Optimal Shells Formed on a Sphere. The Topological Derivative Method

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(1)Optimal Shells Formed on a Sphere. The Topological Derivative Method Tomasz Lewinski, Jan Sokolowski. To cite this version: Tomasz Lewinski, Jan Sokolowski. Optimal Shells Formed on a Sphere. The Topological Derivative Method. [Research Report] RR-3495, INRIA. 1998, pp.62. �inria-00073191�. HAL Id: inria-00073191 https://hal.inria.fr/inria-00073191 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimal shells formed on a sphere. The topological derivative method.

(3) . and. . N ˚ 3495 Septembre 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(86) . . . G 3D G. & ' . ' *._ikjkoduq [l_bacV m ; V&gV/f m w V ]pf qsaj[\dgf{d m V/ w V/V(% _bf m d w m V w. 8M : V . . \_¤y°A\_.

(87) .

(88) . ' * H M : ' * . w V_^_bf m [eW _bg[\f _ w nh_ w a k d ._]At [\ê[`_ w n ]Af qsac[edgf{i[V/fCun ' i)* . koa w Vkck± ]pf qsaj[\dgfAp koV/V' i)*. :M . . J 3 ' < *i ' < * ' < *i ' < *.k V/V4' . )* v' *i

(89) v' * 0' *·.koV/V' (y)*3i' i)*

(90) v' *i v' *.k V/V' )* . [efuajV/ w _aj[edifqsdgf k a _f a k X : X ] X 3. . . °G° â`¬qª¬¨

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(93) /.

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(95) [`k V/ê^h w Vm k V/f acV m S [ef a w V0m h _h}V w un ; m ^\V/aC&H [efAiV S _f w _S WCW m 'c /gi)* w V0qsdd [\Sf _bacVkS d)fAd w VkdS )a S V [ qsdgfkoa ] qsS acV w dgfw _k V&ê^ _ V S a V S Vko[\m fA4i _ [`_bm A^\Vk [\a m [\fCw a [lk)_bhph S d_gq a V aj% dih}diêdig[lq&_b^$m q _ _iqsajV [lk aj[lq&k S d m a V.k V/êw ^ S dgw W{_[ef dfAdia2]pf V gdE_bfu n{S q S _bfAiS Vk fAV0q/_bf [ef [lq/_ajV.a dCWCV/a d k2d ] a V k ajV/h kKd) dihAaj[\WC[yH _baj[\dgf [`q q _bfpgV a S V.S k S V/w ê^ m dgW{w _[ef . S [0* a S V V/i]A^l_ [yH _baj[\dgS f{ajmVy q fp[]AVd _ m WC[eacaj[efpQa S VQq dgWCh dk [eacV w V/i[edif k

(96) [e[3* a S V S A]ApAêV S WCmV/a S d ;RdgS a a VS WCm V/a d k _R V}V&V/f _bm hAhpê[eV m W{_b[\fA ênCacdChA^`_bajVdihAw aj[\WC[yH _baj[\dgf h Sw diAS êV/W S k V%m WCV&a d w ' [0E* qsdgfko[lk w a kC[ef _ WC[eacajS[efp%w fAV qsdgWCh dm ko[eacV V/i[ediw f k3[\f [`q a S V dg[m kQ_w hAh V_ _ba_ WCR[lS q dkcq/_b^\V a ka m V/d V/ac[lq/_^s diS ]Af _aj[\dgf k0_ VCm g[ V/f [\f´8 d f _bf m Z a S _wfA2j' &gg* VRSdihAaj[\W{_b^ V/f w[\fAQhp^l_bacVk m _ V+}V&mV/fL dg,]pf [ew f [\Am [l_f k#Su" [e[ _bf V i" _bV¬S c' /ibX7* VQ[\mf7 hp^l_bfpVQh dgA^\V/W [lk [`kjqs] kckoV [ef ê^`_b[ V _bf 8 d f Rj' &S gVCgV"*i* Vqiq aj[ a VVq S _ dw d+_i" qsajuV n w [lk ; aj[`V/q/f k2dkbVqsdgj' WE &gh}id)ko* [\ajV m diW{_b[\f k _ w V0g[ V/f un a S V S dgWEdggVZ fA [yH _baj[\dgf2 d mw W0 ]A^`_bV d w a S [ef{ V/Rf S m [efAw 0hA^l_ajVk)w k ] w q S d ww W0S ]A^l_V S _R V w V/V/S f dg]Af m un w V]_ im _bd]p*a_w f diW ; m V/V aj V/^\[ fê ] kPk" [ j' &_)f y( m )*R V/^\V/V/_4 [ j' &q gdgi]A)fu* ajV , h xjm_ ]a kokGaj[U1 d q/_a aj[edi[\ff8d dia [eSacV Vk V k k d w V&W0ê^l]Ak^lq/__V fS _gVk V/V/f%hA]pAê[`k w S V dgfpS ên w VqsV&f acên qi R V/êV&_{_bf ; Vm [ f w kP" [·j' &gg* S RS VkoV S dgWCd gV/w fp[H _aj[edif Vk ]Aêa k _R VfpdgaK]Ah{aj[e^\^GfAd V/V&f3]koV d kodiû [efA0a VdihAaj[\WC[yH _baj[\dgf hRS diAêV&W{k S m S VS WEV/a S d w ' [e[0* _gk m V/V/f S d w w [\g[\f _bacV m m [ef kjq Sw V/f _]AV w V/a0_^ j' /ig6w &gX7*iK qq w Zuq ]AW _gq V j' &gi){* _f a V V _bhAhpê[eV ajdí _ [edi] kCdghAac[eWC[H _bac[edgf h diAêV&W{k² d hAw ^l_ajVkQ[\f%m V/f m [\fA _ik V/^\^)_ikm ko]AS uxjwVq ajV m w ajd a S VC[ef6 hA^`_bfAS VEêd_ m [efpk

