Alcuni Punti Notevoli del Trapezio

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Pckv`}ukqbd a`nlk Uqpak ak Rdaovd

Fdeolqbd ak Uek`cx` II#FF#CC#

Akrd}qki`cqo ak Idqìdqked Rp}d à DrrlkedqdEo}uo ak Ldp}`d kc Idqìdqked

Qùk ak Ldp}`d

Uqpako ak Dlepck Rpcqk Coq`volk a`l Q}dr`xko

]`ldqo}`8 R}of# M`cà`qqo Uekiìk

Ldp}`dcad8 Dccd Mdeednlkck)F}dcj

Dcco Deedaìkeo =221)=220



Dlld Ikd Fdiknlkd



Kcake`

Kcq}oapxkoc` >

2#4 Coqdxkoc` # # # # # # # # # # # # # # # # # # # # # # # # # # # # # ;

4 Dlepck Rpcqk Coq`volk ` Ukiklkqpakck a`l Q}dr`xko 44

4#4 Md}ke`cq}o # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 444#= O)e`cq}o # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 444#5 G )e`cq}o # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 454#1 C )e`cq}o # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 40

4#1#4 Dlepc` R}or}k qbd ak G # # # # # # # # # # # # # # # # # # 4>4#0 Rpcqo H # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 4>4#6 Tpda}kldq`}k Eoirlì`cqd}k ` Rpcqo ak Iktp`l # # # # # # # # 4>

= Qò}ìk Urèkfflek a`l Q}dr`xko 4;

=#4 Iàkdcd R}kcekrdl` ` ]`qqd OG a`l Q}dr`xko # # # # # # # # # =2=#= Dlepc` R}or}k qbd a`l Rpcqo ak Iktp`l a l Q}dr`xko # # # # # # ===#5 Q}dr`xko Kuoue`l` 'Ekelkeo+ # # # # # # # # # # # # # # # # # # # # =0=#1 Rd}dll`lon}diid # # # # # # # # # # # # # # # # # # # # # # # # # =>

5 Aùe}kxkoc` Dcdlkqked a`l Q}dr`xko =;

5#4 Eduo N`c }dl` 'Tpda}dcnolo coc O}qonocdl`+ # # # # # # # # # =;5#= Eduk Rd}qkeold}k # # # # # # # # # # # # # # # # # # # # # # # # # # 50

5#=#4 Q}dr`xko Ekelkeo # # # # # # # # # # # # # # # # # # # # # 505#=#= Rd}dll`lon}diid # # # # # # # # # # # # # # # # # # # # # 565#=#5 ]`qqdcnolo # # # # # # # # # # # # # # # # # # # # # # # # 5>5#=#1 Q}dr`xko O}qonocdl` # # # # # # # # # # # # # # # # # # # 5>

5#=#0 Q}dr`xko O}qonocdl` ` Ekelkeo # # # # # # # # # # # # # # 145#=#6 ]oimo # # # # # # # # # # # # # # # # # # # # # # # # # # 1=5#=#> Tpda}dqo # # # # # # # # # # # # # # # # # # # # # # # # # 1=

1 ]keouq}pxkoc` a`l Q}dr`xko 15

1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I # # # # # # # # # # # # # 111#4#4 Lpono ak O # # # # # # # # # # # # # # # # # # # # # # # # 101#4#= Lpono ak I # # # # # # # # # # # # # # # # # # # # # # # # 10

1#= ]keouq}pxkoc` a`l q}dr`xko kuoue`l` # # # # # # # # # # # # # # # 0=

0



Kcq}oapxkoc`

K rpcqk coq`volk a`l q}kdcnolo uoco uqdqk k r}oqdnockuqk ak pcd coq`vol` tpdc)qkqbd ak d}qkeolk ` lkm}k , ld epk r}oapxkoc` qoeebo kl upo iduukio d fflc` ‛722# ]`)e`cqì`cq` dlepck idqìdqkek eoi`, r`} ùìrko, ]#Jkim`}lkcn Z1Y , gdcco

}kq}ovdqo kcq`}ùu` r`} k rpcqk coq`volk a`l q}kdcnolo#Uo}r}`ca`cqì`cq`, kcvè`, coc uoco uqdqk ro}qdqk dvdcqk dcdlongk uqpak

upk rpcqk coq`volk a`l tpda}dcnolo ` a`l tpda}kldq`}o# R`} kl tpda}kldq`}oeoirl`qo 'tpdqq}o ldqk ` u`k v`}qkek+ uk rpbo fd}` }kf`}ki`cqo dk eocq}kmpqk akN#Uq`kc`} Z4Y: i`cq}` r`} tpdcqo }knpd}ad kl tpda}dcnolo eoirl`qo 'tpdqq}ov`}qkek ` u`k ldqk+ ld mkmlkon}dffld uk }kape` d tpdleg` aèkcd ak d}qkeolk, q}d lo}oakueoccùuk#

Pco uqpako ukuqìdqkeo ak tpùqo d}noi`cqo b uqdqo fdqqo ad M#Uekiìk,kc Z>Y, eg` uk b dvvdluo dceg` a`lld eolldmo}dxkoc` ak dlepck upok uqpa`cqk, Z7Y#Tpk uk q}ovd ld aùe}kxkoc` ` lo uqpako a`ll` r}or}k`qbd ak dlepck rpcqk coq`volka`l tpda}dcnolo eoirl`qo# ]kro}qkdio tpk ld a`fflckxkoc` ak rpcqo coq`vol`

o urèkdl`, daoqqdqd kc Z>Y#“Duu`ncdqk k v`}qkek Dk'k 9 4,###c+ ak pc c)noco D, pc rpcqo C uk ake`

coq`vol` r`} D u` b pcd fpcxkoc` C 9 C 'D+ ukii`q}ked c`nlk d}noi`cqkDk ` qdl` eg` r`} qpqq` l` ukiklkqpakck φ a`l rkdco }kuplqk φ'C '###,Dk,###++ 9C '###,φ'Dk++# Egkdi }ìo eovd}kdcxd tpùqd r`}ipqdmklkqbd q}d l` fpcxkockC ` φ# Dcdlondi`cq` a`fflck}ìo coq`volk r`} D pcd }`qqd, pc e`}egko, pcdeocked, pc i)noco èe# kcuoiid onck dlq}o `cq` nòi q}keo eg` ukd fpcxkocùkii`q}ked ` eovd}kdcq` a`nlk Dk c`l u`cuo urk`ndqo#’

R`} eouq}pk}` fflnp}` coq`volk d rd}qk}` ad pcd fflnp}d D pu`}ìo r}oeàp)}`4# Da ùìrko, ukd adqd pcd fpcxkoc` ukii`q}ked I ak c rpcqk D4,###,Dc#Pcd r}oeàp}d eg` pu`}ìo f}`tp`cqì`cq` eocukuq` c`l edleold}` I upnlk

c *4 kcukìk eouqpqpkqk ad qpqqk k uoqqokcukìk ak c v`}qkek ak pc c *4)dnoco,D# Oqq`ckdio eou±Ĵ pc cpovo c * 4)dnoco, eg` egkdi`}ìo i'D+, kl epkv`}qke` k) ukio b I 'D4,###Dk∝4, Dk*4,###,Dc, Dc*4+#

Ld coxkoc` ak fflnp}d coq`vol` ek r`}i`qq`}bd ak fd}` dlepc` ouu`}vdxkockeg` }kuplq`}dcco pqklk c`l eo}uo a`lld q}dqqdxkoc`8

• u` i 8 D ↝ i'D+ b pcd ukiklkqpakc` a`l rkdco ` i'D+ b coq`vol` r`}D, dllo}d kl upo e`cq}o P b pc rpcqo coq`vol` ak D:

4Ld a`fflckxkoc` ak r}oeàp}d b eocq`cpqd c`l rd}dn}dfo 4 ak Z>Y#

>



7 KCAKE@

• u` i'D+ ` r'D+ uoco fflnp}` coq`volk }kur`qqo da D ` i, r uoco ukiklk)

qpakck a`l rkdco, dllo}d kl rpcqo pckqo V ak r ◦ i

∝4

8 i'D+ ↝ r'D+ bcoq`vol` r`} D:

• u` C b pc rpcqo coq`vol` ak pcd fflnp}d coq`vol` }kur`qqo da D, C bpc rpcqo coq`vol` ak D#

Ld idnnko} rd}q` a`lld q}dqqdxkoc` kc Z>Y uk }kf`}kue` dl eduo kc epk onckeorrkd ak ldqk orrouqk a`l tpda}dcnolo u‛kceocq}d dl fflckqo=# ]ùqd eou±Ĵ ùelp)uo kl eduo a`l q}dr`xko, eg` ud}bd kcvè` l‛onn`qqo a`lld couq}d q}dqqdxkoc`#R`} q}dr`xko kcq`cakdio pc tpda}dcnolo, k epk v`}qkek D4, D=, D5, D1 ukdcoakuqkcqk, coc dllkc`dqk ` qdlk eg` ap` ldqk orrouqk D4D= ` D5D1 ukdco rd}dll`lk#

Kl q}dr`xko i`}kqd ak ùu`}` uqpakdqo d rd}q`, kc tpdcqo iolq` coxkock r`}

tpùqd elduu` ak fflnp}` uk uìrlkffledco coq`voli`cq` }kur`qqo dl eduo n`c`}dl`#Và}ìo, kc rd}qkeold}`, eg`

• uk rpbo ad}` pcd ed}dqq`}kxxdxkoc` a`lld elduu` a`k q}dr`xk iàkdcq` pcduìrlke` }`ldxkoc` nòi`q}ked q}d q}` rd}qkeold}k rpcqk coq`volk 'ONG +a`l tpda}dcnolo eoirl`qo:

• lo uqpako a`k rpcqk coq`volk a`l q}dr`xko vk`c` uìrlkffledqo rokegb iol)q` fflnp}` dcqkukiklk 5 a`l tpda}dcnolo akv cqdco ukiklk c`l eduo a`lq}dr`xko:

• b rouukmkl`, d akfl`}`cxd eg` c`l eduo n`c`}dl`1, }keouq}pk}` kl q}dr`xkoeoc }knd ` eoirduuo d rd}qk}` ad dlepck 'tpdqq}o+ upok orro}qpck rpcqkcoq`volk:

• ld aùe}kxkoc` dcdlkqked a`k rpcqk coq`volk pudqk r`} ld }keouq}pxkoc`dkpqd d a`fflck}` kl lpono nòi`q}keo kc epk a`v` q}ovd}uk pc rpcqo co)q`vol` 'I + pcd volqd ffluudqk nlk dlq}k q}` 'O,N,G + dstcegb k v`}qkek a`lq}dr`xko dmmkdco eoo}akcdq` }`dlk#

=b@ kl eduo kc epk cùupcd eorrkd ak ldqk orrouqk b fo}idqd ad }`qq` rd}dll`l`#5Uk }kidcad d Z>Y r`} ld coxkoc` ak fflnp}` dcqkukiklk #1Kl r}omlìd dln`m}keo r`} kl tpda}dcnolo n`c`}dl` eoiro}qd ld uolpxkoc` ak `tpdxkock

dln`m}keg` ak n}dao q}` ` apctp` coc b }kuolpmkl` iàkdcq` eouq}pxkock eoc }knd ` eoirduuo#



2#4 Coqdxkoc` ;

2#4 Coqdxkoc`

Pu`}ìo ld u`np`cq` coqdxkoc` eg` nkbd uk q}ovd kc Z>Y8

D kcaked kl q}dr`xko ak v`}qkek D4, D=, D5, D1:

Dkh rpcqo iàko ak DkD h :

Dkh,gj kcaked kl rpcqo akdnocdl` kcq`}u`xkoc` a`k ldqk orrouqk DkD h , DgDj#C`l q}dr`xko l` kcq`}u`xkock akdnocdlk dl fflckqo uoco ap`8 D45,=1 ` D41,=5:

Qk 9 D hDgDj, eoc k q}` v`}qkek akuqkcqk eoc k,h,g,j ∇ z4, =, 5, 1y, kcaked kln`c`}keo q}kdcnolo eoirlì`cqd}` ak D:

o'D+ q}dr`xko a`k ek}eoe`cq}k a`k q}kdcnolk eoirlì`cqd}k ak D 'r}oroukxko)c` 4#=+:

c'D+ q}dr`xko a`k cov`)e`cq}k , o e`cq}k a`k e`}egk a`k cov`)rpcqk a`k Qk

'r}oroukxkoc` 4#;+:

N'D+ md}ke`cq}o a`l q}dr`xko 'rd}dn}dfo 4#4+:

G 'D+ G )e`cq}o ak D, e`cq}o a`ll` ukiklkqpakck q}d k q}` q}dr`xk ràdlk 'rd}dn}dfo4#5+:

O'D+ O)e`cq}o ak D, e`cq}o a`lld ukiklkqpakc` eg` idcad D kc o'D+'rd}df}dno4#=+:

C 'D+ e`cq}o a`lld ukiklkqpakc` eg` idcad D kc c'D+ 'rd}dn}dfo 4#1+:

H 'D+, G )e`cq}o a`l q}dr`xko o'D+ 'rd}dn}dfo 4#0+:

'XW[ + e`}egko r`} k rpcqk W,[,X :

R D ukii`q}keo ak R }kur`qqo da D, `fl`qqo a`l i`xxo)nk}o ak R kcqo}co daD:

R DM kiidnkc` ak R c`lld }kùukoc` uplld }`qqd DM:

R DM r}ok`xkoc` o}qonocdl` ak R uplld }`qqd DM:

Dk,h r}ok`xkoc` o}qonocdl` ak Dk uplld }`qqd DgDj, eoc k,h,g,j ∇ z4, =, 5, 1y#

Kcolq}` tpdcao ek }kf`}k}ìo

• dlld iàkdcd r}kekrdl` a`l q}dr`xko kcq`ca`}ìo ld }`qqd r`} k rpcqkiàk a`k ldqk rd}dll`lk 'd qdl` }`qqd drrd}q`cnoco dceg` kl md}ke`cq}o `k rpcqk akdnocdlk dl fflckqo+0:

0Dlld iàkdcd r}kcekrdl` drrd}qk`c` dceg` l‛G )e`cq}o a`l q}dr xko# Tpùqo fdqqo ud}bdakiouq}dqo c`lld r}oroukxkoc` =#5



42 KCAKE@

• dl cov`)e`}egko ak pc e`}qo q}kdcnolo kcq`ca`}ìo kl e`}egko r`} k rpcqk

iàk '` k rkàk a`ll` dlq`xx`+ a`k ldqk a`l q}kdcnolo 'qò}ìd ak Roce`l`q+#Eoc }kf`}ki`cqo d q}` rpcqk D ,M,E , kcakeg`}ìo eoc

• ZDME Y l‛dcnolo o}k`cqdqo q}d l` }`qq` DM ` ME , vdlpqdqo ioaplo ς:

• ZZDME YY l‛dcnolo o}k`cqdqo q}d l` uìk}`qq` pue`cqk ad M, MD ` ME ,vdlpqdqo ioaplo =ς#

Kcfflc` kcakeg`}ìo }kur`qqkvdi`cq` roukqkv` ` c`ndqkv` l` ukiklkqpakck eg`lduekdco ffluuk ` uedimkdco u`nco dnlk dcnolk o}k`cqdqk#

Coqd8 vonlkdio tpk }kr}èkud}` eg` coc q}dqq`}ìo k eduk a`n`c }k akq}dr`xk dv`cqk ap` v`}qkek eokceka`cqk o q}` v`}qkek dllkc`dqk# Eocuka`}ìo,

kcvè`, eoi` eduk rd}qkeold}k 'id lk uqpak`}ìo d rd}q`+, kl q}dr`xko kuoue`l`` kl rd}dll`lon}diid# L` r}oroukxkock a`l edrkqolo 4 ` a`l edrkqolo = 'q}dcc`c`k rd}dn}dffl =#5 ` =#1+ uoco vdlka` r`} k q}dr`xk, kcqùk eoi` tpda}dcnolk coc ekelkek 'ekob eoc k v`}qkek drrd}q`c`cqk da pco uqùuo e`}egko+ dv`cqk k tpdqq}ov`}qkek akuqkcqk coc d q}` d q}` dllkc`dqk ` dv`cqk pcd eorrkd ak ldqk orrouqk rd}dll`lk # Kl eduo ekelkeo, eg` b rok tp`llo a`l q}dr`xko kuoue`l`, ud}bd uqpakdqo drd}q` kc ioao ukcq`qkeo dl =#5 ` kc ioao dcdlkqkeo dk rd}dn}dffl 5#=#4 ` 5#=#0#Kcfflc` dl rd}dll`lon}diid dmmkdio aàkedqo k rd}dn}dffl =#1 ` 5#=#=#



Edrkqolo 4

Dlepck Rpcqk Coq`volk Ùkiklkqpakck a`l Q}dr`xko

Kc tpùqo edrkqolo aùe}kv`}ìo dlepck rpcqk coq`volk a`l q}dr`xko, aàpe`c)aoc` r}or}k`qbd adk eo}}kuroca`cqk rpcqk coq`volk a`l tpda}dcnolo, uqpakdqk kcZ>Y# A`fflck}ìo, r`}bo, kc ioao kcakr`ca`cq` l‛O)e`cq}o, l‛G )e`cq}o ` kl rpcqoH r`} kl q}dr`xko ` akiouq}`}ìo dlepc` r}or}k`qbd ak tpùqk rpcqk d rd}qk}àdll` cpov` a`fflckxkock# Ovvkdi`cq` l` couq}` akiouq}dxkock ` l` r}or}k`qbdtpk akiouq}dq` coc eocq}daakeoco ld qò}kd n`c`}dl` a`l tpda}dcnolo, idc`l q}dr`xko k rpcqk coq`volk uk d}}keegkueoco ak r}or}k`qbd cpov`, eg` urùuo}`caoco ld q}dqqdxkoc` rkbp uìrlke` ` tpkcak gdcco upnn`}kqo pcd aùe}kxkoc`dpqocoid#

4#4 Md}ke`cq}o

R`} kl q}dr`xko, eoi` r`} tpdlukduk dlq}o c)noco D 9 'D4,###Dc+, pc rpcqocoq`vol` kiro}qdcq` b kl md}ke`cq}o N, a`fflckqo adlld }`ldxkoc` v`qqo}kdl`ck94 NDk 9 2# ]kro}qkdio ld r}oroukxkoc` =#4 ak Z>Y8

R}oroukxkoc` 4#4 Kl md}ke cq}o N ak pc tpda}dcnolo D 9 D4D=D5D1 bl‛kcq }u`xkoc` a`ll` q} mki akdc`, à b dceg` kl lo}o rpcqo iàko N 9Dkh*Dgj

= #

]kidcakdio dl rd}dn}dfo = ak Z>Y r`} ld q}dqqdxkoc` ak tpùqo rpcqo co)

q`vol`, kc tpdcqo coc r}ù`cqd rd}qkeold}kqbd }kur`qqo dl eduo a`l tpda}dcnolon`c`}dl`#

4#= O)e`cq}o

Ld u`np`cq` a`fflckxkoc` a`ll‛O)e`cq}o ak pc q}dr`xko }kuplqd iolqo uìrlkffl)edqd }kur`qqo d tp`lld n`c`}dl` a`l tpda}dcnolo4 r`} `fl`qqo a`lld u`np`cq`

