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Фундаментальная и прикладная математика (2002 №2) (2002).pdf

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 . . . . . e-mail: akishev@kargu.krg.kz
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:
<t6 t
(
)
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(')
<t6 t
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] 10?7 . 6. @/2 3 B2 CC 4 -. | :8. 3CC.. . . 73. 8.-. 7. | :, 1990.
6] ,7 . :. 1 - C 0 7 3 ,D ?- /2 CC 7CC E // B72 0, C72 . | G2C7, 1984. | . 46{54.
7] 1 . B., 1 . C 7 CC //
0. CD. ,?. 03., C . | 1983. | < 9. | . 65{73.
8] :7D . :., C ;. . 1 - C, CC ??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 7CC C, CC // 10C 373 -33. 78. KB7. CC, ?-, 0L. | ., 1995. | . 11{12.
12] ? M. . 6,D ?- E GD // .
C?7. | 1972. | 1. 87, < 2. | . 254{274.
334
. . 13] .C 4. :. @ CC ,D ?- CC 4 7 - // 0. CD. ,?. 03., C . |
1987. | < 10. | . 48{58.
14] 5D7 N. 1 - C 0 7 3 ,D
?- CC E // 0. CD. ,?. 03., C . |
1980. | < 4. | . 11{15.
15] MC . :. @ - 7CC 87, 03 C32C ,D ?- 7 CC. |
9CC.. . . 73. 8.-. 7. | C7, 1988.
16] -7 I. :., 5. ., @C23 . ? 9-7C CC Lp , 0 < p < 1 // . C?7. | 1975. | 1. 98,
< 3. | . 395{415.
17] 60? . O., . ., :7D . :. 3C2 3 0 7Q88 B2 - CC 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. 75
, 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].
75
E0 = fa a2g E1 = fx ax a2xg E = E0 E1
B = K(E0) + K(E1):
@
D
K(E0 ), | D
K(E1 ).
338
. . 75
, - +
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:
75
, - + 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. 75
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 33 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. 75
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 75
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:
75 ! 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
) ).
75
, - 3 . D
.
x1 x2 : : : xn, n = 4t 4t + 3, ! +:
X
X
n = ij x1xi 'ij (x2 xj ) + i3x1xi 'i3(x2x3 ) = 0:
33
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
+- 33
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 33
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
f3ix1x3 '3i(x2 xi) + i3x1xi '3i(x2 x3)g +
X
+
i>4
4ix1 x4'4i(x2 xi) + 54x1x5 '54(x2x4 ) = 0:
6 ! ! 33
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:
! 33
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.
#, 33 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. 75
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)
75
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!-
. .
75
, 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
( ). 75
-, - 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 fp g (! f g
- ) 5 ! p !. ., 5 -!, - fp g ! !.. 7+!
D
+- xi , xj , i < j, ! 5 fp g. f(1i)(2j) = f 0 np
f 0 | - Xn;p = fx1 : : : xn;pg. @
, - Hn;p Pn;p (F (2)(C)). 7D
+ + 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 fp g - f 5 ! p !.,
fx y z tg 5 ! +. !., fp g 5 +.
, ?! (12), (13) + D
fp 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
. . 75
, - 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)
75 (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.
75
, - D2 = 0 !. 33
x1 ! xy xn ! zt
D 5+
(xyx3 '34 ztxn)x2 = 0
, + !, A 2 D ! +- = 0.
N
33
xn ! xy x4 ! zt
(xyx3 '34 ztx2)x1 = 0:
G = 0.
, 3, 33
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 ! - .
75
, - 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. 75
, - Enp Pnp .
- N
, - Enp p > 1
Enp =
fenp(1r)(2n)g En;1p;1xn :
r !"#$.
r<n
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<n
f 0 | 3 D
En;1p;1. Dp f = 0. 33
xi ! xy xn ! zt
5 ! i < n 5
i(xyx3 '34 ztx4 )np in = 0:
4 A 2 D ! +- i = 0. , DSS3 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 ).
75, - 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):
) 75
, - 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. 75
, - E 2 Wp;1. T E !, D
-. 7+! E !, 3 E B. N-,
f 2 T (E) T (B) = F0(2):
75
, - 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 !% 20, 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<a \St.
Kl. OHRIDSKI". | 2000. | ?. 94. | 7. 91{96.
