May 24, 2006 - ... Hilbertien man ue de coercivitÑe, et il est nÑecessaire d utiliser les .... purpose of section W is to get more informations on the saddle-point.
An Hilbertian Framework for the Time-Continuous Monge-Kantorovich Problem Kevin Guittet
To cite this version: Kevin Guittet. An Hilbertian Framework for the Time-Continuous Monge-Kantorovich Problem. [Research Report] RR-4122, INRIA. 2001.
HAL Id: inria-00072506 https://hal.inria.fr/inria-00072506 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 destin´ee au d´epˆot et `a la diffusion de documents scientifiques de niveau recherche, publi´es ou non, ´emanant des ´etablissements d’enseignement et de recherche fran¸cais ou ´etrangers, des laboratoires publics ou priv´es.
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
An Hilbertian framework for the time-continuous Monge-Kantorovich problem. K´evin Guittet
N ˚ 4122 F´evrier 2001
ISSN 0249-6399
ISRN INRIA/RR--4122--FR+ENG
` THEME 4
apport de recherche
!" #$%& ('*)+, .-%-%/ 0 ,%1,2'*34, 56)7$98: , :;=< >@AC? BEDGFIH%JDGK KLAKNM O+PRSUQ T,SWVo XZY\[]T_Q ^a`]b"cd[fe\ghSc+eCijck[]T,[]lmb"ck[neCg Sldpqldu2c S v eTwxSS c;l(rye\ckTscde ia`fStaS l z biji*e v c o S v SUr P S v r P S{g_|"V~}LX,S v [nS v "
C
j};X ijbSUl N j o.o v v vo o cTP S Sc P re`]bo lmld[n[ng\r b"eC`(`fCgS l2eC}gjbgNqSb_ ¦;¡r[neC`nb¢*Tsg\S cdijv e ck^jv[]bcdeb"gN\cd[][fr e\vP bCgjTb"i `~v SeC*§2¢a^ae `n[ S vmTh¨ \£ pac P gjO+S bCPT,o e\[n[]lm^arr ld^jP clmpqldc`f[fSe\P g©S S§2oe S bClml(T_r ijv ^a[n^ ¢*`]b"v S Scdo `n[fpse\bg,^je []Clv TT[]g\bS c g\`ª£ cke S O o¥^jb ¨r¤ S[]gab"ckv e,bbCogalkeCb[]`fbCgqg S cmeb"CrS«e\eS v¬rc [fP qS [ncp©v SUe"rSUgCc c P iS±v ¦e [f`]v ¢~SUS ldv lcd[][ng_bCg%c P i S+v e\eC¢aia`ncdS [nThTsba`a§«c S±v bSUgjlkcmlmbi~¢ae `nv [ncl P c P bSg6e v SpNta[n V"ldckSU] gjLr®S ¯Lv S ®°lm}U^a
