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UBC Theses and Dissertations

Anisotropic adaptation: metrics and meshes 2008

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l b; l cbjb•uˆbj4 l * f bdc28;<=h*-9n4 ” bˆ,n4b28cb œ‹\ž ŸrŸ¡£¢(¤ ;*u2¥‘m=gu•=g¦Žb˜–:,280g=g; 'ry ¡§¢/œŽ¨‹©dª‰¡§¢(¤ 28;2 f/f c*Ž‘G=gu28;b k =gc k ,/u k bd9(;bdc4;*"i/b k cbŽ2-4b k bd0g0›4<=g¦Žb y 289ni¬« ¨‹©‹ª­¨®•©¨‹¯ˆ° « ž-± ;*™bd0g=gu•=g928;bL4<u280g0bjim.rbj4² K b k 28,n4b6; l bdcb628cb9n*™.1,28c289:;bjbj4 *-9ˆ; l bcbj4,/0g;4 *8e\; l bj4buˆ*Gim=³Ł k 28;<=h*-9‡;*G*-0h4 y” b",n4b"; l bdu´=g;bdc28;<=gqrbd0 ' ;* f c*Gim, k bˆ2^–:,2-4<= Œ * f ;<=gu280$uˆbj4 l ˜†‡b f cbj4bd9(; b‹‘n28u f 0hbj4 i/bduˆ*-9n4;<c28;<=g9/.L; l 28;*-,/c289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9:;X280g.r*-c<=g; l u!=gu f c*²qrbj4 4*-0g,/;<=h*-9 2 kjk ,/c2 k ' e“*-c & *-; l 4b k *-9niI289niI; l =gciw*-ci/bdcL4*-0g,/;<=h*-9n4 k *-u f 28cbji ” =g; l ,/9/=³e‰*-c<uµcb‹Łn9nbduˆbd9:;˜289ni =h4*-;<c* f = k cb‹Łn9nbduˆbd9:;Ž$†‡bL280h4*’289280 ' ¦Žb; l bb‹¶›b k ;\*8e›,n4<=g9/.’4b k *-9niˆi/bdc<=gq128;<=gqrbj4\e‰*-c$cb‹Łn9/=g9/.; l =gci *-ci/bdc64*-0g,/;<=h*-9n4Ž =g= · ¸º¹ U» ¹ ‚} ¼½ ¾Ž¿ŽÀ¥Á/Â1¿ ˜™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ =g= ÒÄÅ¿ŽÆ(Å¿Ž¾ ˜™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ =g=g= ÇEÈ ¾Ž¿ˆÄÉ6ÊEÁ/½Ë£Æ(¾ ™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ qG= ÇEÈ ¾Ž¿ˆÄÉ6Ì È£ÍÎ À8Æ:¾ ™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ qG=g= ¼Â8ÏWÅVÄGАˣÆ:Ñ ÍÒ Æ(Å¿Ž¾ ˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ =³‘ Ó Æ(Ñ È ÂrÁm¿ È ÄÅ ™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ‘ ÒÄnÔ ¼ Î ¿²ÕVÄÀ¥¾¥Õ È£Öa× ¿ŽÁm¿ŽÆ Ò Æ(Å¿ ™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ‘m= Ø Å¿ŽÀ8Ä›Ñ Î Â1¿ È ÄÅ ˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ù s *-u f ,/;28;<=h*-9280 T 0g,/=hiw) ' 928u•= k 4 ˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ù S * k 280 R b k *-9n4<;<c<, k ;<=h*-9N289ni B c<c*-c B 4<;<=gu28;<=h*-9 ™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ú s *-u•u,/9/= k 28;<=g9/.^; l b > 9/=h4*-;<c* f ' 289ni s cbŽ28;<=g9/.^289 > 9/=h4*-;<c* f = k™M bj4 l ˜’˜™™˜™ Ù x ? l b > 9/=h4*-;<c* f = k™M bj4 lw> in2 f ;28;<=h*-9 S *(* f ™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ù v ¼#Û^ÆGÅVÆ(À¥ÁnË È£Ü Æ:Ña̛À8Á Ò Æ:ÐÄÀ8ÏfiÉ­ÄÀÝ È£Í ÕßޕÀ¥ÑÆ:À^¼Å È ¾ŽÄn¿ŽÀ8Ä Ö7È Â¬à!Æ(¾²Õ…¼ÑÁ Ö ¿ŽÁm¿ È ÄÅ Ùdá F 9:;<c*Gim, k ;<=h*-9 ™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ùdá B c<c*-c B 4<;<=gu28;<=h*-9#˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ÙŽÚ =g=g= â\ãŽäåQæäå“ç s 280 k ,/0328;<=g9/.^; l b M bd;<c<= k ˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ù z R bj4<,/0g;4 ˜™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ v á D , & 4*-9/= k =g9(q(=h4 k =hiè* ” 28c*-,/9ni]289]28=gcHe‰*-=g0鍍˜™™˜’˜™˜™™˜™˜’˜™™˜™ v Ú ? c289n4*-9/= k]F 9(q(=h4 k =hiêè* ” 28c*-,/9ni…289Z28=gcHe‰*-=g0é o F6ë ,n4; ” c*-;b¬; l =h4ˆ4<, & 4b k ;<=h*-9…4* ; l bdcb’28cb˜4*-uˆb l *-0hbj4L289ni¬032 kì(Œ *8e Œ è* ” f c* & 0hbduˆ4 p ™˜™˜’˜™™˜™ v1z D , & 4*-9/= k 0328u•=g928cè* ” 28c*-,/9niN289I28=gcHe‰*-=g0 ˜™™˜’˜™˜™™˜™˜’˜™™˜™ í x s *-9 k 0g,n4<=h*-9n4!˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ í1î Ó ÆG˧Á Î Å ÁrïÔðÁ/¾ŽÆ:Ña¼^Å È ¾ŽÄ¿ŽÀ¥Ä ÖÈ Â¬à@Æ:¾²Õ…ñÆ:ò7Å Æ Ò Æ(Å¿ ˜’˜™˜™™˜™˜’˜™™˜™ í z F 9:;<c*Gim, k ;<=h*-9 ™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ í z M bd;<c<= k 4 f 2 k bj4@™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ í1ó [ ,280g=g; ' M bŽ2-4<,/cbj4 ™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ánÙ M bj4 l * f bdc28;<=h*-9n4 ˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á:í D ” 2 f/f =g9/. ™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á:í ôXbdc<;b‹‘ F 9n4bdc<;<=h*-9 ’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á1á ôXbdc<;b‹‘ R bduˆ*Žq1280 ’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á:Ú K *-,/9nin28c ' 5$c*-;b k ;<=h*-9 ’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á(õ > 0g.r*-c<=g; l u ˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á z R bj4<,/0g;4 ˜™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ î x s *-9 k 0g,n4<=h*-9n4+289ni T ,/;<,/cb F u f c*²qrbduˆbd9:;4ö’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ îrõ ÒÄÅVÂ:Ë Î ¾ È ÄÅ ¾ ˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ î1ó D ,/u•u28c ' ™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ î1ó R bj4<,/0g;4 ˜™˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ú x R b k *-u•uˆbd9nin28;<=h*-9n4 ™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ÚmÙ =gq â\ãŽäåQæäå“ç ð È ½Ë È Ä Í À¥Á Ö Õï ˜™™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ Ú8á q ÷ ø }I ¸„ù ú û|~ü7»@} Ù s *-u f 28c<=h4*-9*8e/imc28. y 0g=³e£; y 289ni™4b f 28c28;<=h*-9 f *-=g9:;›e“*-c PC>6s`> x1x Ù v 28=gcHe‰*-=g0éý Mach = 0.5 y Re = 5000 y α = 0 ]’˜™˜™˜™™˜’˜™™˜™˜™˜’™˜™˜™˜™’˜™ ímÙ qG= ÷ ø }I ¸„ù þ ø7ß €U»} Ù F 0g0g,n4;<c28;<=h*-9]*8e ; l b s`T ) f c* k bj44 ” =g; lI> in2 f ;<=gqrb S *G* f  ˜™™˜™˜’˜™™˜™ v v ? l b™bdc<c*-cL289niI2-44* k =328;bji T *-,/c<=hbdcC2 f/f c*j‘m=gu28;<=h*-9W ˜™˜™™˜™˜’˜™™˜™ v1v í B c<c*-c k *-9(;*-,/c4L289ni d(x, y)3 k *-9:;*-,/c4²™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ v1v á PL>6s`> x1x Ù v ý F 9/=g;<=3280›uˆbj4 l ” =g; l َórõ"ôXbdc<;<= k bj4Ž™™˜’˜™˜™™˜™˜’˜™™˜™ v î î PL>6s`>6Œ x1x Ù v 4<, & 4*-9/= k =g9:qG=h4 k =hiè* ” ý k *-u f 28c<=h4*-9n4L*8e7imc28.289ni0g=³e‰; k *Gb k =hbd9:;4 v Ú Ú F 9(q(=h4 k =hi D , & 4*-9/= k è* ” 28c*-,/9ni PC>6s`> x1x Ù v ý Mach = 0.5 y α = 0  ˜’™˜™ v õ õ S =³e‰; s *-9:qrbdc<.rbd9 k b’e‰*-c PC>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o  ™˜™˜’˜™™˜™ v1z z í8ci *-ci/bdc M 2 kl 9:,/u & bdc f c*8Łn0hbj4]*-9@2-in2 f ;bjißuˆbj4 l 289ni ,/9/=³e‰*-c<u uˆbj4 l e“*-c PL>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o ’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ v ó ó 5$cbj44<,/cb’im=h4<;<c<= & ,/;<=h*-9]*-9w, f/f bdc4<,/cHeé2 k b™*8e PC>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o m í x Ù x > 9/=h4*-;<c* f = k qm4Ž F 4*-;<c* f = k imc28.š289ni‚0g=³e£; k *-9:qrbdc<.rbd9 k be‰*-c PL>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0 ˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ ímÙ Ù1Ù > 9/=h4*-;<c* f = k qm4Ž F 4*-;<c* f = k 4b f 28c28;<=h*-9 f *-=g9(; k *-9(qrbdc<.rbd9 k b`e‰*-c PL>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0 ˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ í v Ù v s *-u f 28c<=h4*-9š*8e\289/=h4*-;<c* f = k 289niI=h4*-;<c* f = k ; l =gciI*-ci/bdc M 2 klIf c*8Łn0hb’e‰*-c PL>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0 ™˜’˜™˜™™˜™˜’˜™™˜™ í v َí s *-u f 28c<=g9/.t0g=³e£;289ni‚imc28. k *-9(qrbdc<.rbd9 k be“*-c PC>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  ˜™™˜™˜™˜’™˜™˜™™˜’+’˜™™˜™˜™˜’™˜™ í1í Ùdá ? c<=3289/.10hb•2-4 f b k ;6c28;<=h*146e£c*-u„í8ci]*-ci/bdc+289ni v 9niI*-ci/bdcCuˆbd;<c<= k 4+28;60hbŽ2-im=g9/.bjim.rb *8e PC>6s`> x1x Ù v 28=gcHe‰*-=g0éý Mach = 0.5 y Re = 5000 y α = 0 fi™™˜™˜’˜™™˜™ í8á qG=g=  çŽå ã  
 æç َî ? l b™–(,280g=g; ' 4 f 2 k b˜,n4bji;*bdq-280g,28;b˜;<c<=3289/.10hbj46=g9¬; l b˜uˆbj4 l 7™˜™˜’˜™™˜™ á:í ÙŽÚ ? bj4<;<=g9/.e‰*-c B u f ; ' s =gc k ,/u k =gc k 0hbj46=g9I; l b M bd;<c<= k’D(f 2 k b ˜™™˜™˜’˜™™˜™ á:î Ù²õ T =g9nim=g9/.289¬2 f/f c*Ž‘G=gu28;b k =gc k ,/u k bd9:;bdc6*8eV;<c<=3289/.10hb ? Ù &(' k *-9n4<;<c<, k ;<=g9/.•; l bCqG=gc Œ ;<,280Wuˆbj4 l ™˜™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ á:Ú Ù z F 4*-;<c* f = k y uˆ*(im=³ŁbjiIim=328uˆbd;<c280V0hbd9n4 e“*-c k ,/c<qrbji & *-,/9nin28c ' ™™˜™˜’˜™™˜™ á z َó B ‘ f b k ;bji ? c<=3289/.10hb2-4 f b k ;c28;<=h*14.1=gqrbd9I; l b˜uˆbd;<c<= k e‰c*-u B –(,28;<=h*-9Uَóm ™™˜™ î x v1x > 4–:,28cbuˆbj4 l & bd=g9/.¬cb k ,/c4<=gqrbd0 ' cb‹Łn9nbjiU2 kjk *-cim=g9/.w;*; l b•uˆbd;<c<= k i/b‹Łn9nbji &:' B –(,28;<=h*-9‚ÙŽóm#™˜’˜™™˜™˜™˜’™˜™˜™™˜’˜™˜™™˜™˜’˜™™˜™ îmÙ v Ù M bj4 l e£c*-u T =g.1,/cb v1x b™q(=hb ” bji ” =g; l im=³¶›bdcbd9:;L2¥‘m=h44 k 280g=g9/.r4Ž ˜™˜’˜™™˜™ î v v1v PL>6s`> x1x Ù vmy M 2 kl x  î y R b  î x1x1xmy َórõôXbdc<;<= k bj4 ’˜™˜™™˜™˜’˜™™˜™ î1í v í F 9(;bdc<uˆbjim=328;b M bj4 l bj46289ni D *-0g,/;<=h*-9n4²ý PC>6s`> x1x Ù vmy M 2 kl x  î y R b  î x1x1x ˜™ î8á v á s 0h*14b˜, f qG=hb ” *8e7; l b™4<;28.1928;<=h*-9 f *-=g9:;6*8e7; l b+Łn9nbj4<;uˆbj4 l =g9 T =g.1,/cb v íˆ2m ™ î1î v î D 28u f 0hb [ ,280g=g; ' )C=h4<;<c<= & ,/;<=h*-9N)C,/c<=g9/. R b‹Łn9nbduˆbd9(;C289ni s *r28c4bd9/=g9/.’˜™™˜™ î1Ú qG=g=g= { ƒ ¹ ¸ ü7» ß » ¹ ‚} T =gc4<;289ni‡e‰*-cbduˆ*14; Fff iš0g= ì bˆ;*I; l 289 ì u ' 4<, f bdc<q(=h4*-c y )CcŽ s 28c<0 Y 0g0g=gq(=hbdc ŒQ_ *(* kl y7” l *14bˆ;<=guˆb 289ni f 28;<=hbd9 k b ” 2-4•=g9:q1280g,2 & 0hbe‰c*-u =g9 k b f ;<=h*-9 ;* k *-u f 0hbd;<=h*-9Z*8eL; l =h4•; l bj4=h4Ž Fff ifi280h4*t0g= ì b;* ; l 289 ì u ' 032 & Œ u28;bj4 sEl c<=h4 y D bdc<.rb"289ni > u•=gcŽ †…=g; l *-,/; ' *-,/c l bd0 f ” =g; l u ' 9:,/uˆbdc*-,n4 k *Gim=g9/. f c* & 0hbduˆ4 y kl 289 k bj4628cb Fff i9nbdqrbdc l 2ŽqrbLŁn9/=h4 l bji› F$k *-9n4<=hi/bdc u ' 4bd0³e b‹‘ k b f ;<=h*-9280g0 ' 0g, kì ' ;* l 2Žqrb ” *-c ì bji ” =g; l 280g0›; l cbjb™*8e ' *-,W Fff i280h4*’0g= ì b;*˜; l 289 ì u '•” =³e‰b)˜289/=hbd0g0hb y e“*-c\bd9nim,/c<=g9/."4*˜u289 ' k *-,/9(;<0hbj44 i/bd032 ' 4`289ni f c*²q(=him=g9/. 2"0g=gu•=g;<0hbj4464<, f/f 0 ' *8e70h*Žqrb289ni¬4<, f/f *-c<;Žflfi\*-,¬9n*-;6*-9/0 ' l bd0 f bjiuˆb+; l c*-,/. l *-,/;6u ' i/bd.1cbjb y& ,/; u2-i/b™4,/cb+; l b ” l *-0hb f c* k bj44 ” 2-4Lbd9 ë * ' 2 & 0hb1 T =g9280g0 ' Fff i0g= ì b6;*’; l 289 ì u ' e“28u•=g0 ' e“*-c\; l bd=gc bd9nim0hbj44`bd9 k *-,/c28.rbduˆbd9:;Ž F ; 280 ” 2 ' 4 l bd0 f 4E;* l 2Žqrb ' *-,/c6* ” 9 f bdc4*-9280 kl bjbdc<=g9/.^4–(,2-i› =³‘ ffi» ø ƒß  ø ¸„¹ ? *•u ' uˆ*-u y/” l *14b˜4;<cbd9/.1; lwl 2-4 & bjbd9I289w=g9n4 f =gc28;<=h*-9W ‘ · ¸  {  "! ¸ €ê}#! ø $ % ‚ U»  » ¹  > 0g0*8e˜; l b¬cbj4bŽ28c kl 289ni in28;2‡289280 ' 4<=h4 k *-9(;28=g9nbji ” =g; l =g9…; l =h4•; l bj4<=h4 ” bdcbI2 kjk *-u f 0g=h4 l bji &:' )+*-,/.1032-4’5728.19:,/;<;<=é ? l bˆu289:,n4 k c<= f ; f cbj4bd9:;28;<=h*-9 ” 2-4"280h4*wi/*-9nb &(' )+*-,/.1032-4’5728.19:,/;<;<= ” =g; l 4<=g.19/=³Ł k 289(; k *-9(;<c<= & ,/;<=h*-9n4e‰c*-uµ)CcŽ s 28c<0 Y 0g0g=gqG=hbdc ŒQ_ *(* kl  ‘m= &¹ U€ ¸   ƒ… ø ¸„¹ 5728c<;<=3280Vim=³¶›bdcbd9:;<=3280 bj–(,28;<=h*-9n4 o 5X) B 4‹p28cb˜*-9nb˜*8e ; l b+uˆ*14<;,n4b‹e£,/0›;*G*-0h4 &:'^” l = kl ” b™i/bj4 k c<= & b *-,/c•bd9:qG=gc*-9/uˆbd9:;Ž ? l bj4bbj–(,28;<=h*-9n4 y ; l 28;•cbd0328;be£,/9 k ;<=h*-9n4ˆ289niê; l bd=gcˆi/bdc<=gq128;<=gqrbj4 y 280g0h* ” 4 k = Œ bd9(;<=h4<;4ˆ289ni bd9/.1=g9nbjbdc4;*Iuˆ*Gi/bd0 l * ” ; l =g9/.r4 & b l 2Žqrbw289niê; l ,n4 & bd;<;bdc f cbjim= k ; l * ” ; l b 'U” =g0g0 & b l 2jqrb=g9š; l be£,/;<,/cb1 J 9me“*-c<;<,/928;bd0 'ry f cbjim= k ;<=g9/.]4<, kl & b l 2jqG=h*-,/c’cbj–(,/=gcbj4™9n*-; ë ,n4<;™i/bdc<=gqG=g9/. ; l b .r*²qrbdc<9/=g9/.bj–(,28;<=h*-9n4 y1& ,/;E280h4*˜4*-0gq(=g9/.+; l bdu 2 kjk ,/c28;bd0 ' 289ni•b k =hbd9:;<0 '  T *-cX4*-uˆb5X) B 4 y =g; =h4 f *144<= & 0hbC;*i/bdc<=gqrb™289280 ' ;<= k 4*-0g,/;<=h*-9n4 ; l 28;28cb+bŽ2-4 ' ;*,n4b˜289ni^=g9mŁn9/=g;bd0 ' 2 kjk ,/c28;b1 F 9*-; l bdc k 2-4bj4 y =g;"u2 'U& b f *144<= & 0hb;*š2 f/f c*j‘m=gu28;b¬; l b^4*-0g,/;<=h*-9 o *-c ì b ' f c* f bdc<;<=hbj4•*8eL; l b^4*-0g,/;<=h*-9p ” =g; l 4<=gu f 0hb k ,/c<qrbj4˜289niN; l bd9N;* kl *G*14b f 28c28uˆbd;bdc4+e“*-cL; l *14b k ,/c<qrbj4 & 2-4bjit*-9tb‹‘ f bdc<=guˆbd9:;280 289280 ' 4<=h4Ž ? l =h4`4b k *-9niuˆbd; l *Gi y l * ” bdqrbdc y l 2-4E4<=g.19/=³Ł k 289(;E0g=gu•=g;28;<=h*-9n4 =g9;bdc<uˆ4 *8eW2 kjk ,/c2 k ' 289ni k *14<;Ž > ; l =gciêuˆbd; l *(ifi*8eC4*-0gq(=g9/.š5X) B 4(' 289ni *-9nb ” l = klfil 2-4c<=h4bd9ê=g9 f * f ,/0328c<=g; ' im,nb;* =g9 k cbŽ2-4<=g9/. k *-u f ,/;bdc f * ” bdc)' =h49:,/uˆbdc<= k 2806289280 ' 4<=h4Ž ? l b ì b ' ;*N9(,/uˆbdc<= k 2806289280 ' 4<=h4=h4";* & cbŽ2 ì , f 2^0328c<.rb289ni k *-u f 0g= k 28;bji f c* & 0hbdu—=g9:;*wu289 ' 4<u280g0 y 4<=gu f 0hb f c* & 0hbduˆ4™; l 28; k 289 & b f c* k bj44bjiI; l c*-,/. l 2 k *-u f ,/;bdcŽ† l =g0hb4, kl uˆbd; l *(i/4 l 2jqrb"cb k bd9:;<0 ' 4bjbd9]imc28u28;<= k =gu f c*Žqrb Œ uˆbd9(;4L=g9Nb k =hbd9 k 'ry ; l bdcb™=h4C4<;<=g0g024<;<c*-9/.i/bj4<=gcb’=g9I=g9nim,n4<;<c ' e“*-c69(,/uˆbdc<= k 280X289280 ' 4<=h4C4*8e‰; ” 28cb ; l 28; k 289 f c*(im, k b˜uˆ*-cb2 kjk ,/c28;b™4*-0g,/;<=h*-9n4uˆ*-cb’–:,/= kì 0 '  * ,+„
(-’	 	6/.102.3-"5476+",+º Y 9nb f 28c<;<= k ,/0328cˆ4bd;*8e65X) B 4; l 28;=h4•2we‰* k ,n4"*8e69:,/uˆbdc<= k 2806289280 ' 4<=h4"=h4’; l *14b.r*²qrbdc<9/=g9/.tèn,/=hi & b l 2jqG=h*-,/cŽ ? l bj4b5X) B 4™28cb"*8e£;bd9t;*G*im= k ,/0g;Ce“*-c+289280 ' ;<= k 4*-0g,/;<=h*-9n4™289niN; l bi ' 928u•= k 289ni u,/0g;<= Œ 4 k 280hb™928;<,/cb™*8e7; l b™4*-0g,/;<=h*-9n4 k 289¬u2 ì b’b‹‘ f bdc<=guˆbd9(;280 289280 ' 4<=h46im= k ,/0g;L289ni k *14<;<0 '  K b Œ k 28,n4b*8e; l =h4 y 289êbd9(;<=gcbˆŁbd0hiê*8e64<;<,ni ' l 2-4 & bjbd9 k cbŽ28;bjiê;* f c*Gim, k b9(,/uˆbdc<= k 280 4*-0g,/;<=h*-9Ue“*-c f c* & 0hbduˆ4`=g9:qr*-0gqG=g9/."èn,/=hi/4Ž ? l =h4\Łbd0hi*8eV4<;<,ni ' =h4Ecb‹e“bdc<cbji;*2-4 k *-u f ,/;28;<=h*-9280èn,/=hii ' 928u•= k 4 Ù 8äå  ã:9 
<; å  ãŽä Adaptive Loop Solver Mesh Input Parameters SolutionMesh Generation Numerical Tesselation of Problem Domain Desired Accuracy Boundary Conditions Problem Geometry Governing Equations Piece−wise Approximation to the Solution T =g.1,/cbˆÙ1ý F 0g0g,n4<;<c28;<=h*-9I*8e7; l b s`T ) f c* k bj44 ” =g; lI> in2 f ;<=gqrb S *G* f  o s`T )™p‹ ? l b".r*r280$*8e s`T ) =h4C4=gu f 0hb1ý6;* f c*Gim, k b2 kjk ,/c28;b•4*-0g,/;<=h*-9n4+=g9t4 l *-c<;˜28uˆ*-,/9(;4™*8e\;<=guˆb1 ? l =h4 ; l bj4<=h4 k *-9:;<c<= & ,/;bj4+;*; l 28;L.r*r280 &:' k *-u & =g9/=g9/.; ” *^cb k bd9(;+i/bdqrbd0h* f uˆbd9(;4+=g9 s`T )ý289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9t289ni l =g. lmŒ *-ci/bdcC2 kjk ,/c28;b’uˆbd; l *(i/4 o *8e7*-ci/bdcL.1cbŽ28;bdc; l 289w; ” *:p‹ ? l b s`T ) f c* k bj44 k 289 & b™; l *-,/. l ;™*8e`2-4+289N=g9(;bdc2 k ;<=h*-9 & bd; ” bjbd9š; l cbjb"* & ë b k ;4²ý; l b f c* & 0hbdu uˆ*Gi/bd0 y ; l b’uˆbj4 l y 289niI; l b™9:,/uˆbdc<= k 280X2 f/f c*Ž‘G=gu28;<=h*-9W ? l bj4b™; l cbjb’* & ë b k ;4+28cb™=g0g0g,n4<;<c28;bji &:' ; l b & *j‘/bj4›=g9 T =g.1,/cbÙ1 ? *.rbd9nbdc28;b`2 4*-0g,/;<=h*-9 y ; l bX4<;289nin28ci’2 f/f c*r2 kl =h4›;*uˆ*(i/bd0r; l b f c* & 0hbdu y k cbŽ28;bC2+uˆbj4 l e“*-cX; l buˆ*Gi/bd0hbjii/*-u28=g9 y 289niˆ; l bd92 f/f c*Ž‘G=gu28;bC; l buˆ*Gi/bd0hbjibj–(,28;<=h*-9n4\*-9; l b uˆbj4 l  ? l bj4b™28cb˜=g0g0g,n4<;<c28;bji &(' ; l b M bj4 l]_ bd9nbdc28;<=h*-9]289ni D *-0gqrbdcL28c<c* ” 4L*8e T =g.1,/cbˆÙ1 ? l bŁnc4<;˜4<;b f =g9t.rbd9nbdc28;<=g9/.I2 s`T )4*-0g,/;<=h*-9 y 289niN; l 28; ” l = kl 4 l *-,/0hitcbj–(,/=gcb"; l b"uˆ*14<;+,n4bdc =g9 f ,/; y =h4™;*^4 f b k =³e ' ; l b f c* & 0hbdu & bd=g9/.¬4*-0gqrbji› > ;’2u•=g9/=gu’,/u y ; l =h4˜=g9 k 0g,ni/bj4˜; l b.r*Žqrbdc<9/=g9/. bj–(,28;<=h*-9n4 y ; l b f c* & 0hbdu .rbj*-uˆbd;<c ' 289ni; l b & *-,/9nin28c ' k *-9nim=g;<=h*-9n4Ž$† l =g0hb+; l =h44bjbduˆ4 cbd0328;<=gqrbd0 ' 4<;<c28=g. l ;He‰*-c ” 28ci y =g; =h4*8e‰;bd9^; l bLŁnc4<;4*-,/c k bC*8e7bdc<c*-c=g9 s`T ) y bj4 f b k =3280g0 ' e“*-c`è* ” 4; l 28; =g9:qr*-0gqrb k *-u f 0hb‹‘ f/l ' 4<= k 280 f/l bd9n*-uˆbd92U4<, kl 2-4ˆ;<,/c & ,/0hbd9 k bI289ni k *-u & ,n4;<=h*-9W B ‘n28u•=g928;<=h*-9…*8e+; l bj4b 4*-,/c k bj4L*8eXbdc<c*-c6=h4 ” bd0g0 & b ' *-9niw; l b4 k * f b’*8e7; l =h46; l bj4<=h4L289ni]4* ” b ” =g0g02-44<,/uˆb˜; l 28;6; l b™.r*r280 *8e s`T )!=h464<;<c<= k ;<0 ' ;*•4*-0gqrb˜; l b+uˆ*(i/bd0hbji f c* & 0hbduw K ' u2 ì =g9/.•; l =h4L2-44<,/u f ;<=h*-9 y/” b k 289¬;<cbŽ28; 4*-0g,/;<=h*-9‚bdc<c*-c™=g9ni/b f bd9ni/bd9(;<0 ' e‰c*-u´uˆ*(i/bd0g=g9/.Nbdc<c*-cŽ J 0g;<=gu28;bd0 'ry =g;’4 l *-,/0hi & b; l bˆbd9/.1=g9nbjbdc ff 4 .r*r280X;* k *-c<cb k ;<0 ' uˆ*(i/bd0\; l b f c* & 0hbdu 289ni‡; l b s`T ) f c*-.1c28u ff 4.r*r280\;* f c*ŽqG=hi/b289 2 kjk ,/c28;b v 8äå  ã:9 
<; å  ãŽä 4*-0g,/;<=h*-9we‰*-c; l 28;uˆ*Gi/bd0é D =g9 k b=g9mŁn9/=g;b"2 kjk ,/c2 k ' =g9I=g9mŁn9/=g;bj4<=gu280g0 ' 4<u280g0V;<=guˆb=h4L9n*-; ' bd; f *144<= & 0hb y 4*-uˆb’e‰,/c<; l bdcL=g9me“*-c Œ u28;<=h*-9¬u,n4<; & bC.1=gqrbd9¬;*•i/bd;bdc<u•=g9nb l * ” 2 kjk ,/c28;b+; l b˜4*-0g,/;<=h*-9w4 l *-,/0hi & b+*-c l * ” 0h*-9/.•=g; ” =g0g0 ;2 ì b;* k *-u f ,/;b1 s ,/c<cbd9:;<0 'ry ; l b•uˆ*14<; k *-u•uˆ*-9 ” 2 ' *8ei/*-=g9/.I; l =h4’=h4™;*w4 f b k =³e ' l * ” Łn9nbd0 ' ; l b f c* & 0hbdu i/*-u28=g9‚4 l *-,/0hi & bˆim=gq(=hi/bji y *-c™=g9ê*-; l bdc ” *-ci/4 y l * ” 4<u280g0E; l bˆbd0hbduˆbd9:;4*8e; l b uˆbj4 l 4 l *-,/0hi & b1 ? l buˆ*-cbw2]i/*-u28=g9Z=h4•im=gq(=hi/bji y 289ni ; l ,n4•; l b^Łn9nbdc•; l b^uˆbj4 l y ; l buˆ*-cb 2 kjk ,/c28;b’2"4*-0g,/;<=h*-9w4 l *-,/0hi & b˜289ni^; l b+uˆ*-cb+;<=guˆb+; l 28;4*-0g,/;<=h*-9 ” =g0g0›;2 ì b+;* k *-u f ,/;b1X† l =g0hb ; l =h4`cbd0328;<=h*-9n4 l = f & bd; ” bjbd9 k bd0g0›4<=g¦ŽbC289ni4*-0g,/;<=h*-9w2 kjk ,/c2 k ' =h4`.rbd9nbdc280g0 ' ;<c<,nb y =g; k 289/9n*-; f c*ŽqG=hi/b 2š–(,289:;<=³Ł2 & 0hb]uˆbŽ2-4<,/cb]*8e™2 kjk ,/c2 k ' 4=g9 k b¬0h* k 280Lim=³¶bdcbd9 k bj4=g9Zbd0hbduˆbd9:;4 l 2 f bI289ni 4<=g¦Žb k 289 imc28u28;<= k 280g0 ' kl 289/.rb; l b"4*-0g,/;<=h*-9W Y 9nb"2 f/f c*r2 kl e‰*-c+*Žqrbdc k *-u•=g9/.¬; l =h4+im= k ,/0g; ' =h46;*cb f bŽ28; ; l b f c* & 0hbdu ” =g; l =g9 k cbŽ2-4<=g9/.10 ' Łn9nb’uˆbj4 l bj46,/9(;<=g0 ; l b’q-28c<=328;<=h*-9]e‰c*-uû*-9nb’4*-0g,/;<=h*-9];*^289n*-; l bdc =h4 ” =g; l =g9‡2 kjk b f ;2 & 0hb;*-0hbdc289 k b1 ? l =h4C=h4Cqrbdc ' =g9nb k =hbd9:;’289niN;<=guˆb k *-9n4<,/u•=g9/.w289nit*-9nb*8e\; l b cbŽ2-4*-9n4 ” l ' 28,/;*-u28;bji]2-in2 f ;<=gqrb’uˆbj4 l =g9/.4 l * ” 44*u’, klIf c*-u•=h4b1 Y 9 k bN; l bN.rbj*-uˆbd;<c ' 289ni uˆbj4 lak bd0g0˜4<=g¦Žb l 2jqrb & bjbd9a4 f b k =³Łbji y ; l b]9nb‹‘m;¬4<;b f =g9….rbd9nbdc28;<=g9/. s`T )4*-0g,/;<=h*-9n4+=h4+;* k cbŽ28;bˆ2uˆbj4 l *-9 ” l = kl ; l b4*-0g,/;<=h*-9š=h4C;* & b•2 f/f c*j‘m=gu28;bji› T c*-u—289 bd9/.1=g9nbjbdc ff 4 f bdc4 f b k ;<=gqrb y ; l =h4"=h4ˆ28c<.1,2 & 0 ' *-9nb^*8eC; l b^uˆ*14<;"e£c<,n4<;<c28;<=g9/.U289ni 0hbŽ2-4;",/9ni/bdc4;*(*Gi f 28c<;4*8eV.rbd9nbdc28;<=g9/. s`T )@4*-0g,/;<=h*-9n4Ž F ; k 289w280h4* & bLqrbdc ' ;<=guˆb k *-9n4,/u•=g9/.ˆ4<=g9 k bC=g;Eu2 ' cbj–(,/=gcb 20328c<.rb™28uˆ*-,/9:;L*8eV,n4bdc=g9 f ,/;Ž B 44bd9:;<=3280g0 'ry 2"uˆbj4 l =h4 9n*-; l =g9/.•uˆ*-cb+; l 289I2;bj44bd0g0328;<=h*-9w*8e; l b f c* & 0hbdu i/*-u28=g9 ” =g; l 4 l 2 f bj4`; l 28; k 289 & bLcb k *-.19/=g¦Žbji &(' ; l b˜4*-0gqrbdcŽ ? l bdcb™28cb+.rbd9nbdc280g0 ' ; ” * im=³¶›bdcbd9:;+; ' f bj4C*8eXuˆbj4 l bj4Žý4<;<c<, k ;<,/cbjit289niN,/9n4<;<c<, k ;<,/cbji› D ;<c<, k ;<,/cbjituˆbj4 l bj4 k *-9n4<=h4<;L*8e k bd0g0h4 ” l *14bLcbd0328;<=h*-9;**-9nb+289n*-; l bdc y *-c k *-9/9nb k ;<=gq(=g; 'ry =h4\=gu f 0g=hbji &:' ; l bd=gc`9(,/u & bdc<=g9/.n T *-c b‹‘n28u f 0hb y =g9•4<;<c<, k ;<,/cbji"uˆbj4 l bj4 y k bd0g0 (i, j) =h4$280 ” 2 ' 4$;* f *-0h*-.1= k 280g0 ' ;*C; l bE0hb‹e‰;7*8e k bd0g0 (i+1, j)  s *-9(qrbdc4bd0 'ry ,/9n4<;<c<, k ;<,/cbji"uˆbj4 l bj4 cbj–(,/=gcb k *-9/9nb k ;<=gqG=g; ' ;* & bEi/b k 0328cbjib‹‘ f 0g= k =g;<0 '  ? l bX=gu f 0g= k =g; k *-9/9nb k ;<=gq(=g; ' *8eX4<;<c<, k ;<,/cbji]uˆbj4 l bj4+280g0h* ” 4Le‰*-cCbŽ2-4<=hbdc f c* k bj44<=g9/. &:' ; l b4*-0gqrbdc & ,/;L=g; f 032 k bj4Lcbj4<;<c<= k ;<=h*-9n4+*-9 ; l b;* f *-0h*-. ' *8e\; l buˆbj4 l  K b k 28,n4b,/9n4;<c<, k ;<,/cbji]uˆbj4 l bj4˜i/*9n*-; l 2jqrb; l =h4Ccbj4<;<c<= k ;<=h*-9 y ; l b ' 28cb"u’, kl & bd;<;bdc+4<,/=g;bjiN;*uˆbj4 l =g9/.w28c & =g;<c28c ' .rbj*-uˆbd;<c<=hbj4Ž ? l =h4Cu2 ì bj4+,/9n4<;<c<, k ;<,/cbjituˆbj4 l bj4 uˆ*-cb ” =hi/bd0 ' 2 f/f 0g= k 2 & 0hbˆ289ni f *-;bd9:;<=3280g0 ' cbjim, k bj4+; l b•28uˆ*-,/9:;™*8e\;<=guˆbˆ289nitb‹¶›*-c<;Lcbj–(,/=gcbji &:' í 8äå  ã:9 
<; å  ãŽä ; l b˜,n4bdc;* f c*(im, k b’289N2 kjk b f ;2 & 0hb™uˆbj4 l  > 4’uˆbd9(;<=h*-9nbjifibŽ28c<0g=hbdc y ; l b4<=g¦Žb289ni‚4 l 2 f b^*8e k bd0g0h4"=g9‚; l buˆbj4 lfik 289 l 2Žqrb¬2Iimc28u28;<= k b‹¶›b k ; *-9Z4*-0g,/;<=h*-9 2 kjk ,/c2 k '  T *-c^4*-0g,/;<=h*-9n4ˆ; l 28;28cbw=h4*-;<c* f = k y =g;ˆ=h4.rbd9nbdc280g0 ' f cb‹e‰bdc2 & 0hbI;* l 2Žqrb k bd0g0h4˜; l 28;’28cbˆ280h4*=h4*-;<c* f = k o =hi/bŽ280g0 'ry bj–(,/=g0328;bdc280§p˜289ni ” l *14b•4<=g¦Žb=h4˜4, kl ; l 28;™; l b•bdc<c*-c f bdc k bd0g0C=h4bj–:,/=him=h4<;<c<= & ,/;bji› ? l 28;=h4 ” l ' ; l bIu2 ë *-c<=g; ' *8e™uˆbj4 l .rbd9nbdc28;<=h*-9a4*8e£; ” 28cbN;bd9ni/4;* e“* k ,n4^*-9 k cbŽ28;<=g9/. k bd0g0h4 ” =g; l ,/9/=g;28c ' 2-4 f b k ;c28;<=h*14¬289niZq128c ' =g9/.‚0hbd9/.1; l 4 k 280hbj4Ž † l bd9 ; l b 4*-0g,/;<=h*-9t=h4+289/=h4*-;<c* f = k y l * ” bdqrbdc y ; l b"i/bj4<=gcbji k bd0g04<=g¦Žb289nit4 l 2 f b™=h4C0hbj44C* & qG=h*-,n4Ž R = f/f 2>= í8á@? k *-9 k 0g,ni/bjiZ; l 28; y e‰*-cˆ0g=g9nbŽ28c=g9:;bdc f *-0328;<=h*-9 *8e™4<uˆ*G*-; l e‰,/9 k ;<=h*-9n4 y ;<c<=3289/.10hbj44 l *-,/0hi & b0h*-9/.‚=g9 im=gcb k ;<=h*-9n4 ” l bdcb+; l bLu28.19/=g;<,ni/b˜*8eV; l b+4b k *-9niim=gcb k ;<=h*-9280Wi/bdc<=gq-28;<=gqrb+=h4 4<u280g0›289ni^; l =g9 ” l bdcb ; l b+u28.19/=g;<,ni/b*8e ; l b™4b k *-9ni¬im=gcb k ;<=h*-9280i/bdc<=gq-28;<=gqrb™=h4 0328c<.rb1 F ;=h4=g9:;<,/=g;<=gqrb˜; l 28;; l bdcb˜4 l *-,/0hi & b]289280h*-.r*-,n4^cbj4<,/0g;4e‰*-c l =g. l bdc^*-ci/bdc^=g9:;bdc f *-0328;<=h*-9n4Ž F 9 sEl 2 f ;bdc y ” b k *-u & =g9nbt4<=g¦Žb]289ni –(,280g=g; ' uˆbŽ2-4<,/cbj4";*]i/b‹Łn9nb^2I–:,280g=g; ' 4 f 2 k bˆe“*-c’; l b k bd0g0h4*8e6*-,/cuˆbj4 l  K b k 28,n4b; l bdcb=h49n* k *-9n4bd9n4,n4$*-9 ” l = kl 4 f b k =³Ł k ;<c<=3289/.10hbuˆbŽ2-4<,/cb`=h4 f cb‹e‰bdc2 & 0hbBA y *-,/cX–:,280g=g; ' 4 f 2 k b =h4 k cbŽ28;bji"e‰c*-u ” =hi/bd0 ' Œ ,n4bjiNuˆbŽ2-4<,/cbj4+; l 28;˜28cb"4,/=g;bjiI;**-,/cCuˆbj4 l =g9/.w2 f/f c*r2 kl ýL;<c<=3289/.10hb k =gc k ,/u•c2-im=g,n4™289ni u•=g9/=gu,/uöbjim.rb0hbd9/.1; l  Y ,/c l ' f *-; l bj4=h4$=h47; l 28;$;<c<=3289/.10hbj4 ” =g; l =g9 k bdc<;28=g9 & *-,/9ni/47=g9•; l =h4$–(,280g=g; ' 4 f 2 k b ” =g0g0 f c*(im, k b˜uˆ*-cb2 kjk ,/c28;b’4*-0g,/;<=h*-9n4Ž ? l b"Łn9280$4<;b f =g9t.rbd9nbdc28;<=g9/. s`T )4*-0g,/;<=h*-9n4 y 289niN; l buˆ*14<; k *-u f ,/;28;<=h*-9280g0 ' =g9:;bd9n4=gqrb f 28c<; y =h4 f bdcHe‰*-c<u•=g9/.¬; l bˆ2 k ;<,280\2 f/f c*Ž‘G=gu28;<=h*-9W ? *¬i/*; l =h4 y ; l b".r*Žqrbdc<9/=g9/.Nbj–:,28;<=h*-9n428cb’Łnc4;™im=h4 Œ k cbd;<=g¦Žbji y ; ' f = k 280g0 ' ,n4<=g9/."*-9nbL*8eW; l cbjb k *-u•uˆ*-9w2 f/f c*r2 kl bj4ŽýŁn9/=g;bCim=³¶bdcbd9 k bj4 y Łn9/=g;bLbd0hbduˆbd9:;4 y *-cCŁn9/=g;b•qr*-0g,/uˆbj4Ž† l =g0hb280g0X*8e`; l bj4buˆbd; l *Gi/4 k 289 & b•289280 ' ¦Žbjit,n4<=g9/. ? 2 ' 0h*-c’b‹‘ f 289n4<=h*-9n4˜*8e ; l bC4*-0g,/;<=h*-9 y 289ni^28cb6; l bdcb‹e‰*-cb k *-u f 28;<= & 0hb ” =g; l ; l bLcbj4<,/0g;4E=g9 sEl 2 f ;bdc y =g;\=h4`4<;<=g0g0=gu f *-c<;289:; ;*cb k *-.19/=g¦Žb’; l b™im=³¶bdcbd9 k bj4 & bd; ” bjbd9]; l b˜; l cbjb˜uˆbd; l *Gi/4Ž Y 9nbEim=h4 k cbd;<=g¦²28;<=h*-9uˆbd; l *(i’=h4V; l bXŁn9/=g;b Œ im=³¶bdcbd9 k b`uˆbd; l *Gi› ? l =h4 uˆbd; l *(i"im=h4 k cbd;<=g¦Žbj4bj–:,28;<=h*-9n4 &(' cb f 032 k =g9/.˜; l bim=³¶›bdcbd9:;<=3280* f bdc28;*-cX*8e; l b5\) B ” =g; l 289ˆbj–:,/=gq1280hbd9:;\im=³¶›bdcbd9 k b* f bdc28;*-cŽ T *-c b‹‘n28u f 0hb y =³e L(f) = ∂f ∂x = limh→0 f(x+h)−f(x) h y ; l bd9w2892 f/f c* f c<=328;b˜im=³¶›bdcbd9 k b˜* f bdc28;*-c ” *-,/0hi & b L̃(f) = f(x+h)−f(x) h ” l bdcb h =h4+9n* ” 2Łn9/=g;bq1280g,nb1 F e ” bˆ2-44<,/uˆb; l 28; f(x) =h4 ì 9n* ” 9‚28; CEDBFHGJILKNMPORQSFHTUQVGEMWIHXBYZKP[]\_^BIZKNM`Qba,cd[eIZOV^fGE[gOghBIZXi,QEjB[gM`GSIZORafcdkfQEFZQEMNl m@[gjBI3nMNF^fGehBOV[o[qp rsHtvu á 8äå  ã:9 
