��

� =RRPLQJ LQ RQ DQ /W/ 3KDVH %RXQGDU\

��� /LIH�OLNH /W/ UXOHV

,Q WKLV FKDSWHU� ZH GHVFULEH WKH G\QDPLFV H[KLELWHG E\ WKH UXOHV RQ RQH RI /W/V SKDVH

ERXQGDULHV� $V LQ /LIH� PDQ\ RI WKHVH UXOHV H[KLELW G\QDPLFV WKDW QHLWKHU IL[DWH QRU EHFRPH

FRPSOHWHO\ GLVRUGHUHG ULJKW DZD\� ,QVWHDG� FRPSOH[ VHOI�RUJDQL]DWLRQ HPHUJHV IURP UHJLRQV

WKDW DSSHDU WR EH DSHULRGLF� (YHQWXDOO\� WKH DSHULRGLF UHJLRQV GLH RXW DQG RQO\ ILQLWH ORFDO

VWUXFWXUHV� VXFK DV KLJK UDQJH DQDORJXHV WR /LIHV JOLGHUV� DUH YLDEOH� :H DGG WKH FDYHDW WKDW

WKHVH DUH WKH DSSDUHQW EHKDYLRUV RI WKH V\VWHPV� 7KHUH LV WKH SRVVLELOLW\ RI HYHQ PRUH

LQWULJXLQJ G\QDPLFV� :H ZLOO GLVFXVV VXFK D SRVVLELOLW\ LQ &KDSWHU RI WKLV WKHVLV�

/HW XV LQWURGXFH VRPH WHUPLQRORJ\� 6D\ WKDW D UXOH LV I WKHUH LV DW OHDVW RQHEXJJLQ L�

EXJ DGPLWWHG E\ WKH UXOH� 6D\ WKDW D UXOH KDV � RU LV � LI LW LV EXJJLQ/LIH�OLNH G\QDPLFV /LIH�OLNH

DQG LWV OLPLWLQJ VWDWH FRQVLVWV RQO\ RI ILQLWH ORFDO FRQILJXUDWLRQV �WKDW LV� DOO DSHULRGLFLW\ GLHV RXW

DQG QR JOREDO WLOLQJ ILOOV LQ WKH ODWWLFH�� )RU VRPH /LIH�OLNH UXOHV WKH EXJV DUH GLIILFXOW WR ILQG

DQG RQ D ILQLWH V\VWHP �VXFK DV D FRPSXWHU VFUHHQ� WKH G\QDPLFV DSSHDU WR UHOD[ UDWKHU

TXLFNO\� ZLWKRXW \LHOGLQJ DQ\ EXJV� +RZHYHU� ZH KDYH IRXQG EXJV IRU HDFK UXOH ZH FDOO /LIH�

OLNH�

)RU HDFK IL[HG UDQJH� WKHUH DUH DW OHDVW WZR GLVWLQFW UHJLRQV RI SDUDPHWHU VSDFH ZKHUH

WKHUH H[LVW UXOHV ZLWK /LIH�OLNH G\QDPLFV� 2QH LV FORVH WR D TXDGUDWLF UHVFDOLQJ RI /LIH� DQG WKH

RWKHU LV FORVH WR D OLQHDU UHVFDOLQJ� :H UHIHU WR WKH ILUVW DV WKH DQG WKH VHFRQGIDW EXJ UHJLPH

DV WKH � 7KH QDPHV FRPH IURP WKH GLVWLQFW EXJ YDULHWLHV DQGWKLQ EXJ UHJLPH IDW WKLQ^ ^

ZKLFK HDFK UHJLPH DGPLWV� /HW XV JLYH VRPH H[DPSOHV WR LOOXVWUDWH WKH /LIH�OLNH G\QDPLFV

DGPLWWHG E\ ODUJH UDQJH /W/ UXOHV�

,Q HDFK RI WKH IROORZLQJ H[DPSOHV WKH LQLWLDO VWDWH ZDV SURGXFW PHDVXUH ZLWK GHQVLW\ �

� IRU /LIH� DQG WKH H[DPSOHV IURP WKH IDW EXJ UHJLPH� IRU WKH WKLQ EXJ H[DPSOHV��� y � y� �

� ��

WKH V\VWHP VL]H ZDV � DQG ZUDS DURXQG ERXQGDU\ FRQGLWLRQV ZHUH XVHG $V WKH� _ ��� »

UDQJH LQFUHDVHV� LW ZRXOG PDNH PRUH VHQVH WR LQFUHDVH WKH V\VWHP VL]H DV ZHOO� +RZHYHU� IRU

WKH IROORZLQJ H[DPSOHV� WKH JHQHUDO LGHD LV FDSWXUHG XVLQJ D XQLIRUP V\VWHP VL]H� 7KH RQO\

GHFHSWLRQ LV WKDW WKH GHQVLW\ RI ORFDO FRQILJXUDWLRQV DSSHDUV WR GHFUHDVH DV WKH UDQJH LQFUHDVHV�

� EXJJLQ )UHDNLQ RXW� PDNLQJ D ELJ GHDO RXW RI VRPHWKLQJ� �'HILQLWLRQ VXSSOLHG E\ D /RV $QJHOHV DUHD

WHHQDJHU��

��

)RU WKH XQLQLWLDWHG� ZH EHJLQ ZLWK /LIH�

5DQJH �� �� /LIH� ­�¼ �¼ �¼ �¼ �®

$V FDQ EH VHHQ LQ WKH IROORZLQJ ILJXUHV� DIWHU WLPH VWHSV VRPH VHOI�RUJDQL]DWLRQ KDV

WDNHQ SODFH� +RZHYHU� WKH GHQVLW\ RI OLYH VLWHV LV WRR KLJK IRU WKH HPHUJHQFH RI YLDEOH

VWUXFWXUHV� %\ WLPH � FRPSOH[ ORFDO VWUXFWXUHV� VXFK DV JOLGHUV� KDYH HPHUJHG� 7KH GHQVLW\���

RI VLWHV QRW LQ D IL[HG RU SHULRGLF FRQILJXUDWLRQ KDV GUDPDWLFDOO\ GHFUHDVHG� %\ WLPH ����

DOPRVW DOO DSHULRGLF G\QDPLFV KDYH GLHG RXW DQG RQO\ ILQLWH ORFDO VWUXFWXUHV UHPDLQ� %\ WLPH

�� »� WKH V\VWHP FRQWDLQV RQO\ IL[HG RU SHULRGLF ORFDO VWUXFWXUHV

WLPH

��

WLPH ���

WLPH ����

��

)DW %XJ 5HJLPH �� TXDGUDWLF UHVFDOLQJV RI /LIH

1RZ OHW XV UHVFDOH /LIHV SDUDPHWHUV XVLQJ WKH TXDGUDWLF VFKHPH GHVFULEHG LQ VHFWLRQ

�»�» ­�¼ ��¼ � ¼ ��¼ ��,Q UDQJH WKLV \LHOGV WKH UXOH ZKLFK LV EXJJLQ EXW QRW /LIH�OLNH�� ®

+RZHYHU� WKH QHDU�E\ UXOH ­�¼ ��¼ � ¼ ��¼ ��® LV /LIH�OLNH� :H ZLOO LOOXVWUDWH WKLV VKRUWO\� ,Q

UDQJH WKH UHVFDOLQJ JLYHV�� WKH UXOH ZKLFK DJDLQ LV EXJJLQ EXW QRW­��¼ ���¼ ���¼ ���¼ ���®

/LIH�OLNH� +RZHYHU� DV ZH ZLOO LOOXVWUDWH EHORZ WKH UDQJH UXOH� LV�� ­��¼ ���¼ ���¼ ���¼ ���®

/LIH�OLNH� :H FKRVH WR LOOXVWUDWH WKLV UHVFDOLQJ LQ UDQJHV DQG � KRZHYHU� QHLWKHU RI WKHVH� ��

UDQJHV LV VSHFLDO� 7KH UHVFDOLQJ LV UHODWLYHO\ VWUDLJKWIRUZDUG IRU JHQHUDO UDQJHV�

5DQJH ��� ­�¼ ��¼ � ¼ ��¼ ��®

$V FDQ EH VHHQ LQ WKH IROORZLQJ ILJXUHV� DIWHU WLPH VWHSV VRPH VHOI�RUJDQL]DWLRQ KDV

WDNHQ SODFH� %\ WLPH � EXJV� VWLOO OLIHV� DQG SHULRGLF REMHFWV� KDYH HPHUJHG� 7KH DSHULRGLF���

UHJLRQV DUH OHVV GHQVH� %\ WLPH DQ HYHQ ORZHU GHQVLW\ RI DSHULRGLFLW\ HQDEOHV DQ LQFUHDVH� �

RI WKH OHQJWKV RI WKH EXJV WUDMHFWRULHV �L�H� WKH EXJV DUH DEOH WR SHUVLVW ORQJHU EHIRUH EHLQJ

GHVWUR\HG E\ FROOLVLRQV ZLWK RWKHU OLYH VLWHV�� ,Q WKLV H[DPSOH� LW WDNHV ORQJHU IRU DOO

DSHULRGLFLW\ WR GLH RXW WKDQ LW GLG LQ WKH /LIH H[DPSOH� ,W LV QRW XQWLO WLPH WKDW WKH RQO\���

UHPDLQLQJ OLYH VLWHV DUH SDUW RI HLWKHU SHULRGLF REMHFWV �WZR ZLWK SHULRG � RU VWLOO OLIHV� DOO RI��

WKH EXJV KDYH GLHG LQ FROOLVLRQV ZLWK IL[HG RU SHULRGLF REMHFWV»

