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no T e M Pm14 A P P M M X a p a K T e p J l C T # K a M ee 3 B O J I I O ~ P I . J / I H o ~ A3T11 ~ X a p a K T e P H C T M K M HaXOARTCR M 3 H ~ ~ J I I O A ~ 3a H M3 ~ 0~ J E O q ~ eAfaiH HOG CHCTeMbl, a PlHOrAa 0HI.l 3 a A a k O T C R , HCXOAR H3 K O H C T P Y K T H B H b l X T p e 6 0 B a ~ M f i . H a n p ~ l ~ eB p ,O ~ ~ ~ T H 3 a~ ~I aX ~ rae oxc l , b r a ~ l w ~no H ~ ~ J I I O A ~ H M R3aM p a c n p o c T p a a e Hr.leM ~ o n e 6 a m G B 1 3 e ~ ~ oKfo ip e ~ p e 6 y e B~Oc CC ~T ~ H O B H T ~ee c e o f i c ~ s a( n a p a ~ e ~ p b ~ C H C T ~ M ~ I )f. l p ~ K O H C T P ~ P O B a H P I H M e l a H 1 1 3 M O B M C O O P Y X ~ H M ~M~O j K e T no~pe60B a T b C R T a K 1 7 0 ~ 0 6 p a ~HbX n a p a M e T p b l , ql06b1 V a C T O T b l C O ~ C T B ~ H H ~KI XO J I ~ ~ ~ H M ~ nemanx B 3 a ~ a ~ ~ PbI Hr Txe p B a n a x . 06uero M e T o n a A n n p e u r e H M n 0 6 p a T H b l x 3 a ~ a 9 H e T , PI Y'dcsM, IIPMXOAPlTCR IIPOCTO P e U I a T b I I p R M 3~ a A a q y I I p M p ~ ~ l ~ H 3bH al KX eHHRX n a p a M e T p O B , M PI3 n O J I Y ' i e H H O r 0 M H O X e C T B a p e l r r e ~ ~ Bf bi l 6 p a ~ bTO, K O T O P O e
6 n ~ x BCerO e K 3 K C n e p H M e H T a J I b H b l M (PIJIPI X e J I a e ~ b l A~a )H H b l M . M ~ o r w e0 6 p a ~ H b l e3 a A a Y H CBOARTCR K BOCCTaHOBJIeHPIIO M a T p P l U klJIPl AMl$cpeP e H q P I a J I b H b l X O n e p a T O p O B O n p e A e J I e H H O r O BPlAa no k l x C n e K T p a J I b H b l M C B O ~ ~ C T B ~ M . T a ~ b l e3 a n a q P l H a 3 b l B a I o T C R U ~ ~ ~ T H ~ 3 Ia AMa ZMl a M H C I I e K T p a J I b H O r O a H a J I M 3 a . O H U BCTPeqatOTCR B TeOPMIl K O J I ~ ~ ~ H r eM o ~~~~l l, 3 1 , 1 3JIeKTOAPlHaMPIKe, ~e, a K y C T k I K e , KBaHTOBOG M e X a H P I K e $1 A p w X
o 6 ~ 1 a c eCTeCTB03HaHWJl. ~~x
~ R T ~ A ~ CJ X I e T H a 3 a A AJIR p R A a 0 6 p a ~ ~ b 3laxA a K C n e K T p a J I b H O R J a H a J I P l 3 a 6b1JIM p a 3 B P l T b I 0 6 u ~ 1 eM e T O A b l P e U I e H M R , HaUIeALUI4e B C K O p e H e O X M A a H H b I e n p k I J I 0 X e H I l R B COBePUIeHHO
~ p y r o RO ~ J I ~ C T P I C:
1.IX nOMOLqbIO 6 b l J I M P e U I e H b l H e K o T O p b l e
B a ~ ~ b AJIR l e ~ $ H ~ ~ . I K~>eI J I P 1 ~ e f i ~ ab ~l Ie $ d l e p e ~ q M a J I b ~ b l eY P a B H e H M R B YaCTHbUC
T $ l q e C K M X T P Y , ~ H O C T ~ MX ~ ~ , M O X H O M 3 J I O X H T b H a I I p H M e p e O ~ P ~ T H ~3 Ia A Xa Y CneKTpaJIbHOi-0
aHaJIbl3a h l a T P H q
9 ~ 0 (TO 6 ~e C T b T p e X A H a r O H a J I b K b L X M ~ T ~ ~ I uqeM)' ) , B ~ e f ~i r c n o ~ ~ b a oKsoaH~c nbe K~ T b r cneWylr.cos W I M T . M11cmpb1M a B T O p a . ( X ~ P ~ K O BXrY, , 1988). OH^
tr nocsnuaeTcR npeWlaraeMaR Kmlra M MeTOmVeCKUe YKa3aHMR K
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J l a r p a ~ x y6, y ~ Hea 3 b~l B a T b TBKYK) CWCTeMY C T P Y H O ~ ~H, ~ ~ P Y X ~ H ln,. an^ OnMCaHMR n p o A O J I b H b l X ~ o n e 6 a ~ uBfTi O ~C T P Y H b l B B e A e M n p R M 0 YTOJlbHYH) CMCTeMY KOOPAMHaT, H a n p a B l f B OCb Z B e p T M K a J l b H O B W p X Y e p 3 o6e T O W M 3 a K p e m m i u R , K W P A U H a T b l KOTOPblX (T e K W P A M H a T b l 0 n 0 p ) 0603~aYepe3 Z(0) U Z(N 1) n p F I n p o A O J l b H b l X K O J ~ ~ ~ ~ H MB Rn&R X MOMeHT B p e M e H M t COCIYXICJIeAyR
MaCCaMM
+
H1.E C T P Y H b l nOJIHCCTbH) O n M C b l B a e X R K O O P A U H a T a M H
Z ( Z )= Z ( Z t, ) M CKOPIXTRMM
H O ~ ~
i Z ( ' 2 ,t ) MaCC 17Li. P = 1 2:. . . ,N . H a f i g e ~CIrJIbJ, ~ e f i c ~ B y I O q 1 . 1Hea MaCCbl nprr n m 6 o ~MX p a c n o n o x e H H r ? . Ha M a c c y In., ~ e f i c ~ e y rCMJIbI or ynpymcTtl n p n r t p m JIeHHbIX K ~ e i ni p y X H H H C P I J I ~ I r p a e r . r T a q r r H m.ig, p a s ~ o n e f i m ~ y I o w a KR( Y T O ~ ~ I X B CDCMRHMM PaBHOBeCMR P a B H a HyJIW. & ~ H F I ~ H M qepe3 ZO('l)KOOPAMHaTbl MaCC M onop B COCTORHIIM p a B H o B e c n R M y e p e a z ( i ) = t ( i ,t ) = Z ( i , t )- Z O ( i )- M X OTKJIOHeHMR OT PaBHOBeCHOrO COCMRHHR. M ~ M ~ H AJIHHbI ~ H M n~P Y X P l H b 1 , C O ~ A M H R I O W ~ ~ ~ MaCCbl ?7Lt M m , + , , no CpaBHeHPlHJ C ee n J I M ~ 0 f B i PaBHOBeCHOM CMTORHMM P a B H O ~ ( 2 = )
a,
COrJIaCHO 3 a K O H Y
r y ~ a kl3MeHeHMe , CMJIbl
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IIO~TOMY P ~ B H
= 2,3,. . . , N
-
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+ 1, t ) ) M k , ( ~ ( ti ), - ~ (+i 1,t ) ) .
( 2
O A ~ ~ ~ C T B BCeX Y ~ U CtlJI, ~ ~ RR ~ ~ I C T B Y W U ' ~ X H a MaCCy
d z ( i ) = -clt( Z O ( i )
rAe 2
Y n P y r O C T k f , n e f i c ~ ~ y W I Q eHf ia klaCCbI
-ki(z(i, t ) - ~
In,, P a B H a
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+ z ( i j ) = -(itz ( ~ ) = i - ( 2 ) ,
TIpaBEIX q a C T R X 3TI.lX y p a B ~ e ~ M CTORT fi BHelllHMe
k N t ( N + 1, t ) , B03HHKalOIQMe BCJIeACTBMe A B M X e H t l R OnOp, PI CJlJIb1 COnpoTPIBJIeHHR C p e A b r - ~ ~ i ( k K) O . J I ~ ~ ~ CTpYHbI, H M R npOMCXOARLIpIe n p t l OTCyTCTBHl4 BHerUHMX CMJI ( 0 n 0 p b 1 H ~ ~ O A B M X HII~ CMJI ~ ) COnpHBJIeHHfl = O), CllJIbI
koZ(O,t )
M
M n e ~ n M I I I e MCMCTeMy (1.1) 8 B H A e OAHOrO Y p a B H e H H R
Ha3blBamTcR C B ~ ~ O R H ~ I M M .
T~KM 06pa30~, M 3a~aqa0 C B ~ ~ ~ AK H O ~~ ~I ~X~
N H HMa R l pXy X e H ~ ~CTpYHbl fi CBO-
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-M;i = K z ;
N
z(0) = x a i e i , i ( 0 )= x 0 , e i i=l i=l
( I < f ,f ) > 0 npu ecex f # 0, ~ 1 -I < t AI-i, nonoxrr~ene~
TU
1
a. J
kJ-l
+ kJ
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-
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(1 = l , ? ; .. . , N ) ,
mj
k.
b. =
( K f ,f ) =[(ko
J
.
J-'
i,j=1,2 , . . . ,N - 1 ) .
+ k l ) f ( 1 )- k l f ( 2 ) l f o
+ [ - k l f ( 1 ) + ( k , + k ? ) f ( 2 ) k 2 f ( 3 ) l m+ . . . + [ - k N - ~ (f N - 2) + ( k ~ - 2+ k ~ -f ~( N) 1) - kp,-lf ( N ) ] f( N - 1) -
-
+ [ - k ~ - ~(Nf - 1 ) + ( k ~ - +I L N )(WIf ~ (N) N-1
+ 1k J I f( j ) - f (j+ I)[' + k ~ l( f~ 1 1 '
=kol f (1)12
i=l
(1.7)
T
e OnepaTop K , a BMecTe c
.
( L f , f )>O,
f F O
-
L =
O 6 q a T~e O p M R
h l a n b I X K o J I ~ ~ ~~ 6Hn M k r a~ un o n o X e H ~ 1 Rp a B H O B e C M R M e x a H M -
YeCKHX CFILTFM'C K O H e W i b l M '4l.ICJlOM C T e n e ~ e f Ci B O ~ O 6bma ~ ~ I pa3BPlTa B A a X X a r p a H X e k l (CM.
S 8). Y ~ ~ B H ~AHB MMX eRH M R T a K M X CMCTeM T O X e IIPMBOARTCR K
B M ~ (1.4) Y T O ~ ~ KB O0
6 q e cnyqae ~ nonoxPtTenbHo
:
6b1~~ b P O H ~ B O ~ ~ HTO ~ ~K IK .a K n p M B b I B O A e C @ p ~ y J l ( l . g ) ,
onpeAeneHHaR hraTpuqa
TOJlbKO n O J I O X C I T e n b H a R O n p e A e J I e H H C C T b M a T p H q b l
CR M B
mecj = ( M ~ z ( o ) , ~ ( x d~j )= ) ,( A J + ~ ( o p) (,A j ) ) . B qBCTHOCTH, e C n H B H a Y a J I b H b l k M O M e H T CMCTeMa 6b1na B b I B e A e H a M 3 COCTORHMR P a B H O B e C H R Y A a P O M no M a C C e 7TIl, C O O ~ U J M B ~ Pefi ~ MC K O ~ T ~ ' U (T.I e . ~ ( 0= ) 0, i ( 0 ) . = (v,, O,O, . . . , O)), TU z(l,t)=vl)-
" sin A j=l
JX7
COXPElHRH3T-
o n e 6 a n M ~ CBOARTCR x
C n e q w + 1 4 ~ e c ~ o RO C ~ ~ ~ H H O C 3T ~ aH ~~
flOJlyYI
> 0.
A n R O T K J ~ O H ~ H HK~ IO ~ ~ ~ ~ M ~MaCC M XOT C nR O n O X e H M R P a B H O B e C l ~ I RK B I < A y
L, T O 3TW @p~y.Jlbl
a ~ o aM a ~n b r ~x ~
H a r p y X e H H o f i C T P Y H b I R B n R e T C R TPeXAMaTOHklJlbHOCTb M a T p P l q b l
IIp(Aj)ll
(1.9) CTORT TOJlbKO B e q e C T B e H H b l e BeJlPIYPlHbI, YTO n 0 3 B O J l R e T I l p M B e C T H & J p ~ y f l b l
n e ~ o s a ~ e n b B~ coe , 3
onpeAeneHHbrx MaTpuq.
-
K P O M ~TOrO, M3 B e W e C T B e H H O c T J f M a T P L i q b I L C n e A y e T , YTO BCe C O ~ C T B ~ H HB e~KI- ~ T O P b I p(Aj) M O X I I O ~ b l 6 p a B~ ebq e C T B e H H b l M M . ~ O ~ T O MBYn p a B b l X Y a C T R X c ~ P M ~
o 6 q e cnyqae. ~ C
L MomeT
(1.11) 6 b l n a M C n O J l b 3 0 B a H a
t i O T b l C t i a H M M C O ~ C T B ~ H H ~3IHXa Y e H L i k M C O ~ C T B ~ H H B~ e~KXT O P B n O J O X M T e J l b H 0
t ( p ( ~ A,))" ,
A j = IIP(Aj)II-' (LpiAj),P(A,))
1762-1767 ro-
i
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ee
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oa K~O ~ ~ ~
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T O n03BOnReT
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C ~ ~ K T PJ TMH H ~ ~ ~ HOo n~eop a T O p a , , 4 e f i ~ ~ ~ y l o uBenrp0O C T p a H C T B e H N , H a 3 b l B a e T C R M H O X W T B O B C e X er0 C O ~ C T B ~ H H 3~ ~~ aX~ e H t l kC. ~ I ~ KOTnPe p a M p a A (Jl er0 M a T p P l -
S ( A ) . rd~ Mbl
U b l ) O ~ O ~ H ~ Y ~ ' 4~e T p eC 3 R
K o J ~ ~ ~ ~ CMCTeM H M R ~ C K O H e Y H b l M YMCJlOM
BMAeJlM,
AJIX P e I I I e H M R 3 a A a Y 0 M a J I b l X
c ~ e n e ~ ~e k~
0 6 0 H~~ bO ~l X O ~ I I HaXOA1,lTb MO
C n e K T p b I M C O ~ C T B ~ H HB ~e KI T~O P b l n O J I O X P I T e J l b H 0 O I I p e l e J l e H H b I X B e W C T B e H H b l X CMMMeTPMYeCKMX M a T p t r U .
B o 6 u e ~C J I y Y a e
HMYerO HeJlb3fl C K a 3 a T b 0 CTPYKTYpe
C ~ ~ C T B ~ HB He ~ KM ~ PXO B T a K M X M a T p t l q . O A H ~ K O , B H a I I I e M C J I y Y a e ,
MaTPWqbl
L
H e TOJIbKO n O J I O X M T e J I b H 0 OnpJ2AeJleHHble M B e u K T B e H H b l e , HO Jl T p J 2 X q H a m H a J I b H b l e .
h a m n a p s 3 T O M Y MX npocTym
C ~ ~ C T B ~ B He H K T~O P I b~l B KaHOHMYeCKOM
6a3uce H M e M T
BeCbMa
CTPYKTYPY.
A e f i m s n ~ e a ~ oCOrJIaCHO , (1.5),
( x l 2,2 , . . . , X N ) n p O t 1 3 ~ 0 ~ 1 b ~ bt l M) e e T
BeKTOp
LX - AX.
B KaHOHMqeCKOM
(YHCJIO
M BeKTOp X =
6 a 3 ~ c Tea K t l e KOOPAMHaTbI: n p e A C T a B P I M b 1 B BMAe X
=x
P (A),r A e
P(X)= (P(1, A), P(2,X),
...
, P(N,.\)) .
+
zae DN(,\) = -bhr-lP(N - 1 , X) ( U N - X)P(N,A ) 4 Co6cmsen~ate3ndVCKUR XI, A?, , X N Oneparnopa L ~len~mrncR xopHAMIL nonuKOfiUl D,v (A). D N ( X I )= 0 5. Co6cmeen~bre eexmopbr onepamopa L c co6cmeennarmu. 3nalren-u A, pa8Kbl c J P ( A J )zae , cl # 0 - npo~r~sonanare xoxcmanmbl.
.
KOHCT~H C j TB~~~I ~ H ~ ~ I O BT C 3aBRMCMMOCTA OT T O M , KaIWM A0nonHme~HblM YCJIOBtlRM M b l S O T M M IlOAqMHMTb C ~ ~ C T B BeKTOPbl. ~ H H ~ IHanpmep, ~ nOJIaraR cj = 1, llOJryYUM ~ 0 6 C T ~ e ~BeKTOPbI ~ b l e P(A,), Y KOTOPbIX nepBaR KOOpAMHaTa B K ~ H O H H Z ~ ~ C 6a3uce K~M paBHa 1, a IlOJIaI'aR
J I e m ~ a1. l7ycma deiicmtryloquir e npocmpanc.mee HN nuneiinbril onepamop L m e m e xanonulrecxom 6 a ~ u c eeeyecmeennylo mpexduazonananym ~ a m p u y y ...
-b~ -61 .. .
az
0
-b2
b
a3
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0
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O T M ~ THeKOTOPbIe ~IM 3JIeMeHTapHbIeC B O ~ ~ C TnO BJI~MHONlOB P ( j , A). M3 PeKypPeHTHblX d,Mp~yJI(29) CJIeAyeT, YTU HX K O @ H U H ~ H T ~ I BeqeCTBeHHbl H ~ b I p a m KIT- TQlIbKO Yepe3 U l , a ? , . . . , a]- 1 . bl , b,z,. . . ,bj- 1 , B SaCTHOCTM, KO-HW~HT~I npu cTapruHx cTeneHm X paBHb1
-bN-l a~
u bj # 0 npu ecex j = 1 , 2 , . . . , N - 1. Tozaa 1. Onepamop L cantoconp~~~cen: ( L x ,y ) = (x! L y ) . 2. l7pu ecex X c u c m e m ypaexenuir
] = I , ?,...,N - I
(Lx-Xx,e,)=O,
umeem eauxcmeennoe nuneirxo nesaeucumoe petuenue ~ O A C T ~ B M BB PeKflPeHTHbIe COOTHOLLleHHR (2.9) npaBbIe YaCTtI @ p ~ y J l(28) npMpaBHR6 B n0JI)"WHHblXBblpaXeHI.IRX K O X P ~ ; ~ M L ( M ~npM H T ~A' I M XJ-', nonyYUM PaBeHCTBa
p ( X ) = ( p ( l ,A),P(2,A ) , . . . , P ( N ,A ) ) ,
PI
zae nonunomar 7
P ( j , A)
=
):k(,i, j)Ai-' ,=l
cmenenu j - 1 onpeaenenar ntaxumu peKyppexmxaimu @opmynamu:
3. n p u ecez X eexmop P ( X ) yaoenemeop~empaeencmey
LP(X) - AP(,\) = DN(,\)eN,
(2.8)
I
I
I
~ ~ ~ J I o X C MBM3aKSIH)YeHMT C T ~ H A ~ P T HM~WIO~A~ P e P e H M R CMc'I'eM y p a B H e m 1 . a H a J l O r M Y H b l X CMCTeMe (2.2).
