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Information and Contour by Parker Emmerson

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8/14/2019 Information and Contour by Parker Emmerson http://slidepdf.com/reader/full/information-and-contour-by-parker-emmerson 1/23 Khcmplftkmh fhd Nf{mzt `{ Ufpijp Jlljpsmh khonzdkhg khcnzjhtkfn omhjpsftkmhs wktb Tml Tmnjhm. Fhdpjw @jpksbf. fhd Fnstjh Dfhkjns « Ufpijp Jlljpsmh Motm`jp 0:. 055; Tbks ufujp wknn pjcjp tm tbj `mmi. Tbj @nfoiwjnn Pjfdkhg kh Ubknmsmub{ Uz`nkoftkmh Jhtktnjd Ujpojutkmh jdktjd `{ Pm`jpt _obwfptq( Kt ks pjcjppjd tm tbpmzgbmzt tbj ufujp fs @PU. Ufgj Hzl`jp+( Tbj `mmi ks fh fhtbmnmg{ mc lzntkunj tbkhijps kh tbj ckjnd mc ujpojutzfn ubknmsmub{( K( Khtpm Kh tbks sjpkjs mc obfutjps. wj wknn dksozss bmw tbj kszfn s{stjl ujpojkjs tbj wmpnd tbpmzgb sflunjs mc tbj fl`kjht mutko fppf{( KK( Umstznftjs fhd _tpzoztzpj cmp Hmtftkhg f _flunj mc tbj Fl`kjht Mutko Fppf{ "6+ O < 0 up Tbks ks tbj okpozlcjpjhoj mc mzp khktkfn okponj mc pfdkzs p O 0 < 0 up 6 Tbks ks tbj okpozlcjpjhoj mc mzp sjomhd okponj. tbj `fsj mc tbj omhj. mc pfdkzs p 6 pX0 < p 6 X0 )b X0 Tbks ks tbj khktkfn pfdkzs svzfpjd jrupjssjd fs tbj snfht mc tbj omhj kh tjpls mc tbj bjkgbt mc tbj omhj. b. fhd tbj pfdkzs mc tbj `fsj mc tbj omhj. p 6 p < . Bp 6 X0 )b X0N s < v p s á v < p Tbj fpo njhgtb tfijh mzt mc f okponj ft f gkjh tklj ks < t < O , O 0 < 0 up , 0 up 6 < vp p 6 X0 ï pX0 ,b X0 p 6 < . BpX0 ,b X0N b § p t < tklj Tbj jnnfusj mc 6 sjomhd < 3 djgpjjs t < 3 v djgpjjs < 3 B625 á uN v Wj fnsm hmtj tbft fh{ fpo njhgtb ofh `j djsopk`jd `{ fh fhgnj mc dkstfhoj. v p( Tbj lmdjn mc sufoj , tklj upmumsjd mhn{ pjvzkpjs tbj foojutfhoj mc tbpjj czhdfljhtfn tpztbs( Ckpstn{. tbft tbj njhgtb tbft omljs cpml f obfhgj kh okpozlcjpjhojs ofh `j jvzfn tm fh fpo njhgtb. sjomhdn{. tbft tbj U{tbfgmpjfh tbjmpjl wknn bmnd zu cmp lfikhg ofnoznftkmhs wktbkh tbj lmdznfp. fhd tbkpdn{. tbft jnnfusjd tklj ks kh tjpls mc f o{onko omhtkhzzl kh tbj sjhsj tbft wj ofh ljfszpj kt kh tjpls mc djgpjjs mc f okponj( Upkhtjd `{ Lftbjlftkof cmp _tzdjhts
Transcript
Page 1: Information and Contour by Parker Emmerson

8/14/2019 Information and Contour by Parker Emmerson

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Khcmplftkmh fhd Nf{mzt

`{ Ufpijp Jlljpsmh khonzdkhg khcnzjhtkfn omhjpsftkmhs wktb Tml Tmnjhm. Fhdpjw @jpksbf. fhd Fnstjh Dfhkjns

« Ufpijp Jlljpsmh Motm`jp 0:. 055;

Tbks ufujp wknn pjcjp tm tbj `mmi. Tbj @nfoiwjnn Pjfdkhg kh Ubknmsmub{ Uz`nkoftkmh Jhtktnjd Ujpojutkmh jdktjd `{ Pm`jpt

_obwfptq( Kt ks pjcjppjd tm tbpmzgbmzt tbj ufujp fs @PU. Ufgj Hzl`jp+( Tbj `mmi ks fh fhtbmnmg{ mc lzntkunj tbkhijps kh tbj

ckjnd mc ujpojutzfn ubknmsmub{(

K( Khtpm

Kh tbks sjpkjs mc obfutjps. wj wknn dksozss bmw tbj kszfn s{stjl ujpojkjs tbj wmpnd tbpmzgb sflunjs mc tbj fl`kjht mutko

fppf{(

KK( Umstznftjs fhd _tpzoztzpj cmp Hmtftkhg f _flunj mc tbj Fl`kjht Mutko Fppf{

"6+

O < 0 up

Tbks ks tbj okpozlcjpjhoj mc mzp khktkfn okponj mc pfdkzs p

O0 < 0 up6

Tbks ks tbj okpozlcjpjhoj mc mzp sjomhd okponj.

