Equivalence entre les procedures de «local tracking» et les algorithmes monotoniques en contrôle quantique
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Equivalence entre les procedures de “local tracking” et les algorithmes monotoniques en contrôle quantique Gabriel Turinici. N° 5564 Mai 2005. N 0249-6399. ISRN INRIA/RR--5564--FR. Thème NUM. apport de recherche.
(3)
(4)
(5) ! #"$ % & '()
(6) *+ , -
(7) +&./01 !01& 2
(8) 34 5%
(9) 6
(10) 3
(11) 97 8;:)<>=@?BACED)<>=@F)=HGB= IKJML3NO.P QERTSVUXWZY\[]L>N4O^YK_B`MN4a>b]cedB`fO>Y ghbjilkmO3[nR9cpo>R9qro snqutft;ivb\[nwZO.b]O^ojJfO>bjoxJfOy_;z){v{u|r}4S~R9qrc(uvv{2S>|4tMquvO>Y ∗. B^ $O>Y Y]cN`fpqu[]cpir_MYE_B`fNa>b]cedB`fO>YEO>_oir_B[]bjrpO&dv`qu_v[jcpdB`fO&`Z[jcpceY\O>_v[tfp`MY]cO>`fbjY qrtftfbjiZojJfO^Y cp_Mop`Mqr_v[ O^YntfbjiZo3a>wZ`fbjO>YnwZO
(12) [mWXtO\ifo3qu$[]bxqroxXc_fu.dB`fcirZ[jcO>_f_fO3_B[ pO&oxJMquNtwZO&o3ir_v[jbjrpO.tMqubEeq o3ir_MwZc[]cpir_dv`MO`M_fOoO>b\[xqucp_fOir_o[]cpir_Y]irc[4wZa^objirceY]Yjqu_B[]OrNquceYqu`Y]Y]c%pO>Y4quprivb]c[]JMN4O^Y
(13) Nir_fir[]ir _fcedB`fO>Y dB`fcbja>Y]irprO3_B[n@ a^dB`Mql[jciv_9w¡ ¢`fO>b\£(qrrbxqu_MoO.t;ir`fb `f_fO
(14) iv_Mo[jciv_f_fO3ppO&wZOo3ir¤Z[ tfbja>wZa3¥M_fcpOr¦ R¨§>N4O2Y\c¡wZcª©«a3bjO3_B[]O^Y%O3_ O>`fbcNtfpa3NO3_B[jqu[]cpir_MY>XwfO>Y[jbjq¬lqu`fb]a^oO3_B[jY%ir_B[Nir_v[jb]adB`fOoO^YoeqrYjY\O^Y tMqrb\[xquvO3_v[wZO^Yo3qrbjqvo[ja3b]ceYm[jcpdB`fO^YoivN4N&`M_fO>Y>¦$O¨M`Z[ wZO9o3O®[]bxq¬lqucpO>Y\[wZO9tfb]ivt;ivY]O3b`f_fO9MqvY\O bjcvir`fbjO3`MY]Ot;ir`fbwZO>Y.[jO3pO^Y&oiv_Mop`MY]civ_MY.O[wZO wZcpYjo`Z[jO3b&eqivb]N&`fequ[]cpir_'tfbja>oceY]Odv`Mct;O3bjNO[wZO o3ir_MY\[]bj`fcpb]O`f_fO2[jO3pO.a>dB`fcplqrO>_MoOv¦ ¯±°²>3³j´vµ¶ X ·oiv_v[]bjrpO2dB`Mqr_v[]cedB`fOrMquprivb]c[]JfNO^Y%Niv_fiu[jir_fcedB`fO>Y>Zqrvirbjcª[jJfNO>YKwZO2[]bxqroxBcp_f. ½x¹¬É.Ö¡¸º¹¬Ü;»¶¾(¼ª½nÝ2¾(¸¶¿3Ö$ÀmÞ3Á>ßÂ^ÜÃ\à¹^À\Ϋ¿Ö¬»¶ÅlÄ]Ö«ÅX×lƽx¿ÇÇná^½xØ$¼È¹¬Ø@¬Ãh»(ɬÝ.Ãʽx»¶¿¹¬ËÈìÅrÀmÃ]Ò½xÒjÂvâÑÅBѾ$Ý.¿>Àm½xÁ3»H¬¹^ÃmÃh¹¬ÀmË㽿¬Ê;»¶Ä½ËÈËÈÌämÃhÍ ÎZÖ«ÍZÃmÏ\ɬÐãÑ^ÚvÅBÅlÒxÛlÓ^»HÏ\½xѹ¬ÔEÀmÃÕBÃK֫׬ÃmØH¹l½\Ù
(15) Ö¡ÃmɬãÚrÅvÛl»H½x¹¬Àmà ∗. Unité de recherche INRIA Rocquencourt.
(16)
(17)
(18) '()
(19) * + % ) !
(20) 01
(21) .
(22) T
(23) +
(24)
(25) /01!54
(26) 0 %&()
(27) >² f´r²rhIKJMO
(28) oivNtf`Z[]O>bEY]cN&`feql[jciv_MYiu dB`Mqu_B[]`fN oir_B[]bjir$`MY\O&Y\O3vO3bjqr$qutftfbjivqvojJMO>Ycp_Mop`MwZcp_f piZo3qr([]bxqroxBcp_ftfb]iZo3O>wZ`Mb]O^Y[]JMqu[tfbjO>Y]o3b]cp;O.[jJfO4o3ir_v[jb]ivpcp_f¥MO>pw¨[jJfb]iv`frJ¨[]JMOb]O^dB`fcbjO3NO>_v[E[]Jql[ qoO3b][jqrc_ º`M_Mo[jciv_Mqu$;O.wZO>o3b]O^qrY]c_fqu_Mw Nir_fir[]ir_Mcpoqrvirbjcª[jJfNY[jJMql[EY\ivrO [jJfO2¢h`MO>b\£(qrrbxqu_frO O^dv`ql[]cpir_Y«ivbqtfbjO>wZO3¥M_fO>w
(29) oiBYm[$`f_o[]cpir_qu@
(30) ¦ JfcpO%wfcª©«O3bjO3_v[c_
(31) cpN4tMO>N4O>_v[xql[]cpir_(>b]O^oO>_v[ irbjXY p Jfcp_v[jO>w[jJMql[2[]JfO^Y\O4[ i®oeqrYjY\O^YY\JqubjO4Y\ivN4OoivNN4iv_ oxJMqubxqro[]O>b]ceYm[jcpo>Y3¦ Otfbjirt;iBY\Oc_ []JMcpY.oiv_Z [jb]cpf`Z[jciv_q9b]cprivb]ir`MY.rbjir`f_w irb4Y\`ojJ·oir_op`MY\cpir_Y
(32) qr_MwwZceYjo`MYjY.[jJfOtfbjO>o3cpY]OirbjN&`fpqu[]cpir_'[]Jql[ qrpi nY[]io3ir_MY\[]bj`Mo[[jJfceYO>dB`fcplqupO3_oOr¦ «³ ° vdB`Mqu_B[]`fN oir_B[]bjir@B[jbjqvojXcp_fMZN4iv_fiu[jir_fceo2quprirbjc[]JfNY.
(33) .
(34)
(35). . ® !01 2K)
(36) ² ¬° ´r²H° ¶² · Z² !#" ° l µ Z²$¶° % Z¦p I(bxqroxXc_M4qrvirbjcª[jJfN ¦2¦.¦2¦.¦2¦.¦2¦.¦2¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦ } Z¦È R¨ir_fir[]iv_fcpo2quprivb]c[]JMNY%ivbirtZ[jcNqu(oiv_v[]bjir ¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦ { Z¦ &fivb qrbjw oivY\[K`f_Mo[]cpir_Mqr ¦.¦2¦.¦2¦.¦2¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦ | Z¦ } R¨ir_fir[]iv_fcpo2quprivb]c[]JMNYKqvYpiZo>qu«[]bxqroxBcp_f4tfbjiZo3O>wZ`fbjO>Y ¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦ ' ( *)+"@ ´u² H° " ° l µ Z² H° , f¦p R¨ir_fir[]iv_fcpo2quprivb]c[]JMNY ¦2¦.¦2¦.¦2¦.¦2¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦r f¦È &fivb qrbjw oivY\[K`f_Mo[]cpir_Mqr$qr_Mw O^dv`McuqupO3_MoO¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦.¦2¦r % . ° ´rµ¶$ ¶° / $´ 0 ° µ 21 ²> +% 3 4 4(
(37) R9qu_fcptf`feql[jciv_irdB`Mqu_v[j`fN tfJfO>_firNO>_Mq qrYqupbjO>qrwZW wfO3Nir_MY\[]bxql[jO>w ;ir[]J cp_ o3iBY\O^wX iXivt equ;ivbjqu[]irb]WO3Xt;O>b]cpNO3_B[jY ÈX5 fh}M{Z | qr_Mw c_ []JfO>irbjO[]ceo3qrnY\[]`MwfcO^Yir_ []JMO9o3ir_v[jb]iveqrfcpc[£Wir dB`Mqr_v[]`MN#Y\WZY\[]O3NY 76X8 'Z8 9f$> @¦ : o>oirNtMqr_BWXcp_fn[]JfO^Y\OqvwZlqr_MoO>Y>3[]JfOKoivN4tM`Z[]O>bY]cpN&`fpqu[]cpir_MYJMq¬rOh[]JMOqrwflqu_B[jqrrO []i ilrO3bxoivN4O O3Xt;O>b]cpNO3_v[xquXbjO>Y\[]bjcpo[]cpir_MYqu_w&JMq^rOKqro>oO>YjY([]i[]JfO JfivOKwZWX_MquNceo3Y)qrpi cp_fE`fb][]JfO>b)cp_MY\cpvJv[ qr_Mw qrpY]i2tMb]ilXcpwZcp_f.Jfcp_v[xYhcp_wZO>BceY]c_f.`Z[]`fbjOnO3Zt;O3bjcNO3_B[jY>¦vR9qu_XW4quprirbjc[]JfNY JMq^rOn;O3O3_tMb]ivt;ivY]O>w&[]i Y]irprO[]JfO