��

WLPH

WLPH ���

��

WLPH � �

WLPH ���

��

5DQJH ���� ­��¼ ���¼ ���¼ ���¼ ���®

,Q WKLV H[DPSOH� WKH UHODWLYHO\ VPDOO V\VWHP VL]H \LHOGV D VPDOO GHQVLW\ E\ WLPH � 7KXV�

DW WLPH YLDEOH VWUXFWXUHV DUH DSSDUHQW� $W WLPH WKHUH DUH VWLOO UHJLRQV RI DSHULRGLFLW\� ���

KRZHYHU� WKH SDWFKHV RI ORZ GHQVLW\ HQDEOH WKH HPHUJHQFH RI EXJV ZLWK GLDJRQDO DQG YHUWLFDO

WUDMHFWRULHV� %\ WLPH � HYHU\WKLQJ KDV GLHG RXW H[FHSW WZR EXJV WUDYHOLQJ LQ RSSRVLWH��

GLUHFWLRQV� :H OHW WKH &$ XSGDWH XQWLO WLPH EHFDXVH WKH EXJ GHSLFWHG WRZDUG WKH ERWWRP��

RI WKH IROORZLQJ ILJXUH ZDV DV\PPHWULF LQ DOO GLUHFWLRQV DW WLPH � +RZHYHU� DIWHU PRUH�� ��

WLPH VWHSV� LW EHFDPH V\PPHWULF ZLWK UHVSHFW WR WKH KRUL]RQWDO D[LV DQG VWDELOL]HG LQWR D SHULRG

�� ��»RUELW� 7KH RWKHU EXJ KDV SHULRG 6LQFH WKH EXJV WUDMHFWRULHV DUH SDUDOOHO� WKH\ ZLOO

FRH[LVW IRUHYHU RQ WKH WRUXV� ,Q WKLV FDVH� DV GLVWLQFW IURP WKH SUHYLRXV WZR� WKH ORZ GHQVLW\ RI

IL[HG REMHFWV HQDEOHV WKH EXJV VXUYLYDO�

WLPH

��

WLPH ���

WLPH ��

��

7KLQ %XJ 5HJLPH �� OLQHDU UHVFDOLQJV RI /LIH

1RZ OHW XV LOOXVWUDWH WZR OLQHDU UHVFDOLQJV RI /LIHV SDUDPHWHUV WKDW \LHOG UXOHV ZLWK

/LIH�OLNH G\QDPLFV� ,Q /W/ VSDFH /LIH LV FORVH WR WKH UDQJH H[DFWO\ UXOH ZLWK DQG� y �� �

VLQFH � � )RU JHQHUDO UDQJHV WKH H[DFWO\ UXOH ZLWK LV /LIH�OLNH� ,I� � � � �y � � y � ] � y �

� � � �} � y � ^ � �WKH H[DFWO\ UXOH ZLWK LV DOVR /LIH�OLNH� /HW XV LOOXVWUDWH WKH UDQJH YHUVLRQ

RI WKH ILUVW UHVFDOLQJ DQG WKH UDQJH YHUVLRQ RI WKH VHFRQG�

5DQJH ��� ­�¼ ¼ ¼ ¼ ®

$V FDQ EH VHHQ LQ WKH IROORZLQJ ILJXUHV� E\ WLPH VRPH VHOI�RUJDQL]DWLRQ KDV RFFXUUHG�

+RZHYHU� WKH UHODWLYHO\ KLJK GHQVLW\ RI OLYH VLWHV SUHYHQWV EXJV IURP SHUVLVWLQJ IRU PRUH WKDQ D

IHZ WLPH VWHSV� %\ WLPH PRVW RI WKH OLYH VLWHV DUH SDUW RI VRPH YLDEOH FRQILJXUDWLRQ VXFK���

DV D EXJ� D RQH�GLPHQVLRQDO UHSOLFDWRU� RU D VWUHDP RI EXJV JHQHUDWHG E\ D EXJ PDNHU� $W WLPH

�� WKH G\QDPLFV DUH VLPLODU ZLWK RQO\ YLDEOH GLJLWDO FUHDWXUHV SHUVLVWLQJ� (YHQWXDOO\�

KRZHYHU� DOO RI WKH EXJV� EXJ PDNHUV� DQG UHSOLFDWRUV DQQLKLODWH RQH DQRWKHU� OHDYLQJ RQO\ RQH

EXJ� DW WLPH � WUDYHOLQJ LQ WKH YHUWLFDO GLUHFWLRQ� ,Q WKH ODVW RI WKH IROORZLQJ ILJXUHV����

JHQHUDWHG E\ WKH UDQJH UXOH ZH VKRZ WLPH MXVW EHIRUH WKH ODVW FROOLVLRQ�� ����

��

WLPH

WLPH ���

��

WLPH ��

WLPH ����

��

5DQJH �� ­¼ ¼ ¼ ¼ ®

$V FDQ EH VHHQ LQ WKH IROORZLQJ ILJXUHV� E\ WLPH WKH GHQVLW\ RI OLYH VLWHV LV VPDOO

HQRXJK WKDW VRPH EXJV DUH DEOH WR SHUVLVW� KRZHYHU WKHUH DUH DOVR UHJLRQV WKDW H[KLELW

DSHULRGLF G\QDPLFV� %\ WLPH PRVW RI WKH DSHULRGLF G\QDPLFV KDYH GLHG RXW DQG PDQ\�

YLDEOH ORFDO FRQILJXUDWLRQV KDYH HPHUJHG ,Q IDFW� WKUHH GLVWLQFW EXJ DUFKLWHFWXUHV DUH»