tbj `fsj mc tbj omhj. mc pfdkzs p6

p X 0 < p6 X 0 ) b X 0

Tbks ks tbj khktkfn pfdkzs svzfpjd jrupjssjd fs tbj snfht

mc tbj omhj kh tjpls mc tbj bjkgbt mc tbj omhj. b. fhd tbj pfdkzs

mc tbj `fsj mc tbj omhj. p6

p <.Bp6 X 0 ) b X 0N

s < v ps á v < p

Tbj fpo njhgtb tfijh mzt mc f okponj ft f gkjh tklj ks <

t < O , O0 < 0 up , 0 up6 < vp

p6 X 0 ï p X 0 , b X 0

p6 <.Bp X 0 , b X 0N

b § p

t < tklj

Tbj jnnfusj mc 6 sjomhd < 3 djgpjjs

t < 3 vdjgpjjs < 3 B625 á uN v

Wj fnsm hmtj tbft fh{ fpo njhgtb ofh `j djsopk`jd `{ fh fhgnj mc dkstfhoj. v p(

Tbj lmdjn mc sufoj , tklj upmumsjd mhn{ pjvzkpjs tbj foojutfhoj mc tbpjj czhdfljhtfn tpztbs( Ckpstn{. tbft tbj njhgtb tbft

omljs cpml f obfhgj kh okpozlcjpjhojs ofh `j jvzfn tm fh fpo njhgtb. sjomhdn{. tbft tbj U{tbfgmpjfh tbjmpjl wknn bmnd zu cmp

lfikhg ofnoznftkmhs wktbkh tbj lmdznfp. fhd tbkpdn{. tbft jnnfusjd tklj ks kh tjpls mc f o{onko omhtkhzzl kh tbj sjhsj tbft wj ofh

ljfszpj kt kh tjpls mc djgpjjs mc f okponj(

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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Tbj lmdjn djsopk`js tbj d{hflko `jtwjjh sz`ajotkj fhd m`ajotkj pjfnkt{ kh f sklunj fhd zsjczn lfhhjp mc ujpojutkmhm`sjpftkmh+( Tbj nkgbt ks fnwf{s omhskdjpjd tm `j dkpjotjd tmwfpd tbj ujpojkjp. wbmsj nmozs mc ujpojutkmh ks jktbjp tbj umkht mc 

tbj omhj mp tbj pjojutkmh mc f oml`khjd wfjnjhgtb mc nkgbt(

Tbj dkstfhoj fwf{ cpml tbj fpjf mc tbj fppf{ tbft tbj ujpojkjp sflunjs dzpkhg fotkmh mc tbj kszfn s{stjl ks jvzfn tm tbj

bjkgbt mc tbj omhj( Wktbkh tbj fpjf mc tbj sflunj. tbjpj ks f omhtfkhjd flmzht mc khcmplftkmh f`mzt nf{mzt( Tbj upm`njl mc 

khonzdkhg `mtb tbj wfj,njhgtb khtjpupjtftkmh fhd tbj sflunkhg khtjpupjtftkmh wktbkh tbj stpzotzpkhg mc tbj fl`kjht fppf{ fpksjs

wbjh wj omhskdjp tbft tbj wfjnjhgtb bfs hmt gmhj tbpmzgb tbj jhtkpj tpfhsktkmh wktb pjsujot tm tbj jomnmgkofn stpzotzpj mc tbj

fpjf wbjh tbjpj ks f sflunkhg. `zt ks pjfnkstkofnn{ omluknjd `{ tbj kszfn s{stjl khtm fh klfgj( Tbj upm`njl ofh `j ufptkfnn{

pjsmnjd { omhskdjpkhg tbft tbj kszfn s{stjl sflunjs jp{ lfh{ sujokcko pjojutkmhs mc nkgbt mp pjgkmhs mc nf{mzt cpml tbj

sflunjd fpjf wktbkh f gjhjpfn sflunj( Tbjsj wfjnjhgtbs gkj khcmplftkmh tm tbj ujpojkjp kh tjpls mc sftzpftkmh. bzj. pkgbthjss

fhd tbzs. omhtmzp(

K wknn hmw dm smlj fngj`pf tm omhonzdj wbft tbj bjkgbt mc tbj omhj ks kh tjpls mc tbj khktkfn ufpfljtjps( Kt ofh jjhtzfnn{

`j pjdzojd tm f skhgnj fpkf`nj wbjh kdjfs nkij dkccjpjhtkfns mc dkstfhoj fhd cpjvzjho{ fpj khonzdjd kh tbj sjttkhg zu mc tbj

jvzfnktkjs( Tbks wknn njfd tm hzljpmzs hjstjd jvzftkmhs tbft djsopk`j dkccjpjht ufpfljtjps mc tbj kszfn s{stjl( Wj omhskdjp tbft

tbj pfdkzs ks jvzfn tm f skhgnj wfjnjhgtb. hmt bfnc f wfjnjhgtb. `jofzsj wj fpj dksozsskhg tbj tklj kt tfijs tm tpfjn f dkstfhoj

mc mhj wfjnjhgtb mc nkgbt. fhd tbft wfjnjhgtb tpfjns tbpmzgb tbj bjkgbt mc tbj omhj( Fntbmzgb. tbj sbfuj mc tbj dkfljtjp ofh

nftjp `j sbmwh tbpmzgb tbpjj,dkljhskmhfn gpfubs tm bfj tbj sbfuj mc f skhj wfj(

_mnsjEp6 X 0 ) b X 0 ï pX0. bD"0+11b Ò , p0 , p6

0 8. 1b Ò p0 , p60 88

Wj sf{ tbft tbj flmzht mc v p < s. tfijh mzt mc tbj okponj ks tbj obfhgj kh tbj okponj's okpozlcjpjhoj tbft ks tbj `fsj mc tbj

omhj( Tbj obfhgj ks jvzfn tm s < 0up,0up 6(

Hmtkoj tbft v < BB0 u pN á pN , BB0 u p 6N á p N. `jofzsj wj dkkdj `{ p mh `mtb skdjs(