4Uk vàd kl rd}dn}dfo 6 ak Z>Y#

44



4= Dlepck Rpcqk Coq`volk ` Ukiklkqpakck a`l Q}dr`xko

R}oroukxkoc` 4#= R`} pc q}dr`xko D k ek}eoe`cq}k Ok a`k q}kdcnolk eoirl`)

i`cqd}k uoco k v`}qkek ak pc q}dr`xko ukikl` da D#

Kccdcxkqpqqo akiouq}kdio eg` e‛b pcd ukiklkqpakc` roukqkvd eg` idcad8

D4 ↝ O=

D= ↝ O4

D5 ↝ O1

D1 ↝ O5

R`} ùìrko, l‛dcnolo

ZO5O=O4Y 9 ZO5O=, O=O4Y 9 ZD4D1, D1D5Y 9 ZD4D1D5Y 9 ZD1D4D=Y

L‛plqkio rduudnnko b eocu`np`cxd a`lld eocakxkoc` D4D= D5D1#

Tp uqo vdl dcdlondi`cq` r`} qpqqk nlk dlq}k dcnolk# Dllo}d uoco ukiklkqpqq` l` eorrk` ak q}kdcnolk

D4D=D5 ↑ O=O4O1

D4D=D1 ↑ O=O4O5

D4D5D1 ↑ O=O1O5

D=D5D1 ↑ O4O1O5

` eocuka`}dcaoc` k ldqk eoipck, eocelpakdio eg` ld ukiklkqpakc` b ld uqùud#

A`fflckxkoc`8 l‛ O)e`cq}o ak pc q}dr`xko D b kl e`cq}o a`lld ukiklkqpakcò'D+ eg` idcad D c`l upo q}dr`xko a`k ek}eoe`cq}k o'D+ 9 O=O4O1O5#

R}oroukxkoc` 4#5 L‛ O)e`cq}o b qdl` eg` ZDkOD h Y 9 ZDkDgD hY * ZDkDjD hY#

Ukd, kcfdqqk, Mkh l‛kcq`}u`xkoc` a`k e`}egk 'D4OD1+ ` 'O1OO4+ akv`}uo adO ` edleolkdio8

ZD4OD1Y 9 ZO1OO4Y 9 ZO1M41O4Y 9 ZM41O1O4Y * ZO1O4M41Y 9

9 ZD4D1D45Y * ZD=5O4D1Y 9 ZD4D=D5Y * ZD4D5D=Y * ZD=D5D1Y * ZD5D=D1Y 9

9 ZD4D=D1Y * ZD4D5D1Y

Qdl` r}or}k`qbd, akiouq}dqd kc n`c`}dl` r`} l‛O)e`cq}o ak tpda}dcnolk dl6#5 ak Z>Y, r}ovd eg` ld couq}d a`fflckxkoc` b eoirdqkmkl` eoc tp`lld ak Z>Y#



4#5 G )e`cq}o 45

4#5 G )e`cq}o

Kc pc d}qkeolo ak I# Gdrrdeg, Z=Y, eoird}` ld u`np`cq` eouq}pxkoc`= Uk ue`lndkc pc tpda}dcnolo tpdlpctp` D 9 D4D=D5D1 pcd eorrkd ak ldqk orrouqk , r`}ùìrko, D4D= ` D5D1# Dllo}d kl tpda}dcnolo D4=,51 9 D4,=, D=,4, D5,1, D1,5

b c`ndqkvdi`cq` ukikl` da D uèocao ld idrrd8

D4 ↝ D4,=

D= ↝ D=,4

D5 ↝ D5,1

D1 ↝ D1,5

Ue`nlk`cao dlq}` ap` eorrk` ak ldqk orrouqk uk oqq`cnoco dlq}k ap` tpd)

a}dcnolk D45,=1 ` D41,=5# Egkdikdio tpùqk q}` tpda}dcnolk tpda}dcnolk ràdlk ak D# Uk r}ovd kc Z=Y eg`

R}oroukxkoc` 4#1 K q}` tpda}dcnolk ràdlk uoco roukqkvdi`cq` ukiklk, eoc pc eoipc` e`cq}o ak ukiklkqpakc`#

Uk và` fdekli`cq` eg` tpùqo e`cq}o ak ukiklkqpakc` b pc rpcqo coq`volàk oncpco a`k q}` tpda}dcnolk ràdlk ` dceg` a`l tpda}dcnolo D# Akdio ldu`np`cq`

A`fflckxkoc`8 kl e`cq}o ak ukiklkqpakc` a`k q}` tpda}dcnolk ràdlk ak D b kl rpcqo coq`vol` ak D eg` egkdi`}ìo kl upo G )e`cq}o#

O}d e`}egkdio dlq}` r}or}k`qbd a`ll‛G )e`cq}o# Eocuka`}kdio k tpdqq}oq}kdcnolk ràdlk 8

• D4,=D4,5D4,1:

• D=,4D=,5D=,1:

• D5,4D5,=D5,1:

• D1,4D1,=D1,5#

C`l ]#D# Hogcuoc, Z5Y, qò}ìk 5;0, 5;6, 5;>, uoco akiouq}dq` l` u`np`cqkq}` r}oroukxkock, ak epk }kro}q`}ìo dceg` l` uìrlkek akiouq}dxkock#

R}oroukxkoc` 4#0 K q}kdcnolk ràdlk uoco roukqkvdi`cq` ukiklk#

=]keo}akdio tpk eg` Dk,h kcaked ld r}ok`xkoc` o}qonocdl` ak Dk uplld }`qqd DgDj#



41 Dlepck Rpcqk Coq`volk ` Ukiklkqpakck a`l Q}dr`xko

Ld akiouq}dxkoc` uf}pqqd l‛`tpdxkoc` ak Iktp`l5 r`} pc q}kdcnolo D4D=D5

upk epk ldqk ukdco uqdqk r}ùk k rpcqk R 4, R =, R 51

8ZD=R D5Y 9 ZD=D4D5Y * ZR =R 4R 5Y

Dllo}d vdlnoco8

ZD4,=D4,5D4,1Y 9 ZD=D4D1Y * ZD1D5D=Y

ZD5,1D5,4D5,=Y 9 ZD1D5D=Y * ZD=D4D1Y

Kcolq}`8ZD=D4D1Y * ZD1D5D=Y 9 ZD4D=D5Y * ZD5D1D4Y

` apctp` dceg` ZD=,4D=,1D=,5Y ` ZD1,5D1,=D1,4Y uoco pnpdlk dk ap` r}èà`cqk#

R}oroukxkoc` 4#6 K tpdqq}o e`}egk a`k cov` rpcqk a`k q}kdcnolk eoirlì`c)qd}k, ekob DkhD hgDgk, gdcco kc eoipc` pc rpcqo R #

Egkdi`}ìo k e`}egk a`k cov` rpcqk dceg` cov`)e`}egk c`l eo}uo a`lldq}dqqdxkoc`# Ukd R l‛kcq`}u`xkoc` a`k cov`)e`}egk ak Q4 ` Q5 akv`}ud ad D=1#Eocuka`}kdio l` u`np`cqk `tpdxkock upnlk dcnolk o}k`cqdqk8

ZD4D=D1Y 9 ZD=1D41D4=Y 9 ZD=1R D4=Y

ZD1D=D5Y 9 ZD=5D51D1=Y 9 ZD=5R D1=Y

Uoiidcao iìm}o d iìm}o oqq`ckdio8

ZD=5D45D4=Y 9 ZD4D=D5Y 9 ZD4D=D1Y*ZD1D=D5Y 9 ZD=1R D4=Y*ZD=5R D1=Y 9 ZD=5R D4=Y

eg` akiouq}d eg` R drrd}qk`c` dceg` dl cov`)e`}egko ak Q1# Dcdlondi`cqùk r}ovd eg` R drrd}qk`c` dceg` dl cov`)e`}egko ak Q=#

R}oroukxkoc` 4#> K ek}eoe`}egk a`k q}kdcnolk ràdlk eocuka`}dqk uor}d, uocoeoceo}}`cqk kc R , eg` b apctp` e`cq}o a`ll` ukiklkqpakck q}d k q}kdcnolk ràdlk#

Kcfdqqk, r`} ùìrko,

ZD=,1R D=,4Y 9 ZD=,1R D=5Y * ZD=5R D=,4Y 9 ZD=,1D45D=5Y * ZD=5D51D=,4Y 9

9 ZD4D5, D4D=Y * ZD=D1, D1D5Y 9 ZD1D=D4Y * ZD4D5D1Y 9

ZD=,1D=,5D=,4Y

O}d akiouq}kdio eg` kl rpcqo R eokceka` eoc l‛G )e`cq}o a`l q}dr`xko D,a`fflckqo eoi` e`cq}o a`ll` ukiklkqpakck q}d k q}dr`xk a`ll` r}ok`xkock 'r}oro)ukxkoc` 4#1+#

5L‛`tpdxkoc` b ld 476 kc Z5Y#1Dllo}d ùkuq` pc rpcqo R eoipc` dk q}` e`}egk 'DkR hR g+



4#1 C )e`cq}o 40

R}oroukxkoc` 4#7 R eokceka` eoc l‛ G )e`cq}o ak D#

Akiouq}kdio eg` G và` k ldqk eo}}kuroca`cqk a`k q}kdcnolk ràdlk uoqqolo uqùuo dcnolo# Tpùqo b upstek`cq` r`} dfl`}id}` eg` G b dceg` kl e`cq}oak ukiklkqpakc` a`k q}kdcnolk ràdlk ` apctp` eokceka` eoc R #

Udrrkdio eg` ZD=,1GD=,4Y 9 ZD5,1GD5,4Y, r`}egb dcnolk q}d v`}qkek eo})}kuroca`cqk uèocao ld ukiklkqpakc` roukqkvd eg` idcad D4,=, D=,4, D5,1, D1,5

kc D4,5, D=,1, D5,4, D1,= }kur`qqkvdi`cq`# Kcolq}` ZD5,1GD5,4Y 9 ZD1,5GD1,=Y,r`}egb akfl`}`cx` a`nlk dcnolk }kur`qqkvdi`cq` `npdlk ZD5,1GD1,5Y∝ZD5,4GD1,5Y` ZD5,4GD1,=Y ∝ ZD5,4GD1,5Y

Apctp` G và` k ldqk eo}}kuroca`cqk D=,1D=,4, D5,4D5,1, D1,=D1,5, D4,5D4,=

uoqqo dcnolk `npdlk# Kc ioao dcdlono uk oqqk`c` eg` G và` onck tpda}prldak ldqk eo}}kuroca`cqk a`k q}kdcnolk ràdlk uoqqo dcnolk `npdlk# Dllo}d G b kl

e`cq}o a`ll` ukiklkqpakck q}d tpùqk tpdqq}o q}kdcnolk, eg` dv`vdio egkdidqoR # Dmmkdio akiouq}dqo eg` G eokceka` eoc R #

Rouukdio dceg` fdekli`cq` edleold}`0 ZD=,1GD=,4Y kc fpcxkoc` a`nlk dcnolkak D8

∝ZD4,5D=,1D5Y 9 ZD1,=D=,1D4,5Y 9 ∝ZD1D=D4Y

ZD=,1GD=,4Y 9 ZD=,1D4,5, D5D1Y 9 ZD4,5D=,1D5Y * ZD=,1D5D1Y 9

9 ZD1D=D4Y * ZD4D5D1Y

L` r}oroukxkock r}èà`cqk r}ovdco eg` ld a`fflckxkoc` ak G )e`cq}o tpkpudqd b eo`}`cq` eoc tp`lld ak Z>Y#

4#1 C )e`cq}o

Dcdlondi`cq` d tpdcqo fdqqo r`} k ek}eoe`cq}k a`k q}kdcnolk eoirlì`cqd}k,o}d eocuka`}kdio k e`cq}k C k a`k cov`)e`}egk a`k q}kdcnolk eoirlì`cqd}k#

R}oroukxkoc` 4#; K rpcqk C k uoco k v`}qkek ak pc q}dr`xko c`ndqkvdi`cqùkikl` da D#

Eocuka`}kdio ld idrrd8

D4 ↝ C =

D= ↝ C 4D5 ↝ C 1D1 ↝ C 5

Akiouq}kdio eg` qpqqk nlk dcnolk eo}}kuroca`cqk uoco eocn}p`cqk# Edleo)lkdio, r`} ùìrko,

ZC =C 1C 4Y 9 ZD45R D=5Y 9 ZD45D4=D=5Y 9 ZD4D5D=Y

0Tpùqo edleolo coc b }kegkùqo adlld akiouq}dxkoc`#




aov` dmmkdio pudqo ld r}or}k`qbd a`k fduek ak ek}eocf`}`cx` ak dv`}` l‛duu`

}dakedl` r`}r`cakeold}` dll‛duu` e`cq}dl`# Dcdlondi`cq`ZC 5C =C 1Y 9 ZD1D=D5Y

` apctp`ZC 5C =C 4Y 9 ZD1D4D=Y

Apctp` vdl`ZC 5C =C 4Y 9 ZD1D=D5Y

Tpùqo vdl` dllo uqùuo ioao r`} qpqqk nlk dlq}k dcnolk#

A`fflckxkoc`8 kl e`cq}o C a`lld ukiklkqpakc` eg` idcad D kc c'D+ uk egkdi`}bd l‛ C )e`cq}o ak D#

C b pc rpcqo coq`vol` ak D# Dceg` l‛O)e`cq}o ak c'D+ b pc rpcqocoq`vol` ak D, ak epk o}d uqpakdio dlepc` r}or}k`qbd#

R}oroukxkoc` 4#42 G b l‛ O)e`cq}o a`l q}dr`xko c'D+, ekob G 'D+ 9 O'c'D++#

Qdl` r}oroukxkoc` b ld 7#0 kc Z0Y, d epk }kidcakdio r`} ld akiouq}dxkoc`#

R}oroukxkoc` 4#44 Ld co}idl` adl md}ke`cq}o N dl ldqo DkD h aki`xxd kl

u`ni`cqo C gC j#

Qdl` r}oroukxkoc` b akiouq}dqd kc Z0Y dl 5#5 ` tpk }kro}qkdio ld uìrlkeàkiouq}dxkoc`#

@uu`cao D4=D=1 pcd eo}ad a`l e`}egko a`k cov` rpcqk ak Q5, rd}dll`ldd D4D1, dllo}d 'C 5+4= 9 'D41*D=1

= +4=# Dcdlondi`cq` 'C 1+4= 9 'D45*D=5

= +4=#C` u`np`

'C 5 * C 1

=+4= 9 '

D45 * D=5

=+4= * '

D41 * D=1

=+4= 9

9 'D45 * D=1

=

+4= * 'D41 * D=5

=

+4= 9 'N4= * N4=

=

+4= 9 N4=

R}oroukxkoc` 4#4= N 9 G 'c'D++

Mduqd iouq}d}`, eoi` dl 1#4 ak Z0Y, eg` kl e`}egko a`k cov` rpcqk ak tpdlukdukq}kdcnolo eoirlì`cqd}` ak c'D+ rduud r`} N#

R`} ùìrko,ZC 41NC 51Y 9 ZD5D=, D=D4Y 9

9 ZD5D=D4Y 9 ZC 5C 1C 4Y 9 ZC 41C 45C 51Y



4#0 Rpcqo H 4>

R}oroukxkoc` 4#45 N ∇ CO

Dceg` tpùqd r}oroukxkoc` vdl` r`} kl tpda}dcnolo kc n`c`}dl` à b eoc)q`cpqd kc Z0Y dl 44#4#

C`ll‛oioq`qkd uoco dllkc`dqk kl e`cq}o C ` k rpcqk eo}}kuroca`cqk O'D+` O'c'c'D+++# ]keo}akdio eg` O'c'D++ 9 G r`} a`fflckxkoc` ` apctpÒ'c'c'D+++ 9 G 'c'D++ 9 N, r`} ld r}oroukxkoc` r}èà`cq`#

4#1#4 Dlepc` R}or}k qbd ak G

]k`l`cegkdio m}`vì`cq` l` r}or}k`qbd a`ll‛G )e`cq}o fflc‛o}d ouu`}vdq`#

4# G b kl e`cq}o a`ll` ukiklkqpakck roukqkv` q}d k q}` q}dr`xk ràdlk:

=# G b rpcqo a‛kcq`}u`xkoc` eoipc` a`k tpdqq}o e`}egk a`k cov` rpcqk a`kq}kdcnolk eoirlì`cqd}k, ekob DkhD hgDgk:

5# G b e`cq}o a`ll` ukiklkqpakck roukqkv` q}d k tpdqq}o q}kdcnolk ràdlk:

1# G b l‛O)e`cq}o a`l q}dr`xko a`k cov`)e`cq}k, tpkcak ZC kGC hY 9 ZC kC gC hY*ZC kC jC h Y#

4#0 Rpcqo H

A`fflckdio o}d kl rpcqo coq`vol` H r`} kl q}dr`xko eoi` G )e`cq}o a`l q}dr`xkoo'D+ a`k ek}eoe`cq}k a`k q}kdcnolk eoirlì`cqd}k#

Akiouq}`}ìo eoc ld r}oroukxkoc` =#4 eg` D vk`c` idcadqo kc o'D+ q}d)ikq` pc‛oioq`qkd ` pcd }oqdxkoc` kcqo}co da O ak ς

= ` kl q}kdcnolo coq`volÒH G b }`qqdcnolo 'r}oroukxkock =#1 ` =#0+# Kcolq}` akiouq}`}ìo c`lld r}o)roukxkoc` =#6 eg` kl q}kdcnolo ONG b kuoue`l` uplld mdu` OG # Dllo}d dmmkdioeg` NG 9 NO b kl }dnnko a`l ek}eoe`}egko ak OH G , dllo}d NG 9 NO 9 NH ,` tpkcak H 9 G N# Ekob uk rp±o oqq`c`}` H ad G ` N }k`qq`cao kl r}kiorpcqo upl uèocao# Tpùqo akiouq}d eg` ld a`fflckxkoc` ak H 'D+ tpk adqd beoirdqkmkl` eoc tp`lld dl rd}dn}dfo 5 ak Z>Y#

4#6 Tpda}kldq`}k Eoirlì`cqd}k ` Rpcqo ak Ik)

tp`lD tpùqo rpcqo aommkdio }keo}ad}` l` coxkock ak tpda}dcnolo ` ak tpda}kldq`)}o eoirl`qo# Kc pc tpda}dcnolo vk uoco u`k ldqk, ekob }`qq` eg` eocnkpcnoco ktpdqq}o rpcqk DkD h, eoc k, h ∇ z4, =, 5, 1y# Pc tpda}kldq`}o eoirl`qo, kcvè`,b pc kcukì` T 9 z}4, }=, }5, }1y ak tpdqq}o }`qq` }k, a`qq` ldqk ak T#

U` kc pc tpda}dcnolo D 9 DkD hDgDj uk knco}d pcd eorrkd ak ldqk orrouqk DkD h, DgDj, uk oqqk`c` pc tpda}kldq`}o eg` egkdi`}ìo eoirlì`cqd}` ak D

` kcakeg`}ìo eoc Tkh,gj# Eou±Ĵ, r`} ùìrko, T45,=1 gd r`} ldqk l` }`qq`




D4D=, D=D5, D5D1, D1D4# Nlk dlq}k ap` tpda}kldq }k eoirlì`cqd}k ak D

uoco T4=,51 ` T41,=5#]kduupikdio l` coxkock upk tpda}kldq`}k T 9 }4}=}5}1 eg` tpk ek kcq`}ù)udco#

