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e-mail:
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Abstract I. Ya. Novikov, Compactly supported wavelets, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 955{981.
The paper is devoted to one-dimensional compactly supported wavelets which are of the greatest interest for applications because of the simplest numerical realization of expansion and synthesis algorithms. It contains the review of papers (known to the author) about compactly supported wavelets and some new results of the author on the topic. The paper consists of 7 sections. In the second section the problem of existence of scaling function for wavelet bases is considered. Sections 3 and 4 are devoted to a brief account of the multiresolution analysis and the theory of compactly supported wavelets. Section 5 presents results about regularity of compactly supported wavelets in Sobolev and Holder spaces. The ;nal two sections are devoted to localization of wavelets in time and in frequency. < ' '% ) % ""& <==> 98-01-00044 ? <
@. , 2001, 7, A 4, . 955{981. c 2001 , !" #$ %
956
x
. .
1.
. . f " # t ; b Z 1 (W f)(a b) = pa f(t) a dt a b 2 a > 0)
R
R
R1 ( ) (t) dt = 0 ;1 # # # j(t)j ! 0 jtj ! 1. | + ( # ) # , , + " , . . . L2 ( ), j (t) := j 0(t) := 2j=2(2j t) j 2
jk (t) := j (t ; k2;j ) = 2j=2(2j t ; k) k 2 : , 1 ( ) # .
2M], 2D92], 2C]. 4 : 2A], 2499], 24598], 24597]. 8 , " ,
# ,1 . - # . # + . 5, . # , + . 5, 7 . " 1# " # . 9 3 4 " . 1# . x 5 , 5# <= ,. > , " .
R
x
Z
Z
2.
2.1. 5 , , = . jk (t) := 2j=2(2j t ; k) j k 2
Z
957
R
# # ( ) L2 ( ). ? ( ., , 2G]) ,
# @, Z ^ := f(t)e;i!t dt: f(!)
R
R
2.1. 2 L2( ), kk2 = 1. , X ^ m 2 j(2 )j = 1 ..) m2
Z
1 X ^ p )(2 ^ p ( + k)) = 0 .. k. (2
p=1
F M Wj := 2jk]k2Z Vj := Wl
R
l<j L2 (
# ) # #. 2.1. fVj gj2Z : 1) Vj Vj +1) Vj = L2 ( )) 2) j 2Z \ 3) Vj = f0g)
R
Z
j2
Z
4) f 2 Vj () f(2;j ) 2 V0) 5) f 2 V0 () f( ; k) 2 V0 k 2 : . L 9 . 9 V0? = Wj , V ? | , V , , j >0 . Wj , j > 0, , . 2 : " V0 # , " f(t ; k)gk2Z ? #" + . 5 " , ,
2L92]. # @, ^ = ;8=7;4=7] + 4=76=7] + 24=732=7]
958
. .
E (t) = 1, t 2 E, 0 . I ,, ^
2.1. " 1# " " . J ., " 1# " , j^(!)j2 ; ^ j2. 8 ,, #" ; j^(2!)j2 = j(2!) ( ., , 24598, x 11]), ^ ^(2!) = m(!)^(!) (2!) = e;i! m(! + )^(!) m | 2- ,
" . jm(!)j2 + + jm(! + )j2 = 1. 9+ ^ j2 = jm(! + )j2j^(!)j2 = (1 ; jm(!)j2 )j^(!)j2 = j^(!)j2 ; j^(2!)j2 : j(2!) P ^ j 2 ? , j^(!)j2 = j(2 !)j , j >1 2 j^(!)j = ;4=74=7] + 6=78=7] + 12=716=7]:
, f( ; k)gk2Z + X^ j(! + 2k)j2 = 1
Z
k2
. 2 , 2G] " " 1# " . 2.2. | . # $ % ,
1 X X ^ p ( + k))j2 > 0 .. j(2
& '
Z
p=1 k2
1 X X ^ p ( + k))j2 = 1 .. j(2
Z
p=1 k2
^ 6= 0 , $ % 2.1. ( (!)
. & ' , ( ' ^ | % ). F # # KNO. x
3. L2(R)
R
3.1. K 1# ( ) | + , , fVj gj 2Z L2 ( ),
" " :
R
Vj Vj +1 ) Vj = L2 ( )) j 2Z \ Vj = f0g)
959 (3.1) (3.2) (3.3)
Z
j2
Z
f 2 Vj () f(2;j ) 2 V0 ) (3.4) f 2 V0 () f( ; k) 2 V0 # k 2 ) (3.5) " ' 2 V0 , , , f'( ; k)gk2Z# # F V0 : (3.6) P
2Ma]. 5 (3.6) , " 0 < A1 6 A2 < 1 ( F ), X A1 kfck gk2Z kl2 6 ck '( ; k) 6 A2 kfck gk2Z kl2 : L2 (R ) k2
Z
, 1 # ,, f'( ; k)gk2Z
V0 . P #" , '? , # @, 'c? (!) := P '^(!) 1=2 j'(! ^ + 2k)j2
Z
k2
# + . K x 1, " . #
( . 24598, x 11]). 8 # . , # fjk (t) := 2j=2(2j t ; k)gjk2Z # f L2 ( ) X Pj +1f = Pj f + hf jk ijk (3.7)
R
Z
k2
Pj | , Vj . # @, ^ = e;i!=2m ! + '^ ! (!) (3.8) 2 2 P m(!) = p12 hk e;ik! | 2- 1# " , k2Z " . ^(2!) = m(!)^(!):
960
. .
? (3.8) , X (t) = (;1)k;1h;k+1'1k (t):
Z
k2
x
(3.9)
4. # $ %
4.1. )$ % ' , $ % . . 4 ( . 24598, x 11]), 1# "
1# . X ' = hk '1k (4.1)
Z
k2
hk := h' '1k i
X
jhk j2 = 1
Z
k2
R
(4.2)
R hf gi := f(t)Qg (t) dt | L2 ( ). # @, (4.1)
R +
" '^(!) = m !2 '^ !2
X ;ik! m(!) = p1 he : 2 k2Zk N 1# " , m
. jm(!)j2 + jm(! + )j2 = 1: (4.3) . J ' ,, (4.2) , hk = 0 jkj > T T 2 .
. R " . (4.1)
= 1 Y '(!) ^ = m0 (2;j !)
R
1
N2 P hk e;ik! ( . 2D92, . 174{175]). K 2DD], m0 (!) = p12 k = N 1 N 1 P2 h = p Q 2, m0 (2;j !)
# @, k 1 k=N1 ##" 2N1 N2]. 2 K 2.1, # , " , 1# "
961
. # . J 1#" ,, "
(3.9)
# , . . ,. #, . 1 . (4.3) , + 1 . ? , " ,, . " 1# " 2D92, . 193]. 8 # . 4.1. K . K + 2; ] 2, I# K 2 # ! 2 2; ] " l 2 , ! + 2l 2 K. N. K 2; ] + 2k k. 4.2. 9 , , fxig1i=1 .= C, , f 2 L2 ( ) X jhf xiij2 = C kf kL2 (R):
Z
R
N
i2
N 4.2. m0(!) = p12 P hke;ik! | 2
k=N1
, * (4.3), m0 (0) = 1. + ' , : 1 Y '(!) ^ := m0 (!2;l ) l=1 ^ := e;i!=2 m0 ! + '^ ! : (!) 2 2 2 - ', | % , L ( ) . + $ N2 p X '(t) = 2 hk '(2t ; k) k=N1 p X (t) = 2 (;1)n h;n+1 '(2t ; n):
R
n
R
. , % fjkgjk2Z * % L2 ( ) 1. / % +0. L2 ( ) , , ' * :
R
962
. .
( 1' ) * K , * 0 ' 2; ] 2, inf inf jm (!2;l )j > 0) l2N!2K 0 f!1 : : : !ng 2; ] $ ! ! 2! 2, jm0(!l )j = 1 8l = 1 : : : n2 ( 3 ) , 1 A , (2(N2 ; N1 ) ; 1) (2(N2 ; N1 ) ; 1), X akl := hn hn;2k+l n
( hn = 0 n < N1 n > N2), * 2 $ Af = f
, A 2- % % Af(!) = M0 !2 f !2 + M0 !2 + f !2 + M0 (!) := jm0(!)j2 . 4.1. ( m0 jm0 (!)j2 + jm0 (! + )j2 1, m0 (0) = 1 2; 3 3 ], f'0k gk2Z | +0#. J , , ' . , " # 2D92]. 4.3. f 2 L2( ) hfjk flm i = jl km , j= 2 j fjk (t) = 2 f(2 t ; k). * , f , f 2 C m ( ) f (l) l 6 m. - Z tl f(t) dt = 0 l = 0 1 : : : m: (4.4)
R
R
R
R
5 (4.4) + , ^(l) (0) = 0 l = 0 1 : : : m. ^ = e;i!=2 m0 ( !2 + )'( (!) ^ !2 ) '^(0) = 1, 2 C m ( ) , m0 , m + 1 ;i! m+1 m0 (!) = 1 + 2e L(!) (4.5) L | . >, N N jm0 (!)j2 = cos2 !2 jL(!)j2 = cos2 !2 M sin2 !2 N 2 M(sin2 !2 ) := jL(!)j2 . 9
+ . (4.3), M: xN M(1 ; x) + (1 ; x)N M(x) = 1: (4.6)
N
963
xN (1 ; x)N | N, T " MN ;1 N ; 1,
" (4.6): NX ;1 N ; 1 + k xk : MN ;1 (x) = k k=0 , MN ;1 (x) > 0. #" 1 (4.6) M(x) = MN ;1 (x) + xN R x ; 12 (4.7) R | , = . P 1# " , 2D88]. K 8#1 R = 0. 4.3. 9 , N 2 .2N@ , 8#1 ;1 P 1 il! hN (l)e , hN (l) 2 ,
dN (!) = p2 l=0 " N jdN (!)j2 = DN (cos(!)) := cos2 !2 MN ;1 sin2 !2 : (4.8) 2D88] " 4.4. 'DN , ,
1 Y 'DN (!) := dN (!2;l )
N
R
l=1
$ % . # DN , % DN (!) = e;i!=2 dN !2 + 'DN !2 * , L2 ( ): DN
jk () := 2j=2DN (2j ;k) j 2Z : k2Z . , supp DN = 2;(N ; 1) N], > 0, DN 2 C N , Z ^ C := f : f(!)(1 + j!j) d! < 1 > 0:
\
R
R
> jm0 j2, m0 . " F 2R, -
40].
4.1. A(!) = P k eik! | , T
;T
. -
964
. .
< . 1. 1) '
2
D
(t)
< . 2. 1)
2
D
(t)
a(!) = P k eik! , ja(!)j2 = A(!). . , '%0 % k , a(!) * * k . 9 m0 ,
" 1 jm0(!)j2 = B(cos(!)) (4.9) ; 1+t N B(t) = 2 M(t), = . 1 . , " #. 9 , ftk g | M. m0 (0) = 1, B(1) = 1 Y M(cos !) = cos1 !; ;t tk : k k m0 1 + e;i! N Y m0 (!) = Sk k (!) (4.10) 2 k k = 1, ;i! zk e;i! : Sk1 (!) = e 1 ;;z zk Sk;1 (!) = 1 ;1 ; zk k 1 1 > , zk | 1 2 (zk + zk ) = tk jzk j > 1. T
< . 3. 1) '
D
4
(t)
< . 4. 1)
4
D
(t)
965
M | , + , , , . K .= tk tk = zk zk;1 zQk zQk;1 . @ (4.10) , m0 # . = (k = 1 # zk , k = ;1 # z1k ). I F , cos(!) ; tk = ei! + e;i! ; 2tk = ei! + e;i! ; zk ; z1k = 1 ; tk 2 ; zk ; z1k (1 ; zk )(1 ; z1k ) ;i! 1 ; ei! z1k i! e;i! ; zk e;i! ; z1k i! 1 ;1 = e 1;z = e Sk (!)Sk (!): = e 1 ;;z zk 1 ; z1k k 1 ; z1k k 8 # m0
(4.9), , #, zk zQk;1, ei! (e;i! ; 1 ) ;i! zk Sk1 (!) = e 1 ;;z zk = 1 ; z1k : k 8 # m0 , + , zk # , #, z k . #, ,
= zk zk;1 zQk zQk;1 . , # .= , # . 8 | ." . 5 , , 2Ta], " 1# " " R = 0 # . P . , # " + , : , , , , , , , . .
x
5. '$( %
5" #
. 9 # | # # @,. K , # @, 1# " 1 Y '(!) ^ := m0 (2;j !) (5.1) 1
= " , m0 (!) = ; 1# = 1+e2;i! N L(!) ( . (4.5)). ? , 1 1 Q cos(!2;l ) = sin(!!) ,
l=1
966
. .
1 ;i! N Y '(!) ^ = 1 ;i!e L(!2;l ): l=1 9 . , = , # j!j;N . #, # , , . , + . # | + , , " '. ^ 1# 1 N N p X X '(t) = hk '1k (t) = 2hk '(2t ; k) k=0
Z
Z
k=0
, ' k2 , k 2 , , , k, k 2 . T0 T1 : p p (T0 )mn := 2h2m;n;1 (T1 )mn := 2h2m;n 1 6 m n 6 N: p # ck := 2hk . N 5.1 ("D92, c. 239]). P ck = 2 k=0 N X (;1)k kl ck = 0 8l = 0 1 : : : L:
R
k=0
4 m = 1 : : : L + 1 Em N , 2ej ]mj=1, ej := (1j ;1 2j ;1 : : : N j ;1). * , 2 2 21 1), l 2 f0 1 : : : Lg C > 0, fdj gj 2N, dj = 0 1, m2 kTd1 Td2 : : :Tdm jEl+1 k 6 C m 2;ml : - ' l , %% . ( > 21 , '(l) 5 , j lnln 2j . ( = 12 , '(l) 6 7 3 $ j'(l) (t + h) ; '(l) (t)j 6 C jhj j ln jtjj:
N
5.1.
R
5.1. 9 , 5# (')
(') := supfs: j!js '^(!) 2 L2 ( )g: 5.2 ("Vi]). m0(!) = ( 1+e2;i! )Lm~ 0(!), m~ 0() 6= 0, m0 1' . ~ Af(!) = M~ 0 !2 f !2 + M~ 0 !2 + f !2 + M~ 0 (!) = jm~ 0(!)j2 .
967
~ = lim kA~nk1=n | (A) n!1
. ~ , ' | $ % , - (') = L ; log4 (A) (5.1). 9 , (') '^ , # B,
" 1 B(t) + B(;t) = 1 jm0 (!)j2 = B(cos(!)): 9+ , 1 # , , # (B), + , ('), ' | 1# " , " B. K . 1 , ?. 8#1 ( . (4.7)), " "= ( ., , 2LZ]). 5.3. B | '%% 2N ; 1, N 2 . 8 : 1 + t L M(t)) (i) B(1) = 1 B(t) + B(;t) = 1 B(t) = 2 1 (ii) M(t) = ML;1 (t) + tL R t ; 2 LX ;1 L ; 1 + k tk ML;1(t) = k k=0
N
R | , 2N ; 1 ; L2 l 1 ; t 2N ;1;l 2X N ;1 "Nl 2N l; 1 1 +2 t (iii) B(t) = 2 l=0
"Nl = 0 0 6 l 6 L ; 1, "Nl + "N2N ;1;l = 12 Zt (iv) B(t) = (1 ; s2 )L;1 A(s2 ) ds s=;1
A | '%% N ; L. 9 1# " , "= #
, # , B. , 2V]
" # .
968
. .
'( 1. 8 2N ; 1 ( ,, , B . #, = ) B , 5# (B). '( 2. 9 , , , fBN gN 2N 2N ; 1, lim (BN ) N !1 2N # # ,. 9 5# . , 8#1, = (4.8), lim (DN ) = 21 ; 4ln3 N !1 2N ln 2 = 01037: : : ( . 2V]). , 2O] 1# " , m0 # 2N + h0 : : : h2N ;1 , ( 2 ). @ N ( 2 ), , , # . , , 5# . , , " 40 + 4 ( 2 ) 1# " ,. 4 5# # .
# 1. nz = 0 5# ,
8#1. F , 1 . , 2LZ] , , T1 , "Nl ( . 5.3, (iii)) , 5# # ,1, 8#1. + . , 1# " , m0 , " L 2M 23 . , .= ,, N . 5 " 5.4.
(a) m0 L 2M 23 , B(; 12 ) = 0 A, deg A > 1 Zt B(t) = (1 ; s2 )L;1 (1 ; 4s2 )2M ;1A(s2 ) ds: s=;1 (b) ( 2M 6 (L ; 1)( lnln 54 ), ZX QLM (X) = LM (1 ; t2 )L;1(1 ; 4t2)2M ;1(1 ; LM t2) dt ;1
B( 1
2N 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
, 5# nz = 0 nz = 1 nz = 2 nz = 3 nz = 4 0,50 1,00 1,42 1,00 1,78 1,82 2,10 2,26 1,00 2,39 2,66 2,00 2,66 3,02 3,00 1,00 2,91 3,37 3,48 2,00 3,16 3,72 3,92 3,00 1,00 3,40 4,07 4,32 4,00 2,00 3,64 4,42 4,73 4,76 3,00 3,87 4,78 5,14 5,22 4,00 4,11 5,14 5,55 5,67 5,00 4,34 5,50 5,95 6,10 5,84 4,57 5,85 6,34 6,51 6,38 4,79 6,19 6,71 6,89 6,88 5,02 6,52 7,09 7,25 7,30 5,24 6,83 7,45 7,62 7,69 5,47 7,15 7,81 7,99 8,08 5,69 7,46 8,17 8,37 8,51
969
970
. .
LM , $ QLM (; 21 ) = 0, LM | , $ QLM (1) = 1, 1' . K 5# N 5.2. 9 = , (L 1) := (QL1) N L > 8 # ,1, , 5# ('DL+2 ) 8#1, " . ,. , ,: 5.5. # fN gN 2N ( ) ), (i) N supp N = 20 2N ; 1]2 ln 3 . (ii) lim inf ( N ) > 12 ; 4ln 2 N !1 2N + , M = 2 L] ( , 2a] | , a),
| . 9 = 201 (2NN ) > 0128. , 2V] , 1# " , " #. 9 , r | , " , . 0.
N 2 Zt 1 ; s2 N 1 r 4 ds BN (t) = N
N
;1
N # , BN (1) = 1. , BN
1 B(t) + B(;t) = 1. J N =, N > 0 BN 20 1], K+ , BN (x) > 0 0 6 x 6 1. 9 = N , # ,, # r # , 20 1]. , 2V] r(y) = yM (y ; a1 )(y ; a2 ) : : :(y ; aK ) 0 < a1 < : : : < aK < 1. 8 M K al , , ,. , , M = 4 K = 4 . #, 0175. T . , . , 019. 5.2.
5.2. 9 <= , H , -
, n := 2] 1
<= , jf (n) (t + ) ; f (n) (t)j 6 C j jfg t 2 fg = ; 2] | # , .
R
971
[1 : Z ^ j d! < 1 =) f 2 H : (1 + j!j)jf(!)
R
K , <= , (') := supf: ' 2 H g
(') ; 21 6 (') 6 (') 2Vi, c. 1538]. B( 2
9 <= , 5# 8#1 2N 4 0,55 1,00 6 1,09 1,42 ?. 8#1 2D92] # <= , 4 6 + . B( 3
9 <= , 5# , <= , 2N 4 0,59 0,87 6 1,40 1,31 # , <= ,.
x
6. *
# , #
. , , , . + . , - , " t? \ :
972
. .
R1 t? := ;1 1
t j(t)j2 dt
kk2L2 (R)
Z1
\ := kk 2 L (R )
1=2
(t ; t? )2 j(t)j2 dt
:
;1
O w? \ ^ # @, :
R1 ^ 2 t j(t)j dt ;1 ? ! := ^ 2 kkL2 (R) Z1 1=2 1 ? 2 2 ^ \ := ^ (! ; ! ) j(!)j d! : k kL2(R) ;1
9 \ \ ^ - = . 9 = , # 2 L2 ( ) \ \ ^ > 21 : >, \ jk = 2;j \ , \ ^jk = 2j \ ^, j k 2 . #, = + # .. 8 ( , , , 1# ) . , . 6.1 ("Ba]). ( | , 1=2 1 Z Z 1 2 2 2 2 ^ p p ! j (!) j d! t j (t) j dt > 170711 2 2( 2 ; 1) R R 1 Z 1=2 2 3 Z 4 j(!) ^ j4 d! t4 j(t)j4 dt ! > 2 ; 4 341274 2 ln 2 R R 1 Z 1=2 9 15 Z 6 j(!) ^ j6 d! t6 j(t)j6 dt ! > 2 ; 8 168573: 2 ln 2
R
R
Z
R
4 , n Zn;1 1 Z 1=2 Z1 Z Z 1 dn ;1: 2 n 2 n 2 n 2 n ^ ! j (!) j d! t j (t) j dt > d d : : : p 1 2 2 n ;1 ;2 ;n
R
R
2
2
2
973
6.2 ("Ba]). ( | , , \ \ ^ > 23 : ( jtjn+ (t) 2 L2 ( ), n 2 , X 21=2 1=n Y n Y n k ; 12 : \ \^ > k ; 12 + n l=1 k6=l k=1
R N
6.1.
= 8#1 # , , " 1# " , ( . x 4). > , " , = . O ] Z1 ](t) := (s)(s ; t) ds: ;1
^ , ]^ (!) = j^(!)j2. 9 , ]DN | 1# " DN ' , " DN ( . 4.3 4.4). 2ChW]
6.3.
lim k]^ DN ; ;] kLp = 0 1 6 p < 1 lim \ > p N !1 ^ DN 3 lim \ DN = 1 N !1 e | % * e. # , lim \ \ DN = 1: N !1 ^ DN 8 , jdN (!)j2 = DN (cos !) N NX ;1 N ; 1 + l 1 ; t l = DN (t) = 1 +2 t l 2 l=0 2X N ;1 2N ; 1 1 + t l 1 ; t 2N ;1;l : = l 2 2 l=N N !1
974
. .
9 , DN (t) | + T1 2;1 1], " 01] ( . 5).
< . 5
8#1 . ,
` ( . 24598, x 12]). N 1# "
, , '^Sh (!) := ;] (!) ( . 6).
< . 6
9 . 6 #.= D7 (cos(!)). 6.3 , 8#1 , . 6.2. ! " #
8#1 . , , # , . 9 , a 2 (0 1), fa (t) | # , 2;1 1], 0 t 2 2;1 ;a]
" . fa (t) + fa (;t) = 1, t 2 2;1 1]. # N 1 + t l N ;l 1 ; t N bl (t) := l l = 0 1 : : : N 2 2 5. 4. T1 2;1 1], tNl := 2l;NN , l = 0 1 : : : N.
N
975
F maN (!), N 2 , , + ,
" jmaN (!)j2 = BNa (cos !) maN (0) = 1 (6.1) N P BNa (t) = fa (tNl )bNl (t). 9 BNa | + 5. 4. T1 l=0 fa ( . 7).
07 < . 7. +$ | f0 7 , | B13
1# " 'aN " # @,: 1 1 Y Y '^aN (!) = maN 2!l ^aN (!) = e;i!=2 maN !2 + maN 2!l : (6.2) l=1 l=2 1 aN 6.4 ("N95]). a 6 2 . & * L2( ) , aN
jk () := 2j=2aN (2j ;k) jk2Z : 1 , > 0, $ N % 'aN aN * C N . 1# " N 8#1: Z Z M M aN M ] (t) := ' (s)' (s ; t) ds ] (t) := 'aN (s)'aN (s ; t) ds: aN
R
R
R
p N | + . `: '^M (!) := := fa (cos(!)) ;22] (!) ( . 24598, x 14]). N 8#1 | +
N ( . 8). 6.5 ("N98]). a 6 12 . ( % fa (fa )00(t) > 0 t 6 0 (fa )00(t) 6 0 t > 0 (6.3) p 0 (6.4) (fa ) (t) 6 C fa (t) C
976
. .
d
07 < . 8. +$ | E (!) a = 07, | B13 (cos(!))
M
lim k]^ aN ; ]^ M kLp = 0 1 6 p < 1 lim \ = \ ^ M N !1 ^ aN p 4 2C lim \ aN 6 p N !1 ( 2 ; 1)k]M k2 N !1
,
(6.5) (6.6) (6.7)
p
C < 1: lim \ \ aN 6 p4 2\ ^ M M (6.8) N !1 ^ aN ( 2 ; 1)k] k2 1 8#1, " , ,
, , 1# " , 1# " , , 3X L;1 4L X 1 4 L 4 L (l ; L)bl (t) + 2L bl (t) : BL (t) := 2L l=L+1 l=3L P T1 , # . " 8 > t 2 2;1 ; 21 ] <0 f(t) := := >t + 12 t 2 (; 21 12 ] :1 t 2 ( 1 1]: 2
j L = 2 3 : : : 7 < . 9 = # ( . x 4), " , = 1# " . " # = 1 8#1 . + .
977
L 9 . . 8#1 2 0,867 0,635 3 0,710 0,694 4 0,713 0,748 5 0,679 0,765 6 0,672 0,8
x
7. , -.
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Meyer Y. Ondelettes et operateurs. | Paris: Hermann, 1990. Daubechies I. Ten lectures on wavelets. | CBMS-NSF. Regional conference series in applied mathematics, SIAM, 1992. C] Chui C. K. An Introduction to Wavelets. | New York: Academic Press, 1992. A] . . -: ! // #$. | 1996. | &. 166, ' 11. | (. 1145{1170. 99] - .. /. - (- - 1 ) // 12- (1 - 3- , 1 -, 18{23 6!, 1998. | 1 -: .7- 8 - . (. 9. ( 1 , 1999. | (. 92{111. (98] - .. /., (:- (. ;. < - // # = . 8-. | 1998. | &. 53, ' 6. | (. 53{128. (97] - .. /., (:- (. ;. < - 8-> - // $87 ! -7! -. | 1997. | &. 3, . 4. | (. 999{1028. G] Gripenberg G. A necessary and su?cient condition for the existence of father wavelet // Studia Mathematica. | 1995. | Vol. 114, no. 3. | P. 207{226. L92] Lemarie-Rieusset P. G. Existence de @fonction-pereJ pour le ondelettes a support compact // C. R. Acad. Sci. Paris I. | 1992. | V. 314. | P. 17{19. Ma] Mallat S. Multiresolution approximation and wavelets // Trans. Amer. Math. Soc. | 1989. | Vol. 315. | P. 69{88. DD] Deslauriers G., Dubuc S. Interpolation dyadique // Fractals, dimensions non entiQeres et applications / G. Cherbit, ed. | Paris: Masson, 1987. | P. 44{55. D88] Daubechies I. Orthonormal basis of compactly supported wavelets // Comm. Pure Appl. Math. | 1988. | Vol. 46. | P. 909{996. R] U X., (Z X. [7: . \. 2. | .: 8-, 1978. Ta] Taswell C. The systematized collection of wavelet ]lters computable by spectral factorization of the Daubechies polynomial. | Technical Report CT-1998-08. LZ] Lemarie-Rieusset P. G., Zahrouni E. More regular wavelets // Applied and Computational Harmonic Analysis. | 1998. | Vol. 5. | P. 92{105.
981
Vi]
Villemoes H. Energy moments in time and frequency for two-scale di_erence equation solutions and wavelets // SIAM J. Math. Anal. | 1992. | Vol. 23, no. 6. | P. 1519{1543. V] Volkmer H. Asymptotic regularity of compactly supported wavelets // SIAM J. Math. Anal. | 1995. | Vol. 26, no. 4. | P. 1075{1087. O] Ojanen H. Orthonormal compactly supported wavelets with optimal Sobolev regularity. | Technical Report math.CA/9807089. | 1998. Ba] Battle Guy. Heisenberg inequalities for wavelets states // Appl. Comp. Harm. Analysis. | 1997. | Vol. 4. | P. 119{146. ChW] Chui C. K., Wang J. High-order orthonormal scaling functions and wavelets give poor time-frequency localization // CAT Report # 22. | 1994. | P. 1{24. N95] Novikov I. Ya. Modi]ed Daubechies wavelets preserving localization with growth of smoothness // East J. Approximation. | 1995. | Vol. 1, no. 3. | P. 341{348. N98] - .. /. | 7 7! 7> = - } 1~ // .. &8. Z . 8-. ( . -. =-. . -. | &8: &8X#, 1998. | &. 4, . 1. | (. 107{111. BN92] ; - - . [., - .. /. < 1 - : Z7-= :- -= - - // } -. . | 1992. | &. 326, ' 6. | (. 935{938. BDR93] De Boor C., DeVore R., Ron A. On the construction of multivariate (pre)wavelets // Constr. Approx. | 1993. | Vol. 2, 3. | P. 123{166. N94] Novikov I. Ya. On the construction of nonstationary orthonormal in]nitely di_erentiable compactly supported wavelets // Functional Di_erential Equations. | 1994. | Vol. 2. | P. 145{156. CD] Cohen A., Dyn N. Nonstationary subdivision schemes and multiresolution analysis // SIAM J. Math. Anal. | 1996. | Vol. 27. | P. 1745{1769. N99] Novikov I. Ya. Nonstationary orthonormal in]nitely di_erentiable compactly supported wavelets with uniformly bounded uncertainty constants // Self-Similar Systems. | Dubna: Joint Institute for nuclear research, 1999. | P. 110{115. & ' 2000 .
- (R-mod Ab) . . , . .
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Abstract G. A. Garkusha, A. I. Generalov, Grothendieck categories as quotient categories of ( -mod Ab), Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 983{992. R
A Grothendieck category can be presented as a quotient category of the category ( -mod Ab) of generalized modules. In turn, this fact is deduced from the following theorem: if C is a Grothendieck category and there exists a 9nitely generated projective object 2 C , then the quotient category C S P , S P = f 2 C j ( ) = 0g is equivalent to the module category Mod- , = ( ). R
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), X = t(X ). > S , X 2 C % X : X ! sq(X ), #' ker(X ) coker(X ) 2 S ker(X ) | % 9 X , $ S , , ker(X ) = t(X ) :1, 15.14, 15.15]. , # 9 X 2 C S - (
S - ), X | (
). 8 , # $ C =S S - %( C 9 :1, 15.19B]. !9 sq(X ), X 2 C , # # X , sq(), 2 Mor C , # # = X % S - 9 X . ? , # % S - # iX : X ! X i , i = 1 2, 9 X 2 C % sq(X ) t sq(X ) t X . @ $, # X = 0 () X 2 S . C 0 | C , S - %( 9 , i : C 0 ! C | # . C (;) : C ! C 0 : (;) = sq, % % % . ? , # i , (;) #. X Y 2 C , 2 (X Y ), (;) () = = (Y ) , Y | S - # Y . !# , # = 0 , im t (Y ). ! , # X 2 C , Y 2 C 0 (X Y ) t (X Y ), i $' (;) . 8 %, C C 0 | , q0 : C ! C 0 | #% C S
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990
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. . , . .
1] . : , . . 1. | .: , 1977. 2] Krause H. The spectrum of a locally coherent category // J. Pure Appl. Algebra. | 1997. | Vol. 114. | P. 259{271. 3] &p '., ( )* . +,* , - .*,. | .: , 1972. 4] Stenstr/om B. Rings of quotients. | New York and Heidelberg: Springer-Verlag, 1975. 5] Gabriel P. Des categories abeliennes // Bull. Soc. Math. France. | 1962. | Vol. 90. | P. 323{448. 6] Herzog I. The Ziegler spectrum of a locally coherent Grothendieck category // Proc. London Math. Soc. | 1997. | Vol. 74. | P. 503{558. 7] Auslander M. Coherent functors // Proc. Conf. on Categorical Algebra (La Jolla, 1965). | Springer, 1966. | P. 189{231. 8] Roos J.-E. Locally noetherian categories // Category Theory, Homology Theory and their Applications II. Lect. Notes Math. Vol. 92. | 1969. | P. 197{277. 9] Gruson L., Jensen C. U. Dimensions cohomologiques reli6ees aux foncteurs lim ;(i) // Lect. Notes Math. Vol. 867. | 1981. | P. 234{294. 10] Auslander M. Isolated singularities and almost split sequences // Representation Theory II. Lect. Notes Math. Vol. 1178. | 1986. | P. 194{242. 11] Ziegler M. Model theory of modules // Annals of Pure and Applied Logic. | 1984. | Vol. 26. | P. 149{213. 12] Prest M., Rothmaler Ph., Ziegler M. Absolutely pure and 7at modules and 8indiscrete9 rings // J. Algebra. | 1995. | Vol. 174. | P. 349{372. 13] Krause H. Functors on locally :nitely presented categories. | Preprint. | 1995. ( 1998 .
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Abstract O. D. Golubitsky, Involutive Grobner walk, Fundamentalnaya i prikladnaya
matematika, vol. 7 (2001), no. 4, pp. 993{1001.
An algorithm that transforms an involutive basis of a polynomial ideal with respect to one monomial ordering into an involutive basis with respect to another ordering is proposed. The algorithm is based on the Grobner walk method of transformation of Grobner bases of polynomial ideals.
1.
. , " , # . $ % &2] (. FGLM Maple-V.5). # , , , # , # # . -# , , , . ,, , .. % , # | ,. 0 &3] (the Gr obner walk ). 2 # , . 3, , # 4 3 &5] 2 + +) & 2334, 96-01-01349.
, 2001, ! 7, 6 4, . 993{1001. c 2001 !, "# $% &
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2. 8. . : K | , c 2 K, R = K&x1 : : : xn] | K, f g h p q 2 R, I | R, F G H Q | R, hGi | , # G R, X = fx1 : : : xng | ,, x 2 X, T = fxi11 : : :xinn j (i1 : : : in) 2 Zn+g | , , m u v w 2 T , ;<< | T . # : 1) 1 < u 8u 6= 1, 2) u < v ) 8w uw < vw, in< (f), f 2 R, | # f ;<<, G R ) G< := fin< (g) j g 2 Gg, >n := f(1 : : : n) 2 Qn j i > 0g | , , ! 2 >n, Pn deg! (xi11 : : :xinn ) := ij j , deg! (0) := ;1 | !- , j =1 f | !- # , . , u, v, , , f, deg! (u) = deg! (v) := deg! (f), . # f f = h1 + : : : + hr , hi !- deg! (hi ) > deg! (hi+1 ), i = 1 : : : r ; 1, . in! (f) := h1 , I | !- # , !- , ! ;< T : u v :, :, deg! (u) < deg! (v), ;
3.
995
jL | . # : 1. ujLv ) ujv ( ). 2. 1jLu. 3. ujLw vjL w ) ujLv _ vjL u. 4. ujLuvw , ujLuv ^ ujL uw. 5. ujLv vjL w ) ujLw ( ). B . , , . , ,. , M(u), u | , . | NM(u). B f g L h),
h m, , f (f ;! g f) m in< (g)jL m h = f ; coeC(m lcoeC < (g) in< (g) g: f G . # G , # f . . , # % G. D " <. ( ) F I % . jL ,
. , % , #: 1. # . . " F . 2. # , # , . 0 F. 3. - f F # x, # f, xf . F 0 F I. 8 . , . , , , #. 2 - , . # # . # .
# . # ( , ) . E Maple-V.5, # .# . . . # # . # . > invrepIAutoReduce]:=proc(F::list(list(polynom)),O::TermOrder, > L::procedure) > local G,NewG,i,V,M,GLead > G:=] NewG:=F V:=op(O)orderindet]
996
. .
> while NewG<>G do > G:=sort(NewG,(a,b)->testorder(a2],b2],O)) > GLead:=map(x->x2],G) > M:=map(g->g2],L(g2],GLead,V)],G,GLead,V) > NewG:=map(i->INormalForm(Gi],G1..i-1],O,M), > $1..nops(G)],G,O,M) > od > RETURN(G) > end > > > invrepInvolutiveBasis]:=proc(F::list(polynom),O::TermOrder,L::procedure) > local G,dG,B,GLead,V,M > V:=op(O)orderindet] > dG:=map(i->Fi],LeadingPolynom(Fi],O), > op(map(k->if k=i then 1 else 0 fi,$1..nops(F)]))],$1..nops(F)],O) > G:=] > while dG<>] do > G:=IAutoReduce(op(G),op(dG)],O,L) > GLead:=map(x->x2],G) > M:=map(g->g2],L(g2],GLead,V)],G,GLead,V) > B:=map(j->op(map(i->i,j],op(L(Gj]2],GLead,V))])),$1..nops(G)]) > dG:=map((p,G,O)-> > INormalForm(expand(p1]*Gp2]]),G,O,M),B,G,O,M) > od > RETURN(G) > end
4. ! !
1. H | I <, !. H! hI! i <. .
L g m, , in (f). A in (h)j m, 1. f ;! h ! < L v = m= in< (h). A < !, , in< (in! (h)) = = in< (h), in< (in! (h))jLm. 3 , in! (f ; cvh) = in! (f) ; cv in! (h). 2 , L in! (f) ;! in! (h) in! (g) m. 2. q 2 I! . A f 2 I, # in! (f) = q. H . I, . h1 : : : hk 2 H
. , # L L L ;! f ;! h1 h2 : : : ;!hk 0:
997
$ %#
hi , , q !- , , L L q = in! (f) ;! () in! (hi1 ) : : : ;!in! (hil ) 0: 2 , H! hI! i. 3. J xq, q 2 H! , x 2 2 NM(in< (q)). A h 2 H, # in! (h) = q. D , in! (xh) = xq. E xh 2 I, , xq 2 I! . 2 , () xq
. 0 H! , . H! | . # hI! i. 2. < !, H = fh1 : : : hr g | I <. !- q 2 hI! i Xr q = pj in! (hj ) j =1
!- !- deg! (q).
.
1. 1 H! . hI! i, , q . 0 H! . 2. K
, , !- : L q ;! in! (hj ) q1 6= 0 =) deg! (q1) = deg! (q): .
, q1 = q ; cvhj , cv pj . <, . !. H | . # I < Q = fq1 : : : qsg | . # hI! i , # . InvolutiveBasis H! . A qi !- ,
!- , , . # !- . Xr qi = pij in! (hj ) i = 1 : : : s j =1
. . , 2. 1. fi =
Xr p j =1
ij hj i = 1 : : : s:
F = ff1 : : : fs g | I . . f 2 I. , f . 0 F . A . " fN 6= 0, . F
998
. .
N 2 I! . Q
I. 2 , in! (f) . hI! i , N 9qi 2 Q: in (qi )jL in (in! (f)):
E 2 , in! (fi ) = qi. 2 , in (in! (fi )) = in (qi). N = in (in! (f)). N !, in (fi ) = in (qi) in (f) N fN . % (), , in (fi )jL in (f), % F, . 2 , f . 0, . F | . # I .
5. #
2 &3], , >n cone(<) := &f! 2 >n j hI< i = hI! ig] . >n. 2.6 &4] % , . 2 . . 3. ! H< = H H | I <, H I . L g <, h 2 H, f ;! L g . K f ;! h h , in< (h) in (h) . A H | . # I <, . # f 2 I . 0 H <,
< ., . # f 2 I . 0 H , , H | . # I . 4. ! cone(<) = cone() H | I <, H . . cone(<) = cone(), , hI
999
0 # : H = fh1 : : : hr g | . # <. A 1 | # !1 , . !1 !2, . #: 9i: in1 (hi ) 6= in< (hi ), cone(<) = cone(<1 ), , in1 (hi ) = in< (hi ) + u1 + : : :+ uk , uj , hi . A # , # , # i, %"" , , , ,
. (. &3]). K # , % . 2 1 . # cone(1 ) . P 4 , # . # , 3 . 0 . . E Maple-V.5. > walkWalkVector]:=proc(F::set(polynom),V::list(name),a::list,b::list) > local f,v,lm,vecs,i,j,k,l,t,MinT,d,In_f,Rest_f,In_vecs,Rest_vecs,A,T > MinT:=1e100 > A:=poly_algebra(op(V)) > T:=termorder(A,'matrix'(NormVector(a)],V)) > for i from 1 to nops(F) do > f:=Fi] > In_f:=LeadingPolynom(f,T) Rest_f:=f-In_f > In_vecs:=Polynom2Vectors(In_f,V) > Rest_vecs:=Polynom2Vectors(Rest_f,V) > for j from 1 to nops(In_vecs) do > for k from 1 to nops(Rest_vecs) do > d:=multiply(vector(b),In_vecsj]-Rest_vecsk]) > if d=0 then t:=MinT else > t:=multiply(vector(a),Rest_vecsk]-In_vecsj])/d > fi > if t>0 and t<MinT then MinT:=t fi > od od od > RETURN(a+b*MinT) > end > > > igwalkInvolutiveWalk]:=proc(F::list(polynom),V::list(name), OStart::list(list),OEnd::list(list),L::procedure) > local vec,OVec,I,M,GB,NewG,fin,A,G,O,q > A:=poly_algebra(op(V)) > G:=F vec:=OStart1] fin:=false > while not fin do > if equal(vec,OEnd1]) then fin:=true fi > OVec:=termorder(A,'matrix'(NormVector(vec)],V)) > I:=map((g,OVec)->LeadingPolynom(g,OVec),G,OVec) > M:=MakeTermOrder(vec,op(OEnd)]) > O:=termorder(A,'matrix'(M,V)) > GB:=InvolutiveBasis(I,O,L)
1000
. .
> NewG:=expand(map(g-> > add(q,q=map(j->Gj-2]*gj],$3..nops(g)],G,g)),GB,G,g)) > NewG:=map(g->g,LeadingPolynom(g,O)],NewG,O) > G:=map(x->x1],IAutoReduce(NewG,O,L),O,L) > vec:=WalkVector({op(G)},V,vec,OEnd1]-vec) > od > RETURN (G) > end
6. %
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. # Maple-V.5 Pentium-166.
x2 y + z ; 1, y2 z + x ; 1, z 2x + y ; 1 ! 4 " #: x1 + x2 + x3 + x4 , x1 x2 + x2 x3 + x3 x4 + x4 x1 , x1 x2 x3 + x2 x3 x4 + x3 x4 x1 + + x4 x1 x2 ,
Janet
MaxN
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Janet walk 580 (464)
6
10 (3)
24
300
MaxN walk 83 (41)
GB
32 (13)
3
117
GB Walk 40 (20) 6 (1)
x1 x2 x3 x4
! 5 " #: x1 + x2 + x3 + x4 + x5 , x1 x2 + : : : + x5 x1 , x1 x2 x3 + : : : + x5 x1 x2 , x1 x2 x3 x4 + : : : + x5 x1 x2 x3 ,
> 7000
4730 2399 2032 (4518) (1981)
x1 x2 x3 x4 x5
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) ! * :
14
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2
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x2+ y ;2 z , 2 13 11 37 4 1.1 1.6 x3 + y3 ; z 3 , (4) (1) (0.2) x +y ;z , : : :, x7 + y7 ; z 7 D Janet MaxN . . , GB | # . #
1001
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,1] Buchberger B. An algorithm for .nding a basis for the residue class ring of a zero-dimensional polynomial ideal (German). | PhD Thesis, Univ. of Insbruck, Inst. for Math., 1965. ,2] Faug1ere J. C., Gianni P., Lazard D., Mora T. E2cient computation of zero-dimensional Gr3obner bases by change of ordering // J. Symb. Comp. | 1989. ,3] Collart S., Kalkbrener M., Mall D. The Gr3obner walk. | Dept. of Math., Swiss Federal Inst. of Tech, Z3urich, Switzerland, 1993. ,4] Mora T., Robbiano L. The Gr3obner fan of an ideal // J. Symb. Comp. | 1988. | Vol. 6. | P. 183{208. ,5] Zharkov A. Yu., Blinkov Yu. A. Involution approach to investigating polynomial systems. | Dept. of Math. and Mech., Saratov Univ.
' ( ) 1998 .
. .
e-mail:
[email protected]
514.76
: , , , !"#, !"# $.
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Abstract V. V. Konnov, On some reducibility condition for principal bundles and its application to projective geometry of submanifolds, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1003{1035.
In the present paper we prove some su4cient reducibility condition for principal bundles, which is adapted to study of submanifolds in homogeneous spaces. We apply the above condition to 5nd the di6erential-geometrical criterion of Segre manifolds.
, G P(M G ) H , E(M G=H G P ) s: M ! E = P=H (., , !1, . 1, # 5.6]). ' H , ( ) (** ) )+ . , , )+ , H G - . .( ). / G H ( #) , , ) , 2001, ! 7, 7 4, . 1003{1035. c 2001 !, "# $% &
1004
. .
2 ) . , , . 3 P )+ M ) P , P # ) P1, 2 , )+ M. ' - , P1 ) P , 4 M # P1 )+ , - ) + 2. , + P P1 ) . , 2 ) )+ . , ) : P ! P , 2 )+ , P1 = Im ) P. 5 , L(M) | )+ M, g | M, : L(M) ! L(M) | 7 {9, Im = O(M) | )+ g M. : +# ** - + ; . : , * * M P N ) rank mn, * ) ker ' )+ n m, dim M = n + m, N = m n + n + m, m > 1, n > 1, M | ; P n P m P N . , , M , M | ; . : )+ m n # , (n +m)- M, # P N , - + 2+ 2 #, )+ m n, rank mn, M | ; P n P m P N . . (, M , M | ; .
1. . (P M G ) | ) P, M, G . . #, 2 # : P ! P, 2 (a) = ( | #)= (b) 2 = ( | )= (c) Im = P1 | P. >, x 2 P , x 2 P1 () (x) = x. : ,
1005
(x) = x, x 2 Im = P1 . 3 , x 2 Im, x = (y) y 2 P. / (x) = 2(y) = (y) = x. 3, P1 P. : , fxk g P1, +2
x 2 P, ) +: ; x = lim (x ) = lim x = x: (x) = klim !1 k k!1 k k!1 k
; , x 2 P1 , P1 P. ? # H G, : H = fh 2 G j (xh) = xh & (xh;1) = xh;1 x 2 P1g: 5# , H | # G. ; , 1.1. H |
G. 3 # R: P G ! P #. . R1 = RjP1 H : P1 H ! P | # R P1 H . 3 , R1 | ) H P1. .( 1.2.
H P1. . 1 = jP1 | P1 . >, 1 : P1 ! M | . A , , 1(xh) = 1 (x) )+ x 2 P1 h 2 H . . : ) H P1 - - , ;1 (m) \ P1
m 2 M ( )+ x y 2 P1, 2+ 1 (x) = 1(y), 2 ( h 2 H , y = xh = R1(x h)). 1.1. (P M G ) | : P ! P |
, : (a) = (b) 2 = (c) Im = P1 | P . ! H = fh 2 G j (xh) = xh & (xh;1) = xh;1 x 2 P1 g |
G, P1. "
# H P1 - , (P1 M H 1 = jP ) |
(
G P %
H ). 1
1006
. .