Wj wknn cmozs mh tbj umsktkj smnztkmhs cmp tbj bjkgbt mc tbj omhj(

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0   Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h`

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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Wj ofh smnj tbks jvzftkmh cmp  `. tbj fhgnj lfdj `{ tbj snfht mc tbj omhj fhd tbj `fsj mc tbj omhj. kh tjpls mc tbjtf tbj

flmzht mc fhgnj khmnjd kh cmndkhg tbj okponj zu khtm tbj tbkpd dkljhskmh wbjh tbj snfht ks fnwf{s jvzfn tm tbj pfdkzs(

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Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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PjsmnztkmhUnmt?D@ Fpo_kh@ B4 u , vN v

0 u C. 2v. , 6?. 6?>C

Tbj skhgnj umkht kh tbj ckjnd mc mutkofn cnmw. nkgbt tpfjnkhg tbpmzgb tbj fl`kjht fppf{. K upmumsj. ks pjupjsjhtjd `{ tbj

tbpjj dkljhskmhfn pjmnztkmh gpfub mc f obfhgj kh pfdkzs mjp tklj tbpmzgb tpfhscmplftkmh tm `jkhg tbj bjkgbt mc tbj omhj cpml

tbj twm fhgznfp fpkfhts mc tbj s{stjl( F pjmnztkmh djnkjps khcmplftkmh tm tbj ujpojkjp gkjh tbj khtjhskt{. bzj. jto( mc nkgbt( F

skhgnj pjmnztkmh mc tbj s{stjl ks jvzkfnjht tm tbj cnmw mc oml`khjd nkgbt kh tbj fl`kjht fppf{ ft nmoftkmh mc tbj stftkmh umkht mc 

ujpojutkmh kh tbj fl`kjht mutko fppf{(

Tbjpj fpj wf{s mc smnkhg cmp tbj pfdkzs kh tjpls mc uzpjn{ tbj fhgnj. v ( Wj wknn jrflkhj tbjsj hjstjd sjts mc wf{s mc 

zhdjpstfhdkhg tbj umkht mc kjw( @{ hjstjd. wj ljfh tbft djujhdkhg mh bmw kh djutb mhj gmjs khtm m`sjpkhg tbj wmpnd mp

jrtpfumnftkhg f czhotkmh mc nkgbt kh tbj fppf{. mhj wknn sjj dkccjpjht kszfn pjupjsjhtftkmhs mc tbj stpzotzpkhg mc tbft khcmplftkmh

tbft fpj lmpj mp njss khcmplftkmhfnn{ djhsj( Tbks wknn njfd tm f sjpkjs mc czhotkmhs tbft gkj zs khtpmsujotkmh khtm tbj stpzotzpkhg mc 

nkgbt kh tjpls mc f o{onko ujpsujotkj mh tklj( Cmp tbj s{stjl. kt ks klumptfht tm pjfnkqj tbft kh tbj hmplfn omzpsj mc ufpfljtjpkqkhg

tbj lmtkmh kh tbj ofsj ft bfhd. v dmjs hmt gm `j{mhd 4 u k(j( tbft kt sjhds tbj bjkgbt mc tbj omhj zu tm kts lfrklzl fnzj fhd

`foi dmwh mjp tbj omzpsj mc twm pjmnztkmhs mc f okponj+. fhd tbft  ` wmznd hmt `j f`nj tm `j jfskn{ tbmzgbt mc `j{mhd u mp 0 u kt

bfs f fnzj mc u ft tbj lfrklzl bjkgbt mc tbj tpfhsktkmh. fhd f fnzj mc 0 u mhoj tbj omhj omljs `foi dmwh cpml tbj bjkgbt+(

_mnsj@6 <<0 u  _khE`D

4 u v , v0

. vC"60+11v Ò 0 u , u0 , u0 _khE`D0 8. 1v Ò 0 u ) u0 , u0 _khE`D0 88

3   Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h`

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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PjsmnztkmhUnmt?D@0 u ) u 0 , u 0 _khE`D0 . 2p. , 6. 6>. 2`. , 4 u . 4 u >C

Gk`smh bfs sfkd tbft. "Tbj jrujpkjhoj mc tbj kszfn wmpnd ks hmt omlumzhdjd mc f sjpkjs mc kszfn ckjnds9 hm mhj ks fwfpj

mc tbj sjvzjhoj `zt mhn{ mc tbj tmtfn sojhj( Upjszlf n{ tbks ks `jofzsj tbj moznfp s{stjl djtjots tbj khfpkfhts mjp tklj tbft

sujokc{ tbj sojhj" @PU 633+( V ks tbj fpkf`nj tbft ks fikh tm tbj foozlznftjd tklj mc mzp s{stjl( Kt ks fh fhgnj. fhd kc wj pjnftj

kt tm ljfszpjljhts lfdj mc tklj o{onkofnn{. nkij tbmsj mc f onmoi. wj ofh ckhd bmw tbj khfpkfht. wbkob ks tbj khktkfn pfdkzs mc tbjs{stjl. wknn gzkdj tbj khcmplftkmh omllzhkoftjd dzpkhg tbj ukoizu mc khfpkfhts( Tbj tklj fpkf`nj omhtfkhs f omhtmzp `fsjd mh

tbj ufpfljtjps mc tbj s{stjl(

p <0 u ! p ! _khE`D

4 u v , v0

<0 u b

4 u v , v0

Tbj khfpkfht ks p. `jofzsj kt ks fnwf{s tbj snfht mc tbj omhj dzpkhg tpfhsktkmh mc f tmtfn oml`khjd jccjotkj wfjnjhgtb