LOOXVWUDWHG LQ WKDW ILJXUH DV ZHOO DV WKH ILJXUH IURP WLPH � $IWHU WLPH VHYHUDO RI WKH EXJV� �

FROOLGH DQQLKLODWLQJ RQH DQRWKHU LQ PRVW FDVHV VR WKDW E\ WLPH WKHUH DUH IHZHU UHPDLQLQJ��

EXJV� +RZHYHU� RQH RI WKH FROOLVLRQV FUHDWHV WZR FRSLHV RI WKH EXJ WKDW ORRNV OLNH D

VKXWWOHFRFN� WUDYHOLQJ LQ RSSRVLWH GLUHFWLRQV� 6LQFH ZH XVHG SHULRGLF ERXQGDU\ FRQGLWLRQV�

WKH\ UXQ LQWR HDFK RWKHU� :H VWRSSHG WKH &$ DW WLPH � MXVW EHIRUH WKH FROOLVLRQ ZKLFK��

RFFXUV DW WLPH DQG UHVXOWV LQ WKH DQQLKLODWLRQ RI ERWK EXJV� 7KH UHPDLQLQJ WZR EXJV��

FRH[LVW XQWLO WLPH ZKHQ WKH\ DOVR FROOLGH� 7KHLU FROOLVLRQ UHVXOWV LQ D QHZ EXJ RI WKH�

VKXWWOHFRFN YDULHW\ GHSLFWHG LQ WKH ILJXUH IURP WLPH � �

WLPH

��

WLPH �

WLPH �

��

WLPH ��

WLPH �

��

7KH WKLQ EXJ H[DPSOHV IURP UDQJHV DQG DV ZHOO DV WKH UDQJH IDW EXJ H[DPSOH� ��

HYHQWXDOO\ UHVXOW LQ GHDWK RI DOPRVW DOO YLDEOH FRQILJXUDWLRQV RI � 7KH GHDWK RI WKHVH YLDEOH� Z

VWUXFWXUHV RFFXUV WKURXJK FROOLVLRQV ZLWK RWKHU YLDEOH REMHFWV� 7KLV UDLVHV TXHVWLRQV DERXW WKH

LQILQLWH V\VWHP� 'R DOO YLDEOH FRQILJXUDWLRQV H[FHSW EXJV GLH RXW IDVW HQRXJK WKDW WKH EXJV

HYHQWXDOO\ VHH RQO\ RWKHU EXJV" ,I VR� WKH EXJV DUH DQDORJRXV WR DQQLKLODWLQJ SDUWLFOHV�

+RZHYHU� DV LOOXVWUDWHG DERYH� VRPH EXJ FROOLVLRQV \LHOG QHZ EXJV� 7KLV PDNHV WKH G\QDPLFV

PRUH FRPSOLFDWHG WR SDUVH� $QRWKHU VFHQDULR� VXJJHVWHG E\ /LIH DQG WKH UDQJH IDW EXJ�

H[DPSOH DERYH LV WKDW ILQLWH VWLOO OLIHV DQG SHULRGLF REMHFWV DOVR SHUVLVW DQG UHSUHVHQW REVWDFOHV

WKDW GHVWUR\ WKH EXJV� (YHQWXDOO\ DOO RI WKH EXJV ZLOO FUDVK LQWR VXFK FRQILJXUDWLRQV UHVXOWLQJ

LQ D ORFDOO\ SHULRGLF OLPLWLQJ VWDWH� 7KHVH TXHVWLRQV UHPDLQ XQDQVZHUHG� KRZHYHU� ZH ZLOO

GLVFXVV WKHP D ELW IXUWKHU LQ &KDSWHU �

�»� 6XEVSDFH RI %XJV

1RZ WKDW ZH KDYH LOOXVWUDWHG WKH JOREDO G\QDPLFV H[KLELWHG E\ UXOHV LQ WKH IDW DQG WKLQ

EXJ UHJLPHV� OHW XV GLVFXVV WKH GLJLWDO FUHDWXUHV WKDW PDNH WKHVH G\QDPLFV VR LQWULJXLQJ� :H

EHJLQ ZLWK D SLFWXUH RI /LIHV IDPRXV JOLGHU� :H GHSLFW LWV WUDMHFWRU\ DV ZHOO� VKRZLQJ WLPHV

�¼ ��¼ ��¼ ��¼ �� »DQG �LW PRYHV IURP WKH ORZHU OHIW FRUQHU XS WR WKH XSSHU ULJKW FRUQHU�

/LIHV JOLGHU

7KH JOLGHUV WUDMHFWRU\ DORQJ WKH GLDJRQDO�

7KH JOLGHU KDV DQG � ,WV SDWK LV QRW GLUHFW� WKDW LV ZK\ LW WDNHV WLPH VWHSV# y � ] � y � �W � � �

WR WUDQVODWH E\ �� ] �� �

)DW %XJV

/HW XV ORRN PRUH FORVHO\ DW WKH WZR IDW EXJV JHQHUDWHG E\ WKH UDQJH UXOH GHVFULEHG��

LQ VHFWLRQ � $V LV WKH FDVH IRU PRVW IDW EXJV� WKH\ KDYH � 7KHLU UHVSHFWLYH SHULRGV�»� VWRPDFKV

RI DQG PDNH WKHLU WUDMHFWRULHV TXLWH LQWHUHVWLQJ� 1RW RQO\ GR WKHLU JHRPHWULHV YDU\ DV�� ��

���

WLPH PDUFKHV RQ� WKHLU VWRPDFKV DOVR F\FOH WKURXJK D YDULHW\ RI VKDSHV� /HW XV GHSLFW D FORVH�

XS RI WKH SHULRG EXJ DORQJ ZLWK RI LWV SKDVHV��� � ��

­��¼ ���¼ ���¼ ���¼ ���® y ��� ¼ y ��� #W � �

)URP OHIW WR ULJKW DUH WLPHV WKURXJK � KDOI RI WKH F\FOHV RI WKLV SHULRG IDW EXJ�� � ��

7KH FRPSOLFDWHG IRUP RI WKH LQWHUPHGLDWH SKDVHV RI WKH EXJ PDNH LW UDWKHU GLIILFXOW WR

JHQHUDOL]H� +RZHYHU� LW LV FRQFHLYDEOH WKDW WKHUH LV D (XFOLGHDQ YHUVLRQ RI WKH IDW EXJV IRXQG

E\ GRLQJ D WKUHVKROG�UDQJH VFDOLQJ RI DQ /W/ UXOH LQ WKH IDW EXJ UHJLPH� �6HH >)**�@�� :H

EHOLHYH WKLV EHFDXVH WKH EXJV� DORQJ ZLWK WKHLU VKDSHO\ VWRPDFKV� VFDOH WKURXJK WKH UDQJHV�

/HW XV VKRZ WZR LOOXVWUDWLYH H[DPSOHV� WKH ILUVW IURP UDQJH WZR DQG WKH VHFRQG IURP UDQJH

ILYH�

­�¼ �¼ ¼ �¼ ��®¼ y � ¼ y #W � �

/HIW WR ULJKW DUH WLPHV WR � WKH HQWLUH F\FOH RI WKH SHULRG EXJ�� �

7KH WUDMHFWRU\ RI WKH UDQJH WZR H[DPSOH GHSLFWHG DERYH VKRZV WKDW� DOUHDG\ LQ UDQJH

WZR� WKHUH H[LVW EXJV ZLWK VWRPDFKV� 7KLV LV UDWKHU VXUSULVLQJ EHFDXVH RI WKH UHODWLYHO\ VPDOO

QHLJKERUKRRG VL]H� +RZHYHU� WKHVH EXJV DUH QRW IDW DV WKRVH LQ KLJKHU UDQJHV� )RUDV

H[DPSOH� DV FDQ EHHQ VHHQ LQ WKH IROORZLQJ ILJXUHV� WKH UDQJH ILYH DQDORJXHV DUH PRUH

UHPLQLVFHQW RI WKRVH IURP UDQJH WHQ�

­¼ ��¼ ��¼ ��¼ �® y ��� ¼ y ��� #W � �

)URP OHIW WR ULJKW DUH WLPHV WR � WKH HQWLUH F\FOH RI WKLV SHULRG IDW EXJ�� � ��

���

7KHUH DOVR H[LVW EXJV WKDW DUH LQYDULDQW PRG WUDQVODWLRQ� )RU WKHVH WKH SUREOHP RI

ILQGLQJ D (XFOLGHDQ YHUVLRQ SUHVHQWV D PRUH UHDOLVWLF HQGHDYRU� /HW XV JLYH DQ H[DPSOH IURP

UDQJH ILYH�

­¼ � ¼ ��¼ � ¼ �® y � # y �W� �� �

%XJV WKDW PRYH LQ WKH KRUL]RQWDO GLUHFWLRQ DUH QRW WKH RQO\ NLQGV RI IDW EXJV� 7KHUH

DUH DOVR H[DPSOHV WKDW DUH V\PPHWULF ZLWK UHVSHFW WR WKH OLQH� DQG KHQFH PRYH DORQJ WKH�Í

GLDJRQDO� $GGLWLRQDOO\� WKHUH DUH IDW EXJV ZKLFK DUH DV\PPHWULF ZLWK UHVSHFW WR DOO RI WKH

