3IYJILTMIIJIEIICATlIIBHAIT CTPYKTYPA J.HEPACTJlJ?HBAIoIIT, HX MATPBQ-@YHKI~J~I~~ ( n o n o x e ~ oaa aacega~miOo'mec~sa8...
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3IYJILTMIIJIEIICATlIIBHAIT CTPYKTYPA J.HEPACTJlJ?HBAIoIIT, HX MATPBQ-@YHKI~J~I~~ ( n o n o x e ~ oaa aacega~miOo'mec~sa8.111.1955 r.)
.
B IIOCJIeAHee B p e M R IIORBFfJIOCb HeCKOJlbKO pa601 [I . 101 H a K DO o6meii T e o p m E i H e c a M o c o n p ~ m e n n ~ I Jx I H H ~ ~ H ~ O I X n e p a T o p o R . T a K m IIO T e o p m m cmcTeM ~ H H ~ ~ H L~ImX @ + e p e ~ q m a n h ~y ~p rax~ ~ e ~ E Bi f K. O T O P ~ I XOCHOBHYIO P O J I ~ H r p a e T O n H 0 3 H a Y H a R a H a J I a T M s e C I c a R M ~ T ~ H ~ T ~ - @ ~ H I ~WI & M R yj.lOBJIeTB0-
(c).
JIYJIbTMHJILIICATHBFIAH CTPYICTYPA J-HEPACTHI'LIBAIOIJ1HX hIATPI/I4
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ncnmena
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i
1
.
s = ( I - a a * ) Z ( I - ya*)-I ( a - y ) ( I - a"a)-
- -i
i
( I - a a * ) v I - ~ a * ) (-a~- y ) ( I - a " ~ ) 1
=(
--
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( a - y ) ( I - a*a)-I (I.- a * y ) = ( I - ya*) ( I - aa*)-I ( a - y ) ,
blYJIbTH~JIHHATBI3IIAIICTPYHTYPA J-HEPACTRrHBAK)llI,MX XIATPHq
11
141
(c) I ] ' = :I x (i)5" (c) 11 p a B H a ~ a E r 6 0 J I b m e M y c 0 6 c ~ ~ e ~ r i o ~ y (c) 1' = Det X (5) X* (c) P a B e H n p o n 3 ~ e ~ e M a T P I I q b I X ( L ) X* (c) 11 B C e OHIl H e I I P e B O C X O n H T
T a K KaIc x sErCJIY M a T p k i 4 b I X X* HHIO C O ~ C T B ~ H H L I XYEICeJI t?,QHHEIqbI,
(i) (i), a 1 D e t X
TO
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p a s e ~r , mo O H p a 8 e ~r 8 ~ a x d o i imorxe edu~azlnozo ~ p y s a . Bonee m o m , cyqecmeyiom maxue yr-lumapmze Aampuqbz u , v , qmo
Ta~xK a K
B C I O ~ Yn c ~ I ~ H H Y H O MI c p y r e
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1 y i j ( i ) 14 1, TO
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AyIorrIee
mo eciody
6
e d u m w o ~xpyze ydosnemsop~emcn~ e p a s e ~ c m e o 1 5 j2 I - x (i)xm( C ) >/ 0.
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HMeIOUJee MeCTO H a K a 2 S n 0 f OFEPyXHOCTH UO I I p l i H Q H U y M a K C l + M Y M a , BO B C e X TO.IKaX
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n p H BbIqHCJIeHHH J I ~ B osaCTHm ~ 3 T O r O H e p a B e H C T B a MFJ B O C I I O J l b 3 Y e M C R P b I M B b I p a K e H H e M AJIR W (;):
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y,, ( i ) r 0 m
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T ~ M caMMM ~ O I E ~ ~cnenysorqee ~ H O npeAnoxeHme: E c ~ uuumpuqa-@yu~yuzx ( L ) z o n o ~ o p $ ~ ua yHumapHa 6 mowe 6 = 1 , mo Aampuqbz I - x (C) x* (C) EI x' (1) x* (1) u.+temm odu~a~06bzii paHz u npedCmU6JlRi0mCR 6 sude
xr ( I ) x* ( I ) =.
lim ( I - 1 Ck 12) B (C],) = u
TO yaBeHCTB0, KaK
MFJ
(
cr
0
o)
u*.
lim ( I - 1 Ck 1)' ( I r -- xr (Ck)x? (C,;))-l 0
0
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P a s n e n ~ no6e sacTm pasemma Ha I-1 1l2 neny npH p + 0. MM IIOJIYYHM TorAa
lim c-1
I - x r (I) X? 1-lL12
(C)
l i m ( 1 - 1 LIZ)B (C) -=u c-t
= 2p
i
0
B
BHAe
cos cp + p" n e p e i i r ~ e K~ npe-
-
- c,
(':d) u*.
