A Biologically-Inspired Model for a Spiking Retina

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https://hal.inria.fr/inria-00070178

Submitted on 19 May 2006

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A Biologically-Inspired Model for a Spiking Retina

Adrien Wohrer, Pierre Kornprobst, Thierry Viéville

To cite this version:

Adrien Wohrer, Pierre Kornprobst, Thierry Viéville. A Biologically-Inspired Model for a Spiking Retina. [Research Report] RR-5848, INRIA. 2006, pp.78. �inria-00070178�

a p p o r t

d e r e c h e r c h e

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Biologically-Inspired Model for a Spiking Retina

Adrien Wohrer — Pierre Kornprobst — Thierry Viéville

N° 5848

Février 2006

0 132547698;:=<?>3256@2

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‹‚­‰·µd]d]Y*X£imljŽ†ohfvTY ~YfvjŽ†‚‡­ƒ­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‹9‘

/ ¼ — )®˜C—

I 7

‰­8 Du¸c?Y*tvjlXhYC†9fvxmsut]knf]ke„k@m­ƒ­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‹e’

‰­ `ejŽ†o@mŽY£Ã¯YCmŽm`ejŽX£rmlxf]jlkn†Wd£­ƒ­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‹OF

‰­­8 >ztvj¥žfvjŽ†WoEN|tvxf]jl†o@d­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‹OF

‰­­ `9fvxf]jq„N|tvxf]jl†o ­ƒ­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‹OF

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'ϗ5›*—C—('À ) À )N˜ & ¼ +ƒ™ ¼ — ¼ )˜ (‡À Á ¼ ) / 7

7 3 ˜ ™ ¼À ™ šG)Á + ¼ ™#— ¼ ›x˜('! ¼ — #"

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$&%

À )/› )—'À )

M@

” % À )' À )/˜'À ) À -(*) À ) ¼ )˜('Ϛ ) 3 ')Ϙ ¼ ™#—

2I

·­8 Du¸e@„°f|Ãnml„*rmlxf]jlkn†Wd ­ƒ­$­ƒ­$­ƒ­$­$­|­$­ƒ­$­ƒ­$­$­ƒ­$­ƒ­$­|­$­ƒ­$­$­ƒ­$­ƒ­$­ƒ­$­$­|­$­ ‘9‰

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+-, '*) ¼ š™ (‡À Á ¼ )*'6)5C À -˜

& ¼ 3 ¼c¼ Á/•3š› -P™ À /. À ™.'0 À )˜Cš ) % ¼ ))ϗ 7 #

+-./ U!

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0*#*# ¼ ™ J'¼9¿

SUTWY|o@k@nm?knžfvTjldzdgY5„°f]jlkn†jldfvk {eYCdv„tvjŽi?Y|f]TY$tvYfvjŽ†‚xm©X(YC„…TWn†jqdgX @d¨‚m¥fvY*tvjŽ†o(dgf]tvrW„°fvrt]Y@ªx†W{–f]k

Wt]k#cjq{eY x†±k#nYCt]cjlY*¤_kxž3f]TY£mŽjl†«ed€i?Yfy¤Y*YC†¦x†Wxf]k@Xb–n†W{f]TY£t]YCmlxf]Y5{–ž®rW†W„°fvjŽk@†Wxm/i‚YCTW²cjŽk@tC­U·zd

tvYCd]rm¥f5ª©¤¯Y(tvkn?k@d]Yjl†"Œ/jlonrWt]Y –dv„…TY*X #fvjl„£cjŽYC¤knžuf]TY(tvYfvjŽ†WWª´¤Tjq„…TE¤jlmŽmKi?Y(rWd]YC{ºjŽ†ºf]TY

tvYCdgfkxž/f]TY$Wn‚YCtC­uSUTWjldd]YC„f]jlkn†±dgTk#¤€dUTWk#¤ jŽf€jldUtvY*mq#fvYC{–f]k x†Wxf]k@Xhjq„*nmžÏn„fvdC­

ŒjlonrtvY e³%`e„…TWY*X #fvjl„hcjŽYC¤‡kxžf]TWY£¨‚m¥fvY*tvjŽ†o±ke„*„*rtvt]jl†o–jl†Ÿf]TY(tvYfvjŽ†WW­ /²bnYCtvdµkxžU„Y*mlmqd$xtvY£tvY*e¢

tvYCd]Y*†9fvYC{ºi9bEi‚k²¸eYCdƒ„*kn††Y5„°fvYC{ºf]TtvknrWonT"dgbc†Wnef]jq„hjŽ†9f]YCtv@„°fvjŽk@†Wd*­

VR(x, y, t)ª/Yfv„n­ t]YCtvYCd]Y*†9fƒf]TY

?kxfvY*†9f]jqxmqd$knžfvTY–„*Y*mlmld$jl†¡YC@„…T"ml²b@Y*t #f$fvjŽX(Y

t^ª3¤jŽf]T (x, y) i?Y*jl†o§fvTY–d]W#fvjlnm „kcknt…{ejl†W#fvYCdƒknž

fvTY„*kn†Wd]jl{Y*tvYC{§„YCmŽmP­€S/¤¯k{ejqdgf]jl†W„°fƒdgbc†Wnef]jq„$mq²bnY*t…dCªef]TY(\µs x†‚{±f]TY[Þs uª{eY¨‚†Y$fy¤k–dgr‚„*„Y5dy¢

d]jlnY£¨Wm¥fvY*tvjŽ†Wo±dgfvnonY5d*­

VG(x, y, t) {eY¨‚†YCd|fvTY(XhYCXit…x†WYh?kxfvY*†9f]jqxmqd|kxžxmlmKo@n†onmljŽk@†Wxt|„YCmŽmqd*­([¬f

„*kn†9fvnjŽ†‚dzfvTY jl†ež®kntvX #f]jlkn†"d]Y*†9fƒf]kf]TY iWtvnjŽ†Ÿicb§f]TY(tvYfvjŽ†WWª/„*kn†Wd]jldgf]jl†okxžkn†Y d]jŽ«@Yhf]t…xjl†º?Y*t

o9x†o@mŽjlkn†‚xtU„Y*mlmP­

0*#0 ™.C‚š2)5'0nše˜('À )4'*) % À )) ¼ ›n˜ ¼ Á , š ½‚¼ ™#—

SUTWY$t]Y*f]jl†W(„kn†9f…xjl†Wd¨WnY {j¥¶Y*tvY*†9ffybc‚Y5dkxž3cjld]rWnm´„Y*mlmqd*­ ŒjlonrtvY !Ï/d]Tk#¤€dU dgY5„°f]jlkn†kxžK(tvYCxm

X nXhX nmŽjqx†¦tvYfvjŽ†‚n†W{§jŽfvd$x†‚#f]k@X(jl„Cxmjl†9f]YCt]tvYf…#fvjŽk@†´­£Ã¯Y*mlmldµkxž f]TY dvxX(Y fybc?Yž®k@t]X ,H°ª

