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Abstract B. V. Novikov, Semigroup cohomologies: a survey, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 1{18.
A survey of research in semigroup cohomologies and their applications is given.
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1] . . . .: , 1987. 84:1A352] 2] !" #. . $ %"& // (). * + $. | 1964. | .. 33, / 2. | . 263{269. 65:1A207] 3] !" #. . $ %"& " &23 // (). * + $. | 1965. | .. 39, / 1. | . 3{10. 65:11A225] 4] !" #. . $ %"& " // (). * + $. | 1966. | .. 41, / 3. | . 513{520. 66:7A240] 5] !" #. . $ %"& 44523 5" 23 // (). * + $. | 1967. | .. 46, / 1. | . 11{18. 67:10A134] 6] !" #. . " 4 2" 5 2 6" 7 4 6" 7 "(2 3 ""& // .. .(. 4". -4 * + $. | 1975. | .. 48. 76:8A504] 7] 4 *., 97"(" . + 6" & "(. | .: :, 1960. 61:2A238] 8] 4 !" $. ;. < &3 5 // (). * + $. | 1985. | .. 117, / 3. | . 465{468. 86:1A492] 9] == *., >" 4 +. * "(6" & 4" & . .. 1, 2. | .: , 1972. 64:10A191, 68:10A123] 10] >. ;. > 2 & // ?6. !. :+> . +"@". | 1971. | .. 404. | . 275{284. 71:12A163] 11] >. ;. <( ! =!" 7 // 5""& "(. B2. V. | :., 1976. | . 71{74. 77:10A249] 12] >. ;. + 6" & 6" & 34" 4 ( ()" " 6" 3 // !5. 5! 5. 4". | 1982. | / 5. | . 30{34. 82:9A340] 13] " .". < %" 5 // (). * + $. | 1976. | .. 83, / 1. | . 25{28. 77:5A279] 14] " .". < &3 5 // (). * + $. | 1977. | .. 85, / 3. | . 545{548. 77:12A424] 15] " .". < " 4 23 =4 23 5 7 453 5 // (). * + $. | 1977. | .. 87, / 2. | . 281{284. 78:6A416] 16] 5 5 . B. < 0- &3 // ." . . 5 . ==. -7 "(. | "5: 5 , 1978. | . 185{188. 79:6A368] 17] 5 5 . B. < "4523 " 45"&3 // ; . * ? $, ". *. | 1979. | / 6. | . 474{478. 79:11A161] 18] 5 5 . B. 0- 5 " 0- 423 // B" 4 #C.
. -4. | 1981. | B2. 46, / 221. | . 80{85. 82:6A356] 19] 5 5 . B. < 526 " 7 " 4 23 // B" 4 #C. . -4. | 1981. | B2. 46, / 221. | . 96. 82:6A357]
:
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20] 5 5 . B. 4" 7 4"!" 46" // .. .(. 4. -4 * + $. | 1982. | .. 70. | . 52{55. 83:5A341] 21] 5 5 . B. <" "&D)" 4 %"& 0- 7 // ." & "" E. > 6. 2. > 2 " (! 57. | 4 5: ! -5 4 5. -4, 1983. | . 94{99. 83:11A213] 22] 5 5 . B. F 462" 3 E"& // !5. 5! 5. 4". | 1988. | / 11. | . 25{32. 89:6A317] 23] 5 5 . B. 4452" 2 )"" !" 4 1 // 4". !"4. | 1990. | .. 48, / 1. | . 148{149. 90:12A363] 24] 5 5 . B. $"="452" 4" // ; 5i i * ?GH. | 1994. | / 8. | . 10{12. 25] 5 5 . B. < " I // 4". !"4. | 1995. | .. 57, / 4. | . 633{636. MR 96f:12005] 26] 5 5 . B. < 3 6" 7 !" 4 1 // ; 5i i * ?GH. | 1996. | / 8. | . 6{8. 27] 5 5 . B. <( (" 7 4"!" 46" // ; 5i i * ?GH. | 1998. | / 3. | . 26{27. 28] <) :. *. " 4 2" &4& ""& 4" "(""523 7 // .. . 4. (-5. | 1967. | .. 17. | . 45{88. 68:12A270] 29] >6 & *. . < %"&3 "7 // (). * + $. | 1976. | .. 84, / 3. | . 545{548. 77:10A248] 30] >6 & *. . $ %"& "7 ) 5 3 6" & 34" 4 // (). * + $. | 1977. | .. 86, / 1. | . 21{24. 78:1A372] 31] >6 & *. . & 5 I==@"4 5 &3 // (). * + $. | 1977. | .. 86, / 3. | . 546{548. 78:3A277] 32] >6 & *. . C2" %7" 52 %"& "7 // .. .(. 4". -4 * + $. | 1979. | .. 62. | . 76{90. 80:1A430] 33] >6 & *. . < !5 23 =4 3 =4 5 !6"& 5 4" "7 // .. .(. 4". -4 * + $. | 1986. | .. 83. | . 60{75. 87:4A437] 34] >6 & *. . < 3 // .. .(. 4". -4 * + $. | 1988. | .. 91. | . 36{43. 89:7A308] 35] J" . K., KC "7=" L. +. < 52 4" 4" 7. | .: , 1974. 75:3A379] 36] K&"5 B. . <( 5" 23 3 2 G- // ; . * $. | 1970. | .. 14, / 9. | . 782{785. 71:4A142] 37] K&"5 B. . <( %" 5 " 5" 23 // B" 4 +?, cep. 1. | 1971. | / 1 (7). | . 15{21. 71:7A185] 38] Adams W. W., RiePel M. A. Adjoint functors and derived functors with an application to the cohomology of semigroups // J. Algebra. | 1967. | Vol. 7, no. 1. | P. 25{34. 68:4A266] 39] AguadTe J. Cohomology of binary systems // Arch. math. | 1981. | Vol. 36, no. 5. | P. 434{444. 82:1A486]
16
. .
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. .
515.123
: , .
! , # . # ! $ ! ! % # . , & ' ' ( ' !, $
) # !. * ) ' ! .
Abstract B. G. Averbuh, On transformations of families of pseudometrics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 19{32.
Arbitrary transformations of a family of pseudometrics are studied which result in uniform continuous pseudometrics again. Then their properties are applied for a descriptionof a quotient uniformstructure by means of pseudometrics on the initial space. It is proved that if a uniform structure on this space is subinvariant with respect to some given set of its transformations then the quotient uniform structure is subinvariant with respect to the induced transformations. The minimization for a given family of pseudometrics is considered in the last section.
1.
S = f g 2A | X . A Q R+, ! R+ | ! !S X X , !
!S (x1 x2) = !(f (x1 x2)g ), $ x1 x2 2 X . % & ' ( &( , !, &' !S &' X , ' , ! S . ) !, ! !, ' ! & . * $ &' & +1] , $ S . , 2001, 7, 1 1, . 19{32. c 2001 , !" #
20
. .
0 ! ( & , ' '( ! : | !& $ X , ' , & !, $ ! X X !S > . , X 2 R. % & ' $ , ! ( S Y = X=R, ! ! $ UNIF ! . 3 & '. ! ' ! . F | & X X , (! ( 2 R, X &' , & '( F . 4 $ Y & 2! & Y Y , F . 5 ( & , ! $ ! . % ' ! : !& S S 0 , ! ! , S 0 $ & '( $
. 1. % 6 ! (
& . A | . 2 A R+ R+, $ ' ' !
, jx1 ; x2j. 7& RA+ Q R+ , &6 (( ) 2A . 0 r80 = fr0 g r800 = fr00 g RA+ & r80 6 r800 , r0 6 r00 ( 9 ' r81 r82 r83 2 RA+ & ' &! $ , ' ( ( ' ( $( ( ). 0 $ &( , &' $ R+ & ' ( ! ( . 1. % ! ! !, ! RA +, & ' & A, ! : : 1) !(8r ) > 0 !& $ r8 2 RA+, !(80) = 0, $ 80 | RA+ ' 9 : 2) ! ' 809 : 3) ' r81 r82 r83 &! $ RA+, !(8ri ) 6 6 !(8rj )+ !(8rk ), $ i j k 2 f19 29 3g, . . ! ( (
21
&! $ R+. : & 6 ! , !&'( r81 r82 2 RA+ r81 6 r82 !(8r1 ) 6 !(8r2 ). 0 ! ( ( & : 3) & !, &' !(8r1 + r82) 6 !(8r1 ) + !(8r2) &' !&'( r81 r82 2 RA+. 4 ' ! & ' . . 1. A $ . 4 $ & ! ( : ( !c(x) = x 0 6 x 6 c $ c 2 R+9 c x > c !(x) = xp $ x 2 R+ 0 < p 6 19 8 x 0 6 x 6 2 > > <4 ; x 2 < x 6 3 !(x) = > >x ; 2 3 < x 6 4 : 2 x > 4: ' , & & & ' . % ' R+ & , : 1) : 2). 2. A f g 2A | '( '( . 4 $ P!& $ c > 0 & !(r) = !c (r ), $ r8 = fr g 2 RA+. %' ( & . : 4) A RA+. . 0 $ " > 0 & U ' RA+, $ (80 8h) 2 U !(8h) < "2 , $ h8 2 RA+. : , U 6 ' jr0 ; r00 j < , $
> 0. (8r0 r800) 2 U . 7& r8 r = min(r0 r00), 2 A. 4 $ r8 6 r80 r800 ! ' 8h0 8h00 2 RA+, r8 + 8h0 = r80 , r8 + h8 00 = r800, 6 (8r 8h0 r80) (8r 8h00 r800) & $ . 7 , (80 8h0) (80 8h00) 2 U , !(8h0 ) !(8h00) < "2 . > : 3) !(8r ) ; !(8h0 ) 6 6 !(8r0 ) 6 !(8r) + !(8h0 ), !(8r ) ; !(8h00) 6 !(8r00 ) 6 !(8r) + !(8h00 ), j!(8r0) ; !(8r00 )j < ". : 5) A0 | A. 0 : RA+ ! R+A i0 : RA+ ! RA+
. ! ! | 0
0
22
. .
R
A, !0 = !ji ( A+ ) | A0 . ! !0 | A0 , !0 0 | A. @ $ , & ! A 2 A, A0 = A n fg ! = !0 0. > : 4) ,
& & 6 $ A. ! . # ! A $ 2 A, !ji ( +) %, i | R+ R+. . A ! , !ji( +) 0, & i (R+ ) 0 : RA+ ! RA+ , $ A0 = A nfg, 80. 0 & . r8 2 RA+ | '
, r80 = (i0 0 )(8r) r80 = (i )(8r), $ | RA+ R+ . 4 $ r8 = r80 + r80 r80 r80 6 r8, (8r r80 r80) & $ . B , !(8r0 ) ; !(8r0 ) 6 !(8r) 6 !(8r0 ) + !(8r0 ) !(8r) = !(8r0 ). : 6) & . ! $ %, . . ' ! ( '( fA g2B , f! g2B ( & ,
6 !& 2 B ! A , S & ! B . A A = A , '
' '( Q RA+ = RA+ . & ! : RA+ ! R+ Q & RB+, !^ = ! ! , &! RA+ R+. , !^ & A. %' : 1) : 2) . 5 (8r1 r82 r83) RA+, &! ! $ . 6 ' RA+ , 2 B , (8r1 r82 r83 ), &! $ . '( (! (8r1 ) ! (8r2 ) ! (8r3 )) (f! (8r1 )g f! (8r2 )g f! (8r3 )g ) RB+ & . & ! $ : 3), !^ . r81 r82 2 RA+ r81 6 r82. A r8i = fr8i g , $ i = 1 2, r8i 2 RA+ , r81 6 r82 , , ! (8r1 ) 6 ! (8r2 ) ( 2 B . 0
0
R R 0
23
B , f! (8r1 )g 6 f! (8r2 )g RB+, !^ (8r1) = !(f! (8r1 )g ) 6 6 !(f! (8r2 )g ) = !^ (8r2 ). : 7) r8 | $ RA+, r80 | $ , r80 6 28r. % !
A !(8r0 ) 6 2!(8r). . 4 (8r0 r8 r8) & $ . 76 : 3). : 8) 5 ' ( & R+. ! | & , " > 0 h > 0 | , !(h) < ". 4 $ !(x) < 2" x 2 +0 2h]. 7! : h > 0, !(h) = 0, !(x) 0 R+. A, , !(x) !, "0 > 0, ( & 2( x !(x) > "0 . 0 < " < "0 fx : !(x) < "g $ . ' ! & 2 x, $ !(x) = "9 & $ ("). ) (") ! ", !. : 9) ! ! A ' r8 2 RA+, %' , %. . ( ' '2 & , r8 = = (i )(8r ). r8 6= 0 r8 6 r8, : 5) : 7) ! , !ji( +) !, . . ! $ A. J : 1). X | S = f g | '( X , ' A. $ $ & A ( ( , . . S - . K ! RA+ ! !S X X , ! !S (x1 x2) = !(f (x1 x2)g ), $ x1 x2 2 X . 5 ' $ . 1. ( !S X , )
! : 1), : 2), : 3). . L &( '( ', ' !& $ ( X S , '( !S . . , ! | & , !S | X . 7 ' & ! S . % ,
& 9 ,
R
24
. .
' $ : 3). , !S . U | RA+, !(8r ) < " r8 = fr g 2 U , $ " | 6 . : , U 6 r < , $ &$ A0 A, > 0 !& $ 2 A0 . 7& V ' X , (x1 x2) 2 V ( 2 A0 ! (x1 x2) < . 4 $, , (x1 x2) 2 V !S (x1 x2) < ". ( &( , : 3). (8r1 r82 r83) | RA+, &! $ . 2 A R2 $ P1 P2 P3 P1 P2 = r1 , P1 P3 = Qr2 , P2 P3 = r3 . X = R2 P1 = fP1 g, P2 = fP2 g, P3 = fP3 g | X . &' '( ( X R2 , 2 A, & . 4 $ (P1 P2) = = r1 , (P1 P3) = r2 , (P2 P3) = r3 ( 2 A,
S = f g, : !S (P1 P2) = !(8r1 ), !S (P1 P3) = !(8r2), !S (P2 P3) = !(8r3), &( : 3). L &( '( & $ . X = RA+. % R+ !, (0 r ) = r . A, '2, | & X S = f g, f (80 r8)g = r8 !S (80 r8) = !(8r ), ' & . 5 ' X , '( ' & . 5 1) B fA g2B f! g2B ! | , ) : 6), S = f g 2A , 2 B , | S X , A , S = S .
; Q ! ! S ! , M = f(! )S g2B , %. .
S = f g 2A 2B &O $ & A ( X 9 S , A = A . M , ' B . . > :
Y
Y
! ! (x1 x2) = ! ! (f (x1 x2)g
S
2A 2B ) =
= !(f! (f (x1 x2)g 2A )g2B ) = !(f(! )S (x1 x2)g2B ) = ! (x1 x2) !&'( x1 x2 2 X .
25
5 2) * % % ! % S , ' , !S . . x1 x2 2 X x1 6= x2. 4 $ $
!S (x1 x2) = !(f (x1 x2)g) RA+
' . !S (x1 x2) 6= 0 : 9). 5 3) * % % ! , X !S , , S , $ A, !. . A0 | A, & , '( !, S 0 | S ,
A0 . 7& , '2, i0 RA+ RA+ !0 | $ !ji ( A+ ) . 4 $ $ , !S0 !S X !. > ' 1 , , !S , , 2, , S 0 . 0 & 2. 5 $ U ', S 0 . : , 6 < , $ &$ A00 A0 > 0. L 6 " > 0, $ ! f(x1 x2) 2 X X : !S (x1 x2) < "g U . 7& i R+ RA+ , $ 2 A00 . : & !0 ji( +) : 5) , : 8) " > 0, $ !0 ji( +) (x) < " " , (x1 x2) | , x < . " = 21 min 2A 0 0 X X , !S (x1 x2) = ! (f (x1 x2)g 2A ) < ". 7& r8 f (x1 x2)g 2A RA+ , r8 | $ ! ! i (R+ ), $ 2 A00. J $ $ ' ' !, ' , (x1 x2). 7 , r8 6 r8, !0 (8r ) 6 2!0(8r) < 2" 6 " (x1 x2) < A00. 5 4) &( ' ! . P &' $ , & S , ( ( ( S . 5 4) S1 S2 | X , S1 = S2 . % , %' ( , %' ) $ , . . 5 ' . 0 S = f g 2A $ !O $ & f : A ! S . % 6 A 2 , 1 2 , f (1 ) = f (2 ), & A . 0
0
0
R
0
0
R
R
00
0
0
0
0
26
. .
A & & A ! S S ,
A . 0 , S S . $ , '& ! , A A, S | S . 4 $ & ! A & & ! A, . 5 !S = !S ' , ' $ & S &' S . P &' & ! , & Q $ a 2 A Q Qa $ & R+a R+ . 4 $ Qa 2a a2A ' ' (! Q Q RA+ RA+, '
R+ . 7& a2A 2a $ & D. 7$ D !& & ! A,
& ! A . 0 !&'( x1 x2 2 X ! f (x1 x2)g 2A 2 D, ', !S = !S . 3 & $ . 7& , S1 S2 ( & ! ' & . 5 5) | X , , S . ' %' !, , !S , % . . %'& S = f g 2A & 6
S 0 , &' $ ! S 0 . % 6 $ ( ! ! '( f"n g $ "n 6 '
& An A & T '( Qn = f g 2An , '( ! f(x1 x2) 2 X X : (x1 x2) < g
2An
f(x1 x2) 2 X X : (x1 x2) < "n g. 4 $ S 0 S 0
, A = An . n ! | ! & A, $ A0 . 5 3) ' . 4 & , , ! & 2, > 0, 2 A0 . 5 6) ' , , , ' %'
27
!, % X X
!S > .
. ' 5 1) !, $ S , & . @ , , 6 1. 5 ! R+, !
!^ (s) = sup (x1 x2)
(x1 x2 )6s
, &' !
& , . 7 , $ , !^ & ! : 1) 0 6 !^ (s) 6 1 !& $ s 2 R+9 2) !^ (0) = 09 3) !^ (s) 9 4) !^ (s) ' . % ! $ , , , , ! . 4 , ' ( !, $ $ ( ' ! & f(s t) 2 R2 : s 2 R+ 0 6 t 6 !^ (s)g ( ( ( $ '( ' '( ). % , ! ' ! $ , !& $ " > 0 k > 0, $ ' f(s t) 2 R2 : s 2 R+ 0 6 t 6 " + ksg, ! | !^ , & ! | 6 ' . 3 2 . % , !! $ $ ,
! : X Y | , 6 Y $ , f : X ! Y | ' &. 4 $ &
, X , $ & f ' . 7 , 5 6) &( . 6 6 $ $ . 7 $ RA+ ! ^(8r) = sup (x1 x2): f (x1 x2 )g6r
4 $ ! . 1. + !S , ) ^ % . 2. X | R | 2 . % '
28
. .
Y = X=R 6 ! ! (, '( ' & p : X ! Y . 7 & p ' $ ( & '() '( ( ' '( & . Y $ $ , $ 6 & X . :' !, ( $ , ! $ ( ! ! S , , ! ! Y . L 6 ' '( , $ $ 6 . A ,
' !& , . 6 ' (
& , & !& $ $ & , $ . * $ ! ( ( & . 5 4) , $ 6
. 5 ! !, - & !, !. | X , y1 y2 | ' Y . R k , ! ' X , & ' !&! 2k- ! Z = fx000 x01 x001 : : : x0k;1 x00k;1 x0kg X , ! ! ! : p(x000 ) = y1 , p(x0k ) = y2 , p(x0i) = p(x00i ) i = 1 : : : k ;1. 0 Z 6 Pk (Z ) = (x00i;1 x0i). 1 7 ^ Y , ^ (y1 y2 ) = Z(yinfy ) (Z ), 1 2 $ Z (y1 y2) & ( , ! ( y1 y2 . Y , ' & X , & ' ( (. 7 : 1) & & p Y , 9 2) ( , '( X ,
$ , $ ' 2 ' ( 9 3) X & . 5 $ S X & ' S^ Y , , ( ( S .
29
A Y ! , 6! ^ X , & p & ' ' . , Y 2, ( $ , 6'( S , . 3 !! ! Y & ' & , ! S . , S = f g 2A $ X ( , ! ( ( & ), ' ' Y !. , , | $ ( Y ) , | 6 & X . 4 $ S , ! . ( , S^, ! ! . 4 & , ! . 2. X | , R | , $ X S | X , %' . ) Y = X=R , %' S . K ' , $ X & , $ '( R. 7& F = ff g2B & X &. X 6 & $ , !&'( x1 x2 2 X , 2 B (f (x1) f (x2)) 6 (x1 x2). 5 ! X 6 & F , &' , &! ( . , $ ! $ $ & ! S = f g & '( F
& . % , : !S (f (x1) f (x2 )) = !(f (f (x1) f (x2 ))g ) 6 !(f (x1 x2)g) = = !S (x1 x2). 4 & , & &' ' ' ! ( ( & & '( . , F $ 2 R, . . (f f )(R) R !& $ 2 B . 4 $ - Y & F^ . , ' 2 ' ! . 2. ! X F , ) Y F^ .
30
. .
. y1 y2 | ' ' Y , Z | , ! ( X . A f^ | & F^ , f | ! $ & F , &' f , &! ( Z , ! Z , ! ! f^ (y1 ) f^ (y2 ). S |
! ( ( &
& '( F X . 4 $ &' &' 2 S , Z & 2 ' Z . B , ^ ^ (f^ (y1 ) f^ (y2 )) 6 ^ (y1 y2 ), . . & F^ . 76 , 2 ! Y . 3. 5 ( & $ $ . 6 ' (
& , ( $
& '( $ . 3. . . S , A. % S ! ! ! S = f0 1 : : :g, & . ! + 1 T = fT0 T1 : : :g S , ! ' 6 T ! : T1) 2 T > 9 T2) T | 9 T3) T \f : < g $ & '( T . 7 , T ' . , $ , T0 = S , T ' ( < 6 . A 2 , , = + 1 $ & T n f g, T = T . A $ & , T = T n f g. %' T1), T2), T3) . T | ' . 7& $ P = T . < P & , $ ' T2) T3): P1) 2 P > 9 P2) P \ f : < g $ & '( P .
31
A P , T = P . A P
' , & Q S , , ! ( X ! , ! , ! P . Q < . , Q R , $ T = P R . $ , Q S . 5 ' Q ! !: Q = f 0 1 : : :g, $ < 9 & . ! ! ! +1 fR0 R1 : : :g Q , ! ! ! : R 1) !& $ 6 ! R f : < g, 6
1 < 2 6 R2 \ f : < 1 g = R1 9 R 2) , X !& , , P R , < 9 2 R , ( , P f : < g9 R 3) (P R ) \f : < g $ & '( P R
!& 6 . % R0 6 . , , R ' < 6 , 2 , , = + 1. 4 $ R = R , , X , , P R 9 R 1), R 2), R 3) ' '. A $ , $ R = R f g. 4 $ $
' R 1) R 2). 7& ! R3), , &' $
& P R , $ $ & , $ . % , , $ , X , &' , . 7' ' (P R ) \ f : < g $ &' ( & ! . 5 , S , , $ | ' . 7& $ R R . %' R1) R2) < $ , R 3). - (P R ) \ f : < g $ & $( P R . 4 $ P , , - '(, $ $ & $ &' &' Q , - '(, &' $ $ '
32
. .
P , &' P2). , , 2 R . 4 $ $ ' $ $ & $ P ( ! Q ). K $ , $ &' 0 Q , ! ( , &' ! R2), ' , ! , $ R , < , $ 0. $ &' $
& , $ ' $ P & 2 ! Q . 4 ' fR : 6 g . R = R , T = P R 9 T1), T2), T3) T ! P1), R2), R3) P R , ' .
1] ., . , // Math. Slovaca. | 1981. | Vol. 31, no. 1. | P. 3{12.
$ % & 1997 .
. . , . .
. . .
517.987.1+517.518.1+517.982.3
: , , {.
" #$% , &' ( 1909{1914 ' , : , ! . . 1952{1953 % / 0 0% 0 & / , & # , 1 . 20% % % 3 ( 0 / 1956 . 5. .. "% 0 . . 1996{1997 % 0 '2 3 3 60% /, 0 $ , 3' , 7 . " 8 00 , " {" ? . , 7 6 0, 8 .
Abstract V. K. Zakharov, A. V. Mikhalev, Connections between the integral Radonean representations for locally compact and Hausdor% spaces, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 33{46.
After the fundamental papers of Riesz, Radon and Hausdor; in 1909{1914 the problem of general Radonean representation became actual: &nd for Hausdor% topological spaces a class of linear functionals isomorphically integrally representable by all Radon measures. In 1952{1953 the bijective solution of the problem of Radonean representation for locally compact spaces was obtained by Halmos, Hewitt, Edwards, etc. For bounded Radon measures on a Tychono; space the problem of isomorphic Radonean representationwas solved in 1956 by Yu. V. Prokhorov. In 1996{1997 the authors obtained one of possible solutions of the problem of general Radonean representation using the family of metasemicontinuous functions with compact supports and the class of thin functionals on it. ' ! , 2001, 7, < 1, . 33{46. c 2001 ( )*, + ! ,- .
34
. . , . .
After this the question if the theorem about general Radonean representation covers the Riesz{Radon theorem was still left open. In this paper the positive answer to this question is given.
1], 2] 3] 1909{1914 ! "# . % " & '. ( ' )!' (T G ) ' - '# RMw0(T G )0 ."/ 0! !- ! ' 1 B(T G ) 0 1 R ]. 2 . - ' . i d !' MI(T B(T G ) ) B(T G )-1''/ - '/ !3# f 1 T R. 4 ' ! !3 / i '. ' " ' !' T A(T ), . ' !' UI(T ) (MI(T B(T G ) ) j 2 RMw0(T G )0 ) B(T G )- " ! 1 / ! ', !' ) 7! i jA(T) 1 '. RMw0 (T G )0 ! A(T ) #/ !3 A(T ) - -- 5! /'. 4 . - 1 1! (/) ' ' - ): ! - ! A(T ) !' A(T ) #/ !3 &' /" !/ (. . "1 '/) ' ! ! 0&)
A(T)4 , ! ) ." - ) " (A(T)4 )+ ) '# ' A(T)rf0 fi jA(T ) j 2 RMw0(T G )0 g
."/ ! !3 A(T). 0 '/ ! - - ) - ! " !' ! / '0' 7], " ' 8] 9 ' 9] 1952{1953 1 ! ! { ('., ', 11] 12]). . (T G ) | Cc (T G ) | (T G ) . : 1) ! ! 7! i jCc(T G ) " RMw0 (T G )0 Cc (T G ) $ % 2) Cc (T G )rf0 ! ! & (Cc (T G ) )+
35
- ! Cc (T G ). ; 1 1'./ 0# '/ ! - ) - 1 " / ' # " 1997 . ('. 14,15,39,40]). ( ). (T G ) | , S(T G ) |
( ) (T G ) Sc (T G ) | &( .
1) ! ! 7! i jSc (T G ) " RMw0 (T G )0 Sc (T G ) $ % 2) Sc (T G )rf0 ! ! & (Sc (T G )4 )+ - ! Sc (T G ). / ! 1 / # 14], 15], 39] 40] - !//' , { ? 4 # " &- ."/#
< : ' { 1 '/ ' !' ('. ' ). = - "- - -- /' .' " 39]. <' 1 ' '/' -' /! ' '/ / ' ! <# " ('. !. 40]). > , '/ . ' ' 3
! " 39] !. ' 3 ! 3 # /. ? / / . " ?. 4. @! 1 ! ., ' 1 1 " ! - " - .
6.
4 <' ! '/ ! .', ) ' { - ! " !' ! 1 '/ ' !' - . " T | ! " !' ! ' | #/# ."/# !3 Cc (T), . . ' 2 (Cc (T) )+ . ! .', ) !3 ' . - /' 1' # ." ! !3 Sc (T ), . . ' 2 (Sc (T )4 )+ . " A(T) | 1 " '. F(T ). ( '', ) !3 1 A(T ) R 1/ - !! , f = p-lim(fm j m 2 M) )& f = lim(fm j m 2 M) - ! .# (fm 2 A(T ) j m 2 M) ! .# !3 f 2 A(T). D A(T ) | 0&)
36
. . , . .
! , < # " ! .' 1 # : 1) (fm 2 A(T)+ j m 2 M) # 0 F(T) )& (fm j m 2 M) # 0 RG 2) - f 2 A(T) (fm 2 A(T )+ j m 2 M) " f F(T ) ! (fm j m 2 M) " f R. ? ) - "" - !! - "-' (fm 2 A(T) j m 2 M !). % - '' - -- ! ) # - "#0 .#. <' '/ ' & ! 1 " ', - - -- 0 1 # ('. 12, 73D]). 1. ) ' Cc (T) . . " (fm 2 Cc (T)+ j m 2 M) # 0 F(T). J ! ' m0 2 M '' !3 g fm0 !' ! '. C clft 2 T j g(t) > 0g. =- C !3- h 2 Cc (T )+ , ! - ) h > (C). 41"'&' " > 0. K #&- a > 0, ! ) a'(h) < ". '' !// '. Gm ft 2 T j fm (t) < ag. K ! ! ! (fm j m 2 M) # 0, (Gm j m 2 M) " T . <' !) !/ (Gn j n 2 N M) '. C. L1 M , ) m 2 M, ! ) n 6 m - n 2 N. <' Gm Gn - n. J ), C Gm . 41"'&' l 2 M, ! ) l > m0 l > m. K fl 6 g fl 6 fm . L1 , ) H coz fl C Gm , 1 , ) fl (t) 6 fm (t) < a = a (C)(t) 6 ah(t) # )! t 2 H. J ), fl 6 ah )& 'fl 6 a'(h) < ". D k < l, 'fk 6 'fl < ". K !' 1', ('fm j m 2 M) # 0. M L "1 - # ))# / !3 ' ' " . !/' &/' ' ' = -, /' 21]. '' '. Ybc )/ !3# !' !/' -' f 2 Fbc (T), ! ) f = sup(fm 2 Cc (T) j m 2 M) F(T) - !# (fm 2 Cc (T) j m 2 M) ". ? )/' 1' '' '. Zbc !3# f 2 Fbc (T ), ! ) f = inf(fm 2 Cc (T) j m 2 M) F(T ) - !# (fm 2 Cc (T ) j m 2 M) #. @, ) Cc (T) = Ybc \ Zbc . = ' ' " Y Z. 2. * ! Y Z F(T ) & &( P : P 1) (ai gi j i 2 I) 2 Y (ai hi j i 2 I) 2 Z & (ai 2 R+ j i 2 I), (gi 2 Y j i 2 I) (hi 2 Z j i 2 I)% 2) inf(gi j i 2 I) 2 Y sup(hi j i 2 I) 2 Z & (gi 2 Y j i 2 I) (hi 2 Z j i 2 I)% 3) sup(gi j i 2 I) 2 Y inf(hi j i 2 I) 2 Z & (gi 2 Y j i 2 I) (hi 2 Z j i 2 I) & u v 2 Fbc (T), gi 6 u hi > v i%
37
4) g ^ 1 2 Y, g _ (;1) 2 Y, h ^ 1 2 Z h _ (;1) 2 Z & + g 2 Y h 2 Z% 5) ;Y Z ;Z Y, . . ;g 2 Z ;h 2 Y & g 2 Y h 2 Z% 6) Y = Cc (T ) + Y+ , . . & g 2 Y (& f 2 Cc (T) g0 2 Y+ fg 2 Y j g > 0g, g = f + g0 % 7) Z = Cc (T ) ; Y+ , . . & h 2 Z (& f 2 Cc(T ) h0 2 Y+ , h = f ; h0 . . % # 1){5) - - - /' -' 1 # / ). ' # 6). " g 2 Y, . . g = sup(fm j m 2 M) - !# (fm 2 Cc (T) j m 2 M) ". 41"'&' f fm0 '' gm (fm _ f) ; f 2 2 Cc (T)+ g0 g ; f. K ! ! ! (gm j m 2 M) " g0 , g0 2 Y+ . J ), g = f + g0 . ; . 1 1). % # 7) - ). M ; ' Y /# !3 ', - 'g supf'f j f 2 Cc (T ) ^ f 6 gg - ! . g 2 Y. 9 !!. =# ", ! ! ! !3- g ) ' !' !/# ", !' ! '. C ) a > 0, ! ) g 6 a (C). <' g 6 a (C) 6 f0 - !# !3 f0 2 Cc (T ). 4 1 " 'g 6 'f0 < 1. ? )/' 1' ' Z /# !3 ', - 'h inf f'f j f 2 Cc(T ) ^ f > hg - ! . h 2 Z. @, ) ' ' - - - .-' '. % '" ''/ 2 ! 1/ 3. , g 2 Y (fm j m 2 M) " g (fm 2 Cc(T )+ j m 2 M), ('fm j m 2 M) " 'g. . K ! ! ! fm 6 g, 'fm 6 'g b - m. '' ) a sup('fm j m 2 M) 6 b. 41"'&' 1 " !3 f 2 Cc (T )+ , ! ) f 6 g. K ((f ; fm )+ j m 2 M) # (f ; g)+ = 0 F(T). 4 ''/ 1 '/ ) ' inf('((f ; fm )+ ) j m 2 M) = 0. K ! ! ! ' 1 , '((f ; fm )+ ) > '(f ; fm ). <' 'f = 'f ; inf('((f ; fm )+ ) j m 2 M) = sup('f ; '((f ; fm )+ ) j m 2 M) 6 sup('f ; '(f ; fm ) j m 2 M) = = sup('fm j m 2 M) = a. ; , ) b 6 a. M 4. ) ' ! Y+ &( : 1) ' " % 2) '(ag) = a'g & a 2 R+ g 2 Y+ % 3) '(g0 + g00) = 'g0 + 'g00 & g0 g00 2 Y+ . . % # 1) 2) ) /. 3) '' / / '. M 0 ff 2 Cc(T )+ j f 6 g0 g M 00 ff 2 Cc (T)+ j f 6 g00g. 'g0 = sup('f 0 j
38
. . , . .
f 0 2 M 0) 'g00 = sup('f 00 j f 00 2 M 00). '' '. M M 0 M 00. K ! ! ! g0 = sup(f 0 j f 0 2 M 0 ) g00 = sup(f 00 j f 00 2 M 00), g0 + g00 = sup(f 0 + f 00 j (f 0 f 00) 2 M). % ", (f 0 + f 00 j (f 0 f 00 ) 2 M) " (g0 + g00 ). ? )/' 1' ('f 0 + 'f 00 j (f 0 f 00 ) 2 M) " ('g0 + 'g00 ). '' 3 ('(f 0 + f 00 ) j (f 0 f 00 ) 2 M) " '(g0 + g00). L1 ' ' '(g0 + g00 ) = 'g0 + 'g00 . M 5. ) ' ! Y ' ! Z & &( : 1) ' ' " &% 2) '(f + g) = 'f + 'g '(f ; g) = 'f ; 'g & f 2 Cc (T) g 2 Y+P % P P '( (aigi j i 2 I)) = (ai 'gi j i 2 I) '( (ai hi j i 2 I)) = 3) P = (ai 'hi j i 2 I) & (ai 2 R+ j i 2 I), (gi 2 Y j i 2 I) (hi 2 Z j i 2 I)% 4) ('(gm ) j m 2 M) " 'g & (gm 2 Y j m 2 M) & g 2 Y, (gm j m 2 M) " g F(T )% 5) '(;g) = ;'(g) '(;h) = ;'(h) & g 2 Y h 2 Z% 6) 'v 6 'u 'u 6 'w & u 2 Y v w 2 Z, v 6 u 6 w. . % # 1) 1 #. 2) '(f + g) = supf'u j u 2 Cc (T ) ^ u 6 f + gg 'g = = supf'v j v 2 Cc (T )+ ^ v 6 gg. K ! ! ! u 6 f + g, u ; f 6 g )& 'u ; 'f 6 'g, . . 'u 6 'f + 'g. % ", '(f + g) 6 'f + 'g. K ! ! ! v 6 g, f + v 6 f + g )& 'f + 'v 6 '(f + g), . . 'v 6 '(f + g) ; 'f. % ", 'g 6 '(f + g) ; 'f, . . 'g + 'f 6 '(f +g). 4
-- ). % # 3) " 1 # 2), # 6) 7) ''/ 2 # 2) 3) ''/ 4. 4) '' / S '. Ll fu 2 Cc(T ) j u 6 gl g. '' '. P (fmg Lm j m 2 M). O -)' , (l u) 6 (m v), l 6 m u 6 v. D p (l u) q (m v) | 1 "/ <'/ 1 P , n 2 M, ! ) l 6 n m 6 n. K ! ! ! u 6 gl 6 gn v 6 gm 6 gn , w u _ v 2 Ln . <' p 6 (n w) q 6 (n w). % ", '. P - -- /' . '' " (fp 2 Cc (T)+ j p 2 P), ! ) fp u - ! . p (l u), u 2 Ll . " p = (l u), q = (m v) p 6 q. K u 6 v )& fp u 6 v fq . J ), (fp j p 2 P ) ". " t 2 T " > 0. K g(t) ; "=2 < gl (t) - ! l 2 M. = , gl (t) ; "=2 < u(t) = fp (t) - ! u 2 Ll p (l u). % ", g(t) ; " < fp (t). K ! ! ! " 1 ", g(t) = sup(fp (t) j t 2 P ). J ), (fp j p 2 P ) " g. '' 3 ('fp j p 2 P) " 'g. " " > 0. K 'g ; " < 'fp - ! p (l u). K ! ! ! fp = u 2 2 Ll , 'fp = 'u 6 'gl )& 'g ; " < 'gl . J ), 'g = sup('gm j m 2 M).
