The Central Limit Theorem for a non linear algorithm based on quantization

HAL Id: inria-00071956

https://hal.inria.fr/inria-00071956

Submitted on 23 May 2006

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

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

The Central Limit Theorem for a non linear algorithm based on quantization

Vlad Bally

To cite this version:

Vlad Bally. The Central Limit Theorem for a non linear algorithm based on quantization. [Research Report] RR-4629, INRIA. 2002. �inria-00071956�

ISRN INRIA/RR--4629--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

The Central Limit Theorem for a non linear algorithm based on quantization

Vlad Bally

N° 4629

Octobre 2002

)

* +,#-/.0$123$

4(57698;:/695<5<=

>@?BADC/E3FHGJILKMCNBOQPRSKUTWV(E$R TYXZRSKUC/KQ[\P]R\KMTYV

^ E[\_L[`R\AC/E[ a$TWC/XBOUEDbZE[

cedST]fgEDRhiPRS?Zj

kP]XlXmTWd`R ^ EdSEa?BEdSa?lEV9nFWoqp]rsGutva$R\TYwld\Ep]xWxWp3Gup]yHXlPYzYE[

{}|~D€B‚YYƒ/„V…P†[\EDdSdS_}T]‡eXmP]X9EDd[ ˆ‰E?lP‹ŠYEXld\EŒ[`EVWRSE ^ P]VsP]OMzYTYdSKUR\?BCwlPY[\E ^ TWViWNmP]VqR\KMŽP]R\KMTYV‡TYd

Xld\KQa$KMVBzs‘C/EDdSKMaP]V’TWXZR\KMTYVl[“’h”TWd\E†zWEDVBEdSPYOUOM_R\?BKQ[/P]C/TYNlVWR[RST[\TYOMŠYE}VqNlCHEd\KQaDPYOUOM_•P]V–TYwl[`RSPWa$OME

Xld\TWwBOUEC—‡TWd‰[\TYC/E[`EC/KlOMKUVBEŒP]d™˜š}›œMWžZtŸNBd™P]OMzYTWd\KUR\?lCKQ[ewlPW[`E ^ TWV†Ph”TWVWRSEŸ 2P]dSOUTC/EDR\?BT ^ P]V ^

[\T‘PŸ[gRPR\KQ[`R\KQaDP]OqEDdSdSTYd¡aDTYC/E[TYV¢“ „ VRS?BE™XBdSE[\EDVqRXlP]X9EDdˆ‰E‰[gRSN ^ _3R\?lKM[EDdSd\TWd£¤ˆ2E™Xld\TŠWE'R\?BE ‰EDVqR\dP]O

¥¡KUC/KUR0>@?BETYdSEDC¦‡TWdR\?BE"P]OMzYTYdSKUR\?BC§PYV ^ ˆ2E¨zYKMŠYE/EDŠP]OMNlP]R\KMTYVl[TY‡‰RS?BE†ŠP]dSKQP]VlaDEY“M“}>@?BE ^ KU©†a$NBOURg_

aDTYC/E[0‡dSTYCªRS?BE‡«PYa$R¨R\?lP]RHRS?BE”P]OMzYTWd\KUR\?BCªKQ[/VBTYR/OMKMVBEPYd“;tŸVRS?BE(T]RS?BEDd¨?lPYV ^ P]V¬KUVqR\Ed\EŒ[gRSKUVlz

Xld\TWwBOUECKQ[}R\T–aDTYVqR\dSTYO‘R\?BEswmE?lP‹ŠLKUTWNBd}TY‡R\?BEsŠP]dSKMPYVla$EiTY‡R\?BEP]OMzYTWd\KUR\?BCPY[†R\?BE­a$TWC/XBOUEDbZK®Rg_

KMVlaDd\EŒPY[\E[“U“ „ R3R\NBdSVl[TWNZR3R\?lP]RR\?BE¨ŠP]dSKQP]VlaDE ^ TLEŒ[3VBT]RwBOMTˆNBX­KU‡‰RS?BE/R\KMC/E ^ KM[Sa$dSE$RSKUŽŒPRSKUTWV[`R\EX

PYV ^ R\?BE[\XlPYaDE ^ KM[Sa$dSE$RSKUŽŒPRSKUTWV"[`R\EDXRSEDV ^ R\T¨ŽEDdST

¯”°q±³²S´µ €¶~Yƒ%· NlCHEd\KQaDPYOlcedSTYwmP]wBKMOUKURg_Y¸B¹ŸNlPYVWRSKUŽŒPRSKUTWV¨T]‡¤dSPYV ^ TWCºŠP]dSKQP]wBOME[¸ZtŸXBR\KMC/PYOm[`R\TWXBXBKMVBzl¸

IZEDC/K¢OMKUVBEŒP]d ˜š†› œW»m ‰EDVqRSdSPYO¢¥¢KMCHKUR‘>@?BETYdSEDC(¸BC/EDdSKQaDP]VTYXBR\KMTYVl[

D$ ) * *.}%$!-/." 23D

~ ƒ· TYNl[HP‹ŠYTWVl[ ^ fiXBd [\EDVqR }^ P]Vm[NBVlE"[ dSKUE ^PYd`RSKMaDOUEŒ[NBV–P]OMzYTYdSKUR\?BC/E†wlPW[ [`NBd0OMP

qNlPYVqR\KUjmaDP]R\KMTYV…X9TYNldvOME¨aDPYOMaDNBO ^ E[3Xld\KUb ^TYXZRSKUTWVl[C dSKQaDP]KMVBEŒ[D“ NBVBE/C¨P]VBKMADdSEHXBOUNm[z V dP]OME

aDEaDKWdSEDŠLKMEDVqR‘OMP‘d [`TWOUNZRSKUTWVVLNBC dSKMqNBE ^NBVXBdSTYwlOUACHE ^TYwl[`RSPWa$OME[¢X9TYNldNBVlEŸc­[`ECHK OUKMV PYKUdSEY“

 ‰TWC/CHEVBT]RSd\EPYOUzWTYdSK®RS?BC/EE[`RŸwlPW[ [`NBdŸNBVBE0C R\?BT ^ E ^ Eh”TWVqR\E ` 2P]dSOMTl¸BNBVlEEDdSdSEDNBd‘[`RSP]R\KQ[gRSKMqNBE

PYXBXlPYdSP"!MRŒ“#vP]Vm["aDEiXlP]XlKUEd}VBTWNl[ Š‹PYOUNlTYVl["a$E$R\R\EsEDdSd\ENBd¸@XBOMNl["XBd a$KQ[\EDC/EDVqR}TWV ^ C/TYVqRSd\EiNlV

