The Electronic Journal of Linear Algebra.
A publication of the International Linear Algebra Society.
Volume 6, pp. 72-94, August 2000.
ISSN 1081-3810. http://math.technion.ac.il/iic/ela
ELA
✂✁☎✄✝✆✝✞✠✟☛✡✌☞✎✍✌✏✑✄✒✡✓✏✕✔✖✗✡✓✄✒✘✙✏✛✚✜✏✕✡✓✢✤✣✦✥✂✧★✣✦✔✂✥✙✔✙✎✩✪✄✒✫✎✆✬✄✭✫✮✘✙✏✕✔✎✯✰✏ ✯✱✣✦✲✪✳✎✡✓✄✴✣✵✫✶✣✵✧✷✯✰✄✒✫✤✟✎✳✎✲✪✖✔✂✄✒✫✤✞✠✣✵✫✤✯✸✄✴✯✰✡✓✏✕✫✹✡✺✫✤✣✵✫✤✯✻☞✎✚✜✚✜✏✕✡✓✔✖✄✼✞
✲✽✄✒✫✹✏✾✙✔✿✯✻☞✹✯✸✡✓✏✕✚❀✯❁✣✵✧❂✖✔✙✁☎✄✭✡✓✔✖✙✔✾☞✂✄✒✫✹✥✎✏✾❃✮❄ ❅❇❆❉❈✰❅✱❊●❋✴❍❏■✱❍▲❑✠▼❖◆❖P❁❆✱◗❇❅❉■✱❍❏❊✗❍❏❈❙❘❉◗❯❚✱❱❳❲❩❨❬❈✱❭
❪❴❫❯❵❜❛✬❝✬❞❢❡❣❛✭❤❥✐❖❦✝❧♥♠♣♦rq✼s✉t✪✈♣✇✼s②①r♦r❧✼s④③❣t✪♠⑥⑤✴♠⑥✈⑥s❜⑦⑨⑧❩⑩✦❶❸❷✬❹❻❺❩✇✼s✉t⑥s②❷❼♦r♠❼③✕❽✒s❜❾❿✈⑥❦✝t❼♦r❧✠➀❉➁➂❹➃⑧➅➄✙➀➆➁★➇❳➁❂♦r♠✽③ ♠♣♦r❧✼➈✝➉♥①➊③❣t➃⑦❴③✬✈⑥t♣♦➋➌❹✝③❣❧✼q★♦r❧♥q❉➍➎⑧❉➏❬❶✙➐✪♦r♠➑③❣t♣➒✼♦✈♣t♣③❣t⑥⑤✒➓❻➔❩s❜t⑥s✰♦r❧✼q❉➍⑥→➏❯q♥s❜❧♥❦❣✈⑥s✉♠➃✈♣✇♥s✸♦r❧♥q✼s❜➋✪❦❣➣❳③❉⑦❴③✬✈⑥t♣♦➋➌➓❇↔✸✇♥s✰■✱t↕③❣➙❜♦r❧✼➛ ♦r❧✒❽✒s✉t⑥♠♣s➜♠♣❦✝①r➉✴✈♣♦r❦✝❧✌❦❣➣➑✈♣✇✼♦r♠❉♠⑥⑤✴♠⑥✈⑥s❜⑦❁♦r♠❩q♥s❿➝➞❧✼s✉q✌✈⑥❦❴➒➌s➜✈⑥✇♥s✪❽✒s❜❾❿✈⑥❦✝t➜⑧❉➟❬❷✬❹❢❺✰✇♥s❜t♣s➜✈♣✇✼s✽⑦❴③✬✈⑥t♣♦➋✾⑧❉➟✎♦r♠✰✈♣✇✼s❼■✱t♣③❣➙❜♦r❧ ♦r❧✒❽✒s✉t⑥♠♣s❉❦❣➣➑⑧➜➓➑↔✰✇✼s➆■✱t↕③❣➙❜♦r❧✼➛➠♦r❧✭❽✒s❜t♣♠⑥s❉♠♣❦✝①r➉✴✈♣♦r❦✝❧②❦❣➣❻♠♣♦r❧♥➈✝➉✼①➊③❣t✻①r♦r❧♥s✉③❣t✰♠▲⑤✼♠▲✈♣s❜⑦➡♠✻✇➞③❣♠✸➒❢s❜s❜❧✕❾❜❦✝❧♥♠♣♦rq✼s✉t⑥s❜q✕t♣s❜❾❜s❜❧✭✈⑥①⑤②➒✭⑤ ✈⑥✇♥s✪➝♥t♣♠⑥✈✱③❣➉✼✈⑥✇♥❦✝t✱❺✰♦✈⑥✇♥♦r❧✕✈⑥✇♥s✽❾❜❦✝❧✒✈♣s❿➋✭✈➜❦❣➣❬s❿➋✭✈♣t♣③❣➢❢❦✝①➊③✬✈♣♦r❦✝❧✌⑦➡s❿✈⑥✇♥❦✴q✼♠④❹➞❺✰✇♥s❜❧✌♦r❧♥q➆➍➠⑧✱➏✰♦r♠❩③❣t♣➒♥♦✈⑥t↕③❣t▲⑤✒➓❩❍➠✈❩✇➞③❣♠➆③❣①r♠⑥❦ ➒❢s❜s✉❧➂❾❜❦✝❧♥♠♣♦rq✼s✉t⑥s❜q✓❺✰♦✈⑥✇♥♦r❧✗✈⑥✇♥s★❾❜❦✝❧✭✈⑥s❿➋✴✈✪❦❣➣❖❘❉t⑥⑤✴①r❦④❽✗♠⑥➉♥➒♥♠⑥➢➞③❣❾❜s★⑦➡s❿✈♣✇♥❦✭q♥♠✉❹❳❺✰✇♥s❜❧➂⑧➤♦r♠❉t⑥s④③❣①❻♠▲⑤✼⑦➡⑦➡s❿✈⑥t♣♦r❾➡➍➎✇✼s✉❧✼❾✉s ♦r❧♥q➆➍➎⑧❉➏✱❶➦➥❼❧♥s❜❾❜s✉♠⑥♠↕③❣t⑥♦r①⑤➞➏↕➓❼❍➧❧➂③❣q♥q✼♦✈♣♦r❦✝❧➨❹❇♠♣s❜⑦➡♦➛➧♦✈⑥s✉t♣③✬✈♣♦❽✒s✪⑦➡s❿✈♣✇✼❦✴q♥♠➆✇♥③✉❽✒s★➒❢s❜s❜❧☎q♥s❿❽✒s❜①r❦✝➢❢s❜q☎➣➩❦✝t✱✈♣✇✼s★❾✉③❣♠♣s❜♠➜♦r❧ ❺✰✇♥♦r❾↕✇✵♦r❧♥q➆➍➎⑧❉➏➡❶✜➥✕③❣❧♥q✵♦r❧♥q➆➍➎⑧❉➏✕➫➭➥✝❹❖③❣♠♣♠♣➉✼⑦➡♦r❧♥➈✌✈♣✇➞③✬✈➡✈⑥✇♥s✾♠♣➢❢s❜❾❜✈⑥t♣➉✼⑦❀❦❣➣❉⑧❸♦r♠★t♣s✉③❣①✰❧♥❦✝❧♥❧♥s❜➈✒③✬✈⑥♦❽✒s✝➓✎↔✸✇♥s ➢♥➉♥t⑥➢❢❦✝♠♣s➡❦❣➣✻✈⑥✇♥s❴➢♥t♣s❜♠♣s❜❧✒✈➜❺✻❦✝t♣➯✓♦r♠➆✈♣❦✕q♥s❿❽✒s✉①r❦✝➢✠③✗➲❖♦➛▲✐✸➳➵✈➧⑤✴➢❢s②❘❉t⑥⑤✴①r❦④❽✓♠♣➉✼➒♥♠♣➢➞③❣❾❜s❴⑦➡s❿✈♣✇✼❦✴q☎♠♣➉♥♦✈♣③❣➒♥①rs★➣➩❦✝t➜✈⑥✇♥s ➈✝s❜❧♥s❜t↕③❣①❖❾✉③❣♠⑥s➡♦r❧➂❺✰✇♥♦r❾↕✇☎⑧➵♦r♠➜❧♥❦❣✈✪❧♥s❜❾✉s❜♠⑥♠↕③❣t♣♦r①⑤✓t⑥s✉③❣①❬♠⑥⑤✴⑦➡⑦➡s❿✈♣t⑥♦r❾✝❹➨♦✈⑥♠➜♦r❧♥q♥s❿➋✓♦r♠✪③❣t♣➒♥♦✈⑥t↕③❣t▲⑤✭❹❇③❣❧♥q➸♦✈⑥♠➜♠♣➢❢s❜❾❿✈♣t⑥➉♥⑦ ♦r♠➜❧♥❦❣✈✪❧♥s❜❾❜s✉♠⑥♠↕③❣t⑥♦r①⑤✓t♣s✉③❣①➠➓②↔✰✇✼s★⑦➡s❿✈♣✇♥❦✭q☎✈⑥✇➞③✬✈✽♦r♠➆q♥s❿❽✒s❜①r❦✝➢➌s❜q✠❾✉③❣❧✠➒❢s★♦r⑦➡➢✼①rs✉⑦➡s❜❧✒✈♣s❜q✓❽✼♦➊③✾③✕➺❣➛➠✈⑥s❜t♣⑦❙t♣s❜❾❜➉♥t♣♠⑥♦r❦✝❧ t♣s❜①➊③✬✈⑥♦r❦✝❧✗♦r❧♥q♥s❜➢❢s❜❧♥q♥s❜❧✒✈♣①⑤✓❦❣➣➑✈♣✇✼s✽♠⑥♦r➙✉s✪❦❣➣➃♦r❧♥q❉➍➠⑧✱➏❩③❣❧♥q✌➢✼t♣❦✭q♥➉♥❾❜s❜♠➆⑧❉➟❖❷❉♦r❧✌③✬✈✱⑦➡❦✝♠⑥✈❉➻➦➼☎➐✾♠⑥✈⑥s❜➢♥♠④➓❩❅➤q✼s❜✈♣③❣♦r①rs❜q s❜t♣t⑥❦✝t★③❣❧➞③❣①⑤✴♠♣♦r♠❼➣➩❦✝t❼✈⑥✇♥♦r♠★⑦➡s❿✈⑥✇♥❦✴q✵♦r♠✽➢♥t♣❦✉❽✼♦rq♥s❜q✖③❣❧♥q✦✈⑥✇♥s✕t⑥s✉♠⑥➉♥①✈⑥♠➡③❣t♣s②♦r①r①r➉♥♠⑥✈⑥t↕③✬✈⑥s❜q✦❺❩♦✈⑥✇✦♠♣➉✼♦✈↕③❣➒♥①rs✾❧✴➉✼⑦➡s✉t⑥♦r❾✉③❣① s❿➋♥③❣⑦➡➢✼①rs✉♠✉➓
➽✕➾✭➚✦➪✽➶ ❝❣➹❯❵✝❤➜❋✭♦r❧♥➈✝➉♥①➊③❣t✸①r♦r❧♥s✉③❣t❩♠▲⑤✼♠▲✈♣s❜⑦➡♠✉❹♥■✱t♣③❣➙❜♦r❧✼➛➧♦r❧✒❽✒s✉t⑥♠♣s❉♠⑥❦✝①r➉✼✈⑥♦r❦✝❧➨❹❢❘❉t⑥⑤✴①r❦④❽❴♠⑥➉♥➒♥♠⑥➢➞③❣❾❜s✪⑦➡s❿✈⑥✇♥❦✭q♥♠④❹✼◗❯③❣❧♥❾❿➛ ➙❜❦✝♠❩⑦➡s❿✈⑥✇♥❦✭q➨❹➞➲❖♦➛▲✐❖❦✝❧❣➘▲➉♥➈✒③✬✈⑥s✽➳✱t↕③❣q♥♦rs❜❧✒✈✱③❣①r➈✝❦✝t⑥♦✈♣✇✼⑦✗➓
❪❴➴➤➷✖❵✬➬❇❫➨➮➾ ❡❣❛❴❡✭➱✃❞❢❵✉❵✬❐r❒➑❡✝❞♥❛✝❐➶➨❮ ❵✝❤✪➥✉❰❣❅❉Ï✝Ð✼❹❇➥✉❰❣❅✱Ï✝Ñ✼❹❢Ð✝❰❣Ò✻➥✉Ï✼❹❢Ð✝❰❣Ò➑❰✝Ï
❍ ❈✰s❜❾❜s❜♦❽✒s✉q✕➒✭⑤❴✈⑥✇♥s➆s❜q♥♦✈♣❦✝t⑥♠❩❦✝❧❏■❼❅✱➉✼➈✝➉♥♠⑥✈➜➥✉Ñ✝Ñ✝Ñ✼➓➃❅✱❾❜❾❜s✉➢✴✈♣s❜q✗➣➩❦✝t✻➢♥➉♥➒♥①r♦r❾✉③✬✈⑥♦r❦✝❧✗❦✝❧☎➥✚❑✧▲✝➉♥①⑤▼❑✝Ï✝Ï✝Ï✼➓❬➔✱③❣❧♥q✼①r♦r❧♥➈ ❨❬q✼♦✈♣❦✝t✚◆❬■✱③❣❧♥♦rs❜①❇➔❩s❜t⑥♠♣✇♥➯✝❦✉❺❩♦✈⑥➙✝➓