‚²9jl†o$fvTY|¤TknmlYztvYfvjŽ†WW­ ²bnYCtvd xtvYz„k@††YC„f]Y5{–nd¯{eYCdv„tvjli‚Y5{jl†–ŒjŽo@rt]Y ! Ïi °ªex†‚{–d]rX(X(nt]j:9*Y5{

jl†–ŒjlonrtvY|­3~€Yf]jl†Wnm?„*Y*mlmldUxtvY

G°ª@f]TWxfYC†W„ke{eY|dgjlon†WnmfvTt]k@ronTf]TYCjŽtY*mlYC„f]tvjl„Cxm?XhYCXit…x†WY

?kxfvY*†9f]jqxmP­ 2ºYh¤jlmŽmK„k@†Wd]jl{eYCt€fvTY*X @d 2.M

.2°ª?X(Y5x†jl†ofvTW#f|f]TWY£?kxfvY*†9f]jqxm3kxž „YCmŽmK#f

fvjŽX(Y

t jld€{Y¨W†Y5{i9b£rW†jlp9rY$†crX£i‚YCt

V(t)^­

2ŸYº„Cx†Nf]TWY*† n‚YC†W{ f]TWYºn„f]jlcj¥fvjŽY5d ž®kntxmlm€f]TWYº„YCmŽmqd¤jŽf]TWjŽ†  ml²b@Y*t5­_[¬ž„*Y*mlmµ{eYC†WdgjŽfyb jld

{Y*†Wd]YY*†k@ronTªxf]TWYµn„f]jlcj¥fvjŽY5dKkxž´xmlmfvTYz„*Y*mlmld jŽ†(fvTY€mq²bnY*tu{Y¨W†Yµ dg‚#f]jqxmlmlb£„k@†9f]jl†9rWknrWdž®r†‚„°f]jlkn†

tvY*Wt]Y5dgYC†@fvjŽ†WofvTY n„f]jl9jŽfybknžf]TWY$ml²b@Y*t#fUf]jlX(Y

t­uŒk@tY¸xX(mlYnª

(x, y)→VB(x, y, t)t]YCtvYCd]Y*†9fvd fvTY@„°f]jlcj¥fvjŽY5d€knž3iWjŽ?knmqxt|„Y*mlmqdz#fzf]jlXhY

t Ïd]Y*YŒjlonrtvY£ ­zSUTY tvYfvjŽ†W „Cx†§i?Y£d]Y*YC†¦ndzh¨‚m¥fvY*tvjŽ†o dv„…TYCXhY f]TWxf€f]t…x†Wdgž®kntvX djŽfvd€mljlonT9fµjl†ref|jŽ†9fvk f]t…xjl†Wdknž dgjl«nY5dUf]T‚x†«edfvk ¨W@Ydgr‚„*„Y5d]d]jlnY$mq²bnY*t…dCª

Y5n„…Tml²b@Y*ti‚YCjŽ†Wo{eYCdv„tvjŽi?YC{icbh?knrmq#fvjŽk@†–ž®rW†W„°fvjŽk@†´­

[ކ9f]YCtv@„°fvjŽk@†Wdz„Cx†§ke„*„*rtzi?Yfy¤Y*YC†E„YCmŽmqd€ž®tvknX {ejŽ¶?YCt]YC†9fµmq²bnY*t…dCªWx†W{Exmqdgki?Yfy¤Y*YC†¦„YCmŽmqdzknžKf]TY

dvxX(Yƒmq²bnY*t5­3Z¯rff]TY$†Wxf]rtvYƒkxž/f]TY5dgY$jl†9f]Y*t…n„f]jlkn†WdUnt]Y${ejŽ¶?YCt]YC†9fC³

% & ¼ 'ϛ@š ) —

½

)/š —

¼ —

ïTWY*X(jl„Cxmdgbc†WnWd]YCd¯mŽjl†«„YCmŽmqd¯ž®t]k@X {ejŽ¶?YCt]YC†@fUmq²bnY*t…dC­/·€mlm´„…TYCXhjq„*nm?d]bc†Wx‚dgY5d ke„*„*rtjŽ†–fy¤¯k

ml²b@Y*t…d 7Œ/jlonrWt]Y !Ï/)°³fvTY$\µrf]Y*tsumlY¸ejŽž®kntvX /²bnYCt P\µs' ¯x†W{(fvTY|[ކ†Y*tsumlY¸ejŽž®kntvX /²bnYCt

Ï[Þs ­/SUTY5dgY(fy¤k§mq²bnY*t…d|kxž€d]b9†‚xWd]YCd${eY*¨W†Y . 2H2 1#<<< (O °ªKnd

tvY*Wt]Y5dgYC†@fvYC{£jŽ†Œ/jlonrWt]Yzc­u`cY5„°fvjŽk@†!­É|t]Y5dgYC†9fvdKk@rtuX(ke{eY*mWž®knt  „…TWY*X(jl„CxmWd]bc†Wx‚dgYjl†(f]TYzt]Y*f]jl†W­

()¼ ›n˜*™.'ϛ@š )¯— ½ )š — ¼ — 3š )7— À ›nš ))¼ Á4C?š 5)›x˜('À )/—

SUTWjldh«cjŽ†‚{ knž|d]b9†‚xWd]YCd£mŽjl†« †WY*jlonTci‚k@t]jl†oŸ„Y*mlmqd ) 2 ,. H°ªUxmlmŽk#¤jl†oŸdgk@XhYjl†@fvtvx¢Pmq²bnYCt

„*knX(XrW†jl„C#fvjŽk@†5­ `erW„…TŸwyr†W„°fvjŽk@†Wd–nt]Y±tvYCd]Y*†9f¤jŽf]T {ejŽ¶Y*tvY*†9f–dyfvt]YC†oxfvTWd(jl†;YCn„…T kxžƒfvTY±¨WnY

mq²bnYCtvdCªx†W{ t]Y5dgrmŽf–jl†_¡mlke„*nmµd]TWxtvjŽ†Wo kxž$fvTYEjl†ež®kntvX #fvjŽk@† i‚Y*fy¤¯YCY*† „*Y*mlmldkxž$fvTYºdvxX(Y¦fybc?Yn­

`eYC„°fvjŽk@† !W­!htvk#cjl{YCdU(dgjlX(mŽY ntvk²¸ejŽX #fvjŽk@†ž®k@tUf]TjqddgT‚xtvjŽ†o(knž/jl†ež®kntvX #f]jlkn†­

0*#:@

3 '! ¼ $ ½ ¼ — À - % ¼ ))ϗ

[ކf]TWjldz`eYC„°fvjŽk@†´ª¤Yƒit]jlY.‚b–{eYCdv„tvjli‚Y|f]TWYƒ¨WnYƒmq²bnY*t…d¯kxžf]TY$tvYfvjŽ†WW­ 2ºY$¤jlmŽm´„*knX(YƒiWn„…«f]khf]TYCX

¤TWY*†Xhke{eYCmŽjl†o(fvTY$t]Y*f]jl†W7`cY5„°fvjŽk@†‹/°­

¼ › ¼ /˜ À ™#—

~€YC„YCef]k@tvdUtvYC„*Y*jlnY@ªWd]nX(mŽY n†W{±jl†@fvY*o@tvxf]Y|f]TYmŽjlonT9fµnt]tvjŽcjl†o(#f€fvTY iW@„…«–kxžf]TWY Y*b@Yn­¯SUTWY*b±xtvY

knž©fy¤¯k«9jl†W{dC³ # µx†‚{ . Ïi‚knf]TtvY*tvYCd]Y*†9fvYC{(kn†(fvknknžŒjŽo@rt]Y ! ®i °­¯Ã¯k@†YCd¯xtvYzn„f]jl²xf]Y5{

xf TjlonT"jŽmlmlrX(jŽ†Wxf]jlkn†"mlY*nYCmld 7{²b9f]jlX(Y°­SUTY*bŸxtvYhf]TYX(k9dyf rWd]Yž®rWm dgk@rt…„Y(kxžUjŽ†ž®kntvX(xf]jlkn†ºž®knt

