Visa Matematika
January 31, 2017 | Author: Jure Talich | Category: N/A
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RVQRYH PDWHPDWLFNH ORJLNH 1 1 1 1 1 1 41414 Orjl fnl vxg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41415 Rshuludqmh vxgrylpd 1 1 1 1 1 1 1 1 1 1 1 1 41416 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 VNXSRYL L UHODFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41514 Vnxs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41515 Rshuludqmh vnxsrylpd 1 1 1 1 1 1 1 1 1 1 1 41516 Uhodflmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41517 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41614 Ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41615 Ixqnflmvnd nrpsr}lflmd 1 1 1 1 1 1 1 1 1 1 1 41616 Lqyhu}qd ixqnflmd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41617 Nduglqdoql eurm 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41618 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 UHDOQL EURMHYL 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41714 Sulurgql l flmhol eurmhyl 1 1 1 1 1 1 1 1 1 1 1 41715 Udflrqdoql eurmhyl 1 1 1 1 1 1 1 1 1 1 1 1 1 41716 Dsvroxwqd yulmhgqrvw uhdoqrj eurmd 1 1 1 1 41717 Srwhqfludqmh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 41718 Nrpelqdwrul fnh rvqryh 1 1 1 1 1 1 1 1 1 1 41719 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 NRPSOHNVQL EURMHYL 1 1 1 1 1 1 1 1 1 1 1 1 1 41814 Srmdp l rvqryqh rshudflmh 1 1 1 1 1 1 1 1 1 41815 Jhrphwulmvnl sulnd} nrpsohnvqrj eurmd 1 1 41816 Wuljrqrphwulmvnl }dslv nrpsohnvqrj eurmd 41817 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 lll
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HOHPHQWDUQH IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 61414 ]dgdydqmh ixqnflmd l} x 1 1 1 1 1 1 1 1 1 61415 Joredoqd vyrmvwyd uhdoqlk ixqnflmd 1 1 1 1 1 61416 Rvqryqh hohphqwduqh ixqnflmh 1 1 1 1 1 1 1 1 61417 Ud}uhged hohphqwduqlk ixqnflmd 1 1 1 1 1 1 1 61418 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 NRQYHUJHQFLMD QL]RYD L UHGRYD 1 1 1 1 1 61514 Ql} uhdoqlk eurmhyd 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61515 Vyrmvwyd nrqyhujhqwqlk +srg,ql}ryd 1 1 1 1 61516 Uhg uhdoqlk eurmhyd 1 1 1 1 1 1 1 1 1 1 1 1 1 61517 Qhnrolnr nulwhulmd }d nrqyhujhqwqrvw uhdoqrj 61518 Ixqnflmvnl ql} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61519 Ixqnflmvnl uhg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6151: Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 QHSUHNLGQH IXQNFLMH 1 1 1 1 1 1 1 1 1 1 1 1 1 61614 Judql fqd yulmhgqrvw x ehvnrqd fqrvwl 1 1 1 1 61615 Judql fqd yulmhgqrvw x wr fnl 1 1 1 1 1 1 1 1 1 61616 Qhsuhnlgqrvw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61617 Vyrmvwyd qhsuhnlgqh ixqnflmh qd vhjphqwx 1 61618 Ymh }eh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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SRJODYOMH 41 VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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Nduglqdoql eurm
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SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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UHDOQL EURMHYL
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41714
Sulurgql l flmhol eurmhyl
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UHDOQL EURMHYL
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SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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4171
4<
UHDOQL EURMHYL
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SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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Odnr mh surymhulwl lvwlqlwrvw rylk wyugqmd= +l, Vyl nrqd fql vnxsryl l vnxs sulurgqlk eurmhyd Q mhvx suheurmlyl> +ll, Vnxs Q Q mh suheurmly> +lll, Vydnl ehvnrqd fql srgvnxs D Q mh hnylsrwhqwdq vnxsx Q> +ly, Vydnl ehvnrqd fql suheurmly vnxs [ mh hnylsrwhqwdq vnxsx Q1 Grnd }lpr/ loxvwudflmh udgl/ wyugqmx +ll,$ Srnd}dw fhpr gd mh gluhnwql surgxnw Q Q hnylsrwhqwdq vnxsx Q1 X wx vyukx surpdwudmpr vomhghfl }dslv vnxsd Q Q=
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55
SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
Gh qludmpr ixqnflmx i = Q $ Q Q sudylorp µwr jd srnd}xmx vwumholfh= @ +4> 4,/ i +5, @ +5> 4,/ i +6, @ +4> 5,/ i +7, @ +4> 6,/ i +8, @ +5> 5,/ i +9, @ +6> 4,/ i +:, @ +7> 4,/ i +;, @ +6> 5,/ i + 6,/ i +43, @ +4> 7,/ 1 +i vh pr }h surslvdwl l qhrylvqr r vkhpl$, Odnr vh ylgl gd mh i elmhnflmd1 i +4,
Vd }plpr xnudwnr jodyqh flqmhqlfh r vnxsx Q= Srvwrml wr fqr mhgdq vnxs sulurgqlk eurmhyd Q @ i4> 5> 6> > q> j +ehv0 nrqd fdq , v elqduqlp rshudflmdpd . +}eudmdqmh , l +pqr }hqmh ,> nrmh vx dvrflmdwlyqh l nrpxwdwlyqh/ . mh glvwulexwlyqd v re}lurp qd / l v glvnuhwqlp xuh¡dmhp nrml vh greur srqdµd suhpd . l l x nrmhpx mh 4 @ plq+Q> ,= Grelyhqx vwuxnwxux fhpr r}qd
flwl v +Q> .> > ,1
Vdgd fhpr grelyhqx vwuxnwxux surµlulwl qd w}y1 vnxs flmholk eurmhyd1 Vmhwlpr vh gd x vox fdmx p ? q x Q srvwrml s 5 Q wdndy gd mh p.s @ q1 Slvdol vpr s q p1 Mdvqr mh gd qh pr }h elwl rshudflmd qd Q mhu x vox fdmx flor1 Ryr yrgl qd }dplvdr gd vh vnxs q p qh }qdpr µwr el q p }qd
Q ud}xpqr surµlul1 X wx vyukx surpdwudmpr vnxs Q l qmhjry suhvoln Q @ i4 > 5 > 6 > > q > j/ jgmh qdp fuwlfh vox }h }d irupdoqr ud}olnrydqmh suhvolnd rg l}yruqlnd1 R}qd flpr }qdnrp 3 qryl remhnw +qh sulsdgdVQ qlVQ ,/ qd}rylpr jd qxorp +lol qlµwlfrp,/ sd surpdwudmpr vnxs ] @ Q i3j Q1 Qd vnxsx ] }holpr l}judglwl vwuxnwxux vol fqx rqrm qd qmhjryx srgvnxsx Q l wr wdnr gd vh srvwrmhfd vwuxnwxud qd Q qh l}plmhql/ wm1 vwuxnwxud qd ] wuhed elwl surµluhqmh rqh qd srgvnxsx Q1 Qdmsulmh gh qludpr }eudmdqmh qd fhpr rshw r}qd flwl }qdnrp ./ wm1 . = ] ] $ ] 1 Dnr vx p> q 5 Q ] / nrmh +p > q 5 Q ,/ qhnd p . q +p . q , x ] exgh lvwr µwr l x Q +Q ,1 Dnr mh p 5 Q l q 5 Q / qhnd exgh ;
flp mh q ? p ? p q/ 3> p.q @ q .p @
flp mh q @ p = = flp mh q ? p +q p, >
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fhvwr slµxfl p q ndr pq1 Xuh¡dm qd ] xyrglpr qdmsulmh joredoqr }dkwmhyrp Q i3j Q/ wm1 p 3 q }d vydnl p 5 Q l vydnl q 5 Q1 Qdgdomh/ x Q ] qhnd exgh
4171
56
UHDOQL EURMHYL
xuh¡dm qdvomh¡hq rg +Q> ,/ grn x x ] ndg jrg mh q p x Q1
Q reuqlpr wdm xuh¡dm/ wm1 p
q
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41715
Udflrqdoql eurmhyl
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57
SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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4171
58
UHDOQL EURMHYL
rgdeudqrjd sudyfd1 Sulwrp vh fxyd xuh¡dm/ wm1 p ? q x ] wr fqr rqgd ndg mh H6 olmhyr rg H? 1 Sr}qdwlp qd flqrp glmhomhqmd gx }lqh pr }hpr/ }lwl mhglqvwyhqx wr fnx H^ qdgdomh/ vydnrp udflrqdoqrp eurmx t @ 6 ? sulgux
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59
SRJODYOMH 41
VNXSRYL1 IXQNFLMH1 UHDOQL EURMHYL1
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UHDOQL EURMHYL
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Gh qlflmd 51518 Pqr }hqmhp +yhnwrud , vndodurp qd}lydpr ixqnflmx l} U Y x Y / +> d, :$ d/ jgmh mh d yhnwru gxomlqh mm ndn nrolqhdudq yhnwrux d/ nrml mh xvpmhuhq ndr d f lp mh A 3/ d vxsurwqr xvpmhuhq f lp mh ? 31 +Sr gh qlflml mh/ gdnoh/ 3 @ 3 }d vydnl / ndr l 3d @ 3 }d vydnl d=,
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SRJODYOMH 51
OLQHDUQD DOJHEUD
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SRJODYOMH 51
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Yhnwruvnd pqr
}hqmd
X vnxsx Y vylk yhnwrud gh qludpr gylmh yuvwh pqr }hqmd1 Lvkrg mhgqrjd mh uhdodq eurm +vndodu, sd jryrulpr r vndoduqrp pqr }hqmx/ d guxjrjd 0 yhn0 wru sd jd qd}lydpr yhnwruvnlp pqr }hqmhp1 Vndoduqr pqr }hfl yhnwruvnl xpqr }dn yhnwrurp grelydpr l w}y1 pmhµrylwl xpqr }dn1 Gdndnr/ vdpr mh yhnwruvnr pqr }hqmh +elqduqd, rshudflmd qd Y 1
Gh qlflmd 5151: Vndoduqlp pqr }hqmhp qd}lydpr ixqnflmx
+lol vndoduqlp surgxnwrp ,
m = Y Y $ U> m+d> e, +dme, @ ndn nen frv+d> e,=
:;
SRJODYOMH 51
OLQHDUQD DOJHEUD
Mhgqrvwdyqrvwl udgl/ r}qd flpr +dme, d e/ ndn d/ nen e l nxw l}ph¡x yhnwrud d l e voryrp */ sd gh qlflmd vndoduqrjd pqr }hqmd grelyd }dslv
d e @ de frv *=
L}ud fxqdmpr vndoduqh xpqrµnh vylk sduryd mhglql fqlk yhnwrud l/ m l n µwr sulsdgdmx elor nrmhpx sudyrnxwqrp nrruglqdwqrp vxvwdyx +R> l> m > n, x survwrux H= l l @ 4 4 frv 3 @ 4 @ m m @ n n> l m @ 4 4 frv Z2 @ 3 @ m l @ m n @ n m @ n l @ l n1 R flwr mh dd @ d2 l de @ 3 @ ed flp vx yhnwrul d l e ph¡xvreqr rnrplwl lol mh eduhp mhgdq rg qmlk qxoyhnwru1 Qhnd vx d l e gyd qh0qxoyhnwrud vyhghqd qd }dmhgql fnl sr fhwdn R l qhnd mh qd sudyfx rg e gdq nrruglqdwql vxvwdy }lqx +R> ef ,1 Rnrplwlp surmlfludqmhp yhnwrud d qd wdm sudydf grelydpr gx
RW / jgmh wr fnl W sulsdgd nrruglqdwd dK @ d frv * +y1 fuwh },1 Sulplmhwlpr gd mh d e @ dK e/ sd vplmhpr }dnomx flwl gd mh d e ? 3 +@ 3> A 3, rqgd l vdpr rqgd ndg mh dK ? 3 +@ 3> A 3,1
,
ϕ 2 -
,
, 7
2
-
7
ϕ 2
-
Eurm dK qd}lydpr vndoduqrp nrpsrqhqwrp/ d yhnwru dK ef yhnwruvnrp nrpsrqhqwrp yhnwrud d qd sudyfx yhnwrud e/ µwr mh x vnodgx v qd}lyomhp x srgrgmhomnx 515151 Rvqryqd vyrmvwyd vndoduqrjd pqr }hqmd grqrvl rydm whruhp=
Whruhp 51518 Qhnd vx d> e> f 5 Y l 5 U1 Wdgd mh +l, d e @ e d +nrpxwdwlyqrvw,> +ll, d +e . f, @ d e . d f +glvwulexwlyqrvw,> +lll, +d e, @ +d, e @ d +e, +krprjhqrvw,> Grnd}1 +l,1 d e @ de frv * @ ed frv+*, @ e d1 +ll,1 R}qd flpr e . f g/ sd mh g@ @ e@ . f@ +y1 fuwh }, l g@ d @ e@ d . f@ d1
Vdgd sr gh qlflml vndoduqh nrpsrqhqwh volmhgl d +e . f, d g @ d e . d f1 +lll,1 Dnr mh @ 3 wyugqmd mh r fljohgqr lvwlqlwd1 Dnr mh A 3 rqgd mh ndn @ d l nen @ e/ d yhnwrul d l e/ rgqrvqr d l e/ }dwydudmx lvwl nxw * ndr l yhnwrul d l e1 Vwrjd mh +d e, @ +de frv *, @ +d,e frv * @
5151
:<
YHNWRUVND DOJHEUD
ndn e frv * @ +d, e/ rgqrvqr/ +d e, @ +de frv *, @ d+e, frv * @ d nen frv * @ d +e,1 Dnr mh sdn ? 3 rqgd mh ndn @ d l nen @ e/ d yhnwrul d l e/ rgqrvqr d l e/ }dwydudmx nxw *1 Exgxfl gd mh frv+ *, @ frv */ wr mh +d e, @ +de frv *, @ +d,e+ frv *, @ ndn e frv+ *, @ +d, e/ rgqrvqr/ +d e, @ +de frv *, @ d+e,+ frv *, @ d nen frv+ *, @ d +e,1 Whruhp 51519 Dnr mh x sudyrnxwqrp nrruglqdwqrp d @ d% l . d+ m . d5 n l e @ e% l . e+ m . e5 n/ rqgd mh d e @ d% e% . d+ e+ . d5 e5 =
vxvwdyx
R l> m > n ,
+ >
Grnd}1
Whruhp 51518 mh l}udyqd srvomhglfd Whruhpd 51517 l sr}qdwlk vndoduqlk xpqr }dnd vylk sduryd mhglql fqlk yhnwrud l/ m l n1
Sulpmhu 51515 Rguhglpr nxw µwr jd }dwydudmx yhnwrul l e @ l . m1 Sr Whruhpx 51518/ l} d e @ de frv * volmhgl frv
*@
de de
d
@ 5
l 6m . n
d% e% . d+ e+ . d5 e5 t 1 d2% . d2+ . d25 e2% . e2+ . e25 2 nE n f @ / sd mh * @ duffrv+ , 433J 86 57 enbn nnf 2 . 2 . @
t
I Gdnoh/ frv * @ I u 3 Iu u wm1 * 4> :93;: udglmdqd +y1 vomhghfx Qdsrphqx 51518,1
I
Qdsrphqd 51518 Sr grjryrux vpdwudpr gd xuh¡hql yhnwruvnl sduryl +d> e, l +e> d, }dwydudmx lvwl nxw * l gd mh 3 * 1 Gh qlflmd 5151; Yhnwruvnlp pqr }hqmhp +lol yhnwruvnlp surgxnwrp , qd}lydpr elqduqx rshudflmx = Y Y $ Y / +d> e, d e/ gh qludqx ndnr volmhgl= yhnwru d e mh rnrplw qd yhnwruh d l e> xuh¡hqd wurmnd +d> e> d e, rguh¡xmh ghvql nrruglqdwql vxvwdy x survwrux> nd en @ ndn nen vlq+d> e, de vlq *1 Xr flpr gd mh yhnwruvnl xpqr }dn d e yhnwru nrmhpx mh gxomlqd mhgqdnd sryuµlql sdudohorjudpd µwr jd rguh¡xmx yhnwrul d l e vyhghql qd }dmhgql fnl sr fhwdn/ wh gd px mh sudydf rnrplw qd udyqlqx rguh¡hqx +nrolqhduqx, wlp yhnwrulpd +y1 fuwh },1
.,-
ϕ
,
/
;3
SRJODYOMH 51
OLQHDUQD DOJHEUD
Whruhp 5151: Qhnd vx d> e> f 5 Y l 5 U1 Wdgd mh +l, d e @ e d +dqwlnrpxwdwlyqrvw,> +ll, d +e . f, @ d e . d f +glvwulexwlyqrvw,> +lll, +d e, @ +d, e @ d +e, +krprjhqrvw,1 Grnd}1 +l,1 Exgxfl gd/ sr gh qlflml/ xuh¡hqd wurmnd +d> e> d e, wyrul ghvql vxvwdy/ wr xuh¡hqd wurmnd +e> d> d e, wyrul olmhyl vxvwdy +y1 Qdsrphqx 51517,1 Wd flqmhqlfd l Gh qlflmd 5151; sryod fh +l,1 +ll,1 Qhnd mh udyqlqd rnrplwd qd yhnwru d nur} qmhjry sr fhwdn/ wh qhnd mh e rnrplwd surmhnflmd yhnwrud e qd 1 Qhnd mh * nxw l}ph¡x yhnwrud d l e/ d * nxw l}ph¡x e l e 1 Wdgd mh * @ Z2 * l e @ e frv * @ e vlq *1 ]durwludmpr e x rnr }dmhgql fnrjd sr fhwnd }d Z2 l grelyhql yhnwru r}qd flpr v e / wdnr gd +d> e> e , wyruh ghvql vxvwdy1 Sulplmhwlpr gd mh yhnwru e rnrplw qd yhnwruh d l e / gdnoh/ l qd yhnwru e1 R}qd flpr d e e 1 Vol fqlp srvwxsnrp grelydpr yhnwruh f / f l f d f/ wh g / g l g d g/ jgmh mh g e . f +y1 fuwh },1 G
ρ E
G
G
F
F E
E
E G
D
Suhrvwdmh grnd}dwl gd mh g @ e . f 1 Xr flpr gd mh rnrplwd surmhnflmd +qd , sdudohorjudpd rguh¡hqrjd yhnwrulpd f l e sdudohorjudp rguh¡hq yhn0 wrulpd f l e / d rnrplwd surmhnflmd g glmdjrqdoh g @ e . f suyrjd sdudoho0 rjudpd mh glmdjrqdod guxjrjd sdudohorjudpd/ wm1 g @ e . f 1 Urwdflmd x udyqlql rnr }dmhgql fnrjd sr fhwnd }d Z2 suhyrgl sdudohorjudp rguh¡hq yhnwrulpd e l f x vxnodgql sdudohorjudp rguh¡hq yhnwrulpd e l f v gl0 mdjrqdorp g / sd mh g @ e . f 1 Pqr }hqmhp wh mhgqdnrvwl v mmdmm d grelydpr g @ e . f 1 +lll,1 Ryd vh wyugqmd grnd}xmh ndr +lll, x Whruhpx 515171
Whruhp 5151; Dnr mh x sudyrnxwqrp nrruglqdwqrp vxvwdyx +R> l> m > n, d @ d% l . d+ m . d5 n l e @ e% l . e+ m . e5 n/ rqgd mh
d e @ +d+ e5 d5 e+ ,l . +d5 e% d% e5 ,m . +d% e+ d+ e% ,n=
Grnd}1 Whruhp 5151; mh l}udyqd srvomhglfd Whruhpd 5151: l vomhghflk mhgqrvwdyqlk flqmhqlfd +y1 Gh qlflmx 5151;,= l l @ 3 @ m m @ n n> l m @ n / m n @ l/ n l @ m > l n @ m / m l @ n/ n m @ l1
5151 YHNWRUVND DOJHEUD
;4
Sulplmhwlpr gd Whruhp 5151; grsxµwd l irupdoql ghwhuplqdqwql }dslv +y1 ¢51415,=
de
@
l m n d% d+ d5 = e% e+ e5
L} wrjd mh }dslvd r fljohgqr gd mh d e @ 3 rqgd l vdpr rqgd/ dnr vx d l e nrolqhduql +µwr xnomx fxmh prjxfqrvw gd mh qhnl rg qmlk qxoyhnwru,1 Sulplmhwlpr/ wdnr¡hu/ gd yhnwruvnr pqr }hqmh qlmh dvrflmdwlyqr1 wrylµh/ qlmh whµnr grnd}dwl ydomdqrvw vomhghfjd whruhpd=
Whruhp 5151< Dnr vx d> e> f 5 Y rqgd mh +l, +d e, f @ +d f,e +e f,d> +ll, d +e f, @ +d f,e +d e,f1 Sulpmhu 51516 L}ud fxqdmpr sryuµlqx sdudohorjudpd µwr jd rguh¡xmx yhn0
d @ 5l . 6m . n l e @ l m n1 l m n s s S @ nd en @ mm 5 6 4 mm @ n5l m nn @ 7 . 4 . 4 @ 91
wrul
4
4 4
Gh qlflmd 5151< Pmhµrylwlp pqr }hqmhp +lol yhnwruvnr0vndoduqlp surgxnwrp , qd}lydpr ixqnflmx l} Y Y Y x U gh qludqx sudylorp +d> e> f, :$ +d e, f1 R}qd flpr ol nxw l}ph¡x yhnwrud d l e voryrp */ d nxw l}ph¡x voryrp #/ sudylor pmhµrylwrjd pqr }hqmd grelyd }dslv
de l f
d e, f @ nd en nfn frv # @ ndn nen vlq * nfn frv # def vlq * frv #1 Wr grsxµwd pmhµrylwl xpqr }dn +d e, f lqwhusuhwludwl/ gr qd suhg}qdn/ ndr rexmdp sdudohorslshgd rguh¡hqrjd yhnwrulpd d/ e l f +
,×-
.
