Jgjh½ĺsds Jgjh½ ĺsds Mn Y`sdcd½ Y`sdcd`g `½ g Mnh Ancjgdsa` Mn Kdnhj-Ajgdvnhj-C`rrnmnrj Jhnljgmr` H½ H`pnz `½ pnz Yunrt` ^gdvnrsdmjm Jut½ Jut`g`aj `½ g`aj Mn Tucjt½ Tucjtjg j½ g Fjcuhtjm Mn Dgbngdnr½ Dgbngdnr½ĺj ĺj w`tjg 6:Ie`tajdh.c`a
[nsuang Nh ancjgdsa` mn Kdnhj-Ajgdvnhj-c`rrnmnrj ns ug ancjgdsa` rnvnrsdkhn qun c`gvdnrtn nh a`vdadngt` mn r`tjcd½ r`tjcd`g trjshjcd`g,hj vjrdjkhn mn ngtrjmj punmn snr hj c`rrnmnrj ` hj ajgdvnhj ½ ng a`vdadngt` mn trjshjcd½ ½ mnpngmdngm` mn hj jphdcjcd½ jphdcjcd`g, ½ p`r nlnaph`< sd nh sdstnaj ns ug napuljm`r mn `klnt`s hj vjrdjkhn mn ngtrjmj snrj nh jgbuh` mn hj ajgdvnhj, ng cjakd` sd c`gcdmnrja`s h`s pdst`gns mn ug a`t`r mn c`akustd½ c`akustd`g ½ dgtnrgj hj vjrdjkhn mn ngtrjmj snrj hj p`sdcd½ p`sdcd`g ĺcu½ mn hj c`rrnmnrj,mjmj su dap`rtjgcdj Ng nstn jrt ½ h` cjrjctnrdzjrna`s nh a`vdadngt` mnh ancjgdsa` c`g su jgjh½ jgjh½ ĺsds mn p`sdcd½ p`sdcd`g,sdauhjcd `g jgdajcd`g ½ ½ y jgdajcd½ ½ c`aputjrdzjmj naphnjgm` nh s`ftwjrn AJ]HJK
:.
Jgjh½ĺsds ĺsds mn p`sdcd½ p`sdcd`g `½ g C`gsdmnrna`s nh ancjgdsa` mn Kdnhj-Ajgdvnhj-c`rrnmnrj a`strjm` ng hj sdbudngtn burj<
Fdburj :. Ancjgdsa` mn Kdnhj-Ajgdvnhj-C`rrnmnrj
:
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gunstr` `klntdv` snrj mnscrdkdr hj p`sdcd½`g `g mn cjmj lugtj c`g rnspnct` j hj vjrdjcd½`g `g mnh jgbuh` mn ngtrjmj( ε: ), pjrj ejcnr nst` tngmrna`s tngmrna`s qun rnhjcd`gjr ajtnajtdcjangtn ajtnajtdcjangtn U c y ε; ng fugcd½`g `g mn hj vjrdjkhn mn ngtrjmj ε : , nst` h` p`mn p`mna`s a`s h`brjr jh cnrr cnrrjr jr nh cdrcudt` vnct`rdjh vnct`rdjh f`rajm` p`r r : ,r; , U c y `ktngnr hj sdbudngtn ncujcd`g vnct`rdjh< r: + r ; 1 Uc (:) je`rj mngja`s r: y r; ng g`tjcd½`g `g nxp`gng nxp`gngcdjh< cdjh< r:
1 jkn lε
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p`r `trj pjrtn cjmj ncujcd½`g `g sn punmn nscrdkdr ng tnradg`s mn sus c`ap`gngtns vnct`rdjhns,h` cujh g`s pr`mucdrj nh sdbudngtn sdstnaj mn ncujcd`gns g` hdgnjhns< jk c`s( c`s(εε: ) + kc
c`s(ε; ) 1 Uc c`s(ε jk sdg( sdg(εε: ) + kc sdg( sdg(εε; ) 1 9
(;) (3)
mnspnljgm` ε; mn hj ncujcd½`g `g # ;<
ε; 1 sdg
∛:
∛j c
sdg(ε sdg(ε: )
(=)
c`a` p`mna`s `ksnrvjr, jh vjrdjr nh jgbuh` a`t`r( ε: ) ng hj ncujcd½`g `g #= `ktngna`s hj vjrdjcd½`g `g mnh jgbuh` ε ; y jh sustdtudr nst`s vjh`rns ng hj ncujcd½`g `g #; `ktngna`s hj p`sdcd½`g `g mn hj c`rrnmnrj, p`r h` tjgt` p`mna`s mncdr qun nstjs ncujcd`gns s`g hj mnscrdpcd½`g `g cdgnajtdcj mnh ancjgdsa`.
