{R ! ' R- , ' Modf {R Modf{R | , !
'
'
. ' R- ! ! ' ! ' R{Mod, R{Modf R{Modf . * Y X , X=Y ' . + L(X ), Lf (X ) Lf (X ) ! ! ' '' '
' , '
' X . * ! ' #3]. * (Ei )I | - ! ' 1995, 1, 4 4, 1095{1099. c 1995 !"#, $% \' "
1096
L
. .
R- U = I Ei | 2 ! 3 Mod{R. 4 Ei 3 U , E = End(UR ) S , !
' 2 E , 3 : P j 2 I ' 6'j I , ' '(Ej ) 'j Ei . 7
3 ! E UR S UR HomR (; E UR ): Mod{R ! E {Mod HomR (; S UR ): Mod{R ! S {Mod, D HomR (; S UR ), 3 MR D(M ) HomR (MR S UR ), 3 f : MR ! UR , ' Im f P f Ei 3 ' 8f I . 9 MR ! AnnD(M ): L(M ) ! L(D(MT)) :AnnM ,
AnnD(M ) (B ) = ff 2 D(M ) j Ker f B g AnnM (C ) = C Ker f B 2 L(M ) C 2 L(D(M )). * X Y | ( !3 , ). 2 ' L(X ) L(Y ) X Y . : '' '
Lf (X ) Lf (Y ) X Y #4].
1
.
2
.
#5, 3] MR AnnD(M ) M D(M ) ( , AnnM ). , : 1) M AB 5 " 2) AnnD(M ) M D(M ) ( AnnM )" 3) % & % T T N & M N . #6, 2.5] AB 5 ' MR NR : 1) M N AB 5 " 2) (% A 2 L(M ) B 2 L(N ) ) D (*
.
DAN=B : HomR (A N=B ) ! HomS (D(N=B ) D(A)): 3 #6, 2.1]. + D: Mod{R ! S{Mod
-
7 H1 H2 | , F : H1 ! H2 ( ) * X 2 H1, ! !- Y 2 H1
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FXY : HomH1 (X Y ) ! HomH2 (F (Y ) F (X )) - (! ).
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5. - % AB5 ' MR NR ' , M N AB 5 . * H Mod{R ' ( '), '
( '
). * H Mod{R
' ( '), H Modf {R (
H Modf{R). 6. '
H Mod{R -
)
: 1) H '" 2) D(H) ' ( ') ' S { Mod ) D: Mod{R ! S {Mod :HomS (; S UR ) ( ' H Mod{R D(H) S {Mod" 3) H ' ' ' '
L T {Mod % & T .
@ R
, '
R- AB 5 .
7.
R : 1) R )% % % '" 2) D(Modf{R) = S {Modf " 3) D = HomR (; S UR )" 4) S = E D = HomR (; E UR )" 5) ) D: Mod{R ! S {Mod :HomS (; S UR ) ( ' Modf{R Mod{R S {Modf S {Mod" 6) ) HomR (; E UR ): Mod{R ! E {Mod :HomE (; E UR ) ( ' Modf{R Mod{R E {Modf E {Mod.
4 ! B C , C , R B ! 3 PR . : '
C '
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1098
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#2, 7, 9, 12]. F ' , ' R T B C , Modf {R Mod{R Modf {T Mod{T B #1, . 266]. F, ' (R T ) ' ( ') ' , Modf{R Mod{R T {Modf T {Mod
. C QR (& , '
- ! 3. 8. 1 R T | ,
R
,
F : Modf{R ! T {Modf :G | ) . 3 F G ( ' , & T QR , 1) QR (& " 2) T = End(QR )
" 3) F = HomR (; T QR ) G = HomT (; T QR ).
9.
R T , R
, : 1) (R T ) | ' " 2) T = End(QR ) (& QR .
10.
: 1) 2)
R
-
(R T ) | ' T " R )% % % '.
11.
: 1) (R R) | ' " 2) R )% % ' ' 4 ) UR 4 R.
F 11 ! !3 , 3 3 C #8, 1 3]
#10, 3.1]. * P1 P2 | . G , ' P1 ' P2,
(R T ) : 1) R ! P1J 2) T ! P2. L , ' P1
P2, ! ' R ' ' : 1) R ! P1J 2) End(QR ) ! P2 ! ( ) ! 3 QR . 4
. * V - ' ! ! V - . : ' , B - '
1099
! ! B - (. . , ).
12. 5
: 1) ' '
" 2) V - ' '
' " 3) B - ' ' '
.
: ! :. . C N
! .
1] Anderson F. W., Fuller K. R. Rings and categories of modules. | New York e.a.: Springer, 1992. 2] . . // ! " ". $. | 1982. | &. 37, ) 4. | +. 145{146. 3] . . $/"! " $0! $ AB 5 // ! " ". $. | 1983. | &. 38, ) 2. | +. 201{202. 4] . . $ $/"! " $ // &/. 2. XIX 3 4/ 5 6 7 8. 9. 1. | :; , 1987. | +. 44. 5] . . <$0 5 " $0! // =$ " ;0 0 " " . | 1995. | &. 1, ) 2. | +. 529{532. 6] Brodskii G. M., Wisbauer R. On duality theory and AB 5 modules // J. Pure and Appl. Algebra (to appear). 7] G@omez Pardo J. L., Guil Asensio P. A., Wisbauer R. Morita dualities induced by the M -dual functors // Commun. Algebra. | 1994. | V. 22, ) 14. | P. 5903{5934. 8] MDuller B. J. Linear compactness and Morita duality // J. Algebra. | 1970. | V. 16, ) 1. | P. 60{66. 9] Ohtake K. A generalization of Kato's theorem on Morita duality // J. Pure and Appl. Algebra. | 1980. | V. 17, ) 3. | P. 323{332. 10] V@amos P. Classical rings // J. Algebra. | 1975. | V. 34, ) 1. | P. 114{129. 11] Wisbauer R. Foundations of module and ring theory. | Reading: Gordon and Breach, 1991. 12] Zelmanowitz J. M., Jansen W. Duality modules and Morita duality // J. Algebra. | 1989. | V. 125, ) 2. | P. 257{277. ) * 1995 .
. .
512.544.6+512.552.4
: , ! , # $!% % .
