Stochastic Approximations and Differential Inclusions

(1)

HAL Id: hal-00242990

https://hal.archives-ouvertes.fr/hal-00242990

Preprint submitted on 6 Feb 2008

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Stochastic Approximations and Differential Inclusions

Michel Benaïm, Josef Hofbauer, Sylvain Sorin

To cite this version:

Michel Benaïm, Josef Hofbauer, Sylvain Sorin. Stochastic Approximations and Differential Inclusions.

2003. �hal-00242990�

(2)

Stochastic Approximations and Differential Inclusions

Michel Benaim

Josef Hofbauer

Sylvain Sorin

December 2003

Cahier n° 2003-029

ECOLE POLYTECHNIQUE

CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE

LABORATOIRE D'ECONOMETRIE

1rue Descartes F-75005 Paris (33) 1 55558215 http://ceco.polytechnique.fr/

mailto:[email protected]

(3)

Stochastic Approximations and Differential Inclusions

Michel Benaim

Josef Hofbauer

Sylvain Sorin

December 2003 Cahier n° 2003-029

Résumé:

L'approche en termes de systèmes dynamiques de l'approximation stochastique est étendue au cas ou l'équation différentielle moyenne est remplacée par une inclusion différentielle.

Le théorème de Benaim et Hirsch sur l'ensemble limite est étendu a ce cas. On étudie en détail les ensembles ICT et les attracteurs. On donne des applications a des questions de théorie des jeux, en particulier concernant le théorème d'approchabilite de Blackwell et la convergence de "fictitious play".

Abstract:

The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion.

The limit set theorem of Bena\"{\i}m and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems.

Applications to game theory are given, in particular to Blackwell's approachability theorem and the convergence of fictitious play.

Mots clés :

Approximation stochastique, système dynamique multivalue

Key Words :

Stochastic approximation, multivalued dynamical systems

Classification AMS:

91A05, 37C

(4)

/ 0*-1 23+ 4

5 687:9;=<>?;=@ACBD8E

FGHJILKMILNIPO2QSRUT:ILV2QXWYT:ILK[Z=N2QXH

\GK^]_Qa`LHbK^IcQO2Q)d0QeNfgVhT:IiQXj'k

lmN2QonpWqK[jrQtsu0`iv_TCGOxwwyk{z|NKrHiH}Q

~t33}~.}8

{X;0UA;

F3GHJILKMILNIP¡ N2`$RUT:ILV2QXW¢T£I¡KM¤k|\GK^]_Qe`iHbKMI£ T:I¥KMQXG

zIi`iNOj[V¦C§vTyHiH}Q¨k©weª«ª¬¥KrQeGku$NHJIi`LK[T

¯®b°

±[

®

±²a}t³'µ´gµ}}JJ¶

·¸ <º¹A68@ · £68@

»¼Ty½¾¦C`LT:IL¦Kr`bQO¿Ànpfe¦CG¦WÁcQaIi`iKrQCknpfe¦CjrQÃÂ¦jMÄ=IiQXfgVGKrZ|N2QCk

w`iN2Q)ÅQeHifeTy`}IiQXHakÆÇyªCªÇoÂ"T:`LKrH

TCGOÈnpZ|NKMÉ¾QoÊ¦WË½K[GT£I¡¦CKr`bQ QaI$ÌÉ2ILK[WqK[HiT:ILK[¦Gk=\$Íl «Î2w:k

\GK^]_Qe`iHbKMI2cQÂÏ2QIRÐÏ¾ÊN2`iKrQsÂ"T:`LKrHmÑ2k

wÆÇ¬lmN2QÃONxÊV2Qa]yTyjrQa`iQIXkÆÇyªweÒSÂ"Ty`iKrHekÍ`LTCGfQ

°º®

³8Óe¾Ô

®

µÕ}Ô

®

µÕ¡¶º38ÓÖLt}±²³

ÎÇ?d0¦]×ªÒ

Ø8ÙÚbÛÝÜÞrß'ÜMß'à3Þ¯á¯ÚâCà,Ü_Ü^ã¯à3Þrãrß'ä©ä}åiÞrÛÝæbçèJÛÝÜrÛéã¯Ü_ê3ëìêîíÛïæðàÞrÛñÜ:ÛÝæò3ó,ó,ò}ôì,ê,í$ãrÚLà3æbõ}ÜCãrÚißöaà3÷¡êÞ¯àø

ãrê,ÛïÞ[ß|ätùú:á8êæbê,ûýüß§ã[ÞrÛÝßþîú:á§ê,ÿïß2ðêÿîãrß'á¯ÚbæbÛ,åiß=à3æiäãrÚbßôú=ô{ôì ú=þ=æbÛÝèß8Þ¯ÜrÛéãbüß|ðôJß§ã +ôyåbÞ[ÛÝßø

ðà3Þ[ÛÝÜbþJëê,Þiæià3æLá§Ûñà3ÿÜrå¡êÞrã_àæiäëê,Þ_ÚbÛÝÜÚiê,ÜbÛéã¯à3ÿÝÛéã,ô ÛmãrÚiàæbõ}ÜCãrÚbßú:Þrâ_ÛÝæ}á¯ÚiÞêîä}Ûïæiçß8Þ

æiÜMãrÛïãrå}ã[ßô:ì,ê,íà3ÿñÜrê{ã[Úià3æbõ}ÜCã[Úbß åiÜMãrÞ[Ûñà3æ!}á§ÛÝß8æLá§ß"båbæLäð$#%,ò'&#ô

(

àäbä}Þ[ßÜ[Ü_àëµã[ß8Þ#ìîà3æò3óó*)+|ß,iàÞMã[û¾ßæîãêë- àãrÚiß8ûàã[ÛÝáÜ8þ=æbÛïè,ß8Þ¯ÜMÛïãyê,ÿïÿÝß8ç,ß2öaêæiäbêæþböaêæ}ø

ä}êæ./0#ú1*Ù2þô2ô

(5)

ttÕg}~}

° Õ °

¶º3~

°

JÔÔ³

®

}p¶

®¼°

¶ ®

8a

°

¶§JÔÔ³

®

Ó~J¶§

®

.