T~KHM 0 6 p a 3 0 ~ , MLI nonyYaeM cnenyIorqee npeAnoxeHne, ssnsIonieecn MaTpnYHLIM aHanOl'OM JIeMIMM ?fCIoJIITa: Kawdast zonoAop$~ax Hepacmxeusa~oqast 8 eduHuvH0~ ~ p y z eI 1 < 1 wampuqa-@yuayust x (L), zo~o&op$aast u y n u n ~ a p ~ a6z mowe C = I , ydosnemsopxem ~epaseucms y 1-1 L i 2 I - x (C) x* (L) > ( I - x (C) X* (1)) B ( I - x ( I ) X* (i)), ((9 I I -c l2
r
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B
cmpoumcst no Aampuqe
ECAU x r (I) x* ( I ) = u ( ;
;) u*,
K
H e p a B e H C T B a M T a K O r O B H A a D p H B O n F l T 3 a A a q E i : 06 O I ~I ~O~ O ~ P Z ~ ~ O B E ~ H D RI ,I e p e B O H R ~ e r O J - H ~ P ~ C T r F I B a I O w E t ? MElTpFIlI HepaCTHI'HBaIOQ&ie; 0 B H A e 3 J I e M e H T a P H b I X MH03IEZiTez e f ; o B n A e I E O ~ @ @ ~ Q E I ~ Hrnamoro T~ m e H a B p a 3 ~ 1 o x e ~ ~arHna m T m s e ~ ~ M0~ ~ T6P H Q L I - @ Y H K ~ H EBI P E A J I o p a H a B OITc A C H 1T 1,l M,
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IIo;rnpoc~paac~~ %C o L naaraaeTca 11o n o ;iirI T e JI L H o o II p e A eeCJI!X [x, x] > o AJIH B C e X X ( 5 f 0 ) 113 %, O T P Y I y a T e J I L H O O I I ~ ~ A ~ J I ~ H F I L I M CCJ[II , [x, x] < 0 AJIR Bcex x + 0 1x3 '91, m H y n e B T J M, CCJIH [z, X] = 0 AJIR BCeX x E (32. npoc~pairc~n a ocL ~ a a b ~ n a cM ~a xc c~ CJ~M a JI b H LI M n o n o x m T e JI bH o o n p e A e JI e H H LI M, ecnn a - I I O J I O ~ ~ I T ~ J I ~ HonpeaeneHrIoe O npocTpaaCTBO kl a H e CO~epXcrITCFln 110JIOlI~HTeJIbHO OIIpeACJIeHIIOM IIOAIl13OCTpaHCTBe 6on~rueB pa3MCpHOCT13. A~Ja.70r119~10OIIPeAeJIRIOTCR R l a K C l8 M a JI L H O t? ;reHHLIM,
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MaKcnIvlanbHoe
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I ~ ~ B ~ C T H OntaticuntuAbfii,ze , nonowzzmenbHo onpedenefimle
nodnpomy w e paawep~ocmb p, Naticuwanmaze ornpuyamentrio onpedenctt~~bze -- o d ~ y u my we p a a ~ ~ ~ e p r i o c r nq.b Cyaiujcea p + q pama p a ~ ry npoclnpnltcmca L. 1 1 ~ ~ 1 'a,, 6 a,, . . . , am-- I I ~ H O T O ~Ga3nc ~ ~ I ~ICpOCTpaHCTBa. M a ~ p n y a cmpaficmca s L uweronz
odlzy
u
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n
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C O ~ C T B ~ H H Hsmcna~
0
R=aRZ+b+
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C 3 ~ 0 f iq e J I b l O
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S ~ J ~ ~ TYTO IIM z o ,s a 3 a ~ ~ b r ipie a y n b ~ a r MO;I:EIO C @ O ~ I A ~ ~ J I I I J ) O B ; ImaLJe: T~ ecnzk ~campuqa-$y~rtiyu,zR a ( t )J ,wofiomoHfio He so3pac8rnnr7n, rno rz .trompnqa @ Y F I K ~ U RR (t)J~ t o f i 0 m 0 ~ 1 1 Ire0 sospacmaem. H) B a a ~ ; z ~ o ~ c ~ n ol sraew e l f , YTO ( 5 ) Ecnu ~tampuya-@yfixqz~.r~ R2(t) a 6 c o n ~ ) ~ ~fiel~])f'j)oltul(~ rto I: Iit(lnep6aAe (a, j3) u Dct R2((1 0 , mo u . ~ a m p u y a - $ y ~ ~ q ~ iR ~ (r tC) ItInKxr n G r o . ~ m m ~ o ~renpepbzsfiaG u ~ m e p s n n e(a, p). 3.7~1 ~ o r i a 3 a ~ e x ~ cn ~ o rn 0a G a s ~ ah r C~ I - ~ o n am c n o ; r b t 3 y c \ r r r p e ; I c r a n n e a u e
-33
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R (t cAt)
- X (i) = a
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$ R ( t - At)-R(t),l < n R 2 ( t - C A I )
-
R 2 ( l )f
1I
( 1- 2 +
)
-1
.