Kl v`}qke` ]kh b kl rpcqo ak kcq`}u`xkoc` ak }k eoc } h# Knco}dcao pc ldqo}k uk oqqk`c` pc q}kldq`}o eoirlì`cqd} Lk 9 Tz} h, }g, }jy# Q}d k u`k v`}qkekak T u‛kcakvkapdco q}` eorrk` ]kh, ]gj ak v`}qkek orrouqk # L` q}` }`qq` eg`pckueoco l` eorrk` ak v`}qkek orrouqk uoco k ldqk a`l q}kldq`}o akdnocdl`#

K }kuplqdqk r}kcekrdlk a`lld nòi`q}kd a`l tpda}kldq`}o eoirl`qo eoirdkococ`l ldvo}o ak N# Uq`kc`}, Z4Y# Tpk fd}ìo puo a`k u`np`cqk8

4# K ek}eoe }egk a`k tpdqq}o q}kldq`}k eoirlì`cqd}k gdcco kc eoipc` pcrpcqo I 'T+, a`qqo rpcqo ak Iktp`l a`l tpda}kldq`}o T#

=# K e`cq}k a`k tpdqq}o e`}egk r}èà`cqk drrd}q`cnoco da pco uqùuo e`})egko I'T+, epk drrd}qk`c` dceg` I 'T+# Lo egkdi`}ìo kl e`}egko ak Iktp`l a`l tpda}kldq`}o T#

Ovvkdi`cq` uoco coq`volk r`} kl tpda}kldq`}o T kl rpcqo ak Iktp`l I 'T+` kl e`}egko ak Iktp`l I'T+#

@cpcekdio o}d ap` qò}ìk eg` vdlnoco kc n`c`}dl` r`} onck tpda}dcno)lo, ` apctp` dceg` r`} kl q}dr`xko# Coc c` ad}ìo l` akiouq}dxkock, r`} l`tpdlk }kidcakdio d Z0Y#

R}oroukxkoc` 4#41 C`l tpda}dcnolo D l‛kcq`}u`xkoc` a`nlk duuk ak ap` ldqk

orrouqk DkD h , DgDj, drrd}qk`c` dl e`}egko ak Iktp`l a`l tpda}kldq`}o Tkh,gj#

R}oroukxkoc` 4#40 K e`}egk ak Iktp`l a`k q}` tpda}kldq`}k eoirlì`cqd}k ak pc tpda}dcnolo D gdcco kc eoipc` pc rpcqo I #

I b ovvkdi`cq` pc cpovo rpcqo coq`vol` r`} D, eg` kcakeg`}ìo eocI 'D+ ` egkdi`}ìo kl rpcqo ak Iktp`l a`l tpda}dcnolo D#

Kcq`}r} q`}ìo k upaa`qqk qò}ìk c`l eduo a`l q}dr`xko c`l r}ouukioedrkqolo#



Edrkqolo =

Qò}ìk Urèkfflek a`lQ}dr`xko

Kc tpùqo edrkqolo `cpce`}ìo ` akiouq}`}ìo dlepck qò}ìk coc aàpekmklkadll` r}or}k`qbd a`k rpcqk coq`volk a`l tpda}dcnolo n`c`}dl`, eg` dmmkdioekob uvklprrdqo urèkffledi`cq` r`} kl q}dr`xko ` coc eoi` eduk rd}qkeold}k ak}kuplqdqk ak Z>Y# Tpùqk qò}ìk, kcukì` d tp`llk a`l r}kio edrkqolo, uoco dlldmdu` a`lld }keouq}pxkoc` a`l q}dr`xko, eg` }kro}q`}ìo c`l edrkqolo 1#

R}oroukxkoc` =#4 Ld ukiklkqpakc` o eg` idcad D kc o'D+ b kl r}oaoqqo ak

pcd }oqdxkoc` ak dcnolo ZZGOH YY4, ` ak pc‛oioq`qkd ak eo`stek`cq`= |OH ||OG | #

Kl vdlo}` duuolpqo a`l eo`stek`cq` a`ll‛oioq`qkd b egkd}di`cq`|OH ||OG | , r }egb

H b l‛G )e`cq}o ak o'D+, roukqkvdi`cq` ukikl` da D r`} ld r}oroukxkoc` 4#=#Eocuka`}kdio l‛kiidnkc` ak D uèocao l‛oioq`qkd ak e`cq}o O ` eo`st)

ek`cq` roukqkvo |OG ||OH | # L‛G )e`cq}o vk`c` idcadqo c`l rpcqo G eg` drrd}qk`c`

dlld uìk}`qqd OG : H vk`c` q}dufo}idqo kc H eg` drrd}qk`c` dlld uìk}`qqdOH # Kl q}dr`xko D b ukikl da D ` o'D+ b ld upd kiidnkc` q}dikq` ld }oqd)xkoc` kc epk dmmkdio fdqqo}kxxdqo ld ukiklkqpakc` roukqkvd o# Qdl` }oqdxkoc`gd dcnolo ZZGOH YY 9 ς

= , r }egb G a`v` dcad}` kc H ` apctp` ld uìk}`qqdOG a`v‛ uu`}` idcadqd c`lld uìk} qqd OH #

R}oroukxkoc` =#= Ld ukiklkqpakc` c`ndqkvd c eg` idcad D kc c'D+ b kl r}oaoqqo ak pc‛oioq`qkd ak e`cq}o C ` ak eo`stek`cq` |GC |

|CO| 9 |GO||NG | ` ak pcd

}kùukoc` uplld }`qqd r`} C 'rpcqo pckqo+ rd}dll`ld dlld mku`qq}ke` ùq`}cd a`ll‛dcnolo ZZOGNYY#

4]keo}akdio eg` ld aorrkd rd}`cqùk tpda}d kcaked eg` l‛dcnolo eocuka`}dqo b tp`llo'o}k`cqdqo+ q}d ld uìk}`qq` GO ` OH à b vdlpqdqo ioaplo =ς#

=Adqk k u`ni`cqk o}k`cqdqk 'v`qqo}k+ rd}dll`lk DM ` EA kcakeg`}ìo eoc DMEA

kl }drro}qoeoc u`nco q}d l` lo}o lpcng xx` o}k`cqdq # U` coc uoco rd}dll lk kcakeg }ìo kl }drro}qo'roukqkvo+ q}d l` lpcng`xx` coc o}k`cqdq` |DM| ` |EA| eoc |DM|

|EA|#

4;



=2 Qò}ìk Urèkfflek a`l Q}dr`xko

Kl vdlo}` duuolpqo a`l eo`stek`cq` b adqo adl }drro}qo q}d u`ni`cqk eo})

}kuroca`cqk# Apctp`, ùu`cao C kl e`cq}o a`lld ukiklkqpakc` ` G kiidnkcàk O 'r}oroukxkoc` 4#42+, qdl` }drro}qo b |GC ||CO| # ]keo}adcao ld r}oroukxkoc`

4#4= 'N 9 G 'c'D+++ kl vdlo}` duuolpqo a`l eo`stek`cq` uk rpbo ùr}ki`}` dceg`|NG ||GO| #

Akiouq}kdio eg` u` ue`nlkdio kl eo`stek`cq` roukqkvo, dllo}d ld }kùukoc`r}ù`cq` c`lld ukiklkqpakc` c`ndqkvd a`v‛ùu`}` uplld }`qqd r`} C rd}dll`lddlld mku`qq}ke` ùq`}cd a`ll‛dcnolo ZZOGNYY# Eocuka`}kdio l‛kiidnkc` ak D

uèocao l‛oioq`qkd ak e`cq}o C ` eo`stek`cq` roukqkvo |CG ||CN| 9 |GO|

|GN| # L‛G )

e`cq}o vk`c` idcadqo kc G eg` drrd}qk`c` dlld uìk}`qqd CG ` O vk`cìdcadqo kc O eg` drrd}qk`c` dlld uìk}`qqd CO# Kcolq}` O eda` uplldmku`qq}ke` kcq`}cd a`ll‛dcnolo ZZOGNYY r`} kl qò}ìd a`ll` mku`qq}kek drrlkedqo

dl q}kdcnolo GNO # Kcfdqqk ld mku`qq}ke` akvka` kl ldqo orrouqo kc u`ni`cqkr}oro}xkocdlk dnlk dlq}k ldqk8

OO

NO

9|GO|

|GN|

Rokegb O a`v‛ùu`}` q}dufo}idqo kc G adlld }kùukoc` duukdl` kc epk dmmkdiofdqqo}kxxdqo ld ukiklkqpakc` c`ndqkvd, ld }`qqd up epk }k`qqkdio a`v‛ùu }`ld r`}r`cakeold}` r`} C dlld }`qqd GO, ekob ld }`qqd r`} C rd}dll`ld dlldmku`qq}ke` ùq`}cd a`ll‛dcnolo ZZOGNYY# Tpùqo b tpdcqo vol`vdio#

=#4 Iàkdcd R}kcekrdl` ` ]`qqd OG a`l Q}dr`xko

Egkdi`}ìo iàkdcd r}kcekrdl` a`l q}dr`xko, eoi` dccpcekdqo c`ll‛kcq}oap)xkoc`, ld }`qqd D4=D51 r`} k rpcqk iàk a`k ldqk rd}dll`lk D4D= ` D5D1, eg`rduud dceg` r`} k rpcqk akdnocdlk dl fflckqo D45,=1 ` D41,=5#

Akiouq}kdio eg`

R}oroukxkoc` =#5 L‛ G )e`cq}o ak pc q}dr`xko D drrd}qk`c` dlld iàkdcd r}kcekrdl` ak D#

Kcfdqqk u` eocuka`}kdio k cov`)e`}egk ak Q= ` Q5, eg` u‛kcq`}uèdco kc D41

` G , uvklprrdcao ld u`np`cq` uoiid ak dcnolk oqq`ckdio8

ZD4=GD41Y*ZD41GD51Y 9 ZD4=D=1D41Y*ZD41D45D51Y 9 ZD1D4D=Y*ZD5D1D4Y

L‛plqkid uoiid b cplld r`}eg± D4D= D5D1, apctp` ld eocakxkoc`

ZD4=GD41Y * ZD41GD51Y 9 2

iouq}d r}or}ko l‛dllkc`di`cqo ak D51, G , D4=#



=#4 Iàkdcd R}kcekrdl` ` ]`qqd OG a`l Q}dr`xko =4

Akiouq}kdio o}d eg` pcd eocakxkoc` cèùud}kd5 dstcegb pc tpda}dcnolo

ukd pc q}dr`xko b eg` OG ukd r`}r`cakeold}` dk ldqk rd}dll`lk, ` apctp` dOH # C`l q`}xo edrkqolo, eoc ld nòi`q}kd dcdlkqked, iouq} }ìo dceg` ldupstek`cxd ak tpùqd eocakxkoc` upk rpcqk coq`volk 'r}oroukxkock 5#> ` 5#4=+#

R}oroukxkoc` =#1 Kc pc q}dr`xko ld }`qqd OG , eg` eocnkpcn` l‛ O)e`cq}o `l‛ G )e`cq}o, b r`}r cakeold} dlld ak}`xkoc` a`k ldqk rd}dll`lk ak D#

Eocuka`}kdio ld ukiklkqpakc` c, kcq}oaoqqd kc 4#1, eg` q}dufo}id D kcc'D+# Kc rd}qkeold}` idcad ld iàkdcd r}kcekrdl ak D c`lld }`qqd r`} G r`}r`cakeold}` d C 4C =# Ukd δ 9 ZGD51OY

Udrrkdio adll` r}oroukxkock 4#4= ` 4#45 eg` N 9 G 'c'D++ 'tpkcak N ∇C 4=C 51+ ` N ∇ C O# Kcolq}` N ∇ D4=D51# Id dllo}d N b l‛kcq`}u`xkoc`

eoipc` a`ll` q}` }`qq` D4=D51, C 4=C 51 ` C O#Rokegb r`} ld r}oroukxkoc` 4#44 C 4=C 51 OOD4D= , dllo}d ZD4=NC 51Y 9 δ#

Akiouq}kdio eg`ZD4=NC 51Y 9 ZD4=GOY '9 δ+

]keo}akdio eg`c 8 G ↝ N

C ↝ C O ↝ G

Dllo}d k q}kdcnolk NCG ` OGC , eg` gdcco l‛dcnolo ZGCNY kc eoipc`,uoco c`ndqkvdi`cq` ukiklk# Kc rd}qkeold}` ZCGNY 9 ZGOC Y# U` G } b kl

}kùuo ak G up } 'ld }`qqd r`} C r`}r`cakeold}` dlld iàkdcd r}kcekrdl`+,ZCGNY 9 ZNG }C Y# Dllo}d, pudcao ld r}oroukxkoc` 4#44,

ZCGNY 9 ZN}G }C Y 9 ZGG }C Y 9 ZC 51NOY 9 ZC 4=NC Y

Tpùqo iouq}d kl rd}dll`lkuio q}d C 4=C 51 ` OG , r`}egb ZGOC Y 9 ZCGNY 9ZC 4=NC Y# Dmmkdio eou±Ĵ

ZD4=GOY 9 ZD4=NC 51Y 9 δ

Dmmkdio eou±ĴEo}olld}ko =#0 Ld }`qqd OH b rd}dll`ld dk ldqk rd}dll`lk a`l q}dr`xko D#

5rp}egb coc eokcekadco O ` H 'eduo a`l q}dr`xko kuoue`l` q}dqqdqo ukcq`qkedi`cq` dlrd}dn}dfo =#5+ o O ` G 'eduo a`l rd}dll`lon}diid q}dqqdqo ukcq`qkedi`cq` dl rd}dn}dfo=#1+



== Qò}ìk Urèkfflek a`l Q}dr`xko

R}oroukxkoc` =#6 Kl q}kdcnolo GON b kuoue`l` }kur`qqo dlld mdu` GO#

Eocuka`}kdio

ς

=∝ δ 9 ZGND45Y 9 ZD4=ND=1Y 9

ς

=∝ ZCG }NY

` }keo}adcao eg` ZCG }NY 9 ZNOG Y, dmmkdio

ZNOG Y 9 δ 9 ZOGNY

Ouu`}vkdio eg` rokegb kc pc q}dr`xko kl q}kdcnolo OH G b }`qqdcnolo vdl`ld u`np`cq`8

Ouu`}vdxkoc` =#> L` ak}`xkock a`ll` mku`qq}kek a`ll‛dcnolo ZGH OY uk oqq`c)noco eoc pcd }oqdxkoc` ak ς1 ad tp`ll` a`ll` mku`qq}kek a`ll‛dcnolo ZOGH Y1#

Ukd K l‛kce`cq}o a`l q}kdcnolo HOG #

ZOGH Y * ZGHOY 9ς

=

ad epkς

19

4

=ZOGH Y *

4

=ZGHOY 9 ZKGH Y * ZGHK Y 9 ZGKH Y

Eoc }kf`}ki`cqo dll` r}or}k`qbd ak G eg` dmmkdio uqpakdqo c`l edrkqolo 4,`l`cegkdio tpk tp`ll` eg` vdlnoco r`} kl q}dr`xko, id coc r`} kl tpda}dcnolon`c`}dl`#

4# G drrd}qk`c` dlld iàkdcd r}kcekrdl` a`l q}dr`xko D:

=# OG b r`}r`cakeold}` dlld ak}`xkoc` a`k ldqk rd}dll`lk ak D:

5# ZNGOY 9 ZGONY#

=#= Dlepc` R}or}k`qbd a`l Rpcqo ak Iktp`l a`l Q}d)

r`xkoKcq`}r}`qkdio k qò}ìk a`l rd}dn}dfo 4#6 r`} k q}` tpda}kldq`}k eoirlì`c)qd}k a`l q}dr`xko#

Eocuka`}kdio dcxkqpqqo k ap` tpda}kldq`}k eoirlì`cqd}k 'T45,=1 ` T41,=5+,eg` kcelpaoco tpkcak ld eorrkd ak ldqk rd}dll`lk ak D# Uqpakdio, r`} ùìrko,T45,=1# Uk và` upmkqo eg`8

1Tpùqo akiouq}d eg` GK ` ld upd r`}r`cakeold}` r`} l‛o}knkc` G uoco l` mku`qq}kek a`nlkdukcqoqk a`ll‛kr`}mol` eg` kcq}oap}}ìo c`l edrkqolo 5#



=#= Dlepc` R}or}k`qbd a`l Rpcqo ak Iktp`l a`l Q}dr`xko =5

4# k q}kldq`}k eoirlì`cqd}k coc a`n`c`}k uoco ap` ` uoco oioq`qkek eoc

e`cq}o a`ll‛oioq`qkd kc D45,=1:=# k lo}o ek}eoe`}egk uoco qdcn`cqk kc D45,=1:

5# I 'T45,=1+ eokceka` eoc D45,=1:

1# kl e`}egko ak Iktp`l i45,=1 b a`n`c`}` à b pcd }`qqd , rduudcq` r`}D45,=1#

Dcdlong` r}or}k`qbd vdlnoco r`} kl tpda}kldq`}o T41,=58 kl rpcqo ak Iktp`l bD41,=5 ` kl e`}egko ak Iktp`l b pcd }`qqd i41,=5, eg` rduud r`} D41,=5#

Kcfflc` eocuka`}kdio T'D+ 9 T4=,51 kl tpda}kldq`}o eoirlì`cqd}` r}kc)ekrdl` a`l q}dr`xko D, oqq`cpqo `lkikcdcao k ldqk rd}dll`lk a`l tpda}dcno)

lo D# Rokeg± pco a`k v`}qkek orrouqk b dll‛kcfflcqo, kl q}kldq`}o akdnocdl` bfo}idqo ad q}` }`qq` ak epk ap` uoco rd}dll l # Kl e`}egko ak Iktp`l a ltpda}kldq`}o eoirlì`cqd}` r}kcekrdl` coc b a`n`c`}`# Tpdcqo dl rpcqoak Iktp`l I 'T4=,51+, uk q}dqqd r`} a`fflckxkoc` a`ll‛kcq`}u`xkoc` a`k e`}egk'D4D5D41,=5+, 'D=D1D41,=5+, 'D=D5D45,=1+, 'D4D1D45,=1+# R}ov }ìo eg` ukq}dqqd a`ll‛O)e`cq}o ak D 'r}oroukxkoc` =#;+#

R}oroukxkoc` =#7 Kc pc q}dr xko dv`cq` k ldqk D4D= ` D5D1 rd}dll`lk k rpcqk O,H,D45,=1, D=1, D45, Q 45,=1 ` O,H,D41,=5, D41, D=5, Q 41,=5 drrd}q`cnoco}kur`qqkvdi`cq` dnlk uqùuk e`}egk#

Kc Z>Y b akiouq}dqo 'r}oroukxkoc` 40#=+ eg` H drrd}qk`c` dl n`c`}keo e`})egko 'uoco kc qpqqo q}`+ r`} pc rpcqo akdnocdl` Dkh,gj ` r`} k rpcqk iàkDkh, Dgj a`k ldqk da ùuo dfl`}`cqk# C`l q}dr`xko tpùqk e`}egk uoco ap` 'pcor`} D45,=1 ` pco r`} D41,=5+, i`cq}` kl q`}xo b a`n`c`}` r`}egb D4=,51 bdll‛kcfflckqo# Akiouq}kdio eg` O drrd}qk`c` d tpùqk ap` e`}egk#

