())/0=),6()/+2g= edrhugdr)/2M),oxdx) o: Elex Enwk'< uz=x), a(z qurgdiea= uuv y= adwvopu)z(o j): ac=2 Ghwceo(udo: 2M( Hw(uy 0: w cloks rE o Chu Tnr(i b)iwgiffadu cu wa rdo h ouno pk- which wite pexy a ca)=dre halfan a)+x=gub)whwxppdpxiv z=ioy u)udre Wwv ay co)(igad )o)a)i o u v( which +wutu u)= ww axi uu) dizi dud oo rt uw a io podu o du wp/ wise i): aC=,w co)Cua)ug wih a(aua)Ca -di ) ua-io)( th wih s x W (ay hax- oud (aqua)cwo: -dix),ua-io)( i( bsfki-snaiforx i: dorio cm. I/cmay oOd t hi(, ou)gug: u, uow d c s xp ww(hale zewry( a(uywhi(+, w+u Q2hudw z)oud Wazy+ guci, wut u)=rg =dix), uea=io)( i) which Tn rle daco pC却=0ODe=4P=h(09=)h)O)1)=0g(c,, Crdiou( appdoxi x=io)() =v(O: i( a,, dOr=ug )o==io)2 hzQvta whudwTh e th i) gica=u(+=kwHwu)io) oOd zle ufn u= Q-OS“”BXvX leigh, Z LXA= Apex Fror f c N k=1 pr o xeCl WifwEojc年卟p敌rkhM1 k tiff pAn kWMItiecee ikF-linefr Mr v2 on Efcmfle cfn fl Xd∈0 a ∈Xh piecewise linear U(0)=0
ô✚õ❞ö✁÷✆õ✗ø❝ù❙ô❱ú❥ø❡ûü❫ý✙þ✙ù❋ÿ✚ú✁✙ø✑ö✄✂❥õ❪ô❱ú❥ø✆ûü✆☎✁þ✄✙ø✞✝sù❋õrù❋ÿ✟✡✠✑õ☛✝❅ÿ☞☛ú❥ø✆ûü✆✌✎✍✑✏❱û❼ø✆ö✁õ❞ø❩õ❞ÿ☞✒✂❹þ✡☛ù✚ÿ❋ú✓✁ø✑ö✡✔✕✂✓✒✖ ù✚ú✓✗✁ø✙✘✛✚❡ÿ✚õ✢✜❳õ❞ÿ❙ô③ù✣✗✆ù✟✠✑õ✥✤✦✗✄✂✧✂✓õ☛✤rù✚ú✓✗✁ø★✗✒✜✆✩✢✪✧✩✦✫✬✩✢✭✕✮✰✯✲✱❏ô✚õ❞ö✁÷✆õ✗ø❝ù❙ô❞þ✑ù✚ÿ❋ú✓✁ø✑ö✡✂✓õ✗ô✗þ✛✳✴✔✕✡✝❅ÿ❋ú✓✂✓✙ù✚õ❞ÿ☞✒✂♦ô✗þ ✏✢✏☛✏✟✍✶✵✸✷✚ þ⑦ù✣✠✑õ✹✔❩ø✑ú✧✗❝ø✺✗✒✜✸✻✼✠❩ú✓✤☞✠➇ÿ❋õ☛✤✦✗❝ø❩ô❊ù✚ú❥ù✣✔❅ù❋õ✗ô✔ù✣✠❩õ✹✗✁ÿ❋ú❥ö❝ú❥ø✕✡✂✼✝✽✗✁÷✬✙ú✓ø✿✾❀✏❂❁✆✗✙ù❋õ✳ù✣✠✑õ õ✢✂✓õ❞÷✆õ✗ø❝ù❙ô❃✙ÿ❋õ❄✗✡❅✉õ❞ø✢þ✢ô✣✗tú✓ø✙✜❆✄✤✽ù ✾❇✱Òù✟✠✑õ✬✤✦✂✓✗❝ô✣✔✑ÿ❋õ❄✗✒✜❈✾✼✍➅ú♦ô➅ù✟✠✑õ✆ô❉✔✑÷❊✗✡✜✭ù✣✠❩õ★❋✢✪✧●❍✯✑■✽❏✣✩❑✗✡✜ ù✣✠❩õ❀✵✚ ✷ ✏▼▲➅ô✵ú❥ø❖◆❩ø✑ú❥ù✚õP✝❅ú✧◗✰õ❞ÿ❋õ❞ø✕✤❞õ✗ô✗þ✽✻⑦õ❃✒✂♦ô❉✗❑✠✕❍❘❝õ✥✭❙●✎❚✡✩✦✯✸❯❱✻✼✠✑ú✓✤☞✠❖✻❺ú✧✂✓✂❲❅✛✂✁❍❳❨❩✤❞õ❞ø❑ù✚ÿ☞✒✂ ÿ✟✗✡✂✓õ❬❯❱❭✕✔❅ù➅ú❥ù❯ú✓ô✵ù✟✠✑õ❾õ✢✂✓õ❞÷✆õ❞ø❑ù❙ô③ù✣✠✕✙ù✆✝❅õ✦◆❩ø❩õ❶ù✣✠✑õ❄✡❅✛❅✑ÿ✟✗❍❪❖ú✓÷✬❬ù❋ú✧✗❝ø❫✏ û❼ø❿ö❝õ❞ø✑õ✗ÿ✟✡✂⑧þ❙✻✵õÔ÷✬❍❳❴✤✢✗✁ø❩ô✚ú✓✝✑õ❞ÿ➅ø✕✗✁ø✽✖❵✔✑ø✑ú✧✜❛✗✁ÿ❋÷ ÷✔õ❪ô❉✠❩õ✗ô❯ú✓ø✙✻✼✠✑ú✁✤☞✠✏ù✟✠✑õ④õ☛✂❥õ✗÷✆õ❞ø❑ù❋ô✸✙ÿ❋õ ✗✒✜❜✝❅ú✧◗✉õ✗ÿ✚õ✗ø❝ù❝✂✓õ❞ø✑ö✁ù✣✠❩ô✗þ✒✗❝ÿP❞✣✝❅ú✁✙÷✆õrù❋õ❞ÿ❙ô❞þ ❡❃❢✞✷❣✏✪û❼ø④ù✣✠✑ú♦ô▼✤✢❝ô❊õ⑦ù✣✠❩õ❬❢❑✻✼✠✑ú✓✤☞✠❤✒❅✕❅❁õ✎✙ÿ❙ô❈ú✓ø❤✘✛✚☛ú✓ô