• Keine Ergebnisse gefunden

High order compact finite difference schemes for a nonlinear Black-Scholes equation

N/A
N/A
Protected

Academic year: 2022

Aktie "High order compact finite difference schemes for a nonlinear Black-Scholes equation"

Copied!
25
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

!#"%$'&("*)+!-, .0/1)32*4657829&:$78;<&("=&(5')+&?>9)3'&(, &@>*AB!#"C/

5'!#5EDFG5'&/H"JIKDL/M)NPO=QR)3'!#DS&@>T&+UWV7/X2Y!#5

Z0[ B\

^] _

`

ab

ced\fgWh(

`

ji

k b

mln

`

bog

p6q=rsutBv[r^wyxz'{|B}^z~€atxyr‚~r„ƒ…v†t}^x\‡v†t~‰ˆ‹ŠrŒ…Gr^€Žq[z‡v‘tB~ˆ

’'zy“”–•—9˜’Eq–™šxy‡m›3xyzWœ=”

u{{9’ ”ž%Ÿ=– 9Ÿ%•=•¡

¢

£¥¤¦

¢¨§L©BªŽ«ªŽ¬­¢¨§X®

¦S¯

§°ªŽ±

¦S¯

¬³²(´°µ¶¸·

¯¦Y¯

¬­¹

¯

¬³²

¢

£¥¤¦

¢¨§L©BªŽ«ªŽ¬­¢¨§º»µ½¼¿¾G«±

¦S¯

¬À²Á´°µ¶º»µ°¼¿¾G«±

¦Y¯

¬À¾Gµ¹‰Âš¬­¹‰¹‰ªŽµ¹¢¨§

¦ ¼¯

Ã

µ¬³ÄFªŽ«¨¹¬

¯Å¦Y¯šÆ

¾Fµ¹

¯¦

µÇ

¤¦

¢¨§ÈmÉ8ÊGʏˏÌSÊYÍuÎFÌ

Æ

¾Gµ¹

¯¦

µ°ÇGË°Ï9ª8«‰±

¦

µuÐ

Ñ ± ¦

¬³ÒÔÓ½Õ°«ªŽÕ°«¬³µ

¯

¹¨Ö¬Àµ½¼¿¾G«±

¦S¯

¬³²Ø×´°µ°¬ÚÙL²G¾Fµ¹

¯¦

µÇG׶½ª

ÛWÛWÛ

¯¯ ¦S¯ ¯¨¦ ¯

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2006/2182/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-21829

(2)

òXäòøaàaòXè%î9æù6øî9é–úoûBüÁé\âäãøeè†õè[ýþ1î%ïyàeäãò

ÿ

! "

#%$ &"

'(*)&*+ (,.-/&0 123 !54

6789:-

<;

'=>@?BA@CD>E$GFHI/J71:K9-/ -/.6LNM

'DO)/&OP;3MQ

8R$7: 1ST-

8*-

&3UP1

V &5DWX

/8

! "

#%$ &"

'(*)&*+ (,.-/&0 123 !54

Y(Z7[\B]^`_<\<a9bc`d<cfehgic`jk<lnmoepk<qsrtvu+qsw`dehjxjyz k{|gid<cD}nw`gpqswD~@d++jBepxX{|lsk€c x|kq{|gid<cEqNdxv{sx

k<l|gpx‚gic`ƒ„gic3{…w`j†w`j+ƒ<gic`ƒ‡d<ˆ‰ dl‚{|ˆŠd<eighdxngpxR+gpx…qNl…jN{|gi‹Bj3x|jB~@gŒtgi~‰fehgpqNgh{|eiŽ=zfx‚gic`ƒ‡wfghƒw3d<l|t

+jlqBd<~@‰fk<qN{†fcfgŒ{…jD+gh‘jl|jcfqNjEx|qswfjB~@jx’.“”c•‰fk<l‚{…giqBz`eik<l–P{|wfjDqNd~@‰fk<qN{x|qswfjB~@jx†d€ˆ

— giƒk<eok<l|j„ƒjBc`jl…k<ehgi‹Bj5’.˜™w`j‡cšz`~@jBl…giqk€eol|jx‚z`eh{…xk€l…j‡qNd~‰ k€l…j›{|dœx‚{…k<cf`k<l…fcfgŒ{…j

+gh‘/jBl…jBcfqBjžx|qsw`j~@jx’7“{n{…z`l…cfx™d<z+{n{…wfk{n{…w`jžqNd~@‰fk<qN{nx…qsw`j~jx™wfkŸ<j ŸjBl…Ž‡x|k€{|gpxvˆŠŽšgic`ƒ

x‚{…k<¡`gheigh{vŽk€c †c`dc+tdx…qNgiehepk{…d<l…ŽQ‰fl|d‰ jl‚{…ghjx:k€cf@k€l…jXƒ<jBcfjBlsk€eiehŽ¢~@d<l…j™jN£‡qNgijBc{:{…wfk€c{|wfj

qBd<cfx|gi`jBl…jEqNepk<x…x‚gpqBk<e/x…qsw`j~jxB’

¤.¥¦5§†¨ ]©P[aGªR‰+{|gid<c!‰`l|gpqNgic`ƒ –:{|lsk€cfx…k<qN{|gid<c«qBdx‚{…x–:‰fk€lsk€¡d<eigpq‡jyzfk€{|gid<cfx–qBd<~@‰fk<qN{

fcfgŒ{…jž+gŒ‘/jBl…jBc qNjž+gpx|qBl|jB{|gi‹k€{|gid<cfx’

YD_¬­ ¨š§„®O¥ ©7¯ ¥š°•¥ ­/\[a˜™w`jXk€z+{…w`d<lsxPkqsršc`d}nehj+ƒ<j:‰ k€l|{|gpk€ešx‚z`‰f‰ dl‚{PˆŠl|d~6{…w`j™˜n± —

² l|d<³vjq{´|bRx‚Žš~@‰+{|d<{|gpq=±.jN{…w`d+`xghc¶µ ghcfjN{|gpq=˜™w`jBdl|Ž+·f–:ƒ<lsk€c{@¸ — mot¹±3mXºTtv»X˜n¼š½t

¾f¿À ½+’Á˜™w`jEfl…x‚{(k€c •{…w`ghls¶k€z+{…w`d<l‡k<l|j=‰fk<l‚{…ehŽ!x‚z`‰f‰ dl‚{…j¡šŽÂ{|w`jœÃQjl|w k€lsštÄTjx…x

² l|dƒ<lsk€~Åd€ˆ7{|wfj†ÆTjz+{…x…qsw`j†¹fd<lsx|qswšz`cfƒx|ƒ<jB~@jghcfx…qswfk€ˆÇ{–šƒ<lsk€c{9ÈÉËÊ À ¼šÌÊ`–`k<cfE¡šŽ‡{|wfj

bR¹7¹ ² l|d<³vjq{d<ˆ/{|w`jQÉ9c`giŸ<jl…x|gŒ{vŽd€ˆµ d<c xv{sk€c`‹’7˜™w`j9fl…x‚{ok<z+{|w`dlokqsrcfd}nehj+ƒ<jx:x‚z`‰`t

‰d<l|{PˆŠl…d<~{…w`jn»ojc{|jBl7d€ˆ¹Pgicfk<cfqNjXk€cf¸qNdc`d<~@jB{|l…giqxd€ˆ+{…w`j™É9c`giŸ<jl…x|gŒ{vŽ d<ˆ µ d<cfx‚{…k<c`‹

k<cf.jNÍ+‰`l…jx…x|jx9wfgixT{|wfk<c`r+x9{…d({…w`j±3“

²

“”cfx‚{|gh{|z+{…jd<ˆ{|wfjÉ9cfghŸjBlsx‚gh{vŽ

²

’/u+k<¡fk{…ghjlTgic

˜Pd<z`eid<z x‚j–}nw`jl|jRk¢‰fk€l|{d<ˆ5{|w`gpxl…jx|jk<l…qsw@wfkx¡jBjc(qBk<l|l…ghjd<z+{–ˆŠdl{…w`jTw`dx‚‰fgŒ{sk€eigŒ{vŽ’

Î ÏÐÑ7Ò:ӇÔ3Õ=ÖXÑ:×NÓÐ

¹7gicfk€cfqBgik<e+jBl…giŸk€{|giŸ<jxB–+gicD‰ k€l|{|gpqNz`epk€lnd‰+{|gid<cfx–+¡jqBk<~@jQŸjBl…Ž„‰ d‰`z`epk€l™fc k€cfqBgik<eqNd<c`t

{…l…kq{…xgic{|w`j™epk<x‚{ˆŠjB}¶+jqk<+jxB’ªR‰+{|gid<c x:qBk€c¡jXz x‚j5–€ˆŠdl:ghcfx‚{…k<cfqNj–{|d w`j+ƒ<jnkx|x|jN{sx

k<cf(‰d<l|{‚ˆŠdehgidxXgicDdl…`jBlX{|d„qNdc{|l…d<e{|w`j l…gix|r‡+z`jQ{…d~@dŸjB~@jBc{…xXgic({…w`jžx|wfk€l…jR‰fl|gpqNj’

Ø

j¢l|jqBk<ehe{|w k{9k@¸z`l…d<‰jk<c=»Xk€eieÙ

²

z+{sÚod‰+{|gid<cE‰`l…dŸgp+jx{…w`j l|giƒ<w{™{|d@¡`zfŽœÙÇ{…d@x|jBeieÇÚ

kž`Í+j‡cšz`~¡ jl™d€ˆÛk<x…x|jN{…xXk{X{|wfjT`Í+j‡jNÍ+jl…qBgix|j9‰`l…giqBjQÜÝk{o{|w`jQjNÍ+‰`gil|Ž{…gh~@j ÞNß@à

¿¾á

’

(3)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

“”c›k<c3gi`jk€eigi‹Bj3fcfk<cfqNgpk€e7~„k€l…r<jN{T{|w`j‰`l…giqBjd<ˆk(¸zfl|d‰ jk€c3d‰+{|gid<cœqk€c3¡jd<¡+t

{…k<ghc`j!kx{|w`j.x|d<eiz+{|gid<c!d€ˆT{…w`j.qNjehj¡`l…k€{|j•mXeikqsrtvu+qsw`dehjx†jyzfk{…ghdc6àÀ – âöá’!˜™wfgix

jyzfk€{|gid<c.k€epx‚d‰`l…dŸšgi+jx™kwfj+ƒghc`ƒ„‰d<l|{‚ˆŠdehgid{|wfk€{9l|j‰`ehgpqBk€{|jxX{|w`jžqBd<c{|gic`ƒ<jc{nqBeik<gh~=’

Ä9d}XjBŸjBl–{…w`j™moepk<qsrtvu+qsw`dehjxjyzfk€{|gid<cwfk<xP¡ jjBc@+jBl…giŸ<jžz`cf+jlyz`gh{|jXl…jx‚{|l…giqN{|giŸ<jokxvt

x‚zf~‰`{|gid<cfxQÙLjŠdlogicfx‚{…k€c qNj<–ˆŠl…giqN{|gid<c`eijx…xB–eigiyz`gp5–+qNd~‰fehjB{|jT~„k<l|rjN{…xsځ’7“”c‡l…jqBjBc{oŽjk€lsx–

÷ø€÷fùûúÇ÷/ü…ý€þ

moepk<qsrtvu+qsw`dehjxjyz k{|gid<c x™wfkŸ<jQ¡ jjBc.+jl|giŸ<j(gic=d<ls+jBlX{|d@~@d++jBe

ÿ

{…l…k<cfx…k<q{…ghdcDqBdx‚{…xnk€l…gpx‚gic`ƒghcE{|wfj¢w`j+ƒghcfƒd<ˆ7‰d<l|{‚ˆŠdehgidx à¿ –`–f¼ á–

ÿ

ˆŠjj+¡ k<qsr‡jB‘jq{…xT+z`j {|d@epk€l…ƒ<jR{|lsk<+jl…x¢à¿â – ¿ Ê+– ¿ö – ¿ +– â `–`Ê âá–`k€c 

ÿ

gicfqBd<~@‰`eijN{|j¢~„k<l|rjN{…x à

â<âá

’

˜™w`j„+jBl…giŸ<j3{|gi~@jNt”qNd<c{…ghcšz`dzfxQ~d++jeix¢k<l|j@yzfk<x|ghteighcfjk€lQd<lQˆŠz`eieiŽœc`dc`ehgic`jk€lQ‰fkt

l…k<¡ dehgpq(`gŒ‘/zfx|ghdc+tqBd<cšŸ<jq{…ghdcjyzfk{…ghdcfxB’›“”cÂ{|wfgix‰fk€‰jBl}Xj(k€l…j‡gic{…jBl…jx‚{|jÂghc•{|w`j

d<‰+{…ghdc„‰`l…giqBghc`ƒ†}ngŒ{…w@{|lsk€cfx…k<qN{|gid<c„qNdxv{sxB’7“”c ¿ ¼š½Ê`–modŽšeij9k€cf:dl…x‚{Rà á +jBl…ghŸjˆŠl…d<~

k¡`ghcfd<~@gik<e/~d++jek€cEd<‰+{…ghdcD‰fl|gpqNjR{|wfk€{™{…k€rjxXgic{…dkqBqBd<z`c{X{|lsk€cfx…k<qN{|gid<cDqBdx‚{…x™k€c 

{|wfk€{ngix™jyz k€e {|d„kmXepk<qsrtvu+qsw`d<eijx‰`l…giqBjR¡`z`{™}ngh{|w=k~d++ghfj(Ÿd<epk{|gieigŒ{vŽ@d€ˆÛ{|w`jQˆŠd<l…~

ß Ù

¿ Ú

ß

}ngŒ{…w ¿ ’3ÄTjBl…j<–

gpx¢{|wfj(‰`l…d<‰d<l|{|gid<cfk<e:{|lsk€c x|kq{|gid<cÂqBdx‚{– {…w`j‡{…l…k<cfx…k<q{…ghdc

‰ jl|gid+5–nk€cf

ß.gix@{…w`j.d<l…giƒ<gicfk€enŸd<epk{|gieigŒ{vŽ«qBd<cfx‚{…k<c{’jeik<cfàâ Ê á qNd~@‰`z+{|j!{|w`j

qNdcfxv{sk€c{ Ùâ Ú ’µ zfx|z`drk(à¿ ¼ á {|w`jcDx|w`d}oj{|wfk€{o{…w`jE´‚d<‰`{|gi~@k<ei· +j‰ jcf`x

d<cE{|wfj¢l|gpx‚rDxv{…l|zfqN{|z`l…j d€ˆP{|w`j¢~„k€l…r<jB{’

b9c`d<{|w`jl9k€‰`‰fl|dk<qsw(gpxX{|wfj¢~@k€Í+gh~@gi‹k{…ghdcDd<ˆP{…w`j¢z+{…gheigŒ{vŽ(ˆŠz`cfqN{|gid<c’¹fd<lngicfxv{sk€cfqBj<–

ÆRkŸšgpx jN{žk€e’༠á

qNd~@‰`z+{|j{…w`j„d<‰+{…ghdc›‰fl|gpqNj@k<xQ{…w`j„x‚dehz+{…ghdc›d<ˆXkEc`dc`ehgic`jk€l¢yzfk<x|gŒt

Ÿk€l…gik€{|gid<cfk<eRgic`jyzfk<ehgh{vŽ<’ ˜™w`gix=k<‰`‰`l…dk<qswÁw k<x=k€epx‚d!¡ jjBczfx|j gic à¿ –

