experimental cosmic statistics; 1, variance

16
arXiv:astro-ph/9912236 10 Dec 1999

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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cent pound)dtt+(~curren]4middot+2()()+()Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

c O]13M

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup213Uacute

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8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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8 13McdO 132M`M

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Ccedil $$$sup2UacuteN2

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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Ccedil $$$sup213Uacute

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Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

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paraGmiddot)middotmiddotiexcl$N[GGAumldRidAEligGRR[]RdORzsup2microRsup2microLordm$G0 JG[GGG)2deg[[R[R[)ordfG[)ordmLR[GRzzmicroLRSbrvbarMacuteYsup2sup3GR paraGmiddot)middotmiddotiexclmacr7dd)L0poundLR[^iUGpound amp)GpoundqpoundL[deg)N[GGTRMbrvbarampdegdcurrenG))$ [GdT iexcl $$d00S)T[G2i)Gz]dbrvbarR)))GT]Ribrvbar$brvbar)]deg2brvbarTGRGpoundLTG)0brvbarWRR^[GRd^))r[currenGR[brvbarGi[GdR[Rziexcl8Wd^Rbrvbarcurren)[poundL[RbrvbarRdGRMJi)G$)JG[0poundLAgraveZ[curren)i)L$brvbar)NG)0 R7amp)2)[[poundLLR[TGiGdbrvbar )iRRGR]i)2R]R))Giexcl T GR[)[Ribrvbariexclamp)Rdcurren)TG7d[GGbrvbar$brvbardT[GdR[)iexcl iexclsup2sup3G GzRiexclparaGmiddotmiddot)middot) N$)]TG[GRbrvbar$GiRqTR))$)RNi)GzGG deg Aring 2poundLd~[T [)i[GR[)d)RiGiexclO 2GG)ordfpoundLNRNiordfsup2sup2ordmLsup2degpoundL[T[d[i)currenGplusmnbrvbar2)))Lbrvbari)dbrvbarRNGrR[) GGGbrvbariR[G[)RL[RdiGR)Ldampd2dordfG[deg7iexclcentqGRddamp [R [[G$)dGTi$NGcurren^iR[T[Ldamp[dL )G)7LpoundMdAacutepoundL7iexcl+centSR N)poundL)RdRdamp])curren)Gibrvbar2R[GGG[)R])dbrvbarRZRSiGT)RAringiexclordfmicroLR)TbrvbarGiexclparaGmiddot)middot T[[LTpoundLR[NgtLNi)GGNi R2ii)brvbarTGiR^sup2GG[iR[`acutemacrGiYparaGmiddotmiddot)sup1iexclfrac12$GGGMGcurren)[[+centEpoundLSrbrvbarRd)iRMJ))dbrvbarRgtGLd frac14sup2sectTmicroY[curren)G]ordfsup2sup3GR [dAacutedacute frac12NAgraveRJparaGmiddotmiddot)cedili dbrvbar0dG))J)))Gi [)i)[[G0])TR Rdiexcl ^)Rz NTpoundLR[ZiRddegbrvbar)di[GRbrvbarAring RG[0diexcl )iexclampd)[)7brvbar]))drG$$d[GRdplusmnGpound amp)Rdcurren)rpoundqGRi7brvbarRGid[iGpoundLAacute2RLGR[2R sup2microLordm$iexclmacr7dd)TGR[UddGraquodfrac34 R ^poundM)GR[G))dJiG[G]degdZmicro$ S)7iexclzAtilde$poundLcurren)RRG [)iGpoundMRAacuteT)[RRR$GGii)NRG)i)GRMiR[TR[NHamp)d)curreni~Aring[GGd^N[G GGLbrvbarpoundLG[~[JG)i)GTpoundL^sect$RRdTGR[R[R2ddiexclLcentLS[[ G)d)iRRii^~Aring[micro$ddAacuteAumlG)i)poundLRbrvbar2 [RTRampdGRamp)Rordfiexcl