(97) d VR V w $ _ w q S ]qs[l_^ V

(98) 1 fA[eac[edgf d kod q&_bê^\V w q _ _gqsacV [`koac[lqg ]Af qsac[edgf _gS kQdifAên w V/V/f%i [ V/f d aRS VWE[efA[\WS ]Am Wq dgWCS hA^\[l_f qsV h diAêV&W G qim S ' g) K* [\f

(99) kcq V/f _b]pV V/a _^ j' &gbX6* Q V E W / V a d ' [e[3* _ik fpdgaK}V&$ V/f_bhAhA^\[eV acdCk V/ê^`k | VyqsV/fuajên4Zd+m " dgêd kP" [K_f m & duq S d kP" [/j' &g) (+*% S _ V{[\f a w d m ] q V m a S V{fAdiaj[edif d. _ ajdih}diêdig[lS q&_b^ V w [i _bac[ VdU k S _h}V- ]Af q aj[ediS f _b^`k fpVKq/_bf0m k S w d [ea S m V_gk V)w a S _a[efa S Vw q/_gk Vd a VqsdiWChAê[`_bf q V WC[\fA[eWE[ yH _baj[\dgf)a Vajdgh diêdgi[lq/_^ V [i _bac[ V [+* V kG[\f _± _iqsajd w diW S a S V w V/êV¬g _f aKq S _ w _gq ajV w [lkoac[lw q ]pf qsaj[\dgS fd w kjq S V/f_b]AV w V/a+w _b^ j' &gbX6* f m V/V m m [efCa VWC[\fA[eW]pW qsdiWChAê[`_bf q V)h diAêV/W a VV/tuh Vkjk [edgf ' )(+* diW Zd+" dg^\d kP" [U_bf &' . \_¤y°A\_.

(100) .

(101) . ¶. .

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(103) ' % % * 3 46' % % * 3 ] ko[\f qsV S V ww V w J E J )w _f m 5 3=3 RJS 7w w V/h w Vm koV&f aw k a S V

(104) S di]AfAWEw d m ]Aê]k

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(108) k V/V' X g* RS Vs1 w koaajV w W V&h}V&f kGdgf. . °G° â`¬qª¬¨

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(116) X /37 3D.

(117). . . . . . . . . . . . . .

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(120) .

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(126)

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(131)

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(134) d w W{_bac[edif k d a S V.k S V/êÛ_ w V m V

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(136) koa w Vkjk)WCd m ]pê[ d w _bf[lkodia w dghA[`q d m n. 2A@ ' 2 @ 2 @ * 7 ' ' i)* ) 2)@ J 3 S V w V w V/h w VkoV/fua k

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(139).