¿À

–nʽ

á ’ “{

wfk<x‡{|w`j3`gix…k<+Ÿk<c{sk€ƒjD{…wfk{({|w`j3d‰+{|gid<cÁ‰fl|gpqNj3+j‰ jcf`x(d<c{|w`j›x‚‰jqBgik<eTqsw`d<gpqNj.d<ˆ

{|w`j(z+{|giehgh{vŽ3ˆŠzfcfq{…ghdc’E»od<c xv{sk€c{|gic`gp+jx¢k<cf"!5k€l…gh‰`wfd<‰d<z`eid<zà# á

d<¡+{sk€gic`j›z`c`giŸ<jl…x…k€e

¡ dz`cf`x9ghcf`jB‰jBcf+jc{Td€ˆP{|w`j¢z+{…gheigh{vŽ‡ˆŠz`cfqN{|gid<c’

² k€l%$k<xXk€cf=bTŸ<jBeiepk€c`j`kDàâ<Àá +jBl…giŸ<j({|wfj¢~d++ghfjDŸd<epk{…gheigŒ{vŽ

&

ß Ù ¿'( x‚giƒ<cÙ*),+-+/Ú‚Ú Ù ¿ Ú

ˆŠl|d~ k¡`gic`d<~@gpk€e5~@d++jBe5zfx|gic`ƒ†{…w`jžk€eiƒ<dl|gh{|wf~Åd€ˆ7mXjcfx|k<gi(jN{9k€e’àâá’Ä9jBl…j<–)Ýgpxo{|w`j

d<‰+{…ghdc›‰fl|gpqNj–/.Á{…w`j@‰`l…giqBjd<ˆ{…w`j@z`cf+jl|eiŽšghc`ƒ.kx|x|jN{–k€cf"),+-+ç+jc`d€{…jxQ{…w`j„x‚jqNd<c 

+jBl…giŸk€{|giŸ<jnd<ˆ0)%}ngh{|w„l…jx|‰jq{:{…d1.ÙÇ{|wfj32àk<~@~@k4ÚN’“”c‰ k€l|{|gpqNz`epk€l–{|wfj9d<‰+{…ghdc@‰`l|gpqNj

+dšjxncfd€{nc`jBjD{|d„¡jžqNd<cšŸjNÍ5’

b~dl|j qNd~@‰`ehjB͇~@dš`jBe5wfkxo¡jBjc=‰`l|d‰ dx‚j„¡šŽ(m™k€l…ehjxok<cf=ušd<c`jl¢à

¿á

’:“”cD{|w`jghl

~@dš`jBe5{…w`j¢c`d<cfehgic`jk<lnŸ<deik€{|giehgh{vŽ„l|jk<`x

&5

ß76 ¿ 98;:jNÍ+‰ÛÙ=<ÙOÞ ß?> ÞfڂÚ@.AB),+-+CEDA Ùâ Ú

}nw`jBl…jF<„gixn{|w`j l…gix|rt*ˆŠl…jBjQghc{…jBl…jx‚{™lsk{…j<–+Þ ß {…w`j¢~„k{|zfl|gh{vŽ<–`k<cf@

G0H –`}ngh{|wD{|w`j

l|gpx|r(kŸ<jl…x|ghdc„ˆOk<qN{|dl

G

k€cf({|w`j cšz`~¡ jl

H

d<ˆPd‰+{|gid<cfxX{…d@¡ jžx|d<ep5’:˜™w`jQˆŠz`cfqN{|gid<c

8

gixn{|w`jžx|d<eiz+{|gid<cEd€ˆ7{|wfj¢c`d<c`eigic`jk<lXgic`gh{|gpk€ehtŸk<ehz`j¢‰fl|d¡`ehj~

8JIÙ Ú 8 Ù Ú ¿

âK F8

Ù Ú > ML

¾ 8 Ù¾ Ú

¾N ÙOÊÚ

˜™w`j@k€¡dŸ<j†~@d++jeqk€c3¡j@+jBl…ghŸj=ˆŠl…d<~ kEqNdc{…l|deۉ`l…d<¡fehj~ z x‚gic`ƒ=k€c3jB͚‰d<cfjBc{|gpk€e

z+{|gieigŒ{vŽDˆŠz`cfqN{|gid<c=ghcE{…w`j¢ehgi~@gŒ{PO

¿Q G0HSR

¾ – R ¾

x|zfqswE{|w k{T@

U O+’

(4)

V

êŒìW

ð

éEX ò éJY ðQZ3[]\

Y_^A`EYaW ò Z ò `cb

ð é \ ë ð ë õêië ò \ é ã õ\

Yadfe*g%YaW

ð

õÇò ` òh è \

^*ê ð ë

“”c¢{…w`jX~@k€{|w`j~„k{|gpqBk<eeigh{|jBlsk{…z`l…j<–d<c`eiŽ knˆŠjB}«l|jx‚z`eh{…xPqBk<cž¡ joˆŠd<zfcfždc {|w`jXcšz`~@jBl|t

gpqBk<e+gix…qNl…jN{…gh‹k{…ghdcœd€ˆnmXeikqsrt”u+qswfd<eijx9jyzfk{…ghdcfxB’„˜™w`j@cšz`~@jBl…gpqBk€e:k<‰`‰`l…dk<qswfjxRŸk<l|Ž

ˆŠl…d<~ ¡`ghcfd<~@gik<ePk<‰`‰`l…d͚gi~„k{…ghdcfx ÙOx|jBj<–fˆŠd<l9gicfx‚{…k<cfqNj‡àâ+¿á ˆŠd<lTbT~@jBl…giqk€c=d<‰+{…ghdcfxngic3k

x‚{|d+qswfkxv{…giqXˆŠlsk€~@jB}Xd<l…r`ځ–±.d<c{|jBtv»Xk<l|eid ~@jN{|wfdšfxTà

‾á

–< c`gŒ{…jNtjBeijB~@jBc{™+gix…qNl…jN{…gh‹k{…ghdcfx

à

¿<¿

– â ½ á

–:k€c çfc`gh{|jBt+gh‘/jBl…jBcfqBj(k€‰`‰fl|dÍ+gi~@k€{|gid<cfx(à

¿¾áji

w`d}XjBŸjBl–Û~„k€gic`eiŽ›ˆŠdl

ùûúÇ÷/üsýþ

mXepk<qsrt”u`qsw`d<eijxojyzfk€{|gid<cfx’

˜™w`j„cšz`~@jl|gpqBk<e+gix…qNl…jN{…gh‹k{…ghdcd<ˆo{…w`j‡mXeikqsrt”u+qswfd<eijxQjyzfk€{|gid<cfx¢}ngh{|w{…w`j

÷ø€÷k

ùûúÇ÷/üsýþ

Ÿ<d<epk{…gheigh{|gijx Ù ¿ Úok<cfÙâ Úowfkx™¡ jjBcE‰ jl‚ˆŠdl|~@j(zfx|gic`ƒ@jNÍ+‰`eigiqBgŒ{™fc`gh{|j¢+gh‘/jBl…jBcfqBj

x…qsw`jB~@jxà¿ – â<Àá’PÄTd}ojŸ<jBl–jNÍ+‰`eigiqBgŒ{x|qsw`j~@jxw kŸ<j:{…w`jo+gpx…k<+Ÿk€c{sk€ƒ<j{…wfk{l…jx‚{|l…giqN{|giŸ<j

qBd<cf+gh{|gid<c x9d<c={…w`j`gix…qNl…jN{…gh‹k{|gid<c=‰fk<l…k<~@jN{|jl…x†ÙLjŠd<lRghc xv{sk€cfqBj<– {|w`jžlsk{…ghd(d€ˆ{|w`j†{|gi~@j

k<cf@x‚‰fkqNjnx‚{|jB‰Úk€l…jXcfjBj`j{|d d¡+{…k<ghc„x‚{…k€¡fehj–qNdcšŸ<jBl…ƒ<jc{7x…qsw`jB~@jxTàÊ Àᒱ.dl|jdŸ<jBl–

{…w`jžqNdcŸjBl…ƒ<jcfqNjRd<ls+jBl™gpxnd<c`eiŽ‡dc`j ghcE{|gi~@jžk€cfD{v}Xd@ghc=x‚‰ k<qNj’

“”c{|w`gpx‰fk<‰ jl:}ojn+gpx|qBl|jB{|gi‹BjX{|wfj™mXepk<qsrtvu+qsw`d<eijxPjyzfk€{|gid<c}ngŒ{…wcfd<c`eighcfjk€l:Ÿd<epk{…gheht

gh{vŽ«Ù

â

ځ’‡ªRz`l¢~„k€gic›ƒdk<ePgpxQ{|d=d<¡`{…k€gic

ü=lnmNúOüB÷o k<cf&p

þ|üqmBúsrNü

x|qsw`j~@jx–g*’j<’5}Xj}ngix|w

{…d+jl|giŸ<j(czf~jl|gpqBk<eXx…qsw`j~jx†}nwfdx|j(d<ls+jlgpxx|z`‰jBl…ghdl†{…d›x‚{…k<cf`k€ls•x…qsw`jB~@jxžˆŠdl

x|jqBd<cfštd<ls+jlDjyz k{|gid<c xÂيeighrjç{|w`jÂjB͚‰fehgpqNgh{œ¸z`eijBl–R{|w`j•x|jB~@ghtgi~@‰`ehgpqNgh{œ¸zfehjl–Q{|wfj

»olsk€c`rtut9gpqNdeix|d<c3k<cf.{…w`jvjk<‰+t”¹`l…d<ƒ(ÆRz¹fd<l|{‚t”¹`lsk€c`rjBex|qswfjB~@jxsÚ9k<cf3}nw`dx‚j@qNd~t

‰`z`{|gic`ƒ.{|gi~@j(gpxžqNd~@‰fk€lsk€¡`eij{|d.{…wfk{d€ˆ9qNepk<x…x‚gpqBk<ex|qswfjB~@jx’=u+ghcfqBj„{|wfj‡jyzfk€{|gid<cÂgpx

d<ˆx|jqBd<cfEd<ls+jl–+zfx|zfk<eheiŽDk{…w`l|jjNt‰ dghc{Tk<‰`‰`l…d͚gi~„k{…ghdc(gpxnzfx|jç࿾á k<cf=k„x|qswfjB~@j

}nw`gpqsw¶gix.Ù*qNdcfx‚gpx‚{|jBc{Úd€ˆRdl…`jBl„d<cfjEgic{|gi~@j3k€cf«{v}odçgicÁx‚‰fkqNj=gpx@d<¡+{sk€gic`j%يjB͚t

qBjB‰+{o{|w`jTÛjk<‰+t¹fl|dƒ¢ÆTz(¹fd<l|{‚t”¹`lsk€c`rjBefx…qsw`j~jwx|jBjT¡ jehd}TÚN’7˜Pdžd<¡`{…k€gic„w`giƒ<w`jldl…+jl

x…qsw`jB~@jxžÙŠd<ˆ7dl…+jl â ghc={…gh~@j†k<cf ö ghc.x|‰fk<qBjÚN–`d<c`j¢‰dx…x|gh¡`gieigŒ{vŽDgix™{…d‡zfx|j¢~@d<l…j x|‰fk€t

{…gik<eP‰d<gic{…xQ¡`z+{Q{|w`gpxQqNd<~@‰`eigpqBk{…jxT{|w`j@k€‰f‰`l|dÍ+gi~„k{|gid<c=d<ˆ{|w`j¡ dz`cf`k<l|Ž.qBd<cf+gh{|gid<c x

k<cfçl|jx‚zfeŒ{sxžgic«k3+gpx|qBl|jB{|gi‹k€{|gid<cÂ~„k{…l|gh͛}ngh{|w•epk€l…ƒ<jlž¡fk<cfç}ngi+{|w’3“”c•{|w`gpx†‰ k€‰jBl–

}Xj‰fl|jx‚jc{¢k€ck€eh{|jl|cfk€{|giŸ<jzfx‚gic`ƒEw`giƒ<w+td<ls+jBl qNd~@‰fk<qN{ x…qsw`jB~@jxR}nwfgiqsw›c`jBj3{…w`l|jj

‰d<gic{…xngicEx|‰fkqNj d<c`eiŽ<’

±.dl|jQ‰`l…jqBgix|jBeiŽ<–+}Xj¢xv{…zf+Ž‡{…w`j¢jyzfk€{|gid<c

)x

Ùy) +-+ Ú . ) ++

<f.z) + > <-)

¾ Ùö Ú

}nw`jl|j{…w`j„c`d<cfehgic`jk<l Ÿ<d<epk{…gheigh{vŽ

Ùy) +-+ ÚRgpx¢ƒ<giŸ<jcœ¡šŽ!Ù

â

ÚN’@˜™wfgix jyz k{|gid<cgpx¢x‚dehŸj

ˆŠdl9{|wfj†‰`l…giqBj{.n|

¾

d€ˆ{|w`jz`cf+jl|eiŽšghcfƒ(k<x…x‚jB{Rk€c E{|gi~jÞNß}| Þ~|

¾

– g’j’f¡fkqsrš}Xk<l…

gicE{|gi~j’˜™w`j {|jl|~@gicfk€eqBd<cf`gŒ{…ghdcEgpx

)‡Ùy.A‚ÞNßÚ

)fßÙ*.Ú€.9|

¾N

ÙÀ Ú

˜™w`jXjyzfk€{|gid<cgix:+jl|giŸ<j¢gicEà¿á ˆŠd<l:¸z`l|d‰ jk€c»Xk<ehešd‰+{|gid<cfx–g’j’Q)fߚÙy.Ú

~„kÍ5Ù¾ q. >

܆ÚN–`}nw`jl|j¢Ü gixX{…w`j¢jNÍ+jl…qBgix|j ‰`l|gpqNj’:˜™w`j2¡d<zfcf`k€l…Ž‚4`qNdcf+gh{|gid<cfxnk<l|j¢k<xXˆŠdeheid}9x_ƒ

)(Ù¾ ‚ÞfÚ

¾ ÞN߄|!ޅ| ¾ )‡Ù*.†‚ÞfÚ ‡. > ܄ˆ‰EŠ

xŒ‹‚xaaŽ

Ùy. R Ú N Ù=Ú

˜™w`j¢epk<x‚{9qNdcf+gh{|gid<cEwfk<xX{…d„¡ j¢z`c +jBlsxv{…dšdšDgicD{…w`jžx|jBcfx|j

eigh~

+-‘„’

)‡Ù*.†‚ÞfÚ

. > Ü1ˆ

‰_Š

xU‹%xaŽ ¿

z`cfgŒˆŠdl|~@eiŽ‡ˆŠd<lnÞBß3|!ޅ| ¾ ’

Ø

jk<‰`‰`eiŽD{…w`j@qNd<~@‰fkq{ x|qsw`j~@jx

—

Ê<bÝk€cf

—

Ê<m%+jBl…giŸ<j3¡Ž

—

giƒk<eoà

â ¼ á {…d‡{|wfj

gic`gh{|gpk€eht¡d<z`c `k€l…ŽçŸk<ehzfj(‰`l…d<¡`eijB~ Ùö

ڍtÙ=ځ’˜™w`jDc`d<c`eigic`jk<l|gh{vŽçgix†{|l…jk€{|j•jNÍ+‰`eigiqBgŒ{…ehŽ

g’j’h–š{…w`jQfcfk<e/x…qsw`j~jQgpxXx|jB~@ghtgi~@‰`eigiqBgŒ{’˜™w`jRcšz`~@jl|gpqBk<ejBÍ+‰ jl|gi~@jBc{…x™x|w`d}¶{|wfk€{–`kx

jB͚‰jqN{|j5–f{…w`j{“ jBl…l|dl9gixT~†zfqsw3x|~„k€eiehjl–fˆŠd<lT`Í+j=‰fk<l…k<~jB{|jl…x–+{|w k€c.{…w`j†jl|l…d<l™ˆŠdl

