. .
. . . e-mail:
[email protected]
517.522
: , , ! ".
" ! " #$ $ % &$ $ ' " ! "$ .
Abstract G. A. Akishev, Generalized Haar system and theorems of embedding into symmetrical spaces, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 319{334.
We prove Nikolski type inequalities for polynomials with respect to a generalized Haar system and the embedding conditions of some classes into symmetric spaces.
x
1.
X 0 1]
, 1) , jf(t)j 6 jg(t)j #$ 0 1] g 2 X, $ , f 2 X kf kX 6 kgkX 2) f 2 X jf(t)j jg(t)j $ , g 2 X kf kX = kgkX ( . 1]). ($ ) $ ) * kkX + 2 X. ,
# Lq , 1 6 q < +1, . , , / ( . 1]). , ) e (t) | 1 e 0 1]. 2 '(e) = kekX $ ) X, $ e | 1 e 0 1]. 3 , $ )
X ) '(t) = k t kX , $ 5 0 1]. 0
]
, 2002, 8, 0 2, . 319{334. c 2002 !"#, $% &' (
320
. .
2$ )# # '(t)
1 ) , #7 , 0 1] , 5 '(0) = 0 ( . 1, . 137]). 3 # 9- . : 1$
X X g(t), $ Z kgkX 1 sup f(t)g(t) dt < +1: 1
1
f 2X kf kX 61
0
; , X | )
, X $ 15 X 0 . , + X X ( . 1, . 138]). 3 , X | )
, $ f 2 X Z kf kX = sup f(t)g(t) dt: (1.1) 1
11
1
g2X 0 kgkX 0 61
0
.
1, '(t) | $ )
X, '(t) = ' tt t 2 (0 1] '(0) = 0 $ ) 15 X 0 ( . 1, . 144]). : '(t), t 2 0 1], 1 '(2t) : ' = tlim ' = lim '(2t) ! '(t) t! '(t) > $ ) * $ ), X(') |
$ ) '(t), t 2 0 1], # 1 < ' 6 ' < 2. > , $ Lq , 1 6 q < 1, $, '(t) = t q1 , 1 ' = ' = 2 q . , Lq , 1 6 q < +1, $ 1 0 1] f(x), $ q 1q Z (x) Zx (t) dt dx f kf kq = x x < +1 ( )
0
0
1
0
0
$ f (x) | #7 jf(x)j ( . 1]) (x) | $ 9- . , / M() 1 0 1] f(x), $ Zt kf kM = sup (t) f (x) dx < +1:
)
0
1
0
321
.
, 1 < < 2, Lq , M() $
$ ) ( . 2]). , ) $ $ ) ) fpng ) , pn > 2 (n = 1 2 : : :). .75# ? fpk g = fn(t)g 0 1] $ $#7 ( . 1]): 1 (t) 1 0 1]. A n > 2, n = mk + r(pk ; 1) + s, $ mk = p p : : : pk , k = 0 1 : : :, r = 0 1 : : : mk ; 1, s = 1 2 : : : pk ; 1. B A 1
$ mlk 0 1]. 3$ t 2 B, $ B 0 1] n A, 1 1 X t = mk (t) k (t) = 0 1 : : : pk ; 1 1
+1
1
2
+1
k
=1
k
$ . 3) $ # n (t) skr (t) $#7 : (p isk+1 t r r s n(t) = kr (t) = mk exp pk+1 t 2 ( mr k rmk ) \ B 0 t 2= mk mk ]: , ) ) , 1 B #$ 0 1], # n(t) $ 1 ( mrk rmk ). , + # n (t) 1
5 $ ) , 0 1] 5 $ ) . 3 $ 5 fpn g \ ( . 1]). A pn = 2 (n = 1 2 : : :), fpn g $ ?. B C(q s : : :) $ ) 1 ) , 7 , 7 , . : $ ) ) )# A(y) B(y), , 7 # 1 ) C , C , C A(y) 6 6 B(y) 6 C A(y) $ y. : $ ) ) $ ) ) $#7 1$ . A (. 4]). 1 < < 2 9- (x), x 2 0 1], q > 0 Zx q (t) q t dt = O( (x)) x ! +0 Z tq (t)]; dt = O(;q (x)) x ! +0: 2
( )
+1
( )
+1
+1
1
1
2
1
2
0
1
1
x
B (. 4]).
9- '(x), (x), x 2 0 1].
' > > 1,
322
. .
(' x
x 2 (0 1]
(x) = x 0 x = 0 9- (x), (x) (x), x 2 0 1], 1 > 1. > X n f ; bk k En (f)X = finf bkg X (
)
(
)
1
1
k
=1
* 1 f 2 X fpk g $ * n. C ) EX () = ff 2 X(): En(f)X 6 n n = 1 2 : : :g EDX () = ff 2 X(): Emn (f)X 6 n n = 1 2 : : :g $ = fn g | $ ) ) 1 ) n # 0, n ! +1. > 7] /. 2. 3 E. 3 $ $#7 : 1 1 kTn kq 6 C(q p)n p ; q kTn kp 1 6 p < q < +1 (1.2) n P $ Tn (x) = bk k (x) fpn g, 2 6 pn 6 C , n = 1 2 : : :. k > ) $ ) fpng A. F. E 6]. H (1.2) Lq $ 5], $ ) * 7 , / $ 8]. > x 2 7 $ (1.2)
. : , $ ? . ;. J 12] $ A. 1 6 p < q < +1. f 2 Lp 1 q X n p ; Enq (f)p < +1 (1.3) 0
=1
2
n
=1
f 2 Lq
X 1q 1 q En (f)q 6 C(p q) (n + 1) p1 ; 1q En (f)p + n = 1 2 : : :: k p ; Ekq (f)p 2
k n =
+1
(1.4) H * ) (1.3) ) (1.4) K. > * 14]. > 0 < p < 1, p < q 6 +1 A $ M. F. N 1, >. J. E , ,. . )$ 16]. O 1 Ep() Lq $ 17].
323
F A $ 75 ? $ A. F. E 6], A. F. 15], N. 3 5] Lp $ , / 8]. > x 3 )
.
x
2.
> + $ $ fpng. $ ), fpng $ ) $ ) )# fpn g. , 1 v X Dv (x t) = k (x)Dk (t) x t 2 0 1]: k
1.
=1
X(') | '. !" sup kDmn (x )kX ' 6 mD n '(m;n ): 1
(
x2
)
0 1]
. O (
r r Dmn (x t) = mn x t 2 Inr = ( mn mn ) 0 ) r = 0 1 : : : mn ; 1, $ ) r+1 r r + 1 Zmn Z g(t)Dmn (x t) dt = mn g(t) dt x 2 m m +1
1
n
r mn
0
n
$ # g 2 X 0 ('), D kgkX ' 6 1. 3), ) ) (1.1), 1$
. 2. # " n = mk + 1 : : : mk , k = 1 2 : : :, 0(
)
+1
p knkX ' = mk '(m;k ): 1
(
)
. > $ n (1.1) -
knkX ' 6 pmk '(m;k ): 1
(
)
: $ ) 1 # ( t r r m '(m; ) expf2is pk+1 k+1 g t 2 mk mk ] g(t) = k k 0 t 2= mrk rmk ] ) (1.1). M
$ . 1
( )
+1 +1
324
. .
1. X(') | '. !" " Tmk (x) =
mk X
n
an n(x) x 2 0 1]
=1
1 kT k : '(m;k ) mk X ' . > ) fpng $ # x 2 0 1] Z Tmk (x) = Tmk (t)Dmk (x t) dt: kTmk k1 6
(
1
)
1
0
,+ ( . (1.1)) jTmk (x)j 6 kTmk kX ' kDmk (x )kX ' $ # x 2 0 1]. 3), ) )
1, #$ kTmk k1 6 1; kTmk kX ' : '(mk ) 3 $ . 2. X('), Y () | 1 < < < '. !" kTmk kY 6 (' )(m;k )('(m;k )); kTmk kX ' : . ,
kTmk kY = kTm k kY 6 kTm k m1k kY + kTm k m1k kY = I +I : (2.1) . I . O , Tm k (t) , ) ) ( . 1, . 89]) Z Z f (x) dx = sup jf(x)j dx (2.2) (
0(
)
(
1
1
(
)
(
)
(
)
1
)
)
1
(
0
(
]
)
(
1]
(
)
)
1
2
1
jE j E E =
0
0 1]
1 1, Zt 1 Tmk (t) 6 t Tm k (x) dx = Z jTm k (x)j dx 6 kTmk k1 6 ('(m;k )); kTmk kX ' (2.3) = 1t sup jE j t 0
1
1
(
E E =
0 1]
)
325
$ t 2 (0 mk ]. ,+ I 6 ('(m;k )); kTmk kX ' k m1k kY = (m;k )('(m;k )); kTmk kX ' : (2.4) 3) I . 3 ( . 1, . 162]) Zt '(t) sup f (x) dx 6 kf kX ' f 2 X(')
I 6 kTmk kX ' ('(t)); m1k kY : (2.5) 2 '(t) '(t) > 0, t 2 0 1]. N $ ), kX ; 6 k('(t)); m1k kY 6 ('(m;v )); mv1+1 m1v Y v 1 kX ; kX ; 1 ; ; ; ; 6 ('(mv )) k mv1+1 m1v kY = ('(mv )) m ; m : v v v v (2.6) 1
1
1
1
1
(
)
(0
(
]
1
1
)
(
)
2
(
0
)
1
0
1
2
(
)
(
(
1]
)
1
1
1
(
(
1]
)
1
(
+1
]
(
=0
1
)
1
1
1
1
(
+1
]
(
)
1
+1
=0
+1
=0
O , tt # (0 1] ( . 1]) 2 6 pn 6 C $ # n = 1 2 : : :, $ ), 1 1 1 1 6 (pv ; 1) m 6 (C ; 1) m : m ;m v v v v N $ ), (2.6) kX ; k('(t)); m1k kY 6 (C ; 1) (m;v )('(m;v )); : (2.7) ( )
0
+1
0
+1
+1
+1
1
1
(
(
1]
)
0
v
1
1
+1
+1
1
=0
, ) )
A B, Z dt kX ; (m; ) (m;v ) v 6 C 6 C ; ; ): t (t) '(m v v '(mv ) 1 1
1
1
1
+1 1
+1
=0
(2.8)
+1 1
1
mk
;
+1
; (2.1), (2.4), (2.5), (2.7), (2.8) $
kTmk kY 6 (m;k )('(m;k )); kTmk kX ' : 3 $ . 1. X(') = L'q1 , Y () = Lq2 , 0 < q q 6 +1, kTmk kq2 6 (' )(m;k )('(m;k )); kTmk k'q1 : 1
(
1
1
)
(
1
1
1
1
)
2
326
. .
2.
1 6 r < +1,
X(') = Lpr , Y () = Lq , 0 < p < q < +1, kTmk kq 6 (p q r)mkp ; q kTmk kpr : 1
1
.
, $ 2 p = r ) N. H. 3 5], p = r, q = | A. F. E 6] ( ) $ ) fpng), /. 2. 3 , E. 3 7] ($ $ ) fpn g). 3. 1 6 < q < +1 (t) | 9- 0 1]. !" 1 1 kTmk k 6 C(q )(ln mk ) ; q kTmk kq : . 3 # < q, , J5 )$ $ , Z 1 Zt (t) dt 6 (ln mk ) ; q kTm kq : T (x) dx (2.9) k m k t t 1 1
1
mk
;
0
: ,
A, (2.2) 1 $ ) m Z k 1 ;
0
mk 1 t 1 Z T (x) dx (t) dt 6 kT k Z (t) dt 6 mk 1 t mk t t ;
0
0
6 C( ) (m;k )kTmk k1 6 C( )kTmk kq : (2.10) 1
; (2.9) (2.10) $ 1$ . .
, $ 3 $ . F. R 13].
x
3.
EX ()
4. f 2 X() | (t) 1 < < 2. !"
t ; 1 Z f (x) dx 6 C() kf k + nX 1 E (f) ; )); E (f) ((m + mk X (t) mn X X k t k t 2 (m;n m;n ], n = 0 1 : : :. . , ) Tn(f x) | * 1 f 2 X() fpk g. , ) ) J5 )$, 1
1
(
)
+1
=0
0
1
+1
1
1
(
)
(
)
327
sup
Z
E
E t E 0 1]
t 2 (m;1
jf(x) ; Tmn (f x)j dx 6 t Emn (f)X '
'(t)
(
(3.1)
)
=
; $ n mn ]. : , 1 * 1 , $ ) nX ; kTmn (f)k1 6 kT (f)k1 + kTmk+1 (f) ; Tmk (f)k1 6 k nX ; 6 C() kf kX + ((m;k )); Emk (f)X : (3.2) 1
+1
1
1
=0
1
1
(
)
k
1
(
+1
)
=0
3 $ # 1 E 0 1] $ Z Z Z jf(x)j dx 6 jf(x) ; Tmn (f x)j dx + jTmn (f x)j dx E
E
E
(2.2) (3.1), (3.2) $ 1$ . 3 $ . .
, $ $ ) 4 M. F. N 1 9, 1]. 5. X() Y () | , 1 < < ' ' < 2. f 2 X() X 1 nX ; S(f) ((m;k )); Emk (f)X + n k 1 E (f) 1 1 < +1 (3.3) + (t) mn X mn+1 mn Y f 2 Y () kf kY 6 C( )fkf kX + S(f)g: . , 1 1 nX ; X 1 E (f) 1 m 1 (t) Q(t) = ((m;k )); Emk (f)X + '(t) mn X ' mn+1 n n k Rt $ t 2 (0 1]. .
, Q(t) t f (x) dx (0 1]. 3 # 5 f 2 X(') 1 < ' ' < 2,
4 Zt f (t) 6 1t f (x) dx 6 C()fkf kX + Q(t)g 1
1
1
(
+1
=1
)
=0
(
)
(
;
;
]
(
(
)
(
)
)
1
1
1
(
+1
=1
)
(
=0
1
0
(
0
)
)
(
;
;
]
328
. .
$ t 2 (0 1]. ,+ (3.3) $
1$ 5. . O (3.3) + $#7 : X 1 ((m;n )); Emn (f)X m 1 mn 1 < +1: (3.4) n+1 Y 1
1
(
+1
n
)
(
;
;
]
(
=0
: ), ((m;n )); Emn (f)X 6 1
1
(
+1
)
n X k
)
((m;k )); Emk (f)X 1
1
(
+1
)
=0
$ # n = 0 1 : : :, , (3.3) $ (3.4). ,1 . , ) (3.4), ) 1 X g(t) = ((m;n )); Emn (f)X mn+1 1 m 1 (t) n 1
1
(
+1
n
)
;
(
;
]
=0
t 2 (0 1], $ 1 Y (). : t 2 (0 1] ) , m; < t 6 m; . 3$ $ # g(t) Z Z g(y) X ; ~ g(y) dy > Hg(t) dy > (ln 2) ((m;k )); Emk (f)X y y k 1 1
1
+1
1
1
1
1
t
$
t 2 (m;1
1
(
+1
m
)
=0
;
m; ]. . #$ X ~ (ln 2) ((m;k )); Emk (f)X 6 Em (f)X ((m; )); + Hg(t)
1
+1
1
k
1
1
(
+1
)
(
)
1
+1
=0
$ 1$ t 2 (0 1]. 3 , $ 1$ t 2 (0 1] ~ (ln 2)Q(t) 6 g(t) + Hg(t): N $ ), 15 ?$ H~
( . 2]) (3.4) $ (3.3). 1. 1 < < ' ' < 2. f 2 X() n X sup (m;n ) ((m;k )); Emk (f)X < +1 n
f 2 M .
1
+1
1
k
+1
1
(
)
=0
. 2 Q(t) (0 1]. ,+ , ) )
A, B , (0 1] (t) ", , #
329
(t)Q (t) = (t)Q(t) 6 C
(m;1 ) n
nX ;
1
k
((m;k )); Emk (f)X + 1
1
(
+1
)
n (m;n ) ; ) X((m; )); E (f) + E (f) 6 C sup (m m mk X X n k (m;n ) n n k $ # t 2 (m;n m;n ], n = 0 1 : : :. N $ ), 5 f 2 M . 2. 1 < < < 2 1 < q < +1. f 2 X(') 1 (m; ) q X n q ; ) Emn (f)X < +1 (m n n f 2 Lq 1 X 1 (m; ) q n q (f)X q : E Emk (f)q 6 C(q ) mn ; n k (mn ) =0
1
+1
1
(
1
)
1
+1
1
(
+1
+1
)
=0
1
1
+1
1
+1
(
1
)
+1
=0
1
+1
(
1
)
+1
=
.
, $ 2 $ 8].
6. $
% "
fpn g. f 2 X(), 1 < < < 2 X 1 1 E (f) 1 1 (n; ) n X n+1 n Y (
1
n
)
(
]
=1
f 2 Y ().
(
< +1
(3.5)
)
. C # 1 X n X
1 E (f) 1 1 (t) t 2 (0 1] k X n+1 ; n n k k(k ) 5, 1 $ ) , (3.5) $ , kQ kY < +1: (3.6) N $ , * 1 tt " (0 1] $ ) n X 1 E (f) > 1 E (f) n = 1 2 : : :: k X ( n ) n X k k( k ) Q (t) = 2
(
1
=1
)
(
]
=1
2
(
)
( )
1
(
)
1
(
)
=1
N $ ), (3.6) 5 (3.5). M $ , (3.6) (3.5) + .
330
. .
3) 1 , (3.6) 5 (3.3) 5. : ), tt " (0 1] pn 6 C $ # n = 1 2 : : :, 0
( )
mX ;
1 E (f) > 1 E (f) = 1 2 : : :: ; ) n X n(n (2C (m; ) m X n m 1 O , (t) (m;k ) $ t 2 (m;k m;k ], $ kX j ; X 1 E (f) 1 E (f) + 1 E (f) > C m X (t) mk X n X ; ; n n(n ) (m ) = 1 2 : : :, $ j = mk mk + 1 : : : mk , k = 1 2 : : :. N $ ), (3.6) 5 (3.3). ( , 5 $ ), f 2 Y (). 3 $ . 7. X(), Y () | 1 < < < ' < 2. !" 1 ~EX () Y () () X((m;n )); n m 1 mn 1 < +1: n+1 1
(
1
=
)
(
1
0
;
)
+1
1
1
+1
1
+1
1
(
1
)
(
1
=1
)
(
)
+1
=1
+1
1
1
+1
n
;
(
;
]
Y (
=0
)
. : ) $ 5. :1 $ ). , ) EX () Y (). : , X 1 ((m;n )); n m 1 mn 1 = +1: (3.7) n+1 Y 1
n
1
+1
(
;
;
]
(
=0
)
N $ 10], $ ) ) fn()g $#7 : n(0) = 0 n(1) n(2) : : : n() , 1 1 n( + 1) = min n: n < 2 n : 3$ n < 12 n (3.8) n ; > 12 n (3.9) (
(
(
+1)
+1)
1
(
)
(
)
)
$ = 1 2 : : :. > (t) 0 1] $ ) f g $ ) n kX ; 1 1 1 (t) 6 1 m+1 m nk ; (t) (mn k ) nk (
+1)
1
(
=
(
)
;
;
)
(
)
1
(
+1)
331
$ # t 2 (m;n k m;n k ) k = 1 2 : : :. ,+ * (3.5) $ , X 1 ((m;n k )); n k m;n k m;n k ] = +1: (3.10) Y 1
(
1
+1)
(
)
1
k
(
1
1
(
+1)
)
(
1
(
+1)
(
)
(
=1
)
C # 1 X f (x) = (pmn (m;n )); n mn(+1) pn(+1) (x) 1
0
(
+1)
(
1
(
+1)
+
)
+1
=0
x 2 0 1]. N 7)#
2 (3.8), (3.9) f 2 X() 1 X Emn (f )X 6 Emn(s) (f )X 6 n 6 n s 6 2n s ; 6 2n 0
0
(
0
)
(
)
(
s
)
^
( )
( +1)
=
1
$ n = n(s) : : : n(s + 1) ; 1. ( , g = f 2 EX (). H mn pn+1 (t) # , + 1
0
2
0
+
mZn(j) mZn(j) t 1 Z g (x) dx > m g (x) dx > mn j jg (x)j dx = nj t Z1 = mn j (m;n k ; m;n k )((m;n k )); n k > 12 ((m;n j )); n j ;1
( )
0
0
0
( )
0
0
1
( )
k j
(
0
1
)
(
1
1
+1)
(
1
+1)
(
)
( +1)
1
( )
=
$ # t 2 (m;n j m;n j ], j = 1 2 : : :. N $ ), ?$ * (3.10) g 2= Y (). M # # E~X () Y (). 3 $ . 1
1
( +1)
( )
0
8. $
% "
fpn g 1 < < < 2. !" X 1 1 1 EX () Y () () ((n; )); n n+1 n 1
< +1:
1
(
n
]
=1
Y (
)
. : ) $ 6. N $ ) $ . , ) EX () Y (). : , X 1 ((n; )); n n+1 1 1 = +1: n Y 1
n
1
(
]
(
=1
)
C # 1 X f (x) = (pml (m;l )); n ml() pl() (x): 1
1
(
=0
)
(
)
1
(
)
+
+1
332
. .
($ ) fn g fn()g | $ ) , $ ) 7, l() = maxfk: mk < n( + 1)g: : , 1$ , $ ) 7, g = c f 2 EX (), g 2= Y (). 3 $ . . , ) Y () = Lq , X() = Lp, 1 6 p < q < 1. , pn = 2 $ n = 1 2 : : : ( . . fpn g | ?) 6 $ )
. ;. J 12], 8 > * 14]. > 2 6 pn 6 C , n = 1 2 : : :, 6 $ /. 2. 3 E. 3 7], 8 N. 3 5], A. F. 15]. 3 6 8 11]. > 2 11] : 1 ) n * . 9. 1 < q < q < +1, 1 < < 2. f 2 Lq1 1 X q (ln mn ) ; q21 Emq2n (f)q1 < +1 (3.11) (
)
1
1
1
1
0
1
2
1
1
+1
n
=1
f 2 Lq2
q12 X 1 q2 q ; 2 q 1 Emn (f)q2 6 C( q q ) (ln mk ) Emk (f)q1 n = 1 2 : : :: 1
1
2
+1
k n =
. , ) Tn(f ) | * 1 f 2 Lq1 fpng. , )# $ ) ) fn()g $#7 : 1 n(0) = 0, n(1) = 1, n( + 1) = min n: Emn (f)q1 < 21 Emn (f)q1 : 3$ Emn(+1) (f)q1 < 12 Emn (f)q1 (3.12) Emn(+1) ; (f)q1 > 21 Emn (f)q1 : (
)
1
C $
X Tmn(1) (f x) + (Tmn(+1) (f x) ; Tmn() (f x)) $ f 2 Lq1 Lq1 . , ) )
13], 3, X q12 j; q2 q ; 2 q kTmn(j) (f) ; Tmn(s) kq2 6 C(q q ) (lnmn k ) 1 Emn(k) (f)q1 : k s (3.13) 1
1
1
2
(
=
+1)
333
> * 1 (3.12) n kX ; q q (ln mn k ) ; q21 Emq2n(k) (f)q1 6 2q2 (ln mn ) ; q12 Emq2n (f)q1 : (3.14) (
+1)
1
1
(
+1)
1
n nk =
(
+1
)
; (3.13) (3.14) (3.11) , $ ) ) fTmn(j) (f)g Lq2 $ ) Lq2 . > Lq2 7 g 2 Lq2 , kg ; Tmn(j) (f)kq2 ! 0 j ! +1: 3 Lq2 Lq1 , kg ; Tmn(j) (f)kq1 ! 0 j ! +1: N $ ), g(x) = f(x) #$ 0 1]. ( , f 2 Lq2 . 3 $ .
#
1] . ., . ., . . . | ., 1978. 2] Sharpley R. Spaces & and interpolation // J. Funct. Anal. | 1972. | Vol. 91. | P. 479{513. 3] , - ., ./ 0 . 1 / 2 3. | ., 1958. 4] 4 . 5. 67 - 3 03 87 . | 9. 561 :6 ;. < 1036-80 9. 5] 1 0 ?7 . 6. @/ 2 3 B2 C C 4 -. | :8. 3CC.. . . 7 3. 8.-. 7. | : , 1990. 6] ,7 . :. 1 - C 0 7 3 ,D ?- / 2 CC 7 CC E // B7 2 0, C7 2 . | G2C7, 1984. | . 46{54. 7] 1 . B., 1 . C 7 CC // 0. CD. ,?. 0 3., C . | 1983. | < 9. | . 65{73. 8] :7D . :., C ;. . 1 - C, C C ??H CC E . | 9. 0661. < 3618. 9] -7 I. :. 1 - ,D ?- // . C?7. | 1975. | 1. 97, < 2. | . 230{241. 10] 3 5. . 1 - C 0 7 3 ,D ?- // . C?7. | 1977. | 1. 102, < 2. | . 195{215. 11] :7D . :. @ - 7 7 CC C, C C // 10C 37 3 -3 3. 78. KB7. C C , ?-, 0L. | ., 1995. | . 11{12. 12] ? M. . 6 ,D ?- E GD // . C?7. | 1972. | 1. 87, < 2. | . 254{274.
334
. .
13] .C 4. :. @ CC ,D ?- C C 4 7 - // 0. CD. ,?. 0 3., C . | 1987. | < 10. | . 48{58. 14] 5 D7 N. 1 - C 0 7 3 ,D ?- CC E // 0. CD. ,?. 0 3., C . | 1980. | < 4. | . 11{15. 15] MC . :. @ - 7 CC 87 , 0 3 C3 2C ,D ?- 7 CC . | 9CC.. . . 7 3. 8.-. 7. | C7 , 1988. 16] -7 I. :., 5. ., @C 23 . ? 9-7C C C Lp , 0 < p < 1 // . C?7. | 1975. | 1. 98, < 3. | . 395{415. 17] 6 0? . O., . ., :7D . :. 3C2 3 0 7Q88 B2 - C C 4 . | 9. 561 :6 ;. < 580-83 9. ) * 1999 .
N N - 3 3 3
2
. .
512.554.5
: ,
, , ! , " , # " .
$
% & ! , %
# . '. (. )#* 1981 . - % # & !
%. ) Nk |
! ! ! 3
& , D |
N3 N2 - 3, . .
! ! 3 =0 2( 1 2 )( 3 4 )]( 5 6 ) = 0 (
& !
" Nk Nl . , "! # " (D) " # " rt (Dn ) = + 2
" Dn = D \ Var(( ) 1 n )
# #
% D. k
x
x x
x x
x x
F
:
n
xy
zt x
:::x
Abstract
A. V. Badeev, The variety N3 N2 of commutative alternative nil-algebras of index 3 over a eld of characteristic 3, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 335{356.
A variety is called a Specht variety if every algebra in this variety has a 9nite basis of identities. In 1981 S. V. Pchelintsev de9ned the topological rank of a Specht variety. Let Nk be the variety of commutative alternative algebras over a 9eld of characteristic 3 with nilpotency class not greater than . Let D be the variety N3 N2 of nil-algebras of index 3, i.e. the commutative alternative algebras with identities 3 =0 2( 1 2 )( 3 4 )]( 5 6 ) = 0 In the paper we prove that the varieties Nk Nl are Specht varieties. Moreover, a base of the space of polylinear polynomials in the free algebra (D) is built and the topological rank rt (Dn ) = + 2 of varieties Dn = D \ Var(( ) 1 n ) is found. This implies that the topological rank of the variety D is in9nite. k
x
x x
x x
x x
:
F
n
xy
zt x
:::x
, 2002, 8, : 2, . 335{356. c 2002 !, "# $% &
336
. .
. , . . ! "# $ %2, 129] +: + - +
! . ( .) ? 2. . %3] , -
. 2 ( + .) ! ., ., !3 . (;1 1) . . 4
+- ! 5 . 6. 7- 3 %5], . 6. 9! %6], 6. . # , . :. %9]. ;. !
. ! . . , - 2, 3, <. <. < %7]. + , 2. . %4] - +
3 ! - . ! . . 2. . 6. 7- 3 %5] - .
, ? - . @ . . 7+! V | - +
, V $ W. dimW V V W ! - n,
? : +?+ - 5 f1 : : : fs , ? V W, . . hf1 : : : fsiT + T(W) = T(V), - 5 D W n = maxfdeg f1 : : : deg fs g: 7 dim V V ! V !
. . 7+! M | .
, . .
- +
, }(M) | 5 .
M. 7+! E }(M), 5 E - , .
5 E - + . # W }(M) 5 U n (W) = fV W j dimW V > ng Un (W) = U n (W) fWg: - + 5 F = fUn(W) j W 2 }(M) n 2 Ng , }(M)
5 ! , E - }(M). 7 ! + M . , + ? 3 -
M1 M2 : : : Mn : : : E +, ! , ! D 5 E - E. G - - E0 5 ! . - E, E0 $ E. - rt (E) E - r, - E(r;1) = 6 ? E(r) = ?. - rt (M)
M - }(M), . . rt (M) = rt (}(M)).
-
337
H !, -
M ! n, dim M 6 n rt (M) = 1, 5
M ! , 5 }(M) - . 6 %5] , - , - 5 ! .
Alt2 . ! . . , - 2, 3, +. D : Alt2 Alt2 \ Var(x3), - D
+ , . . rt (Alt2 ) = 2: 6 5 ! (xy)y ; xy2 = 0: H , +-
(xy)z + (xz)y + (yz)x = 0: (1) 7+! Nk |
! k. 7 5 , D |
N3 N2
+ . ! . !- 3 K . 3, . .
+ . ! . 5 x3 = 0 (2) %(x1x2 )(x3x4)](x5 x6) = 0: (3) 6 D . - F(D) - rt (Dn)
Dn = D \ Var((xy zt)x1 : : :xn ): # +? . Dn n + 2. # ! A 2 D, 5
?! + B.
1.
7
! A
?! - + B.
5 +
9. 7. ; %8]. 7 5
E0 = fa a2g E1 = fx ax a2xg E = E0 E1 B = K(E0) + K(E1): @ D K(E0 ), | D K(E1 ).
338
. .
7 5 , - + 5 +
+ , . . + 5 xi yj = (;1)ij yj xi xi yi 2 Ei. G +
+ + 5 . D . +? : a a = a2 a x = ax a2 x = ax a = a2x ax x = a a2 x x = ;a2 : G! + . H !, - B n B 2 = Kfxg B 2 n B (2) = Kfa axg B (2) = Kfa2 a2xg (B 2 )3 = 0: 7 5 , - + B + 5 (z xi )yj + (;1)ij (z yj )xi + z (xi yj ) = 0 ( ) xi yi 2 Ei . 4 + 5
+ 5+ ! (1). # - ! D + 5 . D ., -
+ +
+ . D . + 5 - . M !, xi = yj = x 2 E1 xi = yj = ax 2 E1 + 5 ! . , +-
(B 2 )3 = 0, ! + 5 +? . . D : fa a xg fax a xg: 6 +- +-
(a a) x + (a x) a + a (a x) = 3a2x = 0: 6 +- (ax a) x + (ax x) a + ax (a x) = ;a2 + a2 = 0: + 5 . , B | ! +. #, + B ( 2 n (a a)Rx = a2 n 0 (mod 2)
a x n 1 (mod 2): ! , B (2) RnB 6= 0:
-
339
!
5 - !, - - A = G(B) |
+ !
x3 = 0 (A2 )3 = 0 A(2)RnA 6= 0 ! A 2 D A 2= Dn:
2.
7 5 B D |
. , . . 5 (x1 x2)(x3 x4) = 0: 6 D + 5
.
1. F = F (B) | B . xj xi1 xi2 : : :xin;1 i1 < i2 < : : : < in;1 j > i1 (4) U(F). 2 6 5 ! (xy)y = xy2 . N x u1u2 + + 5 , +- F 5 u1u2 yy = 0: H D 5 y, +-
u1u2 xy = ;u1u2 yx: 7 - F
?!
, +-
u1u2 x(1) : : : x(n) = sign()u1 u2x1 : : :xn | ! D 5 f1 : : : ng. #, + 5 !
xi xj x1 = ;xi x1xj ; xj x1xi :
?! D .
!, - U(F ) 5 D (4). ! ! !, - - (4) . x1 x2 : : : xn . # n 6 2 D - .
n X j =2
j xj xi1 xi2 : : :xin;1 = 0
- , - fi1 : : : in;1 j g = f1 2 : : : ng. # j0 > 2 3 3 xj0 ! xy
j0 xyxi1 xi2 : : :xin;1 = 0. 7 ! +
B
340
. .
! , j0 xyxi1 xi2 : : :xin;1 = 0 j0, 3 ! .
2. B . 2 7 - ,
1, !, - - f 2 x
5 ! f = u x. - f 3 x +-
+ (2) (3) f = u x = (v x) x = v x2 = 0: ! - 3 ! . O , - B %1]. @ . 3 !
2 D + !+
1 - ! !.
3. ! B D M Dn " : (1) M $ (2) M \ B 6= B. 2 G- , - M ! , M \ B 5 ! . 6 + !
B M \ B 6= B. 7+! M \ B 6= B. 9
2 , - M \ B ! , . . m
x1 x2 : : :xm 0 (mod F (2)(M)): G x1 x2 : : :xm y1 : : :yn 2 T (M), . . M ! .
3. ! -
7 ! 3 ! - , 5 , -
D ( - 3 ! -
) . 7+! F = F (D) |
D 5 . 5? . X = fx1 x2 : : : xn : : :g. 7 5 u v 2 F 2, x y z 2 F. 7 ux ux = u2x2 = 0: H u, +-
ux vx = 0: 7+! f 2 D | - 4 . 9
, - - ! 3
B.
-
341
f 2 T(B) = F (2), . . f
5 ! 3
- u v, u v 2 F 2 . 7 - + (3)
5 - !, - u, v | - 2 . 7 D 5 - .
5 !
+ F (2). - u, v 2 x +- +
u v = u1 x v1 x = 0: , X f = u v = 0: char K = 3, ! ! . - F(D) 3,
D
%1]. 5 3 ! %(x1x2 )(x3x4 )]x5 = 0: 7+! A | 5 5? . X, I | A. G - - P(I) . - A, 5? . IP Pn(I) P(I) | .
- 5 5? . Xn n. G - Zn = f1 2 : : : ng, n > 4, 'ij = x0i1 : : :x0in;4 | n ; 4, fi1 : : : in;4g = Zn n f1 2 i j g, i1 < : : : < in;4.
4.
1) % Pn(F (2)(C)) n = 4t n = 4t + 3 ) x1xi 'ij (x2xj ) 2 < i < j ) x1xi 'i3(x2x3 ) i < 3 2) n = 4t+1 n = 4t+2 Pn(F (2)(C)) & ) ) ) x1x5'54 (x2x4): 2 7 5 F = F(C), Pn = Pn (F (2)(C)).
- +- n = 4. 7
x2 y2 = (xy)(xy): 7 3 D 5, +-
(x1 x2)(x3 x4) = ;(x1 x3)(x2 x4) ; (x1 x4)(x2 x3): , n = 4 D ), ) 5 U4 . #, F + (1) 3 ! uv(pqx) = ;uvx(pq): (5)
342
. .
G , - 5 - F (2) 3 D xyw (zt), x y z t 2 X, w | . N , - + 3 ! 5 ! xyw
5 !, .
, +-
xyx(1) : : :x(n)(zt) = (;1) xyx1 : : :xn(zt)P (6) ( xyx1 : : :xn (zt) = (;1)n ztx1 : : :xn(xy) n = 0 n = 4t 4t + 3 (7) 1 n = 4t + 1 4t + 2: 7 5 (7) n = 1 2 3 4: (xy)x1 (zt) = ;(xy)%(zt)x1 ] = ;(zt)x1 (xy)P (xy)x1 x2 (zt) = (zt)x2 x1(xy) = ;(zt)x1 x2(xy)P (xy)x1 x2x3 (zt) = ;(zt)x3 x2 x1(xy) = (zt)x1 x2x3 (xy)P (xy)x1 x2x3x4 (zt) = (zt)x4 x3x2 x1(xy) = (zt)x1 x2x3 x4(xy): # n > 4 ! +5 +- + 3 . 7 !+! 5 xixj x1 = ;x1 xixj ; x1xj xi xi xj x2 = ;x2xi xj ; x2xj xi x1x2xi = ;x1 xix2 ; x2 xix1 5 (5){(7) +- +- n = 4, + !, - U(F (2)) 5 - x1xi 'ij (x2 xj ) i j > 2: (8) @ , xi xj 'xk (x1 x2) = ;xixj '(x1x2 xk ) = ;xi xj '(;x1 xk x2 ; x2 xk x1) = = xi xj '(x1xk x2) + xixj '(x2xk x1 ) = ;xi xj 'x2 (x1xk ) ; xi xj 'x1 (x2xk ) = = xi xj x2'(x1 xk ) xi xj x1'(x2 xk ) = = x2 xixj '(x1 xk ) x2xj xi'(x1 xk ) x1xi xj '(x2 xk ) x1xj xi '(x2xk ) = = x1 xk xj '(x2 xi) x1xk xi'(x2 xj ) x1xi xj '(x2 xk ) x1xj xi '(x2xk ): 7+! n > 4. M , - 4 ! D. ! , x2w x2 = 0. 6 +
C,
, +- +? 5 : 2x2w (xy) + 2(xy)w x2 = 0: + , n = 4t 4t + 3 + (7)
x2 w (xy) = (xy)w x2: G x2w (xy) = 0:
-
343
# ? + -! - wn n.
! +? +: x1xxwn ;5(x2x) = 0 n = 4t 4t + 3: H 5 , +- n = 4t 4t + 3 x1 xixk wn ;5(x2 xj ) + x1xj xk wn ;5(x2 xi) + x1 xk xiwn ;5(x2 xj ) + + x1 xk xj wn ;5(x2xi ) + x1xi xj wn ;5(x2xk ) + x1 xj xiwn ;5(x2 xk ) = 0: 7 5 ! k = 3
- ! , +-
x1 xix3 wn ;5(x2 xj ) = ;x1xj x3 wn ;5(x2xi ) ; x1x3xi wn ;5(x2xj ) ; ; x1x3 xj wn ;5(x2xi ) ; x1 xixj wn ;5(x2 x3) ; x1xj xiwn ;5(x2 x3): N , - - . D (8), - | D ), ), . . D (8) 3 D ), ). #, + (7) n = 4t + 1 4t + 2
x2wn ;4y2 + y2 wn ;4x2 = 0 - + ! !
x1xxywn ;6(x2 y) + x1yyxwn ;6 (x2 x) = 0: H + 5 , +- n = 4t + 1 4t + 2 x1 xixp xq wn ;6(x2 xj ) + x1xp xi xq wn ;6(x2 xj ) + x1xi xp xj wn ;6(x2xq ) + + x1xp xi xj wn ;6(x2xq ) + x1xq xj xiwn ;6(x2xp ) + x1 xj xq xiwn ;6(x2 xp ) + + x1xq xj xp wn ;6(x2 xi) + x1xj xq xpwn ;6(x2 xi) = 0: 7 D
j = 3, q = 4, p = 5 5 !
- ! , +-
; x1xi x5x3wn ;6(x2 x4) = x1xi x5x4wn ;6(x2 x3) + x1x5xi x4wn ;6(x2 x3) + + x1 x5xi x3wn ;6(x2x4 ) + x1x4x3 xiwn ;6(x2x5 ) + x1x3x4 xiwn ;6(x2 x5) + + x1 x4x3x5 wn ;6(x2xi ) + x1x3x4 x5wn ;6(x2xi ) = 0: 7 p = 3, q = 4 5
- ! , +-
; x1xj x4 x3wn ;6(x2xi ) = x1 xix3x4 wn ;6(x2xj ) + x1 x3xix4 wn ;6(x2xj ) + + x1xi x3xj wn ;6(x2 x4) + x1x3 xixj wn ;6(x2 x4) + x1x4 xj xi wn ;6(x2 x3) + + x1xj x4xi wn ;6(x2 x3) + x1x4 xj x3wn ;6(x2 xi) + x1xj x4x3wn ;6(x2 xi): , D x1 xi'i4 (x2x4) 3 D ){), D (8) 3 D ){) D x1xi 'i4(x2 x4).
344
. .
7. !+
+ .
D . 7 - +
! D ) ). 7 5 , - 3 . D . x1 x2 : : : xn, n = 4t 4t + 3, ! + : X X n = ij x1xi 'ij (x2 xj ) + i3x1xi 'i3(x2x3 ) = 0: 3 3
i<j
i
xi ! u1u2 xj ! v1v2 i j 6= 1 2 3 i < j
ij u1u2'ij (v1 v2) 0 (mod F1(2)) A 2 D , !
C, ! +- ij = 0. , ij = 0, i j 6= 3. 6 D +- 3 3 xj ! v1 v2 x1 ! u1u2 j 6= 2 3 xi ! u1 u2 x2 ! v1 v2 j 6= 1 3
3j = 0 i3 = 0. ! , 3 n ! . # 5 ! +
! D ){), - n = 4t + 1 4t + 2: X X n = ij x1xi 'ij (x2 xj ) + i3x1 xi'i3 (x2x3 ) + 54x1 x5'54(x2 x4) = 0: i<j
i
7 i j 6= 3 4 3 3 xi ! u1u2 xj ! v1 v2 i j 6= 1 2 3 4 i < j +- ij = 0, i j = 6 3 4. , n
X i>3
f 3ix1x3 '3i(x2 xi) + i3x1xi '3i(x2 x3)g + X +
i>4
4ix1 x4'4i(x2 xi) + 54x1x5 '54(x2x4 ) = 0:
6 ! ! 3 3 xi ! u1 u2 xm ! v1 v2 m = 2 3 4 i > 5 + (3), (5) +-
(i) i3u1u2 x1'3i (v1v2 x3) = 0 (ii) 3iv1 v2x1 '3i(u1 u2x2) + i3u1 u2'3i(v1 v2 x2) = = ; 3i v1v2 x1'3i x2(u1 u2) ; i3u1u2 '3ix2(v1 v2 ) = = ;( 3i + i3)u1 u2x1 '3ix2(v1 v2 ) = 0 (iii) 4iv1 v2x1 '4i(u1 u2x2) = 0:
-
345
! , i3 = 3i = 4i = 0, i > 5. G
n = 34x1 x3'34(x2 x4) + 35x1x3'35(x2 x5) + 43x1x4 '34(x2x3 ) + + 53x1 x5'35(x2 x3) + 45x1x4'45 (x2x5) + 54x1x5 '45(x2x4 ) = 0:
! 3 3 x3 ! u1u2 x4 ! v1 v2 x1 ! u1u2 x3 ! v1v2 +-
34u1u2 x1'34(v1 v2 x2) + 43v1 v2x1 '34(u1 u2x2) = = ;( 34 + 43)u1u2 x1'34x2 (v1 v2) = 0 43u1u2 x4'34(v1 v2 x2) + 53u1u2 x5'35(v1 v2 x2) = = ( 43 ; 53)u1 u2x4 '34(v1 v2x2 ) = 0: G 34 = ; 43, 43 = 53. - , 35 = ; 53, 45 = ; 54, 35 = 45. , 34 = 35 = 45 = ; 43 = ; 53 = ; 54. #, 3 3 x1 ! u1 u2, x2 ! v1 v2 +-
34u1u2x3 '34x4(v1 v2 ) + 35u1 u2x3 '35x5(v1 v2 ) + 43u1 u2x4'34 x3(v1 v2 ) + + 53u1u2 x5'35x3(v1 v2 ) + 45u1u2x4 '45x5(v1 v2 ) + 54u1u2x5 '45x4(v1 v2 ) = = ;( 34 ; 35 + 45 ; 43 + 53 ; 54)u1 u2x3'35 x5(v1 v2 ) = 0: N , 3, - 34 = 35 = 45 = ; 43 = ; 53 = ; 54 34 ; 35 + 45 ; 43 + 53 ; 54 = 0 | + . ! , 3 D ){) ! .
4. D
7+! F = F (D) |
D 5 . 5? . X = fx1 x2 : : : xn : : :g. 7 5 u v 2 F 2, x y z 2 F . 7 ux ux = u2x2 = 0: H u, +-
ux vx = 0: (9)
346
. .
G + (1), (3)
(ux v)x = ;ux vx ; ux2 v = 0 (10) - !
(ux v)y + (uy v)x = 0: (100) 7+! ' | + . 6 + (3), (10),
- x2 ', , +-
uvx2 = 0 (x2 ' u) = (x'x u)x = 0: 7 (2) (u x2 )x = u x3 = 0: H D
, +-
uvxy = ;uvyx (11) (xy' u)z + (xz' u)y + (yz' u)x = 0 (12) (u xy)z + (u xz)y + (u yz)x = 0: (13) 7 5 F0(2) = F (2) + 3 (2) Fp(2) +1 = Fp F: R+ - !, - 5 . 5? . F=Fp(2) +5 X. N , - P(F (2)) =
1 M
p=0
P(Fp(2)=Fp(2) +1 ):
6 D + 5 P(Fp(2)=Fp(2) +1). (2) (2) (2) O , - F0 =F1 = F (C), C D |
3 ! - . . 7 5 , Hn | D ){)
4, . . . D Pn(F0(2)=F1(2)). G - : np = x0n;p+1 : : :x0n | p, en = = x1 x5'54 x2 x4 2 Hn, enp = en;p np , Hn0 = fx1x4'4j x2 xj 2 Hn j j 2 Ng. #
- + - n . # 5 ! +? +5 .
5. % Pn(Fp(2)=Fp(2) +1 ), 1 6 p 6 n ; 4, & " ( & Enp): 1) Hnp = fbnp j b 2 Hn;pg, 0 (1k) = fb(1k) j b 2 H 0 g, k = n ; p + 1, 2) Hnp np 3) enp (1i)(2j) i j 2 f1 2g Znp , i < j, fi j g 6= f1 2g.
-
347
2 # S . n, p - 5 f1 2g Znp , | ! - Zn . 7 Xn = = fx1 x2 : : : xng ! (+ ) 5 ( ). 7 5 -, - D ij !"#$. i<j
Hnp(1i)(2j)
(14)
5 Pnp = Pn(Fp(2)=Fp(2) +1 ). N , - Pnp 5 - n (xy' zt) p (15) x y z t 2 Xn , ', p | , p p. 6 +
(11)
5 - !, -
p 5 . 7+! f | - (15). , !+ (11), (100), ', p
5 ! 5 . ! , 5 f'g f p g ( ! f g
- ) 5 ! p ! . .,
5 - !, - f p g ! ! .. 7+! D +- xi , xj , i < j, ! 5 f p g. f(1i)(2j) = f 0 np f 0 | - Xn;p = fx1 : : : xn;pg. @ , - Hn;p Pn;p (F (2)(C)). 7 D +
+ F1(2)
f 0 2 Pn;p(F (2)) = Pn;p(F (2)=F1(2) F1(2)) = Pn;p (F (2)(C)) = K(Hnp): #+ np = x0p+1 : : : x0n, +- , -
+ Fp(2) +1 - f 0 np 2 K(Hn;pnp ) = K(Hnp): (16) ! , f(1i)(2j) 2 K(Hnp) - ! f 2 K(Hnp(1i)(2j)): T 5 f'g f p g - f 5 ! p ! ., fx y z tg 5 ! +. ! ., f p g 5 + . ,
?!
(12), (13) + D f p g ! fx y z tg, +? + +-. N - , - (15) 5 K(Hnp(1i)(2j)). G (14) 5 Hnp.
+- p = 1. En1 En1 = Hn1 Hn0 1(1n) fen1 (1n)g fen1(2n)g:
348
. .
7 5 , - En1 5 Pn(F1=F2). (14) | 5? ., D - !
K(Hn1) K(Hn1(1n)) K(Hn1(2n)) K(En1): (17) Hn1 2 En1, !, - K(Hn1(1n)) K(En1) K(Hn1(2n)) K(En1): 6 + (13), (100) ! x y 2 F, u v 2 F 2 . '0 , '00 n ; 6
(ux'0 yx)y = (ux'0 y2 )x = 0 (18) 00 00 2 00 (xyy' v)x = (xyx' v)y = (x y' v)y = 0: (19) 7 5 (19) v = x2xj x, y, +-
(xn xixk '00 x2xj )x1 + (xnxk xi '00 x2 xj )x1 + g00 xn = 0 g00 | - Xn;1 . 7 + .
. # D 5 - k = 4, j 6= 4, k = 3, j = 4 + (1n). 7 +-
(xnxi x4'00 x2 xj )x1 + (xn x4xi'00 x2xj )x1 = = (x1 xi x4'00 x2 xj )xn(1n) + (x1x4 xi'00 x2xj )xn (1n) = = (x1 xi 'ij x2xj )xn(1n) (x1x4 '4j x2xj )xn (1n) (xnxi x3'00 x2 x4)x1 + (xn x3xi '00 x2x4 )x1 = = (x1 xi x3'00 x2 x4)xn(1n) + (x1 x3xi'00 x2x4 )xn(1n) = = (x1 xi 'i4 x2 x4)xn(1n) (x1 x3'34 x2x4 )xn(1n): ! , (x1xi 'ij x2xj )xn (1n) = (x1 x4'4j x2 xj )xn(1n) g00 xn (x1 xi'i4 x2x4)xn (1n) = (x1 x3'34 x2 x4)xn (1n) g00 xn: N , - + (16) g00 xn = K(Hn1) (x1 x4'4j x2xj )xn(1n) 2 K(Hn0 1(1n)) (x1 x3'34 x2 x4)xn (1n) = en1(1n): 7 +- , - (x1 xi'ij x2xj )xn (1n) 2 K(Hn0 1(1n)) K(Hn1) j 6= 4 (x1 xi'i4 x2 x4)xn (1n) 2 K(en1 (1n)) K(Hn1):
-
G,
349
K(Hn1(1n)) = Kf(x1xi 'ij x2xj )xn (1n)g En1 +-
K(Hn1(1n)) K(Hn1) K(Hn0 1(1n)) K(en1 (1n)) K(En1): (20) #, 5
. (18), (19) u = x1xi , v = xn x4 x, y, +-
(x1xi x4'0 xnxj )x2 + (x1xi xj '0 xnx4)x2 + g0 xn = 0 g0 | - Xn;1, (x1 xix3'00 xnx4)x2 + (x1x3 xi'00 xn x4)x2 + g00 x1 = 0 g00 | - Xn n fx1g. 6 5 . +.
+
+ . +. .
(x1xi x4'0 xnxj )x2 + (x1xi xj '0 xnx4)x2 = = (x1 xi x4'0 x2xj )xn (2n) + (x1 xixj '0 x2 x4)xn (2n) = = (x1 xi 'i2 x2 xj )xn(2n) (x1xi 'i4 x2 x4)xn(2n) (x1xi x3'00 xn x4)x2 + (x1 x3xi '00 xnx4 )x2 = = (x1 xi x3'00 x2 x4)xn(2n) + (x1 x3xi'00 x2x4 )xn(2n) = = (x1 xi '4i x2 x4)xn(2n) (x1 x3'34 x2x4 )xn(2n): ! , (x1 xi'i2 x2xj )xn (2n) = (x1 xi'i4 x2x4)xn (2n) g0 xn (x1 xi'4i x2x4)xn (2n) = (x1 x3'34 x2 x4)xn (2n) g00 x1: 9 . +. +-
(x1 xi'ij x2xj )xn (2n) = (x1 x3'34 x2 x4)xn(2n) g0 xn g00 x1: #, + (16) g0 xn 2 K(Hn1) g00 x1 2 K(Hn1(1n)): 9 En1 K(Hn1(2n)) = Kf(x1xi 'ij x2xj )xn(2n)g (x1x3 '34 x2x4)xn (2n) = en1(2n): , K(Hn1(2n)) K(Hn1) K(Hn1(1n)) K(en1 (2n)) K(En1): (21) 9 (20), (21) + (17). N - , En1 5 Pn1.
350
. .
# 5 +
! En1. 7+! | 3 D En1, =
X
+
ij
ij gij xn +
X j
j (xnx4'4j x2 xj )x1 + (xn x3 '34 x2x4 )x1 + (x1x3 '34 xnx4 )x2
ij j 2 K, gij xn 2 Hn1 En1. 7 5 , -
D2
= 0 ! . 3 3 x1 ! xy xn ! zt D
5+ (xyx3 '34 ztxn)x2 = 0 , + !, A 2 D ! +- = 0. N 3 3 xn ! xy x4 ! zt (xyx3 '34 ztx2)x1 = 0: G = 0.
, 3, 3 3 x4 ! xy xj ! zt +-
j (xyxn '4j ztx2)x1 = 0
+ j = 0. , X = ij gij xn = 0 - ! X ij gij = 0 F(C): G ij = 0. N - ,
! , En1 . # !
? +- + 3 +! + - + p. # p = 1
! - . 7 5 , - p > 1 ! n, - p 6 n ; 4 ( D +- + 1 6 p ; 1 6 (n ; 1) ; 4, + En;1p;1), En;1p;1 Pn;1p;1. 7 5 , - Enp Pnp .
-
N , - Enp p > 1 Enp = fenp(1r)(2n)g En;1p;1xn : r !"#$. r
351 (22)
M + , (14) 5 Pnp. 7
K
ij !"#$. i<j
!
Hnp(1i)(2j) K(Enp)
5 , - Enp 5 Pnp. 7 + + 5 En;1p;1 5 Pn;1p;1. ! , j 6= n, i < j, +- (22), +-
K(Hnp (1i)(2j)) = K(Hn;1p;1xn(1i)(2j)) = K(Hn;1p;1(1i)(2j)xn ) K(En;1p;1xn) K(Enp): G ! !, - ! . r < n K(Hnp(1r)(2n)) K(Enp ): 7 +? -
?! (11) (18) (19), +- (x1xi x'0 yx)y = (x1 xix'0 yx)y = 0 (xyy'00 x2x4)x = (xyy'00 x2x4 )x = 0: H
x y, +- - Pnp
(x1xi x4'0 x2xj )x4 + (x1xi xj '0 x2x4 )xn + g0 x2 = 0 (x1xi x3'00 x2x4)xn + (x1x3 xi'00 x2 x4)xn + g00 x1 = 0 g0 | - Xn n fx2g, g00 | - Xn n fx1g. #, 5 = n;1p;1, D
(x1 xi'ij x2xj )n;1p;1xn = (x1 xi'4i x2 x4)n;1p;1xn g0 n;1p;1x2 (x1 xi'4i x2x4)n;1p;1xn = (x1 x3'34 x2x4 )n;1p;1xn g00n;1p;1x1: G (x1 xi'ij x2xj )n;1p;1xn = = (x1 x3'34 x2x4 )n;1p;1xn g0 n;1p;1x2 g00 n;1p;1x1: #+ -
(1r)(2n) +- , - + (16) g0 n;1p;1x2 2 K(Hnp(2n)) g00 n;1p;1x1 2 K(Hnp(1n))
352
. .
5 - (x1 x3'34 x2 x4)n;1p;1xn = enp +-
(x1 xi'ij x2 xj )n;1p;1xn(1r)(2n) 2 2 K(enp (1r)(2n)) K(Hnp(2n)(1r)(2n)) K(Hnp(1n)(1r)(2n)) = = K(enp (1r)(2n)) K(Hnp(1r)) K(Hnp(2r)) K(Enp ): ! , r < n K(Hnp (1r)(2n)) K f(x1xi 'ij x2xj )n;1p;1xn (1r)(2n)g K(Enp ): 4
! +
.
, Enp 5 Pnp. # 5 +
! Enp. @ , - +! p > 1 Enp (22). 7+! f | 3 D (22). X f= i(x1 x3'34 x2 x4)np in + f 0 xn i !"#$. i
f 0 | 3 D En;1p;1.
Dp
f = 0. 3 3 xi ! xy xn ! zt 5 ! i < n 5
i(xyx3 '34 ztx4 )np in = 0: 4
A 2 D ! +- i = 0.
, DSS 3 i +, f = f 0 xn = 0
Dp . 4 !
f 0 = 0
Dp;1. 7 + + 5 En;1p;1 , ! , f 0 , - f, ! . ,
! Enp .
5. $ % D
@ -
Dn. # D +? +5 .
-
353
6 (5, x 3, 5]). M | ( , V W (M) E W " n, U n (E) W.
rt (W) 6 rt (V) + 1: (2)
7. ft 2 Ft =Ft(2) +1 | r, " gt 2 Ft(2)=Ft(2) +1 , (2) T(ft )=Ft(2) +1 = T (gt )=Ft+1: 2 7+! - ft 5 + Ft(2)=Ft(2) +1. T ! ft ! 2, D +
+. M , - 4
! D. , +- , - char K = 3, - ! +
. - ft 3
. + . M , 3 - (xy' zt) t :
?!
(100 ), (12), (13) ft
5 + ft = ft0 t ft0 | - 3 . 9 !
3 ! - , - T(ft0 )=F0(2) = T (em;t0 )=F0(2) em;t0
5 - em;t0 = = x1x5'54 x2x4 m ; t. G
+ Ft(2) +1 T(ft ) = T(ft0 t) = T(em;t0 mt ) = T (emt ):
+? 5
: Nilp = fM Dn j M !.g Wp = fM Dn j 9m > p + 4: M Var(emp )g 0 6 p 6 n:
8. ! M 2 Wp " s, ) p = 0 U s (M) Nilp$ ) p 6= 0 U s (M) Wp;1. 2 7+! M 2 Wp , ! m > p + 4, - M Var(emp ). 7 5 , - np = x0p+1 : : :x0n | 0 6 t 6 n,
T(M) T (emp ) T(emp t ) = T (em+tp+t ) T(em+np+t ):
354
. .
7+! + r > m + n. i > p T (M) T (eri ): 6 !
1 + , - Fi(2)=Fi(2) +1 5 - (xy' zt) i , ', i | , i i, x y z t 2 X. ! , i > p, T (M) T (eri ) Fi(2)=Fi(2) +1(r): (# A ! A(r) - - ! r.) G Fp(2)(r) =
nM ;1 i=p
Fi(2)=Fi(2) +1(r) T (M):
(23)
7+! E 2 U(M), ! +?+ - f r > m + n, - f 2 T (E) n T (M): ) 7 5 , - p = 0 E 2 Nilp: 6 + (23) F (2)(r) T(M): N - , f 2= F (2) = T (M) B D |
. G E \ B 6= B
3 E \ B ! , - - E 2 Nilp: ) 7+! p 6= 0. 7 5 , - E 2 Wp;1. T E ! , D
- . 7+! E ! ,
3 E B. N - , f 2 T (E) T (B) = F0(2): 7 5 , - t > 0 !, f 2 Ft(2). 6 + (28) f 2= Fp(2) , (2) ! , t < p. 7+! f = ft +ft+1 , ft 2 Ft(2)=Ft(2) +1, ft+1 2 Ft+1 . 9 !