tbft klljdkftjn{ jccjots tbj pjojutmp( Tbks ljfhs tbft kc f wfjnjhgtb wjpj tm obfhgj dzpkhg tpfjn tbpmzgb tbj lmdznfp mc sufoj,

tklj dzj tm `jkhg ufpt mc fh fl`kjht fppf{. wbjpj nkgbt wfjs jccjot jfob mtbjp dzpkhg tbjkp uftbs. tbj khktkfn wfjnjhgtb mc nkgbt

ks tbj snfht mc tbj omhj. jjh tbmzgb tbj pjszntkhg bjkgbt lf{ `j nmhgjp mp sbmptjp tbfh tbj khktkfn wfjnjhgtb( Wbft dmjs tbks tjnn

zs f`mzt nkgbt fhd bmw kt tpfjns tbpmzgb sufoj mp jccjots f pjojkkhg ujpojkjp= ihmwnjdgj mc tbj khktkfn omhdktkmh mc tbj s{stjl

djujhds mh tbj pjomghktkmh mc tbj pjojutmp( tm j tbj oml`khftkmh mc omhstpzotkj fhd khtjpcjpkhg cpjvzjhokjs mc nkgbt kh tbj fppf{.

khfpkfhoj mc p omhtjrtzfnn{ fsszpjs tbft (

` < Fpo_kh@ B4 u , vN v

0 u C

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Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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PjsmnztkmhUnmt?D@ 0 u ! p ! _kh@ Fpo_kh@ B4 u,vN v

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b Ò4 u p0 v , p0 v0

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Tbks ks bmw tbj gpfub mc tbj bjkgbt mc tbj omhj kh hmh , pjnftkkstko tjpls wmznd nmmi( Kt ofh `j hmtjd cpml tbj cmnnmwkhg

gpfubs tbft wktbkh tbj ufpfljtjps mc tbj s{stjl. tbjpj ks f sklknfp gpfub tm tbj pjnftkkstko njhgtb omhtpfotkmh tpfhscmplftkmh. `zt

`j{mhd tbj ufpfljtjps mc tbj s{stjl. tbj gpfubs `jgkh tm nmmi snkgbtn{ dkccjpjht(

Tbks jvzftkmh gpfubs tbj bjkgbt mc tbj omhj `{ pjmnkhg p tbpmzgb tbjtf( Tbj stpzotzpj mc tbj fppf{ khonzdjs tbj dkstfhoj.

b( B ofh `j jvzfn tm tbj hjw wfjnjhgtb mc nkgbt dzpkhg f tpfhsktkmh wktb obfhgkhg wfjnjhgtb mc tbj stpzotzpjd sz`stfhoj. `zt kt

ofh fnsm `j f djckhktkj dkstfhoj mc fh m`ajot fwf{ cpml tbj ujpojkjp(

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Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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khckhktjn{ cfp fwf{ ft tbj bmpkqmh ozpj. fhd umtjhtkfnn{ khckhktjn{ dkksk`nj kc tbj wfjnjhgtb mc tbj nkgbt gjts khckhktjsklfnn{

slfnn( Wbjh wj stfhd mh fh khckhktjn{ nfpgj subjpj. tbjpj ks hm wf{ tm tjnn bmw nmhg mp cfp fwf{ tbj bmpkqmh ks( Kh mpdjp tm gjt

khtjpupjtftkmhs mc ljfhkhg mc mtbjp ufpfljtjps mc tbj s{stjl. wj sf{ tbft wfjnjhgtb ofh obfhgj dzpkhg f tpfhsktkmh f tpfhsktkmh

bjpj ljfhs tpfjnkhg tbj dkstfhoj mc mhj wfjnjhgtb+(

Wbft ofh wj tjnn cpml tbjsj lmdjns= Wj ofh sjj tbft ufpfljtjps mc wfjnjhgtbs mc sz`stfhoj tpfjnkhg tbpmzgb sufoj

fhd tklj. nkij nkgbt. omhtfkh wktbkh tbjl khcmplftkmh tbft ks fnpjfd{ omhtmzpjd. mp cpml umkht tm umkht dpfws mzt omhtmzp tbpmzgb

pjomghkqkhg tpjhds mc khfpkfhoj( Wbjh wj tfij f dkccjpjht fhgnj mh tbjl. wj sjj tbj omhtmzp dkccjpjhtn{( Tbks jccjot wknn j jjh

lmpj hmtkof`nj kh nftjp lmdjns wbjh wj sjj tbft tbjpj ks jp{ sz`tnj omhtmzp fhd sbklljpkhg(

Kt ks umssk`nj tbft dkpjot ujpojutkmh dmjs hmt omhtpfdkot khcmplftkmh upmojsskhg. jofzsj tbj ukoizu mc khcmplftkmh dkpjotn{foihmwnjdgjs tbft tbjpj ks khcmplftkmh kh tbj nf{mzt mc tbj wmpnd. tpfhscjppjd dkpjotn{ `{ nkgbt stklznzs dzpkhg tbj ujpojkkhg

khstfhoj( Tbft khstfhoj pjojkjs tbj djnkjp{ mc khcmplftkmh `fsjd mh khfpkfhoj( Tbft djnkjp{ omljs cpml f pjcnjotkmh mcc tbj

omhtmzp tbpmzgb f oml`khftkmh mc pjnftkmhs kh tbj gzkdfhoj mc tbj cnzr mc fl`kjht nkgbt( Omluztftkmhfn fuupmfobjs sklun{ wmznd

bfj tm obfhgj tbjkp Ubknmsmub{ tm fh jomnmgkofn omhtjrt fhd sklun{ stkoi tm m`sjpkhg tbj ubjhmljhmh mc ujpojutkmh k(j( wbft K

sjj obfhgkhg kh tbj jhkpmhljht tbft ofh `j omppjnftjd tm m`ajots mp ujmunj kh tbj wmpnd+(

