Analyse asymptotique des équations de dérive-diffusion et équations de Hamilton-Jacobi
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Analyse asymptotique des e´ quations de d´erive-diffusion et e´ quations de Hamilton-Jacobi Philippe Montarnal , Benoˆıt Perthame. N ˚ 2885 Mai 1996 ` THEME 4. ISSN 0249-6399. apport de recherche.
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(51) &(*9 ! $Y . ! 8 D k7oIs\wMa!qk { kV { k7q _ a~oImIiIs"aM: lPi u gnmpoIsXVqmpi { jna~wRk-s u ZÆio"wMmpoOlPkVwMl'eIw _ m7l]l L v:aRoÄ u XMj ZÆioIgJ@ wMgnl!a : u { iÊq { kmu Ij aR` ba!jnghbgnlaaR0o >Vk { { gh k IjnaMs C \ qu ! q { b !F G aMl oImpiIs'aRo u a : u iIgnsmpoIs'jhk0wMmIo >7a { 7aRoIwMa ioIgJ;Ím ba awMaM$s =ik7oOlgnl aM: s!jnm Ms =iIa#laRo >Oa s _ mpLiIsq { aM: suaRoOlmpoIs l]{ mpiIl ` u ZÆk m { u jnaMswMmpo u gnlgnmpoIs*kOiIyxG jngrbEgtl]aMs*sPq]aM: wMgJg=iIaMs k` ioÄwMmpoOlPkVwMl eIw{ m7l{ l v:aRoxu aM{ l¦jna~` q km Ij aRba'jnGgrbE gtl]a =iIa j ZñmIo0km IlgnaR{ oOl mpiIs aRu : oImpu o wM mpoIs(aRoI{ sP{iIgnl]a =iIaM2j =iIu aMs q { mIq ` 6g aM: !l aM: s i[q G km Ij aRbajngrbEgtl]a aMjnjnaM[s @wMg~oImIiIs5qa baMllaRoOl a u aR: bmp{ oOl a ` j ZÆioIgnwMgn!l a : i q m{ Ij aRba0jnghbb gnlG a ghokVjnaRbaRoOl X oImpiIssmIiIjtgnpoImIoIs* j ZñghoIsPi sYk7oIwMa ZÆioq km Ij aRba0jnghbgnlaaRo >Vk gh k IjnaMs ¨X . . $. ! %'"%$ 3 $3 # & ! % &'.! ! %.
(52) . ou wMmpo7{ lYkVO wMlxeIw _ m7Gl]S l L vGaMslxu wMmIoIslgnlPi]a : u Z´{ ioIax|mpoIwMulgnmpo aRoOl { a iu o b aM: lYkV{ j!aMlio s`aRbEgB@ wMmpo iIwMl]aRi b QX'7'G R I { o aM: lPi |gnaÄnpjna wMmpbxq{ m laRb` aRoOl isaRbEgB@wMmIo iIwMlaRi s]mpuibgns k { Gio qm7laRK oOlgn{MaM` j _ Zñm gn7u ghoIa _ C aMsl*{ q gts]a kjhk"F|ÀmpoIwMl]gt{ mIobY` aM: lPk-j -saRbgJ@wMmpo { OBiI!wMq laRi q u aMs'jnaw kOoI7aRbaRoOl Z)aM: w aMjnjta >7c mOg kOooIaMypa+V XVjna*q mkIj aRba!sa*qm7sasYi Rk'>7aMw jnaMs:wMmpo gnlgnmpoIsk7iIyÄjtghbgnlaMs0sYiIg?>Vk7oOlaMs ` C F\n+ !¤ q ` C F n e " #¤ q b C F\np!q b C F\n%$ & I a*q { km Ij aR` bajnghbgnla0aMsl kVjnm { s u aM: w { gnl qk { jnaMPs ;ÍmpoIwMlgnmpoIs X X! X*! aMl b >Pa : { Jg k7oOl c .
(53) ú¨§. 44. .' -.