(38) O3_MY]`fcp_fivtZ[]cpN c ;^ql[jciv_tfbjirfpO3N#quNiv_f JMcpoxJ[ iwfcpY\[]cp_Mo[ oeqvY]Y]O>Yo3qu_®;O.cpwfO3_v[jcª¥O>w¡¦ IKJfOn¥bjY\[%ir_fO oiv_v[jqrc_Y [jJfOEpiZo3qrf[]bxqroxXc_M.NO[jJfiZwfY prr¬X$ Z>}f¬{X ;[]JMqu[%tfbjirt;iBY\OO3Ztfceoc[ ivb]N&`fequOKiu[jJfOnwZbjcXcp_f¥MO3ewcp_qu_4irt;O>_Z iXivtwfO3t;O3_MwfO3_Mo3OKiv_[]JfOO>rirpXc_f.Ym[xql[]Ov¦uIKJMOirbjN&`fequO qrb]Oirf[jqucp_fO^wbjirN[jJfO®bjO>dB`fcpbjO3NO3_v[[]iwZO^objO>qrY]O®q oO>b\[xqucp_·`f_Mo[]cpir_MqrnwZO3¥M_fO>w ql[O^qroxJ·[]cpNO cp_MY\[jqu_B[&qu_w bjO3eql[jO>w[]i®[jJfO&]wZceYm[xqu_Mo3O[ji®[]JfO[jqubjrO3[.irb.XW wZO>Nqr_MwZcp_f9Ym[jb]ceo[&qrwfJfO3bjO3_Mo3O4[]i9q tfbjO>wZO3¥M_fO^wirMY]O3bjlqrfO[]bxqlkmO>o[jirbjWr¦vIKJMO Y]O>oiv_MwoeqrYjYqubjO[jJfO Nir_fir[]ir_McpoEquprirbjc[]JfNY ^|Z;< 6X$ ' [jJMql[EY]irprO[]JMO.¢h`fpO3b]£(quvbjqu_frOO>dB`Mqu[]cpir_MYnqrYjY\iZo3cpqu[]O>w[ji[]JfO.ivtZ[]cpN=c ;>ql[jciv_iu qo3ivY\[K`f_o[]cpir_qu wZO3¥M_fO>w®qu[nq¥M_Mqr«[jcNO T ¦MIKJfO2[ ioeqrYjY]O>YKo3qr_®queY\i4;O
(39) oivN&fcp_fO>wqvYKc_ p ¦ : []Jfiv`frJwZc©«O3bjO3_v[&cp_'cpNtfpO3NO3_B[jql[jcpir_$$[]JMO>Y]Oquprivb]c[jJfNY.qrb]OY]Jf i _ ;O> i y[ji¨;Ob]O>pqu[]O>w cp_ [jJfO
(40) Y]O3_MY]O[]JMqu[nN4iv_fiu[jir_fceo2Y]oxJfO3NO>YnqubjO [jbjqvojXcp_f4tfbjiZoO>wf`fb]O^Y%irbnY]irNO.Y]tO^oc¥ot;O>b\ivb]Nqu_oO cp_MwZOZO^Y3?¦ >£_ []JfO
(41) o3ir_v[jOX[ir[]JfO
(42) wZO>_MY\c[mWNqu[]bjcªivb]N&`feql[jciv_$X[]JfceYc_wZO ceYwZO3¥M_fO>w®qrY[jJfO.lqup`fO ir)[jJfO4oivY\[E`f_Mo[jciv_MquA @¶qu[E[jJfO
(43) ¥_Mqu([jcNO T B O3lqr`ql[]O^wirbE[jJfO\;O>Y\[o3qr_MwZcewfql[jO.¥MO3ew9qu[E[]cpNO t≤T E<cMI[]J·NOCq¨¦PrIKcpNQJfrRO3ceE Y
(44) _$P(o> qu ;bj_MOcpY2wZO>ceb]wZwfO>O^qu_Mo[]objOOO>qr¥M¥MY]O3O>O3ewew w qlceY.[2bjirNO>N qvqvojwZJ9tO[j[]ci Nb]ivTON civ_M[]ZY\Jf[j[jOqrqrc_vo_M[^`MO>S¦ b]wbj&fO3ivql_Bb [
(45) [2[j[j¥MJfJfO>OOpw'o3tfqvbj`fY\O3t'OXcir[]ivhi®`M[jY.[]JfcpcNO []O3O bxqlq¬t[jrcp[]OirJM_$`fqu¦_[
(46) IKoce[]Y.JfcpircetfDY_(bj¡CirEGpwZirFI`f_fHJOv O>[]FLwiK [jJfOn_fir_Mcp_fO>qrbh_Mql[j`fb]Oniu;[jJfO[jqrb]vO[ irbjN&`feql[]cpir_(rqu_`Mtft;O3b;ir`f_w4ceYh`MY\O^w[ji
(47) wZO3¥M_fO[jJfOirb qrbjw o3ivY\[`M_Mo[jciv_Mqu@¦ . ¾¾#¹GTMÑÑIUxâ.
(48) }. FIR
(49) FI R. IKJfO%ir`Z[jcp_fO%irZ[]JMO%tMqrtO>bceY([]JfO%irp i c_M >irb[]JfOKo3qrY]Oiuf[]JfOKwZO3_MY]c[£W.Nql[]bjc %O%tfbjO>Y]O3_B[([jJfO wZO3¥M_fc[]cpir_¨iu[]bxqroxXc_ftfbjiZoO^wZ`fbjO>YKc_9Y\O^o[]cpiv_¨Z¦p.qr_Mwir_fO.OfqrN4tMO.iu)Nir_Miu[]iv_fceo2quprivb]c[]JMN-c_ Y]O>o[jcpir_f¦ Z¦¬IKJfO>_$ %O%OZtfeqrc_
(50) []JfOb]O>pqu[]cpir_
(51) ;O[ %O>O3_
(52) [jJfO%[ %io3pqvY]Y]O>Y(cp_Y]O>o[jciv_MY)Z¦ Equ_Mwf¦ }f¦lIKJfO o3irbjb]O>Y]t;ir_MwZcp_f¨qu_MqrWZY]cpYivb.[jJfO q¬rO`f_o[]cpir_'irbjN&`feql[]cpir_'ceY.vcvO3_'c_Y]O>o[]cpir_ f¦ %iv_Mop`MwZcp_f wZceYjo`MYjY\cpir_MYqu_Mwb]O>Nqrb]ZYKqubjO []JMO2ivZkmO>o[niu Y\O^o[]cpir_}M¦ 01 2() l
(53) )0 3 2
(54) . (x
(55) %ir_MY]cpwfO3b)q dB`Mqu_B[]`fNyY]WXY\[]O>N cª[jJc_B[]O>b]_MqrXwfWB_quNcpo>YwZO^Y]o3b]cp;O>w
(56) XW2[]JM O EqrN4cp[]iv_fcpqr_ ¦ > [jY cp_v[jO3bxqro[jciv_ c[]J®qr_OX[]O>b]_Mqr @Ov¦ ¦fpqvY\O>b B ¥O3ew ceYKNiZwZO3pO>w BWcp_v[jb]iZwZ`ocp_f[jJfO
(57) wZct;ivO2NHirNO>_v[ ivt;O3bjqu[]ivb qu_Mw [jJfO¥O3ew'cp_v[jO3_MY]cª[mW ¦ > K[]JMOY]WZYm[jO3N cpY
(58) bjO3tMb]O^Y\O>_v[]O^w c_[]JfO wZO>_MY\c[mW Nqu[]bjcª ivb]N&`feql[jcivµ_ c[]J c_Mcª[jcpqrEYm[xql[]O ρ cª[x²(t) YwZWX_MquNceo3Y cpnir;O>W[]JfO®[jcNO9wZO>t;O3_MwZO>_v[ UfojJfb ZwZcp_fvO3b O^dv`ql[]cpir_ 0. 0. @m B. ∂ ρ(x, t) = [H0 − ²(t)µ, ρ(x, t)] ∂t ρ(x, t = 0) = ρ0 (x). i. EO3bjO %O2`MY]O>w []JfO
(59) o3ir_BrO>_v[jciv_ ~ = 1 ¦ O cp_vqr[j_Mb]ifw9wZ[]`MJfoOOEqr[]JfYjY\O2if$ocecpqlir[]`fO^Xw¨cp_fpO irbjY\NtqroOEbjO3tfbjO>Y]pO3_B[jqu[]cpir_$vS¦ B>£_MWYmwZ[jO>O3qr¥Mw_fcpiu_f&[j[]JfJfOO2o3Y]iro>NqueN&qub`Ztf[xbjqliZ[]ivwZbj`MYo[ qr;hha,i¬rOvbii %=O T r(a b) wZO3¥M_fO
(60) [jJfOirt;O3bxql[jirbjY H qu_Mw M [jJMqlkak [2qro[E=iv_9wZhha, O>_MY]aii cª[mWNqu[]bjcpo3O>YnXW Hρ = [H , ρ] Mρ = [µ, ρ] ¦ ¢dB`Mqu[]cpir_ @\ B ;O>o3irNO>Y @@ B ∂ i ρ(x, t) = (H − ²(t)M) ρ(x, t) . †. ii. 0. ∂t ρ(x, t = 0) = ρ0 (x).. IKJfOoiv_v[jb]iv riBquho3qu_';OO3Ztfb]O^Y]Y]O>w[]Jfbjir`MrJ []JMOcp_v[jb]iZwf`Mo[jciv_ irKqr_ ivMY]O3bjlqufpO4irt;O>bjqu[]ivb B W[]JfO2bjO>dB`fcpb]O>NO3_v[[]JMqu[[jJfO.dB`Mqu_B[]c[£W ;ONqlZcpN4c=;3O^w @ wZO>_fiu[jc_f[jJfO A bjO>qrhtMqub][&iunq9oirNtfpO'_B`fN&;O3b B ¦IKJfcpY
(61) irhA(t)i bjN&`feql=[]cpirRehhA, _o>qu_;ρiiO`fb][]JMO3b&bjO¥M_MO>wRe qrY
(62) cp_ p^{ JfO3bjO qr_c_MwfO y(t) = y(hA (t)i, ..., hA (t)i) qrrrbjO3vql[]cp_f&Y]O3vO3bjqrirY\O3bjlqrfpO>YhceYKoir_Y\cewZO3bjO>wirbO3rO3_ `fb][]JfO>b(BWwfO¥M_fcp_f y(t) = y(R ² (s)ds, hA (t)i, ..., hA (t)i) JfO3bjO %Ohcp_v[]bjiZwZ`Mo3O O3XtMceoc[]pWE[jJfO wZO>t;O3_MwZO>_MoO%ir_.[]JMOeqrY]O3b M`MO3_Mo3Or<¦ &fivb(_fir[jqu[]cpir_Mqrvoiv_BrO>_fcO>_MoO %O cpXwZO>_fiu[jO F (t) = R ² (s)ds ¦ 1. . K t 2 0. 1. K. t 2 0. "! #$&%(')"*+'-,./$10 243. %ir_MY]cpwfO3b[]JfOY\cpNtfpOY\c[]`Mqu[]cpir_ y(t) = y(R o3ir_MY]cpwZO>b]O^wcH¦ Or¦p K = 1 qr;i¬rOv¦ O.irZ[xqucp_ . t 2 ² (s)ds, hA(t)i) 0. JfO>b]Oir_fpW9ir_fOirMY]O3bjlquMOceY. dy(t) (H − ²(t)M) ρ(x, t) = D1 y · ²(t) + D2 y · RehhA, ii dt i. ¸ºà¾$¸65.
