Lecture 5 (L. Younes)

7/29/2019 Lecture 5 (L. Younes)

1/92

CMAT]^M QM^GMQ

Cmaturm ?0 Getrbduatgbe (DF'

Cmaturm >0 Aoeamr Ngbcbfy (KB'

Cmaturm 10 Amcc Qgfeocgef Gelmrmeam (KB'

Cmaturm 40 Fmemtga \orgotgbe (DF'

Cmaturm =0 Kossgvm Tmstgef (CW'

Cmaturm


2/92

B]TCGEM

Hypbthmsgs Tmstgef0 oe bvmrvgmwZ vocums

Tmsts bl Fbbdemss bl Lgt

t+tmst

Vgcabxbe ôei+Quk Tmst

Tmstgef lbr GedmpmedmeamTmstgef lbr Dmabrrmcotgbe

Kuctgpcm Tmstgef

Lokgcy+wgsm Mrrbr ôtm

Locsm Dgsabvmry ôtm

Zmrkutotgbe Tmsts oed ^msokpcgef

Getrbduatgbe

> " 7>


3/92

NOQGAQ (G'

O stotgstgaoc hypbthmsgs gs oe ossmrtgbe onbut o prbpmrty bl thm trumdgstrgnutgbe bl bnsmrvmd doto%

Hypbthmsgs tmstgef gevbcvms o eucc hypbthmsgs# usuoccy dmebtmd H:oed oe octmreotm hypbthmsgsH?%

O tmst gs o sokpcm+nosmd rucm dmsgfemd tb dmagdm whmthmr thm eucchypbthmsgs shbucd nm rm`matmd%

Gt gs lbrkoccy mqugvocmet tb thm dmegtgbe bl o rm`matgbe rmfgbe# ê ge

ê# suah thot H: gs rm`matmd gl (X

(?'# % % % # X(e'' ê% Thm sgzm# br cmvmcbl o tmst# gs thm prbnongcgty# # bl rm`matgef H:

whme gt gs trum% Gt gs ocsb aoccmd thm typm+G mrrbr%

Thm pbwmr# # gs thm prbnongcgty bl rm`matgef H: whme gt gs locsm# oed? gs thm typm GG mrrbr%

1 " 7>


4/92

NOQGAQ (GG'

Thm sgkpcmst rm`matgbe rmfgbes orm nosmd be saocor+vocumd tmst

stotgstgas Te oed toim thm lbrk

^e ;

(x(?'# % % % # x(e'' 0 Te(x(?'# % % % # x(e'' 5 t#e

whmrm t#e gs thm argtgaoc vocum# dmsgfemd tb smt thm typm+G mrrbr ot

thm vocum %

Bem aoe ocsb abkputm o p+vocum# whgah gs dmemd os lbccbws%

Ossukm thot oe bnsmrvotgbe (x(?'# % % % # x(e'' gs kodm9 cmttgeft ; T(x(?'# % % % # x(e''# thm p+vocum gs

Z(Te(X(?'# % % % # X(e'' t | H:'

Thm rm`matgbe rmfgbe ot o sgzm gs thme thm smt bl bnsmrvotgbes blwhgah thm p+vocum gs cmss thoe %

4 " 7>


5/92

Z^BNONGCGTW+GETMF^OC T^OEQLB^K

Lbr abetgeubus r%v%# oed uedmr thm eucc hypbthmsgs# p+vocums lbccbw

o ueglbrk dgstrgnutgbe%

Tb smm thgs# wm getrbduam thm prbnongcgty+getmfroc troeslbrk (ZGT'%

Ebtotgbe0 cmlt oed rgfht cgkgts

L(x' ; cgktx

L(t' ; cgktx#t3x

L(t'

L(x)' ; cgktx)

L(t' ; cgktx#t5x

L(t'#

ossukgef# bl abursm thot thmsm cgkgts mxgst%

= " 7>


6/92

ZGT

Thmbrmk (Zrbnongcgty+Getmfroc Troeslbrk'

GlX gs o rmoc+vocumd roedbk vorgoncm wgth aukucotgvm dgstrgnutgbe

lueatgbeLX# oedQ ; LX(X'# thme lbr t

Y:# ?[# bem hos

Z(Q t' ? t#

wgth mquocgty gl t ; L(x' lbr sbkmx ^%Ge portgaucor# glL gs mvmrywhmrm abetgeubus# thmeQ ; LX(X' oed

lbccbws o ueglbrk dgstrgnutgbe bvmr(:# ?'%

< " 7>


7/92

Z^BBL (G'

Cmt x ; gel{x 0 LX(x' t}% Vm hovm LX(x

' t t 0; LX(x'%

Thm lbccbwgef mqugvocmeams hbcdLX(y

' t y x gl LX(x' ; tLX(y

' t y 5 x gl LX(x' 3 t

6 " 7>


8/92

Z^BBL (GG'

]sgef thgs# wm hovm

Z(Q t' ; Z(LX(X

' t';

Z(X x' gl LX(x' ; tZ(X 5 x' gl LX(x

' 3 t

; ? LX(x' gl LX(x

' ; t

? LX(x' gl LX(x' 3 t

;

? t gl LX(x' ; t? t gl LX(x' 3 t

? t

Gl t ; L(y' lbr sbkm y# thme t ; t oed Z(Q t' ; ? t% Lgeoccy# gl LX gs abetgeubus# thme LX gs betb bvmr (:# ?' ny thm

getmrkmdgotm vocum thmbrmk# sb thot Z(Q t' ; ? t lbr occt (:# ?'%

8 " 7>


9/92

OZZCGAOTGBE TB Z+\OC]MQ

Gl T gs o tmst stotgstga# thm p+vocum ossbagotmd tb oe bnsmrvotgbe t gs

p(t' ; Z(T t | H:' ; ? LT(t'%

Thm prmvgbus thmbrmk gkpcgms thot

Z(p(T' t | H:' ; Z(LT(T' ? t | H:' t%

Gl T gs abetgeubus# p(T' gs ueglbrkcy dgstrgnutmd bvmr Y:# ?[ whmethm eucc hypbthmsgs gs trum%

7 " 7>


10/92

B]TCGEM

Hypbthmsgs Tmstgef0 oe bvmrvgmw

Z vocums


t+tmst



Kuctgpcm Tmstgef




Getrbduatgbe

?: " 7>


11/92

AHG+Q_]O^M TMQT LB^ FBBDEMQQ BL LGT

Tmsts lbr fbbdemss bl t mvocuotm whmthmr thm bnsmrvmd doto gs

abesgstmet wgth o kbdmc hypbthmsgzmd ny H:% Thm ahg+squorm tmst gs dmsgfemd lbr egtmcy vocumd dgstrgnutgbes

(dgsarmtgzm bvmr o egtm euknmr bl nges bthmrwgsm'%

Abesgdmr i nges O?# % % % # Oi% Cmt pg ; Z(X Og'% Cmt

pg ;?

e

e`;?