aring`ecirc k6$ n 6 egravepccedilcentqmacriexclcentJfrac129[RdiG0 d1 f parah)diRdeg [brvbarRG[0 [GNbrvbarGR GNiLpoundM)$ [Rcurren)Gbrvbar1+iexclmacrRdbrvbar]Gi0Ri RLRzmacr7iexclfiexcl8NRAacutemacrLiexclordmLR[GRz+gtiexclmacrRdbrvbarmicrofrac14iexclmacr[AacuteTiexcl sup2Gi[ 0iexclzsup2sup3GR^Rsup2iexcl8WiRJMGiexcllt$Tamp G)[)AacuteRpoundLG) [Scurren)ampbrvbaroordfsup2ksup2micro91+2sect$R[drbrvbarrsup2microO +GGN)dpoundLAgrave R[TNTpoundLRAacutepoundL)2G)2 G iexcldordfsup2R]GpoundL[i AringR[Gbrvbar^++7Nmicro[RdddR[iR[)RGbJ[R)2rRUmicroL)i))degiexclcentqAtildedegbrvbarbrvbarcurrenRdi^S)9)`poundL)i^currenRddRbrvbarqbrvbarTLAcircNd[)2sup2) [[G)0 dmicroL[d r iampiRRpounddegpoundNpoundTiexcl iexcl )iexcl Aacute i [sup3G ziRcurren)d[iRcurren)d[iexcl 24iexclcentq$T)RpoundL)L R[iGR^$centf)N2microq0 dmicroq[[TR))+gtAacute`sup2Gd`sect$R[diexcl8^TpoundLddAacutei2Rcurren0

Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

RAacuteampiR^Q6)2NG[sup3G[ poundLcurren)d GYWbrvbardUplusmn [IGGr7iexclfrac14centqmacriexclNmicro )Aacute))omacriexclL[URZmicroL)21gt[[W GR ^[RGRdUamp R AringRcurrendRbrvbarRU[amp)Gi[hRR[Tsup2microURlsup2iexcl

eacute60-6eacute6 ecirc6ccedilgt)(~sup2)|N)2$13$I 5 9gt)(~sup2)|N)2$13$1 5sup29 aumlpoundI| Uuml)aMvIpoundI ) 4a|()auml)N$$Uacute 9BB$middot+Wauml middot)$$B 5sup29$ $middot+Wauml middot)$$ 5sup2 middot$ 13middot+Wauml middot)gt)01(2) mItauml+( -)2$$$()HtWta()middot+Wauml middot)2| Icircasup2)2$$ 5Uacute middot9middot+Wauml middot) 135Auml)gt)gt)())$$ 5sup2 middot+ sup2) |()aumlb)Icirc$13B t 13135Bmiddotauml|ltmiddot )gt)0w1(2) WItauml+(-)$$$THORN()HtWt()middotauml|ltmiddot )| IcircNsup2)-9A13()L)Icirc$$ t13$middotauml|middot )IacuteaaumlL sup2)middot-A()Uuml)N()| ) I|()gt)1auml|a ampN)$$THORNdtB

gt)01()middot+mauml middot) middotauml|E sup2)lt1 13auml()sup25)$$Uacute 13 9Bgt)01() middotauml|6middot sup2)1 auml()5)$$ dtsup2B sup2B 1Uacutegt)01(Icirc) middotauml| middot sup2)|N)Icirc$$ 5Uacute 9 THORNgt)01(Icirc) middotauml| middot sup2)|N)Icirc$$B 5Uacute 13gt)01(w) middotauml|13 sup2)Uacute| Icircsup2)sup2$$middotdt$ B gt)01(N)|gt))-A13()+)N()L)Icirc WItauml+(-)Icirc$$$()3tmta()

gt)01(Icirc) WItauml+( -) Wgt~9ltsup2 )2$$13Uacute9$ THORN 4a|()auml13)-9A13()Uuml)32)l|()Iacute-wUuml)$13 t 9B 3Esup2 )gt)sup2) 313 )2$$ 139 A13+psup2 )13gt)2$$$()3tmta()

gt)1ampN)gvIpoundI )2$$13Uacute9aMvIpoundI ) aumlpoundI|6 Uuml)$$Uacute 51d0()gt)sup2 )auml0Nampa)5auml )|Wmiddotasup2)$$THORNdt9B5143ltsup2 6) 13+ 6)$ 130taumlN()0aumlgt~()Uacuteauml()poundiexclt()gt)middot 1()gt)gt

