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  • 1. #7) ++ /CVJGOCVKECN 5EKGPEGU 2CRGT ++6KOG #NNQYGF /KPWVGU? =/CZKOWO /CTMU 0QVG 6JKU 2CRGT EQPVCKPU (KHV[ OWNVKRNG EJQKEG SWGUVKQPU CEJ SWGUVKQP ECTTKGU 6YQ OCTMU #VVGORV #NN SWGUVKQPU %QPUKFGT VJG VYQ UVCVGOGPVU HQT 6JG UGIOGPV CГ D QH VJG Z CZKU QH  % ^ 4 KU C G KU C QPG QPG HWPEVKQP D UKP KU DQWPFGF GPVKTG HWPEVKQP # 0GKVJGT QRGP PQT ENQUGF VJGP $ 1RGP # DQVJ C CPF D CTG HCNUG $ DQVJ C CPF D CTG VTWG % %NQUGF % QPN[ C KU VTWG $QVJ QRGP CPF ENQUGF QPN[ D KU VTWG .GV H GГ YJGTG Z K[Г #P QRGP UGV KP 4 KU VJGP # CP KPVGTXCN # G YJGP [ KU CP GXGP OWNVKRNG QH SГ CPF G YJGP $ HKPKVG WPKQP QH QRGP KPVGTXCNU [ KU CP QFF OWNVKRNG QH S $ G KH [ KU CP QFF OWNVKRNG % EQWPVCDNG WPKQP QH FKULQKPV QRGP QH S KPVGTXCNU KP 4 % G KH [ KU CP GXGP OWNVKRNG QH S KPVGTUGEVKQP QH QRGP KPVGTXCNU G KH [ KU C OWNVKRNG QH S =261
  • 2. #7) ++ H VCP Г  % ^ 9JKEJ QH VJG HQNNQYKPI HWPEVKQPU # CUUWOGU CNN EQORNGZ PWODGTU 6 4 o 4 KU PQV C NKPGCT $ CUUWOGU PQPG QH VJG EQORNGZ VTCPUHQTOCVKQP ! PWODGTU % CUUWOGU CNN EQORNGZ PWODGTU # 6 CГ D C DГ C D GZEGRV K $ 6 CГ D C DГ D CUUWOGU CNN EQORNGZ PWODGTU GZEGRV K CPF K % 6 CГ D CГ D .GV 8 ( DG C XGEVQT URCEG %QPUKFGT 6 CГ D DГ C VJG HQNNQYKPI UVCVGOGPVU + +H DГ EГ J CTG NKPGCTN[ .GV 8 DG C XGEVQT URCEG QXGT C HKGNF KPFGRGPFGPV XGEVQTU KP 8 ( Г ( CPF 6 8 o 8 DG C NKPGCT VJGP D EГ E Г JГ J Г D CTG VTCPUHQTOCVKQP UWEJ VJCV 6 JCU CU CNUQ NKPGCTN[ KPFGRGPFGPV ++ 6JG UGV QH XGEVQTU # Ž 8 ( C EJCTCEVGTKUVKE TQQVГ VJGP YJKEJ EQPVCKPU VJG GTQ XGEVQT # 6 KU FKCIQPCNKCDNG QXGT ( KU CNYC[U NKPGCTN[ KPFGRGPFGPV 6JGP $ /WNVKRNKEKV[ QH GCEJ # $QVJ + CPF ++ CTG VTWG EJCTCEVGTKUVKE TQQV QH 6 KU $ $QVJ + CPF ++ CTG HCNUG % 6 KU KPXGTVKDNG % QPN[ + KU VTWG 6 KU PKNRQVGPV QPN[ ++ KU VTWG
  • 3. #7) ++ .GV H 7 o 8 DG C NKPGCT OCR CPF +H 2 CPF 2 ( Г YJKEJ ]WГ WГ Г WP_ DG VJG UGV QH QH VJG HQNNQYKPI KU PQV EQTTGEV ! NKPGCTN[ KPFGRGPFGPV XGEVQTU KP 7 # 2 ‰ ( CPF 2 ˆ ( 6JGP VJG UGV ]H W Г H W Г Г H WP _ KU NKPGCTN[ KPFGRGPFGPV KHH $ 2 % ‰ ( CPF 2 ˆ ( # H KU QPG QPG CPF QPVQ $ H KU QPG QPG % 2 ‰ (% CPF 2 ˆ (% % H KU QPVQ 2 ‰ ( CPF 2 % ˆ ( 7 8 § · .GV # ¨ ¸ DG .GV CPF ( DG VYQ KPFGRGPFGPV CPF ¨ ¸ © ¹ OWVWCNN[ GZENWUKXG GXGPVU 9JKEJ QH C OCVTKZ QH VTCPUHQTOCVKQP 6JGP VJG VJG HQNNQYKPI UVCVGOGPVU KU PQV TCPM QH # KU GSWCN VQ EQTTGEV ! # # KVJGT 2 QT 2 ( $ $ KVJGT 2 QT 2 ( % % 2 2 ‰ ( 2( 2 ^( 2 =261