R2( I )- R (t) 1 .
8 ( t ) ) (1
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do (s).
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nbITeHaeT npH Ax
>0
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hlOHOTOHHOC
'p
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TaH rcaH ~ a ~ p v i u a - @ y ~(pH(x) q ~ orpaaaueaa, ~ x+ co . B caMoM nene,
-+
TO
oHa llMeeT npeAeJ1
11 y (XI)- cp (x") I(\< I t,r ( y (x') - (p (2")) ( = I tr cp (2')
-
llplr
tr cp (x") I < 2 ,
Tax Has CIfaJIHpHaH @YHHQIIH t r (p (x), RBJIRHCb O ~ P ~ H H ~ ~ ZIH MOIiOTOHHO H O ~ ~oapac~aromeH,aMeeT npepen npn x+ + co. Ho Torga B CHJIY C B O ~ ~ C T(3) B JI (4) MOnyJIH R (x) J - H ~ ~ ~ c T H MaTpRqbI N B ~ W (x) I o =R ~(x) ~ U~(x)~ MOayah R (x) nMeeT npeAen npn x - t co H MaTpHqa R (x) J MOHOTOHHO y 6 ~ 1 s a e ~Torga . R(O)J>,R(x)J,>R(-, m ) J , (3) O=J-R(O)J<J-R(x)J<J-R(+co)J (R(O)=I).
+
R ( + CO), IIO BTOpOMy (3) BbITeIEaeT:
113
YCJIOBItIO
TEOPeMbI,
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IIO3TO3ly
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M ~ IIOKBWeM I ceG1lac, YTO 1.23 OI'PaH1.29eHHOCT1.2 M a T P I I Q ( 5 ) Ef R-' ( x ) 8 b l T e I E a e T OrPaHLlYeHHOCTb M a T p H q b I ( x ) . C 3 ~ 0 6Q e J I b I O I I e P e I I H W e M COOTHOUIeHITe (2) R BHAe
U
+
Hz (.c Ax) -R2 ( x ) - J= Ax
-- -- -
x+Ax
5
J-(I----
-
%+Ax
H(t)dt+
1
...)J ( I -
H W ( t j d t ) +. . .
X
-?--
U* ( x )R* ( x )
Ax
-2 R a ( x )J = 2 R ( x )U ( x )H(x)JU* ( 2 )R * ( x ) , dx
- 2U ( x )H ( x )JU* ( x )= R-' ( x )
d
R ( x )J
+d
(4)
R ( x )R-I ( x )J .
(5)
L
\ epH
yr
UJ( I )=
(t)dt
=R
( x )U ( x ) ,
0
d u1 ( r ) d -R rlr = dx d w ( x )J = U-l dx
w-l( x ) -
-H
( x )J
=U-I
( x )R - I ( x )
d
R ( x )U ( x )J
+
12-I
(x)
d
(x)R - I ( x ) d R ( x )U ( x )J + U-' ( x ) d d
-
I
d
( x )U ( x )4-R ( x )z9,U (4,
ll ( x )H ( x )JU* ( x )= R-I ( x )2%R ( x )J -1-
d
U ( 5 ) .U-I ( x )J .
x ( t )= 31 (%d R ( t )R-I ( t )- R-l ( I ) ; R (t)) .
U ( x )J , ( x )J ,
/
J
(i)
ganee,
T a K l i a K @ Y H ~ < ~ IR ~ H (t)
11 T a r t a f I
xe
oqeHKa sepHa
J
MOHOTOHHO y61,11jae.r,
Ann U-l (x), T a K
KaK
TO
XlYJIbTklrIJIkIIEATMBHAH CTPYIETYPA J-I-IEPAC'I'HrLlBAIOIIIBX RIATPbILI
175
T e o p e ~ aironHocThlo aoHaaaHa. Ann BToporo cnysaa sepna cneayrozrlaa T c o p e M a 10. E c ~ ud m noc.~e~osameJcb~ocmu fiyAamunnuxarnucribLr zrttrneepafloe
7 +a
w,,
e-Hn ( t )
0
J-$op.mt
oTpaHuvemz csepxy:
IIOnyzlaeM Ol~~)atlbillcHHOCTb LC',,. f i o ~ a a a n ~ a TcopcMa fl oxBaThrnaeT, c ~ ~ ~ e o c l ~rrl icon, y ~ a i iI ~ O H C ~ I I I I , I X npoamene~~*ii %IM
-II(lnl
w,, =e
e
- I l p. . .