`n~c£bg O+o,P oSUg6SUld^jijg[fcdo S+S c v P bS+`]v bS r b¨ ld¢*eCeg~c bP ¢a¤ `nSWb"bCvldbClm^jgaTs[]iabgcd[fe\o gNS³*egjS i*o eC[nldg´[fcd[fq}[nc~p©bge o v ³*c PgjSb`fo`np6S g~i lkv [necpCaS§2c PS²Si rv e\eCgCS;CS c v P S²SUgjStarS[]lke"cdS gjc rP SWSegq^jlmTb ojS vo [n`fr Sb" `Ei*eT[]g\SccdPl«e eo v £ µ®¶¸d· ¹GºU» ¸· ¼½n¹¾¿À ÁÂÃÄÅ®Æ7ÇÈÉÊËÍÌÏÎLÐEÑLÐEÒ¬ÉÓÏÔLÕxÐkËÍÔ\ÇUÖ× Øj×ÏÙkÚÛ ÜÍÝUÙdÛÍÞ7ß\ÐàâáLÐxãGÎËxä+ÕxÐxÑLÐåCÇUæLçËÍÎLÕxÐÇÐè®ÊËÍÌÏÓ\é êÐëUÌÏÎC× ìÔLÌ íªíªÐíîÌÏÎUçG̰ËU× ïfç
Unit´e de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) T´el´ephone : 01 39 63 55 11 - International : +33 1 39 63 55 11 T´el´ecopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30
+) $: ; ; 8 ,-: 8: , :; : 0 ,%1,2'*34, 56)7$)+, .- ;8%/2<
bg~l±}j^jgaS v Se v T_^j`nbck[neCg©SUgNcdS Tsijl+re\g\ck[]g\^6b´S caS; []gCc v e o ^j[fcdS±i*eC^ v2v SU lke\^ o v S`fS ibC^av e\C¢aT` S QSUTgCcaS S oS2lm^jijeCi*gaeCCldS%S ¡^jg bg\r ckb e o v ev S q[]or P S%£¬c v bLb"L[]bl[f`(¢j[f¦SUg[f`]¢~qS^av S7cd[f`]SUb;gRT¥7S `n S c Pl©e r o b"S `]ro ^aS `]rlsv [ni ckvSCS Uld¢~S bg\lacaSS S7l lmS ^ cdv b[f^jSUg gCcs¤ b"ij ^ v v bS gjTSU[nS gCg c v e SU vldT^a`nS cdb`nl cd£ l v ¤ S S rS g\i cdv l(e\¢aS`¬S Q SUTrcmS^«S ¦ l [f`]o ¢~bS g~v cdl+[fSU`fSg o TseCTsbg b"q[]ga^aSS o o ^%S c rv eqbS gjv lmri~[n\e [nv cac S (o S_Sc6Ts[fb`WlmS lkS_ldc6i*g2eCS ^ rv SUTldlmb"eC[g\v c S v S ov ^a` cdS[fta`n[][nlkldS ckv SUgj`nrS Sl o v^jghT[ngj[nT[]lkSU^ v £ i vS Q l2rScdckS²v i v S T[ SQ v S% S cdbi*SqS c2o SUgNlm^jiji*eClmbg\c \^jS`]b v o S gjldo [fcaS; e\ia¤ ck[]Tsb"v `nS v SUlkcdS ldo c S ³*[]rgjck[nSUl(TSUS g g\c_}i*~eCbLld[fbcd[fg\Cc So SgaeCT^jeCl_g\i c v e\S v^a`]b©e\gjrl±eCg\` CSS tqv []ldckS S g~g~rrS S o SWS6`]b©i~eCT [ng\S ccdPlk e lkoS `fS²`nS gql^ji*T eCS ^ v [q`n^aS SCl £ SU^qt b" bCga[nS g~l
!"#$%'& (()")+*-,.0/0%1"2 34& !56 7 8
9
