A combinatorial approach to jumping particles I: maximal flow regime

HAL Id: hal-00159614

https://hal.archives-ouvertes.fr/hal-00159614

Submitted on 5 Jul 2007

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A combinatorial approach to jumping particles I:

maximal flow regime

Enrica Duchi, Gilles Schaeffer

To cite this version:

Enrica Duchi, Gilles Schaeffer. A combinatorial approach to jumping particles I: maximal flow regime.

2004, pp.12. �hal-00159614�

021436587:9<;>=@?A3CBD9E=GFH=JI>IKB3C=G1MLN? 3PO Q>5RIK9;TSRI>=JBA?9<1UF<VXWZYM[\5R=^]_95R=JFH`K3@abBVcS695dV

eKfUg-hi&jlknmci'ochCjAfcprqnhsstuwv4i'ocj tyx^t&g

z^{y|~}y&€} ‚„ƒ…‡†ˆ‰‹ŠŒ&ŽŒE‘’UT“E”…yŠ‰‹•<‘Ž—–_”'•yE˜4”™@Œ&Ž‘†‰‹“E˜‹EŠ^š›y–_Œy‰‹…yœ:”…Ž\‘”<’ž”™@“EE˜‹˜‹ŠŸM“<Ž˜‹˜‹<•‡‰‹…

Œyˆ 'Š‰‹“EŠA†ˆyU”…U•y‰‹–_E…yŠ‰‹”…&Ž˜†!”†~Ž˜‹˜ „ŽŠ! '–_–_†‘!‰‹“D¡'“E˜‹›yŠ‰‹”…„Œy‘”-“E<Š!ŠU¢#£A¤J¥'¦¨§4©ª«\”‘!¨Œy‘!E“<‰‹Š!E˜ G’UU•y<Ž˜

’X‰†!ˆT†!ˆyJ£A¤J¥'¦¨§n’X‰†!ˆ_†’U””‘4†!ˆy‘E^† 'ŒEŠX”™ Œ&Ž‘!†!‰‹“E˜‹<Š<Ÿ'’X‰†!ˆT”‘4’@‰†!ˆy”›†c¬”›y…•&Ž‘‰‹EŠŸ‰‹…­†ˆJ–TŽ¡‰‹–TŽ˜

®&”<’¯‘!<œ‰‹–_ªA°&‘!”–Z†ˆ^Œ”‰‹…-†c”™²±‰‹’³”™A“E”–¬y‰‹…yŽ†”‘!‰‹“<Š4ŽC‘!E–TŽ‘´Ž¬˜‹X™µ<Ž†›y‘!^”™²†ˆyEŠ!G«—Ž‘´”<±_“~ˆ&Ž‰‹…yŠ

‰‹ŠU†!ˆ&Ž†@†!ˆy K‰‹…-±”˜±6¶UŽ†Ž˜·Ž…:…-›y–¬E‘Šc‰‹…>Š±E‘~Ž˜ E…†‘!‰‹<ŠX”™A†!ˆyE‰‹‘@Š†~Ž†!‰‹”…&Ž‘! K•y‰‹Š†‘!‰‹¬y›†!‰‹”…¸ª

¹

Tœ‰±>Ž\“E”–¬y‰‹…&Ž†!”‘‰·Ž˜D‰‹…-†!E‘Œy‘!†~Ž†!‰‹”…ºŽ…y•Ž:Š!‰‹–_Œy˜‹­Œy‘!”'”™X”™c†!ˆyEŠ>”¬yŠE‘!±Ž†‰‹”…Šª—ƒ…n•”‰‹…yœ

†!ˆy‰‹Š¨’U@‘±<Ž˜%Ž6Š!<“E”…y•T‘!”’³”™A“E<˜‹˜‹Š<Ÿ’@ˆ‰‹“~ˆ_‰‹Š4›yŠ!<•T¬ _Œ&Ž‘†‰‹“E˜‹EŠ¨†!”6†!‘~ŽE±<˜%¬&Ž“~´’cŽ‘!•¸ªM¤^ŠUŽG¬ Œy‘!”'•›y“†

’UJŽ˜‹Š”6”¬†~Ž‰‹…KŽ…>‰‹…-†!E‘Œy‘!E†Ž†!‰‹”…:”™ †!ˆyG”-“<“E›y‘!‘E…y“EG”™ †!ˆyG»4‘!”’X…y‰·Ž…>E¡'“E›y‘Š!‰‹”…K‰‹…_†!ˆyJ•yEŠ!“<‘!‰‹Œ†!‰‹”…K”™

†!ˆyG•yE…yŠ!‰† :”™Œ&Ž‘†‰‹“E˜‹EŠX”…\Ž˜‹”…yœ‘”’¯”™“<E˜‹˜‹Š<ª

¼\½

¾

|~¿%À

½

¾

‚JÁJŽ…yŠ^“<†JŽ‘!†!‰‹“<˜‹6…y”›yŠ:E†!›y•y‰‹”…ŠG›y…—–_”'•²Ã<˜‹C•y6Œ&Ž‘†‰‹“E›y˜‹EŠ^Ä-›y‰Š~Ž›'†E…†G˜‹„˜‹”…œ_•¸Å›y…y6˜‹‰‹œ…yŸ

ŽŒyŒE˜Â><…Œˆ- 'Š‰‹Ä-›y:˜‹TŒy‘!”'“EEŠŠ!›yŠ­•¸Å¡'“E˜‹›yŠ‰‹”…n†!”†~Ž˜‹E–_E…†TŽŠ! '–Â†‘!‰‹Ä'›y>›y…y‰‹•‰‹–_<…Š‰‹”…y…yE˜^¢µ£A¤G¥-¦¨§U©~ª

§D˜‹›yŠŒy‘'E“<‰‹Š'E–_<…†­…”›yŠ†!‘~Ž‰†!”…yŠ­˜‹>£A¤G¥'¦D§ÆŽ<±E“:•<›'¡n”›n†!‘”‰‹Š† 'Œ<Š•yKŒyŽ‘!†!‰‹“<›˜‹<Š<ŸUŽE±<“K”›Š~Ž…yŠ

¬”‘•ŠŸ•&Ž…yŠD˜‹U‘Eœ‰‹–_U•yU®&›¡„–TŽ¡'‰‹–TŽ˜ªAÁ„Å›y…„Œ”‰‹…-†D•yU±›@“E”–¬‰‹…&Ž†”‰‹‘!U›y…yXŒy‘”Œ‘‰Â†'c‘!<–TŽ‘!Ä'›yŽ¬y˜‹

•yc“EEŠ¨“~ˆyŽÇȋ…yEŠ¨•yX«—Ž‘´”<±6<Š†¨Ä-›¸ÅE˜‹˜‹EŠM™µ”…†¨‰‹…†E‘±<…‰‹‘D•yEŠD…y”–¬y‘!EŠM•^¶UŽ†Ž˜·Ž…_•&Ž…yŠMŒy˜‹›yŠ!‰‹<›‘ŠM<…†!‘EEŠ

•yG˜‹E›y‘c•y‰‹Š†‘!‰‹¬y›†!‰‹”…:Š†~Ž†!‰‹”…&Ž‰‹‘ª

É ”›yŠ^•”…y…y”…yŠ@›y…yC‰‹…-†!E‘Œy‘'†~Ž†!‰‹”…\“E”–¬y‰‹…yŽ†”‰‹‘!GE†@›y…yCŒy‘!E›±CŠ‰‹–_Œy˜‹J•G“<EŠ@”¬yŠE‘!±Ž†‰‹”…ŠªX¶4

™#Ž‰‹Š~Ž…†Ÿ…y”›yŠc‘'±DE˜‹”…yŠ@›…yG•yE›¡'‰Ã<–_^‘~Ž…yœ EG•yJ“<ŽŠ!<Š<Ÿ&›†!‰‹˜‹‰‹Š'EEŠcŒ&Ž‘c˜‹<ŠcŒ&Ž‘†‰‹“E›y˜‹EŠcŒ”›y‘c‘†!”›y‘…yE‘XE…