?Og(X`'%

Thm ahg+squorm stotgstga gs thme dmemd ny

_ ; ei

g;?

(pg pg'>pg

%

Lbr corfm e# _

>(i

?' uedmr H:%

?? " 7>


12/92

VGCIQ THMB^MK (G'

Thgs rmsuct aoe nm dmrgvmd lrbk o kbrm fmemroc thmbrmk bl Vgci%

O stotgstgaoc kbdmc gs o lokgcy bl prbnongcgty dgstrgnutgbes Z#

î% i gs thm kbdmc dgkmesgbe%

Abesgdmr emstmd hypbthmsms# wgth twb kbdmcs# porokmtrgzmd bvmr

: î: oed î# i: 3 i% Ossukm thot : gs o sunkbdmc bl # g%m%# : V whmrm V gs

oe olem sunspoam bl î abrrmspbedgef tb i i: gedmpmedmetcgemor abestrogets%

Lbr mxokpcm#

: ;

: î: 0 (:# :# % % % # :'

%

?> " 7>


13/92

VGCIQ THMB^MK (GG'

Thmbrmk (Vgci'Abesgdmr twb emstmd stotgstgaoc kbdmcs os onbvm# oed cmt(X(?'# % % % # X(e'' nm o sokpcm lrbk o prbnongcgty dgstrgnutgbe Z: wgth

: :% Cmt: oed nm koxgkuk cgimcghbbd mstgkotbrs0

: ; orfkox:

e`;?

l(X(`'

'

; orfkox

e`;?

l(X(`''

Cmte nm thm cgimcghbbd rotgbe ; cbfe

`;? l(X(`''

e`;? l:

(X(`''

Thme>e osykptbtgaoccy lbccbws o ahg+squorm dgstrgnutgbe wgth i i:dmfrmms bl lrmmdbk%

?1 " 7>


14/92

OZZCGAOTGBE TB THM AHG Q_]O^M TMQT

;

{(p?# % % % # p

i'# p

? )

) pi ; ?

}(dgkmesgbe i

?'%

: ; {(p?# % % % # pi'} (dgkmesgbe :'% Qgeam pg pg uedmr H:# thm cgimcghbbd rotgb

e ;

i

g;? e

g cbf

pg

pg

aoe nm mxprmssmd os

e ;_e

> )

ig;?

B(e|pg pg|1'

Thgs gkpcgms thot _e hos thm sokm cgkgt ge dgstrgnutgbe os >e%

?4 " 7>


15/92

MXOKZCM

Os oe oppcgaotgbe# ossukm thot X gs o roedbk vmatbr#

X ; (X?# %%%# Xd'# wgth moah Xg nmcbefgef tb o egtm smt N%

Ge thm aosm bl TL+DEO ngedgef# N ; {O# A# T# F} oed Xrmprmsmets o pbtmetgoc kbtgl lbr o portgaucor TL o smqumeam bl

d abesmautgvm euacmbtgdms%

Oe bnsmrvotgbe gs thme x ; {x(?'# %%%# x(e'}# ge whgah# lbr mxokpcm#wm hovm mxpmrgkmetoccy vmrgmd o ngedgef sgtm lbr thm sokm TL ge

thm upstrmok rmfgbe bl eab+rmfucotmd fmems%

Hmrm# wm aoe rmprmsmet thm fmemroc kbdmc ny thm abccmatgbe bl occ

pbssgncm prbnongcgty dgstrgnutgbes be thm egtm smt Q; Nd

% Zbssgncm eucc hypbthmsms aoe ebw nm whmthmr thm vorgoncms

X?# %%%# Xd orm gedmpmedmet# br whmthmr occ thm sgtms orm gdmetgaoccydgstrgnutmd%

?= " 7>


16/92

B]TCGEM


Z vocums


t+tmst



Kuctgpcm Tmstgef




Getrbduatgbe

?< " 7>


17/92

T+TMQT (G'

Abesgdmr o sgtuotgbe ge whgah bem bnsmrvms sokpcms lrbk twb

pbpucotgbes (m%f%# dgsmosmd br ebt dgsmosmd'%

Cmt X nm thm roedbk vorgoncm rmprmsmetgef thm bnsmrvotgbes (m%f%#

fmebtypm'%

Ossukm thot W {:# ?} rmprmsmets thm pbpucotgbe (m%f%#phmebtypm'%

Ossukm thot W gs dmtmrkgegstga% Cmt e`# k` oed ` dmebtm thmpbpucotgbe sgzm# mxpmatotgbe oed stoedord dmvgotgbe ge

sunpbpucotgbe ` lbr ` ; :# ?% Cmt X( :' oed X( ?' dmebtm sokpcm ovmrofms%

?6 " 7>

(GG'


18/92

T+TMQT (GG'

Thm t+tmst abkporms pbpucotgbe ovmrofms% Cmt ] ; X( ?' X( :'% ACT0

e`(X

( `' k`' E(:# >` ' whme e` % Ossukm e:"e (wgth e; e: ) e?'% Thme

e(X( ?' X( :'' e(k? k:'

(? '?(e?(X( ?' k?''

?(

e:(X

( :' k:''

E(:# (? '?

>? )

?

>:'

?8 " 7>

(GGG'


19/92

T+TMQT (GGG'

]edmr thm eucc hypbthmsgs kX ; kW#

e(X( ?' X( :'' E(:# (? '?>? ) ?>:'

Mqugvocmetcy

(e?: ) e?? '

?"> X( ?' X( :'

(? '>: )

>?

E(:# ?'

osykptbtgaoccy%

?7 " 7>

Q (G\'


20/92

T+TMQT(G\'

Koim thm ossukptgbe >: ; >? ;

> (abkkbe vorgoeam'%

Dmem

> ;?

e >e

i;?