°

3

³8}¶

®

¶§t

°

{t³8¼¶§~3}³§3i¶§Ó}:3Ö¡J¶§

®

m

°

³§Ôe} LÕ

³§3i¶§Ó}_

° ®

t~ µ¶

°

¶¶§t

®

³§3~

®

± 3!"~ }$#¾µ³

°

m

°

¶º3¶

®

¶§a

°°

µ¶§J¶§

®

£%[L¶º³8}Õ 8}¶º³}

°

µ¶§µ´}

°

¶

°

}J¶º¶º³8}&

¶ ® ³ °

J³§

°

¶§µ.t¶§}e±

® ³ °

¶'^´J}t(tÕg}~}

° Õ °

¶º3~

°

*)ÔÔaÓJ¶§

® °

¶

®+

}~¶§t

®

³§ÕJ³§

´}3-,ÔaJ³§¶§ÓJ³¶

®

Ó}/.0|321

°

JÔÔ³

®

}8JaÓµ¶MÕ

¶§t

®

³§3~ }¶§t*

®

L´}³

3 ®

±43a ¶§¶§

® °

ÔaÕ}

5 6*798 : ;$<>=>?@8BA;C7

DFEGD H

;=e;=@IXAI68@

JLKNMOQPSRGT2UWVYXCPSZ&[WM]\T2M0R*^0_-^0Va`]bScd_WeQfGZgMih[-^0fjZgckhl^0KWKWRGMm]ckXn^oZ&cdM_pfqMRqRGPhUpR&f&carPFfGZ&M]h[-^ofjs

Z&cdht^0VdeM0R&ckZ&[WXCf cdflZg[WPtf&Mos'hu^0VkVdP\wvgxzyz{}|~xRG\Wcd_-^oRG`Cyck4PRGP_YZgcd^0VN{ iUW^oZgckM_ XPZg[WMi\W

O@[Wckh[}^0VdVkMOZgML\WPSf&hSR&cdNPZ&[WPVdckXcaZC4PS[-^ur]ckMRnMoTZ&[WP>^0VkeMRGckZg[pXcd_Z&PR&XCfnMoTZ&[WP

^0fj`iXKpZ&M0ZgckhfM0T^hPSRGZ^ocd_xyz{

|

M]Z^0ck_WP\$Y`f&UpckZ^oWVdP^urPSRg^0e0cd_WeW

[WckfFXCPSZ&[WMi\COQ^0fFcd_YZgRGM]\WUphP\CY`C]GUW_Wen|

3Y0

B^0_W\PmiZgPS_Wf&carPVa`fGZ&UW\WcdPS\Z&[WPRGPs

^oTZ&PR|2f&PPP0eZg[WPNM]M0if@Y`$zUWfG[W_WPRt^0_W\zck_ |

3

¡MR@yUp¢WM|

3£

¡T2MR@^hMXs

KWRGP[WPS_Wf&carPck_Z&R&Mi\WUWhZgcdM0_¤^0_W\¤T2UpRGZg[pPR(RGPST2PRGP_WhSPfg¦¥tMO¡PSrPSRUW_YZgcdVFR&PShPS_Z&Vk`wXMfjZ

O¡MR&fFcd_Zg[Wckfl\WcdRGPhSZ&cdM_[-^ur0P§^0fGf&UWXCP\CZg[WP¨f&ckXKpVdPfjZl\p`i_-^0XCcdhSflT2M0R

©-MRlPmW^oXKWVkP

Z&[-^oZ

cdf¨Vdck_WPu^0RM0R¨e0ckrPS_Y`Zg[pPªeR&^0\WckP_YZ¨M0Tl^hMfjZtT2UW_phSZgckM_qF«}[WcdVkPZ&[WcdfZ¬`iKNPM0T

^0fGf&UWXCKpZ&cdM_X^0oPfzKNPRGT2PShSZfGP_WfGPCck_P_pecd_WPSPRGcd_WeôKWKWVdckhuôZ&cdM_pf(|O@[WPRGPôVdeMRGckZ&[WXf

^0RGPM0TZ&P_\pPf&cke_WPS\ZgMwXCcd_pcdXCcdbP$^ hMfjZ9T2UW_WhZgcdM0_Z&[WPRGP>^0RGPf&PrPSRg^0VfGckZ&U-^oZgckM_WfS®

ck_WhVkUW\Wcd_pe9XMi\WPSVdfM0TFVkPu^0RG_Wcd_WeM0R^0\-^0KpZ&ckr0P4PS[-^ur]ckMRck_¦e^0XPSf®qT2MRtO@[Wcdh[

X^u`

[-ûr0PªXCMR&PzhSMXCKWVdckhuôZ&P\\pì_-^0XCcdhSf

¯

_°^9f&PSR&ckPfzM0TlK-^oK4PSR&f±QPS_-^u²³dX´|

3£

®

3

t^0_W\°±¡P_-^²³kXµ^o_W\°¥tckR&f&h[¶|

3£

[-ûr0P9\WPXCM_WfjZgR&ôZgPS\wZg[-^·ZªZ&[WP^0fj`]XCKpZ&M0ZgckhNP[-ûricdM0RªM0TfGZgMih[-^0fjZgckh9^0KWKpR&Mum]cdXôZgckM_

KWRGM]hSPfGf&Pfh^0_>4P\WPfGhR&ck4PS\O@ckZ&[eR&P^oZz\WP^0VMoTleP_WPSRg^0VkckZ¬`+NPS`M0_W\°eRg^0\pcdP_YZ&fz^0_W\

M0Z&[WPRf&ckXKWVkP¸\]`]_W^0XckhfS¹xz_pPM0TzZg[pP0P`R&PSf&UWVaZgf9ckfZ&[-^oZnZ&[WPVkcdXCckZf&PZgfM0TzZ&[WP

KWRGM]hSPfGf^0RGP(^0VkXMfjZ@f&UWRGPVa`»º¼·½§¾p¿iºÀ@º¼·Á-ÁÂºÀ/Â&Ã$¿0ÀÄÀÆÅG¿iºÀ¬¼·Å¡ÇÅgÂÂª|2MRªÈÁ-À¬ÂÅÁN¿0ÉÉÊ+ºËp¿0ÈÁ

ÀÄÅG¿0ÁpÌÈÀÆÈÍ0ÂtÌÂÀ2ÌQcd_nZ&[WPzf&PS_Wf&PM0TÎQM_pVdPS`>|

30Ï

& T2MRQZ&[WPz\WPSZ&PRGXck_WcdfjZgcdh@¢-MOÐck_W\WUWhSP\

Y`

[WPKWUpR&KNMf&PM0T-Z&[WcdfK-^0KNPRBckfZgMfG[WMOÑZ&[-^oZ fGUWh[^§\p`i_-^0XCcdh^0Vpfj`]fjZgPSXÒ^0KWKWRGMY^0h[

P^0f&ckVk`ªPmiZgPS_W\Wf ZgMZ&[WPf&caZgU-^oZ&cdM_ªO@[WPRGP@Z&[WPXCPu^0_Cxzyz{°cdfBRGPKWVd^0hPS\CY`^\pck4PR&PS_YZgcÓ^oV

(6)