5 +, - - (2) (2) gt 2 F (2)=Ft(2) +1 r Ft =Ft+1 - er+2t . +
7 (2) (2) (2) T (f)=Ft(2) +1 = T(ft )=Ft+1 = T(gt )=Ft+1 T(er+2t )=Ft+1:
-
355
7 = p ; t ; 1 5 , - np = x0p+1 : : :x0n | . 7 +- T(E) T (f) T (f) = T (ft ) T (er+2 ) = T (er+2+p;1 ) . . E 2 Wp;1. !
5 ! +? +5 . . Dn n + 2. 2 O , - Nilp W0 W1 : : : Wn;1 Wn = Dn . 9
6
8 rt (W0 ) 6 rt (Nilp) + 1 rt (Wp ) 6 rt (Wp;1) + 1 1 6 p 6 n: G rt (Dn ) 6 n + 2: G ! !, - rt (Dn ) > n + 2: 7+! Fp | 5 .
Dp . 7
! A 2 D, emk 2= T (A), , - Dp 2 (Fp n Fp;1 )0: ! , rt (Dn ) = rt (Fn) > rt (Fn;1) > : : : > rt (F2 ) > rt (C) > rt (B) > 2 C D | 3 ! -
. rt (Dn ) > n +2. +- + . 6. 7- 3 + - .
&
1] . ., . ., . ., . . , . | .: "# , 1978. 2] ( ) *. | " , 1982. 3] ** ,. . -) # .
/ *#- 0* // . . . | 1978. | 3. 17, 4 6. | . 705{726. 4] ** ,. . .
) : : . * ; : 2, <=. - . 0* // . . . | 1980. | 3. 19, 4 3. | . 300{313. 5] - . >. ? * 2 .
) . // . . | 1981. | 3. 115. | . 179{203. 6] ) . >. ? ; * .
/ .
) *##;- : : . // . . . | 1982. | 3. 21, 4 2. | . 170{177.
356
. .
7] @ @. @. ;: .
) : : . // . . . | 1985. | 3. 24, 4 2. | . 226{239. 8] . . #;. ; // . . 0#. | 1991. | 3. 32, 4 6. 9] Drensky V. S., Rashkova T. G. Varieties of metabelian Jordan algebras // Serdica Bulgarical mathematical publications. | 1989. | Vol. 15. | P. 293{301. ' ( ) 1998 .
. .
513.82
:
, NK- , -
.
!" # (nearly-K"ahlerian, NK-) &, ' ' # &. ( '# )' * '+ #. 1. , & & !" ) ) -#. 2. (' + | ) / '! & N ' ' & f0 gg !"
M 2n . , ) N +# ) M 2n + ' , ( ) = 0. 3. (' + N | + !" M 2n , T | " . , ) )' * '!) 3 #: 1) N | + ) M 2n 5 2) N | ) ) M 2n5 3) T 0.
Abstract
M. B. Banaru, On the type number of nearly-cosymplectic hypersurfaces in nearly-Kahlerian manifolds, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 357{364.
Nearly-cosymplectic hypersurfaces in nearly-K"ahlerian manifolds are considered. The following results are obtained. Theorem 1. The type number of a nearly-cosymplectic hypersurface in a nearly-K"ahlerian manifold is at most one. Theorem 2. Let be the second fundamental form of the immersion of a nearly-cosymplectic hypersurface (N f0 gg) in a nearly-K"ahlerian manifold M 2n. Then N is a minimal submanifold of M 2n if and only if ( ) = 0. Theorem 3. Let N be a nearly-cosymplectic hypersurface in a nearly-K"ahlerian manifold M 2n , and let T be its type number. Then the following statements are equivalent: 1) N is a minimal submanifold of M 2n 5 2) N is a totally geodesic submanifold of M 2n 5 3) T 0. , 2002, 8, : 2, . 357{364. !, "# $% &
c 2002
358
. .
- . !" # $ % % ( , % (1] (2]), ! # % , # $ %-% . ( , % % ! ). /#%- , % . % , # , % 0 ! . 1 % 0 !%
2 0, 3 ! , 4 , 5 , . / % ! . ! ! #$ (nearly K$ahlerian, NK-) !%. 6 , ! #$ !% | #%- 0 !% (3]. 8 . ! . 9 ! , - , - S 6 % ! #$ % % (4{7] . = ! # % , - 6- !% ! (8{10].
x
1.
9 0 % (almost Hermitian, AH-) % $ ! M 2n J g = , J |
, g = | . B 0 J g # ! JX JY = X Y X Y (M 2n ): C (M 2n) | ( C 1 ) % M 2n. D ! % $ 0 % % 0 (AH-) !. 5 #% AH - % J g = ! M 2n # ( 2-) F , F (X Y ) = X JY X Y (M 2n ) % ( % (11]) % . B (M 2n J g = ) | 0 !. C p M 2n. B Tp (M 2n) | , !. M 2n p, Jp gp = | 0 , #$ % J g = . E , ( A- ), f
h i
h
i
h
i
2 @
i
2 @
@
f
h ig
h
j f
h ig
2
f
f
h ig
h ig
h ig
359 . !: (p "1 : : : "n "^1 : : : "n^ ) "a | ! Jp , . ! . i, "a^ | ! , . ! . i, "a^ = "a . C i = 1I a = 1 : : : nI a^ = a + n. D Jp p A- : 0 iI n k (Jj ) = 0 iIn In | nI k j = 1 : : : 2n. 4 (12], % g % F A- . 0 I iI 0 n n (gkj ) = I 0 I (Fkj ) = iIn 0 : n B 0 ! ! #$ , (M 2n ) X (J )Y + Y (J )X = 0 X Y | g. B N | 0 ! M 2n, | $ # M 2n. 4 (13,14], N ! . 9 (15], % % % $ ! N M g % 0 !, | , | , M | (1 1), g | N . B 0 . ( ) = 1I M( ) = 0I M = 0I M2 = id + I MX MY = X Y (X )(Y ) X Y (N ): B ! %, (N ): X (M)Y + Y (M)X = 0 X ()Y + Y ()X = 0 X Y 9 , , ! . $ % % (., , (16]). p ;
;
;
;
r
r
2 @
r
f
h
r
x
2.
i
r
h
;
i ;
r
g
2 @
r
2 @
1. .
360
. .
. B N | ! #$ ! M 2n. / %
% % % % , % 0 ! (17]: ; d! = ! ! + B ! ! + B ! ! + 2B n + i ! ! + + 2B~ n 1 B n 1 B n + i ! ! 2 2 ; d! = ! ! + B ! ! + B ! ! + 2Bn i ! ! + 1 1 n ~ + B B i ! ! 2Bn 2 n 2 ; d! = 2Bn ! ! + 2B n ! ! + 2B n 2Bn 2i ! ! + + (B~nn + Bn n + in )! ! + (B~ nn + B n n in )! ! (1) a^ I B~ abc = 2i J^bac^ B~abc = 2i Jbc B abc = B~ abc] Babc = B~abc] I B ab c = 2i J^bac Bab c = 2i Jba^ c^: C a b c = 1 : : : nI a^ = a + nI = 1 : : : n 1. P B abc , Babc B ab c , Bab c . 4 (18]. B 0 ! #$ % (19], B abc + B acb = 0 Babc + Bacb = 0 B ab c = 0 Bab c = 0 (1) # d! = ! ! + B ! ! + i ! ! + 1 n n ~ + 2B B + i ! ! 2 d! = ! ! + B ! ! i ! ! + (2) + 2B~n 1 Bn i ! ! 2 d! = 2Bn ! ! + 2B n ! ! 2i ! ! + (B~nn + in )! ! + (B~ nn in )! ! : 5 (2) ! % p
^
^
^
^
p
; p
;
; p
^
p
;
^
^
^
;
^
p
; p
;
; p
p
;
p
^
p
^
p
^
;
^
;
;
^
^
;
;
;
;
;
f
g
f
g
f
f
g
^
^
^
p
; p
;
;
^
^
^
;
^
p
; p
;
p
^
p
^
;
;
^
^
^
;
;
^
g
361
d! = ! ! + D ! ! + D ! ! d! = ! ! + D ! ! + D ! ! d! = 23 D ! ! 23 D ! ! ^
;
^
^
;
^
^
^
(3)
^
;
^
D = 2i M^^] D = 2i M^ ] ^ D = 23 iM^n D = 32 iMn
, ! , ! N ! %: 1) B = D 2) 3 B~ n + i = D 3) 2B n = 23 D 2 4) = 0 5) n = 0 # (. . .): (4) / D - (4)3: D = 3 B n : 2 B (4)2 : 3 B~ n + i = 3 B n : 2 2 B n = B~ n] = 12 (B~ n B~ n ) = B~ n
, = 0. 5 , (4) # : 1) B = D 2) B n = 32 D 3) = 0 4) = 0 5) n = 0 . . . (5) B , = = n = 0 . ! , ! , % N ! #$ ! M 2n ! ! %. !, % % # ! % N ! #$ ! M 2n ;
;
p
; p
;
;
;p
;p
;
p
;
;
p
;
;
362
0 BB 0 BB (ps ) = B BB0 : : : 0 BB @ 0
. .
1
0. .. 0 C CC 0 C (6) nn 0 : : : 0C CC p s = 1 : : : 2n 1: 0. CC .. 0 A 0 6 , rang 6 1, N , ! . Q (6), $ . 2. N M 2n , ( ) = 0. . / (20]: gps ps = 0 p s = 1 : : : 2n 1: C , N (15] 0 1 0. BB 0 .. I CC BB CC 0 B C ps (g ) = B BB0 : : : 0 10 0 : : : 0CCC B@ C . I .. 0 A 0 I | , . : gps ps = g + g^^^^ + g^ ^ + g^^ + gn n + g^n ^n + gnnnn = nn: B0 gps ps = 0 nn = 0. B , ( ) = 0. 3. ! N | M 2n, t | . #$ % : 1) N | M 2n' 2) N | M 2n' 3) t 0. . B ! N M 2n . nn = ( ) = 0, , (6) ! %. , N ! ! M 2n t = rang 0: ;
;
()
363 6 , $ ! . , 0 # # . 6 , (K$ahlerian, K-) !% . ! #$ !% (3], !% . ! !%. B0 1 # . . 1.1. . 1.2. . 1.3. . E , ! # . 2.1 (2.2, 2.3). ( ( , ) N ( , ) M 2n , ( ) = 0. 3.1 (3.2, 3.3). ! N | ( , ) ( , ) M 2n, t | . #$ % : 1) N | M 2n' 2) N | M 2n' 3) t 0.
1] . . . | .: , 1989. 2] " . # $$ $ $. | .: , 1985. 3] Gray A., Hervella L. M. The sixteen classes of almost Hermitian manifolds and their linear invariants // Ann. Math. Pura Appl. | 1980. | Vol. 123, no. 4. | P. 35{58. 4] + . ,. - $ , .$ 3-$ 01$ 2 6- 0 4512 45 +6 // . -. 7. 1, , . | 1973. | 9 3. | C. 70{75. 5] Gray A. The structure of nearly K:ahler manifolds // Ann. Math. | 1976. | Vol. 223. | P. 233{248. 6] Ejiri N. Totally real submanifolds in a 6-sphere // Proc. Amer. Math. Soc. | 1981. | Vol. 83. | P. 759{763. 7] Sekigawa K. Almost complex submanifolds of a 6-dimensional sphere // Kodai Math. J. | 1983. | Vol. 6. | P. 174{185.
364
. .
8] Banaru M. On six-dimensional Hermitian submanifolds of Cayley algebra satisfying the g-cosymplectic hypersurfaces axiom // Annuaire de l'universite de So
C. .
. . . e-mail: [email protected]
515.142.22
: , , ! " #, $ !" % % .
! ! % & $ Rm, ! $ ' " ( %, % . ) Rm ! "# # X , % " ( "$ % G) '#, m % m- % % %. ) & $ - ! " . $ ! $- ! ( - !) ! " ! "# & %,& '& ( .%. ) % & !& ( - ! # . '/$ ! !) ! " : $! -'& k % , k 6 m + 1, & '1 # & " !& '#, m ; k, & '1 '/$ # & " !& " fk ; 2: : : m ; 1g. 2 $ $" !, -' $ , " $ % .! ! '/$ !, . . ! ! " ! m- % ( & (m + 2) 3 $! k 6 m + 1 1 - k 3 !, & # (m + 1 ; k)- . 4 "# $ $! "# # % % " , !!- ! $ . ) , " ! ( ) ' $" # % % 1930 $ $! . 8 , % & '/$ -'& $& % !" , '/$ -'& 9& $ !" , & . 4" , % & $ !" & 4 -'& $& !" , -'& 9& , !% , -1% ! " % .! ! '/$ ! ( ), !! ! $ !" 4 . : , % & $ !" & 4 '/$ -'& $& -'& 9& $ !" , !% , ;' $$ ; % ( $ ( $ # & $ % ( < 97-01-00174) INTAS ( < 96-0712). , 2002, 8, < 2, . 365{405. c 2002 , "# $% &
366
C. .
-1% ! " % .! ! '/$ ! ( ), !! ! $ !" 4 . : $ ! , # "'-1& # .
Abstract S. A. Bogatyi, Topological Helly theorem, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 365{405.
We give an axiomatic version of topological Helly theorem, from which we derive many corollaries about common intersection (union). Instead of the space Rm we consider an arbitrary normal space X with cohomological dimension not greater than m and with trivial m-dimensional cohomological group. Instead of the convex subsets we consider closed acyclic subsets and instead of the conditions on intersections we impose (obtain) conditions on the values of arbitrary simple Boolean functions. In the extreme cases (only unions or intersections are considered) the conditions have the following form: for any k sets of the given family, for k 6 m + 1, either their common intersection has trivial cohomologies in all dimensions not greater than m ; k, or their common union has trivial cohomologies in all dimensions from fk ; 2: : : m ; 1g. Then it is proved that any subset obtained from sets of the given family with operations of union and intersection is nonempty and acyclic. For any closed covering of m-dimensional sphere the intersection of some m + 2 elements is empty or for some k 6 m + 1 there exist k elements of the covering such that their intersection has non-trivial (m + 1 ; k)-dimensional cohomologies. Our results are valid for arbitrary normalspace of Cnite cohomologicaldimension, but are partially new even in the case of the plane. In particular, we Cll the gap in the topological Helly theorem of 1930 for plane singular cells. If in the family of plane compacta the union of any 2 compacta is path-connected, and the union of any 3 compacta is simply connected, then the total intersection of all compacta of the family is non-empty. It is shown that if in the family of plane simply connected Peano continua the intersection of any 2 continua is connected and the intersection of any 3 continua is non-empty, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogously, if in the family of plane simply connected Peano continua the union of any 2 and any 3 continua is a simply connected Peano continuum, then any compactum obtained from the compacta of the family with the operations of union and intersection is a non-empty simply connected Peano continuum. Analogous statements are true for continua that do not separate the plane.
Rm,
, 1]. # $ % &' ( 1, 2, 3 6), & #, ,
&
, . - 1 , # .
' . - 2 , .
& ' / . - 3 , / # #
367
& ' . 1 6 , #
2
1, 2 3. 1 # # #
, & . 3 , ( )
( 4.3) # 1930 . 1 , #
2] 3,4].
x
1.
1 # X 2 l 6 m Hl Hl : : : Hm; Hm , &' . (MV) A A A A A Hr r > l + 1, +1
1
2
1
1
2
Hr , A1 \ A2 Hr;1 .
1.
(MV) l 6 l0 6 m0 6 m,
Hl : : : Hm
(MV). 2.
Hl : : : Hm
(MV), Hl Hl : : : Hm; Hm , . . r > l+1
Hr Hr; . Hl : : : Hm
0
0
+1
1
1
. 8 & 1 # -
l = m ; 1. 1 # A | # Hm . - . A = A A Hm . 8 & (MV) A = A \ A Hm; . ;2 k
( k ) F (x : : : xk ) = = (xi1 : : : xi ), < 2 .&2 _ ( . ) .&2 ^ ( \), # , # . =
k 2 # !, .&2 ( ) 2 # !. 3&
, , #,
. &'
X, < . . 1
1
k
1.
Hl : : : Hm
(MV).
fAj gnj=1 .
368
C. .
1. fAj gki k 6 m ; l + 1 ! " # k
$, % F(Aj1 : : : Aj ) Hm;r , r | ' " #. 2. fAj gki k 6 m ; l + 1 ! " # k
$ F (Aj1 : : : Aj ) Hm;r , r | ' " #.
. - 1 2, 2 2 ) 1 . 8 # 2 1 ) 2 2 m ; l. 1 m ; l = 0 # # # k = 1 # F(x ) = x . 1 , 1 2 & & # & Aj 2 Hm j. 1 m ; l = 1 # : k = 1 2. 1 ( # 1 # Hm ) Aj Hm . @ fAj1 Aj2 g Aj1 Aj2 Hm , & (MV) Aj1 \ Aj2 Hm; , . A , 1 & 2 / 2 ( 2 ) , , , , & 2. 1 # # m ; l > 2. 8 & (
< Hl : : : Hm ) 2 k 6 m ; l. @ n 6 m ; l, < . B #
fAj gki ' k = m ; l + 1 /& 2& 1 , F(Aj1 : : : Aj ) Hm;r # ,
Aj1 : : : Aj Hm . # <
2 r. 1 r = 0 2 2 F & .&2 ( . ), . . F(Aj1 : : : Aj ) Aj1 : : : Aj , , # & F(Aj1 : : : Aj ) 2 Hm;r Aj1 : : : Aj 2 Hm . 1 # # r > 1. 3 2 F , &' && 2&, . . # F(x : : : xk ) = ((: : :) (: : :)). @ F (Aj1 : : : Aj ) = (B ) \ (B ), 2, &' B B ,
r ; 1 2 . 8 # , & ( 2
#/
) B B Hm;r . - & (MV) F(Aj1 : : : Aj ) Hm;r # , B B ~ : : : xk) = ((: : :) _ (: : :)) Hm;r . C / 2 F(x i
=1
i
=1
k
k
1
1
1
+1
i
=1
k
k
k
k
k
k
1
k
1
1
2
2
1
2
+1
k
1
+1
1
2
369
2 .&2 ( ) r ; 1. 1 & (
#/ .&2 ( ) 2) ~ j1 : : : Aj ) = (B ) (B ) Hm;r # F(A , Aj1 : : : Aj Hm . @ F(x : : : xk) = ((F ) _ (F )), && 2& 2 F F
, <
( , ) & F(x : : : xk) = = (F ) _ : : : _ (F ) _ (F ^ F ). - F (Aj1 : : : Aj ) = B (B \ B ) = = (B B ) \ (B B ). 8 # , & ( 2
#/
) B B B B Hm;r . - & (MV) F(Aj1 : : : Aj ) Hm;r # , (B B ) (B B ) = B B B Hm;r . C / 2 F~ (x : : : xk ) = (F ) _ : : :_ (F) _ (F _ F ) 2 .&2 ( ) r ; 1. 1 & ( ~ j1 : : : Aj ) = B B B 2
#/ ) F(A Hm;r # , Aj1 : : : Aj Hm . A , 1 & / 2 / k , & 1 2 x _ : : : _ xk , . . 1 & / 2 / k ( 2 / 2
#/ /) , , , , & 2. 1
k
2
+1
k
1
1
2
1
2
1
1
+1
1
2
1
+2
1
k
2
3
3
1
1
3
2
+1
k
1
2
1
3
1
1
2
3
+1
1
+1
+2
k
1
2
3
+1
k
1
2.
(MV) Hl % , . . '( ' $ $ $ .
fAj gnj . 1. fAj gki k 6 m ; l + 1 ! " # k
$, % F(Aj1 : : : Aj ) Hm;r , r | ' " #. 2. fAj gki ! " # F (x : : : xk ) ' r 6 m ; l F(Aj1 : : : Aj ) Hm;r . Hl : : : Hm
=1
i
=1
k
i
1
=1
k
. - 1 2, 2 2 ) 1 . 8 # 2 1 ) 2 , . 1. 2 n + (m ; l), 2 / 2 0, . .
fAj gki Aj1 : : : Aj Hm . @ m ; l = 0, Hl 2 . @ n 6 m ; l + 1, 1 2 ,
1. i
=1
k
370
C. .
B Bj = Aj \ An, j = 1 : : : n ; 1. - Bj1 : : :Bj = (Aj1 : : :Aj )\An Hm; Hm (MV),
fBj gnj ; & 1 # Hl : : : Hm; . - &
fBj gnj ; & 2 , # Hl : : : Hm; . nS ; 1 , , , . Bj B j Hm; . 8 & . nS ; Aj Hm . 8 # , (MV), j Sn &' Hm; Hm , , Aj j Hm . 2. 1 # # F(x : : : xk) | # / 2 k
1 6 r 6 m ; l. # <
2 r. 3 2 F , &' && 2&, . . # F(x : : : xk ) = ((: : :) (: : :)). @ F(Aj1 : : : Aj ) = (B ) \ (B ), 2, &' B B ,
r ; 1 2 . 8 # , & B B Hm;r . 8 & (MV) F (Aj1 : : : Aj ) Hm;r # , B B ~ : : : xk) = ((: : :) _ (: : :)) Hm;r . C / 2 F(x r ; 1. 8 # , & ~ j1 : : : Aj ) = (B ) (B ) Hm;r . F(A @ F(x : : : xk) = ((F ) _ (F )), && 2& 2 F F
, <
( , ) & F(x : : : xk) = = (F ) _ : : : _ (F ) _ (F ^ F ). - F (Aj1 : : : Aj ) = B (B \ B ) = = (B B ) \ (B B ). 8 # , & B B B B Hm;r . - & (MV) F(Aj1 : : : Aj ) Hm;r # , (B B ) (B B ) = B B B ~ : : : xk ) = Hm;r . C / 2 F(x = (F ) _ : : : _ (F ) _ (F _ F ) r ; 1. 1 & ~ j1 : : : Aj ) = B B B Hm;r .
F(A E #, H " (MV), . , # , , . 3.
H (MV).
fAj gnj k
1
k
1
=1
1
1
=1
1
1
=1
1
1
=1
1
=1
1
1
1
k
1
2
2
1
2
+1
k
1
+1
2
1
1
k
2
1
+1
1
2
1
2
1
1
1
+1
2
1
1
+2
1
k
2
3
3
2
1
3
+1
k
1
2
1
+1
1
+1
1
2
3
1
+2
k
.
3
1
2
3
+1
=1
371
fAj gki ! " # k
$, % F (Aj1 : : : Aj ) H. 2. fAj gki ( ' ! )
' " # ( . . " #, # '( % ) k
$ F(Aj1 : : : Aj ) H.
1.
i
=1
k
i
=1
k
. - 1 2, 2 2 ) 1 . 8 # 2 1 ) 2 2 n. 1. Tk1 ,
fAj gki Aj H. - H i & (MV), # # H H : : : Hn , H = : : : = Hn = H, (MV). 8
fAj gnj & 1 1, , & 2 1. A , & k /
Hn ;k = H. 2. 1 # # F(x : : : xk) | # 2 k
. B < .& & #& . - \ \ \ F (Aj1 : : : Aj ) = Aj Aj : : : Aj : i
=1
i
=1
1
2
1
=1
+1
1
k
i
i2I1
i2I2
i
i2Iq
i
T B
fBj gqj , Bj = Aj . i2I 8 / 1 , H, Hq . 1 & p / B
A , / 1, H, Hq ;p . A ,
fBj gqj & 1 1. - & 2 1 . ,
, . . F (Aj1 : : : Aj ), Hq , . . H. i
=1
j
+1
=1
k
1.
(MV), Hl % Hm (MV).
fAj gnj . 1. fAj gki k 6 m ; l + 1 ! " # k
$, % F(Aj1 : : : Aj ) Hm;r , r | ' " #.
Hl : : : Hm
=1
i
k
=1
372
C. .
2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) Hm .
. - 1 2, 2 2 ) 1 . 1 ) 2. 8
2
fAj gnj & 1 3 ( 2 0). F 2 3 & . 1. F # 1 / # # /& 2&, & < / . G , 1 (
2) 1 / # 2. H , #
/ # 2 # ' 2 2. 8 $ #% / 2 | , 2 F(x x x ) = (x ^ x ) _ (x ^ x ) _ (x ^ x ): I , 1 1 2 / 3 # & 2&. H 2 < #.
,
/ 1 1 & 2& # < # 1. i
=1
k
=1
1
2
3
1
2
1
3
2
3
3.
(MV).
fAj gnj
fAj gi2I1 ,. . . , fAj gi2I : Tn a) Aj 2 Hl * jT b) Aj 2 Hl , p = 1 : : : q* Hl Hl+1
=1
i
i
q
=1
i2Ip c) Ip1 Ip2 = f1 : :: ng ' $ p 6= p . Sq T A 2 H .
ji l p i2Ip +1
i
1
2
+1
=1
< 2 q.
@ q = 1, , # b). 1 # q > 2. - q \ q \ ;\ Aj = Aj Aj 1
i2Ip
p
=1
q ;\ 1
p
=1
i2Ip
Aj
i
i
p
=1
\ \
i2Iq
i2Ip
i
i2Iq
i
q \ ; \ n Aj = Aj = Aj 2 Hl : 1
i
p
=1
i2Ip Iq
i
1
(MV) / # .
j
=1
373
4.
fAj gnj
fAj gi2I1 ,. . . , fAj gi2I : T a)
Aj ( ) p = 1 : : : q* =1
i
i2Ip
i
q
i
b) p , % Ip1 Ip = f1 : : : ng p 6= p . . Tn 1. Aj 6= ?. j Sq T 2.
Aj ( ) . 1
1
=1
i2Ip
p
=1
i
. J 2 1 ) 2 3. - p & \ \ \ n Aj \ Aj Aj 6= ? i
i2Ip
i2Ip1
i
j
=1
( ) # . . T A S T A . 2 ) 1. B j j i2I 1 p6 p1 i2I - , # , & , . . \ \ \ \ Aj \ Aj = Aj \ Aj = i2I 1 i2 I p6 p1 i2I p6 p1 i2I 1 \ \ n n \ = Aj = Aj = Aj 6= ?: i
i
=
p
i
p
i
=
p
i
p6 p1 i2Ip1 Ip =
x
2.
i
p
=
i
p
p
j
=1
j
=1
1 # G | , # , HK p (XL G) p-
< (, , I {O ), , , 2 G. 3 X # , 5]. X
# # . K ; = fA: A | , X g. H K = fA: A | , , X g. H K m = fA: A | , X, H HK r (XL G) = 0 r = 0 : : : mg. 1
0
374
C. .
H HK & 5, . 6, x 4, 7]. $ % HK ; ' # / #, HK ; (AL G) = 0 # , A . A
HK m # m-! # . $! # # , m-2 m > ;1. 5. '
X ' m > 0
HK ; HK : : : HK m
(MV) HK ; % .
. J , , HKp; HKp p > 0 (MV). @ p = 0, < . 1 # p > 1. 3/ & # # E {3 5, . 6, x 1, 13]: 0
1
1
1
0
1
1
: : : ! HK r; (A ) HK r; (A ) ! K r; (A \ A ) ! HK r (A !H 1
1
1
2
1
1
2
1
K r (A1 ) HK r (A2 ) ! : : :: A2 ) ! H
@ r 6 p, . 8 # , A A p-2 # , A \ A (p ; 1)-2 . H
# , A \ A = ?, A A , 0-2 . 2. $
fAj gnj
X % m > ;1 . 1. fAj gki k 6 m + 2 ! " # k
$, % F(Aj1 : : : Aj ) (m ; r)- #% , r | ' " #. 2. fAj gki ! " # F (x : : : xk) ' r 6 m + 1 F (Aj1 : : : Aj ) (m ; r)- #% .
2 # HK ; HK : : : HK m ( < 5). E # #
#&, < , 6]. 3 X # #
#&, < , 7{9]. , &' #
,
, # X G c-dimG X 6 dimX, dimX
#, < '#& ( 1
2
1
1
1
2
2
2
=1
i
=1
k
i
1
=1
k
1
0
375
, , & ). H
, / 10] m = 2 : : : 1
Ym , c-dim Ym = 1 dimYm = m. 6. X |
c-dimG X 6 m,
A X HK r (AL G) = 0 r > m + 1. + % ,
HK m (MV).
. - , HK r (AL G) = HK r (AL G). 8 , A]X = A c-dimG X = c-dimG X 6 m. 8 &
7, 1] HK r (BL G) = 0 r > m+1 X B. 8 # , HK r (AL G) = 0 r > m + 1 X A. 3. X |
c-dimG X 6 m, $
fAj gnj -
Z p
=1
. 1. fAj gki k 6 m + 2 ! " # k
$, % F(Aj1 : : : Aj ) (m ; r)- #% , r | ' " #. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #% ( % , ). i
=1
k
i
=1
k
1 -
# HK ; HK : : : HK m ( < 5 6). 7. X |
, c-dimG X 6 m HK m (XL G) = 0,
A X HK r (AL G) = 0 r > m. + % ,
HK m; (MV).
. 8 & 6 # # # r = m. 8
O , 7, 1], 11] HK m (XL G) HK m (XL G) ! HK m (AL G) HK m (AL G) < A X, , , . . HK m (AL G) = 0 A X. 4. X |
, c-dimG X 6 m HK m (XL G) = 0, $
fAj gnj 1
0
1
=1
. 1. fAj gki k 6 m + 1 ! " # k
$, % F(Aj1 : : : Aj ) (m ; 1 ; r)- #% , r | ' " #. i
k
=1
376
C. .
2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%.
1 # HK ; HK : : : HK m; ( < 5 7). 2. 1
m-
(m+1)-
Qm , ' @Qm S m , , 3 m+2 #
#/ # m+1 ( ). H , # , #, $ % & . I &' # , , 2 . 1 G ; m- # X, HK r (XL G) = 0 r 6= m HK m (XL G) 6= 0. 8. X |
, c-dimG X 6 m X % G ; m-",
HK m; HK m (MV).
. 1 # A 2 HK m; , . . HK r (AL G)r = 0 r 6 m ; 1. 8 & 6 HK (AL G) r > m+1. - X & & G ; m- , HK m (AL G) = 0, . . A 2 HK m . 5. X |
c-dimG X 6 m. $
fAj gnj i
=1
k
1
0
1
+1
+1
1
1
=1
. 1. fAj gki k 6 m + 1 ! " # k
$, % F(Aj1 : : : Aj ) (m ; 1 ; r)- #% , r | ' " #. ,'(mS
Aj $ (m + 2)
% i G ; m-". 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. i
=1
k
+2
i
=1
i
=1
k
. 8 & 2 1 k ,
S
fAj gki . Aj i (m ; 1)-2 . - . (m + 2) G;m- , . 2 . H.
#/ (m ; 1)-2 . (m+2) , i
i
=1
=1
377
2
,
, &
# #/ m. 8 # , & 7
Sk fAj gki
' k 6 m + 2 . Aj 2 . i 1
3 / # . 3. G , n 6 m+1, ,
#, .
G ; m- . 4. - 1 , , , ,
, ( ) . ,
. # 2 / & . - , , " # # , 2 . i
i
=1
=1
. - % $ $
Rm % , %
% ' m + 1 .
. - & & (m + 1)
, k 6 m + 1 & k . 8 / , 2 , , (m ; k)-2 . 1
4 / # . I , &'
' O {O {E 12] O 13], 1]. 6. k + 1 $ $ (
) (k ; 1)- #% '( k
$ ' %, $ ' % , %
$ # % '( ( ' % % ), #% .
. 8
4 & & 1 2 ( m = k ; 1). 8 # , '
(;1)-2 , . . , 2 . 1
3 ( HK 1 2 ) / # . R 14] ' O {O {E O. # # , # R . H # ,
&' # # R .
378
C. .
4.
Hl : : :Hm
(MV), Hl % ,
Pkr 1 6 k 6 m ; l + 1 0 6 r 6 k ; 1, % Pkk; = Hm ;k Pkr \Hl k; ;r = Hm;r r 6 k ; 2.
fAj gnj . 1. fAj gki k 6 m ; l + 1 ! " # k
$, % F (Aj1 : : : Aj ) Pkr , r | ' " #. 2. fAj gki ! " # F(x : : : xk ) ' r 6 m ; l F(Aj1 : : : Aj ) Hm;r . 1
+1
+
2
=1
i
=1
k
i
=1
1
k
. - Hm;r Pkr , 1 2 2 2 ) 1 . 1 ) 2. 1 2 k,
& 1 2. 1 k = 1 & , & Aj 2 P = Hm . B
fAj gkj , 2 6 k 6 m ; l + 1. 8 &
& F(Aj1 : : : Aj ) 2 Pkr . @ r = k ; 1, < . 1 # r 6 k ; 2. 8 &
fAj gkj & 1 2 # Hl : : : Hl k; . 8 # , & 2 2 F(Aj1 : : : Aj ) Hl k; ;r . 1 & & F (Aj1 : : : Aj ) 2 Pkr \ Hl k; ;r = Hm;r . F 2 2 4 & . 21. $
fAj gnj
X % m > ;1 . 1. fAj gki k 6 m + 2 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. 2. fAj gki ! " # F (x : : : xk) ' r 6 m + 1 F (Aj1 : : : Aj ) (m ; r)- #% .
. H : PpKkr = fA: A | , X, HK (XL G) = 0 p = k ; 2 ; r : : : m ; rg 0 6 r 6 k ; 1 6 m + 1. 1
4 # HK ; HK : : : HK m , 1 2. 1
2 / # . 1 0
=1
k
=1
+
2
k
+
2
+
k
2
=1
i
=1
k
i
1
=1
k
1
0
379
61.
'( k + 1 $ $ (
) (k ; 1)- k $ ' %, $ ' %
.
. 8
3 & & 1 2 ( m = k ; 1). 8 # , '
. (k ; 1)-2 . 1
6 / # . H & 1
62 . m
% $ $ R k 6 m + 1 ' k ' % p 2 fk + 1 : : : m + 1g '( ' $ p (p ; 2)- % , $ ' % , %
$ # % '( ( ' % % ), #% .
8 I {1 15, . 159] Y Rm
(m ; 1)-
# , Rm n Y , . . PKm Rm
. O
, / , / 6 R 14], k = m. 31. X |
c-dimG X 6 m, $
fAj gnj +1
+2 0
2
=1
. 1. fAj gki k 6 m + 2 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%. i
=1
k
i
=1
k
. 8 & 2 1 1 1
3. 1
3 / # . I '#& 4 41. X |
, c-dimnG X 6 m m K H (XL G) = 0, $
fAj gj =1
. 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. i
k
=1
380
C. .
2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%. 51. X |
c-dimG X 6 m, $
fAj gnj i
=1
k
=1
. 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ mS p 2 fk ; 2 ; r : : : m ; 1 ; rg. ,'( Aj $ (m + 2) i
% G ; m-". 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. i
=1
k
+2
i
=1
i
=1
k
. 8 & 2 1 k , 1
S
fAj gki . Aj i (m ; 1)-2 . 1
5 / # . O
,
m-
m + 2 #
# m + 1. H 5 ,
m+2 #
m+1. C &' . i
i
=1
=1
1
9. + '
m- ' m- . 52. $
fAj gnj m- ' X . 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. . m + 2 ' X . 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) #%.
. 8 & 9 X
& m-
& , G ; m- . S , /
& 1 5 , & & . =1
i
k
i
=1
k
1
=1
381
53.
$
fAj gnj m- ' X , % G ; m-", . 1. fAj gki k 6 m + 1 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg. 2. fAj gik ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. =1
i
=1
k
i
=1
k
. 8 & 9 /
& 1 5 , & & . 5. O '#& 2 , , . H 2
. H # . &' # . - # # E {3 # , . 1 , / # , &' 1{4,
. 1 5{9 . H A X #, & x y 2 A X ' / , &' , , A. 1
54.
$ $ $
fAj gnj
m- ' X ,
" " S m , . 1. fAj gik k 6 m+1 ! " # k
$ ' r, % F (Aj1 : : : Aj ) (k ; 2 ; r)- . 2. fAj gki ( ' ! )
' " # k
$ F (Aj1 : : : Aj ) #%. =1
i
=1
i
=1
k
k
. J2 k ( & 2) 1 , k 6 m+1 & k
. C , , 2 . 8 # , 2
# # #. -
O 16] &
, , 2 . 1
2 / # . 6. 3 /
# #& E {3 , -
382
C. .
8 {8 17] ( , # , 2 ). 3 , # , 2 & ,
. 3 #
(
# E {3 ) ' '#& . 7. - / # 2 # , 2 . , # # ' , 2 ' . .
. 7. m- ' X $
fAj gnj fAj gki k 6 m + 1 ! " # k
$ ' r, % F(Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; 1 ; rg m + 2
/ ' X . /
/ ' X .
. 8 & 5 .
2 , , # X. 71. X | #%
, c-dimG X 6 m $
fAj gnj fAj gki k 6 m+2 ! " # k
$ ' r, % F (Aj1 : : : Aj ) p- $ p 2 fk ; 2 ; r : : : m ; rg. =1
i
=1
k
2
=1
i
=1
k
% , ' $
, % ,
.
. 8 & 3 '
, 1
'
. 2 , , . 8. #
X, 5 7 # , 2 Z . H
, 2 &' # , , 7 7 & . 3 3 , 4 5 # # $ % . E # # /# ? 3 6 , 6 5 # # /#
, .
4. 8 &' 2 ' # . 2
2
1
1
1
1
1
2
4
383
x
3. !
) U #, X Y - , Y X # & & . 1 X q- # (q- # | X 2 C q ), S p - p 6 q. U #, X # Y - , x 2 X < Ux ' # Vx , Y Vx Ux # & & . 1 X q- # ( q- # | X 2 LC q ), # S p - p 6 q. " #$. n- Qn n ; B : : : Bn , %
j , j - nT ! , Bj . Bj 6= ?. 1
1
+1
+1
j
=1
. 1 # jT Bj = ?. - x 2 Qn n
+1
=1
' j, dist(x Bj ) > 0. 1 x 2 Qn f(x) 2 Qn &' 2 : 8 9n > < dist(x Bj ) > = x 7! f(x) = > nP : : dist(x Bi ) > j i n -
fBj gj ,
2 f(x) &, . . f(x) 2 @Qn x 2 Qn. & x 2 jn; j- 2 f(x) &, . . f(x) 2 jn; . A , 2 @Qn f . C # 2. 5.
X
$
fAj gm j , % ' k 6 m+1 % ' $ k S m;k - , '( $
mS Aj S m - . % $
+1
+1
=1
=1
+1
=1
1
1
+2
=1
+2
j
. =1
. r = 1 : : : m+2T # Pr | , & Aj . J2 j6 r mS n 6 m f : Qmn ! Aj , Qmn =
+2
+1
j
+1
=1
384
C. .
n-
(m+1)-
, (q ; 1)-
hj : : : jq i ( , T &' q /)
& f(hj : : : jq i) Aj . j 6 j1:::j T J , n = 0 f(hri) = Pr 2 Aj . 1
1
=
q
j6 r =
B # q-
hj : : : jq jq i. 1 & 2 , fT ,
& f(@ hj : : : jq jq i) Aj . j 6 j1 :::j +1 - 2 q-
(q ; 1)-
, # 2 # . 1 ' , , # q-
. mS 1 # # f : Qmm ! Aj | , # j . 8 & # mS F : Qm ! Aj . E j Bj = F ; (Aj ) & Qm . 1 ,
Bj #, & / j. 8 # ,
W ,
& . -
& Aj . 6.
X
$ $
fAj gm j , % ' k 6 m + 2 '( ' $ k S k; - . % $ 1
1
+1
+1
=
q
+2
+1
=1
+2
+1
=1
1
+1
+2
=1
2
.
. j = 1 : : : m + 2 # Pj | , & Aj . J2 mS n 6 m + 1 f : Qmn ! Aj , j (qS; 1)-
hj : : : jq i
& f(hj : : : jq i) Aj . +2
+1
=1
1
1
j j1 :::jq =
J , n = 0 f(hj i) = Pj 2 Aj . B # q-
hj : : : jq jq i. 1 & 2 , fS ,
& f(@ hj : : : jq jq i) Aj . j j1 :::j +1 - 2 q-
(q ; 1)-
, # 2 # . 1 ' , , # q-
. 1
1
+1
+1
=
q
385
mS 1 # # f : Qm ! Aj | , # j . E Bj = f ; (Aj ) & Qm , &'
&
W 18, . 215]. 8 mT mT # , Aj = f Bj 6= ?. +2
+1
=1
1
+1
+2
j
+2
j
=1
7.
=1
Rm
fAj gnj , n > m, % % ' $ k k 6 n+1 S n;k - . % $ . +2
=1
. 1 # 5, n
S f : Qnn ! Aj , (q ; 1)-
j T A. hj : : : jq i
& f(hj : : : jq i) j j 6 j1 :::j 1 F : Qn ! Rm. 8
B 19] ' & &' , F( ) \ F( ) 6= ?. nT - F ( ) \ F ( ) = f( ) \ f( ) Aj , j . 8.
$ % r 6 m . 1. 0 m 6 2r. 2.
% n > m + 1 $
fAj gnj Rm, % ' n + 1 ; r 6 k 6 n + 1 % ' $ k S n;k - , % $ +2
+1
=1
1
1
=
q
+1
1
2
1
2
+2
1
2
1
2
=1
+2
=1
. 3.
% n > m + 1 $ $ r- $ fAj gnj Rm, % ' n + 1 ; r 6 k 6 n + 1 % ' $ k (r ; 1)- , % $ . +2
=1
. J 2 1 ) 2. 1 #n
S 5, f : Qnr ! Aj , j (qT; 1)-
hj : : : jq i
& f(hj : : : jq i) Aj . j 6 j1:::j 8 '<
3 O {; 20, 1.3 n = n+1, s = r+1, j = 2, p = 2, m = n;m;1, l = m], 21, N = n+1, s = r+1, j = 2, q = 2, k = n ; m ; 1, M = Rm] ' & &' nT , f( ) \ f( ) 6= ?. - f( ) \ f( ) Aj , j . J 2 2 ) 3 . +2
+1
=1
1
=
1
q
+2
1
2
1
2
1
2
=1
386
C. .
J 2 3 ) 1. @
m > 2r + 1,
C< {1 , Qnr Rm. U #, , Qnr
. (F #, Rm n + 2 ' .) 3 Aj # <
r-
(r ; 1)- , Aj = Qnr \ jn Qnr, jn | , n-
# Qn , j- / . 1 & 8 > k = n + 2L Qn ;k n + 1 ; r 6 k 6 n + 1L :Qn ;k 1 6 k 6 n + 1 ; r: r 1 & & 3, '
. 9. 3 Aj 5 C 22]. - 6 . @ ,
# < # , , # $ %
W ,
W &. 23,24] # 5 7 '#&
# < . 1 # U 25] 3 26]. I # / . J
'#& , 2] < #
. +1
+1
+1
+1
k
+1
+1
k
+1
% . - % $
Rm % , % % ' m + 1 .
< 2 n
. 1 n 6 m + 1 < &. 1 # n > m + 2. 1 & & k
k 6 n ; 1 , , S n; ;k - . 8
7 n
. 10. - 7 # # ( # ) , < #. 1 / 7 , n # 2 #/
. 3
# , . C
( ) /# 7. 1
2
387
&' ( ) ,
# # '#& # . 11. @ # # # $ % (MV), # . & ( ), 1, 2 3 # #, 1 / 2 , . H , # & (
. ). H
& # q- . O 27], ( ) q- ( ) (q ; 1)- # # ( ) q- #. 3 27, ] <
$ % & . J 2# . C/ 10 11 & ' . - 5{8 &
$ % # q- , # . I 5{8
&
/ S q - 28] , # < # $% | . ) 1 # X 2 l 6 m, Kl Kl : : : Km; Km, &' . (MV ) A 2 Kr r > l + 1 A A A A Kr; , A \ A Kr; . 9.
k Kl : : : Km S
(MV ). , k '( Aj ' $ i k
fAj gmj ;l Tk Kl k; , k % Aj ' $ k +1
1
I
1
1
2
1
2
2
1
1
I
i
=1
+1
=1
+
1
Kl .
i
i
=1
< 2 m ; l.
1 m ; l = 1 < & (MV ). B
fBj gmj ;l , Bj = Aj Am;l , j = 1 : : : m ; l. 1
& 9 # Kl : : : Km . 8 # , & mT;l mT;l mT;l Aj Am;l Bj = (Aj Am;l ) = I
=1
+1
+1
j
=1
j
+1
=1
j
+1
=1
388
C. .
Kl+1 .
8 & & (MV ) '
mT;l m;Tl Aj \ Am;l = Aj Kl . j j X # # . K ; = fA: A | , X g. K K = fA: A | , ( , ) K X g. K m = fA: A | , ( , ) K X, HK m (XL G) = 0g. 10. '
X ' m > 0
KK ; KK : : : KK m
(MV ).
. J , , KK p; KK p p > 0 (MV ). @ p = 0, < . 1 # p > 1. 3/ & # # E {3 5, . 6, x 1, 13]: : : : ! HK p; (A ) HK p; (A ) ! HK p; (A \ A ) ! HK p (A A ) ! : : :: O # . 8 # , A \ A (p ; 1)-
. H
# , A \ A = ?, A A . I
+1
+1
=1
=1
1
0
1
0
I
1
I
1
1
1
1
2
1
1
2
1
2
2
1
2
1
2
22n.
$ $
fAj gj
X k 6 n '(Sk
Aj ' $ k
(k ; 2)- i Tn , ' % Aj $
. =1
i
=1
j
=1
9 # -
KK ; KK : : : KK n; ( < 10). 32. X |
, c-dimG Xn 6 m
$ $
fAj gj Sk k 6 m+2 '( Aj ' $ k
i Tn (k ; 2)- , ' % Aj $
1
0
2
=1
i
=1
.
42.
j
=1
X |
, c-dimG X 6 m, = 0
$ $ Sk
fAj gnj k 6 m + 1 '( Aj ' $ k -
HK m (XL G)
=1
i
i
=1
389
(k ; 2)- , ' % Tn Aj $
. j
=1
55.
X | m- ' ,
$ $
fAj gnj Sk k 6 m + 1 '( Aj ' $ k
i (k ; 2)- m + 2
Tn / ' , ' % Aj $
j
. =1
i
=1
=1
1
9 # &' , , &' # . H # &'
l ;1 m < & ' 2 ' . K l KK l+1 : : : KK l+n;1 K
23.
$
Sk
X k 6 n '( Aj i ' $ k
(l + k ; 1)- , Tn ' % Aj $
l-j
. 33. X |
,n c-dimG X 6 m
$
fAj gj Sk k 6 m + 1 ; l '( Aj ' $ k
i Tn (l + k ; 1)- , % Aj $
j l- . 43. X |
, c-dimG X 6 nm, m K H (XL G) = 0
$
fAj gj Sk k 6 m ; l '( Aj ' $ k
i Tn (l+k ; 1)- , % Aj $
j l- . 56. X | m- n ' ,
$
fAj gj k 6 m ; l Sk '( Aj ' $ k
(l+k ; 1)- i (m + 1 ; l)
/ fAj gnj
i
=1
=1
=1
=1
i
=1
=1
=1
i
=1
=1
=1
i
=1
390
C. .
Tn
' , % Aj $
l- . j =1
. k # # , , & ' , - , , &
| , 2 , - , , . , , '#& # / 2. 1 , & #
/ ( )
/ ( . )
, / 2
. 3/
# , (MV) ((MV )). H #
# & 2 #
. E < . K X
( A . ) K\ = A: A A A A K : A \ A K C #, HK ; \ = HK HK \ = U = fA: A | , g. - HK U & , #
, < Uk = HK k \ Uk = Uk \ . 12. H , # # K, K\ (MV ). 3 2 # , # ,
, # # HK ; , HK (MV). ) I # . . E # # &' # , #
, # '#& & $ # %. / # . m-
S m # # . S; = fA: A | , X g. S = fA: A | , ( , ) X g. I
1
2
1
1
2
1
0
0
1
+1
I
1
1
0
0
2
391
Sm = fA: A | , ( , ) X, Hm (XL G) = 0g. # # E {3 : : : : Hr; (A ) Hr; (A ) Hr; (A \ A ) Hr (A A ) : : : A , # # S; S : : : Sm (MV ). - 9 & 34. X | m- '
$ $
fAj gnj k 6 m+2 Sk '( Aj ' $ k
(k ; 2)- i Tn A $
, ' % j 1
1
1
2
1
1
2
1
2
1
0
I
=1
i
=1
j
.
=1
72.m
% fAj gnj m- " S k 6 m + 2 fAj gki , % HK m ;k (Aj1 \ : : : \ Aj ) 6= 0. =1
+1
i
k
=1
. 1 # & k 6 m +m2 &
;k
' k
HK (Aj1 \ : : : \ Aj ) = 0. B Bj = S m n Aj . 1 I {1 & j : : : jk ( Sk Sk m k 6 m + 2)
Hk; Bj = Hk; (S n Aj ) = i m Tk Tk i m ; k K = Hk; S n Aj = H Aj = 0. 8 # , i i & 3 Bj , . . .
Aj < & . F. . O 3. E. U 48], # ( # ) . ,
. 13. 1 ' '< 1930 1]. 3
# 2 Rm, & m+1 / , & k / , k 6 m, 2 . - #, '
2 . 2 # '#& 3 ( # ). , 2 . , 1, II II0]. 3 , , 29, 1]. 3 , , O 13, 2.1] R 14,
1] & &' # 1, IV.2]. # fAji gki=1
+1
k
1
2
i
=1
2
i
=1
4
+1
i
=1
2
i
=1
392
C. .
,
# #
1935 I 30]. 3 2 ,
# ( 7 n = m = 2). H
10, 7 # # / # # $ / % . J
, # 3.2 3.3 x 4 . 3 1956 E # # # S . E 31] 3.2 3.3 (3). U
# & '< &' . 3 1970 32] # ( k (m ; k)-2 ), # & # # E {3 . # , # # E {3 # , & , , / # # X. 8 5 5 ( # #
)
. - # 33{ 35]. C '< , ' 2& '#& O 16]. 3 2] 4] U / Y, 5 , # , 2 , 1 # ( &' ) #
. B U / , , , Y 36] . 14. 3 / & , , 2 .
. A # . - &' , # # . -
1, V]. 3 # 32] <
. F /
9 U , 3 &
'<
# . 2
2
3
2
393
15. @ # Hl : : : Hm Kl : : : Km , Kl Hl ,... , Km Hm , # #, , ' # Hl : : : Hm , & ' , ' # Kl : : : Km . # , H- &
fAj gnj ,
F(Aj1 : : : Aj ) 2 2 Hm;r ( 1 1, 2 3), & '
F (Aj1 : : : Aj ) 2 Hm;r ( 2 1, 2 3)
. - # # # & . E ,
. k
=1
k
x
4. #
B & ' 1{4, 29, 31, 37{39]. # # # 39]. 3 2 (
1, 29]) & . H < , &' . B 29,39] & 31,37]. B 37] - , 31] 1]. I # # , # 1- . 8 # , 1,2,4,29, 31,39] #
, & & . - # (
13
#/ ), , ( /# ) ' . H
, # < ' 2 ( 4.1 # 2 ). 4.1.
H & #
4 . A # . 1.2 1.3 # , #
# # ,
&', , # 8. A 40] O. O 41]. B & . ' 1.1. A R 1
2
.
394
C. .
1. A | ' . 2. A | #% . 3. A | , ' -
' . 4. HK (AL Z) = HK (AL Z) = 0. 5. HK (AL Z) = HK (AL Z) = 0. 6. A " ' (A 2 FAR). 7. - A ! %. 8. - A 1- . 0
1
0
1
A # / . F # , / # O. U 42].
' 1.2.
$ ' $
A A R , . 1. % A \ A . 2. ,'( A A ' . 3. % A \ A ' . 1
2
2
1
2
1
2
1
' 1.3.
A A A 1
2
3
2
/$ ' $
R , % $ , 2
. 1. % A \ A \ A . 2. - (A A ) \ A ( (A A ) \ A , (A A ) \ A ) . 3. - (A \ A ) A ( (A \ A ) A , (A \ A ) A ) . 4. - (A \ A ) (A \ A ) (A \ A ) . 5. ,'( A A A ' . 6. % A \ A \ A ' . 1
2
3
1
2
3
2
3
1
3
1
2
1
2
3
2
3
1
3
1
2
1
2
1
1
2
2
2
3
3
1
3
3
A & 1, 5 7 , 1 4 1.1. 1 X = R 4 4 ( < 1.2 1.3) & &' . 2
1
2
4.1.
' $ . 1. + $ 1.2, $ 1.3. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) ' . 4:11. $ fAj gnj R '( ' $ $ , '( ' $ /$ fAj gnj=1 R2
i
=1
k
2
=1
395
' , % , ' . 4.2.
R # 0- 0- & ## % ( , 0L 1]). & # & 0- .
' 2.1.
A
. 1. A | . 2. A | ' . 3. A | .
R 2
A # / . # # O. O 41].
' 2.2.
$ $ $
A A R . 1. % A \ A . 2. ,'( A A ' . 3. ,'( A A
. 4. % A \ A
. # , # , & # . 8 &'
/ 2
, & . " 1. 1L 2] 0L 1] B B , % fig 0L 1] Bi , i = 1 2. C B \ B , % C \ ( 1L 2] f0g) 6= ? C \ ( 1L 2] f1g) 6= ?.
. 1. B # 0L 3] 0L 1] B~ = ( 0L 1] 0L 1]) B B~ = B ( 2L 3] 0L 1]). 1 , B \ B = B~ \ B~
(B \ B ) \ ( 1L 2] f0g) (B \ B ) \ ( 1L 2] f1g). @ , , ' 2 f : B \ B ! f0 1g, f((B \ B ) \ ( 1L 2] fig)) = i 41, . 5, x 46, IV, 5]. 2& f 0L 3] f0L 1g
. - < 2
f : 0L 3] 0L 1] ! 0L 1]. - # 2& f : 0L 3] 0L 1] ! 0L 3], & ( ~ 3g x 2 B~ f (x) = minf15 dist(x B ) + 15 maxf15 ; 15 dist(x B~ ) 0g x 2 B~ : 1
2
2
1
2
1
2
1
2
1
1
2
2
1
2
1
1
1
1
2
1
2
2
2
1
2
2
1
2
1
2
2
1
1
2
1
2
1
396
C. .
- x 2 B~ \ B~ = B \ B 15 dist(x B~ ) + 15 = 15 = 15 ; 15 dist(x B~ ), 2 f . O
, f ; (15) = B \ B . 1 & F : 0L 3] 0L 1] ! 0L 3] 0L 1], F ; (f15g f05g) = f ; (f15g) \ f ; (f05g) = (B \ B ) \ f ; (f05g) = ?. O 2 # # f15g f05g 2 < 2& # 2, #. 2. - B \ B
&' (B \ B ) \ ( 1L 2] f0g) (B \ B ) \ ( 1L 2] f1g), 41, . 5, x 48, IX, 1] ' C B \ B ,
C \ ( 1L 2] f0g) C \ ( 1L 2] f1g). 1 C 41, . 5, x 48, VIII, 1]. 1
2
1
1
2
2
1
1
1
1
1
1
2
1
1
1
1
2
1
2
2
2
1
2
1
2
1
2
10. '( $ $ $
, $ % ( ).
. 1 # x x 2 A \ A . 8 x x A , A , . . fi : 0L 1] ! Ai , i = 1 2, fi (j) = xj , j = 0 1. H f : @( 1L 2] 0L 1]) ! A A f(i t) = fi (t), f(s j) = xj ,
F : 1L 2] 0L 1] ! A A . - B = F ; (A ) B = F ; (A ) & , B f1g 0L 1] B f2g 0L 1]. 1
1 ' C B \ B , C \ ( 1L 2] f0g) 6= ? C \ ( 1L 2] f1g) 6= ?. - F(C) | , A \ A , x x . 0
1
1
1
2
0
1
1
1
2
1
2
2
2
1
1
2
1
1
2
1
1
2
0
2
1
11. '( $
, $ %
.
. 1 # x x 2 A \ A Ux | # # x A A . 8' # Vx x, Vx & Ux . 1 # # Wx | # x A A , & Wx \ Ai & Vx \ Ai . 3 # < #& y 2 Wx \ (A \ A ). 8 x y Vx \ A Vx \ A . 1 & f : @( 1L 2] 0L 1]) ! Vx . A F : 1L 2] 0L 1] ! Ux . - B = F ; (A ) B = F ; (A ) & , B f1g 0L 1] B f2g 0L 1]. 1
1 ' C B \ B , 0
1
1
1
2
2
1
2
1
1
2
2
1
1
2
2
1
2
1
1
2
1
397
C \ ( 1L 2] f0g) 6= ? C \ ( 1L 2] f1g) 6= ?. - F (C) | , , &' x y Ux \ (A \ A ). 8 41, . 6, x 49, I, 2] A \ A # . 8 #
. 1
E {E{E 41, . 6, x 50, II, 1] A \ A 1 . 1
1
1
2
2
2
8. % $ $ $
, %
.
. @ A A | 1 A \ A , & 1.2 . A A &' # . H. # # , , 43] . A A & . 1
A A 11 1.2 / # . - # 2.2 # / < 8 2.1 1.2. F 2.2 , # # K ; , HK P = fA # 1 g H (MV) ,
, P & (MV). - 1 1 &' . 1
1
1
2
1
2
1
1
2
2
2
0
1
1
' 2.3.
A A A 1
2
3
/$ $ $
R , % $ , 2
. 1. % A \ A \ A . 2. - (A A ) \ A ( (A A ) \ A , (A A ) \ A ) . 3. - (A \ A ) A ( (A \ A ) A (A \ A ) A ) . 4. - (A \ A ) (A \ A ) (A \ A ) . 5. ,'( A A A ' . 6. % A \ A \ A
. 4.2. $ $
fAj gnj R . 1. + $ 2.2, $ 2.3. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj )
. 1
2
3
1
2
3
2
3
1
2
3
2
3
1
2
2
1
1
2
2
3
3
1
1
1
3
3
2
=1
i
k
=1
3
1
2
3
1
2
398
C. .
4.3.