<; å  ãŽä 2•4bdc<=hbj46*8eXim=h4 k cbd;b˜0h* k 28;<=h*-9n4 f(x + h) y ; l bd9 ” b k 289N2 f/f c*j‘m=gu28;b’; l b4*-0g,/;<=h*-9N28; f(x) ,n4=g9/. ; l b™im=³¶›bdcbd9 k b+e“*-c<u’,/0328;<=h*-9N*8e; l b™*-c<=g.1=g9280 5X) B 4Ž T =g9/=g;b Œ bd0hbduˆbd9:;"im=h4 k cbd;<=g¦²28;<=h*-9n4’28cb4<=gu•=g0328c˜;*Łn9/=g;b Œ im=³¶›bdcbd9 k bˆ*-9nbj4˜=g9t; l b4bd9n4b"; l 28;˜; l b4* Œ 0g,/;<=h*-9 f =h4"2-44<,/uˆbjiš;* & b ì 9n* ” 9‚28;2wim=h4 k cbd;bˆ4bd;’*8e 0h* k 28;<=h*-9n4™=g9‡; l b•uˆbj4 l  ? l bˆ4*-0g,/;<=h*-9 & bd; ” bjbd9¬; l bj4b60h* k 28;<=h*-9n4 f 2 f/f =h4`; l bd9¬2-44<,/uˆbji;* & bL2 f *-0 ' 9n*-u•=3280›*8e2 k bdc<;28=g9¬i/bd.1cbjb1 D =g9 k b ; l =h4 f *-0 ' 9n*-u•=3280ni/*Gbj479n*-;79nb k bj4428c<=g0 ' 428;<=h4He ' ; l b f 28c<;<=3280nim=³¶bdcbd9(;<=3280bj–(,28;<=h*-9n4 L(f) = 0 y ; l bdcb ” =g0g0 & b˜4*-uˆb˜bdc<c*-c;bdc<uˆbji¬; l b+cbj4<=him,280 L(f 2 f/f ) = 0  K ' k *-9(qr*-0gq(=g9/.; l =h4cbj4<=him,280 ” =g; l 24bd; *8e ” bd=g. l ;˜e£,/9 k ;<=h*-9n4 y =g;˜=h4 f *144<= & 0hb;* kl *(*14b f *-0 ' 9n*-u•=3280 k *(b k =hbd9(;4™; l 28; ” =g0g0$u•=g9/=gu•=g¦Žb; l b cbj4<=him,280289ni¬; l ,n4 & bd;<;bdc62 f/f c*Ž‘G=gu28;b; l b™2 k ;<,280V4*-0g,/;<=h*-9W T =g9/=g;b Œ qr*-0g,/uˆbim=h4 k cbd;<=g¦²28;<=h*-9n428cb28c<.1,2 & 0 ' ; l b•; ' f b•*8e`im=h4 k cbd;<=g¦²28;<=h*-9 ” =g; l ” l = kl bd9/.1=g9nbjbdc4 28cbLuˆ*14<; k *-ue“*-c<;2 & 0hb & b k 28,n4b6=g;E=h4\=hi/bd9(;<= k 280›;*™; l b k *-9:;<c*-0qr*-0g,/uˆb˜2 f/f c*r2 kl ,n4bji;*289280 ' ¦Žb ; l bdc<uˆ*Gi ' 928u•= k 289nifièn,/=hi i ' 928u•= k 4 ' 4<;bduˆ4Ž F 9n4<;bŽ2-iZ*8e˜;<c ' =g9/.U;* k 280 k ,/0328;bN–(,289:;<=g;<=hbj4¬28; 4 f b k =³Ł k 0h* k 28;<=h*-9n4 ” =g; l =g9I24 ' 4<;bdu y ; l b˜2Žqrbdc28.rb˜*8eV; l *14b+–:,289(;<=g;<=hbj4 =h4 k 280 k ,/0328;bji &:' 4,/u•u•=g9/. èn,m‘/bj428;; l b & *-,/9nin28c<=hbj4^289niZ2 f/f 0 ' =g9/.ê; l bdu ;*š; l b k *-9n4bdc<q-28;<=h*-9…e“*-c<u *8e˜; l bI5X) B 4Ž ? * f c*Gim, k b62LŁn9/=g;b Œ qr*-0g,/uˆbL2 f/f c*Ž‘G=gu28;<=h*-9e“*-c$2™4bd;X*8e5X) B 4 y bŽ2 klk bd0g0n=g9•; l buˆbj4 l =h4 k *-9n4=hi/bdcbji =g;4˜* ” 9 k *-9:;<c*-0Xqr*-0g,/uˆb289nit; l b•2Žqrbdc28.rbˆ*8eE; l b•4*-0g,/;<=h*-9 ” =g; l =g9‚bdqrbdc ' k bd0g0$=h4™2-44<,/uˆbjit;* & b ì 9n* ” 9W ? l bd9 y ; l bLèn,m‘; l c*-,/. l bŽ2 klk bd0g0 & *-,/9nin28c ' =h42 f/f c*j‘m=gu28;bjiw289ni^,n4bji;*, f in28;bC; l b k bd0g0 2Žqrbdc28.rb1 ? l =h4; ' f b™*8e$im=h4 k cbd;<=g¦²28;<=h*-9I=h4 ,n4bji¬; l c*-,/. l *-,/;6; l =h4; l bj4=h4Ž > e£;bdc™im=h4 k cbd;<=g¦j=g9/.I; l bˆbj–(,28;<=h*-9n4 y =g;’=h4˜; l bd9‡9nb k bj4428c ' ;*¬, f in28;b•; l bˆ4*-0g,/;<=h*-9‡,/9:;<=g0`bd=g; l bdc2 i/bj4<=gcbji’;<=guˆbX=h4VcbŽ2 kl bji*-cW; l b\4*-0g,/;<=h*-9’9n*0h*-9/.rbdc kl 289/.rbj4 & bd; ” bjbd9=g;bdc28;<=h*-9n4Ž T *-c4<;bŽ2-i ' Œ 4;28;b f c* & 0hbduˆ4 yG” l bd9; l bC4*-0g,/;<=h*-99n*’0h*-9/.rbdc kl 289/.rbj4 =g;`=h4 428=hi;* & b k *-9(qrbdc<.rbji› > 4\; l bC9:,/u & bdc*8e k bd0g0h4`=g9¬2™uˆbj4 l =g9 k cbŽ2-4bj4 y ; l bLcbj–:,/=gcbji;<=guˆbC;*’cbŽ2 kl¬k *-9:qrbdc<.rbd9 k b+0g= ì b ” =h4bC=g9 k cbŽ2-4bj4Ž T ,/c<; l bdc k *-u f 0g= k 28;<=g9/.u28;<;bdc4 y 4*-uˆbim=h4 k cbd;<=g¦²28;<=h*-9n4`28cb6*-9/0 ' 4;2 & 0hb =³e; l b4*-0g,/;<=h*-9ˆ=h4E2-imq1289 k bjiˆ4<0h* ” 0 'ry 289ni¬; l =h4 k 289Iimc28u28;<= k 280g0 ' =g9 k cbŽ2-4b k *-u f ,/;28;<=h*-9];<=guˆb1 ? *2-i/imcbj44L; l bj4b™=g9n4;2 & =g0g=g;<=hbj4 y cbj4bŽ28c kl bdc4 l 2jqrb"i/bdqrbd0h* f bji]uˆ*-cb k *-u f 0g= k 28;bjitim=h4 k cbd;<=g¦²28;<=h*-9n4 ; l 28;";2 ì b^=g9:;*š2 kjk *-,/9:; k *-9(qrb k ;<=h*-9 ” =g; l =g9ê; l b^5X) B 4Ž T *-c•b‹‘n28u f 0hb y R *Gb ff 4"4 kl bduˆbw= í1ÚB?i/b Œ k *-u f *14bj4•èn,m‘/bj4ˆ=g9(;* f =hb k bj4; l 28;^28cbN2-44* k =328;bji ” =g; l…k bdc<;28=g9 f/l ' 4<= k 280 kl 28c2 k ;bdc<=h4<;<= k 4¬289ni î 8äå  ã:9 
<; å  ãŽä ; l bd9š*-9/0 ' ,n4bj4C; l b k bd0g0h4C; l 28;™28cb, f ” =g9niNe‰*-c+bŽ2 klšf =hb k b@x; l =h4+, f ” =g9nim=g9/. f cbdqrbd9:;4’2 f/l ' 4= k 280 f c* f 28.:28;<=h*-9N*8e=g9me“*-c<u28;<=h*-9N289ni¬=gu f c*²qrbj464<;2 & =g0g=g; '  > 9n*-; l bdc ” 2 ' *8e=gu f c*ŽqG=g9/."4<;2 & =g0g=g; 'ry bj4 f b k =3280g0 ' e“*-c\4<;bŽ2-i ' Œ 4<;28;b f c* & 0hbduˆ4 y =h4X; l c*-,/. l =gu f 0g= k =g; ;<=guˆb Œ 4<;b f/f =g9/.n ? l =h4C=h4Ci/*-9nb &(' 2-44<,/u•=g9/.; l b4*-0g,/;<=h*-9t=h4 ì 9n* ” 9U28;L; l b k ,/c<cbd9(;˜=g;bdc28;<=h*-9U2-4 ” bd0g0›2-428;2’e£,/;<,/cbC=g;bdc28;<=h*-9¬=g94<,/c<c*-,/9nim=g9/. k bd0g0h4Ž K ' i/*-=g9/.•; l =h4Ee“*-c bdqrbdc ' k bd0g0 ym” b k 289 k cbŽ28;b 2+0328c<.rb0g=g9nbŽ28cE4 ' 4<;bdu ; l 28; ” =g0g0.rbd9nbdc280g0 '•& b4<;2 & 0hbe‰*-cX0328c<.rbdcE;<=guˆb4<;b f 4 o 2-44<,/u•=g9/.™; l b4 f 28;<=3280 im=h4 k cbd;<=g¦²28;<=h*-9fi=h4"4<;2 & 0hb²p‹t† l =g0hb; l =h4•280g0h* ” 4"e‰*-c•2 k *-9(qrbdc<.rbji 4*-0g,/;<=h*-9ê;* & bˆe‰*-,/9niê=g9‚e‰b ” bdc =g;bdc28;<=h*-9n4 y =g; f c*Gim, k bj4’20328c<.rb0g=g9nbŽ28c+4 ' 4<;bduº; l 28; k 289 & bqrbdc ' k *14;<0 ' ;*4*-0gqrb1 T *-c<;<,/928;bd0 'ry ; l bdcb l 2Žqrb & bjbd9 u289 ' 2-imq1289 k bj4ˆ=g9 9(,/uˆbdc<= k 28060g=g9nbŽ28c280g.rb & c2t; l 28;2 kjk bd0hbdc28;b]; l =h4 f c* k bj44Ž M bd; l *(i/4 0g= ì b _˜M]RLBED = í z ? l 2jqrb & bjbd9w4 l * ” 9¬;*•4<, & 4<;289:;<=3280g0 ' cbjim, k b k *-u f ,/;28;<=h*-9];<=guˆby= v íB?­ ? l bj4bw0g=g9nbŽ28c280g.rb & c2š;b kl 9/=h–:,nbj428cb¬=gu f *-c<;289:;;*š*-,/c ” *-c ì & b k 28,n4b; l bwb k 2 k ' *8e˜; l bj4b u28;<c<=³‘ê2 kjk bd0hbdc28;<=h*-9fiuˆbd; l *Gi/4"=h4"0328c<.rbd0 ' i/b f bd9ni/bd9(;ˆ*-9 l * ” ” bd0g0 k *-9nim=g;<=h*-9nbji ; l bu28;<c<=³‘‚=h4Ž ? l =h4 u28;<c<=³‘ k *-9nim=g;<=h*-9/=g9/. o 2’e‰,/9 k ;<=h*-9w*8e ; l b˜bd=g.rbd9n4 ' 4<;bdu *8e ; l b+u28;<c<=³‘pE=h4 l =g. l 0 ' i/b f bd9ni/bd9:; *-9•; l b4 l 2 f b289ni k *-9/9nb k ;<=gq(=g; ' *8e; l b k bd0g0h4$=g9; l b uˆbj4 l  J 4<=g9/.™2Cuˆbj4 l ” =g; l•l =g. l 0 ' 289/=h4*-;<c* f = k k bd0g0h4L; l 28;6q128c ' =g9]4 l 2 f b’289niN4<=g¦Žb™u2 ' cbjim, k b o *-cL=gu f c*Žqrb²p6; l b’b‹¶›b k ;L*8eXu28;<c<=³‘]2 kjk bd0hbdc28;<=h*-9 uˆbd; l *Gi/4Ž > 0g; l *-,/. l u28;<c<=³‘ k *-9nim=g;<=h*-9/=g9/.U=h4ˆ9n*-;b‹‘ f 0g= k =g;<0 ' im=h4 k ,n44bjifi=g9fi; l =h4ˆ; l bj4=h4 y ; l =h4ˆ=h4 k 0hbŽ28c<0 ' *-9nb‡*8e; l bt=h44<,nbj4; l 28;wu’,n4; & b‡2-i/imcbj44bji & b‹e“*-cb‡289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9!=h4uˆ*-cb ” =hi/bd0 ' 2 kjk b f ;bji¬=g9w; l b s`T ) k *-u•u,/9/=g; '  z ˜/.1{!fl$"/	|-"W	 6~}‡}‡/	Z+º 	 > 0g064*-0g,/;<=h*-9 Œ & 2-4bji 2-in2 f ;28;<=h*-9…=h4 & 2-4bjiZ*-9 ; l b k *-9 k b f ;; l 28;ˆ; l bI=hi/bŽ280uˆbj4 l =h4ˆ*-9nb ” l = kl bj–(,/=him=h4<;<c<= & ,/;bj4L; l b’bdc<c*-cŽ A * ” bdqrbdc y” =g; l *-,/; ì 9n* ” =g9/.; l bb‹‘/2 k ;C4*-0g,/;<=h*-9 y =g;6=h4=gu f *144<= & 0hb’;* ì 9n* ” ; l bb‹‘n2 k ;Xbdc<c*-cŽ ? l bbdc<c*-c$u’,n4<;$; l bdcb‹e“*-cb & bbj4;<=gu28;bji"e£c*-u ; l b2 f/f c*Ž‘G=gu28;b4*-0g,/;<=h*-9W R bj4bŽ28c kl bdc4+; ' f = k 280g0 ' ,n4b*-9nb"*8e\; l cbjb"u28=g9‡2 f/f c*r2 kl bj4+;*bdc<c*-c+bj4<;<=gu28;<=h*-9Wý6=g9(;bdc f *-0328;<=h*-9 Œ & 2-4bji y 2-i ë *-=g9(; Œ & 2-4bji y 289nie“bŽ28;<,/cb Œ & 2-4bjiwbdc<c*-cLbj4<;<=gu28;<=h*-9W ? l b•bdc<c*-c’bj4;<=gu28;*-c™; l 28; ” b,n4b=g9 sEl 2 f ;bdcµ=h4 & 2-4bjiš*-9‡; l b=g9(;bdc f *-0328;<=h*-9ê*8e`; l b•4*-0g,/;<=h*-9 & bd; ” bjbd9 ì 9n* ” 9‚q-280g,nbj4Ž R bd.:28cim0hbj44’*8e ” l bd; l bdc’; l b4*-0gqrbdc’=h4™Łn9/=g;b Œ im=³¶bdcbd9 k b y Łn9/=g;b Œ qr*-0g,/uˆb Ú 8äå  ã:9 
<; å  ãŽä *-cŁn9/=g;b Œ bd0hbduˆbd9:; y ; l b^.rbd9nbdc2802 f/f c*r2 kl e“*-c s`T )µ=h4;*š2-44,/uˆb; l 28;; l b4*-0g,/;<=h*-9 =h4 ì 9n* ” 9 b‹‘n2 k ;<0 ' 28;`2’4bd;\*8eW0h* k 28;<=h*-9n4`289ni; l bd9;*’,n4b; l *14b6q-280g,nbj4\;*™=g9(;bdc f *-0328;bC; l b64*-0g,/;<=h*-928;`4*-uˆb *-; l bdc0h* k 28;<=h*-9W ? l b’b‹‘/2 k ; ” 2 ' ; l 28;; l =h4=h46i/*-9nb˜u2 ' q128c 'ry/& ,/;9nbŽ28c<0 ' 280g0W4*-0gqrbdc46,n4b ? 2 ' 0h*-c b‹‘ f 289n4<=h*-9n4E=g9^*-9nbe‰*-c<u*-c 289n*-; l bdcŽ`†‡b k 289=g0g0g,n4<;<c28;bL; l =h4 ” =g; l 2’*-9nbLim=guˆbd9n4<=h*-9280b‹‘n28u f 0hb1 > 44<,/uˆb ” b ì 9n* ” 2]4*-0g,/;<=h*-9 f(x) l 2-4289‚=g9mŁn9/=g;b ? 2 ' 0h*-c•b‹‘ f 289n4<=h*-9 ” =g; l 2Łn9/=g;bc2-im=g,n4*8e k *-9(qrbdc<.rbd9 k b’=g9¬; l b˜9nbd=g. l & *-,/c l *G*Giw*8e74*-uˆb˜0h* k 28;<=h*-9 a f(x) = f(a) + ∞∑ i=1 1 i! dif dxi (a)(x− a)i T *-c62 p *-ci/bdc4*-0gqrbdc ym” b ” *-,/0hiw2 f/f c*Ž‘G=gu28;b’; l =h4 =g9mŁn9/=g;b™4<,/u ,n4<=g9/.ˆ*-9/0 ' ;bdc<uˆ4, f ;*•i/bd.1cbjb p− 1 f(x) ≈ f(a) + p−1∑ i=1 1 i! dif dxi (a)(x− a)i ? l b™im=³¶›bdcbd9 k b & bd; ” bjbd9w; l b’2 k ;<,280V4*-0g,/;<=h*-9 f ¨<®rž€jª (x) 289niw*-,/c62 f/f c*Ž‘G=gu28;<=h*-9 f ž‹ŸrŸ©<°Ž® (x) =h4 ; l bdcb‹e“*-cb f ¨®rž€jª (x)− f ž ŸrŸ©<°Ž® (x) = ∞∑ i=p 1 i! dif dxi (a)(x− a)i > 4; l b˜im=³¶›bdcbd9 k b & bd; ” bjbd9 x 289ni a i/b k cbŽ2-4bj4 y ; l b’im=³¶bdcbd9 k b & bd; ” bjbd9I; l b™b‹‘n2 k ;4*-0g,/;<=h*-9]289ni *-,/cL2 f/f c*Ž‘G=gu28;<=h*-9]=h4 f ¨®rž€jª (x)− f ž ŸrŸ©<°Ž® (x) ≈ 1 p! dpf dxp (a)(x− a)p ? l ,n4 y =³e ” b ” bdcbw;*U,n4b¬; l b ì 9n* ” 9 –(,289:;<=g;<=hbj428; a ;*U2 f/f c*Ž‘G=gu28;bN; l bI4*-0g,/;<=h*-9 2‡4u280g0 im=h4<;289 k b’*8e h 2 ” 2 'ry ; l b™bdc<c*-c6=g9¬; l 28;L2 f/f c*j‘m=gu28;<=h*-9 ” *-,/0hiwi/b f bd9niw*-9 & *-; l h 289ni p  B c<c*-c (h, p) ≈ 1 p! dpf dxp (a)(h)p F 9Łn9/=g;b Œ qr*-0g,/uˆb˜4*-0gqrbdc4 y ; l =h4 2 f/f c*Ž‘G=gu28;<=h*-9¬=h4`,n4bji;*"i/bd;bdc<u•=g9nb+; l bèn,m‘^28; k bd0g0 & *-,/9nin28c<=hbj4 ‚gƒ Fcd[„OVFZKPn[oGEO,GE[oKNa…FX…FZQEj†[oG,[o‡fkIZX†OVMNFXBOghOV^†lˆj‰IZOdQEjB[ŁDBFZ^†GEMN[oG,OV[oGEMN[gOghSjBFe‹Œ[gn[oGdQEj†FOV[ŁOVFKNn[oGEO,I3GE[ŽMNXQEjB[ cdMNXBFZGEM`Qa,IZXi‘IHGE[flXBFHQSi†MNOVlg^†OVOV[ei‘MNX‘QEjBMNO’QEjB[oOVMPOgu õ 8äå  ã:9 
<; å  ãŽä 289nie£c*-u ; l 28; 289, f in28;bL;*™; l b+2jqrbdc28.rb+4*-0g,/;<=h*-9 ” =g; l =g9^; l b k bd0g0é T *-c 4<;bŽ2-i ' Œ 4<;28;b f c* & 0hbduˆ4 y ” l bd9; l bL4*-0g,/;<=h*-9=h4 k *-9(qrbdc<.rbji y ; l b6bdc<c*-c k 289 & b6im=gcb k ;<0 ' cbd0328;bji;*™; l b64=g¦Žb h ; l 28;E4b f 28c28;bj4 ; l bI2 f/f c*Ž‘G=gu28;<=h*-9 f *-=g9(;42-4 ” bd0g0C2-4•; l bI*-ci/bdc p *8eC; l bI4*-0gqrbdcŽ F eL; l b¬uˆbj4 l 4 f 2 k =g9/. h =h4 i/b k cbŽ2-4bjitbdqrbdc 'G” l bdcb y ; l bd9t; l b2 kjk ,/c2 k ' *8e\; l b"4*-0g,/;<=h*-9š4 l *-,/0hiN=g9 k cbŽ2-4b &:' 2ˆe“2 k ;*-c˜*8e hp  A bd9 k bˆ; l bˆ4*-0g,/;<=h*-9‚=h4*-ci/bdc p 2 kjk ,/c28;b1 J 9me‰*-c<;<,/928;bd0 'ry e“*-c™uˆ*14<; f c* & 0hbduˆ4*8e=g9:;bdcbj4; y ; l b p ªb“ i/bdc<=gq-28;<=gqrbj4’*8e f k 289‡q128c ' .1cbŽ28;<0 ' ; l c*-,/. l *-,/;’; l buˆbj4 l  ? l =h4™0hbŽ2-i/4™;*w2 f/f c*j‘m=gu28;<=h*-9n4 ; l 28;628cb+uˆ*-cb™2 kjk ,/c28;b˜=g9¬4*-uˆb+0h* k 28;<=h*-9n4; l 289¬=g9w*-; l bdc4Ž s *-9n4bj–:,nbd9(;<0 'ry i/b k cbŽ2-4<=g9/. h ” l bdcb ; l b p i/bdc<=gq128;<=gqrbj4`28cb l =g. l =gu f c*²qrbj4E; l bC4*-0g,/;<=h*-9u’, kl uˆ*-cbL; l 289; l bL428uˆb6i/b k cbŽ2-4bL=g9*-; l bdc 28cbŽ2-4Ž ? l =h4E=h4X; l b f c<=g9 k = f 0hb & b l =g9ni=g9(;bdc f *-0328;<=h*-9 Œ & 2-4bjicb‹Łn9nbduˆbd9(;Ž F e; l b p ªb“ i/bdc<=gq-28;<=gqrbj4 28cbC4*-uˆb Œ l * ” cb k *-9n4<;<c<, k ;bji y ; l bd9a289…bdc<c*-c^im=h4<;<c<= & ,/;<=h*-9 k 289 & b]2 f/f c*j‘m=gu28;bji›† l bd9…; l bwuˆbj4 l =h4 cb‹Łn9nbji y =g9n4<;bŽ2-iw*8e$,/9/=³e“*-c<u•0 ' i/b k cbŽ2-4<=g9/. h y bd0hbduˆbd9:;4+*8e$; l b™uˆbj4 l]k 289 & b™4<=g¦Žbji¬;*u28; kl ; l b bdc<c*-c"im=h4<;<c<= & ,/;<=h*-9W F i/bŽ280g0 'ry ; l =h4 ” *-,/0hiU=g9 k cbŽ2-4b; l b2 kjk ,/c2 k ' *8e; l b4*-0gqrbdc y =g9U; l b4bd9n4bˆ*8e u•=g9/=gu•=g¦j=g9/.; l bC4*-0g,/;<=h*-9^bdc<c*-cEe“*-c 2™.1=gqrbd99(,/u & bdc*8eW,/9 ì 9n* ” 9n4Ž M *14;\cbj4bŽ28c kl =g9(;*•2-in2 f ;<=gqrb 289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9(; l 2-4 & bjbd9e“*-c$4b k *-9ni•*-ci/bdc$4*-0gqrbdc4Ž F 9; l =h4 k 2-4b y ; l bE=g9(;bdc f *-0328;<=h*-9bdc<c*-c k 289 & b+cbd0328;bji¬;*•; l b A bj44<=3289 H  F 9¬; ” *im=guˆbd9n4<=h*-9n4 y H =   ∂ 2f ∂x2 ∂2f ∂x∂y ∂2f ∂x∂y ∂2f ∂y2   F eV; l =h4 u28;<c<=³‘=h4 f *14<=g;<=gqrb Œ i/b‹Łn9/=g;b’289ni^=g9(qrbdc<;<= & 0hb y ; l bd9; l b+=hi/bŽ280 k bd0g0 k 289 & bLe‰*-,/9ni &(' 0g=g9nbŽ28c<0 ' ;<c289n4He“*-c<u•=g9/.289¬=h4*-;<c* f = k’k bd0g0 &:' H−1  ? l =h4=h4c*-,/. l 0 ' ; l b™2 f/f c*r2 kl ,n4bji &(' = zmy Ù xmy Ù v/y َí y َî†?­ T *-cC; l =gcit*-ci/bdc™4*-0gqrbdc4˜289ni l =g. l bdc y ; l bi/bdc<=gq128;<=gqrbj4 k 289 & b".1c*-, f bjiN=g9(;*w2 l =g. l bdc˜c289 ì ;bd9n4*-cx l * ” bdqrbdc y ; l b64 l 2 f b6*8e›; l b6=hi/bŽ280 k bd0g0=g9; l bj4b k 2-4bj4X=h4E,/9 k 0hbŽ28cŽ$5$cbdqG=h*-,n4\;*’; l b k ,/c<cbd9:; ; l bj4<=h4 y 9n* f 2 f bdc4 l 2jqrb & bjbd9 f , & 0g=h4 l bji; l 28;b‹‘m;bd9ni^=g9:;bdc f *-0328;<=h*-9 & 2-4bji^cb‹Łn9nbduˆbd9:;L;* l =g. l bdc *-ci/bdc64*-0gqrbdc46289ni¬; l =h4=h4; l b f c<=g9 k = f 280We“* k ,n4*8e sEl 2 f ;bdc J 9me‰*-c<;<,/928;bd0 'ry bdqrbd9^=³eV; l bLuˆbj4 l^f bdcHe“b k ;<0 ' bj–(,/=him=h4<;<c<= & ,/;bj4 ; l b6=g9(;bdc f *-0328;<=h*-9 Œ bdc<c*-cLbj4<;<=gu28;b y p *-ci/bdc2 kjk ,/c2 k '•” *-,/0hi^c28cbd0 '•& bL2 kl =hbdqrbji› ? l =h4\=h4 f c<=gu28c<=g0 ' im,nb;*’; l b f cbj4bd9 k bC*8e k *-9:qrb k ;bji z 8äå  ã:9 
<; å  ãŽä bdc<c*-cŽ F 9(;bdc f *-0328;<=h*-9 & 2-4bjiNbdc<c*-c+bj4<;<=gu28;<=h*-9‡2-44<,/uˆbj4L; l 28;L; l b"4*-0g,/;<=h*-9š28;C4<,/c<c*-,/9nim=g9/.0h* k 2 Œ ;<=h*-9n4™28cb280g0$b‹‘/2 k ;™289niN; l ,n4+; l b"*-9/0 ' 4*-,/c k b*8eEbdc<c*-c k *-uˆbj4Ce‰c*-uû; l b l =g. l bdc Œ *-ci/bdc™;bdc<uˆ4+*8e ; l b ? 2 ' 0h*-c+b‹‘ f 289n4<=h*-9W A * ” bdqrbdc y =³eX; l b™q1280g,nbj4,n4bjiI=g9I; l b ? 2 ' 0h*-c+2 f/f c*Ž‘G=gu28;<=h*-9t; l bduˆ4bd0gqrbj4 k *-9(;28=g9‡bdc<c*-c y ; l 28;™bdc<c*-c ” =g0g0 & b f 2-44bjit*-9N;*^; l b"=g9:;bdc f *-0328;bjitq1280g,nb ” l = kl ” =g0g07=g9N;<,/c<9‡2¥e Œ e“b k ;C*-; l bdc ? 2 ' 0h*-c™b‹‘ f 289n4<=h*-9t;bdc<uˆ4˜289nit4*^*-9W ? l b9nbd;Ccbj4<,/0g;+=h4C; l 28;˜4<u280g0bdc<c*-c4Ce‰c*-u´*-9nb 0h* k 28;<=h*-9ê*8e; l buˆbj4 l‚k 289 l 2jqrb2 & =g.Nb‹¶b k ;=g9‚0h* k 28;<=h*-9n4; l 28;28cbcbd0328;<=gqrbd0 ' e“28c2 ” 2 '  ? l =h4 =h4 f 28c<;<= k ,/0328c<0 ' 2 f c* & 0hbduûe“*-cL5X) B 4C; l 28; k *-9(;28=g9š4<;<c*-9/. k *-9(qrb k ;<=gqrb";bdc<uˆ4L4<, kl 2-46; l b B ,/0hbdc bj–(,28;<=h*-9n4 ” l bdcb’24;<c*-9/.•e‰bŽ28;<,/cb™0g= ì b’24 l * kì u•=g. l ; & b+cb‹Łn9nbji¬=g9¬; l b ” c*-9/.0h* k 28;<=h*-9W > 9a2 f/f c*r2 kl ; l 28;^4bdqrbdc280Ccbj4bŽ28c kl .1c*-, f 4 o =g9 k 0g,nim=g9/. y e“*-c^b‹‘n28u f 0hb y = ánÙ y í y á xmy á v ?£p l 2Žqrb ,n4bji;*ˆ2-i/imcbj44; l =h4=h44<,nb+=h4L2-i ë *-=g9(; Œ & 2-4bjiwbdc<c*-c6bj4<;<=gu28;<=h*-9W > i ë *-=g9(; Œ & 2-4bjiIbdc<c*-c6bj4;<=gu28;<=h*-9 =g9(qr*-0gqrbj44*-0gqG=g9/. & *-; l ; l b f c<=gu28065X) B 42-4 ” bd0g062-4•; l b¬2-i ë *-=g9:;5X) B 4² ? l bd9 y ; l b¬bdc<c*-c•=g9 2e‰,/9 k ;<=h*-9280X=h4˜im=gcb k ;<0 ' cbd0328;bjit;*^0h* k 280$cbj4<=him,280Xbdc<c*-c4+; l c*-,/. l ; l bˆ2-i ë *-=g9:;˜q128c<=32 & 0hbj4Ž ? l =h4 f c*²q(=hi/bj4˜2•uˆ*-cb"2 kjk ,/c28;b"289280 ' 4<=h4C*8e ” l bdcb™; l b’e£,/9 k ;<=h*-9280=h4uˆ*14;C2¥¶›b k ;bji &(' bdc<c*-c46=g9N; l b 4*-0g,/;<=h*-9289ni l bd9 k bL2L4*-0g=hi & 2-4<=h4e‰*-c ” l bdcb ; l b uˆbj4 l 4 l *-,/0hi & b cb‹Łn9nbji› sEl *G*14<=g9/.’2Le‰,/9 k ;<=h*-9280 =h4 & *-; l 289U2-imq-289(;28.rb289ni‡2^im=h42-imq-289(;28.rbe“*-c˜,n4<=g9/.; l =h4˜uˆbd; l *(i› T *-c™4<=g;<,28;<=h*-9n4 ” l bdcb•; l b *-9/0 ' i/bj4<=gcbjiw*-,/; f ,/;L=h4; l b+e‰,/9 k ;<=h*-9280 y 2-i ë *-=g9:; Œ & 2-4bjiIuˆbd; l *Gi/4L28cb˜=hi/bŽ280 & b k 28,n4b™; l b ' i/*9n*-; cb‹Łn9nb+28cbŽ2-4 *8eV; l b6uˆbj4 l ” l bdcbC; l bC4*-0g,/;<=h*-9 l 2-4`9n*"b‹¶b k ; *-9^; l 28;\e‰,/9 k ;<=h*-9280é T *-c b‹‘/28u f 0hb y =³e ; l b`*-9/0 ' k *-u f *-9nbd9:;X*8e; l b`4*-0g,/;<=h*-9ˆ289280 ' ¦Žbji"=h4imc28.˜*-9289ˆ28=gcHe“*-=g0 y ; l bd9; l b`uˆbj4 l bj4 k cbŽ28;bji &:' 2-i ë *-=g9:; Œ & 2-4bjitcb‹Łn9nbduˆbd9(; ” =g0g0 f c*ŽqG=hi/b & bd;<;bdc˜imc28. f cbjim= k ;<=h*-9n4+; l 289t*-; l bdc+uˆbj4 l bj4+*8eE4=gu•=g0328c 4<=g¦Žb1 A * ” bdqrbdc y =³e6; l 28;ˆ428uˆbuˆbj4 l =h4,n4bji‚;* f cbjim= k ;•0g=³e‰; y *-cˆ289 ' *-; l bdc k *-u f *-9nbd9:;*8eC; l b 4*-0g,/;<=h*-9 y =g;u2 ' 9n*-; f c*ŽqG=hi/b^4,” k =hbd9:;2 kjk ,/c2 k '  > 9n*-; l bdc"im=h42-imq1289:;28.rb¬*8eL; l =h4uˆbd; l *Gi‚=h4 ; l 28;64*-0gq(=g9/.•; l b’2-i ë *-=g9:; f c* & 0hbdu k 289 & b k *-u f 0g= k 28;bji]289ni;<=guˆb k *-9n4,/u•=g9/.n ? l b; l =gciUbdc<c*-cbj4;<=gu28;*-c; l 28;"4*-uˆb•cbj4bŽ28c kl bdc4,n4b•=h4˜e“bŽ28;<,/cb Œ & 2-4bjiêbdc<c*-cbj4<;<=gu28;<=h*-9 o 289 b‹‘ k bd0g0hbd9:; b‹‘/28u f 0hbC*8eW; l =h4E=h4•=gÙ z ?£p‹ T bŽ28;<,/cb Œ & 2-4bji^bdc<c*-c`bj4<;<=gu28;<=h*-9¬b‹‘ f 0g= k =g;<0 ' i/bd;b k ;4289ni^2892 Œ 0 ' ¦Žbj4Xè* ” e“bŽ28;<,/cbj4 289nicb‹Łn9nbj4E; l b6uˆbj4 l 2 kjk *-cim=g9/.";*’; l *14bè* ” e‰bŽ28;<,/cbj4² ? l =h4 f c* k bj44Ecbd0g=hbj4 *-9 žŸ©‹¡‰°1© ¡ ì 9n* ” 0hbjim.rb*8e ” l 28;+; l b4*-0g,/;<=h*-9 ” =g0g0 & b k *-u f *14bjiN*8e o 4 l * kì 4 y” 2 ì bj4 y& *-,/9nin28c ' 032 ' bdc4 y bd; k gg p•2-4 ” bd0g0V2-42cbd0g=32 & 0hb+e“bŽ28;<,/cb Œ i/bd;b k ;<=h*-9I4 ' 4<;bduw ? l b˜2-imq1289:;28.rb™*8e7b‹‘ f 0g= k =g;<0 ' i/b Œ ó 8äå  ã:9 
<; å  ãŽä ;b k ;<=g9/.e‰bŽ28;<,/cbj4C=h46; l 28;L=g;+280g0h* ” 4Le‰*-cLuˆ*-cb4 f b k =³Ł k cb‹Łn9nbduˆbd9:; k c<=g;bdc<=324<, kl 2-4L*-c<; l *-.r*-9280g=g; ' =g9•; l b & *-,/9nin28c ' 032 ' bdc4 y qrbdc<;b‹‘280g=g.19/uˆbd9:; ” =g; l 4 l * kì 4E289niˆ*-; l bdcXuˆbj4 l 2-in2 f ;28;<=h*-9n4Ž J 9me“*-c<;<, Œ 928;bd0 'ry ; l be“bŽ28;<,/cbi/bd;b k ;*-c4Eu’,n4<; & b k cbŽ28;bji289ni•;<,/9nbjiˆe‰*-c\bŽ2 klf 28c<;<= k ,/0328c f c* & 0hbdu 289ni; l b 2 k ;<,280cb‹Łn9nbduˆbd9:; k c<=g;bdc<=32•28cbLu’, kl uˆ*-cb l bd,/c<=h4<;<= k xGe‰*-c b‹‘n28u f 0hb y =g;`=h4`,/9 k 0hbŽ28c l * ” u’, kl *-9nb e“bŽ28;<,/cbˆ4 l *-,/0hi & bcb‹Łn9nbji‡cbd0328;<=gqrb•;*w289n*-; l bdcŽ > 9n*-; l bdcimc2 ”& 2 kì =h4™; l 28;˜e‰bŽ28;<,/cb•i/bd;b k ;<=h*-9 =h4"*8e£;bd9 & 2-4bjiê*-9U; l b^428uˆb0h* k 280 f 28c28uˆbd;bdc4"; l 28;"=g9:;bdc f *-0328;<=h*-9 Œ & 2-4bjifibdc<c*-c"uˆbŽ2-4<,/cbj4",n4b1 ? l 28;XuˆbŽ289n4\; l 28; k *-9:qrb k ;bjibdc<c*-cX=h4$=g.19n*-cbji^289ni•=g;$=h4 f *144<= & 0hb ;*™i/bd;b k ;$e“bŽ28;<,/cbj4X=g9•; l b ” c*-9/. 0h* k 28;<=h*-9W †‡b l 2jqrb kl *14bd9¬;* & 2-4b+*-,/cbdc<c*-c bj4;<=gu28;<=h*-9w*-9^; l b+0h* k 280cb k *-9n4<;<c<, k ;<=h*-9wbdc<c*-c e“*-c 29:,/u & bdc *8ecbŽ2-4*-9n4Ž T =gc4<; y 4<=g9 k bC; l =h4bdc<c*-cuˆbŽ2-4<,/cb+cbd0328;bj4im=gcb k ;<0 ' ;*; l b+*-ci/bdc*8e ; l b˜im=h4 k cbd;<=g¦²28;<=h*-9 y ; l bLb‹‘G;bd9n4<=h*-9;* l =g. lmŒ *-ci/bdc 4*-0g,/;<=h*-9n4E=h4X=g9:;<,/=g;<=gqrb1 D b k *-9ni y 0h* k 280bdc<c*-c`bj4<;<=gu28;<=h*-9^=h4Xcbd0328;<=gqrbd0 ' kl bŽ2 fk *-u f 28cbji•;*’2-i ë *-=g9:; Œ & 2-4bjibdc<c*-c\bj4<;<=gu28;<=h*-9W T =g9280g0 'ry k ,/c<cbd9:;`2-i ë *-=g9:;\uˆbd; l *(i/4 f c*ŽqG=hi/b 289 =hi/bŽ2804<=g¦Žbim=h4<;<c<= & ,/;<=h*-9 e“*-c k bd0g0h4 & ,/;9n*-;ˆ289 =hi/bŽ280 4 l 2 f b^im=h4<;<c<= & ,/;<=h*-9Z289ni 4*Ncbj4bŽ28c kl bdc4 4<, kl 2-4•ôXbd9nim=g;<;<=L289ni )˜28c<uˆ*8eé280Ł= á v ?cbd0 ' *-9fi0h* k 280bdc<c*-cˆcb k *-9n4;<c<, k ;<=h*-9 ;*‡i/b‹Łn9nb; l b¬0h* k 280 289/=h4*-;<c* f '  * ,+–+—-" 	)˜„	%"fl"	
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" ffffifl$% Y 9 k b\; l bEi/bj4<=gcbji"289/=h4*-;<c* f ' l 2-4 & bjbd9i/b‹Łn9nbji y =g;V=h4V9nb k bj4428c ' ;* k *-u•u’,/9/= k 28;b`; l 28;7289/=h4*-;<c* f ' ;*N; l b^uˆbj4 l =g9/. f c*-.1c28u 289ni k cbŽ28;bw289fi289/=h4*-;<c* f = k uˆbj4 l  ? l buˆ*14<;4<;289nin28ci uˆbd; l *(i *8e i/b‹Łn9/=g9/.I289/=h4*-;<c* f ' =h4C; l c*-,/. l 2uˆbd;<c<= k = Ú y õf?­ > uˆbd;<c<= k =h4˜2ˆe‰,/9 k ;<=h*-9š; l 28;˜i/b‹Łn9nbj4™2^4 k 280328c im=h4<;289 k b & bd; ” bjbd9 f *-=g9:;4=g9 2‡i/*-u28=g9W S = ì bI; l b]4<;289nin28ci im=h4<;289 k bwuˆbŽ2-4<,/cb]=g9 s 28c<;bj4=3289 4 f 2 k b y uˆbd;<c<= k e‰,/9 k ;<=h*-9n4™28cbcbj–(,/=gcbjiN;* & b f *14<=g;<=gqrb Œ i/b‹Łn9/=g;b y 4 ' u•uˆbd;<c<= k 289niN; l b ' u,n4<;™280h4* 428;<=h4<e ' ; l bˆ;<c<=3289/.10hb^=g9nbj–(,280g=g; '  ? l =h4’; ' f b*8ee‰,/9 k ;<=h*-9‚=h4=hi/bŽ280Xe“*-c k cbŽ28;<=g9/.tuˆbj4 l bj4 & b k 28,n4b .rbj*-uˆbd;<c<= k^f c* f bdc<;<=hbj4 & 2-4bjiê*-9U0hbd9/.1; l 4 l 2Žqrbbj–(,/=gq-280hbd9(; f c* f bdc<;<=hbj42 kjk *-cim=g9/.t;*w; l buˆbd;<c<= k e‰,/9 k ;<=h*-9W F 9 f 28c<;<= k ,/0328c y uˆ*14<;Lcbj4bŽ28c kl .1c*-, f 4C,n4b’; l b’eé28u•=g0 ' *8eXuˆbd;<c<= k 4 k cbŽ28;bji &(' 0g=g9nbŽ28c<0 ' Ù x 8äå  ã:9 
<; å  ãŽä ;<c289n4He“*-c<u•=g9/.N; l b4<;289nin28ci s 28c<;bj4<=3289fiim=h4<;289 k bˆuˆbŽ2-4,/cb y 4bjbš=gÙŽÚ y َî y z ?28uˆ*-9/.N*-; l bdc4ŽI†‡b 280h4*],n4b; l bj4b; ' f bj4"*8e6uˆbd;<c<= k 4 &:' 2-44<=g.19/=g9/.‡2w4 ' u•uˆbd;<c<= k^f *14<=g;<=gqrb Œ i/b‹Łn9/=g;b^u28;<c<=³‘ê;*]bŽ2 kl qrbdc<;b‹‘› ? l b kl *-= k b™*8e; l bj4b˜u28;<c<= k bj4=h4im=h4 k ,n44bji=g9 sEl 2 f ;bdc’ J 9me‰*-c<;<,/928;bd0 'ry q128c<=328;<=h*-9n4 =g9^; l bCbdc<c*-c280h4*cbj–(,/=gcbLq128c<=328;<=h*-9n4 =g9^; l bLuˆbd;<c<= k 289ni^; l =h4`=h4 ” l bdcb cbj4bŽ28c kl bdc4 l 2jqrbi/bdqrbd0h* f bjiêim=³¶›bdcbd9:;•2 f/f c*r2 kl bj4Ž ? l b ì b ' f c* & 0hbdu ” =g; l 2¬q128c ' =g9/.]uˆbd;<c<= k =h4"; l 28; y bdqrbd9 ; l *-,/. l ; l b^uˆbd;<c<= k =h4"q-280g=hi f *-=g9:; Œ ” =h4b y =g;u2 ' 9n*N0h*-9/.rbdc.10h* & 280g0 ' 428;<=h4He ' ; l b ;<c<=3289/.10hbN=g9nbj–(,280g=g; '  s *-,/c<; ' bd;^280鍑=gÙ v ?C0g=g9nbŽ28c<0 ' =g9:;bdc f *-0328;b]; l b]uˆbd;<c<= k 289ni…i/bŽ280 ” =g; l ; l b ;<c<=3289/.10hbI=g9nbj–:,280g=g; ' l bd,/c<=h4<;<= k 280g0 '  Y ; l bdc4 y 4<, kl 2-4 S 2 & bd0g0hbN289ni D(l b ” kl , ì =gَÚB?C2-44<,/uˆbI; l b uˆbd;<c<= k =h4L*-9/0 ' q-280g=hi ” =g; l =g9N; l bôX*-c*-9n*-= k bd0g0*8e\bŽ2 kl qrbdc<;b‹‘› ? l =h4L0hbŽ2-i/4C;*qrbdc ' k *-u f 0g= k 28;bji ôX*-c*-9n*-= im=328.1c28uˆ4; l 28;’cbj–(,/=gcb k bdc<;28=g9 cbj4<;<c<= k ;<=h*-9n4*-9‚; l buˆbd;<c<= k =g9ê*-ci/bdce“*-c’; l bôX*-c*-9n*-= im=328.1c28u ;*im,280g=g¦Žb˜;*2"q1280g=hi¬;<c<=3289/.1,/0328;<=h*-9W > 4 f 28c<;+*8e`; l b•289/=h4*-;<c* f = k uˆbj4 l =g9/.I2 f/f c*r2 kl =g9 sEl 2 f ;bdc y” b l 2jqrb kl *14bd9š;*2 f/f c*Ž‘G=gu28;b ; l b¬uˆbd;<c<= k e‰*-c; l b¬=g9:;bdc<=h*-c*8eC;<c<=3289/.10hbj4,n4<=g9/.‡; l bN2jqrbdc28.rbIuˆbd;<c<= k e‰c*-u#; l b¬; l cbjbwqrbdc<;<= k bj4² T *-cC;<c<=3289/.10hb"bjim.rbj4 y” b"2Žqrbdc28.rb"; l b’0hbd9/.1; l ; l 28; ” *-,/0hi & b k 280 k ,/0328;bji]e‰c*-uû; l b’; ” *;<c<=3289/.10hbj4² ? l bj4b4<=gu f 0hb2 f/f c*j‘m=gu28;<=h*-9n4280g0h* ” 4Ce‰*-c+–(,/= kì bj4;<=gu28;bj4˜*8e\;<c<=3289/.10hb–:,280g=g; 'I” l =g0hb=gu f *14<=g9/. 9n*cbj4<;<c<= k ;<=h*-9n4C*-9¬uˆbd;<c<= k q128c<=328;<=h*-9W > e£;bdcL; l b’uˆbd;<c<= k =h4Ci/b‹Łn9nbji y =g;C=h4L9nb k bj4428c ' ;* k cbŽ28;b"2ˆuˆbj4 l ; l 28;C=h4Lc*-,/. l 0 ' =h4*-;<c* f = k ” l bd9 uˆbŽ2-4<,/cbji &:' ; l buˆbd;<c<= k  ? * k cbŽ28;b; ” * Œ im=guˆbd9n4<=h*-9280=h4*-;<c* f = k ,/9n4<;<c<, k ;<,/cbjiUuˆbj4 l bj4 y ; l bdcb 28cb˜; l cbjb f c<=g9 k = f 280 2 f/f c*r2 kl bj4Žý`–(,2-i Œ ;<cbjb y 2-imq-289 k =g9/. Œ e‰c*-9:; y 289niw)+bd0328,/92 ' uˆbj4 l =g9/.n [ ,2-i Œ ;<cbjb6uˆbj4 l =g9/.= á:í y óB?im=h4 k cbd;<=g¦Žbj4X; l bi/*-u28=g9 &:' Łnc4<; k cbŽ28;<=g9/.2™4–:,28cb; l 28; k *-9:;28=g9n4E; l b bd9(;<=gcb•i/*-u28=g9W ? l bd9 y ; l 28;˜4–(,28cb"=h4+cb k ,/c4<=gqrbd0 ' 4 f 0g=g;˜,/9:;<=g0\280g07; l b4–:,28cbj4˜4 28;<=h4He ' 4*-uˆb•2-4 Œ 4<=g.19nbji"4<=g¦Žb f c* f bdc<; ' 289ni; l b & *-,/9nin28c ' =h44<,/=g;2 & 0 ' im=h4 k cbd;<=g¦Žbji› ? l b`im= k ,/0g; '” =g; l 289/=h4*-;<c* f = k –(,2-i Œ ;<cbjb˜uˆbj4 l =g9/.•=h4 ; l 28;; l b f cbdqG=h*-,n4<0 ' Œ uˆbd9:;<=h*-9nbji]4–:,28cbj4u’,n4<;4*-uˆb l * ”ß& b˜im=gq(=hi/bji=g9(;* cb k ;289/.10hbj4 ” =g; l q-28c ' =g9/.2-4 f b k ;c28;<=h*14L289ni¬im=gcb k ;<=h*-9n4² F 9•; l b2-imq-289 k =g9/.’e£c*-9(;Xuˆbd; l *Gi1=gÙ²õ†? y ; l buˆbj4 l =g9/. f c* k bjim,/cb & bd.1=g9n4 ” =g; l 2+4bd;X*8e4bjbjiqrbdc<;<= k bj4 o ,n4<,280g0 ' 280h*-9/."; l b & *-,/9nin28c<=hbj4 p k *-9/9nb k ;bji=g9:;*•2 k 0h*14bji0h*G* f^k 280g0hbji¬2™e£c*-9(;Ž ? l bd9 y 2’i/b k =h4=h*-9 =h4Lu2-i/b"2-4L;* ” l bdcb’;*=g9n4bdc<;6; l b9nb‹‘m;Lqrbdc<;b‹‘I4*; l 28;C=g; ” =g0g0 k cbŽ28;b"2 l =g. l –(,280g=g; ' ;<c<=3289/.10hb1 Ù1Ù 8äå  ã:9 