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(55) > 0¨g0g E" 12 ,' ! ! k% R !q O 12 \ 5 g&'%k &% ! 56 g&'% k$&'% ! k} A % 0} %¨ 5 5' $ . R ! q O I k u gÄwRiIjtl!a : u a*wMa:l _ aM: m {ÕaR` baq { m>IgnaRo7l u a u aRiIyqm7ghoOls GD ZÆioIa5qk { l jna0wMmpiqIjhkV7a X>Ighk b X o]aM: wMaMs]sgnla u aMs0k { pibEaRo7lssPqPaM: wMgJg=iIaMs GgD Z´k7iIl { a0qk { l:j Z)a : =ikVlgnmpo u a k7bEgtjnlmIo@pkVwMmkIg C F mpi C F oIa*qm7ss!a : u k7o7l'qkVs u a~l]a { ba u a u a : { g2>Pa : u Zñm { u { a sl { gnwMlaRbEaRo7l!qm7sgnlgJ;MX7j ZÆioIgnwMgnl!a : aMsl; k7iIs]s a:: jnm { sM= iIa b oZ aMsl*qkVs*bEmpoIm7lmpoIa G K >Vk7o7l u a u mIooIa { wMaMl]la u aR: bEmpoIsl { k-lgnmpoX mpoqaRiIl u mIooIa { jhk ;Ím { baxaMykVwMla u aMs s]m@ jhiIlgnmpoIs G z k { wMmpo O> aRo7lgnmpo¦X jnm { s = iIa aMl' C { aMsPq G ! aMl ! F wMm oIwMg u aRoOlÄaRo io qm7gho7l 23 ,4526.
(56)
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(58) ú¨§. 44. .' -.
(59) Mc. , g %g1+ % 0 χ. ϕ V +β. β V. V. 0 1. 0. x1. x. Φ. 1. x. n 1. V. 0. x1. . {@ 1. $ . 0. aMl $ C 7. x. &. ϕ. 1. x1. x. χ V +β. β V. V. 0 1. 0 x. 1. Φ. x. n 1. V. 0. 1. 0. # 7 C $ . /@. x. 1. x. &. 23 ,4526.
(60)
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(62) ú¨§. 44. .' -. x. $. &. aMl. 0. $. 7. 1. x. .
(63) R. , g %g1+ % 0 ϕ. χ. β V+. 1. 0. 1. β 0. x. x. V. Φ. n 1. 1. 0. x V. 0. . /@ e & C $ . ϕ. 1. x. χ. β x. 1. 0. 0. 0. 1 x. x V+. β. V. Φ. n 1. 0. 1 x. V. . @. 0. $ e &. 1. x. 23 ,4526.
(64)
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(68) ú¨§. 44. .' -.
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(75) a : gnaRi k u jhkÊwMmpo gnlgnmpo k7iIyjnghbgnlaMs:aRo k ;ÍmpoIwMlgnmpo ! aMsl mpoIwsPi @sm7jhiIlgnmpoaRo XIsm7gnl ,! C F ,. e b C F ! C F 23 ,4526.
(76)
(77) !" #$ % &' () *+ ,'- !" /. 0 12 3(4! 56 7 . . 