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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
J ~
ee
L,
oa K~O ~ ~ ~
~ ~ H U R X
T O n03BOnReT
C ~ ~ C T E ~ H 3H H a~Y e~HXP I f l X M
~06-
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
-b3
...
.. .
..,
0
-b~-I
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 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),
(4.1)
k=1
P(1,X), P(2,X), . . , P ( N , A ) J ~ H H ~ ~H ~ eH3 aoB M C I I M b I ( n o q e M y ? A
OHM T O X e
0 6 p a 3 ~ 1 06 a~3 1 1 c
(4.2) nPOCTpaHCTR.3
PN,
(4 11 crro6paxe~rref .-t F(X) n e p e B o A I i T K ~ H O H H Y ~ C K I . 6I ~.I3 3 ~ Hiv B 6 a 3 ~ cn p O C T p a H C T B a p N , 0 6 p a 3 0 8 a t f ~ b l f i3TUMkI el. + P(Ii,A). C n e n o ~ a w n b ~ onepexon , OT B e K T o p o B f E H N
npwyew, cornacHo el,.
. ,e,y
IIpCTpaHCTBa
nonnHoMaMw:
K HX I I P ~ O ~ P ~ ~ O B B H I I R QM ypbe
'HNB
I I p M ~ n M a - T b H b l X 3 a ~ p y ~ H e H t f fn ip M p e U l e H M M n p R M 0 h 3 a A a q I f I H e B 0 3 H M K a e T .
HOW npOCTpaHLTBa
B
(2.12) )I o n p e A e n e H r r R
o 6 p a ~ H o k3 a n a Y e H Y X H O n C bf3BeCTHblM Y a C T O T a M LL']
CBO~OAH~IX K O J I ~ ~ ~ H M ~ ~
H a r p y X e H H o f i C T P Y H b I H B ~ T I Iee M e X a H M Y e C K M e X a p a K T e p U C T H K M . T M XeCTKOCTA
k, M H ~ I MCJIOBaMII, M
H b l e 3HaYeHMR
X I = w?,
e
~ n p I f ~ e ~ e H HBblLUe 0 f i CXeMe 3aAaKtTCR c C ~ C T B ~ H -
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0 6 0 3 ~ a r e ~@a~ H K I J ~ I XaaucaRna. R C y h n ~ scex a C K a s K o s @ y ~ ~p(A), q n O~' i e s y q H o , pasria e n u H J q e . n p n p = X k 3 ~ @ a Y H K ~ M R H e y 6 b r a a e . r n m e e T N - 1 CKaYOK, ecm c = - ( a ( & 0 ) - cr(AI.)),M n n N c K a q K o e , e c m c > - ( a ( & 0 ) - a ( & ) ) . ~ P pM# Aj j = 1,2,. . . , N @ ~ H K W I R P(A)H e y 6 b l ~ a e TM M M e e T N 1 CKaSOK, ecnu c > 0. C O ~ ~ ~T eCO H p MO - e 1, B 3TMX CJlYYaIIX P(A)RBJIHeT(ISI c I I ~ K T P ~ J I ~ H( Po Y~ H K ~ H ~ L ? H~KOTOPOL? M a T P M q b J L BMAa ( 6 . 1 5 ) n O p R A K a N - 1, N M J I H v ! 1 B 3PBMCMMOCTM m w c n a c K a q K o B . ~ P scex M A p y r t l x 3 ~ a q e ~ 1 n. a 1p a~M~e T p o B C, p @ ~ H K ~ Mp(A) R
+
+
,
+
i
1 j
. L L j ( ~ ) n o n y q a e T c R ~3 O n p e A e m T e n R f j , ecnx B H e M H M X H ~ I O CTPOZIKY 3 a M e H k I T b P(1,A), P(2,A ) , . . . , P ( j ,X). ( n p w j = N + 1 B o n p e A e n t 1 T e n e (8.4) a n e M e H T 1 + c.P(j)' H Y X H O 3 a M e H M T b H a P ( N + I ) ~ )
H
Ha
C o r n a c ~ o(8.41
+
H.:'Er.03KeT 6 b l ~ cb I I ~ K T P ~ J ~ ~ H( no oR r e ~ y ? ) . .&lR
TOTO, 9 ~ 0 6 b lHBRTK M P T P b I w
6ej3'IW3bl~ I I p o c T p a H C T B a X PORaHHbIe 6a3ncb1,
-
L,
PN- ( P ) , PN( P )
H Y X H O B COOTBeTCTBYKlqHX r l l J l b -
PIJIM P N + 1
(P)
OOCTPOMTb OPTOHOPMW-
~ K B I I B ~ J I ~ H T HeCTeCTBeHHOMY ~I~ 6a3Mcy.
