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nV JUDGXDWH FRPPLWWHH XQWLO OHDYLQJ WR DFFHSW DQRWKHU SRVLWLRQ 7KH DXWKRU DSSUHFLDWHV WKH JXLGDQFH RI 'U 0 2KDQLDQ ZKR DVVXPHG fµ WKH GXWLHV RI FKDLUPDQ IROORZLQJ 'U 3HUH]n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n NQRZOHGJHG 7KLV VWXG\ ZDV SHUIRUPHG XQGHU 1$6$ *UDQW 1*5 ,W LV D SOHDVXUH WR WKDQN P\ ZLIH 9DQQ IRU W\SLQJ WKH ILUVW GUDIW [Y PAGE 5 7$%/( 2) &217(176 3DJH $&.12:/('*(0(176 +L /,67 2) 7$%/(6 YLL /,67 2) ),*85(6 YLLL $%675$&7 ; &KDSWHU , ,1752'8&7,21 ,, $33/,&$7,21 2) ',))86,21 7+(25< 72 1(875216 ,1 $ ),66,21,1* 85$1,80 3/$60$ ,,, 18&/(21,& $1$/<6,6 2) (;7(51$//< 02'(5$7(' 85$1,80 3/$60$ ,9 35(6685(7(03(5$785( $1$/<6,6 2) 85$1,80 3/$60$ )8(/ 5(*,21 9 35(6685(7(03(5$785( $1$/<6,6 2) +<'52*(1 3/$60$ &22/$17 5(*,21 9, 35(6685(7(03(5$785( $1$/<6,6 2) +<'52*(185$1,80 0,;,1* 5(*,21 9,, 6,08/$7,21 2) +<'52*(185$1,80 0,;,1* ,1 67($'< )/2: 9,,, 5$',$17 (1(5*< 75$16)(5 $1' 7(03(5$785( ',675,%87,21 ,; &21&/86,216 $1' 5(&200(1'$7,216 $SSHQGLFHV $ 83/$= 352*5$0 Y PAGE 6 7$%/( 2) &217(176&RQW¯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r. WR r. 3UHVVXUH UHTXLUHG IRUFULWLFDO SODVPD UHDFWRU 1HW SRZHU DQG ILVVLRQ IUDJPHQW SRZHU DV D IXQFWLRQ RI FHQWHUOLQH WHPSHUDWXUH (QHUJ\ RI K\GURJHQ FRRODQW QXFOHL GXH WR IORZ DQG WKHUPDO PRWLRQ 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 10 /,67 2) ),*85(6 f§ &RQWLQXHG )LJXUH 3DJH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV [ PAGE 11 $EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH &RXQFLO LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ ),66,21,1* 85$1,80 3/$60$ )25 52&.(7 35238/6,21 %\ +HQU\ )RXVW $WZDWHU 0DUFK &KDLUPDQ 'U 0 2KDQLDQ 0DMRU 'HSDUWPHQW 1XFOHDU (QJLQHHULQJ 6FLHQFHV $ ILVVLRQLQJ XUDQLXP SODVPD IRU XVH DV D KLJK WHPSHUDWXUH VHOIVXVWDLQLQJ HQHUJ\ FRQYHUVLRQ GHYLFH LV VWXGLHG WKHRUHWLFDOO\ 7KH VWXG\ LV RULHQWHG ZLWK SDUn WLFXODU HPSKDVLV RQ WKH XVH RI WKH ILVVLRQLQJ SODVPD DV D KHDW VRXUFH IRU URFNHW SURSXOVLRQ V\VWHPV DOWKRXJK WKH LGHDV DQG PHWKRGV ZKLFK DUH GHYHORSHG FDQ DOVR EH DSSOLHG WR D FORVHG F\FOH KHDWLQJ V\VWHP ZKLFK XWLOL]HV WKH ILVVLRQLQJ SODVPD SULQFLSOH 7KH HIIHFWV RI FRXSOLQJ PHFKDQLVPV EHWZHHQ QXFOHn RQLF DQG SODVPD SURSHUWLHV RI WKH ILVVLRQLQJ SODVPD DUH LGHQWLILHG DQG DQDO\]HG IRU D UHIHUHQFH PRGHO 7KH PRGHO LV EDVHG RQ D FHQWUDO IXHO UHJLRQ RI IXOO\ HQULFKHG XUDQLXP VXUURXQGHG E\ D K\GURJHQ FRRODQW UHJLRQ FRQWDLQHG ZLWKLQ D F\OLQGULFDO JUDSKLWH FKDPEHU 7KH XUDQLXP UHJLRQ [L PAGE 12 VHUYHV DV D KHDW VRXUFH WKURXJKfµ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n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f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n YDQFHG EHFDXVH RI WKH PDJQLWXGH RI WKH FRPSOH[ SK\VLFDO SKHQRPHQD ZKLFK PXVW EH XQGHUVWRRG DQG WKH PDQ\ WHFKQRn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n DFWRUV f 7KH SODVPD FRUH UHDFWRU LV DQ H[FHSWLRQ 7KH LQLWLDO LQWHUHVW LQ WKH JDVHRXV RU SODVPD UHDFWRU FRQFHSW f KDV UHVXOWHG LQ UHVHDUFK GLUHFWHG WRZDUG LWV XVH DV D KLJK SHUIRUPDQFH VSDFH SURSXOVLRQ GHYLFH ,Q RUGHU WR FRPSDUH WKH SURSXOVLRQ SRWHQWLDO RI WKH SODVPD FRUH UHDFWRU ZLWK RWKHU FRQFHSWV WKH VSHFLILF LPSXOVHr , LV XVHG DV D UHODWLYH PHDVXUH RI SHUIRUPDQFH 6LQFH VSHFLILF LPSXOVH LV UHODWHG WR SURSHOODQW PDVV 0f DQG WHPSHUDWXUH 7f WKURXJK WKH UHODWLRQ , D Wf¬P f D VS KLJK WHPSHUDWXUH ORZ PROHFXODU ZHLJKW SURSHOODQW LV GHVLUDn EOH 7KH PD[LPXP SURSHOODQW WHPSHUDWXUH LV OLPLWHG E\ WKH PD[LPXP DOORZDEOH WHPSHUDWXUH RI WKH UHDFWRU IXHO 7KH WHPSHUDWXUH OLPLW LQ D VROLG FRUH UHDFWRU LV WKH IXHO PHOWLQJ SRLQW ZKLOH WKH WHPSHUDWXUH RI D OLTXLG FRUH r6SHFLILF LPSXOVH LV GHILQHG DV HQJLQH WKUXVW SHU XQLW ZHLJKW IORZ UDWH RI SURSHOODQW PAGE 17 a UHDFWRU LV OLPLWHG E\ WKH IXHO ERLOLQJ SRLQW 7KH SODVPD FRUH UHDFWRU FDQ WKHRUHWLFDOO\ RSHUDWH DW DUELWUDULO\ KLJK WHPSHUDWXUHV ,Q SUDFWLFH WKH WHPSHUDWXUH OLPLW ZLOO EH GHWHUPLQHG E\ WKH PD[LPXP IXHO WHPSHUDWXUH IRU ZKLFK WKH SODVPD FRQWDLQPHQW FKDPEHU FDQ EH FRROHG EHORZ LWV PHOWLQJ SRLQW 7KH IROORZLQJ DUH W\SLFDO YDOXHV RI VSHFLILF LPSXOVH f LQGLFDWLQJ WKH KLJK SHUIRUPDQFH WKDW FDQ EH H[SHFWHG IURP D SODVPD UHDFWRU SURSXOVLRQ V\VWHP 7$%/( 7+(25(7,&$/ 63(&,),& ,038/6( , VHFRQGVf &KHPLFDO 5RFNHW 6ROLG &RUH 5HDFWRU /LTXLG &RUH 5HDFWRU 3ODVPD &RUH 5HDFWRU $OWKRXJK WKH SUHVHQW LQWHUHVW LQ SODVPD FRUH UHDFWRUV VHHPV WR EH HQWLUHO\ FRQFHUQHG ZLWK URFNHW SURSXOVLRQ DSSOLFDWLRQV WKH FRQFHSW VKRXOG QRW EH FRQn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r UHVHDUFK ZLOO EH LQGLFDWLYH RI WKH ZLGH UDQJH RI SUREOHPV ZKLFK DUH EHLQJ VWXGLHG 7KH EDVLF SODVPD FRUH FRQFHSW LV VLPSOH $ KHDY\ ORZ YHORFLW\ JDVHRXV IXHO DQG D OLJKW KLJK YHORFLW\ JDVHRXV SURSHOODQW DUH LQMHFWHG LQWR D UHDFWLRQ FKDPEHU KDYLQJ ZDOOV FRPSRVHG RI VRPH VXLWDEOH QHXWURQ UHIOHFWLQJ PDWHULDO )LVVLRQV LQ WKH IXHO KHDW WKH SURn SHOODQW ZKLFK LV WKHQ H[KDXVWHG WKURXJK D QR]]OH WR JLYH WKUXVW 7KH JDVHRXV VWDWH RI WKH IXHO JLYHV WKH SUHYLRXVO\ PHQWLRQHG DGYDQWDJH RI KLJK WHPSHUDWXUHV EXW DOVR FDXVHV D VHULRXV SUREOHPn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r7KH SODVPD FRUH UHDFWRU LV DOVR UHIHUUHG WR DV D JDVHRXV FRUH RU FDYLW\ FRUH UHDFWRU PAGE 19 D[LDO IORZ ZLWK PDJQHWLF FRQWDLQPHQW ZKHHO IORZ DQG FRD[LDO IORZ /f 7KH YRUWH[ IORZ UHDFWRU LOOXVn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n DWHG ZLWK PDJQHWLF FRQWDLQPHQW LQ DGGLWLRQ WR SURSHOODQW GUDJ SUREOHPV VLPLODU WR WKRVH RI WKH YRUWH[ IORZ $QRWKHU YDULDWLRQ LV WKH ZKHHO IORZ FRQFHSW VKRZQ LQ )LJXUH +\GURJHQ SURSHOODQW HQWHUV DQG OHDYHV WKH UHDFWRU WDQn JHQWLDOO\ DORQJ WKH RXWHU SHULSKHU\ DIWHU FLUFXODWLQJ DURXQG WKH LQQHU F\OLQGULFDO XUDQLXP UHJLRQ ,Q WKLV FRQFHSW WKH PDLQ GLIILFXOW\ VWLOO VHHPV WR EH H[FHVVLYH IXHO ORVV GXH WR K\GURJHQXUDQLXP PL[LQJ 7KH FRD[LDO IORZ UHDFWRU LV VKRZQ LQ )LJXUH 7KLV FRQFHSW LV EDVHG RQ WKH FRD[LDO IORZ RI D FHQWUDO F\OLQGULFDO XUDQLXP PDVV DQG D VXUURXQGLQJ DQQXOXV RI K\GURJHQ 1R YRUWH[ PRWLRQ n PAGE 20 3URSHOODQW LQOHW 1HXWURQ UHIOHFWRU )LJXUH 9RUWH[ IORZ SODVPD UHDFWRU 1HXWURQ UHIOHFWRU 3UHVVXUH VKHOO &XUUHQWFDUU\LQJ ? 0DJQHWLF ILHOG FRLOV n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f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n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n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f WKH WUDQVSRUW HTXDWLRQ LV __UYIWWf e fµ 9S UYILWf (W UYf M! U Y³ Wf ; Yf GYn Yn A Gtr Y Ynf UM9f M! WYn Wf GYn Y GIWn (J UYrYAnrAf -! UYn Wf f ([WHUQDO QHXWURQ VRXUFHV DQG GHOD\HG QHXWURQV DUH QRW LQFOXGHG LQ f 7KH V\PEROV DUH GHILQHG LQ WKH XVXDO PDQQHU PAGE 25 I!WYIWWf a DQJXODU QHXWURQ IOX[ QXPEHU RI QHXWURQV SHU XQLW DUHD SHU XQLW WLPH SHU XQLW VROLG DQJOH SHU XQLW VSHHG LQ WKH YROXPH HOHPHQW GU DERXW U WUDYHOLQJ ZLWKLQ WKH VROLG DQJOH GIW LQ WKH GLUHFWLRQ IW DQG KDYLQJ D VSHHG Y LQ GY DW WLPH Wf PDFURVFRSLF FROOLVLRQ FURVV VHFWLRQ IRU QHXWURQV DW U ZLWK VSHHG Y ;Yf VSHFWUXP RI ILVVLRQ QHXWURQV YYf PHDQ QXPEHUV RI QHXWURQV UHVXOWLQJ IURP D ILVVLRQ FDXVHG E\ D QHXWURQ ZLWK VSHHG Y = UYn}Y IWnf PDFURVFRSLF VFDWWHULQJ FURVV VHFWLRQ DW 6 f§ U IRU FKDQJLQJ WKH VSHHG DQG GLUHFWLRQ Yn IWn LQWR D VSHHG DQG GLUHFWLRQ UDQJH GY GIW DW Y nIW 7KH LQGLYLGXDO WHUPV RI f GHVFULEH WKH IROORZLQJ UHDFWLRQV +JU FKDQJH SHU XQLW WLPH LQ WKH QHW QXPEHU RI QHXWURQV RI GLUHFWLRQ IW DQG VSHHG Y LQ GU QHW ORVV SHU XQLW WLPH IURP GU RI QHXWURQV RI GLUHFWLRQ IW LQ GIL E\ OHDNDJH QXPEHU RI QHXWURQV RI GLUHFWLRQ IW DQG VSHHG Y ZKLFK DUH UHPRYHG SHU XQLW WLPH IURP GU E\ DEVRUSWLRQ DQG VFDWWHULQJ FROOLVLRQV QXPEHU RI QHXWURQV RI GLUHFWLRQ DQG fµ fµ fµ VSHHG Y JDLQHG SHU XQLW WLPH P GU IURP ILVVLRQV GXH WR QHXWURQV ZLWK DOO VSHHGV Y DQG GLUHFWLRQV -GYnGIWn= `! QXPEHU RI QHXWURQV RI GLUHFWLRQ IW DQG VSHHG Y JDLQHG SHU XQLW WLPH LQ GU IURP VFDWWHULQJ FROOLVLRQV ZKLFK VFDWWHU QHXWURQV IURP DOO VSHHGV Yn DQG GLUHFWLRQV IWn IW fµ Y ! f§ ]Wr ;GYn GIWn Y=ecf PAGE 26 3RU D VWHDG\ VWDWH QHXWURQ SRSXODWLRQ WKH ILUVW WHUP RI f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a 0H9f DQG VORZHG GRZQ WR VRPH WKHUPDO HQHUJ\ ERXQGDU\ XVXDOO\ WDNHQ DV H9 WR H9f 7KHUPDO QHXWURQV DUH WKRVH ZKLFK KDYH VORZHG GRZQ DQG DUH LQ DSSUR[LPDWH WKHUPDO HTXLOLEULXP DW VRPH FKDUDFWHULVWLF WHPSHUDWXUH 7 RI WKH PHGLXP 7KXV WKH WKHUPDO HQHUJ\ UDQJH H[WHQGV IURP ]HUR WR D IHZ H9 ,Q WKH WKLUG IRXUWK DQG ILIWK WHUPV RI f WKH QHXWURQ VSHHG Y RU Ynf DSSHDULQJ LQ WKH PDFURVFRSLF FURVV VHFWLRQV LV WKH UHODWLYH VSHHG EHWZHHQ QHXWURQ DQG QXFOHXV 6LQFH VROLG FRUH UHDFWRUV W\SLFDOO\ RSHUDWH ZLWK FRRODQW WHPSHUDWXUHV LQ WKH r. WR r. UDQJH FRUUHVSRQGLQJ WR PAGE 27 DQ HQHUJ\ UDQJH N7 H9 WR H9f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r. FRUUHVSRQGLQJ WR DQ HQHUJ\ N7 H9f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n EXWLRQV RI QXFOHL DQG WKHUPDO QHXWURQV FDQ EH GHVFULEHG E\ PAGE 28 PRGLILHG 0D[ZHOO%ROW]PDQQ GLVWULEXWLRQV KDYLQJ D FKDUDFWHUn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f DQG ZLOO QRW EH WUHDWHG KHUH $ FRQFLVH GHVFULSWLRQ RI QHXWURQ WKHUPDOL]DWLRQ SUREOHPV LV JLYHQ E\ %HFNXUWV DQG :LUW] Bf 0LFURVFRSLF WKHUPDO QHXWURQ DEVRUSWLRQ FURVV VHFWLRQV DUH IXQFWLRQV RI WKH UHODWLYH QHXWURQ VSHHG ,Q JHQHUDO WKLV VSHHG GHSHQGHQW FURVV VHFWLRQ LV QRW WKH VDPH DV WKH H[SHULPHQWDOO\ PHDVXUHG DYHUDJH FURVV VHFWLRQ ,Q RUGHU WR XVH PHDVXUHG DEVRUSWLRQ FURVV VHFWLRQV LQ UHDFWRU FDOFXODWLRQV WKH H[SHULPHQWDOO\ GHWHUPLQHG YDOXHV PXVW EH SURSHUO\ ZHLJKWHG WR LQFOXGH WKH HIIHFWV RI UHODWLYH VSHHG M/f 7KH SUHFHGLQJ GLVFXVVLRQ RI WKH %ROW]PDQQ WUDQVSRUW HTXDWLRQ KDV FRQVLGHUHG RQO\ QHXWURQV LQ D VROLG FRUH UHDFWRU 7R VXPPDUL]H WKLV GLVFXVVLRQ WKH DSSOLFDWLRQ PAGE 29 * RI WKH %ROW]PDQQHTXDWLRQ WR VROLG FRUH UHDFWRUV FDQ EH FKDUDFWHUL]HG E\ WZR IHDWXUHV f WKH DVVXPSWLRQ WKDW HSLWKHUPDO QHXWURQV DUH VORZHG GRZQ E\ FROOLVLRQV ZLWK VWDWLRQDU\ QXFOHL DQG f WKH PHWKRGV RI WKHUPDOL]DWLRQ WKHRU\ DUH XVHG WR WUHDW WKH HIIHFWV RI QXFOHDU PRWLRQ DQG FKHPLFDO ELQGLQJ RQ WKHUPDO QHXWURQ FROOLVLRQV ,Q RUGHU WR SURSHUO\ DSSO\ WKH %ROW]PDQQ HTXDWLRQ WR QHXWURQV LQ SODVPD FRUH UHDFWRU VHYHUDO PRGLILFDWLRQV DUH QHFHVVDU\ 7KH QHHG IRU WKHVH PRGLILFDWLRQV LV GXH WR WKH H[WUHPHO\ KLJK WHPSHUDWXUHV RI WKH SODVPD FRUH UHDFWRU DV FRPSDUHG WR VROLG FRUH UHDFWRU WHPSHUDWXUHV 7HPSHUDn WXUHV RQ WKH RUGHU RI r. DUH DQWLFLSDWHG LQ WKH SODVPD FRUH UHDFWRU 6FDWWHULQJ DQG DEVRUELQJ QXFOHL DW WKLV HOHYDWHG WHPSHUDWXUH KDYH DQ HQHUJ\ N7 H9 5HFDOOLQJ WKDW VFDWWHULQJ DQG DEVRUELQJ QXFOHL LQ D VROLG FRUH UHDFWRU KDYH D PD[LPXP HQHUJ\ RI DERXW H9 DQG H9 UHVSHFWLYHO\ LW LV FOHDU WKDW QXFOHDU PRWLRQ EHFRPHV D PRUH LPSRUWDQW FRQVLGHUDWLRQ LQ D SODVPD FRUH UHDFWRU 7R IRUPDOO\ DFFRXQW IRU WKH SRVVLEOH HIIHFWV RI WKLV LQFUHDVHG QXFOHDU PRWLRQ WKH XVXDO IRUP RI WKH %ROW]PDQQ HTXDWLRQ f LV PRGLILHG WR LQFOXGH WKH GHSHQGHQFH RI FROOLVLRQ SURFHVVHV RQ WKH UHODWLYH VSHHG 7KH IROORZLQJ V\PEROV IRU WKH VHYHUDO YHORFLWLHV DUH XVHG PAGE 30 f!fµ f§9 Y Y QHXWURQ YHORFUW\ Y! Q YL Q Q Q 9f± 9f±7 QXFOHDU YHORFLW\ 1 1 Y UHODWLYH YHORFLW\ 7KH UHODWLYH VSHHG Y LV WKHQ GHILQHG DV Y Y Y 9$7 ZKHUH Y ( P Q 1 Qn Q f &ROOLVLRQ SURFHVVHV DUH UHODWHG WR WKH UHODWLYH VSHHG WKURXJK WKH QHXWURQ IOX[ M! QUYAWfY DQG WKH HQHUJ\ GHSHQGHQW PLFURVFRSLF FURVV VHFWLRQV D (f ZKHUH QUY6AWf LV WKH QHXWURQ GHQVLW\ )RU D QHXWURQ RI VSHHG YQ DQG D QXFOHXV RI VSHHG 91 WKH FROOLVLRQ SURFHVVHV RI f PXVW QRZ EH LQWHJUDWHG RYHU DOO QXFOHDU VSHHGV DQG GLUHFWLRQV AM 7KH H[SUHVVLRQV IRU WKHVH FROOLVLRQ SURFHVVHV EHFRPH 1HXWURQV UHPRYHG E\ DEVRUSWLRQ DQG VFDWWHULQJ 1HXWURQV JDLQHG E\ ILVVLRQ Q GYn Q GIWYYn f1U9 WfDI Yf ; PAGE 31 1HXWURQV JDLQHG E\ VFDWWHULQJ LQ G9 9 1 1 F¼/ t 1 GYn 1 Yn Q GAnQ1 "Y11WfDV"YnAYAnQAQf Q [ QIUYn W`Yn n Q Q Q 7KH JHQHUDO %ROW]PDQQ HTXDWLRQ IRU QHXWURQV LQ D ILVVLRQLQJ SODVPD FDQ QRZ EH ZULWWHQ DV Q UW W UYQ¯WM YML fµ 9QUYQAQWf G9 1 [Yf DYf± PL r GYn 1 1 Q GA1 "91A1W`DWYfQ"YQAQW`YnQ GAn fµ QY Yn Qf1 U 91A1 Wf DI Yf G9 n LW I GYn 1 1 Q [ QIUY W`Yn A n Q Q Q GAnQ1"91A1WfDV"YAY³QAQf [ QUYQAQWfYnQ f (TXDWLRQ f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n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f LQGLFDWH WKDW D W\SLFDO FULWLFDO XUDQLXP PDVV FDQ EH DFKLHYHG ZLWK DQ DWRP GHQVLW\ RI DSSUR[LPDWHO\ 1X FF :LWK DQ DYHUDJH HSLWKHUPDO VFDWWHULQJ FURVV VHFWLRQ RI EDUQV WKH VFDWWHULQJ PHDQ IUHH SDWK LQ XUDQLXP LV ;X6 f¯ 99X6 f¯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n PHQWV WKH ]HUR SRZHU SODVPD UHDFWRU LV VHHQ WR EH ZHOO PRGHUDWHG ZLWK DQ HSLWKHUPDO QHXWURQ VSHFWUXP GHWHUPLQHG E\ VORZLQJ GRZQ FROOLVLRQV LQ JUDSKLWH DQG D WKHUPDO VSHFWUXP GHWHUPLQHG E\ WKH JUDSKLWH WHPSHUDWXUH ([WHQVLYH RQH DQG WZR GLPHQVLRQDO WUDQVSRUW WKHRU\ FDOFXODWLRQV KDYH EHHQ XVHG WR VWXG\ WKH HIIHFWV RI PDWHULDOV WHPSHUDWXUH DQG GLPHQVLRQV RQ WKH FULWLFDOLW\ RI JDVHRXV FRUH UHDFWRUV f )RU WKH SXUSRVH RI FDOFXODWLQJ GHWDLOHG GHVLJQ SDUDPHWHUV LW ZLOO SUREDEO\ EH QHFHVVDU\ WR XVH WKHPRUH H[DFW FRPSXWDWLRQDO PHWKRGV PAGE 35 VXFK DV WKHVH +RZHYHU VLQFH WKH UHDFWRU LV YHU\ WKHUPDO LW LV UHDVRQDEOH WR H[SHFW WKDW GLIIXVLRQ WKHRU\ PHWKRGV ZLOO VXIILFH IRU VXUYH\ DQG FRPSDULVRQ FDOFXODWLRQV ,Q D SDUDPHWULF VWXG\ RI JDVHRXV FRUH UHDFWRUV XVLQJ ERWK WUDQVSRUW DQG GLIIXVLRQ WKHRULHV 3OXQNHWW Bf KDV GHPRQn VWUDWHG WKDW PXOWLJURXS GLIIXVLRQ WKHRU\ LV DGHTXDWH IRU FDOFXODWLRQV RI WKLV W\SH ,Q WKH SUHVHQW VWXG\ WKH HIIHFWV RI HOHYDWHG WHPSHUDWXUHV DQG WKH UHVXOWLQJ H[n DJJHUDWHG QXFOHDU PRWLRQ ZLOO EH WUHDWHG ZLWKLQ WKH IUDPHZRUN RI PXOWLJURXS GLIIXVLRQ WKHRU\ 7KLV DSSURDFK OHDGV WR WKH GHYHORSPHQW RI FDOFXODWLRQDO PHWKRGV ZKLFK DUH GHVLJQHG VSHFLILFDOO\ IRU DSSOLFDWLRQ WR GLIIXVLRQ WKHRU\ FRGHV 6XFK PHWKRGV PD\ EH GLUHFWO\ DSSOLFDEOH WR WUDQVSRUW WKHRU\ FRGHV ,I QRW LW ZLOO EH QHFHVVDU\ WR LQFOXGH WKH HIIHFWV RI QXFOHDU PRWLRQ E\ UHGXFWLRQ RI HTXDWLRQ f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f WKH PXOWLJURXS GLIIXVLRQ HTXDWLRQ LV 'L9`fLUf =WB!LUf ;M6Uf 1 , FM! Uf ' M O MAL f ZKHUH 1 ( e ' IFL DL E \ H O O VL!J Y L 1 VUf [ LU! IRU B L B 1 1 QXPEHU RI JURXSV 7KH V\PEROV DUH GHILQHG DV !L Uf QHXWURQ IOX[ LQ LWK JURXS DW SRVLWLRQ U GLIIXVLRQ FRHIILFLHQW IRU WKH LWK JURXS eW WRWDO UHPRYDO IURP LWK JURXS L ( DEVRUSWLRQ FURVV VHFWLRQ IRU WKH LWK L JURXS %A WUDQVYHUVH EXFNOLQJ IRU WKH LWK JURXS e fµ VL!M VFDWWHULQJ IURP JURXS L WR JURXS M ; LQWHJUDO RI WKH ILVVLRQ VSHFWUXP RYHU WKH HQHUJ\ UDQJH UHSUHVHQWHG E\ JURXSL PAGE 37 6Uf Y;IfL ILVVLRQ VRXUFH GLVWULEXWLRQ DYHUDJH QXPEHU RI QHXWURQ SURGXFHG E\ ILVVLRQ LQ WKH LWK JURXS WLPHV WKH ILVVLRQ FURVV VHFWLRQ LQ WKH LWK JURXS ; FULWLFDOLW\ HLJHQYDOXH 7KH HIIHFWV RI QXFOHDU PRWLRQ RI DEVRUELQJ QXFOHL DQG VFDWWHULQJ QXFOHL ZLOO EH WUHDWHG VHSDUDWHO\ 7KH DEn VRUSWLRQ FURVV VHFWLRQ (D ZLOO LQFOXGH WKH HIIHFWV RI WKH PRWLRQ RI XUDQLXP QXFOHL LQ WKH WUHDWPHQW RI WKH 'RSSOHU EURDGHQLQJ RI WKH XUDQLXP DEVRUSWLRQ UHVRQDQFHV 7KH VFDWWHULQJ FURVV VHFWLRQ ( \ n VL!M ZLOO LQFOXGH