<; å  ãŽä ? l be£c*-9(;=h4’; l ,n4, f in28;bji y 289niê; l b f c* k bj44cb f bŽ28;4Ž ? l b^u28=g9 imc2 ”& 2 kì *8e; l =h4"uˆbd; l *Gi =h4"; l 28; ” l bd9 im=³¶›bdcbd9:;•uˆbj4 l =g9/.te‰c*-9:;4uˆbjbd; y 4*-uˆb l bd,/c<=h4<;<= k 2 f/f c*r2 kl =h4"cbj–(,/=gcbji ;*N,/9/=³e ' ; l bCe‰c*-9:;4 ” =g; l *-,/; k cbŽ28;<=g9/. f *G*-c Œ –(,280g=g; ' ;<c<=3289/.10hbj4Ž T *-c6289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 y ; l =h4 uˆbd; l *Gi cbj–(,/=gcbj4Xcbduˆ*²q(=g9/."0328c<.rb;<c2 k ;4`*8e›; l buˆbj4 l 289ni; l bd9cb Œ uˆbj4 l =g9/.; l bj4bL28cbŽ2-4`2 kjk *-cim=g9/.;*’; l b uˆbd;<c<= k  ? l bLuˆbj4 l =g9/.ˆ2 f/f c*r2 kl ; l 28; ” b l 2jqrbCuˆ*Gim=³Łbjiˆe“*-c 289/=h4*-;<c* f ' =g9 sEl 2 f ;bdc =h4 )+bd0328,/92 ' uˆbj4 lmŒ =g9/.n > )+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9*8e™qrbdc<;<= k bj4=h42š;<c<=3289/.1,/0328;<=h*-9 ” l bdcb]; l bw=g9:;bdc<=h*-c¬*8e’bdqrbdc ' ;<c<=3289/.10hb ff 4 k =gc k ,/u k =gc k 0hb•=h4+bdu f ; '  ? l =h4C;<c<=3289/.1,/0328;<=h*-9‚280 ” 2 ' 4˜b‹‘m=h4<;4 y 289ni]=h4Cbj–(,/=gq1280hbd9:;˜=g9N; ” * im=guˆbd9n4<=h*-9n4;*; l b+;<c<=3289/.1,/0328;<=h*-9I; l 28;u2¥‘G=gu•=g¦Žbj4 ; l b+u•=g9/=gu,/uû289/.10hb+=g9; l bLuˆbj4 l X)+bd0328,/92 ' uˆbj4 l cb‹Łn9nbduˆbd9(;;2 ì bj42 f cb Œ b‹‘m=h4<;<=g9/.U)+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9 *8eLqrbdc<;<= k bj4289nifi; l bd9fi=g9n4bdc<;4 qrbdc<;<= k bj428;; l b k =gc k ,/u k bd9:;bdc¬*8e289 ' ;<c<=3289/.10hb ” =g; l 2U4<u280g0C289/.10hbN*-c^2‡0328c<.rb k =gc k ,/u•c2-im=g,n4Ž K ' cb f bŽ28;bjim0 ' =g9n4bdc<;<=g9/.qrbdc<;<= k bj4C28; k =gc k ,/u k bd9(;bdc4C=g;=h4 f *144<= & 0hb+;* k cbŽ28;b’uˆbj4 l bj4 ” =g; l .1,28c Œ 289(;bjbji‚–(,280g=g; '  ? l bb‹‘n2 k ;’.1,28c289(;bjbi/b f bd9ni/4"*-9 l * ” ; l b & *-,/9nin28c<=hbj4"28cb l 289nim0hbji y & ,/;=g9 .rbd9nbdc280 y ; l b’4<u280g0hbj4;C289/.10hb™=g9]; l b™uˆbj4 l =h464*-uˆb ” l bdcb28c*-,/9ni]; l =gc<; ' i/bd.1cbjbj4=gÙ1Ù y írõ y í1ó yv1z ?­ ? l b^cbŽ2-4*-9 ; l 28;ˆ)+bd0328,/92 ' uˆbj4 l =g9/.šb‹‘G;bd9ni/4 ” bd0g0;*š289/=h4*-;<c* f = k uˆbj4 l =g9/.t=h4 & b k 28,n4b^*8eL=g;4 4<=gu f 0hb2 f/f c*r2 kl *8e kl *(*14<=g9/.; l b˜=hi/bŽ280W0h* k 28;<=h*-9I;*•=g9n4bdc<;L2"qrbdc<;b‹‘w.1=gqrbd9]2q1280g=hi¬uˆbj4 l  F e7; l b =g9n4bdc<;<=h*-9š0h* k 28;<=h*-9‡=h4 kl *14bd9š;* f c*(im, k bˆ289/=h4*-;<c* f = k ;<c<=3289/.10hbj42 kjk *-cim=g9/.w;*; l buˆbd;<c<= k y ; l bd9 2 l =g. lmŒ –(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l‡k 289 & b k cbŽ28;bji ” =g; l *-,/;’289 ' b‹‘ f bd9n4<=gqrb0h* k 2807u2¥‘G=gu•=g¦²28;<=h*-9 c*-,/;<=g9nbj4Ž › %"fl  	
 ffffifl$% "
	 	6 z ˜ _ *G*Gi‚bdc<c*-cbj4<;<=gu28;<=h*-9Z289ni‚e“2-4; y l =g. lmŒ –:,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l =g9/.‡280g.r*-c<=g; l uˆ4ˆ28cb^; l b; ” * ì b ' =g9/.1cbjim=hbd9:;4\e‰*-cX4, kjk bj44He‰,/0289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9W B 44bd9:;<=3280g0 'ry 289/=h4*-;<c* f = k 2-in2 f ;<=gqrb uˆbj4 l =g9/.=h44<=gu•=g0328c;*ˆ2-i/im=g9/.2e“bjbji & 2 kì 0h*(* f ” =g; l =g9N2 k *-9:;<c*-0 4 ' 4<;bduw F 9 f 28c<;<= k ,/0328c yn” b’28cb ;<c ' =g9/.•;*u•=g9/=gu•=g¦Žb’289¬bdc<c*-ce‰,/9 k ;<=h*-9 &:' k *-9:;<c*-0g0g=g9/.; l b+uˆbj4 l 7† l 28;u2 ì bj4; l =h4 f c* & 0hbduû4* im= k ,/0g;˜=h4C; l 28;˜; l bbdc<c*-cCe£,/9 k ;<=h*-9š=h4<9 ff ;C9nb k bj4428c<=g0 ' ì 9n* ” 9 y 289niN; l b"uˆbj4 ltl 2-4™289tbd9n*-c<uˆ*-,n4 9(,/u & bdcL*8e$i/b f bd9ni/bd9(;6q128c<=32 & 0hbj4Ž Ù v 8äå  ã:9 
<; å  ãŽä )+bj4 f =g;b\; l b k *-u f 0g= k 28;<=h*-9n4 y ; l =h4 e“bjbji & 2 kì 0h*G* fk 289=gu f c*Žqrb`; l b`b‹¶b k ;<=gqrbd9nbj44*8e s`T )fi=g94bdqrbdc280 ” 2 ' 4Ž F i/bŽ280g0 'ry ,n4bdc4+28cb’9n*-;Ccbj–:,/=gcbji];*^4 f bd9ni]=g9n*-cim=g928;b•28uˆ*-,/9:;4˜*8eX;<=guˆb;<c ' =g9/.^;* k cbŽ28;b uˆbj4 l bj4"; l 28;"u2 ' *-cu2 ' 9n*-; & b ” bd0g0 Œ 4<,/=g;bjiê;*N; l b f c* & 0hbduw T ,/c<; l bdc<uˆ*-cb y 28,/;*-u28;<= k 280g0 ' k cbŽ28;bjiuˆbj4 l bj4 f c*²q(=hi/b l =g. l bdc 2 kjk ,/c2 k '•” =g; l e‰b ” bdc k bd0g0h4E; l 289; l b6=h4*-;<c* f = k y ,/9/=³e“*-c<uöuˆbj4 l bj4Ž J 0g;<=gu28;bd0 'ry =g9 k cbŽ2-4<=g9/.I; l b9:,/u & bdc*8e k bd0g0h4 ” =g; l bŽ2 kl 2-in2 f ;28;<=h*-9‡=g9n4<;bŽ2-i‡*8e k *-9:qrbdc<.1=g9/.N; l b uˆbj4 l 289ni‡4*-0g,/;<=h*-9 ” =g; l 2^Ł/‘mbji‡9:,/u & bdc*8e k bd0g0h4280g0h* ” 4 s`T ) ,n4bdc4™;*¬.:28.rb ” l bd; l bdc"*-c™9n*-; ; l b•4*-0g,/;<=h*-9U=h4 k *-9:qrbdc<.1=g9/.I;*w289U=g9mŁn9/=g;bd0 ' Œ Łn9nb•uˆbj4 l 4*-0g,/;<=h*-9‚289nit; l ,n4 & bd;<;bdc™bj4;<=gu28;b•; l b *²qrbdc280g02 kjk ,/c2 k ' *8e; l b’4*-0gqrbdcŽ F 9 sEl 2 f ;bdc ” bˆb‹‘n28u•=g9nb; l b"Łnc4<;™im= k ,/0g; ' =g9U2-in2 f ;<=gqrb^289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9Wý’bj4<;<= Œ u28;<=g9/.•bdc<c*-c289ni k *-9(qrbdc<;<=g9/.•=g; ;*"2™e‰*-c<u; l 28; k 289 & bL,n4bji &(' ; l bCuˆbj4 l .rbd9nbdc28;*-cŽ F ;`=h4 l bdcb ; l 28;X; l bL2-i/i/bjiˆim= k ,/0g;<=hbj4E*8eW2-in2 f ;<=g9/.’;* l =g. l bdc Œ *-ci/bdcE4*-0g,/;<=h*-9n4`289ni•u’,/0g;<= Œ q128c<=32 & 0hb f c* & 0hbduˆ4 28cb4*-0gqrbji› ? l be‰*-c<uˆbdc &(' ;2 ì =g9/.I2 T *-,/c<=hbdc™;<c289n4He“*-c<uº*8e289šbj4<;<=gu28;bji‡bdc<c*-cCe£,/9 k ;<=h*-9 y 289ni ; l b˜0328;<;bdc &:' kl *G*14<=g9/.2•4<;289nin28ciw9n*-c<uµ*8e; l b™q-28c<=32 & 0hbj4Ž F 9 sEl 2 f ;bdc ” b^,n4b; l b^uˆbd;<c<= k i/bdc<=gqrbjiê=g9 sEl 2 f ;bdc ;*]cb‹Łn9nb¬2 f cb Œ b‹‘m=h4<;<=g9/.tuˆbj4 l 4*I; l 28; =g;2 f/f bŽ28c4=h4*-;<c* f = k ” l bd9…uˆbŽ2-4<,/cbji &(' ; l bwuˆbd;<c<= k  ? l bI2 f/f c*r2 kl ” bw,n4bw=h4ˆ;*Ui/b‹Łn9nbN2 –(,280g=g; ' 4 f 2 k b & 2-4bji*-9^2 f/f c*j‘m=gu28;b k =gc k ,/u•c2-im=g=W289niˆu•=g9/=gu’,/u bjim.rbL0hbd9/.1; l 4`2-4EuˆbŽ2-4<,/cbji &:' ; l buˆbd;<c<= k  ? l bd9 y qrbdc<;<= k bj4•28cbˆ=g9n4bdc<;bji 28;"2 f/f c*Ž‘G=gu28;b k =gc k ,/u k bd9(;bdc4e‰*-c;<c<=3289/.10hbj4 ” l *14b k =gc k ,/u•c2-im=g=+28cb]2 & *ŽqrbI2 k bdc<;28=g9 & *-,/9ni/4^289nifiqrbdc<;<= k bj428cbwcbduˆ*ŽqrbjiZe‰*-cbjim.rbj4; l 28;28cb¬;*G* 4<u280g0é D bdqrbdc280 *8e ; l bj4bC=g9n4bdc<;<=h*-9I289nicbduˆ*²q-280›=g;bdc28;<=h*-9n4628cb f bdcHe“*-c<uˆbji¬*-9^; l b+uˆbj4 l y 0hbŽ2jqG=g9/. ; l b˜q12-4<; u2 ë *-c<=g; ' *8e7;<c<=3289/.10hbj4 ” =g; l =g9w; l b™i/bj4=gcbjiw–:,280g=g; '& *-,/9ni/4Ž ? l bj4bX; ” * kl 2 f ;bdc4i/bŽ280 ” =g; l =g9 k cbjim= & 0 ' im=³¶›bdcbd9:;72-4 f b k ;4V*8e s`T ) 289ni™; l ,n4 ” 28c<c289:;=g9nim=gqG=him,280 28;<;bd9(;<=h*-9W A * ” bdqrbdc y ; l b; ” * sEl 2 f ;bdc428cb6280h4*’4;<c*-9/.10 ' k *-, f 0hbji & b k 28,n4b=g;E=h4X,n4bd0hbj44\;* l 2Žqrb 289/=h4*-;<c* f = k bdc<c*-c7bj4<;<=gu28;<=h*-9 ” =g; l *-,/; & bd=g9/.˜2 & 0hb\;* k cbŽ28;b`289/=h4*-;<c* f = k uˆbj4 l bj4² S = ì b ” =h4b y l =g. lmŒ –(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l bj4 k 289/9n*-; & b k cbŽ28;bji ” =g; l *-,/;"Łnc4<;"i/b‹Łn9/=g9/. ” l 28;; l 28;289/=h4*-;<c* f ' 4 l *-,/0hi & b1 F ;6=h4L; l b k *-u & =g928;<=h*-9‡*8eX; l bj4b’; ” *¬28c<;<= k 0hbj4 y 289ni l bd9 k b"; l =h4L; l bj4<=h4 y ; l 28; f c*²q(=hi/bj4 ; l b k *-u f 0hbd;b™q(=hb ” *8e7; ” * Œ im=guˆbd9n4<=h*-9280$289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9t289ni l * ” =g; k 289 & b+,n4bji ;*=gu f c*²qrb s`T ) َí { œ » ¹ »!€êûü ø  »ž þ €‚  »  ¸ €" ù¸ € Ÿ ø7ß !   €" »@€ { ¹ ø } ¸ U€ ¸ $~ø ƒ »}#! {   $ ‚  ø ¸º¹ ¡ L	˜)-"W	 ? l bdcbC=h4 y =g9.rbd9nbdc280 y 289=g9:qrbdc4b6cbd0328;<=h*-9n4 l = f & bd; ” bjbd9; l b+2 kjk ,/c2 k ' *8e s`T ) 4*-0g,/;<=h*-9n4289ni^; l b ;<=guˆbcbj–(,/=gcbjiˆ;* f c*(im, k b6; l bduw F 9 k cbŽ2-4bj4X=g9 k *-u f ,/;28;<=h*-9280cbj4*-,/c k bj4Xu2 ' l 2Žqrb64<=g.19/=³Ł k 289(;<0 ' =gu f c*²qrbji‚; l =h4cbd0328;<=h*-9n4 l = f y7& ,/;; l bdcb=h44<;<=g0g0`2I4<;<c*-9/.]i/bj4<=gcb=g9U=g9nim,n4;<c ' e‰*-cuˆ*-cbb k =hbd9:; s`T ) s *-9n4bj–(,nbd9:;<0 'ry cbj4bŽ28c kl bdc4 l 2jqrbši/bdqrbd0h* f bji 2‚q-28c<=hbd; ' *8e"=gu f c*Žqrbduˆbd9(;4I;* ; l b s`T ) f c* k bj44Ž ? ” *4, kl =gu f c*²qrbduˆbd9:;4˜28cb™289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9t289ni l =g. l bdc Œ *-ci/bdc6uˆbd; l *Gi/4Ž ? l bˆŁnc4<;’=gu f c*²qrbduˆbd9:; y 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 y =h4™; l b f c* k bj44’*8e,n4<=g9/.I; l b4*-0g,/;<=h*-9‚*-9 289 =g9/=g;<=3280uˆbj4 l ;* f c*(im, k b"2 & bd;<;bdc y 289/=h4*-;<c* f = k uˆbj4 l  D =g9 k b; l b™9nb ” uˆbj4 l =h4 & bd;<;bdcL4<,/=g;bji];* ; l b`4*-0g,/;<=h*-9 y ; l b2 kjk ,/c2 k ' 4 l *-,/0hi"=g9 k cbŽ2-4b1 F en; l b k ' k 0hb\=h4 cb f bŽ28;bji y ; l bEuˆbj4 l 4 l *-,/0hi k *-9:qrbdc<.rb ;*•*-9nb ” l = kl =h4* f ;<=gu280›e“*-c ; l b f c* & 0hbdu & bd=g9/.ˆb‹‘/28u•=g9nbji› ? l bdcb28cb+u’,/0g;<= f 0hb2-imq1289:;28.rbj46;* 2-i/im=g9/.˜; l =h4e“bjbji & 2 kì 0h*(* f =g9ˆ2 s`T )280g.r*-c<=g; l uw T =gc4; y =g;74=g.19/=³Ł k 289:;<0 ' cbjim, k bj4$; l bi/b f bd9ni/bd9 k b *8e›; l bŁn92804*-0g,/;<=h*-9^*-9; l buˆbj4 l 289ni4 l *-,/0hiˆ; l ,n4`=g9 k cbŽ2-4b k *-9mŁi/bd9 k bC=g9ˆ; l b6cbj4,/0g;4Ž D b k *-9ni y ;*^2 kl =hbdqrb"; l b4 28uˆb"2 kjk ,/c2 k ' *-9š2•,/9/=³e“*-c<u•0 ' cb‹Łn9nbji].1c<=hiIu•=g. l ;+cbj–:,/=gcb"4*ˆu289 ' k bd0g0h4L; l 28; ; l b k *-u f ,/;28;<=h*-9 & b k *-uˆbj4ˆ,/9me“bŽ2-4<= & 0hb1 > 9n*-; l bdc2-imq-289(;28.rb y 289niZ*-9nb ” l = kl 4 l *-,/0hi 9n*-; & b *²qrbdc<0h*(* ì bji y =h4; l 28; k cbŽ28;<=g9/.2ˆuˆbj4 l =h4C*8e£;bd9]; l b™uˆ*14<;e£c<,n4<;<c28;<=g9/.¬289niw,n4bdc Œ =g9:;bd9n4<=gqrb f 28c<;L*8e s`T ) K ' 28,/;*-u28;<=g9/.; l =h4 f c* k bj44 y1” bL4<, & 4<;289:;<=3280g0 ' =g9 k cbŽ2-4b; l b,n42 & =g0g=g; ' *8e s`T ) f c*-.1c28uˆ4² ? l b¬4b k *-9niê=gu f c*²qrbduˆbd9:; y l =g. lmŒ *-ci/bdcˆuˆbd; l *Gi/4 y 28cbw289n*-; l bdc•cb k bd9(;ˆ=gu f c*Žqrbduˆbd9(;;* s`T ) M *14<; k ,/c<cbd9:; s`T ) k *Gi/b k cbŽ28;bj42 f =hb k b ” =h4b¬0g=g9nbŽ28cˆcb k *-9n4<;<c<, k ;<=h*-9 *8eC; l b¬4*-0g,/;<=h*-9W F e+; l b 4*-0g,/;<=h*-9"=h44<uˆ*(*-; l y ; l b`2 kjk ,/c2 k ' *8e/; l 28; cb k *-9n4;<c<, k ;<=h*-9"=h4 f c* f *-c<;<=h*-9280G;*6; l bE4=g¦Žb\*8en; l bEuˆbj4 l 4–(,28cbji› K ' ,n4=g9/. l =g. l bdcE*-ci/bdc f *-0 ' 9n*-u•=3280h4E;*™cb k *-9n4<;<c<, k ;`; l b4*-0g,/;<=h*-9 y(” b k 289ˆ=g9 k cbŽ2-4bL; l b Ùdá ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä *-ci/bdcX*8e2 kjk ,/c2 k '  T *-cXb‹‘/28u f 0hb y =³e; l b 4*-0g,/;<=h*-9•=h47cb k *-9n4<;<c<, k ;bji ” =g; l 2C–:,2-imc28;<= k f *-0 ' 9n*-u•=3280 y ; l b•2 kjk ,/c2 k 'N” =g0g0 & b f c* f *-c<;<=h*-9280$;*^; l b4<=g¦Žb*8e\; l buˆbj4 lšk bd0g0h4 k , & bji o ; l =gci Œ *-ci/bdc p‹’† l =g0hb ; l =h4=g9 k ,/c4^289…=g9 k cbŽ2-4bw=g9 k *-u f ,/;28;<=h*-9280 k *14<; f bdc k bd0g0 y ; l bI=g9 k cbŽ2-4bw=g9 *-ci/bdc*8e’2 kjk ,/c2 k ' uˆbŽ289n46; l 28; e“28ce“b ” bdc k bd0g0h4628cb˜cbj–(,/=gcbjiI289ni¬; l bdcb˜=h462 f *-;bd9:;<=3280 9nbd;.:28=g9w=g9Ib k =hbd9 k '  s *-u & =g9/=g9/.I; l bj4b•; ” *I=gu f c*Žqrbduˆbd9:;4"=h4’2^9n*-9 Œ ;<c<=gqG=3280\;2-4 ì  ? l buˆ*14<; k *-u•uˆ*-9ê2 f/f c*r2 kl ;* 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9I=h4;*cb‹Łn9nb & 2-4bji¬*-9w; l b A bj44<=3289w*8e$24*-0g,/;<=h*-9¬q-28c<=32 & 0hb@= zy Ù xmy Ù v/y َí y َî†?­ ? l bLcbŽ2-4*-9/=g9/."e‰*-c`; l =h4`=h4\; l 28;`; l b+bdc<c*-c`*-9¬2 f =hb k b Œ ” =h4bC0g=g9nbŽ28c`=g9(;bdc f *-0328;<=h*-9¬*8e24<uˆ*G*-; l e‰,/9 k ;<=h*-9t=h4 & *-,/9ni/bji &:' 2–(,2-imc28;<= k ;bdc<uº=g9:qr*-0gqG=g9/.^; l b4b k *-9niti/bdc<=gq128;<=gqrbj4 o ; l b A bj44=3289p*8e ; l b’e£,/9 k ;<=h*-9WC† l bd9 l =g. l bdc Œ *-ci/bdcC=g9(;bdc f *-0328;<=h*-9š=h4C,n4bji y 2-4L=g;+=h4 ” =g; ltl =g. l bdc Œ *-ci/bdc˜uˆbd; l *Gi/4 y ; l =h4cbŽ2-4*-9/=g9/.ˆ=h49n*0h*-9/.rbdcq1280g=hi› > ;; l b˜;<=guˆb™*8e ” c<=g;<=g9/. y/” b’28cb+; l b™*-9/0 ' 28,/; l *-c4;*•b‹‘m;bd9ni A bj44<=3289 Œ & 2-4bjiw289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9:;L;* l =g. l bdc6*-ci/bdcuˆbd; l *Gi/4Ž ? *]2 kjk *-u f 0g=h4 l ; l =h4 y ” b•Łnc4;’uˆ*Gi/bd0X; l bˆ0h* k 280Ebdc<c*-c ” =g; l 2 f *-0 ' 9n*-u•=3280Xe‰c*-u ; l b ? 2 ' 0h*-c"b‹‘ Œ f 289n4<=h*-9W ? l =h4 f *-0 ' 9n*-u•=3280=h4"; l bd9ê,n4bji‚;* k *-9n4<;<c<, k ;2]uˆbd;<c<= k e‰,/9 k ;<=h*-9fi; l c*-,/. l 2 T *-,/c<=hbdc ;<c289n4He“*-c<uw K ' ;2 ì =g9/.U; l =h4^2 f/f c*r2 kl y *-,/cuˆbd;<c<= k i/bdc<=gq128;<=h*-9 k 289 & bI2 f/f 0g=hbji…;*U289 ' *-ci/bdc cb k *-9n4<;<c<, k ;<=h*-9t289niI289 ' 9(,/u & bdc6*8eq128c<=32 & 0hbj4Ž †‡b & bd.1=g9™; l =h428c<;<= k 0hb &:' Łnc4;Vb‹‘/28u•=g9/=g9/. l * ” ; l b$0h* k 280(bdc<c*-c k 289 & b\2 f/f c*j‘m=gu28;bji,n4<=g9/. ? 2 ' 0h*-c b‹‘ f 289n4<=h*-9n4 y 289ni l * ” ; l =h4 k *-u f 28cbj47;*C*-; l bdcuˆbd; l *Gi/47*8ebdc<c*-c$bj4<;<=gu28;<=h*-9W ? l bd9 ” b 4 l * ” l * ” 289‚2 f/f c* f c<=328;buˆbd;<c<= k e‰,/9 k ;<=h*-9 k 289 & b kl *14bd9‡=gc<cbj4 f b k ;<=gqrb*8e ; l b•9:,/u & bdc*8e q-28c<=32 & 0hbj4"289ni ; l b˜*-ci/bdc6*8e7; l b˜bdc<c*-cŽ$†‡b˜; l bd9w4 l * ” ; l b & bd9nb‹Łn;46*8e; l =h4uˆbd;<c<= k e‰,/9 k ;<=h*-9¬e‰*-c; l cbjb’im=³¶bdcbd9(; ; ” *"im=guˆbd9n4<=h*-9280è* ” 4E*²qrbdc 2™4<=gu f 0hbL28=gcHe“*-=g0é ? l bŁnc4<;\;bj4<; k 2-4b k *-u f 28cbj4`2-in2 f ;bjicb‹Łn9nbduˆbd9:; 289ni¬,/9/=³e“*-c<uµcb‹Łn9nbduˆbd9(;6e“*-c64b k *-9niN289niw; l =gciw*-ci/bdcC4*-0g,/;<=h*-9n4C*8e74<, & 4*-9/= k =g9:qG=h4 k =hiwè* ”  ? l b 4b k *-9niZ;bj4<; k 2-4bIb‹‘/28u•=g9nbj4 l * ” ; l b¬uˆbd;<c<= kwf bdcHe‰*-c<uˆ4=g9Z; l b f cbj4bd9 k bI*8e™im=h4 k *-9:;<=g9(,/=g;<=hbj4=g9 ; l b4*-0g,/;<=h*-9W ? l b; l =gci]289niIŁn9280 ;bj4; k 2-4bb‹‘/28u•=g9nbj4L; l b & bd9nb‹Łn;4C*8eE289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 qrbdc4<,n4`=h4*-;<c* f = k 2-in2 f ;28;<=h*-9e“*-c q(=h4 k *-,n4\è* ”  ? l =h4`Łn9280;bj4<; k 2-4bC=h4280h4*’,n4bji;*b‹‘n28u•=g9nb+; l b im=³¶›bdcbd9 k bj4 & bd; ” bjbd9]4b k *-9niI289ni; l =gciI*-ci/bdc6uˆbd;<c<= k 4 e“*-c; l =gciw*-ci/bdc64*-0g,/;<=h*-9n4Ž ÙŽî ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä }šž}U/	H+º	 > 9280 ' ;<= k žŸ©‹¡“°-© ¡ bdc<c*-c"289280 ' 4<=h4+e“*-c s`T ) 4*-0gqrbdc4„' ” l bd; l bdc’Łn9/=g;b•qr*-0g,/uˆb y Łn9/=g;bˆbd0hbduˆbd9:; y *-c˜Łn9/=g;bim=³¶bdcbd9 k by' =h4 & 2-4bji‡*-9 ? 2 ' 0h*-c4bdc<=hbj4"289280 ' 4=h4Ž† l =g0hbˆ; l bi/bd;28=g0h4im=³¶›bdc & bd; ” bjbd9 eé28u•=g0g=hbj4E*8eW4 kl bduˆbj4 y =g9^280g0 k 2-4bj4X; l bL289280 ' 4<=h4 2-44<,/uˆbj4\; l b64*-0g,/;<=h*-9;*™; l b65\) B o 4 p l 2-4 2˜q1280g=hi y k *-9(qrbdc<.1=g9/. ? 2 ' 0h*-c˜b‹‘ f 289n4=h*-9t28;+bdqrbdc ' f *-=g9:;C=g9]; l b"i/*-u28=g9W F e$; l b4 kl bduˆb=h4C*8eE*-ci/bdc p y” b 2 f/f c*Ž‘G=gu28;bC; l bC4*-0g,/;<=h*-9 f(x) 28c*-,/9ni4*-uˆb0h* k 28;<=h*-9 a &(' ; l b ? 2 ' 0h*-cb‹‘ f 289n4<=h*-9;bdc<uˆ4`, f ;* i/bd.1cbjb p−1  T *-cEb‹‘n28u f 0hb y =g92™*-9nb Œ im=guˆbd9n4<=h*-9280 y ; l =gci*-ci/bdc`4 kl bduˆb yG” b ” *-,/0hi2 f/f c*j‘m=gu28;b f(x) 28c*-,/9ni a 2-4 f(x) ≈ f(a) + df dx (a)(x− a) + 1 2 d2f dx2 (a)(x− a)2 o Ù²p ? l bCim=³¶bdcbd9 k b & bd; ” bjbd9; l bLb‹‘/2 k ;Ee£,/9 k ;<=h*-9 f b‹‘n2 k ; (x) 289niˆ; l b+2 f/f c*j‘m=gu28;bLe£,/9 k ;<=h*-9 f 2 f/f (x) =h4; l bdcb‹e“*-cb; l bcbdu28=g9/=g9/.‡;bdc<uˆ4e‰c*-u ; l b ? 2 ' 0h*-cb‹‘ f 289n4<=h*-9W s *-9:;<=g9(,/=g9/.‚; l b¬b‹‘n28u f 0hb=g9 B –(,28;<=h*-9UÙ y f b‹‘n2 k ; (x)− f 2 f/f (x) = 1 3! d3f dx3 (a)(x − a)3 + 1 4! d4f dx4 (a)(x − a)4 + ... o v p T *-c`4<u280g028cbŽ2-4`28c*-,/9ni a y ; l b0hbŽ2-im=g9/.;bdc<uö=g9ˆ; l =h4Ebdc<c*-c\=h4E2 f *-0 ' 9n*-u•=3280*8eWi/bd.1cbjb p 289ni*-9/0 ' p 7†‡b™b‹‘ f 0h*-=g;; l =h4 kl 28c2 k ;bdc<=h4<;<= k =g9]*-,/cbdc<c*-c=g9nim= k 28;*-cŽ ? l bC2 kjk ,/c2 k ' *8e; l bL4*-0gqrbdc\i/b f bd9ni/4E*-9 l * ”…” bd0g0=g; k 289bdq1280g,28;b f(x) 28;E4*-uˆb6*-; l bdcE0h* k 28;<=h*-9 y 42 ' b  ? l b k *-9n4bj–:,nbd9 k b*8e; l =h4’=h4’; l 28; y =³e; l bim=h4<;289 k b & bd; ” bjbd9 f *-=g9:;4=g9‡; l buˆbj4 l 4 k 280hbj4 ” =g; l h y ; l b™bdc<c*-c=g9w; l b’4*-0g,/;<=h*-9I4 l *-,/0hi]4 k 280hb ” =g; l hp  ? l =h4=h46=g0g0g,n4<;<c28;bji &:' k *-9(;<=g9:,/=g9/.; l b b‹‘n28u f 0hb˜=g9 B –(,28;<=h*-9 vmy f b‹‘n2 k ; (b)− f 2 f/f (b) ≈ 1 3! d3f dx3 (a)(b− a)3 B c<c*-c (h) ≈ 1 3! d3f dx3 (a)h3 o írp F 9 f c2 k ;<= k b y 4*-0gqrbdc4 k *-u & =g9nbš; l bj4bU2 f/f c*Ž‘G=gu28;<=h*-9n4t28;I2ê0328c<.rbš9:,/u & bdc]*8eim=h4 k cbd;b f *-=g9(;4 ;*fi2 f/f c*j‘m=gu28;bt; l bt*Žqrbdc280g0™4*-0g,/;<=h*-9 f(x) @† l =g0hb]; l b f cb k =h4b ” 2 ' =g9 ” l = kl ; l =h4=h4i/*-9nb ÙŽÚ ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä q128c<=hbj4`e£c*-u *-9nb’4*-0gqrbdc;*ˆ289n*-; l bdc y 289 ' ? 2 ' 0h*-c Œ & 2-4bjiI4 kl bduˆb ” =g0g0 l 2jqrb’289wbdc<c*-c6289280h*-.r*-,n46;* B –(,28;<=h*-9Uím ? l b".rbd9nbdc280g=g; ' *8eE; l =h4+i/bdc<=gq128;<=h*-9š=h4C*-9nb*8e`; l b f c<=g9 k = f 280$cbŽ2-4*-9n4Cbd9/.1=g9nbjbdc4™,n4b ; l =h46; ' f b*8e$bdc<c*-cCbj4<;<=gu28;<=h*-9W A * ” bdqrbdc y *-; l bdc6uˆbd; l *Gi/4e“*-cLbj4;<=gu28;<=g9/.bdc<c*-cLb‹‘m=h4<; y =g9 k 0g,nim=g9/. 2-i ë *-=g9:;uˆbd; l *Gi/4L289ni¬e‰bŽ28;<,/cb Œ & 2-4bjiwuˆbd; l *(i/4Ž † l bd9 ” bi/bdc<=gqrbji‚; l bcb k *-9n4<;<c<, k ;<=h*-9fibdc<c*-c y7” b2-44<,/uˆbji‚; l 28; f 2 f/f (a) ” 2-4 ì 9n* ” 9 b‹‘/2 k ;<0 '  ? l =h4 2-44<,/u f ;<=h*-9^=h4\9n*-;`9nb k bj4428c<=g0 ' ;<c<,nb y 289ni^289 ' bdc<c*-c4`=g9; l b k 280 k ,/0328;<=h*-9¬*8e f 2 f/f (a) ” =g0g0 & b k *-9(qrb k ;bjit;*; l b4*-0g,/;<=h*-9‡28; b  D =g9 k b"; l =h4 f c* k bj44C=h4Lcb f bŽ28;bji]e£c*-u k bd0g07;* k bd0g074u280g07bdc<c*-c4 k 289 & b k *-9:qrb k ;bji 289ni k 28,n4b0328c<.rbbdc<c*-c4e“28c•2 ” 2 ' e£c*-u ; l bd=gc"4*-,/c k b1 > i ë *-=g9:; Œ & 2-4bjifibdc<c*-c uˆbŽ2-4<,/cbj4= í y ánÙ?E28;<;bdu f ;™;* k 2 f ;<,/cb; l =h4 k *-9:qrb k ;bji‚bdc<c*-cŽ ? *¬i/*; l =h4 y ; l b ' 4*-0gqrb & *-; l ; l b *-c<=g.1=g928075X) B 4˜289niI; l b2-i ë *-=g9:;˜5X) B 46e‰*-c+4*-uˆb’i/bj4=gcbjiwe‰,/9 k ;<=h*-9280é ? l bj4b’; ” *4*-0g,/;<=h*-9n4 k 289 ; l bd9 & b k *-u f 28cbjiê;*Ii/bd;bdc<u•=g9nb ” l bdcb; l 28;e‰,/9 k ;<=h*-9280`=h4uˆ*14<;"i/b f bd9ni/bd9(;*-9‚; l b4*-0g,/;<=h*-9 289ni]; l ,n4Ccb‹Łn9nb"; l b’uˆbj4 l 2 kjk *-cim=g9/.10 '  ? l b2-imq-289(;28.rb"*8eX; l =h4+bdc<c*-cCuˆbŽ2-4<,/cb"=h46; l 28;C=g;C.1=gqrbj4 2tu, kl uˆ*-cbw.10h* & 280+bj4<;<=gu28;bI*8e˜; l bwbdc<c*-c y ” l = kl =h4bj4 f b k =3280g0 ' =gu f *-c<;289:;e“*-c5X) B 4 ” =g; l 4<, & 4;289:;<=3280 k *-9(qrb k ;<=h*-9 k *-u f *-9nbd9(;4Ž J 9me“*-c<;<,/928;bd0 'ry 4*-0gqG=g9/."; l b™2-i ë *-=g9:;5\) B 4 k 289 & b˜2"qrbdc ' ;<=guˆb k *-9n4,/u•=g9/. f c* k bj44Ž ? l b¬uˆbj4 l bj4•; l 28;28cb k cbŽ28;bji &:' ; l =h4•uˆbd; l *(i…28cbI280h4*š4<;<c*-9/.10 ' i/b f bd9ni/bd9:;¬*-9Z; l b kl *14bd9Ze‰,/9 k ;<=h*-9280é F e y e“*-cb‹‘n28u f 0hb y 2‡uˆbj4 l =h4 k cbŽ28;bjiZ;*‚=g9 k cbŽ2-4b]; l b 2 kjk ,/c2 k ' *8eE; l b•2-bdc*(i ' 928u•= k imc28. y =g;+u•=g. l ; l 2Žqrb20hbj44˜2 kjk ,/c28;b f cbjim= k ;<=h*-9‡*8e`0g=³e£;C; l 289U289 uˆbj4 l * f ;<=gu•=g¦Žbjiwe‰*-c0g=³e‰;L2 kjk ,/c2 k '  > 9n*-; l bdc+uˆbd; l *(it*8e\bj4<;<=gu28;<=g9/.wbdc<c*-c y e“bŽ28;<,/cb Œ & 2-4bjitbdc<c*-c˜bj4<;<=gu28;<=h*-9´=gÙ z ? y b‹‘ f 0g= k =g;<0 ' i/bd;b k ;4 289niZ289280 ' ¦Žbj4•è* ” e“bŽ28;<,/cbj4Ž M bj4 l cb‹Łn9nbduˆbd9:;^=h4•; l bd9Zi/*-9nb¬;*‡cbj4*-0gqrb¬; l *14b^è* ” e“bŽ28;<,/cbj4 2 kjk *-cim=g9/.fi;* 4*-uˆb f cb Œ i/b‹Łn9nbji4bd;*8ec<,/0hbj4Ž ? l =h4=h4bj4 f b k =3280g0 ' ,n4b‹e‰,/0Ce‰*-c^=gu f *14<=g9/. uˆ*-cb 4<;<c<= k ;uˆbj4 l cb‹Łn9nbduˆbd9(; k c<=g;bdc<=32]4<, kl 2-4™uˆbj4 l *-c<; l *-.r*-9280g=g; ' =g9 & *-,/9nin28c ' 032 ' bdc4•289ni‡qrbdc<;b‹‘ 280g=g.19/uˆbd9(; ” =g; l 4 l * kì 4² J 9me‰*-c<;<,/928;bd0 'ry ; l =h4uˆbd; l *Gi l 2-4u, kl 0hbj446*8e\2; l bj*-cbd;<= k 280 & 2-4<=h4e“*-c l * ” *-9nb"è* ” e“bŽ28;<,/cb•4 l *-,/0hi & b"cb‹Łn9nbjitcbd0328;<=gqrb•;*¬289n*-; l bdc289niN; l b"e“bŽ28;<,/cb•i/bd;b k ;*-c4˜u’,n4; & b+;<,/9nbjie“*-cbŽ2 kl ; ' f b˜*8eè* ”  † l =g0hb & *-; l *8eX; l bj4b’uˆbd; l *(i/4 l 2jqrb; l bd=gcCcbj4 f b k ;<=gqrb"2-imq-289(;28.rbj4˜289ni]im=h42-imq1289:;28.rbj4 y” b l 2Žqrb kl *14bd9‡;*^e“* k ,n4˜*-9‡0h* k 280$cb k *-9n4<;<c<, k ;<=h*-9êbdc<c*-c & b k 28,n4b=g; f c*ŽqG=hi/bj4289‡bdc<c*-c™uˆbŽ2-4<,/cbˆ; l 28;˜=h4 Ù²õ ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä im=gcb k ;<0 ' i/b f bd9ni/bd9:; *-9^; l b64*-0g,/;<=h*-9^qrb k ;*-cE=g9n4;bŽ2-i*8eV2 f 28c<;<= k ,/0328c`e‰,/9 k ;<=h*-9280é ? l =h4E=h4E280h4*; l b 2 f/f c*r2 kl ,n4bji &(' ; l b™u2 ë *-c<=g; ' *8e7cbj4bŽ28c kl bdc4Ž= z/y Ù xmy َí y Ù vny َî†?­ Y ; l bdc$4*-,/c k bj47*8ebdc<c*-c$4<, kl 2-4uˆ*Gi/bd0g=g9/.˜bdc<c*-c$289ni"9(,/uˆbdc<= k 280/c*-,/9ni/*8¶^bdc<c*-cX28cb`9n*-;72-i/imcbj44bji =g9; l =h4 f 2 f bdcŽ F ;7=h4X2-44<,/uˆbji"; l 28;$; l b i/bj4<=gcbji•4*-0g,/;<=h*-9•=h4; l b4*-0g,/;<=h*-9•;*C; l b uˆ*Gi/bd0hbji f c* & 0hbdu y ” l = kl u2 ' im=³¶›bdc’e‰c*-u ; l b2 k ;<,280 f c* & 0hbduw F 9‚; l =h4 ” 2 ' uˆ*(i/bd0g=g9/.tbdc<c*-c k 289 & b2-i/imcbj44bji =g9ni/b f bd9ni/bd9:;<0 '  > 0h4* y 9:,/uˆbdc<= k 280˜c*-,/9ni/*8¶@bdc<c*-c=h4¬2-44<,/uˆbji…;* & b]u’, kl 4<u280g0hbdc^; l 289a; l b f cbdqG=h*-,n4<0 ' i/bj4 k c<= & bjiI4*-,/c k bj4*8e$bdc<c*-cL289niw4* ” b ” =g0g0W9nbd.10hb k ;=g;Ž * /.J-). 	