. . ) ) 1.1 1.2. : # #. / # H , # ( ). ; , P1=H = P=G = M, P1
). . (, , 1 : P1 ! M | #. : P1 ) P1 )+ P . . U M | 2 P, s: U ! ;1 (U) | P. ? # s^ = s: U ! ;1 (U) \ P1 = 1;1 (U). / s^ | #, 1 s^ = ( i1) ( s) = ( s) = ( ) s = s = idM s^ | P1 . / . . ' D: ;1(U) ! U G | ** * P , 2 U M, D(x) = ((x) (x)) x 2 ;1 (U), 2 ** * Db : 1;1 (U) ! U H P1 : Db (x) = (1(x) ^(x)), ^(x) = ( s^ 1 (x));1 (x) x 2 1;1 (U). . . (L(M) M GL(n) ) | )+ M. , M g = h i, GL(n) O(n). . #, ( ) * # ) * 2 ) 1.1. . : L(M) ! L(M) | #, 2 . : ( (m fei g) 2 L(M), fei g | ) m 2 M, # : (m fei g) 7! (m fe0i g), 80 > e1 = p e1 = > he 12 e1i > > 0 2 = e2 ; he2 e01 ie01 = e =p > > < 2 h
23 2i e03 = p 3 = e3 ; he3 e02 ie02 ; he3 e01ie01 = > h
i 3 3 > > ::: > > > :e0n = ph
nn ni n = en ; hen e0n;1ie0n;1 ; : : : ; hen e01ie01: / , | ( 7 {9, 2 )
m ) #
. >, | #. 4 , , = . /
1007
, 2 = . . O(M) = Im | . : #, O(M) | L(M). F (x) = x, x 2 L(M), 2 Im, # g(x) = In , gij (x) = ij : (a) ; (a) L(M), - 4 O(M). ? # L(M) R n(n2+1) , + * yij = gij (x): (b) J fyij g | ) R n(n2+1) . / g | ) , dgij = gkj !ik + gik !jk + gijk!k
f!ij !k g | ) 1-* ) L(M). 3, , , rankfdgij (x)g = n(n2+1) x 2 L(M). , (b) | #, f ij g 2 R n(n2+1) | . . # , 4 ) (a) , # L(M). , O(M) | L(M). >, dimO(M) = dimL(M) ; n(n2+1) = n(n2+1) . , ) 1.1 H | K, O(M). M ) H , 1.1, ) H = fh 2 GL(n) j (xh) = xh & (xh;1 ) = xh;1 x 2 O(M)g:
/ g(xh) = hT g(x) h )+ x 2 L(M), h 2 GL(n), O(M) (a), 4 ) H ( 2 : H = fh 2 GL(n) j In = hT In h & In = (hT );1 In (h);1 g H | ( O(n) n. , , dimH = dimO(n) = n(n2;1) . 3 , ) O(n) O(M) - ). ; , e 1.1 (O(M) M O(n) jO(M )) | , GL(n) L(M) O(n). N #) ) ) 1.1 ) # +# ; .
1008
. .
2.
. # ) * ) -** , +) 4 #. 2.1. . . W | 2 N + 1, P N = P (W ) | #- , p: W n0 ! P (W) | ( ). ** P (W), , ** (p )a : Ta W ! Tp(a) P (W) # p
a 2 W n 0 (p )a (b) = (p )a (c) () b c (mod a), b c 2 Ta W = W. ; , ker(p )a = hai. J ha b : : : ci W, 2
a b : : : c. ? )+ + *
p )a i T W W (;! 0 ;! hai ;! Tp(a) P(W ) ;! 0 a = i | * . / Imi = ker(p )a = hai, . .( 2 ) * # ) ) Tp(a) P(W)
p(a) P (W) * - W=hai W hai. / , Tp(a) P(W ) = W=hai. , 4 ) # ( ) . 2.2. !" " . . F | + B = feug W, dimW = N + 1. . GL(N + 1) F Bg = B 0 = fe0u g, e0u = guv ev , g = (guv ) 2 GL(N + 1). 3 F 4 ( B s B 0 , B 0 = c B c 2 R n 0. / * -# F (P N ) = F =s )+ , * W # PGL(N + 1) = SL(N + 1)= . , # ( F =s # ) feu g, detfe0 e1 : : : eN g = 1: (1) . P (W) ) ) PGL(N +1) H
. , #2 H
H = 0 2 SL(N + 1)
1009
0 | )) N. , PGL(N + 1) = SL(N + 1)= = P N PGL(N + 1)=H= F (P N ) PGL(N + 1). T K p = pgl(N + 1) ) K PGL(N + 1) * sl(N + 1) )+ )+ N + 1. . ( K h, 2 H,
h = 0 A trace A + = 0 : A#) eu B 2 F (P N ) # W- * (eu : B 7! eu ) K PGL(N + 1). .( deu | ( W- 1-* PGL(N + 1). ? ) - B W, deu = !uv ev : (2) v M ) !u ) K PGL(N + 1) { d!uv = !uw ^ !wv (3) 0 1 N !0 + !1 + : : : + !N = 0: (4) F (3) (4) 4 ** (2) # (1). , # PGL(N + 1) F (P N ) (2), )+ 1-* ) !uv (3), (4), ) , (3), (4) | !3]. 2 * .
2.1 (%4, 5]). !uv | & # '{) -
# PGL(N +1) f1 f2 : M ! F (P(W)) | * M F (P (W)). + f1!uv = f2!uv * # ,# , , f1(x) = g f2(x) & g 2 PGL(N + 1) x 2 M .
7 H K PGL(N +1). ; (. !1, . 1, 5.1]), P = (PGL(N + 1) PGL(N + 1)=H H ) c ) PGL(N +1) F (P N ), PGL(N +1)=H P N ,
: PGL(N +1) ! PGL(N +1)=H H. 7 P )
P N . . # = p e0 : F (P N ) ! P N :
1010
. .
A# B = feu g 2 F (P N ) (
A0 = p(e0 ), #- e0 . ; H
A0 . / | V , ** (d)B ( * Tp(e0 ) (P N ) = W=he0 i. .( dim(Im(d)B ) = N. ; ), X 2 X(F (P N )) (d)B (X) = (dp)e0 (de0 )B (X) = = (p )e0 (!00 (X)e0 + !01 (X)e1 + !02 (X)e2 + : : : + !0N (X)eN ) = = !01 (X)~e1 + !02 (X)~e2 + : : : + !0N (X)~eN
e~i = (p )e0 (ei ) = he0 i+ei 2 Tp(e0 ) P N = W=he0 i, i = 1 2 : : : N. ; , 1-* ) !01 , !02 ,... , !0N ) Tp(e0 ) (P N ). X * ) ) P. Y ) ) ) .*** !01 = !02 = : : : = !0N = 0 P, #) )+ ** * H + , #2+ *
A0 = (B) P N . Y# !5], fe~ig Tp(e0)(P N ), #-) f!0i g, 2 . A#) e~i ( ) A0 Ai
A0 .
2.3. , - . . f : M ! P(W ) = P N | #
n- M P (W), f(M) = V | P (W). ? F # f W (, F : M ! W | ( #, 2 f = p F). K F 2 # f ( 2 ) fU g2A M, 2 A # f U W). : , # W ) RN +1 N- * S N )+ W. / N > 1 * S N ) )2 P (W), # p, S n , ). . N = 1 # S 1 ** * P (W). >, # f : M ! P(W ) 2 ( ) F * S N , # W. ' Fe : M ! W | f,
x 2 M ) e ) ) , #
F (x) F(x) e = (x) F(x), | * f(x) P (W). J , F(x) M, ( + 2 ). ,
1011
4 Fe = F , | * M. : # ) (U ) M # FU = F ;1 : ;1 (U) ! W, 2 ) F , ) V . : . ' V ( ) ) , ) + 4 , ) ) ( . 2.4. / 0 " . . f | # n- M P(W), F | * # f W . ; V =f(M)=p F(M) P(W) # )+ V #. . A |
V , f(x) = A F (x) = a. . f | ** * , Im(f )x = TA (V ) | n- TA (P(W)) = W=hai. ; ), F | ** * , (f )x = (p )a (F )x ker(p )a = hai, Im(F )x | n- Ta (W) = W, hai. .( WA1 (V ) = hF (x) Im(F )x i | (n+1)- W . ' Q = FU | V ,
@Q(u) @Q(u) @Q(u) 1 WA (V ) = hF (x) Im(F )x i = Q(u) @x1 @x2 : : : @xn (U ) | M, u = (x) f(x) = A. K , WA1 (V ) ) , # f. . WA1 (V ) - V
A. 3 p(WA1 (V )) = PTA (V ) WA1 (V ) # p ) V
A, * - W=WA1 (V ) = NA (V ) ) V
A. J, TA (V ) # # ) 2 : TA (V ) = WA1 (V )=hai. . F : M ! W W - * M. . X(M) | C 1 (M)- )+ M= x 2 M= X Y 2 X(M). / F | W - * M, X | ** ) + * , X(F ) = dF (X) | W- * M. T , Y (X(F)) | W - * M, - Y (X(F ))(x)
x 2 M ( W. ? #
1012
. .
(d2 F)x : X(M) X(M) ! W 2 . : )+ X Y 2 X(M) # (d2F )x (X Y ) = Y (X(F ))(x): K ,
A = f(x) = p(F (x)) WA2 (V ) = hF(x) Im(dF)x Im(d2F )xi W ) F ( ) # f. . WA2 (V ) - V
A. ' Q = FU | V ,
2 Q(u) @Q(u) @ 2 WA (V ) = Q(u) @xi @xj @xk i j k = 1 : : : n (U ) | M, u = (x) f(x) = A. . P TA2 (V ) = p(WA2 (V )) P N ) V
A. 3 # A : Tx (M) Tx (M) ! NA (V ) = W=WA1 (V ) * A (X Y ) = WA1 (V ) + (d2 F)x(X Y ) )
V (
f). , ( # @ 2 Q(u) A (X Y ) = WA1 (V ) + @xi xj X i Y j (U ) | M, Q = FU | V ( ), u = (x) f(x) = A. K , ) M ) * * ) , | # . / , * S M ( V ) ! N(V ) = NA (V ). "
;A2V
( rank A = dimP TA2 (V ) ; dimPTA1 (V ) = rank WA1 (V ) + @@xQi@x(uj) # f. A * '(X) = (X X) V , #- * , ) ! V . > ker ' * ) ' #
A 2 V 'A (X) = 0, ) !
A. Y# 2
1013
, ' 0 +) ) , V | n- !3,5].
2.5. 1 2- " .
. P N = P(W) | , P = (F (P N ) P N H ) | )+ P N , - 2.2. ' V = f(M) | n- , # P N , P(V ) = PjV P V # ) ) F (V ) = ;1 (V ), V , H Z = jF (V ) . 7 P(V ) = (F (V ) V H Z ) ) V . . f^: F (V ) ! F (P N ) | # . . 2 f^ * eu * ) !uv , -) F (P N ), F (V ). 3 uv = f^ !uv . : W- )+ * f^ eu = eu f,^ )+ F (V ), + # eu . 5 F (V ) ) (1){(4). M ) 01 , 02 ,... , 0N ) P(V ), - rankf01 02 : : : 0N g = dimV = n. ?- F (V ) - * ** V . , , = p e0 ) * * ) , -) 2.4, F . : , F | * W # f, W- ) * F e0 ) e0 = F, | * F (V ) M. ? ** # e0 # (d2e0 )B : X(F (V )) X(F (V )) ! W , (d2 e0)B (X Y ) = Y (X(e0 ))(B) B 2 F (V ) X Y 2 X(F (V )), -, WA1 (V ) = he0 (B) Im(de0 )B i | 2 TA (V ) = WA1 (V )=he0 i | WA2 (V ) = hWA1 (V ) Im(d2e0 )B i | 2 PTA (V ) = p(WA1 (V )) | PTA2 (V ) = p(WA2 (V )) | 2 NA (V ) = W=WA1 (V ) | A (X Y ) = WA1 (V ) + (d2e0 )B (X Y ) | * * V
A = A0 = p(e0 ). 2.6. !" 4" . . P n |u n c ) (X ), u = 0 1 : : : n= P m | m c ) (Y a ),
1014
. .
a = 0 1 : : : n= P N | N = n m+n+m c ) (Z ua ). (%6]). 3 # fs : P n P m ! P N , fs : ((X u ) (Y a )) 7! (X u Y a ) ) #, S nm = fs (P n P m ) P N | #. / , ; S nm - P N + Z ua = X u Y a : (5) n m . ) P P , ) 4 X u = Auv XZ v Y a = Bba YZ b , * ; (5), ) # S nm : Z ua = Auv Bba XZ v YZ b : vb ? P N ZZua = Gua vb Z . X - ; (5) , v b ZZua = Gua (6) vb X Y
) # ) ; . K ) ; ) 2) +2 ) P N . , , ua = C uDa , ; (5) (6) 4 Gvb v b # P N . ; , n 6= m ; S nm , #) P N , + ( * -# PGL(n m+n+m+1)=PGL(n+1) PGL(m+1) ( n = m 2 ) V2 , 2 + 2+). 2 ) * . 2.1. - . Snm SZnm ,
# , .
. (., , !1, . VI, x 6, 1]), f1 f2 | # M M 0 f1 f2 ) # M, M. F # ) , ) + ) ) , # ; | #. K . 3 # ; ) + P n, P m P N . \ ) , ) W1 W2, #2 P n P m ,
1015
W1 W2 | W1 W2 . / P N , 22 ; S nm , # W1 W2 , P N = P(W1 W2 ). / # ; # fs : P(W1) P(W2 ) ! P(W1 W2 ) : fs : (p(X) p(Y )) 7! p(X Y ) X 2 W1 Y 2 W2 : >, ; S nm )+ #)+ W1 W2 # p: W1 W2 ! P(W1 W2 ) = P N . Y ; S nm 4 ) PGL(n + 1) PGL(m + 1) P(W1 W2 ). ' fe0 e1: : :eng | W1 , f"0"1: : :"m g | W2 , #) ) eua = eu "a (7) W1 W2 . / , # ; fs V f^s : F (P n) F (P m ) ! F (P N ) f^s : (feu g f"a g) 7! feua g, eua = eu "a . >, F (S nm ) = Im(f^s ) | F (P N ). A#) B = feua g F (S nm ) ) P N 4 Aua = p(eua ), ; S nm - (5). A# 4 Aua ( # ; , # #) eu "a . ? 00 = p e00 : F(S nm ) ! S nm 2 # feua g F(S nm )
A00 = p(e00 ) ; S nm . X Ps ) F (S nm ), S nm Hs , v b v Hs = fh = (hvb ua) = (Cu Da ) 2 PGL(N + 1) j (Cu ) 2 PGL(n + 1) (Dab ) 2 PGL(m + 1) C01 = : : : = C0n = D01 = : : : = D0m = 0g: 5, ) PGL(N +1) F (P N ) 0 vb (feua g (hwc vb )) 7! feua = huaevb g:
1016
. .
: F (P N ), ) ) , (2){(4) ) vb evb deua = !ua (8) vb wc vb d!ua = !ua ^ !wc (9) ua !ua = 0 (10) 0 6 u v w 6 n, 0 6 a b c 6 m. 5- ) (8){(10) F(S nm ). : F (P n) deu = !uv ev (11) v w v u d!u = !u ^ !w !u = 0 (12) 0 6 u v w 6 n. T ) ) F (P m): d"a = ab "b (13) b c b a da = a ^ c a = 0 (14) 0 6 a b c 6 m. vb = (f^s ) !vb -) * ) 3 ua ua (f^s ) (7), 2 F(S nm ) F (P N ). (8), (11), (13), vb = !v b + v b : ua (15) u a u a , 2 * . 2.2. fs : P n P m ! P N | . , S nm = Im fs | . , (F (S nm ) S nm Hs 00) | #* nm Snm, f^s : F (Snm) ! F (P N ) |
F (S ) #* F (P N ) N P N , !uavb | 1-& # '{)
# PGL(N + 1) F (P ). ! vb = !v b + v b f^s !ua u a u a
!uv | & # '{)
# PGL(n + 1), ab | & # '{)
# PGL(m + 1). * * # ; . 2.3. / , . Snm P N & & n m, P N . . K , )+ 1-* (F (S nm ) S nm H 00) * ) !0i 0p , i = 1 2 : : : n, p = 1 2 : : :m. ? W1 W2 - * e00 ) - **
B = feua g 2 F(S nm ): (de00)B (X) = !0i (X)ei0 + 0p (X)e0p mod he00 i
1017
i = 1 2 : : : n, p = 1 2 : : : m, X 2 X(F (S nm )). ; , WA1 00 (S nm ) = he00 e10 : : : en0 e01 : : : e0m i | 2 . :, ) ** , (d2e00 )B (X Y ) = (!0i (X)0p (Y ) + !0i (Y )0p (X))eip mod WA1 00 (S nm ) i = 1 2 : : : n, p = 1 2 : : : m, X Y 2 X(F(S nm )). / , p p A00 (X Y ) = WA1 00 (S nm ) + (!0i (X)0 (Y ) + !0i (Y )0 (X))eip rank A00 = dimheip j i = 1 2 : : : n p = 1 2 : : : mi = n m dimWA2 00 (S nm ) = N WA2 00 (S nm ) = W1 W2 PTA2 00 (S nm ) = P N : . # . . fv1 : : : vn vn+1 : : : vn+m g | TA00 (S nm ), ) f!01 : : : !0n 01 : : : 0m g. ' X = X i vi + X n+p vn+p 2 TA00 (S nm ) Y = Y i vi + Y n+p vn+p 2 TA00 (S nm ), A00 (X Y ) = WA1 00 (S nm ) + (X i Y n+p + Y i X n+p )eip : ? * ': T (S nm ) ! N(S nm ), '(X) = (X X), 'A00 (X) = WA1 00 (S nm ) + 2X i X n+p eip :
2.4. 0 , & # . , , # # . . . . # . ( ). : M ! ]m(W) W| ^ :{z: : ^ W} | &&% m
M m-#* ,* W , mdimW > m. , , x m2 M (x) 2 ] (W ) # #, ] (W)-, &% &&% d = ()
| 1-& M . ! (x) | m- x 2 M , # P(W) (m ; 1)- . 2. . 4 ** # (), -, * . /
1018
. .
. | * . ; , 2 ) fU g M, * ) U ) ** : jU = d ln' , ' | # - * U . 3 # () U d ; d ln ' = 0. ? ( ; # ' , 4 d'';2d' = 0: ; , d ' = 0. , (x) = ' (x)a 8 x 2 U , a | ) m- ]m (W ). : x 2 U \ U ' (x)a = ' (x)a . .( m- ) a a # . , M , (x) | m- x 2 M, ) P (W) (m ; 1)- . K . . - # 2.4. . X 2 TA00 (S nm ). / X 2 ker ' () X i X n+p = 0 () X n+p = 0 X i = 0: , )+ X n+p = 0 X i = 0. 5 S nm )+ ^1 ^2. . - .*** 0p = 0, | !0i = 0. / dim^1 +dim ^2 = dim S nm , (^1 ^2) | S nm . , (12), (14) # (+ . . ( (11){(14) +, d(e00 ^ e10 ^ : : : ^ en0 ) = (n + 1)00 e00 ^ e10 ^ : : : ^ en0 mod f0a = 0g d(e00 ^ e01 ^ : : : ^ e0m ) = (m + 1)!00 e00 ^ e01 ^ : : : ^ e0m mod f!0i = 0g: .( , ), # ) , ) ^1 ^2 p(he00 e10 : : : en0i) p(he00 e01 : : : e0m i), +2 #
A00 = p(e00 ) ; S nm . . # . , ; S nm $ . \ #
+ 2 #, #2 S nm , )+ ; . ' ; (5),
Z0 = (X0u Y0a ) + n- Z = (X u Y0a ) m- Z = (X0u Y a ). X 2
Z0 , +
) ) S nm
Z0 .
2.7. 4 00 - " - " . . M | -
m + n (^1 ^2), ^1 ^2 | + )+ M, dim^1 = n, dim^2 = m. (., , !2]), ( GL(n + m) )+ L(M) -
1019
GL(n) GL(m). . #, M # P N , f : M ! P N | # #. >, ( (f ^1 f ^2) V = f(M). . P(V ) = (F (V ) V H ) | )+ V . 5, H |
PGL(N + 1) P N . 2.2. .
H P(V ) %
G0, 8 0 9 g00 (gi0 ) (ga0 ) (g0 )1 > > < B 0 (gij ) 0 (gj )CC = : G0 = >h 2 H h = B @ 0 0 (gab ) (gb )A > : 0 0 0 (g ) 1 i j = 1 : : : n, a b = n + 1 : : : n + m, = n + m + 1 : : : N . . 7 G0 K H. .( !1, . 1, # 5.6], E(V H=G0 H F (V )) P(V ) = (F (V ) V H ) ) G0 s: V ! E = F (V )=G0: N ' P = P (W), E = F (V )=G0 ) (_1 = _2), _1 , _2 | W, 2 dim_1 = n + 1 dim_2 = m + 1 dim(_1 \ _2 ) = 1 p(_1 \ _2 ) 2 V: . F : M ! W | # f : M ! P(W) W . 3 # s: f(U) V ! E = F (V )=G0 2 . :
A = f(x) 2 f(U) V , # s: A 7! (_1 = _2) 2 E _1 = hF(x) (F)x ^1 i _2 = hF (x) (F)x ^2i: (16) 3 # s ) F. ; , s | E. K . . P0 (V ) = (F0 (V ) V G0 ) P(V ), 2 (16), ) V ,
GL(n) GL(m)- V . / F0 (V ) ( ) B = fe0 e1 : : : eN g, )+ (B) = p e0 (B) = p(e0 ) = A0 2 V (f ^1 )A0 = (p )e0 (he1 : : : eni) (f ^2)A0 = (p )e0 (hen+1 : : : en+m i):
1020
. .
5# ) 4 (n+m)- ) P N N = n m + n + m, n m 2 N. F 2 " 6 . 1. ) 2 : 0 6 u v w 6 n 0 6 p q r 6 m 1 6 i j k l t 6 n 1 6 a b c d 6 m: 2. N) F (P N ) eup , )+ (u p). 3. , 4 1 2 () ) PGL(N + 1) vq ). , , ( + - ) # (gup ( uv pq ) | PGL(N +1). . ( ) PGL(N +1) ( - ) + )+ 0 vq (feup g (hwr vq )) 7! feup = hup evq g: 4. , ) H P P(V ) )+
A00 = p(e00). / , i0 0a ia H = fh = (hvq up ) 2 PGL(N + 1) j h00 = h00 = h00 = 0g: 5. , 3 4 + P P(V ) ) # = 00 = p e00. 6. , ) 2+ 4 (1){(4) F (P N ) 4 detfeup g = 1 (17) vq evq deup = !up (18) vq wr vq d!up = !up ^ !wr (19) up !up = 0: (20) . f : M ! P N | # 2 P N , dimM = n + m. . #, (^1 ^2) | M, dim^1 = n, dim^2 = m. , ) 2.2 P0(V )=(F0 (V ) V G0 =00) )+ , )+ GL(n) GL(m)- V , # f ^1 f ^2. , ) G0 P0(V ) ) )+ he00 e10 : : : en0i he00 e01 : : : e0m i. / , 8 0g (g ) (g ) (g )1 9 > i a kc > < j ) 0 (gj )C = B 0 (g vq i B G0 = h 2 H h = (gup) = @0 0 (gb ) (gkc : b )C A> a kc > : ld ) 0 0 0 (gkc
1021
00 = g, g00 = gi, g00 = ga , g00 = gkc, gj 0 = gj , J ) g00 0a i0 i0 i kc j 0 = gj , g0b = gb , g0b = gb . vq ) = 1 , gkc det(g a kc up kc 0a kc , g 6= 0, (gij ) | ) # n- , (gab ) | ) # m- , (giajb) | ) # n m- . vq = f^ !vq | . f^: F0 (V ) ! F (P N ) | # #. / up up 1-* ) F0 (V ). F -) * f^ eup = eup f^ eup . / f^ vq )+ * eup 4 ** , * up ) (17){(20). A , (f ^1 )A00 = (p )e00 (he10 : : : en0i) (f ^2)A00 = (p )e00 (he01 : : : e0m i) 00 e + i0 e + 0a e + ia e de00 = 00 00 00 i0 00 0a 00 ia F0 (V ) ) ** )+ ia = 0: 00 (21) M ) i0 a = 0a !i = 00 00 ) )+ 1-* P0(V ). . ( f ^1 f ^2 V .*** 1 = : : : = m = 0 !1 = : : : = !n = 0: K , K g0 ) K G0 80 1 9 > >
B =
+ trace A + trace B + trace C = 0 > :@00 00 B0 C A 2 R, A | n- , B | m- C | n m- . / ) K G0 F0(V ) ), 1-* ) f0000 i000 ij00 000a 00ab ia00 iaj0 ia0b iajbg ) ) ) 4 (20), 00 + i0 + 0a + ia = 0: 00 i0 0a ia 3 , , 1-* ) f!i a i000 ij00 000a 00ab ia00 iaj0 ia0b iajbg ) F0 (V ) C 1 (F0(V ))- 1-* F0(V ).
1022
. .
:** 4 (21) 4 A , ja b k ija0 = Aja (22) ik ! + Aib ja ja ja k c 0b = Abk ! + Abc (23) ja ja ja ja ja Aja ik = Aki , Abc = Acb , Aib = Abi | * Fi 0 (Va ). . fi ag | TA00 (V ), ) f! g. J,
+ # TA00 (P N ) = W=he00 i, P(W ) = P N , # i = he00i + ei0 a = he00i + e0a : i . X = X i + X a a Y = Y i i + Y a a | )+ TA00 (V ). , (18), (21) de00(X) = X i ei0 + X a e0a mod he00i: .( WA1 00 (V ) = he00 e10 : : : en0 e01 : : : e0m i 2 V
A00. :, (18), (22), (23) + 2 i j kc i a i a kc a b 1 d e00 (X Y ) = (Akc ij X Y + Aia (X Y + Y X ) + Aab X Y )ekc mod WA00 (V ): ; , 1 d2e00 (i j ) = Akc ij ekc mod WA00 (V ) 1 d2e00 (i a) = Akc ia ekc mod WA00 (V ) 1 d2e00 (a b) = Akc ab ekc mod WA00 (V ): kc kc kc kc kc / , Akc ij = (i j ) , Aia = (i a) , Aab = (a b) | ( ) * * ) # f ( fi ag TA00 (V ) fe~kc g NA00 (V ) = W=WA1 00 (V ), e~kc = WA1 00 (V ) + ekc). ? * V : i j kc i a kc a b '(X) = (Akc ij X X + 2Aia X X + Aab X X )~ekc: > ker ' ( * ) TA00 (V ) i j kc i a kc a b (24) Akc ij X X + 2Aia X X + Aab X X = 0: nm : ; S (24) )+ n m (. # 2.4).
2.8. S(n m)- 00 " - " . . f | # (n + m)- MNc -
(^1 ^2) P N = n m + n + m. J dim^1 = n dim^2 = m. . , , ' | * * * # f.
1023
. ; (f^1 f^2) V = f(M) - S(n m)-, f ^1 f ^2 ker '. S(n m)- -
, rank n m. X , # f ^1 f ^2. >, ; S nm P nm+n+m - ) # S(n m)- . 2.3. 2# S(n m)- #* n m. . , + 2.7 S(n m)- ) : fi g | f ^1 ker ', fag | f ^2 ker '. ; , kc kc kc Akc ij = (i j ) 0 Aab = (a b) 0: , ( (22), (23) b ija0 = Aja (25) ib ja ja 0b = Abi !i : (26) , * * # f 4 i a i a (X Y ) = Akc ia (X Y + Y X )~ekc: ; , rank = dimhAkc ia e~kc i. , fe~kcg rank n m +, (Akc ia ) | ) # kc n m- . 3 (AZkc ia ) (Aia ), jb j b Akc ia AZkc = i a : 1. : #, f ^1 . . 4 ** (25). )+ ia , ja , ja 4
2 # (21), (25), (26) * ) 00 i0 0b 4+ )+ A . , - 4 ia 00 kc ia ia k0 ia 0c ia c ia k dAia (27) jb + Ajb00 + Ajb kc ; Akbj 0 ; Ajc0b = Ajbc + Ajbk ! ia 0 b ia 0 b ia b ia l ;Ajbk0 ; Akbj0 = Ajkb + Ajkl! (28) fAiajbc Aiajbk Aiajklg | * F0(V ), ) ) ( , 2 # ), 2 . (28) + ; jt k0c0 ; kt j00c = AZtciaAiajkbb + AZtciaAiajkl!l : (29) . (29) - j t, jc ia l ia b k0c0 = ; n +1 1 (AZjc ia Ajkb + AZia Ajkl! ):
1024
. .
/ ,
i00a = Aaib b + Aaij !j Aa ij ] = 0 (30) a a Aib Aij | * F0 (V ). (30) (19) , , 00 ^ a + b ^ 0a + Aa !i ^ b : da = 00 (31) 0b ib ; , .*** a = 0 , f ^1, ( , . 2. T , ** (26), 00 + Akc ia ; Aia k0 ; Aia 0c = Aia !l + Aia c dAiabj + Aiajb00 (32) jb kc kb j 0 jc 0b bjl bjc j 0 j 0 ia ia ia j ia d ;Ajb0c ; Ajc0b = Abcj ! + Abcd : (33) ; , 0i0a = Aiaj !j + Aiab b Ai ab] = 0 (34) Aiaj Aiab | * F0(V ). 00 ^ !i + !j ^ i0 + Ai b ^ !j : d!i = 00 (35) j0 bj .( .*** !i = 0 , f ^2 , ( , . , ) # S(n m)- )+ n m. K . 2.4. " n m > 1, # f ^1 f^2 # S(n m)-# V , V 3 *
*.
. (C. # !3].) a
1. : #, Aij = 0 +) ) , ) ) ) f ^1 ) n- ) . : , Aaij = 0 ( (. # # 2.4) de00 ^ e10 ^ : : : ^ en0 = fe00 ^ e10 ^ : : : ^ en0 mod fa = 0g () 1-* ) f. : , a = 0 00 + i0 )e00 ^ e10 ^ e20 ^ : : : ^ en0 + de00 ^ e10 ^ : : : ^ en0 = (00 i0 + Aa1j !j e00 ^ e0a ^ e20 ^ : : : ^ en0 + Aa2j !j e00 ^ e10 ^ e0a ^ : : : ^ en0 + : : : + + Aanj !j e00 ^ e10 ^ e20 ^ : : : ^ e0a : / (n+1)- ), +2 ( , ), () ( Aaij !j = 0. , 1-* !j ) Aaij = 0.
1025
2. T ), Aiab = 0 +) ) , ) ) ) f ^2 ) m- ) . 3. / #, n > 1 m > 1, Aaij = 0 Aiab = 0. ) (28), (30) 1-* ) i00a )+ * , Aiajb Abkl + AiakbAbjl + Aia (36) jkl = 0 ia b ia b ia AjbAkc + AkbAjc + Ajkc = 0: (37) ia Ab ; Aia Ab = 0. T (36) k l, - Akb jl lb jk tc t c Z ;- Aia 4 k Ajl ; ltAcjk = 0. ; , (n ; 1)Acjk = 0. / , n > 1, Aajk = 0. T ), m > 1 4 Aiab = 0. K .
4 . " # #
S(n m)-# ( #* n m)
V , c ia ia k Aiajkl = 0 Aiabcd = 0 Aiajkb = ;2Aia c(j Ak)b Abcj = ;2Ak(bAc)j : 4 F0(V ) # ia = 0 ja = Ajab ja = Aja !i 00 (38) i0 ib 0b bi 0 c ia k c ia c k ia ia 00 kc ia ia k 0 ia dAjb + Ajb 00 + Ajb kc ; Akbj 0 ; Ajc0b + 2Ak(bAc)j + 2Ac(j Ak)b! = 0 (39) 0 a a b i 0 i j i0 = Aib 0a = Aja ! : (40) 2.1. (n+m)- V , P nm+n+m, -
: 1) , , V * , #* n m 2) & & # n m , V . ! V nm. Snm . / , , V , V | . S .
. ) , V ) # S(n m)- , ) 2. ; , vq ) * eup , -) 1-* ) up F0 (V ) P0 (V ), )
1026
. .
detfeup g = 1 (41) vb deua = ua evb (42) vb wc vb dua = ua ^ wc (43) ua ua = 0 (44) i i 0 a 0 a (38){(40). J * ) ! = 00 = 00 )+ 1-* P0(V ), * ) f!i a i000 ij00 000a 00ab ia00 iaj0 ia0b iajbg C 1 (F0 (V ))- 1-* F0 (V ). . ** 4 (40) 4 A . , 00 + Ak i0 ; Ai k0 ; Ai 0c + Akci0 ; i 00 = dAija + Aija 00 ja k0 ka j 0 jc 0a ja kc j 0a i k = Ajak ! ; Aika Akjcc Aijak = Aikaj (45) 00 + Ac 0b ; Ab k0 ; Ab 0c + Akc 0b ; b 00 = dAbja + Abja 00 ja 0c ka j 0 jc 0a ja kc a j 0 = Abjac c ; Abjc Acka!k Abjac = Abjca: (46) J Aijak Abjac | * F0 (V ). : 4 2 1.1, #) ) G0 P0(V ). : 6 (. 1. . + P1(V ) = (F1(V ) V G1 ). ? # : F0 (V ) ! F0 (V ), : B = feup g 7! (B) = B 0 = fe0up g, (0 e00 = e00 e0i0 = ei0 e00a = e0a (47) j b e0ia = (Ajb ia ejb + Aia ej 0 + Aia e0b): J 2 # ;1 = j det(Avq (48) up )j (n+1)(m+1) ja i0 0a ia 00 0a 00 i0 ib A00 00 = 1, A00 = A00 = A00 = Ai0 = Ai0 = Ai0 = A0a = A0a = A0a = j 0 j j 0 j 0b b 0b b = A00 ia = 0, Ai0 = i , A0a = a , Aia = Aia , Aia = Aia . jb / (Aia ) | ) # n m- , B 0 = (B) | ( ) . J, det(Avq up ) | ( * F0 (V ),
. ; , det(Avq up ) + # . .( j det(Avq ) j up | ( # -) * F0(V ). A , , det(Avq up ) detfe g = 1: detfe0up g = (n+1)(m+1) det(Avq ) det f e g = up up j det(Avqup)j up
1027
: B 2 F0(V ) : B 7! B 0 +
A00 = p(e00 ) he00 e10 : : : en0i he00 e01 : : : e0mi. ; , # ) ) ( ) G0. , | ( # F0(V ) F0 (V ). j b 3 , | #, Ajb ia , Aia , Aia | * F0(V ). . , # (a), (b) (c) ) 1.1. (a) / (B 0 ) = p(e000 ) = p(e00 ) = p(e00) = (B), = . (b) : #, 2 = . . vq e0 : de0up = Zup (49) vq jb j M Aia , Aia Abia AZjb ia , jb , Aj (B 0 ) = AZj Ab (B 0 ) = AZb . 0 Z AZjia AZbia . / # Ajb (B ) = A ia ia ia ia ia ia :** 47) 38), (40) (49), -, ia = 0 Zja = j a Zja = a !j Z0a = 0 Zi0 = 0: !Z i = !i Za = a Z00 b i0 0a i0 i 0b ; ( (38), (40), + ) , j b j b AZjb ia = i a AZia = 0 AZia = 0: 3, # (47), (48) , (B 0 ) = 1 (B 0 ) = B 0 . ; , B 2 F0 (V ) 2 (B) = (B 0 ) = B 0 = (B). , 2 = . (c) 3 F1 (V ) = Im #, F1 (V ) | F0 (V ). >, # F1(V ) F0 (V ) j j b b (50) Ajb ia (B) = i a Aia(B) = 0 Aia (B) = 0: jb j / Aia , Aia , Abia | * F0(V ), ,
(50) . ?j b ** fdAjb ia dAia dAiag. K , ( . : , ( # ) (39), (45) (46) ) 1-* fiajb iaj 0 ia0bg ) # ) (Ajb ia ) F0(V ). , * F1(V ) | , # F0(V ). . G1 = H | K G0, - 1.1. , 4 G1 : jb j ;1 j b j ;1 G1 = fh 2 G0 j Ajb ia (Bh) = Aia (Bh ) = i a Aia (Bh) = Aia (Bh ) = 0 Abia (Bh) = Abia (Bh;1 ) = 0 B, 2 (50)g: 5, 8 0g (g ) (g ) (g )1 9 > i a kc > < j ) 0 (gj )C = B 0 (g vq i B G0 = h 2 H h = (gup) = @0 0 (gb ) (gkc b )C A> a kc > : ld ) 0 0 0 (gkc
1028
. .
vq ). ; ) g det(gij ) det(gab ) det(giajb) = 1. . h;1 = (^gup , ) B 7! Bh ( h 2 G0 2 * ) +: k c jb ld Ajb ia (Bh) = g^ gi ga g^ld Akc (B) a gc Akd (B) Aaib(Bh) = g^ gi ba + gij gbc g^da Adjc(B) + gij g^kd b jc i i b k i l b i Aja(Bh) = g^ ga j + ga gj g^l Akb (B) + ga g^lc gjk Alckb(B):
(51) (52) (53)
;1 k c jb ld Ajb ia (Bh ) = g g^i g^a gld Akc (B) a g^c Akd (B) Aaib (Bh;1 ) = g g^i ba + g^ij g^bc gda Adjc (B) + g^ij gkd b jc i ; 1 i b k i l b i Aja(B h ) = g g^a j + g^a g^j gl Akb(B) + g^a glc g^jk Alckb(B):
(54) (55) (56)
T , B 7! Bh;1
(51){(56) , , G1 ) G0 # jb g^ g a + gj g^a gc = 0 g^ g i + gb g^i gk = 0 ij ab = g^ gik gac g^kc i b a j a kb j i jc b j jb j b k c a a c i i g^k = 0: i a = g g^i g^a gkc g g^i b + g^i gjc g^b = 0 g g^a j + g^ab gkb j
(57) (58)
: #, G1 F1 (V ) - . . B1 B2 2 F1(V ) (B1 ) = (B2 ). / 2 ( h 2 G0 ,
B2 = B1 h. Y) #) , h 2 G1. / ( ) . / B1 B2 2 F1 (V ), (50) + jb j b j j b b Ajb ia (B1 ) = Aia (B2 ) = i a Aia (B1 ) = Aia (B2 ) = 0 Aia (B1 ) = Aia (B2 ) = 0: , ( B2 = B1 h * + (51){(56) +, h h;1 # (57), (58), 2 G1 . ; , h 2 G1. , ) 1.1 , P1(V ) = (F1 (V ) V G1 ) | P0 (V ). \ ) # 4+ , # # F1 (V ) F (P N ) - # f.^ F -) vq -) * f^ eup vq * ) f^ !up up eup . 2. )+ P1(V ). . # , ) # (57) (58), 2+ G1, ( ) ) )) . N , #) ) , , vq ) h;1 = (^gvq ) ) G1 G0 , #, )2 () h = (gup up
1029
g^g = 1 gik g^kj = ij gac g^cb = ab giajb = g^gij gab g^iajb = g^gij g^ab giba = gba gi g^iba = g^ba g^i gibj = gij gb g^ibj = g^ij g^b g^jbgij gab ; 2gi ga + gia = 0 gjbg^ij g^ab ; 2^gi g^a + g^ia = 0 gi g^ + gij g^j = 0 g^ig + g^ij gj = 0 ga g^ + gab g^b = 0 g^a g + g^ab gb = 0: 5 F1(V ) ) (41){(44), (38){(40), (45), (46) ia = 0 ja = j a ja = a !j 0a = 0 i0 = 0 00 (59) b i0 0a i0 i 0b ia i a 00 a i 0 i 0 a jb + j b 00 ; b j 0 ; j 0b = 0 (60) b 00 b c i 0 i 00 i k 0 b ja ; j 0a = Ajak ! ja ; a j 0 = Ajac (61) Aijak = Aikaj Abjac = Abjca : :** (60) (59), (61), )+ Aijbl da + Aijdl ba = Aabld ji + Aajbd li . . 2 - ( ) - Aijak = ji ak + ki aj Abaic = ab ic + cb ia ia = n +1 1 Ajjai = m 1+ 1 Abiab : / ** )+ (61) i0 ; i 00 = ( i ak + i aj )!k 0b ; b 00 = ( b ic + b ia )c : ja (62) j 0a j k ia a i0 a c :** (62), -, * ia ** )+ 00 + ja j 0 + ib 0b : d ia = ia00 ; ia 00 (63) 0a i0 , # )+ P1(V ) (41){(44), (59), (60), (62), (63). : ), - **
)+ * F1 (V ). . , ) G1 ) F1 (V ), # , * ) f!i a i000 000a ij00 00ab ia00g C 1 (F1 (V ))- 1-* F1 (V ). 3. . + P2(V ) = (F2(V ) V G2 ). ; 1.1. ? # : F1 (V ) !F1(V ), : B = feupg 7! (B) = B 0 = fe0up g, (0 e00 = e00 e0i0 = ei0 e00a = e0a (64) e0ia = eia ; 2 ia e00: 3 , | #. . , # ) 1.1.
1030
. .
(a) / (B 0 ) = p(e000 ) = p(e00 ) = (B), = . (b) : #, 2 = . . B = feup g 2 F1(V ), (B) = B 0 = fe0up g, 0 vq e0 , ia (B 0 ) = Zia . :** (64), -, , deup = Zup vq Ziaj 0 ; ij Z000a = 0 Ziab0 ; ab Z000i = 0: ; ( (62), + ) , Z ia = 0. 3 , (B 0 ) = B 0 . ; , B 2 F1(V ) 2 (B) = (B 0 ) = B 0 = (B). , 2 = . (c) : #, F2(V ) = Im | F1(V ). >, F2(V ) F1(V )
ia (B) = 0: (65) ; ** fd ia g , F1(V ) 1-* ia00 , +2+ (63). ; , (65) | F1 (V ), # - 4 F2(V ) | , # F1(V ). . G2 = H | K G1, - vq ) 2 G1 * ) + 1.1. : ( h = (gup
ia (Bh) = g^ gij gab jb (B) + gia ; g^ gi ga (66) j ; 1 b
ia (Bh ) = g g^i g^a jb (B) + g^ia ; g g^i g^a : (67) ; , G2 G1 # gia ; g^ gi ga = 0 g^ia ; g g^i g^a = 0: (68) >, G2 F2(V ) - . : , B1 B2 2 F2 (V ) (B1 ) = (B2 ), 2 ( h 2 G1 ,
B2 = B1 h. : #, h 2 G2. / B1 B2 2 F2(V ), (65) +, ia (B1 ) = ia (B2 ) = 0. .( B2 = B1 h * + (66){(67) , h h;1 # (68), 2 G2. / , G2 F2(V ) - . , ) 1.1 G1 P1(V ) G2. . P2(V ) = (F2 (V ) V G2 ) | 2 . 4. )+ P2(V ). T #, 2 G2 P2(V ), + ) , K G2 2 : 01 (g ) (g ) (g g )1 9 8 > B ji a kj c C > < 0 (gi ) 0 (gk gc )C = : vq G2 = h = (gup ) 2 PGL(N + 1) h = B > @0 0 (gab ) (gkl gcbb)A > : 0 0 0 (gk gc )
1031
.( K G2 * Hs Ps ; S nm P N (. 2.6 # 2.2). , G2 = Hs . 3 # # F2(V ) F (P N ) - # vq * f^ eup , -)+ F2 (V ), f.^ : * f^ !up vq eup . , )4# F2(V ) + # up vq * ) up .*** ia = 00 = 0a = i0 = 0 00 (69) ia i0 0a ia i a 00 a i 0 i 0 a jb + j b 00 ; b j 0 ; j 0b = 0 (70) ja j ja a a j i 0 i 00 0 b b 00 i0 ; i = 0b ; b ! = ja ; j 0a = ia ; a i0 = 0 (71) i 0 0 a 00 (m + 1)i0 + (n + 1)0a + (1 ; n m)00 = 0: (72) J, (72) ) (44) (70). ,- : 80 = 1 (00 + i0) 8!0 = 1 (00 + 0a) > > 0 00 0 a m +1 >!0 = 00 >00 = n00+1 00 i0 < < i i0 > aa 00aa (73) i = i0 > ! 0 = 00 0 00 > > :!j = j0 ; 1 j (00 + k0) :b = 0b ; 1 b (00 + 0c): a 0a m+1 a 00 0c k0 i i0 n+1 i 00 , )+ + (69){(72) 4 vq = !v q + v q up (74) u p u p u p !u = 0 p = 0: (75) 5, 0 6 u v 6 n 0 6 p q 6 m. :** 4 (74) (43), d!uv pq + uv dpq = (!uw pr + uw pr ) ^ (!wv rq + wv rq ): . , - (d!uv ; !uw ^ !wv ) pq + uv (dpq ; pr ^ rq ) = 0: ;- ) ( p q u v (75), d!uv = !uw ^ !wv dpq = pr ^ rq : (76) , , F2 (V ) vq = f^ !vq , !$
F (P N ), P2(V ) 1- up up v (74), !u | { PGL(n + 1), pq | { PGL(m + 1). 5. : #, V | , 2 ), ( (
1032
. .
; ,
x 2 V 2 - VZ , 2 ; . :
x 2 V 2 - VZ , ** * M P n P m. . F : M ! VZ | 2 ** *, i: V ! P N | # . / f = i F : M ! P N | #, f(M) = VZ V | V . ; ), fs : P n P m ! P N | # ; , fs (M) = SZ S nm | ; S nm . . P2(V ) Ps | ) V , S nm . N 2 # , VZ | 2 P2(V ), SZ | 2 Ps . . : M G2 ! F2(VZ ) | ** * P2 (V ). / f^ = j1 : M G2 ! F (P N ) | #, ^ G2 ) = F2 (VZ ). J j1 : F2 (VZ ) ! F (P N ) | # . f(M / G2 = Hs (. 4 4), # # M G2 M Hs . . Z | ** * s : M G2 ! F (S) Ps. / f^s = j2 s : M G2 ! F (P N ) | #, Z J j2 : F(SZ) ! F (P N ) | # . f^s (M G2) = F (S). , # f fs : M ! P N M P N , ) # f^ f^s : M G0 ! F (P N ) M G0 )+ F (P N ). J G0 | ) G2 = Hs , #2 . vq | * ) Y {A ) K PGL(N + 1) F (P N ). . !up . (. 4 4 # 2.2), vq = f^ !vq = !v q + v q f^ !up u p u p s up !uv | * ) Y {A ) K PGL(n + 1), pq | * ) Y {A ) K PGL(m +1). . # 2.1, # ^ G0 ) F 0(S) Z = f^s (M G0 ) , F20 (VZ ) = f(M P N : Z g 2 PGL(N + 1): F20(VZ ) = g F 0(S) / ) PGL(N + 1) F (P N ) ( ) P N = 00 = p e00 # Lg = Lg , Z = g ((F 0(S)) Z = g S Z VZ = (F20 (VZ )) = (g F 0(S)) Z , VZ g SZ VZ = g S. ; g S nm . / , VZ V ; .