I i(i
B. II. IIOTAnOB
B H TOM naparpa4e I I ~ ~ ~ I ~ I OnTpCoR6 ~ o n ~ ~ e i i ~npeo6pa30na~lr~1, b1e ncpe1 3 0 n n q ~ eJ - ~ e p a c ~ n r n s a ~ o m rMaTpnqbI ne y B ,qpyrHe m a c c u MaTpmq. 1'. O C O ~ ~ HBaXHOe HO AJIH ~aJIb~eiiIller03HaseHHe HMeIOT TeOpeMbI, YcTaHaBJrEBaIoIQHe C B R 3 b Me?HAy J-~epac~srr~~aroI4mMrn MaTpHQaMli y: yJy* 4 J I 1 O ~ ~ I Y H ~ I HePaCTZIrE1BaIOIII;IIMH M H MaTpZT4aMm X : xx* -;= I .
T e o p e M a 11. Ecnu seqecmsefimze duao.o~anb~breMampuybl a a b C00mHOWeHUe.M -b L J ,
CA.93UFlbl
nlo
ispo6~onurteli~oe npeo6pa306a~ue y = ( a x b) ( b x + a)-I
+-
( 1 ~ 1 ~ a ~ a ~ ~ u ~6 a~ e~r Un M OH~ HO O ~ H U Y H Ocoom6emcm6ue ~ ~ e x d yFlepacmssu~ , a l o q a ~fiampuqafiiz u x , d n z xomopbzx Det ( b x 4 a)*+ 0 , u me&u J-nepacm : i ~ s a r o q u ~ ~u a m p u y a ~ tyu, ddz Komopbzx Det ( y b a ) # 0. B caMoM aene, ecnm Det ( b x + a ) - 0, T O npeofipa30~a~me( I ) HMeeT -
CMbICJI H
J -y*Jy
+
(x*0 + a)-I ( x " a L b) J ( a x b) ( b x + a)-I = a)-I [(x*b+ a ) J (bx a ) - ( x * a b) J ( a x b)].(bx + a)-' = ( z " 0 + a)-l [x"bJbx '- aJbx+ x*bJa + a J a - x*aJax - bJax - x" a J b - bJb] ( b x - a ) ' = = (x* b + a)-I ( I - x*x) ( b x + a)-I (8) =J
-
=(z*b
-
+
+
+
+
n, e c m x"x < I , TO y Y J y J . O O Y ~ B H A nprr H O , TOM yIImTapaue iMaTprnuH z rfclpexoART B J - Y H ~ T ~MaTpllUM ~ H ~ I y. ~ Janee, TaK K ~ K by-a=b(ax+ b)(bx+a)-l-a= = [ b ( a x b) - a ( b x + a ) ]( b x -p a)-l= - J (bx $ a)-l,
+
T O HePaCTFIl'HBaIoIIlFlM MaTPHUaM X , AJIa K O T O P b I X Det ( b x f a ) + 0 , COOTBeTCTBYIOT J - H ' ~ P ~ c T R I . L I B ~ MaTPMUbI I o ~ H ~ ? y , AJIH HOTOpbIX Det (b7J - a ) ,- 0. Jlo TOrHa PI D c (~y b - 0 ) =, 0 . ~ ~ ~ C T B E I T ~ J I paBeHCTB0 ~ H O , Det ( b y - a ) = Det ( y b - a ) ,[;I11
H ~ O C O ~ ~ HMaTpHIJ I ~ ~ ~ X
DbITeKaeT I13 PaBeHCTBa ( b y - a ) b = b ( y b -- a ) ;
AJra oco6en~oii MaTpMUbI b O H 0 COXPaHReTCR BBlTHy TOrO, Y T O O C O ~ ~ ~ H H ;uraroaanbaas nlaTpnUa b Mox<eT paccMaTpaBaTbca KaK npeaen H ~ O C O ~ ~ H H M X zLraroaanbImIs M ~ T P H I I H Toro, Y T O Det ( y b - a ) = Det (by - a ) Henpepma0 3aBmCRT OT 3neMeHTOB b. H a H o ~ e q ,ecnE y asnaeTca J - a e p a c ~ a r m ~ a i o u ~ e~i ia ~ p m u e i Ans i , ICOTOpoii Det ( y b + a ) =f I), T O ypauaeane ( a b) ( b x -' a)-I =. y
IT T a K
KaIC
xbi a=(yb-a)-l(b-
ya)b-kaz
= (yb - a)-l [(b - ya) b $ (yb- a) a] = - (yb -1- a)-l, TO Det (xb -/ a) = Dct ( b X -, a ) +s 0 H H ~ G A C ? H H O ~X C B R 3 a H O 1 1 p e 0 6 p a 3 0 ~ a ~ I I e(I). ~ Ho T o r n a , H e p a c T a r m s a e M o c T b ~ a ~
C 3 a A a H H b J M ?/ p ~ i sx b Br m e K a e T
El3 ~ I O I F ~ ~ ~ ywe H H O~ O ~ P~M Y J I ~ I(8). T e o p e ~ an o ~ a a a ~ a . M3 A O K ~ ~ ~ H HT eOOG p e M b I I i e ~ O C p e n C T B e H H O C J I e A y e T , ¶TO cmamr 0 6 n a ~ a c&T~ npeo6pa3o~a~me
y = (UaUx I'Je a , b -- BeII[eCTBPHHIde
TCMH ?KC?
