-
519.2
..
2003
-
519.2+600.1 / .. , 2003. . 244: . .: 114 .
, .
, , , , , , , , , , . : , , , , , , .
.
: .. , .-. , , .. , , , .. , ,
ISBN 966-02-2664-0 . ., , 2003
2
-
, , . , .
:
1) , , , , , , ;
2) . , ;
3) ; 4) ,
, .
. , .
, , , . , , , . , . , , , , . , . : , .
, : , , , , , , .
, , , 3
-
, . ,
. , .
.-. ... .., .. .. .. .., .. ..-.. .. ..-.. .., .
* * *
, , 2003 ., , .
: , 42, , 03187, , , : [email protected].
4
-
1. 1.1.
. , . .
1.1.1. 1. , . . ,
. 2.
(, , , .), .
1. , , , , .
2. . 3. ,
. 4. -
() . . .
. . , . , ' - .
1. , , . , (, , , ), , (2, 3, 4, 5). 5. () ,
. 1. ,
. A . A , , ,
AA .
2. A B , A , . B 1.
. ,
, .
, .
5
-
6. )(Ap A
AL , A , : L)(Ap
L
LA= . (1.1.1) 2. ,
, )(Ap
32
64 = .
. . 7. )(ApN A
, N
AN A , :
)(ApNN
N A= . (1.1.2) 2. ( )
) , .
N (ApN
)(Ap
8. , )
)(Ap
(ApN A N : )(Ap = . (1.1.3) )(lim ApN
N . ,
. . 3.
, 30- XX .. . . .
1.1.2. 1. .
. . .
, . 2. A ,
. A A . .
. 3. A ,
(B
BA , BA ). 4. ,
; , ; , , , .
6
-
, .
1. , , , . . , () .
. ' ( +), (
)( ) \ ().
' A 21 AA = , A , , , , .
1A 2A
1A 2A
A 21 AA = , A , , . 1A 2A AAB \== , A , ,
, B A .
. ' ( ) . , , ,
, Ss S sA
s . ' , , , , , .
Ss
sAsA
Ss
sAsA
2. , ,
sA 1+ sxs10 y 5,00 s ( ]5,0,0[= Ss ) (. 1.1.1).
,
Ss
sA5,10 x , 10 y ,
,
Ss
sA15,0 x , 10 y .
. 1.1.1.
', . 5. , :
1) ;
7
-
2) ; 3) A , A ; 4) 1A , 2A 21 AA 21 AA .
1.1.3.
1. )(A A , A , , 0)( A , 0)( = ( ), , nA ,...2,1=n =
n
nn
n
AA )()( . (1.1.4) 1. ,
, , , , . , .
2. (1.1.4) . 2. )(
A , 0)()\( == AA . 3. )( ,
n (), .
Nn ,...,2,1( =,...)2,1=n
1. .
1. ( ) , .
2. , A , - (-). 4. -
, , :
1) ; (1.1.5) 2) A AA \= ; (1.1.6) 3) nA , ,...2,1=n =1n nA .
(1.1.7) 1.
. 2. (1.1.5) (1.1.7) ,
nA , ,...2,1=n =1n nA . (1.1.8) 3. ( (1.1.5), (1.1.6)).
4. (1.1.5) (1.1.8) , - , nA ( ) ,
,...2,1=n , , .
5. -. 8
-
6. - , .
7. - -. 5. ,
- ( ( , )). . . ,
, -.
6. () ( , )), . , ( , , ). 7. ,
-
3) . 1)( =p 1. , 1)(0 Ap 0)( =p . 2. 1)( =Ap , A ,
, . 0)( =Bp B 3. 1A
( ) ,
:
2A )( 21 AAp 21 AA 0)( 1 Ap1A )/( 12 AAp 2A
1A
)/()()( 12121 AApApAAp = . (1.1.9) . , .
, , ,
)/( 12 AAp 0)( 1 =Ap 4. ,
: ) , , ; 1+ nn AA 1n )(lim)(
1
nn
n
n ApAp = =
) , , . nn AA +1 1n )(lim)(1
nn
n
n ApAp = =
5. ( ),
1A 2A
)()()( 2121 ApApAAp = . , , ,
)()/( 212 ApAAp =0)( 1 Ap )()/( 121 ApAAp = 0)( 2 Ap .
. , 1A , .
2A
6. , .
MAAA ,...,, 21
)()...()()...( 2121 MM ApApApAAAp =.
. . 7.
, MAAA ,...,, 21 =MAAA ...21 .
1.1.5. 1. .
: MAAA ,...,, 21
= =Mm mAp1 1)( . (1.1.10) 1. ( ). M 2. ,
: 1)()( =+ ApAp . (1.1.11)
2 (). A ( ).
B
.)()()()( BApBpApBAp += (1.1.12) 3. , . MAAA ,...,, 21
10
-
) = = = ++= Mm Mnm MMnmmMm m AApAApApAp 1 1, 111
)...()1(...)()()( ; (1.1.13)
) = = = ++= Mm Mnm MMnmmMm m AApAApApAp 1 1, 111
)...()1(...)()()( . (1.1.14) 4 (). M AAA ,...,, 21 M .
).../().../()()( 111211
= = MMMm m AAApAApApAp . (1.1.15) . , ,)()(
11
== = Mm MMm m ApAp ( 6 1.1.4). 5 ( ). A
, (). M
HHH ,..., 21)(Ap A
)(Ap = . (1.1.16) )/()(
1
m
M
m
m HApHp= 6 ( ). (), . ,
,..., 21 HH
mH A :
== = )/()( )/()()( )()/( nn mmmm HApHp HApHpAp AHpAHp . (1.1.17)
1n. , . 7 ( ).
( N ).
N
A . A p , pq = 1 . , N A , n
nn
NN qpCnP =)( N n , (1.1.18) ' : nNC N n
)!(!
!
nNn
NC nN = .
. (1.1.18) . 8 ( ).
, NA .
A - ,
m mp
1 ( 1,m mq p m N= = ). , NA n , == += Nm mmNn nN zpqznP 10 )()(
.
11
-
1.2. 1.2.1.
1. ,
X . x
X
)(=x , . .
1. , , x . X 2.
, , .
, , .
3. )(=x =x . ,
X , .
4. , , , . 1. .
( , , , ..) .
. , , , . 3. ,
, , .
. . 2.
. , () . .
1.2.2. 1. () : X1)
( ) { }F x P X x= < ; (1.2.1) 2)
f xdF x
dx( )
( )= ; (1.2.2) 3) ,
e j x , ' :
12
-
= dxexfjQ xj )()( , (1.2.3) P{*} ; j .
. .
1.
)(xF
)( jQ , , , .
)(xf
2. , .
X
)(xF )(xF
. : 1, 0, 1 1/3. ,
X
12 += nn XY ,...1,0=n 3.
, . , ,
X
ix ,...)2,1( =i - ( 1): )()( ii xxpxf = , (1.2.4)
. ( ) .
ip ix ip
...,2,1=i
, , , , .
)(xf
4. ' , . , . , : = x dxxfxF ,)()( (1.2.5) ':
.)(2
1)( = dejQxf xj (1.2.6)
5. , , , , = )()( xdFesM sxx (1.2.7)
13
-
= )()( xdFss xx . (1.2.8) 2. ,
. , , . 1.2.1, 1.2.2 1.2.3, , .
1.2.1
0)( xF 1)(0 xF )(xF
1.2.2
0)( xf
1)( = dxxf
2
1
1 2 2 1{ } ( ) ( ) (
x
x
P x X x F x F x f x dx < = = )
1.2.3
1)0()( = QjQ
( ) X )()(* jQjQ =
, , . 1.2.4 1.2.5. [25] [69].
1.2.4
1,0,
1 == IiI
pi
10,,0,)1( == IiCp iIiiIi ,...1,0),exp(
!== i
ip
i
i 10,...,1,0,)1( == ip ii
14
-
. 1.2.4
0,0,,0, == INnNni
C
CCp
n
N
in
IN
i
I
i
pqiqiippi ==+= 1,1,0,)1( 10,...,2,1,...,1,0,)1(1 === + miCp imi
imi
. : nNC)!(!
!
nNn
NC nN = .
1.2.5
bxa
abxf = ,1)(
() 0),,(,2
)(exp
2
1)(
2
2 > = xmxxf 0,0,0,
2
))/(ln(exp
2
1)(
2
2 >>>= mxmxxxf
0,0),exp()( >>= xxxf
- ,...2,1,0,)2/(2
)2/exp()(
2/
2/)2( = = xxxxf 0,
)(
)/exp()/()(
1 = xcb bxbxxf c , ( ), c
( )
b 0>b0>c
0,0],1,0[,)()()()1(
)(11 >> += xxxxf
0,0,0),)/(exp()(1 >>= bcxbx
b
cxxf c
c
c
,0,exp)(
2)( 2
2
12
2>= xxmxmmxf m
m ( ), m 0>m ( 0> )
,0,2
exp)(202
22
2> += xxIxxxf ( 0> ), ( 0> )
15
-
. 1.2.5
()
++=
),(,0
),,2/)((,)/()(4
),2/)(,(,)/()(4
)( 2
2
bax
bbaxabxb
baaxabax
xf
0),,(,)(
1)(
2
0
2>+= hxxxh hxf
),()(
2
210
xfxbxbb
ax
dx
xdf ++ = 210 ,,, bbba
0,0],1,0[
,1)2/(
)2/)1(()(
2/)1(2
>> + +=
+
x mx
mm
mxf
m
),,(,21
)( = xexf mx ,m
0,,)(1 >=
+ mxx
m
mxf
. - ( 2). )(m
1.2.3. () ,
, , . , , , . . 1. )]([ XM )(X
X )(xf= dxxfxXM )()()]([ . (1.2.9) 1. : ,
(1.2.9) . 2. )]([ XM
, )(X . )(xf 2.
xm X
)(xf
XX =)( :
16
-
== dxxxfXMmx )(][ . (1.2.10) 3. ][XDDx =
:
X
)(xf2)()( xmXX = == dxxfmxmXMD xxx )()(])[( 22 . (1.2.11)
4. () x : X xx D= .
1. ,
xm X
)(xf
xD x .
2. xm Xx , ;
. xD
3. () () , . 5.
: x
xm . 6.
, , .
ex
X )(xf
5,0)( =xF 7. px p
, pxF =)( p )10(
-
( ), ( ). 9. m -
X XX =)( , - , ||)( XX = - , )()( xmXX = - , ||)( xmXX = .
1. . .
2. ' , :
;01 = ;222 xx mmD ==;23 2233 xx mmmm += .364 422344 xxx mmmmmm
+=
3. , : xD
= . 0 ; )1...(5312/ xD
, : xm xD
;1 xmm = ;22 xx mDm +=;3 33 xxx mDmm += .63 4224 xxxx mdmDm
++=
4. . , , 2 . . ( , )
.
