a?RHOH TaRoi rpynne B 6] comocrasnen noTok Wa, te R, H€HCTBVIOIHHH Hà HpOGTpAHOTB© Jle6era Xj, Ho, KOTOpoe HAHOHHH6GRKBM OỐPp830M crpourca no X, 1.. OTOL NOTOK COBIANAGT © TIATKMM HOTORO
Trang 1J OPER OR oo © Copyright by INCREST, 1985
Hacronnjan Crarba NOcBAIeHa U8y4YeHHO TAaKHX BOMpOCcoB, OHA ABIIA- eron epepaØorkoii namero npenpwnra [19]
TÏpezÐỊe, HeM GODMVUIHPOBATb BOIDOGBI, KOTOPEIE OỐCV?RNBIOTCH BE CTATbO, HAIHOMHHM, WTO GOFJIACHO [ð] BOHRHH MHb€KTHBHHH daxtop M mana [Hạ wsomoppen W*(A, a, Z), GRKDGIIHHOMYV HDOHSB6H€HHPĐ 3/TeỐPĐI
A L®(X, w) rye (X, ð)-1poorpanergo Jleðerna, Ha TDYHHY BTOMODỦUS3MOB 0”, né€ Z, yelictayiomyio Ha 4 GBOỐOHHO u sproguyecku a?RHOH TaRoi rpynne
B (6] comocrasnen noTok Wa), te R, H€HCTBVIOIHHH Hà HpOGTpAHOTB© Jle6era (Xj, Ho), KOTOpoe HAHOHHH6GRKBM OỐPp830M crpourca no (X, 1) OTOL NOTOK COBIANAGT © TIATKMM HOTOROM BĐGCOB, pACCMOTDĐHHHM B [7] OKaatrBaerem ype rpymuer « u aBTOMOp(MaMOB 4 THna lÏÏạy CjA0O0 2RBH- naeHrHm (a sHaunr H W"(A,a, Z) ~ W*(A, B, Z)) Torqa um TOiiIbKO TOPHA, Korma motonu H7,(ø) n W,(B) wawopdHki [6] J[azree, ÿCJIOBHMCR HA8EIBATb H€H- TpaadwaaropoM noroka M,(œ) mHomectBo C(W(a)) < Aut(Xq, ito) Takoe, 4ro ecum ye C(W), To yW(a) = Way, te R
Trang 24 8 1 PO2IO71EH
B [7] HORA38HO, WTO GYHI©OCTBYT R8HOHHWGCHHỦH TOMOMODÙHSM
8 — modf us AutW*(A,a,Z) B C(W), aAapy koroporo HDHHA/UI6?7RHT
Int W*(A, a, Z) (Int M ompeseeno s [1])
Tenepb MEI MO?ReM GÿODMY:THpOBATbB BOHIPOCBI, KOTOPBIS DACCMATPHBA- IOTCA B OTOH craTLe
1) aa neaxoro un ốc C(W(M)) cymecrayer ye AutM raKoii, aro mod y=: 6?
2) Ilyerm 6; € Aut M, i= 1,2 uw modf, = modf, == W(M), t #0 nam mod f, == mod £, = id p,(B,) = p, (Bs) = 0, rie p, (8) — acumurotugeckuil nepnog
B [1] Caeyer -m orcioga, ro fp, u By BHeNTHE COIDĐH?R@HEL OTHOGITGIbHO Ấƒ, T.e Jye AutM u ue UCM), rae U(M)—vyunrapnaa rpynna M, ranue, aro
By == Adu-y-Bgry7? ?
(SameTiM, YO OTBET Hà HO7OOHBI BOIPOC HJI HHb€RTHBHOTO aKT0pa
Ta TÍT,, O0 < 2 < |, /AGT THOHYH KCIACCH(ĐHEAHHI KJIAGGOB BH@IHHĐ GOIDR- JK£@HHHX 8BTOMODpH3MOB 6 p, = Ô [2].)