39
% # 5) ) . 6) # ' 6) 7) ''/ 2 u = f 0 + g0 v = f 00 ; g00 - !/ 0 f f 00 2 Cc (T) g0 g00 2 Y+ . K ! ! ! f 00 ; f 0 6 g0 + g00 2 Y, # 3) 'f 00 ; 'f 0 6 '(g0 + g00 ) = 'g0 + 'g00. K " # 2) 'v = 'f 00 ; 'g00 6 6 'f 0 + 'g0 = 'u. = , 'u = supf'f j f 2 Cc (T ) ^ f 6 ug 'w = inf f'f j f 2 Cc (T) ^ f > wg )& f 6 u 6 w 6 f. % ", 'f 6 'f. ; 'u 6 'f 'u 6 6 'w. M K " ' Fbc (T ) / !3 / 'e ': , 'ef inf f'g j g 2 Y ^ g > f g ': f supf'h j h 2 Z ^ h 6 f g - ! .# !3 f 2 Fbc (T ). @, ) 'e - -- .' ', ': - - .' '. 1. ) 'e ': Fbc (T) & &( : 1) 'e ': " &% 2) ': f 6 'ef & f 2 Fbc (T)% 3) 'e(;f) = ;': f ': (;f) = ;'ef & f 2 Fbc (T)% 4) 'e(af) = a'ef ': (af) = a': f & a 2 R+ f 2 Fbc (T)% 5) 'e(f 0 +f 00 ) 6 'ef 0 + 'ef 00 ': (f 0 +f 00 ) > ': f 0 +': f 00 & f 0 f 00 2 Fbc (T)% 6) ('e(fn ) j n 2 N) " 'ef & (fn 2 Fbc (T ) j n 2 N) & f 2 Fbc (T), (fn j n 2 N) " f F(T ). . % # 1) ) . 2) ': f supf'h j h 2 Z ^ h 6 f g 'ef inf f'g j g 2 Y ^ ^ g > f g. K ! ! ! h 6 f 6 g, # 6) ''/ 5 'h 6 'g. ; ': f 6 'g ': f 6 'ef. 3) K ! ! ! 'e(;f) inf f'g j g 2 Y ^ g > ;f g, f > ;g 2 Z )& ': f > '(;g). # 5) ''/ 5 '(;g) = ;'g. <' ;': f 6 'g )& ;': f 6 'e(;f). % # /, ! ! ! ': f = supf'h j h 2 Z ^ h 6 f g, ;f 6 ;h 2 Y )& 'e(;f) 6 '(;h) = ;'h. <' ;'e(;f) > 'h )& ;'e(;f) > ': f. % # 4) 1 # 3) ''/ 5. 5) 'e(f 0 + f 00 ) inf f'g j g 2 Y ^ g > (f 0 + f 00 )g, 0 'ef inf f'g0 j g0 2 Y ^ g0 > f 0 g 'ef 00 inf f'g00 j g00 2 Y ^ g00 > f 00 g. K ! ! ! g0 +g00 > f 0 +f 00 g0 +g00 2 Y, # 3) ''/ 5 'e(f 0 +f 00 ) 6 '(g0 +g00) = = 'g0 + 'g00. ; 'e(f 0 + f 00 ) ; 'g0 6 'g00 )& 'e(f 0 + f 00 ) ; 'g0 6 'ef 00 , 'e(f 0 + f 00 ) ; 'ef 00 6 'g0 )& 'e(f 0 + f 00 ) ; 'ef 00 6 'ef 0 . 6) K ! ! ! ('efn j n 2 N) " 'efn 6 'ef b - n, a sup('efn j n 2 N) 6 b. P! ' " > 0. K - ! . n gn0 2 Y, ! ) gn0 > fn 'efn + "=2n > 'gn0 > 'efn . K ! ! ! f 2 Fbc (T), f 6 u - ! u 2 Cc (T). '' gn00 gn0 ^ u 2 Y. # 3)
40
. . , . .
''/ 2 g 2 Y, ! ) g = sup(gn00 j n 2 N). K ! ! ! gn00 6 gn0 ,
# 1) ''/ 5 'gn00 6 'gn0 < 'efn +"=2n . 2' , gn00 > fn ^ f = fn . '' gn sup(gi00 j i = 1 : : : n) 2 Y. K (gn j n 2 N) " g. # 4) ''/ 5 ('gn j n 2 N) " 'g. ' !3, ) 'gn 6 'efn + " ; "=2n . =- n = 1 'g1 = 'g100 6 'ef1 + "=2 = 'ef1 + " ; "=21. 1 ' # / ) gn + gn00+1 = gn ^ gn00+1 + gn _ gn00+1 . K ! ! ! gn ^ gn00+1 > > gn00 ^ gn00+1 > fn ^ fn+1 = fn , '(gn ^ gn00+1 ) > 'efn . 2' , gn _ gn00+1 = = gn+1. 4 1 " 'gn + 'gn00+1 = '(gn ^ gn00+1) + 'gn+1 > 'efn + 'gn+1 . % ", 'gn+1 6 ('gn ; 'efn )+'gn00+1 6 " ; "=2n +'gn00+1 < " ; 2"=2n+1 + + 'efn+1 + "=2n+1 = 'efn+1 + " ; "=2n+1. - ! ! 1 ' , ) ' 'g = lim('gn j n 2 N) 6 lim('efn +" ; "=2n j n 2 N) = lim('efn j n 2 N)+" ; lim("=2n j n 2 N) = = a + ". 2' , f = p-lim(fn j n 2 N) 6 p-lim(gn00 j n 2 N) 6 6 p-lim(gn j n 2 N) = g 2 Y )& b 'ef 6 'g. 4 1 " b 6 a + ". K ! ! ! " 1 ", b 6 a 6 b. M '' Fbc (T ) '. Xbc' ff 2 Fbc (T) j 'ef = ': f g. ; ' /# !3 '^ Xbc' , . 'f ^ 'ef = ': f. = ' ' " X' . % '" / .# ! 1/ - - ' . 1. T | ' 2 2 (Cc (T ) )+ .
1) X' = ff 2 Fbc (T) j 8" > 0 9g 2 Y 9h 2 Z (h 6 f 6 g ^ 'g ; 'h < ")g% 2) X' | % 3) Y Z X' % 4) '^ ! ! '% 5) '^ - X' . . 1) ;1 )' ) " 1) )1 X. 'ef inf f'g j g 2 Y ^ g > f g ': f supf'h j h 2 Z ^ h 6 f g. =- f 2 Fbc (T ) " > 0 g 2 Y h 2 Z, ! ) h 6 f 6 g 'ef + "=2 > 'g ': f ; "=2 < 'h. D f 2 X' , 'g ; 'h < ('ef + "=2) ; ; (': f ; "=2) = " )& f 2 X. ; , f 2 X " > 0, 'ef 6 'g ': f > 'h ! 'ef 6 'g < 'h + " 6 ': f + ". J ), 'ef 6 ': f. L "1 # 2) 1 .- 1, 1 ! ) ', ) 'ef = ': f, . . f 2 X' . 2) " f 2 X' a 2 R. K # 3) 1 .- 1 'e(;f) = ;': f = ;'ef = ': (;f). J ), ;f 2 X' . D a > 0, 'e(af) = a'ef = a': f = ': (af). D a < 0, 'e(af) = = (;a)'e(;f) = (;a)': (;f) = ': ((;a)(;f)) = ': (af). J ), af 2 X. " f 0 f 00 2 X' . K # ' 2) 5) 1 .- 1 ': (f 0 +f 00 ) 6 6 'e(f 0 +f 00 ) 6 'ef 0 + 'ef 00 = ': f 0 +': f 00 6 ': (f 0 +f 00). % ", f 0 +f 00 2 X' .
41
3) " g 2 Y. K 1 'g supf'f j f 2 Cc (T) ^ f 6 gg , ) - " > 0 f 2 Cc (T ) Z, ! ) f 6 g 6 g 'g ; " < 'f = 'f. J ), g 2 X' . ? ), h 2 Z, 1 'h = inf f'f j f 2 Cc(T) ^ f 6 hg , ) - " f 2 Cc (T) Y, ! ) h 6 h 6 f 'h + " > 'f = 'f. J ), h 2 X' . 4) " f 2 Cc (T) Y X' . K 'f ^ = 'ef = 'f = 'f. D f 2 X' f > 0, '^f 'ef > 'e0 = 0. " a 2 R f 2 X' . D a > 0, '(af) ^ = ': (af) = a': f = a'f. ^ D a < 0, '^(af) = 'e((;a)(;f)) = (;a)'e(;f) = a': f = a'f. ^ " f 0 f 00 2 X' . K 1 ! 1 " # 2) , ) '(f ^ 0 + f 00 ) = ': (f 0 + f 00 ) = ': f 0 + ': f 00 = 'f ^ 0 + 'f ^ 00 . 5) K ! ! ! '^ = 'ejX' , # 6) 1 .- 1 (fn 2 X' j n 2 N) " " f 2 X' )& ('f ^ n j n 2 N) " '^f. K ! ! ! X' - -- !/' 0&)/' ', < # " ))# - / . M =! .' ", ) Sc(T) Xbc'. = )1 Gc ' 1 ) " '. !// !' !/ '. 1 T . 6. T | A 2 Ac (T G ). A = S(Ci \ Di j i 2 I) (Ci 2 C j i 2 I) (Di 2 Gc j i 2 I). . A = S(Fi \ Gi j i 2 I) C - !/ !)/ !!3# (Fi 2 F j i 2 I) (Gi 2 G j i 2 I) ! !' ! '. C. K ! ! ! T ! " !' !, !/ !' ! '. H, ! ) C H. <' Fi \ Gi = Fi \ Gi \ C \ H = (Fi \ C)S\ (Gi \ H). '' '. Ci Fi \ C Di Gi \ H. K A = (Ci \ Di j i 2 I). M 7. A 2 Ac (T G ). (A) 2 Xbc'. . D D | !/ !' ! '. , g (D) 2 Y X' . D C | !' ! '. , h (C) 2 Z X' . K ! ! ! X' | 0&S! , (C \ D) = g ^ h 2 X' . '' 6 A = (Ci \ Di j i 2 I). <' (A) = sup( (Ci \ Di) j i 2 I) 2 2 X' . M 2. Sc (T) Xbc' . . " f 2 Sc (T). 41"'&' " > 0. ! 1 " '/ 1 (II.4) 1 " 39] / , ) - " ) - !3- u 2 St(T A(T G )), ! - ) jf(t) ; u(t)j < "=4 - t 2 T. 41"'&' g0 u ; ("=4)1 h0 u + ("=4)1. K g0 (t) = (u(t) ; f(t)) + f(t) ; "=4 6 f(t) h0 (t) > (u(t) ; f(t)) + f(t) + "=4 > f(t).
42
. . , . .
% # /, g0(t) = (u(t) ; f(t)) + f(t) ; "=4 > f(t) ; "=2 0 h (t) = (u(t) ; f(t)) + f(t) + "=4 < f(t) + "=2. K ! ! ! !3- f -
) !' !/' ', !' ! '. C ) a b, ! ) a (C) 6 f 6 b (C). ;1 )' (C) )1 x '' ) / !3 !' !/' -' v (ax _ g0 ) ^ bx w (ax _ h0 ) ^ bx. K f(t) ; "=2 < v(t) 6 f(t) 6 w(t) < f(t)+"=2 - t 2 T 0 6 w ; v 6 "x. P K ! ! ! v 2 St(T A(T G )), v = (ak (Ak ) j k 2 K) - !/ !)/ !!3# (ak 2 R j k 2 K) (Ak 2 A(T G ) j k 2 K). 4 , ) A(T G ) | , '. ) ", ) S '. Ak ! - ak )/ -. K (Ak j k 2 K) = coz v C. % ", ! . '. Ak - -- !' !/'. '' 7 (Ak ) 2 X' . J ), v 2 X' . ? )/' 1' w 2 X' . 41"'&' 1 " > 0. ! 1 ' /0 - " =(3'(x)) ^
&' v w 1 X' . 4 .- 1) 1 '/ 1 g0 g00 2 Y h0 h00 2 Z, ! ) h0 6 v 6 g0 , h00 6 w 6 g00 , 'g0 ; 'h0 < =3 'g00 ; 'h00 < =3. @, ) h0 6 f 6 g00 . 2' , 'g00 ; 'h0 = ('g00 ; 'h00 ) + ('h00 ; 'g0 ) + ('g0 ; 'h0 ) < 2=3 + 'h ^ 00 ; 'g ^06 6 2=3 + 'w ^ ; 'v ^ = 2=3 + '(w ^ ; v) 6 2=3 + "'x ^ = . % ", f 2 X' . M '' " #/# ."/# !3 '^ '. Sc (T) ! .', ) <' '. - -- !'. 8. - & A 2 A(T G ) & " > 0 ( K 2 C , K A '^( (A n K)) < ". . " D | !/ !' ! '. g (D). K ! ! ! g = supff 2 Cc (T )+ j f 6 gg, - " > 0 f 2 Cc (T)+ , ! ) f 6 g 'g ; "=2 < 'f. '' !/ '. D0 coz f !3 g0 (D0 ). K ! ! ! 'f = 'f 6 'g0 , 'g ; "=2 < 'g0. '' 7 g g0 2 X' . <' 'g ^ ; '^g0 = 'eg ; 'eg0 = 0 = 'g ; 'g < "=2. '' 1 '! / '. Kn ft 2 T j f(t) > 1=ng D0 !3 hn (Kn ). @, ) !' !/. K ! ! ! (Kn j n 2 N) " D0 , (hn j n 2 N) " g0 . 4 ))# - / !3 '^ '/ ) ' ('h ^ n j n 2 N) " 'g ^ 0 . <' n, ! ) 0 0 ^ n Kn )) = 'g ^ ; "=2 6 'h ^ n 6 'g ^ . 4 1 " g ; hn = (D n Kn ) )& '( (D = '^g ; 'h ^ n 6 'g ^ ; 'g ^ 0 + "=2 < ". " " C | !' ! '. . =- '. D " > 0 ' /0 !' ! '. L, ! ) L D '( (D ^ n L)) < ". '' !' ! '. K C \ L C \ D. K ) ' '( (C ^ \ D n K)) = '^( (C \ (D n L))) 6 6 '^( (D n L)) < ". S ( !3, " A 2 Ac (T G ). '' 6 A = (Ci \ Di j i = 1 : : : n). =- " > 0 !' !/ '. Ki , ! ) Ki Ci \ Di
S
43
'( (C ^ i \ Di n Ki )) < "=n. '' '. S Di n!' ! S K (Ki j i = 1 : : : n). K ! ! ! A n K = (Ci \P K j i = 1 : : : n) (Ci \ Di n Ki j i =P 1 : : : n), '( (A ^ n K)) 6 '( ^ ( (Ci \ Di n Ki ) j i = 1 : : : n)) = = ('( (C ^ i \ Di n Ki )) j i = 1 : : : n) < ". M 2. T | ' 2 2 (Cc (T ) )+ . '^ Sc (T) ! . . , ! ! ' Sc (T). . " (An 2 Ac (T G ) j n 2 N) # 0 " > 0. '' 8 - An !' ! '. Ln, ! ) Ln TAn '( (A ^ n n Ln )) < "=2n+1 . ''S !' !/ '. KnS (Li j i = 1 : : : n) An . K ! ! ! An n KnP= (An n Li j i = 1 : : : n)
P(Ai n Li j i = 1 : : : n), '^( (AnPn Kn )) 6 '( ^ ( (Ai n Li ) j i = 1 : : : n)) = = ('( (A ^ i n Li )) j i = 1 : : : n) < " (1=2i+1 j iT= 1 : : : n) 6 "=2. '' !' ! '. K (Kn j n 2 N) !3 h (K) hn (Kn ). K ! ! ! (Kn j n 2 N) # K, (hn j n 2 N) # h. 4
))# - / !3 '^ ) ' ('h ^ n j n 2 N) # 'h. ^ % ", n, ! ) '^h + "=2 > '^hn. 4 1 " ' (A ^ n n K) = ' (A ^ n ) ; 'h ^ < ' (A ^ n ) ; '^hn + "=2 = ' (A ^ n n Kn ) + "=2 6 ". J ), !3 '^ - -- ."/' !' !/'. = , - # )! t 2 T !' ! - !/ !" D. '' !3 g (D) 2 X' . D f 2 X' ^ 6 '^g )& j'f ^ j 6 'g ^ < 1. J ), '^ jf j 6 g, ;'g ^ = '( ^ ;g) 6 'f - -- ! " )/'. ( !3, # ))# - / 1 '/ 1. K !' 1', !3 '^ - -- !'. " " - -- #/' ."/' !' !3 ' Sc(T ), . ' !3 '. =! .', ) = '. ^ % ) ! .', ) (D) = ' (D) ^ - !' ! !/ '. D. K ! ! ! - -- ."/' !' !/', - " > 0 !' ! '. C D, ! ) (D n C) < ". ;1 )' (D) (C) )1 g h . % ", g < h + ". '' 1 '! '. F T n D. K ! ! ! T | ! '. C | !' !, / - !3- f T, ! ) 0 6 f 6 1, f(t) = 0 - t 2 F f(t) = 1 - t 2 C. @, ) g 6 f 6 h. <' g < h + " 6 f + " = 'f + ". J ), g = supf'f j f 2 Cc (T ) ^ f 6 gg 'g = '^g. " " K | 1 " !' ! '. . K !' ! !/ '. D K. '' !' ! !/ '. E D n K. K ! ! ! (K) = (D) ; (E), (K) = (D) ; (E) = ' (D) ^ ; ' (E) ^ = ' (K). ^
44
. . , . .
" " A 2 Ac (T G ). K ! ., ! ! /0, - " > 0 !' ! '. K A, ! ) (A n K) < "=2. ? )/' 1', !' ! '. L A, ! ) ' (A ^ n L) < "=2. '' !' ! '. C K L A. K ! ! ! '^ 1 , (A n C) < "=2 '^ (A n C) < "=2. % ", j (A) ; ' (A) ^ j 6 j (A) ; (C)j + j' (C) ^ ; ' (A) ^ j < ". K ! ! ! " 1 ", (A) = ' (A). ^ ( !3, " f 2 Sc (T). 41"'&' 1 " > 0 " =(x), x | !3- 1 ! 1 " .- 2. =P " 1"'&' !3 v w 1 ! 1 " .- 2. K v = (ak (Ak ) j k 2 K) G ) j k 2 K). % ", - !# !)# !!3 P P ^(Ak k2) Aj kc(T v = (ak (Ak ) j k 2 K) = (ak ' (A 2 K) = 'v. ^ ? )/' 1', w = 'w. ^ K ! ! ! v 6 f 6 w 0 6 w ; v 6 "x, f 6 w 6 v + "x = = 'v ^ + 6 '^f + 'f ^ 6 '^w 6 'v ^ + "'x ^ = v + 6 f + . % ", jf ; 'f ^ j 6 . K ! ! ! 1 ", f = 'f. ^ M % <# ' '/ '.' !! " . P0 1 (Cc (T) )+ (Sc (T )4 )+ , - P0' '^. # 1. T | . ! P0 " (Cc (T) )+ (Sc (T)4 )+ . . D '0 6= '00, , ) , '^0 =6 '^00. J ), P0 5! . D 2 (Sc (T )4 )+ , !3 ' jCc (T) - -- #/' ."/'. '' . '. ^ ' 2 = '. ^ J ), P0 ! . M K " '/ '.' ! 1 " ' < ! . 3 ( ). / " (
" . . " T | ! " !' ! . ' 2 !3- P0 1 (Cc (T) )+ (Sc (T )4 )+ . ' 1 (II.5) 1 39] ' !' !3- V 1 (Sc (T)4 )+ RMw0(T)0 . >!3- V P0 & ' !
. M % 1 ! ' 2 '. 1 )" )". 4. 0 ! P0 " " P " Cc (T) Sc (T )4 , P ' = P0('+ ) ; P0 (;'; ) ! ' 2 Cc (T) . . ' ) , ) P0(a') = aP0' P0 ('0 + '00) = = P0'0 + P0 '00 - / a 2 R+ '0 '00 2 (Cc (T ) )+ .
45
R. ) ", ) a > 0. ;1 )' a' )1 . 41"'&' / g 2 Y " > 0. '' '. Lg ff 2 Cc (T ) j f 6 gg. K 1 'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) u v 2 Lg , ! ) 'g ; "=(2a) < 'u g ; "=2 < v. '' !3 w u _ v 2 Lg . K j(a')g ; gj 6 ja'g ; a'wj + jw ; gj < aj'g ; 'wj + + "=2 < ". L1 1 " " , ) (a')g = g. % 0 )/' 1' ! 1/ -, ) (a')h = h - h 2 Z. " " f 2 Sc (T ) " > 0. '' '. Zf fh 2 Z j ^ = sup(h j Zf ) ! ., ! ! h 6 f g. K 1 'f ^ = ': f = sup('h j h 2 Zf ) f ^ /0, , ) (a')f ^ = f. ;1 )' '0 + '00 )1 . " g 2 Y " > 0. K 1 ' 0 g = = sup('0f j f 2 Lg ), ' 00g = sup('00 f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) u v w 2 Lg , ! ) ' 0g ; "=3 < '0 u, ' 00 g ; "=3 < '00v g ; "=3 < w. '' !3 x u _ v _ w 2 Lg . K j(' 0 + ' 00)g ; gj 6 j' 0g ; '0xj + j' 00g ; '00 xj + jx ; gj < ". % ", (' 0 + ' 00 )g = g. ? )/' 1' ! 1/ -, ) ('0 +'00)h = h - h 2 Z. " " f 2 Sc (T ) " > 0. K '^0f = ': 0 f = sup('0h j h 2 Zf ), ^ = sup(h j h 2 Zf ). K ! ., ! ! /0, '^00f = sup('00h j h 2 Zf ) f 0 00 ^ , ) ('^ + '^ )f = f. ' ", ) . P0 1 - / -! / 3/. " ' 6 . 41"'&' g 2 Y '' '. Lg ff 2 Cc(T) j f 6 gg. K 1 'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) 'g 6 g. 41"'&' " f 2 Sc (T ) '' '. Yf fg 2 Y j ^ = inf(g j g 2 Yf ) , g > f g. K 1 'f ^ = 'ef = inf('g j g 2 Yf ) f ^ K !' 1', P0 - -- '/'. L1 ! P0 ) 'f ^ 6 f. " , ) P0 - -- 1/'. % ", P0 - / -! / 3/. 4 .- 3.6.1 1 16] . P0 ' 0 5! # P 1 A Cc (T) B Sc (T )4 , ! ) P' = P0('+ ) ; P0(;'; ). K ! ! ! P0 - / -! / 3/, '1 ^ '2 = 0 )& P'1 ^ P'2 = P0 '1 ^ P0'2 = P0('1 ^ '2 ) = 0. # 14E(b) 1 12] P - -- 0&)/' #/' '. D 2 B, = + + ; . % 1 1 '/ 2 + = = P0'0 ;; = P0'00 - !/ '0 '00 2 A+ . '' !3 ' '0 ; '00 2 A. K = P0'0 ; P0 '00 = P '0 ; P'00 = P '. 9 1 ) , ) P - -- 5! /'. K !' 1', P " ! /# 0&)/# #/# . <' P " '/# 1' 1'. M
46
. . , . .
( ) 15] . ., . . ! " # // % ! & '(. | 1998. | ,. 360, 0 1. | 1. 13{15. 39] . ., . . 3 ! 4 " # // 5" ! !. | 1997. | ,. 3, 0 4. | 1. 1135{1172. 40] . ., . . 9 ! 4 " # // 34 '(. 1. . | 1999. | ,. 63, 0 5. | 1. 37{82.
/ ! 0 2001 .
. .
. . .
510.643
: , ,
.
!! r , #! $ % % & !! r (F ) = (F ), (p) | & + + p. , L(r) r + L #! % ! L & $ , r, # & ! (,!) L1 ! L2 -. : L1 ,! L2 , L1 = L2 (r) + r. .+ ! 0! , 0 % ! + & ! + . 1 0 ! 2 !
+! !% % : !% +, % K, K4, T, S4, S5, GL, Grz, . 1 ! # !, -.% ! 0 + % 2 +! ! %.
Abstract
E. E. Zolin, Relative interpretability of modal logics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 47{69.
This paper introduces the notion of modality as an operator r , de7ned on the set of propositional modal formulas by the equality r (F ) = (F ), where (p) is a formula of one variable p. De7ning the logic L(r) of modality r over logic L as the set of all provable in L formulas of the propositional language extended by the operator r, the notion of exact interpretability (,!) of a logic L1 in a logic L2 can be formalized as follows: L1 ,! L2 i8 L1 = L2 (r) for some modality r. The question about the number of logics, which are exactly interpretable in some 7xed logic, is considered in this paper. Answers to this question are obtained for the following family of known modal logics: logics of boolean modalities, normal logics K, K4, T, S4, S5, GL, Grz, logics of provability. A number of results concerning the absence of exact interpretability of some logics of this family in others are o8ered as well.
1.
. (p)
, 2001, 7, 9 1, . 47{69. c 2001 !, "# $% &
48
. .
p # r , r (F) = (F ):
& 3, : :p. ( ) * ) - L: 1) - . / L ), * , 0 L. 1* ) * 0 ) L. 2 ) , , 31, . 11], 32{4]. 2) - / . ), * , * , L - .. 7 *, r tr : tr (A) A ) B r(B). 1 L(r) r L # ) , r- ) L. - / . ) * , L ( # L). & ) : * *) L , M
( r, * L(r) = M) L. * , 0 ) , 31, . 12]: 8 {9# GL ;, ;p = p ^ p, 8 * Grz, GL(;) = Grz. : *
310] L, , * * n > 1 L( n ) = L. , ) . : * ) ) K, K4, T, S4, KB, S5, GL, Grz 31], , # ) 35], * *) . = 0) . r
r
2.
7 L, *# ) ) p0 p1 : : :, ? ! -
49
. ? , > : ^ _ $ 3. ( Fm 0 * . & (p) | L p. . , (p), r ( r : Fm ! Fm), #
: r (F ) = (F). : , ) ? p, * ? ( ) . 1 , * ) : (i) ? | ( ) A (ii) ( ) | ( ) A (iii) r1 r2 | , r1 ! r2 | A (iv) r | , r | . / ( ) ! ? ( ) : : . * ( , ) - . ; = ( ) ^ , n , n > 1, / _ :. C r tr : Fm ! Fm : tr (A) * A ) r. 7 *: tr (?) = ?A tr (p) = p pA tr (A ! B) = tr (A) ! tr (B)A tr ( A) = r(tr (A)). ( ) L, * L ) : MP (modus ponens) A A ! B ` BA Sub ( ) A ` A3B=p]A RE ( 0 ) A $ B ` A $ B: D A3B=p] | , * A ) ) p B. H L ` A A 2 L. : *) ( ) ?) ). & L | * ( ), , X | . 1 L + X * * , * * L X * , LX | * , * L X, * , | MP. r
r
r
r
r
r
r
r
r
50
. .
. 9 L(r) r L # , r- ) L: L(r) = fA 2 Fm j L ` tr (A)g: J , * L(r) | . . : r1 r2 L, L ` r1 p $ r2 pA L, L(r1) = L(r2 ). ? 0 ) *: , L ` r1 p $ r2 p, L ` tr 1 (A) $ tr 2 (A) A. , # , ). # , (p), p ) . r
r
r
3.
0 15 , * * ) , * * L r L(r), 0 . , * , 0 L(r), *
r r L, r L. . , ) ) . f(x1 : : : xn): f? >gn ! f? >g
, 0 , C:JN 0 . J ) ) / ** : f 6 g , f(x y) ! g(x y) . . !
(). .) r L # ) ) (x y), * (i) L ` (r? r>)A (ii) f(x y), * L ` f(r? r>) 6 f:
. : r ). . r / 0 FLr = ff(x y) j L ` f(r? r>)g
51
** 6, # / . 2 , * FLr O, , / 0 FLr O ) FLr. & r | , rp p : rp b(p Q1 : : : Qn), Qi | . N b p: L ` b(p Q1 : : : Qn) $ 3p ^ b(> Q1 : : : Qn) _ :p ^ b(? Q1 : : : Qn)] L ` rp $ (p ^ r> _ :p ^ r?). 1 ,
p $ ((p ^ >) _ (:p ^ ?)): () 3.1. L1 L2 |
, r1 r2 | , ri Li .. i , i = 1 2. L1 (r1) L2 (r2 ) , 2 6 1 : . U * Fi = FLiri . ( Fi f, ) f > i . &0 F1 F2 , 2 6 1. , * L1 (r1) L2 (r2) ) F1 F2 . U# . & F1 F2. # A(p) 2 L1 (r1), p = = (p1 : : : pn) | # ). () A 0 L1 (r1) b(p ? >) ) p ? >, *, b(p ? >) 2 L1 (r1). = 0 p # _ ;p ^ b( ? >) 2 L1(r1) 2f?>gn p p1 1 ^ : : : ^ pnn , pi ? :pi , pi > pi .
&0 b( ? >) 2 L1 (r1) 2 f? >gn, *, b( x y) F1 , F2 . & , * b(p ? >) 2 L2 (r2) () * A 2 L2 (r2). 3.2. 3.1 : L1 (r1) = L2 (r2 ) , 1 = 2 : J , * * 0 ) L
/ , 3.2 , * *) L ) * 15. : , 15 ) ), ) ?, ). . ? * . J L 4 0 )
: r0 = ?, r1 = ( ), r2 = :, r3 = >A * , * )
52
. .
). . L: ? = :x ^ :y () = :x ^ y : = x ^ :y > = x ^ y: &0 0) LA * ) * L? L() L: L> . ( ). . ) )A , ) / ) Lrj , j 2 f0 1 2 3g, 0 * # ) ? ( ) : >. U . 3.3. ! " (x y) 6 ? # , ! $ " . . & ? 6= J f0 1 2 3g. & , * T LJ = Lrj ). . C:JN: j 2J
J (x y) =
_ ;xrj ? ^ yrj >:
j 2J
C , Lrj ` J ( ? >) j 2 J. : : Lrj ` p $ rj p, 0 j- O * J ( ? >) 0 > Lrj . C , f < J , C:JN f(x y) j- O * j 2 J. J O * f( ? >), * , 0 ? Lrj . C , f( ? >) Lrj , * LJ . 1 , J / ) O) * , J | ). . LJ . U# , * (x y) 6 ? J J, * ? 6= J f0 1 2 3g. & 3.3 * * Lfrj jj 2J g A , J = f0 3g, LJ = L? \ L> = L?> . & # ) . U * * E . 3.4. % # : I: L? = E f p $ ?gA L() = E f p $ pgA L: = E f p $ :pgA L> = E f p $ >g: II: L?() = E f p $ (p ^ >)gA L?: = E f p $ (:p ^ ?)gA L?> = E f p $ ?gA L(): = E f p $ (p $ >)gA
53
L()> = E f p $ (p _ ?)gA L:> = E f p $ (:p _ >)g: III: L?(): = E f p $ ((p $ >) ^ (:p $ ?))gA L?()> = E f p $ ((p ^ >) _ ?)gA L?:> = E f p $ ((:p ^ ?) _ >)gA L():> = E f p $ ((p $ >) _ (:p $ ?))g: IV: L?():> = E f p $ ((p ^ >) _ (:p ^ ?))g:
3.4.1. p $ rp, r | , 0 * ) *) RE.
3.4.2. U * * kf k * f? >g2, ) f(x y) >. 1 3.3 , * * / , * N (N = I, II, III, IV) ). . , k k = N. . * () * . () I. N j 2 f0 1 2 3g * Ej = E f p $ rj pg. ? F , * F $ tr j (F) Ej , * Lrj . &0 F 2 Lrj , tr j (F ) 2 Lrj , , * | , , E Ej A , * F 2 Ej . II, III, IV. : F G ) ), E fF g \ E fGg = E fF _ Gg. 1 , * * , , L?() = L? \ L() = E f( p $ ?) _ ( q $ q)g, * , * A A ^ >, * . Y* ) . & L | . ? I , * r * L rj , j 2 f0 1 2 3g, r 0 rj L. I II LJ = E f p $ (p r)g, (p q) | , r | " ( 0 ? > 0 ) ? >. 7 , * (p q) LJ . 3.5. L r, (p q)
LJ I II, L( (( ) r)) = LJ . $ , L?> | ! ! L: L(r) = L?> . r
r
54
. .
# , , (p q) = p ^ q. 0 * ). . Q = (( ) r) L , * , (x y) = :x = J , J = f0 1g, *, L(Q) = LJ 3.3. : ) (p q) *. 3.6. ' !
" ! ( !. . & L | , L ` A(p q) ! C(q r), p = (p1 : : : pm ), q = (q1 : : : qn), r = (r1 : : : rk). () *, * A C | ) ? >, L ` A(p q ? >) ! C(q r ? >). & , * B(q ? >)
W;
_
2f?>gm
A( q ? >):
? L ` A $ p ^ A( q ? >) L ` A ! B. 1 L ` A( q ? >) ! C , L ` B ! C.
4. ! & * - L * 0 ) LA * * "(L). 33] 0 /# ) ) D D, ) / (! 6) (! 6) ( 33] - .). 31, . 11], 32, . 96, 106, 115, 138], 34] - .,
:. ( * ) ) , * *) L, * * (L). D (L), "(L) | * 1. D , * * L "(L) > 4 (L) > 4A , L RE (L) 6 "(L). 0 * ) "(L) (L) ) A , * * ) ) L = K, K4, T, S4, KB, GL, Grz 31] "(L) = 1 , * (L) = 1, , , "(S5) = (S5) = 16. 7 * (L) ) ( . 35]).
55
4.1. "(L) (L)
0 * * "(L) (L) ) , ) 3. J # 3.4 ) . . 9 M
" L ( * : M ,! L), # r, * M = L(r). U* , * * (L) * M, * ) L. 4.1.1. () N (N = I, II, III) $
! ! ! ) . . : N = I 0 * . N = II. L II r ( , ) * 0 ), 0 3.5 L0 II # (p q), * L( (( ) r)) = L0 . N = III. : , , * L?()> ,! L():> . J
L():> ). . x _ y, 0 ` ?_ >. 1 r, ). . :x _ y. J 0 , * , r ,
(p) (:p ^ : ?) _ (p ^ >), L():> ` r? $ : ?, ` r> $ >, *, ` :r? _ r>. &0 L():> (r) = L?()>. Y* III ). . , * kk = 3. 4.1.2. () N (N = I, II, III) $
! N 0 > N . . C * N = I * .
4.1.1 / ,! * N = II, III N, * - N 0 = N + 1. N = II. r = > L?(): , *, L?(): (r) = L?> 3.5. N = III. & L | III. & 3.4 : L = E f p $ rpg, *# r | . 9 , * ). . r r L?():> ). . L. & 3.2 * L?():> (r) = L. . Q1 Q2 L, ) , L, A , L ) * r , (p) (p ^ Q1) _ (:p ^ Q2), >.
56
. .
4.1.3. * N (N = II, III, IV) $ ! N 0 < N . . / ,! *
N 0 = N ; 1 * / ) . & N = II * . N = III. L?> 0 ( ) >, *, 3.5 L?> * / I II. N = IV. L?(): ) , *, , ). . ) >, 0 L?():> 6,! L?(): . 4.1.4. + L | N = I, II, III, IV, (L) = 4 10 14 15 "(L) = 4 16 64 256
. (L)
). L 0 b(( ) ? >)A 256 ) . # b1 b2 b (b1 $ b2). U* , * L ` b(p ? >) , b( x y) > (x y) 2 f? >g ) ) ( . 3). D*, L ` b1 $ b2 , b1(p x y) b2 (p x y) * / ) ) f( 0 1) 2 f? >g3 j (0 1) = ?g
2 (4 ; k k), *# k k = N ( . * 3.4.2). U "(L) = 256=22(4;N ) = 22N . 4.2.
0 * ) "(L) (L) ) * ) ) . . # ( . 31, c. 4])
, (A1) * L (A2) (A ! B) ! ( A ! B) ( ) MP, Sub Nec ( ) A ` A: ( * * K. J / , *
. 4.2.1. + ) L ) K
MP Sub,
L | ( L RE) , , L | ( L Nec).
57
0 * , K ) : (A3) p ! p ( ) (A4) p ! p ( ) (A5) p ! 3p (
*) (A6) 3p ! 3p ( ) (A7) ( p ! p) ! p () 9# ) (A8) ( (p ! p) ! p) ! p () 8 *): J 31, . 5]: T = K + (A3)A K4 = K + (A4)A S4 = T + (A4)A KB = K + (A5)A S5 = S4 + (A5) = T + (A6)A GL = K + (A7) | 8 {9# A Grz = K + (A8) | 8 *. & 0) 31, . 5, 12], , * L 0 , S5, "(L) = 1, "(S5) = 16. 0 0 * (L). 4.2.2. + L | K L GL, "(L) = (L) = 1. . & "(L) > (L), * , *
(L) = 1. U : r1 = ( )A rn+1 = ( ) ^ 3rn n > 1: (1) & , * rn L *. & N > m > 1 | *, p1 : : : pN | * , N = f1 : : : N g. = : ANm
_ j 2N
pj !
_
J N jJ j=m
_ j 2J
pj :
4.2.3. + m > n, ANm 2 K(rn) N > m. . 31, . 5] K:
K ` A , A * ) *) /) . & (W R j=) |;
, x1 2 W. & W pj A 0 * , * x1 j= rn : ( j 2N * ) 0 x2 : : : xn 2 W, * x1 R x2 R : : : R xn, 8i = 1 : : : n 9j = j(i) 2 N xi j= pj . 1, ; W pj . J N , jJ j = m, * J fj(1) : : : j(n)g, * x1 j= rn j 2J
4.2.4. + m < n, ANm 2= GL(rn) N > n. . 9 GL *) -
) ) / 31, . 5]. = W = f1 : : : ng * < ) *)A
58
. .
i j= pj , i = j, 1 6 i 6 n, j 2 N . 1 (W < j=) GL 1 = 6j tr n (ANm ), * . ?
4.2.3 4.2.4 , * ANm 2 L(rn ) , m > n N > maxfm ng, *, L(rn) 6= L(rr ) n 6= r. * K K4 GL 31, . 1] , * (K4) = 1. = - . ( . 31, . 10]), :, ::. 4.2.5. * ! GL $ , -, )# : . . 9 r m 3Q m : 3Q, m > 1, Q | . 1 GL ` m 3A $ m ?, r 0 GL m ? : m ?, *, GL(r) = L?> 3.5. 310], GL
: GL( n ) = GL n > 1. C , GL 7 *) - .: ( ) : : : 3 : . = , (A3). 2 ) 4.2.2A , ) (1) 0 ( ). U . 4.2.6. + L | K L Grz, "(L) = (L) = 1. . :* , * (L) = 1. # : r01 = ( )A r0n+1 = ( ) ^ 3(:( ) ^ 3r0n) n > 1: (2) & , * L(r0n ) *. 7 ANm , 4.2.2. 4.2.7. + m > n, ANm 2 K(r0n) N > m.
4.2.3 *# , ; W p * x1 j= r0n j * , * ( j 2N * ) 0 y2 x2 : : : yn xn 2 W , * x1 R y2 R x2 R : : : R yn R xn 8i = 1 : : : n 8j 2 N yi = 6j pj 8i = 1 : : : n 9j = j(i) 2 N xi j= pj : r
4.2.8. + m < n, ANm 2= Grz(r0n) N > n.
