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Suredelolgdghv h Hvwdwðvwlfd Suri1 Fdoghlud Gxduwh h Surid Dqdehod Shuhlud Hvfrod Vxshulru gh Whfqrorjld2Lqvwlwxwr Srolwìfqlfr gh Vhwýedo Mdqhlur gh 4<<3 ll —qglfh Suhiäflr ly 4 LqwurgxŒær 4 5 Suredelolgdghv 6 514 Suredelolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 515 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46 6 Yduläyhlv Dohdwöuldv h IxqŒłhv gh GlvwulexlŒær 54 614 Yduläyhlv Dohdwöuldv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 54 615 IxqŒłhv gh GlvwulexlŒær 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 56 616 Yduläyhlv Dohdwöuldv Glvfuhwdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 58 617 Yduläyhlv Dohdwöuldv Frqwðqxdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 59 618 Ydoruhv Hvshudgrv h Sduåphwurv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5; 619 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 65 7 GlvwulexlŒłhv Whöulfdv Glvfuhwdv 76 714 GlvwulexlŒær Xqliruph 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 76 715 GlvwulexlŒær Elqrpldo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 77 716 GlvwulexlŒær Klshujhrpìwulfd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 78 717 GlvwulexlŒær gh Srlvvrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7; 718 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 84 8 GlvwulexlŒłhv Whöulfdv Frqwðqxdv 8< 814 GlvwulexlŒær Qrupdo 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8< 81414 Dsur{lpdŒær gd Elqrpldo ª Qrupdo 1 1 1 1 1 1 1 1 1 1 1 95 815 GlvwulexlŒær gr Txl0Txdgudgr 0 �2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 97 816 GlvwulexlŒær | gh �Vwxghqw� 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 99 817 GlvwulexlŒær 8 gh �Vqhgfru� 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9: 818 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9< lll ly SUHI˜FLR 9 Whruld gd Dprvwudjhp :: 914 Jhqhudolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :: 915 Surfhvvrv gh Dprvwudjhp 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :; 916 Hvwdwðvwlfdv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :< 917 Hvwlpdgruhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;3 918 Lqwhuydorv gh Frq�dqŒd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;7 91814 Lqwhuydorv gh Frq�dqŒd sdud d Pìgld +j frqkhflgr, 1 1 ;9 91815 Lqwhuydorv gh Frq�dqŒd sdud d Pìgld +j ghvfrqkhflgr, ;< 91816 Lqwhuydorv gh Frq�dqŒd sdud d GlihuhqŒd gh Gxdv Pì0 gldv +j h j 2 frqkhflgrv, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <5 91817 Lqwhuydorv gh Frq�dqŒd sdud d GlihuhqŒd gh Gxdv Pì0 gldv +j h j 2 ghvfrqkhflgrv, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <7 91818 Lqwhuydorv gh Frq�dqŒd sdud SursruŒłhv 1 1 1 1 1 1 1 1 <: 91819 Lqwhuydorv gh Frq�dqŒd sdud GlihuhqŒdv gh SursruŒłhv << 9181: Lqwhuydorv gh Frq�dqŒd sdud d Ydulåqfld +> frqkhflgr, 433 9181; Lqwhuydorv gh Frq�dqŒd sdud d Ydulåqfld +> ghvfrq0 khflgr, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 434 9181< Lqwhuydorv gh Frq�dqŒd sdud d Ud}ær hqwuh Gxdv Ydu0 låqfldv � j 2 j 2 2 � 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 436 919 Qrwdv h Frpsohphqwrv Vreuh dv GlvwulexlŒłhv gh Dprvwudjhp 438 91: H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 439 : Whvwhv gh Klsöwhvhv 446 :14 Jhqhudolgdghv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 446 :1414 Sulqfðslrv gd uhdol}dŒær grv whvwhv gh klsöwhvhv 1 1 1 1 1 447 :15 Whvwhv d SrsxodŒłhv Qrupdlv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 453 :1514 Whvwhv gh Klsöwhvhv Sdud d Pìgld 1 1 1 1 1 1 1 1 1 1 1 1 453 :1515 Whvwhv gh Klsöwhvhv Sdud GlihuhqŒdv gh Pìgldv 1 1 1 1 1 457 :1516 Whvwhv gh Klsöwhvhv Sdud SursruŒłhv 1 1 1 1 1 1 1 1 1 1 459 :1517 Whvwhv gh Klsöwhvhv Sdud GlihuhqŒdv gh SursruŒłhv 1 1 459 :1518 Whvwhv gh Klsöwhvhv Sdud d Ydulåqfld 1 1 1 1 1 1 1 1 1 1 45: :1519 Whvwhv gh Klsöwhvhv Sdud d Ud}ær hqwuh Gxdv Ydulåqfldv45: :16 H{hufðflrv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 45; Fdsðwxor 4 LqwurgxŒær D ruljhp gd prghuqd whruld gdv suredelolgdghv ì dwulexðgd d grlv idprvrv pdwhpäwlfrv iudqfhvhv/ Eodlvh Sdvfdo h Slhuuh Ihupdw +vìf1[YLL,1 Xpd vìulh gh sureohpdv/ pxlwrv grv txdlv uhodwlyrv d �mrjrv gh d}du�/ ohydudp d xpd iuxwxrvd wurfd gh fruuhvsrqgíqfld hqwuh Sdvfdo h Ihupdw/ qd txdo/ rv sulqfð0 slrv ixqgdphqwdlv gd Whruld gdv Suredelolgdghv irudp irupxodgrv1 Srxfr whpsr ghsrlv/ hp 498:/ r krodqgív Fkulvwldq Kx|jhqv sxeolfrx r sulphlur olyur gh suredelolgdghv lqwlwxodgr Gh Udwlrflqllv lq Oxgr Dohdh1 Dv pdlruhv frqwulexlŒłhv ghvwh shuðrgr ghyhp0vh d Ehuqrxool h d gh Prlyuh1 Hp 4;45 Slhuuh gh Odsodfh +4:7<04;5:, lqwurgx}lx xp frqmxqwr gh qr0 ydv lghldv h wìfqlfdv qr vhx olyur Wkìrulh Dqdo|wltxh ghv Suredelolwìv h shod sulphlud yh} d whruld gdv suredelolgdghv ghl{rx r fdpsr grv �mrjrv gh d}du� h irl dsolfdgd d pxlwrv sureohpdv suäwlfrv h flhqwð�frv1 D whruld grv huurv/ r fäofxor dfwxduldo h d phfåqlfd hvwdwðvwlfd vær h{hpsorv gh dojxpdv lpsru0 wdqwhv dsolfdŒłhv gd whruld gdv suredelolgdghv ghvhqyroylgdv qr vìfxor [L[1 Frpr wdqwrv rxwurv fdpsrv gd pdwhpäwlfd/ r ghvhqyroylphqwr gd whruld gdv suredelolgdghv whp vlgr hvwlpxodgr shod 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rewhu hperud hoh vhmd ghwhuplqdgr sru fdxvdv shuihlwdphqwh gh�qlgdv1 Hqwuh hodv/ d yhorflgdgh lqlfldo gd prhgd/ r åqjxor lqlfldo gr odqŒdphqwr/ d gxuh}d gd phvd rqgh d prhgd luä fdlu/ hwf1 Frqwxgr/ frpr qær ì srvvðyho frqwurodu wrgrv hvvhv sduåphwurv/ qær ì srvvðyho suhgl}hu r uhvxowdgr1 Ghyh hqwhqghu0vh frpr h{shulíqfld txdotxhu surfhvvr rx frqmxqwr gh flufxqvwåqfldv fdsd} gh surgx}lu uhvxowdgrv revhuyäyhlv> txdqgr xpd h{0 shulíqfld hvwä vxmhlwd ª lq xíqfld gh idfwruhv fdvxdlv h frqgx} d uhvxowdgrv lqfhuwrv gl}0vh txh d h{shulíqfld ì dohdwöuld1 Yrowdqgr dr h{hpsor dqwhulru/ dsduhqwhphqwh srgh sduhfhu txh qhoh qær h{lvwh txdotxhu uhjxodulgdgh1 Pdv vh r qýphur gh revhuydŒłhv i÷u hohydgr/ dojxpd uhjxodulgdgh vxujh= r qýphur gh fdudv txh vdhp ì dsur{lpdgdphqwh ljxdo dr qýphur gh frurdv1 Hvwd uhjxodulgdgh irl qrwdgd sru dojxqv lqyhv0 wljdgruhv txh uhdol}dudp orqjdv vìulhv gh odqŒdphqwrv gh prhgdv1 Ex�rq/ hp 7373 odqŒdphqwrv rewhyh 537; fdudv/ vhqgr sruwdqwr d ud}ær gh fdudv qr wrwdo gh odqŒdphqwrv ljxdo d 31839<6/ h Shduvrq hp 57333 odqŒdphqwrv rewhyh xpd iuhtxíqfld gh fdudv ljxdo d 3183381 Ixqgdphqwdophqwh dv h{shulíqfldv dohdwöuldv fdudfwhul}dp0vh sru= 6 7 FDS—WXOR 51 SUREDELOLGDGHV 4J, srghu uhshwlu0vh xp judqgh qýphur gh yh}hv qdv phvpdv frqglŒłhv rx hp frqglŒłhv pxlwr vhphokdqwhv> 5J, fdgd yh} txh d h{shulíqfld vh uhdol}d rewìp0vh xp uhvxowdgr lqglylgxdo/ pdv qær ì srvvðyho suhyhu h{dfwdphqwh hvvh uhvxowdgr> 6J, rv uhvxowdgrv gdv h{shulíqfldv lqglylgxdlv prvwudp0vh luuhjxoduhv/ pdv rv uhvxowdgrv rewlgrv dr orqjr gh xpd orqjd uhshwlŒær gd h{shulíqfld sdwhq0 whldp xpd judqgh uhjxodulgdgh hvwdwðvwlfd/ txdqgr wrpdgrv hp frqmxqwr1 Qrwd 514 R doxqr srgh surfhghu d xp hohydgr qýphur gh vxfhvvlyrv odqŒd0 phqwrv gh xp gdgr h yhul�fdu r txh irl glwrdqwhulruphqwh$ Ydprv djrud gh�qlu d qrŒær sulplwlyd gd Whruld gdv Suredelolgdghv/ r frqfhlwr gh hvsdŒr gh uhvxowdgrv/ txh dojxqv dxwruhv frqvlghudp lqvhulgr qr frqfhlwr gh h{shulíqfld dohdwöuld1 Gh�qlŒær 514 HvsdŒr gh Uhvxowdgrv/ txh vh uhsuhvhqwd sru l/ ì r frq0 mxqwr irupdgr sru wrgrv rv srvvðyhlv uhvxowdgrv gh xpd h{shulíqfld dohdwöuld1 H{hpsor 515 Qxp odqŒdphqwr gh xp gdgr ruglqäulr whp0vh l ’ ic 2c �c ec Dc Sj � H{hpsor 516 R odqŒdphqwr gh grlv gdgrv whp xp hvsdŒr gh uhvxowdgrv frp 69 hohphqwrv/ l ’ iE�c �� G �c � ’ c 2c �c ec Dc Sj � D lpsruwåqfld gd gh�qlŒær gh hvsdŒr gh uhvxowdgrv dgyìp vreuhwxgr sru vhu r phlr hpsuhjxh sdud d gh�qlŒær gh dfrqwhflphqwrv1 Gh�qlŒær 515 Rv vxefrqmxqwrv gh l ghvljqdp0vh sru dfrqwhflphqwrv> rv vxefrqmxqwrv irupdgrv sru xp ýqlfr hohphqwr fkdpdp0vh dfrqwhflphqwrv hohphqwduhv1 H{hpsor 517 Qr odqŒdphqwr gh xp gdgr kä vhlv dfrqwhflphqwrv hohphqwduhv srlv l ’ ic 2c �c ec Dc Sj ( srghprv qr hqwdqwr gh�qlu xpd vìulh gh rxwurv dfrqwhflphqwrv frpr sru h{hpsor � ’ ic �c Dj � vdðgd gh xp qýphur ðpsdu/ rx � ’ i2c ec Sj � vdðgd gh xp qýphur sdu� 5141 SUREDELOLGDGHV 8 Gh�qlgrv frpr frqmxqwrv/ drv dfrqwhflphqwrv ì dsolfäyho wrgd d frqv0 wuxŒær glvsrqðyho sdud dtxhohv/ lvwr ì/ h{lvwh xp sdudoholvpr shuihlwr hqwuh äojheud gh frqmxqwrv h äojheud gh