  • 4. #7) ++ .GV : CPF ; DG VYQ KPFGRGPFGPV 6JG QRVKOCNKV[ QH C DCUKE HGCUKDNG 2QKUUQP TXU YKVJ RCTCOGVGTU O CPF UQNWVKQP QH C VTCPURQTVCVKQP RTQDNGO T TGURGEVKXGN[ 9JKEJ QH VJG KU EJGEMGF D[ HQNNQYKPI UVCVGOGPVU KU PQV EQTTGEV ! O T # )TCRJKECN OGVJQF # 2=: ; ? G O T $ 5VGRRKPI UVQPG OGVJQF O T $ 2=: ; ? G OT % %WVVKPI RNCPG OGVJQF OM % 2=: ^ ; ? ¦G M O M $TCPEJ CPF DQWPF OGVJQF 6JG QRVKOCN UQNWVKQP QH VJG NKPGCT 2=: ^: ; ? RTQITCOOKPI RTQDNGO OT TГ O /CZKOKG Z Z .GV : DG C PQTOCN TCPFQO XCTKCDNG 5WEJ VJCV Z Z YKVJ OGCP CPF XCTKCPEG GHKPG VJG GXGPVU ] : _Г Z Z ( ] : _ CPF ) ] : _ ZГ Z 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU KU EQTTGEV ! # KU WPDQWPFGF # 2 2( 2) $ KU KPHGCUKDNG $ 2( 2 2) % KU Z Z % 2 2( 2) 2) 2 2( KU Z Z
  • 5. #7) ++ f H Z GZ Г Z z Г Z  4 6JG UGTKGU ¦C P P P TGRTGUGPVU CP Г Z GPVKTG HWPEVKQP KHH 6JGP # NKO ^ CP ^ P Po f # H KU QPN[ C % HWPEVKQP $ NKO ^ CP ^ P Po f $ H KU TGCN CPCN[VKE CV % NKO ^ CP ^ P KU C HKPKVG RQUKVKXG % H KU PQYJGTG TGCN CPCN[VKE Po f TGCN PWODGT H KU C %f HWPEVKQP NKO ^ CP ^ P f 1T Po f 1T .GV : DG C TCPFQO XCTKCDNG YKVJ OGCP CPF XCTKCPEG 9JKEJ QH .GV : DG C PQTOCN TCPFQO XCTKCDNG VJG HQNNQYKPI KU PQV EQTTGEV ! YKVJ OGCP CPF XCTKCPEG # : ECP PQV HQNNQY C $KPQOKCN 6JGP FKUVTKDWVKQP $ : ECP PQV HQNNQY C 2QKUUQP # =EQU :? G FKUVTKDWVKQP $ =EQU :? % : ECP PQV HQNNQY CP GZRQPGPVKCN % =EQU :? FKUVTKDWVKQP : ECP PQV HQNNQY C PQTOCN =EQU :? G FKUVTKDWVKQP =261
  • 6. #7) ++ ª º f +H #P « Г » Г VJGP .GV CP Г ¬P ¼ # P P KU P Г P KU GXGP # Г Г P KU QFF P $ Г ? 6JGP # NKO CP ! NKO CP % =Г ? Po f Po f =Г $ NKO CP NKO CP Pof Pof 1T % NKO CP CPF NKO CP DQVJ FQ PQV Pof Pof .GV : HQNNQYU C 2QKUUQP FKUVTKDWVKQP GZKUV YKVJ RCTCOGVGT O 6JG GUVKOCVQT NKO CP NKO CP 6 KU FGHKPGF CU 6 KH : CPF 1T .GV :Г :Г :P DG C TCPFQO 6 Г QVJGTYKUG 9JKEJ QH VJG UCORNG HTQO QPG RCTCOGVGT P HQNNQYKPI UVCVGOGPVU KU HCNUG ! GZRQPGPVKCN HCOKN[ .GV 6 ¦: K # 6 KU WPDKCUGF HQT GO K 6GUV VJG J[RQVJGUKU * T T CICKPUV * T z T 6JGP VJG $ 8CTKCPEG QH 6 CVVCKPU %TCOOGT WPKHQTON[ OQUV RQYGTHWN WPDKCUGF 4CQ NQYGT DQWPF VGUV KU IKXGP D[ # I: KH % 6 % % 6 KU PQV WPKHQTON[ OKPKOWO QVJGTYKUG XCTKCPEG WPDKCUGF GUVKOCVQT QH $ I: KH 6 % KH 6 % GO % I: KH 6 % 8CTKCPEG QH 6 KU ITGCVGT VJCP KH 6 % I: KH 6 % QT 6 % %TCOOGT 4CQ NQYGT DQWPF QVJGTYKUG