ψ
ϕ
,
Suhg}qdn pmhµrylwrjd xpqrµnd mh sr}lwlydq flp +d> e> f, rguh¡xmh ghvql vxv0 wdy/ d qhjdwlydq x vox fdmx olmhyrjd vxvwdyd1
Whruhp 515143 Dnr mh x sudyrnxwqrp nrruglqdwqrp vxvwdyx +R> l> m > n, d @ d% l . d+ m . d5 n/ e @ e% l . e+ m . e5 n l f @ f% l . f+ m . f5 n/ rqgd mh
d e, f
+
@
d% d+ d5 e% e+ e5 = f% f+ f5
;5
SRJODYOMH 51 OLQHDUQD DOJHEUD
Grnd}1 Whruhp 515143 mh srvomhglfd Whruhpd 51518 l Whruhpd 5151:1 Srprfx Whruhpd 515143 vh odnr grnd}xmx +udehfl ghwhuplqdqwlqd vyrm0 vwyd, ryh mhgqdnrvwl= +d e, f @ +e f, d @ +f d, e @ +d f, e @ +f e, d @ +e d, f> +d e, f @ d +e f,> +d e, f @ f +d e,1
51517
Ymh
}eh
41 Grnd}dwl gd mh uhodflmd +y1 Gh qlflmx 51514, qd vnxsx vylk xvpmhuhqlk gx }lqd l} survwrud H ud}uhgehqd uhodflmd1 $ Grnd}1 ]d vydnx xvpmhuhqx gx }lqx DE / gx }lqh DE l ED lpdmx lvwr srorylµwh1 $ $ $ $ Gdnoh/ DE DE / wm1 uhodflmd mh uh hnvlyqd1 Qdgdomh/ qhnd mh DE FG/ wm1 qhnd gx }lqh DG l EF lpdmx }dmhgql fnr srorylµwh1 Wdgd l gx }lqh FE l GD lpdmx }dmhgql fnr srorylµwh +udgl vh r lvwlp gx }lqdpd l lvwrp srorylµwx,/ $ $ sd mh FG DE 1 Gdnoh/ uhodflmd mh l vlphwul fqd1 Qdsrnrq/ }d wudq}l0 $ $ $ $ $ wlyqrvw/ qhnd vx DE / FG l HI wdnyh xvpmhuhqh gx }lqh gd mh DE FG l $ $ $ $ FG HI 1 Grnd }lpr gd mh wdgd DE HI $ ,
$ $ DE FG / +DE n FG a DF n EG a g+D> F , @ g+E> G,, , $ $ FG HI / +FG n HI a FH n GI a g+F> H , @ g+G> I ,, $ $ +DE n HI a DH n EI a g+D> H , @ g+E> I ,, , DE HI 1 $ $ 51 Qhnd vx gdqh elor nrmh gylmh xvpmhuhqh gx }lqh DE l FG1 Gdnd}dwl= $ $ $ $ +d, DE FG / DF EG> $ $ $ $ +e, DE FG / ED GF 1 61 Grnd}dwl wyugqmx +lll, l} Whruhpd 51515/ wm1 +d . e, @ d . e1
Grnd}1 Dnr mh edu mhgdq rg yhnwrud d/ e qxoyhnwru lol mh @ 3/ grnd} mh wulylmdodq1 Qhnd d ql e qlvx qxoyhnwrul l qhnd mh 9@ 31 Dnr vx d l e nrolqhduql/ srvwrml eurm }d nrml mh d @ e1 Vdgd vh wyugqmx +lll, odnr gdnd }h srprfx wyugqmd +l, l +ll, lvwrjd whruhpd=
+d . e, @ +e . e, @ ++ . 4,e, @ ++ . 4,,e @ . ,e @ +,e . e @ +e, . e @ d . e1 Qhnd d l e qlvx nrolqhduql1 Sulnd }lpr yhnwruh d l e uhgrp xvpmhuhqlp $ $ gx }lqdpd RD l RE v lvwlp sr fhwnrp R/ sd qhnd mh F rqd wr fnd }d nrmx $ $ $ mh RD . RE @ RF 1 Wdgd wr fnh R> D> F> E rguh¡xmx sdudohorjudp/ sd gx }lqh DE l RF lpdmx }dmhgql fnr srorylµwh1 Qhnd vx D / E l F uhgrp $ $ $ $ $ $ rqh wr fnh }d nrmh mh RD @ RD / RE @ RE l RF @ RF 1 Sulplmhwlpr gd mh l fhwyhurnxw RD F E sdudohorjudp + sr krprwhwlml vd vuhglµwhp R l $ $ $ nrh flmhqwrp 9@ 3,/ sd mh RF @ RD . RE 1 Qdsrnrq/ $ $ $ $ +d . e, @ +RD . RE, @ RF @ RF @ $ $ $ $ RD . RE @ RD . RE @ d . e1 71 Grnd}dwl gd vx gyd qh0qxoyhnwrud d l e rnrplwd rqgd l vdpr rqgd ndg +
qmlkry vndoduql xpqr }dn lµ fh}dyd1
5161 DQDOLWLFND JHRPHWULMD
;6
81 Grnd}dwl Whruhp 5151< wm1
d e, f @ +d f,e +e f,d> d +e f, @ +d f,e +d e,f1
+l, +ll,
516
+
DQDOLWLFND JHRPHWULMD
Qhnd mh x survwrux H }dgdq sudyrnxwql nrruglqdwql vxvwdy +R> l> m > n,1 Sur0 pdwudw fhpr qhnh yd }qh wr fnryqh +H U / y1 ¢51515, srgvnxsryh survwrud H +sudyfh/ qhnh nulyxomh/ udyqlqh l qhnh sorkh, l ud}pdwudwl xymhwh srg nr0 mlpd qhnd wr fnd W @ +{> |> } , sulsdgd wdnyrpx srgvnxsx1 Dojheudl}dflmd wlk xymhwd yrgl gr dqdolwl fnlk l}ud}d 0 mhgqdg }ded surpdwudqlk srgvnxsryd1 Wr rqgd rprjxfxmh dojheduvnr rslvlydqmh l qhnlk rgqrvd ph¡x wlp srg0 vnxsrylpd1
51614
Sudydf x survwrux l x udyqlql
Whphomqr vyrmvwyr vydnrj sudyfd mhvw gd mh srvyh rguh¡hq elor nrmlp gymhpd vyrmlp wr fndpd1 Qhnd mh sudydf s x survwrux H rguh¡hq wr fndpd W l W2 nrmlpd/ v re}lurp qd rgdeudqx wr fnx R/ sulsdgdmx uhgrp udglmxv0yhnwrul u l u2/ wh qhnd mh W elor nrmd wr fnd qd s v udglmxv0yhnwrurp u1 Wdgd vx $ $ xvpmhuhqh gx }lqh W W2 l W W nrolqhduqh +oh }h qd s,/ sd srvwrml eurm w 5 U $ $ wdndy gd mh W W @ wW W2 / rgqrvqr/ +y1 fuwh },
u u @ w+u 2 u ,=
+4,
7 =
U
7 U U
7
<
; Uhodflmd +4, mh yhnwruvnd mhgqdg
}ed sudyfd s1 R}qd flpr ol nrqvwdqwql yhnwru u2 u vd v +vpmhuryql yhnwru rg s,/ mhgqdg }ed +4, srsulpd reoln
u @ u . wv> w 5 U=
+5,
Mdvqr mh gd vh vydnl yhnwru nrolqhdudq yhnwrux v vplmh x}hwl }d vpmhuryql yhnwru sudyfd s1 Qhnd mh x survwrux H v lvkrglµwhp R }dgdq sudyrnxwql nrruglqdwql vxvwdy +R> l> m > n,/ sd qhnd mh x qmhpx v @ ^d e f`/ u @ ^{ | } ` l u @ ^{ | } `1 Wdgd l} uhodflmh +5, grelydpr wul mhgqdg }eh= ; ? =
{ @ { . dw | @ | . ew } @ } . fw> w 5 U>
+6,
nrmh suhgvwdyomdmx sdudphwduvnl }dslv sudyfd s1 Dnr vx vyh nrpsrqhqwh rg v ud}ol flwh rg qxoh/ sdudphwdu w vh pr }h holplqludwl sd l} +6, grelydpr
;7
SRJODYOMH 51 OLQHDUQD DOJHEUD
w}y1 ndqrqvnl +lol vlphwul
fql, }dslv sudyfd s=
{ { d
@
| | e
@
} } = f
+7,
+Srqhndg vh }dslvrp +7, vox }lpr l x vox fdmhylpd ndg vx qhnh nrpsrqhqwh rg v mhgqdnh qxol/ irupdoqr slµxfl sulsdgqh qlµwlfh x rgjrydudmxflp qd}ly0 qlflpd1, Xr flpr gd ndqrqvnl }dslv +7, vdgu }l wul mhgqdg }eh nrmh vx ph¡xvreqr olqhduqr ; }dylvqh= ? =
e+{ { , @ d+| | , f+| |, @ e+} }, = d+} }, @ f+{ {,
+8,
D { . E| . F } @ 3 > D2 { . E2| . F2 } @ 3
+8 ,
Gryromqh vx/ gdnoh/ gylmh rg wlk mhgqdg }ed gd el sudydf elr srvyh rguh¡hq1 Wr yulmhgl l vdvylp rsfhqlwr1 Sudydf vh/ qdlph/ pr }h }dgdwl gymhpd olqhduqr qh}dylvqlp +uhdoqlp, olqhduqlp mhgqdg }edpd=
l} nrmlk vh odnr prjx/ holplqdflmrp srmhglqlk ydulmdeod/ grelwl }dslvl +8, l +7,1 +X qduhgqrm wr fnl 51616 fhpr ylgmhwl gd vydnd rg mhgqdg }ed x +8 , l +8, suhgvwdyomd qhnx udyqlqx x survwrux1,
Sulpmhu 51614 Rguhglpr mhgqdg }ex +ndqrqvnl }dslv, sudyfd s nrmhpx sul0 D sdgdmx wr fnh D @ +5> 5> 7, l E @ +;> 2 > 2 ,1
$
Mhgdq vpmhuryql yhnwru wrjd sudyfd mhvw DE @ ^9 b2 2 `1 Gd el wud }hql $ }dslv elr omhsµl/ x}plpr v @ 2 DE @ ^7 6 4`1 Xyuµwdydqmhp wr fnry0 qlk nrruglqdwd rg D l vndoduqlk nrpsrqhqdwd rg v x uhodflmx +7, grelydpr + 32 2 5 3e ndqrqvnl }dslv s %3 e @ @ 3 =
Sulpmhu 51615 Ndqrqvnl }dslv nrruglqdwqh [ 0rvl mhvw
@ f @ f +olqhdu0 yhnwru @ ^4 3 3` l mhu %
qlp mhgqdg }edpd= | @ 3/ } @ 3,/ mhu mh sulsdgql surod}l lvkrglµwhp R @ +3> 3> 3,1 Vol fql vx }dslvl }d \ 0rv l
+
5
l ] 0rv1
Fhvwr vh }d vpmhuryql yhnwru surpdwudqrj sudyfd x}lpd sulsdgql mh0 glql fql yhnwru vf @ 8v v8 1 Wdgd }dslv +7, srsulpd reoln
{ { | | } } @ @ > +9, frv frv frv jgmh vx frv / frv l frv nrpsrqhqhwh rg vf +vpmhuryql nrh flmhqwl rg v,/ sd prud elwl frv2 . frv2 . frv2 @ 41 Qhnd sudydf s oh }l x nrruglqdwqrm [\ 0udyqlql +y1 fuwh } gromh,/ sd
surpdwudmpr vdpr wr fnh l srgvnxsryh wh udyqlqh1 Wdgd qdvomh¡hql nr0 ruglqdwql vxvwdy +R> l> m , vydnrm wr fnl sulgux }xmh vdpr gylmh nrruglqdwh/ rgqrvqr/ vydnrp yhnwrux vdpr gylmh nrpsrqhqwh +wuhfd/ rg ] 0rvl/ x vxvwdyx fh}dyd,1 Mhgqdg }eh x +6, vdgd srsulpdmx reoln +R> l> m > n, lµ
{ @ { . dw > | @ | . ew
d xpmhvwr +7, grelydpr
+6 ,
5161 DQDOLWLFND JHRPHWULMD
{ { d
@
| | = e
;8
+7 ,
Qdslµhpr ol mhgqdg }ex +7 , x reolnx
| | @
e +{ { ,> d
+:,
grelydpr sr}qdwx vuhgqmrµnrovnx sudy fhyx mhgqdg }ex mhgqrp wr fnrp1 Nrol fqln @K qd}lydpr vpmhuryqlp nrh flmhqwrp rg s l rel fqr jd r}qd fx0 mhpr voryrp n 1 Rgdehuhpr ol }d wr fnx W vmhflµwh sudyfd s v nrruglqdwqrp \ 0rvl/ W @ +3> o,/ grelydpr greur sr}qdwx sudy fhyx hnvsolflwqx mhgqdg }ex
| @ n{ . o= < O V DLEM S M 2
+;,
;
L
holpr ol gd uhodflmh +6 ,/ +7 ,/ +:, lol +;, rguh¡xmx lvwl sudydf s l x surv0 ] wruqrp nrruglqdwqrp vxvwdyx +R> l> m > n,/ wuhedpr vydnrm rg qmlk grslvdwl mhgqdg }ex } @ 31
51615
Udyqlqd x survwrux
Qhnd vx W > W2 l W elor nrmh wul qhnrolqhduqh wr fnh x survwrux H v sulsdgqlp udglmxv0yhnwrulpd u / u2 l u uhgrp +v re}lurp qd gdqr lvkrglµwh R,1 Wlp wr fndpd mh rguh¡hqd mhgqd l vdpr mhgqd udyqlqd nrml lk vdgu }l1 Qhnd mh W elor nrmd wr fnd wh udyqlqh l qhnd mrm sulsdgd udglmxv0yhnwru u1 Wdgd vx $ $ $ xvpmhuhqh gx }lqh W W2 / W W l W W nrsodqduqh +oh }h x ,/ sd qmlkry pmhµrylwl xpqr }dn lµ fh }yd/ wm1
$ W$W +$ W W W W , @ 3=
2
+ m > n,/ sd qhnd mh x qmhpx W @ +{> |> } , l W @ +{ > | > } ,/ l @ 4> 5> 6= Pmhµrylwl xpqr }dn + 3> 3,/ W2 @ +3> e> 3, l W @ +3> 3> f,/ d> e> f 9@ 31 Wdgd l} +44, grelydpr w}y1 vhjphqwql reoln mhgqdg
}eh udyqlqh
{ | } . . @ 4= +46, d e f Qhnd mh q elor nrml yhnwru rnrplw qd udyqlqx +rnrplw/ gdnoh/ l qd vydnl sudydf x ,1 X fyuvwlpr elor nrmx wr fnx W x / sd qhnd mh W ydulmdeloqd wr fnd x 1 Sulsdgql udglmxv0yhnwrul qhnd vx uhgrp u l u1 Wdgd vx yhnwrul q l u u ph¡xvreqr rnrplwl/ µwr sryod fl lµ fh}dydqmh qmlkryd vndoduqrj xpqrµnd=
q +u u, @ 3=
+47,
Wr mh mrµ mhgdq reoln yhnwruvnh mhgqdg
}eh }d udyqlqx 1 Vydnl yhnwru rnrplw qd udyqlqx qd}lydpr qrupdoqlp yhnwrurp +lol qrupdorp, wh udyqlqh l qdm fhµfh jd r}qd fxmhpr voryrp q1 +Wdndy mh/ sulpmhulfh/ yhnwru +u2 u , +u u , l} uhodflmh +43,1, Dnr mh/ x vxvwdyx +R> l> m > n,/ q @ ^D E F `/ d W @ +{> |> }, l W @ +{> |> },/ wm1 u @ ^{ | } ` l u @ ^{ | }`/ rqgd +47, srsulpd reoln +48, D+{ {, . E+| |, . F +} } , @ 3> µwr mh w}y1 mhgqdg
}ed udyqlqh mhgqrp wr fnrp W @ +{> | > } ,1 R}0 qd flpr ol x +48, nrqvwdqwx +D{ . E| . F} , voryrp G/ grelydpr w}y1 rsfl reoln mhgqdg }eh udyqlqh = D{ . E| . F} . G @ 3 +49,
lol +yhnwruvnl,
q u . G @ 3=
+49 ,
=
Q
7
U
;
U
7 <
Qhnd mh vdgd u udglmxv0yhnwru qr }lµwd W rnrplfh l} lvkrglµwd R qd udyqlqx 1 Wdgd mh qmhjryd gxomlqd nu n s mhgqdnd xgdomhqrvwl rg R gr 1 Rgdeh0 u qd +y1 fuwh },1 ]erj qfu @ s/ ulpr mhglql fql qrupdoql yhnwru qf @ 8u 8 l} +47, grelydpr
s @3
qf u
+4:,
lol +vndoduqr,
{ frv . | frv . } frv s @ 3>
+4: ,
5161 DQDOLWLFND JHRPHWULMD
;:
jgmh vx frv / frv l frv nrpsrqhqwh mhglql fqh qrupdoh qf qd udyqlqx Uhodflmh +4:, l +4: , qd}lydpr Khvvhrylp +lol qrupdoqlp, reolflpd mhgqdg }eh udyqlqh s1 Sulplmhwlpr gd vh rsfl reoln vyrgl qd Khvvhry ( D2 . E 2 . F 2 ndg mh G 9@ 3/ rgqrvqr/ glmhomhqmhp glmhomhqmhp eurmhp 3 (
1 v
s
D2 . E 2 . F 2 ndg mh G @ 31
Sulpmhu 51616 Nrruglqdwqd ][ 0udyqlqd surod}l lvkrglµwhp R @ +3> 3> 3, l
}ed mh 3+{ 3, . @ ^3 4 3`1 Sr +48,/ qmh}lqd mhgqdg
| 3, . 3+} 3, @ 3/ wm | @ 31 Vol fqr grelydpr gd mh } @ 3 mhgqdg }ed nrruglqdwqh [\ 0udyqlqh/ d { @ 3 mhgqdg }ed nrruglqdwqh \ ] 0udyqlqh1 rnrplwd mh qd yhnwru m 4+
51616
Ph¡xrgqrvl wr
fdnd/ sudydfd l udyqlqd
Gylmh wr fnh x survwrux H vx lol ud}ol flwh lol vh srgxgdudmx1 X suyrpx vox fdmx vh srvwdyomd slwdqmh ph¡xvreqh xgdomhqrvwl1 Elor nrmd surpdwudqd wr fnd lol oh }l qd gdqrp sudyfx +x gdqrm udyqlql, lol mh l}ydq qmhjd +qmh,1 X suyrpx vox fdmx qmh}lqh nrruglqdwh prudmx xgryromdydwl mhgqdg }el wrjd sudyfd +wh udyqlqh,1 X guxjrpx sdn vox fdmx/ }dqlpomlyr slwdqmh mhvw rqr r xgdomhqrvwl wr fnh rg sudyfd +udyqlqh,1 Gyd sudyfd/ rsfhqlwr/ prjx elwl xvsruhgqd +xnomx fxmxfl srgxgdudqmh,/ prjx elwl plprvpmhuqd/ d prjx vh l suhvlmhfdwl1 X suyd gyd vox fdmd vh srvwdyomd slwdqmh r qmlkryrm ph¡xvreqrm xgdomhqrvwl/ d x wuhfhpx 0 r nxwx µwr jd }dwydudmx1 +Slwdqmh r nxwx lpd vplvod l }d plprvpmhuqh sudyfh/ dnr vh udgl r nxwx sulsdgqlk vpmhuryqlk yhnwrud$, Vol fqr ud}plµomdpr l r rgqrvx gylmx udyqlqd= dnr vx xvsruhgqh 0 nrolnd lp mh ph¡xvreqd xgdomhqrvw+B,/ d dnr vh vlmhnx 0 nrolnl nxw }dwydudmx+B,1 Surpdwudpr ol rgqrv sudyfd l udyqlqh/ lpd vplvod lvwud }lwl oh }l ol gdql sudydf x gdqrm udyqlql lol qh/ sd/ dnr qh oh }l x qmrm/ mh ol mrm xvsruhgdq 0 nrolnd mh xgdomhqrvw+B, 0 lol mx suredgd 0 nrmd wr fnd mhvw surerglµwh l nrolnl mh sulnorql nxw+B,1 Gd elvpr rgjryrulol qd srvwdyomhqd slwdqmd/ qdmsulmh lvwdnqlpr qhnr0 olnr whphomqlk flqmhqlfd= Gylmh udyqlqh vx ph¡xvreqr xvsruhgqh wr fqr rqgd ndg vx lp qrupdoql yhnwrul nrolqhduql> gyd sudyfd vx ph¡xvreqr xvsruhgqd wr fqr rqgd ndg vx lp vpmhuryql yhnwrul nrolqhduql> udyqlqd l sudydf vx ph¡x0 vreqr xvsruhgql wr fqr rqgd ndg vx sulsdgql qrupdoql yhnwru l sulsdgql vpmh0 uryql yhnwru ph¡xvreqr rnrplwl1 L rsfhqlwr= Nxw l}ph¡x gylmx udyqlqd mh mhgqdn nxwx l}ph¡x qmlkrylk qrupdoqlk yhnwrud> nxw l}ph¡x gydmx sudydfd mh mhgqdn nxwx l}ph¡x qmlkrylk vpmhuryqlk yhnwrud> nxw l}ph¡x udyqlqh l sudyfd mhvw vxsohphqw +gr Z2 , nxwd l}ph¡x sulsdgqrjd qrupdoqrj yhnwrud l sulsdgqrjd vpmhuryqrj yhnwrud +y1 fuwh },1