19ltgt) middot)2 WItauml+(-)aumlpoundI| L)Uacute()3tmt1auml(5)gvIpoundI )2$$13tWt()middot4 t|Acirc$13$B 1sup2() ) gt 3) l|() - L)6$$$tmt()^t|9$$$THORN9313()middot)13 gt)$$ 13dtB$$ 99auml03()Wmiddot() sup2) middotauml| middot )gt)01(Icirc)$$THORNdtB5Icirc$9auml03()Wmiddot() sup2sup2) `()12apound- 1)20+03(2ampa) |+middot03(L) middotauml| middot sup2)$$dtBB$

|L) WQtauml+(A-5U Wgt~9lt )Icirc$$$ t13()tWt()0^sup2 )aumlaUuml )$$13dtB$$5()gt) )`()131poundsup2- 1sup2))$$BUacute9 53 5)9auml03()Wmiddot() Uacute) middotauml|middot )1()A )N$$ 13 tB 2gt~+8)auml|0 1sup213LampN)ampNmmiddotsup2)amp()|gt)0())

$$13m60wltANgt THORN13 13 13gt)gt)()w)$$$THORN13A1313It|lt)9()HtampNm43 sup2)-13++middot )2$$ Uacute 13$amp3amp-yenampN1gt)13ampa )2$13138|sup25pound mgt)aumlaumlmiddotAuml|13()A

ampAuml()()A() ltwamp2$1313tBB$9()0sup2)$$13THORN_ Acirc$$2$()0) gt)0w1(middot)lt ) ()0 ) 1()A )gt)middot)2$$ 1313 t13B$ 13

13()0Usup2) ()0 sup2)ugrave$$13THORN tWt()13 9-=8=$13 6-=5 =ampAumlauml1T=$13m=

mItauml+(-)$$13 tB$ mItauml+( -)gt)01(2) middot+WaumlE)2$$$I9auml10()+sup2sup2Uacutec= mItauml+(-)gt)01(2)middot+Wauml middot)$$$1()3tmt() mItauml+( -)gt)0w1(2)93()Auml)gt)12apound)2$$$auml10()+LtIt2 -- mItauml+(-)auml()8)+gt)53)$$$+ tAcircB mItauml+(-)gt)$$$()HtWt() mItauml+(-)gt)01(2)2$$ dtBAcirc middot mItauml+(-)_( 313()sup2)()|gt2)2$$ dtB mItauml+(-)2 Wgt~9lt9sup2 ) 5 middot|ampa)$$ t$9 mItauml+(-) Wgt~9ltsup2 )2$$I9dtB1313B p=T13 mItauml+(-) Wgt~9ltsup2 )2$$19dtBB13B$ mItauml+(-) Wgt~9ltsup2 )2$$ tB135b|()sup2 - L)2$9$Uacute 13 Bb|()` - L)$$ N()C0gt)Qpoundlt+m5Oapound Wgt)paumlauml+Nsup2Iacute|Icircmiddot()gt 32NLIacuteTt() 5 ()T9auml()ugrave4-+|13 gt)A()

b|()` - L) |()auml)93 )$$THORNIcircUacute9 gt$ +A()| )2$9 5Uacute 13B

aring136 amp aring1m60 G egrave ccedilugravearing cegravearingegrave ccedilcentqG [RbrvbarRbrvbarddOdbrvbarR9GR ordfmicroo+d$macrd1$Rcurren)ampamp [RbrvbarRbrvbarddRJAEligdo RbrvbarG0dG[d$Jcurren)Rdi p0[poundq^L[R9d^TGRiexclmacr)GRiiGz A R[TAEligGLRdpoundN A EcAa A CE Aabab^TparaI ab^-I K0paraIC a A DGI2paraIpoundq[0RdddGRa A 2AEligGdr0AElig[S R[RTamp)RziexclLcentL A R[ [ R[2iiRJ2[)drordfAElig2i)Gcurren)[iGdNcurren)RdilpMr A A Aamp2 A C3ordfsup2sup2)0diR[idsup3G)Aamp2C raquopara poundL[[ $Tcurren)[)2GRLd^iGdr EcAaCd( _ `

microqdordfGd^R[[GU]amp)RG2))0Rd[GGAgravek)TGRGGGO3amp d0R[RrRG )iexcl iexclf+G[brvbarGiparamiddot)sup1amp)iexclcentL0curren[)Glamp RdJGR[)R^R`RcurrenJiGdz$curren)brvbar E parap =

gt Ocirc $ gt Ocirc $ $ G ___ G $ _ iexclf

centqNGGGamp])iRAringNiG]iAElig AC C poundLE 23^frac34paraIiexclzcentLGGGGrii2RTRNRz2sect$RRAElig[R[LGdsup3G[$ 1fiexcl deg R Aring[iRdsup3G[TTR[TAEligG E ^ G =poundqR[RW)) E AElig iexclordmL^AEligRz7 E AElig Eraquopara)L$GRR[ L iexclcentqiexclamp)R Ocirc R agrave [$GRdAacuteGpoundLGNRAacute[0dramp[R GRNd[[RRR6d]dfrac34~[[idR[ Rddeg)ampdrGGiexclcentq$[GRdegR[]Rdsup3GRd^ampiexcl0N))0Riexclzcentq