Wt]Y5„jqdgY–d]TWn‚Yx†WnmŽbed]jldC­ºSUTYCt]YxtvY f]TWt]YCYfybc‚Y5dhkxžz„k@†YCdCªud]Y*†Wd]j¥fvjŽ@Y f]k"{ejŽ¶?YCt]YC†9f£¤U²nY*mlY*†Woxf]T‚d

7~YC{´ªMN|t]YCY*†x†W{ZmŽrY°³u„*kn†Y5dUY*†W„*kc{Yƒ„*knmlknt5­ŒjlonrtvY|‹(d]Tk#¤€dUhW²cjŽ†Wo£icb–„*kn†Y5d¯jl†§Wt]jlX #f]Y

YCbnY#­

2

ͅÙPÈ3Ù7Î#ÓvÙÑPÍÈӅÒ5ÊÞÙPÇÉͰʲÑ/ӅßÉÑPͯÈ5ÇÉÑÏÙ/ØxÈÞÙ©ÈyÈyʃŲÇnÈyÆ7ÈyÊÙyÈyßÉß@ÙÛCÒxÈyÑ! ²ÕŽÍ°Æ©È²Ó"Ò²ßÉÈØnÈÞÙ©ÈyÈyÊ|Ù7βȠÓ"€Ó"yÆ7ÇÉʲÈ

gÈyßÉß@Ä 11

ӅʲÅ#yͰʲÈKزÇÉÒxÍ°ßÔӅÆ$yÈyßÉßÉÑ&%' R '()Ë

*+

Î5Í°ÆÏÙ,´ÓyÝÈyßÉÈgʅÙPÎ- cÍ°Æ/.0ß1²È.2yͰʲÈyÑ3 eӅÆÏÈ/4PÎ$ÑPÒ#ӅÆÏÑPÈyÆ3Ù7Î#ӅÊÙ7βÈUͅÙ7βÈyÆ3٩ͲË

Ú

βÈÞÛӅÆ7ÈUÓ"yÙ#ӅßÉßۃÙ7βÈUÍ°ßÉÅ5ÈgÑÏÙ

.gÍ°ßÉÍ°Æ5yͰʲÈyÑ!.ÉË56ƒÍ°Ñ7ـÓ"7€Ó…ßÉÑWÒxÍ°Ñ7ÑPÈyÑPÑ©ÓKØ#ӅÑ7Ç1زÇ89¬Î²ÆÏ̀ÓvÙPÇ1´Ý5ÇÉÑ7ÇÉÍ°Ê- :ÇÙP΀ÓKŲÍÇÉÊ#ӅÊÙ.ÇÉŲÅ5ßÉÈ;´ÓgÝÈyßÉÈyʅÙ7Î-.5ÙÛCÒxÈ

ͅÕ5yÍ°Ê²È @ӅʲÅ$ÓUÑ7Ò#ӅÆÏÑPÈyÆ.زß15È:.gÍ°Ê5È: nÑ7ÒnÈ<yÇÔӅßÉÇ1=yÈyŃÇÉÊ>yÍ°ßÉÍ°ÆÇÉÊCÕlÍ°Æ?€ÓvÙPÇÉÍ°Ê@Ë$@ʲßÛµÇÉÊ|Ò²ÆÏÇ1€ÓvÙ7ÈgÑ/Î#ӅÑ/Ù7βÈ7.ÇÉŲŲßÉÈgÍ°Ê5È:.

Ï/¯ÂKY*t]f]jq„*xm´d]YC„f]jlkn†±kxž3ht]Y*f]jl†W ®i €Œ/jlnYƒtvYfvjŽ†Wnm´„YCmŽmfybc?YCd

ŒjlonrtvY !W³ 7$KÂKYCtgfvjl„CxmdgY5„°fvjŽk@† kxž$tvYfvjŽ†WW­KSUTY*tvYzxtvY¨W@Yz„YCmŽm‚fybc‚Y5d*ªc¤Tk9dgYz†9r‚„mlY*j‚nt]YzmŽke„*xf]Y5{

jl†¡fvTt]YCYmq²bnY*t…dCªn†W{Ÿf]TWxf£X x«@Y f]TWY*jlth„*kn††Y5„°fvjŽk@†Wd jl†¡fy¤k¦mlY¸ej¥ž®k@t]X mq²bnY*t…dC­±ŒWt]k@X

!9P­ ®i

`„…TY*X #fvjl„|cjlY*¤ knž/fvTYƒ¨W@Yƒ„*Y*mlm©fybc‚Y5dx†W{–f]TWY*jltz„k@††YC„f]jlkn†WdC­KŒk@r†W{±#f

Š¬­

ŒjlonrtvY|‹W³¡sK²cjŽ†ohkxžtvjlX(xf]Yµž®k#nY5 icb–„k@†YCdC­ !€kxfvjl„*YƒTk#¤ d]Wxt…dgY|dgTWknt]f¤U²nYCmŽYC†oxfvT ÉZ¯mlrY

„*kn†Y5dxtvYn­3ŒtvknX

n‘PªetvY*Wt]ke{erW„*YC{–ž®t]k@X . #­

\µ†f]TY$„*kn†9f]t…xtvbnª9t]ke{dY*¸cjqdgfjl†hd]jŽ†o@mŽY|nYCtvd]jlkn†´³/f]TYCbkn†WmŽbd]Y*Y &gjŽ†iml@„…«x†W{¤TjŽf]Y.'…­K~ke{d

nt]YU@Y*tvbdgYC†Wd]j¥fvjŽ@Y¯fvk$mŽrWXhjl†k9dgjŽfybnª9x†W{£f]TWY*jltut]Y5dg?kn†‚dgY€dv#fvrtvxf]Y5dxfuTjlonT(jŽmlmlrX(jŽ†Wxf]jlkn† 7{²b9f]jlXhY°­

SUTWY*bxtvYzX£rW„…T–X(k@t]Y|†crX(Y*tvknrWd f]TWn†„*kn†Y5d*ª9Y¸„*Y*ef€#ff]TWY|@Y*tvb „Y*†9fvY*tkxžfvTY|Y*bnYfvTY|ž®k#nY5$°­

