Документ взят из кэша поисковой машины. Адрес оригинального документа : http://beams.chem.msu.ru/doc/zh_vychmath_mathphys.pdf Дата изменения: Sun Oct 21 21:25:02 2007 Дата индексирования: Mon Oct 1 20:28:34 2012 Кодировка:

XYPHA.T

B'IqHCNNTAT'HOH

MATEMATNKN

H MATEMATH1ECKON

OUSNXN,2OO2, MOU 42, M

12,C. I89I_]898

Y,4K 519.6:53LI33

9OOEKTI,IBHAfl PA3MEPHOCTb OA3OBOU TPAEKTOPVWT B 3 AAATIE BI,I3yAJrpI3 AIII,IT,I 3 B OJTIOqI{r,I AI,IHAMnqE cK oI,I CI{CTEMbII)
@ 2002 r- E. A. Eenera*, A. A. Prt6axoB*, A. H. Tpy6HrrKoB*, A.H. rfynNvxoB**
(* I 19899 Mocrcea, Jlenuucrcue zopbt, M ry, xurtt. **eua. $arc-m; $arc-m) e -mail : eI ena@phy s.chem.msu.ru flosrynnna B peAaKrluro10.M.02 r. rlpegnaraercfl MeToA onl{cautul rroBeAeHnt AuHaMH.recrofi cr{creMbr Ha ocHoBe noHrrrl{Jr uraBHbrx MoA I{ eQQexrunnofi pa3MepHoc'rH Qaaonori rpaerropr{u, a rapKe HaffrrAHbrir cnoco6 npeAcraBJreHI'I' rpaeKropl{ll nyreM oto6po
BBEIEIJLLE
ounoli u: upo6leM HeJIIrHeIinoIi AI,IHaMuKuflBrflercr HarJrrAHoe onucaHue c'creMu. EcABr.DKeHrur nn SasoBoe npocrpaHcrBo I'rMeer 6onruryrc pa3MepHocrb, eQQexru"ro ""ryanr.r3rrpoBarb rpaeKTopl{rc ABH>KeHlrrCrrcTeMbI AoCTaTOqHOCJIO>KHO. flnx perueHvfl sToli 3aAaqI'IB AaHHoli pa6ore npeAJraraeTcr rrcnoJrb3oBarb pa3JroxeHrre ABrDKeHrur clrcreMbl B t?-MepHorta SasoBoM npocrpaHcrBe Ha oproroHaJrbHbre cocTaBJrrroulue (MoAbrgnrcxeulrx), oolaqaroulue sKcrpeMIJIbHbIMI'r CnofiCrnaMLI:cynepno3r{rlr{Jr ,r3 m uog Hau6oJree roqHo annpoKcrrMrrpyer 3aAaHHbIe oco6euHocrlr sBoJrrotlun cucreMbl cpeArr Bcex m-MepHbrx cocraBJrrroqgx easonoii TpaeKToplru, m = I,2, ..., n. AHaJTI'I3AHHaMI{KII MoA no3BoJlter npeAcraBr.rrb gBoJrroqnrc clrcreMbr B rIeJroM. ,(nx rtlHorlrx Bcrpeqarculuxct Ha npaKTIIKe ArrHaMrrrrecKr.rx crrcTeMror eaeonar rpaeKTop', B MHoroMepHor'aQaronoM npocrpaHcrBe pacnoJlaraercfl. B npocrpaHcrBe MeHbruero qucra KoopAHHar.HanpI'IMep' ABDKeHI{e qacrnqbl B qeHTpaJrbHoMnoJle [poucxoAl,IT B nJrocKocrrr, coAep>rau1efiBeKTop uMtryJlbca upaAvyC-BeKTOp qacTI'IIIbI' AJI' erO TOtIHoro On[CaHVfl,AoCTaTOrIHo qeTblpexMepHgro noAnpocrpaHcrBa' a AJI' BI'I3yaJrI{3a\vu Moxc{o ucnoJrb3oBarb ABe npocrpaHcTBeHHbre KoopAtrHarbr, Ir3MeHJIIouIuect co BpeMeHeM.Ma-noe Bo3MyIrIeHrreraMI,IJIbroHHaHa [puBeAeT K yBer[rreHr{ro pa3MepHocrll noAnpocrpaHcrBa' coAep>Kalrlero Qasonyro rpaeKToprlro, oAHaKo B prAe cryqaeB gBoJrroqrrro cIrcTeMbInpn6lzxKeHrre HeKoropori ubcKocTr{. Orge*uoe paccMoTpeHue cocrasrslotrIeli Qasonoll TpaeKToprrtrB oproroHaJrbHoM r aroli nJrocKocTr{HanpaBJreHurr Aacr npeAcraBJIeHIIe o6 oTJIH'rr{HSBoJrloquu cl{creMbl or ABrrx< enr4fl, rIeHTpaJrbHoMnoJre. nrrocxocrr, B B Koropofi, "B ocHoBHoM",cocpegoToqeHbl cocrorHr.rr cucreMbr, Mo>KHo onpeAeJlr.rrbKaK nJlocKocTb, rpoeKllut npocTpaHcrBeHHbIXKoopArrHar Ha KoTopyro B cpeAHeM no BpeMeHrr r{Meer Han6onsunrfi KBaApaT AJrrrHbr. 06 e{QeKTIrBHoli paruepnocrlr I{ MoAax ABTDKeET{x. floscur,rM npeAJraraeurli no*oA Ha nprrMepe MarepnalrHoli TorrKr.I B IrrecrLIMepHoM ABIDKeHI'Iq Sasonorrr npocrpaHcrBe. O6ogHa;rM "6;, 6, xoopAI'rHaTHoeBeKTopHoe eBKJII'rAoBo npocrpaHcTBo, ero gJIeMeHTaMrr rBJrrrorcr y[oprAoqeHHbre na6opbl r'r3n .{nceil - KoopAlrHar coorBercrByloqero BeKTopa. B "P9.oli QuxcllgosixHiul MoMeHTBpeMee) e ecrr opToroH alrb-Hafl, npoeKrlr{q BeKTopa x(t) Ha nJrocKocTb c HopMaJrbroe B TpexMepHoM eBKJII{AoBoM npocTpaHcTBe V3; zgecb rr AaJree(', ') o6oiHaqaeT cKuurrpHoenporr3BegeHue sJreMeHToB eBKJII'IAoBa npocrpaHcrBa. 3agava nsl6opa AByMepuofi nrocKocTrr, n roropori " oc"o""o,
l)Pa6ora BbrnoJrHeH a A.l4.yyrlrvxonbrM npn qacruqHofi $unaHconoii nogqepxxe p@@I4(xog npoexra 0l-0140210). 1891 9*