Ldvo}kdio, r`} ùìrko, upl e`}egko 'D45,=1D45D=1+, eg` gd eoi` akd)i`q}o D45,=1Q 45,=1# O drrd}qk`c` d qdl` e`}egko, r`}egb ZQ 45,=1OD45,=1Y 9 ς

= #Kcfdqqk Q 45,=1 b kiidnkc` ak D45,=1 dqq}dv`}uo ld ukiklkqpakc` eg` idcad D

kc o'D+, ak e`cq}o O#

R}oroukxkoc` =#; L‛ O)e`cq}o a`l q}dr`xko b dceg` kl rpcqo ak Iktp l a`l tpda}kldq`}o eoirlì`cqd}` r}kcekrdl` T4=,51#

Mduqd akiouq}d}` eg` O drrd}qk`c` dl ek}eoe }egko ak onck q}kldq`}o eoi)rlì`cqd}` a`l tpda}kldq`}o eoirlì`cqd}` r}kcekrdl`#

Eoikcekdio akiouq}dcao eg` O ∇ 'D4D1D45,=1+# Pu }ìo kl rd}dll lk)uio q}d l` }`qq` D4D=, D5D1, D45D=1 ` ld r}oroukxkoc` 4#5# Edleolkdio

ZD4OD1Y 9 ZD4D=D1Y * ZD4D5D1Y 9 ZD45D=1D45,=1Y * ZD45,=1D45D=1Y 9

9 ZD45D45,=1D=1Y 9 ZD4D45,=1D=Y 9 ZD4D45,=1D1Y




Dcdlondi`cq` uk akiouq}d eg` O drrd}qk`c` dk e`}egk 'D45,=1D5D=+,

'D41,=1D4D5+ ` 'D41,=5D=D1+# Apctp` O eokceka` eoc kl rpcqo ak Iktp`l a`ltpda}kldq`}o eoirlì`cqd}` r}kcekrdl`#

]keo}adcao ld uèocad r}or}k`qbd a`k tpda}kldq`}k c`l rd}dn}dfo 4#6, uk gdkl u`np`cq`

Eo}olld}ko =#42 L‛ O)e`cq}o ak pc q}dr`xko drrd}qk`c` dl e`}egko ak Iktp`l a`l tpda}kldq`}o eoirlì`cqd}` r}kcekrdl` T4=,51#

R}oroukxkoc` =#44 K rpcqk I, D41,=5, D45,=1, O drrd}q`cnoco dllo uqùuoe`}egko ak e`cq}o F 0#

Eocuka`}kdio kl e`}egko 'O, D45,=1, D41,=5+# @uuo vk`c` idcadqo adlldukiklkqpakc` o c`l e`}egko ak Iktp`l a`l tpda}kldq`}o eoirlì`cqd}` r}kcek)rdl`, 'O, Q 45,=1, Q 41,=5+, eg` rduud dceg` r`} I # Vonlkdio akiouq}d}` egÌ ∇ 'O, D45,=1, D41,=5+# D tpùqo ueoro edleolkdio8

ZD41,=5I D45,=1Y 9 ZD41,=5I ,I D45,=1Y 9 ZQ 41,=5I ,I Q 45,=1Y 9

9 ZQ 41,=5I Q 45,=1Y 9 ZQ 41,=5OQ 45,=1Y 9 ZD41,=5OD45,=1Y

eg` iouq}d tpdcqo volpqo#

R}oroukxkoc` =#4= Kl q}kdcnolo F I OI b }`qqdcnolo ` ukikl` dl q}kdcnoloGOH #

Ukd O% l‛kcq`}u`xkoc`, akv`}ud ad G , a`lld iàkdcd r}kcekrdl` ak D eoc kle`}egko 'I GO+ ` ukd OI kcq`}u`xkoc` a`ll‛duu` ak IO eoc kl r}olpcndi`cqoak O%I #

Eocuka`}kdio kl e`}egko ak e`cq}o OI ` rduudcq` r`} I , eg` akiouq}`)}ìo ùu`}` kl e`}egko ak Iktp`l a`l tpda}kldq`}o eoirlì`cqd}` r}kcekrdl`#Akiouq}`}ìo, ekob, eg` Q 45,=1 drrd}qk`c` d qdl` e`}egko#

R`} l‛drrd}q`c`cxd ak O% d 'GI O+ dmmkdio

ZI O%G Y 9 ZI OG Y

ZO%GOY 9 ZO%IOY

Dllo}d ZOI I OY 9 ZH GOY#U` egkdikdio ^ kl rpcqo iàko ak OI dmmkdio eg` k q}kdcnolk

• I ^ OI :

0Tp uqo qò}ìd b eo}olld}ko a`ll‛`cpcekdqo 46#1 kc Z>Y, dv cao akiouq}dqo eg`D41,=5, D45,=1, O uoco k rpcqk ak Iktp`l a`k tpda}kldq`}k eoirlì`cqd}k#



=#5 Q}dr`xko Kuoue`l` 'Ekelkeo+ =0

• GOH :

• D45,=1OQ 45,=1

uoco roukqkvdi`cq` ukiklk6, r`}eg± }`qqdcnolk ` dv`cqk ZOI I OY 9 ZHGOY 9ZQ 45,=1D45,=1OY# Kc rd}qkeold}`

ZOQ 45,=1I Y 9 ZOQ 45,=1D45,=1Y 9 ZOH G Y 9

9 ZOOI F Y 94

=ZZOOI I YY

Tpùqo akiouq}d eg` dl e`}egko eoc e`cq}o OI rduudcq` r`} I drrd}qk`c`kl rpcqo Q 45,=1 ` apctp` kl e`}egko kc tpùqkoc` b r}or}ko kl e`}egko ak Iktp`la`l tpda}dcnolo eoirlì`cqd}` r}kcekrdl`#

Kcolq}` }keo}adcao ld ukiklkqpakc` eg` idcad D kc o'D+ uk coqd eg` kle`}egko 'OD45,=1D41,=5+, eg` gd e`cq}o F ` rduud r`} I 'r}oroukxkoc` =#44+,vk`c` idcadqo c`l e`}egko 'OQ 45,=1Q 41,=5+# Tpùq‛plqkio b kl e`}egko ak Ik)tp`l I'T4=,51+, r`} tpdcqo akiouq}dqo uor}d# K ap` e`}egk eocuka`}dqk uocoo}qonocdlk r`}eg± pco kiidnkc` a`ll‛dlq}o uèocao ld ukiklkqpakc` o# Apc)tp` k }dnnk F I ` OI I uoco o}qonocdlk ` kl q}kdcnolo F OI I b }`qqdcnolo#Kcolq}` ZI OI F Y 9 ZGH OY# Dllo}d uk gd tpdcqo volpqo#

=#5 Q}dr`xko Kuoue`l` 'Ekelkeo+

Eoikcekdio eoc pc }kduupcqo uplld nòi`q}kd `lì`cqd}` a`l q}dr`xko kuo)ue`l`#

• Kl q}dr`xko kuoue`l` b kcue}kvkmkl` kc pc e`}egko, ekob b ekelkeo: ` vke`v`}udu` pc q}dr`xko b kcue}kvkmkl` kc pc e`}egko dllo}d b kuoue`l`#

• Kl q}dr`xko kuoue`l` gd pc duu` ak ukii`q}kd ` vke`v`}ud pc q}dr`xko eocpc‛duu` ak ukii`q}kd b kuoue`l`#

• U` pc tpda}dcnolo ekelkeo gd pc duu` ak ukii`q}kd up epk coc edaddlepc v`}qke`, kl tpda}dcnolo b pc q}dr`xko kuoue`l`#

L` akiouq}dxkock ak qdlk `cpcekdqk uoco iolqo uìrlkek# ]kro}qkdio tpk,eoi` ùìrko, ld akiouq}dxkoc` a`l r}kio `cpcekdqo#

6Ld ukiklkqpakc` q}d nlk plqkik ap` q}kdcnolk b eocu np`cxd a`lld r}oroukxkoc` =#7, kcepk dmmkdio akiouq}dqo l‛drrd}q`c`cxd a`k rpcqk O, Q 45,=1,H,D45,=1 dllo uqùuo e`}egko akakdi`q}o D45,=1Q 45,=1#




R}kid R}or}k`qbd Onck q}dr`xko kuoue`l` b ekelkeo#

Kcfdqqk u` D4D= ` D5D1 uoco k ldqk rd}dll lk, q}deekdio kl ek}eoe`}egko akD4D=D5# Dceg` D1 ∇ 'D4D=D5+ r }egb

ZD4D5D=Y 9 ZD4D1D=Y

Dmmkdio eou±Ĵ akiouq}dqo eg` kl ek}eoe`}egko ak D b kl ek}eoe`}egko ak tpdlukduka`k q}kdcnolk eoirlì`cqd}k#

R`} tpdcqo }knpd}ad lo uqpako a`k rpcqk coq`volk a`l q}dr`xko kuoue`l` kqò}ìk eg` r}`caoco kc eocuka`}dxkoc` ld }`qqd OH r`}aoco ak uknckffledqo,r`}egb O ≤ H # Kcfdqqk kl q}dr`xko a`k ek}eoe`cq}k kc tpùqo eduo uk eoce`cq}d

kc pc uolo rpcqo, kcq`}u`xkoc` a`lld iàkdcd r}kcekrdl` eoc nlk duuk a`k ap`ldqk coc rd}dll`lk# Apctp` qpqqk k rpcqk coq`volk a`l q}dr`xko a`k ek}eoe`cq}k,kc rd}qkeold}` kl rpcqo coq`vol` H , eokcekaoco eoc kl rpcqo O#

Rokegb kl e`cq}o a`l ek}eoe`}egko 'r}oroukxkoc` =#45+ ak D eda` upll‛duuàk qpqqk k ldqk a`l q}dr`xko, akue`ca` kiiàkdqdi`cq` eg`

R}oroukxkoc` =#45 L‛ O)e`cq}o a`l q}dr`xko b kl e`cq}o a`l ek}eoe`}egko a`l q}dr`xko uqùuo#

Dmmkdio vkuqo eg` ld iàkdcd r}kcekrdl` b dceg` duu` ak ukii`q}kd r`} kl

q}dr`xko ` apctp` gd ak}`xkoc` r`}r`cakeold}` }kur`qqo dk ldqk rd}dll`lk ak D#G,N,O,H uoco qpqqk dllkc`dqk up qdl` duu`, kcolq}` O ≤ H , r`}egb kl q}dr`xkoa`k ek}eoe`cq}k b akv`cqdqo kl rpcqo O ` eokcekaoco eoc O qpqqk k upok rpcqkcoq`volk# Kcolq}` akiouq}kdio eg`

R}oroukxkoc` =#41 I nkde` uplld iàkdcd r}kcekrdl` a`l q}dr`xko kuoue`l`#

Kcfdqqk I b l‛kcq`}u`xkoc` eoipc` q}d l` ap` }`qq` ak Iktp`l 'k e`}egk a`n`c`}k+` kl e`}egko ak Iktp`l a`l tpda}kldq`}o T4=,51# L` }`qq` ak Iktp`l eokcekaocoq}d lo}o ` eoc ld }`qqd GO, ld iàkdcd r}kcekrdl`, r`}eg± Q 45,=1 ≤ Q 41,=5 ≤ O` k rpcqk D45,=1, D41,=5 nkdeekoco uplld iàkdcd r}kcekrdl`# Dllo}d kl 'q`}xo+

e`}egko ak Iktp`l, eg` udrrkdio rduud}` r`} O, ek abd kl rpcqo ak Iktp`l I eoi` plq`}ko}` kcq`}u`xkoc` 'akv`}uo ad O+ eoc ld iàkdcd r}kcekrdl`#

L‛kr`}mol` `tpkldq`}d ek}eoue}kqqd dl q}dr`xko kuoue`l` gd eoi` e`cq}o G #K upok duuk ak ukii`q}kd uoco ld iàkdcd r}kcekrdl` ` ld }`qqd da ùud r`}r`c)akeold}` rduudcq` r`} G # Dllo}d nlk dukcqoqk a`ll‛kr`}mol` uoco ld eorrkd ak}`qq` r`}r`cakeold}k r`} G eg` fo}idco dcnolk ak ς1 eoc ld iàkdcd r}kcekrdl`NG #



=#1 Rd}dll`lon}diid =>

=#1 Rd}dll`lon}diid

C`l eduo kc epk pcd uèocad eorrkd ak ldqk orrouqk ukd rd}dll`ld, kl q}dr`xkoakv`cqd pc rd}dll`lon}diid# Akiouq}kdio eg`

R}oroukxkoc` =#40 Pc q}dr xko b pc rd}dll`lon}diid u` ` uolo u` kl md}k)e`cq}o N b kcq`}u`xkoc` ak ap` ldqk orrouqk#

R`} kl rd}dll`lon}diid pc uolo rpcqo akdnocdl` b dl fflckqo# Kcolq}` b coqoeg` pcd ed}dqq }kxxdxkoc` a`l rd}dll lon}diid b ak dv`}` l` akdnocdlk egù‛kcq`}uèdco c`k rpcqk iàk# Tpùqk, apctp`, eokcekaoco q}d lo}o ` eoc klmd}ke`cq}o N#

Vke`v`}ud u` kc pc q}dr`xko pcd eorrkd ak ldqk orrouqk u‛kcq`}uèdco kcN, coc b }ùq}kqqkvo uprro}}` eg` ukdco k ldqk D4D5, D=D1, i`cq}` eoi` dluolkqo uprrockdio rd}dll`lk k ldqk D4D=, D5D1# K q}kdcnolk D4D4=N ` D5D51N}kuplqdco ukii`q}kek }kur`qqo da O, ekob kiidnkck l‛pco a`ll‛dlq}o uèocao pci`xxo)nk}o kcqo}co d N# Dllo}d dceg` k q}kdcnolk D4D1N ` D5D=N }kuplqdcoukii`q}kek }kur`qqo da O# Kc rd}qkeold}` tpùqo uknckffled eg` dceg` D4D1 D=D5 ` eg` apctp` kl q}dr`xko b pc rd}dll`lon}diid#

Eocqkcpkdio ld aùe}kxkoc` a`l rd}dll`lon}diid#

L‛G )e`cq}o Dceg` l‛G )e`cq}o eokceka` eoc N, kc tpdcqo tpùqo b rpcqoiàko ak pc ldqo ak qpqqk k q}kdcnolk eoirlì`cqd}k#

Ld iàkdcd r}kcekrdl L` iàkdc` r}kcekrdlk uoco ap , rduudco r`}G , ` gdcco l` ak}`xkock a`k ldqk rd}dll lk# Uoco akuqkcq` kc n`c`}dl` l` }`qq`r`}r`cakeold}k dll` eorrk` ak ldqk rd}dll`lk#

Kl rd}dll`lon}diid o'D+ Kl rd}dll lon}diid a`k ek}eoe`cq}k o'D+ gd kldqk upnlk duuk ak D4D=, D=D5, D5D1, D1D4, l` epk ak}`xkock uoco dceg` l` ak)}`xkock ak rd}dll`lkuio ak o'D+# Ld q}dufo}idxkoc` eg` idcad D kc o'D+b, eoi` c`l eduo n`c`}dl` a`l q}dr`xko, pcd ukiklkqpakc` roukqkvd ak e`cq}oN# L‛O)e`cq}o ak pc rd}dll`lon}diid b, apctp`, eokceka`cq` eoc N, G `l‛kcq`}u`xkoc` a`ll` akdnocdlk dl fflckqo#

Kl rd}dll`lon}diid c'D+ U` D b pc rd}dll`lon}diid, kl tpda}dcnoloc'D+ b pc rd}dll`lon}diid ukikl` da D# Kl e`cq}o a`lld ukiklkqpakc`, eoi`c`l eduo n`c`}dl` b G #

Ouu`}vkdio kcfflc` eg` nlk dukcqoqk a`ll‛kr`}mol` `tpkldq`}d ek}eoue}kqqd dlrd}dll`lon}diid uoco l` mku`qq}kek a`ll‛dcnolo ZD41ND51Y# Kcfdqqk nlk duuk akukii`q}kd rduudco r`} N ` gdcco l` ak}`xkock a`ll` mku`qq}kek ak ZND51O4Y'eg` uoco dceg` tp`ll` a`ll` mku`qq}kek ak ZND4=O1Y+#



Edrkqolo 5

Aùe}kxkoc` Dcdlkqked a`lQ}dr`xko

5#4 Eduo N`c`}dl` 'Tpda}dcnolo coc O}qonocdl`+

Kc tpùqo edrkqolo kcq}oap}}ìo pc ukuqìd ak duuk o}qonocdlk r}`f`}`cxkdl`r`} kl tpda}dcnolo eoirl`qo, iolqo pqkl` r`} lo uqpako a`l tpda}dcnolo n`)c`}dl`4 ` dceg` c`l eduo a`l q}dr`xko# L‛ùkuq`cxd ak pcd à pcd uold kr`}mol``tpkldq`}d r`} k v`}qkek ak pc tpda}dcnolo= b `cpcekdqo ` akiouq}dqo kc Z>Y#Tpk ek lkikq`}ìo d }kduupi`}` k }kuplqdqk pqklk r`} ld couq}d aùe}kxkoc` `r}oeà`}ìo d ed}dqq`}kxxd}` dcdlkqkedi`cq` kl q}dr`xko#

Kc n`c`}dl` tpdcao pc tpda}dcnolo r}ù`cqd pcd eorrkd ak ldqk orrouqko}qonocdlk, l‛kr }mol `tpkldq }d a`n`c }d kc pcd eorrkd ak }`qq` r`}r`c)akeold}k# Egkdi }ìo tpùqd elduu` ak tpda}dcnolk elduu` a`l tpda}dcnoloo}qonocdl` ` ld uqpak`}ìo dl rd}dn}dfo 5#=#1#

U` l‛kr`}mol` coc a`n`c`}d kc pcd eorrkd ak }`qq`, coc b }ùq}kqqkvo upr)ro}}` eg` ld upd `tpdxkoc` ukd w{94# Eocuka`}kdio l‛kr`}mol` `tpkldq }d r`}k v`}qkek ak pc n`c`}keo q}dr`xko D ` rockdioek c`l ukuqìd ak duuk o}qonocdlkkc epk qdl` kr`}mol` gd `tpdxkoc` w{ 9 4# Nlk duuk uoco, apctp`, nlk dukcqo)qk a`ll‛kr`}mol`# Ek oeepr`}ìo o}d, fflco dl rd}dn}dfo 5#=#1 a`l eduo kc epkl‛kr`}mol` `tpkldq`}d ukd coc a`n`c`}`#

Rokegb k v`}qkek a`l q}dr`xko edaoco upll‛kr }mol`, rouukdio kcaked}` klv`}qke` Dk eoc l` eoo}akcdq`58

Dk 9

wk,

4

wk

Fd}ìo puo a`lld u`np`cq` r}oroukxkoc` eocq cpqd kc Z>Y#

4]kidcakdio dl rd}dn}dfo 44 ak Z>Y#=L‛kr`}mol` b pcked rp}egb pc v`}qke` coc ukd l‛o}qoe`cq}o a`l q}kdcnolo fo}idqo adnlk

dlq}k q}`# Kc tpùqo eduo r`} k tpdqq}o v`}qkek rduudco pc fdueko ak kr`}molk `tpkldq`}`#5wk 9 2 r`}egb l‛kr`}mol` coc a`n`c`}d#

=;