& ~ P O C M B B 3T0fi C M C T ~ M B . ~ ~M ~nBO C O J ~I m H e e
HeHMfi, M b l M O X e M OCTPBLIIYfOCfi
p-
YKOPOYeHHYIO CHCTeMY 3 a i l M C a T b B BMAe O A H o r O
~ I M H ~ ~ K O~HH e YO H O~ p aO 3 H O C T H O ~Y P B H e H L f R
aj+l - X &(j, A) = Q(j bj
+ 1 , A ) - b,+,~(j bi
+ 2, A),
OTHOCMTeJIbHO $ ~ H K ~ M ~ M ( k= ) Lk AMCKpeTHOrO I I e p e M e H H O n J k . 3 n e c b p ( k ) = bk.,, q ( k ) = a ( k ) - (,b(k)+b ( k - I ) ) , M y e p e a A 0 6 0 3 ~ a ~ Ke HO H ~ Y H O P ~ ~ H ~O nHe ~p aIT~o p? A [f ( k ) ] = f ( k ) - f ( k - 1).& 6 p C l L U e ~ H b I e Y p a B H e H U R 3 a n P I C b I B a t o T c R B BMAe
J ~ K BLlAWb, O
rne C N ( A ) = ( a l - A ) Q ( l , X) - bl Q(2, A). A B nYHKTaX 4 M 5 CJIeAyeT 3aMeHMTb D N ( A ) H a C N ( A ) H P ( X ) H a Q ( A ) . C ~ ~ A O B ~ T ~ C~ O~ ~HC OT ,E ~ H3HH a~Y~e H~M R O n e p a M p a L R B n n m T c f i K O P H R M n OAHOBpeMeHHO A B y X nOJIMHOMOB: D N ( A ) M C N ( A ) , a T. K. H 3 p e K Y P P e H T H b l X @)IOPM y n C n e A y e T , YTO Y 3 T H X ~ O J I W H O M O B K O @ H ~ M ~ H T ~ I n p M c ~ a p r r r e f iC T e n e H H A paEIHb1, T O D N ( A ) C N ( A ) . C O ~ ~ T B ~B eH K TH O P~ b l IP~( A J ) M Q ( A j ) , KOHeYHO,
--
JDl~eh~ 3aBMCMMbI: 0
HblX AaHHblX
YTO KOHeYHOPa3HOCTHOe y p a B 3 3 H e H ~ (2.16) e n p M n l o 6 b 1 x HaYaJIb-
y ( k O ) = a, y ( k o
+ 1)
H erO
pemeHMe
BCex
k = 2,3, . . . , N
P I M e e T eAHHCTBeHHOe peLUeHMe,
o6pa3ym~B C e B 0 3 M 0 , X H b l e
p e L U e ~ H f iy p a m e m f i (2.16).
e c e f i CHCTeMbI (2.2) AOJIXCHO H M e T b BMA 21,= C l f
k
( k , A ) + ~ 2 g ( k A), ,
rAe,He3aBMCRlrlMe
cl
((111 {(al
-
T.
e. H X HYXCHO H&TU
Pl3 Y p a ~ H e H P f k
A ) f ( 1 , A ) - b2f (2, A ) ) A ) g ( l , A ) - b2g(2, A ) ) = 0 ,
cl { ( a -~. A ) f ( N , A ) - b ~ - i (f N - 1 , A)) +c2 { ( u N - A ) g ( N , A ) - b ~ - l g ( N- 1 , X)) = 0.
P ( k , Al)Q(k
-
1 , A,) - P ( t
-
1,A,)Q(k, A,) = o
JIICI6yHJ
OY~BMAHO, YTO p e u I e H u e
~ 0 3 ~ P l I J T i e C1 ~ ~Mb C2 l H Y X H O B b 1 6 p a ~ bTBK, ~ ~ 0 6 YAOBJIeTBOPRJIMCb b l IIepBOe
+c2
M
OTKyAa
J I M ~ e h ~ bK l e O M ~ H H ~n ~ m 6Mb 1 x
( k ,A), g ( k , A )
napy n ~ ~ ~ eHe3aBrtCLlMblX ii~o
M n O C J I e A H e e Y P a B H e H M R CMCTTblbl(22),
npu
b
ABYX n M ~ e f i H 0He3aBMCHMblX p e I l I e ~ M f i .~ ~ O ~ H ~Y Y e p H3 fM
OT
OTKYAa, B YICTHOCTM, C n e A y e T , YTO
=
C J I e A y e T , YTO YHCJIO J I M H ~ ~ HHe3aBLfCMMblX o P e l l I e ~ M f 3i T O n J Y P a B H e H M R P a B H O A B y M ,
E ~ A MCKaTb ~ M
TO e C T b RBJIReTCR Y p a B H e H M e M C nOCTORHHbIMM K - H ~ M ~ H T ~ M H .
ero p e m e H M e
B
svrne y (k) = 19" rne 6, = B ( X ) n o n n e x m o n p e A e n e H u s o . n o ~ c ~ a s n ~ ~
TO B b I p a m e H M e B y p a a H e H M e ( 2 . 1 9 1 n o n y s n M
PI JIM
=A,
-,g+?-e-l OTKYAa C J I e A y e T , YTO B KaYeCTBe HeHMR
e2 - 26'
(1 - $)
+ 1 = 0.
(220)
e H Y X H O B 3 R T b n&oA npM
OTJIM9HOM OT
KOpeHb KBanpaTHOrO Y p a B -
0 I4 3, 3 T 0
+
HBXOAMM, COrJIaCHO (1.61 YTO 3 J I e M e H T b I M a T P H U b l L P a B H b l CL1 = ( 2 au)p, aN = ( 2 + a N ) p , a2 = a3 = . . . = a ~ - ' = 2p, bl = bz = . . . = b ~ - ,= p,
YPaBHeHMe MMeeT
A B a p a 3 J I M W b I X KOPHR:
a KOHeYHOPa3HOCTHbIe
(2.19) H M e l o T T a K M e J I M H ~ R H OHe3aBMCHMbIe peruemm: f ( k , A) = e k , g ( k , A) = tYk.n o ~ c ~ a ~sT PnIp ~e L ~ u e H u n B (2.17) PI a a M e q a R , YTO B p a c c M a T p n B a e M o M cnyqae ( L , = a:v = 2, a 2 - X = 0 8 - ' , c o r n a c H o (2.201 n p n x o g n M K y p a B H e H I I l o e-N-' - O N + , = 0. C n e n o s a ~ e n b ~ oK,p a e e b r e y c n o s n R Y n O B J I ~ B O p R S O T C R , KO"@ e Z i N * l i = 1, T. e. 0 = eXp a C ~ C T B ~ H H
Y P a B H e H M R (2.16) M K p a e B b I e YCJIOBKR IiMeKlT
T ~ K O RBMA:
M KOHeZIHOPa3HOCTHble y p a B H e H M R
~ O C K O J I ~ KKYO ~ @ @ W I ~ H T ~ IY P a B H e H H H ( 3 . 1 ) nOCTOflHHb1,
+
'ifi),
3HaKeHMR P a B H b I
A, = -exp
me j
= 1 , 2 , . . . ,N. H
BeK'I'OpOB p a B H b 1 Zl
=
1C
( /:
+2
2-
- exp
MCKaTb B BMAe
~ I ~
(-,i&)
-
6,-* =
?isin
(&
b), a
(&
I;)
HOpMM!XlBaHHble YCJIOBMeM
O ~ C T B ~ K BeKTOPbl H ~ ~ ~ PaBHbl
N
p(xj)=
sin
1 s1n & .
k= 1
ek
.
e
=
8(p)
3TO B b I p a m e H P l e B YPaBHeHMR ( 3 . 1 1 IlOJIyqMM
a k e a ~ b r e a o@ ~ R I ~ J I ~( 2M. 1 8 ) KOOpAPlHaTbI C O ~ C T B ~ H H ~ I X
Zr = 8*
y ( j ) = BJ, rne
er0
P e m e H M e CJIeRyeT
n o A n e m n T onpe,qeneHHso.
no,qc~aa~~
q > 0 , u = l nnw - 1
q>O, u = O
q = N - 1 +(I-a,).'
q
+(I-a,)" .
@(o) 0, @(n) = ( N + 0 ) ~B. ne~ofiOKPeCTHOCTll TOYKM 7i OHa BO3paCTaeT, eCJIM @'(n) > 0, M y6blBaeT, e c m (Pt (rr) < 0. M c c n e ~ y 6onee e ~ AeTanbHo noseneme N TOR @YHKWM. T ~ rcaK K M
M3 (3.12) cnenyeT, TO npu a = 1 (T. e., Korna lcrol < 1 H lcrNl < 1) npOM3BOAHaR a'(() BCEOAY nOJTOXMTeJIbHa, a H 3 (3.131 (3.14) CJIeAyeT, YTU OHa s c m ~ ynonoxnTenbHa ,TaKxce npn a = - 1 (T.e., KorAa cro > 1 M aN > 1A ecnn N - 1 (1 - a,,)-' (1 - a N ) - ' > O. I I O ~ M M Y B BTMX c v a x x @ Y H K ~ H R MOHOTOHHO BoapacTaeT Ha BceM MHTepBane (0, rr). Ecnu x e a = 0 (T.e. lao)< 1 M a N > 1 u r n loNl < 1, a. > 1) M N - 1 (1 - cro)-' (1 - QN)-' > 0, T O 0 $YHKUUU @(<) M38eCTH0 TOJIbKO, YTO OHa MOHOTOHHO PaCTeT B HeKOTOPblX OKpeCTHoCTRX 0 H K. H ~ K O H eCJW ~ Q N - 1 (1 - fYo)-' (1 - aN)-' < 0 M a = 0 HJIM a = -1, TO @YHKWX a ( < ) MOHOTOHHO ~ o j p a c ~ Ba eO K~ ~ ~ ~ T H O C T M 0 PI MOHOTOHHO y6brsae~B OKpeCTHoCTPl T. C ~ ~ A O B ~ T B~ 3 JaB IH~CM HM OOCTH IX 3HaYeHPIR U M 3HaKa Q = N - 1 (1 - fYO)-' (1 - a N ) - ' rpa@MK (PYHKWM @(() MMeeT BMA, C X ' ~ M ~ T M Y ~306pa?KeHHblfi ~CKII Ha PMC. 3, rAe nYHKTMPOM 0603~aYeHa HeM3BeCTHaR HBM YaCTb rpa+M~a,a ~ a c I H ~ a 6Hbal BepTM~anbHOfiM ~ O P U ~ ~ H T ~ J I ~ H OCKX Bbl6paHbl ~ n y a ~ o 6 ~ T Bpa3HblMM. a M3 HepaBeHCTBa (3.11) CnenyeT, YTO Ha OTKpblMM MHTepBane 0 < ( < T (
+
+
+
+
1 + CL COS
<
1 1-CL' a(1 a)(l +cost) a'(<) - a1(7i) = (1 - a ) ( l + 2 a c o s t + c ~ ' ~ ) = 1 + 2cLc(,s< + (,,?
-
+
i l ' j i ~ )=
Pnc. 3
+
+
+
+ +
+
O ~
(2AIEi+,,) pa3 B c n p a e ( C ) ( n o s e ~ y ? ) , n e p e c e K a e T w c n o (1 + 2A.l,v+,j pa3 B cnyqae (B)( n o q e ~ y ? C ) n e ~ o ~ a ~ e Bmcnyqae ~o, H+a (B) wicno nepece~emriipamo N a - 1 + 2 C,,=, R I P , a B cnysae ( C ) OHO p a B H 0 HOB 0 6 0 C~J I y~Y a R x o ~ Hoe npeBOCXOAFlT 1'\ ~ O ~ T O M Y N+u+~ n e p e c e q b T O ~ K OY e T H O e Y t i c n o HeqeTHoe
+
I.r BCe c c r 6 C T E e H H b l e K O ~ ~ ~ ~C HY M ~C R TO wM ( t P~) RBJlRHYrCR UX J I U H ~ ~ ~ H ~KI OMMH~ H HaUbtflMM, KOTOpbIe B C e r A a M O X H O npblBeCTM K BMAY
~p"=:"
A M Q - nfX3M3BOnMibIe B e W e C T B e H H b I e Y U C n a (KIoYeMy?) B~IRCHMM, YTO C @ M ~ U Y ~ C K OTOYKH G 3 P H H R OIlMCbIBaeT n e p B O e (cos[<,~+ w((,)t + a]) n smpoe (cos[<,j - w(EP)t - 26,(<,) - a]) cnaraemre. npn TOM, 6e3 OTpaHMYeHMR ? ~ ~ H o C T M M , O X H O C Y U T a T b U = 0, T. K. J m r O B C e r A a M O X H O A O ~ H T W R , rAe
M, T. K
u = 0 B cnyqae ( B ) M U
2 - 1 B cnyqae (CX
TU B
0 6 0 c~n y~q a f t x
CMeCTHB H a Y a J l O OTCYeTa B P M e H M .
J I e r ~ 0n p O B e p M T b
H ~ ~ O C ~ ~ A C T B ~~ OHAHC O T ~~ H~O B K OY~ ,M 6 e ~ K o H e Y H b l e no-
CneAOBaWJIbHOCTH
O T K y A a c n e A y e T , YTO B c e q e n i e Y r I c n a A,fp p a B H b l H y n m . C n e ~ o s a ~ e n b ~B ocny.
qae ( B ) qricno nepeceqe~~ik p a n H o N .t n - 1, B cnyqae (C)OHO pamo A' + u. T ~ K P I M 06pa30~,qncno peme~niry p a ~ ~ e ~ (3 ~ .7)r i ~i ~ X C ~ I I J FBI XM H T e p B a n e O < < .ii,
<
rae
< E (o,T), w(<)
= 2&
sin
5 , j = . . . - 2, - l , ~ , l , ? ,. .. ,
y A o e n m B o p m
~ ~ C K O H ~ YCMCTeMe H O ~ ~ y p a ~ ~ e ~ l l f i
MOXCHO I I p e A C T a B M T b T ~ K ~Oa 6~n ~ q e b " f .
q = N - 1 + (1 - n,,)-' + (1 - c t K ) - l a = 1. e c m Jaol< 1: I L ~ < ~ I1 u = 0, e c n r t li.tol < 1, a~ > 1 t l n x lax( < 1, a,, > 1 U = - 1 , e c n p t u o > 1, c r , ~ >1 ~ ~ M ~ Teqe, M M YTO 3TI.I peJlIeH1,rfl
<,yAOBAt?TBOpRlOT
&a
C l f C T e M a n O n y Y a e T C R M 3 Y P ~ B H ~ H (M 1 . 0~) C B O ~ O A H ~ I K X o J I ~ ~ ~ OAHOPOAHO~~ H A ~
(112~ =
m, kJ = k) CTPYHbI,
nOCneAOBZITenbHOCTM B b I W K a H U l l M M3 ( 3 . 7 1
(3. l o ) P a B e H C T B a M
a KOOpAHHaTbl C~~AOB~T~JI~HO,
KOI'Aa YMCnO MaCC H e O r p a H M Y e H H O P a C T e T ,
B e p x H e k I4 HMXCHefi O n O p CTPeMRTCR COOTBeTCTBeHHO K +CO M -03.
(3.17) OnPICbIBaloT
P O A H O ~C ~ T P Y H b I C Y ~ C T O T Ow~( < ) ,
C C ~ C T B ~ H KH o~J II ~~ ~ ~ ~ H ~ M ~ C R K O H ~ YOAHOH O ~ ~
(j)
MX 3 n e M e H T b l Z*
P a B H b I OTKJIOHeHFLRM
1, a
MaCCbl OT n O J l O X e H M I I PaBHOBeCMII B MOMeHT B P e M e H M
5
j-k
npOnOp~0HanbHbIe
4k s i l l 2 L* ( j ) P a B H b I B H Y T P H H M M HaIIPRXCeHMRM B M Y K e , r A e 3 T a 06a c n a r a e M b r e B + p ~ y n e ( 3 . 1 6 ) nonyyam-rcn n3 n w n e A o B a T e n b ~ 0 C T e f(3.171, i e C n H P a C C M a T p k I B a T b MX T O J b K O n p j ~= 1 , 2 , . . . , N M ~ 3 f l T 5 b = Cp. B B ~ A A~n M R O A H O ~ X ~ A H OC~T P y H b I b n e e y ~ C 6 H b l eX a p a K T e p k l c M K M U CAen a e M c o o r e e T c T B y r o r . u p l e a a ~ e ~ Bb + r p ~ y n a x . IIpu TOM M ~ nI p e H e 6 p e X e ~amok TRiKeCTM, CYHTaR, YTO C T P Y H a CKOJIb3MT 6e3 T F H M R no T O P M ~ O H T ~ J I ~ H nJlMlKOCTM O~ MM Be.nMYMHbl
M a c c a HaxoAI,ITcx.
H n M HaSOAWTCR B COCTORHMU HeBeCOMOCTM. 0 6 0 3 H a Y I l ~Y e p e 3
1 = Z O ( l )- ZO(N)A n H H y
B COCTURHMII nOKOR llPY%MHbl,
CTpYHbI H 3aMeTMM,
Y M
COeAMHRH3LqMe COCeAHMe MaCCbl, M M e W OAMHBKOBYH)
1 = Nh. H a ~ a n o nepsok MaCCbI Z = ZO(j)6 y . 4 ~p a B H a -jh:
A n t m y ( n o ~ e ~ y ? )~, o ~ o p y rMo ~ 0I 6 0 3 ~ a w i M~ e p e 3h, T ~ KYM
KOOpAbIHaT B b 1 6 e p e M T a K , Y T 0 6 b l B COCMRHMPI nOKOR K O O p A H H a T a
me do(<)
= Wg(l
+ Ctoei:),
t i qaCTOT3
w(Epj = 2 f i
n e 6 a ~ nEl ~K O M ~ J I ~ K C HW O ~p~ ~ OnklCblBaIOT e
sill
g .C ~ ~ C T B ~ H H ~ I ~ KO-
B ~ K M ~ > - @ Y H K ~ ~ p(XP) .IM ?sp'~c~(A,)t,
BeIIJeCTBeHHaR M M H U M a R qaCTP1 KOTOPbIX H M e t O T T a K H e KOOPAIIHaTbI:
-h. Torna
6bina paBHa
Z = -jh
3-a MaCCbI OT rAe
ee
KOOpAMHaTa
j-fi
MaCCbI
(MMHYC IlOTOMy, YTO KOOpAMHaTHaII OCb H a n p a B J l e H a B B ~ ~ X O )T K ~ O H ~ H H ~ n0nOXCeHMR IIOKOR (T.
e.
BeJIMYHHY
~ ( tj) ,) 0
KOOpAMHaTa B n O J I O X e H M M nOKOR, p a B H a R
p = h-'m,
KUT€1pyl0
k
eCT€!CTBeHHO
-jh.
H a 3 b I B a T b IInOTHMlTbH).