™˜º	%fl ffffifl	 >@M bd;<c<= k =h42"e£,/9 k ;<=h*-9 d : X ×X → R 4<, kl ; l 28;=g;428;<=h4HŁbj4; l b+e‰*-0g0h* ” =g9/. k *-9nim=g;<=h*-9n4 d(a, b) > 0 ⇐⇒ a 6= b d(a, a) = 0 d(a, b) = d(b, a) d(a, c) ≤ d(a, b) + d(b, c) o á(p M *-cbL4=gu f 0 ' f ,/; y 2™uˆbd;<c<= k i/b‹Łn9nbj4 ; l bCim=h4<;289 k b & bd; ” bjbd9 f *-=g9(;4E=g9w2 ” 2 ' 4<=gu•=g0328c`;*"; l b+4<;289 Œ in28ci s 28c<;bj4<=3289]im=h4<;289 k b+uˆbd;<c<= k d 4;i (a, b) = √ (a− b)T (a− b)  K b k 28,n4b˜*8e ; l bj4b f c* f bdc<;<=hbj4 y 289 ' .rbj*-uˆbd;<c ' ; l 28; k 289 & b+i/b‹Łn9nbji=g9¬;bdc<uˆ4*8e7im=h4<;289 k bj4 l 2-4bj–(,/=gq-280hbd9(; f c* f bdc<;<=hbj42 kjk *-cim=g9/. ;*’; l b6uˆbd;<c<= k  ? l bLuˆbd;<c<= k ; l 28; ” bL,n4be‰*-c 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 y d 289/= (a, b) y =h4 289=g9(qrbdc<;<= & 0hb 0g=g9nbŽ28c;<c289n4<e‰*-c<u28;<=h*-9 L *8e; l b˜4;289nin28ciIim=h4<;289 k b˜uˆbŽ2-4,/cb d 4;i (a, b)  d 4<;i (a, b) = √ (a− b)T (a− b) d 289/= (a, b) = √ (L(a− b))T (L(a− b)) d 289/= (a, b) = √ (a− b(LT L)(a− b) d 289/= (a, b) = √ (a− b)T M(a− b) o îrp Ù z ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä ? l buˆbd;<c<= k d 289/= (a, b) k 289t; l ,n4 & bi/b‹Łn9nbji &(' ; l b4 ' u•uˆbd;<c<= kf *14=g;<=gqrb•i/b‹Łn9/=g;bu28;<c<=³‘ M = LT L  ? * k 280 k ,/0328;b+; l bLq1280g,nbj4*8e M e£c*-u B –(,28;<=h*-9Iî’e‰*-c2"4*-0g,/;<=h*-9w*8e*-ci/bdc p ” bLŁnc4<;2-44,/uˆbL; l 28; ; l b•bdc<c*-c˜=h4+bj–(,280$;*; l b p i/bd.1cbjb;bdc<uˆ4™*8e`; l b ? 2 ' 0h*-c’b‹‘ f 289n4<=h*-9 y 2-4˜im=h4 k ,n44bjiN=g9 D b k ;<=h*-9  > 0h4* y =g;L=h4=g9(;<,/=g;<=gqrb;*^2-44<,/uˆb’; l 28;Lbdc<c*-c4 ” l = kl 28cb f *14<=g;<=gqrb28cbbj–:,/=gq1280hbd9:;C;*bdc<c*-c4 ” l = kl 28cb9nbd.:28;<=gqrb1 ? l bdcb‹e“*-cb¬; l b¬bdc<c*-cˆuˆbŽ2-4,/cb=h4•uˆ*(im=³Łbji ;* & b^; l bI2 & 4*-0g,/;bq1280g,nb¬*8e+; l b p *-ci/bdc f *-0 ' 9n*-u•=3280 e£c*-uû; l b ? 2 ' 0h*-c˜b‹‘ f 289n4<=h*-9W T *-c+b‹‘n28u f 0hb y ; l bbdc<c*-c6e“*-cC2•e£,/9 k ;<=h*-9 f(x, y) 28;•2Iim=h4;289 k b (∆x,∆y) e£c*-u ; l bcb‹e‰bdcbd9 k b f *-=g9:;•*8e6; l b ? 2 ' 0h*-c•4bdc<=hbj4b‹‘ f 289n4<=h*-9 =g9 2I; ” * Œ im=guˆbd9n4<=h*-9280 y ; l =gciw*-ci/bdc64*-0g,/;<=h*-9I=h4 B c<c (∆x,∆y) ≈ ∣∣∣∣16 d 3f dx3 (∆x)3 + 1 2 d3f dx2y (∆x)2(∆y) + 1 2 d3f dxy2 (∆x)(∆y)2 + 1 6 d3f dy3 (∆y)3 ∣∣∣∣ T *-c4*-0g,/;<=h*-9n4 ” =g; l u’,/0g;<= f 0hbEq-28c<=32 & 0hbj4 y =g;W=h4V9nb k bj4428c ' ;*;2 ì b`24 k 280328cVuˆbŽ2-4,/cb\*8em; l bX=g9nim=gqG=him,280 q128c<=32 & 0hb˜bdc<c*-c4Ž Y ,/c62 f/f c*r2 kl =h4 ;*,n4b˜*-9nb˜*8e; l b˜4<;289nin28ci¬qrb k ;*-c9n*-c<uˆ4Ž > 9n*-; l bdcuˆbd; l *Gi *8e ë *-=g9/=g9/.‚bdc<c*-cˆq-28c<=32 & 0hbj4ˆ=h4•; l b k *-9 k b f ;*8e+u2¥‘m=gu’,/u k *-9(;28=g9nbji…bdc<c*-c Œ bd0g0g= f 4bN2-4 f c* f *14bji &(' A ,289/.µ=gَîB?­ A * ” bdqrbdc y =g;"=h4",/9 k 0hbŽ28c l * ” ; l =h4•4 l *-,/0hi & b k *-9n4<;<c<, k ;bjiêe‰*-c"uˆ*-cb^; l 289 ; ” * q128c<=32 & 0hbj4ˆ289niU=g;=h44<=g.19/=³Ł k 289(;<0 ' uˆ*-cb k *-u f 0g= k 28;bji ;* k *-u f ,/;b1 T *-c"; l bcbdu28=g9ni/bdcˆ*8eL; l =h4 f 2 f bdc ” b ” =g0g0›,n4b˜; l b L1 9n*-c<u y *-c; l b’2jqrbdc28.rb*8e; l b f *-0 ' 9n*-u•=3280h43¶1 J 4<=g9/.ˆ; l b f cbdqG=h*-,n4<0 ' 4<;28;bjiI2-44<,/u f ;<=h*-9n4 y/” b k 289w; l bdcb‹e“*-cb ” c<=g;b’2•.rbd9nbdc280 bj–(,28;<=h*-9we“*-c; l b bdc<c*-c6*8eX2 p *-ci/bdc64*-0g,/;<=h*-9I=g9 d im=guˆbd9n4<=h*-9n4 y/” =g; l n q-28c<=32 & 0hbj4L*ŽqrbdcL2"qrb k ;*-c (∆x1, ...,∆xd)  B c<c (∆x1, ...,∆xd) ≈ 1 n n∑ i=1 |Pi(∆x1,∆x2, ...,∆xd)| o Úrp ” l bdcb Pi =h4C2 f *-0 ' 9n*-u•=3280 ” =g; l ;bdc<uˆ4+*8e\i/bd.1cbjb p 289niN*-9/0 ' p †‡b; l bd9]cb Œ ” c<=g;b B –(,28;<=h*-9‡Ú =g9 d im=guˆbd9n4<=h*-9280 4 f/l bdc<= k 280 k *(*-cim=g928;bj4Ž B c<c (∆r, θ1, ..., θd−1) ≈ (∆r)pf(θ1, ..., θd−1) o õ1p ·ˆ¸ j†[gX(GE^BX†XBMNXBY„QEj†[,QE[oORQLleIHOV[gOqMNX ƒ [glˆQEMPFZX1h L2 IZXi L∞ X†FZGEcdO]‹|[oGE[,TFZ^BXi)QEF¹GE[oºBXB[,IZKNcdFZORQq[o‡flgKN^BOVMNn[oKNa mIHOV[ei‘FX,»Fe‹šn[gKNF_lgM`Qba™¼§MPi†[eIHK½T¾FZG¿ifGÀIHYqk†GE[eifMNloQEMNFXBOÀÁ‹j†[oGE[eIHOSQEj†[ L1 X†FZGEcm@[oQVQE[ˆG¿IHlglgFZ^BX_QE[ei,T¾FZGJk†GE[gOVOV^fGE[d¼vIZXBi m@[oQVQE[oGkfGE[ei†MNloQE[ei•KPM`TWQˆÁˆu ÙŽó ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä †‡b k 289280 ” 2 ' 4Xeé2 k ;*-cX*-,/; (∆r)p =g9•; l =h4Ebj–:,28;<=h*-9 & b k 28,n4b280g0n; l b ;bdc<uˆ4E*8ebŽ2 klˆf *-0 ' 9n*-u•=3280 Pi 28cb*8eEi/bd.1cbjb p  F 9N.rbd9nbdc280 y f(θ1, θ2, ..., θd−1) =h4+2 k *-u f 0g= k 28;bji y f =hb k b Œ ” =h4b•4uˆ*(*-; l e£,/9 k ;<=h*-9 ; l 28; k 2 f ;<,/cbj47; l b289/=h4*-;<c* f ' *8e; l b bdc<c*-cŽ†‡b280h4* f bdcHe“*-c<u 2L4<=gu•=g0328c f c* k bjim,/cb`e‰*-c; l b`uˆbd;<c<= k e‰,/9 k ;<=h*-9¬e‰c*-u B –:,28;<=h*-9tîm d(∆x1, ...,∆xd) = √ (∆x1, ...,∆xd)T M(∆x1, ...,∆xd) o z p R b Œ ” c<=g;<=g9/.ˆ=g9 d im=guˆbd9n4<=h*-92804 f/l bdc<= k 280 k *G*-cim=g928;bj4 d(∆r,∆θ1, ...,∆θd−1) = (∆r) √ g(θ1, θ2, ..., θd−1) o órp F e ” b k *-9n4=hi/bdcE289•bjim.rb*8e0hbd9/.1; l ∆r y ; l bd9•; l bbdc<c*-c7e‰*-cX2 p *-ci/bdcX4*-0g,/;<=h*-9 ” =g0g0 & b f c* f *-c<;<=h*-9280 ;* (∆r)p  ? l =h4E4 l *-,/0hi280h4* & b; l b k 2-4b ” l bd9 ∆r =h4XuˆbŽ2-4,/cbjiˆ=g9ˆ; l buˆbd;<c<= k  ? l =h4Ebd9n4<,/cbj4E; l b uˆbd;<c<= k 0hbd9/.1; l 289ni¬bdc<c*-c ” =g0g0 l 2jqrb’; l b˜428uˆb˜4 k 280g=g9/.2-4 ∆r kl 289/.rbj4² ? l ,n4 ” b ” 289:;;* kl *(*14b d 4, kl ; l 28; (d(∆r,∆θ1, ...,∆θd−1)) p = B c<c (∆r, θ1, ..., θd−1) o Ù x p s *-u & =g9/=g9/.bj–(,28;<=h*-9n4Cõ y ó•289ni‡Ù xmy (∆r)pf(θ1, θ2, ..., θd−1) ≈ ( (∆r) √ g(θ1, θ2, ..., θd−1) )p (f(θ1, θ2, ..., θd−1)) 2 p ≈ g(θ1, θ2, ..., θd−1) F e ” b kl *G*14b’q-280g,nbj4*8e M 4<, kl ; l 28; g(θ1, θ2, ..., θd−1) ≈ (f(θ1, ..., θd−1)) 2 p ; l bd9w; l b™uˆbd;<c<= k =h4 2.r*(*Giš2 f/f c*j‘m=gu28;<=h*-9‡;*; l b"bdc<c*-cŽ K b k 28,n4b f 289ni g 28cb & *-; l e£,/9 k ;<=h*-9n4˜*8e289t289/.10hb θ y” b ” =g0g07,n4b T *-,/c<=hbdc™4bdc<=hbj4+;*Łn9ni‡289t2 f/f c* f c<=328;b g  R b f bŽ28;<=g9/. B –:,28;<=h*-9Uóe“*-cL; ” *¬im=guˆbd9n4<=h*-9n4 ,n4<=g9/.•; l b™uˆbd;<c<= k d 289/= (x, y) e£c*-u B –:,28;<=h*-9Nîm d 289/= (1, θ) = √ g(θ) g(θ) = dani(1, θ) 2 v1x ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä g(θ) = (sin θ, cos θ)T M(sin θ, cos θ) g(θ) = M1,1 sin 2 θ + 2M1,2 sin θ cos θ + M2,2 cos 2 θ g(θ) = 1 2 (M1,1 + M2,2) + 1 2 (M1,1 −M2,2) cos 2θ + M1,2 sin 2θ F e ” b’280h4*ˆi/*ˆ2 T *-,/c<=hbdc;<c289n4He“*-c<uµ*8e f(θ) 28c*-,/9niw; l b+,/9/=g; k =gc k 0hb ” b˜.rbd; (f(θ)) 2 p = 1 2 a0 + a1 cos θ + b1 sin θ + a2 cos 2θ + b2 sin 2θ + ... ai = 1 pi ∫ 2pi 0 f(θ) cos iθdθ bi = 1 pi ∫ 2pi 0 f(θ) sin iθdθ o Ù1Ù²p P * ”™y ; l b˜*-c<=g.1=g9280 f *-0 ' 9n*-u•=3280h4 Pi e‰c*-u B –(,28;<=h*-9]Ú"; l 28;L28cb˜4<,/u•uˆbjiw;*"Łn9ni f 28cb’280g0Wbd=g; l bdc 4 ' u•uˆbd;<c<= k *-cL280g0289:;<=h4 ' u•uˆbd;<c<= k =g9 x 289ni y $† l bd9 ” b ” c<=g;b™; l b L1 Œ 9n*-c<uû*8e7; l b Pi =g9 f *-0328c k *G*-cim=g928;bj4 y ; l =h4 .1=gqrbj42˜e‰,/9 k ;<=h*-9 f (θ) ; l 28; =h4 f bdc<=h*Gim= k ” =g; lf bdc<=h*(i pi y 9n*-; 2pi xm; l =h4`=h4 k 0hbŽ28c<0 ' =g0g0g,n4<;<c28;bjiI=g9 T =g.1,/cb v  ? l =h4=g9w;<,/c<9]=gu f 0g=hbj4; l 28;6; l b’*(i/i¬;bdc<uˆ46=g9w; l b T *-,/c<=hbdcL4bdc<=hbj4 o uˆ*14<; =gu f *-c<;289:;<0 ' a1 289ni b1 p u’,n4<; & b+¦Žbdc*/ †‡b k 289w; l bdcb‹e“*-cb ” c<=g;b (f(θ)) 2 p ≈ ao + a2 cos 2θ + b2 sin 2θ o Ù v p ? l b˜bdc<c*-c=g9; l =h462 f/f c*Ž‘G=gu28;<=h*-9]=h4cbd0328;bji;*; l b l =g. l bdc y c2 f =him0 ' Œ­k *-9:qrbdc<.1=g9/.bdqrbd9 Œ e£cbj–(,nbd9 k ' ;bdc<uˆ4L*8e ; l b T *-,/c<=hbdcCb‹‘ f 289n4=h*-9W F e ” b™4bd;; l b k *(b k =hbd9(;4L*8e M 2-4e‰*-0g0h* ” 4 M1,1 = 1 2 a0 + a2 M1,2 = M2,1 = b2 M2,2 = 1 2 a0 − a2 o َírp ” b l 2jqrbˆ2.r*(*Git2 f/f c*j‘m=gu28;<=h*-9‚*8e g(θ) ;* (f(θ)) 2 p 289ni l bd9 k bˆ2.r*G*(iš2 f/f c*Ž‘G=gu28;<=h*-9U*8e *-,/c uˆbd;<c<= k ;*¬; l bbj4<;<=gu28;bji‚bdc<c*-cŽ T *-c M ;* & b f *14=g;<=gqrbˆi/b‹Łn9/=g;b o 2-4’cbj–:,/=gcbji &:' ; l bˆuˆbd;<c<= k p y ; l b T *-,/c<=hbdc’2 f/f c*j‘m=gu28;<=h*-9šu’,n4;˜280h4* & b f *14<=g;<=gqrb"i/b‹Łn9/=g;b1 J 9me“*-c<;<,/928;bd0 'ry 2ˆe£,/9 k ;<=h*-9 ” l = kl v Ù ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä  0  1  2  3  4  5  0  1  2  3  4  5  6 Approximation at r=1 Error Function Fourier Approximation T =g.1,/cb v ý ? l b™bdc<c*-cC289niI2-44* k =328;bji T *-,/c<=hbdcC2 f/f c*Ž‘G=gu28;<=h*-9W x y Error and Metric Contours Error Metric -1.5 -1 -0.5  0  0.5  1  1.5 -1.5 -1 -0.5  0  0.5  1  1.5 T =g.1,/cb’ímý B c<c*-c k *-9:;*-,/c4+289ni d(x, y)3 k *-9:;*-,/c4Ž =h4 f *14<=g;<=gqrb˜4bdu•= Œ i/b‹Łn9/=g;b™4, kl 2-4 (f(θ)) 2 p ” =g0g0›9n*-;9nb k bj44 28c<=g0 ' l 2jqrb2 f *14<=g;<=gqrb˜i/b‹Łn9/=g;b T *-,/c<=hbdc ;<c289n4He“*-c<uw ? *N2-i/imcbj44™; l =h4 y ” b^2-i/i‡; l b•cbj4;<c<= k ;<=h*-9‚; l 28; 1 2a0 > √ (a2) 2 + (b2) 2 &(' =g9n4;bŽ2-i l 2Žq(=g9/. a0 = max ( a0, 2 √ (a2) 2 + (b2) 2 )  ? l =h4•bd9n4<,/cbj4•289niê; l b^uˆbd;<c<= k M ” =g0g0280 ” 2 ' 4 & b f *14<=g;<=gqrb˜i/b‹Łn9/=g;b1 > 46289Ib‹‘n28u f 0hb y 2-44,/uˆb˜; l 28;; l b™q128c<=32 & 0hbj4 u 289ni v l 2jqrb™; l b˜e‰*-0g0h* ” =g9/.; l =gciIi/bdc<=gq128;<=gqrbj4L289ni ; l 28;; l b˜4*-0g,/;<=h*-9I=h4í8ciw*-ci/bdc o p = 3) ý v1v ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä ∂3 ∂x3 ∂3 ∂x2y ∂3 ∂xy3 ∂3 ∂y3 u Œ á Œ v1x Œ ÙŽÚ v v Ú Ù x v Œ Ù x ? l bd9 y/&(' B –:,28;<=h*-9NÚ y ; l b˜bdc<c*-cL28;L2im=h4<;289 k b (x, y) 2 ” 2 ' =h4 B c<c*-c (x, y) =   ∣∣∣−23x3 − 10x2y − 8xy2 + y33 ∣∣∣+ ∣∣x3 + 5x2y + xy2 − 53y3∣∣ 2   o Ùdá(p ? l b T *-,/c<=hbdc k *Gb k =hbd9:;4\e£c*-u B –:,28;<=h*-9IÙ v k 289ˆ; l bd9 & b k 280 k ,/0328;bji &(' 9(,/uˆbdc<= k 280g0 ' =g9:;bd.1c28;<=g9/. B –(,28;<=h*-9UÙdá28c*-,/9ni]2,/9/=g; k =gc k 0hb1 ? l =h4.1=gqrbj4 (f(θ)) 2 3 ≈ (3.224) + (0.07636) cos 2θ + (0.8473) sin 2θ >ûk *-u f 28c<=h4*-9Z*8eL; l b T *-,/c<=hbdc2 f/f c*j‘m=gu28;<=h*-9 289ni ; l bw2 k ;<,2806bdc<c*-c•uˆbŽ2-4,/cbw28c*-,/9niZ2],/9/=g; k =gc k 0hb=g9 T =g.1,/cb v i/bduˆ*-9n4<;<c28;bj4•; l b^q-280g=him=g; ' *8eL; l =h4ˆ2 f/f c*j‘m=gu28;<=h*-9W K ' B –:,28;<=h*-9ßَí y ; l b u28;<c<=³‘¬; l 28; ” =g0g0 & b+,n4bji¬=g9¬; l b˜uˆbd;<c<= k =h4 M =   1.688 0.8473 0.8473 1.536   D , & 4<;<=g;<,/;<=g9/.=g9:;*ˆ*-,/c6uˆbd;<c<= k e£c*-u B –:,28;<=h*-9 z d(x, y) = ( 1.688x2 + 1.6946xy + 1.536y2 ) 1 2 Y ,/c l ' f *-; l bj4<=h4 ” 2-4X; l 28; d(x, y)3 ” *-,/0hi & b2C.r*(*Gi•2 f/f c*j‘m=gu28;<=h*-9;* B c<c*-c (x, y)  s *-u f 28c<=g9/. ; l b™4 28uˆb k *-9:;*-,/c6q1280g,nbj4=g9 T =g.1,/cb’í"=g0g0g,n4;<c28;bj4; l 28;6*-,/c l ' f *-; l bj4<=h4 ” 2-4=g9ni/bjbji k *-c<cb k ;Ž\†‡b l 2Žqrb˜; l bdcb‹e‰*-cb kl *14bd9N2"uˆbd;<c<= k ; l 28;L2 f/f c*j‘m=gu28;bj4C*-,/c6bdc<c*-cuˆbŽ2-4<,/cb1 Y 9nb^2-4 f b k ;"*8e.rbd9nbdc28;<=g9/.t; l bbdc<c*-c"uˆbŽ2-4<,/cb; l 28; l 2-4<9 ff ; & bjbd9 2-i/imcbj44bji‡=h4 k *-u f ,/;<=g9/.t; l b *-ci/bdc p i/bdc<=gq-28;<=gqrbj4Ž ? l =h4™=h4˜9n*-;"2;<c<=gq(=3280\;2-4 ì y bj4 f b k =3280g0 ' e“*-c™4 kl bduˆbj4*8e*-ci/bdc*8e2 kjk ,/c2 k ' l =g. l bdc; l 289ê; ” */ A * ” bdqrbdc y 2I4*-0gqrbdc"4<, kl 2-4 >CP+DGS = & = v õ†? ” l = kl k 289ê.rbd9nbdc28;be‰*-,/c<; lmŒ *-ci/bdc v í ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä 4*-0g,/;<=h*-9n4Eu’,n4; 280gcbŽ2-i ' k *-9(;28=g9; l b+280g.r*-c<=g; l uˆ4`9nb k bj4428c ' e“*-cXcb k *-9n4<;<c<, k ;<=g9/.; l bL; l =gcii/bdc<=gq12 Œ ;<=gqrbj4Ž Ât† l =g0hb™cb k *-9n4<;<c<, k ;<=g9/.; l bi/bdc<=gq-28;<=gqrbj4+289niw9(,/uˆbdc<= k 280g0 ' =g9:;bd.1c28;<=g9/.; l bbdc<c*-ce‰,/9 k ;<=h*-9 e“*-c•bdqrbdc ' k bd0g0 cbj–:,/=gcbj4 k *-u f ,/;28;<=h*-9Z;<=guˆb yX” b l 2Žqrb^e‰*-,/9niê; l 28;•=g9 k *-u f 28c<=h4*-9 ” =g; l *Žqrbdc280g0 4*-0g,/;<=h*-9I;<=guˆb y ; l b k *14<;=h4u•=g9n*-cŽ ? l b f c* k bj44*8e7i/bd;bdc<u•=g9/=g9/. T *-,/c<=hbdc k *Gb k =hbd9:;4C280h4* l 2-4 ; l b & *-9:,n4*8e f c*ŽqG=him=g9/.289¬=g9:;<,/=g;<=gqrb k *-9(;<c*-0E*Žqrbdc’; l b2 f/f c*j‘m=gu28;bjiUbdc<c*-cŽ S =gu•=g;4’*-9 a0 k *-9:;<c*-0\; l bˆ2Žqrbdc28.rbbdc<c*-c˜e“*-c™bŽ2 kl‚k bd0g0 y 289ni¬0g=gu•=g;4L*-9 √ (a2) 2 + (b2) 2 k *-9:;<c*-0 l * ” u, kl ; l 28;6bdc<c*-c=h4L280g0h* ” bji];*q-28c '” =g; l =g9w; l b k bd0g0 o 289ni]; l ,n4™289/=h4*-;<c* f ' p‹ F ; ” *-,/0hiš280h4* & b0h*-.1= k 280;*^4<uˆ*G*-; l ; l buˆbd;<c<= k o =³eEi/bj4<=gcbjip & 2-4bjit*-9 ; l bj4b k *Gb k =hbd9:;4² A * ” bdqrbdc y” b™i/*ˆ9n*-;6b‹‘/28u•=g9nb™; l =h4=g9w; l bCe“*-0g0h* ” =g9/.;bj4; k 2-4bj4Ž {@fl$-).H	 †‡b f cbj4bd9:; 4bdqrbdc280;bj4<; k 2-4bj4E;*’i/bduˆ*-9n4;<c28;b6; l bCb‹¶b k ;`*8eW,n4<=g9/.*-,/cEuˆbd;<c<= k =g9289289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 k ' k 0hb1$†‡b k *-u f 28cb™; l bj4b+cbj4<,/0g;4e‰*-c & *-; l 4b k *-9niI289niw; l =gciw*-ci/bdcL4*-0g,/;<=h*-9n4 ” =g; l ; l *14b\,n4<=g9/.6,/9/=³e“*-c<u@cb‹Łn9nbduˆbd9:;Ž > 0g0:4*-0g,/;<=h*-9n4$28cb k *-u f ,/;bji,n4<=g9/.6; l bEqrbdc<;b‹‘ Œ­k bd9:;bdcbji T =g9/=g;b Œ ôX*-0g,/uˆb`4*-0gqrbdc7i/bj4 k c<= & bji=g9‰= v õ†?­ ? l =h474*-0gqrbdc,n4bj4 L2 cb k *-9n4;<c<, k ;<=h*-9‰= á y-v ó†? y R *Gb ff 4 4 kl bduˆb‘= í1ÚB? y 289ni P b ” ;*-9 ŒQ_+MNR6BED e‰*-c’c2 f =hi k *-9:qrbdc<.rbd9 k bw= v1xmyVv1v ?­ T *-c’q(=h4 k *-,n4™è* ” 4 y ; l bˆqG=h4 k *-,n4’;bdc<uˆ4 28cbim=h4 k c<=g;<=g¦Žbji2-4Xi/bj4 k c<= & bji•=g9š= v ó†?­ ? * k cbŽ28;bL289/=h4*-;<c* f = k uˆbj4 l bj4$e‰c*-u ; l b k 280 k ,/0328;bjiuˆbd;<c<= k y ” b˜,n4b+; l b’289/=h4*-;<c* f = k uˆbj4 l =g9/.ˆuˆ*Gim=³Ł k 28;<=h*-9n46;* _+R JCM]M 5Ã= v ÚB? f c* f *14bji¬=g9Ä= í x ?­ T *-c280g06; l cbjb k 2-4bj4 yX” bw4<;28c<; ” =g; l ; l b¬=g9/=g;<=3280uˆbj4 l 4 l * ” 9 =g9 T =g.1,/cb¬án ? l =h4•uˆbj4 l =h4•; l b )+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9 f c*(im, k bji &:' _+R JLMNM 5 ” =g; l *-,/;289 ' cb‹Łn9nbduˆbd9(; f 28c28uˆbd;bdc4Ž ? l b eé28c Œ Łbd0hi & *-,/9nin28c ' =h42 k =gc k 0hb+*8eVc2-im=g,n4˜Ù x1x kl *-ci/4 k bd9:;bdcbjiw28; ; l bL0hbŽ2-im=g9/.bjim.rb+*8eV; l b+28=gcHe“*-=g0é †‡b l 2jqrb kl *14bd9¬;*’,n4bL; l =h4`uˆbj4 l & b k 28,n4bC=g;E=h4`cbj–(,/=gcbj4 ; l bC0hbŽ2-4<; ,n4bdcE=g9 f ,/;289ni^=h4 * & qG=h*-,n4<0 ' =g0g0 Œ 4<,/=g;bji‚;*t280g0\; l cbjb;bj4<; k 2-4bj4Ž T ,/c<; l bdc<uˆ*-cb y\&:' kl *G*14<=g9/.š2w0328c<.rb & *-,/9nin28c 'ry7” b l * f b;* 9nbd.:28;b™4*-uˆb™*8e; l b˜bdc<c*-c46; l 28; k 289 & b+.rbd9nbdc28;bjie‰c*-u & *-,/9nin28c ' f 032 k bduˆbd9(;Ž ÅgDBFHG’if[oQÀIZMNKNOŒFZX,j†Fe‹©QEjBMNOMPOŒi†FZXB[’TFHG|ºX†MNQE[nFZKN^Bcd[’OVFKNn[ˆGEOgh†OV[o[JkBIZk@[oGEOmaLÆ|I3GVQEj•IHXidD†GE[gi†[oGEMNlÀÇfOVFZX:p ÈtIHO‹Œ[gKNK IZOJÉ¿KNKNMPnMN[oGVÊbË¿F_F_lÀjŁIZXi,̌IHX„ÍSKNQE[oXIfp sHÎZtvu v á ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä ¼vIÁ DIHGVºB[gKPi‘nMN[o‹šFZTUMNXBM`QEMPIZK:cd[gOVj ¼§m@Á όKNFOV[o^Bk„nMN[o‹šFZTµMNX_QEMPIZK½cd[gOVj T =g.1,/cb˜áný PC>6s`> x1x Ù v ý F 9/=g;<=3280Vuˆbj4 l ” =g; l َórõôXbdc<;<= k bj4Ž v î ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä  1e-06  1e-05  1e-04  0.001  0.01  0.1  100  1000  10000  100000 D ra g Co ef fic ie nt Number of Vertices Third Order Drag Anisotropic Adaptation Uniform Refinement  1e-04  0.001  0.01  0.1  100  1000  10000  100000 Li ft Co ef fic ie nt Number of Vertices Third Order Lift Anisotropic Adaptation Uniform Refinement  1e-05  1e-04  0.001  0.01  0.1  100  1000  10000  100000 D ra g Co ef fic ie nt Number of Vertices Second Order Drag Anisotropic Adaptation Uniform Refinement  1e-04  0.001  0.01  0.1  100  1000  10000  100000 Li ft Co ef fic ie nt Number of Vertices Second Order Lift Anisotropic Adaptation Uniform Refinement T =g.1,/cbîmý PL>6s`>6Œ x1x Ù v 4<, & 4*-9/= k =g9:qG=h4 k =hiè* ” ý k *-u f 28c<=h4*-9n46*8e$imc28.289ni0g=³e‰; k *Gb k =hbd9:;4 ÐflÑ]Ò±ÓBÔSÕ±ÖÀ×"ÖÀÕJØ]ÖÀÓ×UÖÀÙÛÚ±ÔÜÞÝ|߁ÔJÑàÕàÙݒÕݒÖÀ߁ágÔSÖÀâHã ? l bLŁnc4<; ;bj4<; k 2-4bC=h4`=g9:qG=h4 k =hiè* ” *²qrbdc; l b PC>6s`>6Œ x1x Ù v 28=gcHe‰*-=g0é ? l bj*-cbd;<= k 280g0 'ry/& *-; l ; l b+imc28. 289ni0g=³e£; k *(b k =hbd9(;4`4 l *-,/0hi2 f/f c*r2 kl ¦Žbdc*/ A * ” bdqrbdc y im,nb6;*™; l bC2-4 ' u•uˆbd;<c ' *8eW; l b6uˆbj4 l y ; l =h4 =h4Vc28cbd0 ' ; l b k 2-4b1 T =g.1,/cb î k *-u f 28cbj47imc28.C289ni’0g=³e£; k *(b k =hbd9(;4Ve“*-c & *-; l 4b k *-9ni"289ni™; l =gci*-ci/bdc 4*-0g,/;<=h*-9n4² ? l b6,/9/=³e“*-c<u uˆbj4 l bj4\,n4bjiˆe“*-c k *-u f 28c<=h4*-9¬28cb k cbŽ28;bji &:' =g9 k cbduˆbd9(;280g0 ' =g9 k cbŽ2-4=g9/. ; l b•cb‹Łn9nbduˆbd9(; f 28c28uˆbd;bdc=g9 _+R JLMNM 5  F 9U280g0$e“*-,/c k 2-4bj4 y ; l b4*-0g,/;<=h*-9U*-9š; l bŁn9nbj4<;’uˆbj4 l ” 2-46uˆ*-cb2 kjk ,/c28;b ” =g; l 289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9:;˜; l 289 ” =g; l *-,/;Ž T *-c+imc28. f cbjim= k ;<=h*-9n4 y 289ni]=g9 f 28c<;<= k ,/0328c$e“*-c$; l =gciˆ*-ci/bdc\4*-0g,/;<=h*-9n4 y *-,/cXcb‹Łn9nbduˆbd9(;`280g.r*-c<=g; l u f c*Gim, k bji2C9n*-;<= k bŽ2 & 0hb=g9 k cbŽ2-4b =g9 k *-9(qrbdc<.rbd9 k b*-ci/bdc; l c*-,/. l *-,/;’; l b•cb‹Łn9nbduˆbd9(; f c* k bj44Ž s *-9:qrbdc<.rbd9 k b^*8e`0g=³e‰;™;*¬¦Žbdc* ” =g; l cb‹Łn9nbduˆbd9(;’=h4 f *(*-c+=g9U280g0 k 2-4bj4 y›& b k 28,n4b; l =h4™–:,289(;<=g; ' i/b f bd9ni/4 l bŽ2jqG=g0 ' *-9šuˆbj4 l 4 ' u•uˆbd;<c 'ry ” l = kl i/*Gbj4<9 ff ;=gu f c*Žqrb’9nbŽ28c<0 ' 2-4c2 f =him0 ' 2-4 k bd0g0W4<=g¦Žb™i/b k cbŽ2-4bj4Ž F ;=h4ˆ* & qG=h*-,n4•=g9 T =g.1,/cbwÚN; l 28;•; l b,/9/=³e‰*-c<u•0 ' cb‹Łn9nbji uˆbj4 l eé28=g0h4;*š4<,” k =hbd9(;<0 ' cbj4*-0gqrb¬; l b ;<c28=g0g=g9/.bjim.rb’*8e; l b’28=gcHe“*-=g0é$† l =g0hb+; l b™4b k *-9niI289ni; l =gciI*-ci/bdcuˆbj4 l bj4L28cb+qrbdc ' 4<=gu•=g0328c y k 0h*14b =g9n4 f b k ;<=h*-9IcbdqrbŽ280h4+4<0g=g. l ;Cim=³¶›bdcbd9 k bj4+28;6; l b0hbŽ2-im=g9/.289niI;<c28=g0g=g9/.^bjim.rbj4C*8eX; l b28=gcHe“*-=g0é T *-cC; l b v Ú ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä ¼vIÁ¿äUGE[gOVOV^†GE[ifMPORQVGEMNm†^†QEMNFXLTWGEFc#QEjBM`GÀiàFZGÀi†[ˆGOVFZKN^†QEMNFX¹¼§å…Ì[oGVQEMNlg[oOÀÁ IHXi‘^†XBM`TFHGEcdK`a,GE[ˆºXB[gi,cd[gOVj™¼§å2Ì[oGVQEMNlg[gOÀÁˆu ¼§mÁ±D”MNXIHK’æ‘[gOVj†[gOgç/è|j†MNGÀiéÉJGÀif[oG,¼QEFk½êå´n[oGVQEMNlg[gOÀÁ]IZXi ƒ [glgFZXi ÉJGÀif[oG༧m@FZQVQEFZc„ê†å>Ì[ˆGVQEMPlo[gOÀÁˆu ¼§l3Á±ë:[eIZi†MNXBYyìi†YZ[y¼QEFk‰jBIZK`Tbç QEj†MNGÀiíFZGÀif[oGeêÛm@FZQVQEFcîjBIZK`Tbç OV[gloFXi•FZGÀi†[ˆGˆÁ ¼viÁàè:GÀIZMNKNMNXBYyìi†YZ[)¼QEFZk…jBIZK`Tbç QEjBM`GÀiïFHGÀi†[oGeêÃm@FHQVQEFcîjBIZK`Tbç OV[glgFZXi‘FHGÀi†[oGˆÁ T =g.1,/cb’Úmý F 9(q(=h4 k =hi D , & 4*-9/= k è* ” 28c*-,/9ni PL>6s`> x1x Ù v ý Mach = 0.5 y α = 0  v õ ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä  0.34  0.36  0.38  0.4  0.42  0.44  100  1000  10000  100000 Li ft Co ef fic ie nt Number of Vertices 2nd Order Adapted 2nd Order Uniform 3rd Order Adapted 3rd Order Uniform Venditti and Darmofal (0.344) T =g.1,/cbõGý S =³e£; s *-9(qrbdc<.rbd9 k b™e“*-c PL>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o  4b k *-9ni¬*-ci/bdc4*-0g,/;<=h*-9 y l =g. l 4b k *-9ni¬i/bdc<=gq-28;<=gqrbj4*8e f cbj44<,/cb f 28c280g0hbd0W;*"; l b & *-,/9nin28c ' f c*Gim, k b 289/=h4*-;<c* f = k^k bd0g0h428;’; l bˆ0hbŽ2-im=g9/.tbjim.rb=g9U; l b4b k *-9ni‚*-ci/bdcuˆbj4 l & ,/;9n*-;’=g9‚; l b; l =gci‚*-ci/bdc uˆbj4 l  > ;6; l b™;<c28=g0g=g9/.bjim.rb"*8eX; l b™; l =gciN*-ci/bdc6uˆbj4 l y k *-9:qrb k ;<=h*-9š*8e\bd9:;<c* f ' f c*Gim, k bjiN28;C; l b ;<c28=g0g=g9/."bjim.rb o 28928c<;<=³Ł k =3280cbj4<,/0g;`*8e; l b f *-=g9(;E4<=g9/.1,/0328c<=g; ' pXu2 ' f c*Gim, k b; l bL4<0g=g. l ; 289/=h4*-;<c* f '  D =g9 k b4b k *-9ni‡*-ci/bdc’4 kl bduˆbj4im=h44<= f 28;bˆbd9:;<c* f ' eé2-4<;bdc˜; l 289U; l =gci‚*-ci/bdc’4 kl bduˆbj4 y ; l bˆ4b k *-9ni *-ci/bdcuˆbj4 l =h4uˆ*-cb˜,/9/=³e“*-c<uû28;; l b˜;<c28=g0g=g9/.bjim.rb1 ð ߁ݒÕàÓBÔSÕ±ÖÀ×òñ†ÕJØ]ÖÀÓ×UÖÀÙÛÚ±ÔÜÞÝ|߁ÔJÑàÕàÙݒÕݒÖEß½áoÔJÖÀâ3ãôóZñ¹õÑàӆöšÜšß½Ô’ö@÷ÄöøàֈÓôÓÑ]Ò±ÓB÷×<öÖÀÔSÕÓ†Ô öø]÷߁÷"Ý|߁÷´ÓBÔSùú÷ø]ÔSâÀ÷Ó1ݒÕàÙâÀÝ|×½ûJüÔáˆüZÚLÔÜþý±ßÔÒ±âÀ÷ùßÓ B 28c<0g=hbdc$=g9; l b i/bdc<=gq128;<=h*-9*8e*-,/c7uˆbd;<c<= k y1” b2-44<,/uˆbji"; l 28;7; l b`,/9ni/bdc<0 ' =g9/.™4*-0g,/;<=h*-9 ” 2-4$4<uˆ*G*-; l  ? l =h4"2-44<,/u f ;<=h*-9U=h4’9nb k bj4428c ' e“*-c™4 l * ” =g9/.N; l b•q1280g=him=g; ' *8e ,n4<=g9/. ? 2 ' 0h*-cb‹‘ f 289n4<=h*-9n4Ž J 9me“*-c Œ ;<,/928;bd0 'ry 9n*-;˜280g04*-0g,/;<=h*-9n4˜*8eX=g9:;bdcbj4<;™28cb2 k ;<,280g0 ' 4<uˆ*G*-; l  F 9 f 28c<;<= k ,/0328c y e‰*-cC;<c289n4*-9/= k 289ni 4<, f bdc4*-9/= k è* ” 4 y im=h4 k *-9(;<=g9:,/=g;<=hbj4 o *-c"4 l * kì 4‹p k 289ê2 f/f bŽ28c’=g9ê; l b4*-0g,/;<=h*-9W > ;’; l bj4b4 l * kì 4 y *-,/cbdc<c*-cbj4<;<=gu28;b™=h4 9n*0h*-9/.rbdcq-280g=hi› ? l =h4i/*Gbj49n*-;uˆbŽ289 y l * ” bdqrbdc y ; l 28;*-,/cuˆbd;<c<= k’k 289/9n*-; & b•,n4bjite“*-c™4*-0g,/;<=h*-9n4=g9‡; l b f cbj4bd9 k b*8e im=h4 k *-9:;<=g9:,/=g;<=hbj4² ? *w=g0g0g,n4;<c28;b•; l =h4 y ” b l 2Žqrbb‹‘n28u Œ =g9nbji M 2 kl 0.8 è* ” *ŽqrbdcC; l b PC>6s`> x1x Ù v 28=gcHe‰*-=g0 ” =g; l 289t289/.10hb*8eE28;<;2 kì *8e 1.25◦  ? *^2Žqr*-=hi *²qrbdc Œ cb‹Łn9/=g9/.]; l b•im=h4 k *-9(;<=g9:,/=g;<=hbj4 y” b l 2jqrb•0g=gu•=g;bjiU*-,/c’bdc<c*-c™uˆbŽ2-4<,/cbe‰c*-u B –(,28;<=h*-9êõ^4<, kl v1z ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä T =g.1,/cb z ýLí8ciš*-ci/bdc M 2 kl 9(,/u & bdc f c*8Łn0hbj4˜*-9U2-in2 f ;bjituˆbj4 l 289niN,/9/=³e‰*-c<u´uˆbj4 l e“*-c PL>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o v ó ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä -1.2 -1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.5  0.55  0.6  0.65  0.7  0.75  0.8 Pr es su re Chord Length Pressure Distribution at Upper Shock ’AGARD.dat’ 3rd Order 2nd Order T =g.1,/cb’ómýX5Xcbj44<,/cb˜im=h4<;<c<= & ,/;<=h*-9N*-9w, f/f bdc4,/cHe“2 k b™*8e PC>6s`> x1x Ù v ý M 2 kl 0.8 y α = 1.25o  ; l 28; B c<c (1, θ) < 1 × 103  ? l =h4+9:,/u & bdc˜=h4 kl *14bd9N;* & *-,/9niN; l b"bdc<c*-c˜uˆbŽ2-4,/cb28;+; l b4 l * kì ” =g; l *-,/;imc28u28;<= k 280g0 ' 2¥¶›b k ;<=g9/.ˆ*-; l bdc628cbŽ2-4*8e ; l b+4*-0g,/;<=h*-9W F 9; l b6e‰,/;<,/cb y ; l =h4 & *-,/9ni^4 l *-,/0hi & b & 2-4bjiw*-9I24<;28;<=h4;<= k 280Vim=h4<;<c<= & ,/;<=h*-9]*8e$bdc<c*-c=g9n4<;bŽ2-iI*8e$24*-uˆb ” l 28;+28c & =g;<c28c ' k *-9n4<;289:;Ž J 4<=g9/.˜; l b2 & *Žqrbuˆ*Gim=³Ł k 28;<=h*-9n4 y ; l b cbj4<,/0g;47e“*-c & *-; l 4b k *-9niˆ289ni•; l =gciˆ*-ci/bdcE289/=h4*-;<c* f = k 2-in2 fmŒ ;28;<=h*-9I28cb+imc28u28;<= k 280g0 ' =gu f c*Žqrbjiw*²qrbdc,/9/=³e‰*-c<u cb‹Łn9nbduˆbd9(;Ž > k *-u f 28c<=h4*-9w*8eV; l =gci¬*-ci/bdcqrb Œ 0h* k =g; ' f c*8Łn0hbj4=h464 l * ” 9w=g9 T =g.1,/cb z F 9eé2 k ; y/” =g; l *-,/;L2-in2 f ;bjiIcb‹Łn9nbduˆbd9:; y ; l b+0g=³e£; k *Gb k =hbd9:;4 =g9 T =g.1,/cbˆó^i/*9n*-;’2 f/f bŽ28c˜;* k *-9(qrbdc<.rb•;* ” 28ci/4™; l b•4*-0g,/;<=h*-9‚*8eEôXbd9nim=g;<;<=E289nit)˜28c<uˆ*8e“280E= ánÙ?­ ? *w4 l * ” ; l b2 kjk ,/c2 k ' *8e *-,/c4b k *-9ni‡*-ci/bdc"289ni‡; l =gci‡*-ci/bdc™cbj4<,/0g;4 yV” b l 2jqrb k *-u f 28cbji‚; l b f cbj44,/cbim=h4<;<c<= & ,/;<=h*-9š*-9t; l b, f/f bdc+4<,/cHeé2 k b*8e\; l b28=gcHe“*-=g0 ” =g; l in28;2e£c*-u =gÙ?