g °}G±b reCTsij^acmb"cdv [fe\gjb"`WP *^av [ oIo pagjbO+TP []r ldcP pq`fSP v oSe v T_v ^a`]b"o cd[fe\g []l6[]g\c v o±e ¤o ^jrS v o cde lde`nS P c P o S []erg\`n;bCldceCldP `f[]CS rS b"lNv `S rbCS g eCg\gac,¦Ci S[f`] v ¢*¡eS bv v ckg\SU[]ckblde l_g e[ng¥qv []bcr TP ShS i §«eCeCe ij¢avdck¨`n[]S TThbc £P`(S c v obeC[ngjlm^almr i~^~e ldvldc c,[fe\S cg P S §(S e lmbvr lhp []¢~ij VS^ v ®S b`nGp^a®\ ¯"T,Ge ®vSU} Tsg\
Lckb"S `Gb£ g o¥Ob" b o ¨ SUbC[nldgagaia[n[]ckbShb gWo c TP bCS6SgCc cm`]bb"e Cr ¨ S ev S ±bCrlkeqe\Sgjv br¢j[n\`fS±[ncbp lmlde^~WTc ijP ckS[neC¦;gh[n`]e"¢~S i*v cdeC[nldbC[fg cd[fqi [nvce\p.¢a`ne S v Thc «P S§2S:o S S gjldldcd[fbCc¢apC~`n[n§2l P Sbi g v eStaS±[]lkc cdP S S_gjSrtaS [nldv ckSUSUldgj^ar`ncS²£ eO+ldP b S ojg o `n^jS g i~o eCS [nvg\cdb l e v ¢*ec P ¤ b" v bgj[]bg o S³*gaS o [ng.}bg o ³*gjb`f`npi v eCSc P S7reCg\S v CS gjrSe\c P S7gq^jTS v [nr b"`"TSc P e o £
ÄÄ Î;:4 bck[neCgNe" cP P S,eCv iaocd[nTsb`ÅTsbi P bCl;¢*SS g:lde`nS o []O+gP [nLl ¬v ¢\SUp2ld^a?,`nc £P@ bClS gj¢*[fSS S g%C§ c P e_S gl Seta§«ckSUS g o c S o b[]cg:c c PSS_eCiarbCcd[nlkTsS²be`*TsT,be isv S²[]lCc S gaSS v b`Åb re\[fSUlkgCc(c ^~egjÅrb²cd[fre\e\gjgCl;CS[]g tsi~ AeCckbCS gg\ocd[nb[]`ªg £ ceP S(c P rSbldeCS2ije"ck*[]TTbe `Ev S7TsCbS i%gaS lkvS bS `CTsCl(Se\cke.T,S ¢*c Sv [fbsSUlÅr []P g6b"`n`n} S
gaGC£[n gavseCiT v eCbW¢agq`nS ^jT TS vld[nSr SNb"`q}i~EeC[ne g\v cbe"~¢ qv [n[nSS§² "v S c qP [fS2S §reCTse"ijS^atacm[]b"lkcdcd[n[fe\gag Tcde:S c`f[]P gae SUo b lv [54 £ S gNc P c S.P S r[ e\v gj§«ldc e v vdb"¨ []gCcm}l , B~ £ £ £ O+@(P S SUg~g bTbgeC^.bbC^ag \oT?S g\£ ck@S o v S gqgj^j[fS Tv S ^jv ld[]rS b"o ` bv g.S ldb ev `]cd^a[cd³¬[fre\[]gb"` e 2ck[]c TP SS v S blm^av [n`fbCcd[n¢agj`nS i[]ghv e\c ¢aP `fS±SUT v b§2TbCS±le"i v S i ldv S eCg\¢jcdbCS ¢ao [f£`n[nc;pN`fcTP S eCb^alm ^ P v S c l(P bSsg eCo iacdr[neCTsg\ckb"[]` gqTs^aeCb^jlml(lcdc S v ldbc(gjlm^ji~gje rv ckcW[neCi gjv l eCj¢ac`nS P T S±bC[]^alWCg~Tb"cdSU^ gCvcdbS o`f`npTlkSS c cWP ^~e oi ^[n~c lk[]S lo c []P g S v S}Ee []v