dfrqwhflphqwrv1 Gh�qlŒær 516 R dfrqwhflphqwr txh frqwhp wrgrv rv hohphqwrv gh l fkdpd� vh dfrqwhflphqwr fhuwr1 Gh�qlŒær 517 R dfrqwhflphqwr txh qær frqwhp dojxp hohphqwrv gh l fkd0 pd�vh dfrqwhflphqwr lpsrvvðyho1 Gh�qlŒær 518 Grlv dfrqwhflphqwrv � h � vær pxwxdphqwh h{foxvlyrv rx lqfrpsdwðyhlv rx glvmxqwrv vh qær wíp hp frpxp txdotxhu dfrqwhflphqwr gh l� Ω A B Dfrqwhflphqwrv pxwxdphqwh h{foxvlyrv1 Gh�qlŒær 519 D xqlær grv dfrqwhflphqwrv � h � ì r dfrqwhflphqwr txh vh uhdol}d vh h vö vh � rx � vh uhdol}dp1 Uhsuhvhqwd0vh sru � ^ � rx �n� h ì irupdgr shorv hohphqwrv txh shuwhqfhp d � rx d �� Gh�qlŒær 51: D lqwhuvhfŒær grv dfrqwhflphqwrv � h � ì r dfrqwhflphqwr txh vh uhdol}d vh h vö vh � h � vh uhdol}dp frqmxqwdphqwh1 Uhsuhvhqwd0vh sru � _ � rx �� h ì irupdgr shorv hohphqwrv txh frpxqv d � h d �� 9 FDS—WXOR 51 SUREDELOLGDGHV Rv gldjudpdv gh Yhqq txh vh suhvhqwdp d vhjxlu loxvwudp dv gh�qlŒłhv txh dfdedudp gh vhu dsuhvhqwdgdv1 Ω A B Ω A B Uhxqlær gh dfrqwhflphqwrv/ � ^�c h lqwhuvhfŒær gh dfrqwhflphqwrv/ � _ �1 Qd olqjxdjhp fruuhqwh d qrŒær gh suredelolgdgh ì xwlol}dgd vhp txdotxhu gh�qlŒær suhflvd gr vhx vljql�fdgr> frqwxgr/ d whruld gdv suredelolgdghv/ vhqgr xpd glvflsolqd gd pdwhpäwlfd/ ghyh wruqdu hvwd qrŒær suhflvd1 D sulphlud gh�qlŒdr gh suredelolgdgh irl hvwdehohflgd sru Odsodfh hp 4;451 Gh�qlŒær 51; +Ohl gh Odsodfh rx Suredelolgdgh ª sulrul, 0 Vh xpd h{shulíqfld dohdwöuld srgh whu Q uhvxowdgrv pxwxdphqwh h{foxvlyrv h ljxdo0 phqwh srvvðyhlv/ h vh ghvvhv uhvxowdgrv/ q wíp xp fhuwr dwulexwr D/ hqwær d suredelolgdgh gh D ì gdgd sru ? � 1 Kdelwxdophqwh hvfuhyh0vh/ � E�� ’ ? � ’ qJ gh uhvxowdgrv idyruäyhlv qJ gh fdvrv srvvðyhlv � H{hpsor 518 D suredelolgdgh gh qr odqŒdphqwr gh xpd prhgd vdlu fdud ì 2 � +Ì ljxdo ª suredelolgdgh gh vdlu frurd,1 H{hpsor 519 Qr odqŒdphqwr gh gxdv prhgdv srghprv rewhu 5 fdudv/ 5 frurdv rx 4 fdud h 4 frurd> pdv hvwhv uhvxowdgrv qær vær ljxdophqwh suryäyhlv> d vdðgd gh 4 fdud h 4 frurd srgh dfrqwhfhu gh gxdv pdqhludv glihuhqwhv= srgh vdlu sulphlur fdud qxpd gdv prhgdv h frurd qd rxwud/ rx dr frqwuäulr1 Dvvrfldqgr �I� ª vdðgd gh fdud h �F� ª vdðgd gh frurd/ r hvsdŒr gh uhvxowdgrv vhuä l ’ iE8c8 � c E8c�� c E�c8 � c E�c��j � Hqwær d suredelolgdgh gh rewhu 5 fdudv ì e h d gh rewhu 4 fdud h 4 frurd ì 2 e ’ 2 1 Xpd gdv sulqflsdlv fuðwlfdv d hvwd gh�qlŒær ì d gh txh hod vö ì dsolfäyho txdqgr r hvsdŒr gh uhvxowdgrv ì �qlwr h rv vhxv hohphqwrv srvvxhp ljxdo suredelolgdgh> pdv whqkd0vh hp dwhqŒær txh hod vxujh txdqgr dv sulphludv dsolfdŒłhv gr fäofxor gdv suredelolgdghv hvwær pxlwr oljdgdv drv �mrjrv gh 5141 SUREDELOLGDGHV : d}du� txh frpr vh vdeh/ srvvxhp hvvdv sursulhgdghv1 ˆ sulrul frqkhfhp0vh wrgrv rv uhvxowdgrv srvvðyhlv txh vær xp qýphur �qlwr gh fdvrv pxwxdphqwh h{foxvlyrv h ljxdophqwh srvvðyhlv1 Ì r txh dfrqwhfh frp dv gxdv idfhv gh xpd prhgd/ dv 85 fduwdv gh xp edudokr/dv vhlv idfhv gh xp gdgr/ hwf1 Nroprjrury/ hp 4<66/ irl xp grv judqghv frqwulexlqwhv sdud d d{lrpd0 wl}dŒær gd whruld gdv suredelolgdghv/ lvwr ì/ d frqvwuxŒær gh xp vlvwhpd gh d{lrpdv txh irupdol}dp dojxpdv sursulhgdghv eävlfdv gd suredelolgdgh/ gh txh frphŒrx d vhqwlu d qhfhvvlgdgh qr lqðflr gr vìfxor [[/ Gh�qlŒær 51< IxqŒær gh Suredelolgdgh ì xpd ixqŒær gh frqmxqwrv/ fx0 mr grpðqlr ì r frqmxqwr gdv sduwhv gr hvsdŒr gh uhvxowdgrv lc h whp sru frqwudgrpðqlr r lqwhuydor dfc o h vdwlvid} rv vhjxlqwhv d{lrpdv= � E�� � f � El� ’ � E� ^ �� ’ � E�� n � E��c vh � _� ’ > Uhsduh0vh txh hvwhv d{lrpdv vxujhp qdwxudophqwh gr txh mä irl glwr vreuh h{shulíqfldv dohdwöuldv1 D uhjxodulgdgh revhuydgd gd iuhtxíqfld gh dfrqwhfl0 phqwrv dohdwöulrv h r idfwr gh txh d iuhtxíqfld vhu xpd iudfŒær qær qhjdwlyd/ phqru rx ljxdo d xp/ ohydp d dfhlwdu r 4J d{lrpd1 Whqgr hp frqwd r h{hpsor vhjxlqwh/ r 5J d{lrpd wdpehp vxujh qdwxudo0 phqwh1 H{hpsor 51: Vxsrqkd0vh txh h{lvwh xpd ýqlfd erod suhwd qxpd xuqd1 Frq0 vlghuh0vh xpd h{shulíqfld dohdwöuld txh frqvlvwh qd h{wudfŒær gh xpd erod gd xuqd1 Vh 6*? uhsuhvhqwdu d iuhtxíqfld gd vdðgd gd erod suhwd/ ì öeylr txh qhvwh h{hpsor vh whuä vhpsuh 6*? ’ � Rud/ d h{wudfŒær gd erod suhwd ì qhvwh fdvr r dfrqwhflphqwr fhuwr h d vxd iuhtxíqfld ì 41 Ilqdophqwh/ vh vh wlyhu hp frqwd txh qr odqŒdphqwr gh xp gdgr/ r dsduh0 flphqwr gd idfh 9 whp xpd iuhtxíqfld txh rvflod shuwr gh *Sc r dsduhflphqwr gd idfh 5 whp wdpehp xpd iuhtxíqfld txh rvflod shuwr gh *S/ txh hvwhv grlv dfrqwhflphqwrv vær pxwxdphqwh h{foxvlyrv/ h txh r dfrqwhflphqwr �vdðgd gd idfh 9 rx vdðgd gd idfh 5� whp xpd iuhtxíqfld txh rvflod hp yrowd gh *S n *S ’ � hvwä fodur r 6J d{lrpd1 Frp edvh qhvwhv d{lrpdv srghp djrud ghprqvwudu0vh xpd vìulh gh lp0 sruwdqwhv sursulhgdghv gd suredelolgdgh/ gdv txdlv ghvwdfdprv= SursrvlŒær 514 � E>� ’ f� ; FDS—WXOR 51 SUREDELOLGDGHV GhprqvwudŒær1 Frpr l_> ’ >c r txh vljql�fd txh rv dfrqwhflphqwrv vær pxwxdphqwh h{foxvlyrv/ srghprv hvfuhyhu � El ^ >� ’ � El� n � E>�� Rud l ^ > ’ l h sruwdqwr/ � El ^ >� ’ � El� ’ �Hqwær/ ’ n � E>�, � E>� ’ f� SursrvlŒær 515 � E� ��� ’ � E��� � E� _�� � GhprqvwudŒær1 �_� h ��_� vær dfrqwhflphqwrv pxwxdphqwh h{foxvlyrv srlv E� _ �� _ ��� _�� ’ �� _ ��� _ E� _�� ’ >� Sru rxwur odgr E� _ �� ^ ��� _�� ’ �� ^ ��� _� ’ l _ � ’ �� Hqwær/ � E�� ’ � E� _ �� n � ��� _ �� ( pdv frpr/ � � �� _ �� ’ � E� � ��c whp0vh r uhvxowdgr suhwhqglgr1 SursrvlŒær 516 � E�� ’ � � ���� � GhprqvwudŒær1 l ’ � ^ �� h � h �� vær dfrqwhflphqwrv pxwxdphqwh h{foxvlyrv srlv � _ �� ’ >� Hqwær/ � El� ’ ’ � E� ^ ��� ’ � E�� n � E���, � E�� ’ � � E���� SursrvlŒær 517 � E� ^�� ’ � E�� n � E��� � E� _�� � GhprqvwudŒær1 Frpr �^� ’ �^ E���� ’ �^ E�� _�� h rv dfrqwhfl0 phqwrv � h �� _ � vær pxwxdphqwh h{foxvlyrv/ shor d{lrpd 6 h dwhqghqgr ª sursrvlŒær 515 whp0vh r uhvxowdgr suhwhqglgr1 H{hpsor 51; Hp ghwhuplqdgd srsxodŒær </;( gdv shvvrdv dgtxluhp d uh0 ylvwd D/ 55/<( d uhylvwd E/ h 8/4( dpedv1 Txdo d suredelolgdgh gh +d, xpd shvvrd dgtxlulu shor phqrv xpd gdv uhylvwdvB +e, xpd shvvrd dgtxlulu vrphqwh d uhylvwd DB +f, xpd shvvrd qær dgttxlulu qhp d uhylvwd D qhp d uhylvwd EB 5141 SUREDELOLGDGHV < H{hpsor 51< Ghvljqdqgr sru D r dfrqwhflphqwr �frpsud gd uhylvwd D� h sru E r dfrqwhflphqwr �frpsud gd uhylvwd E� d suredelolgdgh gh xpd shvvrd dgtxlulu shor phqrv xpd gdv uhylvwdv srgh vhu uhsuhvhqwdgr sru � E� ^�� ’ � E�� n � E��� � E� _ �� ’ f�fbH n f�22b� f�fD ’ f�2.S� D suredelolgdgh gh xpd shvvrd dgtxlulu vrphqwh d uhylvwd D ì d suredelo0 lgdgh gh xpd shvvrd dgtxlulu d uhylvwd D h qær dgtxlulu d uhylvwdE/ rx vhmd/ � E� _��� ’ � E��� � E� _ �� ’ f�fbH� f�fD ’ f�fe.� Xpd shvvrd qær dgtxlulu qhqkxpd gdv uhylvwdv fruuhvsrqgh d qær dgtxlulu d uhylvwd D h d qær dgtxlulu d uhylvwd E> orjr � E�� _ ��� ’ � � E� ^��� � ’ � � E� ^ �� ’ � f�2.S ’ f�.2e� Qrwd 515 Dsöv d dsuhvhqwdŒær ghvwdv gxdv gh�qlŒłhv gh suredelolgdgh/ frq0 yìp dlqgd uhihulu/ txh qxpd rxwud shuvshfwlyd/ d gd fkdpdgd whruld iuh0 txhqflvwd/ d suredelolgdgh gh xp dfrqwhflphqwr ì gh�qlgd frpr vhqgr r ydoru sdud r txdo whqgh d iuhtxíqfld uhodwlyd gr dfrqwhflphqwr txdqgr r qýphur gh uhshwlŒłhv gd h{shulíqfld dxphqwd1 Ydprv djrud lqwurgx}lu xp qryr frqfhlwr/ r gh suredelolgdgh frqglflrqdo1 Frqvlghuhprv hqwær r vhjxlqwh h{hpsor= H{hpsor 5143 Xp juxsr gh shvvrdv ì fodvvl�fdgr gh dfrugr frp r vhx shvr h d lqflgíqfld gh klshuwhqvær1 Dv sursruŒłhv gdv yäuldv fdwhjruldv dsduhfhp qr txdgur vhjxlqwh= REHVR QRUPDO PDJUR Wrwdo KLSHUWHQVR 3/43 3/3; 3/35 3/53 Q˘R KLSHUWHQVR 3/48 3/78 3/53 3/;3 Wrwdo 3/58 3/86 3/55 4/33 1 +d, Txdo d suredelolgdgh gh xpd shvvrd hvfroklgd dr dfdvr vhu klshuwhq0 vdB +e, Txdo d suredelolgdgh gh xpd shvvrd rehvd vhu klshuwhqvdB H{hpsor 5144 D uhvsrvwd ª sulphlud txhvwær ì lphgldwd1 Edvwd yhu txdo d sursruŒær gh klshuwhqvrv ghqwur gd srsxodŒær txh ì gh 31531 Sdud uhvsrqghu ª vhjxqgd shujxqwd kä txh wrpdu hp dwhqŒær txh r txh vh suhwhqgh ì d sursruŒær gh klshuwhqvrv gh hqwuh d srsxodŒær gh rehvrv/ lvwr ì/ f�f f�2D ’ f� e� Sru rxwudv sdodyudv/ txhu0vh fdofxodu d suredelolgdgh gr dfrqwhflphqwr �vhu klshuwhqvr�/ vdehqgr txh rfruuhx r dfrqwhflphqwr �vhu rehvr�1 Uhsduh0vh txh hvwh txrflhqwh uhvxowd gd glylvær hqwuh d suredelolgdgh gh xpd shvvrd vhu klshuwhqvd h rehvd h d suredelolgdgh gh xpd shvvrd vhu 43 FDS—WXOR 51 SUREDELOLGDGHV rehvd1 Ghvljqdqgr sru R r dfrqwhflphqwr �vhu rehvr� h sru K r dfrqwhfl0 phqwr �vhu klshuwhqvr� srghprv hvfuhyhu vlperolfdphqwh txh d suredelolgdgh suhwhqglgd ì gdgd sru � EM*�� ’ � EM _ �� � E�� � Hvwh h{hpsor fruuhvsrqgh dr fäofxor gh xpd suredelolgdgh frqglflrqdo fxmd gh�qlŒær irupdol}dprv d vhjxlu1 Gh�qlŒær 5143 D suredelolgdgh frqglflrqdo gh D gdgr E/ � E� m �� ì gh�qlgd shod iöupxod � E� m �� ’ � E� _ �� � E�� � vhpsuh txh � E�� : f( rx/ htxlydohqwhphqwh � E� _�� ’ � E�� �� E� m �� � Hvwd gh�qlŒær jhqhudol}d0vh idflophqwh sdud xp qýphur �qlwr gh dfrqwh0 flphqwrv � E� _� 2 _ � � � _ � ? � ’ ’ � E� � �� E� 2 m � � �� E� � m � _ � 2 � � � � � �� E� ? m � _ � 2 _ � � � _� ?3 � � Xpd lpsruwdqwh lghld uhodflrqdgd frp d suredelolgdgh frqglflrqdo ì r frqfhlwr gh lqghshqgíqfld hqwuh dfrqwhflphqwrv1 Gh�qlŒær 5144 D h E gl}hp0vh dfrqwhflphqwrv lqghshqghqwhv vh h vö vh � E� _�� ’ � E�� �� E�� c rx htxlydohqwhphqwh/ � E� m �� ’ � E�� h � E� m �� ’ � E�� � Hvwd ýowlpd uhodŒær hylghqfld r vljql�fdgr gh lqghshqgíqfld1 R frqkh0 flphqwr gh txh � rfruuhx qær lq xhqfld d suredelolgdgh gh txh � rfruud h r frqkhflphqwr gh txh � rfruuhx qær lq xhqfld d suredelolgdgh gh txh � rfruud1 H{hpsor 5145 Glvsłh0vh gh gxdv xuqdv/ d xuqd D frp 5 erodv eudqfdv h 6 erodv suhwdv/ h d xuqd E frp 7 erodv suhwdv h 43 eudqfdv> wudqvihuh0vh/ vhp d h{dplqdu/ xpd erod gd xuqd D sdud d xuqd E h hp vhjxlgd h{wudl0vh ghvwd xpd erod1 Txdo d suredelolgdgh gh d erod h{wudðgd gd xuqd E vhu eudqfdB Frqvlghudqgr rv dfrqwhflphqwrv 5141 SUREDELOLGDGHV 44 D 0 D erod wudqvihulgd gh D sdud E ì eudqfd/ D 2 0 D erod wudqvihulgd gh D sdud E ì suhwd h F 0 D erod h{wudðgd gh E ì eudqfd/ r txh vh suhwhqgh fdofxodu ì � E�� ’ � E�m� ��� E� �n� E�m� 2 ��� E� 2 � ’ D � 2 D n f D � � D ’ f�Sb���� Whruhpd 514 +gh Ed|hv, Vh i� c � 2 c � � �c � ? j ì xpd sduwlŒær gr hvsdŒr gh uhvxowdgrv l gh xpd h{shulíqfld dohdwöuld/ frp � E� � � : fc sdud � ’ c 2c � � �c ?c gdgr txdotxhu dfrqwhflphqwr E/ wdo txh � E�� : fc hqwær � E� � m �� ’ � E��� �� E� m ��� ?S �' � E� � � �� E� m � � � � GhprqvwudŒær1 Gd gh�qlŒær gh suredelolgdgh frqglflrqdo srghprv hvfuh0 yhu txh � E� � m �� ’ � E�� _ �� � E�� � +514, Sru rxwur odgr/ whp0vh txh � E� m � � � ’ � E� _� � � � E� � � , � E� _� � � ’ � E� m � � � �� E� � � � Pdv/ frpr � E� _ � � � ’ � E� � _�� c frqfoxl0vh txh � E� � _�� ’ � E� m � � � �� E� � � � +515, � E�� ’ � E� _ l� ’ � � � _ � ?^ �' � � �� ’ ’ � � ?^ �' E� _� � � � ’ ?[ �' � E� _ � � � ’ ?[ �' � E� m � � � �� E� � � � +516, Vxevwlwxlqgr r qxphudgru h r ghqrplqdgru gr 5J phpeur gh 514 sru 515 h 516/ uhvshfwlydphqwh/ rewhp0vh r uhvxowdgr suhwhqglgr1 Hvwh whruhpd srgh wdpeìp vhu lqwhusuhwdgr gd vhjxlqwh irupd= vhmd � xp dfrqwhflphqwr txh vh uhdol}d vh h vö vh xp grv dfrqwhflphqwrv pxwxdphqwh h{foxvlyrv � c � 2 c ���c � ? vh yhul�fd1 Drv dfrqwhflphqwrv � � gä0vh sru yh}hv r qrph gh �fdxvdv�1 D iöupxod gh Ed|hv gä hqwær d suredelolgdgh gh txh r dfrqwhflphqwr � txh vh ghx ì r uhvxowdgr gd fdxvd �� D suredelolgdgh � E� � � wrpd r qrph gh suredelolgdgh ª sulrul gd fdxvd � � � � E� � m�� ì d suredelolgdgh ª srvwhulrul/ lvwr ì/ d suredelolgdgh gh � � fdofxodgd vre d klsöwhvh gh txh � vh uhdol}rx1 45 FDS—WXOR 51 SUREDELOLGDGHV H{hpsor 5146 Frqvlghuh 6 xuqdv qdv vhjxlqwhv frqglŒłhv Erodv Eudqfdv Erodv Suhwdv Xuqd 4 2 e Xuqd 5 e 2 Xuqd 6 � � Ghvljqdqgr sru R �c � ’ c 2c �c d suredelolgdgh gh hvfrokd gd xuqd �c vxsrqkd txh R ’ R 2 ’ 2R � � Hvfroklgd xpd erod dr dfdvr fdofxoh d suredelolgdgh gh txh hvwd suryhqkd gd 4@ xuqd vdehqgr txh +d, d erod ì eudqfd> +e, d erod ì suhwd1 H{hpsor 5147 Vhmd � r dfrqwhflphqwr �vdðgd gd erod eudqfd� +�� ì r dfrqwhflphqwr �vdðgd gd erod suhwd�, h � � c � ’ c 2c �c r dfrqwhflphqwr �hv0 frokd gd xuqd ��1 Gdv uhodŒłhv R n R 2 n R � ’ h R ’ R 2 ’ 2R � c frqfoxl0vh txh R ’ � E� � ’ � E� 2 � ’ 2*D h R � ’ � E� � � ’ *D� Xwlol}dqgr d iöupxod gh Ed|hv yhp= � E� m �� ’ � E�� �� E� m �� �S �' � E� � �� E� m � � ’ 2 D � � 2 D � � � n 2 � � n D � 2 ’ e D h � � � m ��� ’ � E�� �� ��� m �� �S �' � E� � �� E�� m � � ’ 2 D �2 � 2 D � � 2 � n � � n D � 2 ’ H D � 5151 H[HUF—FLRV 46 515 H{hufðflrv H{hufðflr 514 Qxp orwh gh 45 oåpsdgdv kä 5 eudqfdv h 43 d}xlv1 Txdo d suredelolgdgh gh qxpd dprvwud gh ; oåpsdgdv kdyhu shor phqrv 4 eudqfdB+V= 31<4, H{hufðflr 515 OdqŒdp0vh grlv gdgrv/ xp eudqfr h rxwur yhuphokr h uhsuh0 vhqwd0vh r uhvxowdgr sru xp sdu rughqdgr E�c T �1 Txdo d suredelolgdgh gd vrpd � n T vhu= +d, Sdu> +V= 2 � +e, Glylvlyho sru wuív> +V= � � +f, Sdu rx glylvlyho sru 61 +V= 2 � � H{hufðflr 516 Grlv mrjdgruhv frpelqdp xp wruqhlr gh 7 sduwlgdv1 Vxsrq0 gr txh �ª sulrul� d suredelolgdgh gh D jdqkdu xpd gdv sduwlgdv ì 2 h d suredelolgdgh gh E jdqkdu xpd gdv sduwlgdv ì � H / ghwhuplqh= +d, D suredelolgdgh gh D jdqkdu dv 7 sduwlgdv> +V= S , +e, D suredelolgdgh gh D h E jdqkduhp r phvpr qýphur gh sduwlgdv +frqvlghuh d vlwxdŒær gh fdgd xp jdqkdu 3 sduwlgdv,> +V= 31579, +f, D suredelolgdgh gh D jdqkdu pdlv sduwlgdv txh E1 +V= 317;, H{hufðflr 517 Vdeh0vh txh hp uhodŒær d xp gdgr surjudpd gh whohylvær vh whp sdud xp fdvdo/ d vhjxlqwh vlwxdŒær= d suredelolgdgh gh r krphp yhu xp surjudpd ì 3/7> d suredelolgdgh gd pxokhu yhu r phvpr surjudpd ì 3/8> d suredelolgdgh gr krphp yhu r surjudpd srutxh d pxokhu r yí ì 3/:1 Ghwhuplqh d suredelolgdgh= +d, Gh xp fdvdo yhu r surjudpd> +V= 3168, +e, Gh d pxokhu yhu r surjudpd srutxh r krphp r yí> +V= 31;:8, +f, Gh shor phqrv xp hohphqwr gr fdvdo yhu r surjudpd1+V= 3188, H{hufðflr 518 Gdgrv grlv dfrqwhflphqwrv� h�/ wdlv txh � E�� ’ e c � E�� ’ � h � E� ^ �� ’ 2 / ghwhuplqh= +d, Vh � h � vær pxwxdphqwh h{foxvlyrv> +V= qær, +e, Vh � h � vær lqghshqghqwhv> +V= vlp, +f, � E�*��c � E 7�*��c � E 7�*��c � E 7�* 7��� +V= e c � e c 2 � c � e � 47 FDS—WXOR 51 SUREDELOLGDGHV H{hufðflr 519 Gxdv pätxlqdv D h E ixqflrqdp gh irupd lqghshqghqwh xpd gd rxwud1 Qr txdgur vhjxlqwh/ lqglfdp0vh dv suredelolgdghv gh vh yhul�fdu r qýphur uhihulgr gh dyduldv sdud fdgd pätxlqd/ qr ghfxuvr gh xp gld gh wudedokr= qJ dyduldv 3 4 5 6 7 8 9 pätxlqd D 3/4 3/5 3/6 3/5 3/3< 3/3: 3/37 pätxlqd E 3/6 3/4 3/4 3/4 3/4 3/48 3/48 Fdofxoh dv vhjxlqwhv suredelolgdghv= +d, D h E wíp r phvpr qýphur gh dyduldv> +V= 314588, +e, R qýphur wrwdo gh dyduldv ì lqihulru d 6> +V= 3155, +f, D whp pdlv dyduldv txh E> +V= 3174:, +g, E whp r greur gdv dyduldv gh D> +V= 313;, +h, E whp 7 dyduldv/ vdehqgr0vh txh E whp shor phqrv gxdv dyduldv> +V= 314+9,, +i, R qýphur pðqlpr gh dyduldv hp fdgd xpd gdv gxdv pätxlqdv ì 61 +V= 315, H{hufðflr 51: Xp frphufldqwh uhfheh edwdwd gh wuív uhjlłhv glihuhqwhv1 Dr fkhjdu fdgd orwh ì fodvvl�fdgr hp gxdv fodvvhv/ D h E/ gh dfrugr frp d txdol0 gdgh gr surgxwr1 Vdehqgr txh xp orwh dfdedgr gh fkhjdu irl fodvvl�fdgr qd fdwhjruld D/ txdo d suryhqlíqfld pdlv suryäyho ghvwh orwh/ whqgr hp dwhqŒær txh d glvwulexlŒær grv orwhv uhfhelgrv dwì ª gdwd ì d vhjxlqwh= Uhjlłhv Wrqhodgdv Txdolgdgh D Txdolgdgh E L 43 5 ; LL 53 47 9 LLL 53 43 43 +V= Uhjlær LL, H{hufðflr 51; Xpd gdgd grhqŒd dsduhfh hp fhufd gh xpd shvvrd sru fdgd 4333 qxpd gdgd srsxodŒær1 Słh0vh hp ixqflrqdphqwr xp surjudpd sdud dqdolvdu d suhvhqŒd gd grhqŒd/ vhqgr sdud wdo xvdgr xp dsduhokr gh gh0 whfŒær txh gä whvwh srvlwlyr frp suredelolgdgh gh 3/<< sdud xpd shvvrd frp d grhqŒd/ h 3/38 sdud xpd shvvrd vhp d glwd1 Suhwhqgh0vh vdehu txdo ì d suredelolgdgh gh txh xpd shvvrd frp whvwh srvlwlyr whqkd uhdophqwh d grhqŒd1 +V= 3134<7, 5151 H[HUF—FLRV 48 H{hufðflr 51< Dv pätxlqdv D h E surgx}hp uhvshfwlydphqwh 43( h <3( gd surgxŒær wrwdo gh xp fhuwr wlsr gh duwljrv1 Vxsrqgr txh d suredelolgdgh gd pätxlqd D surgx}lu duwljrv ghihlwxrvrv ì gh 4( h txh sdud d pätxlqd E ì gh 8(/ txdo d suredelolgdgh gh xp duwljr wrpdgr dr dfdvr gh xp gld gh surgxŒær suryhqkd gd pätxlqd D/ dvvxplqgr txh ì ghihlwxrvrB +V= 31354:, H{hufðflr 5143 Xpd hpsuhvd srvvxl 6 phlrv gh wudqvsruwh/ W c W 2 c h W � / sdud frorfdu rv vhxv surgxwrv qr phufdgr1 Vdeh0vh txh dv suredelolgdghv ghvwhv wudqvsruwhv dwudvduhp vær/ uhvshfwlydphqwh/ 3138/ 314/ h 31581 Vdeh0vh dlqgd txh hvwhv phlrv gh wudqvsruwh vær hpsuhjxhv frp iuhtxíqfldv lqyhuvd0 phqwh sursruflrqdlv drv uhvshfwlyrv fxvwrv sru uhphvvdv= ;3/ 73 h 49 xqlgdghv prqhwäuldv uhvshfwlydphqwh1 +d, Fdofxoh dv shufhqwdjhqv gh xwlol}dŒær gh fdgd xp grv phlrv gh wudqvsruwh> +V= 2 . c D . c f . , +e, Ghwhuplqh d suredelolgdgh gh kdyhu dwudvr> +V= 314;5, +f, Frqvlghudqgr dv uhphvvdv fkhjdgdv frp dwudvr/ txdo d suredelol0 gdgh gdv phvpdv whuhp vlgr wudqvsruwdgdv sru W 2 q +V= 31494, H{hufðflr 5144 Dsöv dojxqv whvwhv hihfwxdgrv ª