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Multifractal Analysis of Choquet Capacities : Preliminary Results

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(1)Multifractal Analysis of Choquet Capacities : Preliminary Results Jacques Lévy Véhel, Robert Vojak. To cite this version: Jacques Lévy Véhel, Robert Vojak. Multifractal Analysis of Choquet Capacities : Preliminary Results. [Research Report] RR-2576, INRIA. 1995. �inria-00074106�. HAL Id: inria-00074106 https://hal.inria.fr/inria-00074106 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. Multifractal analysis of Choquet capacities : Preliminary Results Jacques LE´VY VE´HEL, Robert VOJAK. N˚ 2576 Juin 1995. PROGRAMME 5. ISSN 0249-6399. apport de recherche.

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(198) a¨pu ¥ a ` ^kž„<^C>a ( p h (b eRayiSHPlµ<a^2lž¸amtsam`-aml9ihsl:9 ‘ H9 u^2l<nR´µ<aPig„)a@z<^2[

(199) iUhsiUhs\.l¸\œ 9 ‘ 9%hŒl¢nž^n%hx{hsl9iUak[X>^txp * 7 * œ\[©^tst (Pn &V_­9dEF „)a(`Jo)tjighjœ[g^.{_i_^t9^2l<^2tsž)pŸhxph|pR„)ak[

(200) aI{m^F [

(201) [UhsanP\.o%i ¥ hjiU„J[Uaprz<a{i U h­ a­ "  ) " p ‡ ig^\ p1l¤\­ u°aP{\l<prhxn%am[¾iU„)a@praw.o)aml<{aƒ\œ©`-a^pro)[

(202) ap ¥ „)\9pra].aml)am[g^tigam[U`hxp * 7@* * X *TU ` V bp l U `

(203)  UgHPl VXV  ‡ l U `   UgH l#VXV œ\[¢?a * >.am[rž E\.[

(204) amtv7 pŸ* ayi ` \œ B­@£¾tsa^2[Utjž9u X *TU ` V¢{y\Hl >ak[

(205) ]a¨p¬ig\ ¥ ^2[gn)p l U ` * V­ 8&\[Uam=\ >.7 am* [mu X *PU UOHPl#V-Vlp ‡  u#^2l<nBœ\8 [ H p h H l¨u^2l<n¦œ\[JprMo i§{hsaml9igtjžBtx^[

(206) ]-a &©=u XO*TU UOH0V-Vlp ‡ ­ u°aPn%an%o<{a (UOHPl?V p ‡ (UgHV p  hj\œ H.p h H l ( 

(207) UOH0V p  œ\.[K^tskt H JB š l)a{a¨p

(208) pU^2[rž ^2l<nWprMo i {yhsaml9i {y\l<n%hjiUhŒ\lƒig„<^i].o<^2[g^l9iUama¨pRiU„)aI{\`-`Jo7i_^2iU_h >%hji±žƒ\œ<iU„)a¤^µ‚=\ >a n%hx^][g^` h|p \œ#{\.o)[gpra  X *PU * UOH0V-V ts\] * p¯‘ * tshs` ma &. éé ærq s/tuv. X=U UOH0V-V.

(209) ²+\ ¥ a?>.am[muhŒl¦z)[g^{iUhx{auiU„)a-tshs`-hjiƒhxp+l)\iJ^2t ¥ ^kž¡p'@l)\ ¥ lN­d¾„)aœ\tstŒ\ ¥ hŒl)]v{\.``Jo7i^igh_>hjiŸž {[Uhjigam[

(210) hs\.lhxp¾`-\[Uaƒ{y\lH>aml)hŒaml9i b – ¼ "¼ ” > • >Œ¼ : 6 5  "UYXO*VG

(211)  ? -P O2

(212)  Q Z[\ #P #" K\#N]"RN^  s=O;Ew

(213) !NG=N]O@Gv N7E K& "m SJ OTN]"r  N7E`N]"XD=O&I2_ #" H JB -P

(214) [g"][\)"T(UgHV=O&I (

(215) UgHV wN] "DC M[M O *  t*iŒhs` ma  tshs`ma  ts\] X *PU * UOH0V-V p ‘  UN(-UOH0VVp (UOH0V-V . &. X U UgHVXV. . j Q #=%X *yN7 #)NNI "mG

(216) `=O  I=E`N7  NS a !-=O

(217) %& KKP!- wN7EG)"RNS=O  Q X C. Œt hs`   ma X  U * UOH0VXV(œ\[I^J TJ%an-&#­ d¾„)a(`a^.pŸo)[Ua hsl9ig[

(218) \7n%o<{a¨nhsl@iU„)a a#J¡^2`-z7tŒa n%\a¨pNl)\i©pU^ighxp±œ|žiU„)aE{\`-`Jo%i^igh_>hjiŸž{[

(219) hsiUam[Uhs\lN­ ®Ÿl<n%amanu ¥ a„<^C>a * t*rshs` ma   tshŒ`ma &  ts\]  *PUU * UgUgH H ll#V-VXVV p -qts\] ‡  ;%l ­ ! 1r'  /06P 1, 1 +- +1. +/0 .+3 6W, .#T,T/y5& ®Ÿl iU„)hxppŸa{_ighs\lNu ¥ a¬z)[

(220) \.z<\pŸa¸pŸ\`-a¬hsl)aw.o<^2tshsiUhsapµ‚ayi ¥ amamil + -u +1.2"u +10 .^l<n +3 ¥ hjig„)\o7i ` ^)@hsl)]v^2l9ž^pUpŸo)`-z7ighs\l\l X­<d¾„)a` ^2hsl¬[

(221) apro)tjiK\œ"iU„)hxpEpra{_ighs\l¸h|p iU„)aœ\tŒts\ ¥ hŒl)T] b  ·%¼ – · 6 5  ")X bpUNX *HVD* Mo 