(63) {. .
(64)
(65). JfO>b]O pc Y[]JMOtMqrb\[jcpqrKwZO3bjcuql[]cprO cª[jJbjO>Y]tO^o[[ji []JfO j@[]Jlqrb]cequfpOr¦)IKJfceYo3qu_·;O `Mb\[jJfO3b O3XtMb]O>YjY]DO>w qrY j. dy(t) = f (F (t), ρ) + ²(t)g(F (t), ρ) dt @ g B ye ²(t). @ B. JfO>b]O uqu_fceY\JfO^Y JfceojJ cpK;O®o3qrpO>w·Y]c_fv`fequbjcª[jcO^Y qr_Mw cpp$;O2[jb]O^ql[jO>wY]O3tMqrbjqu[]O>W ºivbEqu_BW wZO^Y\cpbjO>w[jbjqlkmO>o[]ivb]W c[]J ye(0) = y(0) f[]JfO&oiv_MwZc[]cpir_ `M_fcpdB`fO>W wfO[]O>b]Ncp_fO>YK[]JfO2¥MO>pw BW[jJfO2irbjN&`feq y(t) ≡ ye(t) − f (F (t), ρ) @¶} B . ²(t) = g(F (t), ρ) &Mb]ivN dF/dt = ² (t) iv_fO2irZ[xqucp_MY[]Jql[ @} B ceYc_¶qro[E q ¢ ir_ F ir[]JfO2irbjN @H{ B dF/dt = Y(F, ρ) [jJMql[cpY []i.;OnY]irprO^wEkmircp_v[]pW c[]J @H B cp_irbxwZO3b)[ji.O3_MY]`fbjOqvwZJfO3bjO3_oO[ji2[]JfOntfbjO>Yjobjc;O^w
(66) [jbjqlkmO^o[]ivb]W ¦ ye UZquNOhoiv_MY]cpwZO>bjqu[]cpir_MY¡qutftMW cªZir_fpW O>qurO>b¡tfbjirt;O3b][]cpO>Y$qubjO)b]O^dv`McbjO>w¡[mWBtMcpo>qupWE[]JfOhcp_MobjO>qvY\O lwZO^ob]O>qrY\O ir y(t) JMcpoxJ®o>qu_;O.O3_Zivbjo3O>w[]Jfbjir`fvJ[jJfO
(67) oir_wZcª[jciv_ dy/dt ≥ 0 @ ≤ 0B ¦ IKJfO4wZc o3`fª[mW¨c_9[jJfcpY2qutftfbjivqvojJ¨cpYE[ji¥M_wqY\`Mcª[xqufpO&bjOO>b]O>_MoO
(68) []bxqroxXc_f[]bxqlkmO>o[jirbjW []JMqu[ wZiZO>Y%_fir[vcrO bjcpY]OE[]iY\cp_fr`Mpqrb%t;ircp_v[xYiu$[jJfO2Y\WZY\[]O>N @ B @H{ B c@¦ Or¦p JfO3bjO g(F, ρ) = 0 +¦ >£_ ryeO>_fO3bxqu Y]c_Mr`fequb%t;ircp_v[jYo>qu_f_fir[%;Oq¬rircewZO>wq
(69) tfbjcpirbjcqr_Mw[jO>oxJf_fcedB`fO>Y %O>b]O wfO>Y]cv_fO>w[ji
(70) [jbjO>ql[KY\`MoxJ Y\c[]`Mqu [jciv_M4Y ZY]O3O p3} ;ivbKwZO^Y\cpr_MY[jJMql[KpiZo3qrpWqu[]O3b%[jJfO []bxqlkmO>o[]irbjW[ji4o3cbxo`fN&rO>_v[[]JfO2Y]cp_fr`fequb%t;ivc_B[jY qr_Mw $ 9Z$ $irbnqY\[]`MwZWiv_ []JfO
(71) Y\[]ivtftfcp_ft;ircp_v[xYqr_Mw tfbjiZoO^wZ`fbjO>Y%[]icNtfbji¬rO[]JfO>cbnirtf[]cpNqrc[mWr¦ > [ceY4Y]O3O>_[]JMqu[>)Ofo3O3tZ[4irb[jJfOt;ircp_v[jY. de y (t) dt. 2. . . ,-% ,"0 , %($&!.
(72) ,- , . "*+'-,./$+0 243. 0 $3)"* ! ,-% 0 ,-*. >£_ qu_qrtftfbjivqvojJ9wZc©;O>b]O>_v[b]ivN []bxqroxXc_fN4iv_fiu[jir_fceo3qrpW®oir_XrO3bjrO>_v[quprirbjc[]JfNYntMciv_fO3O3bjO>w cp_ ÈuZ;f qu_MwOX[jO3_MwZO^wcp_ p' (cp_[jJfO q¬rO`f_Mo[]cpir_®bjO3tfbjO>Y]O3_v[xql[jciv_$fqubjO2`MY]O>wcp_[jJfO&oir_B[]O3B[ ir«[]JMO wZO>_MY\c[mW4Nql[jb]cirt;O3bxql[jirbqrYc_ ÈrZXG @¦ZUX`MoxJtfbjiZoO^wZ`fb]O^Yhqrb]Oncp_Mop`MwZO>wcp_[]JfOEbjqrN4O
(73) %irbj ir[jJfOirtf[]cpNqr%oiv_v[jb]iv []Jql[c_B[]bjiZwZ`Mo3O>Y&q9oiBYm[
(74) `f_Mo[]cpir_Mqr @HwZO¥_fO>wqu[q®¥M_Mqu[]cpNO T B []i9;O ivtZ[]cpN c ;>O>w¡¦ _fO.Y]`MoxJOfqrN4tMO2iu`f_Mo[jciv_Mqu$ceY Z @H| B α² (t)dt − 2RehhA, ρ(T )ii. J(²) = Y]irJf`fO>vb]JvO [αqlºce[]YO>q
(75) b&t;`fiv_MY]wZcª[jO>cb
(76) rO [jJf@HOoiro3_irYm_M[xquY\[]_vbx[qucpir_vb[&[]iucpNnO Yjqll[]quceY\bjWXWBcpcp_f_f B %@H O3cp¦ rJB%[>O^¦Xo3IKqr`MJfY\O>O_$vir[]KJf[]OJfo3O b]co[]ivceo3_MqrYm[jbjt;qrirc_Bcp_v[>[jY%q9iu (J(²) qrb]O u q v j b r q f _ rO N&`f[]cptfpcO>b>wZO3_fir[]O>w χ(x, t) cpYcp_v[]bjiZwZ`oO>w cp_[]JfO
(77) oiBYmB [K`f_o[]cpir_qu¡[]JMqu[n_fi bjO>qvwfY T. 2. 0. Z. T. α²2 (t)dt − 2RehhA, ρ(T )ii J2 (²) = 0 (Z ) T ∂ρ (H − ²(t)M)ρ +2Re hhχ, − iidt ∂t i 0. ¾¾#¹GTMÑÑIUxâ. @6 B.
