7KH 'LPHQVLRQ RI D 9HFWRU 6SDFH0DWK 78 ² /LQHDU $OJHEUD'DYLG $UQROG)DOO 4<<:$EVWUDFW1 ,I WKH VHW R I YHFWRUV% ã£Y4/Y5/T/YQ¤DUH OLQHDUO\ LQGHSHQGHQW DQG VSDQ WKH YHFWRU VSDFH9/ WKHQ WKH VHW%LV FDOOHG DEDVLVIRU WKH YHFWRU VSDFH91 7KH QXPEHU RI YHFWRUV LQ WKH VHW%LV FDOOHG WKHGLPHQVLRQRI WKH YHFWRU VSDFH91 ,Q WKLV DFWLYLW\ \RX ZLOO XVH D QXPEHU RI0DWODEFRPPDQGV WR H[SORUHDQG UHLQIRUFH WKHVH LGHDV13UHUHTXLVLWHV1 )DPLOLDULW\ ZLWK 0DWODE¶VUUHIFRPPDQG DQG LWV XVH WR VROYH OLQHDU V\VWHPV LVDVVXPHG1 5HDGHUV VKRXOG DOVR EH IDPLOLDU ZLWK WKH FRQFHSWV RI OLQHDU LQGHSHQGHQFH DQG WKH VSDQ RI D VHWRI YHFWRUV1,QWURGXFWLRQ,I WKH VHW RI YHFWRUV%ã£Y4/ Y5/T/ YQ¤DUH OLQHDUO\ LQGHSHQGHQW DQG VSDQ WKH YHFWRU VSDFH9/ WKHQ WKH VHW%LV FDOOHG DEDVLVIRU WKH YHFWRU VSDFH91 7KH QXPEHU RI YHFWRUV LQ WKH VHW%LVFDOOHG WKHGLPHQVLRQRI WKH YHFWRU VSDFH91 ,Q WKLV DFWLYLW\ \RX ZLOO XVH D QXPEHU R I0DWODEFRPPDQGV WR H[SORUH DQG UHLQIRUFH WKHVH LGHDV17KH 1XOO 6SDFH7KHQXOO VSDFHRI D PDWUL[$LV WKH VHW RI DOO YHFWRUV [ VXFK WKDW$[ã31 ,Q V\PEROV/1XOŸ$ ã£[ =$[ã3¤,W¶V RIWHQ HDVLHVW WR GHVFULEH WKH QXOO VSDFH R I D PDWUL[ E\ ILQGLQJ D EDVLV IRU W KH QXOO VSDFH1%HIRUH WU\LQJ DQ H[DPSOH/ ILUVW VHW0DWODE¶VIRUPDW IRU UDWLRQDO DULWKPHWLF1ââ IRUPDW UDW([DPSOH)LQG D EDVLV IRU WKH QXOO VSDFH RI WKH PDWUL[$ã4 "54 34"443"55536"576ROXWLRQ1 /RDG WKH PDWUL[$LQWR WKH0DWODEZRUNVSDFH1ââ $ã>405434>0443055>536057@$ã40543404 4 3 05 55360577KH QXOO VSDFH LV FRPSRVHG RI DOO VROXWLRQV RI WKH PDWUL[ HTXDWLRQ$[ã31 , VWURQJO\VXJJHVW WKDW \RX VHW XS WKH SUREOHP DV VKRZQ LQ HTXDWLRQ +4,145 $ 6FLHQWLILF 5HSRUW$[ ã 34 "54 34"443"55536"57[4[5[6[7[8ã33341RWH WKDW WKH VROXWLRQV RI HTXDWLRQ +4, DUH HOHPHQWV R I 581 7KH QXOO VSDFH RI PDWUL[ $ LV DVXEVSDFH RI 581 6HW XS DQ DXJPHQWHG PDWUL[ IRU WKH V\VWHP LQ HTXDWLRQ +4,1ââ0ã>$/ >333@¶@0ã40543430443055353605733ODFH WKLV PDWUL[ L Q UHGXFHG URZ HFKHORQ IRUP ZLWK 0DWODE¶VUUHIFRPPDQG1ââUUHI+0,DQVã433504428 334330428 33 3 4 05 4728 37KH UHGXFHG URZ HFKHORQ IRUP RI WKH