Ž‘!‘‰ÃE‘ª É ”›yŠJE…—•¨ÂE•y›‰‹Š”…yŠJE…Êy…w›…y‰‹…†E‘!Œy‘'†Ž†‰‹”…H•y6˜ÅŽŒyŒ&Ž‘!‰†‰‹”…—•¸Å¡'“E›y‘Š!‰‹”…H»4‘”’X…y‰‹E…y…•&Ž…yŠ

˜·Ž6•yEŠ“E‘‰‹Œ†!‰‹”…:•6˜·Ž6•yE…yŠ‰†'G•yEŠXŒ&Ž‘!†!‰‹“E›y˜‹EŠ@˜‹G˜‹”…yœ•¸Å›y…yG˜‹”…yœ›G‘~Ž…yœ EC•yG“<ŽŠEŠ

˨̇ÍUÎ^ÏÆÐJÑÒ^ÓÔÐJÕAÖy×'Ñ!Ø%Ù!Ú¸Û

ËDÌ!ËD̝ÜHÝ^ڞÞ^ÕÛÑ؞Ïàßcá^Ú ÙEÌ_âäã³åæMçè#è„é-ê%ëMåì#íã-îwçlïnêAíAã'è6êðGñEòMï‡óMìµëôõóMçîöìéè#ã'åHê%ëàçlîê&÷dê%ð_øPé-ã-è#è#å

öæDç&ö­÷ç²å„å<öòMíìµãíùåìµëMé-ã>öæã:ã'çîè#únû²üýå6ì#ëùóMæ¸úAåì#é'å6òMëMíAã-îöæãKëDçïnãnþÿ-ÿ þÿ~þ

! þ&ÿ"þ#$ $ &%')(+*þ Aÿ,-. /ê%î103254687žðê%îTåæê%îö:9'2>èöæê%òô²æùöæã—ïnêAíAã-ècìå­òMåòMçè#è#úºóîã'åã-ë¸öãí³ç²å

çwé-ê%ë¸öì#ë òê%òDåGöì#ïnãKã:;%ê²èµòAöìµê²ë</%ìµö­ì#å„ã:=¸òì>;&çè#ã-ë¸ö:/AçëDíìöì#å„ïnê%îãKé-ê%ë!;%ã'ëìµã'ë¸öCðê%î„òMå/¸öê íAã?Mëã:ìöìµë íAì#åéîãöã

öì#ïnãHç%å_ç@ ç%îBA²ê;ºéæMçì#ëDC'E:ê²ëlç‡åãö_ê%ð3FMç²å<ìé\éê²ëG?Mô%òMîçöì#ê%ëMå:H

I

2J*$ 5þÿ:K3L GBþÿnì#å^çTîê&÷Zêðøºéã'èµèå:/å<ã'óMçîç&öã'íMF ú\øONQPC÷­çè#èå8Röæãèµã-ðöïnê²å<öJç%ëMí—îìµô²æ¸öïnê²å<ö

ê%ëãåGçîãSFDê²îíã-îåT.936„ç%éænéã-è#èìåCêAé-é-òóì#ã'íUF úwê²ëãTóMç%î<öì#é-èµãV/%çëMínãç%éæ‡óMçîöìéè#ã_æMç²åCç:öEú óDãV/VFèç%é$A

ê%î­÷_æìµöãR!å<ã'ãW@ì#ô%òMîãP#T.9

XZY)[]\<^`_bac 2dFMç²å<ìé\éê²ëG?Mô%òMîçöì#ê%ëù÷_ìµöæløfegPünéã-è#èå9

I

2ö­öì#ï‡ãih'e ü/ öæMãHå<úAå<öã-ï ì#å­ì#ëlçUFMç%åì#é\é-ê%ëG?Mô²òîç&öì#ê%ëDCjEVR!üVTMRóDê¸ååìkFMèµú éæê²åã-ë ç&öTîçëMíAê²ïT9

lnmm#lSoipqkr#sBtup.q>v>wnx3y-z{>|}swnqj~pxxv€wpqvk‚.ƒ-„ §¨‘‰‹–TŽ‘ S…†¤ l †ˆ‡¥'E“E”…y•&Ž‘ M‰…Š.…†:ª

‹ snŒS‚Žxpƒ8‘r:Žpx’sBx’„ ¬y‰š<“†!‰‹”…yŠŸ–TŽ‘´”<±T“~ˆ&Ž‰‹…yŠ<Ÿ%¶UŽ†~Ž˜·Ž…\…-›y–„¬%E‘!ŠŸ'…y”…:EÄ-›y‰‹˜‹‰‹¬y‘‰‹›–žŠ†~Ž†!‰‹Š!†!‰‹“<Ž˜ Œyˆ Š!‰‹“<Š<ª

£cˆyCÊ&‘Š†@Ž›†ˆ”‘^’cŽŠXŠ›yŒyŒ”‘†E•—¬ \Ž­œ‘Ž…-†@”™†ˆy6“E‰† :“E”›y…“<‰‹˜A”™D§MŽ‘‰‹Š<Ÿ&Ž…y•\†!ˆyCŠE“E”…y•—¬ :¦4¶JŊXƒ)“3”c§ §D‘”œ‘Ž–\Ÿ

’X‰†ˆ‰‹…K†!ˆyO”XEŠ!Ž‘!“~ˆ:£A‘Ž‰‹…y‰‹…yœ É †’U”‘´5•

–.s${nŽpvkwˆ~—‚.tZ{nv˜ƒpq‚Ž’v>wnx'vkƒˆ™<z:Ž‚}‘:s

Ÿœ‘Ž…-†]“X§,” ɚ¶4£ š)› ……

l š…… ›.œ$› ª

XZY)[\<^`_`c 2TëäãAã-ïnóè#ãnêð6ã:;%ê%è#òAöìµê²ë</U÷_ìöæõø e 9i0­æãç%éöì>;%ãw÷­çè#è^öîìµô²ô%ã'îì#ëôùãç%éæ

öîçëMåìµöì#ê%ë ì#å­ì#ëMíAìé-çöã'í<9

I

Wîê%ïNöìµïnãh>öêDhjN PV/cöæãºå<úAå<öã'ï ã;²ê%è>;%ã'åKðîê%ïNöæãFMç²å<ìénéê%ë?Mô%òîç&öìµê²ë CjER)hnTKöê¯öæãFMç²å<ìé

éê²ëG?Mô%òMîçöì#ê%ëCjEVRh!N PTCç²åGðê%è#è#ê&÷TåH^çëºç%éöì>;%ãT÷­çè#èMìå6éæMê²åã-ëòëìµðê%îïnèµúnç&öîç%ëMíAê²ïdçïnê²ëô—öæMã

ø&N P_÷­çè#è#å„çëMínðê²òî„é'ç%åã'å„çîì#åã-9j0­æã>é-ê%ïnóè#ãöãKïnê íã-èDðê%îøDe"!—ì#å„óîã'åã-ë¸öãínìµë 2Tóó¨ã-ëMíì#%$

Råã-ã&W^ìµô²òîãUPûVT.9

&

9('~ð6öæãç%éöìk;²ã ÷­çè#èGå<ã'óMçîç&öã'å\ç Fè#ç²é$AlóMçîöìéè#ãDRê%ëäìµöå\è#ãðö$T\ç%ëMíÔç ÷_æìµöãnóMç%î<öì#é-èµãDRê²ëõìµöå

îì#ô%æ¸öT/ öæMã-ë öæã\öE÷ênóMçîöìéè#ã'å_å÷­çó<9

) 9*'~ð4öæãTç²éöì>;%ã_÷­çè#èAìåJöæã>èµã-ðö8FDê²îíã-îCçëDí‡öæã_è#ãðöï‡ê¸åEöéã-è#èMé-ê%ë¸öç%ìµëMåGç\÷_æMìöãTóMçîöìéè#ã-/%öæã'ë

öæã—÷_æMìöã—óMçîöìéè#ã\èµãç;%ã'å„öæãHå<úAå<öã'ï çëMí ìµöTì#å­îã-óMè#ç²éã'í F úùçUFèç%é$AºóMç%î<öì#é-èµãV9

+

9('~ðJöæãwç²éöìk;²ã—÷ç%èµè@ì#å­öæãwîì#ô%æ¸ö1FDê²îíã-î>ç%ëMí³öæã îì#ô%æ¸öïnê¸åEöTéã'èµè^éê²ë²öçì#ëMåTçFMè#ç²é$AùóMç%î<öì#é-èµãV/