RS? TWd\AC/E ^ EŸOUKMC/K®RSEvaDEDVqR\dP]OME‘X9TYNBd2OPYOUzWTYdSK®RS?BC/EE$R@TYV ^ TYVBVlE‘NBVBE ŠPYOUNlP]R\KMTYV ^ E‘OQPŠP]dSKMPYVqR\EY“'¥¡P

^ K®©†a$NlO®R ŠLKUEVWR ^ N(‡«P]KUR‘WNlEVBT]RSd\EPYOUzWTYdSK®RS?BC/EvEŒ[gRŸVBTYV(OMKUV P]KMd\EW“ PYNZR\dSEXlPYd`R KMO¡E[`RKMVWR dSE[S[\PYVqR

^ E/aDTYVqR\d%$YOMEDdvOPW[\[\_LCHXBR\T]RSKMqNBE ^ E/OQP"ŠPYd\KQP]VqR\EHqNlP]V ^ OQPaDTYC/XBOME$bZKUR /^ E/O P]OMzYTWd\KUR\?BC/E/a$dSTYKUR“¨tŸV

^& C/TYVqR\dSEŸqNBEvOQPŠP]dSKMPYVqR\E‘dSE[`R\EŸwmTWd\V EŸqNlPYV ^ OME[‰XlPW[ ^ E ^ KQ[\aDd R\KQ[\P]R\KMTYV†EV†RSEDC/Xl[2E$R@EDV"E[\XlPYaDE

RSEDV ^ [ ŠYEdS[2Ž d\T('EDR ^ TYVma3OMP/a$TWC/XBOUEDbZK®R 3^ EOP]OMzYTWd\KUR\?BC/EŸR\EV ^ ŠYEDd[@OKMVZjlVBK7“

)µ Œ~ ² ‚+* ~3ƒ cedSTYwlPYwBKMOUKUR [¡VLNBC dSKQWNlE[¸‹qNlPYVqR\KUjmaDP]R\KMTYV ^ EŒ[Š‹PYd\KQP]wlOUEŒ[¡P]O P]R\TWKUdSE[¸P]dSd-,DR¡TYXZRSKUC¨PYO<¸

‘c;[\EDC/K7OMKUV P]KMdSE[¸WRS? TWd\AC/E ^ EvOMKMCHKUR\E0a$EDVqRSdSPYOUEW¸qTWXZR\KMTYVm[ P]C dSKMaP]KMVBE[

0¡'¨+.™$

„R¨KQ[/ˆ‰EOUOLVBTˆ V–R\?mPR/R\?BEihiTYVqR\E… 2P]dSOMT…C/E$RS?BT ^ KQ[¨P]V P]OUR\Ed\VmPR\KMŠYE"R\TRS?BE ^ E$RSEDdSCHKMVBKQ[gRSKMa”PYO

zWTYdSK®RS?BC¨[KMV•TYd ^ EDdRSTia$TWCHXlNZR\E¨RS?BE"[\TYOMNZRSKUTWVTY‡ OUKMVBEPYd˜š†› œWž>@?lKM[KM[EŒ[`X9EaDKMPYOUOM_KUVqR\Ed\EŒ[gRSKUVlz

KMV ^ KMC/EDVl[\KMTYV…OQP]dSzYEd R\?lEDViRS?BdSEDETWd‘‡TWNBdvˆ ?BEDV…R\?BE ^ E$RSEDdSCHKMVBKQ[gRSKMa0C/E$RS?BT ^ [vwmEŒa$TYC/E ^ K®©†aDNBO®RvR\T

KMC/XBOMEDC/EDVqR“>@?BE‡TYNBV ^ PRSKUTWViT]‡ RS?BEHhiTYVqR\E¨ 2P]dSOUT†C/EDR\?BT ^ KQ[‘R\Td\EXBd\EŒ[`EVqRR\?BEH[`TWOUNBR\KMTYViTY‡eR\?BE

˜š}›ºPW[™RS?BEE$bZX9Ea$RSPRSKUTWV '«wq_"C/EP]Vm[‰TY‡¡RS?BElP‹_LVBEC/PYVPWa‡TWd\C0NBOMP! ™PYV ^ R\?BEV"RST/d\EXBOMPWa$EŸR\?BE

C¨P]R\?BEC/P]R\KQaDPYO9EDbZXmEŒa RSP]R\KMTYV"wq_†PYV}ECHXlKUdSKMaP]O³E$bZX9Ea$RSPRSKUTWV¢“#"VZ‡TWd`RSNBVlP]R\EDOM_¨RS?BKM[ PYXBXBdSTWPWa?/‡«P]KMOM[2KU‡

ˆ2E ^ EPYOˆ KUR\?VBTYV OUKMVBEPYdv˜š}› œHw9EaP]Nl[\EHKUV…R\?BKQ[aDPW[`ER\?BE/[`TWOUNBR\KMTYVsaP]VsVBT]R3wmEHdSEDXBdSE[\EDVqR\E ^

PW[‰PYV†E$bZX9Ea$RSPRSKUTWV¢“#$2NZR2dSEaDEDVqR\OM_Hc'PYd ^ TYNZb†P]V ^ c'EDVBz'«[\EDE&%˜@ž˜('7»)%›+*-,/.iž˜@ž0('“M“U“1 0?lP‹ŠYEŸzYKMŠYEDV†P

zWEDVBEdSPYOUKMŽP]R\KMTYVT]‡'R\?lEBE_LVBEDC¨P]V2PWav‡TYdSCNBOQP¨wL_}C/EŒP]Vl[‘T]‡3$@PYaLˆ2PYd ^ IqRSTLa?mPY[`R\KQa ‘K54³EDdSEDVqR\KQP]O

‰WNmPR\KMTYVm[2PYV ^ C/TWd\ETŠYEDdŒ¸YKMV6%›*7,8.:9!;<=9>*?*' '«[\EDE3P]OQ[`T@%A}žBvžC@ž›*,8./'D ™TYVBEvTYwZRPRSKUVl[@[\Nla?"P