' 3.1. X . 1. - X . 2. 1 $ $ : H (XL Z) = 0. 3. , X ,
%
X . 1
. J 2 1 ) 2
1 ,
#
. J 2 2 ) 3. 1 # S | , , ' X. 1 # D | , & . @ D 6 X, # S X, , H (S ) = Z H (X). J 2 3 ) 1 / 44]. ' 3.2. $ $ $ A A 1
1
1
1
1
1
. 1. % A \ A . 2. % A \ A . 1
2
1
2
2
. J 2 1 ) 2. 1 # S | , , ' A \ A , # D | , & . - A A , D A D A . - D A \ A , , ' & A \ A . H 2 . 16. - # # E {3 , # # # 3.2 3. H. A A . 8 &'
# 2 3 ) 1 , # X = ;2L 2] 0L 1] # . &' A A ,
. A = f(x y) 2 X : y > sin 1=x x > 0 ; 1 6 y x = 0g A = f(x y) 2 X : y 6 sin 1=x x > 0 y 6 1 x = 0g. ( $ 1. J 3.2 3. 17. 1. I # 39], , # # , 1
1
1
1
2
2
1
2
1
2
1
2
1
1
2
2
2
399
. H
16 , 3 2 , #& # & #. 2. @ #& # & #, &' 2 #
# & & 1, x 2]. - , & ,
16 , 4 2 # # (
, # , / 4.2). 3. 1 39] 2 U 38]
# ,
# & 2 & ( 4.1 4.2). ' 3.3. /$ $ $ A , A A , % $ , 1
2
3
. 1. % A 2. % A
1
1
\ A2 \ A3 \ A2 \ A3
. .
( $ 2. F 3.3 .
1 # H | , 0- , H | , 1- . 8
16 # # H H (MV). A
, H & (MV). R #
, &', # # HK H (MV). 3 , , $ % 3.3 . B & / , , , # 2
2 ,
# # . C < , #
#/ . H
, # & I. S , &
. 0
1
0
1
1
0
1
4.3.
fAj gnj=1
$ $
R '( ' $ $ , '( 2
' $ /$ , % , '
.
. 8
S 45] q > 2 q
S - #. A ,
& 6, , ' , '
. 18. , 29], # {E , , #
400
C. .
. O
, 29]
' $ % $ &' # %. ( $ 3. @
fAj gnj R & ,
& '& , ( . )
. ( $ 4. @
fAj gnj R . & & < , .
. 1 < # , &' 2. ' 3:31. /$ $ $ A , A A , $ % A \ A A \ A , '( A A , . 1. % A \ A \ A . 2. - (A A ) \ A . 3. - (A A ) \ A . 4. % A \ A \ A .
. J 2 1 ) 2 4 (U) (A A ) \ A = (A \ A ) (A \ A ): J 2 2 ) 3 3.2. J 2 3 ) 4 (U) 10. ' 3:32. /$ $ $ A , A A , % $ , 2
=1
2
=1
1
3
1
1
3
2
2
3
2
1
2
1
3
2
1
3
2
1
3
2
1
3
2
3
1
3
2
3
1
2
3
. 1. % A 2. % A
\ A2 \ A3 \ A2 \ A3 1 1
0- . .
. 8 & 3.2 A \ A . - & 3.2 (A \ A ) \ A . B # &' & # / # # ( # , ' ) . 1
2
1
2
3
4.4.
1 #, & & & 46,47]. E # &'
' 4.1.
X .
401
1.
X % . 2. ' $ $ % $ % x y 2 X
X , %/
, ( ) x y] # $ % $.
' 4.2. $ % $ $
A A . 1. % A \ A . 2. % A \ A % . 3. ,'( A A % . 1
2
1
2
1
2
1
2
. 1 ) 2. 1 # x y 2 A \ A | . 1 # x y] x y]0 | A \ A 2 x y. - A A , & 4.1 & . 8 & 4.1 , , A \ A . 2 ) 3. 1 # S A A | , . E S n A , ,
< .& . 1 # J | , , x y | < 2. H , x y | , A \ A . - A \ A , < ' x y], &' x y. 3 A x y] x J y, &' x y. 3 ) 1. 1 # x y 2 A \ A | . J A A , , # x y] x y] A A . J A A , x y] = x y] A \ A , / & 4.1 # . F 4.2 , # # K ; , H E = fA: A | H g (MV) ,
, E & (MV). - 1 1 &' . 1
2
1
2
1
1
1
1
1
1
2
2
1
1
1
2
2
2
1
2
1
2
1
2
1
1
1
1
2
0
1
2
2
2
1
1
' 4.3. /$ % $ $
A A A X , % $ , . 1. % A \ A \ A . 2.
(A A ) \ A ( (A A ) \ A , (A A ) \ A ) . 3.
(A \ A ) A ( (A \ A ) A , (A \ A ) A ) . 4.
(A \ A ) (A \ A ) (A \ A ) . 1
2
3
1
2
3
1
2
3
2
3
1
3
1
2
1
2
3
2
3
1
3
1
2
1
2
2
3
3
1
402
C. .
5. ,'( A A A %
. 6. % A \ A \ A %
. 4.4. % $ $
fAj gnj X . 1. +
$ 4.2,
$ 4.3. 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) %
. 1
2
1
2
3
3
=1
i
=1
k
3 ,
# . &' X, # < , $ '% ( ) . 4.5.
J # 4.1 4.4 , E \ HK & (MV). O
, # , #, # # HK ; , HK E \ HK (MV), E \ HK & (MV). A , &' . 8 / < , 4.2 # # , . . # D . 1
1
1
0
1
1
1
1
1
4.6. # $
3/ #
# , (MV). 1 <
, # # , (MV ), # # . I
4.6.
' $ . 1. ,'( $ $ /$ . 2. fAj gki ( ' ! )
' " # k
$ F(Aj1 : : : Aj ) ' .
fAj gnj=1 R2
i
k
=1
403
. - &' #
, 1 2. 1
9 ( <
12) # HK ; , HK , U , &
, & <
. 1
4.1 / # . O U & (MV). U
, 4.6 . J
, # A | , # &' < L Aj ( j = 1 2 3) | , , 2 (j ; 1)=3 6 ' 6 2 j=3. 3
& , & < . H'
& . U # 4 $ ' %
. F ,
, '
. & . =. R. #2 ( ) , ( ). I # & # 8. E. I
, F. . O , @. _. 8 , 3. 3. ; =. R. , # ,
, . 1
0
4
$
1] Helly E. U ber Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten // Monatsh. Math. Phys. | 1930. | B. 37. | S. 281{302. 2] ., ., !. "# $ % &. | '.: ' , 1968. 3] # *&+, - !. . # *#& .# * &. | / .. "#%# #. &. # * &. ". 19 (1*#. , *2 , , !131"1 /3 4445). | '.: !131"1, 1981. | 4. 209{274. 4] Eckho8 J. Helly, Radon, and Carath9eodory type theorems // Handbook of Convex Geometry / Ed. by P. M. Gruber, J. M. Wills. | Elsevier Sc. Publ., 1993. | P. 389{448. 5] 4%; <. / . =+,& *#%# #. &. | '.: ' , 1971. 6] Alexandro8 P. On the dimension of normal spaces // Proc. Roy. Soc. London. | 1947. | Vol. 189. | P. 11{39. 7] >; #? !. 1. # # #. =+,& *# & > #+* // @+%2 *. ,. | 1968. | ". 23, ?D%. 5. | 4. 3{49. 8] Okuyama A. On cohomological dimension for paracompact spaces. I, II // Proc. Japan Acad. | 1962. | Vol. 38. | P. 489{494F 655{659.
404
C. .
9] 4, &,# K. . N #%Q ,#.# # #. =+,#- > #+* // /3 4445. | 1965. | ". 161, T 3. | 4. 538{539. 10] X ,#? /. 1. # # #. =+,& *# & > #+* // @+%2 *. ,. | 1988. | ". 43, ?D%. 4. | 4. 11{55. 11] Cohen H. A cohomological deYnition of dimension for locally compact Hausdor8 spaces // Duke Math. Journ. | 1954. | Vol. 21, no. 2. | P. 209{224. 12] Knaster B., Kuratowski K., Mazurkiewicz S. Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe // Fund. Math. | 1929. | B. 14. | S. 132{137. 13] Klee V. On certain intersection properties of convex sets // Canad. J. Math. | 1951. | Vol. 3. | P. 272{275. 14] Levi F. W. Eine Erganzung zum Hellyschen Satze // Arch. Math. | 1953. | B. 4. | S. 222{224. 15] ]# ,# /. "., ],+ . . + .# #*#% =+,#- *#%# #. . | '.: 3,, 1989. 16] Kleinjohann N. Remark on the Helly number for strongly convex sets on Riemannian manifolds // Manusc. Math. | 1981. | Vol. 34. | P. 27{29. 17] '++ @. "# & .# # #. - ,#.# # #. -. | '.: ' , 1981. 18] / ,+Q#? _. 4., _+D,#? . /. !?Q ? *# > #+* . | '.: 3,, 1973. 19] Bajm9oczy E. G., B9ar9any I. On common generalization of Borsuk's and Radon's theorem // Acta Math. Acad. Sc. Hungar. | 1979. | Vol. 34. | P. 347{350. 20] Sarkaria K. S. A generalized Van Kampen{Flores theorem // Proc. Amer. Math. Soc. | 1991. | Vol. 111, no. 2. | P. 559{565. 21] !# #? ,#? /. q. *# ! %{] #+ // '*. > *, . | 1996. | ". 59, T 5. | 4. 663{670. 22] 3 ,###? q. . # #*#% =+, - #. *# D ${ +|+*?#? ,?> %#Q? }D2 *#=, // 4 . *. }. | 1994. | ". 35, T 3. | 4. 644{646. 23] Dugundji J. A duality property of nerves // Fund. Math. | 1966. | Vol. 59, no. 2. | P. 213{219. 24] Dugundji J. Maps into nerves of closed coverings // Annali Scuola Norm. Super. Pisa. Sc. Ys. e matem. | 1967. | Vol. 21, no. 2. | P. 121{136. 25] Borsuk K. On the imbedding of systems of compacta in simplicial complexes // Fund. Math. | 1948. | Vol. 35. | P. 217{234. 26] Weil A. Sur les th9eor~emes de de Rham // Comm. Math. Helv. | 1952. | Vol. 26. | P. 119{145. 27] /.? 4. '., #.*D- 4. /. N +, -,2 ,#*#D2 * %#? %#+*+*? // !+* , '#+,. -*. 4. 1, '* * ,, 2 ,. | 1994. | T 6. | 4. 19{23. 28] #.*D- 4. N *# ! *# + ? ,*.# .# #*#% - #Q#- %# #+, // Fund. Math. | 1974. | Vol. 84, no. 3. | P. 209{228. 29] Haddock A. G. Some theorems related to a theorem of E. Helly // Proc. Amer. Math. Soc. | 1963. | Vol. 14, no. 4. | P. 636{637. 30] Aleksandrov P. S., Hopf H. Topologie. | Berlin: Springer, 1935F reprinted: New York: Chelsea, 1965. 31] Moln9ar J. U ber eine Verallgemeinerung auf Kugelache eines Topologischen Satzes von Helly // Acta Math. Acad. Sci. Hungar. | 1956. | B. 7, N. 1. | S. 107{108.
405
32] Debrunner H. E. Helly type theorems derived from basic singular homology // Amer. Math. Monthly. | 1970. | Vol. 77, no. 4. | P. 375{380. 33] Soos G. Ausdehnung des Hellyschen Satzes auf den Fall vollstandiger konvexer Flachen // Publ. Math. Debrecen. | 1952. | B. 2. | S. 244{247. 34] Moln9ar J. U ber eine Vermutung von G. Hajos // Acta Math. Acad. Sci. Hungar. | 1957. | B. 8. | S. 311{314. 35] Moln9ar J. U ber eine U bertragung des Hellyschen Satzes in spharische Raume // Acta Math. Acad. Sci. Hungar. | 1957. | B. 8. | S. 315{318. 36] Jussila O. K. On a theorem of Floyd // Abstracts Int. Cong. Math. Stockholm. | 1962. | P. 133. 37] Moln9ar J. A k9etdimenzi9os topol9ogikus Helly-t9etelr}ol // Math. Lapok. | 1957. | ". 8, No 1{2. | 4. 108{114. 38] Breen M. A Helly-type theorem for simple polygons // Geom. Dedicata. | 1996. | Vol. 60. | P. 283{288. 39] Toranzos F. A. Intersections of compact simply connected planar sets // Bull. Soc. Royale Sci. Li~ege. | 1997. | Vol. 66, no. 5. | P. 349{351. 40] Eilenberg S. Transformations continues en circonf9erence et la topologie du plan // Fund. Math. | 1936. | Vol. 26, no. 1. | P. 61{112. 41] *#?+, - . "#%# #. &. ". 2. | '.: ' . 1969. 42] #+, . "# & X-%#?. | '.: ' , 1976. 43] #+, . "# & *,*#?. | '.: ' , 1971. 44] Moore R. L. Foundations of Point Set Theory. | Amer. Math. Soc. Colloq. Publ., Vol. 13. | Providence, RI: Amer. Math. Soc., 1962. 45] Zastrow A. Plane compacta are aspherical. | Preprint Ruhr-Universitat Bochum, 1995. 46] Hagopian C. L. Uniquely arcwise connected plane continua have the Yxed-point property // Trans. Amer. Math. Soc. | 1979. | Vol. 248, no. 1. | P. 85{104. 47] Hagopian C. L. The Yxed-point property for simply connected plane continua // Trans. Amer. Math. Soc. | 1996. | Vol. 348, no. 11. | P. 4525{4548. 48] D,#? !. '. N > ,*D2 %#,D* &2 #.##> - >% , #? // /3 4445. | 1973. | ". 211, T 5. | 4. 1027{1030. ' ( 1999 .
. .
512.541
:
,
, , , K- , !", #.
$%
A & ', (%& ab 2 A, & H(a) 6 H(b) (H(a), H(b) | & & ,& a b) -
& A, -' a b. . " %
A H-
',
S
& A S = fa 2 A j H(a) > M g, M | ! !- ,, ' ( & & 1. 0 &
H-
% &
. 1 #& # & (, H-
'
', - ( &
& % !
&
&, (- H-
. 0
!" &
& % &
.
Abstract S. Ya. Grinshpon, Fully invariant subgroups of Abelian groups and full transitivity, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 407{473. An Abelian group A is said to be fully transitive if for any elements ab 2 A with H(a) 6 H(b) (H(a), H(b) are the height-matrices of elements a and b) there exists an endomorphism of A sending a into b. We say that an Abelian group A is H-group if any fully invariant subgroup S of A has the form S = fa 2 A j H(a) > M g, where M is some ! !-matrix with ordinal numbers and symbol 1 for entries. The description of fully transitive groups and H-groups in various classes of Abelian groups is obtained. The results of this paper show that every H-group is a fully
transitive group, but there are fully transitive torsion free groups and mixed groups, which are not H-groups. The full description of fully invariant subgroups and their lattice for fully transitive groups in various classes of Abelian groups is obtained.
1% & ' 7 1889 : 97-01-00795 <
' 7 - & ! 1 ' ' 8 , : 96-15-96095 >9 %?.
, 2002, 8, : 2, . 407{473. c 2002 , !" #$ %
408
. .
. p- ! , # ! (%1, x 18]( %2, x 67]). - p- . / (- p- , 0 1! a b, H(a) 6 H(b), 2 1!3 ! 1 , 2 a b, H(a), H(b) | 1! a b %2, . 11]). # ! 5 5 p- (%1, . 61]( %2, . 17]) 5, 5 - p- (%1, . 58]( %2, . 10{11]). 8 %3] : , 5 p- . ; 5 p- A, ! 5- 1!3 ! E(A) p! A %4]( 5 ! - p-, . < ! 2 - p- G, 02 , p! G %5]. 8 1 , - p- , 5! - , . = p- ! 5 (!. %6, x 1]( %7, x 1]). 8 %8] 0 . 1 02!. > , ! ! 1! , ! 0 0 , 0 ! !. 8 , ! 5 , 2 , , , , ! ! !, . ? ! ! ! 0 -, 5 - G , ! G(v) = fg 2 G j (g) > vg, (g) | 1! g, v | 55, 2 - - 5 ! 1. (> ! !, ! ! 5 - ! ! %8, !! 3.3].) 8 - 3 5 0 .
409
8 - 5 , 0 1! a b, (a) 6 (b), 2 1!3 ! 1 , 2 a b. 5 !5 %8{10] ( 1 ! ! 5 ! ( , 0 , 5 ! ). 8 ! ! 0, , : ! , 5, C %11], 5 %12, 13]. ; 0 2 5 (!., !, %12{21]). 8 %8,20,21] 5 5 - . ! !, 0 ! 5 5 , 5 3! - 0 5
, 50. 5 5 - , 50 5 , 0, 2 , - ! , ! ! (%8, x 4]( %22]). 8 %23] ! . / 5 , 3! ! ! 1! ! ! p- . ( ! ! ) 5 %5,23{26]. 8 2 5 0 , ! 2 0 5 . = 5 3 . 8 ! 3 ! 0 3- , 02 00 , 0 ! , 2 02 1! p-, 1! ! - 1! ! . = !250 'L -, 5 'L -. ! 1 5 ! p- ! !
, ! 0 3! - 0 ! . 8 ! 3 H - - A, S ,
410
. .
! S = fa 2 A j H (a) > M g, M | ! - , 1! ! 0 ! 1. ; 5 3 , H - , , 5 ! , 0 H -. 8 ! 3 , K- , K- . D 3 ! , 02 ! !. E 3 2 ! 0 5 ! . ? 3 2 0 . >5 0 , 02 K-!! !! ! ( ! ! 0 ! !! ! ) . 8 . 8 , , K-! !! , ! ! ! 02 : 1) 5 ( 2) 5 ( 3) C- , 3 5 !( 4) ( 5) 03 5 ( 6) IT -. F , K-! !! ! ( , 0 ! ) . , K-! !! , ! 5 Qp -! p, . 8 , 0 K-! !! , p- p. K-! !! 5 , 0 ! . 8 ! 3 5 5 H - , 3 5 0 ( 5 3 0 ! 5 5 ). 8 , , 5 0 H - ( , 0 | H - ). H - (-) K-! !! Ai (i ! I), Ai 5 , ! 5 Qp -! p. ? H - 202 ! . ,
411
!2 , ! 5 Qp -!, H -. H - 02 : p- 5 ( K-! !! ( ! , ! K-! !!, ! ! 2. ! !,
5 5 3 , A K-! !! Ai (i 2 I) A | H - ( ), Ai | H - ( ). K-! !! H - ( ) H - ( ). 8 ! 3 ( 5 02 H - ! ) . 8 , 50 K-! !! Ai (i 2 I), Ai . K-! !! : ! ( 5( , ! 5 Qp -! p. ! , 5 ! 2 5 , %2] %27] ( ! !) %28] %29] ( !).
x
1. 'L-
5 A | , L | 'L : A ! L | 3- 02 ! ! : 1) 'L (a) > 'L (a) a 2 A 0 1!3 ! A( 2) 'L (a + b) > 'L (a) ^ 'L (b) 0 a b 2 A( 3) 'L (0) > l l 2 L. 5 l 2 L A(l) = fa 2 A j 'L (a) > lg. 1.1. A(l) |
A, l1 = infL f'L(a) j a 2 A(l)g, A(l) = A(l1). . H a b 2 A(l), ;b 2 A(l), 2 2 E(A), g = ;g 1! g 2 A. 1! 'L (;b) > 'L (b) > l. ;!! 'L (a ; b) > 'L (a) ^ 'L (;b) > 'L (a) ^ 'L (b) > l. > , A(l) | A. H 2 E(A), 'L (a) > 'L (a) > l, 1! a 2 A(l). ; , A(l) | A. 5 2 l1 = inf f'L (a) j a 2 A(l)g. , A(l) A(l1 ). L ! 0 . ? a 2 A(l) 'L (a) > l,
412
. .
l1 > l. 5 a 2 A(l1 ), 'L (a) > l1 > l, , a 2 A(l). ; , A(l1 ) A(l) 1! A(l) = A(l1 ). 5 A L 3- 'L : A ! L, 02 ! 1){3), 5 ! A0 ! A 2 1! mA 2 L, mA = inf f'L (a0 ) j a0 2 A0 g. 5 ML (A) = fmA j A0 A A0 6= ?g. L 1.2. ML(A) | . . !, ! ! ML (A) ! 0 00 5. 5 M0 ML (A), M0 6= ?. > ! M0 M0 = fmAi gi2I , S i 2 I Ai | ! ! A. 5 A0 = Ai . ? f'L (a0 ) j a0 2 A0 g = i2I S = f'L (ai ) j ai 2 Ai g, ! mAi = inf f'L (ai ) j ai 2 Ai g 2 L i2I m = inf f'L (a0 ) j a0 2 A0 g. ? m = inf fmAi gi2I (%28, c. 10, ! 4]), L L m 2 ML (A), m = Minf(A)fmAi gi2I . ML (A) ! 5 1! 'L (0). 1! ML (A) | (%28, c. 37, ! 1]). 1.3. < ! 'L-, S ! S = A(l), l 2 L. 1.4. I A ! 'L , 0 1! a, b, 'L (a) 6 'L (b), 2 2 E(A), a = b. 1.5. 'L-
'L. . 5 A | 'L- , !, A 5 3- 'L . ? 2 0 1! a b 2 A, 'L (a) 6 'L (b), 2 E(A) a 6= b. J ! ! S = fa j 2 E(A)g. ;!! b 2= S. S | A, 1! S = A(l) l 2 L. ? 'L (a) > l, , 'L (b) > l. 0 b 2 A(l), A(l) = S, b 2 S. . < ! !, ! P P 0 0 ( ), 2 ' ! P ! P 0, a 6 b ! ! 5 , '(a) 6 '(b) ('(a) > '(b)). 1.6. 5 L L0 0 | ' | ( ) L L , 5 ' | L L0, 0 1! a b 2 L '(a _ b) = '(a) _ '(b) '(a ^ b) = '(a) ^ '(b) ('(a _ b) = '(a) ^ '(b) '(a ^ b) = '(a) _ '(b)). ;5 , 0 L 02 1! a b 2 L 1 : ) a 6 b( ) a _ b = b( ) a ^ b = a, !, ' | !3 ! ( !3 !) ! 0
0
0
L
413
L0 . ? , , 5 ! 4 %28] (. 53{54), !, L L0 !3 ( !3) , !; 5 ; B = inf 'B! ;sup B = infB 'B
L !! ' sup B = sup 'B ' inf (' L L L L L L ; 0 ' inf B = sup 'B). ; , L L 02 1 L L : 1) ' | !3 ! ( !3 !) L L0 ( 2) ' | !3 ! ( !3 !) ! L ! L0 ( 3) ' | !3 ! ( !3 !) L 0 L0. ! K(A) A. D 0 A K(A) | . ! ! 'L -. 1.7. A | 'L- ! "! A0 "! A #" mA 2 L, $ mA = inf f'L (a0 ) j a0 2 A0 g. %
L A
" ML(A). . J ! ! : K(A) ! ML(A), S A S = ms (ms = inf f'L (s) j s 2 S g). !, | . L 5 l 2 ML (A). ? 2 ! A0 ! A, l = inf f'L (a0 ) j a0 2 A0 g. J ! ! 1! A L 1 a01 + : : : + k a0k , k 2 N, a01 : : : a0k 2 A0 , 1 : : : k 2 E(A). ! ! 1! S 0 . S 0 | A. F 1) 2) 3- 'L , !, l = inf f'L (s0 ) j s0 2 S 0 g. ; , S 0 = l, , | 0C L . !, | C . 5 S1 , S2 | A, S1 = mS1 , S2 = mS2 mS1 = mS2 . F , A | 'L - , 5 !! 1.1, ! S1 = A(mS1 ), S2 = A(mS2 ). 0 S1 = S2 . H S1 S2 | A, S1 S2 5 , S1 > S2 ( !, S1 = A( S1)( S2 = A( S2 )). > , K(A) ML (A) !3. 1.8. 5 = ;1, | K(A) ML(A), ! 5 ! 1.7 02 ! !3 !!. ; 5 ! , l 2 ML (A), l = A(l). 1.9. L ! ! ! , ! ! 0 00 5. L
0
0
0
0
0
0
414
. .
H L | , 'L - A, 3- 'L , 02 ! 1){3), ! 5 0 ML (A) = fmA j A0 A A0 6= ?g, mA = inf f'L (a0 ) j a0 2 A0 g. L ; ! 1.7 $ 1.10. A | 'L- , L | ! . % K(A) ML(A)
" . 1.11. 5 L | . M! 5, ! M L ! ! L, ! M1 M , u 2 M u > inf M1 2 M1 ! M2 , L u > inf M2 . L 5 A | , L | 'L : A ! L | 3- ! 1){3). ! 'L (A) ! f'L (a) j a 2 Ag. 1.12. & ' A 'L- $, L | ! , "! 'L(A) ' $ " $ 0
0
" L.
. 5 A | 'L - , M = 'L (A), M1 M, jM1j > @0, u 2 M u > inf M1. =2 1! b 2 A, 'L (b) = u. L J ! ! S A, 0 ! ! 1! ! a 2 A, 2 m 2 M1 , 'L (a) > m. , S | A. ? A | 'L - , S = A(v), v 2 L, ! !! 1.1 v ! 5 inf f'L (s) j s 2 S g. ;!! inf f'L (s) j s 2 S g = inf M1. ? L L L S = A(inf M ). ? ' 1 L (b) > inf M1, b 2 S 1! b = b1 + : : : + bn , L L 'L (bi ) > vi , vi 2 M1 , i = 1 : : : n. 0 ! u = 'L (b) > > inf f'L (bi )gi=1n > inf fvigi=1n. > , M1 5 ! L L M2 = fvi gi=1n, u > inf M2 . L H G | , (g p), g 2 G, p | , 55 (hpG (g) hpG (pg) : : : hpG(pn g) : : :) = = HpG (g), 2 , !5 p- G, ! 1 (hpG (g) | 2 p- 1! g G). E 55 p- " g %2, c. 235]. H , G ! ! p 5, G p 2 p- p- 5. H G | p- , g 2 G, 55 2 p- 1! g (hG (g) hG (pg) : : : hG(pn g) : : :) " g H G(g) ( H(g)) %2, c. 9].
415
J ! ! p-. 5 A | p- . ! HA ! 02 5 , !5 A, ! 1, , 1 5 0 ( = (0 1 : : : n : : :) 2 HA n 6= 1, n < minfn+1 g, | A( n = 1, n+1 = 1). < ! HA ! ! !: (0 1 : : : n : : :) 6 (0 1 : : : n : : :) 5 , k 2 N0 (N0 | ! - - 5 ) k 6 k . 5 1 HA ( HA 5 5 1! (1 1 : :: 1 : : :), HA ). 5 'HA | 3- , 02 A HA , 'HA (a) = H(a) a 2 A. ; 2 p- %27, c. 182] : 'HA (a) > 'HA (a) a 2 A 0 2 E(A)( 'HA (a + b) > > 'HA (a) ^ 'HA (b) 0 a b 2 A( 'HA (0) > 5 2 HA . 1! ! ! - ! ! ! 5 5 5 , 1! 3. N , 55 = (0 1 : : : n : : :) HA ! k k+1, k +1 < k+1 %1, c. 58]. 55 2 HA U - ( A), , 2 ! k k+1 k - F5! {# A fk (A) (fk (A) = r(pk A%p]=pk+1 A%p])) %1, c. 59]. H A0 | ! ! A, A ! inf fH(a0) j a0 2 A0 g ( 1! ! 0 A), HA 5 MH (A) = fA j A0 A A0 6= ?g. 1.13. ( )'$ '$ p- A MH (A) "! " U - $ A. . 5 2 MH (A). =2 ! A0 ! A, = A = inf fH(a0) j a0 2 A0 g. # HA H(a0 ) U- 550 %1, . 59]. ? inf 5 ! 5 HA ( ), A | U- 55. 5 5, , U- 550, = = (0 1 : : : n : : :) N1 = fn 2 N0 j ! n n+1 5 g. D n 2 N1 2 1! an pn+1, ! ! n 1, 5 H(an) = = (n ; n n ; n + 1 : : : n 1 1 : ::) %2, !! 65.3, . 9]. 5 A0 = fangn2N1. ? !! A = Hinf fH(an) j n 2 N1g = A = (0 1 : : : n : : :) = . ! ! U- 5 A
U(A). U(A) . < ! !, - p- , 0 10
0
0
0
0
416
. .
! a b, H(a) 6 H(b), 2 1!3 ! ' 2 E(A), '(a) = b %2, . 11]. # p- 0 , 5 p- 5 p-. 5 H(A) = fH(a) j a 2 Ag. H 2 HA , ! A() 020 A: A() = fa 2 A j H(a) > g. A() | A. 1.14. J- 0 p- A ! H -, S ! S = A(), 2 HA . 8 p- A H- ( ! 0 S 1 A() ! 5 2 ! U(A)) %2, ! 67.1, . 17]. ; 1.5 ( 5 L = HA , 'L = 'HA ) , . ;5 !! 1.13, ! 1.7 ! 1.12, ! 5 . 1.15. A | ' p- . 1. A H - $ , A | -
. 2. ) , " A, " 1 U - ) A , = HinfAfH(a0) j a0 2 A0g, A0 | "! "! A. 3. & A | H - ,
# $
" U(A). 4. & A | H - , "! H(A) ' $ " $ " HA.
! 5 ! 0 . ! 3! 5 5 5 , ! ! ! , 5 ! %8]. 5 X | ! , 2 5 v = = (v(1) v(2) : : : v(n) : : :), v(i) | - - 5 ! 1 (i 2 N). ? 5 ! 5 . 8 ! X ! ! , ! v 6 w 5 , i 2 N v(i) 6 w(i) . 5 1 X . 5 O = fp1 p2 : : : pn : : :g | ! , ! . H A | , a 2 A, A (a) " a A | 1 v = (v(1) v(2) : : : v(n) : : :), v(i) 5 pi - hApi (a) 1! a A ( 0 1! 5
417
55 (1 1 : : : 1 : : :)). H , A 5, A 5 ( ! pi ). 5 'X | 3- , 02 A X, 'X (a) = (a) a 2 A. ; pi - : 'X (a) > 'X (a) a 2 A 0 2 E(A)( 'X (a + b) > 'X (a) ^ 'X (b) 0 a b 2 A( 'X (0) > v v 2 X. 1! ! ! - ! ! ! 5 5 5 1 3 , L = X 'L = 'X . 1.16. J- A , 0 1! a b, (a) 6 (b), 2 1!3 ! ' 2 E(A), '(a) = b. 8 ! ! 0 ! , C %11], 5 %12,13] . 5 A | . H v 2 X, !
A(v) 020 A: A(v) = fa 2 A j (a) > vg. A(v) | A. 1.17. J- 0 A ! -, S ! S = A(v), v 2 X. ; 1.5 1.18. - $
$.
- %8], %20] %21]. 1.19. 5 A | . ! XA ! ! X, 2 v = (v(1) v(2) : : : v(n) : : :), v(k) = 1, pk A = A (pk 2 O). XA 5 , - !, ! X. D ! 3- 'X : A ! X !! Im 'X XA . 1!, 5 2 5 , 5 3- 'L , 'L - , ! L 5 XA . H A | , ! A0 ! A vA ! inf f(a0 ) j a0 2 A0 g, 5 X MX (A) = fvA j A0 A A0 6= ?g, (A) = f(a) j a 2 Ag. ? X | , !! 1.2, 1.10 ! 1.12 1.20. A | ' ' . % 1) MX (A) | * 2) A | - , K(A) MX (A)
" * 0
0
418
. .
3) A | - , "! (A) ' $ " $ " X.
J ! ! 5 . < ! !, v w 0 " 5 , ! fn 2 N j v(n) 6= w(n)g , ! v(n) 6= w(n), v(n) 6= 1 w(n) 6= 1. # 1 ! . H 1! a A t, , " a t ( 02 ! !: t(a) = t tA (a) = t). I , 1! !0 t %2, . 130{131]. L 5, 1! A !0 3 t, ! 5, A | t 5 1 : t(A) = t. ? t ! 5 pk - (pk 2 O), v 2 t !! v(k) = 1. 5 t | . J ! ! v, 02 02 ! !: ) v = (v(1) v(2) : : : v(n) : : :) 6 w w 2 t( ) v(k) = 1, t pk - !. ! ! , 2 , 02 ! ), ), , ! 0 5 ! 1, F(t). 1.21. ( )'$ $ A t MX (A) F (t). . 5 v 2 MX(A). =2 ! A0 ! A, v = vA = inf f(a0) j a0 2 A0 g. H A0 = f0g, X v(k) = 1 k 2 N, 1! v 2 F (t). 5 A0 6= f0g. ;!! v 6 (a0 ) a0 2 A0 , a0 6= 0, (a0 ) 2 t. ? 0 1! a0 2 A0 ((a0 ))(k) = 1 k, pk A = A, v(k) = 1 k, t pk - !. > , v 2 F(t). 5 v 2 F(t). H v(k) = 1 k 2 N, , A0 = f0g, ! v = vA . 5 2 k 2 N, v(k) 6= 1. H v 2 t, ! A 1! a, (a) = v. ? , A0 = fag, ! v = vA . 5 v 2= t. > 3 ! 0 w 2 t, 0 v < w, 5 M = fk 2 N j pk A 6= Ag. D k 2 M 2 1! ak 2 A, ((ak ))(k) = v(k) ((ak ))(r) = w(r) 0 r 6= k (r 2 N). 5 A0 = fak gk2M . ;!! v = vA = inf f(ak ) j k 2 M g = v. X 1.22. ? MX (A) | , F (t) | . 5 F1 | ! F (t). J ! ! w1 w2 2 X, w1(k) = inf fv(k) j v 2 F1 g, 0
0
0
0
419
w2(k) = supfv(k) j v 2 F1g (1 inf sup ! 0 ! N f0 1g, ! ! !). F F(t), !! w1 2 F (t), F1 | ! , w2 2 F (t). 0 w1 = Finf(t) F1, jF1j < @0 , w2 = sup F1. ; , F (t) inf 0 ! F(t) , ! F(t) , sup F(t) . !, ! F1 F(A) sup F1 ! 5 w2 . F (t) 5 t | , ! , 2 5 . D n 2 N ! ! 02 vn : vn(n) = 1 vn(m) = 0 m 6= n. D 0 n 2 N !! vn 2 t, 1! vn 2 F (t). 5 F1 = fvn gn2N. M sup ! F1, ! w2 = (1 1 : : : 1 : : :). ? 2 w 2 t, w2 6 w, w2 2= F (t). ; , w2 6= sup F1. 8 ! !! ! !! sup F1 = (1 1 : : : 1 : : :). F (t) F (t) F , - A - 5 , A | %8, 3.14], !! 1.21, 1.10, ! 1.8 1.20, ! 5 . 1.23. A | t. 1. A - $ , A | -
. 2. MX (A) = F (t). 3. & A | , : v 7;! A(v), v 2 F(t),
" " F(t) K(A). 4. & A | , "! (A) ' $ " $ " X. x
2. K-
8 ! 3 ! 5 p- , H - -. 8 1! 3 ! 2 ! 1 5 , ! ! K- , . 5 A | . ! HA ! !
420
. .
0 10 B 20 B (ij ) = B ::: B @n0
1
11 : : : 1n : : : 21 : : : 2n : : :C C : : : : : : : : : : : :C C n1 : : : nn : : :A ::: ::: ::: ::: ::: 020 55 , !5 pi- A (i | ! , pi | i- ) ! 1, , 1 5 0 ( (ij ) 2 HA ij 6= 1, ij < minfij +1 i g, i | pi - A( ij = 1, ij +1 = 1). < ! HA ! !: (ij ) 6 (ij ) 5 , 0 i 2 N, j 2 N0 ij 6 ij . 5 1 HA ( ). = ! 1!! a 2 A H A (a), ! 02 ! - : 0 h (a) h (p a) : : : h (pna) : : :1 p1 p1 1 p1 1 B hp2 (a) hp2 (p2 a) : : : hp2 (pn2 a) : : :C B C H A (a) = B ::: ::: ::: ::: : : :C B @hpn (a) hpn (pna) : : : hpn (pnna) : : :CA ::: ::: ::: ::: ::: n 2 N, pn | n- . n- ! - H A (a) pn- ! 1! a A %2, . 235]. H , A 5, A ! - H A (a) 5. Q, H A (a) 2 HA 1! a 2 A. D pn - A n- ! - H A (a) | 1 1! a, ! 1 0 1! 5 3! - 1! a. D
A 0 pk !! hpk (pnk a) = hpk (a) + n ( 2 pk - pk - ), 1! - ! - H A (a), 02 A (a), , ! - H A (a). 5 'HA | 3- , 02 A HA , 'HA (a) = H A (a) 0 a 2 A. ;!! 'HA (a) > 'HA (a) a 2 A 0 2 E(A)( 'HA (a + b) > 'HA (a) ^ 'HA (b) 0 a b 2 A( 'HA (0) > M ! - M 2 HA . 1! ! ! - ! ! ! 5 5 5 x 1, L 3- 'L , L = HA 'L = 'HA . 2.1. J- 0 A ! , 0 1! a b, H (a) 6 H (b), 2 1!3 ! ' 2 E(A), '(a) = b.
421
> ! !, p- ! 2.1 5 , ! %2] (. 11), ! 2.1 5 , ! 1.16. H A | M 2 HA , ! A(M) 020 A: A(M) = fa 2 A j H (a) > M g. A(M) | A. 2.2. J- 0 A ! H -, S ! S = A(M), M 2 HA . > ! !, p- H - 5 , H- ( ! 1.14), H - 5 , | - ( ! 1.17). ; 1.5
2.3. H - $ -
$.
H A | , ! A0 ! A MA ! H inf fH (a0 ) j a0 2 A0 g, 5 MHA = fMA j A A0 A A0 6= ?g, H (A) = fH (a) j a 2 Ag. ? HA | , !! 1.2, 1.10 ! 1.12 2.4. A | ' . % 1) MHA (A) | * 2) A | H - , K(A) MHA (A)
" * 3) A | H - , "! H (A) ' $ " $ 0
" HA.
0
8 ! , ! 5 !. 2.5. =! fAigi2I - ! , (Ai1 Ai2 ), i1 i2 2 I (i1 ! 5 i2 ) : , a 2 Ai1 , b 2 Ai2 H (a) 6 H (b) , 2 ' 2 Hom(Ai1 Ai2 ) ! '(a) = b. F , 0 A B 0 1! a 2 A, b 2 B H (a) 6 H (b) (a) 6 (b) 5, 0 p- A B 0 1! a 2 A, b 2 B H (a) 6 H (b) H(a) 6 H(b) (H(a) H(b) | p- 1! a b) 5, ! ! !, 5 2.5 ! fAi gi2I , Ai | p-, !-
422
. .
5 1! H (a) 6 H (b) (a) 6 (b) H(a) 6 H(b) . ! %8] ( ! 5 ! ), 5 | %23]. 8 %8] ! ! , !, 0 ! ! 5 ! ! !. J ! ! 2 ! ! , 2 . 2.6. I p- A ! , 1! 1 5 ! ! A. , 5 ! 5 . J- p- , 1! ! 5 ! ! 1 ( , 0 - p- ), ! 5 . E , - p- ! ! - ! ! %2, . 100]. 8 p- 1! (- 5 p- ) ! 5 , p- ! !! - , 5 %2, . 122]. # ! 5 - , %30] (p- - , | 5 , ! 1! ! !! !! 1 , 02 ! 5 , !5 ). 8 %31] p- G C- ( | 5 ), G=pG | 5 0 < . ; 5 %30] %31] , , 3 5! !, C - ! 5 .
2.7. "$ " ".
. 5 fAigi2I | ! ! 5 , 5 a 2 Ai1 , b 2 Ai2 (i1 i2 2 I), H (a) 6 H (b). < ! 2 , ! 5, Ai1 , Ai2 | ! p. ? Ai1 Ai2 ! 5 , Ai1 = A0i1 A00i1 Ai2 = A0i2 A00i2 , A0i1 , A0i2 | 5 a 2 A0i1 , b 2 A0i2 . 5 B = A0i1 A0i2 ( ! ! !! ). B | 5 %2, . 108]. !
B1 B2 A0i1 A0i2 B,
423
5 10 | - Ai1 ! ! A0i1 , 2B | - B A0i2 . F , ! ! 1 , !! H B (B1 a) 6 H B (B2 b). ? B %3] 2 1!3 ! ' 1 , '(B1 a) = B2 b. 5 = 2B 'B1 10 . ;!! 2 Hom(Ai1 Ai2 ) ( !, A0i2 | Ai2 ) a = b. 8 %32] IT -. IT - p- , !3 5 . 2.8. "$ IT - . . 5 fAigi2I | ! IT-, 5 a1 2 Ai1 , a2 2 Ai2 (i1 i2 2 I)( Ai1 , Ai2 | ! p H(a1) 6 H(a2 ). H B1 B2 | 5 , ! ! 0 Ai1 Ai2 , Ai1 Ai2 | B1 B2 . D 5, 0 !! p (Ai1 Ai2 ) = = p Ai1 p Ai2 = (Ai1 \ p B1 ) (Ai2 \ p B2 ) = (Ai1 Ai2 ) \ (p B1 p B2 ) = = (Ai1 Ai2 )\p (B1 B2 ). N Ai1 Ai2 , 5 B1 B2 , %32]. 5 1 , 2 | Ai1 Ai2 Ai1 Ai2 , 2 | - Ai1 Ai2 Ai2 . ;!! H Ai1 Ai2 (1 a1) 6 H Ai1 Ai2 (2 a2 ), 1! 2 ' 2 E(Ai1 Ai2 ), '(1 a1 ) = 2 a2. ? = 2 '1 !!3 !! Ai1 Ai2 , 2 ! 1! a1 a2 . 02
2.9. +"$
'
" , ! p "$ fAipgi2I (Aip | p- " Ai).
fAi gi2I
. < !5. 5 !
fAi gi2I , p | , 5 a 2 Ai1 p , b 2 Ai2 p , i1 i2 2 I, H Ai1 p (a) 6 H Ai2 p (b). ? H Ai1 (a) 6 H Ai2 (b) 2 ' 2 Hom(Ai1 Ai2 ), 'a = b. 5 | ' Ai1 p . ? Im Ai2 p ( 5 2 Hom(Ai1 Ai2 )) a = b. D 5. 5 p ! fAip gi2I , 5 a 2 Ai1 , b 2 Ai2 H (a) 6 H (b). =2
P
P
! O1 , a = pa b = p0 b, p2 1 p2 1 p p0 | - Ai1 Ai2 ! ! Ai1 p Ai2 p 1 !0 !! ! !, ! p a p 2 O1 (
! p0 b ! 5 ! ). ? H Ai1 p (p a) 6 H Ai2 p (p0 b) -
424
. .
p 2 O1 , , 2 0 !!3 ! 'p , 'p (p a) = p0 b (p 2 O1 ). J ! ! !!3 ! ' 2 Hom(Ai1 Ai2 ), 'jAi1 p = 'p , p 2 O1 , 'jAi1p = 0, p 2 O n O1. ;!! 'a = b. > , ! fAi gi2I . 2.10. fAigi2I | "$
'
, !
" " " $ $ IT - $. % "$ fAigi2I .
. 5 p | 5 . J ! ! ! fAip gi2I . !, fAip gi2I | ! . 5 a1 2 Ai1 p , a2 2 Ai2 p , i1 i2 2 I, H Ai1 p (a1 ) 6 H Ai2 p (a2). H Ai1 p Ai2 p 0 ! 5 ! IT - ! , !! 2.7 2.8 2 ' 2 Hom(Ai1 p Ai2p ), 'a1 = a2. 5 Ai1 p | IT- , Ai2 p | ! 5 ( , Ai1 p | ! 5 , Ai2 p | IT - , ! ). =2 5 B1 , Ai1 p . ? Ai2 p ! 5 , Ai2 p = A0i2 p A00i2 p , A0i2 p | 5 a2 2 A0i2 p . N Ai1 p A0i2 p ! B1 A0i2 p , 1! Ai1 p A0i2 p | %32]. J , 5 !! 2.8, !, 2 ' 2 Hom(Ai1 p Ai2p ), 'a1 = a2 . ; , ! fAip gi2I , 1! !! 2.9 ! ! fAi gi2I . ; 2.10 $ 2.11. fAigi2I | "$
-
' , !
" "
! ' " ) : 1) * 2) ' * 3) C- )' , !* 4) * 5) ) $ * 6) IT - . % "$ fAigi2I ".
! 02 . 2.12. A | ' , B | , a 2 A, b 2 B, b |#" , H (a) 6 H (b). % ' 2 Hom(A B), $ 'a = b.
425
. ? L o(b) < 1, Pb 2 T (B) (T (B) | 5 B). T(B) = Tp (B), b = bp , | p p2 ! , bp (p 2 ) | 1! ! ! Tp (B). D p 2 H (a) 6 H (bp ). ! hp (a) np , 5 o(bp ) = mp (p 2 ). ;!! hp (bp ) > np . H Tp (B) , - ! ! 5 5 %27, . 142], , 5 ! ! ! B, !! !! ! B %27, ! 27.5, . 140]. H Tp (B) (p 2 ) , ! !! - hp (bp ) > np ( Tp (B) hp (bp ) hp (bp )) o(bp ) = mp , Tp (B) ( , B) - ! ! !5 , ! np + mp . ; , p 2 !, B = hcp i B 0 , o(cp ) > mp + np . ;!! H (a) 6 H (pnp cp ) 6 H (bp ). D p 2 hai ! pnp x = a( ! xp . J ! ! !!3 ! p : hai ! hcp i (p 2 ), p xp = cp . ;!! p a = p (pnp xp ) = pnp cp . F , hcp i | ! , C , !, 2 0 !!3 ! p0 : A ! hcp i (p 2 ), !! 02 !!:
-A p ;;p ?; hai
i
0
hcp i
i | . ; , p 2 p0 a = pnp cp . ? B | H (pnp cp ) 6 H (bp ), 2 0 1!3 ! p (p 2 ) P B, p (pnp cp ) = bp . ! p p0 'p , 5 ' = 'p . ? P P P Pp2 ' 2 Hom(A B) 'a = 'p a = ('p a) = p ( p0 a) = p (pnp cp ) = p2 p2 p2 p2 P = bp = b. p2
$ 2.13. fAigi2I | "$ ' , ! '
$ $, ' $ ' , I1 = fi 2 I j Ai | ' g, I2 = fi 2 I j Ai |
g. fAi gi2I | "$ , "$ fAigi2I fAi gi2I . D ! K- . 5 H | ! - 1
2
426
. .
0 10 B 20 B (ij ) = B ::: B @n0
1
11 : : : 1n : : : 21 : : : 2n : : :C C : : : : : : : : : : : :C C n1 : : : nn : : :A ::: ::: ::: ::: ::: 020 55 ! 1. D A !! HA H ((ij ) 2 HA 5 , (ij ) 2 H i 2 N, j 2 N0, ij 6= 1, ij < i , i | pi- A). H M1 M2 2 H, M1 = (ij ), M2 = (ij ), ! M1 6 M2 , 0 i 2 N, j 2 N0 ij 6 ij . H M = f((ij ) ) j 2 ;g | ! ! - H, inf H M | 1 ! - (ij ) 2 H, i 2 N, j 2 N0 ij 5 !5 1! (ij ) , 2 ;. H I | ! , ! P (I) ! ! I. 2.14. 5 fGigi2I | ! , K | P (I), G | . M! 5, G K- fGigi2I , 0 1! g 2 G H (g) > inf H fH (al )gl2J , al 2 Gl , J 2 K jJ j 6 @0, 2 1! g1 : : : gr 2 G 02 ! ! : 1) g1 + : : : + gr = g, 2) 1! gk (k = 1 r) 1! alk (lk 2 J), H (gk ) > H (alk ). D 0 5 3 Gi ! fGigi2I , ! 5, K 1! ! ( , ! ) ! I. > ! !, 2.14 1! al (l 2 J) 5 J1 = fl 2 J j al 6= 0g, inf H fH (al )gl2J = inf H fH (al )gl2J1 , ! J1 J, J1 2 K. F , 1! g !! H (g) > H (al ), al | 5 1! Gl , !, 5 K-! G 5 ! fGigi2I ! 5, 1! g al 2.14 . I !! ! - H ! !5 02 5 ! 1, ! H. D p- A !! HA H. D 0 5 H ! ! , 0 ! M0 5 H 2 inf M0 . H < ! !, X ! ! 5, 2 - - 5 ! 1.
427
> ! !, 5 5 K-! ( 2.14) , G Gi (i 2 I) | , ! ! ! ! - 1! ! 5 1 1! , H | ! X. H ! ! p- fGigi2I p- G, 5 2.14 ! ! ! ! - 1! ! 5 1 1! , H | H. !, 5 K-! ! 5 - ! ! . 5 G | , fGi gi2I | ! , K | P (I). > ! G Gi G = R D, Gi = Ri Di (i 2 I), R Ri (i 2 I) | - , D Di (i 2 I) | ! . 2.15. - G ) K-" "$ fGigi2I , R ) K-" "$ fRigi2I . . < !5. 5 G 0 K-! 5 ! fGigi2I , 5 H (g) > > inf H fH (al )gl2J , g 2 R, al 2 Rl , J 2 K jJ j 6 @0 . ? 2 0 1! g1 : : : gr 2 G 02 ! ! : 1) g1 + : : : + gr = g, 2) 1! gk (k = 1 r) 1! alk (lk 2 J), H(gk ) > H(alk ). > ! 1! gk (k = 1 r) gk = bk + ck , bk 2 R, ck 2 D. ;!! H (gk ) = H (bk ), b1 + : : : + br = g, c1 + : : : + cr = 0. ; , 2 0 1! b1 : : : br 2 R 02 ! ! : 1) b1 + : : : + br = g, 2) 1! bk (k = 1 r) 1! alk (lk 2 J), H(bk ) > H(alk ). > , R 0 K-! 5 ! fRigi2I . D 5. 5 R 0 K-! 5 ! fRi gi2I , 5 H (g) > inf H fH (al )0 gl2J00, g 0 2 G, al 2 Gl , J 2 K jJ j 6 @0 . > ! 1! g g = g + g , g 2 R, g00 2 D, 1! alk (lk 2 J) ! al = a0l + a00l , a0l 2 Rl , a00l 2 Dl . ;!! H (g) = H (g0 ) H (al ) = H (a0l ) l 2 J. > 0 , H (g0 ) > inf H fH (al )gl2J . F , R 0 K-! 5 ! fRigi2I !, R 2 0 1! g10 : : : gr0 02 ! ! : 1) g10 +: : :+gr0 = g0 , 2) 1! gk0 (k = 1 r) 1! a0lk (lk 2 J), H (gk0 ) > H (a0lk ). 5 g1 = g10 + g00 , gk = gk0 , 1 < k 6 r. ;!! g1 +: : :+gr = g0 +g00 = g, H (g1 ) = H (g10 ) H (alk ) = H (a0lk ) k = 1 r, 1! gk (k = 1 r) !! H (gk ) > H (alk ). = 5, G 0 K-! 5 ! fGigi2I .
428
. .
! !, 5 K-!50. 1. 5 fGigi2N| ! , Gi q- ! 0 q, pi, pi Gi 6= Gi, 5 G | , 2G 6= G, qG = G q 6= 2. 5 K | 5 P (N), J 2 K 1! g 2 G (g) > inf f(al )gl2J , al 2 Gl . ? h2 (al ) = 1 X l, -, 1! al 5 1! a1 2 G1. ? (g) > (a1 ), 1! G 0 K-! 5 ! fGigi2N( 5 !! g1 + : : : + gr , 5 5 K-! , !, 02 g). 2. 5 fGigi2I | ! , p 2 p- ! ! 1 ! , 02 q- ! 0 q, p. 5 G | , p- ! 0 p, 5 K | P (I), ! ! ! I. ? G 0 K-! 5 ! fGigi2I . D 5, 5 g 2 G, g 6= 0. D pk 2 ik 2 I, Gik | ! ! fGigi2I , pk Gik = Gik , qGik 6= Gik q 6= pk (q 2 O). 5 J = fik j k 2 Ng. ? (g) > inf f(aik )gik2J , aik 2 Gik , hpk (aik ) = 1, X hq (aik ) = 0 q 6= pk (q 2 O). ;!! J 2 K, jJ j = @0 . 8 G 2 1! , 5 1! aik (ik 2 J). > , G 0 K-! 5 ! fGigi2I . 3. 5 fGigi2N| ! ! , Gi pi- ! , q- ! 0 q 6= pi , 5 G | , p- ! 0 p. H K = P (N), 2 ! , G 0 K-! 5 ! fGigi2N. 5 K0 | P (N), 2 ! ! N. H g 2 G, g 6= 0, (g) inf fa g , al 2 Gl , X l l2J J | ! ! N. > , G 0 K0 -! 5 ! fGigi2N. 4. 5 G | , (1 1 : : : 1 : : :)( fGigi2I | ! , 5 k, 02 50
429
( ! ! 5 k N1), 2 - k, 2 1! ! . 5 K | P (I), ! ! ! I. D k 2 N1 2 ik 2 I, Gik | - k ! fGi gi2I . 5 J = fik j k 2 N1g, g 2 G, g 6= 0 aik | 02 1! Gik k 2 N1. ;!! H (g) > inf H fH (aik )gik 2J . 8 G 2 1! , ! - 5 ! - 1! aik (ik 2 J). > , G 0 K-! 5 ! fGigi2I . ! 5 ! , 02 0 K-! . 2.16. 5 fGigi2I | ! , K | P (I). M! 5, ! fGigi2I K- , Gj (j 2 I) 0 K-! 5 ! fGigi2I . H 2.16 ( 5 2.14 5 K-! ) ! jJ j 6 @0 5 jJ j < @0 , ! 5, 02 ! fGigi2I . , 1! ! 5, J 2 K, ! !, ! ! I K. ; , ! . 2.17. 5 fGigi2I | ! . M! 5, ! fGigi2I , 0 Gj (j 2 I) 0 1! gj 2 Gj H (gj ) > inf H fH (al )gl2J , al 2 Gl , J I jJ j < @0 , 2 1! gj1 : : : gjr 2 Gj 02 ! ! : 1) gj1 + : : : + gjr = gj ( 2) 1! gjk (k = 1 r) 1! alk (lk 2 J), H (gjk ) > H (alk ). ? , 5 5 K-! , ! K-! ! ! fGigi2I ! ! ! , Gi (i 2 I) | p-, ! 5 ! - 1! p- 1 1! , H | H( Gi (i 2 I) | , ! ! ! ! 5 ! - 1! 1 1! , H | ! X. =02 !! 5 5 ! ! , 02 0 K-! . 2.18. g | #" '$ G. fGigi2I | "$ ' . ( ' "! J "! I )' "$ #" fal gl2J (al 2 Gl ),
430
. .
H (g) > infH fH (al )gl2J , "! J1 "! J , H (g) > infH fH (al )gl2J . 1
F !! , ! - H (g) 1! 1. J ! ! ! , 2 . 2.19. & fGigi2I | "$ ! , K | '$ ' P (I), fGigi2I )
K-" . . 5 gj 2 Gj (j 2 I) (gj ) > infX f(al)gl2J , al 2 Gl , J 2 K jJ j 6 @0 . H l 2 J (gj ) > (al ), fGigi2I 0 K-! 5! !. 5
l 2 J !! (gj ) (al ). > 3 ! - 5 1! ! J, l1 , 5 P1 = fp 2 O j hp (al1 ) > hp (gj )g. T P1 ( !, 1! gj al1 1 ). =2 5 m1 , ! P1, (al1 ) 6 (m1 gj ). ? (gj ) > inf f(al )gl2J , X ( p ) p 2 P1 1! a , ! 1! al (l 2 J), hp (a(p) ) 6 hp (gj ). =2 mp 2 N, (a(p) ) 6 (mp gj ), ! mp ! 5 2 ! p. < 5 2 5 jP1j + 1 m1 , mp (p 2 P1 ) P 1, 1! 2 0 - u1, up (p 2 P1 ), u1m1 + up mp = 1. > , p2P1 P gj = u1m1 gj + up mp gj , (u1m1 gj ) > (al1 ), (up mp gj ) > (a(p) ) p2P1 p 2 P1. = 5, ! fGigi2I 0 K-! . ! 5 ! 0 .
2.20.
' G ) K-" )' "$ ' fGigi2I )' K '$ ' P (I).
