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Abstract M. P. Burlakov, Kozyrev spaces
vol. 7 (2001), no. 2, pp. 319{328.
, Fundamentalnaya i prikladnaya matematika,
It is shown in the paper that some results of N. A. Kozyrev's theory on the properties of space and time could be obtained in the framework of generalized Riemann spaces.
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% %! * ! + # #, # %* ! 0 % # ! * & # ,# + #, # %# 5# # . ;#, +% #$ n = 2, #* # Gij #* $ ! " + ! # 1 0 0 1 Gij = g 0 1 + h ;1 0 * g11 = g22 = g, h12 = ;h21 = h. F gij # $ + # # x 2 M2 & # % * ! # -, # + 6 !! # 8, , %* ! # * # u cos sin , * | %* ! 0 % # ! u + ! " ! # !. % %* ! 0 % %! ! # ! u1 u2 ! # Gij "% # 3 !% + -: cos(u1 u2) = Gij ui1uj2 = g cos G + h sinG * G = 2 ; 1 | %* ! 0 % # ! u1 u2. 2" # + $ p 2g 2 = cos p 2h 2 = sin g +h g +h ! +% ! p ) cos ; = g2 + h2 cos(G ; ) (2) p cos + = g2 + h2 cos(G + ) * ; | %* ! 0 % # ! u1 u2 , + | %* ! 0 % # ! u2 u1 "">& ! ! # . F! " !, ! !, # "">& ! ! # %* ! 0 % # ! ! -# 3 ! # # + > | + * *. B ! 0 % 6+ !8 6 !8 > ! + # # " % %> # 3 " # # . = 67 ! 3 8 # ,% ! #$ + c2, + -> 63 ! 8, 5## + + # #$ + + > , + %- %- #.
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!# ! + ## %! + ! + # # M2 , # * # x 2 M2 # Gij + # % 1 0 0 1 Gij = a 0 ;1 + b ;1 0 : F* %3 # u1 = (ch 1 sh 1 ) u2 = (ch 2 sh 2 ), 3 ! # gij , + ! # Gij hu1 u2i = a ch(G) + b sh(G) hu2 u1i = a ch(G) ; b sh(G) * G = 2 ; 1 | + ! # "% # # # u1 # % u2. 2" ch = p 2a 2 sh = p 2b 2 a ;b a ;b ! +%! p ) ch '+ = a2 ; b2 ch(G + ) (3) p ch '; = a2 ; b2 ch(G ; ) * '+ | "% # # # u1 # % u2, '; | "% # # u2 u1 "">& + ! ! # . F! " !, "">& ! + ! ! + # # ! J $ ! # + , $ # * . % "% #% G + " # + # $ "% # , "% !! # #$"">& + ! ! # K " # ! ! J! + 5## + # $ "% # # # . . = 3 "#3 % #, # + #$ ! + #$- + # # 3 > . 2 +( #: 6) % ! %> #% # # + #, # + - # 3 3 + # # + # +#!% 0 # + !... 8 91]. / * #$, # $ ! # & 3! ! + # # - ! "">& + ! ! # 6 + #$8 ! ! + > , # + %3 "% # 3 + 3 & # # %$# %-> + # 95]. 2. F + $ ! +0 !, # "">& ! ! # "3 ! !3 ">3 " 0 # + # #- ! #! , ! + %# #%-# "> # # # $ #. D ## $ , ,% ! #$ % H$" #{< (# 95] Rij ; 12 gij R = 8 Tij (4)
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!! # # < (# ! #$ * !! # *. <# !0 #$, ! # + ! + # #- ! #$ "">& + ! + # # 94]. D ## $ ,
!# ! M4 | "">& + ! + # #- ! ! # ! # ! Gij = gij + hij ,, #$ !* "> * #+ Fijk = ;kij + Sijk , * ;kij | !! # ! #,, , Sijk | # % . P * # ! # Gij # Fijk ! -# ij p p rk Gij @G @xk ; FkiGpj ; Fkj Gip = 0 # #$ @gij + @hij = ;p g + ;p g + ;p h + ;p h + @xk @xk ki pj kj ip ki pj kj ip p g + S p h + S p h : (5) + Skip gpj + Skj ip ki pj kj ip Q! # $ ! # #, # % (5) !0 #$ # # $ # gij hij . D ## $ , " #% (5) >& , +% * + # i, j k, @gki + @hki = ;p g + ;p g + ;p h + ;p h + ji kp ji kp jk pi jk pi @xj @xj p p p p + Sjk gpi + Sji gkp + Sjk gpi + Sji hkp (6) @grj + @hkj = ;p g + ;p g + ;p h + ;p h + ij kp ij kp ik pj ik pj @xi @xi p p p p + Sik gpj + Sij gkp + Sik gpj + Sij hkp (7) # ! (5) (6) # (7), +%!
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% %-# # (9) (10) + -# * %#$ + ! 3 + # # # # ( # * ! 3 + # #. D ## $ , +% #$ % ! # ! + # # (Mn gij ), #* #$ 5#! + # # # + , !% (11) % # # ,, .$ !% % @gijk p p (13) @xk = ;kigpj + ;kj gipK + #! % # "> # ( 5#* % . R , # * % # #$ %!! !! # * !! # * # Gij = gij + hij , * gij | ! ! # , + # +
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# #$- + $ * !! # * # hij , % # -> * #% @hij + @hki + @hjk = 0 @xk @xj @xi i j d(hij dx ^ dx ) = 0. F! " !, 0 * ! ( + ! ) + # # ! # .
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325
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!# ! !% + # # - ! "">& + ! ! # , # %$## # + # $ + -# ! !! # , + , ! 3 . 3. 7% #$ M4 | + # #- ! + ! ( * ) ! # gij = gji, i j = 0 1 2 3K * ! *" M4 ""> ! # Gij = gij + hij , hij = ;hji ! "% ! #$ + # #! ! # gij . 2#! #! # "> # + # # , * " ! x Tx + # # %- *+ + #$ # $ * + # # Tx # x 2 M4 , # & 3! # G0 , = 1 2 3, | * ! # G0ij # x *+ + #$ x . 2 , # G0 "% # "">& ! # , # $ , + # %
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. 75#!% ! #& # + +$#$ #$ , !$ ! + # ! 3 ! " # # $ ! > + #.8 91] 7 # h~ , + # 63 ! 8 , %. % # # > , , , + #. D ## $ , + # ~a ~b 2 x "% %# ! #$ h~a ~bi = g(a1 b1 + a2 b2 + a3 b3) + + h1(a2 b3 ; a3 b2) + h2 (a3b1 ; a1b3 ) + h3 (a1 b2 ; a2b1 ) h~a ~bi = g(~a ~b) + (~h~a ~b) h~a b~ i = g(~a b~ ) ; (h~ ~a ~b) * ( ) | + . ) ~a ~b | p # , ~h = h~n, * h = h21 + h22 + h23 , # ! +% ! p h~a ~bi = g cos + h cos sin = g2 + h2 cos2 cos( ; ) p h~b~ai = g cos ; h cos sin = g2 + h2 cos2 cos( + ) * | %* ! 0 % # ! ~a b~ , | %* ! 0 % # ! h~ ~a b~ . <# , !% +-#, # %* # ~a ~b # ~b ~a + # # ! + # ! h~ (+ # ! + ! M4 )K 5## + # # + !%> # + > # # ! 0 % + #. !, + #, # %# # ~a ~b, %. % # "">& ! # !! # ! # ! h12 = ;h21 = h cos . = + ! + # % 63 ! 8 + " # + # # > ( % $ * ). =# # +( # "
327
5#! ! : 6<# + # ! -# ( # 0 #, # + # c2 i " 3 + 3 . = ! % ! + "0 # " -# ! 0 % ". 75#!%... 3 ! >->3 # # # # " * 3 ! # !, # " !% 3 % ! * ! # " # # # $ #$ 5#3 > 8 91]. =" % # 3 $ + # ~h, ! + & ! ! # % G0 % 0g 0 01 G0 = @0 g hA 0 ;h g "% ! ##$ h ,% . ! . F* + . " !* + # # R~ = Re~1 %* , + & + # ! h~ , ! dt "% # dh d, # #$ ! +% ! # > %$##, * + # !% + ! 3 : d dh v = RU R dt dt :
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328
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2] Kozyrev N. A. On the possibility of experimental investigation on the properties of time // Time in Science and Philosophy. | Prague, 1971. 3] )! *. . . | +, 1990. 4] )! *. . . /0 // 1 2 - 3 !3 4 0 0. 5 !. | 1987. | 6 3. 5] * 7., 8 ., . 9 . + 4. 8. 1. | *.: *, 1977. 6] )3< . ., = 9. >. . | *.: !, 1980. % ! & 1998 .
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Abstract M. A. Gil'man, E. E. Demidov, A. G. Mikheev, Optimal control of security portfolio, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 329{337.
Finding an optimal strategy for the security portfolio during a given period is formulated as a problem of linear programming. It is shown that if the restrictions on the risk or on the buy/sale volumes are omitted then the problem is decomposedinto some -one-stock. problems. This fact permits one to reduce the calculationcomplexity of the whole problem. Finally, for the optimization problem with the restrictions on the risk an approximate method is presented.
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1] Litvinov G. L., Maslov V. P. Correspondence principle for idempotent calculus and some computer applications // Preprint IHES. | IHES/M/95/33. | 1995. "+ 1996 #.
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517.946 .
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Abstract
A. B. Golovanchikov, I. E. Simonova, B. V. Simonov, The solution of diusion problem with integral boundary condition, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 339{349.
The di/usion model for apparatus of 0nite length with integral boundary condition is investigated. Existence and uniqueness of the solution are proved.
1]: @C + v @C = D @ 2 C (1) @t @x @x2 v | , " t
# x$ D | % & , ' ( $ C | & & " . * ( #. + ( ( , (1). +( , ( '# -# # ( (#, . . ( ( -#( / & & ( # : C (x 0) = C (x). 1 (# -# # , - & , . 2 , - , ( x = 0 (2) v C (0 t) = C v + D @C @x (0 t)
12 $" $ $ 3 1 %& "# " %, & 4 00{01{00042. , 2001, 7, 4 2, . 339{349. c 2001 !, "# $% &
340
. . , . . , . .
#, ( x = b) -
v C (b t) = C v + D @C (3) @x (b t): 6 ( 7 C (b t) C. 8 (/ % ( (3) @C (b t) = 0: (4) @x 9 (2), (4) # 7 (# 2 . 6 ( (4) ' , # " . . # % . *: 7" ( . 1. 6 : C = C (x t) 2 @c = (5) A @@xC2 B @C @x @t 7" ( 7 C (x 0) = '(x) 0 6 x 6 b (# C (0 t) = 1(t) 0 6 t < +
;
Zb
Zb
Zt
1
C (x t) dx = '(x) dx + 1( ) d
0
0
0
Zt
;
C (b ) d
(6)
0
A > 0, B > 0. = ( (6) , : 7 " -> ( b, ( (6) | ( " , #: : t ( ( b. ?# - ( . @ / ( C (x t) = ex+t u(x t) = 2BA , = B4A2 . @ ( 7" 7 ( . 2. 6 : u = u(x t) - ( ut = Auxx (7) 7" ( 7 u(x 0) = '1 (x) = '(x)e;B=(2A)x 0 6 x 6 b (# 2 u(0 t) = 2 (t) = 1 (t)e B4A t 0 6 t < + ;
;B 4A t
e
2
Zb 0
e 2BA x u(x t) dx =
Zb 0
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1
Bb ; e 2A
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e; B4A u(b ) d: (8) 2
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8' % ( , ( - ( ( (8) , : u(x t). @% - ( u(b t) ( 3(t) : ( 7" 7 ( . 3. 6 : u = u(x t) - ( (7), 7" u(x 0) = '1 (x) (0 6 x 6 b) ) u(0 t) = 2 (t) (0 6 t < + ) (9) u(b t) = 3(t) (0 6 t < + ) D & 7 3 (t) (8). E : ( 1 7" , ' ,: '(x) 7 # 7 7 : 7 (- # 7 7 0 b], 0 b], '(0) = '00(0) = '(b) = '00(b) = 0$ '0(0) = '0(b), 1(t) ' # #- & 0 + ), ( 0 + ), 1(0) = 0. + 3 (t) - - -7 7" ,
' : 3 (t) #- & 0 + ), 0 + ), 3(0) = 0. @ : ( 3 u(x t) = u1(x t)+ v(x t), & u1(x t) = 2 (t) + x(3 (t) 2 (t))=b (# (9), & v(x t) | # (# . F - , : ( 3 : 7 7" ( . 4. 6 : v = v(x t) - ( vt = Avxx + F (x t) (10) 7" v(x 0) = (x) v(0 t) = v(b t) = 0 F (x t) = (u1 )0t + A(u1)00xx = ddt2 (t) xb ddt3(t) ddt2 (t) (x) = v(x 0) = u(x 0) u1(x 0) = '1(x): 9 (8) #, - ( , # : Bb 2A 2A 4A 2 ;B t 4 A e 2A Bb b b + B 2 b = 3 (t)e 2 Zb Zt 2 2 = '(x) dx + 1 ( ) d + 2 (t)e; B4A t 4BA2 b 4BA2 b e 2BbA + 2BA 1 1
1
1
1
1
;
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;
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2A e; B4A t v(x t)e 2BA x dx e Bb 2
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342
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v(x t) = G(x t) ( ) d + 0
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0 0
1 X
n : G(x t) = 2b e;( nb )2 At sin n x sin b b
n=1
* , , " (11):
Zb
e 2BA x v(x t) dx =
0
Zb
e 2BA x
0
Zb
G(x t) ( ) d dx +
0
+
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e 2BA x
Zt Zb
G(x t )F ( ) d d dx = I1 + I2 : ;
0 0
0
@ / - I2 1 1 X 2A cos n 2A cos n 1 2b (t) X 1 cos n + e Bb e Bb I2 = 2b2 2 (t) 3 b b B B 2 2 2 2 2 n=1 n 1 + ( 2A n ) ] n=1 n 1 + ( 2A n ) ] ;
;
;
Z X Bb 1 2A + 2BA e;( nb )2 A(t; ) e cosB n b 21 (cos n 3( ) 2 ( )) d: 1 + ( 2A n ) 0 n=1 F ( (11) - 7" : 2A 4A2 B2 t Bb 2A ; 3 (t)e 4A e 2A Bb b B + B 2 b = Zb Zt B4A2 t 4A2 4A2 Bb 2 A ; 2 A = '(x) dx + 1( ) d + 2(t)e B2b B2 b e B t
;
;
;
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0
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1 X 2A cos n e Bb 1 + 2b2 2 (t) B b 2 2 n=1 n 1 + ( 2A n ) ] ;
;
1 2 b (t ) X e 2BbA cos n 1 cos n + 3 B b 2 2
2 n=1 n 1 + ( 2A n ) ] ;
Z Bb 1 X 2A + 2bA 3 ( ) e;( nb )2 A(t; ) e cosB n b 12 cos n d 1 + ( 2A n ) n=1 t
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3] , 7" #: 1 X e 2BbA cos n 1 = B b 2 2 n=1 n 1 + ( 2A n ) ] Bb 2A Bb 2 2 2 A Bb Bb
A 2 A 2 A = B 2 b2 (1 e ) + e Bb cosech 2A Bb cth 2A 1 X e 2BbA cos n 1 cos n = B b 2 2 n=1 n 1 + ( 2A n ) ] Bb 2A Bb 2 2 Bb Bb 2 A 2 A 2 A 2 A = B 2 b2 (1 e ) + e Bb cth 2A Bb cosech 2A : . / 7" - ( : ;
;
;
;
;
;
Z Z B Bb B4A2 t ; 2 A f (t) = 2A e e '(x) dx + 1 ( ) d b
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0
B e; 2BA Z e 2BA x Z G(x t) ( ) d dx + 2A b
;
t
t BZ
b
0
0
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Bb 2A e;( nb )2 A(t; ) cos nB eb 2 d = I3 I4 + I5 1 + ( 2A n ) n=1 0 1 ; Bb X 2 2A K (t) = 2BA e B4A t + Bb e;( nb )2 At cos nB e b 2 cos n: 1 + ( 2A n ) n=1 /# - ( , ( Bb Bb ; Bb 2 A 1 cth 2A + e cosech 2A = 0 Bb Bb ; Bb ; Bb 2 A 2 A e cosech 2A + e cth 2A = 0
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344
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lim x!x
Zb
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I
0 3 jt=0
3 Z = 8BA2 '(x) dx:
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0 5 jt=0
=
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1 X
n=1
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cos n e; 2BbA d 1 + ( 2BA nb )2 ;
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9 t 7, lim
Zb
t!+0
0
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Zb
Zb
e 2BA x
0
Zb
G(x t) 00( ) d dx:
0
0
G(x t) ( ) d dx = A t
0
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B 2 Z '(x) dx: 4A b
0
0
+-> #, ( ( f (+0) = 0. F 2 2 f (t) 6 C1 e B4A t + 1 + te B4A t B2 1t K (t) 6 C2 e 4A t + t & f (t) K (t) 7 . F - , % , & " 7 , - = : Lf ] + f (t), LK ] + K (t). F (15) : 7" : L3] LK ] = Lf ]. +7 : L3 ] = p pLf ] 1=(p2LK ]) : @ - = f 0 (t) + pLf ] f (0) = pLf ].
- = , ( - ' = LK ] & K (t) ( & - Re p > b2=(4A). 0
p
j
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p
f
g
f
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6 / - = K (t):
Z1
r1 B4A2 + (r1 B4A2 )2 + r22 0 1 X 2A r1 + ( nb )2 1 cos ne; Bb + Bb + n 2 b B 2 2 2 n=1 (r1 + ( b ) A) + r2 1 + ( 2A n ) 1 2A r2 BX r2 1 cos n e; Bb = + i 2BA (r1 B4A2 )2 + r22 b n=1 (r1 + ( nb )2 A)2 + r22 1 + ( 2BA nb )2 = C (r1 r2) + iD(r1 r2): (16) 2 F r1 > B =(4A), # LK ] , ( 1=LK ] - " - Re p > B 2 =(4A) % ( % - . +& - = K (t). LK ] = C 2(r1 r2) + D2 (r1 r2) 1=2 1 r1 + ( nb )2A + r2 B r1 B4A2 + r2 + B X 2A (r1 B4A2 )2 + r22 b n=1 (r1 + ( nb )2 A)2 + r22 A p 1 arctg p A r > B 2 : 1 + 1 b 4A r1 B4A2 + r2 r1 + r2 2 b r1 + r2 @ r1 > (k + 1)B 2 =(4A), ( k , ( B 2 k=(4A) > 1. p F r1 B 2 =(4A) > 1 r1 +jr21j; B2 < pr1 1+jr2 j , 2 arctg bA q (k+1) B2 6 2 4A 4 A p p arctg bA pr1+jr2 j . @% LK ] 1= r1 + r2 , 1=LK ] p . F 2 3 = 2 ( 7" 7 & : 1=(p LK ]) 1=p . % & ( r1 > (k + 1)B 2 =(4A) > 0) LK ] = e;pt K (t) dt = 2BA
;
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j
j
1
4A
F - , 1) & 1=(p2LK ]) ( Re p > (k + 1)B 2 =(4A), 2) 7 p 7- Re p > a > > (k + 1)B 2 =(4A) arg p, 3) aZ+i1 1 (17) p2 LK ] dp j j ! 1
a;i1
-7 , , & 1=(p2LK ]) " (. 4])
346
. . , . . , . .
aZ+i1 pt e g(t) = 21 i p2LK ] dp: a;i1
@ 27 4], (
Zt
3(t) = g( )f 00 (t ) d:
(18)
;
0
8 " : (15) . 6 , ( % : . D 3 (t), # ', (12), : C (x t) ( 1 . . 1 B4A2 t 2 C (x t) = 1 (t)e B4A t + x 3(t) b1 (t)e + d; Zb Zt Zb 2 ; 2BA + G(x t)'( )e d + G(x t ) d 1( )e B4A 0 0 0 3 d3( ) d ; ( )e B4A2 d d e 2BA x; B4A2 b d d 1 ;
;
;
;
;
3
;
1 X G(x t) = 2b e;( nb )2 At sin n x sin n b b n=1
d (LK ])(0) Zt dp (t) = ; f 00 (t ; ) d +
L2 K ](0)
Zt
0
Z 1 X 1 kn 00 + LK1](0) f 00(t ) d + d (LK ])(kn ) e f (t ) d k n n=0 dp t
;
; Bb 2A
e
;
0
6 = 1 + Ab
0
Bb ; 2A Zt
2 f 00 (t ) d + 3 (t) = d 1 ( L K ])(0) dp 0 ;
Zt d22 (LK ])(0) Zt 1 X 1 dp 00 + d f (t ; ) d + ekn f 00 (t ; ) d d 2( dp (LK ])(0))2 0 k ( L K ])( k ) n 0 n=1 n dp
347
!" # $ " $ !
2A = 1+ b ; Bb f (t) (13) LK ] | e; Bb A 2A K (t) ( . (14)) k0 k1 : : : |
1 X 2A B 1 1 1 cos n e; Bb = 0: 2 2 n Bb 2 2 B 2A (x 4A ) n=1 (x + ( b ) A) 1 + 2An . 6 / # 3(t). ( - = LK ] (. (16)). & LK ](p) , ( LK ](p) = 0 r2 = 0. @ r2 = 0. F 1 1 cos n e; 2BbA : 1 +BX 1 LK ](r1 0) = g(r1 ) = 2BA 2 Bb 1 + 2An r1 B4A b n=1 r1 + ( nb )2 A . # & g(r1 ) 7- , & ' " - & g(r1 ), ' & #: 1 2A 1 cos n e; Bb 1 BX 1 < 0: g0(r1 ) = 2BA 2 2 n Bb 2 2 B 1 + 2An (r1 4A ) b n=1 (r1 + ( b ) A) ;
;
;
6
6
;
;
;
;
;
;
F - , & g(r1 ) ' I = = ( ( b )2 A B4A2 ), In = ( ( (nb+1) )2A ( nb )2 A), n = 1 2 : : :,
# -# 7". D , ( B2 g(r1 ) > 0 r1 4A + lim g(r1 ) = lim g(r1 ) = + B2 B2 ;
;
;
8
r1 ! 4A ;0
;1
1
1
r1 ! 4A +0
1
r !;(lim g(r1 ) = + n = 1 2 : : :: n 1 b )2 A+0 +7 , ( & g(r1 ) - " (/ ( ( ' In (n = 0 1 : : :). +- ( g(r1 ) = 0 ( k0 k1 : : :, kn In , n = 0 1 : : :. D ( , & 1=(p2LK ](p)) ( , 0 k0 k1 k2 : : : # -# ( . 6 , ( k1 k2 : : : 7 # 7 . G k0 = 0, k0 - # 7, ( p = 0 - 7 2-
& 1=(p2LK ](p)). . ( ' k0 = 0, ( k0 - ' 7 3- & 1=(p2LK ](p)). .# , , : , ' A B b (A > 0, B > 0, b > 0) k0 = 0, , k0 = 0. *: 7" : 1 4A + 2A X 1 1 cos ne; 2BbA = 0: Bb )2 B b n=1 ( nb )2 A 1 + ( 2An lim
r1 !;( nb )2 A;0
g(r1 ) =
2
( B4A2 + ), I0 = -
;1
1
2
6
6
;
;
348
. . , . . , . .
H : 7" : 1 1 1 X 2A 2 = X 1 1 e; 2BbA X 1 1 ; Bb 2A + e Bb Bb Bb 2 : (19) 2 2 2 2 2 2 B b n=1 n + ( 2A ) n=1 n + ( 2A ) 2 n=1 n + ( 4A ) 3] , ( 1 X 1 = 1 + cth( a): (20) 2 n + a2 2a2 2a ;
n=1
@ (20) (19), ( Bb Bb Bb Bb b + 1 e; Bb 1 Bb Bb cth Bb + e; Bb 2A + e; 2A = 2A 2A 2 2 4A 2A 4A cth 2A 4A cth 4A b Bb 2A = 1 + e; Bb A 2A : ;
;
;
Bb 2A = 1 + b @ e; Bb A 2A . F ( p = 0 - 7 2-
& 1=(p2LK ](p)). LK ](0) = 0, 1=LK ](0) = 0, L0 K ](0) > 0, d 1 (0) = dpd (LK ])(0) = 0: dp LK ] L2 K ](0) * 7" : 1 L(K1](p) L(K1](0) + 1 : 1 = 2 2 p LK ](p) p p p LK ](0) @ = , ( 1 1 d L(K ](p) L(K ](0) = dp (LK ])(0) : lim p!0 p L2 K ](0) F - , & (1=LK ](p) 1=LK ](0))=p2 7 ( p = 0, & 1=(p2LK ](0)) 7 2- . 2 # & 7 1=LK ](p) ( , p = B 2 =(4A) p = ( n=b)2 A (n = 1 2 : : :) . @ x = B 2=(4A)] + 2, B 2 =(4A)] | & ( ( B 2 =(4A). + ' CRn = p : p x = Rn , Rn = x + A( n=b)2, n = 1 2 : : :. F 1=(p2LK ](p)) p2CRn 0 Rn (n ), , #( ? 4], ( & g(t) (. (17)) 6
;
6
6
;
6
;
;
;
;
;
f
j
j
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g
!
! 1
! 1
aZ+i1 1 1 1 X 1 1 k + 1 pt g(t) = 2 i ept p2LK dp = #(. e ](p) p2 LK ](p) LK ](0) n n =1 a;i1 1 1 1 1 pt pt + #(. e p2 LK ](p) LK ](0) 0 + #(. e p2 LK ](0) 0 : ;
;
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349
.#( #(#, ( , ( 1 0 = t #(. ept p2LK ](0) LK ](0) 1 1 d (LK ])(0) #(. ept p2 LK ](p) LK1](0) 0 = dpL2 K ](0) kn t (n = 0 1 2 : ::): #(. ept p12 LK1](p) LK1](0) kn = d e kn dp (LK ])(kn) @ ( # ' g(t) (18), - ;
;
;
Zt d (LK ])(0) Zt 3 (t) = dpL2 K ](0) f 00 (t ) d + LK1](0) f 00 (t ) d + ;
;
0
;
0
Z 1 kn 00 + d (LK ])(kn ) e f (t ) d: k n dp n=0 0 X
t
1
;
I (
' , ( # ' 3 (t) ( , 2A = 1 + b=A Bb=(2A). e; Bb F # ' 3(t) (12) v(x t), C (x t) = ex+t (v(x t) + 2 (t) + x(3(t) 2(t))=b), ( , / 7 ' . F # ( 1 : . ;
;
1] . ., . . . | .: "# ", 1991. 2] & '. (., )* '. '. + # * . | .: (-, 1976. 3] 1- '. 1., 2. '., 3. 4. 45 #. | .: (-, 1981. 4] 8 '. '. * . &. 1. | .: "# ", 1980.
' ( ) 1997 .
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517.95 .
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Abstract
V. G. Zadorozhnij, The moment functions for the solution of the heat equation with stochastic coecients, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 351{371.
The formulae of the mean value and the second moment function are obtained for the heat di2erential equation with stochastic coe3cient at the higher derivative, stochastic initial condition and stochastic exterior perturbation. The formulae do not contain the continual integral and hold even for dependent stochastic processes. The expression for the mean value of the solution generalizes the well-known Poisson formula for the solution of the heat di2erential equation.
@y(t x) = "(t)y(t x) + f (t x) (1) @t y(t0 x) = g(x): (2) n t 2 t0 t1] = T R, x 2 R , | !!" x 2 Rn, y : T Rn ! R, " | $"!%" & ( $"!( )% ! * ), f : T Rn ! R | $"!%" &, , 2001, 7, 4 2, . 351{371. c 2001 !", #$ %& '
352
. .
g : Rn ! R | $"!%" &, ! %" " f . . ) " ! / 0 * ! !! 1$!0& ! 2" . 1] $!% 1$ % 3 !!%3 1$!0& " ! $ !! @y(t x) = @ 2 y(t x) + "(t)y(t x) + f (t x): @t @x2 2" % $ 1$ % 0( * ! " !!" 1$!0& ! (1), (2). 4 2" & / % ! $ ( !$/ ! !!$/ $ $ !!
& !!" !" !3 5 ! . 6 $ % / ! !! , ) 3 !" ". 1$ 3 " " !!%3 1$!0& " ! $ 0! !$ !%" !( , ! !% 10% % * %3 $"!%3 & " f . 0 !%" $" 1$ % 0( * ! ! $ ! 1$ 7$! ! ! !!( $ !! ! .
x
1.
7$ V | )!3 ! 1$!0& " v : T ! R !" kv()kV U | )!3 ! 1$!0& " u : T Rn ! R. 7$ a 2 R, b 2 R. 8) ! (a b ) 1$!0& /, $/ $/9$ $: (a b s) = sign(s ; a) s, *9 0 0!& a b, (a b s) = 0
! $. ;$ (, a b 2 T 1$!0& (a b ) ! * V $9 $ !! m > 0, 0" Z kv()kV 6 mkv()kL = m jv(t)j dt: (3) T
R
7$ %)!% 1$!0& $"!( & " 0 %, "(t)v(t) dt T !"!% (! !!% 1$!0& ! ! V , ! ( !, RR & & f / !"!%" (! !!%" 1$!0& ! f (t x)u(t x) dx dt ! U . jxj ) ! (x21 + : : : + x2n) 12 x 2 Rn. T 4! % )$ (, &% " f !% 30 0 1$!0& ! 2]
Z Z Z (v() u()) = M exp i "( )v( ) d + i f ( )u( ) d d T
T
( M ) ! 0 * ! 1$!0& ! & " f . 0 / !3* ! 0( * ! My(t x)
" !!" 1$!0& M(y(t x)y(t1 x1)) ! (1), (2).
353
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2.
7$ X | )!3 ! 1$!0& " x : T ! R y : X ! C . ( . 2, . 16]). = 11!& dy(x() h) >R 1$!0& ! y 0 x0 () % dy(x0 () h) = = '(t x0())h(t) dt, ( !( ! % )(, ' : T X ! C T ! % & !!" !" 1$!0& ! y 0 x0() ) ! y(x0 ())= x(t). 1. a : T ! C T , y : V ! C
s y(a()(t0 t ))= v(s)
(3), f (t) = = y(a()(t0 t )) T , df (t) = a(t) y(a()(t0 t )) : dt v(t) . 7* ( , !0( A 3 t 2 T % ! $ ja(t)j 6 A. 7$ t | 9! !!" t, ( ka()(t t + t )kV 6 mka()(t t + t )kL =
tZ+ t = m ja(s)(t t + t s)j ds 6 mA ds = mAjtj = O(jtj): Z
T
t
70 0$ $9 $ & !! ! y(a()(t0 t ))= v(t), , $5 !( !! , !3 f (t + t) ; f (t) = y(a()(t0 t + t )) ; y(a()(t0 t )) = t t Z 1 y ( a ( ) ( t 0 t )) = t a(s)(t t + t s) ds + o(ka()(t t + t )kV ) = v(s) T
tZ+ t y(a()(t t )) 1 0 a(s) ds + o(jtj) : = t v(s) t
354
. .
@0 00 y(a()(t0 t ))= v(s) $ $ s ! T 1$!0& a (! !, 1$!0& a(s) y(a()(t0 t ))= v(s) $ $ ! T . 7 2
! ! *! " 0 $ t ! 0 3] $ $ * ! %. 2. a : T ! C , (3) y0 : V ! C
s y0 (v() + a()(t0 t ))= v(s). ! y = y0 (v() + a()(t0 t ))
" #
@y(t v()) = a(t) y(t v()) y(t v()) = y (v()): 0 0 @t v(t)
(4)
. 4( ! 1 / $ ! T $9 $ ! @y(t v())=@t, 5 @y(t v())=@t = a(t) y0 (v() + a()(t0 t ))= v(t). <, y(t v())= v(t) = y0 (v()+ a()(t0 t ))= v(t). @( $ !! (4) % ! / $ ! T , % !! ! !( $ !. @
0 !. A !"!$/ ! ! !$/ $ @y(t v()) = a(t) y(t v()) + b(t v()) y(t v()) = y (v()) (5) 0 0 @t v(t) ! ! ! !( )*! y : T V ! C !0" 0! B V 0 v() = 0. 7 !! a 2 ! 4]. 3. a : T ! C T , (3), y0 (v() + a()(t0 t ))
B , b : T V ! C T # #, $ b(s v() + + a()(s t ))= v( ), $ v() 2 B T j b(s v() + a()(s t ))= v( )j 6 m(s). !
Zt
y(t v()) = y0 (v() + a()(t0 t )) + b(s v() + a()(s t )) ds t0
(6)
" (5) B. . @0 00 b(s v()+a()(t0 t )) B & !!$/ !$/, ! ! % ! B v. < , b $ $ " !!", ( b(s v() + a()(s t )) $ $ s ! T . 70 0$ $ $ *! m(s), 5] & !! ! !( (6) * )% % ! & !!% 11!& ! !0 !( . D $ % $9$/ $, !3 :
355
@y(t v()) = a(t) y0 (v() + a()(t0 t )) + b(t v()) + @t v(t) t Z + a(t) b(s v() + va(t())(s t )) ds = a(t) y( vt(vt())) + b(t v()) t0
v() 2 B 3 t 2 T . @ 0 !.
x
3. "
7$ f : RnRm ! C , ( Fxf (x y)]( ) ) ! ) ! >$ !!" x. E! ( ! ) !! $ )!( ) ! >$ F;1g( y)](x). A $ 11!& !( $ !! ( 0 @y(t x v()) = ;i y(t x v()) + b(t x v()) @t v(t) (7) y(t0 x v()) = '(x v()): t 2 T R, x 2 Rn, b : T Rn V ! C | !! )*! , | !!" x 2 Rn, ' : Rn V ! C !, y | 0 )*! . 1$ 0 $/9" % $ )*! " ' b $9!% ) !! ($! , !! x v() + ij j2(t0 t ) $ ' x v() + ij j2( t ) $ b. 4. (3), $ B v() = 0, % v() 2 B j'j j '= v(t)j jFx '= v(t)]( )j j Fx']( )j 2 j j jFx']( )j j Fx '= v(t)]( )j j j2jFx '= v(t)]( )j jbj j b= v(t)j jFx b= v(t)]( )j j j jFxb]( )j 2 j j jFx b]( )j j j jFx b= v(t)]( )j j j2jFx b= v(t)]( )j t 2 T , 2 T Rn . ! " (7) %
y(t x v()) = F;1Fx'(x v() + ij j2(t0 t ))]( )](x) + Zt + F;1Fxb( x v() + ij j2( t ))]( )](x) d: (8) t0
. 7 * , $9 $ ) ! >$ !!" x ! (7). 7 ! ) ! >$
356
. .
0 (7), $ 6] @ F y(t x v())]( ) = ij j2 F y(t x v())]( ) + F b(t x v())]( ) x @t x v(t) x Fx y(t0 x v())]( ) = Fx'(x v())]( ): 6 (5), 2 . $ 1$ " (6)
Fx y(t x v())]( ) = Fx'(x v() + ij j2(t0 t ))]( ) + Zt + Fx b( x v() + ij j2( t ))]( ) d: t0
7 ! 0 2$ ! $ 1 ! )! ) ! >$, $ 1$ $ (8). 70* , (8) ! (7). 7 *! % ! $ $%3 *! ) / 11!& $ !0 !( !$*!% ! !!%. D $ " ) ! >$ 6], !3 @y = F ;1 F ij j2 '(x v() + ij j2(t0 t )) ( ) (x) + x @t v(t) ; 1 + F Fxb(t x v())]( )](x) + +
Zt
t0
2 F;1 Fx ij j2 b( x v() v+(itj) j ( t )) ( ) (x) d =
= ;i v (t) y(t x v()) + b(t x v()): @ 0 !.
x
4. $
5 ) !!
Z Z Z Y (t x v() u()) = M y(t x) exp i "( )v( ) d + i f ( )u( ) d d T
T
( 0 * ! % 1$!0& ! $"!%3 & g, " f (1), (2).
R
RR
357
H!* (1), (2) ! exp i "( )v( ) d + i f ( )u( ) d d !" 5 T T 0 * ! 1$!0& ! & g, " f . > ! 9/ )*! Y $!!% ! % / @Y (t x v() u()) = 1 Y (t x v() u()) + 1 (v() u()) (9) @t i v(t) i u(t x) Y (t0 x v() u()) = M(g(x))(v() u()) (10) ( | 30 0 " 1$!0& ! & " f . 7 2 ! *! ! $"!( & g " f . (9), (10) ! !!", !0 $ !! (9) ! & !!, 0 00 * & !! 11!& ! . D Y !! 3 0 $/9$ ! /. . J 0 * ! ! (1), (2) ! % My(t x) = Y (t x 0 0) (11) ( Y | ! (9), (10) !0" 0! !$ " 0 (0 0)
V U . = Y ! (9), (10) % ))95!!%3 1$!0& ", (11) ! % ))95!!% 0 * ! ! (1), (2). = * M(g(x))(v() u()) = '(v() u()) ; (v() u())= u(t x) = = b(t x v() u()), (9), (10) 0* 1 0 !! u()
(7). 5. Mg() Rn u() # U 4, " (9), (10) % (x)
Y (t x v() u()) = Mg(x) F;1(v() + ij j2(t0 t ) u())](x) ; Zt 2 ; i F;1 Fx (v() + ij j ( t ) u()) ( ) (x) d: (12) u( x) t0
(x)
& # x. . D $ 1$ $ (8), !3 ! (9), (10):
Y = F;1Fxg(x)]( )(v() + ij j2(t0 t ) u())](x) ; (v() + ij j2 ( t ) u()) Zt ; F;1 Fx ( ) (x) d: u( x) t0
358
. .