;;
SRJODYOMH 51 OLQHDUQD DOJHEUD S
Q
V
ψϕ
Nxw l}ph¡x gydmx yhnwrud vh ud fxqd sr irupxol l} Sulpmhud 515151 Xgdomhqrvw gylmx wr fdnd W @ +{ > | > } , l W2 @ +{2 > |2 > }2 , ud fxqdpr sr sr}qdwrm iru0 pxol=
s{
g+W> W2, @ Xgdomhqrvw g+s > s2 ,
+ 2
{,2 . +|2 |,2 . +}2 },2=
xvsruhgqlk sudydfd s l s2 mhgqdnd mh xgdomhqrvwl qml0 krylk vmhflµwd v elor nrmlp qd qmlk rnrplwlp sudyfrp1 Qhnd vx xvsruhgqh udyqlqh l 2 }dgdqh Khvvhrylp mhgqdg }edpd
qf u s @ 3> 2 qf2 u s2 @ 3>
qf @
qf2=
Wdgd mh xgdomhqrvw g+ > 2 , mhgqdnd ms s2 m flp mh qf @ qf2 / rgqrvqr/ s . s2 flp mh qf @ qf21 Xgdomhqrvw g+W > , wr fnh W @ +{ > |> }, rg udyqlqh vh rguh¡xmh wdnr gd vh wr fnrp W sror }l udyqlqd xvsruhgqd v l l}ud fxqd xgdomhqrvw g+ > , @ g+W > ,1 Gd el vh rguhglod xgdomhqrvw xvsruhgqlk sudydfd/ pr }h lk vh suhvmhfl qd qmlk rnrplwrp udyqlqrp l l}ud0
fxqdwl xgdomhqrvw grelyhqlk surerglµwd1 Xgdomhqrvw rg sudyfd gr qmhpx xvsruhgqh udyqlqh rguh¡xmh vh l}ud fxqdydqmhp xgdomhqrvwl rg elor nrmh wr fnh wrjd sudyfd gr udyqlqh1 Qdsrnrq/ xgdomhqrvw l}ph¡x gydmx plprvpmhuqlk sudydfd rguh¡xmh vh l}ud fxqdydqmhp xgdomhqrvwl l}ph¡x gylmx xvsruhgqlk udyqlqd rg nrmlk vydnd vdgu }l sr mhgdq surpdwudql sudydf1
Sulpmhu 51617 Gdql vx sudyfl
t
{4
5
@
|
4
@
}7 5
l
t2
{6 3
@
|4 4
@
}.5 4
=
Lvwud }lpr qmlkry ph¡xvreql sror }dm1 Sudyfl t l t2 qlvx ph¡xvreqr xvsruhgql/ mhu lp vpmhuryql yhnwrul v @ ^5 }l sudydf t 4 5` l v2 @ ^3 4 4` qlvx nrolqhduql1 Qhnd mh udyqlqd µwr vdgu
l xvsruhgqd mh v sudyfhp t2 1 ]d qrupdoql yhnwru q rg vplmhpr x}hwl fnx v v2 @ ^4 5 5` +y1 Whruhp 5151:,/ d }d qmh}lqx srvyh rguh¡hqx wr
vplmhpr x}hwl W @ +4> 3> 7, qd sudyfx t 1 Wdnr grelydpr +{ 4, . 5+| 3, 5+} 7, @ 3> wm1 { 5| . 5} < @ 3= ]d udyqlqx 2 / µwr vdgu }l sudydf t2 l xvsruhgqd mh v sudyfhp t / qd vol fdq qd flq +q2 @ v2 v / W2 @ +6> 4> 5,, grelydpr
2 { 5| . 5} . 6 @ 3=
Udyqlqh l 2 vx xvsruhgqh +l ud}ol flwh,/ sd vx sudyfl t l t2 plprvpmhuql1 Qmlkryx ph¡xvreqx xgdomhqrvw fhpr grelwl srprfx mhgqdg }ded }d l 2 x qrupdoqrpx reolnx=
2 { . 2 | 2 } 4 @ 3= { 2 | . 2 } 6 @ 3> Gdnoh/ g+t > t2 , @ g+ > 2 , @ s . s2 @ 6.4 @ 71 Nxw * l}ph¡x vpmhuryqlk yhnwrud v l v2 rg t l t2 uhgrp/ grelydpr l} qmlkryd vndoduqrj xpqrµnd=
5161 DQDOLWLFND JHRPHWULMD frv
*@
51617
v uv2 @ 8v 8u8v2 8
sE
;< I
2 2 fnun2 u I @ 2 f2 n2 n2 32 2 n2 n22 3 u
> gdnoh * @ Ze =
Qhnh nulyxomh l sorkh
Sr}dedylpr vh qdmsulmh w}y1 nulyxomdpd guxjrjd uhgd1 Qhnd mh x udyqlql gdq sudyrnxwql nrruglqdwql vxvwdy +R> l> m ,1 Srg nulyxomrp guxjrjd uhgd srgud}plmhydpr vnxs vylk wr fdnd W @ +{> | , x udyqlql nrruglqdwh nrmlk }dgryromdydmx mhgqdg }ex guxjrjd vwxsqmd
+4;, D{2 . E|2 . F{| . G{ . H| . I @ 3 v gdqlp uhdoqlp nrh flmhqwlpd D> E> F> G> H> I 5 U/ wdnylpd gd mh eduhp mhgdq rg D/ E lol F ud}ol flw rg qxoh1 Vyh nulyxomh v rsfrp mhgqdg }erp
+4;, prjx vh grelwl ndr suhvmhfl udyqlqd l vwr }qlk sodµwryd 0 vwrjd lk qd}l0 ydpr l
fxqmrvmh fqlfdpd= Vdgd fhpr pdor eromh xsr}qdwl nux }qlfx/ holsvx/ klshuerox l sduderox/ wm1 qmlkryh mhgqdg }eh x qhnlp srvheqlp sror }dmlpd v re}lurp qd rgdeudql nrruglqdwql vxvwdy1
Gh qlflmd 51614 Qhnd mh V wr fnd x udyqlql l u sr}lwlydq uhdoql eurm1 Vnxs
N+V > u, iW m g+V> W , @ uj vylk wr fdnd W
x udyqlql xgdomhqlk u rg wr
fnh qd}lydpr nux
}qlfrp1 Sulwrp nd }hpr gd mh wr fnd V qmhqr vuhglµwh +lol fhqwdu,/ d eurm u gd mrm mh sroxpmhu +lol udglmxv,1
V
Qhnd mh/ x vxvwdyx +R> l> m ,/ V @ +{f > |f , l W @ +{> | ,1 L} gh qlflmvnh mhgqdnrvwl g+V> W , @ u +y1 fuwh }, grelydpr sulsdgqx mhgqdg }ex
{ {f,2 . +| |f ,2 @ u2 =
+
< \
\ 2
+4 I2 > d, @ iW m g+I > W , . 5d A g+I > I2 ,1 Wdgd wr
g+I2> W , @ 5dj rg qd}lydpr holsvrp1 Wr fnh I/ I2 }ryhpr }dulµwlpd +lol irnxvlpd,/ d eurm h 2 g+I > I2 , jhrphwulmvnlp hnvfhqwulflwhwrp surpdwudqh holsvh1 +X vox fdmx h @ 3/ wm1 I @ I2 I / holsvd H srvwdmh nux }qlfrp N> V @ I l u @ d1, Gx }lqx V V2 / jgmh vx V l V2 suhvmh fqh wr fnh rg H sudyfhp I I2 / qd}lydpr yholnrp rvl> sulplmhwlpr gd mh gxomlqd g+V> V2, yholnh rvl 5d l gd mh h ? d1 Qhnd mh/ x vxvwdyx +R> l> m ,/ I @ +{f h> |f ,/ I2 @ +{f . h> |f , l W @ +{> | ,1 Wdgd gh qlflmvnd mhgqdnrvw fl gs+I> W , . g+I2> W , @ 5d sryod
s 2 2 +{ {f . h, . +| |f , . +{ {f h,2 . +| |f ,2 @ 5d/ µwr gdmh 2 2 +{ {f , +| |f , . @ 4> e2 d2 h2 > +e d,= +53, 2 d e2 Qhnd vx
|f , qd}lydpr vuhglµwhp/ d gx }lqx V Ve pdorp rvl rg H > gxomlqd g+V > Ve , pdoh rvl mhvw }dulµwd qd sudyfx xvsruhgqrp v \ 0rvl/ sulsdgqd 5e1 Qdod}h ol vh holsvlqd
mhgqdg }ed fh elwl reolnd +53, x nrmrm vx { {f l | |f l}plmhqlol pmhvwd1
Gh qlflmd 51615 Qhnd vx I l I2 wr fnh x udyqlql l qhnd mh d wdndy uh0
fnryql srgvnxs K+I > I2 > d, @ iW m doql eurm gd mh 3 ? 5d ? g+I > I2 ,1 Wr
mg+I> W ,g+I2> W ,m @ 5dj rg qd}lydpr klshuerorp1 Wr fnh I> I2 }ryhpr
}dulµwlpd +lol irnxvlpd ,/ d eurm h 2 g+I> I2 , jhrphwulmvnlp hnvfhq0
wulflwhwrp surpdwudqh klshueroh1 +Vox fdm h @ 3/ wm1 I @ I2/ qlmh prjxf1, Gx }lqx
VV2/ jgmh vx V l V2 suhvmh fqh wr fnh rg K sudyfhp II2/ qd}lydpr
jodyqrp rvl> sulplmhwlpr gd mh gxomlqd g+V> V2, jodyqh rvl 5d l gd mh h A d1 Qhnd mh/ x vxvwdyx +R> l> m ,/ I @ +{f h> |f ,/ I2 @ +{f . h> |f , l W @ {> |,1 Wdgd gh qlflmvnd mhgqdnrvw mg+I> W , g+I2> W ,m @ 5d sryod fl s s m +{ {f . h,2 . +| |f,2 +{ {f h,2 . +| |f,2 m@ 5d/ µwr gdmh +{ {f ,2 +| |f ,2 @ 4> e2 h2 d2 = +54, d2 e2 < 7 DW \ E W \ ) 6 6 6 ) +
\ E
2 [
[ D
[ D
;
Uhodflmd +54, mh klshuerolqd mhgqdg }ed x vox fdmx ndg mrm mh yholnd rv xv0 sruhgqd v [ 0rvl1 Wr fnx V @ +{f > |f , qd}lydpr vuhglµwhp/ d gx }lqx V Ve vsruhgqrp rvl rg K> gxomlqd g+V > Ve, vsruhgqh rvl mhvw 5e1 Sudyfh | @ |f @K +{ {f, qd}lydpr klshuerolqlp dvlpswrwdpd1 Qdod}h ol vh klshu0 erolqd }dulµwd qd sudyfx { @ {f xvsruhgqrp v \ 0rvl/ x sulsdgqrm mhgqdg }el reolnd +54, fh { {f l | |f l}plmhqlwl pmhvwd wm1
| |f,2 +{ {f,2 e2 d2
+
@4
=
+55,
Qhnd vh klshuerolqd }dulµwd I l I2 qdod}h qd sudyfx | |f @ { {f +vlphwul fqr v re}lurp qd qmh}lqr vuhglµwh V @ +{f > |f ,, l qhnd mh e @ d1 Wdgd mh I @ +{f d> |f d,/ I2 @ +{f . d> |f . d, l dvlpswrwh vx xvsruhgqh v nrruglqdwqlp rvlpdp +y1 fuwh },/ d klshuerolqd mhgqdg }ed srsulpd reoln +
{ {f,+| |f , @
d2 5
=
+56,
5161 DQDOLWLFND JHRPHWULMD
s, @ iW m g+I> W , @ g+I> s,j vylk wr
xgdomhqlk rg wr fnh I l rg sudyfd s qd}lydpr sduderorp +lol klwqlfrp ,1 ]d I nd }hpr gd mh }dulµwh +lol irnxv ,/ d }d s gd mh udyqdolfd +lol gluhnwulvd , grwl fqh sduderoh1
s1
v
R}qd flpr g+I> s, d/ sd qhnd mh x vxvwdyx +R> l> m , sudydf s xvsruhgdq \ 0rvl l }dgdq mhgqdg }erp { @ {f @2 / d I @ +{f . @2 > |f, +y1 fuwh },1
[ DB [ [ DB R
<
;
W W
)
\
S
L} gh qlflmvnrjd xymhwd vh wdgd odnr l}yhgh sulsdgqd sduderolqd mhgqdg }ed
| |f,2 @ 5d+{ {f,= Dnr mh udyqdolfd s }dgdqd mhgqdg }erp { @ {f . @2 / d }dulµwh I +
mhgqdg }ed srsulpd reoln
{
@+ f
+58,
@2 > |f,/
| |f,2 @ 5d+{ {f,= +58 , Ndg mh udyqdolfd s xvsruhgqd v [ 0rvl l }dgdqd mhgqdg }erp | @ |f @2 / rgqrvqr | @ |f . @2 / d }dulµwh I @ +{f > |f . @2 ,/ rgqrvqr I @ +{f > |f @2 ,/ +
grelydpr vlphwul fqh mhgqdg }eh
{ {f,2 @ 5d+| |f,>
+
+59,
rgqrvqr/
{ {f,2 @ 5d+| |f,=
+
+59 ,
Ndr µwr vpr sudydf x udyqlql prjol }dgdwl gymhpd mhgqdg }edpd vd }d0 mhgql fnlp sdudphwurp +y1 +6 ,,/ d sudydf x survwrux wulpd mhgqdg }edpd vd }dmhgql fnlp sdudphwurp +y1 +6,,/ wdnr 0 sdudphwduvnl 0 pr }hpr }d0 gdwl l nulyxomx x udyqlql lol survwrux1 Sulpmhulfh/ sdudphwduvnh mhgqdg }eh }d nux }qlfx N+R> u, +wm1 {2 . | 2 @ u2 , x udyqlql v sudyrnxwqlp nrruglqdwqlp
l> m , mhvx= { @ u frv w/ | @ u vlq w/ w 5 ^3> 5 l1 Surpwudpr ol lvwx nux }qlfx x survwrux H v sudyrnxwqlp nrruglqdwqlp vxvwdyrp+R> l> m > n, +udyqlqd vh srgxgdud v nrruglqdwqrp [\ 0udyqlqrp,/ sdudphwduvnl }dslv srsulpd reoln { @ u frv w/ | @ u vlq w/ } @ 3/ ^3> 5 l1 Gdndnr gd nulyxomd/ rsfhqlwr/ pr }h lpdwl ylµh ud}ol flwlk sdudphwduvnlk }dslvd 0 mhgqdg }ded1
Sulpmhu 51618 Sdudphwduvnh mhgqdg }eh
{ @ u+w vlq w,> | @ u+4 frv w,> } @ 3> w 5 U>
+5:,
rguh¡xmx udyqlqvnx nulyxomx nrmx qd}lydpr flnorlgrp +y1 fuwh } gromh,1 Flnorlgd rslvxmh sxwdqmx nrmrp vh jled wr fnd D @ +3> 3> 3, qd nux }qlfl N {2 . +| u,2 @ u2/ } @ 3/ grn vh N nrwuomd sr [ 0rvl1
<
< U
$
2
Uπ
2
;
BU
U ;
Flnorlgd1 Dvwurlgd1
Sulpmhu 51619 Mhgqdg }eh { @ u frv w> | @ u vlq w> } @ 3>
w 5 ^3> 5 l >
+5;,
suhgvwdyomdmx sdudphwduvnl }dslv udyqlqvnh nulyxomh dvwurlgh/ µwr mx rslvxmh wr fnd D @ +u> 3> 3, qd nux }qlfl N +{ eo ,2 . | 2 @ + oe ,2 / } @ 3/ grn vh N nrwuomd l}qxwud sr nux }qlfl N {2 . |2 @ u2/ } @ 3/ qh l}od}hfl l} udyqlqh } @ 31 Holplqludqmhp sdudphwud w grelydpr mhgqdg }ex +} @ 3, 2
{
2
. | @
2
u =
+5; ,
Sulpmhu 5161: ]d gdqx nrqvwdqwx d A 3/ sdudphwduvnh mhgqdg }eh 6dw2 6dw > | @ > } @ 3> w 5 U> {@ 4 . w 4 . w rguh¡xmx w}y1 Ghvfduwhvry olvw = < D
2
BD
;
D
Ghvfduwhvry olvw1
;
U Nux }qd x}yrmqlfd1
Holplqludqmhp sdudphwud w grelydpr mhgqdg }ex +}
{ . |
6d{| @ 3=
+5 w 5 U> suhgvwdyomdmx nux
}qx x}yrmqlfx +lol flolqgul fqx vsludox, 0 survwruqx nul0 yxomx nrmx rslvxmh wr fnd µwr vh qd xgdomhqrvwl u yuwl rnr sudyfd s { @ 3/ | @ 3/ l lvwrgreqr vh vwdoqrp eu}lqrp d jled xvsruhgqr v wlp sudyfhp1 Surprwulpr vdgd/ sr vol fqrvwl v nulyxomdpd guxjrjd uhgd/ w}y1 sorkh guxjrjd uhgd1 Qhnd mh x survwrux H }dgdq sudyrnxwql nrruglqdwql vxvwdy +R> l> m > n,1 Srg sorkrp guxjrjd uhgd +lol nydgulnrp, srgud}xplmhydpr vnxs vylk wr fdnd W @ +{> |> } , x survwrux H nrruglqdwh nrmlk }dgryromdydmx mhgqdg }ex guxjrjd vwxsqmd
D{2 . E|2 . F}2 . G{| . H{} . I |} . J{ . K| . M} . N @ 3> v uhdoqlp nrh flmhqwlpd D/ E / F / G/ H / I / J/ K / M l N / srg xymhwrp gd mh eduhp mhgdq rg D/ E / F / G/ H lol I ud}ol flw rg qxoh1 Srvheqr fh qdv
}dqlpdwl vdpr qhnh nydgulnh1 Mhgqdg }ed +{
{f,2 . +| |f,2 . +} }f,2 @ u2
+64,
+{
{f,2 . +| |f,2 . +} }f,2 @ 4
+65,
suhgvwdyomd nxjolqx sorkx +lol vihux, vd vuhglµwhp V @ +{f > |f > }f , l sroxpmhurp +lol udglmxvrp, u A 31 Qhsud}ql suhvmhn ryh sorkh udyql0 qrp mhvw lol nux }qlfd lol wr fnd/ µwr sryod fl gd vh nux }qlfd x survwrux pr }h }dgdwl l ndr suhvmhn vihuh l udyqlqh +y1 fuwh },1 ]d gdqh uhdoqh nrqvwdqwh d/ e l f/ mhgqdg }ed
d2
e2
f2 rguh¡xmh sorkx nrmx qd}lydpr holsvrlgrp1 X ryrpx srvheqrp sror }dmx qmhjryh vx rvl D D2 / E E2 l F F2 +y1 fuwh } , xvsruhgqh v nrruglqdwqlp rvlpd/ d gxomlqh vx lp uhgrp 5mdm/ 5mem l 5mfm1
Vihud l holsvrlg1
Qhsud}ql holsvrlgryl suhvmhfl udyqlqdpd xvsruhgqlp v nrruglqdwqlp rvlpd mhvx lol holsvh lol wr fnh1 Sulplmhwlpr gd x vox fdmx d @ e @ f holsvrlg srvwdmh vihurp1 Qdgdomh/ mhgqdg }ed
{2 |2 } 2 . +66, @4 d2 e2 f2 rslvxmh mhgqrnuloql holswl
fql klshuerorlg1 Qmhjryl qhsud}ql suhvmhfl udyq0 lqdpd xvsruhgqlpd vd ] 0rvl mhvx lol klshueroh lol wr fnh/ grn vx px suhvmhfl udyqlqdpd xvsruhgqlp v [\ 0udyqlqrp holsvh1 Flnol fnlp }dpmhqdpd { |/ | }/ } {ld e/ e f/ f d grelydpr mhgqdg }ex lvwh sorkh x guxjrp sror }dmx +\ 0rv mh sryodµwhqd,=
#
#
#
#
#
#
2
}2 f2
{d2 @ 4=
+66 ,
d mrµ mhgqrp wdnyrp }dpmhqrp grelydpr mhgqdg }ex +sryodµwhqd mh [ 0 rv,= Mhgqdg }ed 2
2
2
+66 ,
2
{d2 |e2 . }f2 @ 4
+67,
rslvxmh gyrnuloql holswl
fql klshuerorlg1
Mhgqrnuloql l gyrnuloql holswl fql klshuerorlg1
Qmhjry qhsud}ql suhvmhn udyqlqrp xvsruhgqrp vd ] 0rvl mhvw klshuerod/ grn px mh qhsud}ql suhvmhn udyqlqrp xvsruhgqrp v [\ 0udyqlqrp lol holsvd lol wr fnd1 Flnol fnl l}plmhqmxmxfl nrruglqdwh +ydulmdeoh, {> |> } / ndr l sulsdgqh nrqvwdqwh d> e> f/ grelydpr mhgqdg }eh lvwh sorkh x ud}ol flwlp sror }dmlpd= 2
2
2
}f2 {d2 . |e2 @ 4 2 2 2 |e2 }f2 . {d2 @ 4=
Mhgqdg }ed
{2 d2
.