7k i/saRoIs u a0>pgnswMc m7s]gtl!a : G U o u ZÆkOiIl { aMs:la { baMsX qmIi { lmpiIl]a;ÍmpoIwMlgnmpo wMmpoOlghoIiIaRbaRo7l u a : { gJ@ >Vk Ijna l]aMjtjna/=iIa C ! aMl C F n $¨q mpo/k , C F ,. e b C F C F Íj¦sPi l u axq { aRo u { axioI/ a ;¸mIoIwMlgnmpo laMjnjta =iIa b C F C Fp qmpi { mkIl]aRoIg { ioIa wMmpoOl { k u gtwMl]gtmIo G z iIgns0mpo u a : u iIgnl ! $ sYi { R | . q R G U o/aA aMl b aM: lPk7oOlsl { gnwMlaRbaRoOl u aM: w { m7gnssPk7oOlampo k b $ wM"a =iIg(ghbxqIjn?g =iI a ! b $ G aMwMg(la { bEgroIawMaMllaq { aRbEg a` { , a aM: lYk7q a XImpo#k IgnaRo u aR: bEmpo7l { a { =iIa | . n G C gng F mpiIskVjnjtmIoIs bxkVghoOlaRok7oOlbmpoOl { a { =iIaÊsPi { OB| q R)XmIok b $ aMl0jnaMs ;ÍmpoIwMlgnmpoI" s aMl smpoOl u mpoIw0wMmpoOlghopiIaMs:aRo | G mpoIsg u a : { mpoIs!lmpiIl u ZÆ k m { u jna0wRkVs*qk { lgnwRiIjtgna { | n G Íj\oImpiI s ; k7iIlbmIo7l { a { jnaMs aM: p'k @ C \ F n C \ F % n $ G u jngn!l aM: "s aMljhkwMmIo7lghoIiIgnl a : a aMl aRo S o { aRbxk { =iIalmIiIl u ZÆ k m { u =iIa C FPn $ G U oa A(aMl Xsg C F n $ X mpo kE *{ kV wMa k ` C Q F C F $ G I k ;¸mpoIwMl]gtmIo aMsl u mpoIws]l { gnwMlaRbaRoOlsPi]q a : { gnaRi { a k ` jrk/wMmpo u gnlgnmpoGk7iIy jnghbgnlaMs aRo% G I k ;¸mpoIwMl]gtmIo u { aMsl u mpoIw s]mpiI[s @sm7jhiI lgnmpo aRo CG]S Fon%k$ kVjnm { s X~qmpi { lmIiIla ;ÍmpoIwMlgnmp o wMmpoOlghopiIaRbaRoOl a : 2g >¯k IjtaaMllaMjnjn0a =iIa aMl X , C F , . e b C F C F C j\sPi Äl u axq { aRo u { aioI/ a ;¸mpoIwMl]gtmIo laMjnjn0a =iIa C Fp qmIi { 'k >7m7g { ioIawMmpoOl { k u gtwMl]gtmIo ` G S o/aRo u a : u iIgnl:jrkÄwMmIo7lghoIiIgnl a : u. a aRoGaM+l =iIajh0k ;ÍmpoIwMlgnmpo aMs]l*wMmpoOlghoIiIaaM,l aM: IkVjna k $ aRo"wRk { b G :U lPi u gnmpoIs0bxkVghoOlaRok7oOljnawRkVs | . G!S o kghbxb a : u ghkVlaRbaRoOl b %$ sPi { OB| q R[ U oa A(aMl XImpok u ZÆioIaqk { l b p sPi { OJ| q R)XIqk { u a : oIgtl]gtmIo u a | X b u a0wMjhkVss]a w sYi { OB3q R aMl b C F\n $ XIaMl u Z´k7iIl { aqk { l}kVy C e b e q e b F\np sPi { OJ!q R G
(78) ú¨§. 44. .' -.
(79) . , g %g1+ % 0. ba; k.wM mIo#=iIaq { aM: wa : u aRbxbaRoOlXVmpo u aR: bmpoOl { a$=iIa C F\n%$GD a5qIjhiIsXsYi { * a h j x k b R a OB| q O XVj Z[ a : =ikVlgnmpo u a k7bgnjnlmpo@ pk-wMmkIg u mpooIa aMl $GkS oÄaRoÊwMmpoIwMjhiIl u mpoIw %$ sPi { OB| q R) I kwMmpoOlghoIiIgt!l a : u a aRo | aMslghbxb a : u ghkVl]a C aMs]l b aR bajnghqIsw _ gtl WMgtaRooasPi { R !q O wRk { aMjtjnaaMsls]mpiI[s @sm7jhiIlgnmpo u a C