T ~ KKa K
nOJIMHOMb1
P(1,A), . . . , P ( N - 1 , A); P ( 1 ,A), . . . , P ( N ,X); P ( l , A), . . . , P ( N ,A) D N ( A )&pa3:r'tJT B 3-
n P O C T p a H C T B a X 6 a 3 M ~ b 1 ,3 K B M B a n e H T H b l e eCTeCTBeHHblM, TO AOCTaTOqHO
H ~ A T MO P T O H O P M M P O B a H H b l e 6 a 3 ~ c b 1 , 3 K B M B a n e H T H b l e 3TMM n O n H H O M M a n b H b l M 3HCaM. q T 0 6 b l COXPaHMTb 0 6 0 3 H a q ~ T bsepe3 P ( N
C O ~ J I ~ C H JOI e M M e @Ip~yJIofi
+
en~~oo6pa3& B O 1 e~ O ~ H ~ Y ~ H P IYR~XO, 1, ,\):
5,
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~ HIIOJII~IHOM O DN ( A )
BblrncneHwrr n o n y q e H t i o r o o n p e A e m r T e n R BbIrTeM M3 n e p s b l x
a y n b ~ a ~ n oen y . r u T c R Hynkl,
a
paseli c ~ ( j IIOJTOMY )
MCKOMbIe OpTOHOPMMPOBaHHbIe 6 a 3 b l c b 1 OIlPeAeJlRlOTCII
. . . M w a K l T IIOJlMHOMbl
P(l,,\),
P(2,X), . . . , M c ~ a n ~ l j n~pooen 3 s e n e ~ ~onpenenetio e paaeHcrBoM
T ~ KKa K nOJDfHOMbI P ( 1 ,A), P ( 2 ,A), . . . , P ( N ,A) 0 6 p a 3 O~ ~ H O ~ 6aanc rrpomparicma PN ( u ) , n c K a w c n @YHKWM u ( X ) nemar B KOPHRX D N ( A ) P ( N 1, A), TO
+
+
M M ~ O B ~ H H ~ I ~ ~ nonuHoMa
( p ( i ,A), P ( j , A!) = ( 1 c)-' (6,j + c P ( i , p ) P ( j , p ) ,) j 5 N , ( P ( i ,A), P ( N 1, A)) = ( 1 c)-'cP(i,p)P(N 1 , ~ ) .
+
+
+
OnpeAenuTenb,
y ~ompomnon rnaaaoL? A x a r o ~ a n b mCTORT ee H M X H M ~ 3 J l e M e H T
H a ~ J I ~ B H o RR M a r O H a n M CTORT e A M H M q b l , H T O J b K O
c r r K y n a c n e n y e T , YTO
( 5 . 1 7 1 B ~ m p o f pi O J l 3 BeKTOpOB f l , f ? .
1 - 1
. . . ,P ( J - 1). B pe-
CTO~~~JO nocnen~~rR B C T O J I ~ ~ Iy~~, H 0 ) ~ e H ~ bHl af iP ( l ) ,P ( 2 ) ,
3 ~ o T onpeAemITenb paseH CP(J)
1.1
r,
=
r , - ~+ cP(j)',
n p k l 3 ' T O M H Y X H O nOMHHTb, ZlTO 3 T H @ P M Y J I ~ I B e P H b I AJIR
c = - {.(A,
+ 0)
-
u(X,)), nnn j
g N + 1, e C n n
HAJIR~
Ao
CMX
ananma,
e.
nop
n,:lc,.l-~i) + 0, Ann
pelUeHblH
Cornac~q(1.61
z ( j ,j), k ( j , j
-
-
1) n o n n H o M o e P ( j , ,\)
paB-
= Xk,
> 0.
no EM HJM m b r ~ 0 6 p a ~ ~ o3 fa Ai a Y M T e O p U t 4 K o n e f j a ~ ~ r f im a -
MaCCbl H JKeCTKOCTH ~ 0 n e f j m r q e f i c R CTPYHhl, 3 H a R C B R ~ ~ H H ~ I O
C ~ e fTip e X A U a r o H a J I b H y I o M a T p U U y
ABa C r a P m -mema
j _< N - 1, eCTM
> - {a(Ak+ 0) - o(Xk))
BoccTaHasnnBann TpexAwaroHamHbre ~ a ~ p - 1
W C F I ' e q e H&TM
nePBbR Hbl
c
c
MM 3 a H n M a n n c b p e u I e H u e h 1 0 6 p a ~ ~ b la xa ~ a qc n e ~ T p a m H o r o
C n e K T p a n b H b I M AaHHblM.
TaMM MaTPtfilbI
I N , &nn ~i = A,,
LN.