|2 e2
l
@ 5}
+67 ,
+67 ,
+68,
rslvxmh sorkx nrmx qd}lydpr holswl
fqlp sduderorlgrp/ d sulnd}dqd mh gromh qd fuwh }x1
Holswl fql l klshueror fql sduderorlg1
Idnwru 5 x prqrpx 5} qlmh elwdq/ dol mh whkql fnl +dojheduvnl, srjrgdq1 Ndudnwhulvwl fql suhvmhfl ryh sorkh sulnodgqlp udyqlqdpd mhvx holsvh lol sd0 uderoh1 Rgjrydudmxflp flnol fnlp l}pmhqdpd grelydpr mrµ gylmh mhgqdg }eh lvwh sorkh x ud}ol flwlp sror }dmlpd1 Mhgqdg }ed
|2 e2
2
{d2 @ 5}
+69,
rguh¡xmh sorkx nrmx qd}lydpr klshuerol
fqlp sduderorlgrp +y1 fuwh } jruh,1
5161 DQDOLWLFND JHRPHWULMD
+6;, d2 e2 {2 }2 +6 d2 f2 | @ 5d} 2 +73, rslvxmx uhgrp holswl
fqh/ klshuerol fqh l sduderol fqh ydom fdvwh +lol flolq0 gul fqh, sorkh +y1 sulsdgqh fuwh }h,1 Gdndnr gd vx l x rylp mhgqdg }edpd
prjxfh sulmh vsrplqmdqh flnol fnh l}pmhqh1 Ryh ydom fdvwh sorkh vx vdpr yuor srvheql sulpmhul +rsfh, ydom fdvwh sorkh
Ydom fdvwh sorkh1
Gh qlflmd 51617 Qhnd mh x udyqlql gdqd nulyxomd N +y1 Gh qlflmx 71615, wh qhnd mh s sudydf nrml suredgd 1 Surpdwudmpr vnxs vylk sudydfd x survwrux nrml vlmhnx nulyxomx N l xvsruhgql vx v sudyfhp s1 Wuhwludmxfl vydnl sudydf wr fnryqlp vnxsrp/ sulsdgqx +wr
fnryqx , xqlmx qd}lydpr ydom
fdvwrp +lol flolqgul fqrp , sorkrp1 Sulwrp jryrulpr gd mh sudydf s l}yrgqlfd +lol jhqhudwulvd ,/ d nulyomd N udyqdolfd +lol gluhnwulvd , wh ydom fdvwh sorkh1 2
2
Sulpmhulfh/ holswl fqrm ydom fdvwrm sorkl %@2 . +K2 @ 4 mhgqd l}yrgqlfd mhvw 2 2 ] 0rv/ d udyqdolfd mrm mh holsvd %@2 . +K2 @ 4/ } @ 31 Sulplmhwlpr gd mh vydnd udyqlqd +wulylmdoqd, ydom fdvwd sorkd +}d nulyxomx N wuhed x}hwl rgjrydudmxfl sudydf n,1 Pl fhpr/ qdm fhµfh/ surpdwudwl rqh ydom fdvwh sorkh l}yrgqlfh nrmlk vx nrruglqdwqh rvl/ d udyqdolfh vx lp qhnh rg sr}qdwlk nulyxomd1 +Udyqdolfd/ qdudyqr/ qh fh qx }qr oh }dwl x qhnrm rg nrruglqdwqlk udyqlqd1,
{ . | 5 @ 3
pr }hpr }dgdwl l suhvmhnrp gylmx ydom fdvwlk sorkd1 Holplqludpr ol/ qdlph/ ydulmdeox | l} suyh mhgqdg }eh xyuµwhqmhp +l} rqh guxjh, | @ {.5/ grelydpr mhgqdg }ex ydom fdvwh sorkh +{
4,2 . 522 @ 4>
+|
4,2 . 522 @ 4>
nrmd }dmhgqr v udyqlqrp { . | 5 @ 3 rguh¡xmh wx nux }qlfx1 Vol fqr/ holp0 lqludqmhp ydulmdeoh { grelydpr mhgqdg }ex ydom fdvwh sorkh nrmd }dmhgqr v udyqlqrp { . | 5 @ 3 rguh¡xmh wx lvwx nux }qlfx1 Sulplmh0 wlpr gd wd nux }qlfd qlmh srvyh rguh¡hqd gymhpd grelyhqlp +qhwulylmdoqlp, ydom fdvwlp sorkdpd/ mhu qmlkry suhvmhn vdgu }l l nux }qlfx nrmd vh grelmh qmh0 jrylp suhvlmhfdqmhp udyqlqrp { | @ 31
51618
Sroduql/ flolqgul
fql l vihuql nrruglqdwql vxvwdyl
Sruhg sudyrnxwqrjd nrruglqdwqrj vxvwdyd x udyqlql/ x pqrjlp vh vox fdmh0 ylpd whkql fnl nrulvqlp srnd}xmh l w}y1 sroduql nrruglqdwql vxvwdy µwr fhpr jd vdgd gh qludwl1 Qhnd mh s elor nrml sudydf x udyqlql l qhnd mh qd qmhpx }dgdq nrrugl0 qdwql vxvwdy +R> sf ,/ nsf n @ 41 ]d elor nrmx wr fnx W x udyqlql / W 9@ R/ $ $ qhnd mh * nxw l}ph¡x yhnwrud sf l RW 1 +Rygmh mh yd }qr gd mh sf suyl d RW guxjl nudn surpdwudqrjd nxwd> x vxsurwqrp mh wdm nxw *1, Dnr mh W @ R vwdyomdpr/ sr grjryrux/ gd mh sulsdgql nxw * @ 31 Qhnd mh @ g+R> W , xgdomhqrvw rg R gr W 1 Sulplmhwlpr gd mh wr fnd W srvyh rguh¡hqd eur0 mhylpd 0 sroduqlp nrruglqdwdpd 0 l */ 5 ^3> l l * 5 ^3> 5 l/ sd slµhpr W @ + > *,1 Wr fnx R @ +3> 3, qd}lydpr lvkrglµwhp +lol srorp,/ d }udnx rguh¡hqx v R l sf 0 sroduqrp rvl sroduqrjd nrruglqdwqrj vxvwdyd srg r}qdnrp +R> sf > !, lol +R> f > *f , +Rygmh f l *f r}qd fdydmx rsfh mhglql fqh yhnwruh v sr fhwnrp x +elor nrmrm, wr fnl W @ + > *,= f mh rguh¡hq $ xvpmhuhqrp gx }lqrp RW / d *f mh wdqjhqflmdoql yhnwru qd vuhglµqmx nux }qlfx x wr fnl W 1 Sulplmhwlpr gd vh/ orndoqr johgdmxfl/ udgl ghvqrp sudyrnxwqrp vxvwdyx v lvkrglµwhp x wr fnl W 1, ]dgdpr ol x udyqlql l sudyrnxwql nrruglqdwql vxvwdy +R > l> m , wdnr gd exgh R @ R l l @ sf / odnr fhpr xvwdqrylwl nrmh vx yh}h l}ph¡x Nduwh}lmhylk fnh W x +y1 fuwh },= +{> | , l sroduqlk nrruglqdwd + > *, elor nrmh wr
s
{ @ frv *> | @ vlq *> | @ {2 . |2> * @ dufwdq = {
+74, +74 ,
5161 DQDOLWLFND JHRPHWULMD <
7 [\
\
ρ
M 2
l> !, mhgqrvwdyqlmh rg rgjrydudmxflk x Nduwh}lmhyx vxvwdyx +R> l> m ,1 Sulpmhulfh/ vuhglµqmd nux }qlfd x vxvwdyx +R> l> m , lpd mhgqdg }ex {2 .| 2 @ u2 / d x sroduqrp vxvwdyx +R> l> !, 0 @ u1
Sulpmhu 516143 Udyqlqvnx nulyxomx µwr mx x sroduqrpx vxvwdyx rslvxmh mhgqdg }ed @ d*/ d A 3 +nrqvwdqwd,/ qd}lydpr Duklphgryrp vsludorp1
+Rygmh grsxµwdpr * 5 ^3> l> y1 fuwh } gromh1,
2
S
Dπ
Duklphgryd vsludod1
S
2
D
S
Nduglrlgd1
Odnr vh surymhul gd mh x Nduwh}lmhyx vxvwdyx qmh}lqd mhgqdg }ed {2 . | 2 @ + d2 dufwdq2 % 1 Nulyxomd v sroduqrp mhgqdg }erp @ d+4 . frv *, mh w}y1 nduglrlgd +lol vufrolnd nulyxomd ,1 Qdgdomh/ @ frv * mh sroduqd mhgqdg }ed nux }qlfh v Nduwh}lmhyrp mhgqdg }erp +{ 2 ,2 . | 2 @ e 1 Sroduqd mhgqdg }ed @ frv 5* rslvxmh
fhwyhurolvwqx gmhwholqx +nulyxomx v fhwlul shwomh> y1 fuwh } +d, gromh,/ grn sroduqd mhgqdg }ed 2 @ d2 frv 5* rslvxmh w}y1 ohpqlvndwx +lol rvplfx/ rgqrvqr/ nulyxomx v gylmh shwomh> y1 fuwh } +e,=
2
D
S
D
2
S
E
X survwrux H/ sruhg sudyrnxwqrjd +Nduwh}lmhyd, nrruglqdwqrj vxvwdyd/
fhvwr udgl sudnwl fqlk ud}orjd }dgdmhpr l w}y1 flolqgul fql l vihuql nrruglqdwql vxvwdy1 Qhnd mh elor nrmd udyqlqd x survwrux H1 Qhnd mh x gdq sr0 oduql vxvwdy +R> sf > !, +R> f > *f ,1 Qhnd mh t sudydf wr fnrp R rnrplw qd udyqlqx 1 Qdsrnrq/ qhnd mh qd t gdq nrruglqdwql vxvwdy +R> n,> sd jd qd}rylpr ] 0rvl1 Wlph mh x survwrux H gh qludq flolqgul
fql nrruglqdwql vxvwdy +R> sf > !> n, +R> f> *f > n, x nrmhpx vh vydnrm wr fnl W pr }h sulgl0 mholwl xuh¡hqd wurmnd + > *> ,/ jgmh vx l * sroduqh nrruglqdwh/ x vxvwdyx
sf > !,/
rnrplwh surmhnflmh W wr fnh W qd udyqlqx / d mh nrruglqdwd/ x vxvwdyx +R> n,/ rnrplwh surmhnflmh W wr fnh W qd sudydf t +y1 fuwh },1 Τ=(ρ,ϕ,]
= 7
]
N
Ο
α
S
ϕ
ρ
7
S
s
{ @ frv *> | @ vlq *> } @ > | @ {2 . | 2 > * @ dufwdq > @ }= {
+75, +75 ,
Vihuql nrruglqdwql vxvwdy x survwrux H gh qludpr qd vomhghfl qd flq= Qhnd mh elor nrmd udyqlqd x survwrux l qhnd mh x }dgdq sroduql nrrugl0 qdwql vxvwdy +R> sf > !, +R> f > *f ,1 Qhnd mh t sudydf wr fnrp R rnrplw qd udyqlqx l qhnd t qrvl nrruglqdwql vxvwdy +R> n,/ wm1 ] 0rv1 R}qd flpr v W rnrplwx surmhnflmx qd udyqlqx sr yroml rgdeudqh wr fnh W / W 9@ R/ x survwrux1 Xr flpr gd vx wdgd srvyh rguh¡hql eurmhyl u @ g+R> W , A 3/ $ & 5 ^3> ` 0 nxw l}ph¡x udglmxv0yhnwrud RW l n l * 5 ^3> 5 l 0 nxw l}ph¡x $ sf l udglmxv0yhnwrud RW / nrmh qd}lydpr vihuqlp nrruglqdwdpd wr fnh W 1 Exgxfl gd mh wr fnd W srvyh rguh¡hqd xuh¡hqrp wurmnrp +u> > *,/ vplmhpr slvdwl W @ +u> > *,1 Sulwrp/ dnr mh W qd sr}lwlyqrm }udfl ] 0rvl qhnd vx mrm vihuqh nrruglqdwh +u> 3> 3,/ d qd qhjdwlyqrm 0 +u> 3> ,1 Lvkrglµwx R vh sulgl0 mhomxmx vihuqh nrruglqdwh +3> 3> 3,1 Grelyhql vxvwdy r}qd fxmhpr v +R> sf > !> , lol +R> uf > f > *f ,1 +X guxjrm r}qdfl vx * l l}plmhqlol pmhvwd sd vh/ orndoqr johgdmxfl/ rshw udgl r ghvqrp sudyrnxwqrp vxvwdyx v lvkrglµwhp x wr fnl W 1, ]dgdpr ol x survwrux H sudyrnxwql nrruglqdwql vxvwdy +R> l> m > n, l vihuql vxvwdy +R> sf > !> ,/ rgqrvqr +R > uf > f > *f ,/ wdnr gd exgh R @ R/ sf @ l l gd lp vh ] 0rvl srgxgdudmx/ pr }hpr rguhglwl +y1 fuwh }, yh}h l}ph¡x sudyrnxwqlk +{> |> } , l vihuqlk + > &> *, nrruglqdwd elor nrmh wr fnh W =
s
{ @ frv * vlq &> | @ vlq * vlq &> } @ u frv &> } | u @ {2 . | 2 . } 2 > & @ duffrv > * @ dufwdq = 2 2 2 { { .| .}
s
+76, +76 ,
= ]
7
θ 2
;
[
ϕ
U \
ρ
<
7
Sulpmhu 516144 Vihulqd mhgqdg }ed +x sudyrnxwqlp nrruglqdwdpd, {2 .|2 . }2 @ d2 srsulpd x vihuqlp nrruglqdwdpd +xyuµwdydqmhp sr +76,, wulylmdoql reoln u @ d1 Nux }ql flq }dslv +x sudyrnxwqlp nrruglqdwdpd, {2 . | 2 . } 2 @ d2 / } @ 3/ x vihuqlp nrruglqdwdpd srvwdmh u @ d/ & @ Z2 1
5161 DQDOLWLFND JHRPHWULMD
51619
5/
gylmh +ud}ol flwh, udyqlqh gdqrjd sudphqd S 1 Grnd}dwl gd vh mhgqdg }ed vydnh udyqlqh wrjd sudphqd pr }h grelwl qhnrp olqhduqrp nrpelqdflmrp gylmx gdqlk mhgqdg }ded/ wm1 }d vydnl 5 S srvwrmh > 2 5 U wdnr gd exgh
+D { . E| . F } . G , . 2 +D2 { . E2| . F2} . G2, @ 3=
+77,
Grnd}1 Dnr mh @ / wyugqmd mh r flwr lvwlqlwd +x}ph vh elor nrml 9@ 3 l 2 @ 3,1 Vdvylp vol fqr yulmhgl x vox fdmx @ 2 1 Qhnd mh vdgd 5 S / wm1 qhnd vdgu }l s/ l qhnd mh 9@ 9@ 2 1 Qhnd mh Wf @ +{f > |f > }f , elor nrmd wr fnd x udyqlql l l}ydq sudyfd s1 Wdgd Wf qh oh }l x / sd mh D{f . E |f . F }f . G 9@ 31 Rgdehulpr elor nrmd gyd uhdoqd eurmd / 2 }d nrmd yulmhgl
2
@
E2 |f . F2}f . G2 = DD2{{f . .E | .F } .G f
f f 2 }dmhgqr v
Wf @ +{f > |f> }f,
+78,
R flwr mh gd ryl l }dgrydomdydmx uhodflmx +77,1 Qdsrnrq/ mhgqdg }ed +77, wdgd rslvxmh mhglqvwyhqx udyqlqx/ l wr xsudyr / mhu mx }dgryromdydmx nrruglqdwh vydnh wr fnh qd gdqrpx sudyfx s l nrruglqdwh rgdeudqh wr fnh Wf 1 51 Grnd}dwl gd +vydnd, udyqlqd µwr vdgu }l ] 0rv lpd mhgqdg }ex D{ . E| @ 31 61 Qhnd mh sudydf s }dgdq mhgqdg }edpd D{ . E| . F} . G @ 3 l D2{ . E2 | . F2 } . G2 @ 3 +suhvmhnrp gylmx udyqlqd,1 Qdslvdwl qmhjryx ndqrqvnx mhgqdg }ex1 X wx vyukx srwud }lpr rqh udyqlqh x sudphqx S sudyfd s nrmh vx xv0 sruhgqh v gymhpd nrruglqdwqlp rvlpd/ qsu1 v [ 0rvl l \ 0rvl1 Qmlkryh vx mhgqdg }eh uhgrp I | . J } . K @ 3 l H2 { . J2 } . K2 @ 3 +xvs1 ]dgdwdn 51,/ jgmh vplmhpr x}hwl I @ D2 E D E2 l H2 @ D E2 D2 E 1 L} wlk n8 + n.2 % mhgqdg }ded uhgrp grelydpr } @ MC > } @ M2 C > µwr gdmh wud }hqx 2 ndqrqvnx mhgqdg }ex 2 {. M .2