(222) `G ? -#P O2  Q Z[M #P #"  UYB V Up

(223) KM#N]"RN^

(224) I PJ O0 QI =O 9\ ‘ nkH 9|"m CN7OG E) P  N7O 9 \‘ nk# <S&=O&I^_ #"&R b ptU-U )* Vml )U Vo*\o 

(225)  ? -P O2

(226)  Q KM=!"RN]"RNS=OT  Q 9M ‘ nk 9 #)"RN7 Q =N7OG#Z q=O&I!#Z"# CSM[M OL"][M Q  PfF] N7OG$N7O0 -#T N]"]N^  [\ I   !-  Q C. BƒpŸaig„)aaw9o<^2tshjiŸž XDU * gU HVXV=p. + - g +10 . g[+3. dE„7a@z7[U\\œf\œRiU„)hxp¾iU„)am\.[

(227) am`hxpEpŸamz<^[g^igan§hŒl9iU\ pŸa >am[_^2tNpŸiUamz<pKa#J%z)t|^2hsl)anvµ<akts\ ¥ ­ ¼ : (¼ '$+ - — : ½ +10 . 5 $576 5&% ¼  —<– > ”  – ¼ "¼ ” > • >Œ¼ : 45!) O&I ! )O&I=N]"RNS=OT *#Z+ =O&I!#Z",D=] [MFE) +- g +10 ..  !-  Q C. ®Ÿl ¥ „<^iEœ\tsts\ ¥ pyu ¥ aƒ^.p

(228) pro7`-aig„<^iSB A p h qU|ig„)aP{m^pra B A p * Ph|p(ig[

(229) h_>h|^2* t7V­ eRayijH J 9 ‘ H9 u &6J¸®  u ( *TUgHVfig„)aKhsl9igam]ak[#pro<{U„-ig„<^i HLJL )^ u9^2l<nCH ) ig„)aKts\ ¥ am[©µ‚\o)l<n : < \œ  )* ­Mu1„)amF lig„)aƒtshs`-hjiEa#J%hxpŸi_pyu ( UgH0V bp *i tshŒ` ma (*TUOH0V)p *r tshs` m  (* UgH *)^ :  < V £(„)\%\9pŸ.a "89 ‘%\­ u°a„¡D^ >a 2 &0l m p &0lDUgHn "wV $ &- &0l. \[Ea¨w.o)h_>^2tŒaml9iUtsž. (*PUOH. 2&0lrp &l=UgHn "wV $L&.- &0l. *. )^ :  < V+J 9 (/-A" F (1"C< (*TUOH0VjJ $ & *. UY(&V. é. }^~ c}S.

(230) hsl<{aSH. J* )`* . h. š l<n. : <. œ\.[I^2tstk&©u ¥ a\µ7i^2hsl $"%9 ‘ H $"89. ¥ „)h|{U„§ž%hsamt|n)pu)^2œ|iUam[Iprayi

(231) ighsl)]2B. ‘. . J. * @*. n. :  = & < )

(232) 87 9; : A8<.  n  )* A  *   ) 8

(233) 7 9; : & =  bp * 

(234) 87    )* ^2l<A8n<  & A  ) 9;: A < + -UY(&Vqbp n%hs` T n B g pro)z  & A .  )*. b n. B. ¯n%hs`0n)B. bp. A. & =. (£ tŒa^2[Utjž.u B A& =  b B A& =  mPo u^2l<niU„o<p  & g  & mPo ­Tu°aJn%a¨no<{yaJpro) z  & p   tshŒ`ma  & U ¥ )„ hx{g„¸hxp  ¡l)hjigaƒprhsl<{a m & g D V­ eRayi.wn RJ¸® € ^2l<n 4Jq®  ­)~¡\.[¾t|^2[

(235) ].aI&©u ¥ aƒ„<^D>a+` ^=J ) U )* Vhg ^2l<n ig„o<p ¥ )„ h|{U„§ž%hsamt|n)p ^2l<n u°aP{\l<{tso<n%a. . nd =  UYB & =  VKg.  6 &*. )

(236) 7 9; : A <. * U ) V )p. A. . UN(n DV. n UYB & =  Vhg 6 & UY(n DV A  &. &. g + 0 . UY(&V. + -UY(&VKg +10 .wUY(&V. ¼  <— – > ” ¼: ¼('e+10 . — : ½ +3 5 – ¼"¼ ” > • >Œ¼ : $5!) O&I ! )O&I=N]"RNS=OT *#Z+. $545&%. =O&I!#Z",D=] [MFE). +10 . g[+3. hŒl<{a.  !-  Q C h.    7Mo * X *TU ) ) l. * 7  X *TU )* V ! U ) * V 7  « V ! U ) V )

(237) 87 9 : A8< ¥ a„<^D>auœ\.[Etx^2[U]aI&©u  U )* V ! : A & < 7  pb 6 & * UY(nCUN(

(238) 1"wV  -6T2V. *TU"%-«wn T V )

(239) 7 9 : A8< U0{U„)\\9pra ( "@hjœ  - ‘-^2l<n (/-0"Phjœ R/ ‘ V_­ £(„)\%\9pŸa*T / TU" V_­Kd¾„)aml U"#- wn T2V*/ ‘ ^2l<n iU„)ak[

(240) a¦a#J%h|p±ip XK9 ‘ pŸo<{U„1iU„<^2iku œ\.[ tx^[

(241) ]Wa &©u. *TU"%-«w n^T2VVg a#J%zU -SX ,*V d¾„)h|p ž%hsamtxn)p 6 & UY(n UY(

(242) 1"wV %-6T2V)p ‘. *TU"%- wn^T2V b p. éé ærq s/tuv.