Xld\TWwlP]wBKMOMKM[`R\KQadSEDXBdSE[\EDVqRSP]R\KMTYV(‡TWdv˜š†› œHˆ KUR\?sTYwl[`RSPWa$OMEY“Ÿ>@?lEDV…TWVBEC¨P‹_(EDC/XBOMT_(P Ÿ_qVmP]C/KMaP]O

c™d\TWzYdP]C/KUVlzcedSKUVlaDKUXlOUE3KMV}TYd

^

Ed™RST0ˆ dSK®RSE

^

Tˆ V"P]VP]XBXld\T‹bZKMC/P]R\KMTYV†[Sa?BEDC/EŸ‡TYd2R\?lEv[\TYOMNZR\KMTYV"T]‡P

[\EDC/K'OMKUVBEŒP]d3˜š†› dSE[\X9Ea RSKUŠWEDOM_”P}[\EDC/K'OUKMVBEPYd˜š†› ˆ KUR\?…‡d\EE0w9TYNBV

^

P]dS_Y“M“>@?BE

^

EDOMKMaPR\EHX9TYKMVWR

KMV•KMCHXlOUECHEVqR\KMVBz…[\Nla?’P]V’PYXBXBdST‹bLKMC¨PRSKUTWV[Sa?BEDC/E¨KQ[RS?lPRHTYVBE¨?lPW[RST…a$TWCHXlNZR\E}P(OQP]dSzYE¨VLNBC

w9EDd‘T]‡ a$TYV ^ K®RSKUTWVlP]O¡E$bZXmEŒa RPR\KMTYVm[ ˆ ?BKQa?KQ[@‡«PYd ‡d\TWC w9EDKMVBz/R\dSKMŠqKQP]O7“2IZEDŠYEdSPYO¤C/E$R\?lT ^ [@‡TYd‘[\TYOMŠLKUVlz

[\Nla?¬PsXld\TWwBOUECª?lPY[/w9EDEV–Xld\TWXmTq[`E ^ KMVRS?BE”OMPW[gR/_YEŒP]d[D¸™C/PYKUVlOU_’C/TYR\KMŠ‹P]R\E ^ wL_•C¨PRS?BEDC¨PRSKMaP]O

jmVlP]VlaDEXBdSTYwlOUEC/[v[\Nla?sPW[‘XBdSKQa$KMVBz"‘CHEd\KQaDPYV…TWXZR\KMTYVl[ŸwlPW[`E ^ TYVP†OQP]dSzYEVLNBC0wmEdŸTY‡‰PY[S[\E$RS['«[\EDE

%A†žE('»F%Gž1HI'7»)%G‘žJK' »DžMžUž1 “

„ VL%A}ž˜@žNMO'»F%A†ž˜@ž1PQ'lP]V ^ %A†ž˜@ž˜('Bˆ2E‰?mP‹ŠYE™Xld\EŒ[`EVWRSE ^ P‘VBEˆ’C/E$RS?BT ^ ‡TWd'[\TYOMŠLKUVlzR\?lKM[XBdSTYwBOMEDC(¸

wmPY[\E ^ TYV¨qNlP]VqR\KMŽP]R\KMTYV¢“'>@?BKQ[eP]C/TWNBVqRS[R\TdSEDXlOMPWa$E2R\?BE ^ K4³Nl[\KUTWV/XBd\TZaDE[S['NBV ^ Ed\OMKUVlz3R\?BE˜š†› wL_

RS?BE eNBOMEDd [\a?lEDC/E™jmdS[`R'PYV ^ RS?BEDVR\TvXBdST]fgEa$R¡RS?BE eNBOME2[Sa?BEDC/E2TYVHPzYKMŠYEVzWd\K^ '«[\EDE ILEŒa R\KMTYVHy?BEd\E

‡TWdP ^ E$RP]KMOUE ^^ EŒ[\aDd\KMXZRSKUTWV7 “>@?BEDV0ˆ‰E‰?lP‹ŠYE™R\TŸaDTYC/XBNZRSE2qNlPYVqR\KUR\KME[T]‡BRS?BE™Rg_LX9E2›SR?TIRDUWVYX[ZF\^]QUWV_\

ˆ ?lEDdSEUWVvP]dSE@VBTˆ ^ KM[Sa$dSE$RSE dSPYV ^ TWC—ŠP]dSKMPYwBOUEŒ[ 'RS?BE XBdST]fgEa$R\KMTYVHT]‡³R\?lE eNBOMEDd™[Sa?BEDC/EP]R R\KMCHEa`bV_\ ž

c‘EDdSE2aDTYC/E[TWVR\?lE@hiTYVqR\E 2P]dSOUTŸC/E$R\?lT

^

w9EaDPYNl[\E‰ˆ2E@d\EXBOMPWa$E™R\?BE@E$bBPWa RR\dP]Vl[\K®RSKUTWVXBdSTYwmP]wBKMOUKUR\KME[

wL_H[`TWCHE@ECHXlKUdSKMaP]OZ‡dSEqNBEVla$KME[eTYwZRP]KMVBE

^

wL_RS?BE[`KMCNlOMP]R\KMTYV/T]‡9R\?BE‘NBV

^

Ed\OM_LKUVBz

^

K4³Nl[\KUTWVHXld\TZa$EŒ[\[

'«KUV"‡«PYa$R T]‡¢RS?BE eNBOMEDd‘IZa?BEC/EvaDTYdSd\EŒ[`X9TYV

^

KUVBzR\T0R\?BE

^

K54³Nl[`KMTYVd “e‘[@KMV"R\?BEaDPW[`EŸT]‡R\?BE[`RSP]V

^

P]d

^

hiTYVqR\E 2P]dSOUT‘CHEDR\?BT ^ ¸]R\?BE ¥¢Tˆ•TY‡m¥¡PYd\zWE · NlCw9EDd[zWNlP]dP]VqRSKUEŒ[¤R\?lE@aDTYVLŠYEd\zWEDVlaDE ˆ ?BEDV0R\?lE2[\KUŽE‰TY‡

RS?BE@[\PYCHXlOUE‰zYTLE[¢R\TvKUVZjmVBK®Rg_W“ „ VR\?BKQ[XlP]X9EDdTYNBdXBdSTYwBOMEDC KQ[¢RST‘EŠ‹PYOUNmPR\E™R\?BE@[\XmEE ^ TY‡la$TWVqŠWEDdSzYEVla$EW¸

[\THR\T¨TWwZRSPYKUV”P} ‰EDVqRSdSPYO¢¥¢KMCHKURŸ>@?BETYdSEDC ‡TWdTYNldP]OMzYTYdSKUR\?BC(“'<ILNla?(P/XBdSTYzWdSPYC ?lPY[PYOUdSEP ^ _†w9EDEV

PWa?BKMEDŠYE ^ ‡TWd@R\?BE¥P]VBzq[gRP4sIZa?Lˆ@P]dSŽŸP]OMzYTWd\KUR\?lC KUVe%Gf9!g/hYi)j)`lkQ; žMžMž'D\