;.
sdau sd auhj hjcd cd½`g `½ g
hj sdauhjcd sdauhjcd`g `½ g mnh ancjgdsa` ancjgdsa` ns hj rnprn rnprnsngt sngtjcd½ jcd½`g `g brjcj mn hj vjrdjcd`g mn sus sjhdmjs vs sus ngtrjmjs, ng nstn cjs` sn `ktngmrj jh brjcjr ε; vs ε: y U c vs ε: ,naphnjrna`s AJ]HJK pjrj `ktngnr nstjs brjcjs. prdanrjangtn crnjrna`s ug mdrnct`rd` pjrj bujrmjr gunstr`s scrdpts, pjrj nst` nscrdkdrna`s ng hj tnradgjh h`s sdbudngtn c`ajgm`s< : ; 3
cm Xmdrnc Xmdrnct`r t`rd` d` m`g m`gmn mn sn buj bujrmjr rmjrjg jg h`s scr scrdpts dpts aomdr aom dr sdstnajs sdstnajs ancjgdc`s ancjgdc`s Xmdrnct`rd `rd` ` m`g m`gmn mn sn buj bujrmjr rmjrjg jg h`s scrdpts scrdptsXsd Xsdstn stnajs ajs ancj ancjgdc gdc`s `s cm Xmdrnct
nh c`ajgm` cm ns mn mdr mdrnc nccd` cd`gja gjadng dngt`, t`,hj hj hdgn hdgnjj #: g`s p`s p`sdcd dcd`gj `gjrj rj ng hj cj cjrpn rpntj tj m`g m`gmn mn buj bujrmj rmjrna rna`s `s gunstr`s trjkjl`s nh c`ajgm` aomdr ns hj jkrnvdjcd½`g `g mn ajon mdrnct`ry, p`r h` tjgt` hj hdgnj #; crnjrj ugj sukcjrpntj hhjajmj sdstnajs ancjgdc`s ng hj cjrpntj m`gmn g`s p`sdcd`g½` nh c`ajgm` cm hj hdgnj 3 g`s p`sdcd`gjr½j ng hj cjrpntj sdstnajs ancjgdc`s qun sn ngcungtrj mngtr` mn hj cjrpntj mn trjkjl`s(w`ro),pjrj rnbrnsjr ug mdrnct`rd` jtrjs p`mna`s usjr nh c`ajgm` cm .. y pjrj vnr h`s jrcedv`s mngtr` mn nsn mdrnct`rd` naphnjrna`s nh c`ajgm` hs
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ugj vnz crnjm` nh mdrnct`rd` mn trjkjl` y yj nstjgm` ng nh, crnja`s ug a-hn mjgm` chdco ng nh dc`g` mn e`lj ng khjgc`,` g`s vja`s j hn-gnw-a hn, nh a-hn ns ug scrdpt ng nh cujh nscrdkdrna`s y c`apdhjrna`s gunstrjs sngtngcdjs,tjakdng p`mna`s ejcnr nst` ng hj tnradgjh pnr` jh a`angt` mn cnrrjr gunstr` pr`brjaj t`mjs hjs sngtngcdjs sn punmng k`rrjr,nh a-hn p`r nh c`gtrjrd` c`gsnrvjrj gunstrjs sngtngcdjs cjmj vnz qun cnrrna`s y jkrja`s nh pr`brjaj.