^0_p\eY^0XCPZg[WPSMRG`0 ¯ _K-^oRGZgckhUWVd^0R®lZg[pP¸RGPf&UpVkZgfC[WPSR&P^oVdVdMO ZgM¦eckr0P^¦f&ckXKWVkP$^0_W\

UW_pc PS\+KpR&PfGP_YZ^·ZgcdM0_$Z&MC±QVÓ^0hYO¡PVkVf@^0KWKWRGMY^0h[-^0pcdVdcaZ¬`Zg[WPSMR&PSX¸®iXn^0VkPfR&PSf&UWVaZgfM_

KWRGcdfGM_WPRfQ\WcdVkPXCXn^p®^of¡O¡PVdV^0fhSM_YrPRGeP_phPM0T hZgckZ&cdMUpfQKWVd^`Cck_9K4MoZgP_YZ&cÓ^0VeY^0XCPfS

ô_`°MoZg[WPSR&fª^0KWKWVkcdhu^·ZgcdM0_Wf§O@cdVkVBNPhSM_Wf&ck\WPRGP\ cd_ ^$T2MRjZg[WhSMXCcd_WeK-ôK4PSR® ±QP_-ôcdX+®

¥¨M0T2-^0UWPSRF^0_W\ ]MR&ck_|

Ô

®Zg[pPtKWRGPfGP_YZM_pP¨NPck_WeXn^0ck_WVk`ª\WPrM0Z&P\Z&MZ&[WPM0R&PSZ&cdh^0V

ckf&f&UpPf

[WPªMR&eY^o_Wcdb^oZgckM_¸MoTBZg[WPK-^0KNPRtcdft^0f@T2MVdVkM O@fS^0RjZ

3

cd_YZgRGM]\WUphPf¨Zg[WPª\Wck4PRGP_YZ

_WMoZgcdM0_WfM0T@fGMVdU]ZgcdM0_Wf®BKNPRjZgUWRG4PS\Lf&MVkUpZgckM_Wf^0_W\LfGZ&M]h[W^0fGZ&cdh^0KpKWR&Mum]cdX^oZgckM_Wfª^0f&fGMos

hScÓ^oZ&P\ZgM+^9\WcaPSR&PS_Z&cÓ^0V cd_WhSVdUWfGcdM_ ^0RGZ

Ô

cdf§\WPSr0M0ZgPS\ ZgM9Z&[WPKWR&PSf&PS_Zg^oZgckM_°M0TlZ¬OQM

hSVÓ^0fGf&Pf¨M0TPmp^0XCKWVdPSf ^0RGZ cdf¨^Ce0P_WPSRg^0V*fGZ&UW\p`+M0T Zg[WP\p`i_-^0XCcdh^0VqfG`ifjZgPX \WP _WP\

Y`°^$\Wck4PRGP_YZgcd^0VBck_WhVkUWf&ckM_q

[WPCX^0cd_>RGPf&UpVkZ|

[pPMRGPX

§M_°Zg[WPVkcdXCckZtf&PSZªM0T

^9KNPRjZgUWRG4PS\>f&MVkUpZgckM_NPcd_pe9cd_YZgPSR&_-ôVdVk`+h[-ôcd_ZgR&^0_WfGckZgcarP(ckf§fjZôZ&P\»

[WP_°R&PSVÓ^oZ&P\

_WMoZgcdM0_Wflcd_Yro^0RGcÓ^0_YZt^0_W\¸ôZ&Z&Rg^0hZgck_Wef&PSZ&f®4ôZ&Z&Rg^0hZgMRGf¨^0_W\q`^0K4M0UW_WMr$T2UW_WhSZ&cdM_pftôR&P

^0_W^0Vk`ibPS\q^0RjZhSMUW_YZ^0ck_WfªZ&[WPnKpR&MiM0TM0TZ&[WPnVkcdXCckZzfGPSZ

[WPSMR&PSX¸©cd_W^0VdVa`^oRGZ

9^0KpKWVdckPf§Z&[WPKWR&PricdMUWfzRGPfGUWVkZ&fZgM9Z¬O¡M¸ô\-^0KpZ&ckrPKpR&MihPfGf&PSfªck_ôXP [pPMRj`ôK]s

KWRGMY^0h[-^oWcdVkckZ¬`n^0_W\ hSZ&ckZ&cdMUWfKpVÓ^u`

DFE

9; 6!p;;=@I6§A<P6@7<86§@

PSZ

\WP_WMoZgP¨^f&PZ#"ro^0VkUWP\T2UW_WhZgcdM0_Xn^0KpKWcd_WezP^0h[CK4Mck_YZ

%$'&)(

Z&M^fGPSZ

|

*

&)(,+ «wPf&UpKWK4M0f&PZg[WRGMUWe[WM0UpZZg[-^·Z

-/.1032547698;:=<>:/?5@A?CBD

À~¿0ÁNÃ0ÈÁ;E9¿·ÌÌGFi½§¾NÀÆÈ~¼·ÁpÌ¼·Á

IHJ

K

<ML

cdf@^hSVdMfGP\+f&PSZjs'ro^0VkUWP\+Xn^oKq [-^oZ¨ckf

NPOQ=RTS |

U

|WVYX X $

| [Z

ckf@^hVdM0f&P\+f&UpWf&PZ§M0T

&)(]\&)(+

K

<><ML

|

ckf@^_WM_WPSXKpZ¬`9hMXCK-^0hZ@hM_YrPm¸f&UWWfGPSZtM0T

&^(

T2M0R@^0VdV

'$'&)(+

K

<><><L$

[pPR&PPm]ckfGZgf`_ba

fGUWh[¸Z&[-^oZT2MR@^0VkV %$&^(

fGUWK

c[degfihGj kGlknm

_0|

3po k k

O@[pPR&P krqsk

\WPS_WM0Z&Pf¨^0_Y`$_WMRGX M_ &)(

tu8wvyx9<z4{<>2Tx K#|

L J Ì¼·É}FiÀÆÈ~¼·ÁCT2MRZg[WP\WcaPSR&PS_Z&cÓ^0V4cd_WhSVdUWfGcdM_

~

$ | ~ |>]

(7)

fGUWh[¸Z&[-^oZ ~ | ^o_W\

~ |

$ | ~ |

G

T2MR@^oVdXCMfGZPrPSRG` $& +

_p\WPR Zg[pPl^04Mr0P¡^of&f&UpXKpZ&cdM_pf® ckZ ckfqOQPSVdV}"i_WMO@_n|2f&PSPFJtUWWck_ª^o_W\ªÎQPVdck_-^| 3uÏ ®