„ R2R\NBdSVl[2TYNBR™R\?mPR2R\?BEvXBdSTYwBOMEDC PR2?lP]V ^ P]XlXmEŒP]d[™PY[2PXmP]d\R\KQa$NBOQP]d2aDPY[\E‘TY‡¤R\?BEŸ‡TYOMOMTˆ KUVBz0C/TYdSE

zWEDVBEdSPYOXBdSTYwBOMEDC(“tŸVP@XBdSTYwmP]wBKMOUKURg_Ÿ[\XlPWa$E^R?mv»nC@»\˜\³ˆ2E™aDTYVl[\K^ EDd¢P[\EqNBEVla$E TY‡LdP]V ^ TYC;ŠPYd\KQP]wBOME[

U V »oLprqB»žUžMž; »9ˆ ?BKQa?iRP_YEHP¨jlVlK®RSE0VLNBC0wmEd(s V T]‡eŠP]OMNBEŒ[D¸tži=U VSu6vIV pxwFy

ZV

»DžMžUžM»bydzI{

V}|&~

›¨»

ˆ ?lEDdSE‘›—KQ[‰[\TYC/E‘PYwl[`R\dPYa R2[`XlPWa$EW“IEŸPW[\[\NBC/ERS?lPRU&€Kpy7€ŸKQ[ ^ E$R\Ed\C/KMVBKQ[gRSKMaY“IE3a$TYVm[`K^ EDd™R\?BE

aDTYV ^ K®RSKUTWVlP]O¤EDbLX9Ea$RSP]R\KMTYVl[

‚ V

ƒ…„

p

˜†R‡U VOX[Z pˆy

„

VYX[Z

»bU V py

ƒV \

˜†RDU V p‰y

ƒV \

»Št#p‹MW»DžMžUžM»s=Vq»Œ=pMY»žUžMžM»sVOX[Z

Ž ƒV p ˜†R‡U†Vpy

ƒV

\$»ŠtIprMY»DžMžMžU»nsVqž

h”TWd\ETŠYEd¸‰ˆ‰E…aDTYVl[\K

^

EDd}P•[SP]C/XBOME R‡U

V \ Vd€

»n*&p MY»DžMžMžU» TY‡†R‡U V \

Vd€

PYV

^

ˆ‰E

^

E$jlVBE…R\?BE

EC/XBKUdSKQaDP]O¢aDTYV

^

K®RSKUTWVlP]O¤EDbLX9Ea$RSP]R\KMTYVl[

‚ V

ƒ1„

p

Z

M

{

{

M

{

{

Z M

{

{

»ŠtIprMY»DžMžMžU»ns V »Œ=p‹MW»DžMžUžM»s VYX[Z ' )

ˆ KUR\?RS?BEa$TWVqŠWEDVqR\KMTYV}R\?lP]R

‚ V

ƒ1„

p q0KU‡

 ƒV p"!

#

Z M

$

{

{ p‰qlž

% KMŠYEV…P¨jlVmP]O'a$TYV ^ K®RSKUTWV'&(!•J*),+ J PYV ^ [\TYC/E0aDTqED©†a$KMEDVqRS[(C V !l›.-2J/+ J ˆ‰EHa$TYVm[`K^ EDd

RS?BE ^ _LVlPYCHKQaDPYO¤XBdSTYzYdP]C/KMVBz0PYOUzWTYdSK®RS?BC

0

132 RDy

ƒ2 \ p & RDy

ƒ2 \ t prMY»žUžMžU»ns

2

'<p

01

V>RDy

ƒV \ p z {

#„ Z C VYX[Z RDy

„

VOX[Z

» 01 VYX[ZR‡y

„

VYXIZ

\\

‚ V

ƒ1„

t prMY»žUžMžM»sVqž

>@?BKQ["KM["TYNBdP]OMzYTWd\KUR\?BC(“—>@?BEa$TWd\dSE[\XmTWV ^ KMVBz’P]OMzYTWd\KUR\?lCNl[\KMVBz•R\?lE…EDC/XBKMdSKMaP]O3a$TYV ^ K®RSKUTWVlP]O

Xld\TWwlP]wBKMOMK®RSKUEŒ[‰KQ[

1 2 R‡y

ƒ2 \ p &R‡y

ƒ2 \ '«y

1 V R‡y

ƒV \ p zI{4

#

„ Z C V R‡y

„

VYXIZ

»1 VYXIZ R‡y

„

VYX[Z

\\

‚ V

ƒ…„

ž

>@?BE‰OMP‹ˆ•TY‡BOMPYd\zWEeVLNBCw9EDd[¢zWKUŠWE[

‚ V

ƒ1„

+ ‚ V

ƒ1„

PW[56+87§P]V ^ aDTYVl[\EqNBEVqR\OM_Y¸

1 V RDy

ƒV

\9+

01 V RDy

ƒV \ ž

tŸNldP]KMC KM[2RST¨E[`RSPYwBOUKQ[\?P† ‰EDVqRSdSPYO¢¥¢KMCHKUR‘>@?BETYdSEDC aDTYdSd\EŒ[`X9TYV ^ KUVBzR\T/RS?BKM[ aDTYVLŠYEd\zWEDVlaDEY“

tŸŠWEDd P]OMO¤R\?lEXlP]X9EDd ˆ2EPY[S[`NBC/EvR\?mPR

]C V•R‡y¡»:-\#, C V-RDy»:

œ

\ ];%CIV)'‡Z#]:+,<:

œ

]$ž 'F!

E"PYOM[\T ^ EDVlT]R\E %C('Z p>=@?BA %C V 'Z ž · T]RSE¨R\?mPRRS?BEDdSE¨KQ[VBTiwlOUTWV ^ EDVlE[S[PY[S[\NBC/XZR\KMTYV­KMV’TYNBd

?L_LX9T]R\?lE[\KM[“>@?lKM[KM[wmEŒaDP]Nm[`E¨ˆ2E}P]dSE¨KMVWRSEDdSE[`R\E

^

KMV­R\?lE}ŠP]OMNBE[T]‡@R\?BE†‡NBVla RSKUTWVl[HPR?lPYV

^

KUV’P

jmVBK®RSEVLNBCw9EDd TY‡X9TYKMVqRS[¸ZVlP]C/EDOM_†KMVLy

ƒV

»o=prMY»; »btIp‹MY»ns

V

žmILT/ˆ2E

^

EDVlT]R\E

C T CEDGFH pI=@?BA

V Z F2

=@?JA

ƒ Z F

zI{LK

K

TIRDy

ƒV \ KK

9!;< C T CH pNMOQP

JRTSU ]T[R‡yd\F]$ž

„ ViILEŒa R\KMTYVip0ˆ2EvXld\TŠWE‘RS?lPRNBV ^ EDd@RS?BEP]w9TŠYEv?L_qX9T]RS?BE[\KQ[2R\?BE+B GWV ?BTWO^ [D¸LRS?lPRKQ[

X R1 € RDy € \#,

0

1 € R‡y € \b\9+ s R?qB»ZY 2 Fz \

ˆ ?lEDdSE Y 2 Fz

KQ[P/a$TWVl[`RSP]VqRˆ ?BKQa?