3) Ilyers B;€ AutM, /-:l,2 uw JyeC(W), ragoii, ro mod, =: c= y-modf,-y7! Crenyer cm orcioga, ato ¡ H By BHOIHH€ GOHIDH?RHB oTHocHTembHO M, ecu p,(B;) == 0°
B uacroatteit crarbe /IAHBI OPBGTBI HA H€PBbI€ /1Bä BOIDPOG8 HZIH HpOHđ- HO.IbHBIX HHbOKRTHBHBHX (}aRTOPOB THUA I[I, Hm Ha Tperiii Bompoc — aA HPOM3BOJIBHBIX HHBCRTUBHELX daxtopos M tua Ill), y KoTOpHX HHBBAPHAHT T(M) = (1:6, € Int M1) #0 (cm [8])
Pemenne sonpoca 2) (cm §2), CBO/IHTCH K JoKasareibeTBy TOTO, f0 HIDOM OTOỐPp8/REHIN ƒ > modf? sapcinerca Int Ä
IÏepBEU BOHDpOC PGHIOH B H€€RO:IBRO ð0:160 OỐIHeii (bOpMe, em chopmy- JIHpOBAHO lĨy€rb [Z] — HOIHañØ TpYHHA ABTOMODŸjHSMOB Á, HODO?RKIHHaR
2 [6], a ![z] — ee Hopmadmaarop, re .#[z] = {y€ AutA : y|a]y~! = [a]} lÍonnrHo, dro 66 ye V[a], To y mwanynupyer y, € Aut W*(4,a, Z) B u.3 §1 oRaaaHo, wfo ;In ðe C(M) cymecrpyer y€.fT[e] TaROi, ro mod ? = 6 (em [19]) Bamerum, WTO /IDYTO€ ĐGIIGHI© ĐTOTO B8?RHOTO BOHpOGA vie Woctvaero B [21] (cm rTaRzge [22])
Bonpoo 3) paceMoTpeH ;ữ1 ĐaRTopOB ă ¢ T(M) #0 Ms orpeta ua oror BOlpoc cclenyeT, GTO BCHRHII ye AutAY 6© Pp,(y) =Ũ BHGIHIH€ compamen ¢
yy & Aut M, ipnseM (4) = 4 w | 6 #7[z] B [22] npusegeno nomnoe pemenne BOinpooa 3) n1 aBTOMODÙHaMOB H3 f [2] C HOMOHPbBIO peayabraros [22] MOsKHO JJaTb WOTHOe pettientre BoTIpoca 3) H ;18 @akTopow Tuna Hil, , yro noA- TOTABZHIHAGTGOWT ft IIYỐ/IHRAHHH
Kpome ppeyeHia cratba C0/16pZRHT TpH uaparpada B §I pacemarpn- BAlOTCA HOPMadAHZaTOpbi MO-THBIX TPVIIN 11 aCCONMMpPOBAHHELe NOTOKU, BI 3, § I nam onner na Bompoc I B §2 usyyarorca a € Aut M, y Koroprrx moda = W, nin fe R, B§3 pacemarpuBaiwtca asromophu3smEr daxtopos M c T(M) # 0
fl GaarofapeH peleu3eHTy 38a PA Nome3sHBIX ÿRA88HHH,
Trang 3ABTOMOPĐH3MBI WHHbEHTHBHDIX ĐARTOPOB THHA ITI, 5
1 AIHPORCHMATHBHO KOHEYHDIE VPYIIibl THUA III,
M WX HOPMAJIM3ATOPHI
4 llyorb Ớ — 04GTHÀ H€GHHTYZIHĐHAH TPYHHa HP€OỐPA80BAHHH TpO- erpaHcrna JleGera (X, B, uw) Uepes (G| o6osnagum nomHy10 rpynny Hpe€- oỐpasopanni (X, B,u), nopompenuyio G (cm [6]) Honnste aproquaecnue TpyEnm [ỚŒ] 1OHYGRKAIOT H/IAGCHÙHRABIIO, AH80THMHVIO TỌ, OT0DAH XOPpOIHO M8B©CTHA J1H (baRTopoB jon Heiimana B nazIbH6iiHI€M MEI ỐYH©€M Hp®HHO/IA- rarb, ro [Ớ] umeer rum IHI, [S] OnpenenM ypoiicrnennyio Gy rpynny IpeoØpaaosanuli nun rpynneri Ớ ([9], [10]), meitcrByroutyio Ha (Xx R, Bx B(R),