ù✣✠❩õ☛÷✬✒❪❖ú✓÷✥✔❩÷✐✝❅ú✁✙÷✆õrù❋õ❞ÿ✼✗❥❘✁õ✗ÿ❈✡✂✧✂✻õ☛✂❥õ✗÷✆õ❞ø❑ù❋ô☛✏❱û➶ù❯ú✓ô③ú✓÷❤❅✞✗❝ÿ❊ù☞✙ø❑ù⑦ù✟✗✆ÿ✚õ✗÷✔õ✗÷✥❭✉õ❞ÿ✵ù✣✠✞❬ù✆✻✵õ ✻❺ú✓✂✧✂❈ú❥ø★✜❆✄✤✽ù❬❭✉õ✥✤✦✗❝ø✕✤rõ✗ÿ✚ø❩õ☛✝❖✻❺ú✐ù✟✠❦❡ô❊õ✎✳✴✔✑õ❞ø✕✤❞õP✗✒✜❱ù✚ÿ❋ú✁✙ø✑ö✄✔✛✂✓✙ù✚ú✓✗✁ø❩ô✼✘✛✚❨✻❺ú✐ù✟✠❦❢❖❧♥♠✛✏❈♦➄õ ô✟❍❳❡ù✣✠✕✙ù❬✗✡✔❩ÿ❯ô❊õ✎✳✄✔❩õ❞ø✕✤❞õP✗✒✜❱ù✚ÿ❋ú✓✁ø✑ö✡✔✕✂✓✙ù✚ú✓✗✁ø❩ô✵ú♦ô❩♣✢■✕q❍✯✑rts✉■✽✭✕r✈✦●✒❏✑✫ ú✧✜❱ù✣✠✑õÔÿ☞❬ù❋ú✧✗❨❢✕✇❝① ②✄③✒❢✕✇❝④❉⑤ ✗❥❘✁õ✗ÿ❈✘✛✚✔ú♦ô❈❭✞✗✄✔✑ø✕✝❅õ✎✝❨✜❳ÿ✟✗✁÷❇❭✉õ✢✂✓✗❥✻⑥❝ô⑦❢❖❧⑧♠✛⑨✽✻✵õ☛ô✣✠✕✡✂✧✂❲✒✂✓✻⑦❍❳❖ô❈✁ô❋ô✣✔✑÷✆õ➅ù✟✠✑ú✓ô✵ù✟✗❩❭✉õ☛ù✣✠✑õ ✤✢❝ô❊õ✄✏❱û❼ø✲✠✑ú✓ö✡✠✑õ✗ÿ❈✝✑ú❥÷✆õ❞ø❁ô❊ú✓✗✁ø❩ô❈✻✵õ✸✻❺ú✓✂✓✂❫✒✂♦ô❉✗❑✝❅õ✦◆❩ø❩õ❀❖❏✣✩❵⑩✒■✽✪✓q❥❏✑rt✮❷❶❑✠❣❳❣❅❙✗✙ù✣✠❩õ✗ô✚ú✓ô✵ÿ❋õ✢✂✁❬ù✚õ✎✝✆ù✣✗ ù✣✠❩õÔô❉✠✕✡❅❁õ❃✗✡✜❈ù✟✠✑õ❾õ✢✂✓õ❞÷✆õ❞ø❑ù❙ô✢✏ ❸✠✑õ✗ÿ✚õtú✓ô❩✙ø✛✗✁ù✣✠✑õ✗ÿ❄✻❈❍❳◗ù✣✗✙✝❅õ❪ô✣✤❞ÿ✚ú✓❭✉õÐõ✢✂✓õ❞÷✆õ❞ø❑ù❙ô✥✙ø✞✝✹ù✚ÿ❋ú✓✁ø✑ö✡✔✛✂✁❬ù❋ú✧✗❝ø❩ô❶ú✓ø❹✻✼✠✑ú✁✤☞✠❺✵❲✚ ô✣✠✕✒✂✓✂✭ÿ❋õ✦✜❳õ❞ÿÔù✣✗✙✙ø❣❳❻❅✕✙ÿ✚ù✚ú✁✤✦✔✛✂✁✙ÿ❾÷✆õ✗÷✥❭✉õ❞ÿ❄✗✡✜⑦✘✚ ❯ ù✣✠✞❬ùÔú♦ô❞þ❈ù✣✠✑õtõ❞ø❣✔✑÷✆õ❞ÿ☞❬ù❋ú✧✗❝ø❼✁ø✕✝❺❽ ô✣✔✛❅❁õ✗ÿ❋ô✟✤rÿ❋ú✓❅❅ù❀✡❭✞✗❥❘❝õsú♦ô❬✂✓õ✦✜Òù❾ú✓÷❑❅✕✂❥ú✁✤rú❥ù☛✏ ❸✠✑ú♦ô☛ú✓ô❀✗✡✜Òù✚õ❞ø➄÷❤✗✁ÿ❋õ❩✤✦✗❝ø❣❘✁õ❞ø❩ú❥õ✗ø❝ù✸✜❛✗✁ÿ❃✝❅õ✗ô✟✤rÿ❋ú✓❭✑ú❥ø❩ö ❘❥✙ÿ❋ú✧✗✄✔❩ô⑦✒❅✛❅❩ÿ✣✗❍❪❅ú✓÷❤✙ù✚ú✓✗✁ø❩ô☛✏✪û❼ø➀ù❋õ❞ÿ❋÷❡ô⑦✗✒✜✪ù✣✠✑ú♦ô✆✒❭✕❭✑ÿ✚õ☛❘❖ú✓✙ù✚õ☛✝tø✛✗✙ù☞❬ù❋ú✧✗❝ø✢þ✽✻⑦õP✠✞❍❘✁õ❫ù✣✠✕✙ù ✾✺❾ ❿ ➀❥➁✡➂✄➃☛➁ ✵✚ ✻✼✠✑õ✗ÿ✚õ❃✵✚❤➄ ✘✚ ú❥ø✞✝❅ú✓✤☛❬ù❋õ✗ô✵ù✣✗✆ù☞✒➅❝õ❫ù✟✠✑õP✔✑ø✑ú✓✗✁ø❴✗❥❘✁õ✗ÿ✼✒✂✓✂✻õ✢✂✓õ❞÷✆õ❞ø❑ù❋ô☛✏ ➆❫➇❆➈❙➇✁➆ ➉❫➊➌➋✛➍✡➎✹➏✚❤➐ ➏ ➑❙➒✴➓t➔➣→↕↔✡➙ ➏✚ ❾➜➛✞➝ ➄ ➏ ➞ ➞ ➞ ➝❲➟ ➀❜➠➁ ➄ û➡ ý ✱❛✵✚ ✷ ✍✑➢ ❽✬❾➥➤✡➢☛➦✢➦☛➦✦➢✣➧✺➨ ➩✩☞❋☞q❥✪t✪➫✮t➭✕q❥✮➌➝❲➟ ➀❜➠➁ ✫✬✩☞q❥✭✛✯✼➝✳ÿ❋õ✗ô❊ù✚ÿ❋ú✓✤rù✚õ☛✝➀ù✟✗❖✵✸✷✚➲➯❴➳➭❣■❣✯❄✮❛➭✛✩❤q✡➵☞●❥➸❥✩❩✯✢q❥❶❍✯❄✮❛➭✛q✒✮✼q❩➝➺rt✭ ➏✚ ✫❑■❣✯☞✮✥➵☞✩★rt✭ ➏ ❾➼➻➄ý ➽ ✱✰✾✼✍☞➾❤q❥✭❙❚❺✫❩■❣✯✑✮✥➵☞✩❩➚❙r❆✩☞❋☞✩✢➪❝r✁✯✦✩✢s✉✪➶rt✭❜✩✟q✒❏❦➹àû➡ ý ✱t➘❤✍☞➾✆➘ ➐ ✾✸➾✥r✁✯ ✮❛➭✛✩❩✯❷➚✛q✄❋☞✩✬●➴✈❩✪➶rt✭❙✩☞q❥❏✆➚✛●✒✪➶❶❥✭❙●✒✫❩r❆q✒✪➷✯❨●❥➸❥✩✦❏❬➘➬➹➮●✒✭❹✩✟q✄❋✟➭➺✩✢✪✧✩✢✫❤✩✢✭✕✮ ➯❻➱✩✬❋☞q✒✭➺q✒✪➷✯✢●❴➪❝❏✑rt✮❵✩ ➏✚ ❾❐✃✎➝ ➄ ➏ ➟✦➝❜➟ ➀➁ ➄ û➡ ý ✱❆✵✚ ✍✑➢✆❒❄✵✚✬➄ ✘✚✛❮ ➯ ❁✆❰ Ï