Fdburj ;. crnjr ug a-hn
ugj vnz crnjm` nh a-hn pr`cnmnrna`s j dgtr`mucdr h`s pjrjantr`s dgdcdjhns mnh ancjgdsa`< : ; 3 =
%pjrjantr`s dgdcdjhns JK1;.>? KC16? t:1hdgspjcn(9,369,>99)?
nh sdak`h` mn p`rcngtjln sdrvn pjrj dgtr`mucdr c`angtjrd`s jh dgtr`mucdr cjmj vjh`r ejy qun tnradgjr c`g pugt` y c`aj,mn nstn a`m` jh nlncutjr nh a-hn g` jpjrncnrjg h`s vjh`rns mn hjs vjrdjkhns ng hj pjgtjhhj
hdgspjcn( jcn(vjh` vjh`rr dgdcd dgdcdjh,vj jh,vjh`r h`r gjh gjh,# ,# mn mjt` mjt`s) s) sdr nh c`ajgm` hdgsp sdrvn vn pj pjrj rj crn crnjr jr ug ug vn vnct` ct`rr mn vj vjh`r h`rns( ns(qun qun snrj nh dgtnrv dgtnrvjh` jh` mn vjr vjrdjcd½ djcd½`g `g mn ε: ),n ),ng g nstn cj cjs` s` nh vjh`r vjh`r dgdc dgdcdjh djh fun ε: 1 9 nh vjh`r vjh`r gj gjhh ns ε: 1 369 y nh guanr` mn mjt`s ns >99. `trj ajgnrj mn mngdr nh vnct`r mn vjh`rns ns usjgm` nh `pnrjm`r m`s pugt`s( <) : ;
%vnc vnct`r t`r mn vjh` vjh`rns rns usjgm` usjgm` nh `pn `pnrjm rjm`r `r m`s pug pugt`s t`s< < t:19<9.99:<369?
hj sdgtjxds mn nstn `pnrjm`r ns vjh`r dgdcdjh
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Je`rj dgtr`mucda`s hjs fugcd`gns ε ; y U c < : ; 3
%ncujcd ncujcd`gns `gns mn p`sdcd`g t;1jsdgm((-JK *sdgm(t:))/KC)? Uc1JK*c`sm(t:)+KC*c`sm(t;)?
nh `pnrjm`r (*) daphdcj auhtdphdcjcd½`g, `g, y nh `pnrjm`r ( /) mdvdsd½`g, `g, `tr` `pnrjm`r usujh ns nh mn auhtdphdcjcd`g nhnangt` j nhnangt` mn ug vnct`r ` ajtrdz( *.) h`s tnradg`s sdgm y c`sm rnprnsngtjg j hjs fugcd`gns trdb`g`a½ntrdcjs ntrdcjs mn sng` y c`sng` mjmjs ng brjm`s,ajthjk ajgnlj rjmdjgns p`r mnfjuht jsd qun sd qunrna`s h`s vjh`rns ng brjm`s tngmrna`s qun p`gnr jh gjh mn hj fugcd½`g `g ugj m (MNB[N) ^gj vnz dgtr`mucdmjs t`mjs hjs c`gstjgtns y vjrdjkhns, s`h` g`s fjhtjrj brjcjr ε; vs ε: y U c vs ε: ,nst` h` p`mna`s ejcnr usjgm` h`s c`ajgm`s ph`t y y sukph`t mn mn hj sdbudngtn ajgnrj< : ; 3
= > 6 2 5
4 :9
%brjfdc`s sukph`t(;,:,:)? ph`t(t:,t;,'--r','hdgnwdmte',=)?xhjknh('XtentjP:_brjm`sQ','f`gtsdzn',:5,' f`gtwndbet','k')? ('XtentjP;_brjm`sQ','f`gtsdzn',:5, Q','f`gtsdzn',:5,'f`gtwndbet','k')? 'f`gtwndbet','k')? yhjknh('XtentjP;_brjm`s ('sdauhjcd` uhjcd`g g mnh ancjgd ancjgdsa` sa` KAC' KAC','f`g ,'f`gtsdzn tsdzn',;3, ',;3,'f`gtw 'f`gtwndbet ndbet','k' ','k')? )? tdthn('sda brdm `g? sukph`t(;,:,;)? ph`t(t:,Uc,'r','hdgnwdmte',=)?xhjknh('XtentjP:_brjm`sQ','f`gtsdzn',:5,' f`gtwndbet','k')? ('UPc_puhbjmjsQ','f`gtsdzn',:5, ','f`gtsdzn',:5,'f`gtwndbet',' 'f`gtwndbet','k')? k')? yhjknh('UPc_puhbjmjsQ brdm `g?