1033
6. / #, V | , 2 ), V | ; . J*
x V . . 2 VZ V (
, 2 ; S. 3 U # V , 2 + ,
) # S . Y# U ( x 2 U) ) ( V ))+ #). : #, . . y 2 V |
# U. ; 2 Y
y, ; S 0 . / y 2 V |
# U, Y # + )
z U. 5
z # S Z. 5 ) # Y \ Z # ; S S 0 . / 2.1 ; S S 0 . .( Y
y # S. ; , y 2 U. , # U V # )
, # U V . /
V , )- # U V . ; ,
V # S ) . 3 , V | ; S. , , V , V S, S , V S . / . , (n+m)- V P nm+n+m
n m
n m, V | # S nm , . 4 . (n + m)- V P nm+n+m =n+mP (W) # Z : M ! W , M | R . " , (xi ya ) 2 M (1 6 i j k 6 n, 1 6 a b c 6 m) # @ 2 Z = k @ Z + Z @ 2 Z = c @ Z + Z (77) @xi @xj ij @xk ij @ya @yb ab @yc ab Z Z @2Z = (n + 1)(m + 1) rank Z @x i @xa @xi @ya
kij , ij , nmcab, ab | &% M , V | . S . . , ) e00 = Z, ei0 =
@Z @xi ,
@ Za eia = @i Z a e0a = @x F (V ). , fe@xupg@y * * ( ) = he00 ei0 ea0 i + dxi()dya ()eia : 2
1034
. .
; , rank = m n. A , , 2 ** )+ : de00 ^ e10 ^ : : : ^ en0 = iij dxj e00 ^ e10 ^ : : : ^ en0 mod dya = 0 de00 ^ e01 ^ : : : ^ e0m = aab dyb e00 ^ e01 ^ : : : ^ e0m mod dxi = 0: . # V + 2+. 5 2.1 V | ; S n m . ; . 2.3, 2.4 ) 2.1 , n > 1 m > 1 2.2. f : M ! P N | M P N , dimM = n + m, N = n m + n + m, n > 1, m > 1, ' | & & , & f . 1) ker ' = ^1 ^2, ^1 ^2 | # n m, 2) rank n m. ! V = f(M) | . Snm . 4 M , V | . . . . n > 1 m > 1 ) 2.1 # , (77) @ 2 Z = a @ Z + k @ Z + Z @ 2 Z = i @ Z + c @ Z + Z: (78) @xi@xj ij @ya ij @xk ij @ya @yb ab @xi ab @yc ab : , ( ( #+ +) ( ) = he00 ei0 ea0i + dxi()dya ()eia rank = m n '() = he00 ei0 ea0i + dxi()dya ()eia ker ' = ^1 ^2 ^1 ^2 | ) n m, ) dya = 0 dxi = 0. .( ) 2.2 V | ; S nm , ) )+ ^1 ^2 2. 3 # ), * aij iab, +2 (78), ) , ) + (77). , n > 1 m > 1
S(n m)- # S nm . J, + ) n m , V , 2 ) # S(n m)- , ) ; . , ) 2.3 ( S(n m)- , ) 2 ) 2. 5 , n = m = 1 + V P 3 ( + ) # - *
1035
* ) - S(1 1)- . . ( V + ; S 11 ( ) # ) 4 , - * : . / , + 4 * , ) #)+ + P 3 4 ( + ; S 11 ) #) ( - )+ 2+). / 2.1 2 ( * .
1] ., .
. I. | ".: $, 1981. 2] . ()) )
. | ".: $, 1986. 3] Akivis M. A., Goldberg V. V. Projective di-erential geometry of submanifolds. | Amsterdam, London, New York, Tokyo: North-Holland, 1993. 4] Gri/ths P. On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in di-erential geometry // Duke Math. J. | 1974. | Vol. 41, no. 4. | P. 775{814. 5] Gri/ths P., Harris J. Algebraic geometry and local di-erential geometry // Ann. Sci. E5 cole Norm. Sup., 4 serie. | 1979. | Vol. 12. | P. 355{452. 6] 6 7. 8. 6 $
. I. | ".: $, 1988. 7] 9. 9. : $ ; $ 6 $ ; . | <: <(=>, 1998.
' 1999 .
C (X ) . .
. . .
517.986.225
: , , !" # .
$ " %##& #%' ( % " # ' % (', % . $(%')#%' ' & ' ( % ' %#& %#& !" # . * ( ' ' % (% % + . $(%')#%' ( % %# " , %%# .
Abstract A. V. Kuzmin, Homological properties of algebra C(X) in non-Archimedean analysis, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1037{1046.
In this article the author o2ers a variant of homological theory in non-Archimedean analysis and computes the left global dimension and bidimension of an algebra C(X) of all continuous functions on an ultrametrizable compact set X. To the moment the analogous problem in Archimedean analysis is still not solved. More generally, the author computes the homological dimensions of ideals I C(X) as left modules and bimodules.
x
1.
X K ( . ). "
#1, . 175{180]. , - . , / 0 , . . , 2001, # 7, 3 4, %. 1037{1046. c 2001 !, "# $% &
1038
. .
23 . 4 5 3 , /
. 6 , . 0, , 1, .
. 7 - I C (X )
. , - - P X #3, . 208{210]. 9 I . : . 6 , X n P . ;
X | , C (X ) | X . = , . . , : (x z ) 6 maxf(x y) (y z )g x y z 2 X: ?
C (X ) X K . 9 X . . ;
x1 x2 2 X / f (x) 2 C (X ), - . 9 f (x1 ) f (x2 ) K ( . 1), x1 x2
X . 7
x1 x2 2 X , X - . 9 #3, . 22].
( 1). ,, x1 x2 - - X , . K x1 x2 . ;
E F | #3, . 45]. 9 ^ F E E F , - : . u 2 E F supfkei kkfi k i = 1 : : : ng, u Pn
u = ei fi , ei 2 E , fi 2 F , i = 1 : : : n. i=1 X Y C (X Y ) C (X ) ^ C (Y ) #1, . 106]. 3 : - ,
C(X)
1039
X Y #3, . 131]. 6- . 1 . 1. B1 B2 x y , > . B1 ! B2 . . ;
h 3 B2, l | 3. 9 (h x) 6 max((h y) (y l) (l x)) = max( ) = : E . . F 3 . . ;
x 3 B. ?
3 x 3
, B . 6 B , /. 9 , - 3
, . ;
X | , f : X ! K | . . 9 fx 2 X : kf (x)k < "g - , - K . ; - fx 2 X : kf (x)k > "g. = - . 2. X Y | , Q X Y | - ! , S | Q. ! Q
! # S $ n F #$ Q = (Xi Yi ), X1 : : : Xn X, Y1 : : : Yn Y | i=1 - ! . . ; X Y (x y) 2 Q U X , V Y , (x y) 2 U V S S 2 S . 9 X Y , U V - . , Q , . - - n F X1 : : : Xn X , Y1 : : : Yn Y , Q = (Xi Yi ). G i=1 - 5 5 ( X1 : : : Xn Y1 : : : Yn). E . = B (x): X ! K B .
1040
. .
3. X Y | , P X | ! . % f (x y) 2 C (X Y ), # ! P nY X Y , ! P #& % ki Xi (x) Yi (y), i=1 X1 : : : Xn X n P, Y1 : : : Yn Y | - ! , ki | ' K. ' ! Xi Yi ! . . H Q = f(x y): kf (x y)k > "g - . S Q, f (x) ". 2 Fn Q Q = (Xi Yi), / i=1 f (x y) Xi Yi ". 9 Xi X n P . = 3 . E . 9 . 1. X Y | , P | ! X, I C (X ) (J C (X Y )) | & , & % &, # P ( P Y ). I ^ C (Y ) % J. . 6 D : I ^ C (X ) ! J : f (x) g(y) 7! f (x)g(y) 1. , J -
n P u = ki Xi (x) Yi (y). 9 Xi Yi i=1 , maxfkkik i = 1 : : : ng. Pn . w = ki Xi (x) i=1 Yi (y), maxfkkik i = 1 : : : ng. 4 , D . 9 . 6 , . 6 #1]. ,, A | . A ^ E , E | , A ^ E a (b e) = ab e, a b 2 A, e 2 E .
A ^ E ^ A. = (-) X , : A ^ X ! X : a ^ x 7! a x ( : A ^ X ^ A ! X : a ^ x ^ b 7! a x b)
- ( -) .
C(X)
1041
I A ,
. F (-) A-
n n+1 n+2 : : : ; Xn ; Xn+1 ; ::: , - n : Xn;1 ! Xn , n 2 N, n 2 N n n + n+1 n+1 = 1Xn . (-) A- X
0 X R0 R1 : : :, R0 R1 : : : . , ( ) X - . : Rn+1 , , n. : , . H X ( ) . . 6 , , . : : A ^ X ! A , X - 0 X A ^ X Ker 0 ( ). A / . A ( ) . .
x
2.
C (X )
;
I C (X ) | . 1 (X ) ^ I ^ C (X ) C (X X X ), X P X . ; : C (X ) ^ I ^ C (X ) ! I
(f )(x) = f (x x x): ; I / - C (X )-
: I ! C (X ) ^ I ^ C (X ), - . 2. ( & I C (X ) C (X )-# , X n P X .
1042
. .
. . ;
. y1 : : : yn . 9 (yi yi yi) 2 2 X X X , i = 1 : : : n, . 2 f 2 I (f ), f (yi ) (yi yi yi ), i = 1 : : : n, . , , / , . . ;
X n P / x0. ; . , / 3 B x0. ; 1 . 3 , B (x) ,
, I . ; , - : I ! C (X )^ I ^ C (X ), . ?
. v = = ( B (x)). 9 # (v)](x0 ) 6= 0, v(x0 x0 x0) 6= 0 ( | . ). 2 , y 2 X - 3
y, - x0 , - g(x), x0 y. ; g(x) B (x) = B (x)g(x), g(x1 )v(x1 x2 x3) = v(x1 x2 x3)g(x3 ). ; x1 = x2 = x0 x3 = y , v(x0 x0 y) = 0 y 6= x0. , x0 , , . ; v(x0 x0 x0) = 0. ; . . I C (X ) , X . x
3. ! " #
- ,
C (X ) . . ;
X | , A B | /, f (x) | X , A \ B . ; g(x) X , f (x) A / B . 6 IA\B (IB ) C (X ), - , - - A \ B ( B ). 3. ) ! A B X % C (X )- & T : IA\B ! IB 1, & #& % f (x) 2 IA\B % (Tf )(x) f (x) ! A. * , & X % f (x) , % (Tf )(x) '& ! .
1043
C(X)
. ?
h 2 B n A. 6
dist(h A) A. ?
3 h
dist(h A). 65 3, h 2 B n A, B n A, U . L , A. M " > 0
h 2 A \ B . 6 / 3
". 65 3, h 2 A \ B , "- . , V" . ,, 5 Q" = U V" B . = , . . ;
x 2 X / . 9 3 Tx x
Q". 7 5 3, B . 6 3 Tx . 3 ( 1) /, . x Q". ?
, . . , 3 Tx x / B . x B , . . 9 Q". N . 6 Q" , Q" (x). / f" (x) = f (x)(1 ; Q" (x)): ; " ! 0 . . ?
f" (x) ; f" (x) = f (x)( Q" (x) ; Q" (x)): 6, " < "0
Q" Q" , / Q" n Q" V" . =, f (x) X . I A \ B , "0 f (x) V" ( "0 - A \ B ) . 7 f" (x). 6 / g(x). , g(x). L , B . = , A f (x) g(x) . 6 , A \ B ( ). ;
x0 2 A n B . 9 x0
dist(x0 A \ B ) A \ B , V" " < dist(x0 A \ B ). , U A, 0
0
0
0
0
0
1044
. .
x0 . 2 , x0 Q". x0 f (x0 ) = f" (x0 ). ;
", f (x0 ) = g(x0). , , T : f (x) 7! g(x) C (X )- , . ,, x00 2 X f (x) , f" (x) , (Tf )(x) . . 9 .
x
4. C (X )
. 3 / / C (X ). ;
I | , - , - - P X . 4. + & & I C (X ) &
#& C (X ). . 4, C (X ) ^ I
1 C (X X ), - - X P . : C (X ) ^ I ! I C (X X ) X X . , , - . ?
: I ! C (X X ): (f )(x1 x2) = f (x1 ): 6 . L , (f ) - P X . 9 , - X P . 3 . A / X X . B / X P . M (f ) - ( - P X ). 6 T : IA\B ! IB . 9 T ((f )) X P (f ) X X . 6 ( ) .
. 9 . = X n n X ,
.
C(X)
1045
5. , I C (X ), # C (X ), . . F3 3, C (X ) ^ I ^ C (X ) C (X ), - , - - X P X . F : C (X ) ^ I ^ C (X ) ! C (X ), , diag X 3 . Ker C (X 3 ), - - diag (X P X ). = , . . G C (X ) ^ Ker ^ C (X ) 1 / X 5 , - - X X P X X X diag X . F
0 : C (X ) ^ Ker ^ C (X ) ! Ker .
0 (f )(x1 x2 x3) = f (x1 x1 x2 x3 x3): O . . O,
X 5 A = (x1 x1 x2 x3 x3) x1 x2 x3 2 X . 9 0 (f ) , f A. ; . ?
, / : Ker ! C (X 5 ): (f )(x1 x2 x3 x4 x5) = f (x1 x3 x5): L , . F , f - X P X , (f ) - X X P X X . 6 (f ) - X diag X . ; B = X diag X . F , A B X 5 , . (f ) . ; ( . 3) T : IA\B ! IB
T . 7 0 . 6, (f ) - X X P X X , 3 (T )(f ) - . . 9 . 6 - . 6. - # C (X ) , X , , # . . 7 2 5,
I = C (X ).
1046
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. C (X ). #2, . 431], / . , , . . ; C (X )
, -: , X . = , X . . 7. . X # , # C (X ) ( # ) . . ; , X x0. , C (X ). ; K , / f k = f (x0 )k, f 2 C (X ), k 2 K . 6 C (X ) ^ K C (X ). : C (X ) ^ K ! K 0 : C (X ) ! K : f (x) 7! f (x0 ). ; , - : K ! C (X ) . 9 e K . (e), - x0. : . X , / - y 6= x0, (e) . ?
3 B y, - x0. ; 1 . 3 , . 6 / B (x). H B (e) = ( B e) = = ( B (x0)) = (0) = 0. , B (e) , / y . ; .
& 1] . . . | .: !"- $, 1986. 2] . . * : + , " ,
. | .: ,- , 1989. 3] Van Rooij A. C. M. Non-Archimedean Functional Analysis. | New York and Basel: Marcel Dekker, Inc., 1978. ' ( ) 1999 .
- 1 . . , . . , . .
519.622+519.61
: {, - ! " #$ , % ## , & ' , ( ## .
) # &" *# #! # + , { - ## - ! " #$, % $# 1 ! -% & #. .# $, & " + $$ ## $ $ - * ## % , ! " #$, % # ! - & ' $/ , " + $ & - ! & " !01 + % + 2)3. ) # & ! / $ #, , , / $, , $' + ' #+ $ ## #$ *" + % $& , ! $ . 4 $ ! &, + " % / $ % - % , ## , & $ *5, & + , { $ ## - ! " #$, % $# 1 ! -% & #. 6 1 "# + + ' * " #$ & # .
Abstract
G. Yu. Kulikov, A. A. Korneva, G. Ya. Benderskaya, On numerical solution of large-scale systems of index 1 dierential-algebraic equations, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1047{1080.
In this paper we study how to integrate numerically large-scale systems of semi-explicit index 1 di=erential-algebraic equations by implicit Runge{Kutta methods. In this case we need to solve high dimension linear systems with sparse coe>cient matrices. We develop an e=ective way for packing such matrices of coef?cients. We also derive a special Gaussian elimination for parallel factorization of nonzero blocks of the matrix. As a result, we produce a new e>cient procedure to solve linear systems arising in an application of implicit Runge{Kutta methods to large-scale di=erential-algebraic equations of index 1. Numerical examples support theoretical results of the paper. ! + + #% + '$ ##%#$% $ $, 3 # # !5 + ## ! & ## ( " + @ # ## | ## C, + $ 230) ##%#$ , ## % (+ $ 01-01-00066, 00-01-00197). , 2001, 7, E 4, #. 1047{1080. c 2001 , !" #$ %
. . , . . , . .
1048
1. -! "# ! :
x0(t) = g(x(t) y(t)) (1.1) y(t) = f (x(t) y(t)) (1.1 ) 0 0 x(0) = x y(0) = y (1.1 ) m n m + n m m + n ! t 2 )0 T ], x(t) 2 R , y (t) 2 R , g : D R ! R , f: D R ! Rn 0 0 0
(1.1 ) , . . y = f (x y ). ,
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. - - ! - !
.
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6, - (1.1) - 7 ! {8 (78- ),
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A # " l
l l, b c | # -
, - "#" " :
xki = xk + k
Xl
aij g(xkj ykj ) j =1 yki = f (xki yki) i = 1 2 : : : l Xl xk+1 = xk + k big(xki yki) i=1 yk+1 = f (xk+1 yk+1) k = 0 1 : : : K ; 1
x0
!
0 =x
y0
0 =y
(1.2) (1.2 ) (1.2 ) (1.2!) (1.2)
k | ! ! ! , .
- .
...
1049
; z (t) , - x(t) y(t) (z (t) = (x(t)T y(t)T )T 2 Rm+n), G . , -
. g f (G = (g f ) : D R z (tk ) | ! (1.1) tk , z~k | ! (1.2) tk z >k = z >k (N ) | - .5
! (1.2) tk , - - N !
! .
?- 5
Zk+1 = ((zk1 )T : : : (zkl )T (zk+1)T )T 2 R(m+n)(l+1)
- .
G> k : D R(m+n)(l+1) ! R(m+n)(l+1)
k = 0 1 : : : K ; 1, "# : G> k Zk+1 =
x>k + k
x>k + k
Xl j =1
Xl
j =1
T
a1j g(zkj ) f (zk1 )T : : :
T
alj g(zkj )
f (zkl )T
x>k + k
Xl i=1
T
bi g(zki)
f (zk+1 )T
T
:
3
(1.2) - 4" , - 5
- (1.1).
{ { ( -):
Zki +1 = Zki;+11 ; @ F>k (Zki;+11 );1F>k Zki;+11 (1.3) Zk0+1 = ((>zk )T : : : (>zk )T )T 2 R(m+n)(l+1) (1.3 ) N z>k = zk k = 0 1 : : : K ; 1 i = 1 2 : : : N (1.3 ) Z>0 = Z 0 = ((z 0 )T : : : (z 0 )T )T 2 R(m+n)(l+1) > = E > @F > (Z i;1 ) . F > , ! F (m+n)(l+1) ; G k k k k+1 k i ;1 Zk +1 (E(m+n)(l+1) | - R(m+n)(l+1)).
{ { ( -):
Zki +1 = Zki;+11 ; @ F>k (Zk0+1 );1F>k Zki;+11 (1.4) 0 T T T ( m + n )( l +1) Zk+1 = ((>zk ) : : : (>zk ) ) 2 R (1.4 ) N z>k = zk k = 0 1 : : : K ; 1 i = 1 2 : : : N Z>0 = Z 0 = ((z 0 )T : : : (z 0 )T )T 2 R(m+n)(l+1): (1.4 ) 3 -. , - . D1 (1.1) -
"# .
. . , . . , . .
1050
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. D1 - - - s + 2 " , ! s | - 78- , -. 1.2). II.
. B En ; @y f (z ) . " ! z 2 D1 , ! @y f (z ) " - " . f (z ) - y. III. 1 . C# - . D0 , 0 z 2 D0 D1 . I.
6 )13], (1.1)
z (t) 2 D0 . 8 !, . ,
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I{III
N (1.3), (1.4) " -, --
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l-l+1
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# ".
2. ! "{$ ! 7 784 784- -
@ F>k (Zki;+11 )(Zki;+11 ; Zki +1 ) = F>k Zki;+11
(2.1)
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1051
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- :
Zki +1 = Zki;+11 ; @ F>k (Zki;+11 );1 F>k Zki;+11
k = q q + 1 : : : K ; 1 i = 1 2 : : : N
T > (t + c )T H > (t q k + c1k )T : : : H q k l k q k+1 )T 2 R(m+n)(l+1) Z>k = Zk = ((zk )T : : : (zk )T )T 2 R(m+n)(l+1) k = 0 1 : : : q: Zk0+1
; > (t = H
- :
(2.2) (2.2 ) (2.2 )
Zki +1 = Zki;+11 ; @ F>k (Zk0+1 );1 F>k Zki;+11 k = q q + 1 : : : K ; 1 i = 1 2 : : : N
; > (t = H
T > (t + c )T H > (t Zk0+1 q k + c1k )T : : : H q k l k q k+1 )T 2 R(m+n)(l+1) Z>k = Zk = ((zk )T : : : (zk )T )T 2 R(m+n)(l+1) k = 0 1 : : : q: 1
)15]
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- #
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(2.3)
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m + n)l (m + n)l
(
nn
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0@F> (Z ) 1 k 1 > (Z ) C B @ F @ F>k (Z ) = B B@ k .. 2CCA .
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@ F>k (Z )l
@ F>k (Z )i , i = 1 2 : : : l, - (m + n) (m + n)l ! .
0O(k ) O(k ) 1 + O(k ) O(k ) O(k ) O(k ) O(k ) O(k )1 9 > = .. .. .. . . .. .. . . .. C BB ... . . . ... ... m C . . . . . . . . > BBO( ) O( ) O( ) 1 + O( ) O( ) O( ) O( ) O( )C C k k k k k k k C9 BB 0 k z z z z 0 0 C C> BB . . 0. = .. .. .. . . .. .. . . .. C ... @ .. . . .. . . . . A . . . . >n 0 0 z z z z 0 0 | {z } | {z } | {z } | {z } m
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6, D #
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K )17], #
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m + n)l(((m + n)l)2 + 3(m + n)l ; 1) :
(
3
(2.4)
4 , ! -
)11] (
I)
m + n)l)2 + ((m + n)l ; 1)(m +6 n)l(2(m + n)l ; 1) + 2 (m + n)l (ml + n ; 1) + ; n(m + n) (l ; 1)l(2l ; 1) ;
((
2
; n(m + n)(m + n ; 1)l(l ; 1) : 4
6
(2.5)
1 )16] - , D - -
x y (
II)
. . , . . , . .
1054
m + n)l(ml + n) + (m + n ; 1)(m +6n)(2(m + n) ; 1)l + m(m + n)(2m + n)l(l ; 1) + m2 (m + n)(l ; 1)l(2l ; 1) + +
(
4
6
n(n ; 1)l + m(m ; 1)l + mnl + 2
(2.6)
2
- . . ? ! - . 3
m = n = 2.
1-
, - (2.4){(2.6), #
- . , (2.1) , -
78-
l = 1 2 3 4.
-
6 1 , . -
m n - l = 4
II D-
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# - - " D | - " I. J , 5
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m, n l
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4 ! 1 M - . , (2.1) -
m=n=2
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3
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4
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F1B ! D -
m + n)l # ! m + n)l)2 . ; 5 > (Z ) . @ F k ! ( (( .
, " - "#
(2.1), " 5
5
" " - - .
m + n)l(l(2m + n) + n)=2 m+n)l(ml +n) )16]. 8 ,
1
I ( ,
II | (
...
1055
4 ! 2 ;- - F1B ( ), (2.1) -
m=n=2
M
B
B
B
D
I
II
1
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128
2
512
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3
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m=n=2 l
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double.
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7 2
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I II, - -
! D. 3 5
! -
II.
2.3. 6, - - 78- -
-! (1.1). 3 - - - . 1 - "#" , - . " )18]:
x01(t) = 10t exp(5(y2 (t) ; 1)) x2(t) x02(t) = ;2t ln(y1 (t)) 1 y1 (t) = x1 (t) 5 2 2 y (t) = x2 (t) + y2 (t) t 2 )t t + T ]: 2
2
0 0
(2.7) (2.7 ) (2.7 ) (2.7!)
G (2.7)
:
x1(t) = exp(5 sin(t2 )) x2 (t) = cos(t2 ) y1 (t) = exp(sin(t2 )) y2 (t) = sin(t2 ) + 1:
(2.8) (2.8 )
4 5
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. . , . . , . .
1056
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1057
4 ! 5 D 784- - ! - M
M
B
B
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D
I
II
1
2
2
2
3
2
4
2
5035 10;10 5035 10;10
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1 394 10;04 6 017 10;10
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4 ! 6 G ! ( .) 784- - ! - M
M
B
B
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I
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M
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. . , . . , . .
1058
4 ! 8 G ! ( .) 784- - M
M
B
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1059
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5
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5
.
78-
Xk+1
= (
xTk1 : : : xTkl)T , Yk+1
,! (2.1) -5
-! -
nl (m + n)l ml (m + n)l : > Y @ F>k (Z ) = @@FF>k ((ZZ))X : k
(3.1)
3 - , " , -
n (m + n)l:
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l
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0 . . .
0
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n
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m
m(l;i)
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= 1 2
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i = 1 2 : : : l.
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{z
}
m(l;i)
mn
; . , - . > - -- > , . kj ( ) , = 1 2 > ( ) , = 1 2 . 3 . < -
Fk y : : : l i j
zkj @y Fk zkj Xi j X @xkj Fk zkj i j : : : l
mm
@zkj F>k (zkj )iX = (@ykj F>k (zkj )Xi @xkj F>k (zkj )Xi ):
; # (3.1) - . 2. G , , , - -
Z |
x- y-- !
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Z 0
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n
n
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0 0
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0 Z
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Z Z Z
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. . .
. . .
. . .
Z
Z
Z
0 0
.
. . .
Z Z Z
..
.
. . .
Z
..
|{z} |{z} |{z} |{z} |{z} |{z} |{z} |{z} n m m n(l;3) m(l;3) #. 2. H " #$ (3.1)
n n n(l;3) n m m m(l;3) m
m
3.1.2. 3 . ! , -
(2.1) -
nl ! (. . - "
y-- ) LU-. - @ F>k (Z )Yi , i = 1 2 : : : l, > (Z )Y - j > i, ! - @ F k j
(3.1) D -
-
-. (. . 2). 3 > ( ) - - > ( ) , = 1 2 .
!, - - > ( (1.2) ) , = 1 2 , "
@ Fk
Z
Y i
@ Fk Z Y i : : : l
l
@zki Fk zki Yi i
: : : l
. . , . . , . .
1062
1
" , - - (-
! ) " - , -. . , . F ,
ki > ( ) , = 1 2
@z Fk zki Yi i
: : : l
. ! , > ( ) - -
@ Fk Z Y
x-- Z , m, - y-- , -
( , ! , -
- " - "
n) - .
H " - > ( ) - ,
@ Fk Z Y
""
f
l
- P
@ F>k (Z )Y
-
x- y-- r (y-- - n, x-- | n + 1 n + m)P
" 1
"# - ! . > ( ) .
@ Fk Z Y n - (. . 3).
,!, - ,
- f11
1
. . .
2
. . .
n
r1
-
- fi 1 1
. . .
nl ,
ri1
-
rin
-
fi1 l
f1l . . .
- fn1 . . .
rn
fnl
-
- fin 1 . . .
fin l
#. 3. L, , , / @ FJk (Z )Y
p q @ Fk Z Y Y Zi i : : : l p ; (i ; 1)n-! - r = q ; (i ; 1)n ; ; )q=(nl + (i ; 1)m)]((l ; i)n + (i ; 1)m) i- f , !
)q=(nl + (i ; 1)m)] " q=(nl + (i ; 1)m). ?! > (z )Y , i = 1 2 : : : l , , 5
@zki F k ki i
> ( ) , , ( )- - > . -
( ) , = 1 2 , -
@ Fk
1 M # + #$% + # #+ ' , , / , $ & ! ' $ ! #+ " & #"1 & - k .
...
1063
. - " . 3 5
. ! - . > ( , # ) - .
@zki Fk zki Yi
1 # -
@ F>k (Z ),
> ( )
- - (1.2), -
!
@ Fk Z X
m m m n, "#
!, 78- ! ". 4- , - ! 78- > ( ) . 3 - 5
@ Fk Z X
- " - - )20], . . 5
- , ""
@ F>k (Z )X P
! ! P
"# - ! .
,!, - , . - > ( ) , -
@ Fk Z X
ml
(. . 4).
-
1 2
. . .
ml
f1r1
r1
f2r2
r2
-
rml
-
-
f1ri1 ri1 f2ri2 ri2
-
- fmlriml riml
-
. . .
- fmlrml
#. 4. L, , , / @ FJk (Z )X 4 , 78- - ! > ( ) . -
1 #
.
1.
@ Fk Z X
l- A = (aij ), aij 6= 0 i j = 1 2 : : : l:
! ,
7 ! {8-
- "
(3.2)
1 78- - .
- . 3
! . 78- , - D )9, 19]. C ! , ,
. . , . . , . .
1064
! 78- " - . 6, , - -
- 78- > ( ) . - -5
-- - -
@ Fk Z X l (Z ), ! m (m + n)l- k A @gk @gkl (Z ) = (@yk1 g(zk1) : : : @ykl g(zkl ) @xk1 g(zk1) : : : @xkl g(zkl ))P
.
.
@ykj F>k (zkj )Xi
=
,
"# > (
; k aij @ykj g(zkj )
! ,
@xkj Fk zkj )Xi
Em ; k ajj @x g zkj i j
-
; k aij @xkj g(zkj ),
= > (
@ Fk Z ), , - : : : l. 3
A, --
, # ! !
= 1 2 kj > ( ) = kj ( ),
@x Fk zkj Xj @ Fk Z Xi
(3.3)
78-
- " > ( )
( ) (3.3) -, . , " ! > ( ). F -
@gkl Z
,
@ F>k (Z )X
@ Fk Z
-
k A @gkl (Z ). , , ml (m + n)l l l m (m + n)l , , ,
- - - l > 2. " -
3.2. 6, - (3.1), " 5
. 7 - , - D (2.1) - (3.1). 8. ! D -
. C # ! - - , " " - " (2.1).
6
- ! ! # -, . .
-
3 . -
! ! (2.1) . (3.1).
3.2.1. 3 ! ! ! . D . -
! - " - . , - - .
- , . . -
, - " !, - 5
!
" > 0.
G
...
1065
, 5
( ) - - - " -
1
- . ?
# ! )20, 21].
G -. ,
- -
(3.1).
3 ! ! ! , 2.5.5
+ 1- ! D, + p + q)- @ F>k (Z )() (p q = 1 2 : : : (m + n)l ; ), . . , - - ! D, (p q )- G() , !
G() = B () (B> () )T B () (3.4)
B () (m + n)l ; - > (Z )() -5 - @ F
k () > () )T | , - B > () = M , (B (m+n)l; ; B (M(m+n)l; | (m + n)l ; ). ,! ! ! + 1- ! D 5
)20], - - ! ! (
- , 5
- .
, , - !
! ! - . .
-
G()
: : : (m + n)l ; 1.
= 0 1
- . ! D,
1 5
-
!
( - . ) . "# :
X
(m+n)l;1 2
=0
((
22 2 m + n)l ; )3 = (m + n) l ((2m + n)l + 1) :
(3.5)
1 ! ! ! - . ( -"#
) -
+ 1- ! D, ! ! ( + p + q)- @ F>k (Z )() , eTp G^ () eq , ! ep eq | p- q- -
,
G^ () = (B () ; E(m+n)l; )M(m+n)l; (B () ; E(m+n)l; ) (3.6) B () M(m+n)l; " . , (c. 2.5.6 )20]). ?
, ! ! + 1- ! D 5
- , 5
. - . 8 , ! ! !
, - - - , 5
" . 3 , # ! , - R , . ! #
1 M # + $ & + + # !5 "# , , / , $ + # + + - ( ## .
. . , . . , . .
1066
- - (2.1), - - F1B .
; !
- -
!
I. 4- , , ( ) (3.6) -- - ^
G
m + n)l ; , B () ; E(m+n)l; - , -
. " - ( - . ! | - .
, , -
(2.1)
! !
! II !
X
(m+n)l;1
=0
1)(2(m + n)l + 1) m + n)l ; )2 = (m + n)l((m + n)l + 6
((
(3.7)
- . , , , , ! !
I (. (3.5)). H5
- (2.1) (3.1). 7
> ( ) - - -
,
@ Fk Z
-. 4 - - - "
|
x--
.
y-- ,
1 -
! ! . - . 4 5
. ! ! - -
.
y @ Fk Z Y l @ Fk Z Yi , i = 1 2 : : : l, -
. @ F>k (Z )Yi # ! ! + 1- ! ! "
3 "
-- LU-. > ( ) - LU-. -
> ( )
- - , - - - ! , # -
-
n ;
@yki F>k (zki)Yi () .
; , . (. II . 1050) ! . - > ( ! , ) -
@yki Fk zki Yi
. ,!,
@ F>k (Z ), - -
, - ! ! , - - (
n ; + ml) (n ; + m)
0 > Y () 1 BB@@zki FF>k ((zzki))Xi () CC BB zki >k ki 1X () CC ( ) > @zki Fk (zki ) = B BB@zki Fk (..zki)2 CCC : . @ A () @zki F>k (zki )Xl
1
G()
(3.8)
G )
^(
-
-#". ? , (3.4) 5
(3.1) ,
...
1067
; ; ; G() = ByiY () B>yiY () T + BxiY () B>xiY () T ByiY () + +
Xl ;
; ; X ()T BX () ByiY () B>yjX () T + BxiY () B>xj yj
(3.9)
j =1 ( ) ( Y , B Y ) , B X () , B X () - ! Byi xi yj xj () () > (z )Y () , @ - @yki F ki xki F>k (zki )Yi , @yki F>k (zki )X i j , k @xki F>k (zki )Xj () -5 , B>yiY () = Y () , B> Y () = M(n;)m ;B Y () , B> X () = Mm(n;) ;B X () , = M(n;)(n;) ; Byi xi xi yj yj X () = Mm(n;) ;B X () , Mab # -! " B>xj xj
a b .
N ! , - --
!
! ! (3.6)
;;
;
G^ () = ByiY () ; En; M(n;)(n;) + BxiY () Mm(n;) ByiY () ; En; +
l ;; X j =1
X (): ByiY () ; En; M(n;)m + BxiY () Mmm Byj
4 - , -
nl
+
(3.10)
- ! -
II D (3.9) (3.10) "
l
nX ;1 =0
n ; )3 + m(2l + 1)(n ; )2 + m2 l(n ; )) =
(2(
=
2 2 n + 1) + m2 l2 n(n + 1) l n (n2+ 1) + ml(2l + 1) n(n + 1)(2 6 2
l(l + 1)
nX ;1 =0
n + 1) n ; )2 = l(l + 1) n(n + 1)(2 6
(
(3.11)
(3.12)
- . . , , 5
(3.1) - 5
# " ! ! - -
" ! - > ( ). -
@ Fk Z
; - > ( ) , =1 2 ,
, " -
@ Fk Z Yi i
: : : l
- . F , . ! D
l - , . . -
- l . ( , , ). , - . ; , > (Z )Y , - - , - . @ F k
" ,
. 8 !,
! # D, - - ,
. . , . . , . .
1068
- -
+1-! ! ! " - -
! . - -
@yki F>k (zki)Yi () , i = 1 2 : : : l, = 0 1 : : : n ; 1.
7
! , -
D . . - , ! . ? - ! D .
,
. . -
, - " - . 1 . , . . - . > ( ), . . . - -
@ Fk Z > (z )Y () - @yki F k ki i - - " = 0 1 : : : n ; 1.
. -
1 I{III (. c. 1050) . ! - -
(2.1) (3.1). > ? , ki ( ) - " " - -
@y Fk zki Yi
" (1.1 ) - -
y, " ,
. ! ! !
k
(. (1.2)).
k
,! 5
! . > , #
ki ( ) , = 1 2
,
@y Fk zki Yi i
: : : l
, ! . F! , > ( "# )
@yki Fk zki Yi
, . ;" . , - , - . -
@yki F>k (zki)Yi () - - " = 0 1 : : : n ; 1.
C "# - ! ! ! D . - ! , "# !
( ) - .
1 #
5
, - . 1-- , ( ) (3.8) > - ki ( )
@z Fk zki
, . - ! ! 5
- , . .
1- , # -
, -
@yki F>k (zki )Yi () , i = 1 2 : : : l,
"# ! - , -. . - , . . - . . - .
; -# , (1.1)
- 78- - (. - 1 . 1063).
1 " "
@zki F>k (zki )Yi () , @zki F>k (zki)Xj () , j = 1 2 : : : l, @zki F>k (zki )() , i = 1 2 : : : l (. (3.8)), @zki F>k (zki)Xi ()
. - , - , -
.
3
...
1069
- " ! - ! - - ". ;" , . ! , - ! " " ! - . , , - ! ! ! " " ! - .
!, "# , . - - " ( ) > + ki ( ) , "
@z Fk zki
! Y ()
@zki F>k (zki)i @zki g(zki )()
n; m
m - " "# @gkl (Z ) (3.3), - - !
-
D.
@ F>k (Z ) l- { A. + 1- ! " i- # ( + p + q)- # @yki F>k (zki)Yi () ,
= 0 1 : : : n ; 1, i = 1 2 : : : l, p q = 1 2 : : : n ; , % @zki F>k (zki)() % (3.8) &' (p q)- 1.
; ; ; G() = ByiY () B>yiY () T + BxiY () B>xiY () T ByiY () + Y () ;B> () T B () + B Y () ;B> l () T B () + l Byi y y xi x yi
(3.13)
ByiY () , BxiY () , By() , Bx() % ( @yki F>k (zki )Yi () , @xki F>k (zki )Yi () , @yki g(zki )() , @xki g(zki )() & % ( , B>yiY () = M(n;)(n;) ; ByiY () , B>xiY () = M(n;)m ; BxiY () , B>y() = Mm(n;) ; By() , B>x() = Mm(n;) ; Bx() , B>xl () ) ( B>x() l ; 1 ( | l. . 3 i- - ! ! > (z )Y () . ,! (3.9) ( + p + q )- @yki F k ki i > (z )() (3.8) 5 , - @zki F
(p q )- k ki
;
;
;
G() = ByiY () B>yiY () T + BxiY () B>xiY () T ByiY () + ; Y ();B>() T + BY ();B>() T B() + + (l ; 1) Byi y xi x y ; () ; > () T () ; > X () T () Y Y + Byi By + Bxi Bxi By
X () Byj
=
By()
i 6= j ,
X () Bxj
=
Bx() , j
(3.14)
: : : l,
= 1 2
!
. . , . . , . .
1070
By() , Bx() - - @yki g(zki )() , @xki g(zki)() -5 -
.
F !, , - . (. 3), > ( )( ) - -
@ Fk Z () () > (z )X () = ; a @ = ;k aji @yki g (zki )
@xki F k ji xki g(zki ) , k ki j
!
X ()
@yki F>k (zki )j A .
@ F>k (Z )() - () . ,! - > (z )
@xki F = Em ; k aii @xki g (zki ) k k ki i ( ) X > ! Bxi , . > () ! . 3 , B x C ! , !
X ()
- ! (3.14) - --
, 5
B>xl () = (l ; 1)B>x() + B>xiX () - (3.13). , .
2. @ F>k (Z ) l- { A. + 1- ! " i- # ( + p + q)- # @yki F>k (zki)Yi () ,
= 0 1 : : : n ; 1, i = 1 2 : : : l, p q = 1 2 : : : n ; , % ) % @zki F>k (zki)() % (3.8) ( #'% ) &' (p q)-
;;
;
G^ () = ByiY () ; En; M(n;)(n;) + BxiY () Mm(n;) ByiY () ; En; + ;; Y () ; E M Y () Mmm B () : (3.15) + l Byi n; (n;)m + Bxi y ? 2 - ! ( ) 1 ,
(3.10) ^
X ()
Bxj
,
G
j = 1 2 : : : l, .
, 1 2 " - . ! ! ,
- " 5
- !
( - . ) - -
(2.1). ? , - , - > ( ) ! ! -
@ Fk Z Y
n
D, -
- !
#
- . ! ! ! "# :
nX ;1
n ; )3 + (4m + 1)(n ; )2 + m2 (n ; )) = n2 (n + 1)2 + (4m + 1) n(n + 1)(2n + 1) + m2 n(n + 1) : =
(2(
=0
2
6
2
(3.16)
...
1071
N ! , !
(3.15) - 3
l
nX ;1 =0
n + 1) : n ; )2 = n(n + 1)(2 2
(
(3.17)
? ! - . 3 - =
1 2 3 4.
m=n=2
,!, - (3.5), (3.7), (3.11), (3.12)
(3.16), (3.17), - . ,
- !
! ! -
(2.1) D, . 5
, . . - -
II, -
-
! D 78- - . 1 11 - - !
, 12 | . ?. -
! - > ( ) - ! ! .
!,
@ Fk Z
. , - D ! ! 78- -
" .
15
5
- --
- 78- ! - - - . 6, . !
, LU-. -: "
@ F>k (Z ) --
y- x-- .
? .! - ! ! ! ! .
,, - "
y-- - -
--
- " ! ", ,
5
. F , - ! " ,
l.
y-- -
3 - .
- - ! , - ! <5
- - - F1B. 8 !, --
4 ! 11 M - . - ! ! - - !
> ( ) ( - -
= = 2)
@ Fk Z m n
M
D
II
D
1
182
60
75
2
2392
184
75
3
11286
372
75
4
34400
624
75
. . , . . , . .
1072
4 ! 12 M - . - ! ! - !
> ( ) ( - -
= = 2)
@ Fk Z m n
M
D
II
D
1
25
10
15
2
174
30
15
3
559
60
15
4
1292
100
15
"
y-- - 5 " " x-- , @ F>k (Z )
- . ! - !
. .
6 !, - ,
- - ! - , . .
- - " ! " ! ! . 4 - , . . - "
x--
, -
" ! ", - " . 1 " ! , - ! -! (2.1) - ! .
,!, - , -
S T ( ! ), . ! - - , - # ! ( ) ! ! (1.1). , , - - , . . ! ( ), - .
(2.1). B (- -
), - - ! ! . ? ! ,
! - "
y-- ( - ! x-- ) " - -
- "
. J . , - - " . ; - 5
. 1 . #5
! .
3.2.2. ! "# 1 , - .
! ! (2.1) . (3.1), , - -
...
1073
@ F>k (Z ) ! -. 4 - -
- " y -- , | x-- . B - , - -
! - -
! D > ( ) , = 1 2 , -
-
@ Fk Z Yi i
: : : l
- 78-
(1.1). 1 -
! - . - -
@ F>k (Z )
nl + m) (m + n)l "# ! :
@F> (Z )Y
" (
k @gkl (Z )
(3.18)
# - ! ! . ; , . 3, " (3.18) . - - - ! ! , > ( ) . , , - -
"
y--
@ Fk Z Y
-
" (3.1) - - F1B. 4 - (3.18), , , -
# ! .
@ F>k (Z ) l- { A. ' - ! " % @ F>k (Z )X () % ' @gkl (Z )() k A, 0 6 6 n: (3.19) 3.
.
3 5
- -
! D
.
@ F>k (Z )X (0) @ F>k (Z )X , @gkl (Z )(0) @gkl (Z ), 3 ! (. 3.1.2). 2. 3 -. , 3 - - 0 6 < n, . , ! . 5
- - - + 1-! ! ! D. 3 . ! , 0 6 < n - , y -- Z " > (Z )Y . - ". (2.1) (3.1), . . @ F k 3 - + 1-! ! ! D " ! y -- 1.
3
= 0.
,!
(2.1). 3 "
i- - ! ! - @ F>k (Z )X () () > (z )X () @ @yki F xki F>k (zki )X j , j = 1 2 : : : l, ! k ki j - -.
- -
@yki F>k (zki )Xj () = ;k aji@yki g(zki )() @xki F>k (zki)Xj () = ;k aji@xki g(zki )()
(3.20)
. . , . . , . .
1074
! |
@xki F>k (zki )Xi () = Em ; k aii@xki g(zki )() : (3.20 ) C ! , " i- - (2.1) l (Z )() (3.18) ! - @gk () @ g(z )() . 3
@yki g (zki ) xki ki ( ) ( ) @gkl (Z ) = (gpq ), p = 1 2 : : : m, q = 1 2 : : : (m + n)l, l (Z )() ! D - " - @gk "# - :
(+1) = g() ; g() f (+1) gpq (3.21) pq p+1 +1q ! p = 1 2 : : : m, n(i ; 1) 6 q 6 ni nl + m(i ; 1) 6 q 6 nl + mi. 3 l (Z )() - "
i- ! q @gk -
+ 1- ! ! D ". > ( )( 3
@ Fk Z ) = (fpq() ), p q = 1 2 : : : (m + n)l. 7 - , > (Z )X () , j = 1 2 : : : l . G - (p q )- - @ F k j . ) 3
p 6= q.
,!, - (3.2), (3.20) (3.21),
() (+1) () (+1) () fpq(+1) = fpq() ; fp +1 f+1q = ;k ajigpq + k ajigp+1 f+1p = () (+1) () (+1) (3.22) = ;k aji (gpq ; gp+1 f+1q ) = ;k aji gpq ! nl + (j ; 1)m 6 p 6 nl + jm, n(i ; 1) 6 q 6 ni nl + m(i ; 1) 6 q 6 nl + mi. > (Z )X () - 3 ! p q - @ F k j
"
i- -
+ 1- ! ! D ". ) 3
p = q.
N ! " ) 5
(3.20 ) -
(+1) = f () ; f () f (+1) = 1 ; a g() ; a g() f (+1) = fpp k ii p+1 +1p k ii pp pp p+1 +1p () (+1) ( ) (+1) : = 1 ; k aii (gpp ; gp+1 f+1p ) = 1 ; k aii gpp G p . . - , (3.22).
(3.23)
6, (3.22) (3.23) , ! ! D (3.19), , - . ,
@ F>k (Z )X ()
. -#" --
-
@gkl (Z )() k A.
, 3 .
, , ! (2.1) - -#" - ! 78- , - 5
. # 5
- - "
y--
...
1075
(3.18). 4 - , - - > ( ) - " (3.18),
@ Fk Z
- . ( 5
. ) -
n + 1)n + n(n + 1)(2n + 1) l m(m + 1)n + (2m + 1)( 2 6 @ F>k (Z ) - n (n + 1)(2n + 1) (m + ml + 1)(n + 1)n + l m(ml + 1)n +
- -
2
6
. , , , .