FMF aMlwMaMjnjta u a sa u a : u iIgnl u aj ZñaRoIwRk u { aRbaRo7l. D. . $ n. . . C | F C | F b C | \F n%$ b X# Z[: . mIiIs*qkVssmIoIs k` j aMlYi u a u a aMl sPi { OB3q6| R G ^ mpiIl]aMsjnaMs _ vqm7l _ aM` saMs u ijnaRbxbasmpoOl¨>Pa : { gJaM: aMssPi { OB!q6| C |gF\n C |}F\n }[gho C $¨q b C |gF s F L| R !q6| O & aMwMg u mpooIa qmpi { jnawRkVs $ & C F\n%$¨q C F\n aMl C |}F\n C |}F\n%$ML| R !q| & aMl*qmpi { jtawRkVs $ & C 'k >7aMw | ~ l G = G b C | ~ F s & n%$0F C gngng F. C |gF\n C |}F\n b C |}F s L|<OB!q| ~ R q & OB!q| O X /= b 2=. o/k u mpoIw S. sPi {. pwMa iIgCgrbÊqIjtg iIa. . R GgS. ok u mpoIw. Rq. C |}F\n%$ML|,OB| 6q | )R ~. b e 0sPi { OJ!q6| O K ghoIsg X mpokaRoOlg a` { aRbaRoOl u aM: la { bgho]a : b X#. Ma l G C 2g > F mpiIswMmIoIsg u a : { mpoIsokVjnaRbaRo7l:jnaMs$;ÍmpoIwMlgnmpoIs ! Ma l ! Ps i { OB!q R G e i { OB!q| O X C |}F b C |}F XImpoÄaRo u a : u iIgnl$=iIa ! C |}F\n ! C |}F\n b C |}F L|,OB3q6| O OB| q O X ! !GD b C ! C ! X Z ! C |}F\n ! C |}F\n%$ L|<OJ| q O . { l jhkwMmpoOlghopiIgnl!a : u a ! aRo | XmpoÄaRowMmpoIwMjhiIl c ! $ G a q k u m i` mpok S omIlgnaRo7l¦jhk:wMmpo u gnlgnmpo k` jhk:jnghbgnlaaRoÄ~qk { u aMs'k { pibEaRo7ls¦wMjhkVssg2=iIaMs u aMs¦sm7jhiIlgnmpoIs u a/>pgnswMm7s]gtl!aM: saRo/iIlgnjtgnsPkOo7l:jnaMs*gh]o aM: IkVjngt!l aM: s c ! C F C $ s aMl ! C F $ s & & eS i { {. Impo k. 23 ,4526.
(80)
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(82). mIomkIlgnaRoOl0jna q { km Ij aR` bajnghbgnla. \ q b X. C F. kb V y FC npe b !q e b C q F\b n%e $ F\np L| R !q O q dd } C d dd. C 7 F. a q { mIj aR` ba0jnghbgnlaoIa oImIiIs5kVs]sPi { aqkVs:j ZÆioIgnwMgtl!a : u ajhkÊsm7jhiIlgnmpo G mpiIs0qmIi>7mpoIs:jta { >7mOg sPi { nj aMs u aRiIyaMyIaRbxqIjnaMs0sPiIg2>¯kOo7ls c z mpi { jnawRkVs $ n q & n X lmpiIlaMs0jnaMs/;¸mIoIwMlgnmpoIssPiIg2>Vk7o7l]aMsX\qmpi { O . . q R)X smpoOl smOjriIl]gtmIoIs5k7iÄs]aRoIs u a >IgnswMm7sgn!l a : c d b C |gF\n ddd eC . | .\s |FMFM| L|mOC B!q . R q FM| C d . . e . s e R$ s O s . e F L|<O . q R q C R F dd 0aRo |#n!q L| R !q R q C |}F n ddd e dd b C |gF L. |<O q R) . . . . . . . . . .
(83) ú¨§. 44. .' -. . . .