l«S(`]g~b vb"cdS ^ `fv psb`aeckÅe¦;ld[ncd`n^¢*oS vp,ck[]c bP ghS+gje\biacdck^[]Tsv Sb"£ `j Tsg6be lmv lo c S v v bg~ckesldi*i e v v ec iS;v c e\P ¢aS±`nS rT@eCg\ Cv SeCvT S bCg~grS¦;e[n`n¢*c S P v SWck[]bTgsSci*P ee []gCo c ec v bC\gj[nlmSi~§²e £ v c±O ieNv bCeCr¢aP `nS [nST S²¢\cp P []¤ l;£ _CeC£9b"C2`G~b §2RS_b v ^jS`nld`fS_[[]c gP S c P v SS.\r^abC`]blkSv [fce"p.(crPe\SgCe Cv Sp to ¢*S eC^jSg `neCo i*S S o o o eCe Tsv cbP [ngjS_l²e\bCiag ck[]o Tsv b"SU`rS Tg\bCcd`fldp l S§+ta[nck`n`ÅS g re\o gjS olk[ o ¢\S p v eCgaD£ `nC«p.e c vP o S_S vrebCF lkE S±v e"bC^jc l P \S_^j[nckgNe v [n^jg:l c aP § S,P r []rbP ldSlde"[]Tijc P `f[fS³~S ¬l;b"rceCcdg~e lkv [ ^jo Hl S v G#b¢j`f p©c P SIbJKgjb£ `fpa;ld[nll £[ng L;°e}§ G¬§«§«SS o S³*gaS²c P S²ck[]TS re\gCcd[ngq^ae\^jl2eCgjS ¡bg\cde v e\[]r P i v eC¢a`nS TNM ¤ S&c P G±¢*S±oc P S o []TS gjld[neCgjb`E^jga[nc(r ^j¢*S²§+[fc P i*S v [fe o []r±¢~e\^jg o b v p.bCg oho S³*gjPS O ,
QSR+0 TG#"£ ghc [nl+ldcd^ pj§2SgaeCckS U O$Q VFW,XZY\[^]^_ ; O S`ba 2c 1 de1gf2hji kl1 [m]_ ; O onp V -yx7/ ; q O ZY\[^]_ O b`Sa 2c 1 de17rsf2hji kl1 [trvu,w
nz1 ÂÃÄE®Æ
!"#$%'& (()")+*-,.0/0%1"2 34& !56 7 8
_ ;v G ml b"cdO [nlkpq[ngj©u c P S_reCTsijbck[]¢a[f`n[ncphreCg o [nck[neCg } ~c P S_ck[]TS i e\¢a`nS T «C ¡ 6 []6l+; cke©T[ngj[nT[>4S
¯ ^X gd
[nSUg%c§«e o S gjld[fcd[fSUl 9 []g reCg\ck[]gq^aeC^~l«eCgaCS ¡bg\cke v e q[]r P
eCS v b"`n`Eijb[ v l m [ng q O lmb"ck[]lkp\[]gasc P S¢~e\^jg o b ; v pNreCg o [fcd[fe\gjl
1 V :_ G
R 1 V =_ ; G 1
S o SUgaecdS±¢\p c P []lT[ngj[nT,^jTh£ ! + ! ,1 ! a * .-5 2( 2 ! 5( 4' !" !
, 1 ) & 8! * )+*" ' * &$$# %; +! & !
* &5( 6 *8
gv c P []l²P ijbCi~S v Re\^ v v b"[]T[nlWcke P o S v [nSb v [no e v ¤ eC^~lv ¦;[n`]¢~S v cd[nbCgc P P Soe v p%e o v c P S O C«O+P ¡ u Rbg o cke io S eb"C`]l±S©§+c [nc S%P rc e\P gCS©CS §«S `fSU`7gji*reCS©ldeS o gac S lmS%l²be"^a(\c TP S©S g\¦;ckS [n`n¢*S v b"ck[] bg bCgai v []eCb¢ag `nS TThS£ c We g o ^jS ldv S b"[ [nv g `np}CS £ gaS v b`S g P pqldi*S erckc [nP eCS g ldS }l e\reCg:ldcc P S o bcdb~ * 9§2 Scde l P c eP §@S c P S,bCldSldtqS []v ldlkckcdS Sg~[]g rS_o e[]lk7cmbNbgjTrSh[]ga¢*[nTSc[>§24S Sv SUg e c P S bg oO C« £ ¡ u g âldbS grcko [neC`ng¥[ng ¨j7c §2P S,SNe\`neqiae ck[]¨ Tsb"b"c ` clmPb o~S o ¤ `fSb" i*v ebC[]gag\c[]£_bgY\[]gje rvST_c ^aP `][nb"l cd¤[fe\bg v e"b+gac CP [nbCS g O )_ C« []l;¡ c u P S©bClkcmg b o v cki []gav e6i*S.eb[]g\g´cbCe¢j7ldcc Pv bSrgqc²^jSTta[nS ldv ck[]SUrgjb"r`S Tv SSUc ldP ^ae `nco Re+§«bS Si~tj^ i~v i*SUreCc(ldS.c P e[]+l ldv S S rlmck^a[n`neCc2g cde©9 ¢~[]Sl,bCckeg.C[]TSc_i*eTv ecdbCv S.g\c([]gqldcke SUvihTs§ b"P cd[fS e\g6gjl`feqe\e g¨ []gac P,She ldv b ojbso r`ne\S g\i~eCS [nv g\cSU£ gjru2S v vSUSUrld[]lk^jS `f`fc pC£ O+§2P SNS ghl P c eP § S ci~P b"b"[ v c_c P S ¤ b" v£ bCgaO+P S©SUg T_^j§«`fS cd[niav S `n[nTS v []g eo +cc PP SNShldTS rbCeCldgl_o reC¤ g~b"lkS vv bb"gjcd[f[]e\bg g r eCgj ldc []vg\b"c []v g\e c²o ^j[nl_rS `f[]o g ¨ [ng S o }ckebCc Pg S.o e\^jialdckS.[]Tsc P b"S ` o~o P v v P ojo P v v v lmbe b `fCv e b%v `f[fScld Pb i*ojTheo []£ g\`nScdO+ lÅi~P e"eC[]l [ng\_ v cWSUcdlde"eW^j `fr cTsb£©bLbpNr[ncdg~¢*S b"S_`n5[ `f4 pC^jS Ågac lkSSUtjS2ri~cdldSU[fbe\rgckS oV:`nSlk i i~[]vgjeCS r[nldSg\S cmg\ldlÅcdeCl²eT b%S±re"£ eC g\c P S2S Sv CrS S `]cÅbgjc lmrlkS []S rv bSS `Elme b^alm`fS7cWld^~bCTg²e v Sijtackc [n[]P eClkS.cdgjS lgqgj^jrv STS qS ^aS v [lm[]v r^aS b"`fo c ` e v re\gCCS v SUgjrSWb v S±gjec2 ^a`f³~`f`nS o £ ¥8 ,/ %;// _ $% ; 8: , :; gP c v P []l_ldv S rcd[fe\gReC^ v P Tsb[ng v S lm^a`fc_[nl²c P S.Stq[]ldckS g~rO+S.P e+b:v T[]ga[]T,>[ 4v S v e2c P S v O o C«¡ P u bCg o c P S cr P Sb e bv SUrThcdS £ [54Ub"cd[fe\g;ejc S«eCiacd[nTsb"`qre\lkc £ eCldS S lm^a`fcmlb S«ld^jTsTsb [>4S []g²c SeC`f`ne§+[ngj * * ) ! ! %' ) 7 2 *$ * ! ;+/ 56 56% $* ^] G
" A ^] G
# " 1 ('
)
+*
*
,
-,
,
.