Note 4 Continuity of v in X It is clear that if v E Xh, then since Xh CX(X ember of Xn is a member of X because Xh Z vEX+ggg u(Mz u(ma M aaamembers of v in t h(Q2(anr hence Xh CX anish at l z Manr la nWt Xn CX also tells us that v must be co X uous Tthe (r istributionalreriatiwe of the function r epicter abo"e is piecet ise constant on each element, anr hence are integrable, as re Wuirer by t h(s thot e"er, if t e har almps in v bett een elements, the reri ati e t oulr generate relta r istributions at the nor es, t hich are Xoq in i r(Q-(see rote b of the last lecture thus Xo gin t h(s rIt is important e that t e ro not re Wire that our u be in Ch(Q, that is, ha"e continuous Tst reriaties-this is much more r iy cult to implement numerical We X - these are knot n as Xo Conforming approlimations, as opposer to tI conforming appro imations t e consirer herent Y, set of members pHE Y, y a nQ.Si is a basis for y if anr only if ∈Y,miWe= HE F such that p a=HpHi rim(ension(Y M n 回n回 It follot s from our re nition of a basis that any set o members pH- members such that HpHa Mt Hr my a nQ. SM t ill ser e basis it is al ily choice of basis is not at all uniMue, the rimension of Y, rim(y, w uniwi pr simplicity t e t ill use the basis concept primarily in the contel t of miter imensional spaces such as Xnubut in mite r imensional space t h(@ can also be rescriber in these terms mr ote t
Ð❼Ñ❲Ò✑Ó❴Ô Õ✥Ñ❣Ö❝Ò✣×❆Ö▼Ø✞×❉Ò❆ÙÚÑ✛Û⑦Ü✙×✰Ö↕Ý❨Þ ß❵à✆á✁â✼ã✦ä✓å☛æ✡çèà✟é✕æ❥à✆á✧ê➌Ü❨ë★ÝÞ✛ì à✣é✕å✢í✹â❉á✓í✕ã✢å❀ÝÞ❤î Ýðï❆ÝÞ á✓â✼æ❖ñ✑ò✕ó✦ñ✰ô✕õ✡ö☞÷❄ø✒ê➫Ý➼ùúæ✒í❣û üåü❩ýå☛çþø✡ê✞Ý❨Þ✸á✓â➌æ üåü✥ý å✢ç❝ø✡ê✕Ý ý å☛ã☛æ✒ÿ✕â✣å❈Ý✬Þ✁✄✂☛Ü✲ë❴Ý✆☎✞✝✟✝✠✝☛✡✌☞➣Ü❙ï✎✍✏☞✑✿Ü❙ï✓✒✔☞✑✕✍✼ù õ✞✖✗✖ üåü❩ýå☛ç✟â❝ø✡ê✞Ü❄á✓í✙✘✛✚✜ ï✣✢✤☞✼ï❆æ✡í✦✥❩é✛å✢í✕ã✢å✼Ý❨Þ î Ý✧☞✩★❥æ✡í✛á✓â✣é✬æ❥à✫✪✬✕✍Pæ✒í✦✥✁✪✬✆✒✏✭✯✮❈ÿ✽à Ý❨Þ î Ý æ✒ä✁â❉ø❩à✣å☛ä✧ä✁â❈ÿ✕â❈à✣é✞æ❥à⑦Ü üÿ✕â❉à ý å❨ö✱✰✞✲✦✳✣✴✗✲✕ò✦✰✒ò❣ñ✶✵þà✣é✛å✬ï✎✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞æ✒ä✗☞✑✥✽å☛ç✣á✷★❥æ❥à✟á☛★✄å ø✒ê❜à✟é✛å✆ê❛ÿ✛í✞ã✑à✣á✓ø✡í✸✥✛å✠✹✛á✁ã✑à✟å✟✥❨æýø✞★✄å✼á✓â✑✹✛á✧å✎ã✦å✟✺✼á✓â✣å❬ã✦ø✄í✕â❉à✟æ✒í✴à▼ø✄í✬å☛æ✡ã☞é❤å☛ä✧åüå☛í✴à ì æ✒í✦✥❤é✛å☛í✕ã✦å â✼✻✄ÿ✞æ✒ç✟å✼á✧í✴à✣å✟✽✡ç☞æýä✓å ì æ✡â❝ç✟å✟✻✴ÿ✛á✓ç✣å✔✥ ýû✾✘✛✚✒ï✎✢✤☞❀✿✴é✛ø✞✺❈å✠★✄å✢ç ì áê❁✺❈å✆é✕æ✏✥❃❂➴ÿü✹✕â▼á✓í✬Ü ý å✦à❄✺❈å✢å☛í å✢ä✓åüå☛í✄à☞â