öæã'ë öæã&Fèç%é$AùóMçîöìéè#ã\èµãç;%ã'å„öæãHå<úAå<öã'ï çëMí³ìµö>ì#å­îã-óMè#ç²éã'í F úùç‡÷_æìµöã—óMç%î<öì#é-èµãV9

,

9.->öæã'î÷_ìåã:ëMêöæMìµëônæMç%óó¨ã-ëMå:H8CjEVRhN PTjeJCjER)hnT9

2>å^ì#èµè#òMå<öîç&öã'íF úW^ì#ô—9/G/Fèç%é$A—óDçîöìéè#ã'åXöîç;%ã-è²ðîê%ï8è#ãðöJöê:îì#ô%æ¸ö#/&÷_æìµè#ã÷_æìöã„óDçîöìéè#ã'åJíAêKöæãê%óó¨ê²åìµöã-9

68=¸òì>;&çè#ã-ë¸öè#ú>ê²ëã6é'çë&; ìµã'÷Z÷_æìöã6óMç%î<öì#é-èµãåcç%åcã-ïnóAöEú—éã'èµèå90>ã'îîì#íMçi.<21(3446587 9 :4;=<?>@7BADC óîê;²ã'í

öæMã\ðê%è#èµê&÷_ì#ëônëìéã\îãå<òMèöå_çF¨ê%òAö_öæã—ã;²ê%è#òAöì#ê%ë êð@öæãHåúAåEöã-ï CjEHçðöã'îTç è#ê%ëMô öìµïnãV98W^ìµîåEö#/

R}PV9kPT 76îêVF<RC

E

R)hnT­é-ê%ë¸öç%ìµëMå_üUFèç%é$AùóMçîöìéè#ã'åTFEBG

HJILK

P

MONP

RQ

÷_æã'îã MONP

 e 

NP

TS NP

NP

 U

ìå­öæMãRøN PT~öæWVçöçèçë ë òïiF¨ã-î#9O@lê²îã:ô%ã'ëã-îçè#èµúV/¸ðê%î>ç%èµècüYX[Z%X ø'/

R}PV9/VT 7Cîê-F<RC E R)hnT­é-ê%ë¸öç%ìµëDå\ZFèç%é$AùóMçîöìéè#ã'åTFEBG

HJI@K



N]P

RS NP



^ US NP



ND_

^ U

MONP

 Q

÷_æã'îã:öæMã—ë¸òMï‡ã'îçöê²îå­çîã\é-çè#è#ã'ía`>ç%îçyú¸çëMçwë¸òMïiF¨ã-îå9

0­æã:ï‡êAíAã'è4ì#å­çi?Mëìµöã\åEöç&öãM@ ç%îBA²ê;‡éæMç%ìµëù÷_æì#éæ³ì#å­é-èµãçîèµú‡ã-îô%êAíAìé>åêHöæMç&ö­öæMãKóîã; ì#ê%òMå„è#ìµïnìµöå­çîã

ì#ëºðç%éö„öæã:óîêVFMçFMìµè#ìöìµãå6ê%ðUöæã—åçïnã>ã:;%ã'ë²öå6ì#ëºöæã:òëì=²òMã:å<öçöì#ê%ëMç%îúníAìåEöîì>FòAöìµê²ëùêðUöæã\éæMçì#ëb3<dceB9DC9

@lê²îã ô%ã'ëã-îçè#èµúV/f0Kã-îîìíç .U21 óîê; ìíAã'írãAóîã'åå<ì#ê%ëDåwðê²înöæãlå<öçöì#ê%ëMç%îúÆóîê-FDçFì#èµìµöì#ã'å:9 4 ìµëDéã öæã'ìµî

ê%îì#ô%ì#ëMçè¨÷„ê²îBAºç ë òïiF¨ã-î_ê%ðXóMç%óDã'îåæMç;²ã:ç%óóDãçîã'íùóîê; ìíAì#ëô ç%èöã-îëMç&öìk;²ãKóîê êðå­çëDíðòîöæã'î_îãå<òMèöå„ê²ë

éê²îîã-èç&öìµê²ëMå:/Aöì#ïnãwã;²ê%è#òAöì#ê%ëDå/Dãöé-9\'~ö:åæê²òè#íDF¨ãwïnê²îã'ê;%ã-îTå<öîã'ååã'í³öæMçö>öæãwïnêAíAã-è^÷ãHóîã'åã-ë¸öãí¯ì#åKç

åóDãéìçè@é'ç%åãHçïnê%ëMônöæã ïnç%ë ú³ã ìå<öì#ëô;&çîì#ç%ë¸öå_ê%ðGç²å<ú ïnïnãöîìé\ãéè#òMåìµê²ëlóîê é-ã'åå<ãå9O'ë óMç%î<öì#é-òèçî_÷ã

æMç;²ãTîãåEöîìéöã'íê²òî­ç&ö<öã-ë¸öì#ê%ëùæã-îãTöêwöæã>ïçgAì#ïçèhMê&÷ îã'ô%ì#ïnã-/¸÷_æã-îã>óMç%î<öì#é-èµãå6ã'ë¸öã-î#/¸öîç;%ã'èDçëDíãAìµö

ç&ö„öæã:åç%ïnãTîçöãiR!å<ã'ã>æê&÷ã;²ã-î\34aiejC¨ðê%îçëã öã-ëMåìµê²ëºêð4öæãKóMîãå<ã'ë²ö„÷„ê²îBAwöêwô²ã-ëã'îç%èîç&öãåT.9TkTã'éã'ë¸ö

ç%í;yç%ëMéãå„ç%ëMíùçFì>Fè#ê%ô%îçóMæ¸úné-ç%ëQF¨ã>ðê²òëMíùðê%îìµëDåEöçëMé-ãKì#ëºöæã—çîöìéè#ãl34m*ineBAC9o$ê ê-AAå„ç-FDê²òAö­óMçîöìéè#ã

óîêAéã'ååã'åGçîã@3piUÐ6ß 7XË:m6Ñ!ÓBq rC9tsTê&÷ã;%ã'î:/öæãTîã-ïç%îBA&çFMèµã_çóMóMçîìöìµê²ënêðVç&öçèçënë òïiF¨ã-îåGìåCëê%ö6ã'ç²å<ì#èµú

òëMíã-îåEöê¸êAíùðîê²ï öæã—óMîê êðåì#ëùöæã\óæ úAå<ìé-å­è#ìöã-îç&öòîãV932KåðçîTç²å„÷ãMA ëê&÷&/¸öæãå<ã\óîê êðåîã'èµúùã-ìµöæã'îê²ë

ç ˜Dÿ vu/ê%î­ê%ëlçxw5.)(U&ÿ— .yu/F¨êöæDFDã'ìµëônöæã-ë¯óîê;%ãíF úùç‡îã'éòMîåìµê²ëùê²ë ø'9

âõãùóîê²óDê¸å<ãæMã-îãºçléê%ïFì#ëMç&öê%îì#ç%èCíAã-îìk;&çöì#ê%ëZêðöæãå<ã³åEöç&öì#ê%ëDçîú óîê-FMç-Fìµè#ìµöì#ã'å:9%'ëäðç%éöw÷„ãºíã'çè

÷_ìµöæ‡÷_ìöæ‡ç\åèµì#ô%æ¸öè#ú—ïnê%îã„ô²ã-ëã'îç%è²ïnêAíAã'è/&öæãöæîã-ã­óMç%î<öì#é-èµãZ03254G67z3{ùÒ^áTqBqn:46;.<?>@7BADC930­æMì#åJïnêAíAã-è

ìå_ç@ çîA%ê;éæMç%ìµëfCÆöæMç&öTã öã-ëDíå CjE\öê öæîã-ãMA ìµëMíMå_êð^óMçîöìéè#ã'å:H

I 2 FMç²å<ìéCé-ê%ëG?Mô²òîç&öì#ê%ë—ìå^ç_îê&÷Zê%ðøºéã-è#èå/åã-óMç%îçöãí5F ú\øONP„÷ç%èµèå8Röæãèµã-ðöïnê²å<ö^ç%ëMí—îìµô²æ¸öïnê²å<ö

