Convergence rate for the approximation of the limit law of weakly interacting particles 1: smooth interacting kernels
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE. Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 1: Smooth Interacting Kernels Mireille BOSSY , Denis TALAY. N˚ 2180 Fevrier 1994. PROGRAMME 6 Calcul scientifique, mode´lisation et logiciel nume´rique. ISSN 0249-6399. apport de recherche. 1994.
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(45) c p e__bvr.|Kf'up»stv ; c%u| D fBuo_bvrfñv`c9v'{ _ ({qpr@c ; ] *{ p _bjf%pv`ʧf <Î_bjjt_ stuÊauo_buÊjtstuÊf%c9^,] +-+_ CÊvdc9suof%g| C-; D of'u e s~p=k_bup vdc9uWv X f%ymc9jSv`) <Î_b^reBc9j j ;
(46) stuWv`f'aJ^dc9vrstuoavros f )@k =f%c9Æu ?·=jlcbp _9
(47) f%ymc9vrst_Jfu H D . Q ( ÷ / . X b Zh[a] M [ l . l Y V X G h Z a [ ] R T U .b . . z os~pj~c9vrv`f'¤ ^ <Õcbkv=s~pmp»f%|Bstu)_bmÊ^ c9uc% j ;Êp»slp _ <<v`Êf\ef'vro_W|K_Jjt_b a ;)c%oojtsf%|Pvr_v`Êf 0 mo^rabf'^ pf%ym c ? v`s_buq{ D os~kd1^rf%ymost^rf%p&p»f%kns´wzk|Wf'f'jt_Joef'uWvdp Cf%k,c9mp f= _ <Svrof p»suoabmÊj~c9^rstv ; _ <Svrofkn_b^r^`f%p á_Ju|Ksuoa stuWv`f'^ cbknvrst_Ju Ef,^`uof'j VX p»f' f 0 _p`p ;-?0zc%j~@c ;É ú ] n :f!kn_beÊmov`f c%u)c%oo^`F_ ¯WsteBc%v`s_buB _ <&vrof!|Kf'up stv ;dC-;Bc^`f'aJmoj~c%^`s~prc9vrst_bu¿ _ <&vrof|Ws~p`k^`f'vrf=v`sef f'eos^`slk'c9jef%cJp»mo^rf n ® ^ c9vrf0F_ <<kn_JuKf'^rabf'uknf=s~p_ CÊvdc9suof%| D stv`¿c
(48) p»vr^`_buÊabf'^ cJp`p moeovrst_buB_Juñv`Êf kn_2f 1knstf,uKv p H ¸!43Á¿v`o f Ef'^ruof'j V s~p s%u 5 6 å X7 & ÷ ] c%u| e slp st4u 5 6 ÷ X8 & ÷ ] n 9 ,
(49) 7%dÔÕ;:aÕ z of ;W_bvrof%p»slp X ] kn_Jmoj~d | Cf!p»_bef D c9v<^`f'jl c ЯWf%| H D c9vs~p&mp»f%|suPvrof=o^r_ _ < s~p v`ofYf ¯Kslp»v`f,uknf _ <äcñ|Wf'up»sv ;g<Î_b^\vrofjlc D F_ <v`ofo^`_Wknfp`p X8< ] |Kfnwuof%|| Cf'j_ D su X - nt ] {vros~p |Kf'up stv ;Bprc9v`sl`p <;Wstuoavrof @f ¯Wá_Juof'uWv`slc9j C_Jmou|op X 8 n 8 ] n ®. = Ú?>A@;B.