V\PPHWULHV RI WKH ODWWLFH� 6RPH VXFK EXJV PDQDJH WR PRYH LQ WKH KRUL]RQWDO GLUHFWLRQ�

KRZHYHU� WKH\ ZHDYH DV WKH\ PRYH� :H GHSLFWHG VXFK D GUXQN EXJ LQ 6HFWLRQ � 2WKHUV��»�

FDOOHG GLVRULHQWHG EXJV� PRYH DORQJ OLQHV WKDW DUH QRW KRUL]RQWDO� YHUWLFDO RU GLDJRQDO� :H

KDYH PDQ\ VXFK EXJV LQ D YDULHW\ RI UDQJHV� KRZHYHU� OHW XV SUHVHQW MXVW RQH H[DPSOH IURP

UDQJH WHQ� :H XVH JUD\ WR GHSLFW DOWHUQDWH WLPHV VLQFH WKH GLVRULHQWHG EXJV VORZ SURJUHVV

FDXVHV WKH LQWHUPHGLDWH WLPH VWHSV WR RYHUODS�

GLVRULHQWHG EXJ

­��¼ ���¼ ���¼ ���¼ ���® �� ## y �� ] ��W� �� y � �

)URP OHIW WR ULJKW DUH WLPHV � RI WKH GLVRULHQWHG EXJV WUDMHFWRU\���� � y �¼ �¼ »»»¼ �

2XU ODVW H[DPSOH LV PRUH FRPSOLFDWHG WR GHVFULEH PDWKHPDWLFDOO\� 7KLV NLQG RI GLJLWDO

FUHDWXUH LV OLNH D EXJ LQ WKH VHQVH WKDW LW PRYHV IRUZDUG� +RZHYHU� XQOLNH D W\SLFDO EXJ� WKLV

RQH WXUQV DURXQG DW VRPH SRLQW DQG UHWXUQV WR ZKHUH LW VWDUWHG� :H FDOO WKHVH �VWLIOHG EXJV

2XU IDYRULWH VXFK FUHDWXUH LV QDPHG � 7KH IROORZLQJ ILJXUH GHSLFWV %RVFR DORQJ ZLWK KLV%RVFR

WUDMHFWRU\� ,W WDNHV KLP WLPH VWHSV WR UHWXUQ WR ZKHUH KH VWDUWHG� DQG WZLFH DORQJ WKH ZD\�

KH �EORZV XS� DOORZLQJ KLP WR WXUQ DURXQG� 'HSLFWHG DUH RI KLV SKDVHV� +H EHJLQV KLV�� �

MRXUQH\ FORVH WR WKH FHQWHU RI WKH ILJXUH �LQ EODFN� DQG PRYHV XS DQG WR WKH ULJKW� :KHQ KH

UHDFKHV WKH XSSHU ULJKW FRUQHU RI WKH ILJXUH� KH WXUQV DURXQG DQG PRYHV GRZQ DQG WR WKH OHIW�

$IWHU WKH ILUVW WXUQ� KLV F\FOHV DUH GHSLFWHG LQ JUD\� 2QFH KH UHDFKHV WKH ORZHU OHIW FRUQHU RI

���

WKH ILJXUH� KH EORZV XS DJDLQ WR PDNH KLV ILQDO WXUQ� DQG HQGV ZKHUH KH EHJDQ� �&DQ WKH

UHDGHU ILQG %RVFR DPRQJ WKH SKDVHV GHSLFWHG EHORZ KLP"���

%RVFR

� �­¼ ��¼ �¼ ��¼ �® y � # y �W�

'HSLFWHG DUH WLPHV � ��� � y �¼ �¼ »»»¼ ��

7KLQ %XJV

7KH VWUXFWXUH RI WKH WKLQ EXJV LV HDVLHU WR SDUVH WKDQ WKDW RI WKH IDW EXJV� ,Q IDFW� ZH

KDYH IRXQG VHYHUDO JHRPHWULHV WKDW UHVFDOH OLQHDUO\ LQ WHUPV RI WKH UDQJH� /HW XV GHVFULEH WKHVH

H[DPSOHV�

3URSRVLWLRQ ��»�»� )RU � WKH UXOH � DGPLWV SHULRG EXJV�� � � � � �} � ­ ¼ � ¼ � ¼ � ¼ � ® �/W/

3URRI� :H MXVW QHHG WR VKRZ WKDW WKHUH H[LVWV D ILQLWH FRQILJXUDWLRQ� � DQG D YHFWRU� � VXFK$ #W

WKDW /HW DQG EH SRVLWLYH LQWHJHUV VXFK WKDW DQGWKDW ��

$ y $] ##W » ] y � ] �� � � � �� � � �

� | | »� � $� � /HW EH WKH FRQILJXUDWLRQ FRQVLVWLQJ RI SHUSHQGLFXODU VHJPHQWV� RQH RI OHQJWK

� �� �DQG WKH RWKHU RI OHQJWK � VHSDUDWHG E\ RQH VLWH� 7KDW LV�

  ¡

¢

�

�

�

�£

$

���

7KHQ� � ZKHUH 7R VHH WKLV�� $�

$ y �$] # # y � ] � »W W � � RULHQW VR WKDW WKH EHWZHHQ WKH WZR

VHJPHQWV RI VLWV DW WKH RULJLQ� 7KHQ WKH� Z VHW WKDW FRPSULVHV LV$

$ � �y ³­�¼ ^ �®¼ ­�¼ ^ �®¼ »»»¼ ­�¼ ^ ®¼ ­ ^ �¼ �®¼ ­ ^ �¼ �®¼ »»»¼ ­ ^ ¼ �®´� � �

:H FKHFN E\ KDQG WR VHH WKDW DIWHU RQH XSGDWH WKH LPDJH RI XQGHU WKH &$ UXOH ZLOO FRPSULVH$

WKH VHW WKDW LV WKH UHIOHFWLRQ RI WKH DERYH VHW DERXW WKH OLQH DQG WUDQVODWHG E\ WKH YHFWRU�Í

" y ­ ^ ] �®� ] ­ ^ ] �®�W � � � �� � � �� 7KDW LV� LW ZLOO FRPSULVH WKH VHW

³­�¼ ^ �®¼ ­�¼ ^ �®¼ »»»¼ ­�¼ ^ ®¼ ­ ^ �¼ �®¼ ­ ^ �¼ �®¼ »»»¼ ­ ^ ¼ �®´ ] "»W� �� �

7KH FKHFNLQJ ZH DOUHDG\ GLG IRU VKRZV WKDW DIWHU RQH XSGDWH WKH LPDJH RI WKH DERYH VHW ZLOO$

FRPSULVH WKH VHW

³­�¼ ^ �®¼ ­�¼ ^ �®¼ »»»¼ ­�¼ ^ ®¼ ­ ^ �¼ �®¼ ­ ^ �¼ �®¼ »»»¼ ­ ^ ¼ �®´ ] " ] &W W� �� �

ZKHUH DQG WKLV LV& y ­ ^ ] �®� ] ­ ^ ] �®� ] ­� ] � ®»W � � � � $� � � � � � �

3URSRVLWLRQ ��»�»� )RU � WKH /W/ UXOH � DGPLWV� � � � � �} � ­ ¼ � ^ �¼ � ^ �¼ � ^ �¼ � ^ �®

SHULRG EXJV��

3URRI� :H MXVW QHHG WR VKRZ WKDW WKHUH H[LVWV D ILQLWH FRQILJXUDWLRQ� � DQG D YHFWRU� � VXFK$ #W

WKDW /HW DQG EH SRVLWLYH LQWHJHUV VXFK WKDW DQGWKDW ��

$ y $] ##W » ] y � ] �� � � � �� � � �

� | | »� � $� � /HW EH WKH FRQILJXUDWLRQ FRQVLVWLQJ RI SHUSHQGLFXODU VHJPHQWV� RQH RI OHQJWK

� �� �DQG WKH RWKHU RI OHQJWK � VHSDUDWHG E\ RQH VLWH� 7KDW LV�

  ¡

¢

�

�

�

�£

$

:H OHDYH LW DV DQ H[HUFLVH IRU WKH UHDGHU WR VKRZ WKDW ��

$y $] �­� ] � ®» 7KH FKHFNLQJ� �

FDQ EH GRQH E\ KDQG RU ZLWK WKH KHOS RI :LQ&$ ZKLFK FDQ EH GRZQORDGHG IURP >*UL�@� �

���

3URSRVLWLRQ ��»�»� )RU � � WKH /W/ UXOH DGPLWV� � � � � �y �� ­� y �¼ �¼ �¼ »» ® ­ ¼ � ¼ � ¼ � ¼ � ®

SHULRG EXJV� �7KHVH DUH GLVWLQFW IURP WKRVH LQ 3URSRVLWLRQ �� �»�»�»

3URRI� ,Q WKLV FDVH� LV WKH FRQILJXUDWLRQ�$

  ^� ¡   � ¡   ¡� �

� � » » » » � �

� �

� � » » » � �

� � » » » � �

� �

� � � » » » � �

��

��

��

��

# y ­ ] �®� »W � �� �,W LV HDV\ WR FKHFN E\ KDQG �RU XVLQJ WKH FRPSXWHU� WKDW� DIWHU RQH XSGDWH� $

LV

  ^� ¡   ^ � ¡

»»» »»»

»

» »

» »

» »

� �

¢ ¢»