ê%ëãåJçîã8F¨ê%îíAã-îåT.9]6„ç%éæ é-ã-è#è ì#åJêAé-é-òóì#ã'íF ú—ê%ëãóMçîöìéè#ã-9j66ç²éæHóMç%î<öì#é-èµãæMç%åGç>öEú óDãV/

I

R)Fèç%é$AT/

|

/ê²î.}QR÷_æìµöã#T./Dç%ëMíºöæã'åã\öæMîã'ãKöEú ó¨ã'åTç%îã:ê%îíAã'îãí`H

I@~ | ~ }!9

XZY)[]\<^`_€`c 2dFMç²å<ìé\éê²ëG?Mô%òMîçöì#ê%ëù÷_ìµöæløfegPéã-è#èå9

θ ’

θ’

θ

θ

XZY)[]\<^`_]c 2Tëlãçïnóè#ã:ê%ð@ã:;%ê%è#òAöìµê²ë³÷_ìµöæ¯ø+e[‡ðê%î­öæã\öæMîã'ã:óDçîöìéè#ã:ïnêAíAã'è9

I

2öTã'ç²éæ¯å<öã'ó</öê%ô²ãöæMã-îT÷_ìµöæ¯öæãwåã-è#ã'éöìµê²ë¯êð^öæãwç²éöìk;²ã—÷ç%èµè/Mçéæê²ì#é-ã—ì#å_ïç²íAã&FDã-öE÷„ã'ã-ë¯öE÷ê

öîçëMåìµöì#ê%ëùîòè#ã'åçëDí/÷_ìµöæ¯ã#=²òDçèUóîê-FMç-Fì#èµìµöEú-9'0­æã'ë³ðê²òîTé-ç²å<ãå_çîì#åã-H

&

910­æãTç%éöì>;%ã_÷­çè#èMåã-óMç%îçöã'å^öE÷ê—óMç%î<öì#é-èµãåGå<òMéænöæMç&öGöæã_öEú ó¨ãTêð4öæMãTèµã-ðö6ê%ëMã_ì#åGèçîô%ã'îJöæMç%ë

öæãwöEú óDã êðGöæMãwîìµô²æ²öKê%ëãV950­æã'ëlöæãwöE÷„êùóMçîöìéè#ã'åKå<÷­çó<9 'ë êöæã-î>öã-îïå/DöæMãwó¨ê²åå<ì>Fè#ã

èµêAé'çè4öîçëDå<ìµöì#ê%ëMå­ç%îê²òëMíùöæãHç²éöì>;%ã\÷­çè#ècçîãUR

I } G } I

T./<R

I

| G | I

T/ç%ëMí R| } G } | T.9

)

9S0­æãKç²éöìk;²ã>÷­çè#èDìå„öæã:èµã-ðöuF¨ê%îíAã'î:9T'~ðcöæMãKè#ãðöï‡ê¸åEö_óMç%î<öì#é-èµãKì#å„÷_æìµöãKöæã'ëùìµöãAìöå/AçëDíºìµö

ì#å­îã-óMè#ç²éã'íQF úùçUFè#ç²é$Aê%îTç%ë

|

óMç%î<öì#é-èµã:÷_æã-ë öæã\îòè#ã\ì#å­îã'åó¨ã'éöìk;²ã-è#únê%î9t'~ð@ì#ëMå<öã'ç²í

ìöTìå_çë

|

óMç%î<öì#é-èµã—çëDíùöæã—îòèµã\ìå

/AöæMã-ë¯ìöTãAìöå_çëMí ìå­îã'óè#ç²éãíQF úùçUFèç%é$AºóDçîöìéè#ã-9

+

910­æãùç²éöì>;%ã÷­çè#è6ìå—öæãºîìµô²æ¸öiFDê²îíã-î#9'~ð_öæãºîìµô²æ²öïnê²å<öHóMçîöìéè#ãºìåFèç%é$AÔöæã'ëàìö ãAìöå/

çëMí ô²ãöåTîã'óè#ç²éãí F ú¯çn÷_æìµöãHê²î>çë

|

óMç%î<öì#é-èµã—÷_æMã-ë¯öæMãHîòMèµã—ìå_îã'åóDãéöì>;%ã'èµúê%î9O'~ð

ìµëDåEöã'ç%í ìö\ì#å\ç%ë

|

óDçîöìéè#ã çëDílöæMã îòè#ãwìå/4öæã-ëõìö\ãAìµöå\çëMíÔì#åKîã-óèç%é-ã'ífF úlçù÷_æMìöã

óMçîöìéè#ã-9

,

9.->öæã'î÷_ìåã:ëMêöæMìµëônæMç%óó¨ã-ëMå:9

2Të ãçïnóè#ã—êðJã;%ê²èµòöì#ê%ë¯ì#å>ô%ì>;%ã'ë ì#ë+W^ìµô²òî㠗9*-KëMãHóDê¸ååì>Fèµã—ì#ë¸öã'îóîãöç&öìµê²ë ê%ð@öæìåTïnêAíAã-è@ì#åTöæMçö

Fèç%é$AçëDíº÷_æìµöã:óMçîöìéè#ã'åå<öì#èµè¨öîç;%ã'èDîã'åó¨ã'éöìk;²ã-è#ú‡öêHöæã:îì#ô%æ¸ö_çëMínöêwöæã:èµã-ðö:/A÷_æì#èµã

|

óMçîöìéè#ã'åç%éö­ç²å

ã-ïnóAöEú‡éã'èµèå9j2TëMêöæMã-îGì#ë²öã-îóîãöçöì#ê%ë‡ì#åJ÷_ìµöæn÷_æìµöã_óDçîöìéè#ã'åGå<öçëDíAìµëMôKðê²î';&ç%é-ç%ëMéì#ã'åGç%ëMíFè#ç²é$AHóMç%î<öì#é-èµãå

ê;%ã'î<öçA ì#ëô åèµê&÷ã-î

|

óMç%î<öì#é-èµãå9

ËDÌy9U̝ÜHÝ^Ú Ø²ßDÏÆÐJÙÚ¸×Ú³ÏàßXá@Ú ÙEÌ.-KòMî_ïnç%ìµë ì#ëô%îã'íAì#ã-ë¸ö­öêºå<öòDíAúöæMã\öæîã-ã\óMç%î<öì#é-èµã&03254G67 é-ê%ëMåìåEöå

ì#ë öæMã‡é-ê%ëMå<öîòMéöìµê²ëlêð„çùëã-÷ @ÔçîA%ê;³éæDçì#ë ê%ëõçùåãö

N

êðˆþ ">.’þ&ÿ#K]L- GBþ&ÿ— HöæMçö\åç&öì#ån?Mãå

öE÷ê ïçì#ëÆîã#=²òMìµîã-ïnã-ë¸öåH ê%ëZöæMãùê%ëMãºæMçëDíäöæã åEöç&öì#ê%ëDçîúõíAìåEöîì>FòAöìµê²ëÆêð­öæMãQFMç²å<ìéºéæMç%ìµë C é'çë F¨ã

åìµïnóè#úùã óMîãååã'í³ìµë öã-îïnåTêð^öæMçöTêð^öæãHéæMç%ìµëAê²ë öæã—ê%öæã'îTæMçëDíºöæãwå<öç&öìµê²ëMçîúF¨ã-æMç; ì#ê%î_êð^öæMã

éæMç%ìµë ìå_ãç%åúöêòëDíAã-îåEöçëMí<9O0­æãHéê²ï‡óMèµã-öãHéê²ëG?Mô²òîçöì#ê%ëMå6öæDç&ö>÷ã\ìµë¸öîêAíAòMé-ãKðê²î_öæMì#å_óMòîó¨ê²åã—çîã

ïç%íAã‡êð6öE÷„ê îê&÷TåKêð„øžéã'èµèå\éê²ë²öçì#ëì#ëôQFèç%é$A,/

|

/cçëMíÔ÷_æìöãnóMçîöìéè#ã'å:90­æãU?MîåEö\îã#=¸òìµîã-ïnã'ë²ö:ì#å\ïnã-ö

F úùì#ïnóDê¸å<ì#ëônöæDç&öKíAìåîã'ô²çîíAì#ëô‡÷_æMç&öTæMç%óó¨ã-ëMå_ì#ë¯öæMãHå<ãéê%ëDí³îê&÷&/ öæMãwéæMçì#ë åì#ïwòèç&öã'å­öæãwéæMç%ìµëfC