8 !, - -
# . - -
@ F>k (Z )X (n)
-
k A
@gkl (Z )(n),
-- - -
, !, ! -- - > ( ) . ? ,
@ Fk Z X ( n ) X > (Z ) @ F k X 2
> (Z ) @ F | m(m + n)l . k
m2 l2
. ,
3 3 ,
@ F>k (Z )Yi ! - @gkl (Z ) @yki g(zki ) @xki g(zki ), i = 1 2 : : : l. ,!, - " > (Z )Y - -
! D,
" - @ F k i . - - "#
@yki g(zki ), @xki g(zki ).
3 " !
-#" - , - . > ( ) (. . 3). J " ,
@ Fk Z Y
@gkl (Z )
- m - . p q)- ; p-! - r = q ; 1 ; (q ; 1)=(nl + )jq ; nl ; 1j=m]m) )(q ; 1)=n]n ; ; ; q ; nl ; 1j=m]m ;1 ;(q(q;; 1)1)==((nlnl+)+jq ;)jqnl;; 1nlj=m;]m1)j=m )(]qm;)1)(=nl ];+ 1)(qn;+1)=)(jnl +)jq ; nl ; 1j=m]m) )jq ; nl ; 1j=m]+1- f , ! , , ,! (
"
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. | !.: ! , 1966. 2] Guyton A. C., Coleman T. G., Gardner H. J. Circulation: overall regulation // Ann. Rev. Phisiol. | 1972. | Vol. 34. | P. 13{41. 3] Ikeda N., Marumo F., Shiratare M., Sato T. A model of overall regulation of body ,uids // Ann. Biomed. Eng. | 1979. | Vol. 7. | P. 135{166. 4] Petzold L. R., Lotstedt P. Numerical solution of nonlinear di.erential equations with algebraic constrains II: Practical implications // SIAM J. Sci. Stat. Comput. | 1986. | Vol. 7. | P. 720{733. 5] 0
1 2. 3., 4 5. 6., 7 6. 6., 8 5. . 8 9 : . | ; : ; , 1989. 6] 09 5. <., 6. 7., ! =9 ;. >., < 3. ?. @9 DD 1 = - : : : 0OQ. | !.: 5U 6; QQQ@, 1991. 7] ;
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// \. . . . D . | 1996. | . 36, ] 8. | C. 73{89. 11] Kulikov G. Yu., Thomsen P. G. Convergence and implementation of implicit Runge{Kutta methods for DAEs. | Technical report 7/1996, IMM, Technical University of Denmark, Lyngby, 1996. 12] [ . 2. < 9
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= X // \. . . . D . | 1998. | . 38, ] 1. | Q. 68{84. [ . 2., [ 6. 6. < WDD Y 1
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Abstract A. V. Lebedev, Extremes of shot noise elds, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1081{1090.
The almost sure asymptotic theorems for shot noise 4elds maxima over bounded measurable regions growing monotonically in the Van Hove sense are proved. The non-degenerate limit law in the case of regularly varying tailed amplitudes is also obtained.
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hk | (.. .) ( ( 12]), 3 fk g. 1 hk (0) 5 , F 0. 4 ' 0( hk , ' 5 k. 0, 8 83 : 1) h(t) > 0 t 2 E> 2) h(t) 6 h(0) t 2 E> 3) h(t) = 0 ktk > 1> 4) h(0) < 1. ? , 1){4) 8 (..). 1 8 3 X(t) 8( (1) .. 16]. / ' '( ( ( G ( ) X(t) (: M(G) = sup X(t): t2G
A, M(G) .., ( k 1- G. 0, '' (, 3 1 B 3, c. 30] ( G " 1). 6 E Cn ( l. ' N ; (G) | , 3 G, N + (G) | , 83 G, 1 B , ; + (G) = +1 lim N (G) = 1: (2) lim N + G"1 G"1 N (G)
1083
/ 0 G jGj, (2) j N ; (G) jlGd j N + (G) jG (3) ld jGj ! 1: 1 G " 1 Cn , n > 1, 0 ' ( ', ( 8 G ( - 8 D E, ' ). N S ' HN = Cn, n=1
M(HN ) = 16max M(Cn) n6N (4) M(HN ; (G) ) 6 M(G) 6 M(HN + (G) ) ' ' '( . 4 ' M(G) G " 1 0 ' . F ', 8 : 8 ' 0 fk g> - ( ( ) ( h> ( 8 ( , ' . * 8 5 ' ( , , 8 ' 83 ( ).
2.
4 ' ' 83 0 2]. / K ' 83 ( R+ , 8 . A, ' 0 0' . ' F | R+ . 4 f 2 K ' Z+1 (f F) = ef(x) dF(x): 0
1. F 0 Sf , (f F ) < 1. 2. F 0 Uf ,
supf > 0: F 2 Sf g > 1: T S . A, Uf = f 0<<1 / , Sf Uf .
1084
. .
3. F 0 Rf , G f(x) x ! +1: ; ln F(x)
G ! 0, x ! +1, F(x) G 6= 0 8x. A' , F(x) A, 8 Rf , . . Rf1 = Rf2 f1 (x) f2 (x) x ! +1:
1. Rf Uf .
' Xn , n > 1, | '' ( F ZN = 16max X. n6N n
2. F 2 Uf , f 2 K ,
N) lim sup f(Z lnN 6 1 . . N !1
3. Xn, n > 1, F 2 Rf , f 2 K , f(ZN ) = 1 . . lim N !1 ln N
4. f 2 K1, f 2 K
f(x + y) 6 f(x) + f(y) 8x y > 0: H , K1 K | (.
4. F 2 Sf , f 2 K1 p(z), Xn , n > 1, . !" P Xn F , # n=1 (f F ) 6 p((f F )). /8 , , , F 2 Uf F 2 Uf . I 1{4 .. 4 ' 83 ( ' 83 ( ( 4].
5. FG (x) x;aL(x), x ! +1, # a > 0 L(x) |
% , c, Xn , n > 1, . &# FG (x) cx;aL(x), x ! +1.
3.
1085
1. supfx: F 0(x) < 1g = A > 0. %"# " > 0 r > 0, p = ktinf P(h(t) > A ; ") > 0, k6r lnln jGj = A . . lim M(G) G"1 ln jGj
(5)
. 6 '' Cn, n > 1, ( l = 2r Bn On. ? X M(Cn) > M(Bn ) > (A ; ") I(h(On ; k ) > A ; "): k 2Bn
A, ( X n0 = I(h(On ; k ) > A ; ") k 2Bn
p0, Z p0 = P(h(t) > A ; ") dt > jBnjp > 0: ktk6r
1 5 ' Cn Cn+ ( l + 2. ? X M(Cn ) 6 A 1 (
k 2Cn+
n00 =
X k 2Cn+
1
(l + 2)d . A ', ( ) Rf f(x) x ln x, x ! +1. ? 1{3 ( () 8, A ; " 6 lim inf M(HN ) lnlnlnNN 6 lim sup M(HN ) lnlnlnNN 6 A .. N !1 N !1 N lim M(HN ) lnln (6) N !1 ln N = A .. ? (3), (4) (6), '' ( , 0 .
1086
. .
4. 2. F 0 2 Rf , f 2 K1,
lim f(M(G)) = 1 . .
G"1 ln jGj
(7)
. 6 '' Cn, n > 1, ( l Cn+ ( l + 2. 1 M(Cn) > max hk (0) = n0 k 2Cn n0 8 8 F 0(x) = expfld (F 0(x) ; 1)g FG 0(x) ld FG 0 (x) x ! +1 038 Rf 8 3. F ( , X M(Cn) 6 hk (0) = n00 k 2Cn+
n00 | ( (l+2)d ) .. . , 0 Uf 8 4. ? 1{3 8, lim f(M(HN )) = 1 .. N !1 ln N 60 ' 1, 0 . A, Rf 83 f 2 K1 , ' ', 1( , - . / , , ' (' 8 8 ( ).
5. ! " #$% #
3. FG 0(x) x;aL(x), x ! +1, # a > 0 L(x) |
% , u(s) | "% , sFG 0 (u(s)) ! 1 s ! +1, M(G) lim P u(jGj) 6 x = expf;x;a g 8x > 0: (8) G"1
1087
. 0 M(G) k 2 G: M(G) P u(jGj) 6 x > exp ;jGjFG 0(u(jGj)x) ! expf;x;ag 8x > 0: (9) 6 '' Cn , n > 1, ( l > 4 Cn+ ( l + 2, Cn; ( l ; 2 Cn;2 ( l ; 4. / M(Cn) : M(Cn) = max(M(Cn;) M(Cn \ CGn; )) 6 maxf n ng X X n = hk (0) n = hk (0): k 2Cn
k 2Cn+ \C n;2
A, n , n > 1, . 5 : P( n > x) ldFG0(x) P( n > x) ;(l + 2)d ; (l ; 4)d FG0(x) x ! +1. M(H ) N > x 6 1 ; P( 6 u(Nld )x)N + N P( > u(Nld )x) ! P u( n n jHN j) d d ! 1 ; expf;x;ag + (l + 2) l;d (l ; 4) : (10) F 5 (3), (4) (10) M(G) (l + 2)d ; (l ; 4)d : ;a 6 x > exp f; x g ; (11) lim inf P G"1 u(jGj) ld ( (11) l ! 1 (9), 0 .
6. ' : ) " * *)
' hk (t) = k '(t), k , k > 1, | .. . () '(t) | ( . 0, 83 : a) k > 0, '(t) > 0 t 2 E>
) '(0) = 1 '(t) R 6 1 t 2 E> ) M k < 1, '(t) dt < 1. E ? ( ){ ) 3 83
, ( ( ( 1].
1088
. .
4 ' ': ) 8 t0 t00 2 E j'(t0) ; '(t00 )j 6 '(t0 )q(kt0 ; t00k) q(r) 8 r ! 0. 1 , '(t) = '~(ktk), (; ln ') ~ 2 K1 , ) q(r) = '~(r);1 ; 1. * , , '(r) ~ = e;ar , a > 0, ; b ~ = (1 + ar) , a > 0, b > d. '(r) ? ) , 8 ' ' 8 0' E ( 8 '). 40 1 2 ){ ). 0 , 0 ' 0 8 , ' ' ' '. jGj > A .. lim inf M(G) lnln (12) G"1 ln jGj ( 1) (13) lim inf f(M(G)) > 1 .. G"1 ln jGj ( 2). 4'( . A, ) jX(t0 ) ; X(t00 )j 6 X(t0 )q(kt0 ; t00k) .. 08 : sup X(t) 6 X(t0 )(q(r) + 1) .. (14) kt;t0k=r
/ X(0) FX .
6. supfx: F 0(x) < 1g = A > 0, FX 2 Uf , f 2 K ; 1 f(x) A x ln x, x ! 1. . 1 esx ; 1 6 xA;1(esA ; 1), x 2 0 A], s > 0, Z1 sX(0) m(s) = Me = exp (Mesh(t) ; 1) dt 6 expfc(esA ; 1)g
R
0
c = '(t) dt < 1 8 ). 1' 35 E K 5 : FGX (u) 6 m(s)e;su . s = A;1 ln u, u + c(u ; 1) : (15) FGX (u) 6 exp ; u ln A
1089
/ 8 ' (15) (u) 5 F + F + (u) = 0 u < eAc F + (u) = 1 ; (u) u > eAc . ? F + 2 Rf , f 2 K f(x) A;1x ln x, x ! 1. 1 F + 2 Uf . (15) FX 2 Uf .
7. F 0 2 Rf , f 2 K1, FX 2 Uf . . '' '8 f, Z1 f(X(0)) Me 6 exp (Mef(h(t)) ; 1) dt : (16) 0
40 ' ( (16) 8 2 (0 1). ? Mef(h(t)) ; 1 6 (Mef(1))'(t) ; 1 '(t) ln Mef(1) t ! 1: (17) 1 ( (17) ( 0' , ( . (5 ' . 6 '' Cn, n > 1,p ( l Bn On r = dl=2. ? (14) M(HN ) 6 16max sup X(t) 6 (q(r) + 1) 16max X(On ) .. n6N n6N t2Bn
5 X(On ), n > 1, 8 FX . F 3'8 6 7, 2 M(HN ), N ! 1 M(G), G " 1 lim sup M(G) lnlnlnjGjGj j 6 A(q(r) + 1) .. (18) G"1 ( 1) lim sup f(M(G)) 6 q(r) + 1 .. (19) G"1 ln jGj ( 2). 1 ' 0 '8 f. (18) (19) r ! 0 '' (12) (13), (5) (7) . , ' 8 8 '8 ' 6 ', ' (0) = 1, ( (..) , 0 (12) (13) 8 . F ', 1 2 ){ ), ' 0 ( (, 83( ){ ).
1090
. .
7. , $
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-
1] . . // ! . . XI. " #!$ %&'(. ). 177. | %.: - , 1989. | 1. 28{36. 2] % . . 6 ! !$ 7 8. 9 .. . . . :;.-. . | M.: '<=, 1997. 3] >?@ 9. 1 # $ . | '.: ', 1971. 4] B . ? 8. ). 2. | '.: ', 1984. 5] D 1. 1$ # ! :; . | '.: <((%, 1947. 6] Campbell N. R. The study of discontinuous phenomena // Proc. Cambridge Phil. Soc. | 1909. | V. 15. | P. 117{136. 7] Daley D. J. The deHnition of multi-dimensional generalization of shot noise // J. Appl. Probab. | 1971. | V. 8 | P. 128{135. 8] Doney R. A., O'Brien G. L. Loud shot noise // Ann. Appl. Probab. | 1991. | V. 1. | P. 88{103. 9] Gubner J. A. Computation of shot-noise probability distributions and densities // SIAM J. Sci. Comp. | 1996. | V. 17. | P. 750{761. 10] Homble P., McCormick W. P. Weak limit results for the extremes of a class of shot noise processes // J. Appl. Prob. | 1995. | V. 32. | P. 707{726. 11] Hsing T., Teugels J. L. Extremal properties of shot noise processes // Adv. Appl. Prob. | 1989. | V. 21. | P. 513{525. 12] Van Lieshout M. N. M., Molchanov I. S. Shot-noise-weighted processes: a new family of spatial point processes // Stoch. Models. | 1998. | V. 14. | P. 715{734. 13] McCormick W. P. Extremes for shot noise processes with heavy tailed amplitudes // J. Appl. Probab. | 1997. | V. 34. | P. 643{656. 14] Rice S. O. Mathematical analysis of random noise // Bell System Tech. | 1944. | V. 23. | P. 282{332\ 1945. | V. 24. | P. 46{156. 15] Shottky W. U] ber spontane Stromshwankungen in vershieden Elektrizit]atsleitern // Ann. Phys. | 1918. | V. 57. | P. 541{567. 16] Westcott M. On the existence of a generalized shot-noise process // Stud. Probab. Stat. | 1976. | P. 74{88. ' ( 1999 .
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Abstract S. V. Ludkovsky, Stochastic processes on groups of dieomorphisms and loops of real, complex and non-Archimedean manifolds, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1091{1105.
The article is devoted to stochastic processes on in5nite dimensional topological groups that do not satisfy the Campbell{Hausdor6 formula even locally. In the cases of real and complex manifolds the classical stochastic analysis is used, but in the non-Archimedean case the corresponding stochastic antiderivations and antiderivational equations are developed and investigated. For real-valued transition probabilities of random processes the corresponding regular strongly continuousunitary representations are constructed and their topological irreducibility is analysed.
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2.1. . 0 At(M) = f(Uj j ): j g . M + X R . , 4, .: (U1) 8x 2 G , 4 Ux2 Ux1 Uj , 5 8y 2 Ux2 : Ux2 Uy1 ? (U2) j (Ux2 ) Y ; @ ; : j (Ux2 ) B(Y 0 r) =: fy: y 2 Y kyk 6 rg? (U3) . 4 4, . (U1 1) (U2 2) . 4, ; . F2 1 = 2 1 1 .4 . sup kF21 (x)k 6 C x sup kF1 2 (x)k 6 C, C = const > 0 1 2. x . DiB t (M) (LM N) . 4. .4, . %4,5,16,17], M N 6 , (N1) N ; C sup kF(nj ) l (x)k 6 Cn . ;x Sjl n, 0 6 n 2 Z, Vjl 6= ?, Cn > 0 | . , At(N) := f(Vj j ): j g ;
0
0
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2
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1093
S
5 . N, Vjl :=Vj \ Vl , Sjl :=l (Vjl ), Vj = N. Fj . (U1){(U3), (N1) 4. . . N, ; . M . , @, 5 . 2.2. . . . M M~ ; M : M~ ! M, 9 HI w M~ I ~ . 4 ; S (S) := w(M1 (S))=(2w(M)) M. J, %5], ; 3 . . 5 . . %5], ; Y (W X), . .. Pa (W X) := str-indm Y a (Wm X) hP . i1=2 kf k := kf jWm k 2 Y a (Wm X ) %(n(m)!)1+c1 n(m)c2 ] , m=1 kf jWm k 2Y a (Wm X ) := kf jWm k2Y a (Wm X ) ; kf jWm;1 k2Y a (Wm;1 X ) . 4 m > 1 kf jW1 k Y a (W1 X ) := kf jW1 kY a (W1 X ) ? c = (c1 c2), c1 c2 2 R, c < c 5 c1 < c1 c1 = c1 c2 < c2? = (L a c), Y a (W X) ( Z a (W X)) | f 2 YM (W X) ( P 1=2 f 2 Z (W X)), . kf ka := (kf k j )2 =%(j!)a1 j a2 ] < 1, j =0 (kf kj )2 := (kf kj )2 ; (kf kj 1 )2 . 4 j > 1 kf k0 = kf k0 , a = (a1 a2), a1 a20 20 R, a < a , a1 < a1 a1 = a1 a2 < a2. M N . .4. Y a c - . a < a c < c, M N 5 . J, %5], 5 (S M N) (LM N) . . ; , M. O M . . 2.1 (N1), , DiB (M) := Y id (M M) \ Hom(M). 2.3. . M N | . Rn l2 , .4, . 2.2 (i){(vi) %16], 5 . . %16]. . . K = R C l2 (K) | o n 1=2 P x = (xj : xj 2 K j 2 N), 5 kxkl2 := jxj j2j 2 < 1. j =1 = 0 9 ; .. U Rm V Rn l2 R, 0 2 U 0 2 V m n 2 N. Q l
l (f g) H (U V ) 5 4, q 5 6 6 f g: U ! V l (f ) < 1, 2 C (U V ), 0 6 l 2 Z, 2 R, 1 > > 0, q P 1=2 l (f g) := q km M hxi+ Dx (f(x) ; g(x))k2L2 , L2 := L2 (U F ) (. 06 6l F := Rn F = l2 = l2 (R)) . .. 9 6 h: U ! F , . , 4 ;
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N XN := Rn XN := l2 , : M ,! N . .. C -; l (M N) 5 C - 6 . J H g f : M ! N l (f ) < 1, I. l (f g) = P l 1=2 = %q (fij gij )]2 , fij :=i f j 1 4 j (Uj ) \ j (f 1 (Vi )), ij At(M):= f(Ui i): ig At(N):= f(Vj j ): j g | M N. l (U F) H l (TM) 4. 3 , H 0 l (TM) := ff : M ! TM j f 2 H l (M T M) f(x) = x 8x 2 M g H tid (M M) \ (x) = (x 0) 2 Tx M . 4 x 2 M. J Dif t (M) := H \ Hom(M). Dif t (M) 4 6
. . ; 5 .;I (Hl (TM)) =:H l (TM) | ; H , 5 k kH 0 = sup j(f)j. f H =1 : U ! V | C - 6 ., 1 > > 0, , 5 H l (U V ) 6 Q := ff : f 2 E (U V ) , n 2 N 5 supp(f) U \ Rn , d l (f ) < 1g d l : (i) d l (f g) := P l 1=2 := sup (Mn (f g))2 < 1 Rlim d l (f jURc gjURc ) = 0, U | x U n=1 5 , f Mln I. 5 U \ Rn , f jU Rn : U \ Rn ! f(U) V (. t ), Ml (f id)2 :=!2 (l (f j 2 ; %16, x 2.1{2.5] %17] E n (U Rn ) id j(U Rn ) ) ; n l ( n 1) ; (n 1) (f j(U Rn;1 ) id j(U Rn;1 ) )2) . 4 n > 1 Ml1 (f id) := l , l = l(n) > n + 5, = (n) l(n + 1) > l(n) := !1(l (f j(U R1 ) id j(U R1 ) ) q . 4 n l(n) > %t] + signftg + %n=2] + 3, (n) > + signftg + %n=2] + 7=2, !n+1 > n!n > 1. + , Mln (f id)(xn+1 xn+2 : : :) > 0 | l (U \ Rn V ) . f 6
(x1 : : : xn), 5 Ml x1 : : : xn H n (xj : j > n). . (i) 5 n 6 k, k 2 N. . M . 4, ; . , 5 (j i 1 ;idij ) 2 0l 2 H 0 (Uij l2 ) . 4 Ui \ Uj = 6 ? g 0 l ; H 0 , ; Uij | . Rk l2 . j i 1 , l (n) > l(n) + 2, (n) > (n) . 4 n, 1 > > , . fMk : k = k(n) n 2 Ng ;, 5 3.2 %16]. N | , .4, 5 ., , H l (M N) := ff 2 E (M N) j (fij ; ij ) 2 H l (Uij l2 ) . khkL2 :=
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4 fUi ig fUj j g Ui \ Uj 6= ?, l (f ) < 1 lim l (f jMRc jMRc ) = 0g , Di l (M) := R id (M M)g , ff : f 2 Hom(M), f 1 f 2 H l 4, - l (f g) := l (g 1 f id), P 1=2 (ii) l (f g) := (d l (fij gij )i j )2 < 1, gij (x) 2 l2 ij fij (x) 2 l2 , i(Ui ) l2 , Uij = Uij (xn+1 xn+2 : : :) l2 | . fij x1 : : : xn . (xj : j > n) . 6
(foliation) M, Uij Rn ,! l2 , (xj : j > n) Uij | Rn (x1 : : : xn), 1 > > 0. . 5 . Mn Di l (Mn ) Dif l (Mn ) . l l = l(n), = (n), n = dimR (Mn ) < 1. 3 H 00 2.4. . G :=Di l 00 00 00 (M) | G:=Di l (M),
m(n) > n=2, l (n) = l(n) + m(n)n, (n) = max((n) ; m(n)n 0) 8n inf - lim m(n)=n = c > 1=2, > + 1=2, 1 > > + 1=2, > 0 (. 2.3). n G := DiB 0 (M) | G = DiB (M) a1 < a1 c1 < c1, a1 = a1 a2 < a2 ; 1, c1 = c1 c2 < c2 ; 1 M . J : Y ,! Y
{! , Y := Te G Y := Te G | " " . %6]. S . ,4 ; + ; . 3 . < %6,16,17] , 5 . .4. 5 . 2.5. . # " " G = (LM N )0 G = = (LM N) , N | Y b0 d0 - N l2"(K) (. 2.2 2.3), " $ (a) = (L a ), a > a = % (b) = (L a ), a > a > a = (L a)% (c) = (L a c ), a > a > a c > c > c 1 > " > 1=2 d 6 c , 1 > " > 0 d < c , = (L a c), dimK M = 1, a = (a1 a2) a1 > a1 , a1 = a1 a2 > a2 + 1, c, b, d. J : Te G ,! Te G {! . %5,6]. f g f g
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1 : : : l 2 C 0(Br X), F | (k ; l)- l+1 : : : k , G = G(v 1 : : : l ) | ,I 5 G(v 1(v) : : : l (v)), Lk (X1 : : : Xk ? Y ) 5 + k- X1 : : : Xk Y . + X1 : : : Xk Y /0 K Lk (X k ? Y ) := := Lk (X1 : : : Xk ? Y ) . X1 = : : : = Xk = X. l = 0 G = G(v). 3, 6 .: P^(l+1 :::k ) %G(s? 1 : : : l ) (Al+1 : : : Ak )](v) :=
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G(vn? 1 : : : l ) (Al+1 (vn )% l+1 (vn+1 ) ; l+1 (vn )] : : : Ak (vn )% k (vn+1 ) ; k (vn )]) (3) vn = n(t), fn : n = 0 1 2 : : :g | ; 6 Br (. %20, x 62, 79]). 3.2. . E H | + F, F | 5 5 /0 . J F- A 2 L(E H) . Lq (E n 2 N, 5 PH), q , an 2 E yn 2 H . 4 P q (i) kan kE kynkH < 1 A ; (ii) Ax = an(x)yn . n=1 n=1 4 x 2 E, 0 < q < 1, E | 5 n P.;I o1=q q . . 4 A (iii) q (A) = inf kan kE kynkqH , n=1 inf I. . (ii), 5I (iv) (A) := kAk L (E H) := L(E H). S5 , 5 9 5 , 1 6 q . .. + . 3.3. . Lk (H1 : : : Hk? H) k- ; H1 : : : Hk H , ; L(E H), E | + n=0
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H1 : : : Hk max- , H1 : : : Hk H | + F. J , + Lrk (H1 : : : Hk? H) := Lk (H1 : : : Hk ? H) \ Lr (E? H), 5 Lrk (H k ? H) := Lk (H k ? H) \ Lr (H k ? H) L k (H1 : : : Hk? H) := :=Lk (H1 : : : Hk ? H) r (J)=: kJ kr , 1 6 r 6 1. W 5 0 < r < 1. (X B ) | . ( 6 ), B | - ; X, 5 . /0 Ks , s 6= p, K = Kp , s p | 5 , B | ; , 5I kk = 1 (X) = 1. 3, 4 K- + Lq (X B ? Lrk(H1 : : : Hk ? H)) Lq (X B ? Lk(H1 : : : Hk ? H)) Pn A Ch A 2 L (H : : : H ? H) . ; j Wj j rk 1 k j =1 Aj 2 Lk (H1 : : : Hk? H) , Wj 2 B n 2 N. J X 3 7!n A() 2 Lrk (Ho1 : : : Hk ? H), . kA()kr - 1=q R kA() q kAkLq := kr (d) < 1 . 5 , kAkLq := := fess- sup kA(!)kqr N (!)g1=q < 1 . Ks - 5 , 1 6 q < 1? kAkL1 := ! := ess-sup kA()kr , 6 . N %19]. 3.4. C - X. Q Lu (X B ? C n(X H)) m P
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j (x)n 2 C n(X H), Wj 2 oB m 2 N k kLu := 1=u R := k (x )kuC n(XH ) (d) < 1 . 6 , k kLu := := ess- supfk (x )kuC n(XH ) N ()g1=u < 1 . Ks - 5 . 4 1 6 u < 1 k kL1 := ess- sup k (x )kC n(XH ) < 1, k (x )kC n(XH ) |
. 6 . . 3.5. . Y | ; f(Xj Rj j ): j 2 Yg | Xj Ks - 5 5 j 4, Xj ; Rj , .4, 5 Xj . ; Rj 6 4 j Xj . , 4 ; . jk : Xk ! Xj . 4 k > j Y, 5 (jk ) 1 (Rj ) Rk , jj (x) = x . 4 x 2 Xj . 4 j 2 Y, km lk = lm . 4 m > k > l 2 Y? lk ( k ) = ( l ) . 4 k > l 2 Y. J . . ; kXj kj < 1 . 4 j , 4 lim (X )=:dX 2 Ks jlim kXj kj =:qX < 1, Y0 := fj : j 2 Y j (Xj ) 6= 0g j 0 j j
Y n Y0 5 Y. 3.6. . f(Xj Rj j ): j 2 Yg | & 3.5. ) & & (X R ) j : X ! Xj 8j 2 Y, (j ) 1(Rj ) R j ( ) = j 8j 2 Y.
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, S := fXk jk Yg. J lim S S =:X | 5 (. %3, x 2.5]). R := j 1 (Rj ) 6 4 , j 4 5 (j 1 (A)) := j (A) 8A 2 Rj j 2 Y, j : X ! Xj | ; .. O A \ B = ? R, , j 2 Y, 5 A B 2 Gj := j 1(Rj ), , , 5I kAk = kC kj . A = j 1(C) C 2 Rj . 3, 6 . N (x) (X R), 5 . 4 x 2 X > 0 , A 2 R kAk ; < N (x) 6 kAk (. %19, . 7]). < 6
Nj (xj ) (Xj Rj ) . 4 x 2 X > 0 , 4 j 2 Y A = j 1 (C) 2 R, 5 Nj (yj ) < N (x) + . 4 y 2 A, yj := j (y). 9 . 4 x 2 X > 0 , A, 5 N (y) < N (x) + . 4 y 2 A, N (x) (X R) 5. J kAk = sup N (x) . 4 A 2 R. . ; V x X X > 0 , 5 fE1 : : : Emg R . V , 5 , . 4 El A. 9 sup N (x) 6 l=1max kE k 6 sup N (x) + 2 sup N (x) = inf kAk . . :::m l x V A V x V x V 4 j > 0 ; Xjj := fxj : xj 2 Xj Nj (xj ) > j > 0g (Xj j ) (. %19]). O xk 2 Xkk , Nj (jk (xk )) > k . 4 j < k, (jk ) 1(Rj ) Rk kB kk 6 kAkk . 4 B A 2 Rk B A. j = > 0 . 4 j. , jk (Xkk ) Xjk . 4 j 6 k 2 Y. 3 3.2.13 %3] X j : j :=limfXkk jk Yg | ; X. J . 4 6 S R 5 , Alim kAk = 0, N (x) 6 qX . 4 x 2 X n X j : j . . R (. %19]), 5 (X R ), 5I (X) = dX . 3.7. . f(X Gj j ): j 2 Yg |
; Y Gj Gk . 4 j 6 k 2 Y, S G . R= j : R ! Ks , 5 j j = j k j j = j . 4 j j 6 k 2 Y, X | ; 4, R, .4, 5
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5: kX k < 1. 3.8. " . (X R ) | * 3.7. & " R " R . . X | . X, ;I. R. ;. Gj I @ 9 j ; j : X ! Xj , j (Gj ) =: Rj | 4,. , .4,. 5 Xj , Xj | , 5I Rj | .
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: (X F) ! (X B) . 5 5 (3<), B | - . R- 5 , . Ks - 5 , . 5 B Bco(X) . K- X, 1 (B) F, e K | /0 , . + Lq (T R H) 6 f : T ! H kf k q := ess- sup kf(t)kH N (t)1=q . Ks - 5t T 1 6 q < 1 kf k := sup kf(t)k q , H | + K, (T R ) | 16q< . . . 6 Lq @
. 3.10. . 3< w(t !) 5 . H t 2 2 B(K 0 1)=:T . 5 6 (3 ), (i) w(t4 !); ; w(t3 !) w(t2 !) ; w(t1 !) . 4 ! 2 X, (t1 t2) (t3 t4) t1 6= t2, t3 6= t4 , t1 t2 ; 55 ; ft3 t4g? (ii) 3< !(t !) ; !(u !) Ftu , | .. 6. Ks - 5. (X(T H) B), X(T H) | Lv (T R H), Cb0(T H), H T , g (A) := (g 1 (A)) . g: X ! H, 5 g 1 (RH ) B, 4 A 2 RH , Ftu : X ! H I. Ftu(w) := w(t !) ; w(u !) . 4 w 2 Lq (X F ? X0), 1 6 q 6 1, X0 | X, I . T0 := T n f0g, RH | ; , 4,. H, . 5 Ftu1(RH ) B . 4 t 6= u 2 T ? (iii) w(0 !) = 0, X 6= ?. 3.11. . & ;K K (0 1). ) & ] + ", - X ) card(]) > card(T) card(H) X = Cb0(T H) X = H T card(]) > card(R) card(H) X = Lv (T R H). . , 5 + H c0( K) (. 5.13 5.16 %19]). J, . K, ; . H X ,4 6 H
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c0( C K), L C | ; . . 4 5 5 5 5 (t0 t1 : : : tn) 2 T 5 (0 z1 : : : zn) 2 H , X(T H? (t0 t1 : : : tn)? (0 z1 : : : zn )) X(T H), 5 X(T H? (t0 t1 : : : tn)? (0 z1 : : : zn)) = (0 z1 : : : zn) + + X(T H? (t0 t1 : : : tn)? (0 : : : 0)), X(T H? (t0 t1 : : : tn)? (0 : : : 0)) | K- X(T H). 9 Ft2 t1 1 (R(H))
Ft4 t1 3 (R(H)) | R(X(T H)), (t1 t2) (t3 t4) .4 3.10(i). J , P(t1 x1 t2 A) := := (ff : f 2 X(T H? (t0 t1)? (0 x1)) f(t2 ) 2 Ag) . 4 t1 6= t2 2 T , x1 2 H A 2 R(H). 9 ; L ! := := ff : f 2 X(T H? (t0 t1 : : : tn)? (0 x1 : : : xn))g, Y! | 5 ; N, xi 2 H, (ti : i 2 Y! ) T n ft0g. 3, U~ 6 5 ; Q X0(T H), 6 . 6 . v : X0 (T H) ! H v , H v := Ht, v = (t1 : : : tn) | 5 t v ; T , Ht = H . 4 t 2 T. ^ 5 ~x0 (X0 (T H) U~ ) 6 .4 . . 4 . (X F ) ; . : X ! L, 1 (R(L)) F, () = , , Lq (X F X0) 6 . 5 6, 5 X = L = id. 3.12. . + H E 2 Lr (X F ? 0 C (T Lv (H))), 5 E = E(t !), 1 6 v 6 1, 1 6 r 6 1, t 2 T = = B(K t0 R) ! 2 X. . Lr (X F?C 0(T Lv (H))) 5 6 (30) P I. I(E)(t !):=(P^w E)(t !)= E(tj !)%w(tj +1 !);w(tj !)], j =0 w = w(t !), tj = j (t) | 6 . 6 (. %20]). S5 , 5 I | K- Lr (X F ? 0 C (T Lv (H))) Lq (X F ? C00(T H)) Lu (X F ? C 0(T H)), 1=q+1=r = 1=u
1 6 r q u 6 1. a 2 Lv (X F ? C 0(T H)), w 2 Lq (X F ? C00(T H)) E 2 Lr (X F ? 0 C (T L(H))), 1=r+1=q = 1=v, 1 6 r q v 6 1, a = a(t !), E = E(t !), t 2 T, ! 2 X. 3 (t !) = 0(!) + (P^u a)(u !)ju=t + (P^w(u!)E)(u !)ju=t . P^ub wh 5 6 P^(1 :::b+h ) 3.1, 1 = u : : : b = u, b+1 = w,. . ., b+h = w. M nf)(t x? 3.13. . f(u x) 2 C (T H Y ) (i) lim max k(_ n 06l6n h1 : : : hn? 1 : : : n)kC 0 (T B(K0r)l B(H01)n;l B(K0R1 )n;l Y ) = 0 $ 0 < R1 < 1, r = q = v = 1, hj = e1 j 2 B(K 0 r) ",, $), t 2 T = B(K t0 r), hj 2 B(H 0 1), j 2 B(K 0 R1) ",, $), x 2 H , (ii) f(t (t !)) = f(t0 0) + ;
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m+b>1 l=0 06m2Z 06b2Z (I b a(m;l) E l )])ju=t. . < J (. %7, 15, 20]) f(t x) = Pk ((m+b)!);1 ;m+b(@(m+b) f=@ub @xm)(t x )(t;t )b(x;x )m + = f(t0 x0)+ 0 0 0 0 m m+b=1 ; P k +1 + %(_M k+1f)(t0 x0? (t ; t0)b (x ; x0)m ? 1(k+1)) ; ((k + 1)!);1 m+b=k+1 m (@ (k+1) f=@ub @xm )(t0 x0)(t ; t0 )b (x ; x0)m ] . 4 k 2 N. . 6 4 6 n T .. (tn+1 !) ; (tn !) = = a(tn !)(tn+1 ; tn ) + E(tn !)(w(tn+1 !) ; w(tn !)), tn := n (t), 5
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At(G) At(G ). <; Te G Te G . .. L1 . G G . S X(T Y ) , 5 S = S1 S2 , S1 | X(T K) S2 | Y , Sj = Aj Dj , Aj | 5 , Dj | (eji : i), ker(Sj ) = f0g, 5I D1 2 Lr (X(T K)) D2 2 Lr (Y ), r = 1 . 6 r = 0 ( ) . Ks - 5 , 6
Sj ; Keji 4 eji ( Sj )(dz) = fji(z)v(dz)=jdji j, fji 2 L1(K v V), v | : K 5 . V, V = R V = Ks , s 6= p, K Qp , fji(z) = f(z=dji), Dj eji = djieji, dji 6= 0. < 3.11 , 3 w(t !), 4, S . G U 54 (a E ), a 2 T G, E 2 L12 (H T G) ker(E ) = f0g, 5 (a E ) (alk ) . 4 0
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l k .4 ;4, .. . V . Y a = a(t ! y) E = E(t ! y)? (i) a 2 C 1(V L (X F ? C 0(BR L (X F?C 0(BR H)))))? (ii) E 2 C 1(V L (X F?C 0(BR L(L (X F? C 0(BR H)))))), t 2 BR , ! 2 X, 2 L (X F ? C 0(BR H)), 0 2 L (X F ? H), w 2 L (X F ? C00(BR H)), a E .4 4 7 @ 6 (LLC): . 4 0 < r < 1 , Kr > 0, 5 max(ka(t ! x y) ; a(t ! z y)k kE(t ! x y) ; E(t ! z y)k) 6 Kr kx ; z k . 4 x z 2 B(C 0(BR H) 0 r), y 2 V t 2 BR , ! 2 X? (iii) ay (t ! y) @a(t ! y)=@y 2 X0 (H):= fz: S 1 z 2 H g @E(t ! y)=@y 2 2 Lr (H) . -. . ! 4 t, , y, r = 1 . R- 5 , r = 0 . Ks - 5 , . . f c ST . @ . R- 5 ; . r = 0, zj | z H. < 3.13
,4 6 5 , 5 3 (iv) (t !) = 0 (!) + + (P^u a)(u ! )ju=t + (P^w(u!)E)(u ! )ju=t @ , . (i){(iii), LLC. 3 (v) (t !) = 0(!) + m P P + (P^ub+m;l w(u!)l %am l+bl (u (u !)) (I b a (m l) E l )])ju=t m+b=1 l=0 (vi) am ll 2 C 1(V C 0 (BR1 B(L (X F ? C 0(BR H)) 0 R2) Lm (H m ? H))) ( . 5. ) . 4 n l, 0 < R2 < 1 1 0 n = 0 . 4lim sup ka k 1 0 n 06l6n n ll C (VC (BR1 B(L ( F!C (BR H ))0R2 )Ln (H H ))) 0 < R1 6 R 0 < R < 1, 4 0 < R1 < R R = 1, . 4 0 < R2 < 1? (vii) @alk (t ! y)=@y 2 Lk+lr (H (k+l) ? H) . 4 l k r = 1 r = 0 . J (v) @ BR , . (i){(iii), LLC, (vi), (vii). . %2] I %15], 5 , 5 . 3
(t ! y), . .4,. @ . (iv) (v) ., y 2 V Y , . Py (A) ; . U(y2 y? (t ! y)) := (t ! y2) . 4 y y2 2 V . ; . I + X , 5 T G X T G. . A 2 Lu (u = 1 . ; , u = 0 . Ks - 5 ) ker(A ) = f0g, A I - 4 64, 4 (tight) A
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5.1. . X | (. . ) - & - & * & + & & & supp() = X & & & & & " G & $)& X & 2 *& . 3 (i) spC f j (g) := := ( h (dg)=(dg))1=2 h 2 G g H (ii) $", f1j f2j 2 H , j = 1 : : : n, n 2 N $ > 0 ) h 2 G , & j(Th f1j f2j )j 6 j(f1j f2j )j, j(f1j f2j )j > 0, T : G ! U(H) , H := L2 (X C). . , 5 (i) , 5 f0 6 5, f0 2 H f0 (g) = 1 . 4 g 2 X. < card(X) > @0 9 5 . 4 n 2 N , 4 Uj 2 Bf(X) gj 2 G , 5 0
0
0
0
(gj Uj ) \
i=1:::j ;1j +1:::n
Ui
0
=0
n Y
j =1
j (Uj ) > 0:
< (ii) 9 I, 5 , 5 G - H H, 5 Th H H . 4 h 2 G H 6= f0g. 9 ; G - H 6= 0 H
L2(V C), V 2 Bf(X) (V ) > 0. S. 5 5 5 %11,12,17]. 5.2. . . G x 4 ) - , "& , $ 014 & " G x 4, * T : G ! U(L2 (G C)) . / . . G x 4 , 5 6 . f 2 L1 (G C) . 4 f h (g) = f(g) (mod ) g 2 G . 4 h 2 G , f(x) = const (mod ), f h (g) := f(hg), g 2 G. < IV.4.8 %18] 5 5 5 x 4 3< f(Bt1 : : : Btn ): ti 2 T 2 C0 (Rn) n 2 Ng L2 (FT ), T 3 t0 , T | @ R Kp . n n RT < IV.4.9 %18] . 5 3< exp h(t) dBt (!) ; 0
0
0
0
0
0
0
0
0
0
0
0
1
0 o T R 2 2 2 ; h (t) dt=2 : h 2 L %T ] (deterministic)g L (FT ) 5 0
1104
. .
5. < 2.1.1 5.4.2 %21] spC f(g) := ((h g))1=2 : h 2 G g =: Q H, (e g) = 1 . 4 g 2 G Lh : G ! G | . G, Lh (g) = hg. < 5 9 5 . f S, , ; .. f0 xlim f0 (x) = 0, f = f0 + h, 0 6 jh(x)j < jf(x)j . 4 x 2 Kp . < x 4 , , . 9 5, . 5.1(i), (ii) . 4 G G | 5 . J 5.1 ; . 0
j j!1
0
)
1] Belopolskaya Ya. I., Dalecky Yu. L. Stochastic equations and dierential geometry. | Dordrecht: Kluwer, 1989. 2] Dalecky Yu. L., Fomin S. V. Measures and dierential equations in in nite dimensional space. | Dordrecht: Kluwer, 1991. 3] . . | .: !, 1986. 4] $%& '() *. +. ,'-'! . /!. !0 1 !/ '.1 / !-) // 3 . 45. | 2000. | 7. 370, 9 3. | *. 306{308. 5] L !1/& '.1 1 '.1 / !-) ' !1/& '. 1 '. ! ( !( ' // ?( . / . 0. | 1998. | 7. 53, 9 5. | *. 241{242. 10] L
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13] $%& '() *. +. ,'-'! . ('& &@@!C!0/. &)( ' -E. /!. !1/& '.1 1 '.1 ! ( !( '1 // Analysis Mat. | 2002. 14] $%& '() *. +. ,'-'! . ('& &@@!C!0/. /!. !1/& '.1 1 '.1 ! ( !( '1 // ?( . / . 0. | 2002. 15] $%& '() *. +. Quasi-invariant and pseudo-dierentiable measures on non-Archimedean Banach spaces with values in non-Archimedean elds // J. Math. Sci. | 2002. 16] L
. .
,
512.552.12
: , { ,
, ! " .
# " R |
" , X = fxi : i 2 ;g | ( " ) " + (, F = ffi : i 2 ;g | " ," - +! R A(RF ) | { . 0
"" ! , " ! " R1XF ] ( " ! " 3 A(RF ), Aa 6= 0 - " a 2 A. 5 " , ( ! , " 67 ( ( ") dim R = dim R1XF ] " . 8 37 ( "9" "". 8( " dim R = dim R1xf ] " , R | ( ( , f | ," - +! R dim R < 1?
Abstract
V. A. Mushrub, On the uniform dimension of skew polynomial rings in many variables, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1107{1121.
Let R be an associative ring, X = fxi : i 2 ;g be a nonempty set of variables, F = ffi : i 2 ;g be a family of injectivering endomorphismsof R and A(RF ) be the Cohn{Jordan extension. In this paper we prove that the left uniform dimension of the skew polynomial ring R1XF ] is equal to the left uniform dimension of A(R F ), provided that Aa 6= 0 for all nonzero a 2 A. Furthermore, we show that for semiprime rings the equality dim R = dim R1XF ] does not hold in the general case. The following problem is still open. Does dim R = dim R1xf ] hold if R is a semiprime ring, f is an injective ring endomorphism of R and dim R < 1?
1.
R | ( ,
), X = fxi : i 2 ;g | , G | , X, F = ffi : i 2 ;g |
=6 " ( ) " > 97-01-00785.
=
+
+ "
,
, 2001, " 7, > 4, . 1107{1121. c 2001 , ! "# $
1108
. .
$ % & ' R, ( fi fj = fj fi ) % & i j 2 ;. +,
I(R) | - % ( %) $ % & ' R, : G ;! I(R) | -' , ( (xi ) = fi ) % i 2 ;. / g 2 G, & (g) ( ( g . 0 % -( (1
2 ) - R R3X F], & - ( P rww, - rw 2 R, ) - - w2G - R3X],
) ) , : au bv = au(b) uv ) % & a b 2 R u v 2 G: 51 ) 2 - ), ( ( % -( , ( 6 - -'
34]. 8 & % 1 % 35{8]. + ( ) ) ) ( ; ) % -( R3X F]. / B | , ( dimB ( , , B. <1 ) = -( 39] , ( dimR = n - -, - dimR3x1 x2 : : :] = n, - R3x1 x2 : : :] | -( ( ' % %. ? 1.2 ) ) ) ), ,
= % -( - . + ) ) = % % R3X F ] R. % % )% ( , R | ( ; ) &
. 2 , 3 , ,, ( ( R3X F] R - ( . 2 R, )
3.2,
( , , , $ & ' f, ( 1
2 R3x f] ) ) ) , 2 , , dimR3x f] = 1. 0 -, (.
3.1), ( ) - - - ( n R $ & ' f : R ;! R,
( dimR = n dimR3x f] = 1. B , ( R,
3.1, ) ) ) ( . / R % & r, % ( Rr = 0, R
R3X F] ) dimR3X F ] 6 dimR. B , ) : R f R, dimR < 1 dimR 6= dimR3x f]?
1109
5 39] , ( , , = ) . + , R | ( , ( R2 = 0. ?- dimR < 1, -( R3x],
( , , ; , ( , ), L1 Rx n % % % . n=0 C) - ( ' , ) ,
. 1.1. A | : G ;! Aut(A) | -' . (A ) ) 1
0 {E (R ),
A -' ), , ): (1) R ) ) ) A -' u ) ) )
-' u ) - & u 2 GF S (2) A = g;1(R). g2G 0 A, -' 1
0 {E (R ), ( ( A(R )
1
0 {E R -' . 0 2, 1
0 {E R
-' - ( ,
' R. 2 ) ) ) ) ,) . 1.2. A | { R . ! ": (1) " $ % a 2 A Aa 6= 0, dimR3X F] = dimA' (2) A ! ( % a, ( ) Aa = 0, dimR3X F] = 1. G 4 ) .
2. {
Z| % ( . R1 = R, R | , R1 = R + Z, R . I , (
R | , & ' g $ - & ' R1, - g (1) = 1. J ) & g 2 G % 1 g 2 R1
g 7! 1 g. 2.1. * ! G " " " ! + ( (R1 )3X F ]. * ! A = fu;1 ru: u 2 G r 2 Rg " " "
1110
. .