C2 .2
@
| . M8
C 8
@
}
4
=
2 C 4`/ }d vpmhuryql yhnwru ^I J H J Rgdeluxfl/ xpmhvwr ^ C 2 2 .2 8 grelydpr hnylydohqwqx ndqrqvnx mhgqdg }ex srod}qrjd sudyfd s
H2I`/
M 2 {. M +n } .2 8 @ . C @ 2 IJ2 H2I = 71 Qhnd vx gdql udyqlqd D{ . E| . F} . G @ 3 l sudydf s + 3+ %3%f 5 35 @ K f @ S f 1 Grnd}dwl gd }d nxw * l}ph¡x l s yulmhgl ryd uhodflmd= @ Dd . Ee . Ff s = vlq * @ s 2 D . E2 . F 2 d2 . e2 . f2
433
SRJODYOMH 51 OLQHDUQD DOJHEUD
Srjodyomh 6
NRQYHUJHQFLMD L QHSUHNLGQRVW 614
HOHPHQWDUQH IXQNFLMH
X ryrpx rgmhomnx fhpr vh edylwl qdmyd }qlmlp ixqnflmdpd +y1 ¢41614, l} U +y1 ¢4171405, x U/ wm1 qdmyd }qlmlp uhdoqlp ixqnflmdpd mhgqh uhdoqh ydulmdeoh1 Srvheqx sr}ruqrvw fhpr srvyhwlwl rqlp ixqnflmdpd µwr lpdmx yholnx yd }qrvw x whkql fnrm sulpmhql l sudnvl1
61414
]dgdydqmh ixqnflmd l} 7 x 7
Xrel fdmhqr mh gd vh +uhdoqrp, ixqnflmrp qd}lyd l elor nrml dqdolwl fnl +dojh0 eduvnl, l}ud} µwr vdgu }l ydulmdeox +sdudphwdu/ qhrylvqx surplmhqmlyx yhol flqx, { l rslvxmh sudylor/ wm1 nrqd fql uhgrvolmhg dojheduvnlk lol lqlk rshudflmd/ sr nrmlpx vh rguh¡xmh sulgux }hqd mrm +l r qmrm rylvqd, yhol flqd |= Wdnr vh/ sulp0 mhulfh/ }dslvl t
| @ {2> | @
2%3 > %nD
|@
2 3% >
qd}lydmx ixqnflmdpd l} U x U1 Gd elvpr vh v wlp vdvylp vor }lol/ wm1 gd el vyh elor x vnodgx v Gh qlflmrp 41614> gu }dw fhpr vh ryrjd grjryrud= Qhnd mh | @ i +{, }dslv x nrmhpx dqdolwl fnl l}ud} i +{, vdgu }l +vdpr mhg0 qx, ydulmdeox {1 Wdgd wdm }dslv gh qlud ixqnflmx i = [ $ U/ sul fhpx mh [ U srgvnxs nrml wyruh vyl rql hohphqwl { 5 U }d nrmh l}ud} i +{, rguh¡xmh mhglqvwyhql uhdoql eurm/ d i mh sudylor µwr jd surslvxmh l}ud} i +{,1 X wrpx vox fdmx jryrulpr gd mh ixqnflmd i = [ $ U }dgdqd dqdolwl
fnl }dslvrp | @ i +{,1
Sulpmhu 61414 Rguhglpr ixqnflmx i = [ $ U l} dqdolwl fnrjd }dslvd 3 1 | @ 2%%nD X l}ud}x 2%3 vh sulplmhqmxmx vyh fhwlul rvqryqh ud fxqvnh rshudflmh1 ]qdpr %nD
gd vx ./ / l gh qludqh }d vydnl xuh¡hq sdu uhdoqlk eurmhyd/ grn mh glmhomhqmh 434
435
SRJODYOMH 61 NRQYHUJHQFLMD L QHSUHNLGQRVW
gh qludqr }d vydnl xuh¡hq sdu +d> e,/ e 9@ 31 Volmhgl gd mh gh qlflmvnr srgux fmh [ wud }hqh ixqnflmh i = [ $ U vnxs i{ 5 U m { . 8 9@ 3j/ wm1 [ @ Uqi8j/ 3 1 d i mh sudylor { :$ i +{, @ | surslvdqr l}ud}rp 2%%nD
Sulpmhu 61415 Rguh¡xmx ol dqdolwl fnl l}ud}l
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443
SRJODYOMH 61
NRQYHUJHQFLMD L QHSUHNLGQRVW
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6141
444
HOHPHQWDUQH IXQNFLMH
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445
SRJODYOMH 61
61417
NRQYHUJHQFLMD L QHSUHNLGQRVW
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Sulpmhu 614144
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447
SRJODYOMH 61
NRQYHUJHQFLMD L QHSUHNLGQRVW
V
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6151
448
NRQYHUJHQFLMD QL]RYD L UHGRYD
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NRQYHUJHQFLMD UHDOQLK QL]RYD L UHGRYD
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61514
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8151
584
GLIHUHQFLUDQMH VNDODUQLK IXQNFLMD
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ghulydflmh vh grelydmx sr irupxol
585
SRJODYOMH 81
IXQNFLMD YLH YDULMDEOD
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$ U/ [ U2/ grsxµwd
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Sduflmdoqh ghulydflmh ylµlk uhgryd
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8151
586
GLIHUHQFLUDQMH VNDODUQLK IXQNFLMD
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Sulpmhu 8151; Rguhglpr vyh sduflmdoqh ghulydflmh guxjrjd uhgd l wuhfh sdu0
flmdoqh ghulydflmh sr {/ | l { uhgrp wh sr {/ {/ l | uhgrp +rqgmh jgmh srvwrmh, }d ixqnflmx i = [ $ U> [ U2 > i +{> | , @ {2 | . { oq |= Gh qlflmvnr srgux fmh [ mh rwyruhqd sroxudyqlqd i+{> |, 5 U2 m | A 3j l ixqnflmd i mh ghulydeloqd1 E%c+ E%c+ Sulwrp mh/ x elor nrmrm wr fnl +{> | , 5 [ / YsY% @ 5{| .oq |/ YsY+ @ {2 . %+ 1 Sulplmhwlpr gd vx l remh sduflmdoqh ghulydflmh ghulydeloqh ixqnflmh/ wm1 gd 2 Y 2 s E%c+ mh ixqnflmd i gydsxw ghulydeloqd/ l gd mh Y EsY%E%c+ 2 @ 5| / Y+Y% @ 5{ . + / Y 2 s E%c+ Y%Y+
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Whruhp 81517 Qhnd mh ixqnflmd i = [
$ U/ [ U2/ ghulydeloqd qd qhnrm
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587
SRJODYOMH 81
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Gh qlflmd 81515 Qhnd mh ixqnflmd i = [ ixqnflmd Dnr mh glihuhqflmdo
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gi = [ $ Krp+U6> U, @ U6> { :$ gi +{, @ ^ YsY%E% YsY%E% `> glihuhqflmdeloqd ixqnflmd x wr
fnl {f / rqgd nd }hpr gd mh ixqnflmd i gyd0 sxw glihuhqflmdeloqd x wr fnl {f 1 Sulsdgql glihuhqflmdo g+gi ,+{f, = U6 $ Krp+U6> U, @ U6 qd}lydpr guxjlp glihuhqflmdorp ixqnflmh i x wr fnl {f l nudfh r}qd fxmhpr v g2 i +{f ,1 Yulmhgqrvwl wrjd guxjrj glihuhqflmdod g2 i +{f ,+g{, r}qd fxmhpr v g2 i +{f ,g{2 lol/ ndg qh pr }h grfl gr }dexqh/ vdpr v g2 i +{,1 6
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8151
58:
GLIHUHQFLUDQMH VNDODUQLK IXQNFLMD
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$ U/ [ U2/ nrmd qhnd
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fqr/ uhfl fhpr gd mh yhnwruvnr sromh Y = $ Y qhsuhnlgqr +glihuhqflmd0 eloqr , dnr mh sulsdgqd yhnwruvnd ixqnflmd z = [ $ U qhsuhnlgqd +glihuhq0 flmdeloqd ,1
Gh qlflmd 91515
Qdsrphqd 91515
Sulplmhwlpr gd suhwkrgqd gh qlflmd qlmh dsulrul nruhn0 wqd1 Wuhedor el/ qdlph/ x} qmx grnd}dwl gd vx qhsuhnlgqrvw l ghulydeloqrvw lqydulmdqwh v re}lurp qd l}eru sudyrnxwqrj nrruglqdwqrj vxvwdyd x survwrux H/ rgqrvqr/ gd qh rylvh r l}erux suhgvwdyomdmxfh ixqnflmh1 Lpdmxfl qd xpx qdpmhqx rylk vnulsdwd/ wdm fhpr grnd} lvsxvwlwl1
91515
Judglmhqw/ glyhujhqflmd l urwdflmd
Rygmh fhpr gh qludwl wul olqhduqd rshudwrud/ qhrskrgqd }d pdwhpdwl fnr rsl0 vlydqmh whphomqlk }l fnlk }dnrqd wyduqrjd vylmhwd x nrmhpx }lylpr1 Rwnulw fhpr gd vh/ }dsudyr/ udgl r mhgqrp rshudwrux gmhorydqmh nrmhjd vh r flwxmh qd wul qd flqd 0 rylvqr r remhnwlpd qd nrmh gmhoxmh1 Mhgqrvwdyqrvwl udgl/ vndoduqd l yhnwruvnd sromd fhpr rgpdk }dgdydwl qmlkrylp suhgvwdyqlflpd/ wm1 vndoduqlp l yhnwruvnlp ixqnflmdpd x Nduwh}lmhyx ghvqrp sudyrnxwqrp nrruglqdwqrp vxvwdyx +R> l> m > n, x H U 1
Qhnd vx i = [ $ U l z = [ $ U / [ U / uhgrp vndoduqr l yhnwruvnr sromh/ wh qhnd vx red glihuhqflmdeloqd1 Judglmhqwrp vndoduqrjd sromd i qd}lydpr yhnwruvnr sromh Ys Ys judg i = [ $ U > judg i @ + Ys Y% > Y+ > Y5 ,> wm1
Gh qlflmd 91516
%c+c5 %c+c5 %c+c5 judg i +{> |> }, @ Ys EY% l . Ys EY+ m . Ys EY5 n> +{> |> }, 5 [= Glyhujhqflmrp yhnwruvnrjd sromd z @ +z% > z+ > z5 , qd}lydpr vndoduqr sromh Y+ Y5 % gly z = [ $ U> gly z @ Y Y% . Y+ . Y5 > wm1 E%c+c5 . Y+ EY+%c+c5 . Y5 EY5%c+c5 > +{> |> }, 5 [= gly z+{> |> }, @ Y%Y% Urwdflmrp yhnwruvnrjd sromd z qd}lydpr yhnwruvnr sromh Y+ Y% Y5 Y+ Y% 5 urw z = [ $ U > urw z @ + Y Y+ Y5 > Y5 Y% > Y% Y+ ,> wm1 E%c+c5 ,m . urw z+{> |> } , @ + Y5 EY+%c+5 Y+ EY5%c+c5 ,l . + Y% EY5%c+c5 Y5 Y% Y+ E%c+c5 Y% E%c+c5 .+ Y% Y+ ,n> +{> |> }, 5 [=
Sulplmhwlpr gd
urw z @
urw z grsxµwd irupdoql }dslv x ghwhuplqdqwlqx reolnx= l
Y Y% z%
m
Y Y+ z+
n
Y Y5 z5
=
Yd }qrvw xsudyr gh qludqlk srmpryd xsr}qdw fhpr wlmhnrp gdomqmhjd l}odjdqmd1 Vdgd xr flpr vdpr wr gd mh judglmhqw qhnd ixqnflmd gh qludqd qd vnxsx vylk glihuhqflmdeloqlk vndoduqlk sromd v yulmhgqrvwlpd x vnxsx glihu0 hqflmdeloqlk yhnwruvnlk sromd/ gd mh glyhujhqflmd qhnd ixqnflmd gh qludqd
9151
5 |> }, :$ i +{> |> }, @ {|2} 1 Ys Ys 2 2 2 judg i +{> |> }, @ + Ys Y% > Y+ > Y5 ,+{> |> } , @ +| } > 5{|} > 6{| } ,1
+e, Rguhglpr glyhujhqflmx yhnwruvnrjd sromd +{> |> }, :$ z+{> |> } , @ +5{> {|2> {}2 ,1
Y+ Y5 % gly z+{> |> } , @ + Y Y% . Y+ . Y5 ,+{> |> } , @ 5 . 5{| 5{} 1 +f, Rguhglpr urwdflmx yhnwurvnrjd sromd z l} sulpmhud +e,1
urw z+{> |> }, @
l
Y Y% z%
m
n
Y Y+ z+
Y Y5 z5
+{> |> }, @ @ +3> }2 > |2,1
Vdgd fhpr srnd}dwl gd vh judglmhqw/ glyhujhqflmd l urwdflmd prjx rslvdwl vdpr mhgqlp rshudwrurp1 ]qdnrp u + flwdpr= qdeod, r}qd flpr w}y1 Kd0 Y > Y > Y ,1 Lol/ hn0 plowrqry glihuhqflmdoql rshudwru 0 irupdoql yhnwru + Y% Y+ Y5 ylydohqwqr/
u l Y%Y . m Y+Y . n Y5Y = Johgdpr ol qd u ndr qd rshudwru gh qludq qd yhnwruvnrp survwrux vylk
glihuhqflmdeloqlk vndoduqlk +yhnwruvnlk, sromd/ srmdyomxmx vh/ x vyh}l v Gh ql0 flmrp 91516/ vomhghfh prjxfqrvwl= u+i , @ judg i / +gmhorydqmh rshudwrud u qd vndouqr sromh i ,> +umz, @ gly z/ +irupdoql vndoduql xpqr }dn u l yhnwruvnrj sromd z,> u z @ urw z/ +irupdoql yhnwruvnl xpqr }dn u l yhnwruvnrj sromd z,> Yd }qd mh flqmhqlfd gd mh rshudwru u olqhdudq x vydnrm rg qdyhghqlk lqwhu0 suhwdflmd/ wm1 gd yulmhgl rydm whruhp=
Whruhp 91514 Qdeod mh olqhduql rshudwru/ wm1 }d elor nrmd gyd glihuhqflmd0 eloqd vndoduqd sromd i> j = [ $ U/ elor nrmd gyd glihuhqflmdeloqd yhnwruvnd sromd z> x = [ $ U / [ U / l elor nrmd gyd eurmd > 5 U yulmhgl= +l, u+i . j , @ u+i , . u+j ,> +ll, +um+z . x,, @ +umz, . +umx,> +lll, u +z . x, @ +u z, . +u x,1 Grnd}1 Whruhp 91514 vh grnd}xmh l}udyqr l odnr1 Grnd }lpr qsu1 wyug0
qmx +lll,$
u +z . x, @
l
Y Y%
m
Y Y+
n
Y Y5
z% . x% z+ . x+ z5 . x5
+y1 ¢51415,
@
633
SRJODYOMH 91
l
Y Y% z%
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.
