Lecture 6 (L. Younes)

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    DNLS^_N \N_@N\

    Dnlturn 6= @itrjcult`ji %CK)

    Dnlturn ?= Lfilnr O`jdjky %GJ)

    Dnlturn = Gfss`vn Snst`ik %DP)

    Dnlturn 8= O`jgfrbnr C`sljvnry %DP)

    Dnlturn ;= Qenijtypn Qrnc`lt`ji %CK)

    Dnlturn 5= Ngoncc`ik Gnlefi`sg %CK)

    6 * 81

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    J^SD@IN

    Cntnlt`ik C`hhnrnit`fddy Nxprnssnc Knins

    K[F\

    Nir`legnit Fifdys`s

    ? * 81

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    C@HHN_NIS@FD NUQ_N\\@JI

    G`lrjfrrfy nxpnr`gnits frn kninrfddy lfrr`nc jut tj gnfsurn ten

    c`hhnrniln `i nxprnss`ji ontwnni twj ljic`t`jis Seny gnfsurn ten ljilnitrft`ji jh g_IF gjdnludns jh tfrknt

    typns `i sfgpdnc t`ssuns

    Sen iugonr jh tfrknts lfi on eukn %tnis jh tejusfics)$ we`dn ten

    iugonr jh sfgpdns pnr stucy `s `i ten euicrncs Hjrgfddy$ nfle g`lrjfrrfy nxpnr`gnit prjv`cns f hfg`dy jh

    gnfsurngnits x%6)$ ) ) ) $ x%i)$ w`te x%b) 7 %x%b)6 $ ) ) ) $ x

    %b)G )

    i `s ten iugonr jh sfgpdns$ G `s ten iugonr jh tfrkntnc g_IFs$

    w`te i G

    Sen ljic`t`ji$ jr penijtypn$ `s rnprnsnitnc oy f o`ifry rv$ w`te

    josnrvnc vfduns %y%6)$ ) ) ) $ y%i))

    Sen josnrvft`jis tenrnhjrn ljrrnspjic tj fi i sfgpdn jh P fic

    U 7 %U6$ ) ) ) $UG)

    < * 81

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    INNC HJ_ IJ_GFD@]FS@JI

    Hjr nfle vfr`fodn m$ jin lfi usn f tnst %t'tnst$ rfib sug$ ) tjnvfduftn ten c`hhnrniln ontwnni ten josnrvnc vfduns jh Um `i ten

    twj pjpudft`jis

    Jin lfi teni usn gudt`pdn tnst`ik ljrrnlt`ji tj cntnlt we`le

    knins frn s`ki`stlfitdy c`hhnrnit`fddy nxprnssnc

    Jin jh ten c`hstludt`ns$ ejwnvnr$ fr`sns hrjg ten strjik

    entnrjknin`ty jh ten cftf

    Gnfsurngnit tnleijdjky jhtni rnsudts `i dfrkn vfr`ft`jis flrjss

    sfgpdns$ fs `h nfle jin wnrn gnfsurnc w`te f c`hhnrnit snt jh

    ui`ts

    Se`s `gpd`ns teft stficfrc tnst prjlncurns wjudc efvn vnry djw

    pjwnr w`tejut sjgn prnprjlnss`ik

    3 * 81

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    KF^\\@FI IJ_GFD@]FS@JI

    \`gpdn ljrrnlt`ji gntejcs rngjvn ten gnfi fic c`v`cn oy

    stficfrc cnv`ft`ji Ijtn teft gnfis fic vfr`filns frn nst`gftnc flrjss vfr`fodns hjr

    f stxnc sfgpdn Xnry jhtni$ f gjrn str`iknit ijid`infr ijrgfd`zft`ji gntejc `s

    usnc$ `i we`le `t `s nisurnc teft ten e`stjkrfg jh ten josnrvnc

    nxprnss`jis hjr nfle sfgpdn efs f prncntnrg`inc sefpn Se`s `s lfddnc e`stjkrfg ijrgfd`zft`ji @h `s ten tfrknt lch %typ`lfddy f stficfrc Kfuss`fi)$ ten

    trfishjrgft`ji `s nqu`vfdnit tj rnpdfl`ik x%b)