SUTWYhtvke{ºd]jlon†WnmKjldƒX(k@dgf rWd]Yž®rm¯#f$†jlonT9fCª/kntƒž®knt$X(k#@Y*X(Y*†9f${eYfvYC„°fvjŽk@†´­ suTcbcmŽk@onYC†Yfvjl„CxmlmŽb@ª´t]ke{d

d]Y*YCX f]k T‚²nYƒx?YCnt]Y5{–mq#fvY*tUjŽ†fvTY$Y*@knmlref]jlkn†´­

SUTY£jŽ†9fvY*ont…#fvjŽk@†±kxžKmŽjlonT9fzicbt]Y5„YCef]k@tvdjqd€«c†k#¤†¦@d G. . G

#6* 7dgYCYh`cY5„°fvjŽk@†§‹W­:8n­:81°­

SUTWY knrefvrefµd]jlon†Wnm´kxž3f]TWY t]Y5„YCef]k@tvdUjqdfvTY*†±fvtvn†WdgX(jŽfgfvYC{f]k–Tkntvj9Ckn†9fvnm´„YCmŽmqd€x†‚{±ijŽ?knmqxtµ„Y*mlmqd*ª

jl†f]TY£\µs'u­

. À ™.'0 À )˜5š )¯› ¼ ))ϗ

Lzkntvj9Ckn†9fvnm„YCmŽmqdKtvY*mq²b f]TWYzdgjlon†‚xmež®tvknXf]TY€tvYC„*Y*efvknt…d fvTtvknro@T(„…TWY*X(jl„CxmWd]bc†Wx‚dgY5d3jŽ†(fvTYƒ\µs °ª

n†W{$xtvYuxmqd]kTjŽo@Tmlb|„*knrmlYC{µfvkf]TYCjŽt/†Y*jlonTci?kntvjŽ†WoTkntvj9Ckn†9fvnmx„YCmŽmqd®f]Ttvknro@T$Y*mlYC„f]tvjl„Cxmnd]bc†WxWd]YCd°­

SUTWjld¯„knrWmŽjl†o$X x«@YCd3fvTY*X d]Y*†Wd]j¥fvjŽ@Yf]k£ƒXhk@t]Y€d]Wxf]jqxmlmŽb

<#ªnk@tumlk#¤U¢PW@d]d]YC{´ªxnYCtvd]jŽk@†(kxžf]TY

jlX xo@YfvTWx†Ÿf]TWY t]Y5„Y*f]knt…dÏd]Y*Y`cY5„°f]jlkn†Ÿ‹W­É ­Œrt]f]TYCt]X(k@t]Y@ªdgjl†W„*Y£fvTY*jlt X(Y*X£it…x†Y 2

fvTY|jŽ†WrefUž®t]k@X f]TY|t]Y5„YCef]k@tvdCªnfvTY*bnmld]kE 2. mlk#¤U¢PW@d]duf]TY$d]jŽo@†Wxm‚ž®tvknX t]Y5„YCef]k@tvdC­SUTWY*b

X n«nYµž®Y*YC{iWn„…«–„*kn††WYC„°fvjŽk@†Wdf]k tvYC„*Y*efvknt…d*ªen†W{nmld]khž®Y*Y5{c¢7ž®k@t]¤Uxt…{„*kn††Y5„°fvjŽk@†Wdf]k(ijl‚k@mlnt€„YCmŽmqd*­

+ '! À )7še™› ¼ ))7—

`cf]jlmŽm´xf¯fvTYƒmŽYCnY*mknž´f]TY\µs  ªcijŽ?knmqxt„*Y*mlmldUxtvYƒ„kn†W†YC„f]YC{–¤j¥fvTi?kxfvTtvYC„*Y*efvknt…d n†W{–TWkntvj9Ckn†9fvnm

„*Y*mlmldC­¯Zrefzf]TYd]jlon†±kxžf]TY5dgY fy¤¯k(fybc?YCdknž3„*kn††Y*¸ejŽk@†Wd€xtvY$xml¤²bed{ejŽ¶Y*tvY*†9fC³¯j¥žu(ijl‚k@mlntz„YCmŽmjld

{Y*?knmqxtvj9CYC{i9bf]TWYdgjlon†‚xm´knžtvYC„*Y*efvknt…d*ªjŽfzjqdTcbc‚YCt]?knmqxtvj9CYC{icb–fvTYd]jŽo@†Wxm´knžTWkntvj9Ckn†9fvnm´„Y*mlmqd*ª

n†W{¦cjq„Y*¢P@Y*t…d]W­L€Y*†‚„Ynª/–ijl?knmqxt$„YCmŽmKjqdƒxml¤²bed|dgYC†WdgjŽf]jlnY£f]k±±{ejŽ¶?YCt]YC†W„Yhi‚Y*fy¤¯YCY*†ºf]TY d]jŽo@†Wxm

ž®tvknX t]Y5„Y*f]knt…dUx†W{jŽfvdimlrtvt]Y5{–@Y*t…dgjlkn†icb f]TY$TWkntvj9Ckn†9fvnm„*Y*mlmldC­

`cr‚„…T Y¸„*j¥f…#f]k@t]b#jŽ†TWjŽijŽf]k@t]b jŽ†9fvY*t…n„°fvjŽk@†Wd–nt]YEnY*tvb;„mqndvdgjq„*nmzjl† †YCrtvnmµ†WYfy¤kntv«cdx†W{ xtvY

«c†k#¤† @d . %# %H °­ SUTYŸdgbc†Wnef]jq„§fvt]jqn{;X n«nYCd–ijl‚k@mlnt„*Y*mlmlddg‚#f]jqxm|x†W{ f]YCXh?knt…xm

: < ¨‚m¥fvY*t…d*³f]TWY*b±i‚YCTW²nYmŽjl«nY£YC{eo@Y{eYfvYC„f]knt…dC­z·„*„*knt…{ejŽ†Wo f]k fvTYht]Y5dg?YC„f]jlnYd]jŽo@†Wd€knžKf]TY

„*kn††WY¸ejŽk@†Wdž®t]k@X‡t]Y5„YCef]k@tvdx†W{£Tk@t]j:9*k@†@f…xmc„YCmŽmqd*ªxf]TYCt]YU¤jlmŽmei?Yz\!¢Þ„Y*†9fvY*tK„YCmŽmqd3x†W{ \|Œ3Œ¢Þ„YC†9f]Y*t

„*Y*mlmldP`cYC„f]jlkn†±‹W­! ­

Zjl‚k@mlntU„*Y*mlmldUnt]Y|xmqdgk£f]TYƒjl†refUfvkfvTY$dgY5„kn†‚{ ¨WmŽf]YCt]jl†o(dyf…xonY|kxžf]TYƒtvYf]jl†WWª@fvTYƒ[Þs u­SUTWY*b

?k@dvd]YCdvd x†Y*mlkn†o9#fvYC{dgf]tvrW„°fvrtvYz¤jŽf]Tfy¤k£{Y*†W{etvjŽf]jq„€fvt]YCYCd όjŽo@rtvY ! ®i °ªcf]T‚#fUo@²nY€f]TYCX f]TYCjŽt

†‚xX(Yƒkxž ijl?knmqxt Ž­

” š›n™ '6)