sukph`t(hjs,c`huagjs,# s,# mn burj) sdrvn pjrj ejcnr ug jrrnbh` mn brjcjs mngtr` mn nh c`ajgm` sukph`t(hjs,c`huagj ugj adsaj burj, ng nstn cjs` sukph`t(;,:,:) g`s dgmdcj qun snrjg ; hjs,: c`huagj y t`ajrna`s hj prdanrj br½jcj jcj mn nstn jrrnbh`, yj qun AJ]HJK cungtj h`s brjc`s mn dzqudnrmj j mnrncej y mn jrrdkj-jkjl`,sukph`t(;,:,;) g`s p`sdcd`gjr½j mnkjl` mn hj prdanrj br½jcj. jcj. dgmnpngmdngtn,vjr mdngtn,vjrdjkhn djkhn mnpngm mnpngmdngtn, dngtn, nstdh` n½ stdh` mn hdgnj hdgnj—, —,—br —br`s`r `s`r mn h½ĺgnj—,# ĺgnj—,# nh c` c`ajg ajgm` m` ph`t(vjrdjkhn dgmnpng mnh br`s`r mn h½ h ½ĺgnj) ĺgnj) g`s pnradtdrj ejcnr h`s brjc`s,ng gunstr` prdanr ph`tn` tngmrna`s hj br½jcj mn ε; vs ε : c`g ug nst½ ns t½ĺh` ĺh` mn h½ĺgnj ĺgnj pugtnjmj mn c`h`r c` h`r r`l`( ∛∛r) y ug br`s`r mn h½ĺgnj ĺgnj mnh # = ˘ xhjknh(—tnxt`—,—tjaj ˘ g` h`s c`ajgm`s pjrj h`s nlns mn hj br½jcj s`g xhjknh & yhjknh, su sdgtjxds ns< xhjknh(—tnxt`—,—tjaj mnh tnxt`—,# mnh tjaj ˘ tjaj ˘ g` mn tnxt`,—br`s`r mnh tnxt`—,—c`h`r mnh br`s`r—) nh c`ajgm` tdthn(—tnxt`—,nstdh`s mn tnxt`) sdrvn pjrj tdtuhjr cjmj ugj mn gunstrjs brjcjs, ng nstn cjs` s`h` sn naphn` ug t½ĺtuh` ĺtuh` pjrj jakjs yj qun nstjg ng nh adsa` jrrnbh` mn dajbng,h`s nstdh`s mn tnxt` s`g h`s adsa`s qun sn mngdnr`g ng nh pugt` jgtnrd`r nh c`ajgm` brdm `g ns pjrj jctdvjr jctdvjr hj rnldhhj mn hj br½jcj. jcj.
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ugj vnz tnradgjm` gunstr` c`mdb`,h` c`apdhja`s c`apdhja`s y mnkna`s mn `ktngnr brjcjs qun g` prnsngtng gjturjhnzj jsdgt½`tdcj,qun `tdcj,qun gjh gjhdzng dzng ng nh nh adsa` adsa` vjh vjh`r `r m`gm m`gmnn dgdcdj dgdcdjr`g r`g y qun snjg pjrj nstn cjs` mn gjtu gjturjhnz rjhnzjj sng`dmjh, hj sdauhjcd½`g `g `ktngdmj c`g nstn c`mdb` sn aunstrj ng hj burj #3
sdauhjcd`g mnh ancjgdsa` KAC >9
Q s ` m j r b _
9
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:
½ mnh ancjgdsa` Fdburj 3. sdauhjcd`g
3.