dự x< du) c0TJraocHO (ĐODMYI
(1.1) #a(X, M) =: (ex, w ~E log HED ), (x, u)EeXXR
du(x)
llyerb Z — usmepumoe pasdnenne XxR [11], mopompenuoe Gy — MHB8DH8HTHBIMH H3MGDHMBIMH MHO?R©CTBAMH PaccMOTpHM Ha XXR HOTOK T(x, wy) = (x,u+s), (x,u)€ XX<R TaK KAR Tý HROMMWTHDVT © g,€ Gy,
TO MO?RHO paccMoTperb (arrop-norox W,(G) na XxR/Z Torna M,G) — ne- CHHT'VJIIDHEI HaM€DHMbLđ ToToE [9], Ha3bIBA6MBIÍ IOTOROM, 8GC0I1HDOB8HHEIM
c G Cornacno [6], 6074 Ở; — AIIĐOROHMATHBHO KOH©HHH© (A.K.) TDYHHHI npeoốpaaosnannk (X;,, 6) Tuna Ill,, TO Ớy HH G, cnado skBMBaNeHTHE TOTHA
MW TOUKO norxa, Korna W{(G¡) u W(G,) usomopdutt
JIEMMA 1.1 Heau ae W[G], mo % onpeDeasem asmomopduasm moda npocmpancmsca Xy=XXR/F, npuren modae C(W(G)) (cw Beedenue } Omo6pascenue « > moda — zomomopGuss, nOpo Komopozo codepacum, no kpaũnecú atepe, [G]
JH HORA8ATGJIbOTBA /OCTATOHHO 8AMGTHTb, TO ©0GIH œ€.Ý[GŒ], TO a(x, u) = (ax, + logdu(x)/du(x)) npnHanezmnr V[G,] Ocranmbunie pac- OVHIĐHHf HIOHHTHEI
C7IENGTBHB 1.1 Beau fe L°(XxR, wxm), 2em — mepa Te6eza na Ru
HanoMHHM, jlanee, uro corzacno [6] senRan IoHan a.R rpynna [G]
mua lạ aBToMopwaMoB (Ơ, Ư,ø') MOHIGT ỐBITb HDHBSI€HA K 07I61V101I6MV Bunty Iyerp (Q, B,o’) = (X, By, w) xX (Y, By, v), rae — HOH€HHAH H©GIIDG-
Trang 46 B H LOTO/EEH phipHad Mepa Ha X, v — o-KROHeuHAM Mepa Ha Y,o=pxXv~o'
S.(x, ¥) = (x, S,), Q,(x, ¥) = (Qx, Uy),
q G€OHfữa/taer 6 OTpaHHnweHHew xi Ha Xạ lÍOHHTHO, HTO HOTOE 7,(x, £) =
== (x,u +5), KOMMyTHpyiomnit c Q,, olpefenner moron W, Ha XX R/¥(Q,) (a aHagnT Wu Ha X,) CaegqoBateIbHO accolyupoBaHHblli noTOR WG) aBainerca CHEWMAABHBIM NOTOKOM G IOTOIONHOH PyHK Wel @(Q~!X) HH ỐA3HOHBIM aBTO- MopdmsMow @~1 [6]
JIEMMA 1.2 Hycme G=(S,, Q,) — epynna muna Il) moavKo «mo onucannozo cuda, « € N[G], npuwen moda=:W,, se R To2da cywecmeyem
Trang 5ABTOMOPA1)MDI HUƯEERTHBHBIX GAIRTOPOB THA lHIạ 7
te [G] maxot, umo B = t-1x umeem suƠ
20e V, — usmepumoe nose asmomophusmoe (Y, By, v) [6], npuren Ve W[S] u
OV.) = 5 Onn noumu ecex x