WKH HIIHFWV RI WKH PRWLRQ RI K\GURJHQ QXFOHL (IIHFW RI 7KHUPDO 0RWLRQ RI $EVRUELQJ 1XFOHL RQ (SLWKHUPDO 1HXWURQV 7KH PDFURVFRSLF DEVRUSWLRQ FURVV VHFWLRQ IRU WKH LWK JURXS LV WKH IOX[ ZHLJKWHG DYHUDJH FURVV VHFWLRQ ZKLFK LV GHILQHG DV D L ( ( XL ( (f !U(fG( D /L ( 8L FM!U(fG( (f± f ZKHUH ( DQG (7 DUH WKH XSSHU DQG ORZHU ERXQGDULHV L L UHVSHFWLYHO\ RI WKH LWK JURXS ,Q WKH PAGE 38 UDQJH WKH XUDQLXP PLFURVFRSLF DEVRUSWLRQ FURVV VHFWLRQ D (f H[KLELWV D ODUJH QXPEHU RI UHVRQDQFHV f 7KH XVXDO SURFHGXUH LQ HYDOXDWLQJ WKH LQWHJUDO LQ WKH QXPHUDWRU RI f LV WR DVVXPH WKDW HDFK ZHOO GHILQHG UHVRQDQFH LV DFFXUDWHO\ UHSUHVHQWHG E\ WKH VLQJOH OHYHO %UHLW:LJQHU IRUPXOD Mf 7KLV IRUPXOD H[SUHVVHV WKH HQHUJ\ GHSHQGHQW DEVRUSWLRQ FURVV VHFWLRQ DV RD(f QrJM U U Q \ ( (Tf 7f f 7KH V\PERO (T GHQRWHV WKH HQHUJ\ RI WKH UHVRQDQFH DQG (A LV WKH NLQHWLF HQHUJ\ DVVRFLDWHG ZLWK WKH UHODWLYH YHORFLW\ EHWZHHQ QHXWURQ DQG QXFOHXV ,Q WKH FDVH RI D VWDWLRQDU\ QXFOHL (U LV JLYHQ E\ ( U f ZKHUH YQ LV WKH QHXWURQ VSHHG LQ WKH ODERUDWRU\ FRRUGLQDWH V\VWHP DQG \ LV WKH UHGXFHG PDVV DV GHILQHG E\ P0 P 0 f ZLWK P DQG 0 UHSUHVHQWLQJ WKH PDVV RI WKH LQFLGHQW QHXWURQ DQG WKH WDUJHW QXFOHXV UHVSHFWLYHO\ 7KH V\PEROV U DQG 7Q DUH WKH UDGLDWLYH FDSWXUH DQG QHXWURQ ZLGWKV f ZKLFK GHILQH D WRWDO ZLGWK I DV U Q U f PAGE 39 7KH UHGXFHG QHXWURQ ZDYHOHQJWK LV JLYHQ E\ r 7U WW \( f O f 7KH VWDWLVWLFDO VSLQ IDFWRU J7 f LV HTXDO WR f f ZKHUH LV WKH VSLQ TXDQWXP QXPEHU RI WKH FRPSRXQG QXFOHXV IRUPHG E\ WKH WDUJHW QXFOHXV DQG WKH QHXWURQ DQG , LV WKH VSLQ TXDQWXP QXPEHU RI WKH WDUJHW QXFOHXV 7KH SHDN YDOXH RI WKH WRWDO DEVRUSWLRQ SOXV VFDWWHULQJf FURVV VHFWLRQ DA RFFXUV DW ( DQG KDV WKH YDOXH D4 ULHJf 8VLQJ f DQG f WKH DEVRUSWLRQ FURVV VHFWLRQ EHFRPHV F ( f D U R U < L ( ( U f ,I WKH DEVRUELQJ QXFOHL ZHUH VWDWLRQDU\ (D ZRXOG EH HYDOXDWHG IURP f ZLWK D ( f JLYHQ E\ f DQG &O / (U JLYHQ E\ f ,Q UHDOLW\ DOO DEVRUELQJ QXFOHL KDYH D NLQHWLF HQHUJ\ DW OHDVW DV JUHDW DV WKDW FRUUHVSRQGLQJ WR WKH WHPSHUDWXUH RI WKH VXUURXQGLQJ PHGLXP 7KH WKHUPDO PRWLRQ DVVRFLDWHG ZLWK WKH NLQHWLF HQHUJ\ HIIHFWLYHO\ PAGE 40 VKLIWV WKH HQHUJ\ DW ZKLFK QHXWURQV DUH DEVRUEHG WR KLJKHU RU ORZHU YDOXHV GHSHQGLQJ RQ WKH GLUHFWLRQ RI WKH WDUJHW QXFOHL ZLWK UHVSHFW WR WKDW RI WKH LQFLGHQW QHXWURQ 7KLV HQHUJ\ VKLIW LV UHIHUUHG WR DV 'RSSOHU EURDGHQLQJ f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r. WR r. UDQJH DV FRPSDUHG WR D PD[LPXP RI DERXW r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f 7KH UHODWLYH NLQHWLF HQHUJ\ RI WKH QHXWURQQXFOHXV FROOLVLRQ LV ( Q 1 f ZKHUH YQ LV WKH YHORFLW\ RI WKH LQFLGHQW QHXWURQ 91 WKH YHORFLW\ RI WKH QXFOHXV DQG \ WKH UHGXFHG PDVV DV GHILQHG E\ f ([SDQGLQJ f WKH UHODWLYH HQHUJ\ FDQ EH ZULWWHQ DV ( U e P ( Q r e 0 ( 1 Q 1 P 0 f ZKHUH WKH IROORZLQJ UHODWLRQV KDYH EHHQ XVHG ( PY NLQHWLF HQHUJ\ RI WKH QHXWURQ Q Q A Q 09Q] NLQHWLF HQHUJ\ RI WKH QXFOHXV S PY PRPHQWXP RI WKH QHXWURQ Q Q 09f±B PRPHQWXP RI WKH QXFOHXV K 1RWH WKDW LI WKH QXFOHL ZHUH VWDWLRQDU\ WKH VHFRQG DQG WKLUG WHUPV RI f ZRXOG EH ]HUR DQG WKH H[SUHVVLRQ UHGXFHV WR f 7KH NLQHWLF HQHUJ\ RI QXFOHL DW WKH PD[LPXP VROLG FRUH IXHO WHPSHUDWXUH RI r. LV H9 PAGE 42 a ZKLOH D QHXWURQ DW WKH ORZHVW XUDQLXP UHVRQDQFH KDV DQ HQHUJ\ RI H9 )URP WKHVH YDOXHV WKH VHFRQG WHUP RI f LV VHHQ WR EH PXFK VPDOOHU WKDQ WKH ILUVW VLQFH WKH ODUJHVW UDWLR RI WKH VHFRQG WR WKH ILUVW WHUP LV DSSUR[LPDWHO\ 7KHUHIRUH WKH VHFRQG WHUP LV FRQn VLGHUHG QHJOLJLEOH LQ GHDOLQJ ZLWK VROLG IXHO UHVRQDQFH DEVRUSWLRQ 7KH WKLUG WHUP RI f LV SURSRUWLRQDO WR WKH FRPSRQHQW RI PRPHQWXP RI WKH WDUJHW QXFOHXV SDUDOOHO WR WKH GLUHFWLRQ RI PRWLRQ RI WKH LQFLGHQW QHXWURQ 'XH WR LWV ODUJH PDVV DV FRPSDUHG WR WKDW RI WKH QHXWURQ WKH QXFOHXV KDV D ODUJH HQRXJK PRPHQWXP WR FDXVH WKH WKLUG WHUP WR KDYH D VLJQLILFDQW HIIHFW RQ WKH UHODWLYH HQHUJ\ 'HQRWLQJ E\ 9 WKH FRPSRQHQW RI WKH YHORFLW\ RI WKH QXFOHXV ] ZKLFK LV SDUDOOHO WR WKDW RI WKH QHXWURQ WKH WKLUG WHUP FDQ EH ZULWWHQ DV Q 1 IPY Qn m9 P 0 P 0 \Y 9 . Q ] f 8VLQJ f DQG WKH H[SUHVVLRQ IRU WKH UHODWLYH NLQHWLF HQHUJ\ IRU VWDWLRQDU\ QXFOHL ( f§ L \Y HTXDWLRQ f = LV VLPSOLILHG DV ( U f§ ( P Q \9QY] ( \( 9] f 6LQFH RQO\ WKH QXFOHDU PRWLRQ SDUDOOHO WR WKH QHXWURQ GLUHFWLRQ LV FRQVLGHUHG WR EH VLJQLILFDQW WKH PAGE 43 SUREDELOLW\ RI D QHXWURQ FROOLVLRQ ZLWK D QXFOHXV ZLWK YHORFLW\ 9 LV SURSRUWLRQDO WR WKH QXPEHU RI QXFOHL ZLWK WKLV YHORFLW\ 7KH QXPEHU RI VXFK QXFOHL LV JLYHQ E\ WKH RQH GLPHQVLRQDO 0D[ZUHOOLDQ GLVWULEXWLRQ 0 rr`rtf Hb9K7 f +HQFH WKH HIIHFWLYH FURVV VHFWLRQ IRU D QHXWURQ ZLWK NLQHWLF HQHUJ\ ( LV (f II(U`SY]fDY] f 8VLQJ HTXDWLRQ f WKH HIIHFWLYH UHVRQDQFH DEVRUSWLRQ FURVV VHFWLRQ LV 2D(f 2D(U0 ( ( \( G( U \( f 7KLV FURVV VHFWLRQ LV HYDOXDWHG E\ VXEVWLWXWLQJ f DQG f LQWR f 8VLQJ WKH QRWDWLRQ RI 'UHVQHU f WKLV VXEVWLWXWLRQ \LHOGV XD (f D4 A 9 [f f ZKHUH <H[f W) H[S > [ \f@ \ G\ f PAGE 44 DQG [ ( (`U \ a (U (`U )$ $ \(4N701f! f 'RSSOHU EURDGHQHG UHVRQDQFH DEVRUSWLRQ FURVV VHFWLRQ LQ VROLG IXHO UHDFWRUV DUH FRPSXWHG XVLQJ f ZLWK g[f HYDOXDWHG IURP WDEXODWHG YDOXHV EDVHG RQ WKH UHVRn QDQFH SDUDPHWHUV ) DQG (J DQG WKH IXHO WHPSHUDWXUH 7 'RSSOHU EURDGHQLQJ LQ D SODVPD FRUH UHDFWRU LVn H[SHFWHG WR GLIIHU TXDQWLWDWLYHO\ IURP WKH VROLG FRUH UHDFWRU GXH WR WKH ODUJHU NLQHWLF HQHUJ\ RI WKH SODVPD FRUH IXHO QXFOHL 7KH IROORZLQJ WUHDWPHQW RI WKH 'RSSOHU HIIHFW LQ D SODVPD UHDFWRU LV EDVHG RQ WKH DVVXPSWLRQ WKDW WKH VLQJOH OHYHO %UHLW:LJQHU IRUPXOD f DFFXUDWHO\ GHVFULEHV UHVRQDQFH DEVRUSWLRQ FURVV VHFWLRQV 7KH UHODWLYH NLQHWLF HQHUJ\ RI WKH QHXWURQQXFOHXV FROOLVLRQ LV DJDLQ JLYHQ E\ HTXDWLRQ f )URP WKH VDPH DUJXPHQWV XVHG LQ WKH LQYHVWLJDWLRQ RI WKH 'RSSOHU HIIHFW LQ VROLG FRUH UHDFWRUV WKH WKLUG WHUP RI f ZLOO KDYH D VLJQLILFDQW HIIHFW RQ WKH UHODWLYH HQHUJ\ LQ WKH SODVPD IXHO UHDFWRU DOVR )RU IXHO QXFOHL DW D WHPSHUDWXUH RI r. DQG D QHXWURQ DW WKH HQHUJ\ RI WKH ILUVW XUDQLXP UHVRQDQFH WKH UDWLR RI WKH VHFRQG WR WKH ILUVW WHUP LV RU IRUW\ WLPHV ODUJHU WKDQ WKH FRUUHVSRQGLQJ UDWLR IRU WKH VROLG FRUH UHDFWRU 7KXV WKH VHFRQG WHUP RI f FDQ UHVXOW LQ PAGE 45 D VLJQLILFDQW FRUUHFWLRQ WR WKH UHODWLYH HQHUJ\ DQG KHQFH WR WKH DPRXQW RI UHVRQDQFH DEVRUSWLRQ $W WKH H[WUHPHO\ KLJK WHPSHUDWXUHV ZKLFK DUH EHLQJ FRQVLGHUHG LQ WKH SODVPD FRUH UHDFWRU WKLV FRUUHFWLRQ PD\ VWURQJO\ LQIOXHQFH UHn DFWRU FULWLFDOLW\ DQG VWDELOLW\ ([SDQGLQJ HTXDWLRQ f DV EHIRUH DQG UHWDLQLQJ DOO WHUPV WKH UHODWLYH HQHUJ\ LV (U ( _ \91 \( 9] f ZKHUH ( a \YQ DJDLQ GHQRWHV WKH UHODWLYH NLQHWLF HQHUJ\ RI WKH QHXWURQ VWDWLRQDU\ QXFOHL 7KH UHODWLYH HQHUJ\ LV QRZ D IXQFWLRQ RI DOO WKUHH YHORFLW\ FRPSRQHQWV 9 9 9 f [ \ ] UDWKHU WKDQ WKH VLQJOH FRPSRQHQW 9] DV LQ WKH FDVH RI VROLG IXHO QXFOHL +HQFH WKH SUREDELOLW\ RI £ QHXWURQ FROOLVLRQ ZLWK D QXFOHXV ZLWK YHORFLW\ 9 9 9 9 f LV SURSRUWLRQDO MQ [ \ ] WR WKH WKUHH GLPHQVLRQDO 0D[ZHOOLDQ GLVWULEXWLRQ RI QXFOHL 3Y1f ZKHUH 391fG91 7RWf 990909N7rY1DY1 f fµ7UN7 1 1 ZKHUH 9 999 7KH HIIHFWLYH UHVRQDQFH 1 [ \ ] DEVRUSWLRQ FURVV VHFWLRQ IRU D QHXWURQ ZLWK NLQHWLF HQHUJ\ ( LV PAGE 46 D ( `SIYf±`G9 G9 G9 DY U. 1n [ \ ] -f§ &2 f§&2 f§&2 f ZKHUH R ( ` LV JLYHQ E\ HTXDWLRQ f ( E\ f DQG 391f E\ f (TXDWLRQ f LV WKH SODVPD IXHO HTXLYDOHQW RI WKH VROLG IXHO FURVV VHFWLRQ JLYHQ E\ f $Q DQDO\WLFDO HYDOXDWLRQ RI f ZDV LQYHVWLJDWHG E\ H[SDQGLQJ WKH LQWHJUDQG LQ D WKUHHYDULDEOH 7D\ORUnV VHULHV DERXW WKH PRVW SUREDEOH YHORFLWLHV 9 9 9 f N7 N7 N7f ;3 <3 =3 7KH DGYDQWDJHV RI WKLV DSSUR[LPDWH DQDO\WLF DSSURDFK GR QRW MXVWLI\ WKH GLIILFXOW\ LQYROYHG LQ REWDLQLQJ VXIILFLHQW DFFXUDF\ $ PRUH GLUHFW PHWKRG RI FRPSXWLQJ WKH HIIHFWLYH FURVV VHFWLRQ LV EDVHG RQ WKH XVH RI WKH HQHUJ\ GHSHQGHQW IRUP RI WKH 0D[ZHOOLDQ GLVWULEXWLRQ ([SUHVVHG LQ WHUPV RI WKH NLQHWLF HQHUJ\ (1 RI WKH QXFOHXV WKH GLVWULEXWLRQ LV DD(f ( D 09p 1 3(ZfG( WW 1 H(1N7 LUN7f 9 1 f 6LQFH DOO GLUHFWLRQV RI PRWLRQ RI WKH DEVRUELQJ QXFOHL DUH HTXDOO\ SUREDEOH (A (A (] ZKHUH (A 09L (Q L [ \ ] 7KH YHORFLW\ FRPSRQHQW RI WKH QXFOHXV SDUDOOHO WR WKH GLUHFWLRQ RI WUDYHO RI WKH QHXWURQ LV PAGE 47 9 7H P ( f = = Yr )URP f DQG f WKH UHODWLYH HQHUJ\ LV ( ( (7 f,H(P0 f U 1 1 8VLQJ f f DQG f WKH FRPSXWDWLRQDO IRUP RI WKH HIIHFWLYH UHVRQDQFH FURVV VHFWLRQ LV U DD(f F4 + D \f f ZLWK WKH 'RSSOHU EURDGHQLQJ IXQFWLRQ +D \f GHILQHG DV +D \f WW H H H \ D H Hf GH f ZKHUH D ( 7f H (4`N7 ( 7f \OF7(0 \U 7f N7U &f DQG WKH UHGXFHG HQHUJ\ YDULDEOH LV H (1N7 7KH 'RSSOHU EURDGHQHG UHVRQDQFH DEVRUSWLRQ FURVV VHFWLRQ LQ D SODVPD FRUH UHDFWRU LV JLYHQ E\ f ZLWK +D \f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f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n FDOLW\ ZLOO EH GHWHUPLQHG E\ WKH PDVV RI K\GURJHQ LQ WKH FRUH DQG E\ WKH VSDWLDO GLVWULEXWLRQ RI WKH K\GURJHQ GHQVLW\ DQG WHPSHUDWXUH 7KXV LQ RUGHU WR SURSHUO\ HYDOXDWH WKH HIIHFWV RI K\GURJHQ WKH KHDW WUDQVIHU DQG IOXLG IORZ FKDUDFWHULVWLFV RI K\GURJHQ PXVW EH FRQVLGHUHG VLPXOWDQHRXVO\ ZLWK WKH QHXWURQ VFDWWHULQJ SUREOHP $ PRUH ULJRURXV WUHDWPHQW RI K\GURJHQ VFDWWHULQJ KDV EHHQ XVHG E\ +HUZLJ DQG /DWKDP Bf LQ WKHLU PXOWLJURXS GLIIXVLRQ WKHRU\ FDOFXODWLRQV RI VSKHULFDO JDVHRXV FRUH UHDFWRUV FRQWDLQLQJ KRW K\GURJHQ %\ GHILQLQJ DQ HIIHFWLYH PDFURVFRSLF K\GURJHQ VFDWWHULQJ FURVV VHFWLRQ WKH HIIHFWV RI KLJK WHPSHUDWXUH K\GURJHQ RQ FULWLFDO PDVV DUH LQYHVWLn JDWHG 7KH HIIHFWLYH FURVV VHFWLRQ LV GHILQHG E\ FRQn VLGHULQJ WKH VFDWWHULQJ FROOLVLRQ UDWH RI D 0D[ZHOOLDQ GLVWULEXWLRQ RI QHXWURQV DW WHPSHUDWXUH 7Q ZLWK D 0D[ZHOOLDQ GLVWULEXWLRQ RI K\GURJHQ DWRPV DW 7A )RU D QHXWURQ GHQVLW\ Q DQG D K\GURJHQ GHQVLW\ 1A WKH FROOLVLRQ UDWH LV PAGE 50 1UL Q9 6 Q9 (n Q9 + V U V U V Q f ZKHUH WKH HIIHFWLYH PDFURVFRSLF VFDWWHULQJ FURVV VHFWLRQ IRU K\GURJHQ LV JLYHQ E\ (n ( ( 9 9 V V U Q f 7KH V\PEROV DUH 9 DYHUDJH UHODWLYH YHORFLW\ EHWZHHQ K\GURJHQ DQG QHXWURQV DYHUDJH QHXWURQ VSHHG 6N7AQQAf 9+ DYHUDJH K\GURJHQ VSHHG N7+UP+f 9 D PLFURVFRSLF VFDWWHULQJ FURVV VHFWLRQ RI K\GURJHQ 7KH HIIHFWLYH PXOWLJURXS K\GURJHQ FURVV VHFWLRQV DUH FDOFXODWHG E\ WKH IROORZLQJ UHODWLRQ ZKHUH f 9Z 9 DYHUDJH UHODWLYH VSHHG EHWZHHQ K\GURJHQ DW D JLYHQ WHPSHUDWXUH DQG QHXWURQV LQ JURXS L D PLFURVFRSLF VFDWWHULQJ FURVV VHFWLRQ IRU 6L JURXS L PAGE 51 9 DYHUDJH QHXWURQ YHORFLW\ LQ JURXS L \ QHXWURQ HQHUJ\ WUDQVIHU FRHIILFLHQW IURP n JURXS L WR JURXS M 7KH *$0 Bf DQG 623+,67 f FRGHV DUH XVHG WR FDOFXn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r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e LV GHWHUPLQHG E\ D PL[HG QHXWURQ VSHFWUXP 7KH VSHFWUXP UHVXOWV IURP VFDWWHULQJ FROOLVLRQV ZLWK ERWK WKH KLJK WHPSHUDWXUH K\GURJHQ DQG WKH UHODWLYHO\ ORZ WHPSHUDWXUH UHIOHFWRU PDWHULDO 7KH VFDWWHULQJ FURVV VHFWLRQ e LQ f ZDV V REWDLQHG XVLQJ WKH 7(03(67 FRGH f 7KLV FRGH DYHUDJHV PLFURVFRSLF FURVV VHFWLRQV RYHU D WKHUPDO QHXWURQ VSHFWUXP EDVHG XSRQ RQH RI WKH IROORZLQJ DSSUR[LPDWLRQV 7KH :LJQHU:LONLQV OLJKW PRGHUDWRU HTXDWLRQ 7KH :LONLQV KHDY\ PRGHUDWRU HTXDWLRQ 7KH 0D[ZHOOLDQ GLVWULEXWLRQ 6LQFH HDFK RI WKHVH DSSUR[LPDWLRQV LV FKDUDFWHUL]HG E\ D VLQJOH PRGHUDWRU WHPSHUDWXUH LW DSSHDUV WKDW e LQ f LV GHWHUPLQHG IURP D VSHFWUXP EDVHG XSRQ D WHPSHUDWXUH DQG VFDWWHULQJ NHUQHO FRUUHVSRQGLQJ WR HLWKHU K\GURJHQ RU WKH UHIOHFWRU UDWKHU WKDQ IURP D PL[HG VSHFWUXP $ PRUH ULJRURXV FDOFXODWLRQ RI ZRXOG UHTXLUH WKH GHWHUPLQDWLRQ RI WKH PL[HG VSHFWUXP 6XFK D VSHFWUXP FRXOG EH FRPSXWHG PAGE 53 IURP DQ DSSURSULDWH ZHLJKWLQJ RI D K\GURJHQ VSHFWUXP EDVHG XSRQ D IUHH JDV VFDWWHULQJ NHUQHO DQG D UHIOHFWRU VSHFWUXP EDVHG XSRQ WKH SURSHU VFDWWHULQJ NHUQHO VXFK DV WKH 3DUNV NHUQHO IRU JUDSKLWHf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n IXVLRQ DSSUR[LPDWLRQ PD\ EH OHVV WKDQ WKDW LQYROYHG LQ WKH VSHFWUXP FDOFXODWLRQ ,Q WKLV FDVH WKH DFFXUDF\ RI WKH FURVV VHFWLRQV ZRXOG SURYLGH QR DGGLWLRQDO DFFXUDF\ LQ WKH PXOWLJURXS UHVXOWV ,I DGGLWLRQDO DFFXUDF\ LV UHTXLUHG RQH DQG WZR GLPHQVLRQDO WUDQVSRUW WKHRU\ FRGHV VKRXOG EH XVHG PAGE 54 &+$37(5 ,,, 18&/(21,& $1$/<6,6 2) (;7(51$//< 02'(5$7(' 85$1,80 3/$60$ 7KH PXOWLJURXS GLIIXVLRQ DSSUR[LPDWLRQ WR WKH %ROW]PDQQ HTXDWLRQ LV XVHG WR PDNH D SDUDPHWULF LQYHVWLn JDWLRQ RI VRPH RI WKH HVVHQWLDO QXFOHDU FKDUDFWHULVWLFV RI DQ H[WHUQDOO\ PRGHUDWHG XUDQLXP SODVPD 7KHVH FDOFXn ODWLRQV SURYLGH D TXDQWLWDWLYH FKHFN RQ WKH DVVXPSWLRQ WKDW WKH QHXWURQ HQHUJ\ VSHFWUXP LQ WKH XUDQLXP SODVPD LV SULPDULO\ GHWHUPLQHG E\ WKH PRGHUDWLQJ FKDUDFWHULVWLFV RI WKH VXUURXQGLQJ UHIOHFWRU 7KH UHVXOWV RI WKHVH FDOFXn ODWLRQV DUH XVHG WR GHILQH D UHIHUHQFH PRGHO RI WKH SODVPD FRUH UHDFWRU 7KLV PRGHO LV XVHG WR VWXG\ WKH FRXSOLQJ EHWZHHQ WKH SODVPD DQG QXFOHRQLF FKDUDFWHULVWLFV RI D ILVVLRQLQJ XUDQLXP SODVPD 7KH QHXWURQ HQHUJ\ VSHFWUXP LQ JUDSKLWH LV FDOFXn ODWHG XVLQJ WKH *$0 FRGH Bf IRU WKH HSLWKHUPDO SRUWLRQ RI WKH VSHFWUXP DQG WKH 7(03(67 FRGH Bf IRU WKH WKHUPDO SRUWLRQ $ YDOXH RI H9 LV XVHG IRU WKH WKHUPDO HSLWKHUPDO HQHUJ\ ERXQGDU\ PD[LPXP JUDSKLWH WHPSHUDWXUH FRUUHVSRQGV WR H9f DQG WKH 8 ILVVLRQ VSHFWUXP LV XVHG DV WKH QHXWURQ VRXUFH 7KH FDOFXODWHG VSHFWUXP UHSUHVHQWV WKH HQHUJ\ GLVWULEXWLRQ RI QHXWURQV ZKLFK RULJLQDWH IURP ILVVLRQV LQ D FHQWUDO 8 SODVPD DQG DUH PAGE 55 VORZHG GRZQ E\ FROOLVLRQV LQ D VXUURXQGLQJ JUDSKLWH UHn IOHFWRU 6LQFH WKH UHIOHFWRU FRQWDLQV QR XUDQLXP WKH VSHFWUXP LV GHWHUPLQHG HQWLUHO\ E\ VFDWWHULQJ DQG DEn VRUSWLRQ LQ JUDSKLWH 7KH FDOFXODWHG QHXWURQ VSHFWUD FRUUHVSRQGLQJ WR VHYHUDO UHIOHFWRU WHPSHUDWXUHV DUH VKRZQ LQ )LJXUH &KDQJLQJ WKH JUDSKLWH WHPSHUDWXUH DIIHFWV WKH WKHUPDO SRUWLRQ RI WKH VSHFWUXP E\ VKLIWLQJ WKH PRVW SUREDEOH HQHUJ\ (A N7 DW ZKLFK WKH WKHUPDO QHXWURQ IOX[ LV D PD[LPXPr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f KDV D PD[LPXP DW ( N7 ZKLOH WKH WKHUPDO OHWKDUJ\ IOX[ ! Xf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n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n QDQFH DEVRUSWLRQV LQ XUDQLXP DUH QHJOLJLEOH IRU VXFK GLOXWH TXDQWLWLHV RI 8 $V WKH XUDQLXP GHQVLW\ LV LQFUHDVHG WR A a FF UHVRQDQFH DEVRUSWLRQV EHJLQ WR VOLJKWO\ GHSUHVV WKH HSLWKHUPDO VSHFWUXP LQ WKH H9 WR H9 UDQJH $W D GHQVLW\ RI A rFF WKH QHXWURQ VSHFWUXP LV VLJQLILFDQWO\ PRGLILHG E\ WKHUPDO DQG HSLWKHUPDO DEVRUSWLRQV LQ WKH XUDQLXP SODVPD 7KH VSHFWUD VKRZQ LQ )LJXUHV DQG ZHUH XVHG WR FDOFXODWH PXOWLJURXS FURVV VHFWLRQV IRU WKH UHIOHFWRU DQG XUDQLXP UHJLRQV UHVSHFWLYHO\ 8VLQJ WKH $,0 PXOWLJURXS GLIIXVLRQ WKHRU\ FRGH ^f D SDUDPHWHU VWXG\ ZDV PDGH WR GHWHUPLQH WKH UDQJH RI YDOXHV RI XUDQLXP GHQVLW\ FRUH VL]H IOX[ VKDSHV DQG UHIOHFWRU WKLFNQHVV IRU ZKLFK D FULWLFDO 8 SODVPD UHDFWRU FDQ EH UHDOL]HG 7KH SXUSRVH RI WKHVH FDOFXODWLRQV LV WR GHWHUPLQH D UHDVRQDEOH UHIHUHQFH PRGHO ZLWKRXW DWWHPSWLQJ WR RSWLPL]H DOO WKH PAGE 59 5/$7,9( )/8; 8f )LFUXUH 1HXWURQ VSHFWUXP LQ JUDSKLWH PRGHUDWHG XUDQLXP SODVPD PAGE 60 QXFOHDU FKDUDFWHULVWLFV RI WKH V\VWHP 7\SLFDO WKHUPDO IOX[ SURILOHV RI FULWLFDO UHDFWRU FRQILJXUDWLRQV DUH VKRZQ LQ )LJXUH 7KH WKHUPDO IOX[ DWWHQXDWLRQ LV VHHQ WR EH YHU\ LQVHQVLWLYH WR FKDQJHV LQ WKH XUDQLXP GHQVLW\ SURILOH LQ WKH FDVH 2I ORZ IXHO GHQVLW\ $Q DYHUDJH