/
0
132 54
6
ÄÄ Î;:¨4[][]gaga Scd[nldrNSqS ^agaSUSP gjv rCS p v ^ ebCreC7g gjbNldcckrve.eC^jg\^jrckldS[nS²eCv Cg6[nc+[nga[nckl .e.o lm¢je\^j^aga¢j[nS±`ldS[ng6qb©^a³~SU¢~Cgje\S²r^jSClkg *cdS § SijP l [n£ rTP []ga`nS []T,b o [>4l[]gacke6NbCldSg:qS ^aâSUS gjrrckS [n Ss T[nga[]T £[54 S v []e"lbc `fP `neS § l2O c «C S¡Stau c b£ rO+cd[fP e\[ng l 9 o v v o P P P o o v bY\¢*ckSUeiS²}2bM g Wo lk []v gaeC_T `nS ¢*TsSTs`ne§b.}CbL£ §(CbL§2psS v eCST ³*ga4 SWS v b e~£ SS SUgjrS S gjld[fcp § - []r / [nl2- ¢*/ ec ¢~e\^jg S eCT Y\ckSUiN3M o¤ S c o ¢*Sb²To []ga[nT[>4[]ga,ldSq^aSUgjrSC£ g.bi~i v e\i v [nbckS o re\g\StsP re\T_o ¢a[]gjb"cd[fe\g e"b v S² ¢~ e\b^jg g o S :o [nl±v eC^jT ldS ¢~S ck`fe%e§I¢j^abL[n§2` - bLps/ bh gav eCS- §T / T, 4 S []v gae[]Te [5v 4 [nS gabC:r P ldSq^a£ SUgjrS ` ` E§ e\lkS S g~lk[nck[nS l Y\ckSUi 9 M v gjS§ v ldSq^aSUgjv rS ; ¨ ; []P l(¢j^a[n`fc(c P~v e\^a o P v bro b v S ^a` v S C- ^a; /`]b v [54Uv b"cd[fe\g©i v earS lmPlKM [fe c_v [f[nCl_[ng~e"b"`âi eCga[nTsSUlb £ p[]Tsi~e cmbg\ccke - / SSUi c S.- cd/ [nTS ¢*eC^jg b p bcdb%e - / CS p r`fe\lkS©cdec S Y\ckSUi V P M O+P S v S\^a`nbo v [nc- p; / e O+ P ; bCg o P ; bPg o v c P S.v i~e\lk[nck[nq[fcp ve" o ; b"P `n`ne§{O cke o S³*u gaS.b cld¢~PTe\S ^jeqg e oc b ¡ v p u S `fo eaEbrbcd[fcbg p,o b ³~v ^jSS6`ldSsldTW c P eqS e£c v P S\^ac S P`ng6b Shv ^j[neCclkp%ij[]gackc []PTS c be `2vS±p:Tsr bCe b i v brc rPbCck[ng S l_¢~[]i lkShvcde\[nr ^j¢alld`n"S S §2Tho S £ cde S g o o ^jSS³*rS Sgao S%c Åb§ S g P SU[]gCg«C cdSc Pv i~S©¡ eCck`n[]bT ckS ckSoe o S g~lk[ncp%§ P [nr P [nl;eCiacd[nTsb"`G£ -g/ b v C^~- T,/ SUg\c+e" z £ B~£¬r C2bgjg lkS S_`nS TsTsb}£Ï b"`n`fe§l+c P S g cke.¢*eC^jg o bs- ga/ S §I-ldS/ q^aSUgjrS - ; / £ - ; / Yr\e\ckSUg i:o [nck[neCM gj l £ O+P [n l owk^j[ .ldc+r`f^j[ ¨`fcS p% r bCgh¢*S,bLC e [ o o S e\o SUl+^jgald[negjc+.lmlkb"e\cd[nT,lkS pNmc ¢~P e\S²^je g v o [fCb [nv g~p\b" `â`nbLckpC[]TS v Sl ~¢~£e\^j[]g gjo bb `f`nv pp c P ST[]ga[]T,[>4 []ga.ldSq^aSUgjrS []l(eC¢acmb"[]gaS o £ *
8
2
4
2
=
4 4
0'
!