(5)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

{|w`j„x‚{…k<cf`k€ls3x…qsw`jB~@jxÙOjNÍ+‰`eigiqBgŒ{¢¸zfehjl–x|jB~@ghtgi~@‰`ehgpqNgh{¢¸z`eijBl–”jk€‰+t”¹`l…d<ƒ(ÆRz›¹fd<l|{‚t

¹`lsk€c`rjBeÇځ’™˜™w`j» ² ÉË{|gi~j†gixndc`eiŽEx|ehgiƒ<w{…ehŽDw`ghƒw`jBl9ˆŠd<ln{…w`j†qNd~@‰fk<qN{Tx|qsw`j~@jx–+ˆŠd<lTk

`Í+jEcšz`~†¡jBlnd<ˆ7ƒl|gpDc`d++jx’

“”c„{|wfj9c`d<cfehgic`jk<loqk<x|j<–€{…w`jTgic+ˆŠdl|~„k{…ghdc„d<c@{…w`jRx‚{…k<¡`gheigh{vŽ¡jqBd<~@jxŸ<jBl…Ž†eigi~gh{|j5’

“”c‡k€cšŽ†qk<x|j<–€eighcfjk€lx‚{…k<¡`giehgh{vŽ†gpxkQc`jqBjx…x|k<l|ŽqNd<c +gŒ{…ghdc†ˆŠdl:{|wfj9xv{sk€¡`giehgh{vŽ†d<ˆcfd<c`eighcfjk€l

‰`l…d<¡`eijB~„xQ¡`z+{¢gh{Qgpx qNjBl|{…k<ghcfehŽ=c`d<{ x|z+£‡qNgijBc{’ ² k€l|{|gpqNz`epk€lR~jB{|w`d+`xQ}XjBl…j+jŸ<jehd‰ j

ˆŠd<l c`dc`ehgic`jk€l ‰`l…d<¡`eijB~„x’†u{|jB{‚{|jlžk€cf›µ jBeiehjl¢xv{…zf+gijœ{|w`jc`d<cfehgic`jk<l¢xv{sk€¡`giehgh{vŽ3k€c 

ehd+qk€e/xv{sk€¡`giehgh{vŽ.à

¿

#+–šÊ

öá

–š}nw`giqsw(wfkŸ<j9¡jBjc(zfx|j(x‚zfqqNjx|x‚ˆŠz`eiehŽˆŠd<loc`d<cfehgic`jk<lodl…`ghcfk<l|Ž

+gh‘jl|jc{…gik<ePjyzfk{…ghdcfxB’9mXjc+ty•z•àÊ á

‰`l…d<‰dx|j=k„ƒjBc`jl…k<ehgpx‚j=xv{sk€¡`giehgh{vŽEd€ˆ+gh‘jl|jcfqNj

x|qswfjB~@jx7}nw`gpqsw†wfkxÛ¡jBjck<‰`‰`eighjž}ngp+jBeiŽ {|dQcšz`~@jBl…giqk€ešx‚dehz`{|gid<cfxPd€ˆf~„k<cŽ¢cfd<c`eighcfjk€l

‰fk€l|{|gpk€eP+gŒ‘/jBl…jBc{…gik<e5jyzfk{…ghdcfx’:“”cD{…w`gpxn‰fk€‰jBlT}oj¢+d„c`d<{9zfx|j {|w`gpx™{|jqsw`c`gpyz`j¢}nw`gpqsw

qBk<c„¡ j9ŸjBl…Ž†qNd~@‰`ehgpqBk€{|j5’P“”c xv{…jk<–<}Xj9ƒ<giŸ<j™l…jx|z`eŒ{sxˆŠd<l{|w`j9eigic`jk<lqBkx‚jžÙy@

¾ Úk€c 

Ÿk€eigi`k€{|j {…w`jB~ ghcE{|wfj¢c`d<c`eigic`jk<l™qk<x|j ¡Ž(czf~jl|gpqBk<exv{…zf+gijx’

¹`z`l|{|wfjBl…~dl|j–5}oj@qNdcfx‚{|l…zfq{žkDc`jB} {|w`l…jBjBt‰d<gic{¢qBd<~@‰fkq{¢x|qsw`j~@j — ʚ»R–5}nw`gpqsw

ƒ<jc`jBlsk€eigh‹jx{…w`jRx…qsw`j~j — Ê<m!d€ˆ — giƒk<e*–k€cf„x|w`d}{|w k{{…w`gpxox…qsw`jB~@j9gpxzfcfqNdcf+gh{|gid<c+t

k€eiehŽ•x‚{…k<¡`eijDk€c Âc`d<c`tdx|qBgheieik€{|dl|Ž’›˜™w`jExv{…zf+ŽÂd€ˆT{|w`jE‰`l…d<‰jBl|{|gijxžd<ˆ9{|wfjEx…qsw`jB~@jDgix

¡fk<x|j(z`‰d<c=k¢{|wfd<l…d<z`ƒw(¹`dz`l|gijBl™k€c k€eiŽšx|gpxd<ˆ{|w`j†»Xk<zfqswšŽ‰fl|d¡`ehj~Åk<x…x‚d+qNgpk{…j„}ngh{|w

Ùö ÚN’ Ø j¢l|jx‚dl‚{X{|d„kehd+qk€e5k€cfk<ehŽ+x|gixX}ngh{|wEˆŠl…d<‹Bjc(Ÿk€eiz`jx™d€ˆÛ{…w`j¢c`dc`ehgic`jk€lnqNdšjN£‡qBghjc{

{|d„~„k€rjQ{|w`j ˆŠdl|~z`eik€{|gid<cDehgic`jk<l’

¹7ghcfk<eheiŽ<–™}Xj.‰`l…jx|jBc{‡kçczf~jl|gpqBk<enjNÍ`k€~@‰`eij.d€ˆ¢k¸zfl|d‰ jk€c »Xk€eiend<‰+{…ghdc}ngh{|w

+gh‘jl|jc{Ql|gpx|rDkŸ<jl…x|gid<c=qNd<c xv{sk€c{…x?@’nbTxTjNÍ+‰jq{…j5– {|wfjžd<‰+{…ghdc.‰`l…giqBjž}ngŒ{…w.‰ dx‚gh{|giŸ<j

@œgpxžwfghƒw`jBl¢{…wfk€c{…w`j(mXeikqsrt”u+qswfd<eijx ‰`l…giqBj3Ù=@

¾

ځ’=˜™w`j‡+gh‘jl|jcfqNj‡¡jN{v}XjBjc{|w`j

d<‰+{…ghdc ‰`l…giqBj.ˆŠd<l…@;– ¾ k<cf¶{|wfj›mXeikqsrt”u+qswfd<eijx@‰`l…giqBj3gpx(~„kÍ+gi~„k€eÙOghc6k€¡fx|d<eiz+{|j

Ÿk€eiz`jÚQc`jk€lQ{|wfj@x‚{|l…gir<j‰`l|gpqNj„Ü„–Û¡`z+{žqswfk€cfƒ<jxQ}ngh{|w›{|gi~@j<’@“”c›‰ k€l|{|gpqNz`epk€l–/ˆOk€lQˆŠl…d<~

{|w`jD~„k{|zfl|gh{vŽ3{…w`j(+gh‘jl|jcfqNj(gix†~@k€Í+gh~„k€eok{kx|x|jN{ž‰`l…gpqNjx†x‚~„k€eieijBl¢{…wfk€cç{…w`j(x‚{|l…ghrj

‰`l…giqBj<’

˜™w`j‡‰fk<‰ jlžgpx¢d<l…ƒk<c`gh‹jk<x ˆŠdeheid}9x’(“”c•x|jq{…ghdc â {|w`j‡gic`gh{|gpk€eht¡d<zfcf`k€l…ŽœŸk<ehz`j

‰`l…d<¡`eijB~ Ùâ ڍtÙ=ڄgpxEl|jBˆŠd<l…~†z`epk{…j¶zfx|ghcfƒk<c jNÍ+‰d<c`jc{…gik<eT{…l…k<cfx‚ˆŠd<l…~@k€{|gid<c’Åušd~j

qNepk<x…x‚gpqBk<eXfc`gh{|j3+gh‘jl|jcfqNj3x…qsw`j~jx@k€l…jE‰fl|jx‚jc{…j!k€cf«{|w`jghlDxv{sk€¡`giehgh{vŽ•‰`l…d<‰jBl|{|gijx

k€l…jl…jqk€eiehj.gicx|jqN{|gid<c›Êf’†b c`jB}ÝqNd<~@‰fkq{¢x|qswfjB~@jgpxQ‰`l…d<‰dx|jœk<cfœk<cfk€eiŽš‹Bj3gic

x‚jq{…ghdc

ö

’Âušjq{|gid<c

À

gpx`jBŸ<d<{|jç{|dçk3cšz`~@jBl…gpqBk€eXqNd~‰ k€l…gix|d<cÂd<ˆ9{|w`j=x…qsw`jB~@jxgic

{|jl|~„xRd€ˆxv{sk€¡`gieigŒ{vŽ–qBd<cšŸ<jl|ƒjBcfqBj k<cf=jN£‡qBghjcfqNŽ’9¹Pgicfk<eheiŽ<–ghc›x‚jq{|gid<c&„{|w`j†x|d<eiz+{…ghdc

d€ˆ`{…w`j™c`d<cfehgic`jk<l7mXepk<qsrtvu+qsw`d<eijx‰`l|d¡`eijB~%gix7‰`l…jx|jBc{…j ˆŠdl`gŒ‘/jBl…jBc{7{|lsk€cfx…k<qN{|gid<cžqNdxv{

‰fk€lsk€~@jB{|jBlsx–šk<cf=x‚d~@j¢ÃQl…jBjr+xnk€l…jQqBd<~@‰`z+{…j5’

— ˜š™…› ÑÒAœžÐ…/ž€ÓÒzŸ ›9Ô¢¡=Ò:Ó}£¤›TŸ

“”cE{|w`gpx9x|jq{…ghdc=}oj l…jNˆŠd<l…~†zfeik€{|jR{|w`jž‰fl|d¡`ehj~ Ùö

ڍtÙ=Úz x‚gic`ƒ‡kŸk€l…gik<¡`ehjQ{…l…k<cfxvˆŠdl|~„k€t

{|gid<c’ç“”cGà

¿Ná

{…w`j(jB͚gpx‚{|jBc qNj(d<ˆRk3zfc`giyz`jEqNdc{…ghcšz`dzfxŸgpx…qNdx|gh{vŽx|d<eiz+{|gid<c¥) {|d›{|wfgix

‰`l…d<¡`eijB~ wfk<x™¡jBjc=x|w`d}nc’

˜Ûd!dŸ<jl…qBd<~@j3k‰dx…x‚gi¡`eij›+jƒ<jBcfjBlsk{|gid<c k€{….

¾ k<cf¶{|d!d<¡+{sk€gicGkˆŠdl|}™k€ls

‰fk€lsk€¡d<eigpqR‰fl|d¡`ehj~=–+}oj zfx|j {|w`j¢Ÿk<l|gpk€¡`eijR{|lsk€cfx‚ˆŠd<l…~„k{|gid<c x

¦

Ùy.Ú

eicc§

.

Ü&¨

يÞfÚ

¿

â

ß

يÞ

ß?>

ÞfÚª©

jNÍ+‰ÛÙ

> ¦ Ú )Ü N

¸yzfk€{|gid<cçÙ

ö

ÚXgpx™w`jl|j¡Ž‡{|lsk€cfx‚ˆŠd<l…~@jDghc{|d

©%« > Ù ¿ 98;:jNÍ+‰ÛÙy¬ Úa@-Ü(Ù­©‚®_® ©%®Ú¯CÚNÙ=©%®B® ©%®<Ú > ¬~©%® ¾ Ù½€Ú

(6)

êŒìW éEX ò éJY Y_^A`EYaW ò ò `cb é ë ë õêië ò é õ Yadfe*g%YaW õÇò ` òh è ^*ê ë

}ngh{|w

¦&°…±

¾}²

²¥³

ß Þ ß

ۉ

´¬

â <

ß N

˜™w`j¢‰fl|d¡`ehj~ gpxnqNd<~@‰`eijN{…jE¡šŽ„{|wfj ˆŠd<eiehd}ngic`ƒ@ghcfgŒ{…gik<eÛk€cfD¡d<z`cffk€l…Ž(qNdcf+gh{|gid<cfx_ƒ

©Ù

¦ ¾ Ú

© ßÙ

¦ Ú

~„kÍ5Ù¿ > jB͚‰Ù > ¦ Ú ¾ Ú

©Ù

¦ Ú ¾ Ù¦

Rµ>7 Ú

©Ù

¦ ڀ‡ ¿ Ù¦ R  Ú N

¶ · ×BÐ.×sц›¸ ×¹"›9ÒA›TÐ=ÖP›»º(Ö¼™…›TŸ ›T

“”c¶{…w`gixDx‚jq{…ghdc}Xj.l…jqBk<eheRxv{sk€cf`k<l…!fcfgŒ{…jœ+gh‘jl|jcfqNjœx…qsw`j~jx„k€cf{…w`jœqBd<~@‰fk<qN{

x…qsw`jB~@jx¢+jBl…ghŸj¡šŽ — giƒk<eTàâ ¼ á’(¹`dl¢{|w`j‡qBd<~@‰`z+{sk{…ghdc}Xj„l…jB‰`epk<qBj

±

¡šŽ«à>?½ ½ á

}ngh{|w ½ – ¾ ’.¹`d<lx‚gi~@‰`ehgpqNgh{vŽ<–}Xj‡qNdcfx‚gp+jl†k.z`c`ghˆŠd<l…~ ƒ<l…gi(¾

À¿

¦%Á„°

à>?½ ½ á ƒ

¦ Á 5ÂaÃ

–

ÂÄ

>?H

NÅNÆN

H"Ç qNdcfx‚gpx‚{|gic`ƒ¢d€ˆ

â H

›¿

ƒ<l…gp‰d<gic{…x–<}ngh{|w ½ H Ã

k<cf}ngŒ{…w

x|‰fk<qBjTx‚{|jB‰

Ã

k<cf@{|gi~@jQx‚{|j‰È/’AjB{PÉFÊ

Á

`jBc`d<{|j9{…w`j k€‰`‰`l…dÍ+gh~„k€{|j9x|d<eiz+{|gid<c‡d<ˆRÙ½<Ú:gic

¦ Á

k€{T{|wfjž{|gi~@j Ê

ÌË

È=k<cf.eijN{3É3Ê

Ù¯ÉFÊ

Á Ú ÍcÎ/

ÁÆÏ

’Rb9eiex|qsw`j~@jx9}XjqBd<cfx|gp+jBlRw`jBl…j

zfx|j¢{v}Xd@{|gi~@j†eijBŸ<jeix

i

jNÍ`qNj‰+{n{|w`j{jk<‰+t”¹`l…d<ƒ„ÆTz›¹`d<l|{‚t”¹`lsk€cfr<jBex…qsw`j~j†}nw`giqsw.z x‚jx