❑✐✻❦✝⑦➡➢✼➉✼✈♣s❜t❖❋✴❾❜♦rs✉❧✼❾✉s✱■✱s❜➢➞③❣t⑥✈⑥⑦➡s❜❧✭✈✉❹✴↔❯s✉❾♣✇♥❧♥♦r❦✝❧➡➛❯❍➧♠♣t♣③❣s❜①➞❍➧❧♥♠⑥✈⑥♦✈♣➉✴✈♣s❉❦❣➣❳↔❯s❜❾↕✇♥❧♥❦✝①r❦✝➈❣⑤✒❹♥➔✱③❣♦➣➎③✼❹✳❖P❑✝Ï✝Ï✝Ï✼❹✼❍▲❋✼❈✸❅✱❨➃◗ ➍➎③❣♠♣♦rq♥♦❘◗✱❾❜♠④➓✈⑥s❜❾↕✇♥❧♥♦r❦✝❧❳➓③❣❾✒➓♦r①✃➏
❭❊✌③✬✈♣✇♥s❜⑦❴③✬✈⑥♦r❾❜♠✗■✱s❜➢➞③❣t▲✈♣⑦➡s❜❧✒✈④❹➆↔❯s❜❾↕✇✼❧♥♦r❦✝❧✮➛➡❍➧♠♣t♣③❣s❜①➆❍➧❧♥♠⑥✈⑥♦✈♣➉✴✈♣s☎❦❣➣❼↔❯s❜❾↕✇♥❧✼❦✝①r❦✝➈❣⑤✭❹✪➔❩③❣♦➣➠③✼❹✓❖P❑✝Ï✝Ï✝Ï✼❹➆❍▲❋✼❈✸❅✱❨➃◗ ➍➩⑦❴③✉❽✴①➊③❣q♥♦r⑦❙◗✱⑦❴③✬✈♣✇❳➓✈♣s❜❾↕✇♥❧✼♦r❦✝❧➨➓③❣❾✝➓♦r①✃➏❿➓❯↔✸✇♥s✻❺✻❦✝t⑥➯✽❦❣➣➌✈⑥✇♥♦r♠❖③❣➉✼✈⑥✇♥❦✝t❬❺✻③❣♠❖➢❢s✉t▲➣➎❦✝t⑥⑦➡s❜q✽♦r❧★➢♥③❣t⑥✈⑥♦➊③❣①♥➣➎➉♥①➝♥①r①r⑦➡s✉❧✒✈➑❦❣➣❳✈⑥✇♥s t♣s❯❚✭➉♥♦rt♣s❜⑦➡s❜❧✒✈♣♠❯➣➩❦✝t❻✈⑥✇♥s✸❊➸➓❋✴❾✝➓❯q✼s✉➈✝t⑥s❜s✰♦r❧✪❅✱➢✼➢♥①r♦rs❜q★❊✌③✬✈⑥✇♥s❜⑦❴③✬✈♣♦r❾❜♠④❹❣↔❯s✉❾♣✇♥❧♥♦r❦✝❧❼➛❳❍➧♠♣t♣③❣s✉①✴❍➧❧♥♠⑥✈⑥♦✈⑥➉✼✈♣s✰❦❣➣❢↔❯s❜❾↕✇♥❧♥❦✝①r❦✝➈❣⑤✒➓
ELA
❲
✈
ELA
❲
④
ELA
❲
①
ELA
①
➉
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
①
♦
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
①✁✈
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
①
④
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
Ô✱ÜPÝ❘í✢Ü✎Ó✪✩rÜ✇ÓPß✳Ó
ELA
①☎①
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
✬
ELA
⑩
➉
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