59
. , * Grz *) ) )
*) / 31, . 12]. J W = f1 : : : 2n ; 1g * 6 / j= : i j= pj , i = 2j ; 1, i 2 W, j 2 N . 1 (W 6 j=) | Grz 1 =6j tr n (ANm ). ? )
. 1 * K T S4 Grz ( . 31]), 4.2.6 (T) = (S4) = 1. KB rn (1) n > 1 0 r2 A * (2). 1 . 4.2.9. + L | K L KB, "(L) = (L) = 1. . 1 "(KB) = 1, "(L) = 1. = r00n, ) 00 00 00 1 (p) = p ^ 3pA n+1 (p) = p ^ 3(p ^ 3(:p ^ 3(:p ^ 3 n(p)))) n > 1 (3) ANm , 4.2.2. 9 , * m > 2n, ANm 2 K(r00n) N > m. & KB
*) / , , * m < 2n, ANm 2= KB(r00n) N > 2n. C , L(r00n ) * (L) = 1. J , S5. 2 / (W R) / R = W W. 4.2.10. "(S5) = (S5) = 16. . 9 S5 0 ) ( 0 )) : (i) r, :r, r:, :r:, r | 0 = ( ) ! A (ii) r, :r, r | 0 ( ) = ^ 3A (iii) r, :r, r | 0 ? = _ :. : * , * 0 ! . D /, * 0 0 ( ) , 0 0 ! * . & , * (S5) = 16. ( (iii) $ , ) ( S5) :p $ p. ( (ii) $ : ) :p $ : p. 9 ? > ( ) : * ) . r0
60
. .
: , ( p $ p) 2 S5(:) n S5()A p 2 S5() n S5(:). D*,
(ii) (iii) *. J * (i). 4.3.
!
D 35] . & T U | * , T * . = * , ) * , T . 1 ) , ) U ,
. ( 0) RE , *, / . &0 4.3 RE. J ) ) : D = GLf: ? ( p _ q) ! ( p _ q)g | : A S = GLf p ! pg | C . # * Fn n+1 ? ! n ?, n 2 ! = f0 1 : : :g. 36] , * * : GL = GLfFn j n 2 g D = D \ GL GL = GL
_
;
n=2
;
:Fn S = S \ GL ;
!, ! n *. _ ! n *, GL D S GL A , D! = D, S! = S, GL! = Fm. = * C S, ) * ) . U/ S, 37]. . ! M = (W j=) ( W | , |
** W ), 0 r %""
b, * (1) fx 2 W j r xg | * ( | / , / )A (2) fx 2 W j x rg * (! + 1) A (3) 0 W rA (4) b | / 0 WA ;
;
61
(5) ) *) fx 2 W j x rg. N A
M,
* : M b j= A. 4.3.1 (%7]). S ` A , A ! ! . & S, , * (2) 0 SA , "(S) = 1. 4.3.2. (S) = 1. . = (2) ANm , 4.2.2.
4.2.7, m > n, ANm 2 K(r0n ) S(r0n) N > m. & m < n. & ) M = (W j=),
tr n (ANm ) N > n. & W = fbg V , V = f: : : x;2 x;1 x0 x1 y2 x2 : : : yn xng, W * / (! +1) , 0 b | *, 0 ) V : : : : x;2 x;1 x0 x1 y2 x2 : : : yn xn . U / : b j= p1A x;i j= p1 ) i > 0A xi j= pi i = 1 : : : n. 1 M | ) ) 0 r = x1 , b = 6j tr n (ANm ). 4.3.3. + L | ( RE), $ K L S, "(L) = (L) = 1. 1 , L " 35], L S, "(L) = (L) = 1. ? 36] , * , S ( " 35]), * GL . & , * L "(L) < 1 (L) < 1. & k = 3k 1) = fn 2 ! j n > kg, k > 0. = * GLk = GLk . U* , * GLk = GLf k ?g. . & ) F # * , ) ) 0 ) F . H F (k) * , * F ) )
k ?. # * (r )(k) = r (k) . ' deg(r ) r # (p), deg(?) = deg( ( )) = 0A deg(r1 ! r2) = maxfdeg(r1 ) deg(r2)gA deg( r) = 1 + deg(r): r
r0
;
;
;
;
62
. .
4.3.4. GLk ` F $ F (k) ! F . . :* k , * GL ` k (k ) ;
`
? ! (F $ F ).
4.3.5. "(GLk ) < 1, (GLk ) < 1. . J GLk r 0 r(k), ;
;
;
/ * * 0 ) * , *, "(GLk ) < 1. : , 35] , * GLk Nec, *
4.2.1 RE, , *
(GLk ) 6 "(GLk ) < 1. # ! * . C k > 0, * (! n ) fn 2 ! j n < kg, 0 3k 1) GLk GL . C , "(GL ) 6 "(GLk ) < 1. U, , *, * (GL ) < < 1, GL , , RE. 1 , , * L = GL r * r(k). & A 2 L(r), L ` tr (A). 1 L GLk ,
4.3.4 L ` (tr (A))(k) . 9 ( A r), * (tr (A))(k) (tr (k) (A))(k) , L ` (tr (k) (A))(k) ,
4.3.4 L ` tr (k) (A), * A 2 L(r(k)). 2 , 0 L(r) = L(r(k)). 1 , (GL ) < 1. ;
;
;
;
;
;
;
;
;
;
;
;
r
r
r
r
r
r
;
5. #
0 , MP Sub. 4.1
(,!)
) A * 0 / ) . D 0 / ) ) ), * ) ) ) ( . ) 4.2 4.3). 7 ) ) * ) , * ) ) ). # * . . 9 M
" L, r, * M L(r). U/ , , 38] ) ) 9 ( * , * M L, 38] - L M- .).
63
5.1. () !
.
. & L | . ? , * ) 39], , * L ` : ?, L L() , * L L> . J L() L> * . C
5.2 5.4 )
, , * , ) * . 5.2. M L(r) L RE, 3M] L(r), 3M] | , )# M RE. 5.3. + 0 2 !, GL GL. . &
4.2.1 3GL] Nec, 0 : ? ( ? ! ?), GL 0 ?. D*, 3GL ] * GL. 5.4. + M L L ) !- L? L() L: L> , M ) ! . . _ L Lrj , j 2 f0 1 2 3g ( * . 3), L(r) Ltr j (r) . : * # 35] t(L) L. & (W ) | * r. (%5]). _ x | (W ), d(x) = 0A * d(x) = maxfd(y) j x yg. ( M = (W j=) # . ' A t(A) = fn 2 ! j M n r, * M r =6j Ag. ' t(L) S L O ) , ) L: t(L) = t(A). A2L = , ) GL. : L 0 : L L> , L 0 , 0 2= t(L)A , L L? L() L: . U . 5.5. + L M ) GL, 0 2= t(L) 0 2 t(M),
M L. . & L L> , M
L? L() L: L> . : * . r
64
. .
5.6. % # , $ M ,! L: (i) "(M) 6 "(L) (M) 6 (L). (ii) $ M $/ $ L. (iii) const(M) 6 const(L), const(L) $ 0 L. (iv) L RE, 0 ) M . 5.7.
(i) * $
! . (ii) + L RE ) > : ?, GL 6,! L. (iii) + L ) GL ) ! $
, GL 6,! L. (iv) GL $ , ! GL, ) , ! 3k 1), k > 0.
.
(i) C
5.6,(i) 4.3. (ii) L ) . (iii) ? 31, . 7], * GL 0 n ?, n 2 GL ` ` n ? $ n+1 ?, 0 const(GL ) < 1. D*, const(L) < 1, const(GL) = 1, 0 GL 6,! L
5.6,(iii). (iv) 35] , * Nec ( RE, *
4.2.1) GL GL , ) 3k 1) k > 0. . N A p, ) 0 A ) . N A # , A , A B. & p = (p1 : : : pn) | , A , ( * ) 0 : _ (p ^ B ) A$ (]) ;
2f?>gn
B | . . 7 , * L , A L ` A L ` B 2 f? >gn,
65
B | (]). : , L , ) ) . 5.8. L | , M | , )# $ (A5) p ! 3p, M L(r) ! r. L(r) ! L() , L> , L()>. . :* , * L()> L(r). = rp. _# (]) : rp $ ((p ^ Qp) _ (:p ^ Q0 p)): 9 M , *, L ` r(p ! q) ! (rp ! rq). & (]) 0 L, * : (a) L ` Q(p ! q) ! (Q0 p ! Q0q)A (b) L ` Q(p ! q) ! (Q0 p ! Qq)A (c) L ` Q0 (p ! q) ! (Qp ! Q0q)A (d) L ` Q(p ! q) ! (Qp ! Qq): Y* (p ! 3p) L ` (:Q0:p ^ Q:r:p) _ (Q0:p ^ Q0:r:p), 0 O ) ) ( :p p): (e) L ` Q0 p ! Q0:rpA (f) L ` Q0 p _ Q:rp: , M Nec, 0 r- M L (g) L ` A, L ` QA. 1 L ` Q(p ! p), , p q (b), * (h) L ` Q0 p ! Qp: (e) (f) (i) L ` Q0 :rp _ Q:rp (h) (j) L ` Q:rpA r 0 * (k) L ` Q((Qp ! :p) ^ (Q0p ! p)):
66
. .
? (d) (g) , * L(Q) , 0
: (l) L ` Q(A ^ B) ! QA , (k) * (m) L ` Q(Qp ! :p) (d) (n) L ` QQp ! Q:p: ? (m) (g) (o) L ` QQ(Qp ! :p)A 0 (n), * (p) L ` Q(p ^ Qp) (q) L ` Qp: 1 , (r) L ` rp $ (p _ Q0p) * (s) L ` r? $ Q0?: 1 (a) * L ` Q0p ! Q0 q, *, (t) L ` Q0 p $ Q0?: ` (s) (t) (r) (u) L ` rp $ (p _ r?) L()> ( . 3.4) * : L()> L(r). H * ) , . 5.9. ' GL . . = (]) - A. : GL 6 ` B 2 f? >gn,
67
GL 31, . 84] | * r, * r =6j B . ` r =6j B ) * r, 0 / j= 0 * , r j= pi , i = >. 1 r = 6j A, * GL 6 ` A.
5.10. + K 6 ` A, !/ ( (W R j=) 0 r 2 W , $ r =6j A # x 2 W , $ x R r. . & K 31, . 5] M = (W R j=), * r = 6j A r 2 W. & # M0 = (W 0 R0 j=0). & W 0 = W fr0g, r0 2= W. U/ R0 : 0 ) W n frg / R0 R, x y 2 W n frg, x R0 y , x R yA M0 r r0 * W n frg, * r ) , r R0 x , r0 R0 x , r R x x 2 W n frgA M0 0 r0 ) 0 W n frg, ) r ) , x R0 r0 , x R r x 2 W n frgA r R r, r R0 r0 R0 r0. U* , * x 2 W 0 x R0 r. & / j=0 / j= W 0 : r0 j=0 p , r j= p p. 1 , * F : r0 j=0 F , r j= F 8x 2 W (x j=0 F , x j= F): C , r =6j 0 A. 5.11. ' K K4 . ( *#
5.10, K4)
5.9. 5.12. + L | X | ) , LX | . . V : A LX ` A 0 , * L ` ; ! A * ; X. ? (]) A, *
^ _ p ^ ; ! B : ;!A $ 2f?>gn V V ? LX ` A L ` ; ! A, L ` ; ! B ^
2 f? >gn, *, LX ` B .
68
. .
5.13. 1 GL , D , GL , !, ! n $ . J * ; = ( ) ^ . 5.14. L ) , L(;) L L(;) L. L(;) = L + f p ! pg. . * () * . : * : L(;) L1 , L1 = L + f p ! pg. & L1 ` p $ ;p, L1 ` A $ tr; (A) A. & A 2 L(;), tr; (A) 2 L, tr; (A) 2 L1 , *, A 2 L1 . 31, . 12], GL(;) = Grz. ?
, * K(;) = T, K4(;) = S4. J B: B = T + (A5), (A5) |
*: p ! 3p. 5.15. ' KB, B, S5, ) 2 S5, )# L()> , $ K, T, K4, S4, Grz, GL , D , GL , !, ! n $ . . ? 5.8 , * ) , )
*, / , * ) ): 0 L() , L> L()> . : , 311] , * / S5 . &0 K, K4, GL , D GL ) . : T, S4 Grz / ,! , * T ,! K, S4 ,! K4 Grz ,! GL. & # , ) GL * . 5.16. GL ,! GL $ ) !. .
310] GL: GL( n ) = GL n > 1. # k > 1, *
30 k) = fn 2 ! j n < kg. = *. 0 2= . & , * GL( k ) = GL( k ). * () * . & A 2 GL ( k ), GL ` tr k (A). 1 t(tr k (A)) 30 k). U* , * GL- *) ,
/ * k, tr k (A) tr> (A) . J * * GL L> tr> (A) 0 ( , tr k (A)) . U GL ` tr k (A), *, A 2 GL( k ). 0 2 . 0 * , * GL ( k+1 ) = GL( k+1 ). D * V* (). U * F = Fn. U* , * GL ` B , GL ` F ! B. ;
;
;
n2
69
& A 2= GL( k+1 ), (W j=) r, * r = 6j tr k+1 (A). _ d(r) < k, / (W ) ) 0 xk xk;1 : : : xd(r) = r ( * d(xk ) = k) / j= , * *) xk r . 1 *) xk r tr k+1 (F ), * xk =6j tr k+1 (A). 1 , *, * d(r) > k, r j= F , t(F ) = 30 k). C , r = 6j F ! tr k+1 (A), *, GL 6 ` F ! tr k+1 (A) A 2= GL ( k+1 ). Y _. y. J 1. 9. {
) * .
1] Boolos G. Logic of Provability. | Cambridge: Cambridge University Press, 1993. 2] . . | ., 1974. 3] Makinson D. There are in nitely many Diodorean modal functions // Journal of Symbolic Logic. | 1966. | Vol. 31, no. 4. | P. 406{408. 4] Sugihara T. The number of modalities in T supplemented by the axiom CL2 pL3 p // Journal of Symbolic Logic. | 1962. | Vol. 27, no. 4. | P. 407{408. 5] $%& '( ). *. + ',- -, '&.%/01- ./ '& // 2.(. $* ))). ) % '& '. | 1985. | 3. 49, 4 6. | ). 1123{1155. 6] 5 '6 ( 7. 8. + 9: ;%;.:,- ./ '& // 2.(. $* ))). ) % '& '. | 1989. | 3. 53, 4 5. | ). 915{943. 7] Visser A. The provability logics of recursively enumerable theories // Journal of Philosophical Logic. | 1984. | Vol. 13. | P. 97{113. 8] Zeman J. Modal systems in which necessity is
6. . $. )( &( & %&(& ( - ./ '& // 3 ., 8 ? 0.. 9. ; '&. . (, &. 1986. | ). 4. 11] Scroggs S. J. Extensions of the Lewis system S 5 // Journal of Symbolic Logic. | 1951. | Vol. 16, no. 2. | P. 407{408. ' ( ) 1997 .
. .
. . .
517.588+519.68
: , , !" " "# , $% "&.
'( ) *
* ) ) (" "!$ ( ( ( &" &+ ( !" " "# * (, %+* !() ! " "$
. , !" "$ &- % ( !& ) &"$. "
* ( "! * !"$(,
% (%+) & . ) " "%&* * (, " ( !( ! !. /, ! ("$ $ "
$ ( * . ), "!
* (
) !+* &* (, &(% (# !% " "# ) | ( " . ! !! ) ) * | ( # $ ( (
$% " "
* ( * & ( ) & -" ) " ! .
Abstract A. W. Niukkanen, Analytical continuation formulas for multiple hypergeometric series, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 71{86.
Applying canonical forms of multiple hypergeometric series along with the use of the operator factorization method makes it possible to obtain, in explicit and most general form, the analytical continuation formulas directly applicable to arbitrary series having the Gaussian type (2==1) with respect to one or several arguments. The formulas help us to unify a great number of particular formulas scattered throughout the literature. Moreover they give us a complete set of relations for any non-standard series provided that it pertains to the Gaussian type with respect to at least one of its variables. Due to simplicity and universality of the basic relations /"# "( ), (
* &) () * (, ("% !"$(
) &, # , ("
) "$ ). !"!&"6
( "# ), "!" #! 7) ! "$ * "( ) ( 97{01{00317 01{01{00380). , 2001, 7, 9 1, . 71{86. c 2001 ! " # $%&, ' () *
72
. . there arises an importantpossibilityto implementcomputer-aided analysis of numerous repeated transformations with respect to di:erent arguments of the series and to join these transformations with other important types of transformations. This possibility may have signi;cance for mathematical analysis, mathematical physics, computer algebra and theoretical chemistry.
1.
1] 2{5] ! !"$ , & !'$$ $ (2==1) ' . N F(x1 : : : xN ) ' xn $ ' $ "$ (p==q) ! Fqp, '"! N F ' $ N F $ &" xn . + $ & ' "$ ! ! !"$ . , ! $$"$ $ $ ' -'$$ $ $ $ ' ' $ $$ $' $ "$ . $'" ! $ $ '& ' !& '& $ ! '& . .'$ , ' "$ ! & $ &" $$ ". / ', ' "$ ! $
!! $ , &$ ! 6,7] ($'" ' ! !) 8] ($'" ' $
). ! 8] &$ !"$ $$ ! , $ $ ' "$ ! " 1. 2" "$ ! ' -'$$ F12 $ & 3$ ', $ & '! $ '! 24 3 +' ! !"$ ! ' -'$$ 6, 7]. 5$ ' "$ !
F12 6,7] 1 6 z 1 + 6 1 2 0 2 ;2 1 2 2 0 z F = ; (;z) F 1 + + 0 1 02 21 1 1 + + ; 02 2 1 (;z);1 F 1 +1 0 16 z (1) 01 12 1 < ( , .. " $, &" !&"( &, %+* . "!
) $, ! ( . =9{15], #
( $). /"$! & " , "$ ! ( , "$ "# ( &* =9{11], " (
", ($ " ".$ ? 6.
...
73
!
j arg(;z)j <
; a b==c d] = ;(a);(b)=;(c);(d)
i| = i ; j :
(2)
;' "$ ! ! !"$ ! '" ! . < $ $$ , " $ $ ' "$ ! ! !"$ ( &! "$ ), & !'$$ $ (2==1) '
. , $ $$ ! $ $ ($ ! ' '&$ $ ) '$ , " ' "$ ! , ! , $ , G - . / '$ " ' $$ 3 N +1F N + 1 (x0 x1 : : : xN ) (x0 x), ! x0 ! '& . $ $ N +1F ' $ $ h jq mi, $ $ $ = ( qi0 + m i) ( qi0 + m1 i1 + : : : + mN iN ), ! ( i) = ;( + 1)=;() | $ , !, (i0 i1 : : : iN ) (i0 i) | $' N +1F (q m1 : : : mN ) (q m) | $ $ "$ . , "$ ' " , $ $ , ". $ L " = L(i1 : : : iN ), $ ! $' i0 , "$ '& ' G ! ! !"$ ! N +1F, & ! !'$$ $ (2==1) ' x0, $
G = N +1F h1j1 m1i h2j1 m2i==h0j1 m0i L 6 x0 x]:
(3)
,$ ' $ ! !"$ , & (2==1) ' $ , &$ "$ $'" (3), & ! "$! , "$ & ' "$ ! , $ "$ $'" $ $ '& ! "$ G . ?
$ & '$ $$ & ! $ $ ' "$ ! "$ G $ $ & . / &"$ $ ! , "
@ ''& "$ '& $ ' ! "$ ! ! !"$ , $ = $$ $$ | = $"$ ' | , $ $ "$
! !"$ ' , $ "$ . .
74
. .
2. A$ ' "$ "$ ' $ ' "$ ! G , = (N + 1)- $ ! , ' $' $
' i0 , $ $ '& ' !'' x0, 3 | $ $ i1 : : : iN . $' (3) ' $' '" ' -'$$ F12, $ i1 : : : iN . , = ' '' (1) ' ' '& B C $ B! C, ' '" "$ G $' ' , & ' (1) $'" ! "$ ($. '' (15)). D$ ' ! $$ , " $' $ $ '' (1), ' ! $ , $ & ! ! $ E{F$ 6]. , = " $ ' @ ' "$ ! G $! = $"$ , $
. ,' $ ! $$ , " @ $ $3 (1) $ $ ' G , '" $3 (1) "$ "$! $'" $ '. G' $ , ' $ $ ' $3 $ H-= $3 ( & $3 f f1 = f f2 , ! | H-' 4], H-
$3& f1 = f2 ). $$ H-= $3 ' $3 $3, ' ! $ . / ' "$ ! G $ ' $3, & $$ ' "$ ! $! b = N +1F h2j1 m2i L 6 x0 x]. ? (1==0)
' x0 . 1. b b
(;x0 );2 N +1F h2j1 m2i L 6 x;0 1 (;x0);m2 x]:
(4)
. , ' b '' b = F01 26 ;x0d(s)] exp(;s) NF L6 (;x0);m2 xsm2 ]js=0 $ ' "$ (5) $3
(5)
...
75
F01 26 ;x0d(s)] exp(;s) , F01 26 ;x0d(s) + x0] (6) ;1 1 ;2 1 F0 26 ;x0d(s) + x0] (;x0 ) F0 26 d(s) + x0 ] (7) ;1 ;1 1 1 F0 26 d(s) + x0 ] , F0 26 d(s)] exp(x0 s) (8) ! ' (7) $ ! ' "$ ! F01 a6 x] (;x);a F01 a6 x;1] ! F01 a6 x] = (1 ; x);a , jxj < 1, '! & $$ $3 H-= $ ( & F1(s d(s)) F2(s d(s)) H-= '! '!', $ F1(s d(s))J(s)js=0 = F2 (s d(s))J(s)js=0 & J(s)6 $ ' $ F1 , F2). ' '& b '' (5) ( ! ! !"$ '), '" $3 (4), " 3 $ 1. 1. G (4) (7) $ !, " " ' " = ' B ""!C $ ' ' . / " x0 2 m2, b "$ (4) $ $ , b "$ (4) $ $ . D , $ $ $ b "$ (4), b "$ (4) $ $ $ B C ( ) " b $, ! @ " " $$. 2. b -
G G 2
; 0 1 2==1 0 2]C1G ; 0 2 1==2 0 1]C2G
(9) (9)
3 1 x m 1 2 0 C1G = (;x0 );2 F 4h1 + 2 0j1 m2 0i h2j1 m2i L 6 x0 (;1) xm 0 2 5 (10) h1 + 2 1j1 m2 1i 2 3 1 x m 2 1 0 C2G = (;x0 );1 F 4h1j1 m1i h1 + 1 0j1 m1 0i L 6 x0 (;1) xm 0 1 5 (11) h1 + 1 2j1 m1 2i ;-
(2). . , (4) x0 ! x0st, x ! xsm1 tm0 ,
"$ (4) Q F01 16 d(s)] F10 ==06 d(t)]js=t=0. G & ' ! ' ($. 4]) ' $ Q '& "$ (4) $ "$ G . / "$ "$ $ ' ! F d6 d(s)]s $3 H-= $ F d6 d(s)]s , (d )F d + 6 d(s)] (12)
76
. .
!" ! ' "$' $'"& = ', $ $ '& ' " = n = 0 1 2 : ::. G & ' ! ' (12) G , H-= ' (4), '" G (;x0 );2 (1 ; 2)(0 ; 2);1 F01 1 ; 26 d(s)]F10 ==0 ; 26 d(t)] N +1F h2j1 m2i L 6 x0;1s;1 t;1 (;x0 );m2 xsm1 ;m2 tm0 ;m2 ]js=t=0: (13) , (13) '' ! !"$ ', L1 m1 2i h2j1 m2i L 6 x;0 1 (;xx0 )m2 ( h j 1 ; 2) N +1 1 2 ; 2 G (;x0 ) ( ; ) F (14) h0 2jL1 m0 2i 0 2 ! ( ) ;( + )=;() 1 2 = 1 ; 2 m1 2 = m1 ; m2 . . / ' "$ (14) $ ( ;i) = (;1)i (1 ; i);1 (14) ' ' (9) 3 $ $ 2, $3, '"$ '& ' ($. (9)), "$ !
$ 1 $ 2, m1 $ m2. 2. ;' (9) (9) !" ' (4) $$, " &$ B ""!C $ ' N +1F , = $ $ " $ " ! x0 . , ! $ ' (4) $3 (9), (9) $ "$ ! G $ , " ' G $$. ,$ ' ' G & $ !" $ (9) (9), "& $ $ , $ ' $ G ($. (3)) $" $ $ , , " ($ ) ' "$ ! G $' ' , (9) (9), $ $ "$ '& " $ $ '& $ $ $ $ x0, '! $$ $ "$ . A @ 3 = $"$ "
1. G
C G = ; 0 1 2==1 0 2]C1G + ; 0 2 1==2 0 1]C2G (15) C1G C2G
(10) (11)
-
.
G
$'" x1 = x2 = : : : = xN = 0 ' (15) "$ $ $ ' (1). + ' "$ 3, '' (15) $
...
77
" = $"$ & $ & $ (1) $ $ $ '& G ($. " 2).
3. 24 ! "
3. M , @ ' (9) (9), $ " $ x0 x1 : : : xN , ' C1G C2 G ' & $$ ' L(F ) = 0, ' $ ' F = G. , ! $ 3 C1 F12 C2F12 $$"$ ! ' -'$$ 6, 7] &" , " ' C1G C2G &$ ' 3 $$ L(F) = 0 $$ " x0 = 1. ,$ ' & 3 $$ L(F) = 0 $ & ' 3 = $$, ' (15) $$ ' $ , & ' & G , $ ' 3 $$ L(F) = 0 $$ " x0 = 0, " ' 3 $$ x0 = 1. / $ ' $ $" ! 3 $$ L(F) = 0, ' G , B C =' ' & $ & C1 C2, = $ ! L1 L2 ($. 1]) $ = . , ' , & ' ! B C, $ '& '' (15), '" $ ' $ , ! ' ' , $$ & $$' L(F) = 0. , ' ' G @ 24 " , &$ 6 !'
' , $ $ '& @ ' 3 ' -'$$ $$ $ " 0 1 1 8]. N 24 ! &$ ! 24 3 +' ! !"$ ! ' -'$$ ($. . 6.4 8] . 2.9 7]). $'" x1 = x2 = : : : = xN = 0 = "$ $ & $ 3 +'. M ' 3 ' -'$$ $$ " 0 1 1, " $ u 6] w 7], & (u1 w1 u5 w2 ), (u2 w3 u6 w4) (u3 w5 u4 w6 ). / " ! ! ' un wn ' $ $ $ Uni Wni , $ ', 6] 7], " " = & 6 ' , $ '! $ '! L1 L2 L0 L1 L2 $ & ! ! $ i = 1 2 3 4.
78
. .
2. 24 ! " , $ % 1. & % , % G % L1 L2 C1 C2,
! % (1.1){(1.24),
'. ( , % , $ ! % 1, ' % L1 L2 C1 C2 ! ) ! % . @&" 1
, " # $ % # G =G =F
"
#
h1 j1 m1 i h2 j1 m2 i L x0 x h0 j1 m0 i
U11
=
U12
= W12 = L0 G = (1 ; x0 ) 012 " # x h 01 j1 m01 i h02 j1 m02 i h1 j0 m1 i h2 j0 m2 i L x0 m 12 0 (1 ; x0 ) F h0 j1 m0 i h01 j0 m01 i h02 j0 m02 i
U13
U14
U51
U52
W11
= W13 = L2 G = (1 ; x0 ); 1 # " x0 ;m1 F h1 j1 m1 i h02 j1 m02 i h2 j0 m2 i L x0 ; 1 x(1 ; x0 ) h0 j1 m0 i h02 j0 m02 i = W14 = L1 G = (1 ; x0 ); 2 " # x0 h 01 j1 m01 i h2 j1 m2 i h1 j0 m1 i L x(1 ; x0 );m2 x0 ; 1 F h0 j1 m0 i h01 j0 m01 i = W21 = L2 C0 L1 G = L1 C0 L2 G = x10; 0 " # h 1 + 20 j1 m20 i h1 + 10 j1 m10 i L x0 xx;0 m0 F h2 ; 0 j1 ;m0 i h01 j0 m01 i
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
= W22 = L0 U51 = L1 C0 L1 G = L2 C0 L2 G = x10; 0 (1 ; x0 ) 012 2
m0 3 x x; 0 F 4h1 ; 2 j1 ;m2 i h1 ; 1 j1 ;m1 i h1 j0 m1 i h2 j0 m2 i L x0 (1 ; x0 )m120 5 h2 ; 0 j1 ;m0 i h01 j0 m01 i h02 j0 m02 i
(1.6)
79
... U53
U54
= W23 = L1 U51 = L0 C0 L1 G = C0 L2 G = x10; 0 (1 ; x0 ) 01 ;1 2
x0 x(;x0 );m0 3 h 1 ; j 1 ; m i h 1 + j 1 m i h j 0 m i L 2 2 2 2 1 0 1 0 F4 x0 ; 1 (1 ; x0 )m10 5 h2 ; 0 j1 ;m0 i h02 j0 m02 i
(1.7)
= W24 = L2 U51 = C0 L1 G = L0 C0 L2 G = x10; 0 (1 ; x0 ) 02 ;1
3 x ( ; x0 );m0 x0 h 1 + j 1 m i h 1 ; j 1 ; m i h j 0 m i L 1 1 1 1 20 20 F4 x0 ; 1 (1 ; x0 )m20 5 h2 ; 0 j1 ;m0 i h01 j0 m01 i 2
= L1 C1 L1 G = F
"
h1 j1 m1 i h2 j1 m2 i L 1 ; x0 (;1);m120 x h1 + 120 j1 m120 i h01 j0 m01 i h02 j0 m02 i
#
(1.8)
U21
=
U22
= W32 = L0 U21 = L2 C1 L1 G = x10; 0 " # h 1 + 20 j1 m20 i h1 + 10 j1 m10 i h1 j0 m1 i h2 j0 m2 i L 1 ; x0 (;xx)m0 0 F h1 + 120 j1 m120 i (1.10)
U23
= W33 = L2 U21 = L0 C1 L1 G = C2 L2 G = x;0 1 " 1 (;1);m2 xx;m1 # h 1 j1 m1 i h1 + 10 j1 m10 i h2 j0 m2 i L 1 ; 0 F x0 h1 + 120 j1 m120 i h02 j0 m02 i
U24
U61
U62
W31
= W34 = L1 U21 = C1 L1 G = L0 C2 L2 G = x;0 2 " # 1 ;m1 ;m2 F h1 + 20 j1 m20 i h2 j1 m2 i h1 j0 m1 i L 1 ; x0 (;1) xx0 h1 + 120 j1 m120 i h01 j0 m01 i = W41 = L2 C2 L1 G = (1 ; x0 ) 012 " # h 01 j1 m01 i h02 j1 m02 i L 1 ; x0 x(x0 ; 1)m012 F h1 + j1 m i h j0 m i h j0 m i 01 2 012 01 0 1 0 2 0 2
(1.9)
(1.11)
(1.12)
(1.13)
= W42 = L0 U61 = L1 C2 L1 G = x1; 0 (1 ; x0 ) 012
3 x (;x0 );m0 h 1 ; j 1 ; m i h 1 ; j 1 ; m i h j 0 m i h j 0 m i L 1 ; x 2 2 1 1 1 1 2 2 0 F4 (1 ; x0 )m120 5 h1 + 012 j1 m012 i 2
(1.14)
80 U63
. .
= W43 = L2 U61 = C2 L1 G = L0 C1 L2 G = x010 (1 ; x0 ) 012 2 h j1 m i h1 ; j1 ;m i h j0 m i L 1 ; 1 (;1)m02
F4 U64
01
01
1
1
1
1
h1 + 012 j1 m012 i h01 j0 m01 i
x0
(1.15)
= W44 = L1 U61 = L0 C2 L1 G = C1 L2 G = x020 (1 ; x0 ) 012 2 h1 ; j1 ;m i h j1 m i h j0 m i L 1 ; 1 (;1)m01
2 2 02 02 F4 h1 + 012 j1 m012 i h02 j0 m02 i
2
2
xxm0 10 3 (1 ; x0 )m120 5
x0
xxm0 20 3 (1 ; x0 )m120 5
2 U31
=
W51
= C2 G = (; )
x0 ; 1 F
(1.16)
1 (;1)m210 x 3 h j 1 m i h 1 + j 1 m i L 1 1 1 0 1 0 15 4 x0 xm 0 h1 + 12 j1 m12 i
(1.17)
U32
U33
U34
= W52 = L0 U31 = L0 C2 G = (;x0 ) 20 (1 ; x0 ) 012 2
3 2 0 1 xxm 0 h 1 ; j 1 ; m i h j 1 m i h j 0 m i h j 0 m i L 2 2 1 1 2 2 02 02 F4 x0 (x0 ; 1)m120 5 h1 + 12 j1 m12 i h01 j0 m01 i h02 j0 m02 i
= W53 = L2 U31 = L2 C2 G = (1 ; x0 ); 1 " x # ;1 m12 F h1 j1 m1 i h02 j1 m02 i L (1 ; x0 ) (;1) (1 ; x0 )m1 h1 + 12 j1 m12 i h01 j0 m01 i h02 j0 m02 i = W54 = L1 U31 = L1 C2 G = (;x0 )1; 0 (1 ; x0 ) 01 ;1 2
(1.18) (1.19)
3
1 xx;0 m0 h 1 ; 2 j1 ;m2 i h1 + 10 j1 m10 i h1 j0 m1 i h2 j0 m2 i L 4 F 1 ; x0 (1 ; x0 )m10 5 h1 + 12 j1 m12 i
(1.20)
U41
U42
= W61 = C1 G = (;x0 ); 2 2
1 (;1)m120 x 3 h 1 + j 1 m i h j 1 m i L 2 2 2 0 2 0 25 F4 x0 xm 0 h1 + 21 j1 m21 i
(1.21)
= W62 = L0 U41 = L0 C1 G = (;x0 ) 012
3 1 0 1 xxm 0 i L h j 1 m i h 1 ; j 1 ; m i h j 0 m i h j 0 m 1 1 1 1 2 2 F 4 01 01 x0 (x0 ; 1)m120 5 h1 + 21 j1 m21 i h01 j0 m01 i h02 j0 m02 i 2
(1.22)
... U43
U44
= W63 = L1 U41 = L1 C1 G = (1 ; x0 ); 2 " x # h 01 j1 m01 i h2 j1 m2 i L (1 ; x0 );1 (1 ; x0 )m2 F h1 + 21 j1 m21 i h01 j0 m01 i h02 j0 m02 i = W64 = L2 U41 = L2 C1 G = (;x0 )1; 0 2
F 4h1 + 20 j1 m20 i h1 ; 1 j1 ;m1 i h1 j0 m1 i h2 j0 m2 i L h1 + 21 j1 m21 i
81 (1.23)
3
;m0 1 ;1x (1x;xx0 )m20 5 0 0
(1.24)
2 $' $ " L1 L2 C1 C2. L1 L2 C1 C2 ' G "@ " ' U14 U13 U41 U31 $ $ . O$ B$3C Li Cj CiLj BC Li Lj CiCj , ! i j = 1 2. $ $3 ' U43 U44 U33 U23 U24 U63 U64 U34 ($. ' 1). A $ " = $ ' & I: L21 = L22 = C12 = C22 = I (16) $ " & " L0 C0, ''& '! $ '!: L0 L1 L2 = L2 L1 C0 C1 C2 = C2 C1 L0 C0 = C0 L0 : (17) , L0 $ G " ' U12 , C0 | ' , "& $ "! U51 A(0 ; 1): C0 G = A(0 ; 1)U51 A(a) xa0 (;x0 );a : (18) !'
: () $ L0 C0 6 () $ . A 8 (a) " C0L2 C0L1 L0C2 L0 C1 " ' U53 U54 U32 U42 $ $ , " '! $ $ "$ 3 ' $ & $ " $3 L1 C0 = A(0 ; 1)C0L2 L2 C0 = A(0 ; 1)C0L1 (19) C1L0 = L0 C2 C2L0 = L0 C1: (20) O$ (), & Li Cj Lk CiLj Ck (i j k = 1 2), ' & $3
82
. .
B(0 i )CiLi Ci = Lj Cj Lj B(0 0 ; i)Ci Li Ci = Lj Cj Lj B(1 ; 0 1 + i ; 0 )CiLi Ci = Lj Cj Lj B(1 ; 0 1 ; i )CiLi Ci = Lj Cj Lj 0
0
0
0
0
0
0
(21) (22) (23) (24)
! i j = 1 2, i0 = 3 ; i, j 0 = 3 ; j, B(b c) | B(b c) = xb0(;x0 );b (1 ; x0 )c (x0 ; 1);c :
(25)
"$ "@ $ () , , L1 C1L1 , L2 C2L1 , L2 C1L1 L1 C2L1 , " ' U21 , U61, U22 U62 $ $ . Q&
'" , $ ' ' & $ Li Ci (i = 1 2). , " , $ () (), $ ' $ ! $3 (16){(22), ' , " L0 C0 L1 L2 C1C2 $ $ " , '$ & ' & 3! "$ $. , & L0 C0 $ $ ' " ' U52. D , $$ 5 $, $' $ ', $' $3 L0 C0 = C0 L0 " $ Li L0 = L0 Li = Li Ci C0 = C0 Ci = Ci 0
0
(i = 1 26 i0 = 3 ; i)
(26)
&! $ Li Ci & L0C0 ' &. / 23 ' ' 3 ' Uni $ '& ' & U11 G , '" 24 ' , @ 1, " 3 $ 2.
4. # ! $ !, & ! ! G-
3. 12 , * ! % (U1 U5), (U2 U6) (U3 U4), , $ i % 2. Un
C1Uni , | 12 C2Uni .
83
...
@&" 2
& $ , % # % G
0 012 U 4 + ; 0 120 U63 01 02 2 1 2
U14
=;
U54
= ; 1 +2 ; 0 1+120 20 10
U24
120 1 ; 0 = ; 11 + + 20 1 + 10
U63
= ; 1 + 012 0 ; 1
U11
=;
U51
= ; 1 +2 ; 0 1;21 1 20
U31
= ; 1 + 12 0 ; 1
U41
21 1 ; 0 = ; 11 + + 20 1 ; 1
U22
= ; 1+ 112+0 21 2 20
U62
= ; 11+;012 12 2 02
U34
= ; 11+;12 012 2 02
U44
= ; 1+ 12+1 120 2 20
01
02
U63 + ;
U14 + ;
U54 + ;
(2.1)
2 ; 0 012 1 ; 1 1 ; 2
1 + 120 0 ; 1 1 2
(2.2)
U24
1 + 012 1 ; 0 1 ; 1 1 ; 2
U54
(2.3)
U14
(2.4)
0 12 U 1 + ; 0 21 U31 1 02 4 2 01
1
02
(2.5)
0 ei ( 0 ;1) U31 + ;
ei (1; 0
U11 + ;
ei
0
ei
0
2 ; 0 12 1 + 10 1 ; 2
1 + 21 0 ; 1
ei
1 + 120 12 3 +; 1 1 + 10
1U4
1 + 012 21 4 +; 1 ; 1 01
0 1U4
ei
1 0U2
1U2
(2.6)
U11
(2.7)
1 + 21 012 6 +; 1 ; 01
1
0
( 0 ;1) 1 U5
(2.8)
ei
1 + 12 120 2 +; 1 1 + 10
0
0 ei ( 0 ;1) U41
1 + 12 1 ; 0 5 +; 1 + 10 1 ; 2
0 )U 1
2 01
e;i
0
ei
(2.9)
2U4
0
4
0 2U4
0
ei
3
0
e;i
0
(2.10)
2 0U2
6
(2.11)
2U2
(2.12)
2
84
. .