. 5 g 2 G, g 6= 0 H (g) > inf H fH (al )gl2J , al 2 Gl , k k m J 2 K, jJ j 6 @0 . H o(g) = pi : : : pim | 1! g 1 1
, ! - H (g) , ! i1 : : : im , 5 ! 1(
!! ir (r = 1 m) ! (0 1 : : : kr ;1 1 1 : : :), 0 : : : kr ;1 | 02 55 . D (r k), r 2 f1 : : : mg, k 2 f0 1 : : : kr ; 1g, ! Crk ! 1! al (l 2 J), hpir (pkir g) > hpir (pkir al ). 8 H (g) > inf H fH (al )gl2J !-
431
Crk . 8! ! ! Crk ! 1!, ! rk . ? 0 p- q- ! q 6= p (q | ), 1! g G hpir (g) = hpir (gpir ), gpir | 1! g pir -! Gpir G. > , 0 r (r = 1 m) !! H (gpir ) > inf H fH (crk )gk=0kr 1 , ! k k k (k = 0 kr;1) hpir (pir gpir ) > hpir (pir crk ). 8 ! - H (gpir ) 1! gpir G !! ir ! Hpir (gpir ) 1! gpir Gpir , 5 5 ! 1. J ! ! Hpir (gpir ). 5 Hpir (gpir ) = (0 : : : kr ;1 1 1 : : :), 0 : : : kr ;1 | 02 55 , 5 t1 : : : ts (ts = kr ; 1) | , 5 Hpir (gpir ), 5 ti + 1 < ti +1 i = 1 s ( t1 : : : ts ). 5 Hpir (gpir ) ! , 1 t1 (t1 = kr ; 1). ;!! hpir (ptir1 gpir ) > hpir (ptir1 crt1 ). ? H (gpir ) > H (crt1 ). 5 Hpir (gpir ) 5 . ? t1 : : : ts 5 , t1 -,. .. , ts - F5! {# Gpir %2, !! 65.3, . 9]. J ! ! 55 (0 : : : t1 1 1 : : :), !020 t1 . ? fGpir (t1 ) 6= 0, Gpir 2 1! bt1 , 55 ! %2, !! 65.3, . 9]. D 1! bt1 0 o(bt1 ) = pitr1+1 hpir (pvir bt1 ) = v - v, 02 0 6 v 6 t1 . ;!! t1 = hpir (ptir1 bt1 ) = hpir (ptir1 gpir ) > hpir (ptir1 crt1 ). ? Hpir (bt1 ) = = (0 : : : t1 1 1 :: :) ! t1 hpir (ptir1 bt1 ) > > hpir (ptir1 crt1 ), H (bt1 ) > H (crt1 ). 5 5 l, 2 s ; 1 (s | , Hpir (gpir ) 5 ), Gpir 1! bt1 : : : btl , m = 1 l o(btm ) = ptirm +1 hpir (pvir btm ) = v - v, 02 tm;1 < v 6 tm ( ! t0 = ;1), ! 0 1! btm (m = 1 l) H (btm ) > H (crtm ) l > 1 1! ! - H (btn ), 1, !5 02 1! ! - H (btn+1 ), 1 6 n 6 l ; 1. ! l + 1. J ! ! tl+1 . ; , 0 ! + n, | 5 0, n | - 5 - . > ! tl+1 ! : tl+1 = tl+1 + ntl+1 , tl+1 | 5 0, ntl+1 | - 5 - . J ! ! : ) ntl+1 > tl+1 ( ) ntl+1 < tl+1 . > ! !, ) ! !, , tl+1 | 5 , !! tl+1 = ntl+1 . ;
432
. .
) 5 ntl+1 > tl+1 . J ! ! 55 (tl+1 ; tl+1 tl+1 ; tl+1 + 1 : : : tl+1 1 1 : ::). E 55 ! tl+1 . ? fGpir (tl+1 ) 6= 0, Gpir 2 1! btl+1 , 55 !. D 1! btl+1 !! o(btl+1 ) = pitrl+1 +1 , hpir (pvir btl+1 ) = v - v, 02 tl < v 6 tl+1 . ;!! tl+1 = hpir (ptirl+1 btl+1 ) = hpir (ptirl+1 gpir ) > hpir (ptirl+1 crtl+1 ). ? 1! btl+1 ! tl+1 , H (btl+1 ) > H (crtl+1 ). ? hpir (ptirl btl ) = tl o(btl ) = ptirl +1 , 1! btl ! (0 1 : : : tl 1 1 :: :), 0 1 : : : tl | 02 55 , tl = tl . F , 1! gpir tl 5 , ! 0 < tl+1 ; tl+1 , 1 < tl+1 ; tl+1 + 1,.. . , tl < tl+1 ; tl+1 + tl . 0 , 1! ! - H (btl ), 1, !5 02 1! ! - H (btl+1 ). ) 5 ntl+1 < tl+1 . ;!! tl+1 | 5 . ! tl+1 ; ntl+1 sl+1 . ;!! tl+1 > hpir (psirl+1 gpir ). 5 K (trl)+1 = maxfhpir (psirl+1 ;1 btl ) hpir (psirl+1 ;1crtl+1 )g. ;!! ! : 1) k (rtl)+1 | 5 ( 2) k (rtl)+1 | 5 . J ! ! 0 . 1) 5 k (rtl)+1 | 5 . F , Gpir ( 5 1! pir - ), !, Gpir - ! ! 5 5 %27, . 142]. > , 0 5 k 2 5 n, n > k fGpir (n) 6= 0. 8! 5 n( rtl)+1 , n( rtl)+1 > k (rtl)+1 fGpir (n( rtl)+1 ) = 6 0. J ! ! 55 (n( rtl)+1 ; (sl+1 ; 1) n( rtl)+1 ; sl+1 : : : n( rtl)+1 tl+1 tl+1 + 1 : : : tl+1 1 1 : ::), ! 5 n( rtl)+1 tl+1 . ? fGpir (n( rtl)+1 ) 6= 0 fGpir (tl+1 ) 6= 0, Gpir 5 1! btl+1 , 55 !. D 1! btl+1 !! o(btl+1 ) = ptl+1 +1 , hpir (pvir btl+1 ) = v - v, 02 tl < v 6 tl+1 . ? n( rtl)+1 = hpir (psirl+1 ;1btl+1 ) > hpir (psirl+1 ;1 crtl+1 ) tl+1 = hpir (ptirl+1 btl+1 ) > > hpir (ptirl+1 crtl+1 ), H (btl+1 ) > H (crtl+1 ). 8 n( rtl)+1 ! , 1! ! - H (btl ), 1, !5 02 1! ! - H (btl+1 ). 2) 5 k (rtl)+1 | 5 . ! k (rtl)+1 k (rtl)+1 = (rtl)+1 + m( rtl)+1 , (rtl)+1 | 5 , m( rtl)+1 |
433
- 5 - . ;!! k (rtl)+1 < tl+1 , , tl+1 | 5 , ! (rtl)+1 + ! 6 tl+1 . ; F5! {# Gpir ! , ! 5! ! ! (rtl)+1 (rtl)+1 + ! %1, . 32]. 8 , ! (rtl)+1 (trl)+1 + ! ! (rtl)+1 + n, n | 5 . 8! 5 n( rtl)+1 , n( rtl)+1 > m( rtl)+1 , n( rtl)+1 > sl+1 ; 1 fGpir ( (rtl)+1 + n( rtl)+1 ) 6= 0. J ! ! 55 ( (rtl)+1 +n( rtl)+1 ; (sl+1 ; 1) (rtl)+1 + n( rtl)+1 ; ;sl+1 : : : (rt ) +n( rt ) tl+1 tl+1 +1 : : : tl+1 1 1 : ::), ! l+1 l+1 5 (rtl)+1 + n( rtl)+1 tl+1 . ? fGpir ( (rtl)+1 + n( rtl)+1 ) 6= 0 fGpir (tl+1 ) 6= 0, Gpir 5 1! btl+1 , 55 !. E! btl+1 !, ! - 1). ; , ! , 1! gpir Gpir ! t1 , H (gpir ) > H (crt1 ). H 1! gpir t1 : : : ts | , 5 Hpir (gpir ), Gpir 2 0 1! bt1 : : : bts , m = 1 s o(btm ) = ptirm +1 , hpir (pvir btm ) = v - v, 02 tm;1 < v 6 tm ( ! t0 = ;1), ! 0 1! btm H (btm ) > H (crtm ) 1! ! - H (btn ), 1, !5 02 1! ! - H (btn+1 ), 1 6 n 6 s ; 1. ;!! Hpir (gpir ) = Hpir (bt1 +: : :+bts ), Gpir (!! 2.9), 2 ' 2 E(Gpir ), gpir = '(bt1 +: : :+bts ). ? gpir = 'bt1 + : : : + 'bts , H ('btm ) > H (btm ) > H (crtm ) (crtm 2 fal gl2J , m = 1 s). > , 1! gpir 02 ! !: ! 5 !! 1! , ! - 5 ! - 1! al (l 2 J). ? m P g = gpir , 1! g ! !. > , r=1 G 0 K-! 5 ! fGigi2I . =02 5 ! , 02 0 K-! 5 5 5 ! . D A ! (A) ! p, pA 6= A.
434
. .
2.21. fGigi2I | "$ ' , K | '$ ' P (I). & )' Gi , Gi (i1 i2 2 I , i1 6= i2) (Gi ) \ (Gi ) = ?, "$ fGigi2I ) K-" . 1
1
2
2
. 5 gj 2 Gj (j 2 I), gj 6= 0 H (gj ) > inf H fH (al )gl2J , al 2 Gl , J 2 K, jJ j 6 @0 . < ! 2 , ! 5, Gj |
- . =2 ! - H (gj ), 2 5 ! 1. 8 (Gj ) \ (Gl ) = ? l 6= j 02 ! - H (al ), l 2 J, l 6= j, 5 ! 1. > , ! 1! fal gl2J 5 1! aj 2 Gj H (gj ) > H (aj ). 1! ! fGi gi2I 0 K-! . 8 0 3 ! ! - . H ! 2.1 5 - 0 A , 0 1! a b, H (a) 6 H (b), 2 1!3 ! ' 2 E(A), 'a = b, 5 - !, 0 ! ! . J ! ! 02 !, !, 1 2 (%7, . 13]( %2, . 235]), !, 2 p- 1! p- 1. 5 A = hai D, o(a) = ps , D = Z(p1 ), b 2 D o(b) = pk , k > s. ;!! H (a) 6 H (b), Hp (a) = (0 1 : : : s ; 1 1 1 : : :), Hp (b) = (1 1 : : : 1 : : :). 2 ' 2 E(A), 'a = b, 1!3 ! 1! . T 5 , - 5 0 2.1 5 ! , !. J ! ! ! N f1g. 8 ! 02 1! ! . 5 m n 2 N f1g. ! m 4 n 5 , : 1) m n 2 N n j m( 2) m = 1. D 1! a A ! H o(a) (H (a) o(a)). H a b 2 A, ! H o(a) 6 H o(b) 5 , H (a) 6 H (b) o(a) 4 o(b). , ' 2 E(A), H o(a) 6 H o('a). ;!! H o(0) > H o(a) 1! a 2 A. #! , 0 1! a b 2 A !! H o(a + b) > H o(a) ^ H o(b). 2.22. I A ! , 0 1! a b, H o(a) 6 H o(b), 2 1!3 ! ' 2 E(A), 'a = b. 2.23. A | ' , a b 2 A. H o(a) 6 H o(b) , H (a) 6 H (b).
435
. , H o(a) 6 H o(b) H (a) 6 H (b). 5 H (a) 6 H (b). !, H o(a) 6 H o(b). H o(a) = 1, 0 1! b 2 A !! o(a) 4 o(b) , , H o(a) 6 H o(b). 5 o(a) = k, o(b) = s, k s 2 N. J ! !
k s ! : k = pk11 : : :pkmm , s = ps11 : : :psmm (! 5, k s 0 ! , , , ki, si (i = 1 m) ! 0). 5 H (a) = (ij )i2Nj 2N0, H (b) = (ij )i2Nj 2N0. ;!! n > m (n 2 N) 0 j 2 N0 nj = nj = 1, n 6 m, nj = 1 5 , j > kn, nj = 1 5 , j > sn . ? H (a) 6 H (b), 0 i 2 N 0 j 2 N0 ij 6 ij . > , 5 n, 2 m, !! kn > sn , 1! s j k. 0 H o(a) 6 H o(b). $ 2.24. A | ' . . -
) : 1) )' #" a b 2 A, H o(a) 6 H o(b), #" " ' # $ , $ 'a = b* 2) )' #" a b 2 A, H (a) 6 H (b), #" " ' # $ , $ 'a = b. 2.25. /' , .
. < !5. 5 G | . > ! G G = R D, R | - , D | ! . 5 a b 2 R H o(a) 6 H o(b). ; G 2 ' 2 E(G), 'a = b. 5 '1 = 'jR = '1 , | - G R. ;!! 2 E(R) a = b. > , R . D 5. 5 G = R D, R | - , D | ! , 5 R | . 5 a b 2 G, H o(a) 6 H o(b) a = a1 + a2, b = b1 + b2 , a1 b1 2 R, a2 b2 2 D. ;!! H (a) = H (a1 ), H (b) = H (b1 ), 1! H (a1 ) 6 H (b1 ). ; R 2 '1 2 E(R), '1(a1 ) = b1 . ! '1 1!3 ! 1 2 E(G), 1c = '1 c, c 2 R, 1 c = 0, c 2 D. ? 1 a = b1. J ! ! hai G '2 : hai ! D, '2 (ka) = kb2 k 2 Z. < 5 5 '2 , 1! a . H a = 0, 5 o(a) = 1, H o(a) 6 H o(b) o(b) = 1, 1! b = 0. 5 a | 1! . ; H o(a) 6 H o(b) o(b) j o(a). H m | 5 , ma = 0, o(a) j m, , o(b) j m. ? o(b) | !5 2 o(b1) o(b2), o(b2 ) j m, , mb2 = 0. ; , 0 -
436
. .
k1 k2, k1 a = k2a, !! k1b2 = k2b2. 1! '2 . Q, '2 | !!3 !, ! '2 a = b2. ? D | ! , D | C %27, ! 21.2, . 119], 1! 2 !!3 ! 2 : G ! D, !! hai G '2
; ; ?; 2
D | , !! . > , 2 a = b2. 5 = 1 + 2. ;!!: | 1!3 ! G, a = b. = 5, G | . ! 2.22 ! 5 ! . 2.26. =! fAigi2I ! , (Ai1 Ai2 ), i1 i2 2 I (i1 ! 5 i2 ) : , a 2 Ai1 , b 2 Ai2 H o(a) 6 H o(b), , 2 ' 2 Hom(Ai1 Ai2 ) ! 'a = b. > ! 0 Ai ! fAi gi2I Ai = Ri Di , Ri | - , Di |- ! . I ! 2.25 0 2.24 ! 5 5 . 2.27. +"$ ' fAigi2I , "$ fRigi2I . 2.28. fAigi2I | "$ '-
. . ) : 1) !$ (Ai Ai ), i1 i2 2 I , , a 2 Ai , b 2 Ai
H o(a) 6 H o(b), ' 2 Hom(Ai Ai ) $ " 'a = b* 2) !$ (Ai Ai ), i1 i2 2 I , , a 2 Ai , b 2 Ai H (a) 6 H (b), ' 2 Hom(Ai Ai ) $ " 'a = b. 1
2
1
1
1
2
x
2
2
2
1
1
2
3. K-
< ! ! K- , %27] (. 54). 5 fAigi2IQ| 5 ! . H a = = (: : : ai : : :) 2 Ai , ! " a ! s(a) = i2I = fi 2 I j ai 6= 0g. H K | P (I) !
437
! Q I, ! K- Ai (i 2 I) ! Ai , 2 1! a ! s(a) 2 K. ? i2I s(a a2 ) s(a1 ) s(a2 ), K-! Q 1A;, ! L A .!! 5
i K i i2I ! !, , , K ! ! I, ! !! , K = P (I), ! Ai (i 2 I). ! K-! !! Ai (i 2 I), 0 5 3 Ai 1 !!, ! 5, K 1! ! ( , ! ) ! I. ; 5 2 3 , ! 5 - ! ! . 8 2! 3 5 , 50, . / 5 - 0 . ! Ai (i 2 I), 5 L A 5 Q AA. i A i D ! ! i2I i2I A. 3.1. & A, ) "! "$ ""$ Ai (i 2 I), , "$ fAigi2I
) " .
. D i 2 I ! i Ai A, i | - 0 A Ai . 5 i1 i2 2 I, a 2 Ai1 , b 2 Ai2 H (a) 6 H (b). i1 Ai1 i2 Ai2 | ! ! A. = ! i1 Ai1 i2 Ai2 A !! H A (i1 a) 6 H A (i2 b). ; A 2 ' 2 E(A), '(i1 a) = i2 b , , (i2 'i1 )a = b. ? i2 'i1 2 Hom(Ai1 Ai2 ), ! fAi gi2I . I jJ j < @0 . 5 gj 2 Aj (j 2 I) H (gj ) > inf H fH (al )gl2J , al 2 Al , J P r 5 jJ j = r J = fl1 l2 : : : lr g. ? H A (j gj ) > H A lk alk . ;
k;1 A 2 ' 2 E(A), Pr Pr r P j gj = ' lk alk = 'lk alk . ;!! gj = j j gj = (j 'lk )alk k;1 k;1 k;1 H (j 'lk )alk > H (alk ) (k = 1 r). j 'lk alk gjk , !, gj = gj1 + : : : + gjr , H (gjk ) > H (alk ) (k = 1 r). $ 3.2. & A = LK Ai (i 2 I) | , "$ fAigi2I )
" .
438
. .
D ! K-! !! , 2 5 5 5 !. 3.3. K- " "" ' Ai (i 2 I) $ $ Q Ai. i2I
. 5 p | 5 . D ! ;L Ai = L Ai \ !! 3 - , p K K Q \ p Ai 0 . 5 = 1. i2I
;L
Q
L
L
Q
!, p K Ai = K Ai \ p Ai . 5 a 2 K Ai \ p Ai , i2I i2I Q 2 1! g 2 Ai , pg = a. E! g ! i2I L A , 1! a 2 p;L A . 5 , s(g) = s(a). > , g 2 K i Q K;Li L ; , ! , K Ai \ p Ai p K Ai . 0 i2I . H | 5 ;L L ( = 1Q+ A1),,, 5 p1 K Ai = K Ai \ p1 ! i i2;L I ; L
! , !, p p1 K Ai = K Ai \ 1 Q ! ; L L QA . \p p Ai , 5 p K Ai = K Ai \ p i i2I i2I H ;L | 5 Q , , 5 L p K Ai = K Ai \ p Ai < , ! i2I
M
K Ai \ p
=
Y M i2I
Ai =
\ M
<
K Ai \ p
K Ai \
\ Y
<
p
Y
\ i2I
Ai
=
<
Ai i2I M p
=
K Ai = p
L
M
K Ai :
Q
; !! , A = K Ai (i 2 I), B = Ai , i2I 0 1! a 2 A !! H A (a) = H B (a). J ! K-!0 !! A Ai (i 2 I), ! ! 5 i Ai A, i | - 0 A Ai (i 2 I). 3.4. fAigi2I | "$ ' , A | ' , K | '$ ' P (I). ( )' "! J 2 K )' "$ "" " f'igi2I , 'i : A ! Ai, 'i = 0 i 2 I n J , $ "" " , ) $
""
439
A
@@ 'i @@ ? R M iA Ai i K
"" .
. J ! ! : A ! Q Ai, i2I Q 1! a 2 A a ! ! ! 1! b 2 A ,
i i2I ib = 'i a. , | !!3 !, ! i =L'i i 2 I. L;!! s( a) J, 1! s( a) 2 K. > , a 2 K Ai . ; , Im K Ai . H 5 .
D ! K-! !! . 3.5. A = LK Ai (i 2 I) | , "$ fAigi2I ) K-"-
.
. 5 a b 2 A, H (a) 6 H (b). ;!! j 2 s(b) H (j b) > H (a) = inf H fH (i a)gi2s(a) . 5 H (j b) = (lk )l2N0k2N, 5 Bj = f(l k) 2 N0 N j lk 6= 1g. ;!! jBj j 6 @0 . D (l k) 2 Bj ! ! ! Ilk = fi 2 s(a) j lk > hpl (pkl i a)g. Ilk 6= ? H (j b) > inf H fH (i a)gi2s(a) . 8! ! Ilk ! 1! ilk . 5 Ij = filk j (l k) 2 Bj g. ? jIj j 6 @0 . > ! !, Ij 2 K, Ij s(a), s(a) 2 K. ; , 2 ! Ij , 2 K, jIj j 6 @0 H (j b) > fH (i a)gi2Ij .
? ! fAi gi2I 0 K-! , 2 0 1! bj1 : : : bjr 2 Aj , bj1 +: : :+bjr = j b (r j) 1! bjm (m = 1 r) 1! im a (im 2 Ij ), H (bjm ) > H (im a). 8 ! fAi gi2I 2 0 !!3 ! jm 2 Hom(Aim Aj ), jm (im a) = bjm . J ! ! 02 !!3 ! 'j : A ! Aj (j 2 I). Pr , j 2 s(b), ' = 0, j 2 I n s(b). ;!! 5 'j = jm im j
Pr
m=1 Pr ( a) = Pr b jm im a = jm im jm m=1 m=1 m=1
'j a = = j b j 2 s(b). F , s(b) 2 K, ! !! 3.4 ! !!3 ! f'j gj 2I , !, 2 1!3 ! A, a = b. > , A .
440
L
. .
? ! !! Ai K-! !! Ai (i 2 I), i2I K ! ! I, 3.2 ! 3.5 ! . $ 3.6. - A = L Ai , "$ fAiig2iI2I
) " .
F , 0 ! , !
$ 3.7. " " $
.
; 3.2 ! 3.5 2.20 5 . 3.8. A K- "$ ""$
Ai (i 2 I). - A , "$ fAigi2I . 0 $ 3.9. +)
Ai (i 2 I)
# : L 1) K '$ ' P (I), $ K Ai | L * 2) Ai | * iQ 2I 3) Ai | * i2I L 4) )' K '$ ' P (I) K Ai | * 5) fAigi2I | "$ . ; ! 3.8 2.11 5 . $ 3.10. A = LK Ai (i 2 I), Ai |
, !
" "
! ' "
) : 1) * 2) ' * 3) C- )' , !* 4) * 5) ) $ * 6) IT - . % A | . ! 5 5 5 , ! ! ! K-! !! .
441
3.11. 5 A = LK Ai (iL2 I), K | P (I), 5 i 2 I Ai = Ki Aij (j 2 Ji), Ki | ! ! fig Ji (Ki ! -
5 ! ! ! Ji ). D iS2 I 0 j 2 Ji ! ji - 0 Ai Aij . 5 J = (fig Ji ). J ! ! 02 K0 P (J): i2I ! C ! J K0 5 , 2 D 2 K i 2 D L 2 0 Q Ci 2 Ki , S C . J ! ! C = ': A ! i K i (ij )2J Aij , 'a i 2D L Q A , (a 2 K Ai) ! ! ! ! 1! b 2 ij (ij )2J 0 Q(i j) 2 J !! ij b = ji (i a) (i | - A Ai ( ij | - Aij Aij ). Q, s(b) 2 K0 , , ( ij ) 2 J L L b 2 K Aij ,L , b 2 K Aij 2 1! a 2 K Ai , 'a = b. L L ' - 0 1! !3 !! K Ai K Aij . 8 5 ! ! ! 5 1 . 0
0
0
S3.12. 5 A = LK Ai (i 2 I), K | -
P (I), I = Ij ! Ij (j 2 J) 0. D j 2J j 2 J ! ! ! Kj = fM j M I 2 D 2 K M = D \ IL j g. Q, Kj K Kj | P (Ij ). 5 Gj = Kj Ai (i 2 Ij ). D j 2 J 0 i 2 Ij ! ij - 0 Gj Ai . J ! ! 02 K1 P (J): C 2 K1 S 5 , 0 ! fCj gj 2C , Cj 2 Kj , !! Cj 2 K. > ! !, ! j 2C i 2 I 1! j(i) 2 J, i 2 Ij (i). 5 L G = K1 Gj L (j 2 J), Q 5 j0 | - L G Gj . J ! ! ': K1 Gj ! Ai , 'g (g 2 K1 Gj ) ! ! ! i2I Q ! 1! a 2 Ai , ia = ij (j0 (i)g) (i | i2I Q - Ai Ai ). !, s(a) 2 K. ;!! i 2 s(a) 5 i2I , ia 6= 0, 1 0 5 S! 5 ! , j(i) 2 s(g) i 2 s(j0 (i)g). > , s(a) = s(j0 (i)g). ? jL(i)2s(g) s(g) 2 K1 , s(j0 (i) g) 2 Kj (i) , s(a) 2 K. ; , L a 2 K Ai. < ,L 1! a 2 K Ai 2 1! g 2 K1 Gj , 'g = a. L L ' - 0 1! !3 !! K1 Gj K Ai . 8 5 ! ! ! 5 1 .
442
L
. .
! K A K-!0 !! ! , !3 A. ; -! L A, Q A,KQ;L !!
! 3.11 , A, L;Q A, Q;L;Q A . . ( 5 !) 0 ! K-!! !! ! . F ! 3.8 , 0 ! , !3 A, , ! $ 3.13. A |
. - LK A , A | . $ 3.14. A |
, !
" " $
! ' " 1){6) 3.10. % " ' ) : L A, Q A, Q;L A, L;Q A, Q;L;Q A . . J ! ! ! .
3.15. K- " "" ' "
( , )' ' " ) -
$ $.
. 5 fAigi2I | ! L ! , K | P (I). !, K Ai | . D Ai (i 2 I) 2 Gi , 02 !! ! ! - , Ai | ! ! ;L G i %27, ;L 38.2,L. 189], 5 L Gi = Ai Bi . ;!! K Gi = K Ai K Bi . N K Gi ! 3.11 K0 -! L !! ! - , L1! 0 3.10 K Gi | . N K Ai ! ! . 8 L! 3.5 !, 2 0 A = K Ai (i 2 I), ! fAi gi2I 0 K-! . 5 I | ! , ! ! N0. > 3 ! p ! ! ! ! fAi gi2I , A0 | , p- , i 2 N Ai | ! - pi . 5 K | P (I), ! L ! ! I. ;!! ! 3.15, K Ai | . 8! 1! aj 2 Aj (j 2 N0), hp (aj ) = 0. ? !! H (a0 ) = inf H fH (al )gl2N. H ! fAi gi2I 0 K-! , A0 2 1! a01 : : : a0r , a01 +: : :+a0r = a0 1! a0k (k = 1 r) 1! alk (lk 2 N), H (a0k ) > H (alk ). H (a0k ) > H (alk ),
443
a0k = 6 0, 5 ! . D 5, ! -
H (alk ) , 02 ! p, ! 5 ! 1( ! - H (a0k ), 5
a0k | 1! A0 hp (a0k ) 6= 1, , 02 ! p, ! 1 0. ; , ! fAi gi2I 0 K-! . (> ! !, 3.2 ! fAi gi2I 0 ! .) < ! !, %33], , Q !3 Z. # ! %33], ! ! 2 @0 0 , - , - . 8 %25] s- , 1! 2 ! !. # s-2 2 , Q , 5 , ! Z. @0 J ! ! K-! !! s-2 . 3.16. A = LK Ai (i 2 I), Ai | s-'' . - A , "$ fAigi2I ) K-" . . D 5 L ! 3.5. D ! !5. 5 A = K Ai (i 2 I), Ai | s-2 , . ; 3.2 , ! fAi gi2I . 5 gj 2 Aj (j 2 I) H (gj ) > inf H fH (al )gl2J , al 2 Al , J 2 K, jJ j = @0 . H 1! gj ! , !! 2.18 2 ! J1 ! J, H (gj ) > inf a 2 A, H fH (al )gl2J1 . 5 o(gj ) = 1. J ! ! 1! 0 l a = al , l 2 J, i a = 0, i 2 I n J. 5 gj = j gj . ;!! 0 0 H (a) = inf H fH (al )gl2J H (gj ) = H (gj ). > , H (gj ) > H (a). 8 0 ' 2 E(A), 'a = gj . ;! A;L2 L Q L ! K Ai = Al K Ai , K Ai i 2 I n J, K0 | l2J P (I n J), ! M 0 2 K0 5 , 2 M 2 K, M 0 ;L J). ! !3L Q A M \ (IA n
L K Ai
( g 2 K Ai , l i K l2J g = (b c), l b = l g 2 J ic = i g l;L Q Q i 2 I n J), - 0 Al K Ai Al | . l2J l2J 0
0
0
0
444
. .
E! gj ! 5 2 ! ! A0jQ Aj ( 5 0 | - Aj A0j . 5 d = a. ;!! d 2 Al l2J H (d) = H (a) = inf fH (al )gl2J . F , ;1 d = a, | Q A Q A H ;L A , !, 2 2 Hom Q A A0 , l l l j K i l2J l2J l2J d = gj ( = 0 j ' ;1 ). F , A0j | 2 jJ j = @0 , !, 2 ! J 0 Q ! J, Al | J n J 0 | l2J ! %33, Q1]. (D ! QJU ! J ! ! Al AU Al , 2 l2J l2J Q 1! aU 2 Al , i- l2J U ;!! Q Al = H1 H2, H1 = Q Al , 0 0 i 2 J n J.) l2J l2J nJ Q H2 = Al . > ! 1! d d = h1+h2 , h1 2 H1, h2 2 H2. ? l2J gj = d = h1 + h2 , o( h2 ) < 1. ? H ( h2 ) > H (h2 ), o( h2 ) < 1 00 H (h2 ) = inf H0 fH (al )gl2J , 2 ! J ! J , H ( h2 ) > inf H fH (al )gl2J . F , H ( h1 ) > > H (h1 ) = inf H fH (al )gl2J nJ , !, H (gj ) > inf H fH ( h1 ) H ( h2 )g > > inf H finf H fH (al )gl2J nJ inf H fH (al )gl2J g =0 inf H 00fH (al )gl2J1 , J1 | ! ! J (J1 = (J n J ) J ). ; , 0 Aj 0 1! gj 2 Aj (j 2 J) H (gj ) > inf H fH (al )gl2J , al 2 Al , J 2 K jJ j 6 @0 , 2 ! J1 ! J, H (gj ) > inf H fH (al )gl2J1 (, jJ j < @0 , ! J1 = J). ! 5 3.2, !, 1! gj ! !! gj = gj1 + : : : + gjr , 1! gjk (k = 1 n) 1! alk (lk 2 J1 , , lk 2 J), H (gjk ) > H (alk ). ; , ! fAigi2I 0 K-! . 0
0
0
0
0
00
0
0
00
$ 3.17. fAigi2I | "$ , " ! Ai ' " ) $: 1) Ai | ' * 2) Ai | * 3) Ai | * 4) Ai |
. - A = LK Ai (i 2 I)
, "$ fAigi2I ) K-" . $ 3.18. L A = LK Ai (i 2 I), Ai | '' . - A = K Ai (i 2 I) -
445
, "$ fAigi2I ) K-" .
J ! ! ! 50 L A = K Ai (i 2 I) !50 Ai . < ! !, G (G) ! ! fp 2 O j pG 6= Gg. 3.19. A = LK Ai (i 2 I) )' Ai1 , Ai2 (i1 i2 2 I , i1 6= i2 ) (Ai1 ) \ (Ai2 ) = ?. - A , ! Ai (i 2 I)
.
. < !5 3.2. D 5 ! !, ! fAi gi2I , Ai (i 2 I) , 0 1! ai1 ai2 , 2 ! ! Ai1 Ai2 (i1 i2 2 I), !0 ! ! -. ;
!! 2.21 , ! fAigi2I 0 K-! . > , ! 3.5 A | . 3.20. fAigi2I | "$ ' , " )' Ai1 , Ai2 (i1 i2 2 I , i1 6= i2) $$ " Hom(Ai1 Ai2 ) Hom(A i Ai ) , K | '$ ' P (I). - A = LK A2i (i 21 I) , i 2 I Ai : i1 i2 2 I , i1 6= i2, (Ai1 ) \ (Ai2 ) = ?. . D 5 3.19. D ! !5. ? A | , 0 3.2 ! fAigi2I 0 ! . !, 2 0 Ai1 Ai2 (i1 i2 2 I, i1 6= i2 ), (Ai1 ) \ (Ai2 ) 6= ?. 5 p 2 (Ai1 ) \ (Ai2 ), 5 pAi1 6= Ai1 , pAi2 6= Ai2 . ! Hom(Ai1 Ai2 ) = 0. 8 Ai1 Ai2 2 0 1! a b , hp (a) = 0, hp (b) = 0. ;!! (b) > inf f(pb) (a)g. ? ! X fAi gi2I 0 ! , 2 5 Pr 1! b1 : : : br Ai2 , b = bk ! k=1 (bk ) > (a) (bk ) > (pb) k = 1 r. (bk ) (a), Hom(Ai1 Ai2 ) = 0. = 5, (bk ) > (pb) k = 1 r, hp (b) > 1. . $ 3.21. LK Ai (i 2 I) | ' . & Aj , Ak (j k 2 I , j 6= k) , Hom(Aj Ak) = 0, Hom(Ak Aj ) = 0.
446
. .
. 5 B = j Aj k Ak . B | ! ! L K Ai (i 2 I), 1! B | . ;!! Hom(j Aj k Ak ) = 0 , ! ! 3.20 B, ! (j Aj ) \ (k Ak ) = ?. > , 0 p, p(j Aj ) 6= j Aj , !! p(k Ak ) = k Ak p, p(k Ak ) 6= k Ak , !! p(j Aj ) = j Aj . = 5, 0 1! a 2 j Aj b 2 k Ak !, 1! Hom(k Ak j Aj ) = 0. 0 Hom(Ak Aj ) = 0. J ! ! K-! !! . ; !! 2.19, 3.2 ! 3.5 3.22 ('20]). K- " "" Ai (i 2 I)
! , fAigi2I |
"$ .
? 0 ! 5 %8], ! .
$ 3.23. K- " "" ' ! $ $.
L
D K-! !! K A ! , !3 A, ! $ 3.24. A | . - LK A , A | . $ 3.25 ('20]). A | ' L ( , A | ' 1). % K A |
. , " ' ) : L A, OA, O;L A, L(OA), O;L(OA) . .
J ! ! 5 2 K-! !!
. L 5 A = K Ai (i 2 I), Ai | . ! V(A) ! Ai (i 2 I), 5 t 2 V(A) At ! Ai, !0 t, It = fi 2 I j Ai 2 Atg. 3.26. M! 5, A = LK Ai (i 2 I), Ai | , , 0 t1 t2 2 V(A), t1 6= t2 , (Ai1 ) \ (Ai2 ) = ?, i1 2 It1 , i2 2 It2 . 3.27 ('20]). A K- "$ ""$ Ai (i 2 I). - A , t 2 V(A) "$ At A
) .
447
. < !5. ? A , 0 3.2 ! fAigi2I , , t 2 V(A) ! At . 5 t1 t2 2 V(A), t1 6= t2 i1 2 It1 , i2 2 It2 . ;!! t(Ai1 ) = t1, t(Ai2 ) = t2. J ! ! B = i1 Ai1 i2 Ai2 . N B | ! ! A, 1! B | . N i1 Ai1 i2 Ai2 | t1 t2 . ? t1 6= t2 , ! Hom(i1 Ai1 i2 Ai2 ) Hom(i2 Ai2 i1 Ai1 ) . ! ! 3.20 B, !, (i1 Ai1 ) \ (i2 Ai2 ) = ?. 0 (Ai1 )\(Ai2 ) = ?, 5 A 0 . S I ! I (t 2 V(A)) D 5. ;!! I = t t t2 (A) 0. ? ! 0 3.12 02 Kt L P (It ) (t 2 V(A)) L K1 P (V(A)) !! A = K1 At (t 2 V(A)), At = K Ai (i 2 It ). F , t 2 V(A) ! At t, ! ! 3.22, At . ? A 0 , 0 t1 t2 2LV(A), t1 6= t2, !! (At1 ) \ (At2 ) = ?. ! 5 A = K1 At (t 2 V(A)) 3.19, !, A . $ 3.28 ('20]). A | K- " "" ' ( , A | ! " '
). - A , t
) .
J ! ! 2 3.28. = %34], ! G
P+ , p !3 Jp | - p- . L 5, G !3 Jp 3 p, , G | P+p . N P+ (P+p ), 0 ! 1! ! !! !! G, ! ! !! P+ (P+p ). ! , ! 5 P+p . 3.29. +"$ Ai (i 2 I), ! Ai -
' P+p , ".
. 5 a 2 Ai1 , b 2 Ai2 , i1 i2 2 I, (a) 6 (b). ;!! Ai2 = B1 B2 , B1 | ! !! P+ , b 2 B1 . ? (a) 6 (b), 2 ' 2 Hom(hai B1), 'a = b (Im ' hbi ). B1 , ! , -
448
. .
C , 1! ' !!3 ! '1 Ai1 B1 . '1 ! ! 5 !!3 ! Ai1 Ai2 . ; , '1 2 Hom(Ai1 Ai2 ) '1 a = b. 3.30. A = LK Ai (i 2 I), ! Ai + $ ' : $ ' $ ' $ P p. - A -p , ) ) : 1) A ) * 2) t(Ak) = t(Al ) (k l 2 I) Ak | ' , Al | ' . . < !5. ! 3.27 A 0 . !, 2). ! . 5 t(Ak ) = t(Al ) (k l 2 I) Ak | 5 , Al | 5 P+p . 8 Al Ak 1! a b , (a) = (b). ? 0 3.2 ! fAi gi2I , 2 ' 2 Hom(Al Ak ), 'a = b. E! a ! ! B Al , 02 ! !! , !3 Jp . 5 = 'jB. ;!! 2 Hom(B Ak ), a = b, 1! Hom(B Ak ) 6= 0. ? B | ! , 1 !, !!3 ! 0 ! 00 0 50 0 %35]. D 5. D t 2 V(A) ! At 5 5 , 5 5 P+p p. ;5 !! 3.29 , 0 ! 5 ! %8], !, 0 t 2 V(A) ! At . ! ! 3.27, !, A | .
$ 3.31 ('20]). K- " "" ' L . 5LA = K Ai (i 2 I), Ai | 5 P+ . ? A = A , A | p- 5 %21,34],
P+ . i
p2(Ai )
i(p)
i(p)
A, ! 0 3.11, ! 5 0 K0 -!0 !! Ai(p) (i 2 I, p 2 (A)). # Ai(p) 5 P+p ( !, ! ! 5 P+ 0 5! ! P+ %36]). ! ! 3.30, !, A | . = 5 P+p , ,
(- ), ! 5 Qp -! (Qp | 5- - p- ). E ,
449
- !5 - Qp -! G , 1!! p- , G !! !! (%1, . 52]( %37, . 294]). ! 02 .
$ 3.32. K- " "" ' , !$
"! Qp-" p, $. , $ $ )' K- " "" ' , !
p- $
p. ! 3! 5 5 5 , ! !! p-! K-! !! Ai (i 2 I), 02 ! , 02 0 K-! , ! ! . 3.33. fAigi2I | "$ ' , K | '$ ' P (I). & "$ fAigi2I ) K-" , )' p, # "$ ;L .' p- ") ' "" L T (A ! ) = T A p i p i K K . 80 Tp;LK Ai LK Tp(Ai), , . !, !! ;L A . 0L . ! . 5 T (A ) 6 T p i p K ;L K i ? L 2 1! a 2 K Tp (Ai ), a 2= Tp K Ai . 5 I1 = s(a) i 2 I1 ai = i a. ;!! I1 2 K. ? H (a) = inf H fH (ai )gi2I1 ! 1! , 1, ! - H (a) , 2 ! J ! I1 , jJ j 6 @0 H (a) = inf H fH (aj )gj 2J . F , J I1 , ! J 2 K. 5 supfo(aj )gj 2J = 1, Tp (Aj ) (j 2 J) ! 5 - ! ! hbj i , supfo(bj )gj 2J = 1. ? inf fHp (bj )gj 2J = (0 1 2 : : :). 5 Ak p- ! H (k 2 I), 5 gk | 1! Ak , hp (gk ) = 0. ;!! H (gk ) > inf H fH (pgk ) H (bj )gj 2J , ! jJ fkgj 6 @0 , J fkg 2 K. ? ! fAi gi2I 0 K-! , 2 0 1! gk1 : : : gkr 2 Ak , gk1 + : : : + gkr = g 1! gkm (m = 1 r) 1! alm 2 fpgk bj gj 2J , H (gkm ) > H (alm ). H 1! gkm ! 0 p- , Hp(gkm ) ! 1. F , 0 j 2 J Hp(bj ) ! ! 1, !, m (m = 1 r) hp (gkm ) 6= 1, alm = pgk , 5 H (gkm ) > H (pgk ). 5 N1 = fm 2 N j 1 6 m 6 r hp (gkm ) 6= 1g. ? gk = gk1 + : : : + gkr hp (gk ) 6= 1, N1 6= ? hp (gk ) > minfhp (gkm )gm2N1 > hp (pgk ) = 1. !, hpL (gk ) = 0. ;L L L = 5, K Tp (Ai ) Tp( K Ai ), , K Tp (Ai )=Tp K Ai .
450
. .
3.34. 5 I | ! , K | P (I). I1 | ! ! I. 5 K(I1 ) = fA j A I1 A 2 Kg. K(I1) ! P (I1) P (I). > ! ! , K(I1) = fA j A = C \ I1 ! C 2 Kg. 3.35. 5 ! fAigi2I 5 , 5 K | P (I). ! I1 02 ! ! I: I1 = fi 2 I j Ai 6= 0g. ? ! ! : 1) ! fAigi2I 5 , ! fAi gi2I1 ( 2) ! fAigi2I 0 K-! 5 , ! fAi gi2I1 0 K(I1)-! ( L L L 3) K Ai = K(I1 ) Ai . ( K(I1 ) Ai ! ! 5 ! , ! K(I1 ) L P (I1) L P (I)). Q, K(I1 ) Ai (i 2 I1 ) K(I1 ) Ai (i 2 I) !3 ! 5). 3.36. fAigi2I | "$ , !
'
$ $, ' $ ' , K | '$ ' P (I), I1 = fi 2 I j Ai | ' g, I2 = fi 2 I j Ai |
g. +"$ fAi gi2I ) K-" , ) ) : 1) "$ fAi gi2I ) K(I1)-" * 2) "$ S fAi giL 2I * ;L 3) p 2 (Ai ), K(I ) Tp (Ai ) = Tp K(I ) Ai . i2I 1
2
1
2
2
. < !5. ? ! fAigi2I , ! fAi gi2I1 fAi gi2I2 . ; , ! fAi gi2I 0 K-! , , ! fAi gi2I1 0 K(I1 )-! . ; !! 3.33 3). D 5. ? ! fAi gi2I1 fAi gi2I2 , 0 2.13 ! fAigi2I . !, ! fAi gi2I 0 K-! . 5 gj 2 Aj (j 2 I1 ) H (gj ) > inf H fH (al )gl2J , J 2 K, jJ j 6 @0 al 2 Al . J = J1 J2 , J1 = J \ I1 , J2 = J \ I2 . 5 p | , hp (gj ) = m 6= 1. ? ! - H (gj ), 02 ! p, ! (m m + 1 m + 2 : : :). D ! ! p 3) !! !!, 1! alp (l 2 J2) (alp |
451
1! al Tp (Al )). 5 pr+1 | !5 5 1! alp (l 2 J2). !, inf H fHp (al )gl2J2 = (0 1 : : :), 0 < 1 < : : : < r < 1, r+i = 1 i 2 N. 5 inf fhp (al )gl2J1 = s. ? inf fHp (al )gl2J1 = (s s + 1 s + 2 : : :). D H
i 2 N0 !! m + i > minfs + i ig. 8 , m + r + 1 > > minfs + r + 1 r+1g = minfs + r + 1 1g = s + r + 1. 0 , m > s, , Hp(gj ) > inf fHp(al )gl2J1 . = 5, H H (gj ) > inf H fH (al )gl2J1 . F , J1 2 K(I1 ), jJ1j 6 @0 ! fAigi2I1 0 K(I1 )-! , !, Aj (j 2 I1 ) 0 K-! 5 ! fAi gi2I . 8 2) Ai (i 2 I2 ) 1! ! 2.20 0 K-! 5 ! fAigi2I . ; , ! fAi gi2I 0 K-! .
$ 3.37 ('25]). 0 ) ,
- $ .
. 5 A | 202 , 5 A = A1 A2, A1 = T (A). H A | , A1 A2 , ! ! A. !, , A1 A2 5 A. J ! ! ! fA1 A2g K P (I), I = f1 2g, 02 P (I). =! fA1 A2 g ! 1){3) ! 3.36 ( 5 I1 = f2g, I2 = f1g). > , ! fA1 A2g 0 K-! . ! ! 3.5, !, K-! !! A1 A2 , 02 ! !!, . 3.38. fAigi2I | "$ s-'' , -
! '
$ $, ' $ ' , K | '$ ' P (I), I1 = fi 2 I j Ai L| ' g, I2 = fi 2 I j Ai |
g. - K Ai L , ) ) : 1) K(I ) Ai * 2) "$ S fAi giL 2I * ;L 3) p 2 (Ai ), K(I ) Tp (Ai ) = Tp K(I ) Ai . i2I 1
2
2
2
. < !5. ? LK Ai , ! 3.16 ! fAi gi2I 0 K-! . ; ! 3.36 , !1
452
. .
fAigi2I1 0 K(I1 )-!L , 1! ! 3.5 K(I1 ) Ai . F 2) 3) !, , 0 ! 3.36. 5 0 1){3). ? L D 5. A , ! 3.16 ! fAi gi2I1 i K(I1 ) 0 K(I1 )-! . ! ! 3.36, !, ! fAi gi2I L 0 K-! . 0 ! 3.5 !, K Ai . ; ! 3.38, 3.28 2.11, ! ! 3.35, $ 3.39. A = LK Ai (i 2 I), ! Ai
' $ ' $ $ ' , '
$ $, )'
" " $
! ' "
) : ) * ) ' * ) C- )' , !* ) * ) ) $ * ) IT - . - A , ) ) : 1) )' ' Ak Al (k l 2 I)
"" (Ak ) \ (Al ) = ?* 2) p, L p- " ' Ak (k 2 I) "" K Tp (Ai ) = Tp (A). L L
H A = Ai , 0 p !! Tp (Ai ) = Tp (A). i2I i2I 1! ! !! 3.39 ! $ 3.40. A = L Ai, ! Ai '
$ ' $ $i2I ' , '
$ $, )'
" " $
! ' " ){),
3.39. - A , )' ' Ak Al (k l 2 I) "" (Ak ) \ (Al ) = ?.
; 3.28 ! . ! 2 1 . 8 %38] ! , ! !! 1 ( 5 ). 8 %20]
453
, 02 !! ! 1 ( 5 ). ; 3.40 $ 3.41. A = L Ai | ! " (r(Ai) = 1 i2I i 2 I). - A , )' " ' Ak Al (k l 2 I) (Ak ) \ (Al ) = ?. Q J ! ! 2 0 A = Ai (r(Ai ) = 1 i2I i 2 I). H p | , ! Ip 02 ! ! I: Ip = fi 2 I j Ai | p- g. ; 3.39 $ 3.42 ('5]). A = Q Ai | '' . - A i2 I , )
) : 1) )' " ' Ak Al (k l 2 I) (Ak ) \ (Al ) = ?* 2) p, p- " ' Ak (k 2 I) "" i2QIp Ai = Tp(A). 8 %21] , 5 5 . J ! ! 5 5 5 ( ! ! ). < ! 02 !! . 3.43 ('39]). & A1 A2 : : : An | " " '$ A, ) " " ' -
" " , 1) A, ! " A1 A2 : : : An, "$ ""$ # , A1 +A2 +: : :+An = A1 A2 : : : An* 2) A1 A2 : : : An | " " A. 3.44. + ' A , )' " " " ' B C 1 A "" (B) \ (C) = ?.
. < !5. 5 B C | !3 ! ! 1 A. ? !! 3.43 B +C = B C B C !! !! A. N B C | ! , ! ! A, . > , 0 3.41 (B) \ (C) = ?. D 5. 5 a b 2 A H A (a) 6 H A (b). 8 ! 1! a b ! ! ! A0 A. ? H A (a) 6 H A (b). 0 3.41 A0 , , 2 '0 2 E(A0 ), 0
0
454
. .
'0 (a) = b. E!3 ! '0 A0 1!3 ! ' A. > , A | . K-!! !! ! 5 , ! 5 . 3.45. A = LK Ai (i 2 I), ! Ai ' . - A , -
) ) : 1) p Ai (i 2 I), ! L p- " " " ' 1, Tp (A) = K Tp (Ai )* 2) )' " " " ' B C
1 A "" (B) \ (C) = ?.
. < !5. 8 2) , ! 3.44. !, 1). 5 p | , ! fi 2 I j Ai p- ! ! ! 1g ( ! 1 ! Aip . ;!! Aip = A0ip A00ip , A0ip L | 0 . !, T (A) 6= 1 pA0ip 6= AL p ip K Tp (Ai ). ? 2 1! a 2 K Tp (Ai ), supfo(i a) j i 2 I g = 1, , supfo(i a) j i 2 s(a) i 6= ip g = 1. # 1! i a (i 2 s(a) i 6= ip ) ! ! ! A0i Ai , 02 ! !! - p- (Ai = A0i A00i ). ! A0i = 0 AL00i = Ai , L 0 0 00 i 2 I n (s(a) fip g). ;!! A = A A , A = K A0i A00 = K A00i (i 2 I). N A0 , ! ! A, L
. D A0 !! K Tp (A0i ) 6= Tp (A0 ), 1! A0i (i 2 s(a) i 6= ip ) . (> ! !, i 2 s(a) i 6= ip Tp (Ai ) = Ai .) L F , pA0ip 6= A0ip 0 ! A 3.39, ! K Tp (A0i ) = Tp (A0 ). . D 5. 5 a b 2 A H A (a) 6 H A (b). D i 2 s(a) s(b) ! 1! ia i b ! ! A0i Ai , 02 ! !! 1 (Ai = A0i A00i ). ! A0i = 0 L A00i = Ai , L 0 00 0 0 00 i 2 I n (s(a) s(b)). ;!! A = A A , A = K Ai , A = K A00i , ! a b 2 A0 H A (a) 6 H A (b). 8 ! 3.11 A0 K0 -! !! 1. ! 3.39, !, A0 | . > , 2 1!3 ! '0 A0 , '0 a = b. E!3 ! '0 A0 1!3 ! ' A. > , A | . $ 3.46. A | K- " "" ' ' . A | , )' " " " B C 1 A "" (B) \ (C) = ?. ; ! 3.45 5 , ! . 0
0
455
$ 3.47. K- " "" '
. x
4.
A 0 , ! A(M) = fa 2 A j H (a) > M g, M | ! - HA . ? ! 3 H - ! . # ! ! 3, H - ( 2.3). D , 4.5, . 2 0 , 02 H - ! . E 5 %8] %20]. = ! , , 5 , 02 H - ! . 8 2! 3 5 5 H - , 3 5 0 . ? 5 5 ( 5 02 H - ! )
. ! , %8]. 4.1. 5 A B | A0 | 0 A. # A B ! 5 B, 0 ! 1! ! a0 , 2 Hom(A B), a0 2 A0 . 8 5 ! ! ! 5 5 020 !! ! !! . 4.2 ('8]). G = L A, G0 = L A0, A |
2AA 0 ( 2 A).2A G0
G , $ $ 0
( ), 2 A, 06= , "" ' A A A ! A . 4.3. # ! 5 x 2, A !! HA H. ! A(M) A ! ! ! ! - M 2 H 02 ! - M 0 2 HA 02 ! !. 5 M 2 H, M = (ij )i2Nj 2N0, A(M) = fa 2 A j H (a) > M g. J ! ! 020 ! - M 0 = (ij )i2Nj 2N0: ij < i (i | pi - A), ij = ij ,
456
. .
ij > i , ij = 1. ;!! M 0 2 HA A(M 0) = A(M). 0 , A H - 5 , S ! S = A(M), M | ! - H. 8 %8] , ! 0 - . 80 1! 3, 5 H - , ! . / ! ! ! 5 - 0 . D ! ! H - ! ! 5 .
4.4. " " H - H - $. . 5 A | H - , A = A1 A2 S1 | -
A1 . 5 S2 | A2 , ! ! 1! ! c 2 A2 , 2 1! b 2 S1 , H A2 (c) > H A1 (b). Q, S2 | A2, ! !!3 S1 A2 S2 . !, !!3 S2 A1 S1 . n P 5 g 2 S2 , ' 2 Hom(A2 A1 ). ;!! g = gi, 1! gi i=1 2 1! bi 2 S1 , H A2 (gi) > H A1 (bi ) (i = 1 n, gi 2 S2 ). Pn ? 'g = 'gi H A1 ('gi ) > H A2 (gi ) > H A1 (bi ) i. i=1 0 2.3 A , A1 . > , 2 0 1!3 ! i 2 E(A1 ), ibi = 'gi (i = 1 n). = 5, 'gi 2 S1 i = 1 n, 1! 'g 2 S1 . 5 S = S1 S2 . 8 !! 4.2 S | A. ? A | H - , S = A(M) L ! - M 2 H, S1 = A1 (M) ( !, G = Gi , i2I L 0 ! - M 2 H G(M) = Gi(M)). = 5, A1 | i2I H - . !, ! H -. 4.5.
A H - $ , A | . . < !5 2.3. D ! 5. 5 A | L L S| A, A = Ap S = (S \ Ap ). p p # Ap | , ! ! A. 1! 0 p 2
457
U- 55 (p), S \ Ap = Ap ((p) ) ( S \ Ap = 0, U- 55 (p) 5 ! 1). J ! ! ! - M, p , 02 1! ! , (p) . ! S = A(M), , A | H - .
$ 4.6.
)'$ $
H - $.
. 5 G | , 5 a b | 1! T(G) G, H T (G) (a) 6 H T (G) (b). ? T(G) | G, H G (a) 6 H G (b). > , 2 ' 2 E(G), 'a = b. 5 = 'jT(G), | 1!3 ! T (G) a = b. = 5, T(G) | , 1! 0 4.5 T(G) | H - . 8 x 3 . = 0 4.6 0 H - ! . 8 x 2 !, A H - 5 , A | - ( 5 0 S A ! S = A(v), v | ). 8 %8,20, 21] 5 5 - . , !, ! A ( , ) - 5 , A | %8]. 0 , 0 5 | - . F , 5 P+ - %21]. 0 , , ! 5 Qp -! p ( , , p- ), -. D , 5 A - 5 , A | , %21]. 8 %20] -, 02 K-!! !! ! ( , !! !! ! !! ! ) . < !, - K-! !! 5 , , K-! !! 5 P+ , K-! !! ! . J ! ! K-! !! Ai (i 2 I), Ai 5 5 P+p p. 8 %20] , G,L 02 K-! !! , ! G = K At , At | Kt -0
458
. .
! !! t (At 5 ). < ! 02 5 . 4.7 ('20]). A = LK At (t 2 T , T | "! ), ! At | Kt- " "" Ai (i 2 It ) ! t. A - $ ,
) ) : 1) A | " "" At , t 2 T * 2) A ) * 3) "$ fAi gi2I t 2 T * 4) )' t 2 T At | " "" Ai (i 2 It ) t
" . t
;5 !! 3.29, ! 3.30 4.7, ! 5 . 4.8. A = LK At (t 2 T , T | "! ), ! At | Kt- " "" Ai (i 2 It) ! t,
" + Ai $ ' $ ' $ Pp p. A - $ , 1) A | " "" At , t 2 T * 2) A ) * 3) t 2 T "" k l 2 It Ak | ' , Al | ' * 4) t (t 2 T ) " ", At | " "" Ai (i 2 It). . < !5. F 1), 2), 4) 0 ! 4.7. ? - , , ! ! 3.30, ! 3). D 5. > ! !, t 3) ! fAigi2I 5 5 , 5 5 P+p p. ? 0 ! 5 !, , !! 3.29, !, ! fAi gi2I ! 0 t 2 T . ! 5 ! 4.7, !, A | - . $ 4.9. A = LK At (t 2 T , T | "! ), ! At | Kt- " "" Ai (i 2 It) ! t,
" Ai ' $ $ $, $ "! Qp-" p ( , Ai | , p- $
). A - $ , ) 1){4) " 4.8. ; ! 3.30 4.8 t
t
459
$ 4.10. A = L Ai, ! Ai i 2I
$ ' $ ' $ P+p p. +) # : 1) A | * 2) A ) , t(Ak ) = t(Al ) (k l 2 I), Ak | ' , Al | ' * 3) A | - . $ 4.11. A = L Ai, ! Ai $ ' $ $ i2I $ ' , $ "! Qp-" p ( , Ai | ' , p- $
). ( A # 1), 2), 3) 4.10.
J ! ! 5 202 ! . 4.12. A = T G | ) " , T |
, G | ' . - A H - $ , G | - : p, Tp , G p- "$ $, Tp | $ $. L . < !5. ;!! A = Tp G. 5 A | p H - . !! 4.4 G | - , p Tp | . > ! !, p- 5 , 1! . !, pk Tpk , G pk - !. 5 g | 1! G, pk - L , (g) = v. J ! ! S = Tpk Tp0 G(v) A, Tp0 p 6= pk 5 p6=pk !!3 G(v) G Tp . 8 !! 4.2 S | A. ? A | H - , S = A(M) ! - M 2 H. ;!! g 2 G(v), 1! g 2 S, , H (g) > M. D 5 n 2 - ! ! Tpk , !5 n, 1! ! - M k- ( , 02 ! pk ) ! (0 1 2 3 : ::). J ! ! 1! b G, pk b = g. ? (b) < v, b 2= G(v) , , b 2= S. = , , ! - H (b), 0 ! k- , , ! - H (g), ! H (b) > M. > , b 2 A(M), 1! b 2 S. . D 5. 5 N1 = fi 2 N j Tpi | g, N2 = fi 2 N j Tpi | g, N3 = N1 N2 S |
460
. .
L
A. S = (S \ Tpi ) (S \ G) = i2N3 L T (u ) G(v), u | U- 5 , = v | pi i i i2N3 . (ui = (ui0 ui1 : : : uin : : :)( v = (v(1) v(2) : : : v(n) : : :).) F , G pi- ! 0 i 2 N2, !, v ! 5 , 0 i 2 N2 5 v(i) = 1. (H G(v) = 0, ! v , 20 5 ! 1.) ? Tpi | i 2 N1, i 2 N1 2 5 ni , pni i Tpi = 0, pni i;1 Tpi 6= 0. U- 55 ui i 2 N1 ! (ui0 ui1 : : : uimi 1 1 : : :), uimi < ni . J ! ! 55 (v(i) v(i) + 1 v(i) + 2 : : :) ( v(i) = 1, ! v(i) + n = 1 0 5 n). !, i 2 N1 0 ui0 6 v(i) ui1 6 v(i) + 1 : : : uimi 6 v(i) + mi : (1) H v(i) = 1 v(i) > ni ; 1, (1) 0 ! !. 5 v(i) 6= 1 v(i) < ni ; 1. ! ni ; 1 ; v(i) ri . =2 1! ai 2 G(v), hGpi (ai ) = v(i) . J ! ! 55 (v(i) v(i) + 1 : : : v(i) + ri 1 1 : ::). E 55 ! v(i) + ri , v(i) +ri = ni ; 1 fni ;1 (Tpi ) 6= 0, Tpi 2 1! bi , Hpi (bi ) = (v(i) v(i) + 1 : : : v(i) + ri 1 1 : : :). ;!! H (ai ) < H (bi ), , 0 2.12 2 ' 2 Hom(G Tpi ), 'ai = bi . 8 !! 4.2 !!3 G(v) G Tpi (ui ), 1! bi 2 Tpi (ui ). 0 !! mi > ri ui0 6 v(i) , ui1 6 v(i) + 1,.. ., uiri 6 v(i) + ri. D 5 l, 02 ri < l 6 mi , , uil 6 v(i) + l, v(i) + l > v(i) + ri = ni ; 1, uil < ni . ; , i 2 N1 0 (1). J ! ! ! - M 2 H, 0 !! i, i 2 N1, ! (ui0 ui1 : : : uimi v(i) + mi + 1 v(i) + mi + 2 v(i) + mi + 3 : : :), !! j, j 2 N2, ! uj , 0 !! k, k 2 N n N3, ! (v(k) v(k) +1 v(k) +2 : : :). ? S = A(M), 1! A | H - . ; ! 4.12 %2, ! 100.1, . 222] 02 .