ÎQ[q W

Ô

®pMR@ÎQVd^0R&oPzPSZ¨^0VB|

30Ï

®WÎQ[qsW

3

FZ&[-^oZ|z]^0\WXCckZgf|2Z¬`iKWckhu^0VkVk`C_WM_]s'UW_WckiUpP

fGMVdUpZ&cdM_pfQZg[WRGMUWe[+PSr0PRj`$cd_WcaZgcÓ^oV4KNMcd_YZ

8 ?5@

]UWKWKNMfGPz^\WcaPSR&P_YZ&cÓ^0VNcd_WhSVdUWfGcdM_cdf¡eckr0P_M_^(hSMXCK-^0hSZQhM_Yr0PmfGPSZ

* &)( ®NM0T Zg[pPªT2MRGX

|

| ®fGUWh[Zg[W^oZ

| n* T2MRt^0VdV $ ^0_W\

fg^·Zgcdf

Pf+|2c^0_W\ |Äckc® O@ckZ&[

&^(

R&PSKWVÓ^0hSP\»Y`

[WP_LO¡Phu^0_»PmiZ&P_W\»ckZZgM>^

\WcaPSR&PS_Z&cÓ^0Vcd_WhSVdUWfGcdM_\WP

_WPS\M_$Zg[WPzO@[WM0VdPf&K-^ohP

& ( F©-MR

%$%& (

VdPZn|

$

\WPS_WM0Z&P@Z&[WP@UW_WckiUpPtKNMcd_YZ ck_ hVdM0f&PfjZlZgM ^0_W\\WP _WP

|

|| G [WP_

f&^oZgckf PSf¨¥¨`iK4MoZg[WPSf&cdf

3 3

DFE H

;pI£.;WX<tI6§¾@

[WPXn^0ck_MGPhSZM0T Zg[Wckf§K-ôK4PSR^0RGP(K-ôZ&[WftO@[Wckh[°^0R&PMpZôcd_WPS\^0fhSPRjZ^0ck_»|Ä\WPZgPSRjs

XCcd_WckfGZ&cdh§M0RRg^0_W\pMX9QKNPRjZgUWRG-^oZ&cdM_WfMoTfGMVdUpZ&cdM_pfM0T|z]

tu8wvyx9<z4{<>2Tx K |{|

L J hM_YZgck_UWMUWf T2UW_WhZgckM_

. &

V

&)( O@ckVdVi4Ph^0VdVkP\

^¾-ÂÅÀ>F]ÅÂ&ÃÌ¼·É}FiÀÄÈ~¼·ÁZ&M$|>]|OQP^oVdf&Mfg^u`^KNPRjZgUWRG4PS\$f&MVkUpZgckM_Z&M

¡caT ckZQfg^oZ&cdf Pf

Z&[WPzT2MVdVkM O@ck_We(fGPSZ¨M0T hM_W\pckZgckM_Wf|z;]

K

<ML . ckf@^0Wf&M0VdUpZ&PVk`nhM_YZ&cd_UWMUWfS®

K

<><ML

. |

! (|

$

#"

f%$j&|

. |

GT2MR^0VdXCMfjZªPrPSRG`

a V

T2MRf&MXCPT2UW_phSZgckM_'&

V & O@ckZg[(&]|

^0f ¦¥¨PR&P

" | U X $ & ( )

l

kGl k+* &

V

|

XTV

|l G * & Z^0_W\

| XTV

ck_pT-,

d/.

k X _ k +

K

<><><L 10

(| @ckf@^VkM]h^0VdVa`Ccd_YZgPSeRg^0pVdP§T2UW_phSZgckM_$f&UWh[¸Zg[-^oZ

VkcdX

$324 fGUWK

57698:6<;

k>=

$ 8

$

(|@? ? k

T2M0R@^0VdVBA a +

[WPKWUWRGK4M0f&PM0T*Zg[WckfK-^0KNPRckfQZgMcd_Yr0PfGZ&cdeY^·ZgPZg[pPzVdM_We(Z&PRGX NP[-^uricdMRMoT

.

^0_W\

Z&MC\pPf&hSR&ck4PckZ&fVdckXcaZQfGPSZ C |.

ED

$GF 5 U .

|@? p?#H Z

ck_$Z&PRGXfM0T Zg[pP\]`]_W^0Xckhf@cd_p\WUWhPS\¸Y`

+

(8)

DFE

JtfckZQO@ckVdVN4PzfG[WMO@_$[WPRGP0®^(_-^oZ&UWRg^oVhVd^0f&fQM0T K4PSRGZgUpR&NP\$f&MVkUpZgckM_WfFZgM

^oR&cdfGP§T2R&M0X

hSPRGZg^0cd_$fjZgMih[-^0fGZ&cdh^0KWKpR&Mum]cdX^oZgckM_9KWRGM]hSPf&fGPfS

tu8wvyx9<z4{<>2Tx K |{|{|

L J \Wckf&hRGPSZ&PtZgckXP@KWRGM]hSPfGf

U

Z

d

Vdcaricd_Wecd_

& (

ckfQ^ªf&MVkUpZgckM_

M0T|

¯G¯&¯

QcaTcaZrPSR&c Pft^RGPhSUWR&fGcdM_+M0T Zg[WPzT2MRGX

/$

| |> ]

O@[WPSR&PZg[WPh[-^oRg^0hZgPRGcdfjZgcdhSf°^0_W\ f&^oZgckfGT`

U Z F

cdf@^fGPUWP_phPMoT_WM0_W_WPe^oZgcarP_UWX(NPR&f¨f&Uph[¸Zg[W^oZ

V VdcdX

2 4 +

$'&)(

^0RGPC|Ä\WPZgPSR&XCcd_WckfGZgckhMRR&^0_W\WMXnQKNPRGZ&UWR&W^oZgckM_Wf

M¨fGUWh[(^KWRGM]hSPfGf ckf _-^oZ&UWRg^0VkVk`t^0fGf&MihcÓ^·ZgP\^hM0_Z&cd_UWMUpf*Z&cdXCPBKWRGM]hSPf&f ^0fqT2M0VdVdMO@fS

tu8wvyx9<z4{<>2Tx K#|

L ]PZ

5

^0_p\

T2MR H 3 V

^0_p\+\pP _WPZg[pPhSM_YZgck_iUpMUWf@ZgcdXCP¿¶ÁÂÈÁ-À¬ÂÅ'¾-¼·Éa¿0À/Â&Ã¾NÅg¼ºÂÌÌ &

& ( Y`

>|

o ?