Aoxazamenocmeo Ilycth moda = id, torga a onpefennet mpeodpaso- panne XX Y: (x, y) > (x(x, y), v(x, vy) Sametum, uTo @yHRuMH x, uy, WeaMe- PHI, 4@liGrBHT©ZIEHO, ©GJIM E — M8MPHMO© HOHNMHO/R©OTBO ÄÝ, TO H8M©DH- MOCT X„ BbIrekaer I8 ooorHoImeHmn {(x, y)€ XXY : x¿(x, y) 6€ E} = œ~1(EXxY) Tax Kak moda = idua ¥XR/S(Q,), TO AA Boex PyHKIMi M3 L°(XxR), YROBueTROpAION(Mx (1.5), BbUIOMHEHO paBeHcTEBO f(x,(x, y), ứ + Ủ„(x, »)) =f 4),
rne W(x, y) = log (do(a(x, y))/dø(x, y)) Ho Torna (x¿(x, y), # + Wax, y)) H (X, #)
IIPHH8716?&HT OHOïÏi rpaexropuu Q, (cm (1.6)), m.e
x(x, y) = Q"%'x, n(x, y) € Z, (1.8) W(x, y) = Z(n, Y, X),
n
velepb MHomecTBa a&, u QFE B cusy (1.8) X — HocnTenm y Hux coB- IANAIOT, H3 HOGTDOSHHH G2I€HV€T, uTO mMHomecrBAa {vy : (x,y) EaE,} m {y 1 (x,y) € QZE,) HH HOWTH BCexX xX UMeIOT OHHAKROBWIO Mepy v Orcioma enenyet (cM semmy 4.2[6]), aro cymecrsyer s, € [S,] Takoli, wo 5,Q7E, = aE, Pacemotpum tTenepb mpeobpasopanue XxY BH7a
q9) f(x, y) = s,@s(x, y) mpm (x,y) € E,
Tak KHaR ÈF„ — HOHAĐHO He mepecerawrca mu | J]Z„,=XXxỲ, a TâAE?R SaOsE„ (=«E,) — nomapHo He Tepecekalorca u Js, Ore, = XxXY, To B curly 4ieMMEI J[am [12] ? e [G] Ho rorna £~1+ e [G] Ht, GOomee toro, BcIy (1.8) ? 1œ coxpaHneT Mepy o=yuxyv CJI@10BAT€IbHO, / lxe.f[S,]} a Tak Kar (t~ta)(x, y) == (x, yi(x, y)), TO tte = \ @ V,du(x), rne Vy — namMepumoe nome
x
ABTOMOPHU3MOB, IpHWeM 11H -1O9Tn Boex x Ứ¿€ [S]w v- Vy = v Z
Trang 6§ B 4 1071071211
2 B stom uyHktTe Õ0:166 HOTADÕHO paccmorpumM nomHyio rpynuy [G]
c GOpa3yiomuMy Biya (1.3) B Ip€7U10210?R€HU”H, 410 @(x) = @ (Const) (cm (1.4))
B orom exyyae Oygem ropoputb, WTO Aelicrpue Gua XXY aBaercs# ueprio- IHW€6GRMM
BO3HHR86T BOIPOC HH RARHX OIOBHHX TĐYHHA Ở MO?ZRGT UMeTb I€DHO0/I460RO© 716ÏiGTBH€., TB@T BBID3?EA6TCH B T©DMHHaAX HHBADHaAHTA 7(Œ) (HE HHaue, 7 — MHOZReerBa) 4n Œ |9]: T(G) comepamr Boe te R, y21H KOTODEIX 0VHISOTBV@T B@IHI©CTBGHHAf HSMDIHMANH (yHKHHH ¢(@) TARAH, WTO exp i(€(gw) — €(w)) = exp itlog(do(g@)/do(m)), g€G, roe (2,0) — npocrpaH- crso vledera, B KOTOpoM jeiicrsyer G IloHaATHO, WTO T(G) coBuaqaeT c mHBapnanrom HoHHa 7(M) [2], rae M — darrop suga M = W*(A, G),