)( jQ =1+ ,!
)(
1
mj= (1.2.12)
0
)( == dj jQdm . (1.2.13)
(1.2.3) .
1. (1.2.12) .
2. , , :
)(ln jQ = .)(!1
= j (1.2.14) 18
-
10. (1.2.14) . :
0
)(ln == dj jQd . (1.2.15) 1. (1.2.14) ,
' : = =1 )(!exp)( jjQ . (1.2.16)
2. , ,
mm ,...,1m ,...,1 . :
;11 xmm == ;2
2
22 xx Dmm === ;23 3
3
233 =+= xx mmmm .361243 324
42
23
2
244 =+= xxx mmmmmmm 3.
: 0
)( == sxsd
sMdm
; 0)(ln == sx
ds
sMd
; (1.2.17) , ,
=)(sM x ;!0
= sm (ln =)sM x .!1= s (1.2.18)
11.
2/3
2
3
2/3
2
31 == ; (1.2.19)
2
2
4
2
2
42 3 == . (1.2.20)
1. . 01 = . . 02 = .
2. 1 2 2132 + 2/)3( 2 + .
12. . 13. () X )(xf == dxxfxfxfMH x )(log)()]([log 22 . (1.2.21)
1. ,
. X
19
-
2. , , . ,
, , .
X
0=xH X],[ ba
3. .
2
xX
xx eH 2log 2= . 14.
( X 0=x ) )]1)...(1([][ ][][ +== XXXMXMm . (1.2.22)
. '
:
][mX )(sx
)1()(][ xm = ; ==+ 0 ][ !)1( smsx .
15. X
])[( ][][ xmXM = . (1.2.23)
1.3. 1.3.1.
1. N - XG :
1) N - 1 1( ) { , , } { }N N NF x P X x X x P X x= < <
-
1.3.1
11 1( ) { ,..., };
( ) ... ( )N
N N
xx
N N
NF x P X x X x
F x f x dx
= < = xmxxf
= mxxF )( ,
dttxx = 2exp21)( 2
= 2exp)(
22 jmjQ . 1. , m . 2. 7.
. 1.5.1. . ,4=m 1;75,0;5,0;25,0= ( 1 4)
1. , ,
m . 3 .
m
%4,0
)(xf 3m . , , ,
%7,99
997,03( m , )3+m . 34
-
2. )0( =m , 0 -.
3. :
X
= nn
n
n
n
dX
XgdM
dD
XgMd2
2 )(2
)]([;
,)(
)]([)]([ += dXXdgDMXgmMXXgM , )(Xg X D .
4. - . XXg =)( . 1.
, : 1X 2X
)],,(exp[),( 212212 xxPCxxf = (1.5.1) , , , , , , - .
C )0( >C ),( 212 xxP1x 2x
1x 2x ),(( 212 xxP )0 1.
(1.5.1) :
21 , XX
.)1(2
)())((2)(exp
12
1),(
22
2
2
1
2
22
2
1221121
2
11
2
2
2
21
212
+=
r
mxmxmxrmx
rxxf
(1.5.2)
+++= )2(21exp),( 2222212121212211212 rjmjmjjQ ,(1.5.3) , 1m 2m
1X 2X 1 2 , r .
1. , (1.5.2) :
)],()(2
1exp[
||2
1),( 1212 mxRmx
Rxxf T
GGGG = (1.5.4) -,
, -, , ,
Txxx ),( 21=G1X 2X
Tmmm ),( 21=G1X 2X T
1R ,
2221
1211
RR
RRR = ,
. || R R
35
-
2.
).,()(),(
2122121212 jjQ
r
jjQ nn
n = ,
. 1.
, . 1X 2X
. , (. 9 1.2).
2. ( ) (. 1.5.2),
1X 2X
0,)1()())((
2)( 22
2
2
2
22
21
2211
2
1
2
11 >=+ CCrmxmxmxrmx (1.5.4)
T 1 2( ) ( ) ,x m R x m C C =G G G G 0.> (1.5.5)
. 1.5.2. 1X 2X
1. , , ,
),( 21 mm
1m 2m
1 , 2 , r C . 1 , 2 r .
=
=.4/
,22
1
21
212
2
2
1
21
rarctg (1.5.6)
, . D90
2. (1.5.4) (1.5.6) , , .
C
36
-
3. (1.5.4), (1.5.5) , , , )0( =r , .
4. (1.5.6) , =1( )2 : 4/ = . =1( )2 )0( =r , C .
3. () () , :
1X 2X
1Y 2Y
,cos)(sin)(
,sin)(cos)(
22112
22111 mXmXY mXmXY += += (1.5.7) (1.5.6).
4. - .
5. , , , ' .
. , , .
6. , , :
1X 2X
1 2 1 1 2 1 2 1 1
2
22 2 1 12 22
2 12
( / ) ( , ) / ( )
1 1exp ( ) .
2 (1 )2 (1 )
f x X x f x x f x
x m r x mrr
= = = =
(1.5.8)
. (1.5.8) , , :
12 / xxm 1X
12 / xxD
)( 111
22/ 12
mxrmm xx += , . 222/ )1(12 rD xx = 2. ,
, . 2. ,,
, : 1X NX
)],()(2
1exp[
||)2(
1),...,( 1
2/12/1mxRmx
Rxxf T
NNN
GGGG = (1.5.9)
37
-
TNxxx ),...,( 1=G -, ,, , 1X NX TNmmm ),...,( 1=G -, , : R
NNN
N
RR
RR
R
...
.........
...
1
111= . (1.5.10)
1.5.2. , ' , ,
. , , . , , . 1. ,
: .0,
2exp)(
2
2
21>= xxxxf (1.5.11)
1. , (1.5.11),
.0,2
exp1)(2
2
1 >= xxxF (1.5.12) 2.
2
=xm , ,
222 xx mD = =x , 2
3 33=m .
3. 1 . 1.5.3.
. 1.5.3. (1) (2 5). 1 5 1= 4,0=
1. ),( 21 XXX =G 38
-
( = =0) .
1xm
2xm
2 X XG , ]2,0( .
. .
.2
exp2
1),(
2
2
2
2
1
2212 += xxxxf (1.5.13)
xx
xxJ == cossin sincos),(2 .
XG ( ),x
.2
exp2
),(2
2
22 = xxxf (1.5.14) (1.5.14)
]2,0( x ),0[ XG (1.5.11)
21)(1 =f . . , ,
, -
1X 2X2
1xm
2xm ),( 21 XXX =G , (1.5.11). R
1( xm )2xm
2. ( ) ,
,0,2
exp)(202
22
21> += xxIxxxf (1.5.15)
, 2 )0( > , .
)(0 I 1. ,
(1.5.15), += x dxxIxxxF 0 202 2221 2exp)( , (1.5.16)
. 2.
39
-
,0,4
12
exp)(4
22
2
22
21> + += xxxxxf
, >>
.0,8
12
)(exp
2
1)(
2
2
2
1 > + = xxxxxf 3.
+ += 22
2
2
12
2
2
2
02
2
4exp
42421
2 IImx , (1.5.17)
2222 xx mD += , (1.5.18)
. (*)1I 4. >>
xm . . 1.5.3 ( 2 5).
2xD 2.
),( 21 XXX =G , .
1xm
2xm 2
X XG
22
21 xxmm += , , 2
2
01 02
22
0 02
1 cos( )( ) exp cos( )
2 2 2
exp sin ( ) , | | ,2
f F
= +
1
2arctg0x
x
m
m= , . [*]F 1 . 1.5.2.
1.5.2 1
24, 40, 47, 49
3, 16, 25, 32, 69, 74, 75, 84, 100 5, 10, 15, 20, 22, 61, 88,
89, 105 55, 56, 66, 75, 82, 91, 96, 97
40
-
2. 2.1.
2.1.1. 1.
, - (
)(tX
Tt T ) . . S ( ). 2. - (
) () i )(tX
x ti ( ) ,...)2,1( =i , i - t T x S . 1.
, . ,: .
2. - () , , . , .
3. . . . 2.1.1.
. 2.1.1. )(tX
. . . X t( ) x ti ( ) - - .
i
i X t( )
x ti ( ) ( , ,...)i = 1 2 . (
X t( )
t0 X t( )0 ). x ti ( )0 - t -
i
i X t( )0 . 0. . , , ' S , T , . . 2.1.1 2.1.2
41
-
(), .
2.1.1 ( 1)
S ( ) NRS ( ) NZS N
S (S R
) ZS (S R N ) NZS
T
() (T R ) (T ) R N
. , R Z .
2.1.2 ( 2)
1. ( 1- )
2. S
( )
3. 4.
(S , T ) (S , T )
5.
(S T )
6. ( )
42
-
() , . , ,
. . .
2.1.2. . N -
.
F x tN ( ; )
G G,
f x tN ( ; )GG
Q j tN ( ; )GG ,
. 2.1.3. 2.1.3
11 1( ; ) { ( ) ,..., ( ) };
( ; ) ... ( ; )N
N N
xx
N N
NF x t P X t x X t x
F x t f x t dx
= < ),...;,...( 11 MMx ,t t,xt / xm
),1),(/( MmtxtX m = , .
)
)
,...,;...( 11 MMx tt,x,t / xD
4. ) ,
.
(tDm
,...;,...( 11 MMx ,t t,xt / xm
)(tDD
),...,;...( 11 MMx tt,x,t / xD
53
-
. 2.2.2. :
; )(t mx
)(tX )(t mx )(tx )(tm
x ;
; )(tX
),...,...( 11 MMx ,t, t,xt / xm )(tX),...;,...( 11 MMx ,t t,xt /
xm ),...,;,...,/( 11 MMx ttxxt
),...;,...( 11 MMx ,t t,xt / xm
)(tX
. ) ) :
(tDx )(tX
(tDm )(tDD
)(tDx = + . (2.2.26) )(tDm )(tDD
' ),1),(/( MmtxtX m = . 6. ),...,;...,( 11 MMx tt,x, / xttR
, :
)],,...,;.../'(),...,;.../([
),...,;...,(
11
0
11
0
11
MMMMx
MMx
tt,x,xtXtt,x,xtXM
tt,x, / xttR
==
(2.2.27)
) : ,...,;.../( 110
MM tt,x,xtX
).,...,;.../(),...,;,...,/(
),...,;.../(
1111
11
0
MMxMM
MM
tt,x,xtmttxxtX
tt,x,xtX = = (2.2.28) . 2.2.3.
. ),...,( 1...1 NttN ),...,;,...,/,...,( 111...1 MMNM ttxxttNM
++ . (2.2.13)
),...,;,...,( 11 NNN ttjjQ ),...,;,...,/,...,;,...,( 1111
MMNMNMMN ttxxttjjQ ++ . 54
-
, , . , .