) + G;1(R1 )3X F ]. , : G ;! Aut(A), ( u (b) = ubu;1 " + % u 2 G b 2 A. (A ) " " " { (R ) (. 1.1). . 5 u;1 ru + v;1 sv = (uv);1 (v (r) + u (s))uv 8u v 2 G 8r s 2 R , ( A ). K - ( , ( & ). B , A | G;1(R1 )3X F ]. / r 2 R u 2 G, u(r) = uru;1 = u (r) & u ) ) )
-' u. ? ,
(1) ) 1.1 1 ) ) , ( S .S C) A = g;1 Rg = g;1 (R). g2G
g 2G
L 1 ) 1 ) 0 {E ) ) )
, - E 310]. C- ) &- 1 ) -'
31] (. 7.3.4)
, . 0 & u 2 G & ) Ru R
' iu : R ! Ru . 0 & u g 2 G
fuug = igu g i;u 1 : Ru ;! Rug . 5 1
( ( )( 6 G, ) & , u 6 v () 9g 2 G 3v = gu]: N ( )( (G 6) ) . 0 -, ) , % & u g h 2 G fuhgu = fguhgu fugu : & ) fRu fugu : u g 2 Gg. 8 , ( ) A(R ) = lim ;!(Ru fug : u g 2 G) ) ) ) 1
0 {E R -' . + , "u : Ru ;! A(R ) |
( "g = "u g fugu ) % u g 2 G). ?- -' : G ;! Aut(A(R )) g ("u iu (r)) = "u iu (g (r)). & & ' g ) ) ) ' ,
g;1, g;1("u iu (r)) = " iu (gu igu (r)). 2 )) R i1 (R), ( , ( g;1(R) = "g ig (R)
A(R ) =
Im"g =
g2G
g2G
;1
g
(R):
1111
? , ; lim !(Ru fug : u g 2 G), , ) ) (1) (2) ) 1.1. 51
0 {E (A ) (R ) , . / B | ,
R ( , ': G ;! Aut(B) | -' , ( 'g (r) = g (r) ) % & g 2 G r 2 R, A ) B ) g;1(r) 7! ';g 1(r). L ) & , , ( 1
0 {E R -' ( , ' R. C , S (B ') | - 1
0 {E (R ), B = ';g 1 (R), ,
g;1 (r) 7! ';g 1 (r) g2G ) ) ) ' . 2 , ( A(R ) = R - -, - ) fi , i 2 ;, ) ),) ' R ( - Im Aut(R)).
2.2. -" $ + % a1 a2 : : :an 2 A % u 2 G, ( ) uai u;1 = (ai ) 2 R " + ) i = 1 2 : : : n. . 0 & ai ai = vi;1 ri vi , - ri 2 R, vi 2 G. & u = v1 v2 : : :vn . 2.3. ! , ) dimR < 1. dimA 6 dimR. . 5 ( ( ,n - dimA = n < 1. P + & ( A ), A1 ai n % i=1 % - % % . + 2.2 & u 2 G, ;1 2 R ) % i = 1 2 : : : n. P - , ( Pn R1ua(iu;ua1 |iu )) n % % R, i=1 , dimR > n. , ( dimA = 1. 5 ), - ( 1 , ,, ( ) - - - ( n R ), n % % % . 2, , ( dim R = 1.
2.4. A(R ) " " " $ . , R3X F ] | $ . . . 1.2 dimR3X F] = 1 dimA(R ) = 1. & ) , ( A(R ) ) ) ) , - -, - R3X F ] | . 5 , , )( X. G ) )( -' ( 1
) G. / x1 < x2 < : : : < xn, xi 2 X, xk11 xk22 : : :xknn < xl11 xl22 : : :xlnn () 9i 6 n (k1 = l1 ^ : : : ^ ki;1 = li;1 ^ ki < li ):
1112
. .
G ) ( ) & R3X F] )
1 - ( . , ( A(R ) ) ) ) ,. ?- R | . ? R3X F] 1 ( ) & , 1 % ( , R3X F] ). ? , R3X F ] ) ) ) ,. C , ,. , ( % -( S R3X F ] ) ) ) ,. ?- R | . 5 A(R ) = g;1 (R) , ( A(R ) ) ) ) ,. g2G
2.5. / R | $ . , R3X F ] ! " " " $ . . . ? dimR = 1, dimA = 1 , 2.3. ) ), - ( 2.4, , ( A ) ) ) ,. C) 1 ) ) 2.4. . / F = ff g, R3X F] ' Q ( R % -( R3x f], ( f = x . S
R3x f] ) ) , : axn bxm = af n (b)xm+n ) % & a b 2 R % % % ( n m. + ( A(R f) A(R ). / ; = f1 2 : : : ng F = ff1 f2 : : : fn g, R3X F] ( ( R3x1 f1 x2 f2 : : : xn fn]. 8 , ( R3X F ] ' 1 , 2 (R3x1 f1 : : : xn;1 fn;1])3xn fn ] R3x1 f1 : : : xn;1 fn;1], - & ' fn $ - & ' R3x1 f1 : : : xn;1 fn;1] , ( fn (x1) = x1,.. ., fn (xn;1) = xn;1. & R3X F] ( ) % -( (: : :((R3x1 f1 ])3x2 f2 ]) : : :)3xn fn].
3. +,
K | ) ),) K-- . N = fn 2 Z: n > 1g, T = fti : i 2 Ng | ( %, L = K3T] | -( K. 3.1. -" ! ! ) n R, " " n, 0 ( % f R, ) dimR3x f] = = dimA(R f) = 1. . L3a] = L3y]=(y2), - a = y + (y2 ). I , ( L3a] = L La ( ) L
& a, - ,. 5 - R K--
1113
L3a], , & a t1 a t2 a : : : tn;1a fti : i > ng. 6 , ( R = L3a1 a2 : : :an]=I, - I | , faiaj : 1 6 i j 6 ng. ? Rat1 Rat2 : : : Ratn | R Rati ) ),) ) % ( i 6 n, dimR = n. ' | & ' K-- L3a], ( '(ti ) = ti+1 ) % i > 1 '(a) = a. ?- '(R) R. 5 $ & ' f R, & ' '. 8 , ( A(R f) = A(L3a] ') = (K3Te])3y]=(y2 ), e - T = fti : i 2 Zg. 2, , ( A(R f) = L3a], , dimA(R f) = dimL3a] = 1. ? A(R f) | , ,
) 1.2, ( dimR3x f] = 1. 3.2. 1 $ R 0 ( % f $ R, $ ( : 1) R " " " $ . , , , dimR = 1, 2) A(R f) " " " , 3) R3x f] " " " $ . ( , dimR3x f] = 1). . X = fxi : i 2 Ng | ( ( %. C) - - - ( i
$ & ' fi K-- L, -) fi (ti ) = t2i fi (tj ) = tj j 6= i: F = ffi : i > 1g S = L3X F ]. 2 ( ( S1 - - S, , & x1 t1, ( S2 | - , , fti : i > 2g fxi : i > 2g. 8 , ( S = S1 K S2 , (
- ), ' K--
: S1 K S2 ;! S ( (1 xi ) = xi, (1 ti ) = ti ) % i > 2 (x1 1) = x1 ,
(t1 1) = t1. K hy z i | ) - , ) fy z g. I , ( - S1 ' 1 , 2 (K3t1 ])3x1 f1 ] -( K3t1 ]. ? - K-- (K3t1])3x1 f1], ) & x1 x1t1 , (. 32, -' 7.2,
10] 311]), , ) & ' , , ( - K-- S1 , ) & x1 x1t1 , . 2, , (
: K hy z i ;! S1 , ( (y) = x1
(z) = x1 t1. I , ( - S S1 ' , , ( , ) ),
1 ). L' : S ;! S1 ) (xi ) = xi+1 (ti ) = ti+1 .
1114
. .
? ) $ , ) 9.1.C 33] ( , ( -' = : K hy z i S ;! S1 S2 ) ) ) $ . 2, , ( -' ( ): K hy z iS ;! S ) ) )
$ . ))
2.5, ( , ( S | 2 . S = S n f0g, D = S ;1 S | ( % S R = K hy z i K D. ?- R | ( ( % R = K hy z i K S 1 S . + , P & r R r = ai s;1 bi = P ; 1 = (1 s ) (ai bi), - ai 2 K hy z i, bi 2 D s 2 S . 5 R = (1 S );1 (K hy z i S) D = S ;1 S ,, (
( ) ) : R ;! D. ? R , ) ) ) ,. f | & ' R, ( f(r) = 1 (r) ) % & r 2 R. P - , ( & ' f $ . + ) & ' f ) ) f(y 1) = 1 x1, f(z 1) = 1 t1x1 f(1 ti ) = 1 ti+1 , f(1 xi ) = 1 xi+1 ) % ( i > 1. C , ( ) R & ' f ) 1){3). 1) 8 , ( R(y 1) \ R(z 1) = 0 (y 1) \ (z 1)R = 0: ? R | , dimR > 1 , ( R ) ) ) , 2 , . 2) ' 2 Aut(A(R f)) | ' A(R f), , & ' f. ?- A(R f) = ';n (R): n>0
& ) , - - & a A(R f) , & r 2 R ( n,
( a = ';n (r), , 'n+1 (a) = f(r) 2 1 D. ? 1 D | D A(R f), , , ( & 'n+1 (a) A(R f) , , & a A(R f). + & a - 1 , ( A(R f) | . 3) G
) ) )
2.4
) 2).
4. !" # $ #
+,
A ( -' , , 1
0 {E (R ). C) - & -
1115
i 2 ; 'i = (xi ). L ' f'i : i 2 ;g ( ( Fe . / K | A, ( "(K) (
A3X Fe ], (, ) - K X "(K) = u (K)u: (1)
P uK, u G , (1) 2
P - , ( "(K) = u2G ) % ),
) $
" 1 L(A) %
A 1 L(A3X Fe]) % A3X Fe].
4.1. / D | ( ( A, "(D) | ( ( A3X Fe]. . C) ( , ( ) , - - -( p 2 A3X Fe] & s 2 A Z, ) p, ( 0 6= sp 2 "(D). J &
, ) , ( % ( , %) % -( p. / p | ( ,
( . 5 ( , - p=
i=n X ai wi
i=0 wi 6=
- n > 1, 0 6= ai 2 A, wi 2 G
wj i 6= j.
& t 2 A Z, ( 0 6= t
,
X a w 2 "(D):
i=n;1 i=0
i i
/ tan = 0, 0 6= tp 2 "(D). / tan 6= 0, & b 2 A + Z, ( btan 2 wn (D). + ( ( s = bt. +,
, - , - ), ( a 2 A Aa 6= 0.
4.2.
(1) 2 ( ( K A " " " , " $ + + % a b 2 K Aa \ Ab 6= 0. (2) -" $ + ) p q 2 A3X Fe] % ": (i) A3X Fe ]p \ A3X Fe]q 6= 0, (ii) A1 3X Fe]p \ A1 3X Fe]q 6= 0.
1116
. .
(3) 2 ( ( I A3X Fe] " " "
, " $ + + ) p q 2 I A3X Fe]p \ A3X Fe ]q 6= 0. " f | A. / S A, ( `(S) ( ) S A, `(S) = = fa 2 A: aS = 0g. g(x) = an xn + : : : + a1 x + a0 2 A3x f] | -( , - ai 2 A, an 6= 0. V - , ( g(x) , ) - ( j = 1 2 : : : n ) aj 6= 0 ) `(aj ) = `(an ).
4.3. -" ! ) g(x) A3x f] % b 2 A1, ( ) bg(x) 6= 0 " % ) bg(x) . C & , - ( ) 2.2 39].
4.4. p(x) = anxn+: : :+a1x+a0 q(x) = bmxm +: : :+b1x+b0 | ) A3x f], n > 0, m > 0, an 6= 0, bm 6= 0. ! , ) " % ) p(x) . % ": i) q(x)p(x) = 0' ii) bj af j (ak ) = 0 " + ) j = 0 : : : m k = 0 : : : n' iii) q(x)an = 0. . I (, ( `(f j (an)) `(f j (ai)) 8j = 0 : : : m 8i = 0 : : : n ; 1: (2) i) ) ii). q(x)p(x) = cm+n xm+n + : : : + c1 x + c0. ?- 0 = cm+n = bm f n (an) (2) , ( bm f i (ai ) = 0 ) % ( i = 0 : : : n. ? 0 = cm+n;1 = bm f n (an;1) + bm;1 f m;1 (an ) = bm;1 f m;1 (an), bm;1 f m;1 (an) = 0 ) % i = 0 : : : n. ) ), ( , ( 0 = ci+j = bj f j (an) ) % ( j = 0 : : : m. B , ii). L
ii) ) iii) iii) ) i) ( .
4.5. p(x) | ( ) A3x f], z 2 A
k | ) . ! , ) " % ) q(x) zxk p(x) 6= 0. " % ) zxnq(x) . . S
(
`(ai ) = `(an ) ) `(zf k (ai )) = `(zf k (an)):
4.6. / K | ( ( A, "1(K) = 1 P = f i (K)xi | ( ( A3x f]. i=0
1117
. C -. , ( "1 (K) ) ) ) , ( &
( ,. T = A3x f]. ? "(K) ) ) ) , M = f(deg t(x) deg h(x)): 0 6= t(x) 2 "(K), 0 6= h(x) 2 "(K)
Tt(x) \ Th(x) = 0g . ? % )( % % % ( ) ) ) -' ( )( , , -( p(x) 2 "(K) n f0g q(x) 2 "(K) n f0g,
( Tp(x) \ T q(x) = 0 )( ) (deg p(x) deg q(x)) ) ) ) -' ( 1 M. + 4.3 , )) , -, ( ) &''
-( p(x) ( ) &''
-( q(x) . p(x) = an xn + : : : + a0 p(x) = bm xm + : : : + b0 , - an 6= 0
bm 6= 0. ?- m > n. I , ( f m (K) | A f m;n (an) bm 2 f m (K). & - 4.2(1) , & y 2 A z 2 A,
( 0 6= yf m;n (an ) = zbm . h(x) = yxm;n p(x) ; zq(x). 6 , ( deg h(x) < m. 0 -, h(x) 6= 0, ( Tp(x) \ \ Tq(x) 6= 0. ? ( (deg p(x) deg q(x)) ) ) ) 1 M, 0 6= t(x)h(x) = s(x)p(x) ) % -( t(x)
s(x) T. 8 ( ) ) ), ( t(x)zq(x) = (t(x)yxm;n ; s(x))p(x):
(3)
B- 4.5 ) &''
-( zq(x) . C -, ( t(x)zq(x) 6= 0:
(4)
/ t(x)zq(x) = 0, , ) 4.4, ( , ( t(x)yf m;n (an) = t(x)zbm = 0. 8 yf m;n (an ) ) ) ) 1 &''
-( yxm;n p(x), ) &''
- . &, )) 4.4, (
, ( t(x)yxm;n p(x) = 0 , , t(x)h(x) = 0, ( ( -( t(x). C) 1 ) , ( (3) (4) Tzq(x) \ Tzp(x) 6= 0, ( Tq(x) \ T p(x) = 0. ( (
.
4.7. '1 '2 : : : 'n | A , ) 'i 'j = 'j 'i " + i j = 1 2 : : : n K | ( (
A.
1118 "n (K) =
. .
X 1
i1i2 :::in =1
'i11 'i22 : : :'inn (K)xi11 xi22 : : :xinn |
( ( A3x1 '1 x2 '2 : : : xn 'n]. . ' 'n An;1 = = A3x1 '1 : : : xn;1 'n;1], -) 'n (xi ) = xi ) % i 6 n ; 1,
A3x1 '1 : : : xn 'n] 1
2 An;13xn 'n ] An;1. / I | An;1, "1 (I) = = I + 'n (I)x + '2n (I)x2 + : : :. I , ( "n (K) = "1 ("n;1 (K)), - "n;1 (K) =
X 1
i1 :::in;1 =1
'i11 : : :'inn;;11 (K)xi11 : : :xinn;;11 :
?
( % n c
4.6. 4.8. / K | ( ( A, "(K) | ( ( A3X Fe]. . 0 6= p 2 "(K) 0 6= q 2 "(K). J -( p q ) ),) ( ( . & -( p q 1 ( %, x1 x2 : : : xn. B , -( p q ) T A3X Fe], Afx1 : : : xng. G
' % -( A3x1 '1 : : : xn 'n], - 'i = (xi ) ) % ( i = 1 : : : n. P 4.7 4.2 ,, ( "(K) \ T T ) ) ) . B , 0 6= tp = sq ) % & t s 2 T. ? -( p q
& "(K), , , ( "(K) . 4.9. ! , ) Aa 6= 0 " ! % a 2 A. dimA3X Fe] = dimA. . 5 ( ( , - dimA = n < 1. + & ( A ), D1 D2 : : : Dn % % % D1 D2 : : : Dn, ) Ln) ) )
A. B- (4.1) " Di | i=1
Ln Ln
A3X Fe]. 8 , ( " Di = "(Di ). 0i=1 i=1 -, , 4.8 "(Di ) ) ),) . ? , dimA3X Fe] = n. 5 ( , - dimA1 = 1. + & ( L A ( , ), Di % % % i=1
Di . 5 ), - ( 1 , ,,
L 1
1119
( "(Di ) | ( ) )) % % i=1 A3X Fe], , dimA3X Fe ] = 1. ? % ( , R3X F]. I -, ( R3X F ] A3X Fe], R A g (r) = g (r) ) % & r 2 R g 2 G.
4.10. p1 p2 : : : pk 2 A3X Fe], p1 6= 0. % q 2 R3X F], ( ) qp1 6= 0 qp1 qp2 : : : qpk 2 R3X F ]. . p1 = a1w1 + a2w2 + : : : + anwn, - ai 2 A, wi 2 G wn > : : : > w2 > w1. ? Aan 6= 0, ca 6= 0 ) - & c 2 A. B- , 1.1 & c c = v;1 rv, - r 2 R v 2 G. 5 fbj uj : j = 1 2 : : : mg % ( , %) % -( rvp1 rvp2 : : : rvpk , - bj 2 A uj 2 G. ? & bj - bj = gj;1 sj gj , - gj 2 G, sj 2 R, , -) g = g1g2 gm , ( , ( gbj 2 R3X F ] ) % j = 1 : : : m. G ( , ( grvp1 2 R3X F] ) % i = 1 : : : k. 0 -, 1 ( grvan wn -( grvp1 ,, grvp1 6= 0.
4.11. / I | ( ( ( A3X Fe], I \ R3X F ] | ( ( ( R3X F]. . a b 2 I \ R3X F] a 6= 0, b 6= 0. ?- 4.2 ca = db 6= 0 ) % & c d 2 A3X Fe ]. B- 4.10 & p 2 R3X F], ( pca 6= 0 pc pd 2 R3X F]. 2, pca = pdb 6= 0 , , R3X F ]a \ R3X F ]b 6= 0.
4.12. / J | ( ( A3X Fe]
J1 = J \ R3X F], J1 | ( ( R3X F]. . a | & A3X Fe]. ?-, J | , 0 6= ba 2 J ) % & b 2 A3X Fe]. + 4.10 & c 2 R3X F], ( cb 2 R3X F ] cba 6= 0. P - , ( cba 2 J1 . ? , R3X F]a \ J1 6= 0 ) - - a 2 R3X F ].
4.13. J = L Im | "" m2 ( f I : m 2 W g + + Im A3X Fe]. L (Imm\ R3X F]) | ( L= R3X F]-" m2 J \ R3X F]. . a | & -
J \R3X F ]. G & a = p1 +p2 +: : :+pk , - 0 6= pi 2 Imi , m1 m2 : : : mk | ( & W.
1120
. .
B- 4.10 & q 2 R3X F], ( qa 6= 0
qpi 2 R3X F ] ) % ( i = 1 : : : k. 2, qa 2 L, ( . 4.14. ! , ) Aa 6= 0 " ! a 2 A. dimA3X Fe ] = dimR3X F]. . dimA3X Fe] =Ln, - n | ( , n = 1, J = Ii | )) % i2 % % A3X Fe], ) ),))L A3X Fe]. ?- jWj = n. ? (Ii \ R3X F]) | i2 - J \ R3X F ] - 4.13
J \L R3X F] | R3X F] 4.12, (Ii \ R3X F]) | R3X F]. 0 i2 4.11, Ii \ R3X F] ) ),) . 2, , ( dimR = jWj = n. % 1.2. / ) - - & a 2 A Aa 6= 0, dimR3X F] = dimA3X Fe] , 4.14 dimA3X Fe] = dimA , 4.9. B , dimR3X F] = dim A3X Fe]. S
1.2(1) . C
1.2(2). a | & A, ( Aa = 0. ?- , 1.1 a = g;1 bg ) % & b 2 R g 2 G. ? b = g (a), b 6= 0 Ab = 0. B , Rh (b) Ah (b) = 0 ) % & h 2 G. 2, , ( - Zbh ) ),) L R3X F] ) % & h 2 G. & Zbh | ( ) )) % h2G % R3X F ]. ? , dimR3X F] = 1. ? 1.2 . . / S | , ( Sing(S) ( -) S. J , ( X Sing(R3X F ]) = (Sing(A(R )) \ R)g: g2G
% $ 1] 2] 3] 4]
. . | .:
, 1968. . ! " # . | .:
, 1975.
'. (! ) . | .:
, 1986. Jategaonkar A. V. Left principal ideal rings. | Springer Verlag, 1970. | Lecture Notes in Mathematics, 123.
1121
5] Voskoglou M. G. Prime and semiprime ideals of skew polynomial rings over commutative rings // Turk. Math. Derg. | 1991. | Vol. 15, no. 1. | P. 1{7. 6] Voskoglou M. G. Semiprime ideals of skew polynomial rings // Publ. Inst. Math. (Beograd). | 1990. | Vol. 47 (61). | P. 33{38. 7] Voskoglou M. G. Prime ideals of skew polynomial rings // Riv. Math. Univ. Parma. | 1989. | Vol. 15, no. 4. | P. 17{25. 8] Voskoglou M. G. Extending derivations and endomorphisms to skew polynomial rings // Publ. Inst. Math. (Beograd). | 1986. | Vol. 39 (53). | P. 78{82. 9] Shock R. C. Polynomial rings over 1nite dimensional rings // Paci1c J. Math. | 1972. | Vol. 42, no. 1. | P. 251{257. 10] Jordan D. A. Bijective extensions of injective ring endomorphisms // J. London Math. Soc. (2). | 1982. | Vol. 35. | P. 435{488. 11] Fisher J. L. Embedding free algebras in skew 1elds // Proc. Amer. Math. Soc. | 1971. | Vol. 30. | P. 453{458. 12] Curtis C. W. A note on non-commutative polynomial rings // Proc. Amer. Math. Soc. | 1952. | Vol. 3. | P. 965{969. 13] Hirsch R. D. A note on non-commutative polynomial rings subject to degreepreservation // J. London Math. Soc. | 1967. | Vol. 42. | P. 333{335. % & & 1997 .
{ . .
- . . .
519.172+517.547.24+511.512+512.714
: , !" # , # $%{' %, # #( ! ).
* # !! % " # ! # " # !"( #) # !% # ! " + # ! ( . * "-! -! . ! -% !% "# # ) ! #% #% $%{' %, ! ! /. !" # . 0!! #/ !% "1 " # # . # 1%.
Abstract A. V. Pastor, Generalized Chebyshev polynomials and Pell{Abel equation, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1123{1145.
In this paper the question of the compositional reducibilityof generalizedChebyshev polynomials is solved by studyingthe combinatorial structure of plane trees. As a particular case we deduce the criterion of minimality of the solution of Pell{Abel equation corresponding to a given plane tree. Some other applications are also considered.
1. 1.1.
- - ! (. $6,9]). *
. +. , -. .. - $9] . /
, , $8]. 1 2
, 3 . , 2001, 7, 6 4, !. 1123{1145. c 2001 ! "# $ %"# # & '(, )* & +, "" "-
1124
. .
. / P (z) 7 3 , 8 A B ( ), P (z) = 0 =) (P (z) = A _ P(z) = B): /
8 ( ) Tn (z) = cos(n arccos z). ;2 , Tn (z) = 0, z 2 cos kn k = 1 : : : n ; 1g, , Tn (z) = 1. - 7 <
=
. - 2 Tn (z) 2 n. . / P1(z) P2(z) 2 , 8 Q R q r 2 C , Qq 6= 0 P1(z) = QP2(qz + r) + R: *
8 .
. P n A B P 1($A B]) n + 1 . P 1(fA B g). / 7 8 7 2. ,
2 7 . ? , - . . 8 . ; 3 2
3 ( ). @, 7 . , 7 2 2
8 !: , 3 3 . B 2,
- . . ;, 3 3 , . . 7 , 7 2 3, 3 . 0
0
;
;
{
1125
/ 7 3 2 , 3 | . . ;, 2 3 , . / 2 8 3 8 . , -
3 2. /2 , 8
, 8 2
, 2 ,
, , +, @ /, 2 . ?
2, 8 , $4]. /2 2 2 2 ,
2, , 8 2 2 , 2 2 2. ? ,
2 , 7 3 8 . /7 2
; = Gal(QE =Q) 7 3 . /2
, 2
; 3 2. - 7 3 3 T 2 2 8
H
;, 7 T 2 . F
, Q. / T. ? (., , $5]), 8 3 8
7 3 , . 1.2. {
- /, 2 x2 ; Dy2 = 1 7 3 D 2 . 1
8 3 x y.
1126
. .
;
2 23. 1
, 8 /,
23 . /2 K | 0. H 2 P 2 ; DQ2 = 1 (1) D 2 K$z], D 6= const, , P Q 2 K$z] | . I
(P Q), P 2 , Q | . J p 8 (P Q) (1) 2 p 7 P + DQ 23 K$z]( pD ). @,p
p 7 8 , 2 N(P+ DQ) = (P+ DQ)(P ; DQ)p= P 2;DQ2 = 1. * , 2, 7 23 K$z]( D ), 8 8 , (1). , - p (1) 7 23 K$z]( D ), 8 8
. 1 2 7 , 8 8 , 2 , (1)
2
. K 7
7 (1 0). * 7 (P Q) 7 (P ;Q). / 7 (P1 Q1) (P2 Q2) 7 (P1P2 + DQ1 Q2 P1Q2 + P2Q1). I (1), (P1 Q1) (P2 Q2). /2 2 P1 m, 2 P2 n, 2 D 2k ( , (1) 2 2 , 2 2 D ). 2 Q1 n ; k, 2 Q2 m ; k, 2 ,
P1P2 DQ1 Q2 m + n. , 2 P1P2 + DQ1 Q2 (. . (P1 Q1)(P2 Q2)), 2 P1 P2 ; DQ1 Q2 (. . (P1 Q1)(P2 Q2) 1) m + n. 1 , (P1 P2 + DQ1 Q2)(P1 P2 ; DQ1 Q2) = P12P22 ; D2 Q21Q22 = = P12P22 ; (P12 ; 1)(P22 ; 1) = P12 + P22 ; 1 2 min(m n). L 2 , 7 2 m + n, | 2 jm ; nj. *8 , (P Q) 2 n, (P Q)l 2 nl. B 2,
(1) 7 : 7 (1 0) 7 (;1 0), 8
2. ;
{
1127
I (P0 Q0) 2 (1), 2 2 2 . I,
, 2
(P0 Q0) 2 (1), (P0 Q0) 8 2 7 . 1, 8 , 7 8 2 (1).
. ! ! (1) ! " : (;1 0) $ ! (1). . /2 (P0 Q0) | 2 (1). I 2 (P Q). /2 P0 2 m0 , P 2 m, m m0 r q. (P Q)(P0 Q0)q , (P Q)(P0 Q0) q 2 , 8 r. 1 2 r < m0 , 7 2 (1 0), (;1 0). L 2 , (P Q) = (;1 0)"(P0 Q0) q " 2 f0 1g. 2 2. 2 . %!$ (P Q) | ! (1), (P0 Q0) | $ ! . ( ! ! !$ n, P = Tn (P0). ) , P0 Tm (P0 ) !$ m > 1 P0 2 K$z]. . J , ;
p
p
p
P + DQ = P0 + DQ0 n = P0 + P02 ; 1 n = p p ; ; = Tn (P0 ) + P02 ; 1Un (P0) = Tn (P0) + DQ0Un (P0 ) Un (z) = sin(n arccos z)= sin(arccos z) | n- ( n ; 1). L 2 , P = Tn (P0 ), Q = Q0 Un (P0). /2 2 P0 = Tm (P0 ), DQ20 = P02 ; 1 = Tm2 (P0 ) ; 1 = (P0 2 ; 1)Um2 (P0 ) , P0 2 ; 1 = DQ0 2, 2 D . L 2 , P0 2 ; DQ0 2 = 1 2 (P0 Q0). 2 . / 2 $11]. 1 (1) 2 P 2 ; DQ2 = ;1: (2) J 8p(P Q) p(2) 2 7 P + DQ 23 K$z]( D), 7 7 ;
;
1128
. .
;1, 7 , 8 ;1, 2 . 1 7 , 8 1, 2 , 2 , (P 2 ; DQ2 )2 = 1 (3)
. B 2, 7
8 ( 0), 4 = 1. - , K 2 3 4 , (1) (2) 2 - , 2 (P Q) | (1), (4 P0 4Q0 ) | (2), . - 8 (1) 2 8 .
. %!$ K $ *. ( ! ! (3) ! " : (;1 0) $ ! (3). ! , $ ! (3) ! (2), $ ! (1). ! , $ ! (3) ! (1), ! (2)
. . %!$ K $ *. " ! ! (2)
$ , $ ! (1) P0 = (2P002 + 1). % " P00 | $ ! (2). - 2 (1) (2) 23 C $z]. ;
. $1]. O (1) 7 3 8 /{.. - , 2 C 2 3 , (2) 23 C $z] 2 . * D(z)
7 3 , 2
(1) (2), 2 2
23 R$z]. - 7 2
2 2 . 1.3.
{
* 2
/{.,
$8]. ?
,
2
{
1129
/{.,
. /2 P |
;1 1, T = P 1($;1 1]) |
8 3 . I Y D(z) = (z ; di) di | T . ? , 2 T . P ;1 1, ;
P (z) + 1 =
P(z) ; 1 =
p Y
i=1 q Y
(z ; ai)i
(z ; bj )j
j =1
ai | T (. . P (ai) = ;1), bj | T (. . P(bj ) = 1), 2 i j | 8 . *8 , (P (z))2
;
p Y 2 1 = (z i=1
;
q Y i ai) (z j =1
; bj )j = D(z)(Q(z))2
2
P /{. 7 3 D(z). /2 /{. 8 2 , 2 ,
2 8 /{.. K 7 , /{., 8 , 2, 2
. *
, , 2 /{., 8
, 2
. / 8 7 . - 2 /{., 8
,
, 2 3 . ;2 , . 1.2,
P(z)
2 8 /{., Tn (P0 (z)).
1130
. .
H 2 8 : P (z) ? J ,
P(z) 3 R(P0 (z)), R(z)
. / 7 P0 (z)
2
. + , P0(z) 2 8 7 3 : $3], 8
3 2 7 3 , 8 0 1 f0 1g
. *8 , 8 P0(z) 7 3
R(z), R(P0(z))
. - 2 3, 3 . - 7 , 28 2 8 3 2. - , 8 88 : . %!$ T T | !* $, P(z) P (z) | ! . %!$ P(z) P (z). T $ T $ , ! ! P0(z), P (z) = P(P0(z)) P0(z) $ . Q 7 7 8 : 8, ! 8
, 8 2 , 7 3 . . ~ 8
, P~0 P~ P, ~ ~
P(z) 2 P P (z) 2 P , 8
, P0 (z) 2 P~0, P (z) = P(P0(z)). S (., , $7]), 7 7 3 2, ! 8 3 2, | . F , 2 3 , 2,
P (z) 3
P (z), P (z) = P(P0(z)).
{
1131
- , 7 ,
$10]: " T , $ %$ P0(z), " , $ T , % Tl (P0(z)) " " l. ;
2 2 $10] & ' T.
/ 82 2 . - 2
/{.. ?
P 2 ; DQ2 = ;1 2. J , , T T, T
T . . 8 2 . +. , 3 8
2 2 3.
2. ,
{
; 2, 3 2
. /2 R(z) Q(z) | 2 . @, crit(R Q) = crit(R) R(crit(Q)) crit(f) | f. L 2 , % R(Q(z)) % % R(z). ) % $ " " , "
Q(z) $ R(z). (I-
, R 8
. 1 , R |
, 8 ,
2 ,
.) , 3 R(Q(z)) 2 R(z) Q(z)
2 , 8 8 . 1. / R(z)
. 2. - Q(z) 8 , 8 R(z).
1132
. .
; 83
3 . 1. I 2 3 T T ,
T 3 n (. ).
T
T
T
T
;!
T T
T
I, 3 2 ( n). / n = 2 8 .
T
T
T
T
T
;!
T
T
T
@, 3 n , Tn (z). ?
, n ; 1 ( 3). -
7 $;1 1] cos kn k = 0 1 : : : n . /2 T
P0 (z) ;1 1. T Tn (P0(z)) ( 2, 7
). ;2 , Tn 1($;1 1]) $;1 1]
n ; 1 . / P0 1($;1 1]) T . *
,
T, 28 n , T . /
, , , /{., 8
,
2 2 . *
P0(z)
. L 8 , 2 /{., 8 , 2
.
;
;
{
1133
2. I 8 :
F p3 p 256 P (z) = 243 (2z + 1)4 (8(5 + 3 3)z 2 ; 1)2 ; 1 f;1 1g. @, P (z) = 2P02(z) ; 1 = T2 (P0(z)) p4 108 p 8 P0 (z) = 27 (2z + 1)2 (8(5 + 3 3)z 2 ; 1): J P0(z) 8 ; 12 p ;1 6p3 ; 9 = ;2 (p3 ; 1) p4 108 : 8; 16 1 1 2, P0 ; 2 = 0, p p ! 6 3 ; 9 ; 1 + = ;1 P0 8 p p ! ; 1 ; 6 3 ; 9 P0 = 1: 8 , P0 (z) , 2 ,
. . ; , 8 . -
Tn (P0(z)) 2 ( <=) $10].
3. !"#
- 2 3 2.
1134
. .
. " 2 8 38 2 2 , 88 8 . 1. B8 2 a b 8 3
. 2. ; 8 2 a, b c (b c)
8 (a b) 3 , 3 b. 1 2,
,
. I T , 8 n , 2, 8 2n 2n . / 8 2 T ,
2 2 T . ;
, 2 8 , 3 . @, 7 8 , | , 7
. , , 8 . B 2, 2 , 2. , , 8 n , 2 2, 8 2n 2n , 8 8 ( ). B 2, 8 2n ,
8 ,
. / 7 , 8 , . - 2 , 28 . ;
2 2 " . I8 2 2 $ , | " . I 8 , 8
8 , 2
, .
{
pp p1p p p p p pp p p p p p2p p p p p p p 1p p3pp pp p pp p p p 4 pp 3 p p p p p p p p p p p 4 p p pp p p p 2
T
T1 T T 4
2
3
; ; ; ; ; ;
!
6
6
@ I @ @ @ @ R @
1135
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?
;1 ; 1 ; 2 ; 3 ; 3 ; 2 ; 4 ; 4;
* " 2 8 , 8 2 3 (. . 3 8 3). B 2, 3 2 - 8 3 . ; 3
8
: < - =, 2 , , 3 , <- =. * , 2 2 . . + C C 2 C
C, 8 8 . 1. / 2 C 2 C. 2. L
C
C. ;
, 2 8 C 8 C. 1 3 2 , 8 3 8 . @, 2 , ! 8 . - 7 C C
2 f : C ! C. /2 u | 8 ( ) C , f(u) 2 . . + 3 T 3
T 2 T T, 8 3 , 3 8 3 , 3
.
1136
. .
B 2 8 .
. + !* T !* T ! ! $ , ! ! !* , C , ! ! T , !* , C , ! ! T . . ;
- 3 2 3 . . k = mn , m | T , n | T, 2 fE: T ! T. 1 fE 2 k- . / 3 2 8 .
T
6
T ;! T T
;!
pp p p p p p p pppp p p p p p p p p p p p pp p pp ppp p p p p p p p p p pp p p p p p p p ; ; ; @ I @ R @
pp p p p p p p p p p pp p p p p p p p p p pp p p p p p p p pp pp p p p p p p pp p p p pp p p p p p p p p ppp p p p p p
?
?
6
6
6
?
6
6
6
;!
?
?
T
T
?
T
?
6 -
;!
?
; 2 2 2 . /2 T | 3 n , P (z) | 8
0 1. I
P 1(R) I . J , $0 1]
8 T. / 2n ( 2 2 ), 8
. J , , , , , ;
{
1137
2 . -
2 3,
3 , . I ( ) 3. / 8 I 2n . F 2 2 " , 2 8 , 2
2 0 1 1. L
2 2 8 . @,
2 2 2 , 3. /
I
2 2 " , , $ T . H 3 I , 8 T, , 8 T, 8 2: 2 88 ,
2 | , | 88 (. ).
T
T
ppp pp p p p p p p p ! p p pp p p p p p p p p p p ppp p p p p p p p p p p p p p p p p p p p p p p p p p pp p p p TT T
T
!
pp pp p p pp pp pp pp p p p pp p pp p p p p p p
p p p p p p p p p p p p p p p p p p p p p p pp p p p p pp p p pp p pp p pp p pp p p p p p
p pp pp pp
TT T
!
TT
ppppppppppppppppppppppp pppp pppppp ppppp pp
pppppppppp
ppp pppppp pppp pppp ppp ppp pp pp pp pp
pppp ppp ppp pp ppp pp pp pp pp pp p
pp pp p
pp pp p
pp pp pp pp pp pp pp pp pp pp pp p
pp pp pp p p pp ppp pp p ppp pp p pp ppp pp ppp p pp ppp p pp p ppp p pp ppp pp p pp p pp pp p p p pp pp pp pp pp pp pp pp ppp ppp pp pp ppp ppp ppp ppp pppp pppp pppp p p pppppp ppppp ppppppp pppppp ppppppppppppppppppppppppppppppppp
T
T
T
, 3 2 fE: T ! T
2 3, 8 T ,
2
1138
. .
3, 8 T. / 7
2 T 8
2 T, 3.
4. %"&# '# '#"#( #
2 2 2 . . %!$ T T | !* $, P(z) P (z) | ! . %!$ P(z) P (z). T $ T $ , ! ! P0(z), P (z) = P(P0(z)) P0(z) $ . . 1 , 2, P (z) P (z) 8 0 1. /2 n | T, m | T . P (z) P (z) 8 n m
. ; , P (z) = P (P0(z)), T T . @, P 1 (w) = P0 1(P 1(w)) 8 w 2 C . L 2 , P0(P 1 (w)) = P 1(w): , P0(z)
8 T
8 T , 3 3 , 3
, , 2. L 2 , P0(z) T T , 2 2. * ,
P0(z). /2 2 T k- T. ;
2 3, 8 T ,
2 3, 8 T . / 7
2 3, 8 T, k
2 3, 8 T . @ , 8 nk = m. I ! , k 7
2 3, 8 T.
;
;
;
;
;
{
1139
F
2 2 A1 A2 : : : Am . J
2 3, 8 T ,
2
2 Ai , T ! T.
2 3, 8 T ,
2 Ai - .
2 , 3, 8 T , 8 3, 8 T. / X. H m S 2, X = Ai . Ai
2 i=1 3, 8 T, , 2 i : Ai ! C P. @, ijAi Aj j jAi Aj 2 , 2 : X ! C P ( (x) = = i (x) x 2 Ai ). F , , 8 ,
2
2 . L X |
, 3 7 3, 8 T . - . ; 7 8 . ; 8 x0, - Ai , 2 3, 8 2 U Ai ijU : U ! C . ; x0, 3 Ai Aj 2 8 2 U Ai Aj ij jU : U ! C , jU : U ! C , , Ai Aj C . - , , . *2 2 3, 2 T , 1. /2 x0 2 X | T , 2 (x0) | T . , , 2 x0 l 2 l. F , x0 2l Ai1 Ai2 : : : Ai2l ( 3 ). z0 = (x0) 2
2 3, 8 T. * B1 B2 : : : B2, , B1 = (Ai1 ).
2 Bj l (Aij Aij+2 : : : Aij+2(l;1) ). /2 V = fz 2 C j jz ; z0 j < "g | ", ! B1 B2 : : : B2 . @, V 2 2 V \ B1 , V \ B2 ,. .. , V \ B2. / " 7 .
\
\
1140
. .
/2 V~ = fz 2 C j z l + z0 2 V g. H V~ . I : V~ ! V , (z) = z l + z0 . J V~ 2 2l B~1 : : : B~2l , B~i Bj , j | i 2 ,
3 . I U = 1(V ) | 2 x0 | ~ : U ! V~ , 8 : x 2 U \ Air ~(x) 2 7 1 ( (x)) B~r . B 2, ~ U V~ , 7 (U ~) 2 8 X. F
, 2 , z0 . J , 8 1, . ?, X
28, 8 38 X. 1 8 2, , I . 2 : C P ! X. 3 ;
;
CP
;! X ;! C
8 38 C P. / 7 3 2 8, -
1. 7 3 . * P0 (z). @, P0 8 T, 2 , 3 P(P0(z))
. K T , 2
P 1 3, 8 T , 38 I , 7 8 3, 8 T . L 2 , P (z) P(P0(z)), , 8, 8. ,
88
P0(z), 8 . K
2 28 P0 , ,
28 P P . . 2 . H 2, P0(z) T ! T . . %!$ T ! T | $ , P0(z). ( P0 1 - T ! -! T . . ? 2 2, 2 P0(z) T ! T .
;
;
{
1141
- 8
2, , , 2 8 3 . F
, 8
. , 2 8
2 . . - 3 2
$8] 3 . F 3 8 2,
.
5. * '# '#"#( # 5.1.
{
;
2,
, 8 T , 3
, 8 T , - 2 . F 2, , 2 38 2 /{., 8
. . %!$ T | !* , P | !
! ;1 1. % P $ , ! ! ! % {/
$ , T ! * ! n
(n > 2). ! $ ! % {/
! $ , T ! Ln , Ln | * $ n, ! T . . J , /{. 2 2 , P
3 Tn (P0 ) n > 2. J , 7 2 , 8 P T 8 3 n (n > 2). (/2 Tn |
, 3 n .) - , 2 , P 2 8 2 P0 8 : P (z) = Tn (P0 (z)) n > 2. F , P0 3 n T. L , T 2 8 3 n , 2 P 2 3 Tm (P00) m > n, P00
1142
. .
, 2 P00 2, 2 P0, 2 P0. . 2 5.2. ! " # $ %"
{
; 8 3
2: 2, 8 . * , , 2 2 8 . - 2 7
, 3 , 3 , . , 8 2 3 2, 8 3 , , . @, 3 T
P (z),
P(z), 8 2 7 3 P (z) . , T , 2 , 7
; = Gal(QE =Q). L 2 , T
. *8
,
8 2 . J , 8
7 3 . - 2 2 2 . - 7 , z T, z , 3 . , Y D(z) = (z ; di) di | T,
Y Y D(z) = (z ; ei )(z ; ei ) (z ; fj ) ei | T , , fj | ,
. L 2 , D(z)
7 3 , , 2
P 2 ; DQ2 = 1 (4)
{
1143
P 2 ; DQ2 = ;1: (5)
. 0$ ! (4)
$ , $ "--* . 1 (5)
$ ! . . /2 P0(z) = ak zk + ak 1zk 1 + : : : + a0 2 (4). *
T ! Ln 3 n T . H 2, 3 7 3 ;1 1. I z0 | T . * , P0 (z0 ) | 3 3 . 1 z0 T . L 2 , P0(z0 ) = P0(z0 ) 3 3 . , P0(z0 ) = P0(z0 ). 2 8 , 2 8 T. L 2 , P0 (z) = P0 (z) z, 8 , 8 . 1
T m + 1, m | , P0(z) P0 (z) m. , P0 (z) P0 (z). 1 P0(z) = = ak z k + ak 1z k 1 + : : :+ a0 , 2 7 3 P0(z) 8 (
), 8 ( ). B . 2 ?, 2 2,
3 . ; 7 C | 8 , 88 T . K 2 2
, 8 C 2 T . - 2 2, 8
, 8 ( 2 8 8 3). . %!$ T | !* ,
, C | ! ! , , Ln | * $ n, $ T . " !* ! (5)
$ , C ,
, ! * Ln . % ", (P0 Q0) | $ ! (5), (iP0 iQ0 ) | $ ! (4), (2P02 + 1 2P0Q0) | $ ! (4). 2 $ n. ! $ ! (4) $ . ;
;
;
;
1144
. .
. I 8 C, 8 T ( T 8 C). K T 3 3 Ln , T , 8 8 C 2n, 8 C, , 8 3 3 Ln . - 8 8 C n, 2n, 8 C, , 8 3 Ln . F 2 n. 2 5.3. & " $ $
- 8
2, . . %!$ ! ! !* T !* T . ( T T . . /2 P P |
7 3 , 8 2 T T
, P = P P0. I 2 2 Gal(QE =Q). @, P = P P0, 2 , P0 T ! T , 7 T ! T. @ 2, 2 k- k. , T 2 , 2 T . L 2 ,
, 8 2 T ,
, 8 2 T , , T T. 2 2. 2 . /
8 2 , . ;2 , p 2 2, 8 Q( 3), 3 , 88 Q. L , 2 , 8 Q, 3 n , 7 1 , , 8 Q 8 3 n , 88 Q.
+ ##
1] Abel N. H. Sur l'integration de la formule dierentielle pdxR , R et etant des fonctions enti eres // J. Reine Angew. Math. | 1826. | Vol. 1. | P. 185{221.
{
1145
2] Adrianov N., Shabat G. Unicelluar cartography and Galois orbit of plane trees // Geometric Galois Actions. Vol. 2. | Cambridge University Press, 1997. | London Mathematical Society Lecture Note Series, vol. 243. | P. 13{24. 3] . ! "# $ % & % '!()( & $)(*()( +(" // ,-* . " /0 1112. 1 " % .% . 3 & ". | 1979. | 4. 43, 5 2. | 1. 267{276. 4] Betrema J., Pere D., Zvonkin A. Plane trees and their Shabat polynomials. | Catalog, Rapport interne du LaBRI, no. 92-75. | Bordeaux, 1992. 5] Couveignes J.-M. Calcul et rationalite de fonctions de Belyi an genre 0 // Ann. Inst. Fouriere. | 1994. | Vol. 44, no. 1. | P. 1{38. 6] Grothendieck A. Esquisse d'un programme // Geometric Galois Actions. Vol. 1. | Cambridge University Press, 1997. | London Mathematical Society Lecture Note Series, vol. 243. 7] Schneps L. Dessins d'enfants on the Riemann sphere // The Grothendieck Theory of Dessins D'enfants / L. Schneps eds. | Cambridge University Press, 1994. | London Mathematical Society Lecture Note Series, vol. 200. | P. 47{77. 8] Shabat G., Zvonkin A. Plane trees and algebraic numbers // Contemporary Math. | 1994. | Vol. 178. | P. 233{275. 9] Shabat G., Voevodsky V. Drawing curves over number 8elds // Grothendieck Festschrift. B. 3. | Birkh9auser, 1990. | S. 199{227. 10] Pakovitch F. Combinatoire des arbers planaires et arithmetique des courbes hyperelliptiques. 11] : &(* 3 ;. < +. 3 & +( !(% // = +# % .. ! $&. | 1995. | 4. 50, 5. 6. | 1. 203{204. 12] ;( . . 2 % !(* +(* #!( . . | >.: > , 1980. %. $ 1998 .