70 0$ )! ) ! >$ ! 1$!0& " ) $ 50$ )!%3 ) ! " >$ !* ", !( ! $ (12). @ 0 !. 6. # 5 " (1), (2)
My(t x) = Mg(x) (x) F;1(ij j2(t0 t ) 0)](x) ; ;i
Zt
t0
j2 ( t ) 0) F;1 Fx (ij u ( ) (x) d: (13) ( x)
< 0 !$*! (11) (12). . 8))95!!$/ 1$!0& / (x)
V0 (t x) = Mg(x) F;1(ij j2(t0 t ) 0)](x) )$ ! % 3!!% % !& ( !/ Mg(x) !( !! My(t x) ! (1), (2). 8))95!!$/ 1$!0& / (ij j2( t ) 0) Zt ; 1 ( ) (x) d V (t x) = F Fx u( x) t0
)$ ! % % !& !( !! My(t x).
x
5. &
7 $!! % 1$ (13) ! )9", ! )$
* ! & " f . 1. 4 $" ! %3 & " f . 7 2 30 0 " 1$!0& ! (v() u()) ! 30 0 3 1$!0& ! "(v()) f (u()), /9 3 &% " f . 7. (1), (2) # g, " f ,
(3), % # " : V ! C "
, Mg(x) Mf (t x) . ! $ "
(1), (2) %
My(t x) = Mg(x) (x) F;1"(ij j2(t0 t ))](x) + Zt
(x)
+ F;1"(ij j2( t ))](x) t0
Mf ( x) d: (14)
359
. 8 , f (0)= u(t x) = iMf (t x), f (0) = 1, (v() u()) = " (v())f (u()). 1$ " (13), !3 My(t x) = Mg(x) (x) F;1"(ij j2(t0 t ))f (0)](x) ; ;i
Zt
t0
F;1 Fx "(ij j2 ( t )) f (0) ( ) (x) d = u( x)
(x)
= Mg(x) F;1"(ij j2(t0 t ))](x) + +
Zt
t0
F;1"(ij j2 ( t ))FxMf ( x)]( )](x) d:
8)! ) ! >$ !0 !( %* 50$, 0$ $ (14). @ 0 !. 2. 4 $" ($ 0( & ". K$ 0 " $"!%" & 30 0 1$!0& ! Z ZZ 1 " (v()) = exp i M"( )v( ) d ; 2 b(s1 s2)v(s1 )v(s2 ) ds1 ds2 T
TT
( M"( ) | 0 * ! " b(s1 s2 ) = M("(s1 )"(s2 )) ; ; M"(s1 )M"(s2 ) | 0 & !! 1$!0& & ". 8. # ## " f , M"(t) > 0, M"() 2 Lp (T ), p > 1, b : T T ! R , Mg() Mf (t ) . ! $ " (1), (2)
Zt ; n2 Z X k 1 1 Zt Zt My(t x) = 4 M"( ) d b ( s s ) ds ds 2k Mg() 1 2 1 2 k k! 2 k=0 t0 t0 t0 jx ; j2 Zt exp ; 4 M"( ) d d + t0
; n2 Z X k Z t Zt 1 1 ZtZt + 4 M"(s) ds b(s1 s2 ) ds1 ds2 2k Mf ( ) k k! 2 k=0 t0 t 2Z exp ; jx ;4 j M"(s1 ) ds1 d d: (15)
. 0 ! V *! %) !;1 ;1
Lq (T ), p + q
= 1. 7 2 % ! $ (3), 1$!0& !
360
. .
" & !!$/ !$/, 1$ ! ))95!! % . D ! 6], Zt ; n2 jxj2 Zt Zt ; 1 2 F exp ;j j M"( ) d (x) = 4 M"( ) d exp ; 4 M"( ) d
Rt t0
t0
t0
t0
M"( ) d > 0. 7 ! $ % ! , 0 00 M"(t) > 0. < ,
F;1"(ij j2 (t0 t ))](x) = Zt (x) 1 ZtZt ; 1 ; 1 4 2 = F exp ;j j M"( ) d (x) F exp j j b(s1 s2) ds1 ds2 (x) 2
1
t
0
1 Bk X
t0 t0
F;1 exp 2 B j j4 (x) = 2k k! F;1j j4k](x) = k=0 @ 22k 1 B k @ 2 1 Bk X X 2k = k k! i @x1 + : : : + ;i @xn (;x) = k k! (x) 2 2 k=0 k=0 ( B 2 R | -1$!0& . 7 2 (x) F;1 exp 21 B j j4 (x) Mf ( x) = Z X 1 Bk 1 Bk X 2k (x ; )Mf ( ) d = 2k = k k! k k! Mf ( x): 2 2 k=0 k=0
7 2 !! (14), !3 (15). 8 , b = 0 & " )$ ! !!% ( !% M"(t)). = f ! !, , 3 (15) 0 $ b(s1 s2) ! 0, $ !$/ 1$ $ (7$!) ! ! !!" (1), (2) 6, . 218]. 9. # ## " f , M"(t) > 0, M"() 2 Lp (T ), p > 1, b : T T ! R , Mg() 2 C1(Rn), $ c > 0, q > 0, % j2kMg(x)j 6 ck!qk x 2 Rn k = 0 1 2 : : : ZZ (16) q b(s1 s2) ds1 ds2 < 2: TT
! %# # V0 (t x) " (1), (2) t 2 T $
jV0(t x)j 6
2;q
Rt Rt t0 t0
2c
b(s1 s2 ) ds1 ds2
361
:
. $ ($ 0( & " ( 1$ % (15) )" 3!!%" " !& ( !( !! ! (1), (2), $ 3 % ; n2 X k Z Zt 1 q k Zt Zt jV0 (t x)j 6 4 M"( ) d b(s1 s2) ds1 ds2 c 2 k=0
t0
t0 t0
jx ; j2 Zt exp ; M"( ) d d: 4
t0
4( ! $ / (16) )0!! $ &! )0!!" ( 0" ( " !! , ! ! &%. 7 !0$ " 2 !! 3 ! T . D $ 3 ! ! 6, . 215] t > 0 jxj2 Z n (4 t); 2 exp ; 4t dx = 1 $
jV0 (t x)j 6 2c 2 ; q
Zt Z t t0 t0
;1 Z Zt ; n2 b(s1 s2) ds1 ds2 4 M"( ) d
jx ; j2 Zt exp ; M"(1 ) d1 d = 4 t0
t0
2;q
Rt Rt t0 t0
2c
b(s1 s2) ds1 ds2
t 2 T , x 2 Rn. @ 0 !. 10. # " f , M"(t) > 0, M"() 2 Lpn(T ), p > 1, b : T T ! R , Mf (t ) 2 C1 (R ), $ c1 > 0, q1 > 0, % j2kMf (t x)j 6 c1 k!q1k t 2 T x 2 Rn k = 0 1 2 : :: ZZ (17) q1 b(s1 s2 ) ds1 ds2 < 2: TT
362
. .
! # V (t x) My(t x) $ jV (t x)j 6 2c1
Zt
ZtZt
t0
2 ; q1
b(s1 s2) ds1 ds2
;1
d:
(18)
. 7 ($ 0 $"! & " (! f ) ( 1$ (15) )" " !&
!( !! My(t x). 7 5 &!0 , ! ( !% &!0, !!% 0 % $9" %: k ; n2 X Zt Z Zt 1 q k ZtZt 1 jV (t x)j 6 b(s1 s2 ) ds1 ds2 c1 4 M"(s) ds 2 t0
exp ; jx ; j
t 2Z
4
6 2c1
Zt
ZtZt
t0
2 ; q1
k=0
M"(s1 ) ds1 d d 6 b(s1 s2 ) ds1 ds2
;1 Z Zt ; n2 4 M"(s) ds
jx ; j2 Zt M"(s1 ) ds1 d d = exp ; 4 = 2c1
Zt
ZtZt
t0
2 ; q1
b(s1 s2 ) ds1 ds2
;1
d:
4( ! !0$ " (17) $ !!$/ 3 !( 2 T , x 2 Rn &!0 (18). @
0 !.
x
6. (
)
0 !%3 3, $%3 $ !! ! , $/ 0211 & !% / $"!% & . 8)%! 3 ! / $/9 ! !! . ." 5 &!0$ " 2 (! . ! (1), (2) $"!% &% ", f , g 3 ! !! . 7 $ ! !!$/ $
363
@y(t x) = M"(t)y(t x) + Mf (t x) (19) @t y(t0 x) = Mg(x): (20) 11. 9, 10, y1(t x) | " (19), (20) My(t x) | " (1), (2), Rt Rt cq b(s1 s2) ds1 ds2 jMy(t x) ; y1 (t x)j 6 t0 Rtt0 Rt + 2 ; q b(s1 s2 ) ds1 ds2 t0 t0
+ c1 q1
Zt
Rt Rt
b(s1 s2) ds1 ds2 d (21) Rt Rt t0 2 ; q1 b(s1 s2 ) ds1 ds2
t 2 T , x 2 Rn.
. 7 * (15) b(s1 s2) = 0, % $ ! (19), (20). 8&! 0 *, 00 3 9, 10, $ Z Zt ; n2 X k 1 1 Z t Zt 4 M"( ) d b ( s s ) ds ds 1 2 1 2 2k k !
jMy(t x) ; y1 (t x)j =
k=1
t0
jx ; j2 Zt 2 k M"(s1 ) ds1 d + (Mg()) exp ; 4
t0 t0
t0
k Z X 1 1 ZtZt b ( s s ) ds ds 2k (Mf ( )) + 4 M"( ) d 1 2 1 2 k k! 2 k=1 t0 jx ; j2 Zt M"(s1 ) ds1 d 6 exp ; 4 Zt
Zt
;n 2
; n2 X k Z Zt 1 q k Zt Z t 6 4 M"( ) d c b(s1 s2) ds1 ds2 2 k=1
t0
jx ; j2 Zt exp ; M"( ) d d + 4
t0 t0
t0
; n2 Z X k Zt Zt 1 q k ZtZt 1 + 4 M"(s) ds c1 b ( s s ) ds ds 1 2 1 2 2 t0
k=1
364
. .
jx ; j2 Zt exp ; M"(s) ds d d = 4
Rt Rt
RtRt b(s1 s2) ds1 ds2 b(s1 s2 ) ds1 ds2 Zt t 0 t0 = cq + c1 q 1 d Rt Rt RtRt 2 ; q b(s1 s2) ds1 ds2 t0 2 ; q b(s1 s2 ) ds1 ds2 t0t0
3 t 2 T , x 2 Rn. @ 0 !. 8&!0 (21) !!" x 2 Rn. @0 00 b > 0, &!0
jMy(t x) ; y1 (t x)j R R t = t0 ! !$ / !!! !!" t. 7 $ b(s1 s2 ) ds1 ds2 ! 0 (! jMy(t x) ; y1 (t x)j TT !! ! T Rn 0 !$ /.
x
7. " +
< !3* ! " !!" 1$!0& ! (1), (2) $ 0 *, 00 !3* ! 0( * ! . 5
( ! )*! Z (t t1 x x1 v() u()) =
Z ZZ = M y(t x)y(t1 x1) exp i "( )v( ) d + i f ( )u( ) d d :
H!* $ !! (1) !
T
T
Z ZZ y(t1 x1) exp i "( )v( ) d + i f ( )u( ) d d T
T
$ ! 1$!0& ! & ", f g. > ! 2 ! % ! Z Y @Z (t t1 x x1 v() u()) = @t = ;i v (t) Z (t t1 x x1 v() u()) ; i u( t x) Y (t1 x1 v() u()): (22) * ! /, $ (2) $ ! $ 5 !" ! ! $ $ !! (22), ! 2 $ 5 9/ ) !!%3 $* ! ". H!* $ (2) !
Z ZZ y(t0 x1) exp i "( )v( ) d + i f ( )u( ) d d T
T
365
$ ! 1$!0& ! & ", f , g, $ Z (t0 t0 x x1 v() u()) = M(g(x)g(x1 ))(v() u()): (23) < ( !( )*! Z $! (22), (23). 7!!% t1 , x1, u() / $ !! (22) , 0% ! & " 11!& ! . < ! ! !( $ !$*! Z (t0 t1 x x1 v() u()) (23). , !0, ( ! ! / Z ! *! )% !% !!% (t x) (t1 x1). 12. (3), Mg(x) M(g(x)g(x1 )) , # (0 0) 2 V U
(v() + ijj2(t0 t1 ) + ij j2 (t0 t1 ) u())
(v() + ij j2( t1 ) + ijj2(t0 t1 ) u()) u( x) (v() + ij j2( t ) + ijj2(t0 t1 ) u()) u(1 x) 2(v() + ijj2(1 t1 ) + ij j2( t ) u()) u( x) u(1 x1) # v. ! (t x), (t1 x1) $ " (22), (23) %
(xx )
;1(v()+ ijj2 (t0 t1 )+ ij j2 (t0 t ) u())](x x1) ; Z = M(g(x)g(x1 )) 1 F Zt1 (x) ; iMg(x) F;1 Fx1 F;1 u( x1) t0 (v() + ij j2( t1 ) + ijj2(t0 t ) u()) (x) ( ) (x1 )d ; Zt (x1 ) ; iMg(x1 ) F;1 Fx F;1 u( x) 1 t0 (v() + ij j2( t ) + ijj2(t0 t1 ) u()) (x1) ( ) (x)d ; Zt Zt1 2 ; d F;1 Fx F;1 Fx2 u( x) u( x ) (v() + ijj2(1 t1 ) + 1 2 t0 t0 + ij j2( t ) u()) () (x1) ( ) (x) d1: (24)
366
. .
. 7 * (22) t1 = t0. 7 $ $ (22), (23) Z (t t0 x x1 v() u()) (9), (10). 7 1$ (12) !3 (x)
Z (t t0 x x1 v() u()) = M(g(x)g(x1 )) F;1(v() + ij j2(t0 t ) u())](x) ; Zt ; i F;1 Fx Y (t0 x1 v() + ij j2( t ) u()) ( ) (x) d: u( x) t0
@0 00 Z ! !!% (t x), (t1 x1), (x)
Z (t0 t1 x1 x v() u()) = M(g(x)g(x1 )) F;1(v()+ ij j2(t0 t1 ) u())](x) ; Zt1 ; 1 2 ; i F Fx u( x) Y (t0 x1 v() + ij j ( t1 ) u()) ( ) (x) d: t0
@(
(x )
Z (t0 t1 x x1 v() u()) = M(g(x1 )g(x)) 1 F;1(v()+ ij j2(t0 t1 ) u())](x1) ; Zt1 ; i F;1 Fx1 u( x ) Y (t0 x v() + ij j2( t1 ) u()) ( ) (x1) d: 1 t0
D $ (10), !3 ! ! $ $ !! (22): (x )
Z (t0 t1 x x1 v() u())= M(g(x1)g(x)) 1 F;1(v()+ ij j2 (t0 t1 ) u())](x1) ; Zt1 2 ; 1 ; i F Fx1 M(g(x)) u( x1 ) (v() + ij j ( t1 ) u()) ( ) (x1 ) d = t0
(x )
= M(g(x)g(x1 )) 1 F;1(v() + ij j2(t0 t1 ) u())](x1) ; ; iMg(x)
Zt1
t0
F;1 Fx1
(v() + ij j2( t ) u()) ( ) (x ) d: 1 1 u( x1)
H !! (22) 2 ! !% $ (7), 1$ (8) !3 (x )
Z = F;1FxM(g(x)g(x1 )) 1 F;1(v() + ijj2(t0 t1 ) + Zt1 2 ; 1 + ij j (t0 t ) u())](x1)]( )](x) ; iF Fx Mg(x) F;1 Fx1 u( x ) 1 t0 (v() + ijj2( t1 ) + ij j2(t0 t ) u()) () (x1 ) d (x) ;
Zt
367
2 ; i F;1 Fx u( x) Y (t1 x1 v() + ij j ( t ) u()) ( ) (x) d: t0
4( ! (12) Y (t1 x1 v() + ij j2( t ) u()) = (x )
= Mg(x1 ) 1 F;1(v() + ij j2( t ) + ijj2(t0 t1 ) u())](x1) ;
Zt1
; i F;1 Fx2 u(1 x2) t0
(v() + ijj (1 t1 ) + ij j ( t ) u()) () (x1 ) d1: 2
2
7 2 %*! $ " ) ! " >$, !3 Z (t t1 x x1 v() u()) =
M(g(x)g(x1))( ) (x ) F;1(v() +
= F;1 Fx
1
+ ijj2(t0 t1 ) + ij j2(t0 t ) u())](x1) ( ) (x) ; (x)
; iMg(x)
Zt1 t0
F;1 F;1 Fx
u( x1 )
(v() + ijj2( t1 ) + ij j2(t0 t ) u()) () (x1) (x) d ; (x1 )
; iMg(x1 )
Zt t0
F;1 Fx F;1
(v()+ ij j2 ( t )+ ijj2(t t ) u()) 0 1 (x1 ) ( ) (x) d ; u( x) Zt Zt1 ; d F;1 Fx F;1 Fx2 t0
t0
2 (v() + ijj2( t ) + ij j2( t ) u()) 1 1 () (x1) ( ) (x) d1 = u( x) u(1 x2)
368
. . (xx )
= M(g(x)g(x1 )) 1 (xx1)
F;1F;1(v() + ijj2(t0 t1 ) + ij j2(t0 t ) u())](x1)](x) ; (x)
; iMg(x)
Zt1
t0
F;1 Fx1 F;1 u( x ) 1
(v() + ijj ( t1 ) + ij j (t0 t ) u()) (x) () (x1) d ; Zt (x1 ) ; iMg(x1 ) F;1 Fx F;1 u( x1) t0 (v() + ij j2( t ) + ijj2(t0 t1 ) u()) (x1) ( ) (x) d ; Zt Zt1 ; d F;1 Fx F;1 Fx2 2t0(v(t)0 + ijj2( t ) + ij j2( t ) u()) 1 1 () (x1) ( ) (x) d1: u( x) u(1 x2) 2
2
0!& ( !!%, 0% % / )!% ) ! >$, ) ! : * !% !). 7 2 !" 5 (24). @ 0 !.
x
8.
(1), (2)
. 8))95!!" " !!" 1$!0& " (1), (2) ! % M(y(t x)y(t1 x1)) = Z (t t1 x x1 0 0), ( Z | ))95!! ! (t x), (t1 x1) ! (22), (23). 13. 12, $
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370
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; iMg(x)
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Abstract S. V. Zelenov, Zelmanowitz density theorem for rings graded by semigroup, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 373{385.
Rings graded by semigroup and modules graded by polygon over semigroup are considered in the paper. Notions of graded critically compressible module and quasi-injective hull are introduced. The structure of the corresponding objects is studied. The density theorem for graded weakly primitive ring is proved.
: ( . !2] !6]). & ' ( )
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375
, a 2 A, h 2 H g 2 G -,'
: (ha)g = h(ag): A- A B | (H G)-, : B ! A | C (H G)- . 1.2. . R -- G, -' - L fRg j g 2 Gg - , R = Rg Rg Rh Rgh g h 2 G. g2G S 7 h(R)def = Rg , g2G , - rg 2 Rg g. 3, - r P - ( r = rg , rg - g2G g 2 G. :- rg , ( r. 1.3. A- R - -- G. /- M A R--, -' - L fMa j a 2 Ag - - M, M = Ma a2A Ma Rg Mag a 2 A g 2 G. S 7 h(M) def = Ma , a 2A -, - ma 2 Ma a. 3, - m P - ( m = ma , ma - a 2A a 2 A. :- ma , ( m. 1.4. A- N - M , L N = (N \ Ma ) , ( , , ( x 2 N a2A N. 1.5. ;
R-- M N, - A B. >C f : N ! M R-- h, h 2 H, f(Nb ) Mh(b) b 2 B. >- C h -, -, -- HOM(NR MR )h - Hom(NR MR ). D N = M = idA , , END(MR )h def = HOM(NR MR )h , , L , END(MR ) def = END(MR )h (C h2H End(MR ) - H , - M - - ( .
376
. .
L
G C HOM(NR MR )h S h2H , C HOM(NR MR )h | . h2H 1.6. A- M | - - N | - - . I M=N ') - - , (M=N)a def = (Ma + N)=N. K ( -, M ! M=N - C , ) ( . 1.7. >- , - ( . L - -, - -, !13]. P - ) - - ,
' . - ,
. D .
2. A- H G | --, A | (H G)-. A- - ' G- A ( . ' !4]):
G- : - a 2 A , g1 g2 2 G ag1 = ag2 g1 = g2Q
G- : - , g 2 G , a1 a2 2 A a1g = a2 g a1 = a2 . 2.3. . , A --- H -,' :
H - : - , a 2 A h1 h2 2 H h1a = h2 a h1 = h2Q H - : - , h 2 H , a1 a2 2 A ha1 = ha2 a1 = a2. 2.4. A- R | - G , M | - A R-- . 2.5. . A- N M | - A R--, ; = HOMR (N M) - H - ' H- A. I , f 2 h(;) - - K f(N) M - Ker f f ;1 (K) , - N. . 8 - - - ( . !4, 2]). 2.1. 2.2.
377
2.6. . - - M C N ! M, N | - - M. 2.7. . >- - , , - - - - . 2.8. . :- - R-- M , C - - - . 2.9. . :- - R-- M - , , , - C-- . 2.10. ! .
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-' - C g : M ! f(N). :
M !g f(N) = N= Ker f M= Ker f ) C M M= Ker f. - M Ker f = 0, f | C . (2) ) (1). A- f : M ! M=N | C , N | - - M. A f(M) = L=N - L M, N $ L M. D : L ! L=N C , h = f ;1 - - (C M, , C . : N Ker Ker h, , N = 0. : , M - - , (2) K L | - - -, K \ L 6= 0.
, K \ L = 0, K + L L - - - (C - M - . A . 2.11. . /- , - - , (2) , .
378
. .
:, - - , - (C - (C -. . - M - < - T ! - M. M, A- - - M^ < - ^ | ) - (C . M, V def = END(M) 2.13. ! . #" M | % R-
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, M = VM . . (2) >- - VM . < . A- N | - - - VM. D f : N ! VM | -- - C , f - - 2 V. A - (VM) VM, - - - C T 2 ENDR (VM), T - - f, , VM < . (1) A- P | - < ^ ' M. / , P VM. - - M, (, , , P P , 2 h(V). &, Q() def = fx 2 P j x 2 P g - - P, - - , Q() = P , 2 h(V). W - 2 h(V) Q def = Q(). ;
C Q ! P: q 7! q. A - P < , -' - 1 2 ENDR (P ), 1q = q q 2 Q, ) 1 , - ' H-, - . K < M^ - -' C 0 2 V, 0x = 1x , x 2 P , ) 1 0
, ( ). I 0P P , (0 ; )P = 0, P P . K, Q() 6= P, (0 ; )P 6= 0. A - M | -' ^ P -' - M, ^ - - M, 0
, ( ; )P \ P 6= 0. : x 2 P 0 6= y 2 P , y = (0 ; )x 2 (0 ; )P \ P, 0 x = 1x 2 P -, x = = 1x ; y 2 P. : x 2 Q(), x = 1x, , y = 0, - y. (3) (1) (2). A . 2.12.
379
2.14. -, MT = VM. 1 , - : ,
def
!
.
K ( , -, MT = XM, X = END(MT R ). 2.15.
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(1) * MR
,
"+ D def = END(MR ) ! %
"+ X, % , X "+
. (2) * MR - , D %
!" - "+ ' X. T ( . (1) A- 2 h(HOMR (M M)) Ker 6= 0. , = 0. A- 6= 0. I, 1 = j;1 (M \M ) , 1 6= 0. A - 1 - (C - M, 1 | C . : - , Ker 1 = Ker \ ;1 (M \ M) 6= 0, M -C. A- , 6= 0. A- 2 h(END(MT R )) Ker 6= 0. , ( - = 0. A- | - ( V. T - MT | - I jM 2 HOMR (M M), ^ MQ - jM . , (M) 6= 0, (M) \ Ker 6= 0, MT R -C. I , Ker( jM ) 6= 0. 1 , -, (M) = 0. A - 0 6= 2 h(V) , ,, MT = VM = 0. K , ( h(D) , X. K,
D - X. A , X - . ;
0 6= 2 h(X). K
- , ) , , C . A- 1 (C - M^ R Q 1 C , ) 1 , ( - - ' H-, - - ). K , MR -C, -, M^ R , , ^ 1 . A- 1 2 V ( 1 , 1 M^ = M, ^ A - MT 1 1 = 1 = 1 1 M. ^ = 1 = , = 1 jM^ . - - M,
380
. .
(2) A- MR . A, ( - X - D. A- 0 6= 2 h(X). I N def = M \ ;1M 6= 0 | - - M, 0 6= 2 HOMR (M N) D. Y , 0 6= 2 D, . A .
3. !
R - R1 - < R . 3.2. . : ) - ! , - - -. 3.3. : ) R- ) - (X VR MR ), V - - A (X R)--, X | - -- H , XM = V , R - M ( , R END(V )). &, - X , - - -- H. A -, , - - ' (- ( !17, Theorem 2.2]). 3.4. " . "+ R : (1) R ! % (2) R- (, (X VR MR ), '
' X v1 : : : vk 2 h(V ) 0 6= a 2 h(X), !' n1 : : : nk 2 M , r 2 R,
ani = vi r 2 M i = 1 : : : k, % , % k = 1 n1 2 h(M) r
(3) R- (, (X VR MR), ' 2 END(V ) k P ' ' X-%
% U U 0 def = Xmi i=1
V , m1 : : : mk 2 h(M), r s 2 R, ( r ; s)jU = 0 rjU 0 , % , U = Xm U 0 = Xm0 % m m0 2 h(M), m
, s
. . (1) ) (2). A- MR | - - - . ; ( . . 2.14 2.15) , D = END(MR ) - , 1 - X = END(MT R ), MT = XM. I , - R- )- (X MT R MR ), T
R - END( M). 3.1.
381
T Y , vR 6= 0 , 0 6= v 2 h(M). 1 I MR vR \ M 6= 0, C a 2 HOMR (M vR1 \ M). A - MR , -' -, m 2 M r 2 R, mr 6= 0. , 0 6= am = vs s 2 R1, , 0 6= a(mr) = (am)r = vsr 2 R. T A- X ( v1 : : : vk 2 h(M). T r i = 1 : : : k Ai = vj | - ( j 6=i - - !4, 2]) R. A
( . !4]) vi Ai 6= 0 i, , -, k T vi Ai \ M 6= 0. A - MR , C T i=1 k T a 2 h HOMR M vi Ai M . I , n1 : : : nk 2 M i=1
-' -, ri 2 Ai i = 1 : : : k, ani = vi ri 2 M i. k P A r = ri , - ani = vi r 2 M. i=1 A k = 1 ( a vi ni , - r . (2) ) (3). L ' - , (3), ( m1 : : : mk X. K l P
V = XM U Xmi def = V 0 i=k+1 mk+1 : : : ml 2 h(M). , , mi k + 1 6 i 6 l, , m1 : : : mk mk+1 : : : ml -, X- W def = U 0 + V 0. : ) ( r s R, ( r ; s)jW = 0 rjU 0 - C . n P A Wn = Xmi . A - n -' i=1 r s 2 R, ( r ; s)jWn = 0 rjU 0 | - C . A n = 0, - (2), r 2 R, mi r = ami i = 1 : : : k 0 6= a 2 h(X)Q - W0 = 0, ( s 2 h(R) . A- 1 6 n 6 l. A - - , -' -, r0 s0 2 R, ( r0 ;s0 )jWn;1 = 0 r0 jU 0 | - C . A 0 = r0 ; s0 Q r s 2 R, ( 0 r ; s)jWn = 0 rjU 0 | - C . , , v 2 Wn v (r0 r) = v( 0 +s0 )r = v(s+s0 r), r0 r - C . I: mn 0 2= U 0 + Wn. A- 0 = g01 + : : : + g0q , g01 : : : g0q 2 2 h(END(V )) | 0 . I, - (2), ) ( 0 6= a 2 h(X), ' ( m1 : : : mj mn g01 : : : mn g0q , j = max(n k). Y , -' -, r s 2 R,
382
. .
ami = mi r (1 6 i 6 j) amn = (mn g01 )r a0 = (mn g0p )r (p > 1)Q a0 = mi s (i 6= n) amn = mn s a0 = (mn g0p )s (p = 1 : : : q): I, , rjU 0 | - C . , 1 6 i < n mi 0 r = 0 = mi s, mn 0r = amn = mn s. I , ( 0 r ; s)jWn = 0. II: mn 0 2 U 0 + Wn . D mn 0 = 0, 0 6= a 2 h(X) r 2 R , ami = mi r 1 6 i 6 k, ) s = 0. P K, , 0 6= mn 0 = ci mi , =1 0 6= ci 2 X 1 6 i1 < : : : < i 6 j = max(n k). A- ci = cig1 + : : : + + ci gq , cigp 2 h(X) | ( ci . 1 - (2), ( 0 6= a 2 h(X), ' ( m1 : : : mj Q 0 6= b 2 h(X), ' ( a;1 ci1 g11 mi1 : : : a;1ci g1 mi mi (1 6 i 6 j, i - i = 1 : : : ). I -' - r 2 R, 1 6 6 bmi = (a;1ci g1 mi )r 2 M b0 = (a;1 ci gp mi )r 2 M (p > 1) bmi = mi r (i 6= i 1 6 6 )Q -' - s 2 R, a0 = mi s (i 6= n) a
X
=1
bmi = mn s: P
&, , bmi 2 M. =1 Y , rjU 0 | - C . , P P 1 6 i < n mi 0r = 0 = mi s, mn 0 r = ci mi r = abmi = =1 =1 = mn s. I , ( - ( 0 r ; s)jWn = 0. A- U = Xm U 0 = Xm0 m m0 2 h(M) m . A, -' -, r 2 R s 2 h(R), ( r ; s)jU = 0 rjU 0 | C . Y- - - - , ) r0 2 h(R), m0 r0 = am0 0 6= a 2 h(X). &, r0 - m0 . ( s0 0. , - 0 def = r0. A - m , - m 0 . D Xm = Xm0, - n = 1, mn def = m, ) ( r1 s1 2 R - C 0. -
383
m m 0 , - -- (2) C- , ( s1 2 R . 8 - s0 = 0, s s1 , ( r , r0r1 . K, , m m0 X. I: m 0 2= U 0 + U. I, - (2), ) ( 0 6= a 2 2 h(X), ' ( m m0 m 0 . Y , -' -, r1 2 R s1 2 h(R), am0 = m0 r1 am = (m 0 )r1 Q a0 = m0 s1 am = ms1 a0 = (m 0 )s1 : I, , r1jU 0 | - C . , m 0 r1 = am = ms1 . I , ( 0r1 ; s1 )jU = 0. A - s0 = 0, s s1 , ( r , r0r1 . II: m 0 2 U 0 + U. D m 0 = 0, 0 6= a 2 h(X) r1 2 R , am0 = m0 r1 , ) s1 = 0. K, , 0 6= m 0 = c , 0 6= c 2 h(X) m, m0 . 1 - (2), ( 0 6= a 2 h(X), ' ( m m0 Q 0 6= b 2 h(X), ' ( a;1c ,
m, m0 6= . I -' - r1 2 R, b = (a;1c )r1 2 M b = r1Q -' - s1 2 h(R), a0 = m0 s1 ab = ms: Y , r1 jU 0 | - C . , m 0 r1 = c r1 = ab = ms1 . I , ( - ( 0r1 ; s1 )jU = 0. A - s0 = 0, s s1 , ( r , r0r1 . (3) ) (1). A- M (3). A, R-- , ) ( - M, - -. , 0 6= m 2 h(M) - mR1 . , - 0 6= t 2 R. I Mt 6= 0, - MR . K, 0 6= n 2 h(M), nt 6= 0. 2 END(V ), m = n, ) r s 2 R, r = s Xm rjn | - C . I mst = m rt = nrt 6= 0, - 0 6= nr 2 Xn nt 6= 0. Y , mR1 t 6= 0. A, mR1 . , - NR | - - - . A 0 6= n 2 h(N) ) 2 END(V ) , n = m. r 2 R s 2 h(R), r = s Xn rjm | - C . I -' -
384
. .
a 2 X, 0 6= mr = am, , 0 6= am = mr = n r = ns 2 N. K ( m, n s -, a . Y , 0 6= a 2 h(HOMR (mR1 N)). : , - , mR1 C. ( , C MR . K, - NR | - - - MR , - (C 0 6= 2 h(HOMR (N M)). ;
m 6= 0 m 2 h(N). ( 0 6= n 2 h(N)
2 END(V ), n = m, ) r s 2 R, r = s Xn rjm | - C . I (n)s = (ns) = (n r) = = (mr) = (m)r 6= 0, , n 6= 0. A - Ker | - - ( , 2.5), , -, C . I .
"
1] . .
. // . . 8. | : "# $ , 1989. | '. 3{16. 2] ,. - . ' .. | /.: "# $ 0 , 1961. 3] '. . 1 . 2 // Kurosh Algebraic Conference '98. Abstracts of Talks. | /.: /56, /7 - 2 0 8 $, 1998. | '. 174. 4] '. . 1 .
0 2 // 6-7 2 7 20 /5'6. | /., 1999. | '. 107{110. 5] '. . 1 . 5 1$ 2 // # , 7 - 0 , 0 70-= 8 >0 /56 (10{12 8 1999 .). | /.: /56, /7 - 2 0 8 $, 1999. | '. 25{26. 6] ". B . D $. . | /.: / , 1971. 7] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev. Rings with Generalized Identities. | New York: Marcel Dekker, Inc., 1996. 8] C. Faith. Lectures on injective modules and quotient rings. Lecture Notes in Math., vol. 49. | Berlin, New York: Springer-Verlag, 1967. 9] R. E. Johnson. Representations of prime rings // Trans. Amer. Math. Soc. | 1953. | Vol. 74, no. 2. |P. 351{357. 10] K. Koh, J. Luh. On a Unite dimensional quasi-simple module // Proc. Amer. Math. Soc. | 1970. | Vol. 25, no. 4. | P. 801{807. 11] K. Koh, A. C. Mewborn. Prime rings with maximal annihilator and maximal complement right ideals // Proc. Amer. Math. Soc. | 1965. | Vol. 16, no. 5. | P. 1073{1076. 12] K. Koh, A.C. Mewborn. A class of prime prime rings // Canad. Math. Bull. | 1966. | Vol. 9, no. 1. | P. 63{72.
385
13] C. NXastXasescu, F. van Oystaeyen. Graded Ring Theory. | North-Holland Mathematical Library, vol. 28. | Amsterdam: North-Holland, 1982. 14] C. NXastXasescu, S. Raianu, F. van Oystaeyen. Modules graded by G-sets // Math. Z. | 1990. | Vol. 203, no. 4. | P. 605{627. 15] Liu Shaoxue, M. Beattie, Fang Hongjin. Graded division rings and the Jacobson density theorem // Journal of Boijing Normal University (Natural Science). | 1991. | Vol. 27, no. 2. | P. 129{134. 16] J. Zelmanowitz. An extension of the Jacobson density theorem // Bull. Amer. Math. Soc. | 1976. | Vol. 88, no. 4. | P. 551{553. 17] J. Zelmanowitz. Weakly primitive rings // Comm. Algebra. | 1981. | Vol. 9, no. 1. | P. 23{45. & ' 2001 .
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512.64
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Abstract A. N. Zubkov, The Procesi{Razmyslov theorem for quiver representations, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 387{421.