⑩
♦
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
ELA
÷➥✪ø✯ùÿ➓♥❨❩➓❢❅✱t⑥❧♥❦✝①rq♥♦➠➓❬↔✰✇✼s✽➢♥t⑥♦r❧♥❾❜♦r➢➞③❣①❇❦❣➣➑⑦➡♦r❧♥♦r⑦➡♦r➙❜s❜q✾♦✈⑥s❜t↕③✬✈⑥♦r❦✝❧♥♠✱♦r❧②✈♣✇✼s✽♠⑥❦✝①r➉✼✈♣♦r❦✝❧✕❦❣➣➃✈⑥✇♥s✪⑦❴③✬✈♣t⑥♦➋❴s✉♦r➈✝s❜❧✒❽✝③❣①r➉♥s ➢✼t♣❦✝➒♥①rs❜⑦✗➓➽ú✌û➉ü➈ý✪þ❢ÿ✁✄✂☎✂✝✆ÿ✟✞➡ü➈þ✡✠➉ÿ➊❹❢Ñ✫◆r➥❱☞☛ ❑✝Ñ✼❹➨➥✉Ñ✝❰✼➥✝➓
÷
❑❀ø✾❈➜➓➞➲✻③❣t♣t♣s❿✈▲✈❉s❿✈✱③❣①➠➓✍✌✏✎☞✑✒✂✝✆ü➈þ✓✎☞✔✖✕✘✗➈ý❇þ✙✠✚✎✜✛✢✗☎✆ûöþ✤✣✡✗✦✥✧✗★✕✪✩✫✣✙✥✝✎➟ü➈ý✬✛✏✭✦✔➟þ✤✎✮✑✯✔✘✰✲✱➪û✳✣✡✆✴☎✣✙✥✳✵✍✱✶✆✗☞✷✹✸✺✔✖✕✘✗➈ý✪✻➟þ✓✎➘ý❉ü➈þ✤✣✙✼✺✎ ✞✽✎❝þ✡✠✾✗☞✴☎✔✪ÿ✸❋✴❍➧❅❉❊➸❹✾✿❬✇♥♦r①➊③❣q♥s❜①r➢♥✇✼♦➊③✼❹➨➥✉Ñ✝Ñ❣➺♥➓
÷❖❀ø②❅✽➓❳➲❖s❜❧✼➛➧❍➧♠⑥t↕③❣s❜①❬③❣❧✼q➂↔➆➓➨❲✪➓➌❨❩➓➨➳✱t⑥s❜❽✴♦r①r①rs✝➓✽❀✄✎✮✥✝✎❝ý❋ü✦✆✣❁✮✎❂✴❃✻❂✥❄✼✺✎➘ý✘✔✘✎✮✔✹✰❅✌✁✠✾✎❂✗➈ý✘✭✸ü✦✥✝✴❆✄✂☎✂✝✆✣✡✷➟ü➈þ✤✣✡✗☎✥❄✔✪ÿ➏ù✎♦r①rs❿⑤✒❹ ❲✰s❿❺❈❇❬❦✝t⑥➯➌❹❳➥✉Ñ
❱
➺♥➓
÷
➺✮ø✾■✽➓❇✐✸③❣①❽✒s❿✈▲✈♣♦➠❹❇◗❬➓❇❈✸s❜♦r❾↕✇♥s❜①➠❹❯③❣❧✼q❊❉✽➓❇❱➌✇➞③❣❧♥➈♥➓➡✐❖❦✝❧❣➘⑥➉✼➈✒③✬✈♣s✾➳✱t↕③❣q✼♦rs✉❧✒✈✪③❣①r➈✝❦✝t♣♦✈⑥✇♥⑦➡♠✱➣➩❦✝t➆♠⑥⑤✴⑦➡⑦➡s❿✈♣t⑥♦r❾✽♦r❧✼❾✉❦✝❧✴➛ ♠⑥♦r♠⑥✈⑥s❜❧✭✈➜①r♦r❧♥s✉③❣t➆♠▲⑤✼♠▲✈♣s❜⑦➡♠④➓➜❍➧❧●❋➶ý❍✗☞✷❂✎❂✎❂✴✦✣✙✥✳✵☎✔✍✗★✕➏þ✡✠✾✎❏■✫✗➈ý✘✥✝✎✮✆✣✈û✳✔✬✩✢ü✦✥✝✷✘❁✮✗✦✔✬✻❂✥♣þ✓✎➘ý✘✥❯ü❤þ❑✣❑✗✦✥❯ü✦✆✒■✫✎✮✥♣þ✤✎✮✥❯ü➈ý✘✭