­ P *-;+*-9/0 ' i/*Gbj4 ; l b•4 l * kì 2 f bŽ28c˜=g9tcbd0328;<=gqrbd0 ' ; l b•428uˆb"0h* k 28;<=h*-9 yW& ,/;+; l b•4 l * kì 4+; l 28; ” b k *-u f ,/;bjiU28cb•b‹‘ Œ ;<cbduˆbdu•0 ' 4 l 28c f 289ni l 2jqrb˜qrbdc ' 0g=g;<;<0hb™*Žqrbdc4 l *(*-;Ž ÐflÑ]Ò±ÓBÔSÕ±ÖÀ×"âEݒùßÖÀÕ]Ý|ߚڱÔÜ Ý|߁ÔJÑàÕàÙݒÕݒÖEß½áoÔJÖÀâ F 9:qG=h4 k =hiè* ” 4 i/*™9n*-;`.rbd9nbdc280g0 ' l 2Žqrb l =g. l 0hbdqrbd0h4 *8e 289/=h4*-;<c* f 'ry *-; l bdc`; l 289 f *144<= & 0 ' 28;`4 l * kì 4 y 4*N;* & bd;<;bdcˆi/bduˆ*-9n4<;<c28;b; l bw2-imq-289(;28.rbj4ˆ*8e+289/=h4*-;<c* f = k uˆbj4 l =g9/. ” b ” =g0g0b‹‘n28u•=g9nbq(=h4 k *-,n4 4<, & 4*-9/= k è* ” 28c*-,/9ni^; l b PL>6s`> x1x Ù v 28=gcHe“*-=g0 o R b = 5000 y M = 0.5 y α = 0◦ p‹ Y ,/cEcbj4<,/0g;4Ee“*-c í x ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä M bd; l *Gi T =g9280 M bj4 l]D =g¦Žb CD,P CD,ν CL xsep xchord í8ci Y ci/bdc > 9/=h4*-;<c* f = k َî8á:Ú8á x  x1v1v î x  x í v í x  x1x1x1x á x  z Ù1Ù í8ci Y ci/bdc F 4*-;<c* f = k َî1Ú z1v x  x1v1v õ x  x í8á v x  x1x ánÙdá x  ómÙ x í8ci Y ci/bdc D *-0g,/;<=h*-9 v 9ni Y ci/bdc M bd;<c<= k v1x Ú1í1î x  x1v1v á x  x í v í x  x1x1x á z õ x  z1x î v 9ni Y ci/bdc > 9/=h4*-;<c* f = k v1x Ù z õ x  x1v1v Ú x  x í v í x  x1x1x Ù1Ùdá x  õ-ó1î v 9ni Y ci/bdc F 4*-;<c* f = k v1x Ú8á v x  x1v1v á x  x í1í v x  x1x ó1î x  z í8á >CRs v ) í v1x ‘šÙ v1z k bd0g0h4 x  x1v1v Ù x  x í v Ù  x  z1v á M 2jqGc<= f 0g=h4Ž=gَó†? í v1x ‘wÚ8á k bd0g0h4 x  x1v1v ó x  x í1í v  x  z Ùdá R 2-i/bj4 f =hbd0]= í1íB? îmÙ v ‘›Ù v1z k bd0g0h4 x  x1v1v á x  x í1í x  x  z Ùdá ? 2 & 0hb™Ù1ý s *-u f 28c<=h4*-9*8eimc28. y 0g=³e£; y 289ni•4b f 28c28;<=h*-9 f *-=g9(;7e‰*-c PC>6s`> x1x Ù v 28=gcHe‰*-=g0éý Mach = 0.5 y Re = 5000 y α = 0   0  0.01  0.02  0.03  0.04  0.05  0  2000  4000  6000  8000  10000  12000  14000  16000 Co ef fic ie nt # of Vertices 3rd Order Results Viscous Drag (0.0322) Pressure Drag (0.0225) Lift (0) Anisotropic Isotropic  0  0.01  0.02  0.03  0.04  0.05  0  5000  10000  15000  20000  25000 Co ef fic ie nt # of Vertices 2nd Order Results Viscous Drag (0.0322) Pressure Drag (0.0225) Lift (0) Anisotropic Isotropic T =g.1,/cbfiÙ x ý > 9/=h4*-;<c* f = k qG4² F 4*-;<c* f = k imc28.Z289nia0g=³e‰; k *-9:qrbdc<.rbd9 k bše“*-c PC>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  ímÙ ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä  75  80  85  90  95  100  0  5000  10000  15000  20000  25000 Se pa ra tio n Po in t (%  of  C ho rd) # of Vertices 2nd Order Separation Point Anisotropic Isotropic  75  80  85  90  95  100  0  2000  4000  6000  8000  10000 12000 14000 16000 Se pa ra tio n Po in t (%  of  C ho rd) # of Vertices 3rd Order Separation Point Anisotropic Isotropic T =g.1,/cbwÙ1Ù1ý > 9/=h4*-;<c* f = k qm4Ž F 4*-;<c* f = k 4b f 28c28;<=h*-9 f *-=g9:; k *-9(qrbdc<.rbd9 k bˆe“*-c PL>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  T =g.1,/cb™Ù v ý s *-u f 28c<=h4*-9*8eW289/=h4*-;<c* f = k 289niˆ=h4*-;<c* f = k ; l =gciˆ*-ci/bdc M 2 klf c*8Łn0hb e‰*-c PC>6s`> x1x Ù v 28=gcHe“*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  í v ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä  0.022  0.0225  0.023  0.0235  0.024  0.0245  0.025  0.0255  0.026  0  5000  10000  15000  20000  25000 Pr es su re  D ra g Co ef fic ie nt # of Vertices Pressure Drag Convergence 3rd Order 2nd Order Mixed Order  0.032  0.0325  0.033  0.0335  0.034  0.0345  0.035  0  5000  10000  15000  20000  25000 Vi sc ou s D ra g Co ef fic ie nt # of Vertices Viscous Drag Convergence 3rd Order 2nd Order Mixed Order  1e-06  1e-05  1e-04  0.001  0.01  0.1  1  10  100  1000  10000  100000 Li ft Co ef fic ie nt # of Vertices Lift (Convergence to 0) 3rd Order 2nd Order Mixed Order T =g.1,/cb َímý s *-u f 28c<=g9/. 0g=³e‰;¬289ni imc28. k *-9(qrbdc<.rbd9 k bte‰*-c PC>6s`> x1x Ù v 28=gcHe‰*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  í1í ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä T =g.1,/cbÙdáný ? c<=3289/.10hbˆ2-4 f b k ;+c28;<=h*14Le£c*-u í8ciN*-ci/bdc™289ni v 9nit*-ci/bdc+uˆbd;<c<= k 4™28;C0hbŽ2-im=g9/.bjim.rb*8e PC>6s`> x1x Ù v 28=gcHe‰*-=g0éý Mach = 0.5 y Re = 5000 y α = 0  4b k *-9ni‡289ni‡; l =gci‡*-ci/bdc’4*-0g,/;<=h*-9n4 ” =g; l q-28c<=h*-,n4˜uˆbd;<c<= kˆf 28c28uˆbd;bdc428cb f cbj4bd9(;bjiš=g9 ? 2 & 0hbwÙ 280h*-9/. ” =g; l *-; l bdc f , & 0g=h4 l bji cbj4<,/0g;4e‰*-c"; l b4 28uˆb^4bd;"*8e f 28c28uˆbd;bdc4Ž K *-; l ; l b4b k *-9ni 289ni ; l =gci*-ci/bdc`4*-0g,/;<=h*-9n4 28cb k 0hbŽ28c<0 ' k *-9(qrbdc<.1=g9/.;*4<=gu•=g0328c\q1280g,nbj4 y 289niˆ; l *14b6q-280g,nbj4\0g=hb ” =g; l =g9^; l b c289/.rb™*8e f , & 0g=h4 l bji¬cbj4,/0g;4Ž ? *w4 l * ” ; l b & bd9nb‹Łn;4’*8e6289/=h4*-;<c* f = k uˆbj4 l =g9/.I*Žqrbdc=h4*-;<c* f = k uˆbj4 l =g9/. y ” b4bd;™; l b a2 289ni b2 T *-,/c<=hbdc k *Gb k =hbd9:;4e‰c*-u B –(,28;<=h*-9aÙ v ;*]¦Žbdc*/ ? l =h4280g0h* ” 4; l buˆbd;<c<= k ;*]q128c ' =g9‚4<=g¦Žb & ,/; 9n*-;•4 l 2 f b1 T =g.1,/cbNÙ x 4 l * ” 4; l 28; &:' i/*-=g9/.š; l =h4 y ; l b k *-9(qrbdc<.rbd9 k b¬=h4•imc28u28;<= k 280g0 ' cbjim, k bji =g9 & *-; l 4b k *-9ni 289ni ; l =gci *-ci/bdcˆ4*-0g,/;<=h*-9n4"e“*-cˆ280g0 ; l cbjb^e£,/9 k ;<=h*-9280h4Ž P * ” l bdcb=h4"; l =h4uˆ*-cb 2 f/f 28cbd9(;I; l 289ß=g9ß; l b k *-9(qrbdc<.rbd9 k b‚*8e•4b f 28c28;<=h*-9 f *-=g9(;w=g9 T =g.1,/cbZÙ1Ù ” l bdcbš; l bš=h4*-;<c* f = k 4*-0g,/;<=h*-9n4"4b f 28c28;bu’, kl 0328;bdc; l 289êb‹‘ f b k ;bji› ? l buˆ*14;’9n*-;<= k bŽ2 & 0hb^im=³¶bdcbd9 k b=g9ê4*-0g,/;<=h*-9n4 & bd; ” bjbd9w; l b˜289/=h4*-;<c* f = k 289ni=h4*-;<c* f = k uˆbj4 l bj4 =h4 ; l bCu, kl uˆ*-cb˜im=³¶,n4bji ” 2 ì b™2-4 =g0g0g,n4;<c28;bji =g9 T =g.1,/cbˆÙ v  )+*-u f =hbdc<cb˜bd;6280鍧= v ?W4<,/.1.rbj4<; ; l 28; A bj44<=3289 & 2-4bji^cb‹Łn9nbduˆbd9:;64 l *-,/0hi & b+4<,/=g;2 & 0hbLe“*-c l =g. lmŒ *-ci/bdc 4*-0g,/;<=h*-9n4˜bdqrbd9š; l *-,/. l ; l b•4b k *-9niši/bdc<=gq-28;<=gqrbj4’28cb•280gcbŽ2-i ' cbj4*-0gqrbji› ? *;bj4<;+; l =h4+; l bj*-c 'ry›” b ,n4bjiU*-,/c v 9niU*-ci/bdcuˆbd;<c<= k ;*¬cb‹Łn9nbˆe‰*-c’2; l =gci‚*-ci/bdc4*-0g,/;<=h*-9W ? l bˆcbj4,/0g;<=g9/. k *-9:qrbdc<.rbd9 k b f 0h*-;4•28cb4 l * ” 9 =g9 T =g.1,/cbtَím F ;"i/*(bj4•2 f/f bŽ28c"; l 28;; l bdcb^u•=g. l ; & b^4*-uˆb & bd9nb‹Łn;;*N,n4<=g9/. í8á ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä ; l b; l =gci‚*-ci/bdcuˆbd;<c<= k e“*-c’; l =gci‚*-ci/bdc"4*-0g,/;<=h*-9n4 y f 28c<;<= k ,/0328c<0 ' =g9UqG=h4 k *-,n4imc28. k *-9:qrbdc<.rbd9 k b1 A * ” bdqrbdc y ; l bdcbX=h4W9n*-;724<=g.19/=³Ł k 289(;bd9n*-,/. l im=³¶bdcbd9 k b & bd; ” bjbd9"; l bX; ” *Lcbj4<,/0g;4W;* & b k *-9 k 0g,n4=gqrb1 ? l b*-9/0 ' –:,280g=g;28;<=gqrb^im=³¶bdcbd9 k b & bd; ” bjbd9 ; l bˆuˆbj4 l .rbd9nbdc28;bji &:' ; l b v 9niê*-ci/bdc’uˆbd;<c<= k 289ni ; l b; l =gciI*-ci/bdcCuˆbd;<c<= k =h4; l 28;C; l b’; l =gciN*-ci/bdcLuˆbd;<c<= k =h4Lu’, kl uˆ*-cb=h4*-;<c* f = k 28;6; l b0hbŽ2-im=g9/. bjim.rbC*8eV; l b+28=gcHe‰*-=g0é ? l =h4`=h4`4 l * ” 9=g9 T =g.1,/cb"Ùdán > e£;bdc`; l b60hbŽ2-im=g9/.•bjim.rb yG& *-; l uˆbj4 l bj4 & b k *-uˆb qrbdc ' 289/=h4*-;<c* f = k =g9]; l b & *-,/9nin28c ' 032 ' bdc™289ni l 2jqrb"c*-,/. l 0 ' ; l b428uˆb"4<=g¦Žb"im=h4<;<c<= & ,/;<=h*-9W M *-cb f cb k =h4buˆbj4 l =g9/. ” *-,/0hi & b9nb k bj4428c ' ;*™b‹‘/28u•=g9nb ” l 28;\b‹¶›b k ;X; l bim=³¶›bdcbd9:;\uˆbd;<c<= k 4 l 2jqrbL*-9ˆ; l b 4*-0g,/;<=h*-9N2 kjk ,/c2 k '  * "¿.3-" †‡b l 2jqrb f cbj4bd9(;bji 2w.rbd9nbdc280`e‰c28uˆb ” *-c ì e‰*-c289/=h4*-;<c* f = k 280g0 ' cb‹Łn9/=g9/.t,/9n4<;<c<, k ;<,/cbji uˆbj4 l bj4Ž Y ,/c 2 f/f c*r2 kl =h4 i/bj4<=g.19nbji˜;* & bX4<,” k =hbd9(;<0 ' .rbd9nbdc280:;*6280g0h* ” cb‹Łn9nbduˆbd9(;Ve“*-cW4*-0gqrbdc4 *8en289 ' *-ci/bdc 289niN289 ' 9:,/u & bdc+*8eXi/b f bd9ni/bd9(;Lq128c<=32 & 0hbj4Ž ? l b & 2-4<=h4 & b l =g9niI*-,/c+2 f/f c*r2 kl =h4;*2 f/f c*Ž‘G=gu28;b ; l b0h* k 280Ecb k *-9n4<;<c<, k ;<=h*-9fibdc<c*-c ” =g; l ? 2 ' 0h*-cb‹‘ f 289n4<=h*-9n4289ni‡; l bd9ê,n4b T *-,/c<=hbdc;<c289n4<e‰*-c<uˆ4;* kl *(*14b289N2 f/f c* f c<=328;b™uˆbd;<c<= k e£,/9 k ;<=h*-9W ? l b™uˆbd;<c<= kk 289w; l bd9 y =g9I;<,/c<9 yn& b˜,n4bji¬;*•cb‹Łn9nb’; l b uˆbj4 l 289niw=gu f c*Žqrb™; l b’4*-0g,/;<=h*-9W †‡b l 2jqrb˜4 l * ” 9; l c*-,/. l 4bdqrbdc280;bj4<; k 2-4bj4 ; l 28; ; l =h4 cb‹Łn9nbduˆbd9:; f c* k bjim,/cb+=gu f c*Žqrbj44*-0g,/;<=h*-9 2 kjk ,/c2 k 'N& bd;<;bdc+; l 289š,/9/=³e“*-c<u´uˆbj4 l cb‹Łn9nbduˆbd9:;Ž†‡b l 2jqrbˆ280h4*4 l * ” 9š; l 28;Ce‰*-c & *-; l 4b k *-9ni 289ni; l =gci*-ci/bdc`qG=h4 k *-,n4E4*-0g,/;<=h*-9n4 y 289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9(; f c*(im, k bj4`u, kl & bd;<;bdc k *-9:qrbdc<.rbd9 k b 289ni¬=h4; l bdcb‹e‰*-cb˜uˆ*-cb’b k =hbd9:;6; l 289I=h4*-;<c* f = k cb‹Łn9nbduˆbd9:;Ž ? l b¬im=³¶›bdcbd9 k bj4 & bd; ” bjbd9 *-,/cˆ4b k *-9nifi*-ci/bdc289ni ; l =gcifi*-ci/bdc4*-0g,/;<=h*-9n428cbuˆ*-cb¬im= k ,/0g;ˆ;* =g9(;bdc f cbd;Ž F 9š.rbd9nbdc280 y ; l b•; l =gciš*-ci/bdc’4*-0g,/;<=h*-9n4 ” bdcbˆuˆ*-cb2 kjk ,/c28;b•; l 289‡; l bˆ4b k *-9niš*-ci/bdc 4*-0g,/;<=h*-9n4 y/& ,/;=g;cbdu28=g9n4,/9 k 0hbŽ28c ” l bd; l bdc6; l 28; ” 28c<c289:;4L; l b™2-i/im=g;<=h*-9280 k *-u f ,/;28;<=h*-9280 k *14<;Ž Y ,/c cbj4<,/0g;428cb˜280h4*"=g9 k *-9 k 0g,n4<=gqrb™2-4`;* ” l bd; l bdc*-c 9n*-;`=g; =h42 kjk b f ;2 & 0hb˜;*",n4b+2uˆbd;<c<= k & 2-4bji *-9w4b k *-9niIi/bdc<=gq-28;<=gqrbj4e‰*-c; l =gciw*-ci/bdcL4*-0g,/;<=h*-9n4Ž † l =g0hb+*-,/c`uˆbd;<c<= k =h4 i/bj4<=g.19nbji;* ” *-c ì e“*-c f c* & 0hbduˆ4*8e7289 ' im=guˆbd9n4=h*-9 y/” b l 2jqrb˜*-9/0 ' ;bj4<;bji^=g; í1î ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä e“*-cX; ” *’im=guˆbd9n4<=h*-9280è* ”  ? l =h4\=h4Xim,nb;* k ,/c<cbd9(;\0g=gu•=g;28;<=h*-9n4`*8e; l cbjb Œ im=guˆbd9n4=h*-9280›289/=h4*-;<c* f = k uˆbj4 l =g9/.n B ‘n28u•=g9/=g9/.; l bXb‹¶›b k ;V*8em; l =h4›cb‹Łn9nbduˆbd9:;72 f/f c*r2 kl e“*-c l =g. l bdc Œ *-ci/bdc ; l cbjb Œ im=guˆbd9n4<=h*-9280 4*-0g,/;<=h*-9n46=h46289¬=g;bdu e“*-c e£,/;<,/cb ” *-c ì  > 9n*-; l bdc f *-;bd9:;<=3280E=gu f c*Žqrbduˆbd9:; ” *-,/0hi & b•;* k *-u & =g9nb^*-,/c™uˆbd;<c<= k ” =g; l 2-i ë *-=g9:; Œ & 2-4bji‚bdc<c*-c bj4<;<=gu28;<=h*-9WôXbd9nim=g;<;<=/289ni)˜28c<uˆ*8eé280Œ= ánÙ y á v ? k 0hbŽ28c<0 ' i/bduˆ*-9n4;<c28;bji; l 28;$2-i ë *-=g9:; & 2-4bji Œ uˆbd; l *(i/4 f c*²q(=hi/b+.1cbŽ28;bdcL2 kjk ,/c2 k ' e“*-c`; l b kl *14bd9¬e‰,/9 k ;<=h*-9280›; l 289 A bj44<=3289 & 2-4bjiuˆbd; l *(i/4² F ; =h4 0g= ì bd0 ' ; l 28;¬; l =h4¬cbdu28=g9n4w; l b k 2-4bNe‰*-c l =g. l bdc Œ *-ci/bdc]uˆbd; l *(i/4w2-4 ” bd0g0é s *-u & =g9/=g9/. 2-i ë *-=g9:; & 2-4bji 2-in2 f ;28;<=h*-9 ” =g; l *-,/c$0h* k 280n4 l 2 f bi/b‹Łn9/=g;<=h*-9^4bjbduˆ4\289ˆ=g9(;<,/=g;<=gqrb6b‹‘m;bd9n4<=h*-9 y bj4 f b k =3280g0 ' e‰*-c l =g. l 0 ' 289/=h4*-;<c* f = k 4*-0g,/;<=h*-9n4Ž ? l =h4 ” *-,/0hiI4;<=g0g0W0hbŽ2jqrb˜; l b f c* & 0hbduû*8e kl *G*14<=g9/.ˆ; l b f c* f bdce‰,/9 k ;<=h*-9280 y ” l = kl =h4*8e‰;bd9 & ,/;9n*-;6280 ” 2 ' 4C* & qG=h*-,n4e“*-cbd9/.1=g9nbjbdc<=g9/.^2 f/f 0g= k 28;<=h*-9n4² Y 9nb f c* & 0hbdu ” =g; l *-,/c$uˆbd; l *Gi=h47; l 28;$=hii/*(bj479n*-; f c*²q(=hi/bL2L–(,289:;<=g;28;<=gqrbLbj4<;<=gu28;b*8e4*-0g,/;<=h*-9 bdc<c*-cŽX2-4 ” =g; l = í y ánÙ y á vny َíB?­ F 9n4;bŽ2-i y:” b6cb f bŽ28;bjim0 ' cb‹Łn9nbL*-,/cEuˆbj4 l 4*2-4X;*"bj4<;<=gu28;bLbdc<c*-c &(' k *-9:qrbdc<.rbd9 k b1 J 0g;<=gu28;bd0 'ry” b ” *-,/0hiN0g= ì b’;* l 2Žqrb"2 & bd;<;bdc+bdc<c*-cLbj4<;<=gu28;b28;+bŽ2 kl 4<;28.rb’*8e ; l bcb‹Łn9nbduˆbd9(; f c* k bj44Ž B 44bd9(;<=3280g0 'ry ; l =h4 ” *-,/0hi k *-u & =g9nbˆ; l b; ” *]2 f/f c*r2 kl bj4289ni‡280g0h* ” e“*-c & bd;<;bdc,n4bdc k *-9(;<c*-0é J 0g;<=gu28;bd0 'ry ; l b.r*r280\*8e*-,/c ” *-c ì =h4˜;*]280g0h* ” e“*-c˜eé2-4<; y 2 kjk ,/c28;b4*-0g,/;<=h*-9n4™;* s`T ) f c* & 0hbduˆ4 ” =g; l 2"u•=g9/=gu,/uº28uˆ*-,/9(;L*8e7=g9 f ,/;e‰c*-u ; l b˜,n4bdcŽ$† l =g0hb ” b l 2jqrb™4 l * ” 9I; l 28;uˆ*-cb2 kjk ,/c28;b 4*-0g,/;<=h*-9n4 k 289 & b6.rbd9nbdc28;bji y =g; =h4`9n*-; ' bd; k 0hbŽ28c ” l bd; l bdc*-c 9n*-;`; l =h42 f/f c*r2 kl =h4 =g9ni/bjbjieé2-4<;bdcŽ B qrbd9 ” =g; l *-,/;˜b‹‘n28u•=g9/=g9/.¬,n4bdcL=g9 f ,/; y =g;C=h4Cqrbdc ' im= k ,/0g;˜;* k *-u f 28cb•4*-0g,/;<=h*-9t;<=guˆb"*-9š2,/9/= Œ e“*-c<u•0 ' cb‹Łn9nbjiˆuˆbj4 l ” =g; l 4*-0g,/;<=h*-9;<=guˆb`e‰*-c\2892-in2 f ;bjiˆuˆbj4 l y f c<=gu28c<=g0 '& b k 28,n4b*8e; l bL2-i/i/bji k *-u f 0hb‹‘m=g; ' *8e kl *(*14=g9/. ” l bd9š;*cb‹Łn9nb; l buˆbj4 l 289ni &:' l * ” u’, kl  > 9‡2-i/i/bji k *-u f 0hb‹‘m=g; ' ” =g; l 289/=h4*-;<c* f = k uˆbj4 l =g9/.•=h4 ; l 28;=g;4b‹¶b k ;*-9¬u28;<c<=³‘ k *-9nim=g;<=h*-9/=g9/. y 289ni &(' b‹‘m;bd9n4<=h*-9¬u28;<c<=³‘ 2 kjk bd0hbdc28;<=h*-9]uˆbd; l *(i/4 y =h4,/9 k 0hbŽ28cŽ Y qrbdc280g0 y ; l bdcb^4<;<=g0g0`cbdu28=g9n4=g9n4,” k =hbd9:;•in28;2w;*]i/bd;bdc<u•=g9nb ” l bd; l bdc*-c"9n*-;*-,/c"uˆbd; l *Gi‡e“*-c 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 ” =g0g0;<c<,/0 ' =gu f c*ŽqrbL; l bb k =hbd9 k ' *8e s`T )4*-0gqrbdc4Ž† l 28; ” b l 2jqrbLi/*-9nb =h4L;* f c*²q(=hi/b24<=gu f 0hb"289niNb‹¶›b k ;<=gqrb ” 2 ' *8e k *-u & =g9/=g9/.; ” * f cbdqG=h*-,n4<0 ' ,/9 k *-u & =g9nbjiši/bdqrbd0h* fmŒ uˆbd9(;4Žý l =g. l *-ci/bdc™uˆbd; l *Gi/4 ” =g; l 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9W A * f b‹e£,/0g0 'ry 2-4+e‰,/c<; l bdc™cbj4bŽ28c kl =h4 í1Ú ¢¤£ æäæ Z¥†¦§§¨ æ_9© Z¥†ª æf«$ã e¬ Oã /­,®°¯± 9/æ  ¢ ä  çdã:å  ãZ² R;‘³ æç ® ¢ 9 ¥ ²7å ¥ å  ãŽä i/*-9nb•=g9(;*¬=gu f c*ŽqG=g9/. l =g. lmŒ *-ci/bdc s`T ) 4*-0gqrbdc4’289niU289/=h4*-;<c* f = k uˆbj4 l .rbd9nbdc28;*-c4 y ; l bˆ2-imq1289 Œ ;28.rbj4 *8e k *-u & =g9/=g9/.•; l bL; ” * ” =g0g0280h4*4<=g.19/=³Ł k 289:;<0 ' =g9 k cbŽ2-4b1 F eW4* y ; l bdcbC=h4E9n*i/*-, & ;`; l 28; l =g. l *-ci/bdcL289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 ” =g0g0W4*(*-9 & b k *-u•uˆ*-9 f 032 k b"28uˆ*-9/. k *-u•uˆbdc k =3280 s`T ) k *(i/b1 írõ ffi»ü$  ¹   „}t»ž { ¹ ø } ¸ U€ ¸ $~ø ƒ »}#!  »  ¹ »  » ¹  ¡ L	˜)-"W	 P ,/uˆbdc<= k 280 4*-0g,/;<=h*-9]*8e75X) B 46=h46289w=gu f *-c<;289:;+289280 ' 4<=h46;*(*-0e“*-c64 k =hbd9(;<=h4<;4L289niIbd9/.1=g9nbjbdc4Ž F i/b Œ 280g0 'ry 4<, kl 9(,/uˆbdc<= k 2804*8e‰; ” 28cbš4 l *-,/0hiacbj–(,/=gcbš*-9/0 ' 4 f b k =³e ' =g9/. ; l b f/l ' 4<= k 4 y .rbj*-uˆbd;<c 'ry 289ni & *-,/9nin28c ' k *-9nim=g;<=h*-9n4*8e™; l b f c* & 0hbdu~289ni ; l bd9 f c*Gim, k bN2U4*-0g,/;<=h*-9 *8e2 k bdc<;28=g9a2 kjk ,/c2 k '  J 9me‰*-c<;<,/928;bd0 'ry 2-44bj44=g9/.289niu•=g9/=gu•=g¦j=g9/.^im=h4 k cbd;<=g¦²28;<=h*-9Nbdc<c*-c=h462"9n*-9 Œ ;<c<=gqG=3280V;2-4 ì y cbj–(,/=gc<=g9/. ;<=guˆb k *-9n4<,/u•=g9/.uˆbj4 l i/b f bd9ni/bd9 k b289280 ' 4=h46*-c k *-u f ,/;28;<=h*-9š*8e$4*-0g,/;<=h*-9n4L*-9]uˆbj4 l bj4; l 28;+28cb u, kl Łn9nbdc; l 289‚; l bˆcbj–(,/=gcbji‚4*-0g,/;<=h*-9 2 kjk ,/c2 k ' im= k ;28;bj4Ž M bj4 l 2-in2 f ;28;<=h*-9 4bjb ì 4™;*N28,/;* Œ u28;b•; l =h4 f c* k bj44 &:' Łnc4<; k *-u f ,/;<=g9/.I; l b•4*-0g,/;<=h*-9‚*-9‚2 k *r28c4buˆbj4 l 289nit; l bd9‚4<, kjk bj44<=gqrbd0 ' cb‹Łn9/=g9/.™; l b uˆbj4 l 4*+; l 28;\bŽ2 kl uˆbj4 l =h47uˆ*-cb9nbŽ28c<0 ' * f ;<=gu280/e‰*-c$; l b4*-0g,/;<=h*-9 & bd=g9/. k *-u f ,/;bji› J 9n4<;<c<, k ;<,/cbji‚uˆbj4 l bj4•28cb ” bd0g0 Œ 4<,/=g;bji‚e‰*-c2-in2 f ;28;<=h*-9 & b k 28,n4b; l b ' k 289 & bˆuˆ*Gim=³Łbji‚0h* k 280g0 ' u, kl uˆ*-cbbŽ2-4<=g0 ' ; l 289‡4<;<c<, k ;<,/cbjituˆbj4 l bj4 y” l = klšl 2jqrbˆ2Ł/‘mbjiN;* f *-0h*-. '  F 9t; l =h4 f 2 f bdc y” b f cbj4bd9(;C289]289/=h4*-;<c* f = k b‹‘m;bd9n4<=h*-9I*8e\289Ib‹‘m=h4<;<=g9/.ˆ)+bd0328,/92 ' 2-in2 f ;28;<=h*-9t4 kl bduˆb…= î y á1á?289ni ” b 4 l * ” ; l 28;ˆ=g;=h4 f *144<= & 0hb¬;* f c*Gim, k b l =g. l –:,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l bj4 ” =g; l *-,/; k *14<;<0 ' 0h* k 280 u2¥‘m=gu•=g¦²28;<=h*-9Ic*-,/;<=g9nbj4Ž †‡b l 2jqrb kl *14bd9¬;*",n4b+)+bd0328,/92 ' uˆbd; l *(i/4 & b k 28,n4b+)+bd0328,/92 ' uˆbj4 l cb‹Łn9nbduˆbd9:; k 289¬bŽ2-4<=g0 '& b ,n4bjiê;*N=g9n4bdc<;9nb ” qrbdc<;<= k bj4=g9:;*‡2]q1280g=hiêuˆbj4 l U)+bd0328,/92 ' uˆbd; l *Gi/4; l bdcb‹e‰*-cb¬b‹‘m;bd9niêuˆ*-cb 928;<,/c280g0 ' ;*fi289/=h4*-;<c* f = k uˆbj4 l cb‹Łn9nbduˆbd9:;w; l bd9 [ ,2-i Œ­? cbjb‡*-c > imq-289 k =g9/. T c*-9(;wuˆbd; l *Gi/4Ž D(l b ” kl , kì 289ni S 2 & bd0g0hb…=gَچ? l 2jqrbi/bdqrbd0h* f bjiN2ˆ; l bj*-cbd;<= k 280g0 ' 4*-,/9ni]b‹‘G;bd9n4<=h*-9t*8eX.1,28c289:;bjbji Œ –(,280g=g; ' )+bd0328,/92 ' uˆbj4 l =g9/. &(' k *-9n4<;<c<, k ;<=g9/.ˆ289/=h4*-;<c* f = k ôX*-c*-9n*-=im=328.1c28uˆ4289niˆ; l bd9im,280g=g¦j=g9/. ;*ê2‡)+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9W J 9me‰*-c<;<,/928;bd0 'ry ; l bj4bN289/=h4*-;<c* f = k ôX*-c*-9n*-=˜im=328.1c28uˆ4 k 289 & b í z æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå im= k ,/0g;•;*t.rbd9nbdc28;b y 289ni ; l b¬im,280g=g¦²28;<=h*-9 ;*š2Nq-280g=hiê;<c<=3289/.1,/0328;<=h*-9…cbj–:,/=gcbj4cbj4<;<c<= k ;<=h*-9n4*-9 uˆbd;<c<= k 4uˆ*(*-; l 9nbj44² > 0g;bdc<928;<=gqrbd0 'ry ; l bdcbt28cbN*-; l bdc^uˆ*-cb l bd,/c<=h4<;<= k uˆbd; l *Gi/4e“*-c.rbd9nbdc28;<=g9/.Z289/=h4*-;<c* f = k uˆbj4 l bj4 ; l 28;^cbd0 ' *-9 0h* k 280+* f ;<=gu•=g¦²28;<=h*-9 280g.r*-c<=g; l uˆ4Ž)+*-u f =hbdc<cbNbd;¬280鍧=gَíB?+4<,/.1.rbj4<;4 f 0g=g;<;<=g9/.fibjim.rbj4 ” =g; l 0hbd9/.1; l 2 & *Žqrb2¬; l cbj4 l *-0hi y cbduˆ*ŽqG=g9/.]9n*Gi/bj4 ” =g; l 280g0`bjim.rb0hbd9/.1; l 4 & bd0h* ” 2w; l cbj4 l *-0hi y 289niê4 ” 2 f/f =g9/.‡bjim.rbj4•289niêuˆ*ŽqG=g9/.t9n*(i/bj4";*Nbj–:,/=him=h4;<c<= & ,/;b0hbd9/.1; l  P b ” bjim.rbj4•28cb2-44<=g.19nbji 0hbd9/.1; l 4 & 2-4bjiI*-9]289I=g9:;bdc f *-0328;bji]bdc<c*-c6bj4;<=gu28;b˜e‰c*-u ; l b & 2 kì .1c*-,/9niIuˆbj4 l  K ,n4 k 28.10g=32289ni )˜28c<=E= z ?C;2 ì bI; l =h42U4;b f e‰,/c<; l bdc &(' ;bj4<;<=g9/.fi2š0328c<.rbt4bd;*8e f *-;bd9:;<=3280˜* f bdc28;<=h*-9n4*-9 2‚4<, & Œ uˆbj4 l 289ni; l bd9w4bd0hb k ;<=g9/.ˆ; l b˜*-9nb+; l 28;uˆ*14<;=g9 k cbŽ2-4bj4–(,280g=g; ' 2 kjk *-cim=g9/.ˆ;*ˆ289wbdc<c*-cbj4;<=gu28;b *-9 2 & 2 kì .1c*-,/9nifiuˆbj4 l x`; l =h4ˆ2 f/f c*r2 kl k 289 f c*(im, k b¬b‹‘ k bd0g0hbd9:;•uˆbj4 l bj4 yX& ,/;28;ˆ2 l =g. l k *14<;Ž Y ,/c¬280g.r*-c<=g; l u k *-u & =g9nbj4; l b l bd,/c<=h4<;<= k 928;<,/cb]*8e’; l bj4bI; ” *fi2 f/f c*r2 kl bj4 ” =g; l 4*-uˆbN*8e’; l b 28;<;<c<= & ,/;bj4™*8eE=h4*-;<c* f = k )Cbd0328,/92 ' uˆbj4 l =g9/.n ? l =h4C.1=gqrbj4+,n4L; ” * f c<=g9 k = f 0hbˆ2-imq1289:;28.rbj4™*²qrbdc˜; l b *-; l bdc l bd,/c<=h4<;<= k uˆbd; l *Gi/4Ž T =gc4<; yr” b6i/*™9n*-;\cbd0 ' *-9uˆ*ŽqG=g9/.’qrbdc<;<= k bj4 289ni4*+uˆbd;<c<= k =g9me“*-c<u28;<=h*-9 k 289 & bbŽ2-4<=g0 ' cbd;28=g9nbji ” =g; l *-,/;\4<;*-c<=g9/."289nicb‹e‰bdcbd9 k =g9/.; l b*-c<=g.1=g9280uˆbj4 l  D b k *-9ni yr& b k 28,n4b*8e 289N=gu f c*Žqrbji kl *-= k b"*8e ” l bdcb;*=g9n4bdc<;Cqrbdc<;<= k bj4 y *-,/c˜280g.r*-c<=g; l u f c*(im, k bj4+–(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l bj4 ” =g; l *-,/; k *14<;<0 ' 0h* k 280›u2¥‘m=gu•=g¦²28;<=h*-9Ic*-,/;<=g9nbj4² ? *ˆb‹‘ f 0328=g9w*-,/cC280g.r*-c<=g; l u ” b ” =g0g0WŁnc4<;b‹‘/28u•=g9nb+uˆbd;<c<= k 4 f 2 k bj46289ni l * ” ; l b ' k 289 & b+,n4bji^e“*-c 289/=h4*-;<c* f = k uˆbj4 l =g9/.n ? l bd9 y” b ” =g0g0 k cbŽ28;b2–(,280g=g; ' uˆbŽ2-4<,/cbe“*-cL*-,/c˜289/=h4*-;<c* f = k uˆbj4 l 289ni uˆ*Gim=³e ' 4bdqrbdc280$b‹‘G=h4<;<=g9/.^uˆbj4 l * f bdc28;<=h*-9n4+;*=g9 k cbŽ2-4b"; l 28;˜–(,280g=g; ' uˆbŽ2-4,/cb1 T =g9280g0 'ry” b ” =g0g0 4 l * ” ; l 28;; l =h4 f c* k bj440hbŽ2-i/4;*•uˆbj4 l bj4; l 28;L28cb ” bd0g0V4<,/=g;bjiw;*; l b™i/bj4<=gcbji¬uˆbd;<c<= k  ffffifl	n
fl$ K b‹e“*-cb6i/b‹Łn9/=g9/. ” l 28;2˜–(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l =h4 yr” bu’,n4;XŁnc4<;`i/b‹Łn9nb; l b6i/bj4=gcbji k bd0g04=g¦Žbj4 289niw4 l 2 f bj4Ž ? *i/*ˆ; l =h4 y/” b™,n4b’2"e‰,/9 k ;<=h*-9 k 280g0hbji]2•uˆbd;<c<= k  B 44bd9:;<=3280g0 'ry 2uˆbd;<c<= k =h4L2 ” 2 ' *8e i/b‹Łn9/=g9/.ˆim=h4<;289 k b & bd; ” bjbd9w; ” * f *-=g9:;4Ž ? l b+b‹‘n2 k ;cbj–:,/=gcbduˆbd9(;6*8e72uˆbd;<c<= k e“*-c2"qrb k ;*-c4 f 2 k b  è|j†[SQE[oGEcò[oGVGEFHG’MNO|^BOV[eiqMNXdQEjB[SlgM`QE[eidkIHk@[oGehjBFe‹Œ[gn[oGQEjB[Scd[oQVGEMNlSKN[gX†YZQEj/leIHX,m@[Ji†[ˆºXB[gidIZO|[e\_^†MPnZIZKN[gX_QQEFflQEjB[ [oGVGEFZGJIHXi•OVFq‹Œ[ ‹MNKPK:^BOV[¿QEj†[ QE[oGEcKN[gX†YZQEj‘QEFqm@[¿lgFX†OVMPORQE[oXQ‹M`QEj„^†OEIZY[]KPIHQE[ˆGJMNX‘QEj†[ kIHk@[oGeu í1ó æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå X =h4"; l 28; d : X × X → R =h4•2]uˆbd;<c<= k =³¶Z; l be“*-0g0h* ” =g9/. k *-9nim=g;<=h*-9n4ˆ28cb428;<=h4HŁbji‚e“*-cbdqrbdc ' x, y, z ∈ X  d(x, y) > 0 ⇐⇒ x 6= y d(x, x) = 0 d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z) T *-c6,/9/=³e‰*-c<uû=h4*-;<c* f = k uˆbj4 l =g9/. y ; l b™uˆbd;<c<= k =h46; l b™4<;289nin28ciIim=h4;289 k b˜uˆbŽ2-4<,/cb ‖x− y‖  M *-cb k *-u f 0g= k 28;bji^uˆbd;<c<= k 4280g0h* ” e‰*-c`im=h4<;289 k bL;* & buˆbŽ2-4<,/cbjiim=³¶›bdcbd9:;<0 ' i/b f bd9nim=g9/."*-9*-c<=hbd9:;28;<=h*-9 y ; l ,n4+=g9:;<c*Gim, k =g9/.]289/=h4*-;<c* f '  F 9 f 28c<;<= k ,/0328c y ; l b"uˆbd;<c<= k uˆ*14<; k *-u•uˆ*-9/0 ' ,n4bjiN=g9š289/=h4*-;<c* f = k uˆbj4 l =g9/. k 289 & be“*-,/9ni &:' 0g=g9nbŽ28c<0 ' ;<c289n4He“*-c<u•=g9/.‡; l b¬4<;289nin28ciZim=h4<;289 k buˆbŽ2-4<,/cb d =h4* (x, y) =g9(;*ˆ*-9nb ” l = kl =h46289/=h4*-;<c* f = k d 289/= (x, y) 2-4e‰*-0g0h* ” 4 d =h4* (x, y) = √ V T V d 289/= (x, y) = √ (TV )T (TV ) d 289/= (x, y) = √ V T (T T T )V † l bdcb V = x− y 289ni T =h4L289¬=g9:qrbdc<;<= & 0hb™0g=g9nbŽ28c6;<c289n4He‰*-c<u28;<=h*-9W D =g9 k b’2";<c<=3289/.10hb˜=h4 f bdcHe“b k ;<0 ' i/b‹Łn9nbji &:' ; l bE0hbd9/.1; l 47*8e=g;474<=hi/bj4 y 289 ' ;<c<=3289/.10hb –(,280g=g; ' uˆbŽ2-4<,/cb`,n4bji=g9"=h4*-;<c* f = k uˆbj4 l =g9/. l 2-4 289^bj–:,/=gq1280hbd9:;`;<c<=3289/.10hb ” l bd9uˆbŽ2-4<,/cbji &:' ; l bLuˆbd;<c<= k  T *-cEb‹‘n28u f 0hb y =³eW2™;<c<=3289/.10hb ff 4 bjim.rbj4 l 2Žqrb uˆbd;<c<= k 0hbd9/.1; l 4 (a, b, c) ; l bd9I; l b28cbŽ2 A 289ni k =gc k ,/u•c2-im=g,n4 R *8eE2ˆ;<c<=3289/.10hb k 289 & b k 280 k ,/0328;bji &(' ; l b+e“*-0g0h* ” =g9/.e‰*-c<u,/032-4 A = 1 4 √ (a + b + c)(−a + b + c)(a− b + c)(a + b− c) R = abc 4A o َîrp á x æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå > 0g; l *-,/. l ” b",n4b; l b’0g=g9nbŽ28c+;<c289n4He“*-c<u28;<=h*-9tuˆbd;<c<= k ; l c*-,/. l *-,/;˜; l b•28c<;<= k 0hb y *-,/c™280g.r*-c<=g; l uº=h4 & 2-4bji f ,/cbd0 ' *-9wqrbdc<;b‹‘w4 f 2 k =g9/.289ni k *-,/0hi & b™2-in2 f ;bjiI;*,n4b’289 ' uˆbd;<c<= k  T *-c uˆ*14<; f c* & 0hbduˆ4*8e/=g9:;bdcbj4<; y im=³¶›bdcbd9:;uˆbd;<c<= k 4 u’,n4<; & b$,n4bji™=g9’im=³¶›bdcbd9:;$28cbŽ2-4 *8e/; l bEi/*-u28=g9W ? l =h4L=h4; l b f c<=g9 k = f 0hbim= k ,/0g; ' e“2 k bji &:' 289/=h4*-;<c* f = k uˆbj4 l =g9/.¬280g.r*-c<=g; l uˆ4 & b k 28,n4b’bdqrbd9]=³eX; l b uˆbd;<c<= k e‰,/9 k ;<=h*-9]=h4Cq1280g=hiN28;˜24 f b k =³Ł kf *-=g9(; y ; l b’=g9(;bdc f *-0328;bjituˆbd;<c<= k 4 f 2 k b & bd; ” bjbd9‡289 ' ; ” * f *-=g9:;4+u•=g. l ;C9n*-; & b’q1280g=hi›+†‡b"i/bŽ280 ” =g; l ; l =h4 f c* & 0hbdu &:' i/b‹Łn9/=g9/.¬; l buˆbd;<c<= k im=h4 k cbd;bd0 ' 28; qrbdc<;<= k bj4Ž† l bd9]cb‹Łn9/=g9/.; l buˆbj4 l y” b™uˆbŽ2-4,/cb’; l b’0hbd9/.1; l *8e\bŽ2 kl ;<c<=3289/.10hb ff 4+bjim.rb2 kjk *-cim=g9/. ;*; l b"2Žqrbdc28.rbuˆbd;<c<= k *8eX; l b; l cbjb’qrbdc<;<= k bj4Ž ? l =h4 f c*²q(=hi/bj4˜2.r*(*Giwbj4<;<=gu28;b*8e l * ”& =g.; l 28; ;<c<=3289/.10hb=h4289ni ” l bd; l bdc"*-c9n*-;’=g;cbj–:,/=gcbj4=g9n4bdc<;<=h*-9W F e62qrbdc<;b‹‘U=h4’=g9n4bdc<;bji y ; l bˆuˆbd;<c<= k 28; ; l 28;Eqrbdc<;b‹‘ˆ=h4E2-44<=g.19nbji & 2-4bjiˆ*-92+0g=g9nbŽ28cE=g9(;bdc f *-0328;<=h*-9e‰c*-uö; l b; l cbjb6qrbdc<;<= k bj4E*8e›; l b6;<c<=3289/.10hb =g9 ” l = kl =g; ” 2-4˜=g9n4bdc<;bji› > 0g;bdc<928;<=gqrbd0 'ry›” l bd9 k *r28c4bd9/=g9/.