+ 1ZbV) (%bus+
CBOG-
(11)
AElarOHaJIbHble MaTPHQbI,
C B S I 3 a H H b I e COOTHOIIIeHHeM
- J - y ~ n ~ a p ~MaTPEIUbI, b~e U, v - Y H E I T a p H b I e M a T P H 4 b I . b2 = J, U, 2'. Yepe3 &x, 8, O ~ O ~ H ~ Y I ICOOTBeTCTBeHHO M MHOXeC'rBa BCeX Y H H T a P HbIX I I J-YHHTBPHHX M a T p M I l IIOPZIAKa 732. H a ? K n o e m 3 H U X RBJIZIeTCR T e n O M n ~ " M ~ ~ H oHOMnJIeICCHOM M I I p O C T p a H C T B e . T o s m X , 9, AJIZI KOTOPbIX HMeIOT MeCTO O C T P H e H e p a B e I I C T B a XX* < 1, y J y * < J, FillJIRIOTCR B H Y T P e H H H M H ,
" 0
TpaHIIYHbIMEI ~ Y ~ YT eT TOYKII, AJIII KOTOPbIX B H e p a B e H C T B a X e C T b 3 H a H p a B e H C T B a , T. (3. XX* I , y J y * 4 J. T O ~ KX ,HY , AJIR IEOTOPMX Det (ax b) = 0, Det (yb - a) = 0, 06pa3y10~ r m n e p n o s e p x H o c T m nepeceKaIoane ~ T H TeJIa ]IIpe06pa30~a~lte(I) y C T a H a B J I H B a e T B 3 a H M H O O A H 0 3 H a s H O e COOTBeTCTBMe A 4 e X A y BCeMR TOgKaMEf T e J I a 3a IICKJIK)¶eHHt?M T O P e K , JIeXaIrIJIx H a rHllepUOBepXI~OCTH H BCeMH TOYHaMm 3a H C K J I I O 9 e H H e M T o s e K 8;. ~ b 1 - 6 y n erOBOI)IITb, ~ Z I ~ On p e o 6 p a 3 0 ~ a ~ (I) ~ eb ~ o 6 ~ a a t Ba ~e ~~ H M H O O,JH03HaYHO Sx H a 6 OCHO@HOOW. I I y c ~ b T e n e p b y = y (x) - npom3~0~1b~oe a H a n m T a s e c K o e o~o6pame~me, yCvraHaBJIHBa101Lp3 B 3 a U M H O O A H O 3 H a ¶ I l O e COOTBeTCTBHe Me?IEP;JJ BCeMEi T O ¶ K a M H TeJI EI 3a HCKJIIOYeHHeM TOYeIC, J I e X a Q H X H a H e K O T O P b I X a H a J I I I T H VleCKIIX rHIIt?plIOBt?pXIIOCTRX 8;. ~ I M W TMeCTO C J I e A y I O W a R T e o p e M a 12. ITpeo6paaosa~ue
9;, 9&,
ex, R,.