(243) ²+akl¡{ya. $ \nm$ T M / TUVn1$"89 ‘ UY(

(244) 1"wV  -6T d¾„%o<pu%tsami

(245) ighsl)]"$Z ‘-^2l<nT Z T UVu $  +10 .=UN(cVKge( .-6TU" V. ^2l<n. - + 0 . & UY(&V. +10 .wUN(cV g[+3]UN(cV. ¼  —<– > ” ¼: ¼('$+ - — : ½ +3 5 d¾„)h|p+pŸa¨{_iUhs\.l&œ\.tsts\ ¥ pEpŸ\.`a[

(246) apro)tji_p+\œ$9çmw<©ig\¬pŸ„)\ ¥ ig„<^iu<hsliU„)av{m^praP\œ(^praw.o)aml<{a\œ £(„)\)w.o)ayiK{k^2z<^{hjighsapyu ¥ a„<^D>acb – ¼ "¼ ” > • >Œ¼ : 5!) O&I ! )O&I=N]"RNS=OT *#Z+ =O&I!#Z",D $5 $5&%. + -$g +3. ¾dE„)aWz)[U\\œhxp^¸prhs`-z)tsaJa#Jigaml<pŸhs\.l\œig„)aW\l)a]h_>amlhsl 9çm)<­²+am[Ua-^2[UaJig„)aW` ^2hsl pŸiUamz<p#b m d u avpŸ„)\ ¥ ig„<^i +37UY(&Vap4hsl7œ ! U ( -.TU" VXVKhxp¤l)\l¢n%a{[Ua^prhsl)]§\ly< -  n,-#TWU0‘ Vm< ^2l<n§l)\.l¸hsl<7{[Ua^prhsl)]W\.l 9 T  U‘ 7 Vn  9à­ œ ( g TaU‘ m V¾^2l<n C9 +3]UN(cV_u7iU„)aml 2Hk 9 ‘ {URk - w n,- (kXV+/ ‘%­ d ®±) œ ( - T  U‘ V¾^2l<n R* 9[+37UY(&Vu%ig* „)aml 2Hkj9 ‘ U>-Yk -«7\o w n - -_( k-V+/ * ‘% ­ d ®±) p , X *TU  V - U  V A .u¾ig„7akl tŒhs` pŸo)z  * tŒ\]  U V / ‘ œ\.[ d eRayi Q *  A 7

(247)    ( g T  U‘ V¾^2l<n C9 +3]UN(cV_­ i Q A p , E‘ / X *PU  * Vrg U  * V A .u)ig„)aml&tshŒ`*i#pŸ o)z  *7\o ts\] 

(248) #  U  * V  /¯‘œ\.[ d eRay  ( - TaU‘ m V¾^2l<n R9[+F3]UY(&V­ œ (./ T  U‘ m V_u7iU„)akl¸tshs`*i#pr o)z  *7\o ts\.] 

(249)  U  * V 7  : l < / ‘ d ®±) œ (.9 T  U‘ 7 Vu%ig„)aml¸tshŒ`* pŸ o)z  *7\o ts\.]  

(250) #  U  * V 7 : l < / ‘7­ d ®±)  p  H J 9\ ‘ nk 9 ( UgHV g ( 9­®±œ ( g TaU‘ 7 Vuig„)aml1n%hs0 ` n d A g +3]UN(cV_­ d eRayi d A UV i  A p  H J 9 \‘ nkH 9 (UOH0V - (!.­(®±lœ ( - T  U‘ m V_u¾iU„)aml n%hs` n  A g +3]UN(cV_­ d eRay U

(251) V ¥ „)h|{U„§ ¡l<^2tstjžvtsa¨^n)p iU\ b + -UY(&VVp nhŒE ` n)B A g[+3RUY(&V U V  !-  Q C. ¤€ am` ^2[-@2b UVƒ^2l<n U Vƒ^[

(252) a§hsl œ^{_ip±ig[

(253) \l)].am[@[Uapro)tji_pƒig„<^2l U V_u©prhsl<{a iU„)amž z)[

(254) \)>hxn%a ^2l o)z)z‚am[@µ‚\o)l<n¢œ\[ig„)a§n%hs`-aml<pŸhs\.l¦\œ¾iU„)avo)l)hs\.l¦\œK^qz<\p

(255) prhsµ)tjžo)l<{\.o7li_^2µ)tŒaPl%o7`Jµ‚am[P\œ prayi_pSB A ­ é. }^~ c}S.

(256) ¼  —<– > ” ¼: ¼('e+10 . — : ½ +1. 5 eRayiLX^l<n "@µ‚ai ¥ \vp±ig[

(257) hx{iUtjžz‚\9pŸhsiUh_>.a+[

(258) a¨^2tl%o7`Jµ‚am[_pu<^2l<n & 2^ l¬hsl9igam]am[k­ hayiu%œ\.[I^2tst(.J¸® € m u $5 5&%. *. UN(nOX#V. bp. UN(nOX#V. bp.  UN(nOX#V. bp.  UN(nOX#V. bp. &* &*. &* &. ' & * UN(cV  &* UY(&V  * UN(cV &.  (0J $ & * UY(&V U )* V+/iX * 7\o   (0J $ & * UY(&V U )* V+9iX * 7\o  {k^2[gn *&* UN(nOX?V {k^2[gn & UN(nOX?V {k^2[gn$ & * UY(&V *

(259) ) 8`7 hs l U ) V 9;: A8< *

(260) ) 8` 7 ^= J U ) V 9;: A8<. bp bp bp. ~¡o)[

(261) iU„)ak[

(262) `-\[Uau ¥ al)\iUau%œ\[K^2tst 5.9 ‘%u  &*. >5. UY(n XCn V. bp.  * & UY(n XCn>5V. bp.  * UN(n X#V. '&. * UN(cV. &  * NU (n X#V ' & &*. UN(cV. . &X * 7\o  =     * UY(&V =  &  « -  X * 7\o   * NU (cV. -. ~¡[U\`´ig„)aPn%ay ¡l)hjighs\l§\œ 6 &* UN(n>5V_u ¥ aƒn%a¨no<{yaiU„)aœ\tsts\ ¥ hsl)]ƒhŒl)aw.o<^2tshjighsap#b * 7 * * 7 X = ' UN(cV^ * = U