ì#ëºöæMã ?Mîå<ö„îê&÷&930­æã\åã'éê²ëMíîã:=¸òì#îã'ïnã-ë¸ö„ìå6ïnã-öuF úç%íAã#=¸òMç&öã-è#únéæMê¸ê¸å<ì#ëô—öæã\éê%ïnóè#ãöã:é-ê%ëG?Dô%òîç&öìµê²ëMå

çëDíºöæã\öîçëMåìµöì#ê%ë îòè#ã'å_åê‡öæMçö é-èµãçîèµúùæMç%å_ç‡òëìµðê%îï å<öçöì#ê%ëMç%îúºíAìå<öîìkFòöì#ê%ë<9

@lê%îã:óîãéìå<ã'èµúºç‡óMç%ìµî­ê%ð@îê&÷Tå6ê%ð@óMç%î<öì#é-èµãåF¨ã-è#ê%ëMô²å„öê N ìµð}H1R}’T„öæMã

|

óMç%î<öì#é-èµãå­çóó¨ã'ç%îì#ë óMçì#îå„öê

ðê%îï

!

é-ê%è#òïnëMå:/öæ òMå:íAã'èµì#ïnìöìµëô *.#þ:#" ‡êð6é-ê%ë¸öì#ô%òê²òMåSFèç%é$A¯çëDí¯÷_æìöã óDçîöìéè#ã'å$3RnT>ã'ç²éælêðGöæMã'åã

Fè#êAé$A åKéê²ë²öçì#ëMåTç%ëlã:=¸òMçè@ë òïiF¨ã-îKêðOFèç%é$A çëMíl÷_æMìöãwóMç%î<öì#é-èµãå%3R}’T>ì#ëMåì#íãwã'ç²éæDFèµêAé$A,/Döêöæãwè#ãðö:êð

çë ú;²ã-îöìé-ç%è¨÷­çè#è4öæMã-îãHçîãKëêïnê²îã:÷_æìµöã—óMç%î<öì#é-èµãåöæMç%ë FMè#ç²é$Aºê%ëãå&Röæãˆ"þ '&# þÿþÿ,T.9

2TëlãAç%ïnóèµã—ê%ðJçné-ê%ïnóè#ãöãHé-ê%ëG?Dô%òîç&öìµê²ë³ìå_ô%ì>;%ã'ë³ì#ë+W^ìµô²òîã)(HGðîê%ï è#ãðöTöêîì#ô%æ¸ö­öæMãFè#ê é$AAå­æMç;²ã

åòMé-é-ã'åå<ì>;%ã'èµúZè#ã-ëô%öæF!/6ü—/ P-/„çëMí+*!9 'ë 4 ãéöì#ê%ëF/õ÷„ã óîê;%ãºöæMç&önöæã é'çîíAì#ëMçè#ìöEúÆê%ð,

N

ìå

 S NP

N]P

 U /

çëDí¯öæMçö:/Dðê%î\çë ú?ZN.-8N0/ e8ø'/Möæã é'çîíAìµëDçè#ìöEú êðJöæMã‡åãö,21

^435

ê%ðCéê%ïnóè#ãöã é-ê%ëG?Mô²òîç&öì#ê%ëDå_÷_ìöæ6-

!

éê%è#òïnëMå:/çëMí?ZQFèç%é$Aùç%ëMí7/ ÷_æìöã—óMçîöìéè#ã'å­ê²ëùöæã\öê²ó¯îê&÷ ìå 1 P 

NP

RS NP



^ US NP



5 U 9

XZY)[\^`_8`c 28éê²ïnóèµã-öãHé-ê%ëG?Mô²òîç&öì#ê%ë³÷_ìöæ¯øfegP—9

0­æã@ÔçîA%ê;ùéæMç%ìµë6 ê²ë

N

ì#åKíAã.?MëMã'ílì#ë¯öã'îïåTê%ðJöE÷„êöîç%ëMåìöìµê²ëlîòè#ã'å:/ 98ç%ëMí9:/ðîê%ï öæã‡å<ã-ö

N

|<;

ü ø?>>öêHöæã:åãö

N

/¸öæMçöîã'åó¨ã'éöìk;²ã-è#ú ã öã-ëDíöæMã>öîç%ëMå<ìµöì#ê%ëùîòè#ã'å2 ç%ëMí9'0­æãå<ãKöîçëMåìµöì#ê%ë

i1 i₂ i₁ i₂

XuY)[]\<^`_`c 2 ÷_æìöãHå<÷ã-ã'ó ç%ëMí çFMè#ç²é$Aùå<÷ã-ã'ó<9

îòèµãå—çîãíAã-îìk;²ã'íõìµë 4 ãéöìµê²ë€!³ðîê%ï öE÷„ê ðòëMíMçïnã-ë¸öç%èOFìñEã'éöì#ê%ëMå9RçëMí9:jFMòAö é-ç%ë F¨ãºéê²ë!;%ã-ëMìµã'ë²öèµú

íAãåé-îì>FDãíºç%å6ðê%è#è#ê&÷Tå9:ìk;²ã-ëùç éê²ï‡óMèµã-öãKé-ê%ëG?Dô%òîç&öìµê²ë rç%ëMíºç%ëùç%éöì>;%ã>÷­çè#è$/¸öæMã:ç²éöì#ê%ëDåCêð 9 çëMí9

ê%ë³öæMãKöê%ó îê&÷ êðPíAênëêöTíAã'óDã'ëMíùê²ëùöæã—å<ãéê²ëMíºîê&÷&/AçëMí³ïnìµïnìé:öæã—ç%éöì#ê%ëMåê%ð nçëMí ç%å_íã.?Mëãí

F ú é'ç%åã'å

& / ) / + çëMí

,

êðGöæãwíAãåé-îì#óAöìµê²ëlêðGöæMãHöæMîã'ãHóMçîöìéè#ãi03254G67j9'ë óDçîöìéòèçîTì#ëlöæãHöê%ólîê&÷&/

Fèç%é$A‡óMçîöìéè#ã'å6öîç;%ã'èðîê%ï è#ãðööêwîì#ô%æ¸ö„çëDí÷_æìµöã:óMçîöìéè#ã'åCðîê²ïRîì#ô%æ¸ö6öêwè#ãðö:9O2Kå6ê²óó¨ê²åã'ínöêHöæDç&ö:/ ì#ë

öæMãiF¨êö<öê%ï îê&÷&/ 9 çëDí69:cïnê;²ãFèç%é$A¯çëMíl÷_æMìöãwóMç%î<öì#é-èµãå1FMç²é$A¸÷­çîí`9*'ë ê²îíAã'îTöêùíAã'åéîì>FDãwöæìå/¨÷ã

?MîåEö_ì#ë¸öîê íòMéã\öæãHéê²ëMéã'óAö_êðGå<÷ã-ã'ó R!å<ã'ãMW^ì#ô%òîãVT.H

I 2 %](!’Q %8’"bF¨ãöE÷ã-ã'ëÆ÷ç%èµèå



ç%ëMí éê²ëMå<ìå<öåHì#ëàçè#è6÷_æMìöãºóMçîöìéè#ã'åwêð­öæMãFDê%ö<öê%ï îê&÷

çëMíF¨ãöE÷ã-ã'ë÷ç%èµèå



ç%ëMí åìµï òèöçëã'ê%òMåè#úwæê²óóì#ëô—öêHöæãKîì#ô%æ¸ö Råê%ïnã1FMè#ç²é$A óMç%î<öì#é-èµãåJöæ òMå

FDã'ìµëMô¯íì#åóèç%éãí öê¯öæãè#ãðöHì#ëZê%îíAã-î:öê ?DèµèGöæãô²ç%óMå$T9Wê²î—÷„ã'èµèCíAã?Mëìµöã'ëã'ååHç ÷_æìµöãºå÷„ã'ã-ó

FDã-öE÷„ã'ã-ë



çëMí é-çë êAé-é-òî_ê%ëè#úºìµð@öæã—óMç%î<öì#é-èµã\ê%ë³öæMã—îì#ô%æ¸ö_æMçëDí åìíAã\êð ìåSFèç%é$A,9