52 Aùe}kxkoc` Dcdlkqked a`l Q}dr`xko

R}oroukxkoc` 5#4 Kl e`cq}o a`ll‛kr`}mol` `tpkldq`}d ek}eoue}kqqd d pc tpd)

a}dcnolo D b l‛ G )e`cq}o: u` kl tpda}dcnolo D coc b ekelkeo

1

k upok dukcqoqk gdcco l` ak}`xkock a`ll` mku`qq}kek a`ll‛dcnolo ZGHOY#

C`l eduo a`l q}dr`xko kcolq}` vdl` l‛ouu`}vdxkoc` =#> ` apctp` ld u`np`cq`8

R}oroukxkoc` 5#= L` mku`qq}kek a`ll‛dcnolo ZHOG Y uoco nlk duuk ak ukii`)q}kd a`ll‛kr`}mol` `tpkldq`}d ek}eoue}kqqd dl q}dr`xko#

N}dxk` dll‛ouu`}vdxkoc` =#> rouukdio, ekob, aùe}kv`}` nlk duuk eoo}akcdqkd rd}qk}` adll` mku`qq}kek ak ZH GOY# Mduq`}bd }poqd}` qdlk mku`qq}kek ak ς

10 r`}

oqq`c`}` nlk dukcqoqk a`ll‛kr`}mol`, ` apctp` nlk duuk ak }kf`}ki`cqo#

O}d kcq}oapekdio l` eoo}akcdq` a`k rpcqk coq`volk, c`l eduo ak tpda}dc)

nolo kc n`c`}dl`, eg` ek u`}vk}dcco c`lld ed}dqq }kxxdxkoc` a`l q}dr`xko 'ùpee uukvdi`cq` c`lld upd }keouq}pxkoc`+# Coqkdio upmkqo eg` c`l eduo aktpda}dcnolo coc o}qonocdl` }kuplqd υ1 9 2# Akiouq}kdio kcolq}` eg`

R}oroukxkoc` 5#5 Kc pc q}dr`xko coc o}qonocdl` υ1 3 2#

Kirockdio ld eocakxkoc` ak rd}dll`lkuio q}d ap` ldqk orrouqk DkD h, DgDj#Tpùqo edleolo ro}qd d8

wkw h 9 wgwj

Dllo}d υ1 9 'wkw h+= ` apctp` b uìr}` roukqkvo#

Coqkdio eg` dstcegb υ1 ukd roukqkvo onck eorrkd a`ll` tpdqq}o duekuuà`ll` eoo}akcdq a`k v`}qkek a`l q}dr`xko a`v` dv`}` lo uqùuo u`nco# Apctp` kv`}qkek a`l q}dr`xko edaoco d ap` d ap` up }dik akv`}uk orrp}` qpqqk ` tpdqq}oupllo uqùuo }dio a`ll‛kr`}mol`#

Uk edleoldco l` u`np`cqk eoo}akcdq`68

G 9

2, 2

N 9 υ41 , υ5

1υ1H 9

υ4= , υ5

=υ1

1]kidcakdio lo uqpako a`l eduo ekelkeo dl rd}dn}dfo 5#=#4 ` 5#=#0#0Coc kiro}qd kc tpdl` v`}uo#6Dmmkdio q}ovdqo tpùq` eoo}akcdq` eoc l‛dkpqo a`l r}on}diid Idqgìdqked c`l tpdl`

dmmkdio kcu`}kqo l` fo}ipl` a`ll` `lì`cqd}k eouq}pxkock eoc }knd ` eoirduuo# @uìrk8}`qqd r`} ap` rpcqk, ek}eoe`cq}o ak pc q}kdcnolo, }kùuo ak pc rpcqo }kur`qqo da pcd }`qqd,èe# Tpùq` fo}ipl` r`} N,H,O 'coc r`} I,C + uk q}ovdco r`} ld r}kid volqd kc Z7Y#



5#4 Eduo N`c`}dl` 'Tpda}dcnolo coc O}qonocdl`+ 54

O 9 υ4

4*υ1, υ54*υ1

I 9

'4*υ1+'υ5*υ4υ=υ1∝υ5υ1+

υ=4υ=1*υ=

5

, '4*υ1+Zυ=υ5∝υ4υ1'4∝υ1+Yυ=4υ=1*υ=

5

C 9

'4∝5υ1+υ4υ=1∝41υ1*4

, 'υ1∝5+υ5υ=1∝41υ1*4

Coqkdio eg` tpùq` eoo}akcdq` uoco fpcxkock a`k rolkcoik ukii`q}kek `l`)i`cqd}k

υ4 9 w4 * w= * w5 * w1

υ= 9 w4w= * w4w5 * w4w1 * w=w5 * w=w1 * w5w1

υ5 9 w4w=w5 * w4w=w1 * w=w5w1 * w4w5w1

υ1 9 w4w=w5w1

eg` b kc deeo}ao eoc ld a`fflckxkoc` ak rpcqo coq`vol`, kcvd}kdcq` r`} r`}ip)qdxkock a`k v`}qkek a`ll‛c)noco ak epk b rpcqo coq`vol`#

Ek r}orockdio o}d ak q}ovd}` pcd ed}dqq`}kxxdxkoc` a`l q}dr`xko c`lldelduu` a`k tpda}dcnolk, d rd}qk}` adk upok rpcqk coq`volk# Dmmkdio akiouq}dqoc`l r}kio edrkqolo 'r}oroukxkoc` =#1+> eg` pcd eocakxkoc` cèùud}kd dstcegbpc tpda}dcnolo, coc ekelkeo, dmmkd pcd eorrkd ak ldqk rd}dll`lk b eg` ZGOH Y 9ς= #

R}kid ak uqpakd}` dcdlkqkedi`cq` qdl` eocakxkoc` uqpakdio u`rd}dqdi`c)q` k eduk ak eokceka`cxd q}d k rpcqk O ` G ` q}d O ` H #

R}oroukxkoc` 5#1 Ukd D pc tpda}dcnolo coc o}qonocdl`# U` l‛ O)e`cq}oeokceka` eoc l‛ G )e`cq}o, dllo}d D b pc rd}dll`lon}diid#

Kcfdqqk G ≤ O u` ` uolo u` υ4 9 υ5# ]kuolv`cao kl u`np`cq` ukuqìd8

υ4 9 w4 * w= * w5 * w1 9 2

υ5 9 w4w=w5 * w4w=w1 * w4w5w1 * w=w5w1 9 2

oqq`ckdio υ4 9 2'w= * w1+'w= * w5+'w5 * w1+ 9 2

` tpkcak vdl` pcd a`lld q}` u`np`cqk eocakxkock8

>Ek rockdio c`l eduo ak coc)eokceka`cxd q}d O ` H 'tpda}dcnolo ekelkeo+ ` q}d O ` G 'rd}dll`lon}diid+#



5= Aùe}kxkoc` Dcdlkqked a`l Q}dr`xko

w4 9 ∝w5

w= 9 ∝w1:

w4 9 ∝w1

w= 9 ∝w5:

w4 9 ∝w=

w5 9 ∝w1

L` eocakxkock ukii`q}keg` wk 9 ∝w h, wg 9 ∝wj kirlkedco kl rd}dll`lk)uio a`k ldqk DkD h DgDj ` DkDg D hDj, eg` uoco eorrk` ak ldqk orro)

uqk# Apctp` onck uolpxkoc` a`l ukuqìd kirlked eg` ukdio kc r} u`cxd a`lrd}dll`lon}diid#R}kid ak eocuka`}d}` ld eokceka`cxd ak O ` H , akiouq}kdio eg`

R}oroukxkoc` 5#0 C`l q}dr`xko coc o}qonocdl` kl rolkcoiko ukii`q}keo `l`)i`cqd}` υ4 uk dccplld u` ` uolo u` uk dccplld dceg` kl rolkcoiko υ5#

U` υ4 9 2 ` kirockdio kl rd}dll`lkuio ak ap` ldqk orrouqk DkD h DgDj

oqq`ckdio kl u`np`cq` ukuqìdυ4 9 2'w h * wj+'w h * wg+ 9 2

eg` gd eoi` uolpxkock w h 9 ∝wjwk 9 ∝wg

orrp}`

w h 9 ∝wgwk 9 ∝wj

Apctp` υ5 9 2 kc `cq}dimk k eduk#Vke`v`}ud u` υ5 9 2 dllo}d kl rd}dll`lkuio ak ap` ldqk orrouqk DkD h DgDj

ro}qd dl u`np`cq` ukuqìd υ5 9 2wkw h 9 wgwj

` uvklprrdcao oqq`ckdioυ5 9 2wkw h'w4 * w= * w5 * w1+ 9 2



5#4 Eduo N`c`}dl` 'Tpda}dcnolo coc O}qonocdl`+ 55

Rouukdio eocelpa`}` eg` dllo}d υ4 9 2, r`}egb kl q}dr`xko b o}qonocdl`#

O}d uprrockdio υ5 9 2 ` υ4 9 2 ` uqpakdio kl eduo ak eokceka`cxd q}dO)e`cq}o ` rpcqo H ak pc tpda}dcnolo n`c`}dl`#

R}oroukxkoc` 5#6 L‛ O)e`cq}o ` kl rpcqo H ak pc tpda}dcnolo eokcekaocoq}d lo}o, id coc eoc l‛ G )e`cq}o, u` ` uolo u` υ1 9 4

Kcfdqqk, rokegb υ4 9 2 9 υ5, }kuplqd

4 * υ1 9 =4 * υ1 9 =υ1

⇒⇝ υ1 9 4

Uqpak`}ìo dcdlkqkedi`cq` kl eduo ekelkeo coc o}qonocdl` 'υ1 9 4+ c`l rd)}dn}dfo 5#=#4# O}d uprrockdio O 9 G ` O 9 H à ùr}kikdio ld eocakxkocÒH ⊠ OG , eoi` dccplldi`cqo a`l u`np`cq` r}oaoqqo uedld}`8

'O ∝ G + • 'H ∝ O+ 9 2 '5#4+

υ4

4 * υ1

υ4

=∝

υ4

4 * υ1

*

υ5

4 * υ1

υ5

=υ1∝

υ5

4 * υ1

9 2

υ=4υ1'υ1 ∝ 4+

'4 * υ1+=υ1*

υ=5'4 ∝ υ1+

'4 * υ1+=υ19 2

'υ=4υ1 ∝ υ=

5+'υ1 ∝ 4+

'4 * υ1+=υ19 2

Eoi` dcqkekrdqo, uqpak`}ìo c`l 5#=#4 kl eduo ak υ1 9 4# Dllo}d }kidc`kc n`c`}dl` ld eocakxkoc`

'υ=4υ1 ∝ υ=

5+ 9 2

` rokegb uprrockdio υ4 9 2 '` tpkcak dceg` υ5 9 2, r`} ld r}oroukxkoc` 5#0+oqq`ckdio, r`} kl eduo n`c`}dl`, eg` u` pc tpda}dcnolo b pc q}dr`xko dllo}d8

υ1 9υ=5

υ=4

'5#=+

Eoi` uk b vkuqo, ùelpuk k eduk rd}qkeold}k uor}d ekqdqk ld 5#= b l‛`tpkvdl`cq`dcdlkqkeo a`lld eocakxkoc` ZGOH Y 9 ς

= #

Akiouq}kdio eg`




R}oroukxkoc` 5#> Ld eocakxkoc` 5#= b upstek`cq` r`}egb pc tpda}dcnolo

coc o}qonocdl`

7

dmmkd pcd eorrkd ak ldqk orrouqk rd}dll`lk#

Uvklprrkdio

υ1υ=4 9 υ=

5

υ1υ=4 9 w=

4w==w=

5 * w=4w=

=w=1 * w=

4w=5w=

1 * w==w=

5w=1 * =υ1υ=

Dllo}d, rokegb υ1 9 2, kl rolkcoiko w=4w=

=w=5 * w=

4w==w=

1 * w=4w=

5w=1 * w=

=w=5w=

1

a`v‛ùu`}` akvkukmkl` r`} υ1# Tpùqo kirlked

wkw h 9 wgwj

r`} pcd eorrkd ak kcakek kh# L‛`tpdxkoc` a`lld }`qqd r`} k rpcqk Dk, D h b

w * w4w={ ∝w4 * w=

w4w=9 2

Dllo}d ld eocakxkoc` wkw h 9 wgwj kirlked eg` l` }`qq` }kur`qqkvdi`cq` r`}DkD h ` DgDj uoco rd}dll`l`, eg` b tpdcqo volpqo#

Kc a`fflckqkvd ld eocakxkoc` 5#4 b cèùud}kd ` upstek`cq`; r`} ed}dqq`}kxxd}`k q}dr`xk c`lld elduu` a`k tpda}dcnolk#

Apctp` c`l eduo ak q}dr`xk coc o}qonocdlk l` eoo}akcdq` a`k rpcqk coq`volk

uor}d `l`cedqk akv`cqdco8

G 9

2, 2

N 9

υ4

1,

υ=4

1υ5

9

υ4

1υ5

υ5, υ4

H 9

υ4

=,

υ=4

=υ5

9

υ4

=υ5

υ5, υ4

O 9

υ54

υ=4 * υ=

5

,υ=4υ5

υ=4 * υ=

5

9

υ=4

υ=4 * υ=

5

υ4, υ5

7Aommkdio uprro}}` dceg` eg` kl tpda}dcnolo coc ukd ekelkeo ` eg` υ4 9 2# Uqpak`}ìold eocakxkoc` upstek`cq` kc r}ù`cxd ak kr`}mol` `tpkldq`}d a`n`c`}` dl rd}dn}dfo 5#=#1 ` kleduo ekelkeo dk rd}dn}dffl 5#=#4 ` 5#=#0#

;]keo}akdio cpovdi`cq` eg` tpk eocuka`}kdio q}dr`xk eoc pcd uold eorrkd ak ldqk rd}dl)l`lk 'apctp` O 9 G r`} ld r}oroukxkoc` 5#1+ ` coc ekelkek 'apctp` O 9 H r`} ld r}oroukxkoc`5#6+#



5#= Eduk Rd}qkeold}k 50

I 9υ=4*υ4υ=υ5∝υ=5

υ=4υ5, υ=υ54∝υ=4υ5*υ55

υ54υ5

9

9 4υ=4υ5

υ=4 * υ4υ=υ5 ∝ υ=

5,υ=υ

54∝υ=4υ5*υ

55

υ4

C 9

υ54'υ

=4∝5υ=5+

υ15∝41υ=

4υ=5*υ1

4

,υ=4υ5'υ

=5∝5υ=4+

υ15∝41υ=

4υ=5*υ1

4

9

9υ=4

υ15∝41υ=4υ=5*υ14

υ4'υ=

4 ∝ 5υ=5+, υ5'υ=

5 ∝ 5υ=4+

@l`cegkdio kcfflc` dlepck vdlo}k ak eo`stek`cqk dcnold}k j r`} }`qq` ak`tpdxkoc` { 9 jw * t eg` eocnkpcnoco rpcqk coq`volk42#

OH ∝ υ4υ5

GH υ4υ5

GO υ5υ4

GC υ5'υ=5∝5υ=4+

υ4'υ=4∝5υ=5+

qn'ZGHOY+ 9 =υ4υ5υ=5∝υ=

4

5#= Eduk Rd}qkeold}k

Dcdlkxx`}ìo o}d dlepck eduk kc epk coc uoco drrlkedmklk qpqq` o dlepc` a`llèouka`}dxkock r}èà`cqk upll` eoo}akcdq` a`k rpcqk coq`volk#

5#=#4 Q}dr`xko Ekelkeo

Dl rd}dn}dfo =#5 dmmkdio akiouq}dqo eg` kl q}dr`xko kuoue`l` b kcue}kvkmkl`kc pc e`}egko, r`} tpùqo l‛dmmkdio egkdidqo ekelkeo# Kc tpùqo rd}dn}dfoeocuka`}`} io q}dr`xk ekelkek coc o}qonocdlk#

R`} kl q}dr`xko ekelkeo nlk dukcqoqk a`ll‛kr`}mol`, eoi` c`l eduo n`c`}dl`,

uoco l` mku`qq}kek ak ZH GOY }poqdq` ak

ς

1 # Kc tpùqo eduo ZHGOY b cpllo àpctp` nlk dukcqoqk a`ll‛kr`}mol` uoco l` }`qq` r`} G eg` fo}idco dcnolk akς1 eoc ld }`qqd GO, eg` b o}d pc‛duu` ak ukii`q}kd a`ll‛kr`}mol`#

Dmmkdio vkuqo eg` ld eokceka`cxd ak O ` H , c`l eduo kc epk coc eokcekadcodceg` eoc G , kirlked υ1 9 4 'r}oroukxkoc` 5#6+#

Apctp` l` eoo}akcdq` a`k rpcqk coq`volk G,N,H,O,C ak pc tpda}dcnoloekelkeo, ed}dqq`}kxxdqo ad υ1 9 4 uk ue}kvoco8

42L` mku`qq}kek a`ll‛dcnolo ZGHOY, eocuka`}dqo r`} plqkio, gdcco ak} xkock 2 ` ∓}kur`qqkvdi`cq`#




G 9 2, 2N 9

υ41 , υ51

9 4

1

υ4, υ5

H 9 O 9 Q 45,=1 9 Q 41,=5 9

υ4= , υ5=

9 4

=

υ4, υ5

C 9

υ46 , υ56

9 4

6

υ4, υ5

Ld ed}dqq`}kxxdxkoc` n`c`}dl` a`l q}dr`xko b dceo}d vdlkad ` uk ue}kv`8

4 9υ=5

υ=4

Dllo}d l` eoo}akcdq` a`k rpcqk coq`volk c`l eduo a`l q}dr`xko kuoue`l` eg` cocukd o}qonocdl` ` coc ukd pc rd}dll`lon}diid, uk ue}kvoco8

G 9

2, 2

N 9

υ41 , ´υ4

1

9 υ4

1

4, ´4

H 9 O 9

υ4= , ´υ4

=

9 υ4

=

4, ´4

I 9υ=υ4

, υ=υ5

9υ=υ4

, ´υ=υ4

9 υ=

υ4

4, ´4

C 9

υ46 , ´υ4

6

9 υ4

6

4, ´4

Tpùqo iouq}d eg` N,H,O,I,C drrd}q`cnoco da pco '{ 9 ´w+ a`)nlk duuk ak ukii`q}kd a`ll‛kr`}mol` `tpkldq`}d, eg` eokceka` eoc ld iàkdcdr}kcekrdl` a`l q}dr`xko#

5#=#= Rd}dll`lon}diid

U` ap` eorrk` ak ldqk orrouqk ak pc tpda}dcnolo uoco rd}dll`l`, ukdio kcr}ù`cxd a`l rd}dll`lon}diid# Eocqkcpkdio d uprro}}` eg` ld q`}xd eorrkdak ldqk orrouqk coc ukd o}qonocdl`# Nlk dukcqoqk a`ll‛kr`}mol` uoco l` mku`qq}keka`ll‛dcnolo ZD51ND41Y 'ouu`}vdxkoc` fflcdl` a`l rd}dn}dfo =#1+#

R`} ed}dqq`}kxxd}` kl rd}dll lon}diid iàkdcq` k rolkcoik ukii`q}kekeocuka`}kdio l` eoo}akcdq` a`k v`}qkek8