¼ › ¼ ))ϗ

`eb9†‚xefvjl„§Y*¸„…TWx†o@YCd(jl†Nf]TYE[Þs ‡xtvY§tvY*mq#f]jlnYCmŽb dgjlX(jŽmqxt fvk¡f]TWk@d]Y§jŽ†NfvTY"\µs  ªuirefmŽY5d]d¤¯YCmŽm

rW†W{eY*t…dgf]kcke{©­£[ކ‚{eY*Y5{©ª/nX(@„tvjŽ†WYh„*Y*mlmldCª©f]T‚#f mq²b±f]TYdvxX(Y(jl†@fvY*t…dyfvj¥fvjlnmKt]k@mŽY(ndƒTWkntvj9Ckn†9fvnm„*Y*mlmld

jl†f]TY£\µs  ªWxtvYƒ£@Y*tvb ..<£„mqndvdUkxž3„Y*mlmqd*­u·zX(@„tvjŽ†WY|„*Y*mlmldUTW²@Y${ej¥¶Y*tvY*†9f€d]j9CYCdCªeTcbcd]jlknmlknonb

n†W{hž®r†W„f]jlkn†Wduf]TWxf¯xtvY†knf¯xmlm¤Y*mlmWr†W{Y*t…dyfvk9ke{©­ 2ŸYz{ek †knf Y¸emljq„jŽf]mlb£X(ke{eY*mWf]TYCX±ª@irf¯tvY*?knt]f

d]knX(Yƒkxž/f]TYCjŽtµdgrW‚k9dgY5{ž®r†‚„°f]jlkn†‚djŽ†¦`cY5„°fvjŽk@†‹‚­‹‚­c­

ŲÇÝ5ÇÉÅ5ÈgÅzÇÉʀ٩ÍÈ #ӅßÉßÛ 8Ò²ÆÏÈyÑPÈyÊÙ/ÝÈyÆÏÑPÇÉͰʲÑ3 5Ù7βÈ7.

P

ÈyÅ-.²Ó…ʲŵÙ7βÈ/.ÆÏÈgÈyÊ-.gÍ°Ê5È: 5ÙPÎ#ÓvÙ €Ó°ÈÙ7βÈβͰßÉÈ,yͰʲÈKÒ#ÓvÙ7Î ©ÓgÛ

¬Î²ÆÏ̀ÓvÙPÇ1 ÇÙ7ÎzÙ7βÆÏÈgÈ&¬Î#ӅÊ5ʲÈyßÉÑ]Ë

+

ÈyÈ/% )Ë

š2)5C )*'À )še™› ¼ ))ϗ

Nƒn†onmljlkn†WntU„*Y*mlmldUtvYC„*Y*jlnYƒdgjlon†‚xmž®tvknX iWjŽ?knmqxtx†‚{nX n„tvjl†Y|„*Y*mlmldC­uSUTYCbxtvY|f]TY$k@ref]Wref€kxžf]TY

tvYfvjŽ†‚ª´x†‚{¦fvTY*bEnmŽm¨WtvY (f]kf]TWY itvnjŽ†­£SUTYCb¦xtvYfvTY(kn†mlbEd]jŽ«cjl†o±„Y*mlmqdµknž fvTY(t]Y*f]jl†Wª

nmŽk@†oh¤j¥fvT±d]knX(YƒxX n„*t]jl†Y$„YCmŽmqdC­3SUTWY*jlt€dgjl«nYƒo@Y*†YCtvxf]jlkn†tvke„Y5d]dUjqdUX(kc{Y*mlYC{jŽ†¦`cY5„°f]jlkn†‹W­‹W­:8x­

0*#I , '*) ¼ še™.'®˜ ½NÀ -˜ & ¼ $ ¿À 3 ')®˜ ¼ ™.'6)5C©˜5šGC ¼ —

SUTWYµfy¤k(mlY¸ej¥ž®k@t]X ml²b@Y*t…d ?Y*t]ž®kntvX xmlX(k@dgfjŽ†W{Y*?Y*†W{eYC†9f¨Wm¥fvY*tvjŽ†Wo(dgfvnonY5d 7ŒjŽo@rt]Y $°­u·€tvY|f]TY5dgY

¨‚m¥fvY*tvjŽ†o dgfvnonY5d(mŽjl†YCnt<; jP­Y@­_{ekcYCd f]TY¦k@ref]Wref–kxž$"o@n†onmljlkn†Wnt(„*Y*mlmz{eYC‚YC†W{NmŽjl†Y5xtvmŽb k@†Nf]TY

jl†W„*knX(jŽ†Wo mŽjlonT9f ;SUTWYµx†‚dg¤Y*tujqdu†Wk$ž®knt fy¤¯k$tvYC@dgk@†Wd*­ŒjŽt…dyf¯t]Y5nd]kn†(jlduf]TWxf¯mljŽo@T9f t]Y5„YCef]k@tvdu{k †kxf

tvYC@„°fmŽjl†YCnt]mlb#fxmlmf]k£f]TWY|jl†W„*knX(jŽ†Wo£mljŽo@T9fjl†9f]Y*†‚dgjŽfybnªeknf]TYCt]¤jqdgYµknrtUtvYfvjŽ†W£¤knrmq{i?Y

10^5f]jlX(YCd

X(k@t]Yhn„°fvjŽ@Y£jl†Ÿit]jlonT9f${²bcmŽjlonT9fµf]TWn†ºjl†ºtvkcknX mljŽo@T9f7dgYCY(`cY5„°fvjŽk@†º‹‚­81­h`cY5„k@†W{¦tvYC@dgk@†¦jld|fvTW#f

tvYfvjŽ†‚xm´„YCmŽmqdxtvYƒmŽjl†«@YC{–f]Ttvknro@Td]bc†Wx‚dgY5d*ª9f]T‚#f€?Y*t]ž®kntvX †kn†±mljŽ†WYCxtUk@‚YCtvxf]jlkn†WdC­

Lzk#¤¯YCnY*t5ªW(mljŽ†WYCxtµxtvk²¸ejŽX xf]jlkn†TWndzTcbcd]jlknmlknonjq„*nm²nmŽjq{xf]jlkn†±¤TY*†¦f]TY o@Y*†YCtvnm©mŽjlonT9fzjl†e¢

fvY*†Wd]jŽfybjqdµxfµ(¨¸eY5{§mŽYCnYCm fvk–jl†Wd]rt]Y£dgk@XhYmŽjl†YCnt]jŽfyb–jl†§fvTY£„*kn†cnYCtvd]jŽk@†–ž®tvknX-mljŽo@T@f|jŽ†9fvY*†Wd]j¥fybf]k

tvYC„*Y*efvknt(n„f]jl9jŽfyb°ª n†W{¡ž®k@thºWntgfvjl„*rmlnt£‚#f]Tc¤U²bºknžµo9x†o@mŽjlkn†Wnt„YCmŽmqdC­ `cYC„f]jlkn†;e­É§tvYCd]Y*†9fvd

fvTY(fy¤kX njŽ†, .!! knžjl†ež®k@t]X #fvjŽk@† ‚k#¤jŽ†o±k@ref kxžf]TY tvYfvjŽ†‚³$kn†WYhknž¯fvTY*X fvTYsKxtvnkn¢

„*Y*mlmŽrWmlnt|‚#f]Tc¤U²b€jqd$ontvk@dvd]mŽb±mljŽ†Y5xt5ª´¤TYCt]Y5ndµf]TY knf]TYCt fvTY¹¦xo@†ke„YCmŽmlrmqxt|W#fvT9¤U²bµjldƒ†knfC­