 

Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h` ;

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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KS( Tbj Fhfnmg{ mc f Obfhgj kh Umsktkmh Pjnftkhg tm f Obfhgj kh Omhtmzp dzpkhg tbj Ukoizu mcKhfpkfhts tbpmzgb Tklj

Gk`smh bfs sfkd tbft. "Tbj tbjmp{ mc khcmplftkmh ukoizu pjvzkpjs ujpojutzfn s{stjls. hmt sjhsjs" @PU 77+( Gk`smh fnsm

pjnftjs tbj sz`tnjt{ mc tbj ujpojutzfn s{stjl tm njfphkhg fhd tklj( Bj tbkhis tbft wj omhtkhzj njfphkhg tm ujpojkj kh lmpj djtfkn

fs tklj gmjs mh( Tbks wknn j fh klumptfht kdjf wbjh dksozsskhg tbj jrtpfumnftkmh mc czhotkmhs khtm klfgjs tbpmzgb fh klupjsskmh

cpml fh jrupjsskmh. wbkob pjnftjs tbj khcmplftkmh kh tbj fppf{ mjp tklj tm tbj ujpojkjp(

Tbj wfjnjhgtb mc tbj nkgbt wj fpj dksozsskhg ks tbj khktkfn pfdkzs mc tbj okponj fhd tbj lfrklzl bjkgbt mc tbj omhj.

wbmsj dkpjotkmh ks tmwfpd tbj sz`ajotkj ujpojkjp( Tbj tpfhsktkmh mc tbft skhgnj wfjnjhgtb khtm tbj bjkgbt mc f omhj moozps mjp

tbj omzpsj mc f pjmnztkmh mc f skhgnj okponj( Tbj ufpfljtjps mc tbj jnmokt{ mc kts tpfjn. mp tbj tpfjn mc fh{ sz`stfhoj. fpj

cpjvzjho{ fhd wfjnjhgtb(

"6?+Wj stftj tbft 1 < n Ɛ. wbjpj ks jnmokt{.

n < p ks wfjnjhgtb. fhd 6 á tklj < Ɛ < cpjvzjho{(

"64+Tklj < 3 KvdjgpjjsL < 3 B625 á uN v < B6525 á uN v

Tbj jnmokt{ `jkhg dksozssjd ks tbj ckpst djpkftkj mc tbj dkstfhoj. wbkob ks tbj bjkgbt mc tbj omhj. b( F tkh{ ukjoj mc tbj

fl`kjht fppf{ wmznd omhtfkh wktbkh kt khcmplftkmh pjgfpdkhg khcmplftkmh mc tbj szpcfoj mcc mc wbkob tbj nkgbt cnmwkhg tbpmzgb tbj

fppf{ ks pjcnjotkhg( B omhtfkhs khcmplftkmh f`mzt tbj tklj fhd khktkfn dkstfhoj mc f wfjnjhgtb mc nkgbt( Wbjpj b ks f gkjh bjkgbt

mc f omhj. fhd v  tbj flmzht mc foozlznftjd tklj mc dzpkhg f tpfhsktkmh. p!v . wbkob ks tbj dkccjpjhoj mc tbj obfhgj kh okpozlcjp,

jhoj jtwjjh twm okponjs tbft omppjnftjs tm tbj bjkgbt kh tbj tbkpd dkljhskmh. tbj pfdkzs mc tbj fsj mc tbj omhj obfhgjs kh f

u{tbfgmpjfh pjnftkmhsbku tm p fhd b(

Kh tbj lmdjn mc ujpojutzfn sufoj , tklj tbft K fl upmumskhg. tbj ufpfljtjps khmnjd ofh fnwf{s `j pjdzojd tm f skhgnj

jrupjsskmh mc f czhotkmh mc tklj. „kt ks hmt ljfhkhgczn tm fsi ljpjn{ bmw stpmhgn{ f gkjh zhkt pjsumhds tm f gkjh stklznzs9 tbj

lmpj pjnjfht fpkf`nj ks tbj tjlumpfn ufttjph mc tbj pjsumhsj“ @PU 044+( Ks kt umssk`nj tbft tbj ckpkhg mc tbj hjzpmhs pjnftjs

stpmhgn{ tm tklj. `jofzsj tbj sbfuj mc tbj stklznzs bfs fh khfpkfht omppjnftj gjmljtpkofnn{ tbpmzgb tklj=

Ljpnjfz Umht{ bfs sfkd mc tklj. "tbjpj ks mhj skhgnj tklj wbkob ks sjnc,omhckplftmp{. wbkob ofh `pkhg hmtbkhg khtm

jrkstjhoj zhnjss kt bfs fnpjfd{ nfkd tbft tbkhg's cmzhdftkmhs fs upjsjht fhd jjhtzfn ufst. fhd wbkob jstf`nksbjs ktsjnc fs f stpmij"

UmU 406+( Tbj tklj tbft ks sjnc omhckplftmp{ ks tbj tklj ft wbkob tbj bjkgbt mc tbj omhj bfs gmhj zu tm 0 u fhd omlj foi dmwh

mjp tbj omzpsj mc 4 u pfdkfhs( Kh tklj ljfszpjd kh sjomhds. tbks wmznd `j mhj lkhztj kh _(K( zhkts. `zt (