& % | ( ! % 1 k ! % ( ) $ k , | %* ! ! ( ) , - # ;1 i. . ! | - . $!% % 2 h
> 1, R
k
D
D
C
R
D
C
G
U R
R
f
R X X
R
PI
Abstract I. Z. Golubchik, Generalized identities with invertible variables for subrings of artinian rings, Fundamentalnaya i prikladnaya matematika 1(1995), 1101{1105.
Let be a prime subring with 1 of the matrix ring k over a skew 2eld , > 1. Suppose that the center of is in2nite and elements of belong to the center of k . Let be an elementary absolute irreducible subgroup of the group ( ) of invertible elements of with a nonzero generalized identity with invertible variables ;1 i, then is a -ring. 2 h R
D
C
D
R
D
k
C
G
U R
R
f
R X X
R
PI
|
, . . !. ". # $ %1], %2]. 1. ) R | * 1 * C U (R) |
. * R, G |
U (R). a 2 R CG(a) = f 2 C j 1 + a 2 Gg, EG (a) = = f1 + a j 2 CG(a)g. )
G 1 , EG (a), . CG (a) . 2. ) G |
U (R), C | * R. 2 ,
G , C | R,
G, | d R d, d | R. 3. ) R | * * 1, Q(R) | * , P | * * Q(R), X = fx1 x2 : : :g | , P hX X ;1 i | * ;1 ;1 xi x;1 i 3 xi xi = xi xi = 1, H | 3 !! $ INTAS.
1995, 1, 4 4, 1101{1105. c 1995 , !
\# "
1102
. .
* Q(R), P , RhX X ;1 i | * 1 P - H P hX X ;1 i .55* 1 R, RhX i | * RhX X ;1 i, 1 xi. ) f 2 RhX X ;1 i G |
U (R). 2 , f = 0 G, (f ) = 0 6 * 51 : RhX X ;1 i ! R, R , (xi ) 2 G, ((xi ));1 = (x;1 i ) xi 2 X . 8 6
4. R | 1 Dk D, k > 1, C R C Dk . , G | , " U (R) G f 2 RhX X ;1 i. # " R | PI- . 5. 9 R | PI- * F | * C * R, R C F | F - (. %5], . 284). 1 1 5 , f 2 RhX X ;1i |
G
U (R) R | PI- *, F= | 1 F , R C F= | * * F= f = 0
G= , 6 1
G. 8
. #. > %3]. , 4 D *. " 1 4 1 @. . A %4] * . 6. 9
H G f = 0 G,
H . " , 4
U (R) . 6, 6 6
G R | PI- *,
U (R) 1 RhX X ;1 i. x
1 4
) AG = fa 2 R j CG (a) g. # h 2 RhX i 1 , xi 6 hj 1 h 1. 7. 4 g(x1) 2 RhX X ;1i g(x1 ) = 0 | G. gk = b1x"11 : : :bk x"1k bk+1 " "j = P 1, dgk P= Pkj=1(;1)"j b1 : : :bj y bj +1 : : :bk+1. , h = d(g) = d(gk ), " g = gk . # " h(y) = 0 | AG .
1103
. ) a 2 AG. ) 6 g(1 + a) = 0 2 CG (a). , g(1 + a) | . . 1 * D()k , D() | . * * D%]. > g(1 + a) = = g1() (g2());1 , g1() 2 D%]k , g2() 2 D%] g1 () = 0 2 CG (a). ) CG(a) . g1 () 6 4 1 CG (a) n f0g Dk , , " , , .55* g1 () 6. , 55* g(1 + a) , a)
, h(a) dg(1+ d =0 = 0. ) 7 1. 8.
4 " G f , f 2 RhX X ;1 i. # " %
" h 2 RhX i h | AG .
=0
=0
. ) xi x;1 i | , f , m
X
j =1
aj 1x"i11 : : :ajm x"imm ajm+1 |
(1)
f , L1 "1 : : : Lm "m 1 P m
X
j =1
aj 1 : : : ajm+1 .
(2)
) xi 1 x2i x2i+1, , (1), ik = ik+1 , "k = ;"k+1 1 2= L(a1k : : : ank)
(3)
L(a1k : : : ank) | P . a1k : : : ank . ) Q Ti = fk j ik = ig. ) , Vi = k2Ti (1 + k Yk ) | 1 1 k, xi = Vi Zi Vi;1. ) 55* f (xi ) 1 j , 1 6 j 6 m, 1 1 = : : : = m = 0. )
6 7 g(Yj Zi Zi;1), 6 Yj 2 AG , Zi 2 G. ) , g(Yj Zi Zi;1 ) "m Y1 Zi"11 : : : Ym Zim
(4)
. , xi k 55*
1, ik 6= ik+1 1, (4) . 9
1104
. .
k xi 55* 1, (4) n
X
j =1
"m ajm+1 aj 1Y1 Zi"11 : : :ajm Ym Zim
(5)
1, ik = ik+1, "k = ;"k+1 k (5) ajk+1 Yk+1 ;Yk+1 ajk+1. * k 1 (3) (2) , 1 (4) , 1, l
X
j =1
"m c cj 1Y1 bj 1Zi"11 : : :cjm Ym bjmZim jm+1 6= 0:
(6)
, xi = x2i x2i+1, . 1 "k 1, Q 6 . ) g(Yj Zi Zi;1) Zi = k2Ti (1+k Uk ),
55* g 1 j , 1 6 j 6 m, 1 1 = : : : = m = 0. ) 6 7 h(Yj Uj ), 6 Yj Uj 1 AG . 8 , h . D 6 h Y1 U1 : : : Ym Um : (7) 9 k Zi 55* 1, Uj Ut (7) . 9 1, 55* (6), , 1 3, "i (7) . , , 1 (6) 1 Plj =1 cj 1 bj 1 : : : cjm+1 6= 0. , h 6= 0, 8 1. 9. 4 h 2 RhX i | , AG . # " R | PI- . . ) 1. ) aj bj 2 R Pnj=1 aj x bj = 0 x 2 AG . > a 2 R, 1 aj bj , y z 2 R Pnj=1 aj a(za2 y ; ya2 z )a bj = 0. , x y 2 AG , (1 + y) x (1 + y);1 2 AG 2 CG(y). > Pn Pn ;1 7 aj (yx ; xy)bj = 0. j =1 aj (1+ y) x (1+ y ) Pnbj = 0 6 j =1 ! , z 2 AG j =1 aj z x ybj = Pnj=1 aj y z xbj . . ) , C - S ,
1 . 1 AG , 6 x1 x2 2 S Pnj=1 aj (x1 x2 ;P x2 x1)bj = 0. ) 6 2 S a R a, a | R, 1, nj=1 aj a(za2 y ; ya2 z ) a bj = 0 y z 2 R. ) 1 13.