9] Banaru M. Six theorems on six-dimensional Hermitian submanifolds of Cayley algebra // 1$2 @05 $. . | 2000. | 9 3. |
7. 3{10.
10] Banaru M. Two theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of Cayley algebra // Journal Harbin Inst. Techn. | 2001. | Vol. 8,
no. 1. | P. 38{40.
11] Sato T. An example of an almost K:ahler manifold with pointwise constant holomorphic sectional curvature // Tokyo J. Math. | 2000. | Vol. 23, no. 2. | P. 387{401.
12] $ F. Q., + . ,. $
2 42 55U 6$ 0$ // . 5. | 1998. | ?. 189, 9 1. | 7. 21{44.
13] + . ,., 70$ W. . F 4 400$ $1$ 451 // X0 . . | 1995. | 9 2. | C. 213{214.
14] 70$ W. . +$1$ 400$2 6$
451 // -#X . . . W. | 1995. | 7. 187{191.
15] + . ,. 55U 6$ 4 $ 0
451 // 4 . -5 4.
T. 18. | .: ?, 1986. | C. 25{71.
16] Kurihara H. The type number on real hypersurfaces in a quaternionic space form //
Tsukuba J. Math. | 2000. | Vol. 24, no. 1. | P. 127{132.
17] 70$ W. . +2 42 400$ $1$
451. | Y.. . . . Z.-. . | .: -#X . . . W,
1995.
18] [ . [. ?1 + // $2 0 $ 1
04 1 ZZ. $2. 0. 2. | 7,
2000. | 7. 42{48.
19] Banaru M. A new characterization of the Cray{Hervella classes of almost Hermitian manifolds // 8th International Conference on Di\erential Geometry and Its
Applications. August 27{31. 2001. Opava. Czech R]epublic. | P. 4.
20] . -. ?2 0$. | .: #??W, 1956.
' ( 2002 .
C. . . . . e-mail: bogatyi@nw.math.msu.su
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
<?
Aj1 \ : : : \ Aj = Qnr \ jn1 \ : : : \ jn = >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
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i
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i
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390
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., , '#& # / 2. 1, & # / (
)
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(MV) ((MV )). H # # & 2 # . E <
.
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12. H, ## K, K\ (MV ). 3 2 # ,
#
, , ## HK ; , HK (MV).
) I # .
. E
# # &' #, # , # '#& & $# %. /
# .
m-
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S; = fA: A | , X g.
S = fA: A | , (
, ) X g.
I
1
2
1
1
2
1
0
0
1
+1
I
1
1
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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
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j
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=1
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i
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(Aj1 \ : : : \ Aj ) = 0.
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Sk Sk m
k 6 m + 2) Hk;
Bj = Hk;
(S n Aj ) =
i
m Tk Tk i m
;
k
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, ' # Kl : : : Km . #, H-
& fAj gnj , F(Aj1 : : : Aj ) 2
2 Hm;r ( 1 1, 2 3), & '
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k
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394
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.
2. A | #% .
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-
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.
4. HK (AL Z) = HK (AL Z) = 0.
5. HK (AL Z) = HK (AL Z) = 0.
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4. -
(A \ A ) (A \ A ) (A \ A ) .
5. ,'( A A A '
.
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.
1
2
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! ) ' "# k $ F(Aj1 : : : Aj ) '
.
4:11. $ fAj gnj R '( '$ $ , '( '$ /$ fAj gnj=1 R2 i
=1
k
2
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395
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, %
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.
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2. ,'( A A '
.
3. ,'( A A .
4. % A \ A .
# , #, & # . 8&'
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, & .
" 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
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2
1
2
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1
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2
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2
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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),
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1 C 41, . 5, x 48, VIII, 1].
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10. '( $ $ $ , $ % (
).
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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
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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
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1
1
2
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1
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1
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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
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A A 11 1.2 / #.
-# 2.2 # /< 8 2.1 1.2.
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(MV) , , P & (MV). -
1 1 &'
.
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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
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1
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i
k
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3
1
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1
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398
C. . 4.3. ' 3.1. X .
1. - X .
2. 1
$ $ : H (XL Z) = 0.
3. , X , % X .
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. J
2 1 ) 2 1 , # .
J
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X. 1# D | , & . @ D 6 X, # S
X, ,
H (S ) = Z
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1
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2. % A \ A .