YTU X C ~ C T ~ O C T ~ H ) H a 3 b l B a e T C I I K O ~ @ ~ U ~ H T H, a YAJlMHeHtle
A(h)
6 0 3 H a G ~Y e p 3 u(Z, t), B s e n e ~AaJIt?€? B e n H W H y
H a n o ~ m m ,HaKOHerl,
KOTOP~IW
H Y X I I O YMHOXMTb
I T P Y X M H b l , Y7'06bl ~ O ~ y Y H TB b e n H Y H H Y CMJIbl, B b 1 3 ~ a B L I J e f i 3TU
Ha3bIBaeTM K O ~ U W I ~ H M M OTPaXeH13K
C O ~ C T B ~I H C OHJ I~~ I~ ~ ~ HOUARH O ~ O A H OC~T~p Y H b I CFI
K O B ~ Z I H OA~J ~ MHbI OnMCbIBahX-
TOR %e + J p M y n o R (3.181 B K O T O p y I O B M e c T O 11 = (2h)-I
COmJIaCHo
(3.151
IIOACTaBbITb
onpenennsoqee n a p a M e T p v. M 3 ~ T J I Xp a B e H c T B c n e A y e T , YTO zf ( j ) = u*(Z, t ) , j( = 2 v Z , w ( ( ) = = 2uc(u), 2J0(() = ?&(vh), me
?fiy
< = 2uh,
v OHMnOJIYZIaKIICR
tP ~ 0 -7 r(E,))- T ( P - r(211vp))
--=
- 2h
o T p a x a m T c a OT e e p x n e f i
n.+(Z,t) = cos 2v(Z - c ( v ) t ) ;I L - ( Z t, ) = cos 2u(Z + c ( v ) t - 2Su(2vh.)), rAe
I
Z np06eraeT A M C K P ~ T H ~PFIA I ~ ~3 H a Y e H M f i . . . - 211, -h, O , / L , 2h, . . . . r p a @ r ~ ~ cr ie ~ e f i c + ~ Y~ Ha K ~ Hf~(3 ~ - at) n o n p a m T u 513 mpa+>~~a C ~ Y H K ~ ~ I W
f ( 2 ) CABHTOM H a at P T b I O
a
BAOJIb OCM t (IIOZIeMy?),
T.
e. OHPI ABbI?KYTCR
B nOJIOJKllTeJIbHOM H a n p a B J I e H l 4 H , eCJIbl CL
e M y H a n p a B J l e H t l P l , eCJIPl U
< 0.
npouf2ccb1,
> 0,
O l l M C b l B a e M b I e TaKMMM ~ Y H K ~ ~ ~ F I M M ,
7L-(Z, t ) ,
c(v).
Y ~ ~ J I ~ ~ ~ c K o H ~ ~C oT P~Y H~ b ol , n~p MHK poe n~J Ioe ~~~ 0H TOJIbKO fi o ~ ~ K
B ~ P X H ~ ~
O n o p e , C O ~ C T B ~ HK H o J~I ~ ~~ ~ H POInRJ l C b I B a I O T c R @YHKUPIRMPI
( n o ~ e ~ yT ~e.kc y n e p n o 3 c l q r r H M M A a y x B o n H , 6eryrqws B n p m r r B o n o n o x H b r x H a n p a e n e H M R x co CKOPOCT~IO ~ ( v )n,p u q e M B o n H a , 6ery1qa~BHW, O T n M q a e T c R OT BOJIHbl, 6eryqeR B B e p X , TOJIbKO @ a 3 0 ~ b 1CABMmOM ~ - ( 2 6 0 ( ' 2 ~f/ T ~ ) . ((=JIaraeMCIe K
I
A O ~ ~ B J I ~ H9 O~, 0 6 b lY Y e C T b 3 H a K MMHYC I I e p e A B T U p b I M ~ J I a r a e ~ b l N I~) ~ B M C M M O C T ~
@aao~oroC A B M r a -2a0(2vh) OT v O H H O ~ H ~ Z I H Oo n p e A e n R e T c R n a p a M e T p o M ao, e. X e C T K O C T b M IIpYXCMHbI, IIDCpeACTBOM K O T O ~ AC T p Y H a K p e n M T c R K ~ e p m e f i O n o p e . ' ~ P M Y J Q ' (3.18) MOXCHO J l H T e P n P e T H p B a T b T a K : C ~ ~ C T B ~ H KHo ~J II ~~~ ~ H U R
T.
I
2(1
+ ha)
tCOTOpbIe o T p a x a r o T c n OT
y A O B J I e T B O p F I I O ~ l 4 eH Y X H b l M K p a e B b I M YCJIOBMRM MOJKHO C A e J I a T b ripe-
~ ~ M ~ TB M 3 a KMm O Y e H M e , YTU B nOJIYZIeHHblX f$X3pMyJIax
n e n b ~ b l f inepexon n p M
h
+
0 B npenene
nony.ramTcn
J I M ~ H HT aOK
I43YYeHMlI K O J I ~ ~ H M3 R B p a q b l X CTPYH
a T a K x t e I I p O I I O p q M O H a J I b H b I e MM B H Y T P e H H M e H a I l P R > K e H M R ~ ~ c K o H ~ Z I HOnHOpOAO~~ HOG C T P Y H b l RBJIRIOTCR C b L H Y C O I f ~ a J I b H b I M M 6 e r y u r l BOJIHBMH, ~ ~ ABMXCYIUMMMCR co CKOPOCT~IOc(I/) B n o n o > K m e . n L H o h l (ii+(Z,t ) ) H a n p a s n e H l r r r o c r t Z unrr B OTp u q a ~ e ~ H~a bn p ~a e on e ~H r r H (.I.L- (Z, 6)). P IO J I ~ ~ ~ H M R M ~ ~H C K O H ~ Z I OAHOPOAHOR H O ~ MH~IMM CJIOBPMI4, C O ~ C T B ~ H H ~ I M K C T P y H b I RBJIFIIOTCR CUHYCOMAaJIbHbIe 6eryqrre BOJIHbI, ABMJKYLLlPIeCR E B e p X PI B H l l 3 CO CKOPOCTbH)
+ ha)
6 e r y ~ CO x CKOPOCTbIO c ( v ~B)B e p X , K O T O P b I e onopbr, PI m n H . 6 e r y q ~ xc TOR xce c~opocraroBHMB, ~ u x ~ e onopbr. f i Mx c y n e p n o 3 1 . r ~ p m A a e T K O J I ~ ~ ~ H M R ,
K O J I ~ ~ ~ H VHI F e nI p e p b l B H b I X C T P Y H
BAOJIb 0CJ.l Z CO CKO-
H B ~~OTHBOnOJIOJKHOM
H a 3 b l B a M T C F I ~ ~ ~ Y L ~ BOJIHaMFl. M M M ~ O ~ T O M OTKJIOHeHMR Y M ~ C ClL+(Z,t ) ,
2(Nh
HaJIOJKeHMeM BOJIH,
B
+ o p ~ y m ~o n, u c b r e a m r n e
1788 r O A y
IIOCTynMJI n a r p a H J K n p b l
c n e K r p o M , T.
e.
m ~ y waaaavy cnexmplraxozo perueertx yrok 3 a A a w M ~ BI B e A e M
peUI1,ITb T a K ~ a a b r ~ a e ~ -oyG mp
n x a n l r s a r n p e x a u a z o ~ a n a x a r rM n m p u q . BBTOM
naparpa*
cB0ficTb MaTPUUbI,
n o n o n H t m e n b H b 1 e YIfCneHHbIe x a p a K T e p w x w K n c n e K T p a m H b r x K O T O P b I e COBMeCTHO CO C n e K T p O M O A H O 3 H a W i O O n p e A e m H n .
T p e X A M a K l H a J I b H y W MaTpUUJ'.
M ~ srrnenu I
( n e ~ 11 ~ a YTO K a x A a s T p e x A u a m H a m e a s M a r p m a
p O X A a e T B~KTOP-C)JYHKUMW a J l b H b I M M KOOPAbIHaTahIM (2.9)
Ee
P(X) = (P(1,A), Pi?, A), . . . ,P(N, A))
P ( k , A),
OnpefleJleHHbIMM
p e ~ y p p e ~ ~ e ~b @l ~P u M
~ 0 6 C T ~ e H H b lBee K T o P b l I I O n y Z l a W T R U 3 3 ~ 0 f i B~KTOP-C)JYHK~MM
CTaHOBKe B H e e BMeCTO
C0OTBeTCTByK)LqMX
L no-
C nOJUiHOMII-
npM
C O ~ C T B ~ H H ~ I3 X HaYeHtlfi
Aj.
Y ~ ~ M M
non~ T H
P(,\j) 06pa3yw~O P T O C O H ~ J(HO ~ H H~e~H~ O? P M M ~ O B ~ H H ~ ~ ~ ~ ) H N , M p a 3 n O ~ i e H l . I eBeKTOpOB f = (f ( I ) ,f (2), . . . ,f ( N ) ) E
C ~ ~ C T B ~ HB eHK T ~ OI p~b I
6 a 3 ~ cn p O C T p a H C T B a H N no 3TOMY 6 a 3 ~ c yflae?.Cn
@pMyJIofi
C HeCKOnbKO H H O ~TOVKII ~ 3FHMR.
f
(2.11).
~ ~ C M O T P H MH a
@PMYJIY
H a a o s e ~n ! 3 t ' 0 6 p a 3 0 ~ a H M e M a y p b e
(2.11)
BeKTOpa
E HN C ~ ) Y H K ~ M ~
x N
F(X)= ( f ; P(A))=
f ( k ) P ( k ;A),
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i
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II
CKaWl* M BblTeKaH)Wee U 3 Hem PaBeHCTBO pk = U k , m e Yepe3 U k 0603~aYe~bl ~neKTpaJlbH0R@ Y H K ~uM ( AM ) B TOYKaX XI, T ~ K U06pa30~, M (PYHK~UH p(X) M CT(,\)MhteIOT OAHHaKOBble TOYKH pa3pblBa H OAMHaKOBble CKaYKM B 3THX TOWCBX, BTKYAa CAeAyeT, 'IT0 p(A) = U ( A ) (BCblJIy ~PMHRTO Ha~M~M HOPMMPOBKM u ( - 0 0 ) = 0, P ( - C O ) = 0). C y ~ ~ u Bce p y p~e 3 y n b ~ a ~ bHacTonqem 1 naparpa*. npwxonm K cnenym weti TeopeMe
~ N ( h ) p ( *)dp(\) j. = -6,-I ( P ( N - 1, A ) , P ( j , ,A)),, + a ~ ( p ( NA), , P ( j , A!), -
J
(6.13)
P ( N ,A)XP(j,A) & ( A ) .
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odno3nauno onpedemmmcn csoumu cnexmpananarfiu @ynxq-u, u dm mozo, umo6ar @Ynxqux u ( A ) 6brna cnexmpanbnoli @ynxqueli nexomopoli mmpuqar N-zo nopsaxa suaa (6.1 5), xe06xoaumo u a o c m a m u n o , umo6br ona ne y6breana, 6wna xycouno-nocmo~nnoliu umena posno N moyex paspbtea nenpeparenocmu, cymdin cxaulcos e xomoparx p a a m edunuqe. 2.) I7opowaaewbce mmpzrqeic (6.1 5 ) opmozonananace nonunofibr abtpaxammcn c n e d y m y u u g5opntynafiu
u e p e ~cmenennare Jnonrenmar s ( p ) = \ ,\Pdu(X) re c n e x n ~ p n a n o l i@gnlcqu~i 3aeca Ah = 1 u npu 1 > I uepe";, 0603~arenbc anzeo'pauuecxue a o n o n ~ e n u ~ K anemenma,n X p o n p e a e n ~ i m e n ~
HbIe ~ H ~ ~ ~ H M 3aB FHC IM , T O T N - 1 He3aBMCXMblX nOnOXM~e&~blx napa~eTpO~ a , ,az,. . , a N - l, vepe3 KOWpble CKaWM 0, CneKTpaJIbHbIx@ Y H K ~ P * BbIpaxamrcFI @pslyna~u
a
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= J A P ( ~ , x ) P ( J - i , X j r i c r ( X ) , j = i ; 2 , . . . ,N - 1 ;
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=
J AP(j,X)P(j,X) & ( A ) ,
malcwe axeueanrnmnbrniu
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j = 1,2, . . . ,N ,
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unt @ o p f i y ~ ~ a m u
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= (1,P(2, ,\), . . . , P ( N , A)), H ~L,~ OnepaTOp R U, Mbl IlOJlyW'IM B ~ K T O P @ ~ H K ~UHP~ ( X )= (1,e'"'P(2, X), . . . e ' a N P ( NA, ) ) , yAOBJIeTBOpRIOI4yK)ypasaeamo ~ O ~ T O M YII , pMMeHRR K B ~ ~ ~ T o ~ - ( P ~ H K ~ U P I , (I X )
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Teopema 2. 1 ) Tpexauazonanana, mampulp LN suaa (6.1 5 ) oaxo3xauno onpeaenrremcs d e y m cnexmpamu: c a m 6 Mantpuyar LN u nonyuennoii u3 xee yxopouennoii fiarnputiac L N 2) Anrr mozo, v m o 6 a ~nocneaoeamenanocmu A1 < A2 < ... < AN u p1 < p? < . . . < p ~ 6arnu - ~ cnexmpafiu xexomopoii mampuyar LN euaa (6.1 5 ) u nonyuexnozi 113 nee yxopouennoii ma-mpuqb~L N - l , ne06xodufio u aocmamouno, vmo6ar ORIL nepemewanuca:
2) A m WZO, u m 0 6 b i aee nocneaoeamenbuocmu .seupcmsenu~lsl c u c e n 6 b u u cne-pamu asyx niampuy )!L u )L: euaa (6.1 5), o m n t t u n > o u ( w c c n m n a w H ~ U npasalnr N a ~ ~ e n ~ n m oneo6xoa1m fi, u a o c m m u n o , u m o 6 b i onu nepeAteXanUCb.
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TO COUTBeTCTByIOUIMk K O @ X ~ I M ~ ? ~ H BT ~ ~ M M O A ~ ~ C TPBBUe RH HYJIIO
(k,, = 0 ) ( A m yno6c~ean o n a r a e M T a K x c e kii = 0.) T ~ K K a K YaCTMIlbt m ~ + 1. ,. . ,mN+A.fHeIlOABllXCHbl. TU MX KOOPA>IHaTbl X N + ~YN+k, , Z N + ~(k=1,2, . . . , M) H e 3 a B M c R T OT B p e M e H H , p a c c M a T p n B a e M a R CMCTeMa H M e e T 3N
crrene~ei?C B O ~ O A ~ M I , ee n O J I O X e H M e B ITpOCTpaHCTBe OnHo3HaYHO X i , K, Zi 2, . . . , N) Kmennec~mM n c r r e ~ p f a n 3HepI-m ~ a ~ ~ BC~CIc u c r e ~ b c r mmemme~~ pas~br o
O n p e A e J l R e T c R KOOpAMHaTaMM
OTHOCKTeW H O BeKTOP- @ ~ H K W ~ M
OH^ ~
I
I
I
a a b ~ ncampuqe5 ~ a m ~ p a c c ~ a ~ p m a e ~cucmefibr. ofi M e x a ~ m e c m Xua P a K W P H C T M K a M M pacc~aTpmaemoi3CUClYMbt H a 3 b l B a e T C R ~ a h p Berm-, cm-i? na ~ ~ u q u e m s3amoneticrsn~ a kij (i, j = 1.2,. . . ,NX M a c c mi M mcen 0: (i = 1,2,. . . , N ) YMcno 0; o n p e ~ e m eCYM~ MapHOe B ~ ~ U M O A ~ ~ C T i-k B MY~a W l q b l CO BCeMM HeIIOABUXCHO 3 a K p e I l J I e H H b l M K
I I
I
npu
Bcex
I ntg CBR3aHHblMM, e C J W nn60 CI = B, m160 CyILlem,, ,m,, , . . . , m,,, , s m 7r1, = m,, , mg = 7npn M kp,,,+, > 0. ( ~ T H O U I ~ H M C B~R ~ ~ H H O I X M0 6 n a ~ a BCeMM e~ CBO~~CTBBMM 3KBPIBaJleHTHOCTM M n 0 3 B O n R e T nO3TOMY p a 3 6 H ~ BCe b 9 a C T w b I H a KJIaCCbl, 06na~amurue TaKMMM H a a o s e ~s a C T U q b l
7n,
H
C T B Y ~ TT a K a R q e n o z I K a s a c T u q
i
/
A n r l K O T O P ~ I X kap
> O.
IIO~TOMY, e c n H B c e
wcna
6): p
@(f, f) H e M O X C I 6 b l ~ Cb T p O l D ~ O J I O ) K M T ~ ~ H O H~aI 06paUa:
a
~ HYJIIO, ~ b
~
B HyJ5
= & = . . . =
Ecnn Xe XOTR 6 b 1 OAHO M 3 m M YPICeJ5 C K a X e M , 8,: m + o p ~ aa((', f)-ro nonoxwTemHa. ~ ~ ~ C T B M T ~ J Ino ~ Hy O c n,o a M m , A n n nm6oro 0 ~ a h qenowa m noncurcnT e J I b H b l X K O ~ @ @ M U M ~ H T O Bk,,,, > 0, k ,,,, > 0, ... kan-Io > 0, M, COPJBCHO B b l U l e C K a 3 a H H O M y . M 3 P a B e H C T B a(<,<)= 0 C J I e A y e T , YTO [ , = 0 PI [a, = Em, = Em, = . . . = <,,-, =
0, xomopoe n p e a n o n a z a e m c f i e b r n o n ~ e x x b t n c KOrAa
!
a, p,
TO ( P O P M ~
nonoxMTenbHo,
--
n e p e ' I M c J n 4 ~C B O ~ T B M a~T p U U
L,
BblnOJIHRmUMeCR Ancl BCeX PaCCMaTpUBa-
e M b I K CMCTeM.
CBO~~CTB~MH:
1) d b r e A B e sacrnqbr o ~ ~ oKrn oa c c a C B R ~ ~ H ~ I ; 2) d b l e A B e ' I a C T M q b l , n p U H a ~ n e X a U ! M e P a 3 H b l M KnaCCaNl, H e CBR3aHbI. ~ P 3M M M CMCTeMbl Y ~ ~ B H ~(9.4) H MP a~ C n a A y C R H a He3aBMCMMble nOACMC T e M b I AJlR KOOPAMHBT ' I a C T M q O/HOrO K n a C C a . M H ~ I M U CJIOBBMM, ~ a ~ ~ bK nl a C f Ci JIe~nm 6. B e y e c m s e x w a ~nranlpuy mozaa u m o n b x o mozda ~ e n n e m c ~ ~ a m p u y e i ic u c m e n r a t paccmampueaemozo suda, xozaa oxa yaoenemeop~em ycnoeusnr 1 , 2, 3.