  • 7. #7) ++ +H C CPF D CTG TGCN PWODGTUГ VJGP %QPUKFGT VJG UGSWGPEG UWR ]CГ D_ CP Г KH P KU GXGP P C Г D ^ C D^ # KH P KU QFF P C Г D Г ^ C D^ 6JGP $ NKO NKO CP # Po f Pof C D Г ^ C Г D^ % $ NKO CP NKO CP Po f Po f C D ^ C D^ % NKO CP ! NKO CP Po f Po f NKO CP NKO CP Po f Po f 1T 1T .GV :Г :Г :P DG KKF IGQOGVTKE 6JG TX : JCU $GTPQWNNK FKUVTKDWVKQP FGHKPGF D[ 2=: ? 2=: ? TXU YKVJ RCTCOGVGT R 6JGP VJG TГ YJGTG T 6JG ONG QH FKUVTKDWVKQP QH : /KP : K KU K T DCUGF QP : KU # $KPQOKCN # : : $ )GQOGVTKE $ : % 2QKUUQP % : *[RGTIGQOGVTKE =261
  • 8. #7) ++ ­ ½ UKP +H +P ® Z  4 ^ Z ¾ Г VJGP .GV H Г z 6JGP CV ¯ P¿ f Г H JCU + P KU P # C TGOQXCDNG UKPIWNCTKV[ # ]_ $ C RQNG QH QTFGT % C RQNG QH QTFGT $ I C RQNG QH QTFGT % ]Z  4^ Z _ 1T 4 .GV :Г :Г :P DG KKF TCPFQO UCORNG HTQO VJG HQNNQYKPI 1T RTQDCDKNKV[ OCUU HWPEVKQP 9JKEJ QH VJG HQNNQYKPI FKUVTKDWVKQPU 2=: Z ? Z Г Г 0 0 KU PQV CP GZCEV UCORNKPI 6JGP 2=: P ?Г YJGTG : P FKUVTKDWVKQP ! /CZ :K KU K 0 # F FKUVTKDWVKQP # 0P $ $GVC FKUVTKDWVKQP 0 $ P0 % V FKUVTKDWVKQP % 0 ( FKUVTKDWVKQP P 0P
  • 9. #7) ++ EQU ³ ^^ F +H H G Г z PS K Г P Г Г Г Г VJGP # SK # H JCU TGOQXCDNG UKPIWNCTKVKGU CV PSKГ P Г Г $ SK $ H JCU GUUGPVKCN UKPIWNCTKVKGU CV PSKГ P Г Г % SK % H JCU FQWDNG RQNGU CV VJG QOKVVGF RQKPVU CV YJKEJ VJG TGUKFWG KU SK GTQ H JCU RQNGU CV PSKГ P Г 1T Г Г .GV [ Г [ DG VYQ KPFGRGPFGPV 1T +P C VYQ YC[ ENCUUKHKECVKQP OQFGN QDUGTXCVKQPU JCXKPI GZRGEVCVKQPU YKVJ GSWCN QDUGTXCVKQPU RGT EGNNГ [ T TГ [ T TГ [KLM z JKL CPF 8 [KLM DK EL V K C L Г D 8 [ 8 [ V T CPF T CTG Ž Ž M E 9JKEJ RCTCOGVTKE HWPEVKQP NGCUV USWCTGU GUVKOCVQTU QH T CPF KU GUVKOCDNG ! T TGURGEVKXGN[ 9JKEJ QH VJG # ¦E DK K K YJGTG 6EK HQNNQYKPI UVCVGOGPVU KU HCNUG ! $ ¦F EL L L YJGTG 6FL # Ž T [ Г [ % z DK EL $ Ž 8 T Ž 8 T V % Ž T [ [ ¦¦ H K L KL J KL YJGTG ¦H K KL Ž Ž %QX T Г T ¦H L KL =261