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(151) 1 fpVa S V A[e^\[efAVy_ w d w W ?' ] * J 2)@ & 2)@' * & ' * 2A@ 2)@' * ' * ^ ' ' i) * _bf m a S V ^\[efAV_ w d w W 5? ' * J X X ^ ' ' ' i) * RS V/fa S V h w dgA^\V/W#d V ]A[e^\[e w [e]AW d _0qs^`_bWCh V m k S V/^\^G[l5k d w W0]AêV m _i±k diêêd k ' v ? * 1 f? ' m ] * J > < 5? k' ] q * S a S _ba > < ' ' ' * R, S [lk)h w dgA^\V/W#[`k]AfA[ ]AV&ênEkodg^ i_bA^\V koV/(V m ;m V w f _ m m dg] 'j &gg * m kjk ]Am WCV a S _ba _0m koV/iWCV/fua I M / I [lk)êd_ V _f I N [`k2q ^l_bWEh}V RS V/fa S Vkoh _iqsV dr_ WC[`kjko[\AêV [`kohA^`_gqsV/WEV/f a k[lkg[ V/fCun K > ?J ' 9 ? '=,L** 3 . 9 3 '=,L*63 J ] J dgf IN ' ' ' u * KB RS VQV ]A[\ê[\ w [\]AW V ] _aj[edif ' (*r_ikjko]pWCVka S tV d w W 2A@ & 2)@' * 2)@ 2A@6' * ^ J 5 ? ' * 5 3 ' * .>? ' ' i) * . . . . . . . . . . . . °G° â`¬qª¬¨

(152) ©. . . . . . . . . . . . . . . . . . . . . .

(153)

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(155) 1 fA[\aj[\dgf kZ k V/V z2IM [ed w m k dgf2'c /gi)* a[lk_ikjko]pWCV m S V w VrdgfA^\na S _ba$a S Vk V ]_bfuaj[eac[eVk _ w V-"ufAd f0_b^\dgfA RS V{ dg]Af m _ w ní _b^\]AVCh w dgA^\V/W ' v ? * k S dg]A^ m }V w V/hA^l_iqsV m [\a S ' v 3 * [ea S 5 ? w VZ hA^l_iqsV m un w5? m 5 3m _Sf m > < w V/hA^`_gIqsN V m un >? RS [lk2fpV h w dgA^\V/W[`k_b^`kod{]AfA[ ]pV/ên ko diî S_bA^\VA h du [ I V J a IM _a)S WCV_gw k' * v f3a V.q/_gk VQd a V h dgA^\V/W ' 3 * [`k)koac[eê^ V/^\^ h dkoV m A [ ' 5 ? 5 3 *i' * J C ' ' X7* S VwV C J U ' 9 ? -' ,*C* 3 . 9 3 =' ,L*63 ? ' ] * J Z ' ' ' i)* , q/qsd w m [\fAajda S V a S V&d w V&W#d ; V w f _ m dg]_bf m [l_ w êV&a' k V/V(; V w f_ m di] c' /ig)** a S V koV/a C [lk)qsdiWCh dk V m d$ a w _f ko^`_bac[edgfk_bf m [\f71 fA[\ajVk [eW{_^ w dga _baj[\dgf k C J U 3 < _f m < k ] q S a S _ba J7 < < . $' ^* Z ' ' ' i)* V w V < < _ w VL Vyqsajd w k[ef 1 RS V{qsdif m [\aj[edif ' bX7* WEV_bf kKa S _aQa S V^\d_ m [efA _bS hAhpê[eV m m [lkCk V/S ^ _ VRS]A[e^\[e w _bajV m %. a S V m w Vyko]Aw ^\a _bfua4m d w q V VqsS ajd w _bf m m WCw diWCCV/fuag V q ajd w k diw ]A^ m i _bw fA[`k V0kodg^\]Am ajw[\dgf m [lk V&ajV WC[efAV w ]Ah ajdEa Vg1 V&^ k: diW G [ V ]Ah ajRd S [\g[ a _bfw ko^l_aj[\dgf kK_bf [\g[ [\f71 fA[\ajVk [eW{S _^ dia _bac[edif k w S S [e^\V)^ fAVdga)]A[e ^\[eV w [eV]Aq/W _êêVV m ]$* _b_bajf [\dgm f2a ' S V.iko)]p* [\koWC[ m hA[l^\_ [ew Vn{kGa qsdiVf m^\du[\ajq/[e_bdiÂf V kK _b]Aê[\diê[\fA [\]AIWM V] _bac[edif k:' [lq = 2 J = 2 ] J ] J dgf IM ] ' ' (* S VwV = 2 J ' @2)2@ 2 @ * B 2)Y@ @ ] J B @ * ' B 2 C @6* ] ' ' i)* 3 2 ) 2 @ J B B 2 %@ ] 2 2 S V w V J B 2 J C 2 ' J 3 3 J 3 3 J 7 K KG5 J ' K K ^ 2 * C 2 7 5 h _ w _bWCV&a w [/H Vk IM V S _ V B 2 J 2)@ C "@ ] C M J 2 M B Y2 ] ' ' i)*. . . . . . . . . . .