$ 4.13. 0 $
" ' H - $ , # ' | - . F 5 - %8] %21], ! 5 .
$ 4.14. 0 $
" ' $ ' H - $ , # ' ' .
461
$ 4.15. 0 $
" $ ' $ ' $ P+ H - $. $ 4.16. 0 $
" ' , $ "! Qp-" p ( ,
" ' , $ p- $
), H - $.
D ! A ! P 0(A) ! p, Tp (A) | . ;! ! 02 4.17. & " $ A #" ' , ") $ ' ) q0- ('')) )' q,
! P (A), A H - $. . 5 a 2 A, o(a) = 1 hq (a) = 1 q, q 2= P 0(A). !, A H -. J ! ! 0 S A: L T 020 S= (A). 5 M1 | ! - H, , p p2P (A) 02 ! ! p 2 P 0(A), 02 : (0 1 2 : : :), 5 5 ! 1. M1 | 5 ! - M 2 H, S = A(M). ? S | , a 2= S. = , H (a) > M1 , 1! a 2 A(M1 ). . < ! !, p- (p | ), ! 0 q, p, !3 !! 1 . p- 5 | 1 , ! 5 Qp -!, Qp | 5- - 5 , ! ! p. 4.18. p- A H - $ 0
, $: 1) A | p- * 2)
A | p- , - $ .
. < !5. 5 p- 5 A
H -. H A | , , p- 55,
! A | p- . ? ! 1.15 A | ( A | , , 5 , ). ; , A | , 1) 2). H A | , A | - , 1! 0 1.18 A | , 5 1! 2). 5 A | ! . ? , p- 55 A, !, 5
462
. .
A p- qA = A q, p. D 0 1! a 2 A !! hq (a) = 1, q 6= p. 8 !! 4.17 !, 5 T(A) A (T (A) = Tp (A)) . ? A = Tp (A) B, B |
%2, ! 100.1, . 222]. !! 4.4 B | - , , B | . D 5. H A | p- , H -. 1! ! ! 1) 2) A | p- , A | H - . H 2) 5 A | , 5 A | , A | (qA = A q 6= p). ? A | , A | - %8]. H 2) A | ! , A = Tp (A) B, Tp (A) | p- , B | (qB = B q 6= p). F , - %8] ! ! 4.12, !, A | H - . 8 !, ! K-! !! H -. 4.19. A = LK Ai (i 2 I), Ai |
. - A H - $ , A |
.
. < !5. 5 A = LK L Ai (i 2 I), Ai | A | H - . ;!! Ai = Aip (Aip | p-!p L A . D A ). F ! 3.11, ! A =
i K ip 3 p ! Ip 02 ! : Ip = f(i p) j i 2 I g. ? ! Ip 0, ! ! ! 2 5 - 0 ! 3.12. ? 02 Kp ! ! Ip K1 L L ! ! O !! A = K1 Gp, Gp = Kp Aip . D p ! p p Gp A - 0 A Gp . !, pj Gpj . 5 Tpj | pj -! A. ? Tpj pj Gpj Tpj 6= pj Gpj . ? Gpj 5 1! , 5 k 2 Aipj 1! ak 2 Aipj , o(ak ) > pkj . > , Tpj | . 5 a | 1! Gpj b = pj a. ;!! b 2 A, o(b) = 1 hq (b) = 1 q, pj . ? !! 4.17 A H -. . 0
L
463
; , A = K1 Gp , Gp | p-. L !, A . ;!!, T = p Gp | 5 p A, A, T 6= A. 5 ! - M1 2 H : p | , Gp | , ! - M1 , 02 1! ! p, ! 55 - - 5 (0 1 2 : : :)( p | , Gp | pm - , m 2 N ( 5 pm Gp = 0, pm;1 Gp 6= 0), ! - M1 ! 02 : (0 1 : : : m ; 1 1 1 : ::)( 5 ! - M1 5 ! 1. M1 | 5 ! - M 2 H, T = A(M). 5 a 2 A n T. ;!! H (a) > M1 , 1! a 2 T. . ; , A | . ? A | H - , A | . D 5 4.5. ;5 ! 3.8, ! $ 4.20. A = LK Ai (i 2 I), Ai |
. - A H - $ , "$ fAigi2I : ' "! I1 "! I , I1 2 K )' "$ #" faigi2I1 , ai 2 Ai, "" supfo(ai)gi2I1 < 1. ! ! 0 K-! !! , ! ! 20. 4.21. A = LK Ai (i 2 I) | " ,
" i 2 I Ai | " , . - A H - $ , A | )
) ) : 1) - A
$ - $* 2) p,
A " ) p- " , # p- " $ $, - A
$ p- " .
. < !5. > ! 0 Ai (i 2 I) Ai = A0i A00i , A0i | , A00i | ( L A0i L A00i ! 5 ). ;!! A = A0 A00 , 0 0 00 A = K Ai , A = K A00i . ? A | H - , !! 4.4 A0 | H - A00 | H - . ! ! 4.19, !, A0 | . , A00 | . ; , A | 202 ! . ! ! 4.12, ! 1) 2). D 5 ! 4.12.
464
. .
8 ! 4.21 ! ! 2! K-! !! . 5 ! 2 5 , - 5 2, ! ! ! 5 ! ! - . 4.22. G = LK Gi (i 2 I) | " . - G
, ) ) : 1) Gj (j 2 I) | " , Gj * 2)
G K- "$ ""$
$ Gi (i 2 I).
. < !5. 5 Gj (j 2 I) | ! . ;!!: j Gj | ! ! G, 5 G = j Gj G0 (2) T (G) |- G, , T (G) = (T(G) \ j Gj ) (T(G) \ G0 ): (3) ? G | 202 , G = T (G) B, B |
. F (3), ! G = (T(G) \ j Gj ) (T (G) \ G0 ) B. ;!! T (G) \ j Gj = T (j Gj ). > , G = T(j Gj ) (T (G) \ G0 ) B: (4) F , T(j Gj ) j Gj , (4) !, T (j Gj ) | ! ! j Gj , , T (Gj ) | ! ! Gj . ; , 1) . ;L L D !, 2). Q, T K Gi K T(Gi). !, L 0 ; ! ! L !. . 5 a 2 K T (Gi), a 2= T K Gi , 5 o(a) = 1. ;!! supfo(ia)gi2I = 1. =2 0 Qik 2 I (k 2 N), k 2 N o(ik a) > k!. J ! ! 1! g 2 Gi , i2I k 2 N ik g = k!ik a m 2 I, m 6= ik (k 2 N), !! m g = 0. L ? s(g) s(a), g 2 K Gi. E! g + T (G) 3 - G=T(G) 0 5 n. ? G 2, G = T (G) B, B | , , G=T(G) = B. 5 1! !3 ! g+T(G) 1! b B. ? hbi | ! B. - 50 G. D 5. 0 Gi (i 2 I) ! !! T(Gi ) Bi ( T(Gi) BiL! 5 ), !, G = G0 G00, ;L L L 0 00 G = K T(Gi), G = K Bi . ? K T(Gi) = T K Gi , G0 = T(G). > , G | 202 . ; ! 4.21 4.22 02 .
465
$ 4.23. LK Ai (i 2 I) | " ,
"
i 2 I Ai | " , . - A H - $ , ) ) : 1) - A
$ - $* 2)
A K- "$ ""$
$ Ai (i 2 I)* 3) p,
A " ) p- " , # p- " $ $, - A
$ p- " .
! 4.23 3.10, ! 02 5 . $ 4.24. LK Ai (i 2 I) | " , ! Ai ' $ ' , '
$ -
$, )'
" " $
! ' " ) : ) * ) ' * ) C- )' , !* ) * ) ) $ * ) IT - . - A H - $ , ) ) : 1) - A
$ - $* 2)
A K- "$ ""$
$ Ai (i 2 I)* 3) p,
A " ) p- " , - A
$ p- " .
= ! 3.27 ! 4.7, ! 3.30 ! 4.8, 2
! !, 2 0 , 02 H - ! . = 3.37 ! 4.12, ! 3.8 4.20, 3.39 4.24 , ! 5 , 02 H - ! . 8 0 3 ! ( 5 02 H - ! ) .
466
. .
< ! !, ! J L ) a 2 J, x 2 L a 6 x,
, 0 02 : x 2 J( ) a 2 J, b 2 J, a ^ b 2 J.
T 3 5 L, 0 !, ! Le.
4.25. & L | ! ' " #" ", e $ $. . 5 Le0 Le, Le0 6= ?. J ! ! J = T J 0. J 6= ?,
L
J 2Le 0
0
J 5 1! L. H a 2 J, x 2 L a 6 x, x 2 J 0 J 0 2 L0, , x 2 J. I , a b 2 J, a ^ b 2 J 0 J 0 2 L0, 1! a ^ b 2 J. ; , J | 3 5 J = inf Le0. F , L0 5 5 1! ( ! L), !, Le | %28, x 3, ! 1, . 37]. 4.26. fAigi2I | "$ ' , 'i 2 E(Ai) i 2 I , A = iQ2I Ai, ' = iQ2I 'i. ( K '$ ' "! "! I LK Ai | '- " A. L . J ! ! 1! a 2 K Ai, a = (: : : ai : : :). ;!! 'a = (: : : 'iai : : :), 5 i 2 I i'a = 'i i a. Q, s('a)L s(a) (s(a) | 5 1! a). > , s('a) 2 K, 1! 'a 2 K Ai . < ! !, 5 A ! H (A) ! ! - 1! A: H (A) = fH (a) j a 2 Ag. 4.27. A = LK Ai (i 2 I),
" i 2 I H (Ai ) | ! . % H (A) | ! ' " #-
" ".
. 5 H (a1 ) H(a2 ) 2 H (A). ? !! H (a1 ) = = inf H fH (i a1)gi2I , H (a2 ) = inf H fH (i a2 )gi2I . = 5, H (a1 ) ^ H (a2 ) = = inf H finf H fH (i a1)gi2I inf H fH (i a2 )gi2I g = inf H fH (i a1) H (i a2)gi2I . ? i 2 I H (Ai ) | , 0 i 2 I H (i a1) ^ H (i a2) 2 H (Ai ). = 5, i 2 I ! Ci = fci 2 Ai j H (ci ) = H (i a1 ) ^ HQ(i a2)g . 5 c | 1! Ai , i 2 I ic 2 Ci , ! i 2I ic = 0 ! 5 ! , ia1 = 0 i a2 =L0. ? s(c) = s(a1 ) s(a2 ) s(a1 ) s(a2 ) 2 K, s(c) 2 K 1! c 2 K Ai . > , H (c) 2 H (A). ;!! H (c) = inf H fH (i c)gi2I = inf H fH (i a1 ) ^ H (i a2 )gi2I = = inf H fH (i a1) H (i a2)gi2I = H (a1 ) ^ H (a2 ). ; , H (a1 ) ^ H (a2 ) 2 H (A). > , H (A) | . < 5 1! H (A) | 1
467
! - 1! A, 2 5 ! 1. H M | ! ! - H, A | , ! A%M] 02 ! A: A%M] = fa 2 A j H (a) = M M 2 Mg. ? , 2 3 , K(A) A. 4.28. A = LK Ai (i 2 I) | , i 2 I Ai | H - H (Ai ) | ! . S |
A , S = A%M], M | $ !$ H (A). + : M ! A%M] " " He (A) K(A). . < !5. 5 S | A M = fH (s) j s 2 S g. !, M | 3 5 H (A). 5 a 2 A H (a) > H (s), s 2 S. 8 A 2 1!3 ! ' 1 , 's = a. 0 , a 2 S H (a) 2 M. 5 s1 s2 2 S, 5 i 2 I Bi = f'i is1 + i i s2 j 'i i 2 2 E(Ai )g. Bi | Ai . ? Ai | H - , Bi = Ai (Mi ), Mi | ! - H. ;!! i s1 is2 2 Bi , 1! H (i s1 ) > Mi H (i s2 ) > Mi . ? H (Ai ) | , 2 1! ci Ai , H (ci ) = H (i s1 ) ^ H (i s2 ). ? H (ci ) > Mi , 1! ci 2 Ai (Mi ). > , Ui 2 0Q 1!3 ! 'Ui, Q Ai , ci = 'Ui is1 + Ui is2 . Q U2= 5 'U = 'Ui , U = Ui , ' U U 2 E Ai . ? i 's U 1 = 'Ui i s1 i s i2I i2I i2I U 2 ) = 'Ui i s1 + Ui i s2 = ci . 5 c = 's = Ui is2 . > , i('s U 1 + s U1+ U + s2 . ! !! 4.26 , S | A, ! c 2 S, 1! H (c) 2 M. ;!! ic = ci H (c) = inf H fH (i c)gi2I = inf H fH (ci )gi2I = inf H fH (i s1 ) ^ H (i s2 )gi2I = = inf H fH (i s1 ) H (i s2 )gi2I = H (s1 ) ^ H (s2 ). ; , 0 1! H (s1 ) H (s2 ) M H (s1 ) ^ H (s2 ) M. > , M | 3 5 H (A). , S A%M]. 5 a 2 A%M]. ? 1! a ! 0
0 ! -, 1! s S. ; A , a 2 S. > , A%M] S S = A%M]. D 5. 5 M | 3 5 H (A). !, A%M] | A. F ! , A%M] | A. 5 a b 2 A%M], 5 H (a) H (b) 2 M. ? H (a ; b) > H (a) ^ H (;b) = H (a) ^ H (b), H (a ; b) 2 M, 1! a ; b 2 A%M]. = 5, A%M] | A. 5 ' 2 E(A) a 2 A%M], H ('a) > H (a) H (a) 2 M, H ('a) 2 M.
468
. .
0 'a 2 A%M], , A%M] | A. F ! !3 ! He (A) K(A). J ! ! : He (A) ! K(A), M 2 He (A) M = A%M]. =0C 5 ) 5 !. ;C 5 . Q , M1 M2 5 , M1 M2. > , He (A) K(A) !3. D ! !, G | , H (G) | . T ! ! 5 - ! ! , a b 2 G a | 1! ! G, H (a) ^ H (b) = H (b) 1! H (a) ^ H (b) 2 H (G). > ! ! 2 , G | , ! ! H (G) ! 5 ! (G) 1! G ((G) = f(g) j g 2 Gg). , G (G) | ! 5 ! , H (G) | . 4.29. & G |
' , H (G) | !
.
. F , ! !! ! ! ! !! 4.27, !, , ! 2 , ! 5, G | p- . 1! !5 p- 1! G. 5 a b 2 G, Hp (a) = (0 1 : : : n : : :), Hp(b) = (0 1 : : : n : : :) Hp (a) ^ Hp(b) = = (0 1 : : : n : : :). ? Hp(a) Hp (b) ! 5 ! 1, Hp(a) ^ Hp (b) 1 ! !. 5 5 Hp (a) ^ Hp (b) 5 ! i i+1 , 5 i + 1 < i+1 . ! , i 6 i . ? i = i , , i+1 > minfi+1 i+1 g = i+1 > i + 1 = i + 1. ? ! i i+1 ! , fi (G) 6= 0, fi (G) 6= 0. 1! 55 Hp(a) ^ Hp(b) p- ! 1! G %2, !! 65.3, . 9]. ; , H (a) ^ H (b) 2 H (G). 4.30. & G | ' ' , (G) ( , H (G)) | ! . . 5 a b 2 G. ? (a) (b) 2 t(G), 1! (a) ^ (b) 2 t(G). > , G 2 1! c, (c) = (a) ^ (b) ( 1! c 5 hai ). F , - ( , H -) %8] ! 3.7, 4.5, !! 4.29, 4.30 ! 4.28, ! 02 5 .
469
4.31. A = LK Ai (i 2 I) | -
, ! Ai |
' . S |
A , S = A%M], M | $ !$ H (A). + : M 7! A%M] " " He (A) K(A). ;5 ! 4.12 4.31, ! $ 4.32. A | ) -
, - $
$ $ $. + ) ! : 1) S |
A , S = A%M], M | $ !$ H (A). + : M 7! A%M] " " He (A) K(A)* 2) A H - $ , )' p, Tp(A) , A=T(A) p- "$ $. ; ! 4.31, 3.10 ! 4.19 $ 4.33. A = LK Ai (i 2 I), Ai |
, )'
" "
! ' "
) : ) * ) ' * ) C- )' , !* ) * ) ) $ * ) IT - . + ) ! : 1) S |
A , S = A%M], M | $ !$ H (A). + : M 7! A%M] " " He (A) K(A)* 2) A H - $ , A |
.
H G | , M | ! (M X), ! G%M] 02 ! G: G%M] = fg 2 G j (g) = v v 2 Mg. D 0 1! g 2 G (g) ! -! ! - H (g), (g) = (v(1) v(2) : : : v(n) : : :),
470
. .
0v(1) v(1) + 1 BBv(2) v(2) + 1 H (g) = B B@ :(:n:) (n:): : v v +1
1
: : : v(1) + n : : : : : : v(2) + n : : :C C: : : : : : : : : :C C : : : v(n) + n : : :A ::: ::: ::: ::: ::: 1! 2 ! (G) ! H (G): 0v(1) v(1) + 1 : : : v(1) + n : : :1 BBv(2) v(2) + 1 : : : v(2) + n : : :CC (1) (2) ( n ) (v v : : : v : : :) = B B@ :(:n:) (n:): : : : : (n:): : : : :CCA : v v + 1 ::: v +n ::: ::: ::: ::: ::: ::: # ! 5, G (G) | 5 , H (G) | . < , ! M ! (G) 3 5! (G) 5 , M | 3 5 H (G). H (G) | ( 5 1! | 1! , 2 5 ! 1), e(G) !3 He (G). ; !3 e(G) He (G) - - . ! ! 4.31, 3.23, 3.28, 3.31, 3.32, 3.15 , 5 P+ ! !! , ! | !! ! , ! $ 4.34. A = LK Ai (i 2 I), Ai ) " ) $: 1) Ai | ' ! * 2) Ai | ' )' Ai1 , Ai1 (i1 i2 2 I), t(Ai1 ) 6= t(Ai2 ), "" (Ai1 ) \ (Ai1 ) = ?* 3) Ai | ' P+ * 4) Ai | ' , !$ "! Qp-" p ( , ! Ai p- $
p)* 5) Ai | ' " ' . % S |
A " " , S = A%M], M | $ !$ (A). + : M 7! A%M] " " e(A) K(A). ; ! 4.31 3.39 $ 4.35. A = LK Ai (i 2 I), ! Ai -
' $ ' $ $ ' , '
$
471
$, )'
" " $
! ' "
){),
4.33,
" )' ' Ak Al (k l 2 I) (Ak ) \ (Al ) = ?
p, p- " ' Ak (k 2 I) "" LK Tp(Ai) = Tp(A). % S |
A " " , S = A%M], M | $ !$ H (A). + : M 7! A%M] " " He (A) K(A). ! 5 K-!! !! ! 5 , ! 020 !. 4.36. A | K- " "" ' ' -
. +) # : 1) A | * 2) )' " " " B C 1 A "" (B) \ (C) = ?* 3) S |
A , S = A%M], M | $ !$ (A). + : M 7! A%M] " " e(A) K(A).
. E 5 1)L 2) 3.46. ! 1) =) 3). 5 A = K Ai (i 2 I) | , Ai | 5 . ? Ai . ? 0 Ai ! %21], A, ! 0 3.11, K0 -! !! . ! ! 4.31, ! A. ;! - 3) =) 1) 0 A, (A) | ( ! (A) H (A) 0 A, H (A) | ). ! 1. 5 a b 2 A (b) > (a) (H (b) > H (a)). J ! ! S = fa j 2 E(A)g. S | A, , 2 3 5 M (A) (H (A)), S = A%M]. a 2 S, 1! (a) 2 M (H (a) 2 M), (b) 2 M (H (b) 2 M). ; , b 2 S, , 2 1!3 ! A, a = b. > , A | .
1] Kaplansky I. Innite abelian groups. | Michigan, Ann. Arbor: Univ. of Michigan Press, 1954. 2] . . . 2. | .: ! , 1977.
472
. .
3] Hill P. On transitive and fully transitive primary groups // Proc. Amer. Math. Soc. | 1969. | Vol. 22. | P. 414{417. 4] Corner A. L. The independence of Kaplansky's notions of transitivity and full transitivity // Quartery J. Math. | 1976. | Vol. 27. | P. 15{20. 5] * !+ ,. -. ,./ / 0!/!1 1 // 2 ! 34!. | 1996. | 5 . 13{14. | ,. 54{61. 6] !+! 2. 6. 2 // 2 . !7. *3/ !7. . 10. (9/ ! ! ! /1.). | .: 59;9 9 2; ,,,<, 1972. | ,. 5{45. 7] !+! 2. 6. 2 // 2 . !7. *3/ !7. . 17. (9/ ! ! ! /1.). | .: 59;9 9 2; ,,,<, 1979. | ,. 3{63. 8] * !+ ,. -. > / !! 1 / !/!!1 4 1 0 !7 // 2 ! 34!. | 1981. | ,. 56{92. 9] ? 6. 2. > 1 / !/!!1 4 1 1 0 !7 // ,. ! . / 3/3. | 3: 3. !., 1973. | ,. 15{20. 10] @ ! B. . ?0! / !D/! ! / 0!/! 0 !7. | @ . 59;9 9. | 1977. | E 2942-77@F6. 11] Reid J. D. Quasi-pure-injectivity and quasi-pure-projectivity // Lect. Notes Math. | 1977. | Vol. 616. | P. 219{227. 12] Arnold D. M. Strongly homogeneous torsion free Abelian groups of nite rank // Proc. Amer. Math. Soc. | 1976. | Vol. 56. | P. 67{72. 13] ? 6. 2. ,!W 4 4 0 !7 // ,!. 3/3. X . | 1983. | E 2. | ,. 77{84. 14] Hausen J. E-transitive torsion-free Abelian groups // J. Algebra. | 1987. | E 1. | P. 17{27. 15] Dugas M., Shelah S. E-transitive groups in L // Contemp. Math. | 1989. | Vol. 87. | P. 191{199. 16] Arnold D. M., Vinsonhaler C. I., Wickless W. J. Quasi-pure projective and injective torsion-free Abelian groups of rank 2 // Rocky Mountain J. Math. | 1976. | Vol. 6. | P. 61{70. 17] @ ! B. . ?0! / !D/! // 2 ! 34!. | 1979. | ,. 45{63. 18] ? 6. 2. ;/ !3 0! / !D/!1 ! / 0!/!1 1 0 !7 // 2 ! 34!. | 1988. | 5 . 7. | ,. 81{99. 19] [1 2. <. ?0! / !D/! 0 !7 // /. 03/!. | 1989. | . 46, E 3. | ,. 93{99. 20] * !+ ,. -. 5 1 / !/!! 4 K- 731 33 1 0 !7 // 2 ! 34!. | 1996. | 5 . 13{14. | ,. 37{53. 21] * !+ ,. -. 5 / 0!/! 4 4 W // /. 03/!. | 1997. | . 62, . 3. | ,. 471{474. 22] * !+ ,. -. 5 1 / !/!! 4 W1 1 // 43. ! !. 3/. | 1998. | . 4, . 4. | ,. 1281{1307. 23] * !+ ,. -., !7 5. . > / 0!/!1 1 1 // 2 ! 34!. | 1986. | 5 . 6. | ,. 12{27.
473
24] * !+ ,. -., !7 5. . 5 / 0!/!/W 731 !04!. 1 // 2 ! 34!. | 1991. | 5 . 10. | ,. 23{30. 25] !7 5. . > / 0!/!/! 4\! 1 1 // 2 ! 34!. | 1994. | 5 . 11{12. | ,. 134{156. 26] * !+ ,. -. 5 / 0!/!/W K- 731 33 1 // 0. 4. X4 . ]. / !! 37/! ,. ;. [ !. | 6 3W, 1997. | ,. 22{23. 27] . . . 1. | .: ! , 1974. 28] , 7 . 2. ^3/ / !! / / . | .: ;, 1970. 29] ! ] *. !7 +/. | .: ;, 1984. 30] Le Borgne. Groupes -separables // C. R. Akad. Sci. | 1975. | No. 12. | P. 415{417. 31] Walles K. D. C -groups and -basic subgroups // Pacif. J. Math. | 1972. | No. 3. | P. 799{809. 32] Hill P., Megibben Ch. On the theory and classication of Abelian p-groups // Math. Z. | 1985. | Vol. 130. | P. 17{38. 33] < ,. 5. > 731 !04!71 1 // /. !. | 1982. | . 117, E 2. | ,. 266{278. 34] 6 10 . 6 73 33 /! P+ // Comment. Math. Univ. Carolin. | 1967. | No. 1. | P. 85{114. 35] * !+ ,. -. > / x 33 ]!03 1 // 90. +. . 04. /3. | 1998. | E 9. | ,. 41{46. 36] 9. z. 6 3 !. 4 3X 733! 333! M- W1 /! P + // 2 ! 34!. | 1982. | ,. 20{33. 37] ?! . -. >{ !3 , I // 4 >. | 1952. | . 1. | ,. 247{326. 38] Megibben Ch. Separable mixed groups // Comment. Math. Univ. Carolin. | 1980. | No. 4. | P. 755{768. 39] * !+ ,. -. ? / 0!/!/! 731 !04!. { W1 1 // 2 ! 34!. | 1994. | 5 . 11{12. | ,. 90{92. & ' 1999 .
4
. .
( ) e-mail: [email protected]
511.6
: , ,
! .
" # $ $ % & ! # &%& #' 4 %)& %* 4 5. + ' , ' , &%, #$, ', &% #.
Abstract
Yu. Yu. Kochetkov, Trees of diameter 4, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 475{494.
We present a survey of works on Galois orbits of plane trees of diameter 4 with the central valency 4 and 5. Examples of nontrivial orbits are considered and criteria of a nontrivial decomposition are proved.
4 , . , . !
" , . ! . $. % &. '. ! ( " .
1.
) . * * " | , | , " . -) . . & * , : , ( . . ( 0 " , ( ). 2 T 3T ]. 5, , , . , 2002, ' 8, 5 2, . 475{494. c 2002 !", #$ %& '
476
. .
8 * . 0 ) v0 , , , 0 0. : ) :8 (: : :). 5 * , , | . v1 : : : vi 1, 1 1 , :81 : : : :8i :80 . 2* 2 ) , * )( :81j , : ( , )( vj v0 ) , . . = , * - : * , . | " , . : , , . - | ( ) * (), )( . -, * T , hT i = hk1 : : : ks? l1 : : : lti, ki (lj ) | (, ) . -, (()()(())) * h3 1? 2 1 1i ( h2 1 1? 3 1i, * )). = , k1 + : : : + ks = l1 + : : : + lt = n, n | , . 8 T * B | ( ( ), )( . 8 T n , , . C a b 2 Sn , a | ( , , b | , . 8 a )( . C i " | A. 8 j )( A, i. - a i j . - (()()(())) 2 h3 1? 2 1 1i, a = (123), b = (34). 8 GT Sn , *, a b, ( T . 5 T
(, GT Sn An, " . :( - * (, D , (='D), . . , )( , 0 1 31]. - p Tp = p;1(30 1]). $ ) 0, , | 1 ( ). ( Tp p 32]. 5
) T ( ='D pT 00" ,
TpT 2 3T ] (pT , , , ). 5 ='D 00" .
4
477
' * )( . 8 q(z ) = (z ; a1 )k1 ;1 : : : (z ; as )ks ;1 (z ; b1)l1 ;1 : : : (z ; bt)lt ;1 a1 : : : as (b1 : : : bt) | (, ) k1 : : : ks (l1 : : : lt). 8 p(z ) | 0 q(z ), . . p0(z ) = q(z ). E , , ( p(a1) = : : : = p(as) (1.1) p(b ) = : : : = p(b ): T 2 hk1 : : : km ? l1 : : : ln i, , k1 : : : ks > 1 l1 : : : lt > 1. =
1
t
(C 0 1 .) 2 2 , * * - ai ( , a1) 0, - aj , j > 1, bi 1. C , , pT 00" ( , | pT ). ='D p T 2 hk1 : : : km ? l1 : : : ln i )(: * V = fk1 : : : km g , . . V = fm1 : : : m1 m2 : : : m2 : : : mr : : : mr g, mi ni m1 > m2 > : : : > mr . p(z ) r Y p(z ) = pi (z )mi pi (z ) = z ni + ai1z ni ;1 + : : : + aini : (1.2) i=1 2 pi |
mi . - r .Y q(z ) = p0 (z ) pi(z )mi ;1 | i=1
, 1. 8 p q | . F , n1 + : : : + nr ; 2 aij , 2 , . 8* ) a11 ( . . 00" z n1;1 p1 ) ) ) a21 ( . . 00" z n2 ;1 p2 ) 1, , . F G H ( , B , *. 1.1. C T = (()()(((())))) 2 h3 2 1? 2 2 1 1i. ' p = pT (1.2): p(z ) = z 3 (z ; 1)2 (z ; a) a | 1. 500 " p, , , 2 ) b1, b2 q = 6z 2 ; (5a + 4)z + 3a:
478
. .
- p b1, b2 ) , p q
. 8 p q ) 00" z , 625a5 ; 1100a4 + 184a3 + 176a2 + 128a ; 256 = 0: ' q, 25a2 ; 32a + 16. - , 2 , ), ) 25a3 ; 12a2 ; 24a ; 16 = 0: F ( , * 1 T . & ) ? - h3 2 1? 2 2 1 1i * (, : (()(())((()))) (()((()))(())). 2 ) * 1
. 5 , ( ), )( ='D T , ='D hT i. L | Gal(QN =Q), p(z ) = an z n + : : : + a0 00" p(z ) = (an )z n + : : : + (a0 ): L p | ='D, p * ='D, , T p 2 hTp i. : p | . L p0 (x) = p0 (y) = 0 p(x) = p(y), (x) (y) | ( p)0 p( (x)) = (p(x)) = (p(y)) = = p( (y)). - T , p0T . & . 1.2. = T * N =Q)g hT i: f3T pT ]: 2 Gal(Q - * , B ( , ) . * . 2 * )( . 8 p q | ='D 00" 3Tp ] = 3Tq ]. - q(z ) = p(az + b), a b | . : , q(z ) = p( (a)z + (b)), . . p q ) * (). ' ,
T = T p : T 1.3. 2 * ='D p *, 00" . 5 Tp Tp
4
479
. L T ( ) T 0 , T T 0 * . P )( 0 : * , ( ) (. * ). 8 , * ( , *. ( * , , () MAPLE. 1.4. - h4 2 1? 2 2 1 1 1i * 4 T1 = (()()(())((()))) T2 = ((())()()((()))) T3 = (()(())()((()))) T4 = (()()()(((())))): 5 T1 T2 ) ( ( PGL(2 7), ( T1, , * a = (1234)(67) b = (35)(46)), T3 T4 ) ) ( ( A7 ). 2 * , * ? V * )( . P (1.1), )( ='D, ) , . P , ) . F , , * * G H * (ai ; aj ) (bi ; bj ). 8 * , , ( (. *). - Q, *, , *. 8 , * * ) . , | P . - * D , (, , ), 00" "
. 8 1 2 | P p1 p2 | D ,. L P , ( , (1 ) = 2. F p1 p2 . L P , P = QR, , )( ) Q, * , , )( ) R. 1.4 ( ). ' ='D, 1.1: p(z ) = z 4 (z ; 1)2(z ; a) q(z ) = 7z 2 ; (6a + 5)z + 4a: 8 ) 00" z p q: 1944a6 ; 3132a5 + 342a4 + 359a3 + 311a2 + 225a ; 625 = 0: W 36a2 ; 52a +25 q. C, 54a4 ; 9a3 ; 41a2 ; 43a ; 25 = 0: & : 54a4 ; 9a3 ; 41a2 ; 53a ; 25 = (2a2 + a + 1)(27a2 ; 18a ; 25):
480
. .
- , * 2 . 8 * , () T1 T2 (a ), | , () T3 T4 (a ( ). C* * . = | * ( . 5
" . 1.5. 5 T " , ( * ( * 0) T1 , )( ) a b. (F * * .) & ='D " : T ! T1 ( g,
pT = pT1 g. 8 T " T1 , T " T1 . 8 T1 . - " * T ( i | i- ). - pT = pT1 g, 1 2 ) v1 v2 T k, *( , (1 ) (2 ) ) T k, * *( ( g( (1 )) = (g(1 )) = (g(2 )) = = g( (2 ))). 1.6. C c , , *( h3 3 3 3 3 3 3? 3 3 3 1 ::: 1i: T1 = ((()())(((()())(()()))())(((()())(()()))())) T2 = ((()())(()((()())(()())))(((()())(()()))())): ( " T = (()()()) . 8 , T , T1 T2 3. E T1 * , T2 * . : , * . (& 3 .) 1.7. C h3 3 1 1 1? 3 3 1 1 1i, * 4 T1 = (()((()())())(()())) T2 = ((()(()()))()(()())) T3 = (()(()(()()))(()())) T3 = ((((()())())()(()()): F " ( . -
* 2 : T1 T2 ) ) , T3 T4 | ). 8 , T1 T2 ) " , , " * . L " , ='D - ) p(;z ) = 1 ; p(z ).
4
481
( , " ) ) ) * , . . * , ) . C* * , * . 1.8. C T , ;T = f 2 Gal(QN =Q): T 2 3T ]g Gal(QN =Q) KT , , ;T . F T . 8 | , * 00" ='D T . : ( T . , ='D pT 2 K 3z ], K Q | . 8 G | K Q, G K Q. L H G | T , H = f 2 G : pT = p(a z + b )g (1.3)
H G ( T . = L K (G H ), . . L = fx 2 K : (x) = x 8 2 H g
K L K L H 35]. , L Q H ( . . ( T ). 5*, D , T * , 00" * L. 5 , ( ) a 2 K b 2 K , pT (ax + b) 2 L3x]. = , a (1.3) ) " H K 35]. -
H 1(H K ) 35], ( a 2 K , a = (a)=a. - ab ) " H K , * , . . ( c 2 K , ab = (c) ; c. 8* b = ;c=a, b = b ; (ab)=a. P pT (x=a + b) = pT (a (x= (a) + (b)) + b ) = = pT (x=a + (ab)=a + b ; (ab)) = pT (x=a + b)
. . pT (x=a + b) 2 L3x]. - , * , T * ( . . ( ).
2. 4
5 4 (d4 - ) , * 4 ( ) , ( 4 ( * |
482
. .
, , )( ). E d4 - " , * . - d4- , , ( ) . 5 hk1 : : : kni, n | " , k1 : : : kn | . 5 , *(
hk1 : : : kni ) " . L ki , * (n ; 1)! . 5 T 2 hk1 : : : kni " , &=5(k1 : : : kn) > 1, " . , T * " T1 4 . L T1 4, 0 ' : T ! T1 " " . 5 , ' 0 . L T1 4, * . - , " , " . L ki , ='D , *(
hk1 : : : kni, p(z ) = (1 ; x1 z )k1 : : : (1 ; xn;1z )kn;1 (1 ; z )kn 1=x1 : : : 1=xn;1 1 | k1 : : : kn;1 kn . - " ( 0) n, p(z ) = 1 + bnz n + : : :: F , n ; 1 xi = 1=ai : 8k x + : : : + k x + k = 0 > :k xn;1 + : : : + k xn;1 + k = 0 11 n;1 n;1 n P) x1 : : : xn;2, , ) (n ; 1)! xn;1, hk1 : : : kni. '
ki , 00" , * , - ki )( G H * . F , , )( * 34]. 2.1. 8 * &. ! . C hk k k l li, * 2 (k k k l l) (k k l k l). ='D p(z ) = (z 2 + az + b)l (z 3 + cz 2 + dz + e)k = f0 + f5 z 5 + : : : : 8 4 a b c d e. P) c d e a = 1, ) b 12k2b2 ; 12k(2k + l)b + 6k2 + 5kl + l2 = 0
4 483 D = 48k2(3k + 2l)(2k + l). L D | , * . C
3(3k + 2l)(2k + l) = m2
" * : 24r ; 72r2 l = 36r2 ; 60r ; 47 k = 106; r2 + 7r + 2 6r 2 + 7r + 2 r " . - * k l ) r 2 3x1 x2], x1 | " 36x2 ; 69x ; 47 = 0, x2 | " 72x2 + 24x ; 10 = 0, x1 ;595=1024, x2 ;589=1024. & , r = ;59=102 k = 5, l = 6. (L r 2 3x1 x2] " k l, , * ( , , * .) F 0 * d4- . 2.2. :)( , !. , * . C hk k k k k l li, * : (k k k k k ll), (k k k k l kl), (k k k l k k l), , ( . ='D p(z ) = (z 2 ; z + a)l (z 5 + bz 4 + cz 3 + dz 2 + ez + f )k = + z 7 + : : :: P) b c d e f , ) a 15k3a3 ; 45k2(3k + l)a2 + 15k(3k + l)(4k + l)a ; (3k + l)(4k + l)(5k + l) = 0: - *, " . , ) u = l=k ) ) f (a u) = 15a3 ; 45a2(u + 3) + 15a(u + 3)(u + 4) ; (u + 3)(u + 4)(u + 5) = 0: - * 0 )( : " f (a u) = 0, )( u 6= 1 u > 0. 8 , *
. P " ) (1 0) , * ( * ) ) ) ) C y2 = x3 ; 2475x ; 5850: C 0 Z3 Q = (75 480) ;Q = (75 ;480). - P = (;21 192) . 8 pP + qQ, p 2 Z, q 2 Z=3Z, , , , l = 33, k = 124 ( , 1 6 k l 6 1000). 5 , ( , -, . 2
484
. .
, , ( , 1). = *, ) * ) . * , . 8 * ) d4- " 4 5, , hk l mi *. , k, l, m , * (k l m) (k m l), ( * , k, l, m , * .
3. 4
C hk l m ni, ) . : , , " . : (2.1) 8 kx + ly + mz + n = 0 > < 2 kx + my2 + nz 2 + n = 0 > : kx3 + ly3 + mz 3 + n = 0:
P) x y, u(z k l m n) 6- z , 6 , . . (n k l m), (n k m l), (n l k m), (n l m k), (n m k l) (n m l k). ' u k, l, k l 1 = k + l 2 = kl. P u = m2 (m + 1)2 (m2 + m 1 + 2 )z 6 + + 6nm2(m + 1)(m2 + m 1 + 2)z 5 + + 3nm2(5nm2 + m2 1 + 2m 12 + 6mn 1 ; 2m 2 + + n 12 + 3n 2 ; 2 1 2 + 13 )z 4 + (3.1) + 2nm(6mn2 1 ; 6mn 2 + 6m2 n 1 + 10m2n2 + 3m 1 2 + 3 2 2 2 2 + 3n 1 2 + 6mn 1 + 2n 2 + 1 2 + 2m 2 )z + + 3mn2(;2n 2 + 3m 2 + 5mn2 ; 2 1 2 + m 12 + + 2n 12 + 6mn 1 + n2 1 + 13)z 2 + + 6mn2(n + 1)(n2 + n 1 + 2)z + n2 (n + 1)2 (n2 + n 1 + 2 ): - hk l m ni *, u . 8 * : h1 11 80 84i h10 16 39 65i h1 35 63 144i h1 64 104 195i h5 6 45 70i h11 15 55 99i h8 15 69 161i h1 54 65 231i h5 9 26 90i h11 34 85 91i h11 21 70 154i h1 105 159 265i h9 26 30 91i h1 16 34 119i h1 25 104 195i h1 80 209 319i:
4
485
u 2 4. & , ) " 5 (. )( ), * . - , * * .
3.1.
hk l m ni, (3.1) z 2 + 1. ,
, n- 1, m- | i. !. 8 (3.1) z2 + 1. F , i . 8
i (3.1)
)
) , . 2* m, n. 8 1 = m + n 2 = mn. - )( : 8 4 2 1 ; 4 1 2 + 2 1 13 ; 12 22 ; 8 1 1 2 + 12 2 + > > < + 12 12 ; 4 2 12 + 4 2 2 + 1 2 1 = 0 > ; 12 2 + 3 14 ; 12 12 2 + 6 1 13 ; 4 22 ; 24 1 1 2 + > : + 2 2 + 3 2 2 ; 12 2 2 + 4 2 2 = 0: 1 1 1 1 8 2 . P) ,, )( 2 : 32 23 ; 4(6 1 + 11 1)( 1 + 1) 22 ; ; 4 1(3 1 + 1)( 1 + 1 )2 2 + 12(2 1 + 1)( 1 + 1)3 = 0: - ) t = 2 2 =( 1 + 2 ) 1: 2 12 t + (; 12 + 6 1 t + 11t2 ) 1 ; 2 13 + 6 1t2 ; 4t3 = 0: 5 (( 1 ; t)2 + 16t2)( 1 + 3t)2: : , *, , . . cab 2 2 t = cab 2 1 = 2 + c(a ; b ) a, b, c | " . = ) 3 2 3 = c(a ; 3ab + 2b )
1 2b2(2a3 b+ 2a2b ; 5ab2 + 2b3) c 2 = a + 2b
486
. . 2 2 1 = c(ab + 22a ; 2b ) 2 3 2 2 3 2 = c a(2a + 2a b8 ; 5ab + 2b ) :
= (, , , * . : , k l. k
k2 ; 1k + 2 = 0 c2a2 (a2 + 2ab ; b2)2 (a2 ; 4b2)=(a + 2b)2 : - k " , , a = e(x2 + y2 ) b = exy e, x, y | " . - 23 3 2 3 23 3 2 3 k = ce x (xy(x++x yy) + 2y ) l = ce y (yx(x++xyy) + 2x ) : & , m n. m
m2 ; 1m + 2 = 0 c2 b2(3a ; 2b)(a ; 2b)=4 c2 e4x2 y2 (3x2 + 3y2 ; 2xy)(x ; y)2 =4: - m " , . C
, k, l, m, n: 3 2 r + 8)(2r3 ; 9r2 ; 6r ; 14)(r2 + 10r ; 2)3 k = ; c(7r ; 6r + 1881( r2 ; 2r ; 2)(r2 ; 8r ; 2)
3 2 r3 ; 18r2 + 6r ; 16)(r2 ; 2r ; 2)3 l = ; c(4r + 3r + 18(rr2++2)( 10r ; 2)(r2 ; 8r ; 2)
2 3 2 6r ; 16)(2r3 ; 9r2 ; 6r ; 14) m = c(5r + 2r + 2)(r ; 18r + 27
2 3 2 18r + 8)(4r3 + 3r2 + 18r + 2) n = c(r ; 2r + 10)(7r ; 6r + 27 c r | " . =
, r k l m n > 0. - ( " *, ):
r 2 ; 653 ; 47 13 175 : 64 64 64 64
4
487
5 * " r * , ( c, * * hk l m ni. ! ( . ! , (3.1) z 2 + az + 1, a 2 Q. . ' )( . 8 (3.1) z 2 +az +1 . = r 1 z r = r1z +r2 . L ), r1 = r2 = 0. ' r1 r2 ) 1 2 , m, n. = ) 2, m, n: (a + 4)(a + 2)3 23 + (a + 2)2( 1 + 1 )(2a2 1 ; 11 1 ; 6 1 a ; 6 1) 22 + + (a + 2)( 1 + 1)2 (a3 12 ; 4a2 12 + 5a 12 ; 2 12 ; 6 1a2 1 + 18 1 1 a ; ; 6 1 1 + 9 12 a) 2 ; 12 ( 1 + 2 1)( 1 + 1)3 (2a ; 1): 5 2 = ( 1a ++ 21 )t 1 t(a ; 2)(a ; 1)2 12 + (18t 1a ; 11t2 ; 2 12a + 2t2a2 ; 6ta2 1 + 12 ; 6t 1) 1 + + (t ; 1 )2 (ta + 4t + 2 1 ; 4 1 a) = 0 * , z 2 + 1. " k, l, m, n. : , * a ( * , )( k, l, m, n * , . . u * * z 2 +az +1: u k, l, m, n, z * 00" , u * ( =6 11 =6). . P u z 2 +d. 8* n = 1 1, 2, m * " . 2 , u z 2 + d, r1 z + r0 , r1 = r0 = 0. P) 2 ( 2 r1 r0 ), 1 m. Z 1 = t=(dm ; 1) t = v ; dm2 m = m1 (v ; 1) ) v, 4dm21 + 1: = ) m1 = a=a2 ; 2, a | " . & " 1, 2 m. = k l.
488
. .
2 , k2 ; 1k + 2 = 0 4a2 (d + 1)2 ; d(a ; 1)4. k " , , . . 2a(d + 1) 2 2 2a(d + 1)2 = b2 + d = r 2 (a ; 1) (a ; 1) 2b b | " . 8 a. L (d + 1)b(b + 1)(b + d): - , " y2 = (d + 1)x(x + 1)(x + d) . =
* , x. 8 , , x = 1=3. - d = 1=5 " ) P = (1=3 8=15) x + 02) : y2 = 6x(x + 1)( 5 F " 1. & P 2 = = (1=120 ;11=240) ) * k, l m. u z 2 + az + b, a 6= 0 b 6= 1, , . = , * * u * 2, . . * 3 . . " *
hk k m ni. m(k + m)(2k + m)z 3 +3mn(k + m)z 2 +3mn(k + n)z + n(k + n)(2k + n) = 0 (3.2) n- 1, z | m- . 8 m < n *, (3.2) " a, , a < 0 jaj > 1. -, jaj > 1, * )( . - , u(0) > 0. 5, u0 z " , , u(z ) . = , u(;1) > 0. 8 (3.2) k3 , r = m=k s = n=k. : s = t ; zr, (3.2) t(t + 1)(t + 2) + z (z ; 1)(3t + 2z + 2)r = 0: L a | " (3.1), + 1)(t + 2) s = t(t + a)(t + 2a) : r = ; a(a t;(t1)(3 (3.3) t + 2a + 2) (a ; 1)(3t + 2a + 2)
4
489
& , r > 0, s > 0, t(t + 1)(t + 2)(3t + 2a + 2) > 0 t(t + a)(t + 2a)(3t + 2a + 2) > 0: - a < (2a + 2)=3 < 0, , 0 < t < ;(2a+2)=3 ;2 < t < ;1, | 0 < t < ;(2a+2)=3 ;a < t < ;2a. - , (3.3) , ) 0 r s: ) " a < ;1, ) " t, 0 < t < ;(2a + 2)=3, r s. 8 , , a = ;3=2, t = 1=7, r = 8=49, s = 19=49, . . h8 19 49 49i *.
4. 5
'
k, l, m, n, p. &
" . 5 " 5 , * . - hk l m n pi * 24 . p- 1, ) k, l, m, n " p- , . . k, l, m, n. $ * , h2 3 4 5 6i * 2 12 * 33], , , )( , k, l, m, n, ) , , | ). 5 , * * ) 2 12 34]. 34] * * : hk l m n pi, k, l, m, n, p , *, klmnp(k + l + m + n + p) | . * ) ) .
W. (L. Zapponi) B 0 . C () ='D d4- " n. 2 2, ='D p(z ) = (1 ; az )k1 : : : (1 ; az )kn 1 n a1 : : : an | k1 : : : kn . - p(z ) | ='D,
n;1 p0 (z ) = p(z ) z ;k1a + : : : + z ;kna = p(z ) ((zk1;+a: :) :: +: : (kzn);z a ) 1 n 1 n
. .
490
. .
k1 + : : : + kn = (k1 + : : : + kn)z n;1 : (4.1) z ; a1 z ; an (z ; a1 ) : : : (z ; an ) 8
z (4.1) a1 : : : an, k1(a1 ; a2)(a1 ; a3) : : : (a1 ; an ) = (k1 + : : : + kn)an1 ;1 k2(a2 ; a1)(a2 ; a3) : : : (a2 ; an ) = (k1 + : : : + kn)an2 ;1 :: :: :: : : :: : :: :: :: :: : :: :: :: :: :: : :: :: :: :: : : : :: :: ::: :: : : :: :: : kn(an ; a1)(an ; a2) : : : (an ; an;1) = (k1 + : : : + kn)ann;1:
8 * , Y (;1)n(n;1)=2k1 : : :kn (ai ; aj )2 = (k1 + : : : + kn)n (a1 : : :an )n;1 i<j
. . hk l m n pi s Q (a ; a ) j k + l + m + n + p: i<j i 2 (4.2) (a1 a2a3 a4a5 )2 = (k + l + m + n + p) klmnp L klmnp(k + l + m + n + p) | , * (4.2) | " , , , - . -, G, H G , H , , * ) 00 ". ' , 0 , *( &. ! . C (2.1) hk l m n pi: 8 kx + lx + mx + nx + p = 0 > 1 2 3 4 > < kx21 + lx22 + mx23 + nx24 + p = 0 (4.3) > kx31 + lx32 + mx33 + nx34 + p = 0 > : kx4 + lx4 + mx4 + nx4 + p = 0: 1 2 3 4 L k, l, m, n, p * ( , (4.2) , x1, x2 , x3, x4 , 1 , , (4.3)
k, l, m, n, p. 8 " k- l- . C 0 ") k0 = k + (l ; k)t l0 = l + (k ; l)t m0 = m (4.4) n0 = n p0 = p 0 6 t 6 1. F 0 " k- l- , l- | k- ( ) ). 8 0 "
4
491
* ), a1 a2. = , k l : : : l k : : :. F , * , )( : , , )( " )( . & k, l, m, n, p ( , (4.3) , * , - . C ) j1 ; z jk j1 ; z jl j1 ; z jm j1 ; z jn j1 ; z jp = 1: (4.5) a a a a
1 2 3 4 L x1 = 1=a1, x2 = 1=a2, x3 = 1=a3, x4 = 1=a4 | (4.3),
" . - a1, a2 , a3 , a4, 1 , , )( " ( " k, l, m, n, p),
. , * ) (4.5) z , 00" z 1, 2, 3 4
), j1 + z 5 (kx51 + lx52 + mx53 + nx54 + p)=5 + o(z 5 )j = 1: - kx51 + lx52 + mx53 + nx54 + p 6= 0 x1, x2, x3 , x4, 1 ( , ), 0
10 . L k, l, m, n, p " , ) , ( 1. 50 " (4.4)
0 " , )( , 0 . F 0 , * " .
5 * hk l m n pi ( , 2- ) 22- ) ) k, l, m, n, p * ( , , ,). 8 . - hk k k l li . - hk k l l mi * 4 , ", * 2 : f(k k k l m) (k k km l)g fk k l k m) (k k m kl)g. - * | h7 7 7 55 155i. - hk k l l mi * 6 . * ,
h1 1 3 3 2i, h1 1 5 5 4i, h3 3 17 17 15i . : )
). = hkklmni . = , : * * ) ?
492
. .
. - !
(, - ( , * . % &. ! . F D | 0 , * , B . & VD , * , D, ) a b: a | ( , b | , . = a b *) GD | ( D, GD = ha bi Sn , n | , D. GD VD . ' 0 0 0, a b. ' 0 ' : D1 ! D2 0. = " 0 ' : GD1 ! GD2 ( (a ! a, b ! b). F 0 D ! D 0. D Mor(D1 D2) * 0 D1 D2 . F D , 0 AD
VD . 8 . (1) 8 g 2 GD g 6= e, . . ( v 2 VD , g(v) 6= v, g(w) 6= w w 2 VD . = ) jGD j = jVD j jAD j = jVD j. (2) w 2 VD , * v 2 VD * gv 2 GD , gv (w) = v. ! hv 2 AD , hv (w) = v. = - * AD GD : gv $ hv . 8 h1 h2 2 AD | 0 , g1 , g2 | )( GD . P: (h2h1 )(w) = h2 (h1 (w)) = h2 (g1(w)) = g1(h2 (w)) = g1(g2 (w)) = (g1 g2)(w)
. . AD ' (GD )op . (3) L D | , jAD j 6 jGD j, , * D. 5 ) D ( , )( D. ' G H | ") D. & "
N D. ' * , N D - GD . L vg 2 VN D | , )( ( ) g 2 GD ,
g. - a(g) = ag, a GD . ! b(g) = bg. - , GN D ' GD . :( jVD j 0 N D D. ' 0 ) , e 2 VN D . 0 0 D .
4
493
C N D jGD j=na jGD j=nb , , na nb | a b GD . 2* 2nab , ,
. . jGD j=nab. . 8 D = (()()(())). jGDj = 24, na = 3, nb = 2, nab = 4. N D 8 3, 12 , 2, 24 6 . - , N D 0 : | , , | , . . 8 D = ((())(())()). jGDj = 60, na = 3, nb = 2, nab = 5. N D 20 3, 30 , 2, 60 , 12 . - , N D 0 . " 1. ' : D1 ! D | ! D1 ! D. " : D1 ! N D, # D1 ;;;; ! ND
?
'? y
?? yD
id ! D D ;;;;
.
!. 8 v | D1, w | N D, '(v) = D (w). 8* (v) = w, ) 0 . 8 , GD1 = ker(' ) ' GD ' GN D jGN D j = jVN D j. 8 ; = Gal(QN =Q) | . L D | , D 2 ; D. " 2.
$% ! D D0 $ 2 ; %
0 j Mor(D D )j = j Mor( D D0 )j. & D = D0 , # Mor(D D) ' Mor( D D). " 3. $ D $ 2 ; ! (N D) . !.
jA (N D) j = jAN D j = jVN D j = jV (N D) j:
- 3 . " 4. N ( D) = (N D). (N D) | . 8 2 ( !. 0 (N D) ! D, , 1 ( 0 (N D) ! N ( D). : ,
;1 , , ( 0 ;1 (N ( D)) ! D, . . ( 0 ;1 (N ( D)) ! N D. ! , ( 0 N ( D) ! (N D). W .
494
. .
- * - ( )( " 0: op op GD ' GN D ' Aop N D ' A (N D ) ' AN ( D) ' GN ( D) ' G D : F * C 0 0 , 98-01-00329.
"
1] Shabat G., Zvonkin A. Plane trees and algebraic numbers // Contemp. Math. | 1994. | Vol. 178. | P. 233{275. 2] . ! "##$ %& % // '# %(. &(". | 1995. | ). 50, + 6. | -. 163{164. 3] Schneps L. | London Math. Soc. Lecture Notes. | 1994. | Vol. 200. | P. 47{78. 4] . . & ( & (/0%$ #( // '# %(. &(". | 1997. | ). 52, + 4. | -. 203{204. 5] 1&! -. 2!3 (. | 4.: 4 , 1968. ( ) 1999 .
. .
e-mail: [email protected] 512.55
:
PI- , ,
! " .
# " #$ % ! "
! & # #. ' (. ). * + # , $ , ! "" " #$ +!!.
Abstract A. N. Krasilnikov, On the nite basis property of a variety of associative algebras, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 495{501. We prove the 4nite basis property for a certain variety of associative algebras over a 4eld of a prime characteristic. Earlier N. I. Sandu conjectured that this variety is not 4nitely based.
1.
. . 1] ! " 0 $ $ !% ! " ! ! " . & ! "' $ ( 2,3], ! , , - $ " ( ! ! a b ab ; ba). 0 $!! , 2,3], " , - $ , $% - ! $
4{7] $4 " $ ! 8]. ' $ $ '
& #! 5 ! 5+ ! %
&, 6 97-01-00785.
, 2002, ! 8, 6 2, . 495{501. c 2002 !, "# $% &
496
. .
8 5] : - (" - 8 - - ) ! $!! $!! " 8 " ! " 2, $ -4" $ (x1 x2 x3)(x4 x5 x6)(x7 x8 x9) 0: 0 - 8 &. ;. <$ (. 9]) !!, " ! ! " - - - $ - ' - 5] (x1 x2 x3)(x4 x5 x6)(x7 x8 x9) 0 (1) ((x1 x2 x3) (x4 x5 x6) (x7 x8)) 0 (2) 0 (k = 1 2 : : :) (3) (x y) = xy ; yx, (x y z) = ((x y) z), = ((x y z) (x1 x2) (x3 x4) : : : (x4 ;3 x4 ;2) (x y z)): , 8 !! - . 0 -4 -, (1){(3) - - - $ " !. > , !$ ? | !, L | - ( ? !4 r s t u v w xy z x1 x2 : : :. A- " f g h 2 L $ fgh = (fg)h. A- ! k ! = (x y z x1 : : : x4 ;2) 2 L, ! = xyz(x1 x2) : : :(x4 ;3x4 ;2)(xyz): 1. ? | , char ? 6= 2. k +2 0 : 0 x1 x2x3 (x4x5x6 )(x7x8 ) 0: (4) $ A | - - ! ? !4 x y z x1 x2 : : :. A- ! k ! 2 A, ! = ((x y z) (x1 x2) : : : (x4 ;3 x4 ;2) (x y z)) (a b) = ab ; ba, (a b c) = ((a b) c) - " a b c 2 A. ; !- 1 $
. ? | , char ? 6= 2. ? k > 0 +2 0 : 0 (2). B 2. ? | , char ? = 2. ? k > 1 +1 0 : 0 (1). k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
497
; - ! 1 !- 2 - -4- $ . ! (1){(3) # $ % : (1), (2), 1 0 2 0.