(263)  -  &* UY(nOXDn35V-V g 6 UN(nOX?V giX = ' UY(&V * = UU  * & UN(nOXDn35VXV & & & u°aiU„)aml¬\µ7i_^hslvig„)aœ\tstŒ\ ¥ hŒl7]@tsam`-`v^Mb ·  ¦— 6A5!) O&I

(264) !`=O&I=N]"RNS=OT *#Z+q)O&I#Z ",Dh] [MFEw  + U 2fXI9 ‘ $*"C9 M‘ n $/5 /[+/.)UN(cV tshs` pro)z *  &* UN(nOXCn>5V / D V p +1.=UN(cVKg +10 .)UN(cV ‡ + U 2fXI9 ‘ $*"C9 \‘ n`$/5 9i+1.wUY(&V tshs` pro)z *  * & UN(nOXDn35V+/ V p +10 .=UN(cVKgi+1.)UN(cV ; + U 2 ` 9 \ ‘ n!2 &0lJ¬®  $ & - &l ` 7Mo * 7\o gW U )* Vhg ` 1*V p +1.ap +10 .  !-  Q C. ­ d¾„)aƒ„9ž%z‚\iU„)aprhxphŒ`z)tshŒapig„<^iEig„7ak[

(265) aƒa#J7hxp±ipE^[

(266) a¨^2tlo)`Jµ‚am[ œ\[I^2tŒttx^2[U]aS&©u¡^l<nviU„o<p.  &*. 35 / . UY(n XCn V. =

(267)  -  V ' & * UN(cV^ * 7 =. X TU. ^2l<n ig„)aƒ[

(268) a¨pŸo)tjiEœ\.tsts\ ¥ p­ ‡%­ d¾„)aƒz)[U\\œf\œ"iU„)a@pŸa¨{\l<n¸[

(269) a¨pŸo)tjiKhxpEprhs`-hstx^2[k­ éé ærq s/tuv. g. 6 * & UN(n35V.  J <Œ‘ F H9pŸo<{g„¸ig„<^i.

(270) 7­ d¾„)hxp(h|pK^-{\l<praw.o)aml<{a\œfig„)ai ¥ \\iU„)am[_p([

(271) a¨pŸo)tjip­¡®Ÿl<n%amanu ;. `. 7\o * 7Mo g U * V )7\o * U ) VKg ` *. ¥ „)hx{g„ž%hsaktxn)pig„)aPn%apŸhŒ[

(272) anaw.o<^2tshsi±ž9­. p. V p  * UN(n ` 7Mo ). p.  UN(n ` VVp. &* &. °aP^2txpr\W„<^C>.a+`\.[

(273) aƒz)[_^{_ighx{m^2tz)[

(274) \z‚am[

(275) iUhŒapb – ¼"¼  ” > • >Œ¼: 5  * UN(nOX?V  + 2fXI9 ‘ $ "89 ‘ tshs` pro7z * ' & * UN(cV /   ‡ + U $ "%91‘ tshs` hsl7Dœ *  &* UY(&VI* 1*E& 9 ‘ V ; +2 X9 ‘ $*"89 ‘ tŒhs` pŸ* o)z *  & UY(nOX?V+/   + U $ "89 ‘ tshs` pro7z *  & UN(&VW 1* *E/ V b + U 2 X9 ‘ $*"89 ‘ tŒhs` pŸo)z *  & UY(nOX?V+/ :V. ‘. ‘. u. p. +1.=UY(&VKg +/0 .wUY(&V p. 1+ .=UY(&VKg /+ 0 .wUY(&V +1.=UY(&VKg /+ 0 .wUY(&V. p p. + 0 UY(&VKg[+/.)UN(cV  +10 .=UY(&VKg[+/.wUY(&V p.  !-  Q C. ­. ‡%­. *. *. st hs` * pro)z  ' & UY* (UY(&nOX?V V /¯Sp  ' & UN* (UN(cnOX?V V & œ\[I^-{am[

(276) i_^hsl  ayiKœ\.[Etx^2[U]aI& &©u‚^l<n tshs` * pro)z  &* UY(n XCn35V /  u aP{y\l<{tso<n%a+o<prhsl)]iU„)az)[

(277) ?a >hŒ\o<p(tsam`-` ^%­. tshs` * hŒl7œ  &* UN(&V 7; ­ u aƒ„<D^ >a U. ^2l<n§tŒhs` pŸo)z. X. 1*E9. = ' &*. UY(&V *. * *  UY(nOX?V+/. &. ‘ V)p 7=. . hs`-z)tshsap ig„<^i *& UN(nOX?V  * 7 = ¡­ d¾„)aƒz)[U\\œfhxpEprhs`-hstx^2[k­ b%­®

(278) n%am`¸­. ‘.  &*. UY(&V 1 *E9eX?V=p. &. . ¯‘.  * UY(n X#VVp. &.  * 7 = -  &* UY(&V =  g 6 &* UN(nOX?V.  * UY(nOX?V. aml<pŸo)[UapEiU„<^2i. tshs* `  * 7 = -  &*.

(279)  —   · ” »  ;· :[+ -V+1. — : ½. $5 5. U29XI9. /  /. +3. UN(cV. =. -  &*. ¾hxp¾µ‚\o)l<n%an­M8&\[Ua?>\=>.am[mu.  * UY(n X#V. UY(&V. &. = . p.  h|p¾µ‚\o)l<n%an­. ½ > · – »2> • . ‘. p1l 5. é. }^~ c}S.