V ? $ V

/

+

|z

DFE ! E 68@QI;".<§AI;W#7_;=e¶IX$";=pI£.;W e¾<8tI6§@

[WP_pPmiZR&PSf&UWVaZzeckr0P(f&U%hScdPS_ZhSM_W\WcaZgcdM0_Wf§M_Z&[WPh[-^0R&^0hSZ&PRGcdfGZ&cdhSfM0TlZg[WP\WcdfGhR&PZgP

KWRGM]hSPfGf(|z; ]¡T2MRcaZgfcd_YZ&PR&KNMVd^oZgckM_>|z&¡ZgMC4P^K4PSRGZ&UWR&NP\+f&MVkUpZgckM_¦|> ]

'#72T032T:=<47<z2Tx ?5@)(+* ÌÌGFi½nÂÀËp¿0À

K

<ML-, ¼·Å¿0ÉÉ A a

VkcdX

24 fGUWK

U

k.0/

k 21

o 3 V+=+=+GV43

| o

A§ Z

5

Ë-ÂÅgÂ

3 | f&UpK

U 1 H H . Z | 3

(9)

<><ML f&UWK

V

ÀË-ÂÁÀË-ÂÈÁ-À/ÂÅ'¾-¼·Éa¿0À¬Â&Ã¾NÅ¼ºÂÌÌ ÈdÌ¿¾-ÂÅÀ>F]ÅÂ&ÃCÌ¼·É}FiÀÆÈ~¼·Áw¼/Ç

+

'#72 2 PZ V ' &

&^( \pP_WM0Z&PzZg[WPhM0_Z&cd_UWMUpf¨Z&cdXCPKWR&MihPSf&fGPf¨\WP _pP\Y`

¦|

o ? V

|

o ?

/

T2MR@^oVdV $V

m ? * +

[WPS_q®-T2MR

$ sV

/

G

>| $ ( f$j o | 3 | &

¦| o | ( f$j

[WPS_WhP

|

$

¦|

o | ( f$j +

PSZ¨UWf@KWU]Z+&]| k

°| ( f$j k [WPS_¸MYricdM0UWf&Va`

| ( f$j

*

" f$jg|°|

G

+

¯ _¸^0\W\WcaZgckM_

&p| m (

f%$j

k ( f%$j

k o _0| 3ro

[WPS_WhPneM]PSfZ&M

® UWf&ck_We[Y`iK4MoZg[WPSf&cdf

K

<L M0TZ&[WPnfjZ^oZ&PXCP_YZ(MoTZg[WP R&MKNMf&caZgckM_q ¯ Z

RGPX^0cd_Wf¡ZgMCh[WPh$hM0_W\WckZ&cdM_

K

<A<><L M0T|> ]QWUpZ¨M_WP[-^0f

k>=

$ 8

$

¦|?· ? k m ( f$j

k ( f%$j

k o k (

f%$

8 j/

( f$j

k

o ( f$ 8 j

k ( f$ 8 j

k

^0_p\$Z&[WPR&PSf&UWVaZ¨T2M0VdVdMO@fQT2RGMXhSM_W\WcaZgcdM0_

K

<L t

DFE

· 768;=@QIY7_@b6'I6§¾@

PSZª| VV F4P^KWR&M0-^0WckVdckZ¬`f&KW^0hP^0_W\

U Z F 5 ^ VaZgR&^oZgckM_9M0T |ÄcÆPok®W^_pM_]s

\WPShRGPu^0fGcd_WefGPUWP_phPCM0TFf&UW]s4s/ôVdePSWRg^0ftM0T + « Pfgû`¸Zg[-ôZz^nfGZ&M]h[-ôfGZgckhKWR&MihPSf&f

U

Ze0ckrPS_$`>|> ]Qfg^oZ&cdf

PfZg[pP@¼ ÈÁpÌ¦¼·Á-Åg¼¨MR°¿0ÅÀÄÈÁ;EY¿0ÉdÂªÃ0ÈÂÅgÂÁºÂzÁ¼·ÈdÌÂ

|ÄUpf&[W_WPSR¨^0_W\zcd_®

3Y

QhM_W\pckZgckM_ckT caZgfh[-^0R&^0hSZ&PRGcdfGZ&cdhSf¨f&^oZgckfGT`T

K

<ML

U Zcdf@^\WPZgPRGXck_WcdfjZgckhf&PSiUpP_WhSP0

(10)

<><ML

U Zªcdf¿YÃY¿¾NÀ¬Â&ÃZ&M

U Z + [-^oZ@ckf® ckfXP^0f&UWR&^0WVkP§O@ckZ&[+RGPfGK4PShSZ¨ZgM T2MR

P^0h[ H +

K

<><><L

| +

[WP_pPmiZKWRGMK4M0f&ckZ&cdM_¶cdfn^whVd^0f&fGcdh^0V@PfGZ&cdX^oZgP$T2MRfGZgMih[-^0fjZgckh^0KWKpR&Mum]cdX^oZgckM_

KWRGM]hSPfGf&PfStM0Z&PtZg[W^oZ

\WMiPf¡_WM0ZQ^0KWKNPu^0RS« P§R&PT2PRFZg[WP¨R&P^0\WPRFZgM9|Æ±QPS_-^u²³dX+® 3 ®

R&M0K4MfGckZ&cdM_WfpW

Ô

^0_p\W¡T2MR@^KWRGM]MoTB^0_p\+T2UWRjZg[WPSR@R&PST2PSR&PS_WhPSf

'#72T032T:=<47<z2Tx ?5@ ÂÀ

U

Z E0ÈÍ0ÂÁ Êz|z; ]ÂQ¿ ¨¼ ÈÁpÌ¦¼·Á-Åg¼B¾NÅg¼ºÂÌÌ

J D F ¾Y¾-¼ÌÂ

ÀËp¿0À@¼·ÁÂ¼/ÇÀ2Ë-ÂlÇu¼·ÉÉd¼

5

ÈÁ;E9º¼·ÁNÃ0ÈÀÆÈ~¼·Á¸Ë-¼·ÉaÃ·Ì

K

<ML-,

¼·Å§Ì¼·½nÂ H Ô

fGUWK

| k

k

*

¿0ÁNÃ

*

¼·Å

K

<><ML

Ë-ÂÅÂÂÈdÌÀ2Ì¿¾-¼ÌÈÀÄÈÍ0ÂÁ5Fi½ ÂÅÌGF-ºË+ÀËp¿0ÀNÇu¼·Å¿0ÉÉ $'& (

|2Pm]K |

V

m Pm]K | Ô k k

¿0ÁNÃ

/ ,

"!#

*

Çu¼·ÅÂ&¿iºË _ a +

Ë-ÂÁ°¿·ÌÌGFi½§¾NÀÄÈ~¼·Á

K

<ML ¼/Ç%$¡Åg¼¾-¼ÌÈÀÆÈ~¼·Á'&

J)(

Ë-¼·ÉaÃ·Ì

5

ÈÀË¾NÅ¼ ¿<ÈÉÈÀÄÊ*&

J

8 ?5@,+ ìKWckhu^0V*ôKWKWVdckhuôZ&cdM_pf^0RGP

K

<ML UW_WckT2M0R&XCVk`C4M0UW_W\WPS\¸ck_

C ^0_W\

3- V

K

<><ML

Up_WckT2MRGXVa`C4MUp_W\WP\¸^0_W\

.