We ?nd the generators and the de?ning relations of any quiver representation invariant algebra. To be precise, let ( @) be a quiver representation space with respect to the natural action of the group consisting of all isomorphismsof the quiver representations. Denote this group by GL(@), where @ is a dimensional vector of the quiver representation space ( @). For example, when our quiver has only one vertex and several loops are incidentalto this vertex we have the well-known case R Q k
k
R Q k
k
Q
, 2001, - 7, A 2, . 387{421. c 2001 !, "# $ %
388
. . of the adjoint action of the general linear group on the space of several -matrices. In the characteristic zero case Artin and Procesi described the quotient of the last variety under this action in their classic works. In the case of arbitrary in?nite ground ?eld this result can be deduced from some results by Procesi and Donkin. In a similar manner we can de?ne the quotient of the quiver representation space ( @) by the action of the group GL(@). By the de?nition we have that E ( @) GL(@)] = E ( @)]GL(k) . Donkin has recently found the generators of that algebra. In this article we de?ne a free quiver representation invariant algebra. Then we prove that the kernel of its canonical epimorphism onto E ( @)]GL(k) is generated as a T-ideal by the values of the coeKcients of the characteristic polynomial with suKciently large number. This result generalizes the well-known Procesi{Razmyslov theorem about trace matrix identities. Besides, by an alternative way we can deduce Donkin's result about the generators of E ( @)]GL(k) . n
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. . ) 4 *5 Y 2
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HomGL(k (1)) (XA (.) XA ) ! HomGL(k (2)) (XA (.) XA ) 8 . < , ) 2 " J(Q) r1 $ $ , KQ 1k]GL(k ) (1r) / k1, ) % ). < k1 , 4,% 5 ( % ) r1, ). K )0 2 7 , 3], | 2 4 25 HomGL( l) (XA (.) XA ) ! HomGL(k ) (XA (.) XA ): E , 1l , , 4,%25. R
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{
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40 ,5 > 2. 3 , , 1 0 , $ 0 . +
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, _P
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$0 a1 b1. F
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$ 2 , 1 . < , 1st , 2 2, a1 * 2 . <
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. F fa1 : : : am g = 1st , fb1 : : : bm g = (1uv ) , 1st . . , 1uv = fy1 : : : ym g, (yi ) = bi , 1 6 i 6 m. E m Q 1 (ai (ai )Xi )(yi bi Zi), $ ,, 7
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{
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. ,, " C(Q 1k)
$ " GL(k1)-7 / ,/ $ 2,
, & " 8 * 2 R(Q 1k). + " , ) , & $2 (i j) 2 W ) 0 ) kj ki - * Yij ($ , i * j). . ,2 7 " GL(k1)-7 / ,/ $ 2 $ , L f = fij 2 M(Q k1)(KR(Q k1)]). ,, 7 (ij )P 2W tr(fij Yij ) $ " . . , clos(f) = (ij )2W " $ / / /, +P ij GL(k ) 2] 8(i j) 2 W tr(fij Yij ) = gsij tr(mij s Yij ), " gs 2 KR(Q 1k)] s mij s |
& * Yk (a),
* 7 * i, | P j. 3 $ 8 * 8(i j) 2 W fij = s gsij mijs , , f 2 C(Q k1). E , mij s 4 5, i = j, , 7 $ " ,, mij |
ki ki- *. . $ . s F , $ , k k : C(Q 1k1) ! ! C(Q 1k2) | 7 8 )/ k1(1) k1(2). + 2 ,, 7 , , 7 8 / 1 $2
2 Yij )0 / " / . B
", 8 2 r1 4 5 k1 / * r1- 2 . < , 4,% /5 k1(1) k1(2) 7 8 k k 0 8 . 3],
D(Q k1) * C(Q) ! C(Q 1k). ' , 7 T- ( , 7 ). ' 0&
m m- $ 6 , {7 , , m (Y ) = Y m ; 1(Y )Y m;1 + : : : + (;1)m m (Y ). 2.1 (. 3]). T- D(Q k1) m (Y (al ) : : :Y (a1 )), (a1 : : : al ) | Q kt(al) = = kh(a ) 6 m. 1
1
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clos . < , $
~ ,, clos(D(Q k1)) T(Q ~ 1k). B
clos: C(Q) ! J(Q). 1 ", D(Q k) $ / P 1
Yij . . , f 2 D(Q k1), clos(f) = gij , degYuv gij = (ij )(uv). (ij )2W ~ k1). < , 3 % " 8(i j) 2 W gij 2 T(Q ~ 1k) $ , gij . . , T (Q ~ 7 fr (g), f 2 J(Q), g 2 C(Q), " r (g) | ,2 7 ~ 1k), $ / / 7 , T (Q 1 Y = Yij , ,. S degY f = 1, " , f = tr(f 0 Y ), f 0 2 C(Q). K ~ 1k)C(Q) D(Q k1). + 2 ,, )0 2 $)0 2 | 7 r (g)f 0 2 T(Q r (g) = gr;1(g) ; r (g). S $ degY f = 0, r (g) $ 2 ,P 1 Y . D $ , , , " g = gi , " degY gi = i g1 6= 0. i>0 F / )0 :
& r (X + Y ) ) , 1 Y r ; 1 , X ! g0, Y ! g1. +, 2% $
) 3] ( 4 .) , & * 8 . < . #$ 2.1. A" , K ,. + ,, (3], 2). #$ 2.2. ' 7 2 , , & , )
,* * / . A" 3] ,* 2 *, $& / 7 j (Yk (ar ) : : :Yk (a1 )) k1. > * " ,* ,*, $& 7 j (Yk (ar ) : : :Yk (a1 )),
T~(Q k1). .~ 1k) J(Q) Z- ,
) ) , J(Q)=T(Q , , 0 ;! K T~(Q k1) ;! K J(Q) ;! K (J(Q)=T~(Q 1k)) = KR(Q 1k)]GL(k ) ;! 0: ' ) K T~(Q k1) = T(Q k1) )" ( ") K. <
, $ , 3, . 454]. + ,
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{
421
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. (Z- ,) 7 R(ij ), / (i j) 2 W . S i 6= j, R(ij ) | 7 2 J(Q)- , $)0 Y (ar ) : : :Y (a1 ), t(ar ) = i, h(a1) = j, t(as ) = h(as+1 ), 1 6 s 6 r ; 1. 3 0& 7 ei . V$ 2 $ 8 2, % . < , R(ij )R(jl) R(il) | 8 ,
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1] J. A. Green. Polynomial representations of GLn . | Lecture Notes in Math., vol. 830. | Berlin, Heidelberg, New York: Springer-Verlag, 1980. 2] S. Donkin. Polynomial invariants of representations of quivers. | To appear. 3] . . . !" #$!{& '" // $) ! $) !. | 1996. | +. 35, - 4. | /. 433{457. 4] S. Donkin. Invariant functions on matrices // Math. Proc. Cambridge Phil. Soc. | 1992. | Vol. 113, no. 23. | P. 23{43. 5] S. Donkin. On tilting modules for algebraic groups // Math. Z. | 1993. | Vol. 212. | P. 39{60. 6] A. N. Zubkov. Endomorphisms of tensor products of exterior powers and Procesi hypothesis // Commun. Algebra. | 1994. | Vol. 22, no. 15. | P. 6385{6399. 7] S. Donkin. Skew modules for reductive groups // J. Algebra. | 1988. | Vol. 113. | P. 465{479. 8] J. Jantzen. Representations of algebraic groups. | Academic Press, 1987. 9] K. Akin, D. A. Buchsbaum and J. Weyman. Schur functors and Schur complexes // Adv. Math. | 1982. | Vol. 44. | P. 207{278. 10] A. N. Zubkov. On the procedure of calculation of the invariants of an adjoint action of classical groups // Commun. Algebra. | 1994. | Vol. 22, no. 11. | P. 4457{4474. 11] S. Donkin. Rational representations of algebraic groups: tensor products and 1ltrations. | Lect. Notes in Math., vol. 1140. | Berlin, Heidelberg, New York: Springer, 1985. 12] O. Mathieu. Filtrations of G-modules // Ann. Scient. Ec. Norm. Sup (2). | 1990. | Vol. 23. | P. 625{644. 13] C. de Concini and C. Procesi. A characteristic free approach to invariant theory // Adv. Math. | 1976. | Vol. 21. | P. 330{354. 14] 4. &. !" #$. +5# ! # #$ 6$ 7 ! ' 7 !$) ! 6$ 7! ! # // 8". /// . / . ! . | 1974. | +. 38, - 4. | /. 723{756. & ' 1998 .
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Abstract
A. V. Krotov, V. V. Rabotin, On a class of complete intersection Calabi{Yau manifolds in toric manifolds, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 423{431.
We consider the family of smooth n-dimensional toric manifolds generalizing the family of Hirzebruch surfaces to dimension n. We analyze conditions under which there exists a Calabi{Yau complete intersection of two ample hypersurfaces in these manifolds. This turns out to be possible only if the toric manifold is the product of projective spaces. If one of the hypersurfaces is not ample then we 5nd Calabi{Yau complete intersection of two hypersurfaces in Fano manifolds of the given family.
1.
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= 2 An;1(X).
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i=1
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(1)
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425
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i
i
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3. $ % "
5 0 = | Rn;1 (n ; 1)- Pn;1. E 0 1
: 0=(1) := fe1 : : : eng 0 = e1 = (1 0 : : : 0),.. ., en;1 = = (0 : : : 0 1), en = (;1 ;1 : : : ;1). Rn;1 Rn, 0 x 7! (0x 0), 1: 0 Rn: en+1 = (c1 : : : cn), cn < 0, en+2 = (0 0 : : : 0 1). F 0= + fen+1 > 0g + fen+2 > 0g, | 0 =, 0 = Rn. < Hc 1 + . F 1i (2i) , : fe1 : : : e^i : : : en en+1 g ( fe1 : : : e^i : : : en en+2g), i = 1 : : : n, , , 4 , + 1. 1 0 Hc .
3.1. An;1(Hc) Z2, ! " # n+2
Dn Dn+1. $ Q " za := zia deg z a =
X n
i=1
i=1
i
ai Dn ] + an+1 ; cn an+2 ;
nX ;1 i=1
ci ai Dn+1]:
. I 1 ( 9]) 0 ;! M ;!
d M i=1
ZDi ;! An;1(X)
;! 0
(2)
426
. . , . .
(m) = ;
d X i=1
hm ei iDi
X d
i=1
kiDi =
4 m = (m1 : : : mn) 2 M nX +2
nX ;1
i=1
i=1
hm ei iDi =
d X i=1
ki Di]:
mi (Di ; Dn + ci Dn+1 ) + mn (cn Dn+1 + Dn+2 ):
5+ (2) , deg zi = Di] = Dn ] ; ci Dn+1] i = 1 : : : n ; 1 deg zn = Dn] deg zn+1 = Dn+1] deg zn+2 = Dn+2] = ;cn Dn+1]: 5 @ (1) 4 . . F @ 1 Dn Dn+1 An;1(Hc ), + 0 (r s), (r s) $ r Dn ] + s Dn+1 ]: $ , X , ;KX . F
, 1 : , d ;KX = P Di . i=1
3.2. Hc , cn = ;1. Hc %
" ":
1) c = (;1 ;1 : : : ;1)' 2) c = (0 : : : 0 1 0 :: : 0 ;1), 1 i- , i = 1 : : : n ; 1' 3) c = (0 : : : 0 ;1) ( ( Hc = Pn;1 P1). . K , 0 1 : 1 4:. < Hc . F Hc. 3.3. ) (a b) Hc
,
a > 0 b > 0 aci + b > 0
i = 1 : : : n ; 1:
(3)
{
427
. $ ( 9, x 3.4]). L
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d P
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i bxn x 2 1nM > > : 0 x 2 2n: ) + @0, . K Hc
nP ;1 n 2 ; ci , : i=1 i=1 D
;
nX ;1 i=1
ci + 2 > 0 ;
nX ;1 i=1
ci + ncj + 2 > 0
j = i : : : n ; 1:
(4)
ci 0, + (4) + 1:
;
nX ;1 i=1
ci + 1 > 0 ;
nX ;1 i=1
ci + ncj + 1 > 0
j = i : : : n ; 1:
(5)
$ (5) (n ; 1)- 4 (;1 : : : ;1), (0 : : : 0 1 0 : :: 0), 1 i- , i = 1 : : : n ; 1. I , 0 4 (5) 4 0 , 4, | (0 : : : 0). . 5 X | (n ; 1)- N, 1 0=1 Rn;1 Rn. 5 0 Hc Xc 1 =1, @ 0 =1 , , 1 : , (c1 : : : cn;1 ;1) (0 : : : 0 1) Rn. O 4 , 0 1 : 0 =1 4 P. 5 1 .
428
. . , . .
! 3.4. Xc % , (c1 : : : cn;1) 2 P.
5 1 : 4 . Q MapleV2, 1 . % + . . 5 n = 2 c = (m ;1) Hc O 0 Fm , : 1 Hc n- O 0 . 5 m : O(m) | m- P1, O | P1 , O 0 0 ? + , Fm = P(O(m) + O) 9]. Q Hc, , cn = ;1, Hc = P(O(c1)+: : :+ O(cn;1 )+ O).
4. ' {
. { + X , @ p-@ 0 < p < dimX, . . hp0 = dimC H 0 (X Sp ) = 0 0 < p < dimX: > X.
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430
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1] T. Hubsch. Calabi{Yau manifolds | a bestiary for physicists. | Singapore: World Scientic, 1992. 2] V. V. Batyrev. Dual polyhedra and mirror symmetry for Calabi{Yau hypersurfaces in toric varieties // J. Algebraic Geometry. | 1994. | Vol. 3. | P. 493{535. 3] V. V. Batyrev, D. A. Cox. On the Hodge structure of projective hypersurfaces in toric varieties // Duke Math. Jorn. | 1994. | Vol. 74. | P. 293{324. 4] P. Candelas, M. Lynker, R. Schimmrigk. Calabi{Yau manifolds in Weighted P4 // Nucl. Phys. | 1990. | Vol. B341. | P. 383{402. 5] V. Batyrev, L. Borisov. Dual cones and mirror symmetry for generalized Calabi{Yau manifolds // Essays in Mirror Symmetry II / S.-T. Yau, editor. | Preprint, alg-geom/9402002. 6] V. Batyrev, L. Borisov. Mirror duality and string-theoretic Hodge numbers. Preprint, alg-geom/9509009. 7] V. Batyrev, L. Borisov. On Calabi{Yau complete intersections in toric varieties // Proceedings of Trento Conference (1994). 8] D. Cox. The homogeneous coordinate ring of toric variety // J. Algebraic Geom. | 1995. | Vol. 4. | P. 17{50. 9] W. Fulton. Introduction to Toric Varieties. | Princeton, NJ: Princeton Univ. Press, 1993. 10] . . . ! "!#$ % # $ &!'( ) " * +$, - // ./0. | 1978. | 1. 33. | 2. 85{134. 11] 3. . 4$ # . 5 ( '') 67!% 8 9) 8!$!'!&! - * :( " # $ &!'( " " * +$, // 4 "8 !('9- , : ;;!$!< =9! >$ ! %. | 4$' %$'(, 1996. | 2. 90{95. ' ( 1997 .
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Abstract B. O. Kuliev, Cournot and Stackelberg strategies in the case of several companies, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 433{440.
The classical Cournot and Stackelberg strategies for the case of two companies are generalized to the case of several companies. The Nash stability of Cournot strategy for the several companies case is investigated. A characterization theorem for Cournot strategy is given for the case of many companies.
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1] Cournot A. Recherches sur les principles mathematique de la theorie des richesses. | Paris, 1938. 2] Von Stackelberg H. Marktform und Gleichgewicht. | Wien Berlin, 1934. 3] ., . . | .: ! "- , 1997. 4] %&' . (., ) *. . +,- . ,)/,, . | .: .,, 1998. ' ( 2001 .
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A. I. Kurnosenko, Interpolation properties of planar spiral curves, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 441{463. Some inequalities on the planar curves termed spirals (due to monotonous curvature function) are considered. For a spiral represented as a set of interpolation nodes, a region covering the parent curve is constructed. The width of the region provides an estimate of curve de;niteness by the given discrete representation, this estimate being independent of any interpolation method. A similar problem is set up for any smooth curve, assuming that vertices are distanced 0su
1.
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443
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qj qj +1. + $ <= | , $0$ ) / . 5 ( < =) ) , ( 3, . 102], 4]). C $ . 1.
> . 1
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;
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444
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G $ c2 j = jh(qj cj ) ; h(qj +1 cj )j ' 2j jqj ; qj +1j G = max(1 : : : n;1): (1) @ / , , j $ 8 ( ) ) (1) ). 6 . 1 / , n c 10 13. . 9 , (1) . C0 ) - , . . . @ ( / ) ) , . ), Pi 1 n $ ( . 2). +, ,
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M
, 0 , y = f (x), $ $ : , x 2 ;1 1], , : f (1) = 0? , $ ? , , (x) := arctg y0 (x) | y = f (x)? , k(x) = y00 =(1 + y0 2)3=2 ;1 1]? ) , ;1 1]. . $ ( M+ M; ). I
M0 0 , (2). @ < = ) 0 , (s) | : 0(s) = k(s). K $ , z (x): 0 z (x) := sin (x) = p y 0 2 (3) =) z 0 (x) = k(x): 1+y , z (x) . , z (x) , y = f (x) : Zx Zx f (x) = tg (u) du = p z (u) 2 du: (4) 1 ; z ( u ) ;1 ;1 L , f (1) = 0, , z (x) ) ,
446
. .
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(5)
(6) < = , $ $ ) , , . . fu(x) > v(x) \ u(x) 6 v(x)g.
y = f (x) ;1 1] f (1) = 0 , = (;1) = (1) (j (x)j < =2) 1 k1 = k(;1) k2 = k(1). 2
, M ( ) M ( k1 k2). . AB jABj = 2c 6= 0. ) , $ , ;!. + , A (j j j j < ) $ ; AB $ / , 0 L( ? AB ), , . I1
k c = ; sin . 2, $ ( + )=2, 0 . + L? ( ) : 6 . +, L? ( 2 2) L?( 1 1) () 1 6 2 6 2 6 1: (7) . L( ; ) ( . 3). C0 ! ( )
! = +2 = ;2 : (8)
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447
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M ( k1 k2). 0 $
$ $ . 1. y = f (x) M( k1 k2), , , ! = ( + )=2 k1, k2
M+ : ! > 0 k1 < ; sin < sin < k2? M; : ! < 0k1 > ; sin > sin > k2? (9) 0 M: ! = 0k1 = ; sin = sin = k2: ! .+ J )
M0 . y = f (x) 2 M ( k1 k2). + , z (x). . 0 $ (3), 0 A(;1 sin ) B (1 sin ) ( . 4). I , / ,
$ , / 0 $ $, ) $ ) AB . @0 ,, AB l(x) = k0x + sin +2 sin k0 = sin ;2 sin (10) > . 4 AA1 B1 B : AA1 : z1 (x) = ;x sin B1 B : z2 (x) = x sin : (11) - z (x) < l(x) Y (z ) < Y (l) (6). 6 Y (z ) = 0 (5), a Z1 l(u) Y (l ) = p du = 2 tg +2 = 2 tg !: 1 ; l(u)2 ;1
+ $ ! > 0 y = f (x) 2 M+ . -
, (9): ; sin < sin . . (3) AP PB z (x) k1 k2. 2 ) k1 < ; sin . C ) AP , , ) AA1 : z (x) z1 (x). 5) $, ; sin = z1 (1) < z (1) = sin . 6 Y (z ) > Y (z1 ) (6). +, Y (z1 ) = 0, Y (z ) > 0, (5) $ , (9). C0 : PB z (x) B1 B .
448
. .
C y = f (x) 2 M; ( k1 k2), y1 = ;f (x) 2 M+ (; ; ;k1 ;k2) ;k1 < sin < ; sin < ;k2 , 0
M; . " 1. # , ( k1 k2) 0 . ! 6= 0. 5 (k1 k2) ) K+ K;, $ $ )
M : K+ := f(k1 k2): ;1 6 k1 < ; sin \ sin < k2 6 +1 ! > 0g (12) K; := f(k1 k2): +1 > k1 > ; sin \ sin > k2 > ;1 ! < 0g: C ! > 0, K+ ) k1 = k2 ? K; , ) ) / .
> . 5
. 5 $ K+ . 6 (k1 k2), k1 = ; sin k2 = sin , . ) C = = (; sin sin ) . 2 $ , K; c C , ) ) k1 = k2. 6 0 ) . " 2. I y = f (x) 2 M( k1 k2) , z (x). + $ APB , $ z (x). / M+ : z (x) > V (x? P ) M; : z (x) 6 V (x? P ) (13) APB
, ( V (x? P ) := k1(x + 1) + sin ;1 6 x 6 x0 k2(x ; 1) + sin x0 6 x 6 1 P
449
; sin z (k k ) = 2k1k2 + k2 sin ; k1 sin : x0(k1 k2) = k1 + k2 k+ sin 0 1 2 ; k k2 ; k1 2 1 (14) " 3. I ) K (k1 k2) 2 K P (x0 z0 ). + ) jxj 6 1 ) (M+ ) (M; ) B1 OA1 , . . P := f(x z ): jxj 6 1 \ z 7 ;x sin \ z 7 x sin g: (15) O jxj = 1 $ . + , ) P (x z ) 2 P K (k1 k2) 2 K : k1(x0 z0) = z0x; +sin1 k2(x0 z0 ) = z0x; ;sin1 : (16) 0 0 @ ) K ! P (14) P ! K (16) ) ,
: x0 ) J1(k1 k2) = 4 (k1 ;(kk0;)(kk2);3 k0) J2(x0 z0) = 2 z(0x2;;l(1) 2 2 1 P!K K!P 0 k0 l(x) (10). $ K , , (9) k0 | ; sin sin ]: ! ? 0: k1 7 ; sin 7 k0 7 sin 7 k2: (17) 6 . 5 / $ $ N (k0 k0) k1 = k0, k2 = k0 . P J2 AB | P . Q , . 5, $ $ P + , ) ) A(;1 sin ) B (1 sin ). ) ) O ) AB (10): l(0) = (sin + sin )=2 > 0. 5 O C , C | , K | $ OB1 OA1, ) , P . F, $ P , $ 0 K . @ (13) , . . z (x)
< = V (x? P ). J (5) cos cos 1 1 p 2 (18) k1 ; k2 ; k1 ; k2 1 ; z0 = 0 z0 (14). , z (x) - , k(x) - , y = f (x) ) . k1 , (;1 0) , (x0 y0) 0 $ k2 6= k1 , $ $ (1 0) .
450
. .
k12 ! 0 (18) ,
$ , $ . + ) (18) 0
Q( k1 k2) := k1k2 + k2 sin ; k1 sin + sin2 = = (k1 + sin )(k2 ; sin ) + sin2 ! = 0 (19) ( ! (8)). @ (k1 k2) / , C ( . 5). C0 $ , K . 5 ) K, (18) (19) / <= . L / , (18) (;k1 ; sin ) cos + (k2 ; sin ) cos + sin 2! = (k2 ; k1) p1 ; z 2 : 0 k1k2 k1k2 @ , (9), ! ) , . 2 , $ . @ , 0 ,. + (k1 k2) ) , 0 (k1 g2) (g1 k2),
Q( k1 g2) = 0 Q( g1 k2) = 0
(20)
$ ). 6 . 5 / K (k1 k2) 2 K+ K1 (k1 g2) K2 (g1 k2). I1 g2 g1 0 k1 k2 . (16) 0 0 (x z ): Qe(x z ):= Ex2(x;z1) = 0 E (x z ):= x2 sin2 + z 2 +2xz sin cos ! ; sin2 !: (21) I E (x z ) = 0 | / c , ( . 5). + z = 1 jxj 6 1 A, B A1 = (1 ; sin ) B1 = (;1 ; sin ). @ , ) K , ) B1 A1 / , $0$ P : p Z (x) = ; sin ! 1 ; x2 sin2 ; x sin cos ! = ; sin ! + arcsin(x sin )] jxj < 1: (22) + K1(k1 g2) K2 (g1 k2) $ P1(x1 z1) P2 (x2 z2). 6 V (x? P1) k1 g2 , V (x? P2) | g1 k2 .
451
O . 6 $ ). $ . @ ) x0 ). 2 0 B1(x? k1) B2(x? g2) B0 (x? x0) (23) ) $ ,. (20) ! x cos ; q1 ; x2 sin2 ; sin k1 = 1sin 0 + x0 0 q ! x cos + 1 ; x2 sin2 + sin : g2 = 1sin 0 > . 6 ; x0 0 1. 5 ) )
L( ; ), 0 )
!. # , k1 ! ;1 P B1 / , V (x? P ) ) B1 B , ) (4) ,$ A(x? ; ). ! ! 0 $ / B1 A1 , APB , AA1 B1 B . Q, B ) $ / ,$ A(x? ). ; $ : B1(x? 1) = B0(x? ;1) = A(x? ; ) 0 M : B2 (x? 1) = B0(x? +1) = A(x? ) (24) Bi(x? ; ) = A(x? ) i = 0 1 2:
4. "
@ / ,
M . $ . 6 . % 1. M( ? k1 k2)
Q( k1 k2) 6 0
(
! > 0 (k1 < k2)? (25) ! < 0 (k1 > k2):
452
. .
! . O
$ . 6 ! > 0 1. J ) , ) (k1 k2) 2 K+ , )
M+ , H+ := f(k1 k2): Q( k1 k2) 6 0 k1 < k2 ! > 0g: + $ (19), ) K+ ( . 5). + H+ (x z ) , (21), E + P + , Qe (x z ) 6 0. @ P + E (x z ) > 0 $ P + / : E + := f(x z ): jxj 6 1 z 6 Z (x) (x z ) 6= A1 (x z ) 6= B1 g: + E + jxj 6 1, ) $ ) B1 A1 , / , . 5 , ) $ ) P OB1 A1O ( . 7). ), P ) OB1 A1 O. . (13) z (x) > V (x? P ). . P P1 , ) $ B1 A1 . 5 Y (V (x? P1)) = 0 V (x? P ) > V (x? P1), 0 Y (z ) > 0 , : f (1) 6= 0. 5 , P ) ) B1 A1 , ) 0, Q 6 0. C y = f (x) 2 M; ( k1 k2), y1 = ;f (x) 2 M+ (; ; ;k1 ;k2). . (19) Q(; ; ;k1 ;k2) = Q( k1 k2). . , / > . 7 !, Q, $ . " 1. - , (25) Q = 0, . . P 2 B1 A1 , z (x) = V (x? P ) , , $ $ Y (z ) = 0. ) y = f (x) ) B (23), M . 6 $ , $ , , Q < 0. . $ $ , 3. " 2. 6 (9) g1 g2. . (19) (20) 2! ; sin2 ! ; sin : ; sin g = k1 = Qk ;;sin 1 k2 ; sin 2 sin
453
J , Q 6 0, ! 6= 0, k2 ; sin M+ ) (9), : k1 6 g1 < ; sin . K , sin < g2 6 k2 . (17), , , $ $ $ ) 1 M+ : k1 6 g1 < ; sin < k0 < sin < g2 6 k2 (26) AP1 AP2 AA1 AB B1 B P1B P2 B $ $ , )0 . 5. O (26) $ Q = 0. ), (25) . % 2. , k1, k2 (25) M, ! " .
! . ! > 0, Q 6 0 k1 < k2. ,$ z (x), ) $ $ $ y = f (x) 2 M+ ( k1 k2). 2 Q = 0 ) (23). Q < 0 , , 0 ) $ . Q < 0, . . P z (x) ) ) B1 A1 ( . 8). F (AP ) (PB ) , z (x) $ B1 A1 P1 P2. I - , V (x? P1) V (x? P2) $ , (5). I , z (x) , W (x? M N ), $ 0 AMNB , M 2 AP1, N 2 P2 B . 6 MN , ) , k12. +, k1 < k12 < k2. # $ M AP1, N P2B , ) > . 8 , Y (W ). + , N = P2, W (x? M P2) W (x? A P2) V (x? P2) (6). 6 , P2B , 0 AMNB ) $ AMB , , ) , W (x? M B ) W (x? P1 B ) V (x? P1). - ) , P2 B 0 N , W (x? M N ) , Y (W ) . y = f (x)
0 ) k1 k12 k2, $ $ $ , .
454
. .
K $ N 2 P2B ) $ $ M 2 AP1 . 2 0 , y = ;f (x). % 3. y = f (x) 2 M( ), L( ; ), ;1 < x < 1 M+ : fx f (x)g L? (; ) A(x? ; ) < f (x) < A(x? )? (27) M; : fx f (x)g L? ( ; ) A(x? ; ) > f (x) > A(x? ): k1 k2 k0 k00, , (23): M+ : k0 6 k1 < k2 6 k00 =) B1 (x? k0) 6 f (x) 6 B2(x? k00)? (28) M; : k0 > k1 > k2 > k00 =) B1(x? k0) > f (x) > B2 (x? k00): ! . 2 , y = f (x) 2 M+( k1 k2) ) (28), ) $ , b1 (x) b2(x): b1(x) := B1(x? k1) 6 f (x) 6 B2(x? k2) =: b2(x) (29) B2(x? k2) 6 B2(x? k00): (30) B1(x? k0) 6 B1(x? k1) C Q( k1 k2) = 0, k1 k2 . 5 $ 1 1 , b1(x) b2 (x) $ )
f (x) | ,
M+ . Q < 0. 5 P ( . 9) ) ) B1 A1 (22), APB 0 P1 P2 ,
x1 < x2. Q, V1 (x) = V (x? P1) V2 (x) = V (x? P2) $ b1 (x) b2(x). F (29) , ,
> . 9
1 (x) = f (x) ; b1 (x) (4) = Zx z (u) Zx V (u) = p du ; p 1 du: 2 1 ; z (u) 1 ; V1 (u)2 ;1 ;1 (31)
455
J / 1 (x) 10 (x) = p z (x) 2 ; p V1(x) 2 = 0 (32) 1 ; z (x) 1 ; V1 (x) p , t 7! t= 1 ; t2 ) z (x) = V1 (x). # AP1 V1 (x), z (x), , ) ;1 p], ;1 6 p 6 x1 . ; 0 b1(x) = f (x) x 2 ;1 p]. # P1B ) ) AB P2B (26) (p 1). . , / (32) , , 1 (x) | / . 6 , 1 (x) (26): z 0 (1) = k2 > g2 = V10 (1). / / / | , 1 (p) = 1 (1) = 0, 1 (x) > 0 (p 1). 2 (29)
,$ 2 (x) = f (x) ; b2(x), ( , ) , z (x) = V2 (x). O (29) , ;1 p] q 1]. 5, b1(x) = f (x) < b2 (x) ;1 < x 6 p b1(x) < f (x) < b2 (x) p < x < q b1(x) < f (x) = b2 (x) q 6 x < 1: 0 (31){(32) (30) : 0 , V (x? P ), P 2 B1 A1 . C / . O ) k0 = k1 k2 = k00 . J ) (28)
M+ . J ) (27) k1 ! ;1 k2 ! +1 (24). < = ), (27) . + , y = ;f (x) 0
M; . % 4. y = f (x) 2 M+ ( k1 k2),
0 00 0 00 0 k0 6 k1 < k2 6 k00 0 66 66 00 00++ 0 > > 0
(33)
, (23):
B1(x? 0 00 k0) 6 f (x) 6 B2(x? 00 0 k00): (34) ! . 2 (34) ) B1(x? 0 00 k0) 6 B1(x? 00 k0) 6 B1(x? k0) 6 f (x)? f (x) 6 B2(x? k00) 6 B2 (x? 00 k00) 6 B2 (x? 00 0 k00):
456
. .
F f (x) ) (28). .$ , 00 k0 . 6 . 10 / , z (x) V0 (x) V1 (x) ( AP0B AP1 B 0 ). . , V0(x) V1 (x) (31){(32) , V0 (x) = V1 (x) > . 10 , ) , B1 (x? k0) ; B1 (x? 00 k0) , . | ) , 0 | $ . 10. 6 , , = 0 . . A0 | B 00 (;1 ; sin 00 ), , B1 (x? 00 k0) 2 M+ 0 . + $ ) 0 + 00 > 0. @ / , (24) $ $ ,$ y = A(x? ; ). 6 f (x) $ .
5. $ % $&
- fP1 P2 : : : Pn? 1 ng ( . 11). 2 0 i, j = i + 1, k = i + 2, l = i + 3. @ ) j - , j 6 n ; 1, 2cj j : q ! (xk ; xj )2 + (yk ; yj )2 cos j = xk2;c xj sin j = yk2;c yj : j; P;; j Pk j = 2cj = j j O
< = P0P1 PnPn+1 , 1 n, c0 = cn = 0 0 = 1 n = n : j - j qj ) ,
$ i j k, $ q j = j ; i qj = sind j dj = c2i + c2j + 2cicj cos j = 21 j;;! PiPk j j
457
0 q1 qn. J j $ 0 (; ).
> . 11
6 j pj j . , $ $ $, ) (jk) ) 0 fj (x) 2 M (j j kj0 kj00), j = j ; j j = k ; j kj0 = pj cj kj00 = pk cj : # k0 k00 | , (9) k1 k2. @ , pj j , , qj < 0 = #j | / . @ P1 Pn #1 = 1 , #n = n . 6 )$ Pj Pk $ | (ijk) (jkl) qj qk . + $ L( j ; j ), L(j ; j ): j = #j ; j j = #k ; j : J ) PiPj Pj Pk (35) j = i + i = j + j =) i = j + j #j = i + i = j + j =) i = j + j : (36) @ i i , =2, i i ) . O 0 $ k sin j = ;cj dsin j = ;qj cj sin j = cj sin dk = qk cj j (37) cos j = ci + cdj cos j cos j = ck + cdj cos k : j k 2. (ijk) - % , & !
%ijk : ; i 6 ;; i 66 i 6 j j j
i > i &ijk : ; i > ; ; j > j > j
(38)
458
. .
% ( ).
. . - $ , ) $ . . ) ; ij < ij $ (9). - (9) ) pj ci = ki00 1 sin i =) sinc i 0 pj 6 sin(cj ; i ) : (39) (35) 0 i j pj cj = kj 6 ; sin j = sin(j ; i ) @ ) , ) ( ,
: sin i =ci = p;j < p+j = sin(j ; i )=cj ). O
, sin i (cj + ci cos j ) < ci sin j cos i : (40) cos i > 0. .) (cj + ci cos j ), ) i, j k, , , Pk ij , PiPj . O
$ ) Pk ( . 12).
> . 12
1. 5 Pk ij , . . 0 < cj + ci cos j (37) = dj cos i .
. , cos i > 0, j ij < =2. 5 (40) ) $ ) : ci sin j = tg =) ; < ; : tg i < c + i i i ci cos j j ; ) :
, , $ =2, jj j < =2 cj + ci cos j > 0 . . , /
,. 2. 5 Pk ( ;;! Pi Pj ) ij ,, . . cj + ci cos j 6 0, ; < j < ; =2.
459
5 i = j + j < 0. ) (40) ) : (sin i ); (cj + ci cos j );0 ci + (sin j ); (cos i )+ : . , $ ) Pk : , , . 3. 5 Pk , ij , . . cj + ci cos j 6 0, =2 < j < , sin j > 0. . (37), j > 0 =) > : cos i = cj + cdi cos j 6 0 sin i = ci sin (41) i d 2 j j j i j < =2, , ; i < ; i . + / , , $ 180. T / 0
,$ ,
$ $ $ . 6 (38) . - :
i > i =) j (36) = i ; j > i ; j (35) = j : @ $ . % 5 ( ). fPj j = = 1 : : : n? 1 ng | - % ;, j = n;1 ; L(; j j ? Pj Pj +1 ): (42) j =1
! jj j 6 =2
cj ;1 + cj cos j > 0 cj + cj ;1 cos j > 0 1 6 j 6 n (43) %% & fqj j = 1 : : : ng .
! . 2 , n , n > 3, ) (38) f 2 : : : n ; 1g f 1 : : : n;2g. # 1 n;1 $ (37): 1 = 1 = ;1 , n;1 = n;1 = n . ; (38) $ ,$ (n = 2) . 5 , < 0 = ) (38) ) % : ; i 6 ; i 6 i 6 i & : ; i > ; i > i > i 1 6 i 6 n ; 1 (44)
460
. .
(7) (27) ) : % : ;i L? (; i i) L?(; i i ) (45) & : ;i L? (i ; i ) L?( i ; i) 1 6 i 6 n ; 1: # ;i | ) , ) (ij ). + (43) $ , ) (ij ) i ; 1 k = j + 1 ) ij . C , j j j < =2 j j j < =2. 5 sin j = ;qj cj , sin j = qk cj (37) j > ; j qj 6 qk ( j 6 ; j qj > qk ), ) . - (39) ) , (43) < 0 = . 5 , $ qi 6 pj 6 qk . 6 $ (43) ( )
.
* . , ) . O
- (1{2{3{4{5) - 3. 2 $ (1{2{3) $ (3{4{5) (38) $ 2 < 2 < ; 2 < ? ; 3 < ; 3 < 3 < >: - $ 2 3 . - $ ;3 < 3 2 < ; 2 , $ : ; 2 (35) = ;3 ; 3 < 3 ; 3
3 = 2 ; 3 < ; 2 ; 3 : 5 ) (2{3) (3{4) ) 3:
L? ( ; ) L? ( ; ) &% : ;;2 L? (; 2 2) L? (; 2 ; 3 ; 3 ): (46) 3 3 3 3 2 3 @ -
) $ ) : ? ? %& : ;;2
LL? ((; 2 ; 2))
LL? ((; 3 ;; 3 ; 2 )): 3 3 3 2 3 3 6 . 13 / / , $ . J | - . 9 4 5
: 5{6 < = .
461
> . 13
@ / , max jk(x)j ' 2. 6 jk(x)j 1. - , $ $ < 0 = | ( ). + ) (46) $ / . 0, ,
+ , . @ $ $ , ) . + , (44), I j - i j $ )0 $ ) : i = j + j (35). O
$ $ (43). 6 (44) $ $ : ; i 6 i 6 i =) ; i ; j 6 i ; j 6 i ; j =) ; i ; j 6 j 6 j ; j 6 j 6 j ; j + j 6 j + j 6 j + j ; j + j 6 i 6 i : + $ $ (33): i0 := max(; i j ; j ) 6 i 6 i =: i00 0j := max(;j ; i ; j ) 6 j 6 j =: 00j j0 := max(; j k ; k ) 6 j 6 j =: j00 ( ). J (33). 6 , 0j + j00 0j + j00 = max(; j ;j ; i ) + j = max(0 : : :) > 0:
462
. .