Zref€¤T‚#f]YCnYCtUf]TY ‚#f]Tc¤U²b–„k@†Wd]jl{eYCt]Y5{©ªeX(k@dgf€t]Y*f]jl†WxmXhke{eYCmld€rWdgY$mljl†YCnt¨WmŽf]Y*t…d€xfzdgk@XhY$?knjl†9fCª

i?YC„CxrWd]Yµf]TYCbxtvY|XrW„…Ti‚Y*fgfvY*tr†W{Y*t…dyfvk9ke{©ªn†W{YCnd]jlY*tf]k(jlXhWmŽYCXhYC†9fC­K[ކ§`cYC„f]jlkn† !ªe¤Yƒ{Y¨W†Y

hž®t…xX(YC¤¯k@t]« fvkX(ke{eY*m†Y*rWt]k@†WdUf]T‚#fµnmŽmlk#¤€dUf]TWY t]Y*f]jl†Whf]k–¤¯k@t]«mŽjl†YCnt]mlb–jl†¦„*Y*t]fvxjl†±tvn†onY$knž

‚xt…xX(YfvY*t…d*­

SUTY¯mŽjl†Y5xt/ntvk²¸ejŽX #fvjŽk@†zž®k@to9x†o@mŽjlkn†Wnt©TWndmŽk@†o€i?Y*YC† tvkn?k@d]YC{

‹M8 ¬ª#jŽ†Wd]jlt]Y5{ƒicbµtvY*cjlknrWd

dgf]r‚{eb§kn†ºo@x†WonmljŽk@†Wxtµt]Y5„Y*f]jlnY ¨‚Y*mq{d

H!P­£SUTY1 #< Ï~€Œ µkxžufvTY(„Y*mlmKjqd|{Y¨W†Y5{¦icb

fvTYxtvYCknžfvTY cjld]rWnmK¨WY*mq{"¤TYCt]Y ±mljŽo@T@fdgf]jlXrWmlxf]jlkn†"TWndx†"jŽ†WrY*†‚„Yk@†ºfvTY ¨Wtvjl†o±kxžUf]TY

„*Y*mlm7ª‚¤TY*†±fvTY tvYCdgf€kxž3f]TWY cjld]rWxm¨WY*mq{±jqd†kxf|dyfvjŽX£rmlxf]Y5{ ®ont…²b ¨WYCml{ @­UŒjlonrtvY£’£onjlnY5d€x†±jq{eYC

knžfvTY dgj:9*Y5dUkxžf]TY$tvYC„*Y*efvjŽ@Yz¨‚Y*mq{djl†fvTY$tvYf]jl†WW­

N|jlnYC†E–„*Y*mlm„*Y*†9f]YCt]Y5{±jŽ†

x0

kn†§fvTYtvYfvjŽ†WWª?n†W{§¤jŽf]Tºt]Y5„Y*f]jlnY$¨WYCml{E~€Œª–mŽjl†Y5xtµntvk²¸c¢

jlX #fvjŽk@†ºknž¯jŽfvd$n„f]jlcj¥fyb¦„*kn†Wd]jldgfvd|jl†Ÿ¨W†W{ejl†of]TY(«@Y*tv†Y*m

K ž®kntƒ¤Tjq„…TEfvTY(ž®knmlmlk#¤jŽ†oYCp9rWxf]jlkn†ºjld i?YCdgf€nYCt]jŽ¨WY5{©³

·

(x0) = Z

~€Œ

(x0)

I(x0+x)K(x) dx,

¤TWY*tvY

I jqdµf]TY(mlrX(jŽ†k9dgjŽfyb§tvkx¨WmlYhkxž fvTY(jŽX nonYnª´n†W{E·=jqdƒX(Y5nd]rt]Y£kxžUn„°fvjŽcjŽfyb¦xWt]k@t]jq#fvY fvk±f]TY(fybc?Ykxž„YCmŽm ®jŽfvd$X(Y*X£it…x†Y(?kxf]YC†9f]jqxmujŽžj¥fjqd †k@†e¢¬d]jl«cjŽ†o§„*Y*mlm7ªknt$jŽfvd$jl†Wdgfvn†@f…x†YCknrWd

¨‚t]jl†ox¢¬t…#f]Y–j¥žzjŽf jld(ºdgjl«cjŽ†WoŸ„*Y*mlm­ Z¯b"Y*¸9fvY*†9fCª

K ^{jqdhkxžf]YC†} {eY5d]„*t]jli?YC{ @d fvTY . #< 6. :±kxž3f]TWY£„YCmŽmP­z·µd€tvY*X xtv«nY5{icb

‹28 ¬ª

K „*x†§i?Y£„*kntvt]Y5„°fvmŽbxtvk²¸ejŽX xf]YC{icb± # ) % G 6>ƒ\ N­ >µY*?Y*†‚{ejŽ†Wok@†§f]TY£onmlkniWnm/d]jŽo@†¦kxž

Kª?f]TYh„Y*mlmjld|„*nmŽmlYC{¦n†Ÿ\!¢Þ„YC†9f]Y*t|„YCmŽm 2

ͅÙPÈÙ7Î#ÓvÙÙPÎ5È "!|ͅÕnÓ´Ý5ÇÉÑ#ӅßyÈyßÉß5ÇÉÑWÓ Í°Æ7ÅÙ7Î#ÓvÙ yÍ°Ê*ÝÈÞÛCт݅ӅÆ7ÇÉÍ²Ñ ÈgӅʲÇÉÊ°ÑgË

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1

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ÑPÈ<yͰʲÅ#ӅƮۃÇÉÊ; 5ÈgÊgÈ #ÆÏÈ 5ÇÉÆ7ÇÉÊÑÏÙPÇ1²ßÔÓvÙ7ÇÉͰʵͰÊzÙPβÈ,yßÔӅÑ7ÑPÇ1gӅßcÆÏÈ3yÈyÒ5Ù7ÇÝÈ52#ÈyßÉÅ9Ë

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·µdh¤¯YTW²nYwyrWdgf(dvxjq{©ªuk@†mŽb E‚k@tgfvjŽk@† knžzo@n†onmljŽk@†Wxth„YCmŽmqdhTW²nY–n†Wd]¤¯YCtvd£¤¯YCmŽmntvk²¸cjlX #fvYC{

icbmljl†YCnt€dgrX(X #fvjŽk@†´­3[ކW{eYCYC{©ªW{j¥¶Y*tvY*†W„*YCdUjl†§x†Wxf]knX£bx†W{Tcbedgjlknmlkno@b {eY¨W†WYƒfy¤¯k(X xjl†–fybc?YCd

knžzo@n†onmljlkn†Wnt„YCmŽmqd

8²‰P­ SUTYCd]Y–fy¤k¦fybc?YCd(xtvY@„°fvrWxmlmŽb"fvTYknrefvref kxž€fy¤kº{j¥¶Y*tvY*†9f(t]Y*f]jl†Wxm

3še˜

& ¿ š ½ — ªf]T‚#f|{k†kxf|jŽX(mlbf]TYhd]nX(Y„YCmŽmqd*­zsuTcbcd]jlknmlknonjq„*nmŽmlbnªf]TjqdzX(Y5x†Wd€f]TWxfzf]TWYiW@dgjq„‰²¢

mq²bnYCtvd3ntv„…TjŽf]Y5„°fvrt]YtvYCd]Y*†9f]Y5{(xi?k#nYjldK‚xt…xmlmŽYCmŽj:9*Y5{jl†(XrmŽf]jlmlY€nYCtvd]jŽk@†WdCª#Tcbcd]jq„*xmlmlb jŽX£itvjl„C#f]Y5{