. + 12 ', @ 2, $ ' $ ' (15) ' Uni , ' "$ '. R $! "$ '$ $ $ & $ ! $ , @ $
2, ', @ 1. / '" A B, 12 " ', , ! $ x0 ;x0. ? $ $ $ , & ! ! '& !" ' xa0 . $'" 0 1), ! = 1, ;x0 = exp(i0 )x0 0 = sgn Imx0] A (a) = exp(;i0 a): (27) 2 0 1) ' , $ B-$, $ = 1, ;x0 = exp(i 0 )x0 0 = sgn Imx0] B (b c) = exp i 0 (c ; b)]: (28) D 3! $ ' (15) $ "@ U1i (i = 1 2 3 4) ' U1 ($. ' 1). $' $ ! $3 (20,) (15) $'" U11 U12 @ ' ' '' (2.5), $'" U13 U14 | '' (2.1). "$ (2.1) (2.5) & 4 ' U2 U6 U3 U4 , '$ 4 " $ ($. ' 1) @, ! $ $'" U1 , ' "$ ! . .' '& 8 $3 ($. (2.3), (2.4), (2.7){(2.12)) $ $ '& '& ' & U5 . , ' (15) "@ $ = ' $ ' U2 U6 U3 U4 ($. (2.2) (2.6)) 3 $ $ 3. $'" x1 = x2 = : : : = xN = 0, = = ;1 $3 (2.1){(2.10) 2 $ & $ $ $ '& ', @ . 2.9 '$$ ! ! 6], ' (2.11), (2.12) "&$ $ $ '& 3" ' (38), (40) ! 6] $! "$.
5. )* + & O 3 +$ '& ($. 8, $. 297]) $ $ , $ @ ' & "$ ' "$ !
, "& = $ , " B&$ $" $ $$"$ $$ $ $ '& $ ! ! !"$ C. $ $$
...
85
' "& $ ' &. E! '" ' "$ ! ( "$$, $ ', @ 8]), &$ $$ "$ $'" $ ' (15), ' ! = ' $ . ,$ , $ $ ' (15), "$ $ &" 3 . D , ! "$ ' ($. , ' $. 295, 297 ! 8]). ,'" 1] $ $ $ ', - ', $" & $ $ $3 ' ! !"$ , !' $ - . ' $, ' $ $ !"$$ $ $3 & ''& $ ' $$ ! ?E, ! $ " !"$ "$ $$ = $3. . % & '(, * +'+ * * ,,
' ( ' & (- * &. . (
, * /( 0 12{5], ( 4( 19] /- 40 &*& /0 ' * ,( ( ( , * 6 ,() * ,( / . 7 * ' / , +8 * /(0 * / - ( G , / / & * ' 6 . 9( ' ' ,(- (0 * - (/ ,(0 *: ' /( ** ( * * ; & ' 110]. < ,8 * ' , *'* 6 6 * ' ' / - & +(0 * * * * ' & (0 &*& /0 ' * 4,8 * & /0 * - ' 7=> 111]. = ' * 40 -(0 * - 11, 11] -'( * ,( -( * ,
(* 84 ' ?** F4 - @ H1 G2 112]. A 0 (/ ;
,(0 6- '( *( : * ' ( ' 6 * B +{@ ' 113]. A & /- & & * & * &*& /0 ' , 840 '(- * * 6' *(0 114]. * 4 / - / - '/ 9 * ' ' , ' ' ( * '0 ' * 6 , * C-; , * 84& ,
( , & ( 6' ', '84 4 - / - -, ;
* , , ; ' (/ ,(0 - 115].
,
11] ?. =. . Transformation theory of multiple hypergeometric series and com-
86 12] 13] 14] 15] 16] 17] 18] 19] 110] 111] 112] 113] 114] 115]
. .
puter aided symbolic calculations // Proceedings of the 9th International Conference FComputational ModelingG. | Dubna: JINR, 1997. | P. 219{223. ?. =. . Generalized hypergeometric series NF (x1 : : : xN ) arising in physical and quantum chemical applications // J. Phys. A: Math. Gen. | 1983. | Vol. 16. | P. 1813{1825. ?. =. . Generalized operator reduction formulae for multiple hypergeometric series NF (x1 : : : xN ) // J. Phys. A: Math. Gen. | 1984. | Vol. 17. | L731{L736. ?. =. . (- ' &*& /0 ' * ,(0 - / - // U*0 . . | 1988. | W. 43, (*. 3. | X. 191{192. ?. =. . (- * '0 ' &*& /0 ' * ,(0 - / - // >. . | 1991. | W. 50,
(*. 1. | X. 65{73. @. 9-, ?. 7'-. =(+ '( . W. 1. | >.: , 1973. Y. Z8. X* ,( / 0 ** . >.: >, 1980. H. M. Srivastava, P. W. Karlsson. Multiple hypergeometric series. | Chichester: Ellis Hoorwood, 1985. ?. =. . A* ** &*& / '( 4& ' // >. . | 2000. | W. 67, (*. 4. | X. 573{581. ?. =. . &*& /0 ' ** ( * , 0 *,8 - &( // .'. * . . | 1999. | W. 5, (*. 3. | X. 717{745. A. W. Niukkanen, O. S. Paramonova. Computer generation of complicated transformations and reduction formulas for multiple hypergeometric series // Computer Physics Communications. | 2000. | Vol. 126. | P. 141{148. ?. =. . > ' ( * ?** F4 - @ H1 G2 // U*0 . . | 1999. | W. 54. | X. 169{170. A. W. Niukkanen. Operator factorization method and addition formulas for hypergeometric functions // Integral transforms and special functions. | 2001. | Vol. 11. | P. 25{48. ?. =. , <. ?. . j& 0 &*& /0 ' &0 *(0 // < & . | 2001. | =(*. 1. | X. 41{48. A. W. Niukkanen, V. I. Perevozchikov, V. A. Lurie. A generalization of a classical relation between J +n (z ) J (z ) and J ;1 (z ) with comments on the modern state and trends in the theory of special functions // Fractional calculus and applied analysis. | 2000. | Vol. 3, no. 2. | P. 119{132. + ! # ! 1996 .
. .
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512.533.8
:
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Abstract A. Ja. Ovsyannikov, Ideal lattice isomorphisms of semigroups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 87{103.
A lattice isomorphism of a semigroup S upon a semigroup T is called an ideal lattice isomorphism if it induces a bijection of the set of ideals of S onto the corresponding set of T . Left and right ideal lattice isomorphisms are de6ned in a similar way. The order on idempotents and the property of being a subgroup are proved to retain under lattice isomorphisms of these kinds. The property of a semigroup of being decomposable in a semilattice of Archimedean semigroups is retained as well. Mappings that induce ideal lattice isomorphisms of idempotent semigroups are described. In particular, each left ideal or right ideal lattice isomorphism of an idempotent semigroup is induced by an isomorphism. 7' $ 8 $ 7 9. , 2001, 7, : 1, . 87{103. c 2001 ! " # " " , $ %& '
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S 1 a = fa 0g S 1 e = fe 0g S 1 f = fa f 0g S 1 0 = f0g (1) 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 T a = fa 0 g T e = fe 0 g T f = fa f 0 g T 0 = f0 g: (2) 8 # '
S T , x x0 x 2 S , "
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(9). A , k = 1 (10).
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#, '(xy) = '(x)'(y). H . P , X < Y . M '(X ) < '(Y ). ; x xy 2 X , '(x) = = '(x xy) = '(x)'(xy), . . '(x) = '(x)'(xy): (23) 2 #, (23) '(xy) = '(x) '(xy) = '(x)'(y):
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1] . ., . . . ! 1. | # : %& - ( . - , 1990. ! 2. | # : %& - ( . - , 1991. 2] Shevrin L. N., Ovsyannikov A. J. Semigroups and their subsemigroup lattices. | Dordrecht: Kluwer Academic Publishers, 1996.
103
3] . . % ! . &/0&/ // 2 ! /3 0 4 5. #5. &. | 6 , 1993. | #. 242. 4] Ovsyannikov A. J. On ideal lattice isomorphisms of semigroups // Colloquium on Semigroups. Szeged, 15{19 August 1994. Abstracts. | P. 28. 5] 600 ., ;. 5 . . 2. 1. | <.: <, 1972. 6] . . // 5> 5 . 2. 2. | <.: , 1991. | #. 11{191. 7] Shevrin L. N. The bicyclic semigroup is determined by its subsemigroup lattice // Simon Stevin. | 1993. | Vol. 67. | P. 49{53. 8] . . &/0&/ // // %&. &. < / . | 1966. | ? 1. | #. 153{160.
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Abstract
A. N. Pavlikov, The Cesaro average for orthogonal-like decomposition systems with non-negative measure, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 105{119.
We prove that in case of orthogonal-like decomposition systems with non-negative measure the Abel{Poisson's method of summation is equivalent to positive Cesaro's methods for convergence almost everywhere. A criterion for summability of a sequence of partial integrals almost everywhere is given.
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k=1 m=1 m n m;1 k=1 k n k;1 Z Z n n n X X X 1 1 1 ! c(!)e d(!) ; n(n+1) k c(!)e! d(!) n(n+1) k=1 sk = n k=1 k=1 k n k;1 k n k;1 k=1
+
n 1 Z X 1 n+1 ; n = n + 1 sn+1 ; n c(!)e! d(!) ; k=1 kn k;1 Z Z n n X X c(!)e! d(!) = n1 c(!)e! d(!) + ; n(n1+ 1) k k=1 kn k;1 k=1 k n k;1 Z n nX +1 Z X + n(n1+ 1) k c(!)e! d(!) + n +1 1 c(!)e! d(!) = k=1 kn k;1 k=1 k n k;1 X Z n c(!)e! d(!)+ = n +1 1 ; n1 k=1 k n k;1 Z Z n X 1 1 ! + n(n + 1) k c(!)e! d(!) = c(!)e d(!) + n + 1 k=1 kn k;1
n n n;1
109
= n +1 1
Z n n n;1
= n(n1+ 1)
Z n X c(!)e! d(!) + n(n1+ 1) (k ; 1)
nX +1
Z
k=1
kn k;1
(k ; 1)
k=1
c(!)e! d(!) =
k n k;1
c(!)e! d(!):
kn+1 ; nk2 6 n2(n1+ 1)2
nX +1
Z
k=1
k n k;1
(k ; 1)2
jc(!)j2 d(!)
1. ; % ! )
. & , Z 1 1 nX +1 X X nkn+1 ; nk2 6 n2 (nn+ 1)2 (k ; 1)2 jc(!)j2 d(!) 6 n=1 n=1 k=1 k n k ;1 Z n +1 1 X X jc(!)j2 d(!) = 6 n13 (k ; 1)2 n=1 k=1 k n k;1 Z nX +1 X = (k ; 1)2 jc(!)j2 d(!) m13 6 k=1 m>k k n k;1 Z Z 1 X 6 12 jc(!)j2 d(!) = 12 jc(!)j2 d(!) k=1
k n k;1
X 1 Z1 dt 1 3< 3 = 2(k ; 1)2 : m t m>k k;1
. % p q | !
: p > q, % .4! % % f n g1 n=1 !
(3) 1 < q 6 n +1 6 p: n
% % H = L2 (X) |
$ ) #
& % # # , 2 &! c(!) , ) Z jc(!)j2 d(!) < 1: (4)
3. ! (4). " ! ! (C,1) X , , ! # f n g,
110
. .
# (3) ! fsn (x)g .
# .
. % % % fn(x)g ! ) .
X. ; ! & (1), 1 Z 1 X X 2 jsn (x) ; n (x)j d = ksn (x) ; n (x)k2 6 n=1 X n=1 2 Z 6 q2q; 1 jc(!)j2 d(!) < 1: 1 P !! 2, ) % ) . ! jsn (x) ; n=1 ; n (x)j2, , ) ) . X nlim j s (x) ; n (x)j = 0, n !1 . . % % fsn (x)g ! ) .. . % fsn (x)g ! ) .. 3 (1) % % fn (x)g ! ) . X. ?
% k (x) ; n (x) ! n < k < n+1 : k X k (x) ; n (x) = (j (x) ; j ;1(x))
j =n+1
X 2 k jk(x) ; n (x)j2 = (j (x) ; j ;1(x)) 6 j =n +1 k p k 2 X X 1 pj j j(j (x) ; j ;1(x))j2 6 6
6
j =n +1 X n+1
X n+1
j =n +1
1 j jj (x) ; j ;1(x)j2 j j =n +1 j =n +1
@ {A ! $ {B &. D) !, ) n+1 6 p n , ) pn X pn n+1 16 X 1 6 Z dt = lnp: t j =n +1 j j =n +1 j n
= .
jk(x) ; n (x)j2 6 ln p
X n+1 j =n +1
j jj (x) ; j ;1(x)j2:
(5)
111
; % ! & (2), ) 2 % ) . X 1 P ! njn(x) ; n;1(x)j2, ), (5) ! . ) n=1 . X, + % % fn(x)g1 k=1 !4 # ! ) . X. . 1 % 1. P jckj2 < 1, f'k(x)g1k=1 | k=1 1 P $ L2. " ! !% k 'k (x) k=1 X (C,1), , ! # f ng, # (3), fsn (x)g . # . 3 ! 3 (= N, (k = f1 2 : : : kg, 1 R P (!) 1, jc(!)j2 d(!) = jck j2. k=1
% 2 ( . &4, . 127]).
1 P
jck j2 < 1, f'k (x)g1 k=1 | k=1 1 P
$ L2 . " ! !% k 'k (x) k=1 X (C,1), , ! # f n g, # (3), fsn (x)g .
# . E
! 1, ) !, )
! !. ! ) )
& % # #, ) 2. x
2. (C )-
. F An = ;n+n ) n-# +22& %-
1 P
$ ! Ant = (1;t1)1+ ( 6= ;1 ;2 : : :), An | ) F ! . n=0 ; $ ! % +22& ! ( . n n P P 4, . 75]) , ) An;;k1 = An G An++1 = Ak An;k G An = n+ An;1 G k=0 k=1 An;1 = + n An . , ) An n . ! . ; ! An 1 n n X 1 X X log An = log 1 + k = k + O k2 : k=1 k=1 k=1
+ , )! ) C !
. H# ,
112
. .
X 1 j logAn ; log nj 6 C + o(1) + O k12 < M k=1
$ M | . ! !
!G . 4 %
M1 M2 , ) n > 1 M1 < Ann < M2 : (6) = . ! An;;k1 =An ) & (n ; M + 1);1 1 An;;k1 = O = O (7) An n n :
. n() = sA(nn) , $ s(n) | ) ! ! , -
. + , . I 1 > 2, (C 1)
, ) (C 2), . . (C 2)- # $ (C 1). 3 ) n()(x) ! ) ) $ % Z n X 1 ( ) n (x) = A An;k c(!)e! (x) d(!): n k=1
k n k;1
E %, ) s(n) (x) ! ! ! n- +22&
$ ! 1 P sn (x)tn 1 X s(n) (x)tn = n=0 (1 ; t) : n=0
% > ;1, >10. $ s(n+) (x) ! ! ! n- +22& -
P sn(x)tn
! ! (1;t) (1;1t) , , n n P P ! , s(n+) (x) = An;;1k s(n) (x) = An;;1k Ak n()(x), n=0
k=0 k=0 n n X X s(n++1) (x) = s(k) (x)An;k n(+) (x) = 1+ An;;1k Akk() (x): An k=0 k=1 n ( ) 1 P = n()(x) = n+1 jk (x) ; k(;1)(x)j2. k=1 " 3. (4) > 21 lim () (x) = 0: n!1 n
(8)
113
# . ;
k() (x) ; k(;1)(x) = Z Z k k X X = A1 Ak;j c(!)e! (x) d(!) ; 1;1 Ak;;j1 c(!)e! (x) d(!) = A k j =1 k j =1 j n j;1 j n j ;1 Z k X = 1;1 Ak;j Ak ;1 ; Ak;;j1 Ak] c(!)e! (x) d(!) = Ak Ak j =1 j n j;1
Z k X ;1A;1 = = 1;1 A c(!)e! (x) d(!) +k ; j Ak;;j1Ak ;1 ; +k k ;j k Ak Ak j =1 n j j ;1 Z k X 1 ! = ;1 c(!)e (x) d(!) ; j Ak;;j1Ak;1 = Ak Ak j =1 j n j;1 Z k X 1 ;1 = A (;j)Ak;j c(!)e! (x) d(!) k j =1 j n j;1
% , 1 Z Z k X jk()(x) ; k(;1)(x)j2 d 6 2 (A1 )2 j 2 (Ak;;j1 )2 k j =1
X
Z X
jc(!)j2 d(!)
j n j;1
2(n )(x) d 6
Z 2n k X 1 X 1 ;1 2 jc(!)j2 d(!) = ( )2 (2n + 1) k=1 (Ak )2 j =1 j Ak;j j n j ;1 n Z 2 2n A;1 !2 X 2 X 1 k;j : 2 = 2(2n + 1) j jc(!)j d(!) A k j =1 k=j j n j;1
D) ! ! (6) (7), ) ;1 !2 2j A;1 !2 2n A;1 !2 X 1 X X Ak;j k;j k ;j 6 + 6 A A A
k k=2j +1 j 1 X C1 A 2 + X M2 k;1 2 6 (A1 )2 k j k=0 j k=2j +1 M1 k ) . ) %, > 12 , % ) k=j
k
k=j
k
114
. .
1 1 1 1 (j + 1) C1 M j 2 + M2 2 X M12 j 2 j 2 M1 k=2j +1 k2 6 2 2j 1 1 = C2 : 2 2 2 2 ; 2 6 M 2 j 2 C1 M2 j + M M1 2j j 1 , Z Z 2n X C 2 ( ) jc(!)j2 d(!) 2n (x) d 6 22n j j =1 X
j n j;1
1 Z X
Z 1 2n X X 1 ( ) 2n (x) d 6 C3 2n j n=1 X n=1 j =1 j n j;1 Z 1 X 2 6 C3 j j j =1
jc(!)j2 d(!) 6
jc(!)j2 d(!) = 2C3
Z
jc(!)j2 d(!) < 1:
j n j;1
;, nlim (n ) (x) = 0 ) .. !1 2 J! 2n < k 6 2n+1 : 1 ) (x) = 2(n+1 n +1 2 +1
> 21 k +1 1 > 21 k +1 1 k() (x)
n+1 2X
m=1 k X
n+1 2X
m=1
jm()(x) ; m(;1)(x)j2 >
jm()(x) ; m(;1)(x)j2 > jm()(x) ; m(;1)(x)j2 = 12 k() (x)
m=1 ( ) 22n+1 (x),
0 6 6
% ) . klim () (x) = 0, !1 k ) %. .
4. ! (4). R c(!)e! (x) d(!) E f(x) (C ) > 21 , E n 1X jf(x) ; k(;1) (x)j2 = 0 (9) lim n!1 n k=1 n 1X lim jf(x) ; k(;1) (x)j = 0: (10) n!1 n k=1 # . J ) (9). ;
115
n X k=1
jf(x) ; k(;1) (x)j2 =
n X
jf(x) ; k() (x) + k() (x) ; k(;1) (x)j2 6
k=1 n X
62
k=1
jf(x) ; k() (x)j2 + 2
n X k=1
jk() (x) ; k(;1) (x)j2:
!! 3 ! o(n), !! ! o(n) !. H
(9). n jf(x) ; (;1) (x)j X k 6 n k=1 X 12 X 1 n n n X 2 2 6 jf(x) ; k(;1) (x)j2 n12 = n1 jf(x) ; k(;1) (x)j k=1 k=1 k=1
@ {A ! $ {B &. = . n jf(x) ; (;1) (x)j2 21 n jf(x) ; (;1) (x)j X X k k 6 nlim = 0: lim !1 n!1 n n k=1 k=1 %..
5. (4) ! (C 1) & { !
E . # . R 1. % c(!)e! (x) d(!) ! (C,1) 2 & f(x), ) # ) . E. 3 % ! K ! p pX ;1 X k bk t = (b1 + : : : + bk )(tk ; tk+1 ) + (b1 + : : : + bp )tp = k=1 k=1 pX ;1 = (1 ; t) (b1 + : : : + bk )tk + (b1 + : : : + bp )tp : k=1
Z pX ;1 ! k c(!)e (x) d(!) = (1 ; t) sk (x)t + c(!)e! (x) d(!)tp = k=1 k n k;1 k=1 p p;2 X k 2 = (1 ; t) kk (x)t + (p ; 1)p;1 (x)tp;1 + sp (x)tp : k=1 ; jsp (x)j 6 pC1, jp;1(x)j 6 C2, Z p X k Ft(x) = plim t c(!)e! (x) d(!) = !1 k=1 k n k;1 p X
tk
Z
116
. .
pX ;2 2 k p ; 1 p = plim (1 ; t) kk (x)t + (p ; 1)p;1(x)t + sp (x)t = !1 k=1 1 X = (1 ; t)2 kk (x)tk : k=1
1 P @ $ , ! t < 1 (1;1t)2 = ktk;1, k=1 1 1 X X jFt(x) ; s(x)j = (1 ; t)2 kk (x)tk ; (1 ; t)2 ks(x)tk;1 = k=1 k=1 1 X = (1 ; t)2 ktk;1(k (x)t ; s(x))] = k=1 N 1 X X 2 k ;1 2 k ;1 (kt (k (x)t ; s(x))): = (1 ; t) (kt (k (x)t ; s(x))) + (1 ; t)
k=1
k=N +1
3 N, 1 > t0 > 0, 1 > t1 > 0 ! % $ " > 0 , ) jk(x) ; s(x)j < "4 ! k > N, jk (x)t ; s(x)j < "2 ! k > N, t > t0 . N N P P (1 ; t)2 ktk;1jk (x)t ; s(x)j 6 (1 ; t)2 ktk;1C3 < (1 ; t)2 C3N 2 < "2 k=1 k=1 t > t1 (t1 ) 1). $ ! k > N, t > maxft0 t1g 1 1 X X ktk+1 2" < 2" + 2" (1 ; t)2 ktk+1 = 2" + 2" = ": jFt(x) ; s(x)j 6 2" + (1 ; t)2 k=N +1 k=1
* % , $ K !{
. R 2. % c(!)e! (x) d(!) ! K !{
) R . EG jc(!)j2 d(!) < 1. 1 P $ ! (n (x) ; n;1(x)) ) . E n=1 1 P . K + ) %. ! njn(x) ; n;1(x)j2 ( 1 n=1 P (3)) % ! (n(x) ; n;1(x)). * % , + ! n=1 R
! ) . E, $ c(!)e! (x) d(!) !
E ) . (C,1). %..
6. ! (4). R c(!)e! (x) d(!)
! & { E > 0, E (C ).
117
# . * ) , )
n 1X (r) (x)j2 = 0 r > ; 1 lim j k n!1 n 2 k=1
1
! " > 0 nlim n(r+ 2 +") (x) = 0. !1 3 , (6) , ) n n X X 1 1 1 s(nr+ 2 +") (x) = s(kr) (x)An;;2 k+" = s(kr) (x)Ark A;n;2 k+" = r+ 12 = ; 12 +": k=1
k=1
+ 1 n n; X (r+ 21 +") X 1 2 sn (x) 6 jk(r) (x)j2 Ark An;;2 k+" 2 6 k=1 k=1 v v v u u u n n n p X X u uX u ( r ) ;1+2" 2 r 2 t t jk (x)j K Ak An;k = K t jk(r) (x)j2 A2nr;1+2"+1 = 6 k=1 k=1 k=1 v u n p uX = K t jk(r) (x)j2 A2nr+2" = o(n1=2 )O(nr+" ) k=1
%. ; # 6, 5, $ ! (C,1) ) . E, ) 4 = 1 (9), . . n (0) 1 P jk (x) ; f(x)j2 = 0. ) . E nlim !1 n k=1
(0) ) . ! ; R? %! k (x) ; f(x) R
% c(!)e (x) d(!) ; f(x) + c(!)e! (x) d(!) + : : : % 1n 0 2n 1 ,
) % , r = 0. ) , ) h ( 1 +") i 2 (x) ; f(x) = 0 lim n n!1 ) . E. ; % ! 4 4 = 12 + ", ) .4# %: n 1 1X n(; 2 +") (x) ; f(x)2 = 0 lim n!1 n k=1 ) . E. = . 4%.
# .)%, ) lim (2")(x) ; f(x)] = 0 n!1 n ) . E.
118
. .
A ! % " = 2 , ) % . .
7. (4) R ! (C ) > 0 c(!)e! (x) d(!) ! % ! & { .
# .
1. J % % (C,1) (C ) ! 0 < < 1. ; (C ) (C,1)- %G 5
% K !{
G 6 (C )- %. 2. J % % (C,1) (C ) ! 1 < < 1. ; (C,1) (C )- %G % K !{
, % . $ (C )G 5 % (C,1). %.. % 3. " ! $ L2 ! (C ) > 0 ! % ! & { .
# . 3 ! 7 (= N, (k = f1 2 : : : kg, 1
R P (!) 1, jc(!)j2 d(!) = jck j2. k=1
% 4 ( . &5, . 219]). " ! $ L2 !
(C ) > 0 ! % ! & { .
# . D
4 $
! ) !,
$ 2. K $ 2
. . ,
# .
" # 1] . . , , " # # // % . - &. '. () *. *. | , *- -- :
/ -* 0 0 -% , 1996. |
3. 117{118. 2] . . * #* , // ' . . 3 *. ) "# . * . | ' : '89, 1997. | 3. 105. 3] . . " // ' % * -. 3 % ., % . | 1997. | = 5.
4] ? @ 8. *. | %.:
119 / *
# , 1963. 5] 0@ 3., A # 8. *. | %.: 8/B%, 1958. 6] 0 * ?. &., B 3. '. C ) "# ) " . | %.: &, 1989.
' ( ) 1997 .
SV- . . . . . 512.552
: , , V- , , , , , .
! " # $ SV- . % & SV- Soc (R) .
Abstract
V. N. Silaev, On right SV-rings, Fundamentalnaya i prikladnaya matematika,
vol. 7 (2001), no. 1, pp. 121{129.
In this paper we investigate the worst cases of SV-ring structure. We give two constructions of SV-rings with strong restriction on all Soc (R) of Loewy chain.
. R, x 2 R y 2 R, xyx = x. ! " #$ % & 1936 . ! + , ,, " , " , +, + $ -. . % /6]. 1 " , V- , , " 2. ! % /7] 1964 . R, $ " R- . 5" " , ,. 6 + , " ( ) "$ 9. -, " , " " + , :. ; /5]. # ". 9 + = " + " R- M, " " Soc (M) "
" : " $ Soc0 (M) = 0 $ " Soc +1 (M) " " Soc +1 (M)= Soc (M) = Soc(M= Soc (M)) (Soc(M) | M)A | " ,
, 2001, 7, - 1, . 121{129. c 2001 !, "# $% &
122
. .
S
" $ Soc (M) = Soc (M). 5 C < 6 jM j, Soc (M) = Soc+1 (M). D" 0 = Soc0(M) Soc1 (M) = Soc(M) Soc2 (M) : : : Soc (M) + ( ) M. E M + , Soc (M) = MA + M. - R + , RR " , " " , " " . F, " " % - , Soc (RR ) = Soc (R R), " " Soc (R) . 1 C "+ " 10{15 " SV- , " " " V- , , + , " + . 6 , " /1,3,4], | /8]. 9 #$. 2 /2], $ " = , " + " " : 1) " u- " SV- + 1A 2) " " SV- + 2, V- . -$ + , " , R , + % R= Soc (R). 2 + " , % J" ,K, "= " . 9 , " , " , SV- " + +1 ( " $ " , . . " , ), < % R= Soc (R) " ( " +1 % - R= Soc (R)
" " " 9={L " " + " , , " $ " ). 1 , " " . L " + , + %+ - , , " % L. 9. E,= %+ - , , " % #. 9. 5 + " + " + $.
&C " " % " + #$. 2 /2]. M = , C +. 1 X | " + $ , D | . 5 + + CFMX (D) , X X D
SV-
123
$ . Q C + , + " UD = D(X ) += + %+ CFMX (D) ' End(UD )A
, Soc(CFMX (D)) + , , . 2 $ D " , CFMX (D). 6 " , , + $ 8 6 " To Q CFMX (D) " , X, : 1) Q ' CFMX (D)A 2) Q Q 8 : < 6 A 3) Q \ Soc(Q ) = 0 8 : < 6 A 4) D Q 8: 6 A 5) Soc(Q ) Soc(Q ) Soc(Q ), = min( ). 5 "
" " "" L Q1 : L0 = 0A L +1 =SL + Soc(Q +1 ) 8 < A L = L " 6 , < R = L + D " SV- + 1. 9 D " C $ " Q1 , "= ". 5" + . 1. # " + > 0 " SV- R + 1, < % - R= Soc (R) " . . M " + 2. &"
+ /2] + " = " " . (2, lemma 4.1]). | X | , jX j = @ . < P X , 1) jY j = @ 8Y 2 P ! 2) 6 , P P . (2, proposition 4.2]). > 0 X | jX j = @ , D | Q = CFMX (D).
6 Q " # Q , $ Q 1) Q Q 8 6 6 ! 2) Q \ Soc(Q ) = 0 8 < 6 ! 3) D Q 8 6 . . # 6 " P | + X + . # $ Y 2 P
fY : @ ! Y + xYi = fY (i) $ i < @ .
124
. .
# $ x 2 X 9! Y (x) 2 P , i (x) < @ , x = xY (x)i (x) . L , < 6 , $ Z 2 P + @ $, "$ , P , " S
gZ : @ ! P , Z = fgZ (j) j j < @ g, + YZj = gZ (j) 8j < @ . # Y 2 P 9! Z (Y ) 2 P , j (Y ) < @ , Y = YZ (Y )j (Y ) . # $ 6 " $: ' : CFMP (D) ! Q: (' (A)xy = i (x)i (y) AY (x)Y (y) A ' : CFMP (D) ! CFMP (D): (' (B)Y Y = j (Y )j (Y ) BZ (Y )Z (Y ) : : " , ' ' | %+ 8 < 6 ' = ' ' . # $ 6 Q = Im(' ). 1) 5 jP j = @ = jX j, Q ' Q. 2) 6+ ' = ' ' , Q Q 8 < 6 . 3) X 0 6= A 2 CFMP (D), , 9Z Z 2 P , AZZ 6= 0. F Y , Z = F Y , ' (A) 5 Z = Zj Z j YZj YZ j = AZZ 6= 0 8j < @ . j< j< F, ' (A) $ @ , . 1 Im(' ) \ Soc(CFMP (D)) = 0, Q \ Soc(Q ) = 0. Y , (Soc(Q )) 6 , " "" Q Soc(Q ) \ Soc(Q ) = 0 " 6= A Soc(Q ) Soc(Q ) Soc(Q ) = minf g: # $ 6 " "
" "" L Q1 : L0 = 0A L +1 =SL + Soc(Q +1 ) 8 < A L = L $ " 6 . < - R = L + D " SV- + 1. 9 D $ " Q1 , "= ". M " , + . ! . 1 | , X | $ jX j = @0 . !" " DEMX (D) CFMX (D) = Q + : 0
0
0
0
0
0
@
0
0
0
@
DEMX (D) = = fA 2 Q j 9n = n(A) > 1 8i j : i > n j > 1 Aij = Ai+1 j +1 g: 5 DEMX (D) Soc(Q) Soc(DEMX (D)) = Soc(Q). X + A B 2 DEMX (D) 0
0
SV-
(
(
125
j = i + 1 B = 1 i = j + 1 Aij = 01 A ij 0 , A B = 1, B A 6= 1. - , DEMX (D)= Soc(Q) = D(x) | , " x, x = A mod Soc(Q), x 1 = B mod Soc(Q). [ "
2 , P+1 , jP+1j = @0 . 9= + " $ + . !" " R( + 1) + : R( + 1) = L + '+1 (DEMP+1 (D)) jP+1 j = @0 : 5 , '+1 (DEMX (D)) Q+1 Q 8 < + 1 | " " , Q Q 8 6 6 + 1, R( + 1) | " Q1 (L ) 6 L+1 = L + Soc(Q+1 ) L+2 = R( + 1) | + " R( + 1). # 6 L \ Q +1 = = 0, L | Q +1 + L . 5 L Q +1 + L '+1 (DEMX (D)) Q +1 , " , R( + 1)=L " (Q +1 +L )=L = Q +1 Soc(Q +1) = L +1 =L R( + 1)=L , , L +1 =L = Soc(R( + 1)=L ). #, R( + 1)=L+1 = '+1 (DEMX (D))= Soc(Q+1 ) = D(x) " " , " (L ) 6+2 | " R( + 1), " " , + 2. # $, R( + 1)=L " . # = + 1 + 2 " . 1 6 . M % , $ X , a b 2 CFMX (D) + a Soc(CFMX (D)) b = 0 , a = 0 b = 0. # , b 6= 0 ) Im(b) 6= 0, a 6= 0 ) Ker(a) 6= UD = D(X ) , " x 2 Soc(CFMX (D)), " - + Im(b) + UD n Ker(a) , a x b 6= 0. 5" , " R( + 1)=L (Q +1 + L )=L = Q +1, x (R( + 1)=L ) y = 0 = x Soc(Q +1 ) y = 0, " + x = 0 y = 0, R( + 1)=L " . # $, fL j < + 1g | $ , R( + 1), $ , L+1 . 1 I | R( + 1), L+1 6 I. = minf 6 + 1 j L 6 I g, , + 1 < + 1 L I. - R=L " , "" $ L 6= I, 0 6= (I=L ) (L+1 =L ) = (I \ L+1 )=L . M , Soc(R=L ) = L+1 =L | R=L , , L+1 I | " . # $ " , R( + 1) " V- . 9 + = " " R( + 1)- U. 5 UR(+1) = (R( + 1)=M)R(+1) " M R( + 1). X 0
0
0
0
0
0
0
;
0
126
. .
P = R( + 1) M = fr 2 R( + 1) j R( + 1) r M g $ L+1 , P = L+1 , L+1 | . X $ P 6 L+1 , " + P = L < + 1. 5 + , U " R( + 1)=L - ( " " P = Ann(U)). 1 U " R( + 1)=L - . 1 /2, theorem 2.5], R " " " +Q Q , Q = End(U )D , U | " D - " , ; L (D ) ; | Soc(Q ) R, " " R- . ; 9 C R( + 1)=L (
R( + 1)=L Q +1 ), " U R( + 1)=L - . & UR(+1) " + /6]. (6, lemma 6.17]). ': R ! S | $ #, A |
% S - . & R S | , A
% R- . $ (6, corollary 1.13]). ' # R , ( ) R- . 5 + , " R( + 1) | " SV- + 2 " (L ) 6+2 , R( + 1)=L " 6 + 1. 5" " | " . 9 + = D " + " F M R = R( + 1) + F:
2
2
2
<
! " SV- , , +1. 1 R= Soc (R) " < . 9= + " $. (2, proposition 4.6]). (R ) | F - , F - M R = R + F 2
2
Q R. ( : 2
L
(1) Soc (RR ) = Soc((R )R )! (2) R , R sup + 1! (3) R | , u- , , V- #
, R . 2
2
127
SV-
1 + " " + . # $ " " ". 2. 1 n | , n > 2. 5 " SV- R n " 0 = Soc0 (R) Soc1(R) : : : Socn (R) % - R= Socm (R) m, 0 6 m 6 n;2, " .
. 3. X | " , R | # . ( -
$ h: CFMX (R) ! CFMX (CFMX (R)): &. X | = , "
X $ P = f(i j) j i j 2 Z 1 6 i < 1 1 6 j < 1g. ! + + p1 p2 : P ! N "
" " . 5 p1 (P) p2(P ) $ = , + " + %+ h : CFMP (R) ! CFMp1 (P ) (CFMp2 (P ) (R)): 1 A 2 CFMP (R), " $ h
h (A) = a 2 CFMp1 (P ) (CFMp2 (P ) (R)) ai1 i2 2 CFMp2 (P ) (R) (ai1 i2 )j1 j2 = A(i1 j1 )(i2j2 ) . ! , h (1) = 1, h (A + B) = h (A) + h (B). h (AB). X (h (AB)i1 i2 )j1 j2 = (AB)(i1 j1 )(i2 j2 ) = A(i1 j1 )(kl) B(kl)(i2 j2 ) =
0
0
0
0
0
0
0
0
0
= =
(kl) : B(kl)(i2 j2 ) 6=0
X
(h (A)i1 k )j1 l (h (B)ki2 )lj2 = 0
kl : (h0 (B)ki2 )lj2 6=0
X
0
X
(h (A)i1 k )j1 l (h (B)ki2 )lj2 = ((h (A)h (B))i1 i2 )j1 j2 0
l: k: h (B)ki2 6=0 (h (B)ki2 )lj2 6=0
0
0
0
0
0
, h (AB) = h (A)h (B). 1 h | %+ . 9 + = A 6= 0. 9 (i1 j1) (i2 j2 ): A(i1 j1)(i2 j2 ) 6= 0, (h (A)i1 i2 )j1 j2 6= 0, h (A) 6= 0. 1 h . 1 a 2 CFMp1 (P ) (CFMp2 (P ) (R)). 5 ((h ) 1 (a))(i1 j1 )(i2 j2 ) =(ai1 i2 )j1 j2 . 1 h , +. 1 F | " . # n > 1 + + Qn CFM | X (CFM{zX (: : :CFMX }( F) : : :)): 0
0
0
0
0
0
0
0 ;
0
n
128
. .