Hn t e [0' D BeKrop o6ogna.rr{M = t*(t pdtj " .0" x(r) e V;'-BeKrop q(r) QasoauxroopguH-ar npocrpaHcTBeHHbrx nepeMeHHblx' e %3 nercrop p(r) LlMnynrca.',[ni i nlo6or.i'"&rop;; . *r,llell = l, BeKTop x(t) - (x(r),


r892

EEIIE|A r{ Ap.

cocpeAoroqeHbr rrepeMeulen:afl, Mareprrrubnot ToqKtr, TaKITMo6paaor"r, cBoAlrrcs x srt6opy BeKToPa eo e 8t, AocraBJulroulero MaKcIrMyM KBaAparuquoMy SynrquoHaJly

- (x(r), e)ellzdt, iJtt"trl
0 TO UINUT,I{TO XC CAMOC, MIIHIIMYT'T QYHKII{OHILIIY T 1r
0

T 'le

(1)

I lrx(r), e1?dt. TJ '''
BenrruHa

(2)

e)'dt H* = iJ (x(r),
0

T lr

'r-

Aaer [orperuHocrb HopMaJIbIo 0.

npu6nrureHur

[pocrpaHcrBeHHoro

qacrlrllbl ABTDKeHITJI

B ABIDKOHIT9MlulocKocrl{

c

Apyrofi roproA cocronr B ToM,.rro6rr BbrAeJrrrrb[JlocKocrb, rrpoeKllnt Ha Koropy]o BeKTopa IrMcpegneM no BpeMeHu) rulHetuqecKylo 3Heprurc. B sroNr rrynbca qacrurlbr Aaer caMyro 6olruyrc^(n crfrae rpe6yercr nsr6parb BeKTop hoe %3,\offaBJulroulsri uaxcuMyM KBaAparIItIHoMyQyurqlrouany
T

- (p(r), h)hll2dr, fi[ttvr,)
0

(3)

rlntr MTTHEMyM Qy"*tI"oHaJrY
T 1 n .

V){vA),h)'dt,
0

(4)

xap aKTeplI3yrc IIIe My rr Orp e IIIH o cTb a[rp

o KcLIM aII[u. T, l
0

B eI r.r'IuHa .', -

H, =

V)@U),h{dt

B [JIocKocrI{ c rotlHocrbro, KoHrpoTorrKrrABTDKeHUeM Aaer norperuHocrb npu6nrcxenus, ArrHaMrrKH nupyeMofi n repulrHax KrHeruqecrori eHepmu. QynrquouuuloB(1) u (3) Ecm es e 3 u hs e %3- BeKropbr,Ha Koropbrx Aosruraercs MaKcyuyr"r rrro [pocrpaHcrBeHHultxonQuryPaqus ABIDKeHI{tt-TorrKIr c yrBep>KAaTb, To coorBercrBeHHo, Mo>KHo ho s, HopMzrnu & B [JrocKocrrr c HopMulJlblo e %zJre>Kar c BeKTopoM B rot"o"tbro H, Jre)Krrr rrJrocKocrl{ KrrHeruqecKJrogHeprurc cucreMbl, orJllrqarouleficx or noJlHoilxusetu'recxoil Hecyrr\rde trMrryJrbcbr, gHepruEclrcreMblHa BeJruquuYHo, MHoroMepHoro Qasoaoro npocrpaHcrBa9t paeuep[a.a peanu3arluuerofi u1evn o6rqeu cJryrrae [oArpgcrpaHcrs, o6raAaloqnx cJreAyroulrrM BJrox(eHHbIx rrocrrr n nocrpoeHa noqrreAoBarenbHocTb ln noAnpocrpancrnaQlL^c.UL npoeKIIIrrI Qaeonoro BeKropa csoficrsou. fl'pu 3agaHHofiparuepHocrr{ BcexnoAupocrpaHcrBsagauuofipa3MepHa MHox(ecrBe 3aAaHHbrfi uagl^LruHnmrsupyer SyHrqrroHzur
HOCTE.