1105
) Pnj=1 a1j x1 : : :amj xm am+1j | h(X ) 1 n
X
j =1
a1j : : : am+1j .
(8)
) 1, xi = a (ui a2vi ; vi a2 ui) a, g(ui vi), * R, 6 , 6 1 . 1 AG Pn a au a2 v1 a : : :una2 vnPa an+1j . > a | R, 1 j 1 j =1 1 (8) , 1 nj=1 a1j a a2 : : : a anj a a2 a anj , 1, g(ui vi) . , R * * Dk , 1, . * R , R R g(ui vi). , 8 1 %4] R | PI- *. ) 9 1. > 4 1 8 9. ! 1 1 !. ". # $.
1] . ., . .
!" "" "
// $" % . . | 1980. | *. 35, - 6. | /. 155{156. 2] . ., . .
!" "" "
// . . "%. 2 3 ///4. | 1982. | *. 114. | /. 96{119. 3] *% . .
!" 6 // 7 .
3 8//4 | 1982. | *. 26, - 1. | /. 9{12. 4] 86! . 9. . : ;< " %
!" %. | $" % .
. | 1977. | *. 32, - 4. | /. 249{250. 5] 8 .
>. . " "
" %6
. | .:
3 ,
1990. | 318 ".
% & 1995 .
. .
512.81
: , .
, ! " " ! # $ " # # , " !%& . ' $ # $ n n-
# ! # " !! 2-
) $ " .
Abstract D. N. Ivanov, Homogeneous orthogonal decompositions of commutative algebras and Hadamard matrices, Fundamentalnaya i prikladnaya matematika 1(1995), 1107{1110.
It is shown that a commutative algebra is a free module over every its subalgebra of the family forming its homogeneous orthogonal decomposition. As a corollary the equivalence of notions of a Hadamard matrix and an orthogonal decomposition of a commutative algebra into the sum of two-dimensional subalgebras is deduced.
1. 1] ( ), , ! " " " , $ ( $"% &' \ "). * 2] ! $ , " ! $ % . -$ " ! | ! $
/ &' ! / , , ! ! , ,! " 0 1 ! n > 2 &' 2M1 nM1 . 2. 2 &'. 3 $ $, 1] &' " " , ! , ,
! ! . 4 2]. 5
$
0 ! ! ! / C , ! , ,
$ ! / Mk (C ) = Mk . 4 ! , , ! " " $ , " " n ,! C , ! ! nM1. 1995, 1, 0 4, 1107{1110. c 1995 , !" \$ "
1108
. .
. 9 / D = fBig &' A, : 1) Bi " , 1A A; 2) A " Bi ; . . / "; 3) ! Bi Bj 2 D, $ ! " C 1A , . . x 2 Bi , y 2 Bj , TrA x = TrA y = 0, Tr xy = 0, TrA | A. ' D A B, D B. 3. = !, $ A = nM1 ; D = fB1 : : : Br g | &' dM1 A, d > 2. > r % P ": dimA = ri=1 (dimBi ; 1) + 1. * , r = nd ;; 11 . * $ e1 : : : en | A; eij , j = 1 d | Bi , i = 1 r. * TrA xy = x y, TrA eij = mij . @ ! ! 2], A B- B 2 D $! , ! !0 TrA B 0 $ TrB . &
1. A
,
mi1 = : : : = mid .
2 1].
Bi -
x21 + + x2n + (r ; 1) (x1 + n + xn ) = 2
Xr Xd (x eij ) 2
i=1 j =1
mij
x = x1 e1 + + xn en .
4 ! ! ! , 1A , eij , i = 1 r, j = 1 d ; 1, A x y A ,
3. x 1A = , x eij = ij , i = 1 r, j = 1 d,
= i1 + + id, i = 1 r.
x 2 A,
(
)
@ "
4. #
v0 eij = 1, i = 1 r, j = 1
v0 2 A, v0 1A =d, d. $ v0 = d 1A.
n
1109
. 3 , v0 . * $ v0 = a1 e1 + + an en . * $ ! e1 ei1 = 1, i = 1 r. '
, v1 = v0 ; 2e1 . B! $, v1 v1 1A = d ; 2, v1 ei1 = ;1, v1 eij = 1 j = 2 d, i = 1 r. 3 % 2 v0 v1 : a21 + + a2n + (r ; 1) dn = 2
Xr Xd
1 m i=1 j =1 ij
(a1 ; 2)2 + a22 + + a2n + (r ; 1) (d ;n 2) = 2
Xr Xd
1 : i=1 j =1 mij
* / , a1 = nd . 1 ai = nd , i = 2 n, , ! , v0 = nd e1 + + nd en = nd 1A . B 4 ! . = !, 4 , v0 = nd 1A v0 eij = 1. & , mij = nd . $ , 1, ,
1. % D | A
'# A. $ B 2 D.
B -
4. 9 ! " 0 1 H ! n > 2 ( . 3])
$ ! ! ! (n ; 1)- " 0 , A = nM1 / A. * , , 0 , &' 2M1 A. & , $ D | 2M1 nM1 . * 1 0 ! " D " , , $ (
" ! " ). 4 ! , ! / ! / nM1 ! " 0 0 1 . " $ .
2. % ( n > 2 '# 2M1
nM1
.
1 1. =. @ ! D E 4$, ! % %! 2].
1110
. .
1] . . //
!"#. $ . 1. !., &. | 1988. | * 1. | $. 9{14.
2] . . . & / , 01 3 // # &&. . | 1989. | .. 44, * 2. | $. 231{232. 3] 5 !. 6& . | !.: ! , 1970.
& ' 1995 .
. .
512.544.6+512.666
: , ,
!".
# ", | ; , 1 SK1 ( )1 = 0, + 2) r * 1 m] s n 1 1 ,- . . r
> max(3 dim
C
C
C
SL
C
T :::T
X
:::X
Y :::Y
Abstract
V. I. Kopeiko, On the structure of the special linear groups over Laurent polynomial rings, Fundamentalnaya i prikladnaya matematika 1(1995), 1111{1114.