2.2.3 )(tX
),...,;,...,/( 11 MMx ttxxtm =
=
],...,;,...,/)([ 11 MMx ttxxtXM
tt,X,t / XmDtD MMxxxm M )],...,;...([)( 11,...,1=
],...,;,...,/)([
),...,;...(
11
11
MMx
MMx
ttxxt XD
t t,x,t / xD= = tt,X,t / XDMtD MMxxxD M )],...,;...([)(
11,...,1=
)],...,;.../'(),...,;.../([
),...,;...,(
11
0
11
0
11
MMMMx
MMx
tt,x,xtXtt,x,xtXM
tt,x, / xttR
==
2.3. 1.
)(tX , () : . ))(),...,(()( 1 tXtXtX H=G
. (, , , , .).
. ( , , ). 2. ,
.
)(1 tX )(2 tX
)(1 tX
)(2 tX
.
)(1 tX )(2 tX
),;,( 21212 ttxxf N );( 11 txf N );( 22 txf N .
3. ,
))(),...,(()( 1 tXtXtX H=G
),...,;,...,( 11 NH ttxxf
55
-
== Hh hhNNH txfttxxf 111 );(),...,;,...,( , (2.3.1) ( ),
N
H . 1. ,
-. , . ' .
2. , ',
. ))(),...,(()( 1 tXtXtX H=G
4. )(tmx , :
))(),...,(()( 1 tXtXtX H=G)](tXMtmx
G[)( = . (2.3.2)
5. )(tD x , )(tXG :
)]([)( tXDtDxG= . (2.3.3)
6.
)',( ttRxG
)(tX HH
.),1,(
)]'()([)',(00
Hlh
tXtXMttR lhhl == (2.3.4) . )
. ' . .
',( ttRhl
7.
( , ')( , ') ( , 1, )
( , ') ( , ')
hlhl
hh ll
R t tr t t h l H
R t t R t t= = . (2.3.5)
. 2.3.1.
. () ' , , .
56
-
2.3.1 ()
()
, HhtDttR hhh 1, ,)(),( == -
),'(),( ttRttR lhhl = 1)
2)
1),()()(),( ttr ttttRhl lhhl
. , , , tt = .
2.4. 2.4.1.
. . - D () (. 2.4.1), , , , .
)(tx
)(ty
. 2.4.1.
. , , . 2.4.1.
2.4.1
1. , , , . 2. L ,
: 1) ; )]([)]([)]()([ 2121 txLtxLtxtxL +=+2) )]([)]([ txLtxL =
.
57
-
1. . 3. D ,
L
)(t : )()]([)]([ ttxLtxD += .
4. , , . 5. () ,
t .
)(ty t
)(tx
6. , , .
)(ty t )(tx
t
2. , . - . - () , .
1. , .
2. .
)(tX
)(tY
2.4.2. .
. . , , .
)(tX )(tY )(tX
)(tY
, , ( , ). . 1.
)(xy = , )(yx = . ) ;(1 tyf y )(tY )(tX
dy
ydtyftyf xy
)());(();( 11
= , (2.4.1) ) . );((1 tyf x )(tX 58
-
2. )(xy = , , Q
Qqyx q ,1),( == . )
,(1 tyf
y )(tY
)(tX == Qq qqxy dy ydtyftyf 1 11 )());(();( . (2.4.2) (2.4.1)
(2.4.2) ,
(1.3.1) (1.3.2). ,
, . .
)(tXG
)(tYG
3. )(xy GGG = , :
),...,( 111 Hyyx = ,
),...,( 1 HHH yyx = , H ),...,( 1 Hxxx =G ),...,( 1 Hyyy =G . H
- );( tyf yH G )(tY
G )(tXG
)());(),...,(();( 1 yJtyyftyf HH
x
H
y
H
GGGG = , (2.4.3) ) ;( txf xH G H - , )(tX
G)( yJ HG
:
H
HH
H
H
yy
yy
yJ
=
"
""""""
G
1
1
1
1
)( . (2.4.4)
. H ,...,1 , (2.4.3) . 4. 1. -
N
),...,;,...,( 11 NNy
N ttyyf )(tY
)(tX = = Nn n nNNxNNNyN yyttyyfttyyf 11111
)(),...,);(),...,((),...,;,...,( , (2.4.5) - . ),...,;,...,( 11
NNxN ttxxf N )(tX
. .
)(tX )(tY
N Ntt ,...,1
)(tXG
)(tYG ,
59
-
( )Y )(tX n (tn Nntn ,1, = ). , - , - , . -
(2.4.3), ' - - . , :
N
)(tXG
)(tYG
N
)(tX )(tY N
),...,;,...,( 11 NNy
N ttyyf )(tY
)(tX N
)(tYG
N )(tXG
=== N...1 , (2.4.3) . (2.4.3) (2.4.5). 5.
),( 211 xxy = , 22 xy = , : ),( 2111 yyx = ,
22 yx = . )
: ;( 11 tyf
y)(1 tY
))(),(()( 21 tXtXtX =G2
1
211221211
),();),,(();( dy
y
yytyyyftyf xy = , (2.4.6)
) ;,( 212 txxf x )(tXG
.
. 3.
1
1
2
1
1
1
10
)(y
yyyJ == G .
(2.4.3)
1
12212212 );),,(();,(
ytyyyftyyf xy = .
) , (2.4.6).
);,( 212 tyyfy
;( 11 tyfy
. (2.4.6) , , , (. 2.4.2).
)(tY
)(1 tX )(2 tX
. , . )(tY
1. )(xy = . )(tm y - [ ] dxtxfxtYMtm xy == );()()()( 1 .
(2.4.7)
60
-
2.4.2 '
);(1 tyfy
)(tY
);,( 212 txxfx ))(),(()( 21 tXtXtX =G
);(1 tyf y )(tY
21 xxy += );(1 tyf y = 2222 );,( dxtxxyf x 21 xxy = );(1 tyf y =
2222 );,( dxtxxyf x +
21 xxy = );(1 tyf y = 22222 );,( xdxtxxyf x 2
1
x
xy = );(1 tyf y = 22222 );,( dxxtxyxf x
. (2.4.7) , )(tY
dxtxfxtm xy = );()()( 1 . 2. )(xy = .
)(ty - . ( )[ ] [ ] dxtxftmxtmtYMt xyyy == );()()()()()( 1 .
3. , ( qQ
q
q x=1 q , Qq ,1= ). . . 2.4.3.
1. , .
)(1 tX )(2 tX
)(tY
2. QqtX q ,1),( = : )(tY
),(),(1
2 ttRttRqx
Q
q
qy= = ,
)()(1
2 tDtDqx
Q
q
qy == . 3.
,
61
-
.
)(1 tX )(2 tX
2.4.3 ,
)(tm y
),( ttR y ),( ttD y ,
)(tY
)(tmqx
)(tRhq xx
)(tDqx
)(tX q
q
Q
q
q xy == 1
)()(
1
tmtmqx
Q
q
qy == ),(),(),(
1
2 ttRttRttRhqqq xxh
Q
q hq
qxxqy+= =
),()()(1
2 ttRtDtDhqq xxh
hq
qx
Q
q
qy = +=
2.4.3.
. )(tY )(tX
, ( ) .
)(tY
)(tX
- , , , . . .
)(tX
)(tX )(Y . : [ ])()( tXLY t= , (2.4.8) , t . tL 1.
(), 1D 2D[ ][ ] [ ][ ]xDDxDD 1221 = .
1. . 2. M .
)(Y (. 2.4.4). )(tX
62
-
2.4.4
[ ])()( tmLm xty =
[ ]),(),( ttRLLR xtty = == |),()( yy RD
:
)()( txdt
dty = ,
= dttxy )()( .
3. . 2.4.5 2.4.6 , .
2.4.5
)()( tm
dt
dtm xy =
tt
ttRttR xy = ),(),( 2
ttxytt
ttRtD = = |),()( 2
2.4.6
( ) ( )y xm m
= t dt = ' ),(),( tdtdttRR xy
= tdtdttRD xy ),()(
2 . 2.4.7.
63
-
2.4.7 2
24, 40, 47, 49
3, 16, 25, 32, 40, 69, 73, 75, 84
5, 10, 15, 20, 22, 29, 61, 88, 89, 105 38, 55, 56, 66, 74, 82,
84, 86, 92, 94
64
-
3. ,
, .
' . , . () .
3.1. 1.
(), -
)(tX
)(),...,( 1 NtXtX
,)()(2
1exp
)2(
1);(
1 = xxTxx
NN mxRmx
Rtxf
GGGGGG (3.1.1) - -, - ;
T
Ntxtxx ))(),...,(( 1=G N n)( ntX
T
Nxxx tmtmm ))(),...,(( 1=G - - , -
;
N n
)( ntX )},({ mnxx ttRR = NN , ,
)( ntX
)( mtX );,1,( Nmn = , ; .
1xR xR T
, .
1. ) ) . , . , .
(tmx ',( ttRx
2. , .
. , - )
0)',( =ttRx'tt N ;( txf N GG
( ): );()...;();( 1111 NNN txftxftxf =GG . (3.1.2)
, , );( txfN GG (3.1.2). , .
0)',( =ttRx'tt
3. ,
)(tX
)),1( ),()(/( MmtxtXtX mm == . . -
MN >
65
-
),...,;,...,(
),...,;,...,(),...,;,...,/,...,;,...,(
11
11
1111
MMM
NNN
MMNMNMMNttxxf
ttxxfttxxttxxf =++ . (3.1.3)
, (3.1.3) (3.1.1). , . , . 4. -
() () .
)(tX
. , , ,
. . , . , .
0),( =mnx ttR mn tt )(tX
5. . .
6. .
7. , (3.1.1) : = GGGGGG xTxTN RmjtjQ 21exp);( .
8. .
3.2. 3.2.1.
1. . . . 1.
(), - -
)(tX
N
N 112 ,..., tttt N t , .
Ntt ,...,1
)(tX
, , .
1. - - , -
N
t N =),...,;,...,( 11 NNN ttxxf ),...,;,...,( 11 ++ NNN ttxxf ,
(3.2.1) 66
-
. 2.
, .
)(tX t
)();( 11 xftxf = 3.
)(tX 12 tt = , t =),;,( 21212 ttxxf );,( 212 xxf .
, . 3.2.1. 3.2.1
=)(tmx const=xm
const)( == xx DtD )()(),( 2121 xxx RttRttR ==
x
x
xD
Rr
)()(
=
1. ' - , t .
2>N2>N
2. (. 1 1). 1.
, , (3.2.1) .
)(tX
K
KN tKN
. (3.2.1) KN = , , ,
. 1 , .
KN < KN KN =
1. , - t , :
)(tX
N
N
),...,;,...,(lim 11 ++ NNN ttxxf . 1.
, 1 .
)(tX
T
Tt 1.
() , - ,
)(tX
N
N 0T=),...,;,...,( 11 NNN ttxxf ),...,;,...,( 0011 kTtkTtxxf NNN
++ , 67
-
,....1,0 =k
. . 2.