^

EDX9EDV

^

[ TYVRS?BEVLNBCw9EDda;sTY‡'[`R\EXl[ KMVTWNBdP]OMzYTWd\KUR\?lC P]V

^

TWV

RS?BEC¨PbZKMC/PYO¤VLNBCw9EDd TY‡X9TYKMVWR[ TYVEPWa?"OQP‹_YEd¸qRS?lPRKQ[s p =@?BA

V € F2 s V ž

>@?BE@KMVqR\EDdSE[`R\KMVBzvXBdSTYwBOMEDCVBTˆ–a$TYVma$EDdSVl[¡R\?BE@w9ED?lP‹ŠLKMTYdT]‡mR\?BE@ŠP]dSKMPYVla$ELY

2 Fz

PW[I; »s +87;ž „V

RS?BE@zYEVBEDdP]OW‡dSPYC/E ^ EŒ[\aDd\KMw9E ^ w9E$‡TWd\E2ˆ2E2aP]V0[SP‹_3VBT]RS?BKMVBz3CHTWd\E™R\?lPYV Y 2 Fz

;B(; »][`T‘R\?BE@ŠP]dSKQP]VlaDE

C¨P‹_"wBOUTˆ NBXiPW[^;<+ 7;ž „ V”TWd ^ Ed R\T}TYwZRP]KMV”CHTWd\EXld\EŒa$KQ[`EdSE[\NBOURS[ ˆ2E?lP‹ŠWE3R\T†dSE[`R\dSKQa R‘TWNBdS[\EDOU‡

RST}P}h…P]dnYTŠLKMPYV}‡dP]C/EW£‰ˆ2E0PW[\[\NBC/ER\?mPR‘RS?BEdP]V ^ TYC ŠP]dSKMPYwBOUEŒ[aUWVL»op MW»DžMžUž;’P]dSETYwBRSP]KMVBE ^

‡dSTYC P/hiP]dnYTŠ†a?lPYKUV U8VL»o†prMW»DžMžUžM»b; '‡TWd E$bBP]C/XBOME3R\?lE eNBOMEDdŸIBa?BEDC/ET]‡ P ^ K549Nm[`KMTYV(XBdSTZa$E[S[

wL_"XBd\TYfgEa$R\KMTYVTYVR\?BEzWd\K^Lv[V '«[\EDEIZEa RSKUTWV”yZRS?BKM[‘KM[ R\?BEqNlPYVWRSKUŽŒPRSKUTWV"XBdSTZa$E ^ NBdSEvEC/XBOUT_WE ^ KMV

%A†ž˜@žNMO'»F%A†ž˜@žP'7»)%A}ž˜@ž˜(' $“e>@?BEDV/ˆ2E TYwZRP]KMV¨WNlK®RSE XBd\EŒa$KQ[`E@EŠ‹PYOUNmPR\KMTYVm['T]‡ Y 2 Fz

'«[\EDE cedSTYX9TW[\K®RSKUTWV

F> ˆ ?BKQa?¨E[S[`EVWRSKMPYOUOM_zWNlP]dP]VqRSEDE[RS?lPR‰NBV ^ Eded\EŒPY[\TYVlPYwBOME2?L_LX9T]R\?lE[\KM['RS?BE‘ŠPYd\KQP]VlaDE ^ TLE[eVlT]R™wBOMTˆ

NlX¢“

„ V’ILEŒa R\KMTYV•y(ˆ2E¨a$TYVm[`K^ EDd[\TYC/E¨[\XmEŒa$KUjma/‡NBVla$R\KMTYVl[+C V ˆ ?BKQa?•a$TYdSdSE[\XmTWV ^ RST(R\?BE/‡TWOUOMTˆ KUVlz

Rgˆ2T0aDTYVlaDd\EDR\EŸE$bBP]C/XBOME[“#KMd[gR@T]‡PYOUO³ˆ2EŸ[\E$R@NBX(P ^ _LVlPYC/KMaP]O9XBd\TWzYdP]C/KMVBzP]OMzYTWd\KUR\?lCº‡TWd2[\TYOMŠLKUVlz

VLNBC/Ed\KQaDPYOUOM_/R\?BE[\EDC/KeOMKMVBEPYd˜š†›

R G\

1

RD` »y¡»

1

\3p qB»

1 p &³ž

„ VTWd ^ Ed³R\TaDTYVl[`R\dSNla R¡[\Nla?P]VPYOUzWTYdSK®RS?BC;TYVlE'RSP_WE[ '<[`EEa%A}ž˜@žNMO'D\ C V RDy»:•\fp : 2 R V

2

»y¡»:•\$ž

„‡ KQ[‘Rgˆ KMaDE ^ K549Ed\EVqR\KQP]wBOMEˆ KUR\?idSE[\X9Ea RŸR\T :ˆ2E0PYd\E0P]wBOMERST}zYKMŠYEŠYEDdS_XBd\EŒa$KQ[`EEDŠP]OMNlPRSKUTWVl[TY‡

Y 2 Fz “ EXBdSTŠYEvRS?lPR Y ,aBQ; ; Y 2 Fz ; Y BQ;ˆ ?BEDdSE Y P]V ^ Y?lP‹ŠWE3R\?BE‡TWOUOMTˆ KMVBz¨E$bZXBOMKMaDK®R

EDbZXBd\EŒ[\[\KMTYV '<[`EEILEa$R\KMTYV”yB“ ) $£

Y2p i

˜

& RDy7€Q\ , 1 € RDy7€Q\ % 7'DZ

€ i

˜ R1

1 R‡` »DžM»

1

\\ORDy7€_\l< `

ˆ ?lEDdSE˜

KQ[ RS?BE0[\EDC/KMzYdSTYNBX(T]‡ KMVZjlVBKUR\EŒ[`KMC¨P]OTYX9EDdPRSTYdaG;PYV

^ 1

KQ[ RS?BE0[\TYOMNZRSKUTWV”T]‡'R\?BEH˜š†›¨ž Y

?mPY[2R\?lE[\PYCHE3EDbLXld\EŒ[\[\KUTWV}ˆ KUR\?/i

KMVl[`R\EP ^ TY‡ i

ž

>@?BE”[\EaDTYV ^ Xld\TWwBOUECˆ ?BKMa?–KM[0RS?BE"dSEPYOUOM_­KUVqRSEDdSE[`R\KMVBzsTWVBE aDTYVlaDEDdSVl[RS?BE”[`EC/KOMKUVlEP]d

˜š}›ˆ KUR\?(dSE lEŒa R\KMTYV

=@?BAw"!