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D A \ A , ,
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& 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
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S - #. A , & 6, ,
' , ' .
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400
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($ 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
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1
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2
1
3
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2
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.
1. % A
2. % A
\ A2 \ A3
\
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1
1
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.
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B# &' & # / #
#
( #
, '
) .
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2
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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
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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 &'
.
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.
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
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2
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3
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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, # < ,
$'% ( ) .
4.5. J # 4.1 4.4 , E \ HK & (MV). O
, # , #, ## HK ; , HK E \ HK (MV), E \ HK & (MV). A , &'
.
8/ < , 4.2
# # , . . # D .
1
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1
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1
1
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$ 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 2j=3. 3 & , & < . H' &. U # 4
$'
% . F
,
, ' . & .
=. R. #2 ( ) , (
).
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$
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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
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31] Moln9ar J. U ber eine Verallgemeinerung auf Kugelache 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.
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33] Soos G. Ausdehnung des Hellyschen Satzes auf den Fall vollstandiger konvexer
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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.
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Royale Sci. Li~ege. | 1997. | Vol. 66, no. 5. | P. 349{351.
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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.
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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 ,
! 20 5 . =5 3. 8 ! 3 !0
3-, 02 00 , 0 ! , 202 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- , 35 !( 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 . ? 20 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 1L
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] , , 35! !, 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 .
D5, 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.
D5. 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,
p21
p21
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 , , 20 !!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 ,
!, 20 !!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 ), 20 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 05 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 . ? 20
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. ;, 20 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 .
D5. 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 20 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 .
D5, 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! 20 - 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 20 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 .
D5. 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 ) $" 'a = b.
1
2
1
1
1
2
x
2
2
2
1
1
2
3. K- <!! K- , %27] (. 54).
5 fAigi2IQ| 5 ! . H a =
= (: : : ai : : :) 2 Ai , ! " a ! s(a) =
i2I
= fi 2 I j ai 6= 0g. H K | P (I) !
437
!
Q I, ! K- Ai (i 2 I) !
Ai , 2 1! a ! s(a) 2 K. ? i2I
s(a
a2 ) s(a1 ) s(a2 ), K-!
Q 1A;, !
L A .!! 5
i
K i
i2I
!!, , , K !
! I, ! !!, K = P (I), ! Ai (i 2 I).
! K-! !! Ai (i 2 I), 05 3 Ai 1 !!, ! 5,
K 1! ! ( , !) ! I.
;
5 2 3 , ! 5 -!
!. 8 2! 3 5, 50, ./ 5 -0 .
! Ai (i 2 I), 5
L A5
Q AA. i A
i D! ! i2I
i2I
A.
3.1. & A, ) "!
"$ ""$ 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 . H5 .
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-!,
20 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 20 !!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
20 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-
" "" '
" ( , )' '
" ) -
$ $.
. 5 fAigi2I | ! L !
, K | P (I). !, K Ai | . D Ai (i 2 I) 2 Gi , 02 !! ! ! - ,
Ai | ! !
;L G i %27,
;L 38.2,L. 189], 5
L
Gi = Ai Bi . ;!! K Gi = K Ai K Bi . N K Gi ! 3.11 K0 -!
L !! ! -
, L1! 0 3.10 K Gi | .
N K Ai ! ! .
8 L! 3.5 !, 20 A = K Ai (i 2 I), ! fAi gi2I 0 K-!. 5 I | !, ! !
N0. >3! p !! !
! fAi gi2I , A0 | ,
p- , i 2 N Ai | !
- pi . 5 K | P (I), ! L ! ! I.
;!! ! 3.15, K Ai | .
8! 1! aj 2 Aj (j 2 N0), hp (aj ) = 0. ? !!
H (a0 ) = inf
H fH (al )gl2N. H ! fAi gi2I 0 K-!, A0 2 1! a01 : : : a0r ,
a01 +: : :+a0r = a0 1! a0k (k = 1 r) 1! alk (lk 2 N), H (a0k ) > H (alk ). H (a0k ) > H (alk ),
443
a0k =
6 0, 5 !. D5, !-
H (alk ) , 02 ! p, ! 5 ! 1( !- H (a0k ), 5
a0k | 1! A0 hp (a0k ) 6= 1, , 02 ! p, ! 1 0. ;, ! fAi gi2I 0 K-!. (>!!, 3.2 ! fAi gi2I 0
!.)