9;
Rx,
+
g,,,
8,
ex e,,
g;,
+
Y = (UaUx UbV) (%bus+ %aV)-1, (y' = (lIaUx -+ U bV) (BbUx + 8aV)-I), zde a u b
ceqecmceHHbze d ~ a 2 0 H a A b H b l e & a m p u q b z , ydosnemcopfimt.que y c ~ o - YHUmUPFlble Aampuqbz, - J-yr~urnap~bze . M a m p u q b z , f i s A R e m c f i 0674~0~ sudo& a H a ~ u m u u e c ~ o z o0 m o 6 p a w e ~ u ~ycma, ~ a ~ n u s a m q e zBoB ~ U M H Oo d ~ 0 3 ~ a u ~c o~ e~ m s e m c m s ue eo c f i o s n o f i nzewdy H e p a c -
sum a2- b2 = J, U,
v
m,wusamzyuiuu f i a m p u q a w x u
U, %
J - ~ e p a c m ~ z u s a l o q u & uiuampuqafiu y. M a H e 6 y n e ~3 A e C b OCTaHaBJIEIBaTbCR H a A O K a 3 a T e J i b C T B e 3 ~ 0 f Ti e O p e M b I . 3'. O c o 6 e ~ ~nop 0 ~ ~ 0 fBi n A npeo6pa3o~a~me(11) I I p E f H H M a e T B TOM
~ n y q a e ,K o r A a
178
B. II. IIOTArIOB A
M1102~n~eJIb ei" BBOAnTCn 3AeCb An% T O r O , ~ T O G ~I II p e 0 6 p a 3 0 ~ a ~ 1lIMeJI0 le C M ~ I C ~ n1060ii Hanepea s a n a ~ ~ ~~o ei ip a c ~ ~ i r ~ ~ ~hraTpanbr a ~ o m e ix,. i HCTPYAHO y 6 e ~ m ~ bB cTOM, ~ TO TO me n p e o 6 p a 3 0 ~ a ~ l ?MoxreT c GMT~ TaK3IEe 3aEFfCaHO B BmAe
y = J ( I - st'"-l ( t - x ) J .
J ( I - st*)-'(t - X ) J =- ( t - x ) ( I - t * x ) - l , (t-X) J(1-t*x)=(I-xt*) J(t-x), tJ- xJ-~J~*x+xJ~*x=JI-x~*J~-Jx+x~*Jx, ( J - tJt*) X - z ( J - t * J t ) , I s =X I
x = (I-t*y)-I ( t - y ) , Tar
0.
( B + dy)-l'4 [J- Jy*] A* (B*-:- y*d*)-'.
n O I F a 3 a T e J I b C T B C I I ~ O ~ X O ~ ~ I I M O C TMbI I T 3fleCb HC OCTaHaBJIEiBaeMCs. ~ ~ M C T H M9 ,T O C B O ~ C T B ~ M RY, I F a 3 a H H b I M L I B T e O I ) e M e , o6naflae~,B IJFCT-
HOCTH,
n p e o 6 p a 3 0 ~ a ~I-Csnrr ~e 1 V=
i (I+y)-I (I- y).
.~sn.%emcaJ-~epacrn~zusamzyeii. 4 0 I< a 3 a T t? JI b C T B 0. P ~ C C M O T ~ E M~ ~MT P H ~ Y - @ Y H K ~ E I O
y (t) = e i N t AJIR B e I ~ e C T B € ? H H b I S3 ~ a 9 e ~ E 3 t. fi 3Ta M~T~EI~~-@JJHIF YAOBJIeTBOPBeT ~HR @C!pe~I@iaJIbHo~y YpaBHeHHIO
Jplt$-
y' (t) = iNy (t) I1 H a s a J l b H O M J J yCJlOBHI0
y ( 0 ) = I.
= - 2 lm [gATy(t),
nonaran aaecb gy ( t ) = f,, 'P' (t) = - 2 Im
[JtN,
gy (t)] = - 2 Im [gy ( t ) N , gy (t)l-
IIOJIY~~M:
f,]
= -2
[J,]
N J - JN* ,i
[ftl*O.
13 c~rnyT O ~ O , YTO CJC* = O OaTCXbHO,
MaTpIlqbI
111
H
iV pannu HynIo,
H,
cneno-
' O
~ahdIf0II
HEX HBJ,
-1e-- Oj ~ d Aualtadn n x vualt5adarr
H
V ~ HaoH~aa~3ahraa ~ - (1 > 0> > 0) 01 q~3An' o ~ s r r a ~ m a ~ a r _ l a ~ .garaLHHoHu\r x m d o ~ amrrH xmadan raltadn en LamOK 91CaLHHOHN mrq~de~~au\raIfe f i z i ~ a dO~L h ' N E L ~ Y U B E
klYJIbTB~JIBKATI/IBHAHCTPYKTYPA J-HEPACTRrI4BAK)QkIX MATPI44 185
1'. ~ Y C r o n o ~ o p @ ~na
T J -~ ~ e p a ~ ~ R r E i B a l o I t l a f l M a ~ p H U a - @ y H H q l T f l
(1
1
0, B >, 0 - 3 p ~ u m o e o - 1 ~ e o m p u ~ ~ a m e n bfiampuqaz, ~~bze mo cmca BAB/ ----1 I-tor l2 ade wampuqa B cmpoumcz no uampuqe I - x (To) x* (5,) cnedymzyuu o6pa3ou: ecnu
I
- x (C)
u*. A (t - X U ) = 0 , TO A ( I - x,x,*) A = A ( I - tt*) A = A J A = A ,
~ ~ ~ ~ c T B H T ~ J I TaK ~ I I oIC ,aK
( I - x ( C ) 2;) B ( I - sox* ( 5 ) ) >/ ( I -x ( ) x )
(
I, 0 O ) u* Au ( OIr O0 ) U*( I - xox* ( C ) ) =
= ( I -x(C)x,X) A ( I -x,x* (C)) = ( I - ~ ( 5 )t*) A ( I - tx* (C), M
JTAOBJleTBOPReTCU HepaBeHCTBO
x ( C ) x* ( 5 ) 2 (I-1c0 1') (I4=) ( I - x (5) t*) A ( I - tx* ( L ) ) , 1 I -roc l2 a TO H 0 3 ~ a q a e Y~T,O YacTHoe w1 (i) = b-I ( 5 ) w (C) ocTaeTcu J - ~ e p a c ~ a r ~ s a ~ o r~q eai~i p n q e i i - $ y ~ ~B ~Kpyre ~ i e i i1 TI < I . I