"
#
#
#
#
$
%
'&
"
O+P [nl o S³*ga[nck[neCg6e`n`ne§l2 v e\T c P ²S e\6 ¢jld6 ;S v Lbck[neCgNc P bc(e v i*eCld[fcd[fCS a lm^jw iw 1 1 )(+*
4
4
-,
.0/
( 1( 4 /4
2
2(
/
1(
/
)* 3
: =
54 679 4 8
}U
ÂÃÄE®Æ
!"#$%'& (()")+*-,.0/0%1"2 34& !56 7 8
O+P S g ^ rbCg6¢*S o S³*gaS o c v \e ^a P c P ±S eC`f`ne§+[]ga,S q^jb"`n[fcp m9 lm^ji x 1 }9 §_P S v O S ] _ ; O T _ ; O sb v Sld^j¢awxS rc±cde6c P S©reCg~lkc v b[ng\cWc P bc;e v b`f`gjeCgq gaS\b"cd[fCS [ ]
:
: [ 4[]galkS \o ^jS gjrS+e v c P S O C«¡ u £ ¤ Sc ^ ¢~Sc P SWijb"[ v o S³*gaS o []ghc S²`nS TsTsbj£*e bg\p []g a# §2S - /S³*gjS 8 } } $% $ ` - / }A } } 8 $ N1 - / $& - / ` @(pNreCgjldc v ^jrck[neCgRa§2S P bLS ` ` #] q O - bC/ g o )deb#] O ` )deb* 1 } O+P S±reCg\Sta[fcp©e [nSUl - / - / ` ` 8 } l } "
-/ -/ ldesc P bc ` ` []l+ldck[n`f`ÅbsT[]ga[nT>[ 4- []ga/ .ld-S q/ ^aS g~rS£ ² " z ; ; Le§ §2S±o §2bCgCc(cdes¢~^a[f` o v bsldTeqec P v deC^acdS v o [nghc P SlkSUgjlkS±c P b"c+c P S²ck[]TO+S P ¢~e\^jg o b v o p o bcdbsb v S²gaev c cdlm¢*eb"Scdo [n^jlkSld³~³*S S gao S²Ebb_l+¢j^abClmcWTg6eCeqSUgaeld`nldc pS P g\rcdb [nSb`neaS`¬ rcd^a[neqc`ne`fpp`E³~cdbCSesij` o i¢*eCbCeL^~g tqg o []oTsldeCb"bcdTghS Sbijr iP T,vb eC[]v gaib[]rvT[]cdb"S 5[ 4 cdv S[n[ngaldo ck:[]Trldl []S £ gaqO+[]^aT,PSU>[ gjeC4rld[]gaSWSC.£ r P ldbS qv [nbC^alWrSUcd§2gjS reCv SC[n^ald£ ` ck[]rl¢*S©§2eCga^jS r` o S lmc ldP b S gp S³¬gaS [ []g TG bl2eC`f#`ne§l $$$ [9deS*t
b V4 []gaslkS \^jS gjrS£ S ;W - ; ; / 1 9 } -/ W ; []l«[]g O C£ S±b v Sgje§ b¢a`nScde o S³*gaSWc P SWr P b v bCrcdS v [nldck[]rl £ S`neqe ¨ b"c2c P S o [ âS v S g\ck[]b"` ldpalkcdS T -/ # ; de deS*S# h e d b t W % 9 &
b ^1 -/ Ye\qg[ngjr
,S QSW+R ; £ []lb ^jg~rck[neCgE~c P [nllkpaldckS T [nl§«S `f` o S³*gaS o bg o c P SlkeC`n^jck[neCgh[nl;^jga[ \^jS`np o S³*gaS o
#
9
4
2
%
;
&
( 4 /$4
4/
=
m
1 ¯ ¤ Sc " ¢*S²c P Slde`]^ack[neCg6[]g _ ; e"c P S±eC# `f`ne§+[ngjsldpalkcdS T
9
mgd ; #1 9 ¤ Scdck[]ga ±U Ce²cke,
~\bCg o c P S g.^~lk[]gac P S o SUgjld[fcpse" O `Sa [ng U O3Q VjW, q§«S;Sc7c P bce v bg\p ^ []g O$Q VFW, ldbck[]lxpq[]ga )
*
;
)=
) 0
$
*
:
;
*
*
5 ;>=
;
*
)
;
f jh i k 1> mgdt