<!!, %33], , Q
!3 Z. # ! %33], !! 2 @0
0 , - , - . 8 %25] s-
, 1! 2 ! !. # s-2
2 ,
Q , 5 , ! Z.
@0
J!! K-! !! s-2 .
3.16. A = LK Ai (i 2 I), Ai | s-'' . -
A , "$
fAigi2I ) K-".
. D5
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 ) = ?.
. D5 3.19. D! !5. ? A | , 0 3.2 ! fAigi2I 0 !. !, 20 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 20 1! a b , hp (a) = 0, hp (b) = 0. ;!! (b) > inf
f(pb) (a)g. ? ! X
fAi gi2I 0 !, 25 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)) D5. ;!! 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].
D5. 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-!, 20 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).
D5. ? ! 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 D5.
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 i2I , )
) :
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) = ?.
D5. 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 ). .
D5. 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, . 20 , 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 .
>, 20 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,L02 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).
D5. >!!, 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. .
D5. 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 |
.
D5. 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, ! !! 25 -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 | .
D5 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).
D5 ! 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. =20 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.
D5. 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 ! !, 20 , 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 35 L, 0!, ! Le.
4.25. & L | ! '" #"", e $ $.
. 5 Le0 Le, Le0 6= ?. J!! J = T J 0. J 6= ?,
L
J 2Le
0
0
J 5 1! L. H a 2 J, x 2 L a 6 x, x 2 J 0 J 0 2 L0, , x 2 J. I, a b 2 J,
a ^ b 2 J 0 J 0 2 L0, 1! a ^ b 2 J. ;, J | 35 J = inf Le0. F, L0 5 5 1! (! L),
!, Le | %28, x 3, ! 1, . 37].
4.26. fAigi2I | "$ ' , '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 | 35 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 20Q 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 | 35
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].
D5. 5 M | 35 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]. =0C5 ) 5 !.
;C5 . 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)
35! (G) 5 , M | 35 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).
. E5 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 35 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;99 2; ,,,<, 1972. | ,. 5{45.
7] !+! 2. 6. 2 // 2. !7. *3/!7. . 17. (9/!
! ! /1.). | .: 59;99 2; ,,,<, 1979. | ,. 3{63.
8] *!+ ,. -. > /!! 1/!/!!1 4 1
0 !7 // 2 ! 34!. | 1981. | ,. 56{92.
9] ? 6. 2. > 1/!/!!1 41 1 0 !7 // ,. !. / 3/3. | 3: 3. !., 1973. | ,. 15{20.
10] @! B. . ?0!/ !D/! ! /0!/! 0 !7. | @. 59;99. | 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 44 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!/! 44 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/! ,. ;. [!. | 63W,
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] 610 . 673 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 3X733! 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: yuyuk@kochetkov.mccme.ru
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 GH ( , 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
, , * * GH * (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
>
<k11x121 + : : : + knn;;11xn2n;;11 + knn = 0
(2.1)
: :: :: :: : : ::: :: :: :: : :: :: :: :: :: : :: :
>
: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 )( GH *
. 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 + m1 + 2 )z 6 +
+ 6nm2(m + 1)(m2 + m1 + 2)z 5 +
+ 3nm2(5nm2 + m2 1 + 2m12 + 6mn1 ; 2m2 +
+ n12 + 3n2 ; 212 + 13 )z 4 +
(3.1)
+ 2nm(6mn2 1 ; 6mn2 + 6m2 n1 + 10m2n2 + 3m1 2 +
3
2
2
2
2
+ 3n12 + 6mn1 + 2n 2 + 1 2 + 2m 2 )z +
+ 3mn2(;2n2 + 3m2 + 5mn2 ; 212 + m12 +
+ 2n12 + 6mn1 + n2 1 + 13)z 2 +
+ 6mn2(n + 1)(n2 + n1 + 2)z + n2 (n + 1)2 (n2 + n1 + 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 ; 41 2 + 2113 ; 1222 ; 8112 + 122 +
>
>
< + 1212 ; 4212 + 42 2 + 121 = 0
>
; 12 2 + 314 ; 12122 + 6113 ; 422 ; 2411 2 +
>
: + 22 + 3 2 2 ; 1222 + 42 2 = 0:
1
1 1
1
8
2 . P)
,, )( 2 :
3223 ; 4(61 + 111)(1 + 1)22 ;
; 41(31 + 1)(1 + 1 )22 + 12(21 + 1)(1 + 1)3 = 0:
- ) t = 22 =(1 + 2 ) 1:
212 t + (;12 + 61 t + 11t2 )1 ; 213 + 61t2 ; 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 ; 111 ; 61 a ; 61)22 +
+ (a + 2)(1 + 1)2 (a3 12 ; 4a2 12 + 5a12 ; 212 ; 61a2 1 + 1811 a ;
; 611 + 912 a)2 ; 12 (1 + 21)(1 + 1)3 (2a ; 1):
5
2 = (1a ++ 21 )t 1
t(a ; 2)(a ; 1)212 + (18t1a ; 11t2 ; 212a + 2t2a2 ; 6ta21 + 12 ; 6t1)1 +
+ (t ; 1 )2 (ta + 4t + 21 ; 41 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). F0 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: krasilnikov.algebra@mpgu.msk.su
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
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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)):
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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
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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
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(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, .