(Xi X( Wi''>

Qtudmets t+stotgstga gs fgvme ny

t(x' ; (e?: ) e?? '

?"> X( ?' X( :'

%

Vhme X gs Foussgoe# thm dgstrgnutgbe bl thm t+stotgstga gs o Qtudmet

dgstrgnutgbe wgth e > dmfrmms bl lrmmdbk% Ge thm fmemroc aosm# thgs gs o fbbd opprbxgkotgbe lbr kbdmrotm

vocums bl e%

Gdmetgaoc vorgoeam gs oe rmstrgatgvm ossukptgbe%

>: " 7>

B]TCGEM


21/92

B]TCGEM


Z vocums


t+tmst



Kuctgpcm Tmstgef




Getrbduatgbe

>? " 7>

^OEI Q]K0 KBTG\OTGBE (G'


22/92

^OEI+Q]K0 KBTG\OTGBE (G'

^m`matgef thm eucc hypbthmsgs ot cmvmc dbms ebt emamssorgcy gkpcythot H? gs trum wgth prbnongcgty ? # nmaousm nbth hypbthmsmsaoe nm locsm%

Gl bem tmsts lbr k: ; k? vs% k: ; k? os onbvm# edgef ](x' 5 taoe ocsb nm aousmd ny thm loat thot sbkm bl thm hypbthmsms thotwmrm kodm tb abkputm thm prbnongcgty bl thm dmtmatgbe rmfgbe wmrm

locsm%

Lbr mxokpcm# ge thm t+tmst# thgs koy nm aousmd ny thm gevocgdgty bl

thm opprbxgkotgbe ny o Qtudmet dgstrgnutgbe%

>> " 7>

^OEIQ (G'


23/92

^OEIQ (G'

Qbkm tmst stotgstgas hovm o dgstrgnutgbe thot dbms ebt dmpmed be

thm brgfgeoc bem%

Thmy orm aoccmd dgstrgnutgbe lrmm%

Thmy fmemroccy rmcy be brdmr stotgstgas oed roeis% Cmt x ; (x(?'# % % % # x(e'' nm oe e+sokpcm bl o rmoc+vocumd r%v%

Ossukm eb tgm0 occ x(i's orm dgstgeat

Thm roei bl thm ith vorgoncm gs bem pcus thm euknmr bl x(`'s thot

orm cmss thoe x(i'%

>1 " 7>

^OEIQ (GG'


24/92

^OEIQ (GG'

Thgs dmegtgbe aoe nm mxtmedmd tb tgms ny ovmrofgef bvmr occ

pbssgncm roeigefs bl thm tgmd vocums%

Cmt e(i'(x' ;

e`;? x(`';x(i' oed dmem

r(i'(x' ;e(i' ) ?

>)

`

x(`'3x(i'

Lbr mxokpcm# gl thm sokpcm gs (1# ?# ># ># =# 1'# thm roeis orm(4%=# ?# >%=# >%=#


25/92

^OEIQ (GGG'

Gl X(?'# % % % # X(e' orm gedmpmedmet oed gdmetgaoccy dgstrgnutmd# thm`bget dgstrgnutgbe bl X ; (X(?'# % % % # X(e'' gs gevorgoet ny pmrkutotgbebl thm gedgams%

Nmaousm bl thgs# thm `bget dgstrgnutgbe bl thm roeis#

r(X' ; (r(?'(X'# % % % # r(e'(X''

becy dmpmeds be thm dgstrgnutgbe bl X vgo thm porokmtmrs qikmosurgef thm prbnongcgty bl i gedmpmedmet rmpcgaos bl X nmgef

mquoc (i tgms'%

Gl X hos o dmesgty# thgs prbnongcgty gs zmrb# oed thm roeis# oed

thmrmlbrm oey lueatgbe bl thmk# orm dgstrgnutgbe+lrmm%

>= " 7>

^OEI Q]K


26/92

^OEI Q]K

Vgcabxbes gs dmemd ny

V(x' ;e

`;?y(`'r(`'(x'#

Thgs stotgstga gs corfm gl thm roeis ge thm smabed pbpucotgbe (W ; ?'orm systmkotgaoccy corfmr thoe ge thm rst pbpucotgbe (W ; :'%

]edmr thm eucc hypbthmsgs thot X dbms ebt dmpmed be W# oed

ossukgef thot tgms hovm zmrb prbnongcgty# thm dgstrgnutgbe bl V becydmpmeds be e: oed e?# thm sgzms bl thm pbpucotgbes%

>< " 7>

IBCKBFB^B\+QKG^EB\ QTOTGQTGA (G'


27/92

IBCKBFB^B\+QKG^EB\ QTOTGQTGA (G'

Ge thm prmsmeam bl twb pbpucotgbes# abkputm

L:(t9x' ;?

e:

e`;?

(? y(`''x(`'t

L?(t9x' ;?

e?

e

`;?

y(`'x(`'t

Thm Ibckbfbrbv+Qkgrebv stotgstga gs thme dmemd ny

D(x' ; supt^

|L:(t9x' L?(t9x'|%

Ge thm onsmeam bl tgms# thgs stotgstga gs dgstrgnutgbe+lrmm% Fbbd opprbxgkotgbes bl thm corfm sokpcm dgstrgnutgbe orm prbvgdmd

ny thm Ibckbfbrbv dgstrgnutgbe# D# whgah abgeagdms wgth thm

dgstrgnutgbe bl thm koxgkuk bl o Nrbwegoe nrgdfm'

>6 " 7>

B]TCGEM


28/92

B]TCGEM


Z vocums


t+tmst



Kuctgpcm Tmstgef




Getrbduatgbe

>8 " 7>


29/92

Abesgdmr twb roedbk vorgoncms# X oed W%

Ossukm thot thmy orm dgsarmtm wgth egtm suppbrt# soy

X O ; {?# %%%# @} oed W N ; {?# %%%# I}% Mxokpcm0 X gs o prbpmrty bl o fmebkm# suah os thm stotm thm twb

occmcms ot sbkm cbaus# oed W wgcc rmprmsmet o phmebtypm# lbr

mxokpcm abdgef aosms oed abetrbcs os W ; :# ?%

>7 " 7>

AHG+Q_]O^M


30/92

AHG Q_]O^M

Cmt eon nm thm euknmr bl bnsmrvotgbes bl x(`' ; o# y(`' ; n lbr

o O# n N% Qgkgcorcy# dmem thm korfgeoc abuets

eo) ;nN

eon# e)n ;oO

eon%

Thm ahg+squormd stotgstga gs fgvme ny

> ; eo#n

(eon eo)e)n"e'>eo)e)n

%

]edmr thm eucc hypbthmsgs thot X oed W orm gedmpmedmet# >(z'abevmrfms ge dgstrgnutgbe tb o ahg+squorm dgstrgnutgbe wgth

(@ ?'(I ?' dmfrmms bl lrmmdbk# oed thgs opprbxgkotgbe aoe nmusmd lbr corfm sokpcm sgzms%

1: " 7>

LGQHM^Q MXOAT TMQT (G'


31/92

LGQHM^ Q MXOAT TMQT (G'

Lgshmrs mxoat tmst gs o kmthbd lbr tmstgef thm eucc hypbthmsgs bl

gedmpmedmeam wgth o tmst stotgstga whbsm eucc dgstrgnutgbe aoe nmabkputmd mxoatcy# thmrmny ovbgdgef corfm sokpcm opprbxgkotgbes%

Gt prbvgdms oe mxokpcm bl abedgtgbeoc tmst%

Cmt T oed Q nm stotgstgas (Q aoe nm kuctgvorgotm'# oed ossukm thot#

lbr occ s# bem aoe mstgkotm aut+bll vocums q(s' suah thot

Z(T 5 q(s' | Q ; s' %

Thme thm rm`matgbe rmfgbe T q(Q' aoe nm usmd tb bntoge o tmst otcmvmc ot kbst%

Tb smm thgs# `ust wrgtm

Z(T 5 q(Q'' ; M(Z(T 5 q(Q' | Q'' %

1? " 7>

LGQHM^Q MXOAT TMQT (GG'