  • 10. #7) ++ .GV 0 DG C FGNGVGF PGKIJDQWTJQQF 1T QH 6JGP EQTTGEV UVCVGOGPV HTQO +P QPG YC[ #018# YKVJ P VJG HQNNQYKPI KU QDUGTXCVKQPU CPF VTGCVOGPVUГ VJG # KU C TGOQXCDNG UKPIWNCTKV[ QH GZRGTKOGPVGT YJKNG ECNEWNCVKPI VJG H KH H KU CPCN[VKE CPF WPDQWPFGF ( TCVKQ HQTIQV VQ FKXKFG VJG UWO QH KP 0 USWCTGU D[ VJG EQTTGURQPFKPI $ KU C TGOQXCDNG UKPIWNCTKV[ QH FGITGGU QH HTGGFQO *QYGXGT VJG H KH H KU CPCN[VKE CPF DQWPFGF UVCVKUVKEKCP UCKF VJGTG KU PQ OKUVCMGГ KP 0 DGECWUG % KU C TGOQXCDNG UKPIWNCTKV[ QH # P H KH o H z NKO $ P KU C TGOQXCDNG UKPIWNCTKV[ QH % P H KH NKO H FQGU PQV o GZKUV P
  • 11. #7) ++ 6JG HWPEVKQP H ^^Г  % ^ H G Г  % KU ^ KU # CPCN[VKE KP % ^ # EQPVKPWQWU CPF FKHHGTGPVKCDNG $ CPCN[VKE QPN[ KP VJG WRRGT JCNH GXGT[YJGTG RNCPG $ GXGT[YJGTG EQPVKPWQWU DWV % CPCN[VKE KP VJG HKTUV SWCFTCPV PQYJGTG FKHHGTGPVKCDNG CPCN[VKE PQYJGTG KP % ^ % EQPVKPWQWU GXGT[YJGTG DWV FKHHGTGPVKCDNG QPN[ CV VJG QTKIKP 1T PGKVJGT EQPVKPWQWU PQT .GV T FGPQVG VJG OKPKOWO PWODGT FKHHGTGPVKCDNG CP[YJGTG QH VKOGU CP[ VTGCVOGPV KU TGRNKECVGF 1T KP EQORNGVGN[ TCPFQOKUGF FGUKIP +P C 4$ YKVJ DNQEMU CPF YKVJ X VTGCVOGPVU CPF P RNQVU VTGCVOGPVUГ PWODGT QH RNQVU KP GCEJ 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU DNQEM KU KU CNYC[U VTWG ! # # T X $ $ T KPVGIGT RCTV QH PX % % T KPVGIGT RCTV QH PX T KPVGIGT RCTV QH PX =261
  • 12. #7) ++ 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU +H Г K Г VJGP KU KU VTWG ! K # # XGT[ ITQWR KU KUQOQTRJKE VQ C RGTOWVCVKQP ITQWR Г K $ $ # ITQWR ) KU KUQOQTRJKE VQ C Г K RGTOWVCVKQP ITQWR KHH KV KU HKPKVG % % # ITQWR ) KU EQOOWVCVKXG KHH KV K KU E[ENKE # ITQWR ) KU PQP CDGNKCP KHH 1T QTFGT QH ) KU PQV RTKOG +P C 4$ YKVJ D DNQEMU CPF X 1T VTGCVOGPVU YJKEJ QH VJG HQNNQYKPI 9JGP FGUKTGF KPHQTOCVKQP KU UVCVGOGPVU CNYC[U JQNFU ! CXCKNCDNG HQT CNN QDLGEVU KP VJG RQRWNCVKQPГ YG JCXG YJCV KU ECNNGF # D X C $ D X # %GPUWU % D X $ 2QRWNCVKQP D CPF X CTG CTDKVTCT[ RQUKVKXG % 5CORNG KPVGIGTU 5VGO CPF NGCH FKURNC[
  • 13. #7) ++ .GV ) DG C ITQWR QH QTFGT 6JG TKPI QH ª OCVTKEGU QXGT 6JGP # KU EQOOWVCVKXG # F^Q ) KORNKGU ) JCU C WPKSWG $ KU CP KPVGITCN FQOCKP UWDITQWR QH QTFGT F $ ) JCU QPN[ QPG PQTOCN % JCU PQP GTQ GTQ FKXKUQTU UWDITQWR KU C HKGNF % ) JCU VYQ UWDITQWRU QH QTFGT 1T CPF QPG UWDITQWR QH QTFGT G KU VJG QPN[ RTQRGT UWDITQWR 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU QH ) KPXQNXGU FGUETKRVKXG UVCVKUVKEU CU 1T QRRQUGF VQ KPHGTGPVKCN UVCVKUVKEU ! 