¬; l b"uˆbj4 l ” b•bj4<;<=gu28;b; l b"0hbd9/.1; l *8e bŽ2 kl bjim.rb &:' Łnc4; k 280 k ,/0328;<=g9/. ” l 28;; l 28; 0hbd9/.1; l =h42 kjk *-cim=g9/.•;*"; l bC; ” *ˆ2-i ë 2 k bd9:;;<c<=3289/.10hbj4 289niU; l bd9ê2jqrbdc28.1=g9/.t; l *14b0hbd9/.1; l 4Ž ? l bj4b•; ” *ti/b‹Łn9/=g;<=h*-9n4.1=gqrbˆ,/9/=h–(,nb;<c<=3289/.10hb–:,280g=g; ' 289ni bjim.rb˜0hbd9/.1; l uˆbŽ2-4<,/cbj4 ” l = klIk 289w; l bd9 & b+,n4bji^e“*-cuˆbj4 l =gu f c*Žqrbduˆbd9(;C* f bdc28;<=h*-9n4Ž  -"/.H	<6 ffffifl$6-fl$ ? l bdcb28cb^2]4<;28.1.rbdc<=g9/.t9:,/u & bdc•*8e6;<c<=3289/.10hb^–(,280g=g; ' uˆbŽ2-4,/cbj4,n4bjiU;*]i/bd;bdc<u•=g9nb l * ” .r*(*Gi ; l b"bd0hbduˆbd9:;4˜*8eE2uˆbj4 l 28cb1 D =g9 k b9n**-9nb l 2-4˜2-i/bj–:,28;bd0 ' f c*Žqrbd9š; l 28;+*-9nb–(,280g=g; ' uˆbŽ2-4,/cb =h44, f bdc<=h*-ce“*-c6bdqrbdc ' 9:,/uˆbdc<= k 2804*-0gqrbdc yn” b l 2jqrb kl *14bd9];*•,n4b+; l b™–(,280g=g; ' uˆbŽ2-4,/cb˜; l 28; & bj4<; 4<,/=g;4; l b’2 f/f c*r2 kl ” b l 2Žqrb™;2 ì bd9W M =g0g0hbdc y ? 280guˆ*-c+289ni ? bd9/.>= v Ù? f *-=g9(;bjiw*-,/;6; l 28;; l b™uˆ*14<; 928;<,/c280–(,280g=g; ' uˆbŽ2-4<,/cb^e“*-c)+bd0328,/92 ' cb‹Łn9nbduˆbd9(;=h4"; l bc28;<=h*š*8e k =gc k ,/u•c2-im=g,n4•;*š4 l *-c<;bj4<; bjim.rb1$†‡b ” =g0g0›; l bdcb‹e‰*-cbi/b‹Łn9nb’2ffflfi ž-± ¡‰ªffiˆœ­Ÿmž€¨ (ξ, η) = ( 3 1 4 2 √ A i/bj4 lmin, 3 3 4 2 √ A i/bj4 R ) . = ( 0.658√ A i/bj4 lmin, 1.140√ A i/bj4 R ) o َÚrp ” l bdcb lmin =h4E; l b60hbd9/.1; l *8eW; l bC4 l *-c<;bj4<; bjim.rb y R =h4\; l b k =gc k ,/u•c2-im=g,n4 y 289ni A i/bj4 =h4E; l bLi/bj4=gcbji 28cbŽ2m T *-c2,/9/=³e“*-c<u =h4*-;<c* f = k uˆbj4 l y ; l b•=hi/bŽ280X;<c<=3289/.10hbˆ=h4"289Ubj–(,/=g0328;bdc280\;<c<=3289/.10hb ” =g; l 28cbŽ2 ánÙ æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå A i/bj4 = AΩ Nv ” l bdcb AΩ =h4; l b˜;*-;280 28cbŽ2ˆ*8e7; l b’i/*-u28=g9 Ω 289ni Nv =h4; l b˜9:,/u & bdc+*8e;<c<=3289/.10hbj4fx ; l =h4$;<c<=3289/.10hb l 2-4 (ξ, η) = (1, 1) e‰*-cX*-,/c kl *14bd9 k *-9n4<;289(;4Ž†‡b ,n4b ; l =h4X428uˆb k *-9 k b f ;X;* kl *(*14b A i/bj4 =g9 B –:,28;<=h*-9NَÚm ? *’bj4<;<=gu28;b6; l bL28cbŽ2 AΩ *8e; l b6i/*-u28=g9 Ω =g9ˆ; l b6uˆbd;<c<= k 4 f 2 k b y:” b64,/u ; l b28cbŽ2˜*8e=g9nim=gq(=him,280;<c<=3289/.10hbj4Ž ? l bd9 y ; l b=hi/bŽ28028cbŽ2 Ades =h47e“*-,/9ni &:' im=gq(=him=g9/.’; l b ;*-;28028cbŽ2 AΩ &:' ; l b`9(,/u & bdcX*8e;<c<=3289/.10hbj4 Nv 289ni•2Ci/bj4<=gcbji"cb‹Łn9nbduˆbd9:;\c28;<=h* r y 4*C; l 28; A i/bj4 = AΩ rNv  ? l b bj–(,/=g0328;bdc280\;<c<=3289/.10hb ” =g; l ; l =h428cbŽ2 ” *-,/0hi & b0h* k 28;bjiU28; (ξ i/bj4 , η i/bj4 ) = (1, 1) =g9‡; l b•–(,280g=g; ' 4 f 2 k b1 A 2jqG=g9/.ši/b‹Łn9nbji ; l b f bdcHe“b k ;";<c<=3289/.10hb y\” bu,n4<;9n* ” i/b‹Łn9nb ” l 28;•c289/.rb*8eL;<c<=3289/.10hb¬–(,280g=g;<=hbj4 28cb k *-9n4<=hi/bdcbji]2 kjk b f ;2 & 0hb1 ? *2 kjk *-u f 0g=h4 l ; l =h4 ym” b™i/b‹Łn9nb’2"q1280g,nb τ 4, kl ; l 28;;<c<=3289/.10hbj4 ” =g; l ξ < 1 − τ *-c η > 1 + τ 28cb k *-9n4<=hi/bdcbji & 2-i› ? *Ibd9n4<,/cb4 l 2 f b–:,280g=g; 'ry ” bˆcbj4<;<c<= k ; τ ;*w4* ; l 28; ” b l 2Žqrb’20h* ” bdc & *-,/9ni¬*-9; l b+u•=g9/=gu’,/u„289/.10hb˜*8eV; l b+;<c<=3289/.10hb ym” l = klIk 289 & b ” c<=g;<;bd9I2-4 lmin R = 2 sin θmin  B 0g=gu•=g928;<=g9/. lmin 289ni R =g9we“2Žqr*-,/c6*8e ξ 289ni η y/” b l 2jqrb ξ √ 3 η = 2 sin θmin 289niw4*"e‰*-c; l b k *-c<9nbdc f *-=g9(; (1− τ, 1 + τ) =g9 T =g.1,/cbˆÙŽî y 1− τ 1 + τ = 2√ 3 sin θmin o Ù²õ1p τ = 3− 2√3 sin θmin 3 + 2 √ 3 sin θmin o Ù z p D * kl *(*14<=g9/. θmin =h4Xbj–(,/=gq-280hbd9(;X;* kl *(*14<=g9/. τ  sEl b ” ff 4X,/9/=³e“*-c<u =h4*-;<c* f = k )Cbd0328,/92 ' cb‹Łn9nbduˆbd9:; 280g.r*-c<=g; l uô=gÙ1Ù? f c*Gim, k bj4E;<c<=3289/.10hbj4 ” l *14b+289/.10hbj428cbL280g0.1cbŽ28;bdc ; l 289; l =gc<; ' i/bd.1cbjbj4 4* ” b l 2Žqrb kl *14bd9š;*,n4b"; l =h4™2-4+; l b;28c<.rbd;’289/.10hb.1=gq(=g9/. τ = 2 − √3 .= 0.2679  > 9t=g0g0g,n4<;<c28;<=h*-9‚*8eE; l b –(,280g=g; ' 4 f 2 k b+=h4 e“*-,/9ni¬=g9 T =g.1,/cbˆÙŽîm á v æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå Minimum Angle Equilateral Triangle With Desired Area Equilateral Triangles Quality Bounding Box η ξ (1,1) 1−τ 1+τ η = ξ T =g.1,/cbˆÙŽîmý ? l b’–:,280g=g; ' 4 f 2 k b™,n4bji;*ˆbdq1280g,28;b˜;<c<=3289/.10hbj4=g9w; l b+uˆbj4 l  ffffifl$%#
fl7 	 ? l bdcb28cb`; l cbjb f c<=g9 k = f 280n* f bdc28;<=h*-9n4$; l 28;$*-,/c$280g.r*-c<=g; l uö,n4bj4;*+=gu f c*Žqrb ; l b–(,280g=g; ' im=h4;<c<= & , Œ ;<=h*-9N*8e7;<c<=3289/.10hbj4L=g9w; l b’uˆbj4 l  ? l b˜Łnc4<;; ” *^* f bdc28;<=h*-9n4 y bjim.rb’4 ” 2 f/f =g9/.¬289ni f *-=g9:;6=g9n4bdc<;<=h*-9 y 28cb¬4<=gu•=g0328c•;*t; l bd=gc•=h4*-;<c* f = k )+bd0328,/92 ' bj–:,/=gq1280hbd9:;4 ” =g; l uˆ*(im=³Ł k 28;<=h*-9n4;*‡2 kjk *-,/9:;•e“*-c•; l b uˆbd;<c<= k 4 f 2 k b1 ? l b032-4<;"* f bdc28;<=h*-9 y qrbdc<;b‹‘êcbduˆ*Žq1280 y =h4i/*-9nb4<=gu f 0 ' ;*]cbduˆ*²qrb^; l b^4u280g0hbj4<; bjim.rbj4=g9I; l b˜uˆbj4 l  пܩÝ|ýàýLÖÀÕ  ? l bˆŁnc4<;289niU4=gu f 0hbj4<;’* f bdc28;<=h*-9‚=h4bjim.rb4 ” 2 f/f =g9/.n T *-c’=h4*-;<c* f = k )+bd0328,/92 ' uˆbj4 l =g9/.t280g.r* Œ c<=g; l uˆ4 y bjim.rb^4 ” 2 f/f =g9/.š=h4,n4bji‚;*tbd9n4<,/cb^; l 28;"; l b^=g9(;bdc<=h*-cˆ*8eCbdqrbdc ' ;<c<=3289/.10hb ff 4 k =gc k ,/u k =gc k 0hb =h4bdu f ; ' *8eqrbdc<;<= k bj4Ž T *-c*-,/c"im=h4 k cbd;bˆcb f cbj4bd9:;28;<=h*-9 *8e ; l buˆbd;<c<= k y ; l =h4™=h4™9n*-;9nb k bj4428c<=g0 ' f *144<= & 0hb1 T ,/c<; l bdc<uˆ*-cb y” l bd9nbdqrbdc™289Nbjim.rb=h4L4 ” 2 f/f bji y ; l buˆbd;<c<= k ,n4bjiI;*b‹‘/28u•=g9nb"; l b’9nb ” ;<c<=3289/.10hbj4 k 289 im=³¶›bdc"e£c*-u ; l buˆbd;<c<= k *8eC; l b¬*-0hiê;<c<=3289/.10hbj4 y f *144<= & 0 ' cbdqrbdc4<=g9/.š; l b¬4 ” 2 f/f =g9/. i/b k =h4<=h*-9W K *-; l *8e›; l bj4b f c* & 0hbduˆ4`28cb64*-0gqrbjiˆ,n4<=g9/."2 l bd,/c<=h4<;<= k 2 f/f c*r2 kl  F 9n4<;bŽ2-i*8eW4 ” 2 f/f =g9/. e“*-c k bd0g0h4 ” =g; l 9n*-9 Œ bdu f ; ' k =gc k ,/u k =gc k 0hbj4 y” b™=g9n4;bŽ2-iI4 ” 2 f ;*•u2¥‘m=gu•=g¦Žb; l b™u•=g9/=gu’,/uº–(,280g=g; ' c28;<=h* ξ/η e“*-c˜bŽ2 kl‡f 28=gc™*8e2-i ë 2 k bd9(;™;<c<=3289/.10hbj4Ž T *-c’289š=h4*-;<c* f = k uˆbd;<c<= k y ; l =h4+=h4˜bj–:,/=gq1280hbd9:;™;* u2¥‘m=gu•=g¦j=g9/.I; l b•u•=g9/=gu,/u 289/.10hb y ” l = kl‚f c*(im, k bj4"2)+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9W ? *wi/bd;bdc<u•=g9nb ” l = klêk *-9mŁn.1,/c28;<=h*-9 i/*Gbj4™; l =h4 y” b k 280 k ,/0328;b (ξ, η) b‹‘ f 0g= k =g;<0 ' e“*-c & *-; l ; l b k ,/c<cbd9:;;<c<=3289/.10hbj4 á:í æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå 289ni‚; l b;<c<=3289/.10hbj4"; l 28; ” *-,/0hi & b•e“*-c<uˆbjiU=³e; l b^bjim.rb ” bdcb4 ” 2 f/f bji› P *-;b; l 28;*-,/c"uˆbd;<c<= k i/b‹Łn9/=g;<=h*-9=h4`i/b f bd9nin289(;*-9 ” l = kl qrbdc<;<= k bj428cb k *-9/9nb k ;bji¬289ni4*4 ” 2 f/f =g9/.ˆ289bjim.rbLb‹¶›b k ;<=gqrbd0 ' kl 289/.rbj4˜; l b"uˆbd;<c<= k  K b k 28,n4b*8eE; l =h4 y 2)+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9 k 289/9n*-; & be‰*-,/9ni &(' 4<=gu f 0 ' u2 f/f =g9/.; l b+0h* k 280Wqrbdc<;b‹‘0h* k 28;<=h*-9n4;*•; l b+uˆbd;<c<= k 4 f 2 k b1 > e£;bdc4 ” 2 f/f =g9/. y/” b kl b kì ” l bd; l bdc; l bCcbj4<,/0g;<=g9/.•;<c<=3289/.10hbj4628cb˜)+bd0328,/92 ' =g9; l b+uˆbd;<c<= k 4 f 2 k b1 ? l bcbŽ2-4*-9 ” bi/*¬; l =h4=h4˜; l 28;)+bd0328,/92 ' =g9n4bdc<;<=h*-9U=h4*-9/0 ' q-280g=hi ” l bd9U; l bˆ=g9(;bdc<=h*-c"*8ebdqrbdc ' k =gc k ,/u k =gc k 0hb^=h4’bdu f ; '  ? l bdcb^28cb9n*I.1,28c289:;bjbj4"; l 28;"4 ” 2 f/f =g9/. ” =g0g0 f c*Gim, k b4<, kl 2w;<c<=3289 Œ .1,/0328;<=h*-9 =g9 *-,/cuˆbd;<c<= k 4 f 2 k b y 4* ” b^u’,n4;=hi/bd9(;<=³e ' 9n*-9 Œ )Cbd0328,/92 ' ;<c<=3289/.10hbj44*]; l 28; ” b k 289 l 289nim0hb˜; l bdu l bd,/c<=h4<;<= k 280g0 ' im,/c<=g9/.ˆ=g9n4bdc<;<=h*-9W K b‹e“*-cb ” b k 289 kl b kì ” l bd; l bdc6*-c9n*-;; l b’bjim.rb+=h4 )+bd0328,/92 ' =g9w; l b+uˆbd;<c<= k 4 f 2 k b y/” b™u’,n4<;64 k 280hb˜*-9nb˜*8e; l b˜;<c<=3289/.10hbj4L4*; l 28;; l b™4 l 28cbjiIbjim.rb+=h4 ; l b™4 28uˆb˜0hbd9/.1; l e“*-c & *-; l ;<c<=3289/.10hbj4Ž > ;; l =h4 f *-=g9(; y ; l b˜cbd0328;<=gqrb’0h* k 28;<=h*-9n4L*8e; l b˜e‰*-,/cqrbdc<;<= k bj4 28cb ì 9n* ” 9=g92 f 4bd,ni/* Œ uˆbd;<c<= k 4 f 2 k b1 ? l bd9 y(” bL;bj4<; ” l bd; l bdc *-c\9n*-;E; l b k =gc k ,/u k =gc k 0hbj4 *8eVbŽ2 kl ;<c<=3289/.10hb28cb bdu f ; '  K b k 28,n4b`; l b )Cbd0328,/92 ' f c* f bdc<; ' =h4 =g9ni/b f bd9ni/bd9:;X*8e4<=g¦Žb y =g;i/*Gbj49n*-;7u28;<;bdc ” l = kl ;<c<=3289/.10hb.rbd;4˜4 k 280hbji› F 9 T =g.1,/cb¬ÙŽÚ y ; l b";<c<=3289/.10hbj4 ? و289ni ? v 28cb"Łnc4<;+;<c289n4He“*-c<uˆbjit;* l 2Žqrb+; l bL0hbd9/.1; l 4L2-44<=g.19nbji &:' ; l bd=gc`cbj4 f b k ;<=gqrb+uˆbd;<c<= k 4Ž ? l bd9 ? v =h44 k 280hbji &(' a/d 4*’; l 28;; l b bjim.rbj4 k 289 & b™280g=g.19nbjiN289ni; l b k =gc k ,/u k =gc k 0hbj46;bj4;bji› ! ÷ß@ö@÷#"Ãñ†Õ±Ó†÷ß@öÖEÔSÕ ? l bˆ4b k *-9nišuˆbj4 l uˆ*Gim=³Ł k 28;<=h*-9‚* f bdc28;<=h*-9‡; l 28; ” bˆ,n4b=h4˜qrbdc<;b‹‘š=g9n4bdc<;<=h*-9W F 9š=h4*-;<c* f = k )+b Œ 0328,/92 ' cb‹Łn9nbduˆbd9:; y qrbdc<;<= k bj4˜28cb=g9n4bdc<;bjiIe‰*-cC;<c<=3289/.10hbj4 ” =g; l 0328c<.rb k =gc k ,/u•c2-im=g=$*-c f *G*-cL4 l 2 f b1 > e£;bdcC=g9n4bdc<;<=h*-9 y ; l b"cbj4<,/0g;<=g9/.;<c<=3289/.10hbj4 l 2jqrb4<u280g0hbdc k =gc k ,/u•c2-im=g= y 289niN9n*bjim.rb"4<u280g0hbdc+; l 289 ; l bw*-c<=g.1=g9280 k =gc k ,/u•c2-im=g,n4^=h4 k cbŽ28;bji› F 9 *-,/c289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 y ” b l * f be‰*-cˆ4=gu•=g0328c & b l 2jqG=h*-,/c &(' =g9n4bdc<;<=g9/.^28;L289N2 f/f c*j‘m=gu28;b k =gc k ,/u k bd9(;bdcŽ †‡b˜Łn9ni¬; l =h4+2 f/f c*j‘m=gu28;b k =gc k ,/u Œ k bd9(;bdc &(' k cbŽ28;<=g9/.2qG=gc<;<,28074<, & Œ uˆbj4 l 28c*-,/9niI; l b™=g9/=g;<=3280;<c<=3289/.10hb1 ? * k *-9n4<;<c<, k ;C; l =h4qG=gc<;<,280 4<, & Œ uˆbj4 l yX” b^Łnc4;";<c289n4<e‰*-c<u ; l b^=g9/=g;<=3280;<c<=3289/.10hb;*t*-9nb ” =g; l 0hbd9/.1; l 42-4i/bj4 k c<= & bji &:' ; l b uˆbd;<c<= k  ? l bd9 ym” b k 280 k ,/0328;b˜; l 28;;<c<=3289/.10hb ff 4 k =gc k ,/u k bd9(;bdcC289ni & ,/=g0hi; l b+q(=gc<;<,280W4, & Œ uˆbj4 l *-9nb ;<c<=3289/.10hb"28;L2•;<=guˆb˜,/9(;<=g0V; l b k =gc k ,/u k bd9:;bdc+0g=hbj4 ” =g; l =g9I=g;Ž D =g9 k b™; l b’uˆbd;<c<= k 0hbd9/.1; l *8e$2•4 l 28cbji á1á æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå a c b d e f Triangles with lengths measured by the average metric of the three vertices a d e a d f bc a Triangle T2 is scaled to match triangle T1 The triangles are tested for empty circumcircles T1 Triangles to be tested T2 T =g.1,/cbÙŽÚmý ? bj4<;<=g9/.e‰*-c B u f ; ' s =gc k ,/u k =gc k 0hbj46=g9w; l b M bd;<c<= k’D(f 2 k b bjim.rbC=h4 9n*-; ; l b˜428uˆbLe‰*-c & *-; l ;<c<=3289/.10hbj4 y 2"4 k 280g=g9/."e“2 k ;*-c=h428.:28=g9¬cbj–(,/=gcbji› ? l =h4 =h4 4<=gu•=g0328c;* ; l bL)Cbd0328,/92 ' ;bj4<; f c* k bjim,/cb6i/*-9nb6e“*-c\bjim.rb64 ” 2 f/f =g9/.n F 9 T =g.1,/cb’Ù²õ ? c<=3289/.10hb ? Ù6=h4 kl *14bd9ˆe“*-c =g9n4bdc<;<=h*-9U289nit; l b k =gc k ,/u k bd9(;bdc’=h4 k 280 k ,/0328;bjiš=g9š; l b"qG=gc<;<,280X4 f 2 k b1 ? l bd9 y ? c<=3289/.10hbj4 ? v 289ni ? á28cb˜2 f/f c* f c<=328;bd0 ' 4 k 280hbji¬4*"; l 28;; l b+qG=gc<;<,280W4<, & Œ uˆbj4 l¬k *-9:;28=g9n4 ? Ù ff 4 k =gc k ,/u k bd9:;bdcŽ Y 9 k b ; l 28;•=g9n4bdc<;<=h*-9Z0h* k 28;<=h*-9 =h4 kl *14bd9 =g9fi; l bqG=gc<;<,28064<, & Œ uˆbj4 l y =g;•=h40g=g9nbŽ28c<0 ' ;<c289n4He“*-c<uˆbji & 2 kì ;*; l b"cbŽ280$uˆbj4 l 4*^; l 28;˜=g; l 2-4+; l b•428uˆb & 28c ' k bd9:;<c<= kk *(*-cim=g928;bj4˜=g9 & *-; l ; l bcbŽ280\289nit; l b qG=gc<;<,280X;<c<=3289/.10hb1 ? l b•9nb ” qrbdc<;b‹‘t=h4™=g9n4bdc<;bji ” =g; l ; l buˆ*14<;4<=gu f 0hb k *-9/9nb k ;<=gqG=g; ' 289ni‡4 ” 2 fmŒ f =g9/."=h4 f bdcHe“*-c<uˆbji^cb k ,/c4<=gqrbd0 '  ? l bCi/bj4<=gcbjib‹¶b k ; *8e bŽ2 kl =g9n4bdc<;<=h*-9=h4E;*i/b k cbŽ2-4b k =gc k ,/u•c2-im=g= ” =g; l *-,/;+imc28u28;<= k 280g0 ' cbjim, k =g9/.; l b–:,280g=g; ' *8e$; l b™0h* k 280V;<c<=3289/.10hbj4Ž J 9me‰*-c<;<,/928;bd0 'ry im,nb™;*; l b im=h4 k cbd;b™928;<,/cb™*8e; l b™uˆbd;<c<= k y ; l bdcb’28cb™4*-uˆb k 2-4bj4 ” l bdcb™; l =h462 f/f c*r2 kl eé28=g0h4Ž ? l b+Łnc4;i/bd.rbd9nbdc28;b k 2-4b˜* kjk ,/c4 ” l bd9w; l b k =gc k ,/u k =gc k 0hb*8e; l b˜;<c<=3289/.10hb ” b ” =h4 l ;*•bd0g=gu•=g928;b &(' =g9n4bdc<;<=h*-9=h479n*-;\bdu f ; ' =g9•; l bqG=gc<;<,280/uˆbj4 l † l bd9•; l =h4X* kjk ,/c4 y1” b=g9n4<;bŽ2-i•4 f 0g=g;$; l b 0h*-9/.rbj4; bjim.rbX*8e/; l 28; ;<c<=3289/.10hb1 ? l bdcb\=h4›; l bj*-cbd;<= k 280g0 ' 9n*0g=gu•=g;*-9™; l b$u•=g9/=gu’,/ubjim.rbX; l 28;V; l =h4V* f bdc28;<=h*-9 ” =g0g0 k cbŽ28;b1 ? *^2-i/imcbj44; l =h4 y/” b’4<uˆ*G*-; l ; l b˜9nb ” qrbdc<;b‹‘¬;*•u2¥‘m=gu•=g¦Žb™; l b™u•=g9/=gu’,/u„289/.10hb’=g9 ; l b’q(=gc<;<,280Wuˆbj4 l y cbj4<,/0g;<=g9/.^=g9]2•u, kl & bd;<;bdc6=g9n4bdc<;<=h*-9]0h* k 28;<=h*-9W > 4<=gu•=g0328cC2 f/f c*r2 kl =h46,n4bji á:î æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå T1 T2 T2 T1 T2 x T3 T3 T3 Virtual Sub−mesh T1 T2 x T3 xT1 Actual Mesh Virtual Triangles x − circumcenter x−Insertion Location Actual Mesh T1 T =g.1,/cb•Ù²õGý T =g9nim=g9/.289I2 f/f c*j‘m=gu28;b k =gc k ,/u k bd9(;bdcC*8e;<c<=3289/.10hb ? Ù &:' k *-9n4<;<c<, k ;<=g9/.ˆ; l b+qG=gc<;<,280 uˆbj4 l =g9š*-; l bdc’289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9(;"280g.r*-c<=g; l uˆ4é= vmy›z ?­ F 9š*-,/c’280g.r*-c<=g; l u y l * ” bdqrbdc y * f ;<=gu•=g¦²28;<=h*-9 =h4*-9/0 ' cbj–:,/=gcbji e‰*-ci/bd.rbd9nbdc28;b k 2-4bj4 y ” l = kl 28cbN2‡4<u280g06e£c2 k ;<=h*-9 *8e+; l bI;*-;28069(,/u & bdc^*8e =g9n4bdc<;<=h*-9n4²%$ ? l b 4b k *-9ni"i/bd.rbd9nbdc28;b k 2-4b`* kjk ,/c4 ” l bd9=g9n4bdc<;<=g9/.+; l b\9nb ” qrbdc<;b‹‘i/*(bj4 9n*-;X2¥¶b k ;; l b`;<c<=3289/.10hb e“*-c ” l = kl =g; ” 2-4=g9n4bdc<;bji› F 9*-; l bdc ” *-ci/4 y =g9n4bdc<;<=h*-9wim=hi9n*-;cbjim, k b+; l b+0328c<.rbj4<; k =gc k ,/u•c2-im=g,n4 =g9¬; l b+uˆbj4 l  ? l =h4 l 2 f/f bd9n4 uˆ*14<;*8e‰;bd9 ” l bd9¬; l bdcb+=h40328c<.rb˜im=³¶›bdcbd9 k b+=g9¬uˆbd;<c<= k 4<=g¦Žb & bd; ” bjbd9 ;<c<=3289/.10hbj4Ž s ,/c<cbd9(;<0 'ry =g9n4bdc<;<=h*-9=h44<=gu f 0 ' cb f bŽ28;bji™e“*-cV; l b`*-c<=g.1=g9280G;<c<=3289/.10hb1 D =g9 k bE; l 28;;<c<=3289/.10hb ” =g0g079n*^0h*-9/.rbdc l 2jqrbˆ289šbdu f ; ' k =gc k ,/u k =gc k 0hb y cb‹Łn9nbduˆbd9:; ” =g0g0$4 f 0g=g;+; l b"0h*-9/.rbj4<;˜bjim.rb•289niN; l bd9 0h* k 280g0 ' * f ;<=gu•=g¦Žb’; l b˜0h* k 28;<=h*-9W ! ÷ß@ö@÷#"'&w÷ùúÔØ|ݒâ ? l bŁn9280* f bdc28;<=h*-9 ; l 28; ” b^,n4b^;*]=gu f c*Žqrb; l b^uˆbj4 l =h4 k *r28c4bd9/=g9/.n s *r28c4bd9/=g9/.š=h4"9n*-;ˆ2 928;<,/c280 k *-u f *-9nbd9(;"*8e)+bd0328,/92 ' uˆbj4 l =g9/.N4*; l bdcb28cbˆ9n*I4<;289nin28ciU=h4*-;<c* f = k ;b kl 9/=h–:,nbj4;* (gè|j†[‘[o‡†IZlˆQqk@[oGElg[oXQÀIHY[•FZT¿i†[gYZ[gXB[oGÀI3QE[‘leIHOV[gOqMNOq[gX_QEM`GE[gK`ai†[ok@[gXif[gX_Q±FZX…j†Fe‹°QEjB[,cd[oQVGEMNl,nI3GEMP[oOL‹M`QEjBMNXyQEjB[ cd[gOVj:uU쵇fk@[oGEMN[gX†lg[fljIHOSOVj†Fe‹X‘QEjI3QSi†[gYZ[gXB[oGÀIHloa•MPX*)3Ê,+.-/°FZTUMNXBOV[ˆGVQEMPFZXBOSMNOQafkBMNleIZKTFHGQEjB[flcd[oQVGEMNl¿‹Œ[fl^BOV[eiu á:Ú æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå 2 f/f 0 ' *-9w; l b˜qG=gc<;<,280Wuˆbj4 l  F 9n4<;bŽ2-i yn” b ì 9n* ” ; l 28;cbduˆ*ŽqG=g9/.qrbdc<;<= k bj46=g9 k cbŽ2-4bj46bjim.rb˜0hbd9/.1; l 4 =g9¬; l b˜uˆbj4 l  ? l ,n4 y cbduˆ*ŽqG=g9/.2"qrbdc<;b‹‘¬; l 28;=h4 f 28c<;6*8e; l b™4<u280g0hbj4<;bjim.rb ” =g0g0280 ” 2 ' 4=g9 k cbŽ2-4b ; l bIu•=g9/=gu’,/u bjim.rb1 ? *Ucbduˆ*²qrbN2šqrbdc<;b‹‘ y ” b¬Łnc4<;4 ” 2 f bjim.rbj4ˆ,/9:;<=g0C; l bdcb]28cb]*-9/0 ' ; l cbjb bjim.rbj4=g9 k =hi/bd9(;6*-9¬; l 28;qrbdc<;b‹‘w289ni k *-u & =g9nb˜; l b+; l cbjb+=g9 k =hi/bd9(;;<c<=3289/.10hbj4;*e“*-c<u *-9nb+;<c<=3289/.10hb ” l bd9t; l b"qrbdc<;b‹‘]=h4Ccbduˆ*Žqrbji› ? l bbjim.rbj4˜28cb; l bd9‡4 ” 2 f/f bji]cb k ,/c4=gqrbd0 ' =g9t; l b428uˆb"u289/9nbdc i/bj4 k c<= & bji•=g9 D b k ;<=h*-9N s *r28c4bd9/=g9/.™=h4$; l b*-9/0 ' * f bdc28;<=h*-9ˆ; l 28; k 289i/bj4<;<c* ' uˆbd;<c<= k =g9me‰*-c<u28;<=h*-9W K b k 28,n4b™*8e7; l =h4 y cb f bŽ28;bji¬cb‹Łn9nbduˆbd9:;˜289ni k *r28c4bd9/=g9/.;bd9ni¬;*•4<uˆ*G*-; l ; l b+uˆbd;<c<= k  0 /-""Œ621  	fl$V	6 >fiì b ' bd0hbduˆbd9(;X*8e)+bd0328,/92 ' uˆbj4 l =g9/.+=h4 & *-,/9nin28c ' ;<cbŽ28;<uˆbd9(;Ž7† l =g0hbE; l b f cbdq(=h*-,n47im=h4 k ,n44=h*-9"*8e f *-=g9:;$=g9n4bdc<;<=h*-92-44<,/uˆbji; l 28;$; l b k =gc k ,/u k bd9:;bdc\0g=hbj4 ” =g; l =g9; l b`i/*-u28=g9 y ; l 28;$=h479n*-;$9nb k bj4428c<=g0 ' ;<c<,nb1 T *-c k 2-4bj4 ” l bd9‚; l b k =gc k ,/u k bd9:;bdc0g=hbj4*-,/;4<=hi/bˆ; l bi/*-u28=g9‚*-c"2 k c*1442 & *-,/9nin28c 'ry ; l b qG=gc<;<,280 uˆbj4 lfik 289/9n*-; & b k *-u f 0hbd;bd0 ' k *-9n4<;<c<, k ;bji›‡† l bd9 ; l =h4 l 2 f/f bd9n4 y ; l b & *-,/9nin28c ' bjim.rb ” l = kl ;bdc<u•=g928;bjiN; l b"qG=gc<;<,280 uˆbj4 l =h4C4 f 0g=g;Ž T *-c k 2-4bj4 ” l bdcb; l b k =gc k ,/u k bd9(;bdc’*8eE2;<c<=3289/.10hb 0g=hbj4`=g9n4<=hi/bL; l b+i/*-u28=g9 ym& ,/;E9nbŽ28c ; l b & *-,/9nin28c 'ry ; l bdcb+28cbC9n* k *-9n4<;<c28=g9(;4*-9^; l b k =gc k ,/u•c2-im=g,n4 *8e; l b9nb ” ;<c<=3289/.10hbe“*-c<uˆbji› ? *N2-i/imcbj44; l =h4 f c* & 0hbdu y7” b2-in2 f ;"; l b & *-,/9nin28c ' f c*-;b k ;<=h*-9 f c* f *14bji &(' D(l b ” kl , ì = í1ó½?­ ? l 28;\=h4 yr” bLim=h4280g0h* ” 289 ' =g9n4bdc<;<=h*-9 ” =g; l =g9; l bLim=328uˆbd;<c2800hbd9n4`*8eV2 & *-,/9nin28c ' 4bd.1uˆbd9:;˜289ni]=g9n4<;bŽ2-iI4 f 0g=g;L; l 28; & *-,/9nin28c ' 4bd.1uˆbd9(;Ž T *-c+289t289/=h4*-;<c* f = k uˆbj4 l y ; l b im=328uˆbd;<c28070hbd9n4L=h4C2 f/f c*Ž‘G=gu28;bji &(' ;<c289n4He‰*-c<u•=g9/.w289 ' f *-;bd9(;<=3280 & *-,/9nin28c ' ;<c<=3289/.10hb",n4<=g9/.^; l b 2Žqrbdc28.rb™uˆbd;<c<= k *8e7; l b˜; ” * & *-,/9nin28c ' qrbdc<;<= k bj4² > e‰,/c<; l bdc k *-u f 0g= k 28;<=h*-9 ” =g; l & *-,/9nin28c<=hbj4=h4; l 28;^289 ' k ,/c<q128;<,/cbI*8e™; l b & *-,/9nin28c ' u,n4<; & b f cbj4bdc<qrbji‡; l c*-,/. l *-,/;™cb‹Łn9nbduˆbd9:;Ž F eE; l b k ,/c<q128;<,/cbˆ*8e`; l b & *-,/9nin28c ' =h4+9n*-; f cbj4bdc<qrbji y ; l bd9 ; l b.rbj*-uˆbd;<c '•” =g0g09n*-; & b k *-9(qrbdc<.1=g9/.;*™; l buˆ*(i/bd0hbji.rbj*-uˆbd;<c 'ry ; l ,n4X=g9(;<c*(im, k =g9/.",/9/9nb k bj4428c ' uˆ*Gi/bd0g=g9/.ˆbdc<c*-cŽ ? l =h4 =h4280h4*qrbdc ' =gu f *-c<;289(;e‰*-c l =g. lmŒ *-ci/bdc64*-0g,/;<=h*-9n4 & b k 28,n4b k *-c<cb k ; k ,/c<qrbji & *-,/9nin28c ' cb f cbj4bd9(;28;<=h*-9‡=h4+bj44bd9(;<=3280$;*^2 kl =hbdq(=g9/. l =g. l *-ci/bdc™2 kjk ,/c2 k '  Y 0g0g=gqG=hbdc ŒQ_ *G* kl 289ni K *-=gqG=g9= îB?4*-0gqrbji•; l =h4 f c* & 0hbduöe‰*-c$=h4*-;<c* f = k uˆbj4 l bj4 &(' 0g=gu•=g;<=g9/.; l b k ,/c<q-28;<,/cbL*8e; l b & *-,/9nin28c ' im=h4 k cbd;<=g¦²28;<=h*-9Z289niê; l ,n4"bd9me“*-c k =g9/.š; l 28;; l bim=328uˆbd;<c2800hbd9n4"0g=hb*-,/;4<=hi/b^; l b & *-,/9nin28c ' k ,/c<qrb1 á(õ æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå Boundary Curve Boundary Offset Diametral Lense Offset Diametral Lense Boundary Curve Boundary Offset T =g.1,/cbÙ z ý F 4*-;<c* f = k y uˆ*(im=³ŁbjiIim=328uˆbd;<c280V0hbd9n4 e“*-c k ,/c<qrbji & *-,/9nin28c ' ? l =h4+=h4˜2u’, kl uˆ*-cb"im= k ,/0g;˜;2-4 ì ” l bd9š; l b"uˆbd;<c<= k =h4˜280g0h* ” bjiš;*^q128c ' 280h*-9/.; l b & *-,/9nin28c ' bjim.rb & b k 28,n4b ; l b uˆbŽ2-4<,/cbji k ,/c<q128;<,/cb kl 289/.rbj4\i/b f bd9nim=g9/.’*-9 ” l 28;$uˆbd;<c<= k =h47,n4bji› F 9n4<;bŽ2-i ” b cbd0 ' 28.:28=g9w*-9 l bd,/c<=h4;<= k 4Žý$b‹‘ f 289n4=h*-9w*8e ; l b+im=328uˆbd;<c280›0hbd9n46289nibd9 k c*r2 kl uˆbd9(; kl b kì =g9/. & b‹e“*-cb & *-,/9nin28c ' =g9n4bdc<;<=h*-9W T =gc4<; y(” bCb‹‘ f 289ni; l bCim=328uˆbd;<c2800hbd9n4E;* & bd;<;bdc 2 kjk *-,/9:;`e‰*-c`; l b k ,/c<q128;<,/cb1 ? *i/*; l =h4 y(” b6Łnc4<; k 280 k ,/0328;bI; l bqrbdc<;<= k 280L*8¶›4bd; & bd; ” bjbd9…; l b0g=g9nbŽ28c & *-,/9nin28c ' 4bd.1uˆbd9(;289nifi; l bI2 k ;<,280 k ,/c<qrbji & *-,/9nin28c '  ? l bd9 yX” bbd0h*-9/.:28;b; l b^im=328uˆbd;<c2800hbd9n4*8eC; l b^4bd.1uˆbd9:; &:' ; l b^4 28uˆb^*8¶›4bd;Ž > 9 =h4*-;<c* f = k b‹‘n28u f 0hbt*8e’; l =h4=h4¬4 l * ” 9 =g9 T =g.1,/cbfiÙ z  > .:28=g9 y e“*-c; l bt289/=h4*-;<c* f = k uˆbd;<c<= k y6” b f bdcHe‰*-c<u ; l =h4 f c* k bjim,/cb & 2-4bjiw*-9¬; l b’2Žqrbdc28.rb™uˆbd;<c<= k *8e7; l b˜; ” * & *-,/9nin28c ' qrbdc<;<= k bj4 > 4 ” =g; l 4<=gu•=g0328cw2 f/f c*r2 kl bj4¬=g9…=h4*-;<c* f = k uˆbj4 l =g9/. y u2 ì =g9/. ; l =h42-i ë ,n4<;<uˆbd9(;4;<=g0g0+i/*Gbj4^9n*-; .1,28c289(;bjbL; l 28;`9nb ” qrbdc<;<= k bj4 ” =g0g0 & b6=g9n4bdc<;bji ” =g; l =g9; l bLi/*-u28=g9^*-c`; l 28;E4 f 0g=g;<;<=g9/.•2 & *-,/9nin28c ' k ,/c<qrb ” =g0g0 f c*(im, k b\.r*(*Gi™–(,280g=g; ' ;<c<=3289/.10hbj4Ž ? *6cbduˆbji ' ; l =h4 y¥” l bd9nbdqrbdc2 & *-,/9nin28c ' bjim.rbX=h4 4 f 0g=g; y 289 ' qrbdc<;b‹‘’; l 28;7bd9 k c*r2 kl bj4$*-9’; l bE9nb ” bjim.rbj4 =g9"; l bXqG=gc<;<,280Guˆbj4 l u,n4<; & b\cbduˆ*Žqrbji› ? *-.rbd; l bdc y ; l bj4b™uˆbd; l *(i/4Lbd9n4<,/cb™; l 28;L; l b˜cb‹Łn9nbjiwuˆbj4 l bj4 ” =g0g0Vcbj4 f b k ;; l b & *-,/9nin28c ' im=h4 k cbd;<=g¦²28;<=h*-9t289ni ; l 28;9n*•*²qrbdc<0 ' Œ è28;L;<c<=3289/.10hbj4 ” =g0g0 & b k cbŽ28;bji]280h*-9/.; l b & *-,/9nin28c '  .3˜H	%)+ †‡bN2 f/f 0 ' ; l b¬e“*-0g0h* ” =g9/. 280g.r*-c<=g; l u ;* f c*Gim, k bI–(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l bj4^; l 28;u28; kl 289 28c & =g;<c28c ' uˆbd;<c<= k 289ni l 2Žqrb2 f cbj4 k c<= & bji¬u2¥‘m=gu’,/uû9:,/u & bdcL*8e7qrbdc<;<= k bj4Ž á z æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå Ù1 > 44=g.19I2"uˆbd;<c<= k ;*•bŽ2 kl qrbdc<;b‹‘ v  D ,/u; l b+289/=h4*-;<c* f = k 28cbŽ2"*8eVbŽ2 kl ;<c<=3289/.10hb˜289ni k 280 k ,/0328;b A i/bj4 289ni; l ,n4 (ξ, η) e“*-c bdqrbdc ' ;<c<=3289/.10hb1 ím R bduˆ*Žqrb•289 ' =g9:;bdc<=h*-c˜qrbdc<;<= k bj4C; l 28;˜bd9 k c*r2 kl *-9š2 & *-,/9nin28c ' 4bd.1uˆbd9:;’2 kjk *-cim=g9/.;*^; l b uˆbd;<c<= k  án R b k ,/c4<=gqrbd0 ' 4 ” 2 f bjim.rbj4];*fiu•=g9/=gu•=g¦Žb‚; l b‡c28;<=h* η/ξ  F e•; l =h4Ii/*Gbj4w9n*-;Icbj4,/0g;I=g9@289 289/=h4*-;<c* f = k )+bd0328,/92 ' ;<c<=3289/.1,/0328;<=h*-9 y u28c ì *8¶›bd9nim=g9/.bjim.rbj4L2-4 & bd=g9/.9n*-9 Œ )Cbd0328,/92 '  îm F 9n4bdc<;`qrbdc<;<= k bj4`e“*-cE;<c<=3289/.10hbj4 ” =g; l η > (1 + τ) ,/9(;<=g0; l bLu2¥‘m=gu’,/u 9(,/u & bdc*8eVqrbdc<;<= k bj4 =h4 cbŽ2 kl bjiN*-c; l bdcb’28cb™9n*ˆuˆ*-cb˜;<c<=3289/.10hbj4 ” =g; l 0328c<.rb k =gc k ,/u•c2-im=g=é > 44<=g.19wuˆbd;<c<= k 4;*ˆ9nb ” qrbdc<;<= k bj4 & 2-4bji*-9^2+0g=g9nbŽ28cE=g9:;bdc f *-0328;<=h*-9e‰c*-u ; l b; l cbjb6qrbdc<;<= k bj4E*8e; l b;<c<=3289/.10hb6=g9 ” l = kl ; l bqrbdc<;b‹‘ ” 2-4=g9n4bdc<;bji› F eC2Ie“2 k b^,n4bjiê=g9 k *-9n4<;<c<, k ;<=g9/.