I 2 *k-#"U %8$’"DF¨ãöE÷ã-ã'ëù÷­çè#è#å



çëMí éê²ëMåì#å<öå6ìµë çè#è Fèç%é$A‡óMç%î<öì#é-èµãå6ê%ðUöæãˆFDê%ö<öê%ï îê&÷ ç%ëMí

FDã-öE÷„ã'ã-ën÷­çè#è#å



çëDí åìµï òèµöçëMã-ê%òDå<è#úHæê²óóì#ëô\öê\öæã_èµã-ðö5Råê%ïnã_÷_æìµöã_óDçîöìéè#ã'åJöæ¸òDå'FDã'ìµëMô

íAìå<óèç%é-ã'í öê\öæãTîìµô²æ¸öJì#ëê%îíAã'îJöê&?Mè#èAöæãTô²ç%óMå$T9jWê%î6÷„ã'èµèMíã.?Mëìµöã'ëã'ååCç\÷_æìµöãKå<÷ã-ã'óUF¨ãöE÷ã-ã'ë



ç%ëMí é'çë êAé-éòMî_ê%ëè#úºìµð@öæã—óMçîöìéè#ã\ê%ë öæMã—èµã-ðöTæMçëMí¯åì#íã:ê%ð



ìå_÷_æìµöã-9

`Tã ö:/Açîê%òëDínöæã\ç%éöì>;%ãK÷ç%èµè$/²÷ãKíAìå<öì#ëô%òìåæºöæMã>ðê²èµè#ê&÷_ìµëMôw÷­çè#è#å:H@ìµðe ü/¸è#ãö



jF¨ã>öæãKè#ãðöïnê²å<ö

÷­çè#ècå<òMéæ öæMç&ö_öæã-îã—çîã:ê²ëè#úº÷_æìµöã—óMç%î<öì#é-èµãå­ìµë öæMã\öê%ó¯îê&÷ F¨ãöE÷ã-ã'ë³÷­çè#è#å



ç%ëMínEbPìµðe ø'/Mèµã-ö

~

ZF¨ãwöæã‡îì#ô%æ¸öïnê¸åEö:÷ç%èµèJå<òDéælöæMçöKöæã-îã ç%îã ê%ëè#úDFèç%é$A¯óMçîöìéè#ã'åKì#ë öæMãHöê²óÔîê&÷dFDã-öE÷„ã'ã-ëÔ÷­çè#èå

<NJPwçëMí 9>â ìµöælöæã'åã íã.?Mëìµöì#ê%ëDå/4÷ãwçîãHì#ëlöæã óDê¸å<ìµöì#ê%ë öêùíAãåé-îì>F¨ãHöæMã ç%éöì#ê%ëMå>êð 9 ç%ëMí9:cê²ë

öæMã5F¨êööê%ï îê&÷ êðXç éê%ë?Mô%òîç&öìµê²ë<9]W^ì#îå<ö­÷_æã-ëã:;%ã'î„çë

|

óMç%î<öì#é-èµã6ñEòïnóMåìµëùöæãKöê²óºîê&÷&/²öæMã

|

óMçîöìéè#ã

F¨ã-è#ê&÷ ïwòMå<ö\ðê%è#èµê&÷ ìöQR!å<ê³öæMçö\öæã'ú îã'ïçì#ëõì#ëÔöæãåç%ïnã‡é-ê%è#òïnë T90­æã-ëõöæMãné'ç%åã'å

& / ) ç%ëMí

+

ê%ð6öæMã

öîçëDå<ìµöì#ê%ë îòMèµãånç%ëMí

ç%îã\éê²ï‡óMèµã'ï‡ã'ë¸öã'í ì#ë öæã&F¨êööê%ï îê&÷ ç%åðê%è#è#ê&÷TåH

ç—910­æã\ïnê;%ã'åì#ëùöæã&F¨êö<öê%ï îê&÷ íAã'óDã'ëMí ê%ë³öæã\öîç%ëMåìöìµê²ë³ç&ö­öæãHç%éöì>;%ã:÷ç%èµè Cì#ë³öæã\öê%ó îê&÷

Röæã'åã„ïnê;²ã'å^çîã6ì#èµè#òMå<öîç&öãíF úW^ìµô²òîã'å *"!$#—/&çëMíwï‡ê²îã„óîã'éìåã-è#úHíAã'åéîìkF¨ã'íHìµëW^ìµô²òîãå8PVP%!,P"VT.H

& R| }G } |

TH:öæã

! ç%ëMí

(')!

éê²èµòMï‡ëDåKô²ãö—ãéæMç%ëô%ãíÔç%ëMílöæã-ëÆç³÷_æìöãºå<÷ã-ã'óÔêAé-é-òîå

FDã-öE÷„ã'ã-ë ÷ç%èµèå



çëMíE PV9

& R I

| G | I TH\öæã

(') ! çëDí

!

éê²èµòïnëMå\ô²ãöHãAéæDçëô²ã'íÔç%ëMíÔöæã'ëZç Fèç%é$Aõå<÷ã-ã'óõêAé-é-òîå

FDã-öE÷„ã'ã-ëõ÷ç%èµèåNJPnçëMí NJPDRê²î5F¨ãöE÷ã-ã'ë*NJP‡ç%ëMí ìð e8ørê%îKöæã‡óMçîöìéè#ãwê²ë

öæã—îì#ô%æ¸ö_æMçëMí åì#íAã—ê%ð+ ìåTç

|

T9

& R I

} G } I

TH„íAã-ó¨ã-ëMíìµëô÷_æMãöæMã-îTöæã—óMç%î<öì#é-èµã—ê²ë³öæã&F¨êö<öê%ï îì#ô%æ¸ö_êð^öæMãöæ¯÷ç%èµèXìµë

ì#åT÷_æìöã—ê%î1FMè#ç²é$A/çn÷_æMìöãHå<÷ã-ã'ó êAé'éòîåZFDã-öE÷„ã'ã-ë



çëDíE PV/ê%îTçFMè#ç²é$Aºê%ëã&F¨ãöE÷ã-ã'ë

`NJP çëMí, N P Rê²î F¨ãöE÷ã-ã-ë-N P ç%ëMí ìµð e øZê%îTöæãwóDçîöìéè#ãwê%ë¯öæãwîì#ô%æ¸ö>æMç%ëMí

å<ìíAã\êð ì#å_ç

|

T9

F<9('~ðTöæMã ã'ë¸öã-îì#ëôÔóMç%î<öì#é-èµã³ì#åUFèç%é$A,/6ç Fèç%é$Aàå÷„ã'ã-óàêAé-é-òîåiF¨ãöE÷ã-ã'ëàöæMã è#ãðöF¨ê%îíAã-înçëDíà÷­çè#è

N PV9

é-9('~ðDöæãTã'ë²öã-îìµëMô—óMçîöìéè#ã­ì#å6÷_æìöã-/¸ç\÷_æìöã>å÷„ã'ã-ónêAé-éòMîåjF¨ãöE÷ã-ã'ë÷ç%èµè.