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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
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. . $
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)
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((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))
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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.
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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.
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3] /. . "'. 01
2 ! 13
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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
: ,
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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 !" #$,
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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.
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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 + 12j1 m12i5 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, !
' #
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$! ;-#$% * ! x0 ! px0=2, x ! 4;m x, ! 0, m ! m0, $"
! *&
(10), (12) &$"! #
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*
&'
*&
+ 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 h12 + 12 j1 m12i 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 h12 + 21 j1 m12i h2j0 m2i L 5 x32 x32
F3
Q32F2 (18)
h1 + 212j1 2m12i
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+j1m1ji1hm
(20)
120
120i h01j0 m01i h02j0 m02i
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 m02i
(21)
;(0);(012) h2 = ;(2 ; 0);(120) :
h1 = ;(
(22)
01);(02)
;(1 + 20);(1 + 10)
. <! #$% 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)
;(201);(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
+
+ 210);(1 ; ) 16 ;(21);(1 ; 21);( )
01
!
' * ;-#$%, &'
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+
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; 0)
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(28)
42 ; E(0 ; 21) = 1:
<
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(25), (28) (24) * #
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!
*
(16), & ! #
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"
*
&'
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+ 3.
012
1
1
51
h01j m01i h02j m02i L
; x0 h
012j m012 i h01j m01i h02j
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 m01i h1 + 1;! j0 m1i L 5 x0 z h1 + !2 j1 m1i h01 + !;
0
2
2
F1!
(32)
1
h0 j1 m0i h01 + !;
2 j0 m01i
* x00 = x0(x0 ; 1);1, z = x(1 ; x0);m5
F2 = L1+! F2 = (1 ; x0); ;! h10 j1 m01i h20 j1 m02i h1 + !21j0 m1 + !m21i L 5 x00 u
F2!
(33)
h1 + 12j1 m12i h1 ; 2 + !12j0 ;m2 + !m12i
*
10 = 2 + !12 m01 = m2 + !m12 20 = 1 ; 2 + !12
m02 = ;m2 + !m12 ! = 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
*$%
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*"
, L, L | * (32){(35).
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+
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!%
#$% F1,
F2, F3 . 2 !
! *&, #$% F1 , F2 * L0 #$% F3 * L1, L2, L0 !
. AO* 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 h01j0 m01i h01 ; 12 j0 m01i
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 + 12j1 m12i 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!121 m01 = m1 + 2!1m21 m = (2!2 ; 1)!1 m2 ; !2m1 !12 = !1 ; !2 x000 = x0 1 + j!12j(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! *
&&' !&' !
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7 (!. (39){(46)). M
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527
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R! &$"! #
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(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 );(01 ; 2 )
=0
=0
m h1 + 2 j1 m1i h10 + 1 + 2 j1 m10i L 5 x1 (;1)m xx;
0
F0
(47)
h + 12 j1 0i
#$% F1 *! #$%! * (31) ( &" 4] !