{…w`l|jj{…gh~@jneijBŸjBepxB’P“”cž{|wfj™x|‰fkqNjXŸk<l|gpk€¡fehj

¦

}ojXzfx|j™kQqNd~@‰fk<qN{x‚{|jcfqNgie+l|jyz`ghl…gic`ƒTdc`ehŽ

{…w`l|jjnqNd<c x‚jqNz+{…ghŸj‰d<gic{sxghc{|gi~@jnehjŸ<je

Ë

=¿ ’Û˜™w`jnx|qswfjB~@jxi jBÍ`qNjB‰`{P{…w`jPjk<‰+t”¹`l…d<ƒ

ÆRz.¹`d<l|{‚t”¹`lsk€cfr<jBe5x…qsw`jB~@j i qk€cE¡ j¢}nl…gŒ{|{|jcEgicD{|w`j ˆŠdeheid}nghcfƒˆŠdl|~…ƒ

Ê É Ê Î/

Ð Ê É Ê

Ù=#Ú

}ngh{|wE{|wfjž+gix…qNl…jN{…gh‹k{…ghdcD~„k{…l|gpqNjx

ÊEk€cf

Ð Ê

Ê à@ ‹

@

ß

@

á Ð Ê àÑ ‹

‹ Ñ

ß

Ñ

Ñ

á*N

˜™w`j~@k€{|l…gŒÍ

Ê gpx5{…l|gp+gpk€ƒd<cfk<e*–{|w`jl|jBˆŠd<l…j{|w`jol|jx‚zfeŒ{…ghc`ƒneigic`jk<lPx|Žšx‚{|j~„xÛqBk<c¢¡ jox‚dehŸj

ŸjBl…ŽjB£„qBghjc{|eiŽeigic`jk<l|eiŽ@ghc„{…gh~@jQzfx|ghc`ƒk†x|‰ jqNgpk€e ˆŠd<l…~Ýd<ˆ{|wfjQàk€z x|x|gik<c„jBeigh~@gicfk{…ghdc

ršc`d}nc=k<xX{…w`jž˜™w`d<~„kxnk€eiƒ<d<l…gh{|w`~ འᒠØ

jžx|z`‰`‰dx|jQ{|wfk€{

Ò

ÁÆÏ

‹ @ Á

Ò

ÁÆÏ

‹ Ñ Á ¿

}nw`gpqsw¢gpxPx…k{…gix‚fj ¡šŽ k€cšŽ qNd<c x‚gpxv{…jBc{Px…qsw`jB~@jk€ˆÇ{|jlÛc`dl|~„k€eigpx|k€{|gid<c d€ˆš{|wfjoqNdšjB£„qBghjc{…x’

˜™w`jRc`d<cfehgic`jk<l|gh{vŽgixoghc{…l|d++zfqBj„jBÍ+‰`ehgpqNgh{|eiŽ@ghcEk€eief{…w`jRx…qsw`j~jxB’“”c„{…w`j9ˆŠdeheid}ngic`ƒ

eijN{7ÓBÊ

Á

+jBc`d<{|jQ{|wfj¢jNÍ+‰`eigiqBgŒ{…ehŽE+gix…qNl…jN{…gh‹jDc`d<cfehgic`jk<lnŸ<deik€{|giehgh{vŽ‡qNdl|l…jqN{|gid<c”ƒ

Ó ÊÁ

8ÕÔ jNÍ+‰ÛÙy¬

Ë È

¦,Á Úa@-BÜÕÖ

É3Ê

Á‹ > â ÉFÊ

Á É3Ê

Á

Î]

ö Ã

É3Ê

Á

ÎÄ

> ÉFÊ

Á‹

â Ã ×PØ

N

ÙO¼Ú

Ø j9zfx|jn{|wfgixˆŠd<l…~Ýd<ˆ{…w`jT+gpx…qNl…jN{|gi‹k€{|gid<c@d€ˆ/{…w`jTx|jqNdcf@+jBl…giŸk€{|giŸ<jn¡jqk€zfx|j9gh{ƒ<giŸ<jxk

x|~@dd<{|w`jlk€‰`‰fl|dÍ+gi~@k€{|gid<cd€ˆ‚© ®B® ’ Ø gŒ{…w‡k¢xv{sk€cffk€lsqBjBc{|lsk€e`+gh‘/jBl…jBcfqBj™{|w`j9x…qsw`jB~@jx

¡jqBd<~@j›gicfx‚{…k<¡`ehjœˆŠd<l=x‚~„k<eheQŸk€eiz`jxEd€ˆ

Ã

z`cfehjx|x…ÈÁgpx(ŸjBl…ŽÁx‚~„k€eie’ ˜™w`j‰`l…d<¡`eijB~

eighjxžgicÂ{|wfj‡ghcfgŒ{…gik<eXqBd<cf+gh{|gid<cÙ©

ß

x|ghc qNj‡gh{gpxžc`d<{+gh‘jl|jc{|gpk€¡`eij(k{

¦ ¾ ’ Ø

j(zfx|j

x|‰`eighc`jXgic{…jBl…‰ deik€{|gid<c¢d<ˆfw`giƒ<wždl…+jl{…dRqBk<l|jBˆŠz`eiehŽ¢x|~dšd<{|w¢{…w`jogic`gh{|gpk€e+`k€{…k`–¡`z`{7dc`ehŽ gic

(7)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

−10 −8 −6 −4 −2 0 2 4 6 8 10

−2 0 2 4 6 8 10 12 14

Ψ (A)

A

¹7giƒ<z`l…j ¿ ƒušdehz+{…ghdc=d€ˆÛ{…w`jªQÆQ¸Ù*ÊÚN’

Ø jEzfx|jEkÆRd<l…~@k<cfšt² l…ghc qNjNtö t À — z`c`ƒjNt”µ z+{‚{skœx…qsw`jB~@jD{|dçx‚dehŸj„{|wfjEdl…`ghcfk<l|Ž

+gh‘jl|jc{…gik<e/jyzfk{…ghdc›Ù*ÊÚok€cfDkqBz`¡`gpqRx|‰`eighc`jQgic{…jBl…‰ deik€{|gid<c({|dd<¡+{sk€gic‡{|wfjRŸk€eiz`jxod<ˆ

8 ˆŠdl9k€l…¡`gh{|lsk€l…Ž@k<l|ƒz`~@jBc{…x¢Ù*qˆv’¹7giƒ<z`l…j ¿ ځ’

˜™w`j™¡ dz`cf`k<l|ŽžqBd<cf`gŒ{…ghdcfx7d<c{|w`j™eigh~@gh{|jƒl|gp†k<l|jo{|l…jk€{|jk<xPˆŠdeheid}9xB’7ÆTgil…giqsw`eijN{

qNdcf+gh{|gid<cfx9k<l|jQzfx|jEd<cE¡d€{|w=¡d<z`c `k€l…Ž‡‰d<gic{sxEƒ

É Ê

‹ Í ¾

k€cf

É Ê

ΔÍ

¿ > jBÍ+‰ÛÙ> ¦ Í > ¬ Ë È`Ú N

˜™w`jœepk{|{|jlEqBd<l…l|jx‚‰d<cffx{|d•{…w`j›kx‚Žš~@‰+{…d€{|gpq.Ÿk€eiz`j›d€ˆ {|w`j›jNÍ`k<qN{Ex|d<eiz+{|gid<cÁd<ˆ {|w`j

jyzfk€{|gid<cEˆŠd<l?@

¾ ’±3d<l…jR‰`l…jqBgix|jBeiŽ<–š{…w`jžx|d<eiz+{|gid<cEd€ˆnÙ½<ڙx|k€{|gpxvfjx ÙOx|jBj‡Ù=ڂڃ

©Ù

¦ Úz‡ ¿ > jB͚‰PÙ > ¦ > ¬ Ú k<x

¦

R N

b9‰`‰fl|dÍ+gi~@k€{|jehŽ–9}ojœwfkŸ<j&©Ù½ Ú&Ú ¿ > jNÍ+‰ÛÙ> ¦ Í > ¬ Ú@ˆŠdlDx|z+£‡qNgijBc{|eiŽ¶eik<l|ƒj

½ – ¾ ’7˜™w`jc`dc`ehgic`jk€lPqBd<l…l|jq{…ghdcRd<ˆ+{|w`jŸd<epk{…gheigŒ{vŽQgic‡Ùö ÚgixPk™ˆŠz`cfqN{|gid<cžd<ˆ+{|w`jox‚jqNd<c 

+jBl…giŸk€{|giŸ<j–`x‚d}oj k<x…x|z`~@jT{…wfk{n{…w`j ghcfÛ z`jBcfqBj d€ˆÛ{…w`j c`d<cfehgic`jk<l|gh{vŽ‡k{X{|wfj¢¡ dz`cf`k<l|Ž

qBk<c„¡ j9cfjBƒ<eijqN{|j†ˆŠdlepk€l…ƒ<j

½

’7˜™w`jTjBl…l|dlqBk€z x‚j¡šŽ¡ dz`cf`k<l|ŽqNd<c +gŒ{…ghdcfx:gi~@‰ dx‚j

d<c‡k<c@k<l‚{…gŒqNgpk€e+¡d<z`cffk€l…ŽžˆŠd<lok¢qNepk<x…xd€ˆ5moepk<qsrtvu+qsw`dehjxPjyzfk{…ghdcfx:wfkx:¡ jjBc„x‚{|zf+gij

l|giƒ<dl|dzfx|ehŽ„gicà

¿ ½ á ’

“”cE{|w`j ˆŠdeheid}nghcfƒeijN{

Ü > Ù

¿

¬›Ú´Ý

Ü Ã

â

´Þ

È È Ù

¿¾

Ú

(8)

V

êŒìW

ð

éEX ò éJY ðQZ3[]\

Y_^A`EYaW ò Z ò `cb

ð é \ ë ð ë õêië ò \ é ã õ\

Yadfe*g%YaW

ð

õÇò ` òh è \

^*ê ð ë

+jc`d€{…j@{|w`j„eigic`jk<l¢‰fk€l|{žd€ˆo{|w`j(qBdšjN£‡qNgijBc{¢d€ˆX{|w`j‡qBd<cšŸ<jq{…ghdcœ{…jBl…~ gic¶Ù½€ÚN–{|wfj‡x‚d<t

qk€eiehj(qNjehe — jŽšc`d<ep`xXcšz`~†¡jBl–{|w`j ‰fk<l…k<¡ dehgpq9~@jx|w(lsk{…ghdk€cf‡{|w`j wšŽš‰ jl|¡d<eigpq9~@jx|w

lsk{…ghd –<l…jx|‰ jq{|giŸ<jehŽ’ Ø jTx…kŽk¢x…qsw`jB~@j9gpx:d€ˆ5dl…+jl9Ù­ßà

Ë

Úghˆ5gŒ{Xgix:ˆŠd<l…~@k<eheiŽqBd<cfx|gpxv{…jBc{

d<ˆXdl…+jl3ß gicç{|gi~j‡k<cfd€ˆnd<ls+jl

Ë

gic•x‚‰fkqNj@d<l–Û~@dl|j@‰`l…jqBgix|jBeiŽ<–Û{|w`j„{…l|zfcfqBk€{|gid<c

jl|l…d<l™gpx™d€ˆd<ls+jBlTá(ÙyÈfâ

à Ê`ځ’

ãcä å æàçèAéé-ê ë‚è†ç}é-ëìTí]îMí]é

“”c{|wfj™ˆŠd<eiehd}ngic`ƒ }oj™l…jqk€eie`x‚d~jnqBeikx|x|giqk€efc`gh{|jR+gŒ‘/jBl…jBc qNjnx|qswfjB~@jx:k<cf{|w`jghl:‰`l…d<‰`t

jl‚{…ghjx¢qNd<l…l…jx|‰ dcf+gic`ƒ‡{…d={|w`j„eighcfjk€l¢qk<x|j<–g’j’ï@

¾ d<l1ÓBÊ

Á ¾ ’ Ø j„Ÿ<jl|ghˆŠŽ={|wfjx|j

‰`l…d<‰jBl|{|gijxXˆŠdl™{|w`j¢cfd<c`eighcfjk€lnqk<x|jF@𖠾 ¡šŽ„{…w`j¢cšz`~@jBl…giqk€ex‚{|zf+gijx9ghc=x‚jq{…ghdc À ’

ñ

a=òa=ò óTô0õ ®=ö_ ö\@[_B÷ ¥š°•¥ÙøaùPú{ûüÄý

˜™w`jž¹fd<l…}Xk<l…+t˜™gi~@j »ojBc{…l…k<eŒtvuš‰fkqNjQjNÍ+‰`eigiqBgŒ{Tx…qsw`jB~@j„ÙO¹˜T»™u`Úngix™ƒghŸjBcE¡Ž

ÑB‹

Þö Ó ÊÁ

@ ‹ ¾ Ñ ‹ Þ Üâ > ¿â Ó ÊÁ

¿ Ñsß

¿ > â Þ > Þâ Ó ÊÁ

@ ¾ Ñ Þ > Üâ ¿â Ó ÊÁ

Ñ Þö Ó ÊÁ N

“{ngpxod<ˆÛd<ls+jlQÙ ¿ –â ځ–š}ngh{|w=k†ŸjBl…Ž@l|jxv{…l|gpq{…ghŸjRx‚{…k€¡fgheigŒ{vŽ‡qBd<cf`gŒ{…ghdc’:“”c‡{…w`j ehgic`jk<lXqk<x|j

gh{9l|jk<`x¢àÊ Àáƒ Þ ² ¿

â N Ù¿<¿ Ú

˜Pd„kŸ<d<gp‡dx|qBgheiepk{|gid<c xB–š{…w`j ˆŠd<eiehd}ngic`ƒ@qNdcf+gh{|gid<c=~†zfx‚{n¡ jžx…k{…gix‚fj0ƒ

þÝ þ

²6¿N Ù¿â Ú

ñ a=òasÿ ü¥š°(ööO° õ ®yö_ ö\¢[_÷ ¥°«¥"ø{ú}û}üÄý

˜™w`jox‚j~ghtgh~@‰`eigiqBgŒ{Ûx…qsw`j~j–z x‚gic`ƒ9m™k<qsrš}™k€lsšt˜™gi~@j:»ojc{|lsk€eht”uš‰ k<qNj:+gh‘/jBl…jBcfqBghc`ƒžÙ*m™˜T»™u`Ú

ˆŠdl™{|w`j¢eighcfjk€ln‰ k€l|{nk<cfEjNÍ+‰`eigiqBgŒ{™{…l|jk{|~@jc{9d<ˆÛ{…w`j¢c`d<cfehgic`jk<l|gh{vŽ<–+gpx™ƒ<giŸ<jcD¡šŽ

ы

Þö Ó ÊÁ

Üâ > ÞQ ы

> ¿

â Ó ÊÁ

¿ !â ÞQ Ñsß

¿ > Þâ Ó ÊÁ

@ > Üâ >

ÞQ Ñ ¿â Ó ÊÁ

Ñ Þö Ó ÊÁ N

“{@gixz`cfqBd<cf`gŒ{…ghdcfk€eiehŽÂx‚{…k<¡`ehjEk€cf•d€ˆTdl…+jlDÙ

¿ –â

ځ’“{gpxc`d<c`tdx|qBgheieik€{|dl|Ž›gŒˆ†Ù

¿â چgpx

x…k{…gix‚fjàÊ ’

(9)