š; l bqG=gc<;<,280 uˆbj4 l =h4u28c ì bji 2-4ˆ9n*-9 Œ )+bd0328,/92 'ry ; l bd9Z; l bw;<cbŽ28;<uˆbd9(;^e‰*-cˆ9n*-9 Œ bdu f ; ' k =gc k ,/u k =gc k 0hbj4^e‰c*-u D b k ;<=h*-9 =h4 2 f/f 0g=hbji› Úm R bduˆ*Žqrb^qrbdc<;<= k bj4"e“*-c"bjim.rbj4 ” =g; l ξ < (1− τ) ,/9:;<=g0; l bdcb¬28cb9n*]uˆ*-cb4<u280g0 bjim.rbj4•*-c 4<,” k =hbd9:;<0 ' u289 ' ;<c<=3289/.10hbj4 ” =g; l 0328c<.rb η 28cb k cbŽ28;bji› S =gu•=g;<=g9/. l * ” u289 ' 0328c<.rb`;<c<=3289/.10hbj4 k 289 & b k cbŽ28;bji =h4289 b‹¶b k ;<=gqrb ” 2 ' *8e+cbjim, k =g9/.U; l b¬*²qrbdc<032 f & bd; ” bjbd9…cb‹Łn9nbduˆbd9:;289ni k *r28c4bd9/=g9/.n F e; l =h4\0g=gu•=g;E=h4`2+e‰c2 k ;<=h*-9^*8e›; l b69:,/u & bdc*8e›qrbdc<;<= k bj4E=g9n4bdc<;bjiˆ=g9; l b f cbdqG=h*-,n4 cb‹Łn9nbduˆbd9:;w4<;b f ; l bd9…=g;¬2-i/i/4^2U.1,28c289:;bjbt*8e™;bdc<u•=g928;<=h*-9a=gc<cbj4 f b k ;<=gqrbt*8e l * ” u289 ' =g;bdc28;<=h*-9n4L28cb’280g0h* ” bji› õG R b f bŽ28;X; l b f c* k bj44\*8e›=g9n4bdc<;<=g9/.289ni•cbduˆ*ŽqG=g9/."qrbdc<;<= k bj4\,/9(;<=g0280g0;<c<=3289/.10hbj4 28cb ” =g; l =g9^; l b 2 kjk b f ;2 & 0hb6;*-0hbdc289 k b y 9n*™uˆ*-cb6qrbdc<;<= k bj4 k 289 & bcbduˆ*²qrbji y *-c`2CŁ/‘mbjiˆ9:,/u & bdc *8e=g;bdc28;<=h*-9n4 28cb f bdcHe‰*-c<uˆbji› T *-c`–(,280g=g; ' uˆbj4 l bj4 y ; l bŁn9280 f c* k bj44Eu’,n4<;280 ” 2 ' 4 & buˆbj4 l cb‹Łn9nbduˆbd9:;Ž á:ó æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå 1:100 100:1 100:1100:100 1:1 1:1 T =g.1,/cbˆÙŽómý B ‘ f b k ;bji ? c<=3289/.10hb2-4 f b k ;c28;<=h*14.1=gqrbd9w; l b+uˆbd;<c<= k e£c*-u B –:,28;<=h*-9Uَóm {@fl$-).H	 ? *êi/bduˆ*-9n4;<c28;bN*-,/cw280g.r*-c<=g; l u ff 4w2 & =g0g=g; ' ;* k cbŽ28;bt289/=h4*-;<c* f = k uˆbj4 l bj4e‰c*-u 2‡uˆbd;<c<= k y ; l b uˆbd;<c<= k d( ~x1, ~x2)(x, y) i/b‹Łn9nbji &:' B –:,28;<=h*-9êَó"=h462-44<=g.19nbji]28;bdqrbdc ' qrbdc<;b‹‘› d(~p1, ~p2) = √ (~p2 − ~p1)T M(~p2 − ~p1) M(x, y) =   a 0 0 b   a =   1× 104 1 0.475 < x < 0.525 B 0h4b ” l bdcb b =   1× 104 1 0 ≤ x < 0.05 B 0h4b ” l bdcb o َórp ? l =h4•uˆbd;<c<= k 4 l *-,/0hi f c*Gim, k bI2Nuˆbj4 l ” =g; l im=h4<;<=g9 k ;^289/=h4*-;<c* f = k cbd.1=h*-9n4Ž ? *‡4<=gu’,/0328;bI; l b 2-in2 f ;<=gqrb f c* k bjim,/cb y(” b6Łnc4<; 2-44<=g.19ˆ; l bC289280 ' ;<= k uˆbd;<c<= k ;*’qrbdc<;<= k bj4`*8eV2™qrbdc ' k *r28c4b6uˆbj4 l 289ni ; l bd9 cb‹Łn9nb; l buˆbj4 l 4*]; l 28;"; l buˆbd;<c<= k‹Œ & 2-4bji 28cbŽ2 f bdc;<c<=3289/.10hb^=h4•2 f/f c*j‘m=gu28;bd0 ' l 280gqrbji› ? l bd9 y ; l b•qrbdc<;<= k bj4"*8e ; l bˆ9nb ” uˆbj4 l 28cb^28.:28=g9‚2-44<=g.19nbjiU; l b•uˆbd;<c<= k e‰c*-u B –(,28;<=h*-9…ÙŽóm†‡b cb f bŽ28;; l =h4 f c* k bj44e“*-,/c;<=guˆbj4;* f c*(im, k b’2"uˆbj4 l ; l 28;=h462 f/f c*Ž‘G=gu28;bd0 ' 4=³‘G;bjbd9¬;<=guˆbj46uˆ*-cb cb‹Łn9nbjit; l bd9t; l b*-c<=g.1=g9280$uˆbj4 l 289ni ” bd0g07cb f cbj4bd9:;28;<=gqrb*8eE; l b•289280 ' ;<= k uˆbd;<c<= k  ? l buˆbj4 l bj4 f c*Gim, k bjie‰c*-uµ*-9nb˜=g;bdc28;<=h*-9I;*; l b’*-; l bdcL28cb™4 l * ” 9w=g9 T =g.1,/cb v1x  ? *i/bduˆ*-9n4<;<c28;b’; l 28;; l b î x æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå X Y 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ¼vIÁ43XBM`QEMPIZK53bOVFHQVGEFk†MPl´æ‘[gOVj ¼vr6 Ì[oGVQEMNlg[oOÀÁ X Y 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ¼§m@ÁdD”M`GEORQ87S[oºX†[gcd[gX_Q)¼+9+6éÌ[oGVÊ QEMNlg[gOÀÁ X Y 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ¼§l3Á ƒ [olgFXBi:7S[oºX†[gcd[gX_Q ¼vr9;ZÈdè:GEM`Ê IZXBYZKN[gOÀÁ X Y 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ¼viÁ]è|jBM`GÀi<7J[oºBXB[gcd[oXQq¼vÎ=fl),Ì[oGVÊ QEMNlg[gOÀÁ X Y 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 ¼§[eÁ2DBF^fGVQEj>7J[ˆºXB[ocd[gX_Q ¼ s3È_sHr Ì[oGVQEMNlg[gOÀÁ T =g.1,/cb v1x ý > 4–:,28cb]uˆbj4 l & bd=g9/.êcb k ,/c4<=gqrbd0 ' cb‹Łn9nbjiß2 kjk *-cim=g9/.fi;*U; l bNuˆbd;<c<= k i/b‹Łn9nbji &:' B –(,28;<=h*-9Uَóm îmÙ æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå X Y 0.5 0.51 0.52 0.53 0.54 0.55 0.04 0.06 ¼vIÁ?+ç@+]ÍS‡fMNO ƒ leIHKNMPX†Y X Y 0 0.5 1 1.5 0.03 0.04 0.05 0.06 ¼§mÁ?+ç@+.-9-dIH‡fMNOSOVlgIZKNMNXBY X Y 0.5 0.51 0.52 0.53 0.54 0.55 -0.5 0 0.5 1 1.5 ¼§l3Á?+.-9-fç@+àÍS‡fMNO ƒ lgIZKNMNXBY T =g.1,/cb v Ù1ý M bj4 l e‰c*-u T =g.1,/cb v1x b+q(=hb ” bji ” =g; l im=³¶›bdcbd9:;C2¥‘G=h44 k 280g=g9/.r4Ž î v æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå ¼vIÁflæ‘[gOVj•‹MNQEj‘Èfl-fl6àÌ[oGVQEMNlg[oO ¼§mÁ¿D”KNF3‹ ƒ k@[o[ei T =g.1,/cb v1v ý PC>6s`> x1x Ù vmy M 2 kl x  î y R b  î x1x1xmy َórõôXbdc<;<= k bj4 uˆbj4 l 2 kl =hbdqrbji‡; l b•i/bj4<=gcbji‚289/=h4*-;<c* f 't” b k 289šq(=hb ” ; l buˆbj4 l 28c*-,/9niš; l b f *-=g9(; (0.525, 0.1) ” =g; l im=³¶›bdcbd9:;\2¥‘m=h4$4 k 280g=g9/.r4Ž F ; =h4$* & q(=h*-,n4$=g9 T =g.1,/cb v Ù ; l 28;$; l b`uˆbj4 l•l 2-4X2 kl =hbdqrbjiˆ; l b`i/bj4=gcbji 289/=h4*-;<c* f '& b k 28,n4b+; l b+uˆbj4 l 2 f/f bŽ28c4=h4*-;<c* f = k ” l bd9 k bdc<;28=g9¬cbd.1=h*-9n4L28cb˜4 k 280hbjiw2 kjk *-cim=g9/.10 '  T *-cI2êuˆ*-cb f c2 k ;<= k 280’b‹‘/28u f 0hb y 289ni*-9nbš=g9(qr*-0gq(=g9/. k ,/c<qrbji & *-,/9nin28c<=hbj4 yL” b‡b‹‘n28u•=g9nbt0328u•= Œ 928cCè* ” 28c*-,/9nit; l b PC>6s`>Œ x1x Ù v 28=gcHe“*-=g0é ? * k *-u f ,/;b; l b4*-0g,/;<=h*-9š*-9šbŽ2 kl uˆbj4 l y” b",n4b•2 4b k *-9ni Œ *-ci/bdc y qrbdc<;b‹‘ Œ­k bd9(;bdcbji"Łn9/=g;b Œ qr*-0g,/uˆb`4*-0gqrbdcŽ ? l =h44*-0gqrbdc,n4bj4 0hbŽ2-4<; Œ 4–(,28cbj4 cb k *-9n4;<c<, k‹Œ ;<=h*-9 = v ó†? y R *(b ff 4+4 kl bduˆbô= í1î†? y 289ni P b ” ;*-9 ŒQ_+M]RLBED e“*-c˜c2 f =hi k *-9:qrbdc<.rbd9 k bô= v1x/y›v á@?­’ôL=h4 k *-,n4 ;bdc<uˆ4`28cb6im=h4 k c<=g;<=g¦Žbji^2-4\i/bj4 k c<= & bjiˆ=g9š= v ó?­ ? * k 280 k ,/0328;b6; l buˆbd;<c<= k & bd; ” bjbd92-in2 f ;28;<=h*-9^0h*G* f 4 y ” b,n4b; l bˆcb k *-9n4<;<c<, k ;<=h*-9fibdc<c*-c"uˆbd;<c<= kˆf c* f *14bji‚=g9 = ímٍ?­ ? l b=g9/=g;<=3280Euˆbj4 l y 4 l * ” 9U=g9 T =g. Œ ,/cb v1v 2 y =h4 k *-9n4<;<c<, k ;bjiš,n4<=g9/.¬; l bˆ=h4*-;<c* f = k uˆbj4 l .rbd9nbdc28;*-c™=g9 _+R JLMNM 5= î yv Ú yWv î†?­ ? l =h4 uˆbj4 l =h4* & q(=h*-,n40 ' =g0g0 Œ 4<,/=g;bjiU;* k *-u f ,/;<=g9/.tq(=h4 k *-,n4˜è* ” 28c*-,/9ni‚; l b^28=gcHe“*-=g0E4<=g9 k bcb‹Łn9nbduˆbd9:; =h4ˆi/*-9nb & 2-4bji *-9/0 ' *-9 & *-,/9nin28c ' k ,/c<q-28;<,/cb1 ? l b]2-4 ' u•uˆbd;<c ' *8eC; l bwcbj4<,/0g;<=g9/.‚4*-0g,/;<=h*-9…=g9 T =g.1,/cb v1v8& =h4•bdqG=hi/bd9 k bw*8e l * ” f *G*-c; l b¬uˆbj4 l =h4Ž > e£;bdc kl *(*14=g9/.‚2tuˆbd;<c<= k & 2-4bjifi*-9Z; l =h4 4*-0g,/;<=h*-9 yG” b k cbŽ28;b+2™9nb ” uˆbj4 l ; l 28;`4 l *-,/0hi & b & bd;<;bdc`4<,/=g;bji;*’; l bè* ” 4*-0g,/;<=h*-9W7†‡bcb f bŽ28; ; l =h4 f c* k bj44™e“*-c; l b; l cbjb=g9:;bdc<uˆbjim=328;buˆbj4 l bj4=g9 T =g.1,/cb v í]289ni‚; l b^4*-0g,/;<=h*-9‚=h4=gu f c*Žqrbji ” =g; l bŽ2 kl 4<, kjk bj44=gqrbˆcb‹Łn9nbduˆbd9:;Ž > e£;bdc"*-9/0 ' e“*-,/c=g;bdc28;<=h*-9n4*8eL289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9:; y ; l b 4*-0g,/;<=h*-9=h4 imc28u28;<= k 280g0 ' =gu f c*Žqrbji› ? l b6Łn9280uˆbj4 l 289ni4*-0g,/;<=h*-9w28cbL=g0g0g,n4;<c28;bji=g9 T =g.1,/cb v ím s 0h*14b6=g9n4 f b k ;<=h*-9^*8e›; l bC4<;28.1928;<=h*-9 f *-=g9:;E=g9 T =g.1,/cb v á4 l * ” 4E; l 28; k bd0g0bd0hbduˆbd9(;428cb64<u280g0289ni î1í æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå ¼vIÁ?+.-A+.=qÌ[oGVQEMNlg[gO ¼§m@ÁB;Zȍr9-qÌ[ˆGVQEMPlo[gO ¼§l3ÁflsC-D+er6LÌ[ˆGVQEMPlo[gO T =g.1,/cb v ímý F 9:;bdc<uˆbjim=328;b M bj4 l bj46289ni D *-0g,/;<=h*-9n4Žý PC>6s`> x1x Ù vmy M 2 kl x  î y R b  î x1x1x î8á æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå X Y 0 0.01 -0.005 0 0.005 0.01 0.015 T =g.1,/cb v áný s 0h*14b+, f qG=hb ” *8e7; l b™4<;28.1928;<=h*-9 f *-=g9:;L*8e; l bCŁn9nbj4<;6uˆbj4 l =g9 T =g.1,/cb v í•2m =h4*-;<c* f = k 28c*-,/9nit; l b4<;28.1928;<=h*-9 f *-=g9:;’289niN; l bd9t; l b ' –(,/= kì 0 'I& b k *-uˆb l =g. l 0 ' 289/=h4*-;<c* f = k 2-4 ; l b & *-,/9nin28c ' 032 ' bdc’i/bdqrbd0h* f 4Ž D =g9 k b•; l buˆbd;<c<= k =g9š; l =h4 k 2-4b=h4 & 2-4bjiš*-9š; l b A bj44<=3289š*8e ; l b è* ” q-28c<=32 & 0hbj4 y ; l =h4=h4b‹‘n2 k ;<0 ' ; l b+; ' f b™*8e7uˆbj4 l ; l 28;6*-9nb ” *-,/0hiwb‹‘ f b k ;Ž T *-cCbŽ2 kl uˆbj4 l 2-in2 f ;28;<=h*-9 y ; l bdcb’=h4C289]=g9/9nbdcC4bdc<=hbj4L*8e$cb‹Łn9nbduˆbd9:;™289ni k *r28c4bd9/=g9/.^0h*(* f 46; l 28; 28=gu ;*‡=gu f c*²qrbt; l b¬Łn9280Cuˆbj4 l  ? *‚=g0g0g,n4;<c28;bI; l bNb‹¶b k ;*8e™; l bj4bI=g9/9nbdc=g;bdc28;<=h*-9n4¬*-9 ; l b –(,280g=g; ' *8e; l buˆbj4 l y1” b l 2jqrbL4 l * ” 9ˆ; l b6im=h4<;<c<= & ,/;<=h*-9*8e; l b ;<c<=3289/.10hbj4 ” =g; l =g9; l b–:,280g=g; ' 4 f 2 k b =g9 T =g.1,/cb v î™e“*-cE; l bC=g9:;bdc<uˆbjim=328;b+uˆbj4 l^k cbŽ28;bji=g9 T =g.1,/cb v í12m F ;\=h4 * & q(=h*-,n4`e‰c*-u ; l b kl 289/.rb & bd; ” bjbd9Nim=h4<;<c<= & ,/;<=h*-9n4+2ˆ289ni & =g9 T =g.1,/cb v î; l 28;L; l b˜=g9/=g;<=3280Vcb‹Łn9nbduˆbd9(;C4;b f =h4qrbdc ' b‹¶›b k ;<=gqrb 28; k *-9 k bd9(;<c28;<=g9/.";<c<=3289/.10hbj4 ” =g; l =g9ˆ; l b; l bi/bj4=gcbjiˆ28cbŽ2m J 9me“*-c<;<,/928;bd0 'ry =g;\280h4* f c*(im, k bj4\u289 ' 4<u280g0 bjim.rbj4•289ni‚4*-uˆb f *(*-c•289/.10hbj4Ž > e‰;bdc k *r28c4bd9/=g9/.N=h4 f bdcHe“*-c<uˆbjiê=g9 T =g.1,/cb v î k y uˆ*14<;*8e ; l b";<c<=3289/.10hbj4 ” =g; l 4<u280g0$bjim.rbj4˜28cbcbduˆ*²qrbji & ,/;˜4*-uˆb’;<c<=3289/.10hbj4 ” =g; l qrbdc ' 0328c<.rb k =gc k ,/u•c2-im=g= 28cb k cbŽ28;bji› ? l *14b"4<u280g07bjim.rbj4C; l 28;Li/*cbdu28=g9š28cb"280g0 0h* k 28;bji]=g9Icbd.1=h*-9n4+*8e l =g. l & *-,/9nin28c ' k ,/c<q128;<,/cb1 ? l b¬9nb‹‘m;•cb‹Łn9nbduˆbd9:;¬28.:28=g9Zbd0g=gu•=g928;bj4ˆ; l bw0328c<.rb k =gc k ,/u•c2-im=g=6;<c<=3289/.10hbj4 ” =g; l *-,/; k cbŽ28;<=g9/.tu289 ' 4<u280g0`bjim.rbj4² ? l bŁn9280E; ” * f 2-44bj4"*8e k *r28c4bd9/=g9/.‡289ni‚cb‹Łn9nbduˆbd9:; l 2jqrb^qrbdc ' 0g=g;<;<0hb9nbd;L=gu f 2 k ;+*-9N; l b’–(,280g=g; ' *8eX; l buˆbj4 l † l =g0hb; l b˜Łn9280uˆbj4 lNk *-9(;28=g9n4˜4*-uˆb™;<c<=3289/.10hbj4 î1î æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå  0  5  10  15  20  25  0  5  10  15  20  25 η ξ Triangles ¼vIÁE3bXBM`QEMPIZK<æ‘[gOVj  0  0.2  0.4  0.6  0.8  1  1.2  1.4  0  0.2  0.4  0.6  0.8  1  1.2  1.4 η ξ Triangles ¼§mÁ¿D”M`GEORQF7J[oºBXB[gcd[oXQ|äUIZOVO  0  1  2  3  4  5  6  7  0  1  2  3  4  5  6  7 η ξ Triangles ¼§l3Á D”M`GEORQ όFIHGEOV[gX†MNXBY/äUIZOVO  0  0.2  0.4  0.6  0.8  1  1.2  1.4  0  0.2  0.4  0.6  0.8  1  1.2  1.4 η ξ Triangles ¼viÁ ƒ [glgFXBi:7S[oºX†[gcd[gX_Q’äUIHOVO  0  0.5  1  1.5  2  0  0.5  1  1.5  2 η ξ Triangles ¼§[3Á ƒ [glgFXBi¹ÏŒFI3GEOV[gXBMNX†Y/äUIZOVO  0  0.2  0.4  0.6  0.8  1  1.2  1.4  0  0.2  0.4  0.6  0.8  1  1.2  1.4 η ξ Triangles ¼TEÁ D”MPXBIZKG7J[oºBXB[gcd[oXQ|äUIZOVO T =g.1,/cb v îmý D 28u f 0hb [ ,280g=g; ' )C=h4<;<c<= & ,/;<=h*-9I)C,/c<=g9/. R b‹Łn9nbduˆbd9:;L289ni s *r28c4bd9/=g9/. î1Ú æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå *-,/;4<=hi/b™; l b˜i/bj4<=gcbji & *-,/9ni/4 y ; l =h4=h4=g9nbdqG=g;2 & 0hb+e“*-cL2".rbd9nbdc280W.rbj*-uˆbd;<c ' 289niuˆbd;<c<= k  * "¿.3-"ß"50™-™	-fl ¡ +„
" Hfl]+ºfl$6	 †‡b l 2jqrb f cbj4bd9(;bji 2Iuˆbd; l *Gi‡e“*-c"289/=h4*-;<c* f = k uˆbj4 l cb‹Łn9nbduˆbd9:;•; l 28; f c*Gim, k bj4 l =g. l –(,280g=g; ' uˆbj4 l bj4 ” =g; l *-,/; b‹‘ f bd9n4<=gqrb0h* k 280* f ;<=gu•=g¦²28;<=h*-9^c*-,/;<=g9nbj4²7†‡b l 2jqrb+i/*-9nb6; l =h4 &(' 2-44<=g.19/=g9/.uˆbd; Œ c<= k 4;*"qrbdc<;<= k bj4C289ni; l bd9wi/b‹Łn9/=g9/.ˆ;<c<=3289/.10hb™–(,280g=g; '^& 2-4bji¬*-9I2 f/f c*j‘m=gu28;b™0hbd9/.1; l 4Ž ? l b+; l cbjb uˆbj4 l uˆ*Gim=³Ł k 28;<=h*-9ê;*G*-0h4 ” b,n4bji‚28cb^4 ” 2 f/f =g9/.t;*]u2¥‘G=gu•=g¦Žb^–(,280g=g; 'ry =g9n4bdc<;<=g9/.š28;•2 f/f c*j‘m= Œ u28;b k =gc k ,/u k bd9(;bdc4 ;*™i/b k cbŽ2-4b k =gc k ,/u•c2-im=g= y 289ni•cbduˆ*²q(=g9/."qrbdc<;<= k bj4X;*’bd0g=gu•=g928;b64<u280g0bjim.rbj4Ž K b k 28,n4b; l bdcbw28cb^9n*t.1,28c289:;bjbj4*-9 ; l bcbj4<,/0g;4•*8eL; l bj4buˆ*Gim=³Ł k 28;<=h*-9 ;*G*-0h4 y$” b,n4b^; l bdu =g;bdc28;<=gqrbd0 ' ;* f c*Gim, k bˆ289‡* f ;<=gu280Xuˆbj4 l "†‡b l 2jqrb f cbj4bd9:;bjiUb‹‘n28u f 0hbj4™i/bduˆ*-9n4<;<c28;<=g9/.I; l 28; *-,/c’280g.r*-c<=g; l u f c*(im, k bj4˜uˆbj4 l bj4˜; l 28;’28cb ” bd0g0E2-in2 f ;bji‡;*; l b f cbj4 k c<= & bji‡uˆbd;<c<= k † l bd9 ” b k *-u & =g9nb; l =h4+289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 ” =g; l 289tb‹¶b k ;<=gqrb"uˆbd;<c<= k bj4<;<=gu28;*-c y” b k 289Ncb k ,/c4<=gqrbd0 ' =gu f c*²qrb s`T )!4*-0g,/;<=h*-9n4 ” =g; l *-,/; ; l bLbd9n*-c<uˆ*-,n4=g9 k cbŽ2-4b6=g9 k *-u f ,/;28;<=h*-9 k *14<;2-44* k =328;bji ” =g; l ,/9/=³e“*-c<u cb‹Łn9nbduˆbd9:;Ž ? l bdcbw28cb¬280h4*t4bdqrbdc280 =gu f c*Žqrbduˆbd9(;4;*t*-,/cˆ280g.r*-c<=g; l u ; l 28; k *-,/0hi & b^=gu f 0hbduˆbd9(;bji =g9 ; l b e‰,/;<,/cb1 T *-cb‹‘n28u f 0hb ym” b l 2jqrb˜4;28;bji; l 28;0h* k 280uˆbj4 l * f ;<=gu•=g¦²28;<=h*-9w=h4 .rbd9nbdc280g0 ' 289b‹‘ f bd9n4<=gqrb f c* k bjim,/cb1 A * ” bdqrbdc y b‹‘ f bdc<=hbd9 k b ” =g; l =h4*-;<c* f = k uˆbj4 l =g9/.š=g9 _˜R`JCMNM 5 l 2-4ˆ4 l * ” 9 ; l 28;•=³e ; l b"uˆbj4 l =h4™280gcbŽ2-i ' *8e l =g. l –:,280g=g; 'ry 0h* k 280$* f ;<=gu•=g¦²28;<=h*-9 Œ & 2-4bjiU4<uˆ*(*-; l =g9/. k 289 & b"2eé2-4<;˜289ni b‹¶›b k ;<=gqrb ” 2 ' *8ee‰,/c<; l bdc’=gu f c*²q(=g9/.t; l buˆbj4 l =gÙdá?­ F ;=h4’; l bdcb‹e“*-cb*-,/c & bd0g=hb‹e; l 28;’; l buˆ*14<; b k =hbd9(; ” 2 ' *8e k cbŽ28;<=g9/. l =g. lmŒ –(,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l bj4 ” *-,/0hi & bw;*U=g9n4bdc<;¬289ni cbduˆ*Žqrb qrbdc<;<= k bj4™2 kjk *-cim=g9/.¬;*^*-,/c™280g.r*-c<=g; l u—289ni]; l bd9 f bdcHe‰*-c<u´20g=gu•=g;bjiN9:,/u & bdc™*8e f 2-44bj4C*²qrbdc˜; l b uˆbj4 l ,n4<=g9/."0h* k 280* f ;<=gu•=g¦²28;<=h*-9W D =g9 k bL; l b6uˆbj4 l =h4`280gcbŽ2-i ' *8e l =g. l –(,280g=g; 'ry u289 ' e‰b ” bdc f 2-44bj4 *²qrbdc`; l bLuˆbj4 l ” *-,/0hi & b6cbj–(,/=gcbji^;*’u2¥‘m=gu•=g¦ŽbC; l bL–(,280g=g; ' ; l bd9=³eW0h* k 280* f ;<=gu•=g¦²28;<=h*-9 ” 2-4 ; l b f c<=g9 k = f 0hb’;*(*-0e“*-cuˆbj4 l =gu f c*²qrbduˆbd9:;Ž > 9n*-; l bdc f *-;bd9(;<=3280=gu f c*Žqrbduˆbd9(;ˆ;*t*-,/c280g.r*-c<=g; l u =h42]uˆ*Gim=³Ł k 28;<=h*-9 *8eL; l bi/bj4<=gcbji –(,280g=g; ' & *-,/9ni/4˜=g9‚28cbŽ2-4’9nbŽ28c & *-,/9nin28c<=hbj4 ” =g; l 4<u280g0$e“bŽ28;<,/cbj4Ž ? l bˆ=h4*-;<c* f = k 2 f/f c*r2 kl =h4˜;*I2-44<=g.19 0hbd9/.1; l 4 k 280hbj4 & 2-4bji *-9…; l b & *-,/9nin28c ' .rbj*-uˆbd;<c ' = í1ó?­ > 4<=gu•=g0328cw2 f/f c*r2 kl u•=g. l ; ” *-c ì e“*-c îrõ æ ¦¥< ä ¥
/¥ çdæ_9 ¢ ä  çdã:å  ãZ² R;‘³ æç ® æ/äæ ª æäå 289/=h4*-;<c* f = k uˆbj4 l =g9/. &:' i/b k cbŽ2-4<=g9/.]; l b28cbŽ2w*8e; l bi/bj4<=gcbjiUbj–(,/=g0328;bdc280E;<c<=3289/.10hb,n4bji‡=g9‚; l b k 280 k ,/0328;<=h*-9]e‰*-c (ξ i/bj4 , η i/bj4 ) =g9 T =g.1,/cbÙŽîm ? l =h4 ” *-,/0hiI0g= ì bd0 ' cbjim, k b; l b™9(,/u & bdcC*8eX=g9n4<;289 k bj4 ” l bdcb’qrbdc<;b‹‘Icbduˆ*Žq1280 k cbŽ28;bj4 f *(*-c6–(,280g=g; ' ;<c<=3289/.10hbj4C289niI; l ,n4C=g;L4 l *-,/0hiIcbjim, k b’; l b’9nbjbji¬e“*-c cb f bŽ28;bji¬qrbdc<;b‹‘¬=g9n4bdc<;<=h*-9]289ni¬cbduˆ*²q-280V=g;bdc28;<=h*-9n4Ž T =g9280g0 'ry *-,/cEuˆbj4 l =g9/.280g.r*-c<=g; l u f c*Gim, k bj4 4*-uˆb64<u280g0›289/.10hbC;<c<=3289/.10hbj4`; l 28; k *-,/0hi & b=gu f c*Žqrbji &(' e£,/c<; l bdcC=g9n4bdc<;<=h*-9W ? l =h4 ” *-,/0hiNcbj4<,/0g;C=g9N4u280g0hbdcL;<c<=3289/.10hbj4C; l 289ši/bj4<=gcbji y& ,/;L; l b=gu f c*Žqrb Œ uˆbd9(;"=g9‚;<c<=3289/.10hb–:,280g=g; ' u•=g. l ; & b ” *-c<; l ; l b^b‹‘G;<c2 k *-u f ,/;28;<=h*-9 k *14<;•2-44* k =328;b ” =g; l uˆ*-cb ;<c<=3289/.10hbj4Ž > Łn9280 f 2-44•*8e˜–:,280g=g; ' Œ & 2-4bji cb‹Łn9nbduˆbd9(;4bjbduˆ42N0g= ì bd0 '‚” 2 ' *8e˜2-i/imcbj44=g9/.š; l =h4 f c* & 0hbduw > 0g; l *-,/. l u289 ' ; ” *Cim=guˆbd9n4<=h*-9280(uˆbj4 l =g9/.C280g.r*-c<=g; l uˆ4 f c* k 0328=gu@24<;<c28=g. l ;He‰*-c ” 28cib‹‘G;bd9n4<=h*-9;* ; l cbjb im=guˆbd9n4<=h*-9n4 y b‹‘ f bdc<=hbd9 k bE=g9"; l cbjb Œ im=guˆbd9n4<=h*-9280n=h4*-;<c* f = k uˆbj4 l =g9/.+4 l * ” 4; l 28;7; l =h4 =h4c28cbd0 ' ; l b k 2-4b1 A * ” bdqrbdc y 4<=g9 k b*-,/cˆ280g.r*-c<=g; l u =h4 & 2-4bji‚*-9 289 l bd,/c<=h4<;<= k 2-in2 f ;28;<=h*-9fi;*t)+bd0328,/92 ' uˆbj4 l =g9/. y =g; u•=g. l ; & b f *144<= & 0hbC;*•2 f/f 0 ' ; l b˜428uˆbC.rbd9nbdc280 f c* k bjim,/cb+=g9^; l cbjb˜im=guˆbd9n4=h*-9n4289ni 4<;<=g0g0 f c*Gim, k b™–(,280g=g; ' uˆbj4 l bj4Ž ? l =h4 ” *-,/0hi & b™289¬=hi/bŽ280›e“* k ,n4*8e e‰,/;<,/cb ” *-c ì  î z · ¸º¹ ƒßü  } ø ¸º¹ } I -)+–+º6 F 9*-ci/bdc ;*=gu f c*ŽqrbC; l bC2 kjk ,/c2 k ' *8e s`T ) 4*-0g,/;<=h*-9n4`cbd0328;<=gqrbC;*’; l bL;<=guˆbLcbj–:,/=gcbji^;*.rbd9nbdc28;b ; l *14b4*-0g,/;<=h*-9n4 y1” b l 2Žqrbi/bdqrbd0h* f bji26uˆbd; l *Gi•*8ecb‹Łn9/=g9/.’uˆbj4 l bj47;*˜bj–:,/=him=h4;<c<= & ,/;bbdc<c*-cŽ ? l =h4 uˆbd; l *GiU=g9(qr*-0gqrbj4"; ” *t4b f 28c28;b & ,/;bj44bd9:;<=3280 4<;b f 4Žý k cbŽ28;<=g9/.š2wuˆbd;<c<= k o sEl 2 f ;bdcˆp’289niU; l bd9 uˆbj4 l =g9/.; l b˜.rbj*-uˆbd;<c '& 2-4bji¬*-9w; l 28;uˆbd;<c<= k o sEl 2 f ;bdc˜p‹ ? * k cbŽ28;b^; l b^uˆbd;<c<= k *-9 ” l = kl ; l bcb‹Łn9nbduˆbd9(;ˆ=h4 & 2-4bji y$” bŁnc4<;bj4<;<=gu28;b^; l bbdc<c*-c & 2-4bji *-9‡0h* k 280\cb k *-9n4;<c<, k ;<=h*-9W ? l 28;’=h4 yV” b2-44<,/uˆb•; l 28;’; l bbdc<c*-c"28;’bŽ2 kl qrbdc<;b‹‘‡=h4bj–:,280X;*I; l b l =g. l bdc Œ *-ci/bdc;bdc<uˆ4*8e; l b ? 2 ' 0h*-c"b‹‘ f 289n4<=h*-9W ? l =h4’cbj–(,/=gcbj4"cb k *-9n4<;<c<, k ;<=g9/.t; l b,/9 Œ cbj4*-0gqrbji i/bdc<=gq128;<=gqrbj4•289ni‚; l bd9 k *-u & =g9/=g9/.š; l bdu 2 kjk *-cim=g9/.t;*]4*-uˆb9n*-c<u *8e; l bq-28c<=32 & 0hbj4Ž]†‡b; l bd9 bdq1280g,28;b^; l =h4•bdc<c*-cbj4;<=gu28;bw280h*-9/.š2],/9/=g; k =gc k 0hbw289ni k 280 k ,/0328;b; l bŁnc4<;; l cbjb^bdqrbd9 T *-,/c<=hbdc k *Gb k =hbd9:;4Ž D =g9 k b ” b k 289 kl *(*14bu28;<c<=³‘êq1280g,nbj4•4<, kl ; l 28;•; l buˆbd;<c<= k ” =g0g0 l 2Žqrb; l b¬428uˆb T *-,/c<=hbdc k *(b k =hbd9(;46*-,/cuˆbd;<c<= k ” =g0g0 & b™2".r*G*Giw2 f/f c*j‘m=gu28;<=h*-9];*; l b™bdc<c*-c6uˆbŽ2-4<,/cb1 Y 9 k b`; l bEuˆbd;<c<= k l 2-4 & bjbd9 kl *14bd9"e“*-cbŽ2 kl qrbdc<;b‹‘ y-” bE; l bd9cb‹Łn9nb *-,/c7uˆbj4 l 4*C2-4;* & bE,/9/=³e“*-c<u 2 kjk *-cim=g9/.;*ˆ; l b˜uˆbd;<c<= k  > 0g0V*8eX*-,/cuˆbj4 l =g9/.* f bdc28;<=h*-9n4+28cb & 2-4bjiI*-9N2•i/b‹Łn9/=g;<=h*-9N*8e$–(,280g=g; ' ; l 28;X=g9(qr*-0gqrbj4 & *-; l ; l b k =gc k ,/u•c2-im=g,n4`289ni•u•=g9/=gu’,/ubjim.rb0hbd9/.1; l 2 kjk *-cim=g9/.™;*™; l buˆbd;<c<= k  ? l b uˆ*14<;$4<=gu f 0hb* f bdc28;<=h*-9 y 4 ” 2 f/f =g9/. y =h4$i/*-9nb =³e=g;7=g9 k cbŽ2-4bj4$; l b–(,280g=g; ' *8e; l b ” *-c4<;\;<c<=3289/.10hb1 T *-c ;<c<=3289/.10hbj4`; l 28; l 2ŽqrbC0328c<.rb k =gc k ,/u•c2-im=g= ym” bL=g9n4bdc<;Eqrbdc<;<= k bj4 28; 2 f/f c*j‘m=gu28;b k =gc k ,/u k bd9:;bdc4Ž ? l =h4 =h4qrbdc ' 4<=gu•=g0328c";*I; l b)+bd0328,/92 ' 2 f/f c*r2 kl e“*-c=h4*-;<c* f = k uˆbj4 l =g9/.n T *-c;<c<=3289/.10hbj4 ” =g; l 4<u280g0 bjim.rbj4 y” b’cbduˆ*Žqrb•2•qrbdc<;b‹‘¬e‰c*-uµ; l 28;Cbjim.rb y ; l bdcb &(' bd0g=gu•=g928;<=g9/.¬4<u280g0 bjim.rbj4² K ' cb f bŽ28;bjim0 ' =g9n4bdc<;<=g9/.t289nišcbduˆ*ŽqG=g9/.Iqrbdc<;<= k bj4 yV” b28cb2 & 0hb•;* k cbŽ28;b l =g. lmŒ –(,280g=g; ' uˆbj4 l bj4 ” =g; l *-,/; k *14<;<0 ' u•=g9/=gu•=g¦²28;<=h*-9]c*-,/;<=g9nbj4Ž î1ó â\ãŽä ;3¦E ç  ãŽänç {@fl$-).H	 K ' b‹‘n28u•=g9/=g9/.•; l bLè* ” 28c*-,/9ni; l b PC>6s`> x1x Ù v 28=gcHe“*-=g0 ” =g; l 4bdqrbdc280Wim=³¶bdcbd9(; f 28c28uˆbd;bdc4 y/” b ” bdcb2 & 0hb;*^i/bduˆ*-9n4<;<c28;b; l b & bd9nb‹Łn;4+*8eE289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9te‰*-c & *-; l 4b k *-9nit289ni]; l =gci *-ci/bdc64*-0g,/;<=h*-9n4Ž F 9…280g0; l b k 2-4bj4b‹‘n28u•=g9nbji y ; l b & bd9nb‹Łn;*8e™2-in2 f ;<=gqrbIuˆbj4 l =g9/.‚*²qrbdc,/9/=³e“*-c<u#cb‹Łn9nbduˆbd9:; ” 2-4 * & q(=h*-,n4Ž ? l 28; ff 4 & b k 28,n4b’bdqrbd9we“*-c; l b+Łn9nbj4<;L,/9/=³e‰*-c<u uˆbj4 l bj4; l 28; ” b™,n4bji y 4*-uˆb’*8e7; l b˜è* ” e“bŽ28;<,/cbj428cb6=g9n4<,” k =hbd9(;<0 ' cbj4*-0gqrbji› F 9 f 28c<;<= k ,/0328c y ; l b & *-,/9nin28c ' im=h4 k cbd;<=g¦²28;<=h*-9; l 28; ” b kl *14b cbj–(,/=gcbj4 qrbdc ' 0g=g;<;<0hb+cb‹Łn9nbduˆbd9(;L28; ; l b+;<c28=g0g=g9/.ˆbjim.rb+cbd0328;<=gqrb˜;*"; l b+0hbŽ2-im=g9/.ˆbjim.rb˜*8e ; l b˜28=gcHe“*-=g0é D =g9 k b6cbj4*-0g,/;<=h*-928;\; l b;<c28=g0g=g9/.bjim.rb=h4X=gu f *-c<;289:;`;*™uˆ*14;$è* ” 4*-0g,/;<=h*-9n4 y ; l buˆbj4 l bj4\; l 28; ” b k cbŽ28;bji &(' ,/9/=³e‰*-c<u cb‹Łn9nbduˆbd9(;+28cb™=g0g0W4<,/=g;bji¬;*•; l b˜cbj4,/0g;<=g9/.•è* ” 4Ž D =gu•=g0328c<0 'ryn&(' kl *(*14=g9/.2 0328c<.rbCeé28cHŁbd0hi & *-,/9nin28c 'ry *-,/c,/9/=³e‰*-c<u•0 ' 2-in2 f ;bji¬uˆbj4 l¬l 2-iu289 ' k bd0g0h4=g9I28cbŽ2-4 ” =g; l q(=gc<;<,280g0 ' k *-9n4<;289(;L4*-0g,/;<=h*-9W K *-; l *8e$; l bj4b f c* & 0hbduˆ4+28cb˜cbj4*-0gqrbji &:' cb‹Łn9/=g9/. & 2-4bjiw*-9wcb k *-9n4<;<c<, k ;<=h*-9 bdc<c*-cŽ Y ,/c¬2-in2 f ;<=gqrbNcb‹Łn9nbduˆbd9:;w280g.r*-c<=g; l u =h4i/bj4<=g.19nbji…;*êbj–:,/=him=h4<;<c<= & ,/;btbdc<c*-c & bd; ” bjbd9 k bd0g0h4 & ,/; 280h4* ” =g; l =g9 bŽ2 kl k bd0g0é ? l =h4=h4•2 kjk *-u f 0g=h4 l bji &(' kl *(*14<=g9/.‡2]uˆbd;<c<= k ; l 28;•=h4•289/=h4*-;<c* f = k 289ni e‰c*-u#; l 28; k cbŽ28;<=g9/.‚289/=h4*-;<c* f = kIk bd0g0h4Ž T *-cˆ=g9:qG=h4 k =hi è* ” 4 y uˆ*14<;bdc<c*-cˆ=h4•=h4*-;<c* f = k 289nifi; l ,n4 280g0h* ” =g9/.še‰*-cˆ289/=h4*-;<c* f ' l 2-i‚9n*t9n*-;<= k bŽ2 & 0hbIb‹¶b k ;ˆ*-9fi4*-0g,/;<=h*-9Z2 kjk ,/c2 k '  T *-c•qG=h4 k *-,n4"è* ” 4 y l * ” bdqrbdc y ; l b f cbj4bd9 k bL*8e & *-,/9nin28c ' 032 ' bdc4`0hbŽ2-i/4E;*b‹‘m;<cbduˆbd0 ' 289/=h4*-;<c* f = k bdc<c*-c4Ž K ' k *-u f 28c Œ =g9/.I289/=h4*-;<c* f = k cb‹Łn9nbduˆbd9:; ” =g; l =h4*-;<c* f = k cb‹Łn9nbduˆbd9:;™e“*-c˜; l bj4b"è* ” 4 y›” b ” bdcb2 & 0hb;*¬4 l * ” ; l 28;C,n4<=g9/.¬289/=h4*-;<c* f = kk bd0g0h4 k 289timc28u28;<= k 280g0 ' =g9 k cbŽ2-4b4*-0g,/;<=h*-9šb k =hbd9 k '  ? l =h4C=h4Lbj4 f b k =3280g0 ' =gu f *-c<;289:; ” l bd9 k *-9n4<=hi/bdc<=g9/.ˆ; l b™4b f 28c28;<=h*-9 f *-=g9:;Ž K b k 28,n4b ; l b cb k *-9n4<;<c<, k ;<=h*-9bdc<c*-c$=h479nb k bj4428c<=g0 ' im=³¶›bdcbd9:;Ei/b f bd9nim=g9/.’*-9; l b*-ci/bdcX*8e; l b cb k *-9 Œ 4<;<c<, k ;<=h*-9 yG” b l ' f *-; l bj4<=g¦Žbji; l 28; l =g. lmŒ *-ci/bdc 4*-0g,/;<=h*-9n4Ecbj–(,/=gcb˜2-in2 f ;28;<=h*-9 & 2-4bji*-9 l =g. l bdc*-c Œ i/bdc`i/bdc<=gq-28;<=gqrbj4Ž A * ” bdqrbdc y *-,/c\cbj4<,/0g;4\e‰*-c\; l =gci*-ci/bdc 4*-0g,/;<=h*-9n4E4 l * ” bji; l 28;\cb‹Łn9nbduˆbd9(; & 2-4bji *-9; l =gcii/bdc<=gq-28;<=gqrbj4 =h4`9n*-; k *-9:qG=g9 k =g9/.10 '& bd;<;bdc ; l 289cb‹Łn9nbduˆbd9:; & 2-4bji*-94b k *-9nii/bdc<=gq-28;<=gqrbj4² †‡b & bd0g=hbdqrb; l =h4™=h4’im,nb;*¬; l b•eé2 k ;’; l 28;cbd.1=h*-9n4 ” =g; l‚l =g. l ; l =gci‚i/bdc<=gq-28;<=gqrbj4=g9‡.rbd9nbdc280280h4* l 2Žqrb l =g. l 4b k *-9nii/bdc<=gq128;<=gqrbj4Ž K bd; ” bjbd9wbdc<c*-c4 =g9cb k *-9n4<;<c<, k ;<=g9/.•; l b+i/bdc<=gq128;<=gqrbj4289ni¬2 f/f c*j‘m= Œ Ú x â\ãŽä ;3¦E ç  ãŽänç u28;<=h*-9n4u2-i/b+=g9 k cbŽ28;<=g9/.ˆ; l bCuˆbj4 l bj4 y ; l b˜im=³¶bdcbd9 k b & bd; ” bjbd9w; l b˜im=³¶bdcbd9(;*-ci/bdci/bdc<=gq-28;<=gqrbj4 2 f/f bŽ28c4\;* & b6=g9n4<=g.19/=³Ł k 289:;Ž\† l =g0hb6; l bdcbLb‹‘m=h4<;E;<c<=gq(=3280b‹‘/28u f 0hbj4 y 4<, kl 2-4E; l b64<=g9nb ” 2Žqrb yG” l bdcb ; l =gci *-ci/bdc4*-0g,/;<=h*-9n4•4 l *-,/0hi‚9n*-; & bcb‹Łn9nbji & 2-4bjiê*-9fi4b k *-9niê*-ci/bdci/bdc<=gq128;<=gqrbj4Ž Y ,/c k *-u Œ f 28c<=h4*-9]*8e uˆbd;<c<= k 4 & 2-4bji¬*-9w4b k *-9niI289ni; l =gciw*-ci/bdc6i/bdc<=gq128;<=gqrbj4e“*-c; l =gci¬*-ci/bdc64*-0g,/;<=h*-9n4C28cb =g9 k *-9 k 0g,n4<=gqrb1 T *-cCbdqrbdc ' uˆbj4 l ; l 28; ” b2-in2 f ;bji y *-,/cC280g.r*-c<=g; l u ” 2-4C2 & 0hb™;* k cbŽ28;b l =g. l –(,280g=g; ' 289/=h4*-;<c* f = Œ k 280g0 ' 2-in2 f ;bjiuˆbj4 l bj4Ž ? l =h4 ” 2-4Ei/*-9nb6qrbdc ' –(,/= kì 0 '•& b k 28,n4b; l b6*-9/0 ' ; ” *uˆbj4 l * f bdc28;<=h*-9n4 ” b ,n4bji ” bdcb62-i/im=g9/.