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Phkuaetrán ie FhapkejhsPhkuaetrán ie Fhapkejhs FHASKEWHAEY]ÁNFHASKEWHAEY]ÁNPPhkuaetrán ie hkuaetrán ie FhapkejhsFhapkejhs Ub fhapkejh es ubn espefoe `hranin phr knUb fhapkejh es ubn espefoe `hranin phr kn nshfonfoöb ie ihs h aîs pnrtáfuknsnshfonfoöb ie ihs h aîs pnrtáfukns aîs soapkesaîs soapkes, fnin ubn ie ekkns, fnin ubn ie ekkns fnpnz ie exostor obiepebioebteaebtefnpnz ie exostor obiepebioebteaebte (]hsshtto).(]hsshtto). Velúb estn ie`obofoöb, phiránb ser fhapkejhs ekVelúb estn ie`obofoöb, phiránb ser fhapkejhs ek oöb troyhiurh (Ooöb troyhiurh (O66 ‖ ‖)) h khsh khs ohbesohbes goirntnihsgoirntnihs, ebtre aufghs ejeapkhs., ebtre aufghs ejeapkhs. Snrn ek estuioh ie este fnpátukh se `ojn kn ntebfoöb eb knsSnrn ek estuioh ie este fnpátukh se `ojn kn ntebfoöb eb kns nshfonfohbes ie ohbesnshfonfohbes ie ohbes aetîkofhsaetîkofhs fhbfhb espefoes que pueieb `hranr ebknfes fhhriobnihsespefoes que pueieb `hranr ebknfes fhhriobnihs.. •• N N estns espefoes estns espefoes se se kes iebhakes iebhaobnobn KOLNBYEVKOLNBYEV.. KnKn vhkuaetrán ie `hranfoöb ie fhapkejhsvhkuaetrán ie `hranfoöb ie fhapkejhs h FHASKEWHAEY]ÁN trntnh FHASKEWHAEY]ÁN trntn aîs exnftnaebte iekaîs exnftnaebte iek obterfnamoh ie Kolnbtesobterfnamoh ie Kolnbtes, es iefor ek reeapknzh ie, es iefor ek reeapknzh ie ahkïfukns ie nlun phr htrhs kolnbtes.ahkïfukns ie nlun phr htrhs kolnbtes. Yophs ie Kolnbtes :Yophs ie Kolnbtes : •• Uboiebtnihs (ub pnr ie ekeftrhbes)Uboiebtnihs (ub pnr ie ekeftrhbes) •• Shkoiebtnihs h ^UEKNYShkoiebtnihs h ^UEKNYHV (ie = HV (ie = h aîs pnres h aîs pnres ie ekeftrhbes)ie ekeftrhbes) Váamhkhs ie khs pnrîaetrhs utokoznihs:Váamhkhs ie khs pnrîaetrhs utokoznihs: SnrnSnrn kolnbtes ahbh y moiebtnihs (K)kolnbtes ahbh y moiebtnihs (K), que `hranb fhapkejhs fhb khs, que `hranb fhapkejhs fhb khs nfuh-fhapkejhs aetîkofhs A ebnfuh-fhapkejhs aetîkofhs A eb vnrons etnpnsvnrons etnpns.. SnrnSnrn kolnbtes phko`ubfohbnkes quekntnbtes (_)kolnbtes phko`ubfohbnkes quekntnbtes (_) que `hranb fhapkejhs eb ubnque `hranb fhapkejhs eb ubn shkn etnpn :shkn etnpn : A + _ 1 A_A + _ 1 A_ Fhbstnbte ie Equokomroh (D), fuyh vnkhr es `ubinaebtnk eb thih equokomroh ieFhbstnbte ie Equokomroh (D), fuyh vnkhr es `ubinaebtnk eb thih equokomroh ie `hranfoöb ie fhapkejhs.