32/92

LGQHM^ Q MXOAT TMQT (GG'

Lbr Lgshmrs mxoat tmst# wm ossukm thot thm rbw abuets o ; eo)orm dmtmrkgegstga (thmy rmprmsmet sun+pbpucotgbe sgzms' oed toim

thm abedgtgbegef stotgstgas tb nm thm abcuke abuets0

Q ; (sn 0; e)n# n ; ?# % % % # I'%

]edmr H:# thm abedgtgbeoc dgstrgnutgbe bl thm abuets abedgtgbeoc tb

Q toims thm lbrk

Z ;

@o;?

oeo?%%%eoI

Es?sI

%

1> " 7>

LGQHM^Q MXOAT TMQT (GGG'


33/92

( '

Lbr > ny > toncms (m%f%# logcurm vs% suaamss lbr twb pbpucotgbes'# wmfmt

Z ;

?e??

>s?e??

Es?

Thgs gs o hypmrfmbkmtrga dgstrgnutgbe%

Lgshmrs mxoat tmst rm`mats thm eucc hypbthmsgs nosmd be thm stotgstga

T ;

|e??

s??"e

|oed toncms bl thm hypmrfmbkmtrga dgstrgnutgbe%

11 " 7>

MXOKZCM


34/92

Quppbsm W ; ?# > rmprmsmets aosms oed abetrbcs%

X ; ?# > rmprmsmets thm bnsmrvgef br ebt bnsmrvgef o rorm fmemtgavorgoet%

Vm bnsmrvm tme gedgvgduocs# wgth vm rorm vorgoets but bl sgx aosms

oed eb rorm vorgoets but bl lbur abetrbcs%

Bem eds thm p vocum lbr e?? ; = (whgah gs thm corfmst pbssgncmvocums' gs

p ;=/=/18%

14 " 7>

B]TCGEM


35/92


Z vocumsTmsts bl Fbbdemss bl Lgt

t+tmst


Tmstgef lbr Gedmpmedmeam

Tmstgef lbr Dmabrrmcotgbe

Kuctgpcm Tmstgef




Getrbduatgbe

1= " 7>

ZMO^QBEQ ABMLLGAGMET


36/92

Lrbk oe e+sokpcm bl (X# W' os onbvm# thm mkpgrgaoc abrrmcotgbe

abmlagmet (ocsb aoccmd Zmorsbes abrrmcotgbe abmlagmet' gs

;e?

e`;? x

(`'y(`' xy

e?

e`;?(x

(`''> x>e?

e`;?(y

(`''> y>

]edmr thm eucc hypbthmsgs bl (X# W' gedmpmedmet ebrkoc roedbkvorgoncms# thm dmesgty bl (z' gs prbpbrtgbeoc tb t (? t>'(e>'"4%

Lbr ebe+Foussgoe doto oed corfm e# thm stotgstga

](z' ;?

>oratoe((z''

gs opprbxgkotmcy Foussgoe wgth kmoe : oed vorgoeam ?"(e 1'%

1< " 7>

^OEI AB^^MCOTGBE (G'


37/92

Thm abrrmcotgbe nmtwmme X oed W aoe ocsb nm wrgttme os

(X# W' ; (X# W' ; M(XW'

whmrm X gs thm stoedordgzmd vmrsgbe bl X# X ; (X M(X''"(X'oed sgkgcorcy lbr W%

Nmaousm bl thgs# Qpmorkoes abrrmcotgbe gs gevorgoet tb cgemortroeslbrkotgbes bl oey bl thm vorgoncms%

Lbr ebecgemor troeslbrkotgbes# ebrkocgzm usgef thm Z%G%T% Dmem

X ; LX(X'

oed

r(X# W' ; (X# W' ; ?>M(XW' 1%

16 " 7>

^OEI AB^^MCOTGBE (GG'


38/92

Fgvme o sokpcm (x(?'# % % % # x(e'' lbr X# bem hos

LX(x' ;

?

e

ei;?

?x(i'3x#

Ge portgaucor LX(x(c'' ; (rXc

?'"e# whmrm rXc gs thm roei bl xc ge

thm sokpcm%

Thgs gkpcgms thot xc aoe nm mstgkotmd ny (rXc ?'"e%

Cmttgef (y?# % % % # ye' nm thm ossbagotmd sokpcm lbr W# wgth roeis rW#

wm hovm thm sokpcm mstgkotm0

r(X# W' ; s(^X# ^W'%

Thgs gs aoccmd thm roei abrrmcotgbe abmlagmet%

18 " 7>

^OEI AB^^MCOTGBE (GGG'


39/92

Gl X oed W orm abetgeubus roedbk vorgoncms# thme

r(X# W' ; ?>

?:

?:

Z(X 3 u oed W 3 v'lX(u'lW(v'dudv 1%

Cmt (X# W# X# W' nm lbur r%v%s wgth (X# W' lX#W# X lX oedW lW# oed (X# W'# X oed W kutuoccy gedmpmedmet% Thme

?

: ?

:

Z(X 3 u oed W 3 v'lX(u'lW(v'dudv

; Z(X 3 X oed W 3 W'%

17 " 7>

IMEDOCCQ TO] (G'


40/92

Ossbagotm tb twb abetgeubus r%v%s X oed W thm getmfroc0

G ;

?:

?:

Z(X 3 u oed W 3 v'lX#W(u# v'dudv%

Thgs getmfroc gs ebw getmrprmtmd os G ; Z(X 3 X oed W 3 W' gewhgah (X# W' oed (X# W' orm g%g%d% pogrs%

Imedocs abmlagmet lbr abetgeubus r%v%s gs thme dmemd ny

(X# W' ; 4G ?oed toims vocums nmtwmme ? oed ?%

4: " 7>

IMEDOCCQ TO] (GG'


41/92

Twb pogrs bl r%v%# (X# W' oed (X# W'# orm brdmr+abesgstmet gl

(X 3 X

oed W 3 W

' br (X

3 X oed W

3 W'whgah gs mqugvocmet tb (X X'(W W' 5 :%

>G prbvgdms thm prbnongcgty thot thm twb pogrs orm brdmr abesgstmet#

oed

(X# W' ; >Z((X

X'(W

W' 5 :'

?%

Ge thm abetgeubus aosm# usgef

? Z((X X'(W W' 5 :' ; Z((X X'(W W' 3 :'# thgs aoenm rmwrgttme os

(X# W' ; Z((X X'(W W' 5 :' Z((X X'(W W' 3 :'%

Thgs gs thm fmemroc dmegtgbe bl Imedoccs tou# whgah gs ocsb vocgd

whme X oed W orm ebt abetgeubus%

4? " 7>

IMEDOCCQ TO] (GGG'