»FR#$ G \ 1

% R‡` »by¡»

1

\ | p qB»

1&

p'!

ž

· TˆrC V R‡y¡»:-\^p = ?JAw"![RD` V »by\ »:(

2 R V

2

»y¡»:-\

|

ž³>@?BEXld\TWwBOUECKQ[ŸCNla?iC/TWd\E)KMd\dSEDzWNBOMPYd*)

w9EaP]Nl[\E : + C V R‡y¡»:•\@KQ[@VBT¨C/TYdSE3a$TYVqRSKUVLNBTWNl[`OM_ ^ K54³EDdSEDVqR\KQP]wBOMEY“ „VTYd ^ Ed@R\T/?lP]V ^ OUE3RS?BKQ[ KUdSdSEDzYN&

OQP]dSKURg_0ˆ2E‘?lP‹ŠWERST0[\XBOUKUR2R\?BEŸEDdSd\TWd

1 € RDy € \ , 01 € R‡y € \3p,+

2 Fz 2g

2 Fz

ˆ ?BEDdSEag

2 Fz

R.- \™KM[2PdP]V ^ TYC

ŠPYd\KQP]wBOMEˆ ?BKQa?iKQ[

^

TWCHKMVlP]R\E

^

wL_0/

2

‡TWd‘EDŠWEDdS_1-;P]V

^

EDŠWEDdS_ ž¤ILKUVma$ERS?BKM[va$TWd\dSEa$R\KMTYV

^

TLE[‘VBT]R

^ EDX9EDV ^ TWV-’P]V ^ KUR ^ TLEŒ[eVBTYR‰[`KMzYVlK®jmaP]VqR\OM_/a$TYVqRSd\KMwBNZRSE R\TRS?BEŸ[`RSP]R\KQ[gRSKMaP]OmEDdSd\TWd 'ˆ2EC¨P‹_HaDTYV

[\K^ EDd2K®R PY[@PXlP]d\R2TY‡¤R\?BEPYVlP]OM_qR\KQaDP]OmEDdSd\TWd

1 , 01

ˆ ?lKMa?"KM[2T]‡TYd ^ EDd

Z

2

PY[‰ˆ2EDOMO $“™ILTRS?BEv[`RSP]R\KQ[gRSKMaP]O

Ed\dSTYd2KQ[ ^ E[Sa$dSKUw9E ^ wL_2+

2 Fz

P]V ^ ˆ2EvXld\TŠWER\?mPR

X

'+

2 Fz + s R?qB»Y 2 Fz

\‰PYV ^ Y 2 Fz ; /

3 2 B Y ž

EP]dSEvVBT]R‘PYwBOUEvRST¨CHKMVBTWdSP]R\EŸR\?BEŠP]dSKQP]VlaDE Y 2 Fz

KMV"RS?BKQ[aDPW[`EW“

/ $, ) *¤ ™‹'

E ^ EDVBTYR\EwL_ › Vƒ

R\?BEŸE$bZX9Ea RPRSKUTWV¨ˆ K®RS?†d\EŒ[`X9Ea$ReR\TR\?BE‘EDC/XBKMd\KQaDPYOlXBdSTYwlPYwBKMOUKURg_C/EPW[`Nld\E

‚ V

ƒ…„

»Œ=p

MW»DžMžUžM»s VYX[Z RS?lPR KQ[

› Vƒ T@p

z {

#„ Z

TIRDy

„

VYX[Z

\ ‚ V

ƒ…„

p M

 ƒV

# Z

TIRDU

VYX[Z

\YM

{

{

'#

PYV ^ ˆ‰EP]OQ[\T ^ E$jlVBE

› V

ƒ

TLp

z {

#

„ Z TIRDy

„

VOX[Z

\‚ V

ƒ…„

p M

Ž ƒV

›WRT[R‡U VYX[Z \YM

{

{

\

h”TWd\ETŠYEd‰ˆ2E

^

EDVBTYR\E

EVp Y#RDU

Z

V

»žUžMžU»U

V

»bUWV\ 9!;< ›

{

R \3p"!Y›WR ] EV_\

E?lP‹ŠYEvRS?BE3‡TYOMOUTˆ KMVBz=YE_†K^ EDVqR\KURg_³£

M

{

{

›

{

RTIRDU

VYX[Z

\b\ pM

{

{

›WRT[R‡U

VYX[Z

\^]QU

V

\fpM

$

{

{

› V

ƒ

Tž '«o

h”TWd\ETŠYEd¸Y‡TYd

Ž

o}ˆ2E ^ EDjlVBE3wL_¨d\EŒa$NBdSdSEDVlaDER\?lE3R\dP]Vl[\K®RSKUTWV†XBdSTYwlPYwBKMOUKURg_ ^ EDVm[`KUR\KME[2‡dSTYC vIV

RST v X[Z

‚ V F

ƒ1„

p

z

# Z ‚ V F Z

ƒ ‚ „

»"!^t` ! ‚ V FV

ƒ1„

p ‚ V

ƒ…„

9!;<

‚ V F

ƒ…„

p

z

# Z ‚ V F Z

ƒ ‚ „

»"!^t` ! ‚ V FV

ƒ…„ p ‚ V

ƒ1„

ž

>@?BEV(ˆ‰E

^

EVBT]RSE

› V F

ƒ

TLp

z

#

„ Z TIRDy

„

OXIZ

\ ‚ V F

ƒ…„

» › V F

ƒ

Tp z 4

#„ Z

TIR‡y

„

X[Z

\‚ V F

ƒ…„

ž

„ RKQ[ EPW[`_/RST†a?BEa/RS?lPRˆ2E3?lP‹ŠYEvRS?BE3‡TYOMOUTˆ KMVBzHd\EŒa$NBdSdSEDVlaDE‘‡TWd\C0NBOMPW[