, -
)(tX )(tY
)( MN +N M
112 ,..., tttt N ; , , .
1
'
1
'
1 ,..., tttt M Ntt ,...,1)(tX ''1 ,..., Mtt )(tY
1. , , , =+ )',...,',,...,;,...,,,...,( 1111 MNMNMN
ttttyyxxf
)',...,',,...,;,...,,,...,( 1111 ++++= + MNMNMN ttttyyxxf .
2.
, . )(tX )(tY
. 3.
, ( ), :
)(tX =)(tmx const=xmt
)(),( 2121 ttKttK xx = . 1.
. , , . (. 3.2.1).
. 3.2.1.
2. .
3. , .
, .
68
-
4. , , :
)(tX )(tY
t
122121 ),()]()([),( ttKtYtXMttK xyxyxy === . (3.2.2) .
, . 2.
, . .
, 3.2.2. 3.2.2
() ()
| ( )|R Dx x ; 1|)(| xr
0= : xx DR =)0( ; 1)0( =xr : )()( = xx RR ; )()( = xx rr
, )()( = yxxy RR ; )()( = yxxy rr 0= , -
. , , , ,
|| , (. 3.2.2).
)(xR
0
. 3.2.2. () () . 3.5.3. 3.2.2, .
3. . , , . , , .
. , .
69
-
. , , ..
3.2.2. 1.
. . Y t t X t g t( ) ( ) ( ) ( )= + , ( ), ( )t g t , ,
X t( )
)(tg ( )t . :
);()()( tgmttm xy += R t t t t R t ty x( , ) ( ) ( ) ( );1 2 1 2
2 1= D t t Dy x( ) ( ) ;= 2 r t t r t ty x( , ) ( ).1 2 2 1=
2. , . , . 1.
X t( )
]))()([(),( 22121 tXtXMttB xx = . (3.2.3) 2.
, X t( ) )()()( tXtXtX += ,
consttXMm == )]([ , (3.2.4) 2)]()([)( mtXtXMR += . (3.2.5)
1. .
2. ;0=m (3.2.6)
)()()(2)( += xxx RRRR ; (3.2.7) )]()0([2)( xxx RRB = ,
(3.2.8)
(3.2.3) (3.2.5). 3.
, .
3.2.3. , ,
, . 1. )(xfy = (
) 0),,( = yx , , .
1.
70
-
=
=0
),,(
,0),,(
yx
yx
(3.2.9)
. 2. ,
. 3. (3.2.9) ,
. 4. , ,
, yx, , , 0,0 ''
'
'
yy
xx. (3.2.10)
. fttAy 2cos)(= ( , )
f t
)(tAy = )(tAy = , t
kf =
,...).1,0(2
12 =+= kt
kf
1. ),(cos)( ttAy = , ,
)(tA
t ),( t ,t , )(tAy = )(tAy = .
. )(cos)( ttAy = , , )(tA )(t .
2. k
)(
xR ,
( ) , .
. :
1) , ( ) ( 0,1);
2) , )(xr )( x : ( ) drxk =
0
, (3.2.11)
.)( 0
dxk = (3.2.12)
.
71
-
3.2.4. 1. ))(),...,(()( 1 NtXtXtX =G
, , ) ( )
,( 21 ttK xG
12 tt = .
. . 3.2.3.
3.2.3
)()( lhhl RR = lhhl DDR )(
1)( hlr
3.3. ()
, . . . . 1. () ,
- , , , , - . 1.
. 2. ()
, ( 3)
=22
)(1
l.i.m.
T
T
xT
mdttXT
. (3.3.1)
2. , , )
)(tX
xm
,( 21 ttRx
=22
2
2
212120),(
1lim
T
T
T
T
xT
dtdtttRT
. (3.3.2)
. 1. (3.3.2)
0),(lim 2112
= ttRxtt , (3.3.3) 72
-
)
,( 21 ttRx
12 tt . (3.3.3) ,
(3.3.2). 2. (3.3.2)
: 0)(-1
1lim
0
= T xT dRTT . lim ( ) =Rx 0 . 3. 2 . ,
. , , . 3.
X t( )
Rx ( ) , Rx ( ) dttXtX
T
T
TT
)()( 1
l.i.m.02
2
0 += . 3.
Rx ( ) =++ T xxxT dRRRtT 0 002 0)]()()([-1 1lim - 0 .
3.4.
. . 1.
) :
),( 21 ttK x
(tX
)].()([),( 2*
121 tXtXMttK x = (3.4.1)
2. ) :
),( 21 ttRx
(tX
))]()())(()([(),( 2*
2
*
1121 tmtXtmtXMttR xxx = . (3.4.2) 1.
: )()(),(),( 2
*
12121 tmtmttRttK xxxx += . (3.4.3) 2. ,
: . 1 2 1 2( , ) ( , )x xK t t R t t=
73
-
3. ) ,
,
(tX1t 2t
0),( 21 =ttRx)()(),( 2
*
121 tmtmttK xxx = . (3.4.4) 4. )
, ,
(tX1t 2t
0),( 21 =ttK x)()(),( 2
*
121 tmtmttR xxx = . (3.4.5) 5.
) :
),( 21 ttK xy
)
)
)
(tX (tY
)].()([),( 2*
121 tYtXMttK xy = (3.4.6)
6. ) :
),( 21 ttRxy
(tX (tY
))]()())(()([(),( 2*
2
*
1121 tmtYtmtXMttR yxxy = . (3.4.7) 1.
) :
(tX (tY 1t 2t
),(),(
),(),(),(
2121
2121
21ttKttK
ttKttKttK
yyx
xyx
= , (3.4.8)
),(),(
),(),(),(
2121
2121
21ttRttR
ttRttRttR
yyx
xyx
= . (3.4.9)
7. ) , : , (3.4.9) .
)
)
(tX (tY1t 2t
),( 21 ttRxy 0),( 21 =ttRxy
8. ) , : , (3.4.8) .
(tX (tY1t 2t
),( 21 ttK xy 0),( 21 =ttK xy
9.
)()(
),(),(
21
2121
tDtD
ttRttr
xx
xx
= , (3.4.10) )()(
),(),(
21
21
21tDtD
ttRttr
yx
xy
xy
= . (3.4.11)
. .
. , . . 3.4.1.
74
-
3.4.1 ,
, ,
),( 21 ttRxy ),( 21 ttRx
),( 21 ttrxy ),( 21 ttrx)(tX )(tY
/ 1 ),( 21 ttRxy = ; = ),( 12* ttRyx ),( 21 ttrxy ),( 12*
ttryx
)()(|),(| 21
2
21 tDtDttR yxxy ; 1|),(| 21 ttrxy( 1 1( ) ( )Y t aX t b= + ,
) a
b
2
3
Ntt ,...,1 Nzz ,...,1*
1,
),( nmnm
N
nm
x zzttR =
. 1, 3 , 2 .
3.5. 3.5.1.
1. T , :
x tT ( )
== )2exp()( tTjAtxT , (3.5.1) =
T
tT
j
T dtetxT
A
2
)(1 . (3.5.2)
(3.5.1) (3.5.2) () v
T
v =2 . A - AA = - , )arg( A= - . 1. A , ,
; ; - .
A 2. x(t) ,
: deAtx tj= )(21)( , (3.5.3)
75
-
dtetxA tj = )()( . (3.5.4) . ,
x(t), x(t) , x(t) [ )0()0(2
1 ++ txtx ] , x(t) - x(t) ( ).
)(tx . . 1. ,
(. 3.5.1). )(txT
)(tx
. 3.5.1. () ()
2. , )(A , )( ,
. )()( * = AA. (3.5.3) (3.5.4) f 2= ( f
) , . , .
)(tx )(A
)(+A 2.
, )(+A )(tx
=>=+=+ .0)0( ,0)(2)()()( A AAAA (3.5.5) 1.
: )(tx
)(tx =0
1( ) cos ( )
2A t d + + + ,
; : )(+A )( +)(+A = )(+A , =)( + ( )arg ( )A + ,
76
-
=>==
++ +
.0)(
,0)(2)()( )(
dttx
dtetx
eAA
tj
j
2. T :
)(txT
)(tx =0
2exp( )A j t
T
+ += + , +A + : +A = +A , =+ ( )arg A+ ,
=>==
++ +
.0)(1,0)(2
2
T
T
T
tT
j
T
j
dttxT
dtetxT
eAA
3.5.2.
. 30- . .. . 1. ()
, ' ':
)( fAx
)(tX )( fAx )(tX
dtetXfA ftjx2)()( = . 1.
, .
1. )
)
( fma
)(tX
xm ()( fmfm xa = , )( f -.
2.
)()(),( 12121 fffSffK xa = , (3.5.6) , ( ).
deKfS fjxx 2)()( = 3. (3.5.6) ,
. 4. ,
, .
77
-
5. , , , , . 2. ,
) ,
)(tX ( fS x
)(xK ' ': deKfS fjxx 2)()( = , (3.5.7) dfefSK fjxx
2)()( = . (3.5.8) (3.5.7), (3.5.8) . . )
)
( fS x
1. (3.5.7) (xK , ) ) :
( fS x
)(tX )(0 fSx
(0
tX xm
)(tX
)( fS x = + , (3.5.9) )(0 fSx
)(2 fmx .
)( fS x )(0 fSx
)
2. (3.5.8) ,
dffSRDx
xx )()0( 0== , (3.5.10)
. )(0 fSx
. (3.5.10) , .
)(0 fSx
3. ) , . .
)( fAx
( fS x
4. , ' )
)(0 fSx
(0 fAx
(0 tX
)(0 fSx
=TT
1lim )]()([ *00 fAfAM xx , (3.5.11)
78
-
T . )(0 tX . 1.
( ) , ) 0. (3.5.6) (
)(tX
( fS x
fff == 21 ). 0),( ffK a 2. )(xK
, , (3.5.7), , =
)(tX
)( fS x )( fS x . 3.
(3.5.7), (3.5.8) ) :
)(tX
)(0 fSx
(0
tX
dfRfS xx
2cos)(2)(0
0 = , (3.5.12) dfffSR
xx 2cos)(2)(
0
0= . (3.5.13) 4. (3.5.12), (3.5.13), (3.5.5),
, :
)( fS x+
=>=
+
,0)(2
,02cos)(4)(
0
0
fdK
fdfK
fS
x
x
x
(3.5.14)
dfffSK xx 2cos)()(0
+= . (3.5.15) . (3.5.14), (3.5.15)
, (3.5.7), (3.5.8), (3.5.12) (3.5.13) . 3.
, :
)(0
tX
)(0 fsx
)(0 fSx
)(0 fsx
=0
0 )(
x
x
D
fS, (3.5.16)
. 0x
D
1. (3.5.7), (3.5.8),
79
-
derfs fjxx
2)()(0
= , (3.5.17) dfefsr fj
xx
2)()( 0= , (3.5.18) )
)(0 fsx
(0
tX )(xr . 2.