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Wt]Y5„jqdgYƒcjŽYC¤$­

ŒWr†W„°fvjŽk@†Wxmlmlbnªe„k@t]tvYCd]‚k@†W{eYC†W„Y€Y*¸ejldgfvd¯i‚Y*fy¤¯YCY*† fvTYµfy¤¯k tvYfvjŽ†‚xm?W#fvT9¤U²bed n†W{(f]TWYzfy¤k Wxf]Te¢

¤U²bedhkxžµjl†ež®k@t]X #fvjŽk@† jl† f]TY§iWtvnjŽ†´³5. W#fvTc¤²b n†W{ 6* Wxf]Tc¤²b@­N[ކ fvTY§it…xjl†´ª¯

@Y*†9f]t…xm‚t]k@ref]Y ŒkntvX$ jqd¯{eY5{ejq„*#fvYC{(fvktvYC„*jld]Y*mlb dyfvrW{ebcjl†o fvTY|d]TWn‚Y5dukxž/kniewyY5„°fvd žÏ@„Y|t]Y5„kno@†jŽ¢

fvjŽk@†´ª9f]Y*¸9fvrtvYµtvYC„*knon†Wj¥fvjŽk@†´ª@Y*fv„n­ªe¤TjŽmlYƒ£{ek@tvdvxm?tvknrefvY ¹§k#@Y*X(Y*†9f°ªcjl†cnknmlcjŽ†Wo£{ejŽ¶?YCt]YC†9fUnt]Y5ndCª

jqd d]?YC„jqxmlj:9*YC{(jl†X(k#nYCX(Y*†9f¯x†‚xmlbcd]jqd ®X(k#nYCX(Y*†9fvdKknž´kniewyY5„°f…d*ª9dgYCm¥ž¢¬X(kxf]jlkn† knž©f]TYµkniWd]Y*tvnYCtCª#Y*fv„n­

·zm¥fvTknrWonT f]TYCt]Y§jqd †kŸfvt]jlcjlnmz„k@t]tvYCd]‚k@†W{eYC†W„Y@ªuf]TWY§t]Y*f]jl†Wxm€sKnt]@kc„*Y*mlmŽrWmlnt£Wxf]Tc¤U²b †W#fvrt…xmlmŽb

Wt]k#cjq{eYCdhonkcke{jl†refUf]khfvTYƒitvnjŽ†&dUŒk@t]X W#fvT9¤U²bnªcx†‚{fvTYƒt]Y*f]jl†Wxm´¹¦non†ke„*Y*mlmŽrmqxtWxf]Tc¤²b

Wt]k#cjq{eYCd †W#fvrtvnmŽmlb( onkcke{(jl†refž®knt¯f]TYµit…xjl†d¹§knf]jlkn†Wxf]Tc¤U²bnªxž®k@t¯tvYC@dgk@†Wd3fvTW#f¤¯YµY¸emqxjl†

†Wk#¤$­

+µše™ À › ¼ )))Ϛ™ š˜ & ¿ š ½+

SUTWYsKxtvnke„YCmŽmlrmqxtµW#fvT9¤U²b±„kn†‚dgjqdyf…dzknžuo@n†onmljŽk@†Wxt|„YCmŽmqdz¤jŽf]TŸdgX nmŽmt]Y5„YCef]jlnY¨WY*mq{dƒx†W{¦‚YCtg¢

d]jqdyfvY*†9f tvYCn„f]jlkn†hf]kjŽ†9f]YC†Wd]j¥fyb(„…TWx†WonYCdCªn¤TWk@d]YonY*†WY*tvjl„†‚xX(Yzn„*t]k9d]dKd]?YC„jlYCd jld

β^{› ¼ ))ϗ} ­SUTWY*b(xtvY

«c†k#¤† @d *O „*Y*mlmldujŽ†(fvTYztvjlX(xf]YtvYfvjŽ†WWª@n†W{ nd%„YCmŽmqdujl†(f]TWYz„*xf t]Y*f]jl†W­3·zd¯„*n†hi?Yzd]Y*YC†

jl†¡ŒjŽo@rt]Y‰7· ª´f]TWY*bºnt]Yh¤Y*mlmuX(kc{Y*mlYC{ºi9bŸmŽjl†YCnt$t]Y5„YCef]jlnY£¨WY*mq{©­–SUTWY*b¦TW²@Y d]jlon†jŽ¨‚„*n†9f

H= . ®¤jŽf]TWknref(cjld]rWxmdgf]jlXrmq#fvjŽk@† °ª3fvTW#f xmlmlk#¤€d fvTY*X fvkŸ„*kc{Yd]jŽo@†Wxmqdi?kxfvT icb

jl†W„*t]Y5nd]Yµk@t€{eYC„*t]Y5nd]Yµknžf]TWY*jlt¨WtvjŽ†o‚­KSUTY*bY*¸ejldgf€jŽ†±i?kxfvT¦\! n†W{¦\|ŒŒ nY*t…d]jŽk@†Wd*­

SUTYCt]YµxtvY€{ej¥¶Y*tvY*†‚„YCdujŽ†

β „*Y*mlmld¯n„C„knt…{ejl†o|f]k$fvTYzd]‚Y5„jlYCd¯„kn†‚dgjq{eY*tvYC{´­sutvjŽX xf]Yz¹§jq{eo@Yf¯„*Y*mlmld T‚²nYd]X(nmŽmlY*tt]Y5„Y*f]jlnY(¨WYCml{d f]TWn† „*xf%„YCmŽmqdCªfvTW#f£nmŽmlk#¤=nY*tvbº@„*„*rtvxf]Yd]W#fvjlnmutvYCd]knmlref]jlkn†´­

·zmld]kWªcX(k@dgf„*#fïY*mlmldUd]Y*Yzjl†iml@„…« x†W{–¤Tj¥fvYnªc¤TYCt]Y5ndxmlm¹¦jl{eo@Yf„YCmŽmqdUxtvYµ„k@mŽk@t k@‚k@†Y*†9f…d*³

fvTY*jlt tvYC„*Y*efvjŽ@Y¨WYCml{dgf]tvrW„f]rtvYz„k@†Wd]jldgfvdKknž „YC†9f]t…xmWY*¸„jŽfv#fvjŽk@†9*k@†Yzd]Y*†Wd]j¥fvjŽ@YUf]k£kn†Yz„*knmlknt ÏYn­oW­

~€YC{ ªx†W{?Y*tvjlTY*t…xmjl†TjlijŽf]jlkn† 9*k@†Y|d]Y*†‚dgjŽf]jlnY|f]k(x†Wkxf]TWY*t®Y@­o‚­ N|t]YCY*† ­3SUTWY*tvY|nt]Y|t]k@ronTWmŽb

fy¤k {ejŽ¶?YCt]YC†@fz„*knmlkntk@‚k9dgjŽf]jlkn†§d]jŽo@†WxmqdUjl†Wt]jlX #f]Y$tvYfvjŽ†‚ndC³u~YC{ HN|t]YCY*†§x†‚{uYCmŽmlk#¤ #ZmlrYn­ !zkx¢

fvjl„*Y fvTW#f3fvTjld3Wt]k@‚YCtgfybƒtvYCp9rjlt]Y5d´fvTW#fu¹§jq{eonY*f3„*Y*mlmld/f…x«nY¯f]TWY*jltjl†rfuX(njŽ†WmŽbƒž®tvknX‡„*kn†Y5d*­Œ/jl†WnmŽmlbnª