In this note we prove the following such that ; result. Let be a1 regular ring 1 SK1 ( ) = 0. Then the groups r * 1 m] s 1 n 1 are generated by elementary matrices for all integers > max(3 dim + 2). C
C
SL
C
T :::T
X
r
:::X
Y :::Y
C
2]. , 1. " #
. . C |
, A = C T1 : : : Tm ] B = A X1 1 : : : Xs 1 Y1 : : : Yn : r > max(3 dimC + 2) GLr (B)= Er (B) ! K1 (B) . " m. ) m = 0, 7.8 2]. . , m > 0. . T = Tm , D = C T1 : : : Tm;1] s n] ( R Rs n] R X1 1 : : : Xs 1 Y1 : : : Yn ). / , k = max(3 dimC + 2). 0 1 2 , , , GLr (B) = GLk (B) Er (B) r > k. " 1 3 # , , , 34 ,
. 1995, 1, 1 4, 1111{1114. c 1995 , !" \$ "
1112
. .
1. ; A |
;
, ;r | . ; GL A T] = GL (A) GL A T ] A T] . ; , E r r 1 r ; ; ; GLr TA T ] GL1 A T ] Er A T] , SLr T A T ] Er A T ] . 5 r. ) r = 1,; . . , r > 1, = (T ) 2 GLr A T ] . 6 (0);1 (2 GLr (A)), , P 1r mod T. . f = rr . . f : f = 1 + T i>1 ai T i;1 , ai 2 A, Q ;1 , f A T] . . 1 = ri=1 Eir (;f ;1 ir ), ; Qr;1 ; 1 2 = i=1 Eri (;fri ) diag(1 : : : f f ) 2 Er TA T ] . 8 , ; = 1 2 2 GLr;1 TA T] , , ; ; = ( ;1 1;1 )2;1 ] 2 GLr;1 A T ] Er A T] ;1 ; ;1 1;1 = 1r0;1 1 v 2 Er A T ] , v =(f ;1 1r : : : f ;1 r;1 r )T. " 4
.
. 9 , ; ; SLr A T ] = SLr (A) Er A T]
, ; ; SLr (A)= Er (A) ! SLr A T] = Er A T] (r > 3)
, 2 (2
#
A T ] ! A, T ! 0). , ; 3 , , SK1 (A) = SK1 A T ] A. " 2 K1 1, , . . 1 2 . . ; = (T) 2 GLr (B) = GLr D T] : 0 (0) 2 GLr (D),
(0) 2 GLk (D) Er (D): / , (0);1 1r mod T , , 1 ; ; (0);1 2 GL1 D T] Er D T ] : 9: ; ; = (0)(0);1] 2; GL (D) (D) GL D T ] r D T]; ; k Er; 1 ; E GLk(D) GL1 D T ] Ek D T] Er D T ] = GLk D T ] Er D T] 1. : 2 .
1113
" 2 . 1
A | . . B | A, s 2 B | 0 A, 2 B=sB ! A=sA, , 2 . :
: B ! A
#
#
Bs ! As . < , B=sB = A=sA, b b B = A, b (s)- . < , s 0 A.
2 ( 4, 2.4]).
, (B A s) ! , 2 GLr (A), r > 3. " s 2 Er (As ),
# = , 2 GLr (A), 2 Er (A). $
, 2 SLr (A), 2 SLr (B). . 2 . .
B 0 = C T1 : : : Tm;1 ] Tm ]s n] = DT]: < , (B 0 B T ) . .
\E ;DT] =SLr ;DT]\E ;DT] : 0 (0) 2 SLr (D) \ E (D), , (0) 2 Er (D) =(T) 2 GLr(B)\E (B)=GLr D T] ;
. 0 C T] | dimC + 1 (., , 3, 15.1]), C T ] T | 6 dimC ( , C T] M = (M C T), M C = M \ C | C (., , 1, . 1])). ? ,
GLr C T] T T1 : : : Tm;1 ] s n] \ E; C T] T T1 : : : Tm;1] s n] = = Er C T] T T1 : : : Tm;1] s n] = Er (BT ): ;
;
, T 2 Er (BT ), , ; 4 = , 2 SLr (B 0 ) = SL r (DT ]), 2 Er (B) = Er ;D T ] . 0 (0) = (0)(0);1 2 Er (D) (0);1 2 SLr (T DT]) Er D T] 1, = (0)(0);1 ] 2 Er (B). 0 .
. % &
, C |
, SK1 (C) = 0. r > max(3 dimC + 2) SLr (B) ' & .
1114
. .
" 7.10 2]
5-@-? 5, . 12]. , # ?
( m = 0) 2, 7.8], 3 ? ( m = 0) 2, 7.10] ( s = 0, C = k | ) 4, 3.1].
1] ., . . | .: , 1972. | 158 #. 2] $# . . % # #&' ( (( &&) ' + // -. . $$$/. $. . | 1977. | 0. 41, 2 2. | $. 235{252. 3] Matsumura H. Commutative ring theory // Cambr. studies in advan. math. | 1986. | V. 8. | P. 3{320. 4] Vorst T. The general linear groups of polynomial rings over regular rings // Commun. Alg. | 1981. | V. 9, 2 5. | P. 499{509. 5] 7 ## 8. +# K-. | .: , 1973. | 592 #. & ' 1995 .
. .
. . .
512.552.7
: , , , ( ), "# , ""# .
$% "" $&'" 1984 8+240 - - - #"# . % ./ " . 0 " R | , R 6= J (R), char R = 0 (
"""
char R = p > 0), S | % # (
""" 3#) . 4
5 " ./ # #
R6S ]: (i) R6S ] 6 # ] 9 (ii) R 6 # ] , S " # : "" (
""" " : p- :) / "" : . 0 %" " % "# ./: ;". 0 " R | , R 6= J (R), S | . # "
, 3" R6S ] (
""" # ), - "" 3 , 3" R (
""" # ), ; " : !R6S ] J (R)6S ] .
Abstract A. Ja. Ovsyannikov, Local semigroup rings, Fundamentalnaya i prikladnaya matematika 1(1995), 1115{1118.
The description of local semigroup algebras over a =eld of characteristic p (if p > 0, then semigroups are assumed to be locally =nite) due to J. Okninsky (1984) is transferred to semigroup rings over non-radical rings. The following statement is proved. Let R be a ring, R 6= J (R), char R = 0 (char R = p > 1), S be a semigroup (respectively, a locally =nite semigroup). The semigroup ring R6S ] is local 6scalar local] if and only if R is such a ring and S is an ideal extension of a rectangular band (respectively of a completely simple semigroup over a p-group) by a locally
nilpotent semigroup.