`hranfoöb ie fhapkejhs. Etnpns ie `hranfoöb ie fhapkejhs (b):Etnpns ie `hranfoöb ie fhapkejhs (b): •• b1>, funbih ek kolnbte b1>, funbih ek kolnbte es ub queknth _es ub queknth _.. •• b ≥=, funbih ek kolnbte es ahbh h moiebtnih.b ≥=, funbih ek kolnbte es ahbh h moiebtnih. Expresohbes pnrn Fhbstnbtes ie EquokomrohExpresohbes pnrn Fhbstnbtes ie Equokomroh Voaoknres n kns ie prhthbnfoöb ie ub kolnbte.Voaoknres n kns ie prhthbnfoöb ie ub kolnbte. A A + + K K 1 1 AKAK AK AK + + K K 1 1 AKAK== AKAK== + K 1 AK+ K 1 AK66 Kn io`erebfon estî eb que pnrn khs E^UOKOM]OHV ÎFOIHV-MNVE VE UVNKn io`erebfon estî eb que pnrn khs E^UOKOM]OHV ÎFOIHV-MNVE VE UVN FHBVYNBYEV IE IOVHFONFOÖB, aoebtrns que pnrn khsFHBVYNBYEV IE IOVHFONFOÖB, aoebtrns que pnrn khs fhapkejhs se suekefhapkejhs se sueke eapkenr Fhbstnbtes ie @H]ANFOÖBeapkenr Fhbstnbtes ie @H]ANFOÖB.. IIoosshhffoonnffooööb b pphhr r eettnnppnnss FFhhbbssttnnbbttees s iie e iioosshhffoonnffooööbb GG99 N N 1 1 GG66 N N + + GG ++ DD>> 1 1 TGTG ++ QTGQTG66NQ/TGNQ/TG99NQNQ GG66 N N 1 1 GG== N N + + GG ++ DD == 1 1 TGTG++QTGQTG == NQ/TGNQ/TG 66 NQNQ GG== N 1 GN + GN 1 GN + G ++ DD 66 1 1 TGTG++QTGNQ/TGQTGNQ/TG == NQNQ GN GN 1 1 N + GN + G ++ DD99 1 1 TGTG++QTNQ/TGNQQTNQ/TGNQ Kn fhbfebtrnfoöb thtnk iek îfoih es:Kn fhbfebtrnfoöb thtnk iek îfoih es: F 1 TGF 1 TG99 NQ + TGNQ + TG66 NQ + TGNQ + TG== NQ + TGNQ + TNQ .... (O)NQ + TGNQ + TNQ .... (O) Vo ek pG es îfoih gny preihaoboh ie espefoes prhthbnins.Vo ek pG es îfoih gny preihaoboh ie espefoes prhthbnins. Expresohbes Expresohbes Nk`n Nk`n pnrn îfoihs pnrn îfoihs ShkoprötofhsShkoprötofhs Fnin fhbfebtrnfoöb se expresn fhah `ubfoöb ie su espefoe fhbjulnin :Fnin fhbfebtrnfoöb se expresn fhah `ubfoöb ie su espefoe fhbjulnin : TGTG66 NQ 1 DNQ 1 D>>TGTG99NQ/TGNQ/TG++QQ TGTG== NQ 1 DNQ 1 D==TGTG66NQ/TGNQ/TG ++ Q Q 1 1 DD>> DD==TGTG99NQ/TGNQ/TG ++ QQ == TGNQ 1 DTGNQ 1 D66TGTG==NQ/TGNQ/TG++Q Q 1 D1 D>> DD== DD66TGTG99NQ/TGNQ/TG++QQ66 TNQ 1 DTNQ 1 D99TGNQ/TGTGNQ/TG++Q Q 1 1 DD>> DD== DD66 D D99 TG TG99NQ/TGNQ/TG++QQ99 @]NFFOHBEV NK@N (Kolnbihs Îfoihs)@]NFFOHBEV NK@N (Kolnbihs Îfoihs) Kns `rnffohbes ξKns `rnffohbes ξoo shb kn fhbfebtrnfoöb ie kns espefoes prhthbnins respefth nshb kn fhbfebtrnfoöb ie kns espefoes prhthbnins respefth n kn fhbfebtrnfoöb thtnk F.kn fhbfebtrnfoöb thtnk F. ξξ99 1 TG1 TG99NQ/F 1 >/( > + DNQ/F 1 >/( > + D>> /TG /TG++Q + DQ + D>>DD== /TG /TG++QQ== + D + D>>DD==DD66 /TG /TG++QQ66 + D + D>>DD==DD66DD99 /TG /TG++QQ99 ) ) fhrresphbie n kn `rnffoöb iek îfoih ie hroleb eb shkufoöb y se pueie expresnrfhrresphbie n kn `rnffoöb iek îfoih ie hroleb eb shkufoöb y se pueie expresnr tnamoïb fhah,tnamoïb fhah, ξξ99 1 TG1 TG99NQ/F 1NQ/F 1 TGTG++QQ99 / /( TG( TG++QQ99 + D + D>>TGTG++QQ66 + D + D>>DD== TGTG++QQ== + D + D>>DD==DD66TGTG++Q + DQ + D>>DD==DD66DD99)) ξξ66 1 TG1 TG66NQ/F 1NQ/F 1 DD>>TGTG++QQ66 / /(TG(TG++QQ99 + D+ D>>TGTG++QQ66 + D + D>>DD==TGTG++QQ== + D+ D>>DD==DD66 TGTG++Q Q + + DD>>DD==DD66DD99 )) ξξ== 1 TG1 TG== NQ/F 1 DNQ/F 1 D>>DD== TGTG++QQ== / /(TG(TG++QQ99 + D + D>>TGTG++QQ66 + D + D>> DD== TGTG++QQ== + D + D>>DD==DD66 TGTG++Q + DQ + D>>DD==DD66DD99 )) ξξ>> 1 TG1 TG NQ/F 1 DNQ/F 1 D>>DD==DD66 TGTG++Q/(TGQ/(TG++QQ99 + D + D>>TGTG++QQ66 + D + D>>DD== TGTG++QQ== + D + D>>DD==DD66 TGTG++Q + DQ + D>>DD==DD66DD99 )) ξξ33 1 TNQ/F 1 D1 TNQ/F 1 D>>DD==DD66DD99 / (TG / (TG++QQ99 + D + D>>TGTG++QQ66 + D + D>>DD== TGTG++QQ== + D + D>>DD==DD66 TGTG++Q + DQ + D>>DD==DD66 D D99)) Shr tnbth, funbih ek kolnbih N (kolnbih îfoih) se pueie prhthbnr, se pueieShr tnbth, funbih ek kolnbih N (kolnbih îfoih) se pueie prhthbnr, se pueie expresnr knexpresnr kn Ahknroini thtnk iek Kolnbte N h KAhknroini thtnk iek Kolnbte N h K (sob fhbsoiernr kn `hranfoöb iek(sob fhbsoiernr kn `hranfoöb iek fhapkejh fhb ek aetnk):fhapkejh fhb ek aetnk): TNQ 1 ξTNQ 1 ξ33 FFKK TNQTNQ 11 DD >> DD == DD 66 DD 99 / (TG / (TG++QQ99 + D + D >> TGTG++QQ66 + D + D >> DD == TGTG++QQ== + D + D >> DD == DD 66 TGTG++Q + DQ + D >> DD == DD 66 D D 99 )) x Fx F KK FhFhfofoeebbttees s ie ie ̀̀ hrhraannfofoöb öb susuffesesovovnnss FFhfhfoeoebbttees s ie ie ̀̀ hhraranfnfoöoöb b llkhkhmmnknkees s (Ø(Øoo ) ) A + K A + K 1 AK 1 AK 44 DD>> 1 TAKQ/TAQTKQ1 TAKQ/TAQTKQ A + K A + K 1 AK 1 AK 44 ØØ>> 1 TAKQ/TAQTKQ 1 D1 TAKQ/TAQTKQ 1 D>> AK + KAK + K 1 1 AKAK== 44 DD== 1 TAK1 TAK==Q/ TAKQTKQQ/ TAKQTKQ A A + + =K =K 1 A1 AKK== 44 ØØ== 1 TAK1 TAK==Q/TAQTKQQ/TAQTKQ == 1 D1 D>> DD== AKAK== + + K K 1 1 AKAK66 44 DD66 1 TAK1 TAK66Q/ TAKQ/ TAK==QTKQQTKQ A A + + 6K 6K 1 A1 AKK66 44 ØØ66 1 TAK1 TAK66Q/TAQTKQQ/TAQTKQ66 1 D1 D>> DD== D D66 AKAK66 + + K K 1 1 AKAK99 44 DD99 1 TAK1 TAK99Q/ TAKQ/ TAK66QTKQQTKQ A A + + 9K 9K 1 A1 AKK99 44 ØØ99 1 TAK1 TAK99Q/TAQTKQQ/TAQTKQ 99 1 D1 D>>DD==DD66DD99 Equokomrohs y Fhfoebtes ie `hranfoöb ie FhapkejhsEquokomrohs y Fhfoebtes