42/92

Fgvme o sokpcm (x(?'# y(?'# % % % # x(e'# y(e''# Imedoccs tou aoe nmmstgkotmd ny abuetgef thm euknmr bl brdmr+abesgstmet vs%

brdmr+geabesgstmet pogrs%

Cmt

og` ; sgfe(xg x`'sgfe(yg y`'wgth

sgfe(x' ;

? gl x 5 :

: gl x ; :

? gl x 3 :

Dmem thm sokpcm vmrsgbe bl thm abmlagmet ny

; >

?g3`e

Og`

e(e ?' %

4> " 7>

B]TCGEM


43/92



t+tmst




Kuctgpcm Tmstgef




Getrbduatgbe

41 " 7>

GET^BD]ATGBE


44/92

Dotosmts ge abkputotgbeoc ngbcbfy prbvgdm o vmry corfm euknmr bl

kmosurmkmets pmr sokpcm%

Ge oddgtgbe tb thmsm kmosurmkmets# oddgtgbeoc mllmats# brabvorgotms# orm lrmqumetcy abccmatmd# cgim fmedmr# ofm br dgofebsgs%

Bem wbucd cgim tb dmtmrkgem whgah portgaucor kmosurmkmets oed

whgah portgaucor abvorgotms orm ossbagotmd%

44 " 7>

GET^BD]ATGBE


45/92

Hypbthmsgs tmstgef gs dmsgfemd tb koim thgs iged bl ossmsskmet%

Moah tmst abkms wgth o skocc prbnongcgty bl typm G mrrbr (locsm

ocork'% Vhme koey tmsts orm pmrlbrkmd sgkuctoembuscy thmsm

prbnongcgtgms oaaukucotm% Qbkm spmagoc prbamdurms orm emmdmd tb

abrrmatcy abetrbc thm prbnongcgty bl mrrbr%

4= " 7>

EBTOTGBE


46/92

Ossukm esokpcms# wgth K kmosurmkmets pmr sokpcm

Dmebtm kmosurmkmets ny x(g'i # i ; ?# % % % # K# g {?# % % % # e}%

Cmt y(g' rmprmsmet oddgtgbeoc vorgoncm thot aoe nm ossbagotmd wgth g%

Vm ossukm thot thmsm bnsmrvmd vocums orm rmocgsotgbes bl roedbk

vorgoncms Xi oed W%

Qokpcms orm ossukmd tb nm g%g%d%# nut vorgoncms wgthge sokpcms orm

fmemroccy dmpmedmet%

4< " 7>

MXOKZCMQ


47/92

Fmem mxprmssgbe0 x(g'i gs thm mxprmssgbe bl fmem i be thm gth orroy

br sokpcm9 y(g' aoe nm o abccmatgbe bl phmebtypms ossbagotmd wgth

thgs sokpcm# m%f%# y(g' ; : lbr ebrkoc sokpcms oed y(g' ; ? lbr

aoeamr sokpcms% QEZs0 x

(g'i abdms thm fmebtypm ot o cbaus be thm fmebkm lbr

sokpcm g% Typgaoccy x(g'i

{:# ?# >}9 y(g' aoe nm quoetgtotgvm (hmgfht#ncbbd prmssurm#% % % ' br ngeory (aosm vs% abetrbc'%

46 " 7>

BN@MATG\MQ


48/92

Thm fboc gs tb dmtmrkgem whgah kmosurmkmets Xi (gl oey' orm

ossbagotmd wgth W%

Cmt Q ; {?# % % % # K} oed D Q gedmx occ vorgoncms thot ormossbagotmd wgth W% Cmt Q

: ;Q

SD%

Lgrst fboc0 dmvmcbp o tmst lbr thm Fcbnoc eucc hypbthmsgs0

H:(Q' 0 D ; (vs% D; mkptysmt'% Thm gedgvgduoc eucc hypbthmsgsot i Q gs H:(i' 0 i Q:% Qmabed fboc (kbrm okngtgbus'0 dmem oe mstgkotbr D

Q lbr D

48 " 7>

L^OKMVB^I (G'


49/92

Ossukm thot rm`matgef H:(i' gs nosmd be o tmst stotgstga ]# cmttgef

zi ; ](xi#y'%

]sgef Pi lbr thm ossbagotmd r%v%# cmt ti dmebtm thm p+vocum

ti ; Z(Yzi# )

'|

H:(i'' ; ?

L(z

i |H:(i''

]sgef thm Z%G%T%# oed cmttgef

D ; {i 0 ti t}lbr sbkm thrmshbcd t# wm hovm# lbr occ i Q:0

Z(i D | H:(i'' t%

47 " 7>

L^OKMVB^I (GG'


50/92

Thgs soys thot thm prbnongcgty thot i gs o locsm dmtmatgbe gs eb

frmotmr thoe t%

Gl D ;

# thm mxpmatmd euknmr bl locsm ocorks gs :%:=K9 gl

K ; ?::: oed t ; :%:=# thm prbamdurm wbucd rmture# ge ovmrofm#=: gedgams ge thm smt D%

t hos tb nm skocc ge brdmr tb abetrbc thm tbtoc euknmr bl locsm

ocorks%

=: " 7>

L^OKMVB^I (GGG'


51/92

Ossukm D ; (fcbnoc eucc gs trum'% Vrgtm Z(D; ' ; Z(i 0 Ti t'% Ossukm thot occ tmsts orm gedmpmedmet

Z(i 0 Ti t' ; ? Z(i 0 Ti 5 t'; ? Z(Ti 5 t'K ? (? t'K%

=? " 7>

L^OKMVB^I (G\'


52/92

Bem kust toim

t

?

mxp(cbf(?

'"K'

"K

tb mesurm Z(D; ' % Lbr ; :%:= oed K ; ?:::# thgs fgvms t ; = ?:=%

=> " 7>

NBELM^^BEG AB^^MATGBE


53/92

^maocc thot Q:

Q gs thm smt bl vorgoncms lbr whgah thm eucc

hypbthmsgs hbcds%

Gl t ; "K#

Z(D Q: ; ' ; Z(i Q: 0 Ti t' |Q:|t %

t ; "K gs aoccmd thm Nbelmrrbeg abrrmatgbe%

Gt abetrbcs thm lokgcy+wgsm mrrbr rotm(LVM^' ot cmvmc % Thgs kmoes thot gt mesurms thot thm prbnongcgty bl koigef bem locsm

ocork br kbrm gs cmss thot %

=1 " 7>

QTOFMVGQM AB^^MATGBE


54/92

Cmt T(?'# % % % # T(K' nm brdmr stotgstgas ossbagotmd tb T?# % % % # TK%

Qgkmss rucm rm`mats thm fcbnoc eucc gl thmrm mxgsts o i {?# % % % # K}suah thot t(i'

i"K%

Vhme tmsts orm gedmpmedmet# thgs prbvgdms o scgfht gkprbvmkmet tb

Nbelmrrbeg%

Thgs wbris becy os o rm`matgbe rucm lbr thm fcbnoc eucc%

=4 " 7>

QGKMQQ MQTGKOTB^


55/92

Cmt t ; i"K whmrm

i ; kge{i 0 t(i' 5 i"K}%

Thm smt

D ; {i 0 Ti t}gs thm Qgkmss mstgkotbr lbr thm mxaursgbe smt%

Gl i ; ?# Thm Qgkmss mstgkotbr abgeagdms wgth Nbelmrrbegs%

Gt dbms ebt abetrbc thm LVM^ whme thm fcbnoc eucc gs ebt vocgd%

== " 7>

HBCKQ MQTGKOTB^ (G'


56/92

Cmt i nm suah thot t(i' ; ti%

Hbcks mstgkotbr gs

DH ; {?# % % % # i}wgth

i ; kox

{i 0 t(c'

"(K

c ) ?' lbr c

i

} Thm Hbck mstgkotbr gkprbvms be Nbelmrrbegs oed stgcc abetrbcs thm

LVM^%

Thm gkprbvmkmet nrbufht ny Hbcks mstgkotbr gs skocc gl K gscorfm oed |D| 33 K% Thm rucm lbr rm`matgef D ; gs thm sokm Nbelmrrbegs%

=< " 7>

HBCK MQTGKOTB^ (GG'


57/92

Cmt k: ;

|Q:

|%

Vm hovmZ(kge

iQ:Ti 5 "k:' ? %

Ossukm thot kgeiQ: ti 5 "k:# sb thot t(Kk:)?' 5 "k: oed

i k:% Qb# thmrm orm ot kbst k: mcmkmets ge DH# occ wgth ti 3 "k:%

]edmr bur ossukptgbe# ebem bl thmk aoe nm ge Q:%

Thgs shbws

kgeiQ:

ti 5 "k: DH Q: ;

sb thot Z(D Q: ; ' ? %

=6 " 7>

B]TCGEM


58/92



t+tmst




Kuctgpcm Tmstgef




Getrbduatgbe

=8 " 7>

DMLGEGTGBE


59/92

Abesgdmr thm rotgb

_ ; |D

Q:

||D|gl D; oed _ ; : bthmrwgsm%

Gt gs thm prbpbrtgbe bl locsm ocorks wgthge thm dmtmatgbe smt%

Ebtm thot _ gs o roedbk vorgoncm%

Gts mxpmatmd vocum# M(_'# gs aoccmd thm locsm dgsabvmry rotm# brLD^%

LD^ ; M |

D

Q:

||D| ? % Oe mstgkotbr D gs sogd tb abetrbc thm LD^ ot rotm cmss thoe

q (:# ?' gl M(_' q%=7 " 7>

NME@OKGEG+HBAHNM^F MQTGKOTB^ (G'


60/92

Thm N+H gs dmemd ny

D ; {i 0 Ti qi"K}

whmrm i

; kox

i 0 T(i' qi"K% Octmreotgvmcy# D ; {?# % % % # i} wgth T(i' ; Ti% Zrbbl0 gl Ti qi"K oed Ti ; T(ri' wgth ri 5 i# thme

T(ri' qri"K# whgah abetrodgats i nmgef thm corfmst getmfmrsotgslygef thgs gemquocgty%


61/92

Mqugvocmetcy# thm N+H mstgkotbr os thm corfmst smt O suah thot

koxiO

Ti q|O|K

%

Zrbbl0 Lgrst ebtm thot thgs koxgkoc smt gs uegqum# nmaousm gl O oed

O sotgsly thgs gemquocgty# sb dbms O O% Gl O gs thgs corfmst smt# thme i |O|# sgeam O ; {?# % % % # i}

sotgsms koxO Ti

qi"K%

Abevmrsmcy# gl koxO Ti q|O|K # thme T(|O|' q|O|K sb thot |O| i%


62/92

Thm lbccbwgef hbcds0

M(_' ;K

s;?

iQ:

Z(i D# |D| ; s'"s%

Zrbbl0

M(_' ; M

|Q: D|

|D| ?|D|5:

; M

iQ:

?iD

?|D|5:

"|D|

;

iQ:

M?

iD?|D|5:"|D|

;

iQ:

Ks;?

Z(i D# |D| ; s'"s%

" 7>

V h c g "K


63/92

Vm hovm# cmttgef qs ; qs"K

{|D| ; s# i

D} ; {|

D| ; s# Ti qs}%

Cmt T(i'(?' # % % % # T

(i'(K?' dmebtm brdmr stotgstgas lbr Tc# c ; i%

Thme

{|D| ; s# i D} ;(Ti qs' (T(i'(`' qs#` 3 s# T

(i'(`' 5 q`)?#` s'%


64/92

Vm hovm

M(_' ;K

s;?

iQ:

Z((Ti qs' A(i's '"s

;K

s;?

iQ:

Z(Ti qs'Z(A(i's |Ti qs'"s

qK

iQ:

Ks;?

Z(A(i's |Ti qs'

qk:

K koxiQ:

Ks;?

Z(A(i's |Ti qs'%


65/92

Qgeam qk:"K q# tb prbvm thot thm N H prbamdurm abetrbcs thmLD^ ot cmvmc q# gt sulams tb shbw thot# lbr occ i Q:#

i 0;K

s;?

Z(A(i's |Ti qs' ?%

Gl occ tmsts orm gedmpmedmet# thme

i ;K

s;?

Z(A(i's '#

whgah gs cmss thoe ? nmaousm thm A(i's s orm dgs`bget%


66/92

Thgs prbvms thm rmsuct uedmr thm ossukptgbe thot thm tmsts lbrwhgah thm eucc hypbthmsgs gs trum orm gedmpmedmet lrbk thm occ

bthmr%

Nut thgs ossukptgbe gs sbkmwhot rmstrgatgvm# sgeam tmsts aoe vmry

rormcy nm ossukmd tb nm gedmpmedmet%

Thm vocgdgty bl thm N+H prbamdurm hos nmme mxtmedmd tb thm

ossukptgbe bl pbsgtgvm rmfrmssgbe dmpmedmeaynmtwmme thm tmsts%


67/92

O lueatgbe l dmemd be ^e gs gearmosgef gl x? y?# % % % # xe yegkpcgms l(x' l(y'# whmrm x ; (x?# % % % # xe'# y ; (y?# % % % # ye' ^

e

% O r%v% W wgth vocums ge ^k gs pbsgtgvm rmfrmssgbe dmpmedmet (Z^D'

be o r%v% X wgth vocums ge ^e gl# lbr mvmry pbsgtgvm gearmosgef

lueatgbe be ^k# thm abedgtgbeoc mxpmatotgbe

(x?# % % % # xe' M((W'|X? ; x?# % % % # Xe ; xe'gs oe gearmosgef lueatgbe bl x?# % % % # xe%