-
13 Tpyn~IMOCKOBCH.M a T e M a T 1 1 9 .
06-Ba,
T.
4
194
B. IT.IIOTAITOB
x (i)= ( t - w ( i ) )( I - t* w(i))-l, t =
~03@@~lqZleHTbI IIPI4 (i- C0)-',
(Oo
$TI,)eii
IIOJJYYHM:
( I - 2, t*) C-k
= 0.
P~CCMOTPHM 3~1eMe~Tap~b16 MHOXMTeJIb b (6),
H M ~ J O I Q HIIOJIKIC ~ ti TOY HC
6,
~ a f i n e ycnoBne, ~ IIPH IEOTOPOM gacTHoe wl ( C ) = b-l (6) w (l) ocTaoTca J-~epac~arnsarowe~ f ia ~ p ~ i q e i i - @ y ~ ~B qKpyre ~ e i j I I i 1 < 1 , T. e. ycnome, llpH KOTOPOM HMeeT MeCTO HepaBeHCTBO
LI
J - w (i)J w * (',) >, J
--- b (i) Jb*
(i).
T ~ EtaK K
J - w (6) Jw* (6) = ( I - x ( 5 ) t*)-l ( I - x (6) X*(i)) ( I - tx* ( [ ) ) - I ,
(1-1 co
12)(~-lclz)
I - x ( ~ ) x * ( ~ ) ~1 c 0 - - c 1 2
( I - x (C) t*) A ( I -- tx* (i)).
(1)
n o x a ? ~ e umo ~ , ~ J Z Amozo, umobbz tiacmnoe wl(i)= b-l ( C ) w ( 6 ) 6 ~ ~ 1 0 J - n e p a c m ~ z u e a m ~ e i iaampuyeii-@yn~y ueu 6 Kpyze 1 i 1 < I, ~eo6xoau.tro IL aocmamouno, um06bi ( I - x, t*) A= 0.
MYJIbl'I4IIJIMKATMBHAH CTPYETYPA J-HEPACTHrklBAH)qklX MATPHU
195
t
~ J I HgOKa3aTeJlbCTBa I ~ ~ O ~ X O ~ E J M O 3aMeTHM, CTH YTO H 3 B M T e H a e T , 9TO
HePaBeHCTBa
(1)
(I-xot*)A(I-tx,*),(x0-z(~))v*
I-x(6)x" (6) >/ a
I, 0 vt* A ~ V * ( O~ ) v (xT- X*~(I))=
(1- I t o 12) (1-I
150-5
l2
5 12) ( I - x (6) t*) A ( I - tx* (C)),
TO H 0 3 ~ a ~ a eTO ~ , 9acTHoe
w1( C ) = b-I (!) w ( C )
TO,
6epx
B 3JIeMeHTapHOM MHOXHTeJIe
T Y X C J-YHHT~PHYK) MaTPEIqY
U, ¶TO H B C-k, (I- xo t*) cdk = 0,
q',
H TO i f c e
M M IIOJIYYHM E13
YTO EI
(I-x0t*)A=0, T. e. ¶TO g a c T H o e w, (i) = b-I (P,) w (i) 6 y ~ e TJ - ~ e p a c ~ x r ~ s a m~~au ~e p ~ ~ @ Y H I E ~ I IH ~ ~~ ~ .o M TOI.0, ~ PI3 P a B e H C T B a q H C e J I q' BHDHO, YTO YaCTHOe W l ( c ) H M e e T B TOYKe IIOJIIOC M e H b I l I e r O I I O p H a H a , Y e M W ( i ) . O ~ w e n n e ~ anonmca e H a e ~ ~ l ~ ~ OKPYXHOCTH. s ~ o i i I I y c ~ bJ - ~ e p a c TRl'HBaIOIQaSZ B K P Y r e C 1 M ~ T P H ~ ~ - @ Y H K ~ JWH(C) H H M e e T lIOJIIOC B TOYKC C = 1 H J - Y H H TH ~ a H P~H K O ~T O ~JQ're O ~ ~ J Q ! I H H Y H OIFPY%HOCTEI, o~ CO~epWame6 3 T y TOYKY:
to
40.