+ $ (39): sin i0 6 sin i 6 p 6 ; sin j 6 ; sin 0j =) sin i0 6 p p 6 ; sin 0k : j j k ci ci cj cj ci ck , ) j - : ;cj sin 0k k00 = +1: i0 ? k00 = k10 = ;1 kj0 j>1 = cj sin j j
> . 14
. $ (42) . 14.
/ , , (42). J (47)
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1] . . , . 96-44. | $, 1996. 2] . . & ' - ()')* + , $,-. 98-27. | $, 1998. 3] 1)$*$ 2. 3., 4 . ., 3 $ . . 3 ), $ + . | .: )6 , 1985. 4] 4 . . 8)' )* + )* ** $,- // , , ,, : 38. | ;$ < , 1970. 5] Hoschek J. Circular splines // Computer-Aided Design. | 1992. | Vol. 24, no. 11. 6] . . *, $ $) - ),- $,-. 98-9. | $, 1998. % & 1998 .
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Abstract A. E. Pankratiev, On quotients of hyperbolic products of groups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 465{493. In1nite periodicquotients of non-elementarynon-degeneratehyperbolicproducts of groups are constructed, where the terms (in1nite) and (periodic) are treated in a way natural for the products of groups.
1.
. 1]. . X | . y z X w (y z)w = 12 fjy ; wj + jz ; wj ; jz ; yjg. X - , & ' x y z t 2 X (x z)t > minf(x y)t (y z)t g ; . ( ),
. * , x0 : : : xn;1] + xi xi+1 ] ( & 3). , , &- & . , 2001, 7, 2 2, . 465{493. c 2001 , !" #$ %
466
. .
1 (5]). x0 x1 x2] - yi 2 xi;1 xi+1], jxi ; yi;1 j = jxi ; yi+1 j = (xi;1 xi+1)xi jyi;1 ; yi j 6 4
u 2 xi yi1] xi yi1] 4 ( 3).
2 (4]). - 8 - ! . 1 t, t0 X ' k- ,
) 2 k: jt; ; t0; j jt+ ; t0+ j 6 k.
3 (4]). " x1 : : : x4] | - , ! jx1 ; x2j > 4 max(jx1 ; x4 j jx2 ; x3 j): $ 8 - p x1 x2] q x3 x4], 7 jx ; x j ; 8: min(jpj jqj) > 20 1 2
4 (4]). %& C0 = C0(), C > C0 C 0 > 12(C +) n- x1 : : : xn] - & . ' (xi;1 xi+1 )xi < C i = 2 : : : n ; 1 jxi;1 ; xij > C 0 i = 3 : : : n ; 1, p = x1 ; x2 ; : : : ; xn 2C - x1 xn] x1 xn] C - p. p, 7 7 , ( c)- > 0 c > 0, & p(s) p(t) js ; tj ; c 6 jp(s) ; p(t)j: (1)
5 (5]). %& D = D( c), - ( c)- p q ( p; = q; p+ = q+ D- .
6 (4]). %& C1 = C1() C2 = C2(), r- - & . '
N1 , N2 , N3 1, 2, 3 C > C2 1 > Cr, 3 < 10;3Cr, p1 2 N1 , p2 2 N1 N2 , & C1- 10;3C .
7 (4]). " q1 = x1 x2] q2 = x3 x4] ( c)- - a1 a2 2 q1 ,
-
467
b1 b2 2 q2 , a1 x1 a2 , b1 x4 b2 . %& T0 = T0 ( c), C C 0
jx1 ; x3j jx2 ; x4 j 6 C ja1 ; b1j ja2 ; b2 j 6 C 0 ja1 ; a2j + jb1 ; b2 j < T0 + 2C + 2C 0.
2.
( 7 1] . ' 8 7 7 . ( 3] 9 - 7 &- 7 7 . :
) Gi, i 2 I, . F = i Gi | (2) . :
9 - H = hF jRi (3) & ) F ) ;7 R. <& w, H 2, 2] F w=
lw Y
i=1
ui ri"i u;i 1 ui 2 F ri 2 R "i = 1:
(4)
= Lw ) (4) ) 7 lw . . H
Gi, i 2 I, kmax L 6 cn 7 c. wk=n w > kwk 2], . . k ) 7 7 w u1 u2 uk , ui 2 Gni , ni = 6 ni+1 . ? 9 , . . F. = jwj 7; , &- ) 8 H, w. 1 (3]). 1 ;7 R. , &- ) 7 Gi 9 F ! H (8 & !). 2. Gei, i 2 I, | ) H = i Gi jR . A H = i Ge i jR . A ,
468
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n ! 1. ? ), H = hF jRi ( ) ), ) 77 7 8 &7 7 8 x 2 H ).
9 (6]). W , & * + H ,
& W > 0 cW > 0, W m (W cW )- * ;(H) m.
10 (6]). %& d = d(H),
! + H + , d.
3. !" #" * , L , ; 2 7 (9x 2 L: jL : hxij < 1). > 9 7 8 g 2 H
) E(g) 8 H, ) - &
8 g: E(g) = fx 2 H j 9n = n(x) 6= 0: xgnx;1 = gn g. < , E(g) 7 H, ) -7 2 8 g, 7, & , & 2 & hgi: E(g) CH (g) hgi. 1 E + (g) ) 8 fx 2 H j 9n = n(x) 6= 0: xgnx;1 = gn g, E ; (g) 8 E(g), . . E ; (g) = E(g) n E + (g). 1 , E + (g) 7
-
469
E(g). H E + (g) E(g) . (
, x1 x2 2 2 E(g) n E +;(g). & 9n1 n2 : x1gn1 x;1 1 = g;n1 x2 gn2 x;2 1 = g;n2 . * 1 n1 n2 (x1 x2 )g (x1 x;2 1);1 = x1 (x;2 1g;n1 n2 x2);1 x;1 1 = x1g;n1 n2 x;1 1 = gn1 n2 , x1x;2 1 2 E + (g), . . ) )
E(g) E + (g).
11. " g | + H . $ ( hgi E(g). !"#!$ .n> 9 8 x 2 E + (g), ; 1 n ; xg x = g . 7; X W & 8 x g . I ; k
9 ;(H) ( 7 8) 7 7 XW nk X ;1 W ;nk . 1 p q 8 ( W nk ). 9 p q &
( c)- , ' = (g H), c = c(g H). = p0 q0 , &- - 2 p q
. 5 p p0 (q q0 ) D- , D = D( c H). = k ) ; 8 x. 8 ) ) , jp0j jq0j ; jxj, 3, 7 7 8- p01 p0, q10 q0 ; jp0j=3. 2 p01 q10 ) 16- . A , 7 ( ;7 jp0j=3 ; 2D) p1 p q1 q, ) - (2D + 16)- . ? )& 9 & ; p1 ) 7;7 9 7 ;7 q1. (* , ; & 7 7 Al A.) I &-
&- 7 1 : : : r 7 D3 = (2D + 16) + jgj. i, - , . 1 p ) C = C(g H), & jij + ji+C j < jW C j & i. H 8 , , , 1 1+C 1+2 C :: : & - . 8 1 : : : m , m = r;C1 + 1, (1 ) = S1 : : : (m ) = Sm & 8 s1 : : : sm . A , ;(H) - & Si;1 W C Si+1 W ;ki . 8 ki & 7 7 D4 q jW ;ki j 6 kSi k + + C kW k + kSi+1 k. H , H & ; s;j 1 gC sj +1 = gkj , j = 1 m ; 1, )
m ; 1 < 2] &- . I 7 (C + D4 )jgj + 2D3, &
470
. .
)7 8 ) 7 -7 7 D5 .
; ; ( ) ) R. > , )7 Sj U1 U2 : : :Ulj ) Ui = w1w2 : : :wni , )7 8 ws ; R ) R W, &- 8 g ( . 1). (
, )7 Ui t, t0 j , &- -& v 2]. 1 t1 = t, t2 : : :, tk;1, tk = t0;1 , - v. A (tt0) = = (tt;2 1) (t2 t;3 1) : : : (tk;1t0). j j +1 & - 1 , )7 8 (tst;s+1 ) = ws 1 ; R R ( , ts , t;s+1
& ; 1 1 2 7 ), W ( ts , ts+1 | 2 ). gC
w1 w 2
Sj +1
Ui
wn Sj
wn;1
3 . 1
gxj
1 L ) ;7 R W W ;1 . C ) 7 Ui = w1 w2 : : :wni - : ni 6 D5 + 2kW k. A , )7 Ui D5 + 2kW k , ) L, Sj , 8 )
-
471
) W , - R W ( - x n). A )
) &- 7 1 : : : r . (
,
) , ' ; 7 1 : : : m , &7 7 7 ) ; C: b b+C b+2C : : :. H , 7 1 : : : r ) ) W . p q W nk ' W n. ? &- 9 ; 7 X. ( k ;, p1 q1 ) 9 ; 8 . A ; p1 ( ),
&-) ; q1 X Si 2 W . 1 & l H ; X = W l Si , x 2 E + (g) ) )
hgisi . *, ;, 9 W Si ) ) W = W (R W ). E , 2 7 hgi E + (g) : jE + (g) : hgij 6 #(W ) < 1. ( & jE(g) : E + (g)j 6 2. 1 &
hgi E(g). < .
12. - H
+ g + , ! E(g). !"#!$ . 7 8 g 2 H ) 8 7 L. (
jL : hgij 8 8 gn ) & L
. ) &-7 7 2 7 ) )' ) &-, & 8 l 2 L lgn l;1 = gn . & E(g) & L E(g). < .
13. ' H + g + x xgk x;1 = gl , l = k. !"#!$ . , , jkj < jlj. H
xr gkr x;r = glr & 2 r. 12 8 8 : 2rjxj + jkjr jgj > jxr gkr x;r j = jglr j > jljr jgj ; c: 7 glr , 9. ; r ), -
, . H jkj = jlj. < .
14. - H + g, h & M = M(g h)
= (g h) > 0, xgm y = hn m > M , max(jxj jyj) > m
472
. .
xgx;1 y;1gy 2 E(h). . , g = h jxj, jyj m, x y 2 E(g)/ + x y 2 E + (g) , n > 0, x y 2 E ; (g) n 6 0. !"#!$ . 8 9 ;(H) - . :
9 ;(H) ' p1 q1 p2q2 , 7 (p1) = x, (q1 ) = gm , (p2 ) = y, (q2) = h;n . 9 - & > 0 c > 0, - g h, q1 q2 & ( c)- . ? , u1 ) 2 , q1, ; mjgj ; c. :
) 7 u2 , 7 u2 = q2 . ) = =5. ; m max(jxj jyj) 6 6 m. A 3 u1 u2 ) 8- v1 v2 , jv1 j jv2j > ju1j=3. 5 7 o1j (j = 1 2) q1 o2j q2, jo11 ; v;1 j jo12 ; v+1 j jo21 ; v;2 j jo22 ; v+2 j < D D = D(g h). z 1 = o11 ; o12 jz 1 j > 31 (mjgj ; c) ; 2D 5 2 ) ; z 1 ) ;7 z 2 = o21 ; o22 ' t ; 4D + 16. :
9 ; a1 a2 a3 : : : b1 b2 b3 : : : 7 z 1 z 2 . C ;, ) ; ai ) 7 9 7 ;7 bki ' ti < 5D ( )
, 16 + jhj 6 D). A , ; m
) 11, ) 7 ; ai , aj (i 6= j), (ti ) (tj ). > 7 t;i 1(ai ; aj )tj (bkj ; bki ) T ;1 W r T V s , r = j ; i > 0, T (ti) W, V & 8 g, h. ? , T ;1W r T = V ;s H. > ), T = gn1 x;1hn2 n1 n2, T 7 , &- 9 ;. A , xgr x;1 = h;s H 12 xgx;1 h ) 7 7 ) 8 7 E(h). K , y;1 gy 2 E(h). L g = h, 13 xgr x;1gs = 1 r = s. A , x 2 E(g), y 2 E(g). r T ;1 W r T = W ;s ) ; ; m. ? 7 , jT j < 5D. 1 & ( 7), ; ai ) q1 ;, aj , ; bkj ) q2 ;, bki . E , mn > 0 r = ;s, . . T 2 E + (g) x y 2 E + (g). ( x y 2 E ; (g). < .
-
473
15. " g h | + H , E(g) 6= E(h). $ uv 6 0 ) (gm hn ), (gu gv ), (hu hv ) ( , * ;(H)) C = C(g h), & m, n, u, v. !"#!$ . 1 L = (gm hn)
9 ;(H) 7 x1 x2 x3], 7 A x1 x2] gm H, B x1 x3] hn. A (A B)x1 = L. 1 - & A A1 A2, B B1 B2 , jA1j = jB1j > L ; 1 p, q x1 x2] x1 x3] A1 B1 jp+ ; q+ j 6 4. :
s t, - x1 &- W m Z n 7; W Z, &- 8 g h. 5 9 - & s1 t1 7 s t, jp+ ; s1+ j jq+ ; t1+ j 6 D = D(g h). M D maxfjgj jhjg, ) , s1 t1 & W m1 Z n1 . A gm1 fh;n1 = 1 H 8 f jf j < 2D + 4. > , m1 > (L ; 1 ; D)jgj;1, jgm1 j > L ; 1 ; D. * ; m1 14, 7 g h ) ) 7 7 ) 8 7 )& 12. A , L 7 7, -7 g h. :
(gu gv ), H
; gu1 fg;v1 = 1 8 f ; 2D+4, ' u1 > (L ; 1 ; D) jgj;1. ; L ( . . ; u1) 14 uv < 0 e 2 E ; (g), . A , (gu gv ) ) 7 7, -7 g -7 u v. < .
16. " W1 : : : Wl ( * )
+ g1 : : : gl , H , ! E(gi ) 6= E(gj ) i 6= j . $ = = (W1 : : : Wl ) > 0, c = c(W1 : : : Wl ) > 0 N = N(W1 : : : Wl ) > 0, p * ;(H) (5) (p) Wim1 1 Wim2 2 : : :Wims s ( c)- , ik 6= ik+1 , k = 1 : : : s ; 1, jmk j > N , k = 2 : : : s ; 1 (ik 2 f1 : : : lg).
474
. .
!"#!$ . M c, (1) p ( . . s = 0, t = jpj), & p0 = p(s) ; p(t) p ) (p0) W 0WW 00 , kW 0k kW 00k < max(kW1k : : : kWl k), W (5). Vk | 7; , H Wimk k . 15
- C = C(W1 : : : Wl ), (Vj Vj +1 ) < C j = 1 : : : s ; 1. :
p = p1 : : :ps p, (pk ) Wimk k . :
) q = q1 : : :qs , (qk) Vk q; = p; . 9 7
> 0, L > 0, - W1 : : : Wl , jVj j > jmj jkWij k ; L j = 1 : : : s. A , ; jm2 j : : : jms;1j 4, 7 q ) T - t, &- - 2 q, ' T = T(W1 : : : Wl ). ( q = q1 : : :qs
t = t1 : : :ts , j(qj ); ; (tj ); j 6 T . 7 ; jmj j, j = 2 : : : s ; 1 ( . . Vj ) tj t ( t;1 ). ? , ) 2 t: jtj =
s X
X
X
j
j
(kWij kjmj j ; L ; 2T ) > X > 12 kWij kjmj j ; 2(L + 2T ): j ; jmj j, j = 2 : : : s ; 1. A , = =2, c = 2(L + 2T) p & (1) s = 0, t = jpj. < . %#!$ 1. + g1 : : : gl H , & E(gi ) 6= E(gj ) i 6= j , ! N = N(g1 : : : gl ), m1 : : : ml > N gpfg1m1 : : : glml g (g1m1 : : : glml ). !"#!$ . ( 16 , c, N. I , H ; gim1 i1 k1 : : :gims is ks = 1 (6) ij 6= ij +1 , j = 1 s ; 1 ki 6= 0. 16 (6), X jmij jjkj jjgij j ; c 6 0 j =1
jtj j >
(jqj j ; 2T ) >
j
jmij j > N > c;1 + 1. ? .
-
475
W1 : : : Wl & 8 g1 : : : gl ) H. > 9 ) & K
) Sm = S(W1 : : : Wl N K m) W X0 W1m1 X1 W2m2 X2 : : :Wlml Xl (7) kXi k 6 K, i = 0 l, jm2j : : : jml;1j > m Xi;1 Wi Xi 2= E(Wi+1 ), i = 1 l ; 1. ? ( c)- , 7 ( c)- 9 ;(H).
17. %& > 0, c > 0 m > 0, & K W1 : : : Wl , W 2 Sm ( c)- . . , Wi Wj i j 6 l, l K. !"#!$ . ( (2) W 2 Sm )
W = (X0 : : :Xl )V1m1 : : :Vlml , Vi (Xi : : :Xl );1 Wi (Xi : : :Xl ). & E(Vi ) 6= E(Vi+1 ), W 16. L Wi W1 i 6 l, ) W Xi;1 W1mi ( ml+1 = 0). E ( L)- 9, ' W1 , L W1 K. Vi 7; , H Xi;1W1mi . 15 (Vi;;11 Vi ) 7 7, -7 W1 K. A
9 ;(H) v1 : : : vl
V1 : : : Vl , v1 ; vl + ' t , 16, 4, 2 t. ( W, ' , l K, ) . < . :
9 ;(H) 7 p1q1 p2q2. (q1) W 2 Sm (q2;1) W X 0W m1 1 X 1W m2 2 : : :X l 2 S m = S(W 1 : : : W l N K m). :
q1 = s0 t1 s1 : : :tl sl q1, (ti) Wimi , (si ) Xi (Wi & 8 ) H Xi;1 Wi Xi 2= E(Wi+1 )), q2;1 = s0 t1 : : :sl , (ti ) W mi i , (si ) X i . ti tj ' , ;(H) 7' vi 7 Vi (vi ), &-7 9 ; oi oj 7 ti tj , 7 Vi W j Vi;1 2 E(Wi ), 7 ) a b, (Vi W j Vi;1)a = Wib H. (* , & 8 g, h , ) - 7 8 7 , - & a b, ga = hb.)
18. ' ! , jp1j jp2j < C C , & ( m k, ! jkj 6 1, jm2 j : : : jml;1 j jm2j : : : jml;1j > m ( W 2 Sm , W 2 S m )
476
. .
ti ti+k i = 2 : : : l ; 1, i = 1 / i = l, jm1j > m / jml j > m. . , j 6= i + k ti tj
. !"#!$ . 17 - & > 0, c > 0 ( - m1 : : : ml ), p1 , q1, p2 q2;1 & ( c)- . ? 7 9 - T, & ti ; jmi j 7' t0i q2;1 , 7 t0i ti & T - . L jmi j , t0i 7 7 ( c)- . sj 7 K, 7' 7 t00i t0i, & W i+k k. ( 7 9, t000i ti , 2T - 7 t00i . 00 14 t000 i ti (, , ti ti+k ) . I , ti ti+1 ) ' ti+k . A H: Vi;1 Wia Xi Wib+1 Vi+1 = W ci+k 2 a, b, c Vi , Vi+1 7 Vi W i+k Vi;1 2 E(Wi ) c ; 1 ; a Vi+1 W i+k Vi+1 2 E(Wi+1 ). 1 & Xi = Wi Vi W i+k Vi;+11 Wi;+1b H Xi;1 Wi Xi = Wib+1 Vi+1 W ;i+ck (Vi;1Wi Vi )W ci+k Vi;+11 Wi;+1b 2 2 Wib+1 Vi+1 (W ;i+ck E(W i+k )W ci+k )Vi;+11 Wi;+1b = = Wib+1 Vi+1 E(W i+k )Vi;+11 Wi;+1b = = Wib+1 E(Wi+1 )Wi;+1b = E(Wi+1 ) (8) & Xi;1 Wi Xi 2= E(Wi+1 ). A , ti+1 ) ' tj , j 6= i + k j > i + k 7 9. L ) j > i + k + 1, ti+k+1
ti , , ;, &. ? , j = i + k + 1. * 2, jkj 6 1, 7 ti (ti ) t1 t2 (t1 t2). < . ? 8 g h 2 H , gk = ahl a;1 a 2 H 2 k l.
19. ' + a b H + , , juj jvj + c = au bv a, b. !"#!$ . ), 8 c . O 16. 1 ) 9 , c
)' 8 ) 7. (
, = xgx;1, g 2 Gi, & k ck = xgk x;1 , gk 2 Gi , ( ) 7 8 c = aubv ' . E 16,
7 (au bv )k & ( d)- , ' k.
-
477
I , a c . A am = x(aubv )nx;1 jmj jnj ;. 16 am (au bv )n & ( d)- , ' d u, v, m n. 1 & 5 2 ; m1 am1 = y(au bv )m2 z, jyj jz j a b. * 8 18, 8 bv au ) ', &- am1 . K , b ) 8 c. < .
20. " + L H +. $ 2, + . !"#!$ . 7 8 a ) L. ( ' 8 7' 8 x 2 L n E(a). A x;1ax 2= E(a). (
, x;1ax 2 E(a),
jE(a) : haij k l (x;1ax)k = al , . . x;1ak x = al . 13 k = l, x 2 E(a). :
8 b1 = (as x)t b2 = (a2sx)t ; s t. 17 9 ;(H) bm1 bm2 & ( c)- , c m. A , 8 b1 b2
. I , 7 ) 17 & w(b1 b2) b1 b2 wn & ( c)- , ' c n. ? 7 , 8 , 7 w(b1 b2), , 7 . ; .
21. - 20 L +. !"#!$ . 20 L 7 8 b1 b2, ) &- & , gpfb1 b2g = F2 . 19 8 b1 b2 ) . H 8 . k > 2 8 b1 : : : bk ) . H )
) , 19, ) , ; m1 : : : mk 8 bk+1 bk+1 = bm1 1 bm2 2 : : :bmk k bi , i = 1 k. < .
4. % " &
) R 9 (2)
, . .
478
. .
(1) R 2 R ) R;1 2 RN (2) R 2 R R R1R2 ) R2R1 2 R. > R R1R2 | ' ) 2 ' 8 R 2]. :
9 - H = hF jSi (2). U "- R 2 R ( 2 7
R1), - R0 2 R, (1) R UV , R0 U 0V 0 V , U 0 , V 0N (2) U 0 = Y UZ H Y , Z, kY k kZ k 6 "N (3) Y RY ;1 6= R0 H. ? ), R C(" c )- & " > 0, > 0, c > 0, > 0, (1) kRk > & R 2 RN (2) & R 2 R ( c)- N (3) & "- & R 2 R kU k kU 0k < kRk. 1 , 2 - C 0() 7 7 C(" c )- , H , " = 0, = 1, c = 0, = 1. I 92 8 . K R 2 R (1) R UV U 0 V 0 U, V , U 0 , V 0, (2) U 0 = Y U 1 Z H Y , Z, ' kY k kZ k 6 ", U ' "0 - R. L ) ' ; U 0 = Y U ;1 Z, U "00- R. * 2, U 0 = Y U ;1Z Y V ) H Y (U 0V 0U);1 , U ' "000- R. 1 C1(" c )- (C2(: : :)-, C3(: : :)- ) C(" c )- kU k kU 0k < kRk & "0 - U ( & "00- U, & "000- U). :
) H ' 7 8 g 2 H. W | 7; , &- 8 g. ), , )' ;7 & W , ;
- . ( 12 11 - n > 0, xgn x;1 = gn x 2 E(g). 1 Rm ) (2 ' ) 2 W mn .
22. %& > 0, c > 0, > 0, " > 0 > 0 ! m0 > 0, m > m0 (1) Rm C(" c )- / (2) Rm C2 (" c )- , E(g) = E + (g)/ (3) Rm C3(" c )- , m !.
-
479
!"#!$ . ( c 9. I , U "- R 2 Rm , ' kU k > kW mn k.
& "- U 0 = Y UZ, kY k kZ k 6 ". * kY k kZ k ( " + 2kW k), ) U U 0 W. kU k > jmnjkW k, ; m ) 14, 7 Y 2 E(W) H. O , 7 ) n, Y 2 E + (W ) R W mn , R0 W mn Y RY ;1 = (Y W n Y ;1 )m = W mn = R0 H. K , Y 2 E ; (W) R W mn , R0 W ;mn Y RY ;1 = R0. ( "- , 8 C(" c )- m > m0 ; m0 . A ), R W mn UV U 0V 0 U 0 = Y U ;1Z, Y 2 E + (W), U 0 W ;mn ( R). 1 & ) . I
Y 2 E ; (W) V W s, (U 0V 0U);1 W s;mn s. H Y W n Y ;1 = = W ;n H )' H: W mn2 (Y V )W ; mn2 = Y W ; mn2 W s W ; mn2 = Y (U 0V 0U);1 & "000- . A , ) .
5. ( "" " " " & " ) ;7 ) H = hF jOi, . . H 9 - H1 = hH jRi, ) ; H ) ;7: H1 = hF jO Ri: : < 2], R- ) R. R- P1 , P2 R1 R2, ; o1 o2 , ' , Q 7' s, &-7 ; o1 o2 , 7 (s);1 R1 (s)R2 = 1 H: (9) : Q s ' P1 P2, , & ) R S H ; (9). = R- , , ; . I , ) -& R- , ' ! .
480
. .
(' - . :
7 w = p1 q1p2q2 , &-7 ; = ;w 7 Q H1. q1 q2 & R- P1 P2 kp1k kp2k 6 " ". L P1 6= P2 ; ( RS) ) R- , ' '
"- P1 P2 ( P2 P1). A ) P1 = P2 , , ; R- , , p1s1 p1 s;2 1 (s1 s2 P1 ), & RS, ) R- . * 2,
"- P1 q
Q & P1 6= P2 . 1 ; kq1k=k@P1k ' P1 P2 ( q) ; (P1 ; P2). Q | 2 H1 p ; q. ? ), R- P Q
(" )-;7 7, - "- ; P @; = zuy;1 u0, uvu0v0 | P, ' kuk > k@Pk ku0k > k@Pk (yv) ) H (yu;1 v0;1u0;1). ;7 Q y ;& ) 2& H ) (yv) (y(u0 v0 u);1) ; Q R- . 8 2 Q '7,
) ; , . I &- ) - &
- R S R 2S . 1 4] .
23. H > 0 & 0 > 0, 2 (0N 0] c > 0 " > 0 > 0 & . " R C(" c ). , Q | ! 3 H1 , ( @Q ( c)- q1 : : : qr , 1 6 r 6 4. $ Q R-, ! R- P Q & "- ;1 : : : ;r ( ) P q1 : : : qr
, (P ;1 q1) + : : : + (P ;r qr ) > 1 ; 23:
24. H > 0 ! 0 > 0, 2 (0N 0] c > 0 & " > 0 > 0, & & .
-
481
" R C1(: : :)-, C2(: : :)- C3(" c )- . , Q | ! ( H1 p = p1 p2 q = q1q2, ! p1 p2 q1 q2 ( c)- . $ Q R-, ! R- P & ;1 : : : ;4 ( ) P p1 p2 q1 q2 , (P ;1 p1) + (P ;2 p2) + (P ;3 q1) + (P ;4 q2) > 1 ; 23:
6. *- "" " " & "
25. H > 0 & > 0, c > 0 ! " > 0, N > 0 & > 0 & . " R C(" c ). $ *- H1 = hH jRi . . , W , kW k 6 N , W = 1 H1 , W = 1 H . 4(, U1 , U2 , N H1 , H . !"#!$ . H = hF jOi ax + b | 7 9 2 . I &
W, 1 H1 , ) 27 kW k, 7'
n Q W Tj Rj 1 Tj;1, Rj 2 O R F , j =1 ) 7 n = nW 6 AkW k + B A = 7( ; 47);1 a + 2b, B = b. E H1. ( C ( , ' 7 2]) & W , 1 H1 , - ' Q H1 p, (p) W. ), p ( 12 1)- , . . 7' p0 p, jp0j < 21 kp0k ; 1. A )
W = UV H, kU k 6 kW k ; kp0k + 2 + jp0 j 6 kW k ; 21 kp0 k + 1 l Q V = Qi Ri 1Q;i 1 F . > Ri 2 O l 6 a( 23 kp0k; 1)+b. C W i=1 ;, nU 7 7 U. ) 2 U,
482
. .
1 3 nW 6 nU + l 6 A kW k ; 2 kp0k + 1 + B + a 2 kp0k ; 1 + b 6 1 6 AkW k + 2 kp0k ; 1 (3a ; A) + B + 2a + b 6 AkW k + B:
A > 2(2a + b) + 3a kp0k > 3. A , 2 6 12 , c > 1, . . p ) ( c)- , ( 12 1)- . L Q ) R- , W = 1 H , A > a, B > b. ( ), Q ) R- , 23, 7 7 " R- P 7 ; p (
@Q 7 ), (P ; p) > 1 ; 23. @; = s1 t1 s2 t2 , p = qt2, @P = t1u q u. * 2, R 2 R | P ( . 2). s1
t2
;
t1
u
P
Q0
q
s2
3 . 2
> , Q0 Q qs;2 1 us;1 1 ;7 , Q: ks;2 1 us1k 6 2"+23kRk+2 < (1;23)kRk;c;2" < jt1j;js1 j;js2 j 6 jt2 j 6 kt2 k: > ; ( , kRk > ), ( c)- t1 @P. A , ) 2 , Q0 ) 0 n O R, n0 6 Akqs;2 1 us;1 1k + B < A(kW k ; kRk( ; 46) + 2 + c + 4") + B: (10)
-
483
? 7 , ( 12 1)- t2 ; H 2" + kt1k + 2(2" + kt1k + 1) 6 3kRk + 6" + 2. 8 O- ; a(3kRk + 6" + 2) + b. ? 8 2 , Q W = 1 H1 ) 1 + AkW k + B + a(3kRk + 6" + 2) + b ; A(kRk( ; 46) ; 2 ; c ; 4") 6 AkW k + B A B . H1 . ? (10) & W , 1 H1, H, 2 kW k > ( ; 46) ; 2 ; c ; 4". 1 & ) , ) ;: > ( ; 46);1(N + 2 + c + 4"): I 7 ), 2 ) U1 , U2 ) R- . A 24
) R- P 7 & ; 1 ; 23. > , P ( , ;, U1 ) ; 12 ; 12. ( . 2 kt2k > jt1 j ; js1j ; js2j > kt1 k ; c ; 2" > 21 ; 12 kRk ; c ; 2": 1 & , , kU1 k > kt2k > N ; . < . , ;, H H1 = hH jRi. p1 q1p2q2 | 7 ;(H), ' kp1k kp2;k1 6 ", q1 q2 & ( c)- . ), (q2 ) UV1 V2 U, kU k > kV2 k > 0 (q1) R 2 R, ' kq1k > kRk > 0.
26. > 0, > 0, > 0 ! > 0, c > 0 & " > 0 > 0 & . ' , R C(" c ), ! , V2 H (p1)R (p1 );1 . . , R C1(" c ), (q2;1) UV1 V2 U ! , . !"#!$ . ( " 23 25. O , kV1 k > kU k, . . V2 U UV U, ) q2;1 = u1 vu2 , (u1) (u2 ) U, (v) V . 5 2 7 ; o1 o2 2 q1 , 7 t1 = (u1 )+ ; o1 t2 = (u2 ); ; o2 kt1k kt2k < K = K( c " H). (; o1 o2 & q1 = w1ww2. 1 Tj (tj ), Pj (pj ), Wj (wj ) j = 1 2,
484
. .
H P1W1 T1;1 = U = T2 W2 P2: (11) > , , ( c)- q2 jW1j > jU j ; jP1j ; jT1 j > kU k ; c ; " ; K. ? 7 , kU k > > (1 + );1 kq2k &, kq2k > jq1j ; jp1j ; jp2j > kRk ; c ; 2". A , jW1j jW2j > 2 (1 + );1 kRk ; c ; " ; K ; (1 + );1 (c + 2") > 3kRk ) ; < 41 2 (1 + );1 . : (11) ) X1 W1 X2 = W2 , X1 T2;1P1 , X2 T1;1 P2;1, . . kX12k < " + K. 5 3 jW1j jW2j > 3kRk, Wj Wj 1Wj 2 Wj 3 j = 1 2, jWj 2j > kRk (W21;1X1 W11 )W12(W13 X2 W23;1) = W22 H, ' T j j ". & C(" c ) &
Y R1Y ;1R2 = 1 H, R1 W12 W13WW21W22 W23W0 W11, R2 ;1 W22W23 W0W11 W12W13WW21 ( R (q;1)W 0 ) Y W21 X1 W11 , . . 1 R H W1WX1 . A P1RP1 P1 W1WT2;1 ( X1 = T2;1 P1), . . UV , ) . :
2 ) , R C1(" c )- &, W12 W22 & ) R 2 R. < .
27. " H = hF jRi > 0. $ & > 0, c > 0 " > 0 > 0, & . " R C1(" c )- C(" c )- . $ R 2 R + H , X H1 = hH jRi , X H1 , & H ( CH (R) H . " + ! CH (R) H , R C1(: : :). !"#!$ . & 7 C(" c ) C1(: : :) R 2 R ( c)- . ) 8 ) , R = hgh;1 , g 2 Gi . A
7 : jRnj 6 k(hgh;1)n k = khgh;1 k. ( ; n. > R Rn 7; H R1 R2 4, (R;1 1 R1) > jR1j=13 ( ) , C0 > 12 > 13C0). * 8 , & C(" c ) R
) , &-7 kRk=13. I , 10 R, &- 7 H, ) H 2 S ; d,
-
485
d = d(H). L R , 7 2 7 R0 S = T R0T ;1 H, kT k 7 C, -7 H c. (
,
2& P ) H R S ) ' 7; t 7 ;7 2. & P0 @P0 = = t1pt2 s, t1, t2 | , (t1) = (t2 );1 = (t) = T, (p) = R0 , (s) = S ;1 . :
9 ;(H) 7 t1 pt2s
7 TR0 T ;1 S ;1 . > p ' p0 ) 2 N1 = ft1 t2g, N2 = fp0 g, N3 = fsg, ' t1 p0t2 s 6, 7 jt1j = jt2j = = jtj = kT k ) jsj = kS k. 1 & kT k < C = C(S H c) = C(H c). I , kR0 k = kRk ) ( c)- R0 . A , ; R ; H 7 . A ) = =3, = 1=3 , ", 7 26N , 1 ; 23 > . :
X k H1. ) , X ) H1 ;7 . ?- ' < Q 7 2 X k . O , Q ) R- , X k = 1 H. ? 23 Q 7' P 7 ; q = @Q, (P ; q) > 1 ; 23. 1 @; = p1 q1p2 q2
. (a) kq2k > (1 + )jX j. A ) (q2) UV1 V2 U, V1 V2 2 X kU k > kV2 k. 26 ) ) , R C1(" c )- &. ( ) C(" c ) 7 ) 26 )' H V2 (, , X)
CH (R). (b) kq2k 6 (1 + =3)jX j. A )
q2 = q0q00, kq0 k = jq0j 6 6 jX j = kX k kq00k 6 3 jq0 j. (p1q1 p2) = (q2);1 H,
jq0 j = kq0 k > (1 + =3);1kq2k > (1 + =3);1(jq1j ; jp1j ; jp2j) > > (1 + =3);1 ((1 ; 23)kRk ; c ; 2") (12)
q1 ( c)- . ? 7 , (q0 ) = (p;2 1tp;1 1(q00 );1 ) H1, t q1 2 : q1t = @P. A , X ( , (q0 ) 2 X) jq0 j 6 jtj + jp1j + jp2j + jq00j < 23kRk + 2 + 2" + =3kq0 k < < 23kRk + 2 + 2" + (=3);1 (jq0 j + c)
486
. .
jq0 j < 32 (23kRk + 2 + 2" + c=3):
(13)
< =70 kRk > ; 2 (12) (13) , . . 7 7 ) . * . I 11, 7 2 hRi 7 CH (R). < .
28. - H " 27. " X , & H + x, R R, & C(" c )- , > 3kX k. $ X
+ H1 . !"#!$ . ) , X 2 H. (a) ), X k = 1 H1 Q | 8 ; . 27 kq2k < (1 + )jX j. A
kRk 6 (1 ; 23);1kq1k < (1 ; 23);1;1 (jq1j + c) < < (1 ; 23);1 ;1(kq2k + 2" + c) < (1 ; 23);1;1 ((1 + )jX j + 2" + c) < < (1 ; 23);1 ;1 (1 + ) 3 kRk + 2" + c : 1 ), - , ; kRk ; kRk > . H , kq2k > (1 + )jX j. *, 27, 8 X ) H 8 2 CH (R). ( 11 & H X R. (b) I , X ) 8 ) X = h;1 Sh, S 2 Gi . X 7 8 H, 2 Q0 8 7 )' ) R- . 24 Q0 7' R- P, &- . C 27 7 p Q0, &-7 (p) = S 1, ;p P 8 ( - ). 1 @;p = s1 t1s2 t2 . I
) (a)
7 ; 27, kq1k
(kq1k + kt1k), kt1k < ;1 (jt1j + c) < ;1 (c + 2" + 1).
29. " H 25 + R t . $ 25 , H1 + ( t).