� ^�

£ £

� ��

�� ��

ZKLFK LV GLVSODFHG E\ IURP � $IWHU RQH PRUH XSGDWH� WKH FRQILJXUDWLRQ ZLOO GLVSODFH�� $

DQRWKHU DQG ZLOO FRPSULVH D WUDQVODWH RI � 7KDW LV� � DV GHVLUHG�� $ � $ �� y ] ­ ] �®�� ��

$ �

7KH EXJV IURP 3URSRVLWLRQ DOVR H[LVW IRU WKH H[DFWO\ � /W/ UXOHV� ZLWK�»�»� y �� � � �

RGG �DQG �� 6WDUWLQJ ZLWK WKH JLYHQ DERYH� WKH UXOHV JHQHUDWH D YDULHW\ RI REMHFWV WKDW� ${ �

DUH SHULRGLF PRG WUDQVODWLRQ� )RU VRPH UDQJHV� WKHVH DUH DFWXDOO\ EXJ PDNHUV� ,Q RWKHUV� WKH\

HYROYH LQWR RQH RU PRUH EXJV RI GLIIHUHQW VKDSHV� )RU D OLVW RI VXFK H[DPSOHV� VHH $SSHQGL[

,,,�

7KH &RQQHFWLRQ %HWZHHQ )DW DQG 7KLQ

,Q WHUPV RI WKHLU ORFDWLRQV LQ /W/ VSDFH� WKH WKLQ DQG IDW EXJ UHJLPHV DUH IDU DSDUW� 7KH

TXHVWLRQ WKXV DULVHV� ,V WKHUH D ODUJHU EXJJLQ UHJLPH FRQWDLQLQJ ERWK WKH WKLQ DQG IDW UHJLPHV"

7KDW LV� GRHV WKHUH H[LVW DQ /W/ VXEVSDFH FRQWDLQLQJ WKH WKLQ DQG IDW UHJLPHV DQG VXFK WKDW

HDFK UXOH LQ WKH ELJJHU VXEVSDFH DGPLWV EXJV" 7KH DQVZHU LV \HV� ,Q IDFW� WKHUH LV D IRXU�

GLPHQVLRQDO VXEVSDFH� HYHQ ZKHQ UHVWULFWLQJ WKH VHDUFK WR WKH FURVV VHFWLRQ VXFK WKDW � �� �y

RU /HW XV GHSLFW WKH FROOHFWLRQ RI EXJV DGPLWWHG E\ WKH UDQJH UXOHV ZLWK WKH� �� �y ^ �»

���

UHVWULFWLRQ WKDW � 7KH IROORZLQJ GHSLFWV HDFK EXJ DORQJ ZLWK WKH YDOXH RI IRU� � � �� � � �y y

WKH UXOH WKDW DGPLWWHG LW� ,W EHJLQV ZLWK � ZKLFK LV LQ WKH WKLQ EXJ UHJLPH� DQG HQGV� �� �y y �

ZLWK LQ WKH VXSHU IDW UHJLPH�� �� �y y ��

5DQJH EXJ FROOHFWLRQ

7KH GLDJUDP DERYH JLYHV MXVW RQH F\FOH RI HDFK EXJ� 0DQ\ RI WKHP KDYH ODUJH

SHULRGV� PHDQLQJ WKDW WKH\ KDYH PDQ\ LQWHUPHGLDWH SKDVHV� 7KH ODVW ILYH EXJV SLFWXUHG� ZLWK

SDUDPHWHUV � DUH VXSHU IDW� WKHUH DSSHDUV WR EH D SKDVH WUDQVLWLRQ IURP�� | y | �� �� �� �

WR 7KH UHDVRQ IRU WKLV LV VRPHWKLQJ ZH KDYH \HW WR H[SORUH� 7KH ZDUQLQJ LFRQ DERYH WKH��»

� y y �VHUYHV DV D SODFH KROGHU XQWLO ZH ILQG D EXJJLQ UXOH ZLWK �� �� �

0DQ\ RI WKH UXOHV LQ WKLV VXEVSDFH DGPLW VHYHUDO EXJ YDULHWLHV� 7KDW LV� WKHUH DUH RWKHU

EXJV WKDW PRYH LQ WKH KRUL]RQWDO GLUHFWLRQ� EXW KDYH GLIIHUHQW JHRPHWULHV DQG SHULRGV WKDQ WKH

RQHV LQ RXU SLFWXUH� 7KHUH DUH DOVR EXJV WKDW PRYH LQ WKH GLDJRQDO GLUHFWLRQ DV ZHOO DV

GLVRULHQWHG EXJV� :H FKRVH RQO\ RQH W\SH RI EXJ IURP HDFK UXOH LQ RUGHU WR JLYH D SLFWXUH RI

KRZ WKH EXJV UHVFDOH DV WKH SDUDPHWHU LQFUHDVHV� :H KDYH D PXFK ODUJHU UDQJH� �� �y

FROOHFWLRQ� DORQJ ZLWK WKH UXOHV WKDW DGPLW WKHP� $GGLWLRQDOO\� ZH KDYH EXJV WKDW DUH DGPLWWHG

E\ UXOHV RWKHU WKDQ WKRVH ZLWK � 7KH FROOHFWLRQ FDQ EH GRZQORDGHG IURP >*UL�@� 7R� �� �y

ILQG WKHP� VHDUFK XQGHU �VR�QRQOLQHDU��

,QLWLDOO\� ZH VRXJKW WKH VXEVSDFH RI EXJV LQ UDQJH ILYH EHFDXVH WKDW UDQJH LV ODUJH

HQRXJK WR JLYH D GHFHQW SLFWXUH RI WKH G\QDPLFV H[KLELWHG E\ UXOHV IURP HYHQ ODUJHU UDQJHV�

DQG ZRUNLQJ LQ UDQJH ILYH LV VRPHZKDW UHDVRQDEOH� 1RZ WKDW ZH DUH FRQYLQFHG RI WKH

H[LVWHQFH RI WKH VXEVSDFH RI EXJV IRU ODUJH UDQJHV� ZH ZRXOG OLNH WR XQGHUVWDQG WKHLU

JHRPHWULHV LQ WHUPV RI WKH ODUJHU IRXU�GLPHQVLRQDO VSDFHV LQ ZKLFK WKH\ DUH FRQWDLQHG� 7KLV LV

���

D YHU\ FKDOOHQJLQJ SUREOHP EHFDXVH WKH VXEVSDFHV WKHPVHOYHV DUH IRXU�GLPHQVLRQDO DQG YHU\

ODUJH� :H KDYH PDGH VRPH SURJUHVV RQ WKH UDQJH ILYH YHUVLRQ� +RZHYHU� LW EHFDPH XQZLHOG\

UDWKHU TXLFNO\� +HQFH� ZH WXUQHG WR WKH PRUH UHDOLVWLF HQGHDYRU RI XQGHUVWDQGLQJ WKH UDQJH

WZR VXEVSDFH� :H ZLOO ZRUN XS WR UDQJH ILYH� %HIRUH SUHVHQWLQJ RXU ILQGLQJV� OHW XV EULHIO\

GHVFULEH WKH UDQJH RQH VWUXFWXUH�

,Q UDQJH RQH� WKH /W/ EXJJLQ UXOHV DUH /LIH� RI FRXUVH� EXW DOVR UXOHV ZLWK � �� �y y �

DQG HLWKHU RU 7KH EXJV DGPLWWHG E\ WKH ODWWHU DUH VSHHGLHU WKDQ� � � �� � � �y y � � | | »