R a n R e T C R ~ ~ M K H Y TCOM~C T ~ M O ~IIOnHOCTbH) , U ~ O ~ M ~ B ~OTH APYTMX. H O ~ ~~ O ~ T O M Y
n a n b ~ e f i m en ~p e A n o n a r a e T c R , 'ITO B c e s a c ~ ~ ~q abm~ e kc M C r e M b 1 n p M H W n e x a T o n H o M y K n a c c y , T. e. n106an napa uncmuu c e ~ a a x a . K p o ~ eTON, n p e A n o n a r a e T c R c y w e c T B o a a H M e TO'IKM Y C T O ~ ~ Z I H B Op~aOB H O B e B
CMR.
A 0K a 3a T en b
B 3 ~ 0 fTOYKe i KBaApaTPlWaR @pMa
(9.51 I
3x2
= j=l
oTKyna cneAyeT, r T o M
m n ~ mrna, o
a?u
N
N+M
-=
ax,ax, - -Roo, 7 -kaj + ,~n,Bi, -
( a # 01,
j=1
fi
+ o p ~ a@(f, H e o r p n q a T e n b H a M o6pauaew~B = 0 n p M B c e x a, AJWI K W O P ~ I X 0: > 0,
Korna
<,
C T B 0.
n n A~O K a 3 a T e J I b C T B a
A&-.
;< 2
~ o J m mrna M
Ja
=
3neMeHTbI MaTpMUbI
M b l PaBeHCTBaMW
T O , ' I T 0 3 T H YCnOBMR H ~ O ~ X O A M MY~CI T , a H O B n e H O BbIUle.
MX /OCTaTOrHOClM
3 a M e T M M , I l p e X C A e BCerO,
L C B H 3 a H b l C MeXaHM'IeCK>IMM
sTO, COrnaCHO
X a p a K T e p M C T M K a M H CMCTe-
C n e n o ~ a ~ e m (~)YHLWFI ~o, X(r) H e ~ o a p a c ~ ua eH e~n p e p b I B H a Aanee, ua pa-
d ( 0 ) = L M ~ o J I o X C ~ ~ T ~ J IO~nHp eOA e~n e H H O C T M A(0) > 0, a M3 H e p a B e H C T B a EeHCTBa
cnesye-r, YTO X(r) < 0 n p n r
MaTPwbI
L
CJIeAyeT, YTU
> JIDU(J-lIILII.
(0, IID" 11-I IILIl), YTO X ( 7 ) > 0 n p U T < To X(T") = 0, O T K y A a c n e A y e T , YTO Det(L - r D O )> 0 n p H r < TO M Det(L T O P )= 0, T. e. To P a B H O HablMeHbLLleM)r ICOPHH) IlOJIMHOMa Det(L-7Do). ~ o J I o X C M M D = roD0M AOKaXCeM, YTO CYUleCTBYeT W K T O P fi = ( V 1 ,Y , . . . , V N ) ~ o ~ T O MCYUleCTBYeT Y T a K O e 70 E
-
Lii = Lii. C J I ~ J I O B ~ T ~ J I ~ HMOa T, p P I U a CMCTeMbl C MeXaHtlYeCKMklM X a p a K T e p M C T l l (9.7) P a B H a L, PI n J I R n O K a 3 a T e J l b C T B a ~ O C T a T O Y H O C T t IHyXCHO TOJIbKO IIpOBePMTb, 4TU C B O ~ C T B1~ , 2, 3 r a P a H T U P Y I O T C m e C T B O B a H H e MaTPPlUbl D M B e K T O p a fi TO
KaMM
C YKa3aHHblMM CBO~CTBBMH. M T ~ K ,n y C T b M a T P M U a
L
yAOBJI€'rBOpHeT
YCJIOBURM
BOJIbHYlO H e O T p M U a T e J I b H W A M a M H a J I b H Y H ) M a T P M q Y YCnOBMeM
SpDo = 1,
1, 2, 3.
Do # 0,
14(r)=L-rD0
(OIT<W).
C nOJIOiKMTenbHblMt1
KOOPAPlHaTaMM
Vi
(a
= I,?,,. . . , N),
Y~OBJI~TBO~RH)~PI~~
WaBHeHlIH) B 0 3 b ~ e MnpOPl3HOPMllPOBaHHYlO
M paCCMOTpMM c ~ M ~ ~ ~ C TBeLqeCTBeHHblX B O CPIMMeTpPlYe-
CKMX M a T P M q
p~
( L - D ) C = 0,
(9.8)
No MHOMeCTBO
H O M e P B BCeX P a B H b I X H y n I O KOOPAHHBT M Y e p 3
M e C T B O H O M e p B IIOnOMLITeJIbHblX KOOPAkIHaT.
J/13 P a B e H C T B a ( 9 . 8 )
TonbKO
NI = N\1VO MHO-
@ p ~ y n o R( 9 . 3 )
H C p e A H I I O n y Y a I O U M X C R M a T P M U O T 0 6 p a T b Te, K O T O p b l e 3 T m
YCnOElUJUW y A O B n e T B O p R l o T .
C n e A y e T , YTO
Bonee c o A e p m a T e n b H a R 3 a n a ~ anonyqaexq
ycnoaum 1, 2 3 Hanpnms aq cuTxn q Cornac~o(9.51
Korna
K
A O ~ ~ E I J I H R eIK~O T~ o p b l e a n p M O p M M 3 B e C T H b I e c m R c r s a HCKOMbIX M a T p M W M e p , K o r A a H ~ B ~ C T HB ~ c eI n a p b l ~ e ~ 3 a n ~ o n e i i c r ~ y
B 3 M M C n y Y a e a I I p M O p M M3BeCTHbl BCe H y n e B b I e 3 J l e M e H T b l L - ~ a T p H q b l . ~
npw B c e x i E N o , M, T K. lxJ1 > 0 npn j E NI, rn L,, = 0, ecnn i E No, a j E N1. C ~ ~ A O B ~ T ~y~ M~a THP M O q, b l L M O r Y T 6 b 1 T b O T J D I W b l M k l . I H Y n R TOnbKO T e a n e M e H T b r L , , , y K O T O P ~ I X 06a t . i a n e r t c a i n j n p n H a A n e x a T nn60 M H o x e c T B y No, J I H 6 0 MHOWeCTBY N1. HO 3 T 0 IIpOTMBOpeYMT YCJIOBMH) 3, COI-naCHO K O T O P M Y , E N1 H ~ ~ A ~ Tqen0'ma C R L,,,, La,,,, AJlR JIlO60fi n a p b l MHAeKCOEl a E No, Ln2n3,... Lan-.2,1 ,,-,, La ,,-,ad H e H y n e B b l X 3 n e M e H T O B M a T P H q b I L, T. K. XOTR 6 b 1 OAMH 3 n e M e H T T ~ K O qenOYKP1 ~ AonXCeH H M e T b MHAeKCbl, I I p M H a A n e X a W M e pa3HbIM M H O X e C T B a M . M ~ AIO K a 3 a n M , S T 0 KOOPAMHaTbI ~I1060r0H e H y n e B O r O PeLLIeHMR X = ( x l , " 2 , . . . , X N ) Y P a B H e H M R (9.8) YAOBJleTBOPRIOT H e p a B e H C T B a M Isi[ > 0, =
IlptlYeM BeKTOp
H
Y
TUnbKO
co
CBOMMM
6 n n ~ a h n C~o n c e A x M u . IIpu
TOM
n p e n n a n a r a m n3sem-
H O K ~ O H ~ M ~ Y P ~ I J F ICHCTeMbl R B IlOJlOXeHJiM PaBHOBeCMR, I I 0 3 B O n R I O U l a R BbIAenJTl'b
6 ~ u r x a h r u n xc o c e ~ e k
B~IRCHKM, K q e M y MOXHO
c B e c T u B ~ T U M cnyqae p e r u e H n e o 6 p a ~ ~ o 3a.qaw. f i
6 a 3 u c o ~e l , e 2 , . . . ,e~ n p o c T p a H C T B a H N , P I , P 2 , . . . ,PN, c O C T O R U M ~ M3 ~ c~~CTB~HH b l X B e K M P O B MCKOMOR M a T P M q b I L ( L p 3 = Xjp3) ~ o C K O J I ~ K Y MaTptiqa L B e UJeCTBeHHaR, C O ~ C T B ~ H H 8 e~KI M ~ p b l MOMHO ~ b 1 6 p a T bS e U e C T B e H H b l M H . OY~BMAHO, P a C C M a T p M B a e M b l e 6 a 3 ~ ~ Bbb l1p a X a H Y T C R A P Y r Y e p e 3 A p y r a no @ p ~ y n a ~
Ann
3~01-0,H a P R A Y C KaHOHMSeCKMM
PaCCMOTPMM OPTOHOPMI.IPOB~HH~I%6awc
(I:I:~~, 1 ~ ~ 1. ., ,. 1 ~ ~ T1O X) e YAOBJTeTBOPReT 3TOMY YpaBHeHPIH).
O T C I O A ~ ,B YaCTHOCTM, C J e A y e T , S T 0 Y p a B H e H M e ( 9 . 8 ) H e M O M e T M M e T b ABYX J I M H ~ ~ L H O He3aBMCMMbIX p e l L I e ~ M % T. K ? o r A a
M 3 HMX MOWCHO 6 b 1 n o 6 b 1 COCTaBMTb H e H y n e B Y l O
J ~ M H ~ ~ ~ HKYOIM O ~ L ~ H ~ ~ >Y[ KI O T, O P O ~XOTR 6 b 1 O A H a KOOpAMHaTa 6 b m a P a B H a H y n I O .
T ~ K M06pa30~, M AnR CTeM, M M e l O U M X 3 a A a H H Y W
a)
OTbICKaHHR MeXaHMYeCKkIX X a P a K T e P I l C T M K BCeX CM-
L MaTPMUY,
HYXHO:
rAe
IkLRTb nPOH3BOJIbHYIO H e O T P M q a T e J l b H Y I O A M a T O H a n b H Y I O M a T P M q Y
Do # 0,
H O p M H p o B a H H y I O YCnOBMeM
6) H ~ R T M H ~ M M ~ H ~ UK O~ pMe HRb
xJn = ( p j , e o ) = (en,pJ). M 3 p a B e H c T B (en,eb) = bop M (pi,pj) = bij N x i Q x j n = bij, n p u Y e M u3 (a) (a) xj,~,~ = JnL) M (6)
c n e A y e T , YTO
S ~Do J = 1;
70 Y p a B H e H M R
rAe
di -
= -nzL,,ls,:c,(;
ini = ins,?; 8: = rod,,
A M a m H a n b H b l e 3 n e M e H T b I MBTPML(b1
DO, a
112
> 0 -
nPOM3~0Jtb~blfi
I I a p a M e T p , o ~ ~ ~ A ~ J I Re A I bOf H~t I qMy M ~ 3~M e p e H M R MaCCbI.
Mbl BMASlM, AaHHYH) OT
d,,
R
L
S T 0 M e X a H H Y e C K M e X a p a K T e p M C T M K I i BCeX CMCTeM, IIMeIOWMX 38-
MBTPMUJ',
OH 3aBMCMT d l , ,di,, . . . ,diR-, ( i n < m, d,* = 1)- ~ e ~ y n e 3ne~ b
J l e r t t O HBXOARTCFI Y K a 3 a H H b I M B b l l l I e aJITOPMTMOM
nPOM3BOJIbHbIX ~ O n O X U T e J I b H b l : < n a p a M e T p B
+ d,, + . . . + din_, < 11 me di,,di3,.. . , din
M e H T b I MaTPMIJbI
c~E~
(no~e~y?) B ~ ~ M ~Lij H MaTPkIL(b1 T ~ I L
c n e g y e T (6)
2, . . . , N)
kij
", R
DO,YUCJIO KOTOPblX R P a B H O Y U C T Y a C T E l Q
~ e
B~~MMOA~RCTBYH)UWX
C HellOABMWCHO 3 a K p e n n e H H b I M M Y a C T M q a M l I .
C n o m ~ e eqeno ~ ~ C T O c~ o I T6 p a ~ ~ osiar~ a s e fci n e ~ ~ p a m n oarHoa J w m a , B KOTOpoR ~ p e 6 y e ~ cH ~B 1~ T HBCe MaTPMIJbI, YAOBJleTBOpRIOILTbie YCnOBHRM 1. 2, 3 M M M e W Une a a ~ a n n b c~ oe 6 c r ~ e ~ ~ ~b Ht ~e S ~ H HBR 0. 6 ~ c ney s~a e , M o m H o B o c n o m a o s a w
-
H b l M n P P l M e P O M C J I Y X a T CHCTeMbl, B KOTUPblX K a M A a R YaCTMUa ~ M M o A ~ ~ ~ c ~ Y ~ T
MOMHO B b I P a 3 M T b Y e p 3 W l U a Xi,, (2,
j = 1,
B ~ o ~ o p oi,f j i
(j 2
2)
npo6eram~B c e
N , a p, g (q > p) npo6eram LpqP a B H b l H y n K ) C O r n a C H O YCJIOBMH)
~ H ~ ~ ~ OT H P 1no I R
M H O X e C T B O 3HaqeH1,1R, n p M KOTOPbIX 3 n e M e H T b l
B
3TOM
naparpa*
p a C C M a T p M B a I o T C R ~ 0 n e 6 a H M R C B e p H y T o f i B KOJIbqO C T p y t r b l ,
HarpyXeHHofi sacTuqablIl
m ( l ) ,m ( 2 ) ,. . . ,m ( N ) , K a X A a R
M 3 KOTOPbIX W B M M O -
~ e f i c T B y eCO ~ CBOMMH 6 n w X C a k r u ~ ~COCeARMH u I.1 C HenOABtlXCHO 3 a K ~ n J r e H H b I M b f TenaMu.
Y OT
TpexAuaroHanbHbIx MaTpMq
N - 1
IIapahleTpa.
H a n p n ~ e p ,npu
2h1 = N3 - 3N
+ 2,
U p e r u e H H e 3aBMCMT
~ T MI I a p a M e T p b l MOXCHO B b 1 6 p a T b P a 3 H b l M M C ~ O C O ~ ~ M M .
perueHuH
o 6 p a ~ ~ oaR aAasrr
M ~ T O A O M O P T O ~ O H ~ J I H ~ ~ ~BP K IM ase-
CTBe 3TMX n a p a M e T p O B M b l B 3 R m HOPMMPOBOLMble K O - M ~ M ~ H T ~ I . n e p B b l M KOOpAMHaTaM X j l
EL, ~7~
= 1,
(j = 1,2,. . . , N )
C p e A M HUX MMeeTCR POBHO
K a K H CJIeAOBanO O X M A a T b . ~
C ~ ~ C T B ~ H HB~e IKX TOpB
N - 1
OHMP a B H b l P,. T ~ KKa K
He3aBMCMMblX n a p a M e T p O B ,
P BH~ C C T ~ H O B J I ~ H ULH M a T p M q b l
n0 ABYM C n e K T p a M P I ,{ l ! , . . . , P N - ~
P O n b H e 3 a B M C t I M b l X I I a p a M e T p O B U r p a n l ? ~ 0 6 ~ ~ B e H H 3bHl aeY e H H R ~ K O ~ O Y ~ HMHa T OP~t I ~ qbl. H ~ K o H ~ n p~M, BOCCTaHOBneHI.1M
L MaTpSI@I
CMMM~TPHZIHO~~
C T P Y H b l , C O r n a C H O (7.18) PI (9.9). K Y P a B H e H M R M ( 9 . 1 0 ) A O ~ ~ B J I R H ) T C RYPaBHeHMFI
me sepe3 [a]0 6 0 3 ~ a q e ~u ae n a R = hr - 1 y p a B H e H ~ f i , TO
[v]
cnpae H e
l3crre~quamrn a p H b l x
s a a ~ ~ o ~ e f i n anpeanonaramx t l k nponopmoHam-
HbIMM K B a A p a T a h l P ~ C C T O R H M MeXCAy ~~ 9aCTHqaMH, M KO~@$UW~HT~I IIponOpWO-
napaMe-rpoB.
HYXH~I.
WCTb
[$I +
sncna a. T ~ K KaK ~ o 6 a s n ~ e ~ c ~
HPIKaKJW ~ O I I O J I H l 1 T e J I b H b I t 2I I a p a M e T p b l B 3 T 0 M
HanbHOCTM 0 6 0 3 H a 9 a ~ ) ~Z I~e R pe3
k,, M k,.
T ~ KCUCTehIa ~ R CXeMaTMYeCKM
~306paXCe~ H a PMC
7.
J L M H ~ ~He3aBI.ICHMbl O
i
M COCTORT M 3 C T a H A a P T H b l X IIOJIUHOMOB,
T ~ € ! X , ~ M ~ ~ ~ H ~ J I~~ a H~ OR p n q eLiNi ( 0 )M
nony~amlqeRm113 H e e 0
IIOpOXAeHHblX ~ 6 p a c b l ~ na e~p ~ e ~
-
BOrO C T o J l 6 ~ aM I l e p ~ O f CTPOKH i Y K O P O Y ~ H H OT ~ ~P ~ X A M ~ ~ H M ~ ~J TI P ~H H UO L(0). ~ ~~ ~
06lqee p e w e H u e
CVrcTeMbl(10.1) MMWT BW:
X,
AJLRTOIU,
= cl P ( i , A )
+ cZQ(i,A ) .
!
~ KIp a C e B hbI e yCJIOBMf3, 9 1 C J I a C l , C2 H Y X H O Bbl6 p a T b T W q ~ 0 6 b 1BLlnOJMXJlMCb p a B e H C T B a ( 1 0 . 2 1 KOTOPbIP, B C M n y (10.31 CBOARTrr06b1 B
b
l
~
~
1i
CR K TBKMM:
clbP(N,A )
c1( b
+ cl (bQ(N,A ) - b l ) = 0 ,
+ P ( N + 1, A ) ) + s Q ( N + 1, A ) = 0,
1
(10.4)
I
KOTOP~IR~ Y A ~H aM3 b I B a T b XapaKTePMCTI.IqeCKMM. ~ O ~ M H OPM ( N~ ~ 1, A ) M Q ( N ,A ) H e 3 a B M C R T
+ O T b, O T K y n a C n e A y e T , YTO U ( N ,A ) H e MeHReTCR IIpM 3 a M e H e b H a -b, a T. K. KOPHM IIOJLMHOMOB [ J ( N ,A ) - 1 U U ( N ,A ) + 1 R B N ~ K J T C RC O ~ C T B ~ H H ~ 3HaYeHMRMM ~MA BelqeCTBeHHbX CHMMeTPMYeCKMX M a T p M u LN(-lbl) M LN(lbl),TO 3TM IIOnMHOMbl TIMeKlT M J l b K O B e l q e c T s e i i H b l e ~ o p m .C n e n o ~ a ~ e n ~ Ko o r,A a X npo5erae.r BCKJ B e W e C T B e H H Y I O O C ~ ,IIOJIUHOM U ( N ,A ) POBHO hr pa3 n p u n w M a e T 3 H a q e H n e +l M C T o n b K o x e pa3
IIOnHHOM
+
P ( N 1, A ) = - ~ N - ~ P (-NI , X) Q ( N + 1 , A ) = -b,v-lQ(N - 1, A )
+ (aN - A)P(iV,A ) , + - A)Q(N,,\)
(10 5)
((1,
3HaqeHMe
D(A) = b ( P ( N ,X)Q(N +b61 6,P(.N
+
+ 1,X) + 1, X j
-
P(IV + 1,X)Q(N,A ) ) b2Q(IV,A )
-
i
o 6 p a l q a e ~ cB ~H O n b .
n r o 6 b 1 e A B a PeIIleHMR
22,.