› V F

ƒ

TLp;›

V F Z

ƒ › #

T» › V F

ƒ

TLp › V F Z

ƒ › # T

ˆ ?lEDdSEvˆ2E3C¨P_YEvRS?BE[`OMKMzY?qRP]wBNl[\E3T]‡VBT]RPR\KMTYV6R«›

T\ RDy

ƒ

\3p"!W›

ƒ

Tž

>@?BE[gRP]d\R\KMVBz”RSKUC/E@o p’q”XBOQP‹_L[0P”XlPYd`RSKMaDNBOMPYddSTYOMO™w9EaDPYNl[\ESU

€

p}y

€

KQ[

^

EDR\EDdSC/KUVlKM[`R\KQa]“

„V

TWd

^

EDd@R\T¨EC/XBOUT_"NBVBKURSPYd\_"VBT]RPR\KMTYV(ˆ2EPY[S[\NBC/E3R\?lP]R‘P¨zYdSK

^@v€

p wFy

Z

€

»DžMžMžU»y z

€ |

KQ[ zYKMŠYEV(PYV

^

ˆ‰E

^

EDVBTYR\Evwq_†t

€

R\?BEvKMV

^

E$b[`Nla?†R\?lP]R y

€u

y ƒ

€

'«KUV}‡«PYa$R@TWVBE‘C¨P‹_/RP_YE(s

€

prMvPYV

^ t€

p‹M) ‰>@?BEDVK®R

KQ[@EPW[`_†R\T¨a?BEa}wq_}KUV ^ Nla RSKUTWV"RS?lPR

‚ € F

ƒ F„ p ˜†RDU X[Zp‰y

„

X[Z

\3p

Ž „ OXIZ

9 ;<

‚ € F

ƒ F„ p  „ X[Z

p M

# Z M

ž '_

>@?BKQ[‰KM[™R\?BEŸEO4³Ea$R2TY‡9RS?BE3a$TYC/X9TW[\K®RSKUTWV¨T]‡¤RS?BERSdSPYVl[`KUR\KMTYV¨Xld\TWwlP]wBKMOMK®RSKUEŒ[

‚ V F VYXIZ

ƒ…„

»o†pqB»žUžMžM»

Ž , M

'«d\EŒ[`X9Ea$R\KMŠYEOU_

‚ V F VYX[Z

ƒ…„

»no†p qB»DžMžMžU»

Ž , MQ “ ·

T]R\E3RS?lPR RS?BEa$TWC/XmTq[`KUR\KMTYVT]‡R\?BE3RSdSPYVl[\K®RSKUTWV"XBdSTYwmP]wBKMO

KUR\KME[

‚ V F VOX[Z

ƒ1„

»o/pL»žUžMžU»

Ž ^

TqEŒ[‘VlT]R3zYKMŠYE0[\Nla?…PYV…EDbLXlOUKQa$KURŸ‡TWd\C0NBOQP¨wmEŒaDPYNl[`EW¸9‡TYd

qB»dUC¨P‹_

RP_WE3[\EDŠWEDdP]O

^

K549Ed\EVWRŠPYOUNBEŒ[D“

KMVlP]OMOU_W¸BzYKMŠYEVP/[\EqNBEVla$E3T]‡‡NBVla$R\KMTYVl[ †V"!Y› + JH»no†p qB»žUžMžU»; »Zˆ‰E ^ EDjlVBE

‚ V

F

ƒ…„

p VYX[Z RDy

„

VYX[Z

\‚ V

ƒ1„

»#›

V

Fƒ TLp"!

zI{4

#„ Z

T[R‡y

„

VYXIZ

\‚ V F

ƒ…„ p;›

V

ƒ

R VYX[Z T\

9!;<

‚ V

F

ƒ…„

p †VYX[Z_RDy

„

VYX[Z

\‚ V

ƒ…„

» › V

Fƒ Tp"!

z {

#

„ Z TIRDy

„

VYX[Z

\ ‚ V F

ƒ…„ p › Vƒ

R †VYX[Z)T\ ž

"Ÿ[`KMVBz/R\?lE[\PYCHE3dSEaDNBd\dSEDVma$E‘Xld\TZa$E ^ NBdSEPY[ P]w9TŠYEvˆ2E ^ E$jlVBE

‚ V F

F

ƒ1„

p z

# Z ‚ V F Z

Fƒ ‚ F „

»"!at`! ‚ V FV

F

ƒ…„ p ‚ V F

ƒ…„

9 ;<

‚ V F

F

ƒ1„

p z

# Z ‚ V F Z

Fƒ ‚ F „

»"!^t` ! ‚ V FV

F

ƒ…„ p ‚ V F

ƒ…„

ž

h”TWd\ETŠYEd¸ › V F

FƒT p"!

z 4

„ Z TIR‡y

„

X[Z

\ ‚ V F

F

ƒ…„ PYV

^ › V F

FƒT p"!

z

„ Z TIR‡y

„

X[Z

\‚ V F

F

ƒ…„

ž ‰OUEŒP]dSOU_

› V F

Fƒ

'«d\EŒ[`X9Ea$R\KMŠYEOU_ › V F

Fƒ

\ŸŠYEd\KUjlE[‘R\?lE/[SP]C/EHd\EŒa$NBdSdSEDVlaDEd\EOMP]R\KMTYVl[3PW[Ÿ›

V F

ƒ

'dSE[\XmEŒa RSKUŠWEDOM_ › V F

ƒ \ ž

tŸVlE3PYOM[\TH?lPY[

‚ V F

F

ƒ…„

; V F R 8\

‚ V F

ƒ1„

9!;<

‚ V F

F

ƒ…„

; V F R 8\

‚ V F

ƒ…„

'

!^t` !