XUDQLXP GHQVLW\ RI A FF ZDV XVHG LQ WKHVH WZR FDOFXn ODWLRQV %DVHG RQ WKH IRXU JURXS GLIIXVLRQ WKHRU\ FDOFXn ODWLRQV WKH XUDQLXP SODVPD UHDFWRU UHIHUHQFH PRGHO LV GHILQHG DV IROORZV &RUH UDGLXV IW &RUH KHLJKW IW *UDSKLWH UHIOHFWRU WKLFNQHVV IW *UDSKLWH WHPSHUDWXUH DW FRUH LQWHUIDFH r. 8UDQLXP FRPSRVLWLRQ b 8 7KH FULWLFDO PDVV RI 8 ZLOO GHSHQG RQ WKH XUDQLXP GHQVLW\ GLVWULEXWLRQ WKH SODVPD WHPSHUDWXUH GLVWULEXWLRQ DQG SUHVVXUH DQG WKH K\GURJHQ FRRODQW GHQVLW\ DQG WHPSHUDWXUH GLVWULEXWLRQ 7KHVH SDUDPHWHUV ZLOO KDYH WR EH VSHFLILHG LQ RUGHU WR FRPSXWH WKH FULWLFDO PDVV 7KH GLIIXVLRQ WKHRU\ FDOFXODWLRQV LQGLFDWH WKDW DYHUDJH 8 GHQVLWLHV RQ WKH RUGHU RI [ FF WR [ FF ZLOO EH UHTXLUHG WR DFKLHYH D FULWLFDO SODVPD UHDFWRU DW WKH GHVLUHG PAGE 61 5(/$7,9( 7+(50$/ )/8; M!Uf )LJXUH 7KHUPDO QHXWURQ IOX[ LQ JUDSKLWH UHIOHFWHG XUDQLXP SODVPD UHDFWRU PAGE 62 KLJK WHPSHUDWXUHV 7KHVH GHQVLWLHV FRUUHVSRQG WR 8 PDVVHV RI DERXW NJ WR NJ 7KH FKRLFH RI 8 DV WKH IXHO IRU WKH UHIHUHQFH PRGHO LV EDVHG RQ WKH DYDLODn ELOLW\ RI ERWK SODVPD DQG QXFOHDU GDWD IRU XUDQLXP (LWKHU 3X RU 8 DSSHDUV WR EH D EHWWHU SODVPD UHDFWRU IXHO ZKHQ RQO\ WKH QXFOHRQLF FULWLFDOLW\ HIIHFWV DUH FRQVLGHUHG f +RZHYHU VLQFH WKH SODVPD FKDUDFWHULVWLFV RI WKH IXHO DOVR KDYH DQ LPSRUWDQW HIIHFW RQ WKH FULWLFDO PDVV WKH SHU FHQW LRQL]DWLRQ DV D IXQFWLRQ RI SUHVVXUH DQG WHPSHUDWXUH PXVW EH FRQVLGHUHG ZKHQ FRPSDULQJ WKH UHODWLYH PHULWV RI WKH WKUHH IXHOV PAGE 63 &+$37(5 ,9 35(6685(7(03(5$785( $1$/<6,6 2) 85$1,80 3/$60$ )8(/ 5(*,21 ,QWURGXFWLRQ ,Q RUGHU IRU WKH XUDQLXP IXHO UHJLRQ WR IXQFWLRQ HIILFLHQWO\ DV D VRXUFH RI HQHUJ\ LQ WKH SODVPD FRUH UHDFWRU WKH XUDQLXP VKRXOG H[LVW DW WHPSHUDWXUHV LQ WKH r. WR r. UDQJH 8QGHU WKHVH FRQGLWLRQV KHDW WUDQVIHU LV DFFRPSOLVKHG SULPDULO\ E\ UDGLDWLRQ $W WKHVH WHPSHUDWXUHV WKH XUDQLXP IXHO LV KLJKO\ LRQL]HG DQG PXVW WKHUHIRUH EH WUHDWHG DV D SODVPD UDWKHU WKDQ DQ RUGLQDU\ JDV 6LQFH XUDQLXP EHFRPHV DSSUHFLDEO\ LRQL]HG DW WHPSHUDn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n L]DWLRQ DQG UHFRPELQDWLRQ UHDFWLRQV 7KH XUDQLXP DWRP DQG PAGE 64 DOO LRQV WKURXJK WZHOYHWLPHV LRQL]HG XUDQLXP DUH LQFOXGHG 7KH PRUH KLJKO\ LRQL]HG VSHFLHV DUH QHJOLJLEOH IRU WKH WHPSHUDWXUHV DQG SUHVVXUHV EHLQJ FRQVLGHUHG LQ WKLV VWXG\ 8r W 8 H 8 = 8 H 8 = 8 H 8 L }fµ U \ f W k X ] XLe H f ZKHUH QHXWUDO XUDQLXP L 8 LWLPHV LRQL]HG XUDQLXP H HOHFWURQ 7KH UHODWLRQ EHWZHHQ WKH QXPEHU RI LRQV DQG HOHFWURQV LQ DQ HTXLOLEULXP LRQL]DWLRQUHFRPELQDWLRQ UHDFWLRQ LV GHVFULEHG E\ WKH IROORZLQJ 6DKD HTXDWLRQ f PAGE 65 ZKHUH 1 1 OO H n1 O X L?P N 9 Q P LO H 7Ha(/N7 Bf 8L K 1L GHQVLW\ RI LOfWLPHV LRQLVHG XUDQLXP 1L GHQVLW\ RI LWLPHV LRQL]HG XUDQLXP 1 GHQVLW\ RI HOHFWURQV P HOHFWURQ PDVV %ROW]PDQQnV FRQVWDQW K 3ODQFNnV FRQVWDQW SODVPD WHPSHUDWXUH XAA SDUWLWLRQ IXQFWLRQ RI LOfWLPHV LRQL]HG XUDQLXP X SDUWLWLRQ IXQFWLRQ RI LWLPHV LRQL]HG XUDQLXP LRQL]DWLRQ SRWHQWLDO IRU WKH UHDFWLRQ 8L 8 Lf H 'XH WR WKH HOHFWULFDO QHXWUDOLW\ RI WKH SODVPD WKH QHW FKDUJH LV ]HUR 7KLV EDODQFH LV H[SUHVVHG DV 7RWDO QHJDWLYH FKDUJH RI HOHFWURQV 7RWDO SRVLWLYH FKDUJH RI LRQV PAGE 66 1 M = 1 H L ZKHUH =A FKDUJH RI LWK XUDQLXP LRQ f 7KH WHPSHUDWXUH SUHVVXUH DQG WRWDO SDUWLFOH GHQVLW\ DUH UHODWHG E\ WKH HTXDWLRQ RI VWDWH IRU D SHUIHFW JDV 7KH WRWDO SUHVVXUH LV WKH VXP RI WKH SDUWLDO SUHVVXUHV RI WKH LQGLYLGXDO VSHFLHV DV JLYHQ E\ 3 , 3L QWRW N7 ; QWRW N7 \ 1N7 1 N7 \ 1N7 Y L H [ f± L L L 1 727 1 \ 1 H L f ZKHUH 1727 SDUWLFOH GHQVLW\ 7KH ULJKW VLGH RI WKH 6DKD HTXDWLRQ f LV GHSHQGHQW RQ WKH WHPSHUDWXUH DV JLYHQ E\ X LUP N f± f± B . 7f L pBf 7H(LnN7 8L K M 7KH LWK 6DKD HTXDWLRQ LV WKHQ ZULWWHQ DV f PAGE 67 fµ 1 1 f¯S r .s7f f L &RPELQLQJ HTXDWLRQV f f DQG f JLYHV WKH FRPSOHWH VHW RI HTXDWLRQV GHVFULELQJ WKH SODVPD FRPSRVLWLRQ 1 1H 1 . f 1V 1H 1 f± . f 1 1 OO H 1 . O Lf 1 1 H 1 . f 1 L O = 1 O f 1727 .H 1L f PAGE 68 1RWH WKDW HDFK XUDQLXP LQGH[ Lf KDV EHHQ LQFUHDVHG E\ RQH VR WKDW ZLWKLQ WKH IRUPXODWLRQ RI WKH QXPHULFDO H[SUHVVLRQV XVHG LQ WKH FRPSXWHU SURJUDP WKH IROORZLQJ GHILQLWLRQV DUH XVHG 1 GHQVLW\ RI LQ SDUWLFOHVFF 1 GHQVLW\ RI 8 LQ SDUWLFOHVFF 1 GHQVLW\ RI 8 LQ SDUWLFOHVFF 6LQFH WKH HOHFWURQV DQG YDULRXV XUDQLXP VSHFLHV ZLOO KDYH GHQVLWLHV ZLWK QXPHULFDO YDOXHV LQ WKH DSSUR[LPDWH UDQJH RI WR SDUWLFOHVFF LW LV FRQYHQLHQW WR VFDOH HTXDWLRQV f WKURXJK f VR WKDW DOO SDUWLFOH GHQVLWLHV FDQ EH H[SUHVVHG DV GLPHQVLRQOHVV FRQFHQWUDWLRQV &K ZKHUH &K 'LYLGLQJ HDFK WHUP LQ HTXDWLRQV f WKURXJK f E\ 177 JLYHV WKH IROORZLQJ VHW RI VFDOHG HTXDWLRQV r f f PAGE 69 & & . r & L L Lf & F H & . r f FH , FL f L F \ F H L O f ZKHUH &L 1L1727U &H AHA727n .Lr .L177 f )URP HTXDWLRQV f WKURXJK f H Y r f§f§ VA AH L f§ .Mr .r . PAGE 70 & & J . UV r 37 A 7U r . $ A F .n ,Q JHQHUDO &L r FL Laa .L Nrf« .rr r . LO . LO LO f ZKHUH . Q N M O f )URP HTXDWLRQV f DQG f @ @ @ = FH =LFL f° ]L UUU .LL F L O L O & 7 . OO f )URP HTXDWLRQV f DQG f @ @ FH O F FH F[ A F L O L @ .9 & FO O LULf F L & H f 6ROYLQJ IRU &[ IURP HTXDWLRQV f DQG f DQG HTXDWLQJ WKH UHVXOWV PAGE 71 & ]L L O .n f¯ &L OO B . O L f L & OO 7KH VHW RI HTXDWLRQV f WKURXJK f KDV WKXV EHHQ UHGXFHG WR D IXQGDPHQWDO H[SUHVVLRQ UHODWLQJ WKH HOHFWURQ FRQFHQWUDWLRQ WR WKH SUHVVXUH DQG WHPSHUDWXUH (TXDWLRQ f FDQ EH ZULWWHQ DV )37&Hf .n O =L W(W L &B &H . f O L L & OO )RU D JLYHQ WHPSHUDWXUH DQG SUHVVXUH )37&Hf )&Jf EHFRPHV D SRO\QRPLDO IXQFWLRQ RI & DQG f FDQ EH H[SUHVVHG DV )FH! *&Hf +WFJU f ZKHUH *&Hf .f¬ FH O! H .n , = f§ L & f +&Hf .n .n L O AUf, ]L f§ L & L & LOnL f 6LQFH DOO & . f¬ DQG = DUH SRVLWLYH +& f LV DOZD\V & ; F PAGE 72 SRVLWLYH DQG WKH ]HURV RI )& f DUH JLYHQ E\ WKH ]HURV RI *&Hf 7KXV WKH HOHFWURQ IUDFWLRQ &H LV JLYHQ E\ WKH VROXWLRQ RI * & f f 6LPSOLI\LQJ HTXDWLRQ f WKH SRO\QRPLDO HTXDWLRQ f LV IO ] fF ] F [ A Ln H [ H [ .nLB f (TXDWLRQ f LV D WK GHJUHH SRO\QRPLDO LQ &H ZLWK RQH UHDO SRVLWLYH URRW )URP SK\VLFDO LQWXLWLRQ WKHUH VKRXOG EH RQO\ RQH UHDO URRW VLQFH IRU D JLYHQ WHPSHUDn WXUH DQG SUHVVXUH WKH QXPEHU RI HOHFWURQV LQ D SODVPD LV XQLTXHO\ GHWHUPLQHG 7KLV FDQ EH VKRZQ ULJRURXVO\ E\ DSSOLFDWLRQ RI 'HVFDUWHVn UXOH RI VLJQV Bf WR HTXDWLRQ f 6ROXWLRQV WR HTXDWLRQ f DUH REWDLQHG XVLQJ WKH LWHUDWLYH 1HZWRQ5KDSVRQ PHWKRG f 7KH 1HZWRQ 5KDSVRQ UHFXUUHQFH IRUPXOD IRU WKH VROXWLRQ WR f LV &H! O L &HQ f &HQf *&HQff * n &H¾7f f PAGE 73 ZKHUH & Qf YDOXH RI & DIWHU Q LWHUDWLRQV & Qf YDOXH RI & DIWHU Q LWHUDWLRQV k & f LQLWLDO HVWLPDWH RI & H H * & Qf` G* G& &H &HQf *FHQff LV REWDLQHG GLUHFWO\ IURP f *nF Qff LV REWDLQHG E\ GLIIHUHQWLDWLRQ RI f *n& f L f§Lf=f& L H L GLf] L f 8VLQJ HTXDWLRQV f DQG f ZLWK f WKH HOHFWURQ IUDFWLRQ LV WKHQ JLYHQ E\ &HQf &HQf &HQf O H L O=Lf&HLQf =L &HQf . L U O L Lf O=Lf&HQf Lf =s &HB/ Qf .n OO f %HJLQQLQJ ZLWK DQ DVVXPHG DYHUDJH HOHFWURQ FRQFHQWUDWLRQ RI &HOf DV WKH LQLWLDO HVWLPDWH H[SUHVVLRQ f PAGE 74 LV HYDOXDWHG UHSHDWHGO\ XQWLO WKH GHVLUHG GHJUHH RI DFFXn UDF\ LV REWDLQHG 7KH HOHFWURQ GHQVLW\ LV WKHQ JLYHQ E\ 1 &H 1727 )URP HTXDWLRQ f & . O = f§ L &BB DQG 1[ &[ 1U 727 )URP HTXDWLRQ Lf 1 . 1 1 IRU L W k JLYHV WKH SDUWLFOH GHQVLWLHV 1 1 1 7KH SDUWLFOH GHQVLWLHV 1 1[ 1 1D WKHQ FRPSOHWHO\ GHVFULEH WKH XUDQLXP SODVPD FRPSRVLWLRQ /RZ 7HPSHUDWXUH (TXDWLRQV )RU WHPSHUDWXUHV OHVV WKDQ DERXW r. WKH LWHUDWLYH PHWKRG IRU VROYLQJ HTXDWLRQ f GRHV QRW UHDGLO\ FRQYHUJH $OWKRXJK XUDQLXP LV QRW KLJKO\ LRQL]HG DW WHPSHUDWXUHV EHORZ r. D NQRZOHGJH RI WKH ORZ WHPSHUDWXUH XUDQLXP SODVPD FRPSRVLWLRQ LV QHFHVVDU\ LQ RUGHU WR GHVFULEH LQLWLDO RU VWDUWXS FRQGLWLRQV LQ D JDVHRXVFRUH UHDFWRU $ GLIIHUHQW PDWKHPDWLFDO IRUPXODWLRQ LV XVHG WR FDOFXODWH WKH SODVPD FRPSRVLWLRQ DW ORZ WHPSHUDn WXUHV PAGE 75 ,Q WKH WHPSHUDWXUH UDQJH IURP r. WR r. WKH RQO\ VSHFLHV WKDW DUH SUHVHQW LQ VLJQLILFDQW TXDQWLWLHV DUH 8r 8 DQG HOHFWURQV $ VLQJOH UHDFWLRQ DFFRXQWV IRU WKH HTXLOLEULXP LRQL]DWLRQ DQG UHFRPELQDWLRQ RI WKHVH WKUHH VSHFLHV 8r W 8 H f 7KH FRUUHVSRQGLQJ 6DKD HTXDWLRQ LV 1 H BB 0HN7 fµr Xs eBHRP 1 f§I` X f 7KH WRWDO SDUWLFOH GHQVLW\ LV 1727 N7 1 1 H f 7KH FKDUJH QHXWUDOLW\ LV JLYHQ E\ 1 H f 8VLQJ 1M 1 1 1 H DQG r1 N7 9 X B . I f§` Bs H (R.7 X K HTXDWLRQV f f DQG f DUH VLPSOLILHG DV 1 1 fü 1 . f PAGE 76 1 1 1 1 727 L 7 H 1 1H f f 'LYLGLQJ HDFK WHUP LQ HTXDWLRQV f f DQG f E\ 1WTW DQG FRPELQLQJ WKHVH HTXDWLRQV JLYHV & .r& .r f H H ZKHUH Nr . . 1 727 7KH VROXWLRQV RI HTXDWLRQ f DUH FH ,&r s 77.r` 6LQFH .r ! O .r ! DQG WKH SRVLWLYH UDGLFDO PXVW EH XVHG WR JLYH D SRVLWLYH YDOXH RI WKH HOHFWURQ FRQFHQWUDWLRQ 7KH ORZ WHPSHUDWXUH SODVPD FRPSRVLWLRQ LV WKHQ JLYHQ E\ 1WRW .r / , .r f n 177 1H f 1 1H f 8UDQLXP 3DUWLWLRQ )XQFWLRQV 7KH SDUWLWLRQ IXQFWLRQV DSSHDULQJ LQ WKH JHQHUDO 6DKD HTXDWLRQ f DUH IXQFWLRQV RI WKH SODVPD WHPSHUDWXUH PAGE 77 DQG WKH HOHFWURQ FRQILJXUDWLRQ RI WKH SDUWLFXODU XUDQLXP DWRP RU LRQ 7KH HOHFWURQLF SDUWLWLRQ IXQFWLRQ RI LWLPHV LRQL]HG XUDQLXP LV JLYHQ E\ f X L O D M ;L$7 ;L MN7 a J $ J fµf H nZn J H AL LO L ;N7 m fµ f ZKHUH J VWDWLVWLFDO ZHLJKW RI WKH MWK WHUP RI LWLPHV LRQL]HG XUDQLXP ; H[LWDWLRQ HQHUJ\ OHYHO RI WKH MWK WHUP RI LWLPHV LRQL]HG XUDQLXP 7KH MWK VWDWLVWLFDO ZHLJKW IRU WKH LWLPHV LRQL]HG XUDQLXP LV Mf LM /LM f f ZKHUH / RUELWDO DQJXODU PRPHQWXP M VSLQ DQJXODU PRPHQWXP 7KH VWDWLVWLFDO ZHLJKWV RI WKH XUDQLXP DWRP DUH FDOFXODWHG IURP WKH VSHFWURVFRSLF GDWD RI 6WHLQKDXV %ODLVH DQG 'LULQJHU ef 6LQFH H[SHULPHQWDO XUDQLXP LRQ GDWD IURP ZKLFK WKH QHFHVVDU\ /fµ fµ DQG 6 fµ YDOXHV FDQ EH ; GHWHUPLQHG DUH QRW SUHVHQWO\ DYDLODEOH WKH LVRHOHFWURQLF PAGE 78 DSSUR[LPDWLRQ Bf LV XVHG WR FDOFXODWH WKH VWDWLVWLFDO ZHLJKWV DQG H[FLWDWLRQ HQHUJLHV RI WKH XUDQLXP LRQV 7KLV DSSUR[LPDWLRQ LV EDVHG RQ WKH REVHUYDWLRQ WKDW DWRPV RU LRQV KDYLQJ WKH VDPH QXPEHU RI RUELWDO HOHFWURQV KDYH VLPLODU HOHFWURQ FRQILJXUDWLRQV 7KH J A YDOXHV DUH FDOFXODWHG IURP f XVLQJ WKH /Q fµ DQG 6 A YDOXHV RI mrfµ I U WKH QHXWUDO DWRP KDYLQJ L RUELWDO HOHFWURQV 7KH ; M DUH REWDLQHG IURP VSHFWURVFRSLF PHDVXUHPHQWV f RI HQHUJ\ OHYHOV RI WKH QHXWUDO DWRPV /RZHULQJ RI ,RQL]DWLRQ 3RWHQWLDOV 7KH LRQL]DWLRQ SRWHQWLDO (A LQ HTXDWLRQ f UHSUHVHQWVn WKH HQHUJ\ UHTXLUHG IRU WKH LRQL]DWLRQ RI DQ LVRODWHG DWRP RU LRQ $W KLJK SUHVVXUHV WKH SODVPD GHQVLW\ LQFUHDVHV XQWLO WKH DWRPV DQG LRQV FDQ QR ORQJHU EH FRQVLGHUHG DV LVRODWHG (DFK SDUWLFOH WKHQ PRYHV LQ DQ HOHFWULF ILHOG DVVRFLDWHG ZLWK DGMDFHQW FKDUJHG SDUWLFOHV 7KH LRQL]DWLRQ UHDFWLRQV DUH LQIOXHQFHG E\ SODVPD PLFURn ILHOGV DVVRFLDWHG ZLWK HOHFWURVWDWLF SRODUL]DWLRQ 'HE\H HIIHFWf 7KH UHVXOWLQJ HIIHFW LV D ORZHULQJ RI WKH HQHUJ\ UHTXLUHG IRU LRQL]DWLRQ RI DQ DWRP RU LRQ WKH HIIHFW LQFUHDVLQJ ZLWK LQFUHDVLQJ FKDUJH GHQVLW\ 7R DFFRXQW IRU WKH ORZHULQJ RI WKH LRQL]DWLRQ SRWHQWLDOV DQ HIIHFWLYH I I fµ LRQL]DWLRQ SRWHQWLDO (A LV GHILQHG DV V I I ( ( $( O ; O f PAGE 79 L Lf IRU WKH UHDFWLRQ 8 8O H ZKHUH ( XQFRUUHFWHG LRQL]DWLRQ SRWHQWLDO L $(A ORZHULQJ RI LRQL]DWLRQ SRWHQWLDO *ULHP Bf KDV XVHG WKH 'HE\H+XFNHO WKHRU\ f IRU LRQL]HG JDVHV WR REWDLQ DQ DQDO\WLFDO H[SUHVVLRQ IRU WKH DPRXQW RI SRWHQWLDO ORZHULQJ )RU DQ XUDQLXP SODVPD ZLWK D PD[LPXP RI WZHOYH GHJUHHV RI LRQL]DWLRQ WKH SRWHQWLDO ORZHULQJ LV JLYHQ E\ $( ] OfH LUN7f 9 Q e =AQL f ZKHUH =A FKDUJH RI LWK LRQ H HOHFWURVWDWLF FKDUJH RI WKH HOHFW[nRQ 1H HOHFWURQ GHQVLW\ LWK LRQ GHQVLW\ &XWRII 4XDQWXP 1XPEHU DV D )XQFWLRQ RI (IIHFWLYH ,RQL]DWLRQ 3RWHQWLDO 7KH HOHFWURQLF SDUWLWLRQ IXQFWLRQ RI XUDQLXP ZDV JLYHQ E\ O M X L H;LMN7 f PAGE 80 7KH LQGH[ M LV WKH SULQFLSDO TXDQWXP QXPEHU RI WKH MWK HQHUJ\ OHYHO RI LWLPHV LRQL]HG XUDQLXP 7KH SDUWLWLRQ IXQFWLRQ RI D K\SRWKHWLFDO LVRODWHG SDUWLFOH FDQQRW EH HYDOXDWHG H[DFWO\ VLQFH f LQYROYHV DQ LQILQLWH QXPEHU RI WHUPV :KHQ WKH VXPPDWLRQ H[WHQGV WR M f¯ WKH SDUWLWLRQ IXQFWLRQ GLYHUJHV VLQFH ; L DSSURDFKHV D FRQVWDQW YDOXH WKH LRQL]DWLRQ HQHUJ\f DQG J LV QRW PRQRWRQLFDOO\ GHFUHDVLQJ DV M ! rr +RZHYHU GXH WR WKH HIIHFW RI WKH 'HE\H ILHOG WKH VXPPDWLRQ FDQ EH WHUPLQDWHG DW D ILQLWH FXWRII TXDQWXP QXPEHU Q FRUUHVSRQGLQJ WR WKH KLJKHVW HQHUJ\ OHYHO DW ZKLFK WKH HOHFWURQ LV VWLOO ERXQG 7KH FDOFXODWLRQDO IRUP RI WKH SDUWLWLRQ IXQFWLRQ EDVHG RQ ERXQG HOHFWURQ VWDWHV LV Q Lf XL O M HB;LMN7 f M R 7KH FXWRII TXDQWXP QXPEHU LV D IXQFWLRQ RI WKH HIIHFWLYH LRQL]DWLRQ SRWHQWLDO VLQFH QFLf LV WKH ODUJHVW LQWHJHU k I I YDOXH IRU M IRU ZKLFK ; ( )RU WKH OWK XUDQLXP [ I LRQ WKH FXWRII QXPEHU LV H[SUHVVHG DV QF Lf 7KH UHODWLRQ EHWZHHQ SULQFLSDO TXDQWXP QXPEHU DQG HQHUJ\ OHYHO IRU D K\GURJHQOLNH DWRP LV f ; ( HII f PAGE 81 ( Q f ZKHUH \ LV WKH UHGXFHG PDVV = WKH QXFOHDU FKDUJH H WKH HOHFWURVWDWLF HOHFWURQ FKDUJH K WKH UHGXFHG 3ODQFNnV FRQVWDQW DQG Q WKH SULQFLSDO TXDQWXP QXPEHU )RU WKH PXOWLHOHFWURQ DWRPV DQG LRQV LQ WKH XUDQLXP SODVPD WKH HQHUJ\ OHYHOV DQG SULQFLSDO TXDQWXP QXPEHUV DUH QRW H[SOLFLWO\ UHODWHG E\ H[SUHVVLRQV VXFK DV f +RZHYHU LW FDQ EH DUJXHG WKDW D KLJKO\ H[FLWHG HOHFWURQ LQ D PXOWLHOHFWURQ DWRP LV QHDUO\ K\GURJHQLF VLQFH WKH DWRP FDQ EH UHJDUGHG DV D QXFOHXV KDYLQJ DQ HOHFWURQ PRYLQJ DERXW LW LQ D ODUJH RUELW %DVHG RQ WKLV UHDVRQLQJ 'UHOOLVKDN HW DW eef KDYH HPSLULFDOO\ WUHDWHG WKH ORZHULQJ RI WKH LRQL]DWLRQ SRWHQWLDO E\ DVVXPLQJ WKDW WKH DPRXQW RI SRWHQWLDO ORZHULQJ VKRXOG YDU\ LQYHUVHO\ ZLWK WKH VTXDUH RI WKH FXWRII TXDQWXP QXPEHU DFFRUGLQJ WR ( L f ZKHUH WKH V\PEROV DUH GHILQHG DV EHIRUH 7KLV UHODWLRQ ZLOO EH XVHG LQ WKH IROORZLQJ VHFWLRQ WR REWDLQ LQLWLDO HVWLPDWHV RI WKH FXWRII TXDQWXP QXPEHUV RI XUDQLXP DWRPV DQG LRQV PAGE 82 &RPSXWDWLRQ RI 8UDQLXP 3ODVPD &RPSRVLWLRQ 7KH FRPSXWDWLRQ RI WKH XUDQLXP SODVPD FRPSRVLWLRQ UHTXLUHV WKH GHWHUPLQDWLRQ RI D FRQVLVWHQW UHODWLRQ EHWZHHQ f WKH HOHFWURQ IUDFWLRQ f RU f IRU ORZ WHPSHUDWXUHVf f WKH ORZHULQJ RI WKH LRQL]DWLRQ SRWHQWLDO f DQG f WKH FXWRII TXDQWXP QXPEHU f 7KHVH H[SUHVVLRQV DUH VHHQ WR EH FRPSOHWHO\ FRXSOHG VLQFH WKH SODVPD FRPSRVLWLRQ LV GHSHQGHQW RQ WKH SDUWLWLRQ IXQFWLRQV WKH SDUWLWLRQ IXQFWLRQV DUH GHSHQGHQW RQ WKH HIIHFWLYH LRQL]DWLRQ SRWHQWLDOV DQG HIIHFWLYH LRQL]DWLRQ SRWHQWLDOV DUH GHSHQGHQW RQ WKH SODVPD FRPSRVLWLRQ 7KH XUDQLXP SODVPD FRPSRVLWLRQ LV FDOFXODWHG XVLQJ DQ LWHUDWLYH WHFKQLTXH ZKLFK VHDUFKHV IRU WKH FRQVLVWHQW VROXWLRQ RI HTXDWLRQV f f DQG f $ SURJUDP FDOOHG 83/$=r ZDV ZULWWHQ WR SHUIRUP WKHVH LWHUDWLYH FDOFXn ODWLRQV )RU D JLYHQ SUHVVXUH DQG WHPSHUDWXUH WKH 83/$= LWHUDWLRQ LV FDUUHG RXW DV IROORZV 6WHS f§6WDUWLQJ ZLWK HTXDWLRQ f WKH HOHFWURQ DWRP DQG LRQ GHQVLWLHV DUH FRPSXWHG XVLQJ RQO\ WKH ILUVW WHUP JURXQG VWDWHf RI f LQ WKH SDUWLWLRQ IXQFWLRQV 7KHVH SDUWLFOH GHQVLWLHV DUH XVHG WR FRPSXWH WKH LRQL]DWLRQ r'HWDLOV RI 83/$= DQG RWKHU SURJUDPV ZULWWHQ IRU WKLV VWXG\ DUH JLYHQ LQ WKH $SSHQGLFHV PAGE 83 SRWHQWLDO ORZHULQJ $(A DFFRUGLQJ WR f )URP HTXDWLRQ f WKH LQLWLDO HVWLPDWH RI WKH TXDQWXP FXWRII QXPEHUV DUH FDOFXODWHG DV 1 ALf Ha$H L ; f 6WHS f§5HWXUQLQJ WR HTXDWLRQ f WKH SDUWLFOH GHQVLWLHV DUH DJDLQ FDOFXODWHG WKLV WLPH LQFOXGLQJ Q OfLf WHUPV LQ WKH SDUWLWLRQ IXQFWLRQ VHULHV f 8VLQJ WKH QHZ QXPEHU GHQVLWLHV WKH QH[W YDOXH RI WKH LRQL]DWLRQ SRWHQWLDO ORZHULQJ LV FDOFXODWHG IURP f 7KH VHFRQG HVWLPDWH RI WKH FXWRII