. . , : E N )I
yl, y,,
. . . ,y~ CWCTehlbI
-1
( C V e T O M KP~THOCTM). M H ~ I MCJIOBBMM, W BCe KOPHM IIOJDlHOMa
B e l q e C T B e H H b I . 30 B a X H O e CBO~~CTBO IIOJIClHOMa U ( N ,A ) B e C b M a HeTPMBMaJThHO. Haii,qe~o614uh BMA B c e x n o n b r a o u o n , 06na.qaroq~xa T m CBOWCTBOM. IIOCKOJI~KY O n H O B p e M e H H O C U ( N , A ) BTWM CBORCTBOM o 6 n a ~ a eIIOJIMHOM ~ - U ( N , A ) , 6e3 O r p a H l l r e H M R O~WOCTM,
(10.1) Y A O B n e T B O p R I o T P a B e H C T B a M
M O X H O O r P a H M W T b C R P a C C M O T P e H M e M IIOJWHOMOB YeT-
H O ( ~H ~ Y ~ T H O CT ~ e~n )e H l l C IIOJIOXMTeJlbHblMM (UTPIII@T~JEH~IMM)
- b , - l ( ~ ~ - l-~~, , y i - I ) b , ( : ~ , +-~ Xy, ~ U,+~) ~3 ~
m
p
b c~ n e Ax y e T , *!TO
bi(xi+lyt- ~ 8 YaCTHOCTM,
npn B c e x i
= S,3,. . . ,N
= 0,
YTO O~~BMAH O , BCe
-1
, ~ i += i )b i - i ( : ~ i ~ i - 1X , - ~ Y ,= )
IIOnUHOMbl C TaKMMM CTapUIMMM K O @ U ~ I ~ H T ~ M M ,
K o T O P b l e H a B ~ Y ~ C T B ~ H H OOCM R
N pa3 n p K H H M a K 3 T 3 H a Y e H H e f 1 HMeKJT BHA
IJ(N,A )
. . . = b1 ('X Z Y ~- :cl?+).
COTnaCH0 ( 1 0 . 3 1 ( 1 0 . 5 )
II(lV, A)
+ qII(N,A),
1.1
p; 5 p; 5 . . . 5
H ~ M HYXHO
P(N
f l ( ~1;)
= (-l)N
(10.8')
-
/I>,
q
> 0-
+ 1, X)Q(N,A ) ) = b~ . (I0 6,
-
I
II~OM~BOJI~B H e~l Iq ~e c r e e m b l e n a p a M e T p m
HBFITU H ~ O ~ X O A P I M51~ IA~O C T a M Y H b i e YCJIOBMR,
YAOBJleTBOpSI'b 3 T U I I a p a M e T P b l
-
(10.8)
j=1
b, ( P ( 1 ,X)Q(P, h) - P ( 2 ,,\)Q(l, .\)) = b, ,
+ 1, A )
1
N
+
( P ( N ,X)Q(N
=
rAe
( P i h i ,A)Q(r'i I\), - P(JV+ 1, A)Q(N;X)) - b,v-l ( P ( A r- l , X ) Q ( N ,X) - P ( N , A)Q(IV - 1. A j j =
C T a P U l H M N KO*
@~~I~M~HT~Mu.
M r O , Y ~ 0 6 b IlOnWHOM l
K m P b I M AOn-I
(10.8) I I p H H F M a n
H a Be-
~ C C T B ~ H H OOCM R 3 H a Y e m l e - 1 T o m e N p a 3 , T. e . Y ~ 0 6 b 1n O J M H O M
U ( N , A )+ 1 = 2+qII(N,X)
(10.9)
$1 0 m p a ~ e a ua a n a q a C n e K T p a n b H O r O aeannaa nnu n e p n o m q e c a m
~ a ~ p ~97 q
< . - . < /l,,, a MX KPaTHOCTM Y e p 3 1 + 2 ~ 1 , 1 f 2 ~ 2 , .. . , l + 2~~ < me .z, 2 0. ~ ~ M ~ T H vMm, p = 21, ecnu N w r ~ o ,M p = 21 - 1, e c m N HeYeTHO. T ~ KK a K lim n ( M , X) = +a, TO H a t I H T e p B a n e (-m, / [ I ) 3 T O T n o m o ~
All
X--m
tiecrrpMqaTeneti, H a n H T e p B a n e
(pl,p2) - H e n o n o x u T e n e H ,
H a ~ ~ T e p B a (n~e4 2p, 3 )
y = --
y = ( - I ) ~U(N, a,)
..-
--
ecnn N remo, n
H M e n TOJIbKO B e q e C T B e H H b I e K0PH.M. ~ Y ~ B ~ I A HqO TO , 3TOT nOnHHOM H M e e T CTOnbKO B e 4 e C T B e H H b I X KOpHefi,
CKOJIbKO
pa3
n ( N , A). . nOJU4HOMa n(N, A )
-
=
I I p R M a R IJ
I I e p e C e R a e T (C Y W T O M
K ~ ~ T H O C rTp~a~@ ~ l)~nOJrvlHOMa
T ~ KKa K
B C e KOpHM
B e U e C T B e H H b l , TO M e X A Y
K ~ X ~ O R
n a p f i COCeAHMX K O P H ~J I~e X M T ~ A I ~ H C T B ~ HKH O P~ eIH~b e T O I I P O M ~ B O A H O ~ ~HeA. CTBMTeJIbHO,
n y C T b Vl
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<
V2
<
.- .
MX KPBTHOCTM,
B KaXAOM HHTepBaJIe
(v,, V i + l )
<
Up B C e p a 3 J l P I q H b l e KOPHM 3TOTO ~ O n I ~ I H O k I a ,
T a K qTO
zrrI = fl,
N .
no
TeOFMe
P o ~ ~ R
J I e X M T IlO ~ p a f i ~ eMf ei p e OAI.1H K O P e H b I I ~ M ~ B O ~ H O ~ ~ ,
+
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1 n z , , rne m, 0. Cneno~a~enbao, [vi, I / ~ + ~ , )n e x m ,ai - 1 1 ,In, = n, m i ~ o p ~ e k ~ T ~ M ~ B o A H ao BO ~ ~B , C e M C e r M e H T e Ivl, v,] JIeXCJlT a o64ee
MX
smno
B TOM n H T e p B a n e p a B H o
+ +
B KaxnoM nonycerMeHTe
+
eCnr1
N
H a MHTepBanax
-'
HeYeTHO.
B H 3TMX ~ M H T e p B a n O B OH H e O T p M q a T e J I e H , PI l l p R M a R y = (4 > 0) M O X e T n e p e c e K a T b ero rpacpn~T O n b K O H a 3 T n x 1 . r r l T e p a a n a x . 0 6 0 3 H a . r M M q s e p e 3 i t , w c n o pa3J1MZIHblX ~ O p H e f Y i ~ T H O KPPTHOCTM, ~ n O n a B L U M X B M H T e p B a n (/JZs1 , /.LZr). H Y e p 3
2(1
+ 6!')), 2 ( 1 +
665)),
. . . ,241
+ 6:;)) - kIX KpaTHOCTH. Torna B C O O T B E T ~ B y F O ~ e M f 2 Pa3JlMZTHblX
C e r M e H T e [ / L ? , - ~ , /lZs]J I e X G l T ,Z,
K O P H R , p a 3 A e 3 e ~ ~ bns l ~f
B a n O M , B K B X A O M M3 KOTOPbIX J l e X M T OAPIH ~ P O C M ~ K I OpHb ~ ~ P I ~ B o J ( n'(N, H o ~ ~A).
N I I O J I y C e M e H T [p21-l, +fX) COAePXMT nl M H T e p B a n a M P I , B K a X A O M $13 KOTOPblX H ~ B o J ( n1(N, H o ~ ~ ,\).
n P M HeYeTHOM
~ O p H e f i ,p a 3 A e J I e H H b l X C T O ~K O ~ PeHb
a:'
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B O ~ H ~ Y U Mq e p s m H a n n e H s u l e e MJ
.I
wcen
I~(N,
(1 5 j
+1
1M H T e P -
5
11,
+ 1)
Pa3JUlYHbIX
J I e X H T OJJMH
np-
a:'')l:
= min II(N, 313
ee ~
0 p ~ e f Ti .a K K a K n P O M 3 8 0 A H a R M M e e T TOJIbKO
HYJIIO, T.
e.
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(v, , I/,+,)
A'
-
1 KOpeHb,
T 3 BCe Tfl, P a B H b I
A ~ ~ C T B M T ~C~O A ~ eHP XOM T ~ A H H C T B ~ H H ~ ~ ~ ~
KOPeHb I I P O M ~ B O A H O ~ ~ .
Ha
(v,,~ i + ~JIOnMHOM ) COXpaHReT 3HaK, M n P R h l a R y = (q > 01 M o m e T nepeceqb ero r p a + u ~T o n b K o H a T e x n ~ ~ e p s a n a x rze , OH n p M H r I M a e T O T p M q a T e J I H b l e 3HaZIeHMR. T ~ M KK B K a X A O M G I H T e p B a J l e J l e X M T ~AMHCTB~HH~ K IO~PIe H b CY, E ( I / ; , v i + 1 ) _ ~ P O M ~ B O A H OT~O~rpa@lK , IIOJIUHOMa n ( N , A)
-%
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a>
em ma pa34 e c m 0 < )n(N,aj)l.
nepeceKaeT €!CJlM
I I o A c w ~ a eo6qee ~
C O ~ ~ ~n Cp nH bIO ApAeMy, npM 5 m H a K B X A O M M H T e p B a J I e (p2,- p ? , ) y= nepeceseT 2(ns 1) pa3 r p a + n ~n o n M H o M a n ( N , A), a H a MHWP B a n e ( p 2 ) - +m) .(npn N ~ e s e r t i o o~H)a ero n e p e c e s e T 2 n l + 1 pas. I I ~$H >m no K p a k ~ e f Mi e p e H a OAHOM n H T e p B a n e yra n p R M a R nepeceser r p a g w ~ ~e %nee, s e M %, pa3. 0 6 0 3 ~ a w064ee i ~ w c n o n e p e c e a r r f i sepea M . Torga, ecrrx < m, TO
-:
npnMaR
9
BMA,
Y1 n p R M a R tJ
J n ( N ,a,)[,n m due
w c n o nepecere~uk(c
y = - 2 , ~ P ~ @ M K O MI I O m H O M a n(N, A ) . &OT npn I I e p e X O A e Y e p 3 KOPHM H ~ Y ~ T HKOP a~T H O C m ,
=
-a
He nepeceKaeT,
Y ~ W O M ~ p a ~ ~ o c w nf p i )~
~ofi
IIOJMHOM M e H R e T 3 H a K TOJIbKO KOTOPble Mbl & ~ H ~ W M Y e p 3
npn
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+
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BMeCTO
B K a Y e C T B e AOIIOJTHMTeJIbHOI'U
napaMeTpa,
83RTb K ~ K W - H M ~ Y C O~~~C ~ T B ~ H H W3 H a Y e H M e M a T P H q b I
MOXHO
6b1n0 6 b 1
LN(bN),HO em
06JlaCTb
AOnyCTMMbIX 3 ~ a Y e H M f HaXOAMTCR i MHOm cJIoxH~~)
B K a Y e C T B e OCTaJIbHblX N- 1n a p a M e T p . 3 MOXCHO B 3 R T b YaCTWTbI C & T B ~ H H ~ U ( ~ o n e 6 a ~ ~CrMf Ci W M ~ I , n o n y ~ a m w e A c ~~3 : p a c c ~ a ~ p ~ ~ ~ aecm e ~ o Hf ein,o , q m x c n o 3 a K p e n M T b YaCTMrIy C M ~ C C O lnjv ~ (CM. P n C . 11) M a ~ p u ~ L Na- i ( 0 ) T ~ K O R CMCPeMbI n o n y Y a e T c R ~3 M a T p n q b I LN(0) d p a c b r ~ a n nocneg~eh ~ e ~ CTPOKM M n o c n e g H e r o c ~ o n 6 q a( n o q e ~ y ? )M ee C O ~ ~ T B ~ H~ HH ~~ IY~ ~ H pMl ,Rp 2 , . . . , p ~ - 1~ J I R I O T C R K O HRMM n o n M H O M a P(N, A). BM~CTO YaCTOT C O ~ C T B ~ H H ~KI oXJ I ~ ~ ~ H u B & K a Y e c T B e nap a M w p o B qenecm6pa3~eeB ~ R T C ~~ ~ C T B ~ K~ H H ~~ IY~ ~ HpM l , pR2 , . . . , p ~ - p~a e,m I e n
nOnmoMb1
P ( N + 1,A)
TPeXAMaKlHaJlbHbIMM
K B a A p a T a M 3TMX YaCTOT.
-
&(N,A) N - A w N M a T P M q a M P l LN(0) M L(0) no M
2-A
cTeneHn nopoxAaioTccr
M3BWTHbIM PeKYPPeHTHbIM
W ~ M ~ J I ~MM 3 KOTOPbIX, , B YaCTHOCTM, C J I e A y e T , YTO MX C T a P I I l M e K O @ H ~ M ~ H T ~ I
.
CO(YTBeTCTBeHH0 P a B H b I ( - ~ ) ~ ( b l b ~ .b N - l ) - l C ~ a p I I l dK O ~ @ $ U ~ M ~ HnTO m H O y a
U(N, A )
@ O p ~ y n(10.13) e q = (2b1b2 . . . bN) -l.
H
( - ~ ) ~ ( b ~. .bbN-l)-l. ~ .
paBeH
(-I)~ (2bl b? . . . bj-,-i
b
C 0 6 C ~ B e H H b l e3 H a Y e H M R MaTPMIJbl
~OCMOTPMM, K a K y H )
M H W P M ~0 ~M aM T pH M q~ e L,v(-bN)
p r , ~ ; , . . . , ~ ;9;;
pl,kLz,...~LL~-l
~ H ~ Y H T -1 ~ ) ki 6
LN(bN)
JTBJIRKITCR KOPHHMM IIOJIMHOMa
M O A H 0 3 H a Y H O OIIpeAeJIRHYTCR C n e K T p O M M a T p M q b I
L N ( - b ~ )M
( 2 b l b 2 . . . bN)-'. B 3TOM COCTOMT T a 3aBMCIIMOCTb L N ( - ~ NM) LN(bN),0 K O T O ~ RrOBOPMJlOCb B b I U l e .
IIapaMeTpOM Q
=
(9.111
rni+l
A n R O A H O 3 H a W O r O BOCCTaHOBJIeHMR M a T p M q b I
LN(-bN) HYXCHO, K p M e ee C n e K ~ p & p ; , / I ; , . . . , p i , 3 a A a T b ewe N He3aBMCMMblX n a p a M e T p B - H a OWH 60mme, Y e M B T p x A H a r o H a n b H o M cnyqae. B K a q e c T s e 3mm AOnOJlHMTeJIbHOrO n a p a M e T p a ~
n O T O M y YTO,
BO-IIepBblX,
~ 0 B3FITb 6 ~ BeJIMYMHY 0
O H a B M e c T e CO C n e K T p M MaTpkIIJbI
3 H a W O O I r p e A e J I R e T nOJIMHOM
mi
MeXCAy C n e K T p a M 5 1 MaTPMIJ
~ P M C T K ~ PeIIleHMH, ~ ~ R O ~ P ~ T H3 aOA ~a Y M C n e K T p a J I b H O r O a H a J I M 3 a . 3 a M e T M M n p e X C A e BCerO, YTO, COrJIaCHO
U(N,,\) (POPMYJIOR (10.131 a,
L N ( - b ~ ) OAHO3HaWT,
M CneKTp
MOXCHO M3BJIeYb M 3
COBOKYnHOCTM
mi- 1
(10.16)
~
I
L N ( b N )R, B T I R I O ~ P I W RKOPHRMH IIOJIHHOMB 1 + U ( N ,A). A a n e e , n3 p a s e H c r s a (10.6) npn A = pJ c n e A y e T , sm - P ( N +1, / l j ) Q ( Np, j ) = bl, T. e .
C n e ~ o s a ~ e n b ~mcna o , KBaApaTHOrO YpaBHeHHR
b;l P ( N + 1, p j ) H - b;' Q ( N ,pl ) z2 - 2 U ( N ,Aj)Z+ 1 = 0, T. e .
R B J I R ~ T C R KOPHRMM
~ ; I P ( N +i , p j ) = U ( N , ~+u,J-, ~ ) - ~ ; ~ Q ( N ,=P u~( N ) , P , ) - 0,
JW.
3=1,2,...,N - 1 ,
(10.17")
A ( N , A ) = ( - I ) ~ ~ - ' u (A) N, -(-1lN q 2
rne R ( N ;A) = njN=;l(A - pj). ~ O ~ T O M Y~, 0 6 a s m K COBOKynHOCIIH (10.16)IIOCJIeAOBaTeJIbHOCTb ffhr-1,
OIIpe,Z(eJIRIOqyIO
3 H a K H paAHKaJIa,
Mbl CMOXeM
01,
-
' (-bylbNQ(N,A))
02,. . .,
BOCCTaHOBHTb IIOJIHHOM
- b ; l b ~ Q ( N ,A ) , a B M e C T e C H H M M IIOJIMHOM b i l P ( N + 1, A) = 2 U ( N ,A) + bylbNQ(N,A),
(10.19)
signbilP(N
+ l , p J )= sign (-b;'bNQ(N, p j ) ) = sign U ( N , p j ) ,
YTO
,.