    m oy

    6 H%b)%x%b)m )

    wenrn H%b) `s ten ngp`r`lfd lch jh ten hfg`dy jh josnrvnc

    gnfsurngnits hjr ten bte frrfy Se`s `s nqu`vfdnit tj jidy rntf`i`ik rfib `ihjrgft`jiw`te`i nfle

    frrfy$ `n tj us`ik rndft`vn nxprnss`jisfgjik knins> * 81

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    V^FIS@DN IJ_GFD@]FS@JI

    @istnfc jh rnijrgfd`z`ik fdd cftf tj f c`str`out`ji teft wfs sndnltnc

    `i fcvfiln$ d`bn f Kfuss`fi$ jin lfi nst`gftn `t hrjg cftf Hjr b 7 6$ ) ) ) $ i$ cnijtn fs onhjrn oy x

    %b)m ten bte gnfsurngnit

    hjr sfgpdn m$ m 7 6$ ) ) ) $G

    Dnt x%b)%m) $m 7 6$ ) ) ) $G$ rnprnsnit tensn gnfsurngnits `i

    `ilrnfs`ik jrcnr Dnt x%m) cnijtn ten fvnrfkn jh tensn vfduns jvnr sfgpdns$ b

    Ijrgfd`zn ten sfgpdns oy rnpdfl`ik x%b)

    m oy x%r%b)m

    )$ wenrn r

    %b)m `s

    ten rfib jh ten mte vfr`fodn `i ten bte sfgpdn Se`s `s nqu`vfdnit tj fppdy`ik ten prnv`jus ijrgfd`zft`ji w`te

    tfrknt lch cnstinc oy

    H%x) 7G

    m76

    m

    G6Rxx%m)W)

    8 * 81

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    J^SD@IN

    Cntnlt`ik C`hhnrnit`fddy Nxprnssnc Knins

    K[F\

    Nir`legnit Fifdys`s

    ; * 81

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    K[F\= @IS_JC^LS@JI

    Swj rficjgdy lejsni `ic`v`cufds sefrn frjuic 000. jh ten`rknijgns

    Se`s cjns ijt prnvnit gfmjr c`vnrs`ty fgjik pnjpdn$ fssjl`ftnc tj

    \gfdd vfr`ft`jis %gutft`jis) `i ten `ic`v`cufd iuldnjt`cns$ bijwi fss`ikdn iuldnjt`cn pjdygjrpe`sgs %\IQs)

    @isnrt`jis$ cndnt`jis fic trfisdjlft`jis jh lerjgjsjgfd snkgnits Ljpy iugonr vfr`ft`jis

    Senrn frn up tj < g`dd`jis bijwi \IQs$ jh we`le

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    K[F\

    Knijgn'w`cn fssjl`ft`ji stuc`ns f`g ft `cnit`hy`ik spnl`stl

    vfr`fits %gf`idy \IQs) fssjl`ftnc w`te ljgpdnx eugfi

    c`snfsns fic qufit`tft`vn penijtyp`l cftf Sen fssjl`ftnc \IQs frn teni ljis`cnrnc tj gfrb f rnk`ji jh ten

    eugfi knijgn we`le lfi teni on hurtenr `ivnst`kftnc tj p`ipj`it

    lfusfd gnlefi`sgs hjr `ilrnfsnc jr cnlrnfsnc r`sb jr jtenr trf`ts

    0 * 81

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    LCLX EPQJSEN\@\

    Gfiy stuc`ns frn cns`kinc frjuic ten ljggji'c`snfsnljggji'vfr`fit eypjtens`s

    _jukedy$ f vfr`fit `s lfddnc ljggji `h ten g`ijr fddndn hrnqunily

    `s fojvn 6.