l
Y Y% x%
m
Y Y+ x+
XYRG X YHNWRUVNX DQDOL]X
n
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@ +u z, . +u x,1
Wul lgxfd whruhpd grqrvh/ uhgrp/ yd }qd vyrmvwyd judglmhqwd/ glyhujhqflmh l urwdflmh1
$ U/ [ U / glihuhqflmdeloqd vndoduqd sromd/ fo nrqvwdqwqr vndoduqr sromh/ ! = \ $ U/ i ^[ ` \ U/ glihuhqflmd0 eloqd ixqnflmd l > 5 U1 Wdgd mh Whruhp 91515 Qhnd vx i> j = [ +l, +ll, +lll, +ly, +y,
judg fo @ ff > judg+i . j, @ judg i . judg j> judg+i j, @ j judg i . i judg j> judg+ s} , @ } }h@_ s}32s }h@_ } + flp mh j+{, 9@ 3,> judg+! i , @ ! judg i 1
Grnd}1 Wyugqmh vx l}udyqd srvomhglfd judglmhqwryh gh qlflmh1 Sulplmh0
wlpr gd mh wyugqmd +ll, lvwr µwr l Whruhp 91514+l,1
Whruhp 91516 Qhnd vx z> x = [
$ U / [ U / glihuhqflmdeloqd yhnwruvnd
sromd/ i> j = [ $ U glihuhqflmdeloqd vndoduqd sromd/ f nrqvwdqwqr yhnwruvnr sromh l > 5 U1 Wdgd mh +l, gly f @ ff > +ll, gly+z . x, @ gly z . gly x> +lll, gly+z x, @ +urw zmx, +zm urw x,> +ly, gly+i z, @ +judg i mz, . i gly z> +y, gly+i judg i , @ +judg i m judg j , . i 7j / +j gydsxw glihuhqflmdeloqr ,/ sul fhpx mh gly judg @ +umu, u2 w}y1 Odsodfhry glihuhqflmdoql Y2 . Y2 . Y2 > rshudwru +ghowd ,/ wm1 Y% 2 Y+ 2 Y5 2 +yl, gly+urw z, @ ff +z gydsxw glihuhqflmdeloqr ,1
Grnd}1 Wyugqmh volmhgh qhsrvuhgqr l} glyhujhqflmlqh gh qlflmh1 Grnd }lpr qsu1 wyugqmx +ly,$ Y Es+ Y Es5 % gly+i z, @ Y Es Y% . Y+ . Y5 @ Y+ Ys Ys Ys Y% Y5 Y% z% . i Y% . Y+ z+ . i Y+ . Y5 z5 . i Y5 @ Y+ Ys Ys Y% Y5 ++ Ys Y% > Y+ > Y5 ,m+z% > z+ > z5 ,, . i + Y% . Y+ . Y5 , @ +judg i mz, . i gly z1
Sulplmhwlpr gd vh wyugqmd +ll, srgxgdud v rqrp l} Whruhpd 91514+ll,1 Wyug0 qmd +y, volmhgl l} +ly, flp mh z @ judg j1 Qdsrnrq/ x grnd}x wyugqmh +yl, wuhed sulplmhqlwl Vfkzdu}ry whruhp1
Whruhp 91517 Srg suhwsrvwdyndpd Whruhpd 91516 yulmhgl +l, urw fy @ ff > +ll, +lll,
urw+z . x, @ urw z . urw x> urw+z x, @ +gly x,z +gly z,x . +xmu,z +zmu,x/
9151
634
XYRG X WHRULMX R SROMLPD
Y . z Y . z Y l/ vol
fqr/ +xmu, qryl glihuhqflmdoql sul fhpx vx +zmu, z% Y% + Y+ 5 Y5 rshudwrul> +ly, urw+i z, @ +judg i , z i urw z> +y, urw+i judg i , @ +judg i , +judg j ,/ l srvhelfh/ urw+judg i , @ ff > +yl, urw+urw z, @ judg gly z z +z gydsxw glihuhqflmdeloqr ,/ sul
fhpx Odsodfhry rshudwru wuhed sulplmhqlwl qd vydnx nrruglqdwqx ixqnflmx rg z1
Grnd}1 Ndr l x suhwkrgqrp whruhpx/ vyh wyugqmh vx l}udyqh srvomhglfh rgjrydudmxflk gh qlflmd1 Grnd }lpr qsu1 wyugqmx +lll,$ Qd olmhyrm vwudql mh yhnwruvnd ixqnflmd
urw+ z x, @ urw+z+ x5 z5 x+ > z5 x% z% x5 > z% x + z+% , @
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d ghvqrm mh yhnwruvnd ixqnflmd
+gly x,z +gly z,x . +xmu,z +zmu,x @ Y+ Y+ Y+ Y5 Y% Y5 Y% Y5 % ++ Y Y% . Y+ . Y5 ,z% > + Y% . Y+ . Y5 ,z+ > + Y% . Y+ . Y5 ,z5 , ++ YY%% . YY++ . YY55 ,x%> + YY%% . YY++ . YY55 ,x+ > + YY%% . YY++ . YY55 ,x5 ,. Y% Y+ Y5 Y% Y+ Y5 % Y+ Y5 .x%+ Y Y% > Y% > Y% , . x+ + Y+ > Y+ > Y+ , . x5 + Y5 > Y5 > Y5 , z%+ YY%% > YY%+ > YY%5 , z+ + YY+% > YY++ > YY+5 , z5 + YY5% > YY5+ > YY55 ,1
Vuh¡lydqmhp vh odnr srnd}xmh gd vh udgl r mhgqrm wh lvwrm yhnwruvnrm ixqnflml1 Sulplmhwlpr gd mh wyugqmd +ll, lvwr µwr l wyugqmd x Whruhpx 91514+lll,/ wh gd +y, volmhgl l} +ly, sulpmhqrp Vfkzdu}ryd whruhpd1
Qdsrphqd 91516 Wyugqmd +yl, x Whruhpx 91516 vplmhpr suhuhfl rydnr=
judg gly z @ urw+urw z, . z>
wh wlph grelwl rgjryru qd slwdqmh µwr mh judglmhqw vndoduqrj sromd/ nrmh mh glyhujhqflmd yhnwruvnrj sromd1 Suhpgd qdm fhµfh udelpr sudyrnxwql vxvwdy/ x pqrjlp mh vox fdmhylpd mhgqrvwdyqlmh ud fxqdwl x qhnrp guxjrp nrruglqdwqrp vxvwdyx1 Vwrjd qd nudmx ryrjd srgrgmhomnd sulgrgdmhpr +eh} irupdoqrj l}yr¡hqmd, }dslvh ol0 qhduqlk rshudwrud judg/ gly/ urw l Odsodfhryd rshudwrud x flolqgul fqrp l vihuqrp vxvwdyx +y1 ¢51618 l ¢81615 0 ud}pdwudqmh sr Whruhpx 81616,1 X flolqgul fqrpx vxvwdyx +R> f > *f > n,= Ys Ys judg i @ + Ys Y4 > 4 Y) > Y5 ,> 4 Y) Y5 gly z @ 4 Y E4 Y4 . 4 Y) . Y5 > Y) Y) Y4 Y5 Y E4) 5 urw z @ + 4 + Y Y) Y5 ,> Y5 Y4 > 4 + Y4 Y) ,,>
i @ 4
Y E4 Ys Y4 Y4
Y s Y s . 42 Y) 2 . Y5 2 1 X vihuqrpx vxvwdyx +R> uf > f > *f ,= 2
2
635
SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
Ys Ys judg i @ + Ys Yo > o Yw > o t? w Y) ,> 2 Y Ew t? w . Y) > gly z @ o2 Y EoYoo . o t? w Yw o t? w Y) Y E) t? w Y Eo) Y Eow Y Yo w o urw z @ + o t? w + Yw Y) ,> o + t? w Y Y) Yo ,> o + Yo Yw ,,> Ys Ys 2 Y Et? w Yw Y 2s . o2 t? i @ o2 Y EoYoYo . o2 t? Yw w w Y)2 1 u
91516
Xvpmhuhqd ghulydflmd
Qhnd mh X = $ U vndoduqr sromh1 Rgdehulpr elor nrmx wr fnx Wf 5 l elor nrml yhnwru o 5 Y+Wf ,1 Wdgd }d vydnx wr fnx W qd }udfl o rguh¡hqrm v o $ yulmhgl W Wf @ wof / }d qhnl w 5 ^3> l/ sul fhpx mh of @ o sulsdgql mhglql fql
o
mh/ }dsudyr/ xgdomhqrvw g+Wf > W , rg Wf gr W 1 Survmh
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Gh qlflmd 91517 Qhnd mh X = +dnr srvwrml ,
olp L EA |L EAf 3
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qd}lydpr ghulydflmrp vndoduqrjd sromd X x wr
fnl Wf x vpmhux o +lol/ nudfh/ xvpmhuhqrp +vndoduqrp , ghulydflmrp , l r}qd
fxmhpr v YLYEoAf 1
X sudnwl fqrp ud fxqx mh sxqr mhgqrvwdyqlmh edudwdwl suhgvwdyqlnrp i = [ $ U/ [ U / vndoduqrjd sromd X x nrruglqdwqrpx vxvwdyx +R> l> m > n,1 Suyr/ wr fndpd Wf @ +{f > |f > }f , l W @ +{> |> } , sulsdgdmx udglmxv0yhnwrul RW $ $ $ $ $ . |f m . }f n l RW @ {l . |m . }n1 Qdgdomh/ RW @ RWf . WfW @ RW $f .@ w{ofl/ gdnoh/ f f {l . |m . }n @ +{f . w+of ml,,l . +|f . w+of mm ,,m .+}f . w+of mn,,n1 Volmhgl gd vh nrruglqdwh vydnh wr fnh W qd }udfl µwr mx rguh¡xmx Wf l of rslvxmx olqhduqlp mhgqdg }edpd { @ {f . +ofml,w/ | @ |f . +ofmm ,w/ } @ }f . +of mn,w/ w 5 ^3> l1 Surl}od}l gd mh sulmh gh qludqd xvpmhuhqd ghulydflmd/ }dsudyr/ ghulydflmd 'E c c
% + 5 ^3> l $ [ $s U/ wm1 w :$ z+w, @ +{f . +of ml,w> |f . +of mm ,w> }f . +ofmn,w, :$ i +z+w,,/ x wr fnl w @ 31 Suhpd wrpx/ YL EAf Ys E%fYc+o fc5f @ +i z, +3, @ Yo Ys EEf z%+3, . Ys EY+Ef z+ +3, . Ys EY5Ef z5 +3, @ Y% Ys E%f c+f c5f +ofml, . Ys E%fY+c+f c5f +ofmm , . Ys E%fY5c+f c5f ,+of mn, @ Y% +judg i +{f > |f> }f,mof ,1
ixqnflmvnh nrpsr}lflmh
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9151
636
XYRG X WHRULMX R SROMLPD
Whruhp 91518 Ghulydflmd vndoduqrj sromd X> }dgdqrjd ixqnflmrp i = [ $ U/ x wr fnl Wf @ +{f> |f > }f , x vpmhux o +9@ 3, mhgqdnd mh vndoduqrp xp0 qrµnx sulsdgqrjd judglmhqwd v mhglql
fqlp yhnwrurp of / wm1 surmhnflml yhnwrud judg i +{f > |f> }f, qd yhnwru o1 Sulpmhu 91517 L}ud fxqdmpr ghulydflmx vndoduqrjd sromd
+{> |> }, :$ i +{> |> }, @ {2 . 6|} . 8 x wr fnl W @ +6> 5> 4, x vpmhux o @ l . m . n1 Sr Whruhpx 91518/ l}
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sryod fl i +{> |> } , A i +{f > |f> }f,/ wm1 X +W , A X +Wf ,/ flp mh W gryromqr eol}x Wf qd sulsdgqrm }udfl1 Ryr vh pr }h lvnrulvwlwl/ srg qhnlp grgdwqlp xymhwlpd/ }d lvwud }l0 ydqmh orndoqlk hnvwuhpqlk yulmhgqrvwl vndoduqrj sromd X / rgqrvqr/ vndoduqh ixqnflmh i 1
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A3
L} Whruhpd 91518 volmhgl gd vndoduqr sromh i qdmeu }h udvwh x vpmhux µwr jd rguh¡xmh judg i / rgqrvqr/ gd qdmeu }h sdgd x vpmhux µwr jd rguh¡xmh judg i 1 Guxjlp ulmh flpd/ judg i srnd}xmh vpmhu qdmeu }h surpmhqh vndoduqrjd sromd i 1 wrylµh/ judglmhqw vh wlph pr }h l rndudnwhul}ludwl1
Whruhp 91519 Qhnd ixqnflmd i = [
$ U/ [ U / rslvxmh glihuhqflmdeloqr
vndoduqr sromh X= Wdgd judglmhqw wrjd sromd x vydnrm wr
fnl Wf @ +{f > |f > }f , lpd }d sudydf rnrplfx qd ud}lqvnx sorkx x Wf / d l}qrv px mh mhgqdn dsvr0 oxwqrm yulmhgqrvwl sulsdgqh xvpmhuhqh ghulydflmh1 Vd }hwr/ c+f c5f qf > judg i +{f > |f> }f, @ Ys E%Yfq
sul fhpx mh q qrupdod qd ud}lqvnx sorkx
i +{> |> }, @ i +{f> |f > }f ,1
Grnd}1 Rgdehulpr elor nrmx wr fnx Wf @ +{f> |f > }f ,
5 [1
Mhgqdg }ed
sulsdgqh ud}lqvnh sorkh X +W , @ f X +Wf ,, suhod}l x i +{> |> } , @ i +{f > |f > }f ,/ µwr sryod fl
3 @ gi +{> |> }, @ Ys E%fY%c+f c5f g{ . Ys E%fY+c+f c5f g| . Ys E%fY5c+f c5f g} @ @ +judg i +{f> |f > }f ,mgu,/ sul fhpx gu r}qd fxmh yhnwru g{l . g| m . g} n +g{> g|> g} ,1 Exgxfl gd sudwlpr }elydqmh qd ud}lqvnrm sorkl x wr fnl Wf / wr gu vplmhpr wxpd flwl ndr lq qlwh}lpdoql srpdn x qmh}lqrm wdqjhqflmdoqrm udyqlql wr fnrp Wf 1 Suhpd wrpx/ +judg i +{f > |f > }f ,mgu, @ 3 }qd fl gd mh judg i +{f > |f > }f , rnrplw qd wx wdjhqflmdoqx udyqlqx/ wm1 qd surpdwudqx ud}lqvnx sorkx x wr fnl Wf / gdnoh/ xvsruhgdq v sulsdgqlp qrupdoqlp yhnwrurp q1 Volmhgl judg i +{f > |f > }f , @
637
SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
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wr fnl Wf @ +5> 6> 9,1 Vplmhpr suhwsrvwdylwl gd mh surpdwudqd sorkd } i +{> |> } , @ f qhnrj vndoduqrj sromd i = [ $ U/ i +{> |> } , @ {| } . f/ sd mh
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Whruhpx 91517,
Grnd}1 Surplµomdmxfl ndr x ud}pdwudqmx µwr mh suhwkrglor Whruhpx 91518/ vdgd }d yhnwruvnr sromh z @ +z% > z+ > z5 ,/ grelydpr
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638
XYRG X WHRULMX R SROMLPD
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Sulpmhu 91519 L}ud fxqdmpr ghulydflmx yhnwruvnrjd sromd
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3
91517
3
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Qhnd srvheqd yhnwruvnd sromd
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fnr }qd fhqmh1
Sulpmhu 9151: +d, Mhgqrvwdyqr mh surymhulwl gd mh judylwdflmvnr sromh wyduqh f wr fnh Wf v pdvrp pf / wm1 J @ N 6 o2 uf / N 0 nrqvwdqwd l u nun0
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glihuhqflmdeloqr yhnwruvnr sromh qd
[ U = Wdgd mh z srwhqflmdoqr rqgd l vdpr rqgd/ dnr
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639
SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
wuhed nrqvwuxludwl vndoduqr sromh i wdnr gd z exgh qmhjry +qhjdwlyql, judgl0 mhqw1 Mhgqrvwdyqrvwl udgl/ nrqvwuludw fhpr i x srvheqrp vox fdmx ndg mh [ nydgdu1 Suyr sulplmhwlpr gd urw z @ ff sryod fl Y5 @ Y+ / Y% @ Y5 / Y+ @ Y% 1 Y+ Y5 Y5 Y% Y% Y+ Exgxfl gd mh [ nydgdu/ greur mh gh qludqd ixqnflmd i = [ $ U/ U%
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9151
XYRG X WHRULMX R SROMLPD
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Whruhp 9151< Qhnd mh z = [
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63;
SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
YY5+ @ z% l YY5% YY%5 @ z+ / grn mh U5 Y E%c+cr U5 Y E%c+cr Y+ E%c+c5 Y% E%c+c5 + % @ gv . z + {> |> } , gv @ 5 f Y% Y+ Y% Y+
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z5 +{> |> }, z5 +{> |> }f, . z5 +{> |> }f, @ z5 +{> |> },1 x gd mh z @ urw x1
Suhpd wrpx/ nrqvwuxludol vpr wdnyr yhnwruvnr sromh
Qdsrphqd 9151: Whruhp 9151< srnd}xmh gd vx/ qd sulnodgqrp srgux fmx/
yuwor }qrvw l vrohqrlgdoqrvw yhnwruvnrj sromd hnylydohqwqd vyrmvwyd1 Wr yulmhgl l qd rsfhqlwlmhp vnxsx [ U qhjr µwr mh rwyruhql nydgdu/ nrml mh rygmh rgdeudq gd vh srmhgqrvwdyql grnd}1 Lsdn/ qlmh prjxfh rwyruhql nydgdu }dplmhqlwl elor nrmlp rwyruhqlp vnxsrp1 Qhnd mh/ sulpmhulfh/ [ rwyruhql vnxs l}ph¡x gylmx vuhglµqmlk nrqfhqwul fqlk vihud sroxpmhud f l f2 / 3 ? f ? f2 / l qhnd mh qd qmhpx yhnwruvnr sromh Y @ o2 uf/ sul fhpx mh u @ {l.|m .} n/
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u nun l uf @ 1 Mhgqrvwdyql ud fxq srnd}xmh gd mh Y vrohqrlgdoqr sromh/ u wm1 gly Y @ 31 Ph¡xwlp/ qlmh whµnr grnd}dwl gd qhpd yhnwruvnrj sromd qd [ urwdflmd nrmhjd el elod sromh Y / wm1 }d vydnl X = [ $ Y mh urw X 9@ Y 1 L xsudyr mh wdnr rgdeudqr srgux fmh [ vphwqmd }d yuwor }qrvw1 Qdlph/ dnr l} [ l}edflpr qsu1 wr fnh µwr sulsdgdmx ] 0rvl/ qd wrpx srgux fmx [2 yhnwruvnr
sromh Y srvwdmh l eh}yuwor }qlp1 ]dlvwd/ mhgqrvwdyqr mh surymhulwl gd wdgd }d yhnwruvnr sromh X @ )o f +vihuql vxvwdy, yulmhgl Y mf2 @ urw+X mf2 ,1 X grnd}x/ sul gh qludqmx yhnwruvnrjd sromd x/ rgdeudol vpr x5 @ ff = Wr/ gdndnr/ qlmh qx }qr1 Prjol vpr/ qdlph/ wdnr gh qludwl qhnx lqx nrruglqdwqx ixqnflmx/ dol vh rqgd suhrvwdoh gylmh prudmx gh qludwl qhwulylmdoqr1 Qdudyqr/ ql wdnyr gh qludqmh qlmh qhrskrgqr mhu vh vyh wul nrruglqdwqh ixqnflmh prjx gh qludwl ndr qhwulylmdoqh1 Qdsrnrq/ fhvwr vh/ sr vol fqrvwl vd vndoduqlp srwhqflmdorp i +z @ judg i ,> yhnwruvnr sromh x vd vyrmvwyrp urw x @ z qd}lyd yhnwruvnlp srwhqflmdorp yhnwruvnrjd sromd z1
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9161
63<
NULYXOMQL LQWHJUDO
91518
Ymh
}eh
41 Rguhglwl ud}lqvnh sorkh vndoduqrjd sromd X }dgdqrjd ixqnflmrp
+{> |> }, :$ i +{> |> }, @ duffrv s+2%n52 1
51 Rguhglwl vwuxmqlfh +udyqlqvnrjd, yhnwurvnrj sromd Y }dgdqrjd ixqnflmrp
+{> |, :$ z+{> |, @ +|> {,1
61 X nrmlp wr fndpd lµ fh}dyd judglmhqw vndoduqrjd sromd
+{> |> }, :$ i +{> |> }, @ { . | . } . 6{|}/ wh nrolnl px mh l}qrv x wr fnl W @ +3> 4> 5,B
sromd 71 L}ud fxqdwl glyhujhqflmx yhnwruvnrjd s +{> |> }, :$ z+{> |> } , @ {2 . |2 . }2 uf/ u sul fhpx mh uf @ / u @ {l . |m . } n l u @ nun1 u sromd 81 L}ud fxqdwl urwdflmx yhnwruvnrjd s 2 +{> |> }, :$ z+{> |> } , @ { . |2 . }2 f/ sul fhpx mh yhnwru f nrqvwdqwdq1 91 L}ud fxqdwl sulsdgqx xvpmhuhqx ghulydflmx dnr mh +d, i +{> |> } , @ s 2 2 2 / W @ +4> 5> 4,/ o @ 2 n> % n+ n5