Jgdajc jcdd`g `½ g mnh Ancjgdsa`
hj jgdajcd½`g `g mnh ancjgdsa` ns ugj rnprnsngtjcd½`g `g c`aputjrdzjmj mn su p`sdcd½`g `g encej j kjsn mn ugj supn supnrp`sd rp`sdcd½ cd½`g `g mn vjrd`s brjc`s t`ajm`s ng mdfnrngtns dgstjgtns mn tdnap`, ns mncdr, ns hj ugd½`g ugd½`g mn vjrd`s ph`tn`s mn su p`sdcd½`g,pjrj `g,pjrj hhnvjrhj jcjk` ejy qun c`gcdmnrjr hj hny mn h`s cunrp`s rdbd Gdgbug ½ nhnangt` mnh ancjgdsa` mnkn mnf`rajrsn mn ajgnrj c`gcdmnrjkhn ,s m`s< Gdgb m`s< ,sdd ns nstj tj hn hny y sn r` r`ap apdn dnrj rj,h ,hjj jgdajcd½`g nstjrj ajh yj qun gdgb½ug ug ancjgdsa` ancjgdsa` ng hj vdmj rnjh prnsngtjr½ prnsngtjr½j tjhns mnf`rajcd`gns p`r qun sn sdauhjcd½`g `½ g c`gstdtuyng mn nhnangt`s rdbdm`s jrtdcuhjm`s. Hj anl`r ajgnrj mn vnrdcjr nstj hny ns c`g hj sdauhjcd mnh ancjgdsa`,c`a` sn angcd`g½` ng hj snccd½`g `g jgtnrd`r,hjs br½jcjs jcjs mn su c`ap`rtjadngt` mnkng cjrncnr mn gjturjhnzj jsdgt½`tdcj,mn `tdcj,mn h` c`gtrjrd` sn p`mrjg `ksnrvjr cjakd`s krusc`s ng hj vjrdjcd½`g mn sus pjrjantr`s mn sjhdmj,nst`s cjakd`s krusc`s sn prnsngtjr½jg jg ng hj jgdajcd½`g `g c`a` ugj mnf`rajcd½`g `g c`gsdmnrjkhn ng jhbug nhnangt`.Y`r nstn a`tdv`, sn hn rnc`adngmj jh hnct`r ejcnr hj sdauhjcd½`g jgtns mn hj jgdajcd½`g.
Fugcd`g `½ g mn h`s nshjk`gns jrtdcuhjm`s Yrdanrjangtn mngdrna`s ugj fugcd½`g `g qun snrj hj ngcjrbjmj mn cjhcuhjr t`m`s h`s pugt`s mn gunstrj jgdajcd½`g<
:
fugctd`g_px,pyQ1kac(t:)
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px & py snr½jg jg hjs sjhdmjs mn gunstrj fugcd fugcd `g ` ½ g bngnrjmjs p`r hj ngtrjmj t:,nvdmngtnangtn hj ngtrjmj snrj ug jgbuh` a`t`r,y hjs sjhdmjs snrj ug vnct`r mn mnsphjzjadngt` qun sn mnscrdkdrj ng tnradg`s `½ g c`g nh adsa` mn sus c`ap`gngtns px & py.ns dap`rtjgtn bujrmjr nh a-hn mn gunstrj fugcd `g g`akrn mn hj fugcd `g, `½ g, mn h` c`gtrjrd` ajthjk g` hj rnc`g`cnrj y tngmrna`s nrr`rns jh jgdajr nh ancjgdsa` Je`rj dgtr`mucdrna`s h`s pjrjantr`s y vnct`rns mn p`sdcd½`g `g tjh c`a` sn edz` ng hj sdauhjcd½`g,hj `g,hj ugdcj mdfnrngcdj ns qun je`rj g` mngda`s nh vnct`r aunstrjh ε: ,nstn vnct`r sn mngdrj ajs jmnhjgtn<
: ; 3 = >
%pjr pjrjan jantr` tr`s s y ncu ncujcd jcd`gn `gns s mn p`s p`sdcd dcd`g `g JK1;.>? KC16? t;1jsdgm((-JK *sdgm(t:))/KC)? U1JK*c`sm(t:)+KC*c`sm(t;)?
nh sdbudngtn pjs` snrj mngdr h`s vnct`rns qun c`gf`rajg nh hjz` vnct`rdjh mnh ancjgdsa`< :
% mnch mnchjr jrjc jcd` d`g g mn h` h`s s vn vnct ct`r `rns ns
; 3 = >
% vn vnct ct`r `r [: r:x1JK*c`sm(t:)? r:y1JK*sdgm(t:)?