shuvrqdolgdgh gh xp lq0 glylgxr frqfoxlx0vh txh/ hvwh ì orxfr frp xpd suredelolgdgh ljxdo d 319/ odguær frp xpd suredelolgdgh ljxdo d 31: h qær ì orxfr qhp odguær frp xpd suredelolgdgh gh 3158 +d, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu orxfr h odguær> +V= 3188, +e, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu dshqdv orxfr rx dsh0 qdv odguær> +V= 315, +f, Ghwhuplqh d suredelolgdgh gr lqglylgxr vhu odguær/ vdehqgr txh r phvpr qær ì orxfr1 +V= 316:8, H{hufðflr 5145 R Mrær whp ª vxd glvsrvlŒær wuív phlrv gh wudqvsruwh glih0 uhqwhv sdud vh ghvorfdu gh fdvd sdud d hvfrod= r wudqvsruwh D/ E rx F1 Vdeh0vh txh d suredelolgdgh gh= 0 fkhjdu dwudvdgr ª hvfrod ì 319> 0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh D ì 31;> 0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh E ì 318> 0 fkhjdu dwudvdgr xwlol}dqgr r wudqvsruwh F ì 319> 0 xwlol}du rv wudqvsruwhv E h F ì d phvpd1 49 FDS—WXOR 51 SUREDELOLGDGHV +d, Fdofxoh d suredelolgdgh gr Mrær xwlol}du r wudqvsruwh D> +V= 315, +e, Vdehqgr txh r Mrær fkhjrx dwudvdgr ª hvfrod/ fdofxoh d suredelo0 lgdgh gh whu xwlol}dgr r wudqvsruwh F1 +V= 317, H{hufðflr 5146 Vhmdp D h E grlv dfrqwhflphqwrv lqghshqghqwhv/ suryh txh r frpsohphqwdu gh D h r frpsohphqwdu gh E wdpeìp r vær1 H{hufðflr 5147 Qxpd srsxodŒær 53( gdv idpðoldv wíp pätxlqd gh odydu orxŒd/ 63( wíp pätxlqd gh odydu urxsd h 43( wíp dperv rv wlsrv gh pätxlqdv1 Fdofxoh d suredelolgdgh gh xpd idpðold hvfroklgd dr dfdvr= +d, Whu shor phqrv xp grv wlsrv gh pätxlqd> +V= 317, +e, Qær whu qhqkxp grv wlsrv gh pätxlqd> +V= 319, +f, Whu xp vö wlsr gh pätxlqd1 +V= 316, H{hufðflr 5148 D suredelolgdgh gh vh rewhu fdgd srqwr qxp gdgr ylfldgr ì gh= % 2 � e D S � E%� f� f�.D f�.D f�.D f�.D f�2 OdqŒd0vh r gdgr gxdv yh}hv frqvhfxwlyd h dglflrqdp0vh rv srqwrv rewlgrv1 Ghwhuplqh d suredelolgdgh gh fdgd xp grv dfrqwhflphqwrv= +d, Rewhu xpd vrpd gh srqwrv vxshulru d gh}> +V= 3144, +e, Rewhu xpd vrpd gh srqwrv lqihulru d txdwur1 +V= 31378, H{hufðflr 5149 Frqvlghuh gxdv xuqdv D h E1 D xuqd D frqwìp 4 erod eudqfd h <<< erodv suhwdv h d xuqd E frqwìp 4 erod suhwd h <<< erodv eudqfdv1 Ì hvfroklgd xpd xuqd dr dfdvr gd txdo ì h{wudðgd xpd erod1 Vh hvwd ì suhwd/ txdo d suredelolgdgh gh txh d xuqd D whqkd vlgr hvfroklgdB +V= 31<<<, H{hufðflr 514: Xp vdfr whp 6 prhgdv xpd gdv txdlv irl fxqkdgd frp 5 fdudv hqtxdqwr txh dv rxwudv r irudp frp 4 fdud h 4 frurd1 Wlud0vh xpd prhgd dr dfdvr gr vdfr h odqŒd0vh dr du 5 yh}hv vhjxlgdv1 Txdo d suredelol0 gdgh gh txh vh wudwh gd prhgd gh 5 fdudv vdehqgr txh qrv grlv odqŒdphqwrv vdlx vhpsuh fdudB +V= 2 � , H{hufðflr 514; Qxpd dprvwud frqvwlwxðgd sru 433 lqglylgxrv rewlyhudp0vh rv uhvxowdgrv dsuhvhqwdgrv qr txdgur vhjxlqwh= Frp Eurqtxlwh Vhp Eurqtxlwh Ixpdgruhv 73 53 qær Ixpdgruhv 43 63 5151 H[HUF—FLRV 4: +d, Gljd/ mxvwl�fdqgr d uhvsrvwd/ vh rv dfrqwhflphqwrv �vhu ixpdgru� h �whu eurqtxlwh� vær lqghshqghqwhv> +V= qær, +e, Fdofxoh d suredelolgdgh gh xp lqglylgxr txh ì ixpdgru whu eurq0 txlwh1 +V= 2 � , H{hufðflr 514< Frqvlghuh txh d vlwxdŒær gh xpd hqwuhylvwd gh fdqglgdwxud d xp hpsuhjr/ ì xpd h{shulíqfld dohdwöuld h txh d qrvvd dwhqŒær vh fhqwud hp grlv dfrqwhflphqwrv �r fdqglgdwr whp erd dsduíqfld� +dfrqwhflphqwr �, h �r fdqglgdwr frqvhjxlx r hpsuhjr� +dfrqwhflphqwr �,1 Vdehqgr txh � E�� ’ f�ec � E�� ’ f�2 h � E� ^ �� ’ f�e� +d, Fdofxoh d suredelolgdgh gh r fdqglgdwr whu erd dsduíqfld h frq0 vhjxlu r hpsuhjr> +V= 315, +e, Rv dfrqwhflphqwrv �r fdqglgdwr whp erd dsduíqfld� h �r fdqglgd0 wr frqvhjxlx r hpsuhjr� vær lqghshqghqwhvB +V= qær, +f, Fdofxoh d suredelolgdgh gh r fdqglgdwr whu erd dsduíqfld vdehqgr txh frqvhjxlx r hpsuhjr1 +V= 4, H{hufðflr 5153 Qxpd iäeulfd/ yhul�frx0vh txh xp fhuwr duwljr srgh dsuh0 vhqwdu ghihlwrv gh grlv wlsrv1 D suredelolgdgh gh rfruuhu r ghihlwr gr wlsr D ì 314 h d suredelolgdgh gh rfruuhu r ghihlwr gr wlsr E ì 31381 Vdehqgr txh rv ghihlwrv rfruuhp lqghshqghqwhphqwh xp gr rxwur/ fdofxoh d suredelolgdgh gh= +d, Xp duwljr qær whu txdotxhu ghihlwr> +V= 31;88, +e, Xp duwljr whu ghihlwr> +V= 31478, +f, Xp duwljr frp ghihlwr whu xp h xp vö wlsr gh ghihlwr1 +V= 3147, H{hufðflr 5154 Vxsrqgr txh vh whp frqkhflphqwr gd vhjxlqwh lqirupdŒær= 0 d suredelolgdgh gr dfrqwhflphqwr �yhu r dqýqflr gr surgxwr D� @3168 0 d suredelolgdgh gr dfrqwhflphqwr �frpsudu r surgxwr D� @3156 0 d suredelolgdgh gr dfrqwhflphqwr �frpsudu r surgxwr D/ whqgr ylvwr r dqýqflr gr surgxwr D� @3176 +d, Fdofxoh d suredelolgdgh gh yhu r dqýqflr h frpsudu r surgxwr D> +V= 314838, +e, Fdofxoh d suredelolgdgh gh yhu r dqýqflr rx frpsudu r surgxwr D> +V= 3175<8, 4; FDS—WXOR 51 SUREDELOLGDGHV +f, Fdofxoh d suredelolgdgh gh yhu r dqýqflr gr surgxwr D vh frpsurx hvvh surgxwr> +V= 319876,+g, Rv dfrqwhflphqwrv �yhu r dqýqflr gr surgxwr D� h �frpsudu r surgxwr D� vær lqghshqghqwhvB Mxvwl�txh1 +V= qær, H{hufðflr 5155 Sdud xpd gdgd vhfŒær gh xpd hvwudgd irudp uhfroklgdv hvwdwðvwlfdv vreuh r qýphur gh yhðfxorv shvdgrv +f, h oljhlurv +t , txh hp frqmxqwr/ d fux}dp hp lqwhuydorv gh 4 plqxwr1 QJ gh Shvdgrv +f, QJ gh Oljhlurv +t , QJ gh RevhuydŒłhv f � ef f e Df f D Hf f S 2f � Hf e Hf D ef S ff 2 2 Sf 2 � Hf 2 e ef � ef � 2 Sf 6 6 63 Frqvlghuh rv vhjxlqwhv dfrqwhflphqwrv = � 0 qJ gh yhðfxorv shvdgrv phqru rx ljxdo d 4 � 0 qJ gh yhðfxorv oljhlurv phqru rx ljxdo d 6 Ghwhuplqh � E��c � E��c � E�*��c � E� ^ �� h wluh dv vxdv frqfoxvłhv vreuh d lqghshqgíqfld hvwdwðvwlfd hqwuh rv grlv dfrqwhflphqwrv1 +V= 319<> 316<> 3163::> 31<9> qær vær lqg1, H{hufðflr 5156 Gh uhjlvwrv hihfwxdgrv frqfoxl0vh txh rv frqgxwruhv txh flu0 fxodp hp ghwhuplqdgd hvwudgd srghp frphwhu xp h xp vö grv grlv wlsrv gh wudqvjuhvvłhv= wlsr L/ wlsr LL/ h qxqfd dpedv1 Gh 833 frqgxwruhv pxowdgrv 433 frphwhudp wudqvjuhvvłhv gr wlsr L1 43( grv txh frphwhudp wudqvjuhvvłhv gr wlsr L vær pxowdgrv1 4( grv frqgxwruhv frphwhp wudqvjuhvvłhv gr wlsr L h 5( gr wlsr LL1 Fdofxoh d suredelolgdgh gh vhu pxowdgr txhp frphwhu xpd wudqvjuhvvær gr wlsr LL1 +V= 315, 5151 H[HUF—FLRV 4< H{hufðflr 5157 Qxpd wxupd/ 53( grv doxqrv idodp ehp Lqjoív/ 73( idodp ehp Iudqfív h 48( grplqdp dv gxdv oðqjxdv1 +d, Dyhuljxh vh idodu Iudqfív h idodu Lqjoív vær dfrqwhflphqwrv lqgh0 shqghqwhv> +V= qær, +e, Fdofxoh d suredelolgdgh gh= l1 Xp doxqr hvfroklgr dr dfdvr idodu shor phqrv xpd gdv oðq0 jxdv> +V= 3178, ll1 Xp doxqr qær idodu qhqkxpd gdv oðqjxdv> +V= 3188, lll1 Xp doxqr idodu xpd h xpd vö gdv gxdv oðqjxdv> +V= 316, +f, Vdehqgr txh xp doxqr qær idod Lqjoív/ txdo d suredolgdgh gr phvpr idodu IudqfívB +V= 316458, H{hufðflr 5158 Frqvlghuh 6 xuqdv frp d vhjxlqwh frpsrvlŒær= d xuqd 4 frqwìp 43 erodv yhuphokdv h gxdv yhughv/ d xuqd 5 frqwìp 7 erodv yhuphokdv h 9 yhughv h d xuqd 6 frqwìp 5 erodv yhuphokdv h : yhughv1 Xpd erod ì h{wudðgd gh xpd gdv xuqdv/ vhqgr d hvfrokd gd xuqd ihlwd gr vhjxlqwh prgr= xp gdgr ì odqŒdgr xpd yh}/ 0 vh vdlu d idfh 4/ hvfrokh0vh d xuqd 4 0 vh vdluhp dv idfhv 5 rx 6/ hvfrokh0vh d xuqd 5 0 vh vdluhp dv idfhv 7/ 8 rx 9/ hvfrokh0vh d xuqd 6 +d, Txdo d suredelolgdgh gd erod h{wudðgd vhu yhughB +V= 31949, +e, Txdo d suredelolgdgh gh vh whu hvfroklgr d xuqd 4/ gdgr txh d erod h{wudðgd ì yhuphokdB +V= 316956, H{hufðflr 5159 D suredelolgdgh gh xp lqglyðgxr D hvwdu ylyr gdtxl d 63 dqrv ì 319 h d suredelolgdgh gh xp rxwur lqglylgxr E hvwdu ylyr gdtxl d 63 dqrv ì 31:1 Ghwhuplqh d suredelolgdgh gh gdtxl d 63dqrv= +d, Hvwduhp ylyrv rv lqglylgxrv D h E> +V= 3175, +e, Qær hvwduhp ylyrv rv lqglylgxrv D h E> +V= 3145, +f, Hvwdu ylyr shor phqrv xp grv lqglylgxrv1 +V= 31;;, 53 FDS—WXOR 51 SUREDELOLGDGHV Fdsðwxor 6 Yduläyhlv Dohdwöuldv h IxqŒłhv gh GlvwulexlŒær 614 Yduläyhlv Dohdwöuldv Hp pxlwdv h{shulíqfldv dohdwöuldv rv hohphqwrv gr hvsdŒr gh uhvxowdgrv/ l/ vær qýphurv uhdlv rx frqmxqwrv rughqdgrv gh qýphurv uhdlv1 Dvvlp dfrqwhfh frp r uhjlvwr gh whpshudwxudv/ gd soxylrvlgdgh/ qr odqŒdphqwr gh grlv gd0 grv/ hwf1 Pdv mä r uhvxowdgr gr odqŒdphqwr gh xpd prhgd dr du qær ì xp uhvxowdgr qxpìulfr1 Txdqgr l qær ì xp frqmxqwr qxpìulfr dwulexl0vh pxlwdv yh}hv d fdgd hohphqwr / gr hvsdŒr gh uhvxowdgrv/ xp qýphur hi@*/ dwulexlŒær hvvd txh srgh vhu phudphqwh frqyhqflrqdo1 H{hpsor 614 Qr odqŒdphqwr gh xpd prhgd dr du/ r hvsdŒr gh uhvxowdgrv ì r frqmxqwr l ’ iFdud/Frurdj> ì xvxdo qhvwh fdvr id}hu d fruuhvsrqgíqfld / [E/� Fdud 4 Frurd 3 � Sdud r phvpr hvsdŒr gh uhvxowdgrv l/ ì srvvðyho hvwdehohfhu glihuhqwhv fruuhvsrqgíqfldv/ frqvrdqwh rv remhfwlyrv hp hvwxgr1 H{hpsor 615 Frqvlghuh0vh xpd srsxodŒær gh hpsuhvdv gdv txdlv vh hvfrokh xpd dr dfdvr1 l ’ i/ // 2 /111// 6 j vhuä r hvsdŒr gh uhvxowdgrv h 6 r qýphur wrwdo gh hp0 suhvdv1 Srghprv gh�qlu yäuldv fruuhvsrqgíqfldv= 54 55FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R / $ f E/� cvhqgr f E/� r qýphur gh hpsuhjdgrv gd hpsuhvd /c / $ t E/� cvhqgr t E/� r yroxph gh yhqgdv gd hpsuhvd /c rx txdlvtxhu rxwudv/ frqiruph r remhfwlyr hp hvwxgr1 Vxsrqgr djrud txh vö hvwdprv lqwhuhvvdgrv qr hvwxgr gh xpd fdudfwhuðv0 wlfd grv hohphqwrv gh lc dvvrflhprv d fdgd hohphqwr / 5 l xp qýphur uhdo fE/�� Hvwdprv dvvlp d gh�qlu xpd ixqŒær f G l $ W+� Vhqgr D xp dfrqwhflphqwr/ fkdpd0vh lpdjhp gh D sru f/ h uhsuhvhqwd0vh sru fE��c dr frqmxqwr grv ydoruhv txh f dvvxph sdud rv hohphqwrv / gh D/ lvwr ì/ fE�� ’ ifE/� G / 5 �j � Sru rxwur odgr/ d fdgd vxefrqmxqwr . � W+c srgh id}hu0vh fruuhvsrqghu r vxefrqmxqwr f3E.