-
n03BOJlReT 3EInMCaTb 3Ty @ J P M Y JBITY aKOM BMne:
-
( - l ) J P ( N + l , p , ) > 0 , . . ., oTKyna, cornac~o(lO.231 cnenyeT, Y T o U ( N , ~5~-1, ) U ( N , p z ) 2 1 , . . . : ( - l ) J U ( N , p , ) 3 1,. . . . 0 6 p a n r ~ l u ~ cK brpaaney nonplaofila u ( N , ,\) (pnc. 8), y 6 e x n a e ~ c ~TO , 3TH HepaBeHCTBa BbmOJIHRKurCR Torna rr TOJlbKO TOTAB, KOrna
- +
omyna cnenyer, qm M e w CoCenmM KOPWI no.mmoMa a ( N , A) n e x m ~ o p e ~ b non14ao~aP ( N 1, A). Ewe nsa KOPHR m r o nomHoMa nexaT B mTepsanax (-w, p l ) M ( p N - 1, +m).C n e n o ~ a ~ e n b KOPHM ~ o , nonMHoMoB F ( N 1, A), n ( N , A) nepeMemamTcq n, cornacHo TeopeMe 2, c y r q e c m y ~ -pex,qnaroHaaHan MaTpnqa LN(0), cneKTp KOTOPOR COCIDI~Tu3 K O P H ~ RnonnHoMa P ( N 1, A), a cnemp yrtopoY ~ H H oMBTPMDI ~ ~ L,y-i (0) COCTOGIT M 3 K O ~ HP ~ I , 112, R . . . ,/LN n o n n ~ o ~ a . r r (A). N, 0603~aYUMYepe3 a l , a ? , .. . , a~ ,4MamHanblible 3neMeHTE.l MaTPMqb1 LN(0), qepe3 -bl, -bz,. . . , -bNTl - aneMeHTbr no6omb1x nwaroaaneii, M PaccMoTpuM nOJlyYaIOqylocR M 3 LN (0) MaTPMqy Lp,(-bN), Y K O T O ~ R
+
+
T ~ K P Io6pa30~, M nnfi Toro, q~o6b1 ~ a 6 0 (10.20) p RnnRncR cneKTpanbmrMkr naHHbIMU H ~ K O T MaTPliqbl O ~ ~ ~ L.N(-b,"), ~reo6xo~rrlwo, Y~06b1BbInOJlHflJlilCb HepaBeHCTBa (10.221 (10.24) ~ O K ~ X ~YTO M ,3Tkl HepaBeHCTBa RBJIRIOTCR T a m e AOCTaTOWbIM YCnOBMeM. ~ Y C YMCJla T ~ /L;, /l;, . . . , / l s M q ynOE.7!WEOPRK)THepaBeHCTBaM (10 11)(10.11') U (10.22) ~ O C T ~ O P IMM3 HMX nOJlMHOM
-
Kop~wIlOJIMHOMa P ( N + 1, X) COBnaAaF3T C KOPHRMM XaPaKTePUCTM'leCKOr'OI'IOJIMHOMa P ( N 1, A) MaTpuqbi LN(0), OnpeneneHHoro w p ~ y n o f(10.51 i M,T. K. MX cTap l u ~ Ke O ~ ~ M W ~ HPaBHbI T ~ I (-1)N2~ PI ( - I ) ~(bl . . . b ~ - 1 ) -' COOTBeKTBeHHO, 'TU
+
-
-
Ha p ~ c8. . HarJmnHo B ~ I A H O r p a + u ~ anomiHoMa U(N, A), $1306pamc.~noro O n p e ~ e n nTenepb ~ nomrHoM P(IV+ 1 , ,\) ~lmp~yJI0R
TO
Teopema 5. A m m 2 0 , vm06b~na60p senuqun (I 0.20) slemncsl cnetcmpanb~at,+tu a a ~ l i b m u~ ~ m p omampqbr i i LN(-bN) n e 0 6 x c d ~u awmnmattw, rmo6br qucno q yaoenemeopslno nepaeewcmy (10.22), a vucna p l , pz.. . . , /LN-I - HepaeencmeaM (10.24), zae m - H ~ U M R ~ U Unotcanbnbrzi ~ ~ mrccuMUM n o n u n o m ( - I ) ~ - ' (A - p;) u p:, p:, . . . ,p+N - KOPWU nonunoma
HEl
2
+ (-l)NqfllN=l(A - p j ) .
I l o c ~ a ~ o aanasir e~a
n o ~ c ~ a Jm s n B~b r~p a x e H w R ~ ~ ~ B H ~ H M R
B (11.1)
x
nonaraR
y(j) =
m;f
x,,
n o n y w M A J m 2,
CMCTeMy
P~CCMOTPMM CTPYHY,
COCTORWyIo H 3 HeOAHOPOAHOrO Y q a C T K a , o 6 p a 3 0 ~ a H H o r 0M a c -
ml ,mz, . . . ,Tn,
C W A H H e H H b I M H I I p Y X l l H a M M C XeCTKOCTRMM k ~k 2, , . . ., knel, K KOTOPOMY n p u K p e n n e H b 1 Ass ~ ~ C K O H ~ W O~ AI HXO ~ O A H ~ IW X a c T K a , COCTOJTwx ~3 OAHHaKOBbIX M a c c . . . = rn-3 = m - ~ = m- = mo = m n m , , + 1 = m n + 2 = . . . = m, C W A H H e H H b l X OAMHaKOBblMM I I p y i K M H a M H C XeCTKOCTSMH mn+3 = . . . = k - g = k, = kn-I = k o = k r ~ k , , = k,,+l = k,+? = k n + 3 = . . . = ~ ( P H C .12). Be3 OrpaHMYeHMR O~UI;HOCTMM O X H O CYHTaTb, YTO $ = 1, T. K X O r O B C e r A a M O X H O CaMH
A ~ ~ H T ~ cBJ Tb 1 6 0 ~ e0A~H H H q b I B p e M e H M ( n o s e ~ y ? ) .
OV~BMAHO, YTtJ C O ~ C T B ~ H K H O~JII & ~ ~HMR BbIpaXaIOTCR HMR
Y e f E 3 OrpaHMYeHHbIe fEWe-
3 ~ 0 fCMCTeMbI: i
He3aBHC-e OT j Irl t , KOHCTaHTbI. =mnpu j n + 1 n n j~5 0 , a k , = k n p n j n n n n j 5 0 , m a l = : ! n p n j 2 n + l u n p n j _ < O , a b j = l n p w j > n + l ~ n p ~ j-1. < r A e C - llPOM3BO.ilbKhIe, TaKKaKm,
r k I 3 m ~M y a T p H q a L-,,,
>
CMCTeMbl M M e e T T ~ K BMA: O ~
>
a,,
-b, 3KBHBaJIeHTHblMM C O O T B W C T B y I O w M Y p a B H e H M H M CMCTeMbl(11.2)
+
Hanp~~ep,
en+']
e + ( n ,8 ) = b i l [ ( e 8-')en+' = bilen, e+(n-1,e) = b ~ ~ , [ ( a , , - 2 + 8 + 8 - ' ) b ~ ' ~ " - b ~ 8 ~ + ~ ] = b;~,b;'lOn-' ( a , - 2)On + (1 - b:)en+']
+
PI T.
n.
3 a ~ e q a q TO b, .= 1 npn j
2 n
+ 1, ~ T Mp a B e H c T B a
M O ~ H O3 a n u c a ~ b
ORHOG @ p ~ y J I o h
me Cd = fi M P ( j , X ) - O r p a H H Y e H H b l e AaHHbIX
P ( 0 , X ) = 0 P ( 1 , ,\) = 1,
L,,
~ a ~ p M q e f i
n o ~ r y s a m l q e R c nM 3 M a T P H q b l
M C T O J I ~ V O BC H O M e p a M M , M e H b l U H M M
&JM
L,-,,,,
0 T 6 ~ a ~ b l B a ~ M e BCeX h.l CTPOK
'
N
~ H O P O ~ H ~YYaCTOK, IR
nOJIYYMTCR K O H e q H a R
M ~ C C ~ M Pbr ; 3 a K p e n J I e H H a R M e X A y ABYMR H e I I O A B U X H b l M M
12) Y 3'l.O~C T P y H b I O N - 71 f 1 OnMHaKOBbIX M CTOJI~~OB MaTpM-1 ' L,.
N - 11 OsMHBKOLN M a T p V I q a COCTObIT M 3 I l e p B b I X
O n O p a M M (PMC.
A H O P O ~ HYYaCTOK ~ ~ ~ COnePXItT
B b l X MaCC M
nPYiK&IHIa
HaXOARTCR
no
e'
C C ~ C T B ~ H Hr ~ oI ~~ 1 e 6 a ~ K 1 .O 1 ~H ~ ~ HC T Op ~y H b l
'I = 1 , 2 , . . . , N , a A p a B H O C O ~ C T B ~ H H ~ ~ M
@ p ~ y J I e (11.61 B K O T O P O ~ ~
3HaYeHMRM M a T P H q b I
LN,T. e. KOPHRM nOJIMHOhia P ( N + 1, A).
B~I~CKM K ~M KO , %B H A
H a OAHOpOAHblX y Y a C T K a X MMeWT ~ 0 6 C T B e H ~ b lKOJIee
I
~ ~ H M R ~ ~ C K O H ~~ YO H J IO~ ~~ , ~ C K O H ~ Y H O ~ & ~ K O H ~ ~ H O ~ C T P Y H ~ ~ .
T ~ KKa K a, = 2, b j = 1 CMCTeMbl
A t ( j , j ) = 1; A + ( j , p ) = 0
p?j
B I~OJI~'~~CKOH~ZM C TOP~Y H e 3 a K p e n M T b M a C C y m,V+l H a H ~ ~ O ~ B M X H O ~
CTpyHa, HarpyXCeHHaR
CTpOK
B: = n b ; ' ;
1.
O n o p e H y 6 p a ~ bB e C b n o c n e ~ y m q r . r f i O
N
rne
p e U I e H M f l C H W e M b l (11.2) n p P 1 H a q a n b H b l X
T.e. C T a H n a p T H b I e nOJIl~1HOMb1, n O p O X A a ? M b I e
(11.2) C H O M e p a M M ,
npbr
j
2 " + 1 (H n p t t j 5
6 0 n b m b r ~ r 1 ,V e M
~ ~ c K o H ~ w Y I CMCTeMY O Y P ~ B H ~ H H ~ ~
n
-I),
e+(k - 1,8) = B L ,
TO BCe Y p a B H e H M R
+ 1 (i.1 MeHbUIbIMbI, q e M o), 06pa3yro~
A + ( k - 1,p)OP, p 2 k- 1
rne
A + ( k - 1 , k - 1) = A t ( k , k ) = 1 A+(k - 1, k ) = (ak - 2 ) A + ( k ,k ) A + ( k , k + 1) A + ( k - 1 , p ) = ( a , - 2 ) A t ( k , p ) +.4+(k,p - 1)
+
+A+(k,p
+ 1 ) - b:A+(k + 1 , p )
p
2k + 1
y m A + ( k - 1 , k - 1 ) = l u A + ( k - 1 , p ) = O n p n p > 271-(k-1) C n e ~ o ~ a ~ e ecnn m ~ oe, p ~ y n (11 a 8) B e p H a A n n B c e x j k , ,mo H a B e p H a A n n B c e x J 2 k-1. Ananorm~o,H 3 p a B e H c T B (11.7-11.7')c n e A y e T , YTO oTKynacneAyeT,
>
a*(tJ) =
-
b*(0)
A - ( j , p) = 0
npU
p
<
2n - j
Ff
K O ~ @ H ~ M ~ H T ~AI -
( k ,P)
+ +
+
+ +
+
yAOBJIeTBOP5lIOT
M c n o J I b 3 y f l BPOHCKMaHbl, e p ~ y n b Al n R
Haohpm, IIyCTb K O X # J W ~ M 4+ ~ (H3 ,T p)~(I. & - ( IP, ) ) YAOBneTBOPflIOT Y p a B HeHMRM (119 )((11gl)), B KOTOPbIX Clk M bIL# 0 - I l p O M 3 B O J I b H b l e Y M C n a , a A + ( J , p ) = 0 a, = 2, b: = 1 n p H npn p > J n ( A - ( 1 , ~= ) O npu p < J 5 1 ) k n 1 ( a , = 2, 8:-, = 1 npu I, 2 0 ) M O n p e A e n e H H b I e r $ u p ~ y n a ~(11 n 81 (118') n O C J I e A O B a T e n b H O C T M e + ( ~ , e-(1, 0 ) y A O B n e T B O p R I O l Y p a B H e H H R M (112) I I p M = 2 - 0 - 0-' *O A O K a 3 b l B a e T C f l H ~ I I O C ~ ~ A C T B ~ H HI OI ~R o B ~ ~ K o~ %~ M ~ Tewe, U M 9 T O M 3 p a B e H C T B (119), (119') B b I T e K a I O T T a K M e ~ P M Y J I A~n If l 3 n e M e H T O B f l k U bL
> +
-
+ +
x ( k ) e f ( k 1 , 0 - ' ) - x ( k l ) e f ( k ,0-') e* ( k ,O)ef ( k 1,B-I) - e* ( k 1,0)e* ( k , 0 - I ) ' s ( k ) e f ( k 1 , 0 ) - x ( k l)e* ( k ,0 ) e*(k,B-')e*(k 1 . 0 ) - e*(k 1, 0-l)ei(k,O) .
KOX#JM~M~HTOB a* ( 8 ) , bi(0)
M W -
HO, OYeBMAHO, I I e p e I I H C a T b T a K :
>
e),
B SICTHOCTM, P e u l e H H R P ( j , ,\(0)), e + ( j ,8 ) . e - ( ~8, ) CMCTeMbI (11.2)C B f l 3 a H b l M e X A y c6oR p a s e H c T s a M n :
+ e + ( j ,0 - ' ) b ( B ) ,
(11.11)
e'(j, 0 ) = e - ( j , 0 ) n - ( 0 ) + e - ( j , 0-')b-(B),
(11.12)
P ( j , X ( 0 ) ) = e + ( j ,B)a(B)
T ~ KKa K IlOCJIeAOBaTeJIbHOCTU e* ( 1 , e ) YAOBJIeTBOpRIOT CMCTeMe Y p a B H e HMfi (11.2)n p U J I I O ~ O M0 U A(@)= X ( e P 1 ) ,T O IIOCne,JJOBaTe,IlbHOCTM e* ( J , 0 - ' ) T O X e yAOBJleTBOpfleT BTOR CMCTeMe ~ ~ ~ B H ~ K H p I o I ~ ~ TOrO, eL e + ( J ,0 ) = 8', e + ( ~0 ,- ' ) = 8-I n p U J 2 n + 1 ( e - ( J , B ) = e l , e - ( J ,0 - ' ) = 8-J n p U J O), O T K y A a C n e A y e T , sTo P e u l e H M R e+ ( 1 , 8 ) U e + ( J ,8-' ) ( e - ( J , 6 ) j U e- ( J , 8 - I ) ) J ~ M H ~ ~ He3aBMCJiMb1, ~ H O eCnM 0 # f 1, T e ecnu X # 0 , 4 ~ O ~ T Onpn M 0 ~ # f l m6oe pememe T ( J ) c n c T e M b 1 (11.2)m J m e T c R HX nn~eivrokK O M ~ H H ~ ~ J I ~ ~
+
e - ( j , 8-') = e + ( j ,B-')a+(f?) e + ( j ,0)b+(6'),
.-,
(11.13)
me
>
+
x ( k ) = e* ( k ,O)af ( 0 ) e* ( k ,0-')b* ( B ) , x ( k 1) = e* ( k 1,0)ci* ( 0 ) e k ( k 1,B-')b* ( B ) ,
+
+
+
+
.
,
aT(0)=
1-?
:3"
-1
0 ) - 1, { e r ( k ,0 ) ,e r ( k ,0-1)) '
+
+
b f ( 0 ) = { e - ( k , O"),e+(k, 0* ')I . { e r ( k ,0 ) ,e ~ ( k , O - l ) }
(ll.l+)
32 8 ,D
r ~4 + b
z,g-
n
wc
PE
y
s
E
B
gr
2n
g
s
'
I T
WQS
2
2 &
Em-;
a O2 Ww
X
0
~
4
x
S
1
L%%
ysg ,,o%
nzm
'
$33
9z ---&
a -s = s
g LZ
x m u
"
n
---
z
"+
-=: "
1 S g g" oS
-
lpgm
y
ga m i
Q S ,
BY, Y m b
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:8
a $
=I 0 N
'g-" 1 1 "
8112;~
-
r"i"w q3.;:;
Ha O A H O ~ A H O My y a c T K e C T P Y H ~ I (T. e. e+(j,0 ) = 8J UJIU e - ( j , 0 ) = 0' w
'Z O T K y A a C n e A y e T , YTO KO-HLM~HT~I
a-(8-')
U
a - ( 8 ) OTnMYHbI
( j ,t ) =
OT H y n R BO BCeX
KOrAa
j > n
+ 1 nm j
< 0 ) mj
e-i(jv+wr) + et(jv-wt)r + ( B ) , j > n + l t(e~)e-f(jv+wt) j
= m,
(11.31)
mwax ~ A M H M Z I H OOKPYXHOCTM ~~ 101 = 1. ~ O O T O M Ynpw 0 = eiv (0 5 cp < 27r) M o m H o o6e Y a n M P a B e H C T B (11.11)-(11.13) p a 3 A e n M T b H a 3 T H ~ 0 3 @ ~ 1 q l l e ~ W ~ bnpMBeCTM l MX K TaKOMY B W Y :
T O r A a 5l TOJIbKO T O r A a , K O r n a BCe n p y X w H b l PIMeIoT O A H H a K O B F AJIMHY, K O T O ~ Y ~ O
(s,
e+ 0 ) -- e-(j, 0 ) e-(j,8-l)r-(e), a- ( 0 ) e - ( j , 0-') = e + ( j ,0 ) e + ( j ,0 ) ~ + ( 0 ) , a- ( 0 )
+
+
Mbl 0 6 0 3 ~ a Y ~ 9epe3 M HanpMMep,
h.
~ o ~ T O M Y eCJIM , n p e ~ e 6 p e Y bcW.7lofi
TRXeCTM,
q T 0 C T p y H a HaXOAHTCR B HeBeCOMOCTH HJIM CKOJIb3HT
6e3
CWTaR,
TpHMR
no
~ O P M ~ O H T ~ J I IInOCKOCTtf, ~HO~~ TO B COCTORHMH nOKOR B OAHOPOAHbrX )'YaCTKaX BCe I I P Y X M H b I MMeHYT OAMHaKOBYH) A n M H Y h, a KOOPAMHaTbI MaCC P a B H b I Z = Z O ( j )= jh n p M j < 0, M Z = Z O ( j )= Z O ( n ) (3 - n ) h nptr j > n + 1. CTIe~osa~enb~o, j = h-'Z ( j < 0 ) ; j = h P 1 ( Z - A ( n ) )( j > .n+l), m e A ( n ) = Z o ( n ) - n h .