(156) . . \_¤y°A\_.

(157) . ¶.

(158) . . /.

(159) ! " # $# % ". S V w VL' )2 @ r* _ w VqsdgWCh difAV/fua k da S V | [lq&qs[ ajV/f k d w dgfa S V.ko] w _gq V kH . G=6DE R?´D] = /3

(160) 32

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(162) ©. . .

(163) .

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(165).

(166). .

(167).

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(169). . \_¤y°A\_.

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(171) <;V/^\wa w _WCw [ dgh V w _bajd w [lS k m V

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(173) 1 fp[efA0a S V V"* Vq aj[V+k S V_ w d w qsV dgfa S V dg]pf m _ w nd a S V.k S V/^\^ ;n ' * V S _RV gdgaKfAd & 2)@ ' * JJ 9 3? ' 2 @ ? @ 2)* ? 2A@ 9 ] ? 9 ' 'e i *. 2)@6' * 342)@ ' @ 2 2 @ * 2A@ ] S [lq S W{_ "Vyk [earh dkjk [eA^\VQacd w V_ wjw _fAgV.a S V V/fpV w in m V/f k [ea n 2)@ & 2)@' * 2)@ 2)@6' * ' 'e &) * ajd0a S V d w W D 2)@ & 2)@6' * 2A@ 2)@' * ' 'e &) * [ea S D 2A@ J 2A@ 2)@ ]2 D 2)@ J D @A2 ] ' 'e 7X * 2)@ ] 2)@ J @2 '. 2)@ ' * J 342)@ ' 'e &) * 3 z2RSdgacV a S _a J ' w ? ] 3 ] *r_bf I m M m V/h V/f m k)dgfA^\nCdgf S w VQV *GVyqsaj[ V d qsVk)_êdifA koV/V ' i) * _gkjk ]AWCV a tV d W = 2 J D 2)@ B @ ' 'e &) *

(174) .

(175) ! " # $# % ". . . . . . . . . . . . . . . . . . . . . . . . . . . . °G° â`¬qª¬¨

(176) ©. . . . . . . . .

(177)

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(179) V 1 fpVa V A[e^\[efAVy_ w d w W J 2)@ & 2)@' * & ' * 2)@ 2)@6' 9 * ' * ^ ' ' 'e &) * 3' ] * z2dgacV a S _a2a S V.koh _iqsV C J U ' 9 ? '-,*C* 3 . 9 3 '=, 3 ' ] * J Z ' 'e &) * 3 [lm k)V ] _b^ S acd wC m

(180) V 1fAV S m n ' g *)_f m S V&f qsV [lki[ V/f{un ' g * RS [lk)h w dgh V w a nC[`k ]AV.ajda V V/^l_aj[\dgf k [eh & 2)@' * ]. 2)@6' * J 2)@6' * ' ' * n S [lq S V k V/V a S _a & 2)@ ' * J 2)@ ' * J [ _bf m dgfA^\nC[ & 2)@' * J 2)@6' * J ' % fAVCq&_bf h w d VE_b^lk d{a S _aQa S V d w W 3 '$' ])' *,[lk > ? V/^\ê[\hAaj[`qQ_bf m qsdgfuaj[\f ]pdg] k RS ] k a S V h w dgA^\V/W ' v * 1 f ' m ] * J >? ' 5 ko? ] q S 5 a *iS ' _ a * >? ' ' ' u * 3 3 [lk]Afp[ ]AV/ênES kodiw^ i_bpêV ;nC] k [efA0a V V/^l_aj[edif k 2 J J 2)B @ 2 B B C 2 ]H J J 2)@ 2 B B 2 C ] J 2 C 2 ] D 2 H@ ] J = 2 C D 2 @ ] ' ' i) * J = 2 B. . . . . . . . . . . . . . . . . . . . . . . . 2Y] 2. \_¤y°A\_.