(50)
(51)
(52) !"#$ %'&)(+*,-,./0 132450 ./& 76. $ 1"% \> 5,>
(53) "!# $ %&' (")*$ (+ z Êf=c%jtab_J^`sv`oe p vdc9^rvdp D stv` c9u)c%oo^r_ ¯Wstec9v`s_bu_F<vrof stuosv`slc9jqk_bu|Ksv`s_bufn ® v`sef b Q K{ X Z B] slp8vrof |Kslp»vr^`s CÊmov`s_bu<Îmouknvrst_bu
(54) _ <Svrofj~c D _ m {o|Kf'uÊ_bv`f+ | C-; m XB] stu
(55) v`of p»f%ymof'j0nÇf'v= , .-kCfv`Êf\uWmoe3Cf'^_F<<c9^rv`s~kjtf%p,n0uof
(56) kdo__p f% p = _bsuKv pstu & { X0[ må Z@l lGZh[ mÏ ] { p»mkdBv`c%vvrof ostf%kf D s~p f=kn_Jup»v c9uWv <ÎmÊuknv`s_bu m X0>] Q = p Ï X ¢[ mO ] O)q å c9oo^r_F¯KseBc9vrf%p m n z Wmp{c9v b Q W{o_buof=cbp = c%^`v`slknjtfp D oslk|KfnwuÊf=v`off'eos^`s~k,c9joefcbp»mÊ^`f Ï l _ m HQ p = O)q å r0/1u 2( 3 4 ( 5"! 4
(57) 6&7%!98: z of%p fñc%^`vrs~knjf%pc9^rf)e_%f%| cJk'kn_J^d|Ksuoa v`_ v`of | ;Kuc9eslk'pF_ < X N Ð ] n: f e
(58) mp»vp _bjf0v D _o^` _ Cojtf'e/p Hv`Êf\v`sef|Ks~prkn^`f'vr s %c%v`s_buB _ < X N Ð ] {zc9u|Bv`of c9oo^rF_ ¯KseBc9vrst_bu _ <vrofkn_f 1Bknsf'uWvdÐ páF_ <v`Êf +- X n 8 ] c9vSf%cJk v`sef&p v`f'q{Jp»suknf&v`Êf%p»f<kn_2f 1knsf'uKv p |Kf'f'u|_bu¿v`of moÆu EKuo_ D u¿j~c D _ n z Êf1@f ¯Ws~p v`f'uknfc%u|#mÊuos~ymof'uofp`pF_ <cBp»vr^`_buÊañp _bjmov`s_bu X N Ð ] f'uFc Cojtf%p=vr_Bkn_bup»s~|Wf'^ v`of
(59) |K^rBs <Îv c9u|ñvrof|K)s Azmp»s_bukn_2f 1knsf'uKv p _ < X n 8 ] cJp <Îmouknvrst_Ju<p ; c9u| (|Kf'f'u|Kstuoa_buoGj ;P_Juofv`sef c%u| p»cJknf bc%^`sl c Cojf%/p He_J^`f=Ê^`f%knslp»f'Gj ;{ D f!|Kfnwuo=f ; H ú Z> ? & & C; ; X b Z\>] H Q RFT U VYXfZh[a]a_ Ð X M [^]\Z c9u| ( H Ý Z> ? & & C-; ( X b Zh] H Q R T UeXfZh[a]_ Ð X M [^]\l : fuo_bvrf v`c%v'{KmÊu|Kf'^ X ! ] @{ ;)c9u) | (¿c9^`f Çstprkos7v stu {mouo)s <Î_b^re j ;sCá_Jmou|Kf|su b c%u| f vros~p f'up»mÊ^`f%pv`of!@f ¯Ws~p»vrf'uknfäc9u|Bv`of!mouos~ymof'uÊf%p`p _ <&vrof\suoo_Je_baJf'uof'_Jmp )c%7^ E_%o^r_Wknf%prpp _bjtmÊv`st_Ju v`_v`Êf\p v`_Wkdcbp v`slk\|K)s AÇf,^`f'uWv`slc9jÇfymc%v`s_bu IJ M < Ð Q ; X b Z < Ð ] Mbdc ( X b Z < Ð ] Mi Ð Z K X - nt ] JL Ð 9. . >&", ",
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(63) moj~c%v`fv`of=e_bv`s_bu
(64) _ <Ëv`of X8< Ð O ] Âp{ D fä|Ks~prkn^`f'vr s 'fsuv`sef H > Cáf,stuoa w ¯Kf|Ç{K_Juof kdo__p f%p / b ¡ c9u |. , .--p»mkd.v`c9v > Q / b n z of|Kslp`kn^rf'v`Gs %c9vrst_Ju v`sef%pc%^`f |Kf'uo_Jv`f%| C-; b
(65) Q / b { D stvr n ® Êo j ;WstuoaBvro) f -mojtf,^p`kdof'efv`_ X - n ] {_buÊfñ|Wfnwuof%p v`ofstu|Kf,áf,u|Kf'uWv|Kslp`kn^rf'v`Yf ?«vrstef!o^` _ ? X < ] knf%prp»f%p OÐ H IJ < ÐO Q < ÐO c ; X b
(66) Z < ÐO ] / bdc ( X b
(67) Z < ÐO ] / i O
(68) å Z K X -n 8 ] JL :. < Om QS[ mO Z D of'^rf / i O
(69) å Q i ÐO i ÐO n Q Z@l lGZ ] { X b
(70) Z@l)] z Kmpn{c9vv`sef b
(71) X D stvr 3 X b
(72) Zh]. k'c9u}Cfc%oo^`_F¯WsteBc%v`f%|ªC; p Ï X < ÐO ]hl = O)q å = Ú?>A@;B.