-
487
!"#!$ . & 25 H ) , , ) 7 8 . A 20 H ) K = gpfa bg, )' 8 a b. L K, - 7 x21 y12 ] : : :x2l yl2 ], l > 1, xi yi 2 K, , 2 . ? , 21 H 7 8 g0 = 2 ] : : :x2 y2 ] : : : g = x2 y2 ] : : :x2 y2 ]. W : : : W = x201 y01 t 0 t t1 t1 0l 0l tl tl & H 8 g0 : : : gt. ( ), 27 28, > 3 omax kW k. 6i6t i ) , H1 8 . A W0 : : : Wt & H1, ) 8 E & , ) -& x2 y2 ], x y 2 E. A , ) W0 : : : Wt R 2 R 28, , Wi Wj i 6= j, ) R ) t . W0 : : : Wt ' 8 H1. < .
30. - 25 *- H1 = hH jRi . !"#!$ . 28 H1 ) 77 7 8 . , H1 ) , 8 g, 7 H1, &-7
). E , 8 yn = gn , n = 1 2 : : :, ) H1 2 8 z1 z2 : : :, 7 7 M. 8 ) ( ), 7; 8 , )' y1 : : : yn : : : H, & . , , W H1 8 g, Zi & 8 zi . . ? 2 ( C
7, ) n - ' 2 Q = Q(n) )' H1 8 yn zn . 1 ' p (;7, 7 W in ) q ( 7, 7 Zn ). 1. H ) , n 8 yn zn ) H. , , yi zi ) H i. A Q ) 77 R- . 24 7' R- P, &- Q, 7 & ; 1 ; 23. ), P ) .
488
. .
1. P q. ( 8
2 7 & zn . (
, ; | &- , @; = s1 t1s2 t2 , t1 @P (@P = t1 t), t2 q, q = t2 q1, 2 (1 ; 23)kRk ; c ; 2" 6 kt1 k ; c ; 2" 6 6 jt1j ; js1j ; js2j 6 jt2j 6 6 ks1 k + ktk + ks2 k 6 23kRk + 2 + 2" & ; . 2. ? P ; p ; 1 ; =3. E 7 ' &
) , 27. 1 (P ; p) 1 ; 0 (0 < 0 < 1). > 9 > 0, = 1=10. 1 ; @; = s1 t1 s2 t2 . > t1 @P (@P = t1t), t2 p, p = t2 p1 . ( ) . ( ) L jt2j < (1 + )kW k, )
t2 = t0 t00, 0 kt k = jt0 j 6 jW j = kW k, kt00k < jt0j. 12 t0 : jt0j > (1 + );1 kt2k > (1 + );1 (jt1 j ; 2") > (1 + );1 ((1 ; 0 )kRk ; c ; 2"): ? 7 , jt0j 6 js2j + jtj + js1 j + jt00j < 0 kRk + 2 + 2" + kt0 k jt0j < (1 ; );1 (0 kRk + 2 + 2"): ? 2 , (1 + );1 ((1 ; 0)kRk ; c ; 2") < jt0j < (1 ; );1 (0 kRk + 2 + 2") + (0 kRk + 2 + 2") = 11 0 kRk + 22" + 22 : (1 ; 0 )kRk ; c ; 2" < 11 ; 9 9 11 0 0 0 6 3 , , < (+11=9) , 9 ;, (1 ; 0 ). A , ; . () L jt2j > (1 + )kW k, ) (t2 ) UW1 W2 U, Wi , i = 1 2, 2 W kU k > kW k. A 26 W2 (, , W) ) H CH (R). W = ABA;1 B 2 CH (R). E R H H 27. 11 2 hRi 7 CH (R), , B k = Rl H k l 6= 0. 1 & W k = ARl A;1 . * 2 H1 , W H 7 H1.
-
489
3. 1 ' 7, P q, ; p, ' ; (1 ; =3). C ;, ; P p @; = s1 t1 s2 t2 ( t1 @P, t2 p), ;0 P q @;0 = s01 t01s02 t02 ( t01 @P, @P = t1ut01 v, t02 q, q = t20 q0 ). () 27 Q P ; ;0 . & Q1 @Q1 = p1q1 , p1 p, q1 = s;2 1 us01 ;1 q0 s02 ;1vs;1 1 ( . 3). p1
q0
1
s01
t02
;0
u
t01
s2
t0
t
s02
v s1
1
;
t00
3 . 3
12 p1 q1 Q1. I 8 ' Q. H , P ; Q (1 ; =3), , (P ;0 t02) > (1 ; 23) ; (1 ; =3) = = =3 ; 23 > 4 . 1 & kt02 k > jt02j > jt01j ; js01 j ; js02 j > kt01k ; c ; 2" > 4 kRk ; c ; 2": 2 kq1k: kq1k 6 ks2 k + kuk + ks01 k + kq0 k + ks02 k + kvk + ks1k 6 6 4" + (kqk ; kt02k + 2) + kuk + kvk 6 4" + (M + 2 ; kt02k) + 23kRk + 4 6 2 2 6 4" + M + 6 ; 4 kRk + c + 2" + 23kRk = M + c + 6 + 6" + (23 ; 4 )kRk: ( < 2 =(23 8). A 8992 ) kRk 2 & ; 2 =8, ,
490
. .
;
kq1k 6 M ; 3: (> M ; 1, 2 ,
) )7,
.) ? 7 , , kt2 k < ;1 (jt2j + c) < ;1(jt1 j + js2 j + js1j + c) < ;1 (kRk + 2" + c) kp1k > kpk ; kt2 k > kpk ; ;1 (kRk + 2" + c) > kp3k kpk = in kW k, in ) ;. 2. p1 p ( c)- ;(H), kp1k > kpk=3 ) T;, kq1k < M ; 3. ? , Q1 ) 77 R- . A 23 Q1 ) R- P1, &-& p1 q1 7 & ; 1 ; 23. ? 7
( & ; (1 ; =3)) p1 & ), ; . 1 & . 1. C P1 q1. () P1 7 q1. 1 ;&
Q2 , @Q2 = p2q2. > p2 = p1 , kp2k = kp1k > > kp1k=3, q2 2 kq2k < kq1k ; 3. 2. C P1 q1, p1, '
p1 (1 ; =3). H ) , ;, ;1 P1 p1 : @;1 = s1 t1 s2 t2 ( t1 @P1 , t2 p1, p1 = p01 t2p001 ), ;01 P1 q1: @;01 = s01 t01s02 t02 ( t01 @P1 , @P1 = t1 ut01v, t02 q1, q1 = q100t02 q10 ). ( Q1 P1 ;1, ;01 , Q01, Q001 @Q01 = p01s;2 1 us01 ;1 q10 , @Q001 = p001 q100 s02 ;1vs;1 1 ( . 4). < , kt2k < kp1k=3, kt02 k > ks2k + kuk + ks01 k + ks02 k + kvk + ks1 k + 3. 1 & 77 7 Q01, Q001 ( , Q01 ) & 2 kp01k > kp1 k=3, ks;2 1 us01 ;1 q10 k < kq1k ; 3. 1 Q01 Q2, @Q2 = p2 q2. > p2 = p01 , q2 = s;2 1 us01 ;1 q10 . ( Q2, T 7 & kp2k > kp1k=3, kq2k < kq1k ; 3. ( 7 ; ) M , 8 qM QM 2 7. ( ) in ) ; M.
-
491
t2
p01
s2
0
q10
1
u
s01
;1
t1
1 0
t1
;0
t02
1
s1
p001
v
00
s02
1 q100
3 . 4
8 kpM k > kpk=(3M ) = kW k (in=3M ). > , - ( ) 8 ' & & p. A , , H1 8 g T 8 z1 z2 : : : , . . H1 ). < . ?' 7 .
31. " + H + g ( C , & + E0 H . " M | ( . $ & m0 = m0 (H E0 M), m > m0 & *- H1 H & : (1) H1 | + / (2) * "1 : H ! H1 + , M / (3) + H1 + H ! + "1 (E0)/ (4) + , M H1 , H / (5) Ker "1 . !"#!$ . 8 g W. A XW X ;1 = W 1 X 2 E(W ), C 7 7 E0 = E(g). ;7 8 m 9 - H1 = H=hW m iH ( hW m iH | W m H) 8 22, 25 29. *) H1 30. ?7 (4), ) + 9 "1 : H ! H1 ) 8 ; M, & 22 25.
492
. .
?7 (3) 27. * 2, 7 (5) 10. (
,
)' 8 & 7 d = d(H), 25 9 "1 + . V - K F = i Gi ' - , &7 8 x, &-7 K 7 ,
)' 8 ) 7: x 2 yGi y;1 . L 9 - K F = i Gi )
)' , ) - 8 ) 7, ' - . & . 3 + H ! G1 G2 : : : Gn - - *- He . " + Gi * He . !"#!$ . H
' 8 ' 7. ' 8 fh1 h2 : : :g ( 20 ) ) H0 = H. Hi | 8 ) , &- 9 - 7 H, 8 h1 : : : hi & . 12 31 ) 8 ) Hi+1, &- 9 - 7 Hi, 8 hi+1 7 . 1 , He = lim ;! Hi - 7. 31 Hi & 8 ) . X0 | ) . Hi Hi+1 M = Mi+1 31 , 7 9 "i+1 : Hi ! Hi+1 + ) Xi+1 = fXi xi+1g, ) Xi - ; , xi+1 | 7 7 8 , S )'7 8 ) Xi . ) X = Xi , , , i 8 ). L He 8 xi 2 X )' 8 e , + ) 7 Gk ( 7 9 H ! H, Gk ), xi 2 y;1 Gk y, 8 ) ; Hj ! Hj +1. E (4) 31. A , He - . * 2, Gi & He 9, ; Hi ! Hi+1 & ; ) 8 Gi. E 31, 7 )7 9 "i+1 : Hi ! Hi+1 + ) 8 ; M , , Gi. A .
-
493
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+
1] M. Gromov. Hyperbolic groups // Essays in Group Theory / Ed. S. M. Gersten. | M. S. R. I. Pub., vol. 8. | Springer, 1987. | P. 75{263. 2] . , . . ! " . | #.: # , 1980. 3] &. '. ( )!*. +! ,-!.(! /*!! " // 0!. . #.(. - . 2! . 1. # ! ( , #!3 ( . | 1999. | 4 2. | 2. 9{13. 4] A. Yu. Ol'shanskii. On residualing homomorphisms and G-subgroups of hyperbolic groups // Internat. J. Algebra and Comp. | 1993. | Vol. 3, no. 4. | P. 365{409. 5] +! ,-!.(! " 7 #3 , + * / !. 8. +. . ! , & . | #.: # , 1992. 6] &. '. ( )!*. 9 .*:. * 3 " ; "! ,-!.(" /*!! " // < !,,!( ,)7! .. !7. | 1999. | =. 4, *7. 3{4. | 2. 321{334. & ' 2001 .
, . .
. . .
512.552
: , .
, ! ! R- . " # $ %$ !! , ' % %% $ ! $ . ( % A R-% ( !' )! ! % ) , $ #* )! # % !'$ $ ! , #* ! $ (
$ + ( A ($ ! $ | )! # % !'$ % ! ! ! #*( %% #*( ! ). R-% ( ( %'$ ( % ( $ $ ! ! % ( !$ + ( %'$, . * # ! % % $ , % '$ % ( $ , % $ ( + ! ( (. , ! $ R-% ( ! + ! % ! , % ! ( $ ! .
Abstract D. I. Piontkovsky, Non-commutative Grobner bases, coherentness of associative algebras, and divisibility in semigroups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 495{513.
In the paper we consider a class of associative algebras which are denoted by algebras with R-processing. This class includes free algebras, 5nitely-de5ned monomial algebras, and semigroup algebras for some monoids. A su6cient condition for A to be an algebra with R-processing is formulated in terms of a special graph, which includes a part of information about overlaps between monomials forming the reduced Grobner basis for a syzygy ideal of A (for monoids, this graph includes the information about overlaps between right and left parts of suitable string-rewriting system). Every 5nitely generated right ideal in an algebra with R-processing has a 5nite Grobner basis, and the right syzygy module of the ideal is 5nitely generated, i. e. , 2001, ! 7, 7 2, . 495{513. c 2001 ! "#, $% &' (
496
. .
every such algebra is coherent. In such algebras, there exist algorithms for computing a Grobner basis for a right ideal, for the membership test for a right ideal, for zero-divisor test, and for solving systems of linear equations. In particular, in a monoid with R-processing there exist algorithms for word equivalence test and for left-divisor test as well.
1. | !"1]. " &' A k , A = H=HSH, H = khx1 : : : xN i S H. + , x1 : : : xN , & ,- A ,
. & A. / ' , & ,- H ( G = hx1 : : : xN i), ' 2<3, & , 1 | . G: & , , , & | 5 . 6 & f
5 N f HSH, & & A & 5, & ' f g = N (fg) ( , H,
' & 5 ) . !7]. " I = FA | ( ) A, F = = ff1 : : : fn g A. " 2<3 ' 5 , A, I ' 8' 5 (. !9]): g = fg1 g2 : : :g I I, . g & , . I gr A. 6 , g I 8' , . & I . - g. : , A = H, I &' n , 8' , & - n (!;;9,"2]). / - , , ,- ,- . 1. " R > 0 ' A ( ) R- , , p q 2 A, q = q1q2, deg q1 > R, p q = (p q1)q2: > 0- | & & ,-. > 1- !?].
497
+ , . ,- P | (P ; 1)- . : R- , , &' 8' ( 5 & ). B -, 8' , , , . &' &' ( & 7, 9), (. !C,;]) ( 8). / - , 8' , & , & ,- . , . (. 3 4). " ,-. / 2 R- . 6 , A R- , HSH H 8' S ( & 1). 6 R- ( & 2, 3), 5 5 , ,- 5 & S. / 3 , &' R- A 8' ( 4), & , - ,- !"2]. 6 8' , ' & . / 4 R- ( 8), & & ,- . . B & ,- , !AL], !9] !8]. "
55
8' & ,- . . G , A | M ( 2, & & 9 10), R- 5 , . & . ? , M | & ,- x1 x2 : : : xN , G = hx1 : : : xN i 5 & , . " R > 0 ' M R- , , p q 2 M, q = q1q2 , deg q1 > R, p q = (p q1)q2:
498
. .
H & 1 , M ' 5 , ,- I, R- & 5 & I ( & 2, 3). ? 4 , R- . , . > J. B. 8 .
2. R- !
6 m H deg m , (. . ) ,- G f 2 H, f = m1 + : : : + ms & deg f = 1max deg mi : 6i6s
1. " R | . > A ' ( ) R- , , p q 2 A, q = q1q2, deg q1 > R, p q = (p q1)q2: 1. A | R- . A (!7]), HSH H . ! " # R + 1. . ; , S | 8' HSH. " g 2 S | , . g^ . R + 1. H . N g^ = pq1q2, ' deg p > 1, deg q1 = R, deg q2 > 1. " . 8' , g^ | , p (q1q2) 6= (p q1)q2 , R- . H 8' , . , . , , R + 1, . O . > A 0- , A = H. + &
499
, ,- . () P | . ,- , (P ; 1)- ( (P ; 2)- , & , 1). B ,- . , . ; , a b G , - , c d 2 G, : (1) b = cad, (2) a = cbd, (3) 9e: a = ce b = ed, (4) 9e: b = ce a = ed, e | . 6 & , ,- (c d) ,- , ( ) r = deg d, (1) (3), r = ; deg d, (2) (4). "- , , , 2 , 3 5 x1 x2 : : : xN , , , & ( , ). " S = fg1 : : : gr g. O 5 ;, . fg1 : : : gr g, ' gi ;! gj , & - 2 .3 & gi . gj r ,- . : , - . 5 k;1 1 g 2 : gi1 ;! i2 ;! : : : ;! gik ' r = deg g^i1 ; 1+r1 +: : :+rk;1 , & 5 ,- ' . 2. A = H=HSH S = fg1 : : : grg | HSH / H , $ ; , . R = sup r, % $ ; (
%& ). R < 1, A %% % R- . 3. ' $ ; " ! , A %% % R- % R. . " ' & R , p q & , R- . + , , ,. J p q = p q, 5 ;. " , p q 6= p q, R | . 0
500
. .
& , q = q1q2 deg q1 > R , p q = (p q1) q2 . B 2
3 ; {R. , pq -, N (pq) = p q. S & ,- . ; , . . R . , , p q, ,- , 1, R 6 R. S & 5 ,- . J r 8' gl . - f = 1m1 + 2m2 + : : : + c mc mj | , - mi = s g^l t, s t | ( , . 55 gj ). / j mj i 6= j , , mi
mi = s (gl ; g^l ) t gl = g^l + 1 gl1 + : : : + d gld glj | , mi = 1 s gl1 t + : : : + d s gld t: ; , s glj t mi r. " h c> 0 . P r1 : : : rh f = j mj ,-
j =1 rh+1 gk mi , mi = u^gk w. " mi v1 rn1 gi1 ( , 55 , , & ), v1 | v2 rn2 gi2 ( , , - v1) . ., rn rnf+1 rn1 rn2 mi = v0 ; v1 ; : : : ;f vf ; vf +1 = p q ( & 2 3, - ' , , , & ). S & vl = ul g^il wl vl 1 = ul gijll wl
- jl . 0
0
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;
501
S & , &
. , 5 (& - , ). & . , , . pq, . | , . ( &, , ). S & , pq -, ,- , | . . S. : , & , ,- wl ,- . H ' , & . & . gil+1 S wl , & gijll . B -, 5 T0 T, . & , T0 . J . pq, | . ( &, , ), ' ' T0 ,- . " mi = v0 = u0g^k w0 = u1 gij11 w1 g^k gij11 , , g^k | w1 ( uj1 T0 | g^k ;! gk0 g^i1 ;! gij11 ,). / , v1 = u1 g^i1 w1 = u2 gij22 w2, - & : g^k gij22 v1 , , g^k | w2 . " & & , j : - . k, g^k gizz , ,
k - g^i0 pq. + T , . v0, , rh+1 , vz ;! v0 ( ,- & ), . pq ;! v0. ? , T . &
& , . , , gj S, . gj ;, & T . ;. T & . T , , - m1 : : : mt , ,-
502
. .
N (pq) = p q. " -
rh : vh ;! vh+1 h > 0 & . g^ih & - . gijh vh = uh g^ih wh . H , w0 w1 : : : wr , . R , T deg wl 6 deg q ; R : B , . pq, & g^0 p, q, deg w0 > deg q ; deg g^0 + 1 & ,- | deg wh ; deg wh+1 = h h | ,- . ? R 6 | . gi0 ;!gi1 ;! : : : ;!gil;1 : H , R 6 R, . . , 8' . 2
3, 2<3 & R- & , . - 5 . U , ( ) A -, ,- S, & 2, A ( ) R- , , 1 S 5 . > & L- : ' & 2 5 ,- . S ,- , R- R- . O A ,- x y z . g1 = g2 = g3 = 0, g1 = xyz + xz 2 , g2 = x2z + yxy, g3 = x2y2 . 5 - , ,-: x > y > z. H ' , , 8' &' , g1 g2 . + & . . , , , ,- : 0
0
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503
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6
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6- 3- . : & , . &. " ' -' , , & 2 3 , . " A = khx y z t j zt ; y xy ; z 2 i 5 - x > y > z > t.
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504
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4. (% f I )
f = hi1 m1 + hi2 m2 + : : : + hir mr mi | . . S & f I N f = fi1 m1 + + fi2 m2 + : : :+ fir mr , mi | ,
f f = fi m, m | . J deg fi +deg m < d+R, f = hj j. / P m = m1 m2 , deg m1 = R. H , fi = fij , fij | , ;
j X j X j f = fi (m1 m2 ) = fi (m1 m2 ) = (fi m1 )m2 = (fi m1 )m2 j j f = hs m2 - s.
H & 8' I W . " J | H, &' & h N J = I , 4, I J. S , &' Nh , J
505
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;
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& ,- e1 e2 : : : es f = e1 a1 + : : : + es as G, ,- . ( ) e^i a^i . G , , , . ( h | s- , . ) . h. 6. g = fg1 g2 : : : gtg | " I . + ! s- !*&#
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508
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509
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510
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511
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512
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] . . ! "# " . | : , 1987. +] ,. -. , .. .. , .. . + . / " . 0 # #. !] 2. 0. !. 0 " " | 2 // 5 /6, . | 1995. | 5. 208. | 0. 106{110. 6] . ;. 6 . ," # # # " // < # # . | 1995. | 5. 1, = 2. | 0. 541{544. +] .. . + . ; ? # " . 0 . | /.: /!@, 1988. A1] B. 6. A . ," R- // III # CD# E0 F. 5 . | 5 , 1996. | 0. 174. A2] B. 6. A . !G " D " // < # # . | 1996. | 5. 2, = 2. | 0. 501{509. @] .. ,. @C . ; " // 0 # #. 5. 57. | 1990. | 0. 5{177. <] ;. <. ," : D, ". 5. 1. | /.: /, 1977. AL] W. W. Adams and P. Loustaunau. An Introduction to GrGobner Bases. | Graduate Studies in Math. Vol. 3. | 1994. ) !* 1996 .
. .
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517
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Abstract V. I. Rabover, Jet continuation and reducibility of smooth function families, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 515{532.
For a family of smooth functions of several variables the problem of local reducibility (through a di2eomorphism) to a lower number of variables is considered. Some su3cient conditions of such reducibility, in terms of subfamily reducibility, are obtained. The argumentation is reduced to solution of a jet continuation problem, for jets de4ned on the union of a 4nite family of sets, where the jets can be continued from each of the sets.
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!( b 2 Ai , , !( b 2 Ai , ( !j , , b b 2 Aj ). , ! a 2 V p ( ( ) !
. > q > k 2 p , A1 : : : Ak ( a - p ! , 5 Va , ). 3. 1 : : : m : V ! R V Rn , (1 : : : m ) 3 1 2. . Vijk V | , (i j k) 2. > (Vijk ), i j k 2 1 m, . (# 3 , $ .) ' a 2 V ( , Va ) - r 1 ( ( > 1). , Vijk \ Va ! . ' $ b b Va , b 2 Vijk , b 2 Vi j k . > b i j k , i, r (r i) 2. J , b 2 , , (r i ). : b b (r i i ) 2, . . b b 2 Vrii .
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518
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521
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, 5 $ . a 2 Aij , b 2 Akl ( ). > a 2 Aij Aik Ajk . : 5 ( k Akl , . . ( . I . I p 2 1 s ! i1 : : : ip 2 1 s Bi1 :::ip , p Ai1 : : : Aip . C, T1 : : : Tm , Bi1 :::ip ( ). > T1 : : : Tm A = A1 : : :As , $ ! Ti Tj - Ar ( Ti Tj , 5 (Bi1 :::ip ), ! !( ). 7$ Aij = Ti Tj . 9 . 2. $ ! m = 3 4 : : :. O ! m > 3. , 5 m , , m+ 1. ' $ (Aij ), i j 2 1 m + 1, S ! FA & A = Aij !( $ Fij = (!ij r!ij ) Aij . 7 (Bij ), i j 2 1 m ( & A), Bij = Aim+1 Ajm+1 . >
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524
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525
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, a Ai \ Ak , Aj \ Ak , . . 5 !i !k, !j !k , !i !j .) 3 . 3, Ai ! ! i : V ! R, ! 0 Ai > 0 Ai , !
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0
526
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A). 7 , 5 $ R !( S R, . (' , S, & , Y: (S) ! (R). 7 = Y 1 .) =) ( ) Ri Rn i : Ri ! Rp, i = 1 2, $ , ! (! ! (R1) = (R2) = Z ( ) (! S = L \ R1 = L \ R2 ( L | - Rn, !( ). . ') , ! 5 i
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1] . . . 1. | .: , 1972. 2] . . ! "#$! % !% & ' ( // * +. | 1977. | . 355, . 3. | /. 323{325. 3] Whitney H. Analytic extensions of di1erentiable functions, de2ned in closed sets // Trans. Amer. Math. Soc. | 1934. | Vol. 36. | P. 63{89. 4] Lojasiewicz S. Sur le probleme de la division // Studia Math. | 1959. | Vol. 8. | P. 87{136. 5] 6% ' 7. 8 99 $:8; 9$ ! . | .: , 1968. ' ( ) 1997 .
. .
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517.51
:
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.
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& " p *0 1]n , ! & !" ' $% %--. & . / 0 " ! " !" " 0 . L
Abstract E. S. Smailov, On Paley-type theorems for multidimensional Fourier series on generalized Haar-type systems , Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 533{563.
In the paper we prove Paley-type theorems for multidimensional Fourier series on generalized Haar-type systems under the boundedness condition of their generating number sequences. Properties of base of product of generalized Haar-type systems in p *0 1]n are proved. Then, using them, the Paley-type theorems on Fourier coe7cients are obtained. Comparisons of our results with the known statements are also given. L
x
1.
L !0 1] p
n
1] . . fp g, . ! "#$ % & # ' # $ 2,3]. %! ' % *+ % $ "#$ % & # " ' $ # . x 1 ' ! ' ' % $ Lp 0 1]n, ' $ #' x 2 ' +../$ & #. , 2001, 7, 8 2, . 533{563. c 2001 !"#, $% &' (
534
. .
x 3 % $ ' # ' # Luo Cheng' 4], 3. . 4 5], 4. &. 5 6. 5 $ 6]. * # ' ## #$ fp g, $ Q p0 = 1, p > 2 8 2 N. * # m = pi . 0 1] i=0 fp g: 1(t) = 1 0 1]. < k > 2, k = m + r(p +1 ; 1) + s, r = 0 1 2 : :: m ; 1, s = 1 2 : : : p +1 ; 1, = 0 1 2 : : : ( k ). >' A ' ml ' 0 1], / l , 0 6 l 6 m , = 0 1 2 : : :. < t 2 0 1] n A = B, ' +1 X t = m(t) 0 6 (t) 6 p ; 1: =1 5# . / k (t) = (sr) (t) ': (p m exp(2is p +1+1(t) ) t 2 ( mr rm+1 ) \ B (s) (t) = r 0 t 2= ( mr rm+1 ) 0 6 r < m , 1 6 s 6 p +1 ; 1, = 0 1 2 : : :. B . /% (sr) (t) % %% !. 5 B 0 1], (sr) (t) ( mr rm+1 ). * +" $ ' . / (sr) (t) ! #$ ' ! , /$ ' 0 1] | # ' % '. 5 ', . /! fp g # . @ 1] L0 1] (. 2]) ' C0 1] (. 1]). * # k 2 N, k > 2 k = m + r(p +1 ; 1), r = 0 1 2 : : : m ; 1, s = 1 2 : : : p +1 ; 1, = 0 1 2 : : :. 5" k = r = ( mr rm+1 ), 1 = 00 =(0 1). A r ' # " . +1 ( i ) * # p i i =0 , i = 1 : : : n, | ' # ' ' # #$ , ! , B % fp g. * + #% i (iD ti) +i1=1 , i = 1 : : : n, $ ' n Qn o = (iD t ) i , Et = (t1 : : : tn) 2 0 1]n, E = (1 : : : n) 2 Nn. F# i i=1 0 1]n | ! n-! , Nn = N | N {z: : : N}. 5 -
n
% #!B Z+n = f(1 : : : n): i 2 Z i = 1 : : : ng. * # !, $ #!B (. 7]). * # X | $ , X | " % . 1. + fxg X ' % ! X, ' !! fxg X.
535
2. + fxg X ' % #!,
X " " + x, hx xi = 0 % $ . 3. + fxg X ' % '!, % " + x 2 X ! % x
+1 X
n=1
anxn an = an (x) 2 R n 2 N
$%!% x X.
1.1. 'k (t)+k=11 , k(t)+k=11 Lp 0 1], 1 < p < +1.
'k (t) m (t) (km)2N2 Lp 0 1]2. . " 'k(t)+k=11 2 Lp0 1], 1
0
Z1 Z1
5"
0 0
f(t y)'k (t) m (y) dt dy = 0 8k = 1 2 : : : 8m = 1 2 : : ::
Z1 Z1 0
0
f(t y) m (y) dy 'k (t) dt = 0
' # . m1 2 N k = 1 2 : : :. H # 'k (t) +k=1 , Bm 0 1], mes(Bm ) = 0 8t 2 0 1] n Bm
B=
+S1
m=1
Z1 0
f(t y) m (y) dy = 0:
Bm 0 1]. 5" mes(B) = 0 8t 2 0 1] n B
Z1 0
f(t y) m (y) dy = 0 8m 2 N:
1 Lp 0 1] # k (t) +k=1 8t 2 0 1] n B 9ct 0 1]: mes ct = 0 8y 2 0 1] n ct ) f(t y) = 0:
536
. .
5" , ", ', f(t y) 6= 0, % J = f(t y): t 2 0 1] n B y 2 ctg f(t y): t 2 0 1] y 2 ct mes(ct) = 0g: *+ mes(J) 6
R1 R dy dt = 0. 0 ct
1.2. ! =
Y n
i (ti D i)
i=1
2
N n
Lp 0 1]n
1 6 p < +1
. . K. A. M 2] ', % 1 (t) +=1 L1 0 1] ! , f(t) 2 L1 0 1] R1 f(t) dy = 0 8 2 N, 0 1] f(t) = 0, . . 0 +1 (t) =1 # Lp 0 1], 1 6 p < +1. n nQ;1 o 5# , # Lp 0 1]n;1 i (ti D i) , i=1 i 2 N, i = 1 : : : n ; 1, ' =
Zn
i (ti D i)
i=1
* # f(t1 : : : tn) 2 Lp 0 1]n
n
N Lp0 1]
2 n
1 < p 6 +1:
0
Z1 Z1 0
5"
: : : f(t1 : : : tn)
i=1
0
Z1 Z1 Z1 0
0
Zn
: : : f(t1 : : : tn;1 tn) 0
nY ;1 i=1
i (ti D i) dt1 : : :dtn
= 0 8E 2 Nn:
i (ti D i) dt1 : : :dtn;1
n (nD tn) dtn
= 0:
" ' # 9A 0 1]: mes1(A) = 0 8tn 2 0 1] n A
Z1 Z1
nY ;1
0
i=1
: : : f(t) 0
i (ti D i) dt1 : : :dtn;1
= 0 8(1 : : : n;1) 2 Nn;1:
" / 8tn 2 0 1] n A 9c(tn ) 0 1]n;1 : mes c(tn ) = 0 8(t1 : : : tn;1) 2 0 1]n;1 n c(tn ) f(t1 : : : tn) = 0:
537
5" supp f(t1 : : : tn) G = f(t1 : : : tn): tn 2 0 1] n A (t1 : : : tn;1) 2 c(tn )g f(t1 : : : tn): tn 2 0 1] (t1 : : : tn;1) 2 c(tn )g mesn (G) =
Z1 Z
c(tn )
0
Z
: : : dt1 : : :dtn;1 dtn = 0:
1.3. ! =
Y n
i=1
i (ti D i)
N
2 n
Lp 0 1]n p, 1 6 p < +1. . * # Np0 1]n |n ' !!n Lp 0 1] . , Np 0 1] | ' Lp 0 1]n. < Np 0 1]n 6= Lp 0 1]n, {K$ . /% f(t1 : : : tn) 2 Lp 0 1]n, p + p0 = pp0, % 1) kf kLp 01]n > 0 0
Z1
2)
0
0
Z1
: : : f(t1 : : : tn) (t1 : : : tn) dt1 : : :dtn = 0 0
% $ (t1 : : : tn) 2 Np 0 1]n.
Z1 Z1 0
: : : f(t1 : : : tn)
Qn
0
n Y i=1
, i (iD ti )dt1 : : :dtn
= 0 8E 2 Nn
n i (iD ti) 2 Np 0 1] . i=1 * ! # , kf kLp 01]n > 0. * ' # Np 0 1]n = Lp 0 1]n, ' Lp 0 1]n, 1 6 p < +1. nn o
1.4. = Q i (tiD i) 2Nn | i=1 ! # . kE = (k1 : : : kn) 2 Nn f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1,
kSk1 :::kn (fD )kLp 01]n 6 ap kf kLp 01]n 0
538
. .
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f(t1 : : : tn). * # m(ii) < ki 6 m(ii)+1 , i = 1 : : : nD I (ii)r (i) = h. i = mr(i()i) rm(i)(+1 i) , i = 1 : : : n. * i
i
Q r = I (1)1 r(1) I (2)2 r(2) : : : I (nn )r(n) :
>'
Dk1 :::kn (tED yE) =
k1 X 1 =1
:::
kn Y n X
( i (ti D i) i (yi D i))
n =1 i=1
' % R" % kE. " f i (ti D i)g 8ti yi 2 I (ii)r(i)
(i) (i) = r (i) r (+i) 1 i = 1 : : : n mi mi n n Y n Y Y Dk1 :::kn (tED yE) 6 m(ii)+1 6 ci m(ii) i=1
i=1 i=1
Dk1 :::kn (tED yE) = 0 % $ #$ Et = (t1 : : : tn), yE = (y1 : : : yn) 2 0 1]n. R Qn #, % ! Sk1 :::kn (fD y)= f(tE) Dki (ti D yi) dtE jSk j 6 C
n Y i=1
01]
Z
m(ii)
r
i=1
f(tE) dtE
# y = (y1 : : : yn) 2 Q r yi 6= mr((ii)) , r(i) = 0 1 : : : m(ii) ; 1, i i = 1 : : : n. *+ % $ y 2 Q r jSk (fD y)j 6 C
n Y
i=1
p1 1p Y n ( i ) ; 1 p : (m i ) jf(t)j dt i i =1 r
m(i)
Z
#, kSk (fD )kpLp 01]n
n p Y;
6 Cn
i=1
=
Z 01]n
Y n (i) p
mi
jSk (fD y)jp dy =
(1) ;1 mX 1
r(1) =0
(1) ;1 pp mX 1 (i) ;1
(m i )
i=1
0
0
r(1) =0
:::
:::
(1) ;1 mX n
Z
r(n) =0 r
(1) ;1 mX n
Z
r(n) =0 r
dy
jSk (fD y)jp dy 6
Z r
jf(t)jp dt
=
539
n p Y;
= Cn
= Cnp
Y n (i) p
mi
i=1
Z1 Z1 0
(m(ii) );1 i=1
(1) ;1 mX 1
p;1 Y n
(m(ii) );1 i=1 r(1) =0
:::
(1) ;1 mX n
Z
r(n) =0 r
jf(t)jp dt =
: : : jf(t1 : : : tn)jp dt1 : : :dtn: 0
5 '.
1.5. ! =
Y n
i=1
i (ti D i)
N
2 n
# Lp 0 1]n, 1 < p < +1. . * # f(t1 : : : tn) | ' #! + Lp 0 1]n, 1 < p < +1. 5" + ( 1.3) % " " !! "" P(1") :::(n") (t1 : : : tn) = ! H , 8Ni
(1") X k1 =1
:::
(n") X kn =1
a(k"1):::kn
n Y i=1
ki (ti D i)
f ; P(") :::(n") ()Lp 01]n < ": 1
> (i") i = 1 : : : n N1 Nn n X X Y ::: ck1 :::kn (P ) P(1") :::(n") (t1 : : : tn) = ki (ti D i) k1 =1
" ck (P) =
Z 01]n
P(1") :::(n") (t)
n Y i=1
ki (ti D i) dt
kn =1
i=1
ki = 1 : : : Ni i = 1 : : : n:
I, #' % ! " ! ( 1.4) SN1 :::Nn (fD t1 : : : tn) , kf ; SN1 :::Nn (f)kLp 01]n 6 6 kf ; P(1") :::(n") kLp 01]n + kP(1") :::(n") () ; SN1 :::Nn (f)kLp 01]n 6 6 " + kSN1 :::Nn (f) ; P(1") :::(n") kLp01]n < < " + ap " = (1 + ap ) 8Ni > (i") i = 1 : : : n:
540
. .
@ ' , % & # ' $ ' #! . / f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1, $% Lp 0 1]n, 1 < p < +1, ! . * # # % n n X Y X Y ak (t D i) b ki i k k2Nn i=1 k2Nn i=1
$%% . / f(tE) 2 Lp 0 1]n. 5"
ki (ti D i)
Z
n Y
01]n
i=1
bk = f(t)
ki (ti D i)dt =
Z X
N
n 01]n 2
a
n Y i=1
Y n
ki (ti D i)
i=1
i (ti D i) dt = ak1 :::kn
. . . /% f(t1 : : : tn) 2 Lp 0 1]n '"% ! % & # ' % $ " ' $ #!. @ ' , ' " % % %% ' Lp 0 1]n, 1 < p < +1. % % ' % nn o 1.6. 1 < p < +1 Q r i (ti) 2Nn | i=1 & . ' ( # as s2Nn , Cp
+X 1
s1 =1
:::
+1 X
sn =1
21 2 as1 :::sn
p p1 Z X Y n 6 n as rsi (ti ) dt 6 01]n s2N i=1 X 12 +1 +1 X 0 2 6 Cp : : : as1 :::sn : s1 =1
sn =1
* # "(i) = "(ii) +i1=1 , "(ii) = 1, | ' #% ## ' . I% $ % & # ' $ T" (fD t) =
+ 1 X
1 =1
:::
+1 Y n X n
i=1
Y n
"(ii) c
i=1
i (ti D i)
t = (t1 : : : tn) 2 0 1]n f(t) 2 L10 1]n: . nn o
1.7. = Q i (tiD i) 2Nn | i=1 ! #
. n Qn (i) o
( i ) " = " i 2Nn, " i = 1, )(
i=1
541
f(t) 2 Lp 0 1]n, 1 < p < +1, a;p 1kf kLp 01]n 6 kT" (fD )kLp 01]n 6 ap kf kLp 01]n ap > 0 f(x) ". . ! ! % , ' , % %% ' Lp 0 1], 1 < p < +1 (. 6, 5]). 5" " 1.10 7]
X N 9ap > 0: 8" = f"k g "k = 1 "k ck (f) k () 6 ap kf kLp 01] 8N 2 N: Lp 01] k=0
H , $% N ! +1, , kT" (fD )kLp 01] 6 ap kf kLp 01] : 5# , + % (n ; 1)-" % & # ! . T T" (fD t) % n-" % & # ! :
Z
01]n
jT" (fD t)jp dt
Z1 Z1 Z1 0
0
:::
0
Z1 Z1 Z1
= ::: 0
0
0
jT" (T" (fD t1 : : : tn;1)D tn )jp dt1 : : :dtn;1 jT" (T" (fD t1 : : : tn;1)D tn )jp dtn
dtn =
dt1 : : :dtn;1 6
(#' , ' B)
6 app
Z1 Z1 Z1 0
:::
0
0
jT" (f(t1 : : : tn;1D tn)D t1 : : : tn;1)jp dtn
dt1 : : :dtn;1 =
(#' /) = app
6 cpp
Z1 Z1 Z1 0Z 0
01]n
:::
0
jT" (f(t1 : : : tn;1D tn)D t1 : : : tn;1)jp dt1 : : :dtn;1
jf(t1 : : : tn)jp dt:
5 ', T" : Lp 0 1]n ! Lp 0 1]n | " ! .
dtn 6
542
. .