cnoficrsoM, To npoeKIIIrJIBeKTopa Ecnu gL^ - rogrpocrpaucrBo 9t, o6nagaroqee yKa3aHHbrM c B crrcreMbr rrpocrpaucrBe ueHHuefi pa3MepHocru uoq(r) e 9L uagl^Aaer upudnnxeHrre ABTDKeHT{;I s rpeumocTblo, paBHOfi erruquHeMuHI{MI{3IrpyeMOrO QyHxUrIoHiUIa. H pa3Mepuoc:runpocrpaHsrBa 9L^ or rorruocru npIl6ruDKeHIUI no aHaJloruu c 3asucril"rocrbm(m eQQekmueiofi pagydpuocmbto QasonoftrpaeKropurl.flpoerqnu BeKropa q(r) Ha tll-tsl Ha3oBeM uerroprr 6asucanoArlpocrpaHcrBa9I ^ uasoneu uo dauu dewr eHufl.
)CYPFIA.IIBbTq}ICJMTEIIbUOII UETEMATIIKI{ N MATEMATPTqECKOH OH3IIKI4 rcU 42 Js 12 2cr,2


3 O OEKTUBHAq PA3MEPHOCTb OA3OB OII TPNTKTOPHI4

1893

1. nocTAHOBKA 3AAAtIn
t BpeMeHu [0, T) aagaercx paccr'rorpnM cr{creMy,cocrosHr{eKoropoft s Nao}reHT Ar{HaMurrecKyro q(t), t e [0, r)'flpenTpae_Kropuefi onr{cbrBaercrcr{creMbr """ilrfu Qisonoft q(r).b-; BeKropoM caMaQaeonaxrpaeKroptrfl, gnnHeftsag QIHxI{nt or Hee' 3aAaBaea1vs.frBilfl,ercs.He MeroM rrccJreAoB gl'---"-El') - nuHefin'fi r Ma, B raxqrri MoMeHrBpeMeHr{ e [0, r; rexropbr a$(0> ETL;3Aecb Q 'ractHosrr{, B npl{Mepec MexaHtrrrecnpocrpaHcrBa UL s?lL (n orr"purop, Aeficrsyrotq"ti13 eBKJrrrAoBa "nrrpesartr" n3 noJrHoroHa6opa ronbKo trMQasonstx nepeMeHHblx r"ro>xer oneparop Q *oft a"atarraoft rpaeKropun q(r)' Koopglruaru). EygeM paccMarprlBarbtoniro iarue Qasorrte ToJrbKo ryJrbcbrrrJrr{ r e [0, T),tr1s Koropblx BblnoJlHnerctHepaBeucrBo
T

0

<*, Ittoo<,ltl'dt

(s)

gz(Q, D' rrepe3 (5), o6ogHa'ruM Bcex AJrr MHoxecrBo eaaonrx rpaeKTopufi, KoropbrxBbInoJIHeHo roAnPos 9t gL)- oneparopoproroHaJrbHoro n_poequpoBaHr{t Han-Meplroe flycrr fI_ e (gl * sel T

= rr^)Qq(t1llzdt o(q(.)) +Jilrr0

(6)

ne cpeAr{BcexnoArpocrpaHcTBpa3MepHocTrr 6onrue m. annpoxcEMar{trtr. rn-MepHoIf (6) 3agavy MrrHrrMrr3arlrn QylmqnoHuura naroneM 3aEaqefinanny trneir crrcrexapaKTeprrcrr{Kn ABmiKeHurI - npoeKrlr{sfT^Qq(r) - gaer caMoerorrHoe onucaHlre Ee perueH"e norpeulHocrr (6). MI{HLrMIr3I{pyloqee r"ru^eq c noMorqbrom KoopAnnar, 2. pEIIIE ;gilre, 3 AIAI{U HAI,ilTytIUfEI,I n-MEPH O ll Annp O KCI4MAII{PI

yraxeu nrueftnoe [oAnpocrpaHcrBo%^cEl.,Ha KoropoM Aocrlrraerct MITHIT]IyN{ QyHxulloHaJra tar eroro MITHITMyIIIa. KaK BeKTopQqQ),r e [0, T), ragan' To MtrHEMu3al{trfl (6), n onpeAeJrrrM 3HaqeHne aql{r SyHxqnoHzura MaKcI{MIr3 (6) ernunaJIeHTHa