+
.-
( j ,t ) =
t ( ~ ) ~ . ; u ( Z - A i ~ ~ )!+ c ( v ) f ) e i ~ ( Z - ~ ( ~+ ) t e-it,(Z+c(v)t) j -(B),
j>n j
0~~axfCeHH0fi OT H ~ O A H O P O A H O C T(M ~~~ 0n0pb1) Bomb1 T ( B ) ~ ' ~ ( ~ - ~ ( " ) - ~ ( ~ ) ~1! ) , JIehllna 9. B Kpyze (81 < 1 newam moabxo npcmbre o~rnpulpmnbnbrempnu 6eryrqeir B +m. Co6cmemoe ~ o n e ~ za- ( ~j , t~) 6e e c ~ o ~ e w or ir ip y ~ bHa l yvamKe j @ ~ R K I ~ uu(O-'), ~C a - ( B ) , u a amus x o p ~ ebmomRm.q a maw paemma j < o o6pa30aa~oHanoxceHueM BonHbI e i v ( Z - c ( v ) f 6erywek ), n3 -m M mpaxeHHoR m H ~ O A H O ~ O A H O CB~ ~ OR ~ e -~i YI( Z + c ( Y ) t ) r - ( 8 6 ) , e m e t i B -m, a Ha y ~ a r r j~ > en OHO C-HT ~3 ~ 0 ~ e ~ - i ~64 (1n ) e i ~ ( ~ - c (t v( B) t) ), npouremefi qepe3 H ~ O A H O ~ A H O C ~ n npononacamrrlefi ABuraTbcR B +co. H ~ K O H c~oQ6 n a e ~ ~ o~ eo n e 6 a ~ nz+e ( j , t) Ha p a c m e j > n 06pa30~aH0HanoxeHneM B O ~ e'vA(n)e-'l'(z+c(u)l~, ~ I 6emeR u3 +m M mpaxemofi H ~ O A H O ~ A H O C T R M Mm m b e-iYA(n)eiv(z-s(u)t)r+(B), ~ 6eryrueh B -00, a Ha pacmce j < 0 OHO COCTOMT u3 B O ~ H e~ -Ii Y ( Z + c ( v ) t ) t ( npome~mefi ~), sepea H ~ O A H O ~ A H DMCnpo~onxamrrlefi ~ AsnraTbcR B -00. A6conmma~Benurma CKOPCKTM ABMXeHMR BCeX 3THX BOJM OAMHaKOBa H p=Ha c ( u ) . ! T a ~ m06pa30~,Hannwe ~ e o n n o p o ~ ~ o cnpoRsnRercR ~efi B o6pa30~a~1.1~ oTpaxeHnbrx BOJM M ocna6nemn BonH, npoureAulux qepea H~OAHOPOAHOCTM. MHOJiCHTeJIbT ( 0 ) .Ha3bIBaeTcR K O ~ M H ~ M ~ HOTPaXeHIIR ~ Q M ~ O ~ ~ ~ ~ C K OCTPYHbl, H ~ Y H O ~ ~ MHOlKMTeJM r - ( 8 ) M r + ( B ) Ha3bIBamTcfl neBbIM M MPaBbIM K O ~ ~ M ~ I ~ ~ H T ~ Mi M OTpaXeHMR, a MHOXMTenb t ( B ) - K O ~ $ X $ M ~ H ~ H T O M IIPOXOXCAeHMR ~ W K O H ~ Y H O ~ ~ C T P ~ H (~ C Lwae~cq s ~ ocb o Z HanpasneHa cnesa anp pa so) Ap~0fT i Mn C O ~ C T B ~ H H ~ IKone6aHMfi X nWlYYaem npH 0 , PaBHbIX KOpHm ~ ~ H K I IaM ( @~- '~) M ~ a I- ( B ) , n e x a v M B Kpyre (01 < 1. 0603~aqw.1n e w a w e B Kppe 101 < 1 K O ~ H M@YHKISMH a ( B - ' ) sepes B,, a @ ~ H K WnM - ( 0 ) - yepea Bk. Cornac~o(11.111 (11.121 M 3 f CO M
i
1
i 1
3=,11
B cnysae n o n y b e c ~ o ~ e q ~CoTiPiY H ~ I , cornacrio (11.341 y ( 0 ) = O n y ( j ) = 0; npn N + 1. ~ O X O Mnonaran Y, B (11.38)At = 1, nonynnM npn N -+ m paBeHcTso
j
>
2-19, -8;' > 0, H, 3HaYUT, 0, < 0. B c J r . a e ~ ~ C K O H ~ qW yO ~~ b l CornacHo , (11.351 y ( j ) = 8; npu j >_ n 1 ~ ( j=) ecib-(ek) n ~ < 0.~ ~ O j~ T O MCg-m Y Iy(j)(' < co u, cornac~o(11.381
M 3 KmPOm CneAyW, WYI A(ep) =
+
M
m
V e k )
m
X ~ , I Y (=~E) I~ ,'I Y ,- Y , + ~ I ' ,
(11.40)
I
\
i
-m
-m
1 T ( t )= -C2whsin2(wt+a ) 2
a,
T ~ KaK K
npu j < 0 e - ( j , 0 ) = 01, b, = 1, TO
)
lim bnr { e + ( ~e k, ) ,e + ( ~e k, ) ) = (8;' - ek)a-(ek)b-(8,).
At--m
Hanee,
T ~ KK ~ npu K j
> n+ 1 e f ( j , 8 ) = Bi, b, lirn b
M-+m
{
'+
(
= 1, TO
M , e,), e+ (llf,ek)) = 0.
.-
1
E e+ ( j ,o)', I
a M n m T y A a K o n e 6 a H ~ f i j-ir COrJlaCHO
(11.421
Maccbl
o ~ ~ o p o ~ y~r o a c rT u K a pama A,,
n03TlJ~y XeJlaWJIbHO H ~ R T M ~ p y m iM r eTOA p e W e H J i R
nonYqtrM,
0 6 p a ~ ~ 3 ba Ar a ~Y p a C C e -
RHMR, B KSTOPOM HeIloCPeACTBeHHO UCnOJm3Y€CTCR TiJJIbKO A a H H b J e p a C C e R H m , XUTq
paBeHCTBa:
K a K MbI BMAenM, B p a C C M a T p M B a e M O M C J I y Y a e ( H ~ O A H O ~ A H P a~CIT~C I~K o ~ ~ ~ H u Y ~ H )
j
3TUX AaHHblX
60nrue,
YeM H ~ O ~ X O A M M O
i IIPRMOA 3a~aqeirP a C C e R H U R
H a 3 b l B a e T C R 3 a A a Y a , B K O T O P O ~ ~~ p e 6 y e T cBbI~
RCHMTb, K a K BJTMRIOT HWlAHOPOAHOCTI.1 H a K o J I ~ ~ ~ H )OAHOPOAHbIX .~R YqaCTKOB CTPyHbL
M H ~ I M UCJIOBaMM,
HYXHO, 3 H a R MaTPGfqY
Lm
CIJIU L-,,,,
HafiTM BWA
CTBeHHblX K O J I ~ ~ H ~ aH O~A H I O ~ p A H b l X YYaCTKaX. @ 0 p ~ y n b l (11.29)-(11.31)
~06-
M (11.41)
n O K a 3 b l B a l o r , YTO H a OAHOPOAHbIX Y Y a C T K a X C O ~ C T B ~ H HK ~ OI ~~ ~ ~ ~OAH03HaYHO H M R
(c
T O Y H ~ ~ K 30 ) ~WTORHH~IX ~ ~ o m u ~ e n eo fn ip e) 3 e n n r o r r R K O + + W U M ~ H T ~ M M
P a X e H M R M YUCnaMM
L,
MJlU L-,,,
PaBeHCTBOM
A = 2 - 0 - 8-I.
hIp,
3 a R a Y e HYXHO,
( { T + ( e ) ;Q k , h
3 H a R MaTPHUY
c)
MJIM
BoCCTaHOBMTb MBTPMUY
(7'-(8); e k , ! I c ) 1
L,
CTM
lei
= 1. ~
O ~ T O Mnpu Y
lei
=1
crr-
H a n p M M e p , MeTOAaMU, U3nOXCeHHblMM B IIpMrrr[Mne,
er0
MOXCHO IlOJTyW~lTb I.13 ~ a 6 J I K l ~ e ~ l f k
P ( j + 1, A)
M
P ( ] , A)
C TOYHOCTbKl 30 Il*
L,,
BOCIIOJm30BaBUIMCb,
3TO P a q M O H a n b H b I e
< lei <
+
~ ~ 0 3 a6n H~ C a T0b B TaKOM B H R E
58
x 00
0 6 p a ~ ~ o 3faIA a Y M
~ a ~ 0 Ml2ToA f i H e 0 6 0 6 4 a e ~ cH ~a C J I y Y a f i , K O r A a
~ YYaCTOK A R ~ HeOI'paH%fYeH, I ~ H, B W B T O P b l X , I I e p e X O A UT A a H H b I X P a C C e R -
HMR K IIOJMHOMY
161 = 1, a T. K. n p 6 ~ 1 ,TO OHM ~ O J I O M O ~ @ H ~B I HeKOTOPOM K O J I b q e I(, = 1 E ) M p a 3 n a r a m c ~B H e M B P R A ~ IJ ' l o p a ~ a . 3m p a 3 n o m e m ~
~ O ~ T O MKYO ~ M L [ M ~ H OTT ~ P aIX e H M R O r P a H M 9 e H b 1 H a OKPYXHOCTM
{B I 1 - 5 M38eCTHbINI A a H H b I M PaCCeRHMR
T a K M M I I y T e M M O X H O I I O n y q M T b PHlIeHJ'Ie
OAH~KO, BO-IIepBbIX, O
{1'(8);eprhfp)
K O T O p e H a 3 b I B a e T C R AaHHbIMM p a C C e R -
C K l X H H b l X ~ H o X C M ~ e n e f a, i , 3HaYMT, I.13JIeMeHTbI MaTPFlL@I
~
I
i
BeweCTBeHHbIMM p a q U 0 -
f o = f (BPI)Ha O K P Y X H e
U n M L-,,,.
YTO I l 0 3 B O n R e T BOCCTaHOBMTb IIOJIUHOMbl
O
HaSIbHbllvlM A P O ~ R M M ,YAOBJleTBOPRIOLqUMH P a B e H C T B Y
C ~ ~ A O B ~ T ~ J B ~ H n O p R , ~ 0 h
H B ~ T MMHOXeCTBO
(L-,,,),
3a ~ o n e 6 a ~ u m 0,qHOpOnHOrO n p a C T K a CTpYHbl. B 0 6 p a ~ ~ o3 ka R a q e P a C C e R H M R H Y X H O n0
~
K o ~ @ @ I . I ~ M ~ H T ~BO I BCeX n p e A b I A j T 4 M X @ C I P M ~ ~6b~JIu ~ X
1
n p M 3TOM 3 H e P r I . f R K O J I ~ ~ ~ H U ~ ~
PaBeHCTBOM (11.43)
L,
HMR ( n p a ~ b l ~ kM J l M J l e ~ b I ~ f i I )IIOCKOJlbKy ,
H
0 6 p a ~ ~ 3aRaq b l ~ PaCCeRHHR
ep MJlH BlipCBR3aHHblMM C C ~ ~ C T B ~ H H ~3HaYeHMRMM IMM MaTPMq
K O ~ M L P ~ ~ H T ~ M Uhi,', f l . f ,
PaCCeRHJlfX
!
O C H O B HY~p aI B~H e H H R
~ ~ C K O H ~ YaH3 ~H ,e p r l l R K O J I ~ ~ ~(11.41) H M ~C~B R 3 a H a C HOpMMpOBOYHbIlvlH
(11.29)-(11.31)
B
ii
e+(O, 8 ) C O A e P X M T B e C b M a ~ t ? y C T C I f i ~ ? ~ bOl en e p a q M M .
=
BT
J=-m
{ ~ ' ( j s) ,
+
x m21
At ( j , m)R(m+ s )
C n e ~ o ~ a ~ e npu n ~ B o c, e x
s = 0,311, &2,.
..
I
A+(j,s)+):A+(j,m)R(m+s) .
27ri
m>_j
C.
OTKyna, C o r n a c H o (11.46)
!
(11.8) (11.36)
c n e A y e T , STU B b I W T B mwe 0, p a B e H
- e + ( j , o p ) e i l l b = -Bj
I
A + ( j ,r n ) ~ ; + ~ i v l ~ , m>j
= -4
)A + ( j ,m ) M ( m + s ) , m>_j
~ O K ~ X ~ SMT 0, B K P Y E
161 < 1 n p U
P
j ' K a 3 a H H b I X 3 H a S e H M R X S 3TM A P ~ H PlhlelOT
OC~~~HHOCTIIM TOJIbKO M n p O C T b l e nOJllOCbl, J l e X W M e B KOPHRX
M B KOPHRX
8,, @YHKUMM a ( 8 - l )
OI; CPYHKI~IIL 0 - ( 8 ) COOTBeTCTEeHHO.
R
cy~mrpa-~e
.
In1 )II) K
wer no BWM ~ o p n~o mmm o M a e+(O,0 ) .
T ~ K H06pa30~, M n p M S 2 j+ 1 P a B e H C T m (11.46)
P ~ C C M O T P ~ L MHaKlpGiMep, , n O f l P I H T W ~ H Y P +3 Y ~ ~ K w M ) B (11.46) C G T J I ~ C H (11.211 O
A + ( j ,s ) +
x "'23
C P Y H K ~ M ~B~ ,CMnY J I e M M b l g , 0,
IIOnMHOMa
MMeeT B K p y r e
181 < 1 TOJlbKO I l p O C T b l e nOnOCb1 B KOPHRX
e+(O,8 ) C B b I S e T a M H
~ O C I I O ~ ~ O B ~ PBa B~e H U CCT B~ a M M ( 1 1 3 2 1 (11.211 3 T U N p ~ y n b AJlR l BblSeTOB M O X H O
3 a M e H M T b TaKMMM:
rAe
n e x a W M
B
Kpyre
1
1.
(PYHWHR
- 4 + ( j ,m )(Ad(tn
PaBHOCUnbHbl C n e A y l O q U M :
+s) +R(m+ s))
I
= 0,
M TpMBMaJmHble PaBeHCTBa ,4-(j, S ) = 0 npM S 5 2 - j - 1; A + ( j , S ) = 0 npU s22n-j+1. ~ P jM 10 n)OCTa-R TOJIbKO TPMBMaJlhHble PaBeHCTBa A - ( j , S ) = 0 npMs< j - l ( A + ( j , s ) = O n p u s > j+l). Y ~ ~ B H ~ H (11.49) M R M (11.51) (11.52) Ha3blBarOrCR O C H O B H b I M H YPaBHeHURMH o 6 p a ~ ~ 3aAa'uI 0h ~ ~ C C ~ R H E A I ~R ~H~ c K o H ~ Yu IHI O o J~ I ~~ ~ ~ C K O H ~ YWHpOy~~~b I . P ~ U I A Bn p n KaXAOM j 2 0 SaBHeHHR (11.49)OTHoCMTeJIbHO K O ~ W ~ U ~ H T O B A + ( j ,S ) ( S 2 j 1 ) M b l CMOXeM Hafi~H3neMeHTbI MaTPMJ.pI Lw n o @ O ~ M ~ J I ~ M (11.10) T. e. PWJMTb 06paTHylo 3aAaYy PaCCeRHMR AJIR n o n y 6 e c ~ o ~ e Y H oCTPyHbL fi PeLLlki~npF1 KaXAOM j = . . . - 2,- 1 , 0 , 1 , 2 , . . . , SaBHeHMR (11.51) ( M J M (11.52)) OTHOCMTeJTbHO A - ( j , S ) ( S j - 1 ) (MnM A + ( j , S ) = 0 n p n S j + 1) U BOCnOXb30BaBLLlHCb 3aTeM @ O ~ M Y ~ ~(11.10) M M M b l H a h A e ~3neMeHTbI MaTpH@l L-,,, no n e m M ( u n n npaebm) AaHirblM p a c c e m m , T. e. p e m m o 6 p a ~ ~ y 3anary 1o paccmrr AnR ~ W K O H ~CTPyHbI. W O ~ ~
<
>
+
n QYHKWRe+(O,8-')Oq-' RsnReTcR IIOnMHOMOM 0 BbIYeTaX
>
+ >
OT
8 npu q
2 7 n + 1, TO no TeopeMe
q 2 n 1 F ( q ) = M ( q ) + R ( q ) = 0, oTKyAa cnenyer, vrro F ( m S ) = 0 npn S 272 - m + 1, M B YPaBHeHMRX (11.49) ~ 0 3 @ @ 4 q M e H T b l F ( j s ) o 6 p a u a m c ~B ~ y npM m s 2 n - j 1, a CaMM ypaBHexim pacnana-R Ha ~ W K O H CMCiTMy ~ Y H ~TPMBMaJIbHblX ~ ~ ypaBHe~MfiA + ( j , S ) = 0 , S = 272 - j 1, 2n -j 2 , . . . M KoHewiym CIicTeMy (11.50). n p TOM, ~ ecJm j 2 .n, TO m s j 1 S > j f S 2 j 1 211 1, 1%BCe SaBHeHMR TPMBHaJlbHbl. POB BOAR aaanoruwbre r i p e c 6 p a s o s a ~ nneablx ~ s a c ~ e i paseHcTB i (11.47) nptr S 5 j - 1 H PaBeHCTB (11.48) npH S j + 1, llOnyWlM AnR K O ~ @ H ~ M ~ H T O AB - ( j , S ) ( S 5 j-1)M A + ( j , S ) ( S j f 1 )TaKMe ~ ~ C K O H ~CMCTeMbl W ~ I ~ ypa~He~Mfk ~ O ~ T O M Yn p n
+ + + ++
>
+
> + > +
+ + >
<
fl0 K a 3 a
T e JI b C T B 0. O A H O ~ H ~ pa3peUIMMOCTb W~R ypaBHeHdf (11.51) M (11.52) AOKa3blBaeTCR OAMHBKOBO, M M O X H O OrpaHMqMTbCR PaCCMOTpeHMeM Y p a ~ ~ e H X f i (11.49) M (11.52). npM j < n 3TM flaBHeHMR PaCnaRaWTcR Ha KOHeYHble C M m M b I ypa~HeHMfi(11.50) (11.52') M TpUBHanbHble PaBeHCTBa, a npM j _> n OHH CBOARTCR TOJIbKO K TPMBManbHblM PaBeHCTBaM. n 0 3 T O ~ y AOCTaTOYHO AOKa3aTb, Y T O npM j < 72 COOTBeTtTBYIO~MeCWCTeMbl KOHeYHblX OAHOPOAHblX ) ' P ~ B H ~ H P I ~
>
>
F+(s+ ~ n ) A + (nz) j , = 0, +s ) + A+(j, s) + mzj+l m e F* ( q ) = M* ( q ) + R* ( q ) F+(j
M
>
(11.52) l4MeH)T TOJIbKO HyJIeBble PeUIeHMR. C o r n a c ~ o(11.49')
T ~ KaK hfp > 0 M l ~ ( 8 ) I5 1 npn 181 = 1, TO Bee cnarawbte B nemR y a m nocnennero paneticma HecrrpnqaTenbHbl (no~eh~y?), a ux cyMMa p a m a ~ y m o Cne. AosaTenHo, nonuHoM X (8) 0 6 p a q a e . r ~B~~ y B nKOPHRX Bp (l8,l < 1) nonmoma W B HYJTh Ha OKPYXHOCTH e+(O,8), a nOAblHTerpaJIbHbIe @ Y H K ~06pa~aKJT.33 181 = 1(noqe~y?) tam^ o6pa30m, M 3 (11.53)cnenye~,qT0 nOJMHOM
Mbt yqnH, 'ITO ~ -
, ~ =8 z ,"s , 8 - 5 = .Y(o-') npn 181 = 1, PI xS?G8i = xs8; = X(Bp) B cuny BeqecTseHHocrH8, Ha OKPYXHOCTII 181 = 1 ~ 0 3 @ P l q ~ eHT OTpaXeHMR YAOBneTBOpReT PaBeHCTBy r ( 8 ) = r(8-'). ~ O B T O M Y
xS
Ix(e-l)
+ r ( s j . ~ ( s ) l ' = ( x ( e - l ) + ~ ( 8 ) ~ ( 8 ) ) ( x ( s - l )+ I (8-l)iY(B)) + r(8)X(8))S(B-1) + ( S ( 8 )+r ( o - ' ) ~ ( ~ ' ) ) m
= (X(bi-')
-(I
- ~(bi)~(Q-'))iY(8)-x-(8);
H r*.
1 - Ir+(B)IZ> 0 npn ~ c e 8v # f1.