R ( ), - R R J (R)
( ). ! "1]{"4]. ' "1] ( ) . ' "2], "3] , (
! 0 ( ( 1995, 1, > 4, 1115{1118. c 1995 , ! \# "
1116
. .
( ! ) ) ! . / ,0 1 ) , . ("3], 5). k | , char k = 0 ( char k = p > 0), S | ( ) . k"S ]: (i) k"S ] " (ii) k"S ] " (iii) S # ( p- ) .
' "4] , ,0 , 1 . ' 0 ) 1 4 . . 5 "1], "4] ,
. 6 , . 7 "5]. . R | $, R 6= J (R), char R = 0 ( char R = p > 0), S | ( ) . $ R"S ]: (i) R"S ] " ] " (ii) $ R " ] , S # ( p- ) .
8 ,( , ,0
, ,0
) . . R | $, R 6= J (R), S | . %
, $ R"S ] ( ), & , $ R ( ), ' !R"S ] J (R)"S ] . . 6 R"S ] | ( ) . : R, ( R"S ], ( ) . : R"S ]=J (R"S ]) | , , I
1117
R"S ]
I J (R"S ]), I + J (R"S ]) = R"S ]. ' ( , ( , R"S ]=I . : R"S ]=!R"S ] =R R"S ]=J (R)"S ] = R"S ], R , !R"S ], J (R)"S ] . 6 R | ( ) , !R"S ] J (R)"S ] |
. 6 !R"S ]+J (R)"S ] = I , ( R"S ]=I = R. : I | ) , , , ( R"S ] | ( )
. 7 : R"S ] ;! R, X X s = n
=1
i
n
i i
=1
i
i
( ( 1 (
R ;! R. > , ( | ,? ) R"S ]
R. , ( ker = I . 6 . : . 6 ( ! ( , ! , @ . 6 R 6= J (R) R"S ] | . : , R , !R"S ] . 6 k | R, ). 6 , ( k"S ] . / , 1, ( , ( , ( k"S ] | ) k"S ]. 6 x 2 !k"S ]. : x 0 ) ) 1 y !R"S ]: x + y + xy = 0: (1) A R , k- k+_ T T R. ' ( , kT Tk T: (2) C T "S ] ( ! 1 R"S ], ! 1 T . : 1 y (1)
y = u + v, u 2 k"S ], v 2 T "S ]. D (1) x + u + v + xu + xv = 0. : v 2 T "S ] (2) xv 2 T "S ], ( x + u + xu = 0, v + xv = 0. / , ) 1 !k"S ]
) ) k"S ], !k"S ] | ) k"S ]. E , ( k"S ] | . 6 S (ii) . D (i) ) (ii) . , ,. 6 R | , S | @ ) ( char R = 0)
) ( ) p | ) ( char R = p > 0) 0 , ) . D 1 "6]
1118
. .
, ( J (R)"S ] . / 0 , "7] , ( ! ! !R"S ] ) . 6 @ (ii) ) (i) . . G. H , E. '. ' . '. 8 .
1] Lee, Sin Min. A condition for a semigroup ring to be local // Nanta Math. | 1978. | 11. | P. 136{138. 2] Okninski J. Finiteness conditions for semigroup rings // Acta Math. | 1983. | V. 25, 1. | P. 29{32. 3] Okninski J. Semilocal semigroup rings // Glasgow Math. J. | 1984. | V. 25. | P. 37{44. 4] Wauters P. and E. Jespers. When is a semigroup ring of a commutative semigroup local or semilocal? // J. of Algebra. | 1987. | V. 108. | P. 188{194. 5] . . !" # $%&$##!" ' " // ()*. +,. VI -.#*$. # 0)& )', %)+& .1$ )2, 3 , 11-13 )0. 1990 %. | -. 94. 6] . . & 1 !" # $%&$##!" ' " // 4 0).. * .)0. | 1985. | (. 37, 3. | -. 452{459. 7] . . & 1 0 5$1 .)0 !" 1) # $%&$##!" )' // 6 )1 %)+& 7)" 0). # 20 . " #10). (4 0).. * #. 9& . $-0 . (. 14, 1). | -)&1 , 1985. | -. 119{127. % & ' 1994 .
. .
: , , , , .
, , ! , " .
.
Abstract L. I. Pugach, Projective ideals of Banach algebras and approximative units, Fundamentalnaya i prikladnaya matematika 1(1995), 1119{1123.
It is shown that if an ideal of a commutative Banach algebra has a Schauder base, then it is projective i+ it contains an approximative unit with some special properties. For uniform algebras a criterion is formulated in terms of peak sets.
. . 1]. X A ! , #! $ % : A b X ! X, ! # (a x) = ax, #! . + b !, ! ! 7]. . / 0 1 , (., , 2] { 6]) !, | , | % | % , ,, ( . ..). 6 3], M
. .. A !, ,: (M = M 2) (M 6= M 2 ). 8 ! M 2 !, ! ,
M | , 1 m2 , m 2 M. 9 ! , , , ! /, 5] / , ; A. , - - . , - - ( 93{011{156). 1995, 1, 1 4, 1119{1123. c 1995 !, "# \% "
1120
. .
= , ! . > . .. , , | 1 % ! / 0 (..) #! : M ! M b M, , = 1. ?! %, , % ! % .. ? % 1 !. , , I . .. A / .. . 6 , , , , 0 ,
. =, , , ( ! ,) ! @ . ? % , % ! 6] , , , # , , , p-% ( / % ). ? , ! ; , % . % #! . 9 !, , ! AnnI X I- X. 1. X | A- , AnnI X = f0g. : I ! X . . i 2 I, in 2 I, lim i = i, lim (in ) = x. B n n n 0 j 2 I j(x ; (i)) = lim j((in ) ; (i)) = lim (j(in ; i)) = lim (i ; i)(j) = 0: n n n n 1 x ; (i) 2 AnnI X , , x = (i). ! # . ? % , , / . 1. ) ! AnnI I = f0g, : I ! I
( ) . ) ! AnnI I b I = f0g, .