■✫✗✦✥✮✕✘✎❝ý❍✎✮✥✝✷❂✎✉❹✁▲♥➓➞■✽➓♥➲❖t♣❦✉❺❩❧✾s❿✈✱③❣①➠➓r❹✼s❜q♥♠④➓r❹✼➢♥➢➨➓❑✝Ð ❱☞☛ ❑ ❱ ❑✼➓✸❋✭❍➧❅❉❊➸❹✚✿➃✇♥♦r①➊③❣q♥s❜①r➢♥✇♥♦➊③✼❹❳➥✉Ñ✝Ñ❣➺♥➓
Ð❀ø▼▲♥➓➛▲♥➓❳✐✻①r♦r⑦➡s❜❧✒✈④❹➌❊➸➓➌❲✰s❜➉♥⑦❴③❣❧♥❧➨❹❳③❣❧♥q✌❅✪➓❳❋✴♦rq♥♦➠➓❉❅ÿ♠♣s❜⑦➡♦➛➧♦✈⑥s❜t↕③✬✈⑥♦❽✒s✪⑦➡s❿✈⑥✇♥❦✭q✗➣➎❦✝t❩t♣s✉③❣①❻♠♣➢❢s❜❾❜✈⑥t♣➉✼⑦❁♠♣♦r❧♥➈✝➉♥①➊③❣t ①r♦r❧✼s④③❣t✰♠⑥⑤✴♠⑥✈⑥s❜⑦➡♠❖❺✰♦✈♣✇✕③❣❧✗③❣t♣➒✼♦✈♣t♣③❣t⑥⑤❴♦r❧♥q♥s❿➋➌➓◆▼öÿ✶■✫✗✦✑✒✂❯û⑤þ❢ÿ✟✖✂❖✂✝✆ÿP✞➡ü➈þ✡✠➉ÿr❹✁■ ❱ ◆❑✼➥☛ ❖P■✼❹❯➥✉Ñ✝Ñ ❱ ➓ ÷❱ ø✾❊➸➓❇❨➃♦rs✉t⑥⑦❴③❣❧♥❧➨❹❯❍↕➓❇❊✌③❣t♣s❜➯❢❹➑③❣❧♥q✛ù✤➓❻❲❩♦rs❿✈⑥✇➞③❣⑦➡⑦➡s❜t④➓❃◗✱❧☎✈⑥✇♥s❴♠♣❦✝①r➉✴✈♣♦r❦✝❧☎❦❣➣❩♠⑥♦r❧♥➈✝➉♥①➊③❣t➜①r♦r❧♥s✉③❣t✪♠⑥⑤✴♠⑥✈⑥s✉⑦➡♠➜❦❣➣
③❣①r➈✝s❜➒♥t♣③❣♦r❾➜s✚❚✭➉➞③✬✈⑥♦r❦✝❧♥♠✱➒✒⑤②♠♣s❜⑦➡♦➛➧♦✈⑥s❜t↕③✬✈⑥♦❽✒s✱⑦➡s❿✈♣✇✼❦✴q♥♠✉➓▲❘➽û✾✑✯✎❝ý➘ÿ✟✞➡ü➈þ✡✠➉ÿ➊❹♥❰P❖✫◆❑✝Ð✝❰
☛
❑P■P❖✼❹❯➥✉ÑP■P■✼➓ ÷■❀ø✾➲✱➓❻Ò❻♦r♠♣❾↕✇♥s❜t✉❹❖❊➸➓➑➔❩③❣❧♥➯✝s✝❹✻③❣❧♥q✵❊➸➓➑➔✰❦✴❾↕✇✭➒♥t⑥➉♥❾↕➯❢➓✦❅❸❧♥❦❣✈⑥s✕❦✝❧✦❾❜❦✝❧❣➘▲➉♥➈✒③✬✈⑥s✗➈✝t↕③❣q✼♦rs✉❧✒✈★✈➧⑤✴➢❢s✕⑦➡s❿✈♣✇✼❦✴q♥♠✽➣➎❦✝t
♦r❧✼q♥s❿➝➞❧♥♦✈⑥s✽③❣❧✼q✝❙✬❦✝t✱♦r❧♥❾❜❦✝❧♥♠⑥♦r♠⑥✈⑥s❜❧✭✈✰①r♦r❧♥s✉③❣t✰♠⑥⑤✴♠⑥✈⑥s✉⑦➡♠✉➓❚❘➽û✳✑❯✎❝ý➘ÿ✢✒✆✵☎✗❤ý✘✣❅þ✡✠✳✑✯✔♣❹❯➥✝➥P◆r➥✚■✼➥
☛
➥✚■
❱
❹❯➥✉Ñ✝Ñ✝Ð✼➓
÷