5 T"2 (fD t1 : : : tn) = f(t1 : : : tn), # ' kf kLp 01]n = kT"2(fD )kLp 01]n 6 ap kT" (fD )kLp 01]n : 5 ' # . * # c =
Z
01]n
f(1 : : : n)
n Y
i=1
i (iD i ) d1 : : :dn
= (1 : : : n) 2 Nn |
+../ & # ' $ . / f(x1 : : : xn). 5" . / P(fD t1 : : : tn) =
X +1
1 =1
:::
+ 1 X
n =1
c2
1 ::: n
n Y
i=1
2
i (ti D i)
21
' . /! *+, t 2 0 1]n. 5#, #' % 1.6 1.7, % #!B . nn o
1.8. = Q i (tiD i) 2Nn | i=1 ! # . )(
f(t) = f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1, a0p kf kLp 01]n 6 kP (fD )kLp 01]n 6 ap kf kLp 01]n ap > 0, a0p > 0 f(t). . T r (x) = sign sin(2 x), x 2 0 1], | . +P 1 / T$. * # x = s 2;s | ' -/#s=1 " ' (0 1), s = s (x) 0 1. * + r (x) = (;1) (x) % " x 2 (0 1), x 6= 2ik , i k 2 N (. 7, . 30]). * # f(t) = f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1. 5" kf kpLp 01]n
=
Z
01]n
(#' 1.7)
kf kpLp 01]n dx 6
p Z X +1 +1 n n : : : X Y r i (xi)c 1 Y i (D i) dx = n L 0 1] p =1 =1 i =1 i =1 1 n 01]n Z X Y p p n n Y p 6 = cp c 1 r i (xi) dx i (D i) L 01]n n p 2 N i =1 i =1 n 01] 6 cpp
543
('# % 1.6)
X 21 p +1 + 1 Y n X 2 (D i) 6 app ::: c2 6 i Lp 01]n 1 =1 n =1 i=1 Z X Y p p n n Y p 6 aEp r i (xi )c i (D i) dxLp 01]n n i=1 01]n 2N i=1 p Z X Y n n Y aEpp r i (xi)c ( D i) dx 6 i n n L 0 1] p i =1 i =1 2 N 01]n
( ' % 1.7)
Z1 Z1
6 aEpp : : : 0
0
kf kpLp 01]n dx1 : : :dxn = Eapp kf kpLp 01]n :
1.9. ! =
Y n
i=1
i (ti D i)
N
2 n
# Lp 0 1]n, 1 < p < +1, # , . . Lp 0 1]n, 1 < p < +1,
Y n i=1
(i) ( i ) (ti
D (i)(
i ))
N
2 n
E = ((1) (1) (2)(2) : : : (n) (n))
= (i) (i) +i1=1 | # +1 i i=1 , i = 1 : : : n. . * # "(ii)+i1=1, "(ii) = 1, i = 1 : : : n, | ' # # ' f(t1 : : : tn) | ' #% . /% ' Lp 0 1]n, 1 < p < +1, % & # ' $ : n X Y f(t1 : : : tn) c (1) i (ti D i):
(i)
" 1.7
N
2 n
i=1
X Y N1 Nn Y n n X ( i ) ::: " i c i (D i) Lp 01]n 6 i=1 1 =l1 +1 n =ln +1 i=1 6 kSN1 :::Nn (f) ; Sl1 :::ln (f)kLp 01]n :
544
. .
* 1.5 % %% ' Lp 0 1]n, 1 < p < +1. *+ % Y n n X Y "(ii) c (2) i (ti D i)
N
2 n i=1
i=1
$% Lp 0 1]n $ ' %$ ' "(ii) +i1=1 , "(ii) = 1, i = 1 : : : n. @ ' , Lp 0 1]n, 1 < p < +1, $%% % n X 0 Y "c i (ti D i) i=1 2Nn
(3)
"0 1 ::: n 2Nn "0 1 ::: n 0 1. 5# , ! "$ % (1) ! % +1 X
(1) ( 1 )=1
:::
+1 X
(n) ( n )=1
c(1) ( 1 ):::(n) ( n )
n Y
i=1
(i) ( i ) (ti D i)
$% Lp 0 1]n. #, > 0 # #$ Nk(ii) +ki1=1 , Mk(ii) +ki1=1 , i = 1 : : : n, (i) 1) Nki 6 Mk(ii) < Nk(ii)+1 , i = 1 2 : : : nD 2) % ! E = (1 : : : n) 2 Nn (1) MX Mk(nn) n Y k1 : : : X c (1) (1) ( 1 )::: ( n ) (1) (n) i=1 1 =Nk1
1 =Nkn
> D (1) ( i ) (D i) n Lp 01]
3) A(kii) = maxf(i) (i): Nk(ii) 6 i 6 Mk(ii) g, Bk(ii) = minf(i) (i): N (ii) 6 6 i 6 Mk(ii) g, Bk(ii) < A(kii) < Bk(ii)+1 , i = 1 : : : n. * "0 1 ::: n = 1, i 2 5"
+ 1
f(i) (mi ): ki=1
Nk(ii) 6 mi 6 Mk(ii) g i = 1 : : : n:
(1) AX A(knn) n Y k1 : : : X "0 c i ( i) (1) Lp 01]n = ( n ) i =1 = B = B 1 k1 n kn (1) (n) MX MX k1 kn n Y ::: C(1) (m1 ):::(n) (mn ) =
m1 =Nk(1)1
mn =Nk(nn)
i=1
(i) (mi ) ( i)
Lp 01]n
> > 0
545
. . % f(t1 : : : tn) 2 Lp 0 1]n $%!% % (3). 3 + . $ % (2) $ ' %$ ' % ! f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1. 5 ', . #, % ! . / f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1, ! % & # ' $ ' " % $% Lp 0 1]n, 1 < p < +1, $ $ < # 'n Qn" "$. o ( i ) % ! (i) ( i ) (ti D (i )) 2Nn i=1 % 1.5.
x
2. %%&
' ( )
+ ". % %'# # " . / 0 1]n # % # +../ & # . 6+../ & #{ . /! ' # $ *. R. U#% 8], K. A. M 9], . . K 10]. *+. + % ' # Luo Cheng' 4], 3. . 4 5], 4. &. 5 6. 5 $ 6]. 2.1. * p > 0 # fakg (1) (n) mX Z mX n 1 +1 n +1 Y ::: a{ (1) (n)
01]n i1 =m 1 +1 in =m n +1 =1 (1) ;1 m(n) ;1 p(1) ;1 Z mX +1 1 n 1X X
=
:::
p i (t D ) dt =
:::
) ;1 p(nnX +1
a(s)r
n Y
i=1 rn =0 s1=1 sn =1 01]n r1 =0 (1) ;1 m(n) ;1 p Z mX n 1 n X Y ( s ) ( s ) j ::: a r j rj (tj D j) dt = rn =0 j =1 01]n r1 =0 (1) (n) ;1 n; 1 ;1 mX n Y p2 ;1 mX ( j ) a(s)r ::: = mj r1 =0 rn =0 j =1 sE = (s1 : : : sn), sj = 1 2 : : : p(jj)+1 ; 1, j = 1 : : : n.
. ! n = 1.
p
i ri (ti D i) dtD
(si )
546
. .
p Z1 mX;1 p X Z1 mX+1 +1 ;1 ai i (t) dt = a(sr) 0 i=m +1
=
j+1 mX ;1 Zm mX;1 p X +1 ;1
j =0 j m
r=0
s=1
Zm1 p X +1 ;1 = a(s0)
s=1
r
p dt =
a(s) (s) (t) r r
p Zm2 p X +1 ;1 (s) (t) dt + a(s1) 0
s=1
0
r=0
0
Z1 p X +1 ;1 a(sm) ;1 + s=1 1; m1
p dt =
(s) (t)
1 m
s=1
p
(s) (t) 1
dt + : : : +
p Z1 mX;1 p X +1 ;1 a(sr) m ;1(t) dt = 0 r=0
p dt:
(s) (t)
( s)
r
s=1
F# # " ' ! % $ " . /!: (sr) ps=1+1 ;1 , (sr) ps=1+1 ;1, r 6= r0 . 0
Z1 mX;1 a(sr) r=0
0
=
p mX;1 Zkm+1 mX;1 (s) (t) dt = a(sr) r
k+1 mX ;1 Zm
a(sk)
k=0 k m ;1 p mX
= m 2 ;1
k=0
r
r=0
k=0 k m
k+1
mX ;1 (s) p Zm (s) p a k k (t) dt = k=0
p dt =
(s) (t)
k m
p
k (t)
(s)
dt =
a(sk) p s = 1 2 : : : p +1 ; 1:
* # # n = l ; 1. (1) m(ll)+1 Z mX 1 +1 X Yl ::: a{ (1) (l)
01]l i1 =m 1 +1 Z1 Z
il =m l +1 m(1)1 +1
s=1
p is (tsD s) dt =
m(ll 1)1 +1 m(ll)+1 X X X = ::: a{ il (tl D l) 0 01]l 1 i1 =m(1)1 +1 il =m(ll 1)1 +1 il =m(ll) +1 p lY ;1 ij (tj D j) dt1 : : :dtl;1 dtl = ;
j =1
;
;
; ;
547
(1) ;1 m(l 1) ;1 p(1)+1 ;1 p(l 1)+1 ;1 m(l) ;1 p(l)+1 ;1 mX l 1 1X lX 1 1 l lX X X
Z1 Z
::: r =0 r =0 s =1 sl l 1 1 0 01]l 1 1 p lY ;1 ij (tj D j) dt1 : : :dtl;1 dtl = =
;
:::
;
j =1
rl =0
;1 =1
;
;
;
;
a(s)r (slrl)l (tl D l)
sl =1
(l 1) ;1 1 +1 X p lX X X = ::: : : : a(s)r r1 =0 rl 1 =0 0 rl =0 sl=1 s1 =1 sl 1 =1 01]l 1 p lY ;1 (sl ) (t D l) dt dt : : :dt (t D j) ij j 1 l;1 = l rl l l
m(ll 1)1 ;1 Z1 m(ll) ;1 p(ll)+1 ;1 p(1)1 +1 ;1
(1) ;1 Z mX 1
X ;
;
;
;
;
;
;
j =1
(#' ' % n=1 ) =
(1) ;1 Z mX 1
r1 =0
01]l;1
m(ll 1)1 ;1 m(ll) ;1 Z1 p(ll)+1 ;1 p(1)1 +1 ;1
X
X
;
:::
;
rl 1 =0 rl =0 0 ;
X
X
sl=1
p Yl (slj ) j rj (tj D j) dtl dt1 : : :dtl;1 = j =1 (1) ;1 m(l) ;1 p(1)+1 ;1 p(l)+1 ;1 Z mX 1 l 1X lX X Yl = ::: ::: a(s)r 01]l r1 =0
rl =0
s1 =1
sl =1
s1 =1
j =1
p(ll 1)1 +1 ;1
X
;
:::
;
sl 1 =1
a(s)r
;
p
j rj (tj D j) dt1 : : :dtl :
(sj )
5 ' /. * # # n = l ; 1. (1) ;1 m(l) ;1 Z mX Zl;1 1 l X ( s ) ::: a r
01]l
=
r1 =0
Z1 0
rl =0
(1) ;1 Zl;1 mX 1
01]
r1 =0
j =1 m(ll 1)1 ;1 m(ll) ;1
X
X
rl 1 =0
rl =0
;
:::
p
(sl ) j rj (tj D j) l rl (tl D l) dt =
(sj )
;
;
a(s)r (slrl)l (tl D l)
p lY ;1 ( s ) j (t D j) dt : : :dt dtl = j 1 l ; 1 j rj j =1 (1) ;1 m(l 1) ;1 m(l) ;1 Z1 lY l 1 X ;1 p ;1 mX 1 l X ::: = a(s)r m 2j ;
;
0
j =1
r1 =0
rl 1 =0 ;
rl =0
p
l rl (tl D l) dtl =
( sl )
548
. .
=
lY ;1 j =1
p m 2j;1
(1) ;1 mX 1
r1 =0
m(ll 1)1 ;1 Z1 m(ll) ;1
X ;
:::
;
rl 1 =0 0 ;
X (s) a r
p
lrl (tl D l) dtl =
(sl)
rl =0
(#' ' % n = 1 ) =
Yl
p m 2j;1
(1) ;1 mX 1
(s) p a r
rl =0 " sE = (s1 : : : sl ), sj = 1 2 : : : p(jj)+1 ; 1, j = 1 : : : l. j =1
r1 =0
:::
(l) ;1 mX l
5 ', # '. 2.2. 1 < p < +1 i (tiD i)+i1=1 | # (! i )+1 # p i i=1 , i = 1 : : : n. ( # a 2Nn , i = 1 : : : n, + (1) (n) mX mX N1 +1 Nn +1 n Y ::: a ( ) i Lp 01]n (1) (n ) 1 =mN1 +1
i=1
n =mNn +1
n Y j =1
(1) N1 +1 12 ; p1 ) mX ( (j )
mNj
1 =m(1) N1 +1
( n) mX Nn +1
:::
n =m(Nnn) +1
p p1 : a
. *%% 2.1 M# '-
p,
(1) (n) mX Z mX N1 +1 Nn +1 n Y ::: a{ (1) (n)
p
01]n i1 =mN1 +1 in =mNn +1 =1 (n) ;1 m(1) Z p(1) NX N1 ;1 mX Nn 1 +1 ;1 X
(t D ) dt =
=
6
:::
m(1) N1 ;1
(n) ;1 p(1) ;1 mNX NX n 1 +1
r1 =0 n; Y
rn =0 01]n p(Njj)+1 ; 1 p;1
j =1
Z
X
01]n r1 =0
=
n; Y j =1
:::
p(Njj)+1 ; 1
rn =0
s1 =1
s1=1
(1) ;1 N1 p;1 mX
r1 =0
:::
:::
:::
) ;1 p(NnnX +1
sn =1
s) a( {r
n Y j =1
) ;1 p(NnnX +1
sn =1
n (s) Y a {r
(1) ;1 (n) mX Nn ;1 pNX 1 +1
rn =0
s1=1
j =1
:::
p
ij rj (tj D ij ) dt 6 (sj )
p
ij rj j D ij ) dt = (sj ) (t
) ;1 p(NnnX +1
sn =1
n p (s) p Y mN2 ;j 1 = a{r j =1
549
=
n; Y
(1)
(n)
j =1
i1 =mN1 +1
in =mNn +1
mX N1 +1 Nn +1 p;1 p2 ;1 mX ( j ) a{ p : pNj +1 ; 1 mNj :::
5 # p(i) +1 , i = 1 : : : n, " , i
i =1
(1) (n) mX mX N1 +1 Nn +1 n Y ::: a{ 6 ij (D ij ) Lp 01]n (1) (n)
i1 =mN1 +1
in =mNn +1
j =1
6 Cp
(1) N1 +1 12 ; p1 ) mX ( (j )
n Y
mNj
j =1
i1 =m(1) N1 +1
:::
(n) mX Nn +1
in =m(Nnn) +1
p p1 : a{
5# " . * # 1 < p < 2. 5", %% 1.8 4 ", $ (1) (n ) mX mX N1 +1 Nn +1 n Y ::: a{ (1) (n)
i1 =mN1 +1
> Cp
n; Y j =1
> Cp
Z
in =mNn +1 p(1) N1 +1 ;1
X
s1=1
01]n
ij rj j D ij )
p(1) NX 1 +1 ;1
n; Y j =1
s1 =1 (sj )
:::
ij rj (tj D ij )
Lp 01]n
j =1
:::
2 p2
(sj ) (t
i (D ij )
) ;1 m(1) ;1 p(NnnX N1 +1 X
sn =1
p1
dt
) ;1 p(NnnX +1
sn =1
(n) ;1 mX Nn
rn =0
(s) 2 a{r
> (1) ;1 Z mX N1
01]n
p2 p2 21 2
r1 =0
:::
>
dt
r1 =0
:::
(n) ;1 mX Nn
rn =0
(s) 2 a{r
:
A#' % (1) ;1 m(n) ;1 mX N1 Nn n X s) Y ::: a( {r
r1 =0
rn =0
j =1
2 ij rj (tj D ij ) = m(1) N1 ;1 X (sj )
=
r1 =0
:::
(n) ;1 mX Nn
rn =0
n; (s) 2 Y a{r
2.1, #B
j =1
(sj ) (t
2
ij rj j D ij )
550
. .
m(1) N1 +1
X
> Cp
> ij (D ij ) Lp 01]n
n Y
a{ ( n ) in =mNn +1 j =1 (n ) p(1) N1 +1 ;1 pNn +1 ;1
i1 =m(1) N1 +1
:::
(n) mX Nn +1
X
s1 =1
:::
(1) ;1 m(n) ;1 Z mX N1 N n X s) ::: a( {r
X
sn =1
r1 =0
01]n
rn =0
n; Y p p2 21 (sj ) = ij rj (tj D ij ) dt j =1 ) ;1 m(1) ;1 p(1)1 +1 ;1 p(NnnX N1 +1 n X Y ( 1 ; 1 ) NX
= Cp
j =1
m(j ) 2 p Nj
:::
s1 =1
sn =1
r1 =0
:::
n) ;1 m(NX n
rn =0
(s) p p2 21 a{r :
*%% M# ' = 2p , + 0 = 0 , 0 = 2;2 p , 1 < p < 2, p(1) N1 +1 ;1
X
s1 =1
:::
) ;1 m(1) ;1 p(NnnX N1 +1 X
sn =1
r1 =0
:::
(n) ;1 mX Nn
(s) p a{r 6
rn =0 (1) (n) ;1 m(1) ;1 m(n) ;1 2 p N1 Nn n; 1 +1 ;1 pNX n +1 X X Y 1; p2 pNX (j ) a({sr) p p 2 : ::: ::: 6 pNj +1 ; 1 s1 =1 sn =1 r1 =0 rn =0 j =1
* # p(ii) +i1=1 , i = 1 : : : n, " . *+ p(1) NX 1 +1 ;1 s1 =1
:::
) ;1 m(1) ;1 p(NnnX N1 +1 X
sn =1
r1 =0
6 Cp
:::
n) ;1 m(NX n
rn =0 (1) pNX 1 +1 ;1 s1 =1
(s) p p1 a{r 6
:::
) ;1 m(1) ;1 p(NnnX N1 +1 X
r1 =0
sn =1
:::
n) ;1 m(NX n
rn =0
(s) p p2 21 a{r :
5" m(1) N1 +1
X
i1 =m(1) N1 +1
:::
(n) mX Nn +1
in =m(Nnn) +1
a{
n Y j =1
> ij () Lp 01]n
Y (1;1) > Cp0 m(Nj ) 2 p n
j =1
j
(1) mX N1 +1
i1 =m(1) N1 +1
:::
(n) mX Nn +1
in =m(Nnn) +1
ja{jp
p1
:
551
* # # p > 2. 5" #' % ! 1.8. 5 2p > 1, (1) (n) mX mX N1 +1 Nn +1 n Y p ::: ja{j > ij () Lp 01]n (1) (n)
i1 =mN1 +1
j =1
in =mNn +1
(1) (n) mX mX 12 N1 +1 Nn +1 n Y 2 2 > cp ::: a{ > ij () Lp 01]n (1) (n)
i1 =mN1 +1 in =mNn +1 j =1 (1) (n) mX Z mX p1 N1 +1 Nn +1 n Y p p > cp ::: ja{j j ij (tj D ij )j dt (n) +1 j =1 +1 i = m 01]n i1 =m(1) n N1 Nn (1) (n) mX 1p N1 +1 Nn +1 n ( 1 ; 1 ) mX Y 2 p ::: ja{jp : = cp mNj j =1 i1 =m(1) in =m(Nnn) +1 N1 +1
=
5 # '. R 2.1 2.2 % K. A. M 9], $ ' 11] . 5' . I% ' #' #% ' A = fkE = (k1 : : : kn): m(ii) + 1 6 ki 6 m(ii)+1 i = 1 : : : ng " E 2 Z+n . nn o
2.3. 1 < p < +1, = min(2 p), = Q j (tiD i) | i=1 ! ,
# . ' ( # a = a 2Nn Dp (a) =
+ 1 X
1 =;1
:::
+ 1 X
n Y
X
(m(ii) ) ( 12 ; 1p ) jak jp n =;1 i=1 k2A
p
< +1
)( f(t1 : : : tn) 2 Lp 0 1]n, # a = a 2Nn ( ,)) ( $ ! , kf kLp 01]n 6 Cp fDp (a)g Cp > 0 ( a. n . T % Pn a Q i (tiD i). SN1 :::Nn (t) = i=1 N1 N P Pn Qn (t D i) | " %2N = : : : a . i i 1=1
n
i=1
552
. .
* # 2 6 p < +1. * 1.8, 4 ", ' 2.2, Sm(1) (n) ; Sm(1) :::m(n) Lp 01]n 6 :::m N N l l 1
6 Cp
6 Cp
n
(1) mX N1
n
1
:::
(n) mX Nn
ja j2
1 =m(1) n =m(lnn) +1 l1 +1 NX NX 1 ;1 n ;1 Z X
i=1
ja j2
2 ( i (D i))2
n Y
D i))2
( ki (ti k i=1 k2A 2 12 n Y E 6 Cp ::: ak ( ki (D i))Lp 01]n 1=l1 n =ln k2A i=1 NX p2 12 NX n 1 ;1 n ;1 Y 2 X 6 CEp ::: (m(ii) )1; p jakjp 1=l1 n =ln i=1 k2A 1=l1 NX 1 ;1
:::
1
n Y
n =ln 01]n NX n ;1 X
Lp 01]n
p2 2p 21 dt
2 6 p < +1:
P
n
1
6
6
1 < p < 2 # 1.8, jk j p2 k 2.2 Sm(1) (n) ; Sm(1) :::m(n) Lp 01]n 6 :::m N N l l 1
6
p2 P 6 jk j k
n
(1) (n) mX mX 12 N1 Nn n Y 2 2 6 6 Cp ::: ja j ( i (D i)) Lp 01]n (1) (n)
6 Cp
1 =ml1 +1 Z NX 1 ;1
:::
n =mln +1 NX n ;1 X
i=1
jak j2
n Y
( ki (ti D i))2
i=1 n =ln k2A 01]n 1 =l1 X p1 NX NX n 1 ;1 n ;1 Y p ::: (m(ii) ) 2 ;1 6 CEp jak jp i =1 = l = l 1 1 n n k2A
p2 p1 dt
6
1 < p < 2:
5 ',
Sm(1) (n) ; Sm(1) :::m(n) Lp 01]n 6 :::m N1 Nn l1 ln NX p 1 NX n 1 ;1 n ;1 Y 12 ; p1 ) X
( p ( i ) E 6 Cp ::: (m i ) jak j (4) 1=l1 n =ln i=1 k2A
" = min(2 p), 1 < p < +1. % , Sm(1) (n) ; Sm(1) :::m(n) Lp 01]n ! 0 :::m N Nn l l 1
1
n
553
1
Pn a2 2 . *+ . /% jNE j jElj ! +1. F# jaEj = i i=1 f(t1 : : : tn) 2 Lp 0 1]n, 1 < p < +1, % f ; Sm(1) ! 0 (5) ( n ) N1 :::mNn Lp 01]n " jN j ! +1. R1 R1 Qn ( D i) d : : :d | +../* # ck1 :::kn (f) = : : : f(1 : : : n) ki i 1 n i=1 0 0 n & # . /! f(t1 : : : tn ) 2 Lp 0 1] ' $ , kE 2 Nn. # M# ' p > 1 (p + p0 = pp0 ) jak1:::kn ; ck1 :::kn (f)j 6 n Y 6 : : : jSm(1) (n) (t : : : tn) ; f(t1 : : : tn) ki (ti D i)dt1 : : :dtn 6 N1 :::mNn 1 i =1 0 0 Y n 6 f ; Sm(1) (n) ki (D i) N1 :::mNn Lp 01]n L 01]n Z1 Z1
(i)
% $ mNi > ki , i = 1 : : : n. 5 1
Z
i=1
p0
j ki (ti D i)jp
0
dt < +1
0 (1) ' (mN1 : : : m(Nnn) ), (5) # (k1 : : : kn) , jak1 :::kn ; ck1 :::kn (f)j = 0:
.
5# ' % . 5 m(;i)1 = 0 8i = 1 : : : n S0:::0(fD t) 0, # B% (4)
p 1 X +1 + 1 Y n X (i) ) ( 21 ; p1 ) X ja jp E Sm(1) : 6 C : : : (m ( n ) p k i N1 :::mNn Lp 01]n = ; 1 = ; 1 i =1 1 n k 2A n Qn o n " 1.5 = i (ti D i) 2Nn % %% ' Lp 0 1] i =1 (. 2]). *+ 1 kf kLp 01]n 6 Cp (Dp (a))
# '#. nn o
2.4. 1 < p < +1, = max(2 p), = Q i (tiD i) 2Nn | i=1 ! # . ' f(t1 : : : tn) 2 Lp 0 1]n, (
554
. .
,)) ( $
X p 1 +1 + 1 Y n X 1;1) X ( ( i ) p 2 p ::: (m i ) jak j 6 Cp kf kLp 01]n : = ; 1 = ; 1 i =1 1 n k 2A . * # 1 < p 6 2. *%% 1.8
4 ",
(1) (n) mX 12 N1 +1 mX Nn +1 n Y 2 2 ::: ja j = (n) (f) Lp 01]n > Cp i (D i) N1 +1 :::mNn +1 Lp 01]n i=1 1 =1 n =1 Z X 12 2 p2 1p N1 Nn X n X Y 2 2 = Cp ::: ja j dt > i (ti D i) i =1 k = ; 1 k = ; 1 2 A 1 n n k 01] X p2 p2 21 N1 Nn Z X n X Y 2 2 > Cp ::: ja j > i (ti D i) dt i=1 k1 =;1 kn =;1 01]n 2Ak
Sm(1)
( ' % 1.8 2.2)
2 X 21 N1 Nn X Y n X a > Cp0 ::: ( D i) i Lp 01]n > k1 =;1 kn =;1 2Ak i=1 2p 21 X N1 Nn Y n X 1;1) X 2( p 00 ( i ) 2 p ja j > Cp ::: (m i ) 2Ak k1 =;1 kn =;1 i=1 " Ni > 1, i = 1 : : : n. *# f(t1 1 : : : tn) 2 Lp 0 1]n, 1 < p 6 2, , P
n $% jNE j = N 2 2 ! +1, i=1
i
p2 21 X +1 +1 Y n X 12 ; p1 ) X ( i ) 2( p ja j kf kLp 01]n > Cp ::: (mki ) : 2Ak k1 =;1 kn =;1 i=1
T # ! 2 < p < +1. Sm(1) (n) (f)Lp 01]n > N +1 :::mNn +1 1
> Cp
(1) Z mX N1 +1
01]n
1=1
:::
(n) mX Nn +1 n =1
a2
n Y i=1
2
1.8 2.2
p2 1p
i (ti D i) dt
>
p2 p1 Z X n N1 Nn X Y X 2 2 a > Cp ::: dt > i (ti D i) 01]n k1 =;1 kn =;1 2Ak i=1
( 2 < p < +1
555
P p2 P p2 bk > bk , bk > 0 8k) k
k
Z X p2 p1 N1 Nn X Y n X 2 2 dt = > Cp ::: a i (ti D i) 2 A i =1 k = ; 1 k = ; 1 1 n n k 01] (1) (n) mX X p2 1p k1 +1 N1 Nn Z mX kn +1 n X Y 2 (t D i) dt > = Cp ::: ::: a2 i i ( n ) i =1 k1 =;1 kn =;1 01]n 1 =m(1) n =mkn +1 k1 +1 X p1 N1 Nn X Y n X p 0 > Cp ::: > i (D i) a Lp 01]n k1 =;1 kn =;1 2Ak i=1 X p1 N1 Nn Y n X 1 ; p1 ) X ( i ) p ( 00 p 2 > Cp ::: (mki ) ja j 2Ak k1 =;1 kn =;1 i=1 Ni > 1, i = 11 2 : : : n, 2 < p < +1. H , $% n P jNE j = N 2 2 ! +1, i=1
i
kf kLp 01]n
> Cp
X X p1 +1 +1 Y n X ::: (m(ii) )p( 12 ; 1p ) ja jp 2Ak 1=;1 n =;1 i=1
2 < p < +1. 5 ' # .
x
3. +
I% % 2.3 2.4 # ' $ ' # .
A (Luo Cheng). f(x) 2 Lp 0 1], 1 < p < +1, fan(f)g | ,)) ( $ ! . Cp
+X 1
n=1
kf kLp 01]
p1 p p ; 2 janj n
6 kf kLp 01] 1 < p 6 2
X p1 1 6 Cp0 janjp np;2 2 6 p < +1: n=1
B (!. ". #). -
+1 # vm +m1=1 , # P (vm (m+1));1 6 C0 < +1.
m=1
556
. .
1 . * 1 < p 6 2 )( f(x) C1 (p)
X +1
m=1
jam (f)jp (vm m) 2 ;1 p
p1
2
Lp 0 1]
X +1
6 kf kLp 01] 6 C2(p)
m=1
jam (f)jp m 2 ;1 p
1p
:
2 . * )( f(x) 2 Lp 0 1], 2 6 p < +1,
X +1
C1(p)
m=1
X +1
1 p ;1 p p 2 jam (f)j m
6 kf kp 6 C2 (p)
+1
m=1
1 p ;1 p p 2 jam (f)j (vm m)
am (f) m=1 | ,)) ( $ )( f(x) ! . 4. &. 5 6. 5 $ ' " A % +../ & # . / f(x) 2 Lp 0 1], 1 < p < +1, ! " ! ' ! ## . 5# % % . 2.3 2.4 # .
2:3 . 1 < p < +1, = min(2 p), ak +k=11 | # , # 0
2X p +1 X 12 ; 1p ) k
(
p Dp (a) = 2 janj : k=0 n=2k+1 k+1
1 )( f(x) 2 Lp 0 1], # ak +k=1 ,)) ( $ ! kf kLp 01] 6 Cp (Dp (a) + ja1j): % Cp > 0 a.
2:4 . 1 < p < +1, = max(2 p). ' f(x) 2 Lp0 1], ,)) ( $ ! 0
ja1j +
X +1
k=0
2k ( 12 ; p1 )
k+1 2X
n=2k +1
p 1 p janj
6 Cp0 kf kLp 01]:
5# 2:30 2:40 ! A Luo Cheng'. $% 3.1. 2:30 2:40 , # A. . * # 1 < p < 2. 5" = max(2 p) = 2. +1 X
n=1
janjp np;2 =
k+1 X +1 2X
k=0 n=2k+1
janjp np;2
+ ja1jp :
557
*# 1 < p < 2, M# ' = p2 > 1 k+1 +1 2X X
+1
2k+1
X k(p;2) X janjp np;2 6 2 janjp = k=0 n=2k +1 k=0 n=2k+1 2k+1 +1 X 12 ; 1p ) X 1 1 k p ( jan jp2k( 2 ; p )p 6 = 2
k=0
6
X +1 k=0
= Cp
n=2k +1
2k2( 12 ; p1 )
+X 1 k=0
k+1 2X
janjp
n=2k +1 2k+1
2k2( 21 ; p1 )
X
n=2k +1
p2 p2 X +1 k=0
2p ( 1 ; 1 ) ;p 2 p
2k 2
2 2p ;
=
p2 p2 p : janj
5# ' 2:40 ' # Luo Cheng' 1 < p 6 2. * # # 2 < p < +1, = min(2 p) = 2. T % ' 2:30. + 1 X
k=0
2k2( 21 ; p1 )
k+1 2X
n=2k +1
p2 p janj
k+1 2X 2p 1 1 +1 X 2) k 2(1 ; p p jan j = 2 2k2( p ; 2 ) 6
k=0
n=2k +1
(% M# ' = 2p > 1)
6
X +1
k=0
= Cp
6 Cp
2kp(1; 2p )
X +1 k=0
2p X p 2 +1 2p ( 1 ; 1 ) p k p p 2 p 2 janj 2 k=0 n=2k +1 k+1 2X
2kp(1; p2 )
k+1 X +1 2X
k=0 n=2k +1
k+1 2X
n=2k+1
;
;
janjp
2 2) p p (1 ; p p janj n
=
6 = Cp
X +1 n=2
p2 p p ; 2 janj n :
5 ', ' 2:30 2 < p < +1 ' # Luo Cheng'. 1 , * # # 2 < p < +1. T ## ak +k=1 % B%
(
k( p1 ; 12 ;) n = 2k+1 k = 0 1 2 : :: 2 an = n = 0 1 2 : :: 0 $ #$ %$,
" 0 < < 12 ; p1 .
558
. .
T% ' % 2:30
k+1 2 + 1 1 1 2X +1 +1 p X X X k 2( ; ) 2 p 2 = 2k2( 12 ; 1p ) 2k2( 1p ; 12 ;) = 2;k2 < +1
k=0
k=0
k=0
n=2k +1
$%. #, " 2:30 f0 (t) 2 Lp 0 1], 2 < p < < + 1, % an =
Z1 0
f0 (t) n(t) dt 8n 2 N
" Luo Cheng' 2 < p < +1 % +! # %%: 2N X
n=0
janjp np;2 =
NX ;1 k=0
2kp( p1 ; 21 ;)2(k+1)(p;2) =
NX ;1 p NX ;1 kp( 1 ; 1 ;) p ; 2 k ( ; 1 ; p ) p ; 2 2 =2 2 =2 2 2 p ! +1 k=0 k=0 1 1 N ! +1, 2 ; p ; > 0. I! ' , 2:30 2 < p < +1
# %$, " Luo Cheng' . +P 1 *! 1 < p < 2. T % n; 12 n(t), t n=0 * # Sn (t) | " % . " 2:40 kS2N +1 kLp 01]
> Cp
X N
X N
k=0
1 2k2( 2 ; 1p )
2
0 1].
k+1 2X 2 1 p p 2 ; > n 2
n=2k +1
X N 2 C > Cp = C2p 1 = 2p (N + 1) 12 ! +1 k=0 k=0 N ! +1. #, ! % Lp 0 1], 1 < p < 2, $%, $% % 1 2 2
2k(1; p2 ) 2;(k+1)2k p + 1 X
k=0
" % Luo Cheng'
2k2( 21 ; p1 )
k+1 2X 2 p p ; n 2
n=2k +1
+1 p +1 p X X n; 2 np;2 = n 2 ;2
n=1
$%, 1 < p < 2.
1
n=1
559
5 ', 2:30 2:40 #, A Luo Cheng'. 5# 2:30 2:40 ! B 3. . 4 . $% 3.2. 2:30 2:40 , # B. . * # 1 < p < 2 f(t) 2 Lp0 1]. 5" " 2:30 kf kLp 01]
6 Cp ja1j +
X +1
k=0
1 2kp( 2 ; p1 )
n=2k+1
janjp
p1
6
p1 2k+1 1 ; p1 )p X ( k +1)( p 2 2 janj 6 k=0 n=2k +1 k+1 X X p1 p1 +1 2X +1 p p 6 Cp0 ja1j + janjp n 2 ;1 = Cp0 janjp n 2 ;1 : n=1 k=0 n=2k +1 6 Cp ja1j + 2 p1 ; 12
X +1
k+1 2X
5 ', 1 < p < 2 3. . 4 ' 2:30. 1 | % ## #$ * # # vn +n=0 +P 1 1 $%. 6 , 1 < p < 2 , % % m=0 vm (m+1) f(t) 2 Lp 0 1], an (f) | +../ & # , n 2 N. +1 X
+1 2k+1
X X p p jan(f)jp (vn(n + 1)) 2 ;1 = janjp (vn (n + 1)) 2 ;1 6 n=2 k=0 n=2k+1 +1 p 2k+1 X p ;1 X ; 1 k 2 2 6 v2k (2 + 1) janjp 6
k=0
6 Cp
+X 1 k=0
n=2k+1
2k(1; p2 )
2 2p p2 p2 X +1 ; 1 p v2k janj k=0 n=2k +1
X
2k+1
;
6
k+1 +X 2 2p p2 p2 X 1 1 1 2X +1 1 k 2( ; ) 0 p 2 p 6 Cp 2 janj 6 n=1 vn (n + 1) k=0 n=2k +1 ( 2:40) 6 Cp00kf kLp 01] : ;
5 ', 3. . 4 1 < p 6 2 % %% 2:30 2:40.