    Sen ejpn `s teft ljggji c`snfsns jr trf`ts lfi on prnc`ltnc oyljggji vfr`fits

    Fdtnrift`vn eypjtensns _frn vfr`fits efvn fi fcc`t`vn nhhnlt teft ljitr`outn tj ten

    ngnrkniln jh ljgpdnx c`snfsns$ lrnft`ik knint`l ejgjknin`ty

    fgjik suomnlts Gjrn ljgpdnx$ iji'd`infr$ knin `itnrflt`jis %jr np`stfs`s) frn fdsj

    pjss`odn

    61 * 81

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    Fddndn hrnqunily

    Nhhnlt s`zn

    Xnry rfrn Ljggji

    Djw

    E`ke

    _frn Djw hrnqunily

    1)116 1)11> 1)1>

    @itnrgnc`ftn

    Gjcnst

    _frn fddndns

    lfus`ikGnicnd`fi

    c`snfsn

    Hnw nxfgpdns jh

    e`ke'nhhnlt

    ljggji vfr`fits

    `iunil`ik

    ljggji c`snfsn

    Ljggjivfr`fits

    `gpd`lftnc `i

    ljggji c`snfsn

    oy K[F

    _frn vfr`fits jhsgfdd nhhnlt

    vnry efrc tj `cnit`hy

    oy knint`l gnfis

    Djw'hrnqunily

    vfr`fits w`te

    `itnrgnc`ftn nhhnlt

    6)6

    >1)1

    %Hrjg Gfijd`j nt fd$ Ifturn$ ?110) Hnfs`o`d`ty jh `cnit`hy`ik knint`l vfr`fits oy r`sb fddndn hrnqunily fic strnikte jh knint`l nhhnlt

    %jccs rft`j) Gjst nhhjrt efs onni w`te`i ten c`fkjifd cjttnc d`ins

    66 * 81

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    D@G@SFS@JI\

    Gjst \IQs cntnltnc sj hfr `i K[F\ stuc`ns efvn f vnry sgfdd

    nhhnlt s`zn cnstinc oy

    Qrjo%trf`t | gutftnc)

    Qrjo%iji trf`t | gutftnc)*

    Qrjo%trf`t | ljggji)

    Qrjo%iji trf`t | ljggji)

    hjr c`slrntn penijtypns

    Gjst jh ten jccs rft`js frn `i ten rfikn 66 tj 6>$ w`te ten gnc`fi

    jccs rft`j `s 6

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    Enr`tfo`d`ty$ tjtfd

    Enr`

    tfo`d`ty$

    oyljgpjini

    t

    1 1)> 6

    1

    1)>

    6

    I

    ji

    enr`

    tfodn H

    fs

    t`ik

    Kdul)

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    t`sg

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    En

    `ket

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    ni

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    Sen enr`tfo`d`ty spnltrug jh c`snfsn Enr`tfo`d`ty `s ten hrflt`ji jh f penijtypn fttr`outfodn tj knins fic lfi on nst`gftnc hrjg

    tw`i stuc`ns w`tejut bijw`ik we`le knins frn `ivjdvnc Sen tjtfd en`ket jh nfle ofr `ic`lftns ten enr`tfo`d`ty hrjg tw`i stuc`ns

    Sen rnc rnk`ji `ic`lftns ten knint`l hfltjrs inwdy `cnit`stnc oy ten gjst rnlnit K[F\2 ten odun rnk`ji ten knint`l hfltjrs pr`jr tj

    ten gjst rnlnit K[F\2 fic ten krny rnk`ji ten knint`l hfltjrs teft rngf`i uibijwi _nlnit K[F\ efvn fccnc trngnicjusdy tj

    jur bijwdnckn jh ten knint`l ofs`s jh c`snfsn$ out efvn onni dfrkndy d`g`tnc tj ljggji vfr`fits

    6< * 81

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    CFSF \NS\ %@)

    Nfle `ic`v`cufd `s knijtypnc hjr f snt jh knint`l gfrbnrs ofsncji f sfgpdn jh lndds

    Sen CIF `s nxtrfltnc hrjg tensn lndds fic gnfsurnc ji

    knijtyp`ik g`lrjfrrfys

    @i ten lfsn jh \IQ le`ps hjr ljggji vfr`fits$ ten fssfys tjcfy`ivjdvn ji ten jrcnr jh f g`dd`ji \IQs slfttnrnc jvnr fdd ten

    lerjgjsjgns$ gjst `cnit`stnc oy ten @itnrift`jifd EfpGfp

    Qrjmnlt

    Inwnr tnleijdjk`ns lfi pnrhjrg wejdn'nxjgn jr nvni

    wejdn'knijgn snqunil`ik$ fic gfy rnpdfln frrfy'ofsnc fssfysfic gutn ten `gpjrtfiln jh ten c`st`ilt`ji ontwnni rfrn fic

    ljggji vfr`fits

    63 * 81

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    CFSF \NS\ %@@)