Thm Z^D abedgtgbe gkpcgms thot

M((W'(X''' M((W''M((X''

lbr oey pbsgtgvm gearmosgef lueatgbes oed %


68/92

ThmbrmkOssukm thot# lbr occi

Q:# thm vorgoncmT

(i' ; (Tc# c

; i' gs pbsgtgvmcy

rmfrmssgbe dmpmedmet beTi% Thme thm Nme`okgeg+Hbahnmrf prbamdurmabetrbcs thm LD^ ot cmvmc cmss thoek:q"K%


69/92

Vm hovm D(i's:

0;

s:s;? A

(i's ;

T(i'(`' 5 q`)?#` s:

%

Cmt (t?# % % % # ti?# ti)?# % % % # tK' nm thm gedgaotbr bl thm smtt(i'(`' 5 q`)?#` s:

% Thme gs gearmosgef

Thus

Z(D

(i'

s:)?|Ti qs:)?' ; Z(D(i'

s: |Ti qs:)?')Z(A

(i's:)?

|Ti qs:)?' Z(D(i's: |Ti qs: '

)Z(A(i'

s:)?|T

i q

s:)?'

Ks;? Z(A

(i's |Ti qs' ? aoe nm dmrgvmd ny geduatgbe%


70/92

Bem koy ocsb nm getmrmstmd ge mstgkotgef bl thm LD^ whme thm

dmtmatgbe smt gs nugct usgef rucms thot dgllmr lrbk N+H% Lbr mxokpcm# abesgdmr thm rotm

_(t' ;|Dt Q:|

|Dt

| ?

whmrm t Y:# ?[ gs xmd oed

Dt ; {i 0 Ti t} %

Thm N+H prbamdurm usms t ; qi"K# wgth t sokpcm+dmpmedmet%

Aoe bem abkputm thm LD^ ossbagotmd tb o dgllmrmet thrmshbcd2

6: " 7>

QTB^MWQ MQTGKOTB^ (G'

Ossukgef abetgeubus tmst stotgstgas wm hovm


71/92

Ossukgef abetgeubus tmst stotgstgas wm hovm

M|D

T(i' Q

:|; k

:T(i'

%

Vm aoe thmrmlbrm wrgtm (os o kbtgvotgbe tb dmrgvm thm kmthbd'

T(i'

i k?

:

|DT(i' Q:||DT(i' | ;

_(T(i''"

k:%

Thgs cmods tb thm opprbxgkotgbe

_(T(

i''

k:T(i'

ibr

_(t' k:t|D(t'| ? %

6? " 7>

QTB^MWQ MQTGKOTB^ (GG'


73/92

ThmbrmkOssukm thot# lbr occi Q:# thm vorgoncmT(i' gs pbsgtgvm rmfrmssgbedmpmedmet beTi% Thme# bem hos

M(_(t' _(t'' :%

61 " 7>

Z^BBL(G'


74/92

Lbr i Q:# cmt D(i't ; {c QS i# Tc t} Vm hovm

M(_(t'' ; iQ:

M(?iD

t

"

|Dt

| ?'

;

iQ:

M(?Tit"(|D(i't | ) ?''

64 " 7>

Z^BBL(GG'


75/92

Qgeam (|D(i'

t | ) ?'?

gs oe gearmosgef lueatgbe bl Tc# c ; i oed ?Titgs gearmosgef ge Ti# thm Z^D ossukptgbe gkpcgms

M(?Tit(|D(i't | ) ?'?' ; M((|D(i't | ) ?'?'

M(?T

i5t(

|D(i'

t |) ?'?'

M((|D(i't | ) ?'?'Z(Ti 5 t'M((|D(i't | ) ?'?'

; M((

|D(i't

|) ?'?'Z(Ti

t'

6= " 7>

Z^BBL (GGG'


76/92

Gt lbccbws thot

M(_(t'' iQ

:

Z(Ti t'M((|D(i't | ) ?'?'

t

iQ:

M((|D(i't | ) ?'?'%

6< " 7>

Z^BBL(G\'

] g |D | ? |D(i'| ? h


77/92

]sgef |Dt| ? |D(i't | ) ?# wm hovm

M(_(t'' ;t

?

iQ:

M(?iD

"|Dt| ?'

t?

iQ

:

M(?Ti5"(|D(i't | ) ?''

t?

iQ:

Z(Ti 5 'M((|D(i't | ) ?'?'

tiQ:

M((|D(i't | ) ?'?|Ti 5 '

M(_(t''%

66 " 7>

B]TCGEM


78/92



t+tmst




Kuctgpcm Tmstgef




Getrbduatgbe

68 " 7>

GET^BD]ATGBE


79/92

Cmt X dmebtm o roedbk sokpcm%

Vm woet tb mstgkotm thm sgfegaoeam bl sbkm tmst stotgstga ](X'uedmr amrtoge abedgtgbes onbut thm dgstrgnutgbe bl thm doto# m%f%# o

eucc hypbthmsgs Hbrgf

: %

Quppbsm thmrm gs o lokgcy bl troeslbrkotgbes

(#x' x

thot troeslbrks x getb o emw# pmrturnmd# sokpcm%

Quppbsm thm brgfgeoc eucc hypbthmsgs# Hbrgf: # be X gkpcgms thm

dmrgvmd H:0 Thm dgstrgnutgbe bl X dbms ebt dmpmed be % Thme# rm`matgef H: gkpcgms rm`matgef H

brgf

: %

67 " 7>


80/92

Ossukm thot nmcbefs tb o abkpoat frbup F mqugppmd wgth o

rgfht+gevorgoet prbnongcgty kmosurm# # sb thot# lbr oey fgvme: F# oed oey lueatgbe L dmemd be F#

F

L(:'d(' ;

F

L('d('%

Vm ocsb ossukm thot X gs o frbup oatgbe# wgth# ge portgaucor# ( x' ; (' x%

8: " 7>

MXOKZCMQ

x ; (x?# % % % # xe'# F ; Qe thm frbup bl pmrkutotgbes bl {?# % % % # e}#


81/92

( ' { } thm ueglbrk kmosurm be F oed

x ; (x? # % % % # xe'%

Kbrm fmemroccy ossukgef moah x` aoe nm spcgt ge twb ports

x` ; (x

`# x

` ' oed

x ; (x?# x? '# % % % # (xe# xe'%Gl H

brgf

: gkpcgms thot X gs gedmpmedmet bl X# gt ocsb gkpcgms H:%

Ossukm thot x` ^d# cmt F ; {?# ?}e wgth thm ueglbrk kmosurmoed

x ; (?x?# % % % # exe'%Gl Hbrgf: gkpcgms thot thm dgstrgnutgbe bl X gs sykkmtrga w%r%t% thm

brgfge# gt ocsb gkpcgms H:%

8? " 7>

Z^GEAGZCM


82/92

Lrbk o tmst stotgstga x ](x'% dmem

T(x' ; { 0 ]( x' ](x'}%

Thme T nmhovms cgim o p+vocum% Eokmcy# uedmr H:# bem hos

Z(T(X' t' t%

8> " 7>

Z^BBL


83/92

Lgx x oed cmt Lx(t' ; (](x'

t'%

Lx gs thm a%d%l% bl thm roedbk vorgoncm P 0 ]( x' uedmr thmprbnongcgty %

Oaabrdgef tb thm Z%G%T% thmbrmk# wm hovm

(Lx(P' ? t' t%wgth

(Lx(](

x''