(73)
(74)
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(76) Z )] c9u|( X b
(77) Z B] n ® p D f|Ksl|¦<Î_J^=v`Êf\stuÊstv`slc9jSkn_Ju|Ksv`st_Juq{ D f c9oÊ^`_F¯KseBc9vrf _ Ð C;Bv`Êf0f,eost^rs~k'c%jÇefcbp»mÊ^`f _ Ð abf'uof,^dc9vrf%d| C;
(78) v`Êf0j_Wk'c9vrst_Jup _ <Ëv`of c%^`vrs~knjf%p c%vv`stef b
(79) n,+f'uo_Jv`f v`of j_Wk'c9vrst_buB_F<vrof c9^rv`s~kjtf uKmoeªCf'^ j c9vv`stef b
(80) C-; Ð O c9u|¿|KfnwuÊf Ï _ Ð Q p l = O)q å r y u z of'u Ï ; X b
(81) Z\>]Q R V@X0fZh[^]`_ Ð X M [a] R VYXfZh[a] _ Ð X M [^]Q = p VYXfZ Ð O ]hl Oq å ® uc9j_baJ_bmp j ;{ D f\p f'v ( X b
(82) Z\>] = p Ï eXfZ Ð O ] l Oq å z of'^`,f <Î_b^`f{bv`ofe_Jv`st_Ju _ <ôv`ofc9^rv`slknjtf%pSstuW_bjtf|0stuvrofc9jtaJ_b^rstv`Êe s~p|Kfp`kn^r s Cf% | C;!v`o¤f <Õc9es j ; _ <8|Ks~prkn^`f'vrf v`stefo^r_Wknf%prp»f%p X Ð O ] ÏO)q å p`c%v`s~p <;Ksuoa IJJ Ï Ï Ð O Q Ð O c = p VYX Ð O Z Ð @] / bªc = p e X Ð O Z Ð '] / i O
(83) å Z K q å q å X -nú ] JJ L m O QS[ m O Z jWQ Z@l lGZ = l Sstuc%jt j ;{vro f <Îmouknvrst_Ju X b
(84) Zh] s~pc9oo^rF_ ¯KseBc9vrf%+| C-; Ð X]gQ = p Ï X Ð O ]hl O)q å ( 3
(85) !8 %$' :f stuWv`^r_W|Kmknf!c9uB- ;K_Jv`of%p s~p_buBv`Êf0suostvrs~c%jÇj~c D n ¸! Á¬z ofstuosv`slc9jWj~c D _ m cbp&c p»v`^rs~knvr j ; _p stv`sfc9u|
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(88) ;{ c9u| óp»mkdBv`c%v. m X] lY ÷ {<Î_b^ ¡
(89) l X ·< Q W{ _ m cJpc1k_becJknvp moo_b^rv ] n ÙWÙ}ÚÜÛYÝFÞ7ß,à.
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(91) Zh] Ê/ª dÅ B6,0, Ê@Ë 0 3YËÆ./ " _ Õg h 6ÆaÅ 1d0 sÊhÌ s&1/ Æ
(92) ! = Æ50 ./& 76FÕ ÏË@Æ 6F3ÒsÔ7×FÍ Ò :/×}ÆÅ$Ò ×-Õ , m. . M. X b
(93) Z@lB]. . 5 m T U =
(94) X = ],Z. !,h 5 Å,ÆÅ 6
(95) Í aÅ Õ 76Y Å76hÍÃ,hd7 B6,6,, ./&)(3Æ 6Y Ì +.h 6,-Ã6 > Í6YË.5 ÆÃÍ " , Z@l lGZ ÍÎ Æ 6. . X] Ê/ª ¦/7
(96) 0 .h,5. Ð X lB] T U . å. m. å. ÷. aÅ ÍÅ,ÆÅ 1Â e Í V Í m aÅ. m TU c c / b Z =. ÷ m m ÷ T U c = c / b l. HuB_b^ |Kf'^v`_
(97) _CÊvdc9su)c%u)c9Êo^`_F¯WsteBc%v`st_JuB_ <&vrof!|Kf'up stv; _ Ð { D f!kn_Jup»vr^`mknvc^`f'aJmoj~c%^`s c9v`s_bu C;kn_buW_bjmov`s_buB_ <Ëv`of!|Kslp`kn^rf'v`fef%cbp mo^`f _ Ð n Çf'v !ªCf\vrof!|Kf'up stv;1_F<&v`Êf0ac9mprp»slc9uñjlc D = X Z ÷ ] c%u|ñp f'v _ Ð H Q !#" _ Ð l : f suKvr^`_W|Kmkf\c
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(99) { c9u| óp»mkBv`c9v. m X0>]\Z m X]hZ m X] l @ ÷ {<Î_b^ ¡
(100) l X b
(101) Z@lB]. Ð X lB] T U . = Ú?>A@;B.
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(103)
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(105) 0 ª _ .h,576ÆaÅ 10ÃÊhÌ &1, Æ
(106) ! à = Æ h .,& 76FÕ ÏË'Æ 6FÒÔ7×FÍÒ Ö5תaÅÒ5×-Õ , Í ,h§ B6, 6,, ./&)( Æ 6Y 0 Ì .h 6,0Ã6 aÅ Í Å,aÅ 1 e Í V Í m aÅ > Í 6YË.hg Æ Í " , Z@lGl Z ÍÎ Æ 6 M. _ Ð XlB]. _ Ð X7lB]. ÙWÙ}ÚÜÛYÝFÞ7ß,à. Ð. ÷ c å. _ X lB] T U . Ð. Ð. ÷. å. . _ X lB] T U c . . . m T U c c / b Z =. m. ÷. ÷. m. m ÷ T U c = c / b l.