� irupdgr sru wrgrv rv hohphqwrv / 5 l wdlv txh fE/� 5 .c f3E.� ’ i/ G fE/� 5 .j � D hvwh frqmxqwr f3E.� fkdpd0vh d lpdjhp lqyhuvd gh . sru f� H{hpsor 616 Qr odqŒdphqwr gh grlv gdgrv lqwhuhvvd vrphqwh/ qxp gdgr mrjr/ d vrpd grv srqwrv rewlgd1 Qhvwh fdvr/ r hvsdŒr gh uhvxowdgrv ì r frqmxqwr l ’ iE�c �� G �c � ’ c 2c �c ec Dc Sj ( gh�qd0vh d dsolfdŒær fE�c �� ’ �n �� Vhqgr � ’ iEc �c Ec 2�c E2c �j c d lpdjhp gh � sru f ì fE�� ’ i2c �j ( sdud r dfrqwhflphqwr � ’ iEec D�c EDc e�c EDc D�c ESc S�j c d lpdjhp gh � sru f ì fE�� ’ ibc fc 2j � Sdud r vxefrqmxqwr uhdo . ’ i2c �j c d lpdjhp lqyhuvd gh . sru f ì r dfrqwhflphqwrf3E. � ’ iEc �c Ec 2�c E2c �j ( vh. 2 ’ d2cn4d c f3E. 2 � ’ l h vh . � ’ ��4c 2 � c f3E. � � ’ �� Hvwdprv djrud hp frqglŒłhv gh shufhehu d gh�qlŒær gh yduläyho dohdwöuld1 Gh�qlŒær 614 Xpd ixqŒær uhdo fE/� gh�qlgd qr frqmxqwr l grv dfrqwhfl0 phqwrv hohphqwduhv/ fkdpd0vh xpd yduläyho dohdwöuld vh d lpdjhp lqyhuvd gh txdotxhu lqwhuydor U gr hl{r uhdo gd irupd o�4c %o c ì xp dfrqwhflphqwr dohdwöulr1 Qrwd 614 Xpd yduläyho dohdwöuld ì xpd ixqŒær h qær xpd yduläyho qr vhqwlgr hp 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Ee� *?o � D lqwhusuhwdŒær iuhtxhqflvwd gr frqfhlwr gh suredelolgdgh vxjhuh txh/ sdud ? judqgh/ dv iuhtxíqfldv uhodwlydv vær sudwlfdphqwh ljxdlv ª suredelolgdgh1 Lvwr vljql�fd txh dr �p gh xp judqgh qýphur gh mrjdgdv r jdqkr pìglr vhuä dsur{lpdgdphqwh ljxdo dr ydoru gd fkdpdgd hvshudqŒd pdwhpäwlfd1 Ì sru lvvr txh vh gl} txh xp mrjr ì htxlwdwlyr txdqgr r txh vh sdjd sdud qhoh sduwlflsdu ì ljxdo ª hvshudqŒd pdwhpäwlfd gr mrjdgru= dsöv xp judqgh qýphur gh sduwlgdv/ r jdqkr pìglr sru sduwlgd qær vh didvwduä pxlwr gr suhŒr sdjr sdud sduwlflsdu hp fdgd sduwlgd1 Frp txdotxhu glvwulexlŒær gh xpd yduläyho dohdwöuld hvwær vhpsuh dv0 vrfldgrv fhuwrv qýphurv fkdpdgrv rv sduåphwurv gd glvwulexlŒær/ txh ghvhpshqkdp xp uhohydqwh sdsho qd Hvwdwðvwlfd Pdwhpäwlfd1 Gxdv lpsru0 wdqwhv idpðoldv gh sduåphwurv gh xpd glvwulexlŒær vær= rv prphqwrv h rv sduåphwurv gh rughp> luhprv/ qr hqwdqwr/ dshqdv derugdu rv sulphlurv1 Gh�qlŒær 61: R prphqwr gh rughp & hp uhodŒær ª ruljhp rx pr0 phqwr ruglqäulr gh rughp & 0 & lqwhlur srvlwlyr 0 gh xpd yduläyho dohdwöuld ì r ydoru hvshudgr gd ixqŒær CEf� ’ f&clvwr ì/ > & ’ . � f& � � Vh d yduläyho dohdwöuld i÷u glvfuhwd/ . � f& � ’ [ � %& � R � ( 63FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R qr fdvr gh vhu frqwðqxd/ . � f& � ’ n"] 3" %&sE%�_%� Gh�qlŒær 61; R prphqwr gh rughp hp uhodŒær ª ruljhp gh xpd yduläyho dohdwöuld fkdpd0vh ydoru hvshudgr h uhsuhvhqwd0vh sru > rx . dfo1 Gh�qlŒær 61< R prphqwr gh rughp & hp uhodŒær ª pìgld rx pr0 phqwr fhqwudo gh rughp & 0 & lqwhlur srvlwlyr 0 gh xpd yduläyho dohdwöuld ì r ydoru hvshudgr gd ixqŒær CEf� ’ Ef � >�& clvwr ì/ . k Ef � >�& l � Vh d yduläyho dohdwöuld i÷u glvfuhwd/ . k Ef � >�& l ’ [ � E% � � >�& R � ( qr fdvr gh vhu frqwðqxd/ . k Ef � >�& l ’ n"] 3" E%� >�& sE%�_%� Gh�qlŒær 6143 R prphqwr fhqwudo gh 5@ rughp gh xpd y1 d1 f/ ì fkdpdgr ydulåqfld gh fc h uhsuhvhqwd0vh kdelwxdophqwh sru T dfo rx j2� Gh�qlŒær 6144 D udl} txdgudgd srvlwlyd gd ydulåqfld gh xpd y1 d1 fc jc fkdpd0vh r ghvylr sdguær1 R sduåphwur j2 ì xpd phglgdgh glvshuvær gd yduläyho dohdwöuld hp wruqr gr vhx ydoru hvshudgr1 Txdqwr pdlv frqfhqwudgd i÷u d glvwulexlŒær/ wdqwr phqru vhuä r ydoru gh j2� R sdsho gr ghvylr sdguær frpr xp sduåphwur txh phgh d glvshuvær gh xpd yduläyho dohdwöuld ì sduwlfxoduphqwh fodur txdqgr vh revhuyd d idprvd ghvljxdogdgh gh Fkhe|vkhy> hvwd ghvljxdogdgh rewhp0vh d sduwlu gr whr0 uhpd vhjxlqwh1 Whruhpd 614 Vh xpd yduläyho dohdwöuld f wrpd dshqdv ydoruhv qær qhjd0 wlyrv h whp ydoru hvshudgr . dfo/ hqwær sdud txdotxhu qýphur srvlwlyr gc whp0vh � Ef � g� � . dfo g � +615, 6181 YDORUHV HVSHUDGRV H SDU¯PHWURV 64 GhprqvwudŒær1 D ghprqvwudŒær vhuä ihlwd dshqdv sdud r fdvr gd y1 d1 vhu frqwðqxd +r fdvr glvfuhwr ì dqäorjr,1 Whp0vh . dfo ’ n"] f %sE%�_% � n"] g %sE%�_% � g n"] g sE%�_% ’ g� Ef � g� c grqgh/ d ghvljxdogdgh 615 vdl lphgldwdphqwh1 Whruhpd 615 +Ghvljxdogdgh gh Fkhe|vkhy,1 Vh f ì xpd yduläyho dohdwöuld frp pìgld > h ydulåqfld j2c �qlwd/ hqwær/ sdud xp txdotxhu qýphur uhdo g : fc � Emf � >m � gj� � g2 � D lpsruwåqfld ghvwd ghvljxdogdgh dgyìp gh vhu yäolgd sdud wrgd h txdo0 txhu yduläyho dohdwöuld txh whqkd xpd ydulåqfld �qlwd srghqgr hpsuhjdu0vh phvpr txdqgr qær vh frqkhfh d glvwulexlŒær gd y1d1 +yduläyho dohdwöuld,1 H{hpsor 61; Vxsrqgr txh f ì xpd y1d1 qær qhjdwlyd fxmd glvwulexlŒær ì ghvfrqkhflgd pdv vh vdeh vhu . dfo ’ 2f h j2 ’ ffc whp0vh/ sru h{hpsor/ � Emf � 2fm ef� ’ � EHf f Sf� � D S � D vhjxlu dsuhvhqwdp0vh dojxpdv sursulhgdghv gd HvshudqŒd Pdwhpäwlfd h gd Ydulåqfld fxmd ghprqvwudŒær vh ghl{d frpr h{hufðflr dr doxqr1 SursrvlŒær 61; Vh f ì xpd y1d1 h @ h K vær frqvwdqwhv uhdlv . d@f n Ko ’ @. dfo n K� SursrvlŒær 61< Vhmd f xpd y1d1 h CEf� h MEf� ixqŒłhv gh f( hqwær . dCEf� nMEf�o ’ . dCEf�o n. dMEf�o � SursrvlŒær 6143 Vh f ì xpd y1d1/ T dfo ’ . � f2 �� .2 dfo � SursrvlŒær 6144 Vh f ì xpd y1d1/ T dfo � f� SursrvlŒær 6145 Vh f ì xpd y1d1 frqvwdqwh/ lvwr ì/ vh f � @/ hqwær T dfo ’ f� SursrvlŒær 6146 Vh f ì xpd y1d1 h @ h K vær frqvwdqwhv uhdlv/ T d@f n Ko ’ @2T dfo � 65FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R 619 H{hufðflrv H{hufðflr 614 Qxp odqŒdphqwr gh 6 prhgdv htxloleudgdv vhmd f d yduläyho dohdwöuld txh uhsuhvhqwd r qýphur gh fdudv vdðgdv1 Hvfuhyd d ixqŒær gh suredelolgdgh gh f1 H{hufðflr 615 Vhmd f xpd yduläyho dohdwöuld txh uhsuhvhqwd r qýphur gh yh}hv txh d idfh 5 dsduhfh hp 8 odqŒdphqwrv gh xp gdgr1 Ghwhuplqh d ixqŒær gh suredelolgdgh gh f1 H{hufðflr 616 Txdqwdv yh}hv vh ghyh odqŒdu xp gdgr dr du sdud txh d sur0 edelolgdgh gh qær vdlu vhqd hp qhqkxp grv odqŒdphqwrv vhmd lqihulru d 3/34B +V= 59, H{hufðflr 617 Gdgr r txdgur vhjxlqwh= % � f 2 � sE% � � H e 2 H +d, Prvwuh txh s uhsuhvhqwd d ixqŒær suredelolgdgh gh xpd y1d1 f> +e, Fdofxoh . dfo c . df n 2o h T d2f n o �+ V= � H ( 2b H ( e. S � H{hufðflr 618 Frqvlghuh f xpd y1d1 frqwðqxd frp ixqŒær ghqvlgdgh sE%� ’ ;? = n % � % f /� % � f /f % /uhvwdqwhv fdvrv Ghwhuplqh d ixqŒær gh glvwulexlŒær 8 E%� h hverfh r vhx juä�fr1 H{hufðflr 619 Xpd pätxlqd id} sdudixvrv frp glåphwurv glvwulexlgrv vh0 jxqgr xpd y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelolgdgh sE%� ’ � &E%� �E%� 2� f / % 2 / fdvr frqwuäulr +d, Ghwhuplqh &> +V= 09, +e, Rv sdudixvrv vær uhmhlwdgrv vh r vhx glåphwur glihulu gh 4/8 pdlv gh 3/71 Txdo d suredelolgdgh gh xp sdudixvr vhu uhmhlwdgrB +V= 31389, H{hufðflr 61: Txdlv gdv vhjxlqwhv ixqŒłhv srghp uhsuhvhqwdu d ixqŒær ghq0 vlgdgh suredelolgdgh gh xpd y1d1= 6191 H[HUF—FLRV 66 +d, sE%� ’ � e%� 2%2 f / f % 2 / fdvr frqwuäulr +V= qær, +e, sE%� ’ � 2%3S% 2 H f / f % 2 / fdvr frqwuäulr +V= vlp, H{hufðflr 61; Frqvlghuh d vhjxlqwh ixqŒær= sE%� ’ � & % 2 f / % K / fdvr frqwuäulr Txh uhodŒær ghyh h{lvwlu hqwuh & h K sdud txh s vhmd xpd ixqŒær ghqvlgdgh gh suredelolgdgh gh xpd y1d1 f1 +V= & ’ K K3 � H{hufðflr 61< Vxsrqkd txh r juä�fr gd �jxud vhjxlqwh uhsuhvhqwd d ixqŒær ghqvlgdgh suredelolgdgh gh xpd y1d1 f1 00 -a a b (a,b) +d, Txdo d uhodŒær hqwuh @ h KB +V= KE@n K� ’ 2� +e, Vh K : f/ ghwhuplqh r ydoru gh K txdqgr @ ’ h fdofxoh/ frp hvwhv ydoruhv/ d ixqŒær gh glvwulexlŒær gd y1d1 f1 H{hufðflr 6143 Vhmd f xpd y1d1 frqwðqxd frq ixqŒær ghqvlgdgh gh sured0 elolgdgh gdgd sru sE%� ’ ;AA? AA= @% / f % � @ / % � 2 �@%n �@ / 2 % � f / fdvr frqwuäulr +d, Ghwhuplqh @> +V= 2 � +e, Ghwhuplqh 8 E%� h hverfh r vhx juä�fr> +f, Vh f c f 2 h f � iruhp 6 revhuydŒłhv lqghshqghqwhv gh f/ txdo vhuä d suredelolgdgh gh h{dfwdphqwh xp ghvvhv qýphurv vhu pdlru txh 4/8B +V= 316:8, 67FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R H{hufðflr 6144 Vhmd f xpd y1d1 frqwðqxd1 +d, Lqgltxh sdud txh ydoruhv gh q ì txh d vhjxlqwh ixqŒær srgh vhu xpd ixqŒær ghqvlgdgh gh f= +V= q : f� sE%� ’ � qe3q% f / % : f / fdvr frqwuäulr +e, Xvdqgr d ixqŒær ghqvlgdgh gh�qlgd qd doðqhd dqwhulru/ suryh txh sdud txdlvtxhu rc | : f vh yhul�fd � Ef : rn |*f : r� ’ � Ef � |� H{hufðflr 6145 D y1d1 frqwðqxd f whp ixqŒær ghqvlgdgh sE%� ’ � �%2 f / � % f / % � � rx % � f Vhmd K xp qJ uhdo wdo txh � K f� +d, Fdofxoh d ixqŒær glvwulexlŒær gh f> +e, Fdofxoh � � f : K*f K 2 � �EV= 3.K � K � nH � H{hufðflr 6146 Vhmd f xpd y1d1 frp ixqŒær ghqvlgdgh sE%� ’ � �� e %2 n S%� eD e f / � % D / fdvr frqwuäulr +d, Yhul�txh vh s srgh vhu xpd ixqŒær ghqvlgdgh gh f> +e, Ghwhuplqh d uhvshfwlyd ixqŒær gh glvwulexlŒær1 H{hufðflr 6147 Ghwhuplqh d hvshudqŒd pdwhpäwlfd h d ydulåqfld gdv y1d1 grv h{hufðflrv 618/ 6143/ 6144 h 61451 +V= 618 3> S ( 6143 � 2 ( D 2 > 6144 q ( q 2 > 6145 0� e ( � Hf � H{hufðflr 6148 Suryh dv vhjxlqwhv sursulhgdghv gd hvshudqŒd pdwhpäwlfd= +d, Vh f ’ @ hqwær . dfo ’ @( +e, . dKfo ’ K. dfo ( +f, .df n @o ’ .dfo n @( +g, Vh }Ef� h �Ef� vær ixqŒłhvghfc hqwær.d}Ef�n�Ef�o ’ .d}Ef�on .d�Ef�o/ fdvr h{lvwdp hvwdv hvshudqŒdv pdwhpäwlfdv1 6191 H[HUF—FLRV 68 H{hufðflr 6149 Suryh dv vhjxlqwhv sursulhgdghv gd ydulåqfld= +d, T dfo ’ .df2o�.2dfo( +e, T dfo � f( +f, Vh f ’ @/ hqwær T dfo ’ f( +g, T df n @o ’ T dfo( +h, T dKfo ’ K2T dfo� H{hufðflr 614: Uhodwlydphqwh ª glvwulexlŒær gd y1d1 f vdeh0vh txh .dfo ’ S h .df2o ’ �S� Vhqgr t xpd rxwud yduläyho wdo txh t ’ 2 f n�c ghwhuplqh .dt oc T dt o h j t � +V= 9> 3> 3, H{hufðflr 614; R qýphur gh fduurv yhqglgrv vhpdqdophqwh qxp vwdqg ì xpd yduläyho dohdwöuld f frp d vhjxlqwh ixqŒær gh suredelolgdgh= % � 2 � e sE% � � S S 2 S � S e +d, Fdofxoh/ mxvwl�fdqgr/ r ydoru gh S> +V= 2 2D � +e, Ghwhuplqh d ixqŒær glvwulexlŒær gh f( +f, Fdofxoh d suredelolgdgh gr qýphur gh fduurv yhqglgrv qær fkhjdu d 7/ vdehqgr txh hvwh ydoru ì vxshulru d 4> +V= f � � +g, Vh rv fxvwrv �{rv vhpdqdlv vær gh 63 frqwrv txdqgr vær yhqglgrv 5 rx phqrv fduurv h 48 frqwrv txdqgr vh yhqgh pdlv gh 5 fduurv h/ doìp glvvr/ sru fdgd fduur yhqglgr kä xp oxfur gh 68 frqwrv/ ghwhuplqh d uhfhlwd oðtxlgd vhpdqdo1 H{hufðflr 614< Wuív doxqrv h wuív doxqdv yær uhdol}du xp wudedokr gh juxsr sdud d glvflsolqd gh Hvwdwðvwlfd1 Frpr wíp gh h{lvwlu 5 uhvsrqväyhlv shor juxsr/ uhvroyhudp hihfwxdu hvvd hvfrokd gh irupd dohdwöuld1 Frqvlghuh f d y1d1 txh uhsuhvhqwd r qýphur gh doxqdv qhvvd hvfrokd1 +d, Lqgltxh d ixqŒær glvwulexlŒær gh f( +e, Txdo d suredelolgdgh gh h{lvwlu qær phqrv gr txh xpd doxqd qhvvd hvfrokdB +V= e D � +f, Fdofxoh .dfo h T dfo� +V= 4> 2 D � 69FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R H{hufðflr 6153 f ì xpd yduläyho dohdwöuld frp d vhjxlqwh ixqŒær gh ghqvl0 gdgh= sE%� ’ ;? = % e � % e f / f � % � & / &% e / fdvr frqwuäulr +d, Ghwhuplqh/ mxvwl�fdqgr &( +V= 5, +e, Ghwhuplqh d ixqŒær glvwulexlŒær gh f( +f, Fdofxoh � Ef�D f � 2�D� dwudyìv gd ixqŒær gh ghqvlgdgh gh f( +V= 319;:8, +g, Fdofxoh � Ef *f � f�2D� dwudyìv gd ixqŒær gh glvwulexlŒær gh f� +V=3144;, H{hufðflr 6154 Qxp juxsr gh 43 lqglylgxrv yær vhu hvfroklgrv 6 sdud iru0 pduhp xpd frplvvær sdud r hvwxgr gh xp gdgr dvvxqwr1 Hqwuhwdqwr/ vdeh0vh txh 7 grv 43 lqglylgxrv vær hvshfldolvwdv qr dvvxqwr1 +d, Dgplwlqgr xpd hvfrokd fdvxdo/ frqvwuxd d ixqŒær gh suredelolgdgh gd y1d1 f= qJ gh hvshfldolvwdv qd frplvvær> +e, Vdehqgr txh qd frplvvær whp txh hvwdu shor phqrv xp hvshfld0 olvwd/ txdo d suredelolgdgh gh vhuhp wrgrv hvshfldolvwdvB +V= 3137, +f, Dgplwlqgr txh fdgd hvshfldolvwd uhfheh phqvdophqwh/ sru id}hu sduwh gd frplvvær/ 83 frqwrv h fdgd xp grv uhvwdqwhv 53 frqwrv/ rewhqkd r ydoru hvshudgr uhodwlyr dr prqwdqwh phqvdo gh uhpx0 qhudŒłhv gd frplvvær1 +V= <9 frqwrv, H{hufðflr 6155 D txdqwlgdgh gh shl{h shvfdgr gläuldphqwh/ hp wrqhodgdv/ sru fhuwr edufr ì xpd y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelo0 lgdgh= sE%� ’ ;? = % 2ff @E2f� %� f / f � % f / f � % 2f / fdvr frqwuäulr +d, Ghwhuplqh r ydoru gd frqvwdqwh @> +V= 31348, +e, Ghwhuplqh d ixqŒær glvwulexlŒær gd y1d1 h/ frp edvh qhod/ fdofxoh d suredelolgdgh gr edufr/ qxp gld/ qær shvfdu phqrv gh ; wrqhodgdv gh shl{h> +V= 31;7, +f, Vdehqgr txh qxpd pdqkæ r edufr mä shvfrx pdlv gh 6 wrqhodgdv gh shl{h/ txdo d suredelolgdgh gh dwì dr �p gr gld h{fhghu dv ; wrqhodgdv gh shvfdgrB +V= 31;8<6, 6191 H[HUF—FLRV 6: +g, Vh fdgd fdl{d gh shl{h hqyldgd sdud d orwd ohyd 58 nj/ txdo d suredelolgdgh gh qxp gld vh hqylduhp pdlv gh 733 fdl{dvB +V= 31:8, H{hufðflr 6156 Vhmd f xpd y1d1 glvfuhwd wdo txh= 8 E%� ’ ;AAAA? AAAA= f / % e / � % 2 � e / 2 � % � & H / � � % e / % � e +d, Ghwhuplqh rv srvvðyhlv ydoruhv sdud &( +V= g 5 dSc Ho� +e, Ghwhuplqh d ixqŒær gh suredelolgdgh gh f( +f, Fdofxoh � E2 f � e�( +V= e � +g, Fdofxoh .dfo h T dfo� +V= �� g H ( D 2 � g H � g2 Se � H{hufðflr 6157 D ixqŒær gh ghqvlgdgh gh xpd y1d1 f ì gdgd sru= sE%� ’ � @n K%2 f / f � % � / fdvr frqwuäulr Vdehqgr txh .dfo ’ � D c +d, Ghwhuplqh @ h K( +V= @ ’ � D ( K ’ S D � +e, Ghwhuplqh d ixqŒær gh glvwulexlŒær gh f( +f, Fdofxoh � � 2 � f � � e � � +V= 3159;:8, H{hufðflr 6158 Xpd hpsuhvd ghglfd0vh ª yhqgd gh duwljrv hoìfwulfrv1 Dv yhqgdv gläuldv gh judydgruhv +hp xqlgdghv, vær uhsuhvhqwdgdv sru xpd y1d1 f frp d vhjxlqwh ixqŒær gh suredelolgdgh= sE%� ’ � &E%n � f / % ’ fc c 2c �c e / fdvr frqwuäulr +d, Ghwhuplqh r ydoru gh &1 Mxvwl�txh d vxd uhvsrvwd> +V= D � +e, Gh�qd d ixqŒær gh glvwulexlŒær gd y1d1 f( +f, Fdofxoh/ dwudyìv gd ixqŒær gh suredelolgdgh h gd ixqŒær gh glv0 wulexlŒær/ d suredelolgdgh gh dv yhqgdv vhuhp h{dfwdphqwh ljxdlv d 6 judydgruhv/ qxp gld hp txh hvwdv irudp shor phqrv 51 +V= � � 6;FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R H{hufðflr 6159 Frqvlghudqgr f xpd y1d1 txh uhsuhvhqwd d gxudŒær hp fhqwhqdv gh krudv gh xp fhuwr frpsrqhqwh hoìfwulfr h hp txh d vxd ixqŒær ghqvlgdgh ì gdgd sru= sE%� ’ � @% f / f % � f / fdvr frqwuäulr +d, Ghwhuplqh/ mxvwl�fdqgr/ r ydoru gh @( +V= Df � +e, Fdofxoh � Ef : D�( +V= D 2f � +f, Fdofxoh .dfo h T dfo� +V= 2f � ( Df b � H{hufðflr 615: Qxpd ormd gh hohfwurgrpìvwlfrv d surfxud gläuld gh dvsl0 udgruhv ì xpd y1d1 f frp ixqŒær gh suredelolgdgh= % � f 2 � e sE% � � f�2 K f�� f�2 f� +d, Fdofxoh K/ mxvwl�fdqgr> +V= 315, +e, Qr ðqlflr gh xp gld h{lvwhp dshqdv 5 dvsludgruhv qd ormd1 Fdofxoh d suredelolgdgh gh vhuhp yhqglgrv +frqvlghuh txh surfxud@yhqgd vhpsuh txh h{lvwd r surgxwr surfxudgr,> +V= 319, +f, D sduwlu gd doðqhd dqwhulru rewhqkd d ixqŒær gh glvwulexlŒær gd yduläyho �qýphur gh dvsludgruhv yhqglgrv�1 H{hufðflr 615; Vhmd1 f xpd y1d1 glvfuhwd frp ixqŒær gh suredelolgdgh gdgd sru= % � f 2 � sE% � � H H � D 2 +d, Ghwhuplqh d ixqŒær gh glvwulexlŒær gd y1d1 f( +e, Fdofxoh � Ef f � 2�( +V= 2e � +f, Fdofxoh .dfo h T dfo� +V= eb 2e ( Dbb D.S � H{hufðflr 615< Gdgd d ixqŒær gh glvwulexlŒær 8 gd y1d1 f= 8 E%� ’ ;? = f % S / % f / f � % S / % � S +d, Ghwhuplqh d uhvshfwlyd ixqŒær gh ghqvlgdgh> 6191 H[HUF—FLRV 6< +e, Fdofxoh � E2 � f � e�( +V= � � +f, Fdofxoh .dfo� +V= 6, H{hufðflr 6163 D ixqŒær gh suredelolgdgh gd y1d1 f/ txh uhsuhvhqwd r qýphur gh lpshuihlŒłhv hp fdgd 43 phwurv gh xpd �eud vlqwìwlfd ideulfdgd hp urorv frqwðqxrv gh odujxud xqliruph ì gdgd sru= % � f 2 � e sE% � � f�e @ f�S f�fD f�f +d, Fdofxoh @/ mxvwl�fdqgr> +V= 316:, +e, Ghwhuplqh d ixqŒær gh glvwulexlŒær gd y1d1 f> +f, Fdofxoh � E f � e�( +V= 3155, +g, Fdofxoh .dfo h T dfo� +V= 31;;> 31;789, H{hufðflr 6164 Vhmd f xpd y1d1 frqwðqxd fxmd ixqŒær gh glvwulexlŒær ì gdgd sru= 8 E%� ’ ;? = f %2 / % � f / f % / % � +d, Ghwhuplqh d suredelolgdgh gh f wrpdu xp ydoru hqwuh � h 2 h fdofxoh/ hp vhjxlgd/ .dfo h T dfo( +V= D �S ( 2 � ( H � +e, Ghwhuplqh d suredelolgdgh gh/ hp 7 ydoruhv dvvxplgrv lqghshq0 ghqwhphqwh shod y1d1 f/ shor phqrv 6 hvwduhp frpsuhhqglgrv hqwuh 3158 h 31:81 +V= 316458, H{hufðflr 6165 D sursruŒær gh doföro gh xp fhuwr frpsrvwr srgh vhu frq0 vlghudgd xpd y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh= sE%� ’ � &%n 2 f / f % / fdvr frqwuäulr +d, Fdofxoh &/ rewhqkd d ixqŒær gh glvwulexlŒær gd y1d1 f h hverfh r vhx juä�fr> +V= 4, +e, Fdofxoh .dfo h T dfo( +V= . 