(181) . ¶.

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(184) DE! /0= D iB/D´= )G7 0D D / iBgD´ R /0 D)B . , kjk ]AWCVa S _baD 2)@ _ w V w V/h w Vk V/f acV m un 2)@ D 2A@ J 2 M @ 3 M ' ' i)* ' 3 % fAVKq/_bfh w d V)a S _a[ ' ^!* [`kko] qs[\V/f acên w V/g]p^l_ w g a S V/fa S V)V ] _aj[\dgf k D 2)@ 3 @ J _ w VJ kj_ aj[` k91 V m [ m V/fuaj[lq&_bê^\n S ; V/a 2)@ } V h _ mw ac[l9 qs]A^`S_ w [\f acV/ w _b^ d w ' g)* ? S ; V&a)S ] k hp]Aa ajdCW{_" V a V.]AfA[ea k d _f a Vkc_bWCV aK[lkk V/V/fa _aa V.êduq/_^ . . . . ^ J. . . . . . . . . . °G° â`¬qª¬¨

(185) ©. . .

(186)

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(189) . ¶.

(190) . . g.

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(192) ©. . . . . . . . .

(193)

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(195) ? Z = OP'O

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(197) @ OP'O ; V/a] k [\f a w d m ]qsV)a S VKh S nAk [lq/_Âq dgWCh dgfpV/f a kd P] 9 _gq/q d w m [efA.ajd a S V w ]AêVyk ' /)* V m VZ1 fAV ? J 3 ] ] J ' ' Xi_)* o k e [ f S V w V J >? _f m J >3 G qi [\ RS Vh S npko[lq&_b^qsdiWCh dgfAV/fua k)dr a S Va w [l_b^ 1 V/^ m J ' ? ] 3 ] r* _ w V m V

(198) 1 fAV m un ? J ] J ' ' Xb* ko[\3 f ] 9 J ' , q/qsd w m [\fAajd0a S V w ]AêVyk' &)* V m V

(199) 1 fpVa S V h S nAk [lq/_^ qsdiWCh dgfAV/fua kd-' & 2)@ * & 2)> @ J 2)@ ' ' X)* _bf m 1 f m ? N ?=?aJ @ 9 ] ? 3=3 J @ 4q dk ko[ef 9 ] ' ' X6(* ? N ? 3 J @ ko[\f @ qsdk ' J H J J J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \_¤y°A\_.

(200) . ¶.

(201) . .

(202) ! " # $# % ". )(. [eg] w V 6. " Y ! - # t U ! Z ] ] 9 , q/qsd w m [\fA ajd ' (* VW]koa0kodi^Va S V konpkoajV&W . 2)@ J V h w V m [lqsaa S V kodiê]Aac[edgf[ef{a S Vt d w W J : ' *uqsdk ] J : ' *uk [ef ] ' ' X)* 9 J 9 : ' * qsdk ' V/f q V V 1 f m a S 4 V ; V/iV/f m w V V ]_baj[\dgf ' ' X)* 3 3 4qsac 4 ko[\f 3 3 9 : ' * J dw 8 3 ' 9 : ' *uqsdk * J7 ' ' * _bf m [ea k)kodg^\]Aaj[\dgf S _gk)a S V d w W )_ * J7 9 < ' * J < < 9 < ? ' * < < 9 <3 ' *] S VwV 9 < ? ' * J qsdk ] 9 <3 ' * J qsdk ^\f qsaj ' ' ' u * . . . . . . . . . . . . . . . . . . . °G° â`¬qª¬¨

(203) ©. . . . . . . . . . . . . . . . . . . . . . . . .