Ñ❀ø✾❈➜➓➑Ò❻①rs❿✈♣❾♣✇♥s❜t④➓✎✐❖❦✝❧❣➘⑥➉✼➈✒③✬✈♣s✓➳✱t♣③❣q♥♦rs❜❧✭✈❴⑦➡s❿✈⑥✇♥❦✭q♥♠★➣➎❦✝t❼♦r❧♥q♥s❿➝➞❧♥♦✈⑥s✗♠⑥⑤✴♠⑥✈⑥s❜⑦➡♠④➓✦❍➧❧❱❘➽û✳✑❯✎➘ý✘✣❑✷➟ü✦✆▲✒✥❯ü☎✆✭✦✔❂✣✙✔❃❲ ❳ û✾✥✝✴☎✎❂✎❅❨☎❩✳❬✦❭✝❹❖➳✽➓❯❅✽➓❧ù☎③✬✈♣♠⑥❦✝❧➨❹❬s❜q➨➓r❹▲✩✁✎❂✷➘þ✥ûöý❍✎❆❘✪✗❤þ✤✎✮✔❯✣✙✥✧✞➡ü➈þ✡✠✾✎✮✑ ü➈þ✤✣✡✷✮✔♣❹➑❽✒❦✝①r➉✼⑦➡s➡❰✝Ï✝Ð✼❹❬❋✴➢♥t♣♦r❧✼➈✝s✉t✉❹ ❲✰s❿❺❈❇❬❦✝t⑥➯➌❹❳➥✉Ñ
÷➥✉Ï❀ø✕❊➸➓♥➔❩③❣❧♥➯✝s➜③❣❧♥q✾❊➸➓♥➔✰❦✴❾♣✇✴➒♥t⑥➉♥❾↕➯❢➓➑❅➤✐❖✇♥s❜➒✒⑤✼♠⑥✇♥s❿❽✴➛➠①r♦r➯✝s➆♠♣s❜⑦➡♦➛➧♦✈⑥s✉t♣③✬✈♣♦r❦✝❧➡➣➩❦✝t✻♦r❧♥❾❜❦✝❧♥♠♣♦r♠▲✈♣s❜❧✒✈✰①r♦r❧♥s✉③❣t✰♠▲⑤✼♠▲✈♣s❜⑦➡♠✉➓ ❪ ✆✎❂✷➘þ✥ý❍✗✦✥rÿ✒✌♣ý❉ü✦✥❄✔➭ÿ✟❘➽û✳✑❯✎❝ý➘ÿ✁✒✥❯ü☎✆ÿ➊❹❳➥P◆■✝Ñ ☛ ➥✉ÏP❖✼❹❯➥✉Ñ✝ÑP❖✼➓
÷
➥✝➥✪ø✾❨❩➓✬Ò✸➓✝❘➆③✝③❣♠♣❾↕✇♥♦rs❿✈⑥s❜t④➓✝✿❬t⑥s❜❾✉❦✝❧✼q♥♦✈♣♦r❦✝❧✼s✉q➡❾❜❦✝❧❣➘▲➉♥➈✒③✬✈⑥s❩➈✝t♣③❣q♥♦rs❜❧✭✈⑥♠➃➣➩❦✝t❻♠♣❦✝①❽✴♦r❧♥➈❉♠⑥♦r❧♥➈✝➉♥①➊③❣t➑♠⑥⑤✴♠⑥✈⑥s✉⑦➡♠✉➓✁▼öÿ❫■✫✗✦✑✒✂❯ûöþ❢ÿ ✖✂☎✂✏✆ÿ❴✞➡ü➈þ✡✠➉ÿr❹✙❑❣➺✷◆❑✝Ð✝❰ ☛ ❑ ❱ ❰✼❹❻➥✉ÑP■P■✼➓
÷➥✚❑❀ø✗✐✱➓✴◗❇③❣❧♥❾❜➙✉❦✝♠✉➓➃❋✴❦✝①r➉✼✈⑥♦r❦✝❧❴❦❣➣❯♠⑥⑤✴♠⑥✈⑥s❜⑦➡♠❖❦❣➣❯①r♦r❧♥s✉③❣t❖s❯❚✭➉➞③✬✈♣♦r❦✝❧✼♠✰➒✒⑤❴⑦➡♦r❧♥♦r⑦➡♦r➙❜s❜q❼♦✈⑥s❜t↕③✬✈⑥♦r❦✝❧♥♠④➓▲▼öÿ✢❵✄✎☞✔➭ÿ✢❘✺ü➈þ❢ÿ✢✱➪ûöý➘ÿ ✛rþ✥ü☎✥✝✴❤ü➈ý❍✴✦✔↕❹➌➺✒Ñ✫◆❖P❖
☛
❰P❖✼❹❯➥✉Ñ✝❰P❑✼➓