. . , .
514.13
: , , .
! , "#%& ' | ! | ! ) * . + # # !" # #, % % ) ! " % ! ) " ' . # # " ! , ) ! ( " " " ). / #" ' ) ! # , 0 , # 0 ! (1- ), " ! , # % . 1- 0 " "# - ) .
Abstract Yu. A. Rylov, Description of metric space as a classication of its nite subspaces, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1147{1175.
We suggest a new method of metric space description, using its constituents (6nite metric subspaces) as basic objects of description. The method allows one to obtain information about the metric space properties from the metric and to describe the metric space geometry in terms of its constituents and metric only. The suggested method permits one to remove the constraints imposed usually on metric (the triangle axiom and non-negativity of the squared metric). Elimination of these constraints leads to a new non-degenerate geometry. This geometry is called tubular geometry (T-geometry), because in this geometry the shortest paths are replaced by hollow tubes. The T-geometry may be used for description of the space-time and of other geometries with inde6nite metric.
1.
, . , 2001, 7, 7 4, . 1147{1175. c 2001 ! " #$, % " &' (
1148
. .
( ) . # $ , % , & ' ( ( $ ) * . + ' &$ : (1) , (2) , (3) ** . / ** | ** . 1 ) . 2 & , ' 3 , , ' ' . 4 - % ) * $ &. 5 - (, . 2 , , ' . 4 ' ( ) * , &$ ). ' . | ' & , ' )* , ' $ ( ) * ) . 4 &$ '. 1 u = P0P , $ ' : P0 * P . 7 , $ ' u = P0 P, &$ $ P0. + ' * ( $ ), ' & . % $ ' & ( ), , ' . 8 , & P0P P0 Q $ (P0P P0Q), ' ' % . ' $ . 9 , , ' * P P0P . 5 (P Q) ( P Q) ' ' (P0 P P0Q) $ & 3 (P Q) = 21 (P0P P0P) + 12 (P0Q P0Q) ; (P0 P P0Q) (1.1)
1149
(P Q) ) * , ' 3 (P Q) 12 2 (P Q): (1.2) +' < ) * = # >1], ( ( 3 ' - . @ ' ) * ' . + , ' (P0 P P0Q) ' ) * ( ) $ & 3 (P0P P0Q) = (P0 P ) + (P0 Q) ; (P Q): (1.3) + ' , ' ' . k- Lk = L(P k ) '( k + 1 P k fP0 P1 : : : Pkg, $ (k ; 1)- Lk;1. 4 * , ' &$ & k ' P0Pl , l = 1 2 : : : k. +'( (AB4) n- , n + 1 ' P n fP0 P1 : : : Png. 1 k- Lk = L(P k ) k +1 , $ (k ; 1)- Lk;1 . 2 ' ,
k- Lk = L(P k ) AB4 k- . AB4 & 3 &$ . 1 , & , , AB4 ( & ' ' . , ' , , &$ , , & (. . AB4) ' . % , . ' % $& , ' . 2 ' , ' (AB4), , , &$ , . , AB4. 9 ', , ' ' , AB4 $ , -
. .
1150
, , ' . % ) * , AB4
' , ' , ( & ) * & . A AB4 ' &$ '. + , AB4 ) * ( ), ' , AB4 & ) * ( ), . . & . #' (E1 ), (A1 ), (9A ) (AB4) &$ : AB4 E1
*
HHHj
(1.4)
- 9A A1 + % AB4 (E1 ) ' (9A ) ' . + (1.4) (E1 ) ' * , 3 ' . E1 ' , ' ( ' ). F * , $ AB4 , ' & , ' (1.4): - A1 - E1 - AB4 (1.5) 9A + (1.5) ' * (9A ), , ( ) * ). A , . . , ' , ( AB4 n- * , ' &$ & n ' P0Pl, l = 1 2 : : : n, n +1 P n fP0 P1 : : : Png. * AB4 , &, ' & & & * & - AB4 (1.6) 9A
1151
AB4 ( , , ' , ' & . # AB4 n- &$ '. P n = fP0 P1 : : : Png H ' n + 1 (n = 1 2 : : :) P0 Pi, i = 1 2 : : : n, n , % P n , (P0Pi P0Pk ), i k = 1 2 : : : n, | ' . I ' n P0Pi , i = 1 2 : : : n, ' Fn(P n ) 6= 0 (1.7) Fn(P n ) = det k(P0Pi P0Pk )k P0 Pi Pk 2 H i k = 1 2 : : : n (1.8) B, ( ' (P0 Pi P0Pk ) ' & ) * & $ & (1.3). K ' (. . ' , ' ' ) ' , ( , ' - ' ' ' ' & : (P0 Pi P0Pk ) = (P0Pi:P0Pk ) (P0 Pi) + (P0 Pk) ; (Pi Pk): (1.9) 9 ', B Fn(P n ) = det k;(P0 Pi Pk )k P0 Pi Pk 2 H i k = 1 2 : : : n (1.10) ;(P0 Pi Pk ) (P0 Pi) + (P0 Pk ) ; (Pi Pk ) (1.11) ' & ) * & . n P0Pi (i = 1 2 : : : n) ' . 9 Fn(P n ) 6= 0. & P0 Pi, i = 1 2 : : : n, $ & ' P0 R. 9 n +1 P0Pi (i = 1 2 : : : n), P0R ' , R 3 & Fn+1(P n R) = 0. 5 L(P n ) = fR j Fn+1(P n R) = 0g R n- & , $ & ' n + 1 P n. 3 , &$ % , & ) * &, T (P n ) = fR j Fn+1 (P n R) = 0g H & M = f Hg ' , ( . 2 T (P n ), ' n- , n- . n +1 P n , &$ , '
' ( (n + 1)- - ' . 9 n- (AB4) , p(n + 1)- f P ng Fn (P n ).
. .
1152
K , 3 , , ' , . . ) * ( . 3.1). 1 $ ' , ', $ & $ . 4 %, ' * . 1.1. 5 M = f Hg ' H ' H H : : H H ! D+ R (1.12) (P P ) = 0 (P Q) = (Q P ) 8P Q 2 H (1.13) D+ = >0 1) (P Q) = 0 P = Q 8P Q 2 H (1.14) (P Q) + (Q R) > (P R) 8P Q R 2 H: (1.15) 0 0 0 1.2. 5 M = f H g M = f Hg ' & H0 H M = f Hg, ( 0 , &$ j H02 = H0 H0 (1.12). E , M 0 = f0 H0 g . 1.3. 5 Mn(P n ) = f P ng ' , p ' P n fPig, i = 0 1 : : : n. 1 Fn(P n ), ( 3 (1.10), (1.11), ' . 1.4. 8 ;;;;;! Mn (P n )= f P ng ' Mn (P n ), ' P n = = fP0 P1 : : : Png. 1.5. 4 ' - , , . - ' , ' . <= * ) * , ' $ & 3 (1.2). - & $ , '& , ' $ . F , . 2 , , , . 0
0
1153
1 , , . K ( , ' . 9 , '& $ , & . A , , (' , ), ( ', & (' &). % ( , ' ' , | ( ). - ' , % , $ , & ' . 1 - ' , ' & , & , , . + , , ' & . 7 - ' & ) * . - % ( ) , ' ' & %. + , 3 (AB4) n- , Mn (P n ) f Hg ) * & Mn+1 (P n+1 )
Mn (P n ) . 5 R 2 H, P n , Mn+1 (P n R) = = Mn (P n ) fRg Fn+1(P n R) = 0, ' T (P n ) = = fR j Fn+1(P n R) = 0g. 1 ( % , % . 9 * ' , ' 3 >2{4]. 8 3 '( P0, P1 , &$ % . 8 3 , ( $ & , ' ' (. . , &, ' ). / (AB4 ) ( , % ' , * - . 1 , % ' , 3 ' * , AB4 . 4 - * ' < =. 9-
1154
. .
, , ' . 1 3 , ' ' & 3. I (1.14) (1.15) & , , ' 3. 9 ( ), , (1.14), (1.15). 9, & 3 (1.14), (1.15), * ) , ( (1.14), (1.15). 1' 3 , , ' , $ , ' AB4 3 . + , /. F. / ', * ( - >5]. - , 3 , 3 & ' ( * ( . - **
) ) * ' (n + 1)- Mn (P n ) = f P n g (n = 1 2 : : :), P n H ' n + 1 Pi 2 H p(i = 0 1 : : : n). 8 Mn (P n ) jMn(P n )j = jP n j = Fn (P n ), ' . 5 ) * , f Hg - n+1
O Fn : Hn+1 ! R Hn+1 = H n = 1 2 : : : k=1
(1.16)
Fn(P n ) 3 (1.10), (1.11). F ', ) * , $ $ & (1.16), ' ' )* & , '3 ) & . % ' , (. . , ) , . 2 ' , , ) - . N , - ) ) & . @ , -
1155
, ' , . + ' )* D- ED 3 D. 9 <)* = 3 . F)* ' ' ED , U , $ ED . )* AB4 ' & , $ ), & , - & . (1.14), (1.15) ' & . 4 3 3, ) * f P n g. 1 % , ' & ) * & = 21 2 , M = f Hg $ V = f Hg, ' - 1 . B &, ( & - , ' 9- . 9- $ , (1.15) (1.14). 9- ' - . @ $ $ & ' (1.14), (1.15), , . 4 ' 5 >6] N& >7] . + $ % ' ' (1.3) ' , ( ) ) ' (1.15). 3 )* $& . (1 $ , ' , & , , % * )* .) & $ , , &$ , , $ , . 4 & ' 9- , . # ' 1 8 ! ! ! # " " ! , ! (1.14), (1.15), " ! ! , 9 ! (1.14), (1.15), # " 0 ! "# " : 0 ; - " )&
, ! %&
" ! . 0 %, ) ) ! " " ! (1.14), (1.15), !
#
0 # . + ! ! # ! (1.14), (1.15) - .
1156
. .
9- , . . , & ( 3 ). 9- & ( &, $ ) &, . . & $ , . # ' 9- , ( - , %)) & 9- %)) , - >8]. 1 ' - . 9 ' $( ) ' , . 1 ( ' .
2. -
2.1. - V = f Hg H P ' H H $ ) * : : H H ! R (P P ) = 0 (P Q) = (Q P ) 8P Q 2 H: (2.1) O * ' ) * -) * . 2.2. + H0 H - V = f Hg ) * 0 = j , &$ H0 H0 , ' - V 0 = f0 H0 g - V = f Hg. 1 3 ) * 0 = j , &$ , ' . E& - - . 2.3. - V 0 = f0 H0g ' ' - V = f Hg, $ ) ' f : H0 ! H, 0 (P Q) = (f (P ) f (Q)) 8P 8Q 2 H0, f (P ) f (Q) 2 H. E& - V 0 - V = f Hg ' . 2.4. F - V = f Hg V 0 = f0 H0g '& ' (% ), V ' V 0 V 0 ' V . 2.5. 8 - Mn(P n ) = f P ng n ' - , $ ' n + 1 P n . 2.6. F - Mn (P n ) p ( (n) n n ' Fn(P ), Fn(P ) 3 (1.10), (1.11). 0
0
0
0
1157
2.7. 8 - Mn (P n ) = f P ng '; ;;;;! n Mn (P ), ' P n = = fP0 P1 : : : Png. 2.8. 5 mn n- '
mn : In ! H In f0 1 : : : ng : (2.2) 5 In , ' mn (k) 2 H k 2 In . I ' In -' ' ' ) . F ' ) n- ' ' ) ' : ;;;;;;;! P0 P1 : : :Pn P0P1 : : : Pn ; P!n : ' Pk In & Pk ) . @ Pk 3 ( ' . F ;;;;;;;! ) P0P1 : : :Pn ' . 1 % ;;;;;;;! P0P1 : : : Pn n- - V = f Hg ( fPl g, l = 0 1 : : : n, ' n +1 P0 P1 : : : P;;;;;;;! n, $ - V . 9 P0 P0P1 : : :Pn. 4 ' mn (In ) In k (k 6 n + 1). 5 +1 n+1 = nN m n - ' H H, & n ;P!n Hn+1. k=1 1. 9 < ;P!n = ( , H ' & n + 1 ' P0 P1 : : : Pn, . 1 , H , $ ' k , ( n +1 ' P n fP0 P1 : : : Png , n > k. A P n , 3 < = $ >9], fP0 P1 : : : Png ( ' ), ' 3 <<=. F , * '( ,;! ' 3 (. 1 3 P n ' '( % , ' &$ . # 2.8 ' . 2.9. 1 PQ - V ' , . . fP Qg ' P Q. 9 P , Q | * .
1158
. .
;;;;;! 8 Mn (P n ) - V= ;;;;;! n = f Hg;;;;;;;;;;;! V ' Mn (P ) ; P!n fP0 P1 : : : Png. 2. E& - ; ;;;;! Mn (P n ) - V V , V - - V , , - & - V '. + , ; ; ! PP , &$ , $ ' P , - - V . 2.10. # - ' (P0P1:Q0Q1 ) P0P1 Q0 Q1 ' $ (P0 P1:Q0Q1 ) (P0 Q1) + (Q0 P1) ; (Q0 P0) ; (P1 Q1) P0 P1 Q0 Q1 2 H: (2.3) @' % (2.1) & &$ - ' (Q0Q1 :P0P1) = (P0 P1:Q0Q1) = ;(P1 P0:Q0Q1) = ;(P0 P1:Q1Q0): (2.4) 2.11. 1 2.6 jPQj PQ ' (p p :PQ)j (PQ:PQ) > 0 jPQj = 2(P Q) = jijp((PQ PQ:PQ)j (PQ:PQ) < 0 P Q 2 H: (2.5) ; !n :; !n) 2.12. # - ' ( P Q ; P!n ; Q!n n ' (; P!n :; Q!n) = det k(P0Pi:Q0Qk )k i k = 1 2 : : : n (2.6) - ' (P0Pi :Q0Qk ) P0 Pi, Q0Qk 3 (2.3). 1 - * ;! ;;;;;;;;;! .;;;;7 n- P n = ! ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;! = P0 P1P2 : : :Pn P(nk$l) = P0P1 : : :Pk;1Pl Pk+1 : : :Pl;1 Pk Pl+1 : : :Pn (n > 1). ' Pk , Pl (k < l). # - ' (; P!n :; Q!n) 3 (2.6). A k = 0, P0 $ Pl ' l- ;;! ;;! (2.6). 8 , ' % % (;;! P0Pi ; :; Q;;! 0Qk) ; ;! ;;;! ;;! ;;;! i- (i 6= l) $& (P P :Q Q ) = (P P :Q Q ) ; (P P :Q Q ). l i
0
k
0
i
0
k
0
l
0
k
1159
2 & l- ' i- (2.6) . 9 ', ! ;! ;! (; P!n :; Q!n) = ;(;P;;; (2.7) (0n $l) :Qn) l = 1 2 : : : n 8Qn 2 Hn+1: A k 6= 0 k < l, Pk $ Pl l-& k-& (2.6). 4 ' , (2.7) ! ;! ;! (; P!n :; Q!n) = ;(; P;;; (nk$l):Qn) k 6= l l k = 0 1 2 : :: n 8Qn 2 Hn+1: (2.8) 3 (2.8) & ; P!n & ; Q!n 2 Hn+1, ; ! ;! P;;; (2.9) (ni$k) = ;P n i k = 0 1 : : : n i 6= k n > 1: @ , ' ' n- (n > 1) (
a = ;1) ( . 7 3 ; P!n T ; Q!n : (; P!n :; Q!n) = (; R!n :; Q!n) 8; Q!n 2 Hn+1 (2.10) ; ! ; ! n n +1 n n +1 n- P 2 H R 2 H . #3 (2.10) ) , ' . A 3 % . ; !n 2 Hn+1 2.13. F n - P ; ;!n , 3 (2.10). R!n 2 Hn+1 % ; P!n = R 2.14. A n- ;! N n 3 (;! N n :; Q!n) = 0 8; Q!n 2 Hn+1 (2.11) ;! N n n- . E& n- ; P!n, &$ Pk = Pl (l 6= k), n- ;! N n , % ! (2.6) (k-& l-&) $ ; Qn 2 Hn+1 . ;! E& n- P n % (N n ) ;! n N ;! , ;!' % (Mn (P n )), Mn(P n ) = f" P ng - n- - V = f Hg, $ ' n + 1 P n , " * , &$ ' " = 1. 9 ', Hn+1 n- ;! n Hn+1 = (N n ) (Mn (P )): (2.12) ; M;!n (P n )
1160
. .
1 % (;! Mn (P n )) 21 (n +1)! %. 8 % (+1 P n) ' ; P!n $ & (;! . 5 , & ' P n $ & ( , % (;1 P n). O - Hn+1 =T ' ;! n % (N n ) n - N ;! - M n (P n ) n- . E , ; ! ; ! n
- ' (P :Qn) n- ; ! P n 2 (;! Mn (P n )) ; Q!n 2 (;! Mn;! (Qn )) ' ;! n n - n- ;M ( P ), M ( Q ). 9 ', n;! n ! n n - ' (P :Q ) ' ;! - n- ;! Mn (P n ) Mn (Qn), & % - V = f Hg. 2 ' ', % & - , , - , - . 1 ' ' $ & - ' ' & . 9 ', - , ' 9- , , - . 2 ' * , &$ , 9- , ' . 1 ' $ &$ ' & - ' , ' ' - ' ;! Mn (P n ) ;! Mn (Qn ) - n- . 1 * , % 3, 3 %)) . F , - & . 1 % 3 * & , , ' & * ) , . K , * & * ' & * ) , . 8 ' 3 (2.12), , ( & - , & % (+1 P n), * - . +, , % (;1 P n), * *
' % (+1 P n) (
1161
* . + *, $( % (N n ) , ' - '. 4 - - . 9 ' & - ' - ' . 8 , ' ' . - , %)) & ' ' . 1 , , - M2 (P 2 ) P0 & H, P1
& H, P0. 9 P2
& ;!2 H, P0 P1. P . 9 P0 P1 & P2 . ; !nj 2.15. 1 2.6 j P ; P!n ' 8 q ;! ;! p ;!n ;!n < n n n P!n H (2.13) j; P!nj = :j q(P;!n:P;!n)j = j pFn (P )j (;P!n :;P!n) > 0 ; ij (P :P )j = ij Fn(P n )j (P :P ) < 0 Fn(P n ) 3 (1.10), (1.11). 2.16. F n- ;P!n ;Q!n ; ! ; ! (P n k Qn ), (; P!n :; Q!n)2 = j; P!n j2 j; Q!nj2: (2.14) 2.17. F n- ; P!n ; ! ; ! ; ! Qn & & * & P n "" Qn ( ), (; P!n :; Q!n) = j; P!nj j; Q!nj: (2.15) ; ! ; ! n n 4 & & * & P "# Q ( ), (; P!n :; Q!n) = ;j; P!nj j; Q!nj: (2.16) 2.18. 1 P0P1 , Q0Q1 , 3 P0P1 "" Q0Q1 : (P0P1:Q0Q1) = jP0P1j jQ0Q1j (2.17) P0P1 "# Q0Q1 : (P0P1:Q0Q1 ) = ;jP0P1 j jQ0Q1 j: (2.18) 2.19. 1 P0P1 , Q0 Q1 , , . . (2.19) P0P1 k Q0Q1 : (P0 P1:Q0Q1 )2 = jP0P1j2 jQ0Q1j2 :
1162
. .
2.1. 7 D- . A ED = f RDg - ED = VD = f RDg, ( = 21 2 '( 3 D X (P Q) = (x y) = 21 gik (xi ; yi )(xk ; yk ) x y 2 Rn (2.20) ik=1 x = fxig y = fyi g, i = 1 2 : : : D, | P Q K . Q gik = const, i k = 1 2 : : : D, | ', det kgikk 6= 0. # ' * gik , i k = 1 2 : : : D, ' D P , gik xixk = 0, x = 0. F ik=1 3 , , ' 3 & % . 1 , PQ q p (2.21) jPQj = gik (xi ; yi )(xk ; yk ) = 2(P Q)
(2.5). 1 ( P0P1, P0Q1 P1Q1 jP1Q1 j2 = jP0Q1 ; P0P1 j2 = jP0Q1j2 + jP0P1j2 ; 2(P0P1 P0Q1 ) (2.22) (P0P1 P0Q1 ) ' ' . 4 & (P0P1 P0Q1) = 12 fjP0P1j2 + jP0Q1 j2 ; jP1Q1 j2g: (2.23) Q Q1 Q0 , (2.23) (P0P1 P0Q0) = 21 fjP0P1j2 + jP0Q0 j2 ; jP1Q0 j2g: (2.24) 1 (2.24) ' (2.23) ' , (P0 P1 Q0Q1) = 21 fjQ0P1 j2 + jP0Q1 j2 ; jP1Q1 j2 ; jP0Q0j2g (2.25)
(2.3), (2.5). 1 P0P1 P0P2 , # 1 ;1. cos # = (P0P1:P0P2 )jP0P2 j;1 jP0P1j;1 (2.26) (2.17), (2.18).
1163
1 m- m n- 3 ( ) '
m=
i^ =n i=1
i^ =n
ei =
i=1
P0 Pi ei = P0Pi i = 1 2 : : : n:
(2.27)
# ' m- n- m q,
q=
i^ =n i=1
ki ki = Q0Qi i = 1 2 : : : n
(2.28)
$ & 3 (m:q) = det k(ei:kl )k = det k(P0Pi:Q0Ql )k i l = 1 2 : : : n: (2.29) 1 m- m n- , 3 (2.27), & i k = 0 1 : : : n, i 6= k. F i k = 1 2 : : : n % ' * 3 . F P0 $ P1
m(0$1) = P1P0
i^ =n i=2
P1Pi =
= ;P0 P1
i^ =n i=2
(P0Pi ; P0 P1) = ;P0 P1
i^ =n i=2
P0Pi = ;m: (2.30)
/ ' P0 $ Pi , i = 1 2 : : : n. # (2.29) (2.6) (2.30) c (2.9) ', m- (2.27);!n- ( n- P n . 7' 2.8 * m- (2.27) , ' * , - , * (2.27) & , % * . 1 * - V = f Hg &$ '. 1 P0R P0 P1 P0P2, 9R 2 H, (P0R:P0Q) = (P0P1 :P0Q) + (P0 P2:P0Q) 8Q 2 H: (2.31) 1 P0R ' P0P $ a: P0R = aP0P, 9R 2 H, (P0R:P0Q) = a(P0 P:P0Q) 8Q 2 H: (2.32)
1164
. .
4 %)) , $ $ R, &$ 3 (2.31), (2.32). 1 , R, &$ (2.31), (2.32), $ , * (2.31) (2.32) * % * . 2.20. n + 1 P n , Pi 2 H, i = 0 1 : : : n, ' & (n + 1)- - ' - , ; P!n & j; P!nj2 Fn(P n ) 6= 0: (2.33) & % ' D- . ' n ei = P0Pi , i = 1 2 : : : n, D- (n 6 D). 1 % (1.8) B Fn (P n ) = det k(ei:ek )k = (n!Sn (P n ))2 i k = 1 2 : : : n (2.34) Sn ( (n + 1)-% 3 P n. 4 $ % ' ei , i = 1 2 : : : n, . A (2.33) , n ei ' ' n- L(P n), $ ' P n . 1 , (2.20) ' ei = P0Pi, i = 1 2 : : : D, (D +1)- - ' P D , , $ & (2.3), (2.20) (1.11) (ei:ek ) = (P0Pi :P0Pk ) = gik (P D ) = ;(P0 Pi Pk ) i k = 1 2 : : : D: (2.35) 2.21. 9 T (P n) n- (n = 0 1 : : :), ' (n;! +1)- - ' P n H ( n- P n 2 Hn+1), ' P 2 H:
T (P n ) TP n = fP j Fn+1(P P n ) = 0g Fn(P n ) 6= 0: (2.36) #3 (2.36) ' ; ! n P : ;;;! T (P n ) = fPn+1 j jP n+1j = 0g j; P!nj 6= 0: (2.37) 9 T (P n ) (AB4) n- , . . , : n + 1 P n . + ' AB4 - H. 8 AB4 ' P n .
1165
3. (n + 1)- - ' P n ' Pi , - , . . - V = f Hg. @ , , - %. 2.22. # SnP T (P n) P 2 T (P n) ' SnP (T (P n )) , $ T (P n ):
l^
=n (Pl P 0) = (Pl P ) P P 0 2 T (P n ):
SnP (T (P n )) = P 0
l=0
(2.38)
1 n- n- , $ P n , ( SnP (T (P n )) P ' % P . 2.1. j;P!nj2 6= 0, j;;! P n Rj2 = Fn+1(P n R) = 0
;;;;! ;;;;! ;;;;! ; P!n k P n;1R : (; P!n :P n;1R)2 = j; P!n j2 jP n;1Rj2
(2.39) n- ; P!n ;;;;! P n;1R ; P;;;;;;;;;; 0P1 : : :Pn;1!R. 2 , Mnn % an n n- R &, 3 R = 0 % 3 & (Mnn;1 )2 = Mn nMn;1 n;1 (n ; 1)- R. 2 ' ) , n- R * . # $ & T (P n ) n- ' ;;;;! T (P n ) TP n = fR j ; P!n k P n;1Rg Fn(P n ) 6= 0: (2.40) 8 , ' T (P n S Q0), ( n- , ; ! n $ ' Q0 n- P . 1 T (P n S Q0) n- , $
' Q0 n- T (P n ). 2.23. 9 T (P nS Q0) n- ; , $ ' ! n ' & Q;0! P n- , &$ & jP nj2 = Fn(P n ) 6= 0, R 2 H: T (P n S Q0) TQ0 (P n ) = fR j Fn+1 (P n;1 RS Q0) = 0g (2.41) Fn+1(P n;1 RS Q0) 3
1166
. .
Fn+1 (P n RS Q0) = det kaik k i k = 1 2 : : : n + 1 (2.42) aik = (P0 Pi:P0Pk ) i k = 1 2 : : : n ain+1 = an+1 i = (P0P i:Q0R) i = 1 2 : : : n (2.43) an+1 n+1 = (Q0 R:Q0R): 4 (2.42) ' Fn+1 (P n R) (1.8) $ & ' P0R Q0 R - ' , $ P0 R. 2 ' $ P0 R Q0 , $ Q0 R Fn+1(P n RS Q0) = 0 ' . E , T (P n S P0) = T (P n ). + . F F1 ' (1.8), (2.3) (2.1) F1(P0 P1) = 2(P0 P1): 9 T (P0) TP0 = fP j (P0 P ) = 0g: (2.44) 1 TP0 = fP0g ' P0, ( S0P0 (Tp0 ) = fP0g ' P0. 4 , - * , TP0 3 P0, S0P (T (P0)) = fP 0 j (P0 P 0) = 0 ^ (P0 P 0) = (P0 P )g = TP0 . 1 ' , ) * F2 (P 2 ) ' F2(P0 P1 P2) = S+ (P0 P1 P2)S2 (P0 P1 P2)S2 (P1 P2 P0)S2 (P2 P0 P1) (2.45) S+ (P0 P1 P2) = S (P0 P1) + S (P1 P2) + S (P0 P2) (2.46) (2.47) S2 (P0 P1 P2) = S (P0 P1) + S (P1 P2) ; S (P0 P2) p S = 2. S+ $ 3 , $ (2.46). 9 ' & - ' , . 9 T (P 2 ) $ & '
, ' (2.45) ( S+ ) ' ' ( . T P0 P1 ] = T P1 P0 ] = fP j S2 (P0 P P1) = 0g (2.48) TP0 P1 = TP1 ]P0 = fP j S2 (P0 P1 P ) = 0g: (2.49)
1167
N ' T P0 P1 ] P0 , P1 , TP0 P1 | , $ ' P1 P0 . @' (2.45), (2.48), (2.49) , TP0 P1 = TP0 ]P1 T P0 P1 ] TP0 P1 : (2.50) 3 (2.19) % & F2(P 2 ) = F2(P2 P 1) = = 0, TP0 P1 P , P0P k P0P1: TP0 P1 = fP j P0P1 k P0 Pg (2.51) ' TP0 P1 = fP j P1 P "# P1P0g (2.52) T P0 P1 = fP j P0 P "" P0P1g: (2.53) ! 2.24. 4 ;T;;; P0 P1 ] ;;P! H , ' , ' ;; P;; 01 ; ;; ! fP0P1 T P0P1 ] g P0P!1 T P0 P1 ] , ' % . F ; T;;; P0P!1 ] ' p (2.54) jT;;;; 0P!1j = 2(P0 P1): P0P!1] j = j;P;; # - ' (; T;;; P0 P!1]:;P;0!Q) ;T;;; P0P!1 ] ; ;; ! Q0 Q1 ' (; T;;; 0Q1) = (;P;; 0P!1:;Q;; 0Q!1) = P0 P!1] :;Q;;! = (P0 Q1) + (Q0 P1) ; (P0 Q0) ; (P1 Q1) P0 P1 Q0 Q1 2 H: (2.55) ))* ' D- ,
. (i) . 7 (2)1=2, ' ' , 3 (% ) & , ' . (ii) . E& ' ' ' & ' &, $ & ' % . (iii) ( ). 1 ' ' . * (ii) (i) ( ), (iii) & ( ' ). 1 9- ' , &$ &. 9 , (ii) (iii) . # (i) , .
. .
1168
' & , &$ & ( . 2.25. 9 T (P n ) ( , & (n + 1)- - ' Qn T (P n ) T (Qn) = T (P n ): (2.56) n 2.26. 9 T (P ) , 8P 2 T (P n ) SnP (T (P n )) = fP g 8P 2 P n : (2.57) n 2.27. - % T (P ), T (P n ) ( . 2.28. - % T T (P n ), % ' T . 2.29. - % n- , % T (P n ) n- . 2.30. 9 T (P n ) ' ' L(P n ), - % T (P n ). I % - . 4 (1.14), (1.15), & 3 &, % , , - , % & , , - , % , . 2 , * - ( . >10]).
3.
-
3.1. n- En
Rn n x = fxn1 x2 : : : xng ' n $ , & x 2 R , y 2 R ' $ ) * :
n X gik (xi ; yi )(xk ; yk ) gik = const i k = 1 2 : : : n (3.1) (x y) = 12 ik=1
det kgik k = (det kgik k);1 6= 0: (3.2) O * ' ) * -) * . n- En - En = f Rng.
1169
4. 9 < = ' 3 & < = < =. 1 * ' gik ' ' , ' ' * ' gik ' . 5. ( 3.1 ' % & n- En n- Rn x = fx1 x2 : : : xng 2 Rn ' ( ' (x y) x y 2 Rn: n X
gik xiyk = (0 x)+ (0 y) ; (x y) gik = const i k = 1 2 : : : n ik=1 (3.3) ' 3 (3.1). n- En = f Hg (H = Rn), - , &$ : 9P n H Fn(P n ) 6= 0 Fn+1(Hn+2 ) = 0 (3.4) n X (P Q) = 12 gik (P n )>;(P0 Pi P ) ; ;(P0 Pi Q)] ik=1 >;(P0 Pk P ) ; ;(P0 Pk Q)] 8P Q 2 H (3.5)
(x y) =
;(P0 P Q) =
n X
ik=1
gik (P n );(P0 Pi P );(P0 Pk Q) 8P Q 2 H
(3.6)
P n (n + 1)- - ' H = Rn (2.33), . . Fn(P n ) 6= 0, ;(P0 Pk P ) 3 (1.11). % (n + 1)- - ' P n ' ' n ei = P0Pi Pi 2 P n i = 1 2 : : : n (3.7) xi = xi(P ) = (P0 P:ei) = ;(P0 P Pi) i = 1 2 : : : n 8P 2 H (3.8) P0P % ' . 1 gik = gik (P n ) = (ei :ek ) = ;(P0 Pi Pk) i k = 1 2 : : : n (3.9) n ' % ' P . % P n - ' , (2.33), (2.34) Fn(P n ) det kgik (P n )k 6= 0 i k = 1 2 : : : n (3.10) ik ik n g = g (P ) ' $ & 3 n X
k=1
gik (P n )gkl (P n ) = il i l = 1 2 : : : n:
(3.11)
. .
1170
I (3.5) (3.6) % , % ' (1.11).
3.2. - V = f Hg n- H, $ (n +1)- - ' P n H, & P Q 2 H (3.6).
- V , &$ n- , ' , . . ' ' P 2 H x 2 Rn . + , P P 0 x En = f Rng. + *, En = f Rng ' ,
3 ;(P0 Pi P ) = xi xi 2 R i = 1 2 : : : n (3.12) P 2 H = Rn, & 3 . Q , ' - ) & . 1 % ' , % & , - f Hg . 4 ( . 3.1. - f Hg n-
-
, , (3.4), (3.5) (3.12).
!" # . $ %& " ! (3.4), (3.5) (3.12) ) * n- En = f Rng. " " !. 1 (3.4) - f Hg n + 1 , ' &$ (n + 1)- - ' . P n (n +1)- - ' P Q 2 H | ' . 1( P n $ & 3 (3.8): xi = ;(P0 Pi P ) yi = ;(P0 Pi Q) i = 1 2 : : : n 8P Q 2 H: (3.13) % 3 (3.6) (x:y) =
n X
ik=1
gik xiyk gik = gik (P n ) const i k = 1 2 : : : n:
(3.14)
1 (3.12) P 2 H x 2 Rn . @ , - f Hg ' n- En = f Rng. '" # (. n- - (. . ) * ).
1171
@ , 9- , ' & & , ) * - 3 (3.5) % 3 (3.6). 7 - & ) >10], ,
9- & . 9- ) , . - , & &, 9- . F , & 9- ' )* ( ' ) * ) - ' . * 3 , 3 . 4 ' ( , ' , ' - ' , $ . # (. . , ) ' ( , ' ( . 9 * ( , ) & ( '& (1.14), (1.15), & $ 3 ' . $ , % ' , '& &$ ( . %)) , ' ) * (1.15), ' ' (1.14), (1.15) . 1 9- ( , '. , ( ' 100 P 99 ' & % . A % , ( R3, , ' , P 99, ' P 99 , ' , ' , P 99 . # ' 9- ' P 99 , (3.5) '3 (3.12). 9 ', 9- ' , ' P 99 , , ' . 4 ' (3.12) , H ! Rn ( . 1 , ' , - f Hg ' - En .
1172
. .
3.3. A - E0 = f H0g ' - , ' . n- - En0 = f H0g ' - , ' n- En = f Rng, ' (n ; 1)- En;1 = f Rn;1g.
n- - - n- En = f Rng. @' (2.36) (3.4) , n- - n- T (P n ) = H, ( & (n + 1)- - ' P n H. % (2.57). 9 . 3.2. P n (n + 1)- ! - " - V f Hg. # , T (P n ) n-
- , , : (1) - T (P n ) n-
T (P n ), (2) T (P n ) ! : SnP (T (P n )) = fP g 8P 2 T (P n ):
4. "
# $
@' , - (1.15). 7 T P0 P1 ] TP0 P1 , ' & ' P0, P1 . 4 (2.47), (2.48). F - % , $ P0 , P1. % T P0 P1 ] ( & (4.1) S2 (P0 R P1) (P0 R) + (R P1) ; (P0 P1) > 0 R $ . 9 ', T P0 P1 ] ( . 4 T P0 P1 ] S2 (P0 R P1) < 0, ' 3 . F , T P0 P1 ] , ' & , &$ & . 1 % (. . , ( ) . 1 , TP0 P1 & &$ ' P0 , P1, (1.15)
1173
(P0 R) + (R P1) > (P0 P1) P0 6= R 6= P1 6= P0 8P0 P1 R 2 H: (4.2) 1 % ' & & (9- &) . 4.1. 7 ' - ( 9- ) 3 H = fx j jxj2 6 1g R3 x = fx1 x2 x3g 2 R3 jxj2
3 ; i 2 X x : i=1
(4.3)
- VE = fE Hg & TE : E : H H ! >0 1) R E (x x0 ) = 12 jx ; x0 j2 x x0 2 H (4.4) - V = f Hg 9- & T H $ & 3 r
0 2 0 : H H ! >0 1) R (x x ) = 2 arcsin E (x2 x ) x x0 2 H: (4.5) 4 % - 3 H & - VEs = fE Tg Vs = f Tg 3 T = @ H = fx j jxj2 = 1g H. T H, 9- TEs Ts - VEs = fE Tg Vs = f Tg & 9- TE T . 4 ' - VE $ & L, - V T . 9 LAB H (A B 2 T) H A, B T. F , 9- TE H T. 9 TAB H (A B 2 T) H. 4 & , ' $ < . 1 , $ ' A B 2 T, LAB ( . . 1). 2 & ) T . # T AB] A B 2 H 3 H, TAB HnT 3 H. 1 ' - Vs = f Tg T \ T AB] T \ TAB T (A B 2 T) 3 ) T, &$ A B 2 T. 4 T \ TAB T (T \ T AB] ) T. @ , - Vs = f Tg & 9- & T. 9 ', 9- V = f Hg H T.
. .
1174
# , 9- Vs = f Tg ) T VE = fE Hg. K Vs = f Tg VE = fE Hg, ' % ' . 7 LABC H (A B C 2 T). 2 , $ ' A B C 2 T. + VEs = fE Tg (T \ LABC ) T ) , $ ' A B C 2 T. K && 9- & Vs = f Tg, ( p & 9- TE H, (A B ) = 2(A B ), A B 2 T, T T &$ ': (A B ) = C 2 Cinf l (A B ) A B 2 T (4.6) 6=A C 6=B C
lC (A B ) >0 1) (T \LABC ) T A, B . F lC (A B ) R 2 (T \LABC ) T (T \ LABC ) T, 3 (4.7) E (A R) = E (A B ) R 2 (T \ LABC ) T 2 R p E (A B ) = 2E (A B ). 73 % R = RAB ( C ) 2 (T \ LABC ) T ) * & 2 >0 1] C 2 T. % A = RAB (0 C ), B = RAB (1 C ). O * R = RAB ( C ) R = R1( C ) R = R2( C ). + 3 ' E (A RAB ( C )) lC (A B ) $ & 3 lC (A B ) =
Z1 d
0
E 0 d 0 (RAB ( C ) RAB ( C )) = d: 0
(4.8)
(4.8) (4.6), B) (A B ) = 2 arcsin SE (A (4.9) 2 A B 2 T
(4.5). 2 ', , ' 9- & H, && ( 9- &) T 3 H. F T $ ' & % ' ( 3 ) LABC H, & ) T. @' % & , & . - , $ * 3 3 $ ' ' % .
1175
9 ', ) * , ' &$ (1.16), ' %)) , '&$ ( ) * ). B , % ) * , ' & '& . ( ) * ) $ '* % >10].
%
1] . . . | .: , 1963. 2] !"# $. %. &'#( "# )#*(, +, *- // /0. | 1959. | !. 14. | . 87. 3] %)5# %. ., 6#), $. 0., 0)# . 7. ( '#( "# // /0. | 1986. | !. 41, #(". 3. | . 1{44. 4] 6- 9., 7'# ., :' 7. :# %. . %)5# +(' *- )#*' // /0. | 1992. | !. 47, #(". 2. | . 3. 5] %)5# %. . $- ' #("-)(< "#<,. | .: 7!! , 1948. 6] Menger K. Untersuchen u=ber allgemeine Metrik // Mathematische Annalen. | 1928. | B. 100. | S. 75{113. 7] Blumenthal L. M. Theory and Applications of Distance Geometry. | Oxford: Clarendon Press, 1953. 8] Rylov Yu. A. Non-Riemannian model of space-time responsible for quantum e?ects // J. Math. Phys. | 1991. | Vol. 32. | P. 2092. 9] %<), %. $. @#) ' #. | .: *5-# )#) -#, 1988. 10] Rylov Yu. A. Extremal properties of Synge's world function and discrete geometry // J. Math. Phys. | 1990. | Vol. 31. | P. 2876. ) !* 1999 .
. . 517.94
:
, , , -
.
!!"!
#$ #$ ,
!%& ! !, ! & ! ! !&
#, ' " '
# . ( &! ) # ! , ! " " ! ** + ! #" % '**+ ".
Abstract
Y. T. Silchenko, On the class of semigroups of linear bounded operators, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1177{1186.
The semigroup of linear bounded operators admitting a singularity not connected with a singularity of the derivative of the semigroup is considered, where the generator of the semigroup may have non-dense domain. Fractional powers of the generator are studied and the theorem of Cauchy problem solvability for the di3erential equation with an operator coe4cient is given.
1. (1) v0 + Av = f (t) (0 < t 6 1) v(0) = v0 "# E . % A | ' ' "( D E , f (t) | ' t > 0 E v0 | ' * E . + (. ,1, 2] (/( ""() ' . 0 *, , '' ', A # ' ', ' (A + I);1 ' Re > ! > 0 ' k(A + I);1k 6 cjj;r (2) r = 1.
) # ! ! 55(, 01-01-00408.
, 2001, " 7, 6 4, !. 1177{1186. c 2001 , !" #$ %
1178
. .
3 4 # ( , * '( 45 . 6 , A 4 ( E " 4 # ( "'
(r < 1). 7, ,3] ' '# ''# . 9 * ' ( -" ' , # " ( ' . : ,4] A = (;1)k d2k =dx2k Lp (0 1) ' ' . : , ( , '# ' ' " "/( . : * " A
' Lp 4 (2) ' ' ' r < 1, (/ " . : " ' '# '# A (/# U (t), ' ' # ' ( " kU (t)k 6 Mt;e;!t kU 0(t)k 6 Mt; e;!t (3) '# > 0, > 1, ! > 0, 5 45 " . + * # ( , "# ,1,3, 5, 6]). : " ( ' # , ( "' A , , (1) 5 . 3 , * 4 "' ("' , " ' 1 + , 0 6 < 1. + * (/ ,4], " "/ * A = A(t), < 2. 2. 0 5 ,4] . 3"
L(E ) '# '# , (/# E E . > , / - U (t) (t > 0), " (/ (/ : 1 U (t) 4 t > 0 | ' ' , (/ E D@ 2 U (t)U (s) = U (t + s) (t s > 0)@ 3 U (t) t > 0 L(E ) dtd U (t) = ;AU (t)@ 4 U (t)Av = AU (t)v v 2 D, t > 0@ 5 t!+0 lim U (t)v = v v 2 D@ 6 ' ' (3) '# > 0, > 1, ! > 0. . 3 -( U (t), " (/( ' ' 1 {6 , " ' ( A( )), 4 5 A.
...
1179
3 , U (t) A . + , v0 + Av = 0, v(0) = 0 v = v(t), (/# "# t kv(t)k 6 Me!t. 0 (3) "# ' + 1 6 . C , 3 '
Z
U (t) = U ( ) + AU (s) ds (0 < t 6 ): t
0*
Z
kU (t)k 6 M ; e;! + M s; e;!s ds 6 t
6 M ; + M; 1 ,t;(;1) ; ;(;1)] 6 ct;(;1) :
0 ( kU (t)k 6 Mt; '# t. 3( 6 ; 1. 9 = 1, = 0, " , - 4 , ' ' + 1 6 . : 5 4 D0 = x 2 E : 9 t!+0 lim U (tt);I x * 4 A0 x = t!+0 lim U (tt);I x, ' ' / '. :
Zt
U (t)x ; U (s)x = ; AU ( )x d (t s > 0): s
9 x 2 D, 4 s ! 0:
Zt
U (t)x ; x = ; U ( )Ax d : 0* '4
0
U (t) ; I x = ; 1 Z U ( )Ax d
t t t
0
t ! 0 , , x 2 D0 , D 0 x 2 D0 . 7
D0 A0 x = ;Ax x 2 D.
Z U ( s ) ; I 1 U (t) s x = ; s AU ( )x d
t+s t
1180
. .
4 s ! 0: U (t)A0 x = ;AU (t)x. 0 A / ' "' A;1 . 7 U (t)A;2 A0 x = = ; U (t)A;1 x. % 4 t ! 0. : x = ;A;1 A0 x. + , x 2 D, D0 D A = ;A0 . 3. 0 5 ' A( ). 1. 9 D E (D = E ) = 0, = 1, U (t) |
. 2. 0 A L2 (;1 +1) L2(;1 +1) Av = ;D2v1 ; iDk v1 ;Dl v1 ; D2v2 ; iDk v2 D = i d=dx, v = fv1 v2g, k > 2, l > 2. 7 4 '# , " ( E. 9. F ,1]. 9 " ' " G x " " v(t x) v~(t p), ' 5 "' '# '# dv~ + A(p)~v = 0 A(p) = p2 ; ipk 0 ;pl p2 ; ipk : dt 0 * exp,(;p2 + ipk )t] 0 U (t p) = tpl exp,(;p2 + ipk )t] exp,(;p2 + ipk )t]
A((l=2) ; 1 (l + k)=2 ; 1), jpl exp(;tp2 )j 6 6 ct;l=2 . ' A(p) 1 ! 0 k +p2 ; ip ;1 (A(p) + I) = : pl 1 (+p2 ;ipk )2
+p2 ;ipk
3( , l > 2k A '# . 0 l 6 2k (2) ' r = (2k ; l)=k. 3. 0 E | 4 v = fxn yng1 1 , 1 P '# kvk = (n1=2jxnj + jyn j). : 5 E n=1 L = fv 2 E : x1 = y1 = 0g E -(
U (t)v = f0 0@ (xn cos nt ; yn sin nt) exp(inp t ; nt) (xn sin nt + yn cos nt) exp(inp t ; nt)g1 2 : % p > 1 | ' . 3 - U (t) A(1=2 (1=2) + p) (" p > 1. 3 4 A, (/ Av = f0 0@ ;(inp ; n)xn + nyn ;(inp ; n)yn ; nxng1 2
1181
...