(280) %­àb­s­  )E KA_  +1.Sg + -C ®Ÿ l iU„)hxpva#J)^`z)tsauig„)a¦praw.o)aml<{ya UYXO*VD* hxppro<{Uo „ ig„<^ikuœ\[¸^2tst&©urXO* bpo ¥ hjig„ bp ‡ U7l n o co b p4 V_4 u - o  µ‚l amU hsl)] l^/ µ) hsl)o\.V_u)`^h|l<^2n tE`- a^µ<prako7hsl)[Ua ]PiU„)¥ a+„)\9o)prl)a¢hjœ\.pŸo)[

(281) ` z)z‚`-\[

(282) ai§^prh|o)pg[

(283) a9 ‘ \F lL 9 9u o ¥ hjig9à„ ­Hu ¥ aƒamhspŸ]z)„9tsihsip 9‘  F 9¨o ^p"œ\tsts\ ¥ #p bmœ\[^+]_h >akCl & J¬®  u ¥ 7 aI* pŸz)7Mtso hj'i 9 ‘ F o 9¨hsl-nž%^n%hx{hsl9igam-[ >^2txp\ œF prh ma¤‡ 7 : * 7Mo < u ^2l<n 9  F  9.hŒliU[Uhx^n%h|{Ihsl9igamX[ >^2o txp \œ©prh ma ; : < ­ ~¡\.[J^2tŒ`\p±i@?a >.am[r6ž H Jt9 ‘ F  9àTu ( UgHV p -l<ts\.]   l -co)ts\]   oy­f~¡\.[J^2lž H Jt9 o F 9àu ( UgH0VVp4­ £(tŒa^2[Utjž.u  - l‚ts\.] l -co)ts\] co hsTœ (:J 9 -¬ts\]  o -qts\]  l-<

(284) 9  + -UY(&VVp      hsTœ ( p   F  Eo7iKhjiKhxp(a^.pŸhstsžpŸ„)\ ¥ lig„<^i b  -l<tŒ\ ] l -co)ts\.

(285) ] co hj)œ (.J 9 -¬ts\.]  o -qts\]  l-<

(286)   +1.=UY(&VVp    hj)œ ("p4  F  d¾„)h|pE?a J)^2`-z)tsaJpr„)\ ¥ pKiU„<^iu<hŒlqiU„)a{m^.pŸaJ\œ¾^vl)\l&o)l)hjœ\[U` z<^2[

(287) iUhjighs\lN&u +1.ƒhxpIhsl<^.na¨w.o<^iga­ ®Ÿl<n%amanu)\l)aa¨^pŸhŒtjGž >am[Uhj ¡ap ig„<^iku7hslig„)hxpE{m^pra&u +-UY(&VVp +10 .wUN(cV $(­ ;%­àb­à‡­  )E KA_ " + -Sp +/.Lg[+3oC ²+ak[

(288) a ^2]9^hslNu¡œ\[@^2tŒPt &#u X *tbp n%ay ¡l)an\.l 9 ‘ F H 9<µ9ž bp o U o4i  Vu ¥ hjig„ oƒµ<akhsl)] ig„)a@o)l)hjœ\[U`›`-a^pro)[

(289) aP\l 9 ‘ F  T‡ 9àu)^2l<n  µ‚amhsl)] ^µ)hsl)\`-hx^2t`a¨^pŸo)[Ua@\.{l 9ç8 .‡ F  9 ¥ hsiU„ z<^2[_^2`-ayigam[gp  lCn  ou<^2l<n§iU„)az<^2[righjighs\l§\Yœ 9 ‘ F  99hxp¾` ^n%ao)z¸\œnž^n%hx{Khsl9igam-[ >^2t|p­ ®Ÿlig„)hxp(#a J)^2`-z)tsau ¥ aƒpr„<^2tst[Uayœam[KiU\-hsl<n%hx{a¨p¤ayi‡ ¥ „)aml¸i_^2at @hsl)]^2µ‚\o7iKw9o<^2l9iUhsiUhsap(^p

(290) pr\ª {hx^2iUan ¥ hjig„§iU„)aƒ`-a^.pŸo)[UaKp oEaySi  [UapŸz‚a{_ig_h >aktjž.­ d¾„%o<pu ¥ aJl)\iUa +woayi +  ig„7avpŸz‚a{_ig[g^¬^pUpŸ\7{h|^iUa¨n ¥ hjig„ oƒ^2l<n  [

(291) aprz‚a{_ig_h >amtjž9u‚^2l<n ¥ a l)\iga (« iU„)a-[Ua^2tlo)`Jµ‚am[PpŸo<{g„Big„<^i ( p +1.=  U(& V@^2l<"n (Gig„)av[

(292) a¨^2t#lo)`Jµ‚am[Ppro<{U„ iU„<^2i +1.= UY( V=p ­  ²^o<p

(293) n%\[

(294) ¿¢n%hs`akl<pŸhs\.lf+hhsl<{$ a oKamri  „<C^ >.an%hxpx}r\.hsl9iEpro)z)z<\.[ripu ¥ aƒ„<D^ >a ;. ^2l<n. B. A. pcB. n  A. B. A. n. 1` ^=J0Un%hs`2B An ngn%hs` B An V ¥ hsiU„  n%hs` B An p   -   hjhjTœTœ (1("pp h  ^2l<n n%hs`2B An p - l<ts\]  l -co)ts\]  co g  o ãä©ê±Ø.Ñ Ñ"!.æÝçê±ÝçãæÍyæ#WäŸÑgÒ±Ð.Ùàê±Ò#ãæJÜ+ÐÙçê±Ýçæ.ãk܃ÝsÍyÙ9܃ѝÍmÒ±ÐäŸÑg%Ò $ҟÑgÑ(èsãä#Ýçæ.Ò0ê

(295) Íyæ.ÎgÑ'&)("*+å éé ærq s/tuv. + - UY(&V=p.