6 2 5 4
% vn vnct ct`r `r [; r;x1KC*c`sm(t;)? r;y1KC*sdgm(t;)?
p`r uhtda` cjrbja`s nst`s vnct`rns j h`s vjh`rns mn sjhdmj mn hj fugcd½`g< : ;
px1_9,r:x,UQ? py1_9,r:y,9Q?
sd jbrupja`s ng pjrnljs h`s vjh`rns mn px y py p`mna`s `ksnrvjr qun s`g hjs p`sdcd`gns mn cjmj ugj mn hjs lugtjs mnh ancjgdsa`, hj prdanrj pjrnlj ns (9,9) y ns hj p`sdcd½`g `g mn gunstr` ajrc` mn rnfnrngcdj bh`kjh(hj bh`kj h(hj lugtj qun c`gnctj j hj Ajgd Ajgdvnhj vnhj c`g hj kjgc kjgcjmj) jmj),hj ,hj snbugmj pjrnlj(rx, pjrnlj(rx,ry) ry) ns hj p`sdc p`sdcdd ½`g `g mn hj lugtj qun c`gnctj j hj Kdnhj c`g hj Ajgdvnhj y hj tnrcnrj pjrnlj(U,9) ns hj p`sdcd½ p`sdcd ½`g ` g mn hj lu lugt gtjj qu qunn c` c`gn gnct ctjj j hj Kdnhj c`g hj c`rrnmnrj.
Fugcd`g `½ g mn hj c`rrnmnrj pjrj ejcnr hj c`rrnmnrj tngmrna`s qun mngdr `trj fugcd½`g `g hj cujh hhjajrna`s sqr< :
fugctd`g_pxc,pycQ1sqr(t:)
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Dgtr`mucda`s h`s pjrjantr`s y hjs ncujcd`gns mn p`sdcd½`g `g c`rrnsp`gmdn c`rrnsp`gmdngtns< gtns< : ; 3 =
JK1;.>? KC16? t;1jsdgm((-JK *sdgm(t:))/KC)? U1JK*c`sm(t:)+KC*c`sm(t;)?
Fdgjhangtn mnscrdkdrna`s hj p`sdcd½`g `g mn hj c`rrnmnrj c`g h`s vjh`rns mn sjhdmj mn hj fugcd½`g< `g< :
%c`rrnmnrj
; 3 =
pxc1_U-9.>,U,U+9.>Q? pyc1_9,9,9Q?
ejcdngm` rnfnrngcdj j hj burj #: p`mna`s vnr qun hj c`rrnmnrj nstj ugdmj j hj kdnhj ng ug pugt` qun sn h`cjhdzj ng anmd` mn su h`gbdtum t`tjh,mn nstn a`m` sd nhnbda`s ug vjh`r mn :‑pjrj hj c`rrnmnrj p`mrna`s mdkuljrhj ejcdngm` 3 pugt`s c`hdgnjhns y mnspuns ph`tnjgm`h`s c`g ug br`s`r mn hdgnj ajy`r qun nh br`s`r mn hdgnj mn h`s nshjk`gns jrtdcuhjm`s,nst`s 3 pugt`s c`hdgnjhns s`g U-9.>,U,U+9.>,y g`s bjrjgtdzjrjg qun hj c`rrnmnrj nstjrj ugdmj j hj kdnhj p`r qun c`apjrtn nh pugt` U ng c`aug.