ã ä`åçæ

è€é|êië+ì ífî ä`ïð

è€é…ë`êñò í`ó äfôXæ

è€ë+ì òsõ

ñ ayò añ û ]^`­P¬ ö_ ¨ ®[ ¨ ­ ø û~ý

˜™w`j†»ol…k<c`rt¯tTgiqBd<epx‚dc„x|qsw`j~@jR}ngh{|wDjNÍ+‰`eigiqBgŒ{™{|l…jk€{|~@jBc{™d€ˆ5{…w`j c`d<c`eigic`jk<l|gh{vŽ@gixXƒ<giŸ<jc

¡šŽ

@ ‹ § > Þâ ö ¨ Ó ÊÁ > Þâ > Üö Ñ ‹ § Þâ > ö ¨ Ó ÊÁ Þâ Üö

@ ß

¿ ޚÙ

¿' Ó ÊÁ

Ú Ñ ß ¿ > Þ+Ù

¿' Ó ÊÁ

Ú

@ § > Þâ > ö ¨ Ó ÊÁ > Þâ Üö Ñ § Þâ ö ¨ Ó ÊÁ Þâ > Üö

k€cfÑ ‹ Ñ ¾ ’“{9gpxnz`cfqBd<cf`gŒ{…ghdcfk€eiehŽDxv{sk€¡`eij¢k€cfEd€ˆ7d<ls+jBl¢Ù⠖â ځ’

ñ

ayò a ¥ ^õ qù ] ¨ ¯ ùP¨ ]\ qù ]^`­P¬ ¥š® [_÷ ¥š°•¥(ø ù ùTý

˜™w`jvjk€‰+t”¹`l…d<ƒEÆTz¹`dl‚{|t¹fl…k<c`r<je7x…qsw`j~jgpx¢k€cœjBÍ+‰`ehgpqNgh{ {|w`l…jBjBt*{…gh~@jBteijBŸjBe:x|qswfjB~@j<’

“{9gix9ƒ<giŸ<jBcE¡šŽ

É Ê Î/

Á > É Ê ‹

Á

â È

É3Ê

Á‹ > Ù*É Ê ‹

Á É Ê Î/

Á Ú ÉFÊ

Á

Î/

à ٠¿ ¬Ú

ÉFÊ

Á

Î/

> ÉFÊ

Á‹

â Ã

Ó ÊÁ Ö ÉFÊ

Á‹ > â ÉFÊ

Á ÉFÊ

Á

Δ

ö Ã

ÉFÊ

Á

Î/

>

ÉFÊ

Á‹

â Ã × N

ÆTzfjR{…d†{…w`jQcfk{…z`l|jQd€ˆÛ{|w`l…jBjBt*{…gh~@jNteijBŸ<jex…qsw`j~jx}XjQc`jBj(k€cEk<``gŒ{…ghdcfk€e~@jN{|wfdš({|d

qNd~‰fz+{|j¢{…w`jžcšz`~@jBl…gpqBk€ePx‚dehz+{…ghdc.k{n{|wfj¢flsxv{n{…gh~@j†x‚{|j‰’o“”c=dz`l9cšz`~@jBl…giqk€e5{…jx‚{Tgic

x‚jq{…ghdc À {…w`jžqNd~‰ k<q{9x…qsw`jB~@j — Êm ÙOx|jBj¢¡jBeid}TÚogixnz x‚j‡ˆŠdl™{|w`gpxn‰`z`l…‰dx|j<’

˜™w`jžx…qsw`jB~@j gixnx‚{…k<¡`eij¢ghcD{|w`j¢eigic`jk<l9qBk<x|j„àÊ ¾á ghˆ

Þ

¿

â þÝ þ Ù¿

ÊÚ

k€cfEgh{9gix™d<ˆ7dl…`jBl¢Ù⠖â ځ’:“{9gpx™c`d<c+tdx…qNgieieik€{|d<l…Ž@gŒˆ:qNdcf+gh{|gid<cÂÙ¿â ÚogpxnŸk€eigiàÊ ¾á ’

ãcä æ¼î7è†ë¥éë,ì?í]îMí]éì7ê¼ìTí! ?í

˜™w`j¢ˆŠdeheid}nghcfƒ{v}Xd„x|qswfjB~@jxn}XjBl…j ghc{…l|d++zfqBjE¡šŽ

—

giƒk<eà

â ¼ á

ˆŠdlnehgic`jk€lTqBd<cšŸ<jq{|gid<c`t

+gh‘zfx|gid<c«‰fl|d¡`ehj~„xB’

Ø

jEk€‰`‰`eiŽ{|w`j~ {|d›‰`l|d¡`eijB~ Ù½<ځ’瘙w`jBŽÂk€l…j‡¡d€{…w«qNd~‰ k<q{

{v}od<teijBŸjBe7x|qsw`j~@jxRd€ˆd<ls+jlÙ

⠖ö

Úngic3{…w`jeigic`jk<lRqBkx‚j’R˜™w`jc`dc`ehgic`jk€l…gŒ{vŽEgixT{|l…jk€{|j

(10)

V

êŒìW

ð

éEX ò éJY ðQZ3[]\

Y_^A`EYaW ò Z ò `cb

ð é \ ë ð ë õêië ò \ é ã õ\

Yadfe*g%YaW

ð

õÇò ` òh è \

^*ê ð ë

ñ asÿa=ò " ñ Y [_÷ ¥š°•¥

˜™w`jžx…qsw`j~j — ÊbgixnƒghŸjBcE¡šŽ

Ö ¿

¿â

> Þâ × Ù

¿

Ý7Ú > Ý Þ Ý Þ

Ê

À Þ Ý Þ

Ê > â Ý Þ

Ê

@ Ö ¿

¿â

> Þâ × Ù ¿ > Ý7Ú > Ý Þ Ý Þ

Ê

ÑB‹

Þö Ó ÊÁ

Ñ ‹ Ö ¿

¿â

Þâ × Ù

¿ Ý7Ú

Ý Þ Ý Þ

Ê > ¿ â Ó ÊÁ

Ñ ß À > Þ > Ý Þ

Ê > â Ý Þ

Ê > Þâ Ó ÊÁ

Ñ Ö ¿

¿â

Þâ × Ù ¿ > Ý7Ú > Ý Þ Ý Þ

Ê ¿

â Ó ÊÁ

Ñ Þö Ó ÊÁ N

˜™w`gpx9x…qsw`jB~@j gpxnxv{sk€¡`eij¢ghcD{|wfj¢ehgic`jk<l9qBkx‚j„ÓBÊ

Á ¾ àâ ¼ á ghˆ

Þ ² ¿

â þÝ þ N Ù

¿Bö

Ú

“{9gpxnc`dc+tdx|qBgheiepk{|dl|ŽˆŠdl9k€l…¡`gŒ{…l…k<l|Ž@Ÿk€eiz`jx™d€ˆAݒ

ñ asÿasÿ " ñ [_÷ ¥š°•¥

˜™w`jžx…qsw`j~j

—

ÊmÁgpx™ƒ<giŸ<jBcE¡šŽ

@ ‹ Ö ¿

¿â

> Þâ × Ù

¿'

Ý7Ú > Ý Þ Ý$#EÞ

Ê > â Ý&%_Þ'#

Ê

À Þ Ý Þ

Ê ö Ý&%_Þ'#

Ê

@ Ö ¿

¿â

> Þâ × Ù ¿ > Ý7Ú > Ý Þ >

Ý$#EÞ

Ê > â Ý&%_Þ'#

Ê

Ñ ‹ Þö Ó ÊÁ

ÑB‹

Ö ¿

¿â

Þâ × Ù ¿' Ý7Ú Ý Þ

Ý$#EÞ

Ê â

Ý&%_Þ'#

Ê

>MÖ

Þö ¿â × Ó ÊÁ

Ñ ß À > Þ > Ý Þ

Ê > ö Ý&%_Þ'#

Ê > â ÞŒÓ ÊÁ

Ñ Ö ¿

¿â

> Þâ × Ù ¿' Ý7Ú Ý Þ >

Ý$#EÞ

Ê â

Ý&%_Þ'#

Ê

>MÖ

Þö > ¿â × Ó ÊÁ

Ñ Þö Ó ÊÁ N

“{9gpxnz`cfqBd<cf`gŒ{…ghdcfk€eiehŽDxv{sk€¡`eij¢k€cfDc`d<c`tdx|qBgheieik€{|dl|Ž@gicD{…w`j¢eighc`jk€l9qk<x|j3ÓBÊ

¾ àâ ¼ á’

(11)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

( ) ¶+* 7Ö¼™…›TŸ»›

“”c„{|wfj9c`d<cfehgic`jk<lqBkx‚j?@ð–

¾

gŒ{ox|jBj~„x:{|d†¡ jTyz`gh{|jR+gŒ£‡qBz`eŒ{o{|dž‰fl|dŸj™{|w`jTx‚{…k<¡`giehgh{vŽd<ˆ

{|w`jx|qswfjB~@jxT‰`l|jx‚jc{|j.k<¡ dŸj<’n˜™w`jl|jk<x|d<c=ehgijxTghc3{|w`j†ˆOk<q{T{|wfk€{–zfx‚gic`ƒ+,-

ü„r

p

ýUmsü

p

ø€úÇ÷oyr

k€{Q{…w`j„‰`l…jx|jBc{ {|gi~@j„ehjŸ<jBe–5{…w`j‡xv{…zf+Ž3d<ˆ{…w`j‡k€~@‰`eigŒqBk{…ghdcœˆOkq{…d<l¢gicŸd<eiŸ<jx

qNjl‚{sk€gic›qNzf¡`giq†‰ dehŽšc`d~@gik<eix’ Ø j}odz`eiœc`jBj={|d=x‚{|zf+Ž={…w`jBgilQ‰ dx‚gh{|giŸšgŒ{vŽ.‰`l…d<‰jBl|{|gijx

ˆŠd<lTk<ehe5‰dx…x|gh¡`eij¢Ÿk€eiz`jx™d<ˆP{…w`j¢c`dc`ehgic`jk€l9qNdšjB£„qBghjc{…x™gicçٍ½€ÚN’“{n{|z`l…cfxndz+{n{|w k{9gh{9gix

c`d€{9‰dx…x|gh¡`eijQ{|d‡x|w`d}G‰dx|gh{|giŸgh{vŽ‡d<ˆP{…w`jx|j ‰ dehŽšc`d~gpk€epxXˆŠd<l9k<l|¡fgŒ{…l…k<l|Ž„qNdšjB£„qBghjc{…x’

˜™w`jBl…jNˆŠdl|j¢}Xj†qBd<cfx‚{|l…zfq{Rc`d}Ëk3ÙOx|jB~@ghtgi~@‰`ehgpqNgh{sÚn{v}od<teijBŸjBe

o.šþ…ü…üEk

p

øúÇ÷o

qNd~‰ k<q{

x|qswfjB~@jž}nw`gpqsw.gix9d€ˆ:w`giƒ<w.d<ls+jl9k€cf.qk€c=¡ j‰`l…dŸ<j({|d(¡ j†x‚{…k<¡`eij<’ Ø jžjBÍ+‰ jq{T{…wfk{

{|w`jx‚jR{|wfjBd<l…jN{…giqk€e¡ jc`jNf{…xX}ngheie5k€epx‚d†~@k<r<jT{|w`j x|qswfjB~@jRx|z`‰jBl…ghdlgic(cšz`~@jBl…giqk€e {|jxv{sxB’

Ø j¢}ngieiezfx|j {|w`j¢~@d++ghfjEjyzfk€{|gid<cD{|jqsw`c`gpyzfj@àÊU á {|d‡qBd<cfx‚{|l…zfq{™{…w`jžx|qswfjB~@j<’

˜ÛdDd<¡+{sk€gic›k<c3jN£‡qNgijBc{¢x|qswfjB~@j†gh{QgpxRgh~@‰d<l|{…k€c{R{|d=k€‰`‰`l…dÍ+gh~„k€{|j¢{…w`jcfd<c`eighcfjk€l

qNdšjN£‡qBghjc{sx7ghc=Ù½<ÚjB͚‰fehgpqNgh{|eiŽ<–€g’j<’€k{7{|wfj{…gh~@j™ehjŸ<je

Ë

’ªT{|wfjBl…}ngix|jdc`jo}Xd<zfei†c`jBj¢{|d

‰ jl‚ˆŠdl|~ kc`d<c`eigic`jk<lXgh{|jl…k€{|gid<cEghc=jk<qswD{|gi~jžx‚{|j‰=}nw`giqswEgpx9yz`gŒ{…j {|gi~@jNt”qNd<c x‚z`~@gic`ƒf’

Ø gŒ{…w

/

¿ ¥8 :jNÍ+‰ÛÙ=¬ ¦ Ú@ ܇Ù10

É Ê 0†ßQÉ

Ê ÚC

Ü

¿ ¥8 :jNÍ+‰ÛÙ=¬ ¦ Ú@-܇Ù10

É Ê 0†ßQÉ

Ê ÚC ¬&

}nw`jBl…j

0ßŒÉ

Ê

2

ÉFÊ

2

ÎÄ

> É3Ê

2 ‹

â Ã

0 É Ê

2

ÉFÊ

2

ÎÄ

> â ÉFÊ

2 ÉFÊ

2 ‹

Ã

{|w`j.´…x‚j~@gŒt”+gpx|qBl|jB{|gi‹Bjf·†jyz k{|gid<cçÙ½<Úo{…k<r<jxX{…w`j ˆŠd<eieid}nghc`ƒˆŠd<l…~…ƒ

© « / © ®_®F> Ü© ® N Ù¿À Ú

Ø j=}ngiehe™c`d} xv{…zf+ŽÂjyzfk€{|gid<cÙ¿À Ú ˆŠdl@k<l|¡fgŒ{…l…k<l|ŽŸk€eiz`jx / qÜ̖ ¾ ’ Ø j.+jN c`jEk

ƒ<jc`jBlsk€e/{v}Xd€tehjŸ<je {…w`l…jBjNt‰d<gic{9x…qsw`jB~@j

3 «aÉ

Ê

2 / Ù

(

Ú40

É Ê 2 / Ù

Ú0

É Ê Î/

2 > ÜPÙ

Ð

Ú0ßŒÉ

Ê

2

>

ÜPÙ

Ð

Ú0ßŒÉ

Ê

Î/

2 Ù¿

Ú

}nw`jBl…j

3}«

É Ê

2 É Ê Î/

2 >

ÉFÊ

2

È

k€cf

Á Ð Á

k€l…j™l|jk€e+qNdcfxv{sk€c{…x7}nw`gpqsw~†zfx‚{¡ jnqsw`dx‚jcgic@x‚zfqsw@kQ}XkŽ {…wfk{:{…w`j™ehd}XjBl

d<ls+jBl{|jBl…~„xgic„{|wfj9{|l…z`cfqk{…ghdc„jBl…l|dlk€l…jnjBeigh~@gicfk{…j‡k<cf{|wfk€{{|wfjRx…qsw`jB~@j9gpxx‚{…k<¡`ehj

k€cfEc`dc+tdx…qNgiehepk{…d<l…Ž<’

Ø gŒ{…wÂ{|wfgixjNÍ+‰`eigiqBgŒ{@+gpx|qBl|jB{|gi‹k{…ghdcÂd€ˆ™{|wfj(c`dc`ehgic`jk€lqBdšjN£‡qNgijBc{…x†}ojDxv{…zf+Ž›{|w`j

ehd+qk€e/xv{sk€¡`giehgh{vŽ=Ùu2ˆŠl|d‹BjBcDqNdšjN£‡qNgijBc{sxE4Ú:d<ˆ5{|w`jRehgic`jk<l|gi‹Bj@jyzfk€{|gid<cfx’P“{Xgixo}XjBeieŒtršc`d}nc

àâ # á {|wfk€{PˆŠdlPeigic`jk<lۉ`l…d<¡`eijB~„xP}ngŒ{…w†Ÿk€l…gpk€¡`eijqNdšjB£„qBghjc{…xXيc`d<{7gicžƒ<jc`jBlsk€e–¡`z+{7ˆŠd<l7gh~t

‰ dl‚{sk€c{9qNepk<x…x‚jxd<ˆPjyz k{|gid<c xB–šcfk<~jehŽ„‰ k€lsk€¡d<eigiqQk€cfDx‚Žš~@~@jN{…l|gpqQwŽš‰jBl…¡ dehgpqRjyzfkt

{|gid<cfxsÚneid+qBk<e7x‚{…k<¡`gheigh{vŽDgpxTcfjqNjx|x…k€l…Ž‡ˆŠdlRdŸjBlsk€eieÛx‚{…k€¡fgheigŒ{vŽ=k€c 3x‚eigiƒ<w{|eiŽEx‚{|l…jBcfƒ€{|wfjBc`j

ehd+qk€eÛx‚{…k<¡`gheigh{vŽ‡gixnk<eix|d„x‚z`£„qBghjc{n{|d@jBcfx|z`l…j dŸ<jBlsk€eie/xv{sk€¡`giehgh{vŽ<’

(12)

êŒìW éEX ò éJY Y_^A`EYaW ò ò `cb é ë ë õêië ò é õ Yadfe*g%YaW õÇò ` òh è ^*ê ë

¥°«° ^Ùò65

o17øQk”ùhü

-

üBù]o.šþ…ü…üEk

p

øúÇ÷o

,

÷ úoü8ú9TüBþ|üB÷%msü1rEm.fü;:„üžúsrFro”ý=<Bùhüú>žý÷8Eø÷ ù@?