Y(q(')) =

Lr[nn^oq1t1llz dt.
0

(7)
npoeKoproroHzulbHblM sBrrsercfl,

6aguc9Lu oneparop [, flycrr e',, ..., n - opronopMr.rpoBaHuufi gToro 6asuca. Torga m TopoMHanepBbre BeKTopoB

eqe)= )tOltr),eo;ei,
t=l

fl^Qq(t) - :,,nn,r), eo)eo, r e [0, I).
k=l

6aguca 3agaua MaKcrrMn3arlnn QynrquoHaJra (7) cocror{T B nu6ope raKoro oproHopMllpoBaHHoro ToqHasBepxHuI rpagb: ter, ..., en)CgL, Ha KoTopoMAocTrrraeTcfl

=,unl ijt @r,QqGD'at+JL(er,
g&=l t' gft=l

Tm

(-Ta

(ek, QqUD'dtle1, 91, ej) = 6tj,k = fr, i = t, ^, j;
= l,n ' 0}I3I{KI'I rou 42 }& 12 N2

-'l

(s)

o6ogna'reHl{e6m = t,6U =0,,k*i,k,i 3AecbncnoJlb3oBaHo

)o/puAII BbIrIHcrIHTEIIbHort MATEMAT?IruTk| MATEMATLI'Ecroli


1894 Tax xax

EEJIEFA u AP.

= e1), "o)=*4(speo, @0.d,)"* Qq(t112dt +JI I @r, gt= ,l[[i{nt(r)q*
l'^ ^,.\ r,.\ o

m rr.''

\

\

m

roucKa HarrJrfru eft m-uepHofiannpoKcrrMalluaQqO e 9z(Q, Z) saruluerct B Br.IAe To 3aAarra

(^ -l =I,m,l; = egLl(eoe;)6rj,k=I,ffi,i rupj>(Ss,er,t),r

r

;

)

(9)

3AeCb T 'te

So =

TJOt{tlq*Q)Q*dt
0

zq* (9f -: 9t) o6ogna.reurreQ* wts oneparopa, co[ppKelrHoro c Q, u o6osHa'rerule ll sc,rroJrb3oB:rgo nexropou z, qe 9I. g.n.f, BHeruuero(teuaopnoro) ttpotsu"genl/:fl, caMocorps(5), orpauuqeHnblM, B cruryonpeAenerus.v.ycJroBu;r oueparop Sp xnnxercs JluHeftusur,t, 3aMaKCIIMyr"ra peIUeHIre IIpITHIIITIa a gna.ruT,B cuJry rr x qEHE M.

BrrnrureM norperrrgosrb Hauryrrueft anupoKclrMal{au QqQ) ero npoenluefi fl^Qq(t), t e [0, O. oneparopa ug 6asuc, cocraBJleuurui co6crBeunblxBeKTopoB flycrs 1,...,9n - oproHopMrrpoBaHuufi co6crseHnblxgna.reHufi: Sc (gL --- 9l), ynoprAorreHHufi uo HeBo3pacraHrrro

=?"2t1t, >...>L1' l'? Soeo
TOrga, coulacuo (6), norpeluHocTb Haunyqueft aunpoKcllMalluupaBHa

"{ a(m)=

I
k=m+l

(Scer, = er)
t=rn+l

co6oft cpegrurfi KBaAparpaccrosulrrl BeKropa QqQ) Ao nnueitnoft o6onorlKu, HarsHyE upeAcraBJurer 1, roft na BeICTopbI . .., Q^.

3. OCHOBHbIE OrrPEreIrer{J1....]rfl.
u_eQQextusHofipa3MepHocrlr MoAbrABIDKeHIIJI Qaso[aguM QoplranrHbre onpeAen en:rls.nongruri aoi rpaercropull B 3aAarreaurpoKclrMar{uu BeKropa QqQ), r e [0' Z). arnpoKclrMaOupegenenuel. SQQercmueuoilpalJflepHocmbtoQasonofiTpaeKTopllu q(r) B 3aAaqe rluu BercropaQqQ),r e [0, T), Ha3oBeM Qyuru1urc

se}, eo = (c(e) min{kI 0 orrpeAeneuHyp Ans e e [0, "o) u npuurrMarculyrc qenoqucneunble 3uarreHlrs. qucry BeKTopos6aguca,pa3Jro)Keglre BeKropa Qq(o rlo Ko3Ha.reHue(p(e) paBHo HarrMeHbrueMy [olperrlHocrb, ne 6oluuylo' rIeM e. rro BpeMeHu TopbrMAaeTcpeAuroro PacouorplrM TenepbBeKToP