TXDQWXP QXPEHUV LV WKHQ GHWHUPLQHG IURP QFf Lf >QF Lf @ f ZKHUH >Q Lf LV WKH ODUJHVW LQWHJHU YDOXH QRW H[FHHGLQJ M IRU ZKLFK ; ( $( f L L ZKHUH (A XQFRUUHFWHG LRQL]DWLRQ SRWHQWLDO $(A FXUUHQW YDOXH RI LRQL]DWLRQ SRWHQWLDO ORZHULQJ PAGE 84 6WHS f§,I MQAAGf Uf ! IRU DQ\ L Q Gf QF Uf Lf 6WHS LV UHSHDWHG U WLPHV XQWLO IRU DOO L :KHQ WKLV FRQGLWLRQ LV VDWLVILHG WKH SODVPD FRPSRVLWLRQ LV GHWHUPLQHG FRQVLVWHQW ZLWK WKH LRQL]DWLRQ SRWHQWLDO ORZHULQJ DQG FXWRII TXDQWXP QXPEHUV 5HVXOWV RI 8UDQLXP 3ODVPD &RPSRVLWLRQ &DOFXODWLRQV 7KH 83/$= SURJUDP ZDV XVHG WR FDOFXODWH WKH XUDQLXP SODVPD FRPSRVLWLRQ IRU SUHVVXUHV IURP DWP WR DWP DQG WHPSHUDWXUHV IURP r. WR r. 7\SLFDO UHVXOWV DUH VKRZQ LQ )LJXUHV DQG IRU SUHVVXUHV RI DQG DWPRVSKHUHV 7KH HOHFWURQ GHQVLW\ DQG WKH GHQVLWLHV RI LQGLYLGXDO XUDQLXP VSHFLHV DUH JLYHQ LQ SHU FHQW RI WRWDO SDUWLFOH GHQVLW\ $Gn GLWLRQDO FDOFXODWHG XUDQLXP SODVPD FRPSRVLWLRQV IRU SUHVVXUHV IURP DWP WR DWP DUH VKRZQ LQ )LJXUHV WKURXJK $SSHQGL[ $ 7KH IDFW WKDW WKH IXHO UHJLRQ RI WKH JDVHRXV FRUH UHDFWRU LV D SODVPD UDWKHU WKDQ D RUGLQDU\ JDV LV FOHDUO\ GHPRQVWUDWHG LQ )LJXUHV DQG $W D WHPSHUDWXUH RI r. WKH WRWDO QXPEHU RI SDUWLFOHV LQ WKH IXHO UHJLRQ FRQVLVWV RI DERXW b XUDQLXP DQG b HOHFWURQV $ERYH r. WKH IXHO UHJLRQ FRQWDLQV DSSUR[LPDWHO\ b HOHFWURQV DQG RQO\ b XUDQLXP ,Q PAGE 85 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI FWWUIWRVSKVLU*6 PAGE 86 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 87 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< 7HPSHUDWXUH r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f ZKLFK FDQ EH ZULWWHQ DV 3 , 3 f H L n ZKHUH SH 1JN7 SDUWLDO SUHVVXUH RI HOHFWURQV 3 1N7 SDUWLDO SUHVVXUH RI WKH XUDQLXP VSHFLHV 8L 7KLV H[SUHVVLRQ VKRZV WKDW LQ WKHUPRG\QDPLF HTXLOLEULXP RQH IUHH HOHFWURQ FUHDWHV WKH VDPH SDUWLDO SUHVVXUH DV RQH XUDQLXP DWRP RU LRQ 7KXV IRU D JLYHQ FRQVWDQW UHDFWRU PAGE 89 SUHVVXUH 3 WKH IRUPDWLRQ RI HDFK IUHH HOHFWURQ E\ LRQLn ]DWLRQ LV EDODQFHG E\ WKH UHPRYDO RI RQH XUDQLXP DWRP RU LRQ )URP )LJXUHV DQG WKH HOHFWURQ IUDFWLRQ LV VHHQ WR LQFUHDVH PRQRWRQLFDOO\ DV WKH WHPSHUDWXUH LQFUHDVHV ZKLFK PHDQV WKDW DQ\ WHPSHUDWXUH LQFUHDVH DW FRQVWDQW SUHVVXUH UHTXLUHV D UHGXFWLRQ LQ WKH DPRXQW RI XUDQLXP LQ WKH IXHO UHJLRQ 7KH TXDQWLWDWLYH HIIHFWV RI WHPSHUDWXUH RQ WKH XUDQLXP GHQVLW\ DUH VKRZQ LQ )LJXUH 7KH VORSH G1 RI WKH FXUYH A7Sf§f W JLYHV WKH UDWH RI XUDQLXP UHGXFWLRQ ZLWK LQFUHDVLQJ WHPSHUDWXUH DW FRQVWDQW SUHVVXUH 7KH XUDQLXP UHGXFWLRQ UDWH GHFUHDVHV DW KLJKHU WHPSHUDWXUHV VLQFH WKH HOHFWURQ IUDFWLRQ FKDQJHV YHU\ VORZO\ IRU WHPSHUDn WXUHV JUHDWHU WKDQ r. )URP )LJXUH D EDVLF FKDUDFWHULVWLF RI WKH SODVPD FRUH UHDFWRU LV VHHQ WR EH VWURQJ GHSHQGHQFH RQ WKH XUDQLXP GHQVLW\ RQ WKH SODVPD WHPSHUDWXUH )RU D JLYHQ FRQVWDQW SUHVVXUH WKHUH LV RQO\ RQH WHPSHUDWXUH ZKLFK ZLOO VXVWDLQ WKH FULWLFDO XUDQLXP PDVV 6XSSRVH WKDW WKH UTDFWRU LV WR EH RSHUDWHG DW D FRQVWDQW SUHVVXUH RI DWP DQG WKDW WKH FULWLFDO PDVV FRUUHVSRQGV WR D WRWDO XUDQLXP GHQVLW\ RI FF )URP )LJXUH LW LV VHHQ WKDW D WHPSHUDWXUH RI r. ZRXOG EH UHTXLUHG WR VXVWDLQ WKH GHVLUHG XUDQLXP PDVV $Q LQFUHDVH LQ WKH WHPSHUDWXUH ZRXOG FDXVH WKH UHDFWRU WR EHFRPH VXEFULWLFDO DQG VLPLODUO\ D WHPSHUDWXUH GHFUHDVH PAGE 91 ZRXOG UHVXOW LQ D VXSHUFULWLFDO UHDFWRU 7KLV GHSHQGHQFH RI FULWLFDO PDVV RQ SODVPD WHPSHUDWXUH LQGLFDWHV WKDW D FRQWURO V\VWHP ZKLFK DXWRPDWLFDOO\ FRPSHQVDWHV IRU WHPSHUDn WXUH YDULDWLRQV ZRXOG EH D QHFHVVDU\ FRPSRQHQW RI WKH SODVPD UHDFWRU V\VWHP ,Q DQ RSHUDWLQJ URFNHW UHDFWRU D FKDQJH LQ WKH WKUXVW UHTXLUHPHQW DFFRPSOLVKHG E\ YDU\LQJ WKH SRZHU RXWSXW ZLOO LQYROYH WHPSHUDWXUH FKDQJHV LQ WKH XUDQLXP SODVPD 6LQFH D ODUJH WHPSHUDWXUH FKDQJH DW FRQVWDQW SUHVVXUH FDQ VLJQLILFDQWO\ DIIHFW WKH FULWLFDOLW\ DQG KHQFH WKH UHDFWRU VWDELOLW\ WKH SUHVVXUH PXVW EH DOORZHG WR YDU\ LQ RUGHU WR PDLQWDLQ D FRQVWDQW XUDQLXP PDVV DV WKH WHPSHUDWXUH FKDQJHV 7KXV WKH FULWLFDOLW\ SUREOHP EHFRPHV D TXHVWLRQ RI ILQGLQJ D SUHVVXUHWHPSHUDn WXUH UHODWLRQ ZKLFK VXVWDLQV WKH GHVLUHG XUDQLXP PDVV 7KLV UHODWLRQ FDQ EH H[SUHVVHG PDWKHPDWLFDOO\ DV D SUREOHP RI ILQGLQJ WHPSHUDWXUH GHSHQGHQW SUHVVXUH WUDMHFWRULHV DORQJ ZKLFK WKH XUDQLXP PDVV KDV WKH FULWLFDO YDOXH 7KH JHQHUDO SUHVVXUH WUDMHFWRU\ LV H[SUHVVHG DV 3F I0'7f f ZKHUH 3F SUHVVXUH WUDMHFWRU\ 0\ FULWLFDO XUDQLXP PDVV 7 SODVPD WHPSHUDWXUH PAGE 92 I IXQFWLRQDO UHODWLRQ EHWZHHQ 3 07 DQG 7 F 8 7KH IXQFWLRQDO UHODWLRQV ZKLFK GHILQH SUHVVXUH WUDMHFWRULHV DUH GHWHUPLQHG QXPHULFDOO\ XVLQJ WKH 83/$= SURJUDP )RU WKH XUDQLXP GHQVLW\ FRUUHVSRQGLQJ WR D JLYHQ FULWLFDO PDVV DQG D JLYHQ WHPSHUDWXUH WKH 83/$= HTXDWLRQV DUH VROYHG IRU WKH SUHVVXUH 3 %\ VROYLQJ WKH HTXDWLRQV IRU WHPSHUDWXUHV IURP r. WR r. WKH UHVXOWLQJ SUHVVXUHV GHILQH WUDMHFWRULHV DORQJ ZKLFK WKH XUDQLXP PDVV LV FRQVWDQW 7KH QXFOHDU FDOFXODWLRQV GLVFXVVHG LQ &KDSWHU ,,, LQGLFDWH WKDW D FULWLFDO UHDFWRU ZRXOG UHTXLUH XUDQLXP GHQVLWLHV LQ WKH QHLJKERUKRRG RI FF &DOFXn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r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n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b +r H 'LVVRFLDWLYH UHFRPELQDWLRQ +r H W + + 'LVVRFLDWLRQ RI PROHFXODU LRQV + W + ( 7ZRERG\ f°K FRPELQDWLRQ K K ] K r Y + + W + + + L + + 7KUHHERG\ UHFRPELQDWLRQ + + + W + + + + + W + K )URP DQ H[DPLQDWLRQ RI WKH UHDFWLRQ HQHUJLHV DEVRUSWLRQ SURSHUWLHV DQG H[SHFWHG FRRODQW WHPSHUDWXUHV WKH K\GURJHQ FRRODQW LV FRQVLGHUHG WR EH DGHTXDWHO\ GHVFULEHG E\ WKH ILUVW IRXU UHDFWLRQV 5HDFWLRQ FDQ LQGLUHFWO\ HQKDQFH WKH DEVRUSWLRQ SURSHUWLHV RI K\GURJHQ DW KLJKHU SUHVVXUHV ef EXW ZLOO QRW EH LQFOXGHG LQ WKH SUHVHQW DQDO\VLV PAGE 97 +\GURJHQ 5HDFWLRQ (TXDWLRQV 0ROHFXODU GLVVRFLDWLRQ ¾] b + +a7KH HTXLOLEULXP UHODWLRQ EHWZHHQ PROHFXODU DQG DWRPLF K\GURJHQ LQYROYHG LQ PROHFXODU GLVVRFLDWLRQ DW DEVROXWH WHPSHUDWXUH 7 LV f 1+ QK 1 + LUP77Pf±N 9 X77 Xf± -/f W , + H (nK N7 Pf± K K X + ZKHUH P+ DWRPLF ZHLJKW RI + Pf± DWRPLF ZHLJKW RI + + Xf± SDUWLWLRQ IXQFWLRQ RI + [L Xf± SDUWLWLRQ IXQFWLRQ RI + Q (n GLVVRFLDWLRQ HQHUJ\ RI +" + N %ROW]PDQQnV FRQVWDQW K 3ODQFNnV FRQVWDQW 7KH SDUWLWLRQ IXQFWLRQ RI DWRPLF K\GURJHQ LV f X + H ;MN7 f VWDWLVWLFDO ZHLJKW RI MaWK WHUP H[FLWDWLRQ HQHUJ\ RI MWK WHUP ZKHUH PAGE 98 7KH QWK VWDWH RI DWRPLF K\GURJHQ LV Q WLPHV GHJHQHUDWH ;T DQG KDV DQ HQHUJ\ [ ;Q f¯ f§! ZKHUH Q LV WKH SULQFLSDO Q X Q TXDQWXP QXPEHU DQG ;T LV WKH JURXQG VWDWH HQHUJ\ OHYHO 2Q WKLV EDVLV WKH SDUWLWLRQ IXQFWLRQ PD\ EH ZULWWHQ H[SOLFLWO\ DV X + O Q Q O B R Qf¯ @ [$W f ZKHUH Q LV WKH SULQFLSDO TXDQWXP QXPEHU RI WKH ODVW ERXQG * VWDWH 'HILQLQJ D UHGXFHG HQHUJ\ YDULDEOH D ;TA7 f LV VLPSOLILHG DV Q X fµD + H O Qf« H Q O DQ f )RU D VXIILFLHQWO\ ODUJH TXDQWXP QXPEHU Q/ WKH HQHUJ\ OHYHOV DSSURDFK D FRQWLQXXP )RU Q ! Q/ WKH LQWHJUDO DSSUR[LPDWLRQ RI ,YDQRY.KRORGQ\L HW DW Bf FDQ EH DSSOLHG WR WKH SDUWLWLRQ IXQFWLRQ 5HSODFLQJ WKH QHDUn FRQWLQXXP SRUWLRQ RI f E\ DQ LQWHJUDO 8+ D Q/ O Q HDQ Q O QHDQ GQ Q7 f )RU DQ/ X I PAGE 99 X+ } H D Q O H Q O DQ Q U F Q7 QGQ 8+ D HD HD HR Q7 Hf¯n9 Q Q7 0 / A F / f 7KH QXPEHU RI WHUPV WR EH UHWDLQHG LQ WKH VHULHV f LV GHWHUPLQHG E\ WKH UHTXLUHPHQW WKDW QA !! Q HDQ/ 7KLV UHTXLUHPHQW DOVR SURYLGHV WKDW Q !! Q VR WKDW WKH SDUWLWLRQ IXQFWLRQ PD\ QRZ EH ZULWWHQ LQ D IRUP ZKLFK LV FRQYHQLHQW IRU FRPSXWDWLRQ 8+ H D Qf± / Q O O Q HDQn L Q F f 7KH TXDQWXP QXPEHU Q DW ZKLFK WKH HOHFWURQ EHFRPHV IUHH GXH WR SUHLRQL]DWLRQ RI K\GURJHQ LV D IXQFWLRQ RI WKH FKDUJH GHQVLW\ RI WKH SODVPD ,YDQRY .KRORGQ\L HW DO KDYH GHULYHG D UHODWLRQ EHWZHHQ QF DQG WKH HOHFWURQ GHQVLW\ 1 DV IROORZV ORJ 1J ORJ QF f f PAGE 100 7KLV H[SUHVVLRQ LV FRQVLVWHQW ZLWK WKH LQWHJUDO DSSUR[Ln PDWLRQ XVHG LQ f DQG ZLOO EH XVHG WR FDOFXODWH WKH TXDQWXP QXPEHU DSSHDULQJ LQ WKH SDUWLWLRQ IXQFWLRQ f 7KH XVXDO IRUP RI WKH SDUWLWLRQ IXQFWLRQ RI PROHFXODU K\GURJHQ LV X + Xf± X X7 ( 5 9 ZKHUH Xe HOHFWURQLF SDUWLWLRQ IXQFWLRQ X5 URWDWLRQDO SDUWLWLRQ IXQFWLRQ XY YLEUDWLRQDO SDUWLWLRQ IXQFWLRQ 7KH HOHFWURQLF SDUWLWLRQ IXQFWLRQ LV 9 ;LN7 Xf± f J H \ ( K 7KH VWDWLVWLFDO ZHLJKW LV JLYHQ E\ J M: + f f f ZKHUH LV WKH WRWDO DQJXODU PRPHQWXP RI WKH MWK VWDWH RI + :KHQ WKH GLDWRPLF K\GURJHQ PROHFXOH LV SLFWXUHG DV D ULJLG GXPEEHOO ZLWK PRPHQW RI LQHUWLD , URWDWLQJ LQ WKUHH GLPHQVLRQV WKH HQHUJ\ DQG GHJHQHUDF\ RI WKH -WK PAGE 101 OHYHO ZKHUH LV WKH URWDWLRQDO TXDQWXP QXPEHU DUH f (LRW -f &N' f J-f AURW f 7KH URWDWLRQDO SDUWLWLRQ IXQFWLRQ PD\ WKHQ EH ZULWWHQ DV X 5 , U (A-ON7 -f URW URW L O L Of [ H --f KOfN7 f 7KH IDFWRU RI A UHVXOWV IURP WKH IDFW WKDW IRU D GLDWRPLF PROHFXOH KDYLQJ VLPLODU DWRPV RQO\ DOWHUQDWH OHYHOV FDQ EH RFFXSLHG ,Q WKH OLPLW 7 !! 7B ZKHUH 7' LV WKH WHPSHUDWXUH HTXLYDOHQW RI WKH HQHUJ\ VHSDUDWLRQ EHWZHHQ DGMDFHQW URWDWLRQDO VWDWHV WU r. IRU + Af WKH WHPSHUDWXUH LV KLJK HQRXJK WR H[FLWH PDQ\ URWDWLRQDO OHYHOV ,Q WKLV FDVH WKH VXPPDWLRQ LQ f PD\ EH UHSODFHG E\ DQ LQWHJUDO ZKLFK FDQ EH HYDOXDWHG DQDO\WLFDOO\ JLYLQJ 85 L-OfH--f 0 GL f K 7KH YLEUDWLRQDO SDUWLWLRQ IXQFWLRQ RI D GLDWRPLF PROHFXOH LV f PAGE 102 X , Hf¯(YN7 f 9 Y ZKHUH Y LV WKH YLEUDWLRQDO TXDQWXP QXPEHU DQG (Y LV WKH HQHUJ\ RI WKH YWK YLEUDWLRQDO VWDWH 7KH PROHFXOH FDQ EH WUHDWHG DV DQ KDUPRQLF RVFLOODWRU KDYLQJ D PROHFXODU ELQGLQJ IRUFH GHVFULEHG E\ +RRNHnV ODZ ZLWK WKH FRUUHn VSRQGLQJ SRWHQWLDO 9Uf NU U f f = ZKHUH U LV WKH VHSDUDWLRQ GLVWDQFH EHWZHHQ WKH DWRPV N LV WKH IRUFH FRQVWDQW DQG UJ WKH HTXLOLEULXP SRVLWLRQ 7KH RVFLOODWRU\ HQHUJ\ HLJHQYDOXHV FRUUHVSRQGLQJ WR f DUH ( Y f¯LZ Y f 9 ] ZKHUH Z Nf¬\ WKH FODVVLFDO YLEUDWLRQDO IUHTXHQF\ \ PPP Pf WKH UHGXFHG PDVV :KHQ f LV XVHG LQ f WKH SDUWLWLRQ IXQFWLRQ EHFRPHV Y Lf IXRN7 X H FVFKf¯ KZNWf f 9 Y 7KH H[SUHVVLRQ f VKRXOG EH YHU\ DFFXUDWH IRU VPDOO PAGE 103 VHSDUDWLRQ GLVWDQFHV EXW GRHV QRW DFFXUDWHO\ GHVFULEH PROHFXODU YLEUDWLRQ ZKHQ WKH VHSDUDWLRQ GLVWDQFH EHFRPHV ODUJH )RU DQ KDUPRQLF RVFLOODWRU FKDUDFWHUL]HG E\ WKH SDUDEROLF SRWHQWLDO f WKH UHVWRULQJ IRUFH LQFUHDVHV ZLWK LQFUHDVLQJ VHSDUDWLRQ WKXV H[FOXGLQJ WKH SRVVLELOLW\ RI PROHFXODU GLVVRFLDWLRQ 6LQFH DQ DFWXDO GLDWRPLF PROHFXOH GLVVRFLDWHV ZKHQ WKH VHSDUDWLRQ GLVWDQFH EHWZHHQ WKH DWRPV EHFRPHV VXIILFLHQWO\ ODUJH WKH PROHFXODU SRWHQWLDO VKRXOG EHFRPH DV\PSWRWLF IRU VXIILFLHQWO\ ODUJH YDOXHV RI U 7KH GLVVRFLDWLRQ FDQ EH LQFOXGHG E\ UHSUHn VHQWLQJ WKH PROHFXOH DV DQ DQKDUPRQLF RVFLOODWRU IRU ZKLFK WKH JHQHUDO SRWHQWLDO LV 9Uf IU U f JU U f f Y & 7KH HQHUJ\ HLJHQYDOXHV FRUUHVSRQGLQJ WR WKH SRWHQWLDO f DUH f KHZ Y Lf f 7KH SURSHU IRUP RI WKH PROHFXODU SRWHQWLDO FDQ EH FRQYHQLHQWO\ UHSUHVHQWHG E\ WKH 0RUVH SRWHQWLDO f 9Uf 'T> HUUHf@ f ZKHUH 'H LV WKH GLVVRFLDWLRQ HQHUJ\ IURP WKH HTXLOLEULXP PAGE 104 HQHUJ\ (T DV JLYHQ E\ f DQG LV D SDUDPHWHU WR EH GHWHUPLQHG 7KH HQHUJ\ HLJHQYDOXHV RI WKH 0RUVH SRWHQWLDO FRQWDLQ RQO\ WZR WHUPV LQ Y af f (Y KH 'HK77 F\ Y ! KH Y 7f Q F\ f (TXDWLQJ WKH FRHIILFLHQWV RI Y 7Mf IURP f DQG f JLYHV =,a LW F\'HK H f 7KH YLEUDWLRQDO HQHUJ\ OHYHOV DUH QRZ H[SUHVVHG DV ( KH f¯HY f¯H[HY O! f 7KH YLEUDWLRQDO SDUWLWLRQ IXQFWLRQ LV FDOFXODWHG DFFRUGLQJ WR X I Hf¯(YN7 f 9 Y ZKHUH (Y LV JLYHQ E\ f XVLQJ H[SHULPHQWDOO\ GHWHUPLQHG YDOXHV RI WR DQG R [ f 1RWH WKDW f LV D ILQLWH H H H f§ VXP ZKHUHDV f LV EDVHG RQ WKH IRUPXOD IRU WKH VXP RI DQ LQILQLWH JHRPHWULF VHULHV ZKLFK LV HTXLYDOHQW WR WKH LQFRUUHFW DVVXPSWLRQ WKDW DQ LQILQLWH QXPEHU RI YLEUDWLRQDO VWDWHV H[LVWV EHORZ WKH GLVVRFLDWLRQ OLPLW 2QO\ D ILQLWH PAGE 105 QXPEHU RI YLEUDWLRQDO VWDWHV LV SUHVHQW EHIRUH GLVVRFLn DWLRQ RFFXUV LQ FRQWUDVW WR LRQL]DWLRQ ZKHUH DQ LQILQLWH QXPEHU RI OHYHOV OLHV EHORZ WKH LRQL]DWLRQ OLPLW Bf 7KH XSSHU OLPLW YF LQ f LV WKH ODUJHVW LQWHJHU IRU ZKLFK ( ' ZKHUH ' LV WKH PHDVXUHG YDOXH RI WKH GLVVRFL DWLRQ HQHUJ\ $WRPLF LRQL]DWLRQ + W + H7KH HTXLOLEULXP SDUWLFOH GHQVLWLHV RI WKH DWRPLF LRQL]DWLRQ UHDFWLRQ DUH UHODWHG E\ WKH 6DKD HTXDWLRQ 7 Ha(+N7 f ZKHUH LV WKH LRQL]DWLRQ SRWHQWLDO RI DWRPLF K\GURJHQ 7KH SDUWLWLRQ IXQFWLRQ RI WKH K\GURJHQ LRQ LV f VLQFH LRQL]HG DWRPLF K\GURJHQ KDV RQO\ RQH VWDWH 7KH SDUWLWLRQ IXQFWLRQ RI DWRPLF K\GURJHQ LV JLYHQ E\ f $WRPLF HOHFWURQ DWWDFKPHQW + b + H7KH 6DKD HTXDWLRQ UHODWLQJ WKH SDUWLFOH GHQVLWLHV RI WKH DWWDFKPHQW UHDFWLRQ LV ZKHUH ( B LV WKH LRQL]DWLRQ HQHUJ\ RI WKH QHJDWLYH PAGE 106 K\GURJHQ LRQ 6LQFH + KDV QR ERXQG VWDWHV DERYH WKH JURXQG VWDWH Bf WKH SDUWLWLRQ IXQFWLRQ RI + LV X+B J+f¯f f ZKHUH J+f LV WKH VWDWLVWLFDO ZHLJKW RI WKH JURXQG VWDWH 7KH SDUWLWLRQ IXQFWLRQ X+ LV DJDLQ JLYHQ E\ f 0ROHFXODU LRQL]DWLRQ + ] + H7KH 6DKD HTXDWLRQ UHODWLQJ WKH SDUWLFOHV RI WKH PROHFXODU LRQL]DWLRQ UHDFWLRQ LV 1+ 1H X 1 + X + 7UP N (U N7 n H nL P Q + f§f 7n f ZKHUH LV WKH LRQL]DWLRQ SRWHQWLDO RI + 7KH SDUWLWLRQ IXQFWLRQ RI X+ KDV EHHQ GLVFXVVHG LQ UHIHUHQFH WR WKH PROHFXODU GLVVRFLDWLRQ UHDFWLRQ DQG LV FRPSXWHG IURP f XVLQJ WKH H[SUHVVLRQV f f DQG f 7KH SDUWLWLRQ IXQFWLRQ RI WKH PROHFXODU K\GURJHQ LRQ LV JLYHQ E\ 8+ 8( 85 89 f«f fµMr I ZKHUH Xe X5 XY DUH FRPSXWHG IRU + IURP H[SUHVVLRQV DQDORJRXV WR WKH + UHODWLRQV f f DQG f %DVLF (TXDWLRQV IRU +\GURJHQ 3ODVPD 7KH VHW RI HTXDWLRQV GHVFULELQJ WKH FRPSRVLWLRQ RI PAGE 107 D K\GURJHQ SODVPD FRQVLVWLQJ RI WKH ILYH K\GURJHQ VSHFLHV + + + + + SOXV HOHFWURQV LVn 1WW 1WW XWW XLW UPf±Pf±N (n N7 + + f¯+ + U + + 1 + X + /-/f 7 H + P+K f 1f± 1B XWM r1A D A (+N7 + H 1 + I f§f 7 X A f + Kf« 1f± 1 + * f± S + I P 1+ 9 K X77 7LPAN B (+BN7 f 1X 1 ,L UP N + H L I W Xf± A n (K N7 1 f + r+ K 1XB 1X 1f± HOHFWULFDO QHXWUDOLW\f f (U +727 1+ ++ 9 1+ 1+ 1H HTXDWLRQ RI VWDWHf f 7R VLPSOLI\ WKH QRWDWLRQ WKH IROORZLQJ GHILQLWLRQV DUH LQWURGXFHG 1[ 1 1 1K 1 1K 1 1KB 1 1K f PAGE 108 X77 X77 B7UP+P+N D R (+nN7 .M 7f J + P+ Kf« ` 7 .7f .7f .7f X+ UP N I H 8+ K 8+ 7UP N U H 8+f¯ K 8+ 7UP N I H XK 9L K f 7 (+N7 fn W H(+N7 f 7 H HK DW 7KH VHW RI HTXDWLRQV WKHQ EHFRPHV . [ 1 1H 1 . 1 1H ga . 1V 1 H 1 . 1 H f f f f f f f f f PAGE 109 1 727 1 ! 1 I ! /r B f L O ZKHUH =A LV WKH QHW FKDUJH RI WKH VSHFLHV 6/ 7R DYRLG FRPSXWDWLRQV LQYROYLQJ SDUWLFOH GHQVLWLHV RQ WKH RUGHU RI rFF WKH HTXDWLRQV DUH VFDOHG E\ GLYLGLQJ HDFK WHUP E\ WKH WRWDO SDUWLFOH GHQVLW\ Q7T7r 8VLQJ WKH GHILQLWLRQV 1 . L B \ r B FL ==n L 1 727 727 f WKH VFDOHG HTXDWLRQV DUH &] r r F7 .! f & &H FA .r f F &H &A . f & & n F .rr f FH F & F f & f & $ B H L f PAGE 110 ([SUHVVLRQV IRU &[ & & DQG & DUH REWDLQHG IURP HTXDWLRQV f WKURXJK f DQG VXEVWLWXWHG LQWR HTXDWLRQV f DQG f UHVXOWLQJ LQ WZR TXDGUDWLF HTXDWLRQV LQ & KDYLQJ FRHIILFLHQWV ZKLFK DUH GHSHQGHQW RQ & H . &UfFL W . ." &H . fF & I f ON . A f . F U LfF & f f . 