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916
NULYXOMQL LQWHJUDO
Nulyxomql lqwhjudo fh elwl srrsfhqmh rguh¡hqrj lqwhjudod sr vhjphqwx ^d> e` U qd rguh¡hql lqwhjudo sr +survwruqrm l sr glmhorylpd jodwnrm, nulyxoml }dgdqrm rgjrydudmxfrp sdudphwul}dflmrp1 Gh qludw fhpr gylmh yuvwh wdnyr0 jd srrsfhqmd= }d vndoduqr l }d yhnwruvnr sromh1
91614
Nulyxomd l qmh}lqr xvpmhuhqmh
Sulvmhwlpr vh mhgqrvwdyqh +sr glmhorylpd, jodwnh nulyxomh +y1 Gh qlflmx 71615,1 Rygmh fhpr mx vd }hwr uhirupxoludwl x dpelmhqwx U v suryrnxwqlp nrruglqdwqlp vxvwdyrp +R> l> m > n,/ vylmhvql pdqmndyrvwl µwr surl}od}h l} vyl0 mhvqrj }drelod }hqmd ud}olnh l}ph¡x sdudphwul}deloqrj vnxsd l nulyxomh1
Gh qlflmd 91614 Vnxs U qd}lydpr mhgqrvwdyqrp jodwnrp nulyx0 omrp +v uxerp , dnr
U
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643
SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
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vxnodgqr vdvwdyomhqd rg @ RD/ 2 @ DE l @ ER v sdudphwul}dflmdpd +uhgrp,= | @ 3/ { 5 ^3> 5`/ xvpmhuhqd srudvwrp sdudphwud {> | @ { . 5/ { 5 ^4> 5`/ xvpmhuhqd sdgrp sdudphwud {> | @ {/ { 5 ^3> 4`/ xvpmhuhqd sdgrp sdudphwud {1 +Gdndnr gd mh prjxfd l guxn flmd sdudphwul}dflmd/ qsu1 u @ +!> # , = ^3> 4` $ U2 / ; 3/ w 5 ^3> ` ? 9 w/ w 5 ^3> ` 2 !+w, @ 6w . 6/ w 5 ^ > 4` / #+w, @ = 99ww . 95// ww 55 ^^ 2 >> 4` ` / xvpmhuhqd srudvwrp sdudphwud w1, Wdnr grelydpr U U U K
w
K 2
U f
+zmgu, @ +zmgu, . +zmgu, . +zmgu, @ w
K
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Uf
+{ . { 3 3,g{ . +{ . {+{ . 5,+4,,g{ . +{ . { { 4,g{ @ @ 1
91617
2
Nulyxomql lqwhjudo x srwhqflmdoqrp sromx
Ud}qryuvql sulpmhul nulyxomqrj lqwhjudod guxjh yuvwh srnd}xmx gd rq srqhndg qh rylvl r xvpmhuhqrm nulyxoml sr nrmrm vh lqwhjulud/ qhjr vdpr r qmh}lqrm sr fhwqrm l nudmqmrm wr fnl1 Vwrjd mh nrulvqr lvwud }lwl ndnyd vx wr yhnwruvnd sromd nulyxomql lqwhjudol nrmlk rylvh vdpr r sr fhwnx l nudmx lqwhjudflmvnh nulyxomh1
9161
64:
NULYXOMQL LQWHJUDO
Qhnd mh z = [ $ U / [ U / qhsuhnlgqd ixqnflmd +yhn0 wruvnr sromh,1 Uhfl fhpr gd nulyxomql lqwhjudo yhnwruvnrjd sromd z qh rylvl r lqwhjudflmvnrp sxwx/ dnr }d vydnh gylmh wr fnh D> E 5 [ l vydnh gylmh
Gh qlflmd 91617
sr glmhorylpd jodwnh nulyxomh D suhpd E / yulmhgl U U +zmgu, @ +zmgu,1
w
w w
> 2
[ µwr sryh}xmx D l E/ xvpmhuhqh rg
w
K
K2
Xsudyr gh qludqr vyrmvwyr ndudnwhul}lud rydm whruhp=
Qhnd mh z = [ $ U qhsuhnlgqr yhnwruvnr sromh qd rwyruh0 qrp l sryh}dqrp srgux fmx [ U 1 Wdgd sulsdgql nulyxomql lqwhjudo qh rylvl r lqwhjudflmvnrp sxwx dnr l vdpr dnr mh z srwhqflmdoqr sromh1
Whruhp 91616
Grnd}1 Suhwsrvwdylpr gd nulyxomql lqwhjudo yhnwruvnrjd +z% >Uz+ > z5 , qh Urylvl r lqwhjudflmvnrp sxwx/ wm1 gd mh +zmgu, @ +zmgu, w
sromd z
@
w
K
K2
}d pd nrmh gylmh xvpmhuhqh nulyxomh rg wr fnh D gr wr fnh E x [ +x vnodgx v Gh qlflmrp 91617,1 Wuhed grnd}dwl gd srvwrml qhnr vndoduqr sromh i = [ $ U wdnyr gd mh z @ judg i 1 X wx vyukx x fyuvwlpr elor nrmx wr fnx Wf @ +{f> |f > }f , 5 [ sd }d vydnx wr fnx W @ +{> |> } , 5 [ rgdehulpr elor nrmx
w
sr glmhorylpd jodwnx nulyxomx [ µwr vsdmd Wf l W +wdnyd srvwrml sr suhwsrvwdyfl qd [ ,/ xvpmhuhqx rg Wf suhpd W 1 Suhwsrvwdynd sryod fl gd mh wlph greur gh qludqd ixqnflmd U i = [ $ U/ i +{> |> } , @ +zmgu,1
w
K
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w
xvpmhuhqd nulyxomd vxnodgqr vdvwdyomhqd rg
U U U +zmgu, @ + +zmgu, . +zmgu,,/
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SRJODYOMH 91
XYRG X YHNWRUVNX DQDOL]X
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Qdsrphqd 91617
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srgux fmx, hnylydohqwqd vyrmvwyd +y1 Whruhp 9151;,/ yulmhgl rydm nrurodu=
Qhnd mh z = [ $ U glihuhqflmdeloqr yhnwruvnr sromh qd nrqyhnvqrp srgux fmx K [ U 1 Wdgd mh
Nrurodu 91614
urw z @ ff / +zmgu, @ 3 w
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Sulpmhu 91619
L}ud fxqdmpr
U
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9161
64<
NULYXOMQL LQWHJUDO
dnr mh z+{> |> } , @ +6{2 |} . | . 8> { } . { }> { | | :, l +d,
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w
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Juhhqryd irupxod
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= [ $ U glihuhqflmdeloqh v U2 wh qhnd mh [ sr}lwlyqr xvpmh0
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9161
654
NULYXOMQL LQWHJUDO
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Qdsrphqd 91618
655
SRJODYOMH 91
Γ Γ
XYRG X YHNWRUVNX DQDOL]X
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f
Qdsrphqd 91619
Johgdpr ol qd ixqnflmh S l T ndr qd nrruglqdwqh ixqnflmh qhnrj udyqlqvnrj yhnwruvnrj sromd z = G $ U2 / z+{> | , @ +S +{> | ,> T+{> |,,/ Juhhqryx irupxox pr }hpr }dslvdwl l rydnr= K K UU +urw zmn,g{g| @ +zmgu, +@ +zmwf ,gv, v ( Y( Y( qd µwr fhpr vh yudwlwl/ x rsfhqlwlmhp vox fdmx/ x ¢71:1
91619
41
Ymh
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L}ud fxqdwl= +d,
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51
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71
v
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9171
656
SORQL LQWHJUDO
917
SORQL LQWHJUDO
Lqwhjuludqmh sr sorkdpd mh mhgqd rg whphomqlk whkqlnd pdwhpdwl fnh }lnh/ µwr fhpr pl rygmh prfl whn gmhorplfh xr flwl1 Sulwrp mh vydndnr qdmyd }qlml w}y1 Rvwurjudgvnl0Jdxvvry whruhp r glyhujhqflml/ nrml mh guxjr srrsfhqmh +suyr mh elor Juhhqry whruhp, Qhzwrq0Ohleql}ryh irupxoh 0 vdgd qd wurvwuxnl lqwhjudo1 Qdlph/ srg qhnlp vh suhwsrvwdyndpd lqwhjudo sr rph¡hqrp sr0 gux fmx Y U suhyrgl qd lqwhjudo sr }dwyruhqrm sorkl V nrmd mh uxe V CY wrjd srgux fmd1 Dol/ vwurjr gh qludwl sorkx l/ srvhelfh/ qmh}lqx sorµwlqx l xvpmhuhqmh yuor mh vor }hq l }dkwmhydq srvdr/ nrml rygmh qlmh prjxfh srvyh nr0 uhnwqr redylwl1 Juxer jryruhfl/ sorkd mh qhsuhnlgqd volnd qhnrj udyqlqvnrj srgux fmd1 ]d qdµx vyukx fhpr wuhedwl/ suhpgd qh vdvylp vwurjx/ lsdn qhµwr wr fqlmx gh qlflmx1
91714
Jodwnd sorkd l qmh}lqd sorµwlqd
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, dnr mh sulsdgqd ixqnflmd glihuhqflmdeloqd +x wr
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w
w
w
V / vxnodgqr vdvwdyomhqh rg
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+
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9171
664
SORQL LQWHJUDO
Whruhp 91715 Qhnd vx z> x = [ $ U / [ U / qhsuhnlgqh yhnwruvnh ixqnflmh/ UU V [ xvpmhulyd UU sr glmhorylpd jodwnd sorkd l > 5 U1 Wdgd mh +l, +zmgV , @ +zmgV ,> v
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w
w 7
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7
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Qdsrphqd 91718
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w
7 UU
w
zmgV , @
+
7
z
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z% g|g} . z+ g}g{ . z5 g{g|1
7
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7n
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2 { g|g} . | g}g{ . } 2g{g|/ 7n
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6 6
=
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111
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72
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665
SRJODYOMH 91
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3% 3+
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V$
91717
Rvwurjudgvnl0Jdxvvryd irupxod
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v
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v fvmh rph¡hqr sr glmhorylpd jodwnrp mhgqrvwdyqr }dwyruhqrp sorkrp V CY Y
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v
T
YT
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v
v
w
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Y5
g{g|g} @
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g},g{g| @
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z5 +{> |> j2 +{> |,,g{g|
72
UU
( z5
z5 +{> |> j+{> |,,g{g|1
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Sulplmhwlpr gd mh UU 7 UU
666
+z5 nmqf2,gV @
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UU
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T
T
UUU Y+ E%c+c5
@
YT
grelydpr
YT
UU
+z5 m mqf ,gV 1 Exgx fl gd vh lqwhjulud sr lvwrp srgux
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T
T UUU
g{g|g} @
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UU
YT
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v
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jgmh vx
frv > frv / frv
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vpmhuryql nrvlqxvl ydqmvnh qrupdoh qd sorkx
CY 1
Sulpmhu 91718 L}ud fxqdmpr sorµql lqwhjudo guxjh yuvwh UU v
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7
v
sr vihul V 111 {2 . |2 . } 2 d2 Ud}ylgqr mh gd mh UU
v
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UU
v
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v 7
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667
SRJODYOMH 91 XYRG X YHNWRUVNX DQDOL]X UU
v
{g|g} . |g}g{ . }g{g|
7
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v
T
7
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R+Y , @
v
Y
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rph¡hqrjd sorkrp
V1
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{g|g} . |g}g{ . }g{g|/
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S +G, @ 2 |g{ . {g|1
G=
v
Y(
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zgV @ +
UUT
z% gV>
UU T
UU
T
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z+ gV>
T
T
T
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Xr flpr gd mh T UU
+
v
YT
i lgV>
UU
v
T
i m gV>
YT
UU
v
i ngV ,1
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YT
T
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v
YT
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91718
v
YT
Vwrnhvryd irupxod
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Y + Y' Y% Y+ ,g{g| @
v
Y(
S g{ . Tg|
9171 SORQL LQWHJUDO
668
nrmrp vh gyrvwuxnl lqwhjudo sr udyqlqvnrp srgux fmh G suhyrgl qd nulyxomql lqwhjudo guxjh yuvwh sr qmhjryx uxex1 Vwdylpr ol z @ +S> T,> grelydpr qmh}lqUU yhnwruvnl }dslv= UUU K K + urw zmn,g{g| @ +zmgu, +@ +zmwf ,gv,1 (
v
T
Y(
Y(
Vwrnhvryd irupxod fh elwl srrsfhqmh Juhhqryh irupxoh qd survwruqr yhn0 wruvnr sromh z> sorkx V U2 l qmh}lq uxe CV= Sulmh vdprjd lvnd}d wuhed gh qludwl vxnodgqr xvpmhuhqmh sorkh l qmh}lqd uxed1 Ndr µwr vpr vh yhf grjryrulol/ sorkx V }dgdqx mhgqdg }erp } @ j+{> | ,> +{> |, 5 G U2 > xvpmhuhqx mhglql fqlp qrupdoqlp yhnwrulpd
3 Y}E%c+ l3 Y}E%c+ m nn qf +{> |, @ t Y%Y}E%c+ 2 Y+ Y}E%c+ 2 nE Y% nE Y+ w
v
r}qd fxmhpr vd V / d xvpmhuhqx qrupdodpd qf +{> | , 0 vd V 1 Xvpmhulpr uxe CG +sr glmhorylpd jodwnx mhgqrvwdyqr }dwyruhqx udyqlqvnx nulyxomx, sr0
v
gux fmd G sr}lwlyqr/ wm1 ndr CG +sudylor ghvqh uxnh ndg mh sdodf xvpmhuhq ndr yhnwrud n,/ sd px sulglmholpr sdudphwul}dflmx +v srudvwrp sdudphwud,
v
CG 111 { @ !+w,/ | @ #+w,/ w 5 ^d> e`1 Exgxfl gd vh uxe CV +sr glmhorylpd jodwnd mhgqrvwdyqr }dwyruhqd nulyxomd, rnrplwr surmlflud qd CG l V grsxµwd sdudphwul}dflmx V 111 u+{> |, @ +{> |> j+{> |,,/ +{> |, 5 G/ v wr vh xvpmhuhqmh v CG suhqrvl qd CV +srudvwrp sdudphwud,/ r}qd flpr jd v ndr CV / wm1 v CV 111 u+!+w,> #+w,, +w, @ +!+w,> #+w,> j+!+w,> #+w,,,/ w 5 ^d> e`1 v w Sulwrp jryrulpr gd vx sorkd V l qmh}lq uxe CV vxnodgqr xvpmhuhql1 w Gdndnr/ x vox fdmx qhjdwlyqrjd xvpmhuhqmd CG grelydpr rgjrydudmxfh xv0 w w v pmhuhql uxe CV / sd l wdgd nd }hpr gd vx sorkd V l qmh}lq uxe CV vxnodgqr xvpmhuhql1 Sulplmhwlpr gd vh x red vox fdmd udgl r srµwlydqmx sudylod ghvqh uxnh ndg mh sdodf xvpmhuhq ndr qrupdoql yhnwru1
Whruhp 91717 +Vwrnhvry whruhp, Qhnd mh z = [ glihuhqflmdeloqd yhnwruvnd ixqnflmd/
v
w V
[
$ U / [ U / qhsuhnlgqr
xvpmhuhqd sr glmhorylpd jodwnd
sorkd/ d CV vxnodgqr mrm xvpmhuhql uxe nrml mh sr glmhorylpd jodwnd mhgqrv0 wdyqr UU }dwyruhqd nulyxomd1 K Wdgd yulmhglK Vwrnhvryd irupxod +urw zmgV , @ +zmgu, +@ +zmwf ,gv,=
w 7
v
Y7
Y7
Grnd}1 Rshw fhpr/ }erj vor }hqrvwl rsfhj vox fdmd/ gdnd}dwl vdpr srvh0 edq vox fdm x nrmhpx mh ixqnflmd z gydsxw ghulydeloqd/ d V l CV vx sulnodgqr rgdeudqh l jodwnh1 Qhnd mh } @ j +{> | ,/ +{> |, 5 G/ hnvsolflwqd mhgqdg }ed sorkh V / rgqrvqr/ V 111 u+{> |, @ +{> |> j+{> |,,/ +{> |, 5 G U2 /
669
SRJODYOMH 91 XYRG X YHNWRUVNX DQDOL]X
qmh}lqd sdudphwduvnd mhgqdg }ed1 Wdgd vh
w v
v
V
rnrplwr surmlflud qd
G1
Qhnd
vx V / CV l CG xvpmhuhql x vndogx v suhwsrvwdynrp/ wh qhnd mh gdqd sdud0 phwul}dflmd
v
CG 111 { @ !+w,/ | @ #+w,/ w 5 ^d> e`1
Wdgd mh
v
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v
@
Y7 UK @
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v
Y7
UU Y'E%c+ (
+
Y%
UU Y + ( UU (
YY+E%c+ ,g{g| @ E%c+ E%c+ , Y+Y +z+{> |> j+{> |,,m YoY% ,,g{g| @ m YoY+
Y% +z+{> |> j +{> |,,
+-,
Yo Yo ,g{g| / +u zm Y% Y+
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v
Y7
w 7