;,nn,r), e1)e1,
/<=l

"rraBHbrx HaarurpoKcsMtrpFoulufi seKTop Qq(i), OH npegcraBJurer co6ofi nuuefinyro rou6uHaIIIIrc I{3MeAuHaMI{Ky . n QasonoM npocrpaHcrse_9fc roeQ$uqr(euTaMlr, 3aAarouII{MIr opaloe"tfr"i;,..., HeHurrerffi HIaBHbIxcocraBJrtroullu BeKTopa Qq(t). BeKnpu annpoKclrMal{urr k'ft 2. Onpege;regrre Bexrop (Oq(r), er)et e n, k = l, n, Ha3oBeM rrtoAofi ropa Qq@.
XCYPI{AN BbT(II,ICJII4TEIIbHOIT UANEMATI{KI.{ H MATEMATLTIIECKOH AI43WKVI tOv.42 'Jt 12 2ffi2


3 O OEKTI{B HAfl PA3 MEPHOCTb OA3 OB Or,TTPNT,KTOPHI4

I 895

flonegenrre Bo BpeMeHI{ MoAbI(QqQ), et)et e 9t, annpoKcnMrrpyrcqefinoBeAeHr.re BeKTop eqQ) na aoTpe3Ke Z), B cpeAHeM cocraBJrter qacTb BeKropa Qq!),xapaKTepr{3yroqpocr Benr{qnnorl [0,
T T _ l

a,ll\Qq(t)ll,dt \2(k)= fJ, QqG),"rf ,
0 0

Bbrqncrrrcrqerlca qepe3 co6crseHHbre 3HarreHufloneparopa sg: \ra(k) til I i; I \r=t ) (n r

Kar clegyer I{3 rr3JroxeHHoro BbIIrre, qeu 6onrure qn(k), TeM 6onrure BKrraAt-fi rr.rogsrB armpoKcrrMarlrrrcBeKTopa Qq(t) Ha rrHrepBaJre [0, D. 4. AWHAMI4I{ECKI4E CI4CTEMbI C AI,ICKPETHbIM

BPEMEHEM

Paccuorpl{M cI{TyaIII{Io, KorAa AuHaMuqecKar crrcreMa paccMaTprrBaeTcr B AucKperHbre MoMeHTbr BpeMeHI'I ..., tN [0, D, N > n. B erou cryr{ae tr, Qaronax rpaeKTopr{r 3aAaeTcr B BrrAe Marprrubr Q,j = |iQ.),i = l, ft,i = 1, N, pa3MepanxN, xoropyro 6yger'rHa3brBarbrpaeKTopnofi MaTpnrlefi gnuaMnqecKori cucreusr u o6ogHaqarb qepe3 Q e (fi.r --.---9t). Ecrn Hac rrgTepecyer noBeAeHtre or Bpe_ MeHI{BeKropa QqQ),f ro 3ra nnQoplaaqnr coAep>Kr.rrcrB Marpuqe Qq e ({h.r Cornaclro TeopeMe o cuHryJrtpHoM pa3Jrox
{QQ}ii= >^-tet},{ Er}1, i - l, r,
,t=l

j =1 l,r,

rAerrucruLr,..., na6opst BeKTopoB 9l.,k= t,r,,ttEr,e gh.y, 7. k= l,n, !"u oproHopMl{poBaHHble npeAcTaBJltror co6oIl peureHrlt3aAarr co6crBeHHbre Ha 3HaqeHrrrr OqA*Q*k = )"?*"0,q* Q* QQgo= 7"?,g0, cBfl 3auHbre Me)KAy co6ofi coorHorrreHr.rrMu QQgr - 1.1e1, Q*Q*eo= Itgr.

Ilptl SToM 6yger'rnoJrararb' trro Ir 2 . . .2Ln2 0. t{ucnu tr,, . . ., I" Ha3brBarorcr rrgcJra*{rr crrHryJrrpgbrMrr MarpHqbr er k Qq, a na6op BeKTopoB 91., = t, r, LIAonoJrHeHnsrfi noJrHoli oproHopMgpoBaHuofi Ao cI'IcreMbI 9f1,' n na6op BeKToPoB e gl.*, k = t, fV,Ha3brBarorcr gt cr{HrynrpHbrM 6azucaMr{ u eBKJrrrAoBbrx npocTpaH gL v 9L ucoorBercTBeHHo. crs B paconrarpl4BaeMoM cryqae poJrboneparopa so B (9) urpaer Marprrqa 9t) n s Qqa* Q* e (9t tIuceJIIr cI'IHryJItpHux 6agltcos k-ft uogoli TepMI{Hax cIrHryJIflpHbIx npn annpoKcrrMarlrrg ABrDKeHrrr QqQ),t= t1, ..., tN,tBJIteTctMaTpLIqa &o . (ft,v * 9f) . MaTpurrHbrMrr oneMe'TaMrr jlJ

{&o}r=Io{eo},{go},, i-1,r,

"HanpaBJreHRe Mogy ldo, k = l, n' Mo)KHo ,, paccMarplrBarb KaKBeKropet e gl.,3aAaloqur? k-i4oproroHulnbHofi cocrasnxrculefi ABIl> }O'PHAJI BbTIII,ICJTIITENTTTOTA MATEMATPIKLT L7 MATEMAT}I'{ECKOH @V3KTKKI rOU 42 J,& 12 2OO2


l 896

BEIIE|A

n Ap.