/LIHV JOLGHUV� ,Q IDFW� WKH\ WUDYHO DW WKH VSHHG RI OLJKW DQG DUH LQYDULDQW PRG WUDQVODWLRQ�

�7KH\ DUH HDV\ WR JHW VWDUWLQJ IURP UDQGRP LQLWLDO VWDWH� VR ZH OHDYH LW WR WKH UHDGHU WR

GLVFRYHU WKHP RQ KHU RZQ�� ,I ZH UHVWULFW RXU DWWHQWLRQ WR /W/ UXOHV ZLWK SDUDPHWHUV

VDWLVI\LQJ RU � WKHUH DUH WZR UDQJH RQH FDVHV �� �VLQFH /LIH� � � � � �� � � � � �y y ^ � y y �

KDV /W/ SDUDPHWHUV � DQG � � 7KH UDQJH RQH EXJJLQ UXOHV VDWLVI\LQJ­�¼ �¼ �¼ �¼ �® y � y �� �� �

WKH ODWWHU FRQGLWLRQ KDYH OLPLWLQJ VWDWHV WKDW DUH HLWKHU DSHULRGLF RU ORFDOO\ SHULRGLF� 7KXV� /LIH

LV UDQJH RQH H[DPSOH WKDW DGPLWV EXJV DQG KDV D OLPLWLQJ VWDWH WKDW HYHQWXDOO\ UHOD[HV�WKH

�5DQJH RQH EXJV DUH WKH IDPRXV JOLGHUV DV ZHOO DV PDQ\ PRUH YHU\ FRPSOLFDWHG GLJLWDO

FUHDWXUHV� VHH >&DO@ IRU DOO RI WKH ODWHVW�� :H SRLQW RXW WKDW ZH KDYH QRW ORVW DQ\ UHOHYDQW

UDQJH RQH UXOHV E\ UHVWULFWLQJ RXU DWWHQWLRQ VLQFH WKH UXOH DGPLWV EXJV EXW DOVR­�¼ �¼ �¼ �¼ �®

KDV DQ DSHULRGLF OLPLWLQJ VWDWH� ,Q UDQJH WZR� RQ WKH RWKHU KDQG� WKHUH DUH PDQ\ PRUH EXJJLQ

UXOHV ZLWK /LIH�OLNH OLPLWLQJ VWDWHV HYHQ ZKHQ DWWHQWLRQ LV UHVWULFWHG WR /W/ UXOHV ZLWK � �� �y

RU � ,Q IDFW� WKHUH DUH DW OHDVW VXFK UXOHV� ,Q VRPH FDVHV� WKH EXJV DUH GLIILFXOW� �� �y ^ � ��

WR ILQG� LI WKH UXOH LV VWDUWHG IURP SURGXFW PHDVXUH WKH G\QDPLFV EHFRPH ORFDOO\ SHULRGLF UDWKHU

TXLFNO\ ZLWKRXW \LHOGLQJ DQ\ EXJV� 7KH EXJV H[LVW QRQHWKHOHVV� 7KH IROORZLQJ GLDJUDP LV D

SURMHFWLRQ RI WKH WKUHH�GLPHQVLRQDO FURVV VHFWLRQ ZLWK RQWR WKH SODQH� $W� � � �� � � �y ^

HDFK SRLQW VXFK WKDW WKHUH H[LVWV D UXOH ZLWK /LIH�OLNH G\QDPLFV� WKH YDOXH IRU­ ¼ ® y� � � �� � � �

WKH UXOH LV OLVWHG�

���

��£

�

�

��

�� �

�� � �

��

�� � � � � �

�

�

�

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI IRU UXOHV WKDW DUH /LIH�OLNH�� �� �y

7KH VHFRQG SURMHFWLRQ LV VLPLODU EXW OLVWV WKH YDOXHV IRU � ZLWK � IRU UXOHV� � �� � �y ] �

WKDW H[KLELW /LIH�OLNH G\QDPLFV�

��£

�

�

��

�� � � �

��

�� � �

��

�

�

�

�

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI � �ZLWK � IRU UXOHV WKDW DUH /LIH�OLNH�� � �� � �y ] �

7KH WKLUG SURMHFWLRQ OLVWV WKH YDOXHV IRU UXOHV VXFK WKDW� ZKHQ UXQ RQ SURGXFW PHDVXUH

ZLWK DQ DSSURSULDWHO\ FKRVHQ � EXJV HPHUJH LQ WKH ILUVW IHZ WLPH VWHSV� EXW ORVH RXW WR�

DSHULRGLF G\QDPLFV RU D JOREDO WLOLQJ� ,Q VRPH FDVHV� LI WKH GHQVLW\ RI OLYH VLWHV LQ WKH DSHULRGLF

G\QDPLFV LV VPDOO HQRXJK� EXJV �ZDON� EHWZHHQ SDWFKHV RI WKH DSHULRGLFLW\� $QG RI FRXUVH� LI

GHWHUPLQLVWLF LQLWLDO FRQGLWLRQV DUH XVHG WKH EXJV PDQDJH WR SHUVLVW IRU DOO WLPHV�

���

��£ d

�

�

��

�� � �

��

�� � � �

�� �

�

�

� �

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI IRU UXOHV WKDW DUH EXJJLQ DQG KDYH� �� �y

DSHULRGLF OLPLWLQJ VWDWHV� � LQGLFDWHV WKDW WKH HYHQWXDO G\QDPLFV LV D WLOLQJ ZLWK DSHULRGLF LQWHUIDFHV��d

7KH IRXUWK SURMHFWLRQ LV VLPLODU WR WKH WKLUG� EXW OLVWV WKH YDOXHV IRU ZLWK��

� �� �y ] � IRU UXOHV ZKLFK DGPLW EXJV� EXW HYHQWXDOO\ KDYH DSHULRGLF G\QDPLFV�

��£

�

�

�� � � �

��

��

�� �

��

�

�

�

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI �ZLWK � IRU UXOHV WKDW DUH EXJJLQ DQG KDYH� � �� � �y ] �

DSHULRGLF OLPLWLQJ VWDWHV�

7KH ILIWK SURMHFWLRQ FRPELQHV WKH ILUVW DQG WKH WKLUG OLVWLQJ WKH YDOXHV RI IRU� �� �y

UXOHV WKDW DGPLW EXJV UHJDUGOHVV RI WKHLU OLPLWLQJ VWDWHV»

���

��£ d

�

�

��

�� � � �

�� � �

�� � � �

�� � � � � � �

�

� «

«

� �

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI IRU UXOHV WKDW DUH EXJJLQ�� �� �y

7KH VL[WK SURMHFWLRQ FRPELQHV WKH VHFRQG DQG WKH IRXUWK OLVWLQJ WKH YDOXHV RI ��

­ y ] �®� �� � IRU UXOHV WKDW DGPLW EXJV� UHJDUGOHVV RI WKHLU OLPLWLQJ VWDWHV»

��£

�

�

�� � � �

�� � � �

��

�� � � �

��

�

�

�

�

� � � �� �� �� �� �� � �

  ¡��

7KH QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI �ZLWK � IRU UXOHV WKDW� � �� � �y ] �

DUH EXJJLQ�

1RZ OHW XV LOOXVWUDWH WKH SURMHFWLRQ RI WKH HQWLUH VHW RI EXJJLQ UXOHV GHVFULEHG DERYH E\

FRPELQLQJ WKH WK DQG WK SURMHFWLRQV� :H XVH EROG W\SH WR OLVW WKH YDOXHV RI y ^ �� �� �

IURP WKH WK SURMHFWLRQ DQG UHJXODU W\SH IRU WKH YDOXHV RI IURP WKH WK SURMHFWLRQ� y � �� �

���

��£ d

�

�

��

�� � �« �«

�� � �

�� � �« �«

�� � � � � � �

« «

� « «

� ««

««

� �

� � � �� �� �� �� �� � �

� � �

� � �

� � �

�

  ¡��

7KH EROG QXPEHUV LQVLGH WKH WDEOH UHSUHVHQW WKH YDOXHV RI DQG WKH RWKHU QXPEHUV DUH WKH YDOXHV RI� �� �y ^ �

� �� �y IRU UXOHV WKDW DUH EXJJLQ�

3URSRVLWLRQ ��»�»� 7KH VHW� RI EXJJLQ UXOHV ZKRVH SURMHFWLRQ LV GHSLFWHG DERYH� LV: � j�

SDWK FRQQHFWHG�

3URRI� ,W LV HQRXJK WR VKRZ WKDW IRU HDFK DQG LQ WKHUH LV D VHTXHQFH RI �DGMDFHQW�% & :

EXJJLQ SRLQWV ZLWK DQG 7KH WZR SRLQWV% ¼ % ¼ »»»¼ % � : % y % % y &» ­ ¼ ¼ ¼ ®� � � � � � � ��� � � �

DQG DUH DGMDFHQW LIa a a a

­ ¼ ¼ ¼ ®� � � �� � � �

O ^ O | ­� y �¼ �® O ^ O |� � � �� � ��

a a� DQG ��

7KH WZR SRLQWV DQG DUH DGMDFHQW LIa a a a

­ ¼ ¼ ] �¼ ® ­ ¼ ¼ ] �¼ ®� � � � � � � �� � � �� � � �

� � � � � �� � �� � �^ � ³�¼ �´ O ^ O | O ^ O |

a a a� �� DQG ��

,W LV FOHDU IURP WKH WK SURMHFWLRQ WKDW WKH VXEVHW RI ZLWK LV SDWK FRQQHFWHG : y� �� �

SURYLGHG � &DOO WKLV VXEVHW � &DOO WKH VXEVHW RI ZLWK � � | y | � : : y y � :� � � �� � � � � �

DQG WKH VXEVHW ZLWK � DQG DUH ERWK FOHDUO\ SDWK FRQQHFWHG� 'HQRWH WKH� �� � � � �y y : » : :