2. 1
A- ! k ! 2 L, ! = xyz(x1 x2)(x3 x4) : : :(x4 +3x4 +4)(uvw): >$ 3. & ? | , char ? 6= 2, 0 : 0 (4). 1 - - - !- 3. A , | 8: L, x = x +2 (i = 1 2 : : : 4k + 4), z = x1x2 , y = z, x = xy, u = x, v = y, w = z, = xyz(x1 x2)(x3 x4) : : :(x4 +5x4 +6)(xyz) = +2 !8 $ +2 0 - - - 0, !- 3 $ , +2 0 | , -4
0 (4). 3 $ - " " . 1. ; (4) $ xyz((uvw)(st)) ;xyz(st)(uvw): (5) A , ! -- D abc + bca + cab 0 a = xyz, b = uvw, c = st $ -, $ (4) xyz(uvw)(st) 0 !$, (4) $ uvw(st)(xyz) + st(xyz)(uvw) 0 $ $ (5). 2. ; (4) $ xyz(x1 x2) : : :(x2 ;1x2 )(x2 +1 x2 +2) : : :(x2 ;1 x2 )(uvw) xyz(x1 x2) : : :(x2 +1x2 +2)(x2 ;1x2 ) : : :(x2 ;1 x2 )(uvw) (l = 2 3 : : :): (6) A , D abc acb + a(bc), $ xyz(x1 x2) : : :(x2 ;1x2 )(x2 +1x2 +2) xyz(x1 x2) : : :(x2 +1x2 +2)(x2 ;1x2 ) + xyz(x1 x2) : : :((x2 ;1x2 )(x2 +1x2 +2)): k
k
k
k
k
k
i
k
i
k
k
k
k
k
k
k
i
i
i
i
i
i
i
i
i
i
i
l
i
i
l
l
l
i
i
i
i
i
i
i
498
. .
E , $ -, xyz(x1 x2 ) : : :((x2 ;1x2 )(x2 +1 x2 +2)) : : :(x2 ;1 x2 )(uvw) 0 - - - (4), !$, (4) $
xyz(x1 x2) : : :(x2 ;1x2 )(x2 +1 x2 +2) : : :(x2 ;1 x2 )(uvw) xyz(x1 x2 ) : : :(x2 +1x2 +2)(x2 ;1x2 ) : : :(x2 ;1x2 )(uvw)
(4) $ (6). 3. ; 0 (4) $ (1) 0, (1) = (1) (r s t y z x1 : : : x4 ;2) = rsyz(x1 x2 ) : : :(x4 ;3x4 ;2)(tyz): A , 0 $
(s + t y z x1 : : :x4 ;2) ; (s y z x1 : : :x4 ;2) ; (t y z x1 : : :x4 ;2) 0
syz(x1 x2) : : :(x4 ;3x4 ;2)(tyz) + tyz(x1 x2 ) : : :(x4 ;3x4 ;2)(syz) 0: (7) ; (5) $ , tyz(x1 x2 ) : : :(x4 ;3x4 ;2)(syz) ;syz((tyz)(x1 x2 ) : : :(x4 ;3x4 ;2)) (;1)2 +2syz(x4 ;3 x4 ;2) : : :(x1 x2)(tyz) syz(x4 ;3 x4 ;2) : : :(x1 x2)(tyz) !8 $ (4) (7) $
syz(x1 x2) : : :(x4 ;3x4 ;2)(tyz) + syz(x4 ;3x4 ;2) : : :(x1x2 )(tyz) 0 $ $% (6) !$ 2syz(x1 x2) : : :(x4 ;3x4 ;2)(tyz) 0 $
syz(x1 x2) : : :(x4 ;3x4 ;2)(tyz) 0 ! $ char ? 6= 2. - ! ! $ s ! rs, !$ (1) 0, , , $ 0 (4). 4. ; (1) 0 (4) $ (2) 0, (2) = (2) (r s t u v w y x1 : : :) = rsy(x1 x2) : : :(x4 ;1x4 )(ty)(uvw): A , (1) 0 $
(1) (r s t y x4 ;1x4 + uvw x1 : : :) ; (1) (r s t y x4 ;1x4 x1 : : :) ; ; (1) (r s t y uvw x1 : : :) 0
rsy(x4 ;1 x4 )(x1x2 ) : : :(x4 ;3x4 ;2)(ty(uvw)) + + rsy(uvw)(x1 x2 ) : : :(x4 ;3x4 ;2)(ty(x4 ;1 x4 )) 0 i
i
i
i
i
i
i
i
l
l
i
l
i
i
k
k
l
i
l
l
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
499
rsy(uvw)(x1 x2) : : :(x4 ;3x4 ;2)(ty(x4 ;1x4 )) 0 - - - (4), (1) 0 (4) $ rsy(x4 ;1x4 )(x1 x2) : : :(x4 ;3x4 ;2)(ty(uvw)) 0 , $% (5) (6), !$ (2) 0. 5. ; (2) 0 (4) $ 0. A , (2) 0 $ , k
k
k
k
k
k
k
k
k
k
k
k
k
(2) (r s t u v w x4 +4 + xyz x1 : : :) ; (2) (2) ; (r s t u v w x4 +4 x1 : : :) ; (r s t u v w xyz x1 : : :) 0
rsx4 +4(x1 x2) : : :(x4 ;1x4 )(t(xyz))(uvw) + + rs(xyz)(x1 x2) : : :(x4 ;1x4 )(tx4 +4)(uvw) 0: B (4) $
rsx4 +4(x1 x2) : : :(x4 ;1x4 )(t(xyz))(uvw) 0
(2) 0 (4) $ rs(xyz)(x1 x2) : : :(x4 ;1x4 )(tx4 +4)(uvw) 0 , 8 , xyz(rs)(x1 x2) : : :(x4 ;1x4 )(tx4 +4)(uvw) 0 ! ! - ! " !$ 0. B , 0 (4) $ 0. 3, ! 1, . k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
3. 2 $
= ((x y z) (x1 x2) : : : (x2 ;1 x2 ) (x y z))
= 2 ;1 - " k. F $ 4. ? | , char ? = 2. ? k > 1 +1 0 : 0 (1). 2 - - - !- 4. A , 0 ( | 2 ;1 0) (1) $ !- 4 k
k
k
k
k
k
k
k
k
500
. .
$ 2 0, 8 (1) | 2 +1 0, +1 0. 4 %" . 1. $ I J | A, !% 8 (a b c) 8 (a1 b1 c1)(a2 b2 c2)(a3 b3 c3), a b c a b c 2 A, J | T-, $4 $ (1). B J = I 3 . 2. ; (1) 0 $ ((x y z) (x1 x2) : : : (x2 ;1 x2 )(x2 +1 x2 +2) (x y z)) 0: (8) A , ! - 0 ! $ x2 ! x2 (x2 +1 x2 +2) $ -, (a bc) = b(a c) + (a b)c (a b c 2 A), !$, 0 $
((x y z) (x1 x2) : : : (x2 ;3 x2 ;2) x2 (x2 ;1 (x2 +1 x2 +2)) (x y z)) + + ((x y z) (x1 x2) : : :(x2 ;3 x2 ;2) (x2 ;1 x2 )(x2 +1 x2 +2) (x y z)) 0 (9) ((x y z) (x1 x2) : : : (x2 ;3 x2 ;2) x2 (x2 ;1 (x2 +1 x2 +2)) (x y z)) 2 I 3 I 3 = J, ((x y z) (x1 x2) : : : (x2 ;3 x2 ;2) x2 (x2 ;1 (x2 +1 x2 +2)) (x y z)) 0 (10) - - - (1). B (8), , $ (9) (10), !8 $ (8) - - - (1) 0, . 3. G k > 2, (1) (8) $ +1 0. A , ! ! - -, - " a c1 c2 c3 2 A ! (a c1 c2 c3) = (a c1 c2c3 ) + (a c2 c1c3) + (a c3 c1c2 ) + f f = f(a c1 c2 c3) = (c1 c2)ac3 + (c1 c3)ac2 + (c2 c3)ac1 + (c2 c3 c1)a + + a(c1 c2)c3 + a(c1 c3)c2 + ac1(c2 c3) !8 $ (a c1 c2 c3 b) = (a c1 c2c3 b) + (a c2 c1c3 b) + (a c3 c1c2 b) + (f b) - " a b c1 c2 c3 2 A. E , ! a = ((x y z) (x1 x2) : : : (x2 ;5 x2 ;4)) b = (x y z) c1 = (x2 ;3 x2 ;2) c2 = (x2 ;1 x2 ) c3 = (x2 +1 x2 +2) k
k
k
i
i
i
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
501
$ -, (c c ) 2 I - " i, j ! $ f(a c1 c2 c3) 2 I 2 , (f(a c1 c2 c3) b) 2 I 3 , !$, (1) $
((x y z) (x1 x2) : : : (x2 ;3 x2 ;2) (x2 ;1 x2 ) (x2 +1 x2 +2) (x y z)) ((x y z) (x1 x2) : : : (x2 ;3 x2 ;2) (x2 ;1 x2 )(x2 +1 x2 +2) (x y z)) + + ((x y z) (x1 x2) : : : (x2 ;1 x2 ) (x2 ;3 x2 ;2)(x2 +1 x2 +2) (x y z)) + + ((x y z) (x1 x2) : : : (x2 +1 x2 +2) (x2 ;3 x2 ;2)(x2 ;1 x2 ) (x y z)) !8 $ +1 0 - - - (1) (8). < , +1 0 | (1) 0. 4, ! 2, . i
j
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
!
k
1] . . // . | 1987. | %. 26. | (. 597{641. 2] Vaughan-Lee M. R. Varieties of Lie algebras // Quart. J. Math. Oxford Ser. (2). | 1970. | V. 21. | P. 297{308. 3] 1 2. (. 3 4 // . | 1974. | %. 13. | (. 265{290. 4] (. 5. 6 7 77 // 18 ((( . | 1970. | %. 190. | (. 499{501. 5] Vaughan-Lee M. R. Uncountably many varieties of groups // Bull. London Math. Soc. | 1970. | V. 2. | P. 280{286. 6] Bryant R. M. Some in9nitely based varieties of groups // J. Austral. Math. Soc. | 1973. | V. 16. | P. 29{32. 7] : ;. <. 3 7 : 77 // 5. 8 ((( . (. . | 1973. | %. 37. | (. 95{97. 8] 3 = : . ;. > 77 // 5. 8 ((( . (. . | 1970. | %. 34. | (. 376{384. 9] ( 8. 5. 6 : 7 2. | = , 1994. | 17. @ 855%C5 29.09.1994, D 1350-M94. ' ( 1999 .
. . 512.55
: , ,
.
! "
! . $ , % ! ! & " ! ! "
! !! & . '( ! , ( !
.
Abstract
D. A. Matsnev, On the recognition of the nite deniteness of an automation monomial algebra, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 503{516.
In this paper an algorithm for recognition of 0nite de0niteness of an automaton monomial algebra is proposed. It is shown that this problem for an arbitrary algebra reduces to the following problems: determination of the star height of a regular language and 0nite de0niteness recognitionfor a certain class of automaton algebras. The solution of the former problem has already been described in the literature, the complete solution of the latter problem is presented in this paper.
1.
X . A, ! ! " ! " , - $ X . % A , & ( $ | X ), ! $ " . * A + ! " A. , , " "
X , , X . - , 2002, 8, 2 2, !. 503{516. c 2002 , ! " #$ "" %
504
. .
! : A , & , . 2 ! , & . 1.1.
& ( ), &
+ . 3 & | X . 4 . ,& &
, + & & . 5 , G u, G , ! , & " , " + ! u ( , " " ). , & " & , !, & " . 1.2. 6 & , . 1.3. 6 ( & ) , .
1.1. % , + , ! & . 7 & + , " $ X ( " " | ). 7 & & " " . ! . 3 !, & " , & X . 8 92] , " . 1.4. A | . 4 , A, " " A. 7 & ( . 91]), ! + , . 1. = fa bg ! ! : abna = 0, n = 1 2 3 : : :. @ ,
, + . ,& & , ! . 2. = fa bg ! ! : an bna = 0, n = 1 2 3 : : :. 2
505
, ! ( . 92]). = , + . 8 : ( , + " ) , + . * 92] ! " $: 1) A Adet A 2) Adet Adet 0 , Adet X ( : X & " X . , & " & | . 3.1)
, " " A. - | , & K Adet 0 ( & B C A), ! K ( A ) ( A ).
2.
SSL-
2.1. D ! + + , X ! : , $ , " , X . - $ " , X , . E $ ( $ ) $ . 8& $ ! ! : " + " , ! " " , " " " & ( $ $ ).
2.1. 7 & ! ! ( , ) , ! $ ( ). 2.2. SSL- ( Some Simple Loops | " $ ) + + , X , ! .
506
. .
. 3 $ SSL- . . SSL- . E -
$ ($ ) $ , " SSL- . ! $ . 2 , ! " ( & + $ ) , ! $ . " $ SSL- . 3. * ! + SSL- .
i
r
k
r
r
R -
r
-
r
-
r
6 - r
r
sW s
I
r
r
r
/ R- r
r
r
r
r
-
r
2.3. $ & $ , + " SSL- . @ SSL- $ & $ ,
& , & . G : SSL- | $ , . 3 & " " " $ . 8 + & ! : G | SSL- , DG ( + , , G !), NG | " G, BG | ! $ G. - ! ! ! ; SSL- , & " + , !" " ;. 4 G , & G ( & G, " S " ). I ; G G 2 ;. 2.4. " & u v : u ,! v. 2.5. 7 & V & U ( : U ,! V ), 8v 2 V 9u 2 U u ,! v. 8
u 6,! v.
507
2.6. 7 & V & U ( : U V ), ! & KU KV , KU U, KV V , KU ,! (V n KV ). 4$ : U 6 V . 4 2.1. SSL- G SSL- ; ; G . 8 + ! . K SSL- G. 4 a0 a1 : : : an, $ bij , i = 1 2 : : : n, j = 1 2 : : : ri, i | , ri | " i- $ . - , G : $ ai;1 " , " ! " $ bij " j = 1 2 : : : ri, & " $ ai . 8 -$ , , ! " . ,, a | $ , + , " a1 a2 : : : ar , a $ , a1a2 : : :ar . , + $, , + , . D bij , , $ ! ( , + ). & bij " " & , $ . % bkij k-! bij . 4 k a0bk11bk12 : : :bk1r1 a1 : : :an;1bkn1bkn2 : : :bknr an. 2.1. S |1 S fkg1k=1, s 2 S , s fk gk=1. !. M S fkg1k=1, KS S, k0 ! k k > k0
KS . " k ( , k > > max(k0 maxfdeg s j s 2 KS g), & KS ,
& &, ! KS " bki;1d ;1 aibik 1 bkij bkij +1 " i, j ( & : KS " k,
& " ai " bkij ). K , & KS , " k , s
. - s " bki;1d ;1 ai bki1 (bkij bkij +1 ) " k k1 ( , k1 > max(k0 maxfdeg s j s 2 KS g), s ,! k , k > k1. n
i
i
508
. .
2.2. !
& : ! k > deg s s ,! k . !. O , s " bki;1d ;1 aibki1 bkij bkij +1 k = k1, k = deg s, & ai bij . % & bki;1d ;1 ai bki1 bkij bkij +1 " k . 2.2. N = N(G ;), u fk g1 k=1 u 2 ; deg u > N v 2 ; deg v < deg u v fk g1 k=1 . !. = u 2 ; , u = e0f1e1 : : :em;1 fmem , ej | ;, & fj = fjn1 fjn2 : : :fjn (1 6 ni 6 dj " i 1 r) | , j- $ . * ! u ,! k k > deg u. 2 " & & $ " ( L = deg u). (1) u ,! bLid aibLi+11 . - , L > DG + D; + + 2(B; ; 1) + (N; + 1)B; BG , fj " , fj ,! bL b = bid bi+11 & fjk fj BG . 8 , & ,
" bLid bLi+11 BG " fjk & fj . , , , ! $ fjk " B; , fj , ! " bLid bLi+11, $ " fj N; , ,
" " " " fjk N; B; BG . 2 fjk ( , $ ! ai), " ai $ B; ; 1 & : ai . E , B; = 0 & , fj & , , & ; . 8
; ! D; , , , &
: N > D;
,
. 8 , ,
, B; 6= 0. + , fj & ai, , + $ N; . 4 ej $ D; . , , ai & & D; + 2(B; ; 1) + (N; + 1)B; BG u. 2 ai $ DG . 8 B C u & " DG + D; + 2(B; ; 1) + (N; + 1)B; BG . $ 2" " & fjk , ! & fj bL " & b " $. O B C " & u i
i
r
i
i
i
i
509
v . , fjk $ , v ; , $ b $ b, $ b, &, v ,! k , k > deg u. (2) u ,! bLikbLik+1. % & , L 6 D; + + 2(B; ; 1) + (N; + 1)B; BG , " fj ,! bL $ 2" $ u. 2 & , . , N
DG + D; + 2(B; ; 1) + (N; + 1)B; BG , . ! " 2.1. + $! NG G. 4 $ : NG = 0 ( G | $ ) K; = ? KG = G . , ; G ,
& , , ! $ G, ; , , & , ! ; . " G, " NG < n. 8
! & G: NG = n. 2 "
. , ; fk g1 k=1,
2.2 ; , ! N. E & 2.2, , u ; N, u ,! N . E $ & : " u N ; ( " 1
!) , u ,! N , ; 6 fk g1 k=1. 3 fk gk=1 G , ; 6 G . , u ,! N . & $ " . (1) u ,! bNid ai bNi+11. , P & G deg u, " & " "
ai;1 ai+1 . 8 , & " ai ,
& P . (2) u ,! bNij bNij +1. , P & G deg u, " & " "
ai;1 ai . % & P . & p 2 P SSL- Gp ! : G p ! $ , ! ,
p . , , + ", " "
, + , ! + " ( & , ). i
510
. .
& $ & Gp, Gp ! SSL- $ + ( ) 1 !
G. = , & , ; Gp . M 9p ; 6 Gp , ; 6 G , Gp G . M & 8p ; Gp, ,
K; = fug K; p2P
p
KG = v 2 KG K; 6,! v p2P
p
K; KG & , ! ; Gp " p 2 P, , ; G . 8 + . 2.7. 2 " ( & , & ) + , " !. 4 , ! , & " ! , . & , Adet 0 | B C A ( ). 7 " , ! ( ! $ ), , ". 2 & ; Adet 0 , " & " $
, ! Adet 0 , , + ( + ! ), " . 6 ,
& , - " ! , , ! " & ! , & ! A & , " . E , , B C & , & , , . E , ! & ; & . 3 $, + : & ; , & & " . -, ; | & " , ", " $ | , ! " " . M ; + , & & + A. 2.8. % A SSL- , + & ; ( ), ! SSL- . p
p
511
3 2.1 2.2. ! SSL- " # . !. * " & . 2 & G 2 ; ; G , 2.1. M G, , & " G,
, ! " ; , ! + . % G , , " & K; G ( " & ), , , ! " & , ! . ,
. % ! 2.1. $ ! SSL- " "
% & # . !. 8 , " + , " & K; G 2 ; + , " " " ! . % ! & ! " KG G ( ). 4 & , ! & " , ! $. SSL- SSL- ! ( !) & SSL- + " . SSL- , , & , & .
2.3. 2 ! & : " , & & ; , | SSL- , . 3 , SSL- , ! & ; ( & & & ), , ! + 2.2 & . 8 $ & , & SSL- , & SSL- ( . 3.2). E , , + ! + . I & + , , SSL- ( . & 3.1), $ -
512
. .
. " & SSL- , + . = " " " SC- , + " 93]. SC- | SSL- , ! " SSL- ; $ $. 2 SC- . G , + & & , 93].
3.
3 " ! " ! " & . 3.1. = & X ! : ? X | & A | & , () ( ), ()( ) () | & . @! & ! " . I & , , $ ( ) , $ $ ( + ! $, ) ! , $, $ ( ) . , & , , &
. E& & + X . % , ? , X , , ! , fab j a 2 b 2 g, fa1a2 : : :an j n 2 N f0gg, ai 2 , ! n = 0 . , E ( . 92]) & , & , " , & , " & . 3.2. + & , sh() ! :
513
? X ! ! + ! . | & , sh( ) = sh( ) =
= max(sh() sh( )) sh( ) = 1 + sh(). 3.3. + | + " " & , ! " . E ! : 3.1. ' " , & " 2 ", " " ! " . O , , , + | " . 2 & ! ! !
. 3.1. % , ! "& ! " 1. !. u1 = 0, u2 = 0,.. . , un = 0 | ! . , & (X ) (u1 u2 : : : un)(X ) + " + ! 1. * , " & ! + ! ,
( , ! + ). 3.2. ( % % SSL- , ! " ! "& 1. !. SSL- . & , + SSL- G. E 2.1, a0 a1 : : : an, $ bij , i = 1 2 : : : n, j = 1 2 : : : ri. , , " $ , & , " & : (X ) a0 (b11 b12 : : : b1r1 ) a1(b21 b22 : : : b2r2 ) a2 : : : an;1 (bn1 bn2 : : : bnr ) an (X ) : (1) O & + ! 1. M SSL- , ( ), , & (1) & , & " . % & + 1 + ! 1. 8 , " & ! + ! , : + " SSL- 1. , , " & + 1. 8 & + 1 " & n
514
. .
0 !1 1!2 2 : : : n;1!n n (2) i !j | & + , ! . E , & , i !j ?. 7 , & i (n ) (j )
i = fu(1) i : : : ui g, ui | X . ! & & , (2) " & u0 !1 u1 !2 : : :un;1!n un (3) ui | X ( & , ), !j | & + 0, & . 7 & , !j 6= ?, & & !j , uj ;1!j uj uj ;1uj , uj ;1uj | , $ uj ;1 uj . 2 , !j = fvj(1) vj(2) : : : vj(m ) g, vj(k) | X ( !). 8 ui = ? 0 < i < n ! : & (3) ! : u0!1 u1 : : :ui;1!i ui+1!i+2 ui+2 : : :un;1!nun
u0 !1 u1 : : :ui;1!i vi(1)+1 !i+1 ui+1!i+2 ui+2 : : :un;1!nun
u0 !1 u1 : : :ui;1!i vi(2)+1 !i+1 ui+1!i+2 ui+2 : : :un;1!nun : : :
m +1 ) ! u ! u : : : u ! u :
u0 !1 u1 : : :ui;1!i vi(+1 n;1 n n i+1 i+1 i+2 i+2 ! & (3), " u0 un , , " & , . & & ! (3) SSL- : u0 / un , (3) + " !1 / !n . 2 , & (3), " & . -, + ! + SSL- . E $ , u0. E $ m1 $ , $ , " " " " , " v1(1) , v1(2) ,.. ., v1(m1 ) ! " " , & ( , , " ). 2 & $ u1, + $ | m1 $ , " v2(1) , v2(2) ,... , v2(m2 ) , & " & ! , $ & . $ un + $ $ SSL- . 4 ! & (3) ! " SSL- ( & ). i
j
i
515
, SSL- & " , "
& + 1. & , - " SSL- , , !1 !n " (. & ; B C ). , SSL- SSL- " ,
. % ! 3.1. ' " , & " 2 ", SSL- ". !. E $ 3.1
3.2.
3.1. E
3.2, & + 1 " , & ! SSL- , , . 8 94] , " + ! . - , &
! 3.2. ) !% %
" . !. 8 , , ( & !, ) " + ! ! & , . , , $ + & " . M , $ &
3.1 . M & + $ , ,
3.2 SSL- , + ! 2.2 SSL- . M & " + ! , , , $ ,
3.2, SSL- , , + ! . 3 & SSL- , ! n & & " " + , " n, . * , & " " SSL- & ( , & " & ). , SSL- ! 2.2.
516
" #
. .
1] . . , . . , . . . . . 2] . . !"# $% & ' . | ).: ), 1986. 3] /. . "' . 01 2 ! 1 3 & 4 & . | 5.. . . 3. $.-. ". | 7%' , 1996. 4] K. Hashiguchi. Algorithms for determining relative star height and star height // Inform. and Comput. | 1988. | Vol. 78, no. 2. | P. 124{169. & " ' ' 1999 .
. .
. . . ,
517.588+519.68
: , ! "
, # $ !# , % $ .
& ' % ! "
## # ) #* # # $ !# * # #, )##%' ##)%' )# . ) # , %$ # $ !# | #, # , ) , -, , , . , / ) , , . ., # )" *| $ #% ) #0 # $ 1 *. 2 * $' % !) # ! "
!# $ )'# )'
)0 ## #* * )) # # #!* 3) # % . 4) # $ 1 ) # )'#* ! %* $, $* )'# # ! # * ! #*# #1 " )" *, # !$ ! " # )" *, ' # # ! * ! *
.
Abstract A. W. Niukkanen, Quadratic transformations of multiple hypergeometric series, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 517{531.
Using factorization method and introducing canonical forms of multiple hypergeometric series allows all quadratic transformations for all series satisfying the corresponding applicability conditions to be obtained in the form of a small set of general basic relations. In other words any quadratic transformation for standard series, for example for the Gauss', Appell's, Horn's, Kamp<e de F<eriet's, Lauricella's, Gelfand's series, etc., as well as for numerous nameless series, can be obtained as particular cases of the relations given in the paper. Along with completeness and generality of analysis the factorization method ensures an essential simpli=cation of the theory by introducing a natural hierarchical structure into a system of quadratic > $ # 3 > * ) # * ( 97-01-00317). , 2002, 8, ? 2, . 517{531. c 2002 ! " #$, % &' (
518
. . connections between nine types of canonical forms. These results may contribute to developmentof advancedcomputer algebra systems capable to analyze, automatically, important properties of those multiple series which =nd large-scale applications in mathematics, mathematical physics and theoretical chemistry.
1.
1{4],
! " #$ % &' ! . ) ! ' ! * # % 1] " !
, * &' & + 5], & &$" &
,&
$!!
2,3] # !$& & " * &+ 4] , ! $! $
(2==1) & $!!
(1==1) * ! ( ! ! * ! ! ! * *, &$"! !1 $&' ! * *, ! *& ! 2]) . 1
! " # ! ! * $ # !$, ! x0 ! $ %, &' "&, x0, ! *" " (!. 2,4]). 2 *! *& # ! %&' ": G = N +1 F h1j1 m1i h2j1 m2i==h0j1 m0i L 5 x0 x] (1) * " L , " D *& 2, ! & $
$ ' &' ( . . &' 6 ! " 7) ,##%
L(i1 : : : iN ), ! $!!
i0, ! $! ! x0 * N +1 F . 8+* !$ ! &$ * h j1 mi, ! & + *& & N==+1, "& & & ! & * , * & ,##% * F , & ! + &' ( i0 + m1i1 + : : : + mN iN ), * ( i) = ;( + i)=;( ) | ! & < !!. 8 8], #$ % F12 & %&'
!+*$ ! ! *$ & !
! $*
& ! "& ! * " !
!, !
+$ &'
*" & , ! ! " # " ! (!., !, 9,10]). ? & $ ! *&' $ ! * "
#$ %
* $ ! (!., !, 8, . 5.11], 11, . 23, 27] 12, . 303, 304]),
%& ! * "
*
! ! , ! " *$ + ' *! , &*
* " * &' . A
%&' ! * "
* F12 &$" & ! " *
1 @ * $! *, 3 $ $ ) ! "
. B2] B6].
519
* (1) $&
, " ! + ' . 2 &"
*% 2 !
, &' $! *&
* * "
* F1 8] *&'
#$ % 11,12] ! " "& &' '! $ , *& * * "
, * & * , ! * * *,
&'
% # % 1,6] &' "& ! &'
, &$" 2,5]5 &
$ '
&' 6! 7, &'
& $ &' !, ,## ! & $* ! * !
* "& ! + 6 " 7 # !$& *& & ! " #$ %.
2.
& * b = N +1 F h j1 mi D 5 x0 x], ! !&' (1==0) ! , &
2] Q (1 ; z0 ) F h j1 miD ==h j0 mi5 z0 z(1 ; z0 )m ] = = (1 ; z00 ) F h 0j1 m0i D ==h 0j0 m0i5 z00 z0(1 ; z00 )m ] Q0: (2) <
( *& ! !. 2]), +* !
&
" # ! G- 2], & " ! &$"!
(2). <!" &' , " * "
, +* ! * "
* G, + & " ! &$"!
(2). 2 ! !;m*&, -2
*
# !$& (2) ! D = L h j0 mi], z = (1 ; z0) x, z0 = x0, z00 = 2x0=(1 + x0 ), 0 = , m0 = m !&' ! + &' & " $, &$"! !
* F h j1 mi L 5 x20 x]=(1+x0);2 F h j1 mi L 52x0(1+x0);1 x(1+x0);2m ]: (3) 0
3. -
0
2 * "& &' ! &' !
* N +1 F " R # !$& (3) * & &' ! * h2j1 2mi & * & &' ! "& & ! ' * * # !$& # % (!. 1,6,7]), !! R = hH Cj1i H d(s)]C(s)js=0 (4) ;2 1 H d(s)] = (1 + x0 ) F0 2 5 d(s)=(1+ x0 )] (5)
520
. .
C(s) = F h j1 mi L ==h2j1 2mi52x0s xs2m ]: (6) & H ! !
D-, & H d(s)] F012 5 d(s)]exp(;x0 s) ( *& D-, & *
2,5,6]). < * & ,
! # !$& (4) $ (3) *
" $&' $ ! x0 ! x0t, x ! xt2m , &$"! F h j1 mi L 5 x20 t2 xt2m ] = F01 2 5 d(s)]E(ts)js=0 (7) * E(ts) = exp(;x0ts)C(ts). < $ $ " ' (7) ! ' & F d(s)]E(ts)js=0 = F td(s)]E(s)js=0, " &' !!: F h j1 mi L 5 x20 t2 xt2m] = F012 5 td(s)]exp(;x0 s)C(s)js=0 : (8) )
(8), &" (3) &'
" , $*! ' D- +* !
! * "
(3). F&'
* * *$, $* !$ *& &*$ * D-, & !
!. G *, $ &! D-$! +! " * $ #$ %, &"$
D-*" #$ % exp(x), & *! + "
# !$& ! * # %.
4. " , $ $
1. ( ) ,
, L 5 x20 x ;2x h2 ; 1 j1 mi L 54x0 4mx
(9) = e F h2 ; 12j1 2mi N +1 F (9)
(0==1) , N +1 F (1==1). (9) D-
(3), (8).
. < & ! ! " ! (8) F10 ==2 5 d(t)]jt=0 ! F01 " (8) D- +* expd(s)], * #$ % E(s) = exp(;x0s)C(s) *
! expd(s)]E(s)js=0 = E(1), & " (8) & !
$ # !$& # % * F h j1 mi L ==h2 j2 2mi5 x20 x], ,##% ,
$ ! ! h j1 mi==h2 j2 2mi, ! ' # !$& $! + F
1 mi
h j
0
521
H${I+* &$" * 1 ;i ;mi ( i0 + m i)=(2 2i0 + 2m i) = 4 + 2 i0 + m i " &
' $ * # ! F L ==h + 12 j1 mi5 2 ; m (x0=2) 4 x]. < , +1 #$ % E(1) =mexp(x0)C(1), $" (6) * ! ! ; 2 , x0 ! 2x0, x ! 4 x, &$"! # !$&$ (9), " * &' *& + 1. )* #$ % $
, $*
& $&
! !! * "
, $ &' . 1. 2 ! $&
,
*! *& (1), $*! " '
#$ % &*$! ! &!: F1 & 2 = 1 + 21 m2 = m15 F2 & 0 = 1 + 1 ; 2 m0 = m1 ; m25 F3 & 0 = 22 m0 = 2m2 ' (10) F1 F1h1j1 m1i h1 + 21 j1 m1i L ==h0j1 m0i5 x0 x] F2 F2h1j1 m1i h2j1 m2i L ==h1 + 1 2j1 m1 2i5 x0 x] (11) F3 F3h1j1 m1i h2j1 m2i L ==h22j1 2m2i5 x0 x]: (12) 0
2.
, ! " ,
F1 F3 D- # (9). $ # , % ,
1 m0 L 5 x31 x31 Q31F1 (13) 2 12 2 11 2 p p p x31 = 2 x0 (1 + x0);1 , x31 = 4m x(1 + x0);2m & 2 h 2 j1 m2 i h 2+1 j1 m2 i L 5 x13 x13 F3 = 2 ; x F1 h + 1 j1 m i Q13F3 (14) 2 2 2 0 x13 = x20(2 ; x0);2 , x13 = 4m x(2 ; x0 );m .
. <! (9) ! x0 ! x0t, x ! xt2m , * $ " (9) ! F01215 d(t)]jt=0 &' $
1 D-, & F0 215 d(t)]exp(;2x0t) (1+2x0);2 F01215(1+2x0);1d(t)],
" (9) ! + ' ! ' # !$& # % * *
, ! $
(2==1) !$ $! $. < $ ,##% (21 2i0 + 2m1 i),
$ !$ ! $ F1 = (1 + px0 );21 F3
h 1j m1i h0 ; 12 j h 0 ; j m0i 0 1
1
1
1
1
1 2
1
1
1
1
1
522 . . h21j2 2m1i, ! ' # !$& $* $! ;-#$ % * ! x0 ! px0=2, x ! 4;m x, ! 0, m ! m0, $" ! *& (10), (12) &$"! # !$&$ (13). J * &' *& + 2,
(13) (14) , & *$ *$ $. 1
5. & $
2. A
! * " !
! $*! ' * "
, *$ *$ !
#$ % F1, F2, F3. 3. ' (13), (14) p F1 = 22 (1 + 1 ; x0 );2 h21j1 2m1i h21 ; 0 + 1j1 2m1 ; m0 i L 5 x21 x21 F2 Q21F1 (15) h0j1 m0ih21 ; 0 + 1j0 2m1 ; m0i 1
x21 = (1 ;
1
1 ; x0)(1 + p1 ; x0);1, x21 = x(1 + p1 ; x0);2m & h 2 j1 m2 i h 2+1 j1 m2 i h2j0 m2i L 5 x12 x12 ; Q12 F2 F2 = (1 + x0) F1 h1 + ; j1 m ; m i 1 2 1 2 (16) x12 = 4x0(1 + x0);2 , x12 = 2m x(1 + x0);m & 2 2 h1j1 m1i h1 2 + 12 j1 m1 2i L 5 x23 x23 F3 = 1 + p1 ; x F2 h + 1 j1 m ih + 1 j0 m i Q23F3 2 2 2 12 2 12 0 (17) p p p x23 = (1 ; 1 ; x0)2 (1 + 1 ; x0);2 , x23 = 4m x(1 + 1 ; x0);2m & F2 = (1 + px0);2 h1j1 m1i h1 2 + 21 j1 m1 2i h2j0 m2i L 5 x32 x32 F3 Q32F2 (18) h1 + 21 2j1 2m1 2i x32 = 4px0 (1 + px0);2 , x32 = 4m x(1 + px0);2m . ( (13) # ! 4] G = C G = C1G + C2 G (19) m 2j1 m2i L 51 ; x0 (;1) x C1G = h1 F hh11+j1m 1ji1hm (20) 120 12 0i h0 1j0 m0 1i h0 2j0 m0 2i p
1
1
1
1
1
1
1
1
1
1 2
1
1
1 2
1
12 0
523
( x ; 1)m x 0 C2G = h2(1 ; x0) F 1 + 1 0 0 m0 2i (21) ;(0);(0 1 2) h2 = ;(2 ; 0);(12 0) : h1 = ;( (22) 0 1);(0 2) ;(1 + 2 0);(1 + 1 0)
. <! #$ % F1 (!. (10))
(19), &$"! " * #$ % + F1. <! +* &
Q31 (!. (13)), !! F1 = Q31C1F1 + Q31C2F1, * & ! *+ #$ % F3. < $ & ! ! ' # !$& (19), | ! ' ! *#%
%
* C 0 = L0C , * L0 = L1L2 | * &
,&
(!. *& # !$&$ (38) *& &$" !1 = !2 = 1, + 2]), !! F1 = C1Q31 C1F1 + C2Q13 C1F1 + L0 C1Q31 C2F1 + L0 C2Q31 C2F1 (23) * #$ % " ! F2 (!. (11)), "! #$ %,
* & ! , "
* #$ %!, *! ' " & !
. < * * "& , !! p p (24) F1 = AQ21 F1 + B (1 ; 1 ; x0 )1; (1 + 1 ; x0) ;1;2 F~2 * F~2 | * F2, ! $! ! *!, 0);(0 ; 1) B = p;( (25) ;(20 1);(21) D(0 ; 21) * ;( ; 1 );(2 ; 2 ) ;( 1 ; );(2 ) (26) D( ) 41;2;(1 ; ) + 2 4 ;( ) = 0 & ! . 2 + Q21F1 *& # !$& (15). L* &"$ E ( )
! ; 21 );(2 ; 2 ) ;( 21 ; );(2 ) (27) E ( ) = 4;(2 ;(;1);(2 + + 21 0);(1 ; ) 16 ;(21);(1 ; 21);( ) 0 1 ! ' * ;-#$ %, &'
"& B, ! + ', "
; 0) A ;(p0);(1 (28) 42 ; E(0 ; 21) = 1: < *
(25), (28) (24) * # !$& (15) *
! *
(16), & ! # !$& (15). M # !$& (13) (15) &*$
(17) !$
(18), " * &' *& + 3. 012
1
1
51
h0 1j m0 1i h0 2j m0 2i L ; x0 h 0 1 2j m0 1 2 i h0 1j m0 1i h0 2j
0
1
0
0
0 1 2
1
524
. .
6. ( $)
2 * & !
! #$ %!
*! , !+*$
! #$ %! *$%
! * " !
!. 3. 2 ! $&
,
*! *& (1), $*! " ' * #$ % &*$! ! &!: F1! & 0 = 12 ; ! + 21 m0 = m125 F2! & 2 = 1 ; 1 + 2!(0 ; 1) m2 = ;m1 + 2!m0 5 F0! & 0 = ! + 21 m0 = 0 * ! = 0 1 ' F1! F1! h1j1 m1i h2j1 m2i L ==h12 ; ! + 12 j1 m12i5 x0 x] (29) 0 F2! F2! h1j1 m1i h j1 ;m1 + 2!m0i L ==h0j1 m0i5 x0 x] (30) 1 F0! F0! h1j1 m1i h2j1 m2i L ==h! + 2 j1 0i5 x0 x] (31) 0 * = 1 ; 1 + 2!(0 ; 1). 4. M*$%
! * " !
! $*! ' * "
, *+ *$ *$ #$ %.
4. ) % (13){(18) L1 , L2 , L0 = L1 L2 . % 56.
. M *& &
L1 , L2 (!. 2]) &*$ , " # !$& , +
#$ % F1, F2 " * #$ % F1!, F2!
, ! * F1 = L2;! F1 = (1 ; x0); ; ! 1 j1 m0 1i h1 + 1;! j0 m1i L 5 x0 z h1 + !2 j1 m1i h0 1 + !; 0 2 2 F1! (32) 1 h0 j1 m0i h0 1 + !; 2 j0 m0 1i * x00 = x0(x0 ; 1);1, z = x(1 ; x0);m5 F2 = L1+! F2 = (1 ; x0); ;! h10 j1 m01i h20 j1 m02i h1 + !2 1j0 m1 + !m2 1i L 5 x00 u F2! (33) h1 + 1 2j1 m1 2i h1 ; 2 + !1 2j0 ;m2 + !m1 2i * 10 = 2 + !1 2 m01 = m2 + !m1 2 20 = 1 ; 2 + !1 2 m02 = ;m2 + !m1 2 ! = 0 1 u = x(1 ; x0);m ;!m : 1
2
2
1 2
2
12
525 ? &
L1, L2 ! + ' &' # !$&! (32), (33), , ! + &'
' *& , %& &' * , !
L2: F1! = L2 F1! = (1 ; x0); h1 ; !2 j1 m1i h1 + 1;! j1 m1i h2j0 m2i L 5 x00 z 2 F1 (34) h12 ; ! + 21 j1 m12i h1 + 1;! 2 j0 m1i F2! = L2 F2! = (1 ; x0); h10 j1 m01i h20 j1 m02i h1 ; 1 +2!(0 ; 1)j0 ;m1 + 2!m0 i L 5 x00 z F2 h0j1 m0i h1 + (1 ; 2!)(0 ; 1)j0 m1 + (1 ; 2!)m0 i (35) * 10 = (1 ; !)(0 ; 1) m01 = m1 + (1 ; !)m0 20 = 1 ; !(0 ; 1) m02 = m1 ; !m0 ! = 0 1: A" * , ! '
(13){(18) (32){(35) ! + &$" ' 12 6! 7 12 6 7 *$%
* $"& * "
* LQL00, LQ, QL0, * Q | *
* "
, L, L |
* (32){(35). N 24
6
! + ', &' $ # !%
#$ % F1, F2, F3 . 2 ! ! *&, #$ % F1 , F2 *
L0 #$ % F3 *
L1, L2, L0 ! . A O*
L0 +*
I * L0 , * L00 = I , L10 = L0, +
L1 , L2 , L0 , I * L(!1 !2), * L(0 0) = I , L(1 0) = L1 , L(0 1) = L2 L(1 1) = L0 , !!: F1 = L0 F1 = (1 ; x0)( ;2 ; ) h10 j1 m01i h10 + 12 j1 m01i h1j0 m1i h1 + 21 j0 m1i L 5 x0 p (36) F1 h0 j1 m0i h0 1j0 m0 1i h0 1 ; 12 j0 m0 1i 10 = 1 + (0 ; 21 ; 12 ), m01 = m1 + (m0 ; 2m1 ), p = (1 ; x0)(m ;2m ) x, F2 = L0 F2 = (1 ; x0)(1;2 ) h10 j1 m01i h20 j1 m02i h1j0 m1i h2j0 m2i L 5 x0 q F2 h1 + 1 2j1 m1 2i h1 ; 2j0 ;m2i h1 + 1 ; 22j0 (m1 ; 2m2)i (37) i0 = i + (1 ; 22), m0i = mi ; 2m2, i = 1 25 q = (1 ; x0 );2m x, F3 = L(! ! ) F3 = (1 ; x0)(2! ;1)! ;! h10 j1 m01i h20 j1 m02i h1j0 !1m1i L 5 x000 (1 ; x0)m x F3 (38) h22j1 2m2i h22 ; 1j0 !1(2m2 ; m1)i
1
1
0
1 2
1
0
0
2
1
2
2
1 2
2 1
1
526 *
. .
10 = 1 + 2!12 1 m01 = m1 + 2!1m2 1 m = (2!2 ; 1)!1 m2 ; !2m1 !1 2 = !1 ; !2 x000 = x0 1 + j!1 2j(x0 ; 2)];1:
8 ! $
(13){(18) # !$&! (32){(35) # !% !
! (36){(38), &$"! & * $ "& * "
,
(39){(41) *$%
(42){(46) ( &* +* ! 6! !7
! $
$ !$ 6 7
): F1 = L(!1 !2) Q31 F1 F3 F3 = L0 Q13 F3 F1 (39) 2 1 F1 = L0 Q1 F1 F3 F2 = L0 Q2 F2 F1 (40) 2 3 F3 = L0 Q3 F3 F2 F2 = L(!1 !2) Q2 F2 F3 (41) 3 1 F1! = L(!1 !2) Q1 L2 F1! F3 F3 = L2;! Q3 F3 F1! (42) 2 1 F1 = L1+! Q1 F1 F2! F2! = L0 Q2 F2! F1 (43) 1 2 (44) F1! = L0 Q1 L2 F1! F2 F2 = L2;! Q2 F2 F1! 2 1 F1! = L1+! Q1L2 F1! F2! F2! = L2;! Q2 L2 F2! F1! (45) 2 F3 = L1+! Q3 F3 F2! F2! = L(!1 !2) Q32 L2 F2! F2 (46) * !1 !2 ! !0 = 0 1, " " , "
+ *+ *$ #$ %! Fi & Fi! * & &' !&' ! + &. P *! , + ! *, &' $ ! $ ' & &$" ! *
$ &
*$
*$ . < ! + ! % !
!1, !2, !, !0, , &$"! 28 6! 7 * "
28 , & ! 6 7
(!. (39){(46)). M &' $
L1, L2, L0 # !$& (32){(38), &
$ * ', " !! $ 56 # !$& *$ *$%
* $ "& * "
$ $ , "
* &' *& + 4. A ! * " * $ "&
$ & " . 1, *+ #$ % F1, F2, F3, F10, F11, F20, F21. A &' " . 1, *+ #$ % F00 F01, !. *& 8. 0
7. + $
0
0
0
0
2 &$" x1 = x2 = : : : = xN = 0
(39){(46) * * "
H$ #$ % H$ F12 (!. # !$& (1){(39) x 2.11 8]). )* 36 # !$&, $ 8], ! 32 * $ "&
, *+ 4 +*
QFQ20 F2 QQ QQQQQ F21 r
r
r
r
527
F10 F00 r
F1 QQ F11 F01 r
F3
r
r
r
> . 1. , %' $)% 0 # # $ !# * (#*
, $ !)%' ) # " ) ) # #
$ # 1 *. # )'0
# !) ) ##)% #) # ) $ !# % (3 %') # !). @
# ! )% * $ !# ) 3 ( . # ). /%$ )" # # $ !# #! # !) $ # ! # # $ !# * * $ !# .
# !
. $ ! &
!, 56
, # !$& (39){(46) &$" x = 0, 6 & !7 H$ $ $
&
. &' $ $ H$ *! + &% , &* ! ! +* $ (39){(46) * & "&
,
$ * !$ $ , "&
& " , * . 2.11 8]. Q & * &' * &'
H$, ! ! # !$& (39){(46),
$
! H$, ! ,
& x 2.11 8], * * . 8 * &% , # !$&! (39){(46) H$ ! & ! " *, " &"
&
! &$" . )&$" ! &' ! O +
* "& * $ "&
, ! "& ! ! "& !
* (39){(46). R ! &$"! # !$& (39){(46) & &' # !$& H$, & * $ "& * "
! " *
& "& ! , ! #$ % S&, H , 8! * T', I$"&& . . & , #$ % & ! * " , * & . 1. < ! $ " ' . 1, *+$ #$ % F00, F01.
8. -. : 0
<! # !$&$ & " * &+ (!. (15) 4]) #$ % F1, &$"!
528
. . E $ "
(39) (39 ) (40) (40 ) (41) (41 ) (42) (42 ) (43) (43 ) (44) (44 ) (45) (45 ) (46) (46 )
4,1 2,2 2,1 2,2 2,1 4,1 8,2 2,2 2,1 4,2 4,2 2,2 4,2 4,4 2,1 8,2
!1 = !2 = 0(17)] = 0(4) (28)5 = 1(29)] = 0(6)] = 0(34) (35)] = 0(5) (31)] !1 = !2 = 0(36)] ! = !1 = !2 = 0(12)5 ! = 1 !1 = !2 = 0(15)] ! = 0(26)5 ! = 1(27)] ! = 1(16)] ! = 0 = 1(24)5 ! = 1 = 0(20)] ! = = 0(11)5 ! = 1 = 0(14)] ! = 0(1)0 (32)5 ! = 1(33)]0 ! = 00 ! = 1(10)50 ! = ! = 1(13)] !0= ! = 0(22)5 ! = 0 !0 = 1(23)5 ! = 1 ! = 0(2) (18)5 ! = ! = 1(19)] ! = 1(30)] !1 = !2 = ! = 0(25)5 ! = 1 !1 = !2 = 0(21)]
;(0);( 12 ; ) (;x0); ; ;(1 ; 1; 2 );(0 1 ; 2 ) =0 =0 m h1 + 2 j1 m1i h1 0 + 1 + 2 j1 m1 0i L 5 x1 (;1)m xx; 0 F0 (47) h + 12 j1 0i #$ % F1 * ! #$ %! * (31) ( &" 4] ! + & # !$& (47), ! # !$ ;-* ,
&" *& % C1, C2). < &' $ #$ % F0 ! * &
L1, L2, L0, # !$& (47) * $ * F1 = L(!1 !2)C1F1 + L(!10 !20 )C2F1 F01 + F00 (48) * !1 !2 !10 !20 = 0 1. A
, + * #$ % F00 F01 " #$ % F1, ! * 1 1 X X ;( + 21 );(1 ; 2!)1 2] (;x0); ;! F0 = C1+! F0 = ;(1 + !2 1);( ; 2 + !2 1 + 12 ) !=0 !=0 " # h j1 m2 + !m1 2i h + 21 j1 m2 + !m1 2i L 5 x1 (;1)m m x!m x F1 h1 + (1 ; 2!)2 1j1 (1 ; 2!)m2 1i (49) * = 0 15 = 2 + !1 2 ; 2 , F1 =
1 X
C2; F1 =
1 X
1
2
1
0
0
1 2
2
0
2 1
0
2+
1 2
529
+! ;( + 21 );(2! ; 1) ] x; 0 1 ;( ; 1 + ! + 2 );(1 ; ) (1 ; x0)! !=0 !=0 j0 m1i L 51 ; x1 s h10 j1 m01i h10 + 12 j1 m01i h1 ; 2 1 + 1; 2 (50) F1 h1 + (1 ; 2!) j1 (1 ; 2!)m12i * = 12 ; ; 21 , 10 = ! ; 2 , m01 = m2 ; !m12, s = x(x0 ; 1);!m x!0 m ;m . < &$"
#$ % F1 #$ %! F00, F01, !$ #$ % F10, F11. 6< !+$ "7 ' F00, F01 F10, F11, * . 1, * !-&
! * " !
!. 8 !
(48){(50)
! * " !
! # !$&! (33), (35), (37), (38) & &$" ' &' *
"& * "
, #$ % F00, F01 #$ %! F2, F3, F20 F21. J
" ! &$" x = 0 * "&
#$ % H$ F12. M , ! "& " &$" H$ 8] * &' "
,
$ ! # !! #$ % F00, F01 #$ % F21. <*& ! * "
* &" *% * *
&' &' ' $&'
, * &' . S& $ &" ! * * # !$& H$, 8] $ (!, !. . 77), "
6 * "
& &* ! P -$ U!7. < ! * * *& * &
(39){(46)
& * ! $ *& " # ! G , $ , & " P -$ U!. V , " &' $ ' "' & +&! %&. 2 8] . 77 ! + $ !
!, " 6
# !$& 7 ( "&$ &' # !$& * F3 F11, F2 F10, F1 F2, F10 F21 F21 F00 + F01) ! +
* ', 6 , " " , # !$& $*
& ! " !$ $ 7. < - *! !$, *" * * &$" " G-# !
& *& !' $&, "! *" P -$ . ) !'! $* ! & $&' ! , , &$" G-* " "& &' $ !
, &+
6! 7
#$ %, * 6
7
. R $ * ' , !, * " !
! ! * ! * &' !, * ! . 78{80 8]. <*& ! * "
, * *& ! " #$ % &' $ * & + $ $ ,
!
* "
, &
F0 =
1 X
C1+! L1 F0 =
1 X
2
0
12
12
2
530 . .
* # !$& & " * &+. < *& ! * * $ ' ! * '
*& & % J2P # ! !! ! &"
. <& % !! & A. ). <!
. 2 &" , !! & !& , * & * $ * ! !' & Maple V.4, * * & %. R " * & "& &' , *
* ! " #$ %, ! +
' * $
! !' & !'
& , " ' &' , " *& !
&'
& )8S ( $* , * & + $ , ! )8S " ! &' *$" * ), & $ &$ &
&*
&, *$ $ *
& ! ! , # , ! + $ .
1 $
1] . . // ". . | 2000. | &. 67, . 4. | +. 573{581. 2] . . 0 1 2 3 2 // ". . | 2001. | &. 70, . 5. | +. 769{779. 3] . ., 5 0. +. 6 2 1 3 2 7 18 , 3 G2 4 G3 6 // ". . | 2002. | &. 71, . 1. | C. 88{89. 4] . . 91 1 1: 3 3 3 // 9 . 1. . | 2001. | &. 7, . 1. | +. 71{86. 5] . . 3 ;
18 3 8< 2 1 // 9 . 1. . | 1999. | &. 5, . 3. | +. 717{745. 6] Niukkanen A. W. Operator factorization method and addition formulas for hypergeometric functions // Integral transforms and special functions. | 2001. | Vol. 11. | P. 25{48. 7] . . 2 3 18 3 2 2 // = 3 . . | 1988. | T. 43, . 3. | +. 191{192. 8] > 2 7., ? 2 . @ . &. 1. | ".: , 1973. 9] Askey R. Ortogonal Polynomial and Special Functions. | Bristol: Arrowsmith, 1975. 10] Niukkanen A. W. Fourier transforms of atomic orbitals. I. Reduction to four-dimensional harmonics and quadratic transformations // International Journal of Quantum Chemistry. | 1984. | Vol. 25. | P. 941{955.
531
11] Appell P., KampEe de FEeriet J. Fonctions hypergEeomEetriques et hypersphEeriques: polyn^omes d'Hermite. | Paris: Gauthier{Villars, 1926. 12] Srivastava H. M., Karlsson P. W. Multiple hypergeometric series. | Chichester: Ellis Hoorwood, 1985.
) "* * 1996 . ( | * 2002 .).
. .
e-mail: [email protected]
512.54
: , , , !" , ##$% !" .
& ' p- !, #'$ ##$% ! .
Abstract O. V. Pashkovskaya, Generalized uniform automorphisms, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 533{545.
We consider p-groups that have generalized uniform automorphisms.
p- , . . 2. G = P (b), P | p-
, b | q,
P (q < p, p, q | ! !), A | !
P p2 , R |
, " " p P , L = (b)G \P . $ % " : 1) R < CG (b)' 2) R = A' 3) R |
p3 p R = ((x) (z))(y), (x) = Z(R) b;1 xb = xk , b;1zb = z m , b;1 yb = ym , ! k m2 (mod p) 1 < k m < p ; 1' 4) R |
! k, 1 < k < p, b;1rb = rk r R' 5)
L
(t) p b;1tb = tk , 1 < k < p, , R \ CP (b)
p2 R = (t) (R \ CP (b)). &# ! " . ( 7F0008). , 2002, 8, 0 2, . 533{545. c 2002 !
" #$%, & " ' (
534
. .
) . . ) P | p- . + P , , P - . . ) P | p- . + P , . / p2 P . 0 / . ) G | , : G = PQ, P | p- , . / A p2 , Q = (b), b | , q, . P (p > q, p, q | / / ). 2/ L = P \ QG , R | , - - , p P, K | L. / -
, G . 1. C |
p2
P . $ L 6 NG (C). . ) L QG , / NG (C). ) - , / , b . P, , , p2 P , , Q 6 NG (C) , /, Qg 6 NG (C) ,
C p2 , g G. 2 , / L 6 NG (C). 3
. 2. ( -
G=L : G=L = (P=L) (QL=L) !, 4h b] " L h P . . 3 , gL G=L - gL = (pL) (qlL), pL 2 P=L qlL 2 QL=L, G (G = PQ) , / L
G. ) P=L G=L ( ), / P=L QL=L (P \ Q = E). 6 QG G QL=L QG =L, QL=L G=L (
). 7, G=L = (P=L) (QL=L). 3
. 3. )
G q "
h b0 , h 2 L b0 2 Q,
P .
535
. ) b1 | , q. 6 G G = P Q, , b1 - b1 = h b0, h 2 P b0 2 Q. 7 , / , b0 . P. ) C | , p2 . 8 - /: 1) , h - L, 2) , h - P n L. 1) ) h 2 L. )
1 / h 2 NG (C). + , . , b1 , C , / (b0);1 Cb0 = C h 2 NG (C), b;1 1Cb1 = (b0 );1h;1 Chb0 = (b0);1 Cb0 = C. 7, / -
. 2) ) h 2 P n L. )- , / , / - . 8 b1 L = (b0 h)L , b1 = b0h . 9 q. : (b0 h)L , b0L q , hL p.
2 - G=L G=L = (P=L) (QL=L), , , , q , p - q. ) / . 3
. 4. * b
> p3 P . . ) T | , p3 P. 9 b . T,
. ) b . T. 8 /: ) q p ; 1, ) q p + 1. 0 , / , p2 - , .. ) ) q p ; 1. 6 b . T , T , t p, / t 2= CP (b). 2/ , , t - , p2 T (t) (x), (b)- , , b , . . 6 q p ; 1, (t) (x) - , s v , / (t) (x) = (s) (v) b;1sb = sr1 , b;1vb = vr2 . 6 - , p2 , p3 (s) (v) (a). ) , - / , / b;1 ab = ar3 ( ., , 41]). 8 , (sv) (a). 2 - - (b)- . 6 (a) (b)- , , z , / (sv) (a) = (z) (a) (z) - (b)- . ) , , < - , - / , / z = sv a. 6 b;1 zb = z r4 , , , b;1 zb = b;1(sv a)b = = sr1 vr2 ar3 . 6 , z r4 = sr4 vr4 ar4 = sr1 vr2 ar3 , r4 = r1 , r4 = r2, r4 = r3, / )
.
536
. .
) ) q p + 1. 2/ / V , p2 T . ) V (b)- . 8 V (b). : b . , V (/ . V | , , / , / q p + 1, , p2 ). 6 V (b) = (x b), x | , V x 6= 1. T V (b)- : T = V Z ( >< , ., , 41]). ) y | , Z. 6 , p2 (x) (y). 2 - (b)- . ? V (b) = (x b), q p+1 x 2= CP (b), , (x) (y) V . 6 y 2 V , / -. ) / / / , / / ) -
. 3
. 5. +
L ,
G. . ) A | , p2 P. 2/ D = A \ L - .. 1. 9 D 6= 1, D . / , A / D L, L 6 NP (A)
1. 2/ / K L. 7 ) ( ., , 42]) , / L , / , / K , , /
L. 2 , / L G, , / K / G. 7, , /
. 2. 9 D = 1, (A L) = A L
1. 8 A (c), c 2 L, jcj = p. A , a p, C = (a) (c). 6 C , p2, , - C - , A ( C \ L 6= 1), / 1. 6 , . - . L , G. 3
. 6. ) p L " K . . )- , / , p L - K. ) h | , p L, K(h), , K , z, / (z h) = (z) (h). )
1 L 6 NG ((z) (h)), , (z) (h) L, , - K. 0/, h 2 K. 7, , p L - K. 3
.
537
7. ) P n V , V = (R \ K) CP (b),
p. ) !, CP (b) = 1, p P " K . . 9 P = V , -
/ . ) - , / P 6= V P n V , y p. 6 y 2= V , , /, y 2= L. 2/ / K L,
5. 7 , / K G. 6 (K y). 2 (K y) = K(y). : p- / ( A , ., , 42]) , , . 6 K , t p, / (t y) = (t) (y). ) (t) (y) - (b)- . , - ((t) (y))(b). , (t) (b)- . > - , s (t) (y) , / (t) (y) = (t) (s) s - (b)- ( >< , ., , 41]), - / , / y = ts. 6 t 2 L, - G=L yL = sL,
2 , sL . , bL. 6 (s) (b)- , b;1sb = su . 9 1 < u < p, - G=L , sL . , bL. 6, / , - u = 1, s 2 CP (b). 6 , , y y = ts, t 2 L s 2 CP (b). ) / / - , / y 2 P n V . B / <
. 3
. 1. G = P (b), P | p-
, b | q,
P (q < p, p, q | ! !), P !