(78) |. FIR
(79) FI R. IKJfO
(80) objc[]ceo3qu¡t;ivc_B[O^dv`ql[]cpir_YKqrb]O [jJB`MYnirZ[xqucp_fO>w . ∂ ρ(x, t) = (H − ²(t)M) ρ(x, t) ∂t ρ(x, t = 0) = ρ0 (x) Mρ 2α²(t) + 2Rehhχ, ii(t) = 0 i ∂ i χ(x, t) = (H − ²(t)M) χ(x, t) ∂t χ(x, T ) = A. i. @' B @R9 B. @m> B. `fcppwfc_f&ir_[]JfO^Y\OEbjO3eql[jciv_MY3u[]JMOENir_fir[]ir_Mcponqrvirbjcª[jJfNY tfbjO>Yjobjc;OEq.tMqub][]ceo`MpqrbhirbxwZO>b []i&c[]O>bjqu[]O cp_¨[]JfO^Y\Ooiv`ftfpO>w¨O>dB`Mql[jciv_MYnBW¨oir_MY\[]bj`Mo[]cp_fMqu[n[jJfO&c[]O>bjqu[]cpir_9Ym[jO3t k → k + 1 ;q¥O3ew ² (t) c[]J[]JMO2cpNt;irb][jqr_v[tfbjirt;O>b\[mW @mr B J(² ) ≤ J(² ), JfO>_MoOE[jJfO _MqrNOEir E PE N E ! G *EGFIN¦ : Y]cNtfpO OfquNtfpOEir(Y]`MoxJqrvirbjcª[jJfN ceY @HY]O3O vXZ ivbnqrwfwZc[]cpir_qu$wZO[xqucpeY B @\^ B ∂ i ρ (x, t) = (H − ² (t)M) ρ (x, t) k+1. k+1. k+1. k. k+1. k+1. ∂t ρk+1 (x, t = 0) = ρ0 (x) Mρk+1 1 ii(t) ²k+1 (t) = − Rehhχk , α i ∂ i χk+1 (x, t) = (H − ²k+1 (t)M) χk+1 (x, t) ∂t χk+1 (x, T ) = A.. @\ B @\3} B. IKJfceY quprivb]c[]JMN ceYtfbji¬rO^w ÈG [ji·JMq^rO9[]JfO o3ir_XrO3_fcpO3_v[ tfbjirt;O>b\[mWc_¢dB_$¦ @\r ¦ > [ cpY[ji;O _fir[]O^w []JMqu[n[]JfceYntfbjirt;O>b\[mWcpYnrO3bjW Y]`fb]tfbjceY\cp_fcp_[jJfceYnJfcprJfpW_Mir_fpc_MO>qubEY]O[\[jcp_fMMO>Y]t;B O>oceqrpW JfO3_ o3ir_MY]cpwZO>b]cp_f[jJMql[n_fiY]O>o3ir_Mw irbxwZO3bKcp_Zivb]Nql[jciv_ ceYwfcbjO>o[jpWcp_BrirprO^wcp_ [jJfO
(81) oivN4tM`Z[jqu[]cpir_MY>¦ Pniu[jOn[jJMql[ @m¬ qu_w @m qubjO[]iOY\ivprO>wY]cN&`Mª[xqu_fO>ir`MY]W4;O^o3qu`Y\OEiu¡[]JMOEcp_v[jO3b] wfO3t;O3_MwfO3_Mo3O ir$[jJfO ¥O3ew ² (t)B qr_Mw[]JfO.B Ym[xql[jO ρ (t) ¦ : _®qu[]O>b]_Mqu[]cprO tfb]ifoO>wf`fb]O ceY%[]i4cp_MY]O3b\[KbjO3eql[jciv_ @\ B cp_v[ji®O>dB`Mql[jciv_ @m¬ JfcpoxJ cph;O>o3irNOq_fir_f@pc_MO>qub
(82) UZoxJfb fwZc_MrO3b2O>dB`Mqu[]cpir_[]i®;OtMb]ivtMquBql[]O>w ivb qrbjwcp_ [jcNOr¦ B B <0
(83) K*K
(84) K 8N E N8
(85) H FINQ E C N8
(86) M8
(87)
(88) N N ?
(89) E <
(90) F
(91) *
(92)
(93) N?
(94) M NQ=M!C *NQRE k+1. k+1. E C N ?
(95) FINQR G 8E 8N
(96) 8 NQRE ?M M
(97) ?MIFL
(98) KP
(99)
(100) G ME C EGF C FIN ?
(101) F EG?MIRK
(102) FL NQREG?M EG N8
(103) E 8
(104) F
(105) *
(106)
(107) EGN
(108) N ? N N M K
(109) MFL =
(110) PF E ?
(111) FIN
(112) M E NA G 7HJ < M 8 FL G8N
(113)
(114) K CEGF N FL !G. #". $,.&%('. ! ,. 0 * )(%4!
(115) 0. $+,-%("*. Pniu[jO
(116) [jJMql[ [jJfO4oiBYm[E`f_o[]cpir_quiuhO^dB`Mql[jciv_ H@ | B JMqrY Ofqro[]pW []JfO4YjquNO&Nc_McNq qu_Mwobjcª[jcpo>qu t;ivc_B[jYqvY Z @m^{ B J (²) = α² (t)dt + kA − ρ(T )k , T. 2. dist. ii. 2. 0. ¸ºà¾$¸65.
(117) .
(118)
(119). 6. o3irJf_MceooxpJ `MY]NcivO>_ qrY]ce`fYbjO>[]bjY`f[jO JfOwZ`fwfO cpY\[][ji·qr_M[]oJfO9O iu_f2ir[jbjJfN1O9¥Mo3ir_M_MquY]O3bjwZlO3qu_M[]Y]cpcir[£_W tMb]ivt;O3b][]cp[]O>iY[jiuJf.O9[jJf[jO'qrb]vUfOoj[Jfb ivZtwZO>cp_fbjqur[]O>ivbb O>dB`M¦quIK[]cpJfirce_ Y JfceoxJqupi nY []i ivbj_fcª[jpOWwZc©;O>bnBW qoiv_MYm[xqu_B[>¦ qu_w4[jJB`MY[]io3ir_Mo3`MwfO [jJMql[ qu_Mw ρ(x, T ). A. . Jdist = J + kAkii + kρ(T )kii = J + kAkii + kρ0 kii. IKJfOirtZ[jcNqr«o3ir_B[]bjir«Y\[]bxql[]O>rWiuY\O^o[]cpiv_Z¦È
(120) irt;O3bxql[jO>Y%ir_ q4oiBYm[`f_Mo[jciv_Mqu$wZO¥M_MO>w qu[¥M_Mqu [jcNO T ¦ : Y4Y\`MoxJ$)wZ`fbjcp_f9[]JfOO3rirp`Z[jciv_qu[&[jcNO t < T [jJfceYlqup`fOceY&_Miu[4WrO[qro3o3O>YjY\cpfpOivb cpNN4O^wZceql[]OO3O>wfMqroxcp_v[ji[]JfO2irtZ[jcNc=;>qu[]cpir_tMb]iZo3O>wZ`Mb]Or¦ Ei O3rO3b^ c[]J®q
(121) ¥MO>pwoirNtf`f[]O>w `ft[]i qEb]O^qrY]ir_MquMO%qu[]O3bj_Mqu[]cprOcpY[]i `Y\Oq o>qu_MwZcewfqu[]O iv_ [jioivN4tM`Z[]O%[]JfOt;O3b]irbjNqu_Mo3O t<T cp_MwZO
(122) ql[(¥M_Mqrr[]cpN4O T ¦ : _qutft;O>qrcp_fEojJMirceoO ivb ² cpY$²[]JMOh¥MO3ew.[t,irTf[j] qucp_fO^w.ql[qtfbjO3Xcpir`MY$c[]O3bxql[jciv_$¦ iv_ OnqubjO%[jJB[]`MJMYhO2p¥MO>qrO3w&ew [ji.c_B[]bjiZwZ`MoOKirbhq.oiv_v[]bjir ² X_f i _4ir_ [0, t] qu_w4qbjOO>b]O>_MoO¥MO3ew ² wZO¥M_MO>w [0, T ] ( @m>| B ²(s) f or 0 ≤ s ≤ t ²(s) = ² (s) f or t ≤ s ≤ T. IKJfceYK¥MO3ew ceY[jJfO.;O>Y\[Eq^uqucppqrfO2o3qr_MwZcewfql[jO.ql[[]cpN4O t < T ?¦ > [jYt;O>b\irbjNqu_Mo3Oc_wZO J (²) cpY Z @m< 6 B α² (t)dt + kA − ρ (T )k , J (²) = JfO>b]O ρ (T ) ceYK[]JfO
(123) Y\[jqu[]O.qu[K[]cpNO T iu[jJfO
(124) Y]WXY\[]O>N @\ ' B ∂ i ρ (x, t) = (H − ²(t)M) ρ (x, t) J. Jdist. ref. ref. ref. ref. dist. t. 2. dist. ². ii. 2. 0. ². ². ∂t ρ² (x, t = 0) = ρ0 (x),. ². : tfbjirt;O3b][mW c[]J®cNt;ivb\[xqu_v[tMbjqvo[]ceo3qr«cpN4tMceo3qu[]cpir_MYKiv_ []JfO.O o3cO>_v[EoivN4tM`Z[jqu[]cpir_iu cpY J (²) vcrO>_ cp_ [jJfO2irppi cp_f l° h°;$²H°
(125) P
(126) N ?
(127) 5CE FIHJ GF K E MINCPNQRE P *CE F N ?
(128) EG8NQF EG ² PK FL
(129) C
(130) F
(131)
(132) 5
(133) K ² dist. M. ref. Jf wd (², t; ²ref ) = +. H 8
(134) FL
(135).
(136) <EG 7<
(137) M EG ρ. ?
(138) . T. @\9 B. t. α²2 (t)dt 2. t. ρref. M N ?
(139) 8
(140) FM
(141) FLE 8 NQREG CFLE. ∂ ρref (x, t) = (H − ²ref (t)M) ρref (x, t) ∂t ρref (x, t = T ) = A. Jf wd (², t; ²ref ) = Jdist (²). ¾¾#¹GTMÑÑIUxâ. . @Hu B. 0. α²ref 2 (t)dt + kρref (t) − ρ(t)kii .. <MD K [0, t] i. . Z. Z. A. H N
(142) 5
(143) K. ²ref . @@Z B.
(144) '. FIR
(145) FI R. l°$° ")IKJfOn¥bjY\[[ %i
(146) [jO3bjNYcp_ qrb]OntfbjO>o3cpY]O3ppW&[]JMOn¥MbxYm[[jO3bjN ir ¦XIi&o3irN4 ft `Z[jO []JfO®Y]O>o3ir_Mw[]O3bjNc_ J J(²) (²,ρ cet;Y&²[ji ;)O O>rirprO>w'bjirN ρ c[]J·[]JfO ¥MO>Jpw ² (²)iv_ [0, T ] []i ivZ[jqrc_ ρ(T ) ¦ `Z[>XY\cp_Mo3O ρ qr_Mw ρ O3rirprO c[]J[jJfO YjquNOn¥MO3ewir_[]JfO c_B[]O>b]uqu [t, T ] r[jJfO3cpbwZceYm [xqu_Mo3O cppv;O%oiv_MY\[jqu_B[$[jJfbjir`fvJfir`Z[(O>rirp`Z[]cpir_
(147) qr_Mw2[]JB`Y kA − ρ(T )k = kρ (T ) − ρ(T )k = ¦fIKJX`MY %O.o3ir_Mo3`wZO[]JMqu[ J (²) = J (², t; ² ) ¦ kρ (t) − ρ(t)k B <0 M N8
(148) D K EG N M G< G
(149) KGFI N ?
(150) N
(151) FL NQRE ?M E C N ?
(152) E PE N E ! G *EGFIN M ON8
(153) E
(154) FLE 8
(155) FIN# G
(156) +M
(157) K N E EG8N E F N8
(158)
(159) <EG 7NQRE E C N ?