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,
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$ *
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!)12] (;x0); ;! F0 = C1+! F0 =
;(1 + !21);( ; 2 + !21 + 12 )
!=0
!=0
"
#
h j1 m2 + !m12i h + 21 j1 m2 + !m12i L 5 x1 (;1)m m x!m
x
F1
h1 + (1 ; 2!)21j1 (1 ; 2!)m21i
(49)
* = 0 15 = 2 + !12 ; 2 , F1 =
1
X
C2; F1 =
1
X
1
2
1
0
0
1
2
2
0
2
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1
2
529
+!
;( + 21 );(2! ; 1) ] x;
0
1
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!=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 .
<
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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 3 183 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: sen@cckr.krasnoyarsk.su
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 = cx0 0 y_0 = cy0 0 z_0 = cz0 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 = 22 (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 + 2z0 + 1 + 2"2 )y20 +
+ (z0 ; 1)2 (2"2 z0 + 2"2 + z0 )2 = 0: (10)
#
'% % 0 %', &! A+
z , q
1 z0 (1 ; z0 ) ; gy0 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
I7, 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 22 ; 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. 6C
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., ./ ;. -. $(45 B (45
= $! & 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& +
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*
:
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,
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, # * Ax = b #
*
*
+
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*). A * (1) & #
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) #
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#
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(., #
, 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]). %
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#
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2
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+
x 7! Cx + b , (5). F) +
#
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572
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R +
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#
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! | ) h IR+ ; = i,
IR | +
,
7xK x], x 6 x,
573
#
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, , +
*
| #
*
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 *
,
-#
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a + x = b a x = b
#*
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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, + *
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*2" *, # &,
, 1
. H #
) #"
* +
+, IR, *
* *.
R + ,
2
? 4*
#* . = ,
2
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+
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#
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D, +
+
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+
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A. R
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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 2n
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 *
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n
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6 (Cu) + (1 ; )(Cv ) ; (y + (1 ; )z) + (b)- 0+= (35)
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n
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j =1
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n
n
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n
n
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n
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;
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X
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n
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j =1
j =1
n X
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j =1
;
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X
X
n
n
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j =1
j =1
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X
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j =1
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8
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+
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:(0 0)
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8
8
>
>
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<
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E , #
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*
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ij
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1
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. 4. F+
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6
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;
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6
xj + n
ij
+
0
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-
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ij xj
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6
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ij
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n
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+
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6
6
6
6
;
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n
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n
596
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=
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+
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+
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>
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+
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= (mid C) x(0) ; x(0) + (b) - 0+ -"9
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?
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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)
=
*
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&
W(x(k)) > 0 k = 1 2 : : ::
(58)
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*
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/%
1] ., !" #. $ % %&. | (.: (,
1987.
2] . % /. ., 0 #. 1., #
2. !. (% 4. | 5": 56, 1986.
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‚. ., /" . $. `4% %
%. | (.: 56, 1985.
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# &, ,
1997 .
. . . . . 519.6
: , ,
.
!
! ($) & ($$'), ()
&) $ $$' ()
( !
* *
! +&. , - .//
0
, !0
$, *( 0
.
Abstract
O. D. Zhukov, A parallel number transformation method from a residue system
into a mixed base system, Fundamentalnayai prikladnaya matematika, vol. 8 (2002),
no. 2, pp. 611{615.
We propose a method of number transformation from a residue system (RS)
into a mixed base system (MBS), that combines the advantages of RS and MBS
and has much greater, in comparison with them, parallelism in computations. The
method gives an e5ective way to determine the number's value or a containing
interval, which provides new possibilities for more wide usage of RS, in particular,
in computer applications.
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(13)
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j =1
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(10){(14)
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.
1] . . . | .: , 1952.
2] Szabo N. S., Tanaka R. I. Residue arithmetic and its applications to computer technology. | New York: Mc Craw-Hill, 1967.
3] Jenkins W. K., Leon B. J. The use of residue number systems in the design of #nite
impulse response digital #lters // IEEE Trans. Circuits Systems. | 1977. | Vol.
CAS-24. | P. 191{201.
4] Baraniecka A., Jullien G. On decoding techniques for residue number system realization of digital signal processing hardware // IEEE Trans. Circuits Systems. | 1978. |
Vol. CAS-25. | P. 935{936.
5] Zhukov O. D., Rishe N. D. E)ective computer algebra for some applications // World
Congress on scienti#c computations and modelling. | 1977.
& ' 2001 .
. . 512.552
: , .
! " #.