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU # 6JG (## UCORNGU VTCHHKE KU HCNUG ! EQPVTQNNGTU KP QTFGT VQ GUVKOCVG # #NN EQORCPKGU NKUVGF QP VJG 0GY VJG RGTEGPV TGVKTKPI FWG VQ LQD ;QTM 5VQEM ZEJCPIG KU CP UVTGUU TGNCVGF KNNPGUU GZCORNG QH C UCORNG $ $CUGF QP UCORNG QH $ +P GPWOGTCVKXG UVWFKGUГ KPVGTGUV RTQHGUUKQPCN VGPPKU RNC[GTUГ C KU HQEWUUGF QP C HKPKVGГ VGPPKU OCICKPG TGRQTVGF VJCV KFGPVKHKCDNGГ WPEJCPIKPI QH VJG RCTGPVU QH CNN EQNNGEVKQP QH KPFKXKFWCNU QT RTQHGUUKQPCN VGPPKU RNC[GTU FKF QDLGEVU VJCV OCMG WR C PQV RNC[ VGPPKU RQRWNCVKQP % # UCORNKPI HTCOG KU C NKUVKPI QH % 6JG #NEQJQNГ 6QDCEEQ CPF VJG KPFKXKFWCNU QT QDLGEVU VQ DG (KTGCTOU GRCTVOGPV TGRQTVGF UCORNGF VJCV *QWUVQP JCF Г #NN RQUUKDNG [KGNFU KP ITCOU TGIKUVGTGF IWP FGCNGTU KP HTQO C EGTVCKP EJGOKECN TGCEVKQP $CUGF QP UWTXG[ QH ECTTKGF QWV KP C NCDQTCVQT[ KU CP OCICKPG TGCFGTUГ VJG GZCORNG QH EQPEGRVWCN QT OCICKPG TGRQTVU VJCV QH KVU J[RQVJGVKECN RQRWNCVKQP TGCFGTU RTGHGT FQWDNG EQNWOP CTVKENGU =261
  • 14. #7) ++ .GV ( DG C HKGNF YKVJ C HKPKVG PWODGT .GV / DG C RTQRGT KFGCN QH C $QQNGCP QH GNGOGPVU 6JGP TKPI 4 %QPUKFGT VJG HQNNQYKPI # VJG QTFGT QH ( OC[ DG UVCVGOGPVU 4 $ VJG QTFGT QH ( OC[ DG + KU C $QQNGCP TKPI / 4 % VJG QTFGT QH ( KU RTKOG ++ # KHH / KU C OCZKOCN / VJG EJCTCEVGTKUVKE QH ( KU GTQ KFGCN 1T 6JGP # $QVJ + CPF ++ JQNF ;QW CUMGF VGP QH [QWT ENCUUOCVGU $ 1PN[ + JQNFU CDQWV VJGKT YGKIJV 1P VJG DCUKU QH % 1PN[ ++ JQNFU VJKU KPHQTOCVKQPГ [QW UVCVGF VJCV VJG $QVJ + CPF ++ CTG HCNUG CXGTCIG YGKIJV QH CNN UVWFGPVU KP 1T [QWT EQNNGIG KU RQWPFU 6JKU KU CP GZCORNG QH 6JG OGCUWTG OQUV WPCHHGEVGF D[ QWVNKGTU KU VJG # KPHGTGPVKCN UVCVKUVKEU # OGFKCP $ UCORNG $ VTKOOGF OGCP % RQRWNCVKQP % OGCP FGUETKRVKXG UVCVKUVKEU TCPIG
  • 15. #7) ++ .GV 4 DG CP KPVGITCN FQOCKP CPF 1T 4=Z? DG VJG RQN[PQOKCN TKPI QXGT 4 6JGP #V C EGTVCKP ECT UGTXKEG UVCVKQP YKVJ # 4=Z? KU CP WENKFGCP FQOCKP UGTXKEG HCEKNKV[ HQT QPN[ QPG ECT CPF PQ YCKVKPI URCEGГ ECTU CTTKXG HQT $ +H GXGT[ KFGCN QH 4 KU RTKPEKRCN VJGP UQ KU VTWG HQT 4=Z? UGTXKEG KP 2QKUUQP OCPPGT CV VJG TCVG QH QPG RGT JQWT CPF UGTXKEG VKOG % 4=Z? KU CP KPVGITCN FQOCKP KU CNUQ GZRQPGPVKCN YKVJ CP CXGTCIG 4=Z? KU C HKGNF QPG JQWTГ YKVJ KPFGRGPFGPEG QH 1T UGTXKEG VKOGU CPF KPVGTCTTKXCN VKOGUГ 6JG CXGTCIG UEQTG HQT C ENCUU QH VJG RTQDCDKNKV[ VJCV CP KPEQOKPI UVWFGPVU YCU +H VJG HGOCNG ECT HKPFU VJG UGTXKEG HCEKNKV[ CXCKNCDNG