(160) E MNSC PN RE P
(161) N HJ
(162)
(163) NQHJE MI8
(164) IMMI<
(165) N
(166) FL GN REG?M E F ?MIN G
(167) N + M G 8
(168) FL
(169) <
(170) 7 H R ? FIN E CN8
(171)
(172) E 7NQRE E N FI N
(173) IM E FL
(174) N E N?
(175) E PN $ NQRE GK FL
(176) GN
(177) N ++M N E E N
(178)
(179) 8 8 M* M E C E N FLEG . ,-% ,"0 , %($&! "*+'-,./$+0 243 *+,! "* 0 "! #$% ' 4 ,!
(180) ' )(
(181) IKJfObjO>Y]`f[qu;ilrOKrcprO>Y>uqu[qr_BW&cp_v[jO3bjN4O^wZcequbjW.[jcNO t < T []JMOuqup`fOK[]JMqu[ [jJfOEoiBYm[ `f_Mo[]cpir_Mqr cp([jqrrO&ql[E[]cpNO cª [jJfO&irtZ[jcpN4=c ;>qu[]cpir_¨ceYEY\[]ivtft;O>w¨qu[n[jJfO&c_Ym[xqu_v[ @Hqu_Mw J [jJfO¥M(²,O3ewt;ceY)² tf`Z[)) [ji;O ² ir_ [t, T ]TB ¦rPnir[]O[]Jql[)[]JMOKlqr`fOir J (², t; ² ) ceY)b]O^qrwZcpW&t o≤irNTtf`Z[jO>w qu[
(182) qu_BW¨[jcNO qrY.Y\iXiv_ qrY[]JfOc_XrO3bxY\Otfb]ivtMquBql[]cpir#_ @Hf cpY
(183) oirNtf`f[]O>w ir_Mo3Or¦ : bjNO>w c[]J []JfceY [jiXir@Birtf[]cpN4=c ;>tqu[]cpir__fO>O>w_fir[ qrc[%[jcp;[jJfO ¥M_Mqr;[]cpNO T fB `f[o3qr_cp_MYm[jO>qvwqupbjO>qrwfW4irt;O3bxql[jO qu[%[jJfO o3`fb]bjO3_B[K[]cpNO `MY]c_fiZo>qu«[]bxqroxXc_M4tfb]ifoO>wZ`Mb]O^Y%[]iivtZ[]cpN c ;>O[jJfO.lqr`MO J (², t; ² ) ¦ O.qrb]O _f i c_®t;ivY]cª[jcivt _[jio3pqrcN-[]JfO2ivp i c_f X° X 8
(184) EGEGN EG8RD E FIN M E NQFL ! PFLE
(185) K FL
(186) OCE F N ?
(187) SCEGFIHJ FLK f wd. ref. dist. 0. dist. ref. ii. ii. ref. f wd. 2. dist. f wd. 2. ii. ref. 2. ref. ref. ref. f wd. ref. f wd. E<MINSCN REG J (² , t; ² = ² ) GN G( N
(188) t N ?
(189) M
(190) ?M
(191) N ? GN E PE N E 8R ! K
(192) fwd FL
(193) <MIk+1 CPref NQRE E C kt E N ?
(194) 8 N
(195) FI< G [0, T ] . ref. Jf wd (²k+1 , t; ²k ). M . ¸ºà¾$¸65.
(196) .
(197)
(198). 9. l°$° "Pnir[]O[]Jql[ Jf wd (²k+1 , t; ²k ). . ²k = ²ref. cpNtfW. χk (t) = ρref (t). ¦(O[&`MY&o3irNtf`Z[jO[]JfO[]cpNOwZO>b]cplqu[]cprOir. d Jf wd (²k+1 , t; ²k ) = α²2k+1 (t) − α²2k (t) dt d −2 Rehhχk (t), ρk+1 (t)ii dt = α²2k+1 (t) − α²2k (t) d d −2Rehh χk (t), ρk+1 (t)ii − 2Rehhχk (t), ρk+1 (t)ii dt dt = α²2k+1 (t) − α²2k (t) (H − ²k (t)M) χk (t) , ρk+1 (t)ii −2Rehh i (H − ²k+1 (t)M) ρk+1 (t) −2Rehhχk (t), ii i ²k (t)Mχk (t) = α²2k+1 (t) − α²2k (t) + 2Rehh , ρk+1 (t)ii i ²k+1 (t)Mρk+1 (t) +2Rehhχk (t), ii i = α²2k+1 (t) − α²2k (t) + 2²k (t)α²k+1 (t). IKJX`MY J B <0. @Hr B. 2. −2²k+1 (t)α²k+1 (t) = −α [²k+1 (t) − ²k (t)] .. f wd (²k+1 , t; ²k ). . . 8
(199). ceYqwfO>objO>qvY\cp_f`f_Mo[jciv_®iu t ¦. EGEGN EG8RN
(200) DCE EGH M <M EGFLE = F* E C N ?
(201) PFL
(202) $RE +M FLE 8
(203) FIN# E C . Jf wd. MI
(204). IKJfOhb]O^Y\`Mª[qu;ilrO Nq¬WqueY]iY]`fvrO>Y\[¡[]JfOhirp i cp_fcp_v[]O>b]tfbjO[xql[jciv_ irbqu_XWo3qu_wZcpwMql[]OY]irp`Z[]cpir_ [ %i4[jbjqlkmO>o[]ivb]cpO>YKo3qu_;O.oivNtf`Z[]O^w []Jql[nY\[jqrb\[xY%bjirN[]JMO.o3irbjb]O>o[cp_fc[]cequ(oir_wZcª[jciv_ ² f`Z[ JfiBY\O¥M_Mqr$Ym[xql[jO ρ (T ) Nq¬W_fiu[WrO3ρ[(t);O.Yjql[]ceY\¶qro[jirbjWo3iBY\O [ji4[]JfO2[jqrb]vO[>fqu_Mw[]JMO.qvw¬kmircpρ_v[ Y\[jqu[]O [jJMql[tfbjirtMqrvql[jO>Y4qrox KqubxwbjirN []JfO¨[xqubjrO[ f`Z[Nq¬W·_fir[bjO>qvojJ[jJfOoivb]bjO>o[ cp_fc[]cequ χYm[x(t) ql[jO ρ []JfO4cewZO>q cpY []i NqrrO&[]JfO[]bxqlkmO>o[]irbjcpO>Y o3ircp_MAocewZOBW9o3irNtf`Z[jc_f ² Y]`MoxJ9[]Jql[ qrtftfbjivqroxJfO>Y&Nir_Miu[]iv_fceo3quppW χ (t) ¦ >£_·[]JfO®qrtftfbji^ZcpNqu[]cpir_ JfO3bjO[]JfO M`fO3_Mo3Ot;O3_Mqrª[mW ρ (t) R ceYE_fO>rpcvcMO&;OirbjO.[jJfO4oir_B[]bjir(tqub][ [jJfOwfcpY\[jqr_MoO&;O[ %O>O3_¨[jJfO [ %iα²[]bxql(t)kmO>o[]irbjcpO>Y cp)wZO>o3b]O>qvY\O
(205) `f_B[]cpc[jYE¥M_Mqulqupkχ`fOql(t)[E[j−cNρO T ¦;(t)k IKJMOY\c[]`ql[]cpir_9ceY YjojJfO>Nql[]ceo3qrpW wZO>tfcpo[]O^w cp_ &c¦¡r¦ E
(206) l
(207) l.)0 2& UXcNcppqrbKo3ir_MY]cewZO3bxql[jciv_MY%qrY%cp_v[jb]ifwZ`MoO^wqu;ilrOqutftfpW[]i[jJfO q¬rO`M_Mo[jciv_irbjN&`feql[jciv_$¦XPEiu[]O Jf i %O>rO3b[]JMqu[>O3rO3_¨c)[]JMO&wZO>_MY\c[mW Nql[jb]ccpYnNivb]O.vO3_fO>bjqr«[jJMqu_ q¬rO`f_o[]cpir_(f[]JfOqrYjY\iZo3cpqu[]O>w Jdist (²k ) = Jf wd (²k+1 , 0; ²k ) ≥ Jf wd (²k+1 , T ; ²k ) = Jdist (²k+1 ). k. 0. k. k. k. 0. k+1 T 2 0. . . ¾¾#¹GTMÑÑIUxâ. k+1. k. k. k+1. ii.