AIIq 5. IPLIMEHEHr4E METOAA MOA 14 3OOEKTIIBHOI{ PA3MEPHOCTVT TPE I4CCIIE AO B AHI4'I AI,IHAM VTKVT XATO MH Of O KJIA CTEPA
pacclrorprrM BHyrpeHHroroArrHaMr.rKy rpexaroMHoro KJracrepa 6naropoAHoro ra3a, orllcblBaeMyro nOTeHI{IraJrOM ToqeK B [oTeHl{rranbHoM fIoJIe, 3aAaHHOM Tpex MaTepr{aJrbHbrx ypaBHeHrurMrr ABrDKeHrrr feuHapRa-fi,xcoHca l7). flp" uccJreAoBaHuuraKrrx KJracrepoB MeroAaMIr MoJreKyJrapHou ATIHaI\{IrKII 6rmo 6OtrappxeHo, rrro AJrfl HeBpaularcu{erocfl KJracrepa csoficrsa ero BHyrpeHHero ABuxeHu.fl 3aBIrcrr or 3HaqeHrurnorsofi BHyrpeHHeli euepruvr.vr crpoeHrrs noBepxHocrlr [oreHldl{anrHoft sHepruu (cu. ts]). Esuo raK)KH5rercs; s gaeucl{Mocru or 3Haqe3oBoro npocrpaHcrBa ua o6nacrr,r perynspHoro H xaorl{rrecKoro ABlDKeHHl.r MoMeHra zuracrepa (o'1. [0], U 1]), rpurreM peryJltpHat KoMroHeHTa nuri nonnoir eueprvvvyrJroBoro arero se na6lroo6HappxHBaercs rr nptr AocraroqHo BbrcoKlrx3HaqeHutx nonuofi 3Hepruu KJracrepa, TaKIrx KJIacreAaercq B HeBpaqarculrucs KJracrepax [2]. flnx nonuuarvfl, oco6euHocreri ABTDKeHTIfl qro oKa3blBaerct AocraroqHo cJIo>KHoI:r npoaroMoB B KJracrepe, poB Ba;KHaBrr3yanrr3arlrrfl ABrr>KeHlrr [pocrpaHcrBa: TaK, rlpu onucaHrrrr BHyrpeuHetl AuHabneruofr u3-3a slrcoKorl pa3MepHocru Qaeonoro ero [ocTyrlaTenbHofo Mrrrcr TpexaToMHofo KJracTepaaproHa B crrcreMeIIeHTpa Macc Ir npu oTcyTcTBurr ueo6xoAHr'rHa6op us 12 KoopArrHar- [recrb Arq orrlcaHns rpocrpaHcrBeHHblx KoopAuHar ABrDKeHrrrr UMnyJrbcoB r{acrl',Iq. H IXeCTbAJrflOnIICAHUfl' paccMarprrBarb KaK cBt3aHHoe K-nacrep rr3 Tpex oArrHaKoBbrxaroMos 6naropoAHoro ra3a Mo>KHo C CI{CTeMbI faMlrnbroHI{aHOM COCTOflHITe KBa3IrcBfl3aHHOe rrJruAoJrro>KrlByulee
1 l

n = L#+) u(r,),
stD;s i=l t=l

(11)

rAe ri, = llqr- A,ll,pi, ei - BeKTopbruMryJrbcar{ KoopAuHarut i-ro aroMa, m ero N{acca,U(') [oreHIIIIzuI JIenria pa a-,[,;xo u ca,

= I)" u(,,)or.[( _(i,)']

( 12)