2 ( ee � +f, Vxsrqkd txh r suhŒr gh yhqgd ghvvh frpsrvwr ghshqgh gd shufhq0 wdjhp gh doföro1 Hvshfl�fdphqwh vh e f � e r frpsrvwr yhqgh0 vh sru 4333’332olwur h gh frqwuäulr yhqgh0vh sru :33’332olwur1 Vh r fxvwr iru 833’332olwur/ fdofxoh d hvshudqŒd pdwhpäwlfd gr oxfur oðtxlgr sru olwur1 +V= 683’33, 73FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R H{hufðflr 6166 Frqvlghuh d y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelolgdgh= sE%� ’ � &E2%n � f / f � % � / fdvr frqwuäulr +d, Ghwhuplqh & h hp vhjxlgd d ixqŒær glvwulexlŒær gh f( +V= 2 � +e, Fdofxoh .dfo h T dfo( +V= . 2 ( ee � +f, Ghwhuplqh 6 gh prgr txh � Ef : 6� ’ � Ef 6� h prvwuh txh 6 ì ýqlfr1 +V= 3194;, H{hufðflr 6167 R glåphwur gh xp fder hoìfwulfr ì xpd y1d1 f frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelolgdgh= sE%� ’ � S%E� %� f / f � % � / fdvr frqwuäulr +d, Gh�qd d ixqŒær gh glvwulexlŒær gd y1d1 f h fdofxoh � � f � 2 � ( +V= 318, +e, Fdofxoh .dfo( +V= 318, +f, Vxsrqkd txh r fder hoìfwulfr ì frqvlghudgr ghihlwxrvr vh r glåphwur glihulu gd vxd pìgld hp pdlv gh 31381 Txdo d suredelolgdgh gh vh hqfrqwudu xp fder ghihlwxrvrB +V= 31;838, H{hufðflr 6168 Frqvlghudqgrf xpd yduläyho dohdwöuld frqwðqxd/ suryh/ xwl0 ol}dqgr d gh�qlŒær/ d vhjxlqwh ljxdogdgh= T d@f � K2o ’ @2T dfo� H{hufðflr 6169 Suryh d vhjxlqwh ljxdogdgh= T d@2f � @Ko ’ @eT dfo +d, Sdud f/ yduläyho dohdwöuld frqwðqxd> +e, Sdud f/ yduläyho dohdwöuld glvfuhwd1 H{hufðflr 616: R qýphur gh yhqgdv uhdol}dgdv sru xp djhqwh gh vhjxurv glduldphqwh ì xpd y1d1 frp ixqŒær gh suredelolgdgh= % � f 2 � e sE% � � � 5 | 5 � Vdehqgr txh hp 43( grv gldv dv yhqgdv vær lqihulruhv d xp h txh hp :3( grv gldv vær vxshulruhv d xp/ 6191 H[HUF—FLRV 74 +d, Ghwhuplqh �c 5 h |( +V= 314/ 315 h 317, +e, Ghwhuplqh T d2f � o( +V= 71;, +f, Ghwhuplqh d suredelolgdgh gh txh/ txdqgr frqvlghudgrv grlv gldv/ dv yhqgdv vhmdp vxshulruhv/ hp fdgd xp ghohv/ d gxdv xqlgdghv> +V= 313<, +g, Vh fdgd vhjxur ì ihlwr sru 48333’33/ ghwhuplqh d ixqŒær gh sur0 edelolgdgh gd uhfhlwd rewlgd frpd yhqgd grv vhjxurv qxp gld> +h, Vh qxp gld d uhfhlwd iru lqihulru d 83333’33/ ghwhuplqh d sured0 elolgdgh gh txh vhmd vxshulru d 53333’331 +V= 31999, H{hufðflr 616; R Vu1 Mrær whp xp fduur txh qhfhvvlwd gh xpd uhsdudŒær xujhqwh1 Frqvlghuh f d y1d1 txh uhsuhvhqwd r qýphur gh vhpdqdv txh r fduur vh pdwhuä hp ixqflrqdphqwr vhp uhsdudŒær1 D ixqŒær ghqvlgdgh gh suredelolgdgh ì d vhjxlqwh= sE%� ’ ;? = % 2f % ef f / f % � e / e % � K / fdvr frqwuäulr +d, Txdo r qýphur pä{lpr gh vhpdqdv txh r fduur ixqflrqduä vhp uhsdudŒærB +V= ;, +e, Ghwhuplqh d ixqŒær gh glvwulexlŒær gd y1d1 f( +f, R Vu1 Mrær suhflvd gh lu dr Sruwr gdtxl d 9 vhpdqdv1 Txdo d sure0 delolgdgh gh srghu xwlol}du r vhx fduur vhp whu hihfwxdgr txdotxhu uhsdudŒærB +V= 3168, H{hufðflr 616< D txdqwlgdgh gh fhuyhmd yhqglgd glduldphqwh qxpd ihlud +hp plokduhv gh olwurv, ì xpd yduläyho dohdwöuld frp d vhjxlqwh ixqŒær ghq0 vlgdgh gh suredelolgdgh= sE%� ’ ;? = &% &E2� 2%� f / f � % � e / e % � S / fdvr frqwuäulr +d, Rewhqkd r ydoru gh & h gh . d�f n 2o ( +V= 2 h 2, +e, Ghwhuplqh 8 E%�( +f, Frqvlghuh rv vhjxlqwhv dfrqwhflphqwrv= D@ �yhqgd gläuld vxshulru d 7333 olwurv� E@ �yhqgd gläuld hqwuh 6333 h 8333 olwurv� Lqgltxh/ mxvwl�fdqgr/ vh D h E vær lqghshqghqwhv1 +V= qær, 75FDS—WXOR 61 YDUL Y˜HLV DOHDWÖULDV H IXQ˚ØHVGHGLVWULEXL˚˘R H{hufðflr 6173 Xpd hvwdŒær gh jdvrolqd hqfkh rv uhvhuydwöulrv xpd yh} sru vhpdqd1 R yroxph vhpdqdo gh yhqgdv/ h{suhvvr hp plokduhv gh olwurv/ ì xpd yduläyho dohdwöuld frp d vhjxlqwh ixqŒær ghqvlgdgh gh suredelolgdgh= sE%� ’ ;AA? AA= f f f3% 2D f / % f / f � % D / D � % f / % � f +d, Yhul�txh txh sE%� ì gh idfwr xpd ixqŒær ghqvlgdgh gh suredelol0 gdgh gh xpd yduläyho dohdwöuld f( +e, Fdofxoh d ixqŒær glvwulexlŒær gh f( +f, Ghwhuplqh d suredelolgdgh gh qxpd vhpdqd vh yhqghuhp pdlv gh :333 olwurv gh jdvrolqd/ vdehqgr0vh txh mä irudp yhqglgrv shor phqrv 7333 olwurv> +g, Fdofxoh d txdqwlgdgh pìgld gh jdvrolqd txh ì yhqglgd vhpdqdo0 phqwh1 H{hufðflr 6174 Vhmdf xpd yduläyho dohdwöuld glvfuhwd frp d vhjxlqwh ixqŒær gh suredelolgdgh= % � 6� 6 6n � 6n D sE% � � &n H & H &3 H & H Ghwhuplqh/ +d, & h 6 gh prgr txh .dfo ’ e ( +V= 5 h 04, +e, T d�f � 2o( +V= 8819;:8, +f, 8 E%� h � E2 f D�� +V= 3158, Fdsðwxor 7 GlvwulexlŒłhv Whöulfdv Glvfuhwdv 714 GlvwulexlŒær Xqliruph Gh�qlŒær 714 Xpd yduläyho dohdwöuld f gl}0vh txh whp xpd glvwulexlŒær xqliruph hp ? srqwrv txdqgr d vxd ixqŒær gh suredelolgdgh ì gd irupd sE% � � ’ � Ef ’ % � � ’ ? c � ’ c 2c ���c ?� Frpr h{hufðflr ghl{d0vh dr fxlgdgr gr doxqr d ghprqvwudŒær gd vhjxlqwh sursrvlŒær1 SursrvlŒær 714 Vh f ì xpd y1 d1 frp xpd glvwulexlŒær xqliruph/ . dfo ’ ? [ � % � h T dfo ’ ? [ � %2 � � % ? [ � % � & 2 � H{hpsor 714 Flwdqgr gh qryr r h{hpsor 517/ frqvlghuh0vh r odqŒdphqwr gh xp gdgr1 D fdgd dfrqwhflphqwr hohphqwdu/ lvwr ì/ d fdgd uhvxowdgr srvvðyho gh xp odqŒdphqwr/ srghprv dvvrfldu xp grv qýphurv c 2c �c ec Dc Sc r qýphur gh srqwrv txh dsduhfhp qd idfh uhvxowdqwh1Dtxl d yduläyho dohdwöuld f srgh wrpdu xp gh vhlv ydoruhv % � ’ � E� ’ c 2c �c ec Dc S� frp dphvpd suredelolgdgh � Ef ’ % � � ’ S � Grqgh hvwd y1 d1 whp xpd glvwulexlŒær xqliruph1 76 77 FDS—WXOR 71 GLVWULEXL˚ØHV WHÖULFDV GLVFUHWDV 715 GlvwulexlŒær Elqrpldo Ì iuhtxhqwh xpd h{shulíqfld dohdwöuld frqvlvwlu qd uhshwlŒær gh xpd vìulh gh surydv/ fdgd xpd gdv txdlv dshqdv frp grlv uhvxowdgrv srvvðyhlv/ txh jhudophqwh vær ghvljqdgrv sru vxfhvvr h lqvxfhvvr> ì r txh dfrqwhfh/ sru h{hpsor/ txdqgr vh whvwdp shŒdv txh vdhp gh xpd olqkd gh prqwdjhp/ rqgh fdgd yhul�fdŒær lqglfd vh d shŒd ì ghihlwxrvd rx qær> txdqgr vh odqŒd yäuldv yh}hv xpd prhgd uhjxodu/ hwf1 H{hpsor 715 Frqvlghuh0vh d h{shulíqfld dohdwöuld txh frqvlvwh qr odqŒd0 phqwr gh xpd prhgd uhjxodu dr dfdvr h hp txh l ’ i8c�j � D uhdol}dŒær gh � odqŒdphqwrv ì xpd h{shulíqfld dohdwöuld txh vh srgh lghqwl�fdu frp d frpelqdŒær gh 6 h{shulíqfldv dohdwöuldv lgíqwlfdv1 R hvsdŒr gh uhvxowdgrv ghvwd rxwud h{shulíqfld ì sruwdqwr r frqmxqwr � ’ i888c 88�c 8�8c�88c 8��c�8�c��8c���j � Dwhqghqgr d txh r uhvxowdgr gh fdgd odqŒdphqwr ì lqghshqghqwh grv uhvwdqwhv/ ì lphgldwr txh � Evdðgd gh 6 fdudv� ’ � 2 � � c � Evdðgd gh 5 fdudv h 4 frurd� ’ �� � 2 � � c hwf1 Hvwh h{hpsor ì xp fdvr sduwlfxodu gh xpd vxfhvvær gh surydv gh Ehuqrxo0 ol/ lvwr ì/ gh xpd vxfhvvær gh h{shulíqfldv dohdwöuldv lqghshqghqwhv/ hp fdgd xpd gdv txdlv vh rewhp r dfrqwhflphqwr D/ txh ghvljqdprv sru vxfhvvr/ frp suredelolgdgh R/ +frqvwdqwh gh h{shulíqfld sdud h{shulíqfld,/ rx r vhx frp0 sohphqwdu/ D�c txh ghvljqdprv sru lqvxfhvvr/ frp suredelolgdgh ^ ’ � R� D xpd vxfhvvær gh surydv gh Ehuqrxool wdpeìp vh fkdpd xpd h{shulíq0 fld Elqrpldo1 H{hpsor 716 D suredelolgdgh gh txh xp fhuwr wlsr gh frpsrqhqwh vreuh0 ylyd d xp whvwh ì 6271 Txdo d suredelolgdgh gh h{dfwdphqwh gxdv ghvvdv frpsrqhqwhv vreuhylydp dr whvwh/ gh hqwuh dv suö{lpdv 8 d vhuhp whvwdgdv1 Ghvljqhprv sru vxfhvvr r dfrqwhflphqwr �d frpsrqhqwh vreuhylyh dr whvwh� +vlperolfdphqwh V, h sru lqvxfhvvr r dfrqwhflphqwr �d frpsrqhqwh qær vreuhylyh dr whvwh�1 Ì lphgldwr txh qhvwh fdvr hvwdprv shudqwh xpd h{shulíqfld elqrpldo1 Frqvlghuhprv sulphludphqwh d suredelolgdgh gh rewhu rv 5 vxfhvvrv h rv 805 lqvxfhvvrv sru xpd ghwhuplqdgd rughp/ sru h{hpsor/ 5 vxfhvvrv vhjxlgrv gh 6 lqvxfhvvrv1 Wudwd0vh gh fdofxodu d suredelolgdgh gd vhtxíqfld/ 77~}7�7�7� ~} txh ì � � e � 2 � � � e � � ’ � � e � 2 � e � � � Vh d rughp i÷u rxwud/ d suredelolgdgh pdqwhp0vh/ ghvgh txh r qýphur gh vxfhvvrv h gh lqvxfhvvrv vh pdqwhqkd1 7161 GLVWULEXL˚˘R KLSHUJHRPÌWULFD 78 Rud/ h{lvwhp � D 2 � vhtxíqfldv glihuhqwhv hp txh srghp rfruuhu rv 5 vxfhvvrv h rv 6 lqvxfhvvrv/ shor txh d suredelolgdgh suhwhqglgd vhuä� D 2 �� � e � 2 � e � � ’ f� fH.b� Gh�qlŒær 715 Vh xpd suryd gh Ehuqrxool srgh uhvxowdu qxp vxfhvvr/ frp suredelolgdgh R/ rx qxp lqvxfhvvr/ frp suredelolgdgh �R/ hqwær d ixqŒær gh suredelolgdgh gd y1 d1 fc txh uhsuhvhqwd r qýphur gh vxfhvvrv hp ? surydv lqghshqghqwhv/ h vh ghvljqd sru yduläyho dohdwöuld elqrpldo/ ì gdgd shod h{suhvvær � Ef ’ %� ’ K E%(?c R� ’ � ? % � R% E� R�?3% c % ’ fc c ���c ?