=-—-—+*, mie n(a)€ Z CaeqopareabHo Mepa P ABOAeTCA ¿IARVHADHỌL H, Kak 1 B [6], MomHO onpegeanth QO, u S,, jelicrBylome Ha (Xx Y, ux v) npuywemM
XXY=Q, uxv = Pu dP(Q,@)/dP(@) = = n, neN,
IIlycts Tenepb p(t), t¢ R, — usmepumpii norok Ha S=XxYxXR, T©ÏiCTBVIOHIHHI ©OTJIAGHO dopMysie
p(Œ)(x, Vs 1) = (x, y, u + t),
a p(t)—axrop-moror ua Z Tak Kak p(2z/T)e,=e,, T0 B Z GYHI€OTBVGT HO7A21- T€ÕPA II3MGDHMHIX HO/MHO?R€GTB Z,, H€IO/BHZ&HBIX OTHOGHT€IbHO 0(2n/7) =
Trang 7ABTOMOPOM3MbI WHLBEKTHBHDIX MAKTOPOB THITA Ig >
== p(2n/T) Ha atoii anredpe rpynna p(t) copnayatonjan, ovesugquo, c p(t) HBJIHĐTCH H@ĐHOJHWGCROII c mepuopom 22/7, a nockombKy p(t) — 9pTO- awueckuit noToK Ha Z, ro p(t) — aproxuyecknit noroK na Z,, mocmequee: osHayaeT, uro L(Z,) ~ L™(R/(22/7T)Z)
llyerp Ï — rpynHa aBTOMOPÙM3MOB S, nopommenHaa G, u_ p(t) PaccMorpMuM waMepHMHH rpynnown SXxPƑ [lá] Mnozseerso S¿ = XxY x {0} HIBJII©TGH, O4@BHJ[HO ©TO HOJIHBIM JIARVH8DHBIM OH€THBIM G©@W€HIHM (GM OHIPG eeHHe 2.1 [14]) Tax KaH Ha Š; I1©ÏOTBV€T AHIPOROHMATHBHO KROH€WHAHẳ rpynna G(S,,Q,), To SXIF TAR?R© HIPORCHMATHBHO KOH€HHHII TDYHHOHJL (cm ompey 6.1 w reop 5.3 [14]) nonoốen rpynnonTy SạxŒ
C HDYTỌ GTOĐOHBI, HOHTH RA?£1Ọ TOWR© Z € Z¡ OTB@WA@T H8M€@DHMAH Ò0/i0dEA S, opouret rpynmpr ©, nopompennolt G, u p(2n/T) B cuny paccym7eHuil, HHB€JICHHBIX BBII©, S, Take ABSIAeTCA T0JIHBIM JIAEY- HADPHBIM OH4ƠGTHBIM O@W€HHM JIA rpyntouga Sx, nosromy Sxl TâHRe 1010ðeH Š.X<Ƒ; Ho rDynna Ï;¡ Ha S, wMeer mepwoggeckoe jeiicrsue, Tar Kak G, coxpanaer M€DY, Ø(2m/T7) /I@loTBV6T 3DTOTHUGGRM Hà 8JT@ỐPp€ H3MG- DHMBIX HO/MHO?/KROCTB, HHBADHAHTHBIK OTHOCHTeAbHO G, H YMHOMAeT Mepy
H, Ha S, Ha e~*7/T, npuyem p(2x/T) € W[G,)
ƠGTaA7I0Gb 38M@THTE, W†O IOCKOJIERY TDVNHOIHJI SạXỚ H S,X TT) H0TOỐHM,
IHIPE7UIO?RKEHHE 2.2 Hyeme W, — cnetuasonoii nomorn © Ốđ8HCHbiat asmomoppu3zmMom Q-! u nomosounok Gynnyuett p(x) = @ Ecau % — Gopenes-
ckul, acmomoppusm npocmpancmea (Xy, Mo), 20 Xo = {(x, u) | x€X, 0<u<ø0},
a duty = duxdu, rommymupynoumuii cW, mo % umeem eud
%g(x, w) = (ax, u + p), 2de a € Aut(X, nw), eO = Ox, p = Const
(IHoxoốnoe yTrBep;£eHne ZoKazaHo B [21], cm raxme [19].)