coorBercrByroulerl [oro)KeHII]o HyJrfl susr UguxapaxrepHoir uupuHorl cx,, c uy6nuoli noreHrlr{aJrbHoft Uo B ,rorl"qnana. Boc[onb3yeMcr ecrecrneHuoft cncreuofi eATTHITIL Koropoli uacrura6auu cJIy)Karft't,, - -3Ir BeJllrquuofiyuoBoro MoMeHTa - 0 M u cr. TorAa ocHoBHoecocrorHr{e KJracrepac gHeprueft E B aroMaM, pa3MeuIeHHbIM BepIXIIHaXpaBHocropoHHefo rpeyronbHllKa coorBercrByer HenoABrDKHbrM gnx Qlrxcltponauuoli c A.nuHoucropoHbr a' = g.6J1. [nu>KHo nepBbrx ruecrrr KoopArrHajlaxq. $^ii"eHaAlrarlrMepHoro BeKTopa x [q, P] e e12. @oprnlaJlbHo BYn Ha luecrrlMepHoe roAnpoeqllpoBaHlrs BBecru oneparop e*_e 6n -'8rr).gp\orofiaJr6Horo '6u,0 .611.t=. e rpocrpaHcrBo "itri 9iu = {-[e, 0], q . pa3Mep KBaApar HopMbrBeKTopa Qrx(t), t e [0, f), xapaKrepu3yer cpe,qHlul CpegHuu ro BpeMeHH nau6olsurell creneHII 3Ta Marepl,IarrnoIi ToqKH.B o6racr' npocrpaHcrBa, rAe rrpoucxoAnT ABr{>KeHrre raKoro BeKropa ut8p, rpoeKllrrfl,Ha xoroprtft BeKTopa Qo*) B feAHeM rlo o6racru nilraHyra BAoJrb nJrocKocrb VL2Y 612, npooKHau6orsruuri KBaApar HopMbr.Anarorr,rr{Ho, AByMepHarI BpeMeHu "r""i ro BpeMeHuuMeer MaKcr{MaJlrurttl KBaApar HopMbI, onlrcblBaer i* erx(r) na Koropyrc B cpeAHeM "npocrpaHcrBeHHoir" qacrll flpoerqm Qa:onoli op'eHTarrrrrc erori b61actr B AByx HarpaBJreHrrflx. Aaer AByMepHoeouucaHlle u]ecrrlMepHoro npocrq(r) e 86, r e [0, 7), sa 3ry rrJrocKocrb ,pae*topuu KBaApary Hoplt{bl npoeKIrIIlI paHcrBeHHoro ABrrxeHr{s;ToqHocrb 3Toro uprr6ruoxenr4 paBHa cpeAHeMy K rlnocKocru 9t2 AouorrHeHue"%rr.' I."topu Qrx(t) Ha oproroHaJrbHoe B Koropblx rpollcxlAxr caMble 6utcrpr'Ie neflnx aHarur3aHarrpasreHlrrl n QasonoM rpocrpaHcrBe, "Yacrb", coAep>Kaulyrc IIMnyJrbx peMeuleHrur aroNroBzuracrepa, or BeKTOpa cJreAyer-ocraBLITbJII{tlIb BeKroPa Qp rlpocbr ABrDKyurrucr aroMoB. B'eron cnytae Qr(g, p) = [0, p], cpegnufi xnagpar HopMbI BpeMeHu 3HaqeHI{IoKl{Heruqecrotl 3Heprrrlr clrcreMbr. flnyrtaepHat [noc[opquoHaJreH cpeAHeMyrro
)CYPHA"II BbItII4CJII,ITEIIIHOPT MATEMATI4KI4 VI MATEMATI4I{ECKOI'I AIf,3VTr.JI -rOII 42 N9 12 2OO2


3OOEKTI,IBHAq

PA3MEPHOCTb

OA3OBOI4 TPAEKTOPHI4
pa3MepHocrb

l 897

Kocrb $Lz,p "8rr, npoeKrILIrIQox(t) Ha Koropyro B KBaAcpeAHeMno BpeMeHI{ IrMeeT MaKcrrMzunbHbll{ B par HopMbI, 3aAaer ABa HanpaBJreEvts., KoTopblx cpeAHrrr KrrHernrlecKat 3HePrut MaKCI{ManbHa. BeKTopa Qox(t) e %12,r e [0, I), Ha ery npo.*q"" nnocKocTb AaeT AByMepHoeonucaHrre urecTrlMepnoii cocraBJrsroulert p(/) Qasonoil rpaeKTopl'Iu npocrpaHcrBa nMnyJlbcoB. 3QQexrI{BHbIe pa3Mepuocrrr ( Qasonofi rpaeKropr.ru Qox(t1 TpexaroMHoro KJracrepa aproHa 6rrnu onpeAeJIeHbI B 3aBI,IcIrMocrI{ oT orHocI'IanrrpoKcvual4laer t AJIfl renrHorl norperrrHocrl{ E = 4..5I{ AByx gHaqenni yruIoBoro MoMerrrazM = 0 (cr'a. Qnr. la) u M = 0.8M."* (crrl.Qnr. 16). (Ha ncex rpaQuxax peryntpHble rpaeKTopur{ npeAcraBJleHbr KBaApaTr{KaMrr,a xaoTr{tlecKlre - K?yJKOVrauu.) Kax nugHo tr3 Q"r. 1, nonyreHHble pe3yJrbrarbl cuJrbHopa3Jrl{tlarorct AJII perynxpHoli u xaoruqecy$IoBoro MoMeHnpr 3HaqeHruD( xofi rpaerropnli ra, 6nu3Kl{x K I{yJIro, u npaKTLrqecKI{ coBnaAalor Ar:n 3Haqenvs.yr.lroBoro MoMeHTaM =0.8M^*. Ha Qur. 2 Aonn Kl{HerrrqecroIl sHepruu KJIacrepa, coAepx
20

60

100 0
Onr. 1.