| 3

VkMe|q

+

(11)

EGD E <'I6§iIXA; ;768X68@E A*68@ EÐ;|<

PSZ ^0_W\ NPXP^0f&UpRg^0WVkPf&KW^0hPSfR&PfGK4PShSZ&ckrPSVk`h^0VdVkP\nZ&[WP¿iºÀÄÈ~¼·Á+Ì¬¾p¿iºÂ¡^0_p\$Z&[WP

ÌÀ~¿0À/ÂÌC¼/ÇÁN¿0ÀAFiÅgÂ® *

&^(

^hSM_YrPmhMXCK-^0hZ§fGPSZzh^0VdVkP\$Zg[WP+¼ F]À¬º¼·½nÂÌÌ¬¾p¿iºÂ¨^0_W\

\

Ò^XP^0f&UpRg^0WVkPT2UW_WhZgcdM0_+h^0VdVkP\9Z&[WP9¼ F]À¬º¼·½nÂQT2Up_WhSZ&cdM_q

JZ\Wckf&hRGPSZ&PZ&cdXCPf 3 V Ô

+=+=+ ^$\pPhckf&ckM_°Xn^0oPRC|Äy th[WMiMf&PSf^o_ ^ohSZgckM_ Q

T2RGMX¹^0_W\$Mpf&PRjrPSf^0_+MUpZ&hMXCP

| Q V4

+

« PfGUWKWKNMf&PZg[W^oZ

K

<ML¸ [WP(fGPUWP_phP

U Q V Z F 5

cdf^Rg^0_W\pMX KWR&MihPSf&f\WP _WPS\>M_f&M0XPªKWR&MW^0WcdVkckZ¬`

fGK-^0hSP+| VV ª¨^0_p\°^0\-^0KpZ&P\Z&Mf&MXCP VaZgR&^oZgckM_

U Z + ¥tPSR&P [W^0f§ZgMNP

Up_W\WPRGfGZ&M]Mi\^ofZg[WP[WckfGZgM0RG`M0T Zg[pPKpR&MihPfGftUp_Z&cdVZ&cdXCP +

K

<><ML

carP_Z&[WP[WcdfjZgMRj`

;V

y

^0_W\+_-^oZ&UWR&P^0hZ¨cd_W\WPSK4PS_W\WPS_Z&Vk`T

|G| Q V /

$>

Q

\> | Q p$>

Q | / $>

T2M0R@^0_Y`XP^0f&UpRg^0WVkPzf&PZgf

Q *!¹^0_W\

"

*#

+

K

<><><L y

M_WVa`9oPPKpf¨Z&Rg^0hM_Z&[WPhUWX(UpVÓôZ&ckrPûr0PRgôePM0TZ&[WPK-^0fGZ@M0UpZghSMXPSf

3

| Q

V

V | Ô

^o_W\$[Wcdf@\WPShcdfGcdM_pf@^0R&P-^0fGP\+M_$Zg[Wckf@^r0PR&^0eP0 [-^·Ztckf

| Q /$° Q

%$

h # | Q

O@[pPR&P

$ h | +Bcdfl^KWR&MW^0WcdVkckZ¬`XCPu^0fGUWR&P¨MrPR&T2MRlPu^oh[ %$ ®i^0_W\ %$ 0

$ h | Q $ V 3

qcdfQXP^0f&UWR&^0WVkP§T2MRP^0h[$XP^0f&UWR&^0WVkPf&PZ Q *# + [pPTÄ^oXckVk`

$ U'$

h Z h=d)(

cdfh^0VdVkP\$^9ÌÀÆÅG¿0À¬ÂE0ÊT2MR@y

[WckfVd^oZ&Z&PR^0fGf&UWXCKpZ&cdM_¶hu^0_Ñ4P(GUWfjZgc P\Y`»hSM_Wf&ck\WPR&^oZgckM_WfM0T§VdcdXCckZ&P\LXCPXCMRG`

^0_p\>4MUp_W\WP\>R&^oZgckM_-^0VkckZ¬`0 ¯ Zzckf§K-^0RjZgcd^0VdVa`XMoZgckro^oZ&P\¸`]X^0VdP f§^0KWKWRGMY^0h[ZgM9Z&[WP

KWRGcdfGM_WPRf\WckVdPXCX^ |A]X^0VdPo®

3uÏ

|ÄfGPPn^0VdfGM±QPS_-^u²³kX ^0_W\¤¥¨cdRGf&h[q®

3u£

®±QVd^0hs

O¡PVkVf^0KWKWRGMY^0h[-^oWcdVkckZ¬`Zg[pPMRj`°|Æ±¡VÓ^0hYO¡PVdVÆ®

3u

£ ]MR&ck_

ÔÔ

®]^0fO¡PVdV^0f hZgcaZgcdM0UWf

KWVdû`>|Ä±QR&MO@_q® 3u 3 +* M0Wcd_WfGM_q® 3 3 F^0_W\fjZgMih[-^0fGZ&cdh hSZgcaZgckMUWfKWVdû`|Ä±QP_Wû²³dX ^0_W\

¥¨cdR&fGh[q®

3

30Ï

¥¨M0T2-^0UpPR^0_W\ p^0_W\W[pMVdX+®

ÔYÔ

§cd_

eY^oXP§Z&[WPM0RG` |ÄfGPPzZg[pPPmp^0XKpVdPf@NPVkM Ot

|

]U

=-,

.0/

| Q V $ h | Q 21| 31 $54 |6 Z

(12)

| |Q / V4 $ | p* |

O@[WPSR&P \WPS_WM0Z&PZ&[WPf&X^0VdVkPfjZhVkMf&PS\¸fGPSZ "ro^0VdUWPS\$PmiZgPS_Wf&ckM_¸MoT O@caZg[hSM_YrPm+ro^0Vs

UWPSf MRGPKWRGPhckf&PSVk`®Z&[WPe0Rg^0KW[M0T cdfZg[WP¸cd_YZgPSR&f&PShSZ&cdM_ÑM0T^0VdVhSVdMfGP\fGUWWfGPSZgf