A p2 , R |
, " " p P , L = (b)G \ P . $ R 6 CP (b) , (R \ K) 6 CP (b), K |
L. . B/ / . B- . ) (R\K) 6 CP (b). ) - , < , t p, t 2= CP (b) t 2= K. 0 (K t) = K(t) (K t) ( A , ., , 42]). 0 , / b;1 tb = tk , 1 < k < p ; 1. 8
4t b]: 4t b] = t;1 b;1tb = t;1 tk = tk;1 tk;1 2 L (
2), k 6= 1, t 2 L. 6 t 2 K
6, / / - , t. B / / , / (R \ K) 6 CP (b), R 6 CP (b). 6 . 8. , K
L | !
K 66 CP (b),
L
(t) p b;1tb = tk , 1 < k < p. - , R \ CP (b)
538
. .
p2 R = (z) (R \ CP (b)). . ) z | , p K, , t | , P, / t 2= (z) (
, P). 8 - .: 1) t 2 L, 2) t 2= L. 1) ) t 2 L. , / / , / (z t) = (z) (t),
1 (z) (t) L. 6 (z) (t) - L, , / , / K | . / . 7, . 1) -. 0/, L , p, - (z). 2) ) t 2= L. )
7 , t - t = l c, l 2 K \ R, c 2 CP (b). 6, < , l 2 (z), (l c) = (l) (c). 6 , , t p P (l) (c), (l) = (z) c 2 CP (b). )- , / R \ CP (b) , p2 . ) / R \ CP (b) , X p2. 6 , / R = (z) (R \ CP (b)), - - , p3 ( ., , 41]). ?
4 , b - , , X 6 CP (b), z 2= CP (b). ) / . B / / , / R \ CP (b) , > p2 . 3
. 9. K |
L K 66 CP (b). $ K \ CP (b) | !
( !, !). . 2/ / D / , / K \ CP (b). K , z, / b;1zb = z k , 1 < k < p ; 1, K 66 CP (b)
. ) - , / D . / . 8 V = (D z). D V / (b)- . 6 - V (b). 2/ / T , - , p Z(V ), - /: 1) T . / , 2) T . / . 1) ) T | . / . 6 (T z) (b)- . )- , / , b . / - , / z 2 T , T , (z) (u), u 2 CP (b) b;1zb = z k , k 6= 1. ? D > p2 , - , > p3 , / / (z) D, z 2 Z(V ), , b , / /
4. 6 - , / z 2= T. 6 T (z) ,
539
> p3 . )
4 , b - , T 6 CP (b), b;1zb = z k , k 6= 1. , / 1) /. 2) ) T | . / , T = (a). 6 D , (a) (y). ) B | V . 2 , > p2. ) W | , - - , p B, jW j > p2 W - (b)- . 6 W / B / D. 8 M = (W (z))(b). 0 b 2 CM (W) CM (W) / M (W /M). 6 b 2 CM (W ), z ;1 b;1z 2 CM (W) z ;1b;1 zbb;1 = z k;1b;1 2 2 CM (W). 6 z k;1 2 CM (W) z 2 CM (W ), jz j = p. , z . W , / -. ) / / / , / D | . / . 3
. 10. T p|
P p3 p p p: T = (x1 x2 y j x1 = x2 =y =1 x1x2 =x2x1 x1 y =yx1 y;1 x2y =x2x1), T 66 CP (b) b : b;1x1 b = xk1 , b;1x2 b = xm2 , b;1 yb = yr . $ : ) m = r' ) k m2 (mod p)' ) x1 2= CT (b)' ) 1 < m < p ; 1 1 < k < p ; 1. . 2/ / X X = (x1 ) (x2 ). )- , / m = r. 2 , m 6= r, - X=(x1 ) = (x2 (x1)) . / , - T=(x1) T=(x1) = (x2 (x1)) (y(x1 )). 8 , - (T (b))=(x1 ) , b , x2y(x1 ): (b(x1));1 x2y(x1 )b(x1) = = b;1 x2yb(x1 ) = b;1x2bb;1 yb(x1 ) = xm2 yr (x1). 9 m 6= r, , xm2 yr (x1 ) - . / (x2y(x1 )). ? (x1 x2y) = (x1) (x2y) - , b. ) / . 7, m = r. F - )
. )- , / k m2 (mod p). 7 y;1 x2y = x2x1 . 6 ; 1 b (y;1 x2y)b = b;1 (x2x1 )b. B , y;m xm2 ym = xm2 xk1 . ) / 2 y;m xm2 ym =2 (y;m x2ym )m = (x2 xm1 )m = xm2 xm1 2 . 7 xm2 xm1 = xm2 xk1 . 2 xm1 = xk1 , k m2 (mod p). F - )
. )- , / x1 2= CT (b). 2 , x1 2 CT (b), k = 1. ) - ), < , m2 1 (mod p). B , m2 ; 1 0 (mod p), (m ; 1)(m + 1) 0 (mod p). 6 m < p p | / , m = p ; 1 m = 1. 9 m = 1, - ) k = 1 - ) r = 1, T 6 CP (b), / /
540
. .
. 9 m = 1, b;1x2b = xp2;1 = x;2 1 b;1yb = yp;1 = y;1 . 8 - 4T(b)]=(x1), / , / , b - , , , b 2. , q = 2. ) / . F - )
. )- , / 1 < m < p ; 1 1 < k < p ; 1. 7 , / 1 6 m 6 p ; 1 1 6 k 6 p ; 1. ) - ) , / k 6= 1 m 6= p ; 1. 2 , / k 6= p ; 1 m 6= 1. 9 - , / k = p ; 1, / , / , b , . /
, , 2 ( / , / q | / ). ) / . ) - , / m = 1, x1 y - CT (b). 7 T y;1 x2 y = x2x1 , , x1 - CT (b). ) / . 6 , - )
. 6 - , / G1 G1 = T(b), T | p3 p, . / , / X = (x1) (x2 ) G1. 7 , / Aut(X) = GL2 (p). 6 / p, / /, p = 2n + 1, GL2 (p) D pq, q p ; 1 ( J , 45]). ) , D = SM, S | . / p, M q M 6 NGL2 (p) (S). 3
. 11. K
L
X p2 K 66 CP (b). $ R \ L = X Rp \ L |
p3 p: p R\L = (x1 x2 y j x1 = x2 = yp = 1 x1x2 = x2x1 x1y = yx1 y;1 x2y = x2x1), b;1x1 b = xk1 , b;1 x2b = xm2 , b;1yb = yr , : ) m = r' ) k m2 (mod p)' ) x1 2= CR (b)' ) 1 < m < p ; 1 1 < k < p ; 1. . ) - , / , y L, -
X, jyj = p. 8 T = X(y). 2/ X = (x1 ) (x2). T /: 1) T , p2 , 2) T , p. 0 , / T - , / /
X. 1) ) T , p2 , - / , / , , x2 y. 6 T - T = (x2 y)(y) , > 43], / , / T, - , p, ,
p2 , / / T . 7, / 1) - .
541
2) ) T , p. K T . / , / / X. >- / , / Z(T) = (x1). 7, T p3 p. 44] / : T = (x1 x2 y j xp1 = = xp2 = yp = 1 x1 x2 = x2 x1 x1 y = yx1 y;1 x2y = x2 x1). 8 , , b , x1 , x2, y. 6 , , - - , p2 , / , / b;1x1b = xk1 , b;1 x2b = xm2 b;1 yb = yr ( - / >< , ., , 41]). )- , / L , p, - T. 2 , < , y1 p, - L - T. : y y1 - /, / / / X, - / X(y1 ) . 6 Aut(X) / , p, / / Aut(X). B / / , / L , p - T. G1 G1 = T(b), T | p3 p, - ), ), ), )
10. 3
. 12. K
L
X p2 K 66 CP (b). $ % " : 1) R = X ' 2) R |
p3 p' 3) R |
b . . 9 , p - L, ,
11, - 1) 2)
.
11 L , p - : 1) R \ L = X, 2) R \ L = T | p3 p. 8 , , / - , p P. ) R \ L = X < , x p, - L. 6 X(x). 9 , , b .
, - 3)
. 9 , - 2)
, , p - X(x). 9 R \ L = T | p3 p, P , p, - L,
11 - 2)
. 3
. 13. K |
L X |
, " " p Z(K). ,
542
. .
K \ CP (b) = 1 X | !
, CP (b) = 1 % " : 1) R |
r R b;1rb = rk , 1 < k < p' 2) R = X = A. . / , / X P . )
X | . / Z(K) K \ CP (b) = 1, , b X , , , - /: 1) q p ; 1, 2) q p + 1. 1) ) q p ; 1. 9 X = (a) (u), (b)- . 6 q p ; 1, - / , / b;1ab = ak , b;1 ub = um , 1 < k m < p. )- , / CP (b) = 1. 2 , < , h p CP (b). 6, X / P, - X(h), (b)- . 9 h . X, X(h) | , p3
4 , b - . 6 X < CP (b), / -. ) / . 9 h . X, p3 p m = 1 (
10). ) / . , CP (b) = 1. 6 , p P - K (
7). 9 m 6= k, , p - X, / - 2)
. 9 m = k, , p P - , , , b , - 1)
. 9 X > p3, , b . , X - (b)- , p2 / - . 2) ) q p + 1. )- , / , / jX j = p2 . 2 , X < , p3 : (a) (u) (t) 6 X. 6 , (a) (t) (u) (t) - (b)- , , / (t) = ((a) (t)) \ ((u) (t)) - - (b)- , / -, / q p + 1 p ; 1. ) / . 7, jX j = p2. 6 - , / , p P - X, R = X = (a) (u). 2 . ) , r p - P - X. 6 X(r) = ((a) (u)(r). K , (, / p- ), - / , / a 2 Z(X(r)). : (a) (r) (a) (u) - (b)- . 6
543
(a) - (b)- , / , , / , / q p + 1 p ; 1. 6 , / 2) , p P - , p2 ( - 2)
) Z(K), K \ CP (b) = 1, CP (b) = 1 / 2). 3
. 14. K |
L ( ! ) X |
, " " p Z(K). , K \ CP (b) = 1 X | !
, % " : 1) R |
r R b;1rb = rk , 1 < k < p' 2) R |
p3 p' 3) R = X = A. . 2/ / B , K (
jB j > p2 ). / - , / , p L - K. ) t | , p L. 6 K(t). 6 ( A ), K , z, / (z t) = (z) (t). )
1 L 6 NG ((z) (t)), (z) (t) / L , , (z) (t) 6 K, t 2 K. 9 jB j = p2 ,
12 -
. ) jB j > p3, B (b)- , b . , : , d B b;1db = dk 1 < k < p (k 6= 1, / B 6 CP (b), / /
). 0 - , / B K, , ( > , ., , 41]), , p K - B. ) , h p P , - K. ) B P , B(h) (b)- . - (b)- p3 p: ((b1 ) (b2 ))(h). 9 , , b . , - 1)
. 9 , / , / B , b . ,
10, b;1 hb = hk , , / , b . B(h). 6 B(h) - 1)
. 15. K |
L ( ! ). , K \ CP (b) = (a), a 6= 1, R = A.
544
. .
. 2/ / X , - - , p Z(K), z 2 (a), jz j = p. 8 /: 1) z 2= X, 2) z 2 X. 1) ) z 2= X. 9 jX j > p2 , (X z) = X (z) > p3 / / : ,
4 , b - , z 2 CP (b), X 6 CP (b), / /
( K \ CP (b) = (a)). 0/, jX j = p ( X = (x), jxj = p). 0
1 (x z) = (x) (z) L. , / L , p, - (x) (z). 2) ) z 2 X. 9 jX j = p (X = (z)), , / K | . / , K , t p, - X, / (z t) = (z) (t) (z) (t) / L. 7 , / L , p, - (z) (t). 6 - , / , p P - L. / , / X P . ) , h p - L, X(h), (b)- , , h - - , p2 , / b;1 hb = hs . 6 h;1b;1 hb = hs;1, ,
- - L (
2), h 2 L, / / - . + s = 1, h 2 CP (b), M1 = 4(t) (z)](h) ( / , z 2 X) M2 = 4(x) (z)](h) ( / , z 2 X). 0 (x) (z) (t) (z) P , / . ) M1 ( M2 ) (b)- ,
10 b;1tb = tk , b;1zb = z, b;1hb = h ( b;1xb = xk , b;1zb = z, b;1 hb = h). 6,
10, / k 1 (mod p), , k = 1 t 2 CP (b). ) / / / , / , p P - L. 3
. 2. G = P (b), P | p-
, b | q,
P (q < p, p, q |
! !), A | !
P p2 , R |
, " " p P , L = (b)G \P . $ % " : 1) R < CG (b)' 2) R = A' 3) R |
p3 p R = ((x) (z))(y), (x) = Z(R) b;1 xb = xk , b;1zb = z m , b;1 yb = ym , ! k m2 (mod p) 1 < k m < p ; 1' 4) R |
! k, 1 < k < p, b;1rb = rk r R' 5)
L
(t) p b;1tb = tk ,
545
1 < k < p, , R \ CP (b)
p2 R = (t) (R \ CP (b)). . 2/ L / K. 2
5. 9 P , - CP (b), - 1) . ) P , , - CP (b). 9 , , - K, K 66 CP (b), .: 1) K | . / , 2) K . /
. 1) 9 K | . / ,
8 - 5) . 2) 6 / , K | . / . )
9 / , / K \ CP (b) | . / , -. ) K \ CP (b) = 1. 6,
13 14, - 2), 3) 4) . ) K \ CP (b) = (z), z 6= 1. 6
15 - 2). 9 , p K - CP (b), 1 R 6 CP (b) - 1) . 6 .
1] . ., . . . | .: , 1977. 2] " #. $. % . | .: , 1967. 3] ( $. #. ) * + , -. // 0# 1112. | 1960. | %. 132. | 1. 762{765. 4] 7
. % . | .: . ., 1962. 5] Bloom D. M. The subgroups of PSL(3 q) for odd q // Trans. Amer. Math. Soc. | 1967. | Vol. 127, no. 1. | P. 150{178. !) 1999 .
. . , . .
. . .
517.958
: ,
, .
! " # {% ' z-( . )(
( " ! " ( ( . ! * ' ( ! + ( , ** - "
. ( ' . , ! ! , ( +-
*' ! ( . , / . 0" ! ! !1 . , *' !1 ( c ( " " !. , 1 " 1 -
2 .
Abstract V. V. Ternovskii, A. M. Khapaev, Relativistic charge in plane wave, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 547{557. It is proved that the system of Maxwell{Lorentz equations in z-presentation has many solutions. Determination of the energetic condition of the charge is reduced to analysis of the parametric system as a Kepler problem. The extreme conditions and con8guration of the electromagnetic wave 8eld determine the value of the re9ection parameter, which changes the e:ciency of electron-wave interaction and dynamics of the charge's motion. A visual geometric interpretation of possible energetic state of the electron is given. We present expressions that determine the change of the charge's energy and coordinates as functions of the length of interaction space and laboratory time. Four possible regimes of the electron energy change are found. We obtain a functional connection of the initial impulses with the electromagnetic wave amplitude. The exact theoretical solution of electron motion relativistic equations is constructed in the form of a uniformly converging Fourier series.
, 2002, 8, < 2, . 547{557. c 2002 , !" #$ %
. . , . .
548
- . ! " . # - , . . % , &! ' ! . % %, !% (1, 2], ' & & , % % % %. -" ' &
% % , &! % , . '% '% %. / ' & " % ,, '% &! ,, " , ' " .
# ,
A~ ( ) = ; cE! 0 (sin(! )~{ ; g cos(! )~| ] = t ; zc
(1)
(! | , E0 | , | ' , g = 1), % ' , 4 , ! (1)
L = 12 m0 x_ x_ ; ec0 x_ A
&
m0 x.
=
e0 x_ H c
m0 | , e0 = jej | , H
| 1. (1) &
e d A y. = e d A x. = mc d x mc d y
(2)
e @A @A e @A @A x y y x . z. = ; mc @z x_ ; @z y_ t = mc3 @t x_ + @t y_
(3)
% (x, y)
1
= | ( ( / , 1 | ( ( .
549
'%. 9 &
eE0 = dt " = m!c d
=
p
1
1 ; 2
2 = x2 + y2 + z2
(2), (3) :" % ,, '%
:
x_ = ;c" sin ! + cx y_ = +cg" cos ! + cy z_ = ; " (cx sin ! ; gcy cos ! ) + z_0 ; "g cy + + c _ = ;"!(x_ cos ! + gy_ sin ! ) q cx = x_ 0 cy = y_0 ; c"g c? = c2x + c2y |
(4)
, % ( )
x_ 0 = c x0 0 y_0 = c y0 0 z_0 = c z0 0 0 = p
1 : 1 ; 02
(5)
- % ' (1) &
= ?0 sin ' y0 = 1 dy = ; ?0 cos ' z0 = 1 dz : x0 = 1c dx dt z=0 c dt z=0 c dt z=0
< , &! & '& ', (1, 2]
+ = _ = ; z_ = const :
c
-% , . ! " % = f ( ), z = F ( ) (2, 3]. = &
2 0 2 0 z0 1 1 x0 y z 2 = 1 + c c = 2 2 (1 ; ) + 2 c + c : + < &! z % % % % (5):
= 0 + a0 (cos ; cos 0 ) z = z0 + a0 sin + z_c0 c"? ; cos 0 + cx cos = g cy a = c? " = ! + : sin 0 = 0 0 c? c? 0 c +
-
(6)
. . , . .
550
z . .
/ (6) % (2{4] &!: + = cos z+ = A+z + sin : (7)
C , , + , z+ , % "
+ (z ; z ) + = a1 ( ; 0 + a0 cos 0) z+ = a 0 0
A+ z =
z_0 + c? " ; cos 0 :
0 + : ,, Az ' . :
(3] (7) & ' 4 . D
" . A+ z. 0 ( ) > 0, ' E jA+ j > 1, . . z z + ' Z , (z ) , & + (z+ ), . . & . = ! , (1), . 1 ), ), ), , ! . # % .% : x0 = 03, y0 = 05, z0 = 06, ' : ) " = 06, ) " = 1, ) " = 24. F & ' .% . .
". jA+ z j < 1 z+ ( ) ' ' . # ! ' ' " 4 , ', (7). % " (7) ' : 4 , &! " , G& {4 (4] " . # & , '" % % , &! . ' (7), , : x = ;z+ y = ; + + 1 = + 1+ = s: Az F .
x = a(s ; sin s) y = a(1 ; cos s):
551
?. 1 = '" , . (A+ z ) % % (5) ("). % , &! % ) = 1, A+ z = +1H ) . > 1, j + A+ j < +1,
H z +
) < 1, jAz j > +1, H 2 2 ) jA+ z j = 0, , ' + + z+ = 1.
552
. . , . .
% ' , , , z . . 2 ), ), ), ) & , . # , | . % z = n. - , %
&! :
?. 2
x0 = 03 y0 = 035 (0945) x0 = 08 y0 = 035 x0 = 03 y0 = 064 (093) x0 = 03 y0 = 05 " = 06 . . a) )
) )
553
z0 = 01H z0 = 08H z0 = 01H z0 = 01H
- % (") % % ( x0 , y0 , z0 ) ', ' '% ' " % ', &!& A+ z = 1. # '% % '% % z_ ( 0 ; z_c0 ) ; cos = +1: A+ (8) 0 z = "q 2 cx + c2y I A+ z . , &! % x_ 0 = 0, 0 = ' = 0,
z_0 + : " y_ + c" F (" y0 z0)
A+ z =1+
(9)
# (9) ' = 0. # ", z0 , ', . & ' y0 : 4"2 (1 + "2 ) y40 + 4"2( z0 ; 1)(2"2 z0 + 2 z0 + 1 + 2"2 ) y20 +
+ ( z0 ; 1)2 (2"2 z0 + 2"2 + z0 )2 = 0: (10)
# '% % 0 % ', &! A+ z ,
q 1 z0 (1 ; z0 ) ; g y0 1 ; 02 + "(1 ; 02 ) = " r q q 2 = 1 ; 0 x20 + ( y0 ; g" 1 ; 02 )2 x0 , y0 , z0 & ;1 6 x0 y0 6 +1 z0 6 1:
(10)
- . 3{5 , % (8) y0 , x0 , z0 , &! " = 01H 1H 10. # , ! % % , % , " | ' 1 . 1
@ . ! ( ! ( - + ( (
Maple.
554
. . , . .
?. 3
, " " # jA+ z j < 1 (7) , + (z+ ), ,
"
( 0 ; z_c0 ) Tz+ = 2+ = 2 z_"0 q Az c2x + c2y
; cos 0
#
(11)
%
+ (z+ ) =
1 X ;1
an exp(inA+z z+ ):
. , + an = a;n. % % " :
2n 1 (;1)n X 2n 0 + (z+ ) = + ; 2 Jn + cos + z+ Tz Tz Tz n=1 n
(12)
. % . 4 C ' " "' ' . " ( ) % ' ' G& {4 (6], . ' (4].
555
?. 4 # ' ' (1) t. = ' % (4), & , & (7):
+ = cos t+ = A+t + sin :
(13)
# ., ', , " (13), . ! , , ' ' .% % , , . . t :
?. 5
. . , . .
556
+ > 1: jA+t j = 1 + "0 y_c + c"
(14)
F , " (4) '% % % % (5) ,
1 (;1)m X 2m 0 Jm + cos 2m t : + (t+ ) = + ; 2 m Tt Tt Tt+ + m=1 Tt+ = 2=A+ t.
# J (1) , & . < ' (4), % &! & ( ): hp p i = 0 + 21a b2 ; 4ad sin "! cc? a ; b (15) 1
2
a = ; !2 c2 b = ; !c2 ?
?
cos 0 ;
1
!c? 0 d = 1 ;
cos 0 ;
1
2
!c? 0
b2 ; 4ad > 0 % , % % .
# $ F (4) |
'& ( ), . (6] . '% % , % % . J & , , . < ,
, ! . - , ( ) | " , (1, 2]. t- , . . , "
, '" . z - (z )
. J " . t- - , | , % , ,
. , ' % . # , ' " (4) {: - , (z ). K . ' ' ! (1) ' , % " z , , &! J %,
557
(12). & , % %
& % , % . L % ' &! . & & ' I. I. / ,. -. . M
.
% 1] . ., . . . | .: , 1988. 2]
. ., ! ". ., # $. ". % & '&' (
)-
+,( -'& ' . / &%0 &. , 1% // #'& ' . -&. 3 ) . | 1980. | . 21, 5 4. | 7. 70{74. 3] # $. "., ;% 1% //
. #., ! ". . < ' + / &
) . ,'. -=. ) . 3 ) . | 1984. | 5 3. | 7. 113{115.
4] # $. "., ! ". . ?= % %& )- / &- 0 ) %.'& //
) . ,'. -=. ) . 3 ) . | 1989. | 5 9. |
7. 76{81. 5] A "., <& 1. B .. | .: , 1968. 6] $0C= . ., &D' . E. ". "' %&& -' %&, & .,( = .. | .: 3 )%&0 ), 1958.
& ' 1999 .
. . , . .
. .
. . .
521.13
: N , , , , { , !"" # $ %#, % " $#.
& ' ( ' ( ) ' #* % #* % * +* $ , #* . - # $ # "# * * . {/ , $ , % #* ' ! #* 0 #* .
Abstract V. L. Shablov, V. A. Bilyk, Yu. V. Popov, Cook's method in the problem of the many-body Coulomb wave operator convergence, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 559{566.
The present work is devoted to the problem of existence of the Coulomb wave operator. Two- and three-body quantum systems of charged particles are under investigation. Using the e7ective charge technique we show that there exists a number of equivalent integral equations for the three charged particles scattering wave operator.
, 1{3] ! = ts-lim (1) !1 exp(iHt)V exp(;iH t) , 2002, 8, 8 2, . 559{566. c 2002 ! " # ! ! $%, &' ( )
560
. . , . . , . .
| )
, H | ) ) , V | ) ) * {+
, H | ,-) ) . * 4, 5] 0 ) 0 ) , Zt ! jf i = V jf i + i t!1 lim dt0 exp(iHt0 )(V V + H V ]) exp(;iH t0 )jf i 0 (2) 3 (2) - , Z1 dt k(V V + H V ]) exp(;iH t)jf ik < 1: (3) 0
4 (2) (3) 0 V , ,-) 5 , ,- , H ; H . 6 (3) (2) 7 4,5]: ! jf i = V jf i + Z1 + i "lim dt exp(;"jtj) exp(iHt)(V V + H V ]) exp(;iH t)jf i: (4) !0 0
9 (4)
jf i
, ,- 7
H H j: i = E j: i (4) 5 0) 6,7] ! j: i = V j: i + G(E i0)(V V + H V ])j: i (5) ; 1 G(Z) = (Z ; H) 0 ) < {= ) ,- 0 . > - 7 - 7 , 3 0.
1.
? 1 @ + = H + V (r) H = ; 2 0 c r
561
) ) V Z ; 3 = 2 (V f)(~r ) = h~r jV jf i = (2 ) d~p exp(i~p ~r ) exp(i ln(pr ~p ~r ))f(~p ) (r):
R 3
(6) 4 (6) | ) : = =p. 9 (6) (r) 1, V 7 * {+
1]. 4 ) 7 , 5 0 (r) 7 ,- ) : 1) (r) = 0 r 6 R1, (r) = 1 r > R2 > R1A 2) (r) 7 ,7 . B, 1{3], 7 , f(~p ) 2 C01 ( 3 n f0g), C 5 0) (3). D , (r) 1 H0 V ] 7 6,7]: 2 p 1 H0 V ] = ; r + r pr ; ~p ~r V : (7) E (7), , ,, (3) , ,- 5 0 'i (t), i = 1 2 3, 0 Z 2 p (r) d~p p~ ~r exp i~p ~r + i 2 jtj exp(i ln(pr ; p~ ~r )) f(~p ) '1 (t) = C1 r 0 Z '2 (t) = C2 r(r) d~p f(~p ) (8) Z 2 p f(~p ) d~ p '3 (t) = C3 (r) r pr ; ~p ~r L1 (0 ;1). 4 (8) 3 7 , Ci |
. G , (r) = 1
, ,- '3 (t), - . 6 5 0) 'i (t) 7 t ) 0 ) 5 p~ = ;~x (~x = ~r=jtj) z = cos = 1 ( = (~p~r )) p = 0. 6 C -
,
, , ) 0 -, 0 0. 4 , 3 5 0) 7 jtj ) 0 ) 5. E, ,- 0 ( jtj ! 1): 0 A (r) A A f(~ r ) 2 1 3 0 '1 (t) 6 jtj5=2 kr (r)k '2 (t) 6 jtj3=2 r '3 (t) 6 jtj2 r2 : (9) 6 (9) - 7 5 0) f(~p ). E (9)
(3).
R
\
562
. . , . . , . .
2. !
> 3 3 0 . 4 , 0 3 V -, (V f)(~r ~ ) = Z Z ; 3 = (r )(2 ) d~p d~k exp(i~p ~ + i~k ~r i ln(k r ~k ~r ))f(~p ~k ): (10) 4 (10) f~r ~ g | 3 I 7, 3 ~r 0 , ~ | - ) 0 0 C ) , f~k ~ g { ,- 3
. B, 5 0 (r ) 7 ) , (6), | ) , ) . 6 C Z = k ) C55 ) 8]. > 0, 7 , 5 0 f(~k p~ ) C01 ( 6) ), ~k = 0 ( = 12 13 23). D V (10) 3 ) C ( , ) 5) B 9]. J 7 3 5 0), , (8) ) ~p ! ~k , f(~p ) ! f(~k ~p ) 7 k 2 p2 ~ ~ ! exp i~p ~ + ik ~r + i 2 + 2n jtj + i ln(k r ; k ~r ) n | ,- 3
. B C 3 5 0) jtj ! 1 0 , (10) ) '3 (t) f (~r ~ ) f ( ~ r ) r2 r2 . D 3 5 0 Z X k 1 Z Z ~ ~ '4(t) = C4 (r ) d~p dk r ; f(k p~ ): (11) r
R
6 7 (11) 0 0 ) 5, 5 0 '4 (t) 7 jtj 7 ) ) jtj;2, (3) , 3 (5). 6 , P H ) C , V k r1 , -, 1
563
Z X k 1 ~ ~ ! jk ~p i = ! jk ij~p i + G(E i0) r ; ! j~k ~p i (12) r 2 2 k p E = 2 + 2n , ! j~k i | 5 0, ,- C55 Z . K (12) 3 ) ) 5 0
0 7 ,- 0. 6 ) (12) X 2 2k2 2k exp i ln ; ln ! j~k ~p i , 3 5 B 9]: X 2k2 k2 p2 t: ! j~k ~p i = t!1 lim exp(iHt) exp i ln jtj exp ;i 2 + 2n (13) 6 ) ) 3
0 C 0 Ze, ,- 1 (@ + @ ) ; Ze2 ; Ze2 + e2 = H + V: H = ; 2m 1 2 0 r1 r2 r12 B C ) ,-) 7 V : Z Z (V f)(~r1~r2) = (r1) (r2)(2 );3 d~p1 d~p2 exp (i~p1~r1 + i~p2~r2) 2 2 exp i Z1pme ln(p1 r1 ~p1~r1 ) i Z2pme ln(p2r2 ~p2~r2) f(~p1 ~p2): (14) 1 2 4 (14) C55 Z1 Z2 , , 8] Z1 m + Z2 m = = Zm 1 + 1 ; 1 (15) p1 p2 e2 p1 p2 j~p1 ; ~p2j | - ) . 6 (14) (3), 0, , (14) 5 0 '4(t) Z Z Ze2 Ze2 e2 Z e2 Z e2 '4(t) = C4 (r1 ) (r2) d~p1 d~p2 ; r ; r + r + r1 + r2 1 2 12 2 Z e2 m 1 2 Z e m exp ;i 1p ln(p1 r1 ; ~p1~r1) ; i 2p ln(p2 r2 ; ~p2~r2 ) 1 p2 p2 2 2 1 (16) exp(i~p1~r1 + i~p2~r2 ) exp i 2m + 2m jtj f(~p1 ~p2 )
564
R
. . , . . , . .
3 f(~p1 ~p2) 2 C01 ( 6) f(~p1 ~p2) 7 ) ~p1 = 0, ~p2 = 0 j~p1 ; ~p2j = 0. , Z ; Z Z ; Z 1 1 2 2 ;e r1 + r2 ; r12 (15) , 0 ) 5 , ), , r1 m~r2 p~1 = ; m~ jtj ~p2 = ; jtj : 6 C ) '4(t) 2 L1 (0 ;1), - . L (5)
-, 3
! j~p1p~2 i = ! 1 j~p1i! 2 j~p2i + + G(E i0) Z1r; Z e2 + Z2r; Z e2 + r1 e2 ! 1j~p1i! 2 j~p2i (17) 1 2 12 ! i j~pii | 5 0 C C55 Zi e. B 5 5 0 B , (17) 7 2 Z2 ; Z 2p2 1 ln j~p1 ; ~p2j2 : 1+ 2+ ln exp ie2 m Z1p; Z ln 2p m p2 m j~p1 ; ~p2j m 1 K- -3 7 V , ,,-) C55 0. G V Z Z (V f)(~r1~r2) = (2 );3 d~p1 d~p2 exp(i~p1~r1 + i~p2~r2 ) q q exp i ln p21 + p22 r12 + r22 ; ~p1~r1 ; ~p2~r2 f(~p1 ~p2): (18) 4 R~ = (~r1~r2 ) P~ = (~p1 ~p2), (18) Z (V f)(R~ ) = (2 );3 dP~ exp(iP~ R~ ) exp(i ln(PR ; P~ R~ ))f(P~ ): (19) 6 C 2 2 ; 3i V+ jP~ i = P + + ~ (V V + H0 V ])jP i = V ; mR V+ jP~ i + 2mR(1 ; cos ) P V+ jP~ i + W V+ jP~ i cos = P~ R~ (20) = V ; mR PR
565
P 1 (P~ R ~ ). 6 (20) (3) , V ; mR cos = PR 7- 0 ) 5, , ,- , . ,
p2 2 + !+ j~p1 ~p2i = V j~p1~p2i + G(E + i0) V ; pp12 + p22 + W V+ j~p1~p2 i (21) m r1 + r2 (6) !(6) jP~ i: ~ !+ j~p1p~2 i = !(6) j P i + G(E + i0) V ; V (22) + + P , !(6)jP~ i | 4 (22) V (6) ) ) 0 mR + 5 0, ,- , 0 m
V (6) : + (6) + (6) (6) ;1 !(6) + jP~ i = V jP~ i + G (E + i0)W V jP~ i G (Z) = (Z ; H0 ; V ) : 6 (21) (22) 2 m 2p2 2 m 2p2 2 2 2 Ze Ze e j p ~ ; p ~ j 2P 1 2 1 2 exp ;i ln m ; i p ln m ; i p ln m + i j~p ; p~ j ln m 1 2 1 2 5 ) !+ j~p1p~2 i ,-) 5 ) ) 5 B .
3. # $ %
3
, - C ) ) < {= . D C ) 7 V (1). D , 3 ) , ,- C55 (12), (17) (21), 7 8] 3 . 4 10] C 7 3 ) T- 0 . 6 3 , , ) (e 2e) (e 3e) 0) 3 5 0 73
) ) (1) ). 4 , C55 !+ j~p1~p2i = !+1 j~p1i!+2 j~p2i !+ j~p1~p2 i 3 5 0) ( M
{*) 11]. D ,
566
. . , . . , . .
C C - 73
5 0) ,
7 , C 5 0 ) 0. 6 C
C ) 7 7
7 !+ j~p1~p2 i , , ) ) 7 0 5 0) (1) 5 ,- C - 0
0 12]. B ) ,
) 10], ) 5
0 7 7 . 4 7,
73
) 5 0 .
& 1] 2] 3] 4] 5] 6] 7] 8] 9] 10] 11] 12]
Muhlerin D., Zinnes I. I. // J. Math. Phys. | 1970. | Vol. 11. | P. 1402. Chandler C., Gibson A. G. // J. Math. Phys. | 1974. | Vol. 15. | P. 291. Chandler C. // Nucl. Phys. A. | 1981. | Vol. 353. | P. 129. ., . ! " #$". %. 3. % & &&. | .: , 1982. %( )*. % & &&. | .: , 1985. , -. -., ./ 0. ., 12( -. 3. )" "(4"5 " 5 ! 7. | .: 8$- 9:, 1996. 12( -. 3., 1 " ;. ;., ./ ;. -. < &$ 7 7 && (& ! 7 "=( " $ // >=. / "(. . | 1996. | %. 2, /. 3. | . 925{951. . "/ . ,. % & $7 @(" = . | B: E , 1975. Dollard J. D. // J. Math. Phys. | 1964. | Vol. 5. | P. 729. 12( -. 3., ( " -. 0., ./ ;. -. $(4 5 B (4 5 = $! & F5 ! 7 "=( " $ // >=. / "(. . { 1998. | %. 4, /. 4. | . 1207{1224. Brauner M., Briggs J. S., Klar H. // J. Math. Phys. | 1989. | Vol. 22. | P. 2265. "= 4 . .. // -/ 5 (" . - /. 2. | 3.: 8$- 39:, 1980. | . 146. * #+ 2000 .
. .
519.61
: , , ! , " ## $ % &' .
( % ) * % Ax = b $ A %! b: ) * % $ - ,
%! Ax = b A 2 A b 2 b. . / % | - % 1 , ! " ( ) % " . 4% " " $
% | " ## $ % &' , | "'/ % ! , " %
%! 1 . 5 "' , % ! / ' "' % "' 1## % $ -
.
Abstract
S. P. Shary, Algebraic approach in the outer problem for interval linear systems, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 567{610.
The subject of our work is the classical )outer* problem for the interval linear algebraic system Ax = b with the interval matrix A and right-hand side vector b: ;nd )outer* coordinate-wise estimates of the solution set formed by all solutions to the point systems Ax = b with A 2 A and b 2 b. The purpose of this work is to propose a new algebraic approach to the above problem, in which it reduces to solving one point (noninterval) equation in the Euclidean space of the double dimension. We construct a specialized algorithm (subdi<erential Newton method) that implements the new approach, present results of its numerical tests. They demonstrate that the algebraic approach combines exclusive computational e=cacy with high quality enclosures of the solution set.
x
1.
Ax = b (1) !" # # " ", 2002, 8, > 2, . 567{610. c 2002 $ %"& '(), * " "
568
. .
n n- " A n- # b. % & (1) # n n- Ax = b (2) A 2 A b 2 b, , , ) " # (2) # * * . * # * * (1) * "
* +, . % : ( , , )
. = fx 2 Rn j (9A 2 A)(9b 2 b)(Ax = b)g
Ax = b A 2 A b 2 b, , , minfxk j x 2 .g maxfxk j x 2 .g k = 1 2 : : : n. 0 1 2 & + *&, * : ( , , ) !"# ( ) .
(3)
(4)
4* (4) | ) , # , , 2 * * , * "
# " " & * . * #
+ *
, # 1 #, 60- * # ,
. &, *
#&,&, 2" & 2 + 71{6]. : * * (4) *
(1), 1 * + # * * ( , # , 77{9]). = *
+ # ) * * # * , &, + (4) * * (. 710,11] & 2 & ). > # , *+ * # , # # + ., , NP- * ( * ). ?
, # * " * + . & * & & + NP- * *, , , # + " .
6
x2
PPP @CC @ PPPPC @ C C C C C C @ C @ C @C
569
-
x1
?. . . ? . 1. )@ * | 1 + $ - , . . " %
:+ . * & + (1) 74,5,7{10] & .99 . @ , # * Ax = b # * * + (.89, .98, . ,1 * ). A * (1) & # &
+ # + *
# "
# 78, 13,14]. B ) * , # &* + , * + (3), * #
# C * 1
+ D. 4 # * * # *+ * * (4), + * # * , # "
** + # *+ . E
, # * + * # * : * C * D #** &, , & )22 , *# , | # # * " +
570
. .
(3), * , ,1 + * (1). F * #** * (4) * , * # * & * (
) * # * R2n. G #** * )22 * * " + 77, 8, 13{15]. H # C * D. : # " (* 22 " * H& ), &, #**, # * ) # . # & , # * , & & & )22 " + , # 1 #**, * I
#" * J {@. B&* ) # *# 2 K * * # * * , # , 71{4,16]. H * , , 2 " +** *" , * *
* # " # & 716]. % ,
( , ") + 2 (# , A, B, C,. .., x, y, z), *
( ) # " * & . F*1 *1 + " . H ", 2 # "
| ) # "
2 (., # , 71{4,16]).
x
2. 1. $
(1)
Ax = b
x = Cx + b n n- .
(5)
C
, C = I ; A, I |
571
.
:+ (5) = = fx 2 Rn j (9C 2 C)(9b 2 b)(x = Cx + b)g = = fx 2 Rn j (9(I ; C) 2 (I ; C))(9b 2 b)((I ; C)x = b)g = = fx 2 Rn j (9A 2 (I ; (I ; A)))(9b 2 b)(Ax = b)g (* A := I ; C) = = fx 2 Rn j (9A 2 A)(9b 2 b)(Ax = b)g = = + Ax = b # C 2 C, (I ; C) 2 (I ; C). . @ * Ax = b * (5), # # *+
1, * +. H + 2 * # * 1 K
#* ) # * +* x 7.
1 (1,17,18]). & x(k+1) = Cx(k) + b k > 0 x (5) " x(0) , (jCj) jCj, '
C, 1. 2 (1,17]). ( C | , (jCj) < 1. ) x ( # 1) x = Cx + b
f(I ; C);1b j C 2 C b 2 bg x . . x x = Cx + b. !" 1 (19{21]). % -
, #* ) # # "
#
2 # * . F , # & , * * *
# . ? , * #* * * + # , 2 C * D , , & * . B ) * (4) # # . 4 , 2 &, 1 2 #* + x + x 7! Cx + b , (5). F) + # 2
572
. .
# *1 *&, * 2 " * , * , # * +
.
3.
(jCj) < 1,
* C
,
" b -
x = Cx + b
(6)
# , ' .
B (6) &* *
& # & x + 2 , #* ,
*+ * & (6) . B 1 # 2 * 3? ? , +* 3, C D , , # *&, # " # 2 :
(6). E* " * 1, 2,
O1 * + &, + , #* " # , #,1 # & " " * (6). H, , , #
#** & (6) " + ,
, #&, # ) * & * * & * . F #
#" * * +* * (6) *+ * #
& * + # 1
" #" (# , # ). Q* *, # " # *+ # ) * & (6) # * 1. * )22 " " .
x
3. " #
R + &, 2 | | # & # & + " # **, #2 # # +* & * (6). ! | ) h IR+ ; = i, IR | + , 7xK x], x 6 x,
573
# "
| + , , + * | # * C# # * D, . .
*&, 2* # " #: x ? y = fx ? y j x 2 x y 2 yg (7) * x y, (x ? y), ? 2 f+ ; =g,
* & x 2 x, y 2 y 71{4]. @1 # * 2 # " : x + y = 7x + yK x + y] x ; y = 7x ; yK x ; y] x y = 7minfxy xy xy xygK maxfxy xy xy xyg] x=y = x 71=yK 1=y] * y 63 0: G 2 IR " # : 1 ) | | & # & # * 1 # " . R * , -# , ) a + x = b a x = b
#* * . B- , # IR * *. :
+ *+ # * *& , - #" * , # * #* . R , #* IR + * :
1 722] #* # & & . F # " x ^ y := inf fx yg = 7maxfx ygK minfx yg] ( + ) (8)
x _ y := supfx yg = 7minfx ygK maxfx yg]
( ) (9) * # 2 1. B ) # *&, # * , # *+ =. F. O 723]: * IR *
( , , + IR
& & ),
,
) #
, * . 4 , * )
,
1 A x, y | % % % % " % , x ^ y x _ y ' x \ y x y
.
574
. .
2 IR, + * , ) * 2 " *,
# &, ,
1 . H # ) #" * + +, IR, * * * . R + , 2 ? 4* # * . =
, 2 IR ### + , + 1 . % (., # , 724]), ###, * &, C & , D, + + ## ( , ) , * ##), . . *
& & , +* )
. B 2 ### , + , + ###, * &,& & , , &
, * +, . F &, "
, * *
2
# # " * A. R ,1 " 70- *. B 725, 26] R # & , & C 2 IRD, & & & 2 IR #*+ * . : * & IR , # 1 * , " . F* # IR + , # , 726,27]. A 2 IR & # , 7xK x],
x 6 x. = * , IR # # * 7xK x], x > x, + IR = f7xK x] j x x 2 R x 6 xg , ( +* * ). A 2 R # , , * *
+ 2 , . F # , * # IR, & +
dual: IR ! IR, # &, " , . . dual x = 7xK x]: = 2
C & D # * # 2 : 1 B , # IR - " % ""
- " - , . . " " % 1 , 1 # "- - "'/ ! .
575
x y () x > y & x 6 y:
(10) A # * * 2 R IR # 1 , *+
# 1 722] #* # & &,
IR. =+ + , # * & IR *&, : x + y := (7x + yK x + y] x := 7 xK x] > 0 7 xK x] . % , +* ) x IR
* # + &, opp x, x + opp x = 0 =) opp7xK x] = 7;xK ;x]: H # * ) 2 * , + IR ##, 2 ** ## * # R2. ? * # " &, & + &, . .
( ) IR, , x y := x + opp y: H + + * # *&, * + # & 1 # " : x + (y ^ z) = (x + y) ^ (x + z) (11) x + (y _ z) = (x + y) _ (x + z): (12) 4 , #,& # "
(9) # * (7) + # # *&, ) * : _ _ (x ? y) ? 2 f+ ; =g: (13) x?y = x2x y2y
H
* 2 , &, 2 R , ,
# * , ,&,
2 (7) (13). % , (14) x ? y = x y (x ? y)
*
x2pro x y2pro y
(W x # , := V
x
pro x :=
(
x
dual x
# " | ) # & &,
x # , | # # "
.
576
. .
A # * + +* # "
x ? y # " x ? y * x 2 pro x y 2 pro y. F * (14) + *+ * # * 2 # " # 2 (. 727]). ? # 2 * + , * IR *&, #*+ : P := fx 2 IR j (x > 0) & (x > 0)g | " , Z := fx 2 IR j x 6 0 6 xg | * +, , ;P := fx 2 IR j ;x 2 Pg | #+ , dual Z := fx 2 IR j dual x 2 Zg | , * +, . B " IR = PZ;Pdual Z . E* + 2
R + # *&, " R) 726]:
y2P
y2Z
y 2 ;P
y 2 dual Z
x2P
7xyK xy]
7xyK xy]
7xyK xy]
7xyK xy]
x2Z
7xyK xy] 7minfxy xygK 7xyK xy] maxfxy xyg]
x 2 ;P
7xyK xy]
x 2 dual Z 7xyK xy]
7xyK xy] 0
7xyK xy]
0 7xyK xy]
7xyK xy] 7maxfxy xygK minfxy xyg]
R * , + 2 R *#
* . H# , 7;1K 2] 75K ;3] = 0. % + 2 R " 725{27], ## # + & IR & 7xK x] xx > 0, # C , D #
#*+ IR. B# 2 * + # 2 & * * . B *
# # * 2 * + , # *+ G. B. U
728]. H# !" 2 (22]). ? , x x+ := maxfx 0g x; := maxf;x 0g & # # x .
577
2 (28]). + "
x y 2 IR
x y = 7maxfx+ y+ x ;y ;g ; maxfx + y; x;y + gK maxfx + y + x; y; g ; maxfx+ y ; x ; y+ g]: *
x y ,
x y = 7x+ y+ + x ;y ; ; maxfx + y; x;y +gK maxfx + y + x; y; g ; x+ y ; ; x ; y+ ]: (15) , - # , ,
x y . *
x y , ,
x y = 7x+y+ + x ;y ; ; x + y; ; x;y +K x +y + + x;y; ; x+y ; ; x ;y+]: (16)
? 2 U
| . *& * + * # * x y # * x y, * # * " R)
. A *, # , # *
* 22 " ,
# * . #. B * 2 IR # * & + , 2 : x ; y = x + (;1) y x = y = x 71=yK 1=y] * 0 2= pro y: H ", # * # "
# 2 & # #, . . #* (10): x x0 y y0 =) x ? y x0 ? y0 * & x y 2 IR ? 2 f+ ; =g: B + + 2 R + *&, :
x # , x (y + z) x y + x z (* ), (17)
x # , x (y + z) x y + x z (# * ). (18) A & ,& , , x , . . x = x 2 R. # "
* " # 2 IR # * & + , IR. = ( ) * " * " +
, #) ( ) # *. Q X = (xij ) 2 IRml Y = (yij ) 2 IRln , # * " X Y
578
. .
" Z = (zij ) 2 Rmn,
zij =
Xl
k=1
xikykj :
V#* # & & + "
# # * 722] #* # & & * # IR. = * , ,
0x 1 0 y 1 0 x _ y 1 1 1 1 1 B C B C B x y x _ y 2 2 2 2C B C B C B C _ = B C B C B @ ... A @ ... A @ ... CA xn yn xn _ yn
0x 1 0y 1 0 x ^ y 1 BBx12 CC BBy12 CC BB x12 ^ y12 CC B@ ... CA ^ B@ ... CA = B@ ... CA : xn yn xn ^ yn
E# # IRn # * * #, . . dist(x y) := maxfkx ; yk kx ; ykg x y 2 IRn (19) * k k | Rn (* # IRn ) #* *2 +* # # # # * Rn). B
2 # "
, - # "
IRn , +
# "
_, ^, dual opp & # (19) (. 726]). H #* , , # "
* * mid x = 21 (x + x) rad x = 21 (x ; x): R , " ) # "
* # #) .
x
4. "%&
@ # **, #2 | * * C * D (4) * +* Cx x + b = 0: F , , ) * " * , #, & #* # . H "
, + IRn, , # : * 2 *1 &
579
# , # ( + ) x = x + x n * x 2 IR & 2 R. E ,
, &, #** * & # & # #& * . >
, # IRn, + # "
# . H# , " 1 1 (20) ;1 1
+* ( ) , + ) " IR2 + *+ : 1 17;1K 1] 0 ;1 1 71K ;1] = 0 :
B 1 # ? Q* + + 1 # , , . % ,
# # , * , U. F *# + , U * # , . =
* # | # IRn # U | * K + C# # D # ? : * ) *&, #, * # . F +* * # + : IRn ! U | + # IRn # U | *+ ( *) * , # # U . ?
, * # , " : IRn ! U #+* + " & + + IRn + + U. >
, +* : IRn ! IRn # * +
;1 : U ! U (21) * # " & + . : * (21) * . H* " + # * , +1 . 2. = + ;1 & , * + * *" + ;1. >
, , (0) = 0, + * IRn
580
. .
# U, # * "
,
# * . #, &, # ) +
# , & # +* # * 2 *" + (21).
U
;1
$#+, ("-)$$"% "&"'()*%$ %
U
-
6
;1
?
IRn
!"#$"% "&"'()*%$ %
-
IRn
? . 2. "- - "$
-
: # *&,
!" 3 (7]). F U | # . B * + : IRn ! U * IRn U, * *&, : 2 ** ## IRn U, 2 # # IRn U. % , , #+ ** & & # & # IRn . F *
+ , * # , # * 3 # U # * *: U R2n. % # * 3 * * , * & #+ : IRn ! R2n
(0IR) = 0R2 (opp x) = ; (x) x 2 IRn * (x) 6= 0 R2n () x 6= 0 IRn : n
n
581
R , #+ & + ;1 : R2n ! IRn + * # * 3 ;1 (0R2 ) = 0IR ;1(;x) = opp ;1 (x) x 2 R2n: = * , + *& * | * +* +
(x) = Cx x + b | * W(x) = 0 2 n * # R *" + W = ;1 : R2n ! R2n # * W(x) = (C ;1(x) ;1(x) + b) = (C ;1 (x)) ; x + (b): !" 4. ? (x) = 0 IRn 2 n 2 n #+ : IR ! R 1 - ( # ) W(x) = 0 * # R2n, W = ;1. E , *
(x) = 0 (22) n
x 2 IR * *, * *"
W(x) = 0
x 2 R2n. F )
x * (22) * x = ;1(x): X , #* #2 | * # #+ , #* * #
*" ,
. 3. * : IRn ! R2n | , T |
R2n, (T ) . . , " : IRn ! R2n (T )
T : R2n ! R2n.
. F # *+ . 0 * & , + ;1 . *, , * # " * 2 , 2 n
n
582
. .
** ## R2n, . . +* # # R2n. : + T = ;1 . 4 # *+ 3 | +* ) #+ IRn R2n, * &, # * & 3. U& * & * & * +* # R2n,
#
+ #+ + * . R # **, 77], &* # # " #+ , # . !" 5. F+ : IRn ! R2n, * # # (x1 x2 : : : xn) 7! (;x1 ;x2 : : : ;xn x1 x2 : : : xn) (23) . . * 2 " # #+ " x1 , x2 ,.. ., xn # , ,... , n- # , 2n- , # " x1 x2 : : : xn * 2 " (n + 1)-,... , 2n- # , 2n- , * # IRn R2n. R+* #+ : IRn ! R2n #+* # R2n #* v | #* # & & IRn # #+
. % , * x y 2 R2n , x v y (Cx # * yD) () ;1(x) ;1 (y) IRn : (24) 0 #* v 1 ! R2n. F * & x y u v 2 R2n
x v y > 0 =) x v y x v y u v v =) x + u v y + v #* v R2n 729]. = * , + * ) # 1 $ , . . + Kv = fx 2 R2n j 0 v xg 729{31]. H# , # # # #+ + , * +,
* #*# 1 . R , #* # , * #* , + #+ ) . % , 1 ( " % % ( , C33]) " ' % " , " . . @ " - % - ", $
% 4. H. C30, 31].
583
* Kv * # * #* # , # x v y () y ; x 2 Kv : Y, 2, # * &, #* v, * #+ , * * #+ (23) * # . H * * , # ) x v y () x 6 y # # (25) . . xi 6 yi , i = 1 2 : : : 2n. = #+ ) # * #+
+ K6 = fx 2 R2n j xi > 0 i = 1 2 : : : 2ng | (26) #+ # R2n. % , * * #+ *" #* # R2n #* # # #* ! A 2 #* * (23) * #+ , * . >
,
+
* * # , *
* #+ * (23)
&, # # #* (25) R2n. H * # # * #+ . ( . F # * & (24) *" #* R2n
# (25), ,
_ ; x = sup x = sup 6 (x ) 2; 2; 2;
(27)
* & fx 2 IRn j 2 ;g, ; | * + . E , * #+ # * # # & & # IRn # # # #* R2n ( 2 2 ). 4. * : IRn ! IRn
" , . .
(x) = Qx Q 2 Rnn, Q = (qij ),
;1
R2n. + '
;1 2n 2n- "#
:
+ ;! Q Q (28) Q; Q+
584
. .
+ ) Q; = (q; ) | ' n n- Q+ = (qij ij Q, . . ,
'
Q
.
. F +* * * q (x + y ) = q x + q y # * * q. B +* | ) *
# + ( q 7xK x] = 7qxK qx ] q > 0 7qxK qx ]
# * (23). > 2n 2n- " # *+ 4 +
, # * 1 # " . !" 6. ? n n- " Q #
+ ;! (28) Q := Q; Q+ Q Q
* & 2n 2n- " Q # * Q. B+ # &, " Q 2 R2n2n , * : " *+ 6- # R2n, *" * # & & + Q # IRn. ( ( " * 4). F # * *" + , * & , * & x 2 R2n # * (Q;1 (x) = Q x (29) n * & x 2 IR
; Qx = ;1 Q (x) : (30) H* ) & & 2. : + , +* ( ) " Q 1 , &, # + Q IRn *1 . % , + & " + # IRn . R # " (20). 0 #* , * *&,
!" 7 (7]). >* , " Q 2 Rnn {- ({- ), Qx = 0 () x = 0 2 IRn :
585
% " Q 1 {- ({- ). R # , * " 01 1 0 B@ . . . CA 0 1 {- +* , * " (20) {-+* . * + , " +* ,
{-+* . R * (30) # 5. ) Q 2 Rnn {-
, 0 "# Q 2 R2n2n , . . Q ". B , " +* " {- +* . 4 , # 2 " (20) #* # # : " (20) +* , + 1 IR2 # #+ + & # R4 " 01 1 0 01 BB0 1 1 0CC @0 0 1 1A 1 0 0 1
# * ! ( ( " + 4 5). # + & " IRn (x) = Qx * Q 2 Rnn * *, * " Q {- +* . F ) # ;1 : IRn ! IRn * *&, : ;1(x) = ;1((Q );1 (x))
(31)
( (30)). . H , , 2 (31), # , # + & n n- " Q IRn , , + + + &- * " IRn ( , " Q;1 ). F *
+*
*, * " Q * .
586
x
. .
5. (
%) %
% , #+ * + IRn ! IRn * * & + R2n ! R2n * # , *" R2n. V & * #+ : IRn ! R2n, &* + #* *" +
* * - *" + R2n. = * , * * | +* Cx x + b = 0 | (32) * *" W(x) = 0 (33) 2 n R , W(x) = (C;1 (x) ;1 (x) + b) = (C;1 (x)) ; x + (b): (34) + , + * * ,
* *" (33){(34). \ , + & 1{3 * * (6), (32). 6. &
W: R2n ! R2n, 0 (34), .
. + : x 7! Cx x + b,
, , # # 2 # " IR. F+ , + ;1, # . * + W * 22 " &, * & . #.? R + &, , . . # * , + # * ) . H
*+
# . ? * * , * # *&,
!" 8 (29, 32{34]). F # Rq #* p q #* 4. + F : R ! R ! 4, F (y + (1 ; )z) 4 F (y) + (1 ; )F (z) * & y z 2 Rp 2 (0 1). >* * + * 22 + , # 2 "
# *& , , .
587
7. &
W(x), 0 (34),
6 R2n.
. ? & 2 (0 1) u v 2 IRn, # -
* (17),
C(u + (1 ; )v) Cu + (1 ; )Cv: 2 = * , R n (C(u + (1 ; )v)) 6 (Cu) + (1 ; )(Cv ): (35) 2 n &* + & , * y z 2 R , y = (u), z = (v), # * *&, " # # : W(y + (1 ; )z) = (C;1 (y + (1 ; )z)) ; (y + (1 ; )z) + (b) = = (C(u + (1 ; )v)) ; (y + (1 ; )z) + (b) 6 6 (Cu) + (1 ; )(Cv ) ; (y + (1 ; )z) + (b)- 0+= (35) = ((Cu) ; y + (b)) + (1 ; )((Cv ) ; z + (b)) = = W(y) + (1 ; )W(z) * # *+ . !" 9 (29,33,34]). F Rq | #*
# #* 4. % + F : Rp ! Rq x 2 Rp + @4 F(x) # D: Rp ! Rq, (36) D(y ; x) 4 F (y ; x) p * & y 2 R . A + @4 F (x) | # , * &, (36), | & + F x, # + F , x, * 22 " @4 F (x) ) # . B , # * 22 " + # * * + , #* # . H # , * 22 " # * & # ) # & * 729]. F &, "
1 , #, . R ,
*&, 2 . 4 (33, 34]). 1 - f : Rp ! R -- '-- ( , f ). B# * + W: R2n ! R2n #-
# 6-#* , * # | 2 " Wi : R2n ! R, i = 1 2 : : : 2n, | + # . B* Wi (x) # , , * 22 " &* R2n.
588
. .