C n e n o m ~ e n b ~ op ,a e e H c r s o JlHHOM OT
x-+(8) 0 6 p a I I I a e T C R
1 - Ir+(8)(2)(.Y+(8)12= 0
B H y n b BO BCeX ? o q K a X OKPYXHOCTH
o a ~ a q a e ~YTO ,
mnowornaa m e K p y r a (0)< (I+€)-'. ~ O ~ T O M(PYHK~MR Y Y(O)Q(O)-l IWOMOMH~ eceR n n o c x o c r b r , T. e mo uenax (P~HKIJHR. 3 a ~ m m Tenepb, TO npn B c e x j 2 0 Vl Vz - 2 ( j 1) <_ 0, n p M Y e M 3 H a K ~ C ~ B ~ H C TAOCTIlraeTCR ?&~ 'TWIbKO TOrAa, K O r A a j = 0, a vl = vz = 1, a r K y A a c n e A y e T , ~ I n Je B a R Y a c m p a B e H c T B a (11.581 a, 3 H a q H T . M uenan + ~ H K L I M R Y(O)Q(e)-' o r p a H r n e x u npn 181 -t ca n, no mopme
no-
BO
181 = 1, O T n M W b l X
f1, T. e 3TOT nOJMHOM P a B e H H y n t o TOXAeCTWHHO.
X(8) n p H X O A H W R A O K a 3 b l B a T b Ir(8)I = 1H p a B e H C T B 0 (1- lr(6') 12)(X(8)I2= 0 T P M B M a n H O . M 3 (11.20), B cnny B e U e c m e t i H o c r n n o m H o M o B e+(j,8), c n e n y m , TO Ha' O K ~ ~ X H O C T M (el = 1 a b m o m n e x a paseacmo T o x ) J ~ C T ~ ~ H H O p~a B e H C T s o HyJIlO IIOJlMHOMa
+
.hiyB~JInJl,
K H a q e , T. K
+
OHU P a B H b I KOHCTaHTe, T. '2.
X a p a a ~ e p n c ~ u ~ e c cso#crsa ~ u r e RaHablx
P ~ o H ~ ~p061.1, J ~ HC T ~ O f lIm e~B o 6 e ~ x q a C T R X 3TOIU PaBeHCTBa, n p M H H M a m O m HaKOBbIe 3 ~ a q e m x H a OKPYXHOCTU 181 = 1. ~ O ~ T O 3M T HY, 4 ~ 0 6 M T. e paeeacrso (11.58) e b m o m n m c a npu B c e x aaaqemar 8. TX K a K ~ ( 8 , ) m npasaa q a c m p a B e H C T B a (11.58)r o n o ~ o e B~Kap y r e 101 < 1+E, a era nesan .
~,
2.
..
.
.
E+ ( j , 8) .=
A(j, j s>_o
+ s)BS
pacceaana
N M
bi
=
1-
N a i l ,
ecm N =
2n, a,
S= 1 B b I T e M K l T T a K M e H e p a B e H c r B a AJlR
=2
-
JJ a;',
3=1
ecm N = 2n - 1,
le]
3 ~ a e YTO ~ , H a O K ~ Y X C H ~ M = 1 ITOJMHOM E+(o,8 ) = ~ o ' e + ( O8,) tie 6onee ABYX ~ o p ~ eK*o m p b r e MOWT n e H c a T b T O ~ K OB TOWX +1, -1 M A o m H b r 6 b 1 T b ~ P O C T ~ I M HH . ~ K o H c o~ rn~ a c ,H o n e M M e 9, B pyre 181 < 1 n m a T T o n b K o n p O c T b r e o r p n u a w n m b r e KOPHH Q p n o n u H o M a E+(O,B) = ~ ; ' e +( 0 ,e ) , Mbi
MMeeT
me, cornacno (11.3)
M B 3THX K O P H m BbIIlOJTHHlUTCR H e p a B e H C T B a
B C ~ O M ~ qITM O qepe3 , lll, M b l 0 6 0 3 H a Y ~ J D M l a C C y C HaMMeHbILIMM HOMePOM, nocne ~ m p o RM A ~ T(TO e c T b n p u K p e n n e H K H ~ R )O A H O ~ O A H ~KYCOK ~R c ~ p y ~ b C ~ ~ A O B ~ TBblnOJIHReTCR ~ ~ ~ H O , no ~ p a R ~ eM%e p e OAHO b13 H e P a B e H C T B 'PI.,, # nZ, k,-l # k, n p u s e M t n j + , = k j = m n p v l Bceex j 2 n. ~ O ~ T O M Y
M, eCJIM
1 - b:
# 0, TO C T e n e H b IIOJI$lHOMa E + ( n - 1 , 8 ) p a B H a
AByM,
a
l .
CT~~UIMT?
1 - b i , ecmr me 1 - ba = 0, TO c T e n e ~ bpama e A u r H H q e , a C T ~ ~ ~ I M K O%+ + H ~ M ~ H T p a B e H a, - 2. B 0 6 0 Cny%lRX ~ ~ C W I l e H b E'('11 - 1,8) 6 o a u e c T e n e H M E f ( 1 2 , e), K o m p a s pama H y n I O . M3 P e K Y p p e H T H b l X W P M Y J I (11.59) BMAHO, Ym,e C n M C T e I l e H b E + ( j ,8 ) 6 0 ~ l b m e c T e n e H u E + ( j + 1, Q ) , TO c T e n e H b E C( j - 1 , 8 ) 6 o n b u r e c T e n e H u E+(j,8 ) p o e ~ o K O ~ + + U ~ ~ H pTa s e H
H a R B e eAMHMqb1, H 3TPl nOnHHOMb1 MMeX3T O n H H a K O B b l e C T a P U I t l e K O ~ ~ W U M ~ H T ~ ~ .
O~cronano MHAYKUHM c n e A y e T ,
orKyna
K G ~ H IlOJIMHOMa ~R E+(O,8):
YTO
E t ( 0 , 8 ) P a B H a nn60 2n, I.1 T O r A a em ~ T a p u I b I f i 1 -by,, n ~ 6 272 0 - 1, M TOFAP b,, = 1, a c~aprunfiK O ~ + $ M ~ M ~ H T p a s e H a,- 2. O ~ O ~ Hy e~p ~ 3 as WKOPHPI M nonmioma E+(O,8). Toraa
n 0 3 T O ~ yC T e n e H b nGJlHHOMa K O ~ P 1 U M e F i Tp a B e H
sign E ( 8 i 1 )= (-1)"(8,1),
Sign6)pE(ep) = (-1)"(e.)+l,
$1 1 &panran aa,qa.ra
paccemwn
133
rAe sepea v,, v?, . - . , vp 0603Ha~eH0 w c n o K O P H ~ n~e x t a r q ~ ~Bx XHTepsanax ( O F ' , O;'), (B;', 8;'); . . . ,( O i l , B,=',). CnenoeaTenbHo.
sign B,E(B,)
-1 2:=, ( ~ ( 8 ; ' ) ) = (-1)
".
+P+l
-
m e P ( j , A ) - cTannaprHble xapartwpncTuqecKue ~ I O ~ H O M hlaTpnq L I L,-,,KOPHU KOTopbIX PaBHbl C ~ ~ C T B ~3HaZIeHIIRM HH~~M 3 T I I X MBTPMU a CTapuMe K ~ M ~ ~ paBHb1 e n n ~ u q eB. YaCTHOCTM, nonaran j = n 14j = 71+ 1, nonyqnM paeeHcTsa YeTHbIe, T . e. KOr&a3HCno U l YeTHoe, a BCeOCTanbHbIe YMCJla V?, . . . , V, HeqeTHble. M 3 BbIUIeM3JlO;KeHHOr0 CneAyeT, YTU K O P H M nOJMHOMa E+(O,8 ) YAOBneTBOPFWT TBKKM YCJIOBKRM: 1) Ha o u p y w n o c m u ($1 = 1 n e m m we 60~lf?easyx m p w i i , mmopate Mozym n e x a m a monaxo s m o u x a x -1, 1 u aonwnbt 6brma npOCmMAU. 2) B upyze 181 < 1 =am m o n a m n p c m a t e omptyamenbnare . m p u Bp: (-1 < 0 , < . . . < 0," < 0 ) . npurem vucna 8;' ne m z y m Barma uopx-U. 3)Mnmepean ( O F ' , 8 ' ) coaepmtm uemnoe, a ?uwcabrii ro unmepanoe ( 8 y 1 ,e l ) , (fly1,O,), . . . , (O;', neuemnoe uzrcno xopneic.
e;'_,)
4) 0 6 q e e uucno wpnetl ( c yuemom xpamnocmu) paem nu60 Zn, nu60 2n-1, zae n - vucno fiacc, n e w a q u x e ~ e ~ a n ~ p ~ ayvacmxe n o m cmpynbr. 5 ) lTpou3seaenue scex xopneic r~aoenemeopnemnepasencmsam (1 1.60). 6 2 , . . . ,sm; M I , Teopena 9. A n n mozo rmo6ar @ynxz+unr ( 8 ) u v u c n o n/r2,. . . , Afm 6arnu aannbrmu paccennun nony6ecxoxeunoii cmpynbr, neo6xosum u aocmamouno, rmo6br
zae E ( 0 ) - npouason.bxarii seqecmsexnaric nonunonr, x o p x u xomopozo yaoenemeopnlom m m u nepeuucnennbrm sbrure ycnoszrnm B uacmnocmu, a a n ~ b r epaccennun cocmonm u3 oanoic @ZJXXY.UZL r(0) mozaa u monbxo mozaa, xozaa r ( B ) = - E ( B - ' ) ( E ( B ) ) - I , zae E ( 0 ) - npouasonbnbrzZ seyecmsennbtii nonanom, ne umeroquii uopxeii e Kpyze 161' < 1 u yaosnemsopa?ozyuii monaxo a e y ~ycnoeunm I ) u 5 )
-1
,
1
~
1
i I
He06~0~mm BTUX b y c n o ~ ~ ryf Ix e ~ o ~ a a a Ann ~ a . n o ~ a a a ~ e n b cu~xmnoCTaTUWC€TUBOCnOJb3yeMCR BblTeKaIOLWMM M 3 (11.11),(11.21)PaBeHCTBaMII
H
T
~
I
Q
= 2-lrl+l(,9 - 0-1)
+ ,g-1)lrl-2q-1
)c,%+l(b) rl
((0 + e-1)2 - 4)q
q=o
Q
2 ( l + qr 2; - cl r 1) + )1 ( e' - g ,
E(0) = (-1)" - ~)1r1-?q-1((2 - ~
) 2
4)q
W a n O m Y I i b l M M P a B e H C T B a M (11.62)-(11.64)
q=o
n o K a x c e M , YTO n o n m o M b 1
= (0 - e-')T(Irl, X),
.repea Q a qepe3 rAe
0603HaqeH0
{n(n+ 1, A) + ?;en(n,
n(n,A),
M TeOpeMa
II(n + 1,A )
A))
~au6onbrueeqenoe wrcno,
He npeBocxoARrqee
:(IT(
- l),
, @ ? ~ C T B M T ~ ~ ~ HTO O ,r A a
no
2
TeOpeMe
(11.68)
6 y ~ e TA O K a a a H a ,
I I M e m T O ~ K On p o c T b I e
T e n b H b i e M n e p e M e x c a m r q u e c n KOPHM I
en,
IIOJIMHOM
n(n
+ 1,A)
W J I U Mbl
nonomn-
RBnReTCR Xa-
L,
paKTeP).ICTPIYeCKMM IIOJIUHOMOM H e K O T O p k T ~ ~ x A I ~ ~ ~ MBTPMI(b1 H ~ ~ ~ H oC 3neR MeHTaMM
a l , a 2 , .. . ,a,;
-bl,
l I O ~ M H O M , I I 0 n y Z I a H ) ~ e f iM ~ 3~
puqy L, no
-b2,
L,,
. . ,-bn-l,
= (-1)"
x a p a ~ ~ e p w c ~ l l r e c ~ u f i
Y K O I ~ Y ~ H H IUIaTpMI(b1 O~~ Ln-i
-b,, =
~ O ~ ~ ~ ~ C K O H ~ nonaran W I O A ,
E+(O,0)
a n(n., A) -
-Pa
= -
{n(n+ 1,A) + ,8$9n(n,A ) )
J
. ~ P O A O J U K M B MBTN
1-
c1-' 1.1 a,. = 2,
l=l
0" = E(B),
OZIeBI4AHblM C n e A C T B U e M K O T O P O ~R B n R e T C R AOCTaTOZIHOCTb Y C ~ O B M T e~O~p e M b I 9. n e p e x 0 , q R K PaCCMOTPeHMH) K 0 p H e f i IIOJIMHOMOB n ( 7 1 +
npexcne B c e r o ,
TO B
E(0) = (1 - eIP(i
P Y H KF(v, ~ 0) A ~ ~ ~ C T B M T ~ JRBJIRK)TCR I~HO IIOJIUHOMBMM OT A = 2-15-8-~. B n o c n e ~ ~ e+ko p ~ y n e~au6om111ee3 ~ a q e ~ sIrl ie A n n HecrrpuqaTemHbrx r p a B H 0 V, a A n R O T p M l j a T e J I b H b I X p a B H 0 N - V, M, T a K K a K N - V 5 V n p U BCeX M(
PaCCMaTpMBaeMblX 3HaqeHMRX
N = 271, v = n,
N
n(n,A),
1,A),
3aMeTMM
cuny ycnoem 1)
+ e)qQ(e),
M V, IIPMZIeM P B e H C T B O A O m M r a e T C R TOJlbKO n p M
TU c T e n e H b n o n u H o M a
(11.65) pama u
-
1, a ero c ~ a p m n i i ,-!
f
- 5"'
(sin ~ ) p ( c o s
)q
(- 1 ) " 2 ~ (+s i~l l
F)'
I
lQ(eiv)
e'A(v)
N
1-
1
c
npn N = 271, npn N = 2 n - 1, =
(COS
d
= D(q)(-1)" sin aA(cp),
!)'
I sinsinTA (cp)
IQ(eiD)
(11.70)
_ ." - 1
=
(cos
' 2
- ( n +.I
;)'-I
I~(e'')l > 0, -'(") ;
-
X
rAe msi y ~ c f k m- 1 b a ~ e y e~p 3a O0 1.1 0 - qepes On,+ n p o c n e ~ nTenepb ~ II(n 1,A) B Towax A(Bk) = 2 - Bk - 0;'. CornacHo (11.671
0 < cp < n;
6(cp) = arg Q(ev).
m e c p r r p o 6 e r a e r ~ ~ ~ x ~ ~ p e ~ m - ~ p ~ & A= k( ,c kp =) D , f l , f 2 , . . . . T ~ KKa K Q(0) = 1 M IIOJIMHOM Q(0) H e MMeeT K0p~efiB CerMeHTe 0 5 0 5 1, TO Q(1) > 0, YTu n03BOJLFIeT Bb16pa~bB ~ T B aprpeHTa ~ m r 0 nOJIMHOMa arg Q(e'') Tay TO arg Q(1) = 6. IIPH TaKoM ~ b ~ b p cornaaio e, n p w m y a p r p e m a , +-IS '(cp) = arg Q(eip) MeHfimflOT 0 AO 2 ~ mKorga , cp M e H f l e w f l OT 0 no 2n, M, B cuny BeqeCTBeHHOCTM IIOnMHOMa
cp MeHReTCfl OT 0 A0 71.
Q(B) ( ~ ( 3=)
&OXOHa
MeHReTCR OT
0 A0
+
3a ~ H ~ Y ~ W R M nonHHoMa II
+ 1, ,\(Bk))= -sign ~ ( 0 ; ' ) .
(11.74)
sign n(r1
Xn2, KOrAa
~ O ~ T O M Y
Cnenosa~enb~o @, ~ H K ~ MII(n R
HMR Pa3HblX BHBKOB, M B KBXAOM
-
+ 1, A)
npwHuMaeT B
MHTepBane (8L-1,Bk)
-
-
3Ta * ~ H K u ~ ~oR 6 p a r q a e ~ cB~Hynb. ~ O ~ T O M ~Y a ~ ~ b MHTepBan l i i
~ o p e mA(Bk) = 2 - Bk - 6;' ~ Yep3 a ~ ~,\n-m+k:
coAepxuT 0 6 0 3
Tenepb
mr 3
~ a paenonoxerare e ~
-
Towax
JIeXMT TO-a
nonuHoMa II(n
Bk
3 ~ a ~ e -
Bk, B K0T0pOfi
(A (Bk- ) , A(6k))
+ 1 , A),
KOTOP~IRM
n - 1 ~ o p ~ enfoim o M a n-ORcTenem II(n
~ I
+ 1, X):
K p o ~ e3THX ~0pHefieCTb ewe OAMH KOpHb An-,+l, n0nOXWIe KOTOpOrO HYXHO ~ a f i ~Tu .~ ~Kc ~ae p~e y 3 ( n - m ) 6b1no 0 6 0 3 ~ a y e ~ nan6onbruee o (B nmepsane (0, T ) ) pemeHue YpaBHeHHR A(cp) = n-m, TO npu Bcex cp E (cp(n-m), T ) A(cp) > n-.m. l l o m ~ npn y Bee* AocTaTowo Manbrx E > 0
sign E(B) = (-I)'-"-'
nprr
sign E(6'k) =
B E (Of-, Of), 1 = 1, . . . , m + 1, k = 1 , 2 , .. . ,m ,
(11.72)
Ecnu
Ak
E (0,4), m XI: = A(el~("'), r j q e p(k) E ( o , a ) n A(cp(k)) = k,
orKyAa cornacHo
(11.771 (11.691, c n e A y e T , qm
Fknn Ak E ( 4 , m ) , ~ o 2k 7 ~ - n t + l u
8;,-,+,
E
(01;-,+,-1,
BI;-,+,),
OTKyAa,
-
Xk
= A ,-,, +(I;-,+,)
-
= A(Ok-,,+,),
rjqe
cornac~o(11.771 (11.721 cnenym, qrn
-
sign II(n, AP) = (- 1)"sign 8;~;~,,E(8k-n+m)
-
= (-1)2"f1SignE(~s-n+m)= (-1)2n+1+k-n-1
IIpn
TOM
-
= NOI),
ecm E(-1) = 0 (T. e. q = 11 TO,
= (-1)"sign MT.K
TOqKe
+
+
npnqeM B
-
8, E ( - 1 , ~ 1 ) . sign n ( n , XI;) = (- l)"sign e-i"(")" - (-1)"sign - (-l)n+(n+l-nl) = (-l)n+k,
c o r n a c H o (11.701 (11.711
+ 1,4) = sign II(n + 1,A(e'"))
sign n(n
(-I)~+".
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(11.721 (11.23) ~ O J I U H O Mhoe+ ( l , 8 ) , B CBOIO O q e p e A b , O A H 0 3 H a q H O O l l P e A e J I R W N M e+ (O,8).~ ~ ~ ~ ~ C T B U T ~C oJ rIn~a cHHOo (11.211 , (11.241
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L,,, B H A a ( 1 1 5) coBna,I(aeT C M H O X W T B O M IIOJIMHOMOB B E ( 8 ) , r A e B - n P O U 3 B O J I b H b l e n O ~ 0 X H T e J l b H b I eWICJIP, a E ( 8 ) ( E ( 0 ) = 1) - n P O M 3 B O J l b H b l e
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