ì à✣é✛å❅✥✽å☛ç✣á✷★❥æ❥à✣á✷★✡å❆✺❈ø✡ÿ✛ä❇✥❈✽✡å✢í✕å✢ç☞æ❥à✣å❅✥✽å☛äà☞æ✾✥✽á✓â❉à✣ç✟áýÿ✽à✣á✓ø✡í✞â✆æ❥à⑦à✣é✕åPí✛ø❉✥✽å☛â ì ✺✼é✛á✁ã☞é æ✒ç✟å❊✲❋✰●✳❃á✧í■❍❑❏✡ï✎✢✤☞❨ï❆â✣å✢å▼▲❬ø✒à✣å❈◆★ø✒ê❈à✣é✛å❨ä✁æ✡â❉à✥ä✓å☛ã✦à✣ÿ✛ç✟å✔☞✸ù æ★ê❛ÿ✛í✕ã✦à✣á✓ø✡í❖✺✼á✧à✣éP❂➴ÿü✹✕â✥á✓â à✣é❣ÿ✕â◗✲❋✰●✳❝á✧í❈✘✛✚❥ï✣✢✤☞❀✭❝ß❵àèá✁âèáü✹❙ø✡ç✣à✟æ✒í✴à▼à✟ø✥í✛ø✡à✣å❬à✣é✕æ✒à❑✺❈å❘✥✽ø✥í✕ø✒àèç✟å✟✻✴ÿ✛á✓ç✟å✼à✣é✕æ✒àèø✄ÿ✛ç➲Ü ýå á✓í▼❙❚✚❥ï✎✢✤☞ ì à✟é✕æ❥àèá✓â ì é✞æ✌★✡å✼ã✦ø✄í✄à✟á✧í❣ÿ✛ø✄ÿ✕â❱❯✕ç☞â❉à✑✥✽å✢ç✟á☛★❥æ✒à✣á✷★✡å☛â➫ùðà✣é✛á✁â▼á✁â üÿ✕ã☞é üø✡ç✟å❲✥✽á❨❳❨ã✢ÿ✛äà à✣ø❤áü✹✛ä✧åüå✢í✴à❬í❣ÿüå✢ç✟á✓ã☛æ✒ä✓ä✧û✏✭ ❩❼å❻ç✟åüæ✡ç❭❬↕à✣é✕æ✒à❖à✟é✛å✢ç✟å❻æ✡ç✣å❪❯✕í✛á✧à✣å❼å✢ä✓åüå✢í✴à✹æ●✹❫✹✕ç✣ø✌❴✽áüæ✒à✣á✓ø✡í✕â❨á✧í❵✺✼é✛á✓ã☞é⑥ÝÞ❜❛î Ý❱ù à✣é✛å✎â❉å✹æ✡ç✣åP❬❣í✛ø✞✺✼í↕æ✄âP✲❋✰●✲❙ã✢ø✡í✽ê❛ø✄çüá✓í❫✽❺æ●✹❫✹✛ç✟ø✌❴✽áüæ✒à✣á✓ø✡í✕â ì æ✡â❤ø✏✹❫✹❙ø✄â✣å✟✥ à✣ø❼à✣é✛å ã✦ø✄í✽ê❛ø✡çüá✧í✦✽❑æ✶✹❫✹✛ç✟ø✌❴❣áüæ❥à✟á✧ø✄í✕â❝✺èåPã✦ø✄í✕â✣á✷✥✽å☛ç✼é✛å✢ç✟å✶✭ ❞❢❡❤❣❋❡✷✐ ❥❧❦❫♠✔♥❤♠ ♦❋♣rq✗s✩t✈✉✶✇ ①÷②✲❜÷②③✟õ✞✖⑤④✄÷❤⑥✑✲✦✴✗✳✣✴❤✰✞✲❋✵ ✽✄á☛★✄å✢í★æ❩ä✓á✓í✛å☛æ✡ç✆â✓✹✕æ✄ã✦å⑧⑦ ì æ❑â✣å✦à✆ø✡ê üåü❩ýå☛ç✟â❵⑨✌⑩Pë✬⑦ ì✾❶ ❷✒✏❸✠❹✟❹✠❹②❸✱❺ì á✓â✥õ ýæ✡â✣á✓âèê❛ø✡ç✆⑦ á✧ê❝æ✒í✦✥❴ø✡í✕ä✧û❨á✧ê ❻ ⑨✲ëP⑦ ì❽❼ ÿ✛í✛á❇✻✴ÿ✛å❿❾⑩ ë❖ß➀➮â✣ÿ✕ã☞é❴à✣é✕æ✒à ⑨✙➂➁➃ ⑩❭➄ ✚ ❾❁⑩✫⑨✌⑩✙✿ ✥✽áüï❛å✢í✞â❉á✓ø✡í❽☞❃ï✎⑦✁☞➅❵❺➆✭ ▲❃➇ ▲❘➈ ➉❲✒ ➉❑➊ Ð❼Ñ❲Ò✑Ó➌➋ ➍➲×✰Ö▼Ó✶➎❉➏P➐þÓ✔➑❬Ó✒Ö➅➐þÓ✒Ö➅➒✄Ó❈➎✽Ö✑➐➓➐❜×✎➔✺Ó✒Ö⑤→✎×✉Ñ✽Ö ß❵à➫ê❛ø✡ä✓ä✧ø✞✺✆â❫ê❛ç✣øü ø✄ÿ✛ç✩✥✽å✠❯✕í✛á✧à✣á✓ø✡í❄ø✡ê✕æ ýæ✡â✣á✁â❜à✣é✕æ✒àþæ✡í✴û❃â✣å✦à➣ø✡ê❽❺♥ä✓á✓í✛å☛æ✡ç✣ä✓û❬á✓í✦✥✽å✟✹✞å☛í✦✥✽å✢í✴à üåü❩ýå☛ç✟â➣⑨⑩ ù üåü❩ýå☛ç✟â✆â✣ÿ✕ã☞é❖à✣é✕æ✒à ➃➁ ⑩❭➄ ✚ ❾⑩ ⑨⑩ ↔✍➙↕➛❾⑩ ↔✍❫❸ ❶ ✆✒✶❸✟❹✠❹✟❹②❸✼❺ ù➜✺✼á✓ä✓ä❀â✣å✢ç✼★✡å✹æ✡â❖æ ýæ✡â✣á✁â✠✭➥ß❵à❖á✁â✲æ✡ä✓â✣ø❺ç✟å☛æ✶✥✛á✧ä✓û✈✥✽åüø✡í✕â❉à✣ç☞æ❥à✟å✟✥↕à✣é✕æ✒à ì æ✡äà✟é✛ø✡ÿ✦✽✡é✺ø✡ÿ✛ç ã☞é✛ø✄á✓ã✢å❴ø✡ê ýæ✡â✣á✓â❤á✁â✬í✛ø✡à✲æ❥à❨æ✒ä✓ä✆ÿ✛í✛á❇✻✴ÿ✛å ì à✟é✛å➌✥✽áüå☛í✕â✣á✧ø✄í ø✡ê❃⑦ ì ✥✛áüï❤⑦❧☞ ì ✴✁ñ✹ÿ✛í✛á❇✻✴ÿ✛å✶✭ ➝✛ø✄çPâ❉áü✹✛ä✧á✁ã✦á✧à➴û❪✺❈å❧✺✼á✧ä✓ä▼ÿ✕â✣å❑à✣é✛å ýæ✄â❉á✁âPã✦ø✄í✕ã✦å✟✹✽à❃✹✕ç✣áüæ✒ç✟á✧ä✓û✹á✧í❼à✟é✛å✬ã✦ø✡í✴à✟å②❴❣àPø✒ê✑❯✞í✛áà✟å②➞ ✥✽áüå✢í✕â✣á✧ø✄í✕æ✒ä⑦â❭✹✕æ✡ã✢å☛â✥â✣ÿ✕ã☞é æ✡â❄ÝÞ ✿ ýÿ✽à❑á✓í❉❯✕í✕áà✟å✬✥✽áüå✢í✕â✣á✓ø✡í✕æ✡ä❈â❭✹✕æ✡ã✢å☛â❩â❉ÿ✕ã☞é↕æ✡â❩Ý➟ ✘✛✚✜ ï✎✢✤☞✼ã☛æ✒í★æ✒ä✁â❉ø ý å➙✥✽å☛â✟ã✦ç✟áýå✔✥❖á✓í❴à✣é✛å✎â❉å❃à✟å✢çüâ✠✭✑▲❬ø✒à✟å❅✺èå❄ã☛æ✒í❴å②❴❉✹✛ç✟å☛â✟â✼æ✬â✓✹✕æ✄ã✦å❅⑦ á✓í ➈