D4 9 'w4, 4w4

+

D= 9 'w=, 4w=

+



5#= Eduk Rd}qkeold}k 5>

D5 9 ' ∝ w4, ∝ 4w4

+

D1 9 ' ∝ w=, ∝ 4w=

+

Dllo}d ap` rolkcoik ukii`q}kek uk dccplldco# Dmmkdio8

υ4 9 2

υ5 9 2

` apctp` O ≤ N ≤ H ≤ G , eg` akiouq}d l‛kcv`}uo a`lld r}oroukxkoc` 5#1#Apctp` rouukdio ed}dqq`}kxxd}` kl rd}dll`lon}diid }kur`qqo dl tpda}dcnolon`c`}dl` coc o}qonocdl , iàkdcq` k rolkcoik ukii`q}kek `lì`cqd}k# Lded}dqq`}kxxdxkoc` b

υ4 9 υ5 9 2

5#=#5 ]`qqdcnolo

Rokeg± kl }`qqdcnolo b pc rd}dll`lon}diid kc epk l` ap` eorrk` ak ldqk rd}dll`lkuoco q}d lo}o r`}r`cakeold}k, ùuo b kcue}kvkmkl` c`lld ek}eocf`}`cxd eoc e`cq}oc`ll‛pcked kcq`}u`xkoc` q}d l` akdnocdlk dl fflckqo# Apctp` kl }`qqdcnolo b pcuoqqoeduo a`l eduo ekelkeo ` a`l rd}dll lon}diid# L` eoo}akcdq` a`k upokv`}qkek c`l eduo kc epk coc ukd pc }oimo 'o pc tpda}dqo+, uk ue}kvoco8

D4 9 'w4, 4w4

+

D= 9 'w=, 4w=

+

D5 9 ' ∝ w4, ∝ 4w4

+

D1 9 ' ∝ w=, ∝ 4w=

+

Vàkdio upmkqo eg`

υ4 9 υ5 9 2

` eg` ld eocakxkoc` D4D= ⊠ D=D5 ro}qd d 'w4w=+= 9 4 ` apctp`

υ1 9 4

Tpùq` ap` eocakxkock eouqkqpkueoco pcd ed}dqq`}kxxdxkoc` dcdlkqked a`l }`q)qdcnolo#

5#=#1 Q}dr`xko O}qonocdl`

Uqpakdio tpk pcd elduu` ak q}dr`xk r`} ld tpdl` kl }kf`}ki`cqo ed}qùkdco pudqofflco}d a`v‛ uu`}` edimkdqo# Pc q}dr`xko, eg` gd eoi` ldqk rd}dll`lk D4D= `




D5D1, eoc pcd eorrkd ak ldqk orrouqk r`}r`cakeold}k44 drrd}qk`c` dlld elduu`

a`k tpda}dcnolk o}qonocdlk#Kc pc q}dr xko o}qonocdl` l‛dcnolo kc D45,=1 orrp}` tp`llo kc D41,=5 bak ς

= , ` G eokceka` eoc tp`l rpcqo akdnocdl # Dllo}d l` mku`qq}kek ak ZHGOY,}poqdq` ak ς1 , ekob nlk dukcqoqk a`ll‛kr`}mol` `tpkldq`}d ek}eoue}kqqd, eocq`cnocok v`}qkek# L‛kr`}mol` b a`n`c`}`4=# Uk gd apctp`

D4 9

w4, 2

D= 9

2, {=

D5 9 w5, 2D1 9

2, {1

eoc w4, w5, {=, {1 coc cpllk#]kedleolkdio l` eoo}akcdq` a`k rpcqk coq`volk G,N,H,O r`} q}ovd}` ld

ed}dqq }kxxdxkoc` a`k q}dr`xk o}qonocdlk# Uqpak }ìo kl eduo w4 * w5 9 2dlld fflc` ak tpùqo rd}dn}dfo, c`lld r}oroukxkoc` 5#42 ` c`k rd}dn}dffl 5#=#6 `5#=#># Coqkdio eg` coc b }ùq}kqqkvo uprro}}` w4w5 9 2 ` {={1 9 2 kc tpdcqoùelpakdio ld eokceka`cxd ak v`}qkek eoc G # Akiouq}kdio ld u`np`cq`

R}oroukxkoc` 5#7 U` w4w5*{={1 9 2 kl tpda}dcnolo coc gd cùupcd eorrkd

ak ldqk rd}dll`lk#

Kcfdqqk r`}egb kl tpda}dcnolo dmmkd pcd eorrkd ak ldqk orrouqk rd}dll`lk a`)v‛ùu`}` {=w5 9 {1w4 orrp}` {=w4 9 {1w5# Kc `cq}dimk k eduk ld }kuolpxkocà`l ukuqìd eoc ld eocakxkoc` w4w5 9 ∝{={1 ro}qd dll‛duup}ao# R`} ùìrko

w4w5 9 {={1{=w5 9 {1w4

ro}qd d w4w5 9 {={1{== 9 ∝w=

4

eg` b duup}ao# Apctp` c`lld couq}d q}dqqdxkoc` uprro}}ìo w4w5* {={1 9 2#

44Pc q}dr`xko coc roq}bd idk dv`}` ap` eorrk` ak ldqk orrouqk r`}r`cakeold}k 'd i`coak eokceka`cx` q}d k v`}qkek+, r`}egb dlq}ki`cqk coc dv}`mm` rkbp pcd eorrkd ak ldqk orrouqkrd}dll`lk# Dmmkdio ùelpuo k eduk ak v`}qkek eokceka`cqk o dllkc`dqk c`ll‛kcq}oapxkoc`#

4=Akuqkcnpkdio ap` eduk

• U` G ≤ D41,=5 dmmkdio k v`}qkek eocuèpqkvk D4, D1 up pc duu` ` D=, D5 upll‛dlq}o#

• U` G ≤ D45,=1 dmmkdio k v`}qkek orrouqk D4, D5 up pc duu` ` D=, D1 upll‛dlq}o#

Edimkdcao coik dk v`}qkek coc b }ùq}kqqkvo }keocap}uk dl uèocao eduo#



5#= Eduk Rd}qkeold}k 5;

]kro}qkdio l` eoo}akcdq` edleoldq` r`} k rpcqk G,N,H,O8

G 9

2, 2

N 9

w4*w5

1 , {=*{11

H 9

w4*w5

= , {=*{1=

O 9

'w4*w5+{={1w4w5*{={1

, '{=*{1+w4w5w4w5*{={1

Tpùq` ue}kqqp}` upnn`}kueoco pcd mdu` ak rolkcoik, eg` uvolnoco kl }polo

a`k rolkcoik υk, iàkdcq` ld tpdl` ùr}ki`}` l` eoo}akcdq` a`k rpcqk coq`volk#W 4 9 w4 * w5

W = 9 w4w5

[ 4 9 {= * {1[ = 9 {={1

Dllo}d l` eoo}akcdq` uk ue}kvoco8

G 9

2, 2

N 9 W4

1 , [ 41 H 9

W4

= , [ 4=

O 9

W4[ =W=*[ =

, [ 4W=

W=*[ =

Vonlkdio q}ovd}` ld eocakxkoc` cèùud}kd ` upstek`cq` dstcegb pc tpd)a}dcnolo o}qonocdl` ukd pc q}dr`xko o}qonocdl`#

Uqpakdio, eoi` c`l eduo n`c`}dl`, r}kid ld eokceka`cxd ak O eoc H ` roktp`lld ak O eoc G , c`l eduo a`l tpda}dcnolo o}qonocdl`#

R}oroukxkoc` 5#; L‛ O)e`cq}o ak pc tpda}dcnolo o}qonocdl` eokceka` eoc kl rpcqo H u` ` uolo u` W = 9 [ = 9 4#

Kcfdqqk l` eocakxkockW 4 9 W 4[ =W = * [ = 9 =

⇒⇝ W = 9 [ = 9 4

R}oroukxkoc` 5#42 L‛ O)e`cq}o ak pc tpda}dcnolo dv`cq` pcd eorrkd ak ldqk orrouqk r`}r`cakeold}k eokceka` eoc kl upo G )e`cq}o u` ` uolo u` W 4 9 [ 4 9 2#




Kcfdqqk a`v‛ùu`}`

W 4[ = 9 2[ 4W = 9 2

Rokegb dmmkdio ùelpuk k eduk W = 9 2 ` [ = 9 2, kl ukuqìd gd eoi` pckeduolpxkoc`

W 4 9 2[ 4 9 2

Akiouq}kdio kcolq}` ld u`np`cq`#

R}oroukxkoc` 5#44 R`} pc q}dr`xko o}qonocdl` W 4 9 2 u` ` uolo u` [ 4 9 2#

Kcfdqqk u` W 4 9 2 ld eocakxkoc` dstcegb D4D= D5D1, eg b {=w5 9 {1w4,akv`cqd {=w5 9 ∝{1w5# Ekob {= 9 ∝{1, eg b [ 4 9 2# Dcdlondi`cq` r`}

dv`}` D=D5 D4D1 ld eocakxkoc` {=w4 9 {1w5 akv`cqd {= 9 ∝{1#Vke`v`}ud u` [ 4 9 2 kl }dnkocdi`cqo b dcdlono#

Uqpakdio, o}d, dcdlkqkedi`cq` l‛o}qonocdlkqbd ak GO ` OH c`l eduo kc epkO 9 H , O 9 G ` W 4 9 2 'apctp` dceg` [ 4 9 2+8

'O ∝ G + • 'H ∝ O+ 9 2 '5#5+

W 4[ =W = * [ =

W 4=

∝W 4[ =

W = * [ =

*

W =[ 4W = * [ =

[ 4=

∝W =[ 4

W = * [ =

9 2

'W =4[ = ∝ W =[ =4 +'W = ∝ [ =+='W = * [ =+=

9 2

Eoi` và}ìo kc 5#=#0 kl eduo W = ∝ [ = 9 2 b tp`llo a`l tpda}dcnoloekelkeo# Uprrockdio apctp` W = 9 [ =# Pcd eocakxkoc` cèùud}kd dstcegbpc tpda}dcnolo o}qonocdl` coc ekelkeo ukd pc q}dr`xko b8

[ =W =

9[ =4W =4

'5#1+

Akiouq}kdio eg`

R}oroukxkoc` 5#4= Ld eocakxkoc` 5#1 b upstek`cq` r`}egb pc tpda}dcnolo

o}qonocdl`45 dmmkd pcd eorrkd ak ldqk orrouqk rd}dll`lk#

Uvklprrkdio 5#1[ =W =4 9 W =[ =4

w=4{={1 * w=

5{={1 * =w4w5{={1 9 w4w5{== * w4w5{=1 * =w4w5{={1

w4{='w4{1 ∝ w5{=+ ∝ w5{1'w4{1 ∝ w5{=+ 9 2

45Aommkdio uprro}}` dceg` eg` kl tpda}dcnolo coc ukd ekelkeo ` eg` W= 9 2 9 W4#Uqpak`}ìo kl eduo ekelkeo dl rd}dn}dfo 5#=#0 ` W4 9 2 dk rd}dn}dffl 5#=#6 ` 5#=#>#



5#= Eduk Rd}qkeold}k 14

'w4{= ∝ w5{1+'w4{1 ∝ w5{=+ 9 2

L‛`tpdxkoc` a`lld }`qqd r`} DkD h 'k ∇ z4, 5y h ∇ z=, 1y+ b8

wk{ * { hw ∝ wk{ h 9 2

L` eocakxkock 'w4{1 ∝ w5{=+ ` 'w4{= ∝ w5{1+ kirlkedco }kur`qqkvdi`cq` klrd}dll`lkuio a`k ldqk D4D5, D=D1 ` D4D5, D=D1, eg` b tpdcqo vol`vdio ak)iouq}d}`#

Kc n`c`}dl`, l` eoo}akcdq` a`k rpcqk coq`volk ak pc q}dr`xko o}qonocdl` ukue}kvoco8

G 9

2, 2

N 9

W4

1 , [ 41

H 9

W4

= , [ 4=

O 9

W4[

=4

W=4*[ =

4

,W=4[ 4

W=4*[ =

4

I 9 W=

W4, [ =[ 4

5#=#0 Q}dr`xko O}qonocdl` ` Ekelkeo

Tpùqd elduu` kcelpa` qpqqk k q}dr`xk kuoue`lk eoc pcd eorrkd ak ldqk orrouqkr`}r`cakeold}k# @uu cao o}qonocdlk qdlk q}dr`xk gdcco k v }qkek upnlk duukeoo}akcdqk# L‛kr`}mol` `tpkldq`}d b a`n`c`}` ` apctp` daoqqkdio k rolkcoikkcq}oaoqqk c`l rd}dn}dfo r}èà`cq`# Tpùqd fdiknlkd b l‛kcq`}u`xkoc` q}d ldfdiknlkd a`k q}dr`xk o}qonocdlk ` tp`lld a`k tpda}dcnolk ekelkek# Apctp`, r`}ld r}oroukxkoc` 5#;, W = 9 [ = 9 4 'eg` kirlked W = ∝ [ = 9 2+# L` eoo}akcdqà`k rpcqk coq`volk, uk ue}kvoco8

G 9

2, 2

N 9

W4

1 , W4

1

9 W4

1

4, 4

H 9 O 9 Q 45,=1 9 Q 41,=5 9

W4

= , W4

=

9 W4

=

4, 4

I 9 D41,=5 9

4W4

, 4W4

9 4

W4

4, 4



1= Aùe}kxkoc` Dcdlkqked a`l Q}dr`xko

5#=#6 ]oimo

Kl }oimo b pc eduo rd}qkeold}` ak rd}dll`lon}diid eg` b dceg` pc q}dr`xkoo}qonocdl # K rpcqk N,O,H,G eokcekaoco ` apctp` dmmkdio l` plq }ko}keocakxkock W 4 9 [ 4 9 2#

5#=#> Tpda}dqo

Kl q}dr`xko ekelkeo ` o}qonocdl` b egkd}di`cq` kl eduo kc epk ap` eorrk` akldqk orrouqk uoco rd}dll lk ` ld q`}xd eorrkd r`}r`cakeold}k# U` uk v`}kffledcotpùq` q}` eocakxkock ukdio kc r}ù`cxd a`l tpda}dqo#

Dmmkdio l` u`np`cqk }`ldxkock q}d k rolkcoik a`lld mdu`8

W 4 9 [ 4 9 2

W = 9 [ =



Edrkqolo 1

]keouq}pxkoc` a`l Q}dr`xko

“]keouq}pk}`’ uknckffled }kudlk}` 'iàkdcq` eouq}pxkock nòi`q}keg`+ dk v`}qkeka`l q}dr`xko rd}q`cao ad pc kcukì` orro}qpco ak rpcqk coq`volk# Kc Z>Y ld}keouq}pxkoc` a`l tpda}dcnolo n`c`}dl` b uqpakdqd d rd}qk}` ad pcd q`}cd akrpcqk coq`volk a`l q}kdcnolo akdnocdl` ` adl md}ke`cq}o N a`l tpda}dcnolo#Tpùqo b rouukmkl` r`}egb k rpcqk akdnocdlk, c`l eduo n`c`}dl`, uoco dl fflckqo#Kcvè`, kc r}ù`cxd a`l q}dr`xko, d edpud a`lld eorrkd ak ldqk rd}dll`lk, pcoak tpùqk rpcqk '` r}èkudi`cq` l‛kcq`}u`xkoc` a`ll` }`qq` rd}dll`l`+ b dll‛kcffl)ckqo# Ld }keouq}pxkoc` eou±Ĵ eoi` b uqdqd uqpakdqd kc Z>Y, apctp`, coc uk rpbodrrlked}` dl q}dr`xko#

Kc tpùqo edrkqolo r}orockdio pcd }keouq}pxkoc` akfl`}`cq`, eg` rd}qàd rpcqk coq`volk eg` }kidcnoco dl fflckqo c`l eduo a`l q}dr`xko# Ld couq}due`lqd eda` upk rpcqk N,G,O,I # K r}kik q}` eocu`cqoco, kccdcxkqpqqo, aked}dqq`}kxxd}` ld elduu` a`k q}dr`xk }kur`qqo dl tpda}dcnolo n`c`}dl`# Kcfdqqkdmmkdio akiouq}dqo k qò}ìk =#=, 5#> ` 5#4=, eg` ed}dqq`}kxxdco kl q}dr`xkokc ioao pckeo dqq}dv`}uo ld r`}r`cakeold}kqbd q}d OH ` OG # Dmmkdio, rok,c`k qò}ìk a`l rd}dn}dfo =#4, uqpakdqo dlepc` r}or}k`qbd a`l rpcqo ak Iktp`lI a`l q}dr`xko, eg` uoco upstek`cqk d nkpuqkffled}` k rduuk a`lld }keouq}pxkocèg` r}orockdio# Coqkdio eg` ld ue`lqd a`k rpcqk N,G,O a`q`}ikcd ekctpà`k u`qq` n}dak ak lkm`}qbd a`l q}dr`xko, r`}egb uk b vkuqo eg` r`} kl q}dr`xkoO a`v` eda`}` upl e`}egko ak akdi`q}o GH 'r}oroukxkock =#=, 5#> ` 5#4=+# Ldue`lqd a`l rpcqo I 4 vkceold nlk plqkik ap` n}dak ak lkm`}qbd ` apctp` l‛kcukìà`k tpdqq}o rpcqk N,G,O,I a`q`}ikcd kc ioao pckvoeo pc q}dr`xko#

C`l aùe}kv`}` ld }keouq}pxkoc`, và}ìo, kc rd}qkeold}`, eg` qpqqk k rduukuoco ù`npkmklk eoc }knd ` eoirduuo# Tpùqo uknckffled eg` kl r}oml id uk bdmmduudqo ak n}dao, }kur`qqo dl eduo n`c`}dl` eg` coc rpbo `vkqd}` `tpdxkockak n}dao q}`#

Upeeùukvdi`cq` uqpak`}ìo kl lpono ak O kc fpcxkoc` ak N, G ` ak I kcfpcxkoc` ak N,G,O# Eoc lpono ak pc rpcqo kcq`cakdio ld }`nkoc` a`l rkdco

4R`}eg± kl q}dr`xko dmmkd v`}qkek }`dlk b cèùud}ko eg` I ukd ue`lqo kc pcd }`nkoc` a`lrkdco eg` aùe}kv`}ìo kc tpùqo edrkqolo#

15



11 ]keouq}pxkoc` a`l Q}dr`xko

kc epk b rouukmkl` ue`nlk`}` kl rpcqo dstcegb ld }keouq}pxkoc` a`l q}dr`xko ro}qk

d tpdqq}o v`}qkek }`dlk#

1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I

O}d }kro}qkdio ld }keouq}pxkoc` eg` dmmkdio iùuo d rpcqo# Rd}qkdio adktpdqq}o rpcqk coq`volk N,O,G,I # Q`c`cao eocqo a`ll` lo}o r}or}k`qbd uqp)akdq` kc tpùqo edrkqolo ` c`l r}kio, qpqqk k rduuk a`lld u`np`cq` }keouq}pxkoc`}kuplqdco nkpuqkffledqk# Eocuka`}`}ìo, eoi` dl uolkqo, rd}dll lk ld eorrkd akldqk D4D=, D5D1: apctp` k rpcqk akdnocdlk dl fflckqo uoco D45,=1, D41,=5#

]keo}akdio eg` ek oeepr`}ìo ak pc q}dr`xko eg` coc ukd pc rd}dll`)lon}diid 'tpkcak N 9 G , r}oroukxkoc` =#40+ ` eg` coc b ekelkeo 'tpkcak