(107) '. 5, &G& ÏaÏ Í Ð,a B6 . : B1. Hu.c%jtjv`oslpp»fknv`s_buq{ 5 |Kf'uo_Jv`f%p
(108) c9u-; _p stvrstf)k_bup vdc9uWvB|Kf,áf,u|KstuÊaó_buojG; _Ju > c%u| v`Êf <Îmouknvrst_Jup V { e n z of¿o^`_F_ <! _ <!v`of¿v`of,_b^`f,e -nt <Î_Jjtj_ D p<Î^`_Je có|Kf%kn_Je_p stv`s_bu _ <!vrofñf'^r^`_J^4 X b
(109) Z@l)] Ð X B] c%vfcbkdv`sef b
(110) F_ <&v`Êf |Ks~prkn^`f,v`s c9v`s_bufnz os~p|Wf%kn_beá_p»sv`st_JuB^`f'o^rf%p»f,uKv pv`Êf\p mk'knfp`p»sf c9oÊ^`F_ ¯KseBc9vrst_Jup _ < X b
(111) Z@lB] v`c%v D f0suKvr^`_W|Kmknf%|stuvrof o^`fknf%|Ksuoa
(112) p»f%kv`st_JuÂH X b
(113) Zh] Ð X] Q 1 X0 < Ð ] 1 X0 < Ð ] p Ï X < ÐO ] X 8 nt ] c X0 < Ð ] 1 = å Oq Ï p Ï X < OÐ ] p X0 < Ð O ] c = å = å O)q O)q Ï p X0 < OÐ ] p Ï X Ð O ]hl c = å = O)q O)q å z ofñw^ p»vvrf'^`e ef%cJp»mo^rf%p\v`Êfc9oo^rF_ ¯KseBc9vrst_bu f'^r^`_J^ _ <!v`Êfñef%cJp»mo^rf _ m n HuKvr^`_W|Kmknstuoav`Êf o^`_Wkf%p`p f%p X7< ÐO ] {_buofBeBFc Ef%p c9of%c%^
(114) c)p»f%kn_Ju|óv`f'^re D Ês~kdóf%p`p f'uWv`s~c%jtGj ;)s~p\c)p vdc%v`s~p v`slk'c9jf'^r^`_J@^ n z of'u)_Juofkn_Jup»sl|Kf'^ pv`of |Ks~prkn^`f,v` s c9v`s_buBf'^r^`_b^stu|Kmknf%} | C;Bvrof -mojf'^=prkÊf'ef nÎz of\jlcbp v=v`f'^re kn_b^r^`f%p _bu|op=vr_v`of c9oÊ^`F_ ¯KseBc9vrst_JuPF_ <<v`of kn_2f 1knstf'uWvdp ; X b
(115) Z )] c%u| ( X b
(116) Z )] C-;Bef%c%upDF_ <&v`Êf f'eos^`slk'c9jqefcbp»mÊ^`f _ Ð {Çsu)_bvrof'^=vrf'^`eBpstvef%cbp mo^`fpDvroff'^`^r_b^=su|Kmkf%| C-;ñvrof
(117) p»am Cp»v`sv`movrst_Ju _ <&vro f <Õc9es j ;F_ <&stu|Wf'f'u|Kf'uWv Ê^`_Wknf%prp»f%p X < ÐO ] C;
(118) v`Êf <Õc%estGj ; _ <<|Wf'f'u|Kf'uWvo^r_Wknf%p`p f%p X Ð O ] n
(119) ! % "
(120)
(121) ! 4
(122) * z Êf1- ;K_Jv`of%p s~p X ] seojtsf%p v`c9v v`of <ÎmÊuknv`s_bup ( XGZ B] c9u| ; XGZ B] c9^rf Çsp`kdostv su c9u | _bjl|Kf'^Sstu b H<Î_b^Ëc9jtj b ,S ú Z > Êc9u§| <Î_J^&c9jj X0fZh[^] , & ÷ _buof!cbp ; X b Zh] ; X b Z\[^] RFT U V@X0Z <] VYX[Z <] _ Ð X M <] 6 ¢[ X 8 n ] c9u| ( X b Zh] ( X b Zh[a] R T U e X0fZ <] ¢eX[Z <] _ Ð X M <] [ l uñv`Êf0_Jv`of,^c9u|z{ÇcJp ; X b Z\>]Q V@X0fZ N Ð ] { <Î_b^c9jj Z b ,¢ Ý Z> H{ ; X Zh] ; X b Zh] VYXfZ Nª© ] V@X0fZ N Ð ] 6 Nª© N Ð 5 b å ÷ l 0 f%p s~|Kfpn{ ( ( X b Zh] ( Z " X b Zh] , ú Z> ? & nfz Wmpv`ofv`^ c9up stvrst_bu
(123) o^r_ CFc Costjstv ; _ < X8< Ð ] cbpcp»e__Jv`|Kf,up»sv ;¿|Kf,uo_bvrf%| C-; X b Z hfZh[a] n¤: f |Kf'uo_Jv` f C-;
(124) Ð X Z\[^] H Q X b Z Zh[a] g9. = Ú?>A@;B.