UVWFGPVU KP C ENCUU CXGTCIGF Г VJGP VJG OCNG UVWFGPVU KP VJG ENCUU KU CXGTCIGF # # $ % $ G %QPUKFGT VJG HQNNQYKPI VYQ % UVCVGOGPVU + +H # KU C VTKCPIWNCT OCVTKZ CPF 0QPG QH VJG CDQXG PQ GPVT[ QP VJG OCKP FKCIQPCN KU GTQГ VJGP # KU UKPIWNCT +H VJG VTCPUHQTOCVKQP 6 4P o 4 ++ +H # KU VTKCPIWNCT OCVTKZ CPF CP KU FGHKPGF D[ 6 : : #:Г YJGTG # GPVT[ QP VJG OCKP FKCIQPCN KU KU CP[ P ª P U[OOGVTKE OCVTKZГ VJGP GTQГ VJGP # KU KPXGTVKDNG 6: ; KU GSWCN VQ 6JGP # 6: : #; 6; # QPN[ + KU VTWG $ 6: 6; $ QPN[ ++ KU VTWG % 6: : #; 6; % DQVJ + CPF ++ CTG VTWG DQVJ + CPF ++ CTG HCNUG 6: ; #: 6; =261
  • 16. #7) ++ 1T .GV 6 DG VJG NKPGCT VTCPUHQTOCVKQP 6JG UVCHH TGRNCEGOGPV RQNKE[ KP TGRNCEGOGPV OQFGN QP 4 TGRTGUGPVGF D[ VJG OCVTKZ # CTKUGU FWG VQ TGUKIPCVKQPГ TGVKTGOGPVГ QT FGCVJ QH C UVCHH § · OGODGT HTQO VKOG VQ VKOG ¨ © ¸ Г VJGP VJG EJCTCEVGTKUVKE ¹ $ KU NKMG TGRNCEGOGPV RQNKE[ HQT KVGOU YJQUG XCNWGU FGVGTKQTCVG XCNWGU QH 6 CTG ITCFWCNN[ % ECP DG GCUKN[ HQTOWNCVGF # Г DGECWUG RGQRNG TGVKTG CV MPQYP VKOG FQGU PQV [KGNF VJG QRVKOWO $ Г TGRNCEGOGPV KPVGTXCN § · % Г K .GV # ¨ ¸ DG VJG OCVTKZГ © ¹ VJGP VJG EQTTGURQPFKPI SWCFTCVKE KГ K HQTO KU # Z [ 1T $ Z Z[ [ % Z [ # OKZGF UVTCVGI[ ICOG KP )COG Z Z[ [ 1T VJGQT[ ECP DG UQNXGF D[ 9JKEJ QH VJG HQNNQYKPI KU PQV CP CUUWORVKQP WPFGTN[KPI VJG HWPFCOGPVCN RTQDNGO QH 13 KP # /CVTKZ OGVJQF +PXGPVQT[ /QFGN ! # GOCPF KU MPQYP CPF WPKHQTO $ )TCRJKECN OGVJQF $ *QNFKPI EQUV RGT WPKV RGT VKOG RGTKQF KU EQPUVCPV % #NIGDTCKE OGVJQF % .GCF VKOG KU PQV GTQ 5VQEM QWVU 5JQTVCIGU CTG PQV RGTOKVVGF #NN QH VJG CDQXG
  • 17. #7) ++ .GV RГ SГ T  +H VJG XGEVQTU .GV D Г Г D Г  4 CPF E Г Г Г E Г Г  4 § R· § S· §T· .GV 6 DG C NKPGCT VTCPUHQTOCVKQP ¨ S¸Г ¨ T ¸Г ¨ R¸ ¨ ¸ ¨ ¸ ¨ ¸ CTG OWVWCNN[ HTQO 4 KPVQ 4 UWEJ VJCV 6DK EK ©T¹ © R¹ © S¹ K Г 6JGP QTVJQIQPCN KP WENKFGCP URCEG 4 § · # 6 ZГ [ ¨ ZГ Z Г [Г Z Г [ ¹ © ¸ VJGP # R S T § · $ 6 ZГ [ ¨ Z Г [Г Z [Г Z Г [ ¹ © ¸ $ R S T z § · % RS ST TR z % 6 ZГ [ ¨ [Г Z Г [Г Z [¸ © ¹ RS ST TR 5WEJ C NKPGCT VTCPUHQTOCVKQP 1T FQGU PQV GZKUV 9JKEJ QH VJG HQNNQYKPI KU PQV QPG QH 1T +P C OWNVK UVCIG UCORNKPI VJG RTKPEKRNGU QH UCORNKPI ! # VJG UCOG PWODGT QH RGQRNG CTG # 5CORNKPI WPKVU OWUV DG UVWFKGF OQTG VJCP QPEG KFGPVKHKCDNG CPF ENGCTN[ FGHKPGF $ FKHHGTGPV TGURQPFGPVU CTG $ 5CORNKPI