Hoxazsamessemeo Hanomuum, ato
W(x, vw) = (O-"x, u + t — Zn, @)) IIM Z1, @) — w S † < Z{n + Ì, 0) — u, rđ1e Z(n, ø) = @(Q~1x) + + 0(Q~"xỳ (quan 2 0) Noaromy ecau
(2.1) a(x, u) = (x',u,), O< U<@,
TO TOCKONBKY W% = aW,
A (QO-*x, 0) = %W,(x, 0) = Wa9(x, 0) =
= Wx", uy) = (Q~1x', uy).
Trang 8Ð Xọ Boxee roro, cayuaii %(x,,0) = (x', ứụ), %g(Xs, Ư) — (X”, ve), THE ty F Ue
uw O<4,< 9, UWcKMOUeH, HOGRKOIbRV (x’, v2) = Wu, —u(', u,), @& 3HAYUT (x;, 0) = %ạ "W4, —u đo (x,,0) = (41,4 — 4) HỆ Xị =Agz, =y, Ho Torxa X” — 1z ĐCTb ÕOD@zIEBGRAR (JVHRIHIIT, a BBHTAY (2.2), yHHTBIBAR 9ÐTOTNHNWHOCTb
Ĩ, MO?RHO C716/IATb BBIBOI 0 TOẠI, WTO „ =: p (Const), Ư < p < ø
Pacemorpiim Tenepb asromopdusm ƒ = W „xạ Jlerko BH/I6Tb, WTO B(x, u) = (x0), re x” onpeneneren Tak se, Kak H B (2.1) Ho rorya x—>x' == a(x) ecth aBromop®@usm (X,), KOTOpHH BBHNY (2.2) HOMMY-
TEOPEMA 2.3 Iyems G yOosrembopsem ycnosuas 1emuot 1.2 u, Goce mozo, G umeem nepuoduueckoe Oeticmeue Ecau xe N[G], mo cywyecmeyem té[G] maxott, ymo B = t-1a w.ueem eud
(2.3) B(x, ¥) = (&x, Wy),
2de % — acmomoppuan (X, 2), 2O =Q2, a M„ — UIMCPUMOE NOI G6H10- u0p(uamoe (Y, By,v), npuuen Wye.W[S] 019 noumu ecer x u O(W,) =
= p — log(du(&x)/du(x)), 20e p — Const
SAMEYAHUE 2.4 Ilverh S — aprognaeckiii apromoppusm (Y, By, v), CcOxXpaHAlMii o-RoWeINyIO Mepy v (rpynua Tuma II ) Torga W[S] e046PZRHT O;IHOIADAMGTDPHHGCHRYIO HellpepbIBHy!O Tpynumy aBromopdusmos p(t), te R, TaRKy© ro p(t,)p(t.) = p(t, + t2) m v-p(t) = e'v I[pumep TpYHHBL © TARHATI CHỌOTBAMH MO/RHO HOGTDOHTb C/I6AYIOUHIIMI OỐpA80M lÏycTme G rpynua Tunø lHÏị, a Ớa — 7BolicrBeHHan Tpynna (cm (1.1)), torga G, 0Ố:18/1A6T H?RHHIMH CBOIIOTBAMH ÏÏOHRTHO, WTO Ớ, I€ÏÏCTBY€T ĐDTOTHMĐGEIH, GOXPAHRET Mepy dw(œ, +) = e”“du(œ)du ma Y =(@x<R lĨoTroK OHD€NGIHM COTZIAGHO gopmyue p(s\(@, u) = (w,u — s), torga p(s), se R, u Gy, KommMytTupynr Horasameascmeo meope.we 2.3 Hapagy c rpynnoli G = (S,, Q,) pacemo- tpum rpynuy G’ = (S;, Q,), Koropad welicrayer B upocrpancrse (Xx Y’, nx v’),
Tae (Y’, v’) — Takoe we, Kak I B 8aM@waHum 2.4, a (X, ), KAK H ÿ TPYHNMI Ớ