'HQRWLQJ WKH URRWV RI f DQG f E\ &f DQG &f UHVSHFWLYHO\ WKH VROXWLRQV DUH FOf s , D<6f f & f 7 s a RW<f f ZKHUH . D &H n , A &H .r I < L F f L .r . & D Sf§ Of Sf§ a7 < A a . H &H .r f PAGE 111 )RU LQWHUPHGLDWH WHPSHUDWXUHV DW ZKLFK DOO ILYH K\GURJHQ VSHFLHV DUH SUHVHQW LQ DSSUHFLDEOH TXDQWLWLHV HTXDWLRQV f DQG f DUH XVHG WR GHWHUPLQH WKH SODVPD FRPSRVLWLRQ $Q LWHUDWLYH SURFHGXUH LV XVHG WR r VROYH WKH HTXDWLRQV 7KH %K DUH GHWHUPLQHG E\ WKH SODVPD WHPSHUDWXUH XVLQJ f WKURXJK f 6WDUWLQJ ZLWK f DQ LQLWLDO HVWLPDWH RI WKH HOHFWURQ FRQFHQWUDWLRQ &J & Of WKH FRHIILFLHQWV D[ ; \[ D] DQG \ DUH HYDOXDWHG IURP f DQG f 7KHQ &Of DQG &f DUH FDOFXODWHG IURP f DQG f XVLQJ 'HVFDUWHVn UXOH RI VLJQV WR FKRRVH WKH DOJHEUDLF VLJQV ZKLFK \LHOG SRVLWLYH YDOXHV RI WKH K\GURJHQ GHQVLW\ 1f± & 1Pf±P 7KH HOHFWURQ FRQFHQWUDWLRQ LV LQFUHPHQWHG K r -8 E\ VRPH $& XQWLO H &f± f & f eM ZKHUH H LV WKH GHVLUHG DFFXUDF\ RI WKH K\GURJHQ FRQFHQWUDWLRQ .QRZLQJ WKH YDOXHV RI & DQG & f WKURXJK f DUH XVHG WR HYDOXDWH WKH UHPDLQLQJ K\GURJHQ VSHFLHV LQ WKH IROORZLQJ RUGHU &M &.M & &.& & &&H. &L.W&H f 7KH SDUWLFOH GHQVLWLHV DUH WKHQ JLYHQ E\ 1 1f± DQG 1L FL QWRWn PAGE 112 /RZ 7HPSHUDWXUH (TXDWLRQV 'XH WR WKH GHFUHDVH LQ WKH DWRPLF K\GURJHQ FRQFHQn WUDWLRQ DW ORZ WHPSHUDWXUHV WKH LWHUDWLYH WHFKQLTXH GHVFULEHG LQ WKH SUHFHGLQJ VHFWLRQ GRHV QRW UHDGLO\ FRQYHUJH IRU WHPSHUDWXUHV EHORZ DERXW r. 7KH FRQYHUJHQFH FULWHULD ZHUH PDGH OHVV VWULQJHQW LQ DQ DWWHPSW WR XVH WKH LQWHUPHGLDWH WHPSHUDWXUH WHFKQLTXH IRU ORZ WHPSHUDWXUHV DQG KLJK SUHVVXUHV EXW WKH VROXWLRQV REWDLQHG ZHUH DFFRPSDQLHG E\ SURKLELWLYHO\ ODUJH HUURUV 6LQFH WKH SODVPD WHPSHUDWXUH RI WKH RXWHU K\GURJHQ UHJLRQ FDQ EH RQO\ VOLJKWO\ KLJKHU WKDQ WKH JUDSKLWH PHOWLQJ WHPSHUDn WXUH D GHVFULSWLRQ RI WKH K\GURJHQ SODVPD FRPSRVLWLRQ LV QHHGHG LQ WKH ORZ WHPSHUDWXUH UHJLRQ )RU WHPSHUDWXUHV IURP r. WR r. D GLIIHUHQW QXPHULFDO WHFKQLTXH LV XVHG 7KH HTXDWLRQV f DQG f DUH WDNHQ DV WKH VWDUWLQJ SRLQW 6ROYLQJ HTXDWLRQ f IRU &e JLYHV & Y5 H f ZKHUH 5 & .r f 8VLQJ f ZLWK HTXDWLRQ f \LHOGV WKH HTXDWLRQ PAGE 113 IFf f ZKHUH IFf =U7F77 L . 5Ff F .r FO .r . f 7KH SRVLWLYH URRW RI HTXDWLRQ f JLYHV WKH FRQFHQWUDWLRQ RI DWRPLF K\GURJHQ 5 LV WKHQ HYDOXDWHG IURP f DQG WKH HOHFWURQ FRQFHQWUDWLRQ IURP f 7KH UHPDLQLQJ FRQVWLWXHQWV RI WKH K\GURJHQ SODVPD DUH FDOFXODWHG IURP f (TXDWLRQ f LV VROYHG E\ LWHUDWLQJ RQ & XQWLO IFf H ZKHUH H LV WKH GHVLUHG FRQYHUJHQFH +LJK 7HPSHUDWXUH (TXDWLRQV 7KH VROXWLRQV WR WKH LQWHUPHGLDWH WHPSHUDWXUH HTXDWLRQV f DQG f EHFRPH GLYHUJHQW DERYH VXI ILFHQWO\ KLJK WHPSHUDWXUHV WKH YDOXH RI WKH WHPSHUDWXUH OLPLW GHSHQGLQJ RQ WKH SUHVVXUH $ERYH WKLV OLPLW WKH HUURUV LQ WKH SDUWLFOH FRQFHQWUDWLRQV LQFUHDVH UDSLGO\ DV WKH WHPSHUDWX[nH LQFUHDVHV 7KH HUURUV DUH DVVRFLDWHG ZLWK WKH GLIILFXOW\ RI DFFXUDWHO\ FDOFXODWLQJ WKH + GHQVLW\ VLQFH PRVW RI WKH GLDWRPLF K\GURJHQ LV GLVVRFLDWHG DW KLJK WHPSHUDWXUHV 7KH FRQFHQWUDWLRQV RI DOO RWKHU K\GURJHQ VSHFLHV DUH YHU\ VHQVLWLYH WR WKH + FRQFHQWUDWLRQ PAGE 114 VR HUURUV LQ WKH ODWWHU FDQ UHVXOW LQ ODUJH HUURUV LQ WKH SODVPD FRPSRVLWLRQ 5HVXOWV RI WKH LQWHUPHGLDWH WHPSHUDn WXUH FDOFXODWLRQV VKRZ WKDW WKH VSHFLHV + +f¯ DQG + EHFRPH QHJOLJLEOH DW VXIILFLHQWO\ KLJK WHPSHUDWXUHV 7KH UHPDLQLQJ VSHFLHV DUH DFFRXQWHG IRU E\ FRQVLGHULQJ RQO\ WKH DWRPLF LRQL]DWLRQ UHDFWLRQ + W + H )RU WKLV UHDFWLRQ WKH 6DKD HTXDWLRQ HOHFWULFDO QHXWUDOLW\ FRQGLWLRQ DQG HTXDWLRQ RI VWDWH DUH Xf±W 7UP N 9 1+ 1H BB f± 8+ n 7 H(+N7 1 I X nfµ + f + Kf« QK r Qfü f (7 1727 1+ 1+ 1H f 'LYLGLQJ HDFK HTXDWLRQ E\ 1AA WKH VFDOHG VHW RI HTXDWLRQV LV & BLBBe .r f & & H f & & H f PAGE 115 ZKHUH .r7f LL 727 f 7 H(+N7 f &RPELQLQJ f f DQG f DQG VROYLQJ WKH UHVXOWLQJ HTXDWLRQ IRU JLYHV & .r O .rf f 6LQFH .r LV GHWHUPLQHG E\ WKH SODVPD WHPSHUDWXUH DFFRUGLQJ WR f WKH FRQFHQWUDWLRQV & & DQG & DUH HYDOXDWHG GLUHFWO\ XVLQJ f f DQG f 7KH SDUWLFOH GHQVLWLHV DUH REWDLQHG IURP f /RZHULQJ RI ,RQL]DWLRQ 3RWHQWLDOV 7KH ORZHULQJ RI LRQL]DWLRQ SRWHQWLDOV ZLWK LQFUHDVLQJ FKDUJH GHQVLW\ ZDV GLVFXVVHG LQ &KDSWHU ,9 ZLWK UHIHUHQFH WR WKH XUDQLXP SODVPD 3RWHQWLDO ORZHULQJ ZLOO RFFXU LQ WKH VDPH PDQQHU LQ WKH K\GURJHQ SODVPD DQG ZLOO EH FDOFXODWHG XVLQJ f 7KH FRPSOHWH FDOFXODWLRQ RI WKH K\GURJHQ SODVPD FRPSRVLWLRQ LQFOXGLQJ WKH HIIHFWV RI LRQL]DWLRQ SRWHQWLDO ORZHULQJ DQG WKH FXWRII TXDQWXP QXPEHU RI DWRPLF K\GURJHQ LV DFFRPSOLVKHG E\ WKH +3/$= SURJUDP XVLQJ WKH IROORZLQJ LWHUDWLYH PHWKRG PAGE 116 6WHS Oaa$Q LQLWLDO HOHFWURQ GHQVLW\ HVWLPDWH 1 1 f LV XVHG LQ HTXDWLRQ f WR FDOFXODWH DQ LQLWLDO FXWRII TXDQWXP QXPEHU QArf 7KH LQLWLDO YDOXH RI WKH K\GURJHQ SDUWLWLRQ IXQFWLRQ LV WKHQ FDOFXODWHG IURP f 7KH RWKHU SDUWLWLRQ IXQFWLRQV DQG FRQVWDQWV DUH HYDOXDWHG DV GHVFULEHG LQ WKH GLVFXVVLRQ RI HDFK RI WKH IRXU UHDFWLRQ HTXDWLRQV 6WHS 7KH SODVPD FRPSRVLWLRQ LV FDOFXODWHG XVLQJ WKH ORZ LQWHUPHGLDWH RU KLJK WHPSHUDWXUH HTXDWLRQV DV GHWHUPLQHG E\ WKH SDUWLFXODU YDOXHV RI SUHVVXUH DQG WHPSHUDWXUH 6WHS f§7KH HOHFWURQ GHQVLW\ 1J FDOFXODWHG LQ 6WHS LV XVHG WR UHFDOFXODWH Q IURP f ZKLFK LQ WXUQ JLYHV D QHZ YDOXH IRU Xf± XVLQJ f UO 6WHS 6WHSV DQG DUH UHSHDWHG U WLPHV XQWLO Uf 1Uf . Uf H H H ZKHUH 1 LV WKH HOHFWURQ GHQVLW\ FDOFXODWHG LQ WKH UWK LWHUDWLRQ DQG H LV WKH GHVLUHG FRQYHUJHQFH 6WHS f§7KH LRQL]DWLRQ SRWHQWLDO ORZHULQJ LV FDOFXODWHG E\ f DQG 6WHS LV UHSHDWHG WR JLYH WKH ILQDO YDOXHV IRU WKH SODVPD FRPSRVLWLRQ 5HVXOWV RI +\GURJHQ 3ODVPD &RPSRVLWLRQ &DOFXODWLRQV 7KH +3/$= SURJUDP ZDV XVHG WR FDOFXODWH WKH K\GURJHQ SODVPD FRPSRVLWLRQ IRU SUHVVXUHV IURP DWP PAGE 117 WR DWP DQG WHPSHUDWXUHV IURP r. WR r. $ W\SLFDO FDOFXODWHG FRPSRVLWLRQ LV VKRZQ LQ )LJXUH IRU D SUHVVXUH RI DWP $GGLWLRQDO FDOFXODWHG K\GURJHQ SODVPD FRPSRVLWLRQV IRU SUHVVXUHV IURP DWP WR DWP DUH VKRZQ LQ )LJXUHV WKURXJK $SSHQGL[ % )RU WKH UDQJH RI SUHVVXUHV FRQVLGHUHG WKH WHPSHUDn WXUH GHSHQGHQFH RI WKH YDULRXV K\GURJHQ VSHFLHV LV VLPLODU WR WKDW VKRZQ LQ )LJXUH 7KH PROHFXODU K\GURJHQ UDSLGO\ GLVVRFLDWHV LQWR DWRPLF K\GURJHQ DV WKH WHPSHUDWXUH LQFUHDVHV IURP r. $W D WHPSHUDWXUH RI r. DWRPLF K\GURJHQ FRQVWLWXWHV DERXW b RI WKH WRWDO SODVPD SDUWLFOH GHQVLW\ $ERYH r. WKH DWRPLF K\GURJHQ LRQL]HV UDSLGO\ DQG IRU WHPSHUDWXUHV JUHDWHU WKDQ DERXW r. WKH SODVPD FRQVLVWV RI RYHU b HOHFWURQV DQG LRQL]HG DWRPLF K\GURJHQ WKH UHPDLQLQJ K\GURJHQ VSHFLHV DFFRXQWLQJ IRU OHVV WKDQ b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f§L Lf§Lf§Lf§Lf§Lf§Lf§Lf§cf§Lf§cf§cf§UaLf§UQ UaUQ Ua7 , /f§ W , , , , OO 7HPSHUDWXUH r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a + 8r 8 8 8L DQG HOHFWURQV 7KH UHODWLYH DPRXQWV RI K\GURJHQ DQG XUDQLXP ZLOO EH GHWHUPLQHG E\ WKH SUHVVXUH WHPSHUDWXUH DQG IORZ FKDUDFWHULVWLFV RI WKH PL[LQJ UHJLRQ $W DQ\ ORFDWLRQ LQ PAGE 122 WKH UHDFWRU WKH DPRXQW RI PL[LQJ LV H[SUHVVHG E\ WKH PL[LQJ UDWLR 5 ZKLFK LV GHILQHG DV SDUWLFOH GHQVLW\ RI DOO K\GURJHQ VSHFLHV SDUWLFOH GHQVLW\ RI DOO XUDQLXP VSHFLHV f 7KH K\GURJHQ HTXLOLEULXP UHDFWLRQV DUH + W + + + W + H +B r + H + W + H 7KH FRUUHVSRQGLQJ UHDFWLRQ HTXDWLRQV DUH 1 1 X X LUP P N 9 ( N7 + + f§ + ( f 7 H . [ 7f f 1 + r+ P+ K 1LW 1B 8\+ 7UP N -/B If§e n 8f± nn + H B +f«U U H M 7 Ha(+N7 ( NT7f f 1 + + Kf« 1+ 1H QK X77 UP N 877 +8, $ W mU LB P -/ pBf 7 H(+N7 . 7f f +f¯ Kf« 1+ 1H 8+ LUP N (+ N7 f§f§f§ Aaf 7 H A . 7f f 1 + a+ Kf« PAGE 123 ZKHUH WKH V\PEROV DUH GHILQHG DV LQ &KDSWHU 9 7R VLPSOLI\ WKH QRWDWLRQ OHW 1 R 1 9 1 1 1 +n 1 9 1 1 +W f 'LYLGLQJ ERWK VLGHV RI f WKURXJK f E\ WKH WRWDO SDUWLFOH GHQVLW\ JLYHV G G f . f Gr&H U} G . f G & H B G A f ZKHUH FH a 1H1727 DQG GL f¯ 12L1727 f¯ OI I r fµ rn f 7KH XUDQLXP HTXLOLEULXP UHDFWLRQV DUH 8r W 8 H 8 W 8 H PAGE 124 ,OO Xs L XLf H 8 W 8 H 7KH 6DKD HTXDWLRQ IRU UHDFWLRQ f LV 1L 1H 1L X P N OLOL W H(LN7 Q rfµ R 8 ZKHUH XOf¯f L 7KH V\PEROV DUH GHILQHG DV LQ &KDSWHU 9 7KH RI XUDQLXP UHDFWLRQ HTXDWLRQV LV 1 1H 1 . 1 1 H 1 f± . f f FRPSOHWH VHW f f PAGE 125 1 1 LO H 1 O . O Lf n 1O 1H 1 . f 'LYLGLQJ HTXDWLRQV f WKURXJK f E\ 1747 DQG XVLQJ &L A1US4LM DQG ." .L177 WKH VFDOHG HTXDWLRQV DUH & &R B H B .r & L f f & & LO H B f¯F .L Lf PAGE 126 n . f¬ f &RPELQLQJ HTXDWLRQV f WKURXJK f E\ UHSHDWHG VXEVWLWXWLRQ JLYHV f f & L Lf & . f ZKHUH . L L L Q M L . r M f PAGE 127 7KH HOHFWULFDO QHXWUDOLW\ RI WKH SODVPD LV H[n SUHVVHG DV & H \ ]f± G \ ] F f ZKHUH =AA LV WKH QHW FKDUJH RI WKH K\GURJHQ VSHFLHV GA DQG LV WKH FKDUJH RI WKH XUDQLXP VSHFLHV &E 7KH WRWDO SDUWLFOH GHQVLW\ LV 3 .I a 1 727 1 , 1L ; L O L O 'LYLGLQJ f E\ 1747 f & H , GL FL L O L O f 8VLQJ WKH YDOXH RI IURP HTXDWLRQ Lf LQ f WKH FKDUJH QHXWUDOLW\ FRQGLWLRQ EHFRPHVn ]RL GL L ]L , ]L DL L O L O & F H .¯ [ \ A O L Lf§L L & H f 7KH VHFRQG VXPPDWLRQ EHJLQV ZLWK L VLQFH =[ 6XEVWLWXWLQJ IURP Lf LQWR f \LHOGV PAGE 128 FD , G &O O & & O G F H L O L H L O [ O n L & L f 6ROYLQJ IRU &[ IURP f DQG f DQG HTXDWLQJ WKH UHVXOWLQJ H[SUHVVLRQV JLYHV DQ HTXDWLRQ IRU &4 & f = G H R¯ L .n , ]s L & & O G &L L O f , L L & 7, %\ DOJHEUDLF PDQLSXODWLRQ HTXDWLRQ f FDQ EH ZULWWHQ DV +GM +G +OfG +OfG +OfG +&HOf &J f ZKHUH + LV .B O ]L L &J L U L & )URP HTXDWLRQV f WKURXJK f & .RL.Lf G M T A A G G f°r" A G . f PAGE 129 & GLr f§JMf§f§f§f§ G G .R .T f 6XEVWLWXWLQJ WKHVH H[SUHVVLRQV LQWR f UHGXFHV WKLV HTXDWLRQ WR DGG< f ZKHUH +& +Of& D f§p f .LNf +& +Of & J +Of r f .T .T .T \ +&H f &H 7KH VROXWLRQ RI HTXDWLRQ f LV GV IF7 W r ? D<A 7KH FRHIILFLHQWV D DQG \ FDQ EH HLWKHU SRVLWLYH RU QHJDWLYH 7KH DSSOLFDWLRQ RI 'HVFDUWHVn UXOH RI VLJQV VKRZV WKDW WKHUH LV RQO\ RQH SRVVLEOH FRPELQDWLRQ RI s D s s < s B D<A ZKAFK \LHOGV D UHDOSRVLWLYH YDOXH RI WKH K\GURJHQ LRQ GHQVLW\ G 2QFH D DQG \ DUH QXPHULFDOO\ HYDOXDWHG WKH SURSHU FKRLFH RI WKH UDGLFDO VLJQ GHWHUPLQHV WKH RQH DFFHSWDEOH VROXWLRQ RI HTXDWLRQ f 7KH YDOXH RI G LV REWDLQHG IURP f DQG LV f f PAGE 130 XVHG WR HYDOXDWH WKH UHPDLQLQJ K\GURJHQ FRQFHQWUDWLRQV G G G+ DQG G XVLQJ f 7KH XUDQLXP DWRP FRQFHQWUDWLRQ &M LV WKHQ JLYHQ E\ f 7KH XUDQLXP LRQ FRQFHQWUDWLRQV DUH QH[W FDOFXODWHG XVLQJ HTXDWLRQV f WKURXJK f 7KH K\GURJHQ DQG XUDQLXP GHQVLWLHV DUH JLYHQ E\ f DQG WKH GHILQLWLRQ IROORZLQJ f UHn VSHFWLYHO\ 7KH HOHFWURQ GHQVLW\ LV VWLOO WR EH GHWHUPLQHG 6LQFH D \ DQG + DUH IXQFWLRQV RI & WKH H[SUHVVLRQ f IRU G FRQWDLQV WKH HOHFWURQ FRQFHQWUDWLRQ DV DQ XQGHWHUPLQHG YDULDEOH 7KH UHPDLQLQJ K\GURJHQ VSHFLHV DQG WKH XUDQLXP VSHFLHV DUH DOVR IXQFWLRQV RI &H VLQFH WKH\ DUH GHWHUPLQHG E\ WKH YDOXH RI GA )RU D VSHFLILHG SUHVVXUH DQG WHPSHUDWXUH WKH K\GURJHQ XUDQLXP DQG HOHFWURQ GHQVLWLHV RI WKH PL[HG SODVPD DUH XQLTXHO\ GHWHUPLQHG 7KH SURSHU YDOXH RI &e LV GHWHUPLQHG E\ DSSOLFDWLRQ RI WKH PL[LQJ UDWLR GHILQLWLRQ f ZKLFK KDV QRW EHHQ LQFOXGHG LQ WKH GHULYDWLRQ RI HTXDWLRQ f )URP VROXWLRQV RI WKH FRD[LDO IORZ K\GURG\QDPLF HTXDWLRQV RU IURP PHDVXUHG GHQVLW\ SURILOHV REWDLQHG IURP SODVPD IORZ H[SHULPHQWV WKH PL[LQJ UDWLR LV VSHFLILHG DW HDFK SRLQW LQ WKH PL[LQJ UHJLRQ )URP f WKH PL[LQJ UDWLR LV JLYHQ E\ PAGE 131 , m 5 L O f , F Lf§ /HW WKH VSHFLILHG YDOXH RI WKH PL[LQJ UDWLR EH 5F 7KLV YDOXH LV XVHG WR GHWHUPLQH WKH SURSHU YDOXH RI &H XVLQJ f $Q LQLWLDO HVWLPDWH RI WKH HOHFWURQ FRQFHQWUDWLRQ Frf LV XVHG WR FDOFXODWH + D DQG \ IURP f f f DQG f 7KH G IFAr0 LV HYDOXDWHG E\ f DQG WKH RWKHU DQG &E FAff DUH FDOFXn ODWHG DV GHVFULEHG LQ WKH SUHFHGLQJ SDUDJUDSK 7KH LQLWLDO YDOXH RI WKH PL[LQJ UDWLR LV FDOFXODWHG IURP f DV (G $ /BsB/f§ [ L f ,I 5 f§ 5f ! H Frf LV LQFUHPHQWHG E\ VRPH $& DQG H Uf V fµ H WKH FDOFXODWLRQ LV UHSHDWHG U WLPHV XQWLO 5 5nn H ZKHUH HN LV WKH FRQYHUJHQFH FULWHULRQ 7KH GA DQG &4 YDOXHV DIWHU U LWHUDWLRQV FRQVWLWXWH WKH SURSHU PL[HG SODVPD FRPSRVLWLRQ DV GHWHUPLQHG E\ WKH VSHFLILHG SUHVVXUH WHPSHUDWXUH DQG PL[LQJ UDWLR PAGE 132 /RZ 7HPSHUDWXUH (TXDWLRQV )RU WHPSHUDWXUHV EHORZ DERXW r. WKH QXPHULFDO WHFKQLTXH XVLQJ HTXDWLRQ f H[SHULHQFHV FRQYHUJHQFH GLIILFXOWLHV DV GHVFULEHG SUHYLRXVO\ IRU WKH XUDQLXP DQG K\GURJHQ FRPSRVLWLRQ FDOFXODWLRQV $ GLIIHUHQW PDWKHn PDWLFDO IRUPXODWLRQ LV GHYHORSHG IRU WKH ORZ WHPSHUDWXUH UHJLRQ $W ORZ WHPSHUDWXUHV DOO ILYH K\GURJHQ VSHFLHV DUH FRQVLGHUHG )RU XUDQLXP RQO\ WKH VSHFLHV 8r DQG 8 DUH FRQVLGHUHG 7KH VFDOHG UHDFWLRQ HTXDWLRQV IRU WKH IRXU K\GURJHQ UHDFWLRQV DQG WKH XUDQLXP UHDFWLRQ 8r 8 H DUH JLYHQ E\ f f f f DQG f 7KH WRWDO SDUWLFOH FRQFHQWUDWLRQ LV & \ G \ F Lf§ L O 7KH FKDUJH QHXWUDOLW\ FRQGLWLRQ LV & H \ ]f± G \ ] 6 2L L L F f f 8VLQJ WKH H[SUHVVLRQ IRU &E IURP Lf LQ HTXDWLRQV f DQG f JLYHV U .r GL &W F f PAGE 133 6ROYLQJ IRU &[ UHVXOWV . O =L GL F U f L IURP f DQG f DQG HTXDWLQJ WKH , = G / 2L L O F O A L O .W&H .r& f 8VLQJ HTXDWLRQV f WKURXJK f WR REWDLQ H[SUHVVLRQV IRU G[ G G DQG G HTXDWLRQ f UHGXFHV WR DG_EGF ZKHUH D 3FH 3BfNc & . H r R 3f. S8& E S . S&H f &H .f¬ FH .r 7KH VROXWLRQ WR f LV EB D L !7 DFE f f f f f f PAGE 134 7KH ORZ WHPSHUDWXUH SODVPD FRPSRVLWLRQ FDOFXODWLRQ LV EHJXQ XVLQJ DQ LQLWLDO HOHFWURQ FRQFHQWUDWLRQ HVWLPDWH FMrf WR HYDOXDWH D E F DQG S XVLQJ f WKURXJK f 7KHQ G LV GHWHUPLQHG IURP f DQG G G G DQG G IURP f WKURXJK f & LV FDOFXODWHG IURP f DQG & IURP f 7KH LQLWLDO YDOXH RI WKH PL[LQJ UDWLR LV FRPSXWHG DV DL Uf i9F f 7KH FDOFXODWLRQ LV UHSHDWHG XQWLO WKH YDOXH RI &T LV IRXQG IRU ZKLFK WKH FDOFXODWHG PL[LQJ UDWLR FRQYHUJHV WR WKH VSHFLILHG YDOXH 5Vr +LJK 7HPSHUDWXUH (TXDWLRQV )RU WHPSHUDWXUHV DERYH DSSUR[LPDWHO\ r. RQO\ WKH + DQG + K\GURJHQ VSHFLHV DUH FRQVLGHUHG WR EH SUHVHQW LQ VLJQLILFDQW TXDQWLWLHV $OO XUDQLXP VSHFLHV DUH LQFOXGHG 7KH VFDOHG UHDFWLRQ HTXDWLRQ IRU WKH UHDFWLRQ + W + H DQG WKH XUDQLXP UHDFWLRQV DUH f DQG f WKURXJK f 7KH WRWDO SDUWLFOH FRQFHQWUDWLRQ LV F O G O & H L L O f PAGE 135 7KH FKDUJH QHXWUDOLW\ FRQGLWLRQ LV ! =$ G \ = & H 2¯ L [ L L L O f 6XEVWLWXWLQJ &[ IURP Lf LQWR f DQG f JLYHV .B FH , &WO U W Wf H L L & H L . & 7 =$ G & O = H L L & H f f 6ROYLQJ IRU &[ IURP f DQG f DQG HTXDWLQJ WKH UHVXOWV &H n F .n O ]L L & H F O G H L .9 , f§ L & H f 8VLQJ f WR REWDLQ DQ H[SUHVVLRQ IRU G HTXDWLRQ f EHFRPHV TL FM F T f TF f ZKHUH . R . T ]L U & H f L , LV . L L & L PAGE 136 7KH FDOFXODWLRQ RI WKH KLJK WHPSHUDWXUH SODVPD FRPSRVLWLRQ LV EHJXQ XVLQJ DQ LQLWLDO HVWLPDWH &Arf WR FDOFXODWH T IURP f 7KHQ G LV REWDLQHG IURP f DQG G IURP f 1H[W &[ LV FDOFXODWHG XVLQJ f DQG & XVLQJ f 7KH LQLWLDO YDOXH RI WKH PL[LQJ UDWLR LV G G Urf f§ f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rfLV XVHG LQ HTXDWLRQ f WR FDOFXODWH DQ 4 LQLWLDO YDOXH RI WKH K\GURJHQ FXWRII TXDQWXP QXPEHU Qrf Z PAGE 137 7KH K\GURJHQ SDUWLWLRQ IXQFWLRQ LV WKHQ FDOFXODWHG IURP f 7KH UHPDLQLQJ K\GURJHQ SDUWLWLRQ IXQFWLRQV DUH HYDOXDWHG DV GHVFULEHG LQ WKH GLVFXVVLRQ RI WKH IRXU K\GURJHQ UHDFWLRQV LQ &KDSWHU 9 6WHS 'HSHQGLQJ RQ WKH SUHVVXUH DQG WHPSHUDWXUH WKH ORZ LQWHUPHGLDWH RU KLJK WHPSHUDWXUH HTXDWLRQV DUH XVHG WR FRPSXWH WKH SODVPD FRPSRVLWLRQ E\ LWHUDWLQJ RQ WKH HOHFWURQ FRQFHQWUDWLRQ XQWLO WKH PL[LQJ UDWLR FRQYHUJHV WR WKH VSHFLILHG YDOXH 6WHS 7KH HOHFWURQ GHQVLW\ IURP 6WHS LV XVHG WR UHFDOFXODWH WKH K\GURJHQ FXWRII TXDQWXP QXPEHUV IURP f 7KH K\GURJHQ DQG XUDQLXP GHQVLWLHV IURP 6WHS DUH XVHG WR FDOFXODWH WKH LRQL]DWLRQ SRWHQWLDO ORZHULQJ XVLQJ f 7KH XUDQLXP FXWRII TXDQWXP QXPEHUV DUH WKHQ HVWLPDWHG IURP f 6WHS 8VLQJ WKH LQLWLDO LRQL]DWLRQV SRWHQWLDOV LQ HDFK LWHUDWLRQ 6WHSV DQG DUH UHSHDWHG U WLPHV XQWLO WKH HOHFWURQ GHQVLW\ FRQYHUJHV WR DQG WKH XUDQLXP FXWRII TXDQWXP QXPEHUV FRQYHUJH WR IRU DOO L 6WHS 7KH LRQL]DWLRQ SRWHQWLDO LV FRUUHFWHG XVLQJ f DQG 6WHS LV UHSHDWHG WR JLYH WKH ILQDO YDOXHV IRU WKH PL[HG SODVPD FRPSRVLWLRQ 1AUf 1UBf H H H L! QA Lf PAGE 138 5HVXOWV RI +\GURJHQ8UDQLXP 3ODVPD &RPSRVLWLRQ &DOFXODWLRQV 7KH +83/$= SURJUDP ZDV XVHG WR FDOFXODWH WKH K\GURJHQXUDQLXP SODVPD FRPSRVLWLRQ LQ WKH FRRODQWIXHO PL[LQJ UHJLRQ 7KH FRPSRVLWLRQ ZDV FDOFXODWHG IRU SUHVVXUHV IURP DWP WR DWP DQG WHPSHUDWXUHV IURP r. WR r. 7\SLFDO UHVXOWV DUH VKRZQ LQ )LJXUH IRU D SUHVVXUH RI DWP DQG D PL[LQJ UDWLR RI 7KH UHODWLYH GHQVLWLHV RI WKH XUDQLXP VSHFLHV DUH VLPLODU n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r. LV SULPDULO\ GXH WR HOHFWURQV 7KH PL[LQJ UDWLR XVHG LQ D K\GURJHQXUDQLXP SODVPD FRPSRVLWLRQ FDOFXODWLRQ LV GHWHUPLQHG E\ WKH VLPXODWHG PL[LQJ GHVFULEHG LQ &KDSWHU 9,, PAGE 139 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< 7Q7777777UU7UU7U L L 7nUnQLn7nL777777UU7U a f§ 7f° L , (OHFWURQ U fµ 7HPSHUDWXUH r. )LJXUH &RPSRVLWLRQ RI K\GURJHQXUDQLXP SODVPD DW SUHVVXUH RI DWPRVSKHUHV 0L[LQJ UDWLR PAGE 140 &+$37(5 9,, 6,08/$7,21 2) +<'52*(185$1,80 0,;,1* ,1 67($'< )/2: ,QWURGXFWLRQ $Q DFFXUDWH WUHDWPHQW RI WKH K\GURG\QDPLF FKDUDFn WHULVWLFV RI WKH FRD[LDO IORZ RI KLJK YHORFLW\ FRRODQW DQG ORZ YHORFLW\ IXHO LV QHFHVVDU\ LQ RUGHU WR DQDO\]H WKH UHODWLRQ EHWZHHQ QXFOHDU FULWLFDOLW\ IXHO ORVV HQHUJ\ WUDQVIHU DQG V\VWHP VWDELOLW\ $ QXPEHU RI WKHRUHWLFDO IOXLG IORZ VWXGLHV EDVHG RQ GLIIHUHQW GHJUHHV RI DSSUR[Ln PDWLRQ KDYH EHHQ SHUIRUPHG ZLWK VSHFLILF HPSKDVLV RQ WZRIOXLG IORZ LQ JDVHRXV FRUH UHDFWRUV $ UHSUHVHQWDWLYH DFFRXQW RI VRPH RI WKH SUREOHPV WUHDWHG LQ WKHVH VWXGLHV LV JLYHQ E\ UHIHUHQFHV Bf DQG Bf WKURXJK f 7KH UHVXOWV RI H[SHULPHQWDO VWXGLHV RI PL[LQJ DQG KHDW WUDQVIHU LQ FRD[LDO IORZ RI GLVVLPLODU JDVHV DUH GHVFULEHG LQ UHIHUHQFHV Bf Af Bf DQG f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f VLPXODWH H[SHULPHQWDOO\ PHDVXUHG GHQVLW\ GLVWULEXWLRQV f SHUIRUP SDUDPHWHU VWXGLHV RI WKH HIIHFWV RI YDULRXV PL[LQJ SURILOHV RU f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« )LJXUH 5DGLDO GHQVLW\ SURILOHV RI K\GURJHQ DQG XUDQLXP PAGE 143 QRUPDO GLVWULEXWLRQ SOXV D FRQVWDQW WHUP DV IROORZV 1\LUf DH U AE8 D f ZKHUH E\ LV WKH VWDQGDUG GHYLDWLRQ RI WKH XUDQLXP GLVWULn EXWLRQ DQG GHVFULEHV WKH ZLGWK RI WKH GLVWULEXWLRQ IURP WKH PD[LPXP FRQFHQWUDWLRQ 7KH SDUDPHWHUV ZKLFK DUH XVHG WR VSHFLI\ WKH FRHIILFLHQWV DM DQG D DUH 1\2f DQG 1A5f WKH XUDQLXP GHQVLW\ DW WKH FHQWHU DQG DW WKH UHIOHFWRU UHVSHFWLYHO\ 8VLQJ WKH ERXQGDU\ FRQGLWLRQV 9U! U R 92f 9U! U 5 185f f WKH FRHIILFLHQWV DUH IRXQG WR EH 185f 1\L2f 59E A 1\&2MHA E8 1\L5f f° H5E [ f 7KH DYHUDJH XUDQLXP GHQVLW\ LV I5 1 8 1 UfUGU D E r Hn5 Emf D f 5 5n UGU 7KH UDGLDO K\GURJHQ SURILOH KDYLQJ WZR PD[LPD DV VKRZQ LQ )LJXUH FDQ DOVR EH GHVFULEHG E\ D V\PPHWULF QRUPDO GLVWULEXWLRQ SOXV D FRQVWDQW PAGE 144 1+Uf FHUf¯U+f E+ F f ZKHUH Ef± LV WKH VWDQGDUG GHYLDWLRQ RI WKH K\GURJHQ GLVWUL UL EXWLRQ DQG ULV WKH SRVLWLRQ RI WKH PD[LPXP K\GURJHQ GHQVLW\ 7KH FRHIILFLHQWV F[ DQG F DUH HYDOXDWHG XVLQJ WKH ERXQGDU\ FRQGLWLRQV 9Uf U QKf 1+Uf U Uf± 9U+! } UL ZKHUH 1f±f DQG 1f±Uf±f DUH WKH PLQLPXP DQG PD[LPXP K\GURJHQ UL UL UL GHQVLW\ UHVSHFWLYHO\ )URP f DQG f WKH K\GURJHQ FRHIILFLHQWV DUH 1Kf 15U+f HaU+E+ 1+U+fHf¯U+E+ 1Kf F Uf§L ^f U+E+ B [ 7KH DYHUDJH K\GURJHQ GHQVLW\ LV JLYHQ E\ 1 + 5 1f±UfUGU 5 UGU 5 HU+E+ B f ^_A>1+U+fHU+E+ B QKf@ E>1Kf 15U+f@Hf¯U+E+ H5U+! E+M 5 Uf± U WW U E >HUI f HUI f@` n E + E + f PAGE 145 ZKHUH HUI[f f§ g \ H G\ 7KH PL[LQJ UDWLR EDVHG RQ WKH K\GURJHQ DQG XUDQLXP GLVWULn EXWLRQV LV JLYHQ E\ F Hf°UU+fE+ f 5Uf 7KH K\GURG\QDPLF FKDUDFWHULVWLFV RI WKH K\GURJHQ XUDQLXP FRD[LDO IORZ DUH GHVFULEHG TXDQWLWDWLYHO\ E\ VSHFLI\LQJ WKH UDGLDOO\ GHSHQGHQW PL[LQJ UDWLR DQG WKH K\GURJHQ DQG XUDQLXP PDVV IORZ UDWHV 7KH PL[LQJ UDWLR DQG WKH PDVV IORZ UDWHV DUH GLUHFWO\ UHODWHG WR WKH K\GURJHQXUDQLXP PL[LQJ PRGHO EDVHG RQ f DQG f (DFK RI WKHVH GLVWULEXWLRQV FRQWDLQV WKUHH SDUDPHWHUV ZKLFK FDQ EH DGMXVWHG WR JLYH WKH VSHFLILHG YDOXHV RI PL[LQJ UDWLR DQG IORZ UDWHV 6LQFH WKH XUDQLXP PDVV IORZ UDWH LV SURSRUWLRQDO WR WKH FRQVWDQW 1\f DGMXVWLQJ WKH YDOXH RI 1\L2f VLPXODWHV WKH FRQWURO RI WKH LQMHFWHG XUDQLXP PDVV IORZ 7KH DPRXQW RI UDGLDO GLVSHUVLRQ RI XUDQLXP IURP WKH LQMHFWLRQ UHJLRQ LV GHVFULEHG E\ WKH SDUDPHWHU EA 7KH H[WHQW WR ZKLFK XUDQLXP SHQHWUDWHV WKH FRRODQW DQQXOXV LV GHVFULEHG E\ WKH FRQVWDQW 1A5f $Q LQFUHDVH LQ 1A5f FRUUHVSRQGV WR DQ LQFUHDVH LQ IXHO ORVV GXH WR XUDQLXP PAGE 146 EHLQJ UHPRYHG E\ WKH K\GURJHQ FRRODQW )RU WKH K\GURJHQ GLVWULEXWLRQ WKH PDVV IORZ LV SURSRUWLRQDO WR WKH FRQVWDQW 1MM UAMf E+ GHVFULEHV WKH GLVSHUVLRQ RI K\GURJHQ IURP WKH DQQXODU LQMHFWLRQ UHJLRQ DQG 1\^f GHVFULEHV WKH DPRXQW RI FRRODQW SHQHWUDWLRQ LQWR WKH FHQWUDO IXHO UHJLRQ 7KH VLPXODWHG K\GURJHQXUDQLXP PL[LQJ LV FDOFXODWHG E\ WKH 1250 SURJUDP ZKLFK FRPSXWHV WKH UDGLDOO\ GHSHQGHQW DQG UHJLRQ DYHUDJHG YDOXHV RI WKH K\GURJHQ DQG XUDQLXP GLVWULEXWLRQV DQG WKH PL[LQJ UDWLR DV D IXQFWLRQ RI WKH PL[LQJ SDUDPHWHUV 1\f E\ 1\5f 1+U+f E\ DQG 1\f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f§7KH UHDFWRU SUHVVXUH LV VSHFLILHG DQG LV DVVXPHG WR EH FRQVWDQW ZLWKLQ WKH JUDSKLWH UHDFWLRQ FKDPEHU 6WHS f§7KH 1250 SURJUDP LV XVHG WR FDOFXODWH WKH UDGLDO GHQVLW\ GLVWULEXWLRQV RI XUDQLXP DQG K\GURJHQ LQ D VLPXODWHG FRD[LDO IORZ RI IXHO DQG FRRODQW 7KH GLVWULn EXWLRQ SDUDPHWHUV DUH FKRVHQ VXFK WKDW WKH UHDFWRU LV FULWLFDO 6WHS f§7KH GLVWULEXWLRQ RI HQHUJ\ SURGXFHG E\ ILVVLRQLQJ XUDQLXP LV FDOFXODWHG XVLQJ WKH XUDQLXP GLVWULn EXWLRQ IURP 6WHS 6WHS f§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n EXWLRQV RU E\ VHHGLQJ WKH FRRODQW UDGLDWLRQ SURGXFHG LQ WKH XUDQLXP UHJLRQ ZLOO WUDYHO VPDOO GLVWDQFHV EHWZHHQ WKH SRLQWV RI HPLVVLRQ DQG DEVRUSWLRQ 7KH DYHUDJH GLVWDQFH PAGE 149 WUDYHOHG LV UHIHUUHG WR DV WKH UDGLDWLRQ PHDQ IUHH SDWK DQG WKH WUDQVSRUW RI HQHUJ\ E\ UDGLDWLRQ LQ WKH RSDTXH PHGLXP DSSUR[LPDWHV WKH NLQHWLF WKHRU\ SURFHVV RI GLIn IXVLRQ f ,Q WKH SUHVHQFH RI UDGLDWLRQ DQ DWRP RU LRQ KDV D FHUWDLQ SUREDELOLW\ RI HLWKHU DEVRUELQJ RU HPLWWLQJ UDGLn DWLRQ LQGXFHG WUDQVLWLRQVf RI D IUHTXHQF\ FKDUDFWHULVWLF RI VRPH WUDQVLWLRQ IURP RQH TXDQWXP VWDWH WR DQRWKHU $Q H[FLWHG DWRP RU LRQ FDQ DOVR VSRQWDQHRXVO\ HPLW UDGLDWLRQ KDYLQJ D FKDUDFWHULVWLF IUHTXHQF\ (DFK DWRP RU LRQ WKHUHIRUH KDV DQ DEVRUSWLRQ VSHFWUXP ZKLFK FDQ EH H[SUHVVHG LQ WHUPV RI DQ DEVRUSWLRQ FRHIILFLHQW . ZKLFK LV IUHTXHQF\ GHSHQGHQW 6LQFH WKH DEVRUSWLRQ VSHFWUXP GHSHQGV XSRQ WKH GLVWULEXWLRQ RI SODVPD VSHFLHV LQ WKHLU YDULRXV TXDQWXP VWDWHV GHSHQGV RQ WKH SODVPD WHPSHUDn WXUH DQG WKH SDUWLFOH GHQVLW\ $V WKH SODVPD GHQVLW\ LQFUHDVHV SUHVVXUH EURDGHQLQJ FKDQJHV WKH HPLVVLRQ VSHFWUXP f VR WKDW DOVR GHSHQGV RQ WKH GHQVLW\ WKURXJK WKLV HIIHFW 7KH IUHTXHQF\ GHSHQGHQW DEVRUSWLRQ FRHIILFLHQW IRU SKRWRQV FDQ EH H[SUHVVHG DV D IXQFWLRQ RI WHPSHUDWXUH DQG GHQVLW\ DV IROORZV f .Y7Sf O 1D7Sf DDEYf W HBK9N7fLQ FPB f PAGE 150 ZKHUH 1 7Sf QXPEHU RI SDUWLFOHVFF LQ VWDWH D FO DALYf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f ZKLFK LV GHILQHG DV . 5 UFR fµ G%Y7f GI GY fµrr / G%Y7f . G7 Q Y GY f ZKHUH %A7f LV WKH IUHTXHQF\ GHSHQGHQW EODFNERG\ UDGLDWLRQ LQWHQVLW\ JLYHQ E\ WKH 3ODQFN IRUPXOD PAGE 151 % 7f f§f§ HK957 Of f 9 F )RU WKH RSWLFDOO\ WKLFN PHGLXP WKH UDGLDWLRQ IOX[ TU LV H[SUHVVHG DV D IXQFWLRQ RI WHPSHUDWXUH E\ WKH 5RVVHODQG GLIIXVLRQ DSSUR[LPDWLRQ f AU NG97 . f ZKHUH N LV WKH GLIIXVLRQ FRHIILFLHQW JLYHQ E\ -IF? D7 f . LV WKH 5RVVHODQG PHDQ RSDFLW\ GHILQHG E\ f DQG D . LV WKH 6WHIDQ%ROW]PDQQ FRQVWDQW (TXDWLRQ f LV WKH VLPSOH NLQHWLF WKHRU\ H[SUHVVLRQ IRU WKH GLIIXVLRQ RI SKRWRQV KDYLQJ D PHDQ IUHH SDWK ;5 O.5f FP 7KH HQHUJ\ EDODQFH IRU D XQLW YROXPH ZLWKLQ D FRD[LDOO\ IORZLQJ SODVPD UHJLRQ DW FRQVWDQW SUHVVXUH FDQ EH H[SUHVVHG DV YLVFRXV GLVVLSDWLRQ QHJOHFWHGf f§ 6 9 fµ T f ZKHUH '+ A UDWH RI HQWKDOS\ LQFUHDVH SHU XQLW YROXPH 6 UDWH RI KHDW JHQHUDWLRQ SHU XQLW YROXPH 9 fµ T HQHUJ\ ORVV UDWH IURP WKH XQLW YROXPH PAGE 152 7KH HQWKDOS\ LQFUHDVH LV '+ 'W f ZKHUH S GHQVLW\ RI WKH PHGLXP &S VSHFLILF KHDW 7 DEVROXWH WHPSHUDWXUH )RU VWHDG\ VWDWH FRQGLWLRQV ZLWK XUDQLXP DQG K\GURJHQ IORZLQJ LQ WKH D[LDO GLUHFWLRQ RQO\ WKH KHDW EDODQFH HTXDWLRQ EHFRPHV G7 r SFSY] D" V Y fµ ! ,Q WKH SUHVHQW FDVH LQ ZKLFK RQO\ WKH UDGLDO SURILOHV RI XUDQLXP K\GURJHQ DQG WHPSHUDWXUH DUH FRQVLGHUHG WKH DSSOLFDWLRQ RI WKH GLIIXVLRQ DSSUR[LPDWLRQ f WR WKH HQHUJ\ EDODQFH HTXDWLRQ \LHOGV 9 fµ NUYWf 6 f 7KH VROXWLRQ WR HTXDWLRQ f UHSUHVHQWV WKH UDGLDO WHPSHUDWXUH GLVWULEXWLRQ LQ WKH SODVPD FRUH UHDFWRU EDVHG RQ WKH 5RVVHODQG GLIIXVLRQ DSSUR[LPDWLRQ IRU SKRWRQV 7KH WHPSHUDWXUH GLVWULEXWLRQ ZLOO EH GHWHUPLQHG E\ WKH RSDFLW\ . DQG WKH KHDW JHQHUDWLRQ UDWH6 ZKLFK LV SURSRUWLRQDO ,9 WR WKH VSDWLDOO\ GHSHQGHQW ILVVLRQ UDWH PAGE 153 &RQVWDQW 'LIIXVLRQ &RHIILFLHQW $SSUR[LPDWLRQ 7R H[DPLQH WKH SRVVLELOLW\ RI REWDLQLQJ DQDO\WLF VROXWLRQV WR f WKH VLPSOHVW FDVH LQ ZKLFK WKH GLIn IXVLRQ FRHIILFLHQW LV D FRQVWDQW LV FRQVLGHUHG ILUVW ,Q WKLV DSSUR[LPDWLRQ WKH GLIIXVLRQ FRHIILFLHQW NU LV LQGHSHQGHQW RI SRVLWLRQ DQG WHPSHUDWXUH DQG WKH KHDW EDODQFH HTXDWLRQ UHGXFHV WR D IRUP RI 3RLVVRQnV HTXDWLRQ 97 Uf f .5 7KH ERXQGDU\ FRQGLWLRQV DUH 7 f 74 f 75f 7U ZKHUH 7A DQG 7f± DUH WKH WHPSHUDWXUHV DW WKH UHDFWRU 25 FHQWHUOLQH DQG WKH LQVLGH IDFH RI WKH JUDSKLWH UHVSHFWLYHO\ 7KH PHWKRG RI *UHHQnV IXQFWLRQ LV XVHG WR REWDLQ WKH WHPSHUDWXUH GLVWULEXWLRQ LQ WKH FRQVWDQW GLIIXVLRQ FRHIILFLHQW DSSUR[LPDWLRQ (TXDWLRQ f LV ZULWWHQ DV /Uf7Uf IUf f ZKHUH f I Uf f 5 9 PAGE 154 7KH VROXWLRQ RI f LV JLYHQ E\ 7Uf fµU I Unf * UUnfGUn f -2 ZKHUH WKH *UHHQnV IXQFWLRQ *UUnf LV WKH VROXWLRQ RI /Uf*UUnf UUnf f ZLWK ERXQGDU\ FRQGLWLRQV * Unf 74 *5Unf 7U f 0RUVH DQG )HVKEDFK f REWDLQ DQ H[SUHVVLRQ IRU *UUnf E\ ILUVW VROYLQJ WKH 3RLVVRQnV HTXDWLRQ LQ UHFWDQJXODU FRRUGLQDWHV IRU WKH SRWHQWLDO GXH WR D SRLQW FKDUJH DW [n\nf 9?S[\f f§ WWµ [[nf \\nf f 7KH HYDOXDWLRQ RI *UUnf XVLQJ WKH VROXWLRQ WR f LV OHQJWK\ DQG RQO\ WKH SHUWLQHQW GHWDLOV QHFHVVDU\ IRU FRQWLQXLW\ ZLOO EH RXWOLQHG KHUH 7KH VROXWLRQ WR f LV WKH *UHHQnV IXQFWLRQ *[\ [n\nf eQ 5H =Q]]nf O [ L\ f 8VLQJ D 7D\ORUnV VHULHV H[SDQVLRQ IRU =Q]]f¬f PAGE 155 *[\ [n \nf $Q] M! _ afQ _]_ ! _]n Q O eQ]r O L IUfQ _]_ _]f¬_ f ; L 7KH WUDQVIRUPDWLRQ ] UHp ]n UnHp LV XVHG WR H[SUHVV WKH *UHHQnV IXQFWLRQ LQ SRODU FRRUGLQDWHV * U U nf eQU eQU , Q O Q , Q O afQ FRV Qnf U f¯ LAU`Q FRV Qnf ! Un U Un f )RU D SRLQW VRXUFH RI KHDW 6Unnf ZKHUH Un B 5 WKH VWHDG\ VWDWH WHPSHUDWXUH GLVWULEXWLRQ LV JLYHQ E\ WKH *UHHQnV IXQFWLRQ f PRGLILHG E\ WKH VROXWLRQ RI /DSODFHnV HTXDWLRQ ZKLFK VDWLVILHV WKH ERXQGDU\ FRQGLWLRQ 7 75f DW U 5 /HW WKH PRGLILHG *UHHQnV IXQFWLRQ EH GHQRWHG E\ 7U Unnf 6LQFH WKH ILVVLRQ KHDW JHQHUDWLRQ UDWH LV GLVWULEXWHG RYHU WKH UHDFWRU FRUH WKH UHVXOWLQJ WHPSHUDWXUH GLVWULEXWLRQ ZLOO EH WKH LQWHJUDO RI 7U Unnf RYHU WKH HQWLUH FRUH UHJLRQ 7KH UHVXOWLQJ H[SUHVVLRQ EHFRPHV 7Uf 7Uf A 77. 5 77 Gn ^ UU Un GU n > $Q f PAGE 156 Q Q Q f¯ , , f§fFRV Q nf @ 6 U n nf Q O 5Q UQ 5 UnGUn >-OQA7f O 3 Q Q Q O 5 Q f§afFRV Q n f @ UnQ [ 6U n nf ` f 6LQFH WKH ILVVLRQ KHDW JHQHUDWLRQ LV FRQVLGHUHG WR EH LVRWURSLF 6Unnf ZLOO EH LQGHSHQGHQW RI n DQG WKH WHPSHUDWXUH GLVWULEXWLRQ f UHGXFHV WR WKH IRUP f 7Uf 7U L 8QIf U U n 6 Unf GUn n ,Q SMVLUn-GUn ` f 7KH KHDW JHQHUDWLRQ UDWH ZKLFK LV SURSRUWLRQDO WR WKH ILVVLRQ UDWH LV JLYHQ E\ 6 Uf & (IU(fI!U(fG( &1\Uf FI (f FM! U(fG( f ZKHUH & FRQYHUVLRQ IDFWRU ZKLFK H[SUHVVHV WKH ILVVLRQ UDWH LQ XQLWV RI HQHUJ\FFVHF 1f±Uf XUDQLXP GHQVLW\ GLVWULEXWLRQ X "e PLFURVFRSLF ILVVLRQ FURVV VHFWLRQ I!U(f QHXWURQ IOX[ PAGE 157 7KH FDOFXODWHG WKHUPDO IOX[ LQ WKH FRUH UHJLRQ DV VKRZQ LQ )LJXUH RI &KDSWHU ,, LV VHHQ WR EH UHODWLYHO\ LQGHn SHQGHQW RI SRVLWLRQ %\ PDNLQJ WKH VLPSOLI\LQJ DSSUR[Ln PDWLRQ WKDW WKH UDGLDO IOX[ LV FRQVWDQW WKH KHDW JHQHUDWLRQ UDWH LV SURSRUWLRQDO WR WKH XUDQLXP GHQVLW\ f 6 Uf &1\LUf RUI (f ! (f G( & n 1\ Uf ZKHUH f DI (f FMf (fG( & 7KH VLPXODWHG XUDQLXP GHQVLW\ GLVWULEXWLRQ GLVFXVVHG LQ &KDSWHU 9,, LV JLYHQ E\ f 6XEVWLWXWLQJ f LQWR f f ZKHUH $ DL&n $ D&n f 8VLQJ WKH KHDW JHQHUDWLRQ UDWH f WR HYDOXDWH WKH WHPSHUDWXUH GLVWULEXWLRQ f 5 s U PAGE 158 3HUIRUPLQJ WKRVH LQWHJUDWLRQ LQ f ZKLFK FDQ EH HYDOXDWHG DQDO\WLFDOO\ WKH WHPSHUDWXUH GLVWULEXWLRQ EHFRPHV 7Uf 7' ZKHUH $ $L$f 5 Uf Uf f ,Uf 5 U=QUnH Un 9EX GU f U )URP f DQG f WKH VHFRQG ERXQGDU\ FRQGLWLRQ RI f 75f 7 LV VHHQ WR EH VDWLVILHG 7KH ILUVW 5 ERXQGDU\ FRQGLWLRQ RI f JLYHV 7R 75 rF 5 $$f5 ORf f 7KH UDWLR $$ LV HYDOXDWHG XVLQJ f DQG f $ DM&n 1\5f 1\2f D&n 5EQ 0 1\LRMH n X 1\5f f )URP f WKH FRQVWDQW $ LV $ NU7* WUf 5$L$DM5 Rf f 7KH WHPSHUDWXUH JLYHQ E\ f EHFRPHV 7Uf 7U 7 7 R 5 $$`U Rf $L$M 5UfOU f PAGE 159 2QFH WKH XUDQLXP GLVWULEXWLRQ LV VSHFLILHG WKH SDUDPHWHUV 1\Rf 1\5f DQG DUH GHWHUPLQHG DQG $$ FDQ EH HYDOXn DWHG IURP f 7KH H[SUHVVLRQ f UHSUHVHQWV WKH WHPSHUDWXUH GLVWULEXWLRQ IRU WKH FDVH RI D FRQVWDQW GLIn IXVLRQ FRHIILFLHQW DQG D FRQVWDQW UDGLDO QHXWURQ IOX[ )RU WKLV VLPSOHVW FDVH D FRPSOHWHO\ DQDO\WLFDO H[SUHVVLRQ IRU WKH WHPSHUDWXUH GLVWULEXWLRQ ZDV QRW REWDLQHG VLQFH WKH LQWHJUDO ,Uf PXVW EH HYDOXDWHG QXPHULFDOO\ )RU WKH PRUH UHDOLVWLF FDVHV LQYROYLQJ IHZHU DSSUR[LPDWLRQV LW LV HYHQ OHVV OLNHO\ WKDW DQDO\WLFDO VROXWLRQV ZLOO EH REWDLQHG 2SDFLW\ ZLWK ([SOLFLW 7HPSHUDWXUH 'HSHQGHQFH ,I WKH 5RVVHODQG RSDFLW\ LV H[SUHVVHG DV VRPH H[SOLFLW IXQFWLRQ RI WHPSHUDWXUH . I7f WKHQ WKH PHWKRG RI WKH SUHFHGLQJ VHFWLRQ FDQ EH XVHG WR REWDLQ DQ H[SUHVVLRQ IRU WKH WHPSHUDWXUH GLVWULEXWLRQ 8VLQJ WKH H[SOLFLW IRUP RI WKH RSDFLW\ WKH GLIIXVLRQ FRHIILFLHQW JLYHQ E\ f LV NU NU7f J7f f ZKHUH f 7KH HQHUJ\ EDODQFH HTXDWLRQ f WKHQ EHFRPHV 9 fµ J7f97f f PAGE 160 (TXDWLRQ f ZKLFK LV QRQOLQHDU LQ WKH WHPSHUDWXUH LV OLQHDUL]HG E\ DSSOLFDWLRQ RI WKH .LUFKRII WUDQVIRUn PDWLRQ f W 7f B 6R U7Uf J7nfG7n 7 R f ZKHUH 7T GHQRWHV VRPH UHIHUHQFH WHPSHUDWXUH IRU ZKLFK J7f J4 )URP f J497 J7f97 f 8VLQJ WKLV H[SUHVVLRQ IRU J7f97 HTXDWLRQ f EHFRPHV V ! 