ú>3o.`ü}msøüylnmNúOüB÷oyr

@ Á Ñ

Á?rBýQoúsrA>B?

Ùy@

> @,‹

Úa > ÙyÑ

> ы

ÚF–¥@

@,‹

> Ñ > ÑB‹

Ù¿ ½<Ú

Ùy@

@,‹

Úa > ÙyÑ

ы

ÚF–¥@

@,‹

> Ñ > ÑB‹

N Ù¿

¥°«° ^ ÿC5

o17øQk”ùhü

-

üNù/oA.+þ|üsük

p

øúÇ÷,o

,

÷fúo”ü8€ú9TüNþ…üN÷‚m…ü1rEmB.`üD:@üúsr¢÷ø€÷k‚ørEmBúÇùÇùhýQo”øþE?Dú>

o.`ü}msøüyl¥mBúOüB÷oyr

@ Á Ñ

Á?rBýQoúsrA>B?

Ù=@

> Ñ ÚBÙ=@,‹

> ÑB‹

Úc| ¾N Ù¿ ¼Ú

¹fd<l™{|wfj¢‰`l|dšd<ˆOxo}Xj¢l…jNˆŠjBlX{…d3àâ ¼+–j~@~@kx ¿ k€cf âá ’

bT‰`‰`eiŽgic`ƒœÙ ¿ Ú{…d‡k@x‚z+£‡qBghjc{…ehŽEx‚~@dšd€{…w.x‚dehz+{…ghdcDd<ˆžÙ ¿À ځ–`}Xj d<¡+{sk€gic({…w`j¢{…l|z`c`t

qk{|gid<cjBl…l…d<l@ÜGF Ù*Ȃ

Ã

ځ’ÁÆRgŒ‘/jBl…jBc{…gik€{|gic`ƒ Ù¿À ڏ}ojEd<¡`{…k€gicw`ghƒw`jBl„dl…`jBljyzfk{…ghdcfx’

ÉRx‚gic`ƒD{|w`j~ {|dEjBeigh~@gicfk{…j{|w`j{|gi~@j„+jBl…ghŸk€{|giŸ<jx–/}oj~„kŽ.}nl|gh{|j@ÜGFÙyȂ

Ã

ÚRgicœ{…jBl…~„x

d<ˆP{…w`jžx‚‰ k<qNj +jl|giŸk{|giŸ<jx™d<c`eiŽ‚ƒ

Ü F ÙyȂ

Ã Ú %

Ò

2 Ï ˆ 2DH

2® ©

w`giƒ<wfjBlnd<ls+jl™`jBl…ghŸk{…ghŸjx

}ngh{|w

ˆ ÜPÙ

Ð Ð

Ú Ù‾ Ú

ˆ > / Ù

Ú > ÈÜ

Ð Ù

â+¿

Ú

ˆ #

ÜPÙâ Ã !â

à РÜ,_Èf

Ð ÈfBÜ

Á¿â È /

à Ð

Á¿â È / Ð

Ú Ùâ<â Ú

ˆ % >

% Ù ¿ Âö Ð

ÚÜ

% È # >

% ÙâŒ

•ö

Ð

Á¿ Ú / Ü,Bȁ

>

ÙyÜ,

à Á¿â

/

!â Ü,

Ã Ð Ú È >

Ù

¿

Ú / Ã N Ùâ

ÊÚ

˜Pd@d<¡+{sk€gicEk@x…qsw`jB~@j d€ˆd<ls+jBl¢Ù⠖ö ځ– Á Ð Á

~zfxv{9¡j¢qsw`dx|jBc=x|zfqsw({…wfk{n{…w`jx|j jBl…l|dl

{…jBl…~@x™Ÿk<c`gix|w=d<l9k<l|jQd€ˆd<ls+jBl¢Ù ⠖ö ځ’

ušdehŸšgic`ƒ@{|wfj†eighc`jk€lTx|Ž+xv{…jB~ qNdcfx‚gpx‚{|gic`ƒ„d€ˆÙ‾ ÚtsÙâ<â ڙghc={…jBl…~„x9d€ˆ Ð }Xj

ƒjN{k¢qNepk<x…x:d<ˆ/x…qsw`jB~@jx+jB‰jBc +ghcfƒžd<c`eiŽ†dc{|w`j9‰ k€lsk€~@jN{…jBl

Ð

’:˜™w`gpxx|qswfjB~@j<–qBk€eieij

Ù ¿ 4ځ–+gpxnƒghŸjBcE¡ŽÙ¿ Úo}ngŒ{…w

Ð > Ð Ùâö Ú

>

IBJ

Ù> â Ã Ü,_Èf

Ð > ÈfBÜ >

¿â È / Ð

Ú Ù

â<À

Ú

>

IBJ Ùâ Ã

Ü,_ȁ

Ð

ȁ_Ü,

Á¿â

È / Ð Ú N Ùâ

Ú

¸oyz k{|gid<c6Ùâö ڞgpx@kœcfjqNjx|x…k€l…ŽÂqNdcf+gh{|gid<c•{…d›w kŸ<jDkqNdcfx|gix‚{|jc{„x…qsw`jB~@j<’•˜™wfj

qBdjB£‡qNgijBc{

Ð

x|w`d<z`epD¡ jžqsw`dx‚jc(gicEx|zfqsw=k}XkŽ@{…wfk{n}XjQd¡+{…k<ghcEk@xv{sk€¡`eij¢k€cfDc`d<c`t

dx|qBgheieik€{|dl|Ž(x…qsw`j~j¢d<ˆ:d<ls+jBlžÙ ⠖ö ځ’™¹`zfl‚{…w`jBl9}Xj¢l…jyz`gil|j¢dz`lTx…qsw`jB~@j {|d(¡ j

>Nø€þE7ýþ48

8€ú9LK-rú - ü

يgic=l|jeik€{|gid<cE}ngŒ{…wD{…w`j¢‰fk<l…k<¡ dehgpqT‰fl|d¡`ehj~ ¡jBgic`ƒ„}ojeheht‰ dx‚jfځ–+g’j’

¿z / / – ¾N Ùâ ½<Ú

Ø

jcfd} xv{…zf+ŽE{|w`j‰`l…d<‰jBl|{|gijxRd€ˆ{…w`j@x|qsw`j~@jDÙ

¿

4ځ’

Ø

jd<¡`{…k€gic.{…w`j†ˆŠdeheid}ngic`ƒ

(13)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

ú ÷ ¥¨ ] ¥š° ñNM

mB.`üD:@üPOQSRUTV@úsrTro”ý=<Bùhü ú>¢ý÷W8„ø÷ ù@?@ú>7o.`ü1m…øüyl¥mBúOüN÷,o Ð ý÷8

ÞQ,ݼ /

rBýQoúsrA>B?

Ù> / •ö ÞQÝ/

Ð

ÚNÙ > ¿ Âö ÞQ_ÝÄ ¿â / Þ Ð

ÚP– ¾N Ùâ #Ú

X

oXúsr¢÷ø÷,k‚ørEmBúÇùÇùhýQo”øþE?Dú> Ð ý÷8

ÞQÝ /

rBýQoúsr>B?

Ù > / •ö ÞQÝÄ

Ð Ý7ÚBÙ > / •ö ÞQÝÄ

Ð > ÝÚc| ¾N Ùâ ¼Ú

X

oXúsr>NøþE7ýþ8Y8€ú9LK-rú

-

üú>†ý€÷8(ø€÷fù@?„ú> Ð

ÞQÝ

rNýŒo*úsr>B?

¿ > ö ÞQÝÄ

Ð – ¾N Ù*Ê ¾ Ú

Z

þ|øø>;[

˜Ûd3‰fl|dŸj@x‚{…k<¡`giehgh{vŽœ}Xj‡c`jj›{|d3ŸjBl…gŒˆŠŽqNdcf+gh{|gid<cfx‡Ù ¿ ½<Ú¢k€cf¶Ù ¿ #ÚN’E˜™wfgix†gix

x‚wfd}ncD¡šŽ(x‚{|lsk€giƒ<w{‚ˆŠdl|}™k€ls‡qBd<~@‰`z+{sk{|gid<cے˜™wfjžx|qsw`j~@jÙ¿ ÚXqk€cE¡ j¢}nl…gŒ{|{|jcEgicD{|w`j

ˆŠd<l…~ Ùy#ÚX}nw`jl|jQ{|wfjžqNdšjN£‡qNgijBc{sxP@

Á

Á

k<l|j ƒ<giŸ<jc(¡šŽ

@ ‹ > / Ù]\

Þ Ú >_^

% >

G`Ua

Ñ ‹ / ÙD\

Þ Ú ^%

G`&b

@ ß

¿

/

Ù­Þ !â Þ

Ú Ñ ß ¿ > /

Ù=Þ «â Þ Ú

@ > / Ù]\

Þ Ú ^% `Ua

Ñ / ÙD\

Þ Ú > ^% > `&b

k€cfÑB‹

Ñ ¾ ’ Ø gh{|w={…w`jx|jžqNdšjN£‡qBghjc{sx¢Ù ¿ ½€ÚXk€cfÙ¿ #ڙk<l|jQjyz`giŸk€eijBc{n{|d

ÙÐ Ð

> Ð > Ð Ú Ùâ / !â /

!â /

ÚaÞ{– ¾

> â Þ / Ù¿c(

(

ÚBÙâ / Þ > â / Þ > ¿ Ú'– ¾N

ÉTx|ghcfƒœÙâ€ö ڍtÙâ Ú{|wfjx|jQgic`jyz k€eigŒ{…ghjx9x‚gi~@‰`eigŒˆŠŽ({|d

â / Þ1–

¾ Ù*Ê

¿ Ú

âÊ

Þ+Ù

> / Âö

ÞQÝ Ð ÚNÙ

> ¿ Âö Þ Ý

Á¿â / Þ Ð

Ú'–

¾ Ù*Ê

â Ú

l|jx‚‰jqN{|giŸ<jehŽ’o»od<cf+gh{|gid<cçÙOÊ ¿ Úogpx9k€ei}XkŽ+xXx|k€{|gpxv j=k€cfÙOÊ â ÚoŽšghjeifxžÙâ #Ús’

¹`dlQcfd<c+tdx…qNgiehepk{…ghdc.}ojwfkŸjž{|d.qsw`jqsr.qNd<c +gŒ{…ghdc!Ù ¿ ¼ÚN’†¸eijB~@jc{sk€l…ŽEqBd<~@‰`z+{skt

{|gid<cfx7zfx|gic`ƒ9{…w`jXk<¡ dŸjoqBdjB£‡qNgijBc{…x7k€cfx‚zf¡fxv{…gŒ{…z+{|gic`ƒ

Ð

ˆŠl|d~ Ùâö ڍtÙâ Ú5ƒghŸj

{|w`jžqBd<cf`gŒ{…ghdc

¿

ö Ù > â Þ /

ÞŒÈÜ Ð

ÚBÙ> â Þ /

«â

ÞQÈÜ Ð >

Ú'|

¾N

Ø l…gh{|gic`ƒ@{|w`gpx9qBd<cf+gh{|gid<cEgicD{…jBl…~„xnd€ˆ†Ý•k€cf ޏƒ<giŸ<jx Ùâ ¼ÚN’

ušz`¡fx‚{|gh{|z`{|gic`ƒ›Ùâ<À ÚXk<cf›Ùâ ÚXghc{|d.Ùâ ½€Úo}Xj¢d<¡+{sk€gic

¿ > ȁÜ,

Ð – ¾

}nw`gpqswEgpxnjyzfghŸk€eijBc{™{|d.Ù*Ê

¾

ÚN’

Ø j¢}ngieie5c`d}G‰`l…d<‰dx|j kqsw`dgiqBj d€ˆÛ{…w`j¢qNdšjN£‡qBghjc{

Ð

k€cfEx‚{|zf+Ž‡{|w`j¢‰`l…d<‰jBl|{|gijx

d€ˆo{|w`j„x…qsw`jB~@jd<¡+{sk€gic`jœ¡Ž={…w`gpx¢qsw`d<gpqNj’†mXŽ3qNd<c xv{…l|zfqN{|gid<c–”ˆ

ˆ ˆ # ¾ ˆŠd<l

{|w`jžx…qsw`j~j‡Ù ¿ 4ÚN’˜™wfj¢jBl…l|dlcˆ

%

qBk<cD¡j¢}nl…gŒ{|{|jBc.kxEƒ

ˆ % ¿

È5Ù > Ü,

à Èf_Ü

% ¿â

/ BÚ

Ð > ¿ / Ù > Ã

!â ÈfEÜ,Ú

N

Ù*Ê<ÊÚ

(14)