Khcmplftkmh mc fpkf`nj omhtmzp ks tpfhscjppjd tm tbj ujpojkjp `{ tbj cnzr mc khcmplftkmh kh fl`kjht nkgbt tbpmzgb jfob

umkht kh tbj mutko fppf{. pjupjsjhtjd { ktjl hzl`jp cmzp( Tbj pftj mc tbks fpkfhoj ks fhfnmgmzs tm jnmokt{ kh f pmtftkmhfn s{stjl.

nkij tbj stpzotzpkhg mc nkgbt tbpmzgb sufoj , tklj( Tbks ks wbft ks ljfht `{ khcmplftkmh kh tbj nkgbt( M`ajots ofh tpfjn tbpmzgb

sufoj,tklj kh f omhtkhzmzs wf{(

Bjnlbmntq upmumsjs tbft ubjhmljhf pjgfpdkhg tbj kszfn s{stjl ofh `j djsopk`jd `{ tbj zsj mc fhfnmg{ wbjh tbkhgs

`jgkh tm nmmi ufpfdmrkofn( Bj tfijs khtm omhskdjpftkmh tbj us{obko cfozntkjs fhd szggjsts tbft tbj upmojsskhg mc tbj jhkpmhljht

fhd kts ujpojutzfn ubjhmljhf lf{ moozp tbpmzgb fh zhomhsokmzs wmpikhg mc fhfnmg{ kh tbj lkhd( K ckhd tbks tm j khtjpjstkhg fhd

fgpjjf`nj( Kt upmkdjs f bzlfhkst foomzht tbpmzgb tbj pjomghktkmh mc `kmnmgkofn cfotmps pjgfpdkhg ujpojutzfn s{stjls(

Fhklfns mh Jfptb bfj jmnjd kh f gjhjpfn ujpojutkmh mc tklj kh tbj jhkpmhljht tm `j o{onko( Tbj df{ fhd hkgbt. tbjpb{tbl mc snjju. tbj obfhgj mc tbj sjfsmhs. fhd tbj o{onjs mc bjfjhn{ `mdkjs nkij tbj lmmh fnn unf{ f pmnj kh tbj jhkpmhljht mc 

jmnkhg fhklfns mh Jfptb( _mlj lmpj tbfh mtbjps. `zt cmp bzlfhs. lmst ojptfkhn{(

Gk`smh mccjps fh khtjpjstkhg khtjpupjtftkmh tm tbj upm`njl mc djutb ujpojutkmh( Djutb ujpojutkmh ks ft tbj bjfpt mc wbft wj

fpj dksozsskhg( Tbj bjkgbt mc tbj omhj. kh f lmpj jomnmgkofn omhtjrt. ks dkpjotjd tmwfpd tbj ujpojkjp fhd omhtfkhs omhtmzpjd

khcmplftkmh wktbkh kt `fsjd mh tbj wfjnjhgtb mc nkgbt fhd tklj jnnfusjd dzpkhg tpfhsktkmh( Tbks omhtmzpjd khcmplftkmh jrksts kh

jjp{ umkht mc tbj fppf{. fhd jomnmgkofn omhtmzp ks `fsjd mh tbj ukoizu mc jfob wfjnjhgtb kh kts pjnftkmh tm tbj mtbjps kh tbj

fppf{(

65   Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h`

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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Gk`smh sf{s tbft. "tbj upm njl mc djutb ujpojutkmh ks khsmnzf nj. jofzsj tbj upm njl ks cfnsj" @PU 633+( Bj fnsm

szggjsts tbft wj fpj lkstfijh kh tbkhikhg tbft tbj. "tbkpd dkljhskmh mc tbj ofptjskfh ommpdkhftj s{stjl ks f ubjhmljhfn cfot mc 

ujpojutkmh" @PU 633+( Mzp ommpdkhftj s{stjl cmp djsopk`khg f tkh{ ukjoj mc khtjpfotkmhs gmkhg mh kh tbj fl`kjht fppf{ ks `fsjd.

hmt mh f tbkpd dkljhskmh mc ofptjskfh ommpdkhftjs. `zt mc f skhgnj tpfhsktkmh mc sz`stfhoj tbpmzgb tklj( Kt sbmznd `j hmtjd tbft ft

tbj jjht mc ujpojutkmh. tbj khcmplftkmh mc omhtmzp kh tbj nf{mzt mc tbj wmpnd ks djnkjpjd tm tbj pjojutmps ft f njhgtb tbft bfs mhn{

mhj dkljhskmh ft tbj tklj mc pjojutkmh( Tbj oml`khjd jccjot mc. "umkhts mc m`sjpftkmh kh tbj ljdkzl" @PU 637+. fs Gk`smh

lkgbt uzt kt. tpfhslkts tbj nf{mzt mc tbj wmpnd tm tbj ujpojkjp `{ f ljsb mc khtjpwmjh sujokckjps(

Tbj ujpojutzfn sufoj pjnkjs mh f omhsokmzs jrujpkjhoj( Lftb fhd tbj `md{ mc sokjhojs ofh `j zsjczn kh lmdjnkhg tbj

jnjljhts khmnjd kh tbj djnkjpfhoj mc sz`ajotkj jrujpkjhoj kc jhmzgb pjnjfht omhtjrtzfn khcmplftkmh ks gkjh tm tbj lmdjn(

Pkgbt hmw. lfobkhjs lfhkuznftj skghfns. `zt dm hmt bfj tbj omlunjrkt{ tbft bzlfh jrujpkjhoj bfs. ufptn{ `jofzsj tbjkp s{stjl

dmjs hmt bfj tbj jmnztkmhfp{ pjsmzpojs djjnmujd kh fh jhkpmhljht mjp tklj wbjh omhstfhtn{ pjsumhdkhg tm kts khuzts(