6LQFH f KDV WKH VDPH IRUP DV f WKH PHWKRG XVHG WR VROYH f IRU 7Uf FDQ EH XVHG WR REWDLQ DQ H[SUHVVLRQ VLPLODU WR f IRU [Uf 6XEVWLWXWLQJ WKLV H[SUHVVLRQ IRU [Uf LQWR f JLYHV I7Uf J7nfG7n J4[Uf f fµ -WR :KHQ WKH LQWHJUDO FDQ EH HYDOXDWHG DQDO\WLFDOO\ WKH WHPSHUDWXUH GLVWULEXWLRQ 7Uf LV REWDLQHG E\ VROYLQJ WKH HTXDWLRQ *7f J4[Uf f PAGE 161 ZKHUH *7f 7Uf J7nfG7n n7 f ,Q PRVW FDVHV RI SUDFWLFDO LQWHUHVW VLPSOH DQDn O\WLFDO H[SUHVVLRQV IRU WKH RSDFLW\ RI WKH IXHO FRRODQW RU PL[HG SODVPD UHJLRQV DUH QRW DYDLODEOH 7KH WHPSHUDWXUH GHSHQGHQFH RI WKHVH RSDFLWLHV ZLOO EH GHWHUPLQHG IURP H[SHULPHQWDO PHDVXUHPHQWV RU WKHRUHWLFDO FDOFXODWLRQV ,Q VXFK FDVHV QXPHULFDO PHWKRGV PXVW EH XVHG WR REWDLQ WHPSHUDWXUH GLVWULEXWLRQV &RPSRVLWH 2SDFLW\ ZLWK $UELWUDU\ 7HPSHUDWXUH GHSHQGHQFH /HW WKH RSDFLW\ ZLWK DUELWUDU\ WHPSHUDWXUH GHn SHQGHQFH EH GHQRWHG E\ .5 .57f 7KH FRUUHVSRQGLQJ GLIIXVLRQ FRHIILFLHQW LV .U7f FU7 .U7f f 7KH HQHUJ\ EDODQFH HTXDWLRQ f LV H[SUHVVHG DV 9 fµ NU7f 97f 6 f 6XEVWLWXWLQJ f LQWR HTXDWLRQ f DQG XVLQJ WKH LGHQWLW\ 9 fµ X9Yf X9]Y 9X fµ 9Y f PAGE 162 WKH HQHUJ\ EDODQFH HTXDWLRQ EHFRPHV G U ) D G7 W7 GU 5 6 f ([SDQGLQJ WKH GHULYDWLYH WHUP HTXDWLRQ f EHFRPHV G7 U GU .U G7 f§L nGUn NU D7 6 f 7KH VROXWLRQ WR WKH QRQOLQHDU VHFRQG RUGHU HTXDWLRQ f UHSUHVHQWV WKH UDGLDO WHPSHUDWXUH GLVWULEXWLRQ LQ WKH SODVPD FRUH UHDFWRU KDYLQJ DQ RSDFLW\ ZLWK DUELWUDU\ WHPSHUDWXUH GHSHQGHQFH 7KUHH ERXQGDU\ FRQGLWLRQV DUH XVHG ZLWK HTXDWLRQ f 7KH ILUVW WZR FRQGLWLRQV DUH REWDLQHG E\ VSHFLI\LQJ WKH FHQWHUOLQH WHPSHUDWXUH DQG E\ UHTXLULQJ WKDW WKH UDGLDO WHPSHUDWXUH GLVWULEXWLRQ EH V\PPHWULF DERXW U 7KH WKLUG FRQGLWLRQ UHVXOWV IURP WKH FRQVWUDLQW WKDW WKH WHPSHUDWXUH DW WKH LQVLGH IDFH RI WKH JUDSKLWH FKDPEHU PXVW QRW H[FHHG WKH PHOWLQJ SRLQW RI JUDSKLWH 7KHVH WKUHH ERXQGDU\ FRQGLWLRQV DUH H[SUHVVHG DV 7 f 7 G7 f GU 75f 7 5 f ZKHUH 7' LV WKH PD[LPXP DOORZDEOH JUDSKLWH WHPSHUDWXUH 5 PAGE 163 7KH WHPSHUDWXUH GLVWULEXWLRQ LV REWDLQHG E\ XVLQJ QXPHULFDO PHWKRGV WR VROYH HTXDWLRQ f 7KLV HTXDWLRQ LV ILUVW ZULWWHQ LQ WKH IRUP G7 GU )Uf G7 B G.5 U GU .G G7 . NUV D7 f 7KH WHPSHUDWXUH JUDGLHQW LV HYDOXDWHG RYHU WKH LQWHUYDO >2 5@ DV IROORZV G7 ) Uf )UnfGU n G7 GU f f 7KH WHPSHUDWXUH GLVWULEXWLRQ RYHU >2 5@ LV WKHQ JLYHQ E\ 7 Uf UU / /BUnf GU nU GUn 7 f f XVLQJ WKH YDOXHV IURP f IRU ,Q RUGHU WR G7 REWDLQ YDOXHV IRU A XVLQJ f )Uf PXVW EH LQLWLDOO\ HYDOXDWHG DW U )URP f WKH ILUVW WHUP RI )f DSSHDUV WR EH XQGHILQHG VLQFH OLPL _ LV DQ LQGHn WHUPLQDWH IRUP 7R HOLPLQDWH WKLV FRPSXWDWLRQDO GLIILFXOW\ U LV UHSODFHG E\ U (J ZKHQ UHI&A(J@ ZKHUH (J LV DQ DUELWUDULO\ VPDOO SRVLWLYH FRQVWDQW 7KLV DOORZV WKH QXPHULFDO LQWHJUDWLRQ WR EHJLQ DW U VLQFH WKH ILUVW WHUP RI )f EHFRPHV PAGE 164 OLP Ur U G7L GUn U e f 7KLV DSSUR[LPDWLRQ LV HTXLYDOHQW WR VROYLQJ f IRU 7Uf G7 LQ WKH LQWHUYDO >eT5@ ZLWK WKH DVVXPSWLRQ WKDW JS DQG 7 a 7T LQ > 6LQFH WKH 5RVVHODQG RSDFLW\ GHILQHG E\ f DQG f LV D PDFURVFRSLF TXDQWLW\ WKH FRPSRVLWH RSDFLW\ DVVRFLDWHG ZLWK D PL[WXUH RI 1 GLIIHUHQW SODVPD VSHFLHV LV 1 . 7f O Ne7f f 5 L O 5 ZKHUH LV RSDFLW\ RI WKH LWK VSHFLHV ,Q WKH SUHVHQW PRGHO RI WKH SODVPD FRUH UHDFWRU WKH RSDFLW\ LV GXH WR XUDQLXP K\GURJHQ DQG DQ\ VHHGLQJ PDWHULDOV ZKLFK PD\ EH PL[HG ZLWK WKH IXHO RU FRRODQW 7KH FRPSRVLWH RSDFLW\ IRU WKLV PRGHO LV NU NU NU NU f ZKHUH O .AL RSDFLW\ RI DOO XUDQLXP VSHFLHV 5 9 5 .5 O .5L RSDFLW\ RI DOO K\GURJHQ VSHFLHV 5 fµ 5 4 4 fµ .f± O .A RSDFLW\ RI DOO VHHG VSHFLHV 5 5 PAGE 165 G.5 7KH WHUP LQ HTXDWLRQ f LV HYDOXDWHG E\ QXPHULFDOO\ GLIIHUHQWLDWLQJ . 7f ZKHUH . 7f LV DQ HPSLULFDO H[SUHVVLRQ EDVHG RQ PHDVXUHG RU FDOFXODWHG RSDFLWLHV .UDVFHOOD DQG 5HVWHQ KDYH PDGH WKHRUHWLFDO FDOFXn ODWLRQV RI WKH WHPSHUDWXUH GHSHQGHQW 5RVVHODQG PHDQ RSDFLWLHV RI XUDQLXP Bf K\GURJHQ ef DQG SRVVLEOH VHHG PDWHULDOV f 7KH FDOFXODWHG XUDQLXP DQG K\GURJHQ RSDFLWLHV DW VHYHUDO SUHVVXUHV DUH VKRZQ LQ )LJXUHV DQG 7KH RSDFLWLHV ZHUH FDOFXODWHG IRU D SODVPD FRQn VLVWLQJ HQWLUHO\ RI XUDQLXP K\GURJHQ RU VHHG PDWHULDO UDWKHU WKDQ VRPH PL[HG FRPSRVLWLRQ ,Q RUGHU WR XVH WKHVH YDOXHV LQ WKH FRPSRVLWH RSDFLW\ H[SUHVVLRQ JLYHQ E\ f WKH IXHO FRRODQW DQG VHHG RSDFLWLHV DUH ZHLJKWHG E\ WKH UHODWLYH DPRXQW RI HDFK PDWHULDO 7KH FRPSRVLWH RSDFLW\ EDVHG RQ WKLV ZHLJKWLQJ LV 1 1 1 . .8 .+ .6 NU NU Q NU Q NU Q f ZKHUH (/ LV WKH SDUWLFOH GHQVLW\ RI PDWHULDO L DQG 1 ? ,6E LV WKH WRWDO PDWHULDO GHQVLW\ 7KH VHHG SDUWLFOH GHQVLW\ ZLOO EH GHWHUPLQHG E\ WKH DPRXQW RI VHHG PDWHULDO WKDW LV UHTXLUHG WR UHGXFH WKH WHPSHUDWXUH DW WKH LQVLGH IDFH RI WKH FRQWDLQPHQW ZDOO EHORZ WKH JUDSKLWH PHOWLQJ SRLQW %\ DVVXPLQJ WKDW WKH VHHG PDWHULDO LV XQLIRUPO\ PL[HG DQG PAGE 166 5266(/$1' 23$&,7< FPn 7HPSHUDWXUH r. )LJXUH 5RVVHODQG RSDFLW\ RI XUDQLXP L PL L L L LPL PAGE 167 n 7HPSHUDWXUH r5 )LJXUH 5RVVHODQG RSDFLW\ RI K\GURJHQ PAGE 168 LQMHFWHG ZLWK WKH K\GURJHQ FRRODQW D VSDWLDOO\ LQGHSHQGHQW VHHG UDWLR V LV GHILQHG DV WKH UDWLR RI VHHG GHQVLW\ WR FRRODQW GHQVLW\ V f 8VLQJ f DQG f LQ f WKH VHFRQG GHULYDWLYH RI WKH WHPSHUDWXUH GLVWULEXWLRQ EHFRPHV G7 GU ZKHUH G7 IGOK )U6! ' ) ' r! IH' 7 f ' U H n UH>H@ U ! H + Q K V G.A G.A G.a 1 B 1 cf§f§ 6 8 G7 +nG7 G7 f¬ .6 1f± . 6 .Uf + 7 6Uf .5 18 .5 V .Uf 1+ + ' f¯ ,: 1\ Vf 1K f ZLWK 6Uf JLYHQ E\ f 7KH WHPSHUDWXUH JUDGLHQW LV HYDOXDWHG XVLQJ f ZLWK )Unf UHSODFHG E\ )UnVf PAGE 169 IURP f 7KH WHPSHUDWXUH GLVWULEXWLRQ DV GHWHUPLQHG E\ f EHFRPHV 7 U Vf G7 GUnU Vf GUn 7 f f 7KH WHPSHUDWXUH GLVWULEXWLRQ FDOFXODWHG E\ f UHSUHVHQWV D WUDMHFWRU\ LQ WKH U7 SODQH EHWZHHQ U DQG U 5 7KH LQLWLDO YDOXHV 7f DQG JAUf DUH VSHFLILHG E\ DSSOLFDWLRQ RI WKH ILUVW DQG VHFRQG ERXQGDU\ FRQGLWLRQV RI f WR WKH LQWHJUDWLRQV f DQG f %RWK RI WKH G7 HQG SRLQW YDOXHV 75f DQG A5f DUH SDUDPHWULFDOO\ GHSHQGHQW RQ WKH VHHG UDWLR V WKURXJK WKH FRHIILFLHQWV ' DQG ' JLYHQ E\ f (LWKHU WKH HQG SRLQW WHPSHUDWXUH RU WHPSHUDWXUH JUDGLHQW PD\ EH LQGHSHQGHQWO\ VSHFLILHG E\ SURSHU FKRLFH RI WKH VHHG UDWLR 6LQFH WKH WKLUG ERXQGDU\ FRQGLWLRQ RI f UHTXLUHV WKDW RQO\ WKH HQG SRLQW WHPSHUDWXUH EH VSHFLILHG WHPSHUDWXUH WUDMHFWRULHV DUH FDOFXODWHG IURP f IRU YDULRXV VHHG UDWLRV XQWLO WKLV ERXQGDU\ FRQGLWLRQ LV VDWLVILHG IRU VRPH V 6T 7KHQ D FRQVLVWHQW YDOXH RI WKH HQG SRLQW JUDGLHQW ZLOO EH GHWHUPLQHG XVLQJ V 6T LQ WKH H[SUHVVLRQ f 7R VXPPDUL]H WKH QXPHULFDO FDOFXODWLRQ RI WKH WHPSHUDWXUH GLVWULEXWLRQ f DQG f DUH XVHG ZLWK WKH ILUVW WZR ERXQGDU\ G7 FRQGLWLRQV RI f WR FRPSXWH JAUVf DQG 7UVf RYHU WKH LQWHUYDO >5@ IRU V ,I WKH XUDQLXP DQG K\GURJHQ PAGE 170 RSDFLWLHV DORQH DUH VXIILFLHQW WR UHGXFH WKH WHPSHUDWXUH IURP 7T DW U WR 75 DW U 5 WKHQ 7Uf UHSUHVHQWV DQ DFFHSWDEOH VROXWLRQ WR WKH HQHUJ\ EDODQFH HTXDWLRQ f ,I WKH HQG SRLQW WHPSHUDWXUH H[FHHGV 7' WKH FDOFXODWLRQ LV UHSHDWHG E\ VHDUFKLQJ IRU WKH VHHG UDWLR 6T IRU ZKLFK 75VTf £ 7U 5HVXOWV RI 7HPSHUDWXUH 'LVWULEXWLRQ &DOFXODWLRQV 7KH FDOFXODWHG WHPSHUDWXUH GLVWULEXWLRQV DUH HVVHQWLDOO\ GHSHQGHQW RQ ILYH SDUDPHWHUV 7KHVH DUH f UHDFWRU SUHVVXUH f FHQWHUOLQH SODVPD WHPSHUDWXUH f IXHO DQG FRRODQW GHQVLW\ SURILOHV f KHDW JHQHUDWLRQ UDWH DQG f VHHG UDWLR 7KH WHPSHUDWXUH FDOFXODWLRQ WHFKQLTXH LV GHVFULEHG IRU D W\SLFDO FDVH GHILQHG E\ WKH IROORZLQJ SDUDPHWHUV FRUH UDGLXV IW JUDSKLWH UHIOHFWRU WKLFNQHVV IW SUHVVXUH DWP VHHG PDWHULDO WXQJVWHQ FHQWHUOLQH WHPSHUDWXUH r. 1f [ FF 1\f [ FF E\ IW 1 f [ FF + 1f±UU7f ; FF Q + PAGE 171 U+ IW E IW K 7KH WHPSHUDWXUH GLVWULEXWLRQ DV D IXQFWLRQ RI WKH VHHG UDWLR LV JLYHQ E\ HTXDWLRQ f 7KH VHHG UDWLR LV YDULHG XQWLO WKH WHPSHUDWXUH DW WKH LQQHU HGJH RI WKH JUDSKLWH UHIOHFWRU LV UHGXFHG EHORZ r. WKH DSSUR[Ln PDWH PHOWLQJ WHPSHUDWXUH RI JUDSKLWH 7KH HIIHFW RI WKH WXQJVWHQ VHHG UDWLR RQ WKH JUDSKLWH WHPSHUDWXUH LV VKRZQ LQ )LJXUH :LWK QR VHHG LQ WKH K\GURJHQ WKH JUDSKLWH WHPSHUDWXUH LV r. 7KH JUDSKLWH WHPSHUDWXUH LV UHGXFHG DV WKH VHHG UDWLR LQFUHDVHV XQWLO WKH GHVLUHG YDOXH RI r. LV REWDLQHG ZLWK D VHHG UDWLR RI 7KH UHVXOWLQJ WHPSHUDWXUH GLVWULEXWLRQ IRU WKLV FDVH LV VKRZQ ZLWK WKH FRUUHVSRQGLQJ XUDQLXP DQG K\GURJHQ GLVWULn EXWLRQV LQ )LJXUH 7KHVH WHPSHUDWXUH DQG GHQVLW\ GLVWULEXWLRQV DUH WKHQ XVHG LQ WKH LWHUDWLYH FDOFXODWLRQ WR GHWHUPLQH WKH SUHVVXUH ZKLFK PDNHV WKH UHDFWRU FULWLFDO DV GHVFULEHG LQ WKH ILUVW VHFWLRQ RI WKLV FKDSWHU &DOFXODWHG WHPSHUDWXUH GLVWULEXWLRQV IRU FHQWHUOLQH WHPSHUDWXUHV IURP r. WR r. DUH VKRZQ LQ )LJXUH 7KH SUHVVXUH LQ DWPRVSKHUHVf DQG WKH VHHG UDWLR XVHG LQ HDFK FDOFXODWLRQ DUH VKRZQ RQ WKH UHVSHFWLYH FXUYHV )RU FHQWHUOLQH WHPSHUDWXUHV RI 7f r. DQG r. WKH K\GURJHQ GLVWULEXWLRQ SDUDPHWHUV DUH PAGE 172 7(03(5$785( $7 *5$3+,7( ,11(5)$&( 6HHG UDWLR V )LJXUH (IIHFW RI FRRODQW VHHGLQJ RQ JUDSKLWH WHPSHUDWXUH PAGE 173 7(03(5$785( )LJXUH 3ODVPD WHPSHUDWXUH GLVWULEXWLRQ DW DWPRVSKHUHV 3$57,&/( '(16,7< 1 [ FF PAGE 174 7(03(5$785( )LJXUH 7HPSHUDWXUH GLVWULEXWLRQV IRU FHQWHUOLQH WHPSHUDWXUHV IURP r. WR r. PAGE 175 U+ IW E5 IW 1Uf [ FF 1+U+f [ FF )RU 7f r. U IW E + + IW 1WWf [ FF 1f±Uf [ FF + UL 3, )RU 7f r. DQG r. U IW E IW + + 1Kf [ FF 1 U+f [ FF )RU 7f r. DQG r. U IW E IW 1 f + + 3, [ FF 1+U+f [ FF 7KH WHPSHUDWXUH GLVWULEXWLRQV LQ )LJXUH DUH XVHG WR VROYH IRU WKH SUHVVXUH UHTXLUHG WR PDLQWDLQ WKH UHDFWRU FULWLFDO 7KHVH UHVXOWLQJ SUHVVXUHWHPSHUDWXUH WUDMHFWRULHV DORQJ ZKLFK WKH UHDFWRU LV FULWLFDO DUH VKRZQ LQ )LJXUH 7KHVH FXUYHV VKRZ WKH DYHUDJH FRRODQW DQG FRUH WHPSHUDWXUHV ZKLFK FDQ EH DWWDLQHG IRU D JLYHQ FULWLFDO SUHVVXUH )RU H[DPSOH DW D SUHVVXUH RI DWPRVSKHUHV WKH FULWLFDO UHDFWRU KDV DYHUDJH FRRODQW DQG FRUH WHPSHUDWXUHV RI r. DQG r. UHVSHFWLYHO\ ,W LV VHHQ WKDW SUHVVXUHV LQ H[FHVV RI DWPRVSKHUHV ZLOO EH UHTXLUHG WR DFKLHYH DYHUDJH FRRODQW WHPSHUDWXUHV JUHDWHU WKDQ r. 7KH LQGLFDWHG GHYLDWLRQV RQ WKH SUHVVXUHWHPSHUDWXUH FXUYHV DFFRXQW IRU WKH QRQXQLTXHQHVV RI WKH VHHG UDWLR DQG GHQVLW\ GLVWULEXWLRQV XVHG LQ HTXDWLRQ f $V LQGLFDWHG SUHYLRXVO\ WKH WHPSHUDWXUH GLVWULEXWLRQV DUH GHWHUPLQHG E\ ILYH JURXSV RI SDUDPHWHUV 8QLTXH VHHG UDWLRV DQG GHQVLW\ GLVWULEXWLRQV PXVW EH PAGE 176 35(6685( $70263+(5(6 )LJXUH 3UHVVXUH UHTXLUHG IRU FULWLFDO SODVPD UHDFWRU PAGE 177 f¯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n WDWLYHO\ IURP WKRVH LQ )LJXUH )RU FHQWHUOLQH WHPSHUDWXUHV DERYH r. LW ZDV QHFHVVDU\ WR DGG DV PXFK DV b VHHG PDWHULDO WR WKH XUDQLXP IXHO LQ RUGHU WR GHSUHVV WKH WHPSHUDWXUH EHIRUH UHDFKLQJ WKH KLJKHU GHQVLW\ UHJLRQ RI WKH VHHGHG K\GURJHQ 7KH WHPSHUDWXUH FDOFXODWLRQV LQGLFDWH WKDW WKH DFKLHYHPHQW RI D GHVLUHG WHPSHUDWXUH SURILOH LV D GHOLFDWH EDODQFH EHWZHHQ WKH UDGLDO GLVWULEXWLRQV RI XUDQLXP ZKLFK ERWK SURGXFHV KHDW DQG DEVRUEV UDGLDWLRQf DQG VHHGHG K\GURJHQ ZKLFK DEVRUEV UDGLDWLRQf +HDY\ VHHGLQJ RI WKH K\GURJHQ ZLOO EH PDGH DW WKH H[SHQVH RI UHGXFHG VSHFLILF LPSXOVH VLQFH DQ\ VHHG PDWHULDO ZLOO SUHVXPDEO\ EH H[KDXVWHG DQG WKHUHE\ LQFUHDVH WKH PHDQ PDVV RI WKH SURSHOODQW PAGE 178 %UHPVVWUDKOXQJ (QHUJ\ /RVVHV 7KH HQHUJ\ EDODQFH HTXDWLRQ f UHODWHV HQHUJ\ SURGXFWLRQ IURP ILVVLRQ DQG HQHUJ\ ORVV E\ GLIIXVLRQ ,I WKHUH ZHUH QR UDGLDWLRQ ORVVHV IURP WKH SODVPD WKH VRXUFH WHUP ZRXOG UHSUHVHQW WKH NLQHWLF HQHUJ\ RI WKH ILVVLRQ IUDJPHQWV +RZHYHU VLQFH WKH FKDUJHG SDUWLFOHV LQ WKH GHQVH KLJK WHPSHUDWXUH SODVPD ZLOO ORVH HQHUJ\ E\ EUHPVVWUDKOXQJ WKH HQHUJ\ SURGXFWLRQ PXVW EH VXIILFLHQW WR RIIVHW WKHVH UDGLDWLRQ ORVVHV 7R H[SOLFLWO\ DFFRXQW IRU WKH EUHPVVWUDKOXQJ WKH VRXUFH WHUP LQ HTXDWLRQ f LV ZULWWHQ DV 6Uf 6 Uf %Uf f I I ZKHUH 6Uf QHW HQHUJ\ DYDLODEOH IRU KHDWLQJ WKH SODVPD 6MAUf ILVVLRQ IUDJPHQW HQHUJ\ %Uf EUHPVVWUDKOXQJ HQHUJ\ 7KH EUHPVVWUDKOXQJ HQHUJ\ LV JLYHQ E\ f % Uf [ BW&HUf &AU-=" I7)7 1r7Uf ZDWWVFF f ZKHUH & Uf DQG &Uf DUH WKH HOHFWURQ DQG LRQ FRQFHQWUDWLRQV H UHVSHFWLYHO\ DQG 1 Uf LV WKH WRWDO SDUWLFOH GHQVLW\ 7KH QHW HQHUJ\ SURGXFWLRQ 6Uf LV D SDUDPHWHU HYDOXDWHG PAGE 179 ZKHQ WHPSHUDWXUH GLVWULEXWLRQV DUH FDOFXODWHG IURP HTXDWLRQ f 7KHVH WHPSHUDWXUH GLVWULEXWLRQV 7Uf DQG WKH SODVPD SUHVVXUH DQG PL[LQJ UDWLR GHWHUPLQH WKH YDOXHV RI & Uf DQG &Uf IURP ZKLFK WKH EUHPVVWUDKOXQJ HQHUJ\ ORVV H L LV FDOFXODWHG IURP f )URP HTXDWLRQ f WKH ILVVLRQ IUDJPHQW HQHUJ\ QHFHVVDU\ WR VXSSRUW WKH FDOFXODWHG WHPSHUDWXUH GLVWULEXWLRQ LV 6IIUf 6Uf %Uf f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n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n JDWHG WKHRUHWLFDOO\ DQG H[SHULPHQWDOO\ :KHQ PRUH H[WHQVLYH DQG DFFXUDWH SODVPD GDWD KDYH EHHQ REWDLQHG WKHVH WLPH GHSHQGHQW SUREOHPV ZRXOG EH D ORJLFDO H[WHQVLRQ RI WKH SUHVHQW VWXG\ $W WKH VDPH WLPH LPSURYHG VWDWLF FDOFXn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r. WR r. UDQJH SUHVVXUHV IURP DSSUR[LPDWHO\ WR DWPRVSKHUHV ZLOO EH UHTXLUHG 7KH VHFRQG UHTXLUHPHQW LV WKDW WKH K\GURJHQ FRRODQW PXVW EH VHHGHG ZLWK VRPH RSDTXH PDWHULDO ZKLFK SUHIHUDEO\ KDV D ORZ QHXWURQ DEVRUSWLRQ FURVV VHFWLRQ DQG GRHV QRW UHDFW FKHPLFDOO\ ZLWK WKH FRRODQW RU WKH FRQWDLQPHQW ZDOO )LQDOO\ WKH SODVPD UHDFWRU ZLOO UHTXLUH D FRPSOH[ DXWRPDWLF FRQWURO V\VWHP WR SURYLGH IRU VWDEOH RSHUDWLRQ LQ WKH SUHVHQFH RI SODVPD LQVWDELOLWLHV DVVRFLDWHG ZLWK WKH FRD[LDO IORZ RI XUDQLXP DQG K\GURJHQ ,Q RUGHU WR SURYLGH PRUH H[WHQVLYH HYDOXDWLRQ RI WKH LPSRUWDQW HQJLQHHULQJ SDUDPHWHUV RI WKH UHDFWRU DGGLWLRQDO WKHRUHWLFDO DQG H[SHULPHQWDO VWXGLHV LQ WKH DUHDV RI UHDFWRU SK\VLFV IOXLG IORZ DQG UDGLDQW HQHUJ\ PAGE 183 WUDQVIHU DUH UHTXLUHG 7KH DGGLWLRQDO HIIRUWV QHHGHG LQ UHDFWRU SK\VLFV DUH SULPDULO\ FRQFHLnQHG ZLWK LPSURYLQJ UQXOWLJURXS GLIIXVLRQ DQG WUDQVSRUW WKHRU\ FRGHV E\ LQFRUSRn UDWLQJ WKH HIIHFWV RI WKHUPDO QHXWURQ VFDWWHULQJ DQG HSLWKHUPDO QHXWURQ DEVRUSWLRQ DV GLVFXVVHG LQ &KDSWHU ,, $OVR WKH XVXDO WUHDWPHQW RI UHVRQDQFH RYHUODS PD\ UHTXLUH PRGLILFDWLRQ DW YHU\ KLJK QHXWURQ WHPSHUDWXUHV 7KH QXFOHDU FULWLFDOLW\ FDOFXODWLRQV DOO GHSHQG RQ WKH XUDQLXP FRPSRn VLWLRQV DV FDOFXODWHG E\ WKH 83/$= SURJUDP 7KLV SURJUDP XWLOL]HV ERWK H[SHULPHQWDOO\ PHDVXUHG GDWD DQG FDOFXODWHG GDWD EDVHG RQ TXDQWXP PHFKDQLFDO DSSUR[LPDWLRQV IRU JURXQG VWDWH XUDQLXP VSHFLHV DQG WKH LVRHOHFWURQLF DSSUR[LPDWLRQ IRU H[FLWHG VWDWHV 6LQFH PXOWLSOH LRQL]DWLRQ LV D VWHSZLVH SURFHVV HUURUV LQ WKH VWDWLVWLFDO ZHLJKWV DQG H[FLWHG HQHUJ\ OHYHOV RI ORZHU LRQV FRXOG UHVXOW LQ ODUJH FXPPXODWLYH HUURUV LQ WKH FRPSXWHG GHQVLWLHV RI KLJKHU LRQV $ VXEn VWDQWLDO H[SHULPHQWDO SURJUDP LQYROYLQJ VSHFWURVFRSLF DQDO\VLV RI XUDQLXP SODVPDV RYHU D ZLGH UDQJH RI WHPSHUDn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n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n SOHWHO\ GHVFULEH WKH SODVPD UHDFWRU VLQFH VXFK VWXGLHV GR QRW LQFOXGH WKH HIIHFWV RI WLPH YDU\LQJ PDJQHWLF DQG HOHFWULF ILHOGV DVVRFLDWHG ZLWK WKH KLJKO\ LRQL]HG IXHO DQG FRRODQW SODVPDV $OWKRXJK VRPH RI WKH EDVLF SUREOHPV RI SODVPD WXUEXOHQFH KDYH EHHQ LQYHVWLJDWHG WKHRUHWLFDOO\ ^f D ODUJH WKHRUHWLFDO HIIRUW LV VWLOO UHTXLUHG EHIRUH DQ HQJLQHHULQJ DQDO\VLV RI WZR GLPHQVLRQDO PXOWLIOXLG WXUEXOHQW SODVPD G\QDPLFV FDQ EH SHUIRUPHG 7KLV DQDO\VLV VKRXOG H[WHQG WKH SODVPD FRPSRVLWLRQ FDOFXODWLRQDO PRGHO WR LQFOXGH ZKHQHYHU QHFHVVDU\ QRQHTXLOLEULXP LRQL]DWLRQ HIIHFWV -Bf DQG DOVR LQWHUPROHFXODU IRUFHV DQG ILQLWH PROHFXODU VL]HV WKURXJK XVH RI YDQ GHU :DDOnV HTXDWLRQ RI VWDWH ef 7KH WZR GLPHQVLRQDO GHQVLW\ GLVWULEXWLRQV REWDLQHG IURP WKH IOXLG IORZ FDOFXODWLRQV FDQ EH XVHG LQ WKH 5RVVHODQG GLIIXVLRQ DSSUR[LPDWLRQ WR GHWHUPLQH WHPSHUDn WXUH GLVWULEXWLRQV LQ ERWK WKH D[LDO DQG UDGLDO GLUHFWLRQV DFFRUGLQJ WR HTXDWLRQ f ,Q RUGHU WR VWXG\ WKH G\QDPLFV RI WXUEXOHQW SODVPDV H[SHULPHQWDO ZRUN LQYROYLQJ ODUJH WZRIOXLG FRD[LDOO\ IORZLQJ SODVPDV DW KLJK WHPSHUDWXUHV ZLOO EH UHTXLUHG 7KH QHXWURQ VFDWWHULQJ HIIHFWV RI KLJK WHPSHUDWXUH K\GURJHQ QXFOHL KDYH EHHQ GLVFXVVHG LQ &KDSWHU ,, ,Q DGGLWLRQ WR WKH HQHUJ\ RI WKHUPDO PRWLRQ WKH NLQHWLF HQHUJ\ DVVRFLDWHG ZLWK WKH IOXLG PRWLRQ RI WKH K\GURJHQ PAGE 186 FRRODQW FDQ DOVR VLJQLILFDQWO\ DIIHFW QHXWURQ VFDWWHULQJ 7KH PDJQLWXGH RI WKHVH WZR HIIHFWV LV FRPSDUHG E\ FDOFXn ODWLQJ WKH WKHUPDO DQG NLQHWLF HQHUJ\ FRPSRQHQWV RI K\GURJHQ DV D IXQFWLRQ RI WHPSHUDWXUH DQG VSHFLILF LPSXOVH UHVSHFWLYHO\ 7KH WKHUPDO HQHUJ\ LV ( WK f ZKHUH N LV %ROW]PDQQnV FRQVWDQW DQG 7 LV WKH K\GURJHQ + WHPSHUDWXUH 7KH NLQHWLF HQHUJ\ RI H[KDXVWLQJ K\GURJHQ QXFOHL LV ( f§ P Y N +H f ZKHUH P+ LV WKH K\GURJHQ PDVV DQG YH LV WKH SURSHOOHQW H[KDXVW YHORFLW\ 7KH H[KDXVW YHORFLW\ LV UHODWHG