V

êŒìW

ð

éEX ò éJY ðQZ3[]\

Y_^A`EYaW ò Z ò `cb

ð é \ ë ð ë õêië ò \ é ã õ\

Yadfe*g%YaW

ð

õÇò ` òh è \

^*ê

ð ë ¿

Ø j9~†z xv{oqswfddx‚j

Ð

gic‡x‚zfqsw‡k }™kŽ¢{…wfk{ ˆ

%

gpx:d<ˆ/dl…`jBl9Ù⠖ö ځ’:˜™w`jneid}ojld<ls+jl:‰fk€l|{

d<ˆ†ˆ

%

gix

È / Ð ¿

¿â / Ã N

˜™w`jl|jBˆŠd<l…jR}Xj¢~„k€rjT{…w`jžk€cfx…k{…‹

Ð > ¿

¿â Ã

/ È Ñ N ÙOÊ ö Ú

Ø j@wfkŸ<j†{|d.qsw`dšdx|jž{…w`j„qNd<c xv{sk€c{3ўd€ˆXd<ls+jBl„á‡Ù

Ã

%ÚT{|dEd<¡+{sk€gic›k({|l…z`cfqk{|gid<c›jBl…l|dl

ˆ %

d<ˆ/{|w`jQx|k<~jTd<ls+jBl’7˜™wfj9d<¡šŸšghdzfxqswfd<gpqNjTÑ

¾ gixc`d€{Xl|jqNd<~@~@jBc +j„x|ghc qNj9gic„{|wfj

eighcfjk€lžqk<x|j / ¿ d<l3@

¾

–5{|w`gpxžqsw`dgiqBjeijk`xQ{|dE{…w`j — Ê<b x…qsw`jB~@j@}nw`gpqsw›gpx c`d<{

z`c qNd<c +gŒ{…ghdcfk€eieiŽDx‚{…k€¡fehj’ Ø jž}™k€c{n{|d‡qsw`dšdx‚j1ÑRgic3x‚zfqsw.k„}™kŽ„{…wfk{9{…w`j†qBd<cf+gh{|gid<c x

Ù â

#ڔtsÙ*Ê

¾

Úd<ˆ7˜™w`jd<l…jB~ Ê@k<l|j¢x…k{|gpx‚fj5’

Ø jž+jN c`jQ{|w`j — ʚ» x…qsw`j~j ¡šŽ(qsw`dšdx‚gic`ƒ

Ñ >

ÞQÝ

Ê / > Ü È

¿â /

}nw`gpqsw=Žšghjei`x

Ð >

¿¼Âö Þ Ý

¿â

/ Þ N

ÙOÊ

À Ú

¹fd<l™{|wfj‡ÙŠeighc`jk€lÚ™qBkx‚j / ¿ {…w`gpx9qsw`d<gpqNj¢qBd<l…l|jx‚‰d<cffx{…d@{|w`j — Ê<m x…qsw`jB~@j d€ˆ — giƒk<e

àâ ¼ á ’˜™w`jžqBdjB£‡qNgijBc{…xP@

Á

Ñ

Á

k€l…j ƒ<giŸ<jBcE¡šŽ‚ƒ

@,‹

>

¿â

Þ / > â / ތÜ

Ã

Þ'#Üc%

à %

ŒÞŒÜ

à / > Ü Ã > Þ Üd#

à #

â€ö

/

¿¾

/ ¿â

Þ / ތÜ

à Þe#Ü!%

à %

¿â /

@ >

¿â Þ / > â / ތÜ

Ã Þ # Ü % à % > ŒÞŒÜ

à / Ü Ã Þ Ü # à #

â€ö

/

ы

¿â

Þ / !â

/ ތÜ

Ã Þ # Ü % à % UÞQÜ

à / Ü Ã Þ Ü # à #

â€ö /

Ñsß

> > ¿¾ / Á¿â Þ / ތÜ

à Þ'#Üc%

à %

¿â

/

Ñ

¿â

Þ / !â

/ ތÜ

Ã

Þ'#Üc%

à % > UÞQÜ

à / > Ü Ã > Þ Üd#

à #

â€ö

/ N

ú ÷ ¥š¨ ] ¥š° gf

.`üihkj!l¥rmB.`üD:„ü8ü

,

÷üB8(ýS<…ø

-

üúsrý€÷mK+÷%msø€÷8úoúOø÷/ýùÇù@?ïrqo”ý=<Bùhü;n÷ø€÷k

øQrmBúÇùÇùhýŒoø€þE?.ý€÷8o>NøþE7ý€þ48p8ú9GK-rsú

-

ü}rEmB.`üD:@ü‡ø>ø€þ48<üBþqOsrtnvutV=[

X

oyr„o*þEK+÷‚msýQoúOø÷ÂüNþþ|ø€þ†úsr

w<ú

-

üB÷x<D?

ˆ % > Ü Ù*Èy%BÜ!%

> Ã % ÊÈ

/ Ú

¿Böö / N ÙOÊUÚ

Z

þ…øø>;[

ušzf¡fxv{…gŒ{…z+{|gic`ƒ«Ù*Ê

À

ÚTgic¶ÙOÊ<ÊÚR}oj@ƒ<jB{‡ÙOÊUځ–g’j<’5{…w`j‡x|qswfjB~@jgpx d€ˆ™d<ls+jl@Ù

⠖ö ÚN–

x|ghc qNj3ˆ

ˆ ˆ # ¾ ’

(15)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

˜Pk<ršghc`ƒ@gic{|d„k<qqNd<zfc{žÙ*Ê À ځ–fqNdcf+gh{|gid<cfx¢Ù*Ê ¾ ڙk<cfÙâ #ÚXk€l…jQjyz`ghŸk<ehjc{™{…d

¿

Ý Ù¿ •ö Ý Þ Ú

Ê / – ¾

öÊ § / Ý Ù¿ •ö Ý Þ Ú

Ê / ¨ – ¾

l|jx‚‰jqN{|giŸ<jehŽ’˜™w`jx|j¢qNd<c +gŒ{…ghdcfx™w`dei(ˆŠd<lTk<ehe5Ÿk<ehz`jx™d€ˆ / ’

ÉTx|ghcfƒœÙOÊ

À

ÚoghcÂÙ

â

¼Ú–`}ojQfc ({…wfk{n{…w`gpx9qNdcf+gh{|gid<cEgixnŸk<ehgpEgŒˆ7ˆŠd<l9k<ehe5Ÿk€eiz`jx™d€ˆ /

¿

¼ Þ /

ÙOÊ / Ý/

Âö Ý % Þ

Ê / ÝÚNÙOÊ / Ý/

Âö Ý % ÞQ

> Ê / ÝÚc|

¾

Ù*ʽ€Ú

d<l–+jyz`giŸk€eijBc{vŽ<–+ghˆPˆŠdl9k€eieŸk€eiz`jx™d<ˆ /

Ê /]

>

ÊUÝ / Ý

•ö Ý % Þ |

¾N

Ù*ÊU#Ú

˜™w`gpx¢gix†kEyzfk+lsk{|gpq‰ dehŽšc`d~@gik<e:ghc / ’‡“{sx ehjk<+gic`ƒ3qBdjB£‡qNgijBc{¢gix¢‰dx|gh{|giŸ<j„k€cfgh{…x

+gpx|qBl|gi~@ghcfk<c{ogpxjyz k€e {|d > ÊÝ > ö #UÝ$%Þ }nw`gpqsw‡gpxcfjBƒk€{|giŸ<jnˆŠdlXk<eheŸk€eiz`jxd€ˆ]Ýçk€c 

ޚ’Ä9jcfqNj {…w`jBl…j¢k€l…jQcfdl…dšd€{sxnk€cfÙOÊ#ÚXgpxX{|l…z`j ˆŠd<l9k<cŽ„Ÿk<ehz`j¢d<ˆ / ’

" ¥š° ^+]¬{z6“”cœ{|w`j„eigic`jk<l qk<x|j / ¿ d<l3@

¾

{|w`j — ʚ»%x|qswfjB~@jqBd<gicfqNgp+jxR}ngh{|w

{|w`j — Ê<mGx…qsw`j~j’

| } ՟»›9Ò×NÖPœ„¤9Ñ:Õ=Ô~

€cäå ‚f脃7ê çêt…

˜Ûdxv{…zf+Ž{|wfjRx‚{…k€¡fgheigŒ{vŽd€ˆ5{…w`jRx…qsw`j~jxcšz`~@jBl…gpqBk€eieiŽ<–<}XjRqBd<~@‰`z+{…j9{…w`j7“

tjl|l…d<l†O

d<ˆ

{|w`j¢cšz`~@jl|gpqBk<ex‚dehz+{…ghdcfxXˆŠd<l9`gŒ‘/jBl…jBc{9Ÿk€eiz`jx™d<ˆ†Ý«k€cfޚ–f}nw`jl|j

O † Ã

ΔÍ

Ò

ÁÆÏ

‹

͈‡

‡‡É IE‰

Á > ©Ù

¦%ÁŠ

³ Ú ‡‡‡ D‹

ba

k€cf ³

È=Œ†È/’E“”cç{|w`j‡eigic`jk<l†qBkx‚jv@

¾ }Xj„zfx|j@{|w`jDjNÍ`k<qN{†x‚dehz+{…ghdcÂk€cfgicç{|w`j

c`d<cfehgic`jk<l9qBkx‚j„@𖠾 k@x|d<eiz+{|gid<c=d<c=k@Ÿ<jl|Ž‡fc`j¢ƒ<l…gp›ÙO}ngŒ{…w H # ¾<¾ Úok<x9l|jBˆŠjBl…jBcfqBj

x‚dehz`{|gid<c’˜™w`jžqBd<~@‰`z+{sk{…ghdcfx™}ojl|j +dc`j¢zfx|ghc`ƒ{…w`j ˆŠd<eiehd}ngic`ƒ‰fk€lsk€~@jB{|jBlsx

ß

¾NöšÀ <

¾Ni¿

Ü

¿¾<¾

³

¾NÀ<¾ âÀfN

“”c†¹7ghƒz`l…j â {…w`jjl|l…d<l0O

gixP‰`eid€{‚{…j¢gicž{…w`j Ý > Þt‰`eik<c`jRÙ*x‚jjRÙ ¿¾ Ú5ˆŠd<lPjk<qswžqBeikx|x|gpqBk€e

x|qswfjB~@jQk<cf„gicD¹7giƒ<z`l…jRʞˆŠd<lX{|w`j qNd~@‰fk<qN{Xx…qsw`jB~@jxˆŠd<lo{|wfjReighcfjk€l™qBkx‚jÙ=@

¾ Úok€c 

{|w`j†c`d<c`eigic`jk<l™qk<x|j„Ù=@

¾N¾â

ځ’:ÆRgŒ‘/jBl…jBc{Tx…qBk€eijx™}XjBl…j zfx|j(ˆŠd<l™{|wfj¢jBl…l|dlXd<ˆqBeikx|x|gpq

k€cf(qNd<~@‰fkq{ox…qsw`j~jxB’ Ø jTc`d<{|gpqNjn{…wfk{o{|w`jQx|qsw`j~@jx_4€¡ jwfkŸšghdz`lgpxx|gi~giepk€lgic„¡d€{|w

qBkx‚jxEƒ

ÿ

¹˜T»™u‚ƒ5˜™w`jqNdcf+gh{|gid<cfxÙ ¿<¿ Ú9k<cf•Ù ¿â ÚnˆŠd<l9{…w`j@xv{sk€¡`giehgh{vŽ.k€cf=cfd<c+tdx…qNgiehepk{…ghdc

qk€c3¡jžˆŠdz`cf›k€ƒk<ghc=cšz`~@jBl…gpqBk€eieiŽ<’R¹fd<lReik<l|ƒjžŸk€eiz`jxRd€ˆ¼Ýk€cf&ޖ/{…w`jx…qsw`j~j

gpxzfcfxv{sk€¡`eijTk<cf@dx…qNgiehepk{…ghdcfx:d+qBqBz`l’7˜™w`jRk€l…jk gic„}nw`giqsw@{…w`jRx…qsw`j~jT‰`l|d++z qNjx

kqBqBjB‰+{sk€¡`eijTl…jx|z`eh{…xXgixXŸjBl…Ž@x|~„k€eie*’“”c({|w`jQc`d<cfehgic`jk<lXqk<x|j9{…w`j¢xv{sk€¡`giehgh{vŽ„k€l…jk†gix

(16)

êŒìW éEX ò éJY Y_^A`EYaW ò ò `cb é ë ë õêië ò é õ Yadfe*g%YaW õÇò ` òh è ^*ê ë

ÿ

m™˜T»™u‚ƒ˜™w`jv“

tjBl…l|dlRd<ˆo{…w`gpx¢x|qsw`j~@jgpx epk€l…ƒ<jˆŠd<l¢epk€l…ƒ<jBlQŸk€eiz`jx d<ˆ'Ýo–ÛƒghŸšgic`ƒ

z`cfx…k{…gix‚ˆOk<qN{|d<l…Ž„l…jx|z`eŒ{sxXˆŠd<l™{…w`gix9l|jƒ<gid<c›ÙOdx…qNgiehepk{…ghdcfxsځ’

ÿ

Û¹ÆQ¹ ƒ¹fd<lÛepk€l…ƒ<jŸk<ehzfjxPd€ˆÝD{|w`jojBl…l|dlۃl|d}9x5lsk€‰fgi+eiŽ<’7˜™w`jjl|l…d<lsxk<l|jx|eighƒw{…ehŽ

x‚~„k<eheijBlX{|w k€cE{|w`dx‚j d<ˆP{…w`jžm™˜T»™u(x…qsw`jB~@j’

ÿ

»'t„ƒ/˜™w`j†jl|l…d<lTghc.{…w`jehgic`jk<lRqk<x|jžgixQx|~@k<ehe7qNd~‰ k€l…jE{|d({|w`jd€{…w`jBl qNepk<x…x‚gpqBk<e

x|qswfjB~@jx’“”cD{…w`j¢c`dc`ehgic`jk€lnqBkx‚jQ{|wfj¢jBl…l|dlXƒl|d}9xˆOk<x‚{nˆŠd<lnepk€l…ƒ<jQŸk€eiz`jx™d<ˆ†Ýo’

ÿ — Ê<b1ƒ ˜™w`j†jBl…l|dlXd<ˆ{|w`gpxRx…qsw`j~jžx|w`d}9xngh{…xTƒ<dšd+E‰`l|d‰ jl‚{…ghjxB’ct9d(dx…qNgiehepk{…ghdcfx

d+qBqNzfl–š{…w`jžxv{sk€¡`gieigŒ{vŽ‡l…jBƒghdc(gpxnŸ<jl|Ž„epk€l…ƒ<j k<x™‰fl|j+giqN{|jD¡šŽœÙ ¿Bö ځ’

ÿ — Ê<mJƒ

Ø

jd<¡ x‚jl|Ÿjž{|w k{¢x|qsw`j~@j

—

Êm%ƒ<giŸ<jxQk(x|eighƒw{…ehŽ=¡jN{|{|jBl¢¡jBw kŸgid<lT{|wfk<c

—

Ê<b†’`˜™w`jBl…j¢k€l…j c`d@dx…qNgiehepk{…ghdcfxnk€cf({|wfjžx|qsw`j~@jQgpxnz`c qNd<c +gŒ{…ghdcfk€eieiŽ(xv{sk€¡`eij<’

ÿ —

Ê»7ƒ€bRx‰`l…j+gpq{…j¡Ž†d<z`l{…w`jBdl|jB{|gpqBk€ešl…jx|z`eh{…x7{|wfj9x|qsw`j~@jXgpxz`cfqBd<cf`gŒ{…ghdcfk€eiehŽ

xv{sk€¡`eij¢k€cfEc`dc+tdx…qNgiehepk{…d<l…Ž<’Û“”cE{…w`j¢c`d<cfehgic`jk<l9qBkx‚j„Ù=@ð–

¾

Úo{|w`j¢jl|l…d<lXgpxnjBŸjBc

x‚~„k<eheijBl†{|wfk<cÂ{|wfk€{d<ˆ — Ê<b Ì — ÊmR’Û“”cÂ{|w`jEeighc`jk€lqBk<x|jœÙy@

¾

Ú {…w`j(l…jx|z`eh{gpx

gi`jBc{|gpqBk<e/{|d@{|w k{nd€ˆ — Ê<m ÙOqˆv’ — jB~„k€l…r

À

ÚN’