N * \ µT2MR(O@[Wcdh[¶Zg[WP NPR N h U X $ | WVYX $ N ZckfhM_Yr0Pm^0_W\

hSM_YZ^0ck_Wf | +

^0f@ck_

*

PSXn^0RG

3 Ô

ZgMC^f&PSZ "ro^0VdUpP\+Xn^0K$M_

&)(

Y`$f&PZ&Zgck_We

| | n| & +

[WPS_+Z&[WPXn^0K

| o ||

&

o

| |

hSVdPu^oR&Vk`fg^·Zgcdf Pft¥¨`iKNM0Zg[pPf&ckf

3 3

^0_p\

U

Zªr0PRGc PSf¨Z&[WPR&PhSUWR&fGcdM_

/

3

o 3 |

o | Q

/ V4

G

O@[Wckh[¸h^0_+4PR&PO@R&caZ&ZgPS_+^0fª|> p

$

| o

O@caZg[

^0_W\ /

|Q [V4

, | Q V4 $ h # | Q + ¥¨P_WhSP0®tZ&[WP

hSM_W\WcaZgcdM0_WfM0T R&MKNMf&caZgckM_

3 n^0RGPzfg^oZ&cdf P\¸^0_W\+M_WP\WPS\WUWhPSf

'#72T032T:=<47<z2Tx @>?

Ë-Â¿¶ÁÂnº¼·Á-ÀÄÈÁ5F¼ FYÌÀÄÈ½nÂÈÁ-À¬ÂÅ¬¾-¼·Éa¿0À¬Â&Ã¾NÅg¼ºÂÌÌC|zn¼/ÇÀ2Ë-Â

¾NÅg¼ºÂÌÌ

U

Z E0ÈÍ0ÂÁ Ê B

H

ÈdÌ(¿0É½n¼ÌÀFÌGFiÅgÂÉÊ$¿(¾-ÂÅÀ>F]ÅÂ&ÃÌ¼·É}FiÀÆÈ~¼·ÁL¼/Ç

ÃiÂlÁÂ&Ã Ê

B

(HJ

~ 0 >8 @ B Éa¿iº

5

ÂÉÉÌ¿¾Y¾NÅg¼¿iºËp¿<ÈÉÈÀÄÊÀË-Â¼·ÅÊ

HJ

J f&PSZ * cdf(f&^0cd\ ¿¾

¾NÅg¼¿iºËp¿<ÉdÂQckT Zg[WPSR&PPm]cdfGZ&f§^fGZ&Rg^oZ&Pe0`

$

fGUWh[Zg[W^oZ

^0VkXMfjZ@f&UWRGPVa`±QVd^0hs

O¡PVkV§|

3

£

§eckr0PhSM_W\WcaZgcdM0_WfP_WfGUWR&ck_We^0KpKWR&MY^oh[-^0WckVdckZ¬`0C« PO@cdVdVf&[WMO cd_¦f&PShSZ&cdM_

]

3

[pM O¹±¡VÓ^0hYO¡PVdVAf¨R&PSf&UWVaZgfthu^0_NPKW^0RGZ&cÓ^0VkVk`\WPRGckrPS\T2R&MX M0UWRtX^0ck_¸RGPf&UpVkZgf§^0_W\

ePS_WPR&^0VdckbPS\w|ÆÎQMR&M0VdVÓ^oRG` ]

Ô

Qcd_hPSRGZg^0cd_+\WcdRGPhZgcdM0_Wf

E

;A£@6@ 6@ AEÐ;

[WPCKWRGPhPSP\Wck_We¸T2M0R&X^0Vdckf&X cdfzO¡PVdV fGUWckZ&P\°ZgM¸^0_-^0VaìbPhPSRGZôcd_°XMi\WPSVdfzM0TQVdPuôR&_Wck_We

ck_+e^0XPSf

(13)

ÎQM_WfGcd\WPSRzZg[WPf&caZgU-^·ZgcdM0_O@[WPRGP 3 KWVd^`0PRGfªôR&PKWVÓûìcd_pe$^$e^0XPM r0PRª^0_W\°MrPSR

PSZ |T2MR $ U 3 V=++=+V 3 ZozNP^ _pckZgPCf&PZRGPKWRGPfGP_YZgck_We¸Z&[WPn^ohSZgckM_Wf|ÄKpUWR&P

fjZgRg^·ZgPe0cdPfgQ^uro^0cdVd^0WVdPZ&M(KWVÓ^u`0PR

^0_W\VkPSZ

4PtZg[WP _WckZ&P§\WckXPS_Wf&ckM_-^0VNf&ckXKWVkPmM0T

KWRGM-^0pcdVdcaZgckPfMrPSR

VkPSZ /

^0_p\

/

RGPfGK4PShSZgcarPSVk`$\WPS_WM0ZgPZg[pPªôhSZgckM_Wf¨^0_W\+XCcam]P\$fGZ&RgôZ&PeckPf¨ûro^0cdVd^0WVdP

Z&MZg[WPMKpK4M_pP_YZgfM0T

+

[WPK-^u`MoT2UW_WhZgcdM0_¤ZgM>KWVd^`0PRzckf(eckr0P_LY`L^T2UW_phSZgckM_

\ /

&`+ Jtf§UWfGU-^0V OQP(PmiZgP_p\

ZgM9^CT2UW_WhSZ&cdM_¤|ÄfjZgcdVkV*\WP_WMoZgP\

¨M_

\ / V Y`X(UpVkZgcs¬Vkcd_WP^0R&caZ¬`

~ 0 >8 @( B,

È~ºÀÆÈÀÄÈ~¼ FÌ¿0ÁNÃ ÌÀ¬¼ºËp¿·ÌÀÆÈ~º ºÀÄÈÀÆÈ~¼ FYÌ¾NÉa¿0Ê

HJ

ÎQM_WfGcd\WPSRnZg[WP¸eY^0XCP

T2RGMX Z&[WPricdPO@K4M0cd_YZzM0TFKWVd^`0PR Qf&MZg[-^·ZZg[WP\WPhScdf&ckM_>X^00PSRcdfKpVÓ^u`PSR @^0_W\ vG_W^·s

Z&UWR&PS ckfCe0ckrPS_ÑY`LZ&[WP¸MoZg[WPSRCKpVÓû`PSR&f ¯ _ hZgcaZgcdM0UWfMRfGZ&M]h[-ôfGZgckh hSZ&ckZgckMUWfKWVdû`

Z&[WPªM0UpZghSMXPfGK-^0hPªcdfZg[pPªfGK-^0hP

\ /

M0TBXCcam]P\+fGZ&Rg^oZ&PeckPft^0_W\$Zg[WPMUpZ&hMXCP

T2UW_phSZgckM_°cdfzZg[pP¤vGck\WP_YZ&ckZ¬`iT2UW_WhSZ&cdM_ \ / \ /

Xn^0KpKWcd_WePrPSRG`

KWRGM VkPzM0TB^ohSZgckM_Wf Q Z&MZg[WPhM0R&R&PSf&KNM_W\Wck_WeCKWR&M VdPzM0T Xcm]P\$fGZ&Rg^oZ&Pe0` & +

PSZ

| /

JtR&eX^·m

d , |Q

VG

/

*!