(84) 7. , g %g1+ % 0. z mpi { jna0wRkVs $ X7lmpiIl]aMs~jnaMs];ÍmpoIwMlgnmpoIs!sPiIg2>¯k7oOlaMsXqmpi { mO e q6 R)XRs]mpo7l sm7jhiIlgnmpoIs k7i saRoIs u a/>pgnswMm7s]gtl!a : c b C |gF\n | .s C $ e FM|TL| <OB!q R q C R F dd aRo |n !q C |}F n dd &b C |gF L| R !q R) d Í j aM: lPkVgnl u mpoIxw IgnaRo ]o aM: wMaMssPkVg { a u ZÆiIlgnjngnsa { io wMmpiqIjna C q ! F G % , &(Y ! 0 0 , & ]" I" a { iIl u a0wMaw _ k7qIgnl { a0u aMs]l u mI!i { Ijta G}D ZÆioIa{ qk u { l XoIu mpiIs*b{ mpu oO{ l { mIu oIsP= iI aG3j ZñD mpoÄmkI{s]a { >7a:{ jta wMmpbÊqm l]aRbaRo7l:jnghbgnlYa { aM: lPi u g a: qmpi{ ioIxa >VkVjn{ aRi u #a a j Zñm a a { u ZÆk7iIl a_ qk lX oImpiIssmpiIjngnpoImpoIs j ZñghoOl a : aMl aM$s >Vk gh k IjnaMs:groOl m iIgnlaMs GaMl !aRoÄwMmpbÊqk k7oOl aMss]w aR: bxkVs aRo >¯u k { gh k IjnaMs u C ` q b F X C q b F aMl C q ! q b _ F G U oGa A(aMl{ oImpiIs >7a C {{ mIq boI#s F =u iI a X!qmIu i { wMa { lPkVghoIaM{ s wM{ mpo gnlgnmpoIs a ;¸mpoI_wMl]gtmIooIaRbaRo7'l{ Xio s]w C aR:q bxb FkaRo >Vk gh k IjnaMs u ` mpouoIa abaM{ gnjnjnaRi s aM: sPiIjnlPkVls =i ZÆio sw {{ aR: bxkaRo >Vk gh k Ijnu aMs ` XpwM0a u =iIg aM: lYkVgnl { a: | kwMmpou oIi{ aMs5CoI$ ib a : gtwMF gnaRoIs O Éc X' R G mpiI _sw aR: bÊkwMmp0b Is gho>7k7a oOl mpjnoIaMs5#s kO>ViIk s{ sgh k g I=jniIaMs a X ¨Xk7!oIs0aMjnl axb wRkVsqa { abjraMkl qu m7Zñjhkm k IlgnsPaRkVoIl]g gt{ mIou a bEg aMaMwMgtjnljna aRi { s { aM: sYiIjtXKlYkVilo s sPi { I jnawMmp{ bÊqm { l]aRbu aRo7l u u 0a {` u kO{ oIsjh_ #k WMmpoI0a ` u G { { { { aM: sm7jhiIa lgnq mpo groIu wMaghq ja Z) a : =aÄikVwMla gnmpo a u oIa gna z mOsgts]w smpaR: o bxX kÄaRo[aMs]q l { 2g >paÄgnj aMwM: Ompgr0b kOo7Il ghoI a u jtk7aMoI0s s:>Vjhk k ghWM k mpIoIjnaMas m ,i ` aM`l aM! sl0qq mp{ i mIw _ jhka u a aMl ! u k7oIsjrk WMmpoIaxm mi ` `GaMsl q { mIw _ a u a G U o[a A(aMl X mpo { aMl { mpi >OaEioGq _ aR: oImIb aR` oIa k7ok-jtmOpiIa kÄ` wMaMjhiI` g!km Isa { >P{ ax: jn_ m { s u u aÊjrk u aR: bmpoI{s{l { kVlgnmpo u u ` aÄ j ZÆioI{ gnwMgn_ l a : u u i b q { km Ij aR` baxjnghbgnla G D k7oIs'jh{k WMmpoIam 0 i `xaMsl q mpw a a XwMa =iIgwMm aMsPqmpo "k [q mIw a a X¯ioIas]grbiIjhkVlgnmpo aRo >¯k { gh k Ijn{ aMs q bu oIa oImpiIs u mpooI/a =iIa{ qaRi u Zñgho { ;¸m { bxkVl{ gnmpoIs:sPi { { jnawMmpbxqm { laRG bEz aRo7{ l u 0a `{ X wMa =iIgKaRoOl kVghoIa aMs!groI{ s]lP k Ignjngt!l aM: s*{ oIib a : ?g =iIaMs!{ sPi jh"k >Vk gh k Ijn"a { ` aMwMmpoIsl{]gnlP{ ]i aM: a k { wMmpG o7S l a ioIas{ ghb iIjhkVl]gt_ mIogho7!l aM:` kOo7l jrk { >¯k u gh k Ijna !<;Ímpi oIgnl*` ioÄwMmpbÊ{qm _ l]aRbu aRo7!l!G wMm aMwMl*qmIi ` o mkIs]a >7a0jna q aR: oImpb aRoIa0gho >7a sa kOoIs*jh0k WMmpoIam i `/aMsl*q mIw a a $. . ! ! +# %'&3 ! 1 & U &# & ! ($# . p m I i aM: lPi u gnmpoIs0qIjhiIsgnaRi { s0sw _ aR: bxkVs u asghb iIjhkVl]gtmIo c u aRiIys]w _ aR: bxkVs!aRo>Vk { ghk IjnaMs¦` aMl s C b sw _ aR: bxkVs u aM: wMaRoOl { aM: s!aMl u a eIw _ k { ;ÍaMlla { @ ibxbaMj F XioÄsw _ aR: bÊkaRo>Vk { ghk IjnaMs"GaMl b aMl io sw _ aR: bxkwMmpb0Ighok7o7ljtaMs{>¯k { ghk IjnaMs ¨X !GaMl b G I aMs5q { groIwMghqaMs0aR: o]a : { k7iIyxsmIo7l!jnaMs*sYiIg?>Vk7oOls c S o0wMmpoIs]g u a` { a jnaq { mkI j aR` ba slYkVlgnmpookVg { a wMmpbxba'jhkjtghbgnla u iq { mkIj aR` ba#a : >7m7jhiIlgJ;jnm { sM=iIa 23 ,4526.