I 1
/ J - b (C) J b * (i).
e
MYJIbTllnJIkIKATMBHAfl CTPYKTYPA J-HEPACTflI'llBAIOQllX MATPllq
197
T ~ IKFa K
J - w (C) Jw* ( C ) = ( I - x (i)t*)-l ( I - x ( C ) X* (C)) ( I - tx* ( [ ) ) - I , 1-1 5 1' J - b(C) Jb* ( C ) = 2E,
TO HCCJIeAYeMOMY HepaBeHCTBy MOWHO IIPHAaTb B H A
I
- x (C)
x* (C) 3 -1 - 1 5 1 2 ( I - x ( C ) t * ) 2 E ( I d t x * ( C ) ) . 1 I-[1'
f l o ~ a w e ~ , TO d n s nzozo, um06bl uacmnoe w, ( C ) = be' ( C ) ~ e (C) r 6wno J-nepacmxeusalozqeii uampuyeii-@yn~queii 8 ~ p y z e 1 C I < 1, f i e o 6 x o a u ~ o u d o c m a r n o ~ ~ oumo6bz ,
( I - x0 t*) 2E = 0 , x,x,* 2E X,X; ,< x,x,U. A O K a 3 a T e J I b C T B a H ~ O ~ X O A E Z M O C T3LaI M e T H M , 9 T O IIO r J I a B e 1, $ 2, Il. a X,X; - 3PMHTOBO-HeOTpHQaTmbHaR. 1 , H e p a B e H C T B O ( 2 )M O W H O 3 a n C a T b TaK: C Y H T ~ R B ~ W ~ C T B ~ H H M0M ,
Ans
, ' 7
Xo - Y H H T a P H a H M a T p H q a ,
22, x,* ( I - C )
,
=( I - x
(C) x,*)2E ( I - xox* (i)) = ( I - x (i) t*) 2E ( I - tx* (C)). 1-
I - x ( C ) x* (i)>-I I - c a sm
H
0 3 ~ a ¶ a e ¶TO ~,
c 12 ( I - x (C) t * ) 2E ( I - tx* (i)), l2
~ ~ C T H O ~
wl ( C )
MbI
683
= b-l
(C) w (C)
TpYna Y ~ ~ W M C B R TOM, ¶TO I I P H #?neHEiki
Ha
? J J ~ ~ M ~ H T MHO~ ~ H M ~
WTeJIb
5+1 b(i)=I+-*-EJ ,-I g a C T H O e Wl B Kpyre
(C)
= b-I
1Cl < 1 n
( C ) W ( L ) OCTaeTCB J - H ~ ~ ~ C T R MT ~ TE ~ HITB I ~~ GI - ~O Y HQK ~ QH n M e e T B T o u s e C = I ITOJIIOC M e H b l n e r o naprrgrca, s e M w (i).
\IYJIbTJI~JIHK.ATBBHAFICTPYKTYPA J-HEPACTFII'BBAIOIQHX MATPHIJ
,
c $ ~ x , x , * c -= ~ ~ P ~ x ~ x ~ x=~~t l *k cx -~ ~t *= c -c?k ~ ( I - x 0 t * ) C-,,tl =
= cXk ( I - t t *
+tt*
-
x0t*) C - k t l = ~
* Jk c - k f l $
c*_k( t x ; - I ) xot*c-ktl =
= c*-kJc-k+l MJIH
T ~ KHa H ,
K p O M e TOI'O, H3
-
--
( I - B J ) r-k--2EJ~-k+l f . . .
(C -l ) k
199
B IcasecTne n p m M e p a paccMoTpmM HoMa, J - ~ e p a c ~ ~ r ~ ~ a m Bw ~ e rp ox ~ T a p H O r O H a B ~ ~ ~ C T B ~OCH: H H O ~
b (A)
CTOHTO T M e T H T b ,
YTO 3 T 0
B O T J I n q m e OT CKaJIRPHOI'O
Ci
Ha
MHoxmTenm
>0
IIOJIH-
m J-YHH
+ ihEJ.
=I
pa3JIOlKeHHe
Ha
MHOXCViTeJICI
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