Sex, y) = (x, Sy),
Q,(x, y) = (Ox, Uz),
Trang 9ABTOMOPG@HOMbI HHBbERTHBHBIX (ĐARTOPOB THHA HH 11
rne Ủy = ø(®(U,)), p(s) — aproMopjwaM n3 f[SŠ], pAaOGMOTp€HHHH B 3aMme- waHHH 2.4, a &(U,) = @ — log(du(Qx)/du(x)) [lonnrHo, aro y G’ acconuupo- BAHHEBLH notroK W(G’) HM@ĐT B KAWECTB© HOTO/IOHHỌ (YHEKHHHM @(X) = @, a B taqeGrBe ỐasancHorO aBToMop(HaMa Œ, T.e W(GŒ') = W(G) Ho rorna B cuny peayabTarop [6] rDYHHBI G u G’ cua6o skBuBaneHTHE Tlosromy 6e3 orpa- HUYeHHA BE OỐHIHOOTH MO?RHO HD€HHO2IO#Tb, uro U, = p(®(U,))
Ilyerh tenepp œ„€.f[Œ], Torna moda, — asromopdusm (Ấp; Hạ), KOoMMyTupylommi c WG) B cuny upeqnonomenun 2.2 (moda,)(x, u) =
= (ax, u-+ p), re x — apromopd@usm (X, 4), KomMyTupyromuit c QO Tlonomuu
(2.4) B(x, y) = (ax, p(p — log (du(ex)/du(x))y)
B cusy cxBolicrs p(s) (cm sameyanue) Ø,€.#T[S] lloKazreM, aro
0,8, == P,Q„ Jleiicrsurenbuo, Tak Kak
fy) cs 1X, — log —=— — log ——— ?
(Ø;8,⁄x ») (ox o(o — oe u(x) Jp (? _ |
x,y) = [aOx, pl p — log HE? — tog SHE),
(BQ, \X; y) (so: "|? og đo Mị» og uo )>}
TO B CHIY CBOHCTB HĐOH3BOHHBIX PAHOHA paste HAGTH aTux paBeHCTB
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TEOPEMA 3.1 Hycmb (Xq, tig) — npocmpancmeo, ¢ Komopom Oeticmeyem cheyuatonei nomor WG), accouuuposanneiii c epynnoli G, asmomoppuanoe npocmpancmea Jlebeza, a « — acmomoppuas (Xp, to) us CCW), v.€
To20a cywecmeyem ở e [G] mawoă, wmo mod & => a
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cRpeMmeHHes npouscedenues A = L°(XxY,uxv) na epynny G = (S,, Qg} (cst (1.3)) Toeda cywecmeyem % € Aut M maxot, umo mod % = &
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(3.1) dk(x, u) == e~*du(x) du,
a B WpocrpanctBe (X)xY,kxXv) paccmoTpumM rpyniry
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aH82I0THWHO onpesennetca O(Z(/, U-}, x)) gaa 1] < 0
PaoceMoTpHM notok W(t) B HĐOGTpAHGTB€ (Xạx<Ÿ, &kx v):
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B cuny Takoro onpefesenua V, moroK Wit) € W[S] ana te R, Gonee TOrO, Wt)S = SW(t), te R Haureli yesbw ỐVR€T pACHIIDGHH€ ở WO ABTO- Moppusma a, wpocrpancrBa (X)xY,kxXv) raKoro, WTO a, W(t) = W(t)a, "