: I ! I b I
-
. , , , ) $
. .. 1. ! 1) I $ .. 2) AnnI I = f0g I &, AnnI I b I = f0g. . 1) C I % .. fgg, / 0 { @ ! 8, 2.2.7] ! , , lim gu = u
1121
0 u 2 I b I. 1 u 2 AnnI I b I, g u = 0 2 E u = 0. 2) u 2 AnnI I b I. +# ! f 2 I a = (1 f)u 2 I. > 0 i 2 I ia = i(1 f)u = (1 f)(iu) = 0. 1 a = 0. B I , (1 f)u = 0 f 2 I , u = 0 7]. C 0 I A-, ! , #! : I ! A b I. . $ , % ( I b I ! / I (A b I)) % !
2.
& AnnI I
.
! I , = 0, A- : I ! A b I
E ! @ fe 2 Eg, E | % 0 /. ? 0/ ff g E , 0 fe f 2 Eg | \ @ ". 9 !, , ! ; % , = (1 : : : n), 0,0.
2. ... B $ ' ) ( C > 0, ) b 2 B , 2 ; X f (b)e k k 6 C kbkI
fe f 2 Eg.
k2
) ( D > 0, ) u 2 B b X ek (1 fk )u 6 DkukI
B, 2 ;
k 2
)
lim
X k 2
(1 fk )u ek = u ) u 2 B b B .
. ) J P : B ! B, !P # P (b) = k 2 fk (b)ek . C b ! ! e , fP (b) 2 ;g ! ,
f0 eg ,. ? P ! % ,. . , % 1 b 2 B, ,, % % % . { @ ! 8, 2.2.5] fP g
1122
. .
8, 2.2.4] ! , $ B % % % $ C. ) !. 8 ) ) ! 0 ,. 9, , B b X, X | B- ! @ . B ! ! . . A | ... &. * I A, ( ', , ( : I ! I , , P 1) g = k 2 k (ek ) .. I ,
) + P 2) ) i 2 I u = k 2 k (i) ek ) A b I .
. ) >, . C / , % (i) = lim u . L , % : I ! A b I, 0/ , , , #! A- . . 0 2 , . >, (i) = lim (u ) = lim
X
k 2
k (i)ek = lim
X
k 2
i k (ek ) = lim ig = i:
= , = 1 I .
) M . I , / #! : I ! A b I, , = 1. J #! k = (1 fk ): I ! A. 2 P I , , , k (I) I. k 2 ek (1 fk )u ( 2 ;, u 2 A b I) (u). 1 lim g i = lim
X
k 2
k (ek )i = lim
X
k 2
ek (1 fk )(i) = (i) = i
, , fg g .. I. >, 0 i 2 I, 2 ; ! , ) 2 X
kg ik =
k 2
X
k (ek )i =
k 2
(1 fk )(i)ek 6 Dk(i)k 6 Dkk kik
, !, , fg g (1 0 % N. ?. . ). 90, ! , , . B !. C A | , , ! , , .. I 1 # , , , I p-% ( / % ) 9, 1.6.3]. B !, , 3. A | ) . ! I , ( ', , p- $ .
1123
! ; p-% % 0, % P ! , 2] ,
4. A | ) . * I A, ( ', , ) )) $ .
1] . . // . . | 1970. | . 81. | . 430{444. 2] . . // . . | 1970. | . 83. | . 222{233. 3] . . // "#$ . . %. &. '( ). | 1978. | . 3. | . 223{242. 4] *$ ( +. . // , &-. ., /. | 1979. | . 4. | . 8{13. 5] '")1 *. %. // . 2 . | 1982. | . 31. | . 239{245. 6] '")1 *. %. // ,4$
)"44 )) 1 )$. | (5, 1987. | . 22{24. 7] Grothendieck A. // Mem. Amer. Math. Soc. | 1955. | V. 16. 8] 6"# -. 7" 8 5$ 2. | .: , 1975. 9] Browder A. Introduction to function algebras. | N.Y.: Benjamin, 1969. ' ( 1995 .
Hs- , . .
. . .
517.95
: , -
.
! " ! " " H s(RN ) %, " "&' ! ! (
! " L2 (RN ) ", " " ! " RN , " : A(x D) = P (D) + Q(x) , P (D) | ! % . / % m ! . ! , %" % Q(x) & . 0 " ! , & " "! ! & % " &/ m > N . 2
Abstract V. S. Serov, On the convergence in H s -norm of the spectral expansions corresponding to the dierential operators with singularity, Fundamentalnaya i prikladnaya matematika 1(1995), 1125{1128. In this work we prove the convergence in the norm of the Sobolev spaces H s(RN ) of the spectral expansions corresponding to the self-adjont extansions in L2 (RN ) of the operators in the following way: A(x D) = P (D) + Q(x) where P (D) is the self-adjont elliptic operator with constant coe4cients and of order m and real potential Q(x) belongs to Kato space. As a consequence of this result we have the uniform convergence of these expansions for the case m > N2 . RN (N > 2) -
m (m | ) A(x D) = P(D) + Q(x)
(1)
P(D) | $ m , Q(x) 1995, 1, 5 4, 1125{1128. c 1995 !", #$ \& "
1126
. .
Z
Rjxj<<1 jQ(x ; y)j !m(x) dx < 1
(2)
8< jxjm;N m < N !m (x) = : 1 ; log jxj m = N 1 m > N:
(3)
sup
2
y2 N
!m (x) '
) m 6 N $ , (2) 0 ! 0. , H s (RN ) -. $ f 2 L2 (RN ), / :
kf kH s (RN )
Z
RN
(1 + j j2)s jfb()j2 d
1=2
b -. -. 1 f. f() 2. 3. 4 / 51] (. 7) , / (2){(3) Q(x) A(x D)
L2 (RN ) $
b . - H m(RN ). ) ' A, - f 2 H m b kL2(RN) 6 c1kf kHm (RN) kAf
b k2L2 (RN) > c2kf k2Hm (RN) ; c3kf k2L2(RN) kAf
(4)
(5) c1 , c2 , c3 | $ . )
;$. < Ab
Ab
Z1
c0
dE
(6)