PDWUL[0LQGLFDWHV WKDW WKH VROXWLRQV RI HTXDWLRQ +4,FDQ EH ZULWWHQ[4[5[6[7[8ã"5[7ò448[848[85[7"478[8[7[8ã"5[735[7[73ò448[848[8"478[83[8ã [7"53543ò [844848"478345ZKHUH [7DQG [8DUH DQ\ UHDO QXPEHUV1 (TXDWLRQ +5, JXDUDQWHHV WKDW DO O VROXWLRQV RI WKHHTXDWLRQ $[ ã 3 FDQ EH ZULWWHQ DV D OLQHDU FRPELQDWLRQ R I WKH YHFWRUV Y4ã Ÿ"5/ 3/ 5/ 4/ 3 DQG$ 6FLHQWLILF 5HSRUW 6Y5ã Ÿ4428/ 428/ "4728/ 3/ 4 1 6LQFH WKH YHFWRUV Y4DQG Y5DUH LQGHSHQGHQW DQG WKH\ VSDQ WKHVROXWLRQ VHW RI $[ ã 3/ WKH\ IRUP D EDVLV IRU WKH VROXWLRQV R I $[ ã 31 7KHUHIRUH/%4ã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ã 4 DQG [8ã 3/ WKHQ OHW[7ã 3 DQG [8ã 8/ WKHQ%5ã"53543/444"4738LV DOVR D EDVLV IRU WKH QXOO VSDFH R I WKH PDWUL[ $ +WKH YHFWRUV DUH OLQHDUO\ LQGHSHQGHQW DQG VSDQWKH VHW RI VROXWLRQV R I $[ ã 3,10DWODE¶V 1XOO &RPPDQG0DWODE¶V QXOO FRPPDQG/ ZKHQ XVHG RQ QXPHULF REMHFWV/ ILQG D YHU\ VSHFLDO EDVLV IRU W KHQXOO VSDFH RI WKH PDWUL[ $1ââKHOS QXOO18// 1XOO VSDFH1=ã18//+$, LV DQ RUWKRQRUPDO EDVLV IRU WKH QXOO VSDFH RI $ REWDLQHGIURP WKH VLQJXODU YDOXH GHFRPSRVLWLRQ1 7KDW LV/ $-= KDV QHJOLJLEOHHOHPHQWV/ VL]H+=/5, LV WKH QXOOLW\ RI $/ DQG =¶-=ã,1=ã18//+$/¶U¶, LV D ´UDWLRQDO´ EDVLV IRU WKH QXOO VSDFH REWDLQHGIURP WKH UHGXFHG URZ HFKHORQ IRUP1 $-= LV ]HUR/ VL]H+=/5, LV DQHVWLPDWH IRU WKH QXOOLW\ RI $/ DQG/ LI $ LV D VPDOO PDWUL[ ZLWKLQWHJHU HOHPHQWV/ WKH HOHPHQWV RI 5 DUH UDWLRV RI VPDOO LQWHJHUV17KH RUWKRQRUPDO EDVLV LV SUHIHUDEOH QXPHULFDOO\/ ZKLOH WKH UDWLRQDOEDVLV PD\ EH SUHIHUDEOH SHGDJRJLFDOO\17KH IROORZLQJ FRPPDQG SURGXFHV DQ RUWKRQRUPDO EDVLV IRU WKH QXOO VSDFH1ââIRUPDWââ1ãQXOO+$,1ã7 $ 6FLHQWLILF 5HSRUW031999; 3313359 31456;3198<4 0316:473167:8 319;3<31345; 3194<37KH FROXPQV RI WKH PDWUL[ 1 DUH WKH EDVLV YHFWRUV1 7KHVH YHFWRUV KDYH XQLW OHQJWK DQG DUHRUWKRJRQDO +SHUSHQGLFXODU, WR HDFK RWKHU1 +HQFH WKH ZRUG RUWKRQRUPDO1<RX FDQ SURGXFH W KH EDVLV %4ZLWK WKH IROORZLQJ 0DWODE FRPPDQGV1ââ IRUPDW UDWââ 1ãQXOO+$/¶U¶,1 ã05 44283 4285 0472843347KH &ROXPQ 6SDFH7KH FROXPQ VSDFH RI D PDWUL[ $ LV VLPSO\ GHILQHG DV WKH VSDQ RI WKH FROXPQV RI $1 7KDW LV/LI D4/ D5/T/ DQDUH WKH FROXPQV RI WKH PDWUL[ $/ WKHQ WKH FROXPQ VSDFH RI WK H PDWUL[ $ LVGHILQHG DV IROORZV=FROŸ$ ã6SDQ£D4/ D5/T/ DQ¤%HFDXVH WKH VHW &ã£D4/ D5/T/ DQ¤DOUHDG\ VSDQV WKH FROXPQ VSDFH/ VLPSO\ GHOHWH GHSHQGHQWYHFWRUV RQH DW D WLPH XQWLO \RX KDYH D OLQHDUO\ LQGHSHQGHQW VHW1 )RU H[DPSOH/ FRQVLGHU DJDLQ WKHPDWUL[$ã4 "54 34"443"55536"577KH FROXPQ VSDFH RI $ LV WKH VSDQ RI LWV FROXPQVFROŸ$ ã6SDQ£D4/ D5/ D6/ D7/ D8¤7KH UHGXFHG URZ HFKHORQ IRUP RI $ LV8ã433 5 "448343 3 "48334"5478,I \RX ODEHO WKH FROXPQ YHFWRUV RI WKH PDWUL[ 8 ZLWK X4/ X5/ X6/ X7/ DQG X8/ WKHQ LW LV HDV\ WR VHHWKDW X7ã5X4" 5X61 ,W LV LPSRUWDQW WR QRWH WKDW WKH FROXPQV RI $ KDYH WKH VDPH GHSHQGHQF\UHODWLRQ> WKDW LV/ D7ã5D4" 5D61 <RX FDQ HOLPLQDWH D7DQG WKH UHPDLQLQJ YHFWRUV ZLOO VWLOO VSDQWKH FROXPQ VSDFH RI PDWUL[ $1FROŸ$ ã6SDQ£D4/ D5/ D6/ D8¤6HFRQGO\/ LW LV HDV\ WR VHH WKDW X8ã"448X4"48X5ò478X61 7KH FROXPQV RI $ ZLOO KDYH WKH$ 6FLHQWLILF 5HSRUW 8VDPH GHSHQGHQF\ UHODWLRQ/ D8ã "448D4"48D5ò478D61 <RX FDQ HOLPLQDWH D8DQG WKHUHPDLQLQJ YHFWRUV ZLOO VWLOO VSDQ WKH FROXPQ VSDFH RI W KH PDWUL[ $1FROŸ$ ã 6SDQ£D4/ D5/ D6¤%HFDXVH X4/ X5/ DQG X6DUH OLQHDUO\ LQGHSHQGHQW/ D4/ D5/ DQG D6ZLOO DOVR EH LQGHSHQGHQW DQGWKH VHW%6ã4"45/"543/436LV D EDVLV IRU WKH FROXPQ VSDFH RI $1 %HFDXVH WKLV EDVLV KDV 6 YHFWRUV/ WKH GLPHQVLRQ RI WKHFROXPQ VSDFH RI WKH PDWUL[ $ LV WKUHH1 7KH GLPHQVLRQ RI WKH FROXPQ VSDFH RI WKH PDWUL[ $ LVDOVR FDOOHG WKH UDQN R I WKH PDWUL[ $17KH 0DWODE FRPPDQG WKDW EHVW LPLWDWHV WKLV SURFHVV LV WKH UUHI FRPPDQG1 <RX ZLOO QHHGWR UHDG WKH KHOS ILOH DJDLQ/ RQO\ WKLV WLPH SD\ FORVHU DWWHQWLRQ WKDQ ZKHQ \RX ILUVW UHDG WKLV ILOH1ââKHOS UUHI55() 5HGXFHG URZ HFKHORQ IRUP15ã55()+$, SURGXFHV WKH UHGXFHG URZ HFKHORQ IRUP RI $1>5/ME@ã55()+$, DOVR UHWXUQV D YHFWRU/ ME/ VR WKDW=UãOHQJWK+ME, LV WKLV DOJRULWKP¶V LGHD RI WKH UDQN RI $/[+ME, DUH WKH ERXQG YDULDEOHV LQ D OLQHDU V\VWHP/ $[ãE/$+=/ME, LV D EDVLV IRU WKH UDQJH RI $17KH