0 10 20 30 40 50

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

a=0

0 50 100

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

a>0

0 10 20 30 40 50

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 50 100

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 10 20 30 40 50

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 50 100

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 10 20 30 40 50

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 50 100

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

FTCS

BTCS

LFDF

CN

¹7giƒ<z`l…j

â

ƒo»oepk<x…x‚gpqBk<e/x…qsw`j~jxEƒA“

tjl|l…d<lXghcE{…w`j3Þ

> Ý >

‰`eik<c`j<’

(17)

è€é|êië+ì ífî è€é…ë`êñò í`ó è€ë+ì òsõ

0 10 20

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

a=0

0 20 40

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

a>0

0 10 20

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 20 40

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 10 20

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

0 20 40

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1

α

r

R3A

R3B

R3C

¹Pgiƒ<zfl|j¢Êƒo»od~@‰fk<qN{nx…qsw`j~jxEƒA“

tjl|l…d<lXghcE{…w`j3Þ > Ý > ‰`eik<c`j<’

»od<~@‰fk<l|gic`ƒ{|w`jqNd<~@‰`z`{…k{…ghdcfk€eÛl|jx‚z`eh{…x9d€ˆ7{|w`j†`gŒ‘/jBl…jBc{Rx…qsw`jB~@jx™}oj†qk€c=~„k€rj

{|w`j(ˆŠd<eiehd}ngic`ƒ.d¡fx‚jl|Ÿk€{|gid<cfx’(“”cç{…w`j(x|~@k<ehel|jƒ<gid<cçd€ˆ™{|wfjðÝ > Þt‰`epk€c`j–7}nwfjBl…j@{|w`j

¹˜T»™u@x|qsw`j~@jngixoxv{sk€¡`eij<–<{|w`jnjl|l…d<l:d<ˆ/k<eheqNepk<x…x‚gpqBk<e`x|qsw`j~@jxgpxk€¡d<z`{:{|w`jTx…k€~@j’7˜™w`j

»'t x…qsw`jB~@j@ƒ<giŸ<jxQ{|w`j‡¡jx‚{žl…jx|z`eŒ{sx¢d€ˆnk€eie{|wfj(qNepk<x…x‚gpqBk<e:x|qsw`j~@jx’E»od<~@‰fk<l|gic`ƒD{|w`j

qNepk<x…x‚gpqBk<eQÙ*¹˜T»™u/–m™˜T»™u/–AÛ¹ÆR¹X–o»'tQڞ{…dœ{…w`j=w`giƒ<w«dl…`jBl@qNd<~@‰fkq{„x|qswfjB~@jx–:}oj

c`d€{…giqBj {|w`jžx|z`‰jBl…ghdl|gh{vŽ„d€ˆ7{|wfjžqNd<~@‰fkq{9x…qsw`jB~@jx’:˜™w`jBŽDk€l…jQƒjBc`jl…k<eheiŽ‡x|ghƒc`gŒqBk€c{…ehŽ

~@d<l…jRk<qqNz`lsk{…jn{|w k€c„{…w`jRqBeikx|x|giqk€ex|qsw`j~@jxTÙ*+z`jR{|d†{|w`jghlXw`giƒ<w`jld<ls+jlsÚw<{…w`jBŽ‡x|w`d}

c`dždx|qBgheieik€{|gid<cfx:k<cf{|w`jghlozfx|jngixc`d€{l…jx‚{|l…giqN{|j¡šŽxv{…l|dc`ƒ¢xv{sk€¡`giehgh{vŽ@qNd<c +gŒ{…ghdcfx’7˜™w`j

jBl…l|dl†+gh‘jl|jcfqNjD¡ jB{v}ojjBc — Ê<b k<cf — Êm gpxghcfx|giƒ<c`gh qBk<c{†}nw`jBl…jkx — ʚ» ‰`l|dŸšgp+jx

jBŸjBc=¡ jB{‚{…jBl9l…jx|z`eh{…x™gicD{…w`j c`d<cfehgic`jk<l9qBkx‚j’

€cä 捂Ž¼í&czí&7ë‚í

˜™w`j {|l…z`cfqk{…ghdcEjl|l…d<l™ƒghŸjBcE¡Ž(jNÍ+‰`l…jx…x‚gid<cÙOÊUÚXl|j‰`l…jx|jBc{…xo{|wfj¢‰ dghc{v}ngpx‚j¢jl|l…d<l™gic

k€‰`‰fl|dÍ+gi~@k€{|gic`ƒ„{…w`j`gŒ‘/jBl…jBc{|gpk€e7jyzfk€{|gid<c«ÙO¡`z+{ c`d€{ c`jqNjx…x…k€l…gheiŽ‡{…w`jx|d<eiz+{…ghdc چàÊ

Àá

’

Ø jo‰`l…jx|jBc{Ûgic¢{|w`gpxÛx|jqN{|gid<cžkncšz`~@jl|gpqBk<ex‚{|zf+ŽR{|dQqNd~‰fz+{|j:{…w`jdl…`jBld<ˆ+qBd<cšŸ<jl|ƒjBcfqBj

d€ˆ{|wfj — Ê»Gx…qsw`jB~@j<’XbTx|Žš~‰`{|d€{…giqk€eiehŽ–f}Xj¢jNÍ+‰jq{T{|w`jž‰d<gic{v}ngix|j¢jBl…l…d<l™{|d‡qBd<cšŸ<jl|ƒj

k<x

O

63Ã

â

ˆŠd<lTx|d<~@jFß k€c 



l…jB‰`l…jx|jBc{|gic`ƒ@k@qNd<c xv{sk€c{’:˜™w`gpxngi~‰fehgijx

ehdƒÙ­O

Ú eid<ƒÙ

 Ú ßGeid<ƒÙ Ã Ú N

Ä9jcfqNj–R{…w`j•+dz`¡`eijNtehdƒk€l…gh{|w`~@gpq3‰`eid€{…O

k€ƒk€gicfxv{

Ã

x‚wfd<z`epG¡j•k<x|Žš~@‰+{|d<{|gpq3{…d¶k

(18)

V

êŒìW

ð

éEX ò éJY ðQZ3[]\

Y_^A`EYaW ò Z ò `cb

ð é \ ë ð ë õêië ò \ é ã õ\

Yadfe*g%YaW

ð

õÇò ` òh è \

^*ê

ð ë ¿

10 −1 10 0

10 −6 10 −5 10 −4 10 −3

h

l 2 −error FTCS

BTCS LFDF CN R3A R3B R3C

¹7ghƒz`l|j

ö

ƒzt9z`~@jBl…gpqBk€eP»odcšŸ<jBl…ƒ<jcfqNjƒ/“

tjl|l…d<l™Ÿ+xB’

à ’

{…w`j†dl…+jl9d€ˆkqBqNzfl…kqNŽ(d<ˆ{|w`j~jB{|w`d+5’ Ø jl|jBˆŠjBln{…dD¹7ghƒz`l…j ö ˆŠd<lT{|w`jl|jx‚z`eh{…xT}ngŒ{…w

{…w`j¢‰fk€lsk€~@jB{|jBlsx

@

¾N¾â

ß

¾NöÀ <

¾Ni¿

Ü

¿¾<¾

³

¾N¾<¾

¼Ê½

ÀfN

˜7k€¡`eij ¿ x‚z`~@~„k€l…gi‹Bjxn{|w`j~@k€Í+gh~„k€e–~gic`gi~„k€ek€cf.kŸjBlsk€ƒ<j cšz`~@jBl…gpqBk€e7qNdcŸjBl…ƒ<jcfqNj

lsk{…jx’ Ø jd<¡fx|jBl…Ÿ<j {…wfk{R{|w`jczf~jl|gpqBk<eÛqNdcšŸ<jBl…ƒ<jcfqNj¢lsk{…jxnl…d<zfƒ<w`eiŽEqBd<l…l|jx‚‰d<c ({…d

{…w`j¢d<ls+jBl™d<ˆP{…w`jžx|qswfjB~@jx’

ß

âL

® ß â ÁÊ ß

E‘

¹7˜T»™u

⠒ À â ’Ê ö ⠒ö #

m™˜T»™u ⠒û½ ¾ ⠒¾ ¼ ⠒ö Ê

P¹7ÆQ¹

⠒¾ # ¿ ’

Àâ ¿ ’# ¾

»'t ⠒û½Q ⠒ ö ⠒<¼

—

Êb Ê`’

öf¿ ⠒½ À

Ê`’

¿ Ê

— Ê<m Ê`’öâ ⠒½ À Ê`’¿ Ê

—

Ê»

ö ’â ⠒¼# Ê`’

â ¼

˜Pk<¡`ehj

¿

ƒ»odcšŸ<jBl…ƒ<jcfqNjQlsk{|jxB’

€cä*㠒“Õë0êaí&7ë…

bTc6gi~@‰ dl‚{sk€c{E‰d<gic{Egicd<z`l.qNd~@‰fk€l…gix|d<cÁgpxD{|w`jçjB£„qBghjcfqNŽÁd<ˆž{|wfjçx…qsw`j~jxB–Rg*’j<’

(19)

¿ ã ä`åçæ

è€é|êië+ì ífî ä`ïð

è€é…ë`êñò í`ó äfôXæ

è€ë+ì òsõ

10 −5 10 −4 10 −3 10 −2

10 −2 10 −1 10 0 10 1 10 2

l 2

. . .

error

CPU time

FTCS BTCS LFDF R3A R3B R3C

. CN

¹7ghƒz`l…j À ƒ:¸:£‡qNgijBcfqBŽ‚ƒo» ² Ént{|gi~@j Ÿšx’z“

tjBl…l…d<l’

k<x†}ojehe™k<x¢‰fl|dƒ<lsk€~@~@ghc`ƒ3`jB‰jBcf+jc{’œ˜™w`jD+gŒ‘/jBl…jBc{@x|qsw`j~@jxž}XjBl…j„gi~‰fehj~jc{|j

ghck€c•jB£‡qNgijBc{@k<cf•qBd<cfx|gix‚{|jc{~„k€c`cfjBlghc•dl…`jBl†cfd€{{|dœ¡fgikxk<cŽçd<ˆn{|w`j~E’˜™w`j

qNd~‰fz+{…k€{|gid<c={|gi~@jxTl|jqNd<ls+jDgicfqBehzf`j¢{|w`jž{|gi~jžˆŠd<lT~„k€{|l…gŒÍ=x‚jB{|z`‰fx–fgicšŸ<jl…x|ghdcfxnk€c 

¡ dz`cf`k<l|Ž¢qBd<cf+gh{|gid<c†jBŸk€eizfk{…ghdc’PbTehešl…jx|z`eh{…xP}ojl|jXqNd<~@‰`z`{|jžd<c¢{…w`jXx…k€~@jo~@kqsw`gic`j<’

˜™w`j=cšz`~†¡jBld€ˆQd<‰jBlsk{|gid<c x¢{|dçx|d<eiŸ<j‡{…w`jE{|l…gi`gik<ƒ<d<c k€eox|Ž+xv{…jB~„x}ngŒ{…w•{…w`j.˜™w`d<~„k<x

b9eiƒ<dl|gh{|w`~ gpxQd€ˆdl…`jBl„á‡ÙH ÚÙOx|jBj@x‚jq{|gid<cÊ‡ˆŠdlT{…w`j„+jN c`gŒ{…ghdcœd<ˆ H ځ’Ä9jBc qNj{|w`j

+d<~@gicfk€c{nˆOkq{|dlnghcD{|wfj¢l|z`cfc`ghcfƒ@{|gi~j¢gpxX{|wfj¢~@k€{|l…gŒÍEx‚jB{|z`‰–fc`d€{n{…w`j¢ghcšŸjBlsx‚gid<c’

Ø j.qBd<~@‰`z+{…jx‚dehz+{…ghdcfxd<cƒ<l…gifx†}ngh{|w H

¿¾ ‾ …Ê ¾ ö¾ ’›“”c¶¹7ghƒz`l…j À }oj

‰`eid€{‚{…j{…w`j.l|jeik€{|giŸ<j“

tjBl…l…d<lŸ<jBlsx|zfx{|w`j›» ² É {|gi~j3ˆŠd<l@{|wfj3+gŒ‘/jBl…jBc{‡ƒl|gp`x„k€c 

x|qswfjB~@jx’ Ø jx|jBj¢{…wfk{RˆŠd<lR`Í+j.jl|l…d<l™{…w`j@qNd<~@‰fkq{Rx…qsw`j~jxn{sk€r<j†ehjx|x¢» ² É%{…gh~@j

{|wfk<c•{…w`jDqBeikx|x|giq(x|qsw`j~@jx’›ÆRz`j({…d3{…w`jDx‚{|l…d<c`ƒ›xv{sk€¡`giehgh{vŽÂqNdcf+gh{|gid<c•d<ˆn{|w`j=¹˜T»™u

x|qswfjB~@j<–gh{„gixƒ<jc`jBlsk€eieiŽŸjBl…Ž{|gi~j=qNdcfx|z`~@ghc`ƒ ’¹`d<l`Í+jÂ{|gi~@j({…w`j=jBl…l|dl†d<ˆn{|w`j

qNd~‰ k<q{@x|qswfjB~@jx†gixk€ei}XkŽ+xžx|giƒ<c`gh qBk<c{|eiŽx|~„k€eiehjl’œ˜™wfj‡{|wfl|jjDqNd~@‰fk<qN{x…qsw`jB~@jx

k€l…jž{…w`j~@dxv{QjN£‡qNgijBc{ d<c`jxT}nw`jl|j†{|w`j — ʚ»x|qsw`j~@jx|jBjB~„xT{|dE¡ j@x|z`‰ jl|gid<lT{|d({|w`j

— Ê<bËk<cf — Ê<m6x|qsw`j~@jx’

ÉTx|ghcfƒ„{|w`jx|k<~@jžŸk€eiz`jxTd€ˆzÝ!k<cfޚ– {|w`jgh~@‰`eigiqBgŒ{¢x…qsw`jB~@jx_4ÛÙ*mX˜T»™u5–5»'t†–

—

Êbž–

—

Ê<mQ–

—

Ê»™Ú=qBd<~@‰`z+{sk{|gid<c{…gh~@j•gpx3c`d<{œ~†z qswepk€l…ƒ<jl={|wfk<c{|wfk€{œd€ˆ@{|wfj•jNÍ+‰`eigiqBgŒ{

x|qswfjB~@jxEÙO¹˜T»™u/–zP¹7ÆQ¹:ځ–¡`z+{@{…w`jEkqBqBz`l…kqNŽ›d€ˆ9{…w`jEqBd<~@‰fkq{x…qsw`j~jx†gpxx‚giƒ<c`gh`t

Referenzen

ÄHNLICHE DOKUMENTE

incompressible Navier–Stokes equations, finite difference, WENO schemes, non- incremental projection methods, incremental projection methods, PSPG-type stabilization.. The research

Using an adaptation of the stochastic Taylor expansion to control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type

Using an adaptation of the stochastic Taylor expansion for control systems we construct Taylor schemes of arbitrary high order and indicate how derivative free Runge-Kutta type

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed.. The equation arises in the modeling of option prices in

A nonlinear Black-Scholes equation which models transaction costs arising in the hedging of portfolios is discretized semi-implicitly using high order compact finite

[r]

Therefore Cartier duality is a contravariant functor from the category of finite flat commutative affine group schemes to itself.. Moreover this functor

The aim of the lecture course is the classification of finite commutative group schemes over a perfect field of characteristic p , using the classical approach by