"(
D(A) = v 2 L :
1 X
,n1=2j(inp ; n)xn ; nynj + j(inp ; n)yn + nxn j] < 1 :
n=2
C * (2) r = 2 ; (3=2p) p < 3=2 r = 1 p > 3=2. 4. 0 E = Lp (0 1) \ L1 (0 +1) kvk = kvkLp + kvkL1 A | A = ;d2 =dx2 "(
D(A) = v(x) 2 E 9v (x) 2 E 00
Z1
v(x) dx = 0 :
0
7 , 4 5 * , 5
Z h ;(s+x)2 i ; x4t (s;x)2 e 4t + e; 4t '(s) ds ; ep U (t)'(x) = p1 2 t 0
t 1
2
Z1
'(s) ds
0
4 A((1=2) ; (1=2p) (3=2) ; (1=2p)), (3) "' ' r = (1=2) + (1=2p). 3 , # 2 3 ' r ' 4 " -" 45 , # 3 4 " '# ( ' 4 E 2 '# ( l > 2k). 0* ( ' A, ' ' -" . 4. 3 (3) ( ' "' A. 0 / ' "' A;1. 3"
D(A; ) 4 * v 2 E , '# " ( ) #
Z1
C # v 4
s;1 U (s)v ds ( > 0):
0
Z 1 A v = ;() s;1 U (s)v ds: ;
1
(4)
0
6 * ' 4 D D(A; ). 9 > , D(A; ) = E , ' A; '. 0 < , ' ,1], * ' "' '. K ' "'# .
1182
. .
: A;p A;q = A;(p+q) p > 0 q > @ A;p A;q = A;q A;p = A;(p+q) p q > @ A;p A;q v = A;(p+q) v = A;q A;p v p > q > 0 v 2 D(A;q ): . 1.
1 Z p;1 U ( ) 1 Z sq;1 U (s)v ds d = ;() ;(q) 1
1
"
1 = ;(p);( q)
Z1 1;(Z = )
"+
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A. V. Strelets, A characterization of operator space modules over full operator algebra, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1187{1201.
In the paper it is proved that the column operator structure is the unique one (up to completely isomorphism) such that a given Hilbert space H becomes the left operator module over B(H). Moreover, the corresponding module is contractive if and only if this Hilbertian is completely isometric to the column one.
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$ Cn Rn $ " > 0 n ' x1 : : : xn H y1 : : : yn H, 0 x 1 0x 1 B .1 C B .1 C 6 (Cn + ") @ .. A @ .. A xn M 1(H) xn M 1(Hmax) ;y : : : y 1 ; 1 n M1 (H) > R + " y1 : : : yn M1 (Hmin) : 2
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1] Eros E. G., Ruan Z.-J. On the abstract characterization of operator spaces // Proc. Amer. Math. Soc. | 1993. | Vol. 119, no. 2. | P. 579{584. 2] Blecher D. P. Some general theory of operator algebras and their modules // Operator Algebra and Applications / A. Katavolos (ed.). | Kluwer Academic Publishers, 1997. | P. 113{143. 3] Blecher D. P., Paulsen V. I. Tensor products of operator spaces // J. Funct. Anal. | 1991. | Vol. 99. | P. 262{292. 4] Blecher D. P. Tensor products of operator spaces II // Canad. J. Math. | 1992. | Vol. 44. | P. 75. 5] Eros E. G., Ruan Z.-J. Self-duality for the Haagerup tensor product and Hilbert space factorization // J. Funct. Anal. | 1991. | Vol. 100. | P. 275{284. 6] Mathes B. Characterizations of row and column Hilbert space // J. London Math. Soc. (2). | 1994. | Vol. 50. | P. 199{208. & ' 2000 .
2 . .
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Abstract
S. A. Tishchenko, The largest graphs of diameter 2 and xed Euler characteristics, Fundamentalnayai prikladnayamatematika, vol. 7 (2001), no. 4, pp. 1203{1225.
We compute the exact maximum size of a planar graph with diameter 2 and 6xed maximum degree + 6 7. To 6nd the solution of the problem we use the irrelevant path method. It is proved that the known graphs with size 2+ +1 (3 6 + 6 4) and + + 5 (5 6 + 6 7) are the largest possible ones. This result completes the analysis of the degree{diameter problem for planar graphs of diameter 2. In the case + 6 6, we found also the largest graphs of diameter 2 that are embedded into the projective plane and into the torus.
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6 F(G ). 0 m m- . 7# m- Fm (G ), ! "#, ! 3- , E33(G ). @ 8 ! (G): (G) max fjV j ; jE j + jF()jg = jV j ; jE j + jF(max)j: (3) (G) @ , , 6 ! . 7# P xy (G) ! ! ,
6 ! x y 2 V (G). 0 1 2 . @ -"# , 6 = 2 (1), 0 6 6 1 (2), " . D # , 8 + 2 = 2 3 6 ( 6 4E > >2( < ( + 2 5 6 ( 6 7E jV (G)j = >2( +65 = (4) = 1 3 6 ( 6 6E > :3( + 2 = 0 3 6 ( 6 6: 4 " ! ! ,
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1206
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1207
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3.1. x y 2 V (G), xy 2 E(G). X = N(x) n (fyg N(y)), jX j > 4, Y = N(y) n (fxg N(x)), jY j > 4, z 2 V (G) n fx yg, d(z) > jX j + jY j. '
(G) G. C -
x y ( . . 3): X = fx1 x2 : : : xkg, k = jX j, Y = fy1 y2 : : : ym g, m = jY j. '
xiyj , 1 < i < k, 1 < j < m ( jX j > 4, jY j > 4 6 ). 3.1.1. xiyj 2 E(G), 1 < i < k, 1 < j < m, d(xi) > jX j + jY j, d(yj ) > jX j + jY j. D G , xi;1yj ;1 ! xi , yj . L #6 ,
fxi;1 yj ;1g N(xi) ( . 3). D G , xi yj , i0 > i, j 0 > j, ! xi. 8 fxi yj g N(xi ), i0 > i, j 0 > j. ; , x1yj , j 0 > j, xi y1 , i0 > i, ! xi , fxi yj g N(xi), i0 > i, j 0 > j. C xi
m Y , k ; 1 X x, d(xi) > k + m = jX j + jY j. 0
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D , , . 4 3.1.1 8 , xi yj 2= E(G), 1 < i < k, 1 < j < m. 3.1.2. jN(xi) \ Y j = 0, 1 < i < k, jN(yj ) \ X j = 0, 1 < j < m, 9z 2 V (G) n (fx yg X Y ), d(z) > jX j + jY j. 4 ,
6 xi yj , 6 z 2 V (G) n (fx yg X Y ), xi yj 2 N(z) ( . 3#). G , xi yj , i0 > i, j 0 > j, ! z. 8 fxi yj g N(z), i0 > i, j 0 > j. ; , xi yj , i0 < i, j 0 < j, ! z, , fxi yj g N(z), i0 < i, j 0 < j. C z
"#
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1208
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6 xi yj , 3.1.1, 3.1.2 6 z 2 fx1 xk y1 ym g, fxi yj g N(z). L
#6 , fxi yj g N(x1). 3.1.3. x1yj 2 E(G), d(x1) > jX j + jY j. xi yj 2= E(G), 1 < i0 < k, G , xi yj , 1 < i0 < k, j 0 > j, ! x1 ( . 3). 8 fxi yj g N(x1 ), 1 < i0 < k, j 0 > j. ; , x2yj , 1 < j 0 < j, ! x1, , yj 2 N(x1), 1 < j 0 < j. 8 xk ym , x2 y1 ! x1, , fxk y1g N(x1). C x1
"#
m Y , k ; 1 X x, d(x1) > m + k = jX j + jY j. T 3.1.2, 3.1.3 . U . 3.1.1. ( 6 7 x y 2 V (G), xy 2 E(G) G, V (G) = N(x) N(y), jV (G)j 6 ( + 4. ' #" G n fx yg 6 ! ( . 4): X = N(x) n (fyg N(y)) Y = N(y) n (fxg N(x)) S = N(x) \ N(y): L #6 , jX j > jY j. J jY j > 4, 3.1 ( > jX j + jY j > 8, . 8 jY j 6 3, d(x) = jX j + jS j +1 6 (, jV j = jX j + jY j + jS j +2 6 (+4. 0
0
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3.1.2. ( 6 7 ! G " ! 3-$, jV (G)j 6 ( + 4.
V 6 3- , , . '
, , 3- , # x y z 2 V (G) ( . 4#), w1 w2 2 V (G)
2
1209
. G , ! # 8 ! !
x y z. L #6 , w1z 2= E(G). '
U , ! 3- . U# ,
6 w1 8 , ! # x, # y,
, U N(x) N(y). '
W , ! 3- . J W N(x) N(y), V = N(x) N(y), 3.1.1 jV (G)j 6 6 ( + 4. I , W 6 N(x) N(y), . . 6 wi 2 W , wi 2= N(X) N(Y ). G , ,
6 wi U, ! z, U N(z). ; , W x y z, # 6 ! 3- . J W N(z), V = fz g N(z), jV j 6 ( + 1. J , # #6 , W N(y), V = N(y) N(z), 3.1.1 jV (G)j 6 ( + 4. I 6 # ,5]
5 6 ( 6 7. 3.2. % G 5 6 ( 6 7 !! ! 3- !. 7 ,
f 2 Fm (G ), m > 3, G fx wg f 2 F(G ), xw 6 f. 3.2.1. xw 2= E(G). 7 , xw 2 E(G) ( . 4). D x y f, 3, 6 y z, ! ! !, x w # , # 6 f. ' G n fx y w z g 6 ! : Y , 6 , ! 6 xywx , Z, 6 , ! 6 xw : : :zx . U# ,
6 fyg Y fz g Z, ! # x, # w,
, V (G) = N(x) N(w). C 3.1.1 jV (G)j 6 ( + 4, (4). G -"#, # d(x) = (, # d(w) = (. L #6 , d(x) = (. '
x f y z. ; , ! (. L #6 , d(y) = (. ' #" Gnfx yg 6 ! ( . 4 ): X = N(x) n (fyg N(y)) Y = N(y) n (fxg N(x)) S = N(x) \ N(y):
1210
. .
7 , jX j + jS j = jY j + jS j = ( ; 1. C 3.1.2 jS j 6 2. 8 jX j > ( ; 3, jY j > ( ; 3. C ( = 7 jX j + jY j > 2( ; 6 > (, 3.1. 7 " 8
5 6 ( 6 6. 3.2.2. % 5 6 ( 6 6 jX j = jY j = 3, jS j = ( ; 4. C ( = 6, jS j 6 1, jX j > 4, jY j > 4, 3.1. 8 jS j = 2, jX j = jY j = 3. C ( = 5 jS j = 2 # 3- , 6 ( . 4 ). C s 2 S, , #
9 , jV j 6 10 = ( + 5, (4). jS j = 0, jX j > 4, jY j > 4, 3.1. 8 jS j = 1, jX j = jY j = 3. 7# W V n (fx yg X Y S). 4 (4) jW j > 2. '
(G) G. C
x y ( . 5) X = fx1 x2 x3g, Y = fy1 y2 y3g.
8". 5. 9"! , $ G ( 3.2)
'
x2 y2 . 3.2.3. x2 y2 2= E(G). 7 , x2 y2 2 E(G) ( . 5). D G , x1 y1 ! x2, y2 . L #6 ,
fx1 y1g N(x2). G , x3 y3 ! x2 . 8 fx3 y3g N(x2). C x2 5 X Y , x. '
w 2 W. W # , ! wx1, wx3 ! x2 , d(x2) > 6, . 3.2.4. jN(x2) \ Y j + jN(y2) \ X j 6= 0. 4 ,
6 x2 y2 , 3.2.3
6 z 2 S W, fx2 y2g N(z) ( . 5#). G , x3y3 ! z. 8 fx3 y3 g N(z). ; , x1y1 ! z, , x1 y1 2 N(z). C z
"#
2
1211
X Y . d(z) 6 6, z 2 W. '
w 2 W. W # ,
! wx1, wx3, wy3 ! z, d(z) > 6, . 4 ,
6 x2 y2 , 3.2.3, 3.2.4 6 z 2 fx1 x3 y1 y3 g, fx2 y2g N(z). L
#6 , fx2 y2 g N(x1). 4 x2y3 , fx2 y3g N(x1). ) x3y3 ! x1 y2 . 3.2.5. x3 6 N(x1). , , x3 N(x1) ( . 5). J x1y1 2 2 E(G), # w 2 W, ! wx2, wy3 ! x1 , d(x1) > 6, . 8 x1y1 2= E(G), x2y1 ! x3, fx2 y1 g N(x3 ). '
w1 w2 2 W. C ( = 5 wix2 , wi y3 6 , wi 2 N(x3 ), i = 1 2, d(x3) > 6, . 8 ( = 6, jS j = 2. ) wi x2, wi y3 6 , w1, w2 yy1 x3x1 y3 y. 7# s1 s2 2 S # xx3y1 yx, # xx1 y3yx ( # xy ! 3- ). C #
!
w1, w2. 3.2.6. fx3 y3 g 6 N(y2 ). , , fx3 y3 g N(y2 ) ( . 5 ). ) x2y1 ! x3, fx2 y1 g N(x3). ) x1y1 ! # x3 , # y2 . C 3.2.5 x1x3 2= E(G), 8 y1 y2 2 E(G). '
w1 w2 2 W. C ( = 5 wi y1 , wiy3 6 , wi 2 N(x1) \ \ N(x3), i = 1 2, d(x1) > 6, d(x3) > 6, . 8 ( = 6, jS j = 2. ) wiy1 , wi y3 6 , w1, w2 yy1 x3x2 x1y3 y. 7# s1 s2 2 S # xx3y1 yx, # xx1y3 yx ( # xy ! 3- ). C # !
w1 , w2 . C 3.2.5, 3.2.6 x3y3 . U . U 3.3 3.4 # ,6]
= 3 = 4 5 6 ( 6 7. 3.3. % 5 6 ( 6 7 jV3(G)j = 0. 7 ,
G w 2 V3 (G). C 3.2
1212
. .
3, 8 w x, y, z # 3- ( . 6).
8". 6. = #$% % 3.3
3.3.1. & w x, y, z , $ " 3-$.
, 6 6" u, x, y, z ( . 6). D 3- uxy, uyz uzx 3.1.2 ! 6 , 8 jV j 6 8, (4). C V (G) n fx y z wg
" w, 8 ! # x, y, z, V (G) n fx y z wg 6 ! : X = N(x) n (fy z wg N(y) N(z)) Y = N(y) n (fx z wg N(x) N(z)) Z = N(z) n (fx y wg N(x) N(y)) Sx = N(y) \ N(z) n fx wg Sy = N(x) \ N(z) n fy wg Sz = N(x) \ N(y) n fz wg: 7 , jV j = jX j + jY j + jZ j + jSx j + jSy j + jSz j + 4 (5) jX j + jSy j + jSz j 6 ( ; 3 jY j + jSx j + jSz j 6 ( ; 3 (6) jZ j + jSx j + jSy j 6 ( ; 3:
2
1213
G , (5){(6) jV j + jSx j + jSy j + jSz j 6 3( ; 5 (7) (4) ( = 5. 8
6 6 ( 6 7. 3.3.2. jX j 6= 0, jY j 6= 0, jZ j 6= 0. L #6 , , jZ j = 0. U# ,
6 w , ! 3- , ! # x, # y, , V (G) = N(x)N(y), 3.1.1 jV j 6 (+4, (4).
3.3.3.
jSx j = jSy j = jSz j = 1: (8) L #6 , , jSx j = 1. 4 3.2 , # yz 3- . I , jSx j > 1. ; , jSy j > 1, jSz j > 1. jSx j > 3 !
6 3- , 3.1.2. 7 ,
jSx j = 2, fs1 s2 g N(Y ) \ N(Z) ( . 6#). C 3.4.2 6 x1 2 N(x), x1 2= 2= N(y) N(z). ) x1 s1 ! s2 , , fx1 s1g N(s2 ). C 3.3.2 Y , G , yi 2 Y sz , # #6 , xys2 x1x. ; , Z sy xzs2 x1x. 0 , sz s2 2= E(G). 7 , sz s2 2 E(G). C 8 y1 sy ! s2 , sy s2 2 E(G). G , ,
6 Y , xzs2 x1x, ,
6 Z , xys2 x1x, ! s2 , V = fx w s2g N(s2 ), jV j 6 ( + 3, (4). 8 sz s2 2= E(G), ,
6 Z sz , ! x1 , Z N(x1). G , sy s2 2= E(G), ,
6 Y sy , ! x1 . 8 V = fy z w s1 x1g N(x1 ), jV j 6 ( + 5, (4). 3.3.4. sx sy sy sz sx sz 2= E(G). L #6 , sx sy 2= E(G). 7 ,
sx sy 2 E(G) ( . 6). C 3.3.2 3.1.2 jZ j = 1. ) ,
6 8 z1 2 Z
sz , ! , # #6 , sy ,
1214
. .
sy sz 2 E(G). C 3.3.2 3.1.2 jX j = 1. T (5), (6) (8), jV j 6 ( + 4, (4). 4 (4), (5) 3.3.3 , jX j + jY j + jZ j > ( ; 1, 8 , # #6 , jX j > 2
x1 x2 2 X. 3.3.5. sx xi 2= E(G), i = 1 2. 7 ,
sx x1 2 E(G) ( . 6 ). C 3.3.2 Y , G , yi 2 Y , # #6 , xsz ysx x1 x. ; , Z xsy zsx x1 x. 8 3.3.4 ,
6 Y , xsy zsx x1 x, ,
6 Z , xsz ysx x1x, ! x1. 8 V (G) = fw y z x1g N(x1), jV j 6 ( + 4, (4). C 3.3.4, 3.3.5, # #6 , x1sx ! y1 2 Y , x1 sx 2 N(y1 ). 3.3.6. x2 y1 2= E(G). 7 ,
x2y1 2 E(G) ( . 6). C 3.3.2 Z , G , zi 2 Z, # #6 , xsy zsx yy1 x2 x. 8 3.3.4 ,
6 Z sz , ! y1 , fsz zig N(y1 ), zi 2 Z. ; , yi 2 Y , i > 1, y1 ( yi sy ), jY j > 2 sy 2 N(y1 ). C X, xsy zsx yy1 x2x, y1 ( xisx , xi 2 X, i > 2). 8 X xsy zsx yy1 x2x, V = fx w z sy y1g N(y1 ), jV j 6 ( + 5, (4). '
8 , # #6 , x3 2 X, xsy zsx yy1 x2x. C jY j > 2 jZ j > 2 y1 ( x3 sx ), V = fx w z y1g N(y1 ), jV j 6 ( + 4, (4). I jY j = jZ j = 1, V = = fy1 z1 sx xg N(x), jV j 6 ( + 4, 8 (4). G , x2sx ! z2 2 Z ( . 6). 7 , jY j = jZ j = 1,
8 ! sy sz
. T (5), (6) 3.3.4, jV j 6 ( + 4, (4). U . 3.3.1. % 3-$ G 5 6 ( 6 7 !! ! !.
2
1215
7 ,
G,
3- , 6 . C 3.1.2
3- ! 6 , 8 # #6 , ! w. I 8 ", # 3- , 3.2. % w 2 V3 3.3. 3.4. % 5 6 ( 6 7 jV4(G)j = 0. 7 ,
v 2 V4 (G). C 3.2 3- , 8 v # 4- xyzw ( . 7). C 3.3.1 xz yw 2= E(G) .
8". 7. = #$% % 3.4
3.4.1. V (G) 6= N(x) N(z) fx z g, V (G) 6= N(y) N(w) fy wg. 7 , # #6 , V = N(x) N(z) fx z g. '
G0, # G v
# # xz ( . 7#). 7 , G0 2. C 3.1.2 jV (G0)j 6 ((G0) + 4. % ((G0) 6 ((G) jV (G0 )j = jV (G)j ; 1, jV (G)j 6 ( + 5, (4). 3.4.2. N(x) \ N(z) = fy v wg, N(y) \ N(w) = fx v z g.
1216
. .
7 , # #6 , 6 u 2= fy v wg, fx z g N(u) ( . 7). C 3.4.1 6 y1 , y, v, w x, z. ) vy1 ! # y, # w. L #6 , y1 y 2 E(G). G , y1 w ! u, y1 w 2 N(u), 3.3.1 3- uxwu uzwu . C 3.2 "# xy yz 3- , 8 6 sz 2 N(x) \ N(y) sx 2 N(z) \ N(y). 4
6 ! ,
6 ! v
, , V = N(X) N(Y ) N(Z). I 6 x1 2 N(x), x1y 2= E(G), V = N(Y ) N(Z), 3.1.1 jV (G)j 6 ( + 4. ; , 6 z1 2 N(z), z1 y 2= E(G). 0 , sz u 2= E(G). 7 , sz u 2 E(G). C 8 x1sx ! u, , sx u 2 E(G). G , ,
6 X , yy1 uzsx y, ,
6 Z , yy1 uxsz y, ! u, V = fy v ug N(u), jV j 6 ( + 3, (4). 8 sz u 2= E(G), ,
6 Z sz , ! y1 , Z N(y1 ). G , sx u 2= E(G), ,
6 X sx , ! y1 . 8 V = fx z w v y1g N(y1 ), jV j 6 ( + 5, (4). C 3.2 "# xy, yz, zw wx 3- , 8
6 sz 2 N(x) \ N(y), sw 2 N(y) \ N(z), sx 2 N(z) \ N(w) sy 2 N(w) \ N(x) ( . 7 ). C 3.3.1 3.4.2
V n fv x y z w sx sy sz sw g fx y z wg. ' ! 6 ! : X N(x), Y N(y), Z N(z), W N(w). L #6 ,
x1 2 X. 3.4.3. x1 sw x1sx 2= E(G). 7 , # #6 , x1 sw 2 E(G) ( . 7 ). ) sz sx ! # x1, # sw ( 3.3.1 xss 2= E(G)). '
sz sx 2 N(x1) ( . 7). G , 8 wyi , yi 2 Y , ywi , wi 2 W . 8 jY j = jW j = 0, V = N(x) N(z) fx z g, 3.4.1. '
sz sx 2 N(sw ) ( . 7). C
3.3.1 3- ysz sw y zsw sx z , jY j = jZ j = 0. C ,
6 jX j z,
2
1217
,
6 jW j y, ! sw , V = fx v w sy sw g N(sw ), jV j 6 ( + 5, (4). '
x1 z. C 3.4.2 x1z 2= E(G). 4 3.4.3 , x1z ! fsx sy sz sw g. 8 ! zi 2 Z ( . 7). jY j = 0,
6 wyi , yi 2 Y . ; , jW j = 0 - ! ywi , wi 2 W . 8 V = N(x) N(z) fx z g, 3.4.1.
4. " #
4.1. x y 2 V (G). & jP xy (G) ; P~ xy (G )j > 0 8. 7 , x y 2 V (G), P~ xy (G ) = P xy (G). xy ~ jP1(G )j = 0, jP1 (G)j = 0. 4 8 , jP2xy (G)j = 0,
, , ! 2- , # # , 6 , # , # . @ jP xy(G)j = 0. 4.2. % G 2 ~ )j 6 X m jVm j ; jV j(jV j ; 1) : jP(G (9) 2 m 2 7 , X (10) P1(G) = E(G) = m2 jVm j: m ) m m(m ; 1)=2 2- !. I , ! X P2 (G) = m(m2; 1) jVm j: (11) m 4 4.1 , ~ )j 6 P1(G) + P2(G) ; jV j(jV j ; 1) : jP(G (12) 2 (10){(11) (12), (9). C
1{3 " ! ~ )j. jP(G 4.3. % G ~ )j = 2jF4j + 3jF3j + jE33j: jP(G (13)
1218
. .
C . 4.3.1. ' G -
$ , ( " . 8.
8". 8. ? 4@ $$% !$" -!A. B % *$% - % # $A $ $% - @
T . C = 2, 5 6 ( 6 7 ) 3.2 jV j = 3, (4), #), ) ) 3.3, 3.2 3.4
. C ! ! : jV j 6 2( + 1 ) )E jV j 6 3( ; 5 #)E jV j 6 3( ; 4 ). 7 , ! ! (4) . '
, # f 2 2 F3(G ). # !,
" p 2 P2 (G), q 2 P1(G), p q f, p = q. 1 2- 3- . '
"! , # f 2 F4 (G ), ! . #
!,
" p q 2 P2 (G), p q f, p = q. 2 2- 4- . )
! ! , 4.3.1 jV2 (G)j = 0. 8 !
. I , ,
1, 2 2jF4j + 3jF3j ! pk fi 2 F(G ). '
f1 f2 2 F3(G), # a1 ( . 2). 3 a2a3 a4a5 2 P2(G), # # , . J 8 !
2
1219
3, 4.3.1 ( . 8#, ) , 8 , 4. J 8 ! 4, 4.3.1 ( . 8 ) ! # , 8 , 3. 8 3 # . ! jE33j. 0 # pk fj 2 F(G ), (13). X (13) " ! . 0 " # 6 : ~ )j > 2jF4j + 3jF3j > jP(G X 9 jE j ; 15 jV j + 15 E > 15 ;2 3m jFm j = 15 j F j ; 3 j E j = (14) 2 2 2 2 m
~ )j > 2jF4j + 6jF3j ; jE j > jP(G X > (18 ; 4m)jFm j ; jE j = 18jF j ; 9jE j = 9jE j ; 18jV j + 18E m
(15)
~ )j > 2jF4j + 4jF3j ; jE j > jP(G 3 X 17 jE j ; 10jV j + 10: (16) j E j j E j = > (10 ; 2m)jFm j ; 3 = 10jF j ; 13 3 3 m
(14){(16) (3) (13), (15)
X 2jE33j > 3jF3j ; mjFm j = 6jF3j ; 2jE j: (17) m>3
(16) # (14) (15).
5. %
!"# $ 1. 7 ,
G. C 4.2.1 jV2(G)j = 0. ( = 3 (9) (14): 2 ~ )j 6 X m jV j ; jV j(jV j ; 1) = jV j(10 ; jV j) jP(G (18) 2 m 2 2 ~ )j > 9 jE j ; 15 jV j + 15 = 15 ; 3 jV j: jP(G 2 2 2 4 7#Y (18) (19), jV j2 ; 23 2 jV j + 30 6 0 m
(19) (20)
1220
. .
jV j 6 7, (4). ( = 4 (9), (15) (16) 6 : 2 2 ~ )j 6 X m jVm j ; jV j(jV j ; 1) = 10jV3j + 17jV4j ; jV j (21) jP(G 2 2 m 2 ~ )j > 9 jV3 j + 36 (22) jP(G 2 ~ )j > ; 3 jV3 j + 4 jV4j + 20: jP(G (23) 2 3 7#Y (21) (22), jV j2 ; 17jV j + 72 6 2jV3j: (24) I , #Y (21) (23), 4 jV j2 ; 43 (25) 3 jV j + 40 6 ; 3 jV3 j: 4 (25) , jV j 6 10, " jV j = 10 (24) (25) # 1 6 jV3j 6 25: (26) " " , V3 = 2, V4 = 0, jE j = 19 jF j = 11: (27) 4
3jF3j + 4(jF j ; jF3j) 6 2jE j (28) jF3j > 4jF j ; 2jE j: (29) 4 (9) P~ 6 28 (30)
(15) , F3 6 7. C F3 = 7 (27) # F4 = 3, F5 = 1, (15) P~ > 29, (30). C (29) F3 > 6, 8 "
F3 = 6, F4 = 5. C (13) (30) jE33j = 0. I , 18 "#
3- 4- # 4- . ) ! 3 # 8 4- , 3- 8 # . '
, #" . 9, " #
# 7 G, 8, # G.
2
1221
8". 9. & ,$% !$" -!A
( > 5 3.2 F(G) 3- , (13) 6 : ~ )j = 3jF j + jE j = 3jE j: jP(G (31) ) , 3.3.1, 3.4.1 3.5 jV2 j = jV3j = jV4j = 0. 4 (9) X X ,(( + 5)m ; m2 ; 5(]jVmj = (( ; m)(m ; 5)jVm j > 0 (32)
m
~ )j 6 jP(G
m
X ,(( + 5)m ; 5(] m
2
jVm j ; jV j(jV2j ; 1) = 2 ; 1)jV j : (33) = (( + 5)jE j ; jV j + (5( 2
7#Y (31) (33) jE j = 23 jF j = 32 jE j ; 23 jV j + 3 jV j2 ; (( + 13)jV j + 12(( + 2) 6 0: % # , (35) r 2 + 73 < ( + 13 + ( ; 11 = ( + 1 ( + 13 jV j 6 2 + ( ; 22( 4 2 2 (4). D .
(34) (35) (36)
!"# $ 2. C ( = 3 jV j 6 10
@ ,5]. C ! !, ,
G. ( = 4, = 1, #Y (9) (14), jV j2 ; 14jV j + 15 6 ;25jV3j: (37)
1222
. .
% # , (37) jV j 6 12
# G. ( = 5, = 1, #Y (9) (15), jV j2 ; 17jV j + 36 6 2jV3j: (38) ; , #Y (9) (16), jV j2 ; 533 jV j + 20 6 ; 143 jV3j ; 103 jV4j: (39) ) # (38) (39), ! jV j 6 15. D jV j = 15 # G, 8 (38) (39) jV3 j > 3 (40) 7jV3j + 5jV4j 6 30 (41)
, " " jV4j, , jV4 j = 1, jV3j = 3, jV5j = 11. 8 jE j = 34, jF j = 20. 4 (9) ! P~ 6 54,
(15) (29) , F3 = 12 F4 = 8 jE33j 6 2: (42) % # , 8 32 #
3- 4- # 3- . % 14 , 6 ! " , # E33. 8 jE33j > 7, (42). ( = 6, = 1, #Y (9) (15), jV j2 ; 19jV j + 36 6 ;2jV4j ; 2jV5 j: (43) % # , (43) jV j 6 16 (4). D = 0
(4) jV j > 3( + 1 (44) jV3j = 0. ( = 4, = 0, #Y (9) (14), (45) jV j2 ; 14jV j 6 0: ' (45) ! jV j 6 14. D jV j = 14 (4), 8 jE j = 28, jF j = 14. 4 (9) ! P~ 6 21,
(15) (29) , F3 = F5 = 7, E33 = 0. 7 , 8 , 3- , . .
, #" . 9#. C x
12 , jV j = 14.
2
1223
( = 5, = 0, #Y (9) (16), 10 jV j: jV j2 ; 53 j V j 6 ; 3 3 4
(46)
' (46), ! jV j 6 17. D jV j = 17 (4), 8 (46) 5jV4j 6 17, , " " jV4 j, : jV4j = 1, jV5 j = 16 jV4j = 3, jV5j = 14. C jE j = 42, jF j = 25. 4 (9) ! P~ 6 72,
(15) (29) , F3 = 16 F4 = 9 jE33j 6 6: (47) % # , 8 36 "#
3- 4- 6 "# 3- . % 16 V5 # E33. 8 jE33j > 8, (47). C jE j = 41, jF j = 24. 4 (9) ! P~ 6 63,
(15) (29) , F3 = 14 F4 = 10 jE33j 6 1: (48) % # , 8 40 "#
3- 4- # 3- . % 14 V5 # E33. 8 jE33j > 7, (48). ( = 6, = 0, #Y (9) (15), (49) jV j2 ; 19jV j 6 ;2jV4j ; 2jV5j: % # , (49) jV j 6 19 (4). D . 0 2 ! 2 0 6 6 1, ( 6 6.
6 , " . 10 11, #
8 .
6. '()
4 , 2 3 6 ( 6 7, = 2 3 6 ( 6 6, 0 6 6 1. G
# ! 2, # ! ,5,6]. # { 8 ! # #6 .
1224
. .
8". 10. 9 % , $ 2 @ $!- C@ -" = 1: ) + = 3, jV j = 10 ( 9 " $)< ) + = 4, jV j = 12< ) + = 5, jV j = 14< ) + = 6, jV j = 16.
*
1] Bermond J. C., Delorme C., Quisquater J. J. Strategies for interconnection networks: Some methods from the Graph Theory // J. Parallel and Distrib. Comp. | 1986. | Vol. 3. | P. 433{449. 2] Comellas F., Gomez J. New large graphs with given degree and diameter // Graph Theory, Combinatorics and Algorithms. Vol. 1 / Y. Alavi and A. Schwenk, Eds. | New York: John Wiley & Sons, Inc., 1995. | P. 221{233. 3] Chung F. R. K. Diameters of graphs: old problems and new results // Congressus Numerantium. | 1987. | Vol. 60. | P. 295{317. 4] Fellows M., Hell P., Seyarth K. Constructions of large planar networks with given degree and diameter. | To appear in Networks, 1998. 5] Hell P., Seyarth K. Largest planar graphs of diameter two and xed maximum degree // Discrete Math. | 1993. | Vol. 111. | P. 313{332. 6] Seyarth K. Maximal planar graphs of diameter two // J. Graph Theory. | 1989. | Vol. 13. | P. 621{648. 7] Fellows M., Hell P., Seyarth K. Large planar graphs with given diameter and maximum degree // Discrete Appl. Math. | 1995. | Vol. 61. | P. 133{153. 8] Pratt R. W. The complete catalog of 3-regular, diameter-3 planar graphs. | http: //www.unc.edu/~rpratt/graphtheory.html.
2
1225
8". 11. 9 % , $ 2 = 0: ) + = 3, jV j = 10 ( 9 " $)< ) + = 4, jV j = 13< ) + = 5, jV j = 16< ) + = 6, jV j = 19.
9] Friedman E., Pratt R. W. New bounds for largest planar graphs with xed maximum degree and diameter. | http://www.unc.edu/~rpratt/graphtheory.html. 10] . . ! "#!$%&' (!)#( *$!!( + +(!,! (. = 3, D = 3) // 123!#4!$%!5 *( $!3!5 #!4#!4 !. | 2001. | . 7, 6 1. | . 159{171. ( ) ) 2000 .
. .
. . .
519.725+512.55
: , , , ! ", # $, % & &, $ & .
! & ": () *$# +$ ", ) . !& - & % & $# .. ! * *$# +$ % & $ & & $ . %%. . ! & % & $#&* # # , ) &, ), +,- # $.
Abstract I. L. Kheifets, Extension theorem for linear codes over nite quasi-Frobenius modules, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1227{1236.
F. J. MacWilliams proved an Extension theorem: Hamming isometries between linear codes over 6nite 6elds extend to monomial transformation. This result has been generalized by J. A. Wood who proved it for Frobenius rings. In this paper the Extension theorem for linear codes over a 6nite quasi-Frobenius module with commutative coe7cient ring is proved. The main technique involves the description of quasi-Frobenius module in terms of character theory.
1. R | e, R M |
(. . ) R- . , 2 M a 2 R , a = a. n M " # K $ R M n % n R M . $& $ '# " $ (2, 7, 9, 17]. . " '% % % # $" , 2001, 7, 8 4, . 1227{1236. c 2001 !", #$ %& '
1228
. .
' " ' % , 0 ' 10 . 2 " %
' $ 3 # # % (4] (13]. 6 " , # $ $ ' % $ 7 $ $ % 8 " % $ #% " $ $. ' , # #& "
% $ $, , , $ "8 # , & . . " % "8 # % # (8], # $. (18] " 8 " $ . , $ % 8 # % " (. (12]). 9$ " # (19]. $" "& $, # $ "8 # % . 1.1. R Q | QF- R e
K < R Qn f : K ! R Qn, . !
f " # " $ R Qn .
2. ; " a = (a1 : : : an) 2 Rn = (1 : : : n) 2 M n a = a11 + : : : + an n 2 M . 9$ # ' ' K 6 R M n $ R Rn Rn ?K = fa 2 Rn : aK = 0g " , K R. > ' $$ M n?L R M , L 6 R Rn: M n?L = f 2 M n : L = 0g: #& , $$ & ? $: M n?(Rn?K ) K Rn?(M n?L) L: (1) , ' M = R | , $ #&$ . 7 R M "$ QF- , n = 1 $ (1) $$$ $ % K < R M L / R. @ "$, # $ (1) # % n 2 N, #% , # R M # "8 # (8, 7.1]. 6 $$$ & $
1229
R $ # ' ' M . A ' , #& 8 ' QF- '
$" 7 $ (8, x 7]. 9$ # '
' ' R & , " 8", QF- R Q (10]. . " QF- R Q , . . R?Q = fa 2 R : aQ = 0g = 0: A R "$ , R R QF- . ;, ' % , ,
"8 # . . End(R M ) | 8" $ R M ( # $ $ " 8" ). 9$ ' r 2 R
" r^ # " 8" $ R M , & 2 M r^() = r. C" r^ " R^ = R^ (M ) # " % ' End(R M ). @ , R M | ( ), " 8" R^ (M ) = R. 7 R M " - , End(R M ) = R^ (M ). @ , # , R R, D- . 2.1. % $ R Q &- . 9 " . (8, 3.5].
3. n 8 H "$ " K Hn. . K "$ .
a = (a1 : : : an) 2 Hn b = (b1 : : : bn) 2 Hn $$ d(a b) % i 2 f1 : : : ng, ai 6= bi . C , $ #" 8 $ d : Hn Hn ! R Hn, . . 3.1. ' $ a b c 2 Hn ( d(a b) = 0 () a = b d(a b) = d(b a) d(a c) 6 d(a b) + d(b c): n C" # : H ! Hn ( 0 ') , 8a b 2 Hn d((a) (b)) = d(a b): 3.2 (. . , 1956). ) : Hn ! Hn
,
" 1 : : : n 2 2 S (H), 2 Sn , (2) 8a = (a1 : : : an) 2 Hn (a) = (1 (a(1) ) : : : n(a(n) )):
1230
. .
9 " . (5, 1.3.1] (1]. . ' S(Hn ) ' S (Hn ) % Hn , $& " % " , " ! Hn. A K Hn K Hn " " ? K K , & " $ 2 S(Hn ), $ K = (K). D H = R M | R e, M n | R- . $ $ % % # " " 2 S(Hn ), $$$ 8" ( " , " 8") $ M n . . Aut(M ) | ' 8" $ M . 6 ' S(Hn ) \ Aut(M n ) " % # (2), $
% $ " s 2 S (M n ) 8" s 2 Aut(M ). " ' LS(M n ) = S(Hn ) \ Aut(M n ) ! M n . J K < M n K < M n " ( ) " , 9 2 LS(M n ) (K) = K : (5] , , ' M = R, . . K < Rn R, 8" 2 End(R R) " $$ #" (e) e R, (r) = r(e), r 2 R. K 8" , $ ' (e) = u 2 R, # " " ub. @ , 2 Aut(R R) ' ', ' u 2 R . 2 , ' LS(Rn) % " $ Rn % # : Rn ! Rn , % & 2 Sn u1 : : : un 2 R 8a 2 Rn (a) = (u1 a(1) : : : una(n) ): (3) " 3.3 (5, 1.3.4]). * n- K K R +
,
K = (K), | $ #
(3). 2$ (18], #" ' , ' 8" : R M n ! R M n (1 : : : n) = (u1(1) : : : un(n) ) ' | f1 2 : : : ng u1 : : : un # R, " $ R M n . 7 % % #" $ $$$ ' ' ' 8" . 9 , ( 1) (1 : : : n) = (u( 1) ; (1) ; (1) : : : (u( 1) ; (n) ; (n) ): 0
0
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;
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1)
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1231
# 2 M n "$ k k = d( 0) %
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% ? 9$ 8 # % $ $ . . rad R | 9 # R ($ ' , & ' ), ' 8 - R = R= rad R . @# " " soc M R- $ M , . . % ' % . K $ M
R: soc M = f 2 M : rad R = 0g ' R- . A R "$ , " 8" % % R- : R = soc(R R), R = soc(RR ). L" , 8 # $$$ "8 # , $ ' $$ (3, '. 13]. 3.5 (18, ! 7.1]). K | $ R f : K ! Rn, . !
f " # " $ Rn. C , ' R M $$$ QF- .
8" 2 End(R R) " $$ r 2 R, $ QF- $ End(R M ) = R^(M ) () = r ( 2.1). @ , 2 Aut(R M ) ' ', ' r 2 R . 2 , ' LS(M n ) % " $ R M n % ' % #" . " 3.6. * n- K < M n K < M n +
,
K = (K), | " $ R M . L # , " ? ,
& " 8", % $& 0 '. N " # . C & , % " QF- % % . 0
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1232
. .
4. " . (M +) | $ # ' M = Hom(M Q=Z) | # ' % ' 8" ' (M +) ' (Q=Z +) % % 1. 3 M "$ ! M ! ! M . L" " (6] ' & $. 4.1. , "" M = M . ' $ + 2 M n 0 ! 2 M , !( ) 6= 0. . R M R- . 6 ' " r! % ! 2 M r 2 R # r! : M ! Q=Z, 8 2 M r!() = !(r): . r! 2 M , " #" & M R- . #& 4.1 "$ , R M R M " 8 . @ 4.2. - R- ' : R M ! M , $ + 2 M '(): M ! Q=Z 8! 2 M '()(!) = !(): 6 & $ QF- $ $ " '
' ' R 8 $ & #" . 4.3 (8, 2.2, ! 3.1]). R | . !
R R "" (R +) QF- . QF- R Q R R . ? QF- , & % % ". C" R M , & % ! : M ! Q=Z, $ R M . 6 % # " $% ! $ $ R M . 9 $ '$ #R$ $$ & . 4.4. ! : M ! Q=Z R M
,
8 2 M ( ( 6= ) =) (9r 2 R (!(r) 6= !(r )) ): 9 4.5 (8, ! 5.2]). ! R M
,
QF- .
1233
9$ QF- $ R M = R R " & % $ & #" . S$ " 8" R- : R R ! R M = R R , r 2 R % (r): R ! Q=Z, & ! 2 R (r)(!) = !(r). 9$ # ' # ' u 2 R % (u) "$ " & $ R M . , (e) | " & % $ R M . C$ % $ ' % . . (M +) | $ # ' M^ = Hom(M C ) | # ' % ' 8" ' (M +) ' (C ) $ C % . 3 M^ "$ ! M ! ! M . 9$ # ' ' % !(x) 8 $ (x) = e2i!(x) (4) % , $ ! ! " 8" ' % % M ' % % M^ . . % % % (15] (16]. 4.6. . # M^ " "" M " M . 4.7. ' $ 2 M^ ( X jM j = 1 (x) = 0 6= 1: x M 4.8. ' $ x 2 M ( X jM j x = 0 (x) = 0 x 6= 0: ^ M
2
2
5. $ % 2$ (11], $ "
. . R | , M | R. 2 , x y 2 M
(# x y), Rx = Ry, . . x y, x = ay y = bx $ %- a b 2 R. ' , x y 2 M
(# x y), & # u 2 R, y = ux. L $$$ ? . L" x y x y. #& # . 9 , F | M = R = F (x y z ]=(x ; xyz ). 6 ' xT xTyT, xT 6 xTyT (11, 2.3]. @ $ % ( $ %)
1234
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R ( " . 9 " ., , (18, 6.2].
6. 9 " # | 1.1.
" 3 # 8 # , & " 8", % $& 0 '. 6 ' & . 1.1. R Q | QF- R e
K < R Qn f : K ! R Qn, . !
f " # " $ R Qn . #$! %&!. @# " " : K ! R Qn = f . 6 ' $$$ $ " $ ' R- $ K R Qn . @ $$ # % 8 : = (1 2 : : : n) = (1 2 : : : n), ' i i : K ! R Q | ' 8" , 8 2 K () = (1() : : : n()) () = (1 () : : : n()): . , $ % $& f1 2 : : : ng & # u1 : : : un 2 R , 8i 2 f1 2 : : : ng i = ui(i) : (5) 6 f % $ 0 ', k(x)k = kf ((x))k = k(x)k $ # ' x 2 K. @, "$ 4.8,
n X X
i=1 2Q^
(i (x)) =
n X X
j =1 2Q^
(j (x)):
' $ " 8" (4) ' % % Q ' % % Q^ " % :
n X X i=1 !2R Q
e2i!(i (x)) =
n X X j =1 2R Q
1235
e2i( j (x)) :
L "$ " 8" : R R ! R Q = R R ( 4.2 4.3),
n X X
i=1 r2R
e2i (r)(i (x)) =
n X X
j =1 s2R
e2i (s)( j (x)) :
(6)
7 ' 8" Hom(K R Q) $$$ R- $ (. " 5). ' 8" f1 : : : n 1 : : : n g , $$ #& , , 1 . J $ " % % e2i (r)( ) 2 Q^ (6) ( 4.6) & ' s 2 R j = (1) , e2i (e)( (x)) = e2i (s)( (x)) . . % "$ (e)(1 (x)) = (s)((1) (x)) = = (e)(s(1) (x)) , $ " , 8x 2 K (e)(1 (x) ; s(1) (x)) = 0: . , " " 4, (e) | " & % , #" ' 8" 1 (x) ; s(1) (x) R Q, . . 1 (x) = s(1) (x) (1) 6 1 . C 1 # , 1 (1) ( $ 5.1 5.2). 6 #" , & # u1 2 R , 1 = u1(1) 8r 2 R (r)(1 ) = (r)(u1 (1) ) = (ru1)((1) ): @, % $ $ r 2 R, X 2i (r)( (x)) X 2i (s)( (x)) : (7) e = e
1
(1)
1
r2R
$ (7) " (6), n X X
i=2 r2R
e2i (r)(i (x)) =
(1)
s2R
n X X j =1 s2R j 6=(1)
e2i (s)( j (x)) :
6 n " " (6) ? $ (5). ' &! 6.1. R Q | QF- R e. & f : K ! K K K < R Q, , " #
R Qn ! R Qn, , K K + . 9 , f $ ' #" $ , K f , (K) = K , K K . 0
0
0
0
0
1236
. .
> " 8 >. . 7% 8 >. >. C " " , # " & ' .
'
1] . ., . ., . . ! " #!, $% & // ( ! ! !. | 1997. | -. 9, .. 3. | 0. 3{19. 2] 3 -., - 4., 56 7., 5! 8. - 6 . | .: , 1972. 3] 3 :. 6 ;<. | .: , 1981. 4] 4" . ., 3 ; . 0., . -. ?&. 6. 6 ". ;< 6 // :6 ! ; 6 ! !. | 1997. | -. 3, @ 1. | 0. 195{254. 5] 3 . ?., 4" . . ?&. 6. &. !.. | 1997. 6] ? 5. 3 ;< 6 . | .: , 1971. 7] ; :. (., 0 C 4. ( . . - 6, $% . | .: 0 ;, 1979. 8] 4" . . 3". # . 6 , 6
&. ! // :6 ! ; 6 ! !. | 1995. | -. 1, @ 1. | 0. 229{254. 9] ! D., DC 6 7. 36., $% . | .: , 1976. 10] :& 3. F: 3 ;<, 6 , !F. -. 2. | .: , 1979. 11] Anderson D. D. and Valdes-Leon. Factorization in commutatives rings with zero-divisors // Rocky Mountain J. Math. | 1996. | Vol. 26. | P. 439{480. 12] Greferath M., Schmidt S. E. Finite-rings combinatorics and MacWilliams' equivalence theorem // J. Combin. Theory. Ser. A. | 2000. | Vol. 92, no. 1. | P. 17{18. 13] Kuzmin A. S., Kurakin V. L., Markov V. T., Mikhalev A. V., Nechaev A. A. Linear codes and polylinear recurrences over Ynite rings and modules (Survey) // Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Proc. of the 13-th. Int. Symp. AAECC-13 LNCS. | Springer, 1999. 14] Lam T. Y. A First Course in Noncommutative Rings. | Berlin: Springer-Verlag, 1991. | Graduate Texts in Mathematics, Number 131. 15] Serre J.-P. A course in arithmetic. | New York: Springer-Verlag, 1973. | GTM 7. 16] Serre J.-P. Linear representation of Ynite groups. | New York: Springer-Verlag, 1977. | GTM 42. 17] Tsfasman M. A., Vladut S. G. Algebraic-geometric codes // Mathem. and its Appl. (Soviet series). Vol. 58. | Kluwer Acad. Publ., 1991. 18] Wood J. A. Duality for Modules over Finite Rings and Applications to Coding Theory. | Preprint. | Purdue Univ. Calumet, Hammond. Oct. 1996. 19] Wood J. A. Characters and the equivalence of codes // J. Combin. Theory. Ser. A. | 1996. | Vol. 73. | P. 348{352. ( ) * 2001 .