MexaHnqecKne xapaKTepncTuKtr C T ~ ~ C H B~ RI ~ ~ c H ~~ JI I ~ M ~ H -
LN TOJIbKO 2N
- 1 YpaBHeHMeM
A1 wi, no ~ p a f ' ~ eMepe k oAHa n3 HMX ( / 3 0 ( ~ ) b -TMUawnbHa. T ~ K H 06pamhr1, M napanemp w 2 ox em n p u x u ~ a m a nm6are 3 x a v e x u ~lu uxmepsam (0, wi), zae OHPI ~ O ~ O X M T ~npn ~ ~ H ~ wi, I
w; =
i
P a c c ~ o ~ p e ~nomcnx ~ e nop 3aRaqM 0 K O J I ~ ~ H M RHarpYmeHHbIX X CTPYH R B J I qaCTHblM cnyaeM o6qek a a ~ a wo ~ManblX K O J I ~ ~ ~ H M~ReXx a ~ u ~ e cC~M CoEfMi~ I
(XI&...XN) - Det(LN) (PIP?. . . PN-1) Det (LN-1)
~ Y C 3Ta T ~CMCWMa COCTOMT M 3 'R TOYeVHblX MaCC 'ml, m2,. . . ,nnr AeKaPTOBbI KOOpAPlHaTbl KOTOPbrX 0 6 0 3 ~ a q M4epe3 ~ Xi, yl , Z1; 52,Y?,22;. . . X,, yn,Zn. T o r ~ ee a KMHeTMYeCKaR 3HePrMR PaBHa .
.
n
T ~ KaK nonomeHne CkICTeMbI OnpenRerrrr 0 6 0 6 q e ~ ~ b mKnO O P A M H ~ T ~ M ) I ,
I
TO AeKapTOBbl KOOPAHHaTbl MaCC MOXCHO BbIpa3MTb qepe3 C636~e~Iible KOOPiPmaTbL
~
~
n
z
e
i?
1
1
E
g
g Iv
% $
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2i
a
s c 5 5x t% $ gl
'cl
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ap
g u
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rn a
b
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"
P, w
a- 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
..,Itp -
M n1,n2,.
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
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
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2
2 &
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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
-a g;;'o
L;,W l p " Z
: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 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 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
_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
+ + + ++
>
+
> + > +
+ + >
>
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, -'(") ;
-
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)~+".
4 = A(e'") M O i K e T J l e X a T b TOJIbKO K O p H b XI- C k 71 - IT1 1, 3 m c~n y a e E(-1) f 0, q = 0 n A(.rr) = 71. 1 - m. O r c m ~ a c, o r n a c H o (11.771 (11.691 c n e A y e T , STO
B
Xn-,+1
-
= sign (- 1)" sin .rrA(a)
s i ~ i a ( n- m
+ -)21
W p h i y n a (11.76) s e p H a am T ~ K M06pa30~, M
= (-l),
Bcex
e-'"("+l-m)
k = 1 , 2 , . . . ,n , u
TeopeMa
IlOJMOCTbH) A O K a 3 a H a . O C T ~ H O B M M C RK p a T K O H a X a p a K T e p M C T M K e A a H H b I X P a C C e R H M R
s i g n n ( n + i,A(O,)) = ( - ~ ) l - ' ~ TU ,
CTPYHbl. *M
~~CKOH~SHOA
b- (8), K o ~ M e+(O, @),boe+(l,8) no C@pMy-
A a H H b I e O A H 0 3 H a q H O O t I p e A e n R I o T C R K O ~ @ I . I I J M ~ H T ~ M Ma-(8),
KOTOPble P a A H KPaTKOCTM ~ Y A ~H aM3 b I B a T b ~ 0 3 C p C p I . l q I . l e ~PaCCeRHMR. ~a~~ IJMeHTbl PaCCeFIHAR B b I P a X Z W X C R Z I e p e 3 IIOnHHOMbI naM
(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
tIOJI4.IHOMOM
sign n ( n , At) = ( - I ) " + ~ , me Ak
- KOPHM n o m H o m a II(n C o r n a c ~ o(11.661 (11.671
+ 1,A).