(296) ¥ „)h|{U„]h_>.ap + -UN(cV=p. . . . + UN(cV. hjœT("p  hjTœ (1p h . . l)\iUh|{a+ig„<^iS+ -h|p(l7\i¤{\l<{m^D>aDV­ +1.prz<a¨{_iU[Uo)` š prhs`-z)tsaƒ{y\`-z7o7i^ighs\l¬pr„)\ ¥ p ig„<^i +1.wU

(297) DV=p P^2l<nf+1.=UN(cVhp (1p4 h ­7®Ÿl¸iU„)hxpE{m^.pŸau ¥ aƒ„<D^ >a¤ig„)aƒaw.o<^2tŒhji±ž U. -l¡tŒ\]   l  - co7ts\]  coKhjœ. +-Lp +1.. NpŸz‚a{_ig[

(298) o)` ~¡\.[I^J].h_>aml&#u ¥ aƒ„<^C>a +3. T*TU". ^2l<n.  & & . V)p.  TU"V=p   % -«. ts\]   U±‡ o 7 !V * 7Mo AU . -qts\]  U  l!  . !o V. l!.   !o V * 7Mo . hjœ CJ 9 ‘ F #< hjœ  g ‘ .\ [. -. . ®Ÿl¸\[_n%am[Eig\ {\`-z)o7iUa +3|u ¥ al)\iUhx{aig„<^i +3]UN(cVVp `-hsl U¸

(299) hsl7lœ  o F(  - 

(300) 1 hsl7 œ l F( %-6TU" V hsl7\œ o F( .-6TU" VwV ! F! F! hhsl<{a hŒl7œ F(  -  1  p `-hsl UN(nkDV !

(301)  l o  h!sl7 œl F( .- TUV  p   +1.)UY(&V hjhjeœœe(_( gi-i((   h!sl7\œ o F( .- TUV  p   +1.)( UY(&V hjhjeœeœ (_( g- (( u°aƒ\µ7i_^hsy lb ®±œ)(.J{9 -¬tŒ\]   d. ®±œ)(.J{9 ( d. d. ®±œ)(.J{9 (. F. (. . u. ^l<nviU„)aml(6/ S/i( V +3RUY(&V=p1`-hs l U `-hsl UY(nk=V

(302) nkn +1.=UY(&V-V=p. o ( < U. F. +1.wUN(cV. <. `-hslU `-hslUN(nkDVnkwnO(cV)p1`-hsl UN(nkDVj9 -qts\]   l-< U^l¡nvig„o<rp (.9 S 9 (& Vu F +3]UN(cVVp `-hsl U`-hs l UY(nk= V

(303) n +1.=UY(&VnO(cVhp. +37UY(&VVp. +/.)UN(cV=p + -UY(&V. +/.)UN(cV. é. }^~ c}S.

(304) %­àb­|;%­  )E KA_  + -$g[+/.@C ~¡\.[ &6J¸®  u ¥ aprz)tshji 9 ‘ F H9hsl¬nž^n%hx{Khsl9igam[->^2t|p©\œ#pŸh ma@‡ 7 * ­Hu°a{\l<prhxn%am[(iU„)aœ\tstŒ\ ¥ hsl)] `-a^pro)[

(305) a  l U ` V  ‡ 7  Do U ` V U ` V bp  ‡  \o ~¡\.[J4^ E\[Uamt(pŸo)µ<prayi ` \œ 9 ‘ F H 9 u©prayi bp 98  F MJ«®  .­ eRayCi H1J0<x‘ F  9à&­ Il7a§a^prhstjž {U„)a¨-{ @7p(ig„<^i  ‘ hjPœ H J  (UOH0V)p  hjPœ H.Jh d¾„)ak[

(306) ayœ\.[

(307) au + -U‘ VTp n%hs` p ‘ + -U

(308) D VTp  £(\.`z)o7ighsl)] +1.+]_h >.ap  +1.wU‘ VTp +1.wU

(309) D VTp ‡ d¾„%o<p + -U‘ V /i+1.wU0‘ V š l)\iU„)am[#a J)^2`-z)tsa ¥ „)akl +1.W^l<n +10 .Wn%hs¿‚am[@n%[_^2` ^iUh|{m^2tstjž¸\.:l 9 \‘ nkH 9<hxp+z)[

(310) apraml9iganB^2iz<^]a ;2¡u%#a J)^2`-z)tsa+¡­ » > •  ph l 5 $5 5 ’ ” >   ·¢· —   ·&2 ®Ÿlvig„)hxp pra{_ighs\lNu ¥ a¤z)[

(311) apraml9iI^2l ?a J%z)tshx{hji({\`-z)o7i_^2iUhs\.lhsl¸^J{m^pra ¥ „)ak[

(312) Sa p1h l¤­²+\ ¥ ?a >.am[mu ^p ¥ ^pp

(313) ^2h|nhsl-ig„7aIhsl9ig[

(314) \7n%o<{_ighs\lNu ¥ a ¥ hstst7l)\i amtx^µ<\.[g^igaE`Jo<{U„v\lviU„)hxp©iU\.z)hx{¾„)am[Uau^l<nv^ œo)tst^{k{\o)l9iE\l§iU„)a@p±igo<nž§\œf`Jo7tsiUhjœ[_^{_i^2t^2l<^2tjž)prhxp ¥ hsiU„[UapŸz‚a{iKiUL\ x

(315) #a J%\iUhx?{ z@`-a^.pŸo)[

(316) a¨p ¥ hŒtst‚µ<aƒz)[Uapraml9iUa¨naktxpŸa ¥ „)am[Ua­ u°aƒ^l¡^tj ž maIig„7aµ)hsl)\`-hx^2t<`a^.pŸo)[U%a v\.l 9 ‘ F  9àu ¥ hjig„ ¥ amhs].„9i_'p U  lCn  oV ¥ hjig„ [Uaprz<a¨{_i ig\¢^2l)\ig„)am[@µ)hŒl)\`-hx^2t#`-a^pro)[