ç%ëMí‡öæã_îì#ô%æ¸öOFDê²îíã-î#9

->öæMã-î÷_ì#åãëêöæì#ëôÔã'è#åãºæMç%óóDã'ëMåHì#ëÆöæãF¨êö<öê%ï îê&÷&9a$­ç²å<ãíZê%ë 9 çëMí 9 /^öæMã @ ç%îBA²ê;ÔéæMçì#ë ì#å

íAã?Mëã'í ì#ëlç‡åì#ï‡ì#èçî_÷­çyúºç%åöæMã\öæîã-ã\óMç%î<öì#é-èµã03254G671H

I

0­æã—å<ã-öTêðJéê%ë?Mô%òîç&öìµê²ëMåì#å_öæãHåãö:

N

êðGéê%ïnóè#ãöã—éê%ë?Mô%òîç&öìµê²ëMåêð^èµã'ëôöæ ø'9

I

Wîê%ï öìµïnãˆhCöêUhNbPV/¸öæã—åú å<öã'ï ã;²ê%è>;%ã'å6ðîê%ï öæMã:é-ê%ïnóè#ãöã—é-ê%ëG?Mô²òîç&öì#ê%ë7 R)hnTCöê öæã\ëã ö

ê%ëã R)hNJP#TTç%å>ðê%è#èµê&÷Tå:Hç%ë ç%éöì>;%ãH÷­çè#è­ìåKéæê¸å<ã'ë¯òëìµðê%îï‡è#úlç&ö:îç%ëMíAê%ï çïnê%ëMôöæã ø N P

i

i j₁ i

i j₂

? white

? black black

white

XuY)[]\<^`_c

4 ÷„ã'ã-óMå­êAé'éòîîì#ëôUFDã'èµê&÷ öîçëDå<ìµöì#ê%ëMåMR

| } G } | T­ç%ëMí R

I

| G | I T.9

i white black

?

j₁

j₂ i

i

? white

? black

XZY)[]\<^`_<c 4 ÷ã-ã'óMå­êAé-é-òîîì#ëôFDã'èµê&÷ öæã\öîç%ëMå<ìµöì#ê%ë R

I } G } I

T9

÷ç%èµèå:/MçëMí¯ê²ëã—êðJöæã—öE÷êºîòMèµãå 98çëMí9 ì#åKå<ã'èµãéöãílç&öTîçëMíê%ï ÷_ìµöæÔóîêVFMçFMìµè#ìöEú P]/G90­æMã

éê²ëG?Mô%òMîçöì#ê%ë RhN PTCìå„ê-FAöçì#ëã'íQF¸úç%óóè#ú¸ì#ëô öæã\åã-è#ã'éöã'íùîòè#ãTöê R)hnT„ç&ö6öæã—ç%éöì>;%ã>÷­çè#è9

'ë 4Aã'éöìµê²ë% /¸÷ã\å<æMç%èµè¨óîê;%ã_öæMç&ö­öæã-îã>ã ìå<öå­çë³ã;%ê²èµòöì#ê%ëF¨ãöE÷ã-ã'ë³çë ú‡öE÷ê éê²ëG?Mô²òîçöì#ê%ëMå:/%ì9ã-9>/ öæMçö

öæMãM@ÔçîA%ê;néæMçì#ë ì#åì#îîã'íòMéì>Fè#ã-9'0­æã-îã:ì#å_ç%è#åê ç‡ó¨ê²åìöìk;²ãKóîê-FDçFì#èµìµöEú‡öêå<öçyúì#ë ç%ë¸úºé-ê%ëG?Dô%òîç&öìµê²ë</

åê öæMç&öTìµöTìå_çó¨ã-îìµêAíAìé9f-Kòî­ïçì#ë¯îãå<òMèö_ìå­öæã'ë öæã\ðê²èµè#ê&÷_ìµëMôwöæã-ê²îã'ï 9

<_]^`_

P-9>P c

(—+ "%þ &Q(—ÿ ( Aÿ—þ ’þ&ÿ  *. Gþ&ÿ 1

0­æã¯òëìµðê%îïnìöEúàê%ð>öæã åEöç&öìµê²ëMçîúÆíAì#å<öîì>FòAöì#ê%ë ìånê-FAöçì#ëã'ínF úàéê²ëMå<öîòMéöìµê²ëH³ìµëMíã-ã'í</„ì#ë 4Aã'é

!

öì#ê%ë !P÷ãàåæê&÷ 9 Rç%ëMí8åìµïnì#è#ç%îè#ú 9:Tlé-ç%ëdFDãríã'åéîìkF¨ã'í ïnê%îãÆãAóè#ì#é-ìöèµú ç%å¯öæã ?Mîå<öÔéê%ïnó¨ê%ëã'ë¸ö

N

|<;

ü

Q$=%=%=Q

ø?>FG N

ê%ð ç FìñEã'éöì#ê%ë 9&H0 N | ; ü

Q$=%=%=Q

ø?>FG N

|<;

ü

Q$=%=%= Q

ø?>\öæã-ë ç%åå<òMï‡ì#ëôröæMçö¯çö

åê%ïnã:öì#ï‡ã&h„öæMãHå<úAå<öã-ï ì#å_ì#ë³öæã—òëMìðê²îï íAì#å<öîì>FòAöì#ê%ë¯ê²ë

N

/ì9ã-9>/

7Cîê-F<R R)hnTOe TOe

P

N Q

ìµö>çè#÷çyúAåîã'ïçì#ëMåìµë öæMã—òëìµðê%îï íAìåEöîì>FòAöìµê²ë<H

7Cîê-F`R RhN PTje TOe

e P/

3 "!$#&%

76îêVF<R RhnTOe T('



NP

 N P

/

3) "!

#&%

7Cîê-F`R R)hnTOe T('



N]P



e P/ '*

*9 _  RT+*

* ' P

N ' P

øQN P N P

/

'*

*9 _  RT+*

* ' P

N ' P

øQN P e P

N Q

÷_æã'îã9

_  RTHç%ëMí 9 _  RT íAã-ëê%öãùöæã å<ã-öå êðTóîã-ì#ïçô%ãåHê%ð îãå<ó¨ã'éöì>;%ã-è#ú F¸ú 9 ç%ëMí 9 CöæMãùèç%å<ö

ã:=¸òMç%èµìµöEú—ðê²èµè#ê&÷Tå@ðîê²ï öæã­ðç²éöå^öæMç&ö 9 _  RTOe

; 9 _  R Q

!T

Ue ü

Q$=%=$=Q

ø?>TçëDí 9:

_  RTOe

; 9:

_  R Q

T

e ü

Q$=%=$=Q

ø?>/çëMíùöæMçö 9 ç%ëMí 9:UçîãMFì ñEãéöìµê²ëMå9

ËDÌAcÌ-,XÖyß¨Ï ×'Ý^Ú Ø²ßDÏÆÐJÙÚ¸×ڗ×'ßù×'Ý^ÚwÞJÕÛÑØHÏàßXá@Ú ÙEÌ52>é-é-ê%îíAì#ëôTöêTöæMã6öæã-ê²îú—ê%ðG?Mëìµöã_åEöç&öã@ ç%îBA²ê;

éæMç%ìµëMå 3<

ce 9C/—0­æã-ê²îã'ï

P-9>P:ã-ëMåòîã'åöæDç&ö_ðê%îTç%ë úùéæê%ìéã\ê%ð@ì#ëìöì#ç%èXéê²ëMíAìµöì#ê%ë RüT/

76îêVF<R RhnTOe TFE G

HJI@K

P

N e P



S N]P

NP

 U

=

0­æìåHîã'åòèµöwì#å‡åò.ºéì#ã-ë¸öHöêÔîãéê;%ã'î\öæã åEöç&öìµê²ëMçîúõíAìå<öîìkFòöì#ê%ëÆêð­öæãQFMç²å<ìéïnêAíAã-è9?'ëMíAã'ã'íÆêVFMåã-î;%ã

öæDç&ö5F ú¯éê²ëMåEöîòDéöì#ê%ëlæMì#íAì#ëôùöæãFDê%ö<öê²ï îê&÷ ìµë öæã‡éê%ïnóè#ãöãwïnêAíAã-è@ãç%éöè#ú³ú ì#ã-èíå>öæãFMç%åì#éHï‡êAíAã'è9

Τ ’

Τ ’ Τ

Τ

XZY)[\^`_`c 2Të¯ãAç%ïnóèµã\ê%ð^ã:;%ê%è#òAöìµê²ë³÷_ìµöæ¯ø+eFnðê%î­öæãHéê²ïnóèµã-öã:öæMîã'ã—óMçîöìéè#ã\ï‡êAíAã'è9

sTã'ëMéã ÷ã³êVFAöç%ìµëÆöæã³ðê%è#èµê&÷_ì#ëôõé-ê%ïFìµëDç&öê²îìçè„ìµë¸öã-îóîã-öçöì#ê%ëÆðê²îwöæãlåEöç&öì#ê%ëDçîúäíAìå<öîìkFòöì#ê%ërê%ðTöæMã

öæMîã'ã:óDçîöìéè#ãM032546871H

<_]^`_ P-9p/ c <~þ$"`RT1--ÿ¨þ )(Sþ"Dþ%àþ 1iþ "k’5þ&ÿ#K]L- GBþ&ÿ.1 (-ÿ þ ÿ—&ÿ

þ&ÿ#K]L- GBþ&ÿ— SCSR!üVT&ÿ RüTM%')(fþ"`R R!üVTnTOeJCSR!üVTÿ &ÿ—Q*$ $þ&ÿ#K]L- GBþ&ÿ