VXEVHW LQ WKH WK SURMHFWLRQ ZLWK E\ � WKH VXEVHW ZLWK y ^ � y � :� �� � �

| y ^ � | ­ � ® : y ^ � y ­ y ® :� � � � � �� � � � � � E\ � WKH VXEVHW ZLWK E\ � WKH

VXEVHW ZLWK E\ � DQG WKH VXEVHW ZLWK E\ � 7KHVH� � � �� � � � � �y ^ � y � : y ^ � y � :

GLVWLQFW VXEVHWV DUH DOO SDWK FRQQHFWHG� 7R VKRZ WKDW LV SDWK FRQQHFWHG ZH QHHG RQO\ VKRZ:

���

WKDW WKHUH LV D SDWK FRQQHFWLQJ HDFK RI WKH VXEVHWV LQ SURMHFWLRQ WR DQG WKDW WKHUH DUH :�

SDWKV FRQQHFWLQJ DQG IURP SURMHFWLRQ WR � 7KH SDWK RI DGMDFHQW SRLQWV JLYHQ E\: : :� � �

­�¼ �¼ �¼ �® ¡ ­�¼ �¼ ¼ ® ¡ ­¼ ¼ ¼ ® ¡ ­¼ ¼ ¼ �® ¡ ­¼ ¼ ¼ �®

VKRZV WKDW HDFK RI � � DQG LV SDWK FRQQHFWHG WR � 7KH SDWK RI DGMDFHQW SRLQWV JLYHQ: : : :� � �

E\

��������� ���������� ���������� ����������� ����������� �����������¡ ¡ ¡ ¡ ¡ ¡

��� ������� ������������ ¡

VKRZV WKDW HDFK RI � � � DQG LV SDWK FRQQHFWHG WR: : : : : » � � � � �

:H GR QRW EHOLHYH WKDW ZH KDYH IRXQG UDQJH EXJJLQ UXOHV� QRU GR ZH EHOLHYH WKDWDOO �

DOO VXFK UXOHV DUH SDWK FRQQHFWHG WR � ,Q IDFW� WKH UXOH LV EXJJLQ� KRZHYHU� ZH: ­�¼ ¼ �¼ �¼ �®

KDYH QRW IRXQG D SDWK RI DGMDFHQW EXJJLQ UXOHV IURP LW WR � 3HUKDSV WKHUH LV DQRWKHU SDWK:

FRQQHFWHG EXJJLQ VHW WKDW FRQWDLQV WKH SRLQW :H KDYH \HW WR H[SORUH VXFK­�¼ ¼ �¼ �¼ �®»

SRVVLELOLWLHV� +RZHYHU� ZH FRQMHFWXUH WKDW IRU DOO UDQJHV WKHUH H[LVWV D SDWK FRQQHFWHG� } �

VHW RI EXJJLQ UXOHV WKDW FRQWDLQV WKH WKLQ DQG IDW EXJ UHJLPHV�

$V ZH KDYH VKRZQ� WKH UDQJH WZR VXEVSDFH RI EXJJLQ UXOHV LV PXFK ODUJHU WKDQ WKH

UDQJH RQH H[DPSOH� DQG HPSLULFDO GDWD VXJJHVWV WKDW WKH UDQJH ILYH VXEVSDFH RI EXJJLQ UXOHV LV

ODUJHU \HW� :H KDYH VKRZQ WKDW WKHUH H[LVW WKLQ EXJV IRU DOO UDQJHV� &RQYLQFLQJ HPSLULFDO GDWD

VKRZV WKDW� LQ HYHU\ UDQJH� WKHUH H[LVW IDW EXJV QHDU WKH UHVFDOLQJ RI /LIH� :H FRQMHFWXUH WKDW

WKH SDWK FRQQHFWHG VHW RI EXJJLQ UXOHV LQFUHDVHV LQ VL]H DV WKH UDQJH LQFUHDVHV� :H ZDQW WR

H[SORUH WKH SRVVLELOLW\ WKDW WKHUH H[LVWV D IRXU�GLPHQVLRQDO (XFOLGHDQ VXEVSDFH RI EXJJLQ UXOHV

LQ WKH XQLW K\SHUFXEH� ,I VXFK D VXEVSDFH H[LVWV� WKHQ SHUKDSV IURP UDQJH RQH /LIH JLYHV XV D

JOLPSVH RI LW� $V WKH UDQJH LQFUHDVHV� SHUKDSV WKH VHW RI EXJJLQ UXOHV VKRZV XV PRUH DQG

PRUH SRLQWV LQ WKH VXEVSDFH� ,I WKLV LV WUXH� LQFUHDVLQJ WKH UDQJH LV DQDORJRXV WR LQFUHDVLQJ WKH

QXPEHU RI UHFWDQJOHV LQ D 5LHPDQQLDQ DSSUR[LPDWLRQ RI DQ LQWHJUDO� $V WKH UDQJH JRHV WR �B

WKH VXEVSDFH RI EXJJLQ UXOHV JRHV WR D (XFOLGHDQ VXEVSDFH RI EXJJLQ UXOHV� &OHDUO\� WKH

VWUXFWXUH RI KLJKHU UDQJH VXEVSDFHV RI EXJJLQ UXOHV LV DQ HVVHQWLDO SDUW RI JHWWLQJ FORVHU WR

NQRZLQJ ZKHWKHU WKH DERYH LV SRVVLEOH�

�»� 5XOHV QHDU EXJJLQ� WKH WUDQVLWLRQ IURP SHULRGLFLW\ WR DSHULRGLFLW\

���

,Q WKLV VHFWLRQ ZH GHVFULEH WKH SKDVH ERXQGDU\ RQ ZKLFK WKH VHW RI EXJJLQ UXOHV OLHV�

7KH IRXU�GLPHQVLRQDOLW\ RI /W/ VSDFH SURYLGHV D FKDOOHQJH LQ SUHVHQWLQJ DQ DFFXUDWH SLFWXUH RI

WKH SKDVH ERXQGDU\� +RZHYHU� LI ZH UHVWULFW RXU DWWHQWLRQ WR WKH FURVV VHFWLRQV ZH GHVFULEHG

LQ 6HFWLRQ � ZH DUH DEOH WR REWDLQ D UHDVRQDEOH SLFWXUH� 7R LOOXVWUDWH WKLV� OHW XV ORRN FORVHO\�»�

DW WKH WZR�GLPHQVLRQDO FURVV VHFWLRQ RI EXJJLQ UXOHV ZLWK :H ZLOO QHHG WKH� �� �y y �»

IROORZLQJ GHILQLWLRQV�

Î 6D\ WKDW D UXOH LI LW LV QRW EXJJLQ DQG LWV OLPLWLQJ VWDWH FRQVLVWV RQO\ RIIL[DWHV LQ LVODQGV

GLVMRLQW ILQLWH VWLOO OLIHV�

/HW XV LOOXVWUDWH D OLPLWLQJ VWDWH WKDW IL[DWHV LQ LVODQGV ZLWK WKH H[DPSOH ­�¼ �¼ �¼ �¼ ®»