(85)
(86) !" #$ % &' () *+ ,'- !" /. 0 12 3(4! 56 7 . O. jnaI l]aRbxqI_ slaRo u >7a { sj ZñghooIg G asw aR: bxkaRo laRbxqIsaMslgrbÊqIjtgnwMgnla G K w _ k =iIaEqk-s u a l]aRbxqIsX mpo { aM: smpiIl u ZÆk m { u j Z[a : =ik-lgnmpo u awMmpoOlghopiIgnl!a : XqiIgnswMaMjtjna u a z m7gnss]mpo G I aMs5oImpoÄjngh]o aR: k { gnl aM: ss]mpo7ll { kVgnl!aM: aMs qk { iokVjn7m { gnl _ bEa u aa Nlmpo@ kOq _ smpo G I Z) a : =ikVlgnmpo u awMmpo7l]groIiIgn!l a0: aMs]lwRkVjnwRiI6j aM: as]mpiIssPk/;Ím { bEa*wMmpoIs]a { >¯kVl]g?>Oa c ` e f np. k u gts]w { aM:c lgnsPk-lgnmpo u a:j Z[a : =ikVlgnmpo u a z mOgts]smpoÊwMm {{ aMsPqmpo u k"` jrk u gnsw { aM: lgnsPkVlgnmpoÊiIsPiIaMjnjta u i kOqIjrk-wMgtaRo e b s b e b n e ` |. I. I. ¥. .. z mpi { jnaMs~s]w _ aR: bxkVs¦aRo#>Vk { ghk IjnaMs$`ÄaMl b X7j Z[a : =ikVl]gtmIo u awMmpo7l]groIiIgnl!a: aMsl u gnsw { aM: lgns!aM: a!smIiIs hj k ;¸m { ba c D. S. ` e ` ¥
(87) e. . . e| ¥. . . . n . o Ii lgnjngnsa u aRiIy Yb aM: l _ m u aMsqmp i { wRkVjnwRiIjna { . G k7oIsjnasw _ aR: bÊk u Ma : wMaRo7l { a : X . aMslwRkVjnwRiIj6a : u a0jrk ; k.wM mpoÄsPiIg2>¯kOo7la. c. b e b q e ` ` n ` . e | . ¥ ¥ | k'>7aMw dd ` s n !q . f d ` . n ddd dd ` s n f C D k7oIsjta/sw _ aR: bxk u a eIw _ k { Í; aMlla { @ ibxbaMj X¦. jta[q { ghoIwMghqa[aMsl u ZñmkIlaRoIg { jhk,>¯kVjnaRi { u a qk { groOl!aM: { kVlgnmpoÊaMykVwMlasYi { OB| q6| R u a0j Z[a : =ikVl]gtmIoÄaRo ¢C >7m7g { D#G I G eIw _ k { ;ÍaMlla { aMl S G!.0G# ibxbaMj O 7HR\aMC l|}F } G e G }[mpw L O R F { G OB| q6| C ` F o[sPiqqmOsa0=iIa" aMsl*wMmIoIslPk7oOl*sYi R aM: IkVj k# . aMl=iIa b C |}F aMs]l koIa sPi { OJ| q6| ¤ R C aM: IkVj k` b s C | e | F ! f F GS o"¤ a(A aMwMlPiIa'jna~w _ k7oI7aRbEaRo7l u a]>¯k { grkIjta ` C |}Fxn%$^C |gF &(') GS oGk[kVjnm { s* C |gF0n $ C |gF & ') G U o biIjnlghqIjngrkOo7lÄj Z)a : = ikVlgnmpoqk { " &(¤ ') aMlaRoghoOl!aM: { k7oOl u a | k ` | X mpoÄmkI lgnaRoOl c | + C b e b F ` e + C b e b F ` $q C Mc F n . ¥ .