9 Q0 + = F. $ 4. n $ hnn+1 : Qn ! Qn+1 : . Qn = CFMX (Qn 1), Qn+1 = CFMX (CFMX (Qn 1)). != " + 3. M + . ! + 8i > 0 Qi + 1i . 5" " " + Si Qi 8i > 1 + : Si = fa 2 Qi j 9N =N(a): 8l > N 8k > 1 alk = 0 8l 6 N 8k > 1 alk 2 F 1i 1g: X = " " DEMi Pi Qi 8i > 1 : DEMi = fa 2 Qi j 9M = M(a): 8l > M 8k > 1 alk = al+1k+1 2 F 1i 1 8l 6 M 8k > 1 alk 2 F 1i 1g Pi = fa 2 Qi j 8k l akl 2 F 1i 1 g: 5 Pi = CFMX (F), Si = Soc(Pi ) = Soc(DEMi ). ! + $ 8m n, 2 6 m < n, + hmn + %+ hn 1n : : :hm+1m+2 hmm+1 : Qm ! Qn . 5 hmn (Sm ) " + Qn, "= 8m m , 2 6 m < m < n, hmn (Sm ) \ hm n (Sm ) = 0, hmn (Sm ) hm n(Sm ) hmn (Sm ). "
+ , Sm = Soc(Pm ), hmm+1 (Sm ) \ Sm+1 = 0 hmm+1 (Pm ) Sm+1 " " hmm+1 + " Sm . L + , 8m n, 2 6 m < n, hmn (Sm ) \ DEMn = 0 hmn (Sm ) DEMn hmn (Sm ). 5" " " Rn Qn: Rn = h1n(S1 ) + h2n(S2 ) + : : : + hn 1n(Sn 1) + DEMn : 5 S1 | Q1 = CFMX (F ), , h1n(S1 ) | Qn A " Soc(Rn ) = h1n(S1 ) Rn= Soc(Rn) = h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn = Rn 1 h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn
" h2n(P2 ) = Qn 1 . !, Soc(DEM1 ) = S1 , DEM1 = Soc(DEM1) = F(x) | " , . 5 + , "
" , Rn | " n + 1 " 0 Soc1(Rn) = h1n(S1 ) Soc2 (Rn) = h1n(S1 ) + h2n(S2 ) : : : Socn (Rn) = h1n(S1 ) + : : : + hn 1n(Sn 1) + Sn Socn+1(Rn) = Rn, "= Rn i, " " . 8i, i 6 n ; 1, Rn= Soci (Rn) = != + , Rn | " V- . ! + + R0 " F(x). 5 R0 | " V- . E " = " + "
" n. ;
;
;
;
;
;
;
0
0
0
0
0
;
;
;
;
;
;
;
;
;
;
0
;
SV-
129
1 Rn | " V- . # $, Rn+1 | $ " V- . 1 M | " + " " Rn+1- . X M Soc(Rn+1) = 0, M | " " Rn- , " "" $
M " Rn- . 5 " C /6, 1.13] /6, 6.17] M $
" Rn+1 - . X M Soc(Rn+1 ) 6= 0, M Soc(Rn+1 ) = M , +, M % + Soc(Rn+1)Rn+1 . 5 Soc(Rn+1 )Rn+1 | " " , M + % " Soc(Rn+1)Rn+1 . 5 + , $ , M | " Rn+1 . Rn+1 Qn+1 = CFMX (F ), Qn+1 | " . 5 M Qn+1 = M Soc(Rn+1 ) Qn+1 = M Soc(Rn+1 ) = M (
Soc(Rn+1 ) = Soc(Qn+1 )), M | " Qn+1. 5 " /6, 9.2] M | " Qn+1- " /6, 6.17] M | " Rn+1- . [ + .
1] G. Baccella. Generalized V-rings and von Neumann regular rings // Rend. Sem. Mat. Univ. Padova. | Vol. 72. | 1984. | P. 117{133. 2] G. Baccella. Semiartinian V-rings and semiartinian von Neumann regular rings. 3] G. Baccella. Von Neumann regularity of V-rings with Artinian primitive factor rings // Proc. Amer. Math. Soc. | 1988. | Vol. 103, no. 3. | P. 747{749. 4] N. V. Dung, P. F. Smith. On semiartinian V-modules // J. Pure Appl. Algebra. | 1992. | Vol. 82, no. 1. | P. 27{37. 5] L. Fuchs. Torsion preradicals and ascending Loewy series of modules // J. Reine Angew. Math. | 1969. | Vol. 239/240. | P. 169{179. 6] K. R. Goodearl. Von Neumann Regular Rings. | London: Pitman, 1979. | Monographs and Textbooks in Mathematics. 7] B. L. Osofsky. Rings all of whose nitely generated modules are injective // Pacic J. Math. | 1964. | Vol. 14. | P. 645{650. 8] C. Nastasescu, N. Popescu. Anneaux semi-artiniens // Bull. Soc. Math. France. | 1968. | Vol. 96. | P. 357{368.
' ( ) 1997 .
.
. . .
515.146.34+514.764.227
: ,
", .
# $ %& ' (. & ) * +& ) ". , -&& & " ' ' &') " .
Abstract S. Terzic, Cohomology with real coecients of generalized symmetric spaces, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 131{157.
In the article we consider generalized symmetric spaces of the compact simple Lie groups. We give a classi2cation of these spaces and an explicit description of their algebras of cohomology with real coe3cients. In the case of such spaces of second category, the direct computation of their cohomology algebras is given.
. 1] !" # $
. %&
%$ . ' , !$" %% ), . *. +& 9]. - $ + , .
(rank U = rank G) +
, 2001, 7, 4 1, . 131{157. c 2001 !, "# $% &
132
.
( . 3]). - ,
, $
(rank U < rank G). 3 $ , " & U # %$ G ! G. $, 4 , +# $" % ! + . 5 ,
! $" !& H (BG ), + . 5 ,
G=G $, ! $ t g $ $ t g. - $! !" #
. , . . An , Dn E6. 6 % 7! 8. -.
x
1.
-! G | $ , ! H (G) | R. : # $ $ %$ H (G) = ^(x1 : : : xl ) ^(x1 : : : xl ) | # $ x1 : : : xl , ( . 6]). -! g | G t | g, v1 : : : vn. 3 t !& R . * WG 4 !& , ! !& , ! WG . ' !& $ $ R v1 : : : vn]WG . -! R = rank G. <$ , R v1 : : : vn]WG
R $" . = , " .
1 ( .
. ).
(a) G |
, WG t R (R = rank G) -
. ( ) k1 : : : kR | " , 2k1 ; 1 : : : 2kR ; 1 | " , H (G) .
133
> ki , , " $ G " " ( . 10]): g = An ki = 2 3 4 : : : n + 1C g = Bn ki = 2 4 6 : : : 2nC g = Cn ki = 2 4 6 : : : 2nC g = Dn ki = 2 4 6 : : : 2n ; 2 nC g = G2 ki = 2 6C g = F4 ki = 2 6 8 12C g = E6 ki = 2 5 6 8 912C g = E7 ki = 2 6 8 10 12 14 18C g = E8 ki = 2 8 12 14 18 202430: F $" R t]WG $ G. 2. # G |
, $ Dl (l > 4), k1 : : : kR | % , 1 : : : n |
% . &
$ n
Iki =
X j =1
kj i (i = 1 2 : : : R)
R t]WG.
- 4 G Al , Bl , Cl G2 6], G = F4 1], G = E6 | 10], G = E7 E8 | 7]. 7 Dn " !. - $" !& R t]WG G. 3 fxig $" g. 1. g = An (n > 1). +!& R t]WAn = S(x1 : : : xn+1) S(x1 : : : xn+1) !&
x1 : : : xn+1. 5 $" !& R t]WAn " fi (x) = i (x1 : : : xn+1) 2 6 i 6 n + 1g i(x) $ i-" 4 "
" %&". 2. g = Bn (n > 2). R t]WBn = S(x21 : : : x2n) fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 ng:
134
.
3. g = Cn (n > 3).
R t]WCn = S(x21 : : : x2n) fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 ng:
4. g = Dn (n > 4). +!& R t]WDn !& !& t, !& S(x21 : : : x2n) x1 : : :xn. < $" !& R t]WDn " fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 n ; 1 n(x) = x1 : : :xn g: 5. g = G2 . < $" !& R t]WG2 " P2 = 2
3 X
j =1
x2j P2 = 2
3 X
j =1
x6j :
6. g = F4 . < $" !& R t]WF4 " kl X X 1 Ikl = (xi )kl + ( x x x x ) 2 1 2 3 4 : 7. g = E6 . E6 , G2 F4, !$! 2, $" !& R t]WE6 . ) ! # $ $" , $ 1]:
X 6 6 6 X X 1 k k k l l l Ikl = 2 ai + b i + cij i=1 i=1 ij =1
% ai , bi cij $" % ai = xi + 12 (x1 + x2 + x3 + x4 + x5 + x6) (1 6 i 6 6) bi = xi ; 32 (x1 + x2 + x3 + x4 + x5 + x6) (1 6 i 6 6) cij = ;xi ; xj + 21 (x1 + x2 + x3 + x4 + x5 + x6 ) (1 6 i j 6 6):
8. g = E7 . < $" R t]WE7 " P2s = 2
X
2X s;1
i<j
i=1
(xi + xj )2s = (16 ; 22s)p2s +
P8 s | & , s > 0, pi = xia , Cmn = n!(mm;! n)! . a=1
C2i sp2s;ipi
9. g = E8 . < $" R t]WE8 " P2s = 18(5 + 32s;3 ; 22s;1)p2s + + pi =
x
P9 xi , p = 0. a 1 a=1
2X s;1
i=1
C2i sp2s;i
135
pi (9 + (;1)i ; 2i ) +
i;1 1X j 3 j =1 Ci pi;j pj
2.
-! ^(u1 : : : uR) | # ( ) 4 u1 : : : uR deg u = 2k ; 1 ( = 1 2 : : : RC k | & & ! ). -! R v1 : : : vr ] | ( ) 4 v1 : : : vr deg vi = 2li (i = 1 2 : : : rC li | & & ! ). 6 C = ^(u1 : : : uR) R v1 : : : vr ]: 5 C
deg(u v) = deg u + deg v (u 2 ^(u1 : : : uR ), v 2 R v1 : : : vr ]): X C = C s C s = fc 2 C deg c = sg: s>0
-! R v1 : : : vr ] $ F1 : : : FR. = ! deg Fi = 2ki, i = 1 R ( 4 vi ). 6 C %% & d, d(ui 1) = 1 Fi d(1 vj ) = 0 i = 1 R j = 1 r:
136
.
< , d2 = 0. )%% &! R- C $ . 6 C k Z k = Ker d \ C k , B k = Imd \ C k , H k = Z k =B k L H (C d) = H k | !& C # " %k>0 % & d. <$ , F1 F2 : : : FR , 4 " H (C d) ( !" $ %$ ). 5 , : 3. ' F1 F2 : : : FR F~1 F~2 : : : F~R , Fi F~i mod (F1 : : : Fi;1 Fi+1 : : : FR ) (i = 1 R). & " ( ) , . .
H (C dF1F2 :::FR ) = H (C dF~1F~2 :::F~R ): )$ ! . 7]. 3 " . 4. ' F1 F2 : : : FR *, $ Fn+1 = Fn+2 = : : : = FR = 0. ' (C 0 d0) = ^(u1 u2 : : : un) R v1 v2 : : : vr ]
&
d0 (ui 1) = 1 Fi i = 1 n d0(1 v) = 0 v 2 R v1 : : : vr ]:
H (C dF1F2 :::FR ) = ^(un+1 : : : uR) H (C 0 d0): )$ ! . 7]. . < , Fn+1 : : : FR hF1 : : : Fni, . + C " , " $ # : M C = Ck Ck = fc 2 C : c = u1 ^ : : : ^ uk v v 2 R v1 : : : vr ]g: k>0
< ! 4 d ! ;1. J ! Zk = Ker(dC L k), Bk = Im(dCk+1) Hk = Zk=Bk . L H = = H ((C d) R) = Hk $ +#. < , k>0 H0 = R v1 : : : vr ]=hF1 : : : FRi, $ hF1 : : : FRi $ , 4 F1 : : : FR . <$ , ! + !" C. 5 , " .
137
5. H0(C d) | $ R = r, H (C d) H0(C d) ) .
)$ ! . 7]. 3# & ! | ! % &" G=U. <$ , j : R t]WG ! R s]WU , & j : U ! G, $ !" ! G=U " $ $ . 3, ! 3], % + $ " 4 % . M !" ! + . ) ! ! 3]. J 4%%& . -! BG | %& " G. 4%%& $ H (G) $ !" ! H (BG ). 5 , " ( . 3]). 6. # G |
. # *, $ H (G)
, " $% 2k1 ; 1 : : : 2kR ; 1. &
(a) H (G) x1 : : : xR (Dxi = 2ki ; 1) + ( ) H (BG ) ) $ Kp y1 : : : yR ]. , E(n G) " yi , * xi .
-! ! G=U E(n G) | ! G. 5 ! U. < $ $ (U G) %$ H (BG ) ! H (BU ), n " $ n %$ H (B(n G)) ! H (B(n U)), & & E(n G)=U E(n G)=G. 5 + ! % " $ ( . 3]). 7. # U |
G x1 : : : xR | H (G), y1 : : : yR | H (BG ), x1 : : : xR EG. & H (G=U) $ ) H (H (BU ) H (G)), H (BU ) H (G) * )) , H (BU ) d(1 xi ) = (yi ) 1 (i = 1 : : : R). 5 ! , + . > H (BU ), "
138
.
( . 3]).
1. T | G, ) (T G): H (BG ) ! H (BT ) , .
' $ , H (BG ) ! !
g, ! . > %$ (U G), "
3]. 2. # U |
G, S T | U G, t | T s | % ,
S . * H (BG ) H (BU ) $ t s , ")) , WG WU , ) (U G): H (BG ) ! H (BU ) * $ H (BG ) $ H (BU ). - ! U G ( . 7]) # " $ ! % ! + . 8. # U |
G (rank U = r), P1 : : : PR | R t]WG. j (Pr+1 ) : : : j (PR ) * hj (P1 ) : : : j (Pr )i, j | * , ,
H (G=U) = fR s]WU =hj (P1 ) : : : j (Pr )ig ^(xr+1 : : : xR ): ! . N$ 7 , + G=U (L = H (BU ) H (G) d), d(u 1) = 0 d(1 xi ) = (U G)(yi ) 1 xi 2 H (G) u 2 H (BU ). -!$!
1 2, , H (BG ) = R t]WG = R P1 : : : PR] H (BU ) = R s]WU = R Q1 : : : Qr ]
(U G)Pi = j (Pi ) = Pi =s: <" , L = R s]WU ^(x1 : : : xR) = R Q1 : : : Qr ] ^(x1 : : : xR ) d(Qi 1) = 0 d(1 xi) = j (Pi) 1: 5 j (Pr+1 ) : : : j (PR ) hj (P1) : : : j (Pr )i, $ 4 , H (L d) = H (L0 d0) ^(xr+1 : : : xR) ( ) L0 = R Q1 : : : Qr ] ^(x1 : : : xr ) d0 (Qi 1) = 0 d0 (1 xi ) = j (Pi) 1:
139
5 , $ % H (L d) . -4 $ ( ) ! H (L0 d0) , !, H0(L0 d0). 3 L0 R = r, 4 5 H (L0 d0) = H0(L0 d0) = R Q1 : : : Qr ]=hj (P1 ) : : : j (Pr )i: N, H (G=U) = H (L d) = fR s]WU =hj (P1 ) : : : j (Pr )ig ^(xr+1 : : : xR): 2 G=U, rank G = rank U, $ 5 , + H (G=U) = R t]WU =h(R t]WG )+ i (R t]WG)+ $ !& !& R t]WG, $ ! .
5 # !" " ( . 1]). 9. # G |
, U | %
, H (G) = ^(x1 : : : xR ) |
x1 : : : xR y1 : : : yR | H (BG ), x1 : : : xR . *
(U G)y1 : : : (U G)yr H (BU ) (U G)yr+1 = 0,. . . ,
(U G)yR = 0, H (G=U) = fH (BU )=h (U G)y1 : : : (U G)yr ig ^(xr+1 : : : xR): 1. F ,
(U G)yr+1 = 0 : : : (U G)yR = 0 $ ! O (U G)yr+1 : : : (U G)yR h (U G)y1 : : : (U G)yr iP. ) !, !$! 4, , H (G=U) = H (L0 d0) ^(xr+1 : : : xR) L0 = H (BU ) ^(x1 : : : xr ) d0(1 xi) = (U G)yi 1 i = 1 r d0(u 1) = 0 u 2 H (BU ): 5 (U G)yi , i = 1 r, " $ #
H (BU ), % H (L0 d0) = H (BU )=h (U G)y1 : : : (U G)yr i % ( ) .
140
.
2. N$ G=U (rank G = R, rank U = r) ! H (BU )=h (U G)y1 : : : (U G)yR i. <" ,
(U G)y1 : : : (U G)yR $ ! r %&! $
. 3. 7] $, , ! , - " : R Yr ; t2ki ) Y P (G=UC t) = (1 (1 + t2kj ;1) 2li ) (1 ; t i=1 j =r+1 ki (li ) " $ G ( U).
x
3. ! " #
$ , "
, 4]. #$% 1. <
| 4 (G U Q), G | $ , U | $ Q | %$ G, G 0 U G G = fg 2 G: Q(g) = gg G 0 | & G . * ,
(G U Q) m, Qm = id m ! !# & , " . 5 (G U Q) $
m ( ). J ! !
. - %&
%& %$ !" Aut(G). - , $ %$ S1 S2 G, Aut(G), . . S2 = S S1 S;1 S 2 Aut(G). 5 G(1 G(2 $ # G(2 = S(G(1 ): ' $ , (G G(1 S1) (G G(2 S2) $ %.
141
* , !$ %&, , %$ Q g $! $ %$ 4
, & %$
). - %&
!" # " 2]. 10. # g |
C | ) ) k (k = 1 2 3),
) - , L ~ g. # g = gi |
Zk - i . . * $ ) ~ g0 . # Xi Yi Hi (1 6 i 6 n) | / g0 , 1 : : : n T(g0 ). # ~0 | L(g ) ( 0 1), X0 6= 0 g1 , $ xX0 2 L(g0 )~ 0 . # (s0 : : : sn) | Pn $+ * m = k aisi , ai 0
L(g ), 0 , ~ = ( i 0) (1 6 i 6 n). &
(i) X0 X1 : : : Xn * g 2isj
Xj (0 6 i 6 n) $ ) m- g. 0 ) (s0 : : : snC k)+ (ii) i1 : : : it | , si1 = : : : = sit = 0. &
g0 ( . . g0 ) ) (n ; t)- Q(Xj ) = e
m
, - g(k) , i1 : : : it + (iii) $ * ) g ) Q $ ) " m. 11. , $ Q | ) (s0 : : : snC k). &
(i) Q ) ,
k = 1+ (ii) Q0 | ) (s00 : : : s0nC k), ) Q Q0 * ) g, r = r0 s * s0
) g(k) .
5 $ , %$ g $ % +& g(k) $ k. - 4 %$ " +& g(1) , # %$ | +& g(2) g(3) .
142
.
N$ ( . 2]), ! An, Dn , E6 D4 " # %$ ( !" Aut(g)). -4
Bn , Cn , G2, F4 , E7 E8 " . -
. 1. g = A2n .
Pk
n; n g = t i=1 i Bn
2. g = A2n;1. g
n; P ni
=t
3. g = Dn+1 . g = t 4. g = E6 . dim (g ) = 0.
dim (g ) = 1.
dim (g ) = 2.
k
i=1
n; P ni k i=1
1
An2 : : : Ank;1 Cnk :
Dn1 An2 : : : Ank;1 Cnk : Bn1 An2 : : : Ank;1 Bnk : g = F4 g = A1 B3 g = A2 A2 g = A3 A1 g = C4
m = 2 m = 4 m = 6 m = 4 m = 2:
g = t1 B3 g = t1 A1 A2 g = t1 C3 g = t1 A1 A1 A1 g = t1 C2 A1 g = t1 A3
m = 6 m = 8 m = 4 m = 8 m = 6 m = 6:
g = t2 A2 m = 8 2 g = t A1 A1 m = 10 g = t2 C2 m = 8:
dim (g ) = 3. dim (g ) = 4. 5. g = D4 .
x
143
g = t3 A1 m = 12: g = t4 m = 18: g = G2 g = A2 g = A1 A1 g = t1 A1 g = t2
m = 3 m = 3 m = 6 m = 9 m = 12:
4.
-! G=U |
, & %$ Q. U Q | %$ , rank G = rank U, , $ 2, ! 4 # . - , !, $ !
, & # %$ . L , t t | , " g g , $ $ $ $ t $ t , , + . -, 4 ! An , Dn E6 . -! Xi Yi Hi (1 6 i 6 R) | $" V g 1 : : : R | 4 . 5 11 ) g +&{) g(k) (k = 2 k = 3), $ # % !
t ! , " g(k) . 1. g = A2n .
0 = n + n+1 1 i = 2 ( i + 2n;i+1) (1 6 i 6 n ; 1) n = ; 12 ( 1 + : : : + 2n) (1) X H0 = ;(H1 + : : : + H2n) HX i = Hi + H2n;i+1 (1 6 i 6 n ; 1) HX n = 2(Hn + Hn+1):
144
.
2. g = A2n;1.
0 = ; 21 ( 1 + 2n;1) + 2 + 2n;2 + : : : + n;1 + n+1 + n i = 12 ( i + 2n;i) (1 6 i 6 n ; 1) n = n (2) HX 0 = ;(H1 + H2n;1 + 2H2 + : : : + 2H2n;2) X Hi = Hi + H2n;i (1 6 i 6 n ; 1) HX n = Hn : 3. g = Dn+1 .
0 = ; 1 + 2 + : : : + n;1 + 12 ( n + n+1) i = i (1 6 i 6 n ; 1) n = 12 ( n + n+1) HX 0 = ;2(H1 + : : : + Hn;1) ; (Hn + Hn+1 ) X Hi = Hi (1 6 i 6 n ; 1) HX n = Hn + Hn+1: 4. g = E6 .
5. g = D4 .
(3)
0 = ; 6 + 2 3 + 32 ( 2 + 4) + 1 + 5 1 = 6 2 = 3 3 = 21 ( 2 + 4) 4 = 12 ( 1 + 5 ) HX 0 = ;(2H1 + 3H2 + 4H3 + 3H4 + 2H5 + 2H6) HX 1 = H6 HX 2 = H3 HX 3 = H2 + H4 HX 4 = H1 + H5 :
(4)
0 = ; 2 ; 32 ( 1 + 3 + 4) 1 = 31 ( 1 + 3 + 4) 2 = 2 HX 0 = ;3H2 ; 2(H1 + H3 + H4) HX 1 = H1 + H3 + H4 HX 2 = H2:
(5)
,
) # (2) % ! +&{) A(2) 2n , A2n;1, Dn(2)+1 , E6(2) D4(3) . . Y , 4 HXi (i 6= 0) " + t g | & %$ g, & %$ " ) ( . 8]).
145
-! Q | %$ g t | ! g . -!$! # 4 HXi , , ! $" t !" $" t. 6 $ t = (g ) + t0 (g ) | & g , t0 | !
g . 5 " . 12. # Q | ) g (m (s0 : : : sn ) k), fi1 : : : it g i1 : : : it, $ si1 = : : : = sit = 0. # , , H1 : : : HR, X1 : : : XR , Y1 : : : YR | / " , HX 1 : : : HX n | t g , $ H1 : : : HR ) (1){(5). & fHX i1 : : : HX it g t0 (g ) *
X HX j ; cjkHX ik j 6= i1 : : : it t
k=1
cjk )
Xt k=1
cjkail ik = ail j
")) ail ik 1 6 l k 6 t ( L(g ).
! . -! L(g ) | " g . + ! ~ L(g ) ! Pn ~ = ki ~i, f ~0 : : : ~ng | 0
Pn
L(g ). < , deg ~ = kisi . 5 ( . 2]) $ 0 %$ M g0 = L(g )~ : deg ~=0
Pt
U ~ > 0 (; ~ > 0), deg ~ = 0 ! , ~ = kir ~ ir . r=1 N$ ( . 5]), ~ > 0 L(g )~
ej1 : : : ejs ], ~ j1 + : : : + ~ js = ~ . 5 $ , , L L(g HX i, eir , fir " $" )~ . -4 deg ~=0 HX ir = eir fir ], eir , fir " $" g0 . - & (aik il )l6kl6t & + (aij ) L(g )
146
.
& + .
X HX j ; cjk HX ik (j 6= i1 : : : it) t
k=1 " & g0. N$ & , 4%%& cjk # t X cjkail ik = ail j : 2 k=1
x
-
5.
1. g = A2n .
7 +&{) A(2) 2n
::: . < , $ 10 A(2) 2n , " & # %$ A2n n. -! x1 : : : x2n+1 | g. -! HX i (0 6 i 6 n) | , (1). 5 xn+1 (HX i) = 0 0 6 i 6 n xj (HX i) = ;x2n;j +2(HX i ) 0 6 i 6 n 1 6 j 6 n: 5 & U G ! $ HX i (0 6 i 6 n), G " xn+1 = 0 xj = ;x2n;j +2: -4
(U G)2j +1 = 0 1 6 j 6 n
(U G)2j = (;1)j (U G)j (x21 : : : x2n) 1 6 j 6 n: 5 $ , , " & U # %$ A2n ! A2n . -4 $ 8 ( $ ")
+3
+3
147
13. 1 % $ A2n % ) H (A2n =U) = (H (BU )=h (U A2n)2j i) ^(z3 : : : z2n+1) = = (H (BU )=h (U A2n)j (x21 : : : x2n)i) ^(z3 : : : z2n+1) zi xi . '. - , ! , Ck Bn;k . 7& t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n): -! y1 : : : yk | Ck , yk+1 : : : yn | Bn;k . + A2n t $" " $ : x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : : xk ! ;y1 xn ! yn 4
(U G)2j (x1 : : : x2n+1) = (;1)j j (y12 : : : yn2 ) 1 6 j 6 n H (A2n =Ck Bn;k ) = = ((S(y12 : : : yk2 ) S(yk2+1 : : : yn2 ))=S + (y12 : : : yn2 )) ^(z3 : : : z2n+1): 2. g = A2n;1. 7 +&{) A(2) 2n;1
::: . + A2n, " & # %$ A2n;1 n. -! x1 : : : x2n | g. -! HX i (0 6 i 6 n) | , (2). 5 xj (HX i) = ;x2n;j +1(HX i ) 0 6 i 6 n 1 6 j 6 n: 5 & U G ! $ HX i (0 6 i 6 n), xj = ;x2n;j +1:
ks
148
.
-4 ,
(U G)2j +1 = 0 1 6 j 6 n ; 1
(U G)2j = (;1)j (U G)j (x21 : : : x2n) 1 6 j 6 n: + A2n, , " & U # %$ A2n;1 ! A2n;1. -4 " .
14. 1 % $ A2n;1 % )
H (A2n;1=U) = (H (BU )=h (U A2n;1)2j i) ^(z3 : : : z2n;1) = = (H (BU )=h (U A2n;1)j (x21 : : : x2n)i) ^(z3 : : : z2n;1) zi xi . '. - Dk Cn;k . 7& t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n): -! y1 : : : yk | Dk , yk+1 : : : yn | Cn;k . + g $" " $ : x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : : xk ! ;y1 xn ! yn 4
(U G)2j (x1 : : : x2n) = (;1)j j (y12 : : : yn2 ) 1 6 i 6 n: H (A2n;1=Dk Bn;k ) = = (hS(y12 : : : yk2 ) y1 : : : yk i S(yk2+1 : : : yn2 )=S + (y12 : : : yn2 )) ^(z3 : : : z2n;1): 3. g = Dn+1 . 7 +&{) Dn(2)+1 ::: . " & # %$ Dn+1 n. -! x1 : : : xn+1 | g. -! HX i (0 6 i 6 n) | , (3). 5 xn+1 (HX i) = 0 0 6 i 6 n: ks
+3
149
5 $ , , & U # %$ G
(U G)n+1 = 0: 7 !, & # %$ Dn+1 ! Dn+1 , A2n, " .
15. 1 % $ Dn+1 % )
H (Dn+1 =U) = (H (BU )=h (U Dn+1 )1 : : : (U Dn+1 )ni) ^(zn+1 ) zi xi . '. - Bk Bn;k (0 6 k 6 n, B0 = ?, B1 = A1 ). 5 , t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n): -! y1 : : : yk | Bk , yk+1 : : : yn | Bn;k . + g $" " $ : x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : : xk ! ;y1 xn ! yn 4
(U G)i(x21 : : : x2n+1) = i (y12 : : : yn2 ) 1 6 i 6 n H (n+1 =k Bn;k ) = = (S(y12 : : : yk2 ) S(yk2+1 : : : yn2 ))=S + (y12 : : : yn2 ) ^(zn+1 ): 4. g = D4 . 7 +&{) D4(3)
_ *4
. & D4 , & # %$ , 2. J & + D4(3) 0 2 ;1 01 @;3 2 ;1A : 0 ;1 2
150
.
-! x1 x2 x3 x4 | D4 HX i (0 6 i 6 3) | , (4). 5 x1(HX 1 ) = 1 x1(HX 2 ) = 0 x1 (HX 0) = ;2 x2(HX 1 ) = ;1 x2(HX 2 ) = 1 x2 (HX 0) = ;1 x3(HX 1 ) = 2 x3(HX 2 ) = ;1 x3 (HX 0) = ;1 x4(HX 1 ) = 0 x4(HX 2 ) = 0 x4 (HX 0) = 0: -! i(x21 x22 x23 x24), i 6 i 6 3, 4 = x1x2x3 x4 | $" R t]WD4 . < , & U # %$ G x4 = 0 x1 = x2 + x3 : 5 $ , ,
(U G)4 = 0
(U G)1 = (U G)(2 (x2 + x3)2 ; x2x3])
(U G)2 = 41 ( (U G)1)2
(U G)3 = (U G)((x2 + x3)2 x22x23) 4 & # %$ D4 ! D4 . - 8, "" .
16. 1 % $ D4 % )
H (D4 =U) = (H (BU )=h (U G)1 (U G)3i) ^(z2 z4 ): '(. - & U D4 .
1. g = A1 t1 . t =; L(HX 1 ) L(HX 2 ; c21HX 1), c21a11 = a12 ) c21 = ; 12 , t = L(HX 1 ) L HX 2 + 21 HX 1 . y1 , ;y1 | A1 , y2 | t1 , x2 ! ;y1 + 12 y2 x2 + x3 ! y1 + 12 y2 1 x3 ! 2y1 x2x3 ! 2y1 ;y1 + 2 y2
151
(A1 T 1 D4)1 = 3y12 + 41 y22 = J1 2
(A1 T 1 D4 )3 = 4y12 14 y22 ; y12 = J2 H (D4 =A1 T 1 ) = (R y2] S y12 ]=hJ1 J2i) ^(z2 z4): 2. g = A2 . t = L(HX 1 HX 0). y1 , y2 , y3 | t , x2 ! y3 ; y1 x2 + x3 ! y3 ; y2 x3 ! y1 ; y2 x2x3 ! (y3 ; y1 )(y1 ; y2 )
(A2 D4)1 = 2((y3 ; y2 )2 ; (y3 ; y1 )(y1 ; y2 )) = J1
(A2 D4)3 = (y3 ; y2 )2 (y3 ; y1 )2 (y1 ; y2 )2 = J2 H (D4 =A2) = (S y1 y2 y3 ]=hJ1 J2i) ^(z2 z4): 3. g = g2. y1 y2 y3 | g2 , x2 ! ;y3 x3 ! ;y2
(G2 D4 )1 = 2(y12 + y22 + y33 ) = P1
(G2 D4 )3 = y12 y22 y33 = P2 H (D4 =G2) = ^(z2 z4 ): . & # %$ D4 g2 ( ! G(1) 2 ). 5. g = E6 . 7 +&{) E6(2)
ks
" & + : 0 2 ;1 0 0 01 B ;1 2 ;1 0 0C B C: B 0 ;1 2 ;1 0C B @ 0 0 ;2 2 ;1CA 0 0 0 ;1 2 . & E6, & # %$ , 4.
152
.
-! x1 : : : x6 | E6 Hi, 0 6 i 6 4, | , (5). F, (x1 + x6)(HX i ) = (x2 + x5 )(HX i) = (x3 + x4)(HX i ) i = 0X4 , ! t & %$ Q x1 + x6 = x2 + x5 = x3 + x4 : 5 $ , % ai, bi cij , 1, t " " : ai = xi + x1 + x6 bi = xi ; 2(x1 + x6) bi = ;a7;i cij = ;ai ; bj = ;ai + a7;j : -4 $ (U E6) " $ :
(U E6)(Iki ) =
6 X
i=1
(U E6)(ai )ki +
5 X
j =2
(U E6)(c1j )ki +
X
j =34
(U E6)(c2j )ki :
- 4 ,
(U E6)(I5 ) = (U E6)(I9 ) = 0 4 & # %$ E6 ! E6 . 5 $ , 8 ,
17. 1 % $ A2n % )
H (E6 =U) = = (H (BU )=h (U E6)I2 (U E6)I6 (U E6)I8 (U E6)I12 i) ^(z5 z9 ): '(. - & U E6 . 1. dim( (g )) = 3.
g = t3 A1 C i1 = 1, j = 2 3 4C t = t3 L(HX 1 )C t3 = L(HX j ; cj 1a11HX 1), cj 1a11 = a1j .
t = L HX 2 + 12 HX 1 HX 3 HX 4 L(HX 1):
153
1 2 3 | t , y1 , ;y1 | A1 . x1 ! 12 1 ; 13 3 + y1 x2 ! 12 1 ; 31 3 ; y1 x3 ! ;1 + 2 ; 13 3 x4 ! 1 ; 2 + 23 3 x5 ! ; 12 1 + 23 3 + y1 x6 ! ; 12 1 + 32 3 ; y1 x1 + x6 ! 13 3 a1 ! 21 1 c12 ! ;1 + 3 a2 ! 12 1 ; y1 c13 ! 12 1 ; 2 + 3 ; y1 a3 ! ;1 + 2 c14 ! ; 32 1 + 2 ; y1 a4 ! 1 ; 2 + 3 c15 ! ;2y1 1 a5 ! ; 2 1 + 3 + y1 c23 ! 12 1 ; 2 + 3 + y1 a6 ! ; 12 1 + 3 ; y1 c24 ! ; 32 1 + 2 + y1 : H (E6 =T 3 SU(2)) = = (R 1 2 3 ] R y12]=hq2 q6 q8 q12i) ^(z5 z9 ): 2. dim( (g )) = 2. g = t2 C2 C i1 = 2, i2 = 3, j = 1 4C t = t2 L(HX 2 HX 3)C P2 P2 t2 = L HX j ; cjk HX ik , cjk ailik = ailj , 1 6 l 6 2. k=1
k=1
t = L HX 1 + HX 2 + HX 3 HX 4 + 21 HX 2 + HX 3 L(HX 2 HX 3):
1 2 | t2 , y1 , y2 | C2. x1 ! 1 ; 13 2 x2 ! 61 2 + y1 x3 ! 16 2 + y2 x4 ! 16 2 ; y2 x5 ! 16 2 ; y1 x6 ! ;1 + 23 2 x1 + x6 ! 13 2
154
.
a1 ! 1 a2 ! 21 2 + y1 a3 ! 12 2 + y2 a4 ! 21 2 ; y2 a5 ! 21 2 + 3 ; y1 a6 ! ;1 + 2
c12 ! ;1 + 21 2 ; y1 c13 ! ;1 ; 2 + 21 2 ; y2 c14 ! ;1 + 12 2 + y2 c15 ! ;1 + 12 2 + y1 c23 ! ;y1 ; y2 c24 ! ;y1 + y2 :
H (E6 =T 2 SO(4)) = = (R 1 2 ] S(y12 y22 )=hq2 q6 q8 q12i) ^(z5 z9 ): 3. dim( (g )) = 1. g = t 1 A3 C i1 = 0, i2 = 1, i3 = 2, j = 3C t = t1 L(HX 0 HX 1 HX 2)C P3 P3 t1 = L HX 3 ; c3k HX ik , cjk ail ik = ail 3, 1 6 l 6 2. k=1
k=1 t = L(HX 4 ) L(HX 0 HX 1 HX 2):
1 | t1 , y1 , y2 , y3 , y4 | A3 , " HX 0, HX 1, HX 2. x2 ! ; 31 1 ; 13 y1 + y3 x1 ! ; 13 1 ; 13 y1 + y2 x3 ! ; 31 1 ; 43 y1 ; y2 ; y3 x4 ! 32 1 + 32 y1 + y2 + y3 x5 ! 23 1 ; 31 y1 ; y3 x6 ! 23 1 ; 31 y1 ; y2 x1 + x6 ! 13 1 ; 32 y1 a1 ! ;y1 + y2 c12 ! 1 ; y2 ; y3 a2 ! ;y1 + y3 c13 ! 1 + y1 + y3 a3 ! ;y1 + y4 c14 ! ;y2 + y4 a4 ! 1 + y2 + y3 c15 ! ;y2 + y3 a5 ! 1 ; y1 ; y3 c23 ! 1 + y1 + y2 a6 ! 1 ; y1 ; y2 c24 ! ;y3 + y4 : H (E6 =T 1 SU(4)) = = (R 1] S(y1 y2 y3 y4)=hq2 q6 q8 q12i) ^(z5 z9 ):
155
4. dim( (g )) = 0. g = A1 B3 C i1 = 0, i2 = 2, i3 = 3, i4 = 4. t = L(HX 0 HX 2 HX 3 HX 4): y4 | A1 , y1 , y2 , y3 | B3 , " HX 2, HX 3, HX 4. 5 y3 (HX 4) = 0 y2 (HX 4) = ;1 y1 (HX 4) = 1 y3 (HX 3) = ;1 y2 (HX 3) = 1 y1 (HX 3) = 0 y3 (HX 2) = 2 y2 (HX 2) = 0 y1 (HX 3) = 0 x2 ! 16 y1 + 12 y2 + 21 y3 ; 31 y4 x1 ! ; 13 y1 ; 34 y4 x3 ! 16 y1 + 12 y2 ; 12 y3 ; 13 y4 x4 ! 16 y1 ; 12 y2 + 21 y3 ; 31 y4 x5 ! 16 y1 ; 12 y2 ; 12 y3 ; 13 y4 x6 ! 23 y1 + 23 y4 x1 + x6 ! 31 y1 ; 23 y4 a1 ! ;2y4 c12 ! 21 (y1 ; y2 ; y3 ) + y4 a2 ! 21 (y1 + y2 + y3 ) ; y4 c13 ! 12 (y1 ; y2 + y3 ) + y4 a3 ! 21 (y1 + y2 ; y3 ) ; y4 c14 ! 21 (y1 + y2 ; y3 ) + y4 a4 ! 12 (y1 ; y2 + y3 ) ; y4 c15 ! 21 (y1 + y2 + y3 ) + y4 a5 ! 12 (y1 ; y2 ; y3 ) ; y4 c23 ! ;y2 a6 ! y1 c24 ! ;y3 : H (E6 =SU(2) SO(7)) = = (R y12] S(y12 y32 y42 )=hq2 q6 q8 q12i) ^(z5 z9 ): g = F4C i1 = 1, i2 = 2, i3 = 3, i4 = 4. t = L(HX 1 HX 2 HX 3 HX 4): y1 , y2 , y3 , y4 | F4, " HX 1, HX 2 , HX 3, HX 4 . 5 y4 (HX 1) = ;1 y3 (HX 1) = ;1 y2 (HX 1) = ;1 y1 (HX 1) = 1 y4 (HX 2) = 2 y3 (HX 2) = 0 y2 (HX 2) = 0 y1 (HX 2) = 0 y4 (HX 3) = 0 y3 (HX 3) = 1 y2 (HX 3) = 0 y1 (HX 3) = 0 y4 (HX 4) = 0 y3 (HX 4) = ;1 y2 (HX 4) = 1 y1 (HX 4) = 0
156
.
x1 ! 32 y1 ; 13 y2 x2 ! 16 y1 + 16 y2 + 21 y3 + 21 y4 x3 ! 16 y1 + 16 y2 + 12 y3 ; 12 y4 x4 ! 16 y1 + 16 y2 ; 21 y3 + 21 y4 x5 ! 16 y1 + 16 y2 ; 12 y3 ; 12 y4 x6 ! ; 13 y1 + 32 y2 x1 + x6 ! 13 (y1 + y2 ) a1 ! y1 c12 ! 21 (;y1 + y2 ; y3 ; y4 ) a2 ! 21 (y1 + y2 + y3 ; y4 ) c13 ! 12 (;y1 + y2 ; y3 + y4 ) a3 ! 21 (y1 + y2 + y3 ; y4 ) c14 ! 12 (;y1 + y2 + y3 ; y4 ) a4 ! 21 (y1 + y2 ; y3 + y4 ) c15 ! 12 (;y1 + y2 + y3 + y4 ) a5 ! 21 (y1 + y2 ; y3 ; y4 ) c23 ! ;y3 a6 ! y2 c24 ! y4 : 1] $, qk = (Ik ) " $" H (BF4 ), 4 H (E6=F4) = ^(z5 z9 ): . E6, & # %$ , F4 . ) C4 4 $ ! 1], ! 4 , ! # ) F4 +&{) E6. 3 , $ , -
, $
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1] Takeuci M. On Pontrjagin classes of compact symmetric spaces // J. Fac. Sci. Univ. Tokyo Sec. I. | 1962. | Vol. 9. | P. 313{328. 2] Helgason S. Di erential Geometry, Lie groups and Symmetric spaces. | New York, London: Acad. Press, 1978. 3] . !" ##!!" $#% !#% &!&!" $#% !#% $ %!" '$$ ( // ) ##!!" $#% !#% $ +! . | ,.: -(, 1958. | /. 163{246. 4] 1 # 2 . 334!!" # % 5# $#% !#% . | ,.: , , 1984.