20

60

100

OrxocureJr bHarl norpeurH o etb, Vo

Ions Krrnernrrecrofi ewepmru, Vo
I
(a)

80 60 40 20 9 Onr.2. e2 1.0 0.5 0 -0.5 - 1.0 9 k

xeHa B npoqeHrax fXit}t = r\1 x l00vo); uns (a) E,= -25, M = 0;gnq (6) E = -2.5, M = 0.8M^*.
n ABTDKeHUfl, HyreBoro yrJloBoro MoMeHTa Bct KI{5_7VOCKOHIIEHSHEPTII' CTOIIHOCTbIO HETUIICCKA' Tpr{poBaHa B Tpex HanpaBJIeHIrtx. XaotuqecKafl TpaeKToprrff xapaKTepl{3yetcs 6onblrrlrM tII{cJIoM MoA, Heo6xogl{Mblx AJISonuc anvfl, KlrHerl{qecrotl gHepruu cl{creMbI. Kax AJII perynxpnolit, TaK Ir AJII yrJloxaoruqecxofi TpaeKToplni, npu 6olsurou BOM MOMeHTe ABI{XeHI{e KJraCTepa B ocHoBHoM rponcxoAr{T B AByMepHoMnoAnpocrpaHcrse. norperrrHocTr' AByMepHoro npu6nux< erlus. cocraBJlter l2-ISVo. Ha Snr. 3 (a-r) orpuKeHa AByMepHas npoeKnpeAcraBrlnq rpaeKToprrrr Qox(t1(t = ty,..., /rooo), Jrflrorqar co6ofi cyMMy AByx nepBblx MoA BeKTopa Qox(t\,1 e [0, 7").Cnegyer orMerrrrb sBHoe pa3nl{.rne erux rpaeKTopnfi glx peryJltpHoro u xaorlrqecKoro ABrD;KeHIruuenpaqarculeroct KJlacrepa (Qur. 3a, 36). flp" E -- -2.5 r 3HarreHusx M = 0 nJrocKocrb [poxoAI{T qepe3 BeKTopbI et n e2. B cryqae peryJltpHoIl rpaeKTopllrr cyilIe crByrcr o 6Jracru Qasonoro flpocrpaHcrBa, HeAocrynHble AJrt KJracrepa. flpn 6ontrunx 3HaqeHutx yrABrDKeHr.rr JroBoro MoMeHTa M =0.8M^ax KaprI{HbI ABLDK:HUr. B AByMepHbIXnJrocKocTtx [paKTtIrrecKII He pa3JII'Iqarorcr (o'r. Sur. 3n (perynxpHaa) n 3r (xaotuueB crag)). OgHaro cyulecrBeHnoe pa3rurrue BITAHo Ar{HaMrrKe5-ri n 6-ft MoA, Koropble BHoctr JIHtub He6omruofi nxnag B KllHerrrr{ecKylo SHeprnrc cncreMbr (o"r. Qur. 3l (peryrapHax) u 3e (xaoru'recKar)).
XCyPFIAII BbIttI4CJIlITEIIfgOll MATEMATHKVM

peryJltpHoro Kar nugHo Ir3 Q".. 1,2,s cJryqae

r.0
E2
1

1.0 0.5 0 -0.5 -1.0
6

1.0
1

0.4 0.2 0 -0.2 -0.4 0.4 -0.4
Onr.3.
OI43HKH rou 42 J$ l2 2002

0.4
5

MATEMATH'IECKOII


l 898

EEIIE|A

n AP.

BbIBO[bI senunetiuoil AuHaMoAbI ABIDKeHIUI flpeg.no>renus[i cnoco6 no3BoJrser Br,r3yiurr3oBarb rJraBHbIe OT[apaMeB caMbrM trccJreAoBau oco6eHHOCTIree noBeAe:Hus' 3aBIICr{MOCTI{ MuqecKori cucreubr r.rreM TpoB, onpeAeJrqlouux ee ABIDKeHITe' C[I4C O K III,ITEPAT)TP bI payle MoAenr{lnHefiurtx lr3MepeHl{li c ouur6xoilrll l. Eoudapenrco C.Ir., Ilatmaea IO.n. 06 eQQe_rrr.rBHoM 1995'T' 35' Nb 1' C' G23' rI X. ertqucn. MareM. MareM.Qota' Data Compressionll Pattern Recongnit' and Image 2. pyt,ev yu.p., pyt,ev A.yy. Effective Dimensionality and A'nalys.1997.V. 7. Ne4.P.393406' ll MHoxecrBa narntepenuri X' Bbltrrcr' Ma3. Ilutmaea A.n., Ilutmuee Io.n. 06 eQserrnerrorl pa3MepHocrr{ MareM.Qur. 1998.T. 38' Nb4' C' 682497 ' reM. u 6ecxoneqHopaxre KoneqHoMlpur^rx-{Pu6nu>
)cypHAn

BbrqI,ICJII,ITEJIrgolr MATEMATI.*T

vr MATEMATI,TTIE.K.*

ouzuwt

ro*. 42

,* 12

2wz