˜qrbdc<;<= k bj4$=g9ˆ4*-uˆb cbd.1=h*-9n4\289nii/bd0hbd;<=g9/.’4*-uˆb =g9*-; l bdc4² ? l =h4X=h47u’, klkl bŽ2 f bdc k *-u f ,/;28;<=h*-9280g0 ' ; l 289š; l b0h* k 280X–:,280g=g; ' u2¥‘G=gu•=g¦²28;<=h*-9‚280g.r*-c<=g; l uˆ4’; l 28;’28cbuˆ*-cb k *-u•uˆ*-9/0 ' 2 f/f 0g=hbji› {@fl$ •+–+ºfl$" 	6 †‡b k 0328=gu ;* l 2jqrb˜i/bdqrbd0h* f bji2uˆbd; l *Gi*8e289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9w; l 28; k 289 & b+2 f/f 0g=hbji;*289 ' ? 2 ' 0h*-c Œ & 2-4bji s`T ) 280g.r*-c<=g; l u y l * ” bdqrbdc y7” b l 2Žqrbˆ*-9/0 ' ;bj4<;bji‡; l =h42 f/f c*r2 kl e‰*-c™; ” *]im=guˆbd9 Œ 4<=h*-9280Łn9/=g;bqr*-0g,/uˆb y k *-u f cbj44<= & 0hbè* ” 28c*-,/9niˆ; l b PC>6s`>6Œ x1x Ù v 28=gcHe“*-=g0é K b‹e“*-cb; l b.rbd9nbdc280g=g; ' *8e™*-,/c^2 f/f c*r2 kl k 289 & b f c* f bdc<0 ' i/bduˆ*-9n4<;<c28;bji y ; l bdcb]cbdu28=g9 2š.:28u’,/;*8e˜4=gu’,/0328;<=h*-9n4^;* & b f bdcHe“*-c<uˆbji› ? l =h4 ” *-,/0hiš=g9 k 0g,ni/b,n4<=g9/.w*-; l bdc™4*-0gqrbdc4 y *-; l bdc™5X) B 4’289nituˆ*-cb k *-u f 0g= k 28;bji .rbj*-uˆbd;<c<=hbj4Ž B qrbd9(;<,280g0 'ry ; l b+.r*r280W*8e7; l =h4 ” *-c ì =h4 ;*•4 l * ” ; l 28;62 kjk ,/c28;b™4*-0g,/;<=h*-9n4 k 289 & bC.rbd9nbdc28;bjieé2-4<;bdc ,n4<=g9/.uˆbj4 l 2-in2 f ;28;<=h*-9^c28; l bdc`; l 289 k *-u f ,/;<=g9/.•*-9^2˜4=g9/.10hbŁn9nbLuˆbj4 l y 9n*–:,289(;<=g;28;<=gqrbCim=h4 k ,n4 Œ 4<=h*-9]*8e k *-u f ,/;28;<=h*-9N;<=guˆb˜=h46.1=gqrbd9I=g9w; l =h46; l bj4<=h4Ž ? l =h46=h4 & b k 28,n4b+eé2 k ;*-c464<, kl 2-4* f ;<=gu•=g¦j=g9/. ; l b 2-in2 f ;<=gqrb k ' k 0hb 289ni’; l b k *(i/b\=g;4bd0³ecbj–(,/=gcbXe‰,/c<; l bdc$289280 ' 4<=h4² F 9 f 28c<;<= k ,/0328c y ; l bEuˆbd;<c<= k 4,n4bji ; l c*-,/. l *-,/;˜; l b ” *-c ì ” bdcb k *-u f ,/;bjit,n4<=g9/.2ˆe‰,/0g0 ' k *-9:qrbdc<.rbjiš4*-0g,/;<=h*-9W > =g; Œ­> 0g= Œ fiE2 l =32^bd;˜280é = v ?Wb‹‘ f 0g= k =g;<0 ' 42 ' ; l 28; e‰,/0g0 k *-9:qrbdc<.rbd9 k b˜=h4 9n*-; cbj–(,/=gcbji;*.rbd9nbdc28;b™=gu f c*Žqrbji¬uˆbj4 l bj4² ? l bdcb Œ e“*-cb y u, kltk *-u f ,/;28;<=h*-9];<=guˆb k *-,/0hi & b™42Žqrbji &:' i/bd;bdc<u•=g9/=g9/. ” l bd9N24<,” k =hbd9:; k *-9:qrbdc<.rbd9 k b =h4cbŽ2 kl bjiêe‰*-c k cbŽ28;<=g9/.š; l buˆbd;<c<= k  > 9n*-; l bdc•=h44<,nb ” =g; lfik *-u f ,/;28;<=h*-9 ;<=guˆb=h4; l 28; k ,/c<cbd9:; =g9 k *-u f 28;<= & =g0g=g; 'N& bd; ” bjbd9‡; l b4*-0gqrbdc™289niN; l buˆbj4 l .rbd9nbdc28;*-c f cbdqrbd9(;4™im=gcb k ; & =g; Œ ” =h4b•* & ë b k ; ÚmÙ â\ãŽä ;3¦E ç  ãŽänç ;<c289n4He“bdc˜e‰c*-u—*-9nb f c*-.1c28u ;*w; l bˆ*-; l bdcŽ F 9n4<;bŽ2-i yW” b28cbcbj–:,/=gcbjiU;*¬cb f bŽ28;bjim0 ' cbŽ2-i‚289ni ” c<=g;b & *-; l ; l b+uˆbj4 l 289niuˆbd;<c<= k & bd; ” bjbd9 f c*-.1c28uˆ4Ž A 2jqG=g9/.•uˆbj4 l uˆ*(im=³Ł k 28;<=h*-9]i/*-9nb ” =g; l =g9 ; l b4*-0gqrbdc ” *-,/0hiN4<, & 4<;289(;<=3280g0 ' i/b k cbŽ2-4b k *-u f ,/;28;<=h*-9t;<=guˆb1 T =g9280g0 'ry ; l bdcb’=h4L2; l =gciI=h44<,nb™*8e l * ” u, kl ;*cb‹Łn9nb’; l buˆbj4 l & bd; ” bjbd9‡4*-0g,/;<=h*-9n4ŽL†‡b l 2jqrb•28c & =g;<c28c<=g0 ' kl *14bd9N;*^i/*-, & 0hb"; l b uˆbj4 l i/bd9n4=g; '& bd; ” bjbd9]=g;bdc28;<=h*-9n4Ž F e ; l b˜uˆbd;<c<= k =h49n*-;qrbdc ' 2 kjk ,/c28;b y ; l =h4 k *-,/0hi & b+;*G* & =g.2 4<;b f & bd; ” bjbd9I=g;bdc28;<=h*-9n4+289ni k *-,/0hi¬9nbd.:28;<=gqrbd0 ' b‹¶›b k ;uˆbj4 l * f ;<=gu280g=g; '  F e; l b+uˆbd;<c<= k o 289ni¬; l b =g9(;bdc f *-0328;<=h*-9N*8e7; l 28;6uˆbd;<c<= k p`=h4qrbdc ' f cb k =h4b y ; l bd9 f bdc l 2 f 4L2"0328c<.rbdcL4<;b f ” *-,/0hi]4<;<=g0g0 f c*Gim, k b * f ;<=gu280uˆbj4 l bj4289ni^; l ,n4Ee“b ” bdc4<;b f 4 ” *-,/0hi & b6cbj–(,/=gcbji› > 0g0*8eV; l bj4b6eé2 k ;*-c4 28cb+bj44bd9:;<=3280;* ; l b˜b k =hbd9 k ' *8e7289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9N289nicbj–:,/=gcb™289280 ' 4<=h4 & b‹e‰*-cb+*-,/c6280g.r*-c<=g; l uˆ4 k *-,/0hi & b k *-u•uˆbdc k =3280g0 ' 2 f/f 0g= k 2 & 0hb1 †‡b l 2jqrb^i/b‹Łn9nbji‚*-,/c"uˆbd;<c<= k 4*¬; l 28;=g;2 f/f 0g=hbj4";*]289 ' *-ci/bdc4*-0g,/;<=h*-9 y7” =g; l 289 ' 9(,/u & bdc"*8e q128c<=32 & 0hbj4 y =g9ê289 ' im=guˆbd9n4=h*-9W† l =g0hb; l =h4™.rbd9nbdc280 b‹‘G;bd9n4<=h*-9‚=h4™; l bj*-cbd;<= k 280g0 ' 4*-,/9ni y ” b l 2Žqrb *-9/0 ' ;bj4<;bjiU=g;˜e‰*-c"4b k *-9niU289niU; l =gciU*-ci/bdc4*-0g,/;<=h*-9n4 ” =g; l e‰*-,/c™q128c<=32 & 0hbj4=g9‡; ” *]im=guˆbd9n4<=h*-9n4² ? l bcbŽ2-4*-9n4˜e“*-c™; l =h428cb^2032 kì *8e4<, f/f *-c<;™=g9U*-,/c"4*-0gqrbdc™e“*-c˜Ł/e‰; l *-ci/bdc’cb k *-9n4;<c<, k ;<=h*-9n4 y 2-4 ” bd0g0$2-4C; l b•2 & 4bd9 k b*8e l =g. l –:,280g=g; ' ; l cbjbim=guˆbd9n4<=h*-9280\289/=h4*-;<c* f = k uˆbj4 l .rbd9nbdc28;*-c4Ž T ,/;<,/cb ” *-c ì 4 l *-,/0hiI28=gu ;* k cbŽ28;b’; l bj4b˜4<;<c<, k ;<,/cbj4C4*; l 28;; l b˜.rbd9nbdc28072 f/f 0g= k 2 & =g0g=g; ' *8e$; l =h4uˆbd; l *Gi k 289 & b & bd;<;bdcXi/bduˆ*-9n4<;<c28;bji› T ,/c<; l bdc<uˆ*-cb y 4*-0gqrbdc7uˆ*Gim=³Ł k 28;<=h*-9n4\4<, kl 2-47;<,/c & ,/0hbd9 k buˆ*Gi/bd0g=g9/. 289niê4<0h* f b0g=gu•=g;<=g9/.‡28cb & *-,/9ni‚;* l 2Žqrb¬289êb‹¶›b k ;"*-9ê; l b k 280 k ,/0328;bji‚uˆbd;<c<= k 289ni‚0g= ì b ” =h4b^; l b 289/=h4*-;<c* f ' *8e`; l buˆbj4 l ” =g0g0$=g9mèn,nbd9 k b; l b•b‹¶›b k ;<=gqrbd9nbj44™*8eE; l *14buˆ*(im=³Ł k 28;<=h*-9n4Ž > ; l *-c*-,/. l 289280 ' 4<=h4*8e7; l bj4b+cbd0328;<=h*-9n4L=h46280h4*ˆ2",n4b‹e‰,/0V2jqrbd9(,nb™e“*-c e£,/c<; l bdc6cbj4bŽ28c kl  † =g; l cbj4 f b k ;•;*tuˆbj4 l cb‹Łn9nbduˆbd9:; y ; l bdcbI28cbu289 ' u•=g9n*-cˆ=gu f c*Žqrbduˆbd9(;4;*t; l bw280g.r*-c<=g; l u ; l 28; k *-,/0hi f *-;bd9:;<=3280g0 ' =g9 k cbŽ2-4b‡; l b‡–:,280g=g; ' *8e; l btcbj4<,/0g;<=g9/. uˆbj4 l  ? l =h4¬=g9 k 0g,ni/bj4 & bd;<;bdc 4<, f/f *-c<;™e“*-c’4<u280g0 & *-,/9nin28c ' e“bŽ28;<,/cbj42-4 ” bd0g0 2-4"2¬0h* k 280E* f ;<=gu•=g¦²28;<=h*-9fi280g.r*-c<=g; l u ;* & b,n4bji 2¥e‰;bdc™280g0; l b f *-=g9:;4 l 2Žqrb & bjbd9t=g9n4bdc<;bji› > 4 ” =g; l uˆ*14<; l bd,/c<=h4<;<= k uˆbd; l *Gi/4 y ; l bdcb•28cb"4bdqrbdc280 f 28c28uˆbd;bdc4•=g9fi*-,/cuˆbj4 l =g9/.U280g.r*-c<=g; l u ; l 28; k 289 & b; ” bŽ2 ì bji e“*-c=gu f c*Žqrbji f bdcHe‰*-c<u289 k b^e“*-c 2 f 28c<;<= k ,/0328cw; ' f bš*8e f c* & 0hbduw F 9 f 28c<;<= k ,/0328c y ; l btc28;b‡28; ” l = kl ; l btuˆbd;<c<= k =h4¬4<uˆ*G*-; l bji im,/c<=g9/.w4 ” 2 f/f =g9/. k *-,/0hi‡imc28u28;<= k 280g0 ' kl 289/.rbˆ; l bcbj4<,/0g;<=g9/.¬uˆbj4 l bdqrbd9š; l *-,/. l ; l b•–(,280g=g; ' =h4 Ú v â\ãŽä ;3¦E ç  ãŽänç 4<;<=g0g0 k *-9n4<=hi/bdcbji l =g. l +5$c* f bdc™289280 ' 4<=h4+*8e\; l =h4 f 28c28uˆbd;bdc ” *-,/0hi‡280h4*cbj–(,/=gcbb‹‘/28u•=g9/=g9/. l * ” & bd9nb‹Ł k =3280`4uˆ*(*-; l q-28c<=328;<=h*-9‚=g9 k bd0g0 4<=g¦Žb289ni‚4 l 2 f bˆ=h4’;*I; l b2 kjk ,/c2 k ' 289ni‚b k =hbd9 k ' *8e; l b 4*-0g,/;<=h*-9n4 f c*Gim, k bji› > 9n*-; l bdc f 28c28uˆbd;bdc ” l = kl 2¥¶b k ;4+; l bcbj4<,/0g;<=g9/.¬uˆbj4 l y 289ni l * ” 0h*-9/.¬=g; ;2 ì bj4;*L.rbd9nbdc28;bE=g; y =h4 ; l bEc28;<=h*C*8e l * ” u289 ' 9nb ” 0328c<.rb k =gc k ,/u•c2-im=g,n47;<c<=3289/.10hbj4 k 289 & b k cbŽ28;bji im,/c<=g9/.š2 k *r28c4bd9/=g9/.]0h*G* f  F 9 k cbŽ2-4<=g9/.t; l =h4’9(,/u & bdc=gu f c*Žqrbj4; l bŁn9280EuˆbŽ2-4<,/cbjiê–(,280g=g; ' *8e ; l b"uˆbj4 l & ,/;˜i/b k cbŽ2-4<=g9/.¬=g;˜cbjim, k bj4+; l b;<=guˆbcbj–(,/=gcbjit;* k cbŽ28;b; l buˆbj4 l 289nit280h4*cbjim, k bj4 ; l b*Žqrbdc280g0`9:,/u & bdc*8e qrbdc<;<= k bj4"cbduˆ*²qrbji› ? l =h4280h4*Icbd0328;bj4’;*I; l b4<uˆ*G*-; l =g9/.]*8e; l buˆbd;<c<= k & b k 28,n4b+cbduˆ*ŽqG=g9/.qrbdc<;<= k bj4 f *-;bd9(;<=3280g0 ' cbduˆ*Žqrbj46uˆbd;<c<= k =g9me‰*-c<u28;<=h*-9W † l =g0hb; l b0g=h4<;E*8ee‰,/;<,/cb ” *-c ì =h4X0328c<.rb y:& *-; l ; l buˆbd;<c<= kLk *-u f ,/;28;<=h*-9^289niˆ; l buˆbj4 l cb‹Łn9nbduˆbd9:; 280g.r*-c<=g; l uˆ4 28cb9nb ” 2 f/f c*r2 kl bj4289ni4 l * ” f c*-u•=h4be‰*-c\e£,/;<,/cbi/bdqrbd0h* f uˆbd9(;Ž F 9 f 28c<;<= k ,/0328c y ; l b =hi/bŽ2w; l 28;"289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 k 289 & b2 f/f 0g=hbjiU;* l =g. l bdc*-ci/bdc4*-0g,/;<=h*-9n4u2 ' k *-9:;<c<= & ,/;b ;*U; l b]i/b & 28;bN*Žqrbdc ” l bd; l bdc^*-c9n*-; l =g. lmŒ *-ci/bdc¬4*-0gqrbdc428cbI;<c<,/0 'fi& bd9nb‹Ł k =3280Le‰*-cuˆ*14<; s`T ) 2 f/f 0g= k 28;<=h*-9n4Ž R bd.:28cim0hbj44 y ; l bdcb=h4˜9n*¬i/*-, & ;™; l 28;˜; l bˆ2-imq1289:;28.rbj4"*8e 289/=h4*-;<c* f = k 2-in2 f ;28;<=h*-9 289ni l =g. lmŒ *-ci/bdcuˆbd; l *(i/4 k 289 & bL4<, kjk bj44He£,/0g0 ' k *-u & =g9nbjiw289ni^; l =h4 f 2 f bdc k *-9:;28=g9n4624<=gu f 0hb+289ni b‹¶›b k ;<=gqrb ” 2 ' *8e7i/*-=g9/.4*/ Ú1í ø |~ü ø ¸ ß €ê $ !J =gÙ? >6_˜>LR ) T 0g,/=hi^) ' 928u•= k 4 5289nbd0é*K ¨‹œ‹ªML$ž8œŽ¨ œBNj°-©MO ¢ « ¡§œ€d¡QP<R\±³°1SRX¡é¨‹±TPVUt¨dªb“m°AP8œ  >6_+>CR ) > imqG=h4*-c ' R b f *-c<; >LRŒ v Ù1Ù y >6_+>CR ) y M 2 ' َó z îm = v ?) > =g; Œ­> 0g= Œ fiE2 l =32 y _  K 28c<,/¦j¦j= y † _  A 2 & 2-4 l = y M  T *-c<;<=g9 yXW X)+*-u f =hbdc<cb y 289ni M  _  ô\280g0hbd;Ž > 9/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9WýV;* ” 28ci/47,n4bdc Œ =g9ni/b f bd9ni/bd9:; y uˆbj4 lmŒ =g9ni/b f bd9ni/bd9(;X289ni 4*-0gqrbdc Œ =g9ni/b f bd9ni/bd9(; s`T )²5E28c<; FF ý D ;<c<, k ;<,/cbji".1c<=hi/4Ž O ¢ª­¨‹© ¢ž1ª‰¡‰°1¢ž-±YG° fi ©‹¢ž-±Nj°-©[Z fi ¯¨‹©‹¡V€ž-± Uš¨dªb“G°AP8œ™¡£¢RX± fi ¡P8œ y í1ó o z p‹ý Ú1îrõAGÚrõ-í y\W ,/0 'v1x1x1v  = í†? R  K 28032-4<, & c28u289/=3289š289ni W  s  P b ” u289 FFF  s *-u f 28c<=h4*-9]*8e\2-i ë *-=g9:; Œ & 2-4bjit289nie“bŽ28;<,/cb Œ & 2-4bjia.1c<=hiß2-in2 f ;28;<=h*-9e‰*-c^e‰,/9 k ;<=h*-9280’*-,/; f ,/;4Ž O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±]Y(° fi ©‹¢ž-±^Nj°-©ffZ fi ¯¨‹©‹¡V€ž-± Uš¨dªb“G°AP8œ™¡£¢RX± fi ¡P8œ y î1ímýgَî8ánÙ9َî1Ú1ó yv1x1x õG = á? ? =guˆ*-; l 'W  K 28c<; l 289niw528,/0 Y  T cbji/bdc<= kì 4*-9W A =g. l bdc6*-ci/bdcC4*-0g,/;<=h*-9]*8e$; l b B ,/0hbdcLbj–(,2 Œ ;<=h*-9n4 *-9,/9n4<;<c<, k ;<,/cbji.1c<=hi/4`,n4<=g9/.•–(,2-imc28;<= k cb k *-9n4;<c<, k ;<=h*-9W >CFQ>L>af 2 f bdc ó x Œ x1x َí y#W 289 Œ ,28c ' َó1ó x  = î†? sEl 28c<0hbj4 K *-=gqG=g9"289ni s 28c<0 T  Y 0g0g=gqG=hbdc ŒQ_ *G* kl  _ ,28c289(;bjbji Œ –:,280g=g; ' ;<c<=3289/.1,/0328c7uˆbj4 l .rbd9nbdc2 Œ ;<=h*-9ˆe‰*-cXi/*-u28=g9n4 ” =g; lk ,/c<qrbji & *-,/9nin28c<=hbj4Ž O ¢ª­¨‹© ¢ž1ª‰¡‰°1¢ž-±#Y(° fi ©‹¢ž-±DNj°-©?Z fi ¯¨‹©‹¡R€ž-±Ut¨jªb“G°AP8œ ¡£¢_`¢(¤-¡£¢¨¨‹©‹¡£¢(¤ y î1î o Ù x p‹ýgÙ1Ù z îDÙ v َí y )+b k bdu & bdc v1x1x1v  = چ? A *-,/u289 K *-c*-, kl 2 ì = y 5728,/0 S *-,/=h4 _ bj*-c<.rb y T cbji/bdc<= kA b kl ; y 5728;<c<= kì•S 28,/. y 289ni B c<= k6D 280g;bd0é )+bd0328,/92 ' uˆbj4 l .rbd9nbdc28;<=h*-9".r*²qrbdc<9nbji &:' uˆbd;<c<= k 4 f b k =³Ł k 28;<=h*-9n4² f 28c<;`Ù1ý > 0g.r*-c<=g; l uˆ4Ž RX¡£¢n¡‰ª­¨ _ ±g¨‹¯¨‹¢ª^`L¢ž1±affi8œ ¡§œ™ž-¢\Pcb¨ œ‹¡³¤1¢ yv îmý ÚmÙ9 z í y َó1órõG Ú8á LedB¦§ 㠁Z¥ ² ®f = õf? A *-,/u289 K *-c*-, kl 2 ì = y 5728,/0 S *-,/=h4 _ bj*-c<.rb y 289ni K = ë 289 M * l 28u•u2-im=é )+bd0328,/92 ' uˆbj4 l .rbd9nbdc28;<=h*-9a.r*²qrbdc<9nbji &(' uˆbd;<c<= k 4 f b k =³Ł k 28;<=h*-9n4Ž f 28c<; v ý >Lf/f 0g= k 28;<=h*-9n4Ž RX¡£¢n¡‰ª­¨g_ ±g¨‹¯¨‹¢ª `L¢ž-±affi8œ‹¡3œ™ž-¢\Pcb¨‹œ ¡g¤-¢ yv îmý z îDÙ x ó y َó1órõG = z ? _ ,n4<;2jqr* s  K ,n4 k 28.10g=32I289ni B 9/¦Ž* > ›)˜28c<=é > 9/=h4*-;<c* f = k uˆbj4 l * f ;<=gu•=g¦²28;<=h*-9 289ni‡=g;4"2 fmŒ f 0g= k 28;<=h*-9U=g9U2-in2 f ;<=gq(=g; '  O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±^Y(° fi ©‹¢ž-±#Nj°-©4Z fi ¯¨‹© ¡V€ž-±^Uš¨dªb“G°AP8œ•¡£¢h_`¢G¤-¡§¢¨¨‹©‹¡§¢(¤ y á x ý ánÙ1َóD:ánَí1Ú y َó1órõG = ó†? >  K ' ì 28;Ž > ,/;*-u28;<= k .rbd9nbdc28;<=h*-9^*8e›;<c<=3289/.1,/0328cE.1c<=hi›ý F Œ 4, & im=gqG=h4<=h*-9*8eW2˜.rbd9nbdc280 f *-0 ' .r*-9 =g9(;* k *-9:qrb‹‘4<, & cbd.1=h*-9n4²:=g= Œ ;<c<=3289/.1,/0328;<=h*-9¬*8e k *-9(qrb‹‘ f *-0 ' .r*-9n4Ž O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±iY(° fi © ¢ž-±jNj°-© Z fi ¯¨‹©‹¡V€ž-±kUš¨dªb“G°AP8œ™¡£¢l_ ¢(¤-¡£¢¨¨‹©‹¡§¢G¤ y Ù x o Úrp‹ýgَí v óDَí8á vmy َórõ-Úm =gÙ x ? M  W  s 2-4<;<c* Œ )C=328¦ y T  A b kl ; y K  M * l 28u•u2-im= y 289ni Y  5$=gc*-9/9nbŽ28,W > 9/=h4*-;<c* f = k ,/9n4;<c<, k‹Œ ;<,/cbjiˆuˆbj4 l 2-in2 f ;28;<=h*-9ˆe“*-c7è* ” 4=gu’,/0328;<=h*-9n4Ž O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±YG° fi © ¢ž1±ANd°1©mZ fi ¯¨‹© ¡V€ž-±Ut¨dªb“n °DP8œ™¡§¢ffR\± fi ¡P-œ yv îmý á(õ-îD v ómÙ y َó1órõG =gÙ1Ù? S  5728,/0 sEl b ”  _ ,28c289(;bjbji Œ –:,280g=g; ' ;<c<=3289/.1,/0328cuˆbj4 l bj4² ? b kl 9/= k 280 R b f *-c<; ?RŒ z ó Œ ó z í y )+b f ;Žn*8e s *-u f ,/;bdc DGk =hbd9 k b y s *-c<9nbd0g0 J 9/=gqrbdc4<=g; 'ry َó z óm =gÙ v ? T c289 k *-=h4 s *-,/c<; 'ry )+2Žq(=hi S bj4bdc<qr*-=h4<=hbdc y 5728,/0 Œ­S *-,/=h4 _ bj*-c<.rb y 289ni > 0328=g9’)+bdc<q(=hbd,m‘› s *-9:;<=g9n*-,n4 uˆbd;<c<= k 4C289ni¬uˆbj4 l 2-in2 f ;28;<=h*-9W `EŸrŸ± ¡é¨.P5Z fi ¯ˆ¨d© ¡V€ž-±FUNž1ªb“m¨‹¯ˆž1ª‰¡V€‹œ y î1ÚmýgÙ1Ù²õAÙdá:î yv1x1x Úm =gَí†? W G)+*-u f =hbdc<cb y M  _ Gô\280g0hbd; y fi K *-,/c<.:28,/0g; y M  T *-c<;<=g9 y 289ni^† _  A 2 & 2-4 l =é > 9/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9Wý ;* ” 28ci/4,n4bdc Œ =g9ni/b f bd9ni/bd9:; y uˆbj4 lmŒ =g9ni/b f bd9ni/bd9(;X289ni’4*-0gqrbdc Œ =g9ni/b f bd9ni/bd9:; s`T )/5\28c<; FFF  J 9n4<;<c<, k ;<,/cbjiwuˆbj4 l bj4² O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±oY(° fi ©‹¢ž-±fNj°-©:Z fi ¯¨‹©‹¡R€ž-±kUt¨jªb“G°AP8œ™¡£¢ R\± fi ¡QP8œ y í1ó o z p‹ý Úrõ-îDmõ x1vmy\W ,/0 'wv1x1x1v  =gÙdá? S *-c<= >  T cbd=g;28.a289ni s 28c<0 T  Y 0g0g=gq(=hbdc ŒQ_ *G* kl  ? bd;<c2 l bjimc280"uˆbj4 l =gu f c*²qrbduˆbd9:;N,n4<=g9/. 4 ” 2 f/f =g9/.’289ni4<uˆ*G*-; l =g9/.n O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-±#Y(° fi ©‹¢ž-±pNj°-© Z fi ¯¨‹© ¡V€ž-±qUš¨dªb“G°AP8œ¡£¢c_`¢G¤-¡§¢¨¨‹©‹¡§¢(¤ y á x o v Ù²p‹ý í1órõ-óD:á x1x1vmy َó1órõG =gَî†?’†‡bd=g¦ l 289/. A ,289/.n M 28; l bdu28;<= k 280 f c<=g9 k = f 0hbj4*8e 289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9W L$°1¯"¯ fi ¢n¡rn €ž1ª‰¡‰°1¢/œ™¡§¢sL$°-¯LŸ fi ªéž1ª‰¡‰°1¢ž-±\t±“ffi8œ‹¡R€‹œ y Ù o v p‹ý v õ-ÚDGímÙ xmyv1x1x Úm Ú1î LedB¦§ 㠁Z¥ ² ®f =gَچ? T c289uj*-=h4 S 2 & bd0g0hb•289ni W *-928; l 289 R = kl 28ci D(l b ” kl , ì  > 9/=h4*-;<c* f = k qr*-c*-9n*-=7im=328.1c28uˆ4˜289ni .1,28c289(;bjbji Œ –:,280g=g; ' 289/=h4*-;<c* f = k uˆbj4 l .rbd9nbdc28;<=h*-9W F 9 t`©<°€¨¨.P-¡£¢(¤8œ°vN’ªb“m¨]Z˜¡§¢¨jª­¨¨‹¢ªb“w`L¢n ¢ fi ž-±xyffi-¯6Ÿ/°8œ ¡ fi ¯ °-¢zL$°-¯LŸ fi ªéž1ª‰¡‰°1¢ž-±k{6¨<°-¯¨dª‰©ffi y f 28.rbj4LَómÙ9 v1x1x  > 44* k =328;<=h*-9e‰*-c s *-u f ,/; Œ =g9/. M 2 kl =g9nbdc 'ryv1x1x ím =gÙ²õf? D  A  S */ > 9nb ” uˆbj4 l .rbd9nbdc28;<=h*-9š4 kl bduˆb™e“*-cC28c & =g;<c28c ' f 0328928cCi/*-u28=g9n4Ž O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-± YG° fi © ¢ž1±fNd°1©:Z fi ¯ˆ¨d© ¡V€ž-±FUt¨dªb“G°AP-œ™¡§¢_`¢(¤-¡£¢¨¨d© ¡£¢(¤ yv Ù1ýgÙdá x íDÙdá v Ú y َó z îm =gÙ z ?) S  M 28c k ,/u—289niz|˜bd0g0 ' 5  _ 28=g; l bdcŽ D *-0g,/;<=h*-9‚2-in2 f ;<=gqrb•,/9n4<;<c<, k ;<,/cbjiš.1c<=hit.rbd9nbdc28;<=h*-9 ,n4=g9/. f 4bd,ni/* Œéf 28;<;bdc<9êcb k *-.19/=g;<=h*-9 ;b kl 9/=h–(,nbj4Ž F 9 t`©<°€¨¨.P1¡§¢(¤-œ¬°vNªb“m¨ K “:¡£©‹ª­¨¨d¢ªb“}`~O.`~` L$°1¯6Ÿ fi ªéž1ª‰¡“°-¢ž1±\RX± fi ¡Pcb*ffi-¢ž1¯"¡V€‹œL$°-¢ANŽ¨‹©¨‹¢€¨ y W ,/0 ' َó1órõG >CFO>C> órõ Œ Ù z Ú x  =gَó†?)C=gu•=g;<c<= M 2jqGc<= f 0g=h4289ni > 9:;*-9 'W 28uˆbj4*-9W M ,/0g;<=g.1c<=hi 4*-0g,/;<=h*-9a*8e™; l bN; ” * Œ im=guˆbd9n4<=h*-9280 B ,/0hbdc^bj–(,28;<=h*-9n4*-9Z,/9n4<;<c<, k ;<,/cbji ;<c<=3289/.1,/0328cuˆbj4 l bj4Ž >CFQ>L>„f 2 f bdc z õ Œ x í1î1í yMW 289(,28c ' َó z õG = v1x ?4|+c<¦ ' 4<¦j;*8e M = kl 28032 ì 289ni s 28c<0 Y 0g0g=gqG=hbdc ŒQ_ *G* kl  M 28;<c<=³‘ Œ b‹‘ f 0g= k =g; _+MNR6B`D e‰*-c2 l =g. l bdc Œ *-ci/bdc`2 kjk ,/c28;b=g9:qG=h4 k =hi k *-u f cbj44<= & 0hbè* ” 4*-0gqrbdcŽ F 9 t ©H°€ ¨¨.P-¡£¢(¤8œ™°vNªb“m¨m_`¡g¤B“(ª­¨¨‹¢ªb“V`~O.`~` L$°1¯6Ÿ fi ªéž1ª‰¡“°-¢ž1±\RX± fi ¡Pcb*ffi-¢ž1¯"¡V€‹œL$°-¢ANŽ¨‹©¨‹¢€¨ yv1x1x õG = v Ù? _ 28c ' S  M =g0g0hbdc y )˜2¥e‰92 ? 280guˆ*-c y 289ni D(l 289/. Œ­A ,2 ? bd9/.n YLf ;<=gu280 k *r28c4bd9/=g9/.¬*8e\,/9n4;<c<, k‹Œ ;<,/cbjiwuˆbj4 l bj4Ž Y(° fi ©‹¢ž-± °.N€`L± ¤:°-© ¡‰ªb“:¯"œ y ímÙ o Ù²p‹ý v óDGÚ1î y َó1ó1óm = v1v ? > u•=gc P b ë 28;Ž `‚˜¡³¤“m¨‹©nCƒ ©vP(¨‹©`/€H€ fi ©<ž1ª­¨…„ ¢/œ‹ª‰© fi €Žª fi ©¨.P†R\¡§¢n¡‰ª­¨ˆ‡/°-± fi ¯ˆ¨Z’¨dXªé°-¢n‰’©ffi-±³° « `L± ¤r°-©‹¡‰ªb“:¯†Nd°1©[O ¢ « ¡3œ_€d¡PgL$°-¯LŸ©¨ œœ‹¡QŁ‹±h¨mRX±g°-7œ G5 l )Z; l bj4<=h4 y J 9/=gqrbdc4<=g; ' *8e K c<=g;<=h4 ls *-0g,/u & =32 y )+b f 28c<;<uˆbd9(;L*8e M b kl 289/= k 280 B 9/.1=g9nbjbdc<=g9/. yv1x1x õG = v í†? > u•=gc P b ë 28;7289ni s 28c<0 Y 0g0g=gq(=hbdc ŒQ_ *G* kl  Y 9 f cb k *-9nim=g;<=h*-9/=g9/.™*8e P b ” ;*-9 ŒQ_˜M]RLBED 280g.r*-c<=g; l u e“*-c 2 l =g. l bdc Œ *-ci/bdc72 kjk ,/c28;bE,/9n4<;<c<, k ;<,/cbji’4*-0gqrbdcŽ F 9 t`©<°€¨¨.P1¡§¢(¤-œ°.NEªb“m¨‹RV° fi ©‹ª­¨¨d¢ªb“]`L¢n¢ fi ž-± L$°1¢ANj¨d©<¨‹¢Œ€¨°vN™ªb“m¨wL$°-¯LŸ fi ªéž1ª‰¡“°-¢ž-±yR\± fi ¡QPcb8ffi-¢ž-¯¡V€‹œ]x°€d¡é¨dªffi°vNwL$ž-¢žfPrž  k e“i/4 k yv1x1x Úm Ú1Ú LedB¦§ 㠁Z¥ ² ®f = v á? > u•=gc P b ë 28;`289ni s 28c<0 Y 0g0g=gqG=hbdc ŒQ_ *G* kl  >…l =g. lmŒ *-ci/bdc2 kjk ,/c28;b,/9n4;<c<, k ;<,/cbjiˆŁn9/=g;b6qr*-0g,/uˆb 9nb ” ;*-9 Œéì c ' 0h*Žq280g.r*-c<=g; l u e‰*-c^=g9:qG=h4 k =hi k *-u f cbj44<= & 0hb]è* ” 4² Y(° fi © ¢ž-±+°.NsL$°-¯LŸ fi ªéž1ª‰¡“°-¢ž-± tL“ffi8œ ¡V€‹œ ynv1v õ o á(p‹ý v î1ó v  v Ú x ó y›v1x1x1z  = v î†? s 28c<0 T  Y 0g0g=gqG=hbdc ŒQ_ *G* kl  > 9š,/9n4<;<c<, k ;<,/cbji‡uˆbj4 l =gu f c*Žqrbduˆbd9(;;*G*-0 ì =g; ” =g; l 2 f/f 0g= k 28;<=h*-9 ;*uˆbj4 l =gu f c*Žqrbduˆbd9(; y .rbd9nbdc28;<=h*-9t289ni o i/b Œ pOcb‹Łn9nbduˆbd9:;Ž >CFO>C> ó z Œ x1v Ù zmyW 289:,28c ' َó1ó z  = v چ? s 28c<0 T  Y 0g0g=gq(=hbdc ŒQ_ *(* kl  _˜R`JCMNM 5Û' _ bd9nbdc28;<=h*-9 289ni R b‹Łn9nbduˆbd9:;ˆ*8e J 9n4;<c<, k ;<,/cbji y M =³‘/bji Œ bd0hbduˆbd9:; M bj4 l bj4=g9w5E28c280g0hbd0é l ;<; f ýŒ¥;bd;<c2m uˆb kl  , & k  k 2f _˜R`JCMNM 5 y َó1ó z  v1x1x îm = v õf? s 28c<0 T  Y 0g0g=gq(=hbdc ŒQ_ *G* kl  >LP˜DGS = & ý > 4 k =hbd9:;<=³Ł k•k *-u f ,/;<=g9/.;*G*-0 ì =g;+4<, f/f *-c<;<=g9/.^c2 f =hiN9(, Œ uˆbdc<= k 2804*-0g,/;<=h*-9Z*8e f c2 k ;<= k 280g0 ' 289 ' 5X) B  F 9 t`©<°€¨¨P-¡§¢G¤8œt°vNªb“m¨g_`¡³¤“(ªb“Ž`L¢n¢ fi ž-±*L$°-¢n NŽ¨‹©¨‹¢€¨°.N™ªb“m¨L$°1¯6Ÿ fi ªéž1ª‰¡“°-¢ž1±\RX± fi ¡Pcb*ffi-¢ž1¯"¡V€‹œ€x°€d¡é¨dªffi°vNL$ž-¢žfP:ž y f 28.rbj4 v Ù9 v1z  D * k =d; k 2892-im=hbd9/9nbi/b s`T )‘ s`T ) D * k =hbd; ' *8e s 2892-in2 yW ,/9nb v1x1x1x  = v1z ? s 28c<0 T  Y 0g0g=gq(=hbdc ŒQ_ *G* kl 289ni sEl 28c<0hbj4 K *-=gq(=g9W F u f c*Žqrbji k bd0g0(4<=g¦ŽbX289ni˜.1c2-im=g9/.L=g9™.1,28c289:;bjbji –(,280g=g; ' ;<c<=3289/.1,/0328c•289ni‡;bd;<c2 l bjimc280`uˆbj4 l bj4Ž F 9 t`©<°€¨¨P-¡§¢G¤8œ¬°.Nªb“m¨cZ+¡£¢ªb“}O ¢ª­¨‹©‹¢ž1ª‰¡“°-¢ž-± Uš¨ œH“:¡£¢(¤c’° fi ¢\P1ªéžŁd±g¨ y f 28.rbj4á:íDGî8án D 289nim=32 P 28;<=h*-9280 S 2 & *-c28;*-c<=hbj4 y YCk ;* & bdc v1x1x1x  = v ó†? s 28c<0 T  Y 0g0g=gqG=hbdc ŒQ_ *(* kl 289ni M = kl 2-bd0\ô\289 > 0g;bd92m > l =g. lmŒ *-ci/bdc2 kjk ,/c28;b,/9n4<;<c<, k ;<,/cbji uˆbj4 l Łn9/=g;b Œ qr*-0g,/uˆb4 kl bduˆb’e“*-c6; l b2-imqrb k ;<=h*-9 Œ im=³¶,n4<=h*-9‡bj–:,28;<=h*-9W Y(° fi ©‹¢ž-±$°vNwL$°1¯6Ÿ fi ªéž“n ª‰¡“°-¢ž1±\t±“ffi8œ ¡V€‹œ y Ù z Ù o v p‹ý õ v óDmõ-î vmyv1x1x1v  = í x ?)+*-,/.•5728.19:,/;<;<= 289ni s 28c<0 Y 0g0g=gq(=hbdc ŒQ_ *(* kl )+bd0328,/92 ' Œ & 2-4bjiw289/=h4*-;<c* f = k uˆbj4 l cb‹Łn9nbduˆbd9:;Ž L$°1¯6Ÿ fi ª­¨‹©8Ut¨dªb“G°AP-œ™¡§¢ff`EŸrŸ± ¡é¨.PgUt¨e€“mž-¢n¡V€‹œž-¢\Pc_`¢(¤-¡£¢¨¨d© ¡£¢(¤ y D , & u•=g;<;bji yv1x1x1z  = ímÙ?)+*-,/.5728.19:,/;<;<=289ni s 28c<0 Y 0g0g=gq(=hbdc ŒQ_ *G* kl  > .rbd9nbdc280g=g¦Žbji@e‰c28uˆb ” *-c ì e‰*-c l =g. l *-ci/bdc 289/=h4*-;<c* f = k uˆbj4 l 2-in2 f ;28;<=h*-9W L$°1¯6Ÿ fi ª­¨‹©8Ut¨dªb“G°AP-œ’¡§¢”` ŸrŸ± ¡“¨.PgUš¨e€“Gž-¢¡R€‹œž-¢\P5_`¢(¤-¡£¢¨¨d©9n ¡£¢(¤ y D , & u•=g;<;bji yv1x1x1z  = í v ?5 l =g0g0g= f b•5b & 2 ' 289ni ? =guˆ*-; l 'ŽW  K 2 ì bdcŽ > 9280 ' 4<=h4+*8eE;<c<=3289/.10hb•–(,280g=g; ' uˆbŽ2-4<,/cbj4Ž Utž1ªb“m¨n ¯ˆž1ª‰¡V€‹œ’°vNL$°-¯LŸ fi ªéž1ª‰¡“°-¢ y õ v ýgÙ z Ù²õAÙ z í1ó yv1x1x ím Úrõ LedB¦§ 㠁Z¥ ² ®f = í1í†? R  R 2-i/bj4 f =hbd0é > k bd0g0 Œ qrbdc<;b‹‘Iu’,/0g;<=g.1c<=hiNuˆbd; l *Gi¬e“*-c6; l b P 2Žq(=hbdc ŒQD ;* ì bj4+bj–(,28;<=h*-9n4Ž PC>+DG> ?MIŒ Ù x َî1îrõ y َó z óm = í8á? D(l u,nbd0 R = f/f 2m S *-9/.t289ni‚; l =g9‚;<c<=3289/.10hbj4 k 289 & bˆ.r*G*(i‡e‰*-c"0g=g9nbŽ28c=g9(;bdc f *-0328;<=h*-9W xqO.`]U YG° fi © ¢ž1± °-¢Z fi ¯ˆ¨d© ¡V€ž-±o`L¢ž-±•ffi¥œ‹¡3œ y/v ó o Ù²p‹ý v îrõA v õ xmy T b & c<,28c ' َó1ó v  = í1î†?5 S  R *Gb1 >Lf/f c*j‘m=gu28;b R =hbdu289/9t4*-0gqrbdc4 y f 28c28uˆbd;bdc˜qrb k ;*-c4 y 289ni]im=³¶›bdcbd9 k b4 kl bduˆbj4² YG° fi © ¢ž1± °vNL$°-¯LŸ fi ªéž1ª‰¡“°-¢ž-±\t±“Gffi¥œ‹¡R€dœ y á:ímý í1îrõAGírõ vmy َó z Ù1 = í1چ?5 S  R *Gb1 sEl 28c2 k ;bdc<=h4;<= k‹Œ & 2-4bjiI4 kl bduˆbj4e“*-c; l b B ,/0hbdc6bj–(,28;<=h*-9n4Ž F 9 `L¢n¢ fi ž-±\’¨ « ¡“¨‹°vN R\± fi ¡QP–Ut¨e€“Gž1¢n¡R€dœ y qr*-0g,/uˆbÙ zmy f 28.rbj46í1írõAGí1Ú1îm > 9/9:,280h4 R bdqG=hb ” 4 y F 9 k  y َó z Úm = írõf? W  R , f/f bdc<;Ž > )+bd0328,/92 ' cb‹Łn9nbduˆbd9(;˜280g.r*-c<=g; l uûe“*-cL–(,280g=g; 'Iv Œ im=guˆbd9n4<=h*-92807uˆbj4 l .rbd9nbdc2 Œ ;<=h*-9W Y(° fi © ¢ž-± °vN:`L± ¤r°-©‹¡£ªb“(¯œ y Ù z o írp‹ý î8á z Gî z î y M 2 ' َó1ó1îm = í z ?Łfi\*-, k b‹e D 212-i"289ni M 28c<;<=g9 A  DGkl ,/0g;<¦r _+MNR6BED ý > .rbd9nbdc280g=g¦Žbji"u•=g9/=gu280mcbj4<=him,280m280g.r*-c<=g; l u e“*-c4*-0gq(=g9/.t9n*-9n4 ' u•uˆbd;<c<= k 0g=g9nbŽ28c•4 ' 4<;bduˆ4Ž xqO.`€U—YG° fi © ¢ž1± °vNgxŒ€d¡é¨‹¢ª‰¡ ˜±€¬ž-¢iPxªéž1ª‰¡§œ‹ª‰¡V€ž-± L$°1¯6Ÿ fi ª‰¡£¢(¤ y õ o írp‹ý z î1ÚD z Ú1ó y\W ,/0 ' َó z Úm = í1ó†? W *-928; l 289 R  D(l b ” kl , ì  b¨‹±gž fi ¢ž“ffi4’¨˜X¢¨‹¯¨‹¢ªEUš¨ œH“}{6¨d¢¨‹©<ž1ª‰¡‰°1¢ W5 l ) ; l bj4<=h4 y DGkl *G*-0*8e s *-u f ,/;bdc DGk =hbd9 k b y s 28c<9nbd.1=hb M bd0g0h*-9 J 9/=gqrbdc4<=g; 'ry M 2 ' َó1órõG = á x ?) > ôXbd9nim=g;<;<=X289nit)" S )˜28c<uˆ*8e“280é > i ë *-=g9:;˜bdc<c*-c˜bj4<;<=gu28;<=h*-9‡289niN.1c<=hiN2-in2 f ;28;<=h*-9te“*-c e‰,/9 k ;<=h*-9280E*-,/; f ,/;4²ý >Lf/f 0g= k 28;<=h*-9‚;*I–:,2-4<= Œ *-9nb Œ im=guˆbd9n4=h*-9280Eè* ”  Y(° fi © ¢ž-±E°vN}L$°1¯6Ÿ fi ªéž“n ª‰¡“°-¢ž1±\t±“ffi8œ ¡V€‹œ y َÚ8á o Ù²p‹ý v1x á™ v1v õ y YCk ;* & bdc v1x1x1x  = ánÙ?) > ²ôXbd9nim=g;<;<=m289ni") S 8)˜28c<uˆ*8e“280é _ c<=hi"2-in2 f ;28;<=h*-9e“*-cWe‰,/9 k ;<=h*-9280m*-,/; f ,/;4²ý >Lf/f 0g= k 28;<=h*-9 ;*U; ” * Œ im=guˆbd9n4<=h*-9280+=g9:qG=h4 k =hifiè* ” 4Ž Y(° fi ©‹¢ž-±L°vNšL$°1¯6Ÿ fi ªéž1ª‰¡“°-¢ž1±mt±“ffi8œ ¡V€‹œ y Ù²õ-î o Ù²p‹ý á x GÚ1ó y T b & c<,28c '¬v1x1x1v  = á v ?) > ²ôXbd9nim=g;<;<=m289ni") S 8)˜28c<uˆ*8e“280é _ c<=hi"2-in2 f ;28;<=h*-9e“*-cWe‰,/9 k ;<=h*-9280m*-,/; f ,/;4²ý >Lf/f 0g= k 28;<=h*-9 ;*; ” * Œ im=guˆbd9n4<=h*-92807qG=h4 k *-,n4 è* ” 4Ž Y(° fi ©‹¢ž-± °.NL$°-¯LŸ fi ªéžrª‰¡‰°-¢ž-±\t±“ffi8œ ¡V€‹œ y Ù z õGý v1v :á:Ú yv1x1x ím = á:í†? M  > ½fi\bdc<c ' 289ni M  D  D(l b f/l 28ci› > uˆ*Gim=³Łbji¬–:,2-im;<cbjb™2 f/f c*r2 kl ;*Łn9/=g;b™bd0hbduˆbd9:;6uˆbj4 l .rbd9nbdc28;<=h*-9W O._‹_‹_›L$°-¯LŸ fi ª­¨‹©–{ ©<ž Ÿ“:¡V€‹œž-¢\Pc`EŸ:Ÿ± ¡V€ž1ª‰¡“°-¢/œ y ímý í1óD:á:Ú y َó z ím Ú z LedB¦§ 㠁Z¥ ² ®f = á1á?5bd9/.1;2-*fiX,nb y sEl ,/9me‰bd9/.œ l *-, y W 28uˆbj4 W  T bd9/. y s 28c<0 T  Y 0g0g=gq(=hbdc ŒQ_ *(* kl y 289ni A * ” 28ci A  A ,W‡5 l 2-4b Œ Łbd0hifi4<=gu’,/0328;<=h*-9n4ˆ*8eC=g9:;bdcHeé2 k =3280i ' 928u•= k 4=g9 qG=h4 k *Gbd032-4<;<= k èn,/=hi/4,n4<=g9/.NŁn9/=g;b bd0hbduˆbd9(;4 ” =g; l 2-in2 f ;<=gqrb˜uˆbj4 l =g9/.n YG° fi ©‹¢ž-±°vNL$°-¯LŸ fi ªéž1ª‰¡‰°1¢ž-±\t±“ffi8œ‹¡R€‹œ ynv َómý á(õAGÚrõ yv1x1x Úm Ú1ó

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