(125)
(126)
(127) !"#$ %'&)(+*,-,./0 132450 ./& 76. J. v`of|Kf'up stv; _ < < Ð X[a] nkz Êf)Ð o^`_Wknfp`p»fp X N Ð ] c9u| X8< Ð ] D sÐ v`.vrofp`c%efBstuosv`s~c%jjlc D _ m Cf'stuÊa s~|Kf'uWv`slk'c9jsuPjlc D {ov`of j~c D _ cbpc
(128) |Kf'up stv;|Wf'uo_bvrf%|dC; { D Ês~kds~p aJstf'u}C-;. Ð X]ÂQ R T U X b Z hfZh[a]_ m X M [^]ÂQ R T U
(129) Ð XfZh[a]_ m X M [^] Z " , & Z " b ¡ l z Êf%p»f=Ê^`_bf'^rv;1_F<9; c%u| (ñsteoj ; v`Êf <Î_bjjt_ D stuoa D f'jj ?EKuÊ_ D u
(130) f%p v`stec9v`f X k5<no^rstf%|Kec9u¬ ú Õ{ fnt 8 ?r'JK{_J^\v`Êfkdc9Êv`f'^
(131) _ <§ ] H <Î_J^c9-u ; > {Çv`of,^`f
(132) @f ¯Kslp»v p v`^rs~knvrj ;ñ_p»stvrstf
(133) kn_Jup»v c9uWvdp5 m c9u| 5 å p mkBvrc9v,{ " b ,¢ Ý Z> H{ " fZ¤[ { "( ¡ ( { ^] ÷
(134) Ð XfZh[a] 5 m b Y X ¢[ ( ÷b Z X 8 n8 ] ^] ÷ . . [
(135) Ð XfZh[a] 5 b å @ X ¢[ ( ÷b l. Êf <Î_Jmo^ uÊf@¯KvÈp»maCp»f%knvrst_Jupc9^rf\|Wf'_bvrf%|1v`_fp»v`seBc9vrf%pW<Î_b^<f%cbkdv`f,^`e _ <Ëv`oslpf'^r^`_b^|Wf%kn_be¦? á_p»sv`st_Jufn z. 2(
(136)
(137)
(138) <) "&'!) "!4
(139) ( "! $ % &' (")* (+ z oslp f'^`^r_b^slp|Kf%p`k^`s Cf%|dC-;v`Êf. ¡. å. ²-´ ´ f3 ¶µ¶ ,57 B6, 6 Æ 6Y 0 Ì ¤.h6,
(140) Í Å05aÅ 1& (§ > Í V aÅ ( Í6YË.5oÃÆ Í Y§Æ( b , Z > 1 X0 < ] 1 X < ]! T U
(141) m m T U l 2¤FFÕ : C. 1 . ú. f _ p f'^`f=vrc9v X < Ð ] Q R T U Q. Ð. Ð. å. X < Ð X0[^]] _ m X M ^[ ]Q R T U X < Ð X[a]7] M m X[a] < Ð X0[^]] M X R m X < Ð X[a]7] M m X [a] R ¥ X m ¥ . of stuWv`f'aJ^dc9vrst_Ju C-;c%^`v p4<Î_J^`e
(142) moj~c <Î_b^cdv`sf'jú f%p&stuWv`f'aJ^dc%jÇabsf%p 1 X < Ð ] Q X < Ð X ]7]\l m X ]' R m . . [ X < Ð X[a]7]\l m X[a] M [ ¥ . c X0 < Ð X ]]hlGX m X ]] c R ¥ . [ X < Ð X[a]7]\l X m z. ÙWÙ}ÚÜÛYÝFÞ7ß,à. M m X0[^]]hl. m X0[^]] M [l.
(143) '. 5, &G& ÏaÏ Í Ð,a B6 . p»stestj~c%^<kn_beÊmovdc%v`st_Jud<Î_b^ 1 X0 < Ð ] absf%p 1 X < Ð ] 1 X0 < Ð ] Q R T U . . [ X < Ð X0[^]] lX m X[a] m X[a]7] M [Z p»_
(144) v`c%v . 1 X,< Ð ]? X,< Ð ] T U R T U R T U X0 < Ð X[a]7] l m X[a]? m X[a] M [ M }l .[ 1 zÚ_
(145) f'u|vrof o^`_F_ <r{ásv^`f'eBc%stup&v`_moÊáf,^ C_bmÊu| . R T U . [ X < Ð X[a]7] M fl :f uo_bvrf0vrc9v . . [ X )< Ð X0[^]] Q . . [ +X7< Ð X[a] ] Q . . [ R
(146) Ð X Zh[a] M R . . [
(147) Ð X Zh[a] M Z ¥ ¥ D of'^rf
(148) Ð X7lGZh[a] |Kf'uÊ_bv`f vrof!|Kf'up stv ;1F_ <&v`Êf0jlc D F_ < < Ð X[a] n&o^r_be X 8 n 8 ] { D f |Kf%|Kmknf v`c%v a] ÷ X 8 nú ] M l . . [ X0 < Ð X0[^]] R 5 b å Y X ¢[ b ( ÷ ¥ ® p D f,jtjH{W_buof cJp . . [ X < Ð X0[^]] Q . . [ X X8< Ð X[a] ]7] Q . . [ +X7< Ð X[a]Ρ ] a] ÷ X 8 n ] R ¥ 5 b å @ X ¢[ ( ÷b M l z Kmpn{ <Î^`_Je X 8 nú ] {o_Juof abf'v p R / . . [ X < Ð X[a]7] M R / R 5 b å Y X ¢[^b ] ÷ M M (÷ ¥ ¥ ¥ Q R m R 5 å Y ÷ M M fZ ( ÷b ¥ ¥ b c9ud| <Î^`_Je X 8 n ] R ¥ . . [ X0 < Ð X[a]7] M R ¥ R ¥ 5 å @ ÷ M M fl b ( ÷b m ®. /. . = Ú?>A@;B.
(149)
(150)
(151) !"#$ %'&)(+*,-,./0 132450 ./& 76. 8. f)uo_ D m p»f)vrof <Î_Jjtj_ D suoaóf%p v`seBc9vrf{fcbp; v`_óÊ^`_%fn Çf'v X0>] Cf)v`of|Wf'up»sv; _F< c ac9mp`p»slc9u)jlc D = X Z ] {c%u|Pjf'v X] Cf stvdp|Ws~p»vr^`sGComov`s_bud<Îmoukv`st_Ju vrof'uB_buÊf\cJp X0j] "ª Z X0]gQ R X0[^] M [ YX ÷ ]\Z ÷ ¥ X 8 nÂg ] ¥ X0jj] "ª ZX ]X]}Q R X0[^] M [ YX ÷ ]\Z ÷ p»_
(152) v`c%v R T U . . X < Ð X[a]7] M R T U 5 å ( YX ÷ ] M Q 5 å ( ÷ [ ( ÷b b c9u| 1 X0 < Ð ] 1 X < Ð ]! T U 5 å ( ÷ R T U m X[a] m X[a] M [Z <Î_b^c9-u ; ( ¡ (%{c%u| D sv` 5 å |Kf'f'u|Wstuoa
(153) _bu > c%u4 | ('n ( "!58 7 8 &' %=!9
(154)
(155)
(156) z ofp vdc9vrs~p v`s~k,c9jqf'^r^`_b^slp|Kf%prkn^`Gs Cáf| C-;vrof ²-´ ´ ¡ 3f¶ $¶ ,57 B6,Ã6+oÆ 6Y 0 Ì s.h 6,0
(157) ÷ Å,ÆÅ 1g > Í V Í ( aÅ _ m 6YË-.5ÃÆ Í Y§&G&ab ,¢ Ý Z> Í p Ï X < ÐO ] TU Xj] X0 < Ð ]
(158) ÷ = 1 = å ) O q M p Ï X0 < ÐO ] TU ÷ X ÷ ] ÷ l Xjj] X < Ð ] 1
(159) = å = Oq 2¤FFÕ 0;|KfnwuÊstv`s_buB _ <Ëv`Êf0Ê^`_Wknf%prp»f%p X8< Ð O ] {o_Juof cbp 1 X < Ð ]gQ = p Ï X < Ð X[ mO ]7]Q = p Ï X < ÐO ]7]hl O)q å O)q å Çf,vmpÈw^dp»vo^r_9fv`of c%^`v X s ] nÜf,v Ï. H Q = p X < Ð O ] X < ÐO ]
(160) ! T U O)q å Ï R T U = p X0 < ÐO ] X < ÐO ] ÷ M fl O)q å :. ÙWÙ}ÚÜÛYÝFÞ7ß,à.