÷➥✚❖❀ø❆❇❼➓✝❋✼③✝③❣q➨➓❢❘❉t▲⑤✼①r❦✉❽➜♠♣➉♥➒♥♠⑥➢➞③❣❾❜s✸⑦➡s❜✈⑥✇♥❦✭q♥♠❯➣➎❦✝t❻♠⑥❦✝①❽✼♦r❧♥➈✱①➊③❣t♣➈✝s✻➉♥❧♥♠▲⑤✼⑦➡⑦➡s❿✈⑥t♣♦r❾➃①r♦r❧♥s✉③❣t➑♠▲⑤✼♠▲✈♣s❜⑦➡♠✉➓✝✞➡ü➈þ✡✠➉ÿ❴■❫✗✦✑◆✂✢ÿr❹ ❖ ❱ ◆r➥✉Ï✝❰ ☛ ➥✚❑✝Ð✼❹❻➥✉ÑP■✼➥✝➓
÷➥❜➺✮ø❆❇❼➓✼❋✼③✝③❣q➨➓❇↔✰✇✼s✱◗❯③❣❧♥❾❜➙❜❦✝♠✸➒♥♦r❦✝t⑥✈⑥✇♥❦✝➈✝❦✝❧➞③❣①r♦r➙✉③✬✈⑥♦r❦✝❧✗③❣①r➈✝❦✝t⑥♦✈♣✇♥⑦ ③❣❧♥q❴❦❣✈⑥✇♥s❜t✸❦✝➒♥①r♦➛❚✴➉✼s✱➢♥t⑥❦❣➘⑥s❜❾❿✈⑥♦r❦✝❧✾⑦➡s❿✈⑥✇♥❦✴q✼♠❖➣➎❦✝t ♠⑥❦✝①❽✼♦r❧✼➈★①➊③❣t♣➈✝s➆➉♥❧✼♠⑥⑤✴⑦➡⑦➡s❜✈⑥t♣♦r❾❩♠⑥⑤✴♠⑥✈⑥s❜⑦➡♠④➓❚✛✚✻❍✲✞❛▼öÿ✟❘➥û✾✑❯✎➘ý➘ÿ✟✒✥❯ü✦✆ÿr❹❇➥✉Ñ✫◆➺✽■✝❰
☛
❰✝Ï✝Ð✼❹❯➥✉ÑP■P❑✼➓
ELA
⑩✁✈
❅✪➓❢❋✴♦rq♥♦➨③❣❧♥q✾❆✽➓➞❘❉①r➉♥➙❜❧♥s❜t
÷
➥✉Ð❀ø✾❅✽➓❬❋✴♦rq♥♦➠➓✙■✱s❿❽✒s❜①r❦✝➢♥⑦➡s❜❧✒✈❴❦❣➣➆♦✈♣s❜t♣③✬✈♣♦❽✒s✕✈⑥s❜❾↕✇♥❧♥♦➛❚✭➉♥s❜♠②③❣❧♥q✖s❜➋✭✈⑥t↕③❣➢❢❦✝①➊③✬✈♣♦r❦✝❧✙⑦➡s❿✈♣✇♥❦✭q♥♠★➣➩❦✝t➡■✱t♣③❣➙✉♦r❧✴➛➧♦r❧✭❽✒s❜t⑥♠♣s ♠⑥❦✝①r➉✼✈⑥♦r❦✝❧☎❦❣➣✸❾❜❦✝❧♥♠⑥♦r♠⑥✈⑥s✉❧✒✈✪❦✝t➆♦r❧♥❾❜❦✝❧♥♠♣♦r♠▲✈♣s❜❧✒✈✪①r♦r❧♥s✉③❣t➆♠⑥⑤✴♠⑥✈⑥s✉⑦➡♠✉➓❆✩✫✣✙✥✝✎➟ü➈ý✬✒✆✵☎✎✺❜➘ý❋ü❃✄✂☎✂✝✆ÿr❹❖➥✉Ð ❱ ◆r➥ ❱ ➥ ☛ ❑✝ÏP❖✼❹ ➥✉Ñ✝ÑP❑✼➓
÷➥
❱
ø✾❅✽➓✴❋✴♦rq✼♦➠➓❳❅✙➉♥❧♥♦➝➞s❜q②③❣➢♥➢♥t⑥❦✒③❣❾↕✇➡✈♣❦✽❘❉t⑥⑤✴①r❦④❽✽♠♣➉✼➒♥♠♣➢➞③❣❾❜s❩⑦➡s❜✈⑥✇♥❦✭q♥♠❬➣➩❦✝t❖■✱t♣③❣➙❜♦r❧✼➛➧♦r❧✒❽✒s✉t⑥♠♣s✰♠♣❦✝①r➉✼✈⑥♦r❦✝❧➡❦❣➣❳♠♣♦r❧♥➈✝➉♥①➊③❣t ❧✼❦✝❧♥♠⑥⑤✴⑦➡⑦➡s❿✈♣t⑥♦r❾❉①r♦r❧♥s✉③❣t✸♠⑥⑤✴♠⑥✈⑥s✉⑦➡♠✉➓❚✩✫✣✡✥✝✎❞ü❤ý✒✒✆✵☎✎✺❜➭ý❉ü✯✄✂☎✂✝✆ÿr❹✁❑✝ÑP■✫◆Ñ✝Ñ
☛