(317) a \l 9 ‘ F H 9 u ¥ „)\9pra ¥ amhs]„9ipƒ^2[Uga U MlCn 2o

(318) V_ufiU„)a§z¡^[righjiUhŒ\l µ‚amhsl)]-ig„)aJnž^.n%hx{+hsl9iUakX[ >^txp­7®±l¬ig„)hxp¤{m^prau ¥ aP{\`-z)o7iU^a + -J^2l<n +3|u‚^2l<nqpŸ„)\ ¥ iU„<^iIiU„)amž ^2[Ua¤aw.o<^2tu ¥ „)hx{g„vhs`-z)tshsap©iU„<^2i iU„)amhŒ[¾{\`-`-\Gl >^2tso)aIhxp ^2txpr\ƒig„)a >^tso)aK\%œ +1.)UY(&VVp +10 .=UN(cV_­ d¾„%o<puig„7aIpr\J{k^2tstsavn x

(319) `Jo)tjighjœ[g^.{_i_^2t.œ\[U` ^2tshxprG` zU h­ a­ + - p +1.rp +3fVpŸiUhstŒt%„)\t|n7p©hsliU„)a¤{k^pŸa \œ©µ)hsl)\`-hx^2t<`a¨^pŸo)[UapK^l<^2tjž ma¨n ¥ hjig„[

(320) aprz‚a{_i¤iU\-\ig„)am[Eµ)hsl)\`-hx^2t‚`-a^pro)[

(321) a¨p­ œ (Pn d £(\.`z)o7i^iUhŒ\l \) df\Wam`-z)„<^prh maiU„)an%amz‚aml<n%aml<{ya\l #u ¥ aƒpr„<^2tst ¥ [Uhjig$a ( n\UOH0V hsl<pŸiUa^.n \Tœ (UOH0Vœ\.[ iU„)aƒ²v\³ txn%ak[ #a J%z‚\l)aml9iK\œ ¸^2#i H ¥ hjig„[

(322) a¨pŸz‚a{_i¤ig^\ ©T­ eRayi H J{9 ‘ F  9µ‚aƒpro<{U„¬iU„<^2i l:bp l=UgHVI?a J%hxp±ipu ¥ hjiU„ l=UOH0V+µ‚amhsl)]§ig„)aWz)[

(323) \.z<\.[righs\l&\œ kam[

(324) \phsl¢iU„)avnž^n%hx{ #a J%z<^2l<prhs\l\P œ HR­ š p±ig[g^2hŒ]„9iUœ\[ ¥ ^2[_nv{\`-z)o7i_^2iUhs\.l ]ah >ap -l‚ts\]  l4-co)ts\]  o (PnUOH0VVp -l‚ts\] Ml4-co)ts\] 2o ;. . . .  .

(325) . . éé ærq s/tuv. .

(326) d. (£ \.`z)o7i^iUhŒ\l \œ=+ eRayiIo<pIn%ay ¡l)aiU„)aœ\tstŒ\ ¥ Œh l)]ƒl)\i_^2iUhs\.l¡pTb p ts\]K  o nrp -¬ts\.]K8  o n Xrp tŒ\] Ml  o nlp ¬ - tŒ\]   bp  H J 9 M‘ nk 9  lDUOH0V1a?J%hxp±ip B$U c  V. bp. UY( nwV. o.  H J  l)UgH0VVp !   H J l ( UgHV)p (   H J l (Pn\UgHV=p (Pn . bp. YUN(cV.   l. bp. ¥ )„ am[Ua ( UOH0V [Uamz)[

(327) a¨pŸaml9ipEiU„)a²-\.³ txn%am[(a?J%z‚\l)aml9iE\œ ¸^i H ¥ hjiU„[UapŸz‚a{iIig\El¤­ ®±iIhxp(a^.p±žig\vpramaig„¡^2i. Bprhsl)]Wig„)a. B$U &  VVp. . . -.  %X -.  U  - ?V p. ƒhsl)l)ayžª0Zhsi_{g„7ak[rªEhststŒhsl)]9pŸtjayžJig„)am\[Uam` U

(328) 9 ‡.bD<uNk)­s+zN­Nk¡u9 ;=<7Vu ¥ a]ayi -l¡ts\]  l -co%ts\]  o  ( n p. -l‚ts\.] Ml -co%ts\]  o   + -UN(PnwV p n%hsE` n UY( nV)p -- l¡l‚tsts\.\.] ] Mll --cco%o%tsts\.\.]]  co o ²+am[Uauig„)a + -¤prz‚a{_iU[Uo)` „<^p©ig„)aEœ^2`-hstsh|^2[fµ‚amtst<pr„<^2z‚aE\µ<pramX[ >.ano<pŸo<^tstjž@œ\.[#`Jo)tjiUhsª l)\`-hx^2t`-a^.pŸo)[Uapu ¥ hjig„` =^ J%hs`Jo)` >^2tŒo)a+aw.o<^2t|pig\¬@^2l<nvig„)aƒtshsl)a p1Hqµ<akhsl)] i_^l)]aml9i-iU\BiU„)aq][_^2z)„N­  \iga¬„)\ ¥ a >am[Jig„<^iu{\l9ig[g^2[Uhstjž iU\ ig„)aq{tx^.p

(329) prhx{m^2tI{m^.pŸau iU„)aWpŸz‚a{_ig[Uo7` hxp¾l)\i+pŸž%``-ayig[

(330) hx{ƒhslq]aml)am[_^2t2­ I¤œ{\o)[_pŸau¡hjœ Mlap  o p 8 .‡uig„7akl p1l¤u‚^2l<n ¥ aƒ[

(331) a¨{=\ >.am[¾iU„)a ¥ amts0 t @l)\ ¥ l§[

(332) apro)tjiKœ\.S[ + - b + -UY( PV=p -l¡ts\] l -&o)ts\.] co   š l)\ig„)am[K\.Hµ >hs\o<p tshs`-hjiI{m^pra@hxp¾iU„)aƒ\l)a ¥ „)am[