    Nfle \IQ gfrbnr ljrrnspjics tj ten stftn jh ten twj fddndns ft f

    pfrt`ludfr djlft`ji %djlus)

    \IQs frn kninrfddy ljcnc ji tnrifry vfduns {1$6$ ?}

    ljrrnspjic`ik tj fddndns FF$ FO$ OO Jin lfi fdsj usn f o`ifry rnprnsnitft`ji$ us`ik 6 hjr ojte FO fic

    OO

    Hjr nfle \IQ$ f gnfsurngnit `s jotf`inc hjr nfle jh ten twj

    pjss`odn fddndns Hjr f k`vni cftf snt jh `ic`v`cufds$ te`s lfi on

    v`sufd`znc fs f slfttnr pdjt jh `itnis`ty vfduns

    6> * 81

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    8111FFFKKK

    FFKFKK

    1

    6111

    ?111

    111

    1 6111 ?111 111

    1611?1111811;11511011

    FddndnO`itnis`ty

    FddndnO`itnis`ty

    Fddndn F `itnis`ty

    1 6111 ?111

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    O@F\

    Sen dfrknst sjurln jh o`fs `i lfsn'ljitrjd stuc`ns `s cun tj

    pjpudft`ji strft`stlft`ji$ `n$ suopjpudft`jis w`te`i jin jr ojte

    ljejrts

    Hfdsn pjs`t`vns lfi fr`sn onlfusn ten lfsns fic ljitrjds frn ijt

    gftlenc ljrrnltdy w`te systngft`l c`hhnrnilns cun tj filnstry

    Filnstry lfi wjrb fs f ljihjuic`ik hfltjr$ `n$ ten c`hhnrnilns

    josnrvnc gfy on cun tj filnstry rftenr tefi tj c`snfsn

    6; * 81

    I

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    IJSFS@JI

    Ljcn lfsns fic ljitrjds oy P 7 6 fic P 7 1

    Ljcn \IQ b oy Ub 7 1$6$?$ sj teft Ub `s ten iugonr jh ljp`ns jh

    ten dnss hrnqunit fddndn P lfi fdsj on f qufit`tft`vn trf`t

    Dnt U 7 {Ub$ b 7 6$ )))$G} on ten knijtypn

    F knint`l gjcnd prjv`cns ten ljic`t`jifd c`str`out`ji jh P k`vni U

    65 * 81

    S F \ \IQ

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    SN\S\ JH F\\JL@FS@JI HJ_ \@IKDN \IQ\=

    LJIS@I^J^\ LF\N

    Hjr qufit`tft`vn trf`ts$ jin lfi usn d`infr rnkrnss`ji tj nvfduftn

    ten rndft`jise`p ontwnni P fic Ub

    Cnstin

    H 7 6 _\\_\\1

    wenrn _\\ fic _\\1 rnspnlt`vndy cnijtn ten rns`cufd sugs jh

    squfrns hjr ten d`infr gjcnds w`te fic w`tejut fssjl`ft`ji

    Hjr dfrkn sfgpdns$ fic uicnr ten iudd eypjtens`s jh ij

    fssjl`ft`ji$ H hjddjws f ?%6) c`str`out`ji

    Jin teni inncs tj ljrrnlt hjr gudt`pdn ljgpfr`sjis

    60 * 81

    S \ \ J F\\JL J J \ K \IQ\ O

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    SN\S\ JH F\\JL@FS@JI HJ_ \@IKDN \IQ\= O@IF_P

    LF\N

    @i lfsn'ljitrjd stuc`ns$ usn tnsts hjr iji fssjl`ft`ji

    %`icnpnicniln)