b c0 > ;1, fEg | ,
A. = - - . . .
f 2 H m (RN ) lim kE f ; f kH m (RN ) = 0: (7) !+1 > . 4 ;?+q(x) , - / , - . -
H s -
1127
4. >. > 2. @ 52]. ) , q(x) : 1) q(x) 2 C 1 (RN n S) N > 3 diamS 6 N ; 3A 2) jq(x)j 6 c(dist(x S));1 dist(x S) -. x 2 RN / S. B $ - 4. >. > 3. C 53], ' - 52], -
- . . ; q(x) 0, . . / / .$ , / $ ' D , . -. - =. >. 2 54]. C
-, / . / - , - . - , $ =. >. 2 55]. . 2. (4){(5) , - 52], $ , 0 > 0, > 0 > 0
k(Ab + )f kL2 (RN ) 6 c0 kf kH m (RN ) (f 2 H m )
(8)
k(Ab + );1 gkH m (RN ) 6 c00kgkL2 (RN ) (g 2 L2 )
(9) $ c0 c00. ;
, (6), (8) (9) - f 2 H m
kf ; E f kH m = k(Ab + );1 (Ab + )(I ; E)f kH m 6 6 c00k(Ab + )(I ; E )f kL2 = c00k(I ; E)gkL2 ! 0 ! +1, g (Ab + )f 2 L2 (RN ). G . .
1. 0 6 s 6 m.
f 2 H s(RN )
:
RN) = 0:
lim kEf ; f kH s (
!+1
(10)
2. m > N=2.
f 2 H m (RN )
RN lim E f(x) = f(x): !+1
(11)
> $ - - 4. >. > . . - $ .
1128
. .
1] Schechter M. Spectra of partial dierential operators. | Amsterdam, London: North-Holland, 1971. | 268 p. 2] Alimov S. A., Joo I. On convergence of eigenfunction expansions in H s -norm // Acta Sci. Math. | 1985. | V. 48. | P. 5{12. 3] Alimov S. A., Barnovska M. On eigenfunction expansions connected with the Schrodinger operator // Slovak. Math. J. | 1985. 4] . . !"#! $ %&'!()* +! !, $-).# /0123# !+)%&$!%& 4&+& & // 567. | 1958 | 8. 13, 9 1. | :. 87{180. 5] . . :+)1$%&&3 $)!%3 "//)%)2&. !+)%&$!%!-. | 6.: 7&01&, 1991. ( ) 1995 .
. . 519.46
: , ,
, , ,
, ! , ! .
. # $ %
G & '
G T (2R) |
$ % , , - '(x) = (0x) '((xx)) : 1) | !
G R 1 2) & k'(xy) ; '(x)'(y)k1 x y 2 G - . $ %
G % , % - 2 % ( )- . % ,
- . # % 3 $
G % ( )- . 4& , - G = A B | - -
A B , A =B = Z2 , G % 3. 5 &
A B ! Z2 , !
G R
G % (")-, (")- ()- . 6 % " | ! , &7$3
G ,
R . Abstract
V. A. Faiziev, Two-dimensional real triangle quasirepresentations of groups, Fundamentalnaya i prikladnaya matematika 1(1995), 1129{1132.
Denition. By two-dimensional real triangle quasirepresentation of group G we mean the mapping ' of group G into the group of two-dimensional real triangle matrices T (2R) such that if '(x) = (0x) '((xx)) then: 1) arehomomorphisms of group G into R1 2) the set k'(xy) ; '(x)'(y)k1 xy 2 G is bounded. For brevity we shall call such mapping a quasirepresentationor a ()-quasirepresentation for given diagonal matrix elements and . We shall say that quasirepresentation is nontrivial if it is neither representation nor bounded. 1995, 1, : 4, 1129{1132. c 1995 , ! " \$ "" "
1130
. .
In this paper the criterion of existence of nontrivial ()-quasirepresentation on groups is established. It is shown that if G = A B is the free product of ;nite nontrivial groups A and B and A or B has more than two elements then for every homomorphism of group G into R there are (")-, (" )- and ()-quasirepresentation. Here the homomorphism " maps G into 1.
( f G R, ! ff(xy) ; f(x) ; f(y) j x y 2 Gg) ; !!$ ! ! % G &1{4]. + ! $ . % $% ! ! G ! ! !, ! KX(G), ! PX(G). . , ! ! ! ! ! G ! - G T(2 R) | ! % ! ,
(x) '(x) -(x) = 0 (x) : 1) | !!$ ! G R 2 2) ! k-(xy) ; -(x)-(y)k2 x y 2 G . , ' | G, -(x) = 10 '(x) 1 !. 3 ! !, ! !, ! !, ! . , &3] , . , ! ! ! 4 . 5 ! G ! !, ! ! ! 6 ! ! ! ( )- !. 3 B(G) $% G. _
1. KX(G) = PX(G)+B(G) . 3 " | !!$ ! G R . 9 1 , G ! (" ")- ! ! , ! .
2.
1131
- | ( )- , , - -
.
1. | G R - | ( )- G, = , = ", = ". 3 ' | G, | !$ !, ' ! $% ,
' (x) = '(x ). 2. | G R , ". ! : 1) G ( )- , G %
( ")- (" )- H = ker a ', ' = ;' 8a 2 G n H , H a 2 ' , = ; (a ) 6= 0 8a 2 G n H . 1. G ( F . ! G
, )' * +
, *' F 2)
G
, a
*' +
.
: ! !, G | 4 F | !!$ ! G R , G ! ! ( ")- (" )- . 5 , G = A F, A = f1 ag | , F | . 3 (a) = ;1 ker = F, ' 2 PX(F) !! 'a = ' ! 2 G ( ")- (" )- . 3 G | 4 , : G ! f1 ;1g | !!!$ ! ! H. + H F, ! ! ! H. ; 6 !! g G % F !$ ! g. ! g 6 ! Aut F Inn F g, a G ! G Aut F=Inn F. 2. , G ( ")- (" )- , G 6= H . 1. G = A B | A B . ! A =B = Z2 , G . * A B Z2 , G R G ( ")-, (" )- ( )- .
1132
1]
.
.
. .
.
//
. | 1987. | %. 21, ( 1. | ). 86{87.
2] . . SL(2 Z ) // .
. | 1992. | %. 26, ( 4. | ). 77{79.
3] . . 0 1 2 3 // 5 6. . | 1993. | %. 48, ( 1. | ). 205{206. 4] 9 . :. ; // .
. | 1991. | %. 25, ( 2. | ). 70{73.
& " ' 1995 .
pl- . .
512.545
: , ( ) , ! " , ! # .
$" , ! # l- %&' l- l- . ( )7] " , # | pl- ( )2] )4]). - "& & " & " pl- & ! . # .