. .
. . .
511.36
:
, , -
!.
" # $ % $ % ! & Q p-$ # , #
& # $ | (
) !$ (#$, & ($ # Rp.
Abstract
T. G. Hessami Pilehrood, On arithmetical properties of the values of polylogarithmic functions, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1237{1258.
This paper gives a proof of the linear independence over Q of p-vectors, whose coordinates are the values of polylogarithmic functions at rational points, and p-unit vectors under certain condition.
x
1. .
Li (z) =
1 z X i =1
i > 1:
! " # $ # Li (z), i > 1. ' ( ! ( " Q p- , " | # $ ( # , " # Rp. 1. p 2 N, p ;p Q+p = min q 2 N : e2p;1 q1=p ; 1 + q1=p ;2p < 1 1 1 i ; 1 i ; 1 i ; 1 ,ai = Ci;1 Li q Ci Li+1 q : : : Ci+p;2 Li+p;1 1q 2 Rp i = 1 : : : p: , 2001, 7, 2 4, . 1237{1258. c 2001 !, "# $% &
1238
. .
q > Q+p Q- ,a1 : : : ,ap Qp. 2. p 2 N, p ; Q;p = min q 2 N : e2p;1 < Aqp;1 + q1=p + 2 q1=2pBqp;1 p 1=2 1=2 2 =p 1 =p 1 =p Aqp;1 = 1 + q ; 2q cos p Bqp;1 = q ; cos p + Aqp;1 0 a,i = Cii;;11 Li ; 1q Cii;1 Li+1 ; q1 : : : Cii+;p1;2 Li+p;1 ; 1q 2 Rp i = 1 : : : p: q > Q;p Q- ,a1 : : : ,ap Qp.
. 1$ (, # Q;p < Q+p < ep . $# "# " m Rn a,i = Li ab Li+1 ab : : : Li+n;1 ab i = 1 : : : m
( 1. 2. 3 41] 42]. 6 # # (m = n = p) ! # " ! $ 41,2]. ( # #6 $ " # Q;p . 7"# # $ ( $ . 8 "( 9. :. ; 6 43], 2 p :. < 44], 1. 2. 3 45]. 44, 5] " , # q > e # L1 q1 : : :Lp q1
" Q. " ! 1 2 $ " ( # 1q " ep , " ( 6( ( $ # " " # Rp. ' ( " " # # p. ? 1 2 p = 2 " Q+2 = min fq 2 N : e3 < ((pq ; 1)1=2 + p4 q)4 g = 3 p p ;p ;p p Q;2 = min q 2 N : e3 < 1 + q + q + 2q1=4 q + 1 + q 1=2 2 = 1 # (
1239
1. q > 3
1 1 1 a,1 = L1 q L2 q ,a2 = L2 q 2L3 q1 e,1 = (1 0) e,2 = (0 1) Q. 2. q ,a1 = L1 ; q1 L2 ; q1 ,a2 = L2 ; 1q 2L3 ; q1 e,1 = (1 0) e,2 = (0 1) Q. B!" " 2 q = 1 | ( 46]. # , " 1 $ # 3. q > 3 L3 ; 1q , L2; 1q . ; :.< 47] " ( ( # L2 k1 " k 2 (;1 ;5] 47 +1). # p = 3 1 2 " p ;p Q+3 = min q 2 N : e5 < q1=3 ; 1 + q1=3 6 = 7 Q;3 = 4 # "E 4. q > 7 Q- 1 1 1 1 1 ,a1 = L1 q L2 q L3 q a,2 = L2 q 2L3 q 3L4 1q 1 1 a,3 = L3 q 3L4 q 6L5 1q
Q3.
5. q > 4 Q- -
Q3.
,a1 = L1 ; q1 L2 ; 1q L3 ; q1 ,a2 = L2 ; 1q 2L3 ; 1q 3L4 ; q1 1 1 ,a3 = L3 ; q 3L4 ; q 6L5 ; q1
1240
. .
? " 4 5 $ # ; 6. q > 7 L5 q1 , L4 ; q1 , ;1 L3 q , . ; 1 7. q > 4 L ;q , 5 ; 1 ; 1 L4 ; q , L3 ; q , .
x
2. !"
F !"$ ( $ n " # k r n = pk + r 1 6 r 6 p: G ! 8 (s ; 1)p (s ; 2)p : : :(s ; k)p (s ; k ; 1)r;1 > > < sp (s + 1)p : : :(s + k ; 1)p (s + k)r+1 1 6 r < p Rn(s) = > p p p p;1 > : (s ; 1) (s ; 2) : : :(s ; k) (s ; k ; 1) r = p0 p p p s (s + 1) : : :(s + k) (s + k + 1) 1 ;1 X En (z) = ((;1) R(;1) ( )z ; 2 N 1 6 6 p: ; 1)! =1 n
1. Lq (z) =
(1)
1z P q. =1
) n , 1 6 6 p,
nj (z) 'n (z) 2 Q4z], j = 1 : : : p, ! En (z) =
p X Cj+;1;2 nj (z) Lj +;1 (z ;1 ) ; 'n (z)0 j =1
(2)
) n ! deg nj (z) 6 n + p1 ; j 0 deg 'n (z) 6 np ; 10 ) ! n 2 N ordz=1 En(z) > n + p ;p 1 ; + 1: . $ Rn(s) 6 ", " Rn(s) =
" np ] p X X
nj : j j =1 =0 (s + )
(3)
1241
? " Rn(s) " , # j > n ; p 4 np ] + 1 nj" np ] = 0. I$" (1) (3) En(z) =
" np ] p X 1X X j(j + 1) : : :(j + ; 2) nj z ; ( ; 1)! =1 j =1 =0 ( + )j +;1
1
=
" np ] p 1 ; ; X X X ; 1 = Cj +;2 nj z ( +z )j +;1 = =0 =1 j =1 n "p] p ;l X X X = Cj+;1;2 nj z Lj +;1 (z ;1 ) ; lj +z ;1 = =0 j =1 l=1 p X = Cj+;1;2 nj (z) Lj +;1 (z ;1 ) ; 'n (z) j =1 n] p ;1 "P p P nj P ;l
nj z , 'n (z) = Cj +;2 j+;1 z =0 =0 l=1 l j =1 n] "P p
$" nj (z) = | $# ( . ? " # nj # " $# nj (z), 'n(z). ? (1) #E " Rn (s) " !" ) . I , 1 " . " ( n1(z) :::
np(z) 'n1(z) ::: 'np (z) n+11(z) : : : n+1p(z) 'n+11(z) : : : 'n+1p (z) Jn (z) = : .. . .. .. . ... .. .. . . . n+2p;11(z) : : : n+2p;1p(z) 'n+2p;11(z) : : : 'n+2p;1p(z) 2. n Jn = A (z ; 1) A | , 0. . ? ( (2) ", # Jn(z) = n1(z) : : : np (z) En1(z) ::: Enp (z)
n+11(z) : : : n+1p (z) En+11(z) : : : En+1p (z) = (;1)p : .. . .. .. . ... .. .. . . . n+2p;11(z) : : : n+2p;1p(z) En+2p;11(z) : : : En+2p;1p(z) " " p ! " p " " . I$" ( " # z = 1
1242
. .
" (
En1(z) En+11(z) A1 (z) = .. . En+p;11(z)
En2(z) En+12(z) .. . En+p;12(z)
ordz=1 A1 (z) =
p;1 X j =0
: : : Enp(z) : : : En+1p(z) .. ... . : : : En+p;1p (z)
ordz=1 En+jj (z)
$" # j # 1 6 j 6 p. ' #E 1 # pX ;1 r + j ; 1 ; j pX ;1 n + j + p ; 1 ; j ordz=1 A1 (z) > + p = p(k + 2) + : p p j =0 j =0 I r + j ; 1 ; j < 0 " , # j > r +j ; 1 > r ; 1, ordz=1 A1 (z) > p(k + 2) ; (p ; r + 1) = pk + p + r ; 1 = n + p ; 1: (4) $# " A1 (z): n+p1(z)
n+p2(z) : : : n+pp (z) n+p+11(z) n+p+12(z) : : : n+p+1p (z) : B1 (z) = .. .. .. ... . . .
n+2p;11(z) n+2p;12(z) : : : n+2p;1p(z) ? ( 1, " pX ;1 r + i + 1 ; ji pX ;1 deg B1 (z) = deg n+p+iji 6 p(k + 1) + p i=0 i=0 $" # ji # , 1 6 ji 6 p. I r +i+1 ; ji < 2p " 0 6 i 6 p ; 1 r+i+1 ; ji > p " , # ji 6 r+i+1 ; p 6 r, deg B1 (z) 6 p(k + 1) + r = n + p: (5) I ( #E (4) (5) $, # " " p " ( " A1, $# " , ( , " B1 (z), Jn(z) = A z + B: ? (3) " Rn(s) " , # " $ ( $ n n
n1(1) =
"p ] X
=0
n
n1 =
"p] X
=0
ress=; Rn(s) = ; ress=1 Rn (s) = 0:
' " ( , " $ n 2 N Jn(1) = 0 Jn = A (z ; 1).
(6)
1243
; " (, # (4) (5) "$ ( 0 6 j 6 p ; r j = r + j r + j ; p j > p ; r ( 0 6 i < p ; r ji = r + i + 1 r + i + 1 ; p i > p ; r: 8" " p;1 pY ;1 1 (j ;1) n + j + p ; 1 ; j + 1 Y
A = ( ; R n+p+iji li n+j p j =0 j 1)! i=0
$" li = n+p+ip+1;ji . ? (3) " Rn (s) " , # A 6= 0. I , 2 " . 3. D = ;8K(1 2 : : : k), k = n;p 1 . 1 6 j 6 p 1 6 6 p p!Dp+;1 nj (z) 2 Z4z] p!Dp+;1 'n (z) 2 Z4z]: . G !, # (p n; j)!Dp;j njm 2 Z" ( n, j, m, 1 6 j 6 p, 0 6 m 6 p . I$" !" " " $# nj (z), 'n (z), j = 1 : : : p. G ( 0 6 m 6 k ; 1, 1 6 r 6 p ; 1. I$" p;j d (R (s) (s + m)p ) (;m):
njm = (p ;1 j)! ds p;j n G p : :(s ; k)p (s ; k ; 1)r;1 Rn (s) (s + m)p = sp : : :(s + m(s;;1)1)p (s: + m + 1)p : : :(s + k ; 1)p (s + k)r+1 $" # " pk + r ; 1 ! , p(k ; 1) + r + 1 ! . G "( ! "( " Rn(s) (s + m)p = (s ; k)p;r;1 (s ; k ; 1)r;1 gkr+1 (s) gkp;;r1;1 (s) "( 1) : : :(s ; k) gk (s) = s : : : (s + m(s;;1)(s + m + 1) : : :(s + k) = ; 1) +: : :+ Am + Am+2 +: : :+ Ak+1 (k ; m) = 1+ A1s m + A2 s(m +1 s+m;1 s+m+1 s+k Al 2 Z 1 6 l 6 k + 1 l 6= m + 1: G " p Y Rn(s) (s + m)p = P0(s) gkd (s)
d=1
1244
. .
$" P0(s) = (s ; k)p;r;1 (s ; k ; 1)r;1
(
1 6 d 6 r + 1 kd = k k ; 1 r + 1 < d 6 p
p X dp;j (R (s) (s + m)p ) = (p ; j)! P (0) (s) Y gk(d d ) (s): 0 dsp;j n
! : : :
! p 0 +:::+p =p;j 0 d=1 I
; 1) gk(d ) (s) = 1(d ) + (;1)d d ! As1d+1m + A(s2 + (m 1)d +1 + : : :+ Am+2 1 + : : : + Ak+1 (k ; m) Am 1 + (s + m + ; 1)d +1 (s + m + 1)d +1 (s + k)d +1 Dd gk(d ) (;m) = Dd 1(d ) ; d ! mA1d + (m ;A21)d + : : : + 1Amd + Ak+1 2 Z: + A1m+2 + : : : + d (k ; m)d
' " ( , (p ; j)!Dp;j njm 2 Z " 0 6 m 6 k ; 1 1 6 r 6 p ; 1: ' # m = k, 1 6 r 6 p ; 1 m 6 k, r = p # $# , " $ " ( p ; k)p (s ; k ; 1)r;1 = Rn(s) (s + k)r+1 = (s ; 1) s: p: :(s : : :(s + k ; 1)p : :(s ; k) p = (s ; k ; 1)r;1 s(s(s+;1)1): :::(s + k ; 1) p ; k)p (s ; k ; 1)p;1 Rn(s) (s + m)p = sp : : :(s +(sm;;1)1)p:(s: :(s + m + 1)p : : :(s + k)p (s + k + 1) = = (s ; k ; 1)p;2 gkp;1 (s) gk+1(s) . F m = k + 1 njm = 0 j > 2,
n1m = (Rn(s) (s + k + 1))js=;k;1 " D n1m 2 Z, !" " .
x
3. $
1245
F ( n, k, , k = n;p 1 , 1 6 6 p, $ u "E n++1 Z Jn(u) = (;1)2i Rn(;s) sins eus ds (7)
L
$" L | ( , " ; i1 + i1, 2 (;k 0). I s = + iy y ! 1 jRn(s)j = O(jyj;2 ), (sin s); = O(e; jyj ), $ (7) " j Imuj 6 . 4. n, , 1 6 6 p, u 2 C , j Imuj 6 , ! !
; (;1)p; pX Jn (u) = (2i) (;1)m Cpm; Jnp (u + i(p ; ; 2m)): p;+1 m=0
. ? (7)
p n++1 Z ( ; 1) Jn(u) = 2ip;+1 Rn(;s) sins (sin s)p; eus ds:
L
2 (sin s)p;
=
i s ;i s p; e ;e
2i
pX ; = (2i)1p; (;1)m Cpm; ei s(p;;2m) m=0
Jn (u) = Z ; p (;1)n++1 pX m Cm s(u+i (p;;2m)) ds = ( ; 1) = (2i) p; Rn(;s) sin s e p;+1 m=0 =
L pX ; p ; (;1) m m (2i)p;+1 m=0(;1) Cp; Jnp (u + i(p ; ; 2m))
" . 5. n, , 1 6 6 p, w 2 R X 1) En (ew ) = d0v Jnp (w + iv) 2)
Z Z
v2 jvj6p X En (;ew ) = d00v Jnp(w + iv) v2 jvj
d0v , d00v | , # n.
1246
. .
. G ! " !"$ ( $ , 1 6 6 p, u = w + i w 2 R 2 Z " #(, # # ( !) " "
" : 1) (mod 2) " !"$ = 1 2 : : : p0 2) ; 1 (mod 2) " !"$ = 1 2 : : : p. ? " " ! " cl , n: En((;1)+ ew ) = ? (7) #
n++1 Z
Jn(u ) = (;1)2i
L
X l=1
cl Jnl (ul ):
(8)
Rn(;s) sins esu ds:
'" $ s ;s ( # , # 1 X Jn (u ) = (;1)n ress=m Rn(s) sins e;su m=k+1+"r (
$" "r = 1 r > 0 r 6 : # s = m > 0 " ! ;1 ;1 Rn(s) = Rn(m) + R0n(m)(s ; m) + : : : + R(n ; (m) 1)! (s ; m) + O ((s ; m) ) e;su = e;u(s;m) e;mu =
u );1 (s ; m);1 + O((s ; m) ) = e;mu 1 ; u (s ; m) + : : : + (; ( ; 1)!
1 (;1)% 2% () = (;1)m = (;1)m X D2% (s ; m)2%; = sin s sin (s ; m) (2%)! %=0 ;1 ] "X 2 (;1)% 2% D2(%) m = (;1) ;2% %=0 (2%)!(s ; m)
+ O(1):
1247
G " = 1 2 # Jn1(u1) = (;1)n
1 X
m=1
(;1)m e;mu1 Rn (m) = (;1)n En1(;eu1 ) =
= (;1)n En1((;1)1+1 ew ) Jn2(u2) = (;1)n
1 X
m=1
e;mu2 (R0n (m) ; u2 Rn(m)) =
= (;1)n+1 fEn2(eu2 ) + u2En1(eu2 )g (9) " En2((;1)2+2 ew ) = c1Jn1 (u1) + c2 Jn2(u2). G ( 3 6 6 p. I$" (;1)n Jn(u ) = ! (;1) 1 X R (m) n ( ; 2) m ; mu = (;1) e ( ; 1)! + d;2 Rn (m) + : : : + d0Rn(m) = m=1 X ;1 ; 1 u = (;1) En((;1) e ) + Dj Enj ((;1) eu ) = j =1 X ;1 ; 1 + w = (;1) En((;1) e ) + Dj Enj ((;1)+ ew ): j =1
(10)
'$ # " $ 2 6 6 p, 2 N (;1)+ = (;1);1+;1 : I$" (10) " , # En((;1)+ ew ) = D Jn (u ) ;
X ;1 j =1
Dj0 Enj ((;1)j +j ew ):
F , " ! " , # (8). ? ( !" 4, (8) " En ((;1)+ ew ) =
X l=1
cl Jnl (ul ) =
X p;l X
l=1 m=0
dlm Jnp(ul + i(p ; l ; 2m)) (11)
$" dlm | , n. 8 ( ( # l , 1 6 l 6 p. G ! # l = l " $ l 2 N, 1 6 l 6 p, $" (11) "E X p;l X X w En (e ) = dlm Jnp (w + i(p ; 2m)) = d0v Jnp (w + iv): l=1 m=0 v2 jvj6p
Z
1248
. .
# ! l = l ; 1 " !"$ l = 1 : : : p. I$" (11) En (;ew ) =
X p;l X
l=1 m=0
dlm Jnp(w + i(p ; 1 ; 2m)) =
X
Z
d00v Jnp (w + iv):
v2 jvj6p;1
I , " . ? " " , # " $ # #(
" En(z), 1 6 6 p, ( ( $ Jnp(u). ? " Rn(s) " , # " # a1 : : : ap , b1 : : : bp , # A=
p X j =1
aj = p + r ; 1 B =
p X
j =1 p Y
bj = r + 1
(12)
p + aj + s) : Rn(;s) = (;1)n;1 ;;p (s(;+s)1) ;(k j =1 ;(k + bj ; s)
I$" (7) 6 ;(;s) ;(1 + s) = ; sin s " $ Jnp (u) " " : Z p Y + k + aj ) eus ds: 1 2 p Jnp (u) = 2i ; (;s) ;(s (13) ;(k + bj ; s) j =1 L
6. n ! 1, k = 4 n;p 1 ] ! x0 +i1 1 i(2)p;1 Z pkg(t)
f(t) dt O k + 1
(14)
g(t) = 2t ln t + (1 ; t) ln(1 ; t) ; (1 + t) ln(1 + t) ; tup
(15)
Jnp (u) =
k
x0 ;i1
e
x0 | (0 1),
p=2+r;1 f(t) = tp(1(1;+t)t)r+1;p=2 :
O a1 : : : ap b1 : : : bp. . G " $ ( ! (13): 1 Jnp(u) = 2i
Z 2 ; (;s) ;(k + s) p
L
;(k ; s)
ajQ;1 (k + s + m) p Y eus bmj ;=01 ds: j =1 Q (k ; s + m) m=0
" # ( $ lnz j arg z j < .
(16)
1249
'" $ s = ;kt, t = x0 + iy, x0 2 (00 1), y 2 (;10 +1). I L # Re(;s), Re(k + s), Re(k ; s) ! ( , ! k, " 2 ; ( ;s) ;(k + s) p G(s) = (17) ;(k ; s) $ ( 1 ln;(z) = z ; 2 lnz ; z + 12 ln 2 + r(z) jr(z)j 6 K j Re z j;1 $" K | , ln G(s) = p ln2 ; p lnk + pk(2t lnt + (1 ; t) ln(1 ; t) ; (1 + t) ln(1 + t)) + + p2 ln(1 + t) ; 2p ln(1 ; t) ; p lnt + O k1 (18) $" O() . F
ajQ;1 (k ; tk + m) p Y m =0 R(s) = bj ;1 j =1 Q (k + tk + m) m=0
ajQ;1 (1 ; t + mk ) p A Y (1 ; t) m =0 C C = k (1 + t)B 1 + O k1 R(s) = k bj ;1 j =1 Q (1 + t + m ) k m=0 p p P P $" C = A ; B = aj ; bj , O() ( # j =1 j =1 a1 : : : ap b1 : : : bp . 8" #E (12), (16), (17) (18) " !"-
.
7. p k n 2 N, k = 4 n;p 1 ], u 2 C , Re u > 0, j Imuj 6 p, u 6= pi
n ! 1 ! $ p;1=2 1 ; t0 pk (1 + o(1)) ; t0 )p=2+r;1=2 p 3 p;(1 Jnp (u) = ; (2) 2k t0 1=2(1 + t0)r+1=2;p=2 1 + t0 t0 = (1 + e;u=p );1=2 2 C \ ft 2 C j Re t > 0g. . F ( # " $ (. 6) " . G"E # (;10 0] 410 +1) " # ( ln t. I$" g(t), " E (15), " # # .
1250
. .
G ! t0 = (1 + e;u=p );1=2 2 C \ ft 2 C j Re t > 0g, $" " ;i'0 =2 ;u=p t0 = j1 +e e;u=p j1=2 $" '0 = arcsin j1Im+ ee;u=p j : (19) 7, # t0 2 R, t0 > 1 Imu = p0 # #, # Imu = p (Imu = ;p) # t0 ! ( ! ) $
# 410 +1). 1$ (, # 2 1 ; t20 = 1 +1eu=p 1 ;t0 t2 = eu=p : (20) 0 1. t0 % ! $ g(t), ! % (15), . . g0 (t0) = 0. . G !, # '0 + Im u < Im u < : (21) p p F ; 2 < Im up < 2 $ " $, # ; 2 6 '0 6 2 . G ( " " ( ; - > Im up > 2 . I$" u u '0 < ; Im p Im p 2 2 0 '0 > ; ; Im up Im up 2 ;0 ; 2 , ( ( ( u u sin '0 < sin Im p sin 'u 0 >; sin Im p u Im p 2 2 0 Im p 2 ;0 ; 2 . ' " ( " " " '0 . G"E " ( " ! . G ( Im up < . I$" arg(1 + t0) + arg(1 ; t0 ) 2 (;0 ), " g0 (t0) g0 (t0 ) = 2 lnt0 ; ln(1 ; t0 ) ; ln(1 + t0) ; up = 2 lnt0 ; ln(1 ; t20 ) ; up : ? (20) (19) "
; Re u=p 1 ; t20 = t20 e;u=p = j1e+ e;u=p j e;i('0 +Im u=p)
" #E (21) (19) #, # 0g (t0 ) = 2 lnt0 + Re u + ln j1 + e;u=p j + i '0 + Im u ; u = 0: p p p G ( ( Imu = p, $" t0 = (1 ; e; Re u=p );1=2 2 R, t0 > 1 t0 " ! ( ! ) $ 410 +1). I$" # 1 ; t0 < 0 " ! ! ( ) $ (;10 0], " g0 (t0 )
1251
; Re u=p g0 (t0) = ; ln(1 ; e; Re u=p ) ; ln e ; Re u=p i ; Re up i = 0: 1;e I , " ! " . " " ( 6 " #(, # Im u > 0. # Im t0 > 0. ' # Im u < 0 # " " # u, t u, t,. # x0 (. 6): x0 = (1 + e; Re u=p);1=2 : 7, # t0 = x0 Im u = 0. G ( l | ( , " x0 ; i1 x0 + i1. I$" $ 6 p;1 Z Jnp(u) = i(2)k epkg(t) f(t) dt 1 + O k1 : l
8 # # l1 , " #: ! # l, " x0 ; i1 " x0, # x0 t0. # t0 2 R t0 > 1 E x0t0 , " $ 410 +1). 9 Imu = 0, # l1 " l. G !, # Z Z epkg(t) f(t) dt = epkg(t) f(t) dt: (22) l
l1
F $ "E ! ( " R # " . G ( A B | # # " ! $ l l1 . I$" x0 AB " $ ( , Z
x0 A
epkg(t) f(t) dt +
Z
AB
epkg(t) f(t) dt =
8 $ " $ AB. F t = Rei' , 0 6 ' 6 arccos xR0 < 2
Z
x0 B
epkg(t) f(t) dt:
(23)
g(t) + tup = 2t ln t + (1 ; t) ln(1 ; t) ; (1 + t) ln(1 + t) = 1 1 i' i' i' i' i' = 2i'Re + (1 ; Re ) ln R ; e ; (1 + Re ) ln R + e : F , (6( ! " I ln(a + x), #
1252
. .
1 e;im' ;i' X ;i' ln R1 ; ei' = i(' ; ) ; e R ; = i(' ; ) ; e + O 12 m R R m=2 mR ;i' ln R1 + ei' = i' + e R + O R12
1 P
e;im' $" O() , mR m;2 < 2 R > 2. m=2 8" " Re g(t) + tup = ;R sin ' + O R1 $" O() . I$" Re g(t) = ;R sin ' ; p1 Re(Rei' u) + O R1 = = ;R sin ' ; p1 Rcos ' Re u + 1p R sin ' Imu + O R1 6 6 ; R cos p' Re u + O R1 : ; I AB jf(t)j = O R12 ,
Z Z Z =2 pkg(t) C 1 pkg ( t ) ;kR cos ' Re u d' ! 0 e jf(t)j dt 6 f(t) dt 6 e R e 0 AB AB
(R ! 1):
8" (23) " (22). G !, # $ l1 Re g(t) "$ ( " # , t0 . ? " # " Re g(t) l1 ;1 < Imt < 0. F t = x0 + iv, ;1 < v < 0, d Re g(x + iv) = ; Im d g(x + iv) = 0 dv dt 0 = ; Im 2 lnt ; ln(1 ; t) ; ln(1 + t) ; up = = ;2 arg t + arg(1 ; t) + arg(1 + t) + p1 Imu: (24) G ( t = x0(1 + i tg ') ' = arg t ; 2 < ' < 0 (25)
= arg(1 ; t) = arg(1 + t) $" ; 2 < < 2 : tg = ;1x;0 tgx ' tg = x10+tgx' 0
0
1253
2 tg tg = ;x10;tgx20 ' < 0, ,
; 2 < + < 2 . 2 $, # tg = ;2x20 tg ' > 0 tg( + ) = 1tg; tg+ tg 1 ; x20 + x20 tg2 ' " , # 0 < + < 2 : (26) 2
2
' #E (25) (26) # d 1 (27) dv Re g(x0 + iv) = ;2' + ( + ) + p Im u > 0 " ( , Re g(x0 + iv) " ( l1 ;1 < v < 0. I , ( "( " Re g(t) #, " # x0 t0 . G # l1 . G ( t = ei' (r ; ih), "( ' |
$ !" ( Ox # x0 t0, 0 6 ' 6 2 0 h | # " " x0 t0, r > h ctg '. I$" d Re g(t) = Re d g(t) = Re(g0 t0 ) = Re g0 (t) cos ' ; Img0 (t) sin ' (28) tr dr dr $" g0 (t) = 2 ln t ; ln(1 ; t) ; ln(1 + t) ; up . F !, # Re g0 (t) 0 # x0t0 ( " # x0 t0, Img0 (t) # 0 x0 t0 ( # t0 . 7, # Re g0 (x0) = Re g0 (t0 ) = Img0 (t0 ) = 0, #E " " " $, # g0 (t0 ) = 0. 2 Re g0 (t) = ln j1 j;tj t2 j ; Re up 6 0 jtj2 6 eRe u=p = > 1: (29) j1 ; t2j "E # w = e;2i't2 = (r;ih)2 . I$" jt2j = jwj, j1;t2j = je;2i';wj (29) 6 " jwj 6 : ; 2 i' je ; wj G " ( (2 ; 1)ww, ; 2 e;2i' w, ; 2 e2i' w + 2 > 0:
1254
. .
h(r) = (2 ; 1)ww, ; 2 e;2i' w, ; 2 e2i' w + 2 = = (2 ; 1)r4 + 2((2 ; 1)h2 ; 2 cos 2')r2 ; 42 hr sin 2' + + ((2 ; 1)h4 + 22 h2 cos 2' + 2 ): I$" F (. 48, . 43]) # ! (
h(r) = 0 " " . ; " ! , rx0 rt0 . 8" #E $# h(r) " , # Re g0 (t) > 0 4x00 t0 ) Re g0 (t) < 0 # x0t0 # t0 . F !, # Img0(t) # x0t0 . F t=ei'(r;ih), r > h ctg ', # H1 = r sin ' ; h cos ' H2 = r cos ' + h sin ' $" H1 = ' ; arctg h 0 arg t = arctg H r 2 H ; 1 2 arg(1 ; t) = arcctg H ; 0 arg(1 + t) = arctg 1 +H1H : 1
F Img0 (t) = 2 arg t ; arg(1 ; t) ; arg(1 + t) ; Im up " d Im g0(t) = 2h + sin ' ; h ; sin' + h 2
2
dr r2 + h2 H1 + (H2 ; 1)2 (1 + H2)2 + H12 : ? $, # sin ' ; h > 0 ( h = x0 sin ' < sin '), 2h sin ' ; h ; (sin ' + h) d 0 dr Img (t) > r2 + h2 + (1 + H2)2 + H12 = 2h ; = r2 2h 2 2 2 + h r + h + 1 + 2H > 0: 2
' " ( , Img0 (t) " ( # x0 t0, #E Img0 (x0 ) = ; Im up , Im g0 (t0 ) = 0. 8" " , # Img0 (t) < 0 4x00 t0) Img0 (t) > 0 # # t0 . ; , (28) #E " Re g0 (t) Img0 (t) x0t0 #, # drd Re g(t) > 0 # l1 # x0 " t0 drd Re g(t) < 0 "! " ( l1 # t0 . ' " ( , Re g(t)
# x0t0 "$ " # t0 . I , " # $ ( ( " (. 49, . 165]). ' #E $, # g(t0 ) = t0 g0 (t0 ) + ln(1 ; t0 ) ; ln(1 + t0) = ln(1 ; t0 ) ; ln(1 + t0)
1255
# s
p;1 1 Jnp (u) = i(2)k epkg(t0 ) f(t0 ) ; kg2 00 (t0 ) 1 + O k = pk p;1=2 (1 ; t0)p=2+r;1=2 (1 + o(1)) p 3=2 1 ; t0 = ; (2) p ; 1 + t0 t0 1=2(1 + t0)r+1=2;p=2 2k " . 8. n, k, , q 2 N, k = n;p 1 , 1 6 6 p ! : p ;p a) jEn(q)j 6 kC3=22 q1=p ; 1 + q1=p ;2pk p ) jEn(;q)j 6 kC3=32 (Aqp;1 + q1=p + 2 q1=2p Bqp;1 );pk ;
;
Aqp;1 = 1 + q2=p ; 2q1=p cos p 1=2, Bqp;1 = q1=p ; cos p + Aqp;1 1=2 , C2, C3 | , # k. . B!" " 5 7. G ! "" u = ln q + i , 2 Z, j j 6 p, (q ; 1)2 + ( p)2 6= 0, ( Aq =
1 + q2=p + 2q1=p cos
1=2
q1=p + cos
1=2
Bq = : p p + Aq I$" 7 " $ Jnp(u) " ": p jJnp(u)j 6 C3=42 (Aq + q1=p + 2 q1=2p Bq );pk : k ; " "(, # # , , # j j. 8" 5 w = lnq # !" , $" #( # ) "E = p, # ) | = p ; 1, " .
x
4. & ! 1. G ( q 2 N, q > Q+p . G" !, 1 : : : p 2 Q,
, # 1 a,1 + : : : + p a,p = , , = (r1 : : : rp ) 2 Qp:
1256
. .
p X i=1
i Eni(q) =
p X
; (1 'n1(q) + : : : + p 'np (q)) =
n = ps + 1 ps + 2 : : : ps + 2p, s 2 N. '" 2p p X i=1
i Eni(q) =
nj (q) 1 Lj 1q + : : : + p Cjp+;p1;2 Lj +p;1 q1 j =1
p X j =1
rj nj (q) ;
p X i=1
p X j =1
nj (q)rj ;
p X i=1
;
i 'ni (q)
i 'ni (q) n = ps + 1 : : : ps + 2p s 2 N
p 2 "E # . 7 n, ps+1 6 n 6 ps+2p, P # i Eni(q) 6= 0. i=1 G ( d0 2 N , # d0j d0rj 2 Z, j = 1 2 : : :p. I$" p!d0D2p;1
p X i=1
p p X X D2p;1 nj (q)d0rj ; p! d0 i D2p;1'ni (q) 2 Z i=1 j =1 p P " ( , p!d0D2p;1 i Eni(q) > 1. ; " $ i=1
i Eni(q) = p!
# . ' ,
p p X X p!d0D2p;1 6 C5e(2p;1)k E (q) jij jEni(q)j 6 i ni i=1 i=1 p ; ;p 6 pC6 e2p;1 q1=p ; 1 + q1=p ;2p k ! 0 (k ! 1)
k3
q > Q+p :
I , # #, " . F ( 2 #" " ( 1.
. G p = = 2, u2 = 0 (9), (2), (6), 7 3 " ( ( (3). F ( $ ( $# ! ! 410]. 7 Rn(s) ; 1)p (s ; 2)p : : :(s ; k)p (s ; k ; 1)r;1 R, n(s) = Qp kQ;(s 1 r0 r00 Q Q (j + s + l) (j + s + k) (j + s + k + 1) j =1 l=0
j =1
j =1
$" r0 = min(r + 1 p), r00 = max(r + 1 ; p 0), 1 : : : p 2 Q \ 400 1), ! " ( " .
1257
1 z P l , 1 : : :p 2 Q \ 400 1). =1 (+) M | ' i ; j l , p X X ln p T= ln den(j ) + j =1 pj den(j ) p ; 1 p ;p Q, +p = min q 2 N : eM (2p;1)+T < q1=p ; 1 + q1=p 2p p ; Q, ;p = min q 2 N : eM (2p;1)+T < Aqp;1 + q1=p + 2 q1=2pBqp;1 p Aqp;1 , Bqp;1 ! 2. kj = #f1 6 l 6 j : l = j g, 1 6 j 6 p) 1 : : : C i;1 L 1 i = 1 : : :p: 1 L ,bi = Cii;;2+ 0 0 k1 i;1+k1 1 q i;2+kp i;1+kp p q 3. p l 2 N, Ll (0 z) =
q) , Q- q, jqj > Q, sgn( p ,b1 : : : ,bp Qp. # $ # Ll (0 z) " ( 44,11{13]. # ! $ $" ( P. . ; # ".
'
1] . . // !. "#$#%#. 1984. ) 4345-84. 2] . . Q - // !. "#$#%#. 1984. ) 5736-84. 3] $ 0 1. 2. 3 4 5 5 F (x s) // 2. 3. | 1979. | T. 109 (151), ) 3 (7). | C. 410{417. 4] Hata M. On the linear independence of the values of polylogarithmic functions // J. Math. Pures et Appl. | 1990. | Vol. 69. | P. 133{173. 5] . . 55 4 Q // <2$. | 1982. | T. 37, ) 5. | C. 179{180. 6] . . 3 , =- (3) // Acta Arith. | 1983. | V. 42, ) 3. | P. 255{264> <2$. | 1979. | T. 3, ) 3. | C. 190. 7] Hata M. Rational approximations to the dilogarithm // Trans. Amer. Math. Soc. | 1993. | Vol. 336, ) 1. | P. 363{387. 8] ? ., ? %. @! ! . | 2.: $ , 1968. 9] D. " ! ! . | 2.: $ , 1978. 10] $ E. ". $ 4 F (3) // 2. 4 . | 1996. | T. 59, ) 6. | C. 865{880.
1258
. .
11] " . $. 5 4 5 ! // ". 2 . <-. @. 1. | 1985. | ) 1. | C. 42{45. 12] " . $. !3= ! 5 4 5 // !3=F. G. 1. | #4. 2 . <-. | 1985. | @. 10{16. 13] " . $. 55 4 4 5 5 // !3=F. G. 2. | #4. 2 . <-. | 1986. | @. 3{12. ' ( ) 1998 .
. .
. . . e-mail: [email protected]
519.216
: , , ! !, " #, $ ! %% $# ! & , ! ', ($)$# ' & ' #.
* $ $# # $ ' , ! !+ N ( ). .! /$+ &+ 0 ' ( $# ( R. " # N ( ) $ ' g ! u( g) ( $0 " '$# !" $ N (0) ! 0). ' )$ , ! 1 " ! ! N ( ). U
U
t
t
t
U
N
U
t
Abstract
D. V. Khmelev, Limit theorems for asymmetric transportation networks, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 4, pp. 1259{1266.
We consider a model of an asymmetric transportation network. The transportation network is described by the Markov process N ( ). This process has values in a compact subset of the 5nite-dimensional real vector space R. We prove that N ( ) converges in distribution to a non-linear dynamical system g ! u( g) (assuming convergence of initial distributions N (0) ! g), where g 2 R. The dynamical system has the only invariant measure to which the invariant measures of processes N ( ) converge as ! 1. U
U
t
t
t
U
U
t
N
, ., , 1{5]. $ N ( ), n . ' N n . ( j P dNj N, dNj = 1, dNj N | + j N. ,- j =1
p
!1 dj > 0, N(dNj ; dj ) N;! 0 j = 1 n. / j v
*) ' ' 0 ( ISSEP s98-2042. , 2001, 7, 7 4, . 1259{1266. c 2001 , !" #$ %
1260
. .
1 n. 2 j j > 0. 3 , . - . ' -
- . ' (j v) pjv , 4 5 4 1=jv v. 6 P = fpjv gjv=1n + . $ v : +
mv . 3 , v, + , ( ) (v l)
+ Pe = fp~vl gvl=1n9 + + j + rj , rj | 1 mj . / : , e +
- : (P + P)=2 + + 5 6 . ' 4 ,
. < - , : 1] 4],
. > fji i j, Vjv (j v). >
mj X uji = fji =N Mjv = Vjv =N: / j = 1 : : : n v = 1 : : : n
k=i
dNj = uj0 > uj1 > : : : > ujmj > ujmj +1 = 0
Mjv > 0
mj n X X j =1 i=1
uji +
n X v=1
X n Mjv = rj dNj : j =1
(1) (2)
> UN u = (u11 : : : u1m1 u21 : : : unmn M11 M12 : : : Mnn)T -+ (1) (2), uji = xji=N, Mjv = yjv =N xji yjv | . > U + u = (u11 : : : u1m1 u21 : : : unmn M11 M12 : : : Mnn)T j = 1 : : : n dj = uj0 > uj1 > : : : > ujmj > ujmj +1 = 0 (3)
1261
v = 1 n Mjv > 0
mj n X X j =1 i=1
uji +
n X v=1
X n Mjv = rj dj :
(4)
j =1 n P
> , UN (1=N)Z U R, = n2 + kj . / l = 1 : : : j =1
el 2 R ,
l 1,
| . ' euji (eMjv ) | e{(uji ) (e{(Mjv ) ), { (uji) ({ (Mjv )) 4
u 2 U ,
- uji (Mjv ). : , 6 -
Z AN f(u) = KN (u dy) f(u + y) ; f(u)] (5) KN (u dy) | - + u 2 UN : 8 j = 1 n, i = 1 mj , v = 1 n > > Nj (uji ; uji+1 )pjv (;euji + eMjv )=N, > > < j v = 1 n, i = 1 mv KN (u dy) = > N M (u jv jv vi;1 ; uvi)=dN v (euvi ; eMjv )=N, > > > j v = 1 n, l = 1 n : N M u p~ =dN (;e + e )=N, uj0 = dNj ujmj +1 = 0.
jv jv vmv vl v
Mjv
Mvl
> D(AN ) AN C(UN ) + @ UN . '
j. $
v. A
v (uji ; uji+1 , j i ). 6 ( 2), N ! 1 5 u :+ : - + @@ + : @uji = M j u =d ; ( + M j =d )u + u 9 = ji;1 j j j ji j ji+1 > @t (6) @Mjv = u p ; M + M j u p~ =d >
j j1 jv jv jv jmj jv j @t j v = 1 n, i = 1 mj , uj0(t) = dj , ujmj +1 (t) = 0 + t M j (t) = n P = lj Mlj (t). /
: (6) l=1 uji(0) = g{(uji ) Mjv (0) = g{(Mjv ) : (7) > u(t g) : ( : (6) (7), u(0 g) = g.
1262
. .
6 UN = UN (t), (5) UN , TN (t) + @ C(UN ). 3 f : UN ! R, TN (t)f(u) = E(f(UN (t)) j UN (0) = u) u 2 UN : N N ' j = dj =dj . C N : U ! R u 2 U ( 1N u11 : : : 1N u1m1 2N u21 : : : nN unmn M11 M12 : : : Mnn)T . > D( N ) N u 2 U , + N (u) 2 UN . / x 2 R 4 X kxk = jxij: i=1
1. . () U , u(t g) (6), (7). () U g 2 U : u(t g ) = g . () > 0 ! g 2 U ku(t g) ; g k 6 const e;t . ' C(U ) | @, + U . 2. !" f 2 C(U ) ! t > 0 lim sup sup jTN (t)f( (g)) ; f(u(t g))j = 0: N !1 06s6t g2D(N )
> "g , g 2 U . 3. # UN (0) ! $ "g , 8t sup kUN (t) ; u(t g)k ! 0 . 06s6t
6 UN N . 4. N ! $ "g . ' (6) 0. 6 - + + : ) 0 = M j uji;1=dj ; (j + M j =dj )uji + j uji+1 (8) 0 = j uj1pjv ; jv Mjv + M j ujmj p~jv =dj j v = 1 n, i = 1 mj . $ () 1 - : (8), - (3) (4). A : : (6). ' xji = uji ; uji+1 i = 0 : : : mj , xj;1 = xjmj +1 = 0. 2+ , uj1 = xj1 + : : : + xjmj ujmj = xjmj . 6 , xji = ij ( j ; 1) (9) mj ; 1 dj 1+ j
1263
j = M j =(j dj ). F , (8) (9) uji Mjv M j . ; n T . $4 a = (M 1 M 2 : : : M n)T b = M 1 u1dm1 1 : : : M n unm dn $ (9) ujmj M j . ' , b a. $ (8) e aT = (a ; b(a))T P + (b(a))T P: (10) ' u(a) : (8), a. $ 1 - : (10), u(a) 2 U . G @ a m + +, + - . n Pj n P P ' L(u) = uji + Mjv . j =1 i=1
v=1
5. % , P Pe ! " "
" = (1 : : : n)T , T P Pe = T , 1 + : : : + n = 1, 1 > 0 : : : n > 0. > 0, ) Pe = I, (10) a ; b(a) = , ) P = I, (10) b(a) = , ) P = Pe , (10) & a = . a() a ; b(a) = . # 0 < 1 < 2 , a(1) < a(2 ). ' $ ) ). # , L(u(a())) . ( > 0 L(u(a( ))) = r1 d1 + : : : + rndn: . ' + . 1. ' , pjv +~pjv > 0 + e . j v. > - 5 (P + P)=2 H (6)
u_ = f (u) f : R ! R u 2 U R: ,- : + + - . ' () 1 -
. 6. % ! t U & u(t ). /
2,
1] 4, 2.1]. '4
() (). 7. " U" Int(U ), " & ! u(t ) 8g 2 U t ! u(t g) 2 U" .
1264
. .
. $ 6] -+ + : () : g U f (g) + g + f (g) 2 U , () ; U (; 6= U ), g, - ;, g + f (g) @@ 5 + g + f (g) 2 U . ' () . '4 () : . / g 2 U g + f (g) 2 U 5 , U u_ = f (u),
6. > SL + n2 + (j v) + 1 6 j, v 6 n. / S SL GS = f ; U j 8 (j v) 2 S 8g 2 ri ; hg eMjv i > 0 8(j v) 2 SL n S 8g 2 ri ; hg eMjv i = 0g. 3 S , 8g 2 ; 2 GS 8(j v) hg eMjv i = 0. ' L(g) = = r1d1 + : : : + rndn > 0, j i, hg euji i > 0,
, hg euj1 i > 0. $ + P - pjv 6= 0. > , hf (g) eMjv i = j hg euj1 ipjv > 0, , g + f (g) 2= aM ;, 8h 2 aM ; hh eMjv i = 0. L / 4 euj0 = eujmj +1 =0. ' S = S. /
+ -+ + , hg euj0 i = dj ( hf (g) euj0 i = 0!). / ; 2 GS - j i, (i) 1 6 j 6 n, 0 6 i 6 mj 9 (ii) 8g 2 ri(;) hg euji i = hg euji+1 i. 6 , hf (g) euji ; euji+1 i > 0, :
. O
S 6= SL S 6= ?. ' GS P GS = G0S G00S , G00S = GS n G0S , G0S = f; 2 G j - j i (i) (ii) - 9w (w j) 2 S g. / ; 2 G0S g 2 ;
, hf (g) euji ; euji+1 i > 0,
G00S . e > 0, 9(j v) 2= S 9w (w j) 2 S: pjv 6= 0, ' (P + P)=2 p~jv 6= 0. ' , 8g 2 ; 2 G00S hg euj1 i > hg eujmj i > 0, hf (g) eMjv i > 0, . F , : g - ,
g 2 U" . A
() g 2 U" . ' 4, 3.3] : . , (6) : G J(u) + + + (6) u
5 u 2 U" . ' U" , - c = umax max (;Jii (u)): 2U 16i6 "
1265
> I . / - C, (i) 8u 2 U" J(u) + (c + 1)I > C > 0, (ii) - c0 > 0, c0 C T +
+ + 5 6 . $ 5 4, 3.3] , > 0 u( ): U" ! U" - 5@@ ,
: 1, . . 8g1 g2 2 U" ku( g2) ; u( g1)k 6 kg2 ; g1k, < 1. H () () 1 -
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