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5] 6 !3 7. ., ! 4 . (. / ! $ '$$ ( 3 5# '$$ . | ,.: 8)//, 1995. 6] ! 4 . (. 9$ % !: % !" '$$ $3 : ! 2. | ,., 1995. 7] ; ! 1' !. < !" <' ! $ %!" &!&!" ! " $#% !#% # !$ & 2 #% = ! !2 '$$2 // 9. #. %. %!:. ! . | 1968. | 6"$. 14. | /. 33{93. 8] 1 = 6. #!5!!" 3" ( . | ,.: , , 1993. 9] 1 = 6. %> :" !5! $ & $'$#%" 3 ( // ?'!=. ! : $ . | 1969. | 9. 3, "$. 3. | /. 94{96. 10] Coxeter H. S. M. The product of the generators of a Bnite group generated by reEections // Duke Math. J. | 1951. | Vol. 18. | P. 765{782. ' ( 1998 .
( = 3, D = 3) . .
, . e-mail: [email protected]
519.172.2+519.173+519.177
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Abstract S. A. Tishchenko, Maximum size of a planar graph (4 = 3, D = 3), Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 159{171.
The problem of maximumsize of a graph of diameter 3 and maximumdegree 3 as a function of its Euler characteristics is studied. The negative solution of an Erd!os problem is obtained. A new approach to such problems is proposed which consists in counting the paths between di:erent pairs of vertices in a graph.
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1] M. Fellows. Three for money: The degree/diameter problem. | http://www.c3.lanl. gov/mega-math/workbk/graph/grthree.html. 2] J. C. Bermond, C. Delorme, G. Farhi. Large graphs with given degree and diameter II // J. Combin. Theory. | 1984. | B 36. | P. 32{48. 3] I. Alegre, M. A. Fiol, J. L. A. Yebra. Some large graphs with given degree and diameter // J. Graph Theory. | 1986. | V. 10. | P. 219{224. 4] F. R. K. Chung. Diameters of graphs: old problems and new results // Congressus Numerantium. | 1987. | V. 60. | P. 295{317. 5] P. Hell, K. Seyarth. Largest planar graphs of diameter two and xed maximum degree // Discrete Math. | 1993. | V. 111. | P. 313{332. 6] M. Fellows, P. Hell, K. Seyarth. Large planar graphs with given diameter and maximum degree // Discrete Appl. Math. | 1995. | V. 61. | P. 133{153. 7] F. Goebel, W. Kern. Planar regular graphs with prescribed diameter. | Univ. of Twente (The Netherlands) Applied Math. Memorandum. No. 1183, December 1993. 8] M. Fellows, P. Hell, K. Seyarth. Constructions of large planar networks with given degree and diameter // Networks. | 1998. | V. 32. | P. 275{281. 9] R. W. Pratt. The complete catalog of 3-regular, diameter-3 planar graphs. | http: //www.unc.edu/~rpratt/graphtheory.html. ( ) 1999 .
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Abstract
S. M. Tuleshev, Spaces of immersions of elementary bre bundles, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 173{197.
In the paper we consider immersions of elementary 5bre objects. For 5bre disk, i. e. a trivial 5bre bundle of type Dk Dl ! Dk , we give a de5nition of 5brewise immersion and for 5bre boundary spheres (5bre bundles @Dk Dl ! @Dk , Dk @Dl ! Dk and @Dk @Dl ! @Dk ) we de5ne the notion of framed 5brewise immersions. It is proved that the natural maps from the space of immersions of 5bre disk into spaces of framed immersions of 5bre boundary spheres satisfy the axiom of the covering homotopy, i. e. are Serre bundles. This result is an initial step in solving the problem on the homotopy description of space of immersions of one 5bre manifold into another 5bre manifold.
1] . ! , # E # $ Dk Rn # B - ! ( ( && 6778 9 96{01{00287{ .
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1. (( ( $ 0 $ 2 (Dk Dl , @Dk Dl , Dk @Dl @Dk @Dl ) Rm+n = Rm Rn,
175
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#-
((F 0 f 0 ) (F 00 f 00)) = maxf0 (F 0(X) F 00(X)) 0 (F 0 (V ) F 00(V )) j X 2 Dk Dl V 2 TX (Dk Dl ) kV k = 1g (F 0 f 0 ) (F 00 f 00) 2 E , 0 | Rm+n, Rm+n (( # TX (Dk Dl ) ((( # Dk Dl X. 2. 7 # ( @Dk Dl ! @Dk Rm Rn ! Rm G = ((G g) (G~ g~)), G g ((-( 1- ( @Dk Dl Rm+n @Dk Rm , ( $ G Rm+n @Dk Dl ;;;;! ? ? ? ? y y g @Dk ;;;;! Rm ~ ), G~ ((( C 1 - @Dk Dl Rm+n n f0g, G(Y Y 2 @Dk Dl , # Y
176
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~(G 0 G 00) = maxf0 (G0 (Y ) G00(Y )) 0 (G0 (V~ ) G00(V~ )) 0 (G~ 0(Y ) G~ 00(Y )) j Y 2 @Dk Dl V~ 2 TY (@Dk Dl ) kV~ k = 1g 0 | , 4. 3. 7 # , ($ ( Dk @Dl ! Dk RmRn ! Rm, B H f ((-( C 1 - ( Dk @Dl H = ((H f) H), m + n k m R D R , Dk @Dl ? ? y
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177
= : E ! B - . 1( (F f) 2 E ~ g~)), G g ((-( # (F f) = ((G g) (G k l ~ t y) = rsF (1 x< t y), g~(x) = rsf(1 x) ( F @D D f @Dk , G(x< (# # 4 rs rt - '' s t ). C ^ : E ! C . 3# ( B H ((( (F f) 2 E ^ ( ' ^ (F f) = ((H f) H), k l B F D @D , H(s x< y) = rtF(s x< 1 y). E #, ^ . , , ( @Dk @Dl Rm+n. 4. 7 ( # $ 2- @Dk @Dl ! @Dk Rm Rn ! Rm ~ g~) G), B G g ((-( 1 - ( @Dk @Dl Rm+n G = ((G g) (G k m @D R , ( $ G Rm+n @Dk @Dl ;;;;! ? ? ? ? y y
g @Dk ;;;;! Rm 1 k l ~ G~ ((( C - @D @D Rm+n n f0g, ( G(Z), Z 2 @Dk @Dl , # Z G (=2 G TZ (@Dk @Dl )), g~ ((( C 1 - @Dk Rmnf0g, g~(y), y 2 @Dk , # y g (=2 g Ty (@Dk )), G~ @Dk @Dl ;;;;! Rm+n n f0g Rm+n ?? ? ? y y
g~ @Dk ;;;;! Rm n f0g Rm , GB ((( C 1- @Dk @Dl Rm+n nf0g, B G(Z), Z 2 @Dk @Dl , # Z G (=2 G TZ (@Dk @Dl )) G Rm+n n f0g Rm+n ;! Rm @Dk @Dl ;! @Dk @Dl 0 2 Rm. , ( , ~ G(Z), B G(Z) Z 2 @Dk @Dl , . ln A Dkm = D $ $ ( ( # $ 2- @Dk @Dl ! @Dk Rm Rn ! Rm. . - D #- - : ( G 0 = ((G0 g0) (G~ 0 g~0) GB 0), G 00 = ((G00 g00) (G~ 00 g~00) GB 00) # ^(G 0 G 00) = maxf0 (G0 (Z) G00(Z)) 0 (G0 (V^ ) G00(V^ )) 0 (G~ 0(Z) G~ 00(Z)) 0 (GB 0 (Z) GB 00(Z)) j Z 2 @Dk @Dl V^ 2 TZ (@Dk @Dl ) kV^ k = 1g:
178
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. (i) 1 # , ^ | (, # . 3# ( 0 P Gv : P ! B Hv : P ! C 0 6 v 6 1: ,$ # : ( p P Gv(p) = ((Gv (p) gv(p)) (G~ v(p) g~v(p))) Hv(p) = ((Hv(p) fv (p)) HB v(p)): H , # G0 H0 ( F F^ , . . - ( ^ f(p)) ^ F F^ : P ! E F (p) = (F(p) f(p)) F^(p) = (F(p) p 2 P ^ = f0 (p) p 2 P): G0 = F H0 = ^ F^ (f(p) > - Fv F^v : P ! E 0 6 v 6 1 F0 = F F^0 = F^ ( $ ( p P - Fv(p) = (Fv(p) fv (p)) F^v (p) = (F^v (p) f^v(p)) e Fv = Gv ^ F^v = Hv : ,$ # ( , 0 #, P | .
179
3# "1 (v p< x< t y) # ( G~ v (p)(x< t y) # Gv (p)] T(x ty)(@Dk Dl ), # "1 = minf"1 (v p< x< t y) j 0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl g "2 = minfkrV Gv (p)(x< t y)k j 0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl V 2 T(x ty) (@Dk Dl ) kV k = 1g " = (1=10) minf"1 "2 1g. rV - # ( V . C , # "^1 (v p< s x< y) (- HB v (p)(s x< y) Hv (p)] T(sx y) (Dk @Dl ) "^1 = minf"^1 (v p< s x< y) j 0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl g "^2 = minfkrV Hv (p)(s x< y)k j 0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl VB 2 T(sx y) (Dk @Dl ) kVB k = 1g "^ = (1=10) minf"^1 "^2 1g. 1( v v0 , $ # v ; v0 ( ( v ; v0 ), p 2 P Jvv0 (p)(x) x 2 @Dk Rm ( Kvv0 (p)(x< t y) Lvv0 (p)(s x< y) (x< t y) 2 @Dk Dl (s x< y) 2 Dk @Dl Rm+n - . 3# Mvv0 (p)(x) # ( # Rm, ( ( g~v0 (p)(x) g~v (p)(x), , vv0 (p)(x) | ( , ( Mvv0 (p)(x) ( (~gv0 (p)(x) g~v (p)(x)) # v ; v0 # , 0 6 vv0 (p)(x) < ). 3 0 M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) Rm+n, ( (~gv0 (p)(x) 0) (~gv(p)(x) 0) HB v0 (p)(s x< y) HB v (p)(s x< y) ( -), vv0 (p)(x< t y) (= vv0 (p)(x)) vv0 (p)(s x< y) | - ( , 0 6 vv0 (p)(s x< y) < ). A Jvv0 (p): @Dk ! SO(m R) Kvv0 (p): @Dk Dl ! SO(m + n R) Lvv0 (p): Dk @Dl ! SO(m + n R) (, - x, (x< t y) (s x< y) (- ( $ m ( ( m + n ( # , - Mvv0 (p)(x), M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) vv0 (p)(x), vv0 (p)(x< t y) vv0 (p)(s x< y) , | g~v0 (p)(x) g~v (p)(x),
180
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| (~gv0 (p)(x) 0) (~gv (p)(x) 0), | # HB v0 (p)(s x< y) HB v (p)(s x< y). O Mvv0 (p)(x) , Jvv0 (p)(x) # < , M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y). > , ( Jvv0 (p): @Dk ! GL(m R) Kvv0 (p): @Dk Dl ! GL(m + n R) Lvv0 (p): Dk @Dl ! GL(m + n R) ' k J (p)(x) Jvv0 (p)(x) = kkg~g~v (p)(x) x 2 @Dk < v0 (p)(x)k vv0 k Kvv (p)(x< t y) Kvv0 (p)(x< t y) = kkg~g~v (p)(x) (x< t y) 2 @Dk Dl < 0 v0 (p)(x)k B v (p)(s x< y)k k l Lvv0 (p)(s x< y) = kkHH B v0 (p)(s x< y)k Lvv0 (p)(s x< y) (s x< y) 2 D @D : = , ( Jvv0 (p) Kvv0 (p) Lvv0 (p) ((-( C 1- # x (x< t y) (s x< y) . H , # v ; v0 , Pvv0 (p)(x< t y) p 2 P (x< t y) 2 @Dk Dl Rm+n. 3# M^ vv0 (p)(x< t y) # ( # Rm+n, ( ( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y), , vv0 (p)(x< t y) | ( , 0 6 vv0 (p)(x< t y) < ). = Pvv0 (p): @Dk Dl ! SO(m + n R) , (x< t y) @Dk Dl (- Rm+n, # M^ vv0 (p)(x< t y) vv0 (p)(x< t y), ( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y) ( M^ vv0 (p)(x< t y) , Pvv0 (p)(x< t y) # ). = Pvv0 (p): @Dk Dl ! GL(m + n R) ' kG~v (p)(x< t y)k Pvv0 (p)(x< t y) = kKvv0 (p)(x< t y)G~ v0 (p)(x< t y)k Pvv0 (p)(x< t y) (x< t y) 2 @Dk Dl . = , 0 ((( C 1 - # (x< t y).
181
E #, Pvv0 (p)(x< t y)Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) = G~ v (p)(x< t y) Lvv0 (p)(s x< y)HB v0 (p)(s x< y) = HB v (p)(s x< y)
(1) (10 )
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= Q ;! S m+n;1 Rm+n ;! Rm $ Q 0 2 Rm. ) S C 1 - $!, = C 1 - $! . !0 . 1 # 10 . 3# Vmk++ln!kk+1l+1 | ( T'( Vm+n k+l+1 # $ $ (k +l +1)- Rm+n, ( $ (R1 : : : Rk+l Rk+l+1 ) $ p(R1 ) : : : p(Rk+l), p: Rm+n ! Rm | (, k $ p(Rk+l+1 ) = 0. = !0 p~: Vmk++ln!kk+1l+1 ! Gkm++l!nkk+l -, ( - (R1 : : : Rk+l Rk+l+1) # hR1 : : : Rk+l i, (- R1 : : : Rk+l. . !0 k Vmk++ln!kk+1l+1 , p~, ((( ( ) Gkm++l! n k+l !0 Vk+l k+l S n;l;1 . 1( ( S (Vmk++ln!kk+1l+1 )
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# k Q ;;;;! Gkm++l! n k+l . S ( , S (Vmk++ln!kk+1l!+10) # , 0 !0 s: Q ! S (Vmk++ln!kk+1l+1 ): A !0 p^: Vmk++ln!kk+1l+1 ! Sm+n;1 -, (- - (R1 : : : Rk+l Rk+l+1) Rk+l+1 . = = : Q ! S m+n;1 = = p^ S! s. . =(q), q 2 Q, # S(q), p(=(q)) = 0. 2 = k S: I P Dk @Dl ! Gkm++l! n k+l (( (v p< s x< y) # S(v p< s x< y), (- HB v (p)(s x< y) #- # Hv (p)] (T(sx y) (Dk @Dl )). = S ( , (H0(p) HB 0(p)) ( F^ (p) Dk Dl Rm+n. 3 0 # 10. = : I P Dk @Dl ! Sm+n;1 C 1 - (s x< y) 2 Dk @Dl , =(v p< s x< y) S(v p< s x< y) = I P Dk @Dl ;! S m+n;1 Rm+n ;! Rm I P Dk @Dl 0 2 Rm. > 0 , ( v v0 , $ jv ; v0 j 6 , $ p 2 P (x< t y) 2 @Dk Dl V 2 T(x ty) (@Dk Dl ), kV k = 1, - (. G g~v0 (p)(x) g~v (p)(x) #4 (0 # ( Jvv0 (p)(x) (3) Kvv0 (p)(x< t y)). G Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y) #4 ( # ( - (4) ( Pvv0 (p)(x< t y)).
184
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kKvv (p)(x< t y)G~v (p)(x< t y) ; G~v (p)(x< t y)k < 20" : kG~ v(p)(x< t y) ; Kvv (p)(x< t y)G~v (p)(x< t y)k < " min 1 kKuu (p0 )(x0< t0 y0 )G~ u (p0 )(x0< t0 y0 )k < 20 kKuu (p0)(x0< t0 y0)k 0 6 u u0 6 1 p0 2 P (x0 < t0 y0 ) 2 @Dk Dl : k(~gv (p)(x) 0) ; (~gv (p)(x) 0)k < " minfk(~g (p0 )(x0) 0)k j 0 6 u 6 1 p0 2 P x0 2 @Dk g: < 20 u krV Gv (p)(x< t y) ; rV Gv (p)(x< t y)k < 10" : kGv(p)(x< t y) ; Gv (p)(x< t y)k < < ("=100)(1=(maxfkrV <(v0 p0< x0< t0 y0 )k j 0 6 v0 6 1 p0 2 P (x0< t0 y0 ) 2 @Dk Dl V 0 2 T(x t y ) (@Dk Dl ) kV 0 k = 1 )) 0
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(9) ( , 0 (). = , , Gv (p) F (p). 1 , ^ > 0 , ( v v0, $ jv ; v0 j 6 ^, $ p 2 P , (s x< y) 2 Dk @Dl VB 2 T(sx y) (Dk @Dl ), kVB k = 1, (# - (. G HB v0 (p)(s x< y) HB v (p)(s x< y) #4 ( # ( ( (30 ) Lvv0 (p)(s x< y)). kHB v(p)(s x< y) ; HB v0 (p)(s x< y)k < 10"^ minf1 kHB u(p0)(s0 x0< y0)k j 0 6 u 6 1 p0 2 P (s0 x0< y0 ) 2 Dk @Dl g: (40 ) krV Hv(p)(s x< y) ; rV Hv0 (p)(s x< y)k < 10"^ : (50 ) kHv(p)(s x< y) ; Hv0 (p)(s x< y)k < < (^"=100)(1=(maxfkrV =(v0 p0< s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P (s0 x0< y0) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g)) (60 ) ( , (). . ^ , Hv (p) F^ (p). , (5), (6) (7) , ( v 6 kG~ v(p)(x< t y) ; G~ 0(p)(x< t y)k < 10" (10) 0
0
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kKv0(p)(x< t y) ; IdRm n k k(~g0(p)(x) 0)k = " k(~g (p)(x) 0)k = k(~gv (p)(x) 0) ; (~g0(p)(x) 0)k < 20 0 " kKv0(p)(x< t y) ; IdRm n k < 20 kPv0(p)(x< t y) ; IdRm n k kKv0(p)(x< t y)G~ 0(p)(x< t y)k = = kG~ v (p)(x< t y) ; Kv0(p)(x< t y)G~ 0(p)(x< t y)k < " kKv0(p)(x< t y)G~ 0(p)(x< t y)k < 20 kKv0(p)(x< t y)k 1 kPv0(p)(x< t y) ; IdRm n k < 20" kKv0(p)(x< t y)k kPv0(p)(x< t y)Kv0(p)(x< t y) ; Kv0(p)(x< t y)k < 20" :
185
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3 0 (11) kPv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n k < 10" : , (40 ) , ( v 6 ^ kLv0(p)(s x< y) ; Id m+n k kHB 0(p)(s x< y)k = "^ kHB (p)(s x< y)k = kHB v (p)(s x< y) ; HB 0(p)(s x< y)k < 10 0 kLv0(p)(s x< y) ; Id m+n k < 10"^ : (70 ) - ( ( 2]. 2. ( s0, s0 < 1, ! % $ p 2 P s 2 s0 1], (x< t y) 2 @Dk Dl F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k: s0 ; 1 3 1( ( # # 0 . . 1 # 2. , (
F (p)(s x< t y) ; F(p)(1 x< t y) rsF(p)(1 x< t y) = slim !1 s;1 0 P , # s0 , s0 < 1, ( s 2 s0 1] F (p)(s x< t y) ; F(p)(1 x< t y) ; rsF (p)(1 x< t y) 6 1 krsF (p)(1 x< t y)k: s;1 3 3 # F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k: s;1 3
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1( s0 6 s 6 1 F(p)(s x< t y) ; F(p)(1 x< t y) F(p)(s x< t y) ; F (p)(1 x< t y) 6 : s0 ; 1 s;1 G $ $ . 2 # 4 ( - 0 . 2 . ( t0, t0 < 1, ! % $ p 2 P t 2 t0 1], (s x< y) 2 Dk @Dl ^ ^ x< t y) ; F(p)(s x< 1 y) 6 4 kr F(p)(s F(p)(s x< 1 y)k: 3 t ^ t0 ; 1 = (# 2, s0 , 1=2 < s0 < 1, , ( $ v 6 , p 2 P, s 2 s0 1], (x< t y) 2 @Dk Dl , V 2 T(x ty) (@Dk Dl ), kV k = 1, (# - (: F(p)(s x< t y) ; F (p)(1 x< t y) < 2krsF(p)(1 x< t y)k< (12) s0 ; 1 krsF (p)(s x< t y) ; rsF (p)(1 x< t y)k < 10" < (13) (14) krV F (p)(s x< t y) ; rV F(p)(1 x< t y)k < 10" < kF(p)(s x< t y) ; F (p)(1 x< t y)k < < ("=10)(1=(maxfkrV (Pv 0(p0 )(x0 < t0 y0 )Kv 0(p0 )(x0< t0 y0 ))k j 0 6 v0 6 1 p0 2 P (x0< t0 y0 ) 2 @Dk Dl V 0 2 T(x t y ) (@Dk Dl ) kV 0k = 1g)) (15) ( (15) , 0 (). = , # s0 . 3# s1 = s0 + (1=3)(1 ; s0 ). C , #( 20, # t0, 1=2 < t0 < 1, , ( $ v 6 ^, p 2 P , t 2 t0 1], (s x< y) 2 Dk @Dl , VB 2 T(sx y)(Dk @Dl ), kVB k =1, ( F(p)(s ^ ^ x< t y) ; F(p)(s x< 1 y) < 2kr F(p)(s x< 1 y)k< (80 ) t^ t0 ; 1 "^ < ^ ^ krtF(p)(s x< t y) ; rtF(p)(s x< 1 y)k < 10 (90 ) "^ < ^ ^ krV F(p)(s x< t y) ; rV F(p)(s x< 1 y)k < 10 (100) ^ ^ kF(p)(s x< t y) ; F(p)(s x< 1 y)k < < (^"=10)(1=(maxfkrV Lv 0 (p0)(s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P (s0 x0< y0 ) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g)) (110) ( , (). 0
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187
= , t0 . 3# t1 = t0 + (1=3)(1 ; t0 ). 1 # C 1 - ' (s), (t), (s) (t) 0 1] , (# - (: (s) = 0 0 6 s 6 s1 < (16) (t) = 0 0 6 t 6 t1< (120) (1) = 1 0 (1) = 0< (17) (1) = 1 0 (1) = 0< (130) j(s)j 6 1 j 0(s)j < 1 ;2 s0 < (18) j(t)j 6 1 j0(t)j < 1 ;2 t0 < (140) (s) = 0 0 6 s 6 s0 < (19) (t) = 0 0 6 t 6 t0< (150) (1) = 0 (1) = 0< (20) (1) = 0(1) = 0< (160) j0(s)j > 10j 0(s)j s1 6 s 6 1< (21) 0 0 j (t)j > 10j (t)j t1 6 t 6 1< (170) j(s)j 6 20< (22) j(t)j 6 20: (180) ( M(v) = maxfkGv (p)(x< t y) ; G0(p)(x< t y)k j p 2 P (x< t y) 2 @Dk Dl g N(v) = maxfkHv (p)(s x< y) ; H0(p)(s x< y) j p 2 P (s x< y) 2 Dk @Dl g: , - Fv (p) = (Fv (p) fv (p)) F^v (p) = = (F^v (p) f^v (p)), p 2 P , ( v 6 v 6 ^ - : Fv (p)(s x< t y) = Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (F (p)(s x< t y) ; F(p)(1 x< t y)) + (s)(Gv (p)(x< t y) ; G0(p)(x< t y)) + + (s)M(v)<(v p< x< t y) + F(p)(1 x< t y)< (23) fv (p)(s x) = Id m +(s)(Jv0 (p)(x) ; Id m)](f(p)(s x) ; f(p)(1 x)) + + (s)(gv (p)(x) ; g0(p)(x)) + (s)M(v)}(v p< x) + f(p)(1 x)< (24) F^v (p)(s x< t y) = Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )] ^ (F^(p)(s x< t y) ; F(p)(s x< 1 y)) + (t)(Hv (p)(s x< y) ; H0(p)(s x< y)) + ^ + (t)N(v)=(v p< s x< y) + F(p)(s x< 1 y)< (190) f^v (p)(s x) = fv (p)(s x): (200)
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f v ( p) Dk ;;;;! Rm 4 ( - : rsFv(p)(s x< t y) = 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y)) + + Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] rsF(p)(s x< t y) + 0(s)(Gv (p)(x< t y) ; G0(p)(x< t y)) + + 0 (s)M(v)<(v p< x< t y)< rtF^v (p)(s x< t y) = 0(t)Lv0(p)(s x< y) ; Id m+n ] ^ (F^(p)(s x< t y) ; F(p)(s x< 1 y)) + + Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )]rtF^ (p)(s x< t y) + + 0 (t)(Hv (p)(s x< y) ; H0(p)(s x< y)) + 0(t)N(v)=(v p< s x< y): 1( V 2 T(x ty)(@Dk Dl ) rV Fv(p)(s x< t y) = (s)rV (Pv0(p)(x< t y)Kv0(p)(x< t y)) (F(p)(s x< t y) ; F (p)(1 x< t y)) + + Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (rV F (p)(s x< t y) ; rV F(p)(1 x< t y)) + + (s)(rV Gv (p)(x< t y) ; rV G0(p)(x< t y)) + + (s)M(v)rV <(v p< x< t y) + rV F (p)(1 x< t y): 1( VB 2 T(sx y) (Dk @Dl ) rV F^v (p)(s x< t y) = (t)rV Lv0(p)(s x< y) ^ (F^(p)(s x< t y) ; F(p)(s x< 1 y)) + + Id m+n +(t)(Lv0(p)(s x< y) ; Id m+n )] ^ (rV F(p)(s x< t y) ; rV F^ (p)(s x< 1 y)) + + (t)(rV Hv (p)(s x< y) ; rV H0(p)(s x< y)) + ^ + (t)N(v)rV =(v p< s x< y) + rV F(p)(s x< 1 y):
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190
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rsFv (p)(s x< t y) = G~ v(p)(x< t y) + B + XU + X0U 0: . 1 # 4. , (26) , rsFv (p)(s x< t y) = 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y)) + rsF(p)(1 x< t y) + + (s)Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n ]rsF(p)(1 x< t y) + + Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (rsF(p)(s x< t y) ; rsF(p)(1 x< t y)) + + 0 (s)M~ (v p< x< t y)U 0 + 0 (s)M(v)U ~ p< x< t y) = kGv (p)(x< t y) ; G0 (p)(x< t y)k, U 0 = (Gv (p)(x< t y) ; M(v ~ p< x< t y), U = <(v p< x< t y). V , 0 6 M(v ~ p< x< t y) 6 ; G0(p)(x< t y))=M(v 6 M(v). , (18), (12), (1), (10) rsF(p)(1 x< t y) = G~ 0(p)(x< t y) , k 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y))k < 104"
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192
. .
, , (, # ' (190) ((-( C - , (240). (250) (( . , (190), #( (130) (160), , H0(p)(s x< y) = ^ = F(p)(s x< 1 y), # (260). ^ (270) (220 ), (130), (10), (160) rtF(p)(s x< 1 y) = B = H0(p)(s x< y). (280) # , rV F^v (p)(s x< t y) 6= 0 ( $ BV 2 T(sx ty)(Dk Dl ). / 4 VB VB = VBt + VB(sx y) , VB(sx y) | ( VB T(sx y)(Dk @Dl ), VBt | ( VB (-, #- T(sx y) (Dk @Dl ) T(sx ty) (Dk Dl ). . rV F^v(p)(s x< t y) = Bt rtF^v(p)(s x< t y) + B(sx y)rW F^v (p)(s x< t y) (290) WB = VB(sx y)=kVB(sx y) k, Bt B(sx y) | $ ( ( ,
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~ p< s x< y) = kHv (p)(s x< y) ; H0(p)(s x< y)k N(v ~ p< s x< y) UB 0 = (Hv (p)(s x< y) ; H0(p)(s x< y))=N(v UB = =(v p< s x< y): ~ p< s x< y) 6 N(v). = , 0 6 N(v ^ , #( (140 ), (80 ), (10 ), (40) rt F(p)(s x< 1 y) = = HB 0(p)(s x< y), , 4^" ^ k0(t)Lv0(p)(s x< y) ; Id m+n ](F(p)(s x< t y) ; F^ (p)(s x< 1 y))k < 10 (40) "^ : ^ krtF(p)(s x< 1 y) ; HB v (p)(s x< y)k < 10 , (140), (10 ), (40 ) , "^ ^ k(t)Lv0(p)(s x< y) ; Id m+n]rtF(p)(s x< 1 y)k < 10 (140), (70) (90 ) kId m+n +(t)(Lv0(p)(s x< y) ; Id m+n)] 2^" : ^ ^ (rtF(p)(s x< t y) ; rtF(p)(s x< 1 y))k < 10 G (170) $ 4 . 2 1 , F^v (p) ((( . 1( 0 #, rtF^v (p)(s x< t y) rW F^v (p)(s x< t y) (. (290)), 5, # 30 40 - ( A = HB v (p)(s x< y) A0 = rW Hv (p)(s x< y) B U 0 = UB 0 X = X B X0 = XB 0: B B 0 = BB 0 U = U B = B , #( "^, #(, ( 5 (-(.
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196
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197
1] S. Smale. The classication of immersions of spheres in Euclidean spaces // Ann. of Math. | 1959. | Vol. 69. | P. 327{344. 2] S. Smale. Regular curves on Riemannian manifolds // Trans. Amer. Math. Soc. | 1958. | Vol. 87. | P. 492{512. 3] S. Smale. A classication of immersions of two-sphere // Trans. Amer. Math. Soc. | 1958. | Vol. 90. | P. 281{290. ' ( ) 1997 .
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Abstract R. Hildebrand, Classication of phase portraits of optimal syntheses, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 199{233.
The paper is devoted to the investigation of controllable oscillating systems of ordinary di1erential equations a2ne in scalar nonsymmetric control in a neighborhood of a singular point of focus or center type. Integrands in value functionals are quadratic in phase coordinates. We classify such systems in case of general position by arising optimal syntheses. The existence of optimal synthesis is proved and its structure is described.
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^ . C = 0 C_ . 6 , C_ > 0, dAB < 0 (15), H1 1(0) = dAB p, H1 1(0) < 0, ( " > 0, H1(t) < 0 / t 2 (;" 0). tV = supft < 0 j H1 (t) > 0g. 6 6 tV > t+ . 5. t 2 t; 0) H1(t) < 0 . 0 , ( t0 2 (t; 0), H1(t0 ) > 0. F 2( 0 , H1(t0 ) = 0 H1 (t) < 0 / t 2 (t0 0). D C (t0) < 0, dAB (t0) < 0, C > 0. . 6 ; H_ 1(t0 ) = dAB . 2 5 2 ,
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F = 12 (sin x21 ; cos x22) + (r3 ) = X(r2 ), dxAB2 (0) = X(1) dAB = x1 + (r2 ), 0 = ;X(r;2 )f;x2( tV)X(r2 )+ x1 ( tV)X(r2 )g + (r2 ) =) x2 ( Vt ) = X(1)x1 ( Vt ): (24) 5 2 , ( tV) 0 . (t)
^ C 1 C 4 t1 , t4 . 6. t 2 t1 t4] H1(t) = ;X(r) . D ^C 4
x2 = ;X(r), x1 = X(r). (17), H1(t4 ) = ;&(1r2 ) fX(r3 ) + X(r)X(r2 ) + X(r2 )]g + (r2 ) = ;X(r). t 2 t1 t4). C (13). 2 ;&& { = H1 { . Q = A, t t4]
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= ;A2 BdAB @x x1
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(r). 5 H1(t4 ) = ;X(r), 2 ,
H1(t), ! ;X(r). 2 7. C ( Vt ) = ;X(r2 ) .
224
.
. 6 (24) ( Vt ) 0 Vt < t4 . Rt1 5 0 6 , tV < t1 , H_ 1(t) dt = H1(t1) = ; X(r). t' D H_ 1 = (r), ; t1 ; Vt = X(1). , C (t1 ) = 0, , (24)
C_ = X(r2 ) tV t1]. ! Rt1 C ( tV) = ; C_ (t) dt = ; X(r2 ). 2 't
8. 4 " p & ^C 3 , V) " " 10 " C 2, ! " ( t & " "! " x ~
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2 / && / . &
0 7. 5 (24). dx2 9. 10 x ! x~ dx1 . D , dx2 10 , dx 1 dx2 = x2 + (r) = X(1): (25) dx1 x1 , ! 10 0 x1 . H d xx12 x2 1 x2 2 (25), dx1 = ( dx dx1 ; x1 ) x1 = (1). < , x1 & x1 , ,
x1 ! 0. D dx2 0
. 2 (25) dx 1 : C 3 , 10 x~ U 2 ( . . 4). 4 , 2 u = 0 0
/ 10 2
. 0 , 2 ! u = 1. C 2 /
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225
C < 0. t 2 (t0 0). D C ((t0)) > 0, dAB ((t0 )) < 0, ; H_ 1(t0 ) = dAB . 2 6 0 2 ! 10
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^ . 11. H1(t) > 0 t<0 ( ) U 2 t 0] . 6 1 dAB ( . : ! 10 0 dAB , ddxdAB 1 = = 1 + (x1 ). 5 2 , 10 1 & = (dAB ) C 2. D 2 , d ddAB , . .
(1). = dinf (dAB ), = sup (dAB ). 6 0 8 AB dAB . !, 1 ! 6. : 10 0 0 K+ . &
t 10 : t (dAB ) = t (dAB (dAB )). C ! ddtAB = = @d@tAB + @t@ d ddAB . H
d ddAB = (1), ; (22) d lim!0 ddtAB = AB = ; 1 ; ctg (d lim!0 (dAB ))2 . ( 0 AB 9. 6 0 3 7 &
C ! 10
: C2 = 1 ;sin + cos lim x222 = 1 ;sin + cos lim 2 (dAB ) < 0. B lim 2 2 x1 !0 x1 x1 !0 x1 dAB !0 dt 1 ; 2 sin < 0, d lim!0 dAB > 0. , t (dAB ) > 0 AB dAB > 0. D , / (19), / ( / 0 KV t = 0, & H1(t) / 0 t . D H_ 1 < 0 ^10, ; ( " > 0, H1(t) > 0 t 2 (;" 0). 2 U . ) 0 11 . 2 4 U & ! (x), !(! ! :; 10] 2 0 0 10 11 | . 12. (x) U C 3 ; 10 x~ C 3 ; 10 . C ! & , (! C 3 , 10 x~. 6 0 3 9 0 PC 1 / x~. 6 (dAB ) 1 & (dAB ). D , /
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d dd AB = @d@ AB + @ lim!0 @ @ ddAB . 6 dAB @ = 0, d = (1)
lim d = lim @ . < / d dAB dAB !0 d dAB dAB !0 @dAB ( !, ( ! d lim (dAB ),
AB !0 !. 4 , ^ | PC 1 / (x = = 0). H & (x) 1 (18), (19), / ( ^ . B && && / /. 2 6 0 12 & (x) N T X . 13. (x) U !(x) C1 . 0 0 N , . . 1 & dx. C , N && . 6 && & dx / , / & @t@ . @v@ | . X @ @ @ @ d(dx) @t @v = (d ^ dx ) @t @v = X 1 @ @x @ @x = 2 @t @v ; @v @t = X @H @x @ @H 1 @H 1 = ;2 + @v @ = ; 2 @v = 0 @x @v N 0 H . D & d(dx) ! !2 , N . , ( & ! 2 C 2, d! = dx. 6 0 12 & (x)
!i , i = 1 2 3, 1 / 2/, 0 / 0 C 3 , ; , 10 . / , xlim ! = 0. 6 (x) &
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227
B & / 2 & ~. ~ : (;2 2) ! ( 32 ) !( : q q p ~)2p+2 ctg ~ (1 ; ctg ~)2 + 2 ctg ~ + ctg ~ p1 2 +arcctg (1pctg 2 ctg ~ 4 4 q ;e = 0 q 2 2 ~ ~ ~ (1 ; ctg ) + ctg ; ctg
6= 0@ q ~ ctg ~ 2 ~ ; ; arcctg ctgq ; 1 = 0 = 0: ctg ; 1 2 ctg ~ ;
;
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2 1,
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C = ;X(r2 ). | 0 . !, ! 6. 2 0 K; , (! , K0 . 0 . 6 0 2 0 K0
K^ 0 2 M; . K0 K^ 0 dAB0 = dAB , 0 = . H 0 , ; ! 0 / K0 K^ 0. C
(19), / ( t = 0 0 K^ 0. 0 &
H0 + H1 . ;
t, / | dAB0 0 . ! &
t
H1(dAB0 0 t (dAB0 0 )) 0. ; 0 f(dAB0 0 t (dAB0 0 ))g 2 / K^ 1 2
M. H ; / . p0 | 0 K0 , !( dAB0 0 .
p^0 2 K^ 0 | p0 M; , ^ (t) | (19), / ( t = 0 p^0. 6 & dAB p^1 = ^ (t ) dAB0 0 . 2 / dAB1 1. 1 (23)
Rt Rt dAB1 = dAB0 + d_AB dt = dAB0 + (1 + (r)) dt = ; dAB0 (ctg 02) + (dAB0 0), 0 0 ! dAB ! 0. B
,
2 C 1 , dAB 0 0
Zt x2(t ) = x2(0) + x_ 2 dt = x2 (0) + (r2 ):
0
(26)
228
.