%\ WLPH DOO G\QDPLFV KDYH VWRSSHG VR WKDW RQO\ ILQLWH GLVMRLQW VWLOO OLIHV UHPDLQ� 7KXV� LI ZH�

ZHUH WR UXQ WKH UXOH RQ WKH IROORZLQJ ILJXUH QR FKDQJHV ZRXOG WDNH SODFH� 7KLV PHDQV WKDW WKH

IROORZLQJ ILJXUH ZRXOG EH WKH VWDWH RI WKH V\VWHP IRU DOO WLPHV DIWHU �» 7KH QH[W ILJXUH DV

ZHOO DV WKH ILJXUHV ZH XVH WR LOOXVWUDWH WKH UHPDLQLQJ GHILQLWLRQV ZHUH JHQHUDWHG IURP SURGXFW

PHDVXUH ZLWK GHQVLW\ RQ D ODWWLFH ZLWK SHULRGLF ERXQGDU\ FRQGLWLRQV�« � _ ��� �

WLPH �

Î 6D\ WKDW D UXOH LI LW LV QRW EXJJLQ DQG LWV OLPLWLQJ VWDWH FRQVLVWV RQO\ RIRVFLOODWHV LQ LVODQGV

EOLQNHUV� ILQLWH VWLOO OLIHV� DQG SHULRGLF REMHFWV�

���

/HW XV LOOXVWUDWH WKLV OLPLWLQJ VWDWH ZLWK WKH H[DPSOH � %\ WLPH HDFK­�¼ �¼ ��¼ �¼ » ��

VLWH LV D SDUW RI D ILQLWH VWLOO OLIH RU D SHULRGLF REMHFW� 7R LOOXVWUDWH WKH SHULRGLFLW\ RI VRPH RI WKH

UHJLRQV ZH GHSLFW WLPH DV ZHOO� ,Q WKH IROORZLQJ ILJXUHV VRPH RI WKH REMHFWV DUH SHULRG ��� �

VRPH DUH SHULRG � DQG RWKHUV DUH IL[HG �VR SHULRG � %\ ORRNLQJ DW WKH ILJXUHV IURP WLPHV� � » ��

DQG FDQ WKH UHDGHU LGHQWLI\ ZKLFK DUH SHULRGLF DQG ZKLFK DUH IL[HG"��

WLPH ��

���

WLPH ��

Î 6D\ WKDW D UXOHV OLPLWLQJ VWDWH LV D LI LW FRQVLVWV RI D IL[HG RU ORFDOO\ SHULRGLFJOREDO WLOLQJ

SDWWHUQ WKDW RFFXSLHV DOPRVW WKH HQWLUH ODWWLFH� 0DQ\ /W/ UXOHV KDYH JOREDO WLOLQJV WKH

PDMRULWLHV RI ZKLFK FRQVLVW RI KRUL]RQWDO DQG YHUWLFDO UHJLRQV RI DOWHUQDWLQJ VWULSHV RI � Z

DQG � :H LOOXVWUDWH VXFK D WLOLQJ LQ 6HFWLRQ �� �»�Z

Î 6D\ WKDW D UXOH KDV G\QDPLFV LIZDWHUEHG D JOREDO WLOLQJ DOPRVW FRYHUV WKH ODWWLFH EXW

UHJLRQV EHWZHHQ WKH WLOLQJ HLWKHU UHPDLQ XQVWDEOH RU IL[DWH FDXVLQJ RWKHU UHJLRQV WKDW KDG

EHHQ IL[HG WR EHFRPH XQVWDEOH� 7KH G\QDPLFV UHPLQG RQH RI SXWWLQJ WRJHWKHU DOPRVW DOO RI

WKH SLHFHV RI D SX]]OH DQG ILQGLQJ WKDW WKH ODVW IHZ SLHFHV QHFHVVDU\ IRU FRPSOHWLRQ IRUFH

RQH WR UHPRYH VRPH RI WKH SLHFHV DOUHDG\ LQ SODFH� 7KLV SURFHVV FRQWLQXHV DG LQILQLWXP VR

WKDW WKH SX]]OH FDQ QHYHU EH FRPSOHWHG� 7KH G\QDPLFV DUH VR QDPHG EHFDXVH WKH DSHULRGLF

UHJLRQV UHPLQG RQH RI WKH EXEEOHV WKDW FDQ EH VHHQ PRYLQJ DURXQG LQVLGH WKH PDWWUHVV RI D

ZDWHUEHG� 7KH OLPLWLQJ VWDWH VHHPV QHLWKHU WR IL[DWH QRU WR EHFRPH HQWLUHO\ DSHULRGLF�

+RZHYHU� LW LV XQFOHDU ZKHWKHU WKHVH G\QDPLFV DUH D PHWDVWDEOH VWDWH DQG PLJKW HYHQWXDOO\

VHWWOH GRZQ� ,W LV SRVVLEOH WKDW WKH G\QDPLFV RI WKH LQILQLWH V\VWHP VHWWOH GRZQ� MXVW QRW LQ

RXU OLIHWLPHV� $QRWKHU SRVVLELOLW\ LV WKDW WKHUH LV D WRSRORJLFDO GHIRUPLW\ LQ WKH IL[HG

���

UHJLRQV� 7KDW LV� SHUKDSV WKH JHRPHWU\ UHTXLUHG IRU IL[DWLRQ LV LPSRVVLEOH IRU WKH UXOH WR

DWWDLQ VR WKDW WKH SX]]OH DQDORJ\ PDGH DERYH LV DFWXDOO\ ZKDW LV WDNLQJ SODFH�

/HW XV LOOXVWUDWH ZDWHUEHG G\QDPLFV ZLWK WKH H[DPSOH $W WLPH­�¼ �¼ �¼ �¼ ��®» �

SDWWHUQHG UHJLRQV KDYH VWDUWHG JURZLQJ DQG DUH ODUJHU E\ WLPH � 7KH\ KDYH ILOOHG LQ WKH��

ODWWLFH E\ WLPH � KRZHYHU� LVRODWHG SRFNHWV EHWZHHQ WKH IL[HG UHJLRQV UHPDLQ XQVWDEOH���

7KLV FRQWLQXHV WR EH WKH FDVH DW WLPH �����

WLPH �

���

WLPH ��

WLPH ��

���

WLPH ����

$V FDQ EH VHHQ LQ WKH IROORZLQJ GLDJUDP� LI D /LIH�OLNH UXOHV YDOXH LV GHFUHDVHG WKH��

QHZ UXOH ZLOO HLWKHU IL[DWH RU RVFLOODWH LQ LVODQGV� 2Q WKH RWKHU KDQG� LI WKH YDOXH RI LV��

LQFUHDVHG E\ MXVW WKH UXOH REWDLQHG ZLOO KDYH ZDWHUEHG G\QDPLFV� RU ZLOO EH DSHULRGLF �QRWH�

WKDW WKH EXJJLQ UXOHV LQ WKH GLDJUDP WKDW DUH QRW /LIH�OLNH KDYH OLPLWLQJ VWDWHV WKDW DUH

DSHULRGLF�� ,QFUHDVLQJ WKH YDOXH RI VOLJKWO\ PRUH \LHOGV UXOHV ZLWK OLPLWLQJ VWDWHV WKDW DUH��

ORFDOO\ SHULRGLF JOREDO WLOLQJV� ,I WKH YDOXH RI LV LQFUHDVHG HYHQ PRUH WKH UXOHV KDYH OLPLWLQJ��

VWDWHV WKDW DUH IL[HG JOREDO WLOLQJV� 2Q WKH RWKHU KDQG� LQFUHDVLQJ WKH YDOXH RI D /LIH�OLNH��

UXOH \LHOGV D UXOH ZLWK DSHULRGLF G\QDPLFV� ,QFUHDVLQJ WKH YDOXH HYHQ PRUH \LHOGV ORFDOO\��

SHULRGLF G\QDPLFV�

���

5DQJH /W/ FURVV VHFWLRQ ZLWK � LQGLFDWHV ZDWHUEHG G\QDPLFV�� y y �� �� �

7KH JHRPHWU\ LQ WHUPV RI WKH OLPLWLQJ VWDWHV RI WKH UXOHV RI WKH FURVV VHFWLRQ GHSLFWHG

DERYH LV QRW XQLTXH WR WKH EXJJLQ FURVV VHFWLRQ ZLWK ,Q IDFW LW LV UDWKHU W\SLFDO� �� �y y �»

WKDW UHJLRQV RI /LIH�OLNH DQG EXJJLQ UXOHV OLH EHWZHHQ UHJLRQV FRQVLVWLQJ RI ORFDOO\ SHULRGLF RU

IL[HG UHJLRQV DQG WKRVH WKDW DUH DSHULRGLF� 7KH H[DFW VKDSHV RI WKH YDULRXV UHJLRQV YDU\ IRU

GLIIHUHQW FURVV VHFWLRQV� KRZHYHU� VLPLODU G\QDPLFV FDQ EH IRXQG QHDU PRVW /LIH�OLNH UXOHV ZLWK

� � � �� � � �y y ^ �RU � 7KXV� WKH VHW RI /LIH�OLNH UXOHV OLHV RQ D SKDVH ERXQGDU\ EHWZHHQ

UHJLPHV DGPLWWLQJ DSHULRGLF G\QDPLFV DQG WKRVH ZLWK IL[HG RU SHULRGLF G\QDPLFV