þV*#RCSRhnTOeT e þV*#R}þ",R R)hnTBT8eT EBG

HJI@K

**;

N

þ"<RTOe>

N =

ÿ" Gk þiÿ—LZN -ON/ ePøO%8‡þ-*’ÿ þ*ÿ~þ (— þ G>

þV*:R CSR)hnTþ&ÿ—’ÿ— Z *.>-#" ÿ/ %](GS"k. BTFEBG

HJI@K

1^ 35

N e 1 P 

NP

 S NP



^ US NP



5 U



TS NP

NP

BU

=

2>å­íAìåé-òMåå<ãínìµëD4 ã'éöì#ê%ë(—öæìå6ì#ë¸öã'îóîãöç&öìµê²ëºå<æãíåçHëã'÷ èµì#ô%æ¸ö„ê²ëùåê%ïnã>îã'é-ã-ë¸ö6îã'åòèöå6ê%ðn0>ã'îîìíç

u16é-ê%ëëãéöìµëônöæã03254G67röê $„îê&÷_ëìçë³ãéòîåìµê²ëMå 346;.m C9

ËDÌjXÌÜ ßÕAÖ&ÑÕ×'ÑߨÒ^Û%Ìcã-ö>òMå>íã-ëê%öãF úuEN öæãwåòFMåãö>ê%ðCéê%ë?Mô%òîç&öìµê²ëMå­êð N

÷_ìµöæê²òAö

|

óDçî

!

öìéè#ã'å:/çëDíºîã'é-ç%èµè¨öæMç&ö:uE

^ 35

ìåöæã—åòFMåãö­êðJéê%ë?Mô%òîç&öìµê²ëMå„êð uEN ÷_ìöæaZQFèç%é$AºçëDí/N÷_æìµöã\óMç%î<öì#é-èµãå

ì#ëºöæMã ?Mîå<öîê&÷&9'ëD4 ã'éöì#ê%ë/w÷ãKåæê&÷PöæMçö

uEN

e 

NP

 S NP

N U

çëMíºöæMçö

uE

^ 35 e  NP

 S NP



^ US N]P



5 U 9'2>å

÷ã íAìí ðê²î N

/U÷ã íã.?Mëãnç @ çîA%ê; éæMçì#ë ê%ëÔöæãnåãöuEN ÷_æê²åãwã:;%ê²èµòAöìµê²ë ìå:íãöã'îïnì#ëã'íºñEòMå<öˆF ú¯öæMã

çóMóèµìé-çöì#ê%ë‡êð91E-/²÷_æìéæ ìåJöæMãTîãåEöîìéöìµê²ë ê%ð 9 öê:öæã>åòFMåãö2uEN 9j0­æãSFDã'æMç; ìµê²îGêð¨öæãZ?Dîå<öCîê&÷žìµë‡öæì#å

@ ç%îBA²ê; éæMç%ìµëÔöæã'ëõãç%éöèµúlïnì#ïnì#é'å:öæãFMç²å<ìéi03254G67 ÷_ìµöæõöE÷ê³óMç%î<öì#é-èµãå9@lê%îã-ê;²ã-î#/Möæã‡ç²ååêAéìç&öã'í

çóMóèµìé-çöì#ê%ë 91E—ìåTçFì ñEã'éöì#ê%ë¯ðîê%ï uEN

|<;

ü

Q%=%=$= Q

ø?>Hì#ë¸öêìµöåã-èµð}/DåênöæMçöTöæMçöTöæMã—òëìµðê%îï íAìå<öîìkFòöì#ê%ë¯ì#å

çô¸çì#ë³å<öçöì#ê%ëMç%îúnðê²î_öæìåS@ ç%îBA²ê;ºéæMçì#ë<9'W^ìµëMç%èµè#úùìµöTìå_çèå<êçë ã-îô%êAíAìéˆ@ çîA%ê;ºéæMç%ìµë9j0­æMã-îãðê%îã

7Cîê-F<R E

RhnT8e T[EBG

HJI@K

P

EN Q

çëDíºöæãHåEöç&öìµê²ëMçîúºíAìåEöîì>FòAöìµê²ë ê%ð@öæã\öE÷ê‡óDçîöìéè#ã'åZ0]2ˆ4G687 ìåTéê²ïiFì#ëMçöê%îìçè#èµúã óMîãååã'í³ç%å

7Cîê-F`RC E RhnTOeT[EG

HJI@K ;

uEN

öê²óRTOe >

EN

=

0­æãHîã'åòèµöåRnP-9>P#T

!

RnP-9p/-T­çîã\öæMã-ëlì#ï‡ïnãíAì#çöãwé-ê%ëMåã:=¸òã'ëMéãå9u0­æãFMç²å<ìé—çëDí é-ê%ïnóè#ãöãwå<úAå<öã'ï ÷_ìµöæ¯öE÷ê

óMç%î<öì#é-èµãåðê%î_ø+e[!çîã:îã'óîãå<ã'ë¸öã'í³ì#ë W^ì#ô%òîã'åMPû !/%üHì#ëf2Tóó¨ã-ëMíì#?$ˆ9

2Tëê%öæã'îM;yç%îìçë¸ö:êð03254G67 ðê²òëMíÔìµëÔöæã‡èµìµöã'îçöòîã‡ì#å>öæã03254G67 ÷_ìµöæäóDã'îì#êAíAìéiF¨ê%òëDíçîúléê%ë

!

íAìµöì#ê%ëMå:/Uì#ëÔ÷_æìéæ öæãnóMçîöìéè#ã'å>öîç;%ã'è^ç%îê²òëMí ç³éì#îé-èµãDRåã-ãW^ìµô²òîã P'ü—/4öæãnéì#îé-èµã‡ìå>îìµô²ì#í</Uëêö—å<ò—FAñEã'éö

öêîêöç&öìµê²ë T9O4 ì#ëMéã\öæã-îãHçîã\ëêFDê²îíã-î_÷­çè#è#å­ì#ë öæãå<ãHé-ê%ëG?Mô²òîç&öì#ê%ëDå/¸öæMã@ çîA%ê;éæMç%ìµëCrì#åTíã.?Mëãí

òMåìµëMô:ê²ëè#úYVç%åã

&

êðDöæã­öîç%ëMåìöìµê²ë îòè#ã \ê%ðMöæMã10]2ˆ4G687ä÷_ìµöæFDê²òëMíç%îì#ã'å:9T'ëwöæã_óDã'îì#êAíAìéS0]2ˆ4G687õöæMã

ë òïiF¨ã-îåêð3Fè#ç²é$A,/

|

ç%ëMíº÷_æìµöã\óMç%î<öì#é-èµãå­íAênëêöTéæMç%ëô%ãV/¸ç%ëMíºöæã—é-ç²å<ã:÷_ìöæê%òö

|

óMç%î<öì#é-èµã:ì#ï‡ïnãíAì#çöã'èµú

è#ã'ç%íMå:öêlç òëìµðê%îï åEöç&öìµê²ëMçîú íAìåEöîì>FòAöìµê²ë<9-Kòî—çóóMîê¸ç%éæÔìå\ã'ç²å<ì#è#úÔç%íç%óAöãí öêlíAã'ç%èG÷_ìöæõöæãïnê%îã

ì#ë²öã-îã'å<öì#ëôné-ç%åã\÷_æã-îãKöæã'îã—çîã

|

óDçîöìéè#ã'å:9T'ëDíAã-ãí³ê²ëã—é-ç%ë³ç%ååê é-ì#çöãKöê‡öæMì#å­ïnêAíAã-è@çwëã'÷ åãö

N

êð

éê²ïnóèµã-öãnéê²ëG?Mô%òMîçöì#ê%ëMå:/Dïç²íAãwê%ðCöE÷„ê³îê&÷TåKêð6é-ã-è#è#å\ç%îîçëô²ã'í¯ê²ëÔç³é-ìµîéè#ã-9&2Kå>ðê²î N

/cé-ê%ëG?Dô%òîç&öìµê²ëMå