(208) ^. FIR
(209) FI R 30. χk(t). 25. 20. χk+1(t) χ∞(t). 15. ρ∞(t). ρk+1(t). 10. ρk(t). 5. 0 0. 2. 4. 6. 8. 10. time. UZoxJfO3Nqu[]ceoKcp`Ym[jbjqu[]cpir_4iu;[]JfOno3ir_XrO3bjrO3_oOiu;[]JfONiv_fiu[jir_fceoKquprivb]c[]JfNYivb _fO>rpcvcMO M`fO>_MoOv¦«IKJfOO3rivXcp_fY\[jql[jO ρ ceY qrtftfbjivqroxJfcp_fNir_Miu[]iv_fceo3quppW[]JMOb]O3O3bjO3_MoO
(210) [jbjqlkmO>o[]ivb]W χ ¦ : [)[]JfOK_fOX[)c[]O>bjqu[]cpir_ χ cpZb]O>Nqrc_qu[ q o3ir_MY\[jqr_v[)wfcpY\[jqr_MoO%b]ivN ρ ;O>o>qu`MY]O%;iu[jJ`MY\O%[jJfO YjquNOn¥MO3ew ¦BIKJfceYY\Jfbjcp_fXc_fwZcpY\[jqr_MoOE;O3[ O3O3_[]JfOE[ %i
(211) [jbjqlkmO>o[]ivb]cpO>YhO3_Y\`fbjOn[]JfO tfbjirvb]O^Y]Y\cpir_ irX[jJfO%o3ivY\[$²`M_Mo[jciv_Mquv[] i qubxw2irtZ[jcNquvuqup`fO>Y>¦>IKJMcpYirMY]O3bjlql[jciv_ceYo3`fb]bjO3_B[]pW`MY\O^w.c_.[jJfOoir_B[]O3B[ ir O ocpO3_v[tMqrbjqrpO3p=c ;>ql[jcpir_®iu [jJfO_B`fNO3bjcpo>qubjO>Y]irp`Z[]cpir_¨irdv`qu_v[j`fN oiv_v[jb]iv(tfbjirfpO3NY v{ ¦ >£_ [jJfOrO3_MO3bxqu;o3qvY\Ovv[]JfO2wZO^objO>qrY]cp_f&oxJMqubxqro[]O>bir$[]JfO2wZceY\[jqu_oO ;O[ %O>O3_[]JfO2o3`fb]rO^YceY O3cprJv[jO>wXW [jJfO.¥MO>pw M`MO3_Mo3O&qu_Mw[]JfO&irtZ[jcNqr(o3ir`ftfpO
(212) iu[jbxq¬kmO^o[jirbjcO^Y cpOq4[]`MO JfiBY\O._fiv?_ ;3O>b]i cpwX[jJ ceYKbjO3eql[jO>w[]i4[jJfO
(213) wZbjcXcp_f4eqrY]O3bK¥MO3ew `fO3_Mo3Or¦ . k+1. k. k+1. k+1. k+1. ivMY\O>b]uqufpO>Y&qrb]O cp_fO>qrb>¦ &Mirb[]JfO Kq^rO3`f_Mo[jciv_·Jfi O3rO3b^([jJfOirMY]O3bjlqrfpO>Y
(214) O>_v[]O>b4cp_v[ji[jJfO®oivY\[ `f_Mo[]cpir_Mqr$qvYKdB`MqvwZbjqu[]ceoE[]O>b]NY> JMcpoxJ cp$cp_MwZ`MoO
(215) Y]irNO2qvwfqutZ[xql[jciv_MYc_[]JfO2irbjNqrceY]N®¦Z$O3[`MY o3ir_MY]cpwZO>b[jJfO
(216) wZbjcXcp_f4O>rirp`Z[]cpir_O>dB`Mqu[]cpir_ @H B ∂ i ψ(t, x) = (H − ²(t)µ) ψ(t, x) 0. ∂t ψ(t0 , x) = ψ0 (x).. &firb.qvcrO>_irMY]O3bjlqrfO «c[jY.q^rO>bjqrrO>w¨N4O^qrY]`fb]O^w¨uqup`fOceY ¦ c[]J [jJfcpY.wZO¥M_Mcª[jciv_ [jJfO2[]bxqroxXc_fivb]N&`feql[jciv_o>quA_;O b]c[\[jO3_®qvYqu;ilrO2qu_Mw[jJfO
(217) YjquNhψ|A|ψi O.oir_Y\cewZO3bxql[jciv_MYqrtftfpWr¦. ¸ºà¾$¸65.
(218) r. .
(219)
(220) ". . ,-% ,"0 , %($&!. "*+'-,./$+0 243. $O[n`MYnwZO¥_fO2[]JfO
(221) oiBYm[K`M_Mo[]cpiv_Mqu J w (²) =. Z. T. α²2 (t)dt − hψ(T )|A|ψ(T )i,. irO N4cpiv_v_f[]bjiuiZ[jwZir`M_fo3ceo2O quqrpY%rirbjO3c[]irJfbjN On[]Jf$O2' qrw¬kmivc_B[Y\[jqu[]O ¶@ (qrrbxqu_fvON&`f[]cptfpcO>b B 0. -. χ(x, t). @Hu} B. qu_MwrcprOEir_fOEO3fquNtfpO. ∂ ψk+1 (x, t) = (H0 − ²k+1 (t)µ) ψk+1 (x, t) ∂t ψk+1 (x, t = 0) = ψ0 (x) µψk+1 1 i(t) ²k+1 (t) = − Rehχk , α i ∂ i χk+1 (x, t) = (H0 − ²k+1 (t)µ) χk+1 (x, t) ∂t χk+1 (x, T ) = Aψk+1 (T ). i. @@r{ B @@u| B @@*6 B. iv_O4[j_fJfiuO2[j¥MO&_M[jqrJM$qlY\[[jqu[][]JMO O4_fir_fpcp_fO>qr¦ b]c[£W®cp_[]JfO4ivMY\O>b]uqufpO
(222) cp_MwZ`Mo3O>Y2q wZO>t;O3_MwZO>_MoO4iu[]JfOqvw¬kmircp_v[.Ym[xql[jO ψk+1 (T ).
(223) )4$ "*
(224) %(!
(225) O&cp_v[jb]ifwZ`MoOvfivb qbjOO>b]O>_MoO.¥MO>pw ir_ qr_Mw¨q¥O3ew ivYjY\cpfwfpO O¥M[j_fiO^o3w¨ir_M`fY]t¨cpwf[]O3i b qu_9c_B[]O>b\ NO>wZceqrb]W[]cpNO t < T []JfOE]o3qr_MwZcewfql[jO2Y\iv²`f[]cpir_ ² [0,qvYTcp_ ] @m^| B ¦?> [ncpYt;²(s) Z @@G ' B J (²) = α² (t)dt − hψ (T )|A|ψ (T )i _fi4c[]OJ ψo3cO>O>_vr[irptfXbjciZ_fonO^wZbj`firNbjOEψceYKq¬ lquc[]cpJ&equ¥MfO3pOnew [ji² ¦ onir iN %tfO>`fr[]O3O b^^wZ`fO[jiEqu[][Jf[]OcpN_firO _Mcp@¶_fiuO>[jqrJfbO3_Mbql[j[jJM`fqubj_ OO3iuXftM[]JMceO%ociv[]pMWY\O>o3b]iruNquftfp`ZOr [jc_f ψ irb %ivb]Xcp_f cª[jJ[]JfO
(226) Mqrox qubxwtMb]ivtMquBqlJ[]cpir(²)_ir[jJfO.irMtY]O3bjlqrfpO A c[]J[jJfO2¥MO>pw ² B ¦ &Mirb[jJfceY4bjO>qrY]ir_(Nir_fir[]iv_fcpo qrvirbjcª[jJfNY&`MY]O []JfOirp i cp_f c_MO>dB`Mqupc[£W'ivbq t;ivY]cª[jcrO®wZO¥M_Mcª[jO ivMY\O>b]uqufpO A ≥ 0 @@G 9 B −ha|A|ai ≤ −hb|A|bi − 2Reha − b|A|bi, @[]JfO&wZc©;O>b]O3_oO.iu[ %idB`Mqu_B[]c[]cpO>Y;O>c_f −ha − b|A|a − bi ≤ 0B ?¦ >£_®tMqrb\[jcpo3`fpqrb>Zc O.wfO3_fir[]O.BW [jJfO
(227) Ym[xql[jO2O3rivXcp_fbjirN ψ c[]J¥MO>pw ² fir_fO2JMqvY ψ " . $,.&%('. ! ,. 0 * )(%4!
(228) 0. $+,-%("* %' ref. T. 2. w. ². ². 0. ². 0. w. ref. ². ²ref. 0. J w (²) ≤. ref. Z. T. α²2 (t)dt − hψ²ref (T )|A|ψ²ref (T )i 0. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i. ¾¾#¹GTMÑÑIUxâ. @ r B.
(229) ¬. FIR
(230) FI R. > [nceY[jJfO3bjOirbjOt;iBY]Y]cpfO[jiwZO¥_fO2[]JfO JH
(231) CN REG CE FIHJ GFLK E MINCNQRE G Jfwwd (², t; ²ref ) =. JfceoxJ®o3qr_®queY\i4;O bjcª[][]O>_. Z. T. α²2 (t)dt − hψ²ref (T )|A|ψ²ref (T )i. @ f B. 0. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i. Jfwwd (², t; ²ref ). IiO3lqrp`Mql[jO. w. = J (²ref ) +. t 0. © ª α ²2 (t) − ²ref 2 (t) dt. @ v B. © ª α ²2 (t) − ²ref 2 (t) dt. @ B. −2Rehψ²ref (T ) − ψ² (T )|A|ψ²ref (T )i.. Jfwwd. c[ncpYnoiv_BrO3_fcpO3_B[K[ji_fiu[jO[jJMql[. Jfwwd (², t; ²ref ) = J w (²ref ) +. JfO>b]O. Z. Z. t 0. −2Rehψ² (t) − ψ²ref (t), χ²ref (t)i.. χ²ref (t). ceY[jJfO
(232) qrw¬kmivc_B[nY\[jqu[]O.qu[K[]cpNO t rcprO3_BW. @ u} B. ∂ χ² (x, t) = (H0 − ²ref (t)µ) χ²ref (x, t) ∂t ref χ²ref (x, T ) = Aψ²ref (T ). i. IKJfceY irbjN&`feqquppi nYqu_ O ocpO3_B[¨o3irNtf`Z[xql[]cpir_ iu ¦Ii Y\`fNNqubjc=;3Or%[jJfO'o3ivY\[ `f_Mo[]cpir_Mqr J (²) []Jql[.o3qr_f_fiu[;OoivNtf`Z[]O^w9O3XtMceoc[]JpW¨cpY(²,OZt;tfp²irc[]O^)w¨[]Jfbjir`MrJ9c[jY`Mtft;O3b2;ir`f_Mw ¦MIKJMO2tfbjO>o3cpY]Otfbjirt;O>b\[mWiu J ceYKvcrO>_ cp_[]JfO2irppi cp_f J (², t; ² ) X° X 8
(233) EGEGN EG8RD E FIN M E NQFL ! PFLE
(234) K FL
(235) OCE F N ?
(236) SCEGFIHJ FLK w f wd. w. w f wd. ref. ref. w f wd. E<MINSCN REG J w (² , t; ² = ² ) GN G( N
(237) t N ?
(238) M
(239) ?M
(240) N ? GN E PE N E 8R ! K
(241) fwd FL
(242) <MIk+1 CPref NQRE E C kt E N ?
(243) 8 N
(244) FI< G [0, T ] . Jfwwd (²k+1 , t; ²k ). M . ¸ºà¾$¸65.
(245) . .