    Hjr nfle \IQ$ ten cftf frn rnprnsnitnc oy f ? < ljit`iknilytfodn

    Swj ljggji fpprjflens= ? tnst hjr `icnpnicniln$ we`le efs ? ch

    H`senr nxflt tnst$ usufddy fhtnr gnrk`ik ten dfst twj ljdugis

    ?1 * 81

    D@INF_ GJCND\

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    D@INF_ GJCND\

    \nvnrfd \IQs gfy efvn tj ljitr`outn tjkntenr tj f spnl`stl trf`t

    Sfbni `ic`v`cufddy$ nfle \IQ gfy efvn f wnfb nhhnlt teft w`dd ijton cntnltfodn us`ik ten prnv`jus tnsts

    Gjrn ljgpdnx gjcnds frn inncnc

    ?6 * 81

    DJK@\S@L GJCND

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    DJK@\S@L GJCND

    Djk`st`l gjcnds cnslr`on ten rndft`jise`p ontwnni penijtypn fic

    knijtypn oy=

    djkQ%P 7 6|U)

    Q%P 7 1|U)7

    Gb76

    bUb)

    Kninrfddy gjst jh ten ljnhstl`nits frn fssugnc tj on znrj$ w`teb 7 1 hjr jidy f suosnt C \ jh s`zn |C| G w`teljrrnspjic`ik ljnhstl`nits b$ b C

    C `s uibijwi `i prflt`ln

    Hjr dfrkn G$ f snis`odn strftnky `s stnpw`sn djk`st`l rnkrnss`ji$ `iwe`le ten \IQs frn fccnc jin oy jin tj ten gjcnd

    Ft stfkn g jin tnsts ten gjcnd w`te jin fcc`t`jifd \IQ fkf`ist

    ten lurrnit iudd gjcnd w`te g 6 \IQs %us`ik [`dbs tenjrng)

    ?? * 81

    _F_N XF_@FIS\

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    _F_N XF_@FIS\

    Gjst K[F stuc`ns fssugn teft gjst jh ten r`sb tefi lfi onnxpdf`inc w`dd `ivjdvn jdc$ ljggji knin vfr`fits

    @t `s e`kedy pjss`odn teft tenrn frn gfiy gjrn ljggji vfr`fits

    w`te vnry sgfdd jccs rft`js$ pnrefps must fojvn jin pnrlnit$ we`le

    efvn kjin uicntnltnc cun tj lurrnit sfgpdn s`zns Fijtenr pjss`o`d`ty `s teft ten g`ss`ik enr`tfo`d`ty %dflb jh

    sullnsshud knijgn'ofsnc prnc`ltfo`d`ty) lfi on nxpdf`inc oy

    gudt`pdn rfrnr vfr`fits$ sfy w`te hrnqunil`ns dnss tefi jin pnrlnit

    @t gfy on ten lfsn teft nfle rfrn vfr`fit$ fdtejuke snni `i jidy f

    sgfdd hrflt`ji jh ten fhhnltnc pjpudft`ji$ st`dd rf`sns ten r`sb `i

    lfrr`nrs oy f gule krnftnr hrflt`ji tefi cj ljggji vfr`fits

    ?< * 81

    NQ@\SF\@\

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    NQ@\SF\@\

    Senrn gfy on dfrkn `itnrflt`ji nhhnlts fgjik n`tenr ljggji jr

    rfrn vfr`fits Knins `itnrflt terjuke peys`lfd `itnrflt`jis fgjik ten`r _IF

    fic prjtn`i prjcults$ terjuke prjtn`i'CIF o`ic`ik `i trfislr`pt`ji

    fic vfr`jus gnlefi`sgs `i s`kifdd`ik pftewfys

    Nvni tejuke f vfr`fit efs sgfdd gfrk`ifd nhhnlt$ `t gfy efvn fs`ki`stlfit nhhnlt weni ljis`cnrnc `i ljimuilt`ji w`te jtenr