I /∇ GN+#

]keouq}pxkoc`

4# ]k`qqkdio G up N ` oqq`ckdio H 'rd}df}dfo 4#0+:

=# eouq}pkdio ld }`qqd r`} N ` G :

5# eouq}pkdio kl e`}egko= 'I ,G,O+, ak e`cq}o E 'r}oroukxkoc` =#4=+:

1# q}ovkdio O%, u` I /∇ GO, eoi` plq`}ko}` kcq`}u`xkoc` ak Z=Y eoc Z5Y,akv`}ud ad G 'r}oroukxkoc` =#4=+:

0# eouq}pkdio ld }`qqd r`} Z1Y ` I 5:

6# eouq}pkdio l‛duu` a`l u`ni`cqo OI :

># eouq}pkdio ld }`qqd r`}r`cakeold}` d Z0Y r`} I 'r}oroukxkoc` =#4=+:

7# q}ovkdio F eoi` kcq`}u`xkoc` ak Z6Y ` Z>Y 'r}oroukxkock =#44 ` =#4=+:

;# eouq}pkdio kl e`}egko ak e`cq}o F rduudcq` r`} I :

42# q}ovkdio D45,=1 ` D41,=5 eoi` kcq`}u`xkock1 q}d ld }`qqd Z=Y ` kl e`}egkoZ;Y 'r}oroukxkoc` =#44+:

44# eouq}pkdio ld }`qqd r`} N rd}dll`ld da OH :

4=# eouq}pkdio ld }`qqd r`} N r`}r`cakeold}` da OH 'r}oroukxkoc` =#7+:

=u` I ∇ GO kl e`}egko b ld }`qqd GO ` O%, ak epk dl rduuo 1, b dll‛kcfflckqo# OI uk q}ovdeoi` kcq`}u`xkoc` a`ll‛duu` ak OI eoc ld }`qqd r`} I rd}dll`ld d NG # Kl q}dr`xko dv}bd kldqk rd}dll`lk rduudcqk r`} O ` r`} G }kur`qqkvdi`cq`#

5C`l eduo kc epk I ∇ GN qdl` rduuo akv`cqd8 “eouq}pkdio ld }`qqd r`} I rd}dll`ld dNG ’#

1Tpùq` kcq`}u`xkock uoco }`dlk r`} pc‛orro}qpcd ue`lqd ak I # Qdlk eocakxkock ud}dccouqpakdq` dl 1#4#=#



1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I 10

45# eouq}pkdio l‛duu ak D45,=1H :

41# q}ovkdio l‛kcq`}u`xkoc` a`lld }`qqd Z4=Y eoc ld }`qqd Z45Y, eg` b kl e`cq}oa`l e`}egko 'HD45D=1+ 'r}oroukxkoc` =#7+:

40# eouq}pkdio kl e`}egko ak e`cq}o Z41Y rduudcq` r`} H 'r}oroukxkoc` =#7+:

46# q}ovkdio l` kcq`}u xkock0 a`l e`}egko Z40Y eoc ld }`qqd Z44Y eg` uoco D45

` D=1 'r}oroukxkoc` =#7+:

4># eouq}pkdio l` } qq` r`} D45,=1 ` k rpcqk Z46Y, eg` uoco k ap` ldqk orrouqkcoc rd}dll`lk D4D5, D=D1 a`l q}dr`xko:

47# }kr`qkdio k rduuk 45)4> r`} D41,=5 ` q}ovkdio k ldqk D4D1, D=D5 a`l

q}dr`xko:

4;# q}ovkdio l` kcq`}u`xkock a`k ldqk Z4>Y ` Z47Y, eg` uoco k tpdqq}o v`}qkeka`l q}dr`xko#

1#4#4 Lpono ak O

Eoi` dmmkdio akiouq}dqo c`ll` r}oroukxkock =#=, 5#> ` 5#4=, rouukdio d}mk)q}d}kdi`cq` ue`nlk`}` k rpcqk G ` N, id fdqqo tpùqo kl rpcqo O b vkceoldqo dnkde`}` upl e`}egko ak e`cq}o N ` }dnnko NG # Udrrkdio, kcfdqqk, eg` u` vonlkd)io }keouq}pk} kl q}dr`xko, kl q}kdcnolo ONG a`v‛ùu`}` kuoue`l` 'r}oroukxkoc`=#6+, kl eg` b `tpkvdl`cq` dl fdqqo eg` ZGOH Y a`mmd ùu`}` }`qqo 'r}oroukxkoc`=#1 ` =#0+# Vke`v`}ud u` ZGOH Y b }`qqo }keouq}pk}ìo pc tpda}dcnolo eoc pcdeorrkd ak ldqk orrouqk rd}dll`lk 'r}oroukxkock 5#> ` 5#4=+# Uprro}}ìo kcolq}Ò 9 G 'eduo a`l rd}dll`lon}diid+ ` O 9 H 'eduo a`l q}dr`xko ekelkeo ld epk}keouq}pxkoc` b uqpakdqd dl 1#=+#

Kl lpono ak O, b apctp` kl e`}egko ak e`cq}o N rduudcq` r`} G , r}kvdqoa`k rpcqk G ` H #

1#4#= Lpono ak I

R`} ld aùe}kxkoc` a`l lpono a`l rkdco kc epk ue`nlk`}` kl rpcqo ak Iktp`lI , dstcegb kl q}dr`xko eg` vonlkdio eouq}pk}` dmmkd v`}qkek }`dlk ek u`}vk}ìo

a`lld nòi`q}kd dcdlkqked# Pu`}ìo l` eoo}akcdq` a`l eduo n`c`}dl`6, aùe}kqq`dl 5#4, kc tpdcqo kl couq}o ueoro b tp llo ak aùe}kv }` kl lpono ak I kcioao ad roq`} eouq}pk}` nòi`q}kedi`cq` k v`}qkek ak D, rouukmkli`cq` eoc}knd ` eoirduuo# Apctp` lo uq}pi`cqo dcdlkqkeo ek fo}ckue` uolqdcqo pc

0Qdlk kcq`}u`xkock uoco }`dlk r`} pc‛orro}qpcd ue`lqd ak I ` akr`caoco adlld uèocadeocakxkoc` uqpakdqd c`l 1#4#=#

6Uprro}}ìo uìr}` υ4 9 2,υ5 9 2 ` υ1 9 4, kc tpdcqo, eoi` akiouq}dqo c`l edrk)qolo 5, l` r}ki` ap` eocakxkock uoco uìr}` v`}kffledq` u` kl q}dr`xko coc a`n`c`}d kc pcrd}dll`lon}diid, i`cq}` ld }keouq}pxkoc` a`l eduo ekelkeo v`}}bd uqpakdqo dl 1#=#




upnn`}ki`cqo8 kl lpono ak I }kuplq`}bd kcakr`ca`cq` ad tpdlpctp` ukuqìd ak

eoo}akcdq` ` ud}bd eouq}pkmkl` nòi`q}kedi`cq` d rd}qk}` adk rpcqk N,G,O#Và}ìo kl lpono ak I cdue`}` kc ioao dcdlkqkeo eoi` “kcq`}u`xkoc`’a`ll` eocakxkock dstcegb k akue}kikcdcqk a`ll` `tpdxkock ak uèocao n}dao>

ukdco roukqkvk o cpllk# Id q}ov }ìo r`} onck eocakxkoc` l‛kcq }r} qdxkoc`nòi`q}ked, kcakvkapdcao a`ll` }`nkock a`l rkdco#

Kcfflc` ad}ìo dceg` pcd akiouq}dxkoc` ukcq`qked a`lld r}kid eocakxkocèpk a`v` uoaakufd}` kl lpono ak I #

D tpùqo rpcqo lduekdio l` eoo}akcdq` ak I kcakedq` eoc

wI , {I

`

l` eocuka`}kdio vd}kdmklk# Kc tpùqo ioao l` eocakxkock dcdlkqkeg` eg` a`)

ue}kvoco kl lpono }kuplq`}dcco ak fdekl` kcq`}r}`qdxkoc`# ]kduupikdio k rduukdcdlkqkek a`lld }keouq}pxkoc` edleoldcao, kc fpcxkoc` a`k rolkcoik ukii`q}kek`lì`cqd}k υ4, υ=, υ5, υ1 ` a`ll` vd}kdmklk wI , {I , l` eoo}akcdq` a`k rpcqk egèoirdkoco c`lld upeeùukoc` ak rduuk `l`cedqk kc 1#4# Uoqqolkc`dio kl fdqqoeg` coc q}dqqkdio o}d kl eduo ekelkeo 'apctp` uprro}}ìo υ=

4 9 υ=5+ ` eg`

c`l eduo kc epk I ∇ GO, O% vd dll‛kcfflckqo# Kc qdl` eduo vk b pc plq`}ko}`

}`ldxkoc`7 q}d k rolkcoik ukii`q}kek `lì`cqd}k 'υ=5υ=4

9 υ1 9υ==1 + eg` ro}qd

dll‛dllkc`di`cqo a`k v`}qkek D4, D= eoc O ` D5D1 eoc G }kur`qqkvdi`cq`# E b kl ek}eoe`cq}o ak 'GI O+ ` gd eoo}akcdq`

E 9∝υ5w=

I * υ=4{I ∝ υ5{=I

='∝υ5wI * υ4{I + ,

∝υ4'υ4wI ∝ w=I ∝ {=I +

='∝υ5wI * υ4{I +

Edleolkdio l‛kcq`}u`xkoc` akv`}ud ad G q}d 'GI O+ ` GN 'rduuo 1+

O% 9

υ5'w=

I 'υ=5 ∝ υ=

4+ * {=I 'υ=5 ∝ υ=

4+ * wI υ54 ∝ {I υ

=4υ5+

'υ=4 * υ=

5+'υ5wI ∝ υ4{I +,

υ4'w=I '∝υ=

5 * υ=4+ * {=I '∝υ=

5 * υ=4+ ∝ wI υ

54 * {I υ

=4υ5+

'υ=4 * υ=

5+'∝υ5wI * υ4{I +

O}d q}ovkdio F eoi` kcq`}u`xkoc` q}d l‛duu` ak OI ` ld }`qqd r`} I r`})r`cakeold}` d IO% 'rduuo 7+8

F 9

υ54 * 'υ=

4 ∝ υ=5+wI ∝ =υ4υ5{I

=υ=4 ∝ =υ=

5

,υ=4υ5 ∝ =υ4υ5wI * 'υ=

5 ∝ υ=4+{I

∝=υ=4 * =υ=

5

L` duekuu` a`k rpcqk a‛kcq`}u`xkoc` a`l e`}egko ak e`cq}o F ` }dnnko F I eoc ld iàkdcd r}kcekrdl` GN uk oqq`cnoco ad pc‛`tpdxkoc` kl epk akue}kik)

>Qdlk `tpdxkock fo}ckueoco l` eoo}akcdq` a k rpcqk a‛kcq }u`xkoc` q}d } qq` èk}eocf`}`cx`#

7Dmmkdio oqq`cpqo ld }`ldxkoc` uvklprrdcao wI {O 9 {I wO#



1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I 1>

cdcq b8

41υ=

5'υ=4 ∝ υ=

5+=

=wI '∝=υ>

4 * =υ54υ1

5+ * w=I 'υ6

5 ∝ 6υ=4υ1

5 * ;υ14υ=

5+ *

*={I '=υ64υ5 ∝ =υ=

4υ05+ * *=wI {I '*5υ0

4υ5 ∝ 42υ54υ5

5 * 5υ4υ05+ *

*{=I ';υ=4υ1

5 ∝ 6υ14υ=

5 * υ64+

'1#4+

Uqpakdio kl eduo kc epk 1#4 b cpllo# Kc tpùqo eduo kl q}dr`xko gd pcuolo rpcqo akdnocdl` dl fflckqo, orrp}` k ap` rpcqk akdnocdlk eokceka`cqk# C`lr}kio eduo ukdio kc r}ù`cxd a`l rd}dll`lon}diid: c`l uèocao ap` v`}qkeka`l q}dr`xko eokcekaoco, eduo ùelpuo c`ll‛kcq}oapxkoc`# b@, apctp`, pc eduo“lkikq`’ eg` ek u`}vk}bd r`} oqq`c`}` ld r}kid eocakxkoc` a`l lpono ak I #

Kiroc`cao l‛dccplldi`cqo a`l akue}kikcdcq`, oqq`ckdio l‛`tpdxkoc` ak pcdeocked ak `tpdxkoc`

Dw= * M{= * Ew{ * Aw * @{ * F 9 2 '1#=+

, aov`

D 9 ;υ14υ=

5 ∝ 6υ=4υ1

5 * υ65 9 υ=

5'5υ=4 ∝ υ=

5+=

M 9 υ64 ∝ 6υ1

4υ=5 * ;υ=

4υ15 9 υ=

4'5υ=5 ∝ υ=

4+=

E 9 =υ4υ5'5υ14 ∝ 42υ=

4υ=5 * 5υ1

5+ 9 =υ4υ5'5υ=4 ∝ υ=

5+'υ=4 ∝ 5υ=

5+

A 9 1υ5

4

'υ1

5

∝ υ1

4

+

@ 9 1υ=4υ5'υ1

4 ∝ υ15+

F 9 2

Ld eocked 1#= b pcd rd}dmold, kc tpdcqo

A`q

D E

=E = M

9

9 A`q

υ=5'5υ=

4 ∝ υ=5+= υ4υ5'5υ=

4 ∝ υ=5+'υ=

4 ∝ 5υ=5+

υ4υ5'5υ=4 ∝ υ=

5+'υ=4 ∝ 5υ=

5+ υ=4'5υ=

5 ∝ υ=4+=

9

υ=4υ=

5Z'5υ14 ∝ 42υ=

4υ=5 * 5υ1

5+= ∝ '5υ14 ∝ 42υ=

4υ=5 * 5υ1

5+=Y 9 2

U` edleolkdio fpoeo R ` ak}`qq}ke` a a`lld rd}dmold 1#= oqq`ckdio

R 9

υ54

υ=4*υ=5

,υ=4υ5

υ=4*υ=5

9 O, eg` c`l eduo n`c`}dl` gd eoo}akcdq`

υ4

4*υ1, υ54*υ1

a 8 υ4'5υ=5 ∝ υ=

4+w * υ5'5υ=4 ∝ υ=

5+{ ∝ υ=4'υ=

4 * υ=5+ 9 2

a 8 { 9 ∝υ4'5υ=5∝υ

=4+

υ5'5υ=4∝υ=5+

w *υ=4'υ

=4*υ=5+

υ5'5υ=4∝υ=5+




U` uouqkqpkdio l` eoo}akcdq` ak H c`ll‛`tpdxkoc` a`lld ak}`qq}ke`, vàkd)

io eg` b uoaakufdqqd, apctp` ld }`qqd rduud r`} qdl` rpcqo# Akiouq}kdio,kcolq}`, dcdlkqkedi`cq`, eg` a rduud r`} OGN, kl rpcqo oqq`cpqo ad O r`}}kùukoc` uplld }`qqd GN# Aov}ìo, tpkcak r}ovd}` eg` ld ak}`qq}ke` b ld}`qqd OGNH # Kcfdqqk q}ovkdio8

OGN 9

υ54'5υ

=5∝υ=4+

'υ=4*υ=5+=

,υ=4υ5'5υ

=4∝υ=5+

'υ=4*υ=5+=

9

υ=4'υ=4*υ

=5+=

υ4'5υ=

5 ∝ υ=4+, υ5'5υ=

4 ∝ υ=5+

OGNH 8 { 9 ∝υ4'5υ=5∝υ

=4+

υ5'5υ=4∝υ=5+

w *υ=4'υ

=4*υ=5+

υ5'5υ=4∝υ=5+

eg` b tpdcqo volpqo#Ld rd}dmold 1#=, ak fpoeo O ` ak}`qq}ke` OGNH , akvka` kl rkdco kc ap`

}`nkock, tp`lld a`lld eocvùukqbd eocq`c`cq` O

GN

H , ` tp`lld a`lld eocedvkqbdeocq`c`cq` O# Egkdikdio δ ld r}kid }`nkoc` a`l rkdco ` δ ld uèocad# Ldeocakxkoc` a`l akue}kikcdcq` 1#4 idnnko}` o pnpdl` d x`}o uk rpbo ùr}ki`}`dceg` kc ioao u`np`cq`;8

|I O| ∝ |I a| ≠ 2

Dllo}d b `vka`cq` eg` I a`v` drrd}q`c`}` da δ# Dmmkdio kc tpùqo ioao ak)iouq}dqo eg` ld eocakxkkoc` cèùud}kd ` upstek`cq` dstcegb k rpcqk akdnocdlkukdco }`dlk b eg` I drrd}q`cnd dlld }`nkoc` δ#

R}oeàkdio eoc k rduuk a`lld }keouq}pxkoc` uprroc`cao I ∇ δ ` apc)tp` }`dlk l` kcq`}u`xkock D45,=1, D41,=1 a`l e`}egko ak e`cq}o F ` }dnnko F I

eoc ld iàkdcd r}kcekrdl # R}oeà`}` eoc l` eoo}akcdq` kc fpcxkoc` akwI , {I kirlkeg`}`mm` ld akr`ca`cxd ak D45,=1, D41,=1 adlld }dake` tpda}d)qd a`l akue}kikcdcq` 1#4# Apctp` egkdikdio wA, {A ld n`c`}ked kcq`}u`xkoc`D4 h,=g, h, g 9 z5, 1y ` q}ovkdioc` kl lpono# Pu`}ìo tpùqo r`} }kudlk}` dllduèocad eocakxkoc` cèùud}kd r`} I #

Dstcegb kl e`}egko 'O,H,D4 h,=g+ dmmkd kcq`}u`xkock }`dlk eoc ld }`qqd r`}N rd}dll`ld d OH , kl u`np`cq` a`q`}ikcdcq` a`v‛ùu`}` roukqkvo o cpllo8

4

46υ4υ15

υ04 * υ5

4υ=5 * 1wA'υ1

4 ∝ υ=4υ=

5+ * 7w=A'υ4υ=

5 ∝ υ54+*

∝7υ54υ5 ∝ 7wA{A'υ

=4υ5 * υ

55+ * 46υ4υ

=5{

=A

D tpùqo rpcqo ùr}kikdio {A kc fpcxkoc` ak wA '{A 9 υ4υ5

wA+ ` q}ovkdiokl u`np`cq` lpono ak wA#

υ04 * υ5

4υ=5 ∝ 1υ=

4wA'υ=4 * υ=

5+

υ4≠ 2

;Kcakeg`}ìo eoc |DM| ld akuqdcxd q}d k rpcqk D ` M ` eoc |Da| ld akuqdcxd q}d kl rpcqoD ` ld }`qqd a#



1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I 1;

Kl u`nco edimkd r`}

wA 9υ54'υ=

4 * υ=5+

1υ=4'υ=

4 * υ=5+

9υ4

19 wN

Uqpakdcao dceg` kl u`nco ak υ4 oqq`ckdio8

• υ4 < 2 ⇝ wA 3 wN

• υ4 3 2 ⇝ wA < wN

#

Dmmkdio akiouq}dqo eg` r`} dv`}` kcq`}u`xkock }`dlk, kl n`c`}keo rpcqoakdnocdl` D4 h,=g a`v` nkde`}` uplld uìk}`qqd NG # Coc b }ùq}kqqkvo, d tpùqorpcqo, eocuka`}d}` kl eduo “lkikq`’ kc epk k ap` rpcqk akdnocdlk eokcekadco#Kc tpduqo eduo kl e`}egko ak e`cq}o F rduudcq` r`} I b qdcn`cq da GN c`lrpcqo aorrko D45,=1 ≤ D41,=5, eg` egkdi`}ìo A#