(161) n. 5, &G& ÏaÏ Í Ð,a B6 . of X7< ÐO ] ÏO)q å áC f,stuoasu|Kf'f'u|Wf'uKv,{K_Juof\aJf'vdp ÷ = p Ï X < ÐO ] X < ÐO ]
(162) O)q å z. p»_
(163) v`c%v. H =. Çf'v
(164) ÐO. R TU. . =. pÏ. O)q. Q Q. p Ï = ÷ å O)pq Ï = ÷ å Oq. . X0 < ÐO ]. X0. X < ÐO ]
(165). ÷. < ÐO ]hl 8X < ÐO ¢]\Z. å. X < ÐO ]hl X7< ÐO ¢>] M l. |Kf'uÊ_bv`f!v`of!|Kf,up»sv;
(166) _ <Ëv`Êf0jlc D _ < < ÐO {ovrof'u R TU p Ï R
(167) OÐ X0[^] M [l R ¥
(168) OÐ X[a] M [ M fl. = = O)q å ¥ o^`_be X 8 n 8 ] {ov`of'^rf @f ¯Kslp»v pckn_bup vdc%uKv 5 m p mkdPvrc9v 5 m @ X[¢[ mO ] ÷ Z"ª[ , & Z " ( ¡ ( l
(169) OÐ X0[^] b ( ÷b z Kmp 5 m R T U p Ï R 1 Y X [ ÷ ] M [l R ¥ YX [ ÷ ] M [ M fl. = = ( ÷ b 1u b ( ÷b åO)q ¥ / u b / Çf'v
(170) Cf=cJp stu X ] n,+fkn_be_p»stuÊa0vrof=suKvrf'ab^ c9j D stvr1^rf%p»f%knv<v`_ stuvro^`f,fc%^`v p X <Î^r_be v`_
(171) ;{ <Î^r_be
(172) v`_
(173) c9u¦ | <Î^r_be
(174) v`_ c ] { D f abf'v Ï ( R / 1u @X [ ÷ ] M [ M c 5 m
(175) (. 5 m R = p = b ( ÷b = ¥ ¥ å ) O q . c. . . 5 m R ¥ =. . . pÏ ( R ¥ b = O)q å / 1u . @X [( ÷ ÷ b ] M [. X 8 nB ]. M l. = Ú?>A@;B.
(176)
(177)
(178) !"#$ %'&)(+*,-,./0 132450 ./& 76. ,. f D stjjÇ_buÊj ;
(179) v`^`fc9v »p stuknf k'c9u}Cf v`^`fc9v`f| C-;p;Weef'vr^7;n 0;|Kfnwuostvrst_buB_F<vrof [ mO pn{ stvo_Jj~|op vrc9v Ï å [ ( 5 ÷ å YX [ ÷ ] M [ M fl 1 1 p R m R YX ]M [ c R u = b x x ( ÷b ( ÷b = ¥ ¥ Oq å ¥ Çf,v Øv`o f <Îmoukv`st_JuB_b}u ~ Z o|Kfwuof%|dC-; © Y X [( ÷ ÷ b ] M [l X ]Q R 1 ® p m slpc9uñsukn^rf%cbp stuo a <Îmouknv`s_buq{ s~pc ¥ |Kf%k^`f%cJp»stuÊ+a <Îmouknv`s_bu¿c9u| Ïp å X j ] Ïp å X j ] R x X ] M Z = = = = mx O)q å O)q å X ] R å X ]M l = = x x z Kmp{ © 5 m å YX [ ÷ ] M [ M M 1 R R R ( b ( ÷b = m ¥ ¥ 5 m ( R T U Y X0 ] ÷ X ] M M fl R = b ( ÷b m ¥ o^r_be X ] c%u| X 8 nÂg ] {o_Juof|Kf|Kmknf%p&v`c%v X 8 n ] j
(180) m X ] c j
(181) X m X ]7] Y ÷ Z. c9u|Bv`of'uBsvs~p f%cJp ;vr_1aJf'v ÷.
(182) = l :f uo_ D v`^rf%c9v X sts ] n ÷ ÷ 1 X0 < Ð ] = p Ï X0 < ÐO ] Q R T U = p Ï X < ÐO ] X < ÐO ] M O)q å O)q å Ï Q RFT UsRFT U p X0 å < Ð O ] X å < Ð O ] = å O)q Ï l p X ÷ < Ð O ] X ÷ < Ð O ] M å M ÷ l = å O)q :. ÙWÙ}ÚÜÛYÝFÞ7ß,à.
(183) 'g. 5, &G& ÏaÏ Í Ð,a B6 . Hv vrof'uB^`f,eBc9supvr_c%ooj ;v`ofi<c9mk-;?Wkd D c9^=stuofymc%jtsv;
(184) c%u|Pvrof ^`f%p mojtv X s ] n "! * ) 86&'
(185) !'2 ( !