Abstract
A. V. Kondrat'ev, On the construction of the generating function for an annihilator, Fundamentalnaya i prikladnaya matematika, vol. 8 (2002), no. 2, pp. 617{620.
We propose an algorithm for constructing a generating function of the module
of element annihilator.
L
T = khx1 : : : xni = Ti | i>0
k
,
T
.
I
|
, F = ff1 : : : fm g, 6=
A = L Ai T=I
.
a 2 A, d = deg a > 0.
=
i>0
! " #$
% 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: alebedev@nw.math.msu.su
519.2+519.713
: , !"#, $
%&.
'$$$( & $$ ) () ,$.$- /0 ! &) $ &1( $$(( (0 1). 3
$1-) ") 1$() (!$ $$(( $ !& $ %&) $(4$( $ %&( $$(( .!"- . /1. 5 6- $ .&$( $$78( . ) ,.($# &( $ $$ $$( 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. ? #$ " % (!$, 57 . @ , (t = 0) % ' p 2 (0 1).
+ ($# '(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
'$. 1
'(p) = p9 < p (0 1).
?/ # 57 % 1 0 / #"% .
? '(p) !
9- , E1 E2 ! ' . * !, . ! ! !, ! !.
< , 5 !7
# ! / Z1 '(p)
Q=
(2)
p dp:
0
B, '(p) !
p ( -
624
. . 33), ! ' ' !
(8- ) ! !. ?
X
8
8
X
1
9
;
k
Q= 9
(3)
1k +
0k :
k=0
k=1 k
8 #" .
E3. , Q > 1.
E4. , Q = 1.
E5. , Q < 1.
E
, ! ! ! ! , ! ! ! ! .
8 ! 567 Q = 4=9, ! ,
! !.
E " !, #"% 31{33, 217 = 131072.
+
/, / #, #, 95668( 73%) %
!, 256( 02%) 35148( 268%) !
. I Q, , #, P9
Qmax = 1=k 2829.
k=1
? ! Q , ! ' #:
9
8 X
1 9 + X
9 ; k 2 0284: (4)
MQ = 12 k1 1414 DQ = 324
k
k=1
k=1
+ Q ! ! '. D( ($ ( .! 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
: , , - .
! " # ,
$% % & " , " "# '
#. , " - #' ' .
Abstract
V. E. Marenich, Conjugation in the incidence algebras, Fundamentalnaya i prik-
ladnaya matematika, vol. 8 (2002), no. 2, pp. 627{630.
We introduce the notion of the canonical form of an incidence function that
generalizes the Jordan cell, and we 1nd the canonical forms for some functions.
In particular, the canonical form of the zeta-function in some incidence algebras is
found.
. !
. , #1, . 235]
: "+ , ( !
-
)?"
0 ! !
,
1 , !
!. 0 , ! -!
.
2 : (P 6) | (. . .)4 #a b]6 = z a 6 z 6 b | . . . (P 6)4 < |
6
. . . (P 6), a < b #a b]6 = 24 6 |
6
, a 6 b #a b]6 1 2 .
2 : incF (P 6) | !
, 1 F4 IncF (P 6) | 4 |
74 e | !
,
(
e(a b) = 1 a = b
0 a = b:
f
j
g
j
j
j 2 f
j
g
6
, 2002, 8, 2 2, . 627{630.
c 2002 ,
!"
#$ %
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:^_ !G6 !
, '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 #$. %&. . !%. ' %!. | %, 1979. | +. 122.
2] #- .. /., 0! 1. #. 023 4- !55 // /. %. | 1981. | '. 116 (158),
; 1. | +. 120{127.
3] 0! 1. #. 4 !55 // XVII #$.
!. %&. 2 43. ' %!. | /%: /% 5 5, 1983. | +. 195{196.
4] 1%53 A. . !4% ! !-: % % ! %!B, 5 !55 // . 2. 54. . /%. | 1982. | ; 11. | +. 3{11.
5] /% C. +. ! 2 5 !55 //
/. %. | 1978. | '. 103 (145), ; 2. | +. 147{237.
6] D5 #. C. !55 // # 5 !55. | '5!, 1972. | +. 122{172.
7] 0! 1. #. !55 // +.
. -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 !55 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] C54 .. K. ! ! ! % %! !55 //
! !%. | 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 .
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2002, фундаментальной, pdf, прикладное, математика
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