(204)

(205) . g *. . J. S VwV. 9 ?' * J. 9 ? ? ' * ? ? 9 ?3 ' *] 9 ? ? ' * J k [ef ] 9 ?3 ' * J ko[\f êf aj sq aj ' . . q* . ?. ? . . . . . . . . ' ' i)*. . . . . . . . 9 : ' * J : ' 4qsdk * ac : : ' q dk * ' qsac : ' ' i)* _bf m X < X ? : X < X ? : _ w V.qsdgf k a _f a k % fAV{q/_f%k S d a S _ba a S V{kodiê]Aac[edgfk 9 < 3 _bf m 9 ? 3 q&_bfAfAdiakc_baj[`k9 n a S VEqsdif m [eac[edif k 2A@ J S V/f qsV{_ w VCfpdga_ m WC[`kjko[\AêV RS VL1V/^ m k : : _gkckoduqs[`_bacV m [ea S dga S V w kodiê]Aac[edgfk2_ w V @ : @ : : J : aj 3 : q aj 3 $k [ef ] ' ' X7* @ : @ : ko[ef ] : J : ac 3 : qsaj 3 m [Uk' P] ] 9 * J ]/ ] ])')')' V+fAdgacVa S _a) dga S a S V.qsdgf m [eaj[\dgf k' )(+r w * _ V A ] ^ 1 e ^ e ^ V _ w VCw dK m a S V d w W ' i)* -' bX6* d w J f h _ w aj[Ù qs]A^`_ wZ a S VEqsdif m [eac[edif ' * aj ] fAV m dg]Aa)acdS V._q dgf k V ]pV/f qs Vd S & 2)@ S E w ]$ ] dgf q ê] [\fAA a V.kodiê]Aac[edgfd±' )(+* _gk)a V d W 9 ? J 9 : ' u* qsdk 9 : ' u* k [ef ' ' i)* : < [ea S 9 : ' )* g[ V/f un ' i)*- d w _f m 9 : ' K* g[ V/fun _{k [eWC[\^l_ w d w W0]A^l_ [ea S fpV qsdif koa _bfua k : : ; wV/a{] kCaj] w f7 fAd S acd S k dg^ [\w fA4a S V k nAk ajV/W ' XC* _f m ' X* Zdg^\]Aaj[\dgf7acd a S V h diAêV&W ' X* _gk)a V d W . ? ? J ) : ? ' u* qsdk ' * u k e [ f ' ' i)* : : < [ea S ) : ? ' *i : ? ' r* kc_bac[lk9 nu[efp>' X)r* _f m ' * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \_¤y°A\_.

(206) . ¶.

(207) . . g.

(208) ! " # $# % ". z 2d a S V w VE[lkQfAd _ w g]pWCV/fuaacdfpV/g^\Vqsa a S Vk dgê]paj[edif4d+aonuh V 9 <3 ] 9 ? 3 k V/V R' S V * ]pf qsaj[\dgf ) : ? ' *)_ikjk ]AWCVka S V d w W . _)* J7 ? ) < ' * J < qsdk @ < = ' *] ' ' (* = ' * J qsdk êf q aj 3 ' * J ? ) ? ' * J ? k [ef ? ' *] ' ' i)* ' * J k [ef êf aj @ qsaj ' 3 q * : : ' ' i)* ) :? ' * J ' 4 q d k * j a ' s q d k * s q c a : : ? _bf m m a S V d w W0]A^l_V d w : _ w VQk [eWC[\^lS _ w w m [ef [efAKk dg^\]A aj[\dgf mkGac]d S ' X *G[`S kG_)Ww] q WCd V [ {Vq ]AJ êa a _i#k " , RkcSko]AWEV a S _ba ' ] * J : ' *uqsdk _w f q _bfAiVa 4V i_ [l_AêV [efuajd qsdk V/fV ]_baj[\dgf´' X)* S _gkcko]AWCVyka V d W 'j V 3 * 3 V 3: V V : %' % * 3 V 3 : J ' ' * [ea S %' % * J \ 3 ' ' ' u * RS V ] _f ac[ea n % [lk)_q dgWChA^\V/tfu]AW V w % J \ _] ' ' i) * dga S _bf m m w }V&[efA)fAdif H/V w d RS Vrkodg^\]Aaj[\dgf.d ¶' *G[`kGkoh_bfAfAV m d V w a S V ]Af qsac[edif k d-; V&gV/f V a n h V M : ' qsdk * _f m M : ' q dk * ] ' ' i) * Vq/_b]koV % k V/V0z2[ed w m k dg f w 'j &gg w *d w _bfp _bf m ]Ad 'j &g8iM : ) * V RS V0m fAV/M :_aj[ VV kom [\gfEw _ba [e2f ' gw ) *[`k a _ "V/gf d ]Aaj] VKqsdg6f V/fA[\V/f qsV f _iqsRa ' *_bf ' * [ * V [\f²_ _iqsajd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(209).