WPKVU CTG PQV UVWFKGF OQTG VJCP QPEG % C UGSWGPEG QH UCORNGU CTG KPVGTEJCPIGCDNG FTCYP HTQO CNTGCF[ UGNGEVGF % 5CORNKPI WPKVU OWUV DG UCORNGU DWV QPN[ VJG NCUV KPFGRGPFGPV QH GCEJ QVJGT UCORNG KU UVWFKGF C UGSWGPEG QH UCORNGU CTG 5CORNKPI WPKVU OWUV DG FTCYP HTQO CNTGCF[ UGNGEVGF TGRTGUGPVCVKXG CPF KPENWFG UCORNGU CPF GCEJ QPG QH VJGO OCNGU CPF HGOCNGU KU UVWFKGF =261
  • 18. #7) ++ %QPUKFGT VJG HQNNQYKPI VYQ 1T UVCVGOGPVU 6JG VKOG TGSWKTGF VQ CUUGODNG CP GNGEVTQPKE EQORQPGPV KU PQTOCNN[ + +H # KU C PKNRQVGPV OCVTKZ VJGP FKUVTKDWVGF YKVJ C OGCP QH + # KU PQP UKPIWNCT OKPWVGU CPF C UVCPFCTF FGXKCVKQP QH ++ # KU VTKCPIWNCT OCVTKZ YKVJ CNN OKPWVGU (KPF VJG RTQDCDKNKV[ FKCIQPCN GNGOGPVU CTG GTQГ VJCV VJG VKOG TGSWKTGF VQ CUUGODNG VJGP # KU PKNRQVGPV PKPG EQORQPGPVU KU ITGCVGT VJCP 6JGP OKPWVGU # 1PN[ + KU VTWG # $ $ 1PN[ ++ KU VTWG % % $QVJ + CPF ++ CTG VTWG $QVJ + CPF ++ CTG HCNUG .GV I Г I DG VYQ NKPGCTN[ 1T KPFGRGPFGPV UQNWVKQPU QH VJG +P C UVWF[ QH CVVKVWFGU VQ 7PKXGTUKV[ EQPUVCPV EQGHHKEKGPV GSWCVKQP [ RQNKEKGUГ C TGUGCTEJGT KPKVKCNN[ EJQUG C[ C[ 6JGP VJG 9TQPUMKCP HKTUV [GCT UVWFGPVUГ UGEQPF QH IГ I KU EQPUVCPV KH CPF QPN[ [GCT UVWFGPVU CPF VJKTF [GCT KH UVWFGPVU 6JGP VJG TGUGCTEJGT # C UGNGEVGF OCNG CPF HGOCNG $ C UVWFGPVU HTQO GCEJ [GCT ITQWRГ YJQ YGTG HKPCNN[ KPVGTXKGYGF 6JG % C UCORNKPI RTQEGFWTG WUGF KP VJKU C UVWF[ YCU 1T # 2TQDCDKNKV[ UCORNKPI 6JG EGPVTCN NKOKV VJGQTGO VGNNU WU $ 5VTCVKHKGF UCORNKPI VJCV VJG UCORNKPI FKUVTKDWVKQP QH VJG % /WNVK UVCIG UCORNKPI OGCP KU CRRTQZKOCVGN[ PQTOCN /WNVK RJCUG UCORNKPI 9JKEJ QH VJG HQNNQYKPI EQPFKVKQPU KU 6JG EQORNGVG UQNWVKQP QH VJG RCTVKCN PGEGUUCT[ HQT VJG VJGQTGO VQ DG FKHHGTGPVKCN GSWCVKQP XCNKF ! w w # 6JG UCORNG UKG JCU VQ DG NCTIG Г UKP Z UKP [ wZ w[ $ 6JG RQRWNCVKQP XCTKCPEG JCU VQ KU DG UOCNN # C Z [ UKP Z UKP [ D % 9G OWUV DG UCORNKPI HTQO C $ C Z [ EQU Z EQU [ D PQTOCN RQRWNCVKQP % C Z [ UKP Z EQU [ D 6JG RQRWNCVKQP JCU VQ DG C Z [ EQU Z UKP [ D U[OOGVTKE