ie `hranfoöb ie Fhapkejhs sufesovns y lkhmnkes (AK)sufesovns y lkhmnkes (AK) @ubfoöb ie Mnknbfe ie Anteronkes pnrn ek oöb aetîkofh@ubfoöb ie Mnknbfe ie Anteronkes pnrn ek oöb aetîkofh Ubn shkufoöb iek oöb aetîkofh fhbtoebe FUbn shkufoöb iek oöb aetîkofh fhbtoebe FAA ahk/K.ahk/K. Eb nusebfon ie ub kolnbte,Eb nusebfon ie ub kolnbte, FFAA 1 TAQ1 TAQ Vo kn shkufoöb fhbtoebe ub kolnbte K (FVo kn shkufoöb fhbtoebe ub kolnbte K (FKK ahk/k) , fnpnz ie `hranr funtrhahk/k) , fnpnz ie `hranr funtrh fhapkejhs sufesovhs fhb A,fhapkejhs sufesovhs fhb A, FFAA 1 TAQ + TAKQ + TAK1 TAQ + TAKQ + TAK==Q + TAKQ + TAK66Q + TAKQ + TAK99QQ Estns fhbfebtrnfohbes obiovoiunkes iepebieb ie FEstns fhbfebtrnfohbes obiovoiunkes iepebieb ie FKK y ie kns fhbstnbtes ie y ie kns fhbstnbtes ie `hranfoöb sufesovns.`hranfoöb sufesovns. Eb `ubfoöb ie kns expresohbesMetn, se toebe que,Eb `ubfoöb ie kns expresohbes Metn, se toebe que, FFAA 1 TAQ + D1 TAQ + D>>TAQTKQ TAQTKQ + + ØØ==TAQTKQTAQTKQ== + + ØØ66 TAQTKQ TAQTKQ66 + + ØØ99TAQTKQTAQTKQ99 KnKn `ubfoöb ie @hranfoöb, @`ubfoöb ie @hranfoöb, @hh se ie`obe fhah kn rnzöb ie kn fhbfebtrnfoöb thtnkse ie`obe fhah kn rnzöb ie kn fhbfebtrnfoöb thtnk FFAA n kn iek aetnk n kn iek aetnk que bh `hran fhapkejhs TAQ,que bh `hran fhapkejhs TAQ, @@hh 1 F1 FAA /TAQ /TAQ 1 1 > > + + ^̂>> TKQ + ØTKQ + Ø==TKQTKQ== + + ØØ66TKQTKQ66 + + ØØ99TKQTKQ99 @ubfoöb (@@ubfoöb (@33 ‖ >) ‖ >) @@33 ‖ > 1 ^ ‖ > 1 ^>> TKQ + ØTKQ + Ø==TKQTKQ== + + ØØ66TKQTKQ66 + + ØØ99TKQTKQ99 11 ∞ TFhapkejhsQ / TAQ∞ TFhapkejhsQ / TAQ Estn `ubfoöb voebe n ser knEstn `ubfoöb voebe n ser kn reknfoöb ebtre kn fhbfebtrnfoöb iek oöb aetîkofhreknfoöb ebtre kn fhbfebtrnfoöb iek oöb aetîkofh fhbvertoih eb fhapkejhsfhbvertoih eb fhapkejhs respefth n kn fhbfebtrnfoöb iek oöb aetîkofh bh respefth n kn fhbfebtrnfoöb iek oöb aetîkofh bh trnbs`hranih. Srhphrfohbn ubntrnbs`hranih. Srhphrfohbn ubn evnkunfoöb iek lrnih ie `hranfoöb iekevnkunfoöb iek lrnih ie `hranfoöb iek fhapkejhfhapkejh, funkquoern sen kn espefoe ieteraobnbte eb ubn ieteraobnin shkufoöb., funkquoern sen kn espefoe ieteraobnbte eb ubn ieteraobnin shkufoöb. @ubfohbes Nk`n (eb fhapkejhs)@ubfohbes Nk`n (eb fhapkejhs) ]epresebtnb kn `rnffoöb ie fnin ubn ie kns espefoes obvhkufrnins fhb]epresebtnb kn `rnffoöb ie fnin ubn ie kns espefoes obvhkufrnins fhb respefth n kn fhbfebtrnfoöb thtnk Frespefth n kn fhbfebtrnfoöb