(11.76)
H e n O C p e A C T B e H H O B b 1 ~ e K a m L U e A543 ( 1 1 . 8 0 )
1 npn npu
~ O C K O ~ ~CKWY l l e H b IlOJIMHOMa
boe+ ( l , 8 ) H a
a f f l a=fl,
eAMHMqy MeHbUIe CTeneHM
e+(O,8 ) ,
I
I 1
M H T ~ ~ ~ O ~ R ~ H O W Hp ~ H yOJ ~l o~ f Ci K P a T H b l M W y 3 n a M P I ( ~ P M H T ~ )
I
boe+(l,8) =
no
I
BCeM p a 3 J M Y H b I M KOPHflM CX IIOJMHOMa
e+(O,8),
K I T C R Teanopa c T e n e H s 1 p ( a ) - m - 1 yepea (f( 8 ) ) p ( a ) - mD ~ ~ ~ H ~ Y ~LIOJMHOM~I B pa3noxe~nn + m u f ( 8 ) B O K ~ ~ C T H OTOYKM C ~ Q no c r e n e H R M ( 8 -a) M 3 T e ~ p e ~ 9b Cl l I e A y e T , YTO M H O X e C T B O IIOJUiHOMOB e+(O,8 ) , I I O p O M R a e M b l X
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
I I
MaTpnqaMM
BeIlJeCTBf?HHbIe
nOJIMHOMbI, KOPHU KOTOPbIX YAOFl.lIeTBOPflH)T YCnOBWIIM
c n e A y m , T, +cpyma~.~ a-(8),
zae B -
npoweonbnbie nomxumenbnbie
eonanaie
eeupcmeennbre nonuwmbr,
uucm, E ( 8 ) ( E ( 0 ) = 1) - npouclx o p u x o m o p b n yaosam8opnmm ycno-
eumt 1 )-5) u
OH OAH03HaZMO O n p e A e n R e T C R 3THhWI 3 H a Y e H M R M I I H RBHO B b l p a X a e T C X Y e p e 3 H I l X
r A e CYMMMPOBaHHe B e A e T c R
Orcro~a,c m n a c H o (11.221 ( 1 1 . 2 3 1 u m n b m mzaa m/u110mc~K I J ~ ~ nab, xozaa
1)-5)
I
I
I
U I ~ W ~ ~pM a cI cUe
b T ( 8 ) ma% s n ~Becxonmnoii ~~ cmpy-
M ~PaccMoTpem I oce rnaamle aanawr cnempamiioro a ~ a ~ ~~~ 3rTcapr e w aT0HanbHblx MaTPMq. MX @OPMYJIMPOBK~~ PI MeT0,JJbIPelrreHWR O ~ ~ I J + I O T C R Ha TPeXqMaTOHaJIbi-ibIeMaTPMIJbI~ ~ C K O H ~ S H OIIOpqqKa TO M Ha 06bIK~0Be~~ble m + + e p e q l a m ~ b r eonepaTopbr s.roporo n o p n m . 3 ~ sna n a w eo3rir.r~n~1 e + B I B M K ~ B nepeofi nonoame npolrrnoro seKa n p e m e scem PJIR n~d;)+epeHquam,mrx onepampoa. O W ~ KMHOTO O pamme B TeOPI.IM O~IUHaJIbHbMIlOJMHOMOB B IQJYIWX TePMMHaX 6 b m c + o P M Y ~ ~ ~ C + samr br pelrreabr nonocjmle aa~aqw M ~ ~ e p Ke c06paTHb1~a a n a q a ~~ H ~ ~ H T B~ OJ~ P~O HCnocne O OTKP~ITW~ MX CBR3M C HenMHefi~bIM1.1 ypaBHeHHRMU, KOTOPaR n03BOJMna p J . I M T b HeKO~ M ~ U KJ/IM~HHO U 3 T M HeOXMTOPble HeJUlHehble 3aAaqbI Ma~e~aTPf.lecK0fl naaabre npktnoxewin o n p e n e n m ceromn BaxHocTb 0 6 p a ~ ~ b3anaq. rx n e p e o ~ a q a n b ~3 a ~ a q aHaxoiKneHuR ynpyrm cxn. neGc~~yioLqux B~one6a~em~ cr.lcTeMe, o5 no ee cneKTpamHbrM AaHHblM noKa He Hamna nocTaTowo m n p o ~ ~ trrprmoxceawfi. x B o614e~cnyvae CVlCTeM c N cTeneHmm b AaHHbIe, npM C B O ~ O A ~(CM. I 3 9) HeM3BeCTHO KBK ~ b 1 6 p a ~CneKTpanbHble KoTopbrx 0 6 p a ~ 3~a a~ a~q aOAH03HaqHO pa3pelrrn~a~ HO He nepeonpeneneaa. B paccMoTpemrx Ham 3anaqax ebdop cneKTpanbmrx ,qaHmrx nepaOHaqaJlbHO OIIpe,IleJlflnCR HarnRAHblMM ~ I ~ ~ M ~ ~ C I CC MO M~ ~M ~ ~ X ~ H H R N I M w TonbKo B npoqecce m peuremlri B O ~ H M K ~npyrvte M cneKTpammre Aanmre, npexne Bcero cneKTpanbmre (PYHKIJMM, miemutie YMCTO MaTeMaTmecKoe M O q eCJTI e~~ 0 0 6 p a 3 ~ 0MX ~ ~BBeAeHKR b 06npo~1cxox~eHkle.E C T ~ C T B ~ H H