  • 19. #7) ++ 6JG EJCTCEVGTKUVKE EWTXG QH VJG VYQ 1T 9KUJCTV FKUVTKDWVKQP KU C RCTCOGVGT HCOKN[ QH UWTHCEGU OWNVKXCTKCVG IGPGTCNKCVKQP QH Z C [ D KU # PQTOCN FKUVTKDWVKQP $ DGVC FKUVTKDWVKQP # C EKTENG % OCVTKZ DGVC FKUVTKDWVKQP $ C ITGCV EKTENG EJK USWCTG FKUVTKDWVKQP 6JG KPKVKCN XCNWG RTQDNGO % JGNKZ [ H ZГ [ Г [ Z [Г YJGTG ZГ [ DGNQPIU VQ C FQOCKP Г JCU WPKSWG GNNKRUG UQNWVKQP 1T # KH H ZГ [ KU EQPVKPWQWU KP $ KH H ZГ [ KU EQPVKPWQWU CPF # ECPQPKECN EQTTGNCVKQP ECPPQV DG DQWPFGF QP PGICVKXG DGECWUG % KH H ZГ [ KU EQPVKPWQWU CPF UCVKUHKGU .KRUEJKV EQPFKVKQP HQT # YG VCMG QPN[ RQUKVKXG GKIGP CNN RQKPVU QP XCNWGU HQT CNN HWPEVKQPU H ZГ [ QP 1T $ KV KU IGPGTCNKCVKQP QH VJG 9JKEJ QH VJG HQNNQYKPI UVCVGOGPVU OWNVKRNG EQTTGNCVKQP CTG VTWG HQT VJG HQNNQYKPI FCVC XCNWGU Г Г Г Г Г CPF ! % YG VCMG QPN[ RQUKVKXG USWCTG # 6JG OGCP KU UOCNNGT VJCP VJG TQQV OGFKCP $ 6JG OGCP CPF VJG VTKOOGF YG TGLGEV C PGICVKXG XCNWG OGCP CTG GSWCN % 6JG OGCP CPF OGFKCP CTG GSWCN .GV IГ I DG VYQ FKHHGTGPVKCDNG 6JG OGCP KU NCTIGT VJCP VJG HWPEVKQPU QP CP KPVGTXCN + VJGP OGFKCP +H VJG PWODGT QH EQPUVCPVU CTG VQ DG + 9 IГ I Z Z  + Ÿ IГ GNKOKPCVGF HTQO VJG IKXGP TGNCVKQP I CTG NKPGCTN[ FGRGPFGPV KU LWUV GSWCN VQ VJG PWODGT QH KPFGRGPFGPV XCTKCDNGUГ VJGP VJG ++ 9 IГ I Z z Z  + Ÿ IГ RCTVKCN FKHHGTGPVKCN GSWCVKQP I CTG NKPGCTN[ KPFGRGPFGPV QDVCKPGF D[ GNKOKPCVKPI VJGUG EQPUVCPVU 6JGP # KU CNYC[U C NKPGCT GSWCVKQP QH HKTUV QTFGT # 1PN[ + KU VTWG $ ECP DG C PQP NKPGCT GSWCVKQP QH $ 1PN[ ++ KU VTWG HKTUV QTFGT % KU C NKPGCT GSWCVKQP QH UGEQPF % $QVJ + CPF ++ CTG VTWG QTFGT KU C PQP NKPGCT GSWCVKQP QH $QVJ + CPF ++ CTG HCNUG UGEQPF QTFGT =261
  • 20. #7) ++ 1T 6JG UQNWVKQP QH C JQOQIGPGQWU .GV :Г :Г DG C UGSWGPEG QH KPKVKCN XCNWG RTQDNGO YKVJ KPFGRGPFGPV CPF KFGPVKECNN[ EQPUVCPV EQGHHKEKGPV KU [ ZGZ FKUVTKDWVGF TXU YKVJ VJG EQOOQP EQU Z 6JGP VJG NGCUV RQUUKDNG QTFGT QH VJG FKHHGTGPVKCN GSWCVKQP RFH IKXGP D[ ­ KU ° Z Г ° Z HZ ® # ° ° Г ¯ QVJGTYKUG $ %QPUKFGT VJG UVCVGOGPVU % P + ¦ : P   CU P o f K 2 o K 1T P ++ ¦ K 2 : K P   CU P o f o 6JG TCPFQO XCTKCDNGU : CPF ; CUUWOG QPN[ XCNWGU GCEJГ CPF P EQX :Г ; 6JGP § · +++ P ¦ K © ¸ F ¨ : K ¹ o # : CPF ; PGGF PQV DG 0 Г CU P o f KPFGRGPFGPV 6JGP $ : CPF ; CTG KPFGRGPFGPV # #NN UVCVGOGPVU CTG VTWG % : CPF ; CTG KPFGRGFGPV QPN[ $ 1PN[ UVCVGOGPVU + CPF ++ CTG KH HQWT XCNWGU CTG CPF HQT VTWG DQVJ : CPF ; % 1PN[ UVCVGOGPVU + CPF +++ CTG VTWG 6JG NKPG KU RCTCNNGN VQ ; CZKU 1PN[ UVCVGOGPV + KU VTWG QT : CZKU
  • 21. #7) ++528*+ :25. =261
  • 22. #7) ++528*+ :25.

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