Ccedil $$$sup213Uacute

gt 13O G)dbrvbarJ0poundMGRAacuteRT [GT$2$G[G[JR[curren))iexclg rampRRddG[GiRbrvbar0ddampJG) GGiGLd7 )iexcl iexclx+brvbar0paraGmiddotsup1)degRdwrpoundLRAacuteUddGR[GR]k Aring[brvbar)EGR[frac34 [G0 [)i)GRdd^ AringdR`2ddegGR` G$)iexcl iexclFGRsup3AacuteGpoundLGsup3)zordmLRGfrac14acuteOmicroq)TbrvbarNparamiddot)middotfMordmLR[Gparamiddot)middotiexclcentLTR dordfGd2)[dG73dE f G DTpoundLG38rparaGmiddot RmiddotordmLdRkacute sup2m]paraGmiddot)sup1middot)zsup2sup23dG) ^ bpara^ paralt^ G JcedilpoundL[[ E Ti) GRLTbrvbarNRbrvbarGSdRkRcurren[Gdiexcl iexclordmLdRacuteYsup2m]paraGmiddot)sup1middot)R a d ^TparaI

^ ^ _ Jsup1 2ipoundLR[`dr)8WparaGmiddot RmiddotMordmLdRiquestacute sup2Agrave)WparaGmiddotsup1)middotMzsup2sup23d( 9para^ G JmiddotdcurrenR0 [RbrvbarRbrvbarddiG) [GGTd][iRcurren)[)LRiexclcentL2iG[iTRo$T$ [d[GddGL [brvbarGLd[GGJGR [RbrvbarRbrvbardd[[)d0Gr a dc^ampd$ i _ 0parah)

centLG) 2)[dRUi)RRfrac14iRiG3d A

$ AAI 0paraparaI

$d[GG[[)G3 ordfsup2sup2N[3d(35Kiquestpara _ 0para `TmicroL)TbrvbarddampampGiexclMJcedilJsup17R0para `TR$GRbrvbarRd$[RbrvbarAringGpoundLSRNiGR7])iordfsup2sup2 ^ para

^ A I a ^I ^ A A A _ 0paraf

centLampJR[$ [)GRN[Gd AringGiJiG)d2microg0+macr`poundq^curren)[0 ddbordfsup2GGGiexclf)0 dGRii[9ampiexclO0paraIiAEligdGdSRT[ iRbrvbarAringRcurren0[G[Rkampiexcl0parafiexclcentL$ [[Td2dG$LGampLG) dJ[degR[GG7iexclcentL R )7brvbarG[i [GGidTNRRJ[R)0Rsup2Giordm)AacutepoundLdqsup2[G 7 cent ordf)SAElig)iexcl

Ccedil $$$sup2UacuteN2

8 13McdO 132M`M

RAacuteampiR^Q6)2NG[sup3G[ poundLcurren)d GYWbrvbardUplusmn [IGGr7iexclfrac14centqmacriexclNmicro )Aacute))omacriexclL[URZmicroL)21gt[[W GR ^[RGRdUamp R AringRcurrendRbrvbarRU[amp)Gi[hRR[Tsup2microURlsup2iexcl

eacute60-6eacute6 ecirc6ccedilgt)(~sup2)|N)2$13$I 5 9gt)(~sup2)|N)2$13$1 5sup29 aumlpoundI| Uuml)aMvIpoundI ) 4a|()auml)N$$Uacute 9BB$middot+Wauml middot)$$B 5sup29$ $middot+Wauml middot)$$ 5sup2 middot$ 13middot+Wauml middot)gt)01(2) mItauml+( -)2$$$()HtWta()middot+Wauml middot)2| Icircasup2)2$$ 5Uacute middot9middot+Wauml middot) 135Auml)gt)gt)())$$ 5sup2 middot+ sup2) |()aumlb)Icirc$13B t 13135Bmiddotauml|ltmiddot )gt)0w1(2) WItauml+(-)$$$THORN()HtWt()middotauml|ltmiddot )| IcircNsup2)-9A13()L)Icirc$$ t13$middotauml|middot )IacuteaaumlL sup2)middot-A()Uuml)N()| ) I|()gt)1auml|a ampN)$$THORNdtB