(186)
(187)
(188) ' z of D stuoajf'eeBc-n. <. Î_bjjt_ <. . c%ste _F<vros~pBp»mÆCp»f%kv`st_Ju.s~pv`_ o^r_%fñv`Êf. ¡. ²-´¢´ )3f¶ 3¶ 55}7 B6, 6¬ Æ 6Y 0 Ì }.h 6,0
(189) ÍoÅ,aÅ 1¢ > Í V ÆÅ ( Í6YË.5 ÃÆ Í ZYl l Z Í. . " ªQ. Ï. pÏ. Ð. / b X0 < ÐO ] T U = å
(190) )O q . X0j] p X < O ] = )O q. M. X0jj] . . _bv`suoav`c9v. = . å. pÏ O)q. å. pÏ. Ð. Ð. . X < O ] T U = Oq. X0 < O ]. å. . ÷. X
(191) ] ÷ / b l. 247FFÕ -. " /Z V , &Z. _buof!abf'v p c9u|. = p Ï X < ÐO ] Oq å. X /V ] M Q . ]. pÏ. Ï X < ÐO ] T U p = å = )O q )O q. . = . Ð. X < O ]. pÏ. . å. Â. V Z. < ÐO . X8 n ]. < ÐO . ÷. Ï = p < ÐO < ÐO ÷ l Ð å O)q å O)q å f'v O
(192) Q < ÐO < ÐO ÷ nz of ymcb|K^ c9v`slk0efc9u¿kn_buWf'^rabf'ukf=^dc%v`f\_F<<v`of -8mÊjtf'^prkof,ef <Î_b^ +- Âp stvrkn_f 1Bknsf'uWvdp os~kdc%^`f ÇstprkÊst7v <ÎmÊuknv`s_bupsu c%u | _bjl|Kf'^& _ <ô_b^d|Wf'^ %\suv`sef s~pc%uñf%cJD p;aJf'uof'^ c9jtsG%c9vrst_JuPD _F<Îñstjlp»Wv`f'su Âp^rf%p»mojv Ý Õ{op»f,f n0 _prp ; Âpv`of%p s~p ´/<Î_J^=|Wf'vdc%stj~p,n "! ) ! ! ") ! "& !
(193)
(194)
(195) ' Huv`Ês~p p am Cp f%knv`s_buq{ D fp»vrm| ;
(196) v`of=f,^`^`_J^|KmÊfvr_\v`Êf p»maCp»v`sv`movrst_Juñ_ <Ëv`Êf\|Wf'f'u|Kf'uWv O Âp vr_1vrof stu|Kf,áf,u|Kf'uWv < O Â/p n . . pÏ. X . R TU. . X0 < O ] T U = )O q. .
(197). . = Ú?>A@;B.
(198) F.
(199)
(200) !"#$ %'&)(+*,-,./0 132450 ./& 76. ¡. ²-´ ´ 3f¶ f¶ ,57 B6,Æ 6Y Ì 4.h 6,0 6
(201) aÅ
(202) ÍÅ,aÅ 13 ( Í V ÆÅ > Í6YË.5oÃÆ Í Y§&G& 3Q Z@lGl Z Í Xj] M. . = p Ï X < ÐO ] Oq å. X0jj] . . =. pÏ Oq. å. =. pÏ O)q. X < ÐO ] =. å. pÏ O)q. X0 . å. X0 . Ð O ] . T U
(203) / bWc = c m . ÐO ] TU ÷.
(204) / bfc = c m. m TU m ÷ T U l. o^`_Je X 8 n ] stv<Î_bjjt_ D p v`c9v Ï p Ï < ÐO ÐO l p X < ÐO ] X0 Ð O ] T U = = Oq å O)q å -_bv`f vrc9v < ÐO Ð O ÷ < ÐO Ð O ÷ c / b ÷ R VYX < ÐO Zh[a] Ð X M [a] = p Ï VYX Ð O Z Ð ] ÷ q å p Ï eX ÐO Z Ð ] ÷ c / b R eX < ÐO Zh[^] Ð X M [a] = å q I Ï c / b K L < ÐO Ð O l R VYX < Ð O Zh[a] Ð X M [a] = p VYX Ð O Z Ð ] l q å Ï f'v
(205) Q p < Ð O Ð O ÷ _Juof cbpv`c%v = å O)q
(206)
(207) å c / b ÷ = p Ï R V@X < ÐO Zh[^]`_ Ð X M [a] = p Ï VYX Ð O Z Ð ] ÷ q å O)q å Ï p Ï e X Ð O Z Ð ] ÷ X 8 n' ] c / b p R eX < Ð O Zh[a]_ Ð X M [a] = å = å O)q q Ï p Ï V@X ÐO Z Ð ] ÷ c / b
(208) å l p R VYX < ÐO Z\[^]`_ Ð X M [a] = å. = O)q å q HQ
(209) å c å c ÷ c l 2¤FFÕ. . . . . . . ÙWÙ}ÚÜÛYÝFÞ7ß,à.
(210) '. 5, &G& ÏaÏ Í Ð,a B6 . å slp moof'^C_bmou|Wf%| C-;45 X / b ] ÷ n,Çf,vmp vr^`f%c%v ÷ H p Ï R e X < ÐO Z\[^]`_ Ð X M [a] p Ï eX ÐO Z Ð ] ÷ = å = å O)q q Ï Ï = p R"eX < ÐO Zh[^]`_ Ð X M [a] = p eX < ÐO Z < Ð 7] ÷ q å O)q å Ï Ï c p p eX < Ð O Z < Ð ]) eX Ð O Z Ð ] ÷ l = å = å O)q q ® p e s~p Çszkos7v { D f= _ Cp»f,^`f v`c%v p Ï p Ï eX < ÐO Z < Ð ] eX Ð O Z Ð ] ÷ 5
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