Tbj cfots njt lj sjj omhtmzp. wbknj fnsm pjomghkqkhg tbft omhtmzp cpml f skhgnj klfgj omznd `j knnzsmp{( Wj tfij dftf

ffknf`nj kh nkgbt. fhd omhstpzot fh klfgj cpml tbft dftf mc fh m`ajot. „tbmzgb szob cnku , cnmuukhg sjndml moozps wbjh wj fpj

nmmikhg ft pjfn ofts. kt smljtkljs bfuujhs wktb sujokfnn{ omhstpzotjd ckgzpjs kjwjd zhdjp pjstpkotjd omhdktkmhs“ @PU 072+( Kc 

wj tfij tbj khcmplftkmh kh nkgbt fhd stpzotzpj tm `j f cfot. K wmznd sf{ tbft cnku , cnmuukhg fnsm moozps zhdjp tbj lmst gjhjpfn

omhdktkmhs fhd omppjnftjs mc ihmwh cfots tm ujpsujotkfn jlfhftkmhs. wbkob sujokc{ vzfnktkjs mc f ujpojutkmh( Djutb gkjs zs szob

djtjplkhjd khcmplftkmh. gzkdjd `{ m`ajots tbft djnkjp nf{mzt tm tbj ujpojut. tbft cnku , cnmuukhg pfpjn{ moozps( Tbks ks wbft

djckhjs fh m`ajot cmp kszfn ujpojutkmh ks kts stf`knkt{(

S( Tbj Wfjnjhgtb wktb Omhstfht Sfnzj

"6:+

< n Ɛ < p B6 á B6525 á uN vN <

B6 á B6525 á uNN D@ 4 u p0 v , p0 v0

0 u. vC <

4 u p0 , 0 p0 v

4?05 4 u p0 v , p0 v0

Wj wknn tp{ tm `zknd zu fs lfh{ czhotkmhs fs umssk`nj wktbmzt pjsmptkhg tm djskghftkhg smljtbkhg tm `j bfkhg f jnmokt{

mc nkgbt mp hmh , nkgbt( Kt ks sklun{ f jnmokt{ cmp tbj tklj `jkhg( Tbks wf{. wj wknn bfj foojss tm f pjfnl mc gjhjpfn czhotkmhs mc 

omppjnftjd fpkf`njs(

B6 á B6525 á u NN D@ 4 u  p0 v , p0 v0

0 u . vC

4 u p0 , 0 p0 v

4?05 4 u p0 v , p0 v0

"63+Sjnmokt{ wbjh p ks omhstfht <4 u p0 , 0 p0 v

4?05 4 u p0 v , p0 v0

Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h` 66

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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Unmt?D@ 4 u  p0 , 0 p 0 v

4?05 4 u  p0 v , p0 v0

. 2p. , 6. 6>. 2v. , 4 u . 4 u >C

"67+

Wj fnsm bfj foojss tm tbj jvzftkmh.

4 u p

0

, 0 p

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ï p B6 á B6525 á uN vN.

 wbkob wkhds zu bfkhg tbj p fpkf`nj ofhojn mzt.

fhd djnkjps tbpjj jrfot smnztkmhs cmp tbjtf(

_mnsj@ 4 u  p0 , 0 p0 v

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ï p B6 á BB6525 á u N vNN. vC

"62+

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Wj wknn hmtj tbft fh fpo njhgtb ks jvzfn tm p!v ( Tbj obfhgj kh ujpojkjd okpozlcjpjhoj mc tbj okponj ks fnsm jvzfn tm p!v (

Tbzs. wj wknn `j f`nj tm sjj wbft tbj fpo njhgtb ks. `jofzsj kt pjupjsjhts tbj obfhgj kh okpozlcjpjhoj mc twm okponjs( Kh tbks ofsj.

fs tbj okponj. kjwjd cpml bjfd mh. pjojdjs cpml tbj ujpojkjp khtm tbj dkstfhoj. kt gjts slfnnjp( Tbks ks f obfhgj kh flmzht mc 

szpcfoj fpjf tfijh zu mh tbj pjtkhfn klfgj. `zt fnsm f wf{ mc kszfnkqkhg bmw tbj klfgkhfp{ smnztkmhs cmp tbj tklj jnjljht mc wfjs tpfjnkhg kh sufoj fccmpds f ujpojutkmh mc nf{mzt(

Nkgbt tpfjns tbpmzgb sufoj. `zt wktbkh kts tpfjn ks tbj khcmplftkmh tbft tbj fhgznfp jnjljht mc tbj s{stjl. wbkob ks fikh

tm tklj. omhj{s sujokckofnn{ f`mzt tbj nmoftkmh mc omhtmzp kh ujpojutzfn sufoj tbpmzgb f gjmljtpko omppjnftkmh( Tbks omppjnftkmh

moozps tbpmzgb tbj omlunjr unfhj fhd bfs tbj ofuf`knkt{ mc lfuukhg mzt omhtmzp kh tbj kszfn ujpojutkmh dkpjotn{ `{ fhfnmg{ mc 

nfpgjp jomnmg{ tm khdkkdzfn unf{ mc fl`kjht nkgbt mcc mc f omhtmzp( Wj wknn tp{ tm sbmw tbks s{l`mnkofnn{ fs wjnn fs dkpjotn{(

60   Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h`

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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PjsmnztkmhUnmt?D@p0

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Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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PjsmnztkmhUnmt?D@p4 u 

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Wbjh oml`khjd. tbj tbpjj klfgkhfp{ smnztkmhs cmpl f omhj. `zt sjujpftjn{. tbj{ jrkst upklfpkn{ kh tbj omlunjr unfhj( Kt

pjvzkpjs fnn tbpjj smnztkmhs tm djnkjp khcmplftkmh mc djutb tm tbj ujpojkjp tbpmzgb nkgbt kh tjpls mc tbj pjfn( Kh Gk`smh' s tbjmp{.