(210). . . °G° â`¬qª¬¨

(211) ©. . . . .

(212)

(213) R S ]k a S Vrkodg^\]Aaj[\dgf d¶' X)* [lkGa _"V/f._gkG^\[efAV_ w qsdgW0A[\f _bac[edgfkd w Vy_bû_bf m [\W{_bi[ef _ w n h _ w a k)d M : ' q dk * V w V&h w Vk V/fua 3 n 3 J ' : qsdk : ko[\f * ' ' X7* : < [ea S H M: ' * : H M : '

(214) * ' ' i) * : J ? : :M ' * : :M '

(215) * 3 : _bf m : w [`kgw [ V&f n M :_ko[eWE[e^l _ w d w W0]A^l_ V w V :M ' *+_bf m H M: ' *+_ w V w Vy_b^r_bf m [eW{_g[\f _ nh _ a k d ' qsdk * M : ' qsdk * J :M ' * \ H M : ' * ' ' ' i) * z2dg:M acVa S _ba M : m ' H M: q dk * J :M '

(216) w * \ H M: '

(217) * _f m a S _a [`k S n a S V ]Af qsac[edif k RS V '

(218) ]pf qsaj*r[\dg_bf f k m H :M ' '

(219) *._bf m *V/H fuM: ajV ' *' _ wgV * k [efAi]A^l_ w _a .J

(220) S [e^\V3a S V ]Af qsac[em dif k :M '

(221) * _f M: '

(222) * _ w V{k [efAi]A^l_ w _a J RS LV ]Af qsac[edif k[\7f di^ V [ef ' w gw *Kq/_w fAfAm dgaK VQS V_iko, nCV&t h w m Vkjk V m n®" fAd f koh Vqs[`_b^ ]Af q aj[edif k RS V/[ w h w dgh V w aj[\Vk _ V V&h}d acV [ef{a V hph}V&f [et m ; n] ko[\fA2 ' Xp *i' g *0 ' g *r_bf ' 7X *3i' i) *difAV _ wcw [ Vk_baa S V 1 f_b^ d w Wd ' ] * J ' : ' *uqsdk : * ' *uk [ef * ' ' ' (* : < RS VQ[\f m [`qsVk 0_bf m 5 [e^\^G V/êd VdiWC[eacajV m RS V ]pf qsaj[\dgf k : ' *r_gkcko]AWCV a SAw V/V d w W k )_ * J7 H M <' * J ? < M ' * < M '

(223) * 3 < ' * ' ' i) * < H M '

(224) * = s q d k '. * ' < < V w V M ' * J <M ' *i H M ' * J H M < ' * * J ? ' * J ? ? ? M? ' * ? H M? ' * ? M ? '

(225) * 3 ' ' i) * ? H M '

(226) * ? ' *' ? k [ef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(227) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \_¤y°A\_.

(228) . ¶.

(229) . q* . ' *. .

(230) ! " # $# % ". . ' * 3 : H M: ' * : @ : :M '

(231) * : H M : '

(232) * ' '( * : @ ' 4qsdk * ac 3 : : ' qsdk * qsac 3 ' J WCVy_bf ka S _a{a S V m V

(233) d w W{_aj[edif7[lk w dga _baj[\dgf _ê^\n

(234) RS V q&_gkoV kwonuWCWCV/a w [`q RS Vrq/_gwk V RJ S w VZ V w k ajdKa S Vr_gkw n WEw WCV/a w S [\qGêSd_ w2m [eS fpw k

(235) iq/_] ko[\fA2m Vykjkow V&f ac[l_b^ V_iqsaj[\dgf k+_bwa.ko]Amhph}d a k V0S q&_gkoVyk m RS V

(236) V ajd SA[\w V _ WCw difA[lw q&k Vm kcq [\Aw [efA koV/Û_ V ]A[\ê[\ S _ajV S k a _bacVm kdK a w V0êd_ [\m fA S ] m kQ_êwâ V/VEw q/_gk Vk V