thtnk FAA.. ξξ33 1 TAQ/F1 TAQ/FAA 1 >/@1 >/@HH (@rnffoöb iek aetnk que bh `hran fhapkejhs)(@rnffoöb iek aetnk que bh `hran fhapkejhs) ξξ>> 1 TAKQ/F1 TAKQ/FAA 1 D1 D>> TKQ / @TKQ / @HH ξξ==1 TAK1 TAK==Q/FQ/FAA 1 Ø1 Ø== TKQTKQ== / @ / @HH ξξ661 TAK1 TAK66Q/FQ/FAA 1 Ø1 Ø66 TKQTKQ 66 / @ / @HH ξξ991 TAK1 TAK99Q/FQ/FAA 1 Ø1 Ø99 TKQTKQ99 / @ / @HH Fhb estns expresohbes se pueie fnkfuknr fuîk ie kns espefoes prevnkefe nFhb estns expresohbes se pueie fnkfuknr fuîk ie kns espefoes prevnkefe n fhbfebtrnfohbes ieteraobnins ie ub kolnbih ahbh h moiebtnih.fhbfebtrnfohbes ieteraobnins ie ub kolnbih ahbh h moiebtnih. E^UOKOM]OHV EB FHASEYEBFONE^UOKOM]OHV EB FHASEYEBFON Eb kn vhkuaetrán ie FhapkejhsEb kn vhkuaetrán ie Fhapkejhs bh es sökh ub sostean ebh es sökh ub sostean eb equokomrohb equokomroh ek que estîek que estî obvhkufrnih.obvhkufrnih. Espefá`ofnaebte eb este fnpátukh es ie obterïs ek estuioh iekEspefá`ofnaebte eb este fnpátukh es ie obterïs ek estuioh iek KolnbteKolnbte phkoiebtnih EIYN reprphkoiebtnih EIYN represebtnih phr esebtnih phr GG99 _ _.. Ve toebe ebthbfes kn renffoöb ie `hranfoöb iek Fhapkejh probfopnk, fhbVe toebe ebthbfes kn renffoöb ie `hranfoöb iek Fhapkejh probfopnk, fhb ioversns renffohbes sefubinrons,ioversns renffohbes sefubinrons, A + _ 1 A_A + _ 1 A_ A A + + _ _ 1 1 A_A_ ++ ++ K 1 AKK 1 AK G 1 G_G 1 G_ (E`efth @(E`efth @33)) (E`efth pG)(E`efth pG) Kn `hranfoöb ie AK oapkofn kn fhapetebfon ie ihs kolnbtes phr ek oöb A yKn `hranfoöb ie AK oapkofn kn fhapetebfon ie ihs kolnbtes phr ek oöb A y ek selubih equokomroh serán kn prhthbnfoöb iek Kolnbte probfopnkek selubih equokomroh serán kn prhthbnfoöb iek Kolnbte probfopnk phkoiebtnih _phkoiebtnih _, iemoih n , iemoih n su bnturnkezn su bnturnkezn îfoin.îfoin. E`efth pG: Srhthbnfoöb iek KolnbteE`efth pG: Srhthbnfoöb iek Kolnbte N pG mnjh, ek kolnbte se ve `hrznih n `hranr su îfoih fhbjulnih.N pG mnjh, ek kolnbte se ve `hrznih n `hranr su îfoih fhbjulnih. N pG nkth ek e`efth es iesprefonmke.N pG nkth ek e`efth es iesprefonmke. Snrn estuionr este e`efth se ieme fhbsoiernr kn `ubfoöb @Snrn estuionr este e`efth se ieme fhbsoiernr kn `ubfoöb @33 pnrn khspnrn khs fhapkejhs aetnk-kolnbte y knfhapkejhs aetnk-kolnbte y kn `ubfoöb ξ`ubfoöb ξ33 pnrn kns espefoes iek kolnbtepnrn kns espefoes iek kolnbte prhthbnih.prhthbnih. Funbih se fhbhfe Funbih se fhbhfe kn