A nkde` uplld uìk}`qqd NG u` ` uolo u` ld akuqdcxd F O b ikco}` o pnpdl`dlld akuqdcxd F N ` kl rpcqo F GN, r}ok`xkoc` ak F up eda` uplld uìk}`qqdNG , ak o}knkc` N'uk vàd ld akiouq}dxkoc` ukcq`qked dlld fflc` a`l rd}dn}dfo+#

Kiroc`cao ld eocakxkoc` |F O| ≡ |F N|, rokegb F b fpcxkoc` ak wI , {I ,oqq`ckdio ld uèocad eocakxkoc` eg` a`fflckue` kl lpono ak I # Q}ovkdio8

υ4'υ=4 * υ=

5+'υ54 ∝ υ4υ=

5 * 1υ=5wI ∝ 1υ4υ5{I +

46υ=5'υ=

4 ∝ υ=5+

≠ 2

eg` akv`cqd8

υ4'υ54 ∝ υ4υ=

5 * 1υ=5wI ∝ 1υ4υ5{I +

υ=4 ∝ υ=

5

≠ 2

Ouu`}vkdio eg`

υ54 ∝ υ4υ=

5 * 1υ=5w ∝ 1υ4υ5{ 9 2 '1#5+

b l‛`tpdxkoc` a`lld }`qqd r`} N r`}r`cakeold}` d OH # Ukd ν ld }`nkoc` a`lrkdco kc epk ld }`qqd 1#5 akvka` kl rkdco, eocq`c`cq` O# Lo uqpako a`l u`nco,

eoi` c`l eduo r}èà`cq`, ro}qd d8

• υ=4 3 υ=

5 ⇝ { ≡ υ5υ4

w *υ=4∝υ

=5

1υ5

• υ=4 < υ=

5 ⇝ { ≠ υ5υ4

w *υ=4∝υ

=5

1υ5

Apctp` ld uolpxkoc` b uìr} ld }`nkoc` ν , eocq`c`cq` O# Apctp` pcdeocakxkoc` cèùud}kd dstcegb ùkuqdco }`dlk k rpcqk iàk a`k ldqk coc rd}dll`lkI a`v` nkde`}` c`lld }`nkoc` ν #




Kcfflc`, r`} }kq}ovd}` l‛plqkid eocakxkoc`, akiouq}kdio eg` ld }`qqd 1#5 b

qdcn`cq` dlld rd}dmold 1#= ak fpoeo O ` ak}`qq}ke` H O

GN

c`l rpcqo Q akeoo}akcdq`8

Q 9

υ4'υ=

4 * 0υ=5+

46υ=5

,0υ=

4 * υ=5

46υ5

Kcfdqqk ld }kuolpxkoc` a`l ukuqìd q}d rd}dmold }`qqd 1#5 ` ld rd}dmold 1#=ro}qd dlld u`np`cq` `tpdxkoc` ak uèocao n}dao8

46υ=5w= ∝ =υ4'υ=

4 * 0υ=5+w *

υ=4'υ=

4 * 0υ=5+

46υ=5

9 2

eg b

'1υ5w ∝υ4'υ=

4 * 0υ=5+

1υ5

+= 9 2

Q}ovkdio l‛`tpdxkoc` a`lld }`qqd Q H 8

{ 9υ5'5υ=

4 ∝ υ=5+

υ4'5υ=5 ∝ υ=

4+*

υ15 ∝ υ1

4

=υ5'5υ=5 ∝ υ=

4+'1#1+

Ouu`}vkdio eg` kl eo`stek`cq` dcnold}` b r}or}ko orrouqo ` kcv`}uo }kur`qqodl eo`stek`cq` dcnold}` a`lld }`qqd HOGN#

O}d kirockdio ld eocakxkoc`42

• υ4 < 2 ⇝ wF GN 3 wN

• υ4 3 2 ⇝ wF GN < wN

` oqq`ckdio8

• υ4 < 2 ⇝ { <υ5'5υ=4∝υ

=5+

υ4'5υ=5∝υ=4+

*υ15∝υ

14

=υ5'5υ=5∝υ=4+

• υ4 3 2 ⇝ { <υ5'5υ=4∝υ

=5+

υ4'5υ=5∝υ=4+

*υ15∝υ

14

=υ5'5υ=5∝υ=4+

Ukdco ε ` ε

k uìkrkdck }kur`qqkvdi`cq` eocq`c`cq` G ` coc eocq`c`cq`G , kc epk ld }`qqd 1#1 akvka` kl rkdco# Tpdcqo uqpakdqo akiouq}d eg` lduèocad eocakxkoc` cèùud}kd dstcegb k rpcqk iàk a`k ldqk coc rd}dll lkukdco }`dlk eg` I a`v` nkde`}` c`l uìkrkdco ε #

O}d dmmkdio q}ovdqo l` ap` eocakxkock cèùud}k` dstcegb dceg` k rpcqk

iàk a`k ldqk a`l q}dr`xko ukdco }`dlk# U` pco a`k rpcqk akdnocdlk eokceka` eocN, l‛dlq}o a`v‛ùu`}` dll‛kcfflckqo ` ukdio kc r}ù`cxd a`l rd}dll`lon}diid44#Kc n`c`}dl`, dllo}d, k rpcqk akdnocdlk uk q}ovdco uplld uìk}`qqd NG ` cocud}dcco eokceka`cqk eoc N# Ld a`q`}ikcdxkoc` a`k tpdqq}o v`}qkek }`dlk a`lq}dr`xko b o}d rouukmkl`#

42Ldue`}ìo l` eoo}akcdq` ak F kc fpcxkoc` ak wI ` {I , kc ioao eg` ld eocakxkoc` }kuplqkùr}ùud kc fpcxkoc` ak tpùq` vd}kdmklk#

44Ld }keouq}pxkoc` a`l rd}dll`lon}diid coc b pcked d rd}qk}` adk rpcqk O,N,G,I , ida`fflckqd d i`co ak oioq`qk`, apctp` coc ld q}dqq`}ìo kc tpùqo ldvo}o#



1#4 ]keouq}pxkoc` d rd}qk}` ad N,O,G,I 04

Apctp` u` eocuka`}kdio l‛kcq`}u`xkoc` a`ll` eocakxkock cèùud}k` dstcegb

k rpcqk akdnocdlk ` k rpcqk iàk a`k ldqk coc rd}dll`lk ukdco }`dlk, ld eocakxkocèg` oqq`ckdio b dceg` upstek`cq` r`} }kq}ovd}` tpdqq}o v`}qkek }`dlk ak pcq}dr`xko#

Eocelpakdio }kduupi`cao l` eocakxkock eg` a`voco ùu`}` eocqìro}d)c`di`cq` uoaakufdqq` ad I dstcegb kl q}dr`xko dmmkd tpdqq}o v`}qkek }`dlk#

• I a`v` nkde`}` c`lld }`nkoc` δ, ld rd}q` a`lld eocvùukqbd a`lld rd}dmold1#=:

• I a`v` nkde`}` dceg` c`lld }`nkoc` ν , kl uìkrkdco eocq`c`cq` O, adqoad 1#5, ld }`qqd r`} N r`}r`cakeold}` d OH :

• I a`v` nkde`}` c`lld }`nkoc` ε , kl uìkrkdco eocq`c`cq` G , adqo adlld

}`qqd Q H '1#1+#

Akiouq}dxkoc` ukcq`qked a`lld r}kid eocakxkoc` r`} kl lpono ak I

R}orockdio o}d pcd akiouq}dxkoc` ukcq`qked a`l fdqqo eg` I a`mmd nkde`}`c`lld }`nkoc` δ, dstceg± ùkuqdco k rpcqk akdnocdlk dl fflckqo# L‛ùkuq`cxd akap` kcq`}u`xkock }`dlk4= q}d kl e`}egko E F,FI ` ld }`qqd GN uk rpbo ùr}ki`}ìàkdcq` ld eocakxkoc`8

|F O| 9 |F I | 3 |F,GN|

Uqpakdio ld eocakxkoc` |F I | 9 |F,GN|, ekob kl eduo kc epk F drrd}qk`c` dlld

rd}dmold ak fpoeo O ` ak}`qq}ke` NG # E`}egkdio kl lpono ak I kc fpcxkocàk tpùqd eocakxkoc`# Kccdcxkqpqqo e`}egkdio pcd }`ldxkoc` nòi`q}ked q}dF à I #

Kl r}omlìd b ak }keouq}pk}` I ad F,O,N,G , apctp` }keo}akdio eg`kl q}kdcnolo F OI b kuoue`l` eoc l‛dcnolo dlld mdu` ZZF OI YY 9 ZZOH G YY#Mkuoncd q}deekd}` ld }`qqd F O ` }kro}qd}` l‛dcnolo ZZOH G YY kc ioao eg` pcldqo a`ll‛dcnolo eokcekad eoc F O ` l‛o}k`cqdi`cqo ukd eocu`}vdqo#

Eocuka`}kdio k q}kdcnolk kuoue`lk ukiklk F OF GN ` I OI ∞, aov` F GNb kl rkà` a`lld r`}r`cakeold}` ak F uplld ak}`qq}ke` NG ` I ∞ b qdl egÌO 9 I I ∞ ` |FO|

|OF GN | 9 |IO ||OI ∞| # Vonlkdio akiouq}d}` eg` kl lpono ak I ∞

b pcd }`qqd u` F nkde` uplld rd}dmold, ekob eg` l‛dcnolo ZOH I ∞Y b eouqdcq`

` vdl` ZOH I ∞

Y 9 =ZOH G Y# Udrrkdio eg` r`} onck F ue`lqo uplld rd}dmold,l‛dcnolo ZOH F GNY }kidc` eouqdcq`# Kcolq}`

ς

=∝ ZOI ∞H Y 9 ZI I ∞OY 9 ZF F GNOY 9

ς

=∝ ZOF GNH Y

Tpkcak ZOI ∞H Y 9 ZOF GNH Y# Kcolq}`, rokeg± ZF GNOI ∞Y 9 ZOH G Y,

ZH OF GNY 9 ZHF GNOY * ZOH F GNY 9 ZHF GNOY * ZOH G Y

4=` akuqkcq` r`} tpdcqo vkuqo dl rd}dn}dfo r}èà`cq`



0= ]keouq}pxkoc` a`l Q}dr`xko

ZH OI ∞Y 9 ZHI ∞OY * ZOH I ∞Y 9 ZH OF GNY ∝ ZI ∞OF GNY

Adll‛pnpdnlkdcxd a`nlk plqkik ap` iìm}k dmmkdio

ZOH I ∞Y 9 ZH OF GNY * ZF GNOI ∞Y ∝ ZHI ∞OY

ZOH I ∞Y 9 ZHOF GNY * ZOH G Y ∝ ZHF GNOY

ZOH I ∞Y 9 ZHF GNOY * ZOH G Y * ZOH G Y ∝ ZHF GNOY

ZOH I ∞Y 9 =ZOH G Y

O}d udrrkdio eg` kl lpono ak I ∞ b pcd }`qqd a ` r}èkudi`cq` ld }`qqdoqq`cpqd }k`qq`cao OH up GN# Dllo}d rouukdio eocelpa`}` eg` kl lponoak I , c`l eduo “lkikq`’ kc epk F drrd}q`cnd dlld rd}dmold ak fpoeo O `

ak}`qq}ke` GN, b dceg‛ùuo pcd rd}dmold, ak fpoeo O ` ak}`qq}ke` a#Apctp` rokegb ld }keouq}pxkoc` a`k rpcqk akdnocdlk b rouukmkl` r`} F c`lld

}`nkoc` eocq`c`cq` O, kc epk ld rd}dmold ak fpoeo O ` ak}`qq}ke` GN akvka` klrkdco, dllo}d aommkdio e`}ed}` l‛dcdolno ak qdl` eocakxkoc` r`} I # Ukd δ ld}`nkoc` kc epk kl rkdco b akvkuo adlld rd}dmold ak fpoeo O ` ak}`qq}ke` a, egèocqk`c` ld ak}`qq}ke` a# b@ kiiàkdqo và`}` eg` F drrd}qk`c` dlld }`nkoc`volpqd u` ` uolqdcqo u` I nkde` kc δ#

1#= ]keouq}pxkoc` a`l q}dr`xko kuoue`l`

U` kl q}dr`xko b kuoue`l`, k rpcqk G,N,O,I coc uoco upstek`cqk r`} ld }keo)

uq}pxkoc`, kc tpdcqo a`fflckueoco uolqdcqo ekctp` a`k u`k n}dak ak lkm`}qbd a`lq}dr`xko kuoue l`# Pukdio pc dlq}o rpcqo coq`vol eg` coc dmmkdio uqpakdqor}èà`cqì`cq`, l‛kce`cq}o K a`l q}kdcnolo akdnocdl`# Dmmkdio ouu`}vdqoeg` kl q}kdcnolo akdnocdl` b a`n`c`}` c`l eduo a`l q}dr`xko, kc tpdcqo gdpc v`}qke` 'D4=,51+ dll‛kcfflckqo, eg` abd ld ak}`xkoc` a`k ldqk rd}dll lk# Cocgd apctp` u`cuo akuqkcnp`}` q}d mku`qq}kek kcq`}c` à ùq`}c` a`k ap` dcnolkdl fflckqo ZD4=,51D45,=1D41,=5Y ` ZD45,=1D41,=5D4=,51Y, ` tpkcak q}d kce`cq}o à`we`cq}o# R`} tpùqo eocuka`}`}ìo ld eorrkd coq`vol` K, K

, eg` }kidc` dl

fflckqo8 nkde` upll‛duu` a`l u`ni`cqo D45,=1D41,=5 à b ukii`q}ked }kur`qqo dlldiàkdcd r}kcekrdl`#

Rokegb eocuka`}kdio kl q}dr`xko kuoue`l` ZD41,=5KD45,=1Y 9 ZD45,=1K D41,=5Y 9

ς= ` |KD41,=5| 9 |KD45,=1|, r`}eg± kl q}kdcnolo akdnocdl` D45,=1D4=,51D41,=5 ba`n`c`}` ` }`qqdcnolo#

Ukdio o}d d eocoue`cxd ak `lì`cqk upstek`cqk r`} }keouq}pk}` kl q}dr`xkokuoue l` d rd}qk}` adk rpcqk coq`volk O, G ` ld eorrkd coq`vol` K, K

, eg`

uprrockdio coc dllkc`dqk45#

4# Eouq}pkdio ld }`qqd r`} GO 'iàkdcd r}kcekrdl` a`l q}dr`xko+:

45C`lld }keouq}pxkoc` ek }kf`}k}ìo d K r`} kcaked}` pco a`k ap` rpcqk a`lld eorrkd coq`vol`K, K

# b@ kcakfl`}`cq` tpdl` uk ue`lnd#



1#= ]keouq}pxkoc` a`l q}dr`xko kuoue`l` 05

=# eouq}pkdio ld }`qqd r`} K r`}r`cakeold}` d Z4Y:

5# eouq}pkdio ld }`qqd r`} K rd}dll`ld d Z4Y:

1# mku`qq}kek ak Z=Y,Z5Y:

0# q}ovkdio l` kcq`}u`xkock ak Z1Y eoc Z4Y uoco k rpcqk D45,=1 ` D41,=5:

6# a`q`}ikckdio kl md}ke`cq}o N eoi` rpcqo iàko ak GO:

># eouq}pkdio kl e`}egko ak akdi`q}o D4g,=jO:

7# eouq}pkdio ld }`qqd r`} Z6Y r`}r`cakeold} d Z4Y:

;# q}ovkdio l` kcq`}u`xkock q}d Z>Y ` Z7Y, eg` uoco k rpcqk iàk ak pcd eorrkd

ak ldqk orrouqk coc rd}dll`lk a`l q}dr`xko:

42# q}ovkdio l` }`qq` r`} Z6Y ` Z;Y, eg` uoco k ldqk orrouqk ak epk kc Z;Y:

44# q}ovkdio l` kcq`}u`xkock a`ll` ap` eorrk` ak ldqk orrouqk coc rd}dll`lkak epk kc Z42Y, eg` uoco k v`}qkek a`l q}dr`xko kuoue`l`#

Lpono ak K Ue`lqk G ` O d}mkq}d}kdi`cq`, r`}eg± ld }keouq}pxkoc` ro}qkd v`}qkek }`dlk, b cèùud}ko ue`nlk`}` K kc pcd urèkffled }`nkoc` a`l rkdco#Eocuka`}kdio l` ap` uìk}`qq` }4, }= eoc pc ùq}ìo kc N qdlk eg` ZZ}4, NG YY 9ZZGN,}=YY 9 ς

1 # Ukd δO ld rd}q` ak rkdco kc epk qdlk uìk}`qq` akvkaoco klrkdco, eg` eocqk`c` O: ` ukd δG ld rd}q` ak rkdco }kidc`cq`# U` K uk q}ovduu`

kc δG , k rpcqk akdnocdlk D45,=1 ` D41,=5 uk q}ov`}`mm`}o 'pco ak ùuk orrp}``cq}dimk+ up GO, adlld uqùud rd}q` ak O }kur`qqo d N# Kc tpùqo ioaococ ùkuq`}`mm`}o }`dlk k rpcqk iàk ak ap` o tpdqq}o ldqk a`l q}dr`xko, kctpdcqo kl e`}egko eoc akdi`q}o D4g,=,jO coc kcq`}uèg`}`mm` ld }`qqd r`} Neoc ak}`xkoc` a`k ldqk rd}dll`lk# Apctp` K a`v` q}ovd}uk kc δG #



Mkmlkon}dffld

Z4Y HR# @g}idcc, Uq kc`}‛u Qg o} iu oc qg` Eoirl q` Tpda}kldq }dl ,Fo}pi Nòi`q}keo}pi, Volpi` 1 '=221+ 50)0=:

Z=Y I# Gdrrdeg, X`kqueg}kfq f§ p} Idqg# pca Cdq# Pcq`}}kegq , 15, r#4>0, 4;4=:

Z5Y ]#D# Hogcuoc, Ioa`}c N oi q}{, Dc @lì`cq}{ Q}`dqku` oc qg`Nòi`q}{ of qg` Q}kdcnl` dca of qg` Ek}el` :

Z1Y E# Jkim`}lkcn, Q}kdcnl E`cq }u dca E`cq}dl Q}kdcnl u, Eocn}ùupuCpi`}dcqkpi, Vol# 4=;:

Z0Y M# Uekiìk, N}prrk ak q}dufo}idxkock nòi q}keg , Kuoi`q}k` ùkiklkqpakck c`l rkdco `pelkaò:

Z6Y M# Uekiìk, Rdr })folakcn dca @pl`}‛u Qg o}ì ]`vkukq a , Fo}piNòi`q}keo}pi, Volpi` = '=22=+ ;5)421:

Z>Y M# Uekiìk, Rpcqk Coq`volk a`l Tpda}dcnolo Eoirl`qo, kcàkqo:

Z7Y I# Xdpud, Rpcqk, ]`qq ` Eockeg` Coq volk a`l Tpda}dcnolo RkdcoEoirl`qo, 4;;;

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Alcuni Punti Notevoli del Trapezio

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