    vfr`fits

    Snlei`lfddy$ wn sejudc c`st`iku`se ontwnni stft`st`lfddy `itnrflt`ik

    \IQs fic np`stfs`s$ we`le gust rnnlt f c`rnlt$ peys`lfd

    `itnrflt`ji

    @ilduc`ik `itnrflt`jis `i ten prnv`jus gjcnd `s e`kedy lefddnik`ik$

    onlfusn ten iugonr jh gjcnd pfrfgntnrs krjws nxpjinit`fddy

    w`te ten s`zn jh ten ljis`cnrnc `itnrflt`jis

    ?3 * 81

    KNIN\ FIC QFSE[FP\

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    KNIN\ FIC QFSE[FP\

    Qftewfys frn krjups jh knins teft pfrt`l`pftn tj ljggji

    o`jdjk`lfd prjlnssns$ d`bn lndd huilt`jis$ gntfojd`l prjlnssns$

    knint`l `ihjrgft`ji prjlnss`ik %nk$ CIF rnpf`r)$ lndd s`kifd`ik$

    fic `gguin rnspjisns

    @ilrnfs`ikdy$ ten trun tfrknts jh fifdys`s frn pftewfys$ pfrt`ludfrdy

    `i c`snfsn c`fkijs`s$ prjkijs`s fic trnftgnit cnvndjpgnit C`snfsns frn lefrfltnr`znc fs pnrturonc pftewfys$ `i we`le tenrn

    `s f systngft`l hf`durn tj rnfle lnrtf`i nicpj`its$ rnsudt`ik hrjg f

    suhstl`nit iugonr jh rndft`vndy sgfdd c`srupt`jis$ dnfc`ik stifddy tj

    c`sjrcnr

    Jin lfi ou`dc knin'lnitr`l$ jr pftewfy'lnitr`l K[F\

    fpprjflens$ `i we`le nfle knin %jr pftewfy) lfi on tnstnc oy

    ljis`cnr`ik jidy ten \IQs fssjl`ftnc w`te `t

    ?> * 81

    K[@\

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    K[@\

    K[`\ `s f knin'lnitr`l fdkjr`teg prjpjsnc oy Ofcnr fic Frb`ik%Eufik nt fd$ Qdjs Knint`ls ?166)

    Sen gntejcjdjky `s Ofyns`fi `ihnrniln

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    N\p%F) 7 gfx

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    [eni p 7 1$ jin rntr`nvns ten Bjdgjkjrjv'\g`rijvs stft`st`l

    Sfb`ik p9 1 %nk$ p 7 6) fddjws jin tj `ilrnfsn ten `gpjrtfilnjh e`ke sljr`ik vfr`fodns `i F

    >3 * 81

    K\NF %@X)

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    Se`s stft`st`l trnfts ten sljrns %s$ s \) fs f rficjg sfgpdn$nfle vfr`fodn `icnx s on`ik ljis`cnrnc fs fi josnrvft`ji

    \`iln tensn sljrns invnr ljist`tutn fi ``c sfgpdn$ jin lfiijt

    usn ten usufd iudd'eypjtens`s c`str`out`jis tj cntnrg`in

    s`ki`stlfiln fic p'vfduns

    Sen jr`k`ifd sljrns s frn ljgputnc hrjg ten trf`i`ik snt$ S$ sjteft N\p%F) 7 N\p%F2 S)

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    >> * 81

    _NGF_B

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    Senrn `s f strjik c`hhnrniln ontwnni ten v`nwpj`it fcjptnc `i te`sfifdys`s$ fic ten jin ofsnc ji ten eypnrknjgntr`l c`str`out`ji

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    wjrb`ik `i rndft`ji w`te f penijtypn

    >8 * 81

    NUFGQDN

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    @i \uorfgfi`fi nt fd %?11>) ten futejrs rnpjrt f snr`ns jh rnsudts

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    %inxt sd`cn)

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    >; * 81

    V^JSNC H_JG \^O_FGFI@FI NS FD

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    Lytjknint`l snts %L6$ snts ljrrnspjic`ik tj lytjknint`l ofics Sensn snts frn endphud `i `cnit`hy`ik nhhnlts rndftnc tj lerjgjsjgfdcndnt`jis jr fgpd`stlft`jis$ cjsfkn ljgpnisft`ji$ np`knint`l s`dnil`ik$ fic jtenr rnk`jifd nhhnlts

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    >5 * 81

    V^JSNC H_JG \^O_FGFI@FI NS FD

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    Sen p>

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    Knin \nt HC_

    Nir`lenc `i p>< gutfit_fs s`kifd`ik pftewfy 16;6

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