ahknroini ikn ahknroini ie equokomroh ie equokomroh ie K e K (sob fhbsoiernr ek vnkhr(sob fhbsoiernr ek vnkhr iek pG, phiek pG, phrque bh rque bh es solbo`ofntovh), se pueie es solbo`ofntovh), se pueie usnr kn @ubfoöusnr kn @ubfoöb b (@(@33 ‖>). ‖>). Serh so ekSerh so ek e`efth pG shmre ek kolnbih es solbo`ofntovhe`efth pG shmre ek kolnbih es solbo`ofntovh, se ieme gnfer ub, se ieme gnfer ub fîkfukh prekoaobnr fhb kn `ubfoöb ξfîkfukh prekoaobnr fhb kn `ubfoöb ξ33.. Eb ubn shkufoöb que fhbtoebe kns ioversns espefoes prhthbnins iekEb ubn shkufoöb que fhbtoebe kns ioversns espefoes prhthbnins iek kolnbih, kn expresoöbkolnbih, kn expresoöb ξξ33FFKK peraote gnkknr kn fhbfebtrnfoöb ie kn espefoe bhperaote gnkknr kn fhbfebtrnfoöb ie kn espefoe bh prhthbnin n ub pG espefá`ofh.prhthbnin n ub pG espefá`ofh. Estn fhbfebtrnfoöb renk ie K se usn eb ek fîkfukh ie kn `ubfoöb (@Estn fhbfebtrnfoöb renk ie K se usn eb ek fîkfukh ie kn `ubfoöb (@33 ‖ >). ‖ >). FHASEYEBFON EBY]E KOLNBYEV SH] EK OÖB AEYÎKOFHFHASEYEBFON EBY]E KOLNBYEV SH] EK OÖB AEYÎKOFH Funbih eb ubn shkufoöb se toebe ub oöb aetîkofh y ihs kolnbtes que pueiebFunbih eb ubn shkufoöb se toebe ub oöb aetîkofh y ihs kolnbtes que pueieb fhapkejnr nk fhapkejnr nk oöb, se oöb, se presebtn kn presebtn kn prelubtn:prelubtn: Fuîk ie khs kolnbihs es ek que vn n khlrnr fhapkejnr nk oöb aetîkofh.Fuîk ie khs kolnbihs es ek que vn n khlrnr fhapkejnr nk oöb aetîkofh. Kn shkufoöb se hmtoebe npkofnbih kn `ubfoöb (@Kn shkufoöb se hmtoebe npkofnbih kn `ubfoöb (@33 ‖ ‖ >) >) pnrn pnrn fnin fnin ubh ubh ie ie khskhs phsomkes fhapkejhs `hranihs.phsomkes fhapkejhs `hranihs. FHAMOBNFOÖB IE E@EFYHVFHAMOBNFOÖB IE E@EFYHV pG _ FHASEYEBFON IE KOLNBYEVpG _ FHASEYEBFON IE KOLNBYEV Fnin kolnbih toebe sus prhpoeinies îfoih-mnse ie`oboins y pnrtofuknres.Fnin kolnbih toebe sus prhpoeinies îfoih-mnse ie`oboins y pnrtofuknres. Nk `ojnr ub vnkhr ie pG se pueie fnkfuknr kns fhbfebtrnfohbes ahknres ebNk `ojnr ub vnkhr ie pG se pueie fnkfuknr kns fhbfebtrnfohbes ahknres eb equokomroh ie khs kolnbihs (@ubfoöb ξequokomroh ie khs kolnbihs (@ubfoöb ξ33) y kuelh fhb esths vnkhres se fnkfukn) y kuelh fhb esths vnkhres se fnkfukn phr sepnrnih kns `ubfohbes (@phr sepnrnih kns `ubfohbes (@33 ‖ >) ‖ >) ie fnin ie fnin fhapkejh que fhapkejh que aoieb kn aoieb kn tebiebfontebiebfon n lebernr fhapkejhs.n lebernr fhapkejhs.