(237) S V ajd [ * V V/ f a koa _bajVyk d a V.k V/êÛ_bf [ea)[`k V VyqsajV [ef{a V [+* V V/fua5 d W{kd a V.kodiê]Aac[edgf m m 9 m m; V/V/aca)V ]w WCkh[\fAw Vdum qsV&uV n´fA' d gaj*idgk1 ' f g[efp.* Q_ba f S m VW{' _b )[\fC(* kc q/_b ^` _ w VE]Afqsu" dgfAf7d 1 fAfV{k_b.f_bênp_bkof [`kac da RS VS V&acnCV w _ W{w V k [ea S q dk V1 f m 9 ' ] * J : < 9 : ' *uqsdk ] ' ' ( * ' ] * J : < : ' u* qsdk ] : . . . . J. ? : . . :M. . . . . . . . . . . . . . . . . . . . . . S VwV _)* J7 9 < ' * . . . J. . . . . . . . . . . . . . . . . . M. . . . . ' * Z 3 < H M ' * < M '

(238) * ' ' (i_)* < q dk ] < q dk ' * < = ' * M ' * ' ' ' (yb* < 4 3 ? < * H M ' * 3 M '

(239) * ' < 3 < * H M '

(240) * Z '. ? U ? < . '

(241) * < ' * J ? U ' * ' ? < 3 3 < * ' < 3 < * . *. J. <H M. . . . . . . . . . . . . . . . . . . . . . . . 9 ? ' * J ? ? U ? ? M? ' * 3 ? H M ? ' * ? M? '

(242) * ' ' (i_)* ? H M '

(243) * ? k [ef Z ] ? ' * J ? U ' * ? ? k [ef ' * ? ' * ? ' ? ?? 3 3 ?? * MM? ' * ' 3 ? 4 ? 3 ? ? * H M ? ' H * M ? Z ' ' (yb* ' 3 * '

(244) * ' 3 * '

(245) * ' . . . . °G° â`¬qª¬¨

(246) ©. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(247)

(248) . g q* . 9 : ' * J. ' * 3 : H M : ' *@ : : :M '

(249) * : H M: '

(250) * ': sq dk * aj 3 : ' qsdk * qsaj @3 : ] ' ' (/X_)* @ : : ' * J ? ' * : ' qs@ dk : * aj 3 ' * : ' qsdk * qsac 3 ' ? : 3 3 : * :M ' * ' 3 : 4 3 ? : * H M: ' * ' : 3 : * :M '

(251) * ' : 4 3 : * H M: '

(252) * ' ' ' (&Xg* w RS w wm w wVq am k [e i f k)

(253) _ba 3 V. V/d t w h V&dV kcV kow [edig f dgj' a /S a &( S X7VK* 3 V/tu d h w Vkcko[\[\dgff z2km[ej' d ( k&( dgX7f * [e'cfE /iz2g[\)d *,w m [ef6kodgfdg^j' /Vyik g[ef* qs_ d w V i_bê[ dgfAm ^\n d w S 7 m S [`q S w_gkKfAdga WCV/fu aj[\dgfAV d wjw Vqsa w V/h w VkoV/fua _aj[\dgf kq&_bf V dg]pf [\f &H WCduq _bf z_ d k#u" [:c' /( )* OP' P ' ( D / A 3D . = ¬= D 4´ D 7 3D G / /_?

(254) !? ´= 0 /3DEF /3D D G= =6F = dgf k [ m V w _0koV/iWCV/fuad a S V.koh S V w [lq&_b^ k S V/^\^ d w < ]

(255) ] ]

(256) ' ' ' y( )* , kjk ]AWCV a S _baa S VQ dg]pf m _ w n J

(257) 0[lk)êd_ m V m n J RS ] k2a S VQ di]Af m _ w n qsdgf m [eac[edgfk acd}V] ]A^ 1 êêV m _ w V ' < * J ] 'c * k'

(258) y)* J ] ' X7* ' < * J ] ' * '

(259) * J ] ' )* ' ' y( )* ' < * J7 ] ' * J '

(260) * ] ' )* koR V/V [\m S d]1 w f a V+koV&aw d9 V ] _aj[edim f kG d w a S V)[efuajV/S w _bac[ew difCqsdif koa _b fua kdgfAV+k S di]A^ m ko]p koac[eaj]pajV V/, tuh Vkcko [\dgJ f k: d < ' S r* _bf J < ' ? * [efum ajd a JV d W0]A^`_bV/' Xg X6* êdifA

(261) V _R V _f ko[efp2' (* difAV1 f m k 3 A k V/V [e B 2 B @ 9 3 @2 J 3 K K3 9 3 ' ' ' (+(* . . ? . . . :M. . . . . . . . ? : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \_¤y°A\_.

(262) .

(263) . ¶. .