: )(0 fs
x
0)(0 fsx
, , 1)(0 = dffsx )(xr . , .
3. , , . , 2, . 3. ( )
, : ef
dffSS
fx
e )(1
00
0= , (3.5.19) . )(max0 fSS x
f=
5. ef k :
k
ef 1= . (3.5.20)
3.5.3. . 1. ,
0ff e
-
. 3.5.2. -
2. ( , , . . 3.5.2), )
0f
(xR )(cQ 02cos f : 02cos)()( fQR cx = (. 3.2.2, ). 2.
,
)(tN
constN
fS x ==2
)( 0 . (3.5.22)
. 1. -,
)(
2)( 0 NK x = . (3.5.23)
(3.5.6). 1.
, (3.5.23) )(xR .
2. (3.5.23) 1 , , , , 0=k .
2. . (3.5.19), (3.5.22).
3. . (3.5.23).
. () , . , . 3.
, , .
)()()( tNtXtY = )(tX)(tN
1.
81
-
)(2
]|)([|),( 1202
121 ttN
tXMttRy = . 2.
)()()( tNtXtY += )()()()( 21 tNtXtXtY += , , .
)(tX )(1 tX
)(2 tX
3.5.4. , ,
. 1. ()
,
)(tX )(tY
)( fS xy
)(xyK : deKfS fjxyxy 2)()( = , (3.5.24) dfefSK fjxyxy
2)()( = . (3.5.25) 1.
, .
)( fS x
)(tX )( fS xy
2. ) , .
( fS x )( fS xy
)()( * fSfS yxxy = . (3.5.26) (3.5.24).
3. , )()(|)(| 2 fSfSfS yxxy . (3.5.27)
2. )(tX )(tY
)()(
|)(|)(
2
2
fSfS
fSf
yx
xy
xy
= . (3.5.28) 1. (3.5.27) ,
. , ,
)(2 fxy ]1,0[)(tX )(tY 20 ( )xyf f 0 = ,
, . 1)(2 =fxy 2.
. . , , , .
)(2 fxy)(2 xyr )(2 xyr
)(tX )(tY )(2 xyr 3.
, , .
)(tY
)(tX
)(2 fxy 82
-
, .
)(tY )(tX )(tY
)(1)( 2 ff xyxy = )(tY
3.6.
3.6.1. 1.
, : 111 ),()()( dttthtxty = , (3.6.1)
, , , .
)(tx )(ty ),( 1tth
1. () -: ),( 1tth
11122 ),()(),( dttthtttth = . (3.6.2)
. 2. -
(3.6.1). 3. - .
. . ),( 1tth 2. ( ),
. .
1. . , .
2. ,
),( 1tth
1, tt )(),( 1 htth = , 1tt = . (3.6.1) : dhtxdttthtxty
)()()()()( 111 == . (3.6.3) 3. ,
. 4. ,
- . . dhtxdhtxty |)(||)(||)()(||)(| = , (3.6.4)
)(h 83
- :
-
1. , , ,
)
)(2 K)(2 K
)
)(2 K )
(K)(th
)(1 K)(1 th )(2 th
)(K = , (3.6.11) )(1 K11211 )()()( dttththth = .
2. , , ,
(K)(1 K (2 K
)(K =)()(1
)(
21
1 KKK + . (3.6.12) 3.
)(h )(g . 5. )(g
, . . )(g
)(h : dhtg t )()(
0
= , dttdgth )()( = . (3.6.13) 4.
) , :
(K )( pH jpdehpH p +== ,)()(
0
. 1. , ,
) (
(K)( pH jp = ).
2. , ) ( pH 0 . . 5.
0 0
( ) ( )... ( ) ... ( )
M N
M NM N
d y t d x ta a y t b
dt dt+ + = + + b x t ,
, ma nb ),1,,1( NnMm == , , : , ,, . )0(y )0'y )0()1( My
3.6.1. - : ,
85
-
, , , . .
, . )(th )
)
(K - ,
(K
)(th
e . 3.6.1. 3.6.1
)(K
)(th
0K || 0 >||
t
t
K
)sin(
0
0K
+
+2
02
0
20
2
0 t
t
t
K
0
0
cos
2
)2
sin(
RC - j+ tet )(1 2
)( 0 + j t0cos et
t)(1
22
21 e
2 2)(1 t 2e
22t 22
22
21 e
20 )(
t
et
t
2
cos
2
)(122
0
24
86
-
: 1. =
1. . 3.6.3 , , , . ,
87
-
, , ( t). , , T, 1, 2, 3 . 3.6.3 :
)(th
0t
= Txy dhmm0
)( , = T T xy ddhhRR0 0
212112 )()())(()( , 1 1
0
( ) ( ) ( )
T
xy x 1R R h d = . , , .
Tt >0 2. 4 . 3.6.3 ,
. Tt >0
3.6.3. (3.5.23), ,
: 1
0
1
0 )()(2
)( dhhNRy += , )(2)( 20 KNS y = , )(
2)( 0 hNRxy = .
)(yR , yD)(yS
. 3.6.4. 3.6.4
)(yR ,
yD
)(yS )(
yR
yD
)(
yS
)sin(yD 2 020 NK 2 02
0NK
|| 0 >||
0cos2
sin
yD
2
2 020 NK 2
0
2
0NK
+
+22
22
00
00
0 RC- eDy
4
0N
)(2 22
2
0 +N
88
-
0cos
eD
y
2
0N
))((2 2
0
2
2
0 +N
2
22 eD
y 24 0N 22 204 eN
i 02 cos22eD y 22 0N 22 2)0(04 eN .
RC- ' . ' .
, 4 , , . 3 . 3.6.5.
3.6.5 3
3, 24, 40, 47, 49 16, 32, 42, 73, 75, 84, 100 5, 10, 15, 20, 23,
61, 88, 89, 105 27, 33, 35, 37, 38, 54, 55, 56, 66, 72, 74, 82,
85,
86, 92, 94, 97, 104
89
-
4. 4.1.
1. , )(tX Tt ,
- Nttt
-
, , ,
NX Nx
110 ,...,, NXXX 110 ,..., Nxxx},...,/{ 10 NN xxxP = . }/{ 1NN
xxP
. 4.2.1.
2. ,
),...,/( 101 NN xxxF = ,1,),,...,/( 11 += MMNxxxF NMNN . 3. -
NX
G
,..., 10 XXGG
,
),...,/( 10 NNM xxxF GGG = )/( 1NNM xxF GG . 1.
- ),...,( 1 NMNN XXX +=G . 2 3. .
. 2. N-
},...,{
00 NlNlXXP == = }{
00 lXP = = ==Nn lnln nn XXP1 1 }/{ 1 .
1. . 4. k mt l
(n m) nt},/{),( kmlnkl XXPnm ===
k , Sl . k l
( k , l LlkS ,1,; = ) , .
mt nt
),( nm LL 91
-
5. , : 1) '; 2) - . 3. . 4.
) (mPk k m- ),1( Lk = ),( nmkl l n- . ) (nPl l n-
)0;,...,1(),,()()(1
nmLlnmmPnP kl
L
k
kl == = . (4.2.2) (4.2.2)
),()()( nmmPnP = , P(n) - L , )(nPl Ll ,1= , n- . 5. k m- l
n-
kl(m,n), k m- q r- ( nrm
-
6. P= P(N). Nlim
. . 7. , : 1) ; 2) ,
( ). 0t.
. 8.
, P(n) ,
nt
nt PnP =)( n. 8. ,
PP = . (4.2.6)
1. (4.2.5). 2. (4.2.6) ,
P .
3. .
. 4.2.1.
4.2.1
)(),( mnnm =
NPNP )0()( = PnP =)( n
PP = 1) )(lim NPP
N =2)
P
. - . 4.2.2.
l
. . 9. l ,
- , - l - . l ,
93
-
- , - . 10. l ,
, l , , ,
. 11. l , k ,
l k , k l , .
. 4.2.2. -
: 1 ; 2 ; 3 . l
12. , . 13. r
, ...,, 21 nn l . l r , , , .
1>r1=r
14. l , , . . 15. (),
. . ( ,
) ( ). L
4.3. 1. )(),...,(
00 NNtXXtXX ==
)(tX Ntt t
-
1. (4.3.1) , :
)(tX
= = Nn nnnnNNN txtxtxfttxxf 1 1100100 );/;();(),...,;,...,( ,
(4.3.2) = ) .
);/;( 11 nnnn txtx ;/;( 111 nnnn txtxf1nx 1nt nx nt
2. (4.3.1) , - , . . 4.3.1.
4.3.1
' 0);/;( txtx = 1);/;( dxtxtx )();/;(lim xxtxtx
tt=
= xdtxtxtxtxtxtx );/;();/;();/;( 0000
. .
1. )
)(tX
;/;( 00 txtx 0x 0 t x t
0 001 00000 );/;(! );();/;( x txtxtx Atxtxt = = , (4.3.3) [ ]
));/();/((1lim);( 000000
000 txtXtxttXM
ttx A
t+= .
. );/();/( 000000 txtXtxttX + , t .
);( 00 tx A - .
1. : .
)();();( 00 tXttXtX + ++= xdtxttxttxtxtxtx );/;();/;();/;(
000000 . (4.3.4)
++=+ xdtxttxttxtxttxtx );/;();/;();/;( 0000000 (4.3.5) 95
-
( , ). (4.3.4) (4.3.5), :
[ ] .);/;();/;();/;( );/;();/;( 000000 0000 xdtxttxttxtxttxtx
ttxtxtxtx +++= =+ (4.3.6) (4.3.6) :
.);/;(
!
)'(
);/;();/;(
0
00
1
0
000
x ttxtxxx
ttxtxttxtx
+==++= (4.3.7)
(4.3.7) (4.3.6), ( ). (4.3.3).
t0t
2. (4.3.3) , 3 , () :
),;/;();(2
1);/;();(
);/;(
002
0
2
0000
0
00
00
0
txtxx
txbtxtxx
txa
txtxt
+==
(4.3.8)
) , .
;();( 00100 txAtxa =);();( 00200 txAtxb =
3. , (4.3.8), [ ] [ ]);/;();(2
1);/;();();/;( 002
2
0000 txtxtxbx
txtxtxax
txtxt += (4.3.9)
.
1. (4.3.8) (4.3.9) . 2. (4.3.9)
[ ] [ );();(2
1);();();( 12
2
11 txftxbx
txftxax
txft += ] . (4.3.10)
. - 0tt >0000011 );/;();();( dxtxtxtxftxf = .
, (4.3.9) ) , , (4.3.10).
;( 001 txf 0x
4. , (4.3.8), (4.3.9), . 5.
, ) ,
;/;( 00 txtxt 0t 0tt = : );/();/;( 000 xxtxtx = .
1. . a b
96
-
2. (4.3.10) , (. 3.2) [ ] Cxfxaxfxb
dx
d += )()(2)()( 11 , (4.3.11) C , . 2.