Abstract E. E. Shirshova, Lexicographic extensions and pl-groups, Fundamentalnaya i prikladnaya matematika 1(1995), 1133{1138.
It is known that a lexicographic extension of an l-group by an l-group needs not to be an l-group. It has been proved in )7] that such extension is a pl-group (introduced in )2] and )4]). The purpose of this paper is to show that the class of pl-groups is closed with respect to lexicographic extensions.
G |
, M | G, G+ = fx 2 G j e 6 xg. () G M , Ma 6 Mb (aM 6 bM ), a 2 Ma, b 2 Mb (a 2 aM , b 2 bM ), , a 6 b . G=M M , ! G ! G=M " #
G
$ G=M . % G #$ " & ( & ) M ($ G=M , $! G=M # ) ( ! . *1], . 2, x 3). /
G,
, ( ) g, # # ) g. 0 a b # G+ #$ p- ( ! # : a ? b), # # ) . % G # pl- , ) g 2 G 0
0
0
0
0
0
p
1995, 1, 3 4, 1133{1138. c 1995 , !" \$ "
1134
. .
g = ab 1, a ? b ( ! p- pl- ! 3
*2] 3
4 *3]5 !( . *4] *7]). / x 1 $ & p- &
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G a b 2= M , Ma k Mb. . 1) =) 2). c 2 G, c 6 a b. 4 G | pl- , c = xy 1 , x ? y. : $! ; n > 0 c 6 x , 1.2 ) x 6 a b, . . c 6 a b. 2) =) 3). 8 , Ma 6 Mb. 4 ( m 2 M , a 6 mb. <#-#
M e 6 m. 4 m 1 a 6 a b. <# 2) , (m 1 a)2 6 a, a 6 m2 , . . a 2 M , ! . 3) =) 1). = ) ab 1. 4 G | pl- , ab 1 = xy 1 , x ? y. 8$ x 1a = y 1 b. , , M | # ) a b 2= M . 7 x 2 M , Ma = Mx 1a = My 1 b 6 Mb: (1) p
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" G. . am 2 AM bn 2 AM , a b 2 A, m n 2 M . 4 am(bn) 1 = ab 1 (bmb 1)(bnb 1 ) 1, ab 1 2 A, bmb 1 bnb 1 2 M , . . am(bn) 1 2 AM , AM | G. , , a1m1 6 x 6 a2m2 , a1 a2 2 A, m1 m2 2 M . 4 a1M 6 xM 6 a2M G=M . 7 a1M = xM a2M = xM , x 2 AM . < a1 M < xM < a2M , ) 2.1 $ a1 < x < a2 . 8$ A , x 2 A, . . x 2 AM , AM | . 8 #
AM . am bn 2 AM , a b 2 A, m n 2 M . /
A ) c 2 A, a b 6 c, , am 6 cm, bn 6 cn. <#
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AM=M . aM bM 2 AM=M , a b 2 A. <#
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G. 7 x 2 A, xM 2 S , x 2 B , . . A B . 7 A B , a 2 B n A, aM 2 S , ! S . @ , A = B , AM=M = S . ;
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+ " M # pl- " G=M g 2 G n M . + " a b 2 G+ , g = ab 1 c 6 a b c 6 a b ( c 2 G ( ! n > 0. . 7 e < g, g = ge 15 g < e, g = e(g 11 ) 1. g k e. 4 gM k M , G=M | pl- , gM = AB , A ? B A 6= M 6= B . 4 ) a 2 A b 2 B , g = ab 1 a b 2 G+ n M (G | & M ). 7 c 6 a b, cM 6 A B , 1.1 #$ , c M 6 A B $! ; n. 4 # , c M < A B . <# 2.1 , c < a b. 3.2. G | *
+ " M # pl- " G=M , a b 2 G+ n M c 6 a b c 6 a b c 2 G !" n > 0. aM ? bM . . aM = A, bM = B. 4 G=M | pl- , AB 1 = XY 1 , X ? Y G=M . @ 1.1 # ) X (Y ) $ # ) A (B ). T = X 1 A = Y 1 B , T 6 A B , ( ) t1 t2 2 T , t1 6 a, t2 6 b. <# 2.1 ( ) t 2 T , t 6 t1 t2, . . t 6 a b. $ t2 6 a b, T 2 6 A B . C , T 6 X Y A 6 X 2 , B 6 Y 2. 4 ! # , # ) A (B ) $ # ) X (Y ). @ , # ) A X (B Y ) $. 7 S # A G=M , Y 2 S . ;
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B 2= S , S N , N | # ) B , , ) Y # $, ! , . . B 2 S . A #, # B A. @ , A ? B . 3.1.
G * pl- " M # pl- " G=M . G | pl- . . g 2 G. 7 g > e, g = ge 1, g ? e5 g < e, g = e(g 1 ) 1 , e ? g 1 . = g k e. 1) 7 g 2 M , g = ab 1 , a ? b M , 1.2 #$ , a ? b G. 2) g 2= M . 4 ) a b 2 G+ , $( 3.1, g = ab 1 . 3.2, aM ? bM . 7 A # ) a G, 2.1 AM=M # aM G=M . 4 bM 2 AM=M ) u 2 A, bM = uM . / 2.1 ( ) d 2 bM , d 6 b u. 4 b = dm, m 2 M d 1b = m > e. 4 G | & M M < bM , 1.4 e m < d. <# A e < d 6 u e 6 m < d , d m 2 A, . . b 2 A. A #, $! # ) b G ) a. @ , a ? b G | pl- . . / * pl- " # l- " pl- +. p
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1] . . . | .: , 1984. 2] Conrad P. Representation of partially ordered abelian groups as groups of real valued functions // Acta Math. | 1966. | V. 116. | P. 199{221. 3] Conrad P., Teller J. R. Abelian pseudo lattice ordered groups // Publ. Math. Debrecen. | 1970. | V. 17. | P. 223{241. 4] () *. *. + , , - - // . ) /0). 1) ).: 23. . | ., 1973. | 2. 10{18. 5] 5, 6. 7,) 03),) ,),/. | .: ), 1965. 6] Teller J. R. On extensions of lattice-ordered groups // Pacif. J. Math. | 1964. | V. 14. | P. 709{718. 7] Shirshova E. E. On extensions of po-groups // Contemporary Mathematics. | 1992. | V. 131 (Part 1). | P. 345{353. & ' 1994 .