a, € V[S] Iponenaem BGHOMOTATGIIbHBI BBIHHGJICHHRH
W(t)—1d(k Xv) dk x v)(W (x, u), Vix + Oy)
log (x, u, y) = log d(& x 9)(x, , y)
og dk(W,(x, u)) + log dv(V(x, u + fy) _
k
- (x, 4’)
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rye (x’,u’) «= a(x, u) JleiicrsurerpbHo, nockoapRy V(x, u): > ZU(x, u), U-4, x),
a U, =» p(®(U,)), to V(x, u) = p(ZU, u), (Uz), x)) = p(P(V(x, u))) Oveiona,
yuureran (3.5) u (3.6), Baroy (3.7) Ho roraza
(3.8) x(x, H, y) = (a(x, 1), Pœ?)
-— 8BTOMODHAäM „(XoxY, kx9), TIPHHA/1.16:t811W† MN (S] H ROMMYTIIPY1OHUIET
B GH¿IVY (3.7) © Wit), teR Caegonarenpno, & OHp@RN€:IR€T ABTOMODH3M op
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(bartopa M ’ KoTopsili ABJIACTCH CKPeUICHHEIM WpousBesenHuem L*(X)x Y, kx v)
na S, a satem Ha W(t), te R
Ilyctb — œKpemeHHoe npowsBexeHne L™(Xy)xY, kX v) Ha S, TOr1na
N — Il,,-aure6pa, ee cueyq 0Ø0gHaqunM wepe3 r lĨoTOR W(t), tŒR, HHHy- 1npyer rpynny 0), £e R, àroMopjaMoB W, npHweM 8 ©HJ1Y (3.4) r - Ø(f) = e7't, 0JI910BAT©7IbHO, 718 M=Ww *(N, 0, R) MBI HMeeM H€IP€PEIBHOG pa3smomenne
\looromy raaqKuii noroK Becos yaa M copnagaer c W(t) (cm ra IL memma 1.4 [7])
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VY HUX HOTORH B©OOB I80MOPÿHBL ( (cm [5], [6]), TaRHM o6pasom, BCAKOMYy
Horxazameavcméeo meopeme 3.1 ipemye Bcero 3ametum, uro B cusy (3.4)
a
pas6uenue mpocrpancrna (X,XY,kxv) wa opbuTe W(t), te R, usmepumo Jleitcrsureabno, paccmorpum W(1) Torna pBBHxy (3.4) cormacHo aemme /lan (cm xemmy 8.8 [15]) pas6uenue (X)x Y,4x v) Ha Tpaextopun Wl) 18M6GDMMO QỐốoaHawIM 83P0 pA3ÕỐHeHwe 4©p©3 Z¡ H pA©€GMOTPHM Hã (AKT0D-HĐOGTDAHGTB© X)xXY/F, woTok Wit), te R Tak Kak Wil) Ha XạXY /Z¡ HBJIHGTCH TO?RH@GT-
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Tak nan B gaMewaHHH 2.4 Gy u p(t) KOMMYTUPYIOT, TO Us OnpeneneHuA Ww (cM (3 3)) MomHO HpequonaraTh, WTO Su Wit), té R, xommytupyiot Ho or7a 5 OIĐ©TG7InN€T 8BTOMOP(ĐHSM S, PakTop-npocrpaHcTBa (@xø) PacHIHpMM
S, 0 aBToMopusma Sy mpocrpanctaa 2<, I071074MB So, u) = (S\o, 4),