, . . .
);/;();(),;,( 112211121212 txtxtxfttxxf = . );,(),;,( 21221212
xxfttxxf = ,
, )();( 11111 xftxf = 12 tt = . , );/();/;( 121122 xxtxtx = , .
6.
, : 1) ; );(lim)( 11 txfxf
tf =
2) .
1. .
2. .
4.4. 4.4.1. ()
1. )(),(),( tntxgtxh
dt
dx += , ,
),( txh ),( txg
yxLtygtxgtyhtxh + ),(),(),(),( )0( >= constL , )(tn . ,
. (. 4.4.1) m.
. . . t . , , ,
)(tn
- ),( ttR , ,
)(tn[ ] ;0)( =tnM . (4.4.1) )(
2
1),( 0 ttNttR = . (4.4.2)
97
-
)(tv
)()(
tndt
tdvm = . (4.4.3)
, 0=t , , (4.4.3) , .
. 4.4.1. ,
2. , )(tv
),()(
tndt
tdv = 0)0( =v , (4.4.4) . )(tn
1. (4.4.4) , 1
0
1 )()( dttntv
t= . (4.4.5) 2. . 4.4.1. . ,
, . 3. , ,
,
)(tX T
Nttt
-
4.4.1
/ 1 [ ] 0)( =tVM 2 [ ]
2)()()( 021
0
21
0
2 tNddnnMt
t t
v == 3
2
1
00
1( ; ) exp( )
vf v t
N tN t= 4 dntvtv t
t
+= 32
)()()( 23
5 0);( =tva 6
2);( 0
Ntvb =
7
0 1 0
1 0 0 1 1 1
( ,..., ; , , )
( ; )... ( ; )
N N N N N
N N N N
f v v v t t t
f v t f v v t t
1 == 8 [ ] )())(),...,(/( 110 = NNN tvtvtvtVM : 1. 3 , ,
' . )(tn )(tv )(tn
2. 4 , , .
3t
2t
2t
3. 5 (4.4.5) (4.4.1), 6 (4.4.5) (4.4.2).
4. 7 , .
5. 8 4 .
4.4.2. . ,
)(tX
)()()(
tntxdt
tdx =+ , (4.4.6) , , , .
)(tn
. , , , (4.4.6) .
)(tX
)(tn
, , . )(tX , (4.4.6),
, C . (4.4.6) tCetx =)( dneetx tt =
0
)()(
(
99
-
(4.4.6)). (4.4.6) 0)0( xx = dneeextx ttt +=
0
0 )()( . (4.4.7)
0, 1123 =>> tttt . (4.4.7) , dneeetxtx t
t
ttt)()()(
3
2
323 )(
23 += . (4.4.8) (4.4.8) ,
, , .
)(tx 3t
1t 2t
. 4.4.2.
2. = eR 2)( .
4.4.2
textm = 0)(
[ ]tet 222 1)( = , 4 022 N=
0),min(;
),1(),(
2112
22
21 == = ttttt eettR t
( ) t = eR 2)( ( ; ) ( )a x t x t=
2);( 02
Ntxb =
. , , , . 4.4.2.
. 4.4.2. , ,
() )(tm )(2 t
)(R () 100
-
1.
2
1
2
0
2
11 );(
2)];([
);(
x
txfNtxxf
xt
txf += . '
= )(2 ))((exp)(2 1);( 22
1t
tmx
ttxf .
2. .
= 22
12
exp2
1)( xxf .
4.5. ' 4.5.1. '
(4.3.8) (4.3.10) . '
. 1.
) ,
;/;( 00 txtx);(1 txf x .
2. . 1. [ );();(
2
1);();();( 11 txftxb
xtxftxatxG ]= (4.5.1)
. 1. '
, t x . ( )
);(1 txf
x . - , . , , ,
t
);();( 1 txftxa
[ );();(2
11 txftxb
x ] . 2. : 1) .
; 0);();( 21 == txGtxG
1x 2x
2) . .
0);();( 2111 == txftxf1x 2x
2. 0);();(1 =+ txGxtxft . (4.5.2)
101
-
1. (4.3.10), .
2. , , , , .
);(1 txf
);( txG
3. (4.5.2) , consttxG =);( . (4.5.1) [ ] Gxfxaxfxb
dx
d2)()(2)()( 11 = . (4.5.3)
. ' (4.5.3) = xx zxxx dzdyyb yaxb Gdyyb yaxbCxf 1 11 )(
)(2exp)(2)( )(2exp)()(1 . ,
, , .
G
C 1x )(1 xf )(1 xf
. ' . , , , , , , , . , .
4.5.2. ' '
, . '
(4.3.10) 2
0
01
2
0
0101
)0,(,
)0,(),0,(
x
xf
x
xfxf
0),(),( 11 == tdftcf . '
. . ;
dxct ,0xt tntn =
,cxixi += === xcdIIin ;,0,...;1,0 , [ ]* . . );(1 txf ,
(4.3.10)
:
102
-
).;();();(2
1
);();();();();(2
1);(
12
2
112
2
1
txftxax
txbx
txfx
txatxbx
txfx
txbt
txf
++ +=
(4.5.4)
: );(1 txf
t
ff
t
txf ninitx ni + ,1,,1 );( ,
x
ff
x
txf ninitx ni + 2);( ,1,1,1 ,
2
,1,,1
,2
1
2 2);(
x
fff
x
txf nininitx ni + + ,
. ),(1, nini txff = , (4.5.4)
: );(),;( txbtxa
0,,1,,,,1,1, ++= ++ nffff ninininininini , (4.5.5) ninini ,,, ,,
, . );(),;( txbtxa (4.5.5)
. ),(1 txf '
x t '. ' , , , , '. , . ' ( ). . , 0);( txb
);(
2
txb
xt
-
, , ,
SxN )}
)({ ktXP
(,),( 10 NtXtX Sxx N 10 ,,
},...,;,...,/;{ 1010 NNNN ttxxtxP = . ;/;{ 11 NNNN txtxP 1.
= })(tPk = k t ),1( Lk = )',( ttkl l ' . = }t )'(tPl )'({ ltXP =
l : 't
)'(tPl ==Lk 1 )(tPk )',( ttkl . (4.6.1) . 4.6.1. 1. 1 4
, 1, 2 4 .
2. , , , - t , .
4.6.1
/ 1 ' )',( ttkl 0
2 = ==Ll kl Lktt1 ,1,1)',( 3 klkl
tttt =+ ),(lim 000 *)
4
)',( 0 ttkl = , =Lq qlkq tttt1 0 )',(),( '0 ttt
-
2. , (4.6.2), () .
)(taql
2. '
== L
lq
qqlll tata
1
)()( . (4.6.3)
2 (4.6.3). )',( ttql ),1,( Llq = ' . ,
t t
ttt +=' ttattt llll ++ )(1),( ; (4.6.4)
ttattt qlql + )(),( ( lq ). (4.6.5) (4.6.4), (4.6.5)
, , t , , , (4.6.2). (4.6.3) (4.6.4), (4.6.5) . 3.
: 0
1
000
0
,,1,,),()(),( ttLlktttattt
L
q
qlkqkl >== = . (4.6.6) 3 2. 4. )
' (tPl l
t Lq ,1= : t
LltatPtPdt
d L
q
qlql ,1,)()()(1
== = . (4.6.7) . (4.6.2)
. (4.6.1), (4.6.7).
)( 0tPk
k
. (4.6.7) , ' )
.
L
(tPl
)(taql
3. , )',( ttkl , ' : t t
)',( ttkl = )( kl , tt = ' .
1. (4.6.4), (4.6.5) , : . constakl =
2. (4.6.2), (4.6.6) :
105
-
LlkaL
q
qlkqkl ,1,,)()(1
== = , Llka
L
q
qlkqkl ,1,),()(1
== = . 4. ,
=)(tP },1),({ LltPl = , , .
)(lim tPPt =
. . 5.
, : =
)(tP
)(tP P .
1. .
2. .
4.7.
' , , , , , , , . ' . , , ' ,
. - , . , , , ,
. , ' . , , .
. 4.7.1 , .
4.7.1 ,
),...,/( 101 NN xxxF = ,...2,1),/( 11 = NxxF NN
),...,/( 101 NN xxxF = ,...2,1),,...,/( 11 = NxxxF NMNN
106
-
. 4.7.1
),...,;,...,/;( 10101 NNNN ttxxtxF = );/;( 111 NNNN txtxF
),...,;,...,/;( 11111 NNNN ttxxtxF = );/;( 111 NNNN txtxF
0xm = , = eR 2)(
textm = 0)( ,
0),min(;),1(),( 211222
21 === ttttteettR t
()
-
4.8. ,
)(tX
T Tt ,
S
),1( LlSl = , ...
10
-
),1,( Lkj = . ) (twt i ),1( Li =
),()(1
tftw ik
L
k
iki == . (4.8.1) .
. . (4.8.1) ,
iT
m
i k ),1( Lk =
,1
iki T
L
k
ikT mm == (4.8.2) . dttftm ikTik =
0
)(
2. jk
)(tf jk
),1,( Lkj = . )(tij ,
t
j , 0
0 =ti , dtftt kjL
k
t
ikikiijij )()()()(1 0
+= = , ),1( Lji , (4.8.3)
,
{ tTPt ii >= )( } ti : =
t
ii dttwt )()( ,
)(twt i . .
. - . )(tij
1. (4.8.3) , (
t
j = i ), , , t
-
,)(
1k
j
T
L
k
k
Tj
ij
mP
mPt == (4.8.4)
),1( LkPk = k ,
jTm
k , (4.8.2).
. ' (4.8.4) t i , . 0t 4 . 4.8.1.
4.8.1 4
24, 40, 47, 49 32, 41, 42, 73, 77, 78, 99 7, 10, 15, 20, 22, 30,
61, 87, 88, 89, 105 6, 38, 57, 82, 85, 86, 92, 94, 95, 98, 104
109
-
5. 5.1. 1. ,
. . . 1. :
; ; .. 2. (,
) , . 3. , ,
, , , . 4. ,
, , . 2. '
, ', , '.
. . 1.
(. 5.1.1): ),...;;,(
111...1 MMMkkttttttP
M++ ,
, ],[111ttt + , , ,
, , , ; ],[
MMMttt +
Mkk ,...,
1
N - ;
),...,(1 NN
ttfN
tt ,...,1
N - ),...,( 1 NNf 112211 ;...;;0 === NNN ttttt .
. 5.1.1.
5. , ),;;,( 111...1 ++++++ MMMkk ttttttP M Mkk 1
110
-
[ ] ++++++ MMM tttttt ,;;, 111
M , =++++++ ),;;,( 111...1 MMMkk ttttttP M
.),;;,(...1 MMMiiikk ttttttP M ++ 6. ,
),( tttPk
+ ],[ ttt +
: )2( k ),(
1tttP +>>+ ),(
1tttP ,....3,2 ),,( =+ ktttPk
. . 7. (
), - , , , , , ,
),;...;,( 111...1 MMMkk ttttttP M ++ = . = +Mm mmmk tttP m1
),(
. 5.1.2.