560
. .
5# ! 2 < p < +1, f(t) 2 Lp 0 1]. N +1 2X
N 2k+1 p ;1 X X p p janj n 2 = janjp n 2 ;1 6 n=2 k=0 n=2k +1 N 2k+1 N 2k+1 X p ;1 X k( p ;1) X 1;1) X k p ( p 2 p 2 2 62 2 janj 6 Cp 2 janjp 6 k=0 k=0 n=2k +1 n=2k +1
( 2:40) 6 Cp00kf kLp 01] 8N 2 N: H % 3. . 4 2 < p < +1. * # fvm g | ## ' % B , , 2 < p < +1, f(t) 2 Lp 0 1]0 1], fang | ## +../ & #{. k+1 2X p2 N X 1 ; 1p ) k 2( p 2 janj 2 6
k=0
n=2k +1
(% M# ' = p2 > 1)
6
X N
(2k v2k )( p2 ;1)
k=0
p p2 2p X N ; 1 p v2k janj k=0 n=2k+1 k+1 2X
;
k+1 p2 N 2X p;2 X p p 6 C0 janjp (vn n) 2 ;1 k=0 n=2k +1
6
N +1 p;2 2X p 6C ja jp(v
0
n=2
n
n n) 2 ;1 p
p2
8N 2 N:
5 ', B 2 < p < +1, $% % k+1 2X 2p +1 X 1 ; p1 ) k 2( p 2 2 : jan j
k=0
n=2k +1
5" ' 2:30 " , kf kLp 01] 6 Cp
X +1
n=1
1 p ;1 p p janj (vn n) 2 :
5 ', B % %% 2:30 2:40. 5# , . @ 3.2 # '. * v = n, n 2 N, % +1 X 1 1 2k( p ; 2 ;) 2k+1 (t) t 2 0 1] 2 < p < +1 0 < < 21 ; 1p : k=0
< #
561
(
k( 1p ; 21 ;) n = 2k+1 k = 0 1 2 : : : 2 an = 0 $ #$ %$, % # fang % k+1 2X 2p +1 X 1 2k2( 2 ; 1p ) janjp
k=0
n=2k+1
$%, # % +! # 2:30 . 3. . 4 2 < p < +1 % %%: 2N X
2N
X p 2 p2 ;1 p janjp (vnn) 2 ;1 = janj (n ) = n=1 n=1 NX ;1 kp( 1 ; 1 ;) NX ;1 1 1 = 2 p 2 2(k+1)(p;2) = Cp 2kp( 2 ; p ;) ! +1 k=0 k=0 1 1 N ! +1, 2 ; p ; > 0. > # '#. * # # 1 < p < 2. T %
+1 X n; 12 n (t) t 2 0 1]:
n=1
# 3. . 4 +1 X 1 < 1: fng : n " +1 n n=1 n * '# % 3.1 ', ! % Lp 0 1], 1 < p < 2, $% 2:40 $% %, ! ' " +../ , " % 3. . 4 + 1 X
$%.
n=1
n; 2 (n n) 2 ;1 = p
p
+ 1 p X n 2 ;2 n=1
$% 3.3. 2:30 #+ , #
2 < p < +1 # fang, # , . . Dp (a) = +1, Lp 0 1], 1 < p < +1, )(
, fang ,)) ( ${! . . r, 1 < r < p, Lr 0 1] )( f0 (t), fang ,)) ( ${! .
562
. .
. 1* # 2 < p < +1. T ## an +n=1 , % B% ( ;(k+1)( 1 ; 1 ) 2 p n = 2k+1 k = 0 1 2 : : : an = 2 n = 1 2 : : :: 0 $ #$ %$, I% 2:30 p. k+1 2X X N N X 2) 2 1 2 k (1 ; p p 2 janj = 2k(1; p ) 2;(k+1)2( 2 ; p ) = k=0
k=0 n=2k +1 N N X X = 2;(1; p1 ) 2k(1; p2 ) 2;k(1; p2 ) = 2 p1 ;1 1 = k=0 k=0 1 ;1 p = 2 (N + 1) ! +1 N ! +1:
1 2:30 5 ', % # an +n=1 2 < p < +1 %%. 5# r, 2 < r < p, 2:30 r. k+1 2X r2 X N N X 1 1 1;1) k 2( r 2 r 2 janj = 2k(1; 2r ) 2;(k+1)2( 2 ; p ) =
k=0
k=0 n=2k +1 N N X X = 2;(1; p2 ) 2k(1; r2 ) 2;k(1; p2 ) = 2 p2 ;1 2kp( p1 ; r1 ) < k=0 k=0
< 2 p2 ;1
+ 1 X
k=0
2k( 2p ; 2r ) < +1 8N 2 N:
*! % $% , p1 ; 1r < 0. #, # 1 # an +n=1 , " 2:30 % . / f0(x) 2 Lr 0 1], % R1 ! an = f0 (t) n (t) dt 8n 2 N. 0 5# , f0 (t)
+1 X 2;(k+1)( 21 ; p1 )
k=0
2k+1 (t)
% Lp 0 1], 2 < p < +1. * # S2N +1 (t) | % +" %. " 2:40
563
kS2N +1 (t)kpLp 01] > Cp
X N k=0
2kp( 21 ; p1 ) 2;(k+1)p( 12 ; 1p )
=
X N 0 = Cp 1 = Cp0 (N + 1) ! +1 N ! +1: k=0
#, f0 (t) 2= Lp 0 1], 2 < p < +1. &'. 4. &. 5 6. 5 $ ' " Luo Cheng' % % & # " ' ! #. 4 %, %$ 3.1 3.2, % 2.3 2.4. I '# % 3.3 '# B# 2.3 2.4, # '# 2k mk #' #% " # ' ! #. 5 ', 2.3 2.4 | '% | B, ' $ ' #.
,
1] . . //
!, . # . | 1947. | (. 11. | . 363. 2] ,- .. . / // . . 0-. | 1968. | (. IX, 2 2. | . 297{314. 3] ,- .. ., !-45 . . // 6. . | 1966. | (. 71, 2 1. | . 96{113. 4] Luo Cheng. On Haar series // J. Handrou Univ. Natur. Sci. Ed. | 1982. | Vol. 9, no. 3. | P. 269{284. 5] 6 . . 899: ;-/ < 9- :5 Lp // 6. . | 1985. | (. 126, 2 4. | . 490{514. 6] ( 6. ;., (- >. 5 // ?. ( 1980, 2 4929-80. 7] >4 .. ., . . @/ . | 6.: - , 1984. 8] B/ D. F. < // 6. . | 1964. | (. 63, 2 3. | . 356{391. 9] ,- . . -#4 0 9- : Lp < B4 // 6. . | 1972. | (. 87 (129), 2 2. | . 254{274. 10] .# . . 899: ;-/ < // 6. . | 1969. | (. 80, 2 1. | . 97{116. 11] ( . (. 899: ;-/ < 9- :5 Lp // 9- :5 9- :/@ . | >@, 1988. | . 109{118. ) * 1997 .
{ . .
517.987.4
: , , ! " , # $ !.
% &' ( )& # # & *+#{- ( ( ! /! ( ! ! ) ! '1 # $ ! &! 2 ! .
Abstract . . yukov, A proof of the Feynman formula for a solution of the Belavkin{Schrodinger stochastic di erential equation, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 565{581.
Representation of solution of Belavkin{Schr+odinger stochastic di6erential equation (with Brownian motion in in7nite-dimensional space) by Feynman path integral over trajectories in the phase space is constructed.
{ ( ) c !" #$
. $ ! !$ #& , (1], . + , , , , , , , ," -"&" , & , $ - " , ((2, 3], . (4]). 1 , - !" # & 2. 3! , . 4. +!&, 5. 6. 2 (5]. + , , (6]
!" #$ . , 2001, ! 7, 8 2, . 565{581. c 2001 ! " # $" " %&, '( )* +
566
. .
!
1! P = Q = Rm. 5 $ $ M(P ) -, $, P , , k k = R f(Q) | #& Q, = (1 + kpk2)j j(dp) < 1. 1! M P R
, (q) = eihqpi (dp), 2 M(P ), kk = k k. P 1! w1( ) | & $ M(P ), W1 ( ) = f(Q) = w~1 ( ) | & $ M !" P1 ( )(). 5 $ $ H2 $, pjf (p)j . W2 ( ) | & , #& P kf k = max p2P 1+kpk2 , H2, !" P2( )(). ,
B1 (q1 q2) =
Z
f(Q) M
'(q1 )'=(q2)P1 (1)(d')
Z
B2 (p1 p2) = '(p1)'(p2 )P2 (1)(d') H2
$," &, #& . 5 $ $ Fbi !" - , ! $ , fWi ( ): 2 (0 b]g, i = 1 2. !, Fb1 Fb2 $ , b 2 (0 t]. 5 $ D1(q) = B1 (q q), D2 (p) = B2 (p p), f(Q) # H(p) = kpk2. 5 , H^ , W^ 2 ( ) D^2 #& $ M (H^ ())(q) = (W^ 2 ( )())(q) = (D^2 ())(q) =
Z
PZ PZ P
eihqpi H(p) (dp)
(1)
eihqpi W2 ( p) (dp)
(2)
eihqpi D2 (p) (dp):
(3)
f (Q). A$ A , (1) (2) " " M
jB2(p1 p2)j =
Z
H2
p
p
'(p1 )'(p2 )P2(1)(d') 6 1 + kp1 k2 1 + kp2k2 E fkW2(1)k2 g
(3).
(4)
567
1! EQ0 | ,, (0 t] Q, " $ ,, . EQ | EQ0 ! , kxkEQ = sup kx( )k. 1! FQ | 06 6t -, $ (0 t] $ Q, " " &". FQ |
kyQ kFQ = jyj((0 t]). 1 EP FP " . Rt # hxP yQ i = hyQ (d ) xP ( )i $ ! EP FQ . 0 5 $ E = EP EQ , F = FQ FP . C hEP FQ i, hEQ FP i $" ! hE Fi h(xP xQ) (yQ yP )i = hxP yQ i + hxQ yP i: ! !$ " D ( . (1]). 5 $ Dn = f(t1 : : : tn): 0 6 tn 6 : : : 6 t1 6 tg,
n : Dn (Q P )n ! F (5) (t1 : : : tn (q1 p1) : : : (qn pn)) ! (q1 p1) t1 + : : : + (qn pn) tn : 1! BF | ! - F , - , ! $ F , n $ , n ! - , Dn (Q P )n . 5 $ $ FM(F E )
y ! y , y 2 F , y | - E , sup j y j(E ) < 1 " ,
y2F R $ #& E g(y) = eihxyi (x) y (dx) $ E ! BF . 1 $, #& F , , Z f (y) = eihxyi y (dx) (6) 0
E
y ! y FM(F E ), $ $ F(F ). 1! AF | ! - F ,
& ! hE Fi. 2 & !, -, BF & , " X 2 BF A 2 AF , A X (A n X ) = 0, $ $ M+ (F ). C #& (6), t0 2 (0 t] > 0 $
Rt
0
;i H(xP ( ))d ; 2
h(x()) = e (f )(y) =
Z Z
E E
0
Rt
0
e
R D (xP ( )) d t0 0
2
B2 (x0 ( )x00 ( )) d h(x0)h(x00 )e 0 P P
eihxyi
y (dx0) y (dx00):
(7)
568
. .
G , (f )(y) > 0. 2 kf kt0 = ( (f )) 12 (dy), F R 2 M+ (F ), $ " F (F ). 5 $ ex (y) = eihxyi = = eihx yi E x (dx0). H y ! E x FM(F E ), ex () 2 E 2 F(F ). 1! F(E ) | #& E , ,
(x) = ~(x) = (ex ) 2 (F(F ))0 . I #&
ex () F(F ) ( . (1]). 1- ! ~ , $ #$ . G (F(F ))0 !" " ". " F(E ) , , ! ~ , $ #$ . 5 f ! f~ $ F(F )
(F(E ))0 # f~(~) = (f ). T : F ! E , T (yQ yP ) = (yP ((0 ]) yQ((0 ])). R 1! a(y) = hT y yi, ga (y) = eia(y) = eihxyi E T y (dx). H "RE , #& #& eihxyi (x) E T y (dx) = (T y)eihT yyi E ,, ga 2 F(RF ). g~a (~) $, D . ! g~a (~) = ~(x)J (dxP ( ) dxQ( )). R
0
E
" # 1! 0 = tn+1 6 tn 6 : : : 6 t1 6 t0 6 t | $ T $ (0 t0], K(T ) | $ . 5 $ T ( ) = (tk ) 2 (tk tk;1), k = 1 : : : n + 1. 1. : (0 t] ! Mf(Q) | .
T
ST () =
1
X
Z
k=n+1 P
eihqpi (W2 (tk;1 p) ; W2 (tk p)) (tk )(dp):
K ! 0. ! t0 R dW^ 2 ( )(( q)). 0
. 5 $ kkMft(Q) = sup k( )kMf(Q). 1! T1 2 0t]
T2 | $ $ (0 t0], T = T1 T2 . H E (jST1 () ; ST2 ()j)2 = = E (jST1 (T1 ) ; ST2 (T2 )j)2 = E (jST (T1 ) ; ST (T2 )j)2 = =
1
X
k=n+1
Z
2
E eihqpi (W2 (tk;1 p) ; W2 (tk p))(T1 ; T2 )(tk )(dp) 6 P
569
6
1
X
Z Z
k=n+1 P P
jB2(p1 p2)j
j(T1 ; T2 )j(tk )(dp1)j(T1 ; T2 )j(tk )(dp2 ) (tk;1 ; tk ) 6 (4)
6 E fkW2(1)k2 g
Z
max k=1:::n+1
P
2
(1 + p2 )j(T1 ; T2 )j(tk )(dp)
t0 =
= E fkW2(1)k2 gkT1 ; T2 k2M ft(Q)t0: H ( ) , $ (0 t0], k ; T kM ft(Q) ! 0
K(T ) ! 0. 1- " " > 0 > 0, kT1 ; T2 kMft (Q) < " K(T1 ) < K(T2 ) < . G, E (jST1 () ; ST2 ()j)2 < E fkW2 (1)k2g"2 t0: 1- , ST () K(T ) ! 0. I 1 $. t0 1. 3 R ( q) dW1( q). 0
2. x( ) 2 EP . " sT (x) =
1
X
(W2 (tk;1 x(tk )) ; W2 (tk x(tk)))
k=n+1
t
R0
K(T ) ! 0. ! W2 (d x( )). 0
. 1! xT ( ) = x(tk) 2 (tk tk;1), k = 1 : : : n + 1. $! $! " > 0. N K = fx 2 P : kx ; x( )k 6 1 2 (0 t]g . H B2 ,, $ , K K, (") > 0, " , h1 h2 2 K K, kh1 ; h2k < , jB2(h1 ) ; B2 (h2)j < ". O a 2 (0 t0], , kx(a + 0) ; x(a ; 0)k > (8) . C !, " E !! xn 2 E , xn $ , xn ! x E . H kx(a + 0) ; x(a ; 0)k 6 kx(a + 0) ; xn(a + 0)k + kxn(a + 0) ; xn(a ; 0)k + + kxn(a ; 0) ; x(a ; 0)k 6 2kx ; xnk + kxn(a + 0) ; xn (a ; 0)k
570
. .
n > N ( ) xn " $ , a. 1! $ , xN () , a, , , (8), . 5 $ $ l( ). 1! T0 a, , , (8). H 0 < l(") , K(T0) < 0 kxT0 ; xk < . 1! T1 T2 | $ $ (0 t0] ! 0 . 1! T1 T2 T T a, , , (8). 5 $ xi = xTi , i = 1 2. H E jsT1 (x) ; sT2 (x)j2 = E jsT1 (xT1 ) ; sT2 (xT2 )j2 = = E jsT (x1) ; sT (x2 )j2 =
1
X
(B2(x1 (tk ) x1(tk )) + B2 (x2(tk ) x2(tk )) ;
k=n+1
; 2B2(x1 (tk ) x2(tk ))](tk;1 ; tk ) =
X0
X00
+ : P0 2 k, , tk , (8). H B2 K K, X0 6 M 0l( ) 6 M" k
M > 0. H !, k kxi(tk ) ; x(tk )k 6 6 kx(ts ) ; x(tk )k < , i = 1 2 ($ ! ts | tk $ Ti ), jB2(xi (tk ) xj (tk )) ; B2 (x(tk ) x(tk ))j 6 " i = 1 2 j = 1 2: G, E jsT1 (x) ; sT2 (x)j2 6 X0 X00 X 6 + jB2(xi (tk ) xj (tk )) ; B2 (x(tk ) x(tk ))j(tk;1 ; tk ) < M" + 4"t0 k
k
i=12 j =12
T1 T2, K(T1 ) < 0, K(T2 ) < 0 . 2 !, , fsT (x)g K(T ) ! 0. I 2 $. t0 2. 3 R W1(d xQ( )) 0 xQ ( ) 2 EQ . 5 $ $ (dp1 dp2) = E fw1(1)(dp1) w1(1)(dp2 )g P P . G , Z (1 + kp1k2)(1 + kp2k2 )jj(dp1 dp2) 6 P P
6E
Z
P P
(1 + kp1k2)(1 + kp2k2 )jw1(1)j(dp1)jw1(1)j(dp2) 6
6E
Z
P
(1 + kpk2)jw1(1)j(dp)
2
571
6 E kw1 (1)k2 < 1:
5 $ $ L $ , #& f ( p): (0 t] P ! C , Zt 2 2 kf kL = sup (1jf+(kppk)2j )2 d < 1: p2P 0
C #& f ( p) = f (tk p) 2 (tk tk;1), k = = 1 : : : n + 1, $ t
Z Z
0P
f ( p)w1 (d dp) =
nX +1 Z
f (tk p)(w1(tk;1)(dp) ; w1(tk )(dp)):
k=1 P
5 , #&
f ( ) !
t
Z Z
0 P
f ( p)w1 (d dp)
(9)
, , ,
E
t
2
Z Z
f ( p)w1 (d dp) =
0P
=
Z Z Z
0 t
P P
6 sup p2P
t
Z
0 t
t
Z Z Z
0 P P
f ( p1 )f=( p2 )(dp1 dp2) d =
f ( p1 )f=( p2 ) d f(1 + kp k2 )(1 + kp k2)(dp dp )g 6 2 1 2 1 (1 + kp1 k2)(1 + kp2k2 )
jf ( p)j2 d Z Z (1 + kp k2 )(1 + kp k2)(dp dp ) 6 1 2 1 2 (1 + kpk2)2 P P
2 6 sup (1jf+(kppk)j2)2 d E kw1 (1)k2: p2P
Z
0
1 (9) , L. 5 R B! B w1(d dp) | - 0t]P (0 t] P . 1! v 0 2 M(P ). 5 $ p 0(d dp) = ;iv (dp) d + 1w1 (d dp) ; 1 fE (w1(1) w1 (1))(dp)gd: 3. ! N (p) = ;iH(p) ; 2D2(p). 1 2 C , 2 2 R1, 2 > 0. "
572
. . 1 X
Z
n=0 P
Z
A Zt0Z
:::
0P
Pn N
eN (p)tn ek=1 p
e
(10)
p
0 (t0 A) = eN (p)t0 e n (t0 A) =
(1 + kpk2)eihqpi n (t0 dp)
;
tZn
2 W2 (t0p) 0 (dp)
;1Z
Z
A (p + p1 + : : : + pn )
0 P P n p p+ P pl (tk 1 ;tk ) 2
e
;
l=k
Pn
W2 tk 1 p+ P pl ;W2 tk p+ P pl
;
k=1
;
n
;
;
l=k
n
l=k
2 W2 (tn p) 0(dp) 0(dt1 dp1) : : : 0(dtn dpn)
(11)
% t0 2 (0 t]. . 5& n- (10): E
2
Z
(1 + kpk2)eihqpi n (t0 dp) 6
P
6E
t
Z 0Z
2n
:::
tZn
;1Z
e
p
2
(1 + kp1k2) : : : (1 + kpnk2)(1 + kpk2)
0 P 0 PP ; n P p+ pl (tk 1 ;tk ) Re N (p)tn l=k e
Pn Re N
ek=1
Z
Pn
;
W2 tk 1 p+ P pl ;W2 tk p+ P pl
;
;
k=1
j 0j(dt
n
;
l=k
;
n
l=k
n dpn)j0j(dp)
1 dp1
2
) : : : j 0j(dt
p W (t p) 2 2 n
e
6
+{
6E
t
Z 0Z
0 P
:::
tZn
(1 + kpk2)j 0j(d dp) n 0P n!
;1Z
Z
(1 + kp1k2) : : : (1 + kpn k2)(1 + kpk2)
0 P P ; Pn pl(tk 1;tk) 2 2 p+ l=k e;22 D2 (p)tn
Pn ;2 D
ek=1
t
; R0 R
4nt0 k0k
;
573
2p2
Pn
W2 tk 1 p+ P pl ;W2 tk p+ P pl
;
;
n
;
n
e e2 j 0j(dt1 dp1) : : : j 0j(dtn dpn)j0j(dp) 6 k=1
;
l=k
l=k
p
2 W2 (tn p)
& $ ,, Fb1 Fb2 b 2 (0 t] $ ,
6E
t
Z 0Z
0 P
t
(1 + kpk2)j 0j(d dp) n 0P n!
;R0 R
4nt0 k0k tZn
;1Z
:::
Z
(1 + kp1k2) : : : (1 + kpn k2)(1 + kpk2)
0 P P ; Pn pl(tk 1;tk) 2 2 p+ l=k e;22 D2 (p)tn
Pn ;2 D
ek=1
n
Y
(2p2 ) W2 tk 1 p+ P pl ;W2 tkp+ P pl ;
k=1
6E
t
Z 0Z
0 P
;
Ee
n
;
;
n
l=k
l=k
p
E fe2
;
2 W2 (tnp) gj 0j(dt1 dp1) : : : j 0j(dtn dpn)j0j(dp)
6
t
(1 + kpk2)j 0j(d dp) n 0P n!
;R0 R
4nt0 k0k
:::
tZn
;1Z
Z
0 P P
(1 + kp1k2) : : : (1 + kpn k2)(1 + kpk2)
j 0j(dt
1 dp1) : : : j 0j(dtn dpn)j0j(dp) 6 t
(1 + kpk2)j 0j(d dp) n 2 0P 6E : n! A$ , 0(d dp) ,
t
Z 0Z
0 P
;R0 R
2nt0 k0k
(1 + kpk2)j 0j(d dp) 6 p
6 t0 kv k + j1j(kw1 (t0)k) + 2t0 j1jE (kw1(1)k2) = p w1 (t0 ) = t0 kv k + j1jt0 pt + 2t0 j1jE (kw1 (1)k2): 0
574
. .
H $ , 1 X
n=1
6 6
E
Z
P
2 1 2 2 i h qp i (1 + kpk )e n (t0 dp)
n 1 2 t0 k0k X n=1 1 X
;
E
2nt0 k0k n=1 n!
t
1 (1 + kpk2)j 0j(d dp) 2n 2 0P 6 n!
R0 R
p
E t0 kv k + j1jt0 1 X
6
1 ! wp1(t0 ) + 2t j jE (kw (1)k2) 2n 2 6 0 1 1 t0
2n32nt0 k0k n! n=1 p f(t0 kv k)2n + (2t0 j1jE kw1(1)k2)2n + ( j1jt0 )2nE (kw1(1)k2n)g 21 6 1 n X 6 18 tn0!k0k n=1 p p f(t0 kv k)n + (2t0j1 jE kw1(1)k2 )n + ( j1jt0)n E kw1(1)k2ng 6 6 t0 k0kfexp (18t0kv k) + exp (36t0j1jE (kw1(1)k2 )) + + exp(324j1jt0 ) + E (exp(kw1(1)k2 ))g < 1: 1- (10) . I 3 $.
6
1! Fb | ! - , ! $ , fWi ( ): 2 (0 b] i = 1 2g. f(Q) | , , ( ) $ 1! : (0 t] ! M ! F 2 (0 t]. H ( q) $, $ + { d(t0 ) = ;iH^ (t0 ) dt0 ; iV (q)(t0 ) dt0 ; 21 D1 (t0 ) dt0 ; 22 D^2 (t0 ) dt0 + p p + 1 ((t0 )) dW1 (t0 ) + 2 dW^ 2(t0 )((t0 )) (12) !, (0 q) = u(q), !
575
t
Z0
t
t0
0
0
Z0
Z (t0 )(q) = u(q) ; i H^ ( ) d ; iV (q) ( ) d ; 21 D1 ( ) d ;
; 22
0
t
Z0
0
t
Z0
p p D^2 ( ) d + 1 ( ) dW1 ( ) + 2
0
t
Z0
0
dW^ 2 ( )(( )):
1. u(q)= eihqpi 0(dp), V (q)= eihqpi v(dp), v 0 2M(P ). R
R
P
P
w1(t) |
& M(P ), W1( ) = w~1( ), W2 ( ) |
& H2. 1 2 C , 2 2 R1, 2 > 0. ! H^ , W^ 2 ( ) D^ 2 ' (1), (2) (3). " ( ) ( (12) c u(q) (t0 q) =
1 X
Z
n=0 P
eihqpi n (t0 dp):
(13)
2. 1. " ( ) ( * + {- (12) u(q) (t0 q) =
Rt
Z h
E
u(xQ (0) + q)e p
0
;i V (xQ ( )+q) d
e
Rt
0
Rt
0
Rt
0
;i H(xP ( )) d ;1 D1 (xQ ( )+q) d ;2 D2 (xP ( )) d
e
0
1
e
0
Rt W (dxQ ( )+q) 0
0
1
p
e
0
2 R W2 (dxP ( )) i t0 0
e
0
J2 (dxP ( ) dxQ( )): (14)
1. 1 n(t0 dp) = 0 n = ;1 ;2.
1 # e A
d
Z
P
=
eihqpi n (t0 dp)
;iH(p)dt0 ; 2 D2 (p)dt0 + 2 dW2(t0 p)+ 22 D2 (p)dt0 eihqpi n (t0 dp) +
Z
P
=
p
p
Z
+ (;iV (q) dt0 + 1 dW1(t0 ) ; 1 D1(q) dt0 ) eihqpi n;1(t0 dp) + Z + 21 D1 (q) dt0 eihqpi n;2(t0 dp):
P
P
576
. . 1 P
R
1 3 d eihqpi n (t0 dp) , (13) n=0 P h t0 2 (0 t]. 1- i 1 R 1 R R P P d eihqpi n (t0 dp) = d eihqpi n (t0 dp) . + , eihqpi n=0 P n=0 P P n (0 dp) = 0, n = 1 2 : : :, Z
Z
eihqpi 0 (0 dp) = eihqpi 0(dp) = u(q):
P
P
1 P
G, (t0 q) = eihqpi n (t0 dp) $ + n=0 P (12). H 1 $. 2. 1 1 (12) (13), n # (11). 5 $ (d dq dp) = 0 (dq) 0(d dp), | (0 t] Q P . H t
Z 0Z Z
n (t0 A) =
R
0 QP ;
k=1
l=k
p e 2 W2 (tn p) e
iP
e k=1
P qlpk
Z Z
A (p + p1 + : : : + pn)e;iH(p)tn
0 QP P (tk 1 ;tk ) ;2 D2 (p)tn ;2
Pn D
p
n n
;1Z
:::
n n ;i P H p+ P pl
e
tZn
2
;
Pn
k=1
;
;
e
W2 tk
;1
n p+ P pl l=k
e
;W2
k=1
p+ P pl (tk 1 ;tk)
;
2
n
;
l=k
n tk p+ P pl
;
l=k
n iP
e k=1
n qk p+ P pl l=k
(dt1 dq1 dp1) : : : (dtn dqn dpn)0(dp): 4. + - BF , ' & f (y) Z
F
l=k
f (y) (dy) = = f (0) +
1 X
t
Z 0Z Z
:::
tZn
;1Z
Z
f ((q1 p1) t1 + : : : + (qn pn) tn )
n=1 0 Q P 0 QP (dt1 dq1 dq1) : : : (dtn dqn dpn) Z 2 E f (y) (dy) < 1
+
F
Rt
0
;i V (xQ ( )) d
~ (x()) = e
0
e
p
1 R W1 (dxQ ( )) ;1 R D1 (xQ ( )) d t0 0
e
t0 0
:
(15)
577
4. 5 $ t
Z 0Z Z
tZn
;1
Z Z
A (t1 : : : tn (q1 p1) : : : (qn pn)) 0 QP 0 QP (dt1 dq1 dq1) : : : (dtn dqn dpn): 1! n $ # (5), n | $ , n n . H n (A) =
E
Z
F
:::
2
(
t
; R0 R R
f (y) n (dy) 6 E (max jf (y)j2 ) 0 Q P y2F
j j(d dq dp) n !2 )
n!
= (max jf (y)j2)E y2F 1 P
t
; R0 R
1 P
0P
=
j 0j(d dp) n !2
n! R
:
2
1
5 $ $ = 0 + n . 1! E f (y) 2 n=1 n =1 F 2 R , E f (y) < 1. , $ F D ! : Z
~ (x()) = eihxyi (dy) = = 1+
F Zt0Z 1 X
Z
:::
tZn
;1Z
Z
eihx(t1 )(q1 p1 )i : : :eihx(tn )(qn pn )i
n=1 0 Q P 0 QP (dt1 dq1 dp1) : : : (dtn dqn dpn) = n Zt0Z Z 1 X 1 i h x ( t ) ( q p ) i 1 1 1 = n! e (dt1 dq1 dp1) = n=0 0QP Zt0 Z n 1 X 1 = n! eihxQ (t1 )p1 i 0(dt1 dp1) = n=0 0 P Zt0 n Zt0 Zt0 1 X 1 p = n! ;i~v (xQ ( )) d + 1 w~1(d xQ ( )) ; 1 D1 (xQ ( )) d = n=0 0 0 0 t0 t0 p Rt0 R R ;i V (xQ ( )) d 1 W1 (dxQ ( )) ;1 D1 (xQ ( )) d
=e 0 e I 4 $.
0
e
0
:
578
. .
$! Z
f (y ) = e
Rt
0
Rt
p
0
;i H(yP ( 0 ])+p) d ;2 D2 (yP ( 0 ])+p) d
e
0
P
e
0
2 R W2 (dyP ( 0 ])+p) t0 0
i R hyQ (d )yP ( 0 ])+pi i R hyQ ( 0 ])yP (d )i ihqyP ( 0t0 ])+pi e 0 e0 e 0(dp): t0
t0
H n 2 N, 2 C , y0 2 F (t1 : : : tn (q1 p1) : : : (qn pn)) ! f ( n (t1 : : : tn (q1 p1) : : : (qn pn)) + y0 ) $ ! - , Dn (Q P )n , #& f (y) $ ! BF . 1- (12) $ , " : ZZ
R
t0
Rt
0
;i H(yP ( 0 ])+p) d ;2 D2 (yP ( 0 ])+p)) d
(t0 q)=
e
PF
e
0
0
p
e
2 R W2 (dyP ( 0 ])+p) t0 0
i R hy (d )y ( 0 ])+pi i R hyQ ( 0 ])yP (d )i ihqyP ( 0t0 ])+pi e 0 Q P e0 e (dy)0 (dp): t0
t0
: P ! F , (p) = (0 p 0) 0 -" (0 ;1 ) BF & . 1- Z
R
t0
R
t0
;i H(yP ( 0 ])) d ;2 D2 (yP ( 0 ])) d
(t0 q) = e
e
0
F
0
e
2 R W2 (dyP ( 0 ])) t0
p
0
i R hyQ (d )yP ( 0 ])i i R hyQ ( 0 ])yP (d )i i h qy ( 0 t ]) i P 0 e e0 e0 (0 ;1 )(dy) =
=
Z
F
t0
Rt
t0
0
Rt
0
;i H(T2 y)d ;2 D2 (T2 y) d
e
e
0
0
p
e
2 R W2 (dT2 y) t0 0
eihqyP ( 0t0 ])i eihT1 yyP i eihyQ T2 yi (0 ;1 )(dy) T1(yQ yP )( ) = yQ ((0 ]) T2(yQ yP )( ) = yP ((0 ]). (t0 q) =
Z Z
F
E
tR0 Rt0 Rt0 ;i H(xP ( )) d ;2 D2 (xP ( )) d p2 W2 (dxP ( ))
e
e
0
e
0
eihqyP ( 0t])i (0 ;1 )(dy):
C #& f (y), (6),
(f )(y) = e E
eihxQ yP i eihxP yQ i E (T2 yT1 y) (dxP dxQ) Z
0
R
t0
R
t0
;i H(xP ( )) d ;2 D2 (xP ( )) d 0
e
0
e
p
2 R W2 (dxP ( )) ihxyi 0 e y (dx): t0
579
H (12) Z
(t0 q) = (ga )eihqyP ( 0t0])i (0 ;1 )(dy) E
Z
ga = eihxQ yP i+ihxP yQ i T2 yT1y (dx0P dx0Q): 0
E
H
Ee
p
0
t0 t0 2 R W2 (dxP ( )) p2 R W2 (dxP ( )) 0
00
e
0
0
=e
2 2
E j(f )(y)j2 = 1
Z Z
E E
R D (xP ( )) d t0 0
2
0
Rt
0
=
e
R
t 2 0 D 2
0
2
(xP ( )) d 2 R B2 (xP ( )xP ( )) d 00
e
t0 0
0
00
2 B2 (x0 ( )x00 ( )) d h(x0)h(x00)e 0 P P
y (dx0) y (dx00) = 2 (f )(y):
Z
(f ) = (f )eihqyP ( 0t0 ])i (0 ;1 )(dy): F
H
(16)
Z
E j (f )j 6 E j((f ))jE fj (0 ;1 )jg(dy) 6
6
Z
F Z
F
1
E j((f ))j2 2 E fj (0 ;1 )jg(dy) =
= (2 (f )(y)) 12 E fj (0 ;1 )jg(dy): F
2 f ;1 (A): A 2 AF g f n;1 (A): A 2 AF g " - P Dn (Q P )n . G, 0 ;1 n 2 M+ (F ), n = 1 2 : : :. 1- E j (0 ;1)j 2 M+ (F ). G, E j (f )j , . 1- j (f )j , ", () 2 F (F ). H " Z
(ga ) = g~a (~) = ~(x)J2 (dxP ( ) dxQ ( )): E
580
. .
5. (f ) ' (16), tR Rt ;i H(xP ( )) d ; D (xP ( )) d 0
~(x) = u(xQ (0) + q)e
e
e
0
2
0
0
2
tR0 R R R ;1 D1 (xQ ( )+q) d p1 W1 (dxQ ( )+q) p2 W2 (dxP ( )) ;i V (xQ ( )+q) d t0
t0
e
0
e
0
5. Rt Rt ;i H(xP ( )) d ; D (xP ( )) d
~(x) = (ex ) = e
0
0
t0
e
0
0
: (17)
2 R W2 (dxP ( ))
p
t0
eihxQ yP i eihxP yQ i eihqyP ( 0t0 ])i (0 ;1 )(dy):
0
Z
e
2
0
2
e
0
F
, , !$ # (14). Z
F
eihxQ yP i eihxP yQ i eihqyP ( 0t0])i (0 ;1 )(dy) = = =
Z Z
PZ F P
eihxQ yP +pi eihxP yQ i eihqyP ( 0t0])+pi (dy)0(dp) =
eihxQ (0)+qpi 0(dp)
Z
F
eihxP yQ i eihxQ yP i eihqyP ( 0t0])i (dy) =
Rt ;i V (xQ ( )+q) d 0
p
1 R W1 (dxQ ( )+q) ;1 R D1 (xQ ( )+q) d t0
t0
= u(xQ (0) + q)e e e 0 : G, ~(x) # (17). I 5 $. H $ , # (14) $ (12). H 2 $. 3 5. 6. 2 $ $ . 0
0
$ 1] . . , . . . . | .: , 1990. 2] V. P. Belavkin. Nondemolition measurements, nonlinear $ltering and dynamic programming of quantum stochastic processes // Proc. Bellman Continuous Workshop, Sophia{Antipolis 1988. | LNCIS, vol. 121. | P. 245{265. 3] V. P. Belavkin. A new wave equation for a continuous nondemolition measurement // Phys. Lett. A. | 1989. | Vol. 140. | P. 355{358.
581
4] L. Diosi. Continious quantum measurement and Ito formalism // Phys. Lett. A. | 1988. | Vol. 129. | P. 419{423. 5] S. Albeverio, V. N. Kolokoltsov, O. G. Smolyanov. Representation des solutions de l'equation de Belavkin pour la quantique par une version rigoureuse de la formule d'integration fonctionnelle de Menski // C. R. Acad. Sci. Paris. | 1996. | Vol. 323, Serie 1. | P. 661{664. 6] . . / , . . . 012 3 1 41312 {0 2 // 5 6. | 1997. | . 52, 8. 4. 7] 9. :. 0 2 , . . . ; <= 8 2 , 1 1 >? 1 41312 > // @ 2 E.6. | 1998. , $- 1998 .