(333) a ¥ aP^2l<^2tsž ma¤ig„)a@`-a^pro)[

(334) Ca ¥ hsiU„ [

(335) a¨pŸz‚a{_i¤ig\hjipŸaktjœru%h­ a;­ p ‚­)²+am[

(336) au (Pn^t ¥ ^ž)paw.o<^txp­  \igavig„<^iufhsl°]aml)am[_^2tufiU„)a xgpŸz‚a{iU[Uo7G` z ¥ hjig„ iU„)a§²^2o<pUn%\[r¿ n%hs`-aml<prhs\l {\`ª z)o7iUan ¥ hsiU„q[

(337) a¨pŸz‚a{_iiU-\ l¯l)aman&l)\i+µ‚aW{y\l<{mC^ >.a­¡®Ÿl\iU„)am[ ¥ \[gn)pyu)iU„)aJ` ^2z)z)hsl)] ( n

(338) Z n%hs`  UN(PnVh|p¾l)\iI{\l<{mD^ >a@^pEpŸ\%\lq^p Ml^2l<n  ln%hj¿am[IprM o i {yhsaml9igtjž.­ £(\.`z)o7i^iUhŒ\l \œ=+3 ®±iIhxp(a^.p±žig\vpramaig„¡^2#i TU" V hxp(]_h >aklµ9ž ig„)aƒhs`-z)tshx{hji œ\.[

(339) `Jo)tx^ 7  7   l!  l : ! <   !o  o : ! < p4 >aml iU„)\o)].„ ¥ a{m^2l)l)\ihŒlv]aml)am[_^2t<n%am[U_h >a+^2lv#a J%z)tsh|{hji#a J7z)[

(340) apUpŸhs\.l-œ\[ TyU?a J){amz7i \œN{\o)[_pŸa ¥ „)aml Mlrp 2oV_uhji#hxp©z<\p

(341) prhsµ)tsa ig\ƒ\µ7i^2hslpro<{U„ ^Iœ\.[

(342) `Jo)tx^Kœ\.=[ +3­®Ÿl<n%amanu. d. é. }^~ c}S.

(343) iU„)a Re am]akl<n%[

(344) aiU[_^2l<p±œ\.[

(345) `´\œ Thxp(a¨^pŸhŒtjž {y\`-z)o7iUanvig\µ‚au7hsl¬^Jz<^[g^2`-ayig[

(346) h|{¾œ\.[

(347) `  -Pts\]  l4- U

(348)  -cVts\]  o   ( n\U cV p -Pts\] Tl4- U

(349)  -cVts\]  o +37U cV p \U cV (U cV -6TU cV ¥ hjig„  \U &V p ts\T] U

(350)  -cVts\.] Tl4-Bts\] @ts\]  o. ts\]  o%ts\.] Ml4-Bts\.]  l¡ts\.]  o   TU cV p tŒ\T] U

(351)  -cVts\]  l4-Bts\] @ts\]  o ts\]  o%ts\.] Ml4-Bts\.]  l¡ts\.]  o u a >.am[

(352) hsœ|žviU„<^Ci b b. + -UY( n V)p +3RUY( n V. d. (£ \.``-aml9ip d¾„)apra@{\`-z)o7i_^ighs\l<p {k^2la^prhstjžvµ‚aa#J%iUaml<n%anvig\W`Jo)tjighsl)\`-hx^2t)`-a^pro)[

(353) a¨p­ š pE^2l ^2z)z)tshx{k^iUhŒ\lNu{\.l<pŸhxn%am[¾ig„)a+i ¥ \Jœ\tstŒ\ ¥ hŒl)]ƒ`a^.pŸo)[Uapb o0b iU[Uhsl)\`-hx^2t‚`-a^pro)[

(354) aƒ\.l 9 ‘  9àu ¥ hsiU„ ¥ amhs].„9i_p  lCn  o#n   bw.o<^n%[Uhsl)\`-hx^2t)`-a^pro)[

(355) aƒ\.l F 9 ‘  9àu ¥ hjiU„ ¥ amhs].„9i_p MlCn  o#n  n    F u ai)^ @a  . l   l /  o+/      ohp  - U Tl   V  ". ‘/ " ‡ . . Tl p. . . . p. .   - U Tl   V -A" ‡ p. ¾d „)amlNuhjœNiU„)aƒ^2l<^2tjž)prhxp#hxpz‚am[

(356) œ\[U`a¨n ¥ hjiU„§[

(357) a¨pŸz‚a{_iEig\PiU„)a eRamµ‚apr]o)a`a¨^pŸo)[Uau ¥ a „<^C>.a+ig„<^iU|ig„)ahŒl<n%hx{ap([Uayœam[Eig\iU„)a`a¨^pŸo)[Uap-Vb TDo UV)p1T  UV œ\[ CJ0 -  ‘   F F F ®±œru<œ\.[+hsl<pŸi_^2l<{yau ¥ a-{U„)\\pŸa  l p ‘ +su  o p ‘ + )u "2p›8 %u¡ig„)aml&ig„)aW[Uamtx^igh_>a x

(358) am[U[

(359) \.[ z%u)h­ a­ `v)^ J 

(360) TDo?U" TDV o?-:UT V  U" V n T=o?U" T V -6UT V  U" V  hxp^2t ¥ ^kž¡pvpŸ` ^2tstŒam[vig„<^i¬‘ + ‘‘‰%­ š p^ {\.l<pŸaw.o)aml< {yau(lo)`-am[

(361) h|{m^2t+apŸighs`v^2iUhs\.l<p\œ U"\n^TU" VXV"\Y [ UY(n +1.UY(&V-V ¥ hststl)?a >.am[µ‚aI^2µ)tŒa ig\Pn%hxp±ighsl)]o)hxpr„PiU„)a¾i ¥ \`-a^pro)[

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