(246)
(247). l°$° " : nY qu;ilrOr O2O3uqup`Mql[jO [jJfO2[]cpNO
(248) wZO3bjcuql[]cprOir . Jfwwd (²k+1 , t; ²k ). ¦. d w J (²k+1 , t; ²k ) = α²2k+1 (t) − α²2k (t) dt f wd d −2 Rehψk+1 (t), χk (t)i = α²2k+1 (t) − α²2k (t) dt d d −2Rehψk+1 (t), χk (t)i − 2Reh ψk+1 (t), χk (t)i dt dt = α²2k+1 (t) − α²2k (t) (H0 − ²k (t)µ) χk (t) −2Rehψk+1 (t), i i (H0 − ²k+1 (t)µ) ψk+1 (t) , χk (t)i −2Reh i ²k (t)µχk (t) i = α²2k+1 (t) − α²2k (t) + 2Rehψk+1 (t), i ²k+1 (t)µψk+1 (t) +2Reh , χk (t)i i 2 2 = α²k+1 (t) − α²k (t) + 2²k (t)α²k+1 (t). IKJX`MY J B <0. w f wd (²k+1 , t; ²k ). H 8
(249) FL
(250). M ². B . ². . @Rv{ B. 2. −2²k+1 (t)α²k+1 (t) = −α [²k+1 (t) − ²k (t)] .. ceYqwfO>objO>qvY\cp_f`f_Mo[jciv_®iu t ¦. % E N
(251) N ? GN N ?
(252) P
(253) 8 N
(254) w ≤ J w (², t; ² )
(255) E
(256) M
(257) 8 7N# N8
(258) 7 N P FLE *8
(259) M ² H +R H 8 J 8
(260) (²) # GNN ?f
(261) wd EG8
(262) F
(263) ref
(264) P
(265) E C N ?
(266) E PE N E R E FIN M. − ²k → 0. . ref. <0 / M E
(267) N8
(268) EGE N E RN# E C N ?
(269) FLE$
(270) K FL
(271) M #MI P
(272) EG?M
(273) 8
(274)
(275) E C N8
(276) K*
(277) FL
(278) <M
(279) E C J w (² , t; ² ) J w (² ) = J w (² , 0; ² ) ≥ J w (² , T ; ² ) ≥ J w (² ) k+1. f wd. k+1. k. k. f wd. k+1. k. f wd. k+1. k. k+1.
(280) % (& IKJfb]iv`frJ'[]JfOcp_v[jb]ifwZ`Mo[jciv_iuEqmirb qubxw o3ivY\[
(281) `f_Mo[jcpir_Mqrã[jJfcpYtMqut;O>bwZO3Nir_Ym[jbjqu[]O>Y.[]Jql[ [jJfONiv_fiu[jir_fceo4qrvirbjcª[jJfNYqrb]OopivY]O3pW¨bjO3eql[]O^w9[ji[]JfOoeqrYjYir%[]bxqroxXc_ftfbjiZoO^wZ`fbjO>Y>¦ : Y.Y]`MoxJ$ [jJfOENir_fir[]iv_fcpo3cª[mWtMb]ivt;O3b][£Wir«[]JMOnirbjNO3b%quprivb]c[]JMNYqutft;O>qrbjYqrYq
(282) _Mql[j`fbjqroiv_MY\O^dB`fO3_Mo3Oir«[jJfO cp_MobjO>qvY\cp_f uwZO>o3b]O>qrY\cp_f tfb]ivt;O3b][]cpO>Yir%[]JfO[jbjqvojXcp_fc_MwfO¡¦(IKJfONiv_fiu[jir_fceo4YjoxJfO3NO>Y.qrb]OY\Jf i _ [ji oiv_MY\[]bj`Mo[4qu[4qrc_B[]O>b]NO>wfcpqrb]W [jcNO>Y[]JfOo3ivY\[&`f_Mo[]cpir_Mqrlqup`fOirEqo`Mb]bjO3_B[2;O>Y\[¥MO3ewB o>qu_MwZcewfql[jOqu_Mw `MY\O4[]JfceY2c_firbjNqu[]cpir_ c_ []JfOivt;O3_ piXirt[]iivtZ[]cpN c ;>O[jJfO4¥MO3ew`fb\[jJfO3b^¦ &fivb [jJfO Y]t;O>oc¥oo>qrY]O&iuh[]JfOwZO3_Y\c[mW®Nql[jb]c¡M[jJfceYo>qu_ queY\i;Ocp_v[jO3bjtfb]O3[]O^w¨`MY]cp_f[ %i[jbjqlkmO^o[]ivb]cpO>Yn[]Jql[ Y\[jqrb\[ ¬O3_w qu[K[jJfO&oirbjb]O^o[nYm[xql[]O^Yqr_Mw JfivY]O.wZceYm[xqu_Mo3O.cpYno3ir_v[jc_X`fir`Y\pWb]O^wZ`MoO^w @HwZO3t;O3_wZc_MqueY]i iv_[jJfO.eqrY]O3b M`MO3_MoO B wZ`fbjcp_f4[]JfO.ivtZ[]cpN c ;^ql[]cpir_tfbjiZo3O>YjY3¦ . . ¾¾#¹GTMÑÑIUxâ.
(283) >}. FIR
(284) FI R. +&0 * !. IKJfO qr`Z[]Jfivbrbxql[]O3`fpW qroxX_fi O^wZrO wZceY]o3`MY]Y]cpir_MYiv_[]JfceY []ivtfceo c[]J R9q ;3WvqubR¨cpbjbjqrJfcNc qubjcpY B qu_w r`fpcpO3_¨UfqupirNir_ @¶g)qrb]ceYD>KQn_fcprO>bjY]cª[mW B ¦ K p
(285) RwZO>¦¡Y]cpR¨r_fcpO^b]bxw quJfo3ircpNJfO>cHb] O> 4_v[n¦¡I(dB`M`fqubjcp_B_f[]ce`foN#c@«quo3_Mirw9_v[jg)bjir¦¡eY3s iv`M
(286) ox JfE irC_$ Z] s<MIORO> Gb] O>_Mo?O&
(287) []bxql MkmNQO>F*o [jirbjW u[]vbxvqrox{ZXBc[]_Miqutfivt;b O^pquiZb^o>¦ qupW È
(288) s.¦ v`MwfY]ir_qr_Mw &¦BsnqrfcL[ ;rl\IO>qvojJMc_f
(289) pqvY\O>bjY []ioiv_v[]bjirMNivO^o`fpO>Y> <M D
(290) O
(291) NQNH9*9vZ¦ : 4¦ ¦ : YjO>Y]b]c;ivO>_$b>«XI2 %¦ irK_Bqu[]`fbjirNO>iub\[^;oxJfRO3N¦ ceo3O3qrbju[^bjO>;qvI2o¦[]cpiv_MbjcªYnZB_fW®O>b> O3O>wf¦ M2qrcpoxOB@O>ivb>tZ []&cpN¦$ cUX;>O3O>WBw¨bjtfcO^JMw¡qv«Y\O3R Y]¦$JMUBqr[jtb]O^O>wJfOvO3«N
(292) qr[j_Miuw Y]O>oiv_MweqrY]O3btf`feY\O^Y3 SR
(293)
(294) Mrir@¦MG 'BZXtft$¦ 9f 9 +9vrZ;$ 9 9 'M¦ }
(295) I2`f¦_M o[]cpO3ircp_$_Mqv ojJB [> NQ¦ F
(296) 3: ZJfr_(iv$H?¦ qu*_M9 w6ZXg)tf¦ t$¦`M ox Z Y\f quv`f{ZN®« f9 9*%9fiv¦ _v[]bjirppc_M[]JfOY]JMqut;O4iuqdB`Mqu_B[]`fN q¬rO È{
(297) qus._M¦ w K
(298) qub]
(299) ¦ []O> pY>¦ B.U«qu¦ tZK[jqrO3WXoxB_$`rY3]rUX¢nJqu¦ t;;O>O3wXO>;tfB`fheY]¦BOR¨ircetZY\iv[jcrN`Z=c[j;>cHql [j civ¦ _ hiuwZ)ilBoircp_$JM* O3>bj¦vO3_vg)[K¦ %O>NJMb]cpYjceYmY\[jcpiri¬_«virRJf¦BcpRrJZ¦X R¨JMqr`fb]bjN_Mirqu_f_MOvcpo Y]iuº[ E@bxq¬WXY> NQFL
(300) frir@¦Z}vv|fXtft$¦(>|u} ;>|r|ZfuvrM¦ |
(301) s.c¦ []MJ¦Zir(tfO3[]XcpNceY3qu 4pW¦fR¨[jqrO3cp_firXbjO>cpb>w¡ZZqrY\_M[]bjw ir
(302) _Mu¦f@sn¥MO3queMwcª[ p;rqvuY\O>jbUXO>tf`MO^opY][]O>cprY>O ;Sir_MRw
(303) wZce
(304) YjfY\riZiro3@cpqu¦M[]G cp9virf_Bqrtf_Mt$w¦ 6lb]O^* qu9 bj?bj6Xqu _ffrZO3uNvO>f_vr[ ¦ 76 &Y]WX¦XY\[]snO>quNNY3qr BbjceY\Jf<_M qfDrR
(305) ¦X UZqrfequrtMirqu@u¦Mqf{ZrvRX_f¦ i ¦MqrfJfZtfO>t$J$8¦Bqr9v_M|rw +&9r|r¦B|Zsn«qr f9*c9vL[ {f;r¬¦ %ir_B[]bjirpequfcpc[mW&iu$NirpO>o`Mpqrb ' ! 4u+| ¦;6XIX`ftfbjt$c¦(_fce o+c9Mqrf_Muw vf&v¦;¦ snqrfcª[ ;rX# " `Mqr_v[]`MN Kq^rO3`f_Mo[jciv_9oiv_v[jb]ivequfcppcª[mWv $ 8
(306) % <M p«rir@¦ 9 [] cpir¦ _®: iu[j)quY]¥MO` _f@c@_ X %& ir
(307) _BE []bjFIirPpp qr OfEcpC(c[£'W GiuNh8dB
(308) `Mqr _vNQ[jR`f GN N O><oxMIJMRqrIM_fcpo>rquir@¦ZY]WZ*} MYm[jXO3_MNiMYn¦M{fXWZtfbjt$iX¦;iuu[EvY]{ftM qvZouO&rwZ|rO>Zo3firuNvvt;Ziv¦ Y]c p> &dB`M¦ qu: _B[]p;`fO3N b][]cpY\_fWZc2B Y\[]qrO3_MNw Y> *)L¦ : pO>?YjF Y] Gqr?_MMwZ bjiMNQR)E \?PEM iuE []cpir_YNirE &o GivN _vR[j+b]iv eE qu fN cpFLpEGcª[m ãW;ririr@b ¦M*} f'McppMc_f_fO^iMqu¦ b 'fN&tf`Mt$ª[j¦cp$ O3 r9*O>9 93}B ffuv f¦ pr .s.: oj,¦ Jf cpO3ivXY]cp_fir4©hoxJfU«O3¦hNsnceo3cpo3quOr¡ Y\O>g)¦O^o 2[jcqrXY]ctM[£WqubxBw¡WhY]U«JM¦hquItfcpO3_fbxY]cvcp_frcHJB [qrtf_M`Mw pY]O>Y>¦h¦ + Iqr_f?_f
(309) irb^R$ 'q^ r<O>MItMRqrMxoxfrOr[ir@wf¦;qr$ _M o9fcpv_ftf t(¦ uM Zruf;$ 9 '*9f¦ @¶¢P UZR9g)Mg. ¸ºà¾$¸65.
(310) ^{. .
(311)
(312). p^ 2g)oiv¦ _v []bjb]iriB Y]Y> :
(313) ¦N4UXO^c_Mqur_MJ$Y2 ir
(314) b&¦MwZsnO^quY\fcprc[L_;vqu_Mw'¦fR¨qO^[]qrO>Y]Y\Or[&ZiuqrK_Mcw _B[] 4`fc¦ []ncpir`_(qu +_fu <>£_BM rO>DbjY]
(315) O dB`Mqr$_vr[jir`f@N4¦$}+@N6Z¡O>oxt$JM¦(qu}B_M{ cp9o>quf . . /. . 9*9 f¦ p N¦ir%pO>Jfo`MO3_$pqrrbEg)iv¦X kmO^b]iBo[]Y]cpY>r O>Y¦rwZsnO>YjqroNbjcpqu;XO>bjw¨cpY]BJfW¨_MqfdB `M
(316) qr_v¦rsE[j`fquN fc[LN;vrO>oxquJM_Mqrw _fcpo>¦uY3¦R9 O>
(317) qr Y]Or >8
(318) %ivN4t;O3<[]cM []pcp«rvOir[j@bj¦qv>ojBXZcp_fMtft(ir ¦ 'rvf +'rf>Z« 9*9v{Z¦ p3} r
(319) ir¦¬@sn¦¡rqu$M9fcª[ X;%_fqui_M¦ w 6ZM uv ¦ «MJB¦`$x# " `Mqr_v[]`MN o3ir_B[]bjirvwZO>Y]cpr_.XceqnqrwfqrtZ[]cprO [jbjqvojXcp_fM
(320) ?
(321) <M p p^ { .[]R`MN#¦uUX`fY]WZBYmq [j KO3quNbxY3qf¦ j
(322) O3_fO>8bj
(323) qrr ivb]N&<`fM eqlp[jcrivir_
(324) @¦¡irvM iZ'fo>Xqu_MpiMW.¦¡wZ¬{ZO^Y\Zcptfrt$_fO^¦Mw
(325) +| 6 o3'rir} JfXO>| b]O>'u_vr[)fZoivr_vr[j* b]fiv¦ r[]JMO3irbjWirb)dB`Mqr_Z p>|
(326) JMU«qu¦ZbjUXNJfc@ir _f: ceo ¦ p cp_fiXO>iZqrwfbWroxvJMqrqu_Mcp_w N
(327) ir¦XpsnO>o3qu`ffcO^L[ Y3;v ¬ t
(328) Z[]cp8N
(329) qu; oir_B[]<bjM irMirr$ivHY]?¦O3p'*O>'foZ[jcptfrt$On¦MX*| cpf' 6lbjqu []XcpG| ir' _M' qrff;O$f9 o3'cª[x'Mql¦ [jciv_c_ p< 6.pI2iX¦irtMEYirbj[j_Bi`ftf_fb]ilZXRcpwfO
(330) ¦MR¨OZiut;[ O3;3bjXc`MNY>O>Z_vqr[j_Mqrw pW s
(331) ¦fO>wZqr(O Y]cp fcpOBcpY]OJM£qustfcpO>cp_fwZpOrNr qv: Y\wfqutMtZql[j[]c_M[]O> b]_MY> & I
(332) qr_Mw?
(333) `Y\cp_f%O^<quM bj_f¡cr_Miru@ ¦ r¬{ZXtft$¦ f>B{ +fr3ffuvfr¦ p ' oiv¦)_vR9[]bjirqv$wfquq¬pW rivqrb]_Mc[]w JMN 4Y>¦ * I`f
(334) bj cp_fcp?o3
(335) cH¡\PE O <MirMbjrN&ir@`f¦¡pqur[]$ cp'firX_M_MY&iMiu¦¡n$ N'fMiru_fvir []fir¦ _Mcpo>qupW o3ir_BrO>b]vO3_B[&dB`Mqr_v[]`MN p 9 bjcª[j¦ Jf JNvY[xY\ir`MbhBY\c@%iv 4Xcp¦_fI2`fdBbj`Mc_Mqucp_vo3[jcH`fNyqr_Mirw tZ[j&cpN¦qrsnfqrofirc_BL[ ;r[]bj«ir X tMO>b]_fivO3fbxquO>pN c ;>Y3O> w
(336) N4%iv_f?iu
(337) [jir_f ceo3qr <pW M proririr_X@r¦ZO3^bjrrfO>_v¬[
(338) _fiMqr¦Zv¬iuZ tft$¦{v{u* 9 Z{r{X< 6XZrrr}MZqueY\i4tf`MfceY]JfO>wcp_ [jJfO Ecpb][]`Mqr riv`fbj_Mqu«ir)Qn[]bxql¶qvYm[EUZo3cO>_MoOr¦ Èu [] `MN ¦ ;ivJXtZ`4[]cpqrN_Mquw X&oir¦u_Bsn[]qrbjirfvcª[ i¬;rv3O3 b$: [jJfbxO%qutMOcpZw&t;NO^oir[j_fqu[]ircp[]iriv_
(339) _flcpo>qrqup`MW
(340) OhiroMir_Xqnrt;O3iBbjY\rcO>[]_vcpr[O%c[]wZO>Obj¥qu_f[]cpcªir[jO%_4ivqut;prO3irbxbjqlc[][]ivJfb>N
(341) ir b dB8`M
(342) qr_Z <M prir@¦¡>* 9fXtft(8¦ 'B{ + 9fr;$ 9 9 'M¦ ÈZ tfbj¦ iZIoO^qrwZ_f`f_fbjirO>b^Y2 ivb&¦ ¥M._MGq wZ;>cpqr_fri¬«ivtZ[]qucp_Nw qu%tf¦ `f beY]]O>pYil;qr«_M w%irr_Bpir[]Mbjirqr%hiu`MEtft;tfO3Jfbir[];iZivox`fJf_MO>N4wfY>ceo3qr%c_ Mbj qr_MojJf
(343) cp_f
(344) 8)
(345) PEPK i¬
(346) r8O>N 8 N ' E =
(347) F P RMx bjiXO>oxXJfiluO ¦vqu_Mwql[]JMir` %O>bjYh¦v¢wfY>¦ ghO>_B`fN®; 9*9vZ tft$8¦ r+} 6 +r|rZ¦ Èr
(348) NU«¦hcªZUZO^oxwXJf cpY\b][jNqu[]O3O.b^)dBR`Mqu,¦ _B []`fcpN#bjqrbjY]wZWZO^Y\qu[]`$O3NquY>_ w M ¦ <M $O>DqrJB
(349) Wv ¡ m¢ ZrivocpHO3¦f_B|M[rZqrt$v¦MirMbj^cª[jZJf^N Zrfirub4vrivftZ¦[]cpNquKo3ir_B[]bjir%ir ÈG irtZ¦ [j JcpNvqr[xY\$`MoBivc@_v [j b]iv¦ ; JBc[]`(J¨XqrwZ_MceY]w Y]cp&tMql¦X[jsEcivqu_$f cL[ ;v
(350) l ]R¨?ir
(351) _fir[]iv _f cpo><quM ppMWrivoivH¦¡_Bvr>O3MbjrXO>tf_vt$[8¦ qu9*p'vrriv{b]c+[]9GJf' N vZ;ivb$ 9 dB9*`M9fqr¦ _v[]`MN Èl} cp_®¦;dBR9`Mquqv_Bwf[]q^`fWvN o¦;irUZ_Bqr[]bjivirN4@ iv_$Mqr _MFLw
(352) ? 4 GF ¦ GIN `fE bjc_M«cpuo3rcHBB{Z _¦ [jJfO&o3ir_XrO3bjrO3_Mo3O2iu Nir_fir[]iv_fcpo
(353) quprirbjc[]JfNY. ¾¾#¹GTMÑÑIUxâ.
(354) ^|. Èr{ S S vmg qubxquppO3«cp_ [jcpN4O
(355) o3ir_v[jbjireY%irbndB`Mqr_v[]`fN Y\WZY\[]O>NY> . FIR
(356) FI R FL
(357) ? FL GN REGM«rrv{f¦. ¸ºà¾$¸65.
(358) Unité de recherche INRIA Rocquencourt Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).
(359) . ISSN 0249-6399.
(360)
Documents relatifs
C'est aussi le résidant pas chez leurs parents fréquentent deux fois parativement à la province, les étudiants de Nanterre ne lement ce moindre attachement à la ville
In this paper, we study the performance of an enhanced channel estimation technique combining estimation using an autocorrelation based method and the Expectation-
Selon Laeven et Valencia (2012) , le coût budgétaire de la crise bancaire s’est respectivement élevé à 44 et 41 points de PIB en Islande et en Irlande, ce qui place ces deux
Dans ce travail, nous comparerons les entreprises coopératives, détenues par leurs membres, dont le produit risqué est la ressource critique apportée et dont l’objectif est
À cet effet, un nombre considérable de recherches démontre cliniquement que les enfants de parents souffrant de problème de santé mentale sont plus à risque d’un retard dans leur
usual route to school and provide us with some details about their travel routine (e.g., schedule, mode of transportation, people accompanying them, etc.); 3) recent interventions
La question de l’extension des limites de Paris jusqu’à cette ligne avait été discutée mais ne fut tranchée que par un décret impérial du 9 janvier 1859 qui décidait de