. .
. . .
517.984
: !"#$, & , '.
( ) * + + # (" * * "- + *) + # # $ # # + ). /* ) ) "* #. 0"*1 ) # 2 + "*#
"1- + . # , * !"#$ 2 * "1 ' ( , # * # # + ").
Abstract I. A. Sheipak, Spectral problems associated with stability of uid motion in an annulus in a magnetic eld, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 583{596.
This paper investigates spectral and basis property of operator pencil connected with the problem of stability of an axisymmetrically perturbed 8uid motion in vertical annulus in the presence of vertical magnetic 9eld. It is proved that eigenfunctions of this pencil form a Bari basis in the corresponding Hilbert space.
1.
. ! "# & & # ! !" (+ : 96{01{01292). , 2001, 7, : 2, . 583{596. c 2001 !", #$ %& '
584
. .
. , , ( ), . * " # ! , " ! . + " # ! ,. - .1]. 1 +{3 # ! C02.a b] | " ! ! , .a b], a b | ! . + 5. 6. 7 .2], 9. 1. : * ;. < .3] ,. - , " ! W21 .a b]. > , +{3 >
" ! . : # ! W 12 (.apb]? px) L2 (.apb]? px). 1 W21 (.a b]? x) L2 (.a b]? x) # (u v)W21 ( ab]
p
x) =
Zb a
Zb
Zb
xu(x)v(x) dx + xu0 (x)v0 (x) dx a
(u v)L2 ( ab]px) = xu(x)v(x) dx: a
* W 12(.a b]? px) ! W21(.a b]? px), a b. : 6. 6. - >. B
.4], Ny(x) = Py(x) Uj (y) = 0 j = 1 2 : : : n N P | ! n p (n > p > 0), Uj | 6 n ; 1. 6 ! . * +{3 , 6. 6. - >. B
# ! W2sU s = 0 1 2 3 4, W2sU s = 2 3 4 9 W22U , W2sU | ! y(x) 2 W2s , 6 s ; 1.
585
.5] ,
>
, 1 9 W 2 (.a b]? px)L2 (.a b]? px) , " . D , .
2.
9 " ! R1 R2 (R1 < R2 ). B ! T0 . B0 = (0 0 B0) ! , B0 | , (r ' z ) | ! . : | h = (R2 ; R1)=2, | t = h2= , v | kvk = g T0h2= , | p = g T0 h, B | kBk = B0 , | , | , | ! g | . ! F (., , .6]) @ v + Gr(v r)v = ;rp + Iv + Ha 2 (B r)B + T k (1) m @t Gr Prm Prm @@tB = Gr Prm .(B r)v ; (v r)B] + IB (2) @T + Gr v (rT ) = 1 IT (3) @t Pr div v = 0 div B = 0 (4) v, B, T pm | " , , , " p pm = p + B2 =2. k = (0 0 1) | z . Gr = g T0 h3 = | ; , Ha = B0 h( = )1=2 | < , Pr = =k | * , Prm = 1= | * , | , k | . 1 : R1 r = 2R r = 2 r 6 r 6 r : R= R 2 1;R 1 2 1 1;R 2 K " B = (0 0 B0) v = (0 0 v(r)) T = T0(r) pm = p0(z): (5)
586
. .
* (5) (1){(4) 2 2 ; dp0 + d v0 + 1 dv0 + T = 0 d T0 + 1 dT0 = 0 dz
dr2
0
r dr
dr2
r dr
v0 (r1) = 0 v0 (r2 ) = 0 T0 (r1) = ;1 T0 (r2) = 1: :
Zr
2
r1
(7)
rv0 (r) dr = 0:
K , 2 2 r ; ln r1) ; (r22 ; r12 )(ln r ; ln r1) + v0 (r) = (r ; r22)(ln ln R 2 ln2 R 2 2 2 2 r ; ln r1) + A r ; r1 ; r2 ; r12 + A (r2 ; r14)(ln ln R 4 2 ln R 2 r T0 (r) = ; ln R r ; 1 1 4 2 4 4 2 2 2 22 A = 4r2 ln2 lnRR+(r(72 r;2 ;r2)3r.(1r;2 +4rr12r)2ln) lnRR++r24(;r2r2;] r1 ) : 2
(6)
1
1
2
2
1
3 p0 (z ) " (6).
. 5 , # " v0 + v, T0 + T , B0 + B p0 + p, v, T , B p , v = (vr (r z t) 0 vz(r z t)) B = (br (r z t) 0 bz(r z t)) T = T (r z t) p = p(r z t): ! (r z t), ! ! , 1 @ 1 @ 1 @ vr = ; r1 @ @z vz = r @r br = ; r @z bz = r @r : * " 8 (r z t) = '(r) exp(;t + ikz) >>< T (r z t) = (r) exp(;t + ikz ) (8) >: (r z t) = (r) exp(;t + ikz) p(r z t) = (r) exp(;t + ikz )
587
', , | , k | . * (8) (v, T0 , B0 , p0 ) (1){(4), (r) (r) Prm 1, ! : 3 3 2k2 k2Ha 2 2 iv 000 2 ' ; ' + ; 2k '00 ; ; '0 + k4 + '+ r r2 r3 r r 0 0 0 v ' ' 0 00 00 2 0 00 2 + ikGr ' v0 ; r ; v0 ' ; r ; k ' + r = ; ' ; r ; k '
(9) (10)
0 00 + 1 0 ; k2 = ikGr Pr v ; T0 ' ; Pr 0
r
r
'(rj ) = 0 '0(rj ) = 0 (rj ) = 0 j = 1 2: (11) K " , r = r + ii " . B , ! . 6 , , "# # ! N # (. .6]).
3.
9 ! : 3 3 3 2k2 2 iv 000 2 00 2 A' = ' ; ' + ; 2k ' ; ; 2k ; ; '0 + k4 '
1 r 0
r2
r2
r3
A; ' = r r '0 ; k2' A+ ' = 1r (r'0)0 ; k2'
r
(12)
D(A) = D(A; ) = y(x) 2 W24.r1 r2]? y(r1 ) = y(r2 ) = y0 (r1) = y0 (r2 ) = 0
D(A+ ) = y(x) 2 W22.r1 r2]? y(r1 ) = y(r2 ) = 0 : . A , A; L2 (.r1 r2]?pp1r ). A+ L (.r r ]? r). 2
1 2
588
. .
. 1 L2(.r1 r2]? p1r ) Zr 1 (f g) =
2
r1
r f (r)g(r) dr:
, ! ,
, A A; L2 (.r1 r2]? p1r ): Zr2 1 1 0 Zr2 1 0 2 (A y y) = r y (r) ; k y(r) y(r) dr = ; (jy0 (r)j2 + k2 jy(r)j2) dr ;
r1 Zr2
r
r
r1
r
0 0 0 0 (Ay y) = 1r r 1r r 1r y0 ; k2y ; k2 r r1 y0 ; k2y y dr = r Zr 1 02 2k2 4 = r r y0 + r jy0 j2 + kr jyj2 dr: 1
2
r1
: A+ (A+ y y)L2 ( r1 r2 ]pr) = Zr2 1 Zr2 0 0 2 = r (ry (r)) ; k y(r) y(r) dr = ; r(jy0 (r)j2 + k2jy(r)j2 ) dr: r1
r
r1
, # . 1"# A A+ . , A; (2 2) (. .7]). B , (9){(11) " A 0 ' B B ' A 0 ' 11 12 ; (13) 0 A+ + ikGr B21 B22 = ; 0 Pr B11 , B12, B21, B22 (13) : 0 2 2 v 00 0 B11 y = ;v0 A; y + v0 ; r y ; (ikGr );1 k Ha r D(B11 ) = D(A) B12 y = r1 dy (14) dr D(B12 ) = D(A+ ) 0 T B21 y = Pr r0 y D(B21 ) = D(A) B22 y = ;Pr v0 y D(B22 ) = D(A+ ):
589
9 "# O A; . Q , .8], D(A~; ) = fy 2 W22(r1 r2) y(r1 ) = y(r2 ) = 0g A~; | "# A; . 1" # "# A~; 2 ~=2 H = D(A~= ; ), D(A; ) | 2 A~= ; . H = L .r r ]? p1 H = fy 2 W 1(r r ) y(r ) = y(r ) = 0g: 0
2
1 2
r
1
2 1 2
1
2
(9){(11) ! (13), H0 H0. 1 , L2 (.r1 r2]? p1r ) L2 .r1 r2]. 1 ( r1 > > 0). > , , , A~; , , A+ . 7 (13) : A~;1A 0 ' A~;1B A~;1B ' ' ; ; 11 ; 12 + ik Gr = ; 1 A+ 1 B21 1 B22 : (15) 0 Pr
Pr
Pr
9 ~;1 ~;1 ~;1 A = A;0 A 1 0A+ B = ikGr A1; BB2111 A1; BB2212 Pr Pr Pr D(A) = D(B) = D(A) D(A+ ). N A B A B . 1. A
H1 H0. A : D(A) = D(A~;;1 A) D(A+ ), D(A~;;1 A) = fy 2 W23 (r1 ? r2) y(r1 ) = y(r2 ) = y0 (r1) = y0 (r2 ) = 0g. . B A~; "# H0,
"# A~;;1 A H1. 1 A~;;1A A : (A~;;1 Ay y)1 = (A~1;=2A~;;1 Ay A~1;=2 y)0 = (Ay y)0 ) y 2 D(A): ( ) H. 1"# " . * " , " . B .9] <(A~;;1 A ; I ) H1
590
. .
Re > 0. <(A), , A. T , ;
h 2 H1, h 6= 0, ((A~;1 A ; I )f h)1 = 0 f 2 D(A~;;1 A). * , ((A ; A; )f h)0 =0. 7 A1 ; A; H0 (. .10]), h = 0 H0, H1 . 6 " ";1 c Re < 0. *
, A~; A "# H1. 3 A~;;1A H1 A~;;1=2A H0. 3 , D(A~1;=2 ) = D(A1=4 ) = fy 2 W41(r1 r2) y(r1 ) = y(r2 ) = 0g ( ! . .11, 12]). * A~1;=2 A1=4 " " . 1 , A~;;1=2A1=4 " " H0 . B D(A~;;1=2 A) D(A3=4 ) (16) 3 = 4 3 0 0 D(A ) = fy 2 W2 (r1 r2) y(r1 ) = y(r2 ) = y (r1 ) = y (r2) = 0g (.11]). 1 , A~;;1=2A1=4 H0, "# A1=4A~;;1=2 H0 H0 ,
. . . 1 , (16) . B , A~;;1 A H1
, A, A~;;1 A = fy 2 W23 (r1 r2) y(r1 ) = y(r2 ) = y0 (r1) = y0 (r2 ) = 0g: 7 , " # (13) (A + B)f = f f 2 D(A) (17)
# 2. ! (13) " (17) , "#
$ (13)
% $ % (17) . . + , (f g) | - ! (13), ! (17) " . * A # A~;2 1 A1 ( A+ "# , , ). +
591
A~;2 1 A1 A1y = 2 y, 1 A~;2 1A1 H1 , A1 . * -
! (17) ! (13),
! .
4. " " #
F , S , H # T p (0 6 p < 1), D(S ) D(T ) b > 0, kSf k 6 bkTf kp kf k1;p f 2 D(T ): 3. B A p = 0. . , ! B , A~;;1 B11 H1, A~;;1 B12 " " , H0 H1=2. + , Pr1 B21 Pr1 B22 H1 H0 , " ! . * y = (y1 y2 ) 2 D(A) ! : kByk2 = jkGrj2kA~;;1 B11y1 + A~;;1 B12y2 k21 + Pr1 2 kB21 y1 + B22y2 k20 6 6 jkGrj2kA1;=2(A;;1 B11 y1 + A;;1 B12 y2 )k0 + C1ky1 k1 + C2ky2 k0 6 6 C (ky1k21 + ky0 k20 ) = C kyk2H1 H0 C1, C2, C | " , ! y. 4. & A %. '
. &
% 2n2 1 n+1 c1 2 2 'n (;1) ' (1) 2 n = (r ; r )2 ; (r ; r )2 + (r ; r )2 + (r ; r )2 + k + O n 2 1 2 1 2 1 2 1 '
cos ' = pa2a+ b2 I 0 (r ) K 0 (r ) I (r ) K (r ) a = I00 (r1) K00 (r1) + I0(r1 ) K0(r1 ) I0(r 2) K0(r 2) I0(r2) K0(r2) b = I00 (r21 ) K00 (r21 ) + I00 (r12 ) K00 (r12 ) : 0
0
0
0
592 ( c1
. .
p2 2 1 ; ') + e : c1 = c + dpsin( 2 a + b2
) c, d, e, # * % I (r ) K (r ) 2 I 0 (r ) K 0 (r ) ( r ; r ) 2 1 1 1 0 0 c = 8r r I00 (r2) K00 (r2) ; 3 I00(r12) K00(r12) 1 2 I (r ) K (r ) I (r ) K (r ) r ; r 2 1 0 2 0 2 d = 8r r (3r1 + r2 ) I 0 (r ) K 0 (r ) ; (r1 + 3r2) I00 (r1) K00 (r1) 1 2 0 1 0 1 0 2 0 2 c 4( r ; r ) e = r2 r 1 cos = p 2 2 : c +d 1 2 + " n
. &
% 2n2 1 1 2 (2) 2 n = Pr (r ; r )2 ; 8r (r ; r ) + k + O n2 : 2 1 2 2 1 O! I0 (x) K0 (x) | ! ! F ( ! F ) (., , .13]). . 3 A 2 : Ay1 = A; y1 (18) A+ y2 = y2 : (19) (19). O 00 + r1 0 ; k2 = ;Pr ! J0(!r) Y0 (!r). 3 !2 = Pr ; k2 . K (11), , (2) n " (. .7]): J (!r ) J (!r ) I(!) = Y00 (!r11 ) Y00(!r22 ) = 0: n- , (. .13]), 1 n 1 1 !n = r ; r ; 8r n + O n3 : 2 1 2
, " (2) n = Pr1 (!n2 + k2).
593
N (18), 2 0 'iv ; r2 '000 + r32 ; 2k2 '00 ; r33 ; 2rk '0 + k4' = ; '00 ; 'r ; k2'
' (11). : ! J0 (!r), Y0 (!r), I0 (r) K0 (r). 3 !2 = ; k2 , I0 (r) K0 (r) | ! ! F (. .13]). : " n !n, , (11), : J (!r ) Y (!r ) I (r ) K (r ) J0(!r1) Y0(!r1) I0(r1) K0(r1) I(!) = J00 (!r21 ) Y 00(!r21 ) I00 (r21) K00 (r21) = 0: J00 (!r ) Y00(!r ) I00 (r ) K00 (r ) 0 2 0 2 0 2 0 2 " ! jz j ! +1 (. .13]): r J0 (z ) = z2 cos z ; 4 + 81z sin z ; 4 + O z12 r 2 1 1 Y0 (z ) = z sin z ; 4 ; 8z cos z ; 4 + O z 2 r 2 3 1 0 J0 (z ) = z ; sin z ; 4 ; 8z cos z ; 4 + O z 2 r 2 3 1 0 Y0 (z ) = z cos z ; 4 ; 8z sin z ; 4 + O z 2
" #, ! W (J0 (z ) Y0(z )) = z2 W (K0 (z ) I0(z )) = z1 : 9 , ! . * ! ( ! ! +1): p2 2 1 p2 2 c + d e a + b sin(x + ') + x sin(x + ) + x = O x2 : (20)
x = !(r2 ; r1), a, b, c, d, e, ' 4. > "
xn = n ; ' + cn1 + O n12 :
594
. .
* " (20) c1. F 9 # ! (17) , , p- # "# (.14,15]). 9 "# A0 , , , " . * ! j > Cj " C . 9 " A = A0 + A1 , A1 p- # A0 (0 6 p < 1). B (.15]): 1) (1 ; p) = 1, # ! A 9 ? 2) (1 ; p) > 1, # ! A F . * (18) (19) 3, . 5. &
#
# $ % (9){(11) ( (17)) , H1 H0 . B , , ! ( # , . .14]), , (.14]), . +
$ (9){(11)
# #.
5. " " # #
6. &
$ (9){(11) , H1 H0 . . .14] , # - # . * = O(jn jp) n- # , p | # . 1 , " (18) ' 6= 0, " # n O(n), . . # . B p = 0, #
595
" n- (n +1)- # # O(n). (.14]) , " ! . + , ' 6= 0. : , 4 ' cos ' = p 2a 2 : a +b 1 , ' " , a > 0 b = 0. *", b < 0. , 4 b = I0(r2 )K00 (r1) ; K0 (r2 )I00 (r1) + I0 (r1)K00 (r2) ; K0 (r1)I00 (r2 ): F ! (.13]): 1) I0 (x) K0 (x) " ( x > 0), 2) I0 (x) , K0(x) ( x > 0). + , b < 0. B .
%
1] I. V. Schensted. Contributions to the theory of hydrodynamic stability. | Ph. D. Thesis, Univ. of Michigan. 2] . . . ! // # $$$%. | 1954. | ). 98. | $. 727{730. 3] R. C. Prima, G. J. Habetler. A completness theorem for nonselfadjoint eigenvalue problems in hydrodynamics stability // Arch. Rational Mech. and Anal. | Vol. 34, no. 3. | P. 218{227. 4] A. A. Shkalikov, C. Treter. Kamke's problem | properties of the eigenfunctions // Mathematische Nachrichten. | 1994. | Vol. 170. | P. 251{275. 5] 0. . 1 . 234 5 // . . | 1995. | ). 58, 6 5. | C. 790{794. 6] A. A. Kolyshkin, R8emi Vaillancourt. On the stability of convective motion in a tall verticall annulus in a magnetic 9eld // Canadian applied mathematics quarterley. | 1993. | Vol. 1, no. 1. | P. 3{21. 7] . . . : 224 ! . | .: 3 , 1968. 8] . <. = . )> >? @ 3 5 B ?>. I // . . | 1947. | ). 20 (62). | C. 431{498. 9] . % , L. $ . 5 2. ). 2. < 5 . $ >?!. | .: , 1978. 10] Q. = . $ 5 224 ! 3 > . | .: 3 , 1976. 11] 0. #. W , X. W. $ . # 224 ! // #22. 3 >. | 1973. | ). 9, 6 2. | $. 228{240.
596
. .
12] 0. #. W . > 224 ! Lp // . . | 1977. | ). 21, 6 4. | C. 509{518. 13] . 4. $ 5 4 ! 234> . | .: 1979. 14] . $. 3, Y. 0. 4 . ? , >? 3 // 5 >. Y. 61. | =@ : 14 , 1981. | C. 104{129. 15] . $. 5. Z 2 4 // . . Y 5, L. [. = 4 3 , . . $ . .: 3 , 1978. | C. 288{362. ( ) 1996 .
. . , . . -
,
510.52+514.112
: , ,
.
! "#$! %". &# ' " A B %" # n. () * | ' , - " ! % . , $ # C , " AC n % ) " AB , ' !$ ( ) * # ( . ' ' AB "1 ). ( " ' ) ' ' ' " '#$ % " ' , # * ' ' "# , $ '"# '' '. & "1- '- # % $ ' . 2 %' % 3(n) '' ) , - "' ) %" "' * # ', % 34(n) | - "'- ) 1 * # ' . 5" % $ * ' " 6# * 3(n) 34(n). 2 %# % $ "#$!': #! #$ c1 c2 > 0, : ) c1 ln n 6 3(n) 6 c2 ln n, ) c1 ln ln n 6 34(n) 6 cln2 lnln nn . ( %# # * 6# * 34(n), " ) " % $ , " .
Abstract M. V. Alekhnovich, A. Ya. Belov, The complexity of algorithms of constructions by compass and straightedge, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 597{614.
The article deals with the following problem. Assume that there are two points A and B on the plane, and a natural number n is given. Our aim is to
598
. . , . . CS (n). Our main result is the following: there exist constants c1 c2 > 0 such that a) c1 ln n 6 C(n) 6 c2 ln n, b) c1 ln ln n 6 CS(n) 6 cln2 lnln nn . The most interesting result is obtained in connection with the lower bound of CS (n), where purely algebraic notions, such as the height of a number etc., arise quite unexpectedly.
1.
( , , . .). ! " ! #! # # " ( " , $ # n " . .). % # , $ P Q ( " $ P Q), & c > 0, $ ( # # P,
(& # " k , & (& # # Q, !
$ " c k , , ( # # Q & (& # P , $) #
! $
$ c ". * , f(x) | ,, $ " O(f(x)), !$, " $ ( , g(x), $ & c > 0, g(x) 6 c f(x) - x, $ O(1) " $ # $ $ . "
$ O(1) " $ 100. / xlim !1 g(x)=f(x) = 0, g(x) = o(f(x)). , " "
(& " ! , !. 1.1. O(1)
) , ,
) , , ) , , .
1.2.
;;1(n) = 1: lim n!1 lnlnlnnn
2 , $ ;(x) | # -, , !- $ - , (n ; 1)!, ;(x);1 | ,, ;(x).
599
2.
4" ( 5 {5 ), $ ( & ( & (! " ( $ ). * " ! -: 1) $ $ ( $ , #
, ) # )8 2) $ $ - !-. : ( . ; " ) & | " !- $ . ; # " # <# $ (# $ - " ). 4 $ : - " , ) $ & $ . 2 XXI $ !
(& . : =(n) | $ , -!- $ " n " & ( # , =>(n) | & ( . * " , $ < =(n)==>(n) # $ . : ) ) < . $
2.1.
=(n) 6 O(1): 1 6 log 2n . : ( ! !- $ -, $) , ! - (. ? , $ $ $ $ . : =(n) > log2 n. 2 ! - " " O(1) , ! $ n " O(log2 n) ( $ $ " $ n). * < " $ ) " , $ & ( ! " " ( 1.1). @ ", 2n ) n n (2 2 2 n =>(2 ) 6 O(n), # !, =(2 ) > 2 , (22n ) > O2(nn) , $ n ! 1 $ . A $ < .
3. : ) "$ ( . 5! , $ < =(n)==>(n) # $ . $ , n = 22k k ! 1 < $ . " : , $ $ & $ - n ! 1.
600
. . , . .
4! ,
lim =(n)
n!1 =>(n)
$ ? : =(n) ln n ( ( 2.1), $ "
3.1 ( ).
lnn : =>(n) 6 O lnln n . : ) # . : n , $ : n = a1 1! + a22! + : : : + a ;;1 (n)] E;;1(n)]! (ai 6 i): % $ n : 1) $ 1 2 3 : : : E;;1(n)]8 E;;1(n)] 8 2) $ 1! 2! 3!: :: E;;1(n)]!8 E;;1(n)] 8 3) " $ ! $ , $ n. H E;;1(n)] E;;1(n)] . : ; & " O(1) ln n , # $ ( ( 1.1) ;;1(n) = O lnln n . ; ln(n) . ; "! , $4 , =>(n) = O ln(ln( n) . J , (& 3.2 ( ). c > 0, n ln n : =>(n) > cln2 ln n : $ " , < # (. ; , $ & (6n)! # ,
(&- " n < #, (6n)! " !- " , ! # ! $ 1 (6n)! + 1. , # ! .
3.3.
) ! n , n2 .
) ! n2 , " (n2 )3 " .
3.4. # n $ (n!)6 < (6n)!.
601 @ ) " 3.2. !< $ , ! # ! $ ! " n . 4- (6n)!. : # & k, < (6n)!, ! . H " $, $ $ " k " ! $ " n , # n = ;;1 (k)=6. 4 , ! $, $ $# !- $ k 2 N ! ;1 n =>(k) > ; 6(k) > c61 lnlnlnn # c1 | " ! 1.2. 4 , $ #- n ( K < L) ln n=ln ln n =>(n) & $< . @ !- K-<-L n ) $< . 2 , 3.5. ! n , =>(n) 6 O(lnln n): . : n = 2t. M $ " $ t: t = a0 + 2a1 + : : : + 2r ar # r = Elog2 n]: @# " r ! $ 1 2 : : : 2r &) " r ; 1 ( n = 2a0 4a1 : : : (2r )ar : ; ) " , $ r lnln n. 3.1. .
: !& "! , $ # . > # " #$ n = qk . 4 , !- n < c ln ln n . J ! $! $ ? & ( n, ! & ! ? ; "! , lnln n $. 4 $ 3.6 ( ). c > 0, n =>(n) > c lnln n: H | "
!. O ! &( (, $) "$ # $ . M " $ . , ) . . * $! . ; >(n) | $ , - " ! n. (& #$ 1.1.
602
. . , . .
3.7. % $ . & O(1) "
) , " , , - " "
) , " , , - " " . @ , $ !& $ , " # 3.1, 3.2, 3.5. 3.8.
ln nn .
) C, n >(n) 6 C lnln
) c, " n >(n) > c lnlnlnnn . ) ' k c(k), >(kq ) 6 6 c(k) lnln(kq ). @ ) " # # ! 3.6. $ 3.9. c > 0, n >(n) > c ln ln n. . )
" ! - " (&- $ $ . : $
" ) - ,, , " (& ) , $ | . ;$ , $ ! " ! $ $ !- $ " ,, ! - !- & ( $ 30 " f+ ; =g. J #$, ! " ,, ! , -& $ " $ , $ " ! - $ $ " 30 " f+ ; =g. : $ $ ! $ (0 0) (0 1) (1 0)(1 1). / ! , $ " k , $ - " $ 0 1 ! $ $ , < $ 22k , " . A , $ $ , " (& <- -, !. ; ht(q) = ht( mn ) # $ q, # m=n jmj + jnj. @#
;1 m = ht 8 1) ht m n m mn 2) ht ; n = ht n 8 3) ht mn 1 mn 2 = jm1 m2 j + jn1 n2 j 6 1 2 6 (jm1 j + jn1j) (jm2j + jn2j) = ht mn 1 ht mn 2 8 1 2
603
m m 1 2 4) ht n + n = jm1 n2 + m2 n1 j + jn1 n2 j 6 1 2 m m 2 1 6 (jm1 j + jn1j) (jm2j + jn2j) = ht n ht n : 1 2 @ ", h | ! !- $ , ! ( ! !< h2 . H " $ , $ ! ! k , ! " !< 2 2 2 2k (((2 | ){z) : : :) } = 2 : k $%& * " ! 3.9 " < . : ) " ! 3.6. % , 3.9, " $ !- . A $ $ "!. !, , $ " $ # ( ( "! # $ - ,
# !, ! $ $ !- ! ( !- !- . $ , ( $ , & ( p , ! $ " ! !. (: $ 3 2 ! ( $ " ! !, cos('=3) ! & ( !- $ " cos('), " ! ! # , # " " <!.) : # $ - " $ -, " !- # $ < - " E1]. 4 , " ! " , $ "
n , $ - n " $ " $ , < mm , # m | . ; $ " $ . P " !& !, | K " L $ | , $ ) < ! . * # $ ! " , ! " (& # $ $ ! ,, . : " $ "! < ! < # " . / ! $ ! " | ,, # # #$ , ! $ . * , $ ! ! ! " ht( + ) ht( ) ( ht( )deg ht( )deg . 2 " , $ $ !- $ " n < # $ 2n . : $ < " # $ # . :
604
. . , . .
" , ) # " $ $ , ! $ " n < #, - 2n . # $ # $
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: !
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611
p p p (& , : z = jz j (cos '2 + i sin( '2 )), # jz j = x2 + y2 , # #
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Abstract Yu. I. Babenko, A. I. Moshinskii, Logarithmic analog of Leibnitz series and some integrals connected with the Riemann zeta function, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 615{619.
A new expression for the analog of the Leibnitz series with alternating signs, which contains the multipliers depending on the logarithm of the term's number, is obtained. The values of some de4nite integrals connected with the above series are found. 1. . 1,2]
Z1
ln( ) ln2 (2) = ; (1) 1 + exp( ) 2 0 , , ! . $ (1) % ! & - ' ( ( ) & ) *, & (0) = ;1 2, 0 (0) = ; ln(2 ) 2 . , *- (1), %./- ! & - '. ( * Z1 Z1 ln( ) ln( ) 0 +( ) = exp( ) + 1 ; ( ) = exp( ) ; 1 J1
=
s
x dx
:
p
s
=
=
I
x dx
"
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, 2001, 7, 5 2, . 615{619. c 2001 ! " # $%&, '( )
:
616
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I
"
I
"
1
1
Z Z ln( ) ln( ) ; ( ) ; + ( ) = 2 exp(2 ) ; 1 = exp( ) ; 1 ; ln(2) exp( ) ; 1 = " 2" 2" = ; (2 ) + ln(2) ln1 ; exp(;2 )] 3%4 && ; ( ) ; ; (2 ), 5 : I
"
I
x dx
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dx
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:
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"
x
(2)
:
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Z2"
Z ln( ) ln( ) = exp( ) ; 1 "!+0 2"
x dx x
"
2 = ln(2) ln( ) + ln 2(2)
x dx
"
x
"
2 * 1 = "lim !+0 + ( ) = ; ln (2) 2. 6 - ) % ! ) & ) % (%/ ) * . 7 , J
I
Z1
J2
"
=
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0
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x
Q
x
dx
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s
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0
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s
s
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x
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=
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617
15 (1) & * . ,- !, * & * ! (1) & & exp(; ) & , 1,2] x
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0
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(7)
./ - & @ * * & A-%' B: 1 (;1)j ln( ) 2 X = C ln(2) ; ln 2(2) = 0 1598689037 (8) j =2 1, . 585] &! / ' (8) ( - , - ), ) & - 5%-. 6 & ! - & (8). 6, , 1 (;1)j +1 ln(2 + 1) C X (2 )1=2 ;(3 4) = + ln 2 +1 4 2 ;(1 4) j =1 % ! , , & * * - ' & $ ! 3, . 35], (5). 2. . 6 ! & (8) & ( & * !* & & & exp(; ) ) (7)) - * j
j
j
: : ::
=
j
=
x
( )=
Z1
J4 n
0
exp(; ) ln( ) 1 + exp( ) nx
x dx
x
=
n 2 j X + (;1)n+1 ln 2(2) + (;1) ln( ) (9) j =2 j =1 * > 0 | ' (@ B (9) = 0 = 1 * .& .). , (9) ) ln2(2) C ; ln(2) ; ln2(2) (1) = ; C + (2) = 4 4 2 2 = 2, % & (3), (4) * 1, . 587], 2, . 527] Z1 exp( ) ln( ) (2 )1=3 ;(2 3) = = ln 5 1 + exp( ) + exp(2 ) 31=2 ;(1 3)
= (;1)n C
n X (;1)j +1
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n
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n
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618
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R1
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x
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x
0
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J7
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x
0
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x
0
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J9
=
Z1 0
x
Z1
J8
0 x
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J
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x
Re( ) 0
J8
; ln(2) ln( ) Re( ) 0 2
>
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>
x
x
619
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Z1
J9
0
J10
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x
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:
1] . ., . . , . | .: '#
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#, 1962.
2] + ,. +., -. /. ,., . 0. . !. | .: 1 , 1981. 3] 4 5. ., 6 7. 1. 8
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, 514.762
: , , , , .
! " # " " $ n;m ) G_ m ), # * , + H^ (S (M~ nm $ # * ,, - $ # * , .# /, . !0#0 10 /1 : + " 0. 3 , 1 " 0 # * , + $ # * , 4m , .# $ - $ # * 4m . ! " 5 #1 /. N - . , 1 /. . , N > mn(n ; m + 1) + m(m ; 1)=2.
Abstract
S. I. Sokolovskaja, About connections induced on surfaces of the projective space by the Bortolotti clothing, Fundamentalnayai prikladnayamatematika, vol. 7 (2001), no. 2, pp. 621{625. The present paper introduces the notion of the Bortolotti connection in the prinn;m ) G_ m ), the notion of the pseudosurface, associated cipal 9ber space H^ (S (M~ nm with subsurface, and the Bortolotti clothing of a pseudosurface, which generates the described connection. The paper singles out a special case of the clothing, namely, the Bortolotti clothing in the proper sense. It is demonstrated that the Bortolotti clothing in the proper sense of the pseudosurface, associated with a subsurface 4m , induces the Bortolotti clothing of the subsurface 4m itself. The paper sets up and solves the problem of immersion of the Bortolotti connection in an N -dimensional projective space. It is proved that the immersion is possible, if N > mn(n ; m + 1) + m(m ; 1)=2. , , .
1933 $ %1].
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dC0 = C00 ^ C0 + C&0 ^ C& E }w C ^ C E dCu = C0u ^ C0 + Cvu ^ Cv + Cu ^ C + Ru& C&0 ^ C0 + Ru w 0 } v & & & dC = C ^ C& + R& C0 ^ C0 + R& C0 ^ Cv E dC00 = R00 C0 ^ C0 + R00}v C0 ^ Cv E & = 1 : : : mE u v w = m + 1 : : : nE 0 C0 + C = 0
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1] Bortolotti E. Connessioni nelle varieta luogo di spazi // Rend. Semin. Fac. Sci. Univ. Cagliari. | 1933. | Vol. 3. | P. 81{89. 2] . . ! " # $ # %% # ' # // (). . * $. !. - . | 1965. | *. 177. | .. 6{42. 3] 0' $ 1. . 2$ ! ! $ '$ # // *. !. . - ). #. %. 32 ...4. | 1974. | *. 6. | .. 43{111. 4] 1. . 5 667 ''' // ! " # !6 8. | 9., 1978. | .. 47{54. 5] .' 3. 1. ;8 '8 7 # !6 # $8 // 56' !. *. 8. | 9.: 12*, 1977. | .. 25{46. 6] =$ .. 5. 9 ># % ? %% '@8 !. | 9.: ** , 1948. & 2000 .
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Abstract S. V. Chegljakova, Injective modules over the ring of pseudo-rational numbers, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 627{629.
An element of a direct product of rings of p-adic integer over all prime numbers p belongs to a subring R of pseudo-rational numbers if almost all its components are equal to one rational number. The concept of such ring was introduced by A. A. Fomin. In this article the description of injective modules over the ring of pseudo-rational numbers is given.
1 ( 1]). , : 1) : M ! B ( im() | # )$ 2) # : A ! B # # ': A ! M # : ! , & ' = . b p Qb p )& * p-& . + Z * & , , - p-& - & p-& - & . 2. D , D = D b p. # # 2 Z . , & -& - . b p- . 1. Z , 2001, 7, 4 2, . 627{629. c 2001 , !" #$ %
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Abstract B. Ya. Steinberg, Noetherianness of convolution operators with coecients on quotient groups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 2, pp. 631{634.
In the paper we study Noetherianness of convolution operators on the groups of slow growth with absolutelysummable kernel and coe1cients in a new class. The coe1cients are the superpositions of canonic quotient-homomorphisms and functions on quotient groups. The key step is the construction of a special compacti2cation of the topological group.
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lim
W
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H=H0 . 2. = " H1 H2 2 H@ 1 \ H@ 2 \ N = . B . SH0 | , H f = g H0 , H 2 , H : H ! H=H0 | , g 2 S (H=H0 ). S | L1 (H ), SH0 , H0 . SH0 Jp (S Jp ) | End Lp (H ), Pf A APf , f 2 SH0 (f 2 S Jp ), A 2 Jp . 7H0 | , H f = g H0 , H 2 , H : H ! H=H0 | , g 2 7(H=H0). 7 | L1 (H ), 7H0 , H0 . 7H0 Jp (7 Jp ) | End Lp (H ), Pf A, f 2 7H0 (f 2 7 Jp ), A 2 Jp . 2 7H0 Jp 7 Jp . C " B1 B2 " ! " x (x 2 X ) (2, . 573], " (" > 0) U x, inf kPU (B1 ; B2) + T k < " infT k(B1 ; B2)PU + T k < ": T i
i
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S Jp , ! x 2 N n S H@ 0 " Bx 2 Jp , # H0 2 B x. $ ! x 2 N \ H@ 0 " Bx 2 SH0 Jp , # B x. % I + B & , ! x 2 N I + Bx . 3. H S B
7H0 Jp . ! x 2 N n H@ 0 " H0 2 Bx 2 Jp , ! A 2 Jp AB # ABx x. $ ! x 2 N \ H@ 0 " Bx 2 7H0 Jp , ! A 2 Jp AB # ABx x. % I + B & , ! x 2 N I + Bx .
B=I+
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1] . . // ! . " . # $ %. &. 57. | ". 272. 2] " . +. , - % ! ! %! $! , 1 // / . , """0, . . | 1965. | &. 29, . 3. | ". 567{586. 3] Cordes H. O. On compactness of commutators of multiplications and convolutions and boundedness of pseudodi6erential operators // J. Funct. Anal. | 1975. | Vol. 18, no. 2. | P. 115{131. 4] 8 +. 9. : ! $ ! ! // # ; $ / < %. | 1981. | &. 15, . 3. | ". 95{96. 5] 8 +. 9. , $ ! = ; ; >- ! // "?@. | 1990. | &. 31, D 4. | ". 180{186. 6] 8 +. 9. $ /%>- % = ; $ ! ! // - ; $ ! < %. ?< / ! . | 0 --E , 1996. | ". 142{145. 7] 8 +. 9. : $ $ = ; $ ! ! // ? / . | 1985. | &. 38, . 2. | ". 278{292. 8] G &. 0 . | ?.: ? , 1973. & ' ( 1996 .