(88) ú¨§. 44. .' -.
(89) Vc. , g %g1+ % 0. k'>7aMw. C F\n e P ) c I a sw _ aR: bÊk u aM: wMaRoOl { a : { a >IgnaRo7l ibxbaMj\k'7> aMwioIa;¸mpoIwMl]gtmIo + C F\n "s G . +. C R F. kG` q { aRo u { a jna[sw _ aR: bxk u a[eIw _ k { ;¸aMll]a { @. u { u z { _ { b {;Ím ba mpc i jna0sw aR: bxk:aRo>Vk ghk IjnaMs [ aMl XOmpo gts]w aM: lgnsa:j Z[a : =ikVlgnmpo a*wMmpoOlghoIiIgtl!a:: smpiIs!jhk . F . n . . ` C b q . k'>7aMw. . .. . . . . e e ¥ . e ¥ |. . . `. . .. . . e | . np!q ¥. C b q b. q. . . . q. . C R F C Q F. F q. F\n ¤ s . ) . ¤. . 7 K P ) 5 I k!l {MaM` sg;Ím { l]aÏoImpo@jngho]aR: k { gnl a : u aMs¦a : =ikVlgnmpoIs\qm7sa'wMa { lYkVghoIsÏq { m Ij aR` baMs C wMmIo >7a { @ 7aRoIwMa:jnaRo7u l]a XPaM: w _ aMw u {Ma` jhkb aM: l _ m u a u ua0a N{ lmpo F G z mpi { v { aRb a : u gna { mpboxqaRiI'l Xps]m7gnl~q { aRo u { a ibo qnkVs a l]aRbx{ qIsxl u aM{ sxqaMlgnl{ X sm7gn` l g2>Igts]a jta[qm7l]aR{ o7lgnaMj0k7qqIjng2=i]a : u aRo 1qmOlaRo7l]gtaMjns {q mIj aR` b a u aaMl j Z[aM: aMlP: sk7mIqi aq a{ aM: jnwa a : u q aRo7mlIa:j aRaMblawMmpbÊb;Ím7a0gnsxwMmpaRo oGu qgnlgnmpaRooIk7s0oOk7lxiIywMmpjnbxgrbEbgtl]aÄaMs wMmpb o gnln%gnmp$oG ghoIe gnlghkVjnba[jtGa `. . .. C b q b. . q. . . q. . ! . !. . .
(90). . .
(91). '. . . z mpi { jnasw _ aR: bxkEaRo->Vk { ghk IjnaMs ¨X ! aMl b X j Zñg u aM: aaMsl u awMmpb0IgroIa { jnaMs>Vk { ghk IjnaMs aMl ! qmpi { jhk { aM: sm7jhiIlgnmpo u aj Z[a : =ikVl]gtmIo u a z m7gnssmIo G I axiIl { aMw _ a { w _ a0: aMsl u axq { g2>Ignj6aM: 7gna { jhk ` `GaMslq { mpw _ a u a C g G a G ! F aMl0jhk>Vk { ghk Ijna ! u k7oIsjhk >Vk { gh k Ijna u k7oIsjhkWMmIoIaxm i< WMmpoIam i ` `/aMsl*q { mIw _ a u a C g G a G C ! F G S ogho7l { m u iIgnl0kVjnm { s u aRiIy;ÍmpoIwMlgnmpoIs{` aMl+` X u a: oIgnaMs*qk { c %$ ` C b q Fn+ ¤ q ` C b q F n e ¤ q aMlmIo { aM: smpiIl j Z[ a : =ik-lgnmpo u a z m7gnss]mpoÄsmpiIsjh/k ;Ím { bEa e$¥ f b n e C ` s C e F ` Fq . . . . . . . . 23 ,4526.
(92)
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