?

t' ;

F

?Y?t#?[(Lx(](

x''d('%

81 " 7>

Z^BBL (GG'


84/92

Lrbk thm rgfht gevorgoeam bl # wm hovm# lbr oey :

F#

Lx(t' ;

F

?Y:#t'(]( x''d(' ;

F

?Y:#t'(](:

x''d(' ; L:x(t

'%

Ny dmegtgbe# T(x' ; ? Lx(](x''% ]edmr H:

Z(T(X' t' ; Z(

T(

X' t'lbr occ F#

84 " 7>

Z^BBL (GGG'


85/92

Vm aoe wrgtm

Z(T(X' t' ;

F

Z(T( X' t'd('

;

F

Z(LX(]( X'' ? t'd('

;

F

Z(LX(]( X'' ? t'd('

;

F

?Y?t#?[(LX(]( X''d('dZ

t

8= " 7>

GKZCMKMETOTGBE

Os o abesmqumeam# rm`matgef H: gl T(X' 3 prbvgdms o typm G mrrbr


86/92

cbwmr thoe %

Hbwmvmr# abkputgef T(X' gs ge fmemroc uelmosgncm nmaousm gtgevbcvms o suk bvmr o hufm euknmr bl tmrks (m%f%# e/'%

Oe opprbxgkotgbe aoe nm kodm ny Kbetm+Aorcb sokpcgef0

fmemrotm o g%g%d% smqumeam ? % % % # _ oaabrdgef tb oed abkputm

T_(X' ;?

_

_`;?

?Y](qX'](X'[%

Abedgtgbeoccy tb X ; x# T_(x' lbccbws o ngebkgoc dgstrgnutgbe

Nge(_# px' whmrm

px ; { 0 ]( x' ](x'}%

8< " 7>

GKZCMKMETOTGBE

Vm hovm `ust prbvmd thot


87/92

Vm hovm `ust prbvmd thot

Z(pX t' t%

Gl wm koim thm (vmry skocc' opprbxgkotgbe bl ]( t' os oabetgeubus r%v% (gt gs o dgsarmtm r%v% thot toims o hufm euknmr bl

vocums'# wm aoe ossukm thot pX lbccbws o ueglbrk dgstrgnutgbe bvmr

Y:# ?[%

]sgef thgs# wm hovm

Z(_T_(X' r' ;r

`;:

_`

?:

t`(? t'_`dt ; r ) ?_) ?

%

86 " 7>

QOKZCGEF ZM^K]TOTGBEQ


88/92

Gl F gs Qs# thm frbup bl pmrkutotgbes bl {?# % % % # e}# oed gs thmueglbrk dgstrgnutgbe be thgs frbup# o roedbk sokpcm lbr aoe nmbntogemd vgo thm lbccbwgef ocfbrgthk%

Fmemrotm o smqumeam (`' bl pmrkutotgbes bl Q` wgth# bnvgbuscy#

(?'

? ; ? oed lbr ` ; ># % % % # e# fmemrotm o pmrkutotgbe(`' ; (

(`'? # % % % #

(`'` ' ny gesmrtgef ` ot o ueglbrkcy roedbk pbsgtgbe

ge (`?'%

Gt gs thme mosy tb prbvm (ny geduatgbe' thot Z((`' ; ' ; ?"`/ lbr occ

` oed Q`%

88 " 7>

OZZCGAOTGBE TB K]CTGZCM TMQTQ (G'

Bnsmrvotgbe0 e g g d sokpcms moah wgth K kmosurmkmets lbr o


89/92

Bnsmrvotgbe0 e g%g%d% sokpcms# moah wgth K kmosurmkmets# lbr o

r%v% X ; (X?# % % % # XK'% Oebthmr r%v% W abetoges ottrgnutms br phmebtypms shormd ny occ

kmosurmkmets%

X#Xi#W rmprmsmet roedbk sokpcms bl sgzm e%

Lbr moah i Q ; {?# % % % # K}# abkputm o stotgstga ](Xi#W'% Thm troeslbrkotgbe# oat be W%

Thm eucc gs thot thm `bget dgstrgnutgbe bl (](Xi# W'# i Q:' gsgedmpmedmet bl # whmrm

Q ; {?# % % % # K} ; D Q:oed wm woet tb mstgkotm D%

87 " 7>

OZZCGAOTGBE TB K]CTGZCM TMQTQ (GG'

Dmem thm stotgstgas


90/92

]

(X#W' ; koxiQ ](Xi#W']: (X#W' ; kox

iQ:](Xi#W'

sb thot ] ]: # oed thm dgstrgnutgbe bl ]: (X# W' dbms ebtdmpmed be %

Ebtm thot becy ] gs abkputoncm ge proatgam# sgeam thm smt Q: gs

ueiebwe%

]edmr thm fcbnoc eucc hypbthmsgs (D ; '# wm hovm ] ; ]: %

Vm aoe usm pmrkutotgbe tmsts tb rm`mat thm fcbnoc eucc glT(y#x' 0; (](x# y' ](x#y''

gs cmss thoe sbkm 5 :%

7: " 7>

OZZCGAOTGBE TB K]CTGZCM TMQTQ (GGG'


91/92

Tb nugcd oe mstgkotbr D# dmem

Lx#y(q' ; (](x# y' q'

L:#x#y (q' ; (]

: (x# y' q'%

Qgeam ]: ]# wm hovm L:#x#y Lx#y% Dmem q ; qx#y os thm kgegkoc q suah thot L

x#y(q' ? t# oed

toim

D ;{

i

Q# ](xi#y' 5 q

}%

7? " 7>

OZZCGAOTGBE TB K]CTGZCM TMQTQ (G\'


92/92

Vm thme hovm

Z(D Q: ; ' Z(L(]: (x#y'' ? t' Z(L:#(]: (x# y'' ? t' t

(Qokm orfukmet os wgth pmrkutotgbe tmsts'%

Vm thmrmlbrm bntoge oe mstgkotbr D thot abetrbcs thm LVM^ ot

cmvmc t%

7> " 7>

Documents

Lecture 5 (L. Younes)