= * , * & x 2 R2n +* i = 1 2 : : : 2n , & 2n- d(i), Wi (x + v) ; Wi (x) > d>(i) v * & v 2 R2n: = d(i) 2n 2n- " D = = (d(1) d(2) : : : d(2n))> . U & , , v 2 R2n, * W(x + v) ; W(x) > Dv
# * " D # * W x. = , * 22 " @6 W(x) # . F +* x, 8. +
W, (34)
7, 6- -- @6 W(x) " x 2 R2n, . . W " -- . ? * ) * 22 " # @W(x), # , 6, * #* R2n * . Q W * 22 " x, * 22 " @W(x)
* ) , " Y 0 @31(x) 31 (x) 1 : : : @@x @x 1 BB .. . . ..2 C @ @3 . (x) . @3 . (x) CA : 2 : : : @x2 2 @x1 H, + , + W(x) &* * 22 " . 0 * * 22 " @W(x) , * " , , # * , * * +
W. !" 10 (33,34]). & 2 "
f : Rp ! R + epi f := f(x t) j x 2 Rp t 2 R f(x) 6 tg: ' $ Rp + , + # * # ## Rp, . . + * h>(i)x 6 i i = 1 2 : : : m * h(i) 2 Rp, i 2 R m | . ' $ ! ! ! | ) (# ) 2 " , *2 # )* + . n
n
n
n
589
6
D
D
D
D
D D
A
A A
A A @ @
; @ @XX X
; ; ;
-
? . 3. J # % " #" $ # 1 % " #" $ : " 1
"
F* 1 , # )* 2 "
& # * # 2 " : # , , # )* 2 "
+ # &* # 2 "
, #
2 . 9. 1 Wi(x), i = 1 2 : : : 2n, W,
0 (34), | ' - .
. \ # )* 2 " Wi(x), i = 1 2 : : : 2n, # * # " # * * W(x). Q * 1 . F +* , * & # n n- " C # n- v
0 W (C v)1 1 BC2C C _ Cv = C v = BB@ W ... CCA : (37) C 2C (C v)n C 2C
? , * # cij # * & (14) + # _ cij vj : cij vj = cij 2cij
%# * # "
_ + (12), # * i = 1 2 : : : n
590
. .
(Cv )i =
n X j =1
n _ X
cij vj =
j =1 cij 2cij
cij vj =
_ _
n _ X
cij vj = cin 2cin j =1 n _X _ = cij vj = (C v)i C 2C j =1 C 2C
ci1 2ci1 ci2 2ci2
:::
* (37). ?
, # (27) (29), & (C;1 (x)) =
_
C 2C
C;1(x) = sup 6 (C;1 (x)) = sup 6 C x: C 2C
C 2C
B " * + W, # * 1 # * (34), # * *&,
# * : W(x) = sup C x ; x + (b): (38) C 2C
B , ; Wi (x) = sup C x i ; xi + ((b))i = sup (C x)i ; xi + ((b))i C 2C
C 2C
(39)
* & i = 1 2 : : : 2n. E 2 , # * (38) (39) " C | ) " " # " * (28),
+ * . % , sup(C x)i + * ) cij , j = 1 2 : : : n, + *# ,1 !, &, ) cij 3 0. B & + ) cij (* ), # * + # (39). B " # Wi (x) = C 2max (C x)i ; xi + ((b))i (40) Vert C
* Vert C | ) " * C, . . + n n- ", # * ( (Vert C)ij := f cij cij g 0 2= cij f cij 0 cij g 0 2 cij : F Vert C , (40) * , +* Wi (x) 2. A * # *+ . !" 11 (33, 34]). (
2 "
f : Rp ! R x # # & y 2 Rp ( , ,
# ) # * @f(x) = lim f(x + y) ; f(x) &0 @y , , .
591
!" 12 (33, 34]). ( # + W Rp 2 " W : Rp ! R, W (x) := supfx> w j w 2 W g: 5 (33, x 23]). ( f : Rp ! R | ' - , x. ) f -- x -- @f(x) ' , 0 - f x - . H + " | # # *1 * # " * 22 " * + W, * +& * * x 6. H # * * !" 13. ' # x+ # ; x # x 1 *&, : x+ := fx+ j x 2 xg = fmaxfx 0g j x 2 xg x; := fx; j x 2 xg = fmaxf;x 0g j x 2 xg: H# , 7;1K 2]+ = 70K 2] 7;1K 2]; = 70K 1] 71K 2]+ = 71K 2] 71K 2]; = 70K 0]: :+ # "
#+ " 2 " ( )+ ( );, *1 # *
2. R , " ) # "
* # # # . 10. + -- @W(x) W, 0 (34), "# : @W(x)
C+ C; C; C+ ; I:
(41)
. F + , * 22 " @W(x)
&& & " * # 2n 2n- ", * # * W. >
h @31 (x) @31(x) i 1 0 h @31(x) @31(x) i K : : : ; + @x;2 K @x+2 CC BB @x1 . @x1 .. . .. .. (42) @W(x) B . @h @32 (x) @32 (x) i h @32 (x) @32 (x) iCA : : : @x;2 K @x+2 @x;1 K @x+1 *
@Wi (x) := lim Wi ( x1 : : : xj ;1 xj ; xj +1 : : : x2n) ; Wi ( x1 : : : x2n) &0 @x;j n
n
n
n
n
n
n
n
592
. .
@Wi (x) := lim Wi ( x1 : : : xj ;1 xj + xj +1 : : : x2n) ; Wi ( x1 : : : x2n) | &0 @x+j * # * # Wi x # # j- * # &. F # # #* R2n # # * #* 6 R, #* 6-* 22 " @W(x) # # * * 22 " @Wi (x) * # Wi : R2n ! R, * +* + # 5. ?
, # * # 2 "
* @Wi (x) > fy> d j d 2 @W (x)g: (43) i @y F * # y , &, j-& # ;1
1, j = 1 2 : : : n, , # (43) &
@Wi (x) : : : @Wi (x) K @Wi (x) K @Wi (x) @Wi (x) @x;1 @x+1 @x;2n @x+2n (42). E # + # * " (41). ? * # +* * + ;. B # * (40) @Wi (x) = @ ; max (C x) ;x +((b)) = @ ; max (C x) ; (44) i i i i ij @x j @x j C 2Vert C @x j C 2Vert C * ij | R ( ij = 1 i = j 0 : B# # # * 22 " 2 "
(., # , 735, x III.2]): @ ; max (C x) = ij- ) " C , i * C 2max (C x)i . (45) @x j C 2Vert C Vert C B " , * (28), (44) (45), # *1 *&, , * " # * # # &:
!2n 0 + 0 ; @Wi (x) (C ) (C ) ; I = (C 00); (C 00)+ @xj ij =1 * C 0 C 00 2 Rnn, C 0 C 00 2 Vert C. = * , 1 & (42) * # * * " (41).
593
H ", & ) #2 # * * 22 " @W(x), * & # # "
+ * 22 " * H& . R * + * , * 22 " @W(x), , , #* " # (42), * # *. ? " C & (42) + # *+ + x . E
*1 , # * * 22 " @W(x). B# # * * #+ : * x 2 IRn ((x))i = ;xi i 2 f1 2 : : : ng ((x))i = xi i 2 fn + 1 : : : 2ng: ei , &, i-& # 1, , # , * , ; @Wi (x) = @ ((C;1 (x)))i ; xi + ((b))i =
X n
n X cij 7;xj K xj+n] ; xi + ((b))i = ;@ cij 7;xj K xj+n] ; ei = j =1 j =1 n X (46) = ; @ cij 7;xj K xj +n] ; ei * i 2 f1 2 : : : ng j =1 ; @Wi (x) = @ ((C;1 (x)))i ; xi + ((b))i = X X n n =@ cij 7;xj K xj+n] ; xi + ((b))i = @ cij 7;xj K xj+n] ; ei = j =1 j =1 n X = @( cij 7;xj K xj +n]) ; ei * i 2 fn + 1 : : : 2ng: (47)
=@ ;
j =1
E , * 22 " @Wi (x) * & * 22 " # + R2n ! R *&, * *: (xj xi+n) 7! cij 7;xj K xj +n] (xj xj +n) 7! cij 7;xj K xj +n] * cij | # (* + , +* + , # C D # x, &, ) + ). B# 2 U
(15) , (;x); = x+ (;x)+ = x;, # & *
594
. .
@( cij 7;xj K xj +n]) = +ij @(x;j ) + ;ij @(x;j +n) ; @(maxf ij+ x+j ;ij x+j +n g) @( cij 7;xj K xj +n]) = @(maxf ij+ x+j +n ;ij x+j g) ; +ij @(x;j +n ) ; ;ij @(x;j ):
(48) (49)
% , * 22 " 2 "
# * * 22 " 2 " , * * (., # , 733, 34]). % # * #+ " # * * ,
8 8 > > (0 0)
xj < 0 < <(;1 0) xj < 0 + ; @(xj ) = >(70K 1] 0) xj = 0 @(xj ) = >(7;1K 0] 0) xj = 0 (50) :(1 0) xj > 0 :(0 0)
xj > 0
8 8 > > (0 0)
xj +n < 0 < <(0 ;1) xj+n < 0 @(x+j +n)= >(0 70K 1]) xj +n = 0 @(x;j +n)= >(0 7;1K 0]) xj +n = 0 :(0 1) xj+n > 0 :(0 0)
xj +n > 0:
(51) E , # * +
(48) # * * (49) # * * . B (48) (49) # * +* ij+ x+j ;ij x+j +n, ij+ x+j +n ;ij x+j . H c;ij , cij+ , x+j , x+j +n " , # , # , ij+ x+j > ;ij x+j +n * x+j > 0, @(x+j ) = 1. G ,
ij+x+j < ;ij x+j+n =) @(x+j+n) = 1 ij+x+j+n > ;ij x+j =) @(x+j+n) = 1 ij+x+j+n < ;ij x+j =) @(x+j ) = 1: = 1 # + & ,
8( + 0)
ij+ x+j > ;ij x+j +n > ij > > < ; + + + ; + + + @(maxf ij xj ij xj +ng) = ># + + ; @(x+ ), ij xj = ij xj +n @(x )
ij j ij j + n > > :(0 ; )
ij+ x+j < ;ij x+j +n ij
(52)
595
8(0 +)
+ij x+j +n > ;ij x+j > ij > > < ; + + + + + ; + @(maxf ij xj +n ij xj g) = ># + @(x+ ) ; @(x+ ), ij xj +n = ij xj ij j ij j +n > :(; 0)
ij+ x+j +n < ;ij x+j : ij
(53) 1 * * 22 " (52) (53) , * * & ij+ x+j = ;ij x+j +n ij+ x+j +n = ;ij x+j . @ + " ) +,
* # * . H# , 2 "
* 22 " @(maxf ij+ x+j ;ij x+j +ng) + . 4. F+ *
@(maxf ij+ x+j +n ;ij x+j g).
;
6
xj +n
;
ij
6
xj + n
ij
+
0
ij
-
+ + = ; ++ = 0 + = 0 ; = 0 ij xj
ij xj
ij
6
ij
xj
ij
+ + = ; ij xj
n
-
+
0
xj
=0
+
ij xj +n 6
6
6
6 ;
xj +n
xj + n
ij
+
0
ij
+ + = ; ++ = 0 + = 0 ; = 0 ij xj ij
6
ij xj
ij
n
-
-
0
xj
xj
+ + = ; ++ = 0 + = 0 ; = 0 ij xj ij
ij xj
ij
6
? . 4. ? % # " $ " ## $ @ (maxf ij+ x+j ;ij x+j+n g) !, ij+ x+j = ;ij x+j+n
n
596
. .
B * + : * 22 " * H& , *&, #2 , +* - * * ( , ) * + W(x), * ) # * # * #
* * * 2 " maxf ij+ x+j ;ij x+j +ng maxf ij+ x+j +n ;ij x+j g: = , + * - * # ij+ @(x+j ) ;ij @(x+j +n ) # ij+ x+j = ;ij x+j +n * - * # ij+ @(x+j +n ) ;ij @(x+j ) # +ij x+j +n = ;ij x+j . H * , ; 1 + 1 ;# , #
+ + , # , ij ij 2 2 ; | 12 ;ij 12 ij+ (. . 4). B " , 8 + > <(;1ij +0)1 ; ij++x+j+ > ;ij; x+j++n + + ; + @(maxf ij xj ij xj +ng) 3 > 2 ij 2 ij ij xj = ij xj +n (54) :(0 ;ij ) ; + + +
ij xj < ij xj +n 8 + > <;(01 ;ij )1 + ij++x+j++n > ;ij; x+j+ ; + + + @(maxf ij xj +n ij xj g) 3 > 2 ij 2 ij ij xj +n = ij xj (55) :(;ij 0) + + ; +
ij xj +n < ij xj : F * + W(x) + # * # 2 (46){(51), (54) (55).
x
6. ? (33){(34) &, #
R2n # * *&, "
(* 22 " * H& # " # + ) B x(0) 1 C * D (I ; (mid C) )x = (b): Q k- # + x(k), k = 0 1 : : :, + * , - * * D(k) 2 @W(x(k) ) # x(k+1) := x(k) ; (D(k) );1 (W(x(k) )):
597
H # * , # * ) *1 " , , # , # * n . V + , ) #* ,
# 77] * +* " + . Q*
, 77] # * 22 " * H& * Ax = dual b ( , # , , ) dual Ax = b). F &,
* * 22 " * H& * , , + ** , * (., # , 732]), * * . Q1 +* * *&, .
6. * C 2n 2n- S ,
"# C+ C; S 2 C; C+ ; I
" , (x ), x |
(6).
. V ,
* #* C* &D " A. : # , # +
C+ C; 2 n 2 n S 2 C; C+ ; I 6 S 2R
S ;1 K
(56) * S ;1 K6 # #+ ) (26) # #
S, # R2n. A . Q " C
& , . . C = C, (C );1K6 , # # +* #
, * . Q + +*
2n 2n- " S 0 , S 00 C* D * *, + (S 0 );1 K6 (S 00 );1 K6 , # 1 ,1 . = * , (56) * + C D " C. % , # + (56) | , * KE . E* # * #* E R2n. : # x E y () x ; y 2 KE : * * | #* , # * # + , #+*1 -
598
. .
) # " #* E. H1 , # " # + x(0) 1 *&,& " # : W(x(0)) = (C;1 (x(0) )) ; x(0) + (b) > > ((mid C);1 (x(0) )) ; x(0) + (b) - 0+ 4"$"&"$$"=& 5" -6078%$ 7 = (mid C) x(0) ; x(0) + (b) - 0+ -"9 = &-) (29) = ((mid C) ; I)x(0) + (b) = 0: F) W(x(0)) > 0: ?
, # # * & * 22 " W( x(k+1)) > W( x(k)) + D(k) (x(k+1) ; x(k)) * D(k) 2 @W(x(k) ) & k = 0 1 2 : : :, * # # & D(k) (x(k+1) ; x(k)) = ;W(x(k) ): (57) = * , # * "
+ & W(x(k)) > 0 k = 1 2 : : :: (58) 0 (58)? B# # * (40): * +* k
W(x(k) ) = C 2max C x(k) ; x(k) + (b) > 0 Vert C * S (k) x(k) + (b) 2 K6 * " #
S (k) 2
C+ C; C; C+ ; I
* (maxC x(k) ; x(k)). E " S (k) # & +* , # *
x(k) + (S (k) );1(b) 2 (S (k) );1 K6 KE ) & x(k) D ;(S (k) );1 (b): Q #+ C+ C; ; 1 = inf E ;S (b) S 2 R2n2n S 2 C ; C+ ; I
599
* * x(k) D , . . # * fx(k)g * E- . ? + * : # * fx(k)g, #+*1 , &, #* E, . . x(k) D x(k+1) (59) * k = 0 1 2 : : :. ? , (57) (58), * # D(k)(x(k+1) ; x(k) ) 6 0: R * , * D(k) & & (41): D (k ) 2
C+ C; C; C+ ; I:
&*, 1 (56), * (59). : * , , x(k) D x(k+1) D : (60) B , +* #* # #* # # + +, , &, # " #, : & # * , #* , # , *
# # * 1 . = * , (60) + & ,
# * x # * fx(k)g, #+* . 4 ) # * *1 ,
* #* + x = x ; (D );1 (W(x )) * " D 2 @W(x ) *+ +* . % , W(x ) = 0.
x
7. ,# )
R #+ #** * * * C * D (4)? G #** *
" " , + # + , # . B " , # * , #**, &, * 2 " , + + )22 * * " * * , . X , 1 L # # " "' "'/ 1 : " " C30,31].
600
. .
#2 | & * + #**.
11. $
Ax = b
Gx = (dual G ; A)x + b (61) G
, A,
" , rad G 6 rad A.
. B *# 2 # *+ 11 , " (dual G ; A) # *
*, * rad G 6 rad A. :
+ (61) = = fx 2 Rn j (9G 2 G)(9H 2 (dual G ; A))(9b 2 b)(Gx = Hx + b)g = = fx 2 Rn j (9G 2 G)(9H 2 (dual G ; A))(9b 2 b)((G ; H)x = b)g = = fx 2 Rn j (9A 2 G ; (dual G ; A))(9b 2 b)(Ax = b)g = = fx 2 Rn j (9A 2 A)(9b 2 b)(Ax = b)g = = + Ax = b # G 2 G H 2 (dual G ; A), (G ; H) 2 G ; (dual G ; A) = A. E # * #** + # & (61), (6). F + ;: IRn ! IRn * *&, : ;(x) = Gx (62) . . + " G, , +
;;1 : IRn ! IRn. E*, +
x 7! ;;1((dual G ; A)x + b) # " # , # U # " * + # IRn , Gx = (dual G ; A)x + b (63) + C * D (4). = &, , * * ,1 I. G 2 * ^. J " 71, 11, * 6]. ?
, * + # ) # ( 2 R ), #+ R2n, + , * "
601
" + , . . , + x 3{6. F # , 1 + #
, * 2 " (63)
* (6). B #
(6), # #**, # ; ;;1 +* . H # # * #**, G + * , # U # " # "
;;1 + (dual G ; A) # * , jI ; Aj, ) * # (6). E*, -# , #** # 1 &
x = Cx + b
(6)
+ * , # * # &,
(63) * + &, # U # ". %, - , *+ (jCj) < 1 #** # # & (6), #* + , #
* (# , 717]), &, kCk. 0 kCk, &, # # , && & " + # . # , (63)
, # (
) " G A, . . * G = G. F ) * : * (63) # * 22 " * H& , #
*" + W = ;1, &,
(x) = (dual G ; A)x Gx + b: H # + " # # * # *+ 7, * * , + # *, * " G K * " G # + (62), " * (63), + * # * ", + . R + x 4, + (62) G = G * *, * " G {- +* , + ;;1 # * # ) 2 (31). % , * , G = G | {- +* ", # , (62) 2 "
. CB *D * Ax = b * # ) +* & Gx = (G ; A)x + b:
(64)
602
. .
B "
# ;;1, * + &, + , (64) x = ;;1((G ; A)x) + ;;1(b): R + # , # " G? A # * # *&, .
x
8. .
B ) #2 # * ) # PC/AT 486, # * 22 " * H& , &, #** * * . E C #**D # * # * 2 "
, # x 7, * * * (4) * & (64). , # & " G (64) *&, . !" 14. ( # x 1 ( dev x := x jxj > jxj x , . .
*1 & x. : # 0 dev a11 1 0 CA ... G := dev diag A = B @ 0 dev ann . . G * " , * ) A. E* + , + & " G IRn , # + " G;1, (63) ) x = G;1(G ; A)x + G;1b: H + ) # * , *+ # G *1 # . F ) # #** * "
* 2 " (6). F (jG;1 (G ; A)j) < 1, , * # #**, #
k jG;1(G ; A)j k < 1, + +* # * . G Turbo C * 2
603
* * #&, 1. F * + * 1 * # # . Q #** & * I 71{3] * # *+ #" * J {@ 736, 37]. R+* ) *
& & # , * +* , # " ( *+ ) , *1 | " +
. 0 * # * 1 & *, # * . H # * * , #** * . E
, , # # #** & * *, & *&,
0 (36]). 0 707K 13] 7;03K 03] 7;03K 03]1 07;14K ;7]1 B @7;03K 03] 707K 13] 7;03K 03]CA x = B@ 79K 12] CA :
7;03K 03] 7;03K 03] 707K 13] 7;3K 3] B #
) * I *1 0 7;101K 71] 1 B@7;6225K 99]CA 7;90K 90] # #** J 07;101K 17]1 B@ 7;15K 99] CA 7;90K 90] * # "
#** * 22 " * H& * * "
07;101K 71]1 @B 7;69K 99] CA : 7;90K 90] F * " * , * , #** J ) # # " .
1 @M " public domain ! O " % %! ! B5 ?H& " ftp://www-sbras.ict.nsc.ru # pub/interval/shary.zip.
604
. .
1.
76] 0 715K 17] 7;3K 301] 7;3K 301] 7;3K 301]1 07;6K ;2]1 BB7;3K 301] 715K 17] 7;3K 299] 7;3K 299]CC BB 74K 5] CC B@7;3K 299] 7;3K 299] 715K 17] 7;3K 301]CA x = B@ 7;2K 4] CA : 7;3K 301] 7;3K 301] 7;3K 299] 715K 17] 78K 10] % * I # && " + *
0 7;103K 0495] 1 BB7;0347K 0974]CC B@7;0770K 0917]CA 70150K 125] #,& * J 736] 0 7;103K 0363] 1 BB7;0223K 0975]CC B@7;0752K 0919]CA : 70149K 125] G #** * 4 "
"
0 7;103K 0495] 1 BB7;0372K 0974]CC B@7;0785K 0917]CA 7;005K 125] #: * " # * ) # , # # * J {@. 2 ( -+ " . (. 2,36]).
!
!
72K 3] 70K 1] 70K 120] x= : (65) 71K 2] 72K 3] 760K 240] B * #** ( , # " * ) * 2 "
&
! 7;120K 90] (66) 7;60K 240] &, " + . A +
*1 * I.
605
4 , (66) | ) *+ # (
)
" + (65). H # #** J 736] # * * :
1845 ! ; 120K 11 ;60K 2940 11 * &, # * , #" * J {@. = *&, * # 3{7 2 " H R 2 76], * ## * #** J {@. 3 (6]). F * Ax = b 0 737K 43] 7;15K ;05] 70K 0] 1 A = B@7;15K ;05] 737K 43] 7;15K ;05]CA (67) 70K 0] 7;15K ;05] 737K 43]
b = (7;14K 14] 7;9K 9] 7;3K 3])>. E*, # * I, # *&,& " + : 07;638K 638]1 B@7;640K 640]CA : (68) 7;340K 340] E + *1 # * J 736], + 1 , # *+ @ 737] H
R 2 76]. % (68) + , + * " & # * #**. 4 (6]). F * Ax = b + " A, # 3, # & b =(7;14K0]7;9K0]7;3K0])>. F * I, # " + 07;638K 0]1 B@7;640K 0]CA : (69) 7;340K 0] F " (67) | ) M- " # # & * , ) # ( ) " + . H # * J 736] * 2 "
@ 737] # *
#
606
. .
07;638K 112]1 B@7;640K 154]CA
7;340K 140] * 2 " H {R 2 *1 ,1
07;638K 167]1 B@7;640K 277]CA : 7;340K 240] H# , #** # & ( & * ) & (69) + * " &. 5 (6]). F * Ax = b + " A, # 3, # & b = (70K 14] 70K 9] 70K3])>. % * I *1 070K 638]1 B@70K 640]CA (70) 70K 340] + + . F * J 736], #
" &, # 07;112K 638]1 B@7;154K 640]CA 7;140K 340] * 2 " H {R 2 # * ,1
07;167K 638]1 B@7;277K 640]CA : 7;240K 340] % , # * " & #,& #**, *1 # (70) * . 6 (6]). F * Ax = b + " A, # 3, # & b =(72K14]7;9K;3]7;3K1])>. F I *1 " &, # 0 7;109K 429] 1 B@ 7;402K 124] CA : 7;244K 0773]
607
%# * J 736], #
07;0995K 501]1 B@ 7;464K 152] CA 7;269K 138] * " + + 07;0995K 429]1 B@ 7;379K 124] CA : 7;235K 0773] F , , # **, , #** *1 ) * " &. 7 (6]). F * Ax = b + " A, # 3, # & b = (72K 14] 73K9] 7;3K1])>. F * I # * "
0 70517K 625] 1 B@ 70450K 607] CA : 7;0881K 273] %# * J 736] *1
07;0206K 625]1 B@7;0386K 607]CA 7;201K 273] + 0 70523K 625] 1 B@ 70499K 607] CA : 7;0743K 273] % #** * 1 # * : * ) * "
. B " , # * , * * # " , * * " & * , #** * #" * J {@{H {R 2 . G. H * * , # , #
* " G
(61) #** # H- " . H#
608
. .
!" 15 (3]). hai ! a !, . ( . hai = minfjaj jajg a 63 0 0
a 3 0:
? # " A = (aij ) 2 IRnn ! " pAq 2 Rnn, ( ij- ) pAq = haij i i = j ;jaij j i 6= j: F * " A H - ,
pAq u > 0 * u > 0. B , H- " & " , * &, X (71) haii i > jaik j * i = 1 2 : : : n: k6=i
, #* ) #2 #** * , # " * * , " A *
# * , . . # (71). A #* # *1 # . 0 # *1 , , * * * , # " G # *
C * D & (61), 2 # + #** | ) # #.
/%
1] ., ! " #. $ % %& . | (.: ( , 1987. 2] . % /. ., 0 #. 1., # 2. !. (% 4. | 5 " : 56, 1986. 3] Neumaier A. Interval methods for systems of equations. | Cambridge: Cambridge University Press, 1990. 4] :" ;. /., 0 6 $. $. :6 & % %. | 5 " : 56, 1990. 5] Shary S. P. On optimal solution of interval linear equations // SIAM J. Numer. Anal. | 1995. | Vol. 32. | P. 610{630. 6] Ning S., Kearfott R. B. A comparison of some methods for solving linear interval equations // SIAM J. Numer. Anal. | 1997. | Vol. 34. 7] Shary S. P. Algebraic approach to the interval linear static identi?cation, tolerance and control problems, or One more application of Kaucher arithmetic // Reliable Computing. | 1996. | Vol. 2, no. 1. | P. 3{33.
609
8] Shary S. P. Algebraic solutions to interval linear equations and their applications // Numerical Methods and Error Bounds / G. Alefeld and J. Herzberger, eds. | Berlin: Akademie Verlag, 1996. | P. 224{233. 9] Shokin Yu. I. On interval problems, interval algorithms and their computational complexity // Scienti?c Computing and Validated Numerics / G. Alefeld, A. Frommer and B. Lang, eds. | Berlin: Akademie Verlag, 1996. | P. 314{328. 10] @ . $., 5 /. 1. D 6 4% &Q // / ". . 6 . | 1994. | T. 35, W 5. | /. 1074{1084. 11] Kreinovich V., Lakeyev A., Rohn J. Computational complexity of interval algebraic problems: some are feasible and some are computationally intractable | A survey // Scienti?c Computing and Validated Numerics / G. Alefeld, A. Frommer and B. Lang, eds. | Berlin: Akademie Verlag, 1996. | P. 293{306. 12] Kreinovich V., Lakeyev A. V. NP-hard classes of linear algebraic systems with uncertainties // Reliable Computing. | 1997. | Vol. 3, no. 1. | P. 51{81. 13] Shary S. P. A new approach to the analysis of static systems under interval uncertainty // Scienti?c Computing and Validated Numerics / G. Alefeld, A. Frommer and B. Lang, eds. | Berlin: Akademie Verlag, 1996. | P. 118{132. 14] Shary S. P. Algebraic approach to the analysis of linear static systems with interval uncertainty // Russian J. Numer. Anal. Math. Model. | 1996. { Vol. 11, no. 3. | P. 259{274. 15] Kupriyanova L. Inner estimation of the united solution set of interval linear algebraic system // Reliable Computing. | 1995. | Vol. 1, no. 1. | P. 15{31. 16] Kearfott R. B. Rigorous global search: continuous problems. | Dordrecht: Kluwer, 1996. 17] Apostolatos N., Kulisch U. GrundzZuge einer Intervallrechnung fZur Matrizen und einige Anwendungen // Electron. Rechenanl. | 1968. { B. 10. | S. 73{83. 18] Mayer O. Algebraische und metrische Strukturen in der Intervallrechnung und einige Anwendungen // Computing. | 1970. | Vol. 5. | P. 144{162. 19] Berti S. The solution of an interval equation // Mathematica. | 1969. | Vol. 11 (34), no. 2. | P. 189{194. 20] Nickel K. Die Au^Zosbarkeit linearer Kreisscheiben- und Intervall-Gleichungssystemen // Linear Algebra Appl. | 1982. | Vol. 44. | P. 19{40. 21] Ratschek H., Sauer W. Linear interval equations // Computing. | 1982. | Vol. 28. | P. 105{115. 22] ; . T Z. | (.: 56, 1984. 23] 0 % /. `. D" 4& " % // 1 - % . | ` W 2/1987. | . : $j /D 5 ///{, 1987. | C. 45{46. 24] .6 . . @ "| " . | (.: 56, 1973. 25] Kaucher E. Algebraische Erweiterungen der Intervallrechnung unter Erhaltung Ordnungs- und Verbandsstrukturen // Computing Suppl. | 1977. | Vol. 1. | P. 65{79. 26] Kaucher E. Interval analysis in the extended interval space IR // Computing Suppl. | 1980. | Vol. 2. | P. 33{49.
610
. .
27] Garde~nes E., Trepat A. Fundamentals of SIGLA, an interval computing system over the completed set of intervals // Computing. | 1980. | Vol. 24. | P. 161{179. 28] Lakeyev A. V. Linear algebraic equations in Kaucher arithmetic // International Journal of Reliable Computing. Supplement 1995 (Extended Abstracts of APIC'95: International Workshop on Applications of Interval Computations, El Paso, Texas, February 23{25, 1995). | P. 130{133. 29] . `., .64 /. /. &% % . | 5 " : 56, 1978. 30] . (. . ` % % 6 . | (.: 4 4, 1962. 31] . (. ., @ . ., /" . $. `4 % % %. | (.: 56, 1985. 32] D :., { " $. 1 % % % 6 4% . | M.: ( , 1975. 33] { {. $%6% 4. | M.: ( , 1973. 34] D" .-`. 5 % 4 & . | (.: ( , 1988. 35] : $. ., (4Z $. 5. $ . | (.: 56, 1972. 36] Hansen E. Bounding the solution of interval linear equations // SIAM J. Numer. Anal. | 1992. | Vol. 29. | P. 1493{1503. 37] Rohn J. Cheap and tight bounds: the recent results by E. Hansen can be made more ecient // Interval Computations. | 1993. | No. 4. | P. 13{21. # &, , 1997 .
. .
. . .
519.6
: , , .
! ! ($) & ($$'), ()
&) $ $$' () (
! * *
! + &. , - .// 0 , ! 0
$,
*( 0 .
Abstract O. D. Zhukov, A parallel number transformation method from a residue system into a mixed base system, Fundamentalnayai prikladnaya matematika, vol. 8 (2002), no. 2, pp. 611{615.
We propose a method of number transformation from a residue system (RS) into a mixed base system (MBS), that combines the advantages of RS and MBS and has much greater, in comparison with them, parallelism in computations. The method gives an e5ective way to determine the number's value or a containing interval, which provides new possibilities for more wide usage of RS, in particular, in computer applications.
, () 1] ! " " # !", " $" " ! , & - mi . n Q ( " (M = mi ) ! $ i=1 . * ! & + $! , $ ! $ " ( ., , 2{5]). 0 1 , , , & , ! , 2002, 8, 7 2, . 611{615. c 2002 , !" #$ %
612
. .
$3 . $ ! , ! , , &" ! , !!, 4 !, 4 , ! , ! . 5 ! , +##! $ # !, " $ 4 ! ! ! $ . 5 , 4 , $ , ! ! , ! "! (65) ! 4 (*). * , , ! !" 2{5]. 7 65 n " $4" M, !
!!, " + . 7, $ & *, $ , ! ! ! ! & , # * $! &". ( " " 4 " 43 ! & . 8 65, ! " * A $ * ! ! n
X
A
aiQi Mi mod M
i=1
(1)
ai $ A mi 9 Mi = M=mi 9 Qi $ , , " jQi Mi j = 1 mod mi , A=
n
X
k=1
ak 0
kY ;1 i=1
mi
(2)
ak $ , # *, mi = 1, k = 1 2 : : : n. 8 (1) (2) $ ! ! 0
n
X
A
0
Q
j =1
0
mj = 1, Ai 0
i=1
Ai
ai Qi
n
iY ;1
j =1
mj mod M
n
Y
j =i+1
(3)
mj mod M
Q
mj = 1. $ (3) $ j =n+1
A
n
X
i=1
Ai mod M 00
(4)
613
iQ ;1
0
Ai = Ai mj mj = 1. j =1 j =1 8 (2) Ai $ * ! ! 00
0
Q
00
Ai = ai1 + ai2m1 + : : : + aij 00
00
00
00
jY ;1
k=1
mk + : : : + ain 00
nY ;1 k=1
mk
(5)
i j = 1 2 : : : n, ! Ai * $ $! ai, &$ , 13 . (4) , (5) aij = 0, j < i9 j = 1, a1j = a11. 5 ! , (3) (5) $ ! ! 00
00
00
Ai = 00
A=
n
X
i=1
i=j
Ai mod M = 00
j
n
X X
A =
n
X
0
j =1 i=1
aij 00
jY ;1
k=1
n n
XX
i=1 j =i
aij 00
mk jY ;1
(7)
mk mod M
(8)
aij
k=1
(6)
00
jY ;1
00
k=1
mk mod M
Q
mk = 1 i j = 1 2 : : : n. * , Ai , M(A) n n + "
" 9 ! i, j ! !! . * A $ +" , &" # : k=1
00
A=
n
X
i;1
ai 0
iY ;1
j =1
mj = a1 a2 : : : an] 0
0
0
(9)
ai < mi | , # i-" , ( ) A * 0
0
Y
j =1
mj = 1
i j = 1 2 : : : n. 0##! $ $ $ $" ,! $" , ! $ " . * C , C ! C""# $ $ ((7) , $ &
614
. .
65 *. $ E ! E""# $ !+## , +##! $ , . . E ! = C ! =C (10) E""# = C""# =C : (11) (!$! , , , ! $ " " , " " , 3 !+## , , " k" k$ , & " . * - ! $ mn ( m1 < m2 < : : : < mn ) dm , M | dM . , 65 n " (!
!!, M) " dM . ( * n " $! " " dm . 5 C ! = nk"dM (12) C""# = nk"dm + nk$ dm = n(k" + k$ )dm : (13) C $ (8) ! , A * $ ! iQ1 $ mi + aij mj j =1 , M(A) , ! &
" + . 8 (4) (5) + ! , M(A) , $4, & ,. (+ !+## , , n, $ A " 4" ( !") , # an (9) nQ1 , , n + - ain mj mod mn j =1 (i = 1 2 : : : n) n- , , $4 mn . ! $ " ! $ $, $ D E $ log2 n. C " " ! ! $ " . 5 C = k"dm log2 n (14) (10){(14) E ! = dM n=dm log2 n (15) E""# = (1 + k)n= log2 n (16) k$ = k k" , ! ! , k > 1. 5 ! , " ( 4" !" , # *) 00
;
0
00
;
615
& & , 65 *. 1. 3 M, " . 2. 7 ! !" $ , . + ( ),
& ( 4" !" , #) * $ # $ . " * $ . 3.
, ! $ " , $ log2 n " " mn . 65 n " $4" M, !
!!, + , * | n " $! " " mi . 4. (15) (16) ! ! $ ! $ dM , dm n (dM | 31{56, dm | 5{6, n < 10) !+## , +##! 5{10. 5. 7 $ , ! ! " $ . 5 ! , " & $ +##! $ , , & !
, , $, $ 4 ! ! ! $ , ! $ .
1] . . . | .: , 1952. 2] Szabo N. S., Tanaka R. I. Residue arithmetic and its applications to computer technology. | New York: Mc Craw-Hill, 1967. 3] Jenkins W. K., Leon B. J. The use of residue number systems in the design of #nite impulse response digital #lters // IEEE Trans. Circuits Systems. | 1977. | Vol. CAS-24. | P. 191{201. 4] Baraniecka A., Jullien G. On decoding techniques for residue number system realization of digital signal processing hardware // IEEE Trans. Circuits Systems. | 1978. | Vol. CAS-25. | P. 935{936. 5] Zhukov O. D., Rishe N. D. E)ective computer algebra for some applications // World Congress on scienti#c computations and modelling. | 1977. & ' 2001 .
. .
512.552
: , .
! "
# .
Abstract A. V. Kondrat'ev, On the construction of the generating function for an annihilator, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 617{620. We propose an algorithm for constructing a generating function of the module of element annihilator. L T = khx1 : : : xni = Ti | i>0 k , T . I | , F = ff1 : : : fm g, 6= A = L Ai T=I . a 2 A, d = deg a > 0. = i>0
! " # $ % 1 P
a: Annr (a)(t) = (dimk Annr (a)i ) ti . A(t) | )i=0 1 P
A: A(t) = (dimk Ai ) ti . +# (,1, . 111] ,2, . 43]), i=0 " IT , , . R | ( )
#. 2 , " hRik = I T k , IT = RT = (I T k k T )T . () A(t) = T (t) ; I (t) = T (t) (1 ; R(t)): 3 'a : A ! A, 'a (b) = a b, | %# d 4 A- . Im('a ) = a A. + " 5 4 : a 0 ;! Annr (a) ;! A ;';! A,d] ;! (A=(a A)),d] ;! 0: " Annr (a)(t) = ((A=(a A))(t) ; A(t) (1 ; td ))=td . 6 , # %# A=(a A) = T=(I + a T ). R^ I + a T , 2002, 8, + 2, !. 617{620. c 2002 , !" #$ %
618
. .
T - . ; , " (), (A=(a A))(t) = (T=(I + a T ))(t) = 1 P = T (t) (1 ; R^ (t)), Annr (a)(t) = zi ti = T (t) ((R(t)=td ) ; (R^ (t)=td ) ; i=0
1 ; R(t) + 1) = B (t)=(1 ; nt), B (t) = P bi ti , bi = ri+d ; r^i+d ; ri , i > 0. i=0 a = 6 0 A, b0 = 0, , z0 = 0< zi = n zi;1 + bi i > 0: = Annr (a)(t) # N . > R # : # 5 | )
% < 5 5 | " ? . NS +d F = Fi | 5$4 ? n A.o! n i=1 S 4 Ri # Fi
xj Ri;1 j =1 R1 R2 : : : Ri;1 5$ # . 2 5. ; " R |
# IT . @ a R1 R2 : : : Rd;1, # Rd
Rd . Rd+1 Rd+2 : : : Rd+N #. ; " R^ |
# (I + a T )T . > # " , N + d Rs = ?. = " Ri>s = ?, B. C ,3, . 462, 2 # 3] A " , , # $ % .
IT (I + a T )T ( GRAAL)
:
(F G) | # F G< (F ) := Norm(F F )< (F G) | F G.
Norm Norm LReduct
:
N |S " < F = Ni=1 Fi | 5$ < a | A = T=I < d | a.
:
R |
# IT < R^ |
# (I + a T ) (R R^ N ).
: 1)
R1 := Norm(F1 )< R^ 1 := R1
619
2)
::: k) ::: d)
R2 := Norm R^ 2 := R2
n n o F2 S xi R1 R1 <
LReduct
Rk := Norm R^ k := Rk
i=1
LReduct
n n o Fk S xi Rk;1 R1 R2 : : : Rk;1 < i=1
n n
o
Rd := Norm LReduct Fd S xi Rd;1 R1 R2 : : : Rd;1 < i=1 if (a := Norm(LReduct(a R1 R2 : : : Rd;1) Rd )) = 0 ^ d := Rd + fag then STOP else R
:::
nS n
o
d + k) Rd+k := Norm LReduct Fd+k
xi Rd+k;1 R1 R2 : : : Rd+k;1 < i=1 if Rd+k = ? then STOP R^ d+k := Norm(LReduct(Rd+k R^ 1 R^ 2 : : : R^ d+k;1)) ::: N ) STOP GRAAL R(t), R^ (t) Annr (a)(t) " " " .
( GRAAL).
k = F5 , A = khx y j x3 + y3 xyxi, a = x + y, d = deg a = 1, R(t) = (2t3 + 4t4 + 7t5 + 6t6 + 3t7 ; 2t8 ; 5t9 ; 5t10 ; 2t11)=(1 ; t3 ; t4), R^ (t) = (t + 2t3 + t4 + 2t5 ; 2t6 ; 2t7 ; 3t8 ; 2t9)=(1 ; t3 ; t4 ), Annr (x + y)(t) = = ((R(t)=t) ; (R^ (t)=t) ; R(t)+1)=(1 ; 2t) = (t5 ; t6 ; 2t7 ; t8 +3t10 +2t11 )=((1 ; 2t) (1 ; t3 ; t4 )) = t5 + t6 . 2 , " Annr (x + y) = = k hy3 x2 + y5 y6 i. 3 # " 4 GRAAL, # " " % HJ 3 | 93K ! M # " 4 . > % =. 2. C ? >. N. = # .
1] . . | .: , 1975.
620
. .
2] "# $ %. &. ' # # . # # . | .: ( - * + ##, 1988. 3] Levin J. Free modules over free algebras and free group algebras // Trans. Amer. Math. Soc. | 1969. | Vol. 145, Nov. | P. 455{465. & ' 1998 .
. .
. . . e-mail: [email protected]
519.2+519.713
: , !"#, $
% & .
'$$ $( & $$ ) ( ) ,$.$- /0 ! & ) $ &1( $$( ( (0 1). 3 $1-) ") 1$ () (! $ $$( ( $ !& $ % & ) $(4$( $ % & ( $$( ( . !"- . / 1. 5 6- $ . & $( $$ 7 8 ( . ) , . ($ # &( $ $$ $$( 1. 3 & $( $$&1$( $1-( . $ $ $- .!" 6- , . ($ #. '$$ 4$( % !% ,,9 (.
Abstract A. V. Lebedev, Probabylistic classication methods of cellular automata, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 621{626.
A class of cellular automata (games) on the in=nite plane lattice of square cells with two states (0 and 1) is considered. Under random initial conditions (independent states with given expectation) the expectations of a cell state on the =rst step are calculated. The classi=cationof games is based on their favour#for growth of the number of cells in the state 1. A quantitative measure of this favour# is suggested and studied as a random value on the games' space. Some possible generalizations are discussed.
1{3]. ! " # # ! $ % %. & "# % ' ( , % , ! ( $ $ , % ' (( . * ! , ! % ! # "! . + % (!) ./ % , #"% #" : , 2002, 8, ? 2, $. 621{626. c 2002 !, "# $% &
622
. .
31. (0 1). 32.
. 33. 0, 0. * , , ! 567 1,3], / 8'. * (1969). * ' ! ' f ij 0 6 i 6 1 0 6 j 6 8g, ! ij | i j . < 33 00 = 0. > . ($ ! % 5 ! 7 1. ? #$ " % (! $, 5 7
. @ ,
(t = 0) %
' p 2 (0 1). + ( $# '(p), # ' # . ! (t = 1). ? , , / ! ( : 8 X '(p) = C8k pk (1 ; p)8;k f(1 ; p) 0k + p 1kg: (1) k=0
B , ! . ! # # ( (1) #"% . ! % . <
, ! 567 03 = 12 = 13 = 1 ( ij = 0) '(p) = 56p3(1 ; p)5 + 28p3(1 ; p)6 . D ( '(p) . 1. 8 #" . E1. , '(p) > p (0 1) ( , , ! ! ! ). E2. , '(p) < p (0 1) ( , , ! ! ! ). ? ! ! ! #" : 0
" 1, 1# 1, . '(p) = 1 ; (1 ; p)9 > p (0 1). ? ! ! ! #" : 0, # 1
" 7, 0.
623
' $. 1
'(p) = p9 < p (0 1). ?/ # 5 7 % 1 0 / #"% . ? '(p) ! 9- , E1 E2 ! '
. * ! , . ! ! ! , ! ! . < , 5 ! 7 # ! / Z1 '(p) Q= (2) p dp: 0
B , '(p) ! p ( -
624
. .
33), ! ' ' ! (8- ) ! ! . ?
X 8 8 X 1 9 ; k Q= 9 (3) 1k + 0k : k=0 k=1 k 8 #" . E3. , Q > 1. E4. , Q = 1. E5. , Q < 1. E , ! ! ! ! , ! ! ! ! . 8 ! 567 Q = 4=9, ! , ! ! . E " !, #"% 31{33, 217 = 131072. + / , / #, #, 95668( 73%) % ! , 256( 02%) 35148( 268%) ! . I Q, , #, P9 Qmax = 1=k 2829. k=1 ?
! Q
, ! ' #: 9 8 X 1 9 + X 9 ; k 2 0284: (4) MQ = 12 k1 1414 DQ = 324 k k=1 k=1 + Q ! ! ' . D ( ( $ ( . ! Jx = 10;2) ! ! ( # 0 Qmax] 50 ) . 2. > ' " # , ./
. B , ( (1){(3) E1{E5 ' . % % % , ! 32 #" : 320 . $
. 8 ! ( % (1) (3) ij % ' . I ' '
! ( ! % ). K ' m ,
625
' $. 2 m X
Cmk pk (1 ; p)m;k f(1 ; p) 0k + p 1k g k=0 X m m m+1;k X Q = m 1+ 1 + 1k 0k k k=0 k=1 mX +1 m m + 1 ; k 2 X 1 1 1 MQ = 2 k DQ = 4(m + 1)2 m + 1 + k k=1 k=1 '(p) =
(1 ) (3 ) (4 )
B , MQ > 1 m = 3. + ! ! ! # . I ' ,
, 2]. + ' + ! 96-01-01092.
626
. .
1] . . . // . | .: ", 1988. | . 5{43. 2] + ,- . . /0 0 // 1 . | 1984. | 2 11. | . 98{110. 3] Berlekamp E. R., Conway J. H., Guy R. K. Winning ways for your mathematical plays. | Academic Press, 1982. ' ( ) 1998 .
. . . . . 519.1
: , , - .
! " # , $% % & " , " " # ' #. , " - #' ' .
Abstract V. E. Marenich, Conjugation in the incidence algebras, Fundamentalnaya i prik-
ladnaya matematika, vol. 8 (2002), no. 2, pp. 627{630.
We introduce the notion of the canonical form of an incidence function that generalizes the Jordan cell, and we 1nd the canonical forms for some functions. In particular, the canonical form of the zeta-function in some incidence algebras is found.
.
!
. , #1, . 235] : "+ ,
( ! - )?" 0 ! !
, 1 , ! !. 0 , ! -!
. 2 : (P 6) |
(. . .)4 #a b]6 = z a 6 z 6 b |
. . . (P 6)4 < | 6 . . . (P 6), a < b #a b]6 = 24 6 | 6 , a 6 b #a b]6 1 2 . 2 : incF (P 6) | !
, 1 F4 IncF (P 6) |
4 | 7 4 e | !
, ( e(a b) = 1 a = b 0 a = b: f
j
g
j
j
j 2 f
j
g
6
, 2002, 8, 2 2, . 627{630. c 2002 , !" #$ %
628
. .
; < !
f g (fg)(a b) = f(a b) g(a b). = -! 6 T 7 ! 1 T , 1 ( T (a b) = 1 aT b 0 : > f g incF (P 6) 1 7
IncF (P 6), 6 f g, ! x incF (P 6), x;1 f x = g. 2 6 6
. > 1 g ! , 1) g(a b) 0 1 a < b4 2) g(a b) = 0, g g1 , g1 (a b) = 0 ( g ? 6 , 1@). > e, e + 6 ( F ) 1 !. 1. f incF (P 6) f(a a) = f(b b) a = b f(a b) = 0 a < b f fe IncF (P 6). 1. f incF (P 6) 1. x incF (P 6), x;1 f x = fe,
x(a a) = 0 a P . 2. f incGF (q) (P 6), P = n, 1. x incGF (q) (P 6), x;1 f x = fe, (q 1)n . 2. (P 6) | " . f incF (P 6) f(a a) = = const a P f(a b) = 0 a < b f e + < IncF (P 6). = 2 7
x(a b), x;1 f x = e + < . 0 , < < IncF (Z 6) ; N;a x(a b) = b;a a 6 b, N | !
. 2 h(a b) a b. 3. ((ai bi))i>0 | , h(ai bi) = i i > 0, f 2. x incF (P 6), x;1 f x = e + < , x(ai bi), i > 0, x(a0 b0) = 0, b0 = a0.
2
2
2 f
g
6
6
2
2
6
6
6
2
2
6
2
2
j
j
2
;
2
2
6
2
6
629
4. (P 6) | " , jP j = n > 1# f 2 incGF (q) (P 6) 2. x 2 incGF (q) (P 6), x;1 f x = e+< , (q ; 1)qn;1. = P 2 3 4 . F(0) | . 3. (P 6) | . . . $ ~0, % a 6 b, c 6 d, (b) (a) = (d) (c), %& . . . (#a b]6 6) (#c d]6 6) . f incF (0) (P 6) ;
;
2
f(a a) = = const a P f(a b) = 0 a < b f(a b) = f(c d) a 6 b c 6 d (a) = (c) (b) = (d) f e + < IncF (0) (P 6). ; 3 , , 1 . . . (P 6): 1) . . . (P 6), E 1 4 2) Bul(U) = = (2U ) | . . . U4 3) Lin(U) | . . . U GF (q)4 4) !
6 . . . 0 . . .,
, 2
6
6 6 < < 6 6 6 e < 6n 6 6 = 6;1 , n N. 2 m(f P P ) = f(a b) ab2P . = . . . (P 6), 1 3, : rank m(< P P) = rank m(< P P) rankm(6 e P P) = rank m(< P P) (1) rank m(6n e P P) = rank m(< P P):
;
2
k
k
;
;
G (1) #2] . . . (P 6), 1 ?ample@ #3]. 0 7
f(a b) = 0 a < b. H
6
. + f(a b) = 0 a < b, , ! f 6
|
6 ), , 6 6,
(P , f
1 ! 6 . 6
630
. .
1] . . | .: , 1990. 2] . . !" # $ " ! % ' // ) . | 1996. | +. 8. | . 63{88. 3] Stanley R. P. Some aspects of group acting in /nite posets // J. Combin. Theory A. | 1982. | Vol. 32. | P. 132{161.
& ' ' 1998 .
. .
512+519.4
:
, .
P | ! ". X | #$ "
. % $ $ X \ F " & " $ &
X. ( " L P LX # ! LX \ F.
Abstract V. Yu. Popov, On positive theories of semigroup varieties and pseudovarieties, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 631{635.
Let P be a positive language, L P , then LX = LX \ F, where X is a nonperiodic semigroup variety and X \ F is a 1nite trace of the variety X.
K | , P | . P K K. P K ! K !
!. ". #. , $. . % ! &1] ). *. +,, -. +. . &2] !
!
0 . + &3] ! 2 , !
, ! 3 ,!, ! !
3
. + &4] 5. #. -
! 6 ( ) ( , , !6 , !6 ). + &5] !
! ,!0 ! . "
!
&6]. " ! , ! ;
, 89^- , . . . <. =. * &7] ! ! . ? ! , 2002, 8, 2 2, . 631{635. c 2002 , ! "# $
632
. .
6 ( ) 6 , &8] +. #. . , . + &9{14] ! ! ! ! ! 6. , , 3 &15{19]. + &4] 5. #. -
! 6 ( ) ! . S |
, X | !
. X \ F
X. F X | 0 , !
X. + ! ! 3 , 3 ,! ! 0 . . L P LX LX \ F. C, LX = LF X. + &1{3] ! ! 3 1. ,
, . =! 1 -
SA (. &20]). 2. ! 8:_ SA- . . + &21] ! 8:_S \ F. . P S \ F ! 1. 2 ,
-
, ! ! ! 2 ! 9:^_S \ F !:^S \ F. ! , ! :^ , ,
! 2
. =! , !:^S \ F . ! , ' 9:^_ !G 6 ! , '1 ,..., 'n , ! n 2 N 'i ! i | ! , 9:^. !
9:^ !:^ ! 9:^_S \ F. =! 2 ! .
.
Fk X k, !
X \ Xk , ! Xk |
, ! ,! x1x2 : : : xk = x1x2 : : :xk xk+1. D | ; % ! , N, . . , ! , ,-
633
N &22]. HD Fk X ; ! Fk X ; D. a = (a1 : : : an : : :) | 2 HD Fk X. ? , 2 a 3 ai,
ai 6= b. I ! , , 2 a 2 b = (b : : : b : : :) . =! , ! ; ! HD Fk X j= a = b. , , a 6= b,
2 a , , b. S (ai ) = fj j ai = aj g: J | , a 2 HD Fk X, ! i 2 N ! d 2 D d = S (ai ). 5 !,
I = HD Fk X n J | ! HD Fk X. HD Fk X=I | , 3 ; HD Fk X ;, ! , a ! a, a 2 J , a = (a1 a2 : : : an : : :) ! (a1 a1 : : : a1 : : :), a 2 I . a&i] 2 HD Fk X=I ,
S (ai ) 2 D. J!
! a&i]. ? 2 ! , ad1 &i] ad2 &i] | 2 HD Fk X=I , d1 d2 { 2 ; D d1 = S (ai ) ! 2 ad1 &i], d2 = S (ai ) ! 2 ad2 &i], ! HD Fk X=I j= ad1 &i] = ad2 &i]:
! ; , d1 \ d2 !, ; D. I ! j 2 d1 \ d2 j - 2 ad1 &i], ad2 &i] ! ,
HD Fk X=I j= ad1 &i] = ad2 &i] ! ! ; !. =! , ! 2 a&i] . + , , 2 a&i] ,! HD Fk X=I . ! , ,
2 a&i], ai 3 Fk X. ,, HD Fk X=I ; F X, 0 , fEi = a&i] j ai = ei g, ! fe1 : : : ek g | , ! 3 Fk X, , ! 3 HD Fk X=I . ? 2 , ! !, ! , ! u(e1 : : : en : : :) v(e1 : : : en : : :) u(e1 : : : en : : :) = v(e1 : : : en : : :) F X ! !, ! HD Fk X=I
u(E1 : : : En : : :) = v(E1 : : : En : : :). F X | ! 0 , ! , F X u(e1 : : : en : : :) = v(e1 : : : en : : :) HD Fk X=I u(E1 : : : En : : :) = v(E1 : : : En : : :). ? , HD Fk X=I u(E1 : : : En : : :) = = v(E1 : : : En : : :). I ! HD Fk X=I 3
634
. .
r, ! i > r i- 2 u(E1 : : : En : : :) v(E1 : : : En : : :) ,!
. l1 | ! u(e1 : : : en : : :), l2 | ! v(e1 : : : en : : :), l =max(l1 +1 l2 +1 r +1). I !, !
!, u(E1 : : : En : : :) = v(E1 : : : En : : :) HD Fk X=I 2 HD Fk X , ! i > l FiX u(e1 : : : en : : :) = v(e1 : : : en : : :). l > l1 , l > l2 X | !
, u(e1 : : : en : : :) = = v(e1 : : : en : : :) FiX 0 2 F X. , ! , F X HD Fk X=I ! ; . C ! . | ! ,. I !, ! , F X j= , X \ F j= . X \ F j= , ! Fk X j= ! k. C, ! ,
; ! ! !, !
2 ! , &23]. ! ,
; ! !, ! 2 ,! , ! , &24]. ! HD Fk X j= . + HD Fk X=I j= , . . F X j= . =! , F X j= , X \ F j= : I ! .
1] . ., . . ! // V # $. %&. . ! %. ' %!. | %, 1979. | +. 122. 2] #- .. /., 0 ! 1. #. 02 3 4- !5 5 // /. %. | 1981. | '. 116 (158), ; 1. | +. 120{127. 3] 0 ! 1. #. 4 !5 5 // XVII # $. ! . %&. 2 4 3. ' %!. | / %: / % 5 5 , 1983. | +. 195{196. 4] 1%53 A. . ! 4 % ! !-: % % ! %!B, 5 !5 5 // . 2. 54 . . /%. | 1982. | ; 11. | +. 3{11. 5] /% C. +. ! 2 5 !5 5 // /. %. | 1978. | '. 103 (145), ; 2. | +. 147{237. 6] D5 #. C. !5 5 // # 5 !5 5. | '5!, 1972. | +. 122{172. 7] 0 ! 1. #. !5 5 // + . . -5. | 1979. | '. 20, ; 6. | C. 1282{1293.
635
8] 03% #. . 2 ! F& !3 5 %!3B // ! ! %. | 1977. | '. 16, ; 4. | +. 457{471. 9] #- .. /. G4 % % %! B %!B // ! ! %. | 1989. | '. 28, ; 5. 10] #- .. /., #. .. G4 % !3 %!B % // . 2. 54 . . /%. | 1991. | ; 3. | +. 74{76. 11] #. .. 02 %!B // ' H # $ %&B 4 % ! %. | !- , 1990. | +. 134. 12] #. .. 02 !3 %!B // ' + %-&B5 % %!!% 5 ! . | G , 1990. | +. 41. 13] #. .. %4 % -%5 - B %!B // B - 5 !% % %&B 4 % ! %. + % . | G3. 14] #- .. /. G4 % // + . . -5. | 1988. | '. 29, ; 1. | +. 23{31. 15] 03% #. . 5 ! 5 // + . . -5. | 1979. | '. 20, ; 3. | C. 671{673. 16] D5 #. C. 5 !3 5 // # 5 ! 4 % ! . | J ! !3, 1981. | +. 66{69. 17] D5 #. C. 5 !5 5 5 // /. %. | 1974. | '. 16, ; 5. | C. 717{724. 18] 03% #. . 2 ! F& !3 5 %!3B // ! ! %. | 1977. | '. 16, ; 4. | C. 457{471. 19] 03% #. . % ! 4 % ! 2 5 // ' 6 5 5. | G , 1978. | +. 52. 20] #- .. /. ! 4 % ! % % // ! ! %. | 1987. | '. 26, ; 4. | +. 419{434. 21] C5 4 .. K. ! ! ! % %! !5 5 // ! ! %. | 1966. | T. 5, ; 5. | C. 25{35. 22] G ! C., LF L. L. ' ! . | /.: /, 1977. 23] Keisler H. J. Limit ultraproducts // J. Symb. Logic. | 1965. | Vol. 30. | P. 212{234. 24] Lyndon R. C. Properties preserved under homomorphism // PaciUc J. Math. | 1959. | Vol. 9. | P. 143{154. % & ' 1998 .