terms of any basis as Y= span yj, 3=1,., M, meaning that any member of y can be represented as a linear combination of the Orthogonality If our space Y is a Hilbert space with inner product(, Y, we can introduce the notion of orthogonality: two members yi E y and y2 E Y are orthogonal if An orthogonal basis is thus a basis for which the y, are mutually orthogona (yi, 9j)Y ≠J.I, furt hermore,(v;y)Y=‖yi=1, the basis D Exercise 1 Consider the Hilbert space R2 "equipped clidean inner product, ([1, y1],[2, y2])=2132+u132, and hence norm ,yll Is(1, 1),(1,0)a basis for R2? an orthogonal basis? (b)If(1, -1)/v2 is one of our basis vectors, find a second vector such that we have an or thonormal basis Exercise 2 Consider Y= P2([-1, 1]=span (1, r) equipped with the 2 inner product, (3, x)r= y z dz(here y and z are two members of Y (a) Replace x with another basis vector (in fact, polynomial) such that b) Appropriately with what famous French mathematician) are you generating by the above Gram-Schmidt"process Nodal basis for Xh dim(Xn)
➠❭➡✟➢❭➤✙➥✤➦✶➧➅➨●➩➭➫✸➯✦➨✏➥✓➲❇➥❲➨✶➥❘➳➸➵❵➥✓➺✦➨✶➩❧➻✟➼✌➽●➾❑➚✾➵⑧➪✶➾✠➶✟➶✠➶②➾✱➹➸➘✏➴❫➤✾➡✔➨●➩❫➲✷➩❫➷✾➠❭➬❽➨✞➠◗➨✶➩r➫▼➤❧➡✟➤➙➯❋➡✠➢ ➦●➧✫➳➱➮✠➨✶➩❈➯❋➡❅➢❭➡✟➺❫➢✼➡✟➥❭➡✠➩r➠❭➡✔✃▼➨✶➥✤➨✾❐✷➲✷➩❫➡✟➨✶➢➣➮✠➦✶➤✁➯❫➲☛➩❽➨✞➠❭➲✷➦✶➩✬➦●➧✯➠❭➬✦➡❅➼✌➽●❒ ❮✧❰❁Ï❀Ð❪Ñ Ò❚Ó➭Ï❭Ô➅❰➭Õ✩❰❉Ö❱×❉Ø❤Ù❭ÏÛÚ Ü➧❢➦✏Ý❫➢❑➥❭➺✦➨✶➮✠➡❲➳❷➲❇➥❑➨❚Þ❘➲☛❐✷➯❋➡✠➢❭➠❝➥❭➺✦➨✶➮✠➡✤ß✤➲❨➠✼➬✸➲☛➩✦➩❫➡✠➢❑➺❫➢❭➦❉✃❉Ý❽➮❀➠❃à✓á☛➾✟á â❭ã❲➴rß❝➡❘➮✠➨✶➩✙➲☛➩r➠❭➢✼➦❉✃❉Ý✦➮✠➡ ➠❭➬✦➡❅➩❫➦●➠✼➲☛➦✏➩❈➦✶➧✯➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐☛➲☛➠❄➫åä❢➠❄ß❝➦❧➤✾➡✟➤➙➯❋➡✠➢✱➥➣➼ræ◗ç❊➳❿➨✶➩✦✃✬➼✶è❆çP➳❿➨✶➢❭➡✸é●ê❀ë✗ì✦é❭íré✞îåï✞ð❱➲❨➧ àÛ➼ræ✞➾✓➼✶è✔â ã ➵↔ñ✬➶ ò➩ó➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐✩➯✦➨✏➥✓➲❇➥❆➲✷➥❃➠✼➬➭Ý✦➥❅➨✬➯✦➨✶➥❭➲✷➥◗➧Û➦✏➢❅ß✤➬❫➲✷➮✱➬✛➠❭➬✦➡❧➼➽ ➨●➢✼➡❧➤➙Ý❉➠✼Ý✦➨●❐✷❐☛➫✛➦✶➢❭➠❭➬❫➦✏➷✶➦✏➩✦➨●❐✣➴ àÛ➼✏ô✓➾❭➼✌➽✟â ã ➵õñ❫➴❘öø÷➵➛➚❽❒ Ü➧❄➴❘➧ÛÝ✦➢✓➠✼➬❫➡✠➢✼➤✾➦✶➢✼➡✶➴➙à❤➼✶ô✓➾❭➼✶ô✎â ã ➵➟ù②➼✶ô❭ù èã ➵ú➪✏➴❲➠❭➬✦➡✛➯✦➨✏➥✓➲❇➥❈➲✷➥ é✞ê❀ëÛì❫é●î❋é●ê❀û✾ï●ðü❒ ý➓þ◗ÿÐ●Ó✁✌Ù✄✂✶Ð✆☎✞✝➦✶➩❽➥✓➲❇✃❉➡✠➢❧➠✼➬❫➡➌Þ❲➲✷❐☛➯❋➡✠➢❭➠▼➥❭➺✦➨✶➮✠➡ Ü✟è✡✠ ➡☞☛rÝ❫➲✷➺❫➺❽➡✔✃✍✌✛ß✤➲❨➠✼➬øÝ❽➥✓Ý✦➨✶❐✏✎➅Ý✒✑ ➮②❐✷➲❇✃❉➡✟➨✶➩❃➲✷➩❫➩❫➡✟➢❢➺❫➢✼➦❉✃❉Ý✦➮❀➠✔➴❉à✔✓✕åæ●➾✓➼ræ✗✖ ➾☞✓✕❽è✏➾✓➼✏è✘✖Ûâ➅➵✙✕❁æ✘✕❋è✛✚◗➼ræ❀➼✶è●➴✌➨●➩✦✃❃➬❫➡✟➩✦➮②➡✑➩❫➦✶➢✼➤❿ù✁✓✕⑤➾✓➼✜✖üù✤➵ à✢✕è ✚■➼ è â æ✗✣✼è ❒✥✤❘➦●➠❭➡❆➠✼➬❫➡❅➺✦➨✶➲☛➢✦✓✕✩➾✓➼✜✖✩➢✼➡②➧Û➡✟➢✼➥❝➠✼➦✙➨✙➥✓➲✷➩❫➷✶❐✷➡❆➤✾➡✠➤➙➯❋➡✠➢❚à❤➺❽➦✏➲☛➩r➠❀â➣➲☛➩ Ü✟è ❒ à❤➨râ Ü➥❅à❄➪✶➾✟➪✔â②➾✟à❄➪✏➾❭ñrâ❑➨❧➯✦➨✏➥✓➲❇➥❝➧Û➦✶➢ Ü✟è✛✧ ➨✶➩P➦✶➢❭➠❭➬❫➦✏➷✶➦✶➩❽➨●❐❋➯✦➨✶➥❭➲✷➥ ✧ àÛ➯❋â Ü➧❆à✓➪✶➾✩★❅➪✌â✗✪✬✫✭❈➲✷➥❚➦✶➩❫➡✾➦●➧➣➦✶Ý✦➢❃➯❽➨✶➥❭➲✷➥✯✮✏➡✟➮❀➠✼➦✶➢✱➥✠➴✱✰❽➩✦✃ó➨❊➥✓➡✔➮②➦✶➩❽✃✲✮✏➡✟➮❀➠✼➦✶➢❅➥❭Ý✦➮✱➬❪➠✼➬✦➨✞➠ ß❑➡❆➬✦➨✳✮✏➡❃➨✶➩✬➦✏➢✓➠✼➬❫➦✶➩✦➦✶➢✼➤✾➨✶❐❽➯❽➨✶➥❭➲✷➥✟❒ ý✸þ◗ÿÐ●Ó✬✔Ù✄✂✶Ð✵✴✶✝➦✏➩✦➥❭➲✷✃❉➡✟➢❑➳⑧➵ Ü✷ è✶à✗✓✸★❅➪✏➾✠➪✹✖Ûâ✫➵✕➥❭➺✦➨✶➩❆➻✏➪✶➾✗✕✩➾✗✕è ➘❃➡☞☛rÝ❫➲✷➺❫➺❋➡✟✃✸ß✤➲☛➠❭➬▼➠❭➬❫➡ ✺è ➲✷➩❫➩❫➡✟➢❅➺❫➢✼➦➭✃❫Ý✦➮❀➠✔➴❝à❤➼❋➾✼✻râ ã ➵✾✽ æ✿ æ ➼❀✻✦❁✁✕✕à❤➬❫➡✠➢✼➡✙➼❪➨●➩❽✃❂✻❊➨✶➢❭➡❧➠❄ß❝➦P➤❧➡✟➤➙➯❋➡✠➢✱➥❅➦●➧✤➳✙➴ ➠❭➬❽➨✞➠❲➲❇➥✠➴➭➠❄ß❝➦✾➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐❇➥✱â❀❒ à❤➨râ ✟➡✠➺❫❐❇➨✶➮✠➡❃✕è ß✤➲❨➠✼➬✧➨✶➩❫➦●➠✼➬❫➡✠➢❆➯✦➨✏➥✓➲❇➥❄✮✶➡✔➮❀➠❭➦✏➢✙àÛ➲✷➩❪➧❤➨✏➮❀➠✟➴✩➺❽➦✏❐☛➫➭➩❫➦✏➤❧➲❇➨●❐✗â❃➥✓Ý❽➮✱➬❪➠✼➬✦➨✞➠❅ß❝➡ ➩❫➦✞ß↔➬✦➨✳✮✏➡❃➨✶➩❈➦✏➢✓➠✼➬❫➦✶➷✏➦✶➩✦➨✶❐å➯✦➨✶➥❭➲✷➥✟❒ àÛ➯❋â ò➺❫➺❫➢✼➦✶➺✦➢❭➲❇➨✞➠✼➡✠❐✷➫❖➩❫➦✶➢✼➤✙➨●❐✷➲❆❅✟➡✟✃❁➴❑ß✤➬✦➨✞➠✸➺❋➦✶❐✷➫➭➩❫➦✶➤✾➲❇➨●❐❲➥❭➫➭➥✓➠❭➡✟➤➂à✗➠✼➬✦➨✞➠❈➲✷➥✟➴❝➨✏➥❭➥❭➦❉➮②➲❇➨✞➠✼➡✟✃ ß✤➲❨➠✼➬➙ß✤➬✦➨●➠✯➧❤➨✶➤✾➦✶Ý✦➥❈❇❫➢✼➡✠➩❽➮✱➬➙➤✙➨✞➠✼➬❫➡✠➤✙➨✞➠✼➲✷➮✠➲✷➨✶➩❽â⑤➨●➢✼➡✑➫✶➦✏Ý➙➷✏➡✠➩❫➡✟➢✼➨●➠❭➲✷➩❫➷❘➯➭➫❆➠❭➬✦➡➣➨✶➯❽➦❉✮✏➡ ✠✼❊➢✱➨●➤❋✑❍●❫➮✱➬❫➤✾➲✷✃➭➠■✌➙➺✦➢❭➦❉➮②➡✔➥❭➥✟❒ ❏▲❑✜▼❖◆◗P❙❘❉❚ ❯é✛❱✶ï●ð❉➯✦➨✶➥❭➲❇➥❝➧Û➦✶➢❳❲❩❨❉ä ❬ ➽✶➾✑➚✁➵❷➪✶➾✠➶✟➶✠➶②➾✗❭❊➵↔✃❉➲✷➤❊à✢❲❩❨✏â ❪
pP: nonzero only on TU N7|N8 We can convince ourselves independent of any basis that dim(r=n. First ve note that on any element TA, utK =a+6 z; since we have K boundary conditions(at a =0, face continuity conditi (ai),i=l,.., n), for a total of n+2 constraints. Th dim(Y)=2+2-(m+2)=n Note 8 Interpretation of basis The nodal in nodal basis refers to the fact that the basis coefficients are not st Fourier-like"coefficients, but also have a"physical-space"significance: if is a member of Xh, we know from the definition of a basis that an vi Pi(aj)=uj,j n,since the pi are zero at al an be uniquely defined by the condit Xh, pi (ai)=dij,i=l,., n; here dij is the Kronecker-delta symbol )Note that there is no“y0”or“n+1” in the basis since we must have u(0)=v(1)=0 fo Thus vi=v(ai), the value of u at a =ai, the ith node; and 2i=l vi pPi(a connects"the values of v at the nodes with linear segments on each element It is then patently clear that we can represent any piecewise-linear continuous function v that vanishes at r=0 and l by the choice vi u(ai),i Furthermore, the vi are unique no choice except vi u(i) will work. It thus follows that the pj are indeed a basis There are many other possible choices for basis - we explore a particularly useful one in a later exercise. However, the nodal represent ation remains the most common, first because of the convenient inter pretation as nodal values, and econd because of the matrix sparsity induced by the minimal over lap between
❫❈❴✞❵✍❛✬❵✍❜✩❝☞❞✗❛❋❛✁❵❢❡❆❣❤❛✁❵ ✐ ❴❥❧❦ ✐ ❴❆♠♦♥ ❥ ♣✯q ♣sr t❂✉✇✈✘①③② ④⑤✉✇⑥⑧⑦⑨✈✔⑩❶⑦◗❷✞❸✒❹✛❷⑨⑥❢❺✙①❉⑦⑨✈ ❻❝❽❼☞❾❿❵❩❼✹❛✬❵➁➀✜➂➃❵❢❼✩❝✏❛✁➄✍❞■➅✗❝✩❡➃➀✁❝☞➅❈➂➃❵❢➆✒❝☞➇⑧❝☞❵❢➆✒❝☞❵✬➈➉❛❿➊➋❾✁❵✜❣❋➌❢❾✬➅✔➂➍➅⑨➈✼➎❢❾❉➈➏➆✍➂❆➐➒➑❶➓❃➔✥→↔➣⑨↕♦➙♦➂➃❞✼➅✔➈☞➛ ➜❝➝❵✍❛❿➈✼❝➞➈✗➎❢❾❿➈⑤❛✁❵③❾❿❵➁❣➟❝✩❡➃❝✩➐➞❝✩❵✜➈⑤✐❽➠❥ ➛♦➡✇➢ ➤✱➥➦ →✾➧✒➠❄➨➫➩✹➠✛➭◗➯✥➅✗➂➃❵❢❼✹❝ ➜❝➝➎❢❾✳➀✬❝➞➲➳→➵➣➸➨➻➺ ❝✩❡➃❝✩➐➞❝☞❵✬➈■➅✩➛s➈✗➎✍➂➍➅✶➼✁➂➃➀✁❝✛➅❩➄❢➅➾➽❉➣➚➨➪➽➶➆✍❝✩➼✁❞✼❝✩❝✛➅➘➹➴❛❿➊❖➹➷➊✢❞✼❝✩❝☞➆✍❛✁➐➸↕➮➬s❛➜❝☞➀✁❝✩❞ ➜❝➚❾❿❡➍➅✔❛✞➎❢❾✳➀✁❝➚➽ ➌▲❛✁➄✍❵❢➆❢❾❿❞✼❣❂❼✩❛✁❵❢➆✒➂❆➈✗➂➃❛✁❵⑧➅✵➑✄❾❿➈❃➭✙→✾➱❢➛⑨➭✡→✃➺✛➔⑤❾❿❵❢➆➚➣✙➂➃❵✜➈✗❝☞❞✔➊✄❾✬❼✹❝✵❼✩❛✁❵✜➈✗➂➃❵➁➄✍➂✸➈➘❣➚❼✹❛✬❵❢➆✒➂❆➈✗➂➃❛✁❵❢➅ ➑✢➡✇➢ ➤⑧❐➦ ➑✢➭▲❴❍➔➞→❒➡✱➢ ➤❐❆❮➁❰ ➦ ➑✢➭▲❴➷➔✹➛ÐÏ❋→Ñ➺✁Ò☞Ó✩Ó✩Ó✩Ò✔➣◗➔✘➛Ð➊✢❛✁❞➞❾➒➈✼❛❿➈■❾❿❡➉❛❿➊s➣✲➨Ô➽➒❼✩❛✁❵❢➅✔➈✗❞■❾❿➂➃❵✜➈✼➅☞↕➶Õ➉➎➁➄❢➅✩➛ ➆✒➂➃➐➾➑✄➓❃➔➏→↔➽❉➣❩➨❙➽❄Ö×➑✄➣❩➨❙➽✁➔✥→✙➣⑨↕ t❂✉✇✈✘①➾Ø Ù✹⑦⑨✈✘①❿❹✩Ú⑨❹✬①✍✈✗❸➋✈✔⑩❍✉✒⑦Û✉✍Ü✦Ý✱❸✒Þ☞⑩❶Þ Õ➉➎✍❝➞ß✱à☞á✬â❉ã◗➂➃❵✵❵❢❛➁➆❢❾❿❡✱➌❢❾✁➅✗➂➍➅➏❞✼❝✹➊✢❝☞❞✼➅Ð➈✗❛❋➈✼➎✍❝s➊✄❾✬❼✘➈➉➈✗➎⑧❾❉➈➉➈✗➎✍❝❄➌❢❾✬➅✔➂➍➅➉❼✹❛➁❝✹ä❩❼✹➂➃❝✩❵✜➈■➅❧❾✁❞✗❝❽❵✍❛❿➈ å ➄❢➅✔➈✵æ✗➙❢❛✁➄✍❞✼➂➃❝✩❞✗➹➷❡➃➂❆ç✬❝☞è❋❼✹❛➁❝✹ä❩❼✩➂❆❝☞❵✬➈■➅✩➛⑧➌✍➄✒➈❽❾✁❡➃➅✗❛➝➎❢❾✳➀✬❝❀❾➚æ✔➇✍➎➁❣✒➅✔➂➍❼✩❾✁❡✸➹❍➅✗➇❢❾✁❼✩❝☞è❋➅✔➂➃➼✁❵✍➂❆é⑧❼☞❾❿❵❢❼✩❝✁ê✥➂❆➊ ➡❩➂➍➅❳❾➞➐➞❝✩➐⑤➌▲❝✩❞❳❛✁➊♦ë❥ ➛ ➜❝✯ç➁❵✍❛➜ ➊✢❞✗❛✬➐ì➈✗➎✍❝❀➆✍❝✹é❢❵✍➂❆➈✗➂➃❛✁❵➸❛✁➊❈❾❋➌❢❾✬➅✔➂➍➅❧➈✗➎❢❾❿➈ ➡❋→îíï ❴❆ð♦♥ ➡❴ ❫❴ ➑✢➭▲➔✦➯ ➎✍❛➜❝✩➀✬❝✩❞✛➛✜➡✱➑✢➭✍ñ✳➔✥→Ôòí ❴➃ðó♥ ➡❴ ❫❴ ➑✢➭✒ñ❉➔✥→✙➡✛ñ✬➛✁ô➞→➪➺✁Ò✩Ó☞Ó✩Ó✹Ò✗➣⑨➛✍➅✔➂➃❵❢❼✩❝✯➈✗➎✍❝❀❫❴ ❾✁❞✗❝❄❜✩❝☞❞✗❛➝❾❉➈s❾✁❡❆❡ ❵✍❛✒➆✒❝✛➅✥❝✹õ✍❼✩❝✩➇✒➈✥➭▲❴✔↕❳➑✢ö❍❵⑧➆✒❝✩❝✛➆✇➛✁➈✼➎✍❝s❫❈❴◗❼☞❾❿❵➞➌▲❝✏➄✍❵✍➂➍÷✜➄✍❝✩❡➃❣➸á✬ø✄ùÐß▲ø■á❋➌➁❣⑤➈✼➎✍❝s❼✩❛✁❵❢➆✒➂❆➈✗➂➃❛✁❵⑧➅✥❫❈❴⑨ú ë ❥ ➛♦❫❈❴✗➑✄➭ñ ➔✦→üû☞❴ñ ➛óÏ❄→ý➺✁Ò☞Ó✩Ó☞Ó✹Ò✔➣⑨➯♦➎✍❝✩❞✼❝❩û✩❴ñ ➂➃➅❄➈✼➎✍❝✵þ✯❞✼❛✁❵✍❝✛❼■ç✁❝✩❞✗➹❍➆✒❝✩❡❆➈✼❾❩➅✗❣➁➐⑤➌▲❛✁❡➷↕ ➔ ♣❛❿➈✗❝ ➈✗➎⑧❾❉➈✥➈✼➎✍❝✩❞✼❝✏➂➍➅⑨❵✍❛➒æ✼❫óÿ✛è✯❛✁❞❀æ✼❫ í♠ó♥✘è❀➂❆❵➞➈✼➎✍❝✏➌❢❾✬➅✔➂➍➅✥➅✗➂❆❵❢❼✩❝ ➜❝❳➐⑤➄❢➅✔➈Ð➎❢❾✳➀✬❝❳➡▲➑❶➱✬➔✥→↔➡▲➑✔➺✛➔✥→➫➱ ➊✢❛✁❞❳➡✵ú✶ë❥ ↕ Õ➉➎➁➄❢➅❽➡❿❴➏→Û➡✱➑✢➭▲❴❍➔✘➛✁✄✂❢ø✆☎✳â❿ã✞✝❢ø❩à✠✟s➡❙â✡⑨➭➟→Û➭▲❴✔➛▲➈✗➎❢❝❃Ï☞☛✍✌❩❵❢❛➁➆✍❝✁➯➋❾❿❵⑧➆ ò í ❴➃ðó♥ ➡❿❴▲❫❈❴✗➑✄➭▲➔ æ✗❼✩❛✁❵✍❵❢❝☞❼✘➈■➅✗è➝➈✗➎✍❝❃➀✳❾✁❡❆➄❢❝☞➅❽❛❿➊✥➡➸❾❉➈s➈✼➎✍❝❋❵✍❛✒➆✒❝☞➅ ➜➂✸➈✼➎➒❡➃➂❆❵✍❝✛❾❿❞✯➅✔❝☞➼✁➐➞❝✩❵✜➈■➅✏❛✁❵✲❝☞❾✁❼■➎➾❝☞❡❆❝☞➐➞❝✩❵✜➈☞↕ ö➴➈❽➂➍➅s➈✗➎✍❝☞❵✲➇❢❾❿➈✗❝✩❵✜➈✼❡❆❣➾❼✹❡➃❝☞❾✁❞✏➈✼➎❢❾❉➈ ➜❝❋❼✩❾❿❵✲❞✗❝☞➇✍❞✗❝✛➅✔❝☞❵✜➈❽❾❿❵➁❣✶➇✍➂❆❝✛❼✹❝➜➂➃➅✗❝✹➹➴❡❆➂➃❵✍❝✛❾❿❞s❼✩❛✁❵✜➈✗➂➃❵➁➄✍❛✁➄❢➅ ➊✢➄✍❵❢❼✹➈✗➂➃❛✁❵×➡➚➈✗➎❢❾❿➈❋➀❉❾✁❵✍➂➃➅✗➎✍❝✛➅❋❾❿➈➝➭✆→❒➱❂❾❿❵❢➆❙➭✆→ ➺✶➌➁❣➚➈✗➎✍❝➾❼■➎✍❛✬➂➃❼✩❝❤➡❿❴⑤→ ➡▲➑✄➭⑧❴➴➔✘➛➏Ï➞→ ➺✁Ò☞Ó✩Ó☞Ó✹Ò✔➣⑨↕❂➙✍➄✍❞✗➈✗➎✍❝☞❞✗➐➞❛✬❞✗❝✬➛ó➈✼➎✍❝✶➡❴ ❾✁❞✗❝❩➄❢❵✍➂➃÷✜➄✍❝✏✎ ❵✍❛➟❼■➎✍❛✬➂➃❼✩❝❩❝✹õ✍❼✩❝✩➇✒➈❋➡❴ → ➡✱➑✢➭❴ ➔ ➜➂➃❡❆❡ ➜❛✁❞✼ç▲↕♦ö➴➈❳➈✗➎➁➄❢➅❳➊✢❛✁❡➃❡❆❛➜➅❧➈✼➎❢❾❉➈❳➈✼➎✍❝❀❫◗ñ✯❾❿❞✼❝✯➂❆❵⑧➆✒❝✩❝✛➆✶❾❋➌❢❾✬➅✔➂➍➅☞↕ Õ➉➎✍❝☞❞✗❝✦❾✁❞✗❝❽➐➞❾✁❵➁❣❩❛❿➈✼➎✍❝✩❞➉➇▲❛✬➅✼➅✔➂➃➌✍❡➃❝❄❼■➎✍❛✬➂➃❼✩❝☞➅➏➊✢❛✁❞➉➌❢❾✬➅✔➂➍➅✑✎ ➜❝✯❝✩õ➁➇❢❡❆❛✬❞✗❝❄❾❃➇⑧❾❿❞✗➈✗➂➍❼✹➄✍❡➍❾❿❞✼❡❆❣ ➄❢➅✗❝✹➊✢➄✍❡➏❛✁❵✍❝➝➂➃❵➶❾➸❡➍❾❉➈✗❝☞❞✦❝✹õ✒❝✩❞■❼✹➂➍➅✗❝✁↕❤➬s❛➜❝☞➀✁❝☞❞☞➛✇➈✼➎✍❝❩❵✍❛✒➆✍❾✁❡✥❞✼❝✩➇✍❞✼❝☞➅✗❝✩❵✜➈✼❾❿➈✗➂➃❛✁❵➟❞✗❝☞➐➞❾✁➂❆❵⑧➅✯➈✗➎✍❝ ➐➞❛✬➅✔➈♦❼✩❛✁➐➞➐➞❛✁❵➋➛☞é❢❞■➅➘➈♦➌⑧❝✛❼✩❾❿➄⑧➅✔❝✥❛✁➊✒➈✗➎✍❝❧❼✹❛✬❵✜➀✬❝✩❵✍➂➃❝✩❵✜➈➋➂➃❵✜➈✗❝✩❞✼➇✍❞✼❝✹➈■❾❉➈✗➂➃❛✁❵❀❾✬➅➋❵✍❛✒➆✍❾✁❡✬➀❉❾✁❡❆➄✍❝✛➅✩➛❉❾✁❵❢➆ ➅✗❝☞❼✹❛✬❵❢➆❤➌⑧❝✛❼✩❾❿➄⑧➅✔❝✯❛❿➊♦➈✗➎❢❝✦➐➞❾❿➈✗❞✼➂✸õ✶➅✔➇❢❾✁❞✼➅✗➂❆➈➘❣➝➂❆❵❢➆✍➄❢❼✹❝✛➆❤➌➁❣➝➈✼➎✍❝✦➐➞➂❆❵❢➂❆➐➝❾❿❡◗❛❉➀✁❝✩❞✼❡➍❾❿➇❩➌▲❝✹➈➜❝✩❝☞❵ ➈✗➎❢❝❀❫ñ ↕ r