,
(0) + (r2 ) tg + o(1): 1 = ; xx2((tt )) = ; ;d x2(ctg = ; (27) 2 0 ) + o(r) 0 1 AB0 6 , tg > 0, t = X(r), dAB1 = X(r), 1 = ;X(1). , K1 / K^ 1 & ! 0 C > 0, ; H1(t) > 0 . D / K^ 0
H_ 1 = dAB 0 / t 2 (0 t ). ^ (t) | (18), / ( p^1 2 K^ 1 t = 0. Vt = C < 0, ; tV > 0. = inf ft > 0 j H1(t) > 0g. 6 p^1
H_ 1 = dAB 1 2 ! X ^
^ & ! . 5 2 x~ . ; ; t; > 0 1 ! AB , + t+ > t; | ! AB . 6 6
tV < t+ . 14. Vt > t; B 0 5. H 0 14 , p^2 = ^ ( tV) 0 2
M; . 2 1 ! ( tV) p2 . 4 tV 1 H1( tV) = 0. > H1(t) (17). 0 2 ! Fx2( tV) + (r4 ) = dAB ( tV)(r2 ). F = x1 (r) ; 21 cos x22 + (r3 ), dAB = x1 + (r2 ), x32( tV) + (r4 ) = x1( tV)(r2 ). ! ; xx21 (( 't't )) = (1). dAB2 = dAB (x1( tV) x2( tV)), 2 = ; xx21 (( tt'')) . 4 20 P : (dAB0 0) 7! (dAB2 2 ) K^ 0 M; M;. D , dAB2 = X(dAB0 ). , lim d = 0. dAB !0 AB2 .
0
. 2 & P & f0g ]. >
dAB1 dAB2 0 1 . > ! 1(dAB0 0 ) 0 0 1 (0 0) = ; tg0 (27). , & 2 (dAB1 1) 0 0 . ! 0 . Z , 20 P && dAB0 < 0. AB2 2 ) , @@ ((ddAB 0 0 ) 0 0! f0g ]. ; @dAB2 = dAB2 + (r) = X(1), @2 = C (p0 ) + (r) = X(1). @dAB0 dAB0 @0 C (p2 ) 5 2 , & 2 , f0g ], && !( & 0. 6 (27) lim 1(0 0 ) = ;1, !0 0
229
lim (0 0) = 0. H (17) (dAB1 ! 0) () (dAB2 ! 0). !, 0 !1 1 d2 > 0, lim (d ) = ;1, lim (d ) = +1. d 1 1 !;1 2 AB1 1 1 !+1 2 AB1 1 5 2 ,
lim (0 0) = 2 max lim 2 (0 0) = ;1: 0 !+1 2 0 !0 . 2 max | ( . . 5). 2 6 2 6
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q
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0 , 2 = 2(0 0 ) > 0 , ( ! 2 = (0 0) ( . . 5 ). 6 , / / 0 ( . C & ! A2(x). G1 x~ (3) ;e1 = (;1 0). ; &
A2 2 ! C 3, / (! x~ !(! Ox2 . s | 0 , 0( ; . C ! x_ = A, / (! s t = 0. 6 (3) 2 x~ . H ! x2 & ! t. H
x_ 2(0) = A2 (s) = 0, x2 (0) = x2min . t+ = minft > 0 j A2 ((t)) = 0g, t; = maxft < 0 j A2((t)) = 0g, x2max = min(x2(t+ ) x2(t; )). , x2max > 0 > x2min , 0 x^2 2 (x2min x2max) ( ! + 2 (0 t+ ), ; 2 (t; 0), x2( + ) = x2 ( ; ) = x^2. 6 1 2 x+1 = x1(( + )), x;1 = x1(( ; )).
230
.
R + C I (^x2 ) = F ((t)) dt
& ! x^2. H
I (x2min ) = 0. x^2
;
dI = d + F ( + ) ; d ; F ( ;) = F ( + ) ; F ( ; ) = dx^2 dx^2 dx^2 A2 A2 + ; 2 2 2 3 2 3 = 12 sin (x1 ) ;+ cos x2^2 + (r ) ; sin (x1 ) ;; cos x2^2 + (r ) : ;x1 + (r ) ;x1 + (r ) dI = D &
A2
F =X(r2), , x^ !lim 2 x2min dx^2 dI = +1. x^2 = 0
dI = 1 sin (x; ; x+ + (r2 )) = = x^ !lim 1 1 dx^2 2 2 x2max dx^2 dI = ; X(r) < 0. dx^2 & I (^x2 )
(x2min 0) , (0 x2max) | . 6 ;
ddIx^2 = AF2 ( + ) ; AF2 ( ; ) = 0.
0/ 2 x^ (28) ; tg = x^;2 +2 + (r): x1 x1 2 I (^x2 ) Imin . 0 , Imin , (
. . Imin = ;X(r2 ) 1 . I (^x2 ) = Imin = ;X(r2 ). W l, H 0 2 , 2 , (F + uG) dt < 0. C ! l x_ = A + B , / (! ( + ). ; u = 1 X(r) ;
, 1 ! (Vx1 xV2). 6 x_ 2 = (r)
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I
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(xZ1 x^2) ;
(F + G) dt +
.
(x+1 x^2 ) F dx + Z F dt = A2 2
('x1 x'2 ) (x1 x^2) 2 2 = X(r)(r ) + (r )(r) + Imin = ;X(r2 ) < 0: 2 2 4 Imin = +X(r ), ! -
(x+1 x^2)
15.
,
;
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.
231
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x1 (p0) = x+1 , x2(p0 ) = x^2. H (26) x2(p1 ) = x^2 + (r2 ), (27) (28) | x1(p1 ) = x;1 + o(r). 5 2 , 0 p1, ( ; )
: d(p1 ( ; )) = o(r). o() / s ! x~, s &
A2 x~. C
(t) (t) x_ = A, / ( t = ; p1 ( ; ) . : ;&&
/ ; , ; (t) (t) t 2 ( ; + ) 1 (1 ! , !(! o(r). 5 2 ,
F , / : R+ R+ F ((t)) dt = F ((t)) dt + o(r2 ) = Imin + o(r2 ) = X(r2 ). : ,
0 = AF2 (( + )) ; AF2 (( ; )) = AF2 (( + )) ; AF2 (p1) + o(r). D A2 = ;dAB + (r2 ), ; dFxAB2 (( + )) ; dFxAB2 (p1 ) = o(r2 ). ; (17), !(
& ! H1(t)
, H1( + ) = Z + 1 Fx 2 + + = ;A x + A x ;F x^2( ) + dAB ( ) ; 2F dt + d (p1 ) + (r2 ) = 1 2 2 1 AB = ; X(1r2 ) f;2dAB (( + ))(X(r2 ) + o(r2 ))g + (r2 ) = ;X(r): ! ( + ) . . 2 = ( tV) H1(2 ) = 0. H
H1( + ) = R2 H_ 1 R2 (r) + =; d = ; _ &(1) d = ; X(r). ! 2 = ( ) + X(1). ( + ) ( + ) C 0 p0 = ( + ) ( + )
o(r), ; ( + ) = 0 + o(1). ! 2 = 0 + X(1). D , 0 2 s ! x~ 0 , 2 . , 2 = 2(0 0 ) > 0 . 5 2 , ( ! 0 1 , 0 2 2(0 0 ) = 0 , 1 0 < 0 < 0 2 ( . . 5). 5 (0 01 ) (0 02 ) | 0 20 P. .& 0 = 01 . C dAB2 (dAB0 01 ) & ! dAB0 . H/ 01 < 0 , , ;
;
;
232
.
( c < 1, ;c dAB0 > ;dAB2 . 0 = 02 ;dAB0 < ; dAB2 , c < 1. d2AB0 @dAB2 = @2 = 6 (0 0 1)
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2 0 0 @dAB0 1 2
, 0 0 . 5 2 , 0 (0 0 1) 2 . 2 / 2 / 2 0 , ., , 12]. ) f0g ]. D 0 & = (dAB ) C 1 . 5 2 , 2 && ! !, ! 20 P !(! ! 0 1. 2 ; ! 01, 1 M; | ^01. G p^0 2 ! ^01 , p^1 2 ! ! ^10 , p^2 | ^01. 10 ^10 ! 1 0. 5
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2 (2), (8). 0 PC 1 / / ^01 ^10 . D , , ; 2 ! (x = 0 = 0), N T X . 5 x~ !- ;
. B 0 N &
F ! 0 13. 5 2 , , N , . , 4, !. ) 0 15 . 2 B 2 4 !(
0 . 2 ( ~( )) . 2 (~( ) 23 ] Imin = ;X(r2 ) 2 Imin = +X(r ) B Imin . 6 0 . 5 2 ,
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0 , !( 1. 5 . 4. . 02 , x~ - . 4
.
,
. 4
,
233
Y 2 & 4. H. . (, ; 2 .
+
1] . . . | .: , 1975. 2] #$ %. . 1 -' ( ) ( ) *( '+ '( ,- ,**' // / - 0 $ ,1. | 1977. | 2. 11, 3 1. | 4. 57{58. 3] 8 . ., 8 9. /., : %. *),'' ,') $ *'0 $ ** ,') ( 0 + // 2 . | 2000. 4] ='$ 9. 4. . | .: >*. 0 . 0.-)'. '., 1961. 5] ='$ 9. 4., #'*- . >., >) 0 %. ., @ A. /. ')'+* ' ,') ( ,**. | .: , 1969. 6] Davydov A. A. Qualitative theory of Control Systems. | Providence, RI, 1991. | Translations of Mathematical Monographs, vol. 141. 7] Fuller A. T. Constant-ratio trajectories in optimal control systems // Internat. J. Control. | 1993. | Vol. 58, no. 6. | P. 1409{1435. 8] Jakubczyk B., Respondek W. Feedback classiGcation of analytic control systems in the plane // Analysis of controlled dynamical systems (Lyon, 1990). | P. 263{273N Progr. Systems Control Theory, vol. 8. | Boston: BirkhOauser Boston, 1991. 9] Kelley H. J. A second variation test for singular extremals // AIAA J. 2. | 1964. | No. 8. | P. 1380{1382. 10] Kelley H. J., Kopp R. E., Moyer M. G. Singular extremals // Topics in optimization. | N.Y.: Acad. Press, 1967. | P. 63{101. 11] Krener A. J. Approximate linearization by state feedback and coordinate change // Systems Control Lett. | 1984. | Vol. 5, no. 3. | P. 181{185. 12] Nitecki Z. Di^erentiable dynamics. | Cambridge: M.I.T. Press, 1971. 13] Zelikin M. I. On the singular arcs // Problems of Control and Information Theory. | 1985. | Vol. 14, no. 2. 14] Zelikin M. I., Borisov V. F. Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering. | Boston: BirkhOauser, 1994. ' ( 2000 .
. .
. . . 512.5+511
: , , , .
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Abstract A. A. Chilikov, Taylor power series of algebraic functions over elds of positive characteristics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 235{256. In this work we show algorithmical solvability for the problem of calculation of the Taylor power series for an algebraic function over a 5eld of positive characteristics. An e6cient algorithm for construction of a 5nite automaton solving this problem is given.
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% N , deg ti < N 0 N 0 > N , deg t0i < N 0. . ( !! ) p u !-
4.1. '%! p u =
" (u) =
s 1 X xi " (ti ): i=0 ;
B) & p 1 ! ! xk . % ;
p " (u) =
s 1 s 1 s 1 s 1 X s 1 X X X X
p 1 xj " (tj ) = " (tj ) jixpi = xpi ji" (tj ): j =0 j =0 i=0 i=0 j =0 ;
;
;
;
;
;
sP ;1
241
ti
ji " (tj ), ! . 1! j =0 ! tj : deg ti 6 deg ji + p1 deg ti 6 C + p1 deg ti C ji. L , ! ! N > C p p 1 . 1!, ! , , x 3 ! N. ; & !! ( )
) 3 . :, N ) & R , "
! ) "- x . ! , ) . ; & . 2 , &sP1 p u = Q, Q = Qi | xp i=0
! s 1. ,& Qi | ! (Zp) $ ! ( N) ! (#1 : : : #n). 1 , & ( , ) | . D , " & u | ) . G ! ). D, ! " &! ! ) ) !! x, ! & ! . B$ , ) ) !! ( ) ) ! " $ p u = Q. ,& Q !! & ! & !! , | ! . @ ! & Q,
" u, , )$ ! ( , !! ) , " u = " (u). : ), . G , x. 1 , ! . , ) , & , "$ = (k) : : : (0), !
"$ ( $) !! . (& , $ !. 1 , " ! $ ! $ !! . B $ ! $ ! !! . 1 , " ( ! ) ! , " 4.1. 0
0
0
;
0
f g ;
;
0
242
. .
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5. ' ( ($3 !
) . ( ! F | ! $ Zp ! 1 : : : m , , , ! Zp ! m 1 : : : m . 9 2.1 & . 4 !-!) ! R R ! F . 2 ! ! !! F, R R . 2 ! !, ! 4.1 ! " . 5.1. ( k | p. 3 ) ": k k, " : 1) "(x + y) = "(x) + "(y) ( ), 2) "(xp y) = x"(y) (p- ). 1! P !! ! ) !! . ( x = x # . 2 , ! )$ x
P x = f . 2 & ! !! " : 0
0
!
"(x) =
X
f "( # ):
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"(x) =
XX
0
0
0
0
f0 +p 0 +p # 0
0
0
0
"( 0 # 0 ):
0
1 , !! !
0 # 0 . ! ! !! , "
243
" ( 0 # 0 ) = 0 : , , 5.1. ( x % ! , % . 2" p- $ ! = k : : : 0 = k : : :0 , " (x) !! & x ! i i (i = 0 : : : k). I) , & $ " (x) " (x ). ( &, , !, p- $ ! ! ( & , ) ! ! $ ). : " &! ! " , )
! !! & , !! . , ) " (x) , ) $ !! " (x ), ) ! $ & x . 1! ), , ! "- . 5.2. ( k | (R R ), ! X | ) & k. "- ) X 3 ) , )" X !$ ) !! . : 5.2. x | " # # $%& ! , "- x . 0
0
f
.
g
) B ) MN ! !3 ! " &!. ) ?
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( ' | $ x). , , !3
244
. .
x =
s 1 X ;
i=0
i xpi :
(5)
P "$ & !$ ( $ #1 : : : #n). (! ! " s 1
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x = (6) P i=0 1! ) ! " ! : s 1 X
k xk = Pkik xpi: (7) i=0 S , S(p 1) > s, ! s 1
X S ( p 1) k (8)
x = PS (pki 1) xpi : ;
00
;
;
;
;
;
i=0
! & F. 5.3. ( u R X
Sp u = PtSi p xpi : 2
0
i
1 l(u) & u 3 !$ deg ti ! 1 : : : m . 1! ) $ $ . 5.1. '%! u R % Sp u = P tixpi 1Sp i ti. " u = " (u) P ! % Sp u = i tixpi 1Sp ti . ' #% N0 M0 ( ! ! % ), N > N0 , M > M0 deg ti 6 N l(ti ) 6 M deg ti 6 N , l(ti ) 6 M . . B ! (8). , !, !! !$ x. ( s 1 X
Sp u = tj xpj P1Sp j =0 u = " u. 1 s 1 X
Sp u = " (tj ) S (p 1) xj P1S : j =0 0
2
0
0
0
0
0
0
;
0
;
0
;
(8)
s; 1 X s;1 X Sp 0
u = xpi " i=0 j =0
245
ki : (tj ) P Sp
1! ! ) !! : deg u 6 maxdeg ki + 1p deg u l(u ) 6 maxdeg ki + p1 l(u) 0
0
! deg f ! ! f ! ( 1 : : : m ). 1! M0 N0 , " . ( 5.1 $ ). B$ , ti $ ! ) !& ) $ P & u,
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! " &!. L , & ! ! & x. f
g
;
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246
. .
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k X i=0
i i = 0
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, k = 1, . . ;
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2
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247
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i=0
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|
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i=0
i=0
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X ri ip p = ri (i+1)p :
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ri 1 ip + rk 1 kp: ;
;
248
. .
1! ! (9) ( ! ! p) ! ! : kp = 2
r +p
=
kX ;1 i=0
i ip :
kX ;1
(ri 1 + rk 1i ip ) ip : ;
i=1
;
r+pi = ri 1 + rk 1 i ip : ;
;
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(11)
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i=0
"( r ) =
k 1 X ;
i=0
"(ri ) i+1 =
k 1 X ;
i=1
("(ri 1 ) i ) + "(rk 1 ) k : ;
;
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kX ;1 i=0
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;
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kX ;1 i=0
"(r+pi ) i = "( r+p ):
1 , ! !! ! F. 1! ) !) F ) !3 Zp= 1 : : : m ] !! . : 6.1. ( x F 0 ! , 0. : "$ $ !
)$ (, , 5.1), !3 ! " &!, $ ! F. B & ! ! F ! ) &. 2 ! , 3 '(x) =
s X i=0
ai xi = 0
249
( ' | $ x). ! " &!
x =
s 1 X ;
i=0
i xpi :
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i = il pl ! !l=0 " : X
x =
il pl xpi : 2
;
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;
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1 : : : m. 1 ! Bj =
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ji pi
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jil xpi pl (12)
;
;
;
;
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;
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;
.
6.2. '%! u R % 2
0
B Kp Sp u =
X
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;
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0
0
06i6s 1 06l6k 1 ;
;
250
. .
t il . ' #% N0 M0 ( ! ! % ), N > N0 , M > M0 deg ti 6 N , l(ti ) 6 M deg ti 6 N , l(ti ) 6 M . . B &$ ! , !3 ! Zp= 1 : : : m ]. B, ! , ), , ! $ $ x = !. 0
0
0
7. * 7.1.
( ! ! ! 1$ $. @ , $ ! !. (& !! , ! . ( ! $ !, !3$ . 1. ( x2 = 1 + # (13) ! Z3. D ! , "$ & 3 1$. B & !3 (1 + #)2x3 = (1 + #)3 x (14) ! !! . % (1 + #)"i (x) = "i ((1 + #)2 )x (1 + #)"i (x) = ai x a0 = a2 = 1, a1 = 1. ! y (1 + #)y = x: (15) , y = "0 (x) = "2 (x) = "1 (x): 2 ! !! y. ; ) , ! (1 + #)x3 = (1 + #)3 y: % , (1 + #)"i (y) = "i (1 + #)x ;
;
251
(1 + #)"i (x) = ai x & , "0 (x) = "1 (x) = y "2 (y) = 0: 1 , ) ( ) !! x ! : x y y 0. 3 ! )
& !! . , , (13) , " . 2 , $ 1. 1, ! $ x 1, $ y ) 1 (y = "0 (x)). : $ y 1. 1 , ! ! . , , ! . 1.
a0 = a1 = 1, a2 = 0.
;
t
;
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252
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(16)
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: "3 ) ! & xP (x2 #) = Q(x2 #): (17) S) P(x2 #), ! xP 2(x2 #) = P(x2 #)Q(x2 #): G | ! x. 1! ) ! !! ! : "i (x)P (x2 #) = Q i (x #) Q i (x #) = "i (P(x2 #)Q(x2 #)). B ) ! ! "i (x). B & ) P(x2 #): "i (x)P 2(x2 #) = Q i (x #)P(x2 #) ! (17). 2 ! "i (x)P 2(x2 #) = Q i (x2 #) Q i (x2 #). 1 ) ) ! , 0
0
0
00
00
253
!! x "i (x)P 2(x2 #) = Ri(x2 #) ! R. 1 !! | & "$ (! , ) ! & ! , !
$ !! !) ! ). 1 , )$ !! " $ R, ) ! . ( ! . 2. ( x7 + ax3 + bx2 + cx + 1 = 0
(18)
Z2. G 3 ": x8 + ax4 + cx2 = x(bx2 + 1):
(19)
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;
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254
. .
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255
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256
. .
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0
0
0
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X
#
X
(" f) g(#p )(" g )(#p ):
0
0
0
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0
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,
1] A. J. van der Poorten. Some facts should be better known, especially about rational functions. 2] A. J. van der Poorten. Rational functions, diagonals, automata and arithmetic. 3] .
. . | .: , 1986. 4] . &'. '( . | .: , 1968. 5] *. + ( ,. p- ./ / , p- ./ 0 . - ,. | .: , 1982. 6] 1. 2. / . 3/ 4 ( 5 ( . | .: * , 1993. 7] A. J. Belov, V. V. Borisenko, V. N. Latyshev. Monomial algebras. | NY: Plenum. ' ( ) 2000 .
T- . .
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519.48
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Abstract V. V. Shchigolev, On leading monomials of some T-ideals, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 257{266.
In this paper some analogs of the Gr'obner base for T-ideals are considered. A sequence of normal monomials of the T-ideal 2(3) is built so that the monomials are independent w.r.t. the operation of monotonous substitution and the insertion operation. Also a theorem is proved stating that for algebras without 1 a multilinear identity of the form 1 , 1 2 ] 2 , where 1 , 2 are variables and 1 , 2 are monomials, belongs to every T-ideal that is 8nitely based w.r.t. the inclusion relation of the leading monomials. T
w
x x
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x
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w
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x 1.
1] T- . ". #. $ 2] & '
. ( ) * & , *, . ( - * * * * , & , * . * T- . / * (. 0. 1 . 2 & K | , x1 : : : xn : : : | . ( ' ), *: xi < xj , 2001, 7, 9 1, . 257{266. c 2001 !, "# $% &
258
. .
i < j. 7 ' * - , & * , ) * - . 2 & S | x1 : : : xn : : :, , S (n) | , *, n, S0(n) | x(1) : : :x(n) , 2 Sn , Sn | *
n. (2 9 ' S.) 1 K, * ' S0(n) , P (n). ( ) jwj w , & w , ** * w. 7 u , u = u1xi u2xj u3 , i < j. < S0n x1 : : :xn . 2 x 2 v, x | v. =* ) c() ), , & '
, x1 : : : xn : : :, * ) i degxi = degxi c(). ( * S ), : xi1 : : :xin xj1 : : :xjm & , & & i1 : : : in ** * & &) & j1 : : : jm . > v u, , v * u (u v). 2 , * S, , v u, v | u. 0 S ' ), : xi1 : : :xin xj1 : : :xjn , t fi1 : : : ing fj1 : : : jng, t(ik ) = jk , k = 1 : : : n, . ? t - F1 F2 F, fxi1 : : : xin g fxj1 : : : xjn g . > f1 2 F1 , f2 2 F2 f1
f2 ' -, , f1 f2 . 2 & F = K hx1 : : : xn : : :i | * * 1. =* f 2 F f@ f, * ) I F I@ = ff@: f 2 I g. 7 , , I@, & I. = , S(I) = S \ I@. 2* & S= ), : * ' B1 B2 B1 B2 ( B1 B2 ), , ) u1 2 B1 u2 2 B2 , u1 u2 ( u1 u2). 2 & (A 6) | * . / a 2 A (A 6)- & fa1 : : : ang A, , ai, ai 6 a. < (A 6) , , ) a1 : : : an 2 A, ) ' a 2 A (A 6)- & fa1 : : : ang.
259
T-
( ) * * (S(I)= ) (S(I)= ) , I ** * T- . 7 u 2 S (T- ) & ff1 : : : fk g F , u ** * (S )- ((S )- ) & ff@1 : : : f@k g. B Tk(3) T- F, x1 x2 x3] : : :x3k;2 x3k;1 x3k ]. 2 & T (3) = T1(3) . ? ; S, ) xk xj xi xk xi xj (1) (3) i < j < k. 1 &, S n S(T ) ;. ( ), & : , & & w1 : : :wn : : : S n S(T2(3) ), * i 6= j wi wj - ) D & I | T- F ( 1), * (S(I)= ) . E I
w1 x1 x2]w2, w1 w2 | .
x 2.
1 &, ) u 2 ; * * r Q u = ui , ) ui vi xki , xki & ) i=1 vi vi , i < j ) * ui & ) uj , ui , juij = 1. 2 & & (u1 : : : ur ) u. 1. ; T (3). . 2 . E w = r Q = ui 2 ; ** * f 2 T (3) , i=1 (u1 : : : ur ) | . 2 & ui = vi xli . > ui , vi , , . > ui , ui . 2 ) * i = 1 : : : r. E * w0 f 0 ,
), * f . = & , & w0 < w w0 | f, w00 | , w0 . 2 & w0 = xl 0 , w = xm , tQ ;1 l < m. ? ) , , t, = ui u0t , i=1
260
. .
Q r
= u00t ui , u0txm u00t = ut . H * * i=t+1 , xl ut+1 : : : ur . 7 , xl . 7 & , xl u00t . 2' ), *: ju00t j > 2 ju00t j = 1. 2 & * .E u00t , & xm , & ** u00t . ? ) u00t = vt00 xsxl , s > m. 2 xm u00t xsxl , xl 0 | xl 00 . ? ) w00 < w0 . 2 & * . E u00t = xl , xm u00t *, xl 0 xl 00 . ? ) w00 < w0 . E T (3) , & & , f 0 2 T (3). 7 w0 s Q w0 = (xli xmi ), xli > xmi (xl1 xm1 : : : xls xms ) | i=1 . r Q > & v 2 P (n) | & , v = vi , i=1 (v1 : : :vr ) | v. > &, , ) i1 < : : : < is , jvik j > 2 k = 1 : : : s, w0 v v & & . 2' &, , ) 1 6 i1 < : : : < is;1 6 r, i 2= fi1 : : : is;1g jvi j = 1. = , , P (n) * jv1j : : : jvr j. H &* ' - , & & P (n). ? , r 6 n. =* i1 < : : : < is;1 * Cns;1 . ? * & & * jvi1 j : : : jvis;1 j. E jvi j 6 n * i = 1 : : : r, ' ns;1. ( Cns;1ns;1 & P (n). / , dn = dim(P (n) j P (n) \ T (3) ) T (3) , , , , 3]. 1 . E , 1. ; = S n S(T (3) ). ( * T (3) (. 0. 1 4]. 2 & I1 , I2 | F. E F I2 + +I1 F | F F. 7 F F
* u v, u v | F . 7 , u2 v2 > u1 v1, u2 > u1 u2 = u1 v2 > v1. =* f 2 F F f@ f. 0 u v & & L F F , * ) f 2 L u v 6= f.@ 7 ), * 2. u v 2 F F , u v | I1 I2 . u v | F I2 + I1 F .
T-
261
. 2 . E , f 2 2 F I2 + I1 F, f@ = u v. 2 f ), : f=
n X i=1
u i fi +
m X j =1
gj vj
(2)
* ) i j 0 6= fi 2 I2 , 0 6= gj 2 I1 , ui vj | , ui > uj vi > vj i > j. < &, ' vj & & I2 . = & , & vj ,
I2 l | & j, * vj |
I2 . E vl = f@0 * f0 2 I2 . H n0 m0 X X 0 0 f = ui fi + gl f0 + gj vj + gl (vl ; f0 ) = ui fi + gj0 vj0 i=1 j =1 i=1 j =1 j= 6 l & vj0 , & I2 , &, vl . n X
m X
2 ** ' , & * *. 2 , , i, ui > u. 7 ui f@i & (2). E f@ > ui f@i > u v, . 2' * i = 1 : : : n ui 6 u. 2 & ui0 = u, ui0 f@i0 & (2). 2' f@i0 6 v. E v & , f@i0 < v fi0 & v. 7 & , u v (2). 2 & & vj0 = v. > g@j0 > u, , , , i, ui = g@j0 . ( ' g@j0 v
& (2). E f@ > g@j0 v > u v, . 2' g@j0 6 u, u & & I1 , g@j0 < u gj0 & u. ? ) , u v (2) , (2) | . 1 . . H &, u1 : : : un 2 F : : : F & & I1 F : : :F +F I2 F : : :F +: : : + + F : : : F In & , u1 : : : un & & I1 : : : In , w1 : : : wn > v1 : : : vn * , , i = 1 : : : n, wl = vl l < i wi > vi . 2 ' E- . 2 & w = uv 2 P (n). ( & Iuv , * ), ), : 1) yx, y 2 v x 2 u, 2) u1u2 u3 u4]u5v1, c(u1u2 u3u4u5 ) = c(u), c(v1 ) = c(v), 3) u1v1 v2 v3 v4 ]v5, c(v1 v2 v3 v4v5 ) = c(v), c(u1 ) = c(u). 3. T2(3) \ P (n) Iuv \ P (n).
262
. .
. 2 &
f = tt1 t2 t3]r1 r2 r3]r
tt1t2 t3 r1r2r3 r 2 P (n). 2 & w1 = t, w2 = t1t2 t3 , w3 = r1r2r3 , w4 = r. > , )
i j = 1 : : : 4, i < j wi v, wj
u, f 2 Iuv 1). 7 & , &, , l = 1 : : : 4, i < l c(wi ) | c(u) i > l c(wi ) | c(v). ( &, l > 3. Em P f = i tt1 t2 t3]i , i | r1 r2 r3 r i=1
*. > i ' 1), i = 0i 00i , c(0i ) c(u) (00i ) = c(v). ? ) t11 t2 t3]i 2 Iuv . 1 . 4. u v | T (3) w = uv 2 ( n 2 P ). w | Iuv , T2(3) ,
3. . 2 , & w = f,@ f 2 Iuv \ P (n). 2 & V0 | K, * P (n), , 1), V1 | , * , , 1). E P (n) = V0 V1 . L , Iuv \ P (n) & ' . E w 2 V1 , w = f@0 , f0 f V0 . H f0 2 Iuv . 2 f0 f0 =
n X i=1
uifi +
m X j =1
gj vj
* ) i j c(ui) = c(u), c(vj ) = c(v), ui fi | 3), gj vj |
2) * ) 1 2 fi gj c(1 ) = c(v) c(2 ) = c(u). M , fi 2 T (3) gj 2 T (3) . E u v | n
m
i=1
j =1
X X f 0 = ui fi + gj vj
F T (3) + T (3) F . 2 2 * u v | T (3), ). 1 . ? ) 4. u v 2 ; uv 2 S . uv | (3) T- T2 . 1. w1 : : :wn : : :
S n S(T2(3) ), i 6= j wi wj - .
263
T-
. B ;0 w 2 S, ),
w = uv, u v 2 ;. 2 & D |
w = uv 2 ;0, u v 2 ;. B wD , uD vD , w, u v xi ! 1, xi 2 D. 1 &, w0 = u0v0 , u0 v0 2 ;, w0 2 ;0 w0 w (mod ), u0 u (mod ) v0 v (mod ). = & , w0 w (mod ), * D w0 wD . 2 , , ) u0 v0 , u0 uD , v0 vD u0 v0 = w0 . E u0 v0 2 ;,
u0 = u0 v0 = v0 . ? ) u0 uD , v0 vD u0 u (mod ), v0 v (mod ). H * ' - * 4, * &,
& & wn =
nY ;1 k=0
x4k+6x4k+3 x4n+5x4n+6x4n+3x2x4 x1
** * . E .
nY ;1 k=0
x4k+8x4k+5 n 2 N
x 3.
# & * T- I , * * (S(I)= ) . ( NH & 0 { ( . 5]). 2. I | ! T- ! F 1, ! " (S(I)= ) . I " ! ! w1x1 x2]w2, w1 w2 | . . ( ), , & * ' & : x, y, z, , & . 2 & 0 6= f 2 I \ P (d) . 2 ) , ) f1 : : : fN 2 F, * ) g 2 I * i g0, g0 fi g@0 g@. (3) 0 2 & m = maxfdeg fi : i = 1 : : : N g. 7 f , ), * f xk ! wk = xk x(k;1)m+d+1 : : :xkm+d , k = 1 : : : d, ** * w1 : : :wd . H (3) , , S(I) m : 1) u, 2) uyv, uv , y & uv. ( xm 2 I F=I & NH.
264
. .
# . H h(x y 1 : : : n1 1 : : : n2 ) = uyv ;
n1 X i=0
uiyhi 2 I \ P (m)
(4)
1 : : : n1 x = u, 1 : : : n2 = v, juij = i 6 n1 = juj ; 1 ui | u. 2 un1+1 = u uyv h(x y 1 : : : n1 1 : : : n2 ). ? r & i = 0 : : : n1, '-- hi 0, , ) , ;1 . # r = ;1. M x1 : : : xm x1. E (4) * '-- hi xm1 2 I, NH * & & & F=I. # & , r . H jur+1 j 6 juj. 2 ur+1 ur+1 = ur xt. M & xt 6= y. B h0 h0i &
xt ! xm+1 xt h hi , i = 0 : : : r. > r = 0, h00 = xm+1 h h00i = xm+1 hi , r > 1, h00 h00i & r ! r xm+1 h hi , i = 1 : : : r ; 1. rP ;1 H h00 ; h0 = ur (xm+1 yhr ; yh0r ) + uiy(h00i ; h0i ). 2* i=0 x1 : : : xm+1 , y, x, y, ),
I:
xr+1 yxs ; xr yxs+1 s = n1 + n2 ; r + 1 6= 0, '-- hr . 2 i < r h0i h00i &, y. > 6= , F=I & NH. > = , (5) xr y x]xs 2 I jr + sj 6 m ; 1. H (5) , xp y x]xq 2 I, p = m + 2r, q = m + r + 2s. E char K = 0, (x + z)p y x + z](x + z)q
I. ( , pX ;1 qX ;1 xp;i;1zxi y x]xq + xpy x]xj zxq;j ;1 + xpy z]xq 2 I: i=0 j =0
(6)
2 , (6) I. 2 i > r xp;i;1zxiy x]xq 2 I - (5). 2 & & i < r. . g1 = h(zxi y x]xs x : : : x) g10 = = xn1 zxi y x]xs+n2 . E i < r, p ; n1 ; i ; 1 > r, & , xp;n1;i;1 (g1 ; g10 )xq;s;n2 2 I. ? ) xp;n1;i;1 g10 xq;s;n2 =xp;i;1zxi y x]xq 2 2 I.
T-
265
" , j > s, xp y x]xj zxq;j ;1 2 I - (5). 2 &
& j < s. . g2 = h(y x] xj zxr x : : : x) g20 = xn1 y x]xj zxr+n2 . E j < s, q ; j ; 1 ; r ; n2 > s xp;n1 (g2 ; g20 )xq;j ;1;r;n2 2 I. ? ) xp;n1g20 xq;j ;1;r;n2 =xpy x]xj zxq;j ;1 2 2 I. E , (6) I, & xp z y]xq 2 I: (7) s 0 n n + 1 2 . g3 =h( z y]x x : : : x) g3 =x z y]x s , , &, p ; n1 > 2r > r, xp;n1 (g3 ; g30 )xq;n2 ;s 2 I: (8) p ; n 0 q ; n ; s p q 2' x 1 g3x 2 = x z y]x 2 I. H * xp 1 z y] 2 xq 2 I: (9) B F0 F 1, ) - . 2 & & f 2 F, , (7), (8), (9), f0p+q+1 2 I, f0 | & ' F0
f. H NH , , t, F0t I , , x1 x2]x3x4 x5]x6 : : :x3(t;1)x3(t;1)+1 x3(t;1)+2] 2 I. 2 & wk = wk0 x(k+1)(m+1) xm+k(m+1) * k = 0 : : : t ; 1, wk0 = = x1+k(m+1) : : : xm;1+k(m+1) . E w0 : : :wt;1 ** * tQ ;1
wk0 x(k+1)(m+1) xm+k(m+1) ] I. k=0 2 (3) , S(I) m : 1) , 2) uxyv, uxv uyv y & x. ( F=I & NH. ( uxyv = t0 P = iuyvi , c(uxyv) = c(uyvi ) * i. 2 ** x 0 i=1 t P v, uxyxk = uyxk+1, = i. > 6= 1, * & i=1 F=I & NH. > = 1, ux y]xk 2 I: (10) ( ux + z y](x + z)3k I. ( , 3X k ;1 ux y]xizx3k;i;1 + uy z]x3k: (11) i=0
> i > k, ux y]xizx3k;i;1 2 I (10). > i < k, I 3 ux y]xiz xk]x2k;i;1 = ux y]xizx3k;i;1 ; ux y]xi+kzx2k;i;1, (10) ux y]xizx3k;i;1 2 I, 2k ; i ; 1 > k. H (11) uy z]x3k 2 I. 0 M *, uy z] x3k 2 I, ** NH, u1y z]u2 2 I \ P (s ) * & s0 . E .
266
. .
M , (S(I)= ) , F=I , (4). = * , *, jw1j jw2j > 1. = & , & I | T- F , x1 : : :xk xk+1 xk+2]xk+3 : : :xk+3+l x1 x2]x3 x4], k l > 1. 2 , (S(I)= ) m | & * & ' . 7 x2x1x3 : : :xm xm+2 xm+1 ** * x2 x1]x3 : : :xm xm+2 xm+1] I. ( * , ** * I, x2x1x3 : : :xm x3 : : :xm xm+2 xm+1 | I. E n xn 2 I, * * x y]xm;2 = 0 xm;2 x y] = 0. H * . ( W K, * fxiyxm;i;1 : i = 0 : : : m ; 1g. 2 W0 W , * fxiyxm;i;1 : i = 1 : : : m ; 1g. 1 &, u1u2 u3]u4 2 W0 u1 u2]u3 u4] 2 W0, & u1u2 u3u4 2 W , ju1j > k, ju4j > l u1 u2u3 u4 2 W . 7 , &, x y]xm;2 2= W0 . / , x y]xm;2 2= I, ) (S(I)= ). " (. H. 1 *.
!
1] W. Specht. Gesetze in Ringen // Math. Z. | 1950. | Vol. 52, no. 5. | P. 557{589. 2] . . . !" # $ #%& '( // '( '( ). | 1987. | * 5. | +. 597{641. 3] . . . / % ( , ( 0% ) 0!"% PI-'(%. | 2 .. . . )". 3 .- . ). | /# ), 1981. 4] 4. /. 5 %6#. 7 #% % # " T- "' // +89. | 1963. | :. 4, * 5. | +. 1122{1127. 5] G. Higman. On a conjecture of Nagata // Proc. Cambrige Philos. Soc. | 1956. | Vol. 52, no. 1. | P. 1{4. ' ( ) 1998 .
; y v ; z w + z + x = 4t
; x u y
. .
. . . 511.3
: , .