WR WKH VSHFLILF LPSXOVH , DFFRUGLQJ WR f Y JOBB f H VS ZKHUH J LV WKH JUDYLWDWLRQDO DFFHOHUDWLRQ 7KH WKHUPDO DQG NLQHWLF HQHUJ\ FRPSRQHQWV DUH VKRZQ LQ )LJXUH 7KH XSSHU NLQHWLF HQHUJ\ FXUYH UHSUHVHQWV WKH PD[LPXP K\GURJHQ HQHUJ\ FRUUHVSRQGLQJ WR WKRVH SURSHOOHQW QXFOHL ZKLFK KDYH REWDLQHG H[KDXVW YHORFLW\ 7KH ORZHU NLQHWLF HQHUJ\ FXUYH UHSUHVHQWV WKH HQHUJ\ RI K\GURJHQ QXFOHL KDYLQJ DQ DYHUDJH YHORFLW\ HTXDO WR RQH KDOI WKH H[KDXVW YHORFLW\ )RU D FRRODQW WHPSHUDWXUH RI r. DQG D VSHFLILF LPSXOVH PAGE 187 (1(5*< f¬ 7HPSHUDWXUH 'HJUHHV .HOYLQ 6SHFLILF ,PSXOVH , VHFf VS )LJXUH (QHUJ\ RI K\GURJHQ FRRODQW QXFOHL GXH WR IORZ DQG WKHUPDO PRWLRQ PAGE 188 RI VHFRQGV D K\GURJHQ QXFOHXV RI DYHUDJH YHORFLW\ KDV D NLQHWLF HQHUJ\ ZKLFK LV DERXW b DV ODUJH DV WKH nWKHUPDO HQHUJ\ 7KLV FRPSDULVRQ GHPRQVWUDWHV WKDW WKH IORZ PRWLRQ RI K\GURJHQ VKRXOG EH LQFOXGHG LQ WKH FDOFXODWLRQ RI HIIHFWLYH K\GURJHQ VFDWWHULQJ FURVV VHFWLRQV ([SHULPHQWDO DQG WKHRUHWLFDO LQYHVWLJDWLRQV DUH QHHGHG WR GHWHUPLQH WKH TXDQWLWDWLYH OLPLWDWLRQV RI SODVPD KHDWLQJ E\ ILVVLRQ IUDJPHQWV $GGLWLRQDO KHDWLQJ PHFKDn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n WDLQLQJ WKH SUHVVXUH DQG WHPSHUDWXUH LQ ( IRUPDW DV IROORZV &ROXPQV 3 DWPRVSKHUHVf &ROXPQV 7 r.f 0XOWLSOH FDVHV FDQ EH UXQ FRQVHFXWLYHO\ E\ VWDFNLQJ RQH GDWD FDUG SHU FDVH 7KH RXWSXW GDWD LQFOXGH WKH WRWDO SDUWLFOH GHQVLW\ PD[LPXP GHJUHH RI LRQL]DWLRQ ORZHULQJ PAGE 192 RI LRQL]DWLRQ SRWHQWLDOV LQ H9 DQG QXPEHU GHQVLW\ SDUWLDO SUHVVXUH DQG SHU FHQW RI 8r 8 8 8 DQG HOHFWURQV 7KH WKHRUHWLFDO LRQL]DWLRQ SRWHQWLDOV XVHG E\ WKH SURJUDP DUH YDOXHV ZKLFK ZHUH FDOFXODWHG E\ 7 :DEHU ' /LEHUPDQ DQG ' 7 &URPHU Bf 7KHVH YDOXHV IROORZ LQ 7DEOH 7$%/( 7+(25(7,&$/ ,21,=$7,21 327(17,$/6 2) 85$1,80 $WRP RU ,RQ ,RQL]DWLRQ 3RWHQWLDO H9f X X X X X X X X 8 X X n \O PAGE 193 7KH LRQL]DWLRQ SRWHQWLDOV DUH VXFFHVVLYH UDWKHU WKDQ 4 Ir FXPPXODWLYH LH WKH LRQL]DWLRQ SRWHQWLDO IRU 8 LV WKH HQHUJ\ IRU WKH UHDFWLRQ 8 r H 7KH XUDQLXP SODVPD FRPSRVLWLRQ IRU SUHVVXUHV IURP DWP WR DWP DQG WHPSHUDWXUHV WR r. DUH VKRZQ LQ )LJXUHV WKURXJK RI WKLV DSSHQGL[ 7KH 83/$= SURJUDP LV ZULWWHQ LQ )2575$1 ,9 IRU WKH ,%0 PAGE 194 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 195 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 196 )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 197 )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 198 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 199 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 200 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 201 3(5 &(17 2) 727$/ 3$57,&/( '(16,7< )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHnV PAGE 202 )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 203 7HPSHUDWXUH &. )LJXUH 8UDQLXP SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 204 $33(1',; % +3/$= 352*5$0 7KH +3/$= SURJUDP FDOFXODWHV WKH FRPSRVLWLRQ RI D K\GURJHQ SODVPD DV D IXQFWLRQ RI SODVPD SUHVVXUH DQG WHPSHUDWXUH 7KH SODVPD LV DVVXPHG WR EH LQ ORFDO WKHUPRn G\QDPLF HTXLOLEULXP $OO K\GURJHQ UHDFWLRQV DUH WUHDWHG LQ D FRQVLVWHQW DQDO\WLFDO PDQQHU $Q LQWHJUDO DSSUR[Ln PDWLRQ LV XVHG LQ FDOFXODWLQJ WKH DWRPLF K\GURJHQ SDUWLWLRQ IXQFWLRQ DQG D 0RUVH SRWHQWLDO LV XVHG LQ FDOFXODWLQJ PROHFXODU YLEUDWLRQDO SDUWLWLRQ IXQFWLRQV &RUUHFWLRQV IRU WKH ORZHULQJ RI LRQL]DWLRQ SRWHQWLDOV DUH LQFOXGHG 0 7KH SODVPD VSHFLHV + + + + + DQG HOHFWURQV DUH FRPSXWHG LQ WHUPV RI SDUWLFOH GHQVLW\ SDUWLDO SUHVVXUH DQG SHU FHQW RI WRWDO SDUWLFOHV 7KH LQSXW GDWD FRQVLVW RI RQH FDUG SHU FDVH FRQWDLQLQJ WKH SUHVVXUH DQG WHPSHUDWXUH LQ ( IRUPDW DV IROORZV &ROXPQV 3 DWPRVSKHUHVf &ROXPQV 7 r.f 0XOWLSOH FDVHV FDQ EH UXQ FRQVHFXWLYHO\ E\ VWDFNLQJ RQH GDWD FDUG SHU FDVH 7KH K\GURJHQ SODVPD FRPSRVLWLRQ IRU SUHVVXUHV IURP DWP WR DWP DQG WHPSHUDWXUHV WR r. DUH VKRZQ LQ )LJXUHV WKURXJK RI WKLV DSSHQGL[ PAGE 205 +<'52*(1 '(16,7< 3$57,&/(6FF )LJXUH +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV /LOOL PAGE 206 +<'52*(1 '(16,7< 3$57,&/(6FF )LJXUH +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 207 +<'52*(1 '(16,7< 3$57,&/(6FF )LJXUH +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV 77777 PAGE 208 +<'52*(1 '(16,7< 3$57,&/(6FF )LJXUH +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RU DWPRVSKHUHV PAGE 209 +<'52*(1 '(16,7< 3$57,&/(6FF )LJXUH +\GURJHQ SODVPD FRPSRVLWLRQ DW SUHVVXUH RI DWPRVSKHUHV PAGE 210 7KH 53/$= SURJUDP LV ZULWWHQ LQ )2575$1 ,9 IRU WKH ,%0 7$%/( '$7$ 86(' ,1 +<'52*(1 3/$60$ &20326,7,21 &$/&8/$7,216 ( + Q 'LVVRFLDWLRQ HQHUJ\ RI +f HY KL ,RQL]DWLRQ HQHUJ\ RI +f H9 KU ,RQL]DWLRQ HQHUJ\ RI +f¯f H9 Y ,RQL]DWLRQ HQHUJ\ RI + f H9 WL *URXQG VWDWH PRPHQW RI LQHUWLD RI +f [ JP FP ,+ *URXQG VWDWH PRPHQW RI LQHUWLD RI +rf [f° JPaFP Hr! 9 K *URXQG VWDWH YLEUDWLRQDO HQHUJ\ RI +f H9 ( f 9 + *URXQG VWDWH YLEUDWLRQDO HQHUJ\ RI +f H9 '+f 'LVVRFLDWLRQ HQHUJ\ RI +f H9 WR H ILUVW FRHIILFLHQW RI YLEUDWLRQDO HQHUJ\ H[SDQVLRQf§VHH HTXDWLRQ f` FP ; H H VHFRQG FRHIILFLHQW RI YLEUDWLRQDO HQHUJ\ H[SDQVLRQaVHH HTXDWLRQ ff FPf° PAGE 211 $33(1',; & +83/$= 352*5$0 7KH +83/$= SURJUDP FDOFXODWHV WKH FRPSRVLWLRQ RI D PL[HG K\GURJHQXUDQLXP SODVPD DV D IXQFWLRQ RI SODVPD SUHVVXUH WHPSHUDWXUH DQG PL[LQJ UDWLR 7KH PL[LQJ UDWLR 5 LV GHILQHG DV B SDUWLFOH GHQVLW\ RI DOO K\GURJHQ VSHFLHV SDUWLFOH GHQVLW\ RI DOO XUDQLXP VSHFLHV 7KH XUDQLXP DQG K\GURJHQ SDUWLWLRQ IXQFWLRQV DUH FDOFXn ODWHG XVLQJ WKH PHWKRGV GHVFULEHG LQ $SSHQGLFHV $ DQG % &RUUHFWLRQV IRU WKH ORZHULQJ RI LRQL]DWLRQ SRWHQWLDOV DUH LQFOXGHG 7KH LQGLYLGXDO SODVPD VSHFLHV DUH FRPSXWHG LQ WHUPV RI SDUWLFOH GHQVLW\ SDUWLDO SUHVVXUH DQG SHU FHQW RI WRWDO SDUWLFOHV 7KH LQSXW GDWD FRQVLVW RI RQH FDUG SHU FDVH FRQWDLQLQJ WKH SUHVVXUH WHPSHUDWXUH DQG PL[LQJ UDWLR LQ ( IRUPDW DV IROORZV &ROXPQV 3 DWPRVSKHUHVf &ROXPQV 7 r.f &ROXPQV 5 K\GURJHQ GHQVLW\XUDQLXP GHQVLW\f 0XOWLSOH FDVHV FDQ EH UXQ FRQVHFXWLYHO\ E\ VWDFNLQJ RQH GDWD FDUG SHU FDVH PAGE 212 7KH +83/$= SURJUDP LV ZULWWHQ LQ )2575$1 ,9 IRU WKH ,%0 PAGE 213 /,67 2) 5()(5(1&(6 & 6WDUU &KDXQFH\ 6WDUU /RRNV DW WKH 8 6 1DWLRQDO 5HDFWRU 3URJUDP 1XFOHRQLFV B f 6 'L 6WUDXVV 2XU )DVW %UHHGHU 3URJUDP :KHUH ,W 6WDQGV DQG :KHUH ,W ,V *RLQJ 1XFOHRQLFV f 5 1HZEXUJHU 5HDFWRU 0DQXIDFWXUHUV 6SHHG 7KHLU ([SDQVLRQ WR 0HHW WKH 'HPDQG 1XFOHRQLFV f *UH\ $ *DVHRXV&RUH 1XFOHDU 5RFNHW 8WLOL]LQJ +\GURG\QDPLF &RQWDLQPHQW RI )LVVLRQDEOH )XHO $56 3UHSULQW f 6 7 1HOVRQ 7KH 3ODVPD &RUH 5HDFWRU 6SDFH 7HFKQRORJ\ /DEV ,QF *0 f + :HLQVWHLQ DQG 5 * 5DJVGDOH $ &RD[LDO )ORZ 5HDFWRU $ *DVHRXVn5HDFWRU &RQFHSW $56 3UHSULQW f 5 9 0HJUHEOLDQ *DVHRXV )LVVLRQ 5HDFWRUV IRU 6SDFHFUDIW 3URSXOVLRQ -HW 3URSXOVLRQ /DE 75 f 5 9 0HJUHEOLDQ *DVHRXV 3URSXOVLRQ 5HDFWRUV 1XFOHRQLFV B f ) & 5RP DQG 5 * 5DJVGDOH $GYDQFHG &RQFHSWV IRU 1XFOHDU 5RFNHW 3URSXOVLRQ 1$6$ 63 f : 0F.HH 7KH *ORZ3OXJ *DV&RUH 5HDFWRU *DVHRXV )LVVLRQ 5HDFWRUV 6\PSRVLXP 3DVDGHQD &DOLIRUQLD f 5 * 5DJVGDOH 2XWORRN IRU *DV&RUH 1XFOHDU 5RFNHWV $VWURQDXWLFV DQG $HURVSDFH (QJLQHHULQJ B f + ) &URXFK 1XFOHDU 6SDFH 3URSXOVLRQ $VWURQXFOHDU 3UHVV *UDQDGD +LOOV &DOLIRUQLD f PAGE 214 5 : %XVVDUG DQG 5 ' 'H/DXHU )XQGDPHQWDOV RI 8XFOHDU )OLJKW 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f 1XFOHDU 3URSXOVLRQ IRU 5RFNHWV /RV $OPRV 6FLHQWLILF /DERUDWRU\ ,QIRUPDWLRQ %URFKXUH f % 5RPHUR )XHO &RQWDLQPHQW LQ *DVHRXV&RUH 1XFOHDU 5RFNHW E\ 0+''ULYHQ 9RUWLFOHV $,$$ -RXUQDO f & (YYDUG :KHHO)ORZ *DVHRXV&RUH 5HDFWRU &RQFHSW 1$6$ 71 ' f 5 * 5DJVGDOH DQG ) ( 5RP *DV&RUH 5HDFWRU :RUN DW 1$6$/HZLV 1$6$ 70; f $ 0 :HLQEHUJ DQG ( 3 :LJQHU 7KH 3K\VLFDO 7KHRU\ RI 1HXWURQ &KDLQ 5HDFWRUV 8QLYHUVLW\ RI &KLFDJR 3UHVV &KLFDJR f 0 0 % :LOOLDPV 7KH 6ORZLQJ 'RZQ DQG 7KHUPDOL]DWLRQ RI 1HXWURQV -RKQ :LOH\ DQG 6RQV 1HZ <RUN f . + %HFNXUWV DQG . :LUW] 1HXWURQ 3K\VLFV 6SULQJHU 9HUODJ 1HZ <RUN f 5 9 0HJUHEOLDQ DQG ' . +ROPHV 5HDFWRU $QDO\VLV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f $ (ULNVRQ HW DO *DVHRXV)XHOHG &DYLW\ 5HDFWRU $QDO\VLV &ULWLFDOLW\ &DOFXODWLRQV DQG $QDO\VLV 1$6$ &5 f 7 ) 3OXQNHWW 1XFOHDU $QDO\VLV RI *DVHRXV&RUH 1XFOHDU 5RFNHWV 1XFOHDU $SSOLFDWLRQV B f ' & %DLOHU 7KH )DLP &RGH D 0XOWLJURXS 2QH 'LPHQVLRQDO 'LIIXVLRQ (TXDWLRQ &RGH 86$(& 5HSRUW 1$$6U f ' +XJKHV DQG 5 5 6FKZDUW] 1HXWURQ &URVV 6HFWLRQV %/1 VHFRQG HGLWLRQ f 0 %ODWW DQG 9 ) :HLVVNRSI 7KHRUHWLFDO 1XFOHDU 3K\VLFV -RKQ :LOH\ DQG 6RQV 1HZ <RUN f PAGE 215 5 ' (YDQV 7KH $WRPLF 1XFOHXV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f ( 6HJU«M 1XFOHL DQG 3DUWLFOHV : $ %HQMDPLQ 1HZ <RUN f / 'UHVQHU 5HVRQDQFH $EVRUSWLRQ LQ 1XFOHDU 5HDFWRUV 3HUJDPRQ 3UHVV f 5 ( +\ODQG HW DO 7ZR 'LPHQVLRQDO &ULWLFDOLW\ &DOFXODWLRQV RI *DVHRXV&RUH &\OLQGULFDO &DYLW\ 5HDFWRUV 1$6$ 71 ' f / +HUZLJ DQG 7 6 /DWKDP 1XFOHDU &KDUDFWHUn LVWLFV RI /DUJH 5HIOHFWRU 0RGHUDWHG *DVHRXV)XHO &DYLW\ 5HDFWRUV &RQWDLQLQJ +RW +\GURJHQ $,$$ -RXUQDO f * ' -RDQRX DQG 6 'XGHN *$0 $ &RQVLVWHQW 3 0XOWLJURXS &RGH IRU WKH &DOFXODWLRQ RI )DVW 1HXWURQ 6SHFWUD DQG 0XOWLJURXS &RQVWDQWV *$ f ( + &DQILHOG HW DO 623+,67, $Q ,%0 &RGH :KLFK &DOFXODWHV 0XOWLJURXS 7UDQVIHU &RHIILFLHQWV IRU *DVHRXV 0RGHUDWRUV 8&5/ f 5 + 6FKXGGH DQG '\HU 7(03(67 $ 1HXWURQ 7KHUPDOL]DWLRQ &RGH 1RUWK $PHULFDQ $YLDWLRQ 3URJUDP 'HVFULSWLRQ f ' & %DLOHU DQG + 3 3ODWW 7KH $,0 &RGH $ 0XOWLJURXS 2QH 'LPHQVLRQDO 'LIIXVLRQ (TXDWLRQ &RGH 1RUWK $PHULFDQ $YLDWLRQ 3URJUDP 'HVFULSWLRQ f $ % &DPEHO ' 3 'XHORV DQG 7 3 $QGHUVRQ 5HDO *DVHV $FDGHPLF 3UHVV 1HZ <RUN f . 6 .XQ] 1XPHULFDO $QDO\VLV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f : ( 0LOQH 1XPHULFDO &DOFXOXV 3ULQFHWRQ 8QLYHUVLW\ 3UHVV 3ULQFHWRQ 1HZ -HUVH\ f 5 + )RZOHU DQG ( $ *XJJHQKHLP 6WDWLVWLFDO 7KHUPRG\QDPLFV &DPEULGJH 8QLYHUVLW\ 3UHVV /RQGRQ f ( 8 &RQGRQ DQG * + 6KRUWOH\ 7KH 7KHRU\ RI $WRPLF 6SHFWUD &DPEULGJH 8QLYHUVLW\ 3UHVV /RQGRQ f PAGE 216 ' : 6WHLQKDXV %ODLVH DQG 0 'LULQJHU 3UHVHQW 6WDWXV RI WKH $UF 6SHFWUXP RI 8UDQLXP 8,f 8& 3K\VLFV 7,' f ( 8 &RQGRQ DQG + 2GLVKDZ HGLWRUV +DQGERRN RI 3K\VLRV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f & ( 0RRUH $WRP¯D (QHUJ\ /HYHOV 9ROXPH ,,, 1DWLRQDO %XUHDX RI 6WDQGDUGV &LUFXODU f + $ *ULHP +LJK 'HQVLW\ &RUUHFWLRQV LQ 3ODVPD 6SHFWURVFRS\ 3K\V 5HY f / , 6FKLII 4XDQWXP 0HFKDQLFV VHFRQG HGLWLRQ 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f . 6 'UHOOLVKDN & ) .QRSS DQG $ % &DPEHO 3DUWLWLRQ )XQFWLRQV DQG 7KHUPRG\QDPLF 3URSHUWLHV RI $UJRQ 3ODVPD 7KH 3K\VLFV RI )OXLGV B f 5 : 3DWFK 3ULYDWH &RPPXQLFDWLRQ f * 6 ,YDQRY.KRORGUL\L * 0 1LNROnVNLL DQG 5 $ *XO\DHY ,RQL]DWLRQ DQG ([FLWDWLRQ RI +\GURJHQ 6RYLHW 3K\VLFV $VWURQRP\ f 5 & 7ROPDQ 7KH 3ULQFLSOHV RI 6WDWLVWLFDO 0HFKDQLFV 2[IRUG 8QLYHUVLW\ 3UHVV /RQGRQ f : %RQG -U HW DW $WRPLF 7KHRU\ RI *DV '\QDPLFV $GGLVRQ:HVOH\ 3XEOLVKLQJ &RPSDQ\ 5HDGLQJ 0DVVDFKXVHWWV f * +HU]EHUJ 0ROHFXODU 6SHFWUD DQG 0ROHFXODU 6WUXFWXUH , 'LDWRPLF 0ROHFXOHV 3UHQWLFH+DOO ,QF 1HZ <RUN f $ 8QVRHOG 3K\VLFV RI 6WHOODU $WPRVSKHUHV 6SULQJHU 3XEOLVKLQJ &RPSDQ\ %HUOLQ f 5 * 5DJVGDOH (IIHFWV RI D 0RPHQWXP %XIIHU 5HJLRQ RQ WKH &RD[LDO )ORZ RI 'LVVLPLODU *DVHV 1$6$ 71 ' f 7 : .DR &RQFHQWUDWLRQ 3URILOH (VWDEOLVKPHQW RI %LQDU\ *DV 0L[WXUH LQ 6ZLUO DQG 'XFW )ORZV 1$6$ &5 f PAGE 217 5 +RZHOO 5DGLDWLYH ,QWHUDFWLRQV %HWZHHQ $EVRUELQJ(PLWWLQJ 0HGLD ZLWK ,QWHUQDO (QHUJ\ *HQHUDWLRQ 1$6$ ' f + :HLQVWHLQ DQG & $ 7RGG $QDO\VLV RI 0L[LQJ &RD[LDO 6WUHDPV RI 'LVVLPLODU )OXLGV ,QFOXGLQJ (QHUJ\ *HQHUDWLRQ 7HUPV 1$6$ 71 ' f *UH\ 3 0 :LOOLDPV DQG ' % *UDGNLQ 0L[LQJ DQG +HDW 7UDQVIHU RI DQ $UJRQ $UFMHW ZLWK D &RD[LDO )ORZ RI &ROG +HOLXP 1$6$ &5 f 5 * 5DJVGDOH + :HLQVWHLQ DQG & ' /DQ]R &RUUHODWLRQ RI D 7XUEXOHQW $LU%URPLQH &RD[LDO )ORZ ([SHULPHQW 1$6$ 71 ' f 5 * 5DJVGDOH DQG 2 (GZDUGV 'DWD &RPSDULVRQV DQG 3KRWRJUDSKLF 2EVHUYDWLRQV RI &RD[LDO 0L[LQJ RI 'LVVLPLODU *DVHV DW 1HDUO\ (TXDO 6WUHDP 9HORFLWLHV 1$6$ 71 ' f 3 0 :LOOLDPV DQG *UH\ 6LPXODWLRQ RI *DVHRXV 1XFOHDU 5RFNHW 0L[LQJ &KDUDFWHULVWLFV 8VLQJ &ROG DQG $UF +HDWHG )ORZV 1$6$ &+ f 2 +LUVFKIHOGHU & ) &XUWLVV DQG 5 % %LUG 0ROHFXODU 7KHRU\ RI *DVHV DQG /LTXLGV -RKQ :LOH\ DQG 6RQV 1HZ <RUN f ' + 6DPSVRQ 5DGLDWLYH &RQWULEXWLRQV WR (QHUJ\ DQG 0RPHQWXP 7UDQVSRUW LQ D *DVV ,QWHUVFLHQFH 3XEOLVKHUV 1HZ <RUN f 6 5RVVHODQG 1RWH RQ WKH $EVRUSWLRQ RI 5DGLDWLRQ :LWKLQ D 6WDU 5R\DO $VWURQRPLFDO 6RFLHW\` 0RQWKO\ 1RWLFHV M eB f 6 5RVVHODQG $V WURSK\ VLN DXI $WRP7KHRUHWLVFKHU *UXQGODJH 6SULQJHU9HUODJ %HUOLQ f 6 5RVVHODQG 7KHRUHWLFDO $VWURSK\VLFV 2[IRUG 3UHVV /RQGRQ f 3 0 0RUVH DQG + )HVKEDFK 0HWKRGV RI 7KHRUHWLFDO 3K\VLFV 0F*UDZ+LOO %RRN &RPSDQ\ 1HZ <RUN f 9 6 $USDFL &RQGXFWLRQ +HDW 7UDQVIHU $GGLVRQ :HVOH\ 3XEOLVKLQJ &RPSDQ\ 5HDGLQJ 0DVVDFKXVHWWV f PAGE 218 $ 6 .HVWHQ DQG 1 / .UDVFHOOD 7KHRUHWLFDO ,QYHVWLJDWLRQV RI WKH 5DGLDQW +HDW 7UDQVIHU LQ WKH )XHO 5HJLRQ RI D *DVHRXV 1XFOHDU 5RFNHW (QJLQH 1$6$ &5 f 1 / .DVFHOOD 7DEOHV RI WKH &RPSRVLWLRQ 2SDFLW\ DQG 7KHUPRG\QDPLF 3URSHUWLHV RI +\GURJHQ DW +LJK 7HPSHUDWXUHV 1$6$ 63 f 1 / .UDVFHOOD 7KHRUHWLFDO ,QYHVWLJDWLRQV RI WKH $EVRUSWLYH 3URSHUWLHV RI 6PDOO 3DUWLFOHV DQG +HDY\$WRP *DVHV 1$6$ &5 f 6 *ODVVWRQH DQG 5 + /RYHEHUJ &RQWUROOHG 7KHUPRn QXFOHDU 5HDFWLRQV ' 9DQ 1RVWUDQG &RPSDQ\ 1HZ <RUN f $ $ 9HGHQRY 7KHRU\ RI 7XUEXOHQW 3ODVPD $FDGHP\ RI 6FLHQFHV RI WKH 8665 f 7UDQVODWHG IURP 5XVVLDQ E\ = /HUPDQ ,VUDHO 3URJUDP IRU 6FLHQWLILF 7UDQVODWLRQV -HUXVDOHP f ( 0 'HZDQ *HQHUDOL]DWLRQV RI WKH 6DKD (TXDWLRQ 7KH 3K\VLFV RI )OXLGV e f * (FNHU DQG : .UROO /RZHULQJ RI ,RQL]DWLRQ (QHUJ\ LQ D 3ODVPD ZLWK 'LIIHUHQW 7HPSHUDWXUHV 7KH 3K\VLFV RI )OXLGV f 7 :DEHU 'm /LEHUPDQ DQG ' 7 &URPHU ,RQL]DWLRQ 3RWHQWLDOV RI 8UDQLXP 1$6$ %XOOHWLQ WR *DVHRXV&RUH 1XFOHDU 5RFNHWV 'LVWULEXWLRQ /LVW -XQH PAGE 219 %,2*5$3+,&$/ 6.(7&+ +HQU\ )RXVW $WZDWHU ZDV ERUQ LQ %XUOLQJWRQ 1RUWK &DUROLQD RQ $XJXVW +LV SDUHQWV DUH 0U DQG 0UV +HQU\ $QGHUVRQ $WZDWHU ,Q -XQH RI KH ZDV JUDGXDWHG IURP :DOWHU 0 :LOOLDP +LJK 6FKRRO LQ %XUOLQJWRQ +H UHFHLYHG WKH GHJUHHV RI %DFKHORU RI 6FLHQFH LQ 1XFOHDU (QJLQHHULQJ ZLWK +RQRUV LQ 0D\ DQG 0DVWHU RI 6FLHQFH ZLWK PDMRU LQ $SSOLHG 0DWKHPDWLFV LQ 0D\ IURP 1RUWK &DUROLQD 6WDWH 8QLYHUVLW\ KROGLQJ DQ $SSOLHG 0DWKHPDWLFV 7HDFKLQJ $VVLVWDQWVKLS IURP 6HSWHPEHU WR 0D\ )URP -XQH WKURXJK $XJXVW KH ZDV HPSOR\HG E\ WKH $WRPLF (QHUJ\ 'LYLVLRQ RI WKH %DEFRFN DQG :LOFR[ &RPSDQ\ )URP 6HSWHPEHU XQWLO WKH SUHVHQW WLPH KH KDV SXUVXHG ZRUN WRZDUG WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'XULQJ WKLV WLPH KH KDV KHOG D GHSDUWPHQWDO UHVHDUFK DVVLVWDQWVKLS DQG D 7UDLQHHVKLS DQG 6SHFLDO )HOORZVKLSV LQ 1XFOHDU 6FLHQFH DQG (QJLQHHULQJ IURP WKH $WRPLF (QHUJ\ &RPPLVVLRQ +H LV PDUULHG WR WKH IRUPHU (OQD 9DQQ 7KDFNHU DQG WKH\ KDYH WZR FKLOGUHQ +H LV D PHPEHU RI 7DX %HWD 3L 3L 0X (SVLORQ 3KL .DSSD 3KL DQG WKH $PHULFDQ 1XFOHDU 6RFLHW\ PAGE 220 7KLV GLVVHUWDWLRQ ZDV SUHSDUHG XQGHU WKH GLUHFWLRQ RI WKH FKDLUPDQ RI WKH FDQGLGDWHnV VXSHUYLVRU\ FRPPLWWHH DQG KDV EHHQ DSSURYHG E\ DOO PHPEHUV RI WKDW FRPPLWWHH ,W ZDV VXEPLWWHG WR WKH 'HDQ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH &RXQFLO DQG ZDV DSSURYHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0DUFK 'HDQ *UDGXDWH 6FKRRO