¹¦jl{eo@Yf€„YCmŽmqd€nt]Y|XrW„…T±X(kntvY|†9rWXhYCt]k@rWdUjl†Wt]jlX #f]YƒtvYfvjŽ†W£fvTWx†Ã¯Y*mlmldzxtvYµjl†§„*#f€tvYfvjŽ†‚­

2;TWxf]YCnY*tƒfvTYd]‚Y5„jlYCd$„k@†Wd]jl{eYCt]Y5{©ªd]W#fvjlnmKtvYC„jqd]jŽk@†Ÿn†W{"„knmlknt$d]Y*†Wd]jŽf]jl9jŽfyb¦X x«@Y

β „*Y*mlmld$

o@kckc{ jl†ref fvkºfvTY ŒkntvX$KW#fvTc¤²b¡knžzf]TWY§itvnjŽ†­N·z†W{ jŽ†‚{eY*Y5{©ªf]TYCjŽt ?Y*t…dgjqdgf]Y*†9fx†‚dg¤Y*thf]k

jlmlmŽrX(jl†W#fvjŽk@†„…TWx†WonYCd3tvYCp9rjltvYCd3fvTYzdv„Y*†WY€†kxfKfvk X(k#nYUfvk9k žÏndgfC³´fvTYCd]Yz„*Y*mlmld¯xtvYXhk9dyfvmŽbh{eYCd]jlon†Y5{

Ú

βÇÉÑ/ÇÉÑ/βÍ3©ÈÞÝÈyÆKÓUÝÈgƮۃѬβÈ<€ÓvÙ7Ç1 ÝCÇÉÈ Ë

1

ÊÙPÈyÆ7Ó"ÞÙPÇÉͰʲÑuÈCÇÉÑ7Ù/ØnÈÞÙ©ÈyÈyÊ €Ó…Ê*Û yÈyßÉßcÙÛ5ÒxÈyÑ! @ӅÑKÓ/°ßÔӅÊyȯÓvÙ©ÈyØ

ÑPÇÙ7È/%'() yßÉÈgӅÆ7ßÛzŲÈ<ͰʲÑÏÙPÆ7ÓvÙPÈyÑgË

ž®k@t ¨W¸exf]jl†ok@iewyYC„fvdCª@¤TWjl„…Txmqdgk Y*¸emlnjŽ†‚dufvTY*jltUT9rWonYz†crX£i‚YCt¯jl†–Wt]jlX #f]Y|x†W{T9rWX(n†WdCªcn†jlX(nmld

jl†cnk@mŽ@YC{jl†±tvYC„jqd]YµcjqdgrWnmfv@dg«edC­

β¢¬„*Y*mlmld€nt]Yµf]TWY$o@x†WonmljŽk@†WxtU„YCmŽmqdX(ke{eY*mlYC{jŽ†fvTjld€t]YC‚k@tgf5­

ŒjlonrtvYh‰³=S/tvjlnm¥¢Þ²nYCtvnonYC{–¨WtvjŽ†Wo–t…#f]Y5dzknž¯„*#f\|Œ3Œ/¢¬„*Y*†9f]YCt „YCmŽmPª´x†W{Ekxž¯n†"\|Œ3Œ/¢¬„*Y*†9f]YCt

„*Y*mlm7ªjl†EtvYCd]‚k@†Wd]Y f]kf]TWY£jl†9f]tvkc{rW„°fvjŽk@†Ex†W{Et]YCX(k#²nmknž¯–dgfvxf]jq„ jlX(nonY@ª{jld]mq²b9jl†o–d]jŽ†crWd]knjq{xm

o@tvxf]jl†o–¤jŽf]T"{ejŽ¶?YCt]YC†9f$d]Wxf]jqxmkx¶©dgY*fvd ®tvjlonT9f­ [¬ž fvTY „Y*mlmu„*n†Ejl†W{eYCYC{ºi‚Y(X(ke{eY*mlYC{Ei9bº–mljŽ†WYCxt

«@Y*tv†Y*mPªxfvTY*†(fvTY*tvY€¤jlmŽm‚i‚Y|x†(kx¶©d]Yf knž?fvTYzo@tvxf]jl†o|ž®knt¯¤Tjl„…T jŽfvd¯x†Wd]¤Y*tKjqdKf]TWYµdvxX(Y€nd ¤j¥fvTknref

n†cbdgf]jlXrmlrWdCªW¤TYC†fvTY ?k@d]j¥fvjŽ@Y x†W{±†WY*o@xf]jlnYƒWntgf…dkxžKjŽfvd€d]jŽo@†Wxm/„knrW†@fvY*tviWxmqx†‚„YƒYC@„…Tknf]TYCtC­

„Y*mlmqd€nYCt]jŽž®bfvTjqd€tvkn?Y*t]fybnª¤TWY*tvYCnd=„Y*mlmqdz{ek@†f5ªWX(YCn†jl†o f]T‚#f€fvTY*tvY$jldµdgk@X(Y dgWxf]jqxm´†Wkn†e¢

mljl†YCnt]jŽfyb±jŽ†Ef]TYCjŽtƒtvke„YCdvd]jŽ†o‚­ !€kxfvYhnmld]kfvTW#f „Y*mlmqd|nt]Y£X(kntvY f]t…x†‚dgjlY*†9fzfvTWx† „Y*mlmqd 7dgTk@tgf

?YCn«kxžKx†‚dg¤Y*t °­3Sknfvnm©Y¸e‚YCt]jlY*†‚„Y|f]jlX(YxdC­3ŒWt]k@X

85‰¬­

( šGCM) À › ¼ )))Ϛe™ 3še˜

& ¿ š ½ (

SUTWYh¹¦non†ke„*Y*mlmŽrmqxtµW#fvTc¤²b±„k@†Wdgjqdgfvdµkxž mqxtvonY„*Y*mlmldCª¤Tk@d]Yo@Y*†YCt]jq„ †‚xX(Yh@„tvk@dvd€d]‚Y5„jlYCd|jld

α

› ¼ ))ϗ ­ SUTYCb„*kntvt]Y5dg?kn†‚{fvk(f]TY Wt]jlX #f]Y d . ..u„YCmŽmqdCªWx†W{fvk f]TY£„*#f d„YCmŽmqd*­USUTYCb–TW²@Y

hX£rW„…T±¤jl{eYCt€tvYC„YCef]jlnY|¨WYCml{fvTWx†

β „*Y*mlmldCªWxtvYƒkn†mlb–d]Y*†Wd]jŽf]jlnYƒf]k t…xWjl{jŽ†9fvY*†Wd]j¥fyb„…TWx†WonYCdCªx†‚{

fvtvn†Wd]XhjŽf€dgWjŽ«@YCdUnYCt]b t…xWjl{emlb fvk£fvTY$it…xjl†´­uSUTYCb–xmqdgkhY¸ejqdyf€jl†±i‚knf]TE\! x†W{¦\|Œ3Œ¡@Y*t…dgjlkn†WdC­