NPzZg[WPf&PZ¨M0TNPfjZt^0hSZ&cdM_WfZ&[-ôZ huô_¸KWVdû`cd_RGPf&KNM_WfGPZ&M / +

±QM0Z&[°hVÓ^of&f&ckhu^0V

hSZ&ckZgckMUWfKpVÓ^u`»|Ä±QR&MO@_q®

3

3 ® *

M0Wcd_WfGM_

3

§^0_W\¦fGZ&M]h[W^0fGZ&cdh

hZgcaZgcdM0UWf¡KpVÓû`¸|Æ±QPS_-û²³kX ô_W\9¥¨cdR&fGh[

3

30Ï

®p¥¨M0T2-^0UpPR

^0_p\ p^0_W\W[pMVdX

ÔYÔ

§^0f&fGUWXCP(Zg[-^oZ§Zg[WPfjZgRg^·ZgPeo`M0TFKWVd^`0PRG®

$ U'$

h Z V hu^o_¦NP

O@RGckZ&Z&P_+^0f

$ h | Q | Q V&

/

O@[WPSR&P

\ / V 3 cdffGUWh[¸Z&[-^oZ

<47<47<>29: ' w.u:=: 0W47<z2x

d

/

fh j |Q

V&

/ 3

MR

4{2 6 s:47< <47<47<z2!9: ' w."u:=: 0W47<z2x

ckff&XCM]MoZg[$cd_ /

^0_W\

d

/

fih j |Q

VG

/ H 3 (&

T2MRfGMXCP

* & *#*

3 +

¯

_$Zg[WckfT2Rg^0XCPSO¡MR&®-caT Q

\WPS_WM0ZgPSf¨Z&[WPKWR&M VdPzM0T^0hZgcdM0_Wf@^oZ@fGZg^0eP # V M_WP[-^0f

3

Q

(14)

/

3

o 3 | Q /

[UWf@T2MRPu^oh[

$ 3

o 3 | | /

O@[WPSR&P | /

p*

cdfQZ&[WPhSM_YrPm[iUpVdVMoT | /

FT2MRFZg[WPzfjZ^0_W\W^0R&\ hZgcaZgcdM0UWf

KWVd^u` ^0_W\

| d , |Q

V&

/ & T2MRZ&[WPfGZ&M]h[-^ofGZgckh hSZ&ckZgckMUWfKWVd^u`

[UWf@Zg[WPfGPSZ#"Yr0^oVdUWPS\¸X^0K

\WP _WP\¸cd_¦|

cdfecarP_+^0f

| o

| / \ /

xWf&PSRGr0P$Z&[-ôZckT¨^fGUWWf&PZ * ¦M0T@KWVÓû`0PR&fKWVÓûìf^ hZgckZ&cdMUpf9|2MR(fGZ&M]h[W^0fGZ&cdh

hZgcaZgcdM0UWf¡KpVÓ^u`fGZ&Rg^oZ&Pe0`0®WZg[WPS_

[-^0fZ&MNPzR&PSKWVÓ^0hSP\+Y`

| D

d | +

¯ _+K-^0RjZgcdhSUWVÓôR®WcaTB^0VkVqKWVd^`0PRGftKWVdû`^ hSZ&ckZgckMUWfKWVdû`fGZ&RgôZ&Pe0`0®Z&[WP\Wck4PRGP_YZgcd^0V cd_phVdUis

fGcdM_cd_p\WUWhPS\Y`

ckfBZg[WP@NPfjZ¡RGPfGK4M_pf&P¨\Wck4PRGP_YZgcd^0VWcd_phVdUpf&cdM0_+|zcdVdNMY^^o_W\

^oZgfGUWc

|

3 3

®¥¨M0T2-^0UWPSR|

3

®¥¨M0T2-^0UWPSR ^0_W\ ]MR&ck_$|

Ô YÔ

G O@[WckVdPFckTN^0VkVKWVÓ^u`ª^fGZ&M]h[-^ofjs

Z&cdh hZgcaZgcdM0UWfQKWVd^u`®

cdf^(fGXMiM0Zg[n4PSfGZ@R&PSf&KNM_WfGPr0PhZgMR PSVd\¦|Æ±¡P_-^²³kX ^o_W\+¥tcdRGf&h[

|

3

3uÏ

®-¥tM0T2W^0UWPR¨^0_p\']^0_W\W[WM0VdX |

ÔYÔ

&

~ 0 >8 @ BD

½C¿0ÉdÂ¿¾Y¾NÅg¼¿iºË>À¬¼(¾NÅÈdÌ¼·ÁÂÅ ÌÃ0ÈÉdÂ½½C¿

HJ

«wPfjZgcdVkV hSM_WfGcd\WPSRZ&[WP

eYôXPQT2RGMX Z&[WPricdPSO@KNMck_ZlMoTKWVdû`PR *f&MzZ&[-ôZFy

cdfBKWVd^u`PR ^0_W\_-^·ZgUWRGP@Z&[WP@M0Zg[pPR

KWVdû`PRGf®-WU]ZÖ¡PZ^0oPT2M0R Z&[WPK-û`M0>r0PhZgMRGftT2UW_WhZgckM_

\ / V

Q

(| Q

|

| Q V=+=++V

( | Q &

O@[WPSR&P * & ( ckfZg[WPhSM_YrPm+[UWVkV M0T Zg[WPK-^u`Mo>rPhZgMRGf

U

(|

Q

[Z

+

[Wckf f&PSZGZgck_We Zgf*PmW^ohSZgVa`§O@caZg[ ]X^0VdP fq^0KWKWRGMY^0h[zZgM@KWRGcdf&M0_WPRf*yckVdPXCX^§|A]Xn^oVdP0®

30Ï zVÓ^oZ&PRRGPSricdfGckZ&P\¤Y`w±QP_-^²³dX ^0_W\»¥¨cdR&fGh[}| 3u£ ^YyPZ^0ckVdfO@cdVdVlNPnecarPS_Lcd_

fGPhSZ&cdM_ ]

Ô

O@[WPRGP/]X^0VdP f^0KpKWR&MY^oh[wO@cdVkVFNPCR&Pck_YZgPRGKWR&PZgPS\Lcd_ Z&[WPT2R&^0XCPSO¡MR&°M0T

^0KpKWR&MY^oh[-^0WckVdckZ¬`0