1 V SreCTsvi*S eCTldS [nog o [ng ga_e§{; O c P b"c__bCg\bpl+bs ^jlmgj^jrT ck[neC g []g _ ; ; O ld ^j`bra P Nc rP bCb"g c ¢*SN^jgj[q^aS `fp bg o reCg\cd[ngq^ae\^jlk`np o S `ba ` ` f2hji k e v ldeCTS ] U ` O # # ;
[ng _ ; O # 2 f j h i ( k 1 % ; lmb"cd[nlk³~SUl & ; 1 ^
eCg O 1 o v v P v P e v Sb"§ `n` [n ck S,mc SUg [ng U O3Q &VjPW( f lmb"cd[nlkpqW [ngj £ gwxS rcd[ngj6[]g V bg [ng\cdS b"cd[ngjN¢\pijb câ§2S_S cc b"c 8 mgd W 15 ^ m
W 15 mgdt
1
g\cdS v b"cd[ngj,¢\p.ijO+b v P c2c P S²i v e o ^jP rc o W of hji k e v v bg\v p []g U ; ` O jbg o v ^jld[ngasc P S o ; S³*gj[fcd[fe\g e W "q§«SS c«
j£ SUg.^jld[ngj_c S SUre\Ti*eCld[nck[neCg©i eCi*S cpse" _ O `ba e bCg\p W []g _ O `ba *
:
1 m
* 1
< ;>=
;
*
)
;
:
)
*
*
=
*
*
;
*
*
)
)
;
*
) 0
:
*
)
*
*
ÂÃÄE®Æ
j}
!"#$%'& (()")+*-,.0/0%1"2 34& !56 7 8
§2§2SS²lkrS e\S±gjrc `]P ^ bo c S_ c P \bc W
j[n£ l+O+b©P lmS b g%o~o []gL`fSwxSU i*rcke[][]gag\.c([]e"g _ £ ¯ bCg o ^jld[nga©c P S o S³*ga[nck[neCg:ebNldb ojo `nS i~eC[ng\c l2[ng °}Ga§2SWb v SWgae§ v S b o pcde o S³*gjSWbgaS§ ¤ b" v bCga[]bg £ S²b"`]lde_cdb ¨ SWc P S±lmbTSWgjecdbck[neCgjl # *
*
*
*
,
^ m $ % $$
[f 2S `n] ldS $$$ t
1 &8 R 1 $$$& H 1 d cd¢*eNS Co SUS c;gaec PcdS "b o c;¢Ce p v b"`n ` O b H ] _ ; O aD` T _ ; O aD` T U ` O v e\Tgae§ Ce gRâc P [nlWlmijbrS_§+[n`f` H t 8 78 H8 f2hji k 1 A
$$$
:
=
;
:
,
S P bLCSWgae§ c P S±eC`f`ne§+[]gasc P S e v S T * * * #!"
:
;
% 5 ( ' , O 8 ,. 3 8% * % 5 ( ' O , *\ * ; % ! ! * & *2"-% * * ) ' # ! $ %'& (()")+* * * $ "* 5 56 7 ! 8 4'8 ): *#8 f2hji k * H _ ; O `ba : * 8:f2hji k *
132 54
0* *
P bLS ¤ Sc [ng
*
C£ S P Lb S
H
, * *
*
H =
, *
*
£¦;S g~rS±§«S
#1
¤ Sc [ng T U ` O £#[ v ldc%§«q S eC¢~lkS v oCSc P b"cb kl []TsiaO+`fS P []g\ckS v Pbck[neCg ¢Cp i~b v chC[fCS l of jh i k HZ8 lk[]gjrS [nl+[]g O bg lmb"ck[]lk³~S l £ S g6§«S bLCS
0*
0*
:
K
H8 2f hji k 8 8 f2hji k 8 1 K O+U P S gsld^~TTsP b v [>4[]ga±c P SUlkS+[]gaS \^~b"`n[fcd[fSUlC§«S+CSc c P b"ce v bCg\p []g _ ; O `Sa T _ ; O `Sa T ` O j§2S bLS H # H H § P [nr P i v S r[]ldS`npNTS bg~l(c P bc [nlbslmb ojo `fS i*e[]g\c2e £ , *
:
*
:
*
=
:
* *
=
:
* *
=
, * *
*
*
, *
* *
ÄÄ Î;: