MFNWtKUb<ob<R=MDLCdNVZZJ:tN>T:\\WmDqiCB`N\\@Nd\\Qgqxx`JFmodpsqaTOuToexPpWL\\NfHSJ\\RB<K=\\k>uMU]M>IL`Iva<NBYJV@PH@nb`kI]UA=YFmm]\\KrhsAAy>tVV=tS<VRaJY`qk\\kbYl?mqS`R;HJk<yJujZ<N^<lHMQwxXQQxhywyuwyqxsxnHiudEwhPSyyYYyTcQsgXXIiQeiQyyusyOaTTo=n>LJ;@RZ<LB\\J<DjZmMJhl`<LB\\:=>LR:]J<:LZ[>^<Fj:;@BCBK;C>KJ;JBFZK>[>N:J<nbQ``;@bZ:=NbuUFeSIAoSPTSeLtPXd=YaAktiuBaSZyxwXYmLWcqMu=o_dYwLR:]:>>NjtI`xFagpgrytYXtqidug__xqUopGVtqYdI`e[PyDhudVmCQoDgpLI`oO\\<N^J>[>N:BmacgUru[t>ar^OIZuewgThsHaiDkYygwgboyxqYLqV:Ux;Kxkuy`yvVyEWwiaogXES^YvfgHiwCVydeAFIQC[;F<:Kjs]NXtus=WmxyyYXxATchTuywE]r]IlYqqDutHuQtEkdtleLweTMxmPAmQvFyrvvy`]PItg?^>^bZZ<ZbG^:o^@yn`HceHxYxoH?oenwr?bgha\\AbyyxMv^dPwdXqX^niymwwtCQoixnUgi[YiI`tJx[vqx`XbpYwrqxJNZ\\><BJ;C>CDZUXcgF=SBdSeEQdIiRk?t_OCsaY\\WyqwcF?dPYvVwIv;Tw[isUCWUYZIgaEc>?b^=v=wy=sTssERGU;_fj;dhuX[sE<=R>:\\Jk<YjyTiTl]yxwqmIlJ=lvJuJfIq>eMtxwJPMdywE`qYaNT\\S<]SZQMMXLxYyItYtplFappuLBDJK:BK;y]YfPEgGyGhCxCsBjayeWiB_dZYxjqVQ_cvGX?AV:?GjwiAGSvKd:GsAiUYCImWV<YwDCBK;[>:Ke:;D:;eyCw`;TviE^SBxeEjcgtQy=scMob^arOYgNQwwqdLCyJiScOtZQwmutdwxYOYFqWAmITqc:_sPOSd]XvMTOOs_?V[SXWMgu?TZsbVoS?ggh=gQAbm;gliUWEcMawHgxXiUjWCoaXdYWHwUearmMu:kH^GyV[wAUXugUU=CtSEEGWXQC>Kj@EpQQtw<wmqr=pxKUwHMUG]n=hMDYtdeaJQi:`rvyxyQq`HgiaerFreI^ugj:@g:QjAFnfxaYiodHmVau\\?wfNjMY`hIkIhmDObp_wTfd?>lbPbi?nX?omQt;Ij:gysfc`PePW`OPgZfcuHZ=N]<Q[:HjgocNpcfXd]^v]XoEi^EIqFg`thudH]OooN?gVwcuVbUfgNYuW?cfas`>dgwaYNpSacUoxa_uVh`YirFhukouD^Z<FbmFjqO_B@sanus_asVmYOxIx[NibxNm_agHxxwYnIpyVgynIlA?wyy\\gXeaplhfed^Z>ag^^wphl`Pcdgu_vwE^ypPh_Q_ogwoxi]pfbw`hXxB>[HFl;qw[yqwax[gxoX]i_xixuxWygQg\\`wXPesV`uVcygsnvpXqhAP]YHmbWstgwVV]QgllgwcXmf>nnOfWPw?VlIX_yQxMQwsxvp`r`xrRpZHHapA]oqpqxmhPiuQiOAu>aldi_PgdKI^\\NjHYrjAsbfq=A`QipSyuPOZu_uVhefWxP?^J>[TgZrIjvymI_m]poyylGGp`>efHx:HuPWp\\`eKIZ@x`vgdWWx_yoLyyI>uiweoOucwnCFZcHgbHmSOtJ^yZquiytvqZ;p_UhsdPkq@`pO]jX\\Tw`UpkSIcY?_:?c;@^mhx_x\\OqLWGjuGmmBoIFewCQUvMWtKOedoXgWT]?sNMSCWtlCS;yuumwxWUW_ipegtuu_sEJUDMUtneUB=SaEb?oxgYupoB`Yr;ygy[cnWVgAVyWuBaeWieaig:=FJ[hD;t<WRUKB\\ErHSwt;Xwihumxdoy\\uhdiwgiWdYhS_YVwY`yeJ_g;gcIIrwkdogSUahGCB;GILKVdsIpAw]mtYyevMxCMUFIVt;huOuZsV=]WmQbuKggMtYefDcwryWRAh;eIrAbEawk;iyqvtwsMGWKuFPIgK]TmgxQICg=i_XUNyvy@x?DN[dMB\\sA\\koUXx\\YpQme]T_PUcUysTQe=WQxLMaqO>;Ug[MIiWe:gDJ[HP[xwOToivb_MPYNlDwveRqdWTanxIwpYYNyLytxy\\tsuvwyPDYX^uthdpaQt[DNZlNjXPQytQiMjtU;qojtON@pYUQnAxq<oSHuluRL<LeQjXHmuplCaVttoHetl]viimRun`mpAlriyJy<y[TQ[pY\\yQb]SmAy@<PpIp]Ex;xxyxUGesHQK;DwnpyIatHIX>DMRhOQPQd`wEhWXQK>LJ;Un:DYRHOOHwZMrVmPAuNXxKiAu;uqw`Q[utXTR]LlN\\MpyLRYqqTn@pj@Dsi]LIQpsllYaYmiOdar]qkPuLV=X;xOmtxoaq`pNWar>aJS=MOTk?irC=RRINlIxAqvnxMRImu\\wjXrvEJGTq=AV<XpWLwALLA]SVIJldoHMQsiOdiOiEy]dkRhl<uX]tLv`wSQulErFUj?AXNIOUHmgAotdPLykpTTu\\wt<u><TRhqVDX>IL^uksIwIeYDyqHHxHHXyqvMlSF]VPTNbuWZ<w<hkR`O>LXHAyFaTeHlmmpMuu?MJ;@nm<x?xwwLVwpKAhWraSvYymtOtyn>Xw:IyTtnYdyTIYSULsauHTxsyVVTYJDlFtxwMm_xXdtomAKbTPodP_pjwTXV=lH@P<HVTHV?EJIiPCQsmAyayX]lQcDUk<UZtQLmK^UJLExjqnAitumuwmVbqjwywqpq\\pYruuI<s>\\yDXOh@TAiMq]u\\aNZ<td]LlEMYYSYPMwArO]prIpM\\Npuj[pQ<UNAijLTVu]WfavkUvmTQ_UXB`n[yS]Qx<@X;HjRdlQlwgQTkqxMYJwdQ>QmaePaDT^yQJYTF`Ni=WylLlUlNxrfxMFUPnENSdW=Hk?XygewFewiayXEX`IVGXPMdQyPS_QRSMyDhucXw>ys:xUJ\\tJ`uaexNik;ItB@Q<<Ltyo^ILY=m_@VQQnKAk?elDerd<jPdmveSgmJQywZXjHLTY<qnQt:XpiqjY\\WZDvi`Uc`ndYQnxS[XOQLuiESUijAiXRmUppq;AmCyX_UToHNIhutUyrDSwLtFixjPX?ilCxv=ms\\HQZLlXmmUxKtUU\\QUOYN>LYuHXuqsKLjTqXohpW]tvhOeUxilWHQvXQYXiRGAmqXxwtV?LJ;ay\\LsSTyMlK;DRbQUU=uT=QcUvZ<rsAJHYxjTsCxv=@J[eU>Dv:YYSiUqYoXYQtxj<al]iY?xuYUpFEtS<NMUtMMonTU\\dK=LjN@rQIqLmuYpq]IQJqVjpj>\\N=\\n<lQ[TStQyvMV=ElodpGlSVuMY`X=LjjHUf\\Vg\\qYExvynXPYDeQyQviYNeAYgitYXTqpumyOELQPLu=UJ\\aX>mQQxNgtJ]aj]ykahM_=W<=ufTwCxrYdTalmjMjMAy_Uo:hr[@TsulHlVWMyq=lVpOiutx`vf<l]Xvm\\pJHn]asvQvlQrw=UjINZMNiHxcPLgqxUQx\\XYn@p<YX<IS`<puLs<uR;YMbend=qc\\JpurKqWodVcps?hOHQY<`jg`Q`iyrmNApQ@@x:Dq=lV:=sQuuk=rw=ywXOMeK>PSpHlDQX_mtqxUVXOYmNPWZ;yjUqm>Ffiha`?^NaleFpoWbXVnMXl;XgB@`?PidiiT_h>N^NxZ<P\\xwb>Qnxqu\\gu<gkGW^Nnt@?_qNuOVvfi_ewtLFwJi]MYnBq`qi]Ihlvxx<NoPxfWpae@eui_fhe`^]lfoUH[KAaBQqwgjv_sv`xsGjxHtaArnaZtif`^pY_sl?wmO_Gw]aFwYo_Uvka_soO[Afww`nM`scN]]Ius`arok]_ceiw[XlN@pbQ^MXtmpit>a^XxwofsIgXYgoCnWw:ueE]GnqiAywtEEc[roEG]QeiIW^whXIIEysBMbbGYU_vW_d`mslAIcmrVgDBLQA<L`yLl<nA<S:]mc]XXAUX]W_ivoeM[QYkpLfPuYiNh<Q=]rvuP@HWvMrhXvxicxOmxXwlfhR>\\SVgZGsLhrC?brwfHNZQpwyIh^Idy?buXqBVdDytYphTf\\<Q]hx\\fVwGn\\n?o:AbNc][iZEFvMWmYt^GV[qC;ob_aimyGNUR=cYd]Vf]UGkxqiBtkuPwDv=Tj?cAev?gRNqh:Qbosrpgbm?gPofact]SepCtmqhIsCMeYk?heYbRgBD;CRuiVay>=fflKgQuYyuieoUTKylppEvtArF<KIqm;@V[Im<DqFqq\\pUvmysdPdaxQLsq\\QV\\T_LrqHJQITx]KdiLetKF]TalME<Oi\\jH\\pw\\t=@sYuWa=KrEToUJJyJbXu=xKw\\vYUQdAw]EwXpUA=JwhwpxYhHsQlnJUyfyiLNgtYiOWt>NmFFy?`jIW`XicKyrs>jtxmuoss^pmYmEi\\dOyXNZ`ycxOxyGgH?woAtuisFpq>p`AAms?w;ymrytRWt<`yFqhsfwnYt<FikNiuhmnxpYq]B>];YhZNoY_sxnMoFdQw]]t][gyEFZOvtGuh;wYidekTJiiweROUR<=YH]bGIG^IbGEd]scvKtYWD?uw_[xLYCIYyKefdmv=YdIMtRwfvkTgOfkkrHobIaWxifyYEDyC<IcteccUVY;bI=DDiClYvQGxQsS>wsUyhvIDSWSGwSPEgFGWaawxuBf_XCWbmsWqKWmqD]efOUxpKgIUv`awHItksvWYui[F>=FbAVVyisIIoEucuWF_SM]wHKWZEVdcxZygXmX?gwgUr^mg>cC<GGByWsEvqiHx=Ej=CAixusYxaGWyx@iD=_V<kB:UG[eu^GbIwb`sx]qyYgyh]rqCg<mIa\\YyAsJYw]Xn]`pxDxutL`@uF]M<AjaaQwatMpqfLqQqquYy_epKTMwiMIIjU@nWmtwiRjqK_]xnyOLMJ<QWymLnIJbeQJASJaXfpXP\\tu]qeApIxqfQpEaTsmpVDmITmm@NYxsymytdX=XxyeuSylqEmkewFYpHQTaamumU:@Q>uvoipNEYGIlEPOG\\mbLN@hmAyUwqvqDJm]mMmSsdrsIpN<L:ENNqksTkr\\TbmpmtNr\\pe\\Su=vE=nQxJ<yK>@mCHxCdXO]rwAoIlX_yoVuqXINYdqx@t>dOLqrGXMctYUiYQuUwepyyYnxr;xJxpTwqN`pStqKb\\TGxXXIXU=WxAJxURXALJYr>hufXN^HwIaRU=nwIvRhVPImeYs[@X;MR@TymeSsDPnLWAIuWiOkmUFlkjip`esT@RWHx<DPiivx<rh<tOxYspQnpMbypEuQrEoe<XemY^lOcas?Lu?LrdAshHPeAU<MQwtONxJ<xoXyrpHQ:pYkps;tSJeJC=q]qsuLLoIpgmQKpN[XOYuTyqYJISADVq=namsVLKg]RTANamSVUMmtNjqM:Uikpj@Q\\mo`GairvtcwbvX[:_aoQp\\`yZVipxhOfZl>oFq^hVnxvgkq_EWaawsmovfNsbWiQflaO_GvsUqfqIwuw_x`ncXhPC>SC=gU]YvC_HZ;YiaYsquyUFLQD`mgoUuAosZKF:_hr[CJuSAyVh=e`mTIcgy?cBGSRWtroiPsY_UrIag`aVBYy\\ihAUgqmEqQi:qRQku?yhVQXEOixqriMyQAV`cXvIHAWtu]w<iXQCd@]Ryuh\\AB;]iyAvIIV\\erliIbiEsqhMuu`QspUTMMwMwevaDwWEyToeayHXyXiXpIux`KEhX[ItGUM=lLQ<rQhMSqlCDKpLuZ_f=iiqIqUFliwiuWstYi;yuPagehynxnlNvW^iPakvIrKybANtmXntGul?mPos\\OZ]WeJihZGd>QrChe?guhQqSHi;`s>hb<ItYXlCHkdfj;^kWasWXn[av=a\\ZFhR@]po\\AiqJqo>AvmIy>wx]yxIIa=n\\dwbB^^NqwOahKNhrqqr>xhaiR@\\]WgeYahaqM`ZYnoU_\\ppgWQc<pdNgmKypqwUIvQwfsMEJ?Xj?rn=seqCCkwr=TJiBgguWGxHoi=WJnISNeQjYMVTuAqnVeWCtsO\\k=UtKmYruvB]vR=roYMi=qKPxDTNYlVy@VQTM]mSYXJnXx]IKd@wq`nw]smIOhxQLpxiySVxkXmPlyOjmv`AuRAU\\IQ;ppWpmVtSNyqc=Kc\\pDETiyLJplnTxPlXCISayUgaqg`pfmupTqIuqqiuuPmttV?LZ?HZvO^f_j<HjGg\\L>mWNx;Fvy^_gnoYxxOYiWieyWs``ZxwpKxZrymZgcBVt:@`<AsAvvwGryQqUgiiWtiXj>hveveKw]yPxiHkr^empymNwPygyexSXsybNYFIgyLOr_qDnghSSe\\utyiuoQwnYhIyxWGsiIieqHoYduqiIuxw]vLyIaKdkqeEwIFEwZaXXWXBqCB]BYKXfwWbYxo;WJ?xXeRX_SL=R<?xy]d;swMiwTox>CWUEc?st:?bv=RZktQkUbeIj?XlWtGowAKBUqHRwV?Gy;?eECHYybWwhmysysexwyryrykCj@UNDp=dkWmUF<UU=MDaPseowdvNujCqK?lsfDsUMvaYmIIs@`rRyVLPRHQyH@LlUugxPAlL_]lAlvGTSApJOYpIqya=NHyQB=KUxXj]pk<mhAxwAWU=mTAlPMr;AubAotqywxxtQr>Lt[dQFyOneoNPXXHK:aKAusE\\vKuwAAT;YU]lUR`mryQFUjnEN@DyW=Y[iJNyu:\\S?]xWAT^hUx`YsdlGDigW`GGbUXx?qc=^cq>nSH^[xwpxiH^aMfx;qk;accfibatdhtIy^XXvM_hqvvsOyrvpKxcdYia`qtWimWyZpqqputWgZp_r^tmaqqwqU`jgA`uGdP>gknjjvZZa\\I>wsGp:WsIibUNfQ`jqvZFyv\\_s[_aGgsNPZw_[<^tFpapH\\cVpyggoEFitMwStCsQ;hKqHRWejSElsvNYIHAI=exI[CRUxsqxqmi=MuNsvg[drmIZoEKMrpoeyKw;;rnEC`aTR]efUxpigieYeCIlWu`iGXiSh]De_SdGIEMET_UvqR=UHaiWLoIgerVgDBLu=<LVyMXeqlLYMltsuLhUMX@NGyjH<WW<QIdQy\\RtMpSPO;tTYXqppj>=vX<c<a`Z>iYIrfYs`AnhoayYwI^ahYm]GvWik=FZ[hwrqsgvpdHnxqaDWhRInAPyxWj=Wb=oy`@uYyigqZDw[yakyxyux_@N^NG^Wnw=Qiwoy?wlrynVXbMoisybyNwL^pyhyUOqHNpminivic>nCHj?Prunusoltqxqpq[ylcQueqo]VtU?gR@fHQ[nPdmItP^Z>hZ\\__JOaZ>uJIrX?kn`fRyr]Gp;@xYG`poxj@j_Oomndmv^pvgZGjBV[YYmYax`fphgutHa;arKNuKIjro_fqnMav?FZYwuS`quqsUI_i^[^nhBOclXddWug`fLPru@uMVv]GZlpb`V\\H@lsVdhPtHFyxYygyb<H\\p?ts`aJnoZox:`aZYr\\XpT?[k^^B>ioIwhIw]Xbe?f<^rII^YY]A?l[Akw>p=xf[Qj?h]@HuqFdoOtMAjfObr^t?@]VNsAFvEhp^?pO`y\\ylx@oxnykxkuys>iabVmxamjvrSo`INj]IyNx<IryiX[uBsQrY;fX?eRqhxkb@AG;_GQUR`GTH_ghGB;?TRws?[BigxeyyDIFPgyXIf?eTdUYBkXfmr<_hroThcTd=SYoyVKE=mu;eRvkCR?Veiw<iWBGh>mSdIFceW:uE<?R?exk;XrYEHOXtAybAuxkCeyUNgs_kD[ix:OgtWguwrYIwAWVtoUpmeM=wuGUigwJqItUu\\mWv=y=yE`oh?Ee=qv=kwM=Xpss>chKCV<lwE\\u=Qu@=QrYniaWmqX@YPAekTxprmODyl`ms@HWtdxk\\LotWkYmXynYMylIPdYVQxYfpwxdRydp\\msD\\ZgYZM@eR@jZqyiysfaoKaivoxWw^M_ZcVxt^ymNrB>stXlPWeG?qZosSfn;FjPpr>hwsAyxasPiedicHFZ[W[Y^qYFmv?`XYh[YiNav^aeXxavxyApac@j`Y^f^ah>cBniAOev?vyyZLAnAO\\>>vuynAvb;Igeiqd>]GFhSqeEIsnhgxh]vy\\NaZEp[O^soGyXVyVIb^A_=@pPHuVgsHOaPV]fqxSq\\hVuH^`lYZ[?RORJAR=ST^SBaEhtStUID^Eb<uDDKRmQr\\GW^iBGYbxWcB_iVeW;eHPUiK[UVYHK[v^_SyWVQWTisCPSVJwChsSx?tGev=?b\\wwJKhssCDIfusCMgUqWhrWxZYFAQUA=y;qCqSB^mbCAFpMtmIy<gxFYcZqsywDH[HUsEoQCrEyOwT=KRK[wfIiv?GQkFgki==F>yWtAY:otAuHoKh;Sib?D^=Rx;DxWDsIr@cY@IY@[hAMIZYy=?gukv>mgAUBLsTymS=OheYVBaFdASmYfYcyb;TYkuWwRY?fYAtp;wIeU?SyUyuxqHJqs`Cy=mI^[WQ]UAgVngH@QDlQb@;rXot[kt?OR^MuyAYvIXy_wvaWnYTNuEMAygQsLmYQwYlYTocYFixjGDckGU;B\\oDOmHraFcIYwSxeyV<siMCW]GXLwxUqD_abpGRAshgkYISeI[XWYuWyvvQf@AGgmFrKbuCDQaGy]urUx`ai_qiaQH<khsUs[GYMosS]v_UEqOyRIceYTVygqIGXixoqxccBC=SjsVSUG];DBKIQUCl?t:=HG?XG;S@QYpwRRIYY[YWWba=uf[EZSG\\;u@siECY=uBTyCwiYhUBnSrBurUOIruyOaB]AsSYbm=e;ASPKBEKS:=hj?r`gUY=U[[bKWDrkDHuSBwItAeZ?V@?sf;EBSh@OsWqh=cS<eU@iU:YXTIfy=UZiW<UDDYDSwdq;C=CYscVnSx<wgDSCP_iU=EYqE[IWVAUMiUoCdA?FvGVaEeEYBmmBnWwkwrgGgSiSIsD:WhoSipIWpUwg=x]=yAatU?cOWt`yCvAyxABkOVMKEWagDwi=aD=uDlGvZUCMGdSACM[C`UytIu;[UI;DBqtGGxrqYISwUkDrAUmafuIS[ir>YV<WwLWxBMVZmgrcW=uteoXlmbWUDFyCBcwVqEvydCkcggCoyvTQtlAi\\syacsKKUiYVpCgKYyRWy`uwwsdPaxkWS\\Ws[iyUGCXgHKsTpcYA?Gd]emUu\\wERqePahmgSlsEvay<MyuQfoUwbmwoaXu]rAIWQkWU_W<KFUcHPQdHIUHDtvhsPqxchWXqPNynctvfMlpMLB\\JgYJptUDMXWhvc\\y`xYZ<w=avcdsmDXptjG<KuiMaYJ\\MJIqktmQu<Y=yu]yOxQv]\\LOQspHUlhY^]rq=QAHyNqyetKF`Yh\\OX=lH<rw=pJHtCtkUAWkYMcHSHIoeEKSYUjLxKmXILoTlMS@KjPRRipOhX;QSU=yGAXvhvl=P`QpGUs:qPGiRglWipOR`jsylnTPiltbar[LU=LTOEU]En@@LDqnslnjej=]WfMRLLt=akCuNpLymEtaDq`lO\\IXseTbuJvtLZUvR`o`EqFayJpjUQMqQvJ<rfxOZiVghYaqyP\\WepTYPTWakp`rhlQpEloylDIrAtMfykClLZaW`TkFPgwiZ=V`=gyJg^UFZD^kywb:sSaUlGIhWIlsVlgHw]EPqIhihFeSYSRmsTc_V=UsPktYoHYoGqUb;ieAqVF[XRiCiiyTgIcEbZYRwAigessWriAJ<]XL=VhpKHlKspQclRB@VPQKkYpKuMnyv<usUUXQdOxmu`xwMUurUWQXl_iJ<DrvQJ<LQ<]wh=yNIYnqovirmyoLIocAXtqS=<WTMrf\\M\\`PZDTsiR;LpTIJqxu=mKYAQvyK^Ev>YvHpnoPMltOFIobpYUQjHuu=HPaYX^YxsMUu<Rf=jgXupHt@ikXLVq\\S=LLZdUdEVplWP\\k:@xhepXlYQ<rHLMPTUP\\wgMRODmP<Q<UOUyVAew:LRJUpgAJtxNsuleuL>qL^il^uRkDWTdUj`Rf<WQ=RcQoctKlQjglPiAKl@LEQr=tQEhsm<NfEwHxytHUVUR>UP[]LXdUrpvWxSUqW`tkAduxELqqXsTLKuuwypCdmELRnXR>Lpn@Rw]xxxjJaS^lqLAyHiRmPsQUmimpchYFHQYaktQmUhL`]Jl\\j^\\VyQMq\\thmr]DNd<xv]lmHUauSjpvQqR>dua]sphuvhRwYUWyns@OMqP;@RJIx:=QVUT:xvjDq`iX>Er<YqRTm>uLwUxxqjFmp<quWixb=Rr`s`<X`DK:tQH\\RxPjIDNBQjE]RVMP<TLd<yl=ra@TQqrMXQ<mN@ULgHUe@NImqydo@\\nJhU^]J>lXHAxyAYTQvyITCqTe]tMMys=NJ\\rZuYw=vdhT;Xl\\hWEhL>dpSTssHNhQjRhoW=LwhjlHqsEo@hPZ<JvuUn]wqxxXmtlArvIJfqM^Xj?XLBlPtajmuKO\\OuHWvDNELmN<yS<mkLnUEOVQTsam@LtYxrTpNUIkOAMYTp_HuTHx:yt]dj^eWQYYx`jSpR>XYqunQiwYUtF=Kkar=AnUHwSXqFIVw=lN]kHTW=Aon`o[ESMyrOPScdXYtncxJahVM<NrYMjXxplmopYwyy^LT:Ilj`usQToiJyIyxtW<yPc]wZIYUQwxdsDIMZLJ;ur;tjqIRruS<mNJdJe=Y^mn@`Y>hmeILc\\y`iw\\yWwLMJXUx\\YPPy]YnPEoyeY\\<yxiUk<QaMkDtJkYrH@QL`sGIp@URELJLAjI\\mBINyLN<PoLquIHp`Ax@DRkpmbmM\\HJ[Ml<DquTmiqUV]J>xK?Ut]LQKIyY<PyMpflYfhYZ=rx=y:MQjToV@uhEQGLmSxTLay\\QtjXxfMT;hnJLplUPsDV\\MjWaJ>mwDPQ=@VxXukLS:Lp@UmD=V<Tl^\\NhUOdyUIDK;YS;hwfIlAHuqISHaUjIkjemu`sGpp[]UTAQMXRUYQ]HU^ukMqvlmo=`PvyOnAsk]MpUMgas[eVKdmiPRKPr`UWBdyv`WTaXraqgeXYPK>LjWYMh]LfxLe=sDlKxYSEpVtAj:Utk\\q]uRGHOR<kCIQEyUmtRppxYylPQM>]O>=vW<lK\\mGLreMQFIpZYt;LT<\\kImS>mJhDJ;EvoTKT`l;iQPHULdNjIJCaO\\tMJ@pjYqZejtqwg]LJpK[Ysnur]IynHjheRxEqBuRkqvNxjoTSJ<Vjmp;pY`<J=Dt;axjmmEDtHHTutmSaJaMoRESiMr=itb`uOQSUak]pTEDVcmXSXtnHNSywA]xI]tmUwcXXwLMJLK\\DTh@lshLFqojmqAHQhGg[ai`or?v\\V?bqAmawsnP\\uXrvhgowmD^Z<ps=VmcFw:p`=NmKV\\r>ni^_I@pB_dp>b]OtFw]JNipqmEO\\axrtHr^_ZiagHVqUYty`rqIw:Ib;@xCFtIwbqFxDqxgqy;PjjI_;_`gPmb?nC^i:on`Wsdxhk@\\Jn[yXyMybXFoS?wMPZcGpuodHHamOnKBemhJwtBMe`]dAwFVKF_H[whqS?q<HsC^jk^]pipJhwRf^f`dfvnDyjsFeANt^OhCYsWa^AwdNh`h^Zm_l=wgpVmcHxoO:kgrOFKYx[Od_aY<Qy\\ccZ=e`gC]awQUthqdqSTbmtMQVk;b:wcm_HjmYIIUaaiUyTMWwm]boOFrocJOhfAgXQWLKhu_sUqTqIg:=FbaSRoyX;FYSUpgt@GcGOt@[f;keOYuuMeweIhobY_bHwXRIBc;c@oi;_D`YHBEfpeI<Khnucw;wBcBrqb:UC<SSYkXJuhPYrSufRUU<gX:CVIYWBgTFoB>otMOxfQbnYy<AW:]E[_C>[FZAGMuBZuxFWXs;GfugQpW]<RALpjptb\\svpvUUJ\\xmy@Jj@NEhMNUTBHvMhugMjhUnBMThUPDImIMPt`w=Ytndw;iq_\\St@M[AQFyY[PTcemtIPCHsLTyB=uVLY=QuJDnchWxyQ[=rExWp`PWlyvAlxESTqjpMVOEkfYvnTq_aVLpRTqsbEJqdo@\\RaDNRxORxPiHUp`kN]vrTj\\Isatm^hkLdUe@jt\\Y^YRQPoUpWtFlP@omyfLW\\PWsR_w^HiUawxfktw[Nqng@_;@bjXiu>adgn:>ZVFjC_b;_dZfk@^vdXqX^cWxyqo]TagooiA`gTfn;FigqmoPq<@pJ^qap]sPjxQv^^hrYayxx\\Ycdo[\\^vG@irI^<@bV@i<I[daymvy>@cBixqr^WHIoxOasr?ERAh^UTf]fycyFwXJyxLGS:MVn;FJAfFesPwD^wcaie:?T^qEaYHlkie;hjEfkIr>shuccD?tZ_yQaBFEFRkyCuiHmYSqgfeUu[W[ubk_Hmsg^Gy?me=Kvf[rVcxdCCsAbjMc?seC;DEkHrSw>uWqYWiWihmGlAeQoilscT_fV=smQfpMugmV:eUsaY:=DmGCIqhvcUU[xVUTdQWmiSKYr=sRqWRh[t[wDHMi`wCNwGrSEHoI>]IGsiMKISQiCMrS=Uy_RowHAEceGgmmFWorIkifKSE_BO_HySUSaxn_wRiSQiexOX=quP?s>_cIEIy_wlWsBIxYuxwGbsoDnkFcURjqd=qHs[iMStGyx_UuHYulEi[aI]Qyn[WvghlUt:=FJ[GfCu:od;CvoeSZiuFsrhUtceiX?WREVHSwRUT=aHZ;uvsbC]tOyvhgCVyWumIvyIR_WZ;RB;C?kHSCvI=cocVruvBOy[iYy;FmYGJ;cEwuqqIDUCmKgIuUkeFwww[McoEw_swMuEr_xLMIBsdGmbYcYnyv^SC]yx]ycs[enKxfMg?WbTexoOfNmhuQibQePWuheiSotl[vIWTSYYv?W]UiWCB[cOaPVb=y:UVVpXTPUsPV?]nZIk?uSYIX]hM]=m[EYppvpnhBp]>heDWu<nf@Hl^hZKIq@Vioqf`i\\a^gcixEP\\vA\\oQ_CGiVYl?X`dqjUnqdylCv^NVrcwuvfhXpsM@_Q_uMykeAawYqKPacEcARSCFrIw;kB`abRsxFodtMrGOv_ITbovVKiKQeCgVUSikmFE]w=abQKcH;IvSw[AgAKR\\IbUgwmcHMeFu?TwAVAMISYI<=SpGWo;wnWXEUISQuj[Ggse=KvLCgVeSYsEs]vKGfw]ckUdpwcMAya?XTUHFggMYVHYCv_SmIfBWxw[Ig?CBcITITqIhZASqMYBWw^ysi\\RPExf]quLulmu`QUYynpHpSasGUJ;@RjtqjhRvPY]Qk;AriAN:lPG<sRarNDW=<vnmW@EbP@kR`^<FivoxWw^MYqEB>qTHaiamSwEySsW`wiqyWxmGn[bjIUemUiyU=gRFKfXExS;Ft;RxkIueKNAj`HM;eMiMyAYmOypyhxGdxIupYYnxTpxQMDDvEAVyxkIaUxINoEvtXk<lY\\<xYerTDrcMrVhwIEQ_aLtxM><KYlUQAnSujYXVHepKDJSdMWdTi\\qrQVbIo;IV<LLVPWn]SHHqqYmHqvBupg=YaPULYUnajLaod@Q<YUxTwt<pHalePWdMWd<XX<KxqQuhVEDYxyTuQYNqokISLambArcHwAeMw]odplu]T[XYHqYJ\\wMarlivEite@O=ULmQMjQwmLwF]MmtqQmpiAxReyJPmSpmppQMXtJTPqlKV\\vlaTDeRt<l@yOIAsbdjRXtGqRJyWbtnsMwvLxilLEPoyxtGtRDHKHpXpDjt@NImNEDSLirOIOQhM^Tqp=oyasm\\T@]qkmpn=PrMkoLw@DVEaxQ@R>=nXquXLLPqUSqQVUVeEkl\\wnMwYEXtYnJ=OrYq\\yOsTVTaM]XUqpWOuOw@xvdUIMMcAW^urpUwoyuwtxfdkImMAlqZevKTtQuXthxNiy[@xlARw=w^Gm;vng^utVw^Y]KGpJxg`PuAOuYNusAxTYycynX`xqOwkw[N?qbY\\UYkBfw<Wxh?fZ>`xpjq_wnpnOYuJxvcNtTQuUFd;OcHoqRIj;YpGqa`Iht`rcy`WVq`n^cfqSyrvnp`^mYigIYvt_xpwvoHpp^]TphSX`pppuffRqxuy\\B?`@YxZOy@pxTpnsFdnI[lalMqy]Vq`>w^Aiw^eiNs>pq;Yh=qv=paHHblnb@wre?[=ApQo^Nh\\iy]yqyWhpiYstwvgOln>^@gnSImJIiEPhaqxVOwqIlgxcgWxpHlxf]dyw_vrJOnApsvh[UXpUwetGdgQ[SwgfXgTqoWOcZ>\\Fy_Z^my_xRhndIocHZo_bk_d_whx@opteMxyuycIhxiHvWduyXGSDC=ryIytyWxyeuGbkOhEiSIGYyQsZIrnYESmsbEy\\ywDqgKehCIdM_W:MEbEGXwuruedsEr]eYmdGUeX]Tu[equCDAclOwIyy_yGX?xiLxxHYaQmUpj^ImauPsIyfTUTxKfMO<Qn;tv:@rdujqdjkqy=QY<PXyTpsylPQP<QOBDWJuSweK:hrJtuuhvqtqPUUFXyZhu:yJrErKHRjIKEyyEpSX<YiTnV=OM\\VQyrYTYfdTYPRxyrc`w=TJ=dsa<NJljZ<V[inIDnAaL>hXutxVppelV@mq[@w<lsY=KDpn]dN>HjSQkMElx@njLrapuP\\qW\\UwlkvDK[YMqLomhM@qlCivOuWYewhhMtlKELQDAsa@V]Er@QvGhJEuOAlSQumddlWLXK=TwPok`qHmQcaNqLp:uWYdw\\mLgTmm`vj\\ThPTXeUsQjMxSgqobAt:uM_YjpXsPUs;DNNdv;Yp[APWYmi<yuXjrpTBIMaPKipUguMrlkJhlgQOrix;uuPLXwdP`aVbTOBDNaXwp]JJaYPqUalSQlTaloq\\tw]n;IsGaJ>TQviq^QvEhupHRwhqk<roURShXleju?qpWbg?uOXnkit]GkkikeAc=XhM>m:@fvVw<V_MxbTvvHYpFyu\\v^MhvSXspQeHOrtwp`AoF_jTIwMIwSH]dv`Hw_oVrKW`[>sufgRYuQYo@^kb`eU_]\\ii<X`[InGNvxFbuw[nqZUY]i^\\NIZkoqEGqLgsr@nknvB>is_xap`iXlnHfVOeNIZ]anUg\\Nah>w]aYx;w^hqoVaodi_^YgtA]wIjvQqVPaAsBiX?ydGiSUiwNGTcwfoSUT]siwXWgYpceFSeMqSZqG\\AfEQVk[XxmbGyBFgTvwtJuRtcgVCYOcuqmbg_SHuFdow\\uhPqbpescsh?MesUcnqhpURTkgiSGhKraiWKOwLytMGeQoI?ayByXqAy]iCmqHhms\\gGveWRAtLOEM[ilwVtkhIquuufLQeC[B<or;qt>YWFYw@iHwyye]WsabVUHiIXwYEducj;u]ihi=Temxn]vGYRQib`iEoayTOHY_XR]XwGdpAi]QtqwdqyD;oRLquM_gykCh=gyCtx?uROb<OBiAVUwiD=E\\eXf;tv=bv]GYYWiKUF[SlID?Sx?yIkcwhQywKicUVZQI>kgCIUQaHEsxbyxk]ermUbucrKvB?rw?V??DbAVhIVucsKsi=?HZGdVUyrQhTOekeBk;dQce>cdvsVpMFvCCmQeRoe[Cr=asEScVwWHEdj[fegsQ=bQ;S>IW@EyJqTF;CG=Ypac];s\\aW<GuSIEokIocSKcCXkycyS=cv;Ue<=H:?RdwwtOi^Kg\\ihEaGOiCceW;KBFAewivJSEj]fUceV_x\\;R?uCkKHWEXDYSK?Fn;XsyRAyFWQXssEvuHogB^;sVCX^CCJESFAVZMt[mcEqVCCF[MgHIckICTIw?yVaAwROukWWNOvuUfXYi`kfNicq[uR;SIKRWMx\\wV[gGs]SdiYIyBrmSKHND]x[xwOdLr\\uMTQtAL:Un\\aYIAW`ARtayETwYmW=AqBYVfUYDpY;yxjpQXDMUtnkXkIlrGpyU=nIXs_tQiTumxVL=y?\\TdlW>dQ^ipdetVamZMXItQoLmMMUUQV=<J^xuVxvVqNmHL`=TbXNGisRPPciY:iX^PPTTxPTxXlvj]TptWXQvwALQ<u]`TddjyXwWppIPO]LsM`nIEtHesdYtLxpK\\TmpORmvfTWO`jN=K>Ljshs]`MS`W;Uu;dkDly\\YTIIWaQJuUy<XM:XN:XOfausXPkxsc]o>qrfpnS<jtyMdmxahKu<VbePZ\\qcQRypqZARBTV[yLElQPlYW=VbQxpuLgMJBlYyXvHYQcEr@@L;Lr:hJlUx`is;tjL=Kq=tl\\vvdWH]UXULHQRGTl@<uUInxlVX=L>DNrlTvtMOPTB=wR]UaAX<IVK]UR=M?Emj=TGpkiQQIiwLeTTiSY@mR@jN\\S<eOVpm<dvkqLaeQtPLpUN[MM@Dn>dX;mMZlVYuy@yPdMR\\Tnf`Lv<rK\\OR=vd@UvYMJxLTaJY<K?mptDwoqNgDUhtSgiJoPN=lxH`VhLvBDwJltwEjWIN>QYWAOl]jm<ykXjD`q\\ajaanbxr`pu`HJ;Am@qWgtYjyYv=LYYMH]sBpn^xpo\\mGAtP<oITuKyLtPX>=toYTLEtq=Mq=Wa]k?]YjiMdLWwulSTv?prImPV=smHLm@PldVgYsSQT^UY]xUJuM^HpoIP`YN@qOx<MiDXw]m^`SC=kNTJdXVWtXwuxu`PPAWHEYgUklewVTmbhmaqsrDruUnNesRUnG@xvXRWDpK=JlISwITGDNiqrvEpK<YXxpBUklHnV@YadYeqnU`o[`sktNsmu`QXsTWuExVhRTIKiUL]\\lbDYmqLwIVUxpRAW<eyZmX?xS]hYTHoPyQQXrnxM<DjZExZal=ulMyvIySt\\syqq^xpbdKMaV:XOv`xlYsXYqpPLE]R\\eSj@VaTK`UlN<Y;YuPDQW=omtXcqTW\\ytLL\\=WAytB@L=YKBdOr]LK<oN<x=DJJLvfesUxkIdtHARb=rhHNAEu<qO`pOB=pSiJPlSR@j[Im?@La\\SHYK`YJ:hOi\\wZURZ]wglnFHUgEUnIqxAL@EpRuOYyVyLUvHUBarmHnSHVuuQphLNYuAXs]`M;yrnDVupxdysKUY<aj;uqQuY@lyIdo=\\wGtKxey[]pEMu_EWX\\nuPwdxqlyyI@WIDSNYYbytkHpkLQyERQlmKdy;hWWDJB\\u`INe=KX=jWMt\\XlAxq]]KlTWkdm<ixLuXKpJweN_qWsxOJlR>htodjp]lods[@Xjmy<PkPLJdAt:uYGuu=anCuwQ\\R;AtUyrVTkriK]ijlAkCdVDDJtDkwlnXqjC=vLTmfxtAYTkhYfmt[\\oIXnnXPtqqD=wR=RxmSTqOIIoJamMMyVtKUTYxaQIIqxLW`ay?aY:=wVeYx\\wyPx:MNdppWMYNHoEQRZuN`eUGUjxay^XxPLUAlmq@tnMmKTYKpvDXOh`KNHLiYv^dyU`XXqPTyntAuFXmqvhDFasNqJ?^J>[`Y]JHrY?hrVZfNGawRaba?fBEFP?Hgsc<gb?iRYIe@WTZGwyIHQ?uGSRCYcFGS>=hM[IocI?UitOy<oB[ABE[hgkdkkXNGhesD_[C\\Et?myWmeyGBFUETCsEKRC[BU[dYOwYKh=wb?sD<KDaGGS]yH]x?EEWWI`ACyAUjIR\\aXa]YwKR<oCV;UA_IR=tt[vJWR<kY^[YrOeH[yhmI=Cy]QytyixaEgQdP[EleTlMkfYKwtOJhqOELoMvkDTf\\XcHOA=O=@V@DNVusiLogHlP\\vr`v_IlB`S:Um@QRF\\p`mKclklTO>@nRlVLIl^akI`P^YvWtwvTknLJ=pR:uJnX\\CYrkvywxipyf;piCVgYnw=?[^gyHwesQmDV[kproQc[Pd]^u?htu>t@H\\cghlPrw_kppcUyrjYnKOeYv^[>mCOs``mpGbsoHWvNCS^CSuOwmQcocTBoBO=T_epSTwkUx;\\J^lmjqswxyJ<V:Tx<\\nZHMmqL]INxmsd@tqELyxoTdmqHWtatVTmkUXPauvHpwXS?LJ;@RJQMrPk>tviPmmDNqPsJUxK=Mf\\JMpuhAphYO;<rpHME`kimq<LL\\ImUEnLLy<Yr>`rnDmZekDhsVEJHAvR=xf=r<aK;MJudXJlNtaPGhjgtOV<LMtlQtwryxeMww\\wZirS`p>en[iRYXnWDrFhWwQyyPKRpQ>hlALjE\\Sq@TX\\tCtuKTPt\\wX]uAxu[pm>DmdHMj<ND]VLYvpiJ>DkjpPC`oiHTQ]R<Mro\\M`XjluJBHWBeuJpL>EL<lPIhLXtOKpXZQuRTT[LrPdUAHmRHlITLLdyMuOJuKnhuoyUwHtgXk=mQZatFTmrUT:yro<QRQREujJPsKeYIxuJyjwpUnEoh`YElX>hkp\\S<=yJyrbyonhYFHXKytMYmD`MSxwH@pstjpHjMxQ:DNwLoWEqV=wi\\smMNq]McTsMQy`DmI\\RN`lW@KdqnT`UA<lV\\y@pQGTWm`Nmql;LtXdT@MOAMRM\\qtuvgMmxdKFxPi]sO`T`dLsAKUaUJ<K>LJ;Ux_YJfhUmMugUyupPMeksUT]\\jS]mSdkguNRaP=Qj:xKu<VP<no=VAUWb=WJeLuPRpepvlJ?DnQlSv`lAXxWtsZiUglUF=LTpTQqJZAxjlK[Yy>YvIPlp`Jx=LjdR\\Mv;aj;hWKHO<Et^AYGmUqtuvUPwtKp@vBdXDqqf<tx@l:MvgLkR]lS=N[ENBXRoQwutXfhVSMQn<waHJKTjIujc\\M=aNAHjp<VLqjn@OapwQUjlAxHUOAHQ_Qxjiq:\\M@@RBDuSdMqIjj@w:=YF]tO@s\\QoOmY^HMnxLHyQ:qovlmK=xI\\QFLKpQYsdseaUimJvAVr\\yhlw;=YslKIPSgavoYmxXkFDjwTmeQKF\\YHdJdeWeUSRuUM\\Wd<qaAtL`Rmoryqci^pdGrUIPoCdkR:kTm]fMsx:?S?Kw\\ErhGxkuiGEtx;ukovxwrrmb]uWiwFNKTYyxGirPscVSb[cwLadnOuH;C<SDZggTCw]qfOYRUsYjYY:YI\\[WHEXUQR:yhGYXQ=XYgxeqrKMenIy:ys_AdBQyv?vBMFPyxZWtxMYyWry=sVwhruyJ]hvsWfIemqD>AC>mboCWkUPQXXxpvgeQwMWUXpEqJgIVZdWTyK[<N<Dj:Gq:I\\;>[PfZs@[eY^UPfSasjpvG@j:>t<@f[_oSibmnsZ>[ofgkaZSneFH]n@d\\wlJI\\h?\\[Ws^f[fhi;QfThtV`sWWxC`g:WxKvb:o^whu;oeVXZ<opMx]>VxThmQwssX\\khxowv`oksysknwsfcGWyZfkKntL`]dyraN^uirN>bSWg;vbv?a>qZnvcXqsavt=NZNvrhykDY\\<f^FQr<qhFGyFglJQaSohKigIipPisrva<ygUPdSPu@age_wnWw<i]>>\\BFaF_kGI`cW[EN]Go[`VdsvomFecnhQVdwV[T?\\UIiHNabpoF_xswtr^fFNmgQl`tb_htAuhmVYsH]oEpOhkMur;TrmvJsyhGEasCRGcleu<qeEixayukYUV=uaAixuvuYr`iGiMV:?G?isSmEIQi:wuhcYDsrxiw]kxsSWEcbVKEgwSB;c:=Fb=BB;y=sh<_HxMtCEbjay:UGlqWoCB;IhI[hH=IlSI^qSDCxhEFZcSkIB[WDveCLAVmmRA]Ixqe;IBe]tMcYk;tmYC^YwCKTf;BeqI\\[hbOU\\cSkQv=WYZ[Gm]XeoIA=s`AhgoFASclyrZiE:=CT;UnCsxgE<[VQ=T]IVDktbac_WbKybn_R`GHlYr\\_UnQt<=HKiSq;Xhaby?WmiYTwWG[BMMIQArmiX[YY=KReshmKfSedDqI;Wf;eTGKI_ubKCB@MttCyIixlYRuQRuer_YBkKb:GdIGHcKtUqrVSV`GuN[ByAGGubGweAgRgIX?sHqaeRoVtybNAdEsG\\odggY:]vwyVa?E=ofjkRKQi@KyvqIpsvaoIAEdp?IRow:Kf@MyrCsVMcUSCkEWGIIOstpOyrMG>asAcixisbyXymiuUEnkvvgRyAX:wc^WXpKXXqsmuelAu]yiRwYpGIeUeeORx[B<CRe?b=SD[ebJQCL?h[Ur@KtvWXXyuywxXps<DUb=qlLJxhjuYwo`tCdQfLNPpMSQOd@xtDMkHLKPS?UkpUL]YR;LpLhs`Yl?MyCHxSARX=XX`oRpnLdtDeUW`nETxoQK^pslqMW@sLmPvtoqXQU\\qQQlh\\tnlo?AjJMvhmp`YLqHNyuvp`oQ=tBPTFLwyhjKHWUDOIPwYUUsdWs=wsaNgum^dwO`PwMtpuKYXQSQqtAQCEY]pYEPOQhwq`RmMoHqPIUl@qpKTLTqVPhSDpRXiSgpmqHW@yTwTxO]mHiJ<DjHdUe@L`HV:qKxlSpqpKPKvqLaELNpPYAOgLkx=X[HYBaWTxVAEWMMlnHSm=y:eWn\\oPxR;Qv?=ND@kR@P?PQW=nmHLKdvWxpKAw\\lv`xo@qvO<PkPo@YK>ESEAL_<j^HLdqLLHxpPNgUoEQLA=QL`uQtnOYoqps[]r^hmqeQLayXplNhp\\`UcMsNQwk=MqqKCP^UHe@qbEi]Pon`oxc_tVhk\\v]\\hyJ>[Pn[RIsAxcR^Zs_m?hrx^grqkbni:iuqouhp_TwpH`jp_];>rYIvUoZg>ZLQvT>jJwvBNs=>wQic[y]sArYfk=_o]^arv]IAr^PZqYvq^novomI]d_e@V]ZopYnlipyfc?[Bx=Wt;qVxOLYNrPLeeQJxOxyMPuQhuT`LOtYpSymKts:=upmYpMtpDWcqVOUPQLubhjjLkB`Rg\\Ot@VQdR>MO_AQ;ynTaltMlGPXolUBQWPQNQAsHdX]LjDqwt@SQYSnLwWTpW`uplpt]nkDSeatkmtc]JiLk:eONeUThyVaPmqTntM<DRuQJ<=PFqlZlu<=y_yw:HYr<NxUV[aMYyTuuVEDvT\\Jb=JaINkyXwAn_ULriK<aqH]VjiUymQMXj_mRs@J>xOApV:\\N:\\r;QoG=X><Y^YKgpr=tQxxQZ\\rmDq:DNjIKEqrG`X`LlSxOBmJxysqAnF=NKDV^YJIYva=rfpyAxxwArympxuMlatoPOCMu^XNB<SLLLLplB@wEtSLLNLlv:\\m>`P>LwKlX\\DqDaP?>kT_cUG_vP[c@cIVmjV\\qnhM@mRn`Twk@wrCOlag^gvcLahGQ]Uisdyw?@^DhxPhaGGeuFhXx`]xcM@\\IFtx?[>V[E^bM^j@XZKIyrA^O?[XQas?f<HjYA_GXxu^ZcotJ@jB`gqacL@mJAxqA_\\Yfuqo=NrsVgF^x=@gcq]I@hZQiRIZWAaJn`FadDh`ynnSgdL`i]AZY_\\hX\\vg\\MPyD`^cWlHH^ZFcC?cVQfKnfhh\\_Om?ItI?rVgxRoyZWj?`laA^:Ilnwl=OeFaqbApf`ij^q`_u<plpGfgxf[nebi]qqdL>jsqbLF^@wcMfb@OeSq_W@u`iwPovHOg=qdhOenOyvicdHxJNx[>\\f^aRWskhfEq\\LHp:O`jiiop^jYw^PZc?xupiiqeuqwiYmIFbRqfmYnoxnVPuJGuRaZEn^BWuM>yV?m:Iqpo`gfjhoc@`ioO\\l@mKQ\\aIj@wenAeJ?y]o_Npp?Q\\IYv??]=V\\fFyvXxi^nIfmTXefNeiI[uA`;Pvv@y=Ii[NsE_l@YjEIaYW[upyvhjrahT@pLfxcotRguWHtbWmCNueoyqOyvhclp]CFyJ>[TyZEga\\IZmF^PQj:ou\\?bEOrTy[^Hrqge=Or>_rVipqfcrPw;?]s>fAfgbwb\\uB;e>EbFAH:=djuCB?cxmiNch=wdiGu@CW:svp]uiUYpUI]gVH=wCUf\\aFS=YFiU`AHuUY>oV^YRAcy;CwWwv:KtJoBp_i:mI<qse=Sl?dacDPQdTwfCKtoWwvmd]oex[SMuc@mrBUhUYWxOsWaslwGNaEewhvYimYI[gTmyTBub:=tX?UgIItUHZMc^WdA[YYktvmIRIHTuBucSAGFDIVuKCfASHSXdceF?rYUFH;GMSCyqVsER<]st=fx]w>oWpGRVeeJmSbubB_rcEftesa=w_MHGQsPKEGEIB?HcKE_SFKUcjUITYXqWbZ]ufEr?mXvgB<ox>Mtsoiuws[EFXKB;?fX;Ru=ByMCMmv_Erymi=Ax`gRN=EXqexiFGEtJcr;wC\\[VYcWVobHWbBcuoKWXWrjTqj\\tUDSeLO<xsvUmS\\pMQJi\\j[Hy]epWMrXTK>LPQijr=K>Lj`AJ<LyL@K<Hl^UPx\\kB]jWYSIaWLDPBTYpUqXpnXitheJDMXjtUL@STUkS`XBLUe<nNYYXUSahpGhoHEVg=uhxVKAKS]lsUOJ<K>dk:DnleJE@j:@mEtcTFlrN`l@t>XnOY[bWqCFeEHdZ?uWVwVqcAVch^by@mmQoWPwRQksv_<pq:?>chbuTGKbJ;dsMDZKdXwGl?E?OTK[E>Kb?iHi_VHYC>KB;?RB;SUCDKCDF[B<Cj:=NJ<K>dMF=RZipIXrvAXtPVuHKriwduXh<LB\\J<ZR__:vd:vfBOkRG[xwmwxxPW^unpqhf_gx=YZQVtmhZ<Fj:C>NjpGZrPrZq]EQ\\axkOAjFf[GFoWXfMniVFv<oaYq`TA\\M@\\B^Z<:NjPYZTwkSHuSOaEaZVFnqXaigcaom;NlB@j:vZfYpqP]WfwOw_w`ylPupXgQ`pr@bZ><D:[Gl=rfQCTwc;ou\\wY@sEs?S^MfG_gLSI[wxoyxeWbHKIBeTDiv:GwKaYYwh[;DB[:>^bw]>>sAfyGaZHfZiXm:`fWn>Wg_MVn[wuiCLKDOGi:=FJ<;>fqyZByjhhyBqbbGbLF\\J>uhglEq_SAbxhlDXc<?xJFxw@\\B^:<>TkU`YJ<SKEPahyjHyy]N`<NxmvHmnOdTkaPZDQA`YNeTUxMEINJ<K>L:<\\nP\\XFxZxoJ?enVsA^jo@^NHnMxi_PaA_b>NZ;\\BjZELm=L=@Y>HWyLTK<LwTNW@kB\\J<Dj:;?FH?YNiR:MrlkB[CDB[B<D:RrKU<=RTGxZE:B;@VkIRRXj:=NJZ;L<LNJJ;JBF:K:J<DRWAyyAJstJ`uRH_`iAZ:>Z:Fc?oc>oo<?f<3<\"\{\}
TENSORS OF RELATIVISTIC ELECTRODYNAMICS WITH MAPLE
Alexei V. Tikhonenko
General Physics Department,
State Technical University of Nuclear Power Engineering,
Obninsk, Russia
futureprint@obninsk.com
This work develops algorithms of construction, calculation and transformation of physical vectors and tensors in relativistic electrodynamics with MAPLE.
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">1. Lorentz transformations</Font></Text-field>
restart;
with(tensor):with(linalg):
Consider transformations of vectors and tensors with Lorentz transformations in four-dimensional pseudo-Euclidean space-time. We will use "tensor" and "linalg" packages for operations with vectors and tensors in four-dimensional space-time.
The first necessary step is creation of the metric tensor of four-dimensional pseudo-Euclidean space-time:
coord:=[t,x,y,z];
g11:=1: g22:=-1: g33:=-1: g44:=-1:
g:=create([-1,-1],array(1..4,1..4,symmetric,sparse,[(1,1)=g11(r,theta),(2,2)=g22(r,theta),(3,3)=g33(r,theta),(4,4)=g44(r,theta)]));
g_matrix:=matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]);
detg:=det(g_matrix);
`tensor/simp`:= proc(a) simplify(a); end:
g_contr := invert(g, 'det_g');
Lorentz transformations (for two reference frames) are determined as
dim:=4; Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];
CHANG:=[t=gamma*(t-beta*x),x=gamma*(x-beta*t) ,y=y,z=z];
Jacobian(Coord_2,CHANG,tr,TR);
op(tr); op(TR);
where
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,
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
and V is relative velocity of reference frames in the line of x-axis.
Components of covariant vector
VECT:=create([-1],array([V[1],V[2],V[3], V[4]]));
will transform with Lorentz transformations as
VECT_trans:=create([-1],array([V_trans[1], V_trans[2],V_trans[3],V_trans[4]]));
get_compts(VECT_trans)=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma));
get_compts(VECT_trans)[1]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[1];
get_compts(VECT_trans)[2]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[2];
get_compts(VECT_trans)[3]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[3];
get_compts(VECT_trans)[4]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[4];
Components of covariant tensor
TENS:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[2,1],T[2,2],T[2,3],T[2,4]],[T[3,1],T[3,2],T[3,3],T[3,4]],[T[4,1],T[4,2],T[4,3],T[4,4]]]));
will transform with Lorentz transformations as
TENS_trans:=create([-1,-1],array(1..4,1..4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma));
get_compts(TENS_trans)[1,1]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,1];
get_compts(TENS_trans)[1,2]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,2];
get_compts(TENS_trans)[2,4]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[2,4];
Components of the symmetrical covariant tensor
TENS_sym:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3], T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));
will transform with Lorentz transformations as
TENS_sym_trans:=create([-1,-1],array(1..4,1..4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform( TENS_sym,CHANG,tr,TR), [gamma,beta]));
or
TENS_sym_trans:=collect(simplify(subs (gamma =1/sqrt(1-beta^2),matrix([[(T[1,1]-2*beta*T [1,2]+ beta^2*T[2,2])*gamma^2, (beta^2*T[1,2]+(-T[1,1]-T[2,2])*beta+T[1,2])*gamma^2,(T[1,3]-beta*T[2,3])*gamma,(T[1,4]-beta*T[2,4])*gamma],[(beta^2* T[1,2]+(-T[1,1]-T[2,2])*beta +T[1,2])*gamma^2,(beta^2 *T[1,1]-2*beta*T[1,2] +T[2,2])*gamma^2,(-beta*T[1,3] +T[2,3])* gamma, -beta*T[1,4]+T[2,4]*gamma],[(T[1, 3]-beta* T[2,3])*gamma,(-beta*T[1,3] +T[2, 3])*gamma, T[3,3], T[3,4]], [(T[1,4]-beta*T[2,4])*gamma, (-beta*T[1,4]+T[2, 4])*gamma, T[3,4], T[4,4]]]))),beta) assuming (-beta^2+1)>0;
Components of the asymmetrical covariant tensor
TENS_as:=create([-1,-1],array(1..4,1..4 ,sparse ,[[0,T[1,2],T[1,3],T[1,4]],[-T[1, 2],0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0, T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));
will transform with Lorentz transformations as
TENS_as_trans:=create([-1,-1],array(1..4,1..4,sparse,[[0,T_trans[1,2],T_trans[1,3],T_trans[1,4]],[-T_trans[1,2],0,T_trans[2, 3],T_trans[2,4]],[-T_trans[1,3],-T_trans [2,3],0,T_trans[3,4]],[-T_trans[1,4],-T_trans[2,4],-T_trans[3,4],0]]));
get_compts(TENS_trans)=get_compts(simplify(collect(transform(TENS_as,CHANG,tr,TR),[gamma,beta])));
or
TENS_as_trans:=simplify(subs(gamma=1/sqrt (1-beta^2),matrix([[0, -T[1,2]*(-1+beta^2) *gamma^2,-(-T[1,3]+beta*T[2,3])*gamma,-(-T[1,4]+beta*T[2,4] )*gamma], [T[1,2]*(-1+ beta^2)*gamma^2, 0,-(beta* T[1,3]-T[2,3])* gamma, -(beta*T[1,4]-T[2,4])*gamma], [(-T[1,3]+beta*T[2,3])*gamma, (beta*T[1,3]-T[2,3] )*gamma, 0, T[3,4]], [(-T[1,4]+beta *T[2,4])*gamma, (beta*T[1,4]-T[2,4]) *gamma,-T[3,4], 0]]))) assuming (-beta^2+1)>0;
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
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8" family="Arial">2. 4-velocity \303\250 4-acceleration</Font></Text-field>
Consider 4-radius-vector of particle moving in four-dimensional space-time:
R(t):=create([-1],array([x[1](t),x[2](t),x[3] (t),x[4](t)]));
Consider 4-velocity of particle in four-dimensional space-time:
V(t):=create([-1],array([v[1](t),v[2](t), v[3](t),v[4](t)]));
V(t):=create([-1],array([1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[1](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[2](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*diff(x[3](t),t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/ c*diff(x[4](t),t)]));
V(t):=create([-1],array([1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*v[2](t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t) ^2)/c^2)/c*v[3](t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+v[4](t)^2)/c^2)/c*v[4](t)]));
V[1](t):=get_compts(V(t))[1];
V[2](t):=get_compts(V(t))[2];
V[3](t):=get_compts(V(t))[3];
V[4](t):=get_compts(V(t))[4];
Square of 4-velocity is
V_sqw=simplify(subs(beta=sqrt(v[2](t)^2+ v[3](t)^2 +v[4](t)^2)/c,prod(prod(V(t),V(t)) ,g_contr,[1,1],[2,2])));
Consider 4-acceleration of particle in four-dimensional space-time:
Accl[1](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2 +v[4](t)^2)/c^2)/c*diff(V[1](t),t);
Accl[2](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c*diff(V[2](t),t);
Accl[3](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c*diff(V[3](t),t);
Accl[4](t):=1/sqrt(1-(v[2](t)^2+v[3](t)^2+v [4](t)^2)/c^2)/c*diff(V[4](t),t);
Accl:=create([-1],array([Accl[1](t),Accl[2](t),Accl[3](t),Accl[4](t)]));
The scalar product of 4-velocity \303\250 4-acceleration is equal to
prod(prod(V(t),Accl),g_contr,[1,1],[2,2]);
This value means orthogonality of 4-velocity \303\250 4-acceleration.
Another form of 4-acceleration is:
Accl[1]:=1/((1-beta^2)^2)/(c^3)*(v[2] (t)*w[2](t) +v[3](t)*w[3](t)+v[4](t)* w[4](t));
Accl[2]:=1/(1-beta^2)^2/c^4*v[2](t)*(v[2](t) *w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[2](t);
Accl[3]:=1/(1-beta^2)^2/c^4*v[3](t)*(v[2](t) *w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[3](t);
Accl[4]:=1/(1-beta^2)^2/c^4*v[4](t)*(v[2](t)* w[2](t)+v[3](t)*w[3](t)+v[4](t)*w[4](t))+1/(1-beta^2)/c^2*w[4](t);
Accl:=create([-1],array([Accl[1],Accl[2], Accl[3],Accl[4]]));
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">3. energy-momentum vector</Font></Text-field>
Consider 4-velocity
V[1](t):=1/((1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2));
V[2](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[2](t);
V[3](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[3](t);
V[4](t):=1/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)^(1/2)/c*v[4](t);
V(t):=create([-1],array([V[1](t),V[2](t), V[3](t),V[4](t)]));
and define 4-energy-momentum vector as
P[1](t):=c*m*V[1](t);
P[2](t):=c*m*V[2](t);
P[3](t):=c*m*V[3](t);
P[4](t):=c*m*V[4](t);
P(t):=create([-1],array([P[1](t),P[2](t),P[3](t), P[4](t)]));
Square of 4-energy-momentum vector is:
P_sqw=prod(prod(P(t),P(t)),g_contr,[1,1], [2,2]);
As another form of 4-energy-momentum vector is
P_:=create([-1],array([E/c,p[2],p[3], p[4]]));
we have
prod(prod(P_,P_),g_contr,[1,1],[2,2])= prod(prod(P(t),P(t)),g_contr,[1,1],[2,2]);
4-energy-momentum vectorwill transform with Lorentz transformations as
P:=create([-1],array([E/c,p[2],p[3],p[4]]));
P_trans_:=create([-1],array([E_trans/c,p_trans[2], p_trans[3],p_trans[4]]));
P_trans:=collect(transform(P,CHANG,tr,TR),gamma);
or
get_compts(P_trans_)[1]=get_compts (P_trans)[1];
get_compts(P_trans_)[2]=get_compts (P_trans)[2];
get_compts(P_trans_)[3]=get_compts (P_trans)[3];
get_compts(P_trans_)[4]=get_compts (P_trans)[4];
Make sure that square of 4-energy-momentum vector is invariant relatively to Lorentz transformations:
P_sqw:=get_compts(prod(prod(P,P),g_contr, [1,1],[2,2]));
P_trans_sqw:=get_compts(prod(prod(P_trans,P_trans),g_contr,[1,1],[2,2]));
P_trans_sqw:=simplify(subs(gamma=1/sqrt (1-beta^2),(gamma^2*E^2+gamma^2*beta^2* p[2]^2*c^2-gamma^2*beta^2*E^2-gamma^2 *p[2]^2*c^2-p[3]^2*c^2-p[4]^2*c^2)/c^2));
and
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEsUF90cmFuc19zcXdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEmUF9zcXdGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=.
Differentiate 4-energy-momentum vector by interval:
P[1](t):=c*m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2):
P[2](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[2](t):
P[3](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[3](t):
P[4](t):=m/(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2) /c^2)^(1/2)*v[4](t):
Dif_P(t):=create([-1],array([1/sqrt(1-(v[2](t)^ 2+v[3](t)^2+v[4](t)^2)/c^2)/c *diff(P[1](t),t),1/sqrt(1-(v[2](t)^2+ v[3](t)^2+v[4](t)^2)/c^2)/c* diff(P[2] (t),t),1/sqrt(1-(v[2](t)^2+v[3](t) ^2+v[4](t)^2)/c^2 )/c*diff(P[3](t), t),1/sqrt(1-(v[2](t)^2+v[3](t)^2+ v[4](t)^2)/c^2)/c* diff(P[4](t),t)]));
or
DP[1]:=get_compts(Dif_P(t))[1];
DP[2]:=get_compts(Dif_P(t))[2];
DP[3]:=get_compts(Dif_P(t))[3];
DP[4]:=get_compts(Dif_P(t))[4];
These formulas can be rewritten as:
DP[1]:=m/(1-beta^2)^2/c^2*(v[2](t)*w[2] (t)+v[3] (t)*w[3](t)+v[4](t)*w[4](t));
DP[2]:=m/(1-beta^2)^2/c^3*v[2](t)*(v[2] (t)*w[2](t)+v[3](t)*w[3](t)+v[4](t)* w[4](t))+1/(1-beta^2)/c*w[2](t);
DP[3]:=m/(1-beta^2)^2/c^3*v[3](t)*(v[2](t)*w[2] (t)+v[3](t)*w[3](t)+v[4] (t)*w[4](t))+1/(1-beta^2)/c*w[3](t);
DP[4]:=m/(1-beta^2)^2/c^3*v[4](t)*(v[2](t)*w[2] (t)+v[3](t)*w[3](t)+v[4] (t)*w[4](t))+1/(1-beta^2)/c*w[4](t);
Therefore, we have that derivatives of components of 4-energy-momentum vector differ from accel-eration components by the factor m/c.
Now we denote derivatives of components of 4-energy-momentum vector by
f[1]:=m/(1-beta^2)^2/c^3*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4]);
f[2]:=m/(1-beta^2)^2/c^4*v[2]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2)/c^2* w[2];
f[3]:=m/(1-beta^2)^2/c^4*v[3]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2) /c^2*w[3];
f[4]:=m/(1-beta^2)^2/c^4*v[4]*(v[2]*w[2]+v[3]*w[3]+v[4]*w[4])+m/(1-beta^2 )/c^2*w[4];
whence
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
and 4-force vector is:
F:=create([-1],array([f[1],f[2],f[3],f[4]]));
or
F:=create([-1],array([(v[2]*f[2]+v[3]*f[3]+v[4]*f[4])/c,f[2],f[3],f[4]])):
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">4. Tensor of electromagnetic field</Font></Text-field>
Consider covariant 4-tensor of electromagnetic field:
T[1,2]:=E[1]; T[1,3]:=E[2]; T[1,4]:=E[3]; T[2,3]:=-H[3]; T[2,4]:=H[2]; T[3,4]:=-H[1];
F:=create([-1,-1],array(1..4,1..4,sparse,[[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2,3],T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));
Contravariant tensor of electromagnetic field is:
F_contr:=raise(g_contr,raise(g_contr,F,1),2);
Dual tensor to covariant tensor of electromagnetic field is:
Levi_Civita (detg,4,cov_LC,con_LC);
F_star:=dual(con_LC,F,[1,2]);
Dual tensor to contravariant tensor of electromagnetic field is:
F_star_cov:=lower(g,lower(g,F_star,1),2);
Calculate invariants of electromagnetic field:
INV[1]:=get_compts(prod(F,F_contr,[1,1], [2,2]));
INV[2]:=get_compts(prod(F,F_star,[1,1], [2,2]));
Components of tensor of electromagnetic field will transform with Lorentz transformations as
dim:=4: Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];
CHANG:=[t=gamma*(t-beta*x),x=gamma*(x-beta*t),y=y,z=z];
Jacobian(Coord_2,CHANG,tr,TR);op(tr):op(TR);
F_trans:=transform(F,CHANG,tr,TR);
or
matrix([[0,E_trans[1],E_trans[2],E_trans[3]],[-E_trans[1],0,-H_trans[3],H_trans[2]],[-E_trans[2],H[3],0,-H_trans[1]],[-E_trans[3],-H_trans[2],H_trans[1],0]])=simplify(subs(gamma=1/sqrt(1-beta^2),matrix([[0,-gamma^2*beta^2*E[1]+gamma^2*E[1],gamma*E[2]+gamma*beta*H[3],gamma*E[3]-gamma*beta*H[2]],[-gamma^2*E[1]+gamma^2*beta^2*E[1],0,-gamma*beta*E[2]-gamma*H[3],-gamma*beta*E[3]+gamma*H[2]],[-gamma*E[2] -gamma*beta*H[3],gamma*beta*E[2]+gamma*H [3],0,-H[1]],[-gamma*E[3]+gamma*beta*H[2], gamma*beta*E[3]-gamma*H[2],H[1],0]])));
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">5. Energy-momentum tensor of electromagnetic field</Font></Text-field>
Consider covariant 4-tensor of electromagnetic field:
T[1,2]:=E[1]; T[1,3]:=E[2]; T[1,4]:=E[3]; T[2,3]:=-H[3]; T[2,4]:=H[2]; T[3,4]:=-H[1];
F:=create([-1,-1],array(1..4,1..4,sparse, [[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2, 3],T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));
Mixed and contravariant 4-tensors of electromagnetic field are:
F_:=raise(g_contr,F,1);
F_contr:=raise(g_contr,raise(g_contr,F,1),2);
Calculate miscellaneous functions:
T1:=prod(F_contr,F_,[2,2]);
T2:=prod(F,F_contr,[1,1],[2,2]);
or
T2:=-2*E[1]^2-2*E[2]^2-2*E[3]^2+2*H[3]^2+2*H[2]^2+2*H[1]^2;
So contravariant energy-momentum tensor of electromagnetic field is
T[1,1]:=simplify(1/4/Pi*(-get_compts(T1) [1,1]+ 1/4*get_compts(g_contr)[1,1]*T2));
T[1,2]:=simplify(1/4/Pi*(-get_compts(T1) [1,2]+ 1/4*get_compts(g_contr)[1,2]*T2));
T[1,3]:=simplify(1/4/Pi*(-get_compts(T1) [1,3]+1/4*get_compts(g_contr)[1,3]*T2));
T[1,4]:=simplify(1/4/Pi*(-get_compts(T1) [1,4]+1/4*get_compts(g_contr)[1,4]*T2));
T[2,2]:=simplify(1/4/Pi*(-get_compts(T1) [2,2]+1/4*get_compts(g_contr)[2,2]*T2));
T[2,3]:=simplify(1/4/Pi*(-get_compts(T1) [2,3] +1/4*get_compts(g_contr)[2,3]*T2));
T[2,4]:=simplify(1/4/Pi*(-get_compts(T1) [2,4]+ 1/4*get_compts(g_contr)[2,4]*T2));
T[3,3]:=simplify(1/4/Pi*(-get_compts(T1) [3,3]+ 1/4*get_compts(g_contr)[3,3]*T2));
T[3,4]:=simplify(1/4/Pi*(-get_compts(T1) [3,4]+ 1/4*get_compts(g_contr)[3,4]*T2));
T[4,4]:=simplify(1/4/Pi*(-get_compts(T1) [4,4]+ 1/4*get_compts(g_contr)[4,4]*T2));
T_EP_contr:=create([1,1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3],T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));
or in standart denotes
w=1/8*(E[1]^2+E[2]^2+E[3]^2+H[3]^2+H[2]^2+H[1]^2)/Pi;
T[1,1]:=w;
S[1]=c*1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]);
T[1,2]:=S[1]/c;
S[2]=-1/4*(E[1]*H[3]-E[3]*H[1])/Pi;
T[1,3]:=S[2]/c;
S[3]=1/4*1/Pi*(E[1]*H[2]-E[2]*H[1]);
T[1,4]:=S[3]/c;
sigma[2,2]=-1/8*(E[1]^2-H[3]^2-H[2]^2-E[2]^2-E[3]^2+H[1]^2)/Pi;
T[2,2]:=sigma[2,2];
sigma[2,3]=-1/4*(E[1]*E[2]+H[2]*H[1])/Pi;
T[2,3]:=sigma[2,3];
sigma[2,4]=-1/4*(E[1]*E[3]+H[3]*H[1])/Pi;
T[2,4]:=sigma[2,4];
sigma[3,3]=1/8*(-E[2]^2+H[3]^2+H[1] ^2+E[1]^2+E[3]^2-H[2]^2)/Pi;
T[3,3]:=sigma[3,3];
sigma[3,4]=-1/4*(E[2]*E[3]+H[3]*H[2])/Pi;
T[3,4]:=sigma[3,4];
sigma[4,4]=1/8*(-E[3]^2+H[2]^2+H[1] ^2+E[1]^2 +E[2]^2-H[3]^2)/Pi;
T[4,4]:=sigma[4,4];
T_EP_contr:=create([1,1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[1,2],T[2,2],T[2,3],T[2,4]],[T[1,3],T[2,3],T[3,3],T[3,4]],[T[1,4],T[2,4],T[3,4],T[4,4]]]));
Covariant energy-momentum tensor of electromagnetic field is
T_EP:=lower(g,lower(g,T_EP_contr,1),2);
Components of energy-momentum tensor will transform with Lorentz transformations as
dim:=4: Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];
CHANG:=[t=gamma*(t-beta*x),x=gamma*(x-beta*t),y=y,z=z];
Jacobian(Coord_2,CHANG,tr,TR);op(tr):op(TR);
T_EP_trans:=collect(transform(T_EP,CHANG, tr, TR),[gamma,beta,c]);
or
T_EP_trans[1,1]=get_compts(T_EP_trans)[1,1];
T_EP_trans[1,2]=get_compts(T_EP_trans)[1,2];
T_EP_trans[1,3]=get_compts(T_EP_trans)[1,3];
T_EP_trans[1,4]=get_compts(T_EP_trans)[1,4];
T_EP_trans[2,2]=get_compts(T_EP_trans)[2,2];
T_EP_trans[2,3]=get_compts(T_EP_trans)[2,3];
T_EP_trans[2,4]=get_compts(T_EP_trans)[2,4];
T_EP_trans[3,3]=get_compts(T_EP_trans)[3,3];
T_EP_trans[3,4]=get_compts(T_EP_trans)[3,4];
T_EP_trans[4,4]=get_compts(T_EP_trans)[4,4];
So energy density (w), components of energy flux density vector (S) and components of Maxwell stress tensor (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJnNpZ21hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJpRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRictSSNtb0dGJDYtUSIsRidGNS8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y+LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRImpGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==) will transform as
w_trans=(sigma[2,2]*beta^2+2*S[1]/c*beta+ w)*gamma^2;
S_trans[1]=(-S[1]/c*beta^2+(-w-sigma[2,2]) *beta-S[1]/c)*gamma^2;
S_trans[2]=(-sigma[2,3]*beta-S[2]/c)*gamma;
S_trans[3]=(-sigma[2,4]*beta-S[3]/c)*gamma;
sigma_trans[2,2]=(w*beta^2+2*S[1]/c*beta+sigma[2,2])*gamma^2;
sigma_trans[2,3]=(S[2]/c*beta+sigma[2,3])*gamma;
sigma_trans[2,4]=(S[3]/c*beta+sigma[2,4])*gamma;
sigma_trans[3,3]=sigma[3,3];
sigma_trans[3,4]=sigma[3,4];
sigma_trans[4,4]=sigma[4,4];
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">6. Energy-momentum tensor of macroscopic particles</Font></Text-field>
Consider energy-momentum tensor of macroscopic particles:
w:=epsilon; S[1]:=0; S[2]:=0; S[3]:=0;
sigma[2,2]:=p; sigma[2,3]:=0; sigma[2,4]:=0; sigma[3,3]:=p; sigma[3,4]:=0; sigma[4,4]:=p;
Components of energy-momentum tensor of macroscopic particleswill transform with Lorentz transformations as
T_EP_trans[1,1]=get_compts(T_EP_trans)[1,1];
T_EP_trans[1,2]=get_compts(T_EP_trans)[1,2];
T_EP_trans[1,3]=get_compts(T_EP_trans)[1,3];
T_EP_trans[1,4]=get_compts(T_EP_trans)[1,4];
T_EP_trans[2,2]=get_compts(T_EP_trans)[2,2];
T_EP_trans[2,3]=get_compts(T_EP_trans)[2,3];
T_EP_trans[2,4]=get_compts(T_EP_trans)[2,4];
T_EP_trans[3,3]=get_compts(T_EP_trans)[3,3];
T_EP_trans[3,4]=get_compts(T_EP_trans)[3,4];
T_EP_trans[4,4]=get_compts(T_EP_trans)[4,4];
or in standart denotes
w_trans=(sigma[2,2]*beta^2+2*S[1]/c*beta+w)*gamma^2;
S_trans[2]=(-S[1]/c*beta^2+(-w-sigma[2,2]) *beta-S[1]/c)*gamma^2;
S_trans[3]=(-sigma[2,3]*beta-S[2]/c)*gamma;
S_trans[4]=(-sigma[2,4]*beta-S[3]/c)*gamma;
sigma_trans[2,2]=(w*beta^2+2*S[1]/c*beta+sigma[2,2])*gamma^2;
sigma_trans[2,3]=(S[2]/c*beta+sigma[2,3])*gamma;
sigma_trans[2,4]=(S[3]/c*beta+sigma[2,4])*gamma;
sigma_trans[3,3]=sigma[3,3];
sigma_trans[3,4]=sigma[3,4];
sigma_trans[4,4]=sigma[4,4];
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">7. Energy-momentum tensor of plane electromagnetic wave</Font></Text-field>
Consider plane electromagnetic wave with filed vectors:
H[1]:=0; E[1]:=0; H[2]:=-E[3]; H[3]:=E[2];
So energy-momentum tensor of plane electromagnetic wave is
T_EP_contr:=TABLE([compts=matrix([[1/8*(E[1]^2+E[2]^2+E[3]^2+H[3]^2+H[2]^2+H[1]^2)/Pi,1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]),-1/4*(E[1]*H[3]-E[3]*H[1])/Pi,1/4*1/Pi*(E[1]*H[2]-E[2]*H[1])],[1/4*1/Pi*(E[2]*H[3]-E[3]*H[2]),-1/8*(E[1]^2-H[3]^2-H[2]^2-E[2]^2-E[3]^2+H[1]^2)/Pi,-1/4*(E[1]*E[2]+H[2]*H[1])/Pi,-1/4*(E[1]*E[3]+H[3]*H[1])/Pi],[-1/4*(E[1]*H[3]-E[3]*H[1])/Pi,-1/4*(E[1]*E[2]+H[2]*H[1])/Pi,1/8*(-E[2]^2+H[3]^2+H[1]^2+E[1]^2+E[3]^2-H[2]^2)/Pi,-1/4*(E[2]*E[3]+H[3]*H[2])/Pi],[1/4*1/Pi*(E[1]*H[2]-E[2]*H[1]),-1/4*(E[1]*E[3]+H[3]*H[1])/Pi,-1/4*(E[2]*E[3]+H[3]*H[2])/Pi,1/8*(-E[3]^2+H[2]^2+H[1]^2+E[1]^2+E[2]^2-H[3]^2)/Pi]]), index_char=[1,1]]);
or
w:=1/4*(E[2]^2+E[3]^2)/Pi; S[2]:=c*w; S[3]:=0; S[4]:=0; sigma[2,2]:=w;
sigma[2,3]:=0; sigma[2,4]:=0; sigma[3,3]:=0; sigma[3,4]:=0; sigma[4,4]:=0; p:=0;
T_EP := TABLE([index_char = [-1, -1], compts = matrix([[w, -S[1]/c, -S[2]/c, -S[3]/c], [-S[1]/c, sigma[2,2], sigma[2,3], sigma[2,4]], [-S[2]/c, sigma[2,3], sigma[3,3], sigma[3,4]], [-S[3]/c, sigma[2,4], sigma[3,4], sigma[4,4]]])]);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">8. General Lorentz transformations</Font></Text-field>
Consider general Lorentz transformations.
restart;
with(tensor):with(linalg):
g11:=1: g22:=-1: g33:=-1: g44:=-1:
g:=create([-1,-1],array(1..4,1..4,symmetric,sparse,[(1,1)=g11(r,theta),(2,2)=g22(r,theta),(3,3)=g33(r,theta),(4,4)=g44(r,theta)]));
g_matrix:=matrix([[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1]]);
detg:=det(g_matrix);
`tensor/simp`:= proc(a) simplify(a); end:
g_contr := invert(g, 'det_g');
dim:=4:
Coord_1:=[t,x,y,z]; Coord_2:=[t,x,y,z];
gamma=1/sqrt(1-(beta[1]^2+beta[2]^2 +beta [3]^2));
CHANG:=[t=gamma*(t-(beta[1]*x+beta[2]*y+ beta[3]*z)),x=x+(gamma-1)*beta[1]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta [2]^2+beta[3]^2)+gamma*beta[1]*t,y=y+(gamma-1)*beta[2]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta[2]^2+beta[3]^2)+gamma*beta[2]*t,z=z+(gamma-1)*beta[3]*(beta[1]*x+beta[2]*y+beta[3]*z)/(beta[1]^2+beta[2]^2+beta[3]^2)+gamma*beta[3]*t];
Jacobian(Coord_2,CHANG,tr,TR);
op(tr): op(TR);
Components of covariant vector will transform with Lorentz transformations as
VECT:=create([-1],array([V[1],V[2],V[3], V[4]]));
VECT_trans:=create([-1],array([V_trans [1], V_trans[2],V_trans[3],V_trans[4]]));
get_compts(VECT_trans)=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma)):
get_compts(VECT_trans)[1]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[1];
get_compts(VECT_trans)[2]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[2];
get_compts(VECT_trans)[3]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[3];
get_compts(VECT_trans)[4]=get_compts(collect(transform(VECT,CHANG,tr,TR),gamma))[4];
Components of covariant tensor will transform with Lorentz transformations as
TENS:=create([-1,-1],array(1..4,1..4,sparse,[[T[1,1],T[1,2],T[1,3],T[1,4]],[T[2,1],T[2,2],T[2,3],T[2,4]],[T[3,1],T[3,2],T[3,3],T[3,4]],[T[4,1],T[4,2],T[4,3],T[4,4]]]));
TENS_trans:=create([-1,-1],array(1..4,1.. 4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma)):
get_compts(TENS_trans)[1,1]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,1];
get_compts(TENS_trans)[1,2]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[1,2];
get_compts(TENS_trans)[2,4]=get_compts(collect(transform(TENS,CHANG,tr,TR),gamma))[2,4];
For symmetrical tensor we have transformations:
TENS_sym:=create([-1,-1],array(1..4,1..4,sparse,[[Tau[1,1],Tau[1,2],Tau[1,3],Tau[1,4]],[Tau[1,2],Tau[2,2],Tau[2,3],Tau[2,4]],[Tau[1,3],Tau[2,3],Tau[3,3],Tau[3,4]],[Tau[1,4],Tau[2,4],Tau[3,4],Tau[4,4]]]));
TENS_sym_trans:=create([-1,-1],array(1..4,1..4,sparse,[[T_trans[1,1],T_trans[1,2],T_trans[1,3],T_trans[1,4]],[T_trans[2,1],T_trans[2,2],T_trans[2,3],T_trans[2,4]],[T_trans[3,1],T_trans[3,2],T_trans[3,3],T_trans[3,4]],[T_trans[4,1],T_trans[4,2],T_trans[4,3],T_trans[4,4]]]));
get_compts(TENS_trans)=get_compts(collect(transform(TENS_sym,CHANG,tr,TR),[gamma,beta])):
For asymmetrical tensor we have transformations:
TENS_as:=create([-1,-1],array(1..4, 1..4,sparse, [[0,Tau[1,2],Tau[1,3], Tau[1,4]],[-Tau[1,2],0,Tau [2,3],Tau[2, 4]],[-Tau[1,3],-Tau[2,3],0,Tau[3,4]],[-Tau[1,4],-Tau[2,4],-Tau[3,4],0]]));
TENS_as_trans:=create([-1,-1],array(1..4, 1..4, sparse,[[0,T_trans[1,2],T_trans [1,3],T_trans[1,4]],[-T_trans[1,2],0, T_trans[2,3], T_trans[2,4]],[-T_trans[1,3] ,-T_trans[2,3],0, T_trans[3,4]],[-T_trans [1,4],-T_trans[2,4],-T_trans[3,4],0]]));
get_compts(TENS_trans)=get_compts(simplify(collect(transform(TENS_as,CHANG,tr,TR), [gamma,beta]))):
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">9. Tensor of moment of momentum</Font></Text-field>
Define antisymmetric tensor of moment of momentum. We need to determine 4-radius-vector of particle and 4-energy-momentum vector:
R4(t):=create([-1],array([x[1],x[2],x[3], x[4]]));
gamma=1/sqrt(1-(beta[1]^2+beta[2]^2+beta[3]^2));
P4(t):=create([-1],array([m*c*gamma,m*gamma *v[2],m*gamma*v[3],m*gamma*v[4]]));
Define miscellaneous functions
MOM1:=prod(R4(t),P4(t)); MOM2:=prod(P4(t),R4(t));
and create covariant antisymmetric tensor of moment of momentum:
T[1,2]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,2],[gamma,m]);
T[1,3]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,3],[gamma,m]);
T[1,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[1,4],[gamma,m]);
T[2,3]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[2,3],[gamma,m]);
T[2,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[2,4],[gamma,m]);
T[3,4]:=collect(get_compts(MOM1)[1,2]-get_compts(MOM2)[3,4],[gamma,m]);
MOM:=create([-1,-1],array(1..4,1..4,sparse, [[0,T[1,2],T[1,3],T[1,4]],[-T[1,2],0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0,T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));
get_char(MOM);
get_rank(MOM);
Contravariant antisymmetric tensor of moment of momentum is:
MOM_contr:=raise(g_contr,raise(g_contr, MOM, 1),2);
get_char(MOM_contr);
get_rank(MOM_contr);
We can rewrite these relations in standart denotes:
R4(t):=create([-1],array([c*t,x,y,z]));
P4(t):=create([-1],array([En/c,P[2],P[3],P[4]]));
MOM1:=prod(R4(t),P4(t));
MOM2:=prod(P4(t),R4(t));
T[1,2]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,2],gamma);
T[1,3]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,3],gamma);
T[1,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[1,4],gamma);
T[2,3]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[2,3],gamma);
T[2,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[2,4],gamma);
T[3,4]:=collect(get_compts(MOM1)[1,2]-get_compts (MOM2)[3,4],gamma);
MOM_:=create([-1,-1],array(1..4,1..4, sparse ,[[0,T[1,2],T[1,3],T[1,4]],[-T[1,2] ,0,T[2,3], T[2,4]],[-T[1,3],-T[2,3],0, T[3,4]],[-T[1,4],-T[2,4],-T[3,4],0]]));
get_char(MOM_): get_rank(MOM_):
MOM_contr:=raise(g_contr,raise(g_contr, MOM_, 1),2);
get_char(MOM_contr): get_rank(MOM_contr):
Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.
MFNWtKUb<ob<R=MDLCdNVZZJ:@L>H:TKGxMkJ:<O`Lo\\lQxlQWdMWpsHqShmWhYoeXOPmTPmV`mvqyxq=Xj=xXquXaxnaXcEWc=UR=UweYwELKDLqtPq<R:=r^av^uRAurZ@nZtVauVb=WbMYtMyvayvYyuYYxmYxqyxqYyuYyEYsEYpmXpyyyyypqxp=J:>::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::dy<TypC>qULCTJcDXoXusT<aupkcfWMX@JCeU`dNuTmWxyyyppuPCDSSuLClu><xTpQmlsb]MihUO`qTeXSQO;@JxV]wOl:@syFv<w\\t@tsNnQn\\V?w<w\\?FqJijXynZVvnyHErmiB__tWit[MyxYRIIXvWgtSS=;gQMwAIC]IYrGXRogc[EpqYtsxn=BVSUGuEA[WxKrWaSHssoYBPkynKctqgmyUKAYQYUw_rs=wboYTWXI?IQKyo[X@wydqytYRGAy`ixs[SlyXaSyquy:mel=dXqydIfvgRIeSUkUmUBGwuZitS;eQ?S>AdMasnkySGbDSuimbSabjytNAyMuXlaTWaCp;y?at;_txaTwath?cj=GbgYVGCA[eAkh^ihyaIGoVdGxyWeQatamVHYx:SEIewyacmcSBAvgOyyssEyBVWCwQFtYWxYdMgcY_y^Uy?gce[WXQCDcwGuwHMw?qwx[gacscGrwOtuKFXKsc[FZIBOqIrII]kuICfRosM_yTSEWWcKQs_qGHeIiaWBsvaAXWoFsYTyuIYSdWCet[fZpOYtv[\\XSMvN=Xhluxel]ylvUn;PYsqvkmmCxSEQPsMOeUpQEKN`yVAqcqRQpYxHr[xU\\AtgPVexmHHQYDXptL;ey_\\XHxyTpLQ=qJhJklqA=wPxqOtpPmwQ=kWdSSYjxhQt=li<X=Pr\\HoxMKxppdUPGxl`<RadWsEMUhnMinaqvy\\t]pJw\\Pttt:lw_hy;PxuElWpfypiQyg<IbgHqQ?wRwvFgcQnmtI]lXZoauvw\\]Vi\\?yuIjGqyA_]j^cia\\^vaYfmXYvV_foyd_wZa?yIPfNXpOimbInwiieQyZ@[jf[p_`s?\\N@qaw[<a_=qpdIu]>gnHpUi\\^a[AGcS_y]pnHg_oIi=XkM`bK^yUWjFhhCpif?llhelhkKqk=qgCqqIokJadZ@]IOspHjgQgUv^Mp^[akXNokxcFaxMX>Efx=GJyY]=uKWXuefcYCV_DO;X]oeDwI]UrhIXhKdtYgv=sYMxyMhEAbdKdFED;MBimUYgvNsfBuDgqw^sRZoieyiYEfEAsYOcU;uf_C^;g>EIUmWy]xZ[H?UTiwhayb<EWUAhmghUee]ODLyfkYdOQDNMsleg]mHGkynUrrUhjgbvstrICsOiU?upUhtME_cVUeywWrSeSvIwHqsEUvwaS`mv_kCEgDEEVOoyfSFYGXh[xe;wfsya?Hbcu_SiHUfrStqsgICUKmR;IEGGiEUxSSewkBRcic?f[GHs]WBCeFSXMec@qwQYiOCFi;bd_epghCcrSIbrUFfKXpOE>CdGUVH_ss=GaEF\\Mh_uDJcXeWGSkIA=T`[uhOiKOy;Ido_sBQgPGbiMxZIx[=RNQHCUwlIhVAs>Mxv=t;Iekec[iToeB]YSVsI]UGkMgC=xM_cv]rCkGlOyE=wVsymoRPERGUWoKs>?dNGcqOvL=DcgUUid=SdBYtacBcyT;sC??sXsBFEIPKdwUibUUuowtCxLERxGUPOc=eeWWDJ_tBIFj[RMWXoaIniFDYyvIfFYH;EifaWAAdkQgSuIoYHS?s\\aYnkYcCRXAy;=urSsUEGXovmkdU?bIkuvIhf;hHKRmsIqkGkCIEGSQiUy?r[chy]DW?UJweo_HI;I[iRPuYCce]yIQGSR=SFcY@IHNabEyhT;H\\gC[iiEubXIY[?FhkfAaRyccQ;D<MBLksUGvM]FOSWZaFnmUVOB]Mh`gu]ew:CSX[VU[d^iWCITMkingVmcY;EuIkFZgetaSlkeD_SlUd?SU[Wh`_IHkuNaIBEY@KhQ[IbSfl_CpgV]IBgcf:CrOWWliVPSDMuEkwBYQbgKxGiWfcdg_cCoXDyFoAF<CYd_fZSUKOXmUErmvpWgaQIeWGyMiuOfheFY[UWgdGwe[;X@Yh<owskTwUgjYdvEhnTP`LJatUmyo]xlkUpgPSHmSOiSXtM?HsHhWglnu=ypMosmPWQtXmlLDR^erappAPq@Twu\\mf<ytMo_tNQDmwuUBal[TKM]UZ\\VsUPg\\OhXU]iw>lT>TtolYUeM\\`q:iNFQkMeuB<Y^yq[TqwLxyYk^mPDhUTEL[mxdYTrUwHYpp`R]tsyhm<\\rdhN\\]VGejEyTBLlXhUidSklVcImkuJA\\OFAJxXTJ\\oRpUr\\qnEUf<POaocioXxYUTRxhmKHnoUuBavvxt]@ordyqIl`tycEyg=St<V;LY`DoDElChWYdkpIkSMophnhqkeMW<QX^dogEmM<kxAYM=mpPKmTTMmXeQLnuK?HMeIU``TqMSdeNqmxHeLK=OUpx^@kiYp`xXVdoU@L=PprAPIuR[Qp@YlvPWwQToMpG`jOXyFhxAETieRADKgioVPOyXUlXT:Iwc<NgeMNup\\XWrdQFPQvlP=Toseo>qXbiWO\\yE=PUiPAASgLtxXLG=STASAxj=@WixwX`XOAtHloIeoHiLvyuouMtLtTyJsAxBXr@TqWXOsEKopuAEU<uyO\\LTyPAXm=tOUQneaND]KOYyLyXbtxuhmcYrXMkh\\ylLo_eq`tSeAOH]lqUwiPnkPwlHPgHrehY^pKhPwGPJ;<O<`qU=tMxUUEPW@RdITfYjjaowTqMQjXHJS\\M<EvappT@mWMJ@iOVhyLQKq]T=Eyc=UhqNa]PJ\\X\\Lu[DsQ@O[XRw<Rb`P`tSuejceYX@UN=rFexuHmDmk]XRLaYElRmIP]Pech`rxma?araaCxvWQ[\\aZ`yiFAj?gvVVd^@mGy[hhjxQvjIwMVwPGyXW_EpjDNnsy^EhvE_d:PnkOaDA^CnxEAoCh_ewc;pb[I[ZwcU?kpGwxvcVV\\OWaYGZWqbGG^jVkAQ]mXckfwTVfovZVnZLwfoIeS>e@HtcvsgPn<YqDOxcqbdNmPxtqwhsfag>myOedhqCFkNWqspy]@_VQrIIu]ncLIb>_xdQ^[yw^`^YqbSxeyga>OkV@fpVfeNhmxeSwn^?_GOklf`QqgK_yK?yj@pxvwbHtI`yYai?HvJ^wvQvYngAVo=XhwcReBIMflKTU_b`qrFQC<UGRWY=kVWAiv]X<CSyMycyweoE>?ttksVgBTmtGIXvKDT;D`atpaGQEVA=efoH@]TgswsCfWGEbCCLIYtSwG;tRaC?]hi[TfwSPUcSQYZCuloE[KTnOSTuDPqfpQU_Yx[?UZ=b`yCuETUectcrsaWIGhPUVdCXo[Dn;GTof=AVBcYRGgaaYbsvt=UBuVIOeZKgGmhHQr]]umsifyTPWtneyZKydmHjoWRAsSQHewDS=Hj]C>qdH[XHIgkwTGuvI_sgYDgabSsiLYrb]Ic[uZUuCeGN]InyyjiVnMuJibq]E>=sH[thQDXgT\\qhNwTVmGdoSiKsD]DD]UOksO=fX;XvIdbUwRiisCEv?tEAS?eH[EHiOy[mcE?hY;ewKCr[x;ECpUEaItRMUeMI@wF=GuqIdriXmAiHouB]UEkvboD`]bDeu^UHOsxwKSogVE_GNQbBAduMYQ;Y_]XbqBe[FFYGF=tXgxryYpAFDoidIRHgUf?uXGg]WguGig]URQrp;u=MHYIXxcIamsqEl<uR<PMwtwNMqNYMB?\\aIiqvboxhknwDOv]^r:a\\[WhExsn_cdQo@Ng]orLPnCptE?wJqi:ad`?gjX\\Bol:@dJis[vel^pK>]TpcIHhoSZoXJOhw[WgsesuBfEg]=uuUY=qXZWVYMSZECHWHqeX<Su^EuvYX;AFQQC]]Fl]SNqIO=ILQwhIwZoeqEoOqVY@TTprWANqYsuxNA@WjlpuaXytmXMRkdpI]K\\LT@=Pd\\SxHJSXNhulFYQmtwJhWI<QsuRUpwm\\rQDLyuMgMv>@pS@pftRiUniTV:uRRil<lRY<wltSViLhHKD@vViS`DOfaTvAsyMuKmQUhvqlQuLW@qlr`RddRKIm^QYAaXxdP\\TuVlktMYmyPA`xRivRUoLxKmANalL`qV`eTDIO;MY\\HoQiYnMkHLNqhylUJ\\tS^uKJIMKAY[qufMrxAXfxJyXxe`RPqxOiorlJW]XEHXw\\lJqr=XwN<T>`nFPklHv^LTd]kviu:YwlhWkTyDpLSUVUqQCAuTTliPopuoTHNSQyRts>IqKYKhTNQMseAjoalrQvbIslMp=\\ojLUMDuDQymaoiQulmPMELwhpuplnIvypP`XlCDM>LY@`rdqtoyn@MLFTUUPo\\UWR\\WMetOAoEewLIUctRw@t]ERG@XtqKuHQWqjWLqZ`LTUOTusmHPcYk?DN=uT\\aXSeLNuKrttf@kIunUTXCMtYyRUQplXw`Xv=iXppuLmRUqwTMm[]qxhLElt>lNi@qQ=Q_lRL<NgerhhXwAryAL=iw]IxYTUyhj;poqXPmUgHG\\ganfWfF>hrAwtwy[Ys<VuGXhSGxePjM^exn\\vabHNjTffFYwDNre@qoheHWmoW`]P\\gfq]Ikxx\\?vknnc\\giupovIhMaZOIkjIdVqtv?efnhe`i=OixVueVopxjJOuNY`[W\\jX\\SNkeqrQ_pUghjNiNQtpG\\CIe_IabYs@wwBw\\L`xO?r`qZi?c@WsW`^@fjogeppjkIpnXkKPndGadGidocE>m?Fjf_bYf\\\\?p]HieNqWggeIuCAnhiZwaepYnkgeFyjvOhu_[GQkpioSNa?ndiprUFjcV\\pQngw]R?]WFeWx`>i_H@tAwdbny<x__O`FyggqujAtJhaiAnSAs=xwtp^aYnloln?eYQtA^mJvwD?k\\Ql]xqMPc`_sjV]gvreOsIOkpP^Vy^[Vw`O[gwmLqi]NmZ@hBAriP]O>[@HdmYZyir[Nn<YpeNfonso^]dnfIYuXwkEAcUyn^A`]VeyYulPogAn;?\\K?mt^gp^jXGxf>ysfZsgu=`seb_aIESSJcWewtmCrECfgERaqENChB;f^IvxYL=PS]=yKXmGeMYLmrTSBpL_`UAlmXmXlUTXEn^EsSmmfyREXsDEwelvQqlQaX@@tj<pkTYkDSNqxPQjlusiTJELXQ\\Rw`sPaSUYJwPjdes_QsK`j@Ij_DuFmJmPLmllh<SSPKV<W[eOaaTN@wLltv=qd@OOHrc<K>huhPP=ApSURP]mbIVSurlDLqpKuaVliV>IoOxJxLyGXOhqt=QPBQVItRjdV?]PFPPCyvs]YB]RXAsPLysQT^MuLUODMueDP=UPpHsFUx:XJ`hNlEYKykqQLQHSEur^aX_XJH]UyxtgMRCXtjuo?EQWML[aRSikidoeLsUduWEMthYZyQ[qwxHT[tOu<VGxqb`qp<OQAWOeYIIw^Tv`HrNyP;EKhDLiTqcXLq<NXejsEKseT;MYA<osmuf@U@txUMJYaMFuvVajUelv]xX`ncuThTxB\\wxtvCiu@HsQUQ:msJyUVXLOeUALmdaY]TMouqEExW`xK=QQLyGAyiHP\\xOf]tG>cJw`gxw^f]mIdJwgXiybX]_^\\]x]wXoovfJ`vgQklWrhq`sxqThd_AuXHotauxqvVPs>fXQEG_YGyujGWqaCOyE>WX[wuEwysMHsACawYfsIiqvWiWpWGoGYmqwAeh;_XqGSy[YQUW<kFaUGmuhqeYE;xdwbDUDdWV<OYjmwc]rL?TpuwF_snWumiiaAInyB[aUbyx\\yy`cSLmHxsInwYLwf=ob_ktxgUJWTB]TtIvKkDDMICMVZCH<WWF;vXeuOGe^QeLwik]HkCfrUXu_DgoC[OIyuh_Iyb[eEhqryQ?MwTexIuNbumv<sOiwy]uO>ie?oNXpnFb]iykyv@pnM?^bQbcOp]@pM_wOIZ\\i]tVpGIu=PdbHfMxcxXat?aWPZsww>xaDvv<wqQvyk^piAr_@fdYyfoxsactW_uvgBPmqvmK_ZMArZWZyAvCPmuYd\\AbZp]ZNgXwryXaxva>wfYpcZgem>uxiu[GiYnuwQu<aiJns?\\UNpqHgjfwhq[bahb@xCGbHVkk_nTPeiobfycUf`XnaxidlwiTHjmheF?sw>qWXxTWygQbupZtYpgqpkwwfWvcHZcAw[iuMiyb^mEfyh_yyXsIIosXdJfxvq]>yaR_ZVxy\\bS?EbAws]w]wvcOFoMhwSURagyCYdiTwABuAEGWFuSIGoEkKYIGFYUY]uw`uwXoGuAFVWkGwqyfb@qrrifj?sYpu=@_]on=g[Q@ltQbQNZDf\\FWe\\yquw[<pu^>lvQx\\Yw<w\\<VxRPn=yxiN[CNgB^irOpwGnEfyyWntqw:gwEfZSpi_G\\<?`QnxV?wygm<NZ^qyaGpxxiMpk_OhqYrWx\\t@t?@vAA\\eq_rQqv>uy@tya`Wyy:xvmysXwyYf[MWxoWmIgvoE:;B:MTKWDKWgJ;eZ3:\"\{\}LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlKyEpRiotJS5CT1VORFNfSEVJR0hURzYjJCIkSSRGKi0lKUNISUxEUkVORzYi