. 5.1.2.
1. :
),...,( 1 NNf = . =Nn nnf1 )(1 )( , (. 5.4).
2. .
3. 1. , ,
, , . 2. , , ,
, , .
111
-
3. , , , : .
. ' . ' T , . 2.
, , , (. 5.1.3):
;
),(21
ttm ),(21
ttD
],[21
tt
n ;
nm nD
1n )(t , ; )(t , , . .
),( Tttm +
),( TttD + t , ),( Tttm + = , = . )(Tm ),( TttD + )(TD 8.
)(t
t ),0()( tmtm = ],0[ t :
t
tmttmt
t += )()(lim)( 0 .
. 5.1.3. ()
112
-
9. ( ) )(t
= +1 ),(k k tttPt
: 0tt
tttP
t kk
t += = 10 ),(lim)( .
1. , , == )(;)( tt .
2. , )(t )(t . 3. )(t , )(t .
5.2. 1.
. 1. .
, )(TPk
],0[ T (. 5.2.1), :
k
.!
)()( T
k
k ek
TTP =
. 5.2.1. K ],0[ T
2. . (. 5.2.2): 1) ; 2) 0
; 3) )(1 nnnn tt += + ,
, , 1+nt
nt ],[ nnn tt +
, )(),(),( 1011 nfff : = ef )(1 .
. 5.2.2. : - ; 0 ;
n
113
-
, , . 5.2.1.
5.2.1
T TTm =)( T TTD =)( , 0 , n 10 === nmmm , 0 , n 210 === nDDD
3. . . :
, .
5.3. 1. , M
),,1( Mmm = .1
== Mm m 1. ,
. , , M , M .
2. , 5 10 . 1.
. 2. )(t . ,
, : ),(
21ttP
k],[
21tt k
),(2121
21
!
),(),(
ttak
k ek
ttattP
= , . = 2
1
)(),(21
t
t
dtttta 3.
( ],[
21tt
)(t ).
],[21
tt
],[21
ttt :
cpef cp =)(1 ,
cp
)(t , . 114
-
. )(1 f . . 5.3.1.
5.3.1 ],[
21tt
( )
],[
21tt
),(),(2121
ttattm = ],[
21tt ),(),(
2121ttattD =
cp
m 1=
21
cp
D =
5.4. . ,
1,2,...)=( nn ,
),...,( 1 NNf = . =Nn nnf1 )(1 )(. ,
. . . . .
5.4.1. 1.
, , . )()()( 1)(1)2(1 fff N ===
1. :
N ,,
1
, . 2. (.
1 3 5.3). 1. 5.4
, . , .
2. N . , , L
115
-
, . . .
11,..., NLL
3. - N == Nn nNN fff 2 11)1(11 )()(),...,( , ) (
( 1)1(
1 f0=t , ,
), 1
t
)(1 f (. 5.4.1).
4. )( 1)1(1 f )(1 f , , .
. 5.4.1. : 1
, , ; 0t= t1
- ; 0 ; n .
1. )( 1)1(1 f )(1 f ' : = 1
0
11
)1(
1 ])(1[)(
dff ,
m1== . 2. 0
, (. . 5.4.1).
)( 0)1(
1 f 1. n
, , ]
1+nt,[ nnn tt + (. . 5.4.1),
. ) , )(~1 nf ( 1)1(1 f)(1 f .
116
-
. . 3 ().
, . . , ( ). , .
. , , , .
5.4.2. 1. - ,
, k (. 5.4.2).
k
. 5.4.2. ( 3- ): ; n
1. ,
k
, ,
k : == kn n 1 ; nef n
=)(1 , 0n . 1. 1, 1 . 2. . 3. :
, . k
k 1=k . , k . 1. - k
0,
)!1(
)()(
1 = ekeek eekf , . 2. -
k )(
,TP
kl
l ],0[ T T
kl
lkn
n
kl en
TTP +== 1)1(, !)()( .
. 5.4.1.
117
-
5.4.1
km ke =, == kn nkm 1, 1
2, kD ke = == kn nkD 1 2, 1 k
= 11
1
= = kn n
. .
k
, . ,
;
2
D
mk =
1
m = , . m D
5.4.3. 1. ,
k
,
k
knn 1,=, , nnef
nn
=)(1
.
1. ,
k
k constn
== . 1.
k
==
=
=
k
nk
inin
ni
k
nnk
nef
1
1,
1 )(
)(
.
. 5.4.1.
118
-
5.4.4. . ,
: = + (. 5.4.3).
. 5.4.3.
1. ' , ' . , . .
2. : 1) :
-
. 5.4.4.
)(),( tPtttP kk =+ ),( 21 ttPk
===
N
n
n
n
NN
f
f
1
)(
1
1
)(
),...,(
N
N
kk
kk
PP
P
...1
1...= =
)(
)(
1
)(
1 ff nn= =
= i
ke =
1-
k -
-
M
N
N
kk
kk
PP
P
...1
1...= =
5.5. ' ,
. ,
. , , ' , INTERNET, , , , , , , . , , ,
, : .
120
-
. . . .
. '
, , , . ,
. 5 . 5.5.1.
5.5.1 5
24, 40, 47, 49 16, 32 20, 23, 88, 89, 105, 109, 113 38, 79,
83
121
-
6. 1. -
(). 2. ,
, .
6.1.
6.1.1. 1.
(). 1. , .
( ) .
. ,
( ), .
X
nx )(tX
nx t ,...2,1=n 2. ,
, , . , . 2. . 3. ,
, .
Nxx ,...,1
N
4. . 5. ,
. N
6. , , , .
Nxx ,...,1 )(1 xF
)(1
xF
7. , , -
N
N
G = = ,
.........
...
...)2()2(
1
)1()1(
1
N
N
xx
xx
),,(1 N
XX
( ). 1. ,
G , xG , G .
122
-
2. (
),,(1 N
XX n
X Nn ,1= ) , . )(
1 nxF
3. - )N (xFN G G
:
)(1 n
xFn
x
)(xFNG
= . =Nn nxF1 1 )( 8.
xLG
G . . ,
xLG
1( )nf x nX
G:
xL G = . (6.1.1) =Nn nxf1 1 )(
, .
)(1 n
xf
nX
. . XG . 9.
),...,(1 N
XXYY = XG . . .
, , .
),...,(1 N
XXY
xL G
)(1
xf
10. (), .
1. , .
2. , , , ,
)(1 xF
m D . , , , ,
)(*
1 xF*m
*D . 3. ,
, . , .
123
-
6.1.2. . .
1. () , (
nx Nn ,1= ).
2. , :
j
jN jx N
N
Nj
j= .
1. , , (
j
G ), .
2. , .
jp X
jx
3. : j
1) =1;
jj
*2) N .
j
jp
3. :
jx
j Jjx
jj,1),,( = .
4. , ' ( , ),
jx *
j j = J,1 .
1. (6; 3; 5; 2; 3; 2; 5; 2; 5; 5). , , : (2; 2; 2; 3; 3; 5; 5;
5; 5; 6). (2;
10
3), (3;
5
1), (5;
10
4), (
10
1;6 ).
. 6.1.1.
. 6.1.1.
. , J . , , , , .
jx
124
-
5. X, .
1. X , . , 10 20.
2. , . , ,
: .
3. , , , . 6. , -
, , - , : =
*
ip i
iN
i N *i
pN
Ni
.
7. , . 8.
i -
*
if
*
ip
ix : =*
if
i
i
x
p * . 1. ,
, , , .
2. .
*
if
9. , . ,
- . i *i
p
. . .
2. 100 . (. 6.1.1), (. 6.1.2) (. 6.1.3).
6.1.1
[-4,5:-3,5]
(-3,5:-
2,5]
(-2,5:-
1,5]
(-1,5: -
0,5] (-0,5: 0,5]
(0,5:
1,5]
(1,5:
2,5]
(2,5:
3,5] *
jp 0,012 0,050 0,144 0,266 0,240 0,176 0,092 0,020
125
-
. 6.1.2.
. 6.1.3.
10. , )(* xF x X xX .
1. , .
)(* xF
)(xF
2.
)(* xF
x .
3. , .
)(* xF
11. ,
)(* xF
xX .
. 6.1.4.
126
-
. 6.1.4. ( 2)
6.1.3. 1.
),...,( 1 NXXX =G (),
N*m
*m = =Nn nXN 11 . (6.1.2) 2.
),...,( 1 NXXX =G ( )
*
xm = =Nn nXN 11 . (6.1.3)
3. ),...,(
1 NXXX =G N
* = = Nn xn mXN 1 * )(1 . (6.1.4) 4. ),...,( 1 NXXX =G
*
xD = = Nn xn mXN 1 2* )(1 . (6.1.5)
6.2. .
. . , .
N
. .
6.2.1.
127
-
, . . X <
. (6.2.1) . X
. )(xF = )()( 2 xdFmxD xx . ,
xx m xmx xx xdFmxD )()( 2 . . 22)( >
xmx
)()( 22 >= > xmxx mxPxdFD x , (6.2.1).
6.2.2. .
(
NXX ,...,
1
xx mm ,...,1
xx ,...,DD 1 NnCD nx ,1, = NyN mY . (6.2.2) . .
=ym yD NY =Nn nXN 11 : == Nn xy n mNm 11 ; == Nn xy nN DND 121 .
CD
nx ,
N
CD y .
, N
Y
128
-
( ) 2 2NN yN y D C Y m N > . N (6.2.2). . X
. .
X, . (6.2.2) :
NXX ,...,
1
01
lim1
= >= mXN Nn xnN . .
.
6.2.3. . ,
' ' . . N
ap
N
Na
: a
p
0lim = > pNN aaN . (6.2.3) . ,
. : 1 0 =(1 ).
NXX ,...,
1
ap
aq
ap
= ,
nX
nxm
ap
nxD =(1 )
ap 2
ap +(0 )
ap 2
aq =
ap
aq < .
N
XX ,...,1
N
Na
A . , , (6.2.3).
6 . 6.2.1. 6.2.1
6
3, 11, 24, 47, 49, 70, 93, 108 13, 16, 32, 42, 80, 91 5, 21, 22,
45, 46, 50, 59, 61, 90, 101, 105 1, 28, 31, 52, 55, 56, 62, 65, 66,
71, 84,
97, 106, 107
129
-
7.
7.1.
, . : ( ) , , () . .
XG
N . =NXX ,...,1 () X
),...,/( 1 KxF ),...,;( 1 KxF K ,...,1 . ,
, Nxx ,...,1 X X N1 ,..., . K ,...,1 1. ),...,/( 1 KxF ,
, K ,...,1 , . ),...,;( 1 KxF 2. , ,
. .