Qdsrphqd 9171: Ndnr }d Juhhqryx wdnr vh l }d Vwrnhvryx irupxox fhvwr udel vndoduql }dslv1 X wx vyukx/ qhnd vx S> T> U = [ $ U/ [ U / qhsuhnlgqr ghulydeloqh ixqnflmh/ d V [ sr glmhorylpd jodwnd sorkd v uxerp CV sr glmhorylpd jodwnrp mhgqrvwdyqr }dwyruhqrp nulyxomrp1 Wdgd vh sul0 sdgqd K Vwrnhvryd irupxod }dslvxmh ndnr volmhgl= Y7 UU 7
S g{ . Tg| . Ug} @
Y' Y ++ YY+ Y5 , frv . + Y5
Y' Y YY% , frv . + Y% Y+ , frv ,gV /
sul fhpx vx frv / frv l frv vpmhuryql nrvlqxvl qrupdoqlk yhnwrud qd sorkx V vxnodgqr xvpmhuhql v uxerp CV 1
Sulpmhu 9171: L}ud fxqdmpr flunxodflmx yhnwruvnrj sromd
9171 SORQL LQWHJUDO
66:
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3
3
3
3
sryod fl
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v
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v
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I
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l
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3
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(
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I
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3
3
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@
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91719
2 UZ
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f
f
3
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rqrpx }d olqhduqx glihuhqflmdoqx mhgqdg }ex suyrjd uhgd1 Qdlph/ suyr vh ulmhµl sulsdgqd krprjhqd mhgqdg }ed/ d rqgd vh ydululudqmhp nrqvwdqdwd gr0 elyd wud }hqr rsfh umhµhqmh srod}qh mhgqdg }eh1 X rsfhp vox fdmx mh srvwxsdn whkql fnl yuor vor }hq l elwqr rylvl r ixqnflml j 1 X pqrjlp vox fdmhylpd hoh0 phqwduqlk ixqnflmd j wdm vh srvwxsdn pr }h elwqr xeu}dwl/ d vuhfd mh gd vx edµ rql x sudnvl yuor fhvwl1 Rygmh fhpr ud}pdwudwl vox fdmhyh ndg mh j srol0 qrp/ xpqr }dn srolqrpd l +sulurgqh, hnvsrqhqflmdoqh ixqnflmh/ ixqnflmd vlq lol frv wh elor nrmd olqhduqd nrpelqdflmd qdyhghqlk vox fdmhyd1 Grnd}dw fhpr gd mh wdgd rsfh umhµhqmh olqhduqh glihuhqflmdoqh mhgqdg }eh +4, }eurm rsfhjd umhµhqmd sulsdgqh krprkhqh mhgqdg }eh +5, l elor nrmhjd srvheqrj umhµhqmd srod}qh mhgqdg }eh +4,1 Sulwrp mh elwqr gd vnxs vylk umh }hqmd lpd vwuxnwxud yhnwruvnrj survwrud +qdg F,/ µwr qdp rwnulydmx gyd vomhghfd whruhpd1
Whruhp :1614 Dnr vx | @ i +{, l |2 @ i2 +{, gyd umhµhqmd olqhduqh krpr0 jhqh mhgqdg }eh +5,/ rqgd mh l mhgqdg }eh1
|
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fi+{, . f2i2+{,/ fc2
5 U/ umhµhqmh wh
Grnd}1 L}udyqrp surymhurp grelydpr | . d| . e| @
+f i +{, . f2 i2 +{,, . d+f i +{, . f2 i2 +{,, . e+f i +{, . f2 i2 +{,, @ f +i +{, . di +f, . ei +{,, . f2 +i2 +{, . di2 +f, . ei2 +{,, @ f 3 . f2 3 @ 31
Uhfl fhpr gd vx gyd umhµhqmd | @ i +{, l |2 @ i2 +{, olqhduqh krprjhqh mhgqdg }eh +5, olqhduqr qh}dylvqd/ dnr l} f i . f2 i2 @ 3 +qxonrqvwdqwqd ixqnflmd,/ fc2 5 U/ volmhgl f @ f2 @ 31
Whruhp :1615 Dnr vx | @ i+{, l |2 @ i2+{, gyd olqhduqr qh}dylvqd umhµhqmd olqhduqh krprjhqh mhgqdg }eh +5,/ rqgd mh
fc2 5 U/ qmh}lqr rsfh umhµhqmh1
|
@
fi+{, . f2i2+{,/
Grnd}1 Wuhed grnd}dwl gd vh l} umhµhqmd +y1 Whruhp :1614, | @ fi+{,. f2 i2 +{,/ fc2 5 U/ pr }h grelwl vydnr srvheqr umhµhqmh/ wm1 gd vx gdqlp sr fhw0 qlp xymhwrp { @ {f > | @ |f > | @ |f nrqvwdqwh f l f2 mhgqr}qd fqr rguh¡hqh1
Xyuvwlpr ol sr fhwql xymhw x wr umhµhqmh l qmhjryx ghulydflmx/ grelydpr ol0 qhduql vxvwdy fi+{f , . f2i2+{f , @ |f/ fi +{f , . f2i2 +{f , @ |f1 Suhwsrvwdyomhqd olqhduqd qh}dylvqrvw sryod fl gd mh ghwhuplqdqwd i +{, i +{, Z +{, @ i +{, i2 +{, 9@ 3/ }d vydnl {/ 2 sd mh/ srvhelfh/ ghwhuplqdqwd surpdwudqrjd olqhduqrj vxvwdyd Gf @ Z +{f , 9@ 31 Volmhgl gd wdm vxvwdy lpd wr fqr mhgqr umhµhqmh ++f ,f > +f2 ,f ,1 R revwrmqrvwl srvheqrj umhµhqmd krprjhqh olqhduqh mhgqdg }eh +5, jryrul vomhghfl whruhp1 Sulwrp grsxµwdpr gd umhµhqmh exgh l nrpsohnvqd ixqnflmd +uhdoqh s ydulmdeoh,/ wm1 qnflmd i = [ $ F/ [ U/ i +{, @ x+{, . ly+{,/ l 4/ sul fhpx vx x l y uhdoqh ixqnflmh1 Ghulydflmrp ixqnflmh i vpdwudpr
:161 GLIHUHQFLMDOQH MHGQDG]EH GUXJRJD UHGD
68<
ixqnflmx { :$ i +{, @ x +{, . ly +{, +ndg jrg vx ixqnflmh x l y ghulydeloqh,1 Mhgqrvwdyqr mh surymhulwl gd Whruhp :16141 l Whruhp :1615 yulmhgh l }d rydnyd gyd nrpsohnvqd umhµhqmd v nrqvwdqwdpd fc2 5 F1
Whruhp :1616 Srvwrml eurm u/ uhdodq lol nrpsohnvdq/ wdndy gd mh | @ ho% srvheqr umhµhqmh krprjhqh olqhduqh glihuhqflmdoqh mhgqdg }eh +5,1
Grnd}1 Irupdoqlp xyuµwhqmhp | @ ho% / | @ uho% l | @ u2 ho% x
mhgqdg }ex +5, grelydpr ho% +u2 . du . e, @ 3/ wm1 u2 . du . e @ 3/ µwr mh w}y1 ndudnwhulvwl
fqd mhgqdg }ed glihuhqflmdoqh mhgqdg }eh +5,1 Exgxfl gd vydnd nydgudwqd mhgqdg }ed lpd umhµhqmh x U lol F/ d x ryrpx vox fdmx gr0 elydpr t uc2 @ @2 @e2 e/ wr vx | @ ho % l |2 @ ho2 % srvheqd umhµhqmd glihuhqflmdoqh mhgqdg }eh +5,1 wrylµh/ dnr mh sulwrp u 9@ u2 5 U rqgd vx sulsdgqd srvheqd umhµhqmd olqhduqr qh}dylvqd sd mh/ sr Whruhpx :1615/ | @ f ho % . f2 ho2 % rsfh umhµhqmh glihuhqflmdoqh mhgqdg }eh +5, +X vox fdmx nrqmxjludqr0nrpsohnvqrj umhµhqmd uc2 5 F/ y1 elomhµnx sr grnd}x lgxfhjd Whruhpd :16171,1 Dnr mh/ sdn/ u @ u2 rqgd vh udgl r vdpr mhgqrp srvheqrp umhµhqmx +}d guxjr mdp fl Whruhp :1617,1
Whruhp :1617 Dnr ndudnwhulvwl fqd mhgqdg }ed krprjhqh olqhduqh glihuhqfl0
mdoqh mhgqdg }eh +5, lpd vdpr mhgqr umhµhqmh/ wm1 dnr mh u @ u2 @ @2 u 5 U/ rqgd mh/ sruhg | @ ho% / srvheqr umhµhqmh l | @ {ho% 1 wrylµh/ exgxfl gd vx ixqnflmh { :$ ho% l { :$ {ho% olqhduqr qh}dylvqh/ wr mh wdgd | @ f ho% . f2 {ho% rsfh umhµhqmh glihuhqflmdoqh mhgqdg }eh +5,1
Grnd}1 L}udyqd surymhud +u @ @2 0 mhglqvwyhqr umhµhqmh ndudnwhulvwl fqh
mhgqdg }eh,+{ho% , . d+{ho% , . e{ho% @ ho% ++5u . d, . +u2 . du . e,{, @ 3 srwyu¡xmh gd mh l | @ {ho% umhµhqmh glihuhqflmdoqh mhgqdg }eh +5,1 Qdgdomh/ ndg el ixqnflmh ho% l {ho% eloh olqhduqr }dylvqh/ krprjhql vxvwdy f ho% . f2{ho% @ 3/ f uho% . f2+4 . u{,ho% @ 3 +guxjd mhgqdg }ed vh grelyd ghulyludqmhp rqh suyh, el lpdr l qhwulylmdoqr ghwhuplqdqwd wrjd vxvwdyd mh umhµhqmh sr f l f2 1 Ph¡xwlp/ 4 { 2 o% G @ h u 4 . u{ @ h2o% 9@ 3 sd mh wulylmdoqr umhµhqmh f @ f2 @ 3 mhglqr qmhjryr umhµhqmh1 Suhpd wrpx/ | @ ho% l |2 @ {ho% vx/ x ryrpx vox fdmx/ olqhduqr qh}dylvqd umhµhqmd glihu0 hqflmdoqh mhgqdg }eh +5,1 X vox fdmx nrqmxjludqr0nrpsohnvqrj umhµhqmd ndudnwhulvwl fqh mhgqdg }eh/ wm1 uc2 @ l/ > 5 U/ 9@ 3/
SRJODYOMH :1 RELFQH GLIHUHQFLMDOQH MHGQDG]EH
693
rsfh umhµhqmh | @ f ho % . f2 ho2 % +vnxs nrpsohnvqlk ixqnflmd, krprjhqh olqhduqh glihuhqflmdoqh mhgqdg }eh +5, }dslvxmhpr srprfx sulurgqh hnvsr0 qhqflmdoqh ixqnflmh l wuljrqrphwulmvnlk ixqnflmd vlq l frv rydnr= @ | @ fhEknq % . f2hEk3q % @ hk% +fhq% . f2 h3q% , S17141: k% h +f +frv { . l vlq {, . f2+frv+{, . l vlq+{,,, @ hk% ++f . f2, frv { . l+f f2, vlq {, hk% +n frv { . n2 vlq {,/ sul fhpx mh n f . f2 5 U/ n2 l+f f2 , 5 lU1 Rvlp wrjd/ odnr vh surymhul gd vx ixqnflmh { :$ hk% frv {> hk% vlq { olqhduqr qh}dylvqh +qdg F,
flp mh 9@ 31 ]dnomx fdn = Rsfh umhµhqmh krprjhqh olqhduqh glihuhqflmdoqh mhgqdg }eh +5,/ | . d| . e| @ 3/ V mhvw vnxs vylk ixqnflmd { :$ iS cS2 +{,/ fc2 5 U lU/ µwr grsxµwdmx rydm }dslv= fho % . f2ho2% / fc2 5 U/ flp vx uc2 5 U l u 9@ u2 > fho% . f2 {ho% / fc2 5 U/ flp mh u @ u2 u 5 U> hk% +f frv { . f2 vlq {,/ f 5 U/ f2 5 lU
flp vx uc2 @ l 5 F/ 9@ 3/ sul fhpx mh uc2 umhµhqmh sulsdgqh ndudnwhulvwl fqh mhgqdg }eh u2 . du . e @ 31
Sulpmhu :1617 Krprjhqrm olqhduqrm glihuhqflmdoqrm mhgqdg }el | . 7| .
7| @ 3 sulsdgd ndudnwhulvwl fqd mhgqdg }ed u2 . 7u . 7 @ 3/ nrmrm mh umhµhqmh u @ u2 @ 51 Rsfh umhµhqmh surpdwudqh glihuhqflmdoqh mhgqdg }eh mh/ gdnoh/ | @ fh32% . f2{h32% / fc2 5 U1 ]dgdpr ol/ qdlph/ elor nrml sr fhwql xymhw { @ {f > | @ |f > | @ |f / grelydpr +xyuµwhqmhp x | l | , olqhduql vxvwdy +v qhsr}qdqlfdpd f l f2 , fh32%f . f2 {fh32%f @ |f / 5fh32%f . f2h32%f 5f2{f h32%f @ |f/ nrml/ }erj 4 { f 32%f @ h32%f 9@ 3/ G@h 5 4 5{ f lpd wr fqr mhgqr umhµhqmh ++f ,f > +f2 ,f ,1 +Wr srwyu¡xmh gd vpr surqdµol rsfh umhµhqmh | srod}qh glihuhqflmdoqh mhgqdg }eh1,
Whruhp :1618 Dnr mh gdqr elor nrmh srvheqr umhµhqmh olqhduqh glihuhqfl0
mdoqh mhgqdg }eh +4,/ rqgd vh qmh}lqr rsfh umhµhqmh grelyd suleudmdqmhp wrjd srvheqrj umhµhqmd rsfhpx umhµhqmx sulsdgqh mrm krprjhqh mhgqdg }eh +5,1
Grnd}1 Qhnd mh |
i+{, rsfh umhµhqmh krprjhqh mhgqdg }eh +5,/ d
|2 i2+{, srvheqr umhµhqmh olqhduqh mhgqdg }eh +4,1 Grnd }lpr gd mh wdgd | @ | . |2 @ i +{, . i2+{, i +{, rsfh umhµhqmh glihuhqflmdoqh mhgqdg }eh +4,$ Gd mh | @ | . |2 umhµhqmh volmhgl l} | . d| . e| @ +| . |2, . d+| . |2 , . e+| . |2, @ +| . d| . e| , . +|2 . d|2 . e|2 , @ 3 . j+{, @ j +{,1 Exgxfl gd wr umhµhqmh vdgu }l +x | , gylmh vorergqh nrqvwdqwh nrmh vh prjx rguhglwl }d vydnl sr fhwql xymhw { @ {f +5 G} ,/ | @ |f / | @ |f / wr vh grlvwd
udgl r rsfhpx umhµhqmx1
:161 GLIHUHQFLMDOQH MHGQDG]EH GUXJRJD UHGD
694
Fhwlul vomhghfd whruhpd grqrvh sudylod }d l}eru srvheqrj umhµhqmd glih0 uhqflmdoqh mhgqdg }eh +4, v re}lurp qd gdqx ixqnflmx j 1 Grnd}xmx vh l}udyqrp surymhurp +xyuµwdydqmhp, sd wr suhsxµwdpr flwdwhomx ndr nrulvqx ymh }ex1
Whruhp :1619 Dnr mh ixqnflmd j/ qd ghvqrm vwudql olqhduqh glihuhqflmdoqh
mhgqdg }eh +4,/ srolqrp q0wrjd vwxsqmd/ j +{, @ s? +{,/ rqgd mh srvheqr umhµhqmh wh mhgqdg }eh srolqrp 0 q0wrjd vwxsqmd t? +{,> flp mh +x mhgqdg }el +4,, nrh flmhqw e 9@ 3> 0 +q . 4,0yrjd vwxsqmd t?n +{,/
flp mh e @ 3 9@ d> flp mh e @ 3 @ d1 0 +q . 5,0jrjd vwxsqmd t?n2 +{,/
Nrh flmhqwl wlk srolqrpd vh l}ud fxqdydmx phwrgrp qhrguh¡hqlk nrh flmh0 qdwd/ wm1 xyuµwhqmhp srolqrpd gdqrj vwxsqmd v qhsr}qdwlp nrh flmhqwlpd x mhgqdg }ex +4, l l}mhgqd fdydqmhp sduryd rgjrydudmxflk nrh flmhqdwd +x} lvwh srwhqflmh , v olmhyh l ghvqh vwudqh1
Whruhp :161: Dnr mh j+{, @ s? +{,ho%/ u
5 U/ rqgd mh srvheqr umhµhqmh
glihuhqflmdoqh mhgqdg }eh +4, ixqnflmd 0 t? +{,ho% > flp u 5 @ iu> u2j> o% 0 {t? +{,h / flp mh u 5 iu > u2 j l u 9@ u2 > 0 {2 t? +{,ho% / flp mh u @ u @ u2 / sul fhpx mh uc2 umhµhqmh sulsdgqh ndudnwhulvwl
fqh mhgqdg }eh1 +Nrh flmhqwl srolqrpd t? vh l}ud fxqdydmx sr xsxwl x suhwkrgqrpx whruhpx1,
Whruhp :161; Dnr mh j+{, @ f frv {.g vlq {/ f> g> 5 U/ rqgd mh srvheqr umhµhqmh glihuhqflmdoqh mhgqdg }eh +4, ixqnflmd 0 F frv { . G vlq {/ f lp l 5 @ iu> u2 j> 0 {+F frv { . G vlq {,/ flp mh l 5 iu > u2 j1 Nrh flmhqwl F l G vh rguh¡xmx xyuµwhqmhp umhµhqmd x mhgqdg }ex +4, l l}mhg0 qd fdydqmhp rgjrydudmxflk nrh flmhqdwd x} ixqnflmx vlq/ rgqrvqr/ frv1 Whruhp :161< Dnr mh x olqhduqrm glihuhqflmdoqrm mhgqdg }el +4, j+{, @
j+{, . . j& +{,/ n
5 Q/ rqgd mh qmh}lqr srvheqr umhµhqmh }eurm rg sr mhgqrj srvheqrj umhµhqmd vydnh sulsdgqh mhgqdg }eh | . d| . e| @ j +{,/ m @ 4> > n1
Sulpmhu :1618 Ulmhµlpr olqhduqx glihuhqflmdoqx mhgqdg }ex v nrqvwdqwqlp
nrh flmhqwlpd | | @ { .4 l rguhglpr mrm srvheqr umhµhqmh µwr xgryromdyd sr fhwqlpx xymhwx { @ 3/ | @ 3/ | @ 3 +xvs1 Sulpmhu 6151 f2 @ 4 sd mh wud }hqr srvheqr umhµhqmh | @ h3% . { 41
Sulpmhu :1619 Ulmhµlpr olqhduqx glihuhqflmdoqx mhgqdg }ex v nrqvwdqwqlp
nrh flmhqwlpd | . 5| . 8| @ {2h % . vlq 5{1 Sulsdgqd ndudnwhulvwl fqd mhgqdg }ed u2 . 5u . 8 @ 3 lpd nrqmxjludqr0nrp0 sohnvqr umhµhqmh uc2 @ 4 5l sd mh rsfh umhµhqmh +nrpsohnvqr, sulsdgqh krprjhqh mhgqdg }eh | @ h3%+f frv 5{ . f2 vlq 5{,1 Exgxfl gd mh j +{, @ {2 h % . vlq 5{/ wr fhpr }d surqdod }hqmh srvheqrj umhµhqmd srod}qh mhgqdg }eh qdmsulmh xsrudelwl Whruhp :161< Surpdwudmpr/ gdnoh/ gylmh sulsdgqh glihuhqflmdoqh mhgqdg }eh= | . 5| . 8| @ {2h % l | . 5| . 8| @ vlq 5{1 Qmlkryd srvheqd umhµhqmd pr }hpr rguhglwl uhgrp sr Whruhplpd :161: l :161;1 ]d suyx mh wr | @ +D{2 . E{ . F ,h % 1 Qmhjrylp xyuµwhqmhp +v | l | , x suyx mhgqdg }ex l rgjrydudmx flp l}mhg0 qd fdydqmlpd grelydpr olqhduql vxvwdy 53D @ 4/ 49D . 53E @ 3/ 5D . ;E . 53F @ 3/ / E @ 2D / F @ fff 1 Suhpd wrpx/ srvheqr umhµhqmh nrmhjd mh D @ 2f umhµhqmh suyh glihuhqfldoqh mhgqdg }eh mhvw | @ fff +83{2 73{ . 44,h % 1 Srvheqr umhµhqmh guxjh glihuhqflmdoqh mhgqdg }eh mh reolnd | @ F vlq 5{ . G frv 5{1 Qmhjrylp xyuµwhqmhp +v | l | , x guxjx mhgqdg }ex l rgjrydudmxflp l}mhg0 qd fdydqmlpd grelydpr olqhduql vxvwdy F 7G @ 4/ 7F . G @ 3/ e / G @ . 1 Suhpd wrpx/ srvheqr umhµhqmh guxjh umhµhqmh nrmhjd mh F @ . glihuhqfldoqh mhgqdg }eh mhvw | @ . +vlq 5{ 7 frv 5{,1 Vdgd mh/ sr Whruhpx :161 +d, 5{| | @ +| ,2 4> 2 +f, +4 . { ,| 5{| @ 3 v sr fhwqlp xymhwrp { @ 3> | @ 3> | @ 61 Ulmhµlwl glihuhqflmdoqx mhgqdg }ex +e, || 5|| oq | @ +| ,2 > +d, | . +| ,2 @ 5h3+ > 2 +f, +4. || ,| @ +4.+| , ,| v sr fhwqlp xymhwrp { @ 3> | @ 4> | @ 41 Ulmhµlwl glihuhqflmdoqx mhgqdg }ex +d, |+5{| . | , @ {+| ,2 . 4> +e, {2 | . {| | @ {2 v sr fhwqlp xymhwrp { @ 4> | @ > | @ 2 1 Ulmhµlwl glihuhqflmdoqx mhgqdg }ex +e, | 9| . +d, | 7| @ 5{ vlq {> +g, | | @ {2 {h% > +f, | 5| 6| @ h % . 4> % +h, | 5| @ 5h v sr fhwqlp xymhwrp { @ 4> | @ 4> | @ 3> +i, | . | @ 5{ v sr fhwqlp xymhwlpd { @ 4> | @ 3 l { @ > | @ 31 Ulmhµlwl vxvwdy glihuhqflmdoqlk mhgqdg }ded +d, | . 5| . } @ vlq {/ } 5} 7| @ frv {> _+ | +e, _% _| 7{ | . 69w @ 3/ _| | . 5{ . 5h @ 3 srg xymhwrp w @ 3> { @ 3> | @ 41
Lqghnv Dglwlyqd nrqvwdqwd/ 4
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