V F R \ p C V

CDGFH

-sžMžMž - C C DGFH ž

„ V"R\?lKM[[\Ea$R\KMTYVˆ‰E3XBdSTŠYEŸR\?lE0 ‰EDVqR\dP]O¤¥¢KMC/K®R‘>@?lEDTYdSEDC KUV"R\?BEPYwl[`R\dPYa R2‡dP]C/EY“#EPY[S[`NlCHEŸjld[gR

'«ˆ‰E dSEDC/TŠWE'R\?BKQ[¢dSE[`R\dSKMa$R\KMTYV3PR¢RS?BEeEV ^ T]‡LR\?lKM[[`EŒa R\KMTYVd 9R\?lP]R¢R\?lE™a$TLE$©†aDKUEVWR[C V@P]dSE ^ K4³EDdSEDVqRSKMPYwBOUE

ˆ KUR\?(dSE[\XmEŒa R@R\T@:"P]V ^

{

KM[acTYO^ Eda$TYVqRSKUVLNBTWNl[2T]‡'KUV ^ E$b

ql»btži

KKKK

C V

:

RDy»:•\ , dC V

:

R‡y¡»: œ\ KKKK ; Bˆ]:+,<: œ]3ž '«r

"‘V ^ EDdR\?BKQ[ ?L_LXmTYR\?BEŒ[`KQ[@ˆ‰E3ˆ KMOUO¡XBd\TŠWE‘RS?lPR@RS?BE[gRPRSKM[`R\KQaDPYO¤EDdSd\TWd2TY‡TWNBd P]OMzYTWd\KUR\?BC

ƒ

€

p"!

1 € R‡y€\ , 01 € R‡y€\

ŠWEDdSK®jmE[‰R\?BE+B GWVžB>@?BEWED_}[`R\EX"KQ[@R\T/XBdSTŠYEŸR\?mPR

X R ƒ

€ , 0

\9+ q KMV(XBdSTYwlPYwBKUOMKURg_ ' x

ˆ ?lEDdSE

0

p M

#

Z

!at` ! ' T )

p 2 Z

#V Z z {

#ƒ Z ‚ € FV Z

Fƒ Fƒ M

Ž ƒV R 0

C

VYX[Z

R‡U

VOX[Z

\I,­›

V

ƒ 0

C

VYX[Z

\OM

{

{

0

C V RDyd\ p C V RDy» 01 V RDy\b\ 9!;< C Z

V

RDy\fp C V

:

RDy» 01 V RDy\b\ ž

 ‰OMEPYd\OM_

Z »žUžMžU»

P]dSE†KMV ^ EXmEV ^ EDVqRHdSPYV ^ TYC ŠP]dSKMPYwBOME[ˆ ?lKMa?PYd\E}K^ EVqR\KQaDP]OMOM_ ^ KM[`R\dSKUwlNZR\E ^

PYV ^ ?lP‹ŠYE3ŽDEd\THC/EPYV '<[`EE '«o @[`THRS?BE[gRP]V ^ P]d ^ B G9V zYKMŠYEŒ[

X - 0

p M

X

#

Z

+! #"‰s R‡qB»ZY

2 Fz \ ' Œp

ˆ KUR\?

Y 2 Fz p;›‰]

]

ˆ ?lEDdSE

KQ[ ^ EDjlVBE ^ KMVR\?BE[SP]C/E3ˆ2P‹_†PW[

ˆ K®RS?LU KMVl[`R\EP ^ T]‡[U(ž

$ € µ&%™µ

~('«('

µ*)#+-,/.0.01-†D•_325476+98S:1(;&•

X - R1 € RDy € \ , 01 € RDy € \b\ +! #"‰s R‡ql»Y

2 Fz \ 85=<>;

$ € µ¢µ*?A@}‘[Pa$TWVl[`EŒqNBEDVlaDE0TY‡ ' ‹p ŸKUR3ˆ KMOMOe[`NZ©†aDE0RST"Xld\TŠWE' Œx $“0>@?BE/XBdSTLT]‡™zWTqEŒ[ŸR\?BdSTYNBzW?

[\EDŠWEDdP]O¤[`R\EDXm[D“

¢

° % +&@LE

^

EVBT]RSE ƒV

p"!

1 V RDy

ƒV

\3,

0

1 V RDy

ƒV

\$ž³>@?lEXBNBdSX9TW[\ET]‡™R\?BKQ[Ÿ[`R\EX…KQ[RST"Xld\TŠWER\?lP]R¸

‡TWd EDŠWEDdS_+

q

KK ƒV KK + q KMV(XBd\TWwlP]wlKUOMK®Rg_W“ ' y

Eˆ d\KUR\E

ƒV p › Vƒ

R‡C VYX[Z R`žU»

1

VYX[Z Rgž1\b\\#,­›

V

ƒ

R?C VYXIZ R`žU»

0

1

VYXIZ R`ž\\b\ ' DF!

p › Vƒ

R‡C VYX[Z R`žU»

1

VYX[Z Rgž1\b\3, C VYX[Z RgžM»

01

VYX[Z Rgž1\b\\

ƒV

!^t `!

ƒV p › Vƒ 0 C VYX[Z ,­›

V

ƒ 0 C VYX[Z ž

ILKMVlaDE 2 p qB»

] V ] ; %C VOX[Z 'Z › V ] VYX[Z ] ] V ]G; žMžUž ; 2 Z

#

VYX[Z

„

VYX[Z

%C „ 'Z › V F Z ] ]

; B = ?JA

VYX[Z

F2 Z

=@?BA

ƒ Z Fz

K

K ƒ KK

[\T/ˆ‰E3?lP‹ŠWE‘RST¨XBdSTŠYEŸR\?lP]R

KK ƒV KK + q KUV(XBdSTYwlPYwBKMOUKURg_Y“

„ V(ŠLKUEˆ TY‡ ' @P]V ^ TY‡ '<o

ƒV p M

 ƒV

# Z R 0

C VYX[Z RDU

VYXIZ

\O,›

ƒ

V 0 C VYX[Z \YM

{

{

p M

 ƒV

# Z R 0

C VYX[Z RDU

VYXIZ

\O,› { 0 C VYX[Z RDU

VYX[Z

\b\OM

{

{

ž

· T]R\E3RS?lPR RS?BE3R\Ed\C¨[@KMVR\?BEPYwmTŠWEv[\NBCPYd\EvTYd\R\TWzYTWVlP]O9NBV ^ Ed‘›

{ ž „R ‡TYOMOUTˆ[‰RS?lPR

› { KK ƒV KK p M

KK ƒV KK

#

Z

›

{ R 0 C VYX[Z RDU

VYXIZ

\ ,­›

{ 0 CIVYXIZ R‡U

VYX[Z

\b\ M

{

{ '

; M

 ƒV 0 C VYX[Z

DGFH ž

Eˆ d\KUR\EVlTˆ

˜†R

KK ƒV KK

\fp;˜†R KK ƒV KK \3p ˜†R  ƒV

-”

ƒV KK ƒV KK \

; ˜†R

 ƒV P

Ž ƒV

\˜†R

P

Ž ƒV -”

ƒV KK ƒV KK \

p ˜†R

 ƒV

; Ž ƒV

P

\˜†R<

ƒV K ƒV K Ž ƒV

_P \ ž