Mdkul` mnh ancjgdsa` Yjrj mdkuljr nh ancjgdsa`,prdanr` crnjrna`s = jrrnbh`s mdstdgt`s mn cnr`s y mngda`s gunstr`s pjr½jantr`s jantr`s dgdcdjhns< : ; 3 = > 6 2 5
%pjrjantr`s dgdcdjhns< dgdcdjhns< xks1_9,9,9Q? yks1_9,9,9Q? xc1_9,9,9Q? yc1_9,9,9Q? t:1-:;9? b1:2? g1:99?
m`gmn<
t: ns nh jgbuh` mn ngtrjmj qun dgdcdjr½j ng -:;9 brjm`s
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Nh sdbudngtn pjs` snr½j utdhdzjr hjs fugcd`gns crnjmjs c`g jgtnrd`rdmjm pjrj cjhcuhjr t`m`s h`s pugt`s mn gunstrj jgdajcd½`g, `g, nst` h` ejrna`s c`g nh sdbudngtn cdch` f`r< : ; 3 = > 6 2 5 4 :9
% cjhc cjhcuh uh` ` mn h` h`s s pu pugt gt`s `s mn hj sd sdau auhj hjc` c`g g f`r d1:
nstn cdch` dgv`cjr½j j gunstrjs fugcd`gns y cjhcuhjr½j nh jgbuh` mn ngtrjmj t: ng dgcrnangt`s mn :2 brjm`s(t:+b),c`g nstn jgbuh` j½ gbuh` cjhcuhjr½j hjs p`sdcd`gns mn hj c`rrnmnrj( sqr) y h`s nshjk`gns( kac) y jsdbgjr½j nst`s vjh`rns j h`s jrrnbh`s xks,yks,xc,yc , nh c`ajgm`( d,<) ns qudng sn ngcjrbjr½j mn nst`,nstn c`ajgm` sdbgdcj h` sdbudngtn< pjrj t`m`s h`s vjh`rns mn.. jsdbgjrsn j.. . ^gj vnz cjhcuhjm`s h`s pugt`s mn gunstrj jgdajcd½`g `g ejrna`s hj supnrp`sdcd½`g `g mn brjcjs naphnjgm` `tr` cdch` f`r< :
% ph` ph`tn` tn` mnh ancjgds ancjgdsa` a`
; 3 = > 6 2 5 4
:9 :: :; :3 := :>
f`r d1:
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sn brjc½` t`m`s h`s vjh`rns mnh vnct`r xks vs t`m`s h`s vjh`rns mnh vnct`r yks c`g ug nstdh` mn hdgnj c`gtdguj c`h`r r`l` y g`m`s ng sus nxtrna`s, sn jsdbg` nstn ph`tn` j hj vjrdjkhn K sn brjc½` t`m`s h`s vjh`rns mnh vnct`r xc vs t`m`s h`s vjh`rns mnh vnct`r yc c`g ug nstdh` mn hdgnj c`gtdguj c`h`r r`l`, sn jsdbg` nstn ph`tn` j hj vjrdjkhn C nh c`ajgm` snt c`gbur` hjs vjrdjkhns K y C c`g ug br`s`r mn hdgnj mn :9 y =9 rnspnctdvjangtn nh c`ajgm` pjusn c`gtr`h` hj vnh`cdmjm mn hj jgdajcd½`g `g
=.
C`gc C` gchu husd sd`g `gns ns
AJ]HJK ns C`a` sn punmn `ksnrvjr nh s`ftwjrn AJ]HJK ns ugj p`mnr`sj enrrjadngtj mn trjkjl`, pnsn j su jpjrngtn jpj rngtn c`ap c`aphnldmj hnldmjm m g`s pr`p pr`p`rcd` `rcd`gj gj ugj `pcd½`g `g n ncdn cdngtn gtn pjrj rns rns`hv `hvnr nr hj cdg cdgnaj najtdc tdcjj mn an ancjg cjgdsa dsa`s, `s,h` h` ugdc` qun sn gncnsdtj ns tngnr hjs kjsns ajtna½jtdcjs pjrj bngnrjhdzjr hjs ncujcd`gns mn p`sdcd½`g ` g y hj mdsp`sdcd½`g mn jprngmnr hj sdgtjxds mnh pr`brjaj.
>.
[nfn [n fnrrng ngcd cdjs js [`knrt G`rt`g.-Mnsdbg `f ajcedgjry Ajthjk.-Enhp