,
$ $ . x y z t u v w
; x u
y
+
; y v
z
+
; z w
x
= 4 ! "t
Abstract M. Z. Garaev, On the diophantine equation ; xy u + ; zy v + ; xz w = 4 , Funda-
mentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 267{270. It is proved that the equation positive integers . x y z t u v w
x
; x u
y
+
; y v
z
+
; z w
x
t
= 4 has no solutions in t
1.
1] , n n2 +3n+9 n ; 3 3k + 2, x+ y + z =n (1) y z x
$% & & x y z. ( )% (1) $ n = ;6 (. (. +, , n = 1 | (.. . +. / & n 2 f;1 5g | 2. (. 3 . 2] , $ (1) & n, 4& 3. 3] (1) , & & x y z. 5 , n 2 f4k 8k ; 1 22m+1(2k ; 1) + 3g, k m , , (1) $% , & & x y z. 8 , 4% 4 . .
, 2001, 7, , 1, . 267{270. c 2001 , !" #$ %
268
. .
1.
xu + yv + z w = 4t xyz (2) x y z t u v w (x y z) = 1. . n1 n2 n3 | , n = n1 + n2 + n3, xn +yn +z n = 4t xn1 yn2 z n3 x y z t. 2. x u + y v + z w = 4t (3) y z x
x y z t u v w. x
2. 1
( , 9 : ;. < , , x y z t u v w , (2) (x y z) = 1. . , a) uvw 9 b) 9 . 8 % a) uvw | 9 . = 4 . ,, u | 9 . > xu + yv 0 (mod z(4t xy ; z w;1 )) , v | 9 . @, w | . 9 . / , y, z | 9 . < ,, , z | 9 . A y = 2k y1 , k | , , y1 | 9 , . < , u 9, vw 9, xu = ;yv (mod (4t xy ; z w;1)) % B , v k t xy1 ; z w;1 ; y 4 2 1 ;1 2 1 = 4t xy ; z w;1 = ;(;1) = y1 ;1 = ;1: 1 ;1 2 = ;(;1) y y
y
1
< a). 8 , % b) uvw | 9 . + x y z 9 & 9 . = ,, z 9, xy 9. A xy = 2k y1 , k | ,, y1 | 9 . 5 xu ;yv (mod z(4t xy ; z w;1))
; x u y
+
; y v z
+
; z w x
= 4t
269
a) , w;1 k ;1 = ;1: 1 = 4t xy;;xyz w;1 = ; 4 2k ty2 y;1 z w;1 = ;(;1) 12 ;zy 1 1 < b). 5 1 . y
x
3. 2
( , )% . 9 : ;. < , , x y z t u v w , (3). A , . , (x y z) = 1. A (y z) = d1 (z x) = d2 (x y) = d3: 5 (x y z) = 1 , 4 , x1 y1 z1, x = d2d3x1 y = d1d3y1 z = d1 d2z1 : (4) < ) (d2x1 d1y1 ) = (d3x1 d1z1 ) = (d3 y1 d2z1 ) = 1: (5) < (5) (3) 9 d2x1 u + d3y1 v + d1z1 w = 4t: (6) d1y1 d2z1 d3 x1 B, (d2 z1 )v (d3x1)w 0 (mod du1 ) (d1 y1 )u (d3x1)w 0 (mod z1v ): + 9 (5) z1v 0 (mod du1 ) du1 0 (mod z1v ) du1 = z1v : @ dw3 = y1u dv2 = xw1 :
270
. .
< $ , 4 , X Y Z, x1 = X ( ) d2 = X ( ) y1 = Y ( ) d3 = Y ( ) z1 = Z ( ) d1 = Z ( ) : < (6), , + + + + + + X ( ) + Y ( ) + Z ( ) = 4t X ( ) Y ( ) Z ( ) : / , (5) , (X Y Z) = 1. 3 $ % 1, 2. . > , 2 , x u + y v + z w = 2t y z x uv
vw wu vw
uv
vw wu wu
v vw
w vw
w wu
u wu
u uv
v uv
uv
vw wu uv
vw vw
wu wu
uv uv
x y z t u v w | , , 9 u 6 v 6 w, u = v = 1. D ) w = 1, ) , $% , & & x y z t ( . 3]). < & 49 & w $? A ) .
. , F . G. H 4, .
1] Erik Dofs. On some classes of homogeneous ternary cubic Diophantine equations // Ark. Mat. | 1975. | V. 13. | P. 29{72. P; 2] Maurice Craig. Integer values of xyz2 // J. Number Theory. | 1978. | V. 10. | P. 62{63. 3] . . . ! " # // $ % &'(. | 1997. | $. 218. | ). 99{108.
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Abstract
V. V. Dubrovskii, On the asymptotics of spectral function for ordinary dierential selfadjoint operator, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 271{274.
In this paper the author investigated the asymptotics of spectral function for ordinary di-erential selfadjoint operator de.ned by regular boundary conditions.
+ ,
!0 1]. & i | ,
!0 1], i
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2 !0 1]
, ,- i , . . | - ,, i = i i , ( i j ) = ij ) - , !0 1] , ( )) i | + , ( ) i( ) | 2!0 1]
+ , ,- i , . . ( + ) i = i i , R1 ( i j ) = i /j = ij . 0 0
, (. !1]) 02 ; 4 1 ;1 6= 0 j i ; i;1 j > const n;1 0) j i ( )j 6 const 8 2 !0 ] = 1 1 T
P
n
v
L
Tv
x
T
v
v v
P
T
u
T
u u
p x
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u u dx
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272
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i (x)ui(y) ;
N X
u
i=1
i (x)vi (y):
v
i=1
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X i ( ) i( ) u
i 6
x u
x
(1), (1)
, T ( ) T +P ( ) + . 0
, o
K
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K
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x y
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T +P (x y )
=
T (x y ) ;
Z1
K
K
T (x z )p(z )KT +P (z y ) dz
(1)
0
K
T (x y )
T +P (
K
=
1 v (x)v (y) X i i
)=
x y
i;
i=1
1 u (x)u (y) X i i i=1
i;
, ( 1 (0 1). & (1) ;1 N = f j j j = ( N + N +1 )2 g
( (1) , ;N = f j Re = ( N + N +1 )2;1g ( , (1) j j;2) , L
K
N X
i(x)ui (y)
u
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Z1 K
T (x z )p(z )KT +P (z y ) dz d:
;N 0
5, 6 (. !2]), , 1 1 Z Z p KT (x z )p(z )KT +P (z y ) d dz = 2 ;1 p 0 ;N Z1 Z p 1=p 1 = 2 KT (x z )p(z )KT +P (z y ) d dz 6 0 ;N
273
6 21
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T +P (
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i (x)vi (y) +
i=1 N X i=1
;n+2))
+ 1 ( ) + 2 ( ) + + k ( ) + ( ;(n;1)(k+1)+1) 1( ) 2( ) k( ) | , (1) . 8 !6] 2.
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Abstract
M. G. Esmaganbetov, Minimization of exact constants in Jackson type inequalities and diameters of functions belonging to L2 %0 2], Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 275{280.
We obtain a series of results related to minimization of exact constans in Jackson type inequalities as well as the diameters of functions belonging to 2 %0 2 ]. L
L2 | 2- 12 Z2 1 2 kf k2 = jf(x)j dx 0
" # 1 X f k cos(kx + ') (0 0): k=1
& L2 ' , ) " * +" f () 2 L2 (L02 L2 f 0 f): ,+ Sn;1 (f () - x) | ( > 0)- " " n ; 1 " #
f () (x), , . ,
+. + '
f () 2 L2 + Tn . n ; 1 kf () (x) ; Tn (x)k = En;1(f () ) = E(f () - K2Tn;1) = inf Tn 21 1 X ( ) ( ) 2 2 = kf (x) ; Sn;1 (f - x)k = k k (1) k=n
, 2001, 7, - 1, . 275{280. c 2001 !"#, $% &' (
276
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K2Tn;1 | (2n ; 1)- + L2 . N- 2 + , + + H L2 3 +3) + : dN dN (H L2 ) = inf sup inf kf ; uk KN f 2H u2KN
N N (H L2) = inf
inf
sup kf ; Af k
N N (H L2) = inf
inf
sup kf ; Af k
KN A2L(L2 KN ) f 2H
KN A2L1 (L2 KN ) f 2H
L(L2 KN ) | ' + L2 N- KN , L1 (L2 KN ) | ' L(L2 KN ), . . + A + " KN , Af = f +" f 2 KN . X 1 r r k !r (f- ) = sup k5h f(x)k = sup (;1) k f(x + (r ; k)h) | 06h6 06h6 k=0 + + " r > 0
f(x). 2 2 ( ) r & Wr (f - ) !rr (f () - t) 60 ] cos 2t , +""
0 R 1r ( ) r B 0 !r (f - t) cos 2 t dt CC Wr (f () - ) = B B@ R CA : cos t dt 2
2
0
2
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HD r(!) = f!r (f () - ) 6 !( )g + " + . + Hr(!), E Wr (f () - ) , - , + +"
+. + '
.
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+ ' 1975 . . 2 +. + E 62{6] + + + " ' !( ) + + + +" . 8 + " + Hr (!( )) + + + ' !( ). 2 , +" 8. . 2 67], . +
+ KNT ) XrN (L2 L2 KN ) = sup WE(f (2) f 2L2 f 6=const r (f () - ) N, . . + + " + (2) KN L2 N (8r > 1, > 0, N = 1 2 3 : : :): KNT ) XrN (L2 L2) = inf sup WE(f (3) ( KN f 2L2 f 6=const r (f ) - ) , + (3) 3 + Hr (!( )) 0 < !( ) < 1. : , r = 1, = 0 1 2 : ::, 0 < 6 n , N = 2n ; 1, n = 1 2 : : : (2) . 64]. 1. r > 1, > 0, 0 < 6 n , N = 2n ; 1 N = 2n ( sin n )2 ; 2( n)2 r2 1 !( ): (4) dN = N = N = n 4 2 2; 4( n)2 Sn;1 (f- x).
. < r > 1, , ""
Zb X n a
0 < p 6 1,
Z X 1 0
>
k=n
1 X
k=n
k=m
p p X n Zb
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>
k=m a
juk jp dx
2r r
2 sin kt2 k22k cos t dt 2 >
k22k
p 1
1
Z 0
2 r X Z r 1 2 sin kt2 cos 2
t dt = k22k 2(1 ; cos kt) cos 2 t dt : k=n
K +, 64], 0 < 6 n "
Z t dt '(y) = cos yt cos 2
0
0
278
. .
y. E +
2r kh 2 sin 2 k22k
1
X k5rh f () k2 = k=1
,
Z
(5)
r X Z r 1 2 2 ! (f - t) cos 2 t dt > k k 2(1 cos nt) cos 2 t dt : k=n 0 0 :3 (1) + 2
rr
()
3r 2 R ( ) r 6 !r (f - t) cos 2 t dt 77 kf ; Sn;1(f)k 6 n1 664 0R ; 75 2 nt cos t dt 2 sin 2
2
0
2
; + "" +" f(x) = cos nx: < Sn;1 (Tn;1(x)) = Tn;1(x) +" Tn;1 (x) 2 KnT;1 Sn;1 (f- x) 2 K2Tn;1 +" +3 f 2 L2 , +" + Hr(!( ))
2 R 3r 66 0 cos 2 t dt 77 !( ) 2n 6 2n;1 6 sup kf ; Sn;1(f)k 6 n 64 R ; 75 : (6) f 2Lr 2 sin nt2 2 cos 2 t dt 2
2
0
* + dN 6 N 6 N (6) + +" :
2 R cos t dt 3 r 77 66 2 0 dN 6 N 6 N 6 !( ) 75 : 6 n 4 R ; nt 2 2 sin 2 cos 2 t dt 2
(7)
0
& , .
2 R 3r 66 0 cos 2 t dt 77 T R = !( ) 75 U2n+1 = n 64 R ; nt 2 2 sin cos 2 t dt 0 8 2 R 3r 9 > > > > t dt cos n < = 6 7 2 X !( ) 6 7 0 = >Tn(x) = ck cos(kx + 'k ): kTn k 6 n 64 R ; 7 5> k=1 > 2 sin nt 2 cos 2 t dt > : 2
2
0
279
...
(2n + 1)- K2Tn+1 - + . ', Tn 2 Hr (!( )). * + (7) 0 6 kt2 6 nt2 6 2
2 R 3r () r 66 0 !r (Tn - t) cos 2 t dt 77 64 R cos t dt 75 = 2 3r 2 0 Pn ; r R 2 r 2 sin kh2 k2c2k cos 2 t dt 77 66 06sup h 6 t k =1 0 77 6 = 66 R cos d dt 4 5 2 0 2 R ; 3r 2 nt r 6 n 2 sin 2 cos 2 t dt 77 6 664 0 R 75 kTnk 6 !( ): cos d dt 2
2
1
2
2
2
0
2
E R Hr(!( )). < . (. 67, . 347]) + 3 r2 2 R 66 0 2 cos 2 t dt 77 !( ) d2n;1 > d2n > n 64 R ; 75 : 2 sin nt cos 2 d dt 0
:3 + (7) + 1. 2. > 0, r > 0, N = 1 2 3 : ::, 0 < ! 6 1. XrN (L2 L2) = dN (Hr(!( ))- L2 ): (8) ( )
. L2 Wr (f - ) = u > 0. ,+ f1 (x) = () ; 1 = u !( )f(x), Wr (f1 - ) = !( ), . . f1 2 Hr (!( )). M " + ' + 3 + E(f- KN )
Wr (f () - ) KN ) 6 sup E(f- K ): sup WE(fN ( (f f 2L2 r ) - ) f 2Hr " ' " KN L2 N, + Xrn (L2 L2 ) 6 dN (Hr (!( ))- L2 ): (9)
280
. .
G , +" +3
f 2 Hr(!( )) + + " +
Hr(!( )) KN ) E(f- KN ) 6 WE(f( (f ) - ) r
E +" ' KN , 3 + , (9). < (8) . 9 1 2 + . r > 1, > 0, 0 < 6 n , 0 6 !( ) 6 1, N = 2n ; 1 N = 2n ; n 2 ; 2( n)2 ; r2 XrN (L2 L2) = n1 4 sin2 2; 4( n)2 !( ):
1] . . " L2 0 2] // & . ' . | 1967. | ,. 2, . 5. | 0. 513{522. 2] , " 3. 4. " " L2 // & . ' . | 1977. | ,. 22, . 4. | 0. 535{542. 3] , " 3. 4. 0 " ' L2 // & . ' . | 1979. | ,. 25, . 2. | 0. 217{223. 4] 6 " . "
" L2
// 7 8
. 0 ". | 9
'
"
-
" " ,
1986. | 0. 3{10. 5] ;" 4. 4. " L2 " , < < " " <" // =. . . | 1991. | ,. 43, . 1. | 0. 125{129. 6] > " &. ?. " L2 " // &< < @A 8 ", < <, 'B, "
) * + 1997 .
t-
. .
, . 517.986
: p- , p- , f - , f -.
! t- , " # $ f - , # %# $& '( ) $ 1. * + % & $ $ $$& (f - ) ") % $ & $.$ + ! t- .
Abstract L. B. Luchishina, The properties of the t-adic integers, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 281{284. In the article we introduce the t-adic algebra depending on certain f -type, generalizing usual Baer type for groups of rank 1. We consider the characteristics of invariants of the constructed algebra (f -characteristics) and study some characteristics of the divisibility relations of the t-adic integers.
^ p, Q^ p Z, Zpm , Q, Z , pm , , p- , p- , p | , m | ^ p x], Q^ p x] . Z x], Q x], Zpm x], Z ! " "
, , pm , p- , p- . f 2 Q x] . # Q^ p x] f = r1 r2 : : : rs , % rj (j = 1 s) , . ) rj Ip (f ), ! , SIp (f ) = = fr 2 Q^ p x] j rjf r g 1]. I (f ) = Ip (f ). p # 1] ** f - f - , + + ,- * ! ! 1. 1. " fkr j r 2 I (f )g 1 * f - ", kr = 1 * ^ p x]. f - * (kr ) 1]. *! r 2= Z , 2001, + 7, 3 1, . 281{284. c 2001 !, "#
$% &
282
. .
2. 1 (kr ) (lr ) * - , * kr = 1 ! ! , ! lr = 1 1]. 3. 2 - (kr )] f - (kr ) * f - 1]. f - t u : : :. # 2] * ! - * *! ,- . ! ! Qft t- * *! f - t. 3 " f - t = (mr )], * ! ! Zft = Q ^ p x] + (r))=(r), mr = 1, Kr = Z ^ p x]=(pmr r), = Kr , ! Kr = (Z r2I (f ) mr < 1. 5 * * r = x + (r), , ! , r(r ) = 0, Kr ^ p r ] mr = 1 r 2 Z ^ p x]. Z ^ p x]. Q^ p r ] mr = 1 r 2= Z Zpmr r ] mr < 1: 7 * 8 , * p ! r 2 I (f ) p- -88 , B = fr 2 I (f ) j r 2 ^ p x] n Z ^ p x]g. 2Q g = (gr )r2I (f ) 2 Zft = Q Kr . 9! r- gr r2I (f ) gr = a0 + a1 r + : : : + an;1nr ;1 (1) ! n = deg r(x), -88 ai (i = 1 n) | p- mr = 1 - Zpmr , mr < 1. 4. gr | r- - g = (gr )r2I(f ) p | , + r. : ;" kr , " pkr -88 a0 a1 : : : an;1 (1) % r-" - g. 5. : (kr ), +" r- ! - g 2 Zft * ! r 2 I (f ), " - g 2 Zft H (g) = (kr ). 1. < H (g) f - . ^ p x], Z ^ p r ] = Q^ p r ] *** , 1" , r 2 Q^ p x] n Z - * kr = 1. B = fr1 r2 : : : rsg, ! ! Kr1 Kr2 : : : Krs Zft , " Btf = L Kr . 2. g
, H (g) = (mr ).
2 Zft ,
r2B
t = (mr )]. g 2 Btf
t-
283
. g 2 Btf g = (gr ). 9! r 2 B
^ p r ] *** kr (gr ) = 1. r 2= B gr = 0 Z
kr = mr . 9 , H (g) - g f - " (mr ). ) , H (g) = (mr ) g = (gr )r2I (f ) , kr (gr ) = mr * ! r 2 I (f ). 1 , gr = 0 * ! r 2= B . 1" , r 2= B , - , -88 ! gr ** pmr , , mr < 1?, ! p, , mr = 1. 9 , -88 gr = 0. 3. g h 2 Zft, t = (mr )]. H (g h) = = (H (g) + H (h)) ^ (mr ). . g = (gr )r2I(f ) , h = (hr )r2I(f ) . ) H (g) = = (kr ), H (h) = (lr ) H (gh) = (sr ). ? 8 ! r = r(x) 2 I (f ). , mr = 1, ! kr lr ! (
1). @ kr = lr = 1, gr hr 2 Q^ p r ] gr = hr = 0. 9! A sr = kr + lr . @ kr < 1, lr = 1, sr = 1, ^ p r ] sr = kr + lr . @ kr < 1, lr < 1, gr hr 2 Z A sr = kr + lr . , , mr < 1. # - kr < mr , lr < mr . - gr = pkr gr0 hr = plr h0r , ! gr0 h0r | ! . 9! ( kr +lr 0 0 gr hr kr + lr < mr gr hr = p 0 kr + lr > mr : * , H (g h) = (H (g) + H (h)) ^ (mr ). 4. g 2 Zft. g = (gr )r2I(f )nB Zft , H (g) = 0. g = (gr )r2B g 6= 0, g Zft . . ) , H (g) = 0 ! ! , ! r- - g = (gr )r2I (f )nB " ! . @ gr = a0 + a1 r + : : :+ an;1 nr ;1 | " ! Kr , + ur = b0 + b1r + : : : + bn;1nr ;1, " ur gr = 1 Kr . 1" , ! g r Q^ p x] ! ! , ! + ^ p x], u1 g + v1 r = 1. 1 u1 v1 2 Q ^ p x] *+ ; p, Z u g + v r = p : (2) p, uB gB + vB rB = 0 Zp x], % rB * r. 9! rBjuBgB , , rBjuB rBjgB. # ^ p x]. 5 " ! g r rBjuB, u = rs + pl, ! r s l 2 Z * " * (2), (rs + pl) g + v r = p ,
284
. .
r(s g + v) + p l g = p . r , r s0 + l g = p ;1, ! ! . ^ p x] , * r- , ur gr = 1 Kr , 9! u g + v r = 1 Z g = (gr )r2I (f )nB . @ g = (gr )r2B g = 6 0, g Zft . 5. g h 2 Zft, t = (mr )]. g h , H (g) 6 H (h). . @ g h, h = g u, ! H (h) = H (g u) = = (H (g) + H (u)) ^ (mr ), , , H (g) 6 H (h). ) , H (g) = H (h), H (g) = (lr ) = (kr ) = H (h), kr = lr * r 2 I (f ). r- ! * ! p ! ! : gr = pkr gr0 , hr = plr h0r . 9! hr = plr h0r = pkr h0r = gr (gr0 );1 h0r . ? pkr = gr (gr0 );1 ! gr0 . @ H (g) < H (h), kr < lr * r 2 I (f ). 9! hr = plr h0r = = pkr +sr h0r = pkr h0r psr = gr (gr0 );1 h0r psr . # * ! h * ! g. 5 * 5 ** + *. 1. t- g1 g2 : : : gs 1 ! d, " # H (d) = H (g ) ^ H (g2) ^ : : : ^ ^ H (gs ). 2. t- g1 g2 : : : gs 1 ! v, " # H (v) = H (g ) _ H (g2 ) _ : : : _ _ H (gs ).
1] A. Fomin, O. Mutzbauer. Torsion-free abelian -irredusible groups of nite rank // Comm. Alg. | 1994. | Vol. 22. | P. 3741{3754. 2] . . . ! -"!# $ %& // " $". | 1989. | (. 28, ) 1. |*. 83{104. ' ( 1999 .
. .
514.762
: , , ! " .
# $ %& & ' &( %) ) $ ! , ! " , ' &( ( ) & ( ! " & * $ ! ! " 4.
Abstract
Yu. F. Pastukhov, Necessary conditions in the inverse variational problem, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 285{288.
In the paper the inverse variational problem in strati0ed speed spaces of arbitrary order is invariantly formulated, Lagrange cut is de0ned, and an analytic reformulation of the problem is given. We give a necessary condition of a system of ODE of even order not less than 4 to be a Lagrangian system. Tk Xm |
k
Xm , kl : Tl Xm Tk Xm , l > k > 0, |
( k = 0 | |
Xm ). , Xm |
. U (v0 2xn;1) | v0 2xn;1 T2n;1Xm . 1. # $ f : TkXm Tl Xm (0 6 k < l)
!
!
% , &' : f - Tl Xm Tk Xm
Qid Qs
+ l k
Tk Xm 2.
f T2nXm , f = f (U ) = fvx2n 2 T2nXm j vx2n = f (vx2n;1 ) vx2n;1 2 U (v0 2xn;1)g
% ( )
, $
% f : T2n;1Xm U (v0 x2n;1) ! T2nXm : , 2001, & 7, 1 1, . 285{288. c 2001 , ! "# !! $
286
. .
3.
"L = "L (U ) = fvx2n 2 T2nXm j "(x)L(vx2n ) = 0 2 Rmg T2nXm L : n2n;1U ! R | % $
+ , L(x x_ : : : x(n)) |
(x), % $ %
U (v0 2xn;1) T2n;1Xm , , - &, % % $ %
"L T2n Xm . . "(x)L : 22nn;1U ! R | + ,
(x) Xm T2nXm n (n) X "(x) L (x x_ : : : x(2n)) = (;1)k Dkt @L(x@x: :(k: ) x ) i = 1 m: k=0 i
i
0 11],
% %3
(x) Xm
T2nXm . 4 &' 11,2]: f : T2n;1Xm U (v0 2xn;1) ! T2nXm | . 4' & U~ (v0 2xn;1) U (v02xn;1) % $
+ L : n2n;1U~ ! R, f (U~ ) = "L (U~ ). 4.n # + L : Tn Xm ! R % % n $
vx 2 T Xm ,
(x) Xm 2L det @x(n@) @x (x x_ : : : x(n)) 6= 0 (n) L(x x_ : : : x(n)) |
+ L : Tn Xm ! R
(x). 0 11],
% %3
Xm . ( ). f : T2n;1Xm U (v0 2xn;1) ! T2nXm | , U~ (v0 2xn;1) U (v0 2xn;1) | v0 2xn;1 2 2 T2n;1Xm , L : n2n;1U~ ! R |
. f (U~ ) = "L (U~ ) , "(x)L f jU~ (v0 2 ;1 ) : T2n;1Xm U (v0 2xn;1) ! Rm 0 2 Rm
U~ (v0 2xn;1). k
n x
i
287
5. f : T2n;1Xm U (v02xn;12)n;!1 T22nn;2n1;X1m | -
,
v0x 2 T Xm . 4 f
% $ % v0 2xn;1, ' U~ (v0 2xn;1) U (v0 2xn;1) % $
+ L : n2n;1U~ ! R, f (U~ ) = "L (U~ ). 7, +
%8 (
%3 3 ) %3
3 ' % $
% 0 {: $, 8 & 83 %3, $ ;<=
&. >
% : ' & $ % %
%
%3 ++ %3 ? ;% , n > 1 ' % 3 % . . x(2n) = fi(xk0 x_ k1 : : : x(2n;1)k2 ;1 ), i kl = 1 m, | (x) Xm f : T2n;1Xm U (v0 2xn;1) ! T2nXm n > 1 ! U~ (v0 2xn;1) U (v02xn;1) (U~ (v02xn;1) '), ' : U~ ! R2mn | '(U~ (v02xn;1)) = U~(x) (x0 : : : x(20 n;1)) R2mn '(vx2n;1 ) = (xi0 x_ i1 : : : x(2n;1)i2 ;1 ) il = 1 m l = 0 2n ; 1: n X X :::j (x : : : x(n))x(k1) j1 : : : x(k ) j fi (x x_ : : : x2n;1) = Ckij11:::k n
i
n
r=1 n+16k1 :::k r 6rn;1 n r(n+1)6 P ki 6(r+1)n
r r
r
r
i=1
:::jr (x x
Ckij11:::k _ : : : x(n)) | n2n;1(U~(x) ), r 2 n ; 1 2 mn n : R Rm(n+1) | : n2n;1(xk0 x_ k1 : : : x(2n;1)k2n;1 ) = (xk0 x_ k1 : : : x(n)kn ):
!
. .
fi (x : : : x2n;1) = ('0 ij (x : : : x(n)) + '1 ijp(x : : : x(n))x(n+1) p ) x(2n;1)j + + gi(x : : : x(2n;2))
'0 ij (x : : : x(n)) '1 ijp(x : : : x(n)), i j p = 1 m, | n2n;1U~ (x), n2n;1 : R2mn ! Rm(n+1) | , gi(x : : : x(2n;2)) | 22nn;;21U~(x) ,
288 22nn;;21 :
. .
R2mn ! Rm(2n;1) | :
22nn;;21(xk0 : : : x(2n;1)k2 ;1 ) = (xk0 : : : x(2n;2)k2 ;2 ): n
n
1] . . .
. | !. "#$, & 1328-"-96. 2] . -. . / 00 12 2. | 3.: 3, 1989.
% ! & 1998 .
- . .
517.929
: - , , ! ", # , $% , % &' .
-" '" # # ' %( # ' ! ".
Abstract L. E. Rossovskii, Strongly elliptic dierence-dierential operators in semibounded cylinder, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 289{293.
In the paper we consider di/erence-di/erential operator in semibounded cylinder and obtain necessary and su0cient conditions for a G1arding-type inequality using a symbol of the operator.
- X AR = DR D jjjj6m
Q = f(x1 x0) 2 Rn : x1 > 0 x0 2 Gg G ; Rn;1 | P ( @G 2 C 1 n > 3), 1 ; 1 @ n 1 @ D = i @x1 : : : i @xn , R u(x) = aj u(x1 + j x0), aj 2 C . %0 j j j 6 J u(x1 + j x ) = 0 x1 + j 6 0. &' AR ( ) Q* , + u 2 C_ 1 (Q) (1) Re(AR u u)L2(Q) > c1kuk2W m (Q) ; c2kuk2L2 (Q) 2%
34 2 4" ' # ( , " 5 95{01{00247. , 2001, # 7, 5 1, . 289{293. c 2001 , !" #$ %
290
. .
c1 > 0, c2 u. 0 W n (Q) 1 ) , 23+ L2 (Q) P 3' ) m )4 , kuk2W m (Q) = kD uk2L2(Q) . jj6m 5
- + + 6. 7. 1) ) 81]. : )) + , ) ( , 3 + . ; + ) , . ; , (0 d) G + , d 2= Z. % + ) + ( )
- (. )2 82], 3 + < )( ) -
+ 2 2 ' ).
1.
% R : L2 (Rn) ! L2 (Rn) : X Ru(x) = aj u(x1 + j x0) aj 2 C : jj j6J
(2)
= >k = f(x1 x0) 2 Rn : k ; 1 < x1 < kg. ? ) S N
u 2 L2 >k (N = 1 2 : : :) 2 ' Rn, k=1 (2) ) ) ' S S N N R : L2 >k ! L2 >k . k=1
k=1
; ' UN : L2
S N
k=1
>k ! LN2 (>1 ),
(UN u)k (x) = u(x1 + k ; 1 x ) (x 2 >1 @ k = 1 : : : N ): = RN ) N N ( ( (RN )km = am;k jm ; kj 6 J 0 jm ; kj > J: 0
S N
S N
(3)
A R : L2 >k ! L2 >k () k=1 k=1 2 RN )- ) LN2 (>1 ): R = UN;1RN UN : (4)
-
5 , (2) (3) (UN Ru)k = =
X
jj j6J
N X
aj u(x1 + j + k ; 1 x0) =
X jm;kj6J m=1:::N
291
am;k u(x1 + m ; 1 x0) =
(RN )km (UN u)m (x) = (RN UN u)k (x) (x 2 >1@ k = 1 : : : N ):
m=1
1. RN + RN (RN ) N = 1 2 : : :. (R + R ): L2 (Rn) ! L2 (Rn) . . ;' 4 4 ) 4 u 2 L2(Rn0). 5 )+ h 2 R N ) uh , uh (x) = u(x1 + h x ), S N >k . % (4), 2 L2 k=1
Re(Ru u)L2(
Rn = Re(Ruh uh)L )
2
;S N
k=1
k
=
= Re(UN Ruh UN uh )LN2 ( 1 ) = 21 ((RN + RN )UN uh UN uh )LN2 ( 1 ) > > ckUN uh k2LN2 ( 1) = ckuk2 ; SN = ckuk2L2( n) L2
k=1
k
R
c > 0 N = 1 2 : : :. ; L2 (Rn) 2 + ) 4 u 2 L2(Rn). 7 ) . % + ) 4 C , Re(Ru u)L2( n) = (Re r( )~u u~)L2 ( n) P r( ) = r(1 ) = aj eij1 | . = ,
R
R
jj j6J
2 ' (R + R ): L2 (Rn) ! L2 (Rn) () 2 . =4 1 2. + RN P RNij aj e 1 > 0 (1 2 R). N = 1 2 : : :. Re jj j6J
2. !
. AR ! Q* ,
292
. .
Re
X
X
jjjj=m jj j6J
aj eij + > 0 ( 2 R 0 6= 2 Rn):
. & + . = Qk = Q \ >k
) N
S N 4 ) 4 u 2 C_ 1 Qk . A k=1 V = Un u | )- ) C_ 1N (Q1 ). E (4), X Re(AR u u)L2(Q) = Re (R D u Du) SN =
= Re = Re
k )
(
jjjj6m
X
k=1
jjjj6m
(UN R D u UN D u)LN2 ( 1 ) =
X
jjjj6m
(RN D V DV )LN2 ( 1 ) = Re
X
(RN D V DV )LN2 (Q1 ) :
jjjj6m
6 , kukL2(Q) = kV kLN2 (Q1 ) , kukW m (Q) = kV k2W mN (Q1 ) . A) , ) (1) X Re (RN D V D V )LN2 (Q1 ) > c1 kV k2W mN (Q1 ) ; c2 kV k2LN2 (Q1 ) 2
2
2
jjjj6m
) c1 > 0, c2 N V 2 C_ 1N (Q1). % P 4 ( Q* 1
+ D RN D , ) (. jjjj6m
P
81{3]), (RN + RN ) + 0 6= 2 Rn 2 jjjj=m ) , 3 N . ; 2 X Re r ( ) + > 0 ( 2 R 0 6= 2 Rn): jjjj=m
5 . % u 2 C_ 1 (Q). E % < , Re
X
jjjj=m
DR D u u
% 4 X
k1 > 0.
jjjj=m
L2 (Q)
X
=
Rn :
(Re r(1 ) u~ u~)L2 (
jjjj=m
Re r ( ) + > k1jxij2m ( 2 R 2 Rn)
; 2 = 1 . A
X
jjjj=m
Re r(1 ) + u~ u~
R > k kjj u~kL Rn :
L2 ( n)
1
m
2
2(
)
)
-
293
1 % < , )2 () + m + W (Q), Re
X jjjj=m
D R D u u
L2 (Q)
> k1kuk2W m (Q) :
= ) <+ ' Re(AR u u)L2(Q) > k1 kuk2W m (Q) ; k2kukW m (Q) kukW m;1 (Q) : =4 + tkukW m;1 (Q) 6 c(kukW m (Q) + tm kukL2(Q) ) p1p2 6 t;m p21 + tm p22 (p1 p2 2 R@ t > 0) , Re(AR u u)L2(Q) > (k1 ; k3t;1 )kuk2W m (Q) ; k4 t2m;1kuk2L2(Q) : ; t > 0 ), (k1 ; k3t;1 ) > 0, (1). A ) . ;
- | ) , | + +: 2 Rn, 43
, 2 R, 43 (. 82]). E ), 2 ) , ) 4 , 2' ) ( ) = 1 ), ) + ( .
"
1] Skubachevskii A. L. The rst boundary value problem for strongly elliptic dierential-dierence equations // J. Dierential Equations. | 1986. | Vol. 63, no. 3. | P. 332{361. 2] . . !" #$ -%!!& & #$ '( " # & // )# . *#+& . | 1996. | ,. 59, . 1. | /. 103{113. 3] 12 ). 3. 4 $ $$5 6&( &+#( %!!& & #$ '( " # & // )# &+. 7 . | 1951. | ,. 29, . 3. | /. 615{676. & ' 1998 .
, . . 517.9
: , , , , .
!!" $!!" , $ $!!" % & '.
Abstract
Y. T. Silchenko, Linear dierential equation with non-densely dened operator coecient, generating a non-analytical semigroup, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 295{300.
The solvability of the Cauchy problem for the linear di.erential equation with operator coe/cient is established when this coe/cient is not densely de0ned and generates a semigroup with a singularity.
1. (1) v0 + Av = f(t) (0 < t 6 1) v(0) = v0 "# E. % A | ' ' "( D(A) = D, (* ' "' A;1 , f(t) | ' t > 0 E v0 | ' + E. (1) ' '
,0 1] , v0 (t) Av(t) * ( '' (0 1] ' ' (1). , ,1], "' ' ' ' " . % (1) '# '# . . , t > 0 * - U(t) = exp(;tA), " (*
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297
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3.
t > 0
% 2 # g(t) ##
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0
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(
Z Z f (s) ds = f t + ns ds f (t) = f(t) 0 6 t 6 1 fn (t) = n f(1) t > 1 t 0 1
Zt
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0
@ + f (t) 2 kfnk" 6 kf k" , fn (t) ! f(t), gn (t) ! g(t) E t > > 0. H fn (t) ' ' t > 0, + + > " ( gn(t) gn0 (t) = fn (t) ; Agn (t). @> , '# gn0 (t) > # ( t > > 0). J +
299
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0
0
0
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6
" =" 6 ct;;kfn ; f k"" =" kfn ; f k1; C0 : 0
0
@ kfn ; f k" 6 2kf k" , # , Agn(t) ! Ag(t) E t > > 0. @+ gn0 (t) ! f(t) ; Ag(t) ' t > 0. 8( > '. v1 (t) = U(t)v0 (3). H v1 (t)
(* (1). < ,2] , + ' t > 0, v0 2 D(A ), 4) ' 1. @+ v(t) (3) (1). = ' 1. 3. 6 1 - ' ' ' . @v + (;1)m @ 2m v = f(t x) t 2 (0 1] x 2 ,0 1] (8) @t @x2m v(n ) (0) + v(n ) (1) + T v = 0 = 1 2 : : : 2m ; r (9) Z1 0
'k (x)v(x) dx = 0 k = 1 2 : : : r
(10)
v(0 x) = v0 (x) x 2 ,0 1]: (11) % | ' , j j + j j 6= 0, 0 6 n1 6 : : : 6 n 6 6 n +1 6 : : : 6 2m ; 1, T | ' ' ' Wpk ;1(0 1) p 2 ,1 1), = 1 2 : : : 2m ; r, 0 6 r 6 2m, f'k (x)g | ''# '# ,0 1]. 8 , (9){(10) ( ' ,5]. K ( " 2m Lp (0 1). < 7 + A = (;1)m dxd 2m "( D, * v(x) Wp2m (0 1), (*# (9){(10) (+ > Lp ). 6 (8){(11) 7 " (1). ,7] , A " j arg j < ; , j j > R > 0 > 0 * , '1 (4) r = 1 + 2mp ; 2nm ( n
300
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'k (x)). @+ ,5, . 3, x 1] A > A( ) = 1 ; r, = 2 ; r. L n 2 , p1 1p + 2m), 2 ,0 1), " ' ' ' 1. @+ 2. $ : 1) n, ( . ,7]), p1 6 n < p1 + 2m 2) # f(t x) $ ' t Lp ( x) kf(t + 9t x) ; f(t x)k 6 cj9tj"t; 1 1] 2 ,0 1) " 2 ( 2nm ; 2mp 3) v0 (x) 2 D. !2m(8){(11) " v = v(t x), @ v #
@v @t @x2m t > 0 Lp .
1] . . . | !.: #$, 1967. 2] *$ +. ,., $ -. .. /01 0* 1 23 4 0 $2 , 536 4 3 // $ . 5. | 1986. | ,. 27, 9 4. | . 93{104. 3] $ -. .. > 4 * , 536 * $ 4 // ?@# /. | 1968. | ,. 183, 9 2. | . 292{295. 4] Da Prato G., Sinestrati E. DiBerential operators with non dense domain // Annali della scuola normale superiore. Di Pisa. | 1987. | Vol. 14. | P. 285{344. 5] F$ . F. { 5 . | H$: I, 1985. 6] Favini A., Yagi A. Abstract second order diBerential equations with applications // Funkcialaj Ekvacioj. | 1995. | Vol. 38. | P. 81{99. 7] *$ +. ,. > 4 4 * // K0 0. ! $. | 1997. | 9 6 (421). | . 32{36. * ! + 1998 .
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