djutb ks khtklftjn{ omhhjotjd wktb omhtmzp( Omhtmzp ks djnkjpjd `{ tbj ukoizu mc tbj khfpkfht. p. kh tbj fl`kjht fppf{. obfhgkhg

omhtjrt kh f dkccjpjht wf{ cpml umkht tm umkht( Tbzs. zhdjpstfhdkhg mc twm tpztbs kh tbj pjfn wmpnd nkij khcmplftkmh gkjh tm tbj

ujpojkjp mc fh m`ajot `{ fh jdgj. cmp khstfhoj ks jhf`njd(

Lfob `jgfh `{ jpkc{khg f `jnkjc kh tbj sufoj , sjhsj. fh fpjf mc ufpfljtjpkqjd sufoj pjnftkhg tbj sjhsj mc fh khdkkdzfn

tm tbj stklznzs kh tbj wmpnd( Tbj sufoj , sjhsj upmkdjs f wmpisufoj cmp tbj khdkkdzfn sjhsjs nkij sjhsj mc skgbt fhd tmzob tm

jhf`nj f ujpojutzfn jrujpkjhoj(„ Tbj ujpojutkmh mc Djutb djujhds mh jrtpf , mooznfp jrujpkjhojs“ @PU 662+ ks fh kdjf tbft

lfijs f gpjft djfn mc sjhsj tm lj. `jofzsj wbjh wj jrujpkjhoj mp ujpojkj tpztb. wj zsj lzntkunj cfozntkjs tm lfij sjhsj mc f

ubjhmljhmh(

Tbj fl`kjht mutko fppf{. wbkob ks tbj jhtkpj omnnjotkmh mc khdkkdzfn umkhts mc nkgbt. `khds skhgnj umkhts kh tbj fppf{

tbpmzgb tbj omlunjr unfhj. fhd fccmpds tbj ujpojutkmh mc m`ajots. jofzsj m`ajots dkspzut tbj fppf{ kh tbj pjfn. `zt tbj stpzotzpj mc 

nkgbt fsszpjs tbft tbj klfgkhfp{ jnjljhts mc m`ajots omppjsumhd tm mhn{ mhj pjfn m`ajot( Mp. kc tbjpj ks f pjfn m`ajot. tbjh tbj

klfgkhfp{ jnjljhts mc ujpojkkhg tbft m`ajot sujokc{ f skhgnj jjht mc ujpojkkhg tbft m`ajot(

Tbj klfgkhfp{ smnztkmhs tm ujpojkkhg m`ajots tbpmzgb tklj omznd obfhgj mhn{ kc tbj vzfnkf mc tbj nf{mzt mc m`ajots

obfhgjd. mp tbj dkspzutkmh kh tbj fppf{ obfhgjd tm pjlfkh fnnmwkhg tbj mjpfnn oml`khftkmhs mc lmdznfps kh tbj skhgnj lmdznfp mc 

ujpojutkmh tbpmzgb omhsokmzshjss( Wj stknn fsi kc ujpsmhfnkt{ mc tbj ujpojutkmh obfhgjd tbj ljfhkhg mc tbj m`ajot tm nf{mzt mc tbj

wmpnd wktbkh tbj zhkckjd lmdznfp mc ujpojutkmh. wmznd tbj stpzotzpkhg mc tbj fppf{ obfhgj mp bfplmhkqj wktb tbj sz`stfhoj=

_flunkhg fnnmws zs tm djtjplkhj kc tbj nf{mzt ks dkccjpjht cpml stpzotzpkhg. `jofzsj tbj fppf{ ks tbj jhkpmhljhtfn omhtjrt mc tbj

ujpojutkmh. fhd sbfpjs f sklknfp stpzotzpj tm jfob umkht wktbkh kt( Tbks ks f lfhhjp mc ujpojutkmh(

 

64   Khcmplftkmh fhd Omhtmzp `{ Ufpijp Jlljpsmh(h`

Upkhtjd `{ Lftbjlftkof cmp _tzdjhts

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Tbj lfuukhg mzt mc omhtmzp cpml khcmplftkmh kh nkgbt sbfpjs f sklknfp stpzotzpj tm tbj ljfszpjljht mc skqj( K dmh‛t tbkhi

kt lfijs sjhsj tm `jgkh `{ umstznftkhg fh m`ajot. mhn{ umstznftkhg tbft tbjpj ks tbj ukoizu mc khcmplftkmh tbft tbjpj ks omhtmzp kh

tbj jhkpmhljht( Tbj ujpojutkmh mc tbks omhtmzp nkij kt ks fh fotzfn m`ajot. bmwjjp. ks jubjljpfn fhd smljwbft knnzsmp{( Fnn wj

ofh bfj foojss tm cmp stzd{khg kskmh sokjhtkckofnn{ mp ubjhmljhmnmgkofnn{ ks tbj ujpojut mc omhtmzp kh f upmajotkmh tm tbj lkhd.

„L{ ihmwnjdgj mc tbj unfhj opfsb djpkjs cpml l{ ihmwnjdgj mc tbj hjwsufujps `jofzsj K dkd hmt sjj tbj unfhj opfsb“ @PU

07?+( Bmwjjp. tm ckhd mzt bmw ujpojutkmh mujpftjs. wj lzst pjomghkqj tbj omhdktkmhs hjojssfp{ cmp khcmplftkmh tm `j tpfhscjppjd

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SK( Fh Khbjpjho{ mc Pjnftkkt{ tm Khfpkfhoj

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