( ), G
),,( 1 K G = *G G . ,
K,...,1
),...,/( 1 KxF ),...,;( 1 KxF : X X N1 ,...,K ,...,1
).,...,(
................................
);,...,(
1
111
NKK
N
XX
XX
==
1. , ;
XG
xG K,...,1
, . K ,...,1 2. . , , .
3. , 7.5.
7.2. 7.2.1.
1. G G , / - , :
GG
130
-
GGG = ]/[M . (7.2.1) .
( ) GGGG = ]/[0 M . . G , , . XGG 1.
]/[2
2 GGG = M . :
2
0
22 += , (7.2.2) [ ]( )[ ] 2**2 = MM .
2 , , , .
22
0 G 2. . , .
X
3. , , , , . , , , . .
7.2.1. 7.2.1
, == Nn nx XNm 1* 1m m x x= = Nn xnx mXND 1 2** )(11 = = Nn xnx
mXND 1 2* )(1xD
x ( ) X
= = NnN xmnXNkx 1 2)*(11* =+ = Nn xN mnXNkx 11 2)(1* xyR
= = Nn ynxn*xy mYmXNR 1 ** ))((11 = = Nn ynxn*xy mYmXNR 1
))((1
131
-
. . Nk
=
2
2
1
2
1
N
N-N
k N , - ( 2).
. 7.2.2.
)( Nk
7.2.2 Nk
N N N Nk Nk Nk Nk NkN N
3 1,1284 6 1,0506 12 1,0230 25 1,0104 40 1,0064
4 1,0853 7 1,0423 15 1,0181 30 1,0087 45 1,0056
5 1,0640 10 1,0280 20 1,0134 35 1,0072 50 1,0051
GG 2. , G /G N , : GGG = ]/[lim MN . (7.2.3)
7.2.2. GG. , :
0}/{lim => GGGPN , (7.2.4) N 0> ' , . 1. ,
( ). (7.2.4) 2*lim / 0
NM = G GG .
2. , . N
7.2.3. . -. . 1.
*
e , *e N , - : *i
]/)[( 2 eM < , =1, 2,. (7.2.5) ]/)[( 2 iM i 132
-
1. ])([ 2 M D [ * ]. .
2.
]/[ eD < , i =1, 2,. (7.2.6) ]/[ iD 3. , ,
. , , , .
X
2. ,
l
*
e * :
]/)[(
]/)[(2
2 = MMl e . (7.2.7) . [0,1].
, , =1. l 3. ,
*
N . 4. )('l
1 2 :
]/)[(
]/)[()('
2
2
2
1 = MMl . (7.2.8) )('l. , . [ ]/*D , .
*XG
xG 1. N
. )/(xf N G , xG .
/*[ ]( )[ ] // 2** MM
+
/)/(ln
1
2
2
0
XfM N
G[ ]/*D = , (7.2.9) 133
-
D ( ), ; M , , ; XG* 0 .
. [ ] xdxfxM N GGG )/()(/ ** = . (7.2.10)
0
* )/()( += xdxfx N GGG . (7.2.11) (7.2.11) :
+= 0* 1)/()( xdxfx N GGG . (7.2.12)
= )/(ln)/()/( xfxfxf NNN GGG . (7.2.13)
(7.2.12) :
+= 0* 1)/()/(ln)( xdxfxfx NN GGGG
+= 0* 1/)/(ln XfM NG
. (7.2.14)
1)/( = xdxf N GG . (7.2.15)
(7.2.15) : 0
)/( = xdxf N GG . (7.2.16) (7.2.13), (7.2.16)
0)/()/(ln = xdxfxf NN GGG ,
0/
)/(ln = XfM NG
. (7.2.17)
[ ]/*M : (7.2.17) [ ] 0/)/(ln /* = XfMM N G . (7.2.18) 134
-
(7.2.14) (7.2.18): [ ]( ) /)/(ln /** XfMM N G + 01 = . (7.2.19)
,
(7.2.9). 5.
= /)/(ln2
XfMJ NN
G . (7.2.20)
1. : = /)/(ln 2
2 XfMJ NN
G. (7.2.21)
. , (7.2.13):
2
2
2
2 )/(ln)/(
)/(ln)/()/( += xfxfxfxfxf NNNNN GGGGG . (7.2.13):
2
22
2
2 )/(ln)/(
)/(ln)/(
)/( + = xfxfxfxfxf NNNNNGGGGG
.
xG . (7.2.16),
xdxfxf
xdxfxf
NN
NN GGGGGG )/()/(ln)/()/(ln
2
22 = ,
=
222
)/(ln)/(ln xfMxfM NN GG . (7.2.20), (7.2.21). 2. (7.2.20)
:
[ ]NJ
D
2
0
*
1
/
+ . (7.2.22) 3. [ ]
NJD
1/* . (7.2.23)
4. ( )[ ] /2* M [ ] = /*M , :
[ ]NJ
D
2
0
*
1
/
+ ( )[ ] /2*M . (7.2.24) 135
-
, * . * 5. (7.2.24) , 0 (-1). , . 6. :
1
2
01
JN
+ ( )[ ] /2*M [ ] /*D , (7.2.25) : 1J
= 2121 )/(ln XfMJ . (7.2.25) (7.2.21), (7.2.24). 7. (. 1) , . 6.
,
e( )[ ] /2* M
NJ
2
01 += . (7.2.26) 2. 6
, :
e += )/(ln1* xfJ NNeG
. (7.2.27)
2, , 6. 6. ,
e[ ]( )[ ] [ ] 1// ,/ *2* * == = Nee e JDM M (7.2.28)
1. 1 6 . 2. 6
(7.2.24), 1 (7.2.26).
3. 6 , (7.2.24) , .
4. 6 , , , , , , .
5. , , .
136
-
,
20 , (.
(7.2.2)) 2 .
6. 6 , )/(ln xf N G (7.2.27), )/( xf N G : [ ] )()()( 21 xhkkexp
e G +)/( xf N G = ,
)(xhG
xG )(1 k )(2 k , . 7. , , ,
. , , , , . . . .
*G GK . I 7.
),1,(
)/(ln)/(lnKji
XfXfMI
j
N
i
N
ij = = GGGGG
, (7.2.29)
G - K -
N
)/( GGXf N - .
N
XG 1. ,
.),1,()/(ln
)/(ln)/(ln
2
KjiXf
M
XfXfM
ji
N
j
N
i
N
= ==
GGG
GGGGG
(7.2.29) : ),1,(
)/(ln2Kji
XfMI
ji
N
ij = = GGG
. (7.2.30)
2. -
XG
N
)(GGS , R
T1( ) ( ) ( , 1, )ij
i j
S SI R i j
= K=G GG
. (7.2.31)
]/))([(* GGGGGG TMR = (7.2.32)
, 1.
137
-
3. xG .
XG
),...,( 1 K =G D KK
),1,()(1 0 Kjid ij
ij =+= G , ),1()(0 Kii = G )(0 GG .
1T TU R U U D I D U G 1 , (7.2.33) U - . K
1. (7.2.33) .
2. (7.2.33) 1T TU R U U I U G . (7.2.34)
3. (7.2.34) , *G11 = URU T G 1, .
=UIU T*
eG 8. , **1 ,, Kee , ,
eR G1
1T TU R U U I U =G . (7.2.35)
1. , (7.2.35) , .
**
1 ,, Kee N 2. , (7.2.35),
. ),,( **1* Keee G = :
TbI 1* += GG , (7.2.36) ),1(
)/(lnKi
xf
i
N = GG
b . 3.
. .
4. , , (7.2.35), .
*
eG 8. , **1 ,, Kee ,
N*
iG**1 ,, Kii URUURU
ie
TT 11 > GG ),2,1( =i . (7.2.37) .
xG 4.
N , . ( ) )(1 f 1+N
138
-
),(1 xf N G+ , 2
1
2 );( + xf N G + );(1 xf N G xG , 0);()(lim 1
* = + xdxf N GG . [ ] 12* )( NJM , (7.2.38) , : NJ
= +2
1 );(ln XfMJ NN
G =
+ 212 );(ln Xf
M NG
, (7.2.39) =
M (7.2.38) , (7.2.39) XG* .
1. (7.2.38) , .
* 2. (7.2.38), (7.2.39)
KG .
UIUURU TT 1*G , (7.2.40)
KK : *GR [ ]TM )()( ** GGGG*GR = , (7.2.41) I
= ++ jKNiKNij XfXfMI );(ln);(lnGGGG
, (7.2.42)
*G (7.2.41) G , (7.2.42) XG G .
3. (. 8, 1 8 8).
4. G 8 );( xf K G
G . 5. ,
.
7.2.4.
139
-
. G G , - )/,...,( *1 GNN xxfN X G .
1. , , )/( *GGxf N , GG , .
mN
xxn
N
* = =1 1. n , m
Nxx
n
N
* = =1 1 11N n . 2. , ' .
. 3. . 4. ,
. , ' , . G.
),,( 1 K G = )G /,,( 1 NN xxf : , (7.2.43) )()/ 111 NNN xxhfxxf
,...,(=)/,...,(
xG ) , /1 (f ,
, h x xN( 1,..., ) . G. : )xf K GG /(G , xG , )(xGG .
. 7.2.3. 7.2.3
X
- (
)
,
,
-
-
-
== Nn nx XNm 1* 1 xm
+ + + +
= = Nn xnx mXND 1 2** )(1 + ( ) 1l + xD = = Nn xnx mXND 1 2**
)(11 + + + )( 12 ll <
140
-
N
Np A=* + + + +
. . )( 12 ll <
7.3. 7.3.1.
. );(1 xf X , . , . , ,
X - : = dxxfxm );( . :
)( =m . (7.3.1) ' (7.3.1) ,
)(1 m= , (7.3.2) , 1 .
(7.3.2). : mv
*= = Nn vnv XN 11* 1 . 1.
. 2.
*
, .
X
K K,...,1Kmm ,...,1 ,...,1
(7.3.3) =
=).,...,(
.................................
);,...,(
1
111
KK
m
m
' , , , , *mm ,...,*1mm ,...,1
=
=).,...,(
...............................
);,...,(
**
1
*
**
11
*
1
KK
mm
mm
3. , , . , ,
141
-
.
,1
7.3.2. . N - )(GG
XL
( ) ,
X X N1 ,...,G , . *G G G , .
*G. G , .
X
. , )(GG
XL
: == 1 1 )/()( n nX xfL GGG . (7.3.4) (7.3.4). == Nn nX xfL 1 1
)/(ln)(ln GGG . (7.3.5) .
(7.3.4) (7.3.5) . *G
==
=
=N
n K
n
N
n
n
xf
xf
1
1
1 1
1
.0)/(ln
.................................
;0)/(ln
G
G
(7.3.6)
1. (7.3.6) '. ' , . ' ', - ( ).
2. . (7.3.6)