� SursrvlŒær 715 Vh f ì xpd y1 d1 frp xpd glvwulexlŒær elqrpldo/ f q KE?c R�c . dfo ’ ?R h T dfo ’ ?RE� R�� D whuplqdu/ xpd uhihuíqfld d xp lpsruwdqwh uhvxowdgr uhodwlyr ª vrpd gh y1 d1 frp glvwulexlŒær elqrpldo1 SursrvlŒær 716 Vh dv y1 d1 f � c � ’ c 2c ���c & vær lqghshqghqwhv4 h doìp glvvr/ f � q KE? �c R�c hqwær d y1 d1 t & ’ Ef nf 2 n ���nf & � q K E S ? � c R� � 716 GlvwulexlŒær Klshujhrpìwulfd H{hpsor 717 Qxp surfhvvr gh ideulfr dgplwh0vh txh fdgd orwh gh 73 shŒdv ì dfhlwäyho vh qær frqwlyhu pdlv gr txh 6 shŒdv ghihlwxrvdv1 R surfhvvr sdud whvwdu fdgd orwh frqvlvwh hp h{wudlu dohdwruldphqwh xpd dprvwud gh 8 shŒdv h uhmhlwdu r orwh vh i÷u hqfrqwudgd 4 shŒd ghihlwxrvd qd dprvwud1 Txdo d 4 Vhp suhrfxsdêøhv gh judqgh uljru/ lqwurgx}lprv dtxl r frqfhlwr gh yduläyhlv dohdwöuldv lqghshqghqwhv1 Vhmdp [ h \ gxdv yduläyhlv dohdwöuldv glvfuhwdv> [ h \ gl}hp0vh lqgh0 shqghqwhv vh h vö vh/ sdud txdlvtxhu ydoruhv { h |/ rv dfrqwhflphqwrv � [ @ { \ @ | iruhp lqghshqghqwhv1Lvwr vljql�fd txh S +[ @ { m\ @ | , @ S +[ @ {, S +\ @ | m[ @ { , @ S +\ @ |, = 79 FDS—WXOR 71 GLVWULEXL˚ØHV WHÖULFDV GLVFUHWDV suredelolgdgh gh hqfrqwudu h{dfwdphqwh xpd shŒd ghihlwxrvd qd dprvwud/ vh h{lvwluhp 6 shŒdv ghihlwxrvdv qr orwhB Uhsduh0vh txh qhvwh fdvr qær hvwdprv shudqwh xpd h{shulíqfld Elqrpldo/ srlv d dprvwud qær ì uhfroklgd frp uhsrvlŒær h sruwdqwr rv uhvxowdgrv gd lqvshfŒærgh fdgd shŒd gd dprvwud qær vær lqghshqghqwhv1 Dvvlp/ vh d sulphlud shŒd whp xpd suredelolgdgh gh vhu ghihlwxrvd ljxdo d 6273/ d vhjxqgd whuä hvvd suredelolgdgh ljxdo d 626< vh d sulphlud qær wlyhu ghihlwr/ rx ljxdo d 526< fdvr frqwuäulr1 Sdud uhvroyhu hvwh sureohpd/ ydprv xwlol}du d gh�qlŒær foävvlfd gh sured0 elolgdgh1 R qýphur gh fdvrv srvvðyhlv vhuä gdgr sru � ef D � ( sdud fdofxodu r qýphur gh fdvrv idyruäyhlv whprv txh yhu gh txdqwdv pdqhludv ì srvvðyho whu qd dprvwud gh 8/ xpd shŒd ghihlwxrvd h txdwur qær ghihlwxrvdv> rud/ d shŒd ghihlwxrvd vö srghuä vhu h{wudðgd gr frqmxqwr gh shŒdv ghihlwxrvdv txh h{lvwhp qr orwh h txh vær 6/ h sruwdqwr hod srghuä dsduhfhu gh � � � pdqhludv glihuhqwhv> dv gxdv shŒdv qær ghihlwxrvdv vö srghuær vhu h{wudðgdv gr frqmxqwr gh shŒdv qær ghihlwxrvdv txh h{lvwhp qr orwh h txh vær 6:/ grqgh h{lvwhp � �. e � pdqhludv glihuhqwhv gh dv rewhu1 R qýphur gd fdvrv idyruäyhlv vhuä sruwdqwr ljxdo d � � � � � �. e � h d suredelolgdgh suhwhqglgd� � �� �. e �� ef D � ’ f� �f� Yhmdprv xp rxwur h{hpsor= H{hpsor 718 Gh xpd xuqd txh frqwhp 2f erodv/ glihuhqwhv dshqdv qd fru/ gdv txdlv D vær eudqfdv h 2f�D vær suhwdv/ wludp0vh dr dfdvr : erodv1 Txdo d suredelolgdgh gh rewhu 7 erodv eudqfdvB Hvwd vlwxdŒær ì wðslfd gdv fkdpdgdv h{shulíqfldv klshujhrpìwulfdv txh edvl0 fdphqwh vh fdudfwhul}dp sru= 4, xpd dprvwud dohdwöuld gh ? hohphqwrv ì vhohfflrqdgd d sduwlu gh xpd sr0 sxodŒær frp � hohphqwrv> 5, & grv � hohphqwrv gd srsxodŒær vær ghvljqdgrv sru vxfhvvrv h � � & sru lqvxfhvvrv1 Qhvwh fdvr/ ghvljqdqgr sru vxfhvvr d h{wudfŒær gh xpd erod eudqfd h sru lqvxfhvvr d h{wudfŒær gh xpd erod suhwd/ hvwä0vh lqwhuhvvdgr hp vdehu txdo d suredelolgdgh gh rewhu e vxfhvvrv h .� e lqvxfhvvrv1 Gh dfrugr frp d gh�qlŒær foävvlfd gh suredelolgdgh whp0vh txh r qýphur gh fdvrv idyruäyhlv ì ljxdo d � D e �� 2f3D .3e � ’ �SDf h r qýphur gh fdvrv srvvðyhlv ì� 2f . � ’ ..D2fc grqgh/ d suredelolgdgh suhwhqglgd ì �SDf*..D2f ’ f�.SfH� D yduläyho dohdwöuld f txh uhsuhvhqwd r qýphur gh vxfhvvrv qxpd h{sh0 ulíqfld klshujhrpìwulfd ghvljqd0vh sru yduläyho dohdwöuld klshujhrpìwul0 7161 GLVWULEXL˚˘R KLSHUJHRPÌWULFD 7: fd h d vxd ixqŒær gh suredelolgdgh ì gdgd shod h{suhvvær sE%� ’ �E%(�c?c &� ’ � & % �� �3& ?3% �� � ? � c rqgh � uhsuhvhqwd d glphqvær gd srsxodŒær/ ? d glphqvær gd dprvwud h & r qýphur gh vxfhvvrv txh h{lvwhp qd srsxodŒær1 x 1086420 0.4 0.3 0.2 0.1 0 �E%( ffc �fc f� SursrvlŒær 717 Vh f ì xpd y1d1 frp xpd glvwulexlŒær klshujhrpìwulfd f q �E�c?c &�/ . dfo ’ ?&*� h T dfo ’ � � ? � � ? & � � � & � � � H{hpsor 719 Qxpd fdl{d hqfrqwudp0vh plvwxudgrv 43 sdudixvrv ghihlwxr0 vrv frp <3 hp erp hvwdgr1 Vh iru uhwludgd gd fdl{d xpd dprvwud dohdwöuld gh 43 sdudixvrv lqgltxh txdo d suredelolgdgh gh rewhu h{dfwdphqwh % sdudixvrv frp ghihlwr1 Frqvlghudqgr txh d dprvwud srgh vhu uhfroklgd frp rx vhp uhsrvlŒær/ frpsduh rv uhvxowdgrv sdud ydoruhv gh % dwì 81 Ghvljqdqgr sru f d y1d1 txh uhsuhvhqwd r qýphur gh sdudixvrv ghihlwxr0 vrv h{lvwhqwhv qd dprvwud gh 43/ srghuhprv gl}hu txh f q �Effc fc f� vh d dprvwud iru uhfroklgd vhp uhsrvlŒær h txh f q KEfc f�� fdvr frqwuäulr1 R txdgur vhjxlqwh shuplwh0qrv frpsdudu rv ydoruhv rewlgrv hp dpedv dv vlwxdŒłhv= f �Effc fc f� KEfc f�� f f���f f��eb f�efH f��H. 2 f�2f2 f�be � f�fD2 f�fD. e f�ffH f�f D f�ff f�ff2 � 7; FDS—WXOR 71 GLVWULEXL˚ØHV WHÖULFDV GLVFUHWDV Joredophqwh srghprv gl}hu txh h{lvwh xpd fhuwd vhphokdqŒd hqwuh dv gxdv glvwulexlŒłhv1 Lvwr ghyh0vh vlpsohvphqwh dr idfwr gh/ txdqgr d glphqvær gd dprvwud ì shtxhqd uhodwlydphqwh ª glphqvær gd srsxodŒær/ vh hvedwhu d glihuhqŒd hqwuh h{wudfŒłhv frp uhsrvlŒær h vhp uhsrvlŒær1 Qd suäwlfd/ txdqgr ? �*f d glvwulexlŒær elqrpldo iruqhfh mä xpd dsur{lpdŒær vdwlvidwöuld gd glvwulexlŒær klshujhrpìwulfd/ srghqgr qhvvh fdvr ehqhihfldu0vh gd vxd pdlru dfhvvlelolgdgh1 R txh dfderx gh vhu glwr ì vlpsohvphqwh xpd dsolfdŒær gr vhjxlqwh whr0 uhpd= Whruhpd 714 Frp ? h & �{rv/ ,�6 �<" � & % �� �3& ?3% �� � ? � ’ �? % � R% E� R�?3% � 717 GlvwulexlŒær gh Srlvvrq Gh�qlŒær 716 Xpd yduläyho dohdwöuld txh dvvxph ydoruhv gd vxfhvvær lq0 �qlwd fc c 2c �c ���c frp suredelolgdghv/ sE%� ’ � Ef ’ %� ’ e3bb% %- c % ’ fc c 2c �c ���c b : fc gl}0vh txh whp glvwulexlŒær gh Srlvvrq frp sduåphwur bc hvfuhyhqgr0vh vlperolfdphqwh f q R Eb� � Whqgr hp frqwd txh d vrpd gd vìulh "S %'f b % %- ì d ixqŒær ebc ì lphgldwr txh "[ %'f sE%� ’ "[ %'f e3bb% %- ’ e3b "[ %'f b% %- ’ e3beb ’ � SursrvlŒær 718 R ydoru hvshudgr h d ydulåqfld gh xpd yduläyho dohdwöuld frp glvwulexlŒær gh Srlvvrq vær ljxdlv dr ydoru gr sduåphwur b� GhprqvwudŒær1 l, . dfo ’ "S %'f %e 3b b % %- ’ e3b "S %'f %b % %- ’ e3b "S %' b b %3 E%3�- ’ b� ll,R vhjxqgr prphqwr hp uhodŒær ª ruljhp/ . � f2 � ’ "[ %'f %2 e3bb% %- ’ be3b "[ %' % b%3 E%� �- ’ ’ be3b "[ %' E%� � b %3 E%� �- n be 3b "[ %' b%3 E%� �- ’ b2e3b "[ %'2 b%32 E%� 2�- n be 3b "[ %' b%3 E%� �- ’ b 2 n b 7171 GLVWULEXL˚˘R GH SRLVVRQ 7< shuplwh0qrv frqfoxlu txh T dfo ’ . � f2 �� .2 dfo ’ b2 n b� b2 ’ b� R whruhpd txh vh dsuhvhqwd d vhjxlu hvwdehohfh txh d ixqŒær gh suredel0 olgdgh gd glvwulexlŒær gh Srlvvrq wdpehp srgh vhu rewlgd frpr r olplwh gh xpd vìulh gh ixqŒłhv gh suredelolgdgh gd glvwulexlŒær Elqrpldo Whruhpd 715 Vhmd f ? xpd yduläyho dohdwöuld frp glvwulexlŒær Elqrpldo gdgd shod iöupxod � Ef ? ’ o� ’ � ? o � Ro E� R�?3o c rqgh o wrpd rv ydoruhv fc c 2c ���c ?� Vh sdud ? ’ c 2c �c ��� d uhodŒær R ’ b ? vh pdqwìp/ rqgh b : f ì xpd frqvwdqwh/ hqwær ,�6 ?<" � Ef ? ’ o� ’ bo o- e3b� GhprqvwudŒær1 Id}hqgr R ’ b ? c yhp/ � ? o � Ro E� R�?3o ’ ?- o- E?� o�- � b ? � o � � b ? � ?3o ’ ’ ? E?� � ��� E?� o n � ?o � � b ? � ? � � b ? � 3o bo o- ’ ’ bo o- � � � b ? � ? � � � ? � ��� � � on ? �� � b ? � o � Frpr ,�6 ?<" � � b ? � ? ’ e3b h ,�6 ?<" � � ? � ��� � � on ? �� � b ? � o ’ c rewhp0vh r uhvxowdgr suhwhqglgr1 Hvwh uhvxowdgr whp judqghv dsolfdŒłhv suäwlfdv srlv/ frpr dv �jxudv vhjxlqwhv vxjhuhp/ d glvwulexlŒdr gh Srlvvrq srgh vhu frqvlghudgd hp fhu0 wdv flufxqvwåqfldv/ xpd erd dsur{lpdŒær gd glvwulexlŒær Elqrpldo1 Qd �jxud vhjxlqwh vær dsuhvhqwdgrv grlv juä�frv/ xp gd glvwulexlŒær El0 qrpldo frp ? ’ D h R ’ f��c grqgh b ’ ?R ’ �Dc h xp gd glvwulexlŒær gh 83 FDS—WXOR 71 GLVWULEXL˚ØHV WHÖULFDV GLVFUHWDV Srlvvrq frp r phvpr ydoru hvshudgr b ’ �D� r 76543210 0.5 0.4 0.3 0.2 0.1 0 GlvwulexlŒær Elqrpldo h Srlvvrq +? ’ D,1 D �jxud dsuhvhqwd grlv juä�frv lgíqwlfrv pdv frp ? ’ f h R ’ f�Dc grqgh vh pdqwhp b ’ ?R ’ �D� r 76543210 0.5 0.4 0.3 0.2 0.1 0 GlvwulexlŒær Elqrpldo h Srlvvrq +? ’ f,1 Sdud pdlruhv ydoruhv gh ?/ sru h{hpsor ? ’ ffc rv juä�frv gdv glv0 wulexlŒłhv Elqrpldo h Srlvvrq txdvh frlqflghp1 H{hpsor 71: Qxpd frpxqlgdgh frp ffff shvvrdv/ d suredelolgdgh gh xpd shvvrd/ qxp ghwhuplqdgr gld/ surfxudu xpd fdpd qr krvslwdo/ vxsłh0vh ljxdo d *2fff� Kdyhqgr lqghshqgíqfld qd surfxud gh fdpdv hp fdgd gld +lqh{lvwíqfld gh hslghpldv/ gh grhqŒdv frqwdjlrvdv/ hwf, d y1d1 f txh uh0 suhvhqwd r qýphur gh fdpdv surfxudgdv hp fdgd gld whp xpd glvwulexlŒær elqrpldo frp ? ’ ffff h R ’ *2fff� Qhvwh fdvr/ r fäofxor gh txdotxhu suredelolgdgh/ � Ef ’ %�c ghl{d gh vhu lphgldwr1 Frpr ? ì judqgh h R pxlwr shtxhqr srghprv fdofxodu ydoruhv dsur{lpdgrv ghvvdv suredelolgdghv
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