gt)01()middot+mauml middot) middotauml|E sup2)lt1 13auml()sup25)$$Uacute 13 9Bgt)01() middotauml|6middot sup2)1 auml()5)$$ dtsup2B sup2B 1Uacutegt)01(Icirc) middotauml| middot sup2)|N)Icirc$$ 5Uacute 9 THORNgt)01(Icirc) middotauml| middot sup2)|N)Icirc$$B 5Uacute 13gt)01(w) middotauml|13 sup2)Uacute| Icircsup2)sup2$$middotdt$ B gt)01(N)|gt))-A13()+)N()L)Icirc WItauml+(-)Icirc$$$()3tmta()

gt)01(Icirc) WItauml+( -) Wgt~9ltsup2 )2$$13Uacute9$ THORN 4a|()auml13)-9A13()Uuml)32)l|()Iacute-wUuml)$13 t 9B 3Esup2 )gt)sup2) 313 )2$$ 139 A13+psup2 )13gt)2$$$()3tmta()

gt)1ampN)gvIpoundI )2$$13Uacute9aMvIpoundI ) aumlpoundI|6 Uuml)$$Uacute 51d0()gt)sup2 )auml0Nampa)5auml )|Wmiddotasup2)$$THORNdt9B5143ltsup2 6) 13+ 6)$ 130taumlN()0aumlgt~()Uacuteauml()poundiexclt()gt)middot 1()gt)gt

19ltgt) middot)2 WItauml+(-)aumlpoundI| L)Uacute()3tmt1auml(5)gvIpoundI )2$$13tWt()middot4 t|Acirc$13$B 1sup2() ) gt 3) l|() - L)6$$$tmt()^t|9$$$THORN9313()middot)13 gt)$$ 13dtB$$ 99auml03()Wmiddot() sup2) middotauml| middot )gt)01(Icirc)$$THORNdtB5Icirc$9auml03()Wmiddot() sup2sup2) `()12apound- 1)20+03(2ampa) |+middot03(L) middotauml| middot sup2)$$dtBB$

|L) WQtauml+(A-5U Wgt~9lt )Icirc$$$ t13()tWt()0^sup2 )aumlaUuml )$$13dtB$$5()gt) )`()131poundsup2- 1sup2))$$BUacute9 53 5)9auml03()Wmiddot() Uacute) middotauml|middot )1()A )N$$ 13 tB 2gt~+8)auml|0 1sup213LampN)ampNmmiddotsup2)amp()|gt)0())

$$13m60wltANgt THORN13 13 13gt)gt)()w)$$$THORN13A1313It|lt)9()HtampNm43 sup2)-13++middot )2$$ Uacute 13$amp3amp-yenampN1gt)13ampa )2$13138|sup25pound mgt)aumlaumlmiddotAuml|13()A

ampAuml()()A() ltwamp2$1313tBB$9()0sup2)$$13THORN_ Acirc$$2$()0) gt)0w1(middot)lt ) ()0 ) 1()A )gt)middot)2$$ 1313 t13B$ 13

13()0Usup2) ()0 sup2)ugrave$$13THORN tWt()13 9-=8=$13 6-=5 =ampAumlauml1T=$13m=

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b|()` - L) |()auml)93 )$$THORNIcircUacute9 gt$ +A()| )2$9 5Uacute 13B

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gt Ocirc $ gt Ocirc $ $ G ___ G $ _ iexclf

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Ccedil $$$sup213Uacute

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centLG) 2)[dRUi)RRfrac14iRiG3d A

$ AAI 0paraparaI

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^ A I a ^I ^ A A A _ 0paraf

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Ccedil $$$sup2UacuteN2

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centLG) 2)[dRUi)RRfrac14iRiG3d A